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\begin{document}
\begin{abstract}
We study global log canonical thresholds on anticanonically
embedded quasismooth weighted Fano threefold hypersurfaces having
terminal quotient singularities to prove the existence of a
K\"ahler--Einstein metric on most of them, and to produce examples
of Fano varieties with infinite discrete groups of birational
automorphisms.
\end{abstract}
\maketitle
\section{Introduction.}
\label{section:intro}
Let $X$ be a Fano variety\footnote{We assume that all varieties
are projective, normal, and defined over $\mathbb{C}$.} of
dimension $n$ that has at most log terminal singularities.
\begin{definition}
\label{definition:threshold} The global log canonical threshold of
the variety $X$ is the number
$$
\mathrm{lct}\big(X\big)=\mathrm{sup}\left\{\lambda\in\mathbb{Q}\ \left|
\aligned
&\mathrm{the\ log\ pair}\ \big(X, \lambda H\big)\ \mathrm{has\ log\ canonical\ singularities}\\
&\mathrm{for\ every\ effective}\ \text{$\mathbb{Q}$-divisor}\ H\ \mathrm{such\ that}\ H\equiv -K_{X}\\
\endaligned\right.\right\}\geqslant 0.
$$
\end{definition}
It follows from \cite{Ti87}, \cite{Na90}, \cite{DeKo01} that the
Fano variety $X$ has an orbifold K\"ahler--Einstein metric in the
case when $X$ has quotient singularities and the inequality
$\mathrm{lct}(X)>n/(n+1)$ holds\footnote{The number
$\mathrm{lct}(X)$ is an algebraic counterpart of the
$\alpha$-invariant introduced in \cite{Ti87}.}.
\begin{example}
\label{example:sextic-double-solid} Let $X$ be a general
hypersurface in $\mathbb{P}(1^{4},3)$ of degree~$6$. Then
$\mathrm{lct}(X)=1$~by~\cite{Pu04d}.
\end{example}
Quasismooth anticanonically embedded weighted Fano threefold
hypersurfaces with terminal singularities are studied
extensively in \cite{CPR}, \cite{ChPa05}, \cite{Ch06e},
\cite{ChPa05h}. In this paper we prove the following~result.
\begin{theorem}
\label{theorem:main} Let $X$ be a general quasismooth hypersurface
in $\mathbb{P}(1,a_{1},\ldots,a_{4})$ of
degree~$\sum_{i=1}^{4}a_{i}$ having at most terminal singularities
such that $-K_{X}^{3}\leqslant 1$. Then $\mathrm{lct}(X)=1$.
\end{theorem}
The proof of Theorem~\ref{theorem:main} is algebro-geometric, but
Theorem~\ref{theorem:main} implies the following result.
\begin{corollary}
\label{corollary:KE}
With\,the\,assumptions\,of\,Theorem\,\ref{theorem:main},\,the\,variety\,$X$\,has\,a\,K\"ahler--Einstein\,metric.
\end{corollary}
It follows from \cite{CPR}, \cite{Pu04d} that
Theorem~\ref{theorem:main} also implies the following result (see
Theorem~\ref{theorem:Cheltsov}).
\begin{corollary}
\label{corollary:Cheltsov} Let $X_{1},\ldots,X_{r}$ be varieties
that satisfy all hypotheses of Theorem~\ref{theorem:main}.~Then
$$
\mathrm{Bir}\Big(X_{1}\times\cdots\times X_{r}\Big)=\Big<\prod_{i=1}^{r}\mathrm{Bir}\big(X_{i}\big),\ \mathrm{Aut}\Big(X_{1}\times\cdots\times X_{r}\Big)\Big>,
$$
the variety $X_{1}\times\cdots\times X_{r}$ is non-rational, and
for any dominant map $\rho\colon X_{1}\times\cdots\times
X_{r}\dasharrow Y$ whose general fiber is rationally connected,
there is a commutative diagram
$$
\xymatrix{
X_{1}\times\cdots\times X_{r}\ar@{->}[d]_{\pi}\ar@{-->}[rr]^{\sigma}&&X_{1}\times\cdots\times X_{r}\ar@{-->}[d]^{\rho}\\
X_{i_{1}}\times\cdots\times X_{i_{k}}\ar@{-->}[rr]_{\xi}&&Y,}
$$
where $\xi$ and $\sigma$ are
birational maps, and $\pi$ is a projection for some $\{i_{1},\ldots,i_{k}\}\subsetneq\{1,\ldots,r\}$.
\end{corollary}
Unlike those of dimension three, no Fano varieties of dimension
four or higher having infinite~groups of birational automorphisms
whose birational automorphisms are well understood have been known
so far. However, we can now easily obtain the following example.
\begin{example}
\label{example:41-41} Let $X$ be a general hypersurface in
$\mathbb{P}(1,1,4,5,10)$ of degree~$20$. Then it immediately
follows from \cite{CPR}, \cite{ChPa05} and
Corollary~\ref{corollary:Cheltsov} that there is an exact sequence
of groups
$$
1\longrightarrow\prod_{i=1}^{m}\Big(\mathbb{Z}_{2}\ast\mathbb{Z}_{2}\Big)\longrightarrow\mathrm{Bir}\Big(\underbrace{X\times\cdots\times X}_{m\ \mathrm{times}}\Big)\longrightarrow\mathrm{S}_{m}\longrightarrow 1,
$$
where $\mathbb{Z}_{2}\ast\mathbb{Z}_{2}$ is the infinite dihedral
group.
\end{example}
The assertion of Theorem~\ref{theorem:main} may fail without
the~generality~assumption.
\begin{example}
\label{example:34} Let $X$ be a hypersurface in
$\mathbb{P}(1,1,2,6,9)$ of degree $18$ given by the equation
$$
w^{2}=t^{3}+z^{9}+y^{18}+x^{18}\subset\mathbb{P}\big(1,1,2,6,9\big)\cong\mathrm{Proj}\Big(\mathbb{C}[x,y,z,t,w]\Big),
$$
where $\mathrm{wt}(x)=\mathrm{wt}(y)=1$, $\mathrm{wt}(z)=2$,
$\mathrm{wt}(t)=6$, $\mathrm{wt}(w)=9$. The hypersurface $X$ has
terminal quotient singularities, and $-K_{X}^{3}=1/6$.
Arguing~as~in the proof of Theorem~\ref{theorem:main}, we see~that
$$
\mathrm{lct}\big(X\big)=\mathrm{sup}\Big\{\lambda\in\mathbb{Q}\
\Big\vert\ \mathrm{the\ log\ pair}\ \big(X, \lambda D\big)\ \mathrm{is\ log\ canonical\ for\ every\ Weil\ divisor}\ D\in\big|-K_{X}\big|\Big\},
$$
which easily implies that $\mathrm{lct}(X)=17/18$ by Lemma~8.12
and Proposition~8.14 in \cite{Ko97}.
\end{example}
Nevertheless, the proof of Theorem~1.3 in \cite{CPR} and the proof
of Theorem~\ref{theorem:main} can also be used to~construct
explicit examples of Fano threefolds to which
Corollaries~\ref{corollary:KE} and \ref{corollary:Cheltsov} can be
applied.
\begin{example}
\label{example:22} Let $X$ be a hypersurface in
$\mathbb{P}(1,2,2,3,7)$ of degree $14$ given by the equation
$$
w^{2}=t^{4}z+y^{7}-z^{7}+x^{14}\subset\mathbb{P}\big(1,2,2,3,7\big)\cong\mathrm{Proj}\Big(\mathbb{C}[x,y,z,t,w]\Big),
$$
where $\mathrm{wt}(x)=1$, $\mathrm{wt}(y)=\mathrm{wt}(z)=2$,
$\mathrm{wt}(t)=3$, $\mathrm{wt}(w)=7$. The hypersurface $X$ has
terminal quotient singularities, and $-K_{X}^{3}=1/6$.
Arguing~as~in the proof of Theorem~\ref{theorem:main}, we see~that
$$
\mathrm{lct}\big(X\big)=\mathrm{sup}\Big\{\lambda\in\mathbb{Q}\
\Big\vert\ \mathrm{the\ log\ pair}\ \big(X, \lambda D\big)\ \mathrm{is\ log\ canonical}\Big\},
$$
where $D$ is the unique Weil divisor in $|-K_{X}|$. Then
$\mathrm{lct}(X)=1$ by Lemma~8.12 and Proposition~8.14 in
\cite{Ko97}. The threefold $X$~has a~K\"ahler--Einstein metric,
and the group $\mathrm{Bir}(X\times X)$~is~finite.
\end{example}
The proof of Theorem~\ref{theorem:main} is based on the results
obtained in \cite{CPR}, \cite{ChPa05}, \cite{Ch06e},
\cite{ChPa05h}, but it is lengthy, because the hypotheses of
Theorem~\ref{theorem:main} are satisfied for general members of
$90$ out of $95$ familes of quasismooth terminal anticanonically
embedded weighted Fano threefold hypersurfaces (see~\cite{IF00}).
For the convenience of the reader, we organize this paper in the
following way:
\begin{itemize}
\item we prove Theorem~\ref{theorem:main} in Section~\ref{section:the-proof} omitting the proofs of Lemmas~\ref{lemma:smooth-points}, \ref{lemma:singular-points}, \ref{lemma:quadratic-involutions};
\item we prove auxiliary technical Lemmas~\ref{lemma:smooth-points}, \ref{lemma:singular-points}, \ref{lemma:quadratic-involutions} in Sections~\ref{section:smooth-points}, \ref{section:singular-points}, \ref{section:quadratics-involutions}, respectively;
\item we consider one important generalization of Corollary~\ref{corollary:Cheltsov} in Section~\ref{section:conic-bundles}.
\end{itemize}
The author would like to thank J.\,Howie, J.\,Koll\'ar,
L.\,O'Carroll, J.\,Park, A.\,Pukh\-likov,~V.\,Sho\-ku\-rov and the
referees for useful comments. The author is grateful to the IHES
for hospitality.
\section{The proof of main result.}
\label{section:the-proof}
Let $X$ be a general quasismooth hypersurface in
$\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree
$d=\sum_{i=1}^{4}a_{i}$~with terminal singularities, and let
$\gimel\in\{1,\ldots,95\}$ be the ordinal number of
$(a_{1},a_{2},a_{3},a_{4})$ in~the~notation of Table~5 in
\cite{IF00}, where $a_{1}\leqslant a_{2}\leqslant a_{3}\leqslant
a_{4}$. Then $-K_{X}^{3}\leqslant 1\iff\gimel\geqslant 6$.
We suppose that $\gimel\geqslant 6$, but there is $D\in|-nK_{X}|$
such that $(X,\frac{1}{n}D)$ is not log canonical, where $n$ is a
natural number. Then to prove Theorem~\ref{theorem:main} it is
enough to derive~a~contradiction, because the class group of the
hypersurface $X$ is generated by the divisor $-K_{X}$.
\begin{remark}
\label{remark:weighted-sum} Let $V$ be a variety, let $B$ and
$B^{\prime}$ be effective $\mathbb{Q}$-Cartier
$\mathbb{Q}$-divisors on $V$ such that the~singularities of the
log pairs $(V, B)$ and $(V, B^{\prime})$ are log canonical, and
let $\alpha$ be a rational number such that
$0\leqslant\alpha\leqslant 1$. Then the log pair $(V,\ \alpha
B+(1-\alpha)B^{\prime})$ is log canonical.
\end{remark}
Thus, we may assume that $D$ is an irreducible surface due to
Remark~\ref{remark:weighted-sum}.
\begin{lemma}
\label{lemma:anticanclass} The inequality $n\ne 1$ holds.
\end{lemma}
\begin{proof}
Suppose that $n=1$. Then the log pair $(X, D)$ is log canonical at
every singular point of the threefold $X$ by Lemma~8.12 and
Proposition~8.14 in \cite{Ko97}. Thus, the equality $a_{1}=1$
holds, because the linear system $|-K_{X}|$ consists of a single
surface in the case when $a_{1}\ne 1$.
The equality $a_{1}=1$ holds for $36$ values of
$\gimel\in\{6,7,\ldots,95\}$, but all possible cases are very
similar. So for the sake of simplicity, we assume that
$\gimel=14$. Then there is a natural double cover $\pi\colon
X\to\mathbb{P}(1,1,1,4)$~branched~over a general hypersurface
$F\subset \mathbb{P}(1,1,1,4)$ of degree $12$.
Suppose that the singularities of the log pair $(X, D)$ are not
log canonical at~some smooth point $P$ of the threefold $X$. Let
us show that this assumption leads to a contradiction.
Put $\bar{D}=\pi(D)$ and $\bar{P}=\pi(P)$. Counting parameters, we
see that $\mathrm{mult}_{\bar{P}}(F\vert_{\bar{D}})\leqslant 2$,
which is a contradiction, because
$(\bar{D},\frac{1}{2}F\vert_{\bar{D}})$ is not log~ca\-no\-ni\-cal
at $\bar{P}$ by Lemma~8.12~in~\cite{Ko97}.
\end{proof}
\begin{lemma}
\label{lemma:smooth-points} The log pair $(X,\frac{1}{n}D)$ is log
canonical at smooth points of the threefold $X$.
\end{lemma}
\begin{proof}
See Section~\ref{section:smooth-points}.
\end{proof}
Therefore, there is a singular point $O$ of the threefold $X$ such
that $(X,\frac{1}{n}D)$ is not canonical at the point $O$. It
follows from \cite{IF00} that $O$ is a singular point of type
$\frac{1}{r}(1,a,r-a)$, where $a$ and~$r$ are coprime natural
numbers such that $r>2a$ (see Table~5~in~\cite{IF00} for the
values of $a$ and $r$).
Let $\alpha\colon U\to X$ be a blow up of $O$ with weights
$(1,a,r-a)$.~Then
\begin{equation}
\label{equation:degree-of-blow-up}
-K_{U}^{3}=-K_{X}^{3}-\frac{1}{r^{3}}E^{3}=-K_{X}^{3}-\frac{1}{ra(r-a)}=\frac{\sum_{i=1}^{4}a_{i}}{a_{1}a_{2}a_{3}a_{4}}-\frac{1}{ra(r-a)},
\end{equation}
where $E$ is the exceptional divisor of $\alpha$. There is a
rational number $\mu$ such that
$$
\bar{D}\equiv \alpha^{*}\big(D\big)-\mu E\equiv -nK_{U}+\big(n/r-\mu\big)E
$$
where $\bar{D}$ is the proper transform of $D$ on $U$. Then it
follows from \cite{Ka96} that $\mu>n/r$.
\begin{lemma}
\label{lemma:negative-K-cube} The inequality $-K_{U}^{3}\geqslant 0$ holds.
\end{lemma}
\begin{proof}
Suppose that $-K_{U}^{3}<0$. Let $C$ be a curve in $E$. Then the
curve $C$ generates an extremal ray of the cone $\mathbb{NE}(U)$.
Moreover, it follows from Corollary~5.4.6 in \cite{CPR} that there
is an~irreducible curve $\Gamma\subset U$ such that $\Gamma$
generates the extremal ray of $\mathbb{NE}(U)$ that is different
from $\mathbb{R}_{\geqslant 0}C$, and
$$
\Gamma\equiv-K_{U}\cdot \Big(-bK_{U}+cE\Big),
$$
where $b>0$ and $c\geqslant 0$ are integers (see Remark~5.4.7 in
\cite{CPR}).
Let $T$ be a divisor in $|-K_{U}|$. Then $\bar{D}\cdot T$ is
effective, because $\bar{D}\ne T$. However we have
$$
\bar{D}\cdot T\equiv -K_{U}\cdot\Big(-nK_{U}+\big(n/r-\mu\big)E\Big)\not\in\mathbb{NE}(U),
$$
because $\mu>n/r$, $b>0$, and $c\geqslant 0$. So we have a
contradiction.
\end{proof}
Taking into account the possible values of
$(a_{1},a_{2},a_{3},a_{4})$, we see that
$\gimel\not\in\{75,84,87,93\}$.
\begin{lemma}
\label{lemma:zero-K-cube} The inequality $-K_{U}^{3}\ne 0$ holds.
\end{lemma}
\begin{proof}
Firstly, suppose that $-K_{U}^{3}=0$ and $\gimel\ne 82$. Then the
linear system $|-rK_{U}|$ does not have base points for $r\gg 0$
and induces a morphism $\eta\colon U\to\mathbb{P}(1,a_{1},a_{2})$
such that the diagram
$$
\xymatrix{
&&&U\ar@{->}[lld]_{\alpha}\ar@{->}[rrd]^{\eta}&&&\\%
&X\ar@{-->}[rrrr]_{\psi}&&&&\mathbb{P}(1,a_{1},a_{2})&}
$$
is commutative, where $\psi$ is a natural projection. The morphism
$\eta$ is an elliptic fibration. Thus
$$
\bar{D}\cdot C=-nK_{U}\cdot C+\big(n/r-\mu\big)E\cdot C=\big(n/r-\mu\big)E\cdot C<0,
$$
where $C$ is a general fiber of $\eta$, which is a contradiction.
Suppose that $-K_{U}^{3}=0$ and $\gimel=82$. Then $X$ is a
hypersurface in $\mathbb{P}(1,1,5,12,18)$ of degree $36$, whose
singularities consist of two points $P$ and $Q$ of types
$\frac{1}{5}(1,2,3)$ and $\frac{1}{6}(1,1,5)$, respectively.
We see that either $P=O$, or $Q=O$. The hypersurface $X$ can be
given by the equation
$$
z^{7}y+\sum_{i=0}^{6}z^{i}f_{36-5i}\big(x,y,z,t\big)=0\subset\mathbb{P}\big(1,1,5,12,18\big)\cong\mathrm{Proj}\Big(\mathbb{C}[x,y,z,t,w]\Big),
$$
where $\mathrm{wt}(x)=\mathrm{wt}(y)=1$, $\mathrm{wt}(z)=5$,
$\mathrm{wt}(t)=12$, $\mathrm{wt}(w)=18$, and $f_{i}$ is a
quasi\-ho\-mo\-ge\-ne\-ous polynomial of degree $i$. Then $P$ is
is given by the equations $x=z=t=w=0$.
Suppose that $Q=O$. Then the linear system $|-rK_{U}|$ has no
base points for $r\gg 0$, which leads to a contradiction as in the
case when $\gimel\ne 82$. So we see that $P=O$.
Let $\bar{S}$ be the proper transform on $U$ of the surface that
is cut out on $X$ by $y=0$. Then
$$
\bar{S}\equiv \alpha^{*}\big(-K_{X}\big)-\frac{6}{5}E,
$$
and the base locus of the pencil $|-K_{U}|$ consists of two
irreducible curves $L$ and $C$ such that the~curve $L$ is
contained in the $\alpha$-exceptional surface $E$, and the curve
$\pi(C)$ is the unique base curve of the~pencil $|-K_{X}|$. Then
$-K_{U}\cdot C=-1/6$ and $-K_{U}\cdot L>0$. We have $\mu\leqslant
n/5$ due to
$$
n/5-\mu=\Big(-K_{U}+\alpha^{*}\big(-5K_{X}\big)\Big)\cdot\bar{S}\cdot\bar{D}\geqslant 0,
$$
because it follows from Lemma~8.12 and Proposition~8.14 in
\cite{Ko97} that $\bar{D}\ne\bar{S}$. However we know that the
inequality $\mu>n/5$ holds by \cite{Ka96}. So again we have a
contradiction.
\end{proof}
Thus, taking into account the
equality~\ref{equation:degree-of-blow-up} and possible values of
$(a_{1},a_{2},a_{3},a_{4})$, we see that
$$
\gimel\not\in\Big\{11, 14, 19, 22, 28, 34, 37, 39, 49, 52, 53, 57, 59, 64, 66, 70, 72, 73, 78, 80, 81, 86, 88, 89, 90, 92, 94, 95\Big\}
$$
by Lemma~\ref{lemma:zero-K-cube}. So the assertion of
Theorem~\ref{theorem:main} is proved for $32$ values
of~$(a_{1},a_{2},a_{3},a_{4})$.
\begin{lemma}
\label{lemma:superrigid} The groups $\mathrm{Bir}(X)$ and
$\mathrm{Aut}(X)$ do not coincide.
\end{lemma}
\begin{proof}
Suppose that $\mathrm{Bir}(X)=\mathrm{Aut}(X)$. Let $\bar{S}$ be a
general surface in $|-K_{U}|$. Then it follows from Lemma~5.4.5
in \cite{CPR} that there is an irreducible surface $\bar{T}\subset
U$ such that
\begin{itemize}
\item the equivalence $\bar{T}\sim c\bar{S}-bE$ holds, where $c\geqslant 1$ and $b\geqslant 1$ are natural numbers,
\item the scheme-theoretic intersection $\bar{T}\cdot\bar{S}$ is an irreducible and reduced curve $\Gamma$,
\item the curve $\Gamma$ generates an extremal ray of the cone $\mathbb{NE}(U)$.
\end{itemize}
The surface $\bar{T}$ is easy to construct explicitly (see
\cite{CPR}), and the possible values for the natural numbers $c$
and $b$ can be found in \cite{CPR}. The surface $\bar{T}$ is
determined uniquely by the point $O$.
Put $T=\alpha(\bar{T})$. Then it follows from Lemma~8.12 and
Proposition~8.14 in \cite{Ko97} that~the~singularities of the log
pair $(X,\frac{1}{c}T)$ are log canonical. Therefore, we have
$D\ne T$.
Let $\mathcal{P}$ be the pencil~gene\-rated by the effective
divisors $nT$ and $cD$. Then the singularities of the log pair
$(X, \frac{1}{cn}\mathcal{P})$~are~not~canonical, which is
impossible due to \cite{CPR}.
\end{proof}
It follows from \cite{CPR} and Lemma~\ref{lemma:superrigid} that
$\gimel\not\in\{11, 21, 29, 35, 50, 51, 55, 62, 63, 67, 71, 77,
82, 83, 85, 91\}$.
\begin{lemma}
\label{lemma:nef-and-big} The divisor $-K_{U}$ is nef.
\end{lemma}
\begin{proof}
Suppose that $-K_{U}$ is not nef. Then it follows from \cite{CPR}
that $\gimel=47$ and $O$ is a singular point of type
$\frac{1}{5}(1,2,3)$. The hypersurface $X$ can be given by the
equation
$$
z^{4}y+\sum_{i=0}^{3}z^{i}f_{21-5i}\big(x,y,z,t\big)=0\subset\mathbb{P}\big(1,1,5,7,8\big)\cong\mathrm{Proj}\Big(\mathbb{C}[x,y,z,t,w]\Big),
$$
where $\mathrm{wt}(x)=\mathrm{wt}(y)=1$, $\mathrm{wt}(z)=5$,
$\mathrm{wt}(t)=7$, $\mathrm{wt}(w)=8$, and $f_{i}$ is a general
quasi\-ho\-mo\-ge\-ne\-ous polynomial of degree $i$. Let $S$ be
the surface on $X$ that is cut out by the equation $y=0$, and
$\bar{S}$ be the proper transform of the surface $S$ on the
threefold $U$. Then
$$
\bar{S}\equiv \alpha^{*}\big(-K_{X}\big)-\frac{6}{5}E,
$$
but the divisor $-3K_{U}+\alpha^{*}(-5K_{X})$ is nef (see
\cite{Ch06e}). Thus, the inequality $\mu\leqslant n/5$ holds due
to
$$
n/5-\mu=\frac{1}{3}\Big(-3K_{U}+\alpha^{*}\big(-5K_{X}\big)\Big)\cdot\bar{S}\cdot\bar{D}\geqslant 0,
$$
because $D\ne S$. However we know that $\mu>n/5$. So we have a
contradiction.
\end{proof}
Thus, the divisor $-K_{U}$ is nef and big, because $-K_{U}^{3}>0$
by Lemmas~\ref{lemma:negative-K-cube} and \ref{lemma:zero-K-cube}.
\begin{lemma}
\label{lemma:discrepancy} The inequality $\mu/n-1/r<1$ holds.
\end{lemma}
\begin{proof}
We only consider the case when $\gimel=58$ and $O$ is a
sin\-gu\-lar point of type $\frac{1}{10}(1,3,7)$, because the
proof is similar in all other cases (cf.
Lemma~\ref{lemma:inequality-for-mu}). Then $X$ can be given~by
$$
w^{2}z+wf_{14}\big(x,y,z,t\big)+f_{24}\big(x,y,z,t\big)=0\subset\mathbb{P}\big(1,3,4,7,10\big)\cong\mathrm{Proj}\Big(\mathbb{C}[x,y,z,t,w]\Big),
$$
where $\mathrm{wt}(x)=1$, $\mathrm{wt}(y)=3$, $\mathrm{wt}(z)=4$,
$\mathrm{wt}(t)=7$, $\mathrm{wt}(w)=10$, and $f_{i}$ is a
quasi\-ho\-mo\-ge\-ne\-ous polynomial of degree $i$. Let $R$ be
the surface on $X$ that is cut out by $t=0$, and $\bar{R}$ be the
proper transform of the surface $R$ on the threefold $U$. Then
$$
\bar{R}\equiv \alpha^{*}\big(-4K_{X}\big)-\frac{7}{5}E,
$$
and $(X, \frac{1}{4}R)$ is log canonical at $O$ by Lemma~8.12 and
Proposition~8.14 in \cite{Ko97}. Then $R\ne D$~and
$$
0\leqslant -K_{U}\cdot\bar{R}\cdot\bar{D}=4n/35-2\mu/3,
$$
because $-K_{U}$ is nef. Thus, we have $\mu\leqslant 6n/35$, which
implies that $\mu/n-1/10<1$.
\end{proof}
So the log pair $(U, \frac{1}{n}\bar{D}+(\mu/n-1/r)E)$ is not log
canonical at some point $P\in E$,~because
$$
K_{U}+\frac{1}{n}\bar{D}\equiv \alpha^{*}\Big(K_{X}+\frac{1}{n}D\Big)+\big(1/r-\mu/n\big)E.
$$
\begin{lemma}
\label{lemma:singular-points} The threefold $U$ is smooth at the
point $P$.
\end{lemma}
\begin{proof}
See Section~\ref{section:singular-points}.
\end{proof}
Thus, the inequality $\mathrm{mult}_{P}(\bar{D})>n+n/r-\mu$ holds.
But it follows from \cite{CPR} that
\begin{itemize}
\item either $d=2r+a_{j}$ for some $j$, and there is a quadratic involution $\tau\in\mathrm{Bir}(X)$ induced~by~$O$,
\item or $d=3r+a_{j}$ for some $j$, and there is an elliptic involution $\tau\in\mathrm{Bir}(X)$ induced by $O$,
\end{itemize}
where $d=\sum_{i=1}^{4}a_{i}$.
\begin{lemma}
\label{lemma:quadratic-involutions} The inequality $d\ne 2r+a_{j}$
holds for every $j\in\{1,2,3,4\}$.
\end{lemma}
\begin{proof}
See Section~\ref{section:quadratics-involutions}.
\end{proof}
Thus, it follows from \cite{CPR} that there is $j\in\{1,2,3,4\}$
such that $d=3r+a_{j}$.
\begin{remark}
\label{remark:curves} Let $V$ be a threefold with isolated
singularities, let $B\ne T$ be effective irreducible divisors on
the threefold $V$, and let $H$~be a nef divisor on the threefold
$V$. Put
$$
B\cdot T=\sum_{i=1}^{r}\epsilon_{i} L_{i}+\Delta,
$$
where $L_{i}$ is an irreducible curve, $\epsilon_{i}$ is a
non-negative integer, and $\Delta$ is an effective one-cycle whose
support does not contain the curves $L_{1},\ldots,L_{r}$. Then
$\sum_{i=1}^{r}\epsilon_{i} H\cdot L_{i}\leqslant B\cdot T\cdot
H$.
\end{remark}
It follows from Lemma~\ref{lemma:quadratic-involutions} that
$\gimel\in\{7, 20, 23, 36, 40, 44, 61, 76\}$ (see \cite{CPR}).
\begin{lemma}
\label{lemma:7-20-36-elliptic} The case $\gimel\in\{7,20,36\}$ is
impossible.
\end{lemma}
\begin{proof}
Suppose that $\gimel\in\{7,20,36\}$. Then $a_{1}=1$, and it
follows from Lemma~\ref{lemma:quadratic-involutions} that
$O$~is~a~singular point of type $\frac{1}{a_{2}}(1,1,a_{2}-1)$.
Then $|-rK_{U}|$ induces a birational morphism $\sigma\colon U\to
V$ such that $\sigma$ contracts smooth rational curves
$C_{1},\ldots, C_{l}$, and $V$ is a hypersurface in
$\mathbb{P}(1,1,a_{3},2a_{4},3a_{4})$ of degree $6a_{4}$, where
$l=d(d-a_{4})/a_{3}$. Let $T$ be the surface in $|-K_{U}|$ that
contains $P$.
Suppose that $P\not\in\cup_{i=1}^{l}C_{i}$. Then it follows from
the proof of Theorem~5.6.2 in \cite{CPR} that there are natural
number $s>0$ and a surface $H\in|-s2a_{4}K_{U}|$ such that
$$
s2a_{4}\big(-nK_{X}^{3}-\mu/(a_{2}-1)\big)=\bar{D}\cdot T\cdot H\geqslant \mathrm{mult}_{P}\big(\bar{D}\big)s>s\big(n+n/a_{2}-\mu\big) s,
$$
which is impossible, because $\mu>n/a_{2}$. So we may assume that
$P\in C_{1}$.
Put $\bar{D}\cdot T=mC_{1}+\Delta$, where $m$ is a non-negative
integer, and $\Delta$ is an effective cycle such that the support
of $\Delta$ does not~contain the~curve $C_{1}$. The curve $C_{1}$
is a smooth rational curve such that $\alpha^{*}(-K_{X})\cdot
C_{1}=2/a_{2}$ and $E\cdot C_{1}=2$.
It follows from \cite{CPR} that there is a surface
$R\in|-a_{3}K_{U}|$ such that~$R$~contains $C_{1}$, but $R$~does
not contain components of the cycle $\Delta$ passing through the
point $P$. Then
$$
a_{3}\big(-nK_{X}^{3}-\mu/(a_{2}-1)\big)=R\cdot\Delta\geqslant \mathrm{mult}_{P}\big(\Delta\big)>n+n/a_{2}-\mu-m,
$$
which implies that $m>a_{3}n/a_{4}$, because $\mu>n/a_{2}$.
Therefore, we have
$$
a_{3}n/a_{4}<m\leqslant\frac{-dnK_{X}\cdot\alpha(C_{1})}{a_{1}a_{2}a_{3}a_{4}}=\frac{dn}{2a_{1}a_{3}a_{4}}
$$
by Remark~\ref{remark:curves}, because
$-K_{X}\cdot\alpha(C_{1})=2/a_{2}$. The inequalities just obtained
imply that $\gimel=7$.
Let $\psi\colon X\dasharrow\mathbb{P}(1,a_{1},a_{2})$ be a natural
projection. The fiber of $\psi$ over $\psi(P)$ consists of two
irreducible components, and one of them is $C_{1}$. Let $Z$ be the
other component of this fiber.~Then
$$
C_{1}^{2}=-2,\ C_{1}\cdot Z=2,\ Z^{2}=-4/3
$$
on the surface $T$. Put $\Delta=\bar{m}Z+\Omega$, where $\bar{m}$
is a non-negative integer and $\Omega$~is~an~effective one-cycle
whose support does not~contain the~curve $Z$. Then
$$
4n/3-2\mu-5\bar{m}/3=\big(Z+C_{1}\big)\cdot\Omega>3n/2-\mu-m,
$$
and $4\bar{m}/3\geqslant 2m-5n/6$, because $\Omega\cdot Z\geqslant
0$. The inequalities just obtained immediately imply that the
inequality $\mu\leqslant n/2$ holds. So we have a contradiction,
because $\mu>n/2$.
\end{proof}
Hence, it follows from Lemmas~\ref{lemma:quadratic-involutions}
and \ref{lemma:7-20-36-elliptic}~that
$\gimel\in\{23,40,44,61,76\}$ and~$d=3r+a_{j}$, where $r=a_{3}>2a$
and $1\leqslant j\leqslant 2$. Then $X$ has~a~singular point $Q$
of type $\frac{1}{\bar{r}}(1,\bar{a},\bar{r}-\bar{a})$~such~that
$$
-K_{X}^{3}=\frac{1}{ra(r-a)}+\frac{1}{\bar{r}\bar{a}(\bar{r}-\bar{a})},
$$
where $\bar{r}=a_{4}>2\bar{a}$ and $\bar{a}\in\mathbb{N}$. It
follows from \cite{CPR} that there is a commutative diagram
$$
\xymatrix{
&U\ar@{->}[ddl]_{\sigma}\ar@{->}[rd]^{\alpha}&&&W\ar@{->}[lll]_{\gamma}\ar@{->}[rrdd]^{\eta}&\\
&&X\ar@{-->}[drrrr]^{\psi}&&\\
V\ar@{^{(}->}[rr]&&\mathbb{P}\big(1,a_{1},a_{2},2a_{4},3a_{4}\big)\ar@{-->}[rr]_{\chi}&&\mathbb{P}\big(1,a_{1},a_{2},2a_{4}\big)\ar@{-->}[rr]_{\xi}&&\mathbb{P}\big(1,a_{1},a_{2}\big),}
$$
where $\xi$, $\chi$, $\psi$ are projections, $\eta$ is an elliptic
fibration, $\gamma$ is a weighted blow up of a point that
dominates the point $Q$ with weights $(1,\bar{a},
\bar{r}-\bar{a})$, and $\sigma$ is a birational morphism that
contracts smooth curves $C_{1},\ldots, C_{l}$ such that $V$ is a
hypersurface in $\mathbb{P}(1,a_{1},a_{2},2a_{4},3a_{4})$ of
degree $6a_{4}$, where $l=d(d-a_{4})/(a_{1}a_{2})$. Let $L$ be a
curve in $|\mathcal{O}_{\mathbb{P}(1,\,a,\,r-a)}(1)|$, where
$E\cong \mathbb{P}(1,a,r-a)$.
\begin{lemma}
\label{lemma:lower-bound-for-mu} Suppose that $P\not\in L$. Then
$\mu>na(r+1)/(r^2+ar)$.
\end{lemma}
\begin{proof}
There is a~unique curve
$C\in|\mathcal{O}_{\mathbb{P}(1,\,a,\,r-a)}(a)|$ such that $P\in
C$. Put $\bar{D}\big\vert_{E}=\delta C+\Upsilon\equiv r\mu L$,
where $\delta$ is a non-negative integer and $\Upsilon$ is an
effective cycle such that $C\not\subset\mathrm{Supp}(\Upsilon)$.
Then
$$
(r\mu-a\delta)/(r-a)=\big(r\mu-a\delta\big)L\cdot C=C\cdot\Upsilon\geqslant\mathrm{mult}_{P}\big(\Upsilon\big)>n+n/r-\mu-\delta,
$$
which implies that $\mu>na(r+1)/(r^2+ar)$, because
$\delta\leqslant r\mu/a$.
\end{proof}
Let $T$ be a surface in $|-K_{U}|$. Then $-K_{U}\cdot T
\cdot\bar{D}\geqslant 0$, which implies that $\mu\leqslant
-na(r-a)K_{X}^{3}$.
\begin{lemma}
\label{lemma:elliptic-involution-T} The point $P$ is not contained
in the surface $T$.
\end{lemma}
\begin{proof}
Suppose that $P$ is contained in the surface $T$. Then $P$ is not
contained in the base locus of the pencil $|-a_{1}K_{U}|$, because
the base locus of the pencil $|-a_{1}K_{U}|$ does not contain
smooth points of the surface $E$. The point $P$ is not contained
in the union $\cup_{i=1}^{l}C_{i}$, because $P\in T$.
The proof of Theorem~5.6.2 in \cite{CPR} implies the existence of
a~surface $H\in|-s2a_{1}a_{4}K_{U}|$~such~that
$$
s2a_{1}a_{4}\big(-nK_{X}^{3}-\mu/a_{2}\big)=\bar{D}\cdot H\cdot T\geqslant \mathrm{mult}_{P}\big(\bar{D}\big)s>s\big(n+n/r-\mu\big) s,
$$
where $s$ is a natural number, which is impossible, because
$\mu>n/r$.
\end{proof}
We have $T\vert_{E}\sim \mathcal{O}_{\mathbb{P}(1,\,a,\,r-a)}(1)$.
Thus, taking into account that $\gimel\in\{7, 20, 23, 36, 40, 44,
61, 76\}$, we see that $\gimel\in\{23, 44\}$ by
Lemmas~\ref{lemma:lower-bound-for-mu} and
\ref{lemma:elliptic-involution-T}, because $\mu\leqslant
-na(r-a)K_{X}^{3}$.
Let $S$ be a surface in $|-a_{1}K_{U}|$ that contains $P$. Then
$\bar{D}\ne S$, because $\mu>n/r$.
\begin{lemma}
\label{lemma:elliptic-involution-exceptional-curves} The point $P$
is contained in $\cup_{i=1}^{l}C_{i}$.
\end{lemma}
\begin{proof}
Suppose that $P\not\in\cup_{i=1}^{l}C_{i}$. Then the proof of
Theorem~5.6.2 in \cite{CPR} implies that
$$
s2a_{1}a_{4}\big(-nK_{X}^{3}-\mu/a_{2}\big)=\bar{D}\cdot H\cdot S\geqslant \mathrm{mult}_{P}\big(\bar{D}\big)s>s\big(n+n/r-\mu\big) s
$$
for some $s\in\mathbb{N}$ and a surface $H\in|-s2a_{4}K_{U}|$,
which is impossible, because $\mu>n/r$.
\end{proof}
We may assume that $P\in C_{1}$. Put $\bar{D}\cdot
S=mC_{1}+\Delta$, where $m$ is a non-negative integer,
and~$\Delta$~is~an effective cycle whose support does not~contain
$C_{1}$. Then it follows from Remark~\ref{remark:curves} that the
inequality $m\leqslant nd/(a_{2}d-a_{2}a_{3})$ holds, because
$-K_{X}\cdot\alpha(C_{1})=(d-a_{3})/(a_{3}a_{4})$.
It follows from the proof of Theorem~5.6.2 in \cite{CPR} that
there is $R\in|-s2a_{4}K_{U}|$~such~that
$$
s2a_{1}a_{4}\big(-nK_{X}^{3}-\mu/a_{2}\big)=R\cdot\Delta\geqslant\mathrm{mult}_{P}\big(\Delta\big)s>s\big(n+n/r-\mu-m\big),
$$
where $s\in\mathbb{N}$. However we have $m\leqslant
nd/(a_{2}d-a_{2}a_{3})$, which implies that $\gimel=23$.
Therefore, we proved that $X$ is a hypersurface
$\mathbb{P}(1,2,3,4,5)$ of degree $14$ and $O$~is~a~singular point
of type $\frac{1}{4}(1,1,3)$. Let $M$ be a general surface in the
linear system $|-3K_{X}|$ that passes through the point $P$. Then
$S\cdot M=C_{1}+Z_{1}$, where $Z_{1}$ is a curve such that
$-K_{U}\cdot Z_{1}=1/5$.~Put
$$
\bar{D}\cdot S=mC_{1}+\bar{m}Z_{1}+\Upsilon,
$$
where $\bar{m}$ is a non-negative integer, and $\Upsilon$ is an
effective cycle, whose support does not contain the curves $C_{1}$
and $Z_{1}$. Then $m<7n/15$ by Remark~\ref{remark:curves}. But
$\mu>n/4$~and
$$
7n/10-6\mu/3-3\bar{m}/5=M\cdot\Upsilon\geqslant\mathrm{mult}_{P}\big(\Upsilon\big)>5n/4-\mu-m,
$$
because $P\not\in Z_{1}$. The inequality obtained implies a
contradiction.
Therefore, the assertion of Theorem~\ref{theorem:main} is
completely proved.
\section{Non-singular points.}
\label{section:smooth-points}
In this section we prove the assertion of
Lemma~\ref{lemma:smooth-points}. Let us use the assumptions and
notation of Lemma~\ref{lemma:smooth-points}. Take an arbitrary
smooth point $P$ of the threefold $X$.
\begin{lemma}
\label{lemma:smooth-points-small-K-cube} Suppose that $a_{4}$
divides $d$, $a_{1}\ne a_{2}$, and $-a_{2}a_{3}K_{X}^{3}\leqslant
1$. Then $\mathrm{mult}_{P}(D)\leqslant n$.
\end{lemma}
\begin{proof}
Suppose that $\mathrm{mult}_{P}(D)>n$. Let $L$ be the base curve
of $|-a_{1}K_{X}|$, and $T$ be~a~surface in the linear system
$|-K_{X}|$. Then $D\cdot T$ is an effective one-cycle, and
$\mathrm{mult}_{P}(L)=1$.
Suppose that $P\in L$. Let $R$ be a general surface in
$|-a_{1}K_{X}|$. Put $D\cdot T=mL+\Delta$, where $m$ is
non-negative integer, and $\Delta$ is an effective cycle whose
support does not contain $L$. Then
$$
-a_{1}\big(n-a_{1}m\big)K_{X}^{3}=D\cdot T\cdot R-mR\cdot L=R\cdot\Delta\geqslant\mathrm{mult}_{P}\big(\Delta\big)>n-m,
$$
which is impossible, because $-a_{1}K_{X}^{3}\leqslant 1$. Thus,
we see that $P\not\in L$.
Suppose that $P\in T$. It follows from Theorem~5.6.2 in \cite{CPR}
that
$$
ns\geqslant -sa_{1}a_{3}nK_{X}^{3}=D\cdot S\cdot T\geqslant\mathrm{mult}_{P}\big(D\big)s>ns
$$
for some $s\in\mathbb{N}$ and some surface $S\in |-sa_1a_3K_{X}|$.
Hence, we see that $P\not\in T$.
Let $G$ be a general surface in $|-a_{2}K_{X}|$ that contains $P$.
Then $G\cdot D$ is an effective~cycle, but it follows from
Theorem~5.6.2 in \cite{CPR} that there are $s\in\mathbb{N}$ and
$H\in|-sa_3K_{X}|$~such that
$$
ns\geqslant-sa_{2}a_{3}nK_{X}^{3}=D\cdot H\cdot G\geqslant \mathrm{mult}_{P}\big(D\big)s>ns,
$$
because $-a_{2}a_{3}K_{X}^{3}\leqslant 1$. So we have a
contradiction.
\end{proof}
\begin{lemma}
\label{lemma:smooth-points-11-34-53-88} Suppose that $a_{4}$
divides $d$, $1=a_{1}\ne a_{2}$, and $-a_{3}K_{X}^{3}\leqslant 1$.
Then $\mathrm{mult}_{P}(D)\leqslant n$.
\end{lemma}
\begin{proof}
Suppose that $\mathrm{mult}_{P}(D)>n$. Then arguing as in the
proof of Lemma~\ref{lemma:smooth-points-small-K-cube}, we see that
the point $P$ is not contained in the base curve of the pencil
$|-K_{X}|$.
Let $T$ be a general surface in $|-K_{X}|$ that contains $P$. Then
Theorem~5.6.2 in \cite{CPR} implies~that there are
$s\in\mathbb{N}$ and $S\in|-sa_3K_{X}|$ such that
$ns\geqslant-sa_{3}nK_{X}^{3}=D\cdot S\cdot T\geqslant
\mathrm{mult}_{P}\big(D\big)s>ns$.
\end{proof}
\begin{lemma}
\label{lemma:smooth-points-a1-a4} Suppose that $a_{1}\ne a_{2}$
and $-a_{1}a_{4}K_{X}^{3}\leqslant 1$. Then
$\mathrm{mult}_{P}(D)\leqslant n$.
\end{lemma}
\begin{proof}
Suppose that $\mathrm{mult}_{P}(D)>n$.
The~proof~of~Lemma~\ref{lemma:smooth-points-11-34-53-88} implies
that $a_{1}\ne 1$. Arguing as~in the proof of
Lemma~\ref{lemma:smooth-points-small-K-cube}, we see that $P$ is
not contained in the~unique surface of~$|-K_{X}|$.
Let $S$ be a surface in $|-a_{1}K_{X}|$ that contains $P$. Then we
may~assume~that $\mathrm{mult}_{P}\big(S\big)\leqslant a_{1}$,
because $P\not\in T$ and $X$ is sufficiently general. Thus, we
have $S\ne D$.
It follows from Theorem~5.6.2 in \cite{CPR} that there are
$s\in\mathbb{N}$ and $H\in|-sa_4K_{X}|$~such~that~$H$~has
multiplicity at least $s>0$ at $P$ and contains no components of
$D\cdot S$ passing through $P$.~Then
$$
ns\geqslant-sa_{1}a_{4}nK_{X}^{3}=D\cdot S\cdot H\geqslant \mathrm{mult}_{P}\big(D\big)s>ns,
$$
because $-a_{1}a_{4}K_{X}^{3}\leqslant 1$. So we have a
contradiction.
\end{proof}
Taking into account the possible values of
$(a_{1},a_{2},a_{3},a_{4})$, we see~that
$\mathrm{mult}_{P}(D)\leqslant n$ whenever
$$
\gimel\not\in\Big\{6,7,8,9,10,12,13,14,16,18,19,20,22,23,24,25,32,33,38\Big\}
$$
by Lemmas~\ref{lemma:smooth-points-small-K-cube},
\ref{lemma:smooth-points-11-34-53-88},
\ref{lemma:smooth-points-a1-a4}. The log pair $(X, \frac{1}{n}D)$
is log canonical at $P$ if $\mathrm{mult}_{P}(D)\leqslant n$ (see
\cite{Ko97}).
\begin{lemma}
\label{lemma:smooth-points-18} Suppose that $\gimel=18$. Then $(X,
\frac{1}{n}D)$ is log canonical at $P$.
\end{lemma}
\begin{proof}
Suppose that the log pair $(X, \frac{1}{n}D)$ is log canonical at
the point $P$. Let us show that this assumption leads to a
contradiction. Note that the inequality $\mathrm{mult}_{P}(D)>n$
holds.
The threefold $X$ is a hypersurface in $\mathbb{P}(1,2,2,3,5)$ of
degree $12$, whose~singularities consist of six points of type
$\frac{1}{2}(1,1,1)$, and a point $O$ of type
$\frac{1}{5}(1,2,3)$. It follows from \cite{Ch06e} that the
diagram
$$
\xymatrix{
&U\ar@{->}[d]_{\alpha}&&W\ar@{->}[ll]_{\beta}\ar@{->}[d]^{\eta}&\\%
&X\ar@{-->}[rr]_{\psi}&&\mathbb{P}(1,2,2)&}
$$
commutes, where $\alpha$ is a weighted blow up of the point $O$
with weights $(1,2,3)$, $\beta$ is a weighted blow up with weights
$(1,1,3)$ of a singular point of type $\frac{1}{3}(1,1,2)$, and
$\eta$ is an elliptic fibration.
Let $C$ be a fiber of the projection $\psi$ that passes through
the point $P$, and $L$ be its irreducible reduced component. We
have $-K_{X}\cdot C=4/5$. But the number $-5K_{X}\cdot L$ is
natural if $L$ contains no points of type $\frac{1}{2}(1,1,1)$.
Then $C=2L$ whenever $C$ contains a point of type
$\frac{1}{2}(1,1,1)$.
Let $T$ be the surface in $|-K_{X}|$, and let $S$ and $\grave{S}$
be general surfaces in $|-2K_{X}|$ that passes through the point
$P$. Then $S$ and $\grave{S}$ are irreducible and $S\supset
L\subset\grave{S}$, but $S\ne D\ne \grave{S}$.
Suppose now that $L$ is contained in $T$. Then $C=2L$ and
$-K_{X}\cdot L=2/5$, but the singularities of the curve $L$
consists of at most double points. Put $D\vert_{T}=mL+\Upsilon$,
where $m$~is~a~non-negative integer, and $\Upsilon$ is an
effective cycle whose support does not contain $L$. Then
$$
2n/5-4m/5=S\cdot\Upsilon\geqslant\mathrm{mult}_{P}\big(\Upsilon\big)\geqslant\mathrm{mult}_{P}\big(D\big)-\mathrm{mult}_{P}\big(L\big)>n-2m,
$$
which implies that $m>n/2$. But $m\leqslant n/2$ by
Remark~\ref{remark:curves}, which implies that $L\not\subset T$.
Suppose that $C=L$. Then $\mathrm{mult}_{P}(L)\leqslant 2$. Put
$D\cdot\grave{S}=\grave{m}C+\grave{\Upsilon}$, where
$\grave{m}$~is~a~non-negative integer and $\grave{\Upsilon}$ is an
effective cycle whose support does not contain $C$. Then
$$
4n/5-8\grave{m}/5=
S\cdot\grave{\Upsilon}\geqslant\mathrm{mult}_{P}\big(\grave{\Upsilon}\big)>n-2m,
$$
which implies that $m>n/2$. But $m\leqslant n/2$ by
Remark~\ref{remark:curves}, which implies that $C\ne L$.
The curve $L$ does not pass through a point of type
$\frac{1}{2}(1,1,1)$, and it follows from the generality of the
threefold $X$ that $C=L+Z$, where $Z$ is an irreducible curve such
that $Z\ne L$.
Put $D\big\vert_{S}=m_{L}L+m_{Z}Z+\Omega$, where $m_{L}$ and
$m_{Z}$ are non-negative integers and $\Omega$ is an effective
cycle whose support does not contain $L$~and~$Z$.
We~may~assume~that $-K_{X}\cdot L\leqslant -K_{X}\cdot Z$, which
implies that either $-K_{X}\cdot L=1/5$ and $-K_{X}\cdot Z=3/5$,
or $-K_{X}\cdot L=-K_{X}\cdot Z=2/5$.
Suppose that $-K_{X}\cdot L=2/5$. Then $L$ and $Z$ are smooth
outside of $O$, and
$$
4n/5-4m_{L}/5-4m_{Z}/5=\grave{S}\big\vert_{S}\cdot\Omega\geqslant\mathrm{mult}_{P}\big(\Omega\big)>n-m_{L}-m_{C},
$$
which implies that $m_{L}+m_{C}>n$. But $m_{L}+m_{C}\leqslant n$
by Remark~\ref{remark:curves}.
Thus, we have $-K_{X}\cdot L=1/5$. The hypersurface $X$ can be
given by an equation
$$
w^{2}z+wg\big(x,y,z,t\big)+h\big(x,y,z,t\big)=0\subset\mathbb{P}\big(1,2,2,3,5\big)\cong\mathrm{Proj}\Big(\mathbb{C}[x,y,z,t,w]\Big),
$$
where $\mathrm{wt}(x)=1$, $\mathrm{wt}(y)=\mathrm{wt}(z)=2$,
$\mathrm{wt}(t)=3$, $\mathrm{wt}(w)=5$, and $g$ and $h$ are
quasihomogeneous polynomials of degree $7$ and $12$,
respectively.
Let $R$ be a surface on the threefold $X$ that is cut out by
$z=0$, and let $\bar{R}$ and $\bar{L}$ be proper transforms of $R$
and $L$ on $U$, respectively. Then~$\bar{R}\cdot \bar{L}<0$, which
implies that $L\subset R\supset Z$,~and the curve $L$ is
contracted by the projection $X\dasharrow\mathbb{P}(1,2,2,3)$ to
a point.
Let $\bar{Z}$ be the proper transform of the curve $Z$ on the
threefold $U$, let $\pi\colon\bar{R}\to R$ be a birational
morphism induced by $\alpha$, and let $\bar{E}$ be the curve on
the surface $\bar{R}$ that is contracted by $\pi$. Then
$$
\bar{L}^{2}=-1,\ \bar{L}\cdot\bar{Z}=\bar{L}\cdot\bar{E}=1,\ \bar{Z}^{2}=-1/3,\ \bar{E}^{2}=-35/6,\ \bar{Z}\cdot\bar{E}=4/3
$$
on the surface $\bar{R}$, which implies that $L^{2}=-29/35$,
$L\cdot Z=43/35$, $Z^{2}=-1/35$ on the surface~$R$.
Suppose that $P\in L\cap Z$. Then $m_{L}+3m_{C}\leqslant 5n$ by
Remark~\ref{remark:curves}, but
$$
n/5+29m_{L}/35-43m_{Z}/35=\Omega\cdot L>n-m_{L}-m_{Z},\ 2n/5-43m_{L}/35+m_{Z}/35=\Omega\cdot Z>n-m_{L}-m_{Z},
$$
which leads to a contradiction. Hence, either $L\ni P\not\in Z$,
or $Z\ni P\not\in L$.
Suppose that $Z\ni P\not\in L$. Then $\Omega\cdot Z>n-m_{Z}$ and
$\Omega\cdot L\geqslant 0$, which implies a contradiction.
Thus, we see that $L\ni P\not\in Z$. Then we have
$$
n/5+29m_{L}/35-43m_{Z}/35=\Omega\cdot L>n-m_{L},\
2n/5-43m_{L}/35+m_{Z}/35=\Omega\cdot Z\geqslant 0,
$$
which implies that $m_{L}<n$. Now it follows from Theorem~7.5 in
\cite{Ko97} that the log pair
$$
\left(R, L+\frac{m_{C}}{n}C+\frac{1}{n}\Omega\right)
$$
is not log
canonical at the point $P$. Then
$\mathrm{mult}_{P}(\Omega\vert_{L})>n$ by~Theorem~7.5 in
\cite{Ko97}, which implies that the inequality $\Omega\cdot L>n$
holds. The inequality $\Omega\cdot L>n$ leads to a contradiction.
\end{proof}
\begin{lemma}
\label{lemma:smooth-points-6} Suppose that $\gimel=6$. Then
$\mathrm{mult}_{P}(D)\leqslant n$.
\end{lemma}
\begin{proof}
Suppose that $\mathrm{mult}_{P}(D)>n$. It follows from
\cite{Ch06e} that the threefold $X$~has two quotient sin\-gu\-lar
points $O_{1}$ and $O_{2}$ of type $\frac{1}{2}(1,1,1)$ such that
there is a commutative~diagram
$$
\xymatrix{
&&&U\ar@{->}[dll]_{\sigma}\ar@{->}[d]^{\alpha}&&Y\ar@{->}[ll]_{\gamma}\ar@{->}[ddr]^{\eta}&&&\\%
&V\ar@{->}[dr]_{\omega}&&X\ar@{-->}[dl]_{\xi}\ar@{-->}[drrr]^{\psi}&&&&&\\
&&\mathbb{P}(1,1,1,4)\ar@{-->}[rrrr]_{\chi}&&&&\mathbb{P}(1,1,1),&&}
$$
where $\xi$, $\psi$ and $\chi$ are projections, $\alpha$ is a blow
up of $O_{1}$ with weights $(1,1,1)$, $\gamma$ is a blow up with
weights $(1,1,1)$ of the point that dominates $O_{2}$, $\eta$ is
an elliptic fibration, $\omega$ is a double cover, and
$\sigma$~is~a~birational morphism that contracts $48$ irreducible
curves $C_{1},\ldots, C_{48}$.
The threefold $U$ contains $48$ curves $Z_{1},\ldots, Z_{48}$ such
that $\alpha(Z_{i})\cup\alpha(C_{i})$ is a fiber of $\psi$ over
the~point $\psi(C_{i})$. Put $\bar{Z}_{i}=\alpha(Z_{i})$ and
$\bar{C}_{i}=\alpha(C_{i})$. Let $L$ be a fiber of the projection
$\psi$~that~passes through the point $P$, and let $T_{1}$ and
$T_{2}$ be general surfaces in $|-K_{X}|$ that contain $P$.
Suppose that $L$ is irreducible. Put $D\cdot T_{1}=mL+\Upsilon$,
where $m$ is non-negative integer and~$\Upsilon$~is an effective
cycle whose support does not contain $L$. Then $m\leqslant n$ by
Remark~\ref{remark:curves}. But
$$
n-m=D\cdot T_{1}\cdot T_{2}-mT_{2}\cdot L=T_{2}\cdot\Delta\geqslant\mathrm{mult}_{P}\big(\Delta\big)>n-m\mathrm{mult}_{P}(L),
$$
which implies that $L$ is singular at the point $P$. Then there
is~a surface $T\in |-K_{X}|$ that~is~singular at the point $P$.
Let $S$ is a general surface in $|-2K_{X}|$ that contains $P$.
Then
$$
2n=D\cdot T\cdot S\geqslant\mathrm{mult}_{P}\big(D\cdot T\big)>2n,
$$
which is a contradiction. Hence, the curve $L$ is reducible.
We have $L=\bar{C}_{i}\cup\bar{Z}_{i}$. Put
$D\vert_{T_{1}}=m_{1}\bar{C}_{i}+m_{2}\bar{Z}_{i}+\Delta$, where
$m_{1}$ and $m_{2}$ are non-negative integers and $\Delta$ is an
effective cycle whose support does not contain $\bar{C}_{i}$ and
$\bar{Z}_{i}$.
In the case when $P\in\bar{C}_{i}\cap\bar{Z}_{i}$, there is $T\in
|-K_{X}|$ such that $T$ singular at $P$, and we can obtain a
contradiction as above. So we may assume that $P\in\bar{C}_{i}$
and $P\not\in\bar{Z}_{i}$. Then
$$
n-m_{1}/2-m_{2}/2=\big(\bar{C}_{i}+\bar{Z}_{i}\big)\cdot\Delta\geqslant\mathrm{mult}_{P}\big(\Delta\big)>n-m_{1}\mathrm{mult}_{P}(\bar{C}_{i})=n-m_{1},
$$
because $\bar{C}_{i}$ is smooth. Hence, we see that $m_{1}>m_{2}$.
But we have
$$
n-m_{1}\leqslant\Delta\cdot\bar{Z}_{i}=n/2-m_{1}\bar{C}_{i}\cdot\bar{Z}_{i}-m_{2}\bar{Z}_{i}^{2}=n/2-2m_{1}+3m_{2}/2<n/2-m_{1}/2,
$$
which gives $m_{1}>m_{2}>n$. But $m_{1}+m_{2}\leqslant 2n$ by
Remark~\ref{remark:curves}. So we have a contradiction.
\end{proof}
Arguing as in the proofs of Lemmas~\ref{lemma:smooth-points-18}
and \ref{lemma:smooth-points-6}, we see that $(X,\frac{1}{n}D)$ is
log canonical at $P$~for
$$
\gimel\in\Big\{7,8,9,10,12,13,14,16,19,20,22,23,24,25,32,33,38\Big\},
$$
which completes the proof Lemma~\ref{lemma:smooth-points}.
\section{Singular points.}
\label{section:singular-points}
In this section we prove the assertion of
Lemma~\ref{lemma:singular-points}. Let us use the assumptions and
notation of Lemma~\ref{lemma:singular-points}. Suppose that $P$ is
a singular point of $U$. Let us derive a contradiction.
The point $P$ is a singular point of type
$\frac{1}{\bar{r}}(1,\bar{a},\bar{r}-\bar{a})$, where $\bar{a}$
and $\bar{r}$ are coprime natural numbers such that
$\bar{r}>2\bar{a}$. Let $\beta\colon W\to U$ be a blow up~of $P$
with weights $(1,\bar{a},\bar{r}-\bar{a})$. Then
$$
-K_{W}^{3}=-K_{X}^{3}-\frac{1}{ra(r-a)}-\frac{1}{\bar{r}\bar{a}(\bar{r}-\bar{a})}.
$$
Let $\breve{D}$ be the proper transform of $D$ on $W$. There is a
rational number $\nu$ such that
$$
\breve{D}\equiv (\alpha\circ\beta)^{*}\big(-nK_{X}\big)-\mu\beta^{*}\big(E\big)-\nu G,
$$
where $G$ is the $\beta$-exceptional divisor. Then
$$
K_{W}+\frac{1}{n}\breve{D}+\big(\mu/n-1/r\big)\breve{E}\equiv
\beta^{*}\Big(K_{U}+\frac{1}{n}\bar{D}+\big(\mu/n-1/r\big)E\Big)-\epsilon G\equiv -\epsilon G,
$$
where $\breve{E}$ is a proper transform of $E$ on the threefold
$W$, and $\epsilon\in\mathbb{Q}$. Then $\epsilon>0$ due to
\cite{Ka96}.
\begin{lemma}
\label{lemma:singular-points-K-cube-is-not-zero} The inequality
$-K_{W}^{3}\ne 0$ holds.
\end{lemma}
\begin{proof}
Suppose that $-K_{W}^{3}=0$. Then it follows from \cite{Ch06e}
that the linear system $|-rK_{W}|$ induces an elliptic fibration
$\eta\colon W\to Y$ for $r\gg 0$. Then $0\leqslant \breve{D}\cdot
C=-\epsilon G\cdot C<0$, where $C$ is a general fiber of the
elliptic fibration $\eta$. So we have a contradiction.
\end{proof}
Thus, it follows from \cite{Ch06e} that either $-K_{W}^{3}<0$, or
$-K_{W}$ is nef and big.
\begin{lemma}
\label{lemma:singular-points-big-not-nef} Suppose that
$-K_{W}^{3}<0$. Then $-K_{W}$ is not big.
\end{lemma}
\begin{proof}
Suppose that $-K_{W}$ is big. Then it follows from \cite{Ch06e}
that we have the following possibilities:
\begin{itemize}
\item the equality $\gimel=25$ holds, and $O$ is a singular point of type $\frac{1}{7}(1,3,4)$;
\item the equality $\gimel=43$ holds, and $O$~is a singular point of type $\frac{1}{9}(1,4,5)$;
\end{itemize}
but both cases are similar. So we assume that $\gimel=43$. Then
$-K_{W}-4\beta^{*}(K_{U})$ is nef (see~\cite{Ch06e}), and there is
a surface $H$ in the linear system $|-2K_{X}|$ such that
$$
\breve{H}\equiv (\alpha\circ\beta)^{*}\big(-2K_{X}\big)-\frac{11}{9}\beta^{*}(E)-\frac{3}{2}G,\\%
$$
where $\breve{H}$ is a proper transform of the surface $H$ on the
threefold $W$. Thus, we have
$$
0\leqslant
\breve{H}\cdot\breve{D}\cdot\Big(-K_{W}-4\beta^{*}\big(K_{U}\big)\Big)=5n/9-11\mu/4-\nu,
$$
which is impossible, because $\nu-n/3+3\mu/4=n\epsilon>0$ and
$\mu>n/9$.
\end{proof}
Let $T$ be a surface in $|-K_{X}|$, and $\mathcal{P}$ be the
pencil generated by the divisors $nT$ and $D$. Then
\begin{equation}
\label{equation:crepant}
\mathcal{B}\equiv -nK_{W}\equiv (\alpha\circ\beta)^{*}\big(-nK_{X}\big)-\frac{n}{r}\beta^{*}\big(E\big)-\frac{n}{\bar{r}} G,
\end{equation}
where $\mathcal{B}$ is the proper transforms of the pencil
$\mathcal{P}$ on the threefold $W$.
\begin{lemma}
\label{lemma:singular-points-nef-big} The divisor $-K_{W}$ is nef
and big.
\end{lemma}
\begin{proof}
Suppose that the divisor $-K_{W}$ is not nef and big. Then
$-K_{W}^{3}<0$, but $-K_{W}$ is not big by
Lemma~\ref{lemma:singular-points-big-not-nef}. Then the
equivalence~\ref{equation:crepant}~almost
uniquely~determines\footnote{For example, it follows from
\cite{ChPa05h} that the equivalence~\ref{equation:crepant} implies
that $n=1$ in the case when $a_{1}=1$.} the pencil $\mathcal{P}$
due to \cite{ChPa05h}.
All possible cases are similar. So we assume that
$\gimel\in\{45,48,58,69,74,79\}$. Then $O$~is~a~singular point of
type $\frac{1}{a_{4}}(1,a_{1},a_{3})$, and $X$ can be given by an
equation
$$
w^{2}z+wf\big(x,y,z,t\big)+g\big(x,y,z,t\big)=0\subset\mathbb{P}\big(1,a_{1},a_{2},a_{3},a_{4}\big)\cong\mathrm{Proj}\Big(\mathbb{C}[x,y,z,t,w]\Big),
$$
where $\mathrm{wt}(x)=1$, $\mathrm{wt}(y)=a_{1}$,
$\mathrm{wt}(z)=a_{2}$, $\mathrm{wt}(t)=a_{3}$,
$\mathrm{wt}(w)=a_{4}$, and $f$ and $g$~ are polynomials.
Let $S$ be a surface that is cut out on the threefold $X$ by
$z=0$, and $\mathcal{M}$ be a pencil generated by the divisors
$a_{2}T$ and $S$. Then it follows from \cite{ChPa05h} that either
$\mathcal{P}=\mathcal{M}$, or $\mathcal{P}=|-a_{1}K_{X}|$.
Suppose that $\mathcal{P}=|-a_{1}K_{X}|$. Then $\mu=n/a_{1}$,
which is impossible, because $\mu>n/a_{4}$.
We see that $\mathcal{P}=\mathcal{M}$. Let $M$ be a divisor in
$\mathcal{M}$, and $\bar{M}$ be its proper transform on $U$. Then
$$
\bar{M}\equiv \alpha^{*}\big(M\big)-\frac{a_{3}}{a_{4}}E
$$
in the case when $M\ne S$, but $\mu>n/a_{4}$. Thus, we see that
$D=S$, but $(X, \frac{1}{a_{2}} S)$ is log~ca\-no\-ni\-cal at the
point $O$ by Lemma~8.12 and Proposition~8.14~in~\cite{Ko97}, which
is a contradiction.
\end{proof}
Taking into account the possible values of
$(a_{1},a_{2},a_{3},a_{4})$, we see~that
$$
\gimel\in\Big\{8,12,13,16,20,24,25,26,31,33,36,38,46,47,48,54,56,58,65,74,79\Big\}
$$
by Lemmas~\ref{lemma:singular-points-K-cube-is-not-zero},
\ref{lemma:singular-points-big-not-nef} and
\ref{lemma:singular-points-nef-big} (see \cite{Ch06e}).
\begin{lemma}
\label{lemma:singular-points-20-25-31-33-38-58-nef} The case
$\gimel\not\in\{12,13,20,25,31,33,38,58\}$ is impossible.
\end{lemma}
\begin{proof}
Suppose that $\gimel\not\in\{12,13,20,25,31,33,38,58\}$. Then
$r=a_{4}$, $r-a=a_{3}$, $\bar{r}=r-a$, $\bar{a}=a$, and
$n\epsilon=\nu-(\bar{r}-\bar{a})(n/r-\mu)/\bar{r}-n/\bar{r}$. We
may assume that $\gimel\ne 24$, because the case $\gimel=24$~can
be considered in a similar way. Then $X$ can be given by the
equation
$$
w^{2}z+wf\big(x,y,z,t\big)+g\big(x,y,z,t\big)=0\subset\mathbb{P}\big(1,a_{1},a_{2},a_{3},a_{4}\big)\cong\mathrm{Proj}\Big(\mathbb{C}[x,y,z,t,w]\Big),
$$
where $\mathrm{wt}(x)=1$, $\mathrm{wt}(y)=a$,
$\mathrm{wt}(z)=d-2a_{4}$, $\mathrm{wt}(t)=a_{3}$,
$\mathrm{wt}(w)=a_{4}$, the point~$O$~is~given by the equations
$x=y=z=t=0$, and $f$ and $g$~are
quasi\-ho\-mo\-ge\-ne\-ous~polynomials. Then
$$
\breve{R}\equiv (\alpha\circ\beta)^{*}\big(-a_{2}K_{X}\big)-\frac{d-r}{r}\beta^{*}\big(E\big)-\frac{\bar{r}-\bar{a}}{\bar{r}} G,
$$
where $\breve{R}$ is a proper transform on $W$ of the surface cut
out on $X$ by $z=0$. Then $\breve{D}\ne\breve{R}$,~and
$$
n\frac{\sum_{i=1}^{4}a_{i}}{a_{1}a_{3}a_{4}}-\frac{\mu\big(d-r\big)}{a\big(r-a\big)}-\frac{\nu\big(\bar{r}-\bar{a}\big)}{\bar{a}\big(\bar{r}-\bar{a}\big)}=-K_{W}\cdot\breve{D}\cdot\breve{R}\geqslant 0,
$$
which implies that $\mu<n/r$, because $\epsilon>0$. However we
know that $\mu>n/r$.
\end{proof}
So, the divisor $-K_{W}$ is nef and big, and
$\gimel\in\{12,13,20,25,31,33,38,58\}$, which implies that
$$
r=a_{4},\ r-a=a_{3},\ \bar{a}=a_{1},\ \bar{r}-\bar{a}=a_{2},\ a_{2}\ne a_{3},\ n\epsilon=\nu+(r-2a)\mu/(r-a)-2n/r
$$
due to \cite{Ch06e}. Then $W$ has a singular point $\bar{P}\ne P$
of type $\frac{1}{\bar{r}}(1,\bar{a},\bar{r}-\bar{a})$ such that
the diagram
$$
\xymatrix{
U\ar@{->}[d]_{\alpha}&&W\ar@{->}[ll]_{\beta}&&V\ar@{->}[ll]_{\gamma}\ar@{->}[d]^{\eta}\\
X\ar@{-->}[rrrr]_{\psi}&&&&\mathbb{P}\big(1,a_{1},a_{2}\big)}
$$
commutes, where $\psi$ is a natural projection, $\gamma$ be a blow
up of the point $\bar{P}$ with weights
$(1,\bar{a},\bar{r}-\bar{a})$, and $\eta$ is an elliptic
fibration. Let $F$ be the exceptional divisor of $\gamma$, and
$\bar{G}$ be the proper transform of the divisor $G$ on the
threefold $V$. Then $F$ and $\bar{G}$ are sections of $\eta$, and
$G\not\ni\bar{P}\not\in \breve{E}$.
It follows from the inequality $-K_{W}\cdot \breve{D}\geqslant 0$
and the proof of Lemma~\ref{lemma:discrepancy} that $\epsilon<1$,
which implies that the log pair $(W,
\frac{1}{n}\breve{D}+(\mu/n-1/r)\breve{E}+\epsilon G)$ is not log
canonical at some point $Q\in G$.
\begin{lemma}
\label{lemma:singular-points-singular-point} The threefold $W$ is
smooth at the point $Q$.
\end{lemma}
\begin{proof}
Suppose that $W$ is singular at the point $Q$. Then $Q$ is a
singular point of type $\frac{1}{\breve{r}}(1,1,\breve{r}-1)$,
where either $\breve{r}=\bar{r}-\bar{a}$, or $\breve{r}=\bar{a}\ne
1$. Let $\omega\colon \breve{W}\to W$ be a blow up of $Q$ with
weights~$(1,1,\breve{r}-1)$, and $\mathcal{H}$ be the proper
transform of $\mathcal{P}$ on $\breve{W}$. Then it follows from
\cite{Ka96} that $\mathcal{H}\equiv -nK_{\breve{W}}$, which
implies that $n=r\mu=a_{1}$ due to \cite{ChPa05h}. However we
know that $\mu>n/r$.
\end{proof}
Thus, it follows from
Lemma~\ref{lemma:singular-points-singular-point} that
$\mathrm{mult}_{Q}(\breve{D})>n-n\epsilon-(\mu-n/r)\mathrm{mult}_{Q}(\breve{E})$.
\begin{lemma}
\label{lemma:singular-points-T} There is a surface $T\in |-K_{W}|$
such that $Q\in T$.
\end{lemma}
\begin{proof}
The existence of a surface $T\in |-K_{W}|$ that passes through the
point $Q$ is obvious in the~case when $a_{1}=1$. Thus, we may
assume that $a_{1}\ne 1$. Then $\gimel\in\{33,38,58\}$, but we
consider only the case $\gimel=38$, because the cases $\gimel=33$
and $\gimel=58$ can be considered in a similar way.
Suppose that $\gimel=38$. Then there is a unique surface $T\in
|-K_{W}|$. Suppose that $Q\not\in T$.
Arguing as in the proof of Lemma~\ref{lemma:lower-bound-for-mu},
we see that $\mathrm{mult}_{Q}(\breve{D})\leqslant
(a_{1}+a_{2})\nu/a_{1}$. Then
$$
\nu\frac{a_{1}+a_{2}}{a_{1}}>n-\big(\mu-n/7\big)-\big(\nu+3\mu/5-2n/7\big),
$$
but $\mathrm{mult}_{Q}(\breve{D})>n+n/r-\mu-n\epsilon$. Thus, we
have $\mu>55n/56-5\nu/2$.
The inequality $-K_{W}\cdot \breve{D}\geqslant 0$ and the proof of
Lemma~\ref{lemma:discrepancy} imply that $\nu\leqslant 10\mu/7$
and $\mu\leqslant 9n/40$, respectively. The hypersurface $X$ can
be given by the equation
$$
w^{2}y+w\Big(t^{2}+tf_{5}\big(x,y,z\big)+f_{10}\big(x,y,z\big)\Big)+tf_{13}\big(x,y,z\big)+f_{18}\big(x,y,z\big)=0\subset\mathrm{Proj}\Big(\mathbb{C}[x,y,z,t,w]\Big),
$$
where $\mathrm{wt}(x)=1$, $\mathrm{wt}(y)=2$, $\mathrm{wt}(z)=3$,
$\mathrm{wt}(t)=5$, $\mathrm{wt}(w)=8$, and $f_{i}(x,y,z)$ is a
quasi\-ho\-mo\-ge\-ne\-ous polynomial of degree $i$. Let
$\breve{S}$ be the proper transform on the threefold $W$ of the
surface that is cut out on $X$ by the equation
$wy+(t^{2}+tf_{5}(x,y,z)+f_{10}(x,y,z))=0$. Then
$$
\breve{S}\equiv \big(\alpha\circ\beta\big)^{*}\big(-10K_{X}\big)-\frac{18}{8}\beta^{*}\big(E\big)-\frac{13}{5}G,
$$
but $\breve{S}\ne\breve{D}$. The divisor $-K_{W}$ is nef. Hence,
we have
$$
0\leqslant -K_{W}\cdot\breve{D}\cdot\breve{S}=3n/4-6\mu/5-13\nu/6,
$$
but $\nu\leqslant 8\mu/5$, which implies $\nu\leqslant 9n/35$. Now
we can easily obtain a contradiction.
\end{proof}
It follows from \cite{Ch06e} that $|-rK_{W}|$ does not have base
points for $r\gg 0$ and induces a birational morphism
$\omega\colon W\to\bar{W}$ such that $\bar{W}$ is a hypersurface
in $\mathbb{P}(1,a_{1},a_{2},2a_{3},3a_{3})$ of degree $6a_{3}$
\begin{lemma}
\label{lemma:singular-points-omega} The morphism $\omega$ is not
an isomorphism in a neighborhood of the point $Q$.
\end{lemma}
\begin{proof}
Suppose that $\omega$ is an isomorphism in a neighborhood of the
point $Q$. Then it follows from the proof of Theorem~5.6.2 in
\cite{CPR} that there is $R\in|-s2a_{1}a_{3}K_{W}|$ such that
$\mathrm{mult}_{Q}(R)\geqslant s$, but $R$ does~not contain
components of the cycle $\breve{D}\cdot S$ that pass through $Q$,
where $s\in\mathbb{N}$. Then
$$
s2a_{3}\big(-na_{1}K_{X}^{3}-\mu/a_{3}-\nu/a_{2}\big)=R\cdot\breve{D}\cdot
T\geqslant\mathrm{mult}_{Q}\big(\breve{D}\cdot T\big)s
>\big(n-\nu-\mu(a_{3}-a_{1})/a_{3}+2n/a_{4}\big)s,
$$
because $Q\not\in\breve{E}$. Now we can derive a contradiction
using $n\epsilon=\nu+(a_{3}-a_{1})\mu/a_{3}-2n/a_{4}>0$.
\end{proof}
It follows from Lemma~\ref{lemma:singular-points-omega} that there
is a unique curve $C\subset W$~that contains $Q$ such~that
$$
-K_{W}\cdot C=0,\ \beta^{*}\big(-K_{U}\big)\cdot C=1/a_{4},\ C\cdot G=1,
$$
which implies that $\gimel\not\in\{33,38,58\}$ by
Lemma~\ref{lemma:singular-points-T}. Hence, we have
$\gimel\in\{12,13,20,25,31\}$.
Put $\breve{D}\cdot T=mC+\Omega$, where $m$ is a non-negative
integer, and $\Omega$ is an effective one-cycle, whose support
does not contain the curve $C$. Then it follows from
Remark~\ref{remark:curves} that
$$
m\leqslant 5n/4-\mu,\ m\leqslant 11n/15-\mu/2,\ m\leqslant 13n/15-\mu,\ m\leqslant 5n/7-\mu/3,\ m\leqslant 2n/3-\mu
$$
in the case when $\gimel=12,13,20,25,31$, respectively. Recall
that $\bar{G}$ is a section of $\eta$.
Let $\mathcal{H}$ be a pencil in $|-a_{2}K_{W}|$ of surfaces
passing through the point $Q$, and $H$ be a general surface in
$\mathcal{H}$. Then $C$ is the only curve in the base locus of
$\mathcal{H}$ that passes through $Q$. Then
$$
-na_{2}K_{X}^{3}-a_{2}\mu/(a_{1}a_{3})-\nu/a_{1}=H\cdot\Omega\geqslant\mathrm{mult}_{Q}\big(\Omega\big)>n-\nu-\mu(a_{3}-a_{1})/a_{3}+2n/a_{4}-m,
$$
which immediately implies that either $\gimel=12$, or $\gimel=13$.
\begin{lemma}
\label{lemma:singular-points-12} The inequality $\gimel\ne 12$
holds.
\end{lemma}
\begin{proof}
Suppose that $\gimel=12$. Let $R$ be a sufficiently general
surface in $|-2K_{W}|$ that contains the~point $Q$. Then
$R\vert_{T}=C+L+Z$, where $L=G\vert_{T}$, the curve $Z$ is
reduced, and $P\not\in\beta(Z)$.
Suppose that $Z$ is irreducible. Then $Z^{2}=-4/3$, $C^{2}=-2$,
$L^{2}=-3/2$ on the surface $T$. Put
$$
\breve{D}\vert_{T}=m_{C}C+m_{L}L+m_{Z}Z+\Upsilon,
$$
where $m_{C}$, $m_{L}$ and $m_{Z}$ are non-negative integers, and
$\Upsilon$ is an effective one-cycle, whose support does not
contain the curve $C$, $L$ and $Z$.
Suppose that $Q\not\in\breve{E}$. Then $m_{C}>2n/3-m_{Z}/3$,
because
$$
5n/6-2\mu/3-\nu=R\cdot\breve{D}\cdot T=m_{L}+m_{Z}/3+R\cdot\Upsilon>m_{L}+m_{Z}/3+3n/2-\nu-2\mu/3-m_{L}-m_{C},
$$
but $4m_{Z}/3\geqslant 2m_{C}-n/3$, because $\Upsilon\cdot
Z\geqslant 0$. Thus, we have
$$
m_{C}>2n/3+m_{Z}/3\geqslant 7n/12+m_{C}/2,
$$
which gives $m_{C}>7n/6$, but $m_{C}\leqslant 5n/6$ by
Remark~\ref{remark:curves}, because
$-K_{X}\cdot\alpha\circ\beta(C)=5/6$.
Thus, we see that $Q\in\breve{E}$. Then $C\subset\breve{E}$ and
$\beta(C)\in|\mathcal{O}_{\mathbb{P}(1,\,1,\,3)}(1)|$, but
$$
5n/6-2\mu/3-\nu=R\cdot\breve{D}\cdot T=m_{L}+m_{Z}/3+R\cdot\Upsilon>m_{L}+m_{Z}/3+7n/4-\nu-5\mu/3-m_{L}-m_{C},
$$
which gives $m_{C}>11n/12-\mu+m_{Z}/3$. We have
$-K_{X}\cdot\alpha\circ\beta(Z)=5/6$ and $Z\cdot\breve{E}=2$, but
$$
4m_{Z}/3\geqslant 2m_{C}+2\mu-5n/6,
$$
because $Z\cdot\Upsilon\geqslant 0$. Thus, we have $m_{Z}>3n/2$,
but $m_{Z}\leqslant n/2$ by Remark~\ref{remark:curves}.
Therefore, the curve $Z$ is reducible. Then $Q\in\breve{E}$ and
$Z=\acute{Z}+\grave{Z}$, where $\acute{Z}$ and
$\grave{Z}$~are~irreducible curves such that
$G\cdot\acute{Z}=G\cdot\grave{Z}=-K_{U}\cdot\beta(\grave{Z})=0$
and $-K_{X}\cdot\alpha\circ\beta(\acute{Z})=7/12$. Then
$$
\acute{Z}^{2}=-4/3,\ \grave{Z}^{2}=C^{2}=-2,\ L^{2}=-3/2,\ L\cdot C=\acute{Z}\cdot C=\acute{Z}\cdot\grave{Z}=\grave{Z}\cdot C=1,\ L\cdot\grave{Z}=L\cdot\acute{Z}=0
$$
on the surface $T$. Put
$\breve{D}\vert_{T}=\bar{m}_{C}C+\bar{m}_{L}L+\bar{m}_{Z}\acute{Z}+\Phi$,
where $\bar{m}_{C}$, $\bar{m}_{L}$, $\bar{m}_{Z}$ are non-negative
integers, and $\Phi$ is an effective cycle, whose support does not
contain $C$, $L$ and $\acute{Z}$. Then
$$
R\big\vert_{T}\cdot\Phi\geqslant\mathrm{mult}_{Q}\big(\Phi\big)>7n/4-\nu-5\mu/3-m_{L}-m_{C},
$$
and $\Phi\cdot \acute{Z}\geqslant 0$. We have
$\beta^{*}(-K_{U})\vert_{T}\cdot\Phi\geqslant 0$. Thus, we see
that
$$
\bar{m}_{C}>11n/12-\mu+\bar{m}_{Z}/3,\ 4\bar{m}_{Z}/3\geqslant \bar{m}_{C}+\mu-5n/6,\ \bar{m}_{C}+\mu\leqslant 5/4-\bar{m}_{Z},
$$
but this system of linear inequalities is inconsistent, which
completes the proof.
\end{proof}
Thus, we see that $\gimel=13$. Then $C\subset\breve{E}$, because
otherwise we have
$$
2\Big(11n/30-\mu/6-\nu/2\Big)=H\cdot\Omega\geqslant\mathrm{mult}_{Q}\big(\Omega\big)>7n/5-\nu-\mu/3-m,
$$
which implies that $m>2n/3$, which is impossible, because
$m\leqslant 11n/15-\mu/2$ and $\mu>n/5$.~Put
$$
\bar{D}\big\vert_{\breve{E}}=\bar{m}C+\Upsilon,
$$
where $\bar{m}$ is a non-negative integer, and $\Upsilon$ is an
effective cycle, whose support does not contain the curve $C$.
Then $\bar{m}\leqslant 5\mu/2$, because we have
$\beta(C)\in|\mathcal{O}_{\mathbb{P}(1,\,2,\,3)}(2)|$ and the
curve $C$ is reduced, where $E\cong\mathbb{P}(1,2,3)$. Then
$\bar{m}\leqslant 11n/12$, because $\mu\leqslant 11n/30$.
The log pair $(W,\ \frac{1}{n}\breve{D}+\breve{E}+\epsilon G)$ is
not log canonical at the point $Q$. Hence, the log pair
$$
\left(\breve{E},\ C+\frac{\nu+\mu/3-2n/5}{n}G\big\vert_{\breve{E}}+\frac{1}{n}\Upsilon\right)
$$
is not log canonical at the point $Q$ by Theorem~7.5 in
\cite{Ko97}. Then
$$
5\mu/3-\nu=\big(\bar{m}C+\Upsilon\big)\cdot C=\Upsilon\cdot C>7n/5-\nu-\mu/3,
$$
because $\mathrm{mult}_{Q}(\Upsilon\vert_{C})>7n/5-\nu-\mu/3$ by
Theorem~7.5 in \cite{Ko97}. Thus, we see that $\mu>7n/10$,~which
is impossible, because $\mu\leqslant 11n/30<7n/10$. The assertion
of Lemma~\ref{lemma:singular-points} is proved.
\section{Quadratic involutions.}
\label{section:quadratics-involutions}
In this section we prove the assertion of
Lemma~\ref{lemma:quadratic-involutions}. Let us use the
assumptions and notation of
Lemma~\ref{lemma:quadratic-involutions}. Suppose that
$d=2r+a_{j}$. To prove Lemma~\ref{lemma:quadratic-involutions} we
must derive a contradiction.
It follows from the equality $d=2r+a_{j}$ that the threefold $X$
can be given by the equation
$$
x_{i}^{2}x_{j}+x_{i}f\big(x_{0},x_{1},x_{2},x_{3},x_{4}\big)+g\big(x_{0},x_{1},x_{2},x_{3},x_{4}\big)=0\subset\mathrm{Proj}\Big(\mathbb{C}[x_{0},x_{1},x_{2},x_{3},x_{4}]\Big),
$$
where $i\ne j$, $a_{i}=r$, $\mathrm{wt}(x_{0})=1$,
$\mathrm{wt}(x_{k})=a_{k}$, $f$ and $g$ are
quasi\-ho\-mo\-ge\-ne\-ous polynomials that do not depend on
$x_{i}$. Put $\bar{a}_{3}=a_{3+4-i}$, $\bar{a}_{4}=a_{i}a_{j}$,
$\bar{d}=2\bar{a}_{4}$. Then there is a commutative~diagram
$$
\xymatrix{
&U\ar@{->}[d]_{\sigma}\ar@{->}[rrrr]^{\alpha}&&&&X\ar@{-->}[d]^{\xi}\\
&V\ar@{^{(}->}[rr]&&\mathbb{P}\big(1,a_{1},a_{2},\bar{a}_{3},\bar{a}_{4}\big)\ar@{-->}[rr]_{\chi}&&\mathbb{P}\big(1,a_{1},a_{2},\bar{a}_{3}\big),}
$$
where $\xi$ and $\chi$ are projections, and $\sigma$ is a
birational morphism that contracts smooth irreducible rational
curves $C_{1},\ldots, C_{l}$ such that $V$ is a hypersurface in
$\mathbb{P}(1,a_{1},a_{2},\bar{a}_{3},\bar{a}_{4})$ of degree
$\bar{d}$~with terminal non-$\mathbb{Q}$-factorial singularities,
where $l=a_{i}a_{j}(d-a_{i})\sum_{i=1}^{4}a_{i}$. Then
$-K_{X}\cdot\alpha(C_{k})=1/a_{i}$.
Let $M$ be the surface that is cut out on the threefold $X$ by
$x_{j}=0$, and $\bar{M}$ be the proper transform of $M$ on the
threefold $U$. Then $M\ne D$ by Lemma~8.12 and Proposition~8.14 in
\cite{Ko97}.
\begin{lemma}
\label{lemma:inequality-for-mu} The inequalities $\mu\leqslant
-a_{j}nK_{X}^{3}(r-a)a/(d-r)\leqslant n(d-r)/(ra_{j})$ hold.
\end{lemma}
\begin{proof}
The divisor $-K_{U}$ is nef. The inequality $\mu\leqslant
-a_{j}nK_{X}^{3}(r-a)a/(d-r)$ follows from
$$
0\leqslant -K_{U}\cdot\bar{M}\cdot\bar{D}=-a_{j}nK_{X}^{3}-\mu\big(d-r\big)/\big(ar-a^{2}\big),
$$
and to conclude the proof we must show that
$-a_{j}nK_{X}^{3}(r-a)a/(d-r)\leqslant n(d-r)/(ra_{j})$.
\newpage
Suppose that $-a_{j}nK_{X}^{3}(r-a)a/(d-r)>n(d-r)/(ra_{j})$. Then
$$
\frac{d-r}{ra_{j}}<-a_{j}K_{X}^{3}\frac{\big(r-a\big)a}{d-r}=\frac{da_{j}(r-a)a}{\big(d-r\big)a_{1}a_{2}a_{3}a_{4}},
$$
but $a_{1}a_{2}a_{3}a_{4}\geqslant a_{j}r(r-a)a$. Thus, we have
$(d-r)^{2}<d(d-2r)$, which is a contradiction.
\end{proof}
We have $E\cong\mathbb{P}(1,a,r-a)$, and
$|\mathcal{O}_{\mathbb{P}(1,\,a,\,r-a)}(1)|$ consists of a single
curve when $a\ne 1$.
\begin{lemma}
\label{lemma:a-not-1} The inequality $a\ne 1$ holds.
\end{lemma}
\begin{proof}
Suppose that $a=1$. Taking into account the possible values of
$(a_{1},a_{2},a_{3},a_{4})$, we see~that
$$
\gimel\in\Big\{6,7,8,9,12,13,16,15,17,20,25,26,30,36,31,41,47,54\Big\},
$$
but we only consider the cases $\gimel=7$ and $\gimel=36$. The
remaining $16$ cases can be~considered in a similar way. So the
reader can easily obtain a contradiction in these cases by
himself.
Suppose that $\gimel=7$. Then $X$ is a hypersurface in
$\mathbb{P}(1,1,2,2,3)$ of degree $8$, which implies~that the
point $O$ is a singular point of type $\frac{1}{3}(1,1,2)$. Let
$S$ be the unique surface in $|-K_{U}|$ that contains the point
$P$. Then $S$ is an irreducible surface, which is smooth at the
point $P$.
The singularities of $U$ consists of singular points $P_{0}$,
$P_{1}$, $P_{2}$, $P_{3}$ and $P_{4}$ of type $\frac{1}{2}(1,1,1)$
such that $P_{0}$ is a singular point of $E$. It follows from
\cite{Ch06e} that there is a commutative diagram
$$
\xymatrix{
&U\ar@{->}[d]_{\alpha}&&Y_{i}\ar@{->}[ll]_{\beta_{i}}\ar@{->}[drr]^{\eta_{i}}&&&\\
&X\ar@{-->}[rrrr]_{\xi_{i}}&&&&\mathbb{P}(1,1,2),&}
$$
where $\xi_{i}$ is a projection, $\beta_{i}$ is a blow of $P_{i}$
with weights $(1,1,1)$, and $\eta_{i}$ is a morphism.
Suppose that $P\not\in \cup_{i=1}^{l}C_{i}$. The proper transform
of $E$ on the threefold $Y_{i}$ is a section of $\eta_{i}$ in
the~case when $i\ne 0$. Hence, there is a~surface~$H\in |-2K_{U}|$
such that
$$
2\big(2n/3-\mu/2\big)=\bar{D}\cdot H\cdot S\geqslant \mathrm{mult}_{P}\big(\bar{D}\big)>4n/3-\mu,
$$
which is a contradiction. So we may assume that $P\in C_{1}$. Then
$-K_{X}\cdot\alpha(C_{1})=1/3$.
Let $Z_{1}$ be an irreducible curve such that
$\bar{M}\vert_{S}=C_{1}+Z_{1}$. Put $L=E\vert_{S}$. Then
$$
C_{1}^{2}=-2,\ Z_{1}^{2}=L^{2}=-3/2,\ C_{1}\cdot Z_{1}=L\cdot C_{1}=1,\ L\cdot Z_{1}=3/2
$$
on the surface $S$. Put
$\bar{D}\vert_{S}=m_{C}C_{1}+m_{Z}Z_{1}+m_{L}L+\Omega$, where
$m_{C}$, $m_{Z}$ and $m_{L}$ are non-negative integers, and
$\Omega$ is an effective cycle, whose support does not contain
$C_{1}$, $Z_{1}$ and $L$. Then
$$
n-3\mu/2+3m_{Z}/2-3m_{L}/2-m_{C}=Z_{1}\cdot\Omega\geqslant 0,\ 3\mu/2-3m_{Z}/2+3m_{L}/2-m_{C}=L\cdot\Omega\geqslant 0,
$$
which implies that $3m_{Z}/2\geqslant 3(\mu+m_{L})/2+m_{C}-n$ and
$3(\mu+m_{L})/2\geqslant 3m_{Z}/2+m_{C}$. Then
$$
4n/3-\mu-m_{L}-m_{Z}=\big(L+C_{1}+Z_{1}\big)\cdot\Omega\geqslant\mathrm{mult}_{P}\big(\Omega\big)>4n/3-\mu-m_{L}-m_{C},
$$
which gives $m_{C}>m_{Z}$ and $4n/3\geqslant \mu+m_{L}+m_{Z}$. So
we see that $m_{Z}\leqslant n/2$ and $m_{C}\leqslant n/2$.
Then it follows from Theorem~7.5 in \cite{Ko97} that the log pair
$$
\left(S,\ C_{1}+\frac{\bar{m}_{L}+\mu-n/3}{n}L+\frac{m_{Z}}{n}Z+\frac{1}{n}\Omega\right)
$$
is not log canonical at $P$, because $m_{C}\leqslant n$. So it
follows from Theorem~7.5 in \cite{Ko97} that
$$
C_{1}\cdot\Omega\geqslant\mathrm{mult}_{P}\Big(\Omega\big\vert_{C_{1}}\Big)>n-m_{L}-\mu+n/3,
$$
which implies that $m_{C}>m_{Z}/2+n/2$, but $m_{C}\leqslant n/2$.
So the case $\gimel=7$ is impossible.
Now we suppose that $\gimel=36$. Then $X$ is a general
hypersurface in $\mathbb{P}(1,1,4,6,7)$ of degree $18$,
and~$O$~is~a~singular point of type $\frac{1}{7}(1,1,6)$. Arguing
as in the case $\gimel=7$, we see that $P\not\in
\cup_{i=1}^{l}C_{i}$, which implies that we may assume that $P\in
C_{1}$. Put $L=C_{1}$.
Let $S$ be a surface in $|-K_{U}|$ such that $P\in S$. Then
$\bar{M}\vert_{S}=L+Z$, where $Z$ is an irreducible curve. Put
$C=E\vert_{S}$. Then the intersection form of $C$, $L$, $Z$ on $S$
is given~by
$$
Z^{2}=C^{2}=-7/6,\ L^{2}=-2,\ Z\cdot L=C\cdot L=1,\ Z\cdot C=5/6,
$$
and $P$ is the intersection point of the curves $L$ and $C$. Put
$$
\bar{D}\big\vert_{S}=m_{L}L+m_{C}C+m_{Z}Z+\Omega,
$$
where $m_{L}$, $m_{C}$ and $m_{Z}$ are a non-negative integers,
and $\Omega$ is an effective cycle, whose support does not contain
the curves $L$, $C$, and $Z$.
It follows from the proof of Theorem~5.6.2 in \cite{CPR} that we
can find $H\in|-s6K_{U}|$ that has multiplicity at least $s>0$ at
the point $P$, but does~not contain~com\-po\-nents of $\Omega$
that pass through the point $P$, where $s$ is a natural number.
Then
$$
s6\big(3n/28-\mu/6-m_{C}/6-m_{Z}/6\big)=H\big\vert_{S}\cdot \Omega\geqslant \mathrm{mult}_{P}\big(\Omega\big)s>\big(8n/7-\mu-m_{L}-m_{C}\big) s,
$$
which implies that $m_{L}>n/2+m_{Z}$, but $m_{L}\leqslant 3n/4$ by
Remark~\ref{remark:curves}. We have
$$
3n/28-\mu/6=-K_{U}\big\vert_{S}\cdot\Big(m_{L}L+m_{C}C+m_{Z}Z+\Omega\Big)\geqslant
-K_{U}\big\vert_{S}\cdot\Big(m_{L}L+m_{C}C+m_{Z}Z\Big)=\frac{m_{C}+m_{Z}}{6},
$$
which implies that $m_{C}+m_{Z}\leqslant 9n/14-\mu$. On the
surface $S$ we have
$$
7\mu/6+7m_{C}/6-5m_{Z}/6-m_{L}=\Omega\cdot C>8n/7-\mu-m_{L}-m_{C},
$$
which implies that $13(\mu+m_{C})/6>8n/7+5m_{Z}/6$. The inequality
$\Omega\cdot Z\geqslant 0$ implies that
$$
2n/7-5\mu/6-m_{L}-5m_{C}/6+7m_{Z}/6\geqslant 0,
$$
which implies that $7m_{Z}/6\geqslant 5\mu/6+m_{L}+5m_{C}/6-2n/7$,
but $m_{Z}\leqslant 3n/8$ by Remark~\ref{remark:curves}.
It follows from Lemma~\ref{lemma:inequality-for-mu} that
$18n/77\geqslant\mu>n/7$. The inequalities obtained
$$\left\{\aligned
&13\big(\mu+m_{C}\big)/6>8n/7+5m_{Z}/6,\\
&21n/48\geqslant 7m_{Z}/6\geqslant 5\mu/6+m_{L}+5m_{C}/6-2n/7,\\
&m_{C}+m_{Z}\leqslant 9n/14-\mu,\\
&3n/4\geqslant m_{L}>n/2+m_{Z},\\
&18n/77\geqslant\mu>n/7,\\
\endaligned
\right.
$$
are inconsistent. So we have a contradiction. Thus, the case
$\gimel=36$ is impossible as well.
\end{proof}
Taking into account the possible values of the quadruple
$(a_{1},a_{2},a_{3},a_{4})$, we see that
$$
\gimel\in\big\{13,18,23,24,27,32,38,40,42,43,44,45,46,48,56,58,60,61,65,68,69,74,76,79\big\}
$$
by Lemmas~\ref{lemma:a-not-1}. Let $T$ be a general surface in
$|-K_{U}|$. Then $T\vert_{E}\in
|\mathcal{O}_{\mathbb{P}(1,\,a,\,r-a)}(1)|$.
\begin{lemma}
\label{lemma:smooth-points-upstairs-in-T} The point $P$ is
contained in the surface $T$.
\end{lemma}
\begin{proof}
It follows from Lemmas~\ref{lemma:lower-bound-for-mu} and
\ref{lemma:inequality-for-mu} that $P\in T$ unless
$\gimel\in\{13,24\}$. Therefore, we may assume that
$\gimel\in\{13,24\}$ and $P\not\in T$. Let us derive a
contradiction.
Let $L$ be the curve in
$|\mathcal{O}_{\mathbb{P}(1,\,a,\,r-a)}(1)|$. Then $P\not\in L$,
because $P\not\in T$. Thus, there is a unique smooth irreducible
curve $C$ in the linear system
$|\mathcal{O}_{\mathbb{P}(1,\,a,\,r-a)}(a)|$ that contains the
point $P$. Put
$$
\bar{D}\big\vert_{E}=\delta C+\Upsilon\equiv r\mu L,
$$
where $\delta$ is a non-negative integer, and $\Upsilon$ is an
effective cycle such that $C\not\subset\mathrm{Supp}(\Upsilon)$.
Arguing as in the proof of Lemma~\ref{lemma:lower-bound-for-mu},
we see that $\delta\leqslant r\mu/a$, which gives $\delta<n$ by
Lemma~\ref{lemma:inequality-for-mu}.
It follows from Theorem~7.5 in \cite{Ko97} that $(E,
\frac{1}{n}\bar{D}\vert_{E})$ is not log canonical at $P$, which
implies that the log pair $(E, C+\frac{1}{n}\Upsilon)$ is not log
canonical at $P$. It follows from Theorem~7.5 in \cite{Ko97}~that
$$
r\mu/\big(r-a\big)\geqslant\big(r\mu-a\delta\big)/\big(r-a\big)=C\cdot\Upsilon\geqslant\mathrm{mult}_{P}\big(\Upsilon\vert_{C}\big)>n,
$$
which implies that $\mu\geqslant n(r-a)/r$, which is impossible by
Lemma~\ref{lemma:inequality-for-mu}.
\end{proof}
It follows from \cite{CPR} that $T\cap E\cap
\cup_{i=1}^{l}C_{i}\ne\varnothing\iff\gimel\in\{43, 46, 69, 74,
76, 79\}$.
\begin{lemma}
\label{lemma:smooth-points-13-24-32-43-46} The case
$\gimel\not\in\{13, 24, 32, 43, 46\}$ is impossible.
\end{lemma}
\begin{proof}
Suppose that $\gimel\not\in\{13, 24, 32, 43, 46, 56\}$. It follows
from the proof of Theorem~5.6.2 in \cite{CPR} that~there are
$s\in\mathbb{N}$ and $H\in |-sa_{1}\bar{a}_{3}K_{U}|$ such that
$\mathrm{mult}_{P}(H)\geqslant s$, but $H$ does not contain
components of the cycle $\bar{D}\cdot T$ passing through $P$ that
are different from the curves $C_{1},\ldots,C_{l}$.
We have $\gimel\in\{69, 74, 76, 79\}$ and
$P\in\cup_{i=1}^{l}C_{i}$, because otherwise we get a
contradiction~using
$$
sa_{1}\bar{a}_{3}\left(-nK_{X}^{3}-\frac{\mu}{a\big(r-a\big)}\right)=\bar{D}\cdot H\cdot T\geqslant \mathrm{mult}_{P}\big(\bar{D}\big)s>\big(n+n/r-\mu\big) s.
$$
We may assume that $P\in C_{1}$.~Put $\bar{D}\cdot
T=mC_{1}+\Delta$, where $m$ is a non-negative integer~number, and
$\Delta$ is an effective cycle, whose support does not contain the
curve $C_{1}$. Then it follows from the proof of Theorem~5.6.2 in
\cite{CPR} that there is a~surface $R\sim
-sa_{1}\bar{a}_{3}K_{U}$~such~that
$$
sa_{1}\bar{a}_{3}\left(-nK_{X}^{3}-\frac{\mu}{a\big(r-a\big)}\right)=R\cdot\Delta\geqslant\mathrm{mult}_{P}\big(\Delta\big)s>\big(n+n/r-\mu-m\big)s,
$$
where $s\in\mathbb{N}$. The inequality obtained is impossible,
because $m\leqslant -a_{i}nK_{X}^{3}$ by
Remark~\ref{remark:curves}.
Suppose that $\gimel=56$. As in the previous case, there is $H\in
|-s24K_{U}|$ such that
$$
s24\big(n/22-\mu/24\big)=\bar{D}\cdot H\cdot T\geqslant \mathrm{mult}_{P}\big(\bar{D}\big)s>\big(12n/11-\mu\big) s,
$$
where $s$ is a natural number. Now we can easily obtain a
contradiction with $\mu>n/r$.
\end{proof}
Thus, to complete the proof of
Lemma~\ref{lemma:quadratic-involutions}, we have to consider the
cases $\gimel=13, 24, 32, 43, 46$ one by one. For the sake of
simplicity, we only consider the cases $\gimel=13$ and
$\gimel=43$, because the remaining cases can be considered in a
similar way.
\begin{lemma}
\label{lemma:43} The inequality $\gimel\ne 43$ holds.
\end{lemma}
\begin{proof}
Suppose that $\gimel=43$. Then $X$ is a general hypersurface in
$\mathbb{P}(1,2,3,5,9)$ of degree $20$, and~$O$~is a singular
point of type $\frac{1}{9}(1,4,5)$. The base locus of $|-2K_{U}|$
consists of two irreducible curves $C$ and $L$ such that $L=T\cdot
E$, and $C$ is the curve among $C_{1},\ldots,C_{l}$ such that
$C\cap L\ne\varnothing$.
Suppose that $P\not\in C$. Then it follows from the proof of
Theorem~5.6.2 in \cite{CPR} that we can find a~surface~$H\in
|-s20K_{U}|$ that has multiplicity at least $s>0$ at the point $P$
and does~not contain components of $\bar{D}\cdot T$ that pass
through $P$, where $s$ is a natural number. Then
$$
s20\big(n/18-\mu/20\big)=\bar{D}\cdot H\cdot T\geqslant \mathrm{mult}_{P}\big(\bar{D}\big)s>\big(10n/9-\mu\big) s,
$$
which implies that $\mu<n/9$, but $\mu>n/10$.
We see that $P\in C$. Then $\bar{M}$ contains $C$ and $L$. Put
$$
\bar{D}\big\vert_{\bar{M}}=m_{1}L+m_{2}C+\Delta,
$$
where $m_{1}$ and $m_{2}$ are a non-negative integers, and
$\Delta$ is an effective cycle, whose support does not contain $L$
and $C$. Then $m_{2}\leqslant n$ by Remark~\ref{remark:curves},
because $\alpha^{*}(-K_{X})\cdot C=1/9$.
The surface $\bar{M}$ is smooth at $P$. So it follows from
Theorem~7.5 in \cite{Ko97} that the log pair
$$
\Big(\bar{M},\ \frac{1}{n}\bar{D}\big\vert_{\bar{M}}+\big(\mu/n-1/9\big)E\big\vert_{\bar{M}}\Big)
$$
is not log canonical in a neighborhood of the point $P$, but
$E\vert_{\bar{M}}=L+Z$, where $Z$ is an irreducible curve that
does not pass through the point $P$. Therefore, the singularities
of the log pair
$$
\Big(\bar{M},\ \big(m_{1}/n+\mu/n-1/9\big)L+C+\frac{1}{n}\Delta\Big)
$$
are not log canonical at the point $P$. So it follows from
Theorem~7.5 in \cite{Ko97}~that
$$
n/9-\mu-m_{1}+m_{2}=\Delta\cdot C\geqslant
\mathrm{mult}_{P}\big(\Delta\vert_{C}\big)>n-m_{1}-\mu+n/9,
$$
because $C^{2}=-1$ and $C\cdot L=1$ on $\bar{M}$. Thus, we have
$m_{2}>n$, which is a contradiction.
\end{proof}
Suppose that $\gimel=13$. Then $r=5$ by Lemma~\ref{lemma:a-not-1}.
The~base locus of the pencil $|-K_{U}|$ consists of two curves
$\bar{C}$ and $\bar{L}$ such that $\bar{C}=E\vert_{T}$, and
$\alpha(\bar{L})$~is~the~base curve of $|-K_{X}|$. Then
$$
\bar{C}^{2}=\bar{L}^{2}=-5/6,\ \bar{L}\cdot\bar{C}=1
$$
on the surface $T$. Put
$\bar{D}\vert_{T}=\bar{m}_{L}\bar{L}+\bar{m}_{C}\bar{C}+\Upsilon$,
where $\bar{m}_{L}$ and $\bar{m}_{C}$ are non-negative integers,
and $\Upsilon$ is an effective cycle, whose support does not
contain $\bar{L}$ and $\bar{C}$. Then
$$
11n/5-11\mu/6=\big(6L+5C\big)\cdot \Big(\bar{m}_{L}\bar{L}+\bar{m}_{C}\bar{C}+\Upsilon\Big)=11\bar{m}_{C}/6+\big(6L+5C\big)\cdot\Upsilon\geqslant 11\bar{m}_{C}/6,
$$
which implies that $\bar{m}_{C}\leqslant 6n/5-\mu$. Thus, we have
$\bar{m}_{C}<n$, because $\mu>n/5$.
Suppose that $P\not\in\bar{L}$. Then it follows from Theorem~7.5
in \cite{Ko97} that the log pair
$$
\Big(S,\ \bar{C}+\frac{\bar{m}_{L}}{n}L+\frac{1}{n}\Upsilon\Big)
$$
is not log canonical in the neighborhood of the point $P$, because
$\bar{m}_{C}+\mu-n/5\leqslant n$, which~implies that the
inequality $\mathrm{mult}_{P}(\Upsilon\vert_{\bar{C}})>n$ holds by
Theorem~7.5 in \cite{Ko97}. Hence, we have
$$
5\mu/6+5\bar{m}_{C}/6\geqslant5\mu/6-\bar{m}_{L}+5\bar{m}_{C}/6=\Upsilon\cdot\bar{C}>n,
$$
which is impossible, because $\bar{m}_{C}\leqslant 6/5-\mu$. Thus,
we see that $P=\bar{L}\cap\bar{C}$.
Put $\bar{D}\vert_{\bar{M}}=m\bar{L}+\Omega$, where $m$ is a
non-negative integer, and $\Omega$ is an effective cycle, whose
support does not contain $\bar{L}$. Then $L^2=1/6$ on the surface
$\bar{M}$. But $m\leqslant n$ by Remark~\ref{remark:curves}.
Arguing as in the case $P\not\in\bar{L}$ we see that
$\mathrm{mult}_{P}(\Omega\vert_{\bar{L}})>n$ by Theorem~7.5 in
\cite{Ko97}, and
$$
11n/30-\mu=\bar{D}\cdot\bar{L}=m/6+\Omega\cdot\bar{L}>m/6+n,
$$
which implies that $m<0$. So we have a contradiction, which
completes the proof of Lemma~\ref{lemma:quadratic-involutions}.
\section{Direct products.}
\label{section:conic-bundles}
Let $X$ be an arbitrary Fano variety with terminal
$\mathbb{Q}$-factorial singularities of Picard rank one, and
$\Gamma$ be a subgroup of the group $\mathrm{Bir}(X)$.
\begin{definition}
\label{definition:untwist} The subgroup
$\Gamma\subset\mathrm{Bir}(X)$ untwists all maximal singularities
if for every linear system $\mathcal{M}$ on the variety $X$ that
has no fixed components there~is $\xi\in\Gamma$ such that
the~singularities of the log pair $(X, \lambda\xi(\mathcal{M}))$
are canonical, where $\lambda\in\mathbb{Q}$ such~that
$K_{X}+\lambda\,\xi(\mathcal{M})\equiv 0$.
\end{definition}
It is well known that the group $\mathrm{Bir}(X)$ is generated by
the subgroups $\Gamma$ and $\mathrm{Aut}(X)$ in the~case when the
subgroup $\Gamma$ untwists all maximal singularities (see
\cite{Ch05umn}).
\begin{definition}
\label{definition:birational-rigidity} The variety $X$ is
birationally superrigid\footnote{There are several similar
definitions of birational superrigidity and birational
rigidity~(see~\cite{CPR},~\cite{Ch05umn}).} (rigid, respectively)
if the~trivial~subgroup (the whole group $\mathrm{Bir}(X)$,
respectively) untwists all maximal singularities.
\end{definition}
The birational rigidity of $X$ implies that there is no dominant
rational map $\rho\colon X\dasharrow Y$ such that
$\mathrm{dim}(Y)\geqslant 1$, and sufficiently general fiber of
the map $\rho$ is rationally connected (see \cite{Ch05umn}).
\begin{example}
\label{example:Pukhlikov} It follows from \cite{Pu04d} that the
variety $X$ is birationally superrigid and $\mathrm{lct}(X)=1$ in
the case when $X$ is one of the following smooth Fano varieties:
\begin{itemize}
\item a general hypersurface in $\mathbb{P}^{r}$ of degree $r\geqslant 6$;
\item a general hypersurface in $\mathbb{P}(1^{m+1}, m)$ of degree $2m\geqslant 6$.
\end{itemize}
\end{example}
\begin{definition}
\label{definition:universally-untwist} The subgroup $\Gamma$
universally untwists all maximal singularities if for every
variety $U$, and every linear system $\mathcal{M}$ on the variety
$X\times U$ that does not have~fixed components, there is a
birational automorphism $\xi\in\Gamma$ such that the log pair
$$
\Big(F,\ \lambda\xi\big(\mathcal{M}\big)\vert_{F}\Big)
$$
has at most canonical singularities, where $F$~is~a sufficiently
general fiber of the natural projection~$X\times U\to U$, and
$\lambda$ is a positive rational number such~that
$K_{F}+\lambda\,\xi(\mathcal{M})\vert_{F}\equiv 0$.
\end{definition}
Let $X_{1},\ldots,X_{r}$ be Fano varieties of Picard rank one with
terminal $\mathbb{Q}$-fac\-to\-ri\-al singularities.~Put
$$
U_{i}=X_{1}\times\cdots\times X_{i-1}\times \widehat{X_{i}}\times X_{i+1}\times\cdots\times X_{r}
$$
and $V=X_{1}\times\cdots\times X_{r}$. Let $\pi_{i}\colon V\to
U_{i}$ be a natural projection.
For every $i\in\{1,\ldots,r\}$,~suppose that
$\mathrm{lct}(X_{i})\geqslant 1$,~and~there is a~subgroup
$\Gamma_{i}\subset\mathrm{Bir}(X_{i})$ that universally~untwists
all maximal singularities. Then the following result
holds\footnote{The assertion of Theorem~\ref{theorem:Cheltsov} is
proved in \cite{Pu04d} for smooth birationally superrigid Fano
varieties.}.
\begin{theorem}
\label{theorem:Cheltsov} The variety $X_{1}\times\cdots\times
X_{r}$ is non-rational, and
$$
\mathrm{Bir}\Big(X_{1}\times\cdots\times X_{r}\Big)=\Big<\prod_{i=1}^{r}\Gamma_{i},\ \mathrm{Aut}\Big(X_{1}\times\cdots\times X_{r}\Big)\Big>,
$$
for any dominant rational map $\rho\colon X_{1}\times\cdots\times
X_{r}\dasharrow Y$ whose general fiber is rationally connected,
there is a commutative diagram
$$
\xymatrix{
X_{1}\times\cdots\times X_{r}\ar@{->}[d]_{\pi}\ar@{-->}[rr]^{\sigma}&&X_{1}\times\cdots\times X_{r}\ar@{-->}[d]^{\rho}\\
X_{i_{1}}\times\cdots\times X_{i_{k}}\ar@{-->}[rr]_{\xi}&&Y,}
$$
where $\xi$ and $\sigma$ are
birational maps, and $\pi$ is a projection for some $\{i_{1},\ldots,i_{k}\}\subseteq\{1,\ldots,r\}$.
\end{theorem}
It is well known that Theorem~\ref{theorem:Cheltsov} is implied by
the following technical result (see \cite{Pu04d}).
\begin{proposition}
\label{proposition:Cheltsov} For every linear system $\mathcal{M}$
on the variety $V$ such that
\begin{itemize}
\item the linear system $\mathcal{M}$ does not have fixed components,
\item the linear system $\mathcal{M}$ does not lie in the fibers of the projections $\pi_{1},\ldots,\pi_{r}$,
\end{itemize}
there are $k\in\{1,\ldots,r\}$, birational map
$\xi\in\prod_{i=1}^{r}\Gamma_{i}$ and a positive rational number
$\mu$ such~that
\begin{itemize}
\item the inequality $\kappa(V, \mu\xi(\mathcal{M}))\geqslant 0$ holds\,\footnote{The number $\kappa(V,\ \mu\xi(\mathcal{M}))$ is a Kodaira dimension of the movable log pair $(V,\ \mu\xi(\mathcal{M}))$ (see \cite{Ch05umn}).},
\item the equivalence $K_{V}+\mu\,\xi(\mathcal{M})\equiv \pi_{k}^{*}\big(D\big)$ holds for some nef $\mathbb{Q}$-divisor $D$ on $U_{k}$.
\end{itemize}
\end{proposition}
\begin{proof}
Let $F_{i}$ be a sufficiently general fiber of $\pi_{i}$. The
subgroups $\Gamma_{1},\ldots,\Gamma_{r}$ universally~untwist all
maximal singularities for every $i=1,\ldots,r$. So there~is
$\xi\in\prod_{i=1}^{r}\Gamma_{i}$ such that the log pairs
$$
\Big(F_{1},\ \mu_{1}\,\xi\big(\mathcal{M}\big)\big\vert_{F_{1}}\Big),\ \ldots,\ \Big(F_{r},\ \mu_{r}\,\xi\big(\mathcal{M}\big)\big\vert_{F_{r}}\Big)
$$
are canonical, where $\mu_{i}$ is a rational number such that
$$
K_{V}+\mu_{i}\,\xi\big(\mathcal{M}\big)\equiv \pi_{i}^{*}\big(D_{i}\big),
$$
where $D_{i}$ is a $\mathbb{Q}$-divisor on $U_{i}$. Then there is
$m\in\{1,\ldots,r\}$ such that $D_{m}$ is nef.
Now arguing as in the proof of Theorem~1 in \cite{Pu04d}, we see
that $\kappa(V, \mu_{k}\xi(\mathcal{M}))\geqslant 0$.
\end{proof}
Let $X$ be a general quasismooth hypersurface in
$\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree
$\sum_{i=1}^{4}a_{i}$ with terminal singularities, where
$a_{1}\leqslant a_{2}\leqslant a_{3}\leqslant a_{4}$. Then $X$ is
a Fano threefold, whose divisor class group is generated by
$-K_{X}$. The possible values of $(a_{1},a_{2},a_{3},a_{4})$ are
given in Table~5 in \cite{IF00}.
There are finitely many non-biregular birational involutions
$\tau_{1},\ldots,\tau_{k}\in\mathrm{Bir}(X)$
explicitly~constructed in \cite{CPR}~such~that the following
result holds (see \cite{CPR}).
\begin{theorem}
\label{theorem:CPR} The subgroup
$\langle\tau_{1},\ldots,\tau_{k}\rangle$ universally untwists all
maximal singularities.
\end{theorem}
Hence, the following two examples follow from \cite{MaMon64},
\cite{ChPa05}, \cite{Pu04d} and Theorems~\ref{theorem:main},
\ref{theorem:Cheltsov}, \ref{theorem:CPR}.
\begin{example}
\label{example:41} Let $X$ be a general hypersurface in
$\mathbb{P}(1,1,4,5,10)$ of degree~$20$, and $U$~be~a~general
hypersurface in $\mathbb{P}(1^{n+1}, n)$ of degree $2n\geqslant
6$. Then $\mathrm{Bir}(X\times
U)\cong(\mathbb{Z}_{2}\ast\mathbb{Z}_{2})\oplus\mathbb{Z}_{2}$.
\end{example}
\begin{example}
\label{example:9} Let $X$ be a general hypersurface in
$\mathbb{P}(1,1,2,3,3)$ of degree~$9$, and $U$~be~a~general
hypersurface in $\mathbb{P}^{r}$ of degree $r\geqslant 6$. Then
$$
\mathrm{Bir}\Big(X\times U\Big)\cong\Big<a,b,c\ \big\vert\ a^2=b^2=c^2=\big(abc\big)^{2}=1\Big>.
$$
\end{example}
It follows from \cite{CPR} that
$\mathrm{Aut}(X)\ne\mathrm{Bir}(X)$ for exactly $45$ values of
$(a_{1},a_{2},a_{3},a_{4})$.
| 8,970
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TITLE: What is meant by the rest energy of non-composite particle?
QUESTION [2 upvotes]: When talking about the rest energy of a composite particle such as a proton, part of the rest energy is accounted for by the internal kinetic energy of its constituent quarks. But what is physically meant by the rest energy of non-composite particles such as quarks?
REPLY [5 votes]: One has to be familiar with four vectors. In the same way as for three vectors the length is an invariant of the vector and is obtained by the dot product of the vector with itself, the "length" of the relativistic four vector is the rest mass, by definition;
the mass $m$ entering the relativistic equation where $p$ is the momentum and $E$ the total energy,
$E^2 - p^2c^2 = m^2c^4$
when $p=0$ is the rest energy.
In a composite particle the invariant mass even when it is at rest is the invariant mass of the four vector composed by adding the four vectors of all the constituent particles . A composite particle displays an effective rest mass.
For a non composite particle, as the electron, the energy when at rest, $p=0$, is its mass.
| 182,142
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IN MEMORIAMS:
Grace M. Schroedel March 21, 1916 - February 2, 2010 Beloved wife, mother, grand- mother, great-grandmother. Seasons pass and time slips by; our remembrance and love for you remains steadfast. You are in our thoughts and hearts always. Your loving family...
MISC. OBITUARIES:
Early Obituary Deadlines The News Tribune Obituaries Desk will be closed Monday, February 15th, 2016 in observance of the holiday. News Tribune obituaries to run Monday, February 15th, or Tuesday, February 16th, will deadline at 3pm on Friday...
IN MEMORIAMS:
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Hal Kunani Aoki 1926 - 2011 Always Loved Never Forgotten Aloha - Mary and Ohana
IN MEMORIAMS:
In Memoriam W. John Sinsheimer Passed away accidentally while swimming in the Pacific Ocean on January 29, 2015 in Wailea, HI, due to a possible rip tide and rough surf. Born March 15, 1932 to Walter and Alysse Sinsheimer in Chicago, IL. His older sister...
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The Eureka Effect
The Eureka Effect
by Nicholas Kollerstrom
Abstract
Are there optimal times when inspiration strikes? Does the cosmos help scientists to gain their Eureka breakthroughs? It is here argued that a collection of such famous dates can be used to test and in fact validate the working of celestial aspects. In particular a hypothesis originally formulated by John Addey concerning the odd-number harmonics 5 and 7 was of central importance. A second part concerns dates when great inventions first began to work, which is a more material and practical thing, and here different harmonics were found to be relevant. An excess of septile aspects has been found to replicate though several different groups.
Introduction
Astrology concerns the moment of birth, and if it is to be applied to or tested on mundane events then one should examine moments of first-beginning or genesis-moments, when something new appears in the world. The progress of modern science offers the ideal opportunity to do this, because it is through the new ideas created by scientists that our modern world has come into being. Only in the history of science do we find absolutely new things coming into existence. The 'Eureka' experience has been well described in Arthur Koestler's book The Act of Creation:
The eureka act proper, the moment of truth experienced by the creative individual, is paralleled on the collective plane by the emergence, out of the scattered fragments, of a new synthesis
It is that 'new synthesis' which here concerns us. Koestler added, "I shall occasionally refer to discovery in its psychological aspect as the Eureka process or Eureka act" (in his chapter 'The Moment of Discovery.')1
That moment was, Koestler explained, an experience of 'bisociation' when two previously unconnected frameworks or 'matrices of logic' came together. I as a science historian decided to look at these moments, to see whether there was any particular celestial signature which they carried. I had been impressed by John Addey's book Harmonics in Astrology, and this gave an indication as to what one might try looking at:
...yet the charts of creative people often seem devoid of anything really noteworthy unless the fifth and seventh series of aspects are considered.2
I collaborated with Mike O'Neill in this project, whereby I would find the moments and he would score the quintile and septile aspects in them. Although the Addey quote had alluded to natal charts, we were here applying it to event charts.
One does not need to understand what it was that dawned upon the scientist, or only in a very general way, to collect these eureka-moments. We are here concerned with the inner experience of the scientist at these great moments, because they tend to get access to what Koestler in his Act of Creation called the 'self-transcending emotions' and these have a key importance for locating the eureka experience in historical time. English-language biographies exist for most of our E-scientists, and they often have a central chapter about the breakthrough experience. As Koestler has described, there will often be a great deal of stress and even despair sometimes which builds up in the individual before that moment arrives.
I believe our first indication that septiles might be important came from analysing a moment when Kepler described a great revelation he had in Graz while lecturing on maths to a nearly empty classroom on 19 July 1595. "It will never be possible for me to describe in words the enjoyment which I have drawn from his discovery..." Kepler wrote. That sense of exultation is a key feature of eureka-moments. We consulted people about the chart for this moment but there seemed nothing special about it, until Mike O'Neill noticed three septile aspects in the sky. Were these the key?
Another 'creative' moment I was then looking at, happened when a sample of plutonium was first created at the Berkeley cyclotron in California by Glenn Seaborg. This dated and timed event had a strong quintile presence. It was not to be sure a Eureka moment but yet suggested that quintile aspects might be important for mentally-creative events.
A first presentation of results was given at the 1987 Astrology Research conference in London3 featuring a list of 18 Eureka moments, showing a significant excess of quintile and septile aspects. We used the orbs suggested by John Addey: he recommended dividing twelve degrees by the 'harmonic number,' thus for quintile aspects one divides by five, to give 2°24' and for septile aspects one has 1°43.' The chance-expected frequencies we estimated as 2.2 and 2.4 per chart respectively, for these orbs, aspects between planets and excluding the lunar nodes.
The same procedure was then repeated for the natal charts of Eureka scientists. These had to have Rodden grade 'A' birth data, and to have been alluded to in Isaac Asimov's Biographical Encyclopaedia of Science and Technology (1975); those were our criteria for inclusion. But also one does really need a biography written about such scientists to ascertain whether or not they had anything resembling a Eureka-moment - these events are subject to considerable mythologising. Thereby we generated a group of eminent scientists of known birth-data, which could be divided into two groups, namely those who had experienced a eureka moment for their theory, and those who had arrived at their views by a more gradual process.
Having done this, we were startled to note that a fairly similar excess of quintile and septile aspects turned up in the natal group, as had been seen in the Eureka moment. The non- eureka group of eminent scientists of known birth-data had a deficit of these two aspects. Members of the non-Eureka group were in no way less eminent than those of the eureka group.
Many of the E-moments are untimed, which does generate the problem with the small orbs here used that their lunar positions are blurred, and some of the lunar aspects scored are doubtful.
Publication
In 1996 Mike O'Neil and I co-authored The Eureka Effect, citing a list of 21 E-moments, ranging from 1672 when Tycho Brahe first spotted a new star in the sky (a supernova) to 1953 when Crick and Watson discovered the structure of the DNA helix one Saturday morning. Experts such as Theodore Landscheidt checked through our list and generally felt that all our moments were valid, although doubt was expressed over one of the moments, when Pluto was discovered in 1930 by Clyde Tombaugh. This was, one person advised us, a moment of 'no special intellectual insight.' Our procedure was, that once we had decided to accept a moment and score it we were then unwilling to change anything, i.e. we didn't 'allow' ourself to subtract moments. At last I felt that my degree in history and philosophy of science had been of some use.
After the 1618 moment of illumination when Kepler discovered his 3rd law of planetary motion, he exclaimed: "Nothing holds me back, I give myself up to sacred frenzy." In contrast with that classic Eureka-moment, his earlier 1595 moment in Graz to which we have alluded was (rightly or wrongly) not included. Because, what then came to him concerned a funny geometrical arrangement to account for the distances between the heavenly spheres, that no-one after him ever took seriously: i.e., there is a general agreement that what came to him there was not 'science', whereas the later discovery of his 3rd law was. Thus we appeal to a consensus. I may not believe in Darwin's theory of evolution by natural selection; however there was a eureka experience when that insight, true or not, came to him (as described by his biographer Howard Gruber) and so it is included. There is agreement, amongst science historians, that that insight is part of 'science.'
I suggest that any qualified science historian asked to produce a list of dateable Eureka moments, eminent enough to have been written up, would necessarily come up with much the same list as has here been presented: accepting a Koestler-definition of the moments, which I think excludes technological innovations. For example, the book Eureka by Edward de Bono4 is just a history of technological innovation, such events being a lot easier to find - but they are not what we are here looking at.
Here is the list as earlier published but with two extra E-moments added: Edison getting his idea for the electric light and Hubble getting his theory of galaxies. No doubt this list is not final, and more will be found.
There was overall a fifty percent excess of the septile aspects, in this group of 23 Eureka-moments, as compared to our best-estimate control group. A somewhat smaller twenty percent of quintiles appeared, in both cases using the 'Addey orbs'. Adding together the excess of quintile and septile aspects gave a chi-squared value of 14.5, and that is significant at 0.00014 or 1 in 7000.
Two New Moments
Two new E-moments included, not included in the 1996 book The Eureka Effect, may here be outlined. The first, concerning Edison's electric light, was described in a book published in 1989:
Eureka moments and instantaneous insights are part of the lore of invention and discovery ... On 8 September 1878, Edison experienced a eureka moment when he discussed with the inventor and industrialist William Wallace the flawed incandescent lamp system of Wallace's collaborator ... After reflection, Edison found these and related insights of his own so promising that he telegraphed an associate, "Have struck a bonanza in electric light.."5
This E-moment is to be contrasted with what we will call invention-moments, when inventions were first made to work: Edison's electric light first reliably glowed a year later on 23 October, 1879. It's rare to have both an E- and I- moment in the life of one individual; Michael Faraday is the only other such case, at any rate for 'found' i.e. datable moment.
The second new eureka-moment, concerning galactic structure, perhaps struck Edwin Hubble as he was walking down Mount Palomar in dawn's early light. He had spent a night up alone in the big new telescope, whose huge reflecting mirror had taken five years to grind. While looking at the Andromeda nebula, he finally understood that it was a galaxy in its own right. An article about this moment, about the night, was entitled The Night the Universe Changed Forever6 and fortunately it gave both the time and date of this E- moment:
Hubble was forced to conclude that the Andromeda nebula was ...at least 300,000 parsecs from earth - the equivalent of a million light years, or more than triple the diameter of Shapley's entire universe [Hubble's mentor, Howard Shapley, had proposed that everything known was contained within the Milky Way]. It was a "eureka" moment. Andromeda's spiral arm was bejewelled by a Cepheid variable. The giant nebula was a sister to the Milky Way, composed of stars by the millions! Hubble crossed out the letter "N" for nova he had previously inscribed on the plate and printed "VAR!" for variable directly beneath it. And though the plate had been taken the previous night, he dated it "6-Oct 1923" to commemorate the moment when his mental tumblers had fallen into place.8
Both of these E-moment descriptions were published after our initial publication and have here been added on.
YearDateTimeScientist Eureka MomentQ,S
1572Nov 21 OS18hTycho BraheHven, DenmarkSupernova3,2
1610Jan 717hGalileo GalileiPaduaDiscovery of Jupiter's moons4,2
1618May 1513hJohannes KeplerPrague3rd Law1,47
1807Oct 614hHumphrey DavyLondonPotassium1,4
1831Aug 2914hMichael FaradayLondonElectromagnetism3,4
1838Sept 2814hCharles DarwinLondonNatural Selection4,5
1846Sept 2323.20Johann GalleBerlinDiscovery of Neptune3,5
1869Mar 114hDimitri MendeleevSt PetersburgPeriodic table0,5
1878Sept 814hThomas EdisonNew JerseyElectric light4,3
1895Mar 2310hWilliam RamsayUCL LondonHelium7,2
1895Nov 818hWilliam RöentgenMunichX-rays2,5
1896Mar 114hHenri BecquerelParisRadioactivity1,2
1915Nov 1813hAlbert EinsteinBaselTheory of Relativity4,3
1921Mar2802hOtto LoewiGraz, AustriaNerve transmission2,3
1923Oct 607hEdwin HubbleMt Wilson, CAGalaxies3,3
1925June 813hWerner HeisenbergGottingenQuantum mechanics4,2
1928Sept 319h50Alexander FlemingLondonPenicillin4,6
1930Feb 1823hClyde TombaughFlagstaff, AZDiscovery of Pluto3,1
1933Sept 1213hLeo SzilardLondonChain reaction3,4
1934Oct 2212hEnrico FermiRomeSlow neutrons0,1
1938Dec 2409hLise MeitnerGoteborg, SwedenAtomic fission1,4
1951April 2612h30Charles TownesWashington DCLaser beams3,7
1953Feb 2810hJames WatsonCambridgeDNA helix1,4
Total quintiles and septiles:Q: 61, S: 84
Chance-expected totals:Q: 50.4, S: 55.4
Percent Excess:Q: 21%, S: 52%
These 23 Eureka-moments can also be of interest as anniversary dates, to stimulate creativity, and as such I had them published in the now-defunct British journal Inventor's World.9
A book published in 2001, The Eureka Effect by D Perkins, had quite a few of the E- moments here designated, plus lot of other such E-moments that are undateable and so not here included. In common with our list, is that the last E-moment given was that of the discovery of the DNA helix structure by James Watson in 1953. This suggests that the absence of any more E-moments in the six decades since then may not be merely some oversight by the present author. Maybe science is more institutionalised today, moved more by big business, and the lone scientist's proverbial eureka cry may now be history?
Having said that, a eureka-cry did appear in newspaper headlines around the world on 24 June, 1993. On the previous morning, Andrew Wiles at Cambridge had solved 'Fermat's Last Theorem', which had remained unsolved for 363 years.10 Our Eureka study had adopted the limitation of not including purely mathematical E-moments, as I did not feel competent to hunt for them. That moment had a Q+S score of {3,3}. That's the most recent claim for an E-moment of which I'm aware.
All of the dated E-moments here listed are by scientists sufficiently eminent, in that they have been described in a biography. It is normal for a scientific biography to have a chapter about the moment of breakthrough and insight and when it happened. One needs to be on guard against a tendency to mythologise the eureka experience, eg the story of Isaac Newton and the falling apple, but such mythic E-moments can never have a date.
Summarising, the above list was created and scored for celestial aspects using three basic references, which in retrospect appear as having been quite sound. Arthur Koestler's book The Act of Creation described the idea of a Eureka-experience, greatly helping us in looking for them; in addition it gave quite a lot of very helpful examples with some dates; the Isaac Asimov reference also gave helpful dates and reliably pointed out biographies which had E- moments, having sections on a thousand or so scientists and inventors; and John Addey's Harmonics in Astrology described the theory of harmonics here used, suggested orbs to use, and recommended the quintile and septile aspects.
The Natal Group
After we published in 1996 the list of eureka and 'non-eureka' scientists, I was a decade or so later able to rummage through my (UCL) college library for non-eureka biographies. This had not been a primary focus initially, and I was able to locate quite a few more of these, increasing the size of that group from 12 to 20. This group will grow continually as astrologers gradually improve their data-archives, and as more biographies of scientists are published. Here is the presently-final group of eminent scientists11 of reliably-known birth time, making a simple binary division into Eureka and non-eureka biographies:
Table 2: Eureka and Non-Eureka eminent Scientists of known birthtime with quintile and septile scores {Q,S}
Eureka Types Non-Eureka Types
Tycho Brahe3.7Nicolas Copernicus1,1
Galileo Galili2,4Andreus Vesalius1,1
Johannes Kepler5,3Robert Hooke2,1
Humphrey Davy2,6John Flamsteed2,1
Louis Pasteur4,0Edmond Halley0,2
Wilheim Röentgen1,3Antoine Lavoisier4,2
Thomas Edison0,6Johann Bode1,1
Alexandre-Edmond Becquerel3,2David Brewster4,4
Nicola Tesla2,3Urbain Le Verrier1,1
Albert Einstein4,3Thomas Huxley3,0
Werner Heisenberg0,2William Crookes3,2
Alexander Fleming5,2Marie Curie1,1
Louis de Broglie4,3Otto Hahn2,2
Enrico Fermi2,3Pierre Joliot-Curie1,0
Charles Townes2,3Linus Pauling1,0
James Watson4,6Emilio Segrè1,0
Hans Bethe1,3
Glenn Seaborg2,3
Paul Ehrlich3,0
Carl Sagan1,3
Total Quintiles, Septiles:43.56Total Quintiles, Septiles:38.29
Expected (n = 16):335.0,38.2Expected (n = 20):43.8,47.8
Excess23%, 46%Excess-13%, -39%
Also, Alexander Bell {3, 2} and G. Marconi {2, 1}.
Once again we see a large excess of quintiles and septiles, this time in the natal group of Eureka-scientists; but very surprisingly, an equal and opposite deficit of these aspects in the non-Eureka group.
The increase in size of the non-E natal group had not in any way decreased its huge septile deficit, which remained at around forty percent. We are astonished to see the group of Eureka-scientists here scoring 240% more septiles than the non-eureka group. That is the largest effect discovered in this investigation. No-one predicted it so we are not at liberty to assign a probability-value.
There were two cases where we were not confidently able to place them in one or the other category, Graham Alexander Bell and Guglielmo Marconi. They both had eureka-ish experiences, of apparatus working. It took a couple years before we developed the follow-on concept of invention-moments, when great inventions first worked, and it became evident that these two were more properly described in a quite different category, as inventors.
So the basic effect, of an excess of quintile and septile aspects, more strongly a septile excess, has replicated though the two groups event and natal, and is in deficit in the non-Eureka natal group.
Comparing the 16 natal Eureka scientists here given in Table 2 with the 14 we published back in 1995, it is evident that the two new cases added both have low scores, namely Werner Heisenberg and Wilheim Röentgen: their Q+S scores were {0,2} and {1,3}. Could this have been some selective prejudice whereby we had earlier omitted these cases? Both E-moments appeared in our originally-published Eureka list. Whatever the reason, I'll take the blame. Their birth-data appear as reliable.
Wilheim Röentgen had a dire E-moment around midnight of 8th November1895 (with seven septiles in the sky) when he saw own hand as a skeleton:
The apparition was so awful that Wilheim Röentgen wondered if he had taken leave of his senses. He could hardly have been more surprised if he had looked into a mirror and no reflection had stared back. He let go of the metal that he had been holding and jumped, startled, by the noise, as it hit the floor.
It was approaching midnight on 8 November, 1895....
After a late meal Röentgen returned to the laboratory. He moved the piece of lead near to the screen, watching its shadow sharpen, and it was then that he dropped it in surprise: he had seen the black shape of the metal held by the hand of a dead man. Pulling himself together he slowly opened his clenched fist and looked astonished once again at the dark skeletal pattern of the bones as his hand moved across the face of the screen. Still doubting what he saw he took out some photographic film for a permanent record. Röentgenhad made one of the most monumental discoveries in the history of science - X-rays... If any single moment marks the start of modern physics and science it is that Friday evening of 8 November in 1895.12
Heisenberg's E-moment was more serene, as he finally came to understand the 'magic matrices' of quantum mechanics:
"I was shocked to the core", recalled Werner Heisenberg, of the time when the principle of quantum mechanics dawned upon him, on 8th June 1925. "I had the feeling that I was seeing through the surfaces of atomic phenomena to their deep underlying basis, which had a remarkable inner beauty. So excited was I that I could not think of sleeping, but remained awake all night and watched the sunrise."13
This happened after he had retired under great mental stress to the tiny island of Heligoland in the North Sea.
Overtones
Figure 1: the family of heptagons
The 23 E-moment charts display overtones of the 7th harmonic, i.e. the 14, 21st etc. harmonics. Thus the 21st harmonic scores a fifty percent excess, just as large as the basic 7th harmonic. Such a claim does involve rather small-orb planetary aspects over centuries gone by. Re-computing these using the modern Jigsaw program (Astrolabe) suggests that modern programs are accurate enough to achieve this, but one would appreciate further corroboration. Such overtones validate the theory of harmonics as described by John Addey, in which they occur somewhat as in a musical note; as well as tending to establish the reality of the 'Eureka effect'. For comparison, regular heptagons and star-heptagons contain different members of this 'family' of aspects. The vertices of the star-heptagon are a 1/14th division of the circle, the basic angle of the 14th harmonic.
Chance-Expected Aspect Frequencies
Expected septile frequency is calculated as follows. The ten planets yield 45 planet-pairs three of which (Venus/Sun, Mercury/Sun and Venus/Mercury) are only ever able to form two septiles between them, because their angular separation remains too small. Only 42 of these pairs form all of the septiles, i.e. the six angular positions which score as a septile (mono, bi- and tri-) excluding the zero position of conjunction. Taking the 'Addey orb' of 12/7°, the likelihood of any one septile chiming is 2x12/(7x360): the factor of two here allows for both approaching and separating aspects. Therefore, the expected number of septiles for a moment in time at that orb is given by 24/(7x360) (42x6+2) = 2.42.
For empirically-generated septile expected frequencies: the eureka moments span 1572-1953, however the great flow of these moments has mainly extended over a century and a half, over the 19th and first half of the 20th century. Here are some randomly-generated expected frequencies using the Jigsaw program (sampling five or ten thousand):
Over 1572-1953, mean 2.40; 1800-1953, mean 2.43; 1820-1930, mean 2.39.
These suggest a close agreement between the theoretically-computed estimated value and those randomly generated by the Jigsaw program, with an uncertainty of less than 1%. Mike O'Neill and I took the value of 2.41 as the best expected frequency (and 2.19 for quintiles). Maybe in retrospect we should have noticed that the natal E-group, spanning a slightly earlier period from 1546-1928, gave on random sampling a slightly lower expected value of 2.39 septiles per chart.14
Concerning quintiles in the natal E-group, let's take two samples: 1546-1800 (n=5), 2.22; 1800-1928 (n=12) 2.19, which gives us more or less the same value: one could possibly prefer 2.20 rather than 2.19 per chart. There is no 'exact' right answer here.
I: When Inventions Worked
To collect these moments when inventions first worked, the main textbooks used were the Readers Digest Inventions That Changed the World and the Biographical Dictionary of Scientists/Engineers and Inventors.
This list of 36 I-moments is almost the same as published in Correlation two decades ago,15 except that the 'Heliospectroscope' which various persons inspecting the list were doubtful about has been removed, and replaced by 'gene therapy'. The latter was first performed at Bethesda hospital in Washington DC on 14 September 1990:
A new era in medicine dawned at Bethesda hospital in Washington D.C. when a four-year old girl was treated by a transfer of genetic material. She was severely immune-deficient from a rare hereditary disease, due to a missing gene, leaving her a helpless prey to diseases. The new invention was a virus cleverly stripped of its harmful capability and having a copy of the missing gene. This gene had earlier been cloned and so it was quite well-known. The virus was capable of entering a human cell and donating this gene to the human DNA. The idea had to pass stringent medical, legal and ethical review boards, because of its far-reaching implications. When permission had been granted, the paediatrician Ken Culver muttered, "Well, here goes" and performed the injection, at 12.52 p.m., September 14th (16.52 GMT). It lasted half an hour. Over the following months the girl's immune system slowly recovered, and she is now living a normal life.16
Often, inventions have no specific date when they first worked. Thus, the invention of the world-wide web by Sir Tim Berners-Lee at CERN laboratories, Geneva over the days before Christmas 1990 had no date; it was a gradual sequence. I corresponded with him, and he firmly explained, "They are frustrated when I tell them, there was no Eureka moment."17
As with the E-moments, these dates can surely be of value to any industrial firms wishing to stimulate an air of inventiveness by using these anniversaries. There is a great excess of British eureka and invention-moments, while the Table also shows that somewhere in the mid-19th century the genius of invention passed over to America.
Computing of expected frequencies is nowadays easy using a research program such as Jigsaw: one generates say ten thousand charts over the interval of Eureka dates and thereby derives the average score of an aspect for any given orb. Such mean values are stable and easily-obtainable for higher-frequency harmonics, e.g. septiles, owing to the small orbs involved; whereas for trines they are liable to vary more with the interval selected, because a single trine aspect between two outer planets may remain in place for decades. Thus any claim made for a trine excess in I-moments has to involve careful modelling of the time- distribution of the moments.
Figure 2: the excess of conjunction, opposition and trine aspects, to each of the planets, in the group of 36 Invention-moments, at 5 degree orb.
Aspects in the I-moment group
There was a huge excess of trine aspects in the I-moment group and also a huge 60% excess of major Uranus aspects (conjunctions, oppositions and trines to 5° orb) in that data, 60 as compared to 36.7 expected. Comparing the different planetary frequencies (see figure), Uranus at the top of the list and Saturn at the bottom well expresses their traditionally understood differences at these moments of revolutionary innovation.
The trine excess in the group of invention-moments was a total surprise. Over the span of two centuries 1800-2000 when the inventions mainly occurred, the trine expected frequency hovered around 1.72 per chart, from which the net excess of trines appeared at 48%. In contrast a deficit of quintiles appeared. The septile excess was smaller than found for the Eureka moments, significant at only quite a low level (1 in 20). Maybe the deficit of quintiles indicates the traditional meaning of this aspect as pertaining to creative, mental activity - surely absent at these grand moments of achievement, which are times of perspiration rather than inspiration.
Figure 3: This shows the same aspects as before, conjunction, opposition and trine, within the same group, but only to Uranus, and scoring per 2degrees of orb.19 There is an excess of 100% within the first two degrees of orb.
Harmonic Theory
The Invention-moments showed a large excess of 14th-harmonic aspects, stronger than the 7th, around +34%, which is curious. The higher harmonic 14th includes all of the 7th harmonic positions, but at half the orb, so overtones of the primary harmonic are bound to show some excess, as they re-sample the same angular positions.
An excess of septiles has thus replicated through several sets of data, in a way that may tend to confirm the 'inspirational' character of this aspect. To quote US astrologer Delphine Jay, 'I like to refer to the septiles as the consciousness-expanding aspect.'20
No differential has ever appeared in these results, between quintiles and biquintiles, nor between mono- bi-, and tri- septiles, which again tents to validate the Pythagorean- numerical approach of Addey, whereby it is the number of the harmonic, eg the number seven, which here 'means' something, rather than the angle as such. Philosophers may wish to discuss for example why the number five should be in deficit at moments when inventions were made to work (-19%), whereas they appear in excess at more mentally- creative moments.
For moments of action, excitement and success, the massive excess of trines, nearly fifty percent (see Table) in the group of Invention-moments seems in accord with traditional views about this aspect. For example, the authors of Mundane Astrology state:
The number three is related to the idea of life, vitality and enjoyment, and hence to what motivates us and moves us to action.
Table 4 - The Excess of Aspects: Trines, Quintiles and Septiles
GroupTrinesQuintilesSeptiles14th H.
E-moments (n=23) 61/50.4 = 21%84/55.4 = 52%
Χ2 = 1486/62.3 = 38%
Χ2 = 9
E-scientists (n=16) 43/35.0 = 23%56/38.6 = 45%
Non-E scientist (n=20) 38/43.8 = -13%29/48.2 = -40%
I-moments (n=36)92/62 = 48%64/79 = -19%107/88 = 22%130/96 = 34%
But there is also the point made by harmonic theorist David Cochrane, that: "The higher harmonics describe more internal, less conspicuously evident traits in the outer world. Lower harmonics describe more externalised traits."21 If perchance that is valid, it would be appropriate to find trines present at these energetic moments, while septiles are stronger at the more inner moments of illumination or inspiration.
Analysis of these Eureka groups has established the basic idea of the working of celestial aspects, where the orbs used from Addey's suggestion were not far off optimal. We should after all expect that these seed- or genesis- moments which have affected the course of history, should express the world-harmony.
Testability of Theory
The discoveries here made suggest predictions, in terms of aspect-totals found within groups e.g. of musical or mathematical eureka-moments, or the corresponding natal groups. I assembled a group of 17 moments of mystical illumination,22 some having time-of-day information, and found that collectively they had an excess of quintile and septile aspects that was somewhat comparable. Surprisingly there was a larger excess of quintiles. These mystic moments displayed a deficit of the even-numbered 4th, 6th and 8th harmonic aspects between planets, which are as it were to do with firm structure, evidently not required at such moments. Again John Addey's way of viewing aspect-harmonics appears as validated by this approach.
In terms of the refutability of the conclusions here presented, it would merely be necessary for someone to gather a group of famous Eureka moments using some other criteria, or of famous invention-moments, and demonstrate that the aspects here described were not present. If that could be done, it would show that the effects here described were a mere consequence of this author's idiosyncratic / biased mode of collecting the data.
Musical groups could be advised of times that had strong quintile and septile aspects (I and M.O. devised a Harmogram program that will show the strength of these aspects as a function of time) to see whether they felt that such times were good for creative work. Or, for the launch of a new invention which can be a risky and expensive business - say for example a new space-ship, one might wish to choose carefully a time of strong trine and septile aspects, and avoid basic Mars-Saturn aspects.
This study has definitively refuted the words of that most careful investigator Michel Gauquelin, that the effect of celestial aspects cannot be proved. The harmonic theory of john Addey is very much in tune with the theory of celestial aspects developed by Kepler in his 1618 Harmonices Mundi23 except that his is more arithmetic-numerical while Kepler's was more musical-geometric. Kepler concluded that the number seven could not be effective, because it was geometrically not constructible and moreover no pleasing musical harmonies could be made using it. This is the sole point on which we are here obliged to disagree with him.24
References:
1 Arthur Koestler The Act of Creation p.1989, Arkana, p.225, 107
2 John Addey Harmonics in Astrology 1976,p.123
3 See Appendix. NK and MO, "The Eureka Effect" The Astrological Journal 1988, 2, pp.90-136.
4 Edward de Bono Eureka: An Illustrated History of Inventions from the Wheel to the Computer 1979.
5 Hughes, T. American Genesis: A Century of Invention and Technological Enthusiasm 1870-1970, Viking 1989, p.75.
6 Gale Christiansen "The Night the Universe Changed Forever" The Griffith Observer June 1997,pp.4-10. I would not normally come across such a journal, but happened to be visiting Diana Rosenberg in New York and she had a copy of it.
7 For a full account, see the author's Eureka! The Celestial Pattern in Moments of Scientific Inspiration 2014.
8 Gale Christianson Edwin Hubble, Mariner of the Nebulae Bristol 1995, p.158.
9 NK "Eureka Moments - Anniversaries that Shaped History" Inventor's World Autumn 1998.
10 Barry Mazer Fermat's Last Theorem 2002,p.270-2.
11 Rightly or wrongly, our list excluded persons where doubt and controversy would arise as to whether or not they were 'scientists' ie Rene Descartes (philosopher) Leonardo da Vinci (artist) and Sigmund Freud (psychoanalyst), all of whom have reliable birth-data and are in the Asimov reference.
12 Frank Close Lucifer's Legacy: The Meaning of Asymmetry 2000, pp.77-79.
13 W. Heisenberg Der Teil und das Ganze Munich 1973,p.78 (reference kindly supplied by Theodore Lanscheidt)
14 I confirmed that the period 1928-1953 did have a higher septile expected value, of 2.52
15 N.K. and Mike O'Neill "Invention-Moments and Aspects to Uranus" Correlation Dec '92, pp.11-23.
16 N.Lemoine and D. Cooper Gene Therapy 1996, p.5.
17 Tim Berners-Lee Weaving the Web 1990
18 NK Eureka, The Celestial Pattern at Times of Historic Inspiration 2013, p.224
19 The horizontal line indicates chance-expected frequency: there are four angular positions for the three aspects, so likelihood of UR being in aspect with any one planet to 2° orb is 4x4/360; given nine planets that form aspects in 36 charts, 16x9x36/360=14.4 expected aspects, per 2° orb
20 D.Jay Practical Harmonics AFA, CA, 1983, p.7
21 D Cochrane Astrology in the 21st Century Florida 2002, p.94.
22 NK "Quintiles, Septiles, and Moments of Mystical Illumination" ISAR journal August 2011 (PDF available online here)
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Jan 22
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Badfish - A Tribute to Sublime
House Of Blues - Dallas, Dallas, Texas United States of America
Friday, January 22, 2016 at 8:00 PM
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Southside Wrestling have announced the final match for their May 11th show, ‘X-Factor’ at the Rushcliffe Arena, Nottingham.
It has been confirmed by Southside Wrestling that Flex Buffington will face Mad Man Manson, one-on-one at the Rushcliffe Arena on May 11th at ‘X-Factor’.
Southside Wrestling announced the match with the following:
Our final match for X-Factor is something completely different, Mad Man Manson returns to Nottingham to take on Flex Buffington in a first time ever one on one.
For those fans who made the trip to our last St Neots show, you will have seen Flex made his debut as RJ Singhs mystery partner to take on El Ligero and Mad Man Manson. As usual with anything involving Mad Man Manson hilarity ensued but when it comes to general wrestling insanity he may have met his match in Flex!
Flex is known for believing he is ‘a body guy’ and loves to pose with his drawn on muscles despite his super heavyweight frame! He was last seen in Southside showing Chris Masters how to work out properly!
We have no idea what will happen when these two characters face each other, but we bet that it will be entertaining!
Photo Credit: Southside Wrestling
The announced match card is as follows:
Match Card
X-Pac & ??? vs Project Ego
Jimmy Havoc vs Mark Haskins
El Ligero vs Tommy End
Young Wolves vs The Hunter Bros
Paul Malen vs Joseph Connors
Pete Dunne vs Stixx
Mad Man Manson vs Flex Buffington
Doors open at 3pm. First bell at 5pm.
Tickets are on sale now and can be purchased here.
The Official Facebook event page can be viewed here.
Source: Southside Wrestling Facebook
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The Woman's Club of Wilmette was established in 1891 and incorporated in 1906. A Junior Auxiliary to the Woman's Club was organized in 1928.
Membership currently includes approxilmately 120 women in the Woman's Club (Seniors), and over 80 Junior Auxiliary members.
Located in an historic building on the North Shore of Chicago, the Woman's Club of Wilmette is an inviting and convenient location for rental events.
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\begin{document}
\maketitle
\begin{abstract}
Recent work in computability theory has focused on various notions of asymptotic computability, which capture the idea of a set being ``almost computable.'' One potentially upsetting result is that all four notions of asymptotic computability admit ``almost computable'' sets in every Turing degree via coding tricks, contradicting the notion that ``almost computable'' sets should be computationally close to the computable sets. In response, Astor introduced the notion of intrinsic density: a set has defined intrinsic density if its image under any computable permutation has the same asymptotic density. Furthermore, introduced various notions of intrinsic computation in which the standard coding tricks cannot be used to embed intrinsically computable sets in every Turing degree. Our goal is to study the sets which are intrinsically small, i.e. those that have intrinsic density zero. We begin by studying which computable functions preserve intrinsic smallness. We also show that intrinsic smallness and hyperimmunity are computationally independent notions of smallness, i.e. any hyperimmune degree contains a Turing-equivalent hyperimmune set which is ``as large as possible'' and therefore not intrinsically small. Our discussion concludes by relativizing the notion of intrinsic smallness and discussing intrinsic computability as it relates to our study of intrinsic smallness.
\end{abstract}
\textbf{Keywords:} intrinsic computability, intrinsic density, asymptotic computation, hyperimmunity, weakly computably traceable
\section{Introduction}
A noteworthy phenomenon in the world of computing is that of problems which are generally ``easy'' to compute but have very difficult worst case instances. This gave rise to the notion of \textit{generic computability}, studied by Kapovich, Myasnikov, Schupp, and Shpilrain \cite{generic} in the context of computing the word problems of finitely generated groups. This notion asserts that a set is computable outside of a ``small'' error set where the algorithm does not answer. The notion of smallness here is that of having asymptotic density $0$:
\begin{definition}
The partial density of $A\subseteq \omega$ at $n$ is
\[\rho_n(A)=\frac{|A\upharpoonright n|}{n}.\]
That is, it is the ratio of the number of things less than $n$ that are in $A$ to what could be in $A$. The upper (asymptotic) density of $A$ is
\[\overline{\rho}(A)=\limsup_{n\to\infty} \rho_n(A)\]
and the lower (asymptotic) density of $A$ is
\[\underline{\rho}(A)=\liminf_{n\to\infty} \rho_n(A).\]
If $\overline{\rho}(A)=\underline{\rho}(A)$, we call this limit the (asymptotic) density of $A$ and denote it by $\rho(A)$.
\end{definition}
Recall that $W_e$ is the domain of the $e$-th Turing machine $\varphi_e$.
\begin{definition}
A set $A$ is generically computable if there is a partial computable function $\varphi_e$ such that $\underline{\rho}(W_e)=1$ and if $\varphi_e(n)\downarrow$, then $\varphi_e(n)=A(n)$. $\varphi_e$ is called a generic description of $A$.
\end{definition}
We think of generically computable sets as being computable ``almost everywhere,'' i.e. there is an algorithm that correctly answers questions on a set of density $1$, but does not answer on a small (density $0$) error set. Here the error set is the set of $n$ on which the description diverges. By changing the behavior of the generic description from diverging to something else, we obtain the other three notions of generic computability.
\begin{definition}
A set $A$ is coarsely computable if there is a total computable function $\varphi_e$ such that $\underline{\rho}(\{n:\varphi_e(n)=A(n)\})=1$. $\varphi_e$ is called a coarse description of $A$.
\end{definition}
For coarse computability, the description is forced to answer every question, but is allowed to give the incorrect answer on the error set. That is, the error set is the set of numbers on which the description and the set disagree.
\begin{definition}
A set $A$ is densely computable if there is a partial computable function $\varphi_e$ such that $\underline{\rho}(\{n:\varphi_e(n)\downarrow=A(n)\})=1$. $\varphi_e$ is called a dense description of $A$.
\end{definition}
For dense computability, the description can both answer questions incorrectly and not answer them on the error set. More specifically, the error set is both the places where the description diverges and the places where it disagrees with the set.
\begin{definition}
A set $A$ is effectively densely computable if there is a total computable function $\varphi_e:\omega\to\{0,1,\square\}$ such that $\underline{\rho}(\varphi_e^{-1}(\{0,1\}))=1$ and $\varphi_e(n)\in\{0,1\}$ implies $\varphi_e(n)=A(n)$. ( $\square$ represents $\varphi_e(n)$ refusing to answer whether $n$ is in or out of the set.)
\end{definition}
Effective dense computability need not answer questions on the error set much like generic computability, but it must refuse to do so outright rather than running for infinite time. (That is, the error set, which is the inverse image of $\square$ under the description, must be computable.) Note that there are some immediate implications among these notions. Effective dense computability implies both coarse computability and generic computability, and both of these imply dense computability. For an overview of the history of these notions, refer to the first section of \cite{dense}.
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One potentially unsettling feature of all four notions of asymptotic computability is that they depend heavily on the way in which information is coded. In fact, Jockusch and Schupp \cite{coarse} give a simple argument that can show every Turing degree contains a set which is effectively densely computable by ``hiding'' an entire set of any degree on a small computable set such as the factorial. (As the other three notions are implied by effective dense computability, the same is automatically true for every notion of asymptotic computability.)
\begin{proposition}
Let $X\subseteq \omega$. Then there is $A\equiv_T X$ which is effectively densely computable.
\end{proposition}
\begin{proof}
Given $X$, let $A=\{n!:n\in X\}$. Then $A$ is clearly Turing equivalent to $X$, and the function
\[ f(n) = \begin{cases}
\square & \mbox{if } n=k!\\
0 &\mbox{otherwise }
\end{cases}
\]
witnesses that $A$ is effectively densely computable.
\end{proof}
Therefore, these notions of being ``almost'' computable are heavily dependent upon how the set is coded: computably re-arranging the elements of a set can break the property of being ``almost computable.'' To combat this, Astor \cite{intrinsicdensity} introduced the notion of \textit{intrinsic density}, a strengthening of asymptotic density. Let $Perm$ be the index set of computable permutations of $\omega$.
\begin{definition}
The absolute upper density of $A\subseteq\omega$ is
\[\overline{P}(A)=\sup\{\overline{\rho}(\pi(A)):\pi\text{ a computable permutation}\}\]
and the absolute lower density of $A$ is
\[\underline{P}(A)=\inf\{\underline{\rho}(\pi(A)):\pi\text{ a computable permutation}\}.\]
If $\overline{P}(A)=\underline{P}(A)$, then we call this limit the intrinsic density of $A$ and denote it by $P(A)$.
\end{definition}
(In particular, if $A$ has intrinsic density $0$, then $\overline{\rho}(\pi(A))=0$ for every computable permutation. Furthermore, $\overline{P}(A)=0$ is enough to ensure $A$ has intrinsic density zero.) Of special interest is the property of having intrinsic density $0$, which has been studied extensively by Astor \cite{intrinsicdensity},\cite{intrinsicsmallness} in relation with other notions of smallness such as immunity. We will refer to sets that have intrinsic density $0$ as \textit{intrinsically small} to ease notation slightly. Technically finite sets meet this definition, but from here on we shall use the term to refer to infinite sets as those are the interesting ones.) We wish to study intrinsically small sets in order to use them as our error sets in an intrinsic version of asymptotic computability which we shall discuss in Section 5.
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\\
One easy observation about intrinsically small sets is that there are more computable functions $f$ such that $\overline{\rho}(f(A))=0$ for all intrinsically small sets $A$ than just the computable permutations. For example, if $\pi$ is a computable permutation, then $2\cdot \pi$ is not a computable permutation but the image of any intrinsically small set under it still has density $0$. The following definition captures the idea of classes of functions preserving smallness.
\begin{definition}
For a class $\mathcal{F}$ of (partial) computable functions from $\omega$ to $\omega$, we say that $A\subset\omega$ is \textit{small for} $\mathcal{F}$ if $\overline{\rho}(f(A))=0$ for every $f\in\mathcal{F}$.
\end{definition}
Notice that $A$ is intrinsically small if and only if it is small for computable permutations. In Section 2, we shall explore which classes of functions $\mathcal{F}$ have the property that every intrinsically small set is small for $\mathcal{F}$. This will give rise to a few questions, which we will study further in Section 3. In Section 4 we shall describe and explore the relativization of intrinsic smallness.
\section{Functions and Intrinsic Density}
We first note that not all intrinsically small sets are small for all computable functions, nor even all total computable functions. To do so, we use the following lemma:
\begin{lemma}
\label{jumpstrategy}
Let $X$ be a set of natural numbers. Suppose that $\{\mathcal{R}_e\}_{e\in\omega}$ is a collection of uniformly $X$-computable infinite sets. Then there is an intrinsically small set $A\leq\emptyset'\oplus X$ such that $A\cap \mathcal{R}_e\neq\emptyset$ for all $e$.
\end{lemma}
\begin{proof}
Note that the index set of injective partial computable functions is $\emptyset'$ computable, as the index set of noninjective partial computable functions is $\Sigma_1^0$. Therefore there is a $\emptyset'$-computable function $f$ such that $\varphi_{f(e)}$ is an enumeration of exactly the injective partial computable functions.
\\
\\
Let $A_0=\emptyset$ and $r_0=0$. Given $A_s$, $R_s$, define $A_{s+1},r_{s+1}$ as follows: Using $X$ as an oracle, find $k$ the least element of $R_s$ with $k>r_{s+1}$, which exists because $R_s$ is infinite. Let $A_{s+1}=A_s\cup\{k\}$. We say $e$ is suitable at stage $s$ if $[0,k]\subseteq dom(\varphi_{f(e)})$ and $[0,2\mathrm{max}(\varphi_{f(e)}(A_{s+1})]\subseteq \mathrm{range}(\varphi_{f(e)})$. Notice that $\emptyset'$ can compute whether or not $e$ is suitable at stage $s$ uniformly in $e$ and $s$ because it can ask finitely many questions about convergence. Now let
\[r_{s+1}=\mathrm{max}\{\varphi_{f(e)}^{-1}(i):e<s\text{ suitable at stage }s,\ i\leq 2\mathrm{max}(\varphi_{f(e)}(A_{s+1})\}+1.\]
Let $A=\bigcup_{s\in\omega} A_s$. By construction, $A\cap R_s\neq\emptyset$ because an element of $R_s$ was added at stage $s+1$. Now let $\pi=\varphi_{f(e)}$ be a computable permutation. Then $\pi$ is suitable at every stage because its domain and range are $\omega$. Now let $k$ be the element added at stage $s+2$ for some $s>e$. Then for every $i\leq 2\mathrm{max}(\pi(A_{s+1}))$,
\[k>r_{s+1}>\pi^{-1}(i).\]
Therefore $\pi(k)>2\mathrm{max}(\pi(A_{s+1}))$. Thus after finitely many elements, each element of $\pi(A)$ is more than double the previous element. It follows immediately that $\overline{\rho}(\pi(A))=0$. As $\pi$ was an arbitrary computable permutation, $A$ is intrinsically small.
\end{proof}
We can now show that there is an intrinsically small set which is not small for total computable functions.
\begin{theorem}
There is a set of intrinsic density $0$ which is not small for total computable functions. That is, there is an intrinsically small set $A$ and a total computable function $f$ such that $\overline{\rho}(f(A))>0$.
\end{theorem}
\begin{proof}
As defined by Jockusch and Schupp \cite{coarse}, let $R_e=\{n:2^e|n\text{ but }2^{e+1}\not|n\}$. Define $f:\omega\to\omega$ via $f(0)=0$ and $f(n)=e$, where $n\in R_e$. (Note that this is well-defined, as the $R_e$'s form a partition of $\omega\setminus\{0\}$.) $f$ is a total computable function.
\\
\\
By Lemma \ref{jumpstrategy}, there is an intrinsically small set $A$ such that $R_e\cap A\not=\emptyset$ for all $e$. Then $f(A)$ is cofinite (in fact it is either $\omega$ or $\omega\setminus\{0\}$), and therefore of intrinsic density $1$. (So $A$ catastrophically fails to have density $0$ under $f$.)
\end{proof}
We see from this example that the failure of injectivity allowed us to cast a wide net in search of elements of $A$ and then group them together to create a set of large density. Below, we shall see that we cannot even limit this to finite inverse images and preserve the property of being intrinsically small. In fact, we cannot even limit this to finite inverse images with uniformly computable size.
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\\
We shall need the notion of a hyperimmune set to do this. Recall that a disjoint strong array is a collection $\{D_{f(n)}\}_{n\in\omega}$ of finite sets coded by a total computable function $f$ and the canonical indexing of finite sets, where the $D_{f(n)}$'s are pairwise disjoint. A set $X$ is \textit{hyperimmune} if for every disjoint strong array $f$, there exists some $n$ with $D_{f(n)}\cap X=\emptyset$.
\begin{theorem}
\label{infinite}
There is an intrinsically small set which is not small for the collection of all total computable functions $f$ such that $f^{-1}(\{n\})$ is finite (and uniformly computable) for all $n$. That is, there is an intrinsically small set $A$ and a total computable function $f$ such that $\overline{\rho}(f(A))>0$ and a total computable function $g$ such that $g(n)=|f^{-1}(\{n\})|$ for all $n$.
\end{theorem}
\begin{proof}
Astor \cite{intrinsicsmallness} proved that the Turing degrees which contain an infinite intrinsically small set are those which are not weakly computably traceable. Kjos-Hanssen, Merkle, and Stephan \cite{highordnc} characterized these degrees as those which are High or DNC.
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It is well-known that there is a binary tree for which all paths are of PA degree. Recall that the PA degrees are exactly the $\text{DNC}_2$ degrees. Therefore, by the hyperimmune-free basis theorem, there is a $\text{DNC}_2$ degree that is hyperimmune-free. (For a review of this information, see Soare \cite{soare}.) This degree contains a set $A$ which is intrinsically small by the result of Astor. As $A$ is hyperimmune free, there exists a disjoint strong array $g$ such that $D_{g(n)}\cap A\neq\emptyset$ for all $n$. Without loss of generality, we can assume that $\mathrm{max}(D_{g(n)})<\mathrm{min}(D_{g(n+1)})$ for all $n$. (Given a disjoint strong array $g$, we can construct a new one $h$ as follows: $D_{h(0)}=D_{g(0)}$, and $D_{h(n+1)}$ is the first cell of the old array whose smallest element is larger than the largest element of $D_{h(n)}$.)
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Define $f:\omega\to\omega$ as follows: If $n\in D_{g(k)}$ for some $k$, let $f(n)=2k$. As $f$ is a disjoint strong array such that $\mathrm{max}(D_{g(n)})<\mathrm{min}(D_{g(n+1)})$, this is computable and well-defined. If $n\not\in\bigcup_{k\in\omega} D_{g(k)}$, then let $f(n)$ be the least odd number not realised as $f(m)$ for some $m<n$. Therefore $f$ is a total computable function with $|f^{-1}(\{n\})|$ finite and uniformally computable. (If $n=2k+1$ is odd, then the inverse image is a singleton. If $n=2k$ is even, then $f^{-1}(\{2k\})=D_{g(k)}$.) Furthermore, as $D_{g(n)}\cap A\neq\emptyset$ for all $n$, $f(A)$ contains all even numbers. Therefore $\overline{\rho}(f(A))\geq\frac{1}{2}$.
\end{proof}
We see that it is much more difficult for a set to be small for non-injective classes of functions. However, both examples relied heavily upon the fact that the functions were not injective. By switching our focus to (mostly) injective classes of functions, we can describe some classes of functions which any intrinsicall small set is small for. First, we provide an easy technical lemma.
\begin{lemma}
Suppose $C$ is an infinite c.e. set. Then there exists an infinite, computable $H\subseteq C$ with $\rho(H)=0$.
\end{lemma}
\begin{proof}
Let $\{c_i\}_{i\in\omega}$ be an enumeration of $C$. Then let $\{h_i\}_{i\in\omega}$ be such that $h_0=c_0$ and given $h_n$, $h_{n+1}=c_j$, where $j$ is the least index with $c_j>h_n+2^n$. Then $H$ is computable because it is a c.e. set with an increasing enumeration, and it clearly has density $0$.
\end{proof}
\begin{theorem}
\label{injective}
Suppose that $A$ is an intrinsically small set. Then $A$ is small for the class of total computable injective functions with computable range.
\end{theorem}
\begin{proof}
We argue by contrapositive: Suppose $f$ is total computable injective function with computable range, and $A$ is a set with $\overline{\rho}(f(A))>0$. Then we construct a computable permutation $\pi$ such that $\overline{\rho}(\pi(A))>0$.
\\
\\
Let $H\subseteq \mathrm{range}(f)$ be a computable set of density $0$. Now define $\pi:\omega\to\omega$ as follows: If $f(n)\not\in H$, $\pi(n)=f(n)$. If $f(n)\in H$, let $\pi(n)$ be the least element of $H\cup\overline{\mathrm{range}(f)}$ not realized in the range of $\pi$ by $m<n$. Then $\pi$ is a computable permutation, and
\[\rho_n(\pi(A))=\frac{|\pi(A)\upharpoonright n|}{n}\geq \frac{|f(A)\upharpoonright n|-|H\upharpoonright n|}{n}=\rho_n(f(A))-\rho_n(H).\]
(The inequality comes from the fact that $\pi$ and $f$ agree on $f^{-1}(\mathrm{range}(f)\setminus H)$.) Therefore, we obtain
\[\overline{\rho}(\pi(A))\geq \overline{\rho}(f(A))-\overline{\rho}(H)=\overline{\rho}(f(a))>0.\]
Therefore $\pi$ is a computable permutation for which $\overline{\rho}(\pi(A))>0$, so $A$ is not intrinsically small.
\end{proof}
Note that simpler proofs of Theorem 2.5 exist which do not require us to create an error set and construct a permutation, however this proof is illustrative of the techniques we shall use for more difficult proofs.
\begin{corollary}
\label{image}
If $A$ is intrinsically small and $f$ is a total computable injective function with computable range, then $f(A)$ is intrinsically small.
\end{corollary}
\begin{proof}
This follows from Theorem \ref{injective} by the fact that $\pi(f(A))=\pi\circ f(A)$ and $\pi\circ f$ is a total computable injective function with computable range because $f$ is.
\end{proof}
\begin{corollary}
\label{join}
If $A$ and $B$ are intrinsically small, then so is $A\oplus B$.
\end{corollary}
\begin{proof}
If $f$ is the function sending $n$ to $2n$, and $g$ is the function sending $n$ to $2n+1$, then by Corollary \ref{image} $f(A)$ and $g(B)$ are both intrinsically small. It is easy to check that the union of two intrinsically small sets is intrinsically small, as the permutation of the union is the union of the images under the permutation. Therefore, $A\oplus B=f(A)\cup g(B)$ is intrinsically small.
\end{proof}
We can improve this result. The use of $H$ in the proof allows us to notice that we can change a subset of density $0$ in the range and not suffer any consequences for preserving intrinsic smallness.
\begin{definition}
A (partial) function $f:\omega\to\omega$ is *-injective, or almost injective, if $\rho(\{n:|f^{-1}(\{n\})|>1\})=0$. That is, a (partial) function is almost injective if the subset of the range where injectivity fails has density $0$.
\end{definition}
\begin{theorem}
\label{almostinjective}
Suppose that $A$ is an intrinsically small set. Then $A$ is small for the class of total computable *-injective functions with computable range.
\end{theorem}
\begin{proof}
We again argue by contrapositive: Suppose $f$ is total computable *-injective function with computable range, and $A$ is a set with $\overline{\rho}(f(A))>0$. Then we construct a total computable injective function $g$ such that $\overline{\rho}(g(A))>0$ and invoke Theorem \ref{injective}.
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\\
Let $H\subseteq \mathrm{range}(f)$ be infinite, computable, and have density $0$. Then define $g(n)=f(n)$ if $f(n)$ has not been realized in $\mathrm{range}(g)$ by some $m<n$, and to be the least element of $H$ not realized in $\mathrm{range}(g)$ otherwise. Then $g$ is injective, as $g(n)$ cannot be in $\mathrm{range}(g\upharpoonright n)$ for any $n$ by construction. Furthermore,
\[\rho_n(g(A))=\frac{|g(A)\upharpoonright n|}{n}\geq \frac{|f(A)\upharpoonright n|-|H\upharpoonright n|-|\{k:|f^{-1}(\{k\})|>1\}\upharpoonright n|}{n}=\]
\[\rho_n(f(A))-\rho_n(H)-\rho_n(\{k:|f^{-1}(\{k\})|>1)).\]
This gives
\[\overline{\rho}(g(A))\geq \overline{\rho}(f(A))-\overline{\rho}(H)-\overline{\rho}(\{k:|f^{-1}(k)|>1\}=\overline{\rho}(f(A))>0.\qedhere\]
\end{proof}
\begin{remark}
While an intrinsically small set is small for the class of total computable *-injective functions with computable range, the image under such functions is not intrinsically small: Take the set $A$ and function $f$ from the proof of Theorem \ref{infinite} and let $g(n)=2^{f(n)}$. Then $g$ is *-injective because its entire image has density zero. However, there is a computable permutation $\pi$ that maps $image(g)$ to the non-factorials and the complement to the factorials. Then $\pi\circ g(A)$ is all but finitely many of the non-factorials and is therefore density one.
\end{remark}
To this point, we've seen that injectivity almost everywhere has been essential in allowing all intrinsically small sets to be small for our class of functions. However, up to this point we've also relied heavily on knowing that the range is computable: if the range is not computable, we may potentially fill in part of the range that $A$ would have been sent to later. In this case, we'd need to shift where the elements of $A$ are sent, potentially sending the density to $0$ in the process. As we'll see below, there are cases in which we can avoid this issue.
\begin{theorem}
\label{density}
Suppose $A$ is a set and $f$ is a *-injective function with $\overline{\rho}(f(A))=q>0$ and $\overline{\rho}(\mathrm{range}(f))-\underline{\rho}(\mathrm{range}(f))<q$. Then there is a *-injective function $g$ with computable range such that $\overline{\rho}(g(A))>0$.
\end{theorem}
\begin{proof}
As $\mathrm{range}(f)$ is c.e., there is a computable subset $H$ of $\mathrm{range}(f)$ with $\underline{\rho}(H)>\overline{\rho}(\mathrm{range}(f))-q$ by Downey, Jockusch, and Schupp \cite{DJS}. In particular,
\[\overline{\rho}(\mathrm{range}(f)\setminus H)\leq \overline{\rho}(\mathrm{range}(f))-\underline{\rho}(H)<q.\]
Define $g:\omega\to\omega$ via $g(n)=f(n)$ if $f(n)\in H$, and $g(n)=0$ otherwise. Notice that $g$ is *-injective, as
\[\{n:|g^{-1}(\{n\})|>1\}\subseteq\{n:|f^{-1}(\{n\}|>1\}\cup\{0\}.\]
Furthermore, $\mathrm{range}(g)=H\cup \{0\}$ is computable. Lastly, notice that
\[\rho_n(g(A))=\frac{|g(A)\upharpoonright n|}{n}\geq\frac{|f(A)\upharpoonright n|-|\{k<n:k\not\in H\text{ and }k\in f(A)|}{n}\geq\]
\[\frac{|f(A)\upharpoonright n|-|(\mathrm{range}(f)\setminus H)\upharpoonright n|}{n}=\rho_n(f(A))-\rho_n(\mathrm{range}(f)\setminus H).\]
By the above fact that $\overline{\rho}(\mathrm{range}(f)\setminus H)\leq\overline{\rho}(\mathrm{range}(f))-\underline{\rho}(H)<q$,
\[\overline{\rho}(g(A))> \overline{\rho}(f(A))-q=q-q=0\]
That is, $\overline{\rho}(g(A))>0$.
\end{proof}
\begin{corollary}
\label{hasdensity}
Suppose that $A$ is an intrinsically small set. Then $A$ is small for the class of total computable *-injective functions whose range has defined density.
\end{corollary}
\begin{proof}
We again argue by contrapositive: Suppose $f$ is total computable *-injective function whose range has defined density, and $A$ is a set with $\overline{\rho}(f(A))>0$. Then by Theorem \ref{density}, as $\overline{\rho}(\mathrm{range}(f))-\underline{\rho}(\mathrm{range}(f))=0$, there is a *-injective function $g$ with computable range such that $\overline{\rho}(g(A))>0$. The result follows by Theorem \ref{almostinjective}.
\end{proof}
By the remark following the proof of Theorem \ref{almostinjective}, we see that the image of an intrinsically small set under a total computable *-injective function whose range has defined density need not be intrinsically small. However if we restrict ourselves to injective functions, can we recover the analogue of Corollary \ref{image}? The same argument does not work, as the image of a c.e. set with defined density under a computable permutation need not have defined density.
\begin{question}
\label{preserve}
If $A$ is intrinsically small and $f$ is a total computable injective function whose range has defined density, then is $f(A)$ intrinsically small?
\end{question}
Additionally, the natural follow-up question to Corollary \ref{hasdensity} remains open. This question is closely related to Question \ref{preserve}.
\begin{question}
\label{question-injective}
Suppose that $A$ is an intrinsically small set. Is $A$ small for the class of total computable *-injective functions? Total computable injective functions?
\end{question}
Notice that if the answer here is yes, then the analogue of Corollary {\ref{image}} for computable injective functions follows immediately from the same argument. Therefore a positive answer yields a positive answer to Question {\ref{preserve}}, and a negative answer to Question {\ref{preserve}} yields a negative answer to Question {\ref{question-injective}}. The opposite direction also seems closely related, but any implications are not immediately obvious.
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\\
Theorems \ref{almostinjective} and \ref{density} help to characterize what must happen in the scenario where the answer to Question \ref{question-injective} is no: The upper and lower density of the range are relatively far apart, allowing small elements of $f(A)$ to show up at late stages after any computable process ``thinks'' $\mathrm{range}(f)$ is done enumerating small elements.
\\
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Corollary {\ref{hasdensity}} can already be used in conjunction with known results. For example, Jockusch (correspondence with Astor) showed that r-maximal sets have intrinsic density (and therefore density) $1$, so the image of any intrinsically small set under a computable injective function whose range is maximal is small.
\section{Hyperimmunity and Intrinsic Smallness}
It is important to note that when studying whether or not certain properties relate to intrinsic smallness, we shall study the sets themselves rather than their degrees: coding tricks can show that every Turing degree contains a set with undefined density. In the c.e. degrees, this set can be taken to be c.e.
\begin{lemma}
\label{degree}
Every Turing degree contains a set $W$ with $\underline{\rho}(W)=0$ and $\overline{\rho}(W)=1$.
\end{lemma}
\begin{proof}
Given $C$, let $D=\{n!:n\in C\}$ and $W=D\cup\bigcup_{n\in\omega}((2n)!,(2n+1)!)$. Then $W\equiv_T D\equiv_T C$, and $\underline{\rho}(W)=0$ because \[\rho_{(2n+2)!}(W)=\frac{|W\upharpoonright (2n+2)!|}{(2n+2)!}\leq \frac{(2n+1)!}{(2n+2)!}=\frac{1}{2n+2}.\]
Conversely, $\overline{\rho}(W)=1$ as
\[\rho_{(2n+1)!}(W)=\frac{|W\upharpoonright (2n+1)!|}{(2n+1)!}\geq \frac{(2n+1)!-(2n)!}{(2n+1)!}=1-\frac{1}{2n+1}.\]
Clearly if $C$ is c.e., then so is $W$.
\end{proof}
We shall see below that additional properties on the starting set $C$ can be recovered in $W$ by modifying the construction.
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\\
We now turn our attention to hyperimmune sets, a competing notion of smallness. Astor \cite{intrinsicdensity} studied the connection between varying notions of immunity and intrinsic density thoroughly. In particular, it is known that hyperimmune sets have intrinsic lower density $0$, and therefore that hypersimple sets have intrinsic upper density $1$. (Hypersimple sets are c.e. sets whose complement is hyperimmune. Recall that hyperimmune sets are infinite by definition, so hypersimple sets are co-infinite.) One question left open in \cite{intrinsicdensity} (later answered by Astor in \cite{intrinsicsmallness} using a degree argument) was whether or not a hypersimple set could have lower density $0$, or at least non-$1$ lower density. The answer is yes, showing that hypersimple sets need not have defined density. We give a constructive proof, showing that every hypersimple set yields a Turing equivalent hypersimple set which has lower density $0$. (That is, every hypersimple set has an equivalent hypersimple set which is ``as small as possible.'')
\begin{theorem}
\label{hypersimple}
Let $C$ be a hypersimple set. Then there is a hypersimple set $W\equiv_T C$ with $\underline{\rho}(W)=0$.
\end{theorem}
\begin{proof}
As $C$ is hypersimple, it has intrinsic upper density (and therefore upper density) $1$. We cannot use the strategy from Lemma \ref{degree} directly, as the resulting set will not even be immune, let alone hyperimmune. To avoid this problem, we shall leave intervals of $C$ intact and introduce gaps between the intervals in noncomputable fashion. Informally, we first wish to shift portions of $C$ over to make large gaps, ensuring that the resulting set has lower density $0$. We then leave an even larger interval of $C$ intact (albeit shifted over finitely much) to ensure that the upper density is $1$. (See Figure \ref{hyperimmunefigure}.) Formally, we shall define c.e. sets $H_i$ and gaps $[u_i, u_i+m_i]$ inductively. Let $H_0=C$. Enumerate $H_0$ until there is a stage $s$ and a number $n$ such that we see $\rho_n(H_0)>\frac{1}{2}$, which exists because $C=H_0$ has upper density $1$. Then let $u_0=n$ and let $m_0$ be the least natural number such that $\frac{u_0}{u_0+m_0}<\frac{1}{2}$.
\\
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Given $H_{e}$ and $[u_e,u_e+m_e]$, define $H_{e+1}$ and $[u_{e+1},u_{e+1}+m_{e+1}]$ as follows: Define $H_{e+1}=(H_e\upharpoonright u_e)\cup (H_e^{\geq u_e}+m_e)$. (For convenience, here $X^{\geq k}$ denotes $\{n\in X: n\geq k\}$, and $X+m=\{n+m:n\in X\}$.) Enumerate $H_{e+1}$ until there is a stage $s$ and a number $n>u_e+m_e$ such that $\rho_n(H_{e+1,s})>1-\frac{1}{e+2}$. Then set $u_{e+1}=n$ and $m_{e+1}$ to be the least natural number such that $\frac{u_{e+1}}{u_{e+1}+m_{e+1}}<\frac{1}{e+2}$. Finally, let $H$ be the set with characteristic function $H(m)=\lim_{n\to\infty} H_n(m)$. Note, first off, that $\bigcup_{e\in\omega} [u_e,u_e+m_e]$ is a c.e. set with increasing enumeration, and hence computable. Furthermore, note that $H$ itself is c.e., as $\lim_{n\to\infty} H_n(m)=H_s(m)$ for any $s$ with $u_s>m$. $\underline{\rho}(H)=0$ as desired, as $\rho_{u_i+m_i}(H)<\frac{1}{i+2}$ for all $i$.
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$H$ itself will not work as the desired $W$: The complement contains the computable subset $\bigcup_{e\in\omega} [u_e,u_e+m_e]$, so it is not even immune, let alone hyperimmune. Therefore, let $W=H\cup\bigcup_{n\in C} [u_n,u_n+m_n]$: that is, enumerate the $n$-th gap into $W$ whenever $n$ enters $C$. Then $W$ is c.e., and we claim that it is hypersimple.
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Recall that the principal function $p_A:\omega\to A$ of a set $A=\{a_0<a_1<a_2<\dots\}$ is the function such that $p_A(n)=a_n$. Also recall that a set is hyperimmune if and only if its principal function is not computably bounded. Suppose that $\overline{W}$ is not hyperimmune. Then it is bounded by some total computable function $f$. However, the total computable function $g$ defined via $g(n)=f(n+\mathop{\Sigma}_{i\leq n} m_i)$ must bound $\overline{C}$: The elements of $\overline{W}$ are the elements of $\overline{C}$ shifted up along with the corresponding gaps. The $n$-th element of $\overline{C}$ is smaller than the $n$-th non-gap element of $\overline{W}$ (as the $n$-th non-gap element of $\overline{W}$ is the $n$-th element of $\overline{C}$ shifted up by the gaps below it), which is at most the $n+\mathop{\Sigma}_{i\leq n} m_i$-th element of $\overline{M}$ because a gap in $\overline{W}$ corresponds to an element of $\overline{C}$ below the gap.
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Thus we have shown that $W$ is a hypersimple set. It is Turing equivalent to $C$ because $\bigcup_{e\in\omega} [u_e,u_e+m_e]$ is computable: $W$ can compute $C$ by ignoring the intervals, and $C$ can clearly compute $H$ and hence $W$.
\end{proof}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[thin] (-1,0) -- (-1,0) node[anchor=north]{$0$} node[anchor=south]{$H_0^{\geq 0}+0$};
\draw[ultra thick] (-1,0) -- (10,0) node[anchor=west]{$C=H_0$};
\draw[ultra thick] (-1,-2) -- (1,-2) node[anchor=north]{$u_0$}
node[anchor=south]{$H_0\upharpoonright u_0$};
\draw[thin] (1,-2) -- (3,-2) node[anchor=north]{$u_0+m_0$}
node[anchor=south]{$H_0^{\geq u_0}+m_0$};
\draw[ultra thick] (3,-2) -- (10,-2) node[anchor=west]{$H_1$};
\draw[ultra thick] (-1,-4) -- (1,-4);
\draw[thin] (1,-4) -- (3,-4);
\draw[ultra thick] (3,-4) -- (5.5,-4) node[anchor=north]{$u_1$}
node[anchor=south]{$H_1\upharpoonright u_1$};
\draw[thin] (5.5,-4) -- (9,-4)
node[anchor=north]{$u_1+m_1$} node[anchor=south]{$H_1^{\geq u_1}+m_1$};
\draw[ultra thick] (9,-4) -- (10,-4) node[anchor=west]{$H_2$};
\node at (5, -5) {.};
\node at (5,-5.1) {.};
\node at (5,-5.22) {.};
\node at (10.5, -5) {.};
\node at (10.5,-5.1) {.};
\node at (10.5,-5.22) {.};
\end{tikzpicture}
\caption{Visualization of the construction of $H$ in Theorem \ref{hypersimple}}
\label{hyperimmunefigure}
\end{center}
\end{figure}
By using $C$ as an oracle rather than an enumeration of $C$, it is clear that this result also applies to co-hyperimmune sets in general, not just hypersimple sets.
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Perhaps the most useful characterization of the hyperimmune sets is that a set is hyperimmune if and only if its principle function is not computably bounded. Recall that the principle function $p_X$ of an infinite set $X=\{x_0<x_1<x_2<x_3<\dots\}$ is the function such that $p_X(n)=x_n$. While Theorem \ref{hypersimple} shows that hyperimmunity and intrinsic smallness are unrelated notions of smallness, we would like to know whether it is possible to provide a simple characterization of intrinsic smallness using principal functions. Perhaps the most natural candidate is that of weak computable traceability from \cite{intrinsicsmallness}, which does provide us with a useful test for intrinsic smallness:
\begin{lemma}
\label{principalfunction}
Suppose that $A$ is not intrinsically small. Then the principle function $p_A(n)$ of $A$ is weakly computably traced, i.e. there are computable functions $g$ and $h$ with $|D_{g(n)}|\leq h(n)$ for all $n$ and $p_A(n)\in D_{g(n)}$ for infinitely many $n$.
\end{lemma}
\begin{proof}
As $A$ is not intrinsically small, there is a computable permutation $\pi$ such that $\overline{\rho}(\pi(A))=q>0$. Define functions $h=\lambda n(n!)$ and $g$ such that $D_{g(n)}=\pi^{-1}([0,n!))$. Then we claim that $g$ and $h$ witness that $p_A$ is weakly computably traced.
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To get a contradiction, suppose this is not the case. Then $p_A(k)\in D_{g(k)}=\pi^{-1}([0,k!))$ for only finitely many $k$. In particular, $\pi(n)\geq n!$ for all but finitely many $n\in A$. This clearly implies that $\overline{\rho}(\pi(A))=0$, however, as $\rho_n(\pi(A))\leq \frac{s+m+1}{m!}$ where $s$ is the number of $k$ for which $p_A(k)\in \pi^{-1}([0,k!))$ and $m$ is the largest number with $m!\leq n$. As $\frac{s+m+1}{m!}$ approaches $0$ in the limit, this contradicts the fact that $\overline{\rho}(\pi(A))=q>0$, so $g$ and $h$ must witness that $p_A$ is weakly computably traced.
\end{proof}
The contrapositive of Lemma \ref{principalfunction} tells us that if the principle function of $A$ is not weakly computably traced, then $A$ is intrinsically small. Unfortunately, Theorem \ref{infinite} tells us that we cannot hope to reverse this in general. However, notice that the proof in fact proves a stronger statement: If $A$ is not intrinsically small, then it is weakly computably traced with witness $h=\lambda n(n!)$. That is, if $p_A$ is not weakly computably traced by $h$, then $A$ is intrinsically small. If this can be reversed, that would characterize the intrinsically small sets.
\begin{question}
Is it the case that if $A$ is intrinsically small, then $p_A$ is not weakly computably traced by $h=\lambda n(n!)$? If it is not the case, is there an intrinsically small set which does not dominate $h$? (I.e. $p_A(n)\leq n!$ infinitely often?)
\end{question}
Of course there are computably dominated intrinsically small sets by Theorem \ref{infinite}, however it is not clear if there are any ``nice'' computable functions (i.e. something naturally occurring in arithmetic or combinatorics) which dominate an intrinsically small set, or even which are not dominated by the principal function of one. Our usual strategy for constructing intrinsically small sets is no help, as it requires arbitrarily large witnesses.
\section{Relative Intrinsic Smallness}
The definition of intrinsic density, and by extension the definition of intrinsic smallness, admits a natural relativization:
\begin{definition}
\label{relativized}
The $X$-absolute upper density of $A\subseteq\omega$ is
\[\overline{P_X}(A)=\sup\{\overline{\rho}(\pi(A)):\pi\text{ an }X\text{ computable permutation}\}\]
and the absolute lower density of $A$ is
\[\underline{P_X}(A)=\inf\{\underline{\rho}(\pi(A)):\pi\text{ an }X\text{ computable permutation}\}.\]
If $\overline{P_X}(A)=\underline{P_X}(A)$, then we call this limit the $X$-intrinsic density of $A$ and denote it by $P_X(A)$.
\end{definition}
It is easy to see that no infinite, co-infinite set $A$ is $A$-intrinsically small, or in fact has $A$-intrinsic density. (One way to observe this is to note that the permutation taking $A$ to the set $W$ in $deg(A)$ from Lemma \ref{degree} is $A$-computable.) Furthermore, given a set $A$, the set of Turing degrees for which $A$ is not intrinsically small is closed upwards and contains the cone above $A$. One may ask if a set is intrinsically small, is it the case that this set is exactly the cone above $A$? The answer is no.
\begin{lemma}
There is an intrinsically small set $A$ and a permutation $\pi\not\geq_T A$ such that $\overline{\rho}(\pi(A))>0$.
\end{lemma}
\begin{proof}
Let $B$ and $C$ be Turing incomparable intrinsically small sets. (These exist given the result of Astor that the degrees containing intrinsically small sets are the degrees which are high or DNC.) Then by Corollary \ref{join}, $A=B\oplus C$ is intrinsically small. Now let $\pi$ be the $B$-computable permutation mapping $\{2n:n\in B\}$ to the non-factorials and the complement to the factorials. Then $\pi(B\oplus C)$ contains the non-factorials, and therefore has density $1$.
\end{proof}
As a corollary, we see that given an intrinsically small set $A$, the set of $X$ for which $A$ is $X$-intrinsically small need not be the degrees strictly below $A$: As $B$ and $C$ in the above proof are Turing incomparable, $B\oplus C$ is strictly Turing above $B$, but is not intrinsically small relative to $B$. However, it is clear that given a set $A$, the collection of Turing degrees of $X$ with $A$ $X$-intrinsically msall is closed downwards.. Must it be a Turing ideal? The following lemma shows the answer is no.
\begin{lemma}
There is an intrinsically small set $A$ and sets $B,C$ with $A$ $B$-intrinsically small and $C$-intrinsically small but not $B\oplus C$-intrinsically small. That is, the set of $X$ for which $A$ is $X$-intrinsically small is not a Turing ideal.
\end{lemma}
\begin{proof}
By the Sacks Splitting Theorem \cite{sacks}, there are low sets $B$ and $C$ such that $B\oplus C\equiv_T \emptyset'$. Therefore a modification of Lemma \ref{jumpstrategy} allows us to obtain a set $A\leq\emptyset'$ which is both $B$-intrinsically small and $C$-intrinsically small. (As $B$ and $C$ are low, $B'\equiv_T C'\equiv_T \emptyset'$, so $\emptyset'$ can enumerate the partial $B$ and $C$ computable injective functions and determine suitability for them.) However, $A$ cannot be $B\oplus C$-intrinsically small because $A\leq_T\emptyset'\equiv_T B\oplus C$.
\end{proof}
Note that although the set of $X$ for which $A$ is $X$-intrinsically small need not be a Turing ideal, Definition \ref{relativized} still makes sense if one considers all $\mathcal{I}$-computable permutations in a Turing ideal $\mathcal{I}$ rather than computable in a set $X$.
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The following lemma allows us to describe the degrees of $X$-intrinsically small sets for certain $X$.
\begin{lemma}
\label{relativizedhighdnc}
Let $X$ be an arithmetical set. Then the Turing degrees which contain an $X$-intrinsically small set $A$ are the $X$-high or $X$-DNC degrees.
\end{lemma}
\begin{proof}
We merely need to check that the proof of Corollary 2.7 from Astor \cite{intrinsicsmallness} relativizes. It is straightforward to check that the proof given by Downey and Hirschfeldt \cite{downeyhirschfeldt} of the result of Kjos-Hanssen, Merkle, and Stephan \cite{highordnc} relativizes: a set $A$ is $X$-weakly computably traceable if and only if it is $X$-high or $X$-DNC.
\\
\\
Using this, the rest of the proof of \cite{intrinsicsmallness} Theorem 2.4 relativizes, and therefore \cite{intrinsicsmallness} Corollary 2.5 does as well. \cite{intrinsicsmallness} Theorem 2.6 also relativizes, which is straightforward to check. To obtain \cite{intrinsicsmallness} Corollary 2.7, Astor employs the following result of Jockusch \cite{jockusch}: Given some property $P$ of some sets of natural numbers, if there is an arithmetical set exhibiting $P$ and $P$ is closed under taking subsets, then the collection of Turing degrees which contain a set exhibiting $P$ is closed upwards. The relativized form of Lemma \ref{jumpstrategy} above yields an $X'$-computable $X$-intrinsically small set $A$. As $X$ is arithmetical, $A$ is arithmetical, so we may apply the result of Jockusch to obtain the relativized form of \cite{intrinsicsmallness} Corollary 2.7.
\end{proof}
There is an obvious gap in Lemma \ref{relativizedhighdnc}. Specifically, can the arithmetical requirement on $X$ be dropped? There are certainly sets $X$ for which there are no arithmetical $X$-intrinsically small sets $A$: If $X=\emptyset^{(\omega)}$, then $X$ computes every arithmetical set and therefore there cannot be an arithmetical $X$-intrinsically small set. An important note here is that the relativization of \cite{intrinsicsmallness} Corollary 2.5 and Theorem 2.6 did not rely on the fact that $X$ was arithmetical, so we already know that $X$-weakly computably traced sets are not $X$-intrinsically small and that any non-$X$-weakly computably traced set computes an $X$-intrinsically small set for even non-arithmetical $X$.
\begin{question}
For which non-arithmetical sets $X$ are the degrees containing an $X$-intrinsically small set those which are $X$-high or $X$-DNC? For which non-arithmetical $X$ are they upwards closed?
\end{question}
A natural question arises from the appearance of $\emptyset^{(\omega)}$: We say a set $A$ is arithmetically intrinsically small if it is $X$-intrinsically small for every arithmetical set $X$. Is there an arithmetically intrinsically small set which is not $\emptyset^{(\omega)}$-intrinsically small?
It turns out that the answer is yes, as $\emptyset^{(\omega)}$ can uniformly compute all of the arithmetical permutations. Therefore a modification of Lemma \ref{jumpstrategy} allows us to construct a $\emptyset^{(\omega)}$-computable set which is arithmetically intrinsically small.
\section{Intrinsic Computability}
Having studied intrinsically small sets, we now turn our attention to their use as error sets in ``almost computable'' settings. Astor \cite{intrinsicdensity} first described four possible variations of ``intrinsic'' generic computability, that is ``intrinsic'' generic descriptions of $A$ which ensure the existence of generic descriptions of $\varphi_e(A)$ for all $e\in Perm$. The four notions differ by how uniformly we can obtain a generic description for a given permutation. We provide the generalizations of each of these notions to the remaining three notions of asymptotic computability mentioned in Section 1, which gives us a total of sixteen separate notions. Throughout this section $x$ will denote an arbitrary element of $\{\text{effective dense, generic, coarse, dense}\}$. We shall begin by describing the strongest of the four notions, which is the most overtly related to our study of intrinsically small sets.
\begin{definition}
$A\subseteq \omega$ is intrinsically $x$-ly computable if there is an $x$ description of $A$ with an intrinsically small error set.
\end{definition}
Astor originally defined this notion as strongly intrinsically $x$-ly computable, however we shorten the definition for the sake of readability.
\\
\\
This is the most natural intrinsic variant of asymptotic computability, as it is obtained by simply requiring the error set to meet a stronger smallness condition. As we shall see, the other three notions introduced in \cite{intrinsicsmallness} are not obtained by simply modifying the error set, but rather by introducing new restrictions on the computation.
\\
\\
We should verify that the intrinsically $x$-ly computable sets are not just the computable sets: clearly the computable sets meet this definition for any $x$, but are there noncomputable examples? It turns out that for the strongest notion, intrinsically effectively densely computable sets, this is not the case:
\begin{lemma}
Suppose that $A$ is intrinsically effectively densely computable. Then $A$ is computable.
\end{lemma}
\begin{proof}
By definition, if $A$ is intrinsically effectively densely computable, then the error set is an intrinsically small computable set. However, no infinite computable set can be intrinsically small, as there is a computable permutation that maps it to the nonfactorials and its complement to the factorials. Therefore, the error set must be finite. As $A$ differs from a computable set by only finitely much, it must be computable.
\end{proof}
Fortunately, the other three do admit noncomputable examples. For generic computability, as mentioned in \cite{intrinsicsmallness}, any c.e. set with intrinsic density $1$, such as a maximal set, is intrinsically generically computable. Similarly, any set of intrinsic density $1$ or $0$ is intrinsically coarsely computable. Notice that any intrinsically generically computable set with defined intrinsic density must have intrinsic density $0$ or $1$ and thus be intrinsically coarsely computable: Let $\varphi_e$ be an intrinsic generic description of $A$. If $\{n:\varphi_e(n)\downarrow=1\}$ is finite, then $A$ has intrinsic density $0$ because $A=\{n:\varphi_e(n)\downarrow=1\}\cup(A\cap\overline{W_e})$ is a union of a finite set with an intrinsically small set. If this set is not finite, then it is an infinite c.e. subset of $A$. Therefore the absolute upper density of $A$ is $1$ because every infinite c.e. set has a computable subset, which can be mapped to the nonfactorials by a computable permutation. As $A$ has defined intrinsic density and its absolute upper density is $1$, it must have intrinsic density $1$. In both cases, $A$ is intrinsically coarsely computable. The following lemma shows that the intrsincially generically computabile sets and the intrinsically coarsely computable sets are not the same, however.
\begin{lemma}
\label{coarseseparation}
There is a intrinsically coarsely computable set which is not intrinsically generically computable.
\end{lemma}
\begin{proof}
By Lemma \ref {jumpstrategy}, there is an intrinsically small set $A$ such that for each infinite c.e. set $W_e$ there exists $a_e\in A\cap W_e$ with $a_e<a_s$ for $e< s$. That is, there is a unique designated element $a_e$ of $A$ for each infinite c.e. set $W_e$. $\emptyset'$ cannot determine if a c.e. set is infinite, but it can ask if there is a large enough element of $W_e$ to continue the construction and put that into $A$ if it exists. This may designate some elements for finite c.e. sets, but this is acceptable.
\\
\\
Now define $B\subseteq A$ by agreeing with $A$ away from the $a_e$'s and diagonalizing against the $e$-th turing machine using $B(a_e)$, i.e. $B(a_e)=1-\varphi_e(a_e)$. (Note that $\varphi_e(a_e)\downarrow$ because $a_e\in W_e$.) Then $B\subseteq A$ has intrinsic density $0$ and cannot be intrinsically generically computable because it disagrees with every turing machine with infinite domain at least once.
\end{proof}
The reverse separation remains open: it is easy to ensure that a given Turing function is not an intrinsic generic description by simply finding one place where it is wrong. However, to ensure that a given Turing function is not an intrinsic coarse description, we must force it to disagree on an infinite set which is not intrinsically small, which is more difficult. The natural strategy is to take an intrinsic generic description $W_i$, say a maximal set, and attempt to change it to diagonalize against the total functions in such a way that the description is still c.e. and its complement is still intrinsically small. The issue arises from our not being able to enumerate all of the total functions using computable indices: there is an enumeration of c.e. indices which contains exactly the computable sets (given an index $e$, enumerate $W_e$ so long as the enumeration is increasing, but do not enumerate smaller elements), but there is no way to distinguish the infinite sets from the finite ones. If we know a given c.e. index $e$ yields an infinite computable set, it is easy to wait for convergence of $\varphi_e$ and diagonalize against it on an infinite computable subset of $W_i$, forcing $\varphi_e$ to not be a n intrinsic coarse description. However if $W_e$ is in fact finite, then we will never see convergence, and failing to converge for the indices of finite sets will make the complement of our new enumeration no longer intrinsically small. If we give up waiting for convergence after some length of time, then there is no guarantee that an infinite computable set will ever enumerate quickly enough to be diagonalized against.
\begin{question}
\label{genericseparation}
Is there an intrinsically generically computable set which is not intrinsically coarsely computable?
\end{question}
One potentially useful result for this question is the result of Arslanov \cite{arslanov} that the only c.e. DNC degree is $\emptyset'$. As mentioned above, we know from \cite{intrinsicsmallness} that the degrees which contain an intrinsically small set are those which are high or DNC. As the domain of an intrinsic generic description is c.e. and can compute an intrinsically small set (its complement), its degree must be high or DNC, and therefore high.
\\
\\
Fortunately, the answer to this question resolves the remaining implications involving intrinsically densely computable sets:
\begin{lemma}
The intrinsically densely computable sets are exactly the intrinsically coarsely computable sets if every intrinsically generically computable set is intrinsically computable, and the intrinsically densely computable sets strictly contain all of the intrinsically generically computable sets and intrinsically coarsely computable sets if this is not the case.
\end{lemma}
\begin{proof}
By Lemma \ref{coarseseparation} there is a set $B$ which is intrinsically coarsely computable but not intrinsically generically computable. Let $A$ be a set which is intrinsically generically computable but not intrinsically coarsely computable. An application of Corollary \ref{join} tells us that $A\oplus B$ is intrinsically densely computable, but it is clear that it cannot be intrinsically coarsely computable or intrinsically generically computable because any intrinsic coarse/generic description of $A\oplus B$ would necessarily yield an intrinsic coarse/generic description of $A$/$B$.
\\
\\
Now suppose that every intrinsically generically computable set is intrinsically coarsely computable, and let $A$ be intrinsically densely computable with witness $\varphi_e$. Then the set $B$ defined via the characteristic function
\[\chi_B(n)=\begin{cases}
\varphi_e(n) & n\in W_e \\
0 & n\in \overline{W_e}
\end{cases}\]
is intrinsically generically computable with witness $\varphi_e$. Therefore it is intrinsically coarsely computable via some total witness $\varphi_i$. Therefore $\varphi_i$ witnesses that $A$ is intrinsically coarsely computable as well because the error set is contained within the union of two intrinsically small sets (the complement of $W_e$ and the error set of $\varphi_i$ on $B$) and thus is intrinsically small.
\end{proof}
The remaining three generalizations of asymptotic computation to the intrinsic setting use a separate idea: Rather than having an intrinsically small error set that ensures the existence of descriptions, we simply assert that descriptions must exist for any computable permutation. Varying the level of uniformity for these descriptions is how we reach three separate notions (Recall that $x\in\{\text{effective dense, generic, coarse, dense}\}$):
\begin{definition}
\mbox{}
\begin{itemize}
\item $A$ is weakly intrinsically $x$-ly computable if $\varphi_e(A)$ is $x$-ly computable for every $e\in Perm$.
\item $A$ is uniformly $x$-ly computable if there is a computable function $f(e,n)$ such that $\lambda n(f(e,n))$ is a(n) $x$ description of $\varphi_e(A)$ when $e\in Perm$.
\item $A\subseteq\omega$ is oracle $x$-ly computable if there is a Turing functional $\Phi_i$ such that $\Phi_i^X$ is a(n) $x$ description of $\varphi_e(A)$ whenever $e\in Perm$ and $X=\mathrm{graph}(\varphi_e)$.
\end{itemize}
\end{definition}
As in the case of the intrinsically $x$-computable sets, Astor's original definitions were ``uniformly intrinsically $x$-ly computable'' and ``oracle intrinsically $x$-ly computable,'' however we shorten these definitions for readability.
\\
\\
It is immediate that all of the straightforward implications from asymptotic computability apply here in each of the three cases, i.e. uniformly coarsely computable sets are uniformly densely computable and so on. Furthermore, it is easy to see that for all $x\in\{\text{effective dense, generic, coarse, dense}\}$, intrinsically $x$-ly computabile sets are uniformly and oracle $x$-ly computable, which both in turn are weakly $x$-ly computable. Furthermore, albeit slightly less trivial, is the fact that oracle $x$-ly computable sets are uniformly $x$-ly computable: Given a Turing functional $\Phi_i$ which witnesses that $A$ is oracle $x$-ly computable, define the partial computable function $f(e,n)$ via $f(e,n)=\Phi_i^{\mathrm{graph}(\varphi_e)}(n)$. Then the definition of oracle $x$-ly computable ensures that this function $f$ witnesses uniformly $x$-ly computable. This means that for a fixed $x$, the four notions form a chain.
\\
\\
As noted in \cite{intrinsicdensity}, it is unclear at first if these notions are all distinct (i.e. whether or not the chain collapses), even when restricting ourselves just to the generic case. Below we shall see that they are not distinct here, although the argument will not generalize to the coarse and dense settings. However, a slight modification of it shall provide a similar but not identical result for the effective dense setting.
\begin{theorem}
\label{genericcollapse}
Suppose that $A$ is oracle generically computable. Then $A$ is intrinsically generically computable.
\end{theorem}
\begin{proof}
Let $\Phi_i$ witness that $A$ is oracle generically computable. Then define the partial computable function $f$ as follows: Note that the set of finite binary strings $\sigma$ which are initial segments of graphs of injective functions is computable. For $\sigma$ in this set, let $f_\sigma$ denote the partial injective function with finite range such that $\mathrm{graph}(f_\sigma)$ is the infinite binary string obtained by adding infinitely many $0$'s to $\sigma$. Compute $f(n)$ by searching for such a $\sigma$ with $n\in \mathrm{range}(f_\sigma)$ and $\Phi_i^{\sigma}(f_\sigma(n))\downarrow$. If one is found, define $f(n)=\Phi_i^{\sigma}(f_\sigma(n))$ for the first such $\sigma$. Otherwise, $f(n)\uparrow$.
\\
\\
First, note that $f(n)\downarrow$ implies $f(n)=A(n)$: If $f(n)\downarrow$, then there is some $\sigma$ such that $\Phi_i^{\sigma}(f_\sigma(n))\downarrow$. As $\sigma$ is an initial segment of the graph of an injective function, $\sigma$ can be extended to $X$ where $X$ is the graph of some computable permutation $\varphi_e$. Then as $\Phi_i$ witnesses that $A$ is oracle generically computable, $\Phi_i^X$ is a generic description of $\varphi_e(A)$, so $\Phi_i^X(\varphi_e(n))\downarrow$ implies
\[\Phi_i^X(\varphi_e(n))=\varphi_e(A)(\varphi_e(n))=A(n).\]
In particular,
\[A(n)=\Phi_i^X(\varphi_e(n))=\Phi_i^\sigma(f_\sigma(n))=f(n)\]
by the finite use principle.
\\
\\
Therefore, it remains to show that the domain of $f$ has intrinsic density $1$. Notice that if $\varphi_e$ is a permutation, then $\varphi_e(dom(f))$ contains $dom(\Phi_i^{\mathrm{graph}(\varphi_e)})$, as if $\Phi_i^{\mathrm{graph}(\varphi_e)}(k)\downarrow$, there is an initial segment $\sigma$ of $\mathrm{graph}(\varphi_e)$ with $k\in \mathrm{range}(f_\sigma)$ that witnesses convergence, and therefore witnesses $f(\varphi_e^{-1}(k))\downarrow$. However, $\underline{\rho}(dom(\Phi_i^{\mathrm{graph}(\varphi_e)}))=1$ as $\Phi_i^{\mathrm{graph}(\varphi_e)}$ is a generic description of $\varphi_e(A)$ and therefore has density $1$. Thus $dom(f)$ has density $1$ under every computable permutation and thus has intrinsic density $1$ as desired.
\end{proof}
\begin{corollary}
\label{effectivedensecollapse}
Suppose that $A$ is oracle effective densely computable. Then $A$ is intrinsically generically computable.
\end{corollary}
\begin{proof}
Construct the description $f$ of $A$ as in the proof of Theorem \ref{genericcollapse}, however instead of searching for convergence, search for convergence to either $0$ or $1$.
\end{proof}
As mentioned above, this argument does not in general apply to oracle coarsely computable sets and oracle densely computable sets. The issue lies in the fact that coarse and dense computation allows for mistakes, so we cannot ensure that any convergent computation is correct.
\\
\\
The remaining implications remain open other than the previously observed chains. The difficulty in separating these notions lies in the fact that the constructed sets cannot be described by building one error set, but rather have a different error set for each computable permutation. More importantly, these countably many computable requirements are heavily interlocked: Consider attempting to construct a weakly intrinsically generically computable set which is not weakly intrinsically coarsely computable. As an example, we may try to define an error set for the identity permutation. However, this defines the membership of the constructed set on a given c.e. set $W_e$. If we wish to diagonalize for a given computable permutation $\pi$, we may find that $\pi(W_e)$ has density $1$, in which case we can't respect $W_e$ and also diagonalize on a set of positive density.
| 15,062
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\begin{document}
\vspace{0.2cm}
\begin{center}
{\Large\bf Transition Probabilities for Flavor Eigenstates of Non-Hermitian Hamiltonians in the PT-Broken Phase}
\end{center}
\vspace{0.2cm}
\begin{center}
{\bf Tommy Ohlsson}~$^{a,~b,~c}$~\footnote{E-mail: tohlsson@kth.se},
\quad
{\bf Shun Zhou}~$^{d,~e}$~\footnote{E-mail: zhoush@ihep.ac.cn}
\\
\vspace{0.2cm}
{\small $^a$Department of Physics, School of Engineering Sciences, KTH Royal Institute of Technology, \\
AlbaNova University Center, Roslagstullsbacken 21, SE-106 91 Stockholm, Sweden \\
$^b$The Oskar Klein Centre for Cosmoparticle Physics, AlbaNova University Center, \\
Roslagstullsbacken 21, SE-106 91 Stockholm, Sweden \\
$^c$University of Iceland, Science Institute, Dunhaga 3, IS-107 Reykjavik, Iceland\\
$^d$Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China\\
$^e$School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China}
\end{center}
\vspace{1.5cm}
\begin{abstract}
We investigate the transition probabilities for the ``flavor" eigenstates in the two-level quantum system, which is described by a non-Hermitian Hamiltonian with the parity and time-reversal (PT) symmetry. Particularly, we concentrate on the so-called PT-broken phase, where two eigenvalues of the non-Hermitian Hamiltonian turn out to be a complex conjugate pair. In this case, we find that the transition probabilities will be unbounded in the limit of infinite time $t \to +\infty$. However, after performing a connection between a non-Hermitian system, which exhibits passive PT-symmetry and global decay, and the neutral-meson system in particle physics, we observe that the diverging behavior of the transition probabilities is actually applicable to the gauge-transformed neutral-meson states, whereas the transition probabilities for physical states are exponentially suppressed by the global decay. We also present a brief review on the situation at the so-called exceptional point, where both the eigenvalues and eigenvectors of the Hamiltonian coalesce.
\end{abstract}
\newpage
\def\thefootnote{\arabic{footnote}}
\setcounter{footnote}{0}
\section{Introduction}
Non-Hermitian Hamiltonians with the joint parity and time-reversal (PT) symmetry have recently attracted a lot of attention~\cite{Bender:2007nj} and very interesting applications have been found for a number of physical systems in particle physics, nuclear physics, optics, electronics, and many others~\cite{review}. In the existing literature, the two-level system with PT-symmetric non-Hermitian Hamiltonians~\cite{Bender:2002yp,Bender:2002vv, Feng:2017, Joglekar:2018, Wang:2019, Roccati:2021} has been extensively investigated, as a simple but instructive example, to explore and clarify all related conceptual issues~\cite{Mostafazadeh:2008pw, Kleefeld:2009vd, Mannheim:2009zj}. However, it is worthwhile to mention that the transition amplitudes and probabilities between the ``flavor" eigenstates have rarely been studied, except for some general discussion in Refs.~\cite{Bagarello1, Bagarello2}.
Since the phenomenon of ``flavor" mixing is quite common in particle physics, such as flavor oscillations of massive neutrinos~\cite{Xing:2019vks} and the neutral-meson system $P^0$-$\overline{P}^0$~\cite{Tanabashi:2018oca} (e.g., $K^0$-$\overline{K}^0$, $D^0$-$\overline{D}^0$, and $B^0$-$\overline{B}^0$), it is intriguing to consider the transitions among ``flavor" eigenstates in the system with PT-symmetric non-Hermitian Hamiltonians~\cite{Ohlsson:2015xsa, Ohlsson:2019noy}. In our previous work~\cite{Ohlsson:2019noy}, we calculated the transition probabilities for the ``flavor" eigenstates in the scenario, where the PT symmetry is always preserved and two eigenvalues of the non-Hermitian Hamiltonian are real, which is known as the \emph{PT-symmetric phase}. In the present work, we aim to extend the previous study in the PT-symmetric phase to the \emph{PT-broken phase}. The primary motivation for such an extension is at least two-fold.
First, by ``flavor" eigenstates of a two-level quantum system with a non-Hermitian Hamiltonian ${\cal H}$, we mean the complete set of basis vectors $\{|u^{}_\beta\rangle\}$ (for $\beta = a, b$), in which the matrix representation of the Hamiltonian is given by ${\cal H}^{}_{\alpha \beta} \equiv \langle u^{}_\alpha|{\cal H}|u^{}_\beta\rangle$ and the left vectors $\langle u^{}_\alpha| \equiv |u^{}_\alpha\rangle^\dagger$ and $\langle u^{}_\beta| \equiv |u^{}_\beta\rangle^\dagger$ (for $\alpha, \beta = a, b$) have been defined as in conventional quantum mechanics. Therefore, the transition amplitudes in the following discussion will be referred to the projection of the time-evolved flavor eigenstates $\{|u^{}_\alpha(t)\rangle\}$ into their initial states $\{|u^{}_\beta\rangle\}$. However, whenever the transition amplitudes ${\cal A}^{}_{\alpha \beta} \equiv \langle u^{}_\beta|u^{}_\alpha(t)\rangle$ are calculated, we will clearly indicate the exact definition of the involved inner product as well as that of the left state vectors. As mentioned, for PT-symmetric non-Hermitian Hamiltonians, the transition amplitudes ${\cal A}^{}_{\alpha \beta}$ and the corresponding probabilities ${\cal P}^{}_{\alpha \beta} \equiv |{\cal A}^{}_{\alpha \beta}|^2$ have been explicitly computed and extensively studied in Ref.~\cite{Ohlsson:2019noy}. Hence, it is a natural continuation to extend the investigation to the PT-broken phase.
Second, in contrast to the PT-symmetric phase, where two eigenvalues of the non-Hermitian Hamiltonian are real, the PT-broken phase will be complicated by a complex-conjugate pair of eigenvalues. As is well known, if PT symmetry is maintained, it is always possible to find a similarity transformation that converts a non-Hermitian Hamiltonian into its Hermitian counterpart~\cite{Mostafazadeh:2008pw, Mannheim:2009zj, Ohlsson:2019noy}. However, this is impossible for the PT-broken phase, rendering it rather different. Hence, in this particular case, the transition probabilities deserve a dedicated study.
The remaining part of this work is organized as follows. In Sec.~\ref{sec: general}, we present the general formalism for the investigation of PT-symmetric non-Hermitian Hamiltonians and summarize the main features of the PT-symmetric phase, the PT-broken phase, and the \emph{exceptional point}, where the transition between these two phases occurs. Then, in Sec.~\ref{sec: broken}, the transition amplitudes and probabilities in the PT-broken phase will be introduced and studied, where the connection between the PT-broken phase and the neutral-meson system is also performed. Finally, in Sec.~\ref{sec: conclusion}, we summarize the main results and draw our conclusions.
\section{General Formalism}\label{sec: general}
For a general discussion about the properties of PT-symmetric non-Hermitian Hamiltonians and their applications, one should be referred to the excellent review by Bender~\cite{Bender:2007nj} and references therein. Particularly, in this work, we focus on the simple two-level system, for which the Hamiltonian is diagonalizable and space-time independent. The space-reflection operator ${\cal P}$ is defined as~\cite{Ohlsson:2019noy}
\begin{equation}
{\cal P} = \left(\begin{matrix} 0 & 1 \cr 1 & 0\end{matrix}\right) \; ,
\end{equation}
and the time-reversal operator ${\cal T}$ is taken to be just the complex conjugation ${\cal K}$, namely, ${\cal T} {\cal O} {\cal T}^{-1} = {\cal O}^*$ for any operators ${\cal O}$ in the Hilbert space. The most general form of the Hamiltonian for the two-level system is given by
\begin{equation}
{\cal H} = \left(\begin{matrix} a & b \cr c & d\end{matrix}\right) \; ,
\end{equation}
where $\{a, b, c, d\}$ are arbitrary complex constants. The PT symmetry of the Hamiltonian system requires that $[{\cal PT}, {\cal H}] = {\bf 0}$, so we have
\begin{eqnarray}
\left({\cal PT} {\cal H}\right) \Psi &=& \left(\begin{matrix} 0 & 1 \cr 1 & 0\end{matrix}\right) \left(\begin{matrix} a^* & b^* \cr c^* & d^*\end{matrix}\right) \Psi^* = \left(\begin{matrix} c^* & d^* \cr a^* & b^*\end{matrix}\right) \Psi^*\; , \label{eq:PTH}\\
\left({\cal H}{\cal PT}\right) \Psi &=& \left(\begin{matrix} a & b \cr c & d\end{matrix}\right) \left(\begin{matrix} 0 & 1 \cr 1 & 0\end{matrix}\right) \Psi^* = \left(\begin{matrix} b & a \cr d & c\end{matrix}\right) \Psi^* \; , \label{eq:HPT}
\end{eqnarray}
where $\Psi$ stands for any vectors in the Hilbert space that the operators are acting on. From Eqs. (\ref{eq:PTH}) and (\ref{eq:HPT}), one can recognize that the PT symmetry of the system implies that $a = d^*$ and $b = c^*$. Therefore, the most general non-Hermitian Hamiltonian ${\cal H}$ actually contains only four degrees of freedom (in terms of the number of real parameters), when it respects the PT symmetry. This is equal to the number of free parameters in the two-level system with the Hermitian Hamiltonian, where $a$ and $d$ are real while $b = c^*$.
For later convenience, we adopt the following parametrization of the most general PT-symmetric non-Hermitian Hamiltonian, viz.,
\begin{equation}
{\cal H} = \left(\begin{matrix} \rho e^{+{\rm i}\varphi} & \sigma e^{+{\rm i}\phi} \cr \sigma e^{-{\rm i}\phi} & \rho e^{-{\rm i}\varphi} \end{matrix}\right) \; , \label{eq:H}
\end{equation}
where all parameters $\{\rho, \varphi\}$ and $\{\sigma, \phi\}$ are real and time-independent. For a recent study on time-dependent parameters in ${\cal H}$, see Refs.~\cite{Grimaudo:2019zli,Ju:2019kso}. With the Hamiltonian in Eq.~(\ref{eq:H}), one can immediately figure out its two eigenvalues
\begin{equation}
\lambda^{}_\pm = \rho \cos \varphi \pm \sqrt{\sigma^2 - \rho^2 \sin^2\varphi} \; .
\label{eq:lambdaPTsymm}
\end{equation}
Under the condition that $\rho^2\sin^2\varphi < \sigma^2$ is satisfied, the two eigenvalues are real. If this condition is not satisfied, namely, $\rho^2 \sin^2\varphi \geq \sigma^2$, we obtain either (i) two complex eigenvalues (if $\rho^2 \sin^2\varphi > \sigma^2$ holds)
\begin{equation}
\lambda_\pm = \rho \cos \varphi \pm {\rm i} \sqrt{\rho^2 \sin^2\varphi - \sigma^2} \; ,
\label{eq:lambdaPTbroken}
\end{equation}
which are complex conjugates to each other, or (ii) a degenerate real eigenvalue $\lambda_\pm = \lambda_0 = \rho \cos \varphi$ with multiplicity 2, since $\rho^2 \sin^2\varphi = \sigma^2$ holds. In Fig.~\ref{fig:plotEigenvalues}, we present the eigenvalues as a function of $\sin \varphi$ for different choices of the ratio of the parameters $\rho$ and $\sigma$.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.6\textwidth]{plotEigenvalues.pdf}
\end{center}
\vspace{-0.6cm}
\caption{Illustration for the real (solid curves) and imaginary (dotted curves) parts of the normalized eigenvalues $\lambda_\pm/\sigma = \xi \sqrt{1- \sin^2 \varphi} \pm \sqrt{1- \xi^2 \sin^2\varphi}$ as functions of $\sin \varphi$ for three different choices of the ratio of the parameters $\rho$ and $\sigma$, i.e., $\xi \equiv \rho/\sigma = 2~\mbox{(red curves)}, 3~\mbox{(orange curves)}, 4~\mbox{(yellow curves)}$, in the \emph{PT-symmetric phase} (none or less shaded areas) and the \emph{PT-broken phase} (shaded areas). The corresponding \emph{exceptional points} are marked by black points (`$\bullet$') and the black thin dashed curve shows the trajectory of the exceptional points.}
\label{fig:plotEigenvalues}
\end{figure}
The eigenvalues are displayed in the PT-symmetric phase ($\rho^2 \sin^2\varphi < \sigma^2$: two real eigenvalues, see Subsec.~\ref{sub:PTsymmetric}) and the PT-broken phase ($\rho^2 \sin^2\varphi > \sigma^2$: two complex-conjugate eigenvalues, see Subsec.~\ref{sub:PTbroken}) as well as the exceptional points are indicated ($\rho^2 \sin^2\varphi = \sigma^2$: one degenerate real eigenvalue, see Subsec.~\ref{sub:EP}). Some helpful comments on the eigenvalues and their corresponding eigenvectors of the non-Hermitian Hamiltonian ${\cal H}$ in Eq.~(\ref{eq:H}) are in order.
\subsection{PT-Symmetric Phase}
\label{sub:PTsymmetric}
Although the PT-symmetric phase has been carefully studied in Ref.~\cite{Ohlsson:2019noy}, we briefly summarize its main features in this subsection in order to make the presentation self-consistent, to clarify our notation, and to easily perform a direct comparison with the PT-broken phase and the exceptional point in the next subsections. As mentioned before, the two eigenvalues in Eq.~(\ref{eq:lambdaPTsymm}) are real if the condition $\rho^2 \sin^2\varphi < \sigma^2$ is fulfilled. This is usually called the PT-symmetric phase of the system. In this case, we write the two eigenvectors corresponding to $E^{}_\pm = \lambda_\pm$ as
\begin{equation}
|u^{}_+\rangle = \frac{1}{\sqrt{2\sin \alpha}}\left(\begin{matrix} e^{+{\rm i}\pi/4} \cdot e^{-{\rm i}\alpha/2} \cr e^{-{\rm i}\pi/4} \cdot e^{+{\rm i}\alpha/2}\end{matrix}\right) \;, \quad |u^{}_-\rangle = \frac{\rm i}{\sqrt{2\sin\alpha}} \left(\begin{matrix} e^{+{\rm i}\pi/4} \cdot e^{+{\rm i}\alpha/2} \cr e^{-{\rm i}\pi/4} \cdot e^{-{\rm i}\alpha/2}\end{matrix}\right) \; , \label{eq:u+u-2}
\end{equation}
where $\cos \alpha \equiv (\rho \sin\varphi)/\sigma$ has been defined and $\phi = 0$ is assumed.\footnote{Note that the definition of the parameter $\alpha$ differs from that in Ref.~\cite{Ohlsson:2019noy}, where $\sin\alpha \equiv (\rho\sin\varphi)/\sigma$. The reason for such a change is to make a coherent presentation in both the PT-symmetric and PT-broken phases.} One can easily verify that ${\cal PT}|u^{}_\pm\rangle = \pm |u^{}_\pm\rangle$, indicating that $|u^{}_\pm\rangle$ are also the eigenvectors of the ${\cal PT}$ operator.
On the other hand, we can apply the bi-orthogonal formalism~\cite{Sachs1963} to the non-Hermitian Hamiltonian in Eq.~(\ref{eq:H}) but with $\phi = 0$ assumed for illustration, and then introduce the so-called metric operator $\eta$~\cite{Mostafazadeh:2008pw}, i.e.,
\begin{equation}
\eta \equiv \sum_{s = \pm} |v^{}_s\rangle \langle v^{}_s| = {\cal P} \left(|u^{}_+\rangle \langle u^{}_+| + |u^{}_- \rangle \langle u^{}_-|\right) {\cal P} = \left(\begin{matrix} \csc\alpha & -{\rm i}\cot \alpha \cr +{\rm i}\cot\alpha & \csc\alpha \end{matrix}\right) \; ,
\end{equation}
where $\langle v^{}_\pm| \equiv |v^{}_\pm\rangle^\dagger$ and $\langle u^{}_\pm| \equiv |u^{}_\pm\rangle^\dagger$. By construction, the relation $\eta {\cal H} \eta^{-1} = {\cal H}^\dagger$ holds, so one can easily prove that there exists a charge-conjugation operator ${\cal C}$ defined as \cite{Bender:2004zz}
\begin{equation}
{\cal C} \equiv {\cal P}^{-1} \eta = \eta^{-1} {\cal P} = \left(\begin{matrix} +{\rm i}\cot\alpha & \csc\alpha \cr \csc\alpha & -{\rm i}\cot\alpha \end{matrix}\right) \; , \label{eq:C}
\end{equation}
satisfying the commutation relation $[{\cal C}, {\cal H}] = {\bf 0}$. This non-Hermitian Hamiltonian system respects both the ${\cal C}$ and ${\cal PT}$ symmetries, and thus, the ${\cal CPT}$ symmetry. Since the ${\cal PT}$-inner product is actually not positive-definite (due to ${\rm det}~{\cal P} = -1 < 0$), it is necessary to introduce the $\eta$- and ${\cal CPT}$-inner products for the definitions of the transition amplitudes and probabilities \cite{Mannheim:2017apd}. More explicitly,
\begin{itemize}
\item {\it The $\eta$-inner product} for any two state vectors $|\psi\rangle$ and $|\chi\rangle$ reads
\begin{equation}
\langle \psi|\chi \rangle^{}_\eta \equiv \langle \psi| \eta |\chi \rangle = |\psi\rangle^\dagger \cdot \eta \cdot |\chi \rangle \; .
\end{equation}
\item {\it The ${\cal CPT}$-inner product} for any two state vectors $|\psi\rangle$ and $|\chi\rangle$ reads
\begin{equation}
\langle \psi|\chi \rangle^{}_{\cal CPT} \equiv \left({\cal CPT}|\psi\rangle\right)^{\rm T} \cdot |\chi \rangle = |\psi\rangle^\dagger \cdot {\cal PC} \cdot |\chi \rangle = \langle \psi|\chi \rangle^{}_\eta \; , \label{eq:CPT}
\end{equation}
where ${\cal PC} = \eta$ from the definition of the ${\cal C}$ operator in Eq.~(\ref{eq:C}) has been used in the last step in Eq.~(\ref{eq:CPT}).
\end{itemize}
Therefore, the $\eta$- and ${\cal CPT}$-inner products are equivalent and we can use either of them to calculate the transition amplitudes and probabilities between two quantum states. These calculations have been performed in Ref.~\cite{Ohlsson:2019noy}.
\subsection{PT-Broken Phase}
\label{sub:PTbroken}
Under the condition $\sigma^2 < \rho^2\sin^2\varphi$, one can check that $[{\cal PT}, {\cal H}] = {\bf 0}$ remains to be valid for the most general form of the Hamiltonian ${\cal H}$ in Eq.~(\ref{eq:H}). This should be the case as we have derived the general form of the PT-symmetric Hamiltonian for arbitrary values of the parameters. However, as we have shown in Eq.~(\ref{eq:lambdaPTbroken}), the Hamiltonian~(\ref{eq:H}) has two complex eigenvalues $E^\prime_\pm = \lambda_\pm$, which in this PT-broken phase are labeled by primes in order to avoid any confusion with the ones in the PT-symmetric phase.
Following the same procedure as in the PT-symmetric phase to calculate the eigenvectors $|u^\prime_\pm\rangle \equiv (a^\prime_\pm, b^\prime_\pm)^{\rm T}$ corresponding to the eigenvalues $E^\prime_\pm$, we obtain
\begin{equation}
|u^\prime_+\rangle = N^\prime_+ \left(\begin{matrix} e^{+{\rm i}\pi/4} \cdot e^{+(\alpha^\prime + {\rm i}\phi)/2} \cr e^{-{\rm i}\pi/4} \cdot e^{-(\alpha^\prime + {\rm i}\phi)/2} \end{matrix}\right) \; , \quad |u^\prime_-\rangle = N^\prime_- \left(\begin{matrix} e^{+{\rm i}\pi/4} \cdot e^{-(\alpha^\prime - {\rm i}\phi)/2} \cr e^{-{\rm i}\pi/4} \cdot e^{+(\alpha^\prime - {\rm i}\phi)/2} \end{matrix}\right) \label{eq:eigenvectors_bp}
\end{equation}
where $\cosh\alpha^\prime \equiv (\rho \sin\varphi)/\sigma$ has been defined. Performing a comparison between Eq.~(\ref{eq:u+u-2}) in the PT-symmetric phase and Eq.~(\ref{eq:eigenvectors_bp}) in the PT-broken phase, one observes the connection between these two cases by simply identifying $\alpha^\prime = -{\rm i}\alpha$ up to the normalization constants and the phase $\phi$. Then, we try to fix the two normalization constants $N^\prime_\pm$ by examining the ${\cal PT}$-inner products of these two eigenvectors, namely,
\begin{eqnarray}
\langle u^\prime_+| u^\prime_+\rangle^{}_{\cal PT} &=& - 2|N^\prime_+|^2 \sin\phi \; , \quad \langle u^\prime_-| u^\prime_-\rangle^{}_{\cal PT} = - 2|N^\prime_-|^2 \sin\phi \; , \label{eq:u+u+u-u-}\\
\langle u^\prime_+| u^\prime_-\rangle^{}_{\cal PT} &=& -2 {\rm i} N^{\prime *}_+ N^\prime_- \sinh (\alpha^\prime - {\rm i}\phi) \; , \quad \langle u^\prime_-| u^\prime_+\rangle^{}_{\cal PT} = +2 {\rm i} N^{\prime *}_- N^\prime_+ \sinh (\alpha^\prime + {\rm i}\phi) \; . \label{eq:u+u-u-u+}
\end{eqnarray}
At first sight, it seems that one can choose proper values of $N^\prime_\pm$ to guarantee the orthogonality conditions $\langle u^\prime_\pm | u^\prime_\mp \rangle^{}_{\cal PT} = 0$. However, as one can observe from Eq.~(\ref{eq:u+u-u-u+}), this is only possible if both $\alpha^\prime = 0$ and $\sin\phi = 0$ hold, or equivalently, at the so-called exceptional point with $\rho^2 \sin^2\varphi = \sigma^2$.
Therefore, for $\rho^2 \sin^2\varphi > \sigma^2$ under consideration, we have to determine the normalization constants $N^\prime_\pm$ by requiring $\langle u^\prime_\pm | u^\prime_\pm \rangle^{}_{\cal PT} = 0$ and $\langle u^\prime_\pm| u^\prime_\mp\rangle^{}_{\cal PT} = + 1$. These requirements differ significantly from those in the PT-symmetric phase. From Eq.~(\ref{eq:u+u+u-u-}) with $\langle u^\prime_\pm | u^\prime_\pm \rangle^{}_{\cal PT} = 0$, we immediately get $\phi = 0$ (or $\phi = \pi$), which is also consistent with our previous convention in the PT-symmetric phase. In addition, from Eq.~(\ref{eq:u+u-u-u+}) with $\phi = 0$ and $\langle u^\prime_\pm| u^\prime_\mp\rangle^{}_{\cal PT} = + 1$, we obtain $N^\prime_+ = e^{-{\rm i}\pi/4}/\sqrt{2\sinh \alpha^\prime}$ and $N^\prime_- = e^{+{\rm i}\pi/4}/\sqrt{2\sinh \alpha^\prime}$, and thus, using Eq.~(\ref{eq:eigenvectors_bp}), we find the two eigenvectors as
\begin{eqnarray}
|u^\prime_+\rangle &=& \frac{e^{-{\rm i}\pi/4}}{\sqrt{2\sinh \alpha^\prime}} \left(\begin{matrix} e^{+{\rm i}\pi/4} \cdot e^{+\alpha^\prime/2} \cr e^{-{\rm i}\pi/4} \cdot e^{-\alpha^\prime/2} \end{matrix}\right) = \frac{1}{\sqrt{2\sinh \alpha^\prime}} \left(\begin{matrix} e^{+\alpha^\prime/2} \cr -{\rm i} e^{-\alpha^\prime/2} \end{matrix}\right) \; , \label{eq:up+}\\
|u^\prime_-\rangle &=& \frac{e^{+{\rm i}\pi/4}}{\sqrt{2\sinh \alpha^\prime}} \left(\begin{matrix} e^{+{\rm i}\pi/4} \cdot e^{-\alpha^\prime/2} \cr e^{-{\rm i}\pi/4} \cdot e^{+\alpha^\prime/2} \end{matrix}\right) = \frac{1}{\sqrt{2\sinh \alpha^\prime}} \left(\begin{matrix} +{\rm i} e^{-\alpha^\prime/2} \cr e^{+\alpha^\prime/2} \end{matrix}\right)\; . \label{eq:up-}
\end{eqnarray}
It is helpful to make some comments on the further connection between the PT-symmetric and PT-broken phases. In the latter case, we have two eigenvalues $E^\prime_\pm = \rho \cos\varphi \pm {\rm i} \sigma \sinh \alpha^\prime$, in which the replacement of $\alpha^\prime = -{\rm i}\alpha$ leads to the two eigenvalues $E^{}_\pm$ in the former case. At the same time, if we replace $\alpha^\prime$ by $-{\rm i}\alpha$ everywhere in Eqs.~(\ref{eq:up+}) and (\ref{eq:up-}), the eigenvectors $|u^\prime_\pm \rangle$ will reduce to $|u^{}_\pm\rangle$ in Eq.~(\ref{eq:u+u-2}).
Nevertheless, given the eigenvectors $|u^\prime_\pm\rangle$ in Eqs.~(\ref{eq:up+}) and (\ref{eq:up-}), one can check that ${\cal PT}|u^\prime_\pm\rangle = |u^\prime_\mp\rangle$ and ${\cal H}|u^\prime_\pm\rangle = E^\prime_\pm |u^\prime_\pm\rangle$, indicating that the energy eigenstates $|u^\prime_\pm\rangle$ are \emph{not} eigenstates of the ${\cal PT}$ operator. This is the reason why this scenario is called the PT-broken phase. However, this is not in contradiction with the fact that $[{\cal PT}, {\cal H}] = {\bf 0}$. Since ${\cal PT}$ is an anti-linear operator and $E^\prime_+ = E^{\prime *}_-$, one should note that ${\cal PT}E^\prime_\pm |u^\prime_\pm \rangle = E^\prime_\mp {\cal PT}|u^\prime_\pm \rangle$. More explicitly, we have
\begin{equation}
{\cal PT} {\cal H} |u^\prime_\pm\rangle = {\cal PT} E^\prime_\pm |u^\prime_\pm \rangle = E^\prime_\mp |u^\prime_\mp \rangle \; , \quad
{\cal H} {\cal PT} |u^\prime_\pm\rangle = {\cal H} |u^\prime_\mp \rangle = E^\prime_\mp |u^\prime_\mp \rangle \; ,
\end{equation}
implying $[{\cal PT}, {\cal H}] = {\bf 0}$. This is quite different from the PT-symmetric phase, in which the two eigenvalues $E^{}_\pm$ are real.
Now, we apply the bi-orthogonal formalism to the non-Hermitian Hamiltonian system in the PT-broken phase. As before~\cite{Ohlsson:2019noy}, we have to find out the eigenvectors of ${\cal H}^\dagger$, namely,
\begin{equation}
{\cal H}^\dagger |v^\prime_\pm\rangle = E^\prime_\pm |v^\prime_\pm \rangle \; . \label{eq:Hdagger_bp}
\end{equation}
The identity ${\cal P}{\cal H}^\dagger {\cal P}^{-1} = {\cal H}$ is still applicable, so we multiply Eq.~(\ref{eq:Hdagger_bp}) on both sides from the left by the ${\cal P}$ operator and then obtain
\begin{equation}
\left({\cal P} {\cal H}^\dagger {\cal P}^{-1} \right) {\cal P} |v^\prime_\pm \rangle = E^\prime_\pm {\cal P}|v^\prime_\pm \rangle \; ,
\label{eq:leftright}
\end{equation}
indicating ${\cal P}|v^\prime_\pm\rangle \propto |u^\prime_\pm \rangle$. Identifying $|v^\prime_\pm \rangle = {\cal P}|u^\prime_\pm\rangle$, we can immediately compute the metric operator $\eta^\prime$, i.e.,
\begin{equation}
\eta^\prime \equiv \sum_{s = \pm} |v^\prime_{+s}\rangle \langle v^\prime_{-s}| = {\cal P} \left(|u^\prime_+ \rangle \langle u^\prime_-| + |u^\prime_-\rangle \langle u^\prime_+|\right) {\cal P} = \left(\begin{matrix} 0 & 1 \cr 1 & 0 \end{matrix}\right)
\end{equation}
and its inverse
\begin{equation}
\eta^{\prime -1} = \sum_{s = \pm} |u^\prime_{+s}\rangle \langle u^\prime_{-s}| = \left(\begin{matrix} 0 & 1 \cr 1 & 0 \end{matrix}\right) \; .
\end{equation}
Note that $\eta^\prime = \eta^{\prime -1} = {\cal P}$ with ${\rm det}~\eta^\prime = -1 < 0$, so it is not positive-definite. In this case, it is impossible to find a Hermitian matrix to convert the non-Hermitian Hamiltonian into a Hermitian one via a similarity transformation. Furthermore, the ${\cal C}$ operator is given by ${\cal C} = {\cal P}^{-1}\eta^\prime = \eta^{\prime -1} {\cal P} = \mathbb{1}_2$, which turns out to be the trivial $2 \times 2$ identity matrix.
Similar to the PT-symmetric phase, we can define the $\eta^\prime$-inner product as well as the ${\cal CPT}$-inner product as follows~\footnote{Strictly speaking, since the operators $\eta^\prime$ and ${\cal CPT}$ (or equivalently ${\cal PT}$) are not positive-definite in the PT-broken phase, it is inappropriate to define the inner product and the norm of state vectors by using these operators. However, since the notion of the ${\cal PT}$-inner product has been widely used in the literature, we follow this definition and implement it to calculate the transition amplitudes and probabilities in Sec.~\ref{sec: broken}. From Eqs.~(\ref{eq:Hdagger_bp}) and (\ref{eq:leftright}), one can observe that the left- and right-eigenvectors are related via $|v^\prime_\pm\rangle = {\cal P}|u^\prime_\pm\rangle$, which means that the usage of the ${\cal PT}$-inner product is equivalent to that of left- and right-eigenvectors in our calculations.}
\begin{itemize}
\item {\it The $\eta^\prime$-inner product} for any two state vectors $|\psi\rangle$ and $|\chi\rangle$ reads
\begin{equation}
\langle \psi|\chi \rangle^{}_{\eta^\prime} \equiv \langle \psi| \eta^\prime |\chi \rangle = |\psi\rangle^\dagger \cdot \eta^\prime \cdot |\chi \rangle \; .
\end{equation}
\item {\it The ${\cal CPT}$-inner product} for any two state vectors $|\psi\rangle$ and $|\chi\rangle$ reads
\begin{equation}
\langle \psi|\chi \rangle^{}_{\cal CPT} \equiv \left({\cal CPT}|\psi\rangle\right)^{\rm T} \cdot |\chi \rangle = |\psi\rangle^\dagger \cdot {\cal PC} \cdot |\chi \rangle = \langle \psi|\chi \rangle^{}_{\eta^\prime} \; ,
\end{equation}
where ${\cal PC} = \eta^\prime = {\cal P}$ has been used in the last step.
\end{itemize}
Therefore, the two inner products are equivalent and we will not distinguish between them. In addition, since the ${\cal C}$ operator is trivial, these two inner products are also identical with the ${\cal PT}$-inner product. However, as we have mentioned, the metric operator $\eta^\prime = {\cal P}$ is no longer positive-definite, and thus, the norm $\langle \psi|\psi\rangle^{}_{\eta^\prime}$ cannot be guaranteed to be positive. In fact, for the energy eigenstates $|u^\prime_+\rangle$ and $|u^\prime_-\rangle$, we have $\langle u^\prime_\pm | u^\prime_\pm \rangle^{}_{\cal PT} = \langle u^\prime_\pm|u^\prime_\pm\rangle^{}_{\eta^\prime} = 0$ and $\langle u^\prime_\pm| u^\prime_\mp\rangle^{}_{\cal PT} = \langle u^\prime_\pm| u^\prime_\mp \rangle^{}_{\eta^\prime} = + 1$. The identity $\eta^\prime {\cal H} \eta^{\prime -1} = {\cal H}^\dagger$, which now coincides with ${\cal P} {\cal H} {\cal P}^{-1} = {\cal H}^\dagger$, indeed leads to a unitary time evolution of the energy eigenstates.
\subsection{Exceptional Point}
\label{sub:EP}
In general, let us investigate the exceptional point (EP) of a non-Hermitian Hamiltonian of an open quantum system on the following form (see, e.g., Ref.~\cite{Ozdemir:2019})
\begin{equation}
{\cal H}_{\rm NH} = \left(\begin{matrix} \omega_1 - {\rm i} \gamma_1 & \kappa \cr \kappa & \omega_2 - {\rm i} \gamma_2 \end{matrix}\right) \; , \label{eq:H_NH}
\end{equation}
where $\omega_1$ and $\omega_2$ are the energies of the states of the system, $\gamma_1$ and $\gamma_2$ are real loss ($\gamma_i > 0$) or gain ($\gamma_i < 0$) parameters (i.e., the widths), and $\kappa$ is the (real) coupling strength between the states of the system, which has the two complex eigenvalues
\begin{equation}
\lambda_\pm = \frac{1}{2} \left[ \omega_1 + \omega_2 - {\rm i} \left( \gamma_1 + \gamma_2 \right) \right] \pm \sqrt{Q}
\end{equation}
with $Q$ being the quadratic expression
\begin{equation}
Q = \kappa^2 + \frac{1}{4} \left[ \left( \omega_1 - \omega_2 \right)^2 - \left( \gamma_1 - \gamma_2 \right)^2 \right] - \frac{{\rm i}}{2} \left( \omega_1 - \omega_2 \right) \left( \gamma_1 - \gamma_2 \right) \; .
\end{equation}
In order for the non-Hermitian Hamiltonian in Eq.~(\ref{eq:H_NH}) to have EPs, the condition $Q = 0$ must be fulfilled and this condition will determine the EPs that give rise to one degenerate eigenvalue.
Now, there are several specific cases that are interesting to study of the Hamiltonian in Eq.~(\ref{eq:H_NH}), which are (i) the case $\omega_1 = \omega_2 = \omega_0$, (ii) a non-Hermitian Hamiltonian consisting of a so-called {\it passive} PT-symmetric Hamiltonian and a Hamiltonian for a {\it lossy} uncoupled system \cite{Joglekar:2018,Ozdemir:2019} with $\omega_1 = \omega_2 = \omega$, $\gamma_1 = \gamma + \chi$, and $\gamma_2 = - \gamma + \chi$, where $\chi$ is a global decay parameter, (iii) a PT-symmetric Hamiltonian with $\omega_1 = \omega_2 = \omega$, $\gamma_1 = - \gamma_2 = \gamma$, and (iv) the case $\omega_1 = \omega_2 = \omega'$, $\gamma_1 = \gamma'$, and $\gamma_2 = 0$. In the following, we describe the four different cases in detail. First, in case~(i), the two complex eigenvalues and quadratic expression reduce to
\begin{equation}
\lambda_\pm = \omega_0 - \frac{{\rm i}}{2} \left( \gamma_1 + \gamma_2 \right) \pm \sqrt{Q}, \qquad Q = \kappa^2 - \frac{1}{4} \left( \gamma_1 - \gamma_2 \right)^2 \; ,
\end{equation}
and therefore, the condition to have EPs is given by $\kappa^2 = (\gamma_1 - \gamma_2)^2/4$. In Ref.~\cite{Roccati:2021}, quantum corrections to non-Hermitian systems at the EPs have been studied. Second, for the non-Hermitian Hamiltonian including the passive PT-symmetric Hamiltonian [i.e., case~(ii)], we obtain
\begin{equation}
\lambda_\pm = \omega - {\rm i} \chi \pm \sqrt{Q}, \qquad Q = \kappa^2 - \gamma^2,
\end{equation}
so the condition for EPs is simply $\kappa^2 = \gamma^2$, which means that variation of a single parameter is sufficient to reach the EPs. In Ref.~\cite{Joglekar:2018}, it has been shown that passive PT transitions in dissipative photonic systems are however not dependent on the existence of EPs. Note that the condition for the EPs is independent of $\chi$ (i.e., the global decay or damping of the system). At the EPs, the Hamiltonian in Eq.~(\ref{eq:H_NH}) will only have damping (described by $\chi$) and be independent of both $\gamma$ and $\kappa$. Thus, at the EPs, the Hamiltonian will effectively be
\begin{equation}
{\cal H}_{\rm NH}^{\rm eff.} = \left(\begin{matrix} \omega - {\rm i} \chi & 0 \cr 0 & \omega - {\rm i} \chi \end{matrix}\right) \; .
\end{equation}
Case~(ii) has been reviewed in Ref.~\cite{Ozdemir:2019} for PT-symmetric systems and their EPs in photonics (see also the review in Ref.~\cite{Feng:2017}), and will also be investigated in this work in connection to the neutral-meson system.\footnote{Note that the dynamics of a loss-loss system is basically equivalent to that of the combination of a loss-gain system and a system with global exponential decay, see, e.g., Ref.~\cite{Ozdemir:2019}.} Third, for the PT-symmetric Hamiltonian [i.e., case~(iii)], the parameter $\chi = 0$ in case~(ii), so we find that
\begin{equation}
\lambda_\pm = \omega \pm \sqrt{Q}, \qquad Q = \kappa^2 - \gamma^2,
\end{equation}
which means that the condition for EPs is again $\kappa^2 = \gamma^2$. In Ref.~\cite{Eleuch:2014}, the EPs in open and PT-symmetric systems have been studied. For example, such a two-state quantum system could exhibit both loss and gain described by the parameters $\gamma_1 = \gamma$ and $\gamma_2 = - \gamma$, respectively, and coupling of the two states via the lattice with the environment. Furthermore, in Ref.~\cite{Wang:2019}, the application of the PT dimer in photonics and phononics, which exhibits EPs, has been discussed. Fourth, in case~(iv), we have
\begin{equation}
\lambda_\pm = \omega' - \frac{{\rm i}}{2} \gamma' \pm \sqrt{Q}, \qquad Q = \kappa^2 - \frac{1}{4} {\gamma'}^2,
\end{equation}
and the EPs are given by the condition $\kappa^2 = {\gamma'}^2/4$. This case is similar to case~(iii), but with $\gamma_2 = 0$, i.e., vanishing gain, which has also been studied in Ref.~\cite{Eleuch:2014}. In addition, the dynamics and EPs of this type of Hamiltonian has been discussed in Ref.~\cite{Wang:2020}.
Note that although case~(iv) is similar to case~(iii), it exhibits a rather different behavior for the degenerate eigenvalue at the EPs, which is more similar to case~(ii) with the important difference that the imaginary part of the eigenvalue is related to the condition for the EPs.
Finally, for the most general PT-symmetric non-Hermitian Hamiltonian in Eq.~(\ref{eq:H}) with $\phi = 0$, which is the same as case~(ii) with $\chi = 0$ or case~(iii), let us give a brief discussion about the EP at $\rho^2\sin^2\varphi = \sigma^2$. The EP can be identified as either the limiting case of $\alpha \to 0$ in the PT-symmetric phase or that of $\alpha^\prime \to 0$ in the PT-broken phase. In either limit, the energy eigenvalues become degenerate $E^{}_\pm$ (or $E^\prime_\pm$) $\to E^{}_0 = \rho \cos\varphi$. Moreover, for the eigenvectors $|u^{}_\pm\rangle$ in Eq.~(\ref{eq:u+u-2}) and $|u^\prime_\pm\rangle$ in Eqs.~(\ref{eq:up+}) and (\ref{eq:up-}), the normalization constants $N^{}_\pm \propto 1/\sqrt{2\sin\alpha}$ and $N^\prime_\pm \propto 1/\sqrt{2\sinh\alpha^\prime}$ are divergent in the respective limits of $\alpha \to 0$ and $\alpha^\prime \to 0$. However, this is an artificial divergence, since $N^{}_\pm$ (or $N^\prime_\pm$) in the limit of $\alpha \to 0$ (or $\alpha^\prime \to 0$) cannot be determined from the ${\cal PT}$-inner products of the relevant eigenvectors. The proper normalization can be taken as $\langle u^{}_0|u^{}_0\rangle = 1$, with $\langle u^{}_0| \equiv |u^{}_0\rangle^\dagger$, so we have
\begin{equation}
|u^{}_\pm\rangle ~ ({\rm or}~|u^\prime_\pm\rangle) \to |u^{}_0\rangle = \frac{1}{\sqrt{2}} \left(\begin{matrix} e^{+{\rm i}\pi/4} \cr e^{-{\rm i}\pi/4} \end{matrix}\right) \; ,
\end{equation}
corresponding to the degenerate eigenvalue $E^{}_0$ at the EP. The rich physics at the EPs and their practical applications have been briefly summarized in Refs.~\cite{Ozdemir:2019, Berry, Heiss:2012dx, Wiersig1, Wiersig2, Sensors}.
Since the time evolution of $|u^{}_0\rangle$ is governed by the Schr\"{o}dinger equation, we have $|u^{}_0(t)\rangle = e^{-{\rm i}E^{}_0 t}|u^{}_0\rangle$, implying that only an overall phase factor will develop and no transitions between any two quantum states are expected. This is also true for the flavor eigenstates $|u^{}_a\rangle = (1, 0)^{\rm T}$ and $|u^{}_b\rangle = (0, 1)^{\rm T}$, which are linear superpositions of the energy eigenstates.
\section{Transitions in the PT-Broken Phase}\label{sec: broken}
\subsection{PT-Inner Product}
\label{sub: PTip}
Since the transition amplitudes and probabilities between two flavor eigenstates in the PT-symmetric phase have been examined in detail in Ref.~\cite{Ohlsson:2019noy}, we now consider the transitions between two flavor eigenstates in the PT-broken phase in this section. In this scenario, the Schr\"{o}dinger equation for the time evolution of the energy eigenstates is
\begin{equation}
{\rm i}\frac{{\rm d}}{{\rm d}t} |u^\prime_\pm(t)\rangle = {\cal H} |u^\prime_\pm(t)\rangle = E^\prime_\pm |u^\prime_\pm\rangle \; ,
\end{equation}
and thus, we have
\begin{eqnarray}
|u^\prime_+(t)\rangle &=& e^{-{\rm i}E^\prime_+ t} |u^\prime_+(0)\rangle = \frac{e^{-{\rm i}\omega t + \gamma t}}{\sqrt{2\sinh \alpha^\prime}} \left(\begin{matrix} e^{+\alpha^\prime/2} \cr -{\rm i} e^{-\alpha^\prime/2} \end{matrix}\right) \; , \\
|u^\prime_-(t)\rangle &=& e^{-{\rm i}E^\prime_- t} |u^\prime_-(0)\rangle = \frac{e^{-{\rm i}\omega t - \gamma t}}{\sqrt{2\sinh \alpha^\prime}} \left(\begin{matrix} +{\rm i} e^{-\alpha^\prime/2} \cr e^{+\alpha^\prime/2} \end{matrix}\right) \; ,
\end{eqnarray}
where the auxiliary parameters $\omega \equiv \rho \cos\varphi$ and $\gamma \equiv \sqrt{\rho^2\sin^2\varphi - \sigma^2}$ have been defined. For reference, we list below the correspondences between the three new parameters $\{\omega, \gamma, \alpha^\prime\}$ and the three original ones $\{\rho, \sigma, \varphi\}$, where $\phi = 0$ has been assumed as in Sec.~\ref{sec: general},
\begin{equation}
\omega = \rho \cos\varphi \; , \quad \gamma = \sqrt{\rho^2\sin^2\varphi - \sigma^2} \; , \quad \alpha^\prime = {\rm arcosh} \left(\frac{\rho\sin\varphi}{\sigma}\right)
\end{equation}
or
\begin{equation}
\rho = \sqrt{\omega^2 + \gamma^2 \coth^2\alpha^\prime} \; , \quad \sigma = \frac{\gamma}{\sinh \alpha^\prime} \; , \quad \varphi = \arccos \left(\frac{\omega}{\sqrt{\omega^2 + \gamma^2 \coth^2 \alpha^\prime}}\right) \; . \label{eq:rsf}
\end{equation}
In the following, we adopt the new set of parameters $\{\omega, \gamma, \alpha^\prime\}$, which can be converted back to the original one by using Eq.~(\ref{eq:rsf}).
To demonstrate the unitary time evolution, we calculate the norms of the time-evolved energy eigenstates to find
\begin{eqnarray}
\langle u^\prime_+(t) | u^\prime_+(t)\rangle^{}_{\cal PT} = |u^\prime_+(t)\rangle^\dagger \cdot {\cal P} \cdot |u^\prime_+(t)\rangle = 0 \; , \\
\langle u^\prime_-(t) | u^\prime_-(t)\rangle^{}_{\cal PT} = |u^\prime_-(t)\rangle^\dagger \cdot {\cal P} \cdot |u^\prime_-(t)\rangle = 0 \; .
\end{eqnarray}
Similarly, one can also verify that $\langle u^\prime_\pm(t)|u^\prime_\mp(t)\rangle^{}_{\cal PT} = +1$, which is time-independent as it should be.
Next, we introduce the flavor eigenstates in which basis the explicit form of the non-Hermitian Hamiltonian is specified. Recall the diagonalization of the Hamiltonian, i.e.,
\begin{equation}
A^\prime {\cal H} A^{\prime -1} = \widehat{\cal H} \equiv \left(\begin{matrix} E^\prime_+ & 0 \cr 0 & E^\prime_- \end{matrix}\right) \quad \Longrightarrow \quad ({\cal H} |w^{}_+\rangle, {\cal H} |w^{}_-\rangle) = (|w^{}_+\rangle E^\prime_+, |w^{}_-\rangle E^\prime_-) \; ,
\end{equation}
where we have written $A^{\prime -1} = (|w^{}_+\rangle, |w^{}_-\rangle)$ with $|w^{}_\pm\rangle$ being two column vectors. Obviously, we can identify $|w^{}_\pm\rangle$ with $|u^\prime_\pm\rangle$ in Eqs.~(\ref{eq:up+}) and (\ref{eq:up-}), since ${\cal H}|u^\prime_\pm\rangle = E^\prime_\pm |u^\prime_\pm\rangle$. Hence, it is easy to derive
\begin{equation}
A^{\prime -1} = \frac{1}{\sqrt{2\sinh \alpha^\prime}} \left(\begin{matrix} e^{+\alpha^\prime/2} & +{\rm i} e^{-\alpha^\prime/2} \cr -{\rm i} e^{-\alpha^\prime/2} & e^{+\alpha^\prime/2} \end{matrix}\right) \; , \quad A^\prime = \frac{1}{\sqrt{2\sinh \alpha^\prime}} \left(\begin{matrix} e^{+\alpha^\prime/2} & -{\rm i} e^{-\alpha^\prime/2} \cr +{\rm i} e^{-\alpha^\prime/2} & e^{+\alpha^\prime/2} \end{matrix}\right) \; ,
\end{equation}
where one can note that $A^{\prime -1} = A^{\prime \rm T}$ and $A^{\prime \dagger} = A^\prime$. Furthermore, it is straightforward to verify that the flavor eigenstates are given by
\begin{eqnarray}
|u^\prime_a\rangle &=& \left(A^{\prime -1}\right)^{}_{a+} |u^\prime_+\rangle + \left(A^{\prime -1}\right)^{}_{a-} |u^\prime_-\rangle = \left(\begin{matrix} 1 \cr 0 \end{matrix}\right) \; , \label{eq:upa}\\
|u^\prime_b\rangle &=& \left(A^{\prime -1}\right)^{}_{b+} |u^\prime_+\rangle + \left(A^{\prime -1}\right)^{}_{b-} |u^\prime_-\rangle = \left(\begin{matrix} 0 \cr 1 \end{matrix}\right) \; , \label{eq:upb}
\end{eqnarray}
which resemble the forms in the PT-symmetric phase. This should be the case as the explicit form of the Hamiltonian in Eq.~(\ref{eq:H}) remains the same in the PT-broken phase. In Eqs.~(\ref{eq:upa}) and (\ref{eq:upb}), $\left(A^{\prime -1}\right)^{}_{\beta s}$ for $\beta = a, b$ and $s = +, -$ denote the matrix elements of $A^{\prime -1}$. One can also prove that the norms $\langle u^\prime_a(t)|u^\prime_a(t)\rangle^{}_{\cal PT} = \langle u^\prime_b(t)|u^\prime_b(t)\rangle^{}_{\cal PT} = 0$ and $\langle u^\prime_a(t)|u^\prime_b(t)\rangle^{}_{\cal PT} = \langle u^\prime_b(t)|u^\prime_a(t)\rangle^{}_{\cal PT} = +1$ are time-independent.
Then, we proceed to compute the amplitudes and probabilities for the transitions between two flavor eigenstates. After some calculations, the transition amplitudes are found to be
\begin{eqnarray}
&~& {\cal A}^\prime_{aa} \equiv \langle u^\prime_a|u^\prime_a(t)\rangle^{}_{\cal PT} = -{\rm i} e^{-{\rm i}\omega t} \frac{\sinh(\gamma t)}{\sinh \alpha^\prime} \; , \label{eq:Apaa}\\
&~& {\cal A}^\prime_{ab} \equiv \langle u^\prime_b|u^\prime_a(t)\rangle^{}_{\cal PT} = e^{-{\rm i}\omega t} \frac{\sinh(\alpha^\prime + \gamma t)}{\sinh \alpha^\prime} \; , \label{eq:Apab}\\
&~& {\cal A}^\prime_{ba} \equiv \langle u^\prime_a|u^\prime_b(t)\rangle^{}_{\cal PT} = e^{-{\rm i}\omega t} \frac{\sinh(\alpha^\prime - \gamma t)}{\sinh \alpha^\prime} \; , \label{eq:Apba}\\
&~& {\cal A}^\prime_{bb} \equiv \langle u^\prime_b|u^\prime_b(t)\rangle^{}_{\cal PT} = -{\rm i} e^{-{\rm i}\omega t} \frac{\sinh(\gamma t)}{\sinh \alpha^\prime} \; , \label{eq:Apbb}
\end{eqnarray}
while the corresponding transition probabilities are defined as ${\cal P}^\prime_{\alpha \beta} \equiv |{\cal A}^\prime_{\alpha \beta}|^2$ (for $\alpha, \beta$ running over $a, b$) and explicitly calculated as
\begin{eqnarray}
&~& {\cal P}^\prime_{aa} = \sinh^2(\gamma t)/\sinh^2 \alpha^\prime \; , \label{eq:Ppaa}\\
&~& {\cal P}^\prime_{ab} = \sinh^2(\alpha^\prime + \gamma t)/\sinh^2 \alpha^\prime \; , \label{eq:Ppab}\\
&~& {\cal P}^\prime_{ba} = \sinh^2(\alpha^\prime - \gamma t)/\sinh^2 \alpha^\prime \; , \label{eq:Ppba}\\
&~& {\cal P}^\prime_{bb} = \sinh^2(\gamma t)/\sinh^2 \alpha^\prime \; . \label{eq:Ppbb}
\end{eqnarray}
One can observe non-conservation of the total probability, i.e., ${\cal P}^\prime_{aa} + {\cal P}^\prime_{ab} \neq 1$ or ${\cal P}^\prime_{ba} + {\cal P}^\prime_{bb} \neq 1$. Moreover, all probabilities in Eqs.~(\ref{eq:Ppaa})--(\ref{eq:Ppbb}) go to infinity for $t \to +\infty$, rendering them to be physically meaningless. However, the metric operator $\eta^\prime = {\cal P}$ is not positive-definite, so we should not expect the sum of transition probabilities to be conserved. One may instead compute the differences between the probabilities, i.e.,
\begin{eqnarray}
{\cal P}^\prime_{aa} - {\cal P}^\prime_{ab} &=& - \left[\sinh^2(\alpha^\prime + \gamma t) - \sinh^2 (\gamma t)\right]/\sinh^2 \alpha^\prime = - \sinh(\alpha^\prime + 2\gamma t)/\sinh \alpha^\prime \; , \\
{\cal P}^\prime_{ba} - {\cal P}^\prime_{bb} &=& + \left[\sinh^2(\alpha^\prime - \gamma t) - \sinh^2 (\gamma t)\right]/\sinh^2 \alpha^\prime = + \sinh(\alpha^\prime - 2\gamma t)/\sinh \alpha^\prime \; , \quad
\end{eqnarray}
which are unfortunately time-dependent. As a remedy for this problem, following the same strategy as in Ref.~\cite{Ohlsson:2019noy}, we construct the ${\cal CPT}$ flavor eigenstates $|\tilde{u}^\prime_a\rangle$ and $|\tilde{u}^\prime_b\rangle$ as follows
\begin{eqnarray}
|\tilde{u}^\prime_a\rangle &=& \frac{1}{\sqrt{2}} \left(|u^\prime_a\rangle + |u^\prime_b\rangle \right) = \frac{1}{\sqrt{2}} \left(\begin{matrix} +1 \cr +1 \end{matrix}\right)\; , \\
|\tilde{u}^\prime_b\rangle &=& \frac{1}{\sqrt{2}} \left(|u^\prime_a\rangle - |u^\prime_b\rangle \right) = \frac{1}{\sqrt{2}} \left(\begin{matrix} +1 \cr -1 \end{matrix}\right)\; ,
\end{eqnarray}
where the expressions for the two original flavor eigenstates $|u^\prime_a\rangle$ and $|u^\prime_b\rangle$ in Eqs.~(\ref{eq:upa}) and (\ref{eq:upb}) have been used. Since the ${\cal C}$ operator is trivial in the PT-broken phase, we can easily prove that ${\cal CPT}|\tilde{u}^\prime_a\rangle = {\cal PT}|\tilde{u}^\prime_a\rangle = + |\tilde{u}^\prime_a\rangle$ and ${\cal CPT}|\tilde{u}^\prime_b\rangle = {\cal PT}|\tilde{u}^\prime_b\rangle = - |\tilde{u}^\prime_b\rangle$. Therefore, the newly-constructed flavor eigenstates are eigenstates of both the ${\cal CPT}$ and ${\cal PT}$ operators. With these ${\cal CPT}$ flavor eigenstates, we repeat the calculations of the transition amplitudes and probabilities, and then obtain the amplitudes $\tilde{\cal A}^\prime_{\alpha \beta} \equiv \langle \tilde{u}^\prime_\beta|u^\prime_\alpha(t)\rangle$ as
\begin{eqnarray}
&~& \tilde{\cal A}^\prime_{aa} \equiv \langle \tilde{u}^\prime_a|u^\prime_a(t)\rangle^{}_{\cal PT} = \frac{1}{\sqrt{2}} \left({\cal A}^\prime_{aa} + {\cal A}^\prime_{ab}\right) \; , \\
&~& \tilde{\cal A}^\prime_{ab} \equiv \langle \tilde{u}^\prime_b|u^\prime_a(t)\rangle^{}_{\cal PT} = \frac{1}{\sqrt{2}} \left({\cal A}^\prime_{aa} - {\cal A}^\prime_{ab}\right) \; , \\
&~& \tilde{\cal A}^\prime_{ba} \equiv \langle \tilde{u}^\prime_a|u^\prime_b(t)\rangle^{}_{\cal PT} = \frac{1}{\sqrt{2}} \left({\cal A}^\prime_{ba} + {\cal A}^\prime_{bb}\right) \; , \\
&~& \tilde{\cal A}^\prime_{bb} \equiv \langle \tilde{u}^\prime_b|u^\prime_b(t)\rangle^{}_{\cal PT} = \frac{1}{\sqrt{2}} \left({\cal A}^\prime_{ba} - {\cal A}^\prime_{bb}\right) \; ,
\end{eqnarray}
and the probabilities $\tilde{\cal P}^\prime_{\alpha \beta} \equiv |\tilde{\cal A}^\prime_{\alpha \beta}|^2$ as
\begin{eqnarray}
&~& \tilde{\cal P}^\prime_{aa} = \tilde{\cal P}^\prime_{ab} = \frac{1}{2} \left({\cal P}^\prime_{aa} + {\cal P}^\prime_{ab}\right) = \frac{1}{2\sinh^2\alpha^\prime} \left[\sinh^2 (\gamma t) + \sinh^2 (\alpha^\prime + \gamma t)\right] \; , \label{eq:Ptpaa}\\
&~& \tilde{\cal P}^\prime_{ba} = \tilde{\cal P}^\prime_{bb} = \frac{1}{2} \left({\cal P}^\prime_{ba} + {\cal P}^\prime_{bb}\right) = \frac{1}{2\sinh^2\alpha^\prime} \left[\sinh^2 (\gamma t) + \sinh^2 (\alpha^\prime - \gamma t)\right] \; . \label{eq:Ptpba}
\end{eqnarray}
Although these probabilities still become infinite in the limit of $t \to +\infty$, one can check that $\tilde{\cal P}^\prime_{aa} - \tilde{\cal P}^\prime_{ab} = 0$ and $\tilde{\cal P}^\prime_{ba} - \tilde{\cal P}^\prime_{bb} = 0$, where the time dependence is completely canceled out. Thus, the introduction of the ${\cal CPT}$ flavor eigenstates basically renormalizes the individual transition probabilities in Eqs.~(\ref{eq:Ptpaa}) and (\ref{eq:Ptpba}) for the flavor eigenstates and reduces the effect of $\eta'$, as discussed in Ref.~\cite{Ohlsson:2019noy}.
It is interesting to note that there is no interference between the two amplitudes ${\cal A}^\prime_{aa}$ and ${\cal A}^\prime_{ab}$ when squaring the modified amplitudes $\tilde{\cal A}^\prime_{aa}$ and $\tilde{\cal A}^\prime_{ab}$ to calculate $\tilde{\cal P}^\prime_{aa}$ and $\tilde{\cal P}^\prime_{ab}$, leading to a simple average of the probabilities in Eq.~(\ref{eq:Ptpaa}). The main reason can be traced back to the amplitudes in Eqs.~(\ref{eq:Apaa}) and (\ref{eq:Apab}), where ${\cal A}^\prime_{aa}$ is purely imaginary, whereas ${\cal A}^\prime_{ab}$ is real up to the same phase factor $e^{-{\rm i}\omega t}$. Similar observations can be made for $\tilde{\cal P}^\prime_{ba}$ and $\tilde{\cal P}^\prime_{bb}$ in Eq.~(\ref{eq:Ptpba}). Therefore, it seems more reasonable to define the ${\cal CPT}$ flavor eigenstates as the final states in the sense of the time-independence of the probability differences.
\subsection{Connection between the PT Symmetry and the Neutral-Meson System}
The non-Hermitian Hamiltonian with complex eigenvalues has been known in particle physics for a long time. As a concrete example, the mixing and oscillation of the neutral-meson system $\{|P^0\rangle, |\overline{P}^0\rangle\}$, such as $K^0$-$\overline{K}^0$, $D^0$-$\overline{D}^0$, and $B^0$-$\overline{B}^0$, can be described by an effective non-Hermitian Hamiltonian~\cite{Lee:1957qq, Lee:1965hi, Bigi:2000yz}
\begin{equation}
{\sf H} = {\sf M} - \frac{\rm i}{2} {\sf \Gamma} \equiv \left(\begin{matrix} M^{}_{11} & M^{}_{12} \cr M^*_{12} & M^{}_{22}\end{matrix}\right) - \frac{\rm i}{2} \left(\begin{matrix} \Gamma^{}_{11} & \Gamma^{}_{12} \cr \Gamma^*_{12} & \Gamma^{}_{22}\end{matrix}\right) \; , \label{eq:HMG}
\end{equation}
where both ${\sf M}$ and ${\sf \Gamma}$ are $2\times 2$ Hermitian matrices. In order to make a distinction between the neutral-meson system and the PT-broken phase under consideration, we have set all $2\times 2$ matrices in the former case in a sans-serif typeface. Without imposing either CPT or CP invariance,\footnote{Note that the C, P, and T transformations, as well as their combinations CPT and CP, for the neutral-meson system should be understood in the same way as in particle physics or relativistic quantum field theories in general.} the time-evolved neutral-meson states can be written as~\cite{Tanabashi:2018oca}
\begin{eqnarray}
|P^0(t)\rangle &=& \left[g^{}_+(t) + z g^{}_-(t)\right]|P^0\rangle - \frac{q}{p} \sqrt{1 - z^2} g^{}_-(t) |\overline{P}^0\rangle \; , \\
|\overline{P}^0(t)\rangle &=& \left[g^{}_+(t) - z g^{}_-(t)\right]|\overline{P}^0\rangle - \frac{p}{q} \sqrt{1 - z^2} g^{}_-(t) |P^0\rangle \; ,
\end{eqnarray}
where $z = 0$ corresponds to the case of either CPT or CP invariance and the relevant time-evolution functions are given by
\begin{equation}
g^{}_\pm(t) \equiv \frac{1}{2} \left[\exp\left(-{\rm i}M^{}_2 t - \frac{1}{2} \Gamma^{}_2 t\right) \pm \exp\left(-{\rm i}M^{}_1 t - \frac{1}{2} \Gamma^{}_1 t\right) \right] \; .
\end{equation}
Note that for $i = 1, 2$, $M^{}_i$ stand for the masses of the energy eigenstates $|P^{}_i\rangle$, while $\Gamma^{}_i$ for the corresponding total decay widths. The masses and decay widths, which all should be positive (i.e., $M_i > 0$ and $\Gamma_i > 0$) for the neutral-meson system, are related to the matrix elements of the effective Hamiltonian with the eigenvalues $\{E^{}_1, E^{}_2\}$ via
\begin{eqnarray}
E^{}_1 &\equiv& M^{}_1 - \frac{\rm i}{2} \Gamma^{}_1 = M^{}_{11} - \frac{\rm i}{2} \Gamma^{}_{11} + pq \left[\kappa + \sqrt{1 + \kappa^2}\right] \; , \label{eq:E1}\\
E^{}_2 &\equiv& M^{}_2 - \frac{\rm i}{2} \Gamma^{}_2 = M^{}_{22} - \frac{\rm i}{2} \Gamma^{}_{22} - pq \left[\kappa + \sqrt{1 + \kappa^2}\right] \; , \label{eq:E2}
\end{eqnarray}
where $\kappa \equiv \left[(M^{}_{22} - {\rm i}\Gamma^{}_{22}/2) - (M^{}_{11} - {\rm i}\Gamma^{}_{11}/2) \right]/(2pq)$ and
\begin{equation}
p^2 \equiv M^{}_{12} - \frac{\rm i}{2}\Gamma^{}_{12} \; , \quad q^2 \equiv M^*_{12} - \frac{\rm i}{2}\Gamma^*_{12} \; .
\end{equation}
The complex parameter $z$ can be expressed as follows
\begin{equation}
z \equiv \frac{\kappa}{\sqrt{1 + \kappa^2}} = \frac{\displaystyle \delta m - \frac{\rm i}{2}\delta \Gamma}{\displaystyle \Delta m - \frac{\rm i}{2}\Delta \Gamma} \label{eq:z}
\end{equation}
with $\delta m \equiv M^{}_{11} - M^{}_{22}$, $\Delta m \equiv M^{}_2 - M^{}_1$, $\delta \Gamma \equiv \Gamma^{}_{11} - \Gamma^{}_{22}$, and $\Delta \Gamma \equiv \Gamma^{}_2 - \Gamma^{}_1$. Now, it is evident that $z = 0$ corresponds to $M^{}_{11} = M^{}_{22}$ and $\Gamma^{}_{11} = \Gamma^{}_{22}$, as implied by the CPT theorem for local quantum field theories.
It is straightforward to calculate the transition amplitudes for $|P^0\rangle \rightarrow |P^0\rangle$ and $|P^0\rangle \rightarrow |\overline{P}^0\rangle$, namely,
\begin{eqnarray}
{\cal A}^{}_{P^0 P^0}(t) &\equiv& \langle P^0|P^0(t)\rangle = g^{}_+(t) + z g^{}_-(t) \; , \\
{\cal A}^{}_{P^0 \overline{P}^0}(t) &\equiv& \langle \overline{P}^0|P^0(t)\rangle = - \frac{q}{p} \sqrt{1 - z^2} g^{}_-(t) \; ,
\end{eqnarray}
where $\langle P^0| \equiv |P^0\rangle^\dagger$ and $\langle \overline{P}^0| \equiv |\overline{P}^0\rangle^\dagger$ have been defined. Accordingly, the corresponding transition probabilities turn out to be
\begin{eqnarray}
{\cal P}^{}_{P^0 P^0}(t) \equiv |{\cal A}^{}_{P^0 P^0}(t)|^2 &=& + \frac{1}{4} \left[e^{-\Gamma^{}_1t} + e^{-\Gamma^{}_2t} + 2 e^{-\Gamma t} \cos (\Delta m t) \right] \nonumber \\
&~& + \frac{1}{4} \left[e^{-\Gamma^{}_1t} + e^{-\Gamma^{}_2t} - 2 e^{-\Gamma t} \cos (\Delta m t) \right] |z|^2 \nonumber \\
&~& + \frac{1}{2} \left(e^{-\Gamma^{}_2t} - e^{-\Gamma^{}_1t}\right) \Re(z) + e^{-\Gamma t} \sin (\Delta m t) \, \Im(z) \; , \label{eq:PP0P0}\\
{\cal P}^{}_{P^0 \overline{P}^0}(t) \equiv |{\cal A}^{}_{P^0 \overline{P}^0}(t)|^2 &=& \frac{|q|^2}{4|p|^2} \left[e^{-\Gamma^{}_1t} + e^{-\Gamma^{}_2t} - 2 e^{-\Gamma t} \cos (\Delta m t) \right] \sqrt{1 - 2 \Re(z^2) + |z|^4}\; , \nonumber\\ \label{eq:PP0bP0}
\end{eqnarray}
with $\Gamma \equiv (\Gamma^{}_1 + \Gamma^{}_2)/2$. Since the decay widths $\Gamma^{}_1$ and $\Gamma^{}_2$ are positive, the transition probabilities ${\cal P}^{}_{P^0 P^0}(t)$ and ${\cal P}^{}_{P^0 \overline{P}^0}(t)$ will vanish in the limit of $t \to +\infty$.
As observed in Ref.~\cite{Ozdemir:2019}, a loss-loss system in photonics can be equivalently described as a PT-symmetric loss-gain system with global exponential decay or amplification. Inspired by this observation of passive PT-symmetry in the loss-loss system, we assume that the most general non-Hermitian Hamiltonian in Eq.~(\ref{eq:HMG}) can be decomposed into a PT-symmetric non-Hermitian Hamiltonian in the PT-broken phase and a lossy term, i.e.,
\begin{eqnarray}
\left(\begin{matrix} M^{}_{11} & M^{}_{12} \cr M^*_{12} & M^{}_{22}\end{matrix}\right) - \frac{\rm i}{2} \left(\begin{matrix} \Gamma^{}_{11} & \Gamma^{}_{12} \cr \Gamma^*_{12} & \Gamma^{}_{22}\end{matrix}\right) = \left(\begin{matrix} \rho e^{+{\rm i}\varphi} & \sigma \cr \sigma & \rho e^{-{\rm i}\varphi} \end{matrix}\right) - \left( \begin{matrix} {\rm i}\chi & 0 \cr 0 & {\rm i} \chi \end{matrix} \right) \; ,
\label{eq:H_pass}
\end{eqnarray}
where the first term in the right-hand side is simply the PT-symmetric Hamiltonian in Eq.~(\ref{eq:H}) with $\phi = 0$ and the second one with $\chi > 0$ represents a global exponential decay. By identifying both sides of Eq.~(\ref{eq:H_pass}), we obtain the following relations
\begin{eqnarray}
M^{}_{11} - \frac{\rm i}{2} \Gamma^{}_{11} &=& \rho \cos\varphi + {\rm i} \rho \sin\varphi - {\rm i}\chi \; , \\
M^{}_{22} - \frac{\rm i}{2} \Gamma^{}_{22} &=& \rho \cos\varphi - {\rm i} \rho \sin\varphi - {\rm i}\chi \; , \\
M^{}_{12} - \frac{\rm i}{2} \Gamma^{}_{12} &=& \sigma \; , \\
M^*_{12} - \frac{\rm i}{2} \Gamma^*_{12} &=& \sigma \; ,
\end{eqnarray}
implying that $M^{}_{11} = M^{}_{22} = \rho \cos\varphi$, $\Gamma^{}_{11} = - 2(\rho \sin\varphi - \chi) $, $\Gamma^{}_{22} = 2(\rho \sin\varphi + \chi)$, $M^{}_{12} = \sigma$, and $\Gamma^{}_{12} = 0$, together with $p = q = \sqrt{M^{}_{12}} = \sqrt{\sigma}$ and $\kappa = -{\rm i}(\rho\sin\varphi)/\sigma$. Using Eqs.~(\ref{eq:E1}) and (\ref{eq:E2}), we immediately find that
\begin{eqnarray}
&~& M^{}_1 = \rho \cos\varphi \; , \quad \Gamma^{}_1 = 2 \left(\chi - \sqrt{\rho^2 \sin^2\varphi - \sigma^2} \right) \; , \\
&~& M^{}_2 = \rho \cos\varphi \; , \quad \Gamma^{}_2 = 2 \left(\chi + \sqrt{\rho^2 \sin^2\varphi - \sigma^2} \right) \; ,
\end{eqnarray}
where it should be noted that $\rho^2\sin^2\varphi > \sigma^2$ and $(1 - \rho^2\sin^2\varphi/\sigma^2)^{1/2} = +{\rm i} \sqrt{\rho^2 \sin^2\varphi - \sigma^2}/\sigma$ has been utilized. The parameter $z$ in Eq.~(\ref{eq:z}) is determined by $\delta m \equiv M^{}_{11} - M^{}_{22} = 0$, $\Delta m \equiv M^{}_2 - M^{}_1 = 0$, $\delta \Gamma \equiv \Gamma^{}_{11} - \Gamma^{}_{22} = -4\rho \sin\varphi$, and $\Delta \Gamma \equiv \Gamma^{}_2 - \Gamma^{}_1 = 4\sqrt{\rho^2 \sin^2\varphi - \sigma^2}$, namely,
\begin{equation}
z = \frac{\delta \Gamma}{\Delta \Gamma} = -\frac{\rho \sin\varphi}{\sqrt{\rho^2 \sin^2\varphi - \sigma^2}} = - \coth \alpha^\prime \; .
\end{equation}
From the previous discussion, one can recognize that $E^{}_1 = E^\prime_+ = \omega + {\rm i}(\gamma - \chi)$ and $E^{}_2 = E^\prime_- = \omega - {\rm i}(\gamma + \chi)$ with $\omega = \rho \cos\varphi$ and $\gamma = \sqrt{\rho^2 \sin^2 \varphi - \sigma^2}$, and thus, we obtain $M^{}_1 = M^{}_2 = \omega$, $\Gamma^{}_1 = 2(\chi - \gamma)$, and $\Gamma^{}_2 = 2 (\chi + \gamma)$. Since $\Gamma^{}_1$ should be identified with a positive decay width, we have $\Gamma^{}_1 > 0$, or equivalently, $\chi > \gamma$. This condition, if expressed in terms of the averaged decay width $\Gamma = 2\chi$ and the decay-width difference $\Delta \Gamma = 4\gamma$, implies that $\Gamma > \Delta \Gamma/2$, and naturally, it also holds that $\Gamma_2 > 0$, since $\chi > \gamma > 0$.
It is also interesting to observe that $z \neq 0$ and $q/p = 1$ are valid in the PT-broken phase under consideration, which cannot be simultaneously true for the ordinary neutral-meson system. At this point, it is helpful to give some remarks on the CPT and CP symmetries in the neutral-meson system, and the ${\cal CPT}$ and ${\cal PT}$ symmetries in the PT-broken phase. Following the convention in Ref.~\cite{Bigi:2000yz}, one can write down the discrete space-time symmetry transformations for the neutral-meson system as
\begin{equation}
{\sf C}|P^0\rangle = -|\overline{P}^0\rangle \; , \quad {\sf P}|P^0\rangle = - |P^0\rangle \; , \quad {\sf T}|P^0\rangle = |P^0\rangle \; ,
\end{equation}
implying that ${\sf CP}|P^0\rangle = |\overline{P}^0\rangle$ and ${\sf CP}|\overline{P}^0\rangle = |P^0\rangle$. Observe that the time-reversal transformation will interchange the initial and final states, which form separately complete bases, and it is same in both the neutral-meson system and the PT-broken phase, i.e., ${\sf T} = {\cal T}$. In the matrix representation, we consider the two flavor eigenstates as $|P^0\rangle = (1, 0)^{\rm T}$ and $|\overline{P}^0\rangle = (0, 1)^{\rm T}$ and their Hermitian conjugated states $\langle P^0| = |P^0\rangle^\dagger = (1, 0)$ and $\langle \overline{P}^0| = |\overline{P}^0\rangle^\dagger = (0, 1)$. It is then straightforward to obtain
\begin{equation}
{\sf C} = \left(\begin{matrix} 0 & -1 \cr -1 & 0 \end{matrix}\right) \; , \quad {\sf P} = \left(\begin{matrix} -1 & 0 \cr 0 & -1 \end{matrix}\right) \; , \quad {\sf CP} = \left(\begin{matrix} 0 & 1 \cr 1 & 0 \end{matrix}\right) \; ,
\end{equation}
where one can observe that the matrix forms of the ${\sf CP}$ and ${\cal P}$ operators are exactly the same. It is not difficult to verify that the CPT or CP invariance in the neutral-meson system guarantees $M^{}_{11} = M^{}_{22}$ and $\Gamma^{}_{11} = \Gamma^{}_{22}$, while CP or T invariance leads to $\Im(M^{}_{12}) = \Im(\Gamma^{}_{12}) = 0$. However, as we have seen, the relations $M^{}_{11} = M^{}_{22}$, $\Gamma^{}_{11} \neq \Gamma^{}_{22}$, and $\Im(M^{}_{12}) = \Gamma^{}_{12} = 0$ hold in the PT-broken phase.
Since the transition probabilities for the flavor eigenstates in the neutral-meson system have been calculated, we can apply them directly to the PT-broken phase. Using Eq.~(\ref{eq:PP0P0}) as well as $\Delta m = 0$ and $\Gamma = 2\chi$, we find that
\begin{eqnarray}
{\cal P}^\prime_{aa}(t) &=& \left[\cosh^2(\gamma t) + \sinh^2(\gamma t) \frac{\cosh^2\alpha^\prime}{\sinh^2\alpha^\prime} + \sinh(2\gamma t) \frac{\cosh\alpha^\prime}{\sinh\alpha^\prime} \right] e^{-2\chi t} \nonumber \\
&=& \frac{\sinh^2(\alpha^\prime + \gamma t)}{\sinh^2\alpha^\prime} e^{-2\chi t} \; ,
\label{eq:Paap}
\end{eqnarray}
and similarly using Eq.~(\ref{eq:PP0bP0}), we obtain
\begin{equation}
{\cal P}^\prime_{ab}(t) = \frac{1}{4} \left(e^{2\gamma t} + e^{-2\gamma t} - 2\right) e^{-2\chi t} \sqrt{(1 - \coth^2 \alpha^\prime)^2} = \frac{\sinh^2 (\gamma t)}{\sinh^2\alpha^\prime} e^{-2 \chi t}\; .
\label{eq:Pabp}
\end{equation}
Comparing the above results with the ones in Eqs.~(\ref{eq:Ppaa}) and (\ref{eq:Ppab}), we realize the additional exponential factor $e^{-2\chi t}$ and the exchange between the expressions of ${\cal P}^\prime_{aa}$ and ${\cal P}^\prime_{ab}$. Such an exchange can be understood by noticing the fact that the ${\cal PT}$-inner product and the ordinary inner product (i.e., the ${\cal T}$-inner product) differ by the parity operator ${\cal P}$ that causes the exchange of the final flavor eigenstates. In addition, in the limit of $t \to +\infty$, one can immediately verify that both ${\cal P}^\prime_{a a}(t)$ and ${\cal P}^\prime_{a b}(t)$ are proportional to $e^{-2(\chi - \gamma)t}$, which approaches zero due to the condition $\chi > \gamma$ for positive decay widths $\Gamma_1 > 0$ and $\Gamma_2 > 0$.
Before concluding this subsection, we make some helpful comments on the PT symmetry and the neutral-meson system. First, if one simply identifies the most general non-Hermitian Hamiltonian in Eq.~(\ref{eq:HMG}) with the PT-symmetric Hamiltonian in Eq.~(\ref{eq:H}) with $\phi = 0$, then it necessarily leads to $\Gamma^{}_1 < 0$ in the PT-broken phase. In such a case, the transition probabilities can be found by setting $\chi = 0$ in Eqs.~(\ref{eq:Paap}) and (\ref{eq:Pabp}). Consequently, the transition probabilities ${\cal P}^\prime_{aa}$ and ${\cal P}^\prime_{ab}$ become infinite in the limit of $t \to +\infty$. For this reason, it seems to be more interesting to consider the passive PT-symmetric Hamiltonian in Eq.~(\ref{eq:H_pass}), for which the masses and decay widths of the energy eigenstates are real and positive. Second, as the time evolution of the neutral-meson states $|P^0(t)\rangle$ and $|\overline{P}^0(t)\rangle$ is governed by the Schr\"{o}dinger equation with the Hamiltonian in Eq.~(\ref{eq:H_pass}), one can perform the gauge transformation~\cite{Ozdemir:2019}
\begin{eqnarray}
\left( \begin{matrix} |P^{\prime 0}(t)\rangle \cr |\overline{P}^{\prime 0}(t)\rangle \end{matrix}\right) = e^{+\chi t} \left( \begin{matrix} |P^0(t)\rangle \cr |\overline{P}^0(t)\rangle \end{matrix}\right)
\end{eqnarray}
such that the Schr\"{o}dinger equation for the gauge-transformed states $|P^{\prime 0}(t)\rangle$ and $|\overline{P}^{\prime 0}(t)\rangle$ is given by
\begin{eqnarray}
{\rm i}\frac{{\rm d}}{{\rm d}t} \left( \begin{matrix} |P^{\prime 0}(t)\rangle \cr |\overline{P}^{\prime 0}(t)\rangle \end{matrix}\right) = \left(\begin{matrix} \rho e^{+{\rm i}\varphi} & \sigma \cr \sigma & \rho e^{-{\rm i}\varphi} \end{matrix}\right) \left( \begin{matrix} |P^{\prime 0}(t)\rangle \cr |\overline{P}^{\prime 0}(t)\rangle \end{matrix}\right) \; .
\end{eqnarray}
Now, it is clear that the transition probabilities calculated in Subsection~\ref{sub: PTip} are applicable to the gauge-transformed states $|P^{\prime 0}(t)\rangle$ and $|\overline{P}^{\prime 0}(t)\rangle$, instead of the physical states $|P^0(t)\rangle$ and $|\overline{P}^0(t)\rangle$. Finally, it is worth pointing out that although only the transition amplitudes and probabilities in the framework of quantum mechanics are aimed for in the present work, the calculations can be performed in parallel for optical beam dynamics in PT-symmetric or PT-broken waveguides. In the latter case, exponential decay or amplification of the optical power takes place and has been experimentally observed~\cite{Feng:2017}.
\section{Summary and Conclusions}\label{sec: conclusion}
The basic properties of non-Hermitian Hamiltonians in both the PT-symmetric and PT-broken phases are interesting and their possible practical applications have recently received a lot of attention. In this work, we have focused on the flavor transitions in the two-level quantum system with PT-symmetric non-Hermitian Hamiltonians. Extending our previous investigation on the PT-symmetric phase with two real eigenvalues, we have considered the PT-broken phase, in which the two eigenvalues are complex conjugates to each other.
First, after solving the eigenvalues and eigenvectors of the non-Hermitian Hamiltonian in the PT-broken phase, we have explicitly constructed the charge-conjugation operator ${\cal C}$ and the metric operator $\eta^\prime$, for which the identities ${\cal C} = \mathbb{1}_2$ and $\eta^\prime = {\cal P}$ are valid. Second, using the ${\cal PT}$-inner product, we have calculated the transition amplitudes and probabilities for the flavor eigenstates, i.e., $|u^\prime_\alpha\rangle \to |u^\prime_\beta\rangle$ for $\alpha, \beta = a, b$. After introducing the ${\cal CPT}$ flavor eigenstates ${\cal CPT}|\tilde{u}^\prime_a\rangle = +|\tilde{u}^\prime_a\rangle$ and ${\cal CPT}|\tilde{u}^\prime_b\rangle = -|\tilde{u}^\prime_b\rangle$ as the final states, we have found that the difference $\tilde{\cal P}^\prime_{aa} - \tilde{\cal P}^\prime_{ab}$ between, instead of the sum $\tilde{\cal P}^\prime_{aa} + \tilde{\cal P}^\prime_{ab}$ of, the corresponding transition probabilities, vanishes and is time-independent. However, the probabilities themselves in the PT-broken phase have been found to be infinite in the limit of $t \to +\infty$, which is totally different from the corresponding result in the PT-symmetric phase. Third, in analogy to the neutral-meson system, we have also calculated the transition probabilities using the ordinary inner product, which is equivalent to the ${\cal T}$-inner product, and observed that the infinite-time behavior of the probabilities originates from the negative decay width $\Gamma^{}_1 = -2\gamma < 0$ of one energy eigenstate in the PT-broken phase. We also make a connection between the neutral-meson system and the passive PT-symmetric Hamiltonian, where an extra global decay term (described by the parameter $\chi > 0$) is present, and thus, $\Gamma^{}_1 = 2(\chi - \gamma) > 0$ and $\Gamma^{}_2 = 2(\chi + \gamma) > 0$. Even in this case, the relations $M^{}_{11} = M^{}_{22}$, $\Gamma^{}_{11} \neq \Gamma^{}_{22}$, and $\Im(M^{}_{12}) = \Gamma^{}_{12} = 0$ are different from those accessible in the ordinary neutral-meson system. For this reason, since the PT-broken phase cannot be used to describe the neutral-meson system, one might have to find practical applications of the PT-broken phase in dynamical systems beyond particle physics. In addition, we have presented a discussion on the exceptional point for several different and interesting cases of non-Hermitian Hamiltonians (including the passive PT-symmetric Hamiltonian with a global exponential decay or amplification) that have applications in various physical systems such as photonics, phononics, and other open quantum and PT-symmetric systems.
Finally, the results that have been presented in this work indicate that the PT-broken phase and the exceptional point have very different properties compared with the PT-symmetric phase and this deserves further exploration. As shown in Refs.~\cite{Wiersig1, Wiersig2, Sensors}, the microcavity sensors prepared at the exceptional point will be much more sensitive to small perturbations, which can be implemented to realize a one-particle detection. In a similar way, the practical applications of the non-Hermitian Hamiltonian in the PT-broken phase may be accomplished only after coupling it to another system. This is the case for the neutral-meson system, where the weak interaction is switched on in order for the neutral mesons to decay. As we have mentioned, realistic applications may be lying beyond particle physics. We leave all these important points for future works.
\section*{Acknowledgments}
T.O.~acknowledges support by the Swedish Research Council (Vetenskapsr{\aa}det) through contract No.~2017-03934 and the KTH Royal Institute of Technology for a sabbatical period at the Uni\-versity of Iceland. The work of S.Z.~was supported in part by the National Natural Science Foundation of China under grant No.~11775232 and No.~11835013, and by the CAS Center for Excellence in Particle Physics.
S.Z.~is also greatly indebted to his family, friends and colleagues for all their support and encouragement during his isolated stay in Huanggang, Hubei, China, where the new corona virus is particularly prevalent and the present work is completed.
\section*{Data Availability}
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Hello,
I am having this issue where an input form looks good on say, 1680x1050 resolution - but when you view the higher resolution monitors, the input form and button start to become misaligned by 2 or 3px. Its driving me nuts!!
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TITLE: Number of binary vectors of a given Hamming weight in a subspace of the Hypercube
QUESTION [5 upvotes]: Let $n$ be a natural number.
Let $U \subseteq \mathbb{F}_2^n$ be a linear subspace of dimension $k$.
What is the maximum number of vectors in $U$ of Hamming weight $\ell$?
The case I am specifically interested in is $k \approx 0.99n$, $\ell = n/2$, where $n$ is a large enough even number. Is this number at most $C \cdot 2^k \cdot \frac{\binom{n}{n/2}}{2^n}$ for some constant $C$, independent of $n$?
REPLY [1 votes]: I just found out that the question (at least with the set of parameters written above) was asked by Ben-Or and answered by Linial and Samordinsky.
Their paper proves that for any constant $r>1/2$ a subspace of dimension $k = rn$ has at most $C_r \cdot 2^k \cdot \frac{\binom{n}{n/2}}{2^n}$ vectors of weight $n/2$.
http://www.cs.huji.ac.il/~nati/PAPERS/lin_codes.pdf
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Favtag is an innovative search engine to allow people to index, classify, and rate the web's content. Traditional search engines scour the internet using complex algorithms to relate certain websites to keywords. Unfortunately, companies have taken advantage of this, to have their websites rank higher under certain keywords. Why should search results be presented based on websites taking advantage of faulty search algorithms?
Favtag is giving back power to end users and taking the power away from companies with the deepest pockets for search engine optimization. Our search engine technology is not perfect and it has vulnerabilities. But, we have very complex systems in place to stop most fraudulent activity.
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\section{Conclusions}
\label{sec:concl}
We have introduced a novel stratified sampling method with adaptive
stratification and sequential, hybridized allocation of samples. For a
fixed stratification, the sample allocation asymptotically approaches
a prescribed linear combination of proportional and optimal
allocation. Letting a fraction of samples be allocated proportionally
adds robustness to the adaptive method, as the stratum standard
deviations are not a priori known but estimated on-the-fly as samples
are added. Moreover, a greedy approach is used to split the stratum
that results in the largest reduction of the stratified sampling
estimator's current variance. Kurtosis dependent estimates are
provided that allow to quantify the probability of drastically
underestimating local strata variances and, thus, of failing to
identify strata that should be refined.
To maintain prescribed stratum sampling rates and for ease of
implementation, a new stratum that arises after re-stratification
should be contained within a single parent stratum, and it should
retain the same geometrical shape. In this work, we suggest using
hyperrectangles or simplices for the stratification, as either of
these shapes can be bisected and the result is two new hyperrectangles
or simplices. In fact, both classes of tessellations allow for a
flexible partition of the stochastic domain.
The proposed method is anticipated to result in significant speedups
for problems where the variability is localized in random space, e.g.,
the PDE solutions describing physical problems with uncertain
parameters and exhibiting steep gradients or discontinuities. In
contrast to, e.g., localized response surface methods based on function
approximations, an advantage of the proposed method is that accurate
identification of steep features is not necessary to obtain good
results. Confining sharp features to strata of small measure is often
sufficient to vastly outperform standard Monte Carlo sampling. This
has been verified experimentally through various test cases,
exhibiting speedups of up to three orders of magnitude compared to
standard Monte Carlo.
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\subsection{Internal language of Markov categories}
\label{sec:internal_language}
There is a strong correspondence between first-order probabilistic programming languages and the categorical models of probability, via their internal languages. The internal language of a CD category~$\C$ has types
\[ \tau ::= X \s I \s \tau \ast \tau \]
where $X$ ranges over objects of $\C$. Any type~$\tau$ can be regarded as an object $\sem \tau$ of $\C$,
via $\sem{X}=X$, $\sem{I}=I$, and $\sem{\tau_1 \ast\tau_2}=
\sem{\tau_1}\otimes \sem{\tau_2}$.
The terms of the internal language are like the language of Section~\ref{sec:gaussian_language}, built from $\letin x t u$, free variables and pairing, but
instead of Gaussian-specific constructs like $\normal()$, $+$, and $\eq$, we have terms for any morphisms in~$\C$:
\[
\infer[(f:\sem{\tau_1}{\otimes} \dots{\otimes} \sem{\tau_n}\to \sem{\tau'}\text{ in $\C$})]{\Gamma\vdash f(t_1\dots t_n) : \tau'}
{\Gamma\vdash t_1:\tau_1\ \dots\ \Gamma\vdash t_n:\tau_n}
\]
Taking $\C=\gauss$ we recover the conditioning-free fragment of the language of Section~\ref{sec:gaussian_language} (\ref{ex:gauss_interpretation}), but the syntax makes sense for any CD or Markov category. A core result of this work is that the full language can be recovered as well for a CD category $\C=\cond(\gauss)$ (\S\ref{sec:cond}).
A typing context $\Gamma={(x_1:\tau_1\dots x_n:\tau_n)}$ is interpreted as $\sem{\Gamma}=\sem{\tau_1}\otimes \dots\otimes \sem{\tau_n}$. A term in context $\Gamma\vdash t :\tau$
is interpreted as a morphism $\sem\Gamma\to \sem \tau$,
defined by induction on the structure of typing derivations.
This is similar to the interpretation of a dual linear $\lambda$-calculus in a monoidal category \cite[\S3.1,\S4]{barberdill}, although because every type supports copying and discarding we do not need to distinguish between linear and non-linear variables. For example,
\begin{align*}&\sem{\letin x t u}=
\sem\Gamma {\xrightarrow{\cpy}} \sem\Gamma\otimes \sem\Gamma
{\xrightarrow{\sem t \otimes \id}} \sem A \otimes \sem\Gamma
{\xrightarrow {\sem u}} \sem B
\\&\sem{\Gamma,x:\tau,\Gamma'\vdash x:\tau}=
\sem\Gamma\otimes \sem\tau\otimes\sem{\Gamma'}
\xrightarrow{\del\otimes\id_{\sem{\tau}}\otimes \del}\sem\tau
\end{align*}
The interpretation always satisfies the following identity, associativity and commutativity equations:
\begin{align}
&\sem{\letin y {(\letin x t u)} v} = \sem{\letin x t \letin y u v} \notag \\
&\sem{\letin x t x} = \sem t\quad\quad \sem{\letin x x u} = \sem u \label{eqn:commutativity} \\
&\sem{\letin x t {\letin y u v}} = \sem{\letin y u {\letin x t v}}\notag
\end{align}
where $x$ not free in $u$ and $y$ not free in $t$. There are also standard equations for tensors~\cite[\S3.1]{statonlevy}, which always hold.
We can always substitute terms for free variables: if we have $\Gamma,x:A\vdash t : B$ and $\Gamma\vdash u : A$
then $\Gamma\vdash t[^u\!/\!_x]:B$.
In any CD category we have
\[\sem{\letin x t u}=\sem{u[^t\!/\!_x]}\text{ if $x$ occurs exactly once in $u$.}\]
In a Markov category, moreover, every term is discardable:
\[\sem{\letin x t u}=\sem{u[^t\!/\!_x]}\text{ if $x$ occurs at most once in $u$.}\]
(It is common to also define a term to be \emph{copyable} if a version of the substitution condition holds when $x$ occurs \emph{at least once} (e.g.~\cite{fuhrmann,kammarplotkin}), but we will not need that in what follows.)
\begin{example}
\label{ex:gauss_interpretation}
The fragment of the Gaussian language without conditioning ($\eq$) is a subset of the internal language of the category $\gauss$.
That is to say, there is a canonical denotational semantics of the Gaussian language where we interpret types and contexts as objects of $\gauss$, e.g. $\sem{\rv}=1$ and $\sem{(x:\rv,y:\rv \otimes \rv)} = 3$. Terms $\Gamma\vdash t:A$ are interpreted as stochastic maps $Ax+b+ \mathcal N(\Sigma)$. This is all automatic once we recognize that addition $(+) : 2 \to 1$, scaling $\alpha \cdot (-) : 1 \to 1$, constants $\underline{\beta} : 0 \to 1$ and sampling $\mathcal N(1) : 0 \to 1$ are morphisms in $\gauss$.
\end{example}
\begin{example} In Section~\ref{sec:cond}, we will show that the full Gaussian language with conditioning ($\eq$) is the internal language of a CD category.
The fact that commutativity~\eqref{eqn:commutativity} holds is non-trivial. It cannot reasonably be the internal language of a Markov category, because conditions $(\eq)$ cannot be discardable. For example there is no non-trivial morphism $(\eq) : 2 \to 0$ in $\gauss$. \end{example}
| 142,568
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War Thunder Cheats and Codes
Welcome to the War Thunder Cheats and Codes page where our team of contributors will help you with a set of cheats, codes, hints, hacks, tips and unlockables.
If you are stuck on something specific and are unable to find any answers on our War Thunder Cheats page
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| 125,466
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Since the end of slavery in America, no workers have been more exploited than. In 1978, Velasquez led more than 2,000 FLOC members in Ohio and Michigan in the largest agricultural labor strike in the history of the Midwest. His organization also worked on behalf of North Carolina workers who pulled tobacco for the suppliers of RJ Reynolds; the worke
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Funding for Moyers & Company is provided in part by:
| 30,753
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.
This is serious and important scholarship on Pico, showing a keen awareness of cutting edge debates on the reception of ancient philosophy in Medieval and Renaissance thought.
Deeply researched and argumentatively cogent, it's well beyond the level of work I've generally seen in an MA thesis, and would be taken seriously as a proposal for a doctoral dissertation in any program concerned with the history of philosophy.
Edward Butler
| 102,197
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)
WASHINGTON (AP) — President Barack Obama will unleash his inner funnyman when he joins political leaders and foes, journalists who cover them and media executives for a night of bipartisan satire at the 128th annual Gridiron Club and Foundation dinner.
The annual winter tradition features musical skits lampooning Republicans and Democrats.
Minnesota Sen. Amy Klobuchar was speaking for the Democrats and Louisiana Gov. Bobby Jindal was the Republicans’ headliner.
Obama also was to make light-hearted remarks during Saturday’s event at a Washington hotel. He skipped it last year.
Except for Grover Cleveland, every president since the Gridiron was founded in 1885 has addressed it. The club is the oldest and most exclusive for Washington journalists. Its motto is “singe but never burn.”
The event is for public consumption, but TV coverage is prohibited.
| 216,624
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TITLE: Systems of multivariate polynomials with less affine roots than roots at infinity.
QUESTION [1 upvotes]: We are considering systems
of $n$ multivariate polynomials in $n$ variables with coefficients from $\mathbb{C}$ (or $\mathbb{R}$):
$p_1(x_1,\ldots,x_n)=0,~~\ldots~~,p_n(x_1,\ldots,x_n)=0$.
We are interested in the common roots of such systems and assume a zero-dimensional solution set (only isolated roots).
Let $d_i$ be the degree of $p_i$ (largest degree of monomials with nonzero coefficient). Then by Bezout's theorem there are $m:=\prod_{i=1}^nd_i$ solutions in the projective space $\mathbb{P}=\mathbb{C}^{n+1}\backslash\lbrace 0\rbrace$, counted with multiplicities.
These solutions split up into $m_a$ affine roots (points in $\mathbb{C}^n$) and $m_p$ projective solutions, which are the solutions of the homogenized system $p_i^h(x_0,x_1,\ldots,x_n)=0$ with $x_0$-component equal zero. Let's call those $m_p$ roots the roots at infinity (since they cannot be scaled back to affine space).
Now the question (sorry if the explanation was too extensive):
Can we find/construct systems of the above class that have strictly less affine roots than roots at infinity, $1<m_a<m_p$, and all roots are simple?
(of course, at least one $p_i$ should have degree$>1$).
We could show that this is not possible for simple cases (e.g., $n=2$, $d_1=d_2=2$) and suspect that this is not possible in general. We are no algebraic geometers and lack the tools to investigate this issue rigorously, so any help / tipps would be appreciated.
REPLY [1 votes]: I am not sure this is what you are looking for.
So, let $\deg p_i=d_i$ and we may assume $d_1\geq d_2\geq \cdots\geq d_n$. I will assume that $d_n\geq 2$, linear equations create issues and am not sure how they can be handled.
We are assuming the homogenization of these have transversal intersections and thus there are $M=\prod d_i$ points in the intersection.
If $l_i$ are the leading forms of the $p_i$, then the points at infinity are the intersection of these $l_i$s. There are $n$ equations in $n-1$ projective space with finite intersection, so after suitable linear change (adding multiples of the last equation to the previous ones) we may assume $l_i , i<n$ intersect in finitely many points and $m_p$ is less than or equal to this number, since it is got by further intersecting with $l_n$. So, by Bezout, we get $m_p\leq \prod_{i=1}^{n-1} d_i=m$, while the total number of points are $M$. Thus, $m_a=M-m_p\geq m(d_n-1)\geq m\geq m_p$.
Here, I write down a linear case which is contrary to the above statements. Take $n=2, p_1=x_1, p_2=x_1+1$. Then $m_p=1, m_a=0$.
| 193,303
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| 229,824
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TITLE: Concentration of very dependent Markov chains
QUESTION [3 upvotes]: Consider the following simple Markov chain $ X_1\to X_2\to\cdots\to X_n $ where each $X_i$ is $\{-1,1\}$-valued and $X_1\sim\mathrm{Unif}(\{-1,1\})$ (such that the chain is stationary).
The flip probability is $\delta$, i.e., $\Pr[X_{i+1} = -x\,|\,X_i = x] = \delta$ for $x\in\{-1,1\}$.
I'm interested in exponentially tight bounds on the tail of $ |\frac{1}{n}\sum_{i=1}^nX_i| $.
If $ \delta $ is very close to $1/2$, then $X_i$'s are weakly dependent and we would expect that $|\frac{1}{n}\sum_{i=1}^nX_i|$ is very close to $0$.
Therefore, the rare event is $|\frac{1}{n}\sum_{i=1}^nX_i|\ge\epsilon$.
There are many bounds in the literature on the tail $\Pr[|\frac{1}{n}\sum_{i=1}^nX_i|\ge\epsilon]$.
If $\delta$ is very close to $0$, then $X_i$'s are strongly dependent and we would expect that $|\frac{1}{n}\sum_{i=1}^nX_i|$ is very close to $1$.
Therefore, the rare event is $|\frac{1}{n}\sum_{i=1}^nX_i|\le1-\epsilon$.
I'm interested in obtaining exponentially tight upper bounds on the tail $\Pr[|\frac{1}{n}\sum_{i=1}^nX_i|\le1-\epsilon]$.
(Note that this looks like anti-concentration on the face of it, but I'm not sure if it's appropriate to call it so, since we're not lower bounding the probability of rare events.)
I looked into the literature and it seems that most existing bounds (e.g., Theorem 1.2 here) behave well in the weakly dependent regime but behave poorly in the strongly dependent regime.
In particular, for the specific parameter regime in my original context, I couldn't even find a bound that beats a trivial union bound: $$\Pr\left[\left|\frac{1}{n}\sum_{i=1}^nX_i\right|\le1-\epsilon\right]\le\Pr\left[\left|\frac{1}{n}\sum_{i=1}^nX_i\right|<1\right] = \Pr\left[\exists i,\,X_i\ne X_{i+1}\right]\le n\cdot\delta.$$
Here I think of $n$ as something like $\frac{1}{\delta}(\log\frac{1}{\delta})^{-3}$.
My questions are:
What is the actual scaling of the above tail as a function of $n$ and $\delta$?
More generally, how much is known about large deviation of strongly dependent Markov chains?
REPLY [0 votes]: You have a $2\times 2$ Markov transition matrix, and you said you want the uniform distribution to be stationary. So the matrix is symmetric, and the off-diagonal element is $\delta$, which you say is small. Note that whether or not you have a switch at any given step are IID Bernoulli-$\delta$ random variables because you have just the 2-state symmetric Markov chain.
So conditional on $k$ switches, their distribution is uniform in the interval.
You are taking $n\ll \delta^{-1}$. Let $\lambda = n \delta$ which is supposed to be small.
The probability to have $0$ switches is $(1-\delta)^n$ which is approximately $e^{-n\delta}$ if $\delta$ is small. Conditional on having $0$ switches, we know $n^{-1} \sum_{i=1}^{n} \mathsf{X}_i$ is 1. But if we condition on having exactly $k$ switches for a $k>1$,then the conditional distribution of $n^{-1} \sum_{i=1}^{n} \mathsf{X}_i$ is not particularly concentrated near $1$ just because of the symmetry of where the switches occur. That is probably why the subset inequality is going to be your best bet, here.
Incidentally, concentration-of-measure is just a term. It does seem to represent the general phenomenon which is most simply demonstrated in the weak law of large numbers for averages of IID random variables. But for Markov chains at times much larger than their mixing times, they have some similar properties to IID random variables. There is large deviation theory for them (which you can find by looking up Donsker and Varadhan). But you are right that for times much less than the mixing time, the actual distribution of which state the Markov chain is in may be concentrated near its starting position.
If $n$ is large and $\lambda = n\delta$ is not too big, then you can approximate well by considering a Poisson process, even if $\lambda$ is very small. Setting $\mathsf{T}_1,\mathsf{T}_2,\dots$ to be the times where the process jumps from one state to the opposite, and rescaling $\mathsf{T}_i/n$ gives an approximation to a Poisson point process on $[0,1]$ with parameter $\lambda$.
In the Poisson process, the probability to have $0$ switches by time 1 is $e^{-\lambda}$. The probability to have exactly $k$ switches in the Poisson approximation is $e^{-\lambda} \lambda^k/k!$.
The question seems interesting and natural about the conditional distribution of the random variable you considered and I suspect it is well-known. But I did not immediately find it when I looked. Let $\mathsf{Y}_n = (1 + n^{-1} \sum_{i=1}^n \mathsf{X}_i)/2$ which gives the fraction of time spent in state $1$ as opposed to $-1$.
Then the approximation for these random variables, $\mathsf{Y}$, is
obtained by taking a Poisson-$\lambda$ point process up to time $1$ and for each $t$ asking how many Poisson events happened prior to $t$. Then you count the total measure of times $t$ such that the number of Poisson events prior to time $t$ is an even number $\{0,2,4,\dots\}$.
It seems like if you condition on exactly $k$ jumps for $k \in \{0,1,2,\dots\}$ you get a Beta distribution $B(a,b)$ where: $a = \lfloor k/2 \rfloor +1$ and $b=\lceil k/2\rceil$. Let us interpret $B(1,0)$ to be the point-mass at $1$. This seems to follow from the symmetry condition of the endpoints somewhat similarly to the usual stars-and-stripes problem. (Sorry for not knowing a good reference.)
| 189,835
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From the unsung hero department
David Nusinow, the current Xorg Debian maintainer, has put a tremendous amount of work in packaging the beast which is known as Xorg. He did the transition from XFree86 to Xorg and manages the transition to modular Xorg 7 now. There are currently some disturbances in the force but I fully agree with this post from Daniel Stone. Thank you David and the XSF, your hard work is greatly appreciated.
| 284,731
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Saints RB Ingram hit with four-game ban for PED violation
New Orleans Saints running back Mark Ingram had the best year of his career in 2017 as a part of the first National Football League backfield in more than four decades to generate two pro bowlers.
For his part, Ingram also enjoyed a stellar 2017 campaign, notching career highs in carries (230), rushing yards (1,124), rushing touchdowns (12), receptions (58) and receiving yards (416). Last season, in a 34-31 victory vs. Washington, Ingram ran for 134 yards and one touchdown. Ingram, 28, will be eligible to return to the team on October 1, after the Saint's Week 4 game against the New York Giants. Ingram is far too important of an offensive piece for the Saints to discount in 2018.
According to ESPN's Darren Rovell, the four-game suspension will cost Ingram $941,176. The Saints went 2-2 in those four games.More news: West Virginia GOP Senate candidate denies claims that comment was racist
More news: Are Analysts Bullish Eland Oil & Gas PLC (LON:ELA) After Last Week?
More news: Trump's peace plan includes giving away parts of Jerusalem
"I feel like I'm only getting better", a joyous Ingram told me following a practice session at this year's Pro Bowl. It provides Kamara with a huge chance to show the Saints, and other potential teams that he can be the man in their backfield. Kamara was also a Pro Bowler last season after rushing for 728 yards and eight touchdowns while also catching 81 passes for 826 yards and five more scores.
The Saints did draft Louisiana Tech running back Boston Scott, however Mouton says he doesn't believe that is Ingram's replacement. Ingram will turn 29 this year, and the Saints are likely to let him play out his contract and test free agency next year.
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| 66,028
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This page contains the news articles from St.Mary's church Washington DC community.
Please check this page for the latest updates and the changes in the scheduled events.
Friday, March 21, 2014
With submission to the will of the Lord and with great sadness we announce the passing of our Holy Father, His Holiness Moran Mor Ignatius Zakka I Iwas, the Patriarch of Antioch and All the East, The Supreme Head of the Universal Syriac Orthodox Church.
| 333,307
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TITLE: disjoint union of projective morphisms is projective?
QUESTION [1 upvotes]: A morphism $f:X \to Y$ of schemes is projective if it factors into a closed immersion
$i:X \to \mathbb{P}_{Y}^n(=\mathbb{P}_{\mathbb{Z}}^n\times_{Spec\mathbb{Z}} Y)$ for some $n$,followed by the projection $\mathbb{P}_{Y}^n \to Y$.
Let $f_{i}:X_{i} \to Y (i=1,2)$ are projective morphism of schemes.
Is it true that $f_{1}\coprod f_{2}:X_{1}\coprod X_{2} \to Y$ is projective?
REPLY [1 votes]: Yes. It suffices to show that $\mathbb{P}^m\coprod\mathbb{P}^n$ can be embedded in to some $\mathbb{P}^r$. Take $r\geq m+n+1$ and include the first copy as the coordinate subspace on the first $m+1$ coordinates and then second copy as the coordinate subspace on the last $n+1$ coordinates.
Writing $f_1\coprod f_2$ as $X_1\coprod X_2 \to \mathbb{P}_Y^{n_1}\coprod \mathbb{P}_Y^{n_2} \to \mathbb{P}^{N}_Y \to Y$ we see the desired result.
| 106,422
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EBRA-CUP, a method to measure cup migration of total hip replacements in standard radiographs has been developed cooperatively by members of the Department of Orthopaedic Surgery and the Institute for Basic Sciences in Engineering / Unit Geometry and CAD at the University of Innsbruck, Austria, since 1984. EBRA-FCA, an extension for measurement of femoral components was added in 1998.
A detailed description of the method was published in: Krismer M., Bauer R., Tschupik J.P., Mayrhofer P.: EBRA: A METHOD TO MEASURE MIGRATION OF ACETABULAR COMPONENTS - TECHNICAL NOTE, J.Biomechanics, Vol. 28, No. 10, pp. 1225-1236, Pergamon, 1995.
Keywords:
| 9,496
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TITLE: Separation of variables and fourier transform
QUESTION [2 upvotes]: This is a more conceptual question than anything else, and I haven't found an answer to exactly my question in previous similar posts.
In some sense performing a separation of variables in a linear differential equation becomes an eigenvalue problem for some linear operator. If the spectrum of the operator is countable then we can use the notions of orthonormal basis in a Hilbert space and prove that the general solution can be written as a sum over the different possible eigenvalues of the solutions we obtained.
Suppose now that the spectrum was "continuous", now of course I would like to use the integral to sum over the different eigenvalues. Under suitable assumptions, this approach is manifestly equivalent to taking the Fourier transform of the original equation and working down, but my question is the following: is there any rigorous way of justifying the backwards approach, i.e. solving the eigenvalue problem and then stating that a general solution should be written as an integral? Physicists use words like "continuous basis" and proceed by analogy with the discrete case, (which works surprisingly well) but is there anyway to justify this?
REPLY [2 votes]: Start with the most basis case of a singular one-dimensional ordinary differential operator
$$
Lf = -f''+qf,\;\;\;\; 0 \le x < \infty.
$$
Assume that $q$ is continuous and real on $[0,\infty)$. You can always impose one linear endpoint condition at $0$ of the form
$$
\Phi_{0}(f)=\cos\alpha f(0)+\sin\alpha f'(0) = 0.
$$
You may or may not need an additional condition at $\infty$ in order to fully define a domain $\mathcal{D}(L)$ on which $L$ is selfadjoint. Physical problems should not require a condition at $\infty$ other than the natural restriction that $f \in L^{2}[0,\infty)$, but general Mathematical problems may require a second condition $\Phi_{\infty}(f)=0$.
For every $\alpha\in\mathbb{R}$, there is a unique eigenfunction $\varphi_{\lambda}$ such that
$$
\varphi_{\lambda}(0)=\sin\alpha,\;\;\varphi_{\lambda}'(0)=-\cos\alpha,
$$
which gives a normalized eigenfunction solution where $\Phi_{0}(\varphi_{\lambda})=0$. Then there is a unique Borel measure $\mu$ such that
$$
f = \int_{-\infty}^{\infty}\left(\int_{0}^{\infty}f(y)\varphi_{\lambda}(y)dy\right) \varphi_{\lambda}(x)d\mu(\lambda), \\
Lf = \int_{-\infty}^{\infty}\left(\int_{0}^{\infty}f(y)\varphi_{\lambda}(y)dy\right)\lambda \varphi_{\lambda}(x)d\mu(\lambda).
$$
The spectral density measure $\mu$ is determined by the condition at $\infty$: there is a different measure for every possible condition at $\infty$, even though the eigenfunctions don't change. You get Parseval's equality
$$
\int_{-\infty}^{\infty}\left|\int_{0}^{\infty}f(y)\varphi_{\lambda}(y)dy\right|^{2}d\mu(\lambda) = \int_{0}^{\infty}|f(y)|^{2}dy.
$$
The Fourier sin and cosine transforms can be derived from this general theory, for example. In general, the measure $\mu$ may have discrete, singular, and absolutely continuous components. So the integrals may reduce to discrete Fourier sums with Fourier integrals, and even allow for possible singular continuous spectrum (singular continuous really should not occur in Physical problems.) This gives rise to a unitary Fourier transform $\mathscr{F} : L^{2}[0,\infty)\rightarrow L^{2}_{\mu}(\sigma)$, where $\sigma$ is the spectrum of $L$, and
$$
(\mathscr{F}f)(\lambda) = \int_{0}^{\infty}f(t)\varphi_{\lambda}(t)dt
$$
And you end with an explicit inversion integral as well. But you can view the iterated integrals as an expansion along a continuous eigenfunction basis that has the appears of orthogonality in the same way that the ordinary Fourier integral expansion appears to be a continuous expansion.
Dirac almost surely modeled his notation after general Sturm-Liouville theory, which was known before Dirac's work in Quantum. There's nothing like this theory for general selfadjoint operators, but this type of formalism works well for Quantum problems where separation of variables can be used to reduce the operators reduce to ordinary differential operators.
When working with two singular endpoints, you can break at a regular point in between, and then tie the problems together with a transition matrix.
Reference: Coddington and Levinson, Theory of Ordinary Differential Equations. Look at the chapters on Singular Sturm-Liouville problems.
| 90,227
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\begin{document}
\title[Purity and flatness in symmetric monoidal closed exact categories]{Purity and flatness in symmetric monoidal closed exact categories}
\author[ E. Hosseini and A. Zaghian]{E. Hosseini and A. Zaghian}
\address{Department of Mathematics, Shahid Chamran University of
Ahvaz, P.O.Box: 61357-83151, Ahvaz, Iran.}
\email{e.hosseini@scu.ac.ir}
\address{Department of Mathematics and Cryptography, Malek Ashtar University of
Technology, P.O.Box: 115-83145, Isfahan, Iran.}
\email{a.zaghian@mut-es.ac.ir}
\keywords{Closed category, exact category, flat object, pure injective object.\\
2010 Mathematical subject classification: 18G55, 18A05, 18A20,
18F20.}
\begin{abstract}
Let $(\CA, \textmd{-}\otimes\textmd{-})$ be a symmetric monoidal
closed exact category. This category is a natural framework to
define the notions of $\otimes$-purity and $\otimes$-flatness. We
show that an object $\CF$ in $\CA$ is $\otimes$-flat if and only if
any conflation ending in $\CF$ is $\otimes$-pure. Furthermore, we
prove a generalization of the Lambek Theorem (\cite{La63}) in $\CA$.
In the case $\CA$ is a quasi-abelian category, we prove that $\CA$
has enough pure injective objects.
\end{abstract}
\maketitle
\section{Introduction}
Let $\CA$ be an additive category. A sequence
$\xymatrix@C0.7pc@R0.9pc{ \CX\ar[r]^f&\CY\ar[r]^g&\CZ}$ in $\CA$ is
said to be a \textit{conflation} if $f$ is the kernel of $g$ and $g$
is the cokernel of $f$. The map $f$ is called an \textit{inflation}
and $g$ is called a \textit{deflation} (\cite[Appendix A]{Ke90}).
For a given class $\CE$ of conflations in $\CA$, $(\CA, \CE)$ is
said to be an \textit{exact} \textit{category} if the following
axioms hold.
\begin{itemize}
\item [(i)] For any object $A\in\CA$, the identity morphism $1_A$ is an inflation.
\item [(ii)] For any object $A\in\CA$, the identity morphism $1_A$ is a
deflation.
\item [(iii)] Deflations (resp. Inflations) are closed under composition.
\item [(iv)] The pullback (resp. pushout) of a deflation (resp. inflation) along an arbitrary morphism exists and yields a deflation (resp. inflation).
\end{itemize}
See \cite[Appendix A]{Ke90} for more details on exact categories.
Assume that $(\CA, \textmd{-}\otimes\textmd{-})$ is a
\textit{symmetric} \textit{monoidal} \textit{closed} exact
category. Then there is a bifunctor
${\CH}om_\CA(\textmd{-},\textmd{-}):\CA^{\mathrm{op}}\times\CA\lrt\CA$
such that for any object $\CG$ in $\CA$ the functor -$\otimes
\CG:\CA\lrt\CA$ has a right adjoint
${\CH}om_{\CA}(\CG,\textmd{-}):\CA\lrt\CA$, i.e. for any pair of
objects $\CF$ and $\CK$ in $\CA$, we have an isomorphism
\begin{align}\label{0}
\mathrm{Hom}_{\CA}(\CF\otimes\CG,\CK)\cong
\mathrm{Hom}_{\CA}(\CF,{\CH}om_{\CA}(\CG,\CK))
\end{align}
which is naturally in all three arguments (see \cite{KM71} for more
details). The bifunctor ${\CH}om_\CA(\textmd{-},\textmd{-})$ is
called the \textit{internal} \textit{hom} on $\CA$.
In some situations, the category $\CA$ does not have enough
projective objects (\cite[Ex III.6.2]{H}). This causes that,some of
the most important theorems in homological algebra do not hold in
general case. The Lazard-Govorov Theorem is one of them. It asserts
that any flat module over a ring is a direct limit of finitely
generated free modules (\cite{La99} and \cite{Go65}). This theorem
has a significance role in the proofs of some important theorems in
homological algebra, especially \cite[Theorem 3]{Sten68} and
\cite[Theorem 2.4]{EG}. In this work we prove a generalization of
the main result of \cite{La63}, \cite[Theorem 3]{Sten68} and
\cite[Theorem 2.4]{EG} in $\CA$ regardless of whether the
Lazard-Govorov Theorem holds in $\CA$ or not (see also
\cite{Em16}). To this end, we need to prove that for any object
$\CF$ in $\CA$, $\CF^+= {\CH}om_{\CA}(\CF,\CJ)$ is a pure injective
object where $\CJ$ is an injective cogenerator for $\CA$. This
approach is very beneficial. Because we can generalize it to the
category of complexes in $\CA$ and deduce the same results for
complexes.
\vspace{.5cm}
\textbf{Setup:} In this work, all categories are symmetric monoidal
closed unless otherwise specified.
\vspace{.5cm}
\section{Purity and flatness in $\CA$}
This section is devoted to the relation between flatness and purity
in $\CA$. A conflation
$$\xymatrix@C-0.9pc{
\CL:\CG'\ar[r]&\CG\ar[r]&\CG''}$$ in $\CA$ is said to be
\textit{pure} if for any object $\CG$ in $\CA$, $\CL\otimes\CG$ is
also a conflation. An object $\CJ$ in $\CA$ is called
\textit{injective} if for any conflation $\CL$ in $\CA$,
$\Hom_{\CA}(\CL,\CJ)$ is a conflation in the ordinary abelian exact
structure of abelian groups. Moreover $\CJ$ is an injective
cogenerator if for any sequence $\xymatrix@C-0.9pc{
\CL:\CG'\ar[r]&\CG\ar[r]&\CG''}$ in $\CA$ where
$\Hom_{\CA}(\CL,\CJ)$ is a conflation of abelian groups then $\CL$
is a conflation in $\CA$. Let $\CJ$ be an injective cogenerator for
$\CA$ and
$(\textmd{-})^+:={\CH}om_{\CA}(\textmd{-},\CJ):\CA\lrt\CA$. We use
the contravariant functor $(\textmd{-})^+$ and prove the following
important interpretation of purity in $\CA$.
\begin{proposition}\label{1}
A conflation $\CL:\xymatrix@C-0.9pc{ \CG'\ar[r]&\CG\ar[r]&\CG''}$
in $\CA$ is pure if and only if $\mathcal{L}^+$ splits.
\end{proposition}
\begin{proof}
If $\CL$ is pure then, $(\CG')^+\otimes\CL$ is a conflation and so,
Hom$_{\CA}((\CG')^+\otimes\CL,\CJ)$ is a conflation of abelian
groups. By \eqref{0}, we have an isomorphism
$$ \mathrm{Hom}_{\CA}((\CG')^+\otimes\CL,\CJ)\cong
\mathrm{Hom}_{\CA}((\CG')^+,{\CH}om_{\CA}(\CL,\CJ))$$ of
conflations of abelian groups. This implies that $\CL^+$ splits.
Conversely, assume that $\CL^+$ splits. Then for any object $\CF$ in
$\CA$, we have a conflation $\mathrm{Hom}_{\CA}(\CF,
{\CH}om_{\CA}(\CL,\CJ))$ of abelian groups. By \eqref{0}, there is
an isomorphism
$$\mathrm{Hom}_{\CA}(\CF,
{\CH}om_{\CA}(\CL,\CJ))\cong\mathrm{Hom}_{\CA}(\CF\otimes\CL,
\CJ)$$of conflations of abelian groups. Since $\CJ$ is an injective
cogenerator then, $\CF\otimes\CL$ is a conflation and so we are
done.
\end{proof}
In the next lemma we show the existence of pure injective objects in
$\CA$. An object $\CE$ in $\CA$ is called \textit{pure}
\textit{injective} if it is injective with respect to pure
conflations.
\begin{lemma}\label{2001}
For any object $\CX$ in $\CA$, $\CX^+$ is pure injective.
\end{lemma}
\begin{proof}
Let $\CL$ be a pure conflation in $\CA$. By \eqref{0}, we have the
isomorphism,
$$\mathrm{Hom}_{\CA}(\CL,
\CX^+)\cong\mathrm{Hom}_{\CA}(\CL\otimes\CX, \CJ)$$ of conflations
of abelian groups. It follows that, $\CX^+$ is pure injective.
\end{proof}
An object $\CF$ in $\CA$ is called \textit{flat} if
-$\otimes\CF:\CA\lrt\CA$ preserves conflations. By the proof of
Lemma \ref{2001}, for a given object $\CX$ in $\CA$, $\CX^+$ is pure
injective. Now, the conditions for proving the main theorem of the
article are available.
\begin{theorem}\label{elh20}
The following conditions are equivalent.
\begin{itemize}
\item[(i)] $\CF$ is a flat object.
\item[(ii)] $\CF^+$ is injective.
\item[(iii)] Any conflation ending in $\CF$ is pure.
\end{itemize}
\end{theorem}
\begin{proof}
$(i)\Rightarrow(ii)$ Let $\CF$ be a flat object and $\CL$ be a
conflation in $\CA$. Then, $\CL\otimes\CF$ is also a conflation.
Apply $\mathrm{Hom}_{\CA}(\textmd{-},\CJ)$ and use the adjoint
property of -$\otimes\CF$ and ${\CH}om_{\CA}(\CF,\textmd{-})$ to
deduce the isomorphism
$$\mathrm{Hom}_{\CA}(\CL\otimes\CF,\CJ)\cong\mathrm{Hom}_{\CA}(\CL,\CF^+)$$
of conflations of abelian groups. This shows the injectivity of
$\CF^+$.
$(ii)\Rightarrow(i)$ Let $\CF^+$ be an injective object. For a
given conflation $\CL$ in $\CA$, $\mathrm{Hom}_{\CA}(\CL,\CF^+)$ is
a conflation of abelian groups and so, by the adjoint property of
-$\otimes\CF$ and ${\CH}om_{\CA}(\CF,\textmd{-})$,
$\mathrm{Hom}_{\CA}(\CL\otimes\CF,\CJ)$ is a conflation of abelian
groups. Since $\CJ$ is an injective cogenerator, then
$\CL\otimes\CF$ is a conflation. This shows the flatness of $\CF$.
$(i)\Rightarrow(iii)$. By $(i)\Leftrightarrow(ii)$, $\CF^+$ is an
injective object. So, for a given conflation
$$\xymatrix@C-0.7pc@R-0.9pc{\CL:\CK\ar[r]& \CG\ar[r]&
\CF}$$ in $\CA$, $\CL^+$ splits and hence by Proposition \ref{1},
$\CL$ is pure.
$(iii)\Rightarrow(i)$. By $(i)\Leftrightarrow(ii)$, it is enough
to show that $\CF^+$ is injective. Let
\begin{align}\label{mess0}
\xymatrix@C-0.7pc@R-0.9pc{\CF^+\ar[r]^f& \CG\ar[r]& \CK}
\end{align}
be a conflation in $\CA$. By the axioms of a closed symmetric
category, we have a morphism
$d_\CF:\CF\lrt{\CH}om_{\CA}(\CF^+,\CF\otimes\CF^+)$ and so by
\eqref{0}, there is a morphism $\lambda_\CF:\CF\lrt\CF^{++}$ in
$\CA$ (see \cite[pp 97-99]{KM71}). This implies that the composition
$\CF^+\lrt\CF^{+++}\lrt\CF^+$ is the identity $1_{\CF^+}$. By the
axiom of an exact category, the top row of the following pullback
diagram
\[\xymatrix@C-0.7pc@R-0.9pc{\CK^+\ar[r]\ar@{=}[d]&\CQ\ar[r]^t\ar[d]^g&\CF\ar[d]^i\\
\CK^+\ar[r]&\CG^+\ar[r]^{f^+}&\CF^{++}}\] is a pure conflation and
hence, it is split ($\CK^+$ is pure injective). Then, we have a
morphism $t':\CF\lrt \CQ$ such that $tt'=1_{\CF}$. If
$g_1=gt':\CF\lrt \CG^+$, then $f^+g_1=f^+gt'=itt'=i$. Consequently,
in the following commutative diagram
\[\xymatrix@C-0.7pc@R-0.9pc{\CF^+\ar[r]^f\ar[d]^{j}&\CG\ar[r]\ar[d]^k&\CK\ar[d]\\
\CF^{+++}\ar[r]^{f^{++}}&\CG^{++}\ar[r]&\CK^{++} }\] $g_1^+kf =
g_1^+f^{++}j = i^+j = 1_{\CF^+}$. It follows that $\CF^+$ is an
injective object.
\end{proof}
\subsection{ Purity and flatness in the category of complexes in $\CA$.}
Recall that a complex in $\CA$ is a cochain
$$\X:\cdots \rt \CX^{n-1} \st{\pa_\X^{n-1}}{\rt} \CX^n \st{\pa_\X^{n}}{\rt}
\CX^{n+1} \rt \cdots$$ in $\CA$ such that for any $n\in\Z$,
$\pa_\X^{n}\pa_\X^{n-1}=0$. The category of all complexes in $\CA$
is denoted by $\C(\CA)$. A complex $\X$ in $\CA$ is called
\textit{acyclic} if for any $n\in\Z$,
$$\xymatrix@C-0.7pc@R-0.9pc{\Ker\pa_\X^n\ar[r]& \CX^n\ar[r]&
\im\pa_\X^n}$$ is a conflation in $\CA$ and it is called
\textit{pure} \textit{acyclic} if for any object $\CG$ in $\CA$,
$\X\otimes\CG$ is acyclic. An acyclic complex
$$\F:\cdots \rt \CF^{n-1} \st{\pa_\F^{n-1}}{\rt} \CF^n \st{\pa_F^{n}}{\rt}
\CF^{n+1} \rt \cdots$$ in $\CA$ is said to be \textit{flat} if for
any $n\in\Z$, $\Ker\pa_\F^n$ is a flat object in $\CA$. The exact
structure on $\CA$ induces an exact structure on $\CA$ as follows. A
sequence $\xymatrix@C0.7pc@R0.9pc{ \X\ar[r]^f&\Y\ar[r]^g&\K}$ in
$\C(\CA)$ is a conflation if for any $n\in\Z$,
$$\xymatrix@C0.7pc@R0.9pc{
\X^n\ar[r]^{f^n}&\Y^n\ar[r]^{g^n}&\K^n}$$is a conflation in $\CA$.
Proposition \ref{1} will enable us to define a notion of purity in
$\C(\CA)$ and prove a generalization of \cite[Theorem 2.4]{EG} in
$\C(\CA)$.
\begin{sdefinition}\label{jimi}
A conflation $\mathbf{L}$ in $\C(\CA)$ is called pure if
$\mathbf{L}^+$ splits.
\end{sdefinition}
Notice that if $\CA$ is locally finitely presented Grothendieck
category with enough projective objects, then the following result
and \cite[Theorem 2.4]{EG} are equivalent (see also \cite{Sten68}).
\begin{stheorem}\label{110}
The following conditions are equivalent.
\begin{itemize}
\item[(i)] $\F$ is a flat complex in $\CA$.
\item[(ii)] $\F^+$ is an injective complex in $\CA$.
\item[(iii)] $\F$ is a pure acyclic complex of flat objects in $\CA$.
\item[(iv)] Any conflation ending in $\F$ is pure.
\end{itemize}
\end{stheorem}
\begin{proof}
The proof is straightforward.
\end{proof}
\subsection{Pure injective objects}
Pure injective objects are one of the most important generalizations
of injective objects which has a significance role in homological
algebra. For instance, they are essential tools in the Swan's
approach on Cup products, derived functors and Hochschild cohomology
({\cite{Sw99}).
Our motivation on this subsection is a question asked by Rosicky in
\cite[Question 1]{Ro09} (see also \cite[Theorem 2.1]{Sw99}). This
question concerning about the existence of enough $\lambda$-pure
injective objects in a locally $\lambda$-presentable additive
category ($\lambda$ is an infinite regular cardinal). It is known
that there is another notion of purity which is different from the
$\lambda$-purity. This purity is known as $\otimes$-purity and
defined in monoidal categories. We are interested to ask
\cite[Question 1]{Ro09} for this purity and find an answer for it.
In this subsection, we show that any symmetric monoidal closed
quasi-abelian category has enough $\otimes$-pure injective objects.
Assume that $\CA$ is a pre-abelian category, that is, an additive
category with kernels and cokernels (see \cite{RW77} and \cite{SW11}
for more details). We know that $\CA$ has a natural structure of an
exact category where conflations are short exact sequences. A
subobject $\CF$ of an object $\CG$ in $\CA$ is called \textit{pure}
if the canonical exact sequence
$$\xymatrix@C-0.7pc{0\ar[r]&\CF\ar[r]&\CG\ar[r]&\CG/\CF\ar[r]&0 }$$ is
pure in $\CA$.
\begin{stheorem}\label{2}
The category $\CA$ has enough pure injective objects.
\end{stheorem}
\begin{proof}
By the axioms of a symmetric monoidal closed category, there is a
morphism $d_\CF:\CF\lrt{\CH}om_{\CA}(\CF^+,\CF\otimes\CF^+)$ and so
by \eqref{0}, we have a morphism $\lambda_\CF:\CF\lrt\CF^{++}$
in $\CA$ (see \cite[pp 97-99]{KM71}). We show that $\lambda_\CF:\CF\lrt\CF^{++}$ is a pure monomorphism.
Let $\CK=\Ker\lambda_\CF$. Then we have the
following commutative diagram\[\xymatrix@C-0.7pc@R-0.9pc{0\ar[r]&\CK\ar[r]^i\ar[d]^{\lambda_\CK}&\CF\ar[d]^{\lambda_\CF}&\\
0\ar[r]&\CK^{++}\ar[r]^{i^{++}}&\CF^{++}}\]with exact rows. Since
$i^{++}$ is a monomorphism then $\lambda_{\CK}=0$. This implies that
$\CK=0$ and so $\lambda_\CF$ is a monomorphism. By Proposition
\ref{1}, it is enough to show that the epimorphism
$(\lambda_\CF)^+:(\CF^{++})^+\lrt\CF^+\lrt 0$ admits a section.
Since $(\CF^{++})^+\cong(\CF^+)^{++}$ then we have the
following commutative diagram\[\xymatrix{\CF^+\ar[r]^{\lambda_{\CF^+}}\ar[dr]_{\mathrm{id}_{\CF^+}}&\CF^{+++}\ar[d]^{(\lambda_\CF)^+}&\\
&\CF^{+}}\]
in $\CA$ (see \cite[pp.
100, diagram (1.3)]{KM71}). So, by Proposition \ref{1},
$\CF\lrt\CF^{++}$ is a pure monomorphism where $\CF^{++}$ is pure
injective by Lemma \ref{2001}.
\end{proof}
This theorem gives another proof for \cite[Theorem 2.1]{Sw99} and
\cite[Corollary 4.6, 4.8]{EEO16} and enable us to prove to prove the
existence of pure injective preenvelope in $\CA$.
\begin{example}
The category can be replaced by any symmetric monoidal closed
Grothenidieck category. For example, the category of modules over an
associative ring, the category of sheaves over an arbitrary
topological space and the category of quasi--coherent sheaves over
an arbitrary scheme (see \cite{H} for the algebraic geometry
background).
\end{example}
For more examples on non-abelian categories see the
\cite{Me18,Me12}.
\begin{stheorem}
The category $\C(\CA)$ has enough pure injective objects.
\end{stheorem}
\begin{proof}
The proof is similar to the proof of Theorem \ref{2}.
\end{proof}
| 134,582
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ERIC Number: EJ125824
Record Type: Journal
Publication Date: 1975
Pages: N/A
Abstractor: N/A
Reference Count: N/A
ISBN: N/A
ISSN: N/A
IQ Tests and the Black Culture
McNiel, Nathaniel D.
Phi Delta Kappan, 57, 3, 209-210, Nov 75. (Author/IRT)
Descriptors: Cultural Differences, Cultural Influences, Cultural Pluralism, Intelligence Tests, Test Bias
Publication Type: N/A
Education Level: N/A
Audience: N/A
Language: N/A
Sponsor: N/A
Authoring Institution: N/A
| 394,826
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Health benefits of drinking red wine/
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According to certain research, consumption of red wine also lowers the risk of lung, prostate, and breast cancer. However, you should take care of the amount of red wine you are drinking to stay safe from these ailments. To keep safe from lung cancer, one glass of red wine each day is a safer bargain. For men, to protect themselves from prostate cancer, you all should drink four to five glasses of red wine every week. Women should always consume a moderate amount of red wine. It keeps you safe from developing breast cancer. Too much wine or any other alcoholic drink might be dangerous. Hence, women should keep their consumption at a moderate level.
Minor, as well as major, colds can also be treated by drinking red wine. It helps you recover from a cold faster and maintains a normal body temperature that is ideal for your body. When a person is suffering from a particular disease or physical abnormality, inflammation or infection can be a common result during a few ailments. Drinking red wine aids in reducing that inflammation because the resveratrol present in the wine possesses anti-inflammatory properties. Red wine also contains another compound, which is known as saponins. It is responsible for lowering the cholesterol level in your body. High levels of cholesterol present in the body can cause blockages in the arteries and blood veins, thereby causing death. Consumption of red wine reduces that risk.
These are some health benefits of drinking red wine. You can easily get a good red wine at Keg n Cork, visit us in store and we'll help you choose a wine that you will enjoy.
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TITLE: Reconstructing unitary representation of Lie group from its generators
QUESTION [7 upvotes]: This question is about the motivation for Weinberg's approach in "The Quantum Theory of Fields" to obtain unitary representations of Lie groups out of its generators.
One is dealing with a Lie group $G$. We have coordinates $\{\theta^a\}$ on a neighborhood of the identity and $T(\theta)$ is the group element with coordinates $\theta$. Group multiplication is encoded in a function $f$ as $$T(\bar{\theta})T(\theta)=T(f(\bar{\theta},\theta)).$$
If $U(T(\theta))$ is a unitary representation on a Hilbert space, the generators of the representation are defined by the expansion $$U(T(\theta))=1+it_a\theta^a+O(\theta^2).$$
The problem then is:
If we know the $t_a$ and how they act on the Hilbert space, how can we find $U(T(\theta))$?
This is dealt with in Weinberg's Appendix 2B:
To prove this theorem, let us recall the method by which we construct the operators corresponding to symmetry transformations. As described in Section 2.2, we introduce a set of real variables $\theta^a$ to parameterize these transformations, in such a way that the transformation satisfy the composition rule (2.2.15): $$T(\bar{\theta})T(\theta)=T(f(\bar{\theta},\theta)).$$
We want to construct operators $U(T(\theta))\equiv U[\theta]$ that satisfy the corresponding condition $$U[\bar{\theta}]U[\theta]=U\left[f(\bar{\theta},\theta)\right].\tag{2.B.1}$$
To do this, we lay down arbitrary 'standard' paths $\Theta_\theta^a(s)$ in group parameter space, running from the origin to each point $\theta$, with $\Theta^a_\theta(0)=0$ and $\Theta_\theta^a(1)=\theta^a$, and define $U_\theta(s)$ along each such path by the differential equation $$\dfrac{d}{ds}U_\theta(s)=it_aU_\theta(s) h^a_{\phantom{a}b}(\Theta_\theta(s))\dfrac{d\Theta^b_\theta(s)}{ds}\tag{2.B.2}$$ with the initial condition $$U_\theta(0)=1,$$ where $$[h^{-1}]^a_{\phantom{a}b}(\theta)=\left[\dfrac{\partial f^a(\bar{\theta},\theta)}{\partial \bar{\theta}^b}\right]_{\bar{\theta}=0}.$$
The basic claim is that if we know the generators $t_a$ of the representation we can find the unitary representation $U[\theta]$ by defining $U_\theta(s)$ through Eq. (2.B.2) and identifying $U[\theta]=U_\theta(1)$.
What is the motivation for Weinberg's approach? What is the motivation to define $U_\theta(s)$ by (2.B.2)? How one would even think about defining this $U_\theta(s)$ through (2.B.2) in order to obtain $U[\theta]$ out of the generators?
REPLY [2 votes]: I believe I have found a coordinate-free answer to this question in which only in the end coordinates are invoked. I'm posting it here too, in case anyone also likes this approach. Corrections are highly appreciated.
There will be a slight shift in notation here. The unitary Lie Group representation will be denoted $\pi : G\to {\rm U}(\cal H)$. Its Lie algebra derived representation will be denoted $d\pi : \mathfrak{g}\to {\operatorname{End}}(\cal H)$.
Straight answer to the question: the motivation to use (2.B.2) to look for $\pi$ given $d\pi$ is that it is the "in-representation" version of the equation saying that along any curve in $G$ the Lie algebra element $X(s)=[R_{\gamma(s)}]_{\ast e}^{-1}\gamma'(s)$ generates a transformation that infinitesimally moves $\gamma(s)$ towards $\gamma(s+\delta s)$.
Given any curve $\gamma :[0,1]\to G$ starting off of the identity its tangent vector can always be seem as the image of something in the Lie algebra by right-translation: $$\gamma'(s)=[R_{\gamma(s)}]_{\ast e}X(s),\quad X(s)=[R_{\gamma(s)}]_{\ast e}^{-1}\gamma'(s).\tag{1}$$
On the contrary to specify any such $\gamma$ we can instead give such $X : [0,1]\to \mathfrak{g}$ and solve (1) with initial condition $\gamma(0)=e$.
The idea is to translate (1) into the representation and find a differential equation for $\pi(\gamma(s))$. We must recall how to define the derivative of $\pi(\gamma(s))$, which is a curve on ${\rm U}(\cal H)$ which has no obvious smooth structure turning it into a Lie group. For that we go with the idea of smooth vectors of a representation. We take ${\cal H}^\infty_\pi$ the space of all $\Psi\in \cal H$ for which $\Pi_\Psi(g)=\pi(g)\Psi$ is smooth. We then define the derivative of $\pi(\gamma(s))$ pointwise on ${\cal H}^\infty_\pi$, i.e.
$$\left[\dfrac{d}{ds}\pi(\gamma(s))\right]\Psi\equiv\dfrac{d}{ds}\left[\pi(\gamma(s))\Psi\right]=\dfrac{d}{ds}\Pi_\Psi(\gamma(s))\tag{2}$$
Since ${\cal H}^\infty_\pi$ may be shown to be dense in the Hilbert space, this defines the derivative of $\pi(\gamma(s))$ everywhere.
Now since $\Pi_\Psi : G\to {\cal H}^\infty_\pi$ we can safely do $$\dfrac{d}{ds}\Pi_\Psi(\gamma(s))=[\Pi_\Psi]_{\ast \gamma(s)}(\gamma'(s))=[\Pi_\Psi]_{\ast \gamma(s)}([R_{\gamma(s)}]_{\ast e} X(s))=(\Pi_\Psi\circ R_{\gamma(s)})_{\ast e}X(s)\tag{3}$$
Now notice that everything happens at fixed $s$, so we are left with the problem of evaluating $(\Pi_\Psi\circ R_g)_{\ast e}Z$ for $Z\in \mathfrak{g}$. To do so we take a short curve $\sigma : (-\epsilon,\epsilon)\to G$ with $\sigma(0)=e$ and $\sigma'(0)=Z$. The obvious such curve is $\sigma(\lambda)=\exp \lambda Z$. We then have $$(\Pi_\Psi\circ R_g)_{\ast e}Z =\dfrac{d}{d\lambda}\bigg|_{\lambda =0}\Pi_\Psi(R_g(\exp \lambda Z))=\dfrac{d}{d\lambda}\bigg|_{\lambda =0}\pi(\exp \lambda Z)\pi(g)\Psi=d\pi(Z)\pi(g)\Psi\tag{4}$$
where the last equality is the definition of the derived representation which may be invoked because if $\Psi$ is a smooth vector so is $\pi(g)\Psi$.
Going back to (3) and (2) this means that $$\left[\dfrac{d}{ds}\pi(\gamma(s))\right]\Psi=d\pi(X(s))\pi(\gamma(s))\Psi,\tag{5}$$
equality for all such smooth vectors then imply equality of the operators and we find $$\dfrac{d}{ds}\pi(\gamma(s))=d\pi(X(s))\pi(\gamma(s)).\tag{6}$$
Eq. (6) is just the "in representation" version of equation (1).
If one introduces coordinates $\theta^a$ centered at the identity it turns out that it is not hard to see that $d\pi(X(s))$ becomes $$d\pi(X(s))=it_ah^a_{\phantom{a}b}(\Theta(s))\dfrac{d\Theta^b(s)}{ds}\tag{7}.$$
Combination of (6) and (7) produces Weinberg's equation (2.B.2) for $U(s)=\pi(\gamma(s))$:
$$\dfrac{d}{ds}U(s)=it_aU(s)h^a_{\phantom{a}b}(\Theta(s))\dfrac{d\Theta^b(s)}{ds}\tag{8}$$
Now Weinberg has the $t_a$, hence he picks standard paths to define each $g\in G$ and uses (8) as a starting point to try defining $\pi(g) = U_g(1)$ where $U_g$ is defined from (8) using the standard path defining $g$.
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\begin{document}
\begin{abstract}
Let $K$ be an algebraically closed field of characteristic zero, and let $\calK := K(t)$ be the rational function field over $K$. For each $d \ge 2$, we consider the unicritical polynomial $f_d(z) := z^d + t \in \calK[z]$, and we ask the following question: If we fix $\alpha \in \calK$ and integers $M \ge 0$, $N \ge 1$, and $d \ge 2$, does there exist a place $\frakp \in \Spec K[t]$ such that, \emph{modulo $\frakp$}, the point $\alpha$ enters into an $N$-cycle after precisely $M$ steps under iteration by $f_d$? We answer this question completely, concluding that the answer is generally affirmative and explicitly giving all counterexamples. This extends previous work by the author in the case that $\alpha$ is a constant point.
\end{abstract}
\keywords{Preperiodic points; $abc$-theorem; unicritical polynomials}
\subjclass[2010]{Primary 37P05; Secondary 37F10, 14H05}
\maketitle
\section{Introduction}\label{sec:intro}
Let $F$ be a field, and let $\phi(z) \in F(z)$ be a rational function, thought of as a self-map of $\bbP^1(F)$. For an integer $n \ge 0$, we denote by $\phi^n$ the $n$-fold composition of $\phi$; that is, $\phi^0$ is the identity map, and $\phi^n = \phi \circ \phi^{n-1}$ for each $n \ge 1$. We say that $\alpha \in \bbP^1(F)$ is \textbf{periodic} for $\phi$ if there exists an integer $N \ge 1$ for which $\phi^N(\alpha) = \alpha$; the minimal such $N$ is called the \textbf{period} of $\alpha$. More generally, we say that $\alpha$ is \textbf{preperiodic} if there exist integers $M \ge 0$ and $N \ge 1$ such that $f^M(\alpha)$ has period $N$; if $M$ is minimal, we say that $(M,N)$ is the \textbf{preperiodic portrait} (or simply \textbf{portrait}) of $\alpha$ under $\phi$. If $M \ge 1$, then we say that $\alpha$ is \textbf{strictly preperiodic}. The \textbf{orbit} of $\alpha$ under $\phi$ is the set
\[
\calO_\phi(\alpha) := \{\phi^n(\alpha) : n \in \bbZ_{\ge 0}\}.
\]
Note that $\alpha$ is preperiodic for $\phi$ if and only if $\calO_\phi(\alpha)$ is finite. We say that a point is \textbf{wandering} if it is not preperiodic.
Let $\calM_F$ denote the set of places of $F$. (If $F$ is a function field, we require the places to be trivial on the constant subfield.) For a place $\frakp \in \calM_F$, let $k_\frakp$ denote the residue field at $\frakp$. Given a rational map $\phi$ and a place $\frakp$, one can consider the reduction of $\phi$ at $\frakp$: Write $\phi(z) = p(z)/q(z)$ with coprime $p,q \in F[z]$, normalized so that all coefficients are integral at $\frakp$, and at least one coefficient is a unit at $\frakp$. Then the \textbf{reduction} of $\phi$ at $\frakp$ is the map $\tilde{\phi}(z) \in k_\frakp(z)$ obtained by reducing the coefficients modulo $\frakp$. We say that $\phi$ has \textbf{good reduction} modulo $\frakp$ if $\deg \tilde{\phi} = \deg \phi$. If $\frakp$ is a place of good reduction for $\phi$, then we say that a point $\alpha \in \bbP^1(F)$ is \textbf{preperiodic for $\phi$ modulo $\frakp$} if the reduction $\tilde{\alpha} \in \bbP^1(k_\frakp)$ is preperiodic for the map $\tilde{\phi}$. We say that $\alpha$ has \textbf{(preperiodic) portrait $(M,N)$ for $\phi$ modulo $\frakp$} if $\tilde{\alpha}$ has preperiodic portrait $(M,N)$ for $\tilde{\phi}$.
If $\alpha$ is not preperiodic for $\phi$, it may still be true that $\alpha$ is preperiodic for $\phi$ modulo $\frakp$ at some place $\frakp$ of good reduction. For example, this will necessarily be true if $F$ has finite residue fields. We now consider the following more specific question regarding preperiodicity modulo places of $F$:
\begin{ques}\label{ques:main}
Let $\phi \in F(z)$. Fix $\alpha \in F$ and integers $M \ge 0$, $N \ge 1$. Does there exist a place $\frakp \in \calM_F$ of good reduction for $\phi$ such that $\alpha$ has preperiodic portrait $(M,N)$ for $\phi$ modulo $\frakp$?
\end{ques}
If the answer to Question~\ref{ques:main} is ``yes," we will say that $\alpha$ \textbf{realizes portrait $(M,N)$ for $\phi$}.
Question~\ref{ques:main} has been studied by multiple authors in the case that $F$ is a number field, dating back to related questions addressed by Bang \cite{bang:1886} and Zsigmondy \cite{zsigmondy:1892} in the late nineteenth century. Much more recently, Ingram and Silverman \cite{ingram/silverman:2009} conjectured that if $F$ is a number field and $\alpha \in F$ is a wandering point for $\phi$, then $\alpha$ realizes all but finitely many portraits for $\phi$. Faber and Granville \cite{faber/granville:2011} later gave counterexamples to this conjecture, noting that if $\phi(z) \in \bbQ(z)$ is totally ramified over all points of period $N$, then a given $\alpha \in \bbQ$ will \emph{fail} to realize portrait $(M,N)$ for all but finitely many $M$. Ghioca, Nguyen, and Tucker \cite{ghioca/nguyen/tucker:2015} subsequently pointed out that if $\phi$ is totally ramified over $\phi^M(\alpha)$ for some $M \ge 1$, then $\alpha$ cannot realize portrait $(M,N)$ for any $N \in \bbN$; their main result (\cite[Thm. 1.3]{ghioca/nguyen/tucker:2015}) is that these are the only obstructions to the analogue of the Ingram-Silverman conjecture in the setting where $F$ is the function field of a curve over an algebraically closed field of characteristic zero. They also claim that the appropriate modification of the Ingram-Silverman conjecture over number fields may be proven, under the assumption of the $abc$-conjecture, by adapting the methods of \cite{ghioca/nguyen/tucker:2015}.
The purpose of the present article is to explicitly describe all exceptions to the result of Ghioca-Nguyen-Tucker in a natural special case. For the remainder of the paper, $K$ will be an algebraically closed field of characteristic zero, and $\calK = K(t)$ will be the rational function field over $K$. Places of $\calK$ correspond naturally to points on $\bbP^1(K)$, and the residue field at each place is isomorphic to $K$. For a point $c \in \bbP^1(K)$, we denote by $\frakp_c \in \calM_\calK$ the place corresponding to $c$.
We take our rational maps to be the \emph{unicritical polynomials}
\[
f_d(z) := z^d + t \in \calK[z]
\]
of degree $d \ge 2$, which have good reduction away from $\frakp_\infty$. For each $c \in K$, we denote by $f_{d,c}$ the specialization of $f_d$ at $\frakp_c$; that is, $f_{d,c}(z) = z^d + c \in K[z]$.
Our main result fully answers Question~\ref{ques:main} with $F = \calK$ and $\phi = f_d$. For what follows, let
\[
\phi_1(z) := -\dfrac{t(z + 1)}{z - (t - 1)} \ \mbox{ and } \ \phi_2(z) := \dfrac{(t+1)(z - 1)}{z + t}.
\]
\begin{thm}\label{thm:main}
Let $K$ be an algebraically closed field of characteristic zero, let $\calK := K(t)$ be the rational function field over $K$, and let $(\alpha,M,N,d) \in \calK \times \bbZ^3$ with $M \ge 0$, $N \ge 1$, and $d \ge 2$. Then there exists a place $\frakp \in \calM_\calK \setminus\{\frakp_\infty\} = \Spec K[t]$ such that $\alpha$ has preperiodic portrait $(M,N)$ under $f_d$ modulo $\frakp$ if and only if $(\alpha,M,N,d)$ does not satisfy one of the following conditions:
\begin{itemize}
\item $M = 1$ and $\alpha = 0$;
\item $(M,N,d) = (0,2,2)$ and $\alpha = -1/2$;
\item $(M,N,d) = (1,1,2)$ and $\alpha \in \calO_{\phi_1}(0) \cup \calO_{\phi_1}(\infty)$;
\item $(M,N,d) = (1,2,2)$ and $\alpha \in \calO_{\phi_2}(0) \cup \calO_{\phi_2}(1/2) \cup \calO_{\phi_2}(\infty)$; or
\item $(M,N,d) = (2,2,2)$ and $\alpha = \pm 1$.
\end{itemize}
\end{thm}
\begin{rem}
The families of counterexamples in the $(1,1,2)$ and $(1,2,2)$ cases were discovered experimentally, and it was unclear whether some dynamical properties of the maps $\phi_1$ and $\phi_2$ could explain their appearance. Tom Tucker later pointed out that $\phi_1$ (resp., $\phi_2$) fixes each of the points in $\calKbar$ of portrait $(1,1)$ (resp., $(1,2)$) for $f_2$ and preserves vanishing at $\frakp_0$ (resp. $\frakp_{-1}$), the unique place at which the totally ramified point $0$ is fixed (resp., has period two) for $f_{2,c}$. Finally, we note that these exceptions have arbitrarily large height: for each $k \ge 0$, the points $\phi_1^k(0)$ and $\phi_1^k(\infty)$ have height $k$, as do the points $\phi_2^k(0)$, $\phi_2^k(1/2)$, and $\phi_2^k(\infty)$ --- see Propositions~\ref{prop:maind2N2} and \ref{prop:maind2N1}.
\end{rem}
As mentioned above, Ghioca, Nguyen, and Tucker consider Question~\ref{ques:main} for general rational maps and for function fields of arbitrary curves. Their main result \cite[Thm. 1.3]{ghioca/nguyen/tucker:2015}, when applied to the case of unicritical polynomials over $\calK$, says the following: For a fixed $d \ge 2$, if $(\alpha,M) \ne (0,1)$, and if $(M,N)$ avoids an effectively computable finite subset $\calZ(d) \subseteq \bbZ_{\ge 0} \times \bbN$, then every $\alpha \in \calK$ realizes portrait $(M,N)$ for $f_d$. Theorem~\ref{thm:main} implies that
\[
\calZ(d) =
\begin{cases}
\{(0,2),(1,1),(1,2),(2,2)\}, &\mbox{ if } d = 2;\\
\emptyset, &\mbox{ if } d \ge 3.
\end{cases}
\]
Moreover, for each $(M,N) \in \calZ(2)$, Theorem~\ref{thm:main} explicitly gives all $\alpha \in \calK$ which do not realize portrait $(M,N)$ for $f_2$.
Theorem~\ref{thm:main} may also be viewed as a natural extension of a previous result of the author:
\begin{thm}[{\cite[Thm. 1.3]{doyle:portraits}}]\label{thm:old}
Let $K$ be as before, and let $(\alpha,M,N,d) \in K \times \bbZ^3$ with $M \ge 0$, $N \ge 1$, and $d \ge 2$. There exists $c \in K$ for which $\alpha$ has portrait $(M,N)$ under $f_{d,c}$ if and only if
\[
(\alpha,M) \ne (0,1) \mbox{ and } (\alpha,M,N,d) \not \in \left\{\left(-\frac{1}{2},0,2,2\right), \left(\frac{1}{2},1,2,2\right), \left(\pm 1, 2,2,2\right)\right\}.
\]
\end{thm}
This is precisely the case of Theorem~\ref{thm:main} in which $\alpha$ lies in the constant subfield $K$. The proof of Theorem~\ref{thm:old} almost exclusively used the geometry of certain dynamical modular curves associated to the maps $f_d$, whereas the proof of Theorem~\ref{thm:main} requires Diophantine methods much like those used in \cite{ghioca/nguyen/tucker:2015}. In particular, the argument for the $d \ge 3$ case of Theorem~\ref{thm:main} provides a completely different proof of the $d \ge 3$ case of Theorem~\ref{thm:old} --- except for the case $M = 0$, for which we simply refer to Theorem~\ref{thm:old} for constant points. The same is true for $d = 2$, except for the cases where $M = 1$ and $N \le 3$, where the Diophantine methods are insufficient for constant points.
We now give a brief overview of the article. In \textsection \ref{sec:prelim}, we collect the main tools required for the proof of the main theorem. In \textsection \ref{sec:reduction}, we prove the $M = 0$ case of Theorem~\ref{thm:main}, and we then show that the problem for $M \ge 1$ may essentially be reduced to $M = 1$.
We prove the general case ($d \ge 3$) of Theorem~\ref{thm:main} in \textsection \ref{sec:deg3}. Focusing on the situation with $M = 1$, we apply the $abc$-theorem for function fields due to Mason and Stothers \cite{mason:1984, stothers:1981} to get a lower bound on the number of places at which $f(\alpha)$ and $f^{N+1}(\alpha)$ agree; we then show that this bound must be greater than the number of places at which either $\alpha$ is periodic or $f(\alpha)$ has period strictly less than $N$, so there must be some place at which $\alpha$ has portrait $(1,N)$. The arguments in this case are quite similar to those used in \cite{ghioca/nguyen/tucker:2015}, though we make modifications based on the specific nature of our maps $f_d$ in order to obtain sufficiently nice bounds.
The case $d = 2$ must be handled separately; this case is discussed in \textsection \ref{sec:deg2}. A technique similar to that used for $d \ge 3$ is used when $N \ge 4$. While this particular method is insufficient when $N = 3$, we are able to prove the result in this case by applying the $abc$-theorem together with properties of the period-three \emph{dynatomic polynomial} associated to the map $z^2 + t$. Unfortunately, the $abc$-theorem can no longer be applied when $N = 1$ and $N = 2$, so we handle these cases with completely different techniques, again appealing to the appropriate dynatomic polynomials. Theorem~\ref{thm:main} is then proven by combining Proposition~\ref{prop:main3} (for the case $d \ge 3$) with Propositions~\ref{prop:maind2N4}, \ref{prop:maind2N3}, \ref{prop:maind2N2}, and \ref{prop:maind2N1} (for $d = 2$ and $N \ge 4$, $N = 3$, $N = 2$, and $N = 1$, respectively).
\subsection*{Acknowledgments}
I would like to thank Tom Tucker for introducing me to this problem, as well as for a number of very helpful discussions over the course of writing this article.
\section{Preliminaries}\label{sec:prelim}
\subsection{Valuations and heights}
Let $\calL$ be a finite extension of $\calK$, which corresponds to a finite morphism of curves $\calX_\calL \to \calX_\calK \cong \bbP^1_K$. For a place $\frakp \in \calM_\calK$, we denote by $\calM_{\calL,\frakp}$ the set of places of $\calL$ that restrict to $\frakp$. Associated to each place $\frakq \in \calM_\calL$ is a valuation $v_\frakq$ and its corresponding absolute value $|\cdot|_\frakq = e^{-v_\frakq( \cdot )}$. When $\calL = \calK$, so that places correspond to points on $\bbP^1(K)$, we abuse notation and write $v_c$, $| \cdot |_c$, and $\calM_{\calL,c}$ for $v_{\frakp_c}$, $|\cdot|_{\frakp_c}$, and $\calM_{\calL,\frakp_c}$, respectively.
We normalize the valuations on $\calL$ so that $v_\frakq(\calL^\times) = \bbZ$; equivalently, if $\pi_\frakq$ is a uniformizer at $\frakq$, then $v(\pi_\frakq) = 1$. Thus, if $\frakp$ is the restriction of $\frakq$ to $\calK$, and if $\alpha \in \calK$, then $v_\frakq(\alpha) = e_{\frakq/\frakp} \cdot v_\frakp(\alpha)$, where $e_{\frakq/\frakp}$ is the ramification degree of $\frakq$ over $\frakp$. This normalization of the valuations also ensures that the product formula holds: For all $\alpha \in \calL^\times$, we have
\[
\prod_{\frakq \in \calM_\calL} |\alpha|_\frakq = 1,
\mbox{ or equivalently, }
\sum_{\frakq \in \calM_\calL} v_\frakq(\alpha) = 0.
\]
For each $\alpha \in \calL$, set
\[
h_\calL(\alpha) = -\sum_{\frakq \in \calM_\calL} \min\{v_\frakq(\alpha),0\}
= -\sum_{\substack{\frakq \in \calM_\calL\\v_\frakq(\alpha) < 0}} v_\frakq(\alpha).
\]
By the product formula, this is equivalent (when $\alpha \ne 0$) to defining
\[
h_\calL(\alpha) = \sum_{\frakq \in \calM_\calL} \max\{v_\frakq(\alpha),0\} = \sum_{\substack{\frakq \in \calM_\calL\\v_\frakq(\alpha) > 0}} v_\frakq(\alpha).
\]
If we consider $\alpha \in \calL$ as a rational map $\calX_\calL \to \bbP^1$, then $h_\calL(\alpha)$ is simply the degree of the map. If $\calL'$ is a finite extension of $\calL$, then $h_{\calL'}(\alpha) = [\calL':\calL]h_\calL(\alpha)$ for all $\alpha \in \calL$. This allows us to give a well-defined \textbf{(absolute) height function} on all of $\calKbar$, given by
\[
h(\alpha) := \frac{1}{[\calL : \calK]} \cdot h_\calL(\alpha)
\]
for any finite extension $\calL/\calK$ containing $\alpha$. Given a rational map $\phi(z) \in \calK(z)$ of degree $d \ge 2$, we also define the \textbf{canonical height} associated to $\phi$:
\[
\hhat_\phi(\alpha) := \lim_{n \to \infty} \frac{1}{d^n} h(\phi^n(\alpha)).
\]
That this is well-defined follows from the fact that $h(\phi(\alpha)) = dh(\alpha) + O(1)$, where the implied constant depends only on $\phi$; see \cite[\textsection 3.2]{silverman:2007}. Note that $\hhat_\phi(\phi(\alpha)) = d\hhat_\phi(\alpha)$ for all $\alpha \in \calKbar$.
We now record a basic height identity for elements of the orbit of a point $\alpha \in \calK$.
\begin{lem}\label{lem:hhat_ineq}
Let $\alpha \in \calK$, and let $d \ge 2$. Then for each $n \ge 1$, the poles of $f_d^n(\alpha)$ are precisely $\frakp_\infty$ and the poles of $\alpha$. Moreover,
\begin{enumerate}
\item if $\frakp$ is a finite pole of $\alpha$, then $v_\frakp(f_d^n(\alpha)) = d^nv_\frakp(\alpha)$;
\item $v_\infty(f_d^n(\alpha)) = d^n \cdot
\begin{cases}
v_\infty(\alpha), &\mbox{ if } v_\infty(\alpha) < 0;\\
-1/d, &\mbox{ if } v_\infty(\alpha) \ge 0.
\end{cases}$
\end{enumerate}
Therefore $h(f_d^n(\alpha)) = d^n \cdot
\begin{cases}
h(\alpha), &\mbox{ if } v_\infty(\alpha) < 0;\\
h(\alpha) + 1/d, &\mbox{ if } v_\infty(\alpha) \ge 0.
\end{cases}$
\end{lem}
\begin{proof}
Since $f_d^n(z)$ is a polynomial in $z$ and $t$, every pole of $f_d^n(\alpha)$ must be equal to $\frakp_\infty$ or a pole of $\alpha$. That the poles of $f_d^n(\alpha)$ are \emph{precisely} $\frakp_\infty$ and the poles of $\alpha$ then follows from parts (A) and (B), which we now prove by induction on $n$.
For $n = 1$, we have $f_d(\alpha) = \alpha^d + t$, so part (A) follows immediately from the ultrametric inequality. Furthermore, since $v_\infty(\alpha^d) \ne -1 = v_\infty(t)$, we have $v_\infty(f_d(\alpha)) = \min\{dv_\infty(\alpha),-1\}$.
Now suppose $n \ge 2$. First, let $\frakp$ be a finite pole of $\alpha$. By the induction hypothesis, $\frakp$ is a pole of $f_d^{n-1}(\alpha)$ of order $d^{n-1}v_\frakp(\alpha)$; applying the $n = 1$ case with $\alpha$ replaced by $f_d^{n-1}(\alpha)$ yields (A). We now consider $\frakp = \frakp_\infty$, in which case the induction hypothesis tells us that
\[
v_\infty(f_d^{n-1}(\alpha)) = d^{n-1} \cdot
\begin{cases}
v_\infty(\alpha), &\mbox{ if } v_\infty(\alpha) < 0\\
-1/d, &\mbox{ if } v_\infty(\alpha) \ge 0.
\end{cases}
\]
Since this quantity is necessarily negative, the $n = 1$ case implies $v_\infty(f_d^n(\alpha)) = d v_\infty(f_d^{n-1}(\alpha))$, which gives us (B).
Finally, we note that for all $n \ge 1$,
\begin{align*}
h(f_d^n(\alpha)) = -\sum_{v_\frakp(f_d^n(\alpha)) < 0} v_\frakp(f_d^n(\alpha))
&= -\sum_{\substack{v_\frakp(\alpha)<0\\\frakp \ne \frakp_\infty}} d^nv_\frakp(\alpha) - d^n \cdot
\begin{cases}
v_\infty(\alpha), &\mbox{ if } v_\infty(\alpha) < 0\\
-1/d, &\mbox{ if } v_\infty(\alpha) \ge 0
\end{cases}\\
&= d^n \cdot \left(-\sum_{v_\frakp(\alpha) < 0} v_\frakp(\alpha) +
\begin{cases}
0, &\mbox{ if } v_\infty(\alpha) < 0\\
1/d, &\mbox{ if } v_\infty(\alpha) \ge 0
\end{cases}
\right)\\
&= d^n \cdot
\begin{cases}
h(\alpha), &\mbox{ if } v_\infty(\alpha) < 0\\
h(\alpha) + 1/d, &\mbox{ if } v_\infty(\alpha) \ge 0.
\end{cases}
\end{align*}
\end{proof}
The following description of the canonical height for points in $\calK$ now follows immediately from the definition.
\begin{cor}\label{cor:height}
Let $\alpha \in \calK$ and $d \ge 2$. Then
\[
\hhat_{f_d}(\alpha) =
\begin{cases}
h(\alpha), &\mbox{ if } v_\infty(\alpha) < 0\\
h(\alpha) + 1/d, &\mbox{ if } v_\infty(\alpha) \ge 0.
\end{cases}
\]
Thus, for all $n \ge 1$, we have $h(f_d^n(\alpha)) = d^n \hhat_{f_d}(\alpha)$.
\end{cor}
\subsection{Dynatomic polynomials for $f_d$}\label{sec:dyn} Throughout the article, we will require certain properties of the dynatomic polynomials for the maps $f_d(z) = z^d + t$. Suppose $x,c \in K$ are such that $x$ has period $N$ for $f_{d,c}(z) = z^d + c$. Then $(x,c)$ is a solution to the equation $f_{d,c}^N(x) - x = 0$. However, this equation is also satisfied whenever $x$ has period dividing $N$ for $f_{d,c}$. We therefore define the $N$th \textbf{dynatomic polynomial} for $f_d$ to be the polynomial
\[
\Phi_N(z,t) := \prod_{n \mid N} (f_d^n(z) - z)^{\mu(N/n)} \in \bbZ[z,t],
\]
where $\mu$ is the M\"{o}bius function. (To ease notation, we omit the dependence on $d$.) The dynatomic polynomials give a natural factorization $f_d^N(z) - z = \prod_{n\mid N} \Phi_n(z,t)$. If $x$ has period $N$ for $f_{d,c}$, then $\Phi_N(x,c) = 0$, and for each $N \ge 1$ the converse is true for all but finitely many pairs $(x,c)$. That $\Phi_N(z,t)$ is indeed a polynomial is shown in \cite[Thm. 3.1]{morton/silverman:1995}; see also \cite[Thm. 4.5]{silverman:2007}. For each $N \ge 1$, we set
\[
D(N) := \deg_z \Phi_N(z,t) = \sum_{n \mid N} \mu(N/n)d^n.
\]
It is not difficult to verify that $\Phi_N(z,t)$ is monic in both $z$ and $t$, that $\deg_t \Phi_N(z,t) = D(N)/d$, and that
\[
\Phi_N(z,t) = z^{D(N)} + \mbox{(terms of lower total degree)}.
\]
In particular, this implies that if $\frakp \in \calM_\calK$ is a pole of $\alpha \in \calK$, or if $\frakp = \frakp_\infty$, then
\begin{equation}\label{eq:poles}
v_\frakp(\Phi_N(\alpha,t)) = \min\left\{D(N)v_\frakp(\alpha), \frac{D(N)}{d}v_\frakp(t)\right\} < 0.
\end{equation}
\begin{lem}\label{lem:htPhiN}
Let $\alpha \in \calK$ and $N \ge 1$. Then $v_\infty(\Phi_N(\alpha,t)) < 0$ and
\[
h(\Phi_N(\alpha,t)) = D(N) \cdot \hhat_{f_d}(\alpha).
\]
In particular, $\Phi_N(\alpha,t)$ vanishes at precisely $D(N) \cdot \hhat_{f_d}(\alpha)$ \emph{finite} places, counted with multiplicity.
\end{lem}
\begin{proof}
Since $\Phi_N(z,t)$ is a polynomial in $z$ and $t$, if $\frakp$ is a pole of $\Phi_N(\alpha,t)$, then $\frakp = \frakp_\infty$ or $\frakp$ is a pole of $\alpha$.
It then follows from \eqref{eq:poles} that the poles of $\Phi_N(\alpha,t)$ are \emph{precisely} $\frakp_\infty$ and the poles of $\alpha$. Therefore
\begin{align*}
h(\Phi_N(\alpha,t))
&= -\sum_{\substack{v_\frakp(\alpha) < 0\\\text{or } \frakp = \frakp_\infty}} \min\left\{D(N)v_\frakp(\alpha), \frac{D(N)}{d}v_\frakp(t)\right\}\\
&= -\sum_{\substack{v_\frakp(\alpha) < 0\\\frakp \ne \frakp_\infty}} D(N)v_\frakp(\alpha) -
\begin{cases}
D(N)v_\infty(\alpha), &\mbox{ if } v_\infty(\alpha) < 0\\
-D(N)/d, &\mbox{ if } v_\infty(\alpha) \ge 0
\end{cases}\\
&= D(N) \cdot
\begin{cases}
h(\alpha), &\mbox{ if } v_\infty(\alpha) < 0\\
h(\alpha) + 1/d, &\mbox{ if } v_\infty(\alpha) \ge 0
\end{cases}\\
&= D(N) \cdot \hhat_{f_d}(\alpha).
\end{align*}
\end{proof}
Finally, we record the following geometric result:
\begin{thm}\label{thm:Y1(N)}
For each integer $N \ge 1$ and $d \ge 2$, the affine plane curve $\{\Phi_N(z,t) = 0\}$ is smooth and irreducible over $K$.
\end{thm}
Theorem~\ref{thm:Y1(N)} was originally proven in the $d = 2$ case by Douady and Hubbard (smoothness; \cite[\textsection XIV]{douady/hubbard:1985}) and Bousch (irreducibility; \cite[Thm. 1 (\textsection 3)]{bousch:1992}, with a subsequent proof by Buff and Lei \cite[Thm. 3.1]{buff/lei:2014}. For $d \ge 2$, irreducibility was proven by Lau and Schleicher \cite[Thm. 4.1]{lau/schleicher:1994} using analytic methods and by Morton \cite[Cor. 2]{morton:1996} using algebraic methods, while both irreducibility and smoothness were later proven by Gao and Ou \cite[Thms. 1.1, 1.2]{gao/ou:2014} using the methods of Buff-Lei. The theorem was originally proven over $\bbC$, but the Lefschetz principle allows us to extend the result to arbitrary fields of characteristic zero.
\subsection{The $abc$-theorem for function fields}
Our main tool for proving the general case of Theorem~\ref{thm:main} is the $abc$-theorem for function fields due to Mason and Stothers \cite{mason:1984, stothers:1981}; see also \cite{silverman:1984} and \cite[Thm. F.3.6]{hindry/silverman:2000}.
\begin{thm}\label{thm:abc}
Let $\calL/\calK$ be a finite extension, and let $g_\calL$ be the genus of $\calL$. Let $u \in \calL \setminus K$, and define $S \subset \calM_\calL$ to be the set of places $\frakq$ for which $v_\frakq(u) \ne 0$ or $v_\frakq(1-u) \ne 0$. Then
\[
h_\calL(u) \le 2g_\calL - 2 + |S|.
\]
\end{thm}
\section{An elementary reduction}\label{sec:reduction}
For the majority of this article, we focus on the case of Theorem~\ref{thm:main} in which $M$ is equal to 1. In this section, we justify this approach: First, we prove the theorem when $M = 0$, and then we show how the $M \ge 1$ case may essentially be reduced to $M = 1$.
\subsection{Periodic points}
In order to have $\alpha$ not realize portrait $(0,N)$ for $f_d$, it must be the case that whenever $\Phi_N(\alpha,t)$ vanishes, so too does $\Phi_n(\alpha,t)$ for some proper divisor $n$ of $N$.
\begin{lem}\label{lem:bifurcation}
Fix integers $N \ge 1$ and $d \ge 2$.
\begin{enumerate}
\item Let $x,c \in K$, and suppose $\Phi_N(x,c) = \Phi_n(x,c) = 0$ for some proper divisor $n$ of $N$. Then
\[
\left. \frac{\partial \Phi_N(z,t)}{\partial z}\right|_{(x,c)} = 0.
\]
\item There are strictly fewer than $D(N)$ elements $c \in K$ for which there exists $x \in K$ with $\Phi_N(x,c) = \Phi_n(x,c) = 0$ for some proper divisor $n$ of $N$.
\end{enumerate}
\end{lem}
\begin{proof}
For part (A), see \cite[Thm. 2.4]{morton/patel:1994}; for part (B), see \cite[Cor. 3.3]{morton/vivaldi:1995}.
\end{proof}
We may now prove the $M = 0$ case of Theorem~\ref{thm:main}.
\begin{prop}\label{prop:periodic}
Let $\alpha \in \calK$, and let $N \ge 1$ and $d \ge 2$ be integers. Then $\alpha$ realizes portrait $(0,N)$ for $f_d$ if and only if $(\alpha,N,d) \ne (-1/2,2,2)$.
\end{prop}
\begin{proof}
For $\alpha \in K$, the result follows from Theorem~\ref{thm:old}, so we assume that $\alpha \in \calK \setminus K$. In this case, we have $\hhat_{f_d}(\alpha) \ge h(\alpha) \ge 1$, so it follows from Lemma~\ref{lem:htPhiN} that the number of places, counted with multiplicity, at which $\Phi_N(\alpha,t)$ vanishes is at least $D(N)$. Now suppose $\frakp_c$ is a place at which $\Phi_N(\alpha,t)$ vanishes, but $\alpha$ has period $n < N$ modulo $\frakp_c$. By Lemma~\ref{lem:bifurcation}(B), there are fewer than $D(N)$ such places, so to prove the proposition it suffices to show that $\Phi_N(\alpha,t)$ vanishes to order one at each such place. By Lemma~\ref{lem:bifurcation}(A), we have
\[
\left. \frac{\partial \Phi_N(z,t)}{\partial z} \right|_{(\alpha(c),c)} = 0.
\]
Here we write $\alpha(c)$ for the reduction of $\alpha$ modulo $\frakp_c$, since this is the image of $c$ under $\alpha$ if we consider $\alpha$ as a rational map. Since the affine curve $\{\Phi_N(z,t) = 0\}$ is smooth, we must also have
\[
\left. \frac{\partial \Phi_N(z,t)}{\partial t} \right|_{(\alpha(c),c)} \ne 0.
\]
This implies that
\[
\left. \frac{\partial \Phi_N(\alpha,t)}{\partial t} \right|_{t = c}
= \left. \frac{\partial \Phi_N(z,t)}{\partial z}\right|_{(\alpha(c),c)} \cdot \alpha'(c) + \left. \frac{\partial \Phi_N(z,t)}{\partial t} \right|_{(\alpha(c),c)} \ne 0,
\]
so $t = c$ is a simple root of $\Phi_N(\alpha,t)$; i.e., $\Phi_N(\alpha,t)$ vanishes to order one at $\frakp_c$, completing the proof.
\end{proof}
\subsection{Strictly preperiodic points} As mentioned previously, we will generally restrict our attention to the case $M = 1$. We now justify this approach.
\begin{lem}\label{lem:reduction}
Fix $\alpha \in \calK$ and integers $M \ge 2$, $N \ge 1$, and $d \ge 2$. Then $\alpha$ realizes portrait $(M,N)$ for $f_d$ if and only if $f_d^{M-1}(\alpha)$ realizes portrait $(1,N)$ for $f_d$.
\end{lem}
\begin{proof}
We simply note that both statements are equivalent to the statement that, at some place $\frakp \in \calM_\calK \setminus \{\frakp_\infty\}$, $f_d^M(\alpha)$ reduces to a point of period $M$ while $f_d^{M-1}(\alpha)$ does not.
\end{proof}
We also record a useful characterization of points in $K$ of portrait $(1,N)$.
\begin{lem}\label{lem:(1,N)}
Fix $x,c \in K$, $d \ge 2$, and $N \ge 1$. Then $x$ has portrait $(1,N)$ for $f_{d,c}$ if and only if $x \ne 0$ and $\zeta x$ has period $N$ for $f_{d,c}$ for some $d$th root of unity $\zeta \ne 1$.
\end{lem}
\begin{proof}
We first note that two points $x$ and $y$ have the same image under $f_{d,c}$ if and only if $y = \zeta x$ for some $d$th root of unity $\zeta$.
Suppose $x$ has portrait $(1,N)$ for $f_{d,c}$. Then $x$ is not periodic, but $f_{d,c}(x)$ has period $N$ and therefore has exactly one preimage $y$ with period $N$. We can write $y = \zeta x$ for some $d$th root of unity $\zeta$, and since $x \ne y = \zeta x$, we must have $\zeta \ne 1$ and $x \ne 0$.
Now suppose $x \ne 0$ and $\zeta x$ has period $N$ for $f_{d,c}$ for some $d$th root of unity $\zeta \ne 1$. Since $\zeta x$ has period $N$, so must $f_{d,c}(\zeta x) = f_{d,c}(x)$. However, since $x \ne 0$ and $\zeta \ne 1$, we have $x \ne \zeta x$. Thus $x$ is not periodic, and therefore $x$ has portrait $(1,N)$.
\end{proof}
\begin{cor}\label{cor:zero}
Fix integers $d \ge 2$ and $N \ge 1$. Then $0$ does not realize portrait $(1,N)$ for $f_d$.
\end{cor}
\section{The degree $d \ge 3$ case}\label{sec:deg3}
In this section, we show that if $d \ge 3$ and $\alpha \ne 0$, then $\alpha$ realizes portrait $(1,N)$ for every $N \ge 1$. We then use this to prove the $d \ge 3$ case of Theorem~\ref{thm:main}; see Proposition~\ref{prop:main3} below.
Fix $\alpha \in \calK^\times$, integers $d \ge 3$ and $N \ge 1$, and a primitive $d$th root of unity $\zeta$. Define the polynomial
\[
\sigma(z) := \frac{1}{\zeta - 1} \left( \frac{1}{\zeta\alpha} z - 1 \right) \in \calK[z],
\]
which maps $\zeta\alpha$, $\zeta^2\alpha$, and $\infty$ to 0, 1, and $\infty$, respectively. Set $\gamma := f_d^N(\alpha)$, and define
\begin{align*}
\calA &:= \{\frakp \in \calM_\calK : v_\frakp(\sigma(\gamma)) \ne 0 \mbox{ or } v_\frakp(\sigma(\gamma) - 1) > 0\};\\
\calB &:= \{\frakp_\infty\} \cup \{\frakp \in \calM_\calK : v_\frakp(\alpha) \ne 0 \mbox{ or } v_\frakp(\Phi_n(\zeta^k\alpha)) > 0 \mbox{ for some } n < N \mbox{ and } k \in \{1,2\}\}.
\end{align*}
\begin{lem}\label{lem:portrait3}
If $\frakp \in \calA \setminus \calB$, then $\alpha$ has portrait $(1,N)$ for $f_d$ modulo $\frakp$.
\end{lem}
\begin{proof}
Let $\frakp \in \calA \setminus \calB$. Since $v_\frakp(\alpha) = 0$, the map $\sigma$ has good reduction --- hence remains invertible --- modulo $\frakp$. Since $\frakp \in \calA$, $\sigma(\gamma)$ reduces to 0, 1, or $\infty$ modulo $\frakp$, which implies that $\gamma$ reduces to $\zeta\alpha$, $\zeta^2\alpha$, or $\infty$ modulo $\frakp$. Since the only poles of $\gamma = f_d^N(\alpha)$ are $\frakp_\infty$ and the poles of $\alpha$, and since such places lie in $\calB$, it must be the case that $f_d^N(\alpha) \equiv \zeta^k\alpha \pmod \frakp$ for some $k \in \{1,2\}$.
Since $\zeta^k\alpha \equiv f_d^N(\alpha) = f_d^N(\zeta^k\alpha)$ (mod $\frakp$) and $\frakp$ is not a pole of $\alpha$, $\zeta^k\alpha$ reduces to a finite point of period dividing $N$ for $f_d$ modulo $\frakp$, and the period must equal $N$ since $\Phi_n(\zeta^k\alpha) \not \equiv 0 \pmod \frakp$ for all $n < N$. Finally, since $\alpha \not \equiv 0 \pmod \frakp$, $\alpha$ has portrait $(1,N)$ modulo $\frakp$ by Lemma~\ref{lem:(1,N)}.
\end{proof}
\begin{lem}\label{lem:nonempty3}
The set $\calA \setminus \calB$ is nonempty.
\end{lem}
\begin{proof}
We first get an upper bound for $|\calA|$. In order to apply Theorem~\ref{thm:abc} with $u = \sigma(\gamma)$, we first verify that $\sigma(\gamma) \not\in K$. Indeed, suppose $\sigma(\gamma) = \lambda \in K$. Then
\[ f_d^N(\alpha) = \gamma = \sigma^{-1}(\lambda) = \zeta((\zeta - 1)\lambda + 1)\cdot \alpha, \]
so $f_d^N(\alpha)$ is a constant multiple of $\alpha$. However, this implies that $h(f_d^N(\alpha)) \le h(\alpha)$, contradicting Lemma~\ref{lem:hhat_ineq}. Therefore $\sigma(\gamma) \not \in K$, so we may apply Theorem~\ref{thm:abc} to get
\[
|\calA| \ge h(\sigma(\gamma)) + 2.
\]
Since $h(\sigma(\gamma)) = h(\gamma/\alpha) \ge h(\gamma) - h(\alpha)$ and $h(\gamma) = h(f_d^N(\alpha)) = d^N \hhat_{f_d}(\alpha)$, we have
\[
|\calA| \ge h(\gamma) - h(\alpha) + 2 = d^N \hhat_{f_d}(\alpha) - h(\alpha) + 2 \ge (d^N - 1)\hhat_{f_d}(\alpha) + 2.
\]
On the other hand, it is straightforward to verify that
\begin{align*}
|\calB| \le 1 + 2h(\alpha) + \sum_{k=1}^2 \sum_{\substack{n \mid N\\n < N}} h(\Phi_n(\zeta^k\alpha))
&= 1 + 2h(\alpha) + \sum_{k=1}^2 \sum_{\substack{n \mid N\\n < N}} \hhat_{f_d}(\zeta^k\alpha)D(n)\\
&= 1 + 2h(\alpha) + 2\hhat_{f_d}(\alpha)\sum_{\substack{n \mid N\\n < N}} D(n)\\
&\le 1 + \hhat_{f_d}(\alpha)\left(2 + 2\sum_{\substack{n \mid N\\n < N}} D(n)\right).
\end{align*}
Combining these bounds on $|\calA|$ and $|\calB|$, we find that $|\calA \setminus \calB| \ge \kappa \hhat_{f_d}(\alpha) + 1$, where
\[
\kappa := d^N - 3 - 2\sum_{\substack{n \mid N\\n < N}} D(n).
\]
To show that $\calA \setminus \calB$ is nonempty, it suffices to show that $\kappa \ge 0$, since $\hhat_{f_d}(\alpha) > 0$ for all $\alpha \in \calK$. Now observe that
\[
\sum_{\substack{n \mid N\\n < N}} D(n) \le \sum_{\substack{n \mid N\\n < N}} d^n \le \sum_{n=1}^{\lfloor N/2 \rfloor} d^n \le \frac{d}{d-1}(d^{N/2} - 1),
\]
and therefore
\begin{align*}
\kappa
\ge d^N - 3 - \frac{2d}{d-1}(d^{N/2} - 1)
&\ge d^N - 3 - d(d^{N/2} - 1)\\
&= d^{N/2 + 1}\left(d^{N/2 - 1} - 1\right) + d - 3.
\end{align*}
This expression is nonnegative for all $d \ge 3$ and $N \ge 2$; on the other hand, when $N = 1$, we have $\kappa = d - 3 \ge 0$. In either case, we conclude that $\calA \setminus \calB$ is nonempty.
\end{proof}
We may now prove Theorem~\ref{thm:main} for the general case $d \ge 3$.
\begin{prop}\label{prop:main3}
Let $(\alpha,M,N,d) \in \calK \times \bbZ^3$ with $M \ge 0$, $N \ge 1$, and $d \ge 3$. Then $\alpha$ realizes portrait $(M,N)$ for $f_d$ if and only if $(\alpha,M) \ne (0,1)$.
\end{prop}
\begin{proof}
For $M = 0$, this follows from Proposition~\ref{prop:periodic}. Corollary~\ref{cor:zero} says that 0 does not realize portrait $(1,N)$ for any $N \ge 1$ and $d \ge 2$, so suppose $\alpha \ne 0$. Letting $\calA$ and $\calB$ be as above, the set $\calA \setminus \calB$ is nonempty, thus $\alpha$ realizes portrait $(1,N)$ for $f_d$, proving the statement for $M = 1$.
Finally, let $M \ge 2$. Since $0 \notin f_d(\calK)$, the point $f_d^{M-1}(\alpha)$ is nonzero, so $f_d^{M-1}(\alpha)$ realizes portrait $(1,N)$ for $f_d$. By Lemma~\ref{lem:reduction}, we conclude that $\alpha$ realizes portrait $(M,N)$ for $f_d$.
\end{proof}
\section{The degree $d = 2$ case}\label{sec:deg2}
We henceforth drop the subscript and write $f = f_2$ and $f_c = f_{2,c}$. To prove Theorem~\ref{thm:main} when $N \ge 4$, we proceed much like in \textsection \ref{sec:deg3}; however, we require different methods when $N \le 3$, so we consider these cases separately.
\subsection{$N \ge 4$}
The proof of Proposition~\ref{prop:main3} in the previous section relied on fixing two preimages of $f_d(\alpha)$ \emph{different from $\alpha$ itself}, then counting the number of places at which $f_d^N(\alpha)$ had the same reduction as one of those two preimages. When $d = 2$, however, there is only \emph{one} preimage of $f(\alpha)$ different from $\alpha$ (namely, $-\alpha$), so we require a minor modification of the technique from \textsection \ref{sec:deg3}.
Fix $\alpha \in \calK$ and $N \ge 1$. Let $\eta \in \bar{\calK}$ satisfy $\eta^2 + t = -\alpha$, set $\calL := \calK(\eta)$, and set $\delta := [\calL:\calK] \in \{1,2\}$. Define the polynomial
\[
\sigma(z) := -\frac{1}{2}\left(\frac{z}{\eta} - 1\right) \in \calK[z],
\]
which maps $\eta$, $-\eta$, and $\infty$ to 0, 1, and $\infty$, respectively. Set $\gamma := f^{N-1}(\alpha)$, and define
\begin{align*}
\calA &:= \{\frakq \in \calM_\calL : v_\frakq(\sigma(\gamma)) \ne 0 \mbox{ or } v_\frakq(\sigma(\gamma) - 1) > 0\};\\
\calB &:= \calM_{\calL,\infty} \cup \{\frakq \in \calM_\calL : v_\frakq(\eta) \ne 0, \ v_\frakq(\alpha) > 0, \mbox{ or } v_\frakq(\Phi_n(-\alpha)) > 0 \mbox{ for some } n < N\}.
\end{align*}
By a proof similar to that of Lemma~\ref{lem:portrait3}, if $\frakq \in \calA \setminus \calB$, then $\alpha$ has portrait $(1,N)$ for $f$ modulo $\frakq$. We now show that there exists at least one such place.
\begin{lem}\label{lem:nonempty2}
The set $\calA \setminus \calB$ is nonempty.
\end{lem}
\begin{proof}
We first get a lower bound for $|\calA|$. By an argument similar to the beginning of the proof of Lemma~\ref{lem:nonempty3}, we have $\sigma(\gamma) \not \in K$, so we apply Theorem~\ref{thm:abc} to get
\[
|\calA| \ge h_\calL(\sigma(\gamma)) - (2g_\calL - 2).
\]
We get a lower bound on $h_\calL(\sigma(\gamma))$ by noting that $h_\calL(\sigma(\gamma)) = h_\calL(\gamma/\eta) \ge h_\calL(\gamma) - h_\calL(\eta)$; since also $h_\calL(\gamma) = \delta \cdot h(\gamma) = \delta \cdot 2^{N-1}\hhat_f(\alpha)$, we have $h_\calL(\sigma(\gamma)) \ge \delta(2^{N-1}\hhat_f(\alpha) - h(\eta))$.
We also require an upper bound on $2g_\calL - 2$. Let $\calR_{\calL/\calK}$ be the ramification divisor of the extension $\calL/\calK$. Since $\calL/\calK$ is generated by $\eta = \sqrt{-(\alpha + t)}$, the only places in $\calM_\calK$ over which $\calL$ may ramify are zeroes and poles of $\alpha + t$. Thus $\deg \calR_{\calL/\calK} \le (\delta - 1)\cdot 2h(\alpha + t) = 4(\delta - 1)h(\eta)$, so it follows from Riemann-Hurwitz that $2g_\calL - 2 \le -2\delta + 4(\delta - 1)h(\eta)$. Therefore
\begin{align*}
|\calA| &\ge \delta(2^{N-1}\hhat_f(\alpha) - h(\eta)) - (-2\delta + 4(\delta - 1)h(\eta)) = \delta \cdot \left( 2^{N-1}\hhat_f(\alpha) -\left(5 - \frac{4}{\delta}\right)h(\eta) + 2 \right).
\end{align*}
On the other hand, it is straightforward to verify that
\begin{align*}
|\calB| \le \#\calM_{\calL,\infty} + 2h_\calL(\eta) + h_\calL(\alpha) + \sum_{\substack{n \mid N\\n < N}} h_\calL(\Phi_n(-\alpha)) &\le \delta + 2\delta h(\eta) + \delta h(\alpha) + \sum_{\substack{n \mid N\\n < N}} \delta h(\Phi_n(-\alpha))\\
&= \delta \left(1 + 2h(\eta) + h(\alpha) + \hhat_f(\alpha)\sum_{\substack{n \mid N\\n < N}} D(n)\right).
\end{align*}
Combining these bounds on $|\calA|$ and $|\calB|$ yields
\[
|\calA \setminus \calB| \ge \delta\cdot\left[ \left(2^{N-1} - \sum_{\substack{n \mid N\\n < N}} D(n)\right)\hhat_f(\alpha) - h(\alpha) - \left(7 - \frac{4}{\delta}\right)h(\eta) + 1 \right].
\]
It remains to show that this bound is positive for all $N \ge 4$ and $\alpha \in \calK^\times$.
First, suppose $v_\infty(\alpha) < 0$. Then $\hhat_f(\alpha) = h(\alpha)$ and $h(\eta) = \frac{1}{2}h(\alpha + t) \ge \frac{1}{2}(h(\alpha) - 1)$. Therefore
\begin{align*}
|\calA \setminus \calB|
&\ge \delta\cdot\left[\left(2^{N-1} - \sum_{\substack{n \mid N\\n < N}} D(n)\right)h(\alpha) - h(\alpha) - \left(7 - \frac{4}{\delta}\right)\frac{1}{2}(h(\alpha) - 1) + 1\right] \\
&= \delta\cdot\left[\left(2^{N-1} - \frac{9}{2} + \frac{2}{\delta} - \sum_{\substack{n \mid N\\n < N}} D(n)\right) h(\alpha) + \left(\frac{9}{2} - \frac{2}{\delta}\right) \right]\\
&\ge \delta\cdot\left[\left(2^{N-1} - \frac{7}{2} - \sum_{\substack{n \mid N\\n < N}} D(n)\right) h(\alpha) + \frac{5}{2}\right],
\end{align*}
where the last inequality follows from the fact that $\delta \in \{1,2\}$. Since $h(\alpha) \ge 0$ for all $\alpha \in \calK$, it suffices to show that the quantity
\begin{equation}\label{eq:coeff}
2^{N-1} - \frac{7}{2} - \sum_{\substack{n \mid N\\n < N}} D(n)
\end{equation}
is non-negative for all $N \ge 4$. We bound the sum just as in the proof of Lemma~\ref{lem:nonempty3} to get
\[
2^{N-1} - \frac{7}{2} - \sum_{\substack{n \mid N\\n < N}} D(n)
\ge 2^{N-1} - \frac{7}{2} - 2(2^{N/2} - 1) = 2^{N/2 + 1}(2^{N/2 - 2} - 1) - \frac{3}{2}.
\]
The rightmost expression is positive for $N \ge 5$ but negative for $N = 4$; however, for $N = 4$ one can show directly that \eqref{eq:coeff} is positive since $D(1) = D(2) = 2$. (Note that \eqref{eq:coeff} is negative for $N \le 3$.) Thus $|\calA \setminus \calB|$ is positive for all $N \ge 4$.
Now suppose that $v_\infty(\alpha) \ge 0$, in which case we have $\hhat_f(\alpha) = h(\alpha) + \frac{1}{2}$ and $h(\eta) = \frac{1}{2}h(\alpha + t) = \frac{1}{2}(h(\alpha) + 1)$. By an estimate similar to the previous case, we find that
\[
|\calA \setminus \calB| \ge \delta\cdot\left[\left(2^{N-1} - \frac{7}{2} - \sum_{\substack{n \mid N\\n < N}} D(n)\right) h(\alpha) + \frac{1}{2}\left(2^{N-1} - \sum_{\substack{n \mid N\\n < N}} D(n) - 3 \right)\right].
\]
Since we have already shown that \eqref{eq:coeff} is positive for all $N \ge 4$, it follows that this expression is positive for all $N \ge 4$ as well.
In both cases, our lower bound on $|\calA \setminus \calB|$ is positive, so $\calA \setminus \calB$ is nonempty.
\end{proof}
The same argument as for Proposition~\ref{prop:main3} yields the $d = 2$, $N \ge 4$ case of Theorem~\ref{thm:main}.
\begin{prop}\label{prop:maind2N4}
Let $(\alpha,M,N) \in \calK \times \bbZ^2$ with $M \ge 0$ and $N \ge 4$. Then $\alpha$ realizes portrait $(M,N)$ for $f_2$ if and only if $(\alpha,M) \ne (0,1)$.
\end{prop}
\subsection{$N = 3$} Since the technique used for periods $N \ge 4$ gives a negative lower bound when $N \le 3$, we again require a different method. For $N = 3$, we consider the third dynatomic polynomial
\begin{align*}
\Phi_3(z,t) &= \frac{f^3(z) - z}{f(z) - z}\\
&= z^6 + z^5 + (3t + 1)z^4 + (2t + 1)z^3 + (3t^2 + 3t + 1)z^2 +
(t + 1)^2z + t^3 + 2t^2 + t + 1.
\end{align*}
The roots of $\Phi_3(z,t)$ in $\calKbar$ are the points of period 3 for $f$, which fall naturally into two 3-cycles. Let $\eta$ be one such root, and let $\calL := \calK(\eta)$; since $\Phi_3(z,t)$ is irreducible, we have $[\calL:\calK] = 6$. Set $\eta_1 := \eta$, $\eta_2 := f(\eta)$, and $\eta_3 := f^2(\eta)$, and note that each $\eta_i$ is a root of $\Phi_3(z,t)$ that generates $\calL/\calK$. Denote by $\calX_\calL$ the normalization of the projective closure of the affine curve $\{\Phi_3(z,t) = 0\}$, and note that the extension $\calL/\calK$ corresponds to the morphism $\calX_\calL \to \bbP^1$ mapping $(z,t) \mapsto t$. In his thesis, Bousch gave a general formula \cite[\textsection 3, Thm. 2]{bousch:1992} for the genera of the curves $\{\Phi_N(z,t) = 0\}$, and in this case we have $g_\calL = 0$. (For an explicit parametrization of the curve $\calX_\calL$, see \cite{walde/russo:1994}.)
By a result of Morton \cite[Prop. 10]{morton:1996} for more general dynatomic curves, $\calM_{\calL,\infty}$ consists of three places, each of which has ramification degree two over $\frakp_\infty$. To describe the ramification at finite places, Morton shows in \cite[p. 358]{morton:1992} that the discriminant of $\Phi_3(z,t) \in \calK[z]$ is given by
\begin{equation}\label{eq:disc}
\disc \Phi_3(z,t) = \Delta_{3,1}^2\Delta_{3,3}^3,
\end{equation}
where $\Delta_{3,1} := \Res_z(\Phi_3(z,t),\Phi_1(z,t)) = -(16t^2 + 4t + 7)$ and $\Delta_{3,3} := -(4t + 7)$. The roots $c$ of $\Delta_{3,1}$ correspond to maps $f_c$ for which one cycle of length three collapses to a fixed point, while the roots of $\Delta_{3,3}$ correspond to maps where the two 3-cycles collide to form a single 3-cycle.
Of particular relevance for us is the polynomial $\Delta_{3,1}$. Let $c_1,c_2 \in K$ be the two roots of $\Delta_{3,1}$. For each $i$, $\calM_{\calL,c_i}$ consists of four places: $\bar{\frakq_i}$, which has ramification degree three, and $\frakq_{i,1}$, $\frakq_{i,2}$, and $\frakq_{i,3}$, each of which is unramified --- see the proof of \cite[Prop. 9]{morton:1996}. The places $\bar{\frakq_1}$ and $\bar{\frakq_2}$ are precisely the places at which $\eta$ has period one; that is, the finite places at which $\eta_1$, $\eta_2$, and $\eta_3$ have the same reduction.
We briefly explain this geometrically: Let $c \in K$ be a root of $\Delta_{3,1}$. Instead of having two 3-cycles, $f_c$ has one 3-cycle $\{x_1,x_2,x_3\}$ and a fixed point $\bar{x}$ which satisfies $\Phi_3(\bar{x},c) = 0$. Thus the only points on $\calX_\calL$ that map to $c \in \bbP^1$ are $(\bar{x},c)$ and $(x_j,c)$ for $j \in \{1,2,3\}$. Since $(\bar{x},c)$ is fixed by the order three automorphism $(z,t) \mapsto (f(z),t)$, this point ramifies over $c$, while the three points $(x_j,c)$ are unramified.
\begin{lem}\label{lem:val2}
Let $c$ be a root of $\Delta_{3,1}$, and let $\bar{\frakq} \in \calM_\calL$ be the unique place ramified over $\frakp_c$. For each $1 \le i < j \le 3$,
\begin{enumerate}
\item $v_{\bar{\frakq}}(\eta_i) = 0$;
\item $v_{\bar{\frakq}}(\eta_i - \eta_j) = 1$;
\item $v_\frakq(\eta_i) = -1$ for each place $\frakq \in \calM_{\calL,\infty}$;
\item There exists a place $\frakq_{i,j} \in \calM_{\calL,\infty}$ such that, for $\frakq \in \calM_{\calL,\infty}$,
\[
v_\frakq(\eta_i - \eta_j) =
\begin{cases}
0, &\mbox{ if } \frakq = \frakq_{i,j};\\
-1, &\mbox{ otherwise}.
\end{cases}
\]
Moreover, $\frakq_{1,2}$, $\frakq_{1,3}$, and $\frakq_{2,3}$ are distinct.
\end{enumerate}
\end{lem}
\begin{proof}
Since $\Phi_3(z,t)$ is monic in $z$, the only poles of $\eta_1$, $\eta_2$, and $\eta_3$ must lie above $\frakp_\infty$. Therefore, to prove part (A), it suffices to show that none of the $\eta_i$ vanish at $\bar{\frakq}$. This follows by noting that $\Phi_3(0,c) \ne 0$.
For each $i,j$, the points $\eta_i$ and $\eta_j$ have the same reduction modulo $\bar{\frakq}$, so $v_{\bar{\frakq}}(\eta_i - \eta_j) \ge 1$. Now, we observe that the product
\[
\prod_{1 \le i < j \le 3} (\eta_i - \eta_j)^2
\]
divides $\disc \Phi_3(z,t)$, which then implies that
\[
\sum_{1 \le i < j \le 3} 2v_{\bar\frakq}(\eta_i - \eta_j) \le v_{\bar{\frakq}}(\disc \Phi_3(z,t)) = v_{\bar{\frakq}}(\Delta_{3,1}^2\Delta_{3,3}^3).
\]
This sum has three terms, and each term is at least 2, hence the sum is at least 6. Also, the polynomial $\Delta_{3,3}$ does not vanish at $\bar\frakq$; moreover, if we factor $\Delta_{3,1}$ into linear factors, only the factor $(t - c)$ vanishes at $\bar\frakq$. This implies that
\[
6 \le \sum_{1 \le i < j \le 3} 2v_{\bar{\frakq}}(\eta_i - \eta_j) \le 2v_{\bar\frakq}(t - c) = 2e_{\bar\frakq/\frakp_c} = 6.
\]
We therefore have equality throughout, so $v_{\bar\frakq}(\eta_i - \eta_j) = 1$ for each $1 \le i < j \le 3$, proving (B).
Now fix $\frakq \in \calM_{\calL,\infty}$ and $1 \le i \le 3$. Note that $v_\frakq(t) = -e_{\frakq/\frakp_\infty} = -2$. If $v_\frakq(\eta_i) < -1$, then an induction argument shows that $v_\frakq(f^n(\eta_i)) = 2^nv_\frakq(\eta_i)$ for each $n \in \bbN$; in particular, we have $v_\frakq(\eta_i) = v_\frakq(f^3(\eta_i)) = 8v_\frakq(\eta_i)$, a contradiction. If instead $v_\frakq(\eta_i) > -1$, then $v_\frakq(f(\eta_i)) = v_\frakq(t) < -1$, so we reduce to the previous case to get a contradiction. Therefore $v_\frakq(\eta_i) = 1$, proving (C).
The only finite zeroes of $\eta_i - \eta_j$ are the two simple zeroes at $\frakq_1$ and $\frakq_2$, so there must be at least two poles of $\eta_i - \eta_j$. By part (C), these must be simple poles lying above $\frakp_\infty$. Comparing the degrees of its zero and pole divisors, $\eta_i - \eta_j$ must have a simple pole at \emph{precisely} two places above $\frakp_\infty$; moreover, if we let $\frakq_{i,j}$ be the remaining infinite place, then $v_{\frakq_{i,j}}(\eta_i - \eta_j) = 0$.
Finally, suppose $(i,j) \ne (i',j')$ but $\frakq_{i,j} = \frakq_{i',j'}$. Reordering the indices if necessary, we may assume that $\frakq_{1,2} = \frakq_{1,3}$. Since $\eta_1 - \eta_2$ and $\eta_1 - \eta_3$ have exactly the same zeroes and poles, with exactly the same orders, it must be that $\lambda := (\eta_1 - \eta_2)/(\eta_1 - \eta_3)$ is constant. This implies that
\[
0 = (\eta_1 - \eta_2) - \lambda(\eta_1 - \eta_3) = \Big(\eta - (\eta^2 + t)\Big) - \lambda\Big(\eta - \big((\eta^2 + t)^2 + t\big)\Big)
\]
for some $\lambda \in K$, contradicting the fact that $\eta$ has degree 6 over $\calK$. We conclude that the places $\frakq_{1,2}$, $\frakq_{1,3}$, and $\frakq_{2,3}$ are distinct, completing the proof.
\end{proof}
\begin{rem}\label{rem:val2}
It follows from Lemma~\ref{lem:val2} and its proof that $h_\calL(\eta_i) = 3$ for each $1 \le i \le 3$, since the only poles of $\eta_i$ are simple poles at the three places lying above $\frakp_\infty$. Similarly, we have $h_\calL(\eta_i - \eta_j) = 2$ for each $1 \le i < j \le 3$.
\end{rem}
Now consider the affine rational map
\[
\sigma(z) := \frac{\eta_2 - \eta_3}{\eta_2 - \eta_1} \cdot \frac{z - \eta_1}{z - \eta_3},
\]
which is constructed to map $\eta_1$, $\eta_2$, and $\eta_3$ to 0, 1, and $\infty$ respectively. Note that
\[
\sigma^{-1}(z) = \frac{\eta_3(\eta_1 - \eta_2)z + \eta_1(\eta_2 - \eta_3)}{(\eta_1 - \eta_3)z + (\eta_2 - \eta_3)}.
\]
\begin{lem}\label{lem:ht_bound}
For all $x \in \calL$, $h_\calL(\sigma(x)) \ge h_\calL(x) - 4$.
\end{lem}
\begin{proof}
By considering $y = \sigma(x)$, it suffices to show that $h_\calL(\sigma^{-1}(y)) \le h_\calL(y) + 4$ for all $y \in \calL$. A standard height argument (see \cite[pp. 90--92]{silverman:2007}) shows that for any $y \in \calL$ we have
\[
h_\calL(\sigma^{-1}(y))
\le h_\calL(y) - \sum_{\frakq \in \calM_\calL} \min \{v_\frakq(\eta_3(\eta_1 - \eta_2)), v_\frakq(\eta_1(\eta_2 - \eta_3)), v_\frakq(\eta_1 - \eta_3), v_\frakq(\eta_2 - \eta_3) \},
\]
so it suffices to show that the quantity
\begin{equation}\label{eq:ht_est}
\sum_{\frakq \in \calM_\calL} \min \{v_\frakq(\eta_3) + v_\frakq(\eta_1 - \eta_2), v_\frakq(\eta_1) + v_\frakq(\eta_2 - \eta_3), v_\frakq(\eta_1 - \eta_3), v_\frakq(\eta_2 - \eta_3) \}
\end{equation}
is equal to $-4$. We now determine the places $\frakq$ at which the `min' term is nonzero.
If the minimum is positive at $\frakq$, then $\frakq \in \{\frakq_1,\frakq_2\}$. By Lemma~\ref{lem:val2}, at each such $\frakq$ and for each $1 \le i < j \le 3$, we have $v_\frakq(\eta_i - \eta_j) = 1 \mbox{ and } v_\frakq(\eta_i) = 0$. Thus the contribution to \eqref{eq:ht_est} at each such place is equal to $1$, so the combined contribution at both places is equal to $2$.
If the minimum is negative at $\frakq$, then $\frakq$ necessarily lies above $\frakp_\infty$. Again applying Lemma~\ref{lem:val2}, $v_\frakq(\eta_i) = -1$ for each $1 \le i \le 3$, and exactly one of $v_\frakq(\eta_1 - \eta_2)$, $v_\frakq(\eta_1 - \eta_3)$, and $v_\frakq(\eta_2 - \eta_3)$ is nonnegative. In particular, this means that $\min\{v_\frakq(\eta_1 - \eta_2), v_\frakq(\eta_2 - \eta_3)\} = -1$, so the combined contribution to \eqref{eq:ht_est} at the three infinite places is $3 \cdot (-2) = -6$. Since we have accounted for all nonzero terms in the sum, we conclude that the expression \eqref{eq:ht_est} is equal to $-4$, as claimed.
\end{proof}
Now let $\alpha \in \calK^\times$ be arbitrary. We will show that $\alpha$ realizes portrait $(1,3)$ for $f$. Define
\begin{align*}
\calA &:= \{\frakq \in \calM_\calL : v_\frakq(\sigma(-\alpha)) \ne 0 \mbox{ or } v_\frakq(\sigma(-\alpha) - 1) > 0\};\\
\calB &:= \calM_{\calL,\infty} \cup \{\bar{\frakq_1}, \bar{\frakq_2}\} \cup \{\frakq \in \calM_\calL : v_\frakq(\eta_i) > 0 \mbox{ for some } i = 1,2,3\}.
\end{align*}
\begin{lem}
If $\frakq \in \calA \setminus \calB$, then $\alpha$ has portrait $(1,3)$ for $f$ modulo $\frakq$.
\end{lem}
\begin{proof}
Let $\frakq \in \calA \setminus \calB$. Since $\frakq \not \in \calM_{\calL,\infty}$, $f$ has good reduction at $\frakq$. Since $\eta$ has period 3 for $f$, it follows that $\eta$ has period 1 or 3 for $f$ modulo $\frakq$. The only poles of $\eta$ are places at infinity, so $\eta \not \equiv \infty \pmod \frakq$; moreover, since $\frakq \not \in \{\bar{\frakq_1},\bar{\frakq_2}\}$ we have
\[
f(\eta) - \eta = \eta_2 - \eta_1 \not \equiv 0 \pmod \frakq,
\]
so $\eta$ cannot have period 1. Thus $\eta$ has period 3 for $f$ modulo $\frakq$.
The zeroes and poles of the coefficients of $\sigma$ all lie in $\calB$, so each coefficient is a unit in the residue field $k_\frakq$. Since also $\eta_1 \not \equiv \eta_3$ (mod $\frakq$), the reduction of $\sigma$ has degree one over $k_\frakq$; i.e., $\sigma$ remains invertible modulo $\frakq$. Thus, since $\sigma(-\alpha)$ reduces to 0, 1, or $\infty$ modulo $\frakq$, $-\alpha$ must reduce to $\eta_i$ for some $i \in \{1,2,3\}$. We have already shown that each $\eta_i$ has period 3 modulo $\frakq$, so the same is true for $-\alpha$. Finally, since $-\alpha \equiv \eta_i \not \equiv 0 \pmod \frakq$, $\alpha$ must reduce to a point of portrait $(1,3)$.
\end{proof}
\begin{lem}\label{lem:d2N3nonempty}
Suppose $h(\alpha) \ge 3$. Then $\calA \setminus \calB$ is nonempty.
\end{lem}
\begin{proof}
We have by Lemma~\ref{lem:ht_bound} that $h_\calL(\sigma(-\alpha)) \ge 14$, so $\sigma(-\alpha) \not \in K$. Applying Theorem~\ref{thm:abc}, it follows that the set $\calA$ has size at least $h_\calL(\sigma(-\alpha)) - (2g_\calL - 2) = h_\calL(\sigma(-\alpha)) + 2$. Again applying the bound in Lemma~\ref{lem:ht_bound} gives
\[
|\calA| \ge (h_\calL(\alpha) - 4) + 2 = 6h(\alpha) - 2.
\]
We also have $|\calB| \le |\calM_{\calL,\infty}| + 2 + \sum_{i=1}^3 h_\calL(\eta_i) = 14$, which implies that $|\calA \setminus \calB| \ge 6h(\alpha) -16$. Therefore $\calA \setminus \calB$ is nonempty when $h(\alpha) \ge 3$.
\end{proof}
We have just shown that if $h(\alpha) \ge 3$, then there exists a place $\frakq \in \calM_\calL$ for which $\alpha$ has portrait $(1,3)$ modulo $\frakq$; choosing $\frakp \in \calM_\calK$ below $\frakq$, the same holds modulo $\frakp$. It remains to prove this in the case $h(\alpha) \le 2$. The constant point case $h(\alpha) = 0$ is covered by Theorem~\ref{thm:old}, so we henceforth assume $h(\alpha) \in \{1,2\}$. To handle these remaining cases --- as well as the $N = 2$ and $N = 1$ cases in Section~\ref{sec:d=2,N=2} --- we use the following consequence of Lemma~\ref{lem:(1,N)}:
\begin{cor}\label{cor:fail}
Let $\alpha \in \calK$ and $N \in \bbN$. The following are equivalent:
\begin{enumerate}
\item The point $\alpha$ does not realize portrait $(1,N)$ for $f$.
\item For every place $\frakp \in \calM_\calK$ at which $\Phi_N(-\alpha,t)$ vanishes, either $\Phi_n(-\alpha,t)$ also vanishes at $\frakp$ for some proper divisor $n$ of $N$, or $\alpha$ also vanishes at $\frakp$.
\end{enumerate}
\end{cor}
\begin{rem}\label{rem:simple}
If $\Phi_N(\beta,t)$ and $\Phi_n(\beta,t)$ both vanish at $\frakp$ for some proper divisor $n$ of $N$, then it follows from the proof of Proposition~\ref{prop:periodic} that $v_\frakp(\Phi_N(\beta,t)) = 1$.
\end{rem}
If $\alpha$ does not realize portrait $(1,3)$ for $f$, then wherever $\Phi_3(-\alpha)$ vanishes, either $\alpha$ or $\Phi_1(-\alpha)$ must vanish as well. We will show that such behavior is impossible when $h(\alpha) \in \{1,2\}$, having already handled all other cases.
\begin{lem}\label{lem:d2N3}
Let $\beta \in \calK$, and suppose $h(\beta) = 1$ or $h(\beta) = 2$. There exists a place $\frakp \in \calM_\calK \setminus \{\frakp_\infty\}$ such that $\Phi_3(\beta,t)$ vanishes at $\frakp$ but $\Phi_1(\beta,t)$ and $\beta$ do not.
\end{lem}
\begin{proof}
Suppose to the contrary that there does not exist such a place. Then, if we fix a place $\frakp = \frakp_c$ for which $v_c(\Phi_3(\beta,t)) > 0$, we must also have $v_c(\Phi_1(\beta,t)) > 0$ or $v_c(\beta) > 0$.
If $v_c(\Phi_1(\beta,t)) > 0$, then it must be that $c \in \{c_1,c_2\}$, where $c_1$ and $c_2$ are as above Lemma~\ref{lem:val2}. Moreover, in this case the order of vanishing of $\Phi_3(\beta,t)$ at $\frakp_c$ must equal one by Remark~\ref{rem:simple}.
By Lemma~\ref{lem:htPhiN}, $\Phi_3(\beta,t)$ vanishes at precisely $6\hhat_f(\beta) \ge 6$ places, counted with multiplicity. Since $\Phi_3(\beta,t)$ can have at most simple roots at $c_1$ and $c_2$, there must be a place $\frakp_c$ at which both $\Phi_3(\beta,t)$ and $\beta$ vanish. In this case, we must have
\[
\Phi_3(0,c) = c^3 + 2c^2 + c + 1 = 0.
\]
Let $C_1$, $C_2$, and $C_3$ be the three roots of $\Phi_3(0,t)$. Since $h(\beta) \le 2$, $\beta$ can have at most two roots; reordering the roots if necessary, we assume that $\Phi_3(\beta,t)$ and $\beta$ both vanish at $C_1$ and possibly $C_2$.
We have put certain restrictions on the places at which $\Phi_3(\beta,t)$ may vanish as well as the order of vanishing at each such place. Let $\rho(\beta) \in K[t]$ denote the numerator of $\Phi_3(\beta,t)$; scaling if necessary, we assume that $\rho(\beta)$ is monic. Set $R := \deg \rho(\beta) = 6\hhat_f(\beta)$.
If $\Phi_3(\beta,t)$ vanishes at $\frakp_{C_1}$ but not $\frakp_{C_2}$, then $\beta = (t - C_1)p/q$ for some $p,q \in K[t]$ with $\deg p \le 1$ and $\deg q \le 2$, and
\begin{equation}\label{eq:nu1}
\rho(\beta) = (t - c_1)^{\epsilon_1}(t - c_2)^{\epsilon_2}(t - C_1)^{R - \epsilon_1 - \epsilon_2}
\end{equation}
for some $\epsilon_1,\epsilon_2 \in \{0,1\}$. If $\Phi_3(\beta,t)$ vanishes at both $\frakp_{C_1}$ and $\frakp_{C_2}$, then $\beta = (t - C_1)(t - C_2)/q$ for some $q \in K[t]$ with $\deg q \le 2$, and
\begin{equation}\label{eq:nu2}
\rho(\beta) = (t - c_1)^{\epsilon_1}(t - c_2)^{\epsilon_2}(t - C_1)^k(t - C_2)^{R - \epsilon_1 - \epsilon_2 - k}
\end{equation}
for some $\epsilon_1,\epsilon_2 \in \{0,1\}$ and $1 \le k \le R - \epsilon_1 - \epsilon_2 - 1$.
The idea is to write $p$ and $q$ as polynomials with indeterminate coefficients, then compare the coefficients of both sides of \eqref{eq:nu1} (resp., \eqref{eq:nu2}). This will determine an affine scheme over $K$; if this scheme is empty, or if the only points on the scheme yield a constant map $\beta$, then we will have completed the proof of the lemma.
We illustrate this computation in one case. Suppose that $\Phi_3(\beta,t)$ vanishes at $\frakp_{C_1}$ but not $\frakp_{C_2}$, and let us suppose that $\deg q = 2$, in which case $R = 15$. Write $\beta = (t - C_1)(p_1t + p_0)/(t^2 + q_1t + q_0)$. Then $\rho(\beta)$ is a polynomial in $t$ with coefficients in $K[p_0,p_1,q_0,q_1]$, and comparing the coefficients of $\rho(\beta)$ with the coefficients of $(t - c_1)^{\epsilon_1}(t - c_2)^{\epsilon_2}(t - C_1)^{15 - \epsilon_1 - \epsilon_2}$ for each pair $(\epsilon_1,\epsilon_2) \in \{0,1\}^2$ yields four different subschemes of $\bbA^4_K = \Spec K[p_0,p_1,q_0,q_1]$. A computation in Magma \cite{magma} verifies that each of these schemes is empty. The proof of the lemma is then completed by a number of similar computations; for the interested reader, the Magma code and output have been included as an ancillary file with this article's arXiv submission.
\end{proof}
Applying Lemma~\ref{lem:d2N3} with $\beta = -\alpha$ shows that if $h(\alpha) \in \{1,2\}$, then $\alpha$ realizes portrait $(1,3)$ for $f$; as mentioned above, we may now conclude that this holds for all $\alpha \in \calK^\times$. Finally, arguing as in the proof of Proposition~\ref{prop:main3}, we have the $d = 2$, $N = 3$ case of Theorem~\ref{thm:main}:
\begin{prop}\label{prop:maind2N3}
Let $\alpha \in \calK$, and let $M \ge 0$. Then $\alpha$ realizes portrait $(M,3)$ for $f_2$ if and only if $(\alpha,M) \ne (0,1)$.
\end{prop}
\subsection{$N \le 2$}\label{sec:d=2,N=2}
In this section, we prove Theorem~\ref{thm:main} in the $d = 2$, $N = 2$ case. The proof for $N = 1$ uses essentially the same technique, so we omit the proof in that case.
Consider the rational map
\[
\phi_2(z) := \frac{(t+1)(z - 1)}{z + t},
\]
with inverse
\[
\phi_2^{-1}(z) = -\frac{tz + (t + 1)}{z - (t + 1)}.
\]
It follows from Theorem~\ref{thm:old} that $0$, $1/2$, and $\infty$ (which is a fixed point for $f$) are the only points in $\bbP^1(K)$ that fail to realize portrait $(1,2)$ for $f$. We will show that the points in $\calK$ that fail to realize portrait $(1,2)$ are precisely the points in the orbits of $0$, $1/2$, and $\infty$ under $\phi_2$:
\begin{prop}\label{prop:(1,2)}
Let $\alpha \in \calK$. Then $\alpha$ does not realize portrait $(1,2)$ for $f$ if and only if
\[ \alpha \in \calO_{\phi_2}(0) \cup \calO_{\phi_2}(1/2) \cup \calO_{\phi_2}(\infty). \]
Moreover, for each $k \ge 0$ we have
\begin{align*}
h(\phi^k(0)) = h(\phi^k(1/2)) = h(\phi^k(\infty)) = k.
\end{align*}
\end{prop}
We begin by giving an alternative description of those points that do not realize portrait $(1,2)$:
\begin{lem}
Let $\alpha \in \calK$. Then $\alpha$ does not realize portrait $(1,2)$ for $f$ if and only if $\alpha = 1/2$ or $\alpha$ satisfies the following conditions:
\[\tag{$*$} \left\{
\begin{split}
&\mbox{$\alpha$ vanishes at $\frakp_{-1}$};\\
&\mbox{$\Phi_2(-\alpha,t)$ vanishes at $\frakp_{-1}$ and possibly at $\frakp_{-3/4}$, but nowhere else}; and\\
&\mbox{if $\Phi_2(-\alpha,t)$ vanishes at $\frakp_{-3/4}$, then $\alpha - 1/2$ also vanishes at $\frakp_{-3/4}$}.
\end{split}
\right.
\]
Moreover, if in this case $\Phi_2(-\alpha,t)$ vanishes at $\frakp_{-3/4}$, then it does so to order 1.
\end{lem}
\begin{proof}
By Theorem~\ref{thm:old}, the only constant points $\alpha \in K \subset \calK$ which fail to realize portrait $(1,2)$ are $\alpha = 0$, which satisfies ($*$), and $\alpha = 1/2$. We henceforth assume $\alpha \in \calK \setminus K$, so that $\hhat_f(\alpha) \ge h(\alpha) \ge 1$.
First, suppose that $\alpha$ does not realize portrait $(1,2)$ for $f$. Let $\frakp_c \in \calM_\calK$ be a place at which $\Phi_2(-\alpha,t)$ vanishes. Then $\Phi_1(-\alpha,t)$ or $\alpha$ also vanishes at $\frakp_c$ by Corollary~\ref{cor:fail}. If $\Phi_1(-\alpha,t)$ vanishes at $\frakp_c$, then $\Phi_2(-\alpha(c),c) = \Phi_1(-\alpha(c),c) = 0$. Thus $z = -\alpha(c)$ satisfies both
\[
\Phi_2(z,c) = z^2 + z + c + 1 = 0 \mbox{ and } \Phi_1(z,c) = z^2 - z + c = 0,
\]
which implies that $c = -3/4$ and $\alpha(c) = \alpha(-3/4) = 1/2$. That $t = -3/4$ is a simple root of $\Phi_2(-\alpha,t)$ follows from Remark~\ref{rem:simple}. Since $h(\Phi_2(-\alpha,t)) = 2\hhat_f(\alpha) \ge 2$, and since $\Phi_2(-\alpha,t)$ has at most a simple root at $\frakp_{-3/4}$, $\Phi_2(-\alpha,t)$ must vanish at some place $\frakp_c \ne \frakp_{-3/4}$. In this case, $\Phi_2(-\alpha,t)$ and $\alpha$ must both vanish at $\frakp_c$; hence $0 = \Phi_2(0,c) = c + 1$,
so $ c= -1$. Therefore $\alpha$ satisfies ($*$).
Now suppose $\alpha$ satisfies ($*$). By Corollary~\ref{cor:fail}, it suffices to show that wherever $\Phi_2(-\alpha,t)$ vanishes, so too must $\Phi_1(-\alpha,t)$ or $\alpha$. Since $\alpha$ satisfies ($*$), $\Phi_2(-\alpha,t)$ vanishes only at $\frakp_{-1}$ and possibly $\frakp_{-3/4}$. The result follows by noting that condition ($*$) forces $\alpha$ to vanish at $\frakp_{-1}$, and if $\Phi_2(-\alpha,t)$ vanishes at $\frakp_{-3/4}$, then condition ($*$) says that $\alpha$ reduces to $1/2$ --- and therefore $\Phi_1(-\alpha,t)$ reduces to $\Phi_1(-1/2,-3/4) = 0$ --- modulo $\frakp_{-3/4}$.
\end{proof}
We now verify that $\phi_2$ and $\phi_2^{-1}$ preserve property ($*$), and that $\phi_2$ behaves nicely with respect to the heights of points satisfying ($*$).
\begin{lem}\label{lem:phi(*)}
Let $\alpha \in \calK$ satisfy property ($*$). Then
\begin{enumerate}
\item $\phi_2(\alpha)$ satisfies ($*$) as well, and
\item $h(\phi_2(\alpha)) = h(\alpha) + 1$.
\end{enumerate}
\end{lem}
\begin{proof}
Since $\alpha$ vanishes at $\frakp_{-1}$, it is clear that $\phi_2(\alpha)$ also vanishes at $\frakp_{-1}$. Also, we have
\[
\Phi_2(-\phi_2(\alpha),t) = \phi_2(\alpha)^2 - \phi_2(\alpha) + (t + 1) = \frac{(t + 1)^2(\alpha^2 - \alpha + (t + 1))}{(\alpha + t)^2} = \frac{(t + 1)^2\Phi_2(-\alpha,t)}{(\alpha + t)^2},
\]
so $\Phi_2(-\phi_2(\alpha),t)$ vanishes at $\frakp_{-1}$; moreover, since $\Phi_2(-\alpha,t)$ \emph{only} vanishes at $\frakp_{-1}$ and possibly to order one at $\frakp_{-3/4}$, the same holds for $\Phi_2(-\phi_2(\alpha),t)$. (Any pole of $(\alpha + t)^2$ is a pole of at least the same order for $(t+1)^2\Phi_2(-\alpha,t)$, so the above expression may only vanish at the zeroes of its numerator.) Finally, if $\Phi_2(-\phi_2(\alpha),t)$ vanishes at $\frakp_{-3/4}$, then so must $\Phi_2(-\alpha,t)$, in which case $v_{-3/4}(\alpha - 1/2) > 0$. Hence
\[
\phi_2(\alpha) - \frac{1}{2} = \frac{(t + 1/2)(\alpha - 1/2) - (t + 3/4)}{(\alpha - 1/2) + (t + 1/2)}
\]
vanishes at $\frakp_{-3/4}$. Therefore $\phi_2(\alpha)$ satisfies ($*$).
We now prove (B). Letting
\[
\gamma := \frac{1}{\phi_2(\alpha)} = \frac{1}{t + 1} + \frac{1}{\alpha - 1},
\]
it suffices to show that $h(\gamma) = h(\alpha) + 1$. Suppose that $\frakp$ is a pole of $\gamma$. Then either $\frakp = \frakp_{-1}$, in which case property ($*$) implies that $v_\frakp(\alpha - 1) = 0$, hence $v_\frakp(\gamma) = -v_\frakp(t + 1) = -1$; or else $\frakp \ne \frakp_{-1}$ is a zero of $\alpha - 1$, in which case $v_\frakp(\gamma) = -v_\frakp(\alpha - 1)$. Therefore
\[
h(\gamma)
= -\sum_{v_\frakp(\gamma) < 0} v_\frakp(\gamma)
= - v_{-1}(\gamma) + \sum_{v_\frakp(\alpha - 1) > 0} v_\frakp(\alpha - 1)
= 1 + h(\alpha - 1).
\]
Since $h(\alpha - 1) = h(\alpha)$, we are done.
\end{proof}
\begin{lem}\label{lem:phi-1(*)}
Let $\alpha \in \calK$ satisfy ($*$), and assume $\alpha \not \in \{0, t + 1, -\frac{t + 1}{2t + 1}\}$.
\begin{enumerate}
\item We have
\begin{align*}
v_{-1}(\alpha) &= 1,\\
v_{-1}(\alpha - (t + 1)) &= 2, \mbox{ and } \\
v_{-1}(t\alpha + (t + 1)) &\ge 3.
\end{align*}
\item The point $\phi_2^{-1}(\alpha)$ also satisfies ($*$).
\end{enumerate}
\end{lem}
\begin{proof}
We first show that $v_{-1}(\Phi_2(-\alpha,t)) \ge 3$. Since $\Phi_2(-\alpha,t)$ can only vanish at $\frakp_{-1}$ and possibly, to order one, at $\frakp_{-3/4}$, we have
\begin{align*}
v_{-1}(\Phi_2(-\alpha,t)) &= h(\Phi_2(-\alpha,t)) - v_{-3/4}(\Phi_2(-\alpha,t))\\
&= 2\hhat_f(\alpha) -
\begin{cases}
0, &\mbox{ if $\Phi_2(-\alpha,t)$ does not vanish at $\frakp_{-3/4}$},\\
1, &\mbox{ if $\Phi_2(-\alpha,t)$ vanishes at $\frakp_{-3/4}$}.
\end{cases}
\end{align*}
Thus $v_{-1}(\Phi_2(-\alpha,t)) \le 2$ if and only if $\hhat_f(\alpha) \le 1$ or $\hhat_f(\alpha) = 3/2$ and $\Phi_2(-\alpha,t)$ vanishes at $\frakp_{-3/4}$. A simple calculation then verifies that the only such $\alpha \in \calK$ satisfying ($*$) are the three values of $\alpha$ excluded from the statement of the lemma. Therefore $v_{-1}(\Phi_2(-\alpha,t)) \ge 3$.
Now, write
\[
\alpha = (t + 1) + \alpha^2 - (\alpha^2 - \alpha + (t + 1)) = (t + 1) + \alpha^2 - \Phi_2(-\alpha,t).
\]
Since $v_{-1}(\alpha) \ge 1$ by assumption, we have $v_{-1}(\alpha^2) \ge 2$, thus $v_{-1}(\alpha) = v_{-1}(t+1) = 1$. This implies that $v_{-1}(\alpha^2) = 2$, and therefore $v_{-1}(\alpha - (t + 1)) = v_{-1}(\alpha^2 - \Phi_2(-\alpha,t)) = 2$ as well. Finally, we write
\[
t\alpha + (t + 1) = \Phi_2(-\alpha,t) - \alpha(\alpha - (t + 1)),
\]
and by what we have already shown, this has valuation at least 3 at $\frakp_{-1}$, proving (A).
We now show that $\phi_2^{-1}(\alpha)$ satisfies ($*$). By part (A), we have
\[
v_{-1}(\phi_2^{-1}(\alpha)) = v_{-1}(t\alpha + (t + 1)) - v_{-1}(\alpha - (t + 1)) \ge 1,
\]
so $\phi_2^{-1}(\alpha)$ vanishes at $\frakp_{-1}$. Now consider
\[
\Phi_2(-\phi_2^{-1}(\alpha),t)
= \left(-\phi_2^{-1}(\alpha)\right)^2 - \phi_2^{-1}(\alpha) + t + 1
= \frac{(t+1)^2\Phi_2(-\alpha,t)}{(\alpha - (t+1))^2}.
\]
Since $\Phi_2(-\alpha,t)$ vanishes to order at least three and $(\alpha - (t+1))^2$ vanishes to order four at $\frakp_{-1}$, it follows that $\Phi_2(-\phi_2^{-1}(\alpha),t)$ vanishes at $\frakp_{-1}$. Moreover, $\Phi_2(-\phi_2^{-1}(\alpha),t)$ can only vanish at $\frakp_{-1}$ and the places at which $\Phi_2(-\alpha,t)$ vanishes, which are only $\frakp_{-1}$ and possibly $\frakp_{-3/4}$ by assumption. (As before, we are using the fact that the above expression cannot vanish at poles of its denominator.)
Finally, suppose $\Phi_2(-\phi_2^{-1}(\alpha),t)$ vanishes at $\frakp_{-3/4}$. This is equivalent to the vanishing of $\Phi_2(-\alpha,t)$, necessarily to order one, at $\frakp_{-3/4}$, in which case $\alpha - 1/2$ also vanishes at $\frakp_{-3/4}$. Thus
\[
v_{-3/4}(\Phi_2(-\phi_2^{-1}(\alpha),t)) = v_{-3/4}(\Phi_2(-\alpha,t)) = 1.
\]
Moreover, we have
\[
\phi_2^{-1}(\alpha) - \frac{1}{2} = -\frac{(t + 1/2)(\alpha - 1/2) + (t + 3/4)}{\alpha - (t + 1)},
\]
which vanishes at $\frakp_{-3/4}$ by our assumptions on $\alpha$. Therefore $\phi_2^{-1}(\alpha)$ satisfies ($*$) as well.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:(1,2)}]
Suppose first that $\alpha$ fails to realize portrait $(1,2)$ for $f$. Since clearly $1/2 \in \calO_{\phi_2}(1/2)$, we will assume that $\alpha \ne 1/2$, so $\alpha$ satisfies ($*$). We proceed by induction on $h(\alpha)$.
By Theorem~\ref{thm:old}, the only constant points $\alpha \in K \subset \calK$ that do not realize portrait $(1,2)$ for $f$ are $\alpha \in \{0,1/2\}$; thus the $h(\alpha) = 0$ case holds. Now suppose $h(\alpha) \ge 1$. Since $t + 1 = \phi_2(\infty)$ and $-(t+1)/(2t+1) = \phi_2(1/2)$ we will assume $\alpha \not \in \{t + 1, -(t+1)/(2t+1)\}$. Then Lemma~\ref{lem:phi-1(*)} says that $\phi_2^{-1}(\alpha)$ satisfies ($*$), hence $h(\phi_2^{-1}(\alpha)) = h(\alpha) - 1$ by Lemma~\ref{lem:phi(*)}. By induction, we have $\phi_2^{-1}(\alpha) \in \calO_{\phi_2}(\delta)$ for some $\delta \in \{0,1/2,\infty\}$, and therefore $\alpha \in \calO_{\phi_2}(\delta)$ as well.
Now suppose $\alpha \in \calO_{\phi_2}(\delta)$ for some $\delta \in \{0,1/2,\infty\}$. Write $\alpha = \phi_2^k(\delta)$ for some $k \ge 0$. We show by induction on $k$ that $h(\alpha) = k$ and that $\alpha$ satisfies ($*$), which is equivalent (for $\alpha \ne 1/2$) to the assertion that $\alpha$ fails to realize portrait $(1,2)$ for $f$.
Since 0 and $1/2$ fail to realize portrait $(1,2)$ for $f$, the statement holds for $k = 0$. The points $\phi_2(0) = -(t+1)/t$, $\phi_2(1/2) = -(t+1)/(2t + 1)$, and $\phi_2(\infty) = t + 1$ all satisfy ($*$), and certainly all three have height equal to one, establishing the $k = 1$ case. Now suppose $k \ge 2$. By induction, $\phi_2^{k-1}(\delta)$ satisfies property ($*$) and has height $k - 1$. By Lemma~\ref{lem:phi(*)}, we conclude that $\alpha = \phi_2^k(\delta)$ satisfies ($*$) and has height $k$, completing the proof.
\end{proof}
In order to prove the more general statement involving points of portrait $(M,2)$ with $M \ge 2$, we require the following:
\begin{lem}\label{lem:(1,2)poles}
Let $k \ge 1$. Then
\begin{align*}
v_\infty(\phi_2^k(0)) = v_\infty(\phi_2^k(1/2)) &= 0; \text{ and }\\
v_\infty(k\phi_2^k(\infty) - t) &\ge 0.
\end{align*}
\end{lem}
\begin{proof}
The map $\phi_2$ reduces to $z - 1$ modulo $\frakp_\infty$, so $\phi_2^k(\delta) \equiv \delta - k$ (mod $\frakp_\infty$) for all $\delta \in K$ and $k \in \bbN$. If we take $\delta \in \{0,1/2\}$, then $\delta - k$ is a nonzero constant for all $k \ge 1$, thus $v_\infty(\phi_2^k(\delta)) = 0$.
Now, for each $k \ge 1$ we set $u_k := k\phi_2^k(\infty) - t$. We show that $v_\infty(u_k) \ge 0$ by induction on $k$. The result clearly holds for $k = 1$, since $u_1 = \phi_2(\infty) - t = 1$, so we assume $k \ge 2$. Then
\begin{align*}
u_k = k\phi_2(\phi_2^{k-1}(\infty)) - t &= k\phi_2\left(\frac{u_{k-1} + t}{k - 1}\right) - t\\
&= \frac{t((k - 1)u_{k-1} - k^2 + 2k) + (ku_{k-1} - k^2 + k)}{u_{k-1} + kt}.
\end{align*}
By induction, we have $v_\infty(u_{k-1}) \ge 0$, and it therefore follows that $v_\infty(u_k) \ge 0$.
\end{proof}
We now prove Theorem~\ref{thm:main} in the case $d = N = 2$.
\begin{prop}\label{prop:maind2N2}
Let $\alpha \in \calK$, and let $M \ge 0$. Then $\alpha$ realizes portrait $(M,2)$ for $f_2$ if and only if $(\alpha,M)$ does not satisfy one of the following conditions:
\begin{itemize}
\item $M = 0$ and $\alpha = -1/2$;
\item $M = 1$ and $\alpha \in \calO_{\phi_2}(0) \cup \calO_{\phi_2}(1/2) \cup \calO_{\phi_2}(\infty)$; or
\item $M = 2$ and $\alpha = \pm 1$.
\end{itemize}
Moreover, for each $k \ge 0$, we have $h(\phi_2^k(0)) = h(\phi_2^k(1/2)) = h(\phi_2^k(\infty)) = k$.
\end{prop}
\begin{proof}
The $M = 0$ and $M = 1$ cases follow from Propositions~\ref{prop:periodic} and \ref{prop:(1,2)}, respectively. We therefore assume $M \ge 2$.
That $\pm 1$ do not realize portrait $(2,2)$ for $f$ follows from Theorem~\ref{thm:old}. Now suppose that $\alpha$ does not realize portrait $(M,2)$ for $f$; equivalently, suppose that $f^{M-1}(\alpha)$ does not realize portrait $(1,2)$ for $f$. By Proposition~\ref{prop:(1,2)}, we have $f^{M-1}(\alpha) = \phi_2^k(\delta)$ for some $k \ge 0$ and $\delta \in \{0,1/2,\infty\}$. Lemma~\ref{lem:hhat_ineq} asserts that any point in the image of $f$ must have a pole at $\frakp_\infty$; since points in the orbits of 0 and $1/2$ under $\phi_2$ do not have poles at $\frakp_\infty$ by Lemma~\ref{lem:(1,2)poles}, we must have $f^{M-1}(\alpha) = \phi_2^k(\infty)$ for some $k \ge 0$. Since the only preimage of $\infty$ is $\infty$ itself, we must have $k \ge 1$.
Set $\beta := f^{M-2}(\alpha)$. Then $f(\beta) = \phi_2^k(\infty)$, so by Lemma~\ref{lem:(1,2)poles} we have that
\[
k\phi_2^k(\infty) - t = kf(\beta) - t = k\beta^2 + (k - 1)t
\]
is regular at $\frakp_\infty$. This implies that $k = 1$, so $f^{M-1}(\alpha) = \phi_2(\infty) = t + 1$. The preimages of $t + 1$ under $f$ are $\pm 1$, and the preimages of $\pm 1$ lie outside of $\calK$. Therefore, if $f^{M-1}(\alpha) = t + 1$ for some $M \ge 2$ and $\alpha \in \calK$, then $M = 2$ and $\alpha = \pm 1$, as claimed.
\end{proof}
As mentioned at the beginning of this section, the proof of the following statement --- the $d = 2$, $N = 1$ case of the main theorem --- uses the same ideas as for Proposition~\ref{prop:maind2N2}, and we therefore omit the proof.
\begin{prop}\label{prop:maind2N1}
Let $\alpha \in \calK$, and let $M \ge 0$. Then $\alpha$ fails to realize portrait $(M,1)$ for $f_2$ if and only if $M = 1$ and $\alpha \in \calO_{\phi_1}(0) \cup \calO_{\phi_1}(\infty)$, where
\[
\phi_1(z) = -\frac{t(z + 1)}{z - (t - 1)}.
\]
Moreover, for each $k \ge 0$, we have $h(\phi_1^k(0)) = h(\phi_1^k(\infty)) = k$.
\end{prop}
Proposition~\ref{prop:maind2N1} is the final case of Theorem~\ref{thm:main}. Therefore, by combining Propositions~\ref{prop:maind2N2} and \ref{prop:maind2N1} with the results of the previous sections, we have proven the main theorem.
\bibliography{C:/Dropbox/jdoyle}
\bibliographystyle{amsplain}
\end{document}
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Gm Truck Wiring Diagram 93 Yukon (Jan 04, 2019) - Thank you for stopping by here. Below is a great graphic for gm truck wiring diagram 93 yukon. We have been looking for this image throughout on -line and it originate from reliable source. If you are searching for any unique option for your own fuse box then this gm truck wiring diagram 93 yukon graphic should be on the top of reference or else you might use it for an optional idea.
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| 221,668
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TITLE: Why should we expect duality to give useful concepts in category theory?
QUESTION [3 upvotes]: Why should we expect the abstract notion of flipping arrows in a category to generate useful concepts from other useful ones? What exactly does flipping the direction of arrows mean and why is it universally useful? It certainly does not seem to be anything meaningful in any concrete category itself(or does it)?
Edit:I can see that duals are useful anyway since they generate free proofs but this would not answer my question.
The main question here is I suppose: Why would one expect dual concepts to be non vacuous before actually taking duals and checking?
REPLY [4 votes]: [Flipping arrows] certainly does not seem to be anything meaningful in any concrete category itself (or does it)?
There are cases where the opposite of a familiar category constructed by arrow-flipping is, or is equivalent to, another category of real interest (rather than an artificial construct): the opposite of the category $\mathbf{Set}$ is equivalent to the category of complete atomic boolean algebras (a surprise when you first hear it, perhaps).
But ok, suppose for the sake of argument that there are relatively few such examples: suppose that flipping arrows only rarely takes from an intrinsically interesting category to some other category we might want to investigate for 'concrete', non-category-theoretic, reasons. This perhaps is perhaps the thought underlying the question.
No matter! Arrow-flipping would still be a useful technique, giving us proofs by duality. For a baby example, if we show that in every category terminal objects are unique up to unique isomorphism, then -- remembering that categories come in pairs related by arrow-flipping -- it will follow by duality, i.e. by arrow-flipping, that every initial object is unique up to unique isomorphism. And so it goes.
The importance of arrow-flipping, i.e. of duality considerations like this, in producing buy-one-get-one-free proofs does not depend on the flipped version of particular interesting categories being themselves interesting. It only depends on categories having duals, interesting or otherwise.
And it can hardly be denied that the resulting duals of useful theorems are very often useful too! To take obvious examples, some familiar constructions are, in category theory terms, products, and other constructions are coproducts. It isn't that the one kind, products, is inhabited and its dual, coproducts, is empty! Likewise some constructions which are familiar pre-categorially turn out to be equalizers, other familiar constructions turn out to be coequalizers. Dualizing theorems about products and equalizers gives us theorems about coproducts and coequalizers -- and there are indeed familiar examples of such things! So yes, both the theorems about products and equalizers and their duals have interesting applications.
| 146,356
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\begin{document}
\title{Statistical Reasoning: Choosing and Checking the Ingredients, Inferences Based
on a Measure of Statistical Evidence with Some Applications}
\author{Luai Al-Labadi\thanks{Department of Mathematics, University of Sharjah},
Zeynep Baskurt\thanks{Genetics and Genome Biology, Hospital for Sick Children}
and Michael Evans\thanks{Department of Statistical Sciences, University of
Toronto}}
\date{}
\maketitle
\noindent\textit{Abstract}: The features of a logically sound approach to a
theory of statistical reasoning are discussed. A particular approach that
satisfies these criteria is reviewed. This is seen to involve selection of a
model, model checking, elicitation of a prior, checking the prior for bias,
checking for prior-data conflict and estimation and hypothesis assessment
inferences based on a measure of evidence. A long-standing anomalous example
is resolved by this approach to inference and an application is made to a
practical problem of considerable importance which, among other novel aspects
of the analysis, involves the development of a relevant elicitation
algorithm.\medskip
\noindent\textit{Key words and phrases}: foundations of statistical reasoning,
model checking, elicitation of priors, checking priors for bias, checking for
prior-data conflict, measuring statistical evidence, relative belief
inferences.\bigskip
\section{Introduction}
It is relevant to ask what characteristics should be required of a theory of
statistical reasoning. The phrase \textit{statistical reasoning} is used here,
as opposed to statistical inference, because there is a logical separation
between how the ingredients to a statistical problem are chosen and checked
for their validity, and the inference step which involves the application of
the rules of a theory of inference to the ingredients. So it is argued in
Section 2 that there are two aspects to a theory of statistical reasoning: (i)
specifying methodology for choosing and checking the ingredients to a
statistical analysis beyond the data and (ii) specifying a theory of inference
using these ingredients that is based on a measure of statistical evidence.
These components correspond to the premises and the argument in logical reasoning.
In Section 3 a specific theory of statistical reasoning, as described in Evans
(2015), that satisfies the criteria developed in Section 2 is reviewed. It is
shown that an application of the theory of relative belief inference resolves
difficulties in a problem that has led to anomalous results for other
theories. The overall aim of the paper is to argue in favor of this approach
to statistical reasoning based on its logical coherence and its utility in applications.
In Section 4 the theory of statistical reasoning is applied to an important
practical problem where some inferential difficulties have arisen, namely, the
problem of determining whether or not there is a relationship between a
binary-valued response variable and predictors. For this, response
$y\in\{0,1\}$ is related to $k$ quantitative predictors $\mathbf{x}
=(x_{1},\ldots,x_{k})$ via $y\sim$ Bernoulli$(p(\mathbf{x}))$ with
\begin{equation}
p(\mathbf{x})=G(\beta_{1}x_{1}+\cdots+\beta_{k}x_{k}) \label{eq1}
\end{equation}
where $G$ is a known cdf and $\mathbf{\beta}=(\beta_{1},\ldots,$ $\beta
_{k})\in R^{k}$ is unknown. This can be regarded as a case-study for the
overall approach and a number of novel results are obtained. Perhaps the
biggest challenge with this model is determining a suitable prior and in
Section 4.1 an elicitation algorithm is developed that improves on earlier
efforts. The bias in the prior is measured in Section 4.2. Model checking is
essential, namely, determining if (\ref{eq1}) holds at least approximately.
Since this is dealt with in Al-Labadi, Baskurt and Evans (2017), that approach
is used here without much discussion. The check for prior-data conflict is
developed in Section 4.3, together with an approach to modifying the prior
when necessary, and relative belief inferences are applied in Section 4.4.
\section{The Foundations of Statistical Reasoning}
When concerned with the foundations of statistics it is reasonable to ask:
what is the purpose of statistics as a subject or what is its role in science?
To answer this, consider a context where an investigator has interest in some
quantity and either wants to know (E) the value of this quantity or has a
theory that leads to a specific value for the quantity and so wants to know
(H)\ if this value is indeed correct and so test the theory. In addition, the
investigator has available data $d,$ produced in some fashion, which it is
believed contains \textit{evidence} concerning answers to (E) and (H). The
purpose of statistical theory is to provide a reasoning process that can be
applied to $d$ to determine what the evidence has to say about (E) or (H),
namely, produce an estimate of the quantity based on the evidence or assess
whether there is evidence either in favor of or against the hypothesized
value. Also, as is widely recognized, estimation and hypothesis assessment
should also produce a measure of the accuracy of the estimate and a measure of
the strength of the evidence for or against the hypothesis. Answering (E)
and/or (H) is called statistical inference and a sound, logical theory of
statistical inference, that contained the minimal ingredients possible, can be
viewed as a major goal of the subject.
Any theory that does not lead to specific answers to (E) and (H) or is
dependent on ingredients or rules of reasoning that are not well-justified, is
unsatisfactory. In the end, the believability of the inferences drawn is
entirely dependent on the soundness of the theory which produced them. So
statistics is not an empirical subject like physics, where conclusions can
also be assessed against the empirical world, but is more like an extension of
purely logical reasoning to contexts where the data does not lead to
categorical answers to (E) and (H) and so produces uncertainty. The view is
taken here that we want to maintain a close relationship between a theory of
statistical reasoning and the theory of logical reasoning. This has a number
of consequences with perhaps the most important being that it implies a
separation of the assessment of the appropriateness of the ingredients
specified to a statistical analysis beyond the data, and the theory producing
the inferences. The ingredients play the role of the premises and the theory
of statistical inference takes the role of the rules of inference used in a
logical argument. The separation of these aspects of a logical argument has
been understood since Aristotle, see Kneale and Kneale (1962).
There are two main theses of the argument developed here (i) all ingredients
to a statistical analysis must be checkable against the data and (ii) the
theory of inference must be based on a measure of statistical evidence. The
rationale for (i) and (ii) are now considered with (ii) discussed first, as it
plays a key role.
The concept of the evidence in the data is clearly of utmost importance to
statistical reasoning. There is no need, however, to provide a measure of the
total evidence contained in the data. For the measure of evidence only has to
deal with (E) and (H) for the quantity of interest. The measure of evidence
must clearly indicate whether there is evidence for or against any specific
value of the quantity of interest being the true value. This follows from\ the
desired relationship with logic, as the rules of logical inference assume the
truth of the premises, so the theory of statistical inference has to be based
on the truth of the ingredients and this implies that one of the possible
values for the quantity of interest is true. A theory of logical reasoning
that could only determine falsity and never truth is not useful and similarly
any valid measure of evidence must be able to indicate evidence in favor as
well evidence against.
Once a measure of statistical evidence is determined, an estimate of the
quantity of interest is necessarily the value that maximizes the measure of
evidence and the accuracy of the estimate can be assessed by looking at the
size of the set of values that have evidence in their favor. The measure of
evidence similarly necessarily determines whether there is evidence for or
against a hypothesized value and the strength of this evidence can be assessed
by comparison with the evidence associated with the other possible values for
the quantity of interest.
Consider now requirement (i). If a satisfactory measure of evidence could be
determined from the data alone, then this would be ideal, but currently this
is not available and it is questionable whether it is even possible. It is
\textit{assumed} hereafter that the data $x\in\mathcal{X}$ can be regarded as
having been produced by some probability distribution on the set $\mathcal{X}$
with unknown density $f.$ If the data was collected via random sampling, then
this assumption seems justified, but it is always an assumption. The density
$f$ is unknown and it assumed that once $f$ is known, then this determines
answers to (E) and ((H). The ingredients are then as follows: it is
\textit{assumed} that $f\in\{f_{\theta}:\theta\in\Theta\},$ a collection of
densities on $\mathcal{X}$ indexed by the parameter $\theta\in\Theta$ called
the statistical model, and it is \textit{assumed} that there is a probability
measure $\Pi$ with density $\pi$ on $\Theta$ that represents beliefs of the
investigator about the true value of $\theta\in\Theta$ and called the prior.
The ingredients correspond to the premises of a logical argument and these may
be true or false.
It can be questioned as to whether both the model and prior are necessary for
the development of a satisfactory theory and certainly minimizing the
ingredients is desirable. But as discussed in Section 3 it seems that a valid
definition of a measure of evidence requires both and again the challenge is
open to develop a satisfactory measure of evidence that uses fewer
ingredients. In particular, the use of a prior is often claimed to be
inappropriate as it is subjective in nature and, as the goal of a scientific
investigation is to be as objective as possible, the prior seems contrary to
that. It needs to be recognized, however, that all ingredients to a
statistical analysis beyond the data are subjective in nature as they are
chosen by the statistician. As discussed in Section 3.2, it is possible to
check both the model and the prior against the (objective) data to determine
whether or not reasonable choices have been made. Also, it is possible to
check whether or not the chosen ingredients have biased the results so that
the inferences obtained are in fact foregone conclusions, namely, could have
been made without even looking at the data. It is our view that checking for
bias and checking for conflict with the data go a long way towards answering
criticisms concerning the subjectivity inherent in a statistical analysis.
Another implication from (i) is that no ingredient can be added to a
statistical analysis unless it can be checked against the data which rules out
the use of loss functions.
It is not clear how the ingredients are to be chosen and guidance needs to be
provided for this. Not much has been written about how the model is to be
chosen but certainly something needs to be said to justify a specific choice
as part of the statistical reasoning argument. Much more has been written
about the selection of the prior and the position is adopted here that it is
necessary to base this on a clearly stated elicitation algorithm, namely, a
prescription for how an expert can translate knowledge into beliefs as
expressed via the prior.
In summary, the desiderata for a theory of statistical reasoning include the
following: a methodology for choosing a model, an elicitation algorithm for
selecting a prior, methodology for assessing the bias in the ingredients
chosen, model checking and checking for prior-data conflict procedures and a
theory of inference based upon a measure of statistical evidence.
\section{A Theory of Statistical Reasoning}
Choosing and checking the ingredients logically comes before inference but it
is convenient to discuss these in reverse order.
\subsection{Relative Belief Inferences}
Consider now a statistical problem with ingredients the data $d,$ a model
$\{f_{\theta}:\theta\in\Theta\},\ $a prior $\pi$ and interest is in making
inference about $\psi=\Psi(\theta)$ for $\Psi:\Theta\rightarrow\Psi$ where no
distinction is made between the function and its range to save notation.
Initially, suppose that all the probability distributions are discrete. This
isn't really a restriction in the discussion, however, as if something works
for inference in the discrete case but does not work in the continuous case,
then it is our view that the concept is not being applied correctly or the
mathematical context is just too general. For us the continuous case is always
to be thought of as an approximation to a fundamentally discrete context, as
measurements are always made to finite accuracy, and the approximation arises
via taking limits. Some additional comments on the continuous case are made subsequently.
As discussed in Section 2, the basic object of inference is the measure of
evidence and what is wanted is a measure of the evidence that any particular
value $\psi\in\Psi$ is true. Based on the ingredients specified, there are two
probabilities associated with this value, namely, the prior probability
$\pi_{\Psi}(\psi)$, as given here by the marginal prior density evaluated at
$\psi,$ and the posterior probability $\pi_{\Psi}(\psi\,|\,d)$, as given here
by the marginal posterior density evaluated at $\psi.$ In certain treatments
of inference $\pi_{\Psi}(\psi\,|\,d)$ is taken as a measure of the evidence
that $\psi$ is the true value but, for a wide variety of reasons, this is not
felt to be correct and Example 1 provides a specific case where this fails.
Also, this measure suffers from the same basic problem of $p$-values, namely,
there is no obvious dividing line between evidence for and evidence against.
Moreover, it is to be noted that probabilities measure belief and not
evidence. If we start with a large prior belief in $\psi$, then unless there
is a large amount of data, there will still be a large posterior belief even
if it is false and similarly, if we started with a small amount of belief.
There is agreement, however, to use the \textit{principle of conditional
probability} to update beliefs after receiving information or data and this is
to be regarded as the first principle of the theory of relative belief.
So how is the evidence that $\psi$ is the true value to be measured? Basic to
this is the \textit{principle of evidence}: if $\pi_{\Psi}(\psi\,|\,d)>\pi
_{\Psi}(\psi)$ there is evidence in favor as belief has increased, if
$\pi_{\Psi}(\psi\,|\,d)<\pi_{\Psi}(\psi)$ there is evidence against as belief
has decreased and if $\pi_{\Psi}(\psi\,|\,d)=\pi_{\Psi}(\psi)$ there is no
evidence either way. This principle has a long history in the philosophical
literature concerning evidence and a nice discussion can be found in Salmon
(1973). This principle doesn't provide a specific measure of evidence but at
least it indicates clearly when there is evidence for or against, independent
of the size of initial beliefs, and it does suggest that any reasonable
measure of the evidence depends on the difference, in some sense, between
$\pi_{\Psi}(\psi)$ and $\pi_{\Psi}(\psi\,|\,d),$ namely, evidence is measured
by \textit{change in belief} rather than belief. A number of measures of this
change have been proposed, see Evans (2015) for a discussion, but by far the
simplest and the one that has the nicest properties is the relative belief
ratio
\begin{equation}
RB_{\Psi}(\psi\,|\,d)=\pi_{\Psi}(\psi\,|\,d)/\pi_{\Psi}(\psi). \label{eqRB}
\end{equation}
So if $RB_{\Psi}(\psi\,|\,d)>(<,=)1$ there is evidence for (against, neither)
for $\psi$ being the true value.\ The use of the relative belief ratio to
measure the evidence is the third and final principle of the theory, what we
call the \textit{principle of relative belief}. The relative belief ratio can
also be written as $RB_{\Psi}(\psi\,|\,d)=m(d\,|\,\psi)/m(d)$ where $m$ is the
prior predictive density of the data and $m(\cdot\,|\,\psi)$ is the
conditional prior predictive density of the data given $\Psi(\theta)=\psi.$
Another natural candidate for a measure of evidence is the Bayes factor
$BF_{\Psi}(\psi\,|\,d)$ as this satisfies the principle of evidence, namely,
$BF(\psi\,|\,d)>(<,=)1$ when there is evidence for (against, neither) $\psi$
being the true value. The Bayes factor can be defined in terms of the relative
belief ratio as $BF_{\Psi}(\psi\,|\,d)=RB_{\Psi}(\psi\,|\,d)/RB_{\Psi}
(\{\psi\}^{c}\,|\,d)$ but not conversely. Note that the relative belief ratio
of a set $A\subset\Psi$ is $RB_{\Psi}(A\,|\,d)=\Pi_{\Psi}(A\,|\,d)/\Pi_{\Psi
}(A)$ where $\Pi_{\Psi},\Pi_{\Psi}(\cdot\,|\,d)$ are the prior and posterior
probability measures of $\Psi,$ respectively. It might appear that $BF_{\Psi
}(\psi\,|\,d)$ is a comparison between the evidence for $\psi$ being true with
the evidence for $\psi$ being false but it is provable that $RB_{\Psi
}(A\,|\,d)>1$ implies $RB_{\Psi}(A^{c}\,|\,d)<1$ and conversely, so this is
not the case. Also, as subsequently discussed, in the continuous case it is
natural to take $BF_{\Psi}(\psi\,|\,d)=RB_{\Psi}(\psi\,|\,d).$
The principle of relative belief leads to an ordering of the possible values
for $\psi$ as $\psi_{1}$ is preferred to $\psi_{2}$ whenever $RB_{\Psi}
(\psi_{1}\,|\,d)>RB_{\Psi}(\psi_{2}\,|\,d)$ since there is more evidence for
$\psi_{1}$ than $\psi_{2}.$ When $\Psi(\theta)=\theta$ this agrees with the
likelihood ordering but likelihood fails to provide such an ordering for
general $\psi.$ It is common to use the profile likelihood ordering even
though this is not a likelihood ordering and this doesn't agree with the
relative belief ordering. It is noteworthy that the relative belief idea is
consistent in the sense that inferences for all $\psi=\Psi(\theta)$ are based
on a measure of the change in prior to posterior probabilities.
The relative belief ordering leads immediately to a theory of estimation. For
basing inferences on the evidence requires that the relative belief estimate
be a value $\psi(d)$ maximizes $RB_{\Psi}(\psi\,|\,d)$ and typically this
value is unique so $\psi(d)=\arg\sup_{\psi\in\Psi}RB_{\Psi}(\psi\,|\,d)$. It
is also necessary to say something about the accuracy of this estimate in an
application. For this a set of values containing $\psi(d)$ is quoted and the
\textquotedblleft size\textquotedblright\ of the set is taken as the measure
of accuracy. Again following the ordering based on the evidence, it is
necessary that the set take the form $\{\psi:RB_{\Psi}(\psi\,|\,d)>c\}$ for
some constant $c\leq\sup_{\psi\in\Psi}RB_{\Psi}(\psi\,|\,d)$ since, if
$RB_{\Psi}(\psi_{1}\,|\,d)\leq RB_{\Psi}(\psi_{2}\,|\,d),$ then $\psi_{2}$
must be included whenever $\psi_{1}$ is. But what $c$ should be used? It is
perhaps natural to chose $c$ so that $\{\psi:RB_{\Psi}(\psi\,|\,d)>c\}$
contains some prescribed amount of posterior probability, so the set is a
$\gamma$-credible region. But there are several problems with this
approach.\ For what $\gamma$ should be chosen? Even if one is content with
some particular $\gamma,$ say $\gamma=0.95,$ there is the problem that the set
may contain values $\psi$ with $RB_{\Psi}(\psi\,|\,d)<1$ and such a value has
been ruled out since there is evidence against such a $\psi$ being true. It is
argued in Evans (2015) that the \textit{plausibility set} $Pl_{\Psi}
(d)=\{\psi:RB_{\Psi}(\psi\,|\,d)>1\}$ be used as $Pl_{\Psi}(d)$ contains all
the values for which there is evidence in favor of it being the true value. In
general circumstances, it is provable that $RB_{\Psi}(\psi(d)\,|\,d)>1$ so
$Pl_{\Psi}(x)\neq\phi.$ There are several possible measures of size but
certainly the posterior content $\Pi_{\Psi}(Pl_{\Psi}(d)\,|\,d)$ is one as
this measures the belief that the true value is in $Pl_{\Psi}(d),$ but also
some measure such as length or cardinality is relevant. If $Pl_{\Psi}(d)$ is
small and $\Pi_{\Psi}(Pl_{\Psi}(d)\,|\,d)$ large, then $\psi(d)$ can be judged
to be an accurate estimate of $\psi.$
It is immediate that $RB_{\Psi}(\psi_{0}\,|\,d)$ is the evidence concerning
$H_{0}:\Psi(\theta)=\psi_{0}.$ The evidential ordering implies that the
smaller $RB_{\Psi}(\psi_{0}\,|\,d)$ is than 1, the stronger is the evidence
against $H_{0}$ and the bigger it is than 1, the stronger is the evidence in
favor $H_{0}.$ But how is one to measure this strength? In Baskurt and Evans
(2013) it is proposed to measure the \textit{strength of the evidence} via
\begin{equation}
\Pi_{\Psi}\left( \left. RB_{\Psi}(\psi\,|\,d)\leq RB_{\Psi}(\psi
_{0}\,|\,d)\,\right\vert \,d\right) \label{eq2}
\end{equation}
which is the posterior probability that the true value of $\psi$ has evidence
no greater than that obtained for the hypothesized value $\psi_{0}.$ When
$RB_{\Psi}(\psi_{0}\,|\,d)<1$ and (\ref{eq2}) is small, then there is strong
evidence against $H_{0}$ since there is a large posterior probability that the
true value of $\psi$ has a larger relative belief ratio. Similarly, if
$RB_{\Psi}(\psi_{0}\,|\,d)>1$ and (\ref{eq2}) is large, then there is strong
evidence that the true value of $\psi$ is given by $\psi_{0}$ since there is a
large posterior probability that the true value is in $\{\psi:RB_{\Psi}
(\psi\,|\,x)\leq RB_{\Psi}(\psi_{0}\,|\,d)\}$ and $\psi_{0}$ maximizes the
evidence in this set. Additional results concerning $RB_{\Psi}(\psi
_{0}\,|\,d)$ as a measure of evidence and (\ref{eq2}) can be found in Baskurt
and Evans (2013) and Evans (2015).
For continuous parameters it is natural to define $RB_{\Psi}(\psi
\,|\,d)=\lim_{\epsilon\rightarrow0}RB_{\Psi}($\newline$N_{\epsilon}
(\psi)\,|\,d)$where $N_{\epsilon}(\psi)$ is a sequence of sets converging
nicely to $\{\psi\}$ as $\epsilon\rightarrow0.$ When the densities are
continuous at $\psi,$ then this limit equals (\ref{eqRB}) so this is a measure
of evidence in general circumstances. Also, it is natural to define the Bayes
factor by $BF_{\Psi}(\psi\,|\,d)=\lim_{\epsilon\rightarrow0}BF_{\Psi
}(N_{\epsilon}(\psi)\,|\,d)$ and, when the densities are continuous at $\psi,$
then $BF_{\Psi}(\psi\,|\,d)=RB_{\Psi}(\psi\,|\,d).$
A variety of consistency results, as the amount of data increases, are proved
in Evans (2015) concerning the estimation and hypothesis assessment
procedures. In particular, when $H_{0}$ is false, then (\ref{eqRB}) converges
to 0 as does (\ref{eq2}). When $H_{0}$ is true, then (\ref{eqRB}) converges to
its largest possible value (greater than 1 and often $\infty$) and, in the
discrete case (\ref{eq2}) converges to 1. In the continuous case, however,
when $H_{0}$ is true, then (\ref{eq2}) typically converges to a $U(0,1)$
random variable. This simply reflects the approximate nature of the inferences
and is easily resolved by requiring that a deviation $\delta>0$ be specified
such that if dist$(\psi_{1},\psi_{2})<\delta,$ where dist is some measure of
distance determined by the application, then this difference is to be regarded
as immaterial. This leads to redefining $H_{0}$ as $H_{0}=\{\psi:$
dist$(\psi,\psi_{0})<\delta\}$ and typically a natural discretization of
$\Psi$ exists with\ $H_{0}$ as one of its elements. With this modification
(\ref{eq2}) converges to 1 as the amount of data increases when $H_{0}$ is
true. Given that data is always measured to finite accuracy, the value of a
typical continuous-valued parameter can only be known to a certain finite
accuracy no matter how much data is collected. So such a $\delta$ always
exists and it is part of an application to determine the relevant value, see
Example 7 here, Al-Labadi, Baskurt and Evans (2017) and Evans, Guttman and Li
(2017) for developments on determining $\delta$.
It is immediate that relative belief inferences have some excellent
properties. For example, any 1-1 increasing function of $RB_{\Psi}
(\cdot\,|\,d),$ such as $\log RB_{\Psi}(\cdot\,|\,d),$ can be used to measure
evidence as the inferences are invariant to this choice. Also, $RB_{\Psi
}(\cdot\,|\,d)$ is invariant under smooth reparameterizations and so all
relative belief inferences possess this invariance property. For example,
MAP\ (maximum a posteriori)\ inferences are not invariant and this leads to
considerable doubt about their validity, see also Example 1. In Evans (2015)
results from a number of papers are summarized establishing optimality results
for relative belief inferences in the collection of all Bayesian inferences.
For example, Al-Labadi and Evans (2017) establish that relative belief
inferences for $\psi$ have optimal robustness to the prior $\pi_{\Psi}$
properties. Also, as discussed in Section 3.2, since the inferences are based
on a measure of evidence a key criticism of Bayesian methodology can be
addressed, namely, the extent to which the inferences are biased can be measured.
Relative belief prediction inferences for future data are naturally produced
by using the ratio of the posterior to prior predictive densities for the
quantity in question. The following example illustrates this and demonstrates
significant advantages for relative belief.\smallskip
\noindent\textbf{Example 1.}\textit{ Prediction for Bernoulli sampling.}
Consider an example discussed in\ Chapter 6 of Diaconis and Skyrms (2018). A
tack is flipped with $x=1$ indicating the tack finishes point up and $x=0$
otherwise, so $x\sim$ Bernoulli$(\theta).$ Suppose the prior is $\theta\sim
U(0,1)$ and the goal is to predict $f$ future observations $(y_{1}
,\ldots,y_{f})$ having observed $n$ independent tosses $(x_{1},\ldots,x_{n})$.
The posterior of $\theta$ is beta$(n\bar{x}+1,n(1-\bar{x})+1),$ the prior
predictive density of $(x_{1},\ldots,x_{n})$ is $m_{n}(x_{1},\ldots
,x_{n})=1/(n+1)\binom{n}{n\bar{x}}$ and the posterior predictive density for
$(y_{1},\ldots,y_{f})$ is
\begin{equation}
m_{n,f}((y_{1},\ldots,y_{f})\,|\,(x_{1},\ldots,x_{n}))=\frac{(n+1)\binom
{n}{n\bar{x}}}{(n+f+1)\binom{n+f}{(n+f)\left[ \frac{n}{n+f}\bar{x}+\frac
{f}{n+f}\bar{y}\right] }}, \label{postpred1}
\end{equation}
which is constant for all $(y_{1},\ldots,y_{f})$ with the same value of
$\bar{y}.$ Maximizing (\ref{postpred1}) gives the MAP predictor of
$(y_{1},\ldots,y_{f}).$ If $n\bar{x}/(n+f)>1/2,$ then the maximum occurs at
$(y_{1},\ldots,y_{f})$ with $\bar{y}=1,$ namely, $(y_{1},\ldots,y_{f}
)=(1,\ldots,1).$ If $n\bar{x}/(n+f)<1/2,$ then the maximum occurs at
$(y_{1},\ldots,y_{f})$ with $\bar{y}=0,$ namely, $(y_{1},\ldots,y_{f}
)=(0,\ldots,0).$ If $n\bar{x}/(n+f)=1/2$, then a maximum occurs at both
$(y_{1},\ldots,y_{f})=(0,\ldots,0)$ and $(y_{1},\ldots,y_{f})=(1,\ldots,1)$.
So using MAP gives the absurd result that $(y_{1},\ldots,y_{f})$ is always
predicted to be either all 0's or all 1's. Clearly there is a problem here
with using MAP.
Now suppose $(x_{1},\ldots,x_{n})=(0,\ldots,0)$ so the prediction is all 0's
and
\[
m_{n,f}((y_{1},\ldots,y_{f})\,|\,(0,\ldots,0))=(n+1)/(n+f+1)\binom{n+f}
{f\bar{y}}.
\]
For fixed $f,$ then $m_{n,f}((y_{1},\ldots,y_{f})\,|\,(0,\ldots,0))\rightarrow
0$ as $n\rightarrow\infty$ whenever $\bar{y}\neq0$ and converges to 1 when
$\bar{y}=0$. Diaconis and Skyrms (2018) note, however, that when $f=n$, then
$m_{n,n}((0,\ldots,0)\,|\,(0,\ldots,0))\rightarrow1/2$ as $n\rightarrow\infty$
and make the comment \textquotedblleft If this is an unwelcome surprise, then
perhaps the uniform prior is suspect.\textquotedblright\ They also refer to
some attempts to modify the prior to avoid this phenomenon which clearly
violates an essential component of the Bayesian approach. In our view there is
nothing wrong with the uniform prior, rather the problem lies with using
posterior probabilities implicitly as measures of evidence, both to determine
the predictor and to assess its reliability.
The relative belief ratio for $(y_{1},\ldots,y_{f})$ is
\begin{equation}
RB((y_{1},\ldots,y_{f})\,|\,(x_{1},\ldots,x_{n}))=\frac{(n+1)\binom{n}
{n\bar{x}}(f+1)\binom{f}{f\bar{y}}}{(n+f+1)\binom{n+f}{(n+f)\left[ \frac
{n}{n+f}\bar{x}+\frac{f}{n+f}\bar{y}\right] }}. \label{relbel}
\end{equation}
With $n=f=20$ and $n\bar{x}=6,$ Figure \ref{diaconisfig} gives the plot of
(\ref{relbel}) as a function of $n\bar{y}.$ The best relative belief predictor
of $(y_{1},\ldots,y_{f})$ is any sample with $f\bar{y}=6$ and $Pl_{n}
(x_{1},\ldots,x_{n})=\{(y_{1},\ldots,y_{f}):f\bar{y}=2,3,\ldots,10\}$ has
posterior content $0.893.$ So there is reasonable belief that the plausibility
set contains the \textquotedblleft true\textquotedblright\ future sample but
certainly there are many such samples. By contrast with MAP,\ a sensible
prediction is made using relative belief.
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.4163in,
width=2.4163in
]
{plot1.eps}
\caption{Plot of the relative belief ratio when $n=20,n\bar{x}=6$ in Example
1.}
\label{diaconisfig}
\end{center}
\end{figure}
For the case when $f=n$ and $(x_{1},\ldots,x_{n})=(0,\ldots,0),$ then
\[
RB((y_{1},\ldots,y_{n})\,|\,(0,\ldots,0))=\frac{(n+1)^{2}}{(2n+1)}\frac
{\binom{n}{n\bar{y}}}{\binom{2n}{n\bar{y}}}=\frac{(n+1)^{2}}{(2n+1)}
{\textstyle\prod_{i=0}^{n-1}}
\left\{ \frac{2-\bar{y}-i/n}{(2-i/n)}\right\}
\]
which is decreasing in $\bar{y}$ and so is maximized for the sample with
$\bar{y}=0.$ Similarly, when $\bar{x}=1$ the predictor is the sample with
$\bar{y}=1.$ So, at the extremes, the predictions based on MAP\ and relative
belief are the same but otherwise there is a sharp disagreement. Also,
\[
Pl_{n}(0,\ldots,0)=\{(y_{1},\ldots,y_{n}):RB((y_{1},\ldots,y_{n}
)\,|\,(0,\ldots,0))>1\}
\]
always contains $(y_{1},\ldots,y_{n})=(0,\ldots,0)$ and for any $c\in(0,1]$
such that $\bar{y}\geq c,$
\begin{align*}
& RB((y_{1},\ldots,y_{n})\,|\,(0,\ldots,0))=\{(n+1)^{2}/(2n+1)\}
{\textstyle\prod_{j=0}^{n-1}}
\left( 1-\bar{y}/(2-j/n)\right) \\
& \leq\{(n+1)^{2}/(2n+1)\}\left( 1-c/2\right) ^{n}\rightarrow0
\end{align*}
as $n\rightarrow\infty.$ Therefore, for any $c\in(0,1],$ there is an $N,$ such
that for all $n>N,$ then $Pl_{n}(0,\ldots,0)$ contains no $(y_{1},\ldots
,y_{n})$ having a proportion of 1's that is $c$ or greater. So $Pl_{n}
(0,\ldots,0)$ is shrinking as $n$ increases in the sense that it contains only
samples with smaller and smaller proportion of 1's as $n$ increases.
The posterior content of the plausibility region equals
\begin{equation}
\sum_{\{n\bar{y}:RB((y_{1},\ldots,y_{n})\,|\,(0,\ldots,0))>1\}}m_{n,f}
((y_{1},\ldots,y_{f})\,|\,(0,\ldots,0))\binom{n}{n\bar{y}} \label{plausprob}
\end{equation}
which equals the sum over all the summands that are greater than
$1/(n+1)\ $and
\begin{align*}
& m_{n,f}((y_{1},\ldots,y_{f})\,|\,(0,\ldots,0))\binom{n}{n\bar{y}}
=\frac{(n+1)}{(2n+1)}
{\textstyle\prod_{i=0}^{n-1}}
\left\{ \frac{2-\bar{y}-i/n}{(2-i/n)}\right\} \\
& =\frac{(n+1)}{(2n+1)}\left\{
\begin{array}
[c]{ccc}
1 & & \bar{y}=0\\
\frac{1}{2} & & \bar{y}=\frac{1}{n}\\
\frac{1}{2}\frac{1-1/n}{2-1/n} & & \bar{y}=\frac{2}{n}\\
\vdots & & \vdots\\
\frac{1}{2}\frac{1-1/n}{2-1/n}\cdots\frac{1-(k-1)/n}{2-(k-1)/n} & & \bar
{y}=\frac{k}{n}\\
\vdots & & \vdots
\end{array}
\right. .
\end{align*}
When $\bar{y}=k/n$ the corresponding term converges to $(1/2)^{k+1}.$ Thus for
all $n$ large enough, the sum (\ref{plausprob}) contains the terms for
$\bar{y}=0,1/n,\ldots,k/n.$ Therefore, for $\epsilon>0$ and all $n$ large
enough, (\ref{plausprob}) is greater than $(1/2)[1+1/2+\cdots+(1/2)^{k}
]-\epsilon=1-(1/2)^{k+1}-\epsilon$ and the posterior content of $Pl_{n}
(0,\ldots,0)$ converges to 1.
So relative belief also behaves appropriately when $f=n$ and $\bar{x}=0$ while
MAP does not. The failure of MAP might be attributed to the requirement that
the entire sample $(y_{1},\ldots,y_{n})$ be predicted. If instead it was
required only to predict the value $n\bar{y},$ then the prior predictive of
this quantity is uniform on $\{0,1,\ldots,f\},$ the posterior of $n\bar{y}$
equals $RB((y_{1},\ldots,y_{f})\,|\,(x_{1},\ldots,x_{n}))/(f+1)$ and the
relative belief ratio for $n\bar{y}$ equals $RB((y_{1},\ldots,y_{f}
)\,|\,(x_{1},\ldots,x_{n})).$ So, as is often the case when the quantity in
question has a uniform prior, MAP and relative belief estimates are the same.
But even in this case there is no natural cut-off for MAP inferences to say
when there is evidence for or against a particular value. The fact that it is
necessary to modify the problem in this way to get a reasonable inference is,
in our view, a substantial failing of MAP. It seems reasonable to suggest that
when an inference approach is shown to perform poorly on such examples, that
it not be generally recommended. Additional examples of poor performance of
MAP are discussed in Evans (2015).
\subsection{Choosing and Checking the Ingredients}
The first choice that must be made is the model and there are a number of
standard models used in practice. There isn't a lot written about this step,
however, and yet it is perhaps the most important step in solving a
statistical problem. It is generally accepted that the correct way to choose a
prior is through elicitation. This means that a methodology is prescribed that
directs an expert in the application area on how to translate their knowledge
into a prior. There are various default priors in use that avoid this
elicitation step, but it is far better to recommend that sufficient time and
energy be allocated for the elicitation of a proper prior. Staying within the
context of probability suggests that a variety of paradoxes and illogicalities
are avoided.
Given the ingredients, the relative belief inferences may be applied correctly
but it is still reasonable to ask if these ingredients are appropriate for the
particular application. If not, then the inferences drawn cannot be considered
valid. There are at least two questions about the ingredients that need to be
answered, namely, is there bias inherent in the choice of ingredients and are
the ingredients contradicted by the data?
The concern for bias is best understood in terms of assessing the hypothesis
$H_{0}:\Psi(\theta)=\psi_{0}.$ Let $M(\cdot\,|\,\psi)$ denote the prior
predictive distribution of the data given that $\Psi(\theta)=\psi.$ Bias
against $H_{0}$ means that the ingredients are such that, with high
probability, evidence will not be obtained in favor of $H_{0}$ even when it is
true. Bias against is thus measured by
\begin{equation}
M(RB_{\Psi}(\psi_{0}\,|\,D)\leq1\,|\,\psi_{0}). \label{biasagainst}
\end{equation}
If (\ref{biasagainst}) is large, then obtaining evidence against $H_{0}$ seems
like a foregone conclusion. For bias in favor of $H_{0}$ consider $M(RB_{\Psi
}(\psi_{0}\,|\,D)\geq1\,|\,\psi_{\ast})\ $where dist$(\psi_{\ast},\psi
_{0})=\delta,$ so $\psi_{\ast}$ is a value that just differs from the
hypothesized value by a meaningful amount. Bias in favor of $H_{0}$ is then
measured by
\begin{equation}
\sup_{\psi_{\ast}\in\{\psi:\text{ dist}(\psi,\psi_{0})=\delta\}}M(RB_{\Psi
}(\psi_{0}\,|\,D)\geq1\,|\,\psi_{\ast}). \label{biasfor}
\end{equation}
If (\ref{biasfor}) is large, then obtaining evidence in favor of $H_{0}$ seems
like a foregone conclusion. Typically $M(RB_{\Psi}(\psi_{0}\,|\,D)\geq
1\,|\,\psi_{\ast})$ increases as dist$(\psi_{\ast},\psi_{0})$ increases so
(\ref{biasfor}) is an appropriate measure of bias in favor of $H_{0}$. The
choice of the prior can be used somewhat to control bias but typically a prior
that makes one bias lower just results in making the other bias higher. It is
established in Evans (2105) that, under quite general circumstances, both
biases converge to 0 as the amount of data increases. So bias can be
controlled by design a priori.
The model needs to be checked against the data. For if the data $d$ lies in
the tails of every distribution in the model, then this suggests model
failure. There are a wide variety of approaches to assessing this and these
are not reviewed here. One general comment is that at this time there do not
seem to exist general methodologies for modifying a model when model failure
is encountered.
The prior can also be checked for conflict with the data. A conflict means
that the observed data is in the tails of all those distributions in the model
where the prior primarily places its mass. For a minimal sufficient statistic
$T$ for the model, Evans and Moshonov (2006) used the tail probability
\begin{equation}
M_{T}(m_{T}(t)\leq m_{T}(t)) \label{conflict}
\end{equation}
to assess prior-data conflict where (\ref{conflict}) small indicates
prior-data conflict. In Evans and Jang (2011a) it is shown that, under general
circumstances, (\ref{conflict}) converges to $\Pi(\pi(\theta)\leq\pi
(\theta_{true}))$ as the amount of data increases. There are a variety of
refinements of (\ref{conflict}) that allow for looking at particular
components of a prior to isolate where a problem with the prior may be. In
Evans and Jang (2011b) a method is developed for replacing a prior when a
prior-data conflict has been detected. This does not mean simply replacing a
prior by one that is more diffuse, however, as is demonstrated in Section 4.1.
\section{Binary-valued Response Regression Models}
The following example, based on real data, is used to illustrate each aspect
of the approach to statistical reasoning recommended here.\smallskip
\noindent\textbf{Example 2. }\textit{Bioassay experiment.}
Table 1 gives the results of exposing animals to various levels in g/ml of a
dosage of a toxin, where $x_{2}$ is the log-dosage and the number of deaths is
recorded at each dosage, see Racine, Grieve, Fluhler and Smith (1986). The
dosages range from $e^{-0.86}=0.423$ to $e^{0.73}=2.075$ g/ml. The logistic
regression model $p(x_{1},x_{2})=G(\beta_{1}+\beta_{2}x_{2})$ is considered
for this data, so $x_{1}\equiv1,G(z)=e^{z}/(1+e^{z}),(\beta_{1},\beta_{2})\in
R^{2}$ and $p(1,x_{2})$ is the probability of death at dosage $x_{2}.$ The
counts $T=(t_{1},t_{2},t_{3},t_{4})$ at the dosages comprise a minimal
sufficient statistic for this problem with observed value $(0,1,3,5).$ The
conditional distribution of $T$ given $(\beta_{1},\beta_{2})$ is a product of binomials.
In Al-Labadai, Baskurt and Evans (2017) a goodness-of-fit test based on this
data was applied for this model using a uniform prior on the space $[0,1]^{4}$
of all probabilities. Relative belief was used to assess the hypothesis that
the model is correct and overwhelming evidence in favor of this model was
obtained and so model correctness is assumed here. One goal is the estimation
of $(\beta_{1},\beta_{2})$ and another is the assessment of the hypothesis
$H_{0}:\beta_{2}=0.$ Acceptance of $H_{0}$ implies that there is no
relationship between the response and the predictor.
\begin{table}[tbp] \centering
\begin{tabular}
[c]{|rcc|}\hline
$x_{2}$ & No. of animals & No. of deaths\\\hline
$-0.86$ & 5 & 0\\
$-0.30$ & 5 & 1\\
$-0.05$ & 5 & 3\\
$0.73$ & 5 & 5\\\hline
\end{tabular}
\caption{Data in Example 1.}
\end{table}
\subsection{Eliciting the Prior}
Elicitation of a prior can be difficult when the interpretation of the
parameters is unclear. For example, with the model (\ref{eq1}) it is not clear
what the $\beta_{i}$ represent in contrast to linear models where they
represent either location parameters or rates of change with respect to
predictors. This leads to attempts to put default priors on these quantities
and there are problems with this approach. For example, suppose
$p(1,x)=G(\beta_{1}+\beta_{2}x),$ where $G$ is the standard logistic cdf and
the prior is given by the $\beta_{i}$ being i.i.d. $N(0,\sigma_{0}^{2})$ where
$\sigma_{0}^{2}$ is chosen large to reflect little information about these
values. In Figure \ref{figlogistic1} we have plotted the prior this induces on
$p(1,1)$ when $\sigma=20.$ This reflects the fact that as $\sigma$ grows all
the prior probability for $p(1,x)$ piles up at 0 and 1 and so this is clearly
a poor choice and it is certainly not noninformative.
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.303in,
width=2.3142in
]
{unif.eps}
\caption{Prior density of of $p(1,x)=G(\beta_{1}+\beta_{2}x),$ $G$ is the
standard logistic cdf, $\beta_{1},\beta_{2}\sim N(0,20^{2})$ and $x=1.$}
\label{figlogistic1}
\end{center}
\end{figure}
The strange behavior of diffuse normal priors has been noted by others.
Bedrick, Christensen and Johnson (1996, 1997), based on Tsutukawa and Lin
(1986),\ make the recommendation that priors should instead be placed on the
$p(\mathbf{x}_{i}),$ as these are parameters for which there is typically
prior information. Their recommendation is that, $k$ of the $\mathbf{x}_{i}$
values be selected and then beta$(\alpha_{1i},\alpha_{2i})$ priors be placed
on the corresponding $p(\mathbf{x}_{i})$ via eliciting prior quantiles$.$ This
results in more sensible priors but depends on the choice of the observed
predictors and it is unclear what kind of priors this induces on the
$\beta_{i}.$
Following Bedrick, Christensen and Johnson (1996, 1997) priors here are
elicited for the probabilities but the approach is different. First, it is not
required that the elicitation be carried out at observed values of the
predictors.\ Rather, it is supposed that there is a set of linearly
independent predictor vectors $\mathbf{w}_{1},\ldots,\mathbf{w}_{k}$ where
bounds can be placed on the probabilities in the sense that $l(\mathbf{w}
_{i})\leq p(\mathbf{w}_{i})\leq u(\mathbf{w}_{i})$ for $i=1,\ldots,k$ with
virtual certainty. By virtual certainty it is meant that for prior probability
measure $\Pi$, then
\begin{equation}
\Pi(l(\mathbf{w}_{i})\leq p(\mathbf{w}_{i})\leq u(\mathbf{w}_{i})\text{ for
}i=1,\ldots,k)\geq\gamma, \label{ineq1}
\end{equation}
where $\gamma$ is chosen to be close to 1. For example, $\gamma=0.99$
certainly seems satisfactory for many applications but a higher or lower
standard can be chosen. The motivation for this is that typically information
will be available for the probabilities such as it is known that
$p(\mathbf{w}_{i})$ is very small (or very large) or almost certainly that
$p(\mathbf{w}_{i})$ is in some specific range. Of course, for some of the
$\mathbf{w}_{i}$ virtually nothing may be known about $p(\mathbf{w}_{i})$ and
in that case taking $[l(\mathbf{w}_{i}),u(\mathbf{w}_{i})]=[0,1]$ is
appropriate. One implication of this is that when the choice is made
$[l(\mathbf{w}_{i}),u(\mathbf{w}_{i})]=[0,1]$ for every $i,$ then the
elicitation procedure should lead to a $\Pi$ that is at least approximately
uniform on the probabilities. The approach to elicitation, via stating bounds
on parameter values that hold with virtual certainty, has been successfully
employed in Cao, Evans and Guttman (2015) to determine a prior for the
multivariate normal model, and in Evans, Guttman and Li (2017) to determine a
prior for the multinomial model.
Another reason for allowing the elicitation procedure to be independent of the
observed $\mathbf{x}_{i}$ is that prior beliefs about $p(\mathbf{x}_{i})$ may
apply equally well about $p(\mathbf{x}_{j})$ for some $j$ simply because
$\mathbf{x}_{i}$ and $\mathbf{x}_{j}$ are close and then it seems that the
correlation between the beliefs should be part of the prior. Modelling such
correlations is harder and hopefully can be avoided by choosing the
$\mathbf{w}_{i}$ carefully. For example, requiring the $\mathbf{w}_{i}$ to be
mutually orthogonal seems like an appropriate way of achieving independence in
many contexts.
The second way in which our approach differs from previous developments is
that $\Pi$ is restricted to the family of multivariate normal priors on
$\mathbf{\beta}$ as this allows us to see directly how (\ref{ineq1})
translates into information about $\mathbf{\beta}.$ For note that
(\ref{ineq1}) is equivalent to
\begin{align}
& \Pi(G^{-1}(l(\mathbf{w}_{i}))\leq G^{-1}(p(\mathbf{w}_{i}))\leq
G^{-1}(u(\mathbf{w}_{i}))\text{ for }i=1,\ldots,k)\nonumber\\
& =\Pi(G^{-1}(l(\mathbf{w}_{i}))\leq\mathbf{w}_{i}^{\prime}\mathbf{\beta}\leq
G^{-1}(u(\mathbf{w}_{i}))\text{ for }i=1,\ldots,k)\nonumber\\
& =\Pi(G^{-1}(l(\mathbf{W}))\leq\mathbf{W\beta}\leq G^{-1}(u(\mathbf{W}
)))\geq\gamma\label{ineq2}
\end{align}
where $\mathbf{W=(w}_{1}\ldots\mathbf{w}_{k})^{\prime}\in R^{k\times
k},l(\mathbf{W})=\mathbf{(}l\mathbf{(w}_{1}),\ldots,l(\mathbf{w}_{k}
))^{\prime}\in R^{k},u(\mathbf{W})=\mathbf{(}u\mathbf{(w}_{1}),\ldots
,u(\mathbf{w}_{k}))^{\prime}\in R^{k}.$ So, if $\mathbf{W\beta\sim
}N\mathbf{(\mu}_{0},\Sigma_{0}),$ then $\mathbf{\beta\sim}N_{k}\mathbf{(W}
^{-1}\mathbf{\mu}_{0},\mathbf{W}^{-1}\Sigma_{0}$\newline$(\mathbf{W}
^{-1})^{\prime})$ and it is clear what this says about $\mathbf{\beta}.$
The task then is to determine $\mathbf{(\mu}_{0},\Sigma_{0})$ so that
(\ref{ineq2}) is satisfied. A natural choice for $\mathbf{\mu}_{0}$ is to put
$\mathbf{\mu}_{0}=G^{-1}(c(\mathbf{W}))$\ where $c(\mathbf{W})=(l(\mathbf{W}
)+u(\mathbf{W}))/2$ \ is the centroid of the $k$-cell $[l(\mathbf{W}
),u(\mathbf{W})].$ For example, when $[l(\mathbf{W}),u(\mathbf{W}
)]=[0,1]^{k},$ then $c(\mathbf{W})=\mathbf{1}_{k}/2,$ where $\mathbf{1}_{k}$
is the $k$-dimensional vectors of ones, which implies $\mathbf{\mu}
_{0}=\mathbf{0}.$ Other choices for $\mathbf{\mu}_{0}$ can be made if there
are good reasons for this.
Given that the $\mathbf{w}_{i}$ have been chosen so that prior beliefs about
the probabilities $p(\mathbf{w}_{i})$ are independent, this implies that the
coordinates of $\mathbf{W\beta}$ are independent and so $\Sigma_{0}
=\,$diag$(\sigma_{1}^{2},\ldots,\sigma_{k}^{2})$ for some choice of the prior
variances $\sigma_{i}^{2}.$ There are, however, typically many choices
satisfying (\ref{ineq2}). For example, taking $\sigma_{i}^{2}=0$ for all $i$
achieves this but clearly this choice does not reflect what is actually known
about the probabilities. As might be expected, the choice of the $\sigma
_{i}^{2}$ is critical and dependent on $G.$ Furthermore, as Figure 1
demonstrates, an injudicious choice results in absurdities.
Since $G^{-1}(u(\mathbf{w}_{i}))-\mu_{0i}>0$ and $G^{-1}(l(\mathbf{w}
_{i}))-\mu_{0i}<0,$ both these values are infinite iff $[l(\mathbf{w}
_{i}),u(\mathbf{w}_{i})]=[0,1]$ and so no information is being introduced via
the prior. In such a case a uniform$[0,1]$ prior on the probability results
and the appropriate normal distribution is determined by approximating the
distribution function $G$ by a normal cdf, see Examples 3, 4 and 5. Suppose
then that at least one of $G^{-1}(u(\mathbf{w}_{i}))$ and $G^{-1}
(l(\mathbf{w}_{i}))$ is finite and so $\sigma_{i}$ satisfies
\begin{equation}
\Phi\left( \frac{G^{-1}(u(\mathbf{w}_{i}))-\mu_{0i}}{\sigma_{i}}\right)
-\Phi\left( \frac{G^{-1}(l(\mathbf{w}_{i}))-\mu_{0i}}{\sigma_{i}}\right)
\geq\gamma^{1/k}, \label{marginal}
\end{equation}
as then independence ensures that (\ref{ineq2}) is satisfied. When both
$G^{-1}(u(\mathbf{w}_{i}))$ and $G^{-1}(l(\mathbf{w}_{i}))$ are finite, the
left side of (\ref{marginal}) has the value 1 when $\sigma_{i}=0,$ is strictly
decreasing to the value 0 as $\sigma_{i}\rightarrow\infty$ and so there are
always values of $\sigma_{i}\geq0$ satisfying (\ref{marginal}). When both
$G^{-1}(u(\mathbf{w}_{i}))$ and $G^{-1}(l(\mathbf{w}_{i}))$ are finite there
is a unique largest solution to (\ref{marginal}), which is the preferred
solution as it best represents the prior information, and it is easily found
numerically by bisection. If $u(\mathbf{w}_{i})=1$ and $l(\mathbf{w}_{i}
)\in(0,1),$ then $\sigma_{i}=(G^{-1}(l(\mathbf{w}_{i}))-\mu_{0i})/\Phi
^{-1}(1-\gamma^{1/k})$ is the solution provided $\gamma>(1/2)^{k}$ which is a
very weak requirement as recall that $\gamma$ represents virtual certainty. If
$u(\mathbf{w}_{i})\in(0,1)$ and $l(\mathbf{w}_{i})=0,$ then $\sigma
_{i}=(G^{-1}(u(\mathbf{w}_{i}))-\mu_{0i})/\Phi^{-1}(\gamma^{1/k})$ is the
solution again provided $\gamma>(1/2)^{k}.$
The following examples consider the situation $l(\mathbf{w}_{i}
))=0,u(\mathbf{w}_{i})=1.$ In this case $\mu_{0i}=0$ and $G^{-1}
(p(\mathbf{w}_{i}))$ will be distributed with cdf $G$ when $p(\mathbf{w}
_{i})\sim U(0,1).$ Generally this leads to a need to approximate $G$ by\ a
normal cdf to obtain a normal prior although no approximation is required in
Example 3.\smallskip
\noindent\textbf{Example 3.} \textit{Probit regression.}
Here $G=\Phi$ and so $G^{-1}(p(\mathbf{w}_{i}))\sim N(0,1)$ when
$p(\mathbf{w}_{i})\sim U(0,1).$ As such $\sigma_{i}=1$ and the standard normal
distribution on $G^{-1}(p(\mathbf{w}_{i}))$ corresponds to no information
about $p(\mathbf{w}_{i})$. When there is no information about any of the
$p(\mathbf{w}_{i}),$ then $\mathbf{\beta\sim}N_{k}\mathbf{(0},\mathbf{W}
^{-1}(\mathbf{W}^{-1})^{\prime})$ which equals the $N_{k}\mathbf{(0}
,\mathbf{I})$ distribution whenever $\mathbf{W}$ is an orthogonal matrix. In
general, however a lack of information about the probabilities leads to a
prior on $\mathbf{\beta}$ that is dependent on $\mathbf{W,}$ namely, dependent
on the values of predictor variables corresponding to the
probabilities.\smallskip
\noindent\textbf{Example 4.} \textit{Logistic regression.}
In this case $G$ is the standard logistic cdf and so $\mathbf{w}_{i}^{\prime
}\mathbf{\beta=}G^{-1}(p(\mathbf{w}_{i}))$ is distributed standard logistic
when $p(\mathbf{w}_{i})\sim U(0,1).$ A well-known $N(0,\lambda^{2})$
approximation to the standard logistic distribution, as discussed in Camilli
(1994), leads to normal priors that are much easier to work with. The optimal
choice of $\lambda,$ in the sense that it minimizes $\max_{x\in R^{1}}
|\Phi(x/\lambda)-e^{x}/(1+e^{x})|$ is given by $\lambda=1.702$ and this leads
to a maximum difference less than $0.009.$ Clearly this error will generally
be irrelevant when considering priors for the probabilities in a logistic
regression problem. So when $\mathbf{w}_{i}^{\prime}\mathbf{\beta}\sim
N(0,1.702^{2})$ then $p(\mathbf{w}_{i})$ is approximately distributed $U(0,1)$
with the same maximum error. Figure \ref{figlogistic2} contains plots of the
density of $p=e^{z}/(1+e^{z})$ when $z\sim N(0,\lambda^{2})$ for various
choices of $\lambda$ and it is indeed approximately uniform when
$\lambda=1.702.$ Using normal probabilities rather than logistic probabilities
leads to relatively small differences, so it seems reasonable to use a normal
prior on $\mathbf{\beta}$ in a logistic regression\textbf{.}
\begin{figure}
[tb]
\begin{center}
\includegraphics[
height=2.1309in,
width=2.3168in
]
{OW6PTE00.eps}
\caption{Plots of the density of $p=e^{z}/(1+e^{z})$ when $z\sim N(0,d^{2})$
and $\lambda=0.5$ (--), $\lambda=1.0$ (- -), and $\lambda=-1.702$ (...).
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ }}
\label{figlogistic2}
\end{center}
\end{figure}
\smallskip
\noindent\textbf{Example 5.} $\mathit{t}$\textit{ regression.}
Suppose that $G$ is taken to be the cdf of $t$ with $\upsilon$ degrees of
freedom. Table 2 presents the optimal choice of $\lambda$ for a $N(0,\lambda
^{2})$ approximation to the $t(\upsilon)$ cdf together with the maximum error.
There does not appear to be much difference in using $t_{\upsilon}$
probabilities instead of normal ones unless $\upsilon$ is quite low.\smallskip
\begin{table}[tbp] \centering
\begin{tabular}
[c]{|c|c|c|c|c|c|c|}\hline
$\upsilon$ & $30$ & $20$ & $10$ & $5$ & $2$ & $1$\\\hline
$\lambda$ & \multicolumn{1}{|l|}{$1.022$} & \multicolumn{1}{|l|}{$1.034$} &
\multicolumn{1}{|l|}{$1.069$} & \multicolumn{1}{|l|}{$1.144$} &
\multicolumn{1}{|l|}{$1.407$} & \multicolumn{1}{|l|}{$1.980$}\\
max error & \multicolumn{1}{|l|}{$0.002$} & \multicolumn{1}{|l|}{$0.003$} &
\multicolumn{1}{|l|}{$0.006$} & \multicolumn{1}{|l|}{$0.013$} &
\multicolumn{1}{|l|}{$0.031$} & \multicolumn{1}{|l|}{$0.058$}\\\hline
\end{tabular}
\caption{Optimal choice of a $N(0,\lambda^2)$ distribution to approximate a $t(\nu)$ distribution.}\label{TableKey copy(1)}
\end{table}
Consider now an application of the elicitation algorithm.\smallskip
\noindent\textbf{Example 6. }\textit{Bioassay experiment (Example 2
continued).}
In this example $k=2.$ To determine the prior it is necessary to choose
$\mathbf{W=(w}_{1}$ $\mathbf{w}_{2}\mathbf{)}\in R^{2\times2}$ and
$[l(\mathbf{W}),u(\mathbf{W})].$ The authors are not experts in bioassay but,
given the range of dosages applied in the experiment, it is reasonable to
suppose that an expert might be willing to put bounds on the probabilities
that hold with prior probability $\gamma=0.99$ when $x_{2}=-0.50$ and
$x_{2}=0.50$ leading to
\[
\mathbf{W=}\left(
\begin{array}
[c]{cr}
1 & -1/2\\
1 & 1/2
\end{array}
\right) .
\]
Let us suppose that an expert believes with virtual certainty that the true
probabilities lie in the intervals $[0.15,0.75],$ when $x_{2}=-0.50,$ and in
$[0.25,0.95],$ when $x_{2}=0.50.$ So the centroid of the $2$-cell
$[0.15,0.75]\times\lbrack0.25,0.95]$ is given by $(0.45,0.60)$ and since
$G^{-1}(p)=\log(p/(1-p))$ for logistic regression, this implies $\mu
_{0}=(G^{-1}(0.45),G^{-1}(0.60))=(-0.20,0.41).$ Also, $[G^{-1}(0.15),G^{-1}
(0.75]=[-1.735,1.099]$ and $[G^{-1}(0.25),G^{-1}(0.95)]=[-1.099,2.944]$ so,
using (\ref{marginal}), the largest values of $\sigma_{1}$ and $\sigma_{2}$
satisfying $\Phi\left( 1.299/\sigma_{1}\right) -\Phi\left( -1.535/\sigma
_{1}\right) \geq\left( 0.99\right) ^{1/2}$ and $\Phi\left( 2.534/\sigma
_{2}\right) -\Phi\left( -1.509/\sigma_{2}\right) \geq\left( 0.99\right)
^{1/2}$ are given by $\sigma_{1}=0.490$ and $\sigma_{2}=0.580.$ Therefore the
prior on $\beta$ is
\begin{align}
& \beta\sim N_{2}(\mathbf{W}^{-1}(-0.20,0.41)^{\prime},\mathbf{W}
^{-1}\text{diag}(0.490^{2},0.580^{2})(\mathbf{W}^{=1})^{\prime})\nonumber\\
& =N_{2}\left( \left(
\begin{array}
[c]{c}
0.105\\
0.610
\end{array}
\right) ,\left(
\begin{array}
[c]{cc}
0.144 & 0.048\\
0.048 & 0.577
\end{array}
\right) \right) . \label{prior}
\end{align}
Figure \ref{priorhists} contains histograms of large samples from the priors
on two extreme probabilities. The shape of the prior is similar for other
values of $x_{2}.$
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.7345in,
width=2.7345in
]
{fighist.eps}
\caption{Density histograms of $p(1,-0.8)\,$(left) and $p(1,0.8)$ (right)
based on a sample of $10^{5}$ from the elicited prior in Example 6.}
\label{priorhists}
\end{center}
\end{figure}
\subsection{Measuring the Bias in a Prior}
Consider applying the approach discussed in Section 3.2 to measuring bias in
the prior derived in Section 4.1 for the bioassay example.\smallskip
\noindent\textbf{Example 7. }\textit{Bioassay experiment (Example 2
continued).}
Consider whether or not there is bias induced by the prior in Example 6 with
respect to the hypothesis $H_{0}:\beta_{2}=0.$ It is necessary to compute
$M_{T}(RB_{2}(0\,|\,T)\leq1\,|\,\beta_{2}=0),$ to measure bias against, and
$\sup_{\beta_{2}\in\{-\delta,\delta\}}M_{T}(RB_{2}$\newline$(0\,|\,T)\geq
1\,|\,\beta_{2}),$ to measure bias in favor, where $RB_{2}(\cdot\,|\,T)$ is
the relative belief ratio function for $\beta_{2}$ based on data $T$ and
$\delta>0$ is such that if $|\beta_{2}|<\delta,$ then practically speaking
$H_{0}$ is considered true. To determine $\delta,$ the more general problem of
what changes in both $\beta_{1}$ and $\beta_{2}$ are deemed irrelevant is
considered. Given the settings used in this experiment, it seems reasonable to
consider $x_{2}$ as restricted to the interval $[-1,1].$ Then, whenever
$|\beta_{1}-\beta_{1}^{\prime}|<\delta$ and $|\beta_{2}-\beta_{2}^{\prime
}|<\delta,$ the difference in log odds satisfies $|\beta_{1}+\beta_{2}
x_{2}-(\beta_{1}^{\prime}+\beta_{2}^{\prime}x_{2})|\leq2\delta$ which implies
that ratio of the odds lies in $(e^{-2\delta},e^{2\delta})$ which for small
$\delta$ is approximately equal to $(1-2\delta,1+2\delta).$ This in turn
implies that the difference in the probabilities is less than $2\delta.$ In
this example we take $\delta=0.01.$
Now $RB(\beta_{1},\beta_{2}\,|\,T)=\{
{\textstyle\prod_{i=1}^{4}}
\binom{5}{t_{i}}p^{t_{i}}(1,x_{2i})(1-p(1,x_{2i}))^{5-t_{i}}\}/m_{T}(T)$
where
\[
m_{T}(T)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\{
{\textstyle\prod_{i=1}^{4}}
\binom{5}{t_{i}}p^{t_{i}}(1,x_{2i})(1-p(1,x_{2i}))^{5-t_{i}}\}\pi
(\beta)\,d\beta.
\]
The relative belief ratio for $\beta_{2}$ is $RB_{2}(\beta_{2}\,|\,T)=\int
_{-\infty}^{\infty}RB(\beta_{1},\beta_{2}\,|\,T)\pi_{1}(\beta_{1}
\,|\,\beta_{2})\,d\beta_{1}$\newline$=m_{T}(T\,|\,\beta_{2})/m_{T}(T)$ where
$\pi_{1}(\cdot\,|\,\beta_{2})$ is the conditional prior density of $\beta_{1}$
given $\beta_{2}\ $which (\ref{prior}) implies is the $N(0.105+0.083(\beta
_{2}-0.610),0.140)$ distribution.
For given $T=(t_{1},t_{2},t_{3},t_{4}),$ the numerator and denominator in
$RB_{2}(0\,|\,T)$ can be estimated via simulation but to calculate the biases
we need to do this for many $T.$ Consider the calculation of $M_{T}
(RB_{2}(0\,|\,T)\leq1\,|\,\beta_{2}=0)$ via the following algorithm and note
that there are only $6^{4}=1296$ values of $(t_{1},t_{2},t_{3},t_{4}
)\in\{0,1,\ldots,5\}^{4}.$\smallskip
\noindent\textbf{Algorithm}
\noindent(i) simultaneously estimate the values $m_{T}(t_{1},t_{2},t_{3}
,t_{4})$ for each $(t_{1},t_{2},t_{3},t_{4})$ via a large sample from
(\ref{prior}) and store these in a table,\newline\noindent(ii) simultaneously
estimate the values $m_{T}(t_{1},t_{2},t_{3},t_{4}\,|\,\beta_{2}=0)$ for each
$(t_{1},t_{2},$\newline$t_{3},t_{4})$ via a large sample from $\pi_{1}
(\cdot\,|\,0)$ and store these in a table,\newline\noindent(iii) using the
values in these two tables estimate $RB_{2}(0\,|\,T)$ for all values of $T$
and then estimate $M_{T}(RB_{2}(0\,|\,T)\leq1\,|\,\beta_{2}=0)$ by summing the
$m_{T}(t_{1},t_{2},t_{3},t_{4}\,|\,$\newline$\beta_{2}=0)$ for those
$(t_{1},t_{2},t_{3},t_{4})$ for which $RB_{2}(0\,|\,T)\leq1.$\smallskip
\noindent The bias in favor can be estimated at $\pm\delta$ in exactly the
same way but in step (ii) replace $\pi_{1}(\cdot\,|\,0)$ by $\pi_{1}
(\cdot\,|\,-\delta)$ and by $\pi_{1}(\cdot\,|\,\delta)$. These computations
were carried out and resulted in the bias against equaling $0.22$ and the bias
in favor equaling $0.77$ at $-\delta$ and $0.78$ at $\delta$. So there is some
bias against $H_{0}$ with this prior but there is appreciable bias in favor of
$H_{0},$ at least when interest is in detecting deviations of size
$\delta=0.01$. For $\beta_{2}=5,$ however, the bias in favor of $H_{0}$ is
$0.006,$ so there is in reality no bias in favor for large values of this
parameter. One could contemplate modifying the prior to reduce the bias in
favor at $\delta=0.01,$ but typically this just results in trading bias in
favor with bias against. The real cure for excessive bias of either variety,
is to collect more data.\smallskip
In general problems the approach to the computations used here will not be
feasible and so alternative methods are required. In certain examples some
aspects of the computations can be done exactly but, in general,
approximations such as those discussed in Nott et al. (2016) will be necessary.
\subsection{Checking and Modifying a Prior}
Consider now checking the prior derived in Section 4.1 for the bioassay
example.\smallskip
\noindent\textbf{Example 8. }\textit{Bioassay experiment (Example 2
continued).}
The tail probability for checking the prior is given by
\begin{equation}
M_{T}(m_{T}(t_{1},t_{2},t_{3},t_{4})\leq m_{T}(0,1,3,5)). \label{priorchk}
\end{equation}
As part of the algorithm discussed in Section 4.2, the values of $m_{T}
(t_{1},t_{2},t_{3},t_{4})$ have been estimated and the proportion of values of
$m_{T}(t_{1},t_{2},t_{3},t_{4})$ that satisfy the inequality gives the
estimate of (\ref{priorchk}). In this example (\ref{priorchk}) equals $0.41$
so there is no prior-data conflict.
If prior-data conflict exists, the methods discussed in Evans and Jang (2011a)
are available to obtain a more weakly informative prior. In this case it is
necessary to be careful as it has been shown in Section 4.1 that simply
increasing the variance of the prior will not necessarily accomplish this. On
the other hand there is the satisfying result that the $N_{2}(\mathbf{0}
,1.702^{2}I_{2})$ prior$,$ where $I_{2}$ is the identity matrix, will avoid
prior-data conflict, so modifying the elicited prior to be closer to this
prior is the appropriate thing to do when a conflict exists.
\subsection{Inferences}
Now consider estimation and hypothesis assessment for the bioassay
example.\smallskip
\noindent\textbf{Example 9. }\textit{Bioassay experiment (Example 2
continued).}
Consider first the assessment of the hypothesis $H_{0}:\beta_{2}=0.$ From the
algorithm the quantity $RB_{2}(0\,|\,(0,1,3,5))$ is available and this
indicates whether there is evidence in favor of or against $H_{0}.$ In this
case $RB_{2}(0\,|\,(0,1,3,5))=0.021$ so there is evidence against $H_{0}$. A
calculation described below gives the value $0.001$ for the strength, so it
seems there is strong evidence against $H_{0}.$
To obtain the joint relative belief estimate of $(\beta_{1},\beta_{2})$ it is
necessary to maximize $RB(\beta_{1},\beta_{2}\,|\,T)$ as a function of
$(\beta_{1},\beta_{2})$ which is the same as the MLE. The plausibility region
for this estimate is then $\{(\beta_{1},\beta_{2}):RB(\beta_{1},\beta
_{2}\,|\,T)>1\}$ and the size and posterior content of this set provide a
measure of the accuracy with which the coordinates of $\beta$ can be
simultaneously known. But it is worth noting that the $i$-th coordinate of
this joint estimate is not necessarily the value that has the greatest
evidence in its favor, rather this is obtained by maximizing $RB_{i}(\beta
_{i}\,|\,T)$ as a function of $\beta_{i}$ with plausibility region
$\{\beta_{i}:RB_{i}(\beta_{i}\,|\,D)>1\}.$ So the evidence approach dictates
that $\beta_{i}$ be estimated by maximizing $RB_{i}(\beta_{i}\,|\,T).$ In
problems where components of a multidimensional parameter are related by some
constraint, then it is clearly necessary to estimate the components
simultaneously, but that is not the case here.
The value of $RB_{i}(\beta_{i}\,|\,T)$ needs to be estimated and, since this
cannot be done for every value of $\beta_{i},$ its value is estimated on a
finite grid. For this let $[L_{i},U_{i}]$ be the effective prior support for
$\beta_{i}$, say containing $0.995$ of the probability, and form the grid
$G_{i}=\{L_{i},L_{i}+\delta,L_{i}+2\delta,\ldots,U_{i}-\delta,U_{i}\}.$ For
each $\beta_{1}\in G_{1}$ estimate $m_{T}((0,1,3,5)\,|\,\beta_{1})$ using a
large sample from $\pi_{2}(\cdot\,|\,\beta_{1})$ which gives $RB_{1}(\beta
_{1}\,|\,(0,1,3,5))=m_{T}((0,1,3,5)\,|\,\beta_{1})/m_{T}((0,1,3,5)).$ It is
then easy to obtain the relative belief estimate $\beta_{1}(0,1,3,5)$ and
plausibility region $\{\beta_{1}:RB_{1}(\beta_{1}\,|\,(0,1,3,5))>1\}.$ The
true relative belief estimate will differ from this estimate by at most
$\delta$ but this difference has been deemed irrelevant. A similar procedure
is carried out for $\beta_{2}$ but now sampling from $\pi_{1}(\cdot
\,|\,\beta_{2})$ to estimate $m_{T}((0,1,3,5)\,|\,\beta_{2}).$ The posterior
density for $\beta_{i}$ satisfies $\pi_{i}(\beta_{i}\,|\,(0,1,3,5))=RB_{i}
(\beta_{i}\,|\,(0,1,3,5))\pi_{i}(\beta_{i})$ and since $RB_{i}(\cdot
\,|\,(0,1,3,5))$ has been computed on the grids,\ these values can be used to
approximate the contents of the plausibility regions via an obvious
quadrature. Similarly, the strengths can be estimated and the strength quoted
above equals $
{\textstyle\sum_{\beta_{2}\in S\text{ }}}
\pi_{2}(\beta_{2}\,|\,(0,1,3,5))\delta$ where $S=G_{2}\cap\{\beta_{2}
:RB_{2}(\beta_{2}\,|\,(0,1,3,5))\leq RB_{2}(0\,|\,(0,1,3,5))\}.$
Implementing this for $\beta_{1},$ the estimate $\beta_{1}(0,1,3,5)=0.11$ was
obtained with plausibility region $[-0.21,0.49]$ having posterior content
$0.35.$ So the range of plausible values for $\beta_{1}$ is not large but
there is not a high belief that the true value is in this interval. Figure
\ref{rbbeta1} is a plot of $RB_{1}(\cdot\,|\,(0,1,3,5)).$
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.4007in,
width=2.4007in
]
{rbbeta1.eps}
\caption{A plot of $RB_{1}(\cdot\,|\,(0,1,3,5))$ over the effective support of
the prior in Example 9.}
\label{rbbeta1}
\end{center}
\end{figure}
An interesting phenomenon occurs when considering the estimation of $\beta
_{2}$. In Figure \ref{rbfig} the left panel plots $RB_{2}(\cdot
\,|\,(0,1,3,5))$ over the effective support of the marginal prior $\pi_{2}$
for $\beta_{2}.$ From this it is clear that the relative belief estimate of
$\beta_{2}$ lies outside this range. Recall, however, that the chosen prior
passed the check for prior-data conflict. The check for prior-data conflict
only tells us, however, that the observed data is consistent with at least
some of the probabilities determined by where the prior places its mass. The
right panel of Figure \ref{rbfig} is a plot of $RB_{2}(\cdot\,|\,(0,1,3,5))$
over a much wider range. Note too that there is an important robustness
property as shown in Al-Labadi and Evans (2017) for $RB_{2}(\cdot
\,|\,(0,1,3,5))$ as it is only weakly dependent on $\pi_{2}.$ In this case
$\pi_{2}$ does not place mass where it appears it should but there is not
enough data to detect the conflict. The relative belief estimate of $\beta
_{2}$ is $\beta_{2}(0,1,3,5)=7.31$ and the plausibility region for $\beta_{2}$
is $[1.14,30.48]$ with posterior content $0.83.$ As such, there is a great
deal of uncertainty concerning the true value of $\beta_{2}.$\smallskip
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.6602in,
width=2.6602in
]
{rbfig.eps}
\caption{Plot of $RB_{2}(\cdot\,|\,(0,1,3,5)$ over the effective support of
the prior (left panel) and over a full range of possible values (right panel)
in Example 9.}
\label{rbfig}
\end{center}
\end{figure}
As long as it is possible to sample from the posterior for a 1-dimensional
parameter, then the computations necessary for the inferences for such a
parameter are feasible. As such, the Gibbs sampling algorithm of Albert and
Chib (1993) is particularly relevant although it is not needed in Example 9.
The harder computations are those involving the various prior predictives but
these do not need to be highly accurate as even one decimal place will
indicate whether there is bias or prior-data conflict.
\section{Conclusions}
Criteria for a satisfactory theory of statistical reasoning have been
developed. Perhaps more should be required, but it seems that those stated are
necessary. A particular approach has been outlined that satisfies these
criteria. An example has shown that this approach can resolve
anomalies/paradoxes that arise via a commonly used methodology. Many other
such instances of resolving inferential difficulties as well as results
establishing optimal performance, have been documented in Evans (2015). An
application of the approach to the problem of binary-valued response
regression has been carried out and it has been shown to lead to a number of
novel insights into such problems.
\section{References}
\noindent Albert, J. H. and Chib, S. (1993) Bayesian analysis of binary and
polychotomous response data. J. of the American Statistical Association 88
(422), 669-679.\vspace{2pt}
\noindent Al-Labadi, L. and Evans, M. (2017) Optimal robustness results for
some Bayesian procedures and the relationship to prior-data conflict. Bayesian
Analysis 12, 3, 702-728.\vspace{2pt}
\noindent Al-Labadi, L., Baskurt, Z and Evans, M. (2017) Goodness of fit for
the logistic regression model using relative belief. J. of Statistical
Distributions and Applications, 4:17.\vspace{2pt}
\noindent Baskurt, Z. and Evans, M. (2013) Hypothesis assessment and
inequalities for Bayes factors and relative belief ratios. Bayesian Analysis,
8, 3, 569-590.\vspace{2pt}
\noindent Bedrick, E. J., Christensen, R., and Johnson, W. (1996) A new
perspective on priors for generalized linear models. J. of the American
Statistical Association, 91, 436, 1450-1460.\vspace{2pt}
\noindent Bedrick, E. J., Christensen, R., and Johnson, W. (1997) Bayesian
binomial regression: predicting survival at a trauma center. American
Statistician, 51, 3, 211-218.\vspace{2pt}
\noindent Cao, Y., Evans, M. and Guttman, I. (2015) Bayesian factor analysis
via concentration. Current Trends in Bayesian Methodology with Applications,
edited by S. K. Upadhyay, U. Singh, D. K. Dey and A. Loganathan, 181-201, CRC
Press.\smallskip
\noindent Camilli, G. (1994) Origin of the scaling constant d = 1.7 in item
response theory. J. of Educational and Behavioral Statistics, 19, 3,
293-295.\vspace{2pt}
\noindent Diaconis, P. and Skyrms, B. (2018) Ten Great Ideas About Chance.
Princeton University Press.\vspace{2pt}
\noindent Evans, M. (2015) Measuring Statistical Evidence Using Relative
Belief. Chapman and Hall/CRC.\vspace{2pt}
\noindent Evans, M., Guttman, I. and Li, P. (2017) Prior elicitation,
assessment and inference with a Dirichlet prior. Entropy 2017, 19(10),
564.\smallskip
\noindent Evans, M. and Jang, G-H. (2011a) A limit result for the prior
predictive applied to checking for prior-data conflict. Statistics and
Probability Letters, 81, 1034-1038.\vspace{2pt}
\noindent Evans, M. and Jang, G-H. (2011b) Weak informativity and the
information in one prior relative to another. Statistical Science, 26, 3,
423-439.\vspace{2pt}
\noindent Evans, M. and Moshonov, H. (2006) Checking for prior-data conflict.
Bayesian Analysis, 1, 4, 893-914.\vspace{2pt}
\noindent Kneale, W. and Kneale, M. (1962) The Development of Logic. Clarendon
Pr.\vspace{2pt}
\noindent Nott, D., Drovandi, C., Mengersen, K. and Evans, M. (2016)
Approximation of Bayesian predictive p-values with regression ABC. To appear
in Bayesian Analysis.\vspace{2pt}
\noindent Racine, A., Grieve, A. P., Fluhler, H. and Smith, A. F. M. (1986)
Bayesian methods in practice: experiences in the pharmaceutical industry (with
discussion). J. of Applied Statistics, 35, 93-150.\vspace{2pt}
\noindent Salmon, W. (1973) Confirmation. Scientific American, 228, 5,
75-81.\vspace{2pt}
\noindent Tsutukawa, R. K. and Lin, H. Y. (1986) Bayesian estimation of item
response curves. Psychometrika, 51, 251-267.
\end{document}
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Recent Posts
Featured Posts
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My top 5 tips for getting through that hot run....
1. Get out there before the sun.
I know it sucks to get up before the world is awake, but trust me avoiding the sun will help keep you a bit cooler. I know for me the humidity will probably still be there, but taking the sun out of the equation will keep you cooler a bit longer. Now since you will be out there before the sun, make sure you are visible, wear reflective clothing, a reflective belt or vest, anything that will make you visible to others out on the road. I usually get up for my long runs on the weekend by 5:30 or 6am so I can make it back by 8 at the latest. The one bonus is you get to see some gorgeous sunrises!
2. Don't overdress.
This may seem like a no-brainer, but this is the time when less is more. A tech tank, some shorts, and you're good to go (and running shoes obviously). And make it tech material to wick the sweat away from you. No cotton (it only ends up being gross and heavy when you get sweaty)! I like to wear my compression socks on runs sometimes, but if it's too hot, I don't. They may only be socks, but they will make you warmer. You can wear your compression socks after your run for recovery. And if you will be out when the sun is out, don't forget to wear a cap or visor. Gotta keep the sun out off of your face.
3. Water, hydration, and more water!
Drink water! Drink water and make sure you are well hydrated before your run. (I'm not saying chug a bunch of water right before you go out there, but get in plenty of water the days before your run). And get plenty of water while you are out there too. Yes, I know that hydration belts stink sometimes and carrying a water bottle is a pain, but do it! Bring your water with you and make sure you will have a few places to refill your water on your route (a water fountain at the park--or my fave, a water cooler Latin cafeteria window--always work). Also, you might want to consider drinking some sports drink too. It will help replenish all the salt you're losing as you sweat. I'm not usually a sports drink fan, but if I'm at a super hot race, I'll take some along with my water. It seems to help. And, when it's super hot, I love throwing some water on the top of my head. So consider a little splash too (or even one of those cold towels). It feels great and does a great job of cooling you down.
4. Slow down.
There is a rule of thumb that says you will run faster when it's cooler, and slower when it's warmer. This isn't the time for PRs or crazy fast miles, a hot run is just about getting in those miles. Keep a good pace, but nothing crazy fast. Start out slow and save your energy. It will probably feel weird that you're not at your normal pace, but running like this will get you through.
5. Run with a friend.
This is a tip that really can be used for any run that's gonna be hard, but get out there with a friend. You will motivate each other and push each other to keep going. I know when I'm visiting Kris, I don't feel like getting out there, but we push each other to do it. Having someone count on you helps to get you out there. And bonus, if you pass out from the heat, your friend will stop your Garmin and drag you home! Just kidding! LOL Now I do a lot of solo runs, but seeing my fit friends out there getting in their runs makes me motivated too. My awesome friend, Jenn, did this year's super hot LA Marathon. I was tracking her and knew it was getting tough, so I sent her messages and gifs to keep her motivated. She told me that was the extra push that kept her moving in the heat. Never underestimate the power of positive peer pressure.
These are just a few tips to get you out there during the heat. I know it's a sweaty crazy mess to get out there in the heat, but you can do it. Get in your miles and you'll be a stronger runner in the end.
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For those who were old enough to have experienced the joy of Mattel’s View-Master, we all knew it was only a matter of time until it made a comeback, due to the existing technology that we now have. And when they finally did come back last year, it of course had virtual reality incorporated into it, as it should. Now, they’re going one step further by collaborating with Google to create an even more updated version of the device, and they showcased it at the New York Toy Fair.
The new View-Master is actually a Google Cardboard solution that seeks to bring “an immersive digital experience for kids.” So if there is no opportunity for kids (or adults for that matter) to go and visit these actual places, they will be able to get the next best thing and experience “spectacular 3D worlds” and explore famous places, landmarks, nature, and even planets. The whole system is composed of the View-Master experience reel, an app, and an Android smartphone. This version, unlike the previous one from last year, will be able to support a wider range of smartphones and will have focal adjustment as well.
When you buy the View-Master viewer when it comes out this fall, it will come with a sample experience reel which will give you an overview of what the “full View-Master experience” will be. This will include a gallery of the best of the View-Master images that we grew up with, a tour inside a space shuttle (which should include a journey into space too), and a “unique destination” that you will experience in 360 degrees.
The View-Master itself (with the said sample experience reel) will cost $29.99. Additional reel packs will be at $14.99 each and will have four themed experience reels in each pack. Mattel will also be curating and creating more reels with other Google Cardboard developers later on. The new View-Master will be available this fall.
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Genshin ImpactThis game is full of many intricacies-game quests. One interesting quest in the game is “the other side of Isle and sea,” which requires the gamer to hut down murals around the archipelago. InOur course of action Genshin ImpactGuide, we will show you how to find other murals within the game. WithoutLet’s not stop there.
Guide On How To Find Other Murals In Geshin Impact
ItIt is important that you note that murals can be found in different places and that there is no specific order that one can follow when searching. WhenIf you see a mural, make sure you take a photo and observe them.-Game photograph of the mural WhenEach primogem is worth 40 points. Mystic Enhancement Ore, 30,000 Mora, and 3 Hero Wits.
List Of Mural Locations In Genshin Impact
1. Minacious Isle Mural
InThe Minacious MuralIf the gamer lands on the teleport point, they must glide eastwards until reaching a rock formation.
WhenThe player should go to the left side of the rock formation. There they will find the murals they are looking for on one of the walls. TheLast thing you should do is to take a picture of the mural.
2. Broken Isle Mural
AnotherA mural can be found in an interesting spot where gamers can find it. Broken Isle Mural. ThisAnother mural that is considered one of the most difficult to paint is mural Genshin Impact. ThisThis is because the player would have to solve a puzzle to reach it. StartingThe player would need a teleport waypoint to get to the rock formation.
ThisThis is where the gamer would find the greatest deal of joy. Dodo King’s painted wall. TheThe next step would be to remove the wall. ItWith the help of a Wind-Blessed HarpastumYou will then be able to solve the puzzle and get the mural. ThisThe mural would be on the right.
3. Twinning Isle Mural
FirstIt is important to point out that the mural location is located in the bottom portion of the island. TheTwo waves of enemies would be required for a gamer to defeat. HilichurlsBefore they can interact with the mural. HilichurlsThe two-way nature of most things is what makes them so important to be prepared.
The Genshin ImpactGamer would need to move from teleport point to the southern portion of the island.
4. Nameless Islet Mural Location In Genshin Impact
FirstIt is important that you note that Nameless IsletThe mural is located on the fog-hidden island. ThisInsel in Genshin Impact can be found in the quest named: “A Trip Through FogAnd Wind”.
ToTo be able reach this island, the player must be there between 10.00 and 14.00. YouYou can reach the island by going on to the twin island and then sailing with the Waverider.
5. Nameless Island Mural Location
TheNameless is the last place to look for a mural. Island Mural location. ThisYou can reach the place with the Twining IsleTeleport waypoint FromFrom this spot, the player can glide to the island’s center.
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\begin{document}
\title{Singular polynomials for the symmetric group and Krawtchouk polynomials}
\author{Charles F. Dunkl\thanks{During the preparation of this paper the author was
partially supported by NSF grant DMS 0100539.}}
\date{16 October 2003}
\maketitle
\begin{abstract}
A singular polynomial is one which is annihilated by all Dunkl operators for a
certain parameter value. These polynomials were first studied by Dunkl, de Jeu
and Opdam, (\textit{Trans. Amer. Math. Soc.} 346 (1994), 237-256). This paper
constructs a family of such polynomials associated to the irreducible
representation $\left( N-2,1,1\right) $ of the symmetric group $S_{N}$ for
odd $N$ and parameter values $-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\ldots$.
The method depends on the use of Krawtchouk polynomials to carry out a change
of variables in a generating function involved in the construction of
nonsymmetric Jack polynomials labeled by $\left( m,n,0,\ldots,0\right)
,m\geq n$.
\end{abstract}
\section{Introduction}
We will study polynomials on $\mathbb{R}^{N}$ with certain properties relating
to the action of the symmetric group $S_{N}$ acting\ as a finite reflection
group (of type $A_{N-1}$). Let $\mathbb{N}$ denote $\left\{ 1,2,3,\ldots
\right\} $ and $\mathbb{N}_{0}=\mathbb{N\cup}\left\{ 0\right\} $; for
$\alpha\in\mathbb{N}_{0}^{N}$ let $\left| \alpha\right| =\sum_{i=1}
^{N}\alpha_{i}$ and define the monomial $x^{\alpha}$ to be $\prod_{i=1}
^{N}x_{i}^{\alpha_{i}}$; its degree is $\left| \alpha\right| .$ Consider
elements of $S_{N}$ as functions on $\{1,2,\ldots,N\}$ then for $x\in
\mathbb{R}^{N}$ and $w\in S_{N}$ let $\left( xw\right) _{i}=x_{w\left(
i\right) }$ for $1\leq i\leq N$; and extend this action to polynomials by
$wf\left( x\right) =f\left( xw\right) $. This has the effect that
monomials transform to monomials, $w\left( x^{\alpha}\right) =x^{w\alpha}$
where $\left( w\alpha\right) _{i}=\alpha_{w^{-1}\left( i\right) }$ for
$\alpha\in\mathbb{N}_{0}^{N}$. (Consider $x$ as a row vector, $\alpha$ as a
column vector, and $w$ as a permutation matrix, with $1$'s at the $\left(
w\left( j\right) ,j\right) $ entries.) The reflections in $S_{N}$ are the
transpositions, denoted by $\left( i,j\right) $ for $i\neq j$, interchanging
$x_{i}$ and $x_{j}$.
In \cite{D1} the author constructed for each finite reflection group a
parametrized commutative algebra of differential-difference operators; for the
symmetric group there is one parameter $\kappa\in\mathbb{C}$ and the
definition is as follows:
\begin{definition}
For any polynomial $f$ on $\mathbb{R}^{N}$ and $1\leq i\leq N$ let
\[
\mathcal{D}_{i}f\left( x\right) =\frac{\partial}{\partial x_{i}}f\left(
x\right) +\kappa\sum_{j\neq i}\frac{f\left( x\right) -\left( i,j\right)
f\left( x\right) }{x_{i}-x_{j}}.
\]
\end{definition}
It was shown in \cite{D1} that $\mathcal{D}_{i}\mathcal{D}_{j}=\mathcal{D}
_{j}\mathcal{D}_{i}$ for $1\leq i,j\leq N$ and each $\mathcal{D}_{i}$ maps
homogeneous polynomials to homogeneous polynomials. A specific parameter value
$\kappa$ is said to be a \textit{singular value} (associated with $S_{N}$) if
there exists a nonzero polynomial $p$ such that $\mathcal{D}_{i}p=0$ for
$1\leq i\leq N$; and $p$ is called a \textit{singular polynomial}. It was
shown by Dunkl, de Jeu and Opdam \cite[p.248]{DJO} that the singular values
for $S_{N}$ are the numbers $-\frac{j}{n}$ where $n=2,\ldots,N,\,j\in
\mathbb{N}$ and $n\nmid j$ ($n$ does not divide $j$). In this paper we
construct singular polynomials for the values -$\frac{j}{N-1}$ ($j\in
\mathbb{N}$ and $N-1\nmid j$) with a new result for the case of $N$ being odd
and -$\frac{j}{N-1}=-l-\frac{1}{2},l\in\mathbb{N}_{0}$. There are conjectures
in \cite[p.255]{DJO} regarding some general properties of the singular
polynomials for $S_{N}$ but these are not as yet established. Here is an easy
example of singular polynomials: let $a_{N}\left( x\right) =\prod
\limits_{1\leq i<j\leq N}\left( x_{i}-x_{j}\right) $, the alternating
polynomial. Then $\left( i,j\right) a_{N}\left( x\right) =-a_{N}\left(
x\right) $ for any transposition ($i\neq j$); further $\frac{\partial
}{\partial x_{i}}a_{N}\left( x\right) =a_{N}\left( x\right) \sum_{j\neq
i}\frac{1}{x_{i}-x_{j}}$. Thus for any $l\in\mathbb{N}_{0}$ we have
$\mathcal{D}_{i}\left( a_{N}\left( x\right) ^{2l+1}\right) =\left(
2l+1+2\kappa\right) \,a_{N}\left( x\right) ^{2l+1}\sum_{j\neq i}\frac
{1}{x_{i}-x_{j}}$ (for each $i$), which shows that $a_{N}\left( x\right)
^{2l+1}$ is singular for $\kappa=-l-\frac{1}{2}$. Irreducible representations
of $S_{N}$ are labeled by partitions of $N$ (see, for example, Macdonald
\cite[p.114]{M}); the polynomial $a_{N}\left( x\right) ^{2l+1}$ is
associated with the representation $\left( 1,1,\ldots,1\right) $ (more
precisely, the span $\mathbb{R}a_{N}\left( x\right) ^{2l+1}$ is an $S_{N}
$-module of isotype $\left( 1,1,\ldots,1\right) $).
Our construction is in terms of nonsymmetric Jack polynomials which are
defined to be the simultaneous eigenfunctions of the pairwise commuting
operators $\mathcal{D}_{i}x_{i}-\kappa\sum_{j<i}\left( i,j\right) $,$\,1\leq
i\leq N$ (details about these may be found in the book by Dunkl and Xu
\cite[Ch.8]{DX}). When $\kappa>0$ these operators are self-adjoint with
respect to the inner product on polynomials defined by
\[
\left\langle f,g\right\rangle _{\mathbb{T}}=c_{\kappa}\int_{\mathbb{T}^{N}
}f\left( x\right) \,\overline{g\left( x\right) }\prod_{1\leq i<j\leq
N}\left| x_{i}-x_{j}\right| ^{2\kappa}dm\left( x\right) ,
\]
where $\mathbb{T}^{N}$ is the $N$-fold complex torus $\left\{ z\in
\mathbb{C}:\left| z\right| =1\right\} ^{N},\,x_{j}=e^{\mathrm{i}\theta_{j}
}$ for $-\pi<\theta_{j}\leq\pi$ (and $1\leq j\leq N$) and the standard measure
is $dm\left( x\right) =\prod_{j=1}^{N}d\theta_{j}$. The constant $c_{\kappa
}$ is chosen so that $\left\langle 1,1\right\rangle _{\mathbb{T}}=1$ (computed
by means of the Macdonald-Mehta-Selberg integral). The nonsymmetric Jack
polynomials are labeled by $\mathbb{N}_{0}^{N}$; in this paper only the labels
$\left( m,n,0,\ldots,0\right) $ will occur.
\section{The Basic Polynomials}
These are the relevant results from Dunkl \cite{D2}. All the polynomials
considered here have coefficients in $\mathbb{Q}\left( \kappa\right) $
(rational functions of $\kappa$ with rational coefficients). The polynomials
$p_{mn}\left( x\right) $ are defined by the generating function
\begin{equation}
\sum_{m,n=0}^{\infty}p_{mn}\left( x\right) s^{m}t^{n}=\left( 1-sx_{1}
\right) ^{-1}\left( 1-tx_{2}\right) ^{-1}\prod_{i=1}^{N}\left( \left(
1-sx_{i}\right) \left( 1-tx_{i}\right) \right) ^{-\kappa} \label{genfun}
\end{equation}
(for\ convergence require $\left| s\right| ,\left| t\right| <\left(
\max_{i}\left| x_{i}\right| \right) ^{-1}$). Note that $p_{mn}\left(
x\right) =x_{1}^{m}x_{2}^{n}$ when $\kappa=0$. In \cite{D2} it was shown that
$\mathcal{D}_{i}p_{mn}=0$ for all $i>2$, $\mathcal{D}_{2}p_{m0}=0,$
$\mathcal{D}_{1}p_{0n}=0$ and
\begin{equation}
\mathcal{D}_{1}p_{mn}=\left( N\kappa+m\right) p_{m-1,n}+\kappa\sum
_{i=0}^{n-1}\left( p_{m+n-1-i,i}-p_{i,m+n-1-i}\right) \label{D1p}
\end{equation}
for $m>n\geq1$, and
\begin{align}
\mathcal{D}_{1}p_{mn} & =\left( \left( N-1\right) \kappa+m\right)
p_{m-1,n}+\kappa p_{n,m-1}\label{D2p}\\
& +\kappa\sum_{i=0}^{m-2}\left( p_{m+n-1-i,i}-p_{i,m+n-1-i}\right) \nonumber
\end{align}
for $n\geq m\geq1$ (the second sum is omitted if $m=1$)$.$ The expression for
$\mathcal{D}_{2}p_{mn}$ can be deduced by interchanging the labels on $p$. The
Pochhammer symbol is $\left( a\right) _{n}=\prod_{i=1}^{n}\left(
a+i-1\right) $ for any $a\in\mathbb{Q}\left( \kappa\right) $ and
$n\in\mathbb{N}_{0}$.
\begin{definition}
\label{wdef}For $m\geq n$ let
\begin{align*}
\omega_{mn} & =p_{mn}+\sum_{j=1}^{n}\frac{\left( -\kappa\right)
_{j}\left( m-n+1\right) _{j-1}}{\left( \kappa+m-n+1\right) _{j}
\,j!}\left( \left( m-n+j\right) p_{m+j,n-j}+jp_{n-j,m+j}\right) ,\\
\omega_{nm} & =\left( 1,2\right) \omega_{mn}.
\end{align*}
\end{definition}
Observe that $\left( 1,2\right) p_{mn}=p_{nm}$, and the coefficients of
$\omega_{mn}$ in the formula are independent of $N$. Also there is the
symmetry property $\left( i,j\right) \omega_{mn}=\omega_{mn}$ for $2<i<j\leq
N$. It can be shown that $\mathcal{D}_{1}x_{1}\omega_{mn}=\left( \left(
N-1\right) \kappa+m+1\right) \omega_{mn}$ and $\left( \mathcal{D}_{2}
x_{2}-\kappa\left( 1,2\right) \right) \omega_{mn}=\left( \left(
N-2\right) \kappa+n+1\right) \omega_{mn}$ for $m\geq n$. By means of the
product rule:
\begin{equation}
\mathcal{D}_{i}\left( fg\right) =f\mathcal{D}_{i}g+\frac{\partial
f}{\partial x_{i}}g+\kappa\sum_{j\neq i}\frac{f-\left( i,j\right) f}
{x_{i}-x_{j}}\left( i,j\right) g, \label{prodrule}
\end{equation}
we can show $\mathcal{D}_{i}x_{i}\omega_{mn}=\left( 1+\left( N-3\right)
\kappa\right) \omega_{mn}+\kappa\left( \left( 1,i\right) +\left(
2,i\right) \right) \omega_{mn}$ and\newline $\left( \mathcal{D}_{i}
x_{i}-\kappa\sum_{j<i}\left( i,j\right) \right) \omega_{mn}=\left( \left(
N-i\right) \kappa+1\right) \omega_{mn}$ for each $i>2$. Thus $\omega_{mn}$
is the nonsymmetric Jack polynomial labeled by $\left( m,n,0,\ldots,0\right)
$). Note that $\omega_{mn}$ is defined whenever $\kappa\notin-\mathbb{N}$,
although $\omega_{mn}=0$ for certain values of $N,m,n$ and $\kappa.$ In fact
this is the key ingredient of our construction.
\begin{theorem}
\label{Dwmn}The following hold for all $\kappa\notin-\mathbb{N}$:
\begin{enumerate}
\item for $m>n$, $\mathcal{D}_{1}\omega_{mn}=\left( N\kappa+m\right)
\omega_{m-1,n}$\newline $+\dfrac{\left( \left( N-1\right) \kappa+n\right)
\kappa}{\kappa+m-n}\left( \omega_{n-1,m}-\dfrac{\kappa}{\kappa+m-n+1}
\omega_{m,n-1}\right) ;$
\item for $m\geq n$, $\mathcal{D}_{2}\omega_{mn}=\left( \left( N-1\right)
\kappa+n\right) \left( \omega_{m,n-1}-\dfrac{\kappa}{\kappa+m-n+1}
\omega_{n-1,m}\right) ;$
\item for $m=n$, $\mathcal{D}_{1}\omega_{nn}=\left( \left( N-1\right)
\kappa+n\right) \left( \omega_{n-1,n}-\dfrac{\kappa}{\kappa+1}\omega
_{n,n-1}\right) .$
\end{enumerate}
\end{theorem}
\begin{proof}
It was shown in \cite[p.192]{D2} that both $\mathcal{D}_{1}\omega_{mn}$ and
$\mathcal{D}_{2}\omega_{mn}$ are in the span of $\left\{ \omega
_{m-1,n},\omega_{n,m-1},\omega_{m,n-1},\omega_{n-1,m}\right\} .$ This implies
that only the coefficients of $p_{m-1,n},p_{n,m-1},p_{m,n-1},p_{n-1,m}$ need
to be calculated. Let $g_{1},g_{2},\ldots$ denote polynomials of the form
$\sum_{j=0}^{n-2}\left( c_{j}p_{m+n-1-j,j}+c_{j}^{\prime}p_{j,m+n-1-j}
\right) $ with coefficients $c_{j},c_{j}^{\prime}\in\mathbb{Q}\left(
\kappa\right) .$ Both formulae (\ref{D1p}), (\ref{D2p}) are used.
Suppose $m>n$, then
\begin{align*}
\mathcal{D}_{1}\omega_{mn} & =\left( N\kappa+m\right) p_{m-1,n}+\kappa
p_{m,n-1}-\kappa p_{n-1,m}\\
& -\frac{\kappa\left( m-n+1\right) }{\kappa+m-n+1}\left( N\kappa
+m+1\right) p_{m,n-1}+g_{1},
\end{align*}
then since $\omega_{m-1,n}=p_{m-1,n}-\frac{\kappa}{\kappa+m-n}\left( \left(
m-n\right) p_{m,n-1}+p_{n-1,m}\right) +g_{2}$ it follows that $\mathcal{D}
_{1}\omega_{mn}-\left( N\kappa+m\right) \omega_{m-1,n}=\frac{\left( \left(
N-1\right) \kappa+n\right) \kappa}{\kappa+m-n}\left( p_{n-1,m}-\frac
{\kappa}{\kappa+m-n+1}p_{m,n-1}\right) +g_{3}$. This proves part (1).
Next suppose $m\geq n$ then
\begin{align*}
\mathcal{D}_{2}\omega_{mn} & =\left( \left( N-1\right) \kappa+n\right)
p_{m,n-1}+\kappa p_{n-1,m}\\
& -\frac{\kappa}{\kappa+m-n+1}\left( N\kappa+m+1\right) p_{n-1,m}+g_{4}\\
& =\left( \left( N-1\right) \kappa+n\right) \left( p_{m,n-1}
-\frac{\kappa}{\kappa+m-n+1}p_{n-1,m}\right) +g_{4},
\end{align*}
and this proves part (2). Part (3) follows from (2) by setting $m=n$ and
applying the transposition $\left( 1,2\right) .$
\end{proof}
The following evaluation formula for $\omega_{mn}\left( 1^{N}\right) $
(where $1^{N}=\left( 1,1,\ldots,1\right) \in\mathbb{R}^{N}$) is a special
case of a general result for nonsymmetric Jack polynomials (see \cite[p.310]
{DX}). Here is a self-contained proof.
\begin{proposition}
\label{val1N}For $m\geq n,$
\[
\omega_{mn}\left( 1^{N}\right) =\frac{\left( N\kappa+1\right) _{m}\left(
\left( N-1\right) \kappa+1\right) _{n}}{\left( m-n\right) !\,n!\,\left(
\kappa+m-n+1\right) _{n}}.
\]
\end{proposition}
\begin{proof}
By the negative binomial theorem $p_{ij}\left( 1^{N}\right) =\frac{\left(
N\kappa+1\right) _{i}\left( N\kappa+1\right) _{j}}{i!\,j!}$. Substituting
this in Definition \ref{wdef} yields
\begin{gather*}
\omega_{mn}\left( 1^{N}\right) =\frac{\left( N\kappa+1\right) _{m}\left(
N\kappa+1\right) _{n}}{m!\,n!}\\
+\sum_{j=1}^{n}\frac{\left( -\kappa\right) _{j}\left( m-n+1\right) _{j-1}
}{\left( \kappa+m-n+1\right) _{j}\,j!}\left( m-n+2j\right) \frac{\left(
N\kappa+1\right) _{m+j}\left( N\kappa+1\right) _{n-j}}{\left( m+j\right)
!\,\left( n-j\right) !}.
\end{gather*}
For now, assume $m>n$ then $\left( m-n+1\right) _{j-1}\left( m-n+2j\right)
=$\newline $\left( m-n\right) _{j}\left( \frac{m-n}{2}+1\right)
_{j}/\left( \frac{m-n}{2}\right) _{j}$ and
\begin{align*}
\omega_{mn}\left( 1^{N}\right) & =\frac{\left( N\kappa+1\right)
_{m}\left( N\kappa+1\right) _{n}}{m!\,n!}\\
& \times\sum_{j=0}^{n}\frac{\left( -n\right) _{j}\left( -\kappa\right)
_{j}\left( N\kappa+1+m\right) _{j}\left( m-n\right) _{j}\left( \frac
{m-n}{2}+1\right) _{j}}{\left( m+1\right) _{j}\left( \kappa+m-n+1\right)
_{j}\left( -N\kappa-n\right) _{j}\left( \frac{m-n}{2}\right) _{j}\,j!}.
\end{align*}
The sum is a terminating well-poised $_{5}F_{4}$ whose value is
\[
\frac{\left( m-n+1\right) _{n}\left( -N\kappa+\kappa-n\right) _{n}
}{\left( \kappa+m-n+1\right) _{n}\left( -N\kappa-n\right) _{n}},
\]
a formula of Dougall (see Bailey \cite[p.25]{B}). The stated formula follows
by using the reversal $\left( a-n\right) _{n}=\left( -1\right) ^{n}\left(
1-a\right) _{n}$. The formula is also valid when $m=n$ (consider $z=m-n$ as a
variable, then the limit of $\left( z\right) _{j}\left( \frac{z}
{2}+1\right) _{j}/\left( \frac{z}{2}\right) _{j}$ as $z\rightarrow0$ is $1$
for $j=0$ and $2\times j!$ for $j\geq1$).
\end{proof}
\section{Restriction to $N=2$}
The main results of this paper revolve around the vanishing of $\omega_{mn}$
for certain values of $N,m,n$ and $\kappa$, but it is also necessary to show
certain $\omega_{mn}\neq0$. This will be accomplished by restricting to $N=2$
(setting $x_{i}=0$ for $i>2$) and finding an explicit formula for $\omega_{mn}.$
\begin{definition}
For $m\geq n$ let
\[
f_{mn}\left( x_{1},x_{2}\right) =\left( x_{1}x_{2}\right) ^{n}\sum
_{j=0}^{m-n}\frac{\left( \kappa+1\right) _{m-n-j}\left( \kappa\right)
_{j}}{\left( m-n-j\right) !\,j!}x_{1}^{m-n-j}x_{2}^{j}.
\]
\end{definition}
\begin{proposition}
For $m\geq n$, $\mathcal{D}_{1}x_{1}f_{mn}=\left( \kappa+m+1\right) f_{mn}$
and \newline $\left( \mathcal{D}_{2}x_{2}-\kappa\left( 1,2\right) \right)
f_{mn}=\left( n+1\right) f_{mn}.$
\end{proposition}
\begin{proof}
It is clear from the generating function (\ref{genfun}) that $\omega
_{m,0}=p_{m,0}=f_{m,0}$ for $m\geq0$. This implies $\mathcal{D}_{1}
x_{1}f_{m,0}=\left( \kappa+m+1\right) f_{m,0}$ and $\mathcal{D}_{2}
x_{2}f_{m,0}=\left( 1+\kappa\left( 1,2\right) \right) f_{m,0}$ (it can be
shown directly from the generating function that $\mathcal{D}_{1}x_{1}
p_{m,0}=\left( \kappa+m+1\right) p_{m,0}$ for $N=2$). By the product rule
(\ref{prodrule})
\begin{align*}
\mathcal{D}_{1}x_{1}f_{mn} & =\left( x_{1}x_{2}\right) ^{n}\mathcal{D}
_{1}x_{1}f_{m-n,0}+x_{1}f_{m-n,0}\frac{\partial}{\partial x_{1}}\left(
x_{1}x_{2}\right) ^{n}\\
& =\left( \kappa+m-n+1+n\right) \left( x_{1}x_{2}\right) ^{n}f_{m-n,0},
\end{align*}
and
\begin{align*}
\left( \mathcal{D}_{2}x_{2}-\kappa\left( 1,2\right) \right) f_{mn} &
=\left( x_{1}x_{2}\right) ^{n}\left( \mathcal{D}_{2}x_{2}-\kappa\left(
1,2\right) \right) f_{m-n,0}+x_{2}f_{m-n,0}\frac{\partial}{\partial x_{2}
}\left( x_{1}x_{2}\right) ^{n}\\
& =\left( n+1\right) \left( x_{1}x_{2}\right) ^{n}f_{m-n,0},
\end{align*}
as claimed.
\end{proof}
Since the joint eigenfunctions of the commuting operators $\mathcal{D}
_{1}x_{1}$ and \newline $\left( \mathcal{D}_{2}x_{2}-\kappa\left(
1,2\right) \right) $ are uniquely determined for generic $\kappa$ (including
$\kappa>0$) we see that $f_{mn}$ is a scalar multiple of $\omega_{mn}$.
Evaluation at $x=\left( 1,1\right) $ determines the constant.
\begin{proposition}
For $N=2,m\geq n,$
\[
\omega_{mn}=\frac{\left( 2\kappa+m-n+1\right) _{n}\left( \kappa+1\right)
_{n}}{\left( \kappa+m-n+1\right) _{n}\,n!}f_{mn}.
\]
\end{proposition}
\begin{proof}
By Proposition \ref{val1N} $\omega_{mn}\left( 1^{2}\right) =\dfrac{\left(
2\kappa+1\right) _{m}\left( \kappa+1\right) _{n}}{\left( \kappa
+m-n+1\right) _{n}\left( m-n\right) !n!}$ while $f_{mn}\left(
1^{2}\right) =\dfrac{\left( 2\kappa+1\right) _{m-n}}{\left( m-n\right)
!}$ (by the Vandermonde sum formula). Since $\left( 2\kappa+1\right)
_{m}=\left( 2\kappa+1\right) _{m-n}\left( 2\kappa+m-n+1\right) _{n}$ this
completes the proof.
\end{proof}
We observe that $f_{mn}\neq0$ provided (as assumed throughout) that
$\kappa\notin-\mathbb{N}$. This leads to the following nontriviality result.
\begin{corollary}
\label{non0}For $N\geq2,m\geq n$ the polynomial $\omega_{mn}\neq0$ provided
that $2\kappa\neq-j$ where $j=m-n+1,m-n+2,\ldots,m$.
\end{corollary}
\section{Some singular polynomials}
These results already appeared in \cite{D2}, and serve as illustration.
\begin{proposition}
For $N\geq2$ and $n\in\mathbb{N}$ such that $N\nmid n$ (equivalently,
$\gcd\left( N,n\right) <N$), $\omega_{n,0}$ is a singular polynomial for
$\kappa=-\frac{n}{N}$.
\end{proposition}
\begin{proof}
Note that $\omega_{n,0}=p_{n,0}$. By formula (\ref{D1p}) $\mathcal{D}
_{1}\omega_{n,0}=\left( N\kappa+n\right) \omega_{n-1,0}$ and $\mathcal{D}
_{i}\omega_{n,0}=0$ for each $i>1$. Further $p_{n,0}\left( 1,0,\ldots\right)
=\frac{\left( \kappa+1\right) _{n}}{n!}$ which is not zero provided
$\kappa\notin-\mathbb{N}$.
\end{proof}
These polynomials have been studied by Chmutova and Etingof \cite{CE} in the
context of representations of the rational Cherednik algebra.
\begin{proposition}
For $N\geq4$ and $n\in\mathbb{N}$ such that $\gcd\left( N-1,n\right)
<\frac{N-1}{2}$, $\omega_{nn}$ is a singular polynomial for $\kappa=-\frac
{n}{N-1}$.
\end{proposition}
\begin{proof}
The condition $\gcd\left( N-1,n\right) <\frac{N-1}{2}$ is equivalent to
excluding the values $\kappa=-j,-j+\frac{1}{2}$ for $j\in\mathbb{N}$. By
Theorem \ref{Dwmn} we have $\mathcal{D}_{1}\omega_{nn}=\left( \left(
N-1\right) \kappa+n\right) \left( \omega_{n-1,n}-\dfrac{\kappa}{\kappa
+1}\omega_{n,n-1}\right) $ which is zero for $\kappa=-\frac{n}{N-1}$,
similarly $\mathcal{D}_{2}\omega_{nn}=0$, and $\mathcal{D}_{i}\omega_{nn}=0 $
for all $i>2$ (for all $\kappa).$ Corollary \ref{non0} shows that $\omega
_{nn}\neq0$ since $2\kappa\notin-\mathbb{N}$.
\end{proof}
Suppose that $M$ is an irreducible $S_{N}$-module of homogeneous polynomials
(that is, $M$ is a linear subspace of the space of polynomials, and is
invariant under the action of each $w\in S_{N}$, and has no proper nontrivial
invariant subspaces) then $M$ is of some isotype (corresponding to an
irreducible representation of $S_{N}$) labeled by a partition $\tau$ of $N$.
\begin{proposition}
\label{mutau}Suppose $\tau$ is a partition of $N$ and the homogeneous
polynomial $f$ is of degree $n$ and of isotype $\tau$, that is, $\mathrm{span}
\left\{ wf:w\in S_{N}\right\} $ is an $S_{N}$-module on which $S_{N}$ acts
by the irreducible representation corresponding to $\tau$, then $\sum
_{i=1}^{N}x_{i}\mathcal{D}_{i}f=\left( n+\kappa\mu\left( \tau\right)
\right) f$, where $\mu\left( \tau\right) =\binom{N}{2}-\frac{1}{2}
\sum_{j=1}^{N}\tau_{j}\left( \tau_{j}+1-2j\right) $.
\end{proposition}
\begin{proof}
It is easy to show that $\sum\limits_{i=1}^{N}x_{i}\mathcal{D}_{i}
f=\sum\limits_{i=1}^{N}x_{i}\frac{\partial f}{\partial x_{i}}+\kappa
\sum\limits_{1\leq i<j\leq N}\left( 1-\left( i,j\right) \right) f$ for any
polynomial $f$. The operator $\sum\limits_{1\leq i<j\leq N}\left( 1-\left(
i,j\right) \right) $ is constant on \newline $\mathrm{span}\left\{ wf:w\in
S_{N}\right\} $, and its value is given by Young's formula,\newline
$\binom{N}{2}-\frac{1}{2}\sum_{j=1}^{N}\tau_{j}\left( \tau_{j}+1-2j\right) $
(see \cite[p.177]{D1}). The Euler operator $\sum\limits_{i=1}^{N}x_{i}
\frac{\partial}{\partial x_{i}}$gives the degree of $f$.
\end{proof}
Thus a necessary condition for a homogeneous polynomial $f$ of isotype $\tau$
to be singular is that $\kappa=-\frac{\deg f}{\mu\left( \tau\right) }.$ The
isotype for $\mathrm{span}\left\{ w\omega_{n,0}:w\in S_{N}\right\} $ when
$\kappa=-\frac{n}{N}$ is $\left( N-1,1\right) $ and $\mu\left( \left(
N-1,1\right) \right) =N$. For the values $\kappa=-\frac{n}{N-1}$with
$2\kappa\notin-\mathbb{N}$ the isotype of $\mathrm{span}\left\{ w\omega
_{nn}:w\in S_{N}\right\} $ is $\left( N-2,2\right) $ and $\mu\left(
\left( N-2,2\right) \right) =2N-2$. It is exactly the filling of the gap at
$\gcd\left( N-1,n\right) =\frac{N-1}{2}$ for $N$ being odd that we consider
in the sequel. Before we leave this section we point out that the singular
polynomials described so far do not depend on $N$ (with the exception just
noted), that is, the singularity property holds for all $N$. This no longer
holds once we consider isotypes corresponding to partitions with more than two parts.
\section{Singular polynomials for half-integer parameter values}
In this section we show that the polynomials $\omega_{\left( 2l+1\right)
\left( m+1\right) ,\left( 2l+1\right) m}$ are singular for $N=2m+1$ and
$\kappa=-l-\frac{1}{2}$, for $l\in\mathbb{N}_{0}$ and $m\in\mathbb{N}.$ By
Theorem \ref{Dwmn} we already know $\mathcal{D}_{i}\omega_{\left(
2l+1\right) \left( m+1\right) ,\left( 2l+1\right) m}=0$ for $i\geq2$.
Thus we need to show that $\omega_{\left( 2l+1\right) \left( m+1\right)
-1,\left( 2l+1\right) m}=0$ for these choices of $N,\kappa$. This will be
done by introducing a new basis of polynomials related to the $\left\{
p_{mn}\right\} $ basis by a linear relation involving Krawtchouk polynomials.
\begin{definition}
\label{qdef}The homogeneous polynomials $q_{mn}$ (for $m,n\in\mathbb{N}_{0}$)
are defined by
\[
\sum_{m,n=0}^{\infty}q_{mn}\left( x\right) u^{m}v^{n}=\sum_{i,j=0}^{\infty
}p_{ij}\left( x\right) \left( u+v\right) ^{i}\left( u-v\right) ^{j},
\]
the generating function converges for $\left| u\right| ,\left| v\right|
<\left( \max_{i}\left| x_{i}\right| \right) /2$.
\end{definition}
We state the basic properties of the symmetric Krawtchouk polynomials (see
Szeg\"{o},\cite[p.36]{Sz}). They are orthogonal for the binomial distributions
with parameter $\frac{1}{2}$. Fix $n\in\mathbb{N}$ then the Krawtchouk
polynomial of degree $m$ (parameters $n,\frac{1}{2}$) with $0\leq m\leq n$ is
given by
\[
K_{m}\left( t;n\right) =\frac{1}{\binom{n}{m}}\sum_{j=0}^{m}\frac{\left(
t-n\right) _{m-j}\left( -t\right) _{j}}{\left( m-j\right) !\,j!}\left(
-1\right) ^{m-j}.
\]
Then the following hold for $0\leq m,l\leq n$:
\begin{enumerate}
\item $K_{m}\left( 0;n\right) =1$, normalization;
\item $\left( 1-s\right) ^{l}\left( 1+s\right) ^{n-l}=\sum_{m=0}^{n}
s^{m}\binom{n}{m}K_{m}\left( l;n\right) $, generating function;
\item $2^{-n}\sum_{t=0}^{n}\binom{n}{t}K_{m}\left( t;n\right) K_{l}\left(
t;n\right) =\delta_{ml}\binom{n}{m}^{-1}$, orthogonality;
\item $K_{m}\left( l;n\right) =\,_{2}F_{1}\left( -l,-m;-n;2\right) $,
hypergeometric polynomial;
\item $K_{m}\left( l;n\right) =K_{l}\left( m;n\right) $, symmetry;
\item $K_{m}\left( n-t;n\right) =\left( -1\right) ^{m}K_{m}\left(
t;n\right) $, parity$.$
\end{enumerate}
Any expansion in $\left\{ p_{n-i,i}:1\leq i\leq n\right\} $ can be
transformed to one in \newline $\left\{ q_{n-i,i}:1\leq i\leq n\right\} $
($n\in\mathbb{N}$) by means of Krawtchouk polynomials.
\begin{lemma}
Suppose $n\in\mathbb{N}$ and $f=\sum_{i=0}^{n}c_{i}p_{n-i,i}$ with
coefficients $c_{i}\in\mathbb{Q}\left( \kappa\right) $, then
\[
f=\frac{1}{2^{n}}\sum_{i=0}^{n}q_{n-i,i}\sum_{j=0}^{n}\binom{n}{j}c_{j}
K_{i}\left( j;n\right) .
\]
\end{lemma}
\begin{proof}
In Definition \ref{qdef} replace $u,v$ by $\frac{s+t}{2},\frac{s-t}{2}$
respectively then $p_{n-j,j}$ equals the coefficient of $s^{n-j}t^{j}$ in
$2^{-n}\sum_{i=0}^{n}q_{n-i,i}\left( s+t\right) ^{n-i}\left( s-t\right)
^{i}=$\newline $2^{-n}\sum_{i=0}^{n}q_{n-i,i}\sum_{j=0}^{n}\binom{n}{j}
K_{j}\left( i;n\right) s^{n-j}t^{j}$. The lemma now follows from the
symmetry relation.
\end{proof}
We use the lemma to show that for special values of $n,i,\kappa$ the
coefficients of $\omega_{n-i,i}$ with respect to $\left\{ q_{n-j,j}:0\leq
j\leq n\right\} $ have a vanishing property: $\omega_{n-i,i}=\sum_{j=0}
^{n}c_{j}q_{n-j,j}$ and $c_{j}=0$ for $j>n-2i$. We will also find a similar
result for $\omega_{n-i,i}+\omega_{i,n-i}$.
\begin{proposition}
\label{w2q0}Suppose $n$ is even, $1\leq i\leq\frac{n}{2}$ and $\kappa
=-\frac{1}{2}\left( n-2i+1\right) $, then $\omega_{n-i,i}=\sum
\limits_{j=0}^{n-2i}c_{j}q_{n-j,j}$ with coefficients $c_{j}\in\mathbb{Q}$.
\end{proposition}
\begin{proof}
Substitute $\kappa=-\frac{1}{2}\left( n-2i+1\right) $ in Definition
\ref{wdef} for $\omega_{n-i,i}$ to obtain
\[
\omega_{n-i,i}=p_{n-i,i}+\sum_{l=1}^{i}\left( \frac{\left( n-2i+1\right)
_{l}}{l!}p_{n-i+l,i-l}+\frac{\left( n-2i+1\right) _{l-1}}{\left(
l-1\right) !}p_{i-l,n-i+l}\right) .
\]
Extract the coefficients of $\omega_{n-i,i}$ with respect to $p_{n-j,j}$ as
follows: for $0\leq j\leq i-1$ replace $l$ by $i-j$ in the first part of the
sum, the value is
\[
\frac{\left( n-2i+1\right) _{i-j}}{\left( i-j\right) !}=\frac{\left(
n-2i+i-j\right) !}{\left( n-2i\right) !\left( i-j\right) !}=\frac{\left(
i-j+1\right) _{n-2i}}{\left( n-2i\right) !};
\]
this is also valid for $j=i$; the coefficient is zero for $i+1\leq j\leq n-i$;
for $n-i<j\leq n$ replace $l$ by $j-n+i$ then
\[
\frac{\left( n-2i+1\right) _{l-1}}{\left( l-1\right) !}=\frac{\left(
l\right) _{n-2i}}{\left( n-2i\right) !}=\frac{\left( j-n+i\right)
_{n-2i}}{\left( n-2i\right) !}=\left( -1\right) ^{n}\frac{\left(
i-j+1\right) _{n-2i}}{\left( n-2i\right) !}
\]
(reversal of the Pochhammer symbol) but $n$ is even, so the value is
$\frac{\left( i-j+1\right) _{n-2i}}{\left( n-2i\right) !}$ just as for
$0\leq j\leq i.$ The expression $\left( i-j+1\right) _{n-2i}$ is a
polynomial in $j$, vanishing at $i+1,i+2,\ldots,n-i$. Thus
\begin{align*}
\omega_{n-i,i} & =\sum_{j=0}^{n}\frac{\left( i-j+1\right) _{n-2i}}{\left(
n-2i\right) !}p_{n-j,j}\\
& =\frac{1}{2^{n}}\sum_{l=0}^{n}q_{n-l,l}\sum_{j=0}^{n}\binom{n}{j}
\frac{\left( i-j+1\right) _{n-2i}}{\left( n-2i\right) !}K_{l}\left(
j;n\right) ,
\end{align*}
by the Lemma. The orthogonality property of $K_{l}$ shows that the coefficient
of $q_{n-l,l}$ vanishes for $l>n-2i$.
\end{proof}
\begin{proposition}
\label{w2q1}Suppose $n$ is odd, $0\leq i<\frac{n}{2}$ and $\kappa=-\frac{1}
{2}\left( n-2i\right) $, then $\omega_{n-i,i}+\omega_{i,n-i}=\sum
_{j=0}^{n-2i-1}c_{j}q_{n-j,j}$ with coefficients $c_{j}\in\mathbb{Q}$.
\end{proposition}
\begin{proof}
It follows from Definition \ref{wdef} that
\[
\omega_{n-i,i}+\omega_{i,n-i}=\sum_{l=0}^{i}\frac{\left( -\kappa\right)
_{l}\left( n-2i\right) _{l}}{\left( \kappa+n-2i+1\right) _{l}\,l!}
\frac{n-2i+2l}{n-2i}\left( p_{n-i+l,l-i}+p_{l-i,n-i+l}\right) .
\]
When $\kappa=-\frac{1}{2}\left( n-2i\right) $ we obtain $\frac{\left(
-\kappa\right) _{l}}{\left( \kappa+n-2i+1\right) _{l}\,}=\frac
{n-2i}{n-2i+2l}$. As before, $\frac{\left( n-2i\right) _{l}}{l!}
=\frac{\left( l+1\right) _{n-2i-1}}{\left( n-2i-1\right) !}$, and replace
$l$ by $i-j$ and $j-n+i$ respectively for the ranges $0\leq j\leq i$ and
$n-i\leq i\leq n$ respectively. Also $\left( j-n+i+1\right) _{n-2i-1}
=\left( -1\right) ^{n-1}\left( i-j+1\right) _{n-2i-1}$ and $n$ is odd. The
polynomial $\left( i-j+1\right) _{n-2i-1}$ in $j$ vanishes at $i+1,\ldots
,n-i-1$. Similarly to the previous proposition we find that
\[
\omega_{n-i,i}+\omega_{i,n-i}=\frac{1}{2^{n}}\sum_{l=0}^{n}q_{n-l,l}\sum
_{j=0}^{n}\binom{n}{j}\frac{\left( i-j+1\right) _{n-2i-1}}{\left(
n-2i-1\right) !}K_{l}\left( j;n\right) ,
\]
and the coefficient of $q_{n-l,l}$ vanishes for $l>n-2i-1$. Because $\left(
i-j+1\right) _{n-2i-1}=\left( i-\left( n-j\right) +1\right) _{n-2i-1}$
the coefficients of $q_{n-l,l}$ also vanish when $l$ is odd.
\end{proof}
We finish the construction of singular polynomials by showing for certain
values of $\kappa,N,n,i$ that the polynomials $q_{n-l,l}$ vanish for $0\leq
l\leq n-2i$. First we consider some partial products in the generating function.
\begin{definition}
The power series $A_{n}\left( u;\kappa\right) ,B_{n}\left( u;\kappa\right)
$ (arbitrary $\kappa\in\mathbb{C},\,\left| u\right| ,\left| v\right|
<\frac{1}{2}$ and $n\in\mathbb{N}_{0}$) are given by
\begin{align*}
\left( 1-\left( u+v\right) \right) ^{-\kappa}\left( 1-\left( u-v\right)
\right) ^{-\kappa} & =\sum_{n=0}^{\infty}A_{n}\left( u;\kappa\right)
v^{n},\\
\left( 1-\left( u+v\right) \right) ^{-\kappa-1}\left( 1-\left(
u-v\right) \right) ^{-\kappa} & =\sum_{n=0}^{\infty}B_{n}\left(
u;\kappa\right) v^{n}.
\end{align*}
\end{definition}
When $\kappa=-l-\frac{1}{2},l\in\mathbb{N}_{0}$ some of the series
$A_{n},B_{n}$ are actually polynomials.
\begin{lemma}
For $n\leq2l$, the functions $A_{n}\left( u;-l-\frac{1}{2}\right)
,B_{n}\left( u;-l-\frac{1}{2}\right) $ are polynomials in $u$ of degree
$2l+1-n,2l-n$ respectively.
\end{lemma}
\begin{proof}
For the first part
\begin{align*}
\sum_{n=0}^{\infty}A_{n}\left( u;\kappa\right) v^{n} & =\left(
1-u\right) ^{-2\kappa}\left( 1-\left( \frac{v}{1-u}\right) ^{2}\right)
^{-\kappa}\\
& =\sum_{j=0}^{\infty}\frac{\left( \kappa\right) _{j}}{j!}v^{2j}\left(
1-u\right) ^{-2\kappa-2j}.
\end{align*}
Thus $A_{n}=0$ if $n$ is odd and $A_{2j}\left( u;-l-\frac{1}{2}\right)
=\frac{\left( -l-\frac{1}{2}\right) _{j}}{j!}\left( 1-u\right) ^{2l+1-2j}
$, which is a polynomial of degree $2l+1-2j$ provided $2j\leq2l$. For the
second part
\begin{gather*}
\sum_{n=0}^{\infty}B_{n}\left( u;\kappa\right) v^{n}=\left( 1-u\right)
^{-2\kappa-1}\left( 1+\frac{v}{1-u}\right) \left( 1-\left( \frac{v}
{1-u}\right) ^{2}\right) ^{-\kappa-1}\\
=\sum_{j=0}^{\infty}\frac{\left( \kappa+1\right) _{j}}{j!}\left(
v^{2j}\left( 1-u\right) ^{-2\kappa-1-2j}+v^{2j+1}\left( 1-u\right)
^{-2\kappa-2j-2}\right) .
\end{gather*}
Thus $B_{n}\left( u;-l-\frac{1}{2}\right) =\frac{\left( -l+\frac{1}
{2}\right) _{j}}{j!}\left( 1-u\right) ^{2l-n}$ (with $j=\left\lfloor
\frac{n}{2}\right\rfloor $), a polynomial of degree $2l-n$ provided $n\leq2l$.
\end{proof}
To be precise the polynomials $A_{n}\left( u;-l-\frac{1}{2}\right) $ are of
degree $\leq2l+1-n$ (being $0$ when $n$ is odd).
\begin{proposition}
\label{q2z}Let $\kappa=-l-\frac{1}{2},l\in\mathbb{N}_{0}$ and suppose that
$n\leq2l$ then $m+n\geq N\left( 2l+1\right) -1$ implies $q_{mn}=0$.
\end{proposition}
\begin{proof}
By combining the generating function (\ref{genfun}) for $\left\{
p_{mn}\right\} $ and Definition \ref{qdef} we obtain
\begin{align*}
\sum_{m,n=0}^{\infty}q_{mn}u^{m}v^{n} & =\sum_{\alpha\in\mathbb{N}_{0}^{N}
}B_{\alpha_{1}}\left( ux_{1};\kappa\right) \left( -1\right) ^{\alpha_{2}
}B_{\alpha_{2}}\left( ux_{2};\kappa\right) \\
& \times\prod_{s=3}^{N}A_{\alpha_{s}}\left( ux_{s};\kappa\right) x^{\alpha
}v^{\left| \alpha\right| }.
\end{align*}
For a fixed $n\leq2l$ the coefficient of $v^{n}$ is the sum over $\alpha
\in\mathbb{N}_{0}^{N}$ with $\left| \alpha\right| =n$. For each $\alpha$
with $\left| \alpha\right| =n$ (implying that each $\alpha_{i}\leq2l$ and
the Lemma applies) the corresponding term is a product of polynomials in $u$
of degree $\left( 2l-\alpha_{1}\right) +\left( 2l-\alpha_{2}\right)
+\sum_{s=3}^{N}\left( 2l+1-\alpha_{s}\right) =N\left( 2l+1\right) -2-n$.
This shows that the coefficient of $u^{m}v^{n}$ vanishes if $m\geq N\left(
2l+1\right) -1-n$.
\end{proof}
\begin{theorem}
Let $\kappa=-l-\frac{1}{2}$ and $N=2m+1$, with $l\in\mathbb{N}_{0}
,m\in\mathbb{N}$ then $\omega_{\left( 2l+1\right) \left( m+1\right)
,\left( 2l+1\right) m}$ is a singular polynomial, of isotype $\left(
N-2,1,1\right) $, and \newline $\left( 1+\left( 1,2\right) \right)
\omega_{\left( 2l+1\right) \left( m+1\right) ,\left( 2l+1\right) m}=0$.
\end{theorem}
\begin{proof}
Let $a=\left( 2l+1\right) \left( m+1\right) ,b=\left( 2l+1\right) m$.
Since $-2\kappa=2l+1<a-b+1$ Corollary \ref{non0} shows that $\omega_{ab}\neq
0$. By Theorem \ref{Dwmn} $\mathcal{D}_{2}\omega_{ab}=\left( 2m\kappa
+b\right) \left( \omega_{a,b-1}-\frac{\kappa}{\kappa+a-b+1}\omega
_{b-1,a}\right) =0 $ for $\kappa=-\frac{b}{2m}=-l-\frac{1}{2}.$ Similarly
$\mathcal{D}_{1}\omega_{ab}=\left( N\kappa+a\right) \omega_{a-1,b}.$ By
Proposition \ref{w2q0} $\omega_{a-1,b}=\sum_{j=0}^{a-1-b}c_{j}q_{a+b-1-j,j}$
(since -$\frac{1}{2}\left( a-b\right) =\kappa$) with some coefficients
$c_{j}\in\mathbb{Q}$. Since $a-1-b=2l$ and $a+b-1=N\left( 2l+1\right) -1$,
Proposition \ref{q2z} shows that each $q_{a+b-1-j,j}=0$ (for $j\leq2l$).
Let $M=\mathrm{span}\left\{ w\omega_{\left( 2l+1\right) \left( m+1\right)
,\left( 2l+1\right) m}:w\in S_{N}\right\} $. By the invariance properties
of $\left\{ \mathcal{D}_{i}:1\leq i\leq N\right\} $ any nonzero element of
$M$ is also singular. In general for $m>n$ the $S_{N}$-module $\mathrm{span}
\left\{ w\omega_{mn}:w\in S_{N}\right\} $, which realizes the representation
of $S_{N}$ induced up from the trivial representation of $S_{1}\times
S_{1}\times S_{N-2}$, decomposes into the isotypes $\left( N-2,1,1\right)
,\left( N-2,2\right) ,\left( N-1,1\right) ,\left( N\right) $ (see
\cite[p.115]{M}). The eigenvalues $\mu\left( \tau\right) $ are $2N,2N-2,N,0$
respectively (see Proposition \ref{mutau}); but the singularity condition
implies $N\left( 2l+1\right) +\kappa\mu\left( \tau\right) =0$ thus the
latter three can not contain singular polynomials for $\kappa=-l-\frac{1}{2}$
(note the degree of the polynomial $\omega_{ab}$ is $N\left( 2l+1\right) $).
This implies that $M$ is of isotype $\left( N-2,1,1\right) $ and hence is of
dimension $\binom{N-1}{2}$.
By Proposition \ref{w2q1} $\omega_{ab}+\omega_{ba}=\sum_{j=0}^{a-b-1}
c_{j}q_{a+b-j,j}$ (since $-\frac{1}{2}\left( a-b\right) =\kappa$ and $a+b$
is odd) with $c_{j}\in\mathbb{Q}$. Since $a-b-1=2l$ and $a+b=N\left(
2l+1\right) $, Proposition \ref{q2z} shows that each $q_{a+b-j,j}=0$.
\end{proof}
It is interesting that the parameters of the singular polynomials just barely
satisfy the various inequalities appearing in the preparatory results.
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\begin{document}
\title[The Crepant Transformation Conjecture]{The Crepant Transformation Conjecture\\ for Toric Complete Intersections}
\author[Coates]{Tom Coates}
\address{Department of Mathematics\\
Imperial College London\\
180 Queen's Gate\\
London SW7 2AZ
\\UK}
\email{t.coates@imperial.ac.uk}
\author[Iritani]{Hiroshi Iritani}
\address{Department of Mathematics\\
Graduate School of Science\\
Kyoto University\\
Oiwake-cho\\
Kitashirakawa\\
Sakyo-ku\\
Kyoto, 606-8502\\
Japan}
\email{iritani@math.kyoto-u.ac.jp}
\author[Jiang]{Yunfeng Jiang}
\address{Department of Mathematics\\
University of Kansas\\
1460 Jayhawk Boulevard\\
Lawrence, Kansas 66045-7594\\
USA}
\email{y.jiang@ku.edu}
\subjclass[2010]{14N35 (Primary); 14A20, 14E16, 14F05, 53D45 (Secondary)}
\keywords{Gromov--Witten theory, Crepant Resolution Conjecture, toric Deligne--Mumford stacks, orbifolds, quantum cohomology, mirror symmetry, Fourier--Mukai transformation, flop, $K$-equivalence, Gamma class, integral structure, variation of GIT quotient, Givental's symplectic formalism, GKZ system, Mellin--Barnes method}
\date{}
\begin{abstract}
Let $X$ and $Y$ be $K$-equivalent toric Deligne--Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for $X$ and $Y$ become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for $X$ and $Y$, with a Fourier--Mukai transformation between the $K$-groups of $X$ and $Y$, via an equivariant version of the Gamma-integral structure on quantum cohomology. We prove similar results for toric complete intersections. We impose only very weak geometric hypotheses on $X$ and $Y$: they can be non-compact, for example, and need not be weak Fano or have Gorenstein coarse moduli space. Our main tools are the Mirror Theorems for toric Deligne--Mumford stacks and toric complete intersections, and the Mellin--Barnes method for analytic continuation of hypergeometric functions.
\end{abstract}
\maketitle
\section{Introduction}
A birational map $\varphi \colon X_+ \dashrightarrow X_-$ between smooth varieties, orbifolds, or Deligne--Mumford stacks is called a \emph{$K$-equivalence}
if there exists a smooth variety, orbifold, or Deligne--Mumford stack $\tX$ and projective birational morphisms
$f_\pm \colon \tX \to X_\pm$ such that $f_- = \varphi \circ f_+$
and $f_+^\star K_{X_+} = f_-^\star K_{X_-}$:
\begin{equation}
\label{eq:common_blowup}
\begin{aligned}
\xymatrix{
& \tX \ar[dr]^{f_-} \ar[dl]_{f_+} & \\
X_+ \ar@{-->}[rr]^{\varphi} & & X_-
}
\end{aligned}
\end{equation}
In this case, the celebrated Crepant Transformation Conjecture of Y.~Ruan predicts that the quantum (orbifold) cohomology algebras of $X_+$ and $X_-$ should be related by analytic continuation in the quantum parameters. This conjecture has stimulated a great deal of interest in the connections between quantum cohomology (or Gromov--Witten theory) and birational geometry: see, for example, \cite{Ruan:crepant,Perroni,Boissiere--Mann--Perroni:1,Boissiere--Mann--Perroni:2,Bryan--Graber--Pandharipande,Bryan--Gholampour:1,Bryan--Gholampour:2,Gillam,Wise,Coates,Lee--Lin--Wang:1,Lee--Lin--Wang:2,Lee--Lin--Wang:ICCM,Iwao--Lee--Lin--Wang,Lee--Lin--Qu--Wang,Chen--Li--Li--Zhao,Chen--Li--Zhang--Zhao,Zhou,Li--Ruan,Gonzalez--Woodward}. Ruan's original conjecture was subsequently refined, revised, and extended to higher genus Gromov--Witten invariants, first by Bryan--Graber~\cite{Bryan--Graber} under some additional hypotheses, and then by Coates--Iritani--Tseng, Iritani, and Ruan in general~\cite{CIT,Iritani,Coates--Ruan}. Recall that a toric Deligne--Mumford stack $X$ can be constructed as a GIT quotient $\big[\CC^m /\!\!/_\omega K\big]$ of $\CC^m$ by an action of a complex torus $K$, where $\omega$ is an appropriate \emph{stability condition}, and that wall-crossing in the space of stability conditions induces birational transformations between GIT quotients~\cite{Dolgachev--Hu,Thaddeus}. Our main result implies the CIT/Ruan version of the Crepant Transformation Conjecture in genus zero, in the case where $X_+$ and $X_-$ are complete intersections in toric Deligne--Mumford stacks and $\varphi \colon X_+ \dashrightarrow X_-$ arises from a toric wall-crossing. We concentrate initially on the case where $X_+$ and $X_-$ are toric, deferring the discussion of toric complete intersections to~\S\ref{sec:introduction_ci}.
\subsection{The Toric Case}
\label{sec:toric_case}
We consider toric Deligne--Mumford stacks $X_\pm$ of the form $\big[\CC^m /\!\!/_\omega K \big]$, where $K$ is a complex torus, and consider a $K$-equivalence $\varphi \colon X_+ \dashrightarrow X_-$ determined by a wall-crossing in the space of stability conditions~$\omega$. The action of $T=(\Cstar)^m$ on $\CC^m$ descends to give (ineffective) actions of $T$ on $X_\pm$, and we consider the $T$-equivariant Chen--Ruan cohomology groups $H_{\CR,T}^\bullet(X_\pm)$~\cite{Chen--Ruan:orbifold_GW}. There is a $T$-equivariant big quantum product $\star_\tau$ on $H_{\CR,T}^\bullet(X_\pm)$, parametrized by $\tau \in H_{\CR,T}^\bullet(X_\pm)$ and defined in terms of $T$-equivariant Gromov--Witten invariants of $X_\pm$. The \emph{$T$-equivariant quantum connection} is a pencil of flat connections:
\begin{equation}
\label{eq:qconn}
\nabla = d + z^{-1} \sum_{i=0}^N (\phi_i \star_\tau) d\tau^i
\end{equation}
on the trivial $H_{\CR,T}^\bullet(X_\pm)$-bundle over an open set in $H_{\CR,T}^\bullet(X_\pm)$; here $z \in \Cstar$ is the pencil variable, $\tau \in H_{\CR,T}^\bullet(X_\pm)$ is the co-ordinate on the base of the bundle, $\phi_0,\dots,\phi_N$ are a basis for $H_{\CR,T}^\bullet(X_\pm)$, and $\tau^0,\dots, \tau^N$ are the corresponding co-ordinates of $\tau \in H_{\CR,T}^\bullet(X_\pm)$, so that $\tau = \sum_{i=0}^N \tau^i \phi_i$.
\begin{theorem}
\label{thm:main_theorem}
Let $X_+$ and $X_-$ be toric Deligne--Mumford stacks, and let $\varphi \colon X_+ \dashrightarrow X_-$ be a $K$-equivalence that arises from a wall-crossing of GIT stability conditions. Then:
\begin{enumerate}
\item the equivariant quantum connections of $X_\pm$ become gauge-equivalent after analytic continuation in $\tau$, via a gauge transformation $\Theta(\tau,z) \colon H^\bullet_{\CR,T}(X_-) \to H^\bullet_{\CR,T}(X_+)$ which is homogeneous of degree zero, regular at $z=0$, and preserves the equivariant orbifold Poincar\'{e} pairing.
\item there exists a common toric blowup $\tX$ of $X_\pm$ as in \eqref{eq:common_blowup} such that gauge transformation $\Theta$ coincides with the Fourier--Mukai transformation
\begin{align*}
\FM \colon K^0_T(X_-) \to K^0_T(X_+) && E\mapsto (f_+)_\star (f_-)^\star (E)
\end{align*}
via the equivariant Gamma-integral structure introduced in \S\ref{sec:integral_structure} below.
\end{enumerate}
\end{theorem}
\noindent Here:
\begin{itemize}
\item The Gamma-integral structure on equivariant quantum cohomology is an assignment, to each class $E \in K^0_T(X_\pm)$ of $T$-equivariant vector bundles on $X_\pm$, of a flat section $\frs(E)$ for the equivariant quantum connection on $X_\pm$. This gives a
lattice in the space of flat sections which is isomorphic to the integral equivariant $K$-group $K^0_T(X_\pm)$. The flat section $\frs(E)$ is, roughly speaking, given by the Chern character of $E$ multiplied by a characteristic class of $X_\pm$, called the $\hGamma$-class, that is defined in terms of the $\Gamma$-function. Part~(2) of Theorem~\ref{thm:main_theorem} asserts that the flat section $\frs(E)$ analytically continues to $\frs(\FM(E))$.
\item The gauge transformation $\Theta(\tau,z)$ will in general be non-constant: it depends on the parameter $\tau$ for the equivariant quantum product, and also on the parameter $z$ appearing in the equivariant quantum connection. When written in terms of the integral structure, however, it becomes a constant, integral linear transformation.
\end{itemize}
\begin{remark}
Throughout this paper, when we consider $K$-equivalence \eqref{eq:common_blowup} of Deligne--Mumford stacks $X_\pm$, $K_{X_\pm}$
means the canonical class as a stack; in general this is different from the ($\QQ$-Cartier)
canonical divisor $K_{|X_\pm|}$ of the coarse moduli space $|X_\pm|$. In particular, we do not require the coarse moduli spaces $|X_\pm|$ to be Gorenstein.
\end{remark}
\begin{remark}
Gonzalez and Woodward \cite{Gonzalez--Woodward} have proved
a very general wall-crossing formula for Gromov--Witten invariants under variation of GIT quotient, using gauged Gromov--Witten theory.
Their result, which is a quantum version of Kalkman's wall-crossing formula, gives a complete description of how non-equivariant
genus-zero Gromov--Witten invariants change under wall-crossing. Thus their theorem must imply the non-equivariant version of the first part of Theorem~\ref{thm:main_theorem}, and the first part of Theorem~\ref{thm:main_theorem_ci}.
Our methods are significantly less general --- they apply only to toric stacks and toric complete intersections --- but give a much more explicit relationship between the genus-zero
Gromov--Witten theories.
\end{remark}
\noindent Theorem~\ref{thm:main_theorem} is slightly imprecisely stated:
we give precise statements, once the necessary notation and definitions are in place,
as Theorems~\ref{thm:global_qconn},~\ref{thm:U},
and~\ref{thm:CTC_qconn} below.
We now explain how Theorem~\ref{thm:main_theorem} implies
the CIT/Ruan version of the Crepant Transformation Conjecture.
\bigskip
The CIT/Ruan version of the Crepant Transformation Conjecture is stated in terms of Givental's symplectic formalism for Gromov--Witten theory~\cite{Givental:symplectic}. In our context, this associates to $X_\pm$ the vector spaces $\cH(X_\pm) := H^\bullet_{\CR,T}(X_\pm)(\!(z^{-1})\!)$ equipped with a certain symplectic form, and encodes $T$-equivariant genus-zero Gromov--Witten invariants via a Lagrangian cone $\cL_\pm \subset \cH(X_\pm)$. The Givental cone $\cL_\pm$ for $X_\pm$ determines the big quantum product $\star_\tau$ on $H^\bullet_{\CR,T}(X_\pm)$, and \emph{vice versa}. The CIT/Ruan Crepant Transformation Conjecture, made in the context of non-equivariant Gromov--Witten theory, asserts that there exists a $\CC(\!(z^{-1})\!)$-linear grading-preserving symplectic isomorphism $\UU \colon \cH(X_-) \to \cH(X_+)$, such that after analytic continuation of $\cL_\pm$ we have $\UU(\cL_-) = \cL_+$. See~\cite{CIT,Coates--Ruan} for more details.
There are various subtle points in the notion of analytic continuation of the (infinite-dimensional) cones $\cL_\pm$, especially under the weak convergence hypotheses that we impose, and some necessary foundational material is missing. Thus we choose to state Theorem~\ref{thm:main_theorem} in terms of the equivariant quantum connections for $X_\pm$ rather than in terms of the Givental cones $\cL_\pm$. The two formulations are very closely related, however, as we now explain. Let $L_\pm(\tau,z)$ denote a fundamental solution for the equivariant quantum connection $\nabla$, that is, a matrix with columns that give a basis of flat sections for $\nabla$. The assignment
\begin{align*}
\tau \mapsto L_\pm(\tau,z)^{-1} \cH_+ && \tau \in H^\bullet_{\CR,T}(X_\pm) && \text{where $\cH_+:= H^\bullet_{\CR,T}(X_\pm) \otimes \CC[z]$}
\end{align*}
gives the family of tangent spaces to the Givental cone $\cL_\pm$. As emphasized in~\cite{CIT}, this defines a variation of semi-infinite Hodge structure in the sense of Barannikov~\cite{Barannikov:qperiods}. The Givental cone $\cL_\pm$ can be reconstructed from the semi-infinite variation as:
\[
\cL_\pm = \bigcup_{\tau} z L_\pm(\tau,z)^{-1} \cH_+
\]
Thus part~(1) of Theorem~\ref{thm:main_theorem} implies the CIT/Ruan-style Crepant Transformation Conjecture whenever it makes sense, with the symplectic transformation $\UU$ defined in terms of the gauge transformation $\Theta$ by $\UU = L_+^{-1} \Theta L_-$. The fact that $\UU$ is independent of $\tau$ follows from the fact that $\Theta$ is a gauge equivalence. The fact that $\UU$ is symplectic (or equivalently, the fact that $\Theta$ is pairing-preserving) follows from the identification, in part~(2) of Theorem~\ref{thm:main_theorem}, of $\Theta$ with the Fourier--Mukai transformation~$\FM$. The Fourier--Mukai transformation is a derived equivalence and thus preserves the Mukai pairings on $K^0_T(X_\pm)$; this implies, via the equivariant Hirzebruch--Riemann--Roch theorem, that $\Theta$ is pairing-preserving. The identification of $\Theta$ with $\FM$ also makes clear that the symplectic transformation $\UU$ has a well-defined non-equivariant limit, since the Fourier--Mukai transformation itself can be defined non-equivariantly.
In terms of the symplectic transformation $\UU$, part (2) of
Theorem \ref{thm:main_theorem} can be rephrased
as the commutativity of the diagram
\[
\xymatrix{
K_T^0(X_-) \ar[r]^{\FM} \ar[d]_{\tPsi_-} & K_T^0(X_+) \ar[d]^{\tPsi_+} \\
\tcH(X_-) \ar[r]^{\UU} & \tcH(X_+)
}
\]
where $\tcH(X_\pm)$ is a variant of Givental's symplectic space and $\tPsi_\pm$ are certain `framing maps' built from the Gamma-integral structure: see Theorem~\ref{thm:U}. This identification of $\UU$ with a Fourier--Mukai transformation was proposed in \cite{Iritani}. Our results also imply Ruan's original conjecture that the quantum cohomology rings of $X_\pm$ are (abstractly) isomorphic, and that the associated $F$-manifold structures are isomorphic. We refer the reader to \cite{CCIT:computing, CIT, Coates--Ruan, Coates, Iritani:Ruan} for discussions on the consequence of these conjectures and several concrete examples.
\subsection{The Mellin--Barnes Method and the Work of Borisov--Horja}
The main ingredients in the proof of Theorem~\ref{thm:main_theorem} are the Mirror Theorem for toric stacks~\cite{CCIT,Cheong--Ciocan-Fontanine--Kim}, which determines the equivariant quantum connection $\nabla$ (or, equivalently, the Givental cone $\cL_\pm)$ in terms of a certain cohomology-valued hypergeometric function called the \emph{$I$-function}, and the Mellin--Barnes method~\cite{Barnes,CdOGP}, which allows us to analytically continue the $I$-functions for $X_\pm$. From this point of view, the symplectic transformation $\UU$ arises as the matrix which intertwines the two $I$-functions (see Theorem~\ref{thm:U}):
\[
\UU I_- = I_+.
\]
On the other hand, components of the $I$-function
give hypergeometric solutions to the Gelfand--Kapranov--Zelevinsky (GKZ) system of
differential equations. The analytic continuation of
solutions to the GKZ system has been studied by Borisov--Horja
\cite{Borisov--Horja:FM}.
They showed that, under an appropriate identification of the
spaces of GKZ solutions with the $K$-groups of the corresponding
toric Deligne--Mumford stacks, the analytic continuation of
solutions to a GKZ system is induced by a Fourier--Mukai
transformation between the $K$-groups.
Our computation may be viewed as a straightforward
generalization of theirs. The differences from their situation
are:
\begin{itemize}
\item[(a)] we work with a fully equivariant version, that is,
the parameters $\beta_j$ in the GKZ system are arbitrary
and we use the equivariant $K$-groups (here
$\beta_j$ corresponds to the equivariant parameter);
\item[(b)] we compute analytic continuation of the $I$-function
corresponding to the \emph{big} quantum cohomology;
in terms of the GKZ system, we do not assume that
lattice vectors in the set\footnote
{Recall that Gelfand--Kapranov--Zelevinsky defined the GKZ system
in terms of a finite set $A \subset \ZZ^d$.
They called it the $A$-hypergeometric system.}
$A$ lie on a hyperplane of height one.
\end{itemize}
\noindent Since we work equivariantly, we can use the fixed point basis in localized equivariant cohomology to calculate the analytic continuation of the $I$-functions. It turns out that analytic continuation via the Mellin--Barnes method becomes much easier to handle in the fully equivariant setting, because we only need to evaluate residues at \emph{simple} poles\footnote{For an example of the complexities caused by non-simple poles, see the orbifold flop calculation in~\cite[\S7]{Coates}.}. It is also straightforward to compute the Fourier--Mukai transformation in terms of the fixed point basis in the localized equivariant $K$-group, and hence to see that analytic continuation coincides with Fourier--Mukai.
Regarding part (b) above, we choose $A$ to be the set $\{b_1,\dots,b_m\} \subset \bN$ of ray vectors of an extended stacky fan~\cite{Borisov--Chen--Smith, Jiang}. Since we do not restrict ourselves to the weak Fano case, and since we work with Jiang's extended stacky fans, the generic rank of the GKZ system can be bigger than the rank of $H^\bullet_{\CR,T}(X_\pm)$. To remedy this, we treat one special variable analytically and work formally in the other variables. In fact, the big $I$-functions are not necessarily convergent in all of the variables, and we analytically continue the $I$-function with respect to one specific variable $\sfy_r$. This amounts to considering an adic completion of the Borisov--Horja better-behaved GKZ system~\cite{Borisov--Horja:GKZ} with respect to the other variables. The analytic continuation in Theorem~\ref{thm:main_theorem} occurs across a ``global K\"{a}hler moduli space'' $\tcM^\circ$ which is treated as an analytic space in one direction and as a formal scheme in the other directions.
\subsection{The Toric Complete Intersection Case}
\label{sec:introduction_ci}
Let $\varphi \colon X_+ \dashrightarrow X_-$ be a $K$-equivalence between toric Deligne--Mumford stacks that arises from a toric wall-crossing, as in \S\ref{sec:toric_case}. Let $\widetilde{X}$ be the common toric blow-up of $X_\pm$ and let $\overline{X}_0$ denote the common blow-down; $\overline{X}_0$ here is a (singular) toric variety, not a stack.
\[
\xymatrix{
& \tX \ar[dr]^{f_-} \ar[dl]_{f_+} & \\
X_+ \ar@{-->}[rr]^{\varphi} \ar[dr]_{g_+} & & X_- \ar[dl]^{g_-} \\
& \overline{X}_0
}
\]
Consider a direct sum of semiample line bundles $E_0 \to \overline{X}_0$, and pull this back to give vector bundles $E_+ \to X_+$, $\widetilde{E} \to \widetilde{X}$, and $E_- \to X_-$. Let $s_+$,~$\tilde{s}$, and~$s_-$ be sections of, respectively, $E_+$,~$\widetilde{E}$, and~$E_-$ that are compatible via $f_+$ and $f_-$ (so $f_+^\star s_+ = \widetilde{s} = f_-^\star s_-$) such that the zero loci of $s_\pm$ intersect the flopping locus of $\varphi$ transversely. Let $Y_+$,~$\widetilde{Y}$, and $Y_-$ denote the substacks defined by the zero loci of, respectively, $s_+$,~$\tilde{s}$, and $s_-$. In this situation there is a commutative diagram:
\begin{equation}
\label{eq:Y_and_X}
\begin{aligned}
\xymatrix{
& \widetilde{Y} \ar[ld]_-{F_-} \ar[rd]^-{F_+} \ar[d]_-{\tilde{\iota}} \\
Y_-\ar[d]_{\iota_-} &\widetilde{X} \ar[ld]^-{f_-} \ar[rd]_-{f_+} & Y_+ \ar[d]^{\iota_+}\\
X_- && X_+
}
\end{aligned}
\end{equation}
where the vertical maps are inclusions, the bottom triangle is \eqref{eq:common_blowup}, and the squares are Cartesian. The $K$-equivalence $\varphi \colon X_+ \dashrightarrow X_-$ induces a $K$-equivalence $\varphi \colon Y_+ \dashrightarrow Y_-$. We now consider the Crepant Transformation Conjecture for $\varphi \colon Y_+ \dashrightarrow Y_-$.
Since the complete intersections $Y_\pm$ will not in general be $T$-invariant we consider non-equivariant Gromov--Witten invariants and the non-equivariant quantum product. (Our assumptions on $X_\pm$ ensure that the non-equivariant theory makes sense.) Denote by $H^\bullet_\amb(Y_\pm)$ the image $\im \iota^\star_\pm \subset H_{\CR}^\bullet(Y_\pm)$, where $\iota_\pm \colon Y_\pm \to X_\pm$ is the inclusion map. If $\tau \in H_{\amb}^\bullet(Y_\pm)$ then the big quantum product $\star_\tau$ preserves the ambient part $H_{\amb}^\bullet(Y_\pm) \subset H_{\CR}^\bullet(Y_\pm)$. We can therefore define a quantum connection on the ambient part:
\[
\nabla = d + z^{-1} \sum_{i=0}^N (\phi_i \star_\tau) d\tau^i
\]
This is a pencil of flat connections on the trivial $H_{\amb}^\bullet(Y_\pm)$-bundle over an open set in $H_{\amb}^\bullet(Y_\pm)$ where, as in \eqref{eq:qconn}, $z \in \Cstar$ is the pencil variable, $\tau \in H_{\amb}^\bullet(Y_\pm)$ is the co-ordinate on the base of the bundle, $\phi_0,\dots,\phi_N$ are a basis for $H_{\amb}^\bullet(Y_\pm)$, and $\tau^0,\dots, \tau^N$ are the corresponding co-ordinates of~$\tau$.
In \S\ref{sec:ambient_part} below we construct an ambient version of the Gamma-integral structure, which is an assignment to each class $E$ in the ambient part of $K$-theory
\[
K^0_\amb(Y_\pm) = \im \iota_\pm^\star \subset K^0(Y_\pm)
\]
of a flat section $\frs(E)$ for the quantum connection on the ambient part $H_{\amb}^\bullet(Y_\pm)$. This gives a lattice in the space of flat sections which is isomorphic to the ambient part of (integral) $K$-theory $K^0_\amb(Y_\pm)$.
\begin{theorem}
\label{thm:main_theorem_ci}
Let $\varphi \colon Y_+ \dashrightarrow Y_-$ be a $K$-equivalence between toric complete intersections as above. Then:
\begin{enumerate}
\item the quantum connections on the ambient parts $H_{\amb}^\bullet(Y_\pm) \subset H_{\CR}^\bullet(Y_\pm)$ become gauge-equivalent after analytic continuation in $\tau$, via a gauge transformation
\[
\Theta_Y(\tau,z) \colon H^\bullet_{\amb}(Y_-) \to H^\bullet_{\amb}(Y_+)
\]
which is homogeneous of degree zero and regular at $z=0$. If $Y$ is compact then $\Theta_Y$ preserves the orbifold Poincar\'{e} pairing.
\item when expressed in terms of the ambient integral structure, the gauge transformation $\Theta_Y$ coincides with the Fourier--Mukai transformation
\begin{align*}
\FM \colon K^0_\amb(Y_-) \to K^0_\amb(Y_+) && E\mapsto (F_+)_\star (F_-)^\star (E)
\end{align*}
given by the top triangle in \eqref{eq:Y_and_X}.
\end{enumerate}
\end{theorem}
\noindent As before, Theorem~\ref{thm:main_theorem_ci} is slightly imprecisely stated: precise statements can be found as Theorems~\ref{thm:global_qconn_ci},~\ref{thm:U_ci}, and~\ref{thm:CTC_qconn_ci} below. Arguing as in~\S\ref{sec:toric_case} shows that Theorem~\ref{thm:main_theorem_ci} implies the CIT/Ruan version of the Crepant Transformation Conjecture for $\varphi \colon Y_+ \dashrightarrow Y_-$ whenever it makes sense, with the corresponding map
\[
\UU_Y \colon \cH_\amb(Y_-) \to \cH_\amb(Y_+)
\]
between the ambient parts of the Givental spaces for $Y_\pm$ being given by:
\[
\UU_Y = (L_+^\amb)^{-1} \Theta_Y L_-^\amb
\]
where $L_\pm^\amb$ are the fundamental solutions for the quantum connections on the ambient parts $H_\amb^\bullet(Y_\pm)$.
The proof of Theorem~\ref{thm:main_theorem_ci} relies on the Mirror Theorem for toric complete intersections~\cite{CCIT:applications}, and on \emph{non-linear Serre duality}~\cite{Givental:equivariant,Givental:elliptic,Coates--Givental,Tseng}, which relates the quantum cohomology of $Y_\pm$ to the quantum cohomology of the total space of the dual bundles $E_\pm^\vee$. Since $E_\pm^\vee$ is toric, it can be analyzed using Theorem~\ref{thm:main_theorem}.
\begin{remark}
The idea of using non-linear Serre duality to analyze wall-crossing has been developed independently by Lee--Priddis--Shoemaker~\cite{Lee--Priddis--Shoemaker}, in the context of the Landau--Ginzburg/ Calabi--Yau correspondence.
\end{remark}
\begin{example}
A mirror $Y$ to the quintic $3$-fold arises~\cite{Greene--Plesser,CdOGP,Batyrev} as a crepant resolution of an anticanonical hypersurface in $X = \big[ \PP^4 / (\ZZ/5\ZZ)^3 \big]$. A mirror theorem for $Y$ has been proved by Lee--Shoemaker~\cite{Lee--Shoemaker}. The variety $Y$ is a Calabi--Yau $3$-fold with $h^{1,1}(Y) = 101$. There are many birational models of $Y$ as toric hypersurfaces, corresponding to the many different lattice triangulations of the boundary of the fan polytope for $X$. Theorem~\ref{thm:main_theorem_ci} implies that the quantum connections (and quantum cohomology algebras) of all of these birational models become isomorphic after analytic continuation over the K\"ahler moduli space (which is $101$-dimensional), and that the isomorphisms involved arise from Fourier--Mukai transformations.
\end{example}
\subsection{A Note on Hypotheses}
Since we work with $T$-equivariant Gromov--Witten invariants of the toric Deligne--Mumford stacks $X_\pm$, we
do not need to assume that the coarse moduli spaces $|X_\pm|$
of $X_\pm$ are projective. We insist instead that $|X_\pm|$ is semi-projective,
i.e.~that $|X_\pm|$ is projective over the affinization
$\Spec(H^0(|X_\pm|,\cO))$, and also that $X_\pm$ contains
at least one torus fixed point. These conditions are equivalent to demanding that $X_\pm$ is
obtained as the GIT quotient $\big[\CC^m /\!\!/_\omega K\big]$ of a vector space by the linear action of a complex torus $K$; they ensure that the equivariant quantum cohomology of $X_\pm$
admits a non-equivariant limit. In particular, therefore, the non-equivariant version of the Crepant Transformation Conjecture
follows automatically from Theorem~\ref{thm:main_theorem}.
We do not assume, either, that the stacks $X_\pm$ or $Y_\pm$ satisfy any sort of positivity or weak Fano condition; put differently, we do not impose any additional convergence hypotheses on the $I$-functions for $X_\pm$ and $Y_\pm$. This extra generality is possible because of our hybrid formal/analytic approach, where we single out one variable $\sfy_r$ and analytically continue in that variable alone. The same technique allows us to describe the analytic continuation of \emph{big} quantum cohomology (or its ambient part), as opposed to small quantum cohomology. In general, obtaining convergence results for big quantum cohomology is hard.
\subsection{The Hemisphere Partition Function}
Recently there was some progress in physics in the exact computation of hemisphere partition functions for gauged linear sigma models. Hori--Romo~\cite{Hori--Romo} explained why the Mellin--Barnes analytic continuation of hemisphere partition functions should be compatible with brane transportation~\cite{Herbst--Hori--Page} in the B-brane category. In the language of this paper, the hemisphere partition function corresponds to a component of the $K$-theoretic flat section $\frs(E)$, and brane transportation corresponds to the Fourier--Mukai transformation. Theorem~\ref{thm:main_theorem} thus confirms the result of Hori--Romo. Note that the relevant equivalence between B-brane categories should depend on a choice of a path of analytic continuation, and that the Fourier--Mukai transformation in Theorem~\ref{thm:main_theorem} corresponds to a specific choice of path (see Figure~\ref{fig:path_ancont}).
\subsection{Plan of the Paper}
We fix notation for equivariant Gromov--Witten invariants and equivariant quantum cohomology in~\S\ref{sec:equivariant_qc}, and introduce the equivariant Gamma-integral structure in~\S\ref{sec:integral_structure}. We establish notation for toric Deligne--Mumford stacks in~\S\ref{sec:notation}. In \S\ref{sec:wall-crossing} we study $K$-equivalences $\varphi \colon X_+ \dashrightarrow X_-$ of toric Deligne--Mumford stacks arising from wall-crossing, constructing global versions of the equivariant quantum connections for $X_\pm$. We prove the Crepant Transformation Conjecture for toric Deligne--Mumford stacks (Theorem~\ref{thm:main_theorem}) in~\S\ref{sec:CRC}, and the Crepant Transformation Conjecture for toric complete intersections (Theorem~\ref{thm:main_theorem_ci}) in~\S\ref{sec:CRC_ci}.
\subsection{Notation} \label{sec:standing_notation}
We use the following notation throughout the paper.
\begin{itemize}
\item $X$ denotes a general smooth Deligne--Mumford stack in~\S\ref{sec:equivariant_qc} and~\S\ref{sec:integral_structure}; it denotes a smooth toric Deligne--Mumford stack in~\S\ref{sec:notation} and later.
\item $T = (\Cstar)^m$.
\item $R_T = H^\bullet_T({\rm pt},\CC)$.
\item $\lambda_j \in H^2_T({\rm pt},\CC) = \Lie(T)^\star$ is the character of $T = (\Cstar)^m$ given by projection to the $j$th factor, so that $R_T = \CC[\lambda_1,\ldots,\lambda_m]$.
\item $S_T$ is the localization of $R_T$ with respect to the set of non-zero homogeneous elements.
\item $\ZZ[T] = K^\bullet_T({\rm pt})$, so that $\ZZ[T] = \ZZ[e^{\pm\lambda_1},\ldots,e^{\pm\lambda_m}]$.
\item $\bmu_l = \{ z \in \Cstar : z^l = 1\}$ is a cyclic group of order $l$.
\end{itemize}
\section{Equivariant Quantum Cohomology}
\label{sec:equivariant_qc}
In this section we establish notation for various objects in equivariant Gromov--Witten theory. We introduce equivariant Chen--Ruan cohomology in~\S\ref{sec:CR}, equivariant Gromov--Witten invariants in~\S\ref{sec:GW}, equivariant quantum cohomology in~\S\ref{sec:QC}, Givental's symplectic formalism in~\S\ref{sec:Givental_cone}, and the equivariant quantum connection in~\S\ref{sec:quantum_connection}.
\subsection{Smooth Deligne--Mumford stacks with Torus Action}
\label{sec:conditions}
Let $X$ be a smooth Deligne--Mumford stack of finite type over $\CC$
equipped with an action of an algebraic torus $T \cong (\CC^\times)^m$.
Let $|X|$ denote the coarse moduli space of $X$ and
let $IX$ denote the inertia stack $X \times_{|X|} X$ of $X$: a
point on $IX$ is given by a pair $(x,g)$ with $x\in X$
and $g\in \Aut(x)$. We write
\[
IX = \bigsqcup_{v\in \sfB} X_v
\]
for the decomposition of $IX$ into connected components. We assume the following conditions:
\begin{enumerate}
\item the coarse moduli space $|X|$ is semi-projective, i.e.~is projective over the affinization $\Spec H^0(|X|,\cO) = \Spec H^0(X,\cO)$; \label{condition:1}
\item all the $T$-weights appearing in the $T$-representation $H^0(X,\cO)$ are contained in a strictly convex cone in $\Lie(T)^*$, and the $T$-invariant subspace $H^0(X,\cO)^T$ is $\CC$; \label{condition:2}
\item the inertia stack $IX$ is equivariantly formal, that is, the $T$-equivariant cohomology $H_T^\bullet(IX;\CC)$ is a free module over $R_T := H^\bullet_T({\rm pt};\CC)$ and one has a (non-canonical) isomorphism of $R_T$-modules
$H_{T}^\bullet(IX;\CC) \cong H^\bullet(IX;\CC) \otimes_\CC R_T$. \label{condition:3}
\end{enumerate}
These conditions allow us to define Gromov--Witten invariants of $X$ and also the equivariant (Dolbeault) index of coherent sheaves on $X$. The first and second conditions together imply that the fixed set $X^T$ is compact. The third condition seems to be closely related to the first two, but it implies for example the localization of equivariant cohomology: the restriction $H^\bullet_T(IX;\CC) \to H^\bullet_T(IX^T;\CC)$ to the $T$-fixed locus is injective and becomes an isomorphism after localization (see \cite{GKM}). Later we shall restrict to the case where $X$ is a toric Deligne--Mumford stack, where conditions (\ref{condition:1}--\ref{condition:3}) automatically hold, but the definitions in this section make sense for general $X$ satisfying these conditions.
\subsection{Equivariant Chen--Ruan Cohomology}
\label{sec:CR}
Let $H_{\CR,T}^\bullet(X)$ denote the even part of the $T$-equivariant orbifold cohomology group of Chen and Ruan. It is defined as the even degree part of the $T$-equivariant cohomology
\[
H_{\CR,T}^k(X) = \bigoplus_{v \in \sfB : k- 2 \iota_v \in 2\ZZ}
H_T^{k-2\iota_v}(X_v;\CC)
\]
of the inertia stack $IX$. The grading of $H_{\CR,T}^\bullet(X)$ is shifted from that of $H_{T}^\bullet(IX)$ by the so-called \emph{age} or \emph{degree shifting number} $\iota_v\in \QQ$ \cite{Chen--Ruan:orbifold}; note that we consider only the even degree classes in $H_T^\bullet(IX)$. (For toric stacks, all cohomology classes on $IX$ are of even degree.) Equivariant formality of $IX$ gives that $H^\bullet_{\CR,T}(X)$ is a free module over $R_T$. We write
\begin{align*}
(\alpha,\beta) = \int_{IX} \alpha \cup \inv^* \beta, && \alpha,\beta \in H_{\CR,T}^\bullet(X)
\end{align*}
for the equivariant orbifold Poincar\'{e} pairing: here $\inv \colon IX \to IX$ denotes the involution on the inertia stack $IX$ that sends a point $(x,g)$ with $x\in X$, $g\in \Aut(x)$ to $(x,g^{-1})$. Since $X$ is not necessarily proper, the equivariant integral on the right-hand side here is defined via the Atiyah--Bott localization formula~\cite{Atiyah--Bott} and takes values in the localization $S_T$ of $R_T$ with respect to the multiplicative set of non-zero homogeneous elements\footnote{Note that $R_T \subsetneq S_T \subsetneq \Frac(R_T)$; we use $S_T$ instead of $\Frac(R_T)$ since we need a grading on $S_T$ later.} in $R_T$.
\subsection{Equivariant Gromov--Witten Invariants}
\label{sec:GW}
Let $X_{g,n,d}$ denote the moduli space of degree-$d$ stable maps to $X$ from genus $g$ orbifold curves with $n$ marked points \cite{Abramovich--Graber--Vistoli:1,Abramovich--Graber--Vistoli:2}; here $d\in H_2(|X|;\ZZ)$. The moduli space carries a $T$-action and a virtual fundamental cycle $[X_{g,n,d}]^{\rm vir} \in A_{\bullet,T}(X_{g,n,d};\QQ)$. There are $T$-equivariant evaluation maps $\ev_i \colon X_{g,n,d} \to \overline{IX}$, $1 \leq i \leq n$, to the rigidified inertia stack $\overline{IX}$ (see \cite{Abramovich--Graber--Vistoli:2}). Let $\psi_i\in H^2_T(X_{g,n,d})$ denote the psi-class at the $i$th marked point, i.e.~the equivariant first Chern class of the $i$th universal cotangent line bundle $L_i \to X_{g,n,d}$. For $\alpha_1,\dots,\alpha_n \in H_{\CR, T}^\bullet(X)$ and non-negative integers $k_1,\dots,k_n$,
the \emph{$T$-equivariant Gromov--Witten invariant} is defined to be:
\begin{equation}
\label{eq:Gromov--Witten}
\correlator{\alpha_1 \psi^{k_1},\dots,\alpha_n \psi^{k_n}}_{g,n,d}^X
= \int_{[X_{g,n,d}]^{\rm vir}}
\prod_{i=1}^n (\ev_i^* \alpha_i) \psi_i^{k_i}
\end{equation}
where we regard $\alpha_i$ as a class in $H_T^\bullet(\overline{IX})$ via the canonical isomorphism $H_T^\bullet(\overline{IX}) \cong H_T^\bullet(IX)$. The moduli space here is not necessarily proper: the right-hand side is again defined via the Atiyah--Bott localization formula and so belongs to $S_T$. Conditions~\eqref{condition:1} and~\eqref{condition:2} in~\S\ref{sec:conditions} ensure that the $T$-fixed locus $X_{g,n,d}^T$ in the moduli space is compact, and thus that the right-hand side of \eqref{eq:Gromov--Witten} is well-defined.
\subsection{Equivariant Quantum Cohomology}
\label{sec:QC}
Consider the cone $\NE(X)\subset H_2(|X|,\RR)$ generated by classes of effective curves and set $\NE(X)_\ZZ := \{ d \in H_2(|X|,\ZZ) : d \in \NE(X)\}$. For a ring $R$, define $R[\![Q]\!]$ to be the ring of formal power series with coefficients in $R$:
\[
R[\![Q]\!] = \left\{ \sum_{d \in \NE(X)_\ZZ} a_d Q^d : a_d \in R \right\}
\]
so that $Q$ is a so-called \emph{Novikov variable}~\cite[III~5.2.1]{Manin}. Let $\phi_0,\phi_1,\dots,\phi_N$ be a homogeneous basis for $H_{\CR,T}^\bullet(X)$ over $R_T$ and let $\tau^0,\tau^1,\dots,\tau^N$ be the corresponding linear co-ordinates. We assume that $\phi_0 = 1$ and $\phi_1,\dots,\phi_r \in H^2_T(X)$ are degree-two untwisted classes that induce a $\CC$-basis of $H^2(X;\CC) \cong H^2_T(X)/H^2_T({\rm pt})$. We write $\tau = \sum_{i=0}^N \tau^i \phi_i$ for a general element of $H_{\CR,T}^\bullet(X)$. The \emph{equivariant quantum product} $\star_\tau$ at $\tau \in H_{\CR,T}^\bullet(X)$ is defined by the formula
\[
(\phi_i \star_\tau \phi_j, \phi_k) = \sum_{d \in \NE(X)_\ZZ} \sum_{n=0}^\infty \frac{Q^d}{n!} \correlator{\phi_i,\phi_j,\phi_k, \tau,\dots,\tau}_{0,n+3,d}^X
\]
or, equivalently, by
\begin{equation} \label{eq:qprod_pushforward}
\phi_i \star_\tau \phi_j = \sum_{d\in \NE(X)_\ZZ} \sum_{n=0}^\infty \frac{Q^d}{n!} \inv^* \ev_{3,*} \left (\ev_1^*(\phi_i) \ev_2^*(\phi_j) \prod_{l=4}^{n+3} \ev_l^*(\tau) \cap [X_{0,n+3,d}]^{\rm vir} \right).
\end{equation}
Conditions~\eqref{condition:1} and \eqref{condition:2} in~\S\ref{sec:conditions} ensure that $\ev_3: X_{0,n+3,d} \to \overline{IX}$ is proper, and thus that the push-forward along $\ev_3$ is well-defined without inverting equivariant parameters. It follows that:
\[
\phi_i \star_\tau \phi_j \in H_{\CR,T}^\bullet(X) \otimes_{R_T} R_T[\![\tau,Q]\!]
\]
where $R_T[\![\tau,Q]\!] = R_T[\![\tau^0,\dots,\tau^N]\!][\![Q]\!]$. The product $\star_\tau$ defines an associative and commutative ring structure on $H_{\CR,T}^\bullet(X)\otimes_{R_T} R_T[\![\tau,Q]\!]$. The non-equivariant limit of $\star_\tau$ exists, and this limit defines the non-equivariant quantum cohomology $\big(H_{\CR}^\bullet(X) \otimes_\CC \CC[\![\tau,Q]\!],\star_\tau\big)$.
\begin{remark} \label{rem:divisoreq_qprod} The divisor equation \cite[Theorem 8.3.1]{Abramovich--Graber--Vistoli:2} implies that exponentiated $H^2$-variables and the Novikov variable $Q$ play the same role: one has
\[
(\phi_i \star_\tau \phi_j, \phi_k) =
\sum_{d \in \NE(X)_\ZZ} \sum_{n=0}^\infty \frac{Q^de^{\<\sigma,d\>}}{n!} \correlator{\phi_i,\phi_j,\phi_k, \tau',\dots,\tau'}_{0,n+3,d}^X
\]
where $\tau = \sigma + \tau'$ with $\sigma = \sum_{i=1}^r \tau^i \phi_i$ and $\tau' = \tau_0 \phi_0 + \sum_{i=r+1}^N \tau^i \phi_i$. The String Equation (ibid.) implies that the right-hand side here is in fact independent of $\tau_0$.
\end{remark}
\subsection{Givental's Lagrangian Cone}
\label{sec:Givental_cone}
Let $S_T(\!(z^{-1})\!)$ denote the ring of formal Laurent series in $z^{-1}$ with coefficients in $S_T$. Givental's symplectic vector space is the space
\[
\cH = H_{\CR,T}^\bullet(X)\otimes_{R_T} S_T(\!(z^{-1})\!)[\![Q]\!]
\]
equipped with the non-degenerate $S_T[\![Q]\!]$-bilinear alternating form:
\[
\Omega(f, g) = - \Res_{z=\infty} (f(-z), g(z)) dz
\]
with $f,g \in \cH$. The space is equipped with a standard polarization
\[
\cH = \cH_+ \oplus \cH_-
\]
where
\begin{align*}
\cH_+ := H^\bullet_{\CR,T}(X) \otimes_{R_T} S_T[z][\![Q]\!]
&&\text{and} &&
\cH_- := z^{-1} H^\bullet_{\CR,T}(X) \otimes_{R_T} S_T[\![z^{-1}]\!] [\![Q]\!]
\end{align*}
are isotropic subspaces for $\Omega$. The standard polarization identifies $\cH$ with the cotangent
bundle of $\cH_+$. The \emph{genus-zero descendant Gromov--Witten potential} is a formal function $\cF^0_X \colon (\cH_+,{-z} 1) \to S_T[\![Q]\!]$ defined on the formal neighbourhood of $-z\cdot 1$ in $\cH_+$ and taking values in $S_T[\![Q]\!]$:
\[
\cF^0_X(-z 1 + \bt(z)) = \sum_{d\in \NE(X)_\ZZ}
\sum_{n=0}^\infty
\frac{Q^d}{n!}
\correlator{\bt(\psi),\dots,\bt(\psi)}_{0,n,d}^X
\]
Here $\bt(z) = \sum_{n=0}^\infty t_n z^n$ with $t_n \in H_{\CR,T}^\bullet(X)\otimes_{R_T} S_T[\![Q]\!]$. Let $\{\phi^i\}\subset H^\bullet_{\CR,T}(X) \otimes_{R_T} S_T$ denote the basis Poincar\'{e} dual to $\{\phi_i\}$, so that $(\phi_i,\phi^j) = \delta_i^j$.
\begin{definition}[\cite{Givental:symplectic, CCIT}]
\label{def:Lag_cone}
\emph{Givental's Lagrangian cone} $\cL_X\subset (\cH,-z1)$ is the graph of the differential $d \cF^0_X \colon \cH_+ \to T^* \cH_+ \cong \cH$. It consists of points of $\cH$ of the form:
\begin{equation}
\label{eq:point_on_cone}
-z 1 + \bt(z) + \sum_{d\in \NE(X)_\ZZ}
\sum_{n=0}^\infty \sum_{i=0}^N
\frac{Q^d}{n!}
\correlator{\frac{\phi_i}{-z-\psi}, \bt(\psi),\dots,\bt(\psi)}_{0,n+1,d}
\phi^i
\end{equation}
where $1/(-z-\psi)$ in the correlator should be expanded as the power series $\sum_{k=0}^\infty \psi^k (-z)^{-k-1}$ in $z^{-1}$.
In a more formal language, we define the notion of a `point on $\cL_X$' as follows. Let $x=(x_1,\dots,x_n)$ be formal parameters.
An \emph{$S_T[\![Q,x]\!]$-valued point} on $\cL_X$ is an element of $\cH[\![x]\!]$ of the form \eqref{eq:point_on_cone} with $\bt(z) \in \cH_+[\![x]\!]$ satisfying
\[
\bt(z)|_{Q=x =0} = 0.
\]
It should be thought of as a formal family of elements on $\cL_X$ parametrized by $x$.
\end{definition}
The submanifold $\cL_X$ encodes all genus-zero Gromov--Witten invariants \eqref{eq:Gromov--Witten}. It has the following special geometric properties \cite{Givental:symplectic}: \emph{it is a cone, and a tangent space $T$ of $\cL_X$ is tangent to $\cL_X$
exactly along $zT$}. Knowing Givental's Lagrangian cone $\cL_X$ is equivalent to knowing the data of the quantum product $\star_\tau$, i.e.~$\cL_X$ can be reconstructed from $\star_\tau$ and vice versa. See Remark~\ref{rem:fundamentalsol_cone}.
\subsection{The Equivariant Quantum Connection and its Fundamental Solution}
\label{sec:quantum_connection}
Let $v\in H_{\CR,T}^\bullet(X)$. The equivariant quantum connection
\[
\nabla_v \colon H_{\CR,T}^\bullet(X) \otimes_{R_T}
R_T[z][\![\tau,Q]\!] \to z^{-1} H_{\CR,T}^\bullet(X) \otimes_{R_T} R_T[z][\![\tau,Q]\!]
\]
is defined by
\[
\nabla_v f(\tau) = \partial_v f(\tau) + z^{-1} v \star_\tau f(\tau)
\]
where $\partial_v f(\tau) = \frac{d}{ds}f(\tau+sv)|_{s=0}$ is the directional derivative. We write $\nabla_i$ for $\nabla_{\phi_i}$ and $\nabla f$ for $\sum_{i=0}^N (\nabla_i f) d\tau^i$. The associativity of $\star_\tau$ implies that the connection $\nabla$ is flat, that is, $[\nabla_i,\nabla_j] = 0$ for all $i$,~$j$. Let $\rho$ denote the equivariant first Chern class (in the untwisted sector):
\[
\rho := c_1^T(TX) \in H^2_T(X) \subset H^2_{\CR,T}(X)
\]
For $\phi \in H^\bullet_{\CR,T}(X)$, we write $\deg \phi$ for the age-shifted (real) degree of $\phi$, so that $\phi\in H^{\deg \phi}_{\CR,T}(X)$. The equivariant Euler vector field $\cE$ and the grading operator $\mu\in \End_\CC(H_{\CR,T}^\bullet(X))$ are defined by
\begin{align}
\label{eq:Euler_mu}
\begin{split}
\cE &:= \sum_{i=1}^m \lambda_i \parfrac{}{\lambda_i}
+ \sum_{i=0}^N
\left(1- \frac{\deg \phi_i}{2}\right) \tau^i \parfrac{}{\tau^i}
+ \partial_\rho \\
\mu(\phi) &:= \left(
\frac{\deg\phi}{2} - \frac{\dim_\CC X}{2} \right) \phi
\end{split}
\end{align}
where $\lambda_1,\dots,\lambda_m\in H^2_T({\rm pt})$ are generators of $R_T$ (see \S\ref{sec:standing_notation}). The grading operator on $H_{\CR,T}^\bullet(X)\otimes_{R_T}R_T[z][\![\tau,Q]\!]$ is defined by
\[
\Gr(f(\tau,z) \phi) = \left(\Big( \textstyle z\parfrac{}{z} + \cE\Big) f(\tau,z)\right)
\phi + f(\lambda,\tau,z) \mu (\phi)
\]
where $\phi \in H_{\CR,T}^{\bullet}(X)$ and $f(\lambda,\tau,z) \in R_T[z][\![\tau,Q]\!]$. The quantum connection is compatible with the grading operator in the sense that $[\Gr, \nabla_i]=\nabla_{[\cE,\partial_{\tau^i}]} = (\frac{1}{2} \deg \phi_i - 1) \nabla_i$, $i=0,\dots,N$. This follows from the virtual dimension formula for the moduli space of stable maps.
\begin{notation}
Let $v\in H^2_T(X)$ be a degree-two class in the
untwisted sector. The action of $v$ on $H^\bullet_{\CR,T}(X)$
is defined by $v \cdot \alpha = q^*(v) \cup \alpha$, where
$q\colon IX \to X$ is the natural projection.
(This coincides with the action of $v$ via the
Chen--Ruan cup product.)
\end{notation}
Consider the flat section equations for $\nabla$, and a fundamental solution
\[
L(\tau,z) \in \End_{R_T}(H^\bullet_{\CR,T}(X)) \otimes_{R_T} R_T(\!(z^{-1})\!)[\![\tau,Q]\!]
\]
determined by the following conditions:
\begin{align}
\label{eq:flatness}
\nabla_i L(\tau,z) \phi & = 0 && \text{for $i=0,\dots, N$}
& &\text{(flatness)} \\
\label{eq:divisor}
\left( v Q \parfrac{}{Q} - \partial_v \right)L(\tau,z)\phi
& = L(\tau,z) \frac{v}{z}\phi && \text{for $v\in H^2_T(X)$}
& & \text{(divisor equation)} \\
\label{eq:initial}
L(\tau,z) |_{\tau=Q=0} & = \id &&
& & \text{(initial condition)}
\end{align}
Here $\phi\in H^\bullet_{\CR,T}(X)$ and $v Q \parfrac{}{Q}$ with $v\in H^2_T(X)$ acts
on Novikov variables as $Q^d \mapsto \<v,d\> Q^d$
(it acts by zero when $v\in H^2_{T}({\rm pt})
\subset H^2_T(X)$).
The flatness equation fixes $L(\tau,z)$ up to
right multiplication by an endomorphism-valued function
$g(z;Q)$ in $z$ and $Q$; the divisor equation implies
that the ambiguity $g(z;Q)$ is independent of $Q$ and
commutes with $v \cup$, $v\in H^2_T(X)$;
finally the initial condition fixes $L(\tau,z)$ uniquely.
The fundamental solution satisfying these
conditions can be written explicitly in terms of (descendant) Gromov--Witten invariants:
\begin{equation}
\label{eq:L_descendant}
L(\tau,z) \phi_i = \phi_i +
\sum_{j=0}^N \sum_{d\in \NE(X)_\ZZ}
\sum_{\substack{n=0 \\ (n\ge 1\text{ if }d=0)}}^\infty
\frac{Q^d}{n!}
\correlator{\frac{\phi_i}{-z-\psi},\tau,\dots,\tau,\phi_j}_{0,n+2,d}^X
\phi^j
\end{equation}
This is defined over $R_T$ (without inverting equivariant parameters) because it can be rewritten in terms of the push-forward along the last evaluation map $\ev_{n+2}$ as in \eqref{eq:qprod_pushforward}.
\begin{proposition}
\label{prop:fundsol}
The fundamental solution $L(\tau,z)$ in \eqref{eq:L_descendant}
satisfies the conditions (\ref{eq:flatness}--\ref{eq:initial}).
Furthermore it satisfies:
\begin{align*}
L(\tau,z) & = \id + O(z^{-1}) & & \text{\rm (regularity at $z=\infty$)} \\
\Gr L(\tau,z) \phi &=
L(\tau,z) \left(\mu - \frac{\rho}{z} \right) \phi
& & \text{\rm (homogeneity)} \\
(\alpha,\beta) & =(L(\tau,-z) \alpha, L(\tau,z)\beta)
& & \text{\rm (unitarity)}
\end{align*}
where $\phi,\alpha,\beta\in H^\bullet_{\CR,T}(X)$.
\end{proposition}
\begin{proof}
The fundamental solution \eqref{eq:L_descendant}
is well-known: see \cite[Corollary 6.3]{Givental:equivariant},
\cite[Proposition 2]{Pandharipande:afterGivental},
\cite[Proposition 2.4]{Iritani}.
The flatness equation follows from the topological recursion
relations as explained in \cite[Proposition 2]{Pandharipande:afterGivental};
the divisor equation follows from the one
\cite[Theorem 8.3.1]{Abramovich--Graber--Vistoli:2} for descendant
Gromov--Witten invariants;
the initial condition is obvious from formula~\eqref{eq:L_descendant}.
Regularity at $z=\infty$ is also clear straight from the definition.
Decompose $\tau = \sigma + \tau'$ where
$\sigma = \sum_{i=0}^r \tau^i \phi_i$
and $\tau' = \sum_{i=r+1}^N \tau^i \phi_i$
(recall that $\phi_0 = 1$ and that $\phi_1,\dots,\phi_r$ induce a
basis of $H^2_T(X)/H^2_T({\rm pt})$).
The string and divisor equations \cite[Theorem 8.3.1]{Abramovich--Graber--Vistoli:2}
imply that one has $L(\tau,z) \phi_i =
S(\tau',z; Q e^{\sigma}) (e^{-\sigma/z}\phi_i)$ with:
\[
S(\tau',z; Q e^\sigma) \alpha = \alpha +
\sum_{j=0}^N \sum_{d\in \NE(X)_\ZZ} \sum_{\substack{n=0 \\
(n\ge 1\text{ if }d=0)}}^\infty
\frac{e^{\<\sigma,d\>}Q^d}{n!}
\correlator{\frac{\alpha}{-z-\psi},
\tau',\dots,\tau',\phi_j}_{0,n+2,d} \phi^j
\]
The dimension axiom shows that $S$ is homogeneous:
$\Gr \circ S(\tau',z; Q e^\sigma) =
S(\tau',z;Q e^\sigma) \circ \Gr$.
The homogeneity
equation for $L(\tau,z)$ follows from this.
The flatness equation, together with
the Frobenius property of $\star_\tau$, shows that
\[
\partial_i
\big(L(\tau,-z)\alpha,L(\tau,z)\beta\big)
= \big(\overline{\nabla}_i L(\tau,-z)\alpha, L(\tau,z) \beta\big)
+ \big(L(\tau,-z) \alpha, \nabla_i L(\tau,z)\beta\big)=0
\]
for $\overline{\nabla}_i = \partial_i - z^{-1} \phi_i\star_\tau$.
Thus the pairing $(L(\tau,-z)\alpha,L(\tau,z) \beta)$ does not depend on
$\tau$. Consider the adjoint
$L(\tau,z)^\dagger$ of $L(\tau,z)$ with respect to $(\cdot,\cdot)$.
Then $(L(\tau,-z)^\dagger \alpha,
L(\tau,z)^\dagger \beta)$ is also independent of $\tau$.
The divisor equation implies that:
\begin{multline*}
\left( v Q \parfrac{}{Q} - \partial_v \right)
(L(\tau,-z)^\dagger \alpha, L(\tau,z)^\dagger \beta) \\
= (-\frac{v}{z} L(\tau,-z)^\dagger \alpha, L(\tau,z)^\dagger \beta)
+ (L(\tau,-z)^\dagger \alpha, \frac{v}{z} L(\tau,z)^\dagger\beta) =0
\end{multline*}
Thus it is also independent of $Q$.
The initial condition at $\tau = Q=0$ now implies
the unitarity $(L(\tau,-z)^\dagger \alpha, L(\tau,z)^\dagger \beta)
= (\alpha,\beta)$.
\end{proof}
\begin{remark}
By solving the differential equations as power series in $\tau$ and $Q$,
we find that the coefficient $L_{d,k}(z)$ of $L(\tau,z)$
in front of $Q^d \tau_0^{k_0}
\cdots \tau_N^{k_N}$ is a rational function in $z$ and $\lambda$,
i.e.~that each matrix entry of $L_{d,k}$
lies in $\Frac(R_T[z]) \cong \CC(\lambda_1,\dots,\lambda_m,z)$.
The Laurent expansion of each matrix element at $z=\infty$ gives an element
of $R_T(\!(z^{-1})\!)$. As a rational function in $z$, $L_{d,k}$
has singularities not only at $z=0$ but also at other finite values
of $z$. This at first sight seems to contradict the fact that the flatness equation
and the divisor equation have singularities only at $z=0$.
In fact there is no contradiction: the equivariant
quantum differential equation is \emph{resonant} at certain values of $z$.
The divisor equation for $L(\tau,z)$ together with the flatness
equation gives
\begin{align*}
v Q \parfrac{}{Q} L(\tau,z) =z^{-1}
\left( L(\tau,z) v - v\star_\tau L(\tau,z) \right)
&&
v\in H^2_T(X)
\end{align*}
and this has a logarithmic singularity at the `large radius limit point' $Q=0$.
The residue at $Q=\tau=0$ is given by the commutator
$[v/z,-]$ and the resonance occurs when
the cup product by $v/z$ has eigenvalues in $\ZZ\setminus\{0\}$.
The coefficients $L_{d,k}(z)$ have poles at resonant $z$.
In non-equivariant Gromov--Witten theory, $v/z$ is always nilpotent and
resonance does not occur.
More generally, if the equivariant parameters $\lambda_i$ are
sufficiently small compared to $|z|$ ($|\lambda_i|\ll |z|$),
resonance does not occur and the coefficients $L_{d,k}(z)$ are regular.
\end{remark}
\begin{remark}[\!\!\cite{Givental:symplectic}]
\label{rem:fundamentalsol_cone}
The fundamental solution $L(\tau,z)$ is determined by the quantum
product $\star_\tau$ via differential equations
\eqref{eq:flatness}--\eqref{eq:initial}.
Then $\tau \mapsto
T_\tau = L(\tau,-z)^{-1} \cH_+$ gives a versal family of
tangent spaces to Givental's cone $\cL_X$. The cone
$\cL_X$
is reconstructed as $\cL_X = \bigcup_\tau z T_\tau$.
\end{remark}
We now study $\nabla$-flat sections $s(\tau,z)$
that are homogeneous of degree zero: $\Gr (s(\tau,z)) = 0$.
By Proposition \ref{prop:fundsol}, if a flat section
$L(\tau,z) f(z)$ is homogeneous of degree zero,
then:
\[
\left(z\parfrac{}{z} + \mu -\frac{\rho}{z}\right) f(z)=0
\]
This differential equation has the fundamental solution:
\[
z^{-\mu} z^\rho = z^{\rho/z} z^{-\mu}
= \exp(\rho \log(z)/z) z^{-\mu}
\]
that belongs to $\End_{R_T}(H_{\CR,T}^\bullet(X)) \otimes_{R_T}
R_T[\log z](\!(z^{-1/k})\!)$ for some $k\in \N$;
here $k$ is chosen so that all the eigenvalues of $k \mu$ are integers.
Note that homogeneous flat sections can be multi-valued
in $z$ (as they contain $\log z$).
We have:
\begin{corollary}
\label{cor:homogeneous_flat_sections}
The sections $s_i(\tau,z) = L(\tau,z) z^{-\mu} z^\rho \phi_i$, $i=0,\dots,N$
satisfy $\nabla s_i(\tau,z) = \Gr s_i(\tau,z) = 0$ and
give a basis of homogeneous flat sections.
They belong to $H_{\CR,T}^\bullet(X)
\otimes_{R_T} R_T[\log z](\!(z^{-1/k})\!)[\![\tau,Q]\!]$
for a sufficiently large $k \in \N$.
\end{corollary}
\section{Equivariant Gamma-Integral Structure}
\label{sec:integral_structure}
In this section we introduce one of the main ingredients of our result: an integral structure for equivariant quantum cohomology. This is a $K^0_T({\rm pt})$-lattice in the space of flat sections for the equivariant quantum connection on $X$ which is isomorphic to the integral equivariant $K$-group $K^0_T(X)$: it generalizes the integral structure for non-equivariant quantum cohomology constructed by Iritani~\cite{Iritani} and Katzarkov--Kontsevich--Pantev~\cite{KKP}. Similar structures have been studied by Okounkov--Pandharipande~\cite{Okounkov-Pandharipande:Hilbert} in the case where $X$ is a Hilbert scheme of points in $\CC^2$, and by Brini--Cavalieri--Ross~\cite{Brini--Cavalieri--Ross} in the case where $X$ is a 3-dimensional toric Calabi--Yau stack. We define the integral structure in~\S\ref{sec:equiv_int_str}. In \S\ref{sec:specialization} we observe that the quantum product, flat sections for the quantum connection, and integral structure continue to make sense when the Novikov variable $Q$ (see~\S\ref{sec:QC}) is specialized to $Q=1$.
The integral structure is defined in terms of a $T$-equivariant characteristic class of $X$ called the $\hGamma$-class. One of the key points in this section is that the $\hGamma$-class behaves like a square root of the Todd class: see equation~\ref{eq:gammaproduct}. When combined with the Hirzebruch--Riemann--Roch formula, this leads to one of the fundamental properties of the integral structure: that the so-called framing map is pairing-preserving (Proposition~\ref{prop:K_framing_pairing} below)
\subsection{The Equivariant Gamma Class and the Equivariant Gamma-Integral Structure}
\label{sec:equiv_int_str}
Let $K_T^0(X)$ denote the Grothendieck group of $T$-equivariant vector bundles on $X$. We write $H^{\bullet\bullet}_{T}(IX) := \prod_{p} H^{2p}_{T}(IX)$. We introduce an orbifold Chern character map $\tch \colon
K^0_T(X) \to H_{T}^{\bullet\bullet}(IX)$ as follows. Let $IX = \bigsqcup_{v \in \sfB} X_v$ be the decomposition of the inertia stack $IX$ into connected components, let $q_v \colon X_v \to X$ be the natural map, and let $E$ be a $T$-equivariant vector bundle on $X$. The stabilizer $g_v$ along $X_v$ acts on the vector bundle $q_v^* E \to X_v$, giving an eigenbundle decomposition
\begin{equation}
\label{eq:E_vf}
q_v^* E = \bigoplus_{0\le f < 1} E_{v,f}
\end{equation}
where $g_v$ acts on $E_{v,f}$ by $\exp(2\pi\tti f)$. The equivariant Chern character is defined to be
\[
\tch(E) = \bigoplus_{v\in \sfB} \sum_{0\le f<1} e^{2\pi\tti f}
\ch^T(E_{v,f})
\]
where $\ch^T(E_{v,f})\in H^{\bullet \bullet}_T(X_v)$ is the $T$-equivariant Chern character. Let $\delta_{v,f,i}$, $1 \leq i \leq \rank(E_{v,f})$ be the $T$-equivariant Chern roots of $E_{v,f}$, so that $c^T(E_{v,f}) = \prod_i (1+ \delta_{v,f,i})$.
These Chern roots are not actual cohomology classes, but symmetric polynomials in the Chern roots make sense as equivariant cohomology classes on $X_v$. The $T$-equivariant orbifold Todd class $\tTd(E) \in H^{\bullet\bullet}_{T}(IX)$ is defined to be:
\[
\tTd(E) = \bigoplus_{v\in \sfB}
\left( \prod_{0<f<1} \prod_{i=1}^{\rank(E_{v,f})}
\frac{1}{1- e^{-2\pi\tti f} e^{-\delta_{v,f,i}}}\right)
\prod_{i=1}^{\rank E_{v,0}}
\frac{\delta_{v,0,i}}{1-e^{-\delta_{v,0,i}}}.
\]
We write $\tTd_X = \tTd(TX)$ for the orbifold Todd class of the tangent bundle.
Recall that, because we are assuming condition \eqref{condition:2}
from \S\ref{sec:conditions}, all of the $T$-weights of $H^0(X,\cO)$
lie in a strictly convex cone in $\Lie(T)^*$.
After changing the identification of $T$ with $(\Cstar)^m$
if necessary, we may assume that this cone is contained within
the cone spanned by the standard characters $\lambda_1,\dots,
\lambda_m$ of $H^2_T({\rm pt}) = \Lie(T)^*$ defined in
\S\ref{sec:standing_notation}.
As is explained in~\cite{Coates--Iritani--Jiang--Segal},
under conditions (\ref{condition:1}--\ref{condition:2}) in \S\ref{sec:conditions}
there is a well-defined equivariant Euler characteristic
\begin{equation}
\label{eq:Euler_char}
\chi(E) := \sum_{i=0}^{\dim X}
(-1)^i \ch^T\big(H^i(X,E)\big).
\end{equation}
taking values in
\[
\ZZ[\![e^{\lambda}]\!][e^{-\lambda}]_{\rm rat}
:= \left\{ f \in \ZZ[\![e^{\lambda_1},\dots,e^{\lambda_m}]\!]
[e^{-\lambda_1},\dots,e^{-\lambda_m}] :
\begin{array}{l}
\text{$f$ is the Laurent expansion} \\
\text{of a rational function in} \\
\text{$e^{\lambda_1},\dots,
e^{\lambda_m}$ at} \\
\text{$e^{\lambda_1} = \cdots = e^{\lambda_m}=0$}
\end{array}\right\}
\]
and we expect that the following equivariant Hirzebruch--Riemann--Roch (HRR)
formula should hold:
\begin{equation}
\label{eq:equiv_HRR}
\chi(E) =
\int_{IX} \tch(E) \cup \tTd_X
\end{equation}
This formula should be interpreted with care. The right-hand side is defined via the localization
formula, and lies in a completion $\hS_T$ of $S_T$:
\[
\hS_T := \left\{
\sum_{n\in \ZZ} a_n : \text{$a_n \in S_T$, $\deg a_n =n$, there exists $n_0\in \ZZ$ such that $a_n = 0$ for all $n< n_0$} \right\}
\]
There is an inclusion of
rings $\ZZ[\![e^{\lambda}]\!][e^{-\lambda}]_{\rm rat} \hookrightarrow \hS_T$ given by Laurent expansion at $\lambda_1 = \cdots = \lambda_m=0$ (see \cite{Coates--Iritani--Jiang--Segal}), and \eqref{eq:equiv_HRR} asserts that $\chi(E)$ coincides with the right-hand side after this inclusion.
We now introduce a lattice in the space of homogeneous flat sections for the quantum connection which is identified with the equivariant $K$-group of $X$. The key ingredient in the definition is the characteristic class, called the \emph{Gamma class}, defined as follows.
Let $E$ be a vector bundle on $X$ and consider
the bundles $E_{v,f}\to X_v$ and their equivariant
Chern roots $\delta_{v,f,i}$, $i=1,\dots,\rank(E_{v,f})$ as
above (see \eqref{eq:E_vf}).
The equivariant Gamma class
$\hGamma(E)\in H^{\bullet\bullet}_{T}(IX)$ is defined to be:
\[
\hGamma(E) = \bigoplus_{v\in \sfB}
\prod_{0\le f<1} \prod_{i=1}^{\rank(E_{v,f})}
\Gamma(1-f + \delta_{v,f,i})
\]
Here the $\Gamma$-function on the right-hand side should be expanded
as a Taylor series at $1-f$, and then evaluated at $\delta_{v,f,i}$. The identity
$\Gamma(1-z) \Gamma(1+z) = 2\pi\tti z e^{-\pi \tti z}/
(1- e^{-2\pi \tti z})$ implies that
\begin{align}
\label{eq:gammaproduct}
\begin{split}
\left[\hGamma(E^*) \cup \hGamma(E)\right]_v
& = \prod_{i,f}
\Gamma(1-\fbar - \delta_{v,f,i}) \Gamma(1-f+\delta_{v,f,i}) \\
& = (2\pi\tti)^{\rank((q_v^*E)^{\rm mov})}
\left[e^{-\pi\tti (\age(q^*E)+ c_1(q^* E))}
(2\pi\tti)^{\frac{\deg_0}{2}} \tTd(E)\right]_{\inv(v)}
\end{split}
\end{align}
where $\cup$ is the cup product on $IX$, $[\cdots]_v$ denotes the component in $H^\bullet_T(X_v)$, $0\le \fbar<1$ is the fractional part of $-f$, $(q_v^*E)^{\rm mov} = \bigoplus_{f\neq 0} E_{v,f}$ is the moving part of $q_v^*E$, $q\colon IX \to X$ is the natural projection, $\age(q^*E) \colon IX \to \QQ$ is the locally constant function given by $\age(q^*E)|_{X_v} = \sum_{f} f \rank(E_{v,f})$, $\deg_0 \colon H^{\bullet\bullet}_T(IX) \to H^{\bullet\bullet}_T(IX)$ is the degree operator defined by $\deg_0(\phi)= 2p \phi$ for $\phi\in H^{2p}_T(IX)$, and $\inv(v)\in \sfB$ corresponds to the component $X_{\inv(v)}$ of $IX$ defined by $\inv(X_v) = X_{\inv(v)}$. Note that $\deg_0$ means the degree as a class on $IX$, not the age-shifted degree as an element of $H^{\bullet\bullet}_{\CR,T}(X)$.
\begin{definition}
\label{def:K_framing}
Define the \emph{$K$-group framing}
\[
\frs \colon K_T^0(X) \to
H^\bullet_{\CR,T}(X)\otimes_{R_T} R_T[\log z](\!(z^{-1/k})\!)[\![Q,\tau]\!]
\]
by the formula:
\[
\frs(E)(\tau,z) = \frac{1}{(2\pi)^{\dim X/2}}
L(\tau,z) z^{-\mu} z^\rho \left( \hGamma_X \cup
(2\pi\tti)^{\frac{\deg_0}{2}} \inv^* \tch(E) \right)
\]
where $k\in \N$ is as in Corollary \ref{cor:homogeneous_flat_sections}
and $\hGamma_X \cup$ is the cup product in $H^{\bullet\bullet}_T(IX)$.
Corollary \ref{cor:homogeneous_flat_sections} shows that the image of $\frs$ is contained in the space of $\Gr$-degree zero
flat sections.
Note that $z^{-\mu}$ maps $H^{\bullet\bullet}_{\CR,T}(X)$
into $H^{\bullet}_{\CR,T}(X)\otimes_{R_T} R_T(\!(z^{-1/k})\!)$.
\end{definition}
For $T$-equivariant vector bundles $E$, $F$ on $X$,
let $\chi(E,F) \in \ZZ[\![e^\lambda]\!][e^{-\lambda}]_{\rm rat}$ denote the equivariant Euler pairing
defined by:
\begin{equation}
\label{eq:Euler_pairing}
\chi(E,F) := \sum_{i=0}^{\dim X} (-1)^i
\ch^T\big(\Ext^i(E,F)\big)
\end{equation}
We use a $z$-modified version $\chi_z(E,F)$ that is
given by replacing equivariant parameters $\lambda_j$ in $\chi(E,F)$
with $2\pi\tti \lambda_j/z$:
\begin{equation}
\label{eq:modified_Euler}
\chi_z(E,F) := (2\pi\tti z^{-1})^{\sum_{i=1}^m \lambda_i \partial_{\lambda_i}}
\chi(E,F) \in \ZZ[\![e^{2\pi\tti \lambda/z}]\!][e^{-2\pi \tti \lambda/z}]_{\rm rat}
\end{equation}
\begin{proposition}[cf.~{\cite[Proposition 2.10]{Iritani}}]
\label{prop:K_framing_pairing}
Suppose that the equivariant HRR formula \eqref{eq:equiv_HRR}
holds.
For $E, F \in K_T^0(X)$, we have
\[
\left(\frs(E)(\tau,e^{-\pi \tti}z), \frs(F)(\tau,z) \right)
= \chi_z(E,F).
\]
\end{proposition}
\begin{proof}
Set $\Psi(E) = \hGamma_X \cup (2\pi\tti)^{\frac{\deg_0}{2}} \inv^*
\tch(E)$.
Using the unitarity in Proposition \ref{prop:fundsol}, we have
\begin{equation}
\label{eq:frs_pairing}
\left(\frs(E)(\tau,e^{-\pi \tti}z), \frs(F)(\tau,z) \right)
= \frac{1}{(2\pi)^{\dim X}} \left(z^{-\mu} e^{\pi\tti \mu}
z^\rho e^{-\pi\tti \rho} \Psi(E), z^{-\mu} z^\rho \Psi(F)\right).
\end{equation}
Write $\lambda \partial_\lambda =
\sum_{i=1}^m \lambda_i \partial_{\lambda_i}$.
Using $(z^{-\mu}\alpha,z^{-\mu} \beta) = z^{-\lambda\partial_\lambda}
(\alpha,\beta)$, $e^{\pi\tti \mu} \rho = -\rho e^{\pi\tti\mu}$,
$(z^{-\rho} \alpha,z^{\rho} \beta) = (\alpha,\beta)$,
we have
\begin{align*}
\eqref{eq:frs_pairing} & =
\frac{z^{-\lambda \partial_\lambda}}{(2\pi)^{\dim X}}
\left(e^{\pi\tti \rho}e^{\pi\tti \mu} \Psi(E), \Psi(F)\right) \\
& = \frac{z^{-\lambda \partial_\lambda}}{(2\pi)^{\dim X}}
\int_{IX} \left( e^{\pi\tti q^* \rho} e^{\pi\tti \mu}
\hGamma_X (2\pi\tti)^{\frac{\deg_0}{2}}\inv^* \tch(E) \right)
\cup \inv^*\left( \hGamma_X (2\pi\tti)^{\frac{\deg_0}{2}}
\inv^* \tch(F) \right)
\\
& =
\frac{z^{-\lambda \partial_\lambda}}{(2\pi)^{\dim X}}
\sum_{v\in \sfB}
\int_{X_v}
e^{\pi\tti q_v^*\rho}
e^{\pi\tti (\iota_{v} - \frac{\dim X}{2})}
\left[\hGamma_X^* \hGamma_X \right]_{\inv(v)}
(2\pi\tti)^{\frac{\deg_0}{2}} \left[\tch(E^*) \tch(F)\right]_v \\
& = z^{-\lambda \partial_\lambda}
\sum_{v\in \sfB}
\frac{1}{(2\pi \tti)^{\dim X_v}}
\int_{X_v}
(2\pi\tti)^{\frac{\deg_0}{2}} \left[
\tch(E^* \otimes F) \cup \tTd_X \right]_v
\end{align*}
where we set $\hGamma_X^* = \hGamma(T^*X)$
and used equation \eqref{eq:gammaproduct} in the last line.
The last expression equals $\chi_z(E,F)$ by the
HRR formula \eqref{eq:equiv_HRR}.
\end{proof}
\begin{remark}
One could also consider the $K$-group framing for \emph{topological} equivariant $K$-theory. For toric stacks, the topological and algebraic $K$-groups coincide.
\end{remark}
\begin{remark}
Okounkov--Pandharipande \cite{Okounkov-Pandharipande:Hilbert}
and Braverman--Maulik--Okounkov \cite{Braverman--Maulik--Okounkov}
introduced shift operators $\bbS_i$ on quantum
cohomology, which induce the shift $\lambda_i \to \lambda_i + z$
of equivariant parameters (see \cite[Chapter 8]{Maulik--Okounkov} for a detailed
description).
Our $K$-theoretic flat sections $\frs(E)$ are invariant under the
shift operators, and our main result suggests that shift
operators for toric stacks
should be defined globally on the secondary toric variety.
\end{remark}
\subsection{Specialization of Novikov Variables}
\label{sec:specialization}
In this section we show that the quantum product, the flat sections for the quantum connection, and the $K$-group framing remain well-defined after the specialization $Q=1$ of the Novikov variable~$Q$. Recall that $\tau^0,\dots,\tau^N$ are co-ordinates on $H_{\CR,T}^\bullet(X)$ dual to a homogeneous $R_T$-basis $\{\phi_0,\dots,\phi_N\}$ of $H_{\CR,T}^\bullet(X)$, and that:
\begin{itemize}
\item $\phi_0=1$;
\item $\phi_1,\dots,\phi_r\in H^2_T(X)$;
\item $\phi_1,\dots,\phi_r$ descend to a basis of $H^2(X) = H^2_T(X)/H^2_T({\rm pt})$.
\end{itemize}
Without loss of generality we may assume that the images of $\phi_1,\dots,\phi_r$ in $H^2(X)$ are nef and integral.
It is clear from Remark~\ref{rem:divisoreq_qprod} that the specialization $Q=1$
of the quantum product is well-defined, and we have:
\[
\phi_i \star_\tau \phi_j \Big|_{Q=1} \in H_{\CR,T}^\bullet(X)
\otimes_{R_T} R_T[\![e^{\tau^1},\dots,e^{\tau^r},
\tau^{r+1},\dots,\tau^N]\!]
\]
As discussed in Remark~\ref{rem:divisoreq_qprod}, the product $\phi_i\star_\tau\phi_j$ is independent of $\tau^0$. We saw in the proof of Proposition~\ref{prop:fundsol} that the fundamental solution $L(\tau,z)$ factorizes as $L(\tau,z) = S(\tau',z; Qe^{\sigma}) e^{-\sigma/z}$. Thus the specialization $Q=1$ makes sense for $L(\tau,z)$ and:
\[
L(\tau,z)\Big |_{Q=1} \in \End\left(H_{\CR,T}^\bullet(X)\right)
\otimes_{R_T} R_T[\tau^0,\tau^1,\dots,\tau^r][\![z^{-1}]\!][\![
e^{\tau^1},\dots,e^{\tau^r}, \tau^{r+1},\dots,\tau^{N}]\!]
\]
The specialization $Q=1$ for homogeneous flat sections $\frs(E)$
in Definition \ref{def:K_framing}
(as well as the homogeneous flat sections $s_i$ in Corollary \ref{cor:homogeneous_flat_sections})
also makes sense and we have
\[
\frs(E)(\tau,z)\Big|_{Q=1}
\in H_{\CR,T}^\bullet(X) \otimes_{R_T}
R_T[\tau^0,\tau^1,\dots,\tau^r,\log z](\!(z^{-1/k})\!)[\![
e^{\tau^1},\dots,e^{\tau^r}, \tau^{r+1},\dots,\tau^{N}]\!]
\]
where $k\in \N$ is such that all the eigenvalues of $k\mu$ are integral.
\section{Toric Deligne--Mumford Stacks as GIT Quotients}
\label{sec:notation}
In the rest of this paper we consider toric Deligne--Mumford stacks $X$ with semi-projective coarse moduli space such that the torus-fixed set $X^T$ is non-empty. This is the class of stacks that arise as GIT quotients of a complex vector space by the action of a complex torus. In this section we establish notation and describe basic properties of these quotients. Good introductions to this material include~\cite[\S VII]{Audin}, \cite{Cox-Little-Schenck} and~\cite{Borisov--Chen--Smith}.
\subsection{GIT Data}
\label{sec:GITdata}
Consider the following data:
\begin{itemize}
\item $K \cong (\Cstar)^r$, a connected torus of rank $r$;
\item $\LL = \Hom(\Cstar,K)$, the cocharacter lattice of $K$;
\item $D_1,\ldots,D_m \in \LL^\vee =
\Hom(K,\Cstar)$, characters of $K$.
\end{itemize}
The characters $D_1,\ldots,D_m$ define a map from $K$
to the torus $T = (\Cstar)^m$, and hence define an action of
$K$ on $\CC^m$.
\begin{notation}
\label{not:cones}
For a subset $I$ of $\{1, 2, \ldots, m\}$, write $\overline{I}$
for the complement of $I$, and set
\begin{align*}
\angle_I &= \big\{ \textstyle\sum_{i \in I} a_i D_i : \text{$a_i \in
\RR$, $a_i > 0$} \big\} \subset
\LL^\vee \otimes \RR, \\
(\Cstar)^I \times \CC^{\overline{I}}
& = \big \{ (z_1,\dots,z_m) : z_i \neq 0 \text{ for } i\in I
\big\} \subset \CC^m.
\end{align*}
We set $\angle_\emptyset := \{0\}$.
\end{notation}
\begin{definition}
\label{def:toricstack}
Consider now a \emph{stability condition}
$\omega \in \LL^\vee \otimes \RR$, and set:
\begin{align*}
\cA_\omega &=
\big\{ I \subset \{1,2,\ldots,m\} : \omega \in \angle_I \big\} \\
U_\omega &=\bigcup_{I\in \cA_\omega}
(\Cstar)^I \times \CC^{\overline{I}} \\
X_\omega &= \big[U_\omega \big/ K \big]
\end{align*}
The square brackets here indicate that $X_\omega$ is the stack quotient
of $U_\omega$ (which is $K$-invariant) by $K$.
We call $X_\omega$ the \emph{toric stack
associated to the GIT data} $(K;\LL;D_1,\ldots,D_m; \omega)$.
We refer to elements of $\cA_\omega$ as \emph{anticones},
for reasons which will become clear in \S \ref{sec:stacky_fan}
below.
\end{definition}
\begin{assumption} \label{assumption}
We assume henceforth that:
\begin{enumerate}
\item $\{1,2,\ldots,m\} \in \cA_\omega$;
\item for each $I \in \cA_\omega$,
the set $\{D_i : i \in I\}$ spans $\LL^\vee \otimes \RR$ over $\RR$.
\end{enumerate}
These are assumptions on the stability condition $\omega$.
The first ensures that $X_\omega$ is non-empty;
the second ensures that $X_\omega$ is a Deligne--Mumford stack.
Under these assumptions, $\cA_\omega$ is closed under
enlargement of sets, i.e.~if $I\in \cA_\omega$ and
$I\subset J$ then $J \in \cA_\omega$.
\end{assumption}
Let $S\subset\{1,2,\dots,m\}$
denote the set of indices $i$ such that $\{1,\dots,m\} \setminus \{i\}
\notin \cA_\omega$. It is easy to see that
the characters $\{D_i : i \in S\}$ are linearly independent
and that every element of $\cA_\omega$ contains $S$
as a subset.
Therefore we can write
\begin{align}
\label{eq:Uomega_factorization}
\begin{split}
\cA_\omega & = \{ I \sqcup S : I \in \cA_\omega'\} \\
U_\omega &\cong U'_\omega \times (\CC^\times)^{|S|}
\end{split}
\end{align}
for some $\cA_\omega' \subset 2^{\{1,\dots,m\} \setminus S}$ and
an open subset $U_\omega'$ of $\CC^{m-|S|}$.
The toric stack $X_\omega$ can be also written
as the quotient $[U_\omega'/G]$ of $U_\omega'$
for $G = \Ker(K \to (\Cstar)^{|S|})$: this corresponds to
the original construction of toric Deligne--Mumford stacks
by Borisov--Chen--Smith \cite{Borisov--Chen--Smith}.
The space of stability conditions $\omega\in \LL^\vee
\otimes \RR$ satisfying Assumption
\ref{assumption} has a wall and chamber structure.
The chamber $C_\omega$ to which $\omega$ belongs is given by
\begin{equation}
\label{eq:ext_amplecone}
C_\omega = \bigcap_{I \in \cA_\omega} \angle_I.
\end{equation}
and $X_\omega \cong X_{\omega'}$ as long as $\omega' \in C_\omega$.
The GIT quotient $X_{\omega'}$ changes when $\omega'$ crosses a codimension-one
boundary of $C_\omega$.
We call $C_\omega$ the \emph{extended ample cone};
as we will see in \S\ref{sec:amplecone} below,
it is the product of the ample cone for $X_\omega$ with a simplicial cone.
\begin{example}
Let $K = \Cstar$, so that $\LL = \Hom(\Cstar,K) \cong \ZZ$.
Let $D_1 = D_2 = 2 \in \ZZ^\vee$, and set
$\omega = 1\in \ZZ\otimes \RR=\RR$.
Then $U_\omega=\CC^2\setminus \{(0,0)\}$,
and $X_\omega$ is the weighted projective stack $\PP(2,2)$.
\end{example}
\subsection{GIT Data and Stacky Fans}
\label{sec:stacky_fan}
In the foundational work of Borisov--Chen--Smith \cite{Borisov--Chen--Smith},
toric DM stacks are defined in terms of \emph{stacky fans}.
Jiang \cite{Jiang} introduced the notion of an
\emph{extended} stacky fan, which is a stacky fan with
extra data.
Our GIT data above are in one-to-one correspondence
with extended stacky fans satisfying certain conditions, as we now explain.
An \emph{$S$-extended stacky fan}
is a quadruple $\mathbf{\Sigma}= (\bN,\Sigma,\beta,S)$, where:
\begin{itemize}
\item $\bN$ is a finitely generated abelian group\footnote{Note that $\bN$ may have torsion.};
\item $\Sigma$ is a rational simplicial fan in $\bN\otimes \RR$;
\item $\beta \colon \ZZ^m \to \bN$ is a homomorphism;
we write $b_i = \beta(e_i)\in \bN$ for the image of the $i$th standard
basis vector $e_i\in\ZZ^m$,
and write $\overline{b}_i$ for the image of $b_i$ in $\bN\otimes \RR$;
\item $S \subset \{1,\dots,m\}$ is a subset,
\end{itemize}
such that:
\begin{itemize}
\item each one-dimensional cone of $\Sigma$
is spanned by $\overline{b}_i$ for a unique
$i\in \{1,\dots,m\}\setminus S$, and
each $\overline{b}_i$ with
$i\in \{1,\dots,m\} \setminus S$ spans a one-dimensional
cone of $\Sigma$;
\item for $i\in S$, $\overline{b}_i$ lies in the support $|\Sigma|$
of the fan.
\end{itemize}
The vectors $b_i$ for $i\in S$ are called \emph{extended vectors}.
Stacky fans as considered by Borisov--Chen--Smith correspond to the cases where $S = \emptyset$.
For an extended stacky fan $(\bN,\Sigma,\beta,S)$,
the \emph{underlying stacky fan} is the triple $(\bN,\Sigma,\beta')$
where $\beta' \colon \ZZ^{m-|S|} \to \bN$ is
obtained from $\beta$ by deleting the columns corresponding to $S
\subset \{1,\dots,m\}$. The toric Deligne--Mumford stack associated to an extended stacky fan $(\bN,\Sigma,\beta,S)$ depends only on the underlying stacky fan.
To obtain an extended stacky fan from our GIT data,
consider the exact sequence:
\begin{equation}\label{eq:exact}
\xymatrix{
0 \ar[r] &
\LL \ar[r] &
\ZZ^m \ar[r]^\beta &
\bN \ar[r] &
0 }
\end{equation}
where the map from $\LL$ to $\ZZ^m$ is given by $(D_1,\ldots,D_m)$
and $\beta \colon \ZZ^m \to \bN$ is the cokernel of the map $\LL \to \ZZ^m$.
Let $b_i = \beta(e_i)\in \bN$ and $\overline{b}_i\in \bN\otimes \RR$
be as above and, given
a subset $I$ of $\{1,\dots,m\}$, let $\sigma_I$ denote the cone in
$\bN\otimes \RR$ generated by $\{\overline{b}_i : i\in I\}$.
The extended stacky fan
$\mathbf{\Sigma}_\omega=(\bN, \Sigma_\omega, \beta,S)$
corresponding to our data
consists of the group $\bN$ and the map $\beta$ defined
above, together with a fan $\Sigma_\omega$ in $\bN\otimes\RR$
and $S$ given by\footnote{This is why we refer to the
elements of $\cA_\omega$ as anticones.}:
\begin{align*}
\Sigma_\omega & = \{\sigma_{I} : \overline{I} \in \cA_\omega\}, \\
S & = \{ i \in \{1,\dots,m\} : \overline{\{i\}}
\notin \cA_\omega\}.
\end{align*}
The quotient construction in \cite[\S2]{Jiang}
coincides with that in Definition~\ref{def:toricstack},
and therefore $X_\omega$ is the toric Deligne-Mumford
stack corresponding to $\mathbf{\Sigma}_\omega$.
Extended stacky fans $(\bN, \Sigma_\omega, \beta,S)$ corresponding to GIT data
satisfy the following conditions:
\begin{itemize}
\item[(1)] the support $|\Sigma_\omega|$
of the fan is convex and full-dimensional;
\item[(2)] there is a strictly convex piecewise-linear function
$f\colon |\Sigma_\omega|
\to \RR$ that is linear on each cone of $\Sigma_\omega$;
\item[(3)] the map $\beta \colon \ZZ^m \to \bN$ is surjective.
\end{itemize}
The first two conditions are geometric constraints on $X_\omega$:
they are equivalent to saying
that the corresponding toric stack $X_\omega$
is semi-projective and has a torus fixed point.
The third condition can be always achieved by adding
enough extended vectors.
Conversely, given an extended stacky fan
$\mathbf{\Sigma}=(\bN, \Sigma,\beta,S)$ satisfying the
conditions (1)--(3) just stated, we can obtain GIT data as follows.
Define a free $\ZZ$-module $\LL$ by the
exact sequence \eqref{eq:exact}
and define $K := \LL\otimes \Cstar$.
The dual of \eqref{eq:exact} is an exact sequence:
\begin{equation}
\label{eq:divseq}
\xymatrix{
0 \ar[r] &
\bN^{\vee} \ar[r] &
(\ZZ^m)^{\vee} \ar[r]&
\LL^{\vee}}
\end{equation}
and we define the character $D_i \in \LL^\vee$ of $K$ to be the image
of the $i$th standard basis vector in $(\ZZ^m)^\vee$ under
the third arrow $(\ZZ^m)^\vee \to \LL^\vee$.
Set:
\[
\cA_{\omega}=\left\{I\subset \{1,2,\cdots,m\}:
S \subset I, \ \text{$\sigma_{\overline{I}}$
is a cone of $\Sigma$}\right\}
\]
and take the stability condition $\omega\in \LL^{\vee}\otimes\RR$
to lie in $\bigcap_{I \in \cA_{\omega}} \angle_{I}$;
the condition (2) ensures that this intersection is non-empty.
This specifies the data in Definition~\ref{def:toricstack}.
\subsection{Torus-Equivariant Cohomology}
\label{sec:equiv_coh}
The action of $T = (\Cstar)^m$ on $U_\omega$ descends
to a $\cQ := T/K$-action on $X_\omega$.
We also consider an ineffective $T$-action on $X_\omega$
induced by the projection $T \to \cQ$.
The $\cQ$-equivariant and $T$-equivariant cohomology of $X_\omega$
are modules over $R_\cQ:= H^\bullet_\cQ({\rm pt};\CC)$
and $R_T := H^\bullet_T({\rm pt};\CC)$ respectively.
By the exact sequence \eqref{eq:exact}, the Lie algebra of
$\cQ$ is identified with $\bN\otimes \CC$ and
$R_\cQ \cong \Sym^\bullet(\bN^\vee \otimes \CC)$.
Let $\lambda_i\in R_T$ be the equivariant first Chern class
of the irreducible $T$-representation given by the
projection $T \cong (\Cstar)^m \to \Cstar$ to the $i$th factor.
Then $R_T = \CC[\lambda_1,\dots,\lambda_m]$.
It is well-known that:
\begin{equation}
\label{eq:Qequiv_cohomology}
H_\cQ^\bullet(X_\omega;\CC) = R_\cQ
[u_1,\ldots,u_m]\big/(\mathfrak{I} + \mathfrak{J})
\end{equation}
where $u_i$ is the $\cQ$-equivariant class Poincar\'e-dual to the
toric divisor:
\begin{equation}
\label{eq:T-invariant_divisor}
\big\{ (z_1,\ldots,z_m) \in U_\omega : z_i = 0 \big\} \big/ K
\end{equation}
and $\mathfrak{I}$ and $\mathfrak{J}$ are the ideals
of additive and multiplicative relations:
\begin{align*}
\mathfrak{I} & = \big\langle \chi - \textstyle\sum_{i=1}^m
\<\chi,b_i\> u_i : \chi\in \bN^\vee\otimes \CC \big\rangle, \\
\mathfrak{J} & = \big\langle \textstyle \prod_{i \not \in I} u_i :
I \not \in \cA_\omega \big \rangle.
\end{align*}
Note that $u_i=0$ for $i\in S$ because the corresponding
divisor \eqref{eq:T-invariant_divisor} is empty (see equation~\ref{eq:Uomega_factorization}). Indeed,
this relation is contained in the ideal $\mathfrak{J}$.
The $T$-equivariant cohomology is given by the extension
of scalars:
\[
H_T^\bullet(X_\omega) \cong H_\cQ^\bullet(X_\omega) \otimes_{R_\cQ}
R_T
\]
where the algebra homomorphism $R_\cQ \to R_T$ is given by
$\chi \mapsto \sum_{i=1}^m \<\chi,b_i\> \lambda_i$
for $\chi\in \bN^\vee \otimes \CC$.
\begin{remark}
We note that the assumptions at the beginning of
\S \ref{sec:equivariant_qc} are satisfied for toric Deligne--Mumford
stacks obtained from GIT data.
First, all the $\cQ$-weights appearing in the $\cQ$-representation
$H^0(X_\omega,\cO)$ are contained in the strictly convex
cone $|\Sigma_\omega|^\vee =
\{ \chi \in \bN^\vee\otimes \RR : \text{$\<\chi,v\> \ge 0$ for all $v\in |\Sigma_\omega|$}\}$.
Second, $X_\omega$ is equivariantly formal
since the cohomology group of $X_\omega$ is generated by
$\cQ$-invariant cycles \cite{GKM}.
Because each component of $IX_\omega$ is again a
toric stack given by certain GIT data (see \S \ref{sec:inertia}),
we have that $IX_\omega$ is also equivariantly formal.
The same conclusions hold for the $T$-action.
\end{remark}
\subsection{Second Cohomology and Homology}
\label{sec:second_cohomology}
There is a commutative diagram:
\begin{equation}
\label{eq:H2_diagram}
\begin{aligned}
\xymatrix{
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\
0 \ar[r] & H^2_{\cQ}({\rm pt};\RR) \ar[r] \ar[d] & H^2_T({\rm pt};\RR) \ar[r] \ar[d] & H^2_K({\rm pt};\RR) \ar[r] \ar@{=}[d] & 0 \\
0 \ar[r] & H^2_{\cQ}(X_\omega;\RR) \ar[r] \ar[d] & H^2_T(X_\omega;\RR) \ar[r] \ar[d] & H^2_K({\rm pt};\RR) \ar[r] \ar[d] & 0 \\
0 \ar[r] & H^2(X_\omega;\RR) \ar@{=}[r] \ar[d] & H^2(X_\omega;\RR) \ar[r] \ar[d] & 0 \\
& 0 & 0 }
\end{aligned}
\end{equation}
with exact rows and columns.
Note that we have $H^2_\cQ({\rm pt};\RR) \cong \bN^\vee \otimes \RR$,
$H^2_T({\rm pt};\RR) \cong \RR^m$, $H^2_K({\rm pt};\RR)
\cong \LL^\vee \otimes \RR$.
The top row of \eqref{eq:H2_diagram} is identified with the exact sequence
\eqref{eq:divseq} tensored with $\RR$.
By \eqref{eq:Qequiv_cohomology}, $H^2_\cQ(X_\omega;\RR)$
is freely generated by the classes $u_i$, $i\in
\{1,\dots,m\} \setminus S$ of toric divisors,
and hence $H^2_\cQ(X_\omega;\RR) \cong \RR^{m-|S|}$.
The leftmost column is identified with the exact sequence
\[
\xymatrix{
0 \ar[r]&
\bN^\vee \otimes \RR \ar[r]&
\RR^{m-|S|} \ar[r]&
\LL^\vee\otimes \RR\big/\sum_{i\in S} \RR D_i
\ar[r] & 0
}
\]
induced by \eqref{eq:divseq}.
In particular we have
\[
H^2(X_\omega; \RR) \cong \LL^\vee \otimes \RR\big/
\textstyle \sum_{i\in S} \RR D_i
\]
where the non-equivariant limit of $u_i$ is identified with
the class of $D_i$.
The homology group $H_2(X_\omega;\RR)$ is identified with
$\bigcap_{i\in S} \Ker(D_i)$ in $\LL\otimes \RR$.
The square at the upper left of \eqref{eq:H2_diagram}
is a pushout and we have:
\begin{align*}
H^2_T(X_\omega;\RR)
& \cong \bigoplus_{i\in \{1,\dots,m\} \setminus S}
\RR u_i \oplus
\bigoplus_{i=1}^m \RR \lambda_i \Big/
\big \langle \textstyle\sum_{i=1}^m \<\chi, b_i\> (u_i - \lambda_i)
\colon \chi \in \bN^\vee \otimes \RR \big\rangle.
\end{align*}
It follows that the middle row of \eqref{eq:H2_diagram}
splits canonically: we have a well-defined homomorphism\footnote
{More precisely $(-\theta)$ gives a splitting of the middle row
of \eqref{eq:H2_diagram}.}
\begin{equation}
\label{eq:theta}
\theta \colon \LL^\vee \otimes \RR \cong H^2_K({\rm pt};\RR)
\longrightarrow H^2_T(X_\omega;\RR)
\end{equation}
such that $\theta(D_i) =u_i-\lambda_i$ and that
\[
H^2_T(X_\omega;\RR) \cong H^2_\cQ(X_\omega;\RR)
\oplus \theta(\LL^\vee\otimes \RR).
\]
The class $\theta(p)$ can be written as the $T$-equivariant first
Chern class of a certain line bundle $L(p)$ associated
to $p$ (see \S \ref{sec:basis_K-theory}).
One advantage of working with $T$-equivariant cohomology instead of $\cQ$-equivariant cohomology
is the existence of this canonical splitting.
We also introduce a canonical splitting of the projection
$\LL^\vee \otimes \RR \to \LL^\vee \otimes \RR/
\sum_{i\in S} \RR D_i \cong H^2(X_\omega,\RR)$.
This is equivalent to choosing a complementary subspace
of $H_2(X_\omega;\RR)$ in $\LL\otimes \RR$.
Take $j\in S$. The corresponding extended vector $\overline{b_j} \in
\bN\otimes \RR$ lies in the support of the fan.
Let $\sigma_{I_j}\in \Sigma$,
$I_j\subset \{1,\dots,m\} \setminus S$ be
the minimal cone containing $\overline{b_j}$ and write
$\overline{b_j} = \sum_{i\in I_j} c_{ij} \overline{b_i}$
for some $c_{ij} \in \RR_{>0}$.
By the exact sequence \eqref{eq:exact}, there exists an
element $\xi_j \in \LL\otimes \QQ$ such that
\begin{equation}
\label{eq:def_xi}
D_i \cdot \xi_j = \begin{cases}
1 & \text{if }i=j; \\
-c_{ij} & \text{if } i \in I_j; \\
0 & \text{if } i\notin I_j \cup \{j\}.
\end{cases}
\end{equation}
Note that one has $D_i \cdot \xi_j = \delta_{ij}$ for $i,j\in S$.
Hence $\{\xi_i\}_{i\in S}$ spans a complementary subspace
of $H_2(X_\omega;\RR) =
\bigcap_{j\in S} \Ker(D_j) \subset \LL \otimes \RR$
and defines a splitting:
\begin{equation}
\label{eq:L_decomp}
\LL\otimes \RR \cong H_2(X_\omega;\RR) \oplus \bigoplus_{j\in S} \RR
\xi_j,
\end{equation}
or, for the dual space,
\begin{equation}
\label{eq:Lvee_decomp}
\LL^\vee \otimes \RR \cong \bigcap_{j\in S} \Ker(\xi_j) \oplus
\bigoplus_{j\in S} \RR D_j
\end{equation}
with $\bigcap_{j\in S} \Ker(\xi_j) \cong H^2(X_\omega;\RR)$.
The equivariant first Chern class of $TX_\omega$ is
given by:
\[
\rho = c_1^\cQ(T X_\omega) = c_1^T(T X_\omega) =
\sum_{i\in \{1,\dots,m\} \setminus S} u_i.
\]
\subsection{Ample Cone and Mori Cone}
\label{sec:amplecone}
Let $D'_i$ denote the image of $D_i$ in
$\LL^\vee\otimes \RR/\sum_{i\in S} \RR D_i \cong H^2(X_\omega;\RR)$.
This is the non-equivariant Poincar\'{e} dual of the toric
divisor \eqref{eq:T-invariant_divisor}, that is, the non-equivariant limit of $u_i$.
The cone of ample divisors of $X_\omega$ is given by
\[
C'_{\omega} = \bigcap_{I\in \cA_\omega'} \angle'_I
\]
where $\cA_\omega'$ was introduced in equation
\eqref{eq:Uomega_factorization} and
$\angle'_I := \sum_{i\in I} \RR_{>0} D'_i$ is an
open cone in $\LL^\vee \otimes \RR/\sum_{i\in S} \RR D_i$
(cf.~Notation \ref{not:cones}).
Under the splitting \eqref{eq:Lvee_decomp} of $\LL^\vee \otimes \RR$,
the extended ample cone $C_\omega$ defined in equation~\ref{eq:ext_amplecone} also splits
\cite[Lemma 3.2]{Iritani}:
\begin{equation}
\label{eq:splitcone}
C_\omega \cong C'_\omega \times \left( \sum_{i\in S} \RR_{>0} D_i
\right) \subset H^2(X_\omega;\RR) \times \bigoplus_{i\in S} \RR D_i.
\end{equation}
The Mori cone is the dual cone of $C'_\omega$:
\[
\NE(X_\omega) = C'^\vee_\omega =
\{ d\in H_2(X_\omega;\RR) : \text{$\eta \cdot d \ge 0$ for all $\eta \in C'_\omega$}\}
\]
\subsection{Fixed Points and Isotropy Groups}
\label{sec:fixed_points}
Fixed points of the $T$-action on $X_\omega$ are
in one-to-one correspondence with minimal anticones, that is, with $\delta \in \cA_\omega$ such that $|\delta| = r$.
A minimal anticone $\delta$ corresponds to the $T$-fixed point:
\[
\big\{(z_1,\ldots,z_n) \in U_\omega : \text{$z_i = 0$ if $i \not \in
\delta$}\big\} \big/ K
\]
We now describe the isotropy of the Deligne--Mumford stack $X_\omega$,
i.e.~those elements $g \in K$ such that the action of $g$ on $U_\omega$
has fixed points. Recall that there are canonical isomorphisms
$K \cong \LL \otimes \Cstar$ and $\Lie(K) \cong \LL \otimes \CC$,
via which the exponential map $\Lie(K) \to K$ becomes
$\id\otimes \exp(2 \pi \tti {-}) \colon \LL \otimes \CC
\to \LL \otimes \Cstar$.
The kernel of the exponential map is $\LL \subset \LL \otimes \CC$.
Define $\KK \subset \LL \otimes \QQ$ to be the set
of $f \in \LL \otimes \QQ$ such that:
\begin{equation}
\label{eq:has_fixed_points}
I_f := \Big\{ i \in \{1,2,\ldots,m\} : D_i \cdot f \in \ZZ \Big\}
\in \cA_\omega
\end{equation}
The lattice $\LL$ acts on $\KK$ by translation,
and elements $g \in K$ such that the action
of $g$ on $U_\omega$ has fixed points correspond,
via the exponential map, to elements of $\KK/\LL$.
\subsection{Floors, Ceilings, and Fractional Parts}
For a rational number $q$, we write:
\begin{align*}
& \text{$\lfloor q \rfloor$ for the largest integer $n$ such that $n \leq q$;} \\
& \text{$\lceil q \rceil$ for the smallest integer $n$ such that $q \leq n$; and} \\
& \text{$\< q \>$ for the fractional part $q - \lfloor q \rfloor$ of $q$.}
\end{align*}
\subsection{The Inertia Stack and Chen--Ruan Cohomology}
\label{sec:inertia}
Recall the definition of the inertia stack $I X_\omega$ from~\S\ref{sec:conditions}.
Components of $I X_\omega$ are indexed by elements of $\KK/\LL$:
the component $X_\omega^f$ of $IX_\omega$ corresponding to
$f \in \KK/\LL$ consists of the points $(x,g)$ in $I X_\omega$
such that $g = \exp(2\pi \tti f)$.
Recall the set $I_f$ defined in \eqref{eq:has_fixed_points}.
The component $X_\omega^{f}$ in the inertia stack
$I X_\omega$ is the toric Deligne--Mumford stack
with GIT data given by $K$, $\LL$, and $\omega$
exactly as for $X_\omega$, and characters
$D_i \in \LL^\vee$ for $i \in I_f$. We have:
\[
X_\omega^f = [\CC^{I_f} \cap U_\omega/K].
\]
The inclusion $\CC^{I_f} \subset \CC^m$ exhibits $X_\omega^f$
as a closed substack of the toric stack $X_\omega$.
According to Borisov--Chen--Smith \cite{Borisov--Chen--Smith},
components of the inertia stack of $X_\omega$ are indexed
by elements of the set $\cbox(X_\omega)$:
\[
\cbox(X_\omega) = \left
\{ v \in \bN : \overline{v} =
\text{$\sum_{i\notin I} c_i \overline{b_i}$ in $\bN\otimes \RR$ for some $I\in \cA$ and $0\le c_i <1$} \right\}
\]
In fact, we have an isomorphism \cite[\S3.1.3]{Iritani}:
\begin{align}
\label{eq:KL_Box}
\KK/\LL \cong \cbox(X_\omega)
&& [f] \mapsto v_f = \sum_{i=1}^m
\lceil - (D_i \cdot f) \rceil b_i \in \bN.
\end{align}
When $j\in S$ and $b_j\in \cbox(X_\omega)$, the element
$-\xi_j \in\LL\otimes \QQ$ defined in
\eqref{eq:def_xi} belongs to $\KK$ and corresponds to $b_j$.
The age $\iota_f$ of the component $X_\omega^f \subset I X_\omega$
is $\sum_{i\notin I_f}\langle D_i \cdot f\rangle$.
The $T$-equivariant Chen--Ruan cohomology of $X_\omega$ is,
as we saw in~\S\ref{sec:CR}, the $T$-equivariant cohomology of the inertia stack
$IX_\omega$ with age-shifted grading:
\[
H_{\CR,T}^\bullet(X_\omega;\QQ) = \bigoplus_{f\in \KK/\LL}
H_T^{\bullet-2\iota_f}\big(X_\omega^f;\QQ\big)
\]
This contains the $T$-equivariant cohomology of $X_\omega$
as a summand, corresponding to the element $0 \in \KK/\LL$;
furthermore the fact that each $X_\omega^f$ is a closed substack
of $X_\omega$ implies that $H_{\CR,T}^\bullet(X_\omega;\QQ)$
is naturally a module over $H_{T}^\bullet(X_\omega;\QQ)$.
We write $\fun_f$ for the unit class in $H_T^0\big(X_\omega^f;\QQ\big)$,
regarded as an element of $H_{\CR,T}^{2\iota_f}(X_\omega;\QQ)$.
Recall that the component $X_\omega^f$ of the inertia stack is
the toric Deligne--Mumford stack with GIT data $(K;\LL;\omega;D_i : i \in I_f)$.
In particular, therefore, the anticones for $X_\omega^f$ are given by
$\{I \in \cA_\omega : I \subset I_f\}$.
$T$-fixed points on the inertia stack $IX_\omega$
are indexed by pairs $(\delta,f)$ where $\delta$
is a minimal anticone in $\cA_\omega$, $f \in \KK/\LL$,
and $D_i \cdot f \in \ZZ$ for all $i \in \delta$.
The pair $(\delta,f)$ determines a $T$-fixed point
on the component $X_\omega^f$ of the inertia stack:
the $T$-fixed point that corresponds
to the minimal anticone $\delta \subset I_f$.
\section{Wall-Crossing in Toric Gromov--Witten Theory}
\label{sec:wall-crossing}
In this section we consider crepant birational transformations $X_+ \dashrightarrow X_-$
between toric Deligne--Mumford stacks
which arise from variation of GIT. We use the Mirror Theorem for toric
Deligne--Mumford stacks~\cite{CCIT,Cheong--Ciocan-Fontanine--Kim} to construct a global equivariant quantum connection over (a certain part of) the secondary toric variety for $X_\pm$; this gives an analytic continuation of the equivariant quantum connections for $X_+$ and $X_-$.
\subsection{Birational Transformations from Wall-Crossing}
\label{sec:birational_transformations}
Recall that our GIT data in \S \ref{sec:GITdata}
consist of a torus $K \cong (\Cstar)^r$,
the lattice $\LL = \Hom(\Cstar,K)$ of
$\Cstar$-subgroups of $K$,
and characters $D_1,\ldots,D_m \in \LL^\vee$.
Recall further that a choice of stability condition
$\omega \in \LL^\vee \otimes \RR$ satisfying
Assumption~\ref{assumption} determines a toric
Deligne--Mumford stack $X_\omega = \big[U_\omega/K\big]$.
The space $\LL^\vee \otimes \RR$ of stability conditions is divided into
chambers by the closures of the sets $\angle_I$, $|I| = r-1$,
and the Deligne--Mumford stack $X_\omega$ depends on $\omega$
only via the chamber containing $\omega$.
For any stability condition $\omega$ satisfying Assumption~\ref{assumption},
the set $U_\omega$ contains the big torus $T=(\Cstar)^m$,
and thus for any two such stability conditions $\omega_1$,~$\omega_2$
there is a canonical birational map
$X_{\omega_1} \dashrightarrow X_{\omega_2}$,
induced by the identity transformation between
$T/K \subset X_{\omega_1}$ and
$T/K \subset X_{\omega_2}$.
Our setup is as follows. Let $C_+$,~$C_-$ be chambers
in $\LL^\vee \otimes \RR$ that are separated by a hyperplane wall $W$,
so that $W \cap \overline{C_+}$ is a facet of $\overline{C_+}$,
$W \cap \overline{C_-}$ is a facet of $\overline{C_-}$,
and $W \cap \overline{C_+} = W \cap \overline{C_-}$.
Choose stability conditions $\omega_+ \in C_+$, $\omega_- \in C_-$
satisfying Assumption~\ref{assumption} and set
$X_+ := X_{\omega_+}$, $X_- := X_{\omega_-}$, and
\begin{align*}
& \cA_\pm := \cA_{\omega_{\pm}} =
\big\{ I \subset \{1,2,\ldots,m\} :
\omega_\pm \in \angle_I \big\}
\end{align*}
Then $C_\pm = \bigcap_{I\in \cA_\pm} \angle_I$.
Let $\varphi \colon X_+ \dashrightarrow X_-$ be
the birational transformation induced by the toric wall-crossing
and suppose that
\[
\sum_{i=1}^m D_i \in W
\]
As we will see below this amounts to requiring that $\varphi$ is crepant.
Let $e \in \LL$ denote the primitive lattice vector in $W^\perp$
such that $e$ is positive on $C_+$ and negative on $C_-$.
\begin{remark}
The situation considered here is quite general. We do not require $X_+$,~$X_-$ to have projective
coarse moduli space (they are required to be semi-projective).
We do not require that $X_+$,~$X_-$ are weak Fano,
or that they satisfy the extended weak Fano condition in \cite[\S 3.1.4]{Iritani}.
In other words, we do not require $\sum_{i=1}^m D_i\in W$ to lie in the
boundary $W \cap \overline{C_+} = W \cap \overline{C_-}$ of
the extended ample cones.
\end{remark}
Choose $\omega_0$ from the relative interior of $W\cap \overline{C_+}
= W \cap \overline{C_-}$. The stability condition $\omega_0$
does not satisfy our Assumption \ref{assumption}, but we can
still consider:
\begin{align*}
\cA_0 & := \cA_{\omega_0} =
\left\{ I \subset \{1,\dots,m\} :
\omega_0 \in \angle_I \right\}
\end{align*}
and the corresponding toric (Artin) stack $X_0 := X_{\omega_0}
=[U_{\omega_0}/K]$
as given in Definition \ref{def:toricstack}.
Here $X_0$ is not Deligne--Mumford, as
the $\Cstar$-subgroup of $K$
corresponding to $e\in \LL$ (the defining equation
of the wall $W$) has a fixed point in $U_{\omega_0}$.
The stack $X_0$ contains both $X_+$ and $X_-$ as open substacks
and the canonical line bundles of $X_{+}$ and $X_-$
are the restrictions of the same line bundle $L_0\to X_0$
given by the character $-\sum_{i=1}^m D_i$ of $K$.
The condition $\sum_{i=1}^m D_i\in W$
ensures that $L_0$ comes from
a $\QQ$-Cartier divisor on the underlying singular
toric variety
$\overline{X}_0 = \CC^m/\!\!/_{\omega_0} K$
associated to the fan $\Sigma_{\omega_0}$.
On the other hand, in \S \ref{sec:FM}, we shall construct
a toric Deligne-Mumford stack $\tX$ equipped
with proper birational morphisms $f_\pm \colon
\tX \to X_{\pm}$
such that the following diagram commutes:
\begin{equation}
\label{eq:crepant_diagram}
\begin{aligned}
\xymatrix{
& \tX \ar[rd]^{f_-} \ar[ld]_{f_+} & \\
X_+ \ar[rd]_{g_+} \ar@{-->}^{\varphi}[rr] & & X_- \ar[ld]^{g_-} \\
& \overline{X}_0 &
}
\end{aligned}
\end{equation}
Then $f_+^\star(K_{X_+})$ and
$f_-^\star(K_{X_-})$ coincide since they
are the pull-backs of a $\QQ$-Cartier divisor on $\overline{X}_0$.
This is what is meant by the birational map $\varphi$ being
\emph{crepant}\footnote
{This notion is also called $K$-equivalence: see the Introduction.}.
Set:
\begin{align*}
M_{\pm} & = \{ i \in \{1,\dots,m\} : \pm D_i\cdot e >0 \}, \\
M_0 & = \{ i\in \{1,\dots,m\} : D_i \cdot e = 0\}.
\end{align*}
Our assumptions imply that both $M_+$ and $M_-$ are non-empty. The following lemma is easy to check:
\begin{lemma}
\label{lem:Apm}
Set:
\begin{align*}
\cA_0^{\rm thin} & := \{ I \in \cA_0 : I \subset M_0\} \\
\cA_0^{\rm thick} & := \{ I \in \cA_0: I \cap M_+ \neq \emptyset,
I \cap M_- \neq \emptyset\}.
\end{align*}
Then one has $M_0\in \cA_0^{\rm thin}$ and
\begin{align*}
\cA_0 & = \cA_0^{\rm thin} \sqcup \cA_0^{\rm thick}, \\
\cA_\pm & = \cA_0^{\rm thick} \sqcup \left\{ I \sqcup J :
\emptyset \neq J \subset M_\pm, I \in \cA_0^{\rm thin}\right\}.
\end{align*}
\end{lemma}
\begin{remark}
\label{rem:circuit}
Let $\Sigma_\pm$ be the fans of $X_\pm$.
In terms of fans, a toric wall-crossing can be described
as a \emph{modification along a circuit} \cite{GKZ:book,Borisov--Horja:FM}, where `circuit' means a minimal linearly dependent set of vectors.
In our wall-crossing, the relevant circuit is $\{b_i: i\in M_+ \cup M_-\}$: we have $\sum_{i\in M_+ \cup M_-} (D_i\cdot e) b_i=0$,
and every proper subset of $\{b_i : i\in M_+ \cup M_-\}$ is linearly
independent.
The partition of the circuit $M_+ \cup M_-$ into $M_+$ and $M_-$
is determined by the sign of the coefficients in
a relation among $\{b_i : i\in M_+\cup M_-\}$.
The modification along the circuit $M_+ \cup M_-$
turns the fan $\Sigma_+$ into $\Sigma_-$: it
removes every cone $\sigma_I$ of $\Sigma_+$ such that
$I$ contains $M_-$ but not $M_+$ and introduces
cones of the form $\sigma_K$ where $K=(I\cup M_+) \setminus J$ for
any non-empty subset $J \subset M_-$.
This description matches with Lemma~\ref{lem:Apm},~\S\ref{sec:stacky_fan}, and~\S\ref{sec:GITdata}.
\end{remark}
There are three types of possible crepant toric wall-crossings:
(I) $X_+$ and $X_-$ are isomorphic in codimension one
(``flop''),
(II) $\varphi$ induces a morphism $X_{+} \to |X_{-}|$
or $X_- \to |X_+|$
contracting a divisor to a toric subvariety (``crepant resolution'')
and
(III) the rigidifications\footnote{See e.g.~\cite{FMN}.} $X_+^{\rm rig}$, $X_-^{\rm rig}$
are isomorphic
(only the gerbe structures change; we call it a ``gerbe flop'').
Define:
\[
S_\pm = \{ i\in \{1,\dots,m\} : \overline{\{i\}}
\not \in \cA_\pm\}.
\]
\begin{proposition}
\label{prop:Spm_classif}
The intersection $S_0 := S_+ \cap S_-$ is contained in $M_0$.
Moreover, one and only one of the following holds:
\begin{itemize}
\item[(I)] $S_+ = S_-$, $\sharp(M_+)\ge 2$ and $\sharp(M_-) \ge 2$;
\item[(II-i)] there exists $i\in \{1,\dots,m\}$ such that
$S_- = S_+ \sqcup \{i\}$, $M_- = \{i\}$ and $\sharp(M_+) \ge 2$;
\item[(II-ii)] there exists $i\in \{1,\dots,m\}$ such that
$S_+ = S_- \sqcup \{i\}$, $M_+ = \{i\}$ and $\sharp(M_-)\ge 2$;
\item[(III)] there exist $i_+,i_- \in \{1,\dots,m\}$ such that
$S_+ = S_0 \sqcup \{i_+\}$,
$S_- = S_0 \sqcup \{i_-\}$, $M_+ = \{i_+\}$
and $M_- = \{i_-\}$.
\end{itemize}
\end{proposition}
\begin{proof}
First we show that $S_0\subset M_0$.
Take $i \in S_0$.
Suppose that $i\in M_+$.
Since $M_0\in \cA_0^{\rm thin}$, we have
$M_0 \cup M_-\in \cA_-$ by Lemma \ref{lem:Apm}.
Thus $\overline{\{i\}} =
M_0 \cup M_- \cup (M_+\setminus \{i\})$ also belongs
to $\cA_-$. This contradicts the fact that $i\in S_-$.
Thus we have $i\notin M_+$, and similarly that
$i\notin M_-$. Hence $i\in M_0$. We have shown
that $S_0 \subset M_0$.
Next we claim that:
\begin{itemize}
\item[(a)] if $S_-\setminus S_+$ is non-empty,
then we have $\sharp(S_- \setminus S_+) = 1$ and
$M_- = S_-\setminus S_+$;
\item[(b)] if $S_- \subset S_+$, then $\sharp(M_-) \ge 2$.
\end{itemize}
Take $i\in S_-\setminus S_+$.
We have $\overline{\{i\}} \in \cA_+\setminus \cA_-$.
Lemma \ref{lem:Apm} implies that an element of
$\cA_+ \setminus \cA_-$ is of the form $I \sqcup J$ with
$\emptyset \neq J \subset M_+$ and $I\subset M_0$,
and in particular does not intersect with $M_-$.
This implies that $\{i\} = M_-$.
Therefore $S_-\setminus S_+ = M_-$ consists of
only one element. This proves (a).
Conversely, if $M_-=\{i\}$,
it follows from Lemma \ref{lem:Apm} that
$\overline{\{i\}}\in \cA_+\setminus \cA_-$
and thus $i \in S_- \setminus S_+$.
This proves (b).
The same claim holds if we exchange $+$ and $-$.
It follows that one and only one of (I), (II-i), (II-ii), (III) happens.
\end{proof}
\begin{proposition}
The loci of indeterminacy of $\varphi$ and $\varphi^{-1}$ are
the toric substacks
\begin{align*}
\bigcap_{j\in M_-} \{z_j=0\}\subset X_+ && \text{and} &&
\bigcap_{j\in M_+} \{z_j=0\}\subset X_-
\end{align*}
respectively. With cases as in Proposition~\ref{prop:Spm_classif}, we have:
\begin{itemize}
\item[(I)] $X_+$ and $X_-$ are isomorphic
in codimension one;
\item[(II-i)] $\varphi$ induces a morphism
$\varphi \colon X_+ \to |X_-|$ that contracts the
divisor $\{z_i=0\}$ to the subvariety
$\bigcap_{j\in M_+} \{z_j=0\}$;
\item[(II-ii)] a statement similar to {\rm (II-i)} with $+$ and $-$
interchanged;
\item[(III)] $\varphi$ induces an isomorphism
$X_+^{\rm rig} \cong X_-^{\rm rig}$
between the rigidifications.
\end{itemize}
\end{proposition}
\begin{proof}
One can check that
$U_{\omega_+} \cap U_{\omega_-} =
U_{\omega_+} \setminus \bigcap_{i\in M_-} \{z_i=0\}
= U_{\omega_-} \setminus \bigcap_{i\in M_+} \{z_i=0\}$
using Lemma \ref{lem:Apm}.
The geometric picture in each case can be seen from
the stacky fans: (I) the sets of one-dimensional
cones are the same;
(II-i) the fan $\Sigma_-$ is obtained by
deleting the ray $\RR_{\ge 0} \overline{b}_i$ from $\Sigma_+$;
$\sigma_{M_+} \in \Sigma_-$ is a minimal cone containing
$\overline{b}_i$;
$\varphi$ contracts the toric divisor $\{z_i=0\}$
to the closed subvariety associated with $\sigma_{M_+}$; (II-ii) similar;
(III) the stacky fan $\mathbf{\Sigma}_-$ is obtained
from $\mathbf{\Sigma}_+$ by replacing $b_{i_-}$ with
$b_{i_+}$; one has $(D_{i_+} \cdot e) b_{i_+} =
- (D_{i_-} \cdot e) b_{i_-}$ by \eqref{eq:exact}
and $D_{i_+} \cdot e + D_{i_-} \cdot e =0$; thus
$b_{i_+}$ and $b_{i_-}$ differ only by a torsion element
in $\bN$.
\end{proof}
\begin{example} \
\begin{itemize}
\item[(I)] Let $a_1,\dots,a_k, b_1,\dots,b_l$ be positive integers such that
$a_1+ \cdots + a_k = b_1+ \cdots + b_l$.
Consider the GIT data given by $\LL^\vee =\ZZ$, $D_1 = a_1,\dots,D_k = a_k$,
$D_{k+1} = -b_1,\dots,D_{k+l}= -b_l$.
If $k,l\ge 2$, we have a flop between
\begin{align*}
X_+= \bigoplus_{i=1}^k \cO_{\PP(a_1,\dots,a_k)}(-b_i)
&& \text{and} &&
X_- = \bigoplus_{j=1}^l \cO_{\PP(b_1,\dots,b_l)}(-a_j).
\end{align*}
\item[(II)] Consider the case where $l=1$ in (I). Setting $d = a_1+ \cdots + a_k = b_1$, we have that $X_+ = \cO_{\PP(a_1,\dots,a_k)}(-d)$ is a crepant (partial) resolution of $|X_-| = \CC^k/\bmu_d$ where $\bmu_d$ acts on $\CC^k$ by the weights $(a_1/d,\dots,a_k/d)$.
\item[(III)] Consider the GIT data given by $\LL^\vee = \ZZ^2$, $D_1 = (1,0)$, $D_2 = (1,2)$, $D_3 = (0,2)$. Take $\omega_+$ from the chamber $\{(x,y) : 0< y < 2x \}$ and $\omega_-$ from the chamber $\{(x,y) : 0< 2x < y \}$. Then we have a ``gerbe flop" between $X_+ = \PP(2,2)$ and $X_- = \PP^1 \times B\bmu_2$.
\end{itemize}
\end{example}
\subsection{Decompositions of Extended Ample Cones}
Recall the decomposition \eqref{eq:Lvee_decomp} of the vector space
$\LL^\vee \otimes \RR$ and the decomposition
\eqref{eq:splitcone} of the extended ample cone.
In the case at hand, we have two (possibly different)
decompositions of $\LL^\vee \otimes \RR$
associated to the GIT quotients $X_+$ and $X_-$:
\begin{equation}
\label{eq:decomp_pm}
\LL^\vee \otimes \RR = \bigcap_{j\in S_\pm}
\Ker(\xi_j^\pm) \oplus \bigoplus_{j\in S_\pm} \RR D_j
\end{equation}
where elements $\xi_j^\pm \in \LL \otimes \RR$, $j\in S_\pm$
are as in \eqref{eq:def_xi} and
$\bigcap_{j\in S_\pm} \Ker(\xi_i^+) \cong H^2(X_\pm,\RR)$.
Under these decompositions, one has
\[
C_\pm = C_\pm' \times \sum_{j\in S_\pm} \RR_{>0} D_j
\]
where $C_\pm' \subset \bigcap_{i\in S_\pm} \Ker(\xi^\pm_i) \cong
H^2(X_\pm;\RR)$ is the ample cone of $X_\pm$.
Let $\overline{C_W} := W \cap \overline{C_+}
= W \cap \overline{C_-}$ be a common facet of $C_+$ and $C_-$,
and write $C_W$ for the relative interior of $\overline{C_W}$.
We now show that these decompositions of
the cones $C_+$,~$C_-$ are compatible along the wall.
\begin{proposition}
\label{prop:wallcone_decomp}
We have $\xi_i^+|_W = \xi_i^-|_W$ for $i\in S_0 = S_+ \cap S_-$
and $\xi_i^\pm|_W = 0$ for $i\in S_\pm \setminus S_\mp$.
Set $\xi_i^W = \xi_i^+|_W = \xi_i^-|_W \in \Hom(W,\RR)$.
Then we have
\[
W' := W \cap \bigcap_{i\in S_+} \Ker(\xi_i^+) = W \cap \bigcap_{i\in S_-}
\Ker(\xi_i^-) = \bigcap_{i\in S_0} \Ker(\xi^W_i)
\]
and so the decompositions \eqref{eq:decomp_pm} restrict to
the same decomposition of $W$:
\begin{equation}
\label{eq:decomp_W}
W = W' \oplus
\bigoplus_{i\in S_0} \RR D_i.
\end{equation}
Under this decomposition of $W$, the cone $C_W$ decomposes as
\[
C_W = C_W' \times \sum_{i\in S_0} \RR_{>0} D_i
\]
for some cone $C_W'$ in $W'$.
With cases as in Proposition \ref{prop:Spm_classif}, we have:
\begin{itemize}
\item[(I)] $C_W'$ is a common facet of $C_+'$ and $C_-'$;
\item[(II-i)] $C_W' = C_-'$, $C_W'$ is a facet of $C_+'$
and $C_- = C_W + \RR_{>0} D_i$;
\item[(II-ii)] $C_W' = C_+'$, $C_W'$ is a facet of $C_-'$
and $C_+ = C_W + \RR_{>0} D_i$;
\item[(III)] $C_W' = C_+' = C_-'$,
$C_+ = C_W + \RR_{>0} D_{i_+}$ and
$C_- = C_W + \RR_{>0} D_{i_-}$.
\end{itemize}
\end{proposition}
\begin{proof}
It suffices to show that $\xi_i^+|_W = \xi_i^-|_W$
for $i\in S_0$ and that $\xi_i^\pm|_W = 0$ for
$i\in S_\pm \setminus S\mp$. The rest of the statements
follow easily.
Suppose that $i\in S_0$.
Recall the definition of $\xi_i^\pm$ in \eqref{eq:def_xi}.
Let $\sigma_{I} \in\Sigma_{+}$ be the minimal
cone containing $\overline{b_i}$.
Then $\overline{I} \in \cA_+$.
If $\overline{I} \in \cA_-$, we have
$\xi_i^+ = \xi_i^-$ by the definition of $\xi_i^\pm$.
Suppose that $\overline{I}\notin \cA_-$.
By Lemma \ref{lem:Apm}, $I$ contains $M_-$
but not $M_+$. We have a relation of the form:
\begin{equation}
\label{eq:b_i_cone}
\overline{b_i} = \sum_{j\in I} c_j \overline{b_j}
\end{equation}
with $c_j>0$.
By adding to the right-hand side of
\eqref{eq:b_i_cone}
a suitable positive multiple of the relation
\[
\sum_{j \in M_+} (D_j \cdot e) \overline{b_j} -
\sum_{j\in M_-} (-D_j\cdot e) \overline{b_j} =0
\]
given by $e\in \LL$ via \eqref{eq:exact}, we obtain a
relation of the form
\[
\overline{b_i} = \sum_{j\in I'} c_j' \overline{b_j}
\]
such that $c'_j>0$ and
$I' = (I \cup M_+) \setminus J$ with $\emptyset \neq J\subset M_-$.
Then $\overline{I'} \in \cA_-$ by Lemma \ref{lem:Apm}
(see also Remark \ref{rem:circuit}).
Note that $c_j = c_j'$ if $j \in I \cap M_0 = I' \cap M_0$.
This implies that $D_j \cdot \xi_i^+ = D_j \cdot \xi_i^-$
for all $j\in M_0$. Since $\{D_j: j\in M_0\}$ spans $W$,
we have $\xi_i^+|_W = \xi_i^-|_W$.
Now suppose that $i\in S_+ \setminus S_-$.
Then $M_+ = \{i\}$ by Proposition \ref{prop:Spm_classif}
and we have a relation $(D_i \cdot e)
\overline{b_i} = \sum_{j\in M_-} (-D_j \cdot e) \overline{b_j}$
given by $e\in\LL$. This implies that $\overline{b_i}$
is contained in the cone $\sigma_{M_-}$ of $\Sigma_+$,
and the definition of $\xi_i^+$ implies that
$D_j \cdot \xi_i^+=0$ for all $j\in M_0$.
Thus $\xi_i^+|_W= 0$.
The case where $i\in S_- \setminus S_+$ is similar.
\end{proof}
\subsection{Global Extended K\"{a}hler Moduli}
\label{sec:global_KM}
Our next goal is to describe a global `moduli space' $\tcM$ and a flat connection over $\tcM$, together with two neighbourhoods in $\tcM$ such that the restriction of the flat connection to one of the neighbourhoods (respectively to the other neighbourhood) is isomorphic to the equivariant quantum connection for $X_+$ (respectively for $X_-$). Thus the equivariant quantum connections for $X_+$ and $X_-$ can be analytically continued to each other. Roughly speaking, the space $\tcM$ will be a covering of a neighbourhood of a certain curve in the \emph{secondary toric variety} for $X_\pm$; in this section we introduce notation for and local co-ordinates on this secondary toric variety.
The wall and chamber structure of $\LL^\vee \otimes \RR$ described in \S\ref{sec:birational_transformations} defines a fan in $\LL^\vee\otimes \RR$, called the \emph{secondary fan} or \emph{Gelfand--Kapranov--Zelevinsky (GKZ) fan}. The toric variety associated to the GKZ fan is called the secondary toric variety. We consider the subfan of the GKZ fan consisting of the cones $\overline{C_+}$, $\overline{C_-}$ and their faces, and consider the toric variety $\cM$ associated to this fan. (Thus $\cM$ is an open subset of the secondary toric variety.) In the context of mirror symmetry, $\cM$ arises as the moduli space of Landau--Ginzburg models mirror to $X_\pm$. It contains the torus fixed points $P_+$ and $P_-$ associated to the cones $C_+$ and $C_-$, which are called the \emph{large radius limit points} for $X_+$ and $X_-$. More precisely, because we want to impose only very weak convergence hypotheses on the equivariant quantum products for $X_\pm$, we restrict our attention to the formal neighbourhood of the torus-invariant curve $\cC \subset \cM$ connecting $P_+$ and $P_-$: $\cC$ is the closed toric subvariety associated to the cone $\overline{C_W} = W \cap \overline{C_+} = W \cap \overline{C_-}$.
Our secondary toric variety $\cM$ is covered by two open
charts
\begin{align}
\label{eq:charts_cM}
\Spec \CC[C_+^\vee \cap \LL] && \text{and} &&
\Spec \CC[C_-^\vee \cap \LL]
\end{align}
that are glued along $\Spec \CC[C_W^\vee \cap \LL]$.
Since the cones $C_\pm$ are not necessarily simplicial,
$\cM$ is in general singular.
For our purpose, it is convenient to use a lattice structure
different from $\LL$ and to work with a smooth cover
$\cM_{\rm reg}$ of $\cM$.
We will define the cover $\cM_{\rm reg}$ by choosing
suitable co-ordinates.
As in \S \ref{sec:fixed_points},
consider the subsets $\KK_\pm \subset \LL\otimes \QQ$:
\[
\KK_\pm := \Big\{ f \in \LL \otimes \QQ : \big\{ i \in \{1,2,\ldots,m\} :
D_i \cdot f \in \ZZ \big\} \in \cA_\pm \Big\}
\]
and define $\tLL_+$ (respectively $\tLL_-$) to be the free $\ZZ$-submodule of
$\LL\otimes \QQ$ generated by $\KK_+$ (respectively by $\KK_-$).
Note that $\tLL_+$ and $\tLL_-$ are overlattices of $\LL$.
\begin{lemma}
\label{lem:integrallattice_decomp}
Set $\tLL_\pm^\vee = \Hom(\tLL_\pm,\ZZ)\subset \LL^\vee$.
We have $D_j \in \tLL_\pm^\vee$ if $j\in S_\pm$.
The decomposition \eqref{eq:decomp_pm} of $\LL^\vee \otimes \RR$ is
compatible with the integral lattice $\tLL^\vee_\pm$: one has
\begin{equation}
\label{eq:lattice_tLvee_decomp}
\tLL^\vee_\pm =
\left(H^2(X_\pm;\RR) \cap \tLL^\vee_\pm \right)
\oplus \bigoplus_{j\in S_\pm} \ZZ D_j
\end{equation}
where we regard $H_2(X_\pm;\RR)$ as a subspace of
$\LL^\vee\otimes \RR$ via the isomorphism $H^2(X_\pm;\RR)
\cong \bigcap_{j\in S_\pm} \Ker(\xi_j^\pm)$.
The lattices $\tLL^\vee_+$ and $\tLL^\vee_-$ are compatible
along the wall; one has (see equation~\ref{eq:decomp_W}):
\begin{equation}
\label{eq:compatible_int_lattice}
W \cap \tLL_+^\vee = W \cap \tLL_-^\vee
= (W' \cap \tLL_\pm^\vee) \oplus \bigoplus_{j\in S_0} \ZZ D_j.
\end{equation}
\end{lemma}
\begin{proof}
Equation \eqref{eq:lattice_tLvee_decomp} holds for both
$X_+$ and $X_-$ and we omit the subscript $\pm$
in what follows.
Since every element in $\cA$ contains $S$,
we have $D_j \cdot f \in \ZZ$ for all $j\in S$ and $f\in \KK$.
This shows that $D_j\in \tLL^\vee$ for $j\in S$.
Thus $\tLL^\vee \supset
(H^2(X;\RR)\cap \tLL^\vee) \oplus \bigoplus_{j\in S} \ZZ D_j$.
Conversely, for $v\in \tLL^\vee$, one has
$v \cdot \xi_i \in \ZZ$ for
all $i\in S$ because $\xi_i \in \KK$.
Then
$w = v- \sum_{i\in S} (v\cdot \xi_i) D_i$ lies in
$\bigcap_{j\in S} \Ker(\xi_j) \cap \tLL^\vee$
and $v = w + \sum_{i\in S} (v\cdot \xi_i) D_i$.
Next we prove \eqref{eq:compatible_int_lattice}.
First we claim that for every element $f\in \KK_+\setminus \KK_-$,
there exists $\alpha \in \QQ$ such that $f + \alpha e \in \KK_-$.
This follows easily from the definition of $\KK_\pm$
and Lemma \ref{lem:Apm}.
It follows from the claim that for any $f\in \tLL_+$, there exists
$\alpha \in \QQ$ such that $f + \alpha e \in \tLL_-$.
Suppose that $v\in W \cap \tLL_-^\vee$.
For any $f\in \tLL_+$, taking $\alpha\in \QQ$ as above,
one has $v \cdot f = v \cdot (f + \alpha e) \in \ZZ$.
Therefore $v \in W \cap \tLL_+^\vee$.
This shows that $W \cap \tLL_-^\vee \subset
W \cap \tLL_+^\vee$. The reverse inclusion follows similarly.
The second equality in \eqref{eq:compatible_int_lattice}
follows from \eqref{eq:lattice_tLvee_decomp}
and Proposition \ref{prop:wallcone_decomp}.
\end{proof}
\begin{remark}
We have $H_2(X_\pm;\RR) \cap \tLL_\pm
= H_2(|X_\pm|;\ZZ)$.
\end{remark}
Set $\ell_\pm = \dim H^2(X_\pm;\RR) = r -\sharp(S_\pm)$
and $\ell = \dim W' = r-1 - \sharp(S_0)$.
We have $\ell \le \min\{\ell_+,\ell_-\}$.
With cases as in Proposition \ref{prop:Spm_classif}, we have:
\begin{itemize}
\item[(I)] $\ell_+ = \ell_-= \ell+1$;
\item[(II-i)] $\ell_+ = \ell+1$, $\ell_-= \ell$;
\item[(II-ii)] $\ell_- = \ell+1$, $\ell_+ = \ell$;
\item[(III)] $\ell_+ = \ell_- = \ell$.
\end{itemize}
Using Lemma \ref{lem:integrallattice_decomp},
we can choose integral bases
\begin{align}
\label{eq:two_bases}
\begin{split}
& \{p_1^+,\dots,p_{\ell_+}^+\} \cup \{D_j : j\in S_+\}
\subset \tLL^\vee_+ \\
& \{p_1^-,\dots,p_{\ell_-}^-\} \cup \{D_j : j\in S_-\} \subset \tLL^\vee_-
\end{split}
\end{align}
of $\tLL_\pm^\vee$ such that
\begin{itemize}
\item $p_1^+,\dots, p_{\ell_+}^+$ lie in the nef cone
$\overline{C'_+} \subset H^2(X_+;\RR)$;
\item $p_1^-,\dots,p_{\ell_-}^-$ lie in the nef cone
$\overline{C'_-} \subset H^2(X_-;\RR)$;
\item $p_i^+ = p_i^- \in \overline{C_W'}$ for $i=1,\dots,\ell$.
\end{itemize}
These bases give co-ordinates on the toric charts
\eqref{eq:charts_cM}.
For $d\in \LL$, we write $\sfy^d$ for the corresponding
element in the group ring $\CC[\LL]$.
The homomorphisms
\begin{align*}
& \CC[C_+^\vee \cap \LL] \hookrightarrow
\CC[y_1,\dots,y_{\ell_+}, \{x_j : j\in S_+\}],
\qquad \sfy^d \mapsto \textstyle \prod_{i=1}^{\ell_+} y_i^{p_i^+ \cdot d}
\cdot \prod_{j\in S_+} x_j^{D_j \cdot d} \\
& \CC[C_-^\vee \cap \LL] \hookrightarrow
\CC[\ty_1,\dots,\ty_{\ell_-}, \{\tx_j : j\in S_-\}],
\qquad \sfy^d \mapsto \textstyle \prod_{i=1}^{\ell_-} \ty_i^{p_i^- \cdot d}
\cdot \prod_{j\in S_-} \tx_j^{D_j \cdot d}
\end{align*}
define the two smooth co-ordinate charts
\begin{align*}
(y_i, x_j : 1\le i\le \ell_+, j\in S_+) && \text{and} &&
(\ty_i,\tx_j: 1\le i\le \ell_-, j\in S_-)
\end{align*}
which are resolutions of (respectively) $\Spec \CC[C_+^\vee \cap \LL]$ and $\Spec \CC[C_-^\vee \cap \LL]$.
We reorder the bases \eqref{eq:two_bases}
\begin{align*}
& \{p_1^+,\dots,p_{\ell_+}^+\} \cup \{D_j : j\in S_+\}
= \{\sfp_1^+,\dots,\sfp_{r-1}^+,\sfp_r^+\} \\
& \{p_1^-,\dots,p_{\ell_-}^-\} \cup \{D_j : j\in S_-\}
= \{ \sfp_1^-,\dots,\sfp_{r-1}^-,\sfp_r^-\}
\end{align*}
in such a way that $\sfp_i^+ = \sfp^-_i\in W$ for $i=1,\dots,r-1$
and $\sfp_r^\pm$ is the unique vector (in each basis)
that does not lie on the wall $W$.
Let
\begin{align*}
\{y_i, x_j : 1\le i\le \ell_+, j\in S_+\}
& = \{ \sfy_1,\dots,\sfy_r\} \\
\{\ty_i, \tx_j : 1\le i \le \ell_-, j\in S_-\}
&= \{\tsfy_1,\dots,\tsfy_r\}
\end{align*}
be the corresponding reordering of the co-ordinates.
Then the change of co-ordinates
is of the form:
\begin{align}
\label{eq:change_of_variables}
\tsfy_i =
\begin{cases}
\sfy_i \sfy_r^{c_i} & 1\le i\le r-1 \\
\sfy_r^{-c} & i=r
\end{cases}
\end{align}
for some $c\in \QQ_{>0}$ and $c_i\in \QQ$.
The numbers $c_i$, $c$ here arise from the transition
matrix of the two bases \eqref{eq:two_bases}.
We find a common denominator for $c$, $c_i$ and
write $c = A/B$ and $c_i = A_i/B$, $1\le i\le r-1$
for some $A, B\in \ZZ_{>0}$ and $A_i\in \ZZ$.
Then $\sfy_r^{1/B} = \tsfy_r^{-1/A}$.
The smooth manifold $\cM_{\rm reg}$ is defined by
gluing the two charts
\begin{align*}
U_+ = \Spec \CC[\sfy_1,\dots,\sfy_{r-1},\sfy_r^{1/B}]
&& \text{and} &&
U_- = \Spec \CC[\tsfy_1,\dots,\tsfy_{r-1},\tsfy_r^{1/A}]
\end{align*}
via the change of variables \eqref{eq:change_of_variables}.
The large radius limit points $P_+\in U_+$ and $P_-\in U_-$ are given
respectively by $\sfy_1= \cdots =\sfy_{r} =0$ and
$\tsfy_1= \cdots = \tsfy_r=0$.
Note that the last variables $\sfy_r$, $\tsfy_r$
correspond to the direction of $e\in \LL$: one has
$\sfy^e = \sfy_r^{\sfp_r^+ \cdot e} = \tsfy_r^{\sfp_r^- \cdot e}$.
The torus-invariant rational curve $\cC_{\rm reg}
\subset \cM_{\rm reg}$ associated to $C_W$ is given by
$\sfy_1= \cdots = \sfy_{r-1} =0$ on $U_+$
and by $\tsfy_1 = \cdots = \tsfy_{r-1}=0$ on $U_-$.
Let $\hcM_{\rm reg}$ be the formal neighbourhood of
$\cC_{\rm reg}$ in $\cM_{\rm reg}$.
Since the global quantum connection is an analytic object,
we need to work with a suitable analytification of $\hcM_{\rm reg}$:
we include analytic functions in the last variable $\sfy_r$
in the structure sheaf
and use the analytic topology on $\cC_{\rm reg} \cong \PP^1$.
The underlying topological space of
$\hcM_{\rm reg}$ is therefore $\PP^{1,\rm an}$;
$\hcM_{\rm reg}$ is covered by two charts
$\hU_+$ and $\hU_-$ with structure sheaves:
\begin{equation}
\label{eq:Upm_str_sheaf}
\cO_{\hU_+} = \cO_{\CC_{+}}^{\rm an}[\![ \sfy_1,\dots,\sfy_{r-1}]\!]
\quad \text{and} \quad
\cO_{\hU_-} = \cO_{\CC_{-}}^{\rm an}[\![ \tsfy_1,\dots,\tsfy_{r-1}]\!]
\end{equation}
where $\CC_{+}$ and $\CC_{-}$ denote the
complex plane with co-ordinates $\sfy_r^{1/B}$ and
$\tsfy_r^{1/A}$ respectively
and the superscript ``an'' means analytic
(space or structure sheaf).
In other words, we can regard $\hcM_{\rm reg}$ as a sheaf
of algebras over $\PP^{1,\rm an}$.
The same construction works over an arbitrary $\CC$-algebra $R$.
We define $\hcM_{\rm reg}(R)$
by replacing the structure sheaves in \eqref{eq:Upm_str_sheaf}
with $(\cO_{\CC_{+}}^{\rm an} \otimes R)[\![\sfy_1,\dots,\sfy_{r-1}]\!]$
and $(\cO_{\CC_{-}}^{\rm an} \otimes R)[\![\tsfy_1,\dots,\tsfy_{r-1}]\!]$.
In the equivariant theory, we use $R = R_T[z] =
H^\bullet_T({\rm pt})\otimes \CC[z]$ for the ground ring.
The global equivariant quantum connection will be defined over $R_T[z]$ and
on (a formal thickening of) a simply-connected open subset of $\PP^{1,\rm an}$
containing $P_+$ and $P_-$.
\begin{remark}
\label{rem:overlattice}
Taking an overlattice $\tLL_\pm$ of $\LL$
corresponds to taking a finite cover of $\cM$.
This is necessary because the power series defining the $I$-function (see~\S\ref{sec:I_function})
is indexed by elements in $\tLL_\pm$.
If one takes into consideration Galois symmetry \cite{Iritani}
of the quantum connection, one can see that the quantum connection
(near $P_\pm$)
descends to the secondary toric variety with respect
to the original lattice $\LL$.
\end{remark}
\subsection{The $I$-Function}
\label{sec:I_function}
Recall Givental's Lagrangian cone introduced
in Definition \ref{def:Lag_cone}.
We consider the Givental cone $\cL_{X_\omega}$
associated to the toric Deligne--Mumford stack $X_\omega$.
Under the decomposition \eqref{eq:L_decomp} of $\LL\otimes \RR$,
we decompose $d\in \LL\otimes \RR$ as:
\[
d = \overline{d} + \sum_{j\in S_\pm} (D_j \cdot d) \xi_j
\]
where $\overline{d}$ is the $H_2(X_\omega;\RR)$-component of $d$.
Define the $H_{\CR,T}^\bullet(X_\omega)$-valued hypergeometric series
$I^{\rm temp}_\omega(\sigma,x,z)
\in H^\bullet_{\CR,T}(X_\omega)
\otimes_{R_T} R_T(\!(z^{-1})\!)[\![Q,\sigma,x]\!]$ by
\begin{equation*}
I^{\rm temp}_\omega(\sigma,x,z) := z e^{\sigma/z}
\sum_{d\in \KK} e^{\sigma\cdot \overline{d}}
Q^{\overline{d}} \prod_{j\in S} x_j^{D_j\cdot d}
\left(
\prod_{j=1}^{m}
\frac{\prod_{a : \<a\>=\< D_j\cdot d \>, a \leq 0}(u_j+a z)}
{\prod_{a : \<a \>=\< D_j \cdot d \>, a\leq D_j \cdot d} (u_j+a z)}
\right)
\fun_{[{-d}]}
\end{equation*}
where $\KK$ is introduced in \S \ref{sec:fixed_points},
$x =(x_j : j\in S)$ and $\sigma \in H^2_T(X_\omega)$ are
variables, and $[-d]$ is the equivalence class
of $-d$ in $\KK/\LL$ (recall from \S\ref{sec:inertia}
that $\KK/\LL$ parametrizes inertia components). The subscript `temp' reflects the fact that we are just about to change notation, by specializing certain parameters, and so this notation for the $I$-function is only temporary.
One can see that the summand of $I^{\rm temp}_\omega$ corresponding
to $d\in \KK$ vanishes unless $d\in C_\omega^\vee$.
Therefore the summation ranges over all $d\in \KK$ such that
$\overline{d}$ lies in the Mori cone
$\NE(X_\omega) = C_\omega'^\vee$ and $D_j \cdot d \ge 0$
for all $j\in S$.
The Mirror Theorem for toric Deligne--Mumford stacks can be
stated as follows:
\begin{theorem}[\!\!\cite{CCIT,Cheong--Ciocan-Fontanine--Kim}]
\label{thm:mirror_thm}
$I^{\rm temp}_\omega(\sigma,x,-z)$ is an
$S_T[\![Q,\sigma,x]\!]$-valued point on $\cL_{X_\omega}$.
\end{theorem}
We adapt the above theorem to the situation of toric wall-crossing.
Let $I_\pm^{\rm temp}$ denote the $I$-function of $X_\pm$.
We introduce a variant $I_\pm$ of the $I$-function which gives a
cohomology-valued function on a neighbourhood of $P_\pm$
in $\hcM_{\rm reg}$. The $I$-function $I_\pm$ is obtained
from $I_\pm^{\rm temp}$ by the following specialization:
\begin{itemize}
\item $Q=1$;
\item for $I_+$, $\sigma = \sigma_+ :=
\theta_+( \sum_{i=1}^r \sfp^+_i \log \sfy_i)
+ c_0(\lambda)$,
\item for $I_-$, $\sigma = \sigma_- :=
\theta_-( \sum_{i=1}^r \sfp_i^- \log \tsfy_i)
+ c_0(\lambda)$.
\end{itemize}
where
$\theta_\pm \colon \LL^\vee\otimes \CC \to H^2_T(X_\pm;\CC)$
are the maps introduced in \eqref{eq:theta} and $c_0(\lambda)
= \lambda_1+ \cdots + \lambda_m$.
Note that we have
\begin{equation}
\label{eq:sigma_p}
\sigma_+
= \sum_{i=1}^{\ell_+} \theta_+(p_i^+) \log y_i -
\sum_{j\in S_{+}} \lambda_j \log x_j + c_0(\lambda)
\end{equation}
since $\theta_+(D_j) = -\lambda_j$ for $j\in S_+$.
More explicitly, one can write $I_+$ as:
\begin{equation}
\label{eq:I+}
I_+(\sfy,z):=
z e^{\sigma_+/z}
\sum_{d \in \KK_+}
\sfy^d
\left(
\prod_{j=1}^{m}
\frac{\prod_{a : \<a\>=\< D_j\cdot d \>, a \leq 0}(u_j+a z)}
{\prod_{a : \<a \>=\< D_j \cdot d \>, a\leq D_j \cdot d} (u_j+a z)}
\right)
\fun_{[{-d}]}
\end{equation}
where recall that $(\sfy_1,\dots,\sfy_r) =(y_i, x_j : 1\le i\le \ell_+, j\in S_+)$
are co-ordinates on
$\hU_+\subset \hcM_{\rm reg}$ and that
\begin{align*}
\sfy^d &= \sfy_1^{\sfp^+_1 \cdot d} \cdots \sfy_r^{\sfp^+_{r}\cdot d}
= \textstyle \prod_{i=1}^{\ell_+} y_i^{p_i^+\cdot d}
\prod_{j\in S_+} x_j^{D_j \cdot d}.
\end{align*}
The $I$-function $I_+$ belongs to the space:
\[
I_+ \in H^\bullet_{\CR,T}(X_+) \otimes_{R_T} R_T[\log \sfy_1,\dots,\log \sfy_r]
(\!(z^{-1})\!)[\![\sfy_1,\dots,\sfy_r]\!].
\]
The series $e^{-\sigma_+/z} I_+(\sfy,z)$ is homogeneous of degree two
with respect to the (age-shifted) grading on $H_{\CR,T}^\bullet(X_+)$
and the degrees for variables given by:
\begin{align}
\label{eq:deg_variable}
\deg z =2 && \text{and} &&
\sum_{i=1}^r (\deg \sfy_i) \sfp_i^+ = 2\sum_{i=1}^{m} D_i
\end{align}
Note that $\deg \sfy_r=0$ because
$\sum_{i=1}^m D_i \in W$.
\begin{remark}
The extra factor $e^{c_0(\lambda)/z}$ in the $I$-function
makes the mirror map compatible with Euler vector fields.
\end{remark}
We now show that $I_+(\sfy,z)$ is analytic
in the last variable $\sfy_r$, so that it defines
an analytic function on $\hcM_{\rm reg}$.
\begin{lemma}
\label{lem:I_analyticity}
Expand the $I$-function as
\[
I_+(\sfy,z) = z e^{\sigma_+/z}
\sum_{k_1=0}^\infty \cdots \sum_{k_{r-1} = 0}^\infty
\sum_{n\in \ZZ}
I_{+; k_1,\dots,k_{r-1},n}(\sfy_r)
\sfy_1^{k_1} \cdots \sfy_{r-1}^{k_{r-1}} z^{n}
\]
Then each coefficient $I_{+;k_1,\dots,k_{r-1},n}(\sfy_r)$ is
a convergent power series in $\sfy_r$ taking values in a homogeneous
component of $H^\bullet_{\CR,T}(X_+)$.
Moreover it can be analytically continued to the universal
covering $\cU_+$ of the space $\{ \sfy_r \in \CC :
\sfy_r^{\sfp^+_r \cdot e} \neq \frc\}$, where
\begin{equation}
\label{eq:conifold_point}
\frc = \prod_{j: D_j\cdot e \neq 0} (D_j \cdot e)^{D_j \cdot e}
\end{equation}
is the so-called ``conifold point''.
\end{lemma}
\begin{proof}
The homogeneity of $I_{+;k_1,\dots,k_{r-1},n}(\sfy_r)$
follows from the homogeneity of the $I$-function mentioned
above. Fix $d\in \KK_+$ such that $k_i =
\sfp_i^+ \cdot d$ for all $i=1,\dots,r-1$
and let $f=[-d] \in \KK_+/\LL$ be the corresponding sector.
The $f$-component of $I_{+;k_1,\dots,k_{r-1},n}(\sfy_r)$
is given by a certain coefficient (in front of some powers of $z^{-1}$)
of the $z^{-1}$-expansion of the series:
\[
\sfy_r^{\sfp_r^+ \cdot d}
\sum_{k\in \ZZ} \sfy_r^{k(\sfp_r^+ \cdot e)}
\left(
\prod_{j=1}^m
\frac{\prod_{a: \<a\> = \<d_j\>, a\le 0} (\frac{u_j}{z}+a)}{
\prod_{a: \<a\> = \<d_j\>, a\le d_j + k e_j}
(\frac{u_j}{z} + a) }
\right) \fun_f,
\]
where $d_j = D_j\cdot d$ and $e_j = D_j \cdot e$.
Since the natural restriction map
$H^\bullet_{T}(X_f) \to H^\bullet_T(X_f^T)$ is injective, it
suffices to check that the restriction of the above series to each
$T$-fixed point is convergent and analytically continues to
$\cU_+$.
Consider a $T$-fixed point in $X_+^f$ corresponding
to a minimal anticone $\delta\in \cA_+$ with $\delta \subset I_f$
(see equation~\ref{eq:has_fixed_points} for $I_f$).
Let $\beta_j$ denote the restriction of $u_j/z$ to the fixed point
and set $\sx = \sfy_r^{\sfp^+_r \cdot e}$.
The restriction of the above series gives:
\[
\Phi(\beta; \sx) =
\sum_{k\in \ZZ} \sx^{k}
\prod_{j=1}^m
\frac{\prod_{a: \<a\> = \<d_j\>, a\le 0} (\beta_j+a)}{
\prod_{a: \<a\> = \<d_j\>, a\le d_j + k e_j}
(\beta_j + a) }
\]
ignoring the prefactor $\sfy_r^{\sfp^+_r \cdot d}$.
Since $\beta_j = \<d_j\>=0$ for $j\in \delta$
and $\delta \cap M_+ \neq \emptyset$,
it follows that the summand vanishes for $k\ll 0$.
We now regard $\{\beta_j : j \notin \delta\}$ as
small\footnote{Note that $\Phi(\beta;\sx)$ have poles at $\beta_j=-a$
for some $a>0$.} complex parameters and prove the analyticity of
$\Phi(\beta;\sx)$.
The conclusion follows by
expanding $\Phi(\beta;\sx)$ in the parameters $\beta_j$ and
evaluating the expansion in the $T$-equivariant cohomology of a point.
By using the ratio test and the fact that $\sum_{i=1}^m e_i =0$,
we see that the radius of convergence of $\Phi$ is positive.
Moreover one sees that $\Phi$ satisfies the differential
equation:
\[
\left[
\prod_{j: e_j>0} \prod_{l=0}^{e_j-1}
\left( e_j \sx \parfrac{}{\sx} + (d_j+\beta_j -l)
\right)
- \sx \prod_{j: e_j<0} \prod_{l=0}^{-e_j-1}
\left( e_j \sx \parfrac{}{\sx} + (d_j + \beta_j - l) \right)
\right] \Phi = 0
\]
which has singularities only at $\sx = 0, \frc, \infty$.
Thus $\Phi$ can be analytically continued to $\cU_+$.
\end{proof}
An entirely parallel statement holds for $I_-(\sfy,z)$.
\begin{remark}
\label{rem:I_analyticity}
Set $I_{+;k_1,\dots,k_{r-1}}(\sfy_r,z) := \sum_{n\in \ZZ}
I_{+;k_1,\dots,k_{r-1},n}(\sfy_r) z^n$.
In the proof of Lemma \ref{lem:I_analyticity}, we
observed that $\Phi(\beta;\sx)$ is analytic for sufficiently
small $\beta_j$.
Therefore, if we expand $I_{+;k_1,\dots,k_{r-1}}(\sfy_r,z)
= \sum_{i=0}^N f_i(\lambda,z,\sfy_r) \phi_i$ for a
suitable $R_T$-basis $\{\phi_i\}$ of $H_{\CR,T}^\bullet(X_+)$,
the coefficient $f_i(\lambda,z,\sfy_r)$ is analytic
on the region $\{(\lambda_1,\dots,\lambda_m, z, \sfy_r)
\in \CC^m \times \Cstar\times \cU_+ :
|\lambda_i| < \epsilon |z|\}$ for some $\epsilon>0$,
where $\lambda_i$ are $T$-equivariant parameters.
\end{remark}
\subsection{Global Equivariant Quantum Connection}
\label{sec:global_qconn}
In this section we use the $I$-function $I_+$
to construct a global
quantum connection on the universal cover
\begin{equation*}
\tcM_+ :=
\left( \left(\hU_+ \setminus
\{\sfy^e =\frc\}\right) \bigr/ \bmu_B \right)\sptilde
\end{equation*}
where $\hU_+$ is the open chart \eqref{eq:Upm_str_sheaf}
of $\hcM_{\rm reg}$ and $\sfy^e = \sfy_r^{\sfp^+_r \cdot e}$
is a co-ordinate on $\hU_+$;
$\bmu_B$ acts on $\hU_+$ by deck transformations
of $\sfy_r^{1/B} \mapsto \sfy_r$.
As in Lemma \ref{lem:I_analyticity},
we denote by $\cU_+$ the universal cover of
$\{\sfy_r \in \CC : \sfy_r^{\sfp^+_r \cdot e} \neq \frc\}$.
The space $\cU_+$ is the underlying topological space of $\tcM_+$,
and $\tcM_+$ is a formal thickening of $\cU_+$.
In a neighbourhood of $P_+$,
the global quantum connection that we will construct can be identified with the equivariant
quantum connection of $X_+$.
The main result in this section is:
\begin{theorem}
\label{thm:global_qconn}
There exist the following data:
\begin{itemize}
\item an open subset $\cU_+^\circ\subset \cU_+$ such that
$P_+ \in \cU_+^\circ$ and that the complement
$\cU_+ \setminus \cU_+^\circ$ is a discrete set;
we write $\tcM_+^\circ = \tcM_+|_{\cU_+^\circ}$;
\item a trivial $H_{\CR,T}^\bullet(X_+)$-bundle
$\bF^+$ over $\tcM_+^\circ(R_T[z])$:
\[
\bF^+ = H_{\CR,T}^\bullet(X_+)\otimes_{R_T}
(\cO_{\cU^\circ_+}\otimes R_T[z])[\![\sfy_1,\dots,\sfy_{r-1}]\!];
\]
\item a flat connection
$\bnabla^+ = d + z^{-1} \bA^+(\sfy)$
on $\bF^+$ of the form:
\[
\bA^+(\sfy) = \sum_{i=1}^{\ell_+} B_i(\sfy) \frac{dy_i}{y_i}
+ \sum_{j\in S_+} C_j(\sfy) dx_j
- \sum_{j\in S_+} \lambda_j \frac{dx_j}{x_j}
\]
with $B_i(\sfy), C_j(\sfy)
\in
\End(H_{\CR,T}^\bullet(X_+)) \otimes_{R_T}
(\cO_{\cU^\circ_+}\otimes R_T)[\![\sfy_1,\dots,\sfy_{r-1}]\!]$;
\item a vector field $\bE^+$ on $\tcM_+(R_T)$, called the Euler vector field, defined by:
\[
\bE^+ = \sum_{i=1}^m \lambda_i \parfrac{}{\lambda_i}
+ \sum_{i=1}^r \frac{1}{2}(\deg \sfy_i) \sfy_i
\parfrac{}{\sfy_i};
\]
\item a mirror map $\tau_+ \colon \tcM_+(R_T) \to H_{\CR,T}^\bullet(X_+)$
of the form:
\begin{align*}
\tau_+ = \sigma_+ + \ttau_+ &&&
\ttau_+\in H^\bullet_{\CR,T}(X_+)\otimes_{R_T} (\cO_{\cU^\circ_+}\otimes R_T)[\![\sfy_1,\dots,\sfy_{r-1}]\!] \\
&&& \ttau_+|_{\sfy_1=\cdots = \sfy_r=0}= 0
\end{align*}
\end{itemize}
such that $\bnabla^+$ equals the pull-back $\tau_+^*\nabla^+$ of the
equivariant quantum connection $\nabla^+$ of $X_+$ by $\tau_+$,
that is:
\begin{align*}
B_i(\sfy) & = \sum_{k=0}^N \parfrac{\tau_+^k(\sfy)}
{\log y_i} (\phi_k \star_{\tau_+(\sfy)}) &&
1\le i\le \ell_+ \\
C_j(\sfy) & =
\sum_{k=0}^N \parfrac{\ttau_+^k(\sfy)}
{x_j} (\phi_k\star_{\tau_+(\sfy)}) &&
j\in S_+
\end{align*}
and that the push-forward of $\bE^+$ by $\tau_+$
is the Euler vector field $\cE^+$ for $X_+$ defined in equation~\ref{eq:Euler_mu}.
Moreover, there exists a global section $\Upsilon^+_0(\sfy,z)$ of
$\bF^+$ such that
\[
I_+(\sfy,z) = z L_+(\tau_+(\sfy),z)^{-1} \Upsilon^+_0(\sfy,z)
\]
where $L_+(\tau,z)$ is the fundamental solution
for the quantum connection of $X_+$
in Proposition~\ref{prop:fundsol}.
\end{theorem}
\begin{remark}
Here the Novikov variables $Q$ in the quantum product and the fundamental solution have been specialized to $1$: see \S\ref{sec:specialization}.
\end{remark}
\begin{remark} An entirely analogous result holds for $X_-$.
\end{remark}
\begin{remark}
The data in Theorem \ref{thm:global_qconn} satisfy
some compatibility equations. The connection matrices $B_i$, $C_i$
are self-adjoint with respect to the equivariant orbifold
Poincar\'{e} pairing $(\cdot,\cdot)$. Furthermore the grading operator
$\bGr^+ = z \parfrac{}{z} + \bE^+ + \mu^+$ on $\bF^+$
(where $\mu^+$ is the grading operator on $H_{\CR,T}^\bullet(X_+)$
defined in equation~\ref{eq:Euler_mu}) satisfies
$[ \bGr^+, \bnabla^+_v] = \bnabla^+_{[\bE^+,v]}$
for any vector field $v$.
These properties are inherited from the quantum connection.
\end{remark}
\begin{remark}[{\cite[Remark 3.5]{Iritani}}]
By construction,
the mirror map $\tau_+$ here depends on how much
we have extended vectors $b_j, j\in S_+$ in the extended
stacky fan. If we add sufficiently many extended vectors,
we can make it submersive near $P_+$ and
Theorem \ref{thm:global_qconn} gives an analytic
continuation of the big quantum cohomology.
In fact we have
\[
\tau(\sfy) = c_0(\lambda) + \sum_{j\in S_+\setminus S_0}
\lambda_j \log x_j +
\sum_{i=1}^{\ell_+} \theta(p_i^+) \log y_i
+
\sum_{j\in S_+}
\alpha_j x_j + \text{higher order terms}.
\]
Here $\alpha_j = \prod_{i\in I_j} u_i^{n_{ij}} \fun_{[-\xi_j]}$,
where $\xi_j\in \KK_+$ is given in \eqref{eq:def_xi},
$I_j\subset \{1,\dots,m\} \setminus S_+$ is such that
$\sigma_{I_j}$ contains $\overline{b_j}$, and
$\overline{b_j} = \sum_{i\in I_j} (n_{ij} + \epsilon_{ij})
\overline{b_i}$ with
$\epsilon_{ij} \in [0,1)$ and $n_{ij} \in \ZZ_{\ge 0}$.
Note that $\fun_{[-\xi_j]}$ corresponds to the Box element
$b_j - \sum_{i\in I_j} n_{ij} b_i\in \cbox(X_+)$.
\end{remark}
\begin{remark}
The logarithmic singularity of $\bnabla^+$
along $\prod_{j\in S_+} x_j=0$ is not very important:
this can be eliminated by shifting the mirror map
$\tau$ by $\sum_{j\in S_+} \lambda_j \log x_j$;
see \eqref{eq:sigma_p}.
\end{remark}
The rest of this section is devoted to the proof of
Theorem \ref{thm:global_qconn}.
First we recall how to compute the quantum connection
of $X_+$ using the $I$-function (cf.~\cite{CCIT:applications}).
By the Mirror Theorem~\ref{thm:mirror_thm},
$I_+^{\rm temp}(\sigma,x,-z)$ is a point on the Givental
cone $\cL_+:= \cL_{X_+}$ for $X_+$.
Recall from Remark \ref{rem:fundamentalsol_cone}
that the cone $\cL_+$ is ruled by its tangent spaces
(multiplied by $z$):
\[
\cL_+ = \bigcup_{\tau \in H^\bullet_{\CR,T}(X_+)} z
L_+(\tau,-z)^{-1} \cH_+.
\]
This implies that one has:
\[
I_+^{\rm temp}(\sigma, x, z) = z L_+(\tau,z)^{-1} \Upsilon^+_0
\]
for some $\tau= \tau(\sigma,x) \in H^\bullet_{\CR,T}(X_+)
\otimes_{R_T} R_T[\![Q,\sigma,x]\!]$ and
$\Upsilon^+_0 \in \cH_+[\![\sigma,x]\!] = H^\bullet_{\CR,T}(X_+)
\otimes_{R_T} R_T[z][\![Q,\sigma,x]\!]$.
The map $(\sigma,x) \mapsto \tau(\sigma,x)$
is called the \emph{mirror map}: this will be determined below.
In Lemma~\ref{lem:I_derivatives} we will construct differential operators
$P_i = P_i(z\partial)$, $i=0,\dots,N$
which depend polynomially on $z$ and on the vector fields $z \partial_v$, $v\in H^2_T(X_+)$,
and $z\partial_{x_j}$, $j\in S_+$, and which satisfy:
\begin{itemize}
\item $\phi_i = z^{-1} P_i I_+^{\rm temp}|_{Q=\sigma=x=0}$,
$0\le i\le N$
are independent of $z$;
\item $\{\phi_i : 0\le i\le N\}$ form a basis of $H^*_{\CR,T}(X_+)$
over $R_T$;
\item $P_0 = 1$.
\end{itemize}
Then:
\begin{equation}
\label{eq:Birkhoff}
\begin{bmatrix}
\vert & & \vert \\
z^{-1}P_0 I^{\rm temp}_+ & \cdots & z^{-1} P_N I^{\rm temp}_+ \\
\vert & & \vert
\end{bmatrix}
= L_+(\tau,z)^{-1}
\begin{bmatrix}
\vert & & \vert \\
\Upsilon^+_0 & \cdots & \Upsilon^+_N \\
\vert & & \vert
\end{bmatrix}
\end{equation}
for $\Upsilon^+_i := P_i( z\tau^*\nabla) \Upsilon^+_0$.
Here $\tau^*\nabla$ is the pull-back of the quantum connection
of $X_+$ via the mirror map $\tau$, and
we used the fact that one has $\partial_v \circ L_+(\tau,z)^{-1}
= L_+(\tau,z)^{-1}\circ (\tau^* \nabla)_v$ for any
vector field $v$ on $(\sigma,x)$-space.
Note that:
\begin{itemize}
\item $\Upsilon^+_i \in \cH_+[\![\sigma,x]\!]$ does not
contain negative powers of $z$;
\item $L_+(\tau,z)$ does not contain positive powers of $z$; and
\item $L_+(\tau,z)= \id + O(z^{-1})$.
\end{itemize}
Thus the right-hand side of \eqref{eq:Birkhoff} can be regarded
as the \emph{Birkhoff factorization} of the left-hand side
(see \cite{Pressley-Segal}), when
we view both sides as elements in the loop group $LGL_{N+1}$
with $z$ the loop parameter.
The properties of $P_i$ listed above ensure that the left-hand side of \eqref{eq:Birkhoff}
is invertible at $Q=\sigma=x=0$, and that its Birkhoff factorization
can be determined recursively in powers of $Q$, $\sigma$ and $x$
(see Lemma \ref{lem:formal_Birkhoff}).
Thus the $I$-function determines $L_+(\tau,z)^{-1}$ as a function
of $(\sigma,x)$, via Birkhoff factorization.
The mirror map $\tau=\tau(\sigma,x)$ is determined by the asymptotics
\[
L_+(\tau,z)^{-1} 1 = 1 + \tau z^{-1} + O(z^{-2})
\]
and $L_+(\tau,z)^{-1}$ determines the pulled-back quantum
connection $\tau^* \nabla$.
We perform the above procedure globally on $\hcM_{\rm reg}$,
using the $I$-function $I_+$ obtained from $I_+^{\rm temp}$
by the specialization $Q=1, \sigma= \sigma_+$.
It will be convenient to assume the following condition.
\begin{assumption}
\label{assump:generation}
The set $\bN \cap |\Sigma_+| = \{ v\in \bN :
\overline{v} \in |\Sigma_+|\}$ of lattice points in the
support $|\Sigma_+|$ of the fan is generated by
$b_j, j=1,\dots,m$ as an additive monoid.
\end{assumption}
\begin{remark}
Assumption~\ref{assump:generation} is harmless: it can be always achieved by adding enough extended vectors to the extended stacky fan and in fact Theorem \ref{thm:global_qconn} holds without this assumption (see Remark \ref{rem:removing_assumption}).
\end{remark}
Recall from \S\ref{sec:inertia} that $H_{\CR,T}^\bullet(X_+)$
is the direct sum of sectors $H_{T}^\bullet(X_+^f)$, $f\in \KK_+/\LL$
and recall from \S \ref{sec:equiv_coh} that each sector
$H_T^\bullet(X_+^f)$ is generated by divisor classes.
Thus we can take an $R_T$-basis of $H_{\CR,T}^\bullet(X_+)$
of the form:
\begin{align*}
\phi_{f,i} = F_{f,i}\left(\theta(p^+_1),\dots, \theta(p^+_{\ell_+})\right) \fun_f
&& \text{$f\in \KK/\LL$, $1\le i \le \dim H^\bullet(X_+^f)$}
\end{align*}
where $F_{f,i}(a_1,\dots,a_{\ell_+}) \in \CC[a_1,\dots,a_{\ell_+}]$
is a homogeneous polynomial.
Recall from \S \ref{sec:inertia} that elements in $\KK_+/\LL$
are in one-to-one correspondence with elements in $\cbox(X_+)$.
Let $v_f \in \cbox(X_+)$ be
the element corresponding to $f\in \KK_+/\LL$.
By Assumption \ref{assump:generation}, there exist
non-negative integers $n_{f,j}$, $j=1,\dots,m$,
such that
\begin{equation}
\label{eq:vf}
v_f = \sum_{j=1}^m n_{f,j} b_j.
\end{equation}
On the other hand, taking a minimal cone $\sigma_f$ in $\Sigma_+$
containing $v_f$, we can write
\[
\overline{v_f} = \sum_{j \notin S_+, \overline{b_j} \in \sigma_f} c_{f,j}
\overline{b_j}
\]
for some $c_{f,j} \in [0,1)$.
We set $c_{f,j} = 0$ if $j\in S_+$ or $b_j \notin\sigma_f$.
Then $\sum_{j=1}^m (n_{f,j}-c_{f,j}) \overline{b_j}=0$
and by \eqref{eq:exact},
there exists an element $d_f \in \LL\otimes \QQ$ such
that $D_j \cdot d_f = n_{f,j} - c_{f,j}$. By definition
of $\KK_+$, $d_f \in \KK_+$ and $[-d_f] = f$ in
$\KK_+/\LL$ by \eqref{eq:KL_Box} and \eqref{eq:vf}.
Set $D_j = \sum_{a=1}^{r} \mu_{ja} \sfp^+_a$
for some $\mu_{ja} \in \ZZ$.
Define differential operators $\calD_j$, $\Delta_f$ as
\begin{align*}
\calD_j & := \sum_{a=1}^{r} \mu_{ja} z \sfy_a \parfrac{}{\sfy_a} \\
\Delta_f &:= \sfy^{-d_f}
\prod_{j=1}^m \prod_{\nu=0}^{n_{f,j}-1}
\left (\calD_j + \lambda_j - \nu z \right).
\end{align*}
\begin{lemma}
\label{lem:I_derivatives}
Let $F_{f,i}$,~$\phi_{f,i}$,~$\Delta_f$ be as above.
Define the differential operator $P^+_{f,i}$ by
\[
P^+_{f,i}:= F_{f,i}\left(
zy_1\parfrac{}{y_1},\dots,
z y_{\ell_+} \parfrac{}{y_{\ell_+}} \right)
\Delta_f.
\]
Then we have:
\[
P^+_{f,i} I_+(\sfy,z) = z e^{\sigma_+/z}
(\phi_{f,i} + O(\sfy)).
\]
\end{lemma}
\begin{proof}
The proof is parallel to \cite[Lemma 4.7]{Iritani}.
Note that the vector field $z y_i \partial/\partial y_i$ applied
to $e^{\sigma_+/z}$ yields the factor $\theta(p_i^+)$
for $1\le i\le \ell_+$ (see equation~\ref{eq:sigma_p}).
Thus it suffices to show that $\Delta_f I_+ = z e^{\sigma_+/z}
(\fun_{f} + O(\sfy))$.
Note that we have
\begin{align*}
\calD_j \left( e^{\sigma_+/z} \sfy^d\right)
& = \left( u_j - \lambda_j
+ z (D_i \cdot d) \right) e^{\sigma_+/z} \sfy^d
\end{align*}
where we use $\sum_{a=1}^{\ell_+} \mu_{ja} \theta(\sfp_a^+)
= \theta(D_j) = u_j -\lambda_j$.
Therefore one has:
\[
\Delta_f I_+ = e^{\sigma_+/z}
\sum_{d\in \KK_+} \sfy^{d-d_f}
\prod_{j=1}^m
\frac{\prod_{a\le 0, \<a\> = \<D_j \cdot d\>} (u_j + az)}
{\prod_{a\le D_j\cdot d - n_{f,j}, \<a\> = \<D_j\cdot d\>}
(u_j + az)}
\fun_{[-d]}
\]
Note that the summand equals $\fun_f$ when $d= d_f$.
We claim that the summand for $d\in \KK_+$ vanishes
if $d-d_f$ does not lie in the dual cone $C_+^\vee$ of $C_+$.
Suppose that $d-d_f\notin C_+^\vee$.
Note that the summand for $d$ contains the factor
$\prod_{j: D_j\cdot d\in \ZZ, D_j \cdot (d-d_f)<0} u_j$.
By the description \eqref{eq:Qequiv_cohomology}
of $H^\bullet_T(X^{[-d]}_+)$, it vanishes in cohomology if
$I = \{j : D_j\cdot d\in \ZZ, D_j\cdot (d-d_f)\ge 0\}\notin \cA_+$.
It now suffices to check $I\notin \cA_+$.
If $I\in \cA_+$, one has $C_+ \subset \angle_I$.
But $d-d_f$ lies in the dual cone of $\angle_I$.
Thus $d-d_f \in C_+^\vee$: this is a contradiction.
The claim follows and the Lemma is proved.
\end{proof}
Applying the differential operators $P^+_{f,i}$, $f\in \KK_+/\LL$,
$1\le i\le \dim H^\bullet(X_+^f)$, to $I_+$,
we obtain a matrix of the form:
\begin{equation}
\label{eq:I_derivative_matrix}
\begin{bmatrix}
& \vert & \\
\cdots & z^{-1} P^+_{f,i} I_+ & \cdots \\
& \vert &
\end{bmatrix}
= e^{\sigma_+/z} \II_+(\sfy,z)
\end{equation}
where $I_+$ is regarded as a column vector
written in the basis $\{\phi_{f,i}\}$ of $H^\bullet_{\CR,T}(X_+)$
and $\II_+(\sfy,z)= \id + O(\sfy)$ is a square matrix.
We may also view $\II_+(\sfy,z)$ as an
$\End(H^\bullet_{\CR}(X_+))$-valued function
via the basis $\{\phi_{f,i}\}$.
By the homogeneity of $e^{-\sigma_+/z}I_+$ and $P^+_{f,i}$,
we find that the endomorphism $\II_+(\sfy,z)$ is homogeneous
of degree-zero with respect to the degree \eqref{eq:deg_variable}
of variables and the grading on $H^\bullet_{\CR}(X_+)$, i.e.~that:
\begin{equation}
\label{eq:I+homogeneous}
\left(z \parfrac{}{z} + \bE^+ + \ad(\mu^+) \right) \II_+(\sfy,z) = 0
\end{equation}
As in \eqref{eq:Birkhoff}, we consider the Birkhoff factorization
of \eqref{eq:I_derivative_matrix}.
Since $e^{\sigma_+/z} = \id + O(z^{-1})$,
it suffices to consider the Birkhoff factorization of $\II_+(\sfy,z)$.
Set:
\[
\gamma(\sfy_r,z) := \II_+(\sfy,z)
\Big|_{\sfy_1 = \cdots = \sfy_{r-1}=0, \lambda_1 = \cdots
= \lambda_m =0}.
\]
By Lemma \ref{lem:I_analyticity}, $z\mapsto \gamma(\sfy_r,z)$
is a loop in $\End(H_{\CR}^\bullet(X_+))$
that depends analytically on $\sfy_r\in \cU_+$.
We first consider the Birkhoff factorization of $\gamma(\sfy_r,z)$.
Since $\gamma(\sfy_r,z)$ is homogeneous,
it is a Laurent polynomial in $z$ and both factors
of the Birkhoff factorization $\gamma(\sfy_r,z) = \gamma_-(z)
\gamma_+(z)$ are also homogeneous if the factorization exists.
Therefore the Birkhoff factorization is equivalent to
the \emph{block LU decomposition} of $\gamma(\sfy_r,1)$:
\begin{align*}
\gamma_-(1) =
\left [
\begin{array}{cccc}
I_{r_1} & & & \\
* & I_{r_2} & &\hsymb{0} \\
\vdots & \vdots & \ddots & \\
*& * & \cdots & I_{r_k}
\end{array}
\right ]
&&
\gamma_+(1) =
\left [
\begin{array}{cccc}
* & *& \cdots & * \\
& * & \cdots & *\\
& & \ddots & \vdots \\
\hsymb{0} & & & *
\end{array}
\right]
\end{align*}
where each block corresponds to a homogeneous component of
$H^\bullet_{\CR}(X_+)$ and $I_r$ denotes the identity
matrix of size $r$.
The block LU decomposition of $\gamma(\sfy_r,1)$ exists if and only if
\[
H = (\gamma(\sfy_r,1) H^{\le p}) \oplus H^{>p}
\]
holds for all $p\in \QQ$,
where $H = H^\bullet_{\CR}(X_+)$ and $H^{\le p}$
(resp.~$H^{>p}$) denotes the subspace of degree
less than or equal to $p$ (resp.~greater than $p$).
This is a Zariski open condition for $\gamma(\sfy_r,1)$.
Since $\gamma(\sfy_r=0,1) = \id$, it follows that $\gamma(\sfy_r,z)$
admits a Birkhoff factorization on the complement
$\cU^\circ_+$ of a discrete set in $\cU_+$.
Clearly one has $P_+ \in \cU_+^\circ$.
\begin{lemma}
\label{lem:formal_Birkhoff}
Let $\gamma(z)\in LGL_{N+1}(\CC)$ be a Laurent polynomial
loop admitting a Birkhoff factorization $\gamma = \gamma_- \gamma_+$.
Let $\Gamma(s,z) \in \End(\CC^{N+1}) \otimes
\CC[z,z^{-1}][\![s_1,\dots,s_l]\!]$ be a formal loop
such that $\Gamma|_{s=0} = \gamma$.
Then $\Gamma(s,z)$ admits a unique Birkhoff factorization
of the form
\[
\Gamma(s,z) = \Gamma_-(s,z) \Gamma_+(s,z)
\]
such that $\Gamma_-(s,z) \in \End(\CC^{N+1})
\otimes \CC[z^{-1}][\![s_1,\dots,s_l]\!]$,
$\Gamma_-(s,\infty) = \id$
and $\Gamma_+(s,z) \in \End(\CC^{N+1})
\otimes \CC[z][\![s_1,\dots,s_l]\!]$.
\end{lemma}
\begin{proof}
It suffices to show that $\Gamma' =
\gamma_-^{-1} \Gamma \gamma_+^{-1}$
admits a Birkhoff factorization $\Gamma' = \Gamma'_-\Gamma'_+$.
Expanding $\Gamma'$ and $\Gamma'_\pm$ in power series in
$s_1,\dots,s_l$, one can determine the coefficients
recursively from the equation $\Gamma' = \Gamma'_- \Gamma'_+$.
\end{proof}
Applying the above lemma to $\II_+(\sfy,z)$, we see that
$\II_+(\sfy,z)$ with $\sfy_r \in \cU^\circ_+$ admits a
Birkhoff factorization
\begin{equation}
\label{eq:Birkhoff_I}
\II_+(\sfy,z) = \bL_+(\sfy,z)^{-1} \Upsilon^+(\sfy,z)
\end{equation}
where
\begin{align*}
& \bL_+(\sfy,z) \in \End(H_{\CR}^\bullet(X_+))
\otimes \cO_{\cU^\circ_+}[z^{-1}][\![\lambda_1,\dots,\lambda_m,
\sfy_1,\dots,\sfy_{r-1}]\!], \\
&\Upsilon^+(\sfy,z) \in \End(H_{\CR}^\bullet(X_+))
\otimes \cO_{\cU^\circ_+}[z][\![\lambda_1,\dots,\lambda_m,
\sfy_1,\dots,\sfy_{r-1}]\!]
\end{align*}
and $\bL_+(\sfy,\infty) = \id$.
Using the homogeneity equation \eqref{eq:I+homogeneous},
we find that the Birkhoff factors $\bL_+$, $\Upsilon^+$
are also homogeneous of degree zero.
Also the chosen $R_T$-basis $\{\phi_{f,i}\}$ of $H_{\CR,T}^\bullet(X_+)$
defines a splitting $H_{\CR,T}^\bullet(X_+) \cong
H_{\CR}^\bullet(X_+) \otimes R_T$, and by the splitting,
one may naturally regard $\bL_+$, $\Upsilon^+$ as
$\End(H_{\CR,T}^\bullet(X_+))$-valued functions.
It follows that:
\begin{align*}
& \bL_+(\sfy,z) \in \End(H_{\CR,T}^\bullet(X_+))
\otimes_{R_T}
(\cO_{\cU^\circ_+}\otimes R_T)[\![z^{-1}]\!]
[\![\sfy_1,\dots,\sfy_{r-1}]\!], \\
&\Upsilon^+(\sfy,z) \in \End(H_{\CR,T}^\bullet(X_+))
\otimes_{R_T} (\cO_{\cU^\circ_+}\otimes R_T)[z]
[\![ \sfy_1,\dots,\sfy_{r-1}]\!].
\end{align*}
Comparing \eqref{eq:Birkhoff_I} with \eqref{eq:Birkhoff},
we obtain
\begin{equation}
\label{eq:fundsol_mirror}
L_+(\tau_+(\sfy),z)^{-1}\bigl|_{Q=1} = e^{\sigma_+/z}
\bL_+(\sfy,z)^{-1}.
\end{equation}
The mirror map $\tau_+(\sfy)$ is given by
$\tau_+(\sfy) = \sigma_+ + \ttau_+(\sfy)$ with
$\ttau_+(\sfy)$ determined by:
\begin{align*}
\bL_+(\sfy,z)^{-1} 1 = 1 + \ttau_+(\sfy) z^{-1} + O(z^{-2}).
\end{align*}
We have $\ttau_+(0)=0$ and $\ttau_+(\sfy) \in
H_{\CR,T}^\bullet(X_+) \otimes
(\cO_{\cU^\circ_+}\otimes R_T)[\![\sfy_1,\dots,\sfy_{r-1}]\!]$.
The first column of \eqref{eq:Birkhoff_I} gives
$I_+(\sfy,z) =z L(\tau(\sfy),z)|_{Q=1} \Upsilon^+_0$, where
$\Upsilon^+_0$ is the first column of $\Upsilon^+$.
(Here we assume that the first column corresponds to
the basis vector
$\phi_{0,1} = 1$ and the differential operator
$P^+_{0,1} =1$.)
\begin{remark}
Equation \eqref{eq:fundsol_mirror} is an equality in the ring:
\[
\End(H_{\CR,T}^\bullet(X_+)) \otimes_{R_T}
R_T[\log \sfy_1,\dots, \log \sfy_r][\![z^{-1}]\!][\![\sfy]\!].
\]
Note that the substitution $\tau=\tau_+(\sfy)$ in $L_+|_{Q=1}$
makes sense: see \S \ref{sec:specialization}.
\end{remark}
By equation \eqref{eq:fundsol_mirror}, $\bL_+(\sfy,z)$ determines
the quantum connection pulled-back by the mirror map $\tau_+(\sfy)$.
Set $\tau_+^* \nabla^+ = d + z^{-1} \bA^+(\sfy)$.
The connection 1-form $\bA^+(\sfy)$ is computed by:
\begin{align*}
\bA^+(\sfy) & := -
z d (\bL_+(\sfy,z) e^{-\sigma_+/z}) e^{\sigma_+/z} \bL_+(\sfy,z)^{-1} \\
& = -z (d\bL_+(\sfy,z)) \bL_+(\sfy,z)^{-1}
+ \bL_+(\sfy,z) (d \sigma_+) \bL_+(\sfy,z)^{-1}
\end{align*}
where the term $d\sigma_+$ gives a logarithmic singularity
(see equation~\ref{eq:sigma_p}):
\[
d\sigma_+ = \sum_{i=1}^{\ell_+} \theta_+(p_i^+) \frac{dy_i}{y_i}
- \sum_{j\in S_+} \lambda_j \frac{dx_j}{x_j}.
\]
Thus the connection form $\bA^+(\sfy)$ is a global $1$-form on $\tcM_+$
satisfying the properties in Theorem \ref{thm:global_qconn}.
\begin{remark}
Note that $\bA^+(\sfy)$ is independent of $z$: in the formal
neighbourhood of $P_+ = \{\sfy_1=\cdots=\sfy_r=0\}$ this follows from
the fact that $d+ z^{-1} \bA^+(\sfy)$ is the pulled-back
quantum connection,
and this is true everywhere by analytic continuation.
\end{remark}
Finally we see that $\bE^+$ corresponds to $\cE^+$.
Choose a homogeneous $R_T$-basis $\{\phi_i\}$ of $H_{\CR,T}^\bullet(X_+)$
such that $\phi_0 = 1$ and $\phi_i = \theta(p_i^+)$
for $1\le i\le \ell_+$ and write $\tau^i_+(\sfy)$ for the $i$th component of $\tau_+(\sfy)$ with respect to this basis.
One needs to check that $\bE^+ \tau^i_+(\sfy) = (1- \frac{1}{2} \deg \phi_i)
\tau^i_+(\sfy) + \rho^i$, where $\rho = \sum_{i=0}^N \rho^i
\phi_i$.
The homogeneity of $\bL_+^{-1}$ shows that $\ttau_+(\sfy)$
is homogeneous of (real) degree two: this implies that
$\bE^+ \ttau_+^i(\sfy) = (1 - \frac{1}{2} \deg \phi_i)
\ttau_+^i(\sfy)$.
If we set $\sigma_+ = \sum_{i=0}^{\ell_+}
\sigma_+^i \phi_i$, we have:
\[
\bE^+ \sigma_+^i =
\begin{cases}
c_0(\lambda)
- \sum_{j\in S_+} (\lambda_j \log x_j + \lambda_j \frac{1}{2} (\deg x_j) )
& i=0 \\
\frac{1}{2} \deg y_i & 1\le i\le \ell_+
\end{cases}
\]
Thus we have:
\[
\bE^+ \tau^i_+(\sfy)
= \begin{cases}
\tau^0_+(\sfy) + c_0(\lambda)
- \sum_{j\in S_+} \lambda_j \frac{1}{2} (\deg x_j)
& i=0 \\
\frac{1}{2} (\deg y_i) & 1\le i \le \ell_+ \\
\left(1 - \frac{1}{2} \deg \phi_i\right) \tau^i_+(\sfy)
& i > \ell_+
\end{cases}
\]
On the other hand, we have
\begin{align*}
\rho & = u_1 + \cdots + u_m = \theta(D_1 + \cdots + D_m) + \lambda_1 +
\cdots + \lambda_m \\
& = \sum_{i=1}^{\ell_+} \frac{1}{2}(\deg y_i) \theta(p_i^+)
+ \sum_{j\in S_+} \frac{1}{2} (\deg x_j) (-\lambda_j)
+ \lambda_1 + \cdots + \lambda_m
\end{align*}
and therefore:
\[
\rho^i = \begin{cases}
c_0(\lambda) - \sum_{j\in S_+} \lambda_j \frac{1}{2} (\deg x_j)& i=0 \\
\frac{1}{2} (\deg y_i) & 1\le i\le \ell_+
\end{cases}
\]
Thus $\bE^+ \tau^i_+(\sfy) = (1- \frac{1}{2} \deg \phi_i)
\tau^i_+(\sfy) + \rho^i$.
The proof of Theorem \ref{thm:global_qconn} is complete.
\begin{remark}
\label{rem:removing_assumption}
For the existence of a global quantum connection
in Theorem \ref{thm:global_qconn} and other main results in this paper,
we do not need Assumption \ref{assump:generation}.
Let us write $\tcM_+^S$ for $\tcM_+$ to emphasize the
dependence on the extension data $S$.
Then one has:
\[
S \subset S' \ \Longrightarrow \
\tcM_{+}^S\subset \tcM_{+}^{S'}
\]
Suppose that an $S$-extended stacky fan does not satisfy
Assumption~\ref{assump:generation}.
By taking a bigger $S'\supset S$, we can achieve Assumption
\ref{assump:generation} and construct a global quantum
connection on $\tcM_+^{S'}$.
Then we obtain a global quantum connection on $\tcM_+^S$
by restriction.
In this way, the global quantum connections
form a projective system over all extension data $S$.
Assumption \ref{assump:generation} ensures that
$\bF^+$ is generated by
a section $\Upsilon_0^+$ and its covariant derivatives.
For the convenience of discussion, we will sometimes
use Assumption \ref{assump:generation} in the rest of the paper,
but this does not affect the final conclusion.
\end{remark}
\section{The Crepant Resolution Conjecture}
\label{sec:CRC}
We now come to the main result in this paper. In Theorem~\ref{thm:global_qconn},
we constructed a global quantum connection $(\bF^+,\bnabla^+, \bE^+)$ for $X_+$
on $\tcM_+^\circ$, where $\tcM_+^\circ$ is an open subset of
the universal cover $\tcM_+$ of $(\hU_+\setminus \{\sfy^e = \frc\})/\bmu_B$.
By applying Theorem~\ref{thm:global_qconn} to $X_-$ rather than $X_+$,
we obtain a global quantum connection $(\bF^-,\bnabla^-, \bE^-)$ for $X_-$
on $\tcM_-^\circ$, where $\tcM_-^\circ$ is an open subset of
the universal cover $\tcM_-$ of $(\hU_-\setminus \{\sfy^e = \frc\})/\bmu_A$.
We now show that these two global quantum connections are gauge-equivalent
on a common covering $\tcM$: the universal cover of
$\hcM_{\rm reg} \setminus \{\sfy^e = 0,\frc,\infty\}$.
\[
\xymatrix{
& \tcM \ar[dl]_{\pi_+} \ar[dr]^{\pi_-} \ar[d] & \\
\tcM_+ \ar[d]
& \hcM_{\rm reg}\setminus \{\sfy^e = 0,\frc,\infty\} \ar[dl]\ar[dr]
& \tcM_- \ar[d] \\
(\hU_+\setminus \{\sfy^e = \frc\})/
\bmu_B &
& (\hU_-\setminus \{\sfy^e = \frc\})/\bmu_A }
\]
Moreover, we show that the analytic continuation of flat sections is induced
by a Fourier--Mukai transformation $\FM \colon K_T^0(X_-) \to K_T^0(X_+)$
through the equivariant integral structure in~\S\ref{sec:equiv_int_str}. We establish the gauge-equivalence of the two global quantum connections in several steps, beginning in~\S\ref{sec:gauge-equivalence} by expressing the gauge transformation involved as a linear sympletomorphism $\UU$ between the Givental spaces for $X_+$ and $X_-$. In~\S\ref{sec:Mellin-Barnes} we use the Mellin--Barnes method to analytically continue the $I$-function $I_+$, deducing from this a formula for~$\UU$. In~\S\ref{sec:FM} we construct a Fourier--Mukai transformation $\FM \colon K_T^0(X_-) \to K_T^0(X_+)$ associated to the toric birational transformation $X_+ \dashrightarrow X_-$. Finally in~\S\ref{sec:FM_match} and~\S\ref{sec:end_of_the_proof} we complete the proof of gauge-equivalence, and of the Crepant Resolution Conjecture in the toric case, by showing that the symplectic transformation~$\UU$ coincides, via the equivariant integral structure, with the Fourier--Mukai transformation~$\FM$.
\subsection{The Global Quantum Connections are Gauge-Equivalent}
\label{sec:gauge-equivalence}
Let $\cU_\pm$ denote the underlying topological space of
$\tcM_\pm$. The space $\cU_+$ is the universal cover
of $\{\sfy_r \in \CC : \sfy_r^{\sfp_r^+ \cdot e} \neq \frc\}$
and $\cU_-$ is the universal cover of
$\{\tsfy_r \in \CC: \tsfy_r^{\sfp_r^- \cdot (-e)} \neq \frc^{-1} \}$.
The underlying topological space of $\tcM$ is the universal cover
$\cU$ of $\cC_{\rm reg} \setminus \{\sfy^e = 0,\frc,\infty\}$.
We have natural maps $\pi_\pm \colon \cU \to \cU_\pm$
and set
\begin{align*}
\cU^\circ & := \pi_+^{-1}(\cU^\circ_+) \cap \pi_-^{-1}(\cU^\circ_-)
\subset \cU \\
\tcM^\circ & := \tcM|_{\cU^\circ}
\end{align*}
where $\cU_\pm^\circ\subset \cU_\pm$ is the open
dense subset from Theorem \ref{thm:global_qconn}.
Note that $\cU \setminus \cU^\circ$ is a discrete set.
Since we use $P_\pm \in \cC_{\rm reg}$ as base points
of the universal covers $\cU_\pm$, we need to specify
a path from $P_+$ to $P_-$ in $\cC_{\rm reg}\setminus\{\sfy^e = \frc\}$
in order to identify the maps $\cU \to \cU_\pm$ between universal covers.
We consider a path in the $\log(\sfy^e)$-plane starting from
$\log(\sfy^e) = -\infty$ and ending at $\log(\sfy^e) = \infty$
such that it avoids $\log(\frc) + 2\pi \tti \ZZ$.
We use a path $\gamma$ as in Figure \ref{fig:path_ancont}
passing through the interval
\[
\bigl ( \log|\frc| + \pi \tti (w-1), \log |\frc| + \pi\tti(w+1) \bigr )
\]
in the $\log(\sfy^e)$-plane, where
$w := -1 - \sum_{j: D_j\cdot e<0} (D_j \cdot e)
= -1 + \sum_{j:D_j \cdot e>0} (D_j\cdot e)$.
\begin{figure}[htbp]
\centering
\includegraphics[, width=0.6\textwidth,bb=135 573 484 729]{path_ancont.pdf}
\caption{The path $\gamma$ of analytic continuation on the
$\log(\sfy^e)$-plane}
\label{fig:path_ancont}
\end{figure}
\begin{theorem}
\label{thm:U}
Let $\cH(X_\pm) = H^\bullet_{\CR,T}(X_\pm) \otimes_{R_T}
R_T(\!(z^{-1})\!)$
denote Givental's symplectic vector space for $X_\pm$ (see~\S\ref{sec:Givental_cone})
without Novikov variables, i.e.~with $Q$ specialized to~$1$.
There exists a degree-preserving\footnote{We use the
usual grading on $H_{\CR,T}^\bullet(X_\pm)$, $R_T =H_T^\bullet({\rm pt})$
and set $\deg z=2$.}
$R_T(\!(z^{-1})\!)$-linear symplectic transformation
$\UU \colon \cH(X_-) \to \cH(X_+)$ such that:
\begin{enumerate}
\item $I_+(\sfy,z) = \UU I_-(\sfy,z)$ after analytic continuation in $\sfy^e$
along the path $\gamma$ in Figure {\rm\ref{fig:path_ancont}};
\item $\UU \circ (g_-^\star v\cup)= (g_+^\star v\cup) \circ \UU$ for all
$v\in H^2_T(\overline{X}_0)$, where $g_\pm \colon X_\pm \to
\overline{X}_0$ is the common blow-down appearing in the
diagram \eqref{eq:crepant_diagram};
\item there exists a Fourier--Mukai transformation
$\FM \colon K_T^0(X_-) \to K_T^0(X_+)$ such
that the following diagram commutes:
\begin{equation}
\label{eq:FM_U}
\begin{aligned}
\xymatrix{
K_T^0(X_-) \ar[r]^{\FM} \ar[d]_{\tPsi_-} & K_T^0(X_+) \ar[d]^{\tPsi_+}\\
\tcH(X_-) \ar[r]^{\UU} & \tcH(X_+)
}
\end{aligned}
\end{equation}
where the vertical map $\tPsi_\pm\colon
K_T^0(X_\pm) \to \tcH(X_\pm)$ is the map
\[
\tPsi_\pm(E) = z^{-\mu^\pm} z^{\rho^\pm}
\left( \hGamma_{X_\pm} \cup (2\pi\tti)^{\frac{\deg_0}{2}}
\inv^* \tch(E) \right)
\]
taking values\footnote{Cf.~Corollary~\ref{cor:homogeneous_flat_sections}.} in the ``multi-valued Givental space'':
\[
\tcH(X_\pm) = H_{\CR,T}^\bullet(X_\pm)
\otimes_{R_T} R_T[\log z](\!(z^{-1/k})\!)
\]
Here $k\in \N$ is an integer such that all the eigenvalues
of $k \mu^{+}, k\mu^-$ are integers.
\end{enumerate}
\end{theorem}
Theorem~\ref{thm:U} will be proved in~\S\ref{sec:Mellin-Barnes} and~\S\ref{sec:FM}.
The Fourier--Mukai kernel will be described in \S\ref{sec:FM}: it is given by
a toric common blow-up $\tX$ of $X_\pm$.
\begin{notation}
\label{not:murho}
In Theorem \ref{thm:U}, $\rho^\pm = c_1^T(TX_\pm)\in H^2_T(X_\pm)$,
$\mu^\pm$ is the grading operator \eqref{eq:Euler_mu}
on $H^\bullet_{\CR,T}(X_\pm)$ and
$\deg_0\colon H^{\bullet\bullet}_T(IX_\pm)
\to H^{\bullet\bullet}_T(IX_\pm)$
is the degree operator as in \S \ref{sec:equiv_int_str}.
\end{notation}
\begin{theorem}
\label{thm:CTC_qconn}
Let $(\bF^\pm, \bnabla^\pm,\bE^\pm)$ be the global
quantum connections for $X_\pm$ over $\tcM_\pm^\circ(R_T[z])$
from Theorem \ref{thm:global_qconn}.
We have that $\bE^+ = \bE^-$ on $\tcM(R_T)$.
There exists a gauge transformation
\[
\Theta \in \Hom\big(H^\bullet_{\CR,T}(X_-), H^\bullet_{\CR,T}(X_+)\big)
\otimes_{R_T} (\cO_{\cU^\circ}\otimes R_T)[z][\![\sfy_1,\dots,\sfy_{r-1}]\!]
\]
over $\tcM^\circ(R_T[z])$ such that:
\begin{itemize}
\item $\bnabla^-$ and $\bnabla^+$ are gauge-equivalent
via $\Theta$, i.e.~$
\bnabla^+ \circ \Theta = \Theta \circ \bnabla^-$;
\item $\Theta$ is homogeneous of degree zero, i.e.~$
\bGr^+ \circ \Theta = \Theta \circ \bGr^-$ with
$\bGr^\pm := z\parfrac{}{z} + \bE^\pm + \mu^\pm$;
\item $\Theta$ preserves the orbifold Poincar\'{e} pairing, i.e.~$(\Theta(\sfy,-z) \alpha, \Theta(\sfy,z) \beta) = (\alpha,\beta)$.
\end{itemize}
Moreover, the analytic continuation of the $K$-theoretic flat sections
in Definition \ref{def:K_framing} (with Novikov variables $Q$
set to be one, see \S \ref{sec:specialization}) is induced
by the Fourier--Mukai transformation:
\begin{align*}
\Theta\Bigl( \frs(E)(\tau_-(\sfy),z) \Bigr) = \frs(\FM(E))(\tau_+(\sfy),z) &&
\text{for all $E \in K_T^0(X_-)$}
\end{align*}
where $\tau_\pm$ are the mirror maps in Theorem~\ref{thm:global_qconn}.
\end{theorem}
\begin{remark}
The symplectic transformation $\UU$ in Theorem~\ref{thm:U} and the gauge transformation
$\Theta$ in Theorem~\ref{thm:CTC_qconn} are related by
\begin{equation}
\label{eq:Theta-U}
L_+(\tau_+(\sfy),z)^{-1} \circ \Theta =
\UU \circ L_-(\tau_-(\sfy),z)^{-1}
\end{equation}
where $L_\pm$ is the fundamental solution for the quantum
connection of $X_\pm$ in Proposition \ref{prop:fundsol}.
The gauge transformation $\Theta$ sends
the section $\Upsilon_0^-\in \bF^-$ to the section
$\Upsilon_0^+\in \bF^+$,
where $\Upsilon_0^\pm$ are as
in Theorem \ref{thm:global_qconn}.
\end{remark}
\begin{remark}
Theorems~\ref{thm:U} and~\ref{thm:CTC_qconn}
can be interpreted as the statement that the symplectic
transformation $\UU$ matches up the Givental
cones $\cL_\pm$ associated to $X_\pm$ after
analytic continuation of $\cL_\pm$:
\begin{equation}
\label{eq:U_cones}
\UU(-z) \cL_- = \cL_+.
\end{equation}
In fact, Remark \ref{rem:fundamentalsol_cone}
suggests that we may analytically continue the Lagrangian
cones by the formula:
\[
\cL_\pm \, \text{``$=$''} \, \bigcup_{\sfy\in \tcM^\circ}
z L_\pm(\tau_\pm(\sfy), -z)^{-1} \cH_+(X_\pm)
\]
and then equation \eqref{eq:Theta-U} would imply \eqref{eq:U_cones}.
As discussed in the Introduction, to avoid subtleties in defining the analytic continuation of Givental cones in the equivariant setting, in this paper we state our results in terms of analytic continuation of the $I$-function (Theorem~\ref{thm:U}) or in terms of the equivariant quantum connection and gauge transformations (Theorem~\ref{thm:CTC_qconn}).
\end{remark}
\begin{remark}
Theorem~\ref{thm:CTC_qconn} implies that the global quantum connections
of $X_+$ and $X_-$ can be glued together to give a flat connection over
$\tcM^\circ$.
This flat connection descends to the formal neighbourhood
$\hcM$ of $\cC$ in the secondary toric variety $\cM$
via Galois symmetry as in Remark \ref{rem:overlattice}.
This global connection, or $D$-module, on
$\hcM$ can be described by explicit GKZ-type differential
equations: it is a completed version of Borisov--Horja's
\emph{better-behaved GKZ system}\footnote
{The better-behaved GKZ system is in general generated by several elements.
In our case, by adding enough extended vectors that
Assumption \ref{assump:generation} is satisfied,
we can make it generated by a single standard generator
$1$, and in this case the better-behaved GKZ system is the same as the
original GKZ system~\cite{GKZ:diffeq}.}
\cite{Borisov--Horja:GKZ}.
In the papers \cite{Iritani, Reichelt--Sevenheck},
the toric quantum connection is described in terms of
GKZ-type differential equations through mirror symmetry.
The $I$-functions $I_\pm(q,z)$ are local solutions to
these differential equations around the large radius limit points.
\end{remark}
\begin{proof}[Proof that Theorem~\ref{thm:U} implies Theorem \ref{thm:CTC_qconn}]
One can easily check that the change of variables
\eqref{eq:change_of_variables} preserves degree, and that
$\bE^+ = \bE^-$.
By Theorem \ref{thm:U}, we have
\begin{equation}
\label{eq:U_Ipm}
I_+(\sfy,z) = \UU I_-(\sfy,z)
\end{equation}
under analytic continuation along the path $\gamma$.
The discussion in~\S\ref{sec:global_qconn}
(see Lemma~\ref{lem:I_derivatives}, equation~\ref{eq:I_derivative_matrix}, and equation~\ref{eq:Birkhoff_I}) yields:
\begin{equation}
\label{eq:ILUpsilon_+}
\begin{bmatrix}
& \vert & \\
\cdots & z^{-1} P_{f,i}^+ I_+ & \cdots \\
& \vert &
\end{bmatrix}
= e^{\sigma_+/z}\bL_+(\sfy,z)^{-1} \Upsilon^+(\sfy,z).
\end{equation}
Similarly, the discussion in~\S\ref{sec:global_qconn}
applied to $X_-$ yields a global section $\Upsilon^-_0$ of $\bF^-$
and a (global) fundamental solution $\bL_-(\sfy,z)e^{-\sigma_-/z}$
for $\bnabla^-= d + z^{-1} \bA^-(\sfy)$ such that:
\begin{equation*}
z^{-1} I_-(\sfy,z) = e^{\sigma_-/z} \bL_-(\sfy,z)^{-1}
\Upsilon^-_0(\sfy,z)
\end{equation*}
Applying the differential operators $P_{f,i}^+$ to $z^{-1} I_-(\sfy,z)$,
we obtain
\begin{equation}
\label{eq:ILUpsilon_-}
\begin{bmatrix}
& \vert & \\
\cdots & z^{-1} P_{f,i}^+ I_- & \cdots \\
& \vert &
\end{bmatrix}
= e^{\sigma_-/z} \bL_-(\sfy,z)^{-1}
\begin{bmatrix}
& \vert & \\
\cdots & P_{f,i}^+(z\bnabla^-) \Upsilon^-_0 & \cdots \\
& \vert &
\end{bmatrix}
\end{equation}
where $P_{f,i}^+(z\bnabla^-)$ is obtained from $P_{f,i}$
by replacing $z\partial_v$ with $z\bnabla^-_v$ for
vector fields $v$.
Let $\tUpsilon^-$ denote the matrix with column vectors
$P_{f,i}^+(z\bnabla^-) \Upsilon^-_0$.
Comparing \eqref{eq:ILUpsilon_+} with
\eqref{eq:ILUpsilon_-} and using \eqref{eq:U_Ipm},
we obtain
\[
e^{\sigma_+/z} \bL_+(\sfy,z)^{-1} \Upsilon^+
=\UU e^{\sigma_-/z} \bL_-(\sfy,z)^{-1} \tUpsilon^-
\]
since $\UU$ is independent of the base variables $\sfy$.
In particular, it follows that $\tUpsilon^-$ is invertible.
Setting $\Theta = \Upsilon^+ (\tUpsilon^-)^{-1}$, we obtain:
\begin{equation}
\label{eq:Theta-U_relation}
\left( e^{\sigma_+/z} \bL_+(\sfy,z)^{-1}\right) \Theta(\sfy,z)
=\UU \left( e^{\sigma_-/z} \bL_-(\sfy,z)^{-1}\right).
\end{equation}
Since $e^{\sigma_\pm/z} \bL_\pm^{-1}$ are fundamental
solutions for $\bnabla^\pm$,
$\Theta$ gives a gauge transformation
between $\bnabla^-$ and $\bnabla^+$, i.e.~
$\Theta \circ \bnabla^- = \bnabla^+ \circ\Theta$.
One may assume that the first columns of $\Upsilon^+$
and $\tUpsilon^-$ are given respectively by $\Upsilon_0^+$
and $\Upsilon^-_0$, and therefore $\Theta(\Upsilon_0^-)
= \Upsilon_0^+$.
Next we see that $\Theta$ preserves the grading and the pairing.
Part (2) in Theorem \ref{thm:U} implies that
$\UU \circ \theta_-(\sfp^-_i) = \theta_+(\sfp^+_i)\circ \UU$
for $i=1,\dots,r-1$, since $\sfp^+_i = \sfp^-_i$ lies on the
wall $W$ for $1\le i\le r-1$.
Therefore
\begin{align*}
e^{-\sigma_+/z} \circ
\UU \circ e^{\sigma_-/z}
& = e^{-\theta_+(\sfp^+_r) \log \sfy_r/z}
\circ \UU \circ e^{\theta_-\left( \sum_{i=1}^r \sfp^-_i \log \tsfy_i
- \sum_{i=1}^{r-1} \sfp^+_i \log \sfy_i
\right)/z} \\
& = e^{-\theta_+(\sfp^+_r) \log \sfy_r/z}
\circ \UU \circ e^{\theta_-( \sfp_r^+) \log \sfy_r /z}
\end{align*}
where we used $\sum_{i=1}^r \sfp^+_i \log \sfy_i =
\sum_{i=1}^r \sfp^-_i \log \tsfy_i$.
This together with \eqref{eq:Theta-U_relation} implies that:
\[
\bL_+(\sfy,z)^{-1}
\Theta(\sfy,z)
= \left(
e^{-\theta_+(\sfp_r^+) \log \sfy_r/z} \circ \UU \circ
e^{\theta_-(\sfp_r^+) \log \sfy_r/z}\right)
\bL_-(\sfy,z)^{-1}
\]
Since $\deg \sfy_r =0$, we know that all of the factors
in this equation except for $\Theta$ are homogeneous
of degree zero; thus $\Theta$ is also homogeneous of degree zero.
The fundamental solutions $e^{\sigma_\pm/z} \bL_\pm^{-1}$
preserve the pairing by Proposition \ref{prop:fundsol}
(we saw in \S \ref{sec:global_qconn} that
they coincide with the fundamental solutions from
Proposition \ref{prop:fundsol} via the mirror maps $\tau_\pm$)
and $\UU$ also preserves the pairing. Thus $\Theta$ preserves the pairing.
Finally we consider the analytic continuation of $K$-theoretic
flat sections. Note that the flat section $\frs(E)(\tau_-(\sfy),z)$
is analytically continued along $\tcM^\circ$ by the right-hand
side of the formula
\[
\frs(E)(\tau_-(\sfy),z) = \frac{1}{(2\pi)^{\dim X_-/2}}
\bL_-(\sfy,z) e^{- \sigma_-/z} \tPsi_-(E)
\]
where $\tPsi_-$ is the map in Theorem \ref{thm:U}.
Using \eqref{eq:Theta-U_relation}, we obtain:
\[
\Theta(\frs(E)(\tau_-(\sfy,z))) = \frac{1}{(2\pi)^{\dim X_-/2}}
\bL_+(\sfy,z) e^{-\sigma_+/z} \UU \left(
\tPsi_-(E) \right)
\]
Part (3) of Theorem \ref{thm:U} shows
that this is equal to $\frs(\FM(E))(\tau_+(\sfy),z)$.
\end{proof}
\subsection{Mellin--Barnes Analytic Continuation}
\label{sec:Mellin-Barnes}
In this section, we compute the analytic continuation of
the $I$-function and determine the linear transformation
$\UU$ in Theorem \ref{thm:U}.
\subsubsection{The $H$-Function}
It will be convenient to introduce another cohomology-valued
hypergeometric function called the $H$-function.
Noting that the $I$-function can be written
in terms of ratios of $\Gamma$-functions:
\[
I_+(\sfy,z):=
z e^{\sigma_+/z}
\sum_{d \in \KK_+}
\frac{\sfy^d}{z^{(D_1+\cdots + D_m) \cdot d}}
\left(
\prod_{j=1}^{m}
\frac{\Gamma\big(1 + \frac{u_j}{z} - \<{-D_j} \cdot d\>\big)}
{\Gamma\big(1 + \frac{u_j}{z} + D_j \cdot d\big)}
\right)
\frac{\fun_{[{-d}]}}{z^{\iota_{[{-d}]}}}
\]
we set:
\[
H_+(\sfy):=
e^{\frac{\sigma_+}{2 \pi \tti} }
\sum_{d \in \KK_+}
\sfy^d
\left(
\prod_{j=1}^{m}
\frac{1}
{\Gamma\big(1 + \frac{u_j}{2 \pi \tti} + D_j \cdot d\big)}
\right)
\fun_{[d]}\]
and similarly for $H_-$.
Formally speaking, $H_+$ belongs to the space:
\[
\prod_{p} \left(
H^p_{T}(IX_+)[\log \sfy_1,\dots,\log\sfy_r] [\![\sfy_1,\dots,\sfy_r]\!]
\right)
\]
Noting that the $T$-equivariant Gamma class of $X_+$ is given by
\[
\hGamma_{X_+} = \bigoplus_{f\in \KK_+/\LL}
\left(
\prod_{j=1}^m \Gamma(1 + u_j - \<D_j \cdot f\>) \right)
\fun_f
\]
we obtain the relationship between the $H$-function and the $I$-function:
\begin{equation}
\label{eq:IH_relation}
z^{-1} I_+(\sfy,z) = z^{-\frac{c_0(\lambda)}{2\pi\tti} - \frac{\dim X_+}{2}}
z^{-\mu^+} z^{\rho^+}
\left(\hGamma_{X_+} \cup (2\pi\tti)^{\frac{\deg_0}{2}}
\inv^* H\left(z^{-\frac{\deg \sfy}{2}} \sfy \right) \right)
\end{equation}
where $\rho^+$, $\mu^+$, $\deg_0$ are as in Notation~\ref{not:murho} and
\[
z^{-\frac{\deg \sfy}{2}}\sfy = (z^{-\frac{\deg \sfy_1}{2}}
\sfy_1 , \dots, z^{-\frac{\deg \sfy_r}{2}}\sfy_r ).
\]
The relationship between $H_-$ and $I_-$ is similar.
\begin{remark}
The $H$-function $H_+$ has an analytic properties analogous
to those of the $I$-function stated in Lemma~\ref{lem:I_analyticity}
and Remark~\ref{rem:I_analyticity}.
Namely $e^{-\sigma_+/(2\pi\tti)} H_+(\sfy)$ is a formal power
series in $\sfy_1,\dots,\sfy_{r-1}$ with coefficients of the form $\sum_{i=0}^N f_i(\lambda,\sfy_r) \phi_i$
where $\{\phi_i\}$ is an $R_T$-basis of $H_{T}^\bullet(IX_+)$
and $f_i(\lambda,\sfy_r)$ is analytic in
$(\lambda_1,\dots,\lambda_r, \sfy_r) \in \CC^m \times \cU_+$.
Notice that the $H$-function has better analytic behaviour with respect
to $\lambda$ since $\frac{1}{\Gamma(x)}$ is an entire function.
The analytic continuation of $H$-functions performed below
should be understood as analytic continuation of the coefficient functions $f_i(\lambda,\sfy_r)$.
\end{remark}
\subsubsection{Restriction of the $H$-Function to $T$-Fixed Points}
Recall that the $T$-fixed points on $X_+$ are indexed by minimal
anticones $\delta\in \cA_+$, and that the $T$-fixed points
on the inertia stack $IX_+$ are
indexed by pairs $(\delta,f)$ with $\delta \in \cA_+$
a minimal anticone and $f \in \KK_+/\LL$
satisfying $D_i \cdot f \in \ZZ$ for all $i \in \delta$.
The minimal anticone $\delta$ determines a $T$-fixed point $x_\delta$
on $X_+$ and
the pair $(\delta,f)$ determines a $T$-fixed point $x_{(\delta,f)}$
on the component $X_+^f$ of the inertia stack $IX_+$.
Let $i_\delta$ and $i_{(\delta,f)}$ denote
the inclusion maps $x_\delta \to X_+$ and $x_{(\delta,f)} \to IX_+$
respectively.
Set $u_j(\delta) = i_{\delta}^\star u_j\in H^2_T({\rm pt})$,
noting that $u_j(\delta) = 0$ if and only if $j \in \delta$.
We have that:
\begin{equation}
\label{eq:restriction_of_H_+}
i_{(\delta,f)}^\star H_+ =
\sum_{d \in \KK_+: [d]=f}
\frac{\sfy^d}{\prod_{j \in \delta} \Gamma\big(1 + D_j \cdot d \big)}
\frac{e^{\frac{1}{2\pi\tti} \sigma_+(\delta)}}
{\prod_{j \not \in \delta}\Gamma\big(1 + \frac{u_j(\delta)}{2 \pi \tti}
+ D_j \cdot d\big)}
\end{equation}
where $\sigma_+(\delta):= i_{\delta}^\star \sigma_+$.
Consider the factor $\prod_{j \in \delta} \Gamma\big(1 + D_j \cdot d \big)^{-1}$
in the summand: since $d \equiv f \mod \LL$ and since $D_j \cdot f \in \ZZ$
for all $j \in \delta$, the term $D_j \cdot d$ here is an integer.
Thus the factor $\prod_{j \in \delta} \Gamma\big(1 + D_j \cdot d \big)^{-1}$
vanishes unless $d \in \delta^\vee$, where
\[
\delta^\vee := \{ d \in \LL\otimes \QQ : D_j \cdot d \in \ZZ_{\ge 0}
\text{ for all } j\in \delta\}.
\]
The $H$-function is a sum over the subset
$\KKeff_+$ of $\KK_+$,
\[
\KKeff_+ = \Big\{ f \in \LL \otimes \QQ :
\big\{ i \in \{1,2,\ldots,m\} : D_i \cdot f \in \ZZ_{\geq 0} \big\} \in \cA_+ \Big\}
\]
which is in general quite complicated, but the restriction
$i_{(\delta,f)}^\star H_+$ of $H_+$ to a $T$-fixed point
in $IX_+$ is a sum over the much simpler set $\delta^\vee$.
\subsubsection{Analytic Continuation of the $H$-Function}
The Localization Theorem in $T$-equivariant cohomology
\cite{Berline--Vergne, Atiyah--Bott, GKM} implies that
one can compute the analytic continuation of $H_+$
by computing the analytic continuation of the restriction
$i_{(\delta,f)}^\star H_+$ to each $T$-fixed point
$x_{(\delta,f)} \in I X_+$.
The restriction $i_{(\delta,f)}^\star H_+$ is a
$H_T^{\bullet\bullet}({\rm pt})$-valued function. During
the course of analytic continuation, we regard the equivariant
parameters $\lambda_1,\dots,\lambda_m$ as generic complex
numbers.
There are two cases:
\begin{itemize}
\item $\delta \in \cA_+ \cap \cA_-$;
\item $\delta \in \cA_+$ but $\delta \not \in \cA_-$.
\end{itemize}
The anticone $\delta$ determines a $T$-fixed point $x_\delta$ in $X_+$,
and in the first case it also determines a fixed point in $X_-$.
In the first case the birational transformation
$\varphi \colon X_+ \dashrightarrow X_-$
is an isomorphism in a neighbourhood of $x_\delta$,
and it is clear from \eqref{eq:restriction_of_H_+}
that $i_{(\delta,f)}^\star H_+ = i_{(\delta,f)}^\star H_-$,
noting that $u_j(\delta)$ is the same for $X_+$ and $X_-$.
In the second case $x_\delta$ lies in the flopping locus of $\varphi$,
and we will see that the analytic continuation of $i_{(\delta,f)}^\star H_+$
is a linear combination of restrictions $i_{(\delta_-,f_-)}^\star H_-$
for appropriate $\delta_- \in\cA_-$ and $f_- \in \KK_-$.
Note that in the second case, $\delta$ has the form $\{j_1,\ldots,j_{r-1}, j_+\}$
with $D_{j_1},\ldots,D_{j_{r-1}} \in W$
and\footnote{Recall that $e \in \LL$ is the primitive lattice vector
in $W^\perp$ such that $e>0$ on $C_+$ and $e<0$ on $C_-$.}
$D_{j+} \cdot e>0$ (see Lemma \ref{lem:Apm}).
\begin{definition} \label{def:next_to_X}
Let $\delta_+ \in \cA_+$ and $\delta_- \in \cA_-$ be minimal anticones.
We say that $\delta_+$ is \emph{next to} $\delta_-$,
written $\delta_+ | \delta_-$, if $\delta_+ = \{j_1,\ldots,j_{r-1},j_+\}$
and $\delta_- = \{j_1,\ldots,j_{r-1},j_-\}$ with
$D_{j_1},\ldots,D_{j_{r-1}} \in W$, $D_{j_+} \cdot e > 0$,
and $D_{j-} \cdot e < 0$. In this case $\delta_+\notin \cA_-$
and $\delta_- \notin \cA_+$.
\end{definition}
\begin{definition} \label{def:next_to_IX}
Let $(\delta_+,f_+)$ index a $T$-fixed point on $IX_+$ and
$(\delta_-, f_-)$ index a $T$-fixed point on $I X_-$.
We say that $(\delta_+,f_+)$ is \emph{next to} $(\delta_-,f_-)$,
written $(\delta_+,f_+) | (\delta_-,f_-)$, if $\delta_+ | \delta_-$
and there exists $\alpha \in \QQ$
such that $f_- = f_+ + \alpha e$ in $\LL\otimes \QQ/\LL$.
\end{definition}
The analytic continuation of $i_{(\delta,f)}^\star H_+$
is a linear combination of $i_{(\delta_-,f_-)}^\star H_-$
such that $(\delta,f)$ is next to $(\delta_-,f_-)$.
\begin{notation}
\label{not:lift}
Fix lifts $\KK_+/\LL \to \KK_+$ and $\KK_-/\LL \to \KK_-$
such that, for any pairs $(d_+, d_-)\in \KK_+\times \KK_-$
with $d_+ - d_- \in \QQ e$, the lifts of $[d_+]$ and $[d_-]$
differ by a rational multiple of $e$.
\end{notation}
\begin{lemma} \label{lem:weights}
Let $\delta_+ \in \cA_+$ and $\delta_- \in \cA_-$ be
minimal anticones such that $\delta_+ | \delta_-$, and
let $j_-$ be the element of $\delta_-$ such that $j_- \not \in \delta_+$.
Then for any $j$, one has:
\[
u_j(\delta_+) =
u_j(\delta_-) + \frac{D_j \cdot e}{D_{j_-} \cdot e} u_{j_-}(\delta_+).
\]
\end{lemma}
\begin{proof}
Write $\delta_- =\{j_1,\dots,j_{r-1},j_-\}$.
Since $D_{j_1},\dots,D_{j_{r-1}},D_{j_-}$ form a basis of
$\LL^\vee \otimes \QQ$, we can write:
\[
D_j = c_1 D_{j_1} + \cdots + c_{r-1} D_{j_{r-1}} + c_- D_{j_-}
\]
Since $D_{j_1}, \dots, D_{j_{r-1}}$ are on the wall, pairing
with $e$ yields:
\begin{equation}
\label{eq:dot_e}
D_j \cdot e= c_- (D_{j_-} \cdot e).
\end{equation}
Applying the homomorphism $\theta_\pm$ from \eqref{eq:theta},
we obtain
\[
u_j-\lambda_j =
c_1 (u_{j_1} - \lambda_{j_1}) + \cdots + c_{r-1} (u_{j_{r-1}} -
\lambda_{j_{r-1}}) +
c_- (u_{j_-} - \lambda_{j_-})
\]
on both $X_+$ and $X_-$.
Restricting to $x_{\delta_+}\in X_+$ and $x_{\delta_-}\in X_-$,
we get the two relations:
\begin{align*}
u_j(\delta_+) -\lambda_j & = - c_1 \lambda_{j_1}
-\cdots - c_{r-1} \lambda_{j_{r-1}} + c_- (u_j(\delta_+) - \lambda_{j_-}), \\
u_j(\delta_-) -\lambda_j & = - c_1 \lambda_{j_1}
-\cdots - c_{r-1} \lambda_{j_{r-1}} - c_- \lambda_{j_-}.
\end{align*}
Comparing the two equations, we get
$u_j(\delta_+) = u_j(\delta_-) +c_- u_{j_-}(\delta_+)$.
The conclusion now follows from equation~\ref{eq:dot_e}.
\end{proof}
\begin{corollary} \
\label{cor:weights}
\begin{enumerate}
\item Let $\delta$ be a minimal anticone such that $\delta\in
\cA_+ \cap \cA_-$. Then
$\sigma_+(\delta) = \sigma_-(\delta)$.
\item Let $\delta_+ \in \cA_+$, $\delta_-\in \cA_-$
be minimal anticones such that $\delta_+|\delta_-$
and let $j_-\in \delta_-$ be an element such that $j_- \notin \delta_+$.
Then:
\[
\sigma_+(\delta_+) = \sigma_-(\delta_-) +
\frac{\log \sfy^e }{D_{j_-}\cdot e}
u_{j_-}(\delta_+)
\]
\end{enumerate}
\end{corollary}
\begin{proof} \
\begin{enumerate}
\item As we discussed, $u_j(\delta)$ is the same for $X_+$ and $X_-$ whenever $\delta \in \cA_+\cap\cA_-$. Therefore $i_\delta^\star \theta_+(D_j) = i_\delta^\star \theta_-(D_j)$ for all $j$. In particular $i_\delta^\star \theta_+(p) = i_\delta^\star \theta_-(p)$ for every $p\in \LL^\vee \otimes \CC$. Setting $p= \sum_{i=1}^r \sfp_i^+ \log \sfy_i = \sum_{i=1}^r \sfp_i^- \log \tsfy_i$, we obtain (1).
\item Lemma \ref{lem:weights} shows that
\begin{equation}
\label{eq:weights_theta}
i_{\delta_+}^\star\theta_+(p) = i_{\delta_-}^\star \theta_-(p)
+ \frac{p\cdot e}{D_{j_-} \cdot e} u_{j_-}(\delta_+)
\end{equation}
for all $p\in \LL^\vee \otimes \CC$.
Setting again $p= \sum_{i=1}^r \sfp_i^+ \log \sfy_i =
\sum_{i=1}^r \sfp_i^- \log \tsfy_i$, we obtain (2).
\end{enumerate}
\end{proof}
\begin{theorem}
\label{thm:analytic_continuation}
Let $(\delta_+,f_+)$ index a $T$-fixed point on $I X_+$.
If $\delta_+ \in \cA_+ \cap \cA_-$ then:
\[
i_{(\delta_+,f_+)}^\star H_+ = i_{(\delta_+,f_+)}^\star H_-.
\]
Otherwise, after analytic continuation along the path
$\gamma$ in Figure \ref{fig:path_ancont}, we have:
\[
i_{(\delta_+,f_+)}^\star H_+ =
\sum_{\substack{(\delta_-,f_-) : \\ (\delta_+,f_+)|(\delta_-,f_-)}}
C_{\delta_+,f_+}^{\delta_-,f_-} \,
i_{(\delta_-,f_-)}^\star H_-
\]
where:
\begin{multline*}
C_{\delta_+,f_+}^{\delta_-,f_-} =
e^{\frac{\pi\tti w}{D_{j_-}\cdot e}
\big(\frac{u_{j_-}(\delta_+)}{2 \pi \tti}
+ D_{j_-} \cdot (f_+-f_-)\big)}
\\
\times \frac{\sin \pi\Big(\frac{u_{j_-}(\delta_+)}{2 \pi \tti}
+ D_{j_-} \cdot (f_+-f_-)\Big)}
{({-D_{j_-}} \cdot e) \sin \frac{\pi}{{-D_{j_-}}\cdot e}
\Big(\frac{u_{j_-}(\delta_+)}{2 \pi \tti} + D_{j_-} \cdot (f_+-f_-)\Big)}
\prod_{\substack{j : D_j \cdot e < 0 \\ j \ne j_-}}
\frac{\sin \pi\Big(\frac{u_j(\delta_+)}{2 \pi \tti} + D_j \cdot f_+\Big)}
{\sin \pi\Big(\frac{u_j(\delta_-)}{2 \pi \tti} + D_j \cdot f_-\Big)}
\end{multline*}
with $w := -1 - \sum_{j:D_j \cdot e<0} D_j \cdot e
= -1 + \sum_{j:D_j\cdot e>0} D_j \cdot e$
and $j_-\in \delta_-$ given by the unique element such that
$D_{j_-} \cdot e <0$.
\end{theorem}
\begin{remark}
The coefficient $C_{\delta_+,f_+}^{\delta_-,f_-}$
does not depend on the choice of lifts $f_+ \in \KK_+$ and $f_-\in \KK_-$
such that $f_+ - f_- \in \QQ e$ (see Notation \ref{not:lift}).
\end{remark}
\begin{proof}[Proof of Theorem \ref{thm:analytic_continuation}]
The first statement follows immediately from \eqref{eq:restriction_of_H_+}
and Corollary~\ref{cor:weights}.
In this case, $i_{(\delta_+,f_+)}^\star H_+$
(respectively~$i_{(\delta_+,f_+)}^\star H_-$)
is a formal power series in
$\sfy_1,\dots,\sfy_{r-1}$ (respectively in~$\tsfy_1,\dots,\tsfy_{r-1}$)
with coefficients that are \emph{polynomials} in $\sfy_r$ (respectively in~$\tsfy_r$),
and the series $i_{(\delta_+,f_+)}^\star H_+$,
$i_{(\delta_+,f_+)}^\star H_-$ match under the change
\eqref{eq:change_of_variables} of co-ordinates.
Consider now
\[
i_{(\delta_+,f_+)}^\star H_+ = e^{\frac{\sigma_+(\delta_+)}{2\pi\tti}}
\sum_{\substack{d \in \delta_+^\vee: \\ [d]=f_+}}
\sfy^d
\frac{1}
{\prod_{j =1}^m\Gamma\big(1 + \frac{u_j(\delta_+)}{2 \pi \tti}
+ D_j \cdot d\big)}
\]
where $\delta_+ \in \cA_+$ but $\delta_+ \not \in \cA_-$.
We can write $d \in \delta_+^\vee$ uniquely as $d = d_+ + k e$
with $k$ a non-negative integer, $d_+ \in \delta_+^\vee$,
and $d_+ - e \not \in \delta_+^\vee$. Then:
\begin{equation}
\label{eq:sum_over_d_+}
i_{(\delta_+,f_+)}^\star H_+ =
\sum_{\substack{d_+ \in \delta_+^\vee :
\\ d_+ - e \not \in \delta_+^\vee \\ [d_+]=f_+}}
\sfy^{d_+}
\sum_{k=0}^\infty
\frac{e^{\frac{\sigma_+(\delta_+)}{2 \pi \tti}}
(\sfy^{e})^k }
{\prod_{j =1}^m
\Gamma\big(1 + \frac{u_j(\delta_+)}{2 \pi \tti} + D_j \cdot d_+
+ k D_j \cdot e\big)}
\end{equation}
Consider the second sum here. This is:
\begin{multline}
\label{eq:inner_sum}
\sum_{k=0}^\infty
{e^{\frac{\sigma_+(\delta_+)}{2 \pi \tti} } (\sfy^e)^k }
\prod_{j : D_j \cdot e < 0}
\frac{(-1)^{k D_j \cdot e}
\sin \pi\big(-\frac{u_j(\delta_+)}{2 \pi \tti} - D_j \cdot d_+\big)}{\pi}
\\
\times
\frac{\prod_{j : D_j \cdot e < 0}
\Gamma\big({-\frac{u_j(\delta_+)}{2 \pi \tti}} - D_j \cdot d_+ - k D_j \cdot e\big)}
{\prod_{j: D_j \cdot e \geq 0}
\Gamma\big(1 + \frac{u_j(\delta_+)}{2 \pi \tti}
+ D_j \cdot d_+ + k D_j \cdot e\big)}
\end{multline}
where we used $\Gamma(y) \Gamma(1-y) = \pi/(\sin \pi y)$.
Thus \eqref{eq:inner_sum} is:
\begin{multline}
\label{eq:inner_sum_as_residue_sum}
\sum_{k=0}^\infty
e^{\frac{\sigma_+(\delta_+)}{2 \pi \tti} }
\Res_{s=k} \Gamma(s) \Gamma(1-s) e^{\pi\tti s} (\sfy^e)^s
\prod_{j : D_j \cdot e < 0}
\frac{e^{\pi\tti s (D_j \cdot e)}
\sin \pi\big(-\frac{u_j(\delta_+)}{2 \pi \tti} - D_j \cdot d_+\big)}{\pi} \\
\times
\frac{\prod_{j : D_j \cdot e < 0}
\Gamma\big({-\frac{u_j(\delta_+)}{2 \pi \tti}} - D_j \cdot d_+ - s D_j \cdot e\big)}
{\prod_{j: D_j \cdot e \geq 0}
\Gamma\big(1 + \frac{u_j(\delta_+)}{2 \pi \tti}
+ D_j \cdot d_+ + s D_j \cdot e\big)} \, ds.
\end{multline}
Consider now the contour integral
\begin{equation}
\label{eq:contour_integral}
e^{\frac{\sigma_+(\delta_+)}{2\pi\tti}} \int_C
\Gamma(s) \Gamma(1-s)
\frac{\prod_{j : D_j \cdot e < 0}
\Gamma\big({-\frac{u_j(\delta_+)}{2 \pi \tti}}
- D_j \cdot d_+ - s D_j \cdot e\big) }
{\prod_{j: D_j \cdot e \geq 0}
\Gamma\big(1 + \frac{u_j(\delta_+)}{2 \pi \tti}
+ D_j \cdot d_+ + s D_j \cdot e\big)}
\left(e^{-\pi\tti w}\sfy^e\right)^s \, ds
\end{equation}
where the contour $C$, shown in Figure~\ref{fig:contour},
is chosen such that the poles at $s=n$ are on the right of $C$
and the poles at $s = {-1}-n$ and at
\begin{align}
\label{eq:interesting_poles}
s = \textstyle \frac{1}{{-D_{j_-}} \cdot e}
\Big( \frac{u_{j_-}(\delta_+)}{2 \pi \tti} + D_{j_-} \cdot d_+ - n \Big)
&& \text{for $j_-$ such that $D_{j_-} \cdot e < 0$}
\end{align}
are on the left of $C$; here $n$ is a non-negative integer.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{contour.pdf}
\caption{The contour $C$}
\label{fig:contour}
\end{figure}
Note that all poles of the integrand are simple.
By assumption we have that $\sum_{j=1}^m D_j \in W$,
and hence that $\sum_{j=1}^m D_j \cdot e = 0$.
Let $\frc\in \CC$ be the conifold point \eqref{eq:conifold_point}.
Lemma~A.6 in~\cite{Borisov--Horja:FM} implies that:
\begin{itemize}
\item the contour integral \eqref{eq:contour_integral}
is convergent and analytic as a function of $\sfy^e$ in the domain
$\{\sfy^e : |\arg(\sfy^e) - w \pi|< \pi \}$;
\item for $|\sfy^e|<|\frc|$, the integral is equal to
the sum of residues on the right of $C$; and
\item for $|\sfy^e|>|\frc|$, the integral is equal to
minus the sum of residues on the left of $C$.
\end{itemize}
The residues at $s = {-1}-n$ vanish,
where $n$ is a non-negative integer: each such residue contains a factor
\[
\prod_{j \in \delta_+} \Gamma\big(1 + D_j \cdot\big(d_+ - (n+1) e\big) \big)^{-1}
\]
and $d_+ - (n+1) e \not \in \delta_+^\vee$, so at least
one of the $\Gamma$-functions is evaluated at a negative integer.
After analytic continuation in $x$, therefore,
\eqref{eq:inner_sum_as_residue_sum} becomes
minus the sum of residues at the poles \eqref{eq:interesting_poles}.
The residue at the pole
\[
p = \textstyle \frac{1}{{-D_{j_-}} \cdot e}
\Big( \frac{u_{j_-}(\delta_+)}{2 \pi \tti} + D_{j_-} \cdot d_+ - n \Big)
\]
is:
\begin{multline}
\label{eq:simplify_this_residue}
-e^{ \frac{\sigma_+(\delta_+)}{2\pi\tti} }
(\sfy^e)^p
e^{\pi\tti p(1 + D_{j_-} \cdot e)}
\frac{\sin \pi\Big(\frac{u_{j_-}(\delta_+)}{2 \pi \tti}
+ D_{j_-} \cdot d_+\Big)}{\sin \pi p}
\frac{1}{({-D_{j_-}} \cdot e)} \frac{(-1)^n}{n!} \\
\prod_{\substack{j : D_j \cdot e < 0 \\ j \ne j_-}}
\frac{e^{\pi\tti p(D_j \cdot e)}
\sin \pi\Big( \frac{u_j(\delta_+)}{2 \pi \tti} + D_j \cdot d_+\Big)}
{\sin \pi\Big(\frac{u_j(\delta_+)}{2 \pi \tti} + D_j \cdot d_+
+ p (D_j \cdot e)\Big)}
\prod_{j : j \ne j_-}
\frac{1}{\Gamma\Big(1+\frac{u_j(\delta_+)}{2 \pi \tti}
+ D_j \cdot d_+ + p (D_j \cdot e)\Big)}
\end{multline}
This simplifies dramatically.
Set $n = k ({-D_{j_-}} \cdot e) + l$ with
$0 \leq l < ({-D_{j_-}} \cdot e)$,
\[
d_- = d_+ + \frac{D_{j-} \cdot d_+ - l}{{-D_{j_-}} \cdot e}e
\]
and $\delta_- = \{j_1,\ldots,j_{r-1},j_-\}$,
where $\delta_+ = \{j_1,\ldots,j_{r-1},j_+\}$
with $D_{j_1}\cdot e = \cdots = D_{j_{r-1}} \cdot e = 0$.
Note that $D_{j_-} \cdot d_- =l \in \ZZ_{\ge 0}$
but $D_{j_-} \cdot (d_- + e) < 0$,
and therefore $d_- \in \delta_-^\vee$ but $d_- + e \not \in\delta_-^\vee$.
Lemma~\ref{lem:weights} implies that:
\[
\textstyle
\frac{u_j(\delta_+)}{2 \pi \tti} + D_j \cdot d_+ + p (D_j \cdot e)
=
\frac{u_j(\delta_-)}{2 \pi \tti} + D_j \cdot d_- -k (D_j \cdot e)
\]
and thus the residue \eqref{eq:simplify_this_residue} is:
\begin{multline}
\label{eq:simplified_residue}
{- e^{\frac{\sigma_-(\delta_-)}{2 \pi \tti} }}
\sfy^{d_- - d_+ - k e}
e^{\frac{\pi \tti w}{D_{j_-}\cdot e} \big(\frac{u_{j_-}(\delta_+)}{2 \pi \tti}
+ D_{j_-} \cdot (d_+-d_-)\big)}
\\
\times
\frac{\sin \pi\Big(\frac{u_{j_-}(\delta_+)}{2 \pi \tti} +
D_{j_-} \cdot (d_+-d_-)\Big)
}
{({-D_{j_-}} \cdot e) \sin \frac{\pi}{{-D_{j_-}}\cdot e}
\Big(\frac{u_{j_-}(\delta_+)}{2 \pi \tti} + D_{j_-} \cdot (d_+-d_-)\Big)}
\\
\times
\prod_{\substack{j : D_j \cdot e < 0 \\ j \ne j_-}}
\frac{\sin \pi\Big(\frac{u_j(\delta_+)}{2 \pi \tti} + D_j \cdot d_+\Big)}
{\sin \pi\Big(\frac{u_j(\delta_-)}{2 \pi \tti} + D_j \cdot d_-\Big)}
\prod_{j=1}^m
\frac{1}{\Gamma\Big(1+\frac{u_j(\delta_-)}{2 \pi \tti}
+ D_j \cdot d_- - k (D_j \cdot e)\Big)}
\end{multline}
where we used
$p = \frac{1}{- D_{j_-} \cdot e}
\left( \frac{u_{j_-}(\delta_+)}{2\pi\tti}
+ D_{j_-} \cdot (d_+- d_-) \right) - k$ and Corollary \ref{cor:weights}.
Let $f_-$ denote the equivalence class of $d_-$ in $\KK_-/\LL$,
noting that $(\delta_+,f_+) | (\delta_-,f_-)$ and that
\[
d_+ = f_+ - f_- + N e + d_-
\]
for some integer $N$. (Here we used Notation~\ref{not:lift}.)
The dependence of \eqref{eq:simplified_residue} on $N$ cancels, giving:
\[
{- e^{\frac{\sigma_-(\delta_-)}{2 \pi \tti}}}
\sfy^{d_- - d_+ - ke}
\frac{1}{\prod_{j=1}^m \Gamma\Big(1+\frac{u_j(\delta_-)}{2 \pi \tti}
+ D_j \cdot d_- - k (D_j \cdot e)\Big)}
\,
C_{\delta_+,f_+}^{\delta_-,f_-}
\]
and minus the sum of these residues gives the analytic continuation
of \eqref{eq:inner_sum_as_residue_sum}.
After analytic continuation in $\sfy^e = \sfy_r^{\sfp^+_r \cdot e}$,
therefore, we have that:
\begin{multline*}
i_{(\delta_+,f_+)}^\star H_+ =
\sum_{\substack{(\delta_-,f_-) : \\ (\delta_+,f_+)|(\delta_-,f_-)}}
\sum_{\substack{d_- \in \delta_-^\vee: \\ d_- + e \not \in \delta_-^\vee \\ [d_-]=f_-}}
\sfy^{d_-}
\sum_{k=0}^\infty
\frac{e^{\frac{\sigma_-(\delta_-)}{2\pi\tti}} (\sfy^e)^{-k} }
{\prod_{j =1}^m\Gamma\big(1 + \frac{u_j(\delta_-)}{2 \pi \tti}
+ D_j \cdot d_- - k D_j \cdot e\big)}
\,
C_{\delta_+,f_+}^{\delta_-,f_-}
\end{multline*}
Comparing with \eqref{eq:sum_over_d_+} gives the result.
\end{proof}
\subsubsection{Analytic Continuation of the $I$-Function and the Symplectic Transformation $\UU$}
\label{sec:U}
Set $\hR_T = H^{\bullet \bullet}_T({\rm pt})$
and let $\hS_T$ be the completion of $S_T$ in \S \ref{sec:equiv_int_str}.
Define an $\hS_T$-linear transformation
$\UU_H \colon H^{\bullet\bullet}_{T}(IX_-)\otimes_{\hR_T} \hS_T
\to H^{\bullet\bullet}_{T}(IX_+) \otimes_{\hR_T} \hS_T$
by
\begin{align}
\label{eq:UH}
\begin{split}
\UU_H (\alpha) =
& \sum_{\substack{(\delta, f) : \\ \delta \in \cA_+ \cap \cA_-}}
(i_{(\delta,f)}^\star \alpha)
\cdot
\frac{\fun_{\delta,f}}{e_{T}(N_{\delta,f})} \\
& + \sum_{\substack{(\delta_+,f_+): \\ \delta_+ \in \cA_+ \setminus \cA_-}}
\sum_{\substack{(\delta_-,f_-): \\ (\delta,f_-) | (\delta_+,f_+)}}
C_{\delta_-,f_-}^{\delta_+,f_-} \cdot (i_{(\delta_-,f_-)}^\star \alpha)
\cdot
\frac{\fun_{\delta_+,f_+}}{e_{T}(N_{\delta_+,f_+})}
\end{split}
\end{align}
where $(\delta,f)$ and $(\delta_+,f_+)$ index $T$-fixed points
in $I X_+$, $\fun_{\delta,f} = i_{(\delta,f)\star} \fun$ and
$N_{\delta,f} := T_{x_{(\delta,f)}} X^f_+$.
Then Theorem \ref{thm:analytic_continuation} can be restated as:
\[
H_+ = \UU_H H_-
\]
Define the linear transformation $\UU$ so
that the following diagram commutes:
\begin{equation}
\label{eq:UH_U}
\begin{aligned}
\xymatrix{
H^{\bullet\bullet}_{T}(IX_-)\otimes_{\hR_T} \hS_T \ar[r]^{\UU_H} \ar[d]_{\tPsi'_-} &
H^{\bullet\bullet}_{T}(IX_+) \otimes_{\hR_T} \hS_T \ar[d]^{\tPsi'_+} \\
H^\bullet_{\CR,T}(X_-) \otimes_{R_T} S_T[\log z](\!(z^{-1/k})\!) \ar[r]^{\UU} &
H^\bullet_{\CR,T}(X_-) \otimes_{R_T} S_T[\log z](\!(z^{-1/k})\!)
}
\end{aligned}
\end{equation}
where the vertical maps are defined by $\tPsi'_\pm(\alpha)=
z^{-\mu^\pm}z^{\rho^\pm}
(\hGamma_{X_\pm} \cup (2\pi\tti)^{\frac{\deg_0}{2}}
\inv^* \alpha)$ and
$k\in \N$ is as in Theorem \ref{thm:U}.
The relationship \eqref{eq:IH_relation} between the $H$-function
and the $I$-function implies part (1) of Theorem \ref{thm:U}:
\begin{equation}
\label{eq:UI}
I_+ = \UU I_-.
\end{equation}
Since the $I$-function contains neither $\log z$ nor
non-integral powers of $z$, it follows that $\UU$ is in fact
a linear transformation:
\[
\UU \colon
H_{\CR,T}^\bullet(X_-) \otimes_{R_T} S_T(\!(z^{-1})\!)
\to
H_{\CR,T}^\bullet(X_+) \otimes_{R_T} S_T(\!(z^{-1})\!)
\]
Diagram \eqref{eq:UH_U} gives that $\UU$ is automatically degree-preserving.
We show that $\UU$ satisfies part (2) of
Theorem \ref{thm:U}.
Noting that $\sfp_i ^+ = \sfp_i^-$, $i=1,\dots,r-1$ are
on the wall $W$, it suffices to show that
$\theta_+(\sfp_i^+) \circ \UU = \UU \circ \theta_-(\sfp_i^-)$
for $1\le i\le r-1$.
This follows from equation \eqref{eq:UI} and the monodromy properties
of the $I$-functions:
\begin{align*}
I_{+} \big|_{\sfy_j \mapsto e^{2\pi\tti} \sfy_j}
& = e^{2\pi\tti \theta_+(\sfp^+_j)/z} I_{+} \\
I_{-} \big|_{\tsfy_j \mapsto e^{2\pi\tti} \tsfy_j}
& = e^{2\pi\tti \theta_-(\sfp^-_j)/z} I_{-}
\end{align*}
for $1\le j\le r-1$.
Note that $\sfy_j \to e^{2\pi\tti} \sfy_j$ corresponds to
$\tsfy_j \to e^{2\pi\tti} \tsfy_j$ under the change
\eqref{eq:change_of_variables} of variables.
It remains to show that:
\begin{itemize}
\item $\UU$ is symplectic;
\item $\UU$ is defined over $R_T(\!(z^{-1})\!)$,
i.e.~that $\UU$ admits a non-equivariant limit.
\end{itemize}
These properties follow from the identification of $\UU_H$ with
the Fourier--Mukai transformation defined in the next section.
We will discuss these points in \S \ref{sec:end_of_the_proof} below.
\subsection{The Fourier--Mukai Transform}
\label{sec:FM}
We now construct a diagram \eqref{eq:common_blowup} canonically associated to the toric birational transformation
$\varphi \colon X_+ \dashrightarrow X_-$,
where $\tX$ is a toric Deligne--Mumford stack and
$f_+$,~$f_-$ are toric blow-ups, and compute the
Fourier--Mukai transformation:
\begin{align*}
\FM \colon K^0_T(X_-) \to K^0_T(X_+) && \FM := (f_+)_\star (f_-)^\star
\end{align*}
In~\S\ref{sec:FM_match} below we will see that this transformation coincides, via the equivariant integral structure in
Definition \ref{def:K_framing},
with the transformation $\UU$ from \S\ref{sec:U}
given by analytic continuation.
\subsubsection{The Common Blow-Up of $X_+$ and $X_-$}
Recall from \S\ref{sec:stacky_fan} that $X_+$ and $X_-$ are defined
in terms of an exact sequence:
\[
\xymatrix{
0 \ar[r] &
\LL \ar[r] &
\ZZ^m \ar[r]^\beta &
\bN \ar[r] &
0 }
\]
where the map $\LL \to \ZZ^m$ is given by $(D_1,\ldots,D_m)$.
This sequence defines an action of $K = (\Cstar)^r$ on $\CC^m$,
and $X_\pm = \big[ U_{\omega_\pm} \big/ K\big]$ for appropriate
stability conditions $\omega_+$,~$\omega_- \in \LL^\vee \otimes \RR$.
Let $b_1,\ldots,b_m$ denote the images
of the standard basis elements for $\ZZ^m$ under the map $\beta$.
Consider now the action of $K \times \Cstar$ on $\CC^{m+1}$
defined by the exact sequence:
\[
\xymatrix{
0 \ar[r] &
\LL \oplus \ZZ \ar[r] &
\ZZ^m \oplus \ZZ \ar[r]^-{\tilde{\beta}} &
\bN \ar[r] &
0 }
\]
where the map $\LL \oplus \ZZ \to \ZZ^m \oplus \ZZ$
is given by $(\tD_1,\ldots,\tD_{m+1})$,
\[
\tD_j =
\begin{cases}
D_j \oplus 0 & \text{if $j < m+1$ and $D_j \cdot e \leq 0$} \\
D_j \oplus ({-D_j} \cdot e) & \text{if $j < m+1$ and $D_j \cdot e > 0$} \\
0 \oplus 1 & \text{if $j=m+1$}
\end{cases}
\]
The map $\tilde{\beta}$ is the direct sum of $\beta$
with the map $\ZZ \to \bN$ defined by the element
\[
b_{m+1} = \sum_{j : D_j \cdot e > 0} (D_j \cdot e) b_j
\]
so the images of the standard basis elements for $\ZZ^m \oplus \ZZ$
under the map $\tilde{\beta}$ are $b_1,\ldots,b_{m+1}$.
Consider the chambers $\widetilde{C}_+$,~$\widetilde{C}_-$,
and~$\widetilde{C}$ in $(\LL \oplus \ZZ)^\vee \otimes \RR$
that contain, respectively, the stability conditions
\begin{align*}
\tomega_+ = (\omega_+,1) &&
\tomega_- = (\omega_-,1) && \text{and} &&
\tomega = (\omega_0, - \varepsilon)
\end{align*}
where $\omega_0$ is a point in the relative interior of
$W \cap \overline{C_+} = W \cap \overline{C_-}$
as in \S \ref{sec:birational_transformations},
and $\varepsilon$ is a very small positive real number.
Let $\tX$ denote the toric Deligne--Mumford stack
defined by the stability condition $\tomega$.
\begin{lemma}
\label{lem:Atilde}
Recall the notation $\cA_\pm$, $\cA_0$, $\cA_0^{\rm thick}$,
$\cA_0^{\rm thin}$, $M_0$, $M_\pm$ in Lemma \ref{lem:Apm}.
The set of anticones for the stability conditions $\tomega_\pm$,
$\tomega$ are given by
\begin{align*}
\cA_{\tomega_\pm} & =
\left\{ I \sqcup \{m+1\} : I \in \cA_\pm \right\} \\
\cA_{\tomega} & =
\left\{ I \sqcup \{ m+ 1\} : I \in \cA_0^{\rm thick}
\right\} \sqcup
\left\{I \in\cA_0^{\rm thick} : I \cap M_0 \in \cA_0^{\rm thin}
\right\}.
\end{align*}
\end{lemma}
\begin{proof}
Straightforward.
\end{proof}
\begin{lemma}
\label{lem:toricblowup}
We have the following statements.
\begin{enumerate}
\item The toric Deligne--Mumford stack corresponding
to the chamber $\widetilde{C}_+$ is $X_+$.
\item The toric Deligne--Mumford stack corresponding
to the chamber $\widetilde{C}_-$ is $X_-$.
\item There is a commutative diagram as in \eqref{eq:common_blowup} , where:
\begin{itemize}
\item $f_+ \colon \tX \to X_+$ is a toric blow-up,
arising from wall-crossing from the chamber $\widetilde{C}$ to $\widetilde{C}_+$;
and
\item $f_- \colon \tX \to X_-$ is a toric blow-up,
arising from wall-crossing from the chamber $\widetilde{C}$ to $\widetilde{C}_-$.
\end{itemize}
\end{enumerate}
\end{lemma}
\begin{proof}
In view of \S\ref{sec:GITdata}, the description of $\cA_{\tomega_\pm}$
in Lemma \ref{lem:Atilde} proves (1) and (2).
The birational transformations $f_+ \colon \tX
\dashrightarrow X_+$ and $f_- \colon \tX \dashrightarrow X_-$
determined by the toric wall-crossings are each morphisms
which contract the toric divisor defined by
the $(m+1)$-st homogeneous co-ordinate.
Indeed, $f_+$ is induced by the identity
birational map $U_{\tomega} \dashrightarrow U_{\tomega_+}$,
and a point $(z_1,\dots,z_m, z_{m+1}) \in U_{\tomega_+}$ is equivalent
to the point $(z_1 z_{m+1}^{l_1},\dots,z_m z_{m+1}^{l_m},1)
\in U_{\omega_+} \times \{1\}$
under the action of the $\Cstar$-subgroup of $K\times \Cstar$
corresponding to $e \oplus 1 \in \LL \oplus \ZZ$,
where we set $l_i := \max(-D_i \cdot e, 0)$ for $1\le i\le m$.
Therefore $f_+$ is induced by a morphism
\begin{equation}
\label{eq:morphism_f+}
U_{\tomega} \to U_{\omega_+} \qquad (z_1,\dots,z_m,z_{m+1})
\mapsto (z_1 z_{m+1}^{l_1},\dots, z_m z_{m+1}^{l_m})
\end{equation}
which is equivariant with respect to the group homomorphism
(quotient by the $\Cstar$-subgroup given by $e\oplus 1$)
\begin{equation}
\label{eq:grouphom_f+}
\phi_+ \colon
K\times \Cstar \to K \qquad (g,\lambda) \mapsto g \cdot \lambda^{-e}.
\end{equation}
Using Lemma \ref{lem:Atilde}, one can easily check that the map
\eqref{eq:morphism_f+}
indeed sends $U_\tomega$ to $U_{\omega_+}$.
We obtain a similar description for $f_-$
by considering the $\Cstar$-subgroup given by $0\oplus 1 \in \LL \oplus \ZZ$
instead of $e\oplus 1$.
\end{proof}
\begin{remark}
Torus fixed points on $\tX$ lying on the exceptional
divisor $\{z_{m+1}=0\}$ of $f_\pm$ correspond to minimal
anticones $\tdelta \in \cA_{\tomega}$ such
that $\tdelta \in \cA_0^{\rm thick}$ and
$\tdelta \cap M_0\in \cA_0^{\rm thin}$.
These minimal anticones take the form
\[
\tdelta = \{j_1,\dots,j_{r-1}, j_+, j_-\}
\]
where $D_{j_1},\dots,D_{j_{r-1}} \in W$,
$D_{j_+} \cdot e>0$ and $D_{j_-} \cdot e <0$.
The birational morphism $f_\pm$ maps
the corresponding torus fixed point $x_{\tdelta}\in \tX$
to the torus fixed point $x_{\delta_\pm}\in X_\pm$ with
\[
\delta_+ = \{j_1,\dots,j_{r-1}, j_+\}\in \cA_+, \quad
\delta_- = \{j_1,\dots,j_{r-1}, j_-\} \in \cA_-.
\]
Torus fixed points on $\tX$ lying away from the exceptional
divisor $\{z_{m+1}=0\}$ corresponds to minimal anticones
$\tdelta\in \cA_\tomega$ of the form
$\tdelta = \delta \cup \{m+1\}$, $\delta \in \cA_0^{\rm thick}
= \cA_+ \cap \cA_-$.
The morphisms $f_\pm$ are isomorphisms in neighbourhoods
of these fixed points, and the torus fixed point $x_\tdelta$ maps to
the fixed point $x_\delta$ in $X_+$ or in $X_-$.
\end{remark}
\begin{remark}
The stacky fan $\widetilde{\mathbf{\Sigma}}$ for $\tX$ is obtained
from the stacky fans $\mathbf{\Sigma}_\pm$ for $X_\pm$ by adding
the extra ray
$
b_{m+1} = \sum_{j : D_j \cdot e > 0} (D_j \cdot e) b_j
$
where
\[
\sum_{j : D_j \cdot e > 0} (D_j \cdot e) b_j
= \sum_{j : D_j \cdot e < 0} ({-D_j} \cdot e) b_j
\]
is a minimal linear relation (or circuit) in $\Sigma_\pm$,
see Remark \ref{rem:circuit}.
So our discussion here is a rephrasing in terms of GIT data
of the material in~\cite[\S5]{Borisov--Horja:FM}.
\end{remark}
\subsubsection{A Basis for Localized $T$-Equivariant $K$-Theory}
\label{sec:basis_K-theory}
Recall that $T=(\Cstar)^m$ acts on $X_\pm$ through
the diagonal $T$-action on $\CC^m$.
We consider the $T$-action on $\tX$ induced from the
inclusion $T = T\times \{1\} \subset T \times \Cstar$
and the $(T\times \Cstar)$-action on $\CC^{m+1}$.
Then all the maps in the diagram \eqref{eq:common_blowup}
are $T$-equivariant.
The $T$-equivariant $K$-groups $K^0_T(X_\pm)$, $K^0_T(\tX)$
are modules over $K^0_T({\rm pt})$, which is the ring
$\ZZ[T]$ of regular functions (over $\ZZ$) on the algebraic torus $T$.
The $T$-invariant divisor $\{z_i=0\}$
on $X_\omega$ defined in \eqref{eq:T-invariant_divisor}
determines a $T$-equivariant line bundle $\cO(\{z_i=0\})$
on $X_\omega$, and we denote the class of this line bundle in
$T$-equivariant $K$-theory by $R_i$.
For the spaces $X_+$, $X_-$, and $\tX$
we write these classes as
\begin{align*}
& \text{$R^+_1,\ldots,R_m^+\in K^0_T(X_+)$} &
& \text{$R^-_1,\ldots,R_m^- \in K^0_T(X_-)$} && \text{and} &
& \text{$\tR_1,\ldots,\tR_{m+1}\in K^0_T(\tX)$.}
\intertext{Let us write:}
& S_j^+ := (R_j^+)^{-1} &
& S_j^- := (R_j^-)^{-1} && \text{and} &
& \tS_j := \tR_j^{-1}
\end{align*}
An irreducible $K$-representation
$p\in \Hom(K, \Cstar) = \LL^\vee$ defines a line bundle
$L(p)\to X_\omega$:
\[
L(p) = U_{\omega} \times \CC\big /(z,v) \sim (g \cdot z, p(g) v), \ g\in K
\]
This line bundle is equipped with the $T$-linearization
$[z,v] \mapsto [t \cdot z, v]$, $t\in T$ and thus defines a class
in $K_T^0(X_\omega)$.
We write $L_+(p)$ for the corresponding line bundle on $X_+$
and $L_-(p)$ for the corresponding line bundle on $X_-$.
We have
\[
R_i^\pm = L_\pm(D_i) \otimes e^{\lambda_i}
\]
where $e^{\lambda_i}\in \CC[T]$ stands for the irreducible
$T$-representation given by the $i$th projection $T\to \Cstar$.
In particular we have $c_1^T(L_\pm(p)) = \theta_\pm(p)$ for the map
$\theta$ in \eqref{eq:theta}.
Similarly a character $(p,n)\in \Hom(K\times \Cstar, \Cstar) =
\LL^\vee \oplus \ZZ$ defines a $T$-equivariant line
bundle $L(p,n)\to \tX$ and we have:
\begin{align*}
& \tR_i = L(\tD_i)\otimes e^{\lambda_i} & 1\le i\le m \\
& \tR_{m+1} = L(\tD_{m+1}) = L(0,1)
\end{align*}
The classes
$L_\pm(p)$ (respectively the classes $L(p,n)$) generate the equivariant $K$-group
$K^0_T(X_\pm)$ (respectively $K^0_T(\tX)$) over $\ZZ[T]$.
Let $\delta_- \in \cA_-$ be a minimal anticone, $x_{\delta_-}$
be the corresponding $T$-fixed point on $X_-$,
$i_{\delta_-} \colon x_{\delta_-} \to X_-$ be the inclusion
of the fixed point, and $G_{\delta_-}$ be the isotropy group
of $x_{\delta_-}$.
We have that $x_{\delta_-} \cong B G_{\delta_-}$,
and that $i_{\delta_-}^\star R_i = 1$ for $i \in \delta_-$.
A basis for $K^0_T(X_-)$, after inverting non-zero elements
of $\ZZ[T]$, is given by:
\begin{equation}
\label{eq:fixed_point_basis}
\big\{(i_{\delta_-})_\star \varrho :
\text{$\varrho$ an irreducible representation
of $G_{\delta_-}$, $\delta_- \in \cA_-$}\big\}
\end{equation}
We need to specify a $T$-linearization on $(i_{\delta_-})_\star \varrho$.
Choosing a lift $\hvarrho \in \Hom(K,\Cstar) = \LL^\vee$
of each $G_{\delta_-}$-representation $\varrho \colon G_{\delta_-} \to \Cstar$,
we write any element in \eqref{eq:fixed_point_basis} in the form:
\begin{equation*}
\label{eq:fixed_point_basis_using_Koszul}
e_{\delta_-,\varrho} := L_- (\hvarrho)
\prod_{i \not \in \delta_-} \big(1 - S_i^-\big)
\end{equation*}
Then $\{e_{\delta_-,\varrho}\}$ is a basis for the localized $T$-equivariant
$K$-theory of $X_-$. There is an entirely analogous basis
$\{e_{\delta_+,\varrho}\}$ for the localized
$T$-equivariant $K$-theory of $X_+$.
We will describe the action of the Fourier--Mukai transform
in terms of these bases.
\subsubsection{Computing the Fourier--Mukai Transform}
Consider the diagram \eqref{eq:common_blowup} and
the associated Fourier--Mukai transform
$\FM \colon K^0_T(X_-) \to K^0_T(X_+)$. In this section we prove:
\begin{theorem} \label{thm:Fourier--Mukai}
If $\delta_- \in \cA_-$ is a minimal anticone such that $\delta_- \in \cA_+$ then
\[
\FM(e_{\delta_-,\varrho}) = e_{\delta_-,\varrho}
\]
where on the left-hand side of the equality $\delta_-$ is regarded as
a minimal anticone for $X_-$ and on the right-hand side $\delta_-$
is regarded as a minimal anticone for $X_+$.
If $\delta_-$ is a minimal anticone in $\cA_-$
such that $\delta_- \not \in \cA_+$ then
$\FM(e_{\delta_{-},\varrho})$ is equal to
\[
\frac{1}{l}
\sum_{t\in \cT}
\left(
\frac{1-S_{j_{-}}^+}{1-t^{-1} }
\cdot
L_+(\hvarrho) t^{\hvarrho \cdot e}
\cdot
\prod_{\substack{j\notin \delta_{-}\\ D_j \cdot e <0}}
(1-S_j^+)
\cdot
\prod_{\substack{i\notin \delta_{-} \\ D_i\cdot e\ge 0}}
\big(1-t^{-D_i \cdot e}S_i^+\big)
\right)
\]
where $j_-$ is the unique element of $\delta_-$ such
that $D_{j_-} \cdot e<0$, $l = -D_{j_-} \cdot e$ and
\[
\cT =\left\{\zeta \cdot (R_{j_{-}}^+)^{1/l} :
\zeta \in\bmu_l \right\}.
\]
\end{theorem}
\begin{remark}
We have
\[
\frac{1}{l} \sum_{t\in \cT} t^n =
\begin{cases}
(R_{j_-}^+)^{n/l} & \text{if $l$ divides $n$;} \\
0 & \text{otherwise.}
\end{cases}
\]
Thus $\frac{1}{l} \sum_{t\in \cT} f(t)$ makes
sense as an element $K_T^0(X_+)$ for a Laurent polynomial
$f(t)$ in $t$. Note that each summand appearing
in the formula for $\FM(e_{\delta_-,\varrho})$ is in fact a Laurent
polynomial in $t$, since the factor $1-t^{-1}$ divides
$1- S_{j_-}^+ = 1 - t^{-l}$.
\end{remark}
Borisov--Horja have computed how non-equivariant versions of
the classes $R_i^-$ change under
pullback~\cite[Proposition~8.1]{Borisov--Horja:K}.
We have parallel results in the equivariant setting.
\begin{proposition}
\label{pro:pullback}
For $p \in \LL$, we have:
\begin{align*}
f_-^\star(L_-(p)) = L(p, 0)
&& \text{and} &&
f_+^\star(L_+(p)) = L(p,-p\cdot e) \\
\intertext{Let
$k_i := \max(D_i \cdot e,0)$ and $l_i := \max({-D_i \cdot e},0)$.
Then:}
f_-^\star R_i^- = \tR_i \tR_{m+1}^{k_i}
&& \text{and} &&
f_+^\star R_i^+ = \tR_i \tR_{m+1}^{l_i}.
\end{align*}
\end{proposition}
\begin{proof}
These statements follows from the description of $f_\pm \colon
\tX \to X_\pm$ in the proof of Lemma \ref{lem:toricblowup};
see \eqref{eq:morphism_f+} and \eqref{eq:grouphom_f+}.
\end{proof}
We now analyze the push-forward of classes
supported on torus fixed points of $\tX$.
\begin{proposition}
\label{pro:pushforward}
Consider minimal anticones
\begin{align*}
\tdelta &=\{j_1,\ldots,j_{r-1},j_+,j_-\} \in
\cA_\tomega && \text{for $\tX$} \\
\delta_+ &= \{j_1,\ldots,j_{r-1},j_+\} \in \cA_+ && \text{for $X_+$}
\end{align*}
such that $\{j_1,\dots,j_{r-1},j_+,j_-\} \subset \{1,\dots,m\}$,
$D_{j_1} \cdot e = \cdots = D_{j_{r-1}} \cdot e = 0$,
$D_{j_-} \cdot e<0$
and $D_{j_+} \cdot e > 0$.
Let $i_{\tdelta} \colon BG_{\tdelta} \to \tX$ and
$i_{\delta_+} \colon BG_{\delta_+} \to X_+$
denote the inclusions of the corresponding $T$-fixed points
and let $f_{+,\tdelta} \colon BG_{\tdelta} \to B G_{\delta_+}$
denote the map induced on the fixed points:
\[
\xymatrix{
x_\tdelta \ar@{=}[r]& BG_{\tdelta} \ar[r]^{i_\tdelta} \ar[d]_{f_{+,\tdelta}} & \tX \ar[d]^{f_+} \\
x_{\delta_+} \ar@{=}[r]& BG_{\delta_+} \ar[r]^{i_{\delta_+}} & X_+
}
\]
\begin{enumerate}
\item The map $f_{+,\tdelta}$
exhibits
$BG_{\tdelta}$ as a $\bmu_l$-gerbe over $BG_{\delta_+}$,
where $l = {-D_{j_-}} \cdot e$.
\item We have:
\[
(f_{+,\tdelta})_\star (i_{\tdelta})^\star L(p,n) =
\begin{cases}
(i_{\delta_+})^\star L_+(p) (R_{j_{-}}^+)^{(p\cdot e + n)/l}
& \text{if $l$ divides $p\cdot e + n$;} \\
0 & \text{otherwise.}
\end{cases}
\]
\item Let $g$ be a Laurent polynomial in $m+1$ variables. Then:
\[
(f_{+,\tdelta})_\star (i_\tdelta)^\star
L(p,n)
g\big(\tR_1,\ldots,\tR_{m+1}\big)
=
(i_{\delta_+})^\star
\frac{1}{l}
\sum_{t\in \cT}
L_+(p) t^{p\cdot e + n}
g\big(t^{-l_1} R_1^+,\ldots,t^{-l_m} R_m^+,t\big)
\]
\end{enumerate}
\end{proposition}
\begin{proof}
The stabilizers $G_\tdelta$ and $G_{\delta_+}$
are given, as subgroups of $K\times \Cstar$ and $K$,
by
\begin{align*}
G_\tdelta & = \{(g,\lambda) \in K\times \CC^\times :
\text{$D_j(g) \lambda^{-D_j \cdot e} = 1$ for all $j\in \delta_+$,
$D_{j_-}(g) = 1$} \} \\
G_{\delta_+} & = \{ h \in K : \text{$D_j(h) = 1$ for all $j \in \delta_+$}\}
\end{align*}
where we regard $D_j$ as a character of $K$.
The homomorphism $G_\tdelta \to G_{\delta_+}$ is
induced by $\phi_+ \colon (g,\lambda) \mapsto h= g \cdot \lambda^{-e}$
in \eqref{eq:grouphom_f+}.
The kernel of the homomorphism is
$\{(\lambda^{e},\lambda): \lambda\in \bmu_l\}$
and we obtain an exact sequence:
\[
\xymatrix{
1 \ar[r] & \bmu_l \ar[r] & G_{\tdelta} \ar[r] & G_{\delta_+} \ar[r] &1
}
\]
Therefore $f_{+,\tdelta}$ exhibits $BG_{\tdelta}$ as a
$\bmu_l$-gerbe over $BG_{\delta_+}$.
For part (2), notice that $(f_{+,\tdelta})_\star$ maps
a $G_{\tdelta}$-representation to its $\bmu_l$-invariant part.
The character $(p,n)\in \Hom(K\times \Cstar,\Cstar)$ induces
a $\bmu_l$-character $\lambda \mapsto \lambda^{p\cdot e + n}$
via the inclusion
$\bmu_l \subset G_{\tdelta} \subset K\times \Cstar$.
Therefore $(f_{+,\tdelta})_\star (\iota_{\tdelta})^\star L(p,n)$
vanishes if $l$ does not divide $p \cdot e + n$.
On the other hand, Proposition \ref{pro:pullback}
gives $(f_+)^\star R_{j_-}^+ = \tR_{j_-} \tR_{m+1}^l$
and hence, if $l$ divides $p\cdot e +n$,
\begin{align*}
(f_{+,\tdelta})^\star (i_{\delta_+})^\star
L_+(p) (R_{j_-}^+)^{(p\cdot e+n)/l}
& = (i_{\tdelta})^\star L(p, -p\cdot e) (\tR_{j_-})^{(p\cdot e+n)/l}
(\tR_{m+1})^{p\cdot e + n} \\
& = (i_{\tdelta})^\star L(p,-p \cdot e) (\tR_{m+1})^{p\cdot e + n}
= (i_{\tdelta})^\star L(p,n).
\end{align*}
Therefore the Projection Formula gives
$(f_{+,\tdelta})_\star (i_{\tdelta})^\star L(p,n)
= (i_{\delta_+})^\star
L_+(p) (R_{j_-}^+)^{(p\cdot e+n)/l}$.
This proves (2).
For part (3) it suffices to take $g$ to be a monomial:
$g(\tR_1,\dots,\tR_{m+1}) = \prod_{i=1}^{m+1} \tR_i^{n_i}$.
In this case:
\begin{equation}
\label{eq:Lpn_g}
L(p,n) g(\tR_1,\dots,\tR_{m+1}) =
L(p+ \textstyle\sum_{i=1}^{m} n_i D_i,
n + n_{m+1} - \sum_{i=1}^m n_i k_i ) \otimes e^{\sum_{i=1}^m n_i \lambda_i}
\end{equation}
Part (2) can be restated as:
\[
(f_{+,\tdelta})_\star (i_{\tdelta})^\star L(p,n)
=(i_{\delta_+})^\star \frac{1}{l} \sum_{t\in \cT}
L_+(p) t^{p\cdot e +n}
\]
Combining this with \eqref{eq:Lpn_g} yields (3).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:Fourier--Mukai}]
Suppose first that $\delta_- \in \cA_+ \cap \cA_-$.
Then, as discussed, $\varphi$ gives an isomorphism
between neighbourhoods of the fixed points corresponding
to $\delta_-$. Thus $\FM(e_{\delta_-,\varrho}) = e_{\delta_-,\varrho}$.
Suppose now that $\delta_- \in \cA_-$ but $\delta_- \not \in \cA_+$,
so that $\delta_- = \{j_1,\ldots,j_{r-1},j_-\}$ with
$D_{j_1} \cdot e = \cdots = D_{j_{r-1}} \cdot e = 0$
and $D_{j_-} \cdot e < 0$. Proposition~\ref{pro:pullback} gives:
\[
(f_-)^\star e_{\delta_-,\varrho}
L(\hvarrho, 0)
\prod_{i \not \in \delta_-} \big(1 - \tS_{m+1}^{k_i} \tS_i\big)
\]
where the index $i$ in the product satisfies $i\le m$.
This restricts to zero at a fixed point $x_{\tdelta} \in \tX$
unless $x_{\tdelta}$ is in $f_+^{-1}(x_{\delta_-})$, that is,
unless $\tdelta$ has the form $\delta_- \cup \{j_+\}$
with $D_{j_+} \cdot e > 0$.
The Localization Theorem in $T$-equivariant $K$-theory~\cite{Coates--Iritani--Jiang--Segal} gives:
\begin{equation}
\label{eq:actually_a_polynomial}
(f_-)^\star e_{\delta_-,\varrho} =
\sum_{\tdelta}
(i_{\tdelta})_\star (i_{\tdelta})^\star
\left[
\frac{
L(\hvarrho,0)
\prod_{i \not \in \delta_-} \big(1 - \tS_{m+1}^{k_i} \tS_i\big)
}
{(1-\tS_{m+1})
\prod_{j \not \in \delta_-, j \ne j_+} (1-\tS_j)
}
\right]
\end{equation}
where $i$,~$j\le m$ and the sum runs over $\tdelta = \delta_- \cup \{j_+\}$
such that $D_{j_+} \cdot e > 0$.
Restricted to such a $T$-fixed point,
$\tS_{j_+}$ becomes trivial, so the numerator
in \eqref{eq:actually_a_polynomial} contains a factor
$(i_{\tdelta})^\star (1-\tS_{m+1}^{k_{j_+}})$
that is divisible by $(i_{\tdelta})^\star (1-\tS_{m+1})$.
Thus \eqref{eq:actually_a_polynomial} depends polynomially
on $\tS_{m+1}$. Now:
\begin{align*}
(f_+)_\star (f_-)^\star e_{\delta_-,\varrho}
& =
\sum_{\delta_+ : \delta_+ | \delta_-}
(i_{\delta_+})_\star
(f_{+,\tdelta})_\star
(i_\tdelta)^\star
\left[
\frac{
L(\hvarrho, 0)
\prod_{i \not \in \delta_-} \big(1 - \tS_{m+1}^{k_i} \tS_i\big)
}{
(1-\tS_{m+1})
\prod_{j \not \in \delta_-, j \ne j_+} (1-\tS_j)
}
\right] \\
&=
\sum_{\delta_+ : \delta_+ | \delta_-}
(i_{\delta_+})_\star (i_{\delta_+})^\star
\left[ \frac{1}{l} \sum_{t \in \cT}
\frac{
L_+(\hvarrho) t^{\hvarrho\cdot e}
\prod_{i \not \in \delta_-} \big(1 - t^{l_i-k_i} S_i^+\big)
}
{(1-t^{-1}) \prod_{
j \not \in \delta_-, j \ne j_+} (1-t^{l_j} S_j^+) }
\right]
\end{align*}
where we used part (3) of Proposition~\ref{pro:pushforward}.
This is:
\[
\sum_{\delta_+ : \delta_+ | \delta_-}
(i_{\delta_+})_\star (i_{\delta_+})^\star
\left[
\frac{ \frac{1}{l} \sum_{t \in \cT}
\frac{1-S_{j_{-}}^+}{1-t^{-1}}
\cdot
L_+(\hvarrho) t^{\hvarrho\cdot e}
\cdot
\prod_{j \not \in \delta_- }
\big(1 - t^{-k_j} S_j^+\big)
}
{
\prod_{j \not \in \delta_+} (1 - S_j^+)
}
\right].
\]
Applying the Localization Theorem again gives the result.
Here we need to check that the restriction of
\[
\frac{1-S_{j_{-}}^+}{1-t^{-1}}
\cdot
\prod_{j \not \in \delta_-}
\big(1 - t^{-k_j} S_j^+\big)
\]
to the fixed point corresponding to $\delta \in \cA_+ \cap \cA_-$
vanishes. If there exists $j\in \delta$ with $j\notin \delta_-$
and $D_j \cdot e \le 0$ then the restriction vanishes since
$i_\delta^\star S_j^+= 1$.
Otherwise one has $\delta \setminus \delta_- \subset M_+$.
In this case $j_-\in \delta$ and there exists $j_0\in \delta
\cap M_+$. Thus the restriction contains the factor
\[
i_\delta^\star \left[ \frac{1-S_{j_-}^+}{1-t^{-1}}
(1 - t^{-D_{j_0} \cdot e} S_{j_0}^+) \right]
= i_\delta^\star
\left[ (1-S_{j_-}^+) \frac{1-t^{-D_{j_0} \cdot e}}{1-t^{-1}} \right]
\]
which vanishes.
\end{proof}
\subsection{The Fourier--Mukai Transform Matches With Analytic Continuation}
\label{sec:FM_match}
We now show that the analytic continuation formula
in Theorem~\ref{thm:analytic_continuation} matches
with the Fourier--Mukai transform in Theorem~\ref{thm:Fourier--Mukai}.
More precisely we show:
\begin{theorem}
\label{thm:UH-FM}
Let $\UU_H$ be the linear transformation in \S \ref{sec:U}
given by the analytic continuation of $H$-functions.
Then $\UU_H$ induces a map $\UU_H \colon
H^{\bullet\bullet}_{T}(IX_-) \to H^{\bullet\bullet}_{T}(IX_+)$
and the following diagram commutes:
\begin{equation}
\label{eq:FM_UH}
\begin{aligned}
\xymatrix{
K_T^0(X_-) \ar[r]^{\FM} \ar[d]_{\tch} & K_T^0(X_+) \ar[d]_{\tch} \\
H^{\bullet\bullet}_{T}(IX_-) \ar[r]^{\UU_H} & H^{\bullet\bullet}_{T}(IX_+)
}
\end{aligned}
\end{equation}
\end{theorem}
We start by computing the Chern characters
of certain line bundles. It is easy to see that:
\begin{align*}
\tch(L_\pm(\hvarrho)) & = \bigoplus_{f\in \KK_\pm/\LL}
e^{2\pi\tti\hvarrho\cdot f} e^{\theta_\pm(\hvarrho)} \fun_f\\
\tch(S_j^\pm) & = \bigoplus_{f\in \KK_\pm/\LL}
e^{-2\pi\tti D_j \cdot f} e^{-u_j} \fun_f
\end{align*}
In view of this, we define
\[
\tch(t) := \bigoplus_{f\in \KK_+/\LL}
\zeta e^{2\pi\tti D_{j_-} \cdot f/l} e^{u_{j_-}/l}\fun_f
\]
for $t= \zeta (R_{j_-}^+)^{1/l}\in \cT$ appearing
in Theorem \ref{thm:Fourier--Mukai}.
Here we fix lifts $\KK_+/\LL \to \KK_+$,
$\KK_-/\LL \to \KK_-$ as in Notation \ref{not:lift}
and identify $f \in \KK_+/\LL$ with its lift
in $\KK_+$.
\begin{lemma}
\label{lem:chernchar_matching}
Suppose that $(\delta_+,f_+)$ indexes a $T$-fixed point on $X_+$,
that $(\delta_-,f_-)$ indexes a $T$-fixed point on $X_-$,
and that $(\delta_+,f_+)|(\delta_-,f_-)$.
Let $j_-\in \delta_-$ be the unique index such that $D_{j_-} \cdot e<0$
and write $l = {-D_{j_-}} \cdot e$.
Setting $t = e^{- 2\pi\tti D_{j_-} \cdot f_-/l}
(R_{j_-}^+)^{1/l}$, we have:
\begin{align}
\label{eq:t_chern}
\begin{split}
i_{(\delta_+,f_+)}^\star
\tch\left(L_+(\hvarrho) t^{\hvarrho\cdot e}\right) & =
i_{(\delta_-,f_-)}^\star \tch\left(L_-(\hvarrho)\right)
\\
i_{(\delta_+,f_+)}^\star
\tch \left(S_j^+ t^{-D_j \cdot e} \right)
& = i_{(\delta_-,f_-)}^\star
\tch(S_j^-)
\end{split}
\end{align}
We also have:
\begin{align}
\label{eq:ch_e_deltavarrho}
i_{(\delta_+,f_+)}^\star
\tch\left[
L_+(\hvarrho) t^{\hvarrho\cdot e}
\prod_{j\notin \delta_-} (1- t^{-D_j \cdot e} S_j^+)
\right]
& =
i_{(\delta_-,f_-)}^\star
\tch(e_{\delta_-,\varrho}) \\
\intertext{and:}
\label{eq:coefficient_match}
i_{(\delta_+,f_+)}^\star \orbich \left[ \frac{1-S_{j_{-}}^+}{l (1-t^{-1})}
\cdot \prod_{\substack{j\notin \delta_{-}\\
D_j \cdot e <0}}\frac{1-S_j^+}{1-S_j^+t^{{-D_j} \cdot e}}
\right]
& = C_{(\delta_+,f_+)}^{(\delta_-,f_-)}
\end{align}
where $C_{(\delta_+,f_+)}^{(\delta_-,f_-)}$
are the coefficients appearing in Theorem \ref{thm:analytic_continuation}.
\end{lemma}
\begin{proof}
This is just a calculation.
Recall from Notation \ref{not:lift} that
$f_- = f_+ + \alpha e$ for some $\alpha \in \QQ$.
Then $D_j \cdot (f_+ - f_-) = {- \alpha} D_j \cdot e$
and $D_{j_-} \cdot (f_+ - f_-) = l \alpha$.
The formulae \eqref{eq:t_chern} easily follow from
Lemma \ref{lem:weights} and \eqref{eq:weights_theta}.
The formula \eqref{eq:ch_e_deltavarrho} is an easy consequence of
\eqref{eq:t_chern}.
To see \eqref{eq:coefficient_match}, we calculate, using \eqref{eq:t_chern},
\begin{align*}
\text{LHS} & = \frac{1}{l}
\frac{1- e^{-u_{j_-}(\delta_+) - 2\pi \tti(D_{j_-} \cdot f_+)}}
{1- e^{- \frac{1}{l}( u_{j_-}(\delta_+) + 2\pi\tti D_{j_-} \cdot (f_+-f_-))}}
\prod_{\substack{j\notin \delta_-\\ D_j \cdot e<0}}
\frac{1- e^{-u_j(\delta_+) - 2\pi\tti D_j\cdot f_+}}
{1- e^{-u_j(\delta_-) - 2\pi\tti D_j \cdot f_-}} \\
& = \frac{1}{l}
\frac{\sin \pi
\left( \frac{u_{j_-}(\delta_+)}{2\pi\tti} + D_{j_-} \cdot (f_+-f_-)
\right)}
{\sin \frac{\pi}{l} \left( \frac{u_{j_-}(\delta_+)}{2\pi\tti}
+ D_{j_-}\cdot (f_+ - f_-) \right) }
\prod_{\substack{j\notin \delta_- \\ D_j \cdot e<0}}
\frac{\sin \pi \left( \frac{u_j(\delta_+)}{2\pi\tti} + D_j \cdot f_+\right) }
{\sin \pi \left ( \frac{u_j(\delta_-)}{2\pi\tti} + D_j \cdot f_- \right)} \\
& \quad \times e^{
- \frac{1}{2}(1-\frac{1}{l}) (u_{j_-}(\delta_+ ) + 2\pi\tti
D_{j_-}\cdot (f_+-f_-) ) +
\sum_{j\notin \delta_-, D_j\cdot e<0}
\left(
\frac{1}{2}(u_j(\delta_-) - u_j(\delta_+)) + \pi\tti D_j \cdot (f_- - f_+)
\right)}
\end{align*}
where we used the fact that $D_{j_-} \cdot f_- \in \ZZ$.
Using Lemma \ref{lem:weights} again to calculate the
exponential factor, we arrive at the expression for
$C_{(\delta_+,f_+)}^{(\delta_-,f_-)}$
in Theorem \ref{thm:analytic_continuation}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:UH-FM}]
We first show that the commutative diagram holds over $\hS_T$.
Then it follows that $\UU_H$ has a non-equivariant
limit, as $\FM$ does.
Consider the element $e_{\delta,\varrho}
\in K_T^0(X_-)$ with $\delta \in \cA_+ \cap \cA_-$.
Theorem \ref{thm:Fourier--Mukai} and the definition
\eqref{eq:UH} of $\UU_H$ show that
\[
\tch( \FM ( e_{\delta,\varrho}) ) = \tch(e_{\delta,\varrho})
= \UU_H ( \tch(e_{\delta,\varrho})).
\]
Consider now $e_{\delta_-,\varrho}\in K_T^0(X_-)$ for
$\delta_-\in \cA_-\setminus \cA_+$.
It is clear that $\tch(\FM(e_{\delta_-,\varrho}))$ is supported only
on fixed points $x_{(\delta_+,f_+)}\in IX_+$ such that
$\delta_+| \delta_-$.
By the definition \eqref{eq:UH} of $\UU_H$, it suffices
to show that:
\begin{equation}
\label{eq:i_ch_FM}
i_{(\delta_+,f_+)}^\star \tch\left(\FM(e_{\delta_-,\varrho})\right)
= \sum_{\substack{f_-\in \KK_-/\LL: \\
(\delta_+,f_+) | (\delta_-,f_-)}}
C_{\delta_+,f_+}^{\delta_-,f_-} \cdot
i_{(\delta_-,f_-)}^\star \tch(e_{\delta_-,\varrho})
\end{equation}
We may rewrite the result in
Theorem \ref{thm:Fourier--Mukai} as
\begin{equation}
\label{eq:FM_restated}
\FM(e_{\delta_-,\varrho}) =
\frac{1}{l}
\sum_{t\in \cT}
\left(
\frac{1-S_{j_-}^+}{1-t^{-1}}
\prod_{\substack{j\notin \delta_- \\ D_j \cdot e<0}}
\frac{1-S_j^+}{1-t^{-D_j\cdot e} S_j^+}
\cdot
L_+(\hvarrho) t^{\hvarrho\cdot e}
\prod_{i\notin \delta_-}
(1- t^{-D_i \cdot e} S_i^+)
\right)
\end{equation}
We have a one-to-one correspondence between the
index of summation $f_-$ in \eqref{eq:i_ch_FM}
and the index of summation $t\in \cT$ in \eqref{eq:FM_restated}
given by
\[
f_- \ \longleftrightarrow \ t = e^{-2\pi\tti D_{j_-}\cdot f_-/l} (R_{j_-}^+)^{1/l}
\]
where $j_-\in \delta_-$ is the unique element satisfying
$D_{j_-} \cdot e<0$ and $l = - D_{j_-}\cdot e$.
Therefore \eqref{eq:i_ch_FM} follows from \eqref{eq:FM_restated},
\eqref{eq:ch_e_deltavarrho} and \eqref{eq:coefficient_match}.
The Theorem is proved.
\end{proof}
\subsection{Completing the Proof of Theorem \ref{thm:U}}
\label{sec:end_of_the_proof}
Combining the commutative diagrams \eqref{eq:UH_U}
and \eqref{eq:FM_UH}, we obtain the commutative diagram
\eqref{eq:FM_U} in Theorem \ref{thm:U}.
Since the Fourier--Mukai transformation $\FM$ can be defined
non-equivariantly, $\UU$ also admits a non-equivariant limit.
Finally we show that $\UU$ is symplectic, i.e.~that $(\UU(-z) \alpha, \UU(z)\beta)
= (\alpha,\beta)$ for all $\alpha$,~$\beta$.
Since $\FM$ is induced by an equivalence of derived categories~\cite{Coates--Iritani--Jiang--Segal},
it preserves the Euler pairing $\chi(E,F)$
given in \eqref{eq:Euler_pairing}.
The proof of Proposition \ref{prop:K_framing_pairing}
shows that the vertical maps $\tPsi_\pm$ in \eqref{eq:FM_U}
preserve the pairing in the sense that:
\[
\left(\tPsi_\pm(E)|_{z\to e^{-\pi\tti} z}, \tPsi_\pm(F) \right)
= \chi_z(E,F).
\]
The commutative diagram \eqref{eq:FM_U} now shows that
$\UU$ is symplectic. This completes the proof of Theorem~\ref{thm:U}.
\begin{remark}
The reader who would prefer to prove that the transformation $\UU$ is symplectic without using the machinery of derived categories can argue as follows. It suffices to show
that the transformation $\FM$ preserves the Euler pairing on the equivariant $K$-groups.
Using Grothendieck duality, one finds that the adjoint of $\FM$
with respect to the Euler pairing is given by
\[
\FM^*(\alpha) = f_{-\star} ( (f_-^\star K_-^{-1})\tK \otimes f_+^\star \alpha)
\]
(see, for example,~\cite[Lemma 1.2]{Bondal--Orlov:semiorth})
where $K_-$ and $\tK$ are respectively the canonical line bundles
of $X_-$ and $\tX$.
It suffices to see that $\FM^*$ corresponds to
the analytic continuation along the path inverse
to $\gamma$ in Figure \ref{fig:path_ancont}.
Consider the Fourier--Mukai transformation in the
reverse direction:
\[
\FM' = f_{-\star} f_+^\star \colon K_T^0(X_+) \to K_T^0(X_-)
\]
Exchanging the roles of $X_+$ and $X_-$ in Theorem \ref{thm:U},
we find that $\FM'$ corresponds to the analytic continuation
along the path $\gamma'$ in Figure \ref{fig:path_rev_ancont}.
We claim that the difference between the paths $\gamma^{-1}$
and $\gamma'$ exactly matches with the difference between
the Fourier--Mukai transformations $\FM^*$ and $\FM'$.
Using $\tK = \tS_1 \cdots \tS_{m+1}$, $K_- = S_1^- \cdots S_m^-$
and Proposition \ref{pro:pullback}, we have
\[
(f_-^\star K_-^{-1})\tK = L(0,w)
= (f_+^\star L_+(-w p)) \otimes (f_-^\star L_-(wp))
\]
for $p\in \LL$ with $p\cdot e = 1$. Therefore
\[
\FM^* = L_-(wp) \circ \FM' \circ L_+(-wp).
\]
On the other hand, one can check that the monodromy of the $K$-theoretic flat
section $\frs(E)$ with respect to the shift $T_{-2\pi\tti w p} \colon
\log \sfy^d \mapsto \log\sfy^d - 2\pi\tti w (p\cdot d)$
corresponds to $E \mapsto L_\pm(wp)\otimes E$
(cf.~the Galois action in \cite[Proposition 2.10 and equation~61]{Iritani}),
and we have the equality
\[
\gamma^{-1} = T_{-2\pi\tti w p} \circ \gamma' \circ T_{2\pi\tti w p}
\]
in the fundamental groupoid of $\{\sfy^e \in \Cstar : \sfy^e \neq \frc\}$. This shows that $\FM^*$ corresponds to the analytic continuation along $\gamma^{-1}$.
\begin{figure}[htbp]
\centering
\includegraphics[bb=131 545 483 729]{path_rev_ancont.pdf}
\caption{The paths $\gamma'$ and $\gamma^{-1}$}
\label{fig:path_rev_ancont}
\end{figure}
\end{remark}
\section{Toric Complete Intersections}
\label{sec:CRC_ci}
We now turn to the Crepant Transformation Conjecture for toric complete intersections. Consider toric Deligne--Mumford stacks $X_\pm$ of the form $\big[\CC^m /\!\!/_\omega K \big]$, where $K = (\Cstar)^r$ is a complex torus, and consider a $K$-equivalence $\varphi \colon X_+ \dashrightarrow X_-$ determined by a wall-crossing in the space of stability conditions~$\omega$ as in~\S\ref{sec:wall-crossing}. We use notation as there, so that $\LL = \Hom(\Cstar,K)$ is the lattice of cocharacters of $K$; the space of stability conditions is $\LL^\vee \otimes \RR$; and the birational map $\varphi$ is induced by the wall-crossing from a chamber $C_+ \subset \LL^\vee \otimes \RR$ to a chamber $C_- \subset \LL^\vee \otimes \RR$, where $C_+$ and $C_-$ are separated by a wall $W$. Consider characters $E_1,\ldots,E_k$ of $K$ such that:
\begin{equation}
\label{eq:conditions_on_line_bundles}
\parbox{0.92\textwidth}{
\begin{itemize}
\item each $E_i$ lies in $W \cap \overline{C_+} = W \cap \overline{C_-}$;
\item for each $i$, the line bundle $L_{X_+}(E_i) \to X_+$ corresponding to $E_i$ is a pull-back from the coarse moduli space $|X_+|$;
\item for each $i$, the line bundle $L_{X_-}(E_i) \to X_-$ corresponding to $E_i$ is a pull-back from the coarse moduli space $|X_-|$;
\end{itemize}
}
\end{equation}
where $L_{X_\pm}(E_i)$ are the line bundles on $X_\pm$ associated to the character $E_i$ in~\S\ref{sec:basis_K-theory}. Let:
\begin{align*}
E_+ := \bigoplus_{i=1}^k L_{X_+}(E_i) &&
E_- := \bigoplus_{i=1}^k L_{X_-}(E_i)
\end{align*}
Let $s_+$,~$s_-$ be regular sections of, respectively, the vector bundles $E_+ \to X_+$ and $E_- \to X_-$ such that:
\begin{itemize}
\item $s_+$ and $s_-$ are compatible via $\varphi \colon X_+ \dashrightarrow X_-$;
\item the zero loci of $s_\pm$ intersect the flopping locus of $\varphi$ transversely;
\end{itemize}
and let $Y_+ \subset X_+$, $Y_- \subset X_-$ be the complete intersection substacks defined by $s_+$,~$s_-$. The birational transformation $\varphi$ then induces a $K$-equivalence $\varphi \colon Y_+ \dashrightarrow Y_-$. In this section we establish the Crepant Transformation Conjecture for $\varphi \colon Y_+ \dashrightarrow Y_-$.
\subsection{The Ambient Part of Quantum Cohomology}
\label{sec:ambient_part}
Under our standing hypotheses on the ambient toric stacks $X_\pm$, the complete intersections $Y_\pm$ automatically have semi-projective coarse moduli spaces, and so the (non-equivariant) quantum products on $H_{\CR}^\bullet(Y_\pm)$ are well-defined. Thus we have a well-defined quantum connection
\begin{equation}
\label{eq:qconn_non_equivariant}
\nabla = d + z^{-1} \sum_{i=0}^N (\phi_i \star_\tau) d\tau^i
\end{equation}
where $\star_\tau$ is the non-equivariant big quantum product, defined exactly as in~\eqref{eq:qprod_pushforward}. This is a pencil $\nabla$ of flat connections on the trivial $H_{\CR}^\bullet(Y_\pm)$-bundle over an open set in $H_{\CR}^\bullet(Y_\pm)$; here, as in the equivariant case, $z \in \Cstar$ is the pencil variable, $\tau \in H_{\CR}^\bullet(Y_\pm)$ is the co-ordinate on the base of the bundle, $\phi_0,\dots,\phi_N$ are a basis for $H_{\CR}^\bullet(Y_\pm)$, and $\tau^0,\dots, \tau^N$ are the corresponding co-ordinates of $\tau \in H_{\CR}^\bullet(Y_\pm)$, so that $\tau = \sum_{i=0}^N \tau^i \phi_i$. We consider now a similar structure on the \emph{ambient part} of $H_{\CR}^\bullet(Y_\pm)$, that is, on:
\[
H_{\amb}^\bullet(Y_\pm) := \im \iota^\star_\pm \subset H_{\CR}^\bullet(Y_\pm)
\]
where $\iota_\pm \colon Y_\pm \to X_\pm$ are the inclusion maps. If $\tau \in H_{\amb}^\bullet(Y_\pm)$ then the big quantum product $\star_\tau$ preserves $H_{\amb}^\bullet(Y_\pm)$~\cite[Corollary~2.5]{Iritani:periods}, and so \eqref{eq:qconn_non_equivariant} restricts to give a well-defined quantum connection on the ambient part of $H_{\CR}^\bullet(Y_\pm)$. The restriction of the fundamental solution $L_\pm(\tau,z)$ for \eqref{eq:qconn_non_equivariant}, defined exactly as in \eqref{eq:L_descendant}, gives a fundamental solution $L^\amb_\pm(\tau,z)$ for the quantum connection on the ambient part.
There is also an ambient part of $K^0(Y_\pm)$, given by $K^0_\amb(Y_\pm) := \im \iota_\pm^\star$, and an ambient $K$-group framing (cf.~Definition~\ref{def:K_framing})
\[
\frs \colon K_\amb^0(Y_\pm) \to
H^\bullet_{\amb}(Y_\pm)\otimes \CC[\log z](\!(z^{-1/k})\!)[\![Q,\tau]\!]
\]
given by
\[
\frs(E)(\tau,z) = \frac{1}{(2\pi)^{\dim {Y_\pm}/2}}
L^\amb_\pm(\tau,z) z^{-\mu} z^\rho \left( \hGamma_{Y_\pm} \cup
(2\pi\tti)^{\frac{\deg_0}{2}} \inv^* \tch(E) \right)
\]
where $\mu$ and $\rho$ are the grading operator and first Chern class for $Y_\pm$, $k \in \N$ is such that the eigenvalues of $k \mu$ are integers, and $\hGamma_{Y_\pm}$ is the non-equivariant $\hGamma$-class of $Y_\pm$. As in~\S\ref{sec:integral_structure}, the image of $\frs$ is contained in the space of flat sections for the quantum connection on the ambient part of $H_{\CR}^\bullet(Y_\pm)$ which are homogeneous of degree zero.
\subsection{$I$-Functions for Toric Complete Intersections}
Recall from~\S\ref{sec:I_function} that the GIT data for $X_+$ determine a cohomology-valued hypergeometric function $I_+$. The $I$-function $I_{X_+} := I_+$ is a multi-valued function of $\sfy_1,\ldots,\sfy_r$, depending analytically on $\sfy_r$ and formally on $\sfy_1,\ldots\sfy_{r-1}$, defined near the large-radius limit point $(\sfy_1,\ldots,\sfy_r) = (0,\ldots,0)$ in $\hcM_{\rm reg}$. The GIT data for the total space of $E_+^\vee$ (regarded as a non-compact toric stack) is obtained from the GIT data for $X_+$ by adding extra toric divisors ${-E_1},\ldots,{-E_k}$. It is easy to see that the corresponding $I$-function $I_{E_+^\vee}$ is also a multi-valued function of $\sfy_1,\ldots,\sfy_r$, depending analytically on $\sfy_r$ and formally on $\sfy_1,\ldots\sfy_{r-1}$, which is defined near the same large-radius limit point $(\sfy_1,\ldots,\sfy_r) = (0,\ldots,0)$ in $\hcM_{\rm reg}$. The global quantum connections for $X_+$ and $E_+^\vee$ were constructed, in \S\ref{sec:global_qconn}, using the $I$-functions $I_{X_+}$ and $I_{E_+^\vee}$. We now introduce a closely-related $I$-function, defined in terms of GIT data for $X_+$ and the characters $E_1,\ldots,E_k$, that will allow us to globalize the quantum connection on the ambient part of $H^\bullet_\CR(Y_\pm)$.
With notation as in~\S\ref{sec:I_function}, except with $u_i$ now denoting the non-equivariant class Poincar\'e-dual to the $i$th toric divisor \eqref{eq:T-invariant_divisor} and with $v_j \in H^2(X_+)$, $1 \leq j \leq k$, given by the non-equivariant first Chern class of the line bundle corresponding to the character $E_j$, define a $H_{\CR}^\bullet(X_+)$-valued hypergeometric series
$I^{\rm temp}_{X_+,Y_+}(\sigma,x,z)
\in H^\bullet_{\CR}(X_+) \otimes\CC(\!(z^{-1})\!)[\![Q,\sigma,x]\!]$ by:
\begin{multline*}
I^{\rm temp}_{X_+,Y_+}(\sigma,x,z) = z e^{\sigma/z}
\sum_{d\in \KK} e^{\sigma\cdot \overline{d}}
Q^{\overline{d}} \prod_{j\in S} x_j^{D_j\cdot d}
\left(
\prod_{j=1}^{m}
\frac{\prod_{a : \<a\>=\< D_j\cdot d \>, a \leq 0}(u_j+a z)}
{\prod_{a : \<a \>=\< D_j \cdot d \>, a\leq D_j \cdot d} (u_j+a z)}
\right) \\
\times
\left(
\prod_{j=1}^{k}
\prod_{a=1}^{E_j \cdot d} (v_j+a z)
\right)
\fun_{[{-d}]}
\end{multline*}
Note that for each $d \in \KK$ and each $j \in \{1,2,\ldots,k\}$, $E_j \cdot d$ is a non-negative integer. (The subscript `temp' here again reflects the fact that this notation for the $I$-function is only temporary: we are just about to change notation, by specializing certain parameters.) Under our hypotheses \eqref{eq:conditions_on_line_bundles} on the line bundles $L_{X_+}(E_j)$, we have a Mirror Theorem for the toric complete intersection $Y_+$:
\begin{theorem}[\!\!\cite{CCIT:applications}]
\label{thm:ci_mirror_thm}
$\iota_+^\star I^{\rm temp}_{X_+,Y_+}(\sigma,x,-z)$ is an
$\CC[\![Q,\sigma,x]\!]$-valued point on $\cL_{Y_+}$.
\end{theorem}
We define the $I$-function $I_{X_+,Y_+}$ to be the function obtained from $I^{\rm temp}_{X_+,Y_+}$ by the specialization $Q=1$, $\sigma = \sigma_+ :=
\theta_+( \sum_{i=1}^r \sfp^+_i \log \sfy_i)$ where $\theta_+$ is as in \eqref{eq:theta}. Thus:
\[
I_{X_+,Y_+}(\sfy,z):=
z e^{\sigma_+/z}
\sum_{d \in \KK_+}
\sfy^d
\left(
\prod_{j=1}^{m}
\frac{\prod_{a : \<a\>=\< D_j\cdot d \>, a \leq 0}(u_j+a z)}
{\prod_{a : \<a \>=\< D_j \cdot d \>, a\leq D_j \cdot d} (u_j+a z)}
\right)
\left(
\prod_{j=1}^{k}
\prod_{a=1}^{E_j \cdot d} (v_j+a z)
\right)
\fun_{[{-d}]}
\]
where $(\sfy_1,\dots,\sfy_r)$ are as in~\S\ref{sec:I_function}. Repeating the analysis in Lemma~\ref{lem:I_analyticity} shows that $I_{X_+,Y_+}$, just like $I_{X_+}$ and $I_{E_+^\vee}$, is a multi-valued function of $\sfy_1,\ldots,\sfy_r$ that depends analytically on $\sfy_r$ and formally on $\sfy_1,\ldots\sfy_{r-1}$, defined near the large-radius limit point $(\sfy_1,\ldots,\sfy_r) = (0,\ldots,0)$ in $\hcM_{\rm reg}$. The arguments in~\S\ref{sec:global_qconn} can now be applied verbatim to $I_{Y_+} := \iota_+^\star I_{X_+,Y_+}$, and thus we construct a global version of the quantum connection on the ambient part $H^\bullet_\amb(Y_+)$, defined over the base $\tcM_+^\circ$. The analog of Theorem~\ref{thm:global_qconn} holds, with the same proof:
\begin{theorem}
\label{thm:global_qconn_ci}
There exist the following data:
\begin{itemize}
\item an open subset $\cU_+^\circ\subset \cU_+$ such that
$P_+ \in \cU_+^\circ$ and that the complement
$\cU_+ \setminus \cU_+^\circ$ is a discrete set;
we write $\tcM_+^\circ = \tcM_+|_{\cU_+^\circ}$;
\item a trivial $H_{\amb}^\bullet(Y_+)$-bundle
$\bF^+$ over $\tcM_+^\circ(\CC[z])$:
\[
\bF^+ = H_{\amb}^\bullet(Y_+)\otimes
\cO_{\cU^\circ_+}[z][\![\sfy_1,\dots,\sfy_{r-1}]\!];
\]
\item a flat connection
$\bnabla^+ = d + z^{-1} \bA^+(\sfy)$
on $\bF^+$ of the form:
\[
\bA^+(\sfy) = \sum_{i=1}^{\ell_+} B_i(\sfy) \frac{dy_i}{y_i}
+ \sum_{j\in S_+} C_j(\sfy) dx_j
\]
with $B_i(\sfy), C_j(\sfy)
\in
\End(H_{\amb}^\bullet(Y_+)) \otimes
\cO_{\cU^\circ_+}[\![\sfy_1,\dots,\sfy_{r-1}]\!]$;
\item a vector field $\bE^+$ on $\tcM_+$, called the Euler vector field, defined by:
\[
\bE^+ = \sum_{i=1}^r \frac{1}{2}(\deg \sfy_i) \sfy_i
\parfrac{}{\sfy_i};
\]
\item a mirror map $\tau_+ \colon \tcM_+ \to H_{\amb}^\bullet(Y_+)$
of the form:
\begin{align*}
\tau_+ = \iota_+^\star \sigma_+ + \ttau_+ &&&
\ttau_+\in H^\bullet_{\amb}(Y_+)\otimes\cO_{\cU^\circ_+}[\![\sfy_1,\dots,\sfy_{r-1}]\!] \\
&&& \ttau_+|_{\sfy_1=\cdots = \sfy_r=0}= 0
\end{align*}
\end{itemize}
such that $\bnabla^+$ equals the pull-back $\tau_+^*\nabla^+$ of the
(non-equivariant) quantum connection $\nabla^+$ on the ambient part of $H^\bullet_{\CR}(Y_+)$ by $\tau_+$,
that is:
\begin{align*}
B_i(\sfy) & = \sum_{k=0}^N \parfrac{\tau_+^k(\sfy)}
{\log y_i} (\phi_k \star_{\tau_+(\sfy)}) &&
1\le i\le \ell_+ \\
C_j(\sfy) & =
\sum_{k=0}^N \parfrac{\ttau_+^k(\sfy)}
{x_j} (\phi_k\star_{\tau_+(\sfy)}) &&
j\in S_+
\end{align*}
and that the push-forward of $\bE^+$ by $\tau_+$
is the (non-equivariant) Euler vector field $\cE^+$ on the ambient part $H^\bullet_{\amb}(Y_+)$.
Moreover, there exists a global section $\Upsilon^+_0(\sfy,z)$ of
$\bF^+$ such that
\[
I_{Y_+}(\sfy,z) = z L_+^\amb(\tau_+(\sfy),z)^{-1} \Upsilon^+_0(\sfy,z)
\]
where $L_+^\amb(\tau,z)$ is the ambient fundamental solution from \S\ref{sec:ambient_part}
\end{theorem}
\begin{remark}
Here, as in Theorem~\ref{thm:global_qconn}, the Novikov variable $Q$ has been specialized to $1$.
\end{remark}
\begin{remark}
Entirely parallel results hold for $Y_-$.
\end{remark}
\subsection{Analytic Continuation of $I$-Functions}
\label{sec:ci_analytic_continuation}
To prove the Crepant Transformation Conjecture in this context, we need to establish the analog of Theorem~\ref{thm:U}. To do this, we will compare the analytic continuation of the $I$-functions $I_{X_\pm,Y_\pm}$ with the analytic continuation of $I_{E_\pm^\vee}$. Let $T = (\Cstar)^m$ denote the torus acting on $X_\pm$, and $\widetilde{T} = (\Cstar)^{m+k}$ denote the torus acting on $E_\pm^\vee$. The splitting $\widetilde{T} = T \times (\Cstar)^k$ gives $R_{\widetilde{T}} = R_T [\kappa_1,\ldots,\kappa_k]$ where $\kappa_j$, $1 \leq j \leq k$, is the character of $(\Cstar)^k$ given by projection to the $j$th factor of the product $(\Cstar)^k$. We regard $\widetilde{T}$ as acting on $X_\pm$ via the given action of $T \subset \widetilde{T}$ and the trivial action of $(\Cstar)^k \subset \widetilde{T}$, so that:
\begin{align*}
\ZZ[\widetilde{T}] = \ZZ[T] [e^{\pm\kappa_1},\ldots,e^{\pm\kappa_k}]
&& \text{and} &&
K^0_{\widetilde{T}}(X_\pm)
= K^0_T(X_\pm) \otimes_{\ZZ[T]} \ZZ[\widetilde{T}]
\end{align*}
\begin{lemma}
\label{lem:FM_E_X}
The Fourier--Mukai transformations
\begin{align*}
\FM \colon K^0(X_-) \to K^0(X_+) &&
\FM \colon K^0(E_-^\vee) \to K^0(E_+^\vee)
\end{align*}
coincide under the natural identification of $K^0(X_\pm)$ with $K^0(E_\pm^\vee)$. The same statement holds equivariantly.
\end{lemma}
\begin{proof}
Consider the fiber diagram:
\[
\xymatrix{
& \widetilde{E}^\vee \ar[ld] \ar[rd] \ar[d] \\
E_-^\vee\ar[d] &\widetilde{X} \ar[ld]^-{f_-} \ar[rd]_-{f_+} & E_+^\vee \ar[d]\\
X_- && X_+
}
\]
where the bottom triangle is \eqref{eq:common_blowup} and the top triangle is the analog of \eqref{eq:common_blowup} for $E_\pm^\vee$,
and apply the flat base change theorem.
\end{proof}
Let $\UU_{E^\vee}$ be the symplectic transformation from Theorem~\ref{thm:U} applied to $E_\pm^\vee$.
Combining Lemma~\ref{lem:FM_E_X} with Theorem~\ref{thm:U} gives a commutative diagram:
\begin{equation}
\label{eq:U_is_independent_of_kappa_1}
\begin{aligned}
\xymatrix{
K^0_{\widetilde{T}}(X_-) \ar@{=}[d] \ar[r]^{\FM} & K^0_{\widetilde{T}}(X_+) \ar@{=}[d] \\
K^0_{\widetilde{T}}(E_-^\vee) \ar[d]_{z^{-\mu_-} z^{\rho_-} \hGamma_{E_-^\vee} \cup (2\pi\tti)^{\frac{\deg_0}{2}} \inv^* \tch({-})} \ar[r]^{\FM} &
K^0_{\widetilde{T}}(E_+^\vee) \ar[d]^{z^{-\mu_+} z^{\rho_+} \hGamma_{E_+^\vee} \cup (2\pi\tti)^{\frac{\deg_0}{2}} \inv^* \tch({-})} \\
\tcH(E_-^\vee) \ar[r]^{\UU_{E^\vee}} & \tcH(E_+^\vee)
}
\end{aligned}
\end{equation}
where $\rho_\pm \in H^2_{\widetilde{T}}(E_\pm^\vee)$ is the $\widetilde{T}$-equivariant first Chern class of $E_\pm^\vee$ and $\mu_\pm$ are the $\widetilde{T}$-equivariant grading operators. Recall that
\begin{align*}
\Gamma_{E_\pm^\vee} = \Gamma_{X_\pm} \Gamma(E_\pm^\vee)
&& \rho_\pm = \rho_{X_\pm} + c_1^{\widetilde{T}} (E_\pm^\vee)
\end{align*}
and that the Chern roots of $E_\pm^\vee$ are pulled back from the common blow-down $\overline{X}_0$ of $X_\pm$. Part~(2) of Theorem~\ref{thm:U} thus implies that we can factor out the contributions of $\Gamma(E_\pm^\vee)$ and $c_1^{\widetilde{T}} (E_\pm^\vee)$ from the vertical maps in \eqref{eq:U_is_independent_of_kappa_1}, replacing the vertical arrows by:
\[
z^{-\mu_{X_\pm}} z^{\rho_{X_\pm}} \hGamma_{X_\pm} \cup (2\pi\tti)^{\frac{\deg_0}{2}} \inv^* \tch({-})
\]
This proves:
\begin{lemma}
The transformations $\UU_X \colon \cH(X_-) \to \cH(X_+)$ and $\UU_{E^\vee} \colon \cH(E_-^\vee) \to \cH(E_+^\vee)$ coincide under the natural identifications of $\cH(X_\pm)$ with $\cH(E_\pm^\vee)$. In particular, $\UU_{E^\vee}$ is independent of $\kappa_1,\ldots,\kappa_k$.
\end{lemma}
The $I$-functions $I_{X_+,Y_+}$ and $I_{E_+^\vee}$ are related\footnote{An analogous relationship holds between $I_{X_-,Y_-}$ and $I_{E_-^\vee}$.} by:
\[
I_{E_+^\vee}(\sfy)\Big|_{\lambda=0,\kappa={-z}} =
e^{\pi \tti c_1(E_+^\vee)/z} I_{X_+,Y_+}(\pm\sfy)
\]
where the subscript on the left-hand side denotes the specialization:
\begin{equation}
\label{eq:specialization}
\begin{cases}
\lambda_i=0 & 1 \leq i \leq m \\
\kappa_j = {-z} & 1 \leq j \leq k
\end{cases}
\end{equation}
and the $\pm$ on the right-hand side denotes the change of variables:
\begin{align}
\label{eq:pm}
&\log \sfy_i \mapsto \log \sfy_i - \pi \tti \sum_{j=1}^k l_{ij} && 1 \leq i \leq r
&& \text{with} && E_j =\sum_{i=1}^r l_{ij} \sfp_i
\end{align}
The specialization \eqref{eq:specialization} is given by a shift $\bbS \colon \kappa_j \mapsto \kappa_j - z$ in the equivariant parameters followed by passing to the non-equivariant limit. Note that the change of variables \eqref{eq:pm} maps $\sfy^d$ to $(-1)^{{-c_1}(E_+^\vee) \cdot d} \sfy^d$.
Recall from Theorem~\ref{thm:U} that, after analytic continuation, we have $I_{E_+^\vee} = \UU_{E^\vee} I_{E_-^\vee}$. Since $\UU_{E^\vee}$ is independent of $\kappa_j$,~$1 \leq j \leq k$, it follows that $\UU_{E^\vee}$ commutes with the shift $\bbS$. Since the Chern roots of $E^\vee$ are pulled back from the common blow-down $\overline{X}_0$ of $X_\pm$, it follows that{
\[
\UU_{E^\vee} \, e^{\pi \tti c_1(E_-^\vee)/z} = e^{\pi \tti c_1(E_+^\vee)/z} \, \UU_{E^\vee}
\]
Setting $\lambda = 0$ and $\kappa_j = {-z}$ in the equality $I_{E_+^\vee} = \UU_{E^\vee} I_{E_-^\vee}$, and replacing $\cH(E_\pm^\vee)$ and $\UU_{E^\vee}$ with their non-equivariant limits
\begin{align*}
\cH(E_\pm^\vee) := H^\bullet_\CR(E_\pm^\vee) \otimes \CC(\!(z^{-1})\!) && \text{and} &&
\UU_{E^\vee} \colon \cH(E_-^\vee) \to \cH(E_+^\vee)
\end{align*}
we find that
\[
I_{X_+,Y_+} = \UU_{E^\vee} I_{X_-,Y_-}
\]
after analytic continuation. Thus:
\[
I_{X_+,Y_+} = \UU_{X} I_{X_-,Y_-}
\]
after analytic continuation.
\subsection{Compatibility of Fourier--Mukai Transformations}
\label{sec:FM_compatibility}
For the analogue of part~(3) of Theorem~\ref{thm:U}, we need to compare the Fourier--Mukai transformation associated to $X_+ \dashrightarrow X_-$ with the Fourier--Mukai transformation associated to $Y_+ \dashrightarrow Y_-$. This is a base change question (cf.~Lemma~\ref{lem:FM_E_X}), but this time we do not have flatness. By assumption, we have:
\begin{equation}
\label{eq:FM_Y_setup}
\begin{aligned}
\xymatrix{
& \widetilde{Y} \ar[ld]_-{F_-} \ar[rd]^-{F_+} \ar[d]_-{\tilde{\iota}} \\
Y_-\ar[d]_{\iota_-} &\widetilde{X} \ar[ld]^-{f_-} \ar[rd]_-{f_+} & Y_+ \ar[d]^{\iota_+}\\
X_- && X_+
}
\end{aligned}
\end{equation}
where the vertical maps are inclusions, the bottom triangle is \eqref{eq:common_blowup} and the top triangle is the analog of \eqref{eq:common_blowup} for $Y_\pm$. The substacks $\tilde{Y}$ is defined by the vanishing of a section $\tilde{s} \colon \widetilde{X} \to \widetilde{E}$, where $\widetilde{E} \to \widetilde{X}$ is the direct sum of line bundles
\[
\widetilde{E} := \bigoplus_{i=1}^k L_{\widetilde{X}}(E_i)
\]
The line bundles $E_-$,~$\widetilde{E}$, and~$E_+$ are all canonically identified via $f_-^\star$ and $f_+^\star$, since they are all pulled back from the common blow-down $\overline{X}_0$ of $X_\pm$. The section $\widetilde{s}$ coincides both with the pullback of the section $s_+$ via $f_+$ and with the pullback of the section $s_-$ via $f_-$. Since the zero loci of $s_\pm$ are assumed to intersect the flopping locus transversely, $\tilde{s}$ is a regular section of $\widetilde{E}$ and the substack $\widetilde{Y} \subset \widetilde{X}$ is smooth.
\begin{lemma}
\label{lem:FM_Y_X}
The following diagram commutes:
\begin{equation}
\label{eq:FM_Y_commutes}
\begin{aligned}
\xymatrix{
K^0(X_-) \ar[r]^{\FM} \ar[d]_{\iota_-^\star} & K^0(X_+) \ar[d]^{\iota_+^\star} \\
K^0(Y_-)_\amb \ar[r]^{\FM} & K^0(Y_+)_\amb
}
\end{aligned}
\end{equation}
where the top horizontal arrow is the Fourier--Mukai transformation $(f_+)_\star (f_-)^\star$ from~\eqref{eq:FM_Y_setup}, and the bottom horizontal arrow is the Fourier--Mukai transformation $(F_+)_\star (F_-)^\star$ from~\eqref{eq:FM_Y_setup}.
\end{lemma}
\begin{proof}
The pullback along $f_+$ of the Koszul resolution of $\cO_{Y_+}$ in $X_+$ gives the Koszul resolution of $\cO_{\widetilde{Y}}$ in $\widetilde{X}$. This implies that, in the right-hand square in \eqref{eq:FM_Y_setup}, $\widetilde{X}$ and $Y_+$ are Tor-independent over $X_+$~\cite[Tag 08IA]{stacks-project}. Tor-independent base-change~\cite[Tag 08IB]{stacks-project} now implies that:
\[
(F_+)_\star \circ \tilde{\iota}^\star = \iota_+^\star \circ (f_+)_\star
\]
Since $F_-^\star \circ \iota_-^\star = \tilde{\iota}^\star \circ f_-^\star$, it follows that
\[
(\iota_+)^\star (f_+)_\star (f_-)^\star = (F_+)_\star (F_-)^\star (\iota_-)_\star
\]
which is the result.
\end{proof}
\begin{remark}
This argument in fact proves that the analog of diagram~\eqref{eq:FM_Y_commutes} for derived categories is commutative, but we only need the statement at the level of $K$-theory.
\end{remark}
\subsection{Completing the Proof} Denote by $\UU_X$ the transformation from the non-equivariant version of Theorem~\ref{thm:U} applied to $X_\pm$. This is a map $\UU_X \colon \cH(X_-) \to \cH(X_+)$
between the non-equivariant Givental spaces for $X_\pm$:
\[
\cH(X_\pm) := H^\bullet_\CR(X_\pm) \otimes \CC(\!(z^{-1})\!)
\]
Let us remark again that the Chern roots of $E_\pm$ are pulled back from the common blow-down $\overline{X}_0$ of $X_\pm$; the second part of Theorem~\ref{thm:U} therefore gives:
\begin{equation}
\label{eq:intertwiner_U_X}
\UU_X \hGamma(E_-) = \hGamma(E_+) \UU_X
\end{equation}
The results from \S\ref{sec:ci_analytic_continuation} and \S\ref{sec:FM_compatibility} combine to give a commutative diagram:
\[
\xymatrix{
K^0(X_-) \ar[rr]^{\FM} \ar[rdd] \ar[ddd]_{\iota_-^\star} && K^0(X_+) \ar[rdd] \ar'[dd]_-{\iota_+^\star}[ddd]\\
\\
& \tcH(X_-) \ar[rr]^<<<<<<<<<{\UU_X} \ar[ddd]_{\iota_-^\star} && \tcH(X_+) \ar[ddd]_{\iota_+^\star}\\
K^0(Y_-)_\amb \ar'[r]^-{\FM}[rr] \ar[rdd] && K^0(Y_+)_\amb \ar[rdd] \\
\\
& \tcH(Y_-)_\amb \ar@{..>}[rr] && \tcH(Y_+)_\amb \\
}
\]
where $\tcH(Y_\pm)_\amb$ is the ambient part of the multi-valued Givental space:
\begin{equation}
\label{eq:multi-valued_ambient}
\tcH(Y_\pm)_\amb := H_{\amb}^\bullet(Y_\pm) \otimes \CC[\log z](\!(z^{-1/k})\!)
\end{equation}
with $k$ as in the statement of Theorem~\ref{thm:U}, and:
\begin{itemize}
\item the top diagonal maps are the $K$-theory framing maps from Definition~\ref{def:K_framing} but with $\hGamma_{X_\pm}$ replaced by $\hGamma_{X_\pm,Y_\pm} := \hGamma_{X_\pm} \hGamma(E_\pm)^{-1}$;
\item the bottom diagonal maps are the ambient $K$-group framing maps from \S\ref{sec:ambient_part}.
\end{itemize}
Here:
\begin{itemize}
\item the top face is commutative, by Theorem~\ref{thm:U} and \eqref{eq:intertwiner_U_X};
\item the back face is commutative, by Lemma~\ref{lem:FM_Y_X};
\item the sides are commutative, by the definition of the framing maps;
\end{itemize}
and we want to define the dotted arrow so that all faces commute. Define $\UU_Y \colon \tcH(Y_-)_\amb \to \tcH(Y_+)_\amb$ to be the unique map such that the bottom face commutes. Chasing diagrams shows that the front face commutes also. Since $I_{X_+,Y_+} = \UU_X I_{X_-,Y_-}$ after analytic continuation and since $I_{Y_\pm} := \iota_\pm^\star I_{X_\pm,Y_\pm}$, we conclude that $I_{Y_+} = \UU_Y I_{Y_-}$ after analytic continuation.
\begin{theorem}
\label{thm:U_ci}
Consider the ambient part
of the (non-equivariant) Givental space for $Y_\pm$ with the Novikov variable $Q$ specialized to~$1$:
\[
\cH(Y_\pm)_\amb = H^\bullet_{\amb}(Y_\pm) \otimes \CC(\!(z^{-1})\!)
\]
Regard $\cH(Y_\pm)_\amb$ as a graded vector space, where we use the age-shifted grading on $H_{\amb}^\bullet(Y_\pm)$ and set $\deg z=2$.
There exists a degree-preserving
$\CC(\!(z^{-1})\!)$-linear transformation
\[
\UU_Y \colon \cH(Y_-)_\amb \to \cH(Y_+)_\amb
\]
such that:
\begin{enumerate}
\item $I_{Y_+}(\sfy,z) = \UU_Y I_{Y_-}(\sfy,z)$ after analytic continuation in $\sfy^e$
along the path $\gamma$ in Figure {\rm\ref{fig:path_ancont}};
\item $\UU_Y \circ (g_-^\star v\cup)= (g_+^\star v\cup) \circ \UU_Y$ for all
$v\in H^2(\overline{X}_0)$, where $\overline{X}_0$ is the common blow-down of $X_\pm$ and $g_\pm \colon Y_\pm \to
\overline{X}_0$ is the composition of the inclusion $\iota_\pm \colon Y_\pm \to X_\pm$ with the blow-down $X_\pm \to \overline{X}_0$;
\item there is a commutative diagram
\[
\xymatrix{
K^0(Y_-)_\amb \ar[r]^{\FM} \ar[d] & K^0(Y_+)_\amb \ar[d]\\
\tcH(Y_-)_\amb \ar[r]^{\UU_Y} & \tcH(Y_+)_\amb
}
\]
where $\FM$ is the Fourier--Mukai transformation given by the top triangle in \eqref{eq:FM_Y_setup} and the vertical arrows are the ambient $K$-group framing defined in \S\ref{sec:ambient_part}.
\end{enumerate}
If $Y_\pm$ is compact then $\UU_Y$ intertwines the (possibly-degenerate) symplectic pairings on $\cH(Y_\pm)_\amb$.
\end{theorem}
\begin{proof}
Everything has been proved except the statement that, if $Y_\pm$ is compact, then $\UU_Y$ intertwines the pairings on $\cH(Y_\pm)_\amb$. But:
\begin{align*}
\Big( \UU_Y(-z) \iota_-^\star \alpha, \UU_Y(z) \iota_-^\star \beta \Big)_{Y_+}
&=
\Big( \iota_+^\star \UU_X(-z) \alpha, \iota_+^\star \UU_X(z) \beta \Big)_{Y_+} \\
&= \Big( \UU_X(-z) \alpha, e(E_+) \UU_X(z) \beta \Big)_{X_+} \\
&= \Big( \UU_X(-z) \alpha, \UU_X(z) e(E_-) \beta \Big)_{X_+} && \text{by Theorem~\ref{thm:U}(2)}\\
&= \Big( \alpha, e(E_-) \beta \Big)_{X_-} && \text{since $\UU_X$ is pairing-preserving} \\
&= \Big( \iota_-^\star \alpha, \iota_-^\star \beta \Big)_{Y_-}
\end{align*}
\end{proof}
\begin{remark}
\label{rem:degeneracy}
If $Y_\pm$ is compact then the Givental space for $Y_\pm$ has a well-defined symplectic pairing, but the restriction of this pairing to the ambient part is non-degenerate if and only if $(\iota_\pm)_\star \colon H^\bullet_{\amb}(Y_\pm) \to H^\bullet_{\CR}(X_\pm)$ is injective. This would hold by the Lefschetz Theorem if $E_\pm$ were a direct sum of ample line bundles, but in our situation the line bundles are always semi-ample and the question is more subtle. Injectivity holds when $Y_\pm$ is a regular semiample hypersurface by a result of Mavlyutov~\cite[Theorem~5.1]{Mavlyutov}.
\end{remark}
Theorem~\ref{thm:U_ci} is the analog, for toric complete intersections, of Theorem~\ref{thm:U}. The analog of Theorem~\ref{thm:CTC_qconn} also holds:
\begin{theorem}
\label{thm:CTC_qconn_ci}
Let $(\bF^\pm, \bnabla^\pm,\bE^\pm)$ be the global
quantum connections for the ambient parts $H^\bullet_{\amb}(Y_\pm)$ over $\tcM_\pm^\circ(\CC[z])$
from Theorem \ref{thm:global_qconn_ci}.
We have that $\bE^+ = \bE^-$ on $\tcM$.
There exists a gauge transformation
\[
\Theta_Y \in \Hom\big(H^\bullet_{\amb}(Y_-), H^\bullet_{\amb}(Y_+)\big)
\otimes \cO_{\cU^\circ}[z][\![\sfy_1,\dots,\sfy_{r-1}]\!]
\]
over $\tcM^\circ(\CC[z])$ such that:
\begin{itemize}
\item $\bnabla^-$ and $\bnabla^+$ are gauge-equivalent
via $\Theta_Y$, i.e.~$
\bnabla^+ \circ \Theta_Y = \Theta_Y \circ \bnabla^-$;
\item $\Theta_Y$ is homogeneous of degree zero, i.e.~$
\bGr^+ \circ \Theta_Y = \Theta_Y \circ \bGr^-$ with
$\bGr^\pm := z\parfrac{}{z} + \bE^\pm + \mu^\pm$;
\item if $Y_\pm$ are compact then $\Theta_Y$ preserves the (possibly-degenerate) orbifold Poincar\'{e} pairing
on $H^\bullet_{\amb}(Y_\pm)$, i.e.~$(\Theta_Y(\sfy,-z) \alpha, \Theta_Y(\sfy,z) \beta) = (\alpha,\beta)$.
\end{itemize}
Moreover, the analytic continuation of flat sections coincides, via the ambient $K$-group framing defined in~\S\ref{sec:ambient_part}, with the Fourier--Mukai transformation:
\begin{align*}
\Theta_Y\Bigl( \frs(E)(\tau_-(\sfy),z) \Bigr) = \frs(\FM(E))(\tau_+(\sfy),z) &&
\text{for all $E \in K^0(Y_-)_\amb$}
\end{align*}
where $\tau_\pm$ are the mirror maps from Theorem~\ref{thm:global_qconn_ci}.
\end{theorem}
Theorem~\ref{thm:CTC_qconn_ci} follows from Theorem~\ref{thm:U_ci} exactly as Theorem~\ref{thm:CTC_qconn} follows from Theorem~\ref{thm:U}.
The transformation $\UU_Y$ in Theorem~\ref{thm:U_ci} and the gauge transformation
$\Theta_Y$ in Theorem~\ref{thm:CTC_qconn_ci} are related by
\[
L_+^\amb(\tau_+(\sfy),z)^{-1} \circ \Theta_Y =
\UU \circ L_-^\amb(\tau_-(\sfy),z)^{-1}
\]
where $L_\pm$ is the ambient fundamental solution from~\S\ref{sec:ambient_part}.
The gauge transformation $\Theta_Y$ sends
the section $\Upsilon_0^-\in \bF^-$ to the section
$\Upsilon_0^+\in \bF^+$,
where $\Upsilon_0^\pm$ are as
in Theorem \ref{thm:global_qconn_ci}.
\section*{Acknowledgements}
We thank Alessio Corti, Alexander Kasprzyk, Yuan-Pin Lee, Nathan Priddis and Mark Shoemaker for a number
of useful conversations.
This research was supported by a Royal Society University Research Fellowship; the Leverhulme Trust; ERC Starting Investigator Grant number~240123; EPSRC Mathematics Platform grant EP/I019111/1; JSPS Kakenhi Grant Number 25400069; NFGRF, University of Kansas; and Simons Foundation Collaboration Grant 311837.
\bibliographystyle{plain}
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\end{document}
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\begin{document}
\maketitle
\begin{abstract}
It has been discovered that the Kadomtsev-Petviashvili(KP) equation governs the distribution of the fluctuation of many random growth models, in particular, the Tracy-Widom distributions appear as special self-similar solutions of the KP equation. We prove that the anti-derivative of Baik-Rains distribution, which governs the fluctuation of the models in the KPZ universality class starting with stationary initial data, satisfies the KP equation. We start from a determinantal formula of the generating function of the KPZ equation, which satisfies the KP. Then we observe that the equation still holds in the large time limit.
\end{abstract}
\section{Introduction}
The fluctuation for the KPZ universality class depends on the initial data. Let $h(x,t)$ be the solution of Kardar-Parisi-Zhang equation:
\begin{equation}
\partial_th(x,t) = \frac{\lambda}{2}(\partial_xh(x,t))^2+\nu \partial_x^2h(x,t)+\sqrt{D}\xi(x,t)
\end{equation}
\noindent $\mu >0,\lambda, D\neq 0$ are fixed parameters. $\xi(x,t)$ is the Gaussian space-time white noise
\begin{equation}
\mathbb{E}(\xi(x,t)\xi(y,s))= \delta(x-y)\delta(t-s)
\end{equation}
The equation is ill-posed because the quadratic non-linear term cannot make sense for a realization of a solution. A typical solution $h(x,t)$ looks like a Brownian motion in variable $x$. One way to make sense of the equation is through Hopf-Cole transformation.
The Hopf-Cole solution of the KPZ equation is defined to be: $h(t,x) = -\log z(t,x)$, where $z(t,x)$ is the solution of the stochastic heat equation with multiplicative white noise:
\begin{equation}
\partial_tz(t,x) = \frac{1}{2}\partial_x^2z(t,x) + z(t,x)\xi(t,x)
\end{equation}
which is well-posed interpreted as It$\hat{\text{o}}$ integral.\\
\\
\indent All the universal fluctuation behavior can be observed on large space and time scales under the KPZ scaling:
\begin{equation}
h_\epsilon(t,x) = \epsilon^{1/2}h(\epsilon^{-3/2}t,\epsilon^{-1}x)
\end{equation}
The fluctuation depends on the initial condition. If the initial condition is $z(0,x) = \delta_0$,which corresponds to $h(0,x) = -\infty$ if $x \neq 0$ and $h(0,x) = 0$ if $x = 0$, i.e. KPZ starting from narrow wedge initial condition, we observe that when $t \rightarrow \infty$:
\begin{equation}
-2^{1/3}t^{-1/3}(h(t,2^{1/3}t^{2/3}x)-2^{-1/3}t^{1/3}x^2-\frac{t}{24}-\log\sqrt{2\pi t}) \rightarrow \mathcal{A}_2(x)
\end{equation}
If the initial condition is $z(0,x) = 1$, we observe:
\begin{equation}
-2^{1/3}t^{-1/3}(h(t,2^{1/3}t^{2/3}x)-\frac{t}{24}) \rightarrow \mathcal{A}_1(x)
\end{equation}
If the initial condition is $z(0,x) = e^{B(x)}$, where $B(x)$ is the two-sided Brownian motion with $B(0) = 0$, we observe:
\begin{equation}
-2^{1/3}t^{-1/3}(h(t,2^{1/3}t^{2/3}x)-\frac{t}{24}) \rightarrow \mathcal{A}_{\text{stat}}(x)
\end{equation}
$\mathcal{A}_1(x),\mathcal{A}_2(x),\mathcal{A}_{\text{stat}}(x)$ are stochastic processes whose finite dimensional distributions are given by Fredholm determinant. These are the conjectured processes which govern the long-time fluctuation of models which belong to the KPZ universality class. $\mathcal{A}_1(x)$ is a stationary process, whose one-point distribution is the Tracy-Widom GOE distribution. The one point marginal of $\mathcal{A}_2(x)-x^2$ is given by the Tracy-Widom GUE distribution. The one point marginal of $\mathcal{A}_{\text{stat}}(x)$ is given by the Baik-Rains distribution\\
\\
\indent In the paper \cite{kp}, it was stated that the GUE and GOE Tracy-Widom distributions are seen to arise as special similarity solutions of the scalar Kadomtsev-Petviashvili(KP) equation:
\begin{equation}\label{kpequation}
\partial_t\phi+\phi\partial_r\phi+\frac{1}{12}\partial_r^3\phi+\frac{1}{4}\partial_r^{-1}\partial_x^2\phi = 0
\end{equation} In this paper, we explain how the Baik-Rains distribution can be seen as a similarity solution of the KP equation. We first explain how the GUE and GOE distributions arise as similarity solutions of the KP equation, then give the definition of Baik-Rains distribution and then state our main results.\\
\begin{example}
\textbf{Tracy-Widom GUE distribution}\cite{kp}:
If we look for a self-similar solution of (\ref{kpequation}) of the form
\begin{equation}
\phi(t,x,r) = t^{-2/3}\psi(t^{-1/3}r+t^{-4/3}x^2)
\end{equation}
This turns (\ref{kpequation}) to
\begin{equation}
\psi'''+12\psi\psi'-4r\psi'-2\psi = 0
\end{equation}
If we look for solutions of the form $\psi = -q^2$, then the above equation becomes the Painl\'eve II equation:
\begin{equation}
q'' = rq+2q^3
\end{equation} in this way we just recover the GUE distribution, since $F_{\text{GUE}} = \exp\{-\int_s^{-\infty}du(u-s)q^2(u)\}$\\
\end{example}
\begin{example}
\textbf{Tracy-Widom GOE distribution}\cite{kp}:
If we look for a self-similar solution of (8) of the form
\begin{equation}
\phi(t,r) = (t/4)^{-2/3}\psi((t/4)^{-1/3}r)
\end{equation}
then the above equation becomes:
\begin{equation}
\psi'''+12\psi'\psi-r\psi'-2\psi=0
\end{equation}
If we look for solutions of form $\psi = \frac{1}{2}(q'-q^2)$, we get the Painl\'eve II equation again, thus we recover the
\begin{equation}
F_{\text{GOE}}(r) = \exp\{-\frac{1}{2}\int_r^{\infty}q(u)du\}F_{\text{GUE}}(r)^{1/2}
\end{equation}
\end{example}
Now let's look at the definition of the Baik-Rains distribution.
\begin{definition}\cite{scaling}
Ai is the Airy function. For $\tau, s \in \mathbb{R}$, define
\begin{gather}
\hat{\Phi}_{w,s}(x)=\int_{\mathbb{R}_{-}}dze^{wz}K_{\text{Ai,s}}(z,x)e^{ws}\\
\hat{\Psi}_{w,s}(y) =\int_{\mathbb{R}_{-}}dze^{wz}\text{Ai}(y+z+s)\\
\rho_s(x,y) = (\boldsymbol{I}-P_0\boldsymbol{K}_{\text{Ai,s}}P_0)^{-1}(x,y)
\end{gather}
\noindent $P_0(x) = \boldsymbol{I}_{x\geq 0}$ is the projection operator, the shifted Airy kernel
\begin{equation}
\boldsymbol{\hat{K}_{\text{Ai,s}}}(x,y) = \int_0^\infty d\lambda\text{Ai}(x+\lambda+s)\text{Ai}(y+\lambda+s)
\end{equation}
the Tracy-Widom GUE distribution can be written as
\begin{equation}
F_{\text{GUE}}(s)=\det(\boldsymbol{I}-\boldsymbol{P}_0\boldsymbol{K}_{\text{Ai,s}}\boldsymbol{P}_0)
\end{equation}
then we define the function $g(s,w)$ which appear as a component in the Baik-Rains distribution
\begin{equation}
g(s,w) = e^{-\frac{1}{3}w^3}[\int_{\mathbb{R}^2_{-}}dxdye^{w(x+y)}\text{Ai}(x+y+s)+\int_{\mathbb{R}^2_{+}}dxdy\hat{\Phi}_{w,s}(x)\rho_s(x,y)\hat{\Psi}_{w,s}(y)]
\end{equation}
Finally the Baik-Rains distribution is defined to be
\begin{equation}
F_\tau(r) = \frac{\partial}{\partial r}(
g(r+\tau^2,\tau/2)F_{\text{GUE}}(r+\tau^2))
\end{equation}
\end{definition}
Notice that the function $g(s, w)$ is also derived in solving the PNG model, using the Riemann-Hilbert technique, given as solutions of a set of differential equations. We also present this equivalent definition here\cite{limiting}: Let $u(x)$ be the solution of the Painl\'eve II equation
\begin{equation}
u_{xx} = 2u^3+xu
\end{equation}
with the boundary condition
\begin{equation}\label{painleve2}
u(x) \sim -\text{Ai}(x) \text{ as } x \rightarrow+\infty
\end{equation}
$v(x)$ is defined to be
\begin{equation}
v(x)= \int_{\infty}^x(u(s))^2ds
\end{equation}
then the Tracy-Widom distribution can be defined in terms of $u$ and $v$. Set
\begin{equation}
F(x) = \exp(\frac{1}{2}\int_x^{\infty}v(x)ds) = \exp(-\frac{1}{2}\int_x^\infty(s-x)(u(s))^2ds),\qquad E(x) = \exp(\frac{1}{2}\int_x^{\infty}u(s)ds)
\end{equation}
then
\begin{gather}
F_{\text{GUE}}(x) = F(x)^2 = \exp(\int_x^\infty(s-x)(u(s))^2ds)\\
F_{\text{GOE}}(x)= F(x)E(x)
\end{gather}
then $F_\tau(r)$ is defined to be $F_\tau(r) = H(r+\tau^2;\tau/2,-\tau/2)$, where
\begin{equation}
H(x;w,-w) = \{y'(x,w)-y(x,w)v(x))\}F_{GUE}(x) = \partial_x(y(x,w)F_{\text{GUE}}(x))
\end{equation}
where
\begin{equation}\label{functiony}
y(x,w) = (2u^2+x-4w^2)a(x;w)a(x;-w)-(u'+2wu)b(x;w)a(x;-w)-(u'-2wu)a(x;w)b(x;-w)
\end{equation}
Here functions $a(x;w),b(x;w)$ arise in the Painl\'eve II Riemann-Hilbert problems. In this paper, we do not need the precise definition of $a(x;w),b(x;w)$, thus we skip the definition here. What we need is the following identities:
\begin{align}\label{identities}
\partial_x a(x,w) &= u(x)b(x,w)\\
\partial_x b(x,w) &= u(x)a(x,w) - 2wb(x,w)\\
\partial_w a(x,w) &= 2u(x)^2a(x,w) - (4wu(x)+2u'(x))b(x,w)\\
\partial_w b(x,w) &= (-4wu(x)+2u'(x))a(x,w) + (8w^2-2x-2u(x)^2)b(x,w)\\
a(x,w) &= -b(x,-w)e^{\frac{8}{3}w^3-2wx}\\\label{identitieslast}
b(x,w) &= -a(x,-w)e^{\frac{8}{3}w^3-2wx}
\end{align}
\noindent It is proven in \cite{scaling} that $y(s,w) = g(s, w)$, thus two definitions we presented are the same. In the following part of the paper, we will identify functions $g(s,w)$ and $y(s,w)$.\\
\\
\indent The main result we discovered about $F_\tau(r)$ is:
\begin{theorem}\label{thm1}
$F_\tau(r)$ is defined to be the partial derivative of $
g(r+\tau^2,\tau/2)F_{\text{GUE}}(r+\tau^2)$ in $r$. If we consider certain scaling form of this anti-derivative of $F_\tau(r)$, which is\begin{equation}
(\partial_r^{-1}F_{t^{-2/3}x})(t^{-1/3}r)= F_{\text{GUE}}(t^{-1/3}r+t^{-4/3}x^2)g(t^{-1/3}r+t^{-4/3}x^2,t^{-2/3}x/2)
\end{equation} its logarithmic derivative $\phi_{br}(x,t,r) = \partial_r^2\log(\partial_r^{-1}F_{t^{-2/3}x})(t^{-1/3}r)$, satisfies the KP equation:
\begin{gather}
\partial_t\phi_{br} + \phi\partial_r\phi_{br}+\frac{1}{12}\partial_r^3\phi_{br}+\frac{1}{4}\partial^{-1}_r\partial_x^2
\phi_{br} = 0
\end{gather}
\end{theorem}
\begin{remark}
Now we know $(\partial_r^{-1 }F_{t^{-2/3}x})(t^{-1/3}r) = F_{\text{GUE}}(t^{-1/3}r+t^{-4/3}x^2)g(t^{-1/3}r+t^{-4/3}x^2,t^{-2/3}x/2)$ and $ \partial_r^2\log F_{\text{GUE}}$ also satisfies the KP equation, we denote it as $\phi_{gue} = \partial_r^2\log F_{\text{GUE}} $, so the Thm. 1 is equivalent to that $\psi(x,t,r) = \partial_r^2\log g$ satisfies the following equation:
\begin{gather} \label{modifiedkp}
\partial_t\psi + \psi\partial_r\psi+\frac{1}{12}\partial_r^3\psi+\frac{1}{4}\partial^{-1}_r\partial_x^2
\psi+\phi_{gue}\partial_r\psi+\psi\partial_r\phi_{gue} = 0
\end{gather}
In this paper, we will prove that $g$ satisfies (\ref{modifiedkp}), which implies Thm. 1.
\end{remark}
\noindent There are two ways we will show that Thm. \ref{thm1} is true. One way is to directly substitute equation (\ref{functiony}) into the equation (\ref{modifiedkp}). Using the identities in (\ref{identities})-(\ref{identitieslast}), we found that equation (\ref{modifiedkp}) holds. The other way is that by observing the Baik-Rains distribution govern the fluctuation of large time scale of the KPZ equation, we use the results that the generating function of the KPZ equation satisfies KP \cite{doussal}, taking limit in $t$, obtaining Thm. \ref{thm1}. But part of the second method is still non-rigorous.\\
\\
\section{Proof of the Theorem 1}
A key fact we will use is the following theorem\cite{kp}:
\begin{theorem}
If a function can be written in the Fredholm determinant form, i.e. $F(x,t,r)= \det(\boldsymbol{I}-\boldsymbol{K})_{L^2[0,\infty)}$, and if the integral kernel $\boldsymbol{K}(u,v,x,t,r)$ satisfies the following relations:
\begin{equation}
\begin{aligned}\label{kernelrelation}
\partial_r\boldsymbol{K} = (\partial_u+\partial_v)\boldsymbol{K}\\
\partial_t\boldsymbol{K} = -\frac{1}{3}(\partial_u^3+\partial_v^3)\boldsymbol{K}\\
\partial_x\boldsymbol{K} = (\partial_u^2-\partial_v^2)\boldsymbol{K}
\end{aligned}
\end{equation}
then $\phi(x,t,r) = \partial_r^2\log F$ satisfy the scalar KP equation:
\end{theorem}
\begin{equation}
\partial_t\phi + \phi\partial_r\phi+\frac{1}{12}\partial_r^3\phi+\frac{1}{4}\partial^{-1}_r\partial_x^2
\phi = 0
\end{equation}
\begin{remark}
The fact that equation (\ref{kernelrelation}) leads to the KP equation was discovered several times \cite{zs74}\cite{po89}, but its appearance in the context of random fluctuation interfaces was discovered in \cite{kp}
\end{remark}
\noindent It is checked in \cite{kp} \cite{doussal} that if $h(x,t)$ is the solution of the KPZ equation with the half-Brownian initial data or narrow wedge initial data, their generating functions $G(t,x,r) = \mathbb{E}[\exp\{-e^{h(t,x)+\frac{t}{12}-r}\}]$ can be written in the Fredholm determinant form, having a kernel which satisfying equation (\ref{kernelrelation}). It is also checked in \cite{doussal} that if $h(x,t)$ is the solution of the KPZ equation with initial condition to be the drifted Brownian motion, its generating function also satisfies the KP equation, by studying the moments of $e^{h(x,t)}$. Here we checked it using a different method, which agreed with the results in \cite{doussal} \\
\\
\begin{theorem}\cite{height}
Let $z_{b,\beta}(t,x)$ denote the solution to the stochastic heat equation with initial data $z(0,x) = \exp(B_{b,\beta}(x))$, where $B_{b,\beta}(x)$ is a two-sided Brownian motion with drift $\beta$ to the left of 0 and drift $b$ to the right of 0, with $\beta > b$, that is, $B_{b,\beta} = \boldsymbol{1}_{x\leq 0}(B^l(x)+\beta x)+\boldsymbol{1}_{x >0}(B^r(x)+bx)$, where $B^l:(-\infty,0] \rightarrow \mathbb{R}$ is a Brownian motion without drift pinned at $B^l(0)=0$ and $B^r:[0,\infty) \rightarrow \mathbb{R}$ is an independent Brownian motion pinned at $B^r(0) = 0$, then for $S >0$\\
\begin{equation}\label{expectationK}
\mathbb{E}[2(Se^{\frac{x^2}{2t}+\frac{t}{24}}z_{b,\beta}(t,x))^{\frac{\beta-b}{2}}K_{-(\beta-b)}(2\sqrt{Se^{\frac{x^2}{2t}+\frac{t}{24}}z_{b,\beta}(t,x)})] = \Gamma(\beta - b)\det(\boldsymbol{I}-\boldsymbol{K}_{b+\frac{x}{t},\beta+\frac{x}{t}})_{L^2(\mathbb{R}_{+})}
\end{equation}
where $K_\nu(z)$ is the modified Bessel function of order $\nu$ and the kernel on the right-hand side is given by\\
\begin{equation}
\boldsymbol{K}_{b,\beta}(x,y) = \frac{1}{(2\pi i)^2}\int dw\int dz\frac{\sigma \pi S^{\sigma(z-w)}}{\sin(\sigma\pi(z-w))}\frac{e^{z^3/3-zy}}{e^{w^3/3-wx}}\frac{\Gamma(\beta-\sigma z)}{\Gamma(\sigma z-b)}\frac{\Gamma(\sigma w-b)}{\Gamma(\beta-\sigma w)}
\end{equation}
where\\
\begin{equation}
\sigma = (2/t)^{1/3}
\end{equation}
The integration contour for $w$ is from $-\frac{1}{4\sigma}-i\infty$ to $-\frac{1}{4\sigma}+i\infty$ and crosses the real axis between $b$ and $\beta$. The other contour for $z$ goes from $\frac{1}{4\sigma}-i\infty$ to $\frac{1}{4\sigma}+i\infty$, it also crosses the real axis between $b$ and $\beta$ and it does not intersect the contour for $w$
\end{theorem}
\noindent The left hand-side of (\ref{expectationK}) is exactly the generating function, since for $\alpha > a$, we have\cite{height}:\\
\begin{equation}
\mathbb{E}[2(uz(\tau,N))^{\frac{\alpha - a}{2}}K_{-(\alpha - a)}(2\sqrt{uz(\tau,N)})] = \Gamma(\alpha - a)\mathbb{E}[e^{-uz(\tau,N)\gamma}]
\end{equation}
where $\frac{1}{\gamma}$ has gamma distribution with parameter $\alpha - a$.\\
Thus we have:
\begin{equation}
\mathbb{E}[\exp\{-Se^{\frac{x^2}{2t}+\frac{t}{24}+h_{b,\beta}(t,x)+\log \gamma}\}] = \det(\boldsymbol{I}-\boldsymbol{K}_{b+\frac{x}{t},\beta+\frac{x}{t}})_{L^2(\mathbb{R}_{+})}
\end{equation}
For $S = e^{\tau^2+r}$, where $\tau$ is related to $x,b,t$ as $x = bt+\frac{2\tau}{\sigma^2}$, and when $b = 0$, we observe that $\boldsymbol{K}_{b+\frac{x}{t},\beta+\frac{x}{t}}$ satisfy relation (\ref{kernelrelation}). The reason that we make this specific choice of $S$ and $\tau$ will be clear from the later context, this is the scaling that gives the Baik-Rains distribution. We will do some transformations on the integral kernel so that equation (\ref{kernelrelation}) becomes obvious while the operator remains unchanged. When $S = e^{\tau^2+r}$,
\begin{align}
\boldsymbol{K}_{b+\frac{x}{t},\beta +\frac{x}{t}}(u,v) = \frac{1}{(2\pi i)^2}\int dw\int dz\frac{\sigma \pi e^{-(z-w)(\tau^2+r)}}{\sin(\sigma \pi(z-w))}\frac{e^{z^3/3-zv}}{e^{w^3/3-wu}}\frac{\Gamma(\beta +\frac{x}{t}-\sigma z)}{\Gamma(\sigma z-b-\frac{x}{t})}\frac{\Gamma(\sigma w-b-\frac{x}{t})}{\Gamma(\beta+\frac{x}{t}-\sigma w)}
\end{align}
Now let $z = \sigma z - \frac{x}{t},w = \sigma w- \frac{x}{t} $, then
\begin{align}
\boldsymbol{K}_{b+\frac{x}{t},\beta +\frac{x}{t}}(u,v) = \frac{1}{(2\pi i)^2}\int dw\int dz\frac{ \pi e^{-(z-w)(\tau^2+r)/\sigma}}{\sin( \pi(z-w))}\frac{e^{t(z^3+3z^2x/t+3zx^2/t^2)/6-\frac{1}{\sigma}(z+\frac{x}{t})v}}{e^{t(w^3+3w^2x/t+3wx^2/t^2)/6-\frac{1}{\sigma}(w+\frac{x}{t})u}}\frac{\Gamma(\beta - z)}{\Gamma( z-b)}\frac{\Gamma(w-b)}{\Gamma(\beta- w)}
\end{align}
Let $u = \frac{1}{\sigma}u, v= \frac{1}{\sigma}v$ and conjugate $e^{(v-u)x/t}$, and then change $r = \frac{1}{\sigma}r$ it becomes,
\begin{align}
\boldsymbol{K}_{b+\frac{x}{t},\beta +\frac{x}{t}}(u,v) = \frac{1}{(2\pi i)^2}\int dw\int dz\frac{ \pi e^{-(z-w)(b^2t/2+bx)}}{\sin( \pi(z-w))}\frac{e^{(tz^3+3z^2x)/6-z(v+r)}}{e^{(tw^3+3w^2x)/6-w(u+r)}}\frac{\Gamma(\beta - z)}{\Gamma( z-b)}\frac{\Gamma(w-b)}{\Gamma(\beta- w)}
\end{align}
When $b= 0$, the only term contain $x,t,r,u,v$ is $\frac{e^{(tz^3+3z^2x)/6-z(v+r)}}{e^{(tw^3+3w^2x)/6-w(u+r)}}$, which clearly satisfies equation (\ref{kernelrelation}). When $b \neq 0$, it will have extra terms when differentiating $t,x$ coming from $e^{-(z-w)(b^2t/2+bx)}$,which fail the equation (\ref{kernelrelation}). The reason that we can directly take derivative under the integral sign is the following lemma:\\
\begin{lemma}\cite{height}
Let $f(z,\zeta)$ be a complex function in two variables and suppose that
\begin{enumerate}
\item $f$ is defined on $(z,\zeta)\in A \times C$ where $A$ is an open set and $C$ is a contour
\item For each $z\in A$, define the contour $\gamma = \{z+re^{it}: 0\leq t\leq 2\pi\}$ with a sufficiently small r such that also the disc around z with radius r lies in A. Suppose that for each $z \in A$\\
\begin{equation}
\int_C\int_\gamma|f(u,\zeta)||du||d\zeta| < \infty
\end{equation}
\item For each $\zeta \in C, z \rightarrow f(z,\zeta)$ is analytic in A
\item For each $z \in A, \zeta \rightarrow f(z,\zeta)$ is continuous on C
\end{enumerate}
Then \begin{equation}
F(z) = \int_Cf(z,\zeta)d\zeta
\end{equation}
is analytic in A with $F'(z) = \int_C\frac{\partial}{\partial z}f(z,\zeta)d\zeta$
\end{lemma}
\noindent It can be easily seen that condition 2 is satisfied. Since $e^{z^3/3}$ decay along $C_z$ as $e^{-c|\text{Im}(z)|^2}$, $e^{w^3/3}$ decay along $C_w$ as $e^{-c|\text{Im}(w)|^2}$. Using gamma ratio formula, as $|z| \rightarrow \infty$,
\begin{equation}
\left| \frac{\Gamma(\beta -\sigma z)}{\Gamma(\sigma z-b)}\right| \simeq |z|^{\beta + b-2\sigma\text{Re}(z)}
\end{equation}
Similarly we have the same bound for large $w$. Thus if we integrate $\beta$ on some finite contour, we have the same polynomial bounds. Thus the whole integrands decay exponentially on the contour, so condition (2) is satisfied.\\
\\
\indent In order to get the formula for stationary initial data, we need to take the limit as $\beta \rightarrow b$ and set $b= 0$. To do so, we need to rewrite the kernel $\boldsymbol{K}_{b,\beta}$. Two contours in the integral kernel of $\boldsymbol{K}_{b,\beta}$ intersect the real axis between the pole at $b/ \sigma$ and $\beta / \sigma$, so when $\beta \rightarrow b$, two contours will collide. Hence by using the residue theorem, we cross the pole at $b/ \sigma$ with the $w$ integration contour
and cross the pole at $\beta / \sigma$ with the $z$ integration contour, both manipulations resulting in a residue term. We have\cite{height}
\begin{align}
\boldsymbol{K}_{b,\beta} &= \boldsymbol{\bar{K}}_{b,\beta}+q_{b,\beta}(x)r_{\beta}(y)\frac{1}{\sigma\Gamma(\beta-b)}+r_{-b}(x)q_{-\beta,-b}(y)\frac{1}{\sigma\Gamma(\beta-b)}\\
&+\frac{\sigma\pi S^{\beta - b}}{\sin(\pi(\beta - b))}r_{-b}(x)r_{\beta}(y)\frac{1}{\sigma^2\Gamma(\beta - b)^2}
\end{align}
where
\begin{align}
\boldsymbol{\bar{K}}_{b,\beta} &= \frac{1}{(2\pi i)^2}\int_{-\frac{1}{4\sigma}+i\mathbb{R}} dw\int_{\frac{1}{4\sigma}+i\mathbb{R}} dz\frac{\sigma \pi S^{\sigma(z-w)}}{\sin(\sigma\pi(z-w))}\frac{e^{z^3/3-zy}}{e^{w^3/3-wx}}\frac{\Gamma(\beta-\sigma z)}{\Gamma(\sigma z-b)}\frac{\Gamma(\sigma w-b)}{\Gamma(\beta-\sigma w)}\\
q_{b,\beta}(x)&= \frac{1}{2\pi i}\int_{-\frac{1}{4\sigma}+i\mathbb{R}}dw\frac{\sigma \pi S^{\beta-\sigma w}}{\sin(\pi(\beta-\sigma w))}e^{-w^3/3+wx}\frac{\Gamma(\sigma w-b)}{\Gamma(\beta- \sigma w)}\\
r_b(x) &= e^{b^3/(3\sigma^3)-bx/\sigma}
\end{align}
\noindent Notice that the only difference between $\boldsymbol{K}_{b,\beta}$ and $\boldsymbol{\bar{K}_{b,\beta}}$ is that they have different contour. We can write
\begin{equation}
\boldsymbol{K_{b,\beta}} = \boldsymbol{\bar{K}_{b,\beta}}+\sum_{i=1}^3f_i(x)g_i(y)
\end{equation}
with suitable $f_i,g_i$, then for Fredholm determinant, we have the following formula
\begin{equation}\label{midkernelidentity}
\det(\boldsymbol{I}-\boldsymbol{K_{b,\beta}}) = \det(\boldsymbol{I}-\boldsymbol{\bar{K}_{b,\beta}})\det[\delta_{i,j}-\langle(\boldsymbol{I}-\boldsymbol{\bar{K}_{b,\beta}})^{-1}f_i,g_j\rangle]_{i,j=1}^3
\end{equation}
Here $\boldsymbol{\bar{K}}_{b,\beta}$ also satisfies relation (\ref{kernelrelation}) if $S = e^{\tau^2+r}$, because the only difference between $\boldsymbol{\bar{K}}_{b,\beta}$ and $\boldsymbol{K}_{b,\beta}$ is that they have different contour, for which the lemma 4 still apply. We denote $\phi = \partial_r^2\log\det(\boldsymbol{I}-\boldsymbol{K_{b,\beta}}),\bar{\phi} = \partial_r^2\log\det(\boldsymbol{I}-\boldsymbol{\bar{K}_{b,\beta}})$ and $\alpha = \partial_r^2\log \det[\delta_{i,j}-\langle(\boldsymbol{I}-\boldsymbol{\bar{K}_{b,\beta}})^{-1}f_i,g_j\rangle]_{i,j=1}^3$, we have $\phi = \bar{\phi} + \alpha$, and both $\phi, \bar{\phi}$ satisfy the KP equation\\
\begin{gather}
\partial_t\bar{\phi} + \bar{\phi}\partial_r\bar{\phi} + \frac{1}{12}\partial_r^3\bar{\phi}+\frac{1}{4}\partial_r^{-1}\partial_x^2\bar{\phi}=0\\
\partial_t(\bar{\phi}+\alpha) + (\bar{\phi}+\alpha)\partial_r(\bar{\phi}+\alpha) + \frac{1}{12}\partial_r^3(\bar{\phi}+\alpha)+\frac{1}{4}\partial_r^{-1}\partial_x^2(\bar{\phi}+\alpha)=0
\end{gather}
combine these two equations, we obtain an equation for $\alpha$ which involves $\bar{\phi}$:\\
\begin{equation}
\partial_t\alpha + \alpha\partial_r\alpha + \frac{1}{12}\partial_r^3\alpha+\frac{1}{4}\partial_r^{-1}\partial_x^2\alpha+\alpha\partial_r\bar{\phi}+\bar{\phi}\partial_r\alpha=0
\end{equation}
This is the first place where we obtain equation (\ref{modifiedkp}). Now we want to see how equation (\ref{midkernelidentity}) leads us to the Baik-Rains distribution.\\
\\
\\
In the limit as $\beta \rightarrow b$, we have the following results
\begin{theorem}\cite{height}
Let $b+\frac{x}{t} \in (-\frac{1}{4},\frac{1}{4})$ be fixed. For the kernel $\boldsymbol{K}_{b,\beta}$, we have
\begin{equation}\label{xilimit}
\lim_{\beta \rightarrow b}\frac{1}{\beta - b}\det(\boldsymbol{I}-\boldsymbol{K}_{b+\frac{x}{t},\beta+\frac{x}{t}}) = \frac{1}{\sigma}\Xi(S,b+\frac{x}{t},\sigma)
\end{equation}
where\\
\begin{equation}
\begin{aligned}
\Xi(S,b,\sigma) &= -\det(\boldsymbol{I}-\boldsymbol{\bar{K}_b})[\frac{b^2}{\sigma^2}+\sigma(2\gamma_E+\ln S)\\&+
\langle(\boldsymbol{I}-\boldsymbol{\bar{K}_b})^{-1}(\boldsymbol{\bar{K}_b}r_{-b}+q_b),r_b\rangle+\langle(\boldsymbol{I}-\boldsymbol{\bar{K}_b})^{-1}(r_{-b}+q_b),q_{-b}\rangle]
\end{aligned}
\end{equation}
Where $\gamma_E$ represents Euler-Mascheroni constant, and for $b+\frac{x}{t} \in (-\frac{1}{4},\frac{1}{4}),\boldsymbol{\bar{K}}_{b+\frac{x}{t}} = \boldsymbol{\bar{K}}_{b+\frac{x}{t},b+\frac{x}{t}}, q_{b+\frac{x}{t}} = q_{b+\frac{x}{t},b+\frac{x}{t}}$\\
\end{theorem}
\noindent For simplicity, we will define
\begin{gather}
A(S,b,\sigma) = \det(\boldsymbol{I}-\boldsymbol{\bar{K}_b})\\
B(S,b,\sigma) =[\frac{b^2}{\sigma^2}+\sigma(2\gamma_E+\ln S)+
\langle(\boldsymbol{I}-\boldsymbol{\bar{K}_b})^{-1}(\boldsymbol{\bar{K}_b}r_{-b}+q_b),r_b\rangle+\langle(\boldsymbol{I}-\boldsymbol{\bar{K}_b})^{-1}(r_{-b}+q_b),q_{-b}\rangle]\\\label{phi}
\phi(x,t,r) = \partial_r^2\log A(S,b+\frac{x}{t},\sigma)\\\label{psi}
\psi(x,t,r) = \partial^2_r \log B(S,b+\frac{x}{t},\sigma)
\end{gather} .\\
Since formula (\ref{xilimit}) is obtained by taking limits of right-hand side of equation (\ref{midkernelidentity}), thus we speculate that the same system of PDEs will also hold for (\ref{xilimit}). What we need to show is that the limit of a sequence of functions satisfy the same PDE.\\
We have the following results about the Fredholm determinant:
\begin{lemma}\cite{height}
The Fredholm determinant $\det(\boldsymbol{I}-\boldsymbol{K_{b,\beta}}), \det(\boldsymbol{I}-\boldsymbol{\bar{K}_{b,\beta}})$ are analytic functions of the parameter $b$ and $\beta$ as long as $b<\beta$
\end{lemma}
\noindent The reason that we require $b < \beta$ simply because otherwise the kernel $\boldsymbol{K}_{b,\beta}$ is not well defined.\\
we want to show in the limit, two parts of right-hand side of (\ref{xilimit}) are both analytic. For the first part, we show that $\det(\boldsymbol{I}-\boldsymbol{\bar{K}_{b+\frac{x}{t},\beta+\frac{x}{t}}}) \rightarrow \det(\boldsymbol{I}-\boldsymbol{\bar{K}_{b+\frac{x}{t},b+\frac{x}{t}}})$ uniformly on compact set. Since
\begin{equation}
\det(\boldsymbol{I}-\boldsymbol{\bar{K}_{b+\frac{x}{t},\beta+\frac{x}{t}}}) = \sum_{n=0}^{\infty}\frac{(-1)^n}{n!}\int_0^\infty\cdots\int_0^\infty dx_1\cdots dx_n\det[\bar{\boldsymbol{K}}_{b,\beta}(x_i,x_j)]_{i,j=1}^n
\end{equation}
We need to show $\bar{\boldsymbol{K}}_{b,\beta} \rightarrow \bar{\boldsymbol{K}}_{b,b}$ uniformly on compact set. That simply because for the integrand of the kernel, $\frac{\Gamma(\beta - \sigma z)}{\Gamma(\beta-\sigma w)} \rightarrow \frac{\Gamma(b - \sigma z)}{\Gamma(b-\sigma w)}$ uniformly on compact set. For the other part, $[\frac{b^2}{\sigma^2}+\sigma(2\gamma_E+\ln S)+
\langle(\boldsymbol{I}-\boldsymbol{\bar{K}_b})^{-1}(\boldsymbol{\bar{K}_b}r_{-b}+q_b),r_b\rangle+\langle(\boldsymbol{I}-\boldsymbol{\bar{K}_b})^{-1}(r_{-b}+q_b),q_{-b}\rangle]$, the only part that is non-trivial is whether $(\boldsymbol{I}-\bar{\boldsymbol{K}}_{b})^{-1}$ is analytic. We need the following theorem which gives an explicit formula for the inverse which can be found\cite{peter}:
\begin{theorem}
Let $\boldsymbol{K}$ be a continuous kernel, let $D = \det(\boldsymbol{I}+\boldsymbol{K})$, and suppose that $D \neq 0$. Then the operator $\boldsymbol{I}+\boldsymbol{K}$ is invertible, and its inverse is $\boldsymbol{I}-D^{-1}\boldsymbol{R}$, where the integral kernel of $\boldsymbol{R}$ is
\begin{equation}
R(x,y) = \sum_0^{\infty}\frac{1}{k!}\int\cdots\int K\begin{pmatrix}
x, & x_1, & \cdots, & x_k \\
y, & x_1, & \cdots, & x_k
\end{pmatrix}dx_1\cdots dx_k
\end{equation}
\end{theorem}
\noindent From Thm. 7 we can conclude that if $K(x,y)$ is an analytic function, so is $(\boldsymbol{I}-\boldsymbol{K})^{-1}$.
\noindent Thus we conclude without fully justification that $\phi(x,t,r) = \partial_r^2\log A(S,b+\frac{x}{t},\sigma)$ satisfies equation (\ref{kpequation}) and $\psi(x,t,r) = \partial^2_r \log B(S,b+\frac{x}{t},\sigma)$ satisfies equation (\ref{modifiedkp}). In order to make the argument complete, we need to prove that the limit of partial derivatives converge uniformly.
\\
\indent In the large time limit, we have the following results\cite{height}:
\begin{equation}
\lim_{t \rightarrow \infty}\Xi(e^{-\frac{\tau^2+r}{\sigma}},\tau\sigma,\sigma) = g(r+\tau^2,\tau/2)F_{\text{GUE}}(r+\tau^2)
\end{equation}
Here the relation of $\tau$ to $x,t$ is:
\begin{equation}
x= -bt +\frac{2\tau}{\sigma^2}, \sigma =(\frac{2}{t})^{1/3}
\end{equation}
The reason for having this scaling is that under this scaling, we have the following result:
\begin{theorem}\cite{height}
Let $b\in (-\frac{1}{4},\frac{1}{4})$ be fixed and consider any $\tau \in \mathbb{R}$. Define $\sigma = (2/t)^{1/3}$ and consider the scaling $x= -bt +\frac{2\tau}{\sigma^2}$. Then, for any $r \in \mathbb{R}$
\begin{gather}
\lim_{t\rightarrow \infty}\mathbb{P}(\frac{h_b(t,x)+\frac{t}{24}(1+12b^2)-2^{1/3}b\tau t^{2/3}}{(t/2)^{1/3}} \leq r) = F_\tau(r)
\end{gather}
\end{theorem}
\noindent When $b=0$, we have $\sigma^2x = 2\tau$. We can see that as $t \rightarrow \infty$, $x$ must goes to the infinity in the speed $x \sim t^{2/3} $ so that $\tau$ can be a meaningful number. For this reason, we want to take the limit in the following way: we rewrite all the results for $h_\epsilon(t,x) = \epsilon^{1/2}h(\epsilon^{-1}x,\epsilon^{-3/2}t)$, the equation scales to:
\begin{equation}
\partial_t h_\epsilon = \frac{1}{2}(\partial_x h)^2+\epsilon^{1/2}\frac{1}{2} \partial_x^2 h+\epsilon^{1/4}\xi
\end{equation}
Then we plug $\epsilon^{-1}x \rightarrow x,\epsilon^{-3/2}t \rightarrow t, \epsilon^{-1/2}r \rightarrow r$ into $\Xi(e^{-\frac{\tau^2+r}{\sigma}},\tau\sigma,\sigma)$, which we denoted as $\Xi_\epsilon$ then take $\epsilon$ goes to 0, we have
\begin{equation}
\lim_{\epsilon \rightarrow 0}\Xi_\epsilon(e^{-\frac{\tau^2+r}{\sigma}},\tau\sigma,\sigma) = g(\sigma r+\tau^2,\tau/2)F_{\text{GUE}}(\sigma r+\tau^2)
\end{equation}
Now $\Xi$ is a function involve $\epsilon$. We write $\phi^\epsilon(x,t,r) = \phi(\epsilon^{-1}x,,\epsilon^{-\frac{3}{2}}t,\epsilon^{-1/2}r),\psi^\epsilon(x,t,r) = \psi(\epsilon^{-1}x,,\epsilon^{-\frac{3}{2}}t,\epsilon^{-1/2}r)$, where $\phi,\psi$ are defined in (\ref{phi}),(\ref{psi}). Equation (\ref{kpequation}), (\ref{modifiedkp}) re-scale as follows:
\begin{gather}
\epsilon^{5/2}\partial_t\phi^\epsilon +\epsilon^{5/2}\phi^\epsilon\partial_r\phi^\epsilon + \frac{1}{12}\epsilon^{5/2}\partial_r^3\phi^\epsilon+\epsilon^{5/2}\frac{1}{4}\partial_r^{-1}\partial_x^2\phi^\epsilon=0\\
\epsilon^{5/2}\partial_t\psi^\epsilon + \epsilon^{5/2}\psi^\epsilon\partial_r\psi^\epsilon + \epsilon^{5/2}\frac{1}{12}\partial_r^3\psi^\epsilon+\epsilon^{5/2}\frac{1}{4}\partial_r^{-1}\partial_x^2\psi^\epsilon+\epsilon^{5/2}\psi^\epsilon\partial_r\phi^\epsilon+\epsilon^{5/2}\phi^\epsilon\partial_r\psi^\epsilon=0
\end{gather}
Thus $\phi^\epsilon,\psi^\epsilon$ satisfy equation (\ref{kpequation}),(\ref{modifiedkp}). We conjecture without fully justification that in the limit as $\epsilon \rightarrow 0$, $\phi^\epsilon \rightarrow \partial^2_r\log F_{\text{GUE}}(\sigma r+\tau^2)$ satisfies (\ref{kpequation}), $\psi^\epsilon \rightarrow \partial^2_r\log g(\sigma r+\tau^2,\tau/2)$ satisfies (\ref{modifiedkp}), which is our Thm. \ref{thm1}. \\
\\
\section{Direct Verification}
Now we are going to directly check that Let $B = \partial^2_r\log y( t^{-1/3}r+t^{-4/3}x^2,\frac{1}{2}t^{-2/3}x)$, we use $y'$ to denote $\partial_1 y$ which is the partial derivative with respect to the first variable. $\phi = \partial_r^2\log F_{\text{GUE}}(t^{-1/3}r+t^{-4/3}x^2) = -t^{-2/3}u^2$, $u$ is defined in (\ref{painleve2}).
\begin{align}
\partial_rB &= t^{-3/3}(\tfrac{y'''}{y}-\tfrac{3y'y''}{y^2}+\tfrac{2y'^3}{y^3})\\
B\partial_rB &= t^{-5/3}(\tfrac{y'''y''}{y^2}-\tfrac{3y'y''^2}{y^3}+\tfrac{5y'^3y''}{y^4}-\tfrac{y'''y'^2}{y^3}-\tfrac{2y'^5}{y^5})\\
\phi\partial_rB &= t^{-5/3}(-u^2)(\tfrac{y'''}{y}-\tfrac{3y'y''}{y^2}+\tfrac{2y'^3}{y^3})\\
B\partial_r\phi &= t^{-5/3}(-2uu')(\tfrac{y''}{y}-\tfrac{y'^2}{y^2})\\
\partial_r^2B &=t^{-4/3}(\tfrac{y^{(4)}}{y} -\tfrac{3y''y''}{y^2}-\tfrac{4y'y'''}{y^2}+\tfrac{12y'^2y''}{y^3}-\tfrac{6y'^4}{y^4})\\
\partial_r^3B &=t^{-5/3}(\tfrac{y^{(5)}}{y}-\tfrac{10y''y'''}{y^2}-\tfrac{5y'y^{(4)}}{y^2}+\tfrac{20y'^2y'''}{y^3}+\tfrac{30y'y''^2}{y^3}-\tfrac{60y'^3y''}{y^4}+\tfrac{24y'^5}{y^5})\\
\partial_tB &= -\tfrac{2}{3}t^{-5/3}(\tfrac{y''}{y}-(\tfrac{y'}{y})^2)-t^{-2/3}(\tfrac{y'''}{y}-\tfrac{3y'y''}{y^2}+\tfrac{2y'^3}{y^3})(\tfrac{1}{3}t^{-4/3}r+\tfrac{4}{3}t^{-7/3}x^2)\\
&- t^{-2/3}(\tfrac{\partial_2y''}{y}-\tfrac{y''\partial_2y}{y^2}-\tfrac{2y'\partial_2y'}{y^2}+\tfrac{2y'^2\partial_2y}{y^3})\tfrac{1}{3}t^{-5/3}x\\
\partial_r^{-1}\partial_x^2B &= 2t^{-5/3}(\tfrac{y''}{y}-\tfrac{y'^2}{y^2})+4x^2t^{-9/3}(\tfrac{y'''}{y}-\tfrac{3y'y''}{y^2}+\tfrac{2y'^3}{y^3})+\tfrac{1}{4}t^{-5/3}\partial_2^2(\tfrac{y'}{y})\\
+&2xt^{-7/3}(\tfrac{\partial_2y''}{y}-\tfrac{y''\partial_2y}{y^2}-\tfrac{2y'\partial_2y'}{y^2}+\tfrac{2y'^2\partial_2y}{y^3})
\end{align}
Now plugin every term into equation (\ref{modifiedkp}), it becomes:
\begin{align}
&\partial_tB +B\partial_rB + \tfrac{1}{12}\partial_r^3B+\tfrac{1}{4}\partial_r^{-1}\partial_x^2B + \bar{\phi}\partial_r B+ B\partial_r \bar{\phi} \\ &= t^{-5/3}(\tfrac{1}{12}(\tfrac{y^{(5)}}{y}-\tfrac{5y'y^{(4)}}{y^2}+\tfrac{2y''y'''}{y^2}-\tfrac{6y'y''^2}{y^3}+\tfrac{8y'^2y'''}{y^3})-\tfrac{1}{6}(\tfrac{y''}{y}-\tfrac{y'^2}{y^2})\\
&+\tfrac{1}{16}(\tfrac{\partial^2_2y'}{y}-\tfrac{2\partial_2y'\partial_2y}{y^2}-\tfrac{y'\partial_2^2y}{y^2}+\tfrac{2y'(\partial_2y)^2}{y^3})+\tfrac{1}{3}w(\tfrac{\partial_2y''}{y}-\tfrac{y''\partial_2y}{y^2}-\tfrac{2y'\partial_2y'}{y^2}+\tfrac{2y'^2\partial_2y}{y^3})\\
&-\tfrac{1}{3}x(\tfrac{y'''}{y}-\tfrac{3y'y''}{y^2}+2\tfrac{y'^3}{y^3})-u^2(\tfrac{y'''}{y}-\tfrac{3y'y''}{y^2}+2\tfrac{y'^3}{y^3})-2uu'(\tfrac{y''}{y}-\tfrac{y'^2}{y^2}))
\end{align}
multiplied by $y^3t^{5/3}$,right-hand side becomes
\begin{align}
\label{equationsofy}
&\tfrac{1}{12}(y^{(5)}y^2-5y'y^{(4)}y+2y''y'''y-6y'y''^2+8y'^2y''')-\tfrac{1}{6}(y''y^2-y'^2y)\\
&+\tfrac{1}{16}(\partial^2_2y'y^2-2\partial_2y'\partial_2yy-y'\partial_2^2yy+2y'(\partial_2y)^2)+\tfrac{1}{3}w(\partial_2y''y^2-y''\partial_2yy-2y'\partial_2y'y+2y'^2\partial_2y)\\
&-\tfrac{1}{3}x(y'''y^2-3y'y''y+2y'^3)-u^2(y'''y^2-3y'y''y+2y'^3)-2uu'(y''y^2-y'^2y)
\end{align}
Then we compute the derivative of $y$, in the following expressions, if we omit the variables, then it just means that's the variable in the definition; $a(-w), b(-w)$ represents $a(x,-w),b(x,-w)$:
\begin{align}
y &= (2u^2+x-4w^2)aa(-w)-(u'+2wu)ba(-w)-(u'-2wu)ab(-w)\\
y' &= aa(-w)\\
y'' &= uba(x,-w)+uab(-w)\\
y''' &= (u'-2wu)ba(-w)+4u^2aa(-w)+(u'+2wu)ab(-w)\\
y^{(4)} &=12uu'aa(-w)+(4u^3+u''+4wu'+4w^2u)ab(-w)\\
&+(u''+4u^3-4wu'+4w^2u)ba(-w)\\
y^{(5)} &= (12u'^2+16uu''+16u^4+16w^2u^2)aa(-w)\\
&+ (24u^2u'+u'''+6wu''+12w^2u'+8wu^3+8w^3u)ab(-w)\\
&+(24u^2u'+u'''-6wu''-8wu^3+12w^2u'-8w^3u)ba(-w)\\
\partial_2y &= -8waa(-w)+2uab(-w)+(-2u)ba(-w)\\
\partial_2^2y &= (-8-16uu')aa(-w)+(16w^2u-16wu'+8u^3+4ux)ab(-w)\\
&+ (16w^2u+16wu'+8u^3+4ux)ba(-w)
\end{align}
Observe that the derivative of $a(x,w),b(x,w)$ behave similarly like $\sin, \cos$, in the sense that the derivatives of $a(x,w),b(x,w)$ are certain combination of $a(x,w),b(x,w)$ themselves. All the partials of $y$ is in the form of $c_1aa(-w)+c_2ab(-w)+c_3ba(-w)$, where $c_1,c_2,c_3$ are coefficients consists of $u,w$ and derivative of $u$ (There is no $bb(-w)$ because $bb(-w) = aa(-w)$ by equation (\ref{identitieslast})). Finally if we plugin the all the partials into equation (\ref{equationsofy}), there are terms: $a^3a^3(-w), a^3a^2(-w)b(-w), a^3a(-w)b^2(-w),a^3b^3(-w),b^3a^3(-w)$ etc, the coefficients before every terms will be canceled to 0 using the following relations:
\begin{gather}
u_{xx} = 2u^3+xu\\
u_{xxx} = 6uu_x+u+xu_x
\end{gather}
\section*{Acknowledgment}
I am very grateful to my supervisor Professor Jeremy Quastel for suggesting this problem to me. He gave me many invaluable guidance and discussions on this topic, besides he gave me many important suggestions on my writing of the paper.
\bibliographystyle{alpha}
\bibliography{pde_baikrains}
\end{document}
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\begin{document}
\baselineskip=15.5pt
\title{Moduli of filtered quiver representations}
\author{Sanjay~Amrutiya}
\address{Department of Mathematics, IIT Gandhinagar,
Near Village Palaj, Gandhinagar - 382355, India}
\email{samrutiya@iitgn.ac.in}
\author{Umesh~V.~Dubey}
\address{ Harish-Chandra Research Institute, HBNI,
Chhatnag Road, Jhunsi, Allahabad - 211019, India}
\email{umeshdubey@hri.res.in}
\subjclass[2000]{Primary: 14D20, 16G20}
\keywords{Moduli spaces, Filtered objects, Representations of quivers}
\date{}
\begin{abstract}
\noindent In this paper, we give a construction of the moduli space of filtered
representations of a given quiver of fixed dimension vector with the appropriate
notion of stability. The construction of the moduli of filtered representations uses
the moduli of representations of ladder quiver. The ladder quiver is introduced using
a given quiver and an $A_n$-type quiver. We also study determinantal
theta functions on such moduli spaces.
\end{abstract}
\maketitle
\section{Introduction}
In \cite{Ki94}, A. King has used geometric invariant theory to construct moduli of
quiver representations. Since the category of filtered representations may not be an
abelian category, we cannot apply the methods of A.~King \cite{Ki94} directly for
the construction.
Once we suitably choose the stability parameter, we get a quasi-abelian category of
semistable representations having a fixed slope.
We generalize the approach of \cite{Ki94} in the filtered representation category
setup via the concept of slope stability studied by Andr\'e \cite{An09} for
quasi-abelian categories.
The moduli construction in this note will be used in the forthcoming work on the
functorial moduli construction of parabolic sheaves. Besides this, it is
interesting to investigate some geometric properties of moduli of filtered quiver
representations, in general.
We first review some basic properties of filtered representations of a quiver $Q$
in Section \ref{sec-2}. We introduce the ladder quiver $\QAl$ (see Section
\ref{ladder_quiver}), an admissible ideal in the path algebra of a ladder quiver.
We describe the category of filtered representations of a quiver $Q$ in terms of
representations of the ladder quiver with relations.
We study the slope functions introduced by Andr\'e \cite{An09} in Section \ref{sec-rank-fn}.
Using the concept of a maximal flag, we generalize the definition of $S$-equivalence
for filtered representations. We also give some examples of slope functions.
In Section \ref{sec-git-mc}, we construct the moduli space of filtered quiver
representations (see Theorem \ref{main-theorem}). We prove that there is a canonical
morphism from this moduli space to the moduli space of representations of a ladder
quiver. For an appropriate choice of slope function, we show that this canonical
morphism is an open immersion (see Corollary \ref{cor-open-immersion}).
In the final Section \ref{sec-det-fns}, we discuss a characterisation of semistability of filtered
representations in terms of determinantal theta functions using the results of
\cite{DW2}. We also study the corresponding morphism on such moduli spaces into a
projective space.
\noindent \textbf{Notations:}
Let $\mathbb{K}$ be an algebraically closed field.
\begin{itemize}
\item $A$\text{-mod} = the category of finitely generated (left) modules over an algebra $A$.
\item $\catQAl$ = the category of ($\mathbb{K}$-linear) representation of $\QAl$.
\item $\catchain$ = the category of ($\mathbb{K}$-linear) representation of $\QAl/I$,
i.e., the category of
representation with the admissible ideal of relations $I$ defined in \ref{ladder_quiver}.
\item $\catfilt$ = the category of ($\mathbb{K}$-linear) filtered representation of $Q$
of length at most $\ell$.
\item $\Sch$ = the category of all $\mathbb{K}$-schemes.
\end{itemize}
\section{Filtered quiver representations}\label{sec-2}
Let $Q = (Q_0, Q_1, s, t)$ be a (finite) quiver without oriented cycles, and let
$\mathbb{K}Q$ be its path algebra which is a finite dimensional $k$-algebra.
We will some time ignore the maps $s$ and $t$ when it is clear from the context,
for example, in the case of linear quiver $\mathbf A_\ell$ the vertex set is
$\{ 1, \ldots, \ell \}$ with the set of arrows $\{ 1, \ldots, \ell-1 \}$ and the
map $s$ (respectively, $t$) is the canonical inclusion (respectively, the shift
$j \mapsto j +1$). The abelian category $\mathbb{K}Q$-mod is equivalent to the
category of $\mathbb{K}$-linear representations of $Q$, see \cite{Re08} for more
details. We will also use $M(v)\,, v \in Q_0,$ (respectively, $M(a); a \in Q_1$)
to denote a vector space (respectively, a linear map) of a representation
corresponding to the $\mathbb{K}Q$-module $M$ at the vertex $v$ (respectively,
the arrow $a$) under this equivalence.
\begin{definition} \label{ladder_quiver}\rm{
If $Q= (Q_0, Q_1, s, t)$ is a quiver and $\mathbf A_\ell =
(\mathbf A_{\ell 0}, \mathbf A_{\ell 1})$ is a linear quiver with $\ell$ vertices.
Then the ladder quiver is defined as
$$
\QAl := ({\mathbf A_{\ell 0}} \times Q_0, {\mathbf A_{\ell 0 }} \times Q_1
\sqcup {\mathbf A_{\ell 1}} \times Q_0, s_\ell , t_\ell ).
$$
To simplify the notation, let $\alpha_i^a := (i , a) \in
{\mathbf A_{\ell 0}} \times Q_1$ for $i = 1, \ldots, \ell$
(or $i \in \mathbf A_{\ell 0}$) and $\beta_j^v := (j , v) \in
{\mathbf A_{\ell 1}} \times Q_0$ for $j=1, \ldots, (\ell-1)$
(or $j \in \mathbf A_{\ell 1}$) represents the arrows in the ladder quiver $\QAl$.
Then the source map for ladder quiver is defined as
$s_\ell (\alpha_i^a) = (i, s(a)), s_\ell (\beta_j^v) = (j, v)$ and the target map
is defined as $t_\ell (\alpha_i^a) = (i , t(a))$ and $ t_\ell (\beta_j^v) = (j +1 , v) $.
Now using this notation for arrows, we can define the admissible ideal $I$ as the
ideal generated by
$ \beta_k^{t(a)} \alpha_k^a - \alpha_{k +1}^a \beta_k^{s(a)}$, where $a\in Q_1$ and
$k = 1, \ldots, (\ell - 1)$ (or $k \in \mathbf A_{\ell 1}$).
}
\end{definition}
We shall denote by $\mathbb{K}(\QAl)$ the path-algebra of the quiver $\QAl$.
\begin{remark}\rm{
The definition of ladder quiver is a special case of more general notion of tensor
product of two quivers defined in \cite[Section 3]{Ke13}
\footnote{We are grateful to B. Keller for pointing out this remark after the first
version of this manuscript was put up on Arxiv.}. Moreover, the tensor product of
path-algebras $\mathbb{K}\mathbf{A}_\ell\otimes_{\mathbb{K}} \mathbb{K}Q$ is isomorphic
to the algebra $\mathbb{K}(\QAl)/I$.
}
\end{remark}
\begin{convention} \label{sink_identification}\rm{
We will identify the vertex set $Q_0$ embedded in the vertex set of ladder quiver as
$ \{\ell\} \times Q_0$, where the vertex $\ell$ is the sink of a linear quiver.
}
\end{convention}
We will now give some examples of ladder quivers of a given quiver.
\begin{example} \rm{
The quiver $\mathbf{A}_\ell \times \mathbf{A}_3$ can be described as follows:
$$
\xymatrix{
\bullet \ar[d]_{\alpha_1^1} \ar[rr]^{\beta_1^1} && \bullet \ar[d]_{\alpha_2^1} & \cdots & \bullet
\ar[d]_{\alpha_{\ell-1}^1} \ar[rr]^{\beta_{\ell-1}^1} && \bullet \ar[d]_{\alpha_\ell^1} \\
\bullet \ar[rr]_{\beta_1^2} \ar[d]_{\alpha_1^2} && \bullet \ar[d]_{\alpha_2^2} & \cdots & \bullet
\ar[rr]_{\beta_{\ell-1}^2} \ar[d]_{\alpha_{\ell-1}^2} && \bullet \ar[d]_{\alpha_\ell^2} \\
\bullet \ar[rr]_{\beta_1^3} && \bullet & \cdots & \bullet \ar[rr]_{\beta_{\ell-1}^3} && \bullet
}.
$$
}
\end{example}
\begin{example} \rm{
Let $Q = (Q_0, Q_1, s, t)$ be a quiver defined as follows:
The set vertices
$
Q_0 = \{ v_1, v_2, v_3, v_4 \}
$
and the set of arrows
$
Q_1 = \{ a, b, c, d \}
$
with the source and target maps $s, t\colon Q_1 \ra Q_0$ given by
\[
\begin{array}{cc}
s(a) = v_1, & t(a) = v_2 \\
s(b) = v_1 & t(b) = v_3 \\
s(c) = v_2 & t(c) = v_4 \\
s(d) = v_3 & t(d) = v_4
\end{array}
\]
By re-labeling the vertex set of the ladder quiver $\mathbf{A}_3 \times Q$ as
$$
(\mathbf{A}_3 \times Q)_0 = \{v_1^1, v_1^2, v_1^3, v_1^4, v_2^1, v_2^2, v_3^3, v_4^4, v_3^1, v_3^2, v_3^3, v_3^4,
v_4^1, v_4^2, v_4^3, v_4^4\},
$$
we can described it as follows:
\[
\xymatrix@C=2.0cm{
& v_1^1 \ar[rr]^{\beta_1^{s(a)}} \ar[dl]_{\alpha_1^a} \ar[dd]^(.2){\alpha_1^b}|\hole
&& v_1^2 \ar[rr]^{\beta_2^{s(a)}} \ar[dd]_(.7){\alpha_2^b}|\hole \ar[dl]_{\alpha_2^a}
&& v_1^3 \ar[dd]^{\alpha_3^b} \ar[dl]_{\alpha_3^a} \\
v_2^1 \ar[rr]^(.75){\beta_1^{t(a)} = \beta_1^{s(c)}} \ar[dd]_{\alpha_1^c}
&& v_2^2 \ar[dd]^(.25){\alpha_2^c} \ar[rr]^(.75){\beta_2^{t(a)} = \beta_1^{s(c)}}
&& v_2^3 \ar[dd]^(0.25){\alpha_3^c} \\
& v_3^1 \ar[rr]^(0.25){\beta_1^{t(b)} = \beta_1^{s(d)}}|\hole \ar[dl]_{\alpha_1^d}
&& v_3^2 \ar[rr]^(0.25){\beta_2^{t(b)} = \beta_2^{s(d)}}|\hole \ar[dl]_{\alpha_2^d}
&& v_3^3 \ar[dl]^{\alpha_3^d} \\
v_4^1 \ar[rr]^{\beta_1^{t(d)} = \beta_1^{t(c)}}
&& v_4^2 \ar[rr]^{\beta_2^{t(d)} = \beta_2^{t(c)}}
&& v_4^3
}
\]
}
\end{example}
The filtered objects of an abelian category has been studied in \cite{S99, SS16}.
We will focus on category of quiver representations. We will also relate filtered
objects with representations of a ladder quiver.
\begin{definition}\rm{
A filtered representation of $Q$ of length at most $\ell$ is an increasing filtration
of length $\ell$ in the abelian category of representations of the quiver $Q$ over
$\mathbb{K}$. We will denote by $M$ the filtered representation, where $M_k$ is a
representation of $Q$ such that $M_{k - 1} \subseteq M_k$ is a subrepresentation,
for $k = 2, \ldots , \ell$ .
}
\end{definition}
\begin{remark}\rm{
We can realize the category of representations of quiver $Q$ inside the category of
filtered representations by taking $M_k = 0$ for $k \leq (\ell - 1)$ (cf. Convention \ref{sink_identification}). The filtered representation can be related to $\Z$-filtered
representation of Schneiders \cite[Definition 3.1.1]{S99}, by putting $M_k = M_\ell$
for $k \geq (\ell + 1)$ and $M_k = 0$ for $k \leq 0$.
}
\end{remark}
Next, we shall describe the category of filtered quiver representations as certain
functor category. This is also useful in getting the structure of quasi-abelian
category on the category of filtered quiver representations of a given quiver.
Let $\Lambda = \{1, 2, \dots, \ell\}$ be a pre-ordered set. We denote by
$\mathrm{Fct}(\Lambda, \mathbb{K}Q\text{-mod})$ the category of functors from
$\Lambda$ to $\mathbb{K}Q\text{-mod}$ \cite{SS16}. We can immediately see the
following equivalence of categories.
\begin{lemma}\label{lemma-1}
The categories $\mathrm{Fct}(\Lambda, \mathbb{K}Q\text{-mod})$ and $\catchain$
are equivalent, where $\catchain$ is the category of representations of $\QAl$ with
an admissible ideal of relations $I$ \rm{(see Definition \ref{ladder_quiver})}.
\end{lemma}
Let $\mathrm{F}_\Lambda(\mathbb{K}Q\text{-mod})$ be the full subcategory of
$\mathrm{Fct}(\Lambda, \mathbb{K}Q\text{-mod})$ consisting of filtered objects.
Then, the natural inclusion functor
\begin{equation}\label{eq-iota}
\iota\colon \mathrm{F}_\Lambda(\mathbb{K}Q\text{-mod}) \ra
\mathrm{Fct}(\Lambda, \mathbb{K}Q\text{-mod})
\end{equation}
is an embedding. Moreover, the inclusion functor $\iota$ has a left adjoint
\begin{equation}\label{eq-kappa}
\kappa \colon \catchain \ra \catfilt
\end{equation}
with $\kappa \circ \iota \simeq \id_{\mathrm{F}_\Lambda(\mathbb{K}Q\text{-mod})}$
(see \cite[Proposition 3.5]{SS16}). We shall use this adjunction in Section
\ref{sec-det-fns}. We shall also need following basic properties of the category of
filtered objects.
\begin{proposition}\label{prop: stable kernels}
The subcategory $\mathrm{F}_\Lambda(\mathbb{K}Q\text{-mod})$ of
$\mathrm{Fct}(\Lambda, \mathbb{K}Q\text{-mod})$ is stable by sub-objects. In particular,
the category $\mathrm{F}_\Lambda(\mathbb{K}Q\text{-mod})$ admits kernels and the
functor $\iota$ commutes with kernels. Moreover, the category
$\mathrm{F}_\Lambda(\mathbb{K}Q\text{-mod})$ is a quasi-abelian category
\end{proposition}
\begin{proof}
See \cite[Proposition 3.3]{SS16} or \cite[Proposition 3.1.17]{S99}.
\end{proof}
Under the equivalence of Lemma \ref{lemma-1}, to give an object $M$ of
$\mathrm{F}_\Lambda(\mathbb{K}Q\text{-mod})$
is equivalent to give a representation $M$ of $\QAl$ such that $M(r) = 0$ for all
$r\in I$ and $M(a)$ is injective for any arrow in the copies of $\mathbf{A}_\ell$
in $\QAl$. We shall denote by $\catfilt$ the full subcategory of $\catchain$
consisting of filtered quiver representations of $Q$ of length at most $\ell$.
We shall denote by $\iota : \catfilt \to \catchain$ the above embedding.
We get the following corollary of Proposition \ref{prop: stable kernels}.
\begin{cor}
The category $\catfilt$ is a quasi-abelian category. In fact, the category $\catchain$
can be identified with the (left) abelian envelope of $\catfilt$ via the functor $\iota$.
\end{cor}
\begin{proof}
The proof follows from \cite[Theorem 3.9]{SS16} (see also \cite[Proposition 3.1.17]{S99})
for the first part, and from \cite[Corollary 3.1.29]{S99} for the second part.
\end{proof}
By \cite[Proposition 1.2.14]{An09}, there exists a full strict subcategory $\mathcal{T}$
of $\catchain$ such that for every object $B\in \catchain$, there exists
$B_{\mathrm{tor}} \in \mathcal{T}$ and $M \in \catfilt$ and a short exact sequence
\begin{equation} \label{cotilting torsion extension}
0\ra B_{\mathrm{tor}}\ra B\ra M\ra 0 \;.
\end{equation}
Moreover, if $\mathrm{rk}(B) = 0$, then $B \in \mathcal{T}$ using \cite[Remark 1.2.15]{An09}.
\begin{lemma}\label{tf resolution}
There is a short exact sequence for any object $B$ of $\catchain$,
\[
0 \to N' \to N \to B \to 0
\]
where $N$ and $N'$ are filtered representations.
\end{lemma}
\begin{proof}
To get the epimorphism from filtered representation $f : N \to B$, define
$$
N(k , v) := \bigoplus_{j = 1}^k B(j , v)
$$
and, vertical maps are defined as in $B$ and horizontal maps are the canonical inclusions.
The epimorphism $f_{(k , v)}$ is defined using sum of maps
$B(j , v) \to B(k , v)$ defined by
$$
B(\beta^v_{k-1}) \circ \cdots \circ B(\beta^v_j)\,,
$$
whenever $j < k$ and for $j=k$ take identity map on that component.
Now, we can check that this gives a surjection from $N$ to $B$. Take $N' := \ker f$.
Then, $N'$ is again a filtered representation. We can check that $N'(1 , v) = 0$ and
for $k >1$, $N'(k , v)$ is a vector subspace generated by
$$
(0, \ldots, 0, -b_j, 0, \ldots, 0, [B(\beta^v_{k-1}) \circ \cdots \circ B(\beta^v_j)] (b_j) )
$$
for $j = 1, \ldots, k-1$ and $b_j \in B_j$.
\end{proof}
We can now describe the category $\mathcal{T}$ explicitly using the Convention
\ref{sink_identification}.
\begin{proposition}
Let $B$ be an object of $\catchain$.
The representation $B$ is in $\mathcal T$ if and only if $B(\ell , v) = 0$ for each
vertices $(\ell , v)$ of the quiver $\QAl$.
\end{proposition}
\begin{proof}
We can see that all objects with $B(\ell , v)=0$ for vertices $(\ell , v)$ satisfies the
conditions of cotilting torsion pair. That is, there is an epimorphism from filtered
representation to $B$ and there is a short exact sequence \ref{cotilting torsion extension},
as orthogonality $\Hom{B_{\mathrm{tor}}}{M} = 0$ is easy to check. The epimorphism follows
from Lemma \ref{tf resolution}.
Define
\[
B_{\mathrm{tor}}(w) =
\left\{\begin{array}{ll}
\ker (B(\beta^v_{\ell-1}))\,, & w = (\ell - 1, v) \\
0\,, & w = (\ell, v) \\
[B(\beta^v_{\ell-1}) \circ \cdots \circ B(\beta^v_j)]^{-1}(B_{\mathrm{tor}}(\ell-1 , v))\,, &
w = (j, v)\;, 1\leq j < \ell -1
\end{array}\right.
\]
We can check that this will define a torsion sub-object and quotient of this will be
filtered representation. This completes the proof.
\end{proof}
Recall, the \emph{kernel} of a morphism $f : M \to N$ in the category $\catfilt$
is defined as $\ker(f)_{(j, v)} := \ker(f_{(j, v)})$ with induced maps for each arrows.
The \emph{cokernel} of a morphism $f : M \to N$ in the category $\catfilt$ is defined as
$$
\mbox{coker}(f)_{(j, v)} := \mbox{Im}(M((j, v))\ra \mbox{coker}(f_{\ell, v}))
$$
with induced maps for each arrows (see \cite[Corollary 3.6]{SS16} or
\cite[Proposition 3.1.2]{S99}). In case, a sub-object (respectively, quotient object)
is kernel (respectively cokernel) of a morphism, then it is called a strict monic
(respectively, strict epi).
We can observe that there are monic in $\catfilt$
which are not strict monic. We can check that the monic morphism $f$ is strict monic,
if $\iota(\mbox{coker}(f)) = \mbox{coker}(\iota(f))$.
More generally, we get the following definition of \emph{strict} filtration using the
concept of \emph{strict} monic.
Let $M$ be an object of $\catfilt$. A filtration
$$
0 = M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_{k-1} \subset M_k = M
$$
of $M$ is called \emph{strict} filtration if each inclusion maps are \emph{strict}
monic morphisms of $\catfilt$ i.e. each quotient satisfies
$\iota(M_i / M_{i-1}) = \iota(M_i)/\iota(M_{i-1}) $.
We recall the following definition of \cite[Definition 1.2.6]{An09}.
\begin{definition}\label{maximal flag} \rm{
A strict filtration
$$
0 = M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_{k-1} \subset M_k = M
$$
is called a flag of length $k$ on $M$ if $M_i \neq M_{i - 1}$ for $1 \leq i \leq k$.
A flag of maximal length (on $M$) is called the maximal flag (on $M$).
}
\end{definition}
Notice that, a flag is \emph{maximal} if all the quotients $M_{i + 1} / M_i$ have no
proper nonzero strict sub-objects. Hence, in particular, if all the quotients of a
flag are simple objects then it is a maximal flag. Now, using lemma \cite[Lemma 1.2.8]{An09},
we can prove that any object $M$ of $\catfilt$ has a unique maximal flag on $M$.
Therefore, we get a well defined associated graded object of a maximal flag on $M$
\begin{equation}\label{max gr}
\mathrm{gr^{max}}(M):= \bigoplus_{i= 1}^k M_i/M_{i-1}.
\end{equation}
We now recall definition of projective objects which will be used later in Section
\ref{sec-det-fns}.
\begin{definition}
A filtered representation $P$ is called projective (respectively, strongly projective)
if for any strict (respectively, arbitrary) epimorphism $ M \to N$ the associated map
$\Hom[\catfilt]{P}{M} \to \Hom[\catfilt]{P}{N} $ is surjective.
\end{definition}
Dually, we can also define the injective objects in $\catfilt$, see
\cite[Section 1.3.4]{S99}.
The projective objects in an exact category is related to the projective objects
of its abelian envelope. In particular, for filtered representations, we get the
following results.
\begin{proposition}\label{filtered projective}\cite[Proposition 1.3.24]{S99}
An object $P$ of $\catfilt$ is projective if and only if $\iota(P)$ is projective in
$\catchain$. The category $\catfilt$ has enough projectives and any projective object of
$\catchain$ is same as $\iota (P)$ for some projective object $P$ of $\catfilt$.
\end{proposition}
\begin{proof}
Since $\catfilt$ is a strictly full subcategory and the left abelian envelope $\catchain$
has enough projective objects, the assertions immediately follow from
\cite[Proposition 1.3.24]{S99}.
\end{proof}
Recall that the indecomposable projective modules in $\catchain$ are in bijection
with the vertices of $\QAl$ and the module corresponding to a vertex $w$ is just
$P'_w = P_w/IP_w$, where $P_w$ is an indecomposable projective representation of
$\QAl$ corresponding to a vertex $w$ in $\QAl$. The indecomposable projective
modules in $\catchain$ are characterized by the property that for each object
$N'\in \catchain$ and for each vertex $v$ of $\QAl$, we have
$$
\Hom[\catchain]{P'_w}{N'} = N'(w),
$$
where $N'(w)$ is a vector space assigned to a vertex $w$ by the representation
$N'$ of $\QAl$. Now using the Proposition \ref{filtered projective}, the objects
$P_w'$ will become the projective indecomposable objects of $\catfilt$.
More precisely, the objects $P_w'$ are filtered representations, and they give the
complete set of projective indecomposable objects of $\catfilt$.
\section{Rank function and slope stability}\label{sec-rank-fn}
In this section, we first discuss the notion of degree and rank functions on the
category of filtered quiver representations following \cite{An09}. The concept
degree and rank functions provides a good definition of semistability for filtered
quiver representations to construct the moduli of filtered quiver representations
using GIT. More precisely, the corresponding slope semistability for
filtered quiver representations provides a good description of the closed points
of the moduli scheme. At the end of the section, we also give two kinds of slope
filtrations for which the corresponding moduli schemes have quite a different
description of closed points.
It is known that the Grothendieck group $K_0(\mathbb{K}Q)$ of the path algebra
$\mathbb{K}Q$ is a free abelian group of finite rank, and the rank of $K_0(\mathbb{K}Q)$
is precisely the cardinality of $Q_0$.
In fact, the algebra $\mathbb{K}Q$ is a hereditary algebra and simple modules are
in bijection with the vertex set $Q_0$. We get $K_0(\mathbb{K}Q) \simeq \Z Q_0$,
where $\Z Q_0$ is the free abelian group generated by the set $Q_0$. Hence, we can
identify an equivalence class of each $\mathbb{K}Q$-module $M$ in $K_0(\mathbb{K}Q)$
with the dimension vector of corresponding representation of the underlying quiver $Q$,
which we denote by $\dv(M)$.
Similar identification exists for any quiver with admissible ideal of relations
\cite[Section 4]{Ki94}. We can also identify the set of all additive functions
$\Hom{K_0(\mathbb{K}Q)}{\Z}$ with $ \Z^{Q_0}$. These additive functions are used to
give the definition of $\theta$-stability by A. King \cite{Ki94}.
Since $\catfilt$ is a quasi-abelian category, using a result of Schneider
\cite[Proposition 1.2.35]{S99}, we can get the isomorphism
$$
K_0(\iota): K_0(\catfilt) \simeq K_0(\catchain).
$$
Hence, we can identify $\Hom{K_0(\catfilt)}{\Z}$ with $ \Z^{Q_0}$ too, and therefore
any additive function on filtered representations is in bijection with additive
functions on representations of the ladder quiver.
Let $\mathrm{sk}(\catfilt)$ be the skeleton of $\catfilt$, that is, the set of
isomorphism classes of objects of $\catfilt$.
\begin{definition}\cite[Def. 1.2.9]{An09}\label{def-rk-fn}\rm{
A \emph{rank function} on $\catfilt$ is a function
$$
\mathrm{rk}\colon \mathrm{sk}(\catfilt) \ra \mathbb{N}
$$
which is additive on short exact sequences and takes the value $0$ only on the
zero object.
}
\end{definition}
The rank function will extend to give the additive map
$$
\mathrm{rk} \colon K_0(\catfilt) \to \Z ; (d_w) \mapsto \sum_{w\in (\QAl)_0} r_w d_w \,
$$
with positivity conditions, $\sum_{j = k}^l r_{(j, v)} > 0$ for each $1 \leq k \leq l$.
Clearly, if all $r_w > 0$ then these conditions are satisfied and we get the notion of
positive additive function in the sense of Rudakov \cite{Ru97}.
Given a $\Theta\in \Gamma := \Z^{(\QAl)_0}$, we get a function $\mathrm{sk}(\catfilt)\ra \Z$
given by
$$
\Theta(\dv(M)):= \sum_{w\in (\QAl)_0} \Theta_w \dv(M)_w \,.
$$
The additive function $\Theta$ is also called degree function. Hence, we get a group
homomorphism
$$
\Theta \colon K_0(\catfilt)\ra \Z\,.
$$
We define a slope function $\mu \colon \mathrm{sk}(\catfilt)-\{0\}\ra \mathbb{Q}$
as follows:
$$
\mu_{\Theta, \mathrm{rk}}(M):= \frac{\Theta(\dv(M))}{\mathrm{rk}(\dv(M))}\,.
$$
\begin{definition}\rm{
We say that a representation $M$ of a quiver $Q$ is $\mu_{\Theta, \mathrm{rk}}$-semistable,
if for all non-zero subrepresentations $M'$ of $M$, we have
$$
\mu_{\Theta, \mathrm{rk}}(\dv(M'))\leq \mu_{\Theta, \mathrm{rk}}(\dv(M))
$$
If the inequality is strict for all proper non-zero subrepresentations $M'$ of $M$,
then we say that $M$ is $\mu_{\Theta, \mathrm{rk}}$-stable.
}
\end{definition}
Since $K_0(\catfilt) \simeq K_0(\catchain)$, we get an extension of the rank function
$\mathrm{rk}$ to $K_0(\catchain)$. This extension of rank function may take value $0$
on some non-zero objects in $\catchain$. In view of this, we may not get slope function
on $K_0(\catchain)$ as an extension of $\mu_{\Theta, \mathrm{rk}}$. However, we can
choose an appropriate $\theta\in \Gamma$ so that we can relate the
$\mu_{\Theta, \mathrm{rk}}$-stability with the $\theta$-stability (defined in \cite{Ki94}).
Recall that for given any stability parameter $\theta\in \Gamma$, an object $M$ of
$\catchain$ is called $\theta$-semistable (respectively, $\theta$-stable) if
$\theta(\dv(M)) = 0$ and for any non-trivial sub-object $M'$ of $M$, we have
$\theta(\dv(M'))\geq 0$ (respectively, $\theta(M') > 0$).
Fix a $\Theta\in \Gamma$ and a dimension vector $\dv$. Let $\mathrm{rk}$ be a rank
function as in the Definition \ref{def-rk-fn}. Once we fix the slope
$\mu = \Theta(\dv) / \mathrm{rk}(\dv)$, then we can define
$$
\theta := (\theta_w)_{w\in (\QAl)_0} =
(\Theta(\dv)r_w - \Theta_w \mathrm{rk}(\dv))_{w\in (\QAl)_0}.
$$
\begin{proposition}
Any object $M$ of $\catfilt$ having slope $\mu$ is $\mu_{\Theta,\mathrm{rk}}$-semistable
(respectively, $\mu_{\Theta,\mathrm{rk}}$-stable) if and only if $ \iota(M)$ is
$\theta$-semistable (respectively, $\theta$-stable).
\end{proposition}
\begin{proof}
Let $M$ be an object of $\catfilt$ having slope $\mu$. If $M$ is
$\mu_{\Theta,\mathrm{rk}}$-semistable , then
for any non-zero subobject $M'$ of $M$, we have
$\mu_{\Theta,\mathrm{rk}}(M') \leq \mu_{\Theta,\mathrm{rk}}(M)$.
That is, we have
$$
\frac{\Theta(\dv(M'))}{\mathrm{rk}(\dv(M'))} \leq
\frac{\Theta(\dv(M))}{\mathrm{rk}(\dv(M))} = \mu =
\frac{\Theta(\dv)}{\mathrm{rk}(\dv)}\,.
$$
Now,
\[
\begin{array}{ll}
\theta(M')
& = \displaystyle \sum_{w\in (\QAl)_0} (\Theta(\dv)r_w - \mathrm{rk(\dv)\Theta_w})\dv(M')_w \\
& \\
& = \Theta(\dv)\mathrm{rk}(\dv(M')) - \mathrm{rk}(\dv)\Theta(\dv(M'))\\
& \\
& \geq 0\,.
\end{array}
\]
In view of Proposition \ref{prop: stable kernels}, we can conclude that $ \iota(M)$
is $\theta$-semistable. Conversely, if $ \iota(M)$ is $\theta$-semistable, then by
reversing the above arguments, we can see that $M$ is $\mu_{\Theta,\mathrm{rk}}$-semistable.
\end{proof}
\begin{proposition}\label{abelian-condition}
Given $\Theta\in \Gamma$ such that the degree function is non-negative on torsion
subcategory, the full subcategory $\catfilt(\mu)$ consisting of zero object and the
$\mu_{\Theta, \mathrm{rk}}$-semistable filtered representations of fixed
slope $\mu$ is a quasi-abelian category.
\end{proposition}
\begin{proof}
Assume that $\Theta$ is chosen in such a way that for each epi-monic $M \ra M'$,
we have $\mu_{\Theta, \mathrm{rk}}(M) \leq \mu_{\Theta, \mathrm{rk}}(M')$. Then, we
get a slope function in the sense of \cite[ Definition 1.3.1]{An09}. This is also
equivalent to choosing a degree function which is non-negative on objects of torsion
subcategory, see \cite[Corollary 1.4.10]{An09}. The result follows using
\cite[Lemma 1.3.9]{An09}.
\end{proof}
Recall that the $\theta$-semistable objects in $\catchain$ forms an abelian full
subcategory of $\catchain$. It is both Noetherian and Artinian, and hence we
have Jordan-H\"older filtration for each $\theta$-semistable object in $\catchain$.
\begin{proposition} \label{strict JH}
If there is a strict filtration of any object $M$ in $\catfilt(\mu)$ with the
successive quotients $\mu_{\Theta,\mathrm{rk}}$-stable, then the filtration obtained
after applying $\iota$ is a Jordan-H\"older filtration of $\iota(M)$.
\end{proposition}
\begin{proof}
Since the filtration is a strict filtration, and hence the successive cokernels are
preserved under the functor $\iota$. We can observe that the category $\catfilt(\mu)$
is closed under sub-objects inside $\catchain$, where the stability on $\catchain$
is given by an appropriate $\theta$. Hence, $\mu_{\Theta, rk}$-stable objects in
$\catfilt$ goes to $\theta$-stable objects in $\catchain$ via the functor $\iota$.
Now using the fact that the stable objects are simple objects and hence after applying
$\iota$, we get Jordan-H\"older filtration.
\end{proof}
\begin{definition}\rm{
Given an object $M\in \catfilt(\mu)$, a strict filtration
$$
0 = M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_{k-1} \subset M_k = M
$$
such that all the quotients $M_i/M_{i-1}$ are $\mu_{\Theta, \mathrm{rk}}$-stable
objects in $\catfilt$ having the same slope, is called the strict Jordan-H\"older
filtration.
}
\end{definition}
Since any stable object is nonzero, a strict Jordan-H\"older filtration gives an
example of maximal flag (see Definition \ref{maximal flag}) in the category
$\catfilt(\mu)$. Using the Proposition \ref{strict JH}, the Jordan-H\"older theorem
is valid for strict Jordan-H\"older filtrations in $\catfilt(\mu)$.
Hence, we get the well defined associated graded objects of a strict Jordan-H\"older
filtration. The associated graded object of a strict Jordan-H\"older filtration is
defined as
$$
\mathrm{gr}(M):= \bigoplus_{i= 1}^k M_i/M_{i-1}
$$
and it depends only on $M$.
An object $N$ is called $\mu_{\Theta, \mathrm{rk}}$-polystable
if $N \simeq \mathrm{gr}(M)$ for some object $M$ with a strict Jordan-H\"older filtration.
\begin{definition}\rm{
We say that two objects $M$ and $N$ in $\catfilt(\mu)$ are $S^{JH}$-equivalent if
both objects have strict Jordan-H\"older filtration and the associated graded
objects $\mathrm{gr}(M)$ and $\mathrm{gr}(N)$ are isomorphic. We say that $M$ and
$N$ in $\catfilt(\mu)$ are $S$-equivalent if $\mathrm{gr^{max}}(M)$ and
$\mathrm{gr^{max}}(N)$ are isomorphic.
}
\end{definition}
Now using the above definition, we get the following Corollary of Proposition \ref{strict JH}
\begin{cor} \label{cor: gr comm}
The restriction of the functor $\iota$ \rm{(see \eqref{eq-iota})} to
$\catfilt(\mu)$ takes $S^{JH}$-equivalent objects to $S$-equivalent objects.
Moreover, $\iota (\mathrm{gr}(M)) \simeq \mathrm{gr}(\iota(M))$.
\end{cor}
\begin{proof}\label{rem-s-equivalence}
Let $M$ and $N$ be two $S^{JH}$-equivalent objects in $\catfilt(\mu)$. Since
$\mu_{\Theta,\mathrm{rk}}$-stability and $\theta$-stability are same for objects
of $\catfilt$, we can define S-equivalence with respect to $\theta$-stability for
objects of $\catfilt(\mu)$. Using Proposition \ref{prop: stable kernels}, we know
that sub-objects, and hence filtrations of $M$ (respectively, $N$) coincide via
$\iota$, and hence Jordan-H\"older filtration does not change under $\iota$. Hence,
if we view the objects $M$ and $N$ in $\catchain$, then they remain
$S$-equivalent in $\catchain$ with respect to the notion of $\theta$-stability.
\end{proof}
\begin{remark}\rm{
We should mention here that it may be possible that $M$ and $N$ are not $S^{JH}$-equivalent in
$\catfilt(\mu)$, but they are $S$-equivalent in $\catchain$. This is also reflected in the
GIT picture (see Section \ref{sec-git-mc}).
}
\end{remark}
\begin{remark}\rm{
Let $\Theta \in \Gamma$ as in Proposition \ref{abelian-condition}, then there exists a functorial slope decreasing strict
filtration (slope filtration \cite[Proposition 1.4.6, Theorem 1.4.7]{An09}) which will be called Harder-Narasimhan filtration
(or HN-filtration) of filtered representations. Moreover,
the HN-filtration of $\iota(M)$ is same as image of HN-filtration of $M$.
}
\end{remark}
\begin{example}\rm{
Define a rank function
$\mathrm{rk}_{\dim}\colon \mathrm{sk}(\catfilt) \ra \Z$
by
$$
\mathrm{rk}_{\dim} (M):= \sum_{v\in Q_0}\dim M(v) +
\sum_{(j, v) \in \mathbf{A_{\ell 1}} \times Q_0} \mathrm{rank} \beta_j^v\,
$$
where $M(v) := M(\ell, v)$ (see Convention \ref{sink_identification}).
In other words, we have $r_w = 1$ for all $w\in (\QAl)_0$. The extension of this rank function
$\mathrm{rk}_{\dim}$ on $K_0(\catchain)$ is precisely the $\dim$ function which is given by
$$
\dim (\dv) := \sum_{w \in (\QAl)_0} d_w \,.
$$
This can also be seen using the description of $N$ and $N'$ as in Lemma \ref{tf resolution} by
computing the dimensions, namely
$$
\dim N = \sum_{j = 1}^\ell (\ell - j +1) \dim B_j
$$
and
$$
\dim N' = \sum_{j = 1}^\ell (\ell - j) \dim B_j.
$$
Using the additivity of $\mathrm{rk}_{\dim}$, we can check that $\mathrm{rk}_{\dim}(B) = \dim (B)$.
In particular, $\mathrm{rk}_{\dim}(B) = \dim (B) = 0$ if and only if $B \simeq 0$ for each object $B$ of $\catchain$.
Since the function $\mathrm{rk}_{\dim}$ is nonzero on some torsion classes, we can not use
\cite[Corollary 1.4.10]{An09} to get the abelian category structure on $\catfilt(\mu)$.
}
\end{example}
\begin{example}\label{ex:sink slope}\rm{
We can define another positive additive function given by dimension at vertices $(\ell , v)$, say
$\mathrm{rk}_s (B) := \sum_{v \in Q_0} r_v \dim B(\ell , v)$, where $r_v > 0$.
The arguments given in the above example (using Lemma \ref{tf resolution}) shows that $\mathrm{rk}_s (B) = 0$
if and only if $ B$ is isomorphic to an object of
$\mathcal T$. We can also define the slope using this new positive additive function, say
$\mu_s (B) := \Theta(\dv(B)) / \mathrm{rk}_s(B)$. Now using \cite[Corollary 1.4.10]{An09}, we can get the
abelian category structure on $\catfilt(\mu)$ with respect to the stability $\mu_s$, if $\Theta\in \Gamma$ is
chosen as in the Proposition \ref{abelian-condition}. Hence, in this case, the Jordan-H\"older
theorem holds in $\catfilt(\mu)$. Consequently, the notion of $S^\mathrm{JH}$-equivalence and $S$-equivalence
coincides, in such cases.
}
\end{example}
\begin{remark}\rm{
If rank function is zero for all objects of torsion subcategory, then it is of the form $\mathrm{rk}_s$ of example \ref{ex:sink slope}.
}
\end{remark}
\section{GIT and Moduli construction}\label{sec-git-mc}
In this section, we consider the moduli problem for filtered quiver
representations with the slope stability using rank function
(see Definition \ref{def-rk-fn}).
We first recall some basic definitions.
Let $\Sch^\mathrm{op}$ be the opposite category of $\mathbb{K}$-schemes and $\Set$
be the category of sets. For a scheme $Z$, its functor of points
$$
\underline{Z}\colon \Sch^\mathrm{op}\ra \Set
$$
is given by $X\mapsto \mathrm{Hom}(X, Z)$. By Yoneda Lemma, every natural transformation
$\underline{Y}\ra \underline{Z}$ is of the form $\underline{f}$ for some morphism of schemes
$f\colon Y\ra Z$.
\begin{definition}\cite[\S 1, p. 60]{Si94}\rm{
Let $\M \colon \Sch^\mathrm{op}\ra \Set$ be a functor, $\Ms$ a scheme and
$\psi\colon \M\ra \underline{\Ms}$ a natural transformation. We say that
$(\Ms, \psi)$ co-represents $\M$ if for each scheme $Y$ and each natural
transformation $h\colon \M\ra \underline{Y}$, there exists a unique $g\colon \Ms\ra Y$
such that $h = \underline{g}\circ \psi$, that is the following diagram
\[
\xymatrix{
\M \ar[d]_\psi \ar[rd]^h \\
\underline{\Ms} \ar[r]_{\underline{g}} & \underline{Y}
}
\]
commutes.
}
Let $\M_1, \M_2 \colon \Sch^\mathrm{op}\ra \Set$ be two functors. A morphism of
functors $g\colon \M_1 \ra \M_2$ is said to be a local isomorphism, if it induces
an isomorphism of sheafification in the Zariski topology.
\end{definition}
Note that if $g\colon \M_1 \ra \M_2$ is a local isomorphism, then the condition that
$(\Ms_1, \psi_1)$ co-represents $\M_1$ is equivalent to the condition that
$(\Ms_2, \psi_2)$ co-represents $\M_2$ \cite[\S 1, p. 60]{Si94} or \cite[Lemma 4.7]{AK07}.
\subsection{Representation spaces and GIT}\label{subsec-git}
Let us fix $\Theta\in \Gamma$ and the dimension vector $\dimv{d}$ for the quiver
$\QAl$. Let
$$
\rs := \bigoplus_{a\in (\QAl)_1}
\Hom[\mathbb{K}]{\mathbb{K}^{d_{s(a)}}}{\mathbb{K}^{d_{t(a)}}}
$$
be the space of representations of $\QAl$ having dimension vector $\dimv{d}$.
Let $\rsc$ be the closed subset of $\rs$ consisting of $(\alpha_a)$ satisfying the
relation in $I$. Let $\rsf$ be an open subset of $\rsc$ consisting of
$(\alpha_a)\in \rsc$ such that for any $a\in \mathbf A_{\ell 1} \times Q_0$, we have
$\alpha_a$ injective map.
There is a natural action of the group
$$
G(\QAl):= \displaystyle \prod_{w\in (\QAl)_0} \mathrm{GL}(\mathbb{K}^{d_w})
$$
on $\rs$ such that the isomorphism classes in $\catQAl$ correspond to the orbits in
$\rs$ with respect to this action. Moreover, $\rsc$ and $\rsf$ are invariant under
this action. Note that $\rsc$ is a closed subscheme of $\rs$, while $\rsf$ is a
locally closed subscheme of $\rs$.
Let $\rs^\mathrm{ss}$ (respectively, $\rsc^\mathrm{ss}$, $\rsf^\mathrm{ss}$) be the
open subset of $\rs$ (respectively, $\rsc$, $\rsf$) which is
$\mu_{\Theta, \mathrm{rk}}$-semistable locus in $\rs$ (respectively, $\rsc$, $\rsf$).
Let $G:= G(\QAl)/\Delta$, where
$\Delta:= \{(t\mathbf{1}_{\mathbb{K}^{d_w}})_{w\in (\QAl)_0}\,|\, t\in \mathbb{K}^*\}$
and there is an action of this group $G$ on $\rs$.
Let $\chi_{\theta} \colon G\ra \mathbb{K}^*$ be the character defined by
\begin{equation}\label{eq-chi-theta}
\chi_{\theta}((g_w)):= \prod_{w\in (\QAl)_0}
\det(g_w)^{\big(\Theta(\dv)r_w - \mathrm{rk}(\dv)\Theta_w\big)}
\end{equation}
where $\Theta\in \Gamma$.
\begin{proposition}
As a special case of \cite[Proposition 3.1 \& 3.2]{Ki94}, we have the following:
\begin{enumerate}
\item A point $x\in \rs$ is $\chi_{\theta}$-semistable if and only if the corresponding object
$M_x\in \catQAl$ is $\mu_{\Theta, \mathrm{rk}}$-semistable.
\item Two points $x, y \in \rs^\mathrm{ss}$ are GIT-equivalent (with respect to $\chi_{\theta}$)
if and only if the corresponding objects $M_x$ and $M_y$ are $S$-equivalent in $\catQAl$
(with respect to $\theta$-stability).
\item A point $x\in \rsc$ is $\chi_{\theta}$-semistable if and only if the corresponding object
$M_x\in \catchain$ is $\mu_{\Theta, \mathrm{rk}}$-semistable.
\item Two points $x, y \in \rsc^\mathrm{ss}$ are GIT-equivalent (with respect to $\chi_{\theta}$)
if and only if the corresponding objects $M_x$ and $M_y$ are $S$-equivalent in $\catchain$
(with respect to $\theta$-stability).
\end{enumerate}
\end{proposition}
\subsection{Moduli functors}
A flat family of $\mathbb{K}(\QAl)$-modules over a connected scheme $S$ is a
locally-free sheaf $\sheaf{F}$ over $S$ together with a $\mathbb{K}$-algebra
homomorphism $\mathbb{K}(\QAl)\ra \mathrm{End}(\sheaf{F})$.
On the other hand, a flat family of representations of $\QAl$ is a representation
of $\QAl$ in the category of locally-free sheaves over $S$. The equivalence between
the $\mathbb{K}(\QAl)$-modules and the representations of $\QAl$ extends naturally
to families.
Consider the functor
$$
\mf(\dv) \colon \Sch^\mathrm{op}\ra \Set ~(\mbox{respectively, }
\mtf(\dv) \colon \Sch^\mathrm{op}\ra \Set)
$$
defined by assigning to each $\mathbb{K}$-scheme $S$ the set of isomorphism classes
of flat families over $S$ of $\mu_{\Theta, \mathrm{rk}}$-semistable
(respectively, $\mu_{\Theta, \mathrm{rk}}$-stable) representations of $\QAl$ having
dimension vector $\dimv{d}$.
There is a natural functor $h\colon \fp{\rs^\mathrm{ss}} \ra \mf(\dv)$
(defined by $(f\colon S\ra \rs^\mathrm{ss}) \mapsto [f^*\mathbb{M}]$, where $\mathbb{M}$
is a tautological family on $\rs^\mathrm{ss}$), which induces a local isomorphism
$\tilde{h}\colon \fp{\rs^\mathrm{ss}}/\fp{G} \ra \mf(\dv)$.
This reduces the problem to the existence of good quotient of $\rs^\mathrm{ss}$ by $G$.
It is proved in \cite{Ki94} that the GIT quotient $\pi \colon \rs^\mathrm{ss}\ra \ms(\dv)$
exists and it is a good quotient.
\begin{theorem}\cite{Ki94}
There exist a projective variety $\ms(\dv)$ over $\mathbb{K}$ which co-represents
the moduli functor $\mf(\dv)$. In particular, the closed points of $M$ corespond to
the $S$-equivalence classes of representations of $\QAl$ having dimension vector $\dimv{d}$.
Moreover, there exists a quasi-projective variety $\mst(\dv)$ co-representing the moduli
functor $\mtf(\dv)$ whose closed points correspond to isomorphism classes of stable
representations.
\end{theorem}
Consider the functor
$$
\mfc(\dv) \colon \Sch^\mathrm{op}\ra \Set ~(\mbox{respectively, }
\mtfc(\dv) \colon \Sch^\mathrm{op}\ra \Set)
$$
defined by assigning to each $\mathbb{K}$-scheme $S$ the set of isomorphism classes
of flat families over $S$ of $\mu_{\Theta, \mathrm{rk}}$-semistable
(respectively, $\mu_{\Theta, \mathrm{rk}}$-stable) objects of $\catchain$ having
dimension vector $\dimv{d}$.
By restricting the tautological family $\mathbb{M}$ on $\rs^\mathrm{ss}$ to
$\rsc^\mathrm{ss}$, we get a natural functor $\fp{\rsc^\mathrm{ss}} \ra \mfc(\dv)$.
By \cite[Proposition 5.2]{Ki94}, it induces a local isomorphism
$\tilde{h}_\mathrm{rel}\colon \fp{\rsc^\mathrm{ss}}/\fp{G} \ra \mfc(\dv)$ reducing
the problem to the existence of good quotient $\rsc^\mathrm{ss}$ by $G$. It is proved
in \cite[\S 4]{Ki94} that the the GIT quotient
$\pi_\mathrm{rel} \colon \rsc^\mathrm{ss}\ra \msc(\dv)$ exists and it is a good quotient.
\begin{theorem}\cite{Ki94}
There exist a projective variety $\msc(\dv)$ over $\mathbb{K}$ which co-represent the
moduli functor $\mfc(\dv)$. In particular, the closed points of $\msc(\dv)$ correspond
to the $S$-equivalence classes of objects of $\catchain$ having dimension vector $\dimv{d}$.
Moreover, there exists a quasi-projective variety $\mstc(\dv)$ co-representing the
moduli functor $\mtfc(\dv)$ whose closed points correspond to isomorphism classes
of stable representations.
\end{theorem}
The following result stated for ladder quiver, however, remains valid for any finite
quiver (without oriented cycles) with admissible ideal.
\begin{proposition}\label{prop: rel embedding}
There is a closed set-theoretic embedding $\psi \colon \msc(\dv) \ra \ms(\dv)$. If
characteristic of $\mathbb{K}$ is zero, then $\psi$ is closed scheme-theoretic embedding.
In positive characteristic, the morphism $\psi$ is scheme-theoretic embedding on the
stable locus $\mstc(\dv)$.
\end{proposition}
\begin{proof}
Since $\pi_{\rm rel}$ is a categorical quotient, we get the following commutative
diagram of topological spaces
\[
\xymatrix{
\rsc^\mathrm{ss} \ar[d]_{\pi_\mathrm{rel}} \ar[r]^j
& \rs^\mathrm{ss} \ar[d]^\pi \\
\msc(\dv) \ar[r]_\psi & \ms(\dv)
}.
\]
Since $j$ is a closed embedding, $\pi$ and $\pi_{\rm rel}$ are good quotients, we can
see that the map $\psi$ is a closed embedding.
Note that $\msc(\dv) = \pi(\rsc^\mathrm{ss})$ in characteristic zero, using Reynold's
operator and in characteristic $p$, we can use properties of geometric quotient on
stable locus to get the scheme theoretic embedding (cf. \cite[Proposition 6.7]{AK07}).
\end{proof}
Now, using $\mu_{\Theta, \mathrm{rk}}$-semistability, we define moduli
functor for filtered quiver representations. Consider the functor
$$
\mff(\dv) \colon \Sch^\mathrm{op}\ra \Set ~(\mbox{respectively, }
\mtff(\dv) \colon \Sch^\mathrm{op}\ra \Set)
$$
defined by assigning to each $\mathbb{K}$-scheme $S$ the set of isomorphism classes of
flat families over $S$ of $\mu_{\Theta, \mathrm{rk}}$-semistable (respectively,
$\mu_{\Theta, \mathrm{rk}}$-stable) objects of $\catfilt$ having dimension vector
$\dimv{d}$.
In the remaining part of this section, we reduce the problem of co-representability
of this moduli
functor to the existence of GIT quotient, and describe the closed points of the
corresponding moduli space (co-representing scheme) using the results of previous
sections.
\subsection*{Moduli construction}
The following Proposition is key to give an algebraic description for two points in the
representation space $\rsf$ to be GIT equivalent (see Proposition \ref{prop:S-eq}).
\begin{proposition}\label{prop-1-ps-filtration} Let $x\in \rsf$. Then,
there is a surjection from the set
$$
\{1\text{-}PS\; \lambda\colon \mathbb{K}^*\ra G\; \mbox{such that}\;
\lim \lambda(t)\cdot x\; \mbox{exist in}\; \rsf \}
$$
to
$$
\{\mbox{strict filtrations of}\; M_x\; \mbox{in}\; \catfilt \} .
$$
\end{proposition}
\begin{proof}
Let $\lambda\colon \mathbb{K}^*\ra G$ be an one-parameter subgroup of $G$ such that the
$\lim_{t\rightarrow 0} \lambda(t)\cdot x = y$ exist in $\rsf$. Following \cite{Ki94},
we have a weight space decomposition
$$
M_x(w) = \bigoplus_{n\in \Z} M_x(w)^n
$$
for each vertex $w\in (\QAl)_0$, where $\lambda(t)$ acts on the weight space $M_x(w)^n$
as multiplication by
$t^n$. For each arrow $a$, we have $M_x(a) = \bigoplus M_x(a)^{mn}$, where
$$
M_x(a)^{mn}\colon M_x(s(a))^n\ra M_x(t(a))^m\,.
$$
Moreover, we have
$$
\lambda(t)\cdot x = \bigoplus t^{m-n}M_x(a)^{mn}\,.
$$
Since the limit $\lim_{t\rightarrow 0} \lambda(t)\cdot x$ exist, we have
$M_x(a)^{mn} = 0$ for all $m < n$. Let
$$
M_x(w)^{\geq n} := \bigoplus_{m\geq n} M_x(w)^m\,.
$$
Then, $M(a)$ gives a map $M_x(s(a))^{\geq n}\ra M_x(t(a))^{\geq n}$ for all $n$.
These subspaces determine subrepresentations $M_{x_n}$ of $M_x$ for all $n$.
Since $y\in \rsf$, it follows that the corresponding filtration
$$
\cdots \supseteq M_n \supseteq M_{n+1} \supseteq \cdots
$$
where $M_n = M_x$ for $n \ll 0$ and $M_n = 0$ for $n \gg 0$, is strict filtration
of $M_x$ in $\catfilt$. We also have
$$
M_y \cong \bigoplus_{n\in \Z} M_n/M_{n+1}\,.
$$
Conversely, suppose we have an strict filtration
$$
\cdots \supseteq M_n \supseteq M_{n+1} \supseteq \cdots
$$
of $M_x$. This will determine a one-parameter subgroup $\lambda$
(by reversing the proof of first part or using the weight space decomposition at
each vertex) such that the $\lim_{t\rightarrow 0} \lambda(t)\cdot x = y$ exist
and $M_y \cong \bigoplus_{n\in \Z} M_n/M_{n+1} \in \catfilt$.
\end{proof}
Now onwards, we fix $\Theta\in \Gamma$ as in the Proposition \ref{abelian-condition}.
Using Hilbert-Mumford criterion, we have the following:
\begin{proposition}\label{prop:semistab}
A point $x\in \rsf$ is $\chi_{\theta}$-semistable (respectively, $\chi_{\theta}$-stable)
if and only if the corresponding object $M_x\in \catfilt$ is
$\mu_{\Theta, \mathrm{rk}}$-semistable (respectively, $\mu_{\Theta, \mathrm{rk}}$-stable).
\end{proposition}
\begin{proposition}\label{prop-closed-orbits}
Let $x\in \rsf^{\mathrm{ss}}$. Then, the orbit of $x$ in $\rsf^{\mathrm{ss}}$ is
closed if and only if the corresponding filtered representation $M_x$ is isomorphic
to $\mathrm{gr^{max}}(M)$ in $\catfilt$.
\end{proposition}
\begin{proof}
Suppose that the orbit of $x$ in $\rsf^{\mathrm{ss}}$ is closed. If the corresponding
filtered representation $M_x$ does not admit any strict filtration in $\catfilt$, then
we are done. If there exists a strict filtration of $M_x$, then it also admits a maximal
flag. By Proposition \ref{prop-1-ps-filtration} and \cite[Proposition 2.6]{Ki94}, we can
conclude that $M_x$ is isomorphic to the associated graded of a maximal flag of $M_x$.
This proves that $M_x$ is isomorphic to $\mathrm{gr^{max}}(M)$. The converse is immediate
from Proposition \ref{prop-1-ps-filtration} and \cite[Proposition 2.6]{Ki94}.
\end{proof}
\begin{proposition} \label{prop:S-eq}
Two points $x, y\in \rsf^{\mathrm{ss}}$ are GIT equivalent, i.e.,
$$
\overline{Gx}\cap \overline{Gy}\cap \rsf^{\mathrm{ss}} \neq \emptyset
$$
if and only the corresponding
objects $M_x$ and $M_y$ are $S$-equivalent in $\catfilt$.
\end{proposition}
\begin{proof}
By \cite[Proposition 2.6]{Ki94}, if $x$ and $y$ are GIT equivalent, then there exist
one-parameter subgroups $\lambda_1$ and $\lambda_2$ such that the integral pairing
$\langle \chi_{\theta}, \lambda_1 \rangle = \langle \chi_{\theta}, \lambda_2 \rangle = 0 $
and $\lim_{t\ra 0} \lambda_1(t)\cdot x$ and $\lim_{t\ra 0} \lambda_2(t)\cdot y$ are in
the same closed $G$-orbit in $\rsf^{\mathrm{ss}}$. Since $\lim_{t\ra 0} \lambda_1(t)\cdot x$
corresponds to the associated graded of a filtration in $\catfilt$, it follows that $M_x$
and $M_y$ have isomorphic maximal filtrations and hence they are $S$-equivalent.
Conversely, if $M_x$ and $M_y$ are $S$-equivalent, then by Proposition
\ref{prop-1-ps-filtration} there exist one-parameter subgroups $\lambda_1$ and
$\lambda_2$ such that $\lim_{t\ra 0} \lambda_1(t)\cdot x$ and
$\lim_{t\ra 0} \lambda_2(t)\cdot y$ are in the same closed $G$-orbit in $\rsf^{\mathrm{ss}}$.
\end{proof}
The proof of the following Proposition is inspired from \cite[Theorem 4.5]{AK07}.
\begin{proposition} \label{prop:localiso}
There is a local isomorphism $\tilde{h}_\mathrm{fil} \colon
\fp{\rsf^{\mathrm{ss}}}/\fp{G} \ra \mff(\dv)$.
\end{proposition}
\begin{proof}
Let $f\colon S\ra \rsf^{\mathrm{ss}}$ be a morphism of $\mathbb{K}$-schemes, and let $\mathbb{M}_\mathrm{fil}$ be
the restriction of the tautological family to $\rsf^{\mathrm{ss}}$. We define
$$
h_\mathrm{fil}\colon \fp{\rsf^{\mathrm{ss}}}\ra \mff(\dv)\; ~\quad \mbox{by}\;
(f\colon S\ra \rsf^{\mathrm{ss}})\mapsto [f^* \mathbb{M}_\mathrm{fil}]
$$
We have the following commutative diagram
\[
\xymatrix{
\fp{\rsf^{\mathrm{ss}}} \ar[r]^\iota \ar[d]_{h_\mathrm{fil}} & \fp{\rsc^{\mathrm{ss}}}
\ar[d]^{h_\mathrm{rel}} \ar[dr]^\pi & \\
\mff(\dv) \ar[r]_j & \mfc(\dv) \ar[r]_{\tilde{h}_\mathrm{rel}} & \fp{\rsc^{\mathrm{ss}}}/\fp{G}
}
\]
For any $\mathbb{K}$-scheme $S$, the map $\fp{\rsf^{\mathrm{ss}}}(S) \ra
\mff(\dv)(S)\times_{\mfc(\dv)(S)} \fp{\rsc^{\mathrm{ss}}}(S)$
defined by
$$
(f\colon S\ra \rsf^{\mathrm{ss}}) \mapsto (h_\mathrm{fil}^S(f), \iota \circ f)
$$
is a bijection. Hence, the square in the above diagram is Cartesian. This will induces
the following Cartesian diagram
\[
\xymatrix{
\fp{\rsf^{\mathrm{ss}}}/\fp{G} \ar[r] \ar[d]_{\tilde{h}_\mathrm{fil}} &
\fp{\rsc^{\mathrm{ss}}}/\fp{G}
\ar[d]^{\tilde{h}_\mathrm{rel}} \\
\mff(\dv) \ar[r]_j & \mfc(\dv)
}
\]
This proves that $\tilde{h}_\mathrm{fil} \colon
\fp{\rsf^{\mathrm{ss}}}/\fp{G} \ra \mff(\dv)$
is a local isomorphism.
\end{proof}
Now, we can state and prove our main result.
\begin{theorem}\label{main-theorem}
There exist a quasi-projective variety $\msf(\dimv{d})$ (respectively, $\mstf(\dimv{d}))$
which co-represents the moduli functor $\mff$
(respectively, $\mathfrak{M}_{\mathrm{fil}}^\mathrm{s}$).
\begin{enumerate}
\item The closed points of the moduli space $\msf(\dimv{d})$ (respectively, $\mstf(\dimv{d})$)
correspond to the $S$-equivalence classes of $\mu_{\Theta, \mathrm{rk}}$-semistable
(respectively, $\mu_{\Theta, \mathrm{rk}}$-stable) objects of $\catfilt (\mu)$ which have
dimension vector $\dimv{d}$.
\item There exists a canonical morphism $\phi \colon \msf(\dv)\ra \msc(\dv)$.
\end{enumerate}
\end{theorem}
\begin{proof}
The moduli space $\msf(\dimv{d})$ is defined as the GIT quotient of $\rsf$ by $G$, hence
it is a good quotient of $\rsf^{\mathrm{ss}}$ by $G$. In other words, the natural
transformation
$$
\fp{\rsf^{\mathrm{ss}}}/\fp{G} \ra \fp{\msf(\dimv{d})}
$$
corepresents the quotient functor $\fp{\rsf^{\mathrm{ss}}}/\fp{G}$.
By \cite[Lemma 4.7]{AK07}, there exists a unique natural transformation
$\eta\colon \mff(\dv)\ra \fp{\msf(\dimv{d})}$ such that the following diagram
\[
\xymatrix@!=3.5pc{
\fp{\rsf^{\mathrm{ss}}}/\fp{G} \ar[d]_{\tilde{h_\mathrm{fil}}}
\ar[rd]^{\fp{\pi_{\mathrm{fil}}}} &\\
\mff(\dv) \ar[r]_{\eta} & \fp{\msf(\dimv{d})}
}
\]
commutes. This shows that $\msf(\dv)$ corepresents the moduli functor $\mff(\dv)$.
By Proposition \ref{prop:S-eq}, it follows that the closed points of the moduli space
$\msf(\dimv{d})$ correspond to the $S$-equivalence classes of objects of the category
$\catfilt$.
By the universal property of categorical quotients, we get a canonical morphism
$$\phi \colon \msf(\dv)\ra \msc(\dv)$$
such that the following diagram
\[
\xymatrix{
\rsf \ar[r] \ar[d]_{\pi_{\mathrm{fil}}} & \rsc \ar[d]^{\pi_{\mathrm{rel}}} \\
\msf(\dv) \ar[r]_\phi & \msc(\dv)
}
\]
commutes.
\end{proof}
It should be notice here that there are two types in the description of closed points
of $\msf(\dimv{d})$, which are as follows:
\begin{enumerate}
\item[(a)] A closed point of $\msf(\dimv{d})$ is said to be of \textbf{Type-1} if
it correspond to the $S^\mathrm{JH}$-equivalence class of an object in $\catfilt(\mu)$.
\item[(b)] A closed point of $\msf(\dimv{d})$ is said to be of \textbf{Type-2} if
it correspond to the $S$-equivalence class of an object in $\catfilt(\mu)$ which is
not $S^\mathrm{JH}$-equivalence class.
\end{enumerate}
Let $R^{\mathrm{JH}}_\mathrm{fil}$ be a subscheme of $\rsf^\mathrm{ss}$ whose closed
points are consisting of all those points $x$ for which the corresponding filtered
representations have strict Jordan-H\"older filtrations.
\begin{proposition}
The scheme $R^{\mathrm{JH}}_\mathrm{fil}$ is a $G$-invariant open subscheme of
$\rsf^\mathrm{ss}$, and it admits a good quotient
$$
\pi_{\mathrm{JH}}\colon R^{\mathrm{JH}}_\mathrm{fil}\ra M_{\mathrm{fil}}^\mathrm{JH}(\dv)
$$
such that the following diagram is Cartesian:
\[
\xymatrix{
R^{\mathrm{JH}}_\mathrm{fil} \ar[r] \ar[d]_{\pi_{\mathrm{JH}}} & \rsc^\mathrm{ss}
\ar[d]^{\pi_\mathrm{rel}} \\
M_{\mathrm{fil}}^\mathrm{JH}(\dv) \ar[r]_\eta & \msc(\dv)
}
\]
where $\eta$ is an open immersion.
\end{proposition}
\begin{proof}
Since $\rsf^\mathrm{ss}$ is $G$-invariant open subset of $\rsc^\mathrm{ss}$,
it follows that
$$
M_{\mathrm{fil}}^\mathrm{JH}(\dv):=
\msc(\dv) - \pi_\mathrm{rel}(\rsc^\mathrm{ss} - \rsf^\mathrm{ss})
$$
is an open subset of $\msc(\dv)$. Then, it is easy to see that
$R^{\mathrm{JH}}_\mathrm{fil} = \pi_\mathrm{rel}^{-1}(M_{\mathrm{fil}}^\mathrm{JH}(\dv))$
is a $G$-invariant open subscheme of $\rsf^\mathrm{ss}$. Hence, we get a good quotient
$\pi_{\mathrm{JH}}\colon R^{\mathrm{JH}}_\mathrm{fil}\ra M_{\mathrm{fil}}^\mathrm{JH}(\dv)$
which is the restriction of $\pi_{\mathrm{rel}}$ to an open subscheme $R^{\mathrm{JH}}_\mathrm{fil}$.
\end{proof}
\begin{proposition}\label{prop-JH-immersion}
There exist an open immersion $\sigma \colon M_{\mathrm{fil}}^\mathrm{JH}(\dv)\ra \msf(\dv)$ such that
the following diagram
\[
\xymatrix{
R^{\mathrm{JH}}_\mathrm{fil} \ar[r]^i \ar[d]_{\pi_{\mathrm{JH}}} & \rsf^\mathrm{ss}
\ar[d]_{\pi_\mathrm{fil}} \ar[r]^j & \rsc^\mathrm{ss} \ar[d]_{\pi_\mathrm{rel}} \\
M_{\mathrm{fil}}^\mathrm{JH}(\dv) \ar[r]_\sigma \ar@/_2pc/[rr]_\eta & \msf(\dv)
\ar[r]_\phi & \msc(\dv)
}
\]
is commutative;
where $i$ and $j$ are open embeddings, $\phi$ is a canonical morphism as in the
Theorem \ref{main-theorem}, and all three vertical morphisms are good quotients.
\end{proposition}
\begin{proof}
The existence of $\sigma$ and $\eta = \phi \circ \sigma$ follows from the universal properties. The proof of open immersion is
similar to the case of $\eta$ as above.
\end{proof}
\begin{cor}\label{cor-open-immersion}
If $\mathrm{rk}$ is zero on torsion classes, then the canonical morphism
$$\phi \colon \msf(\dv)\ra \msc(\dv)$$
is an open immersion. In this case, the closed points of $\msf(\dv)$ are precisely of
\textbf{Type-1} which correspond to $S^{\mathrm{JH}}$-equivalence classes of objects
of $\catfilt(\mu)$.
\end{cor}
\begin{proof}
If $\mathrm{rk}$ is zero on torsion classes, then we have
$R^{\mathrm{JH}}_\mathrm{fil} = \rsf^\mathrm{ss}$ (cf. Example \ref{ex:sink slope}).
The result now follows from Theorem \ref{main-theorem} and Proposition
\ref{prop-JH-immersion}.
\end{proof}
\begin{remark}\leavevmode \rm{
\begin{enumerate}
\item If we consider the trivial quiver in place of $Q$, then the filtered moduli
space $\msf(\dv) = \msc(\dv)$ is just a point for any increasing sequence of dimension
vector $d_1 \leq d_2 \leq \ldots \leq d_\ell$. The filtered moduli space $\msf(\dv)$
is empty for (non-increasing) other dimension vectors. This follows from the
semi-simplicity of representations of $A_\ell$ quivers.
\item If $\ell = 1$, then we are in the situation of \cite{Ki94} and hence for any
$\Theta\in \Gamma$, we get projective moduli spaces.
\item In general, the moduli spaces $\msf(\dv)$ may not be projective varieties.
For certain choice of stability parameter $\Theta$ and dimension vector $\dv$, we
may get projective moduli space of filtered representations of $Q$. In particular,
if the open set $\rsf$ is same as $\rsc^\mathrm{ss}$, then the GIT quotient $\msf(\dv)$
is projective.
For example, let us consider the quiver $S :=A_2 \times A_2$, i.e.,
\begin{equation*}
\quad \xymatrix@C=3.0pc{
v_1 \ar[r]^{\beta_1} \ar[d]_{\alpha_1} & v_2 \ar[d]^{\alpha_2} \\
w_1 \ar[r]_{\beta_2} & w_2
}
\end{equation*}
where the set of vertices $S_0 = \{v_1, v_2, w_1, w_2\}$.
Let us take $\dv = (1, 1, 1, 2)$ and $\Theta = (1, 0, 1, -1)$. With direct computations,
one can check that the representation $M$ of $S$ having dimension vector $\dv$ satisfying
the relation $\alpha_2\beta_1 - \beta_2\alpha_1$ is $\Theta$-semistable if and only if
$M(\beta_1)$ and $M(\beta_2)$ are injective maps. Hence, in this case, we have
$\rsf = \rsc^\mathrm{ss}$.
\end{enumerate}
}
\end{remark}
\section{Determinantal theta functions}\label{sec-det-fns}
In this section, we assume that the characteristic of the field $\mathbb{K}$ is zero.
We will use results of \cite{DW1, DW2} to give an explicit embedding of moduli of
filtered quiver representations in a projective space.
First, we recall the definition of semi-invariants $\bar{c}^{N'}$ for quiver with
relations.
Let $N'$ be an object in $\catchain$. Let
$$
\tilde{P_1}\ra \tilde{P_0} \ra N' \ra 0
$$
be the minimal presentation of $N'$ in $\catchain$, see \cite[Section 2]{DW2} for
more details.
For any object $M'$ in $\catchain$, the semi-invariant $\bar{c}^{N'}$ is defined by
$\bar{c}^{N'}(M')$ to be the determinant of the matrix
$$
\Hom[\catchain]{\tilde{P_0}}{M'}\ra \Hom[\catchain]{\tilde{P_1}}{M'}
$$
whenever it is a square matrix.
Let $\rsc^1, \dots, \rsc^m$ be the irreducible components of $\rsc$. The semi-invariants
$\bar{c}^{N'}$ is defined on the components $\rsc^j$ of $\rsc$ for which
$$
\dim \Hom[\catchain]{\tilde{P_0}}{M'} = \dim \Hom[\catchain]{\tilde{P_0}}{M'}
$$
while $\dim \Hom[\catchain]{N'}{M'} = 0$ for general $M'\in \rsc^j$.
In view of \cite[Theorem 1]{DW2}, we may work with the faithful components of $\rsc$.
Recall that a component $\rsc^j$ of $\rsc$ is called \emph{faithful} if every element
of $\mathbb{K}(\QAl)$ acting trivially on every module from $\rsc^j$ is in the admissible
ideal $I$. Let $\rsc^\mathrm{ff}$ be the union of faithful components of $\rsc$.
\begin{proposition}\label{det-prop-1}
A representation $M_x$ corresponding to a point $x\in \rsc^\mathrm{ff}$
is $\mu_{\Theta, \mathrm{rk}}$-semistable if and only if there is an object
$N'\in \catchain$ of projective dimension one with the minimal projective resolution
\begin{equation}\label{eq-det-1}
0 \ra \tilde{P}_1 \stackrel{\gamma'}{\ra} \tilde{P}_0 \ra N'\ra 0,
\end{equation}
where $\tilde{P}_0 = \bigoplus_{w\in (\QAl)_0} U_{0_w}\otimes P'_w$ and
$\tilde{P}_1 = \bigoplus_{w\in (\QAl)_0} U_{1_w}\otimes P'_w$ are projective modules
in $\catchain$, such that the following holds:
\begin{enumerate}
\item\label{c1} For each $w\in (\QAl)_0$, we have
$\dim U_{0_w} - \dim U_{1_w} = n(\Theta(\dv)r_w - \mathrm{rk}(\dv) \Theta_w)$
for some positive integer $n$.
\item The map
$$
\Hom[]{\gamma'}{M_x}\colon \Hom[]{P'_0}{M_x}\ra \Hom[]{P'_1}{M_x}
$$ is invertible, i.e. $\Theta_{\gamma'}(M_x):= \det \Hom[]{\gamma'}{M_x} \neq 0$.
\end{enumerate}
\end{proposition}
\begin{proof}
Assume that $M_x$ is $\mu_{\Theta, \mathrm{rk}}$-semistable, i.e., the point
$x$ is $\chi_{\theta}$-semistable, and hence there exist a semi-invariant $f$ with
weight $\chi_{\theta}^n$ such that $f(x) \neq 0$.
By \cite[Theorem 1]{DW1}, there exists a representation $N$ of $\QAl$ with the
minimal projective resolution
$$
P_1 \stackrel{\gamma}{\ra} P_0 \ra N\ra 0
$$
having the Euler form $\langle \dim M_x, \dim N\rangle = 0$ such that
$c^{N}(M_x) \neq 0$. From the proof of \cite[Proposition 1]{DW2}, we have a minimal
projective presentation
$$
\tilde{P_1} \stackrel{\gamma'}{\ra} \tilde{P_0} \ra N'\ra 0,
$$
where $N' := N/IN$, and a determinant semi-invariant $\bar{c}^{N'}$ is the restriction
of $c^{N}$ on the faithful component. By \cite[Theorem 1]{DW2}, the
$\mathbb{K}(\QAl)/I$-module $N'$ is of projective dimension one.
Since the determinant semi-invariant $c^N$ has weight $\chi_{\theta}^n$, it follows
from \eqref{eq-chi-theta} that
$$
\dim U_{0_w} - \dim U_{1_w} = n(\Theta(\dv)r_w - \mathrm{rk}(\dv) \Theta_w)
$$
for each vertex $w\in (\QAl)_0$.
Conversely, assume that there is an object $N'\in \catchain$ of projective dimension
one with the projective presentation \eqref{eq-det-1} satisfying the given conditions.
Then, $\Theta_{\tilde{\gamma}}$ can be viewed as a semi-invariant of weight
$\chi_{\theta}^n$. To see this, for given multiplicity vector spaces
$\U= \{U_{0_w}, U_{1_w}\}_{w\in (\QAl)_0}$ satisfying (\ref{c1}), one can consider
the associated character
\begin{equation}\label{chi-U-defn}
\chi_U\big((g_w)_{w\in (\QAl)_0}\big) :=
\prod_{w\in (\QAl)_0} \det(g_w)^{(\dim U_{0_w} - \dim U_{1_w})} \,.
\end{equation}
This proves that $M_x$ is $\mu_{\Theta, \mathrm{rk}}$-semistable.
\end{proof}
\begin{remark}\rm{
In the above result, if we drop the condition on vector spaces $\U$ as in (\ref{c1}),
then what we can conclude is that the point $x$ is $\chi_U$-semistable which, in general,
may not correspond to the $\mu_{\Theta, \mathrm{rk}}$-semistability (see \eqref{eq-chi-theta}).
}
\end{remark}
\begin{cor}\label{det-cor-1}
A representation $M_x$ corresponding to a point $x\in \rsc^\mathrm{ff}\cap \rsf$ is
$\mu_{\Theta, \mathrm{rk}}$-semistable if and only if either
there is an object $N''\in \catfilt$
of projective dimension one with a minimal projective resolution
\begin{equation}\label{eq-det-2}
0 \ra \tilde{P_1} \stackrel{\tilde{\gamma}}{\ra} \tilde{P_0} \ra N''\ra 0,
\end{equation}
\rm{or} there exists an epi-monic $\tilde{P_1} \stackrel{\tilde{\gamma}}{\ra} \tilde{P_0}$
in $\catfilt$; where $\tilde{P_0}$ and $\tilde{P_1}$ are projective objects in $\catfilt$
as in Proposition \ref{det-prop-1} (see Proposition \ref{filtered projective} for
justification of this definition), such that the map
$$
\Hom[]{\tilde{\gamma}}{M_x}\colon \Hom[]{\tilde{P_0}}{M_x}\ra \Hom[]{\tilde{P_1}}{M_x}
$$ is invertible, i.e. $\Theta_{\tilde{\gamma}}(M_x):= \det \Hom[]{\tilde{\gamma}}{M_x} \neq 0$.
\end{cor}
\begin{proof}
Recall that the inclusion functor $\iota \colon \catfilt \ra \catchain$ has a left
adjoint
$$
\kappa \colon \catchain \ra \catfilt
$$
(see \cite{SS16}) and being left adjoint $\kappa$ is a right exact functor. Since
$\kappa$ has an exact right adjoint functor $\iota$, it carries projective objects
to projective objects. In fact, the indecomposable projective modules $P'_w$ are
projective objects in $\catfilt$ (see Proposition \ref{filtered projective}).
Hence, if $\kappa(N')$ is a non-zero object in $\catfilt$, then the functor $\kappa$
takes any presentation of the form \ref{eq-det-1} to a presentation of the form
\ref{eq-det-2}, where $N'' = \kappa(N')$.
i.e., $\tilde{\gamma} = \kappa(\gamma')$, for $\gamma'$ as in the Proposition
\ref{det-prop-1}. In fact, we have the identification
$$
\Hom[\catfilt]{\kappa(\gamma')}{M} \cong \Hom[\catchain]{\gamma'}{\iota(M)}
$$
using the adjunction between $\iota$ and $\kappa$.
If $\kappa(N') = 0$, then we get an epi-monic
$\tilde{P_1} \stackrel{\tilde{\gamma}}{\ra} \tilde{P_0}$ satisfying the
required conditions.
\end{proof}
\begin{proposition}
There exists an ample line bundle $\lambda_\U(\dv)$ on the moduli space $\msc(\dv)$.
\end{proposition}
\begin{proof}
Let $M$ be a flat family of objects of $\catchain$ having dimension vector $\dv$ over
a scheme $S$.
For non-zero vector spaces $\U= \{U_{0_w}, U_{1_w}\}_{w\in (\QAl)_0}$ satisfying
\begin{equation}\label{eq-U-condn}
\dim U_{0_w} - \dim U_{1_w} = (\Theta(\dv) r_w - \mathrm{rk} (\dv) \Theta_w)\,,
\end{equation}
we define a line bundle
$$
\lambda_\U(M) := \Big(\det \bigoplus_w
\Hom[]{U_{0_w}}{M(w)} \Big)^{-1}\bigotimes \Big(\det \bigoplus_w
\Hom[]{U_{1_w}}{M(w)} \Big)
$$
on $S$.
From \eqref{eq-U-condn}, we have
\begin{equation}\label{eq-theta-1}
\sum_{w\in (\QAl)_0} (\dim U_{0_w} - \dim U_{1_w})d_w = 0.
\end{equation}
For any map
$\gamma' \colon P'_1 \ra P'_0\,,$
where $$P'_0 = \displaystyle \bigoplus_{w\in (\QAl)_0} U_{0_w}\otimes P'_w ~\mbox{and}~
P'_1 = \displaystyle \bigoplus_{w\in (\QAl)_0} U_{1_w}\otimes P'_w\,,$$
we get a (global) section $\Theta_{\gamma'}(M):= \det \Hom[]{\gamma'}{M}$ of
$\lambda_\U(M)$ on $S$.
If $M$ is a flat family of $\mu_{\Theta, \mathrm{rk}}$-semistable objects of
$\catchain$ having dimension vector $\dv$ over a scheme $S$, then we get a formal
line bundle on the moduli functor $\mfc(\dv)$.
Let $\mathbb{M}$ be a tautological family of $\mu_{\Theta, \mathrm{rk}}$-semistable
objects of $\catchain$ on $\rsc^\mathrm{ss}$. Then by \eqref{eq-theta-1}, it follows
that $\lambda_\U(\mathbb{M})$ is a $G$-linearized line bundle.
To conclude, using Kempf's descent criterion, that the $G$-linearized line bundle
$\lambda_\U(\mathbb{M})$ descends to a line bundle on the moduli space $\msc(\dv)$,
we need to verify that for each point $x\in \rsc^\mathrm{ss}$ in closed orbit, the
isotropy group of $x$ acts trivially on the fibre over $x$. To see this, let
$x\in \rsc^\mathrm{ss}$ is in closed orbit.
Then, the corresponding module $M_x$ in $\catchain$ is
$\mu_{\Theta, \mathrm{rk}}$-polystable, i.e.,
$$
M_x \cong \bigoplus_{k=1}^m L_k \otimes M_k\,,
$$
where $M_k$ are non-isomorphic $\mu_{\Theta, \mathrm{rk}}$-stable objects of dimension
vector $\dv_k$ with the slope
\begin{equation}\label{eq-slope-equal}
\frac{\Theta(\dv_k)}{\mathrm{rk}(\dv_k)} = \mu_{\Theta, \mathrm{rk}}(M_k) =
\mu_{\Theta, \mathrm{rk}}(M)
= \frac{\Theta(\dv)}{\mathrm{rk}(\dv)}
\end{equation}
and $L_k$ are multiplicity vector spaces.
Note that
$$
\lambda_\U(M_x) \cong \bigotimes_{k=1}^m (\det L_k)^{u_k} \otimes \lambda_\U(M_k)^{\dim L_k}\,,
$$
where $$u_k = \sum_{w\in (\QAl)_0} (\dim U_{0_w} - \dim U_{1_w})d_{k_v}.$$
Note that
\[
\begin{array}{ll}
\sum_{w\in (\QAl)_0} (\dim U_{0_w} - \dim U_{1_w})d_{k_w}
& = \sum_{w\in (\QAl)_0} \big(\Theta(\dv)r_w - \mathrm{rk}(\dv) \Theta_w\big)d_{k_w} ~\quad \quad
(\mathrm{by}~~ \eqref{eq-U-condn} )\\
& \\
& = \Theta(\dv)\mathrm{rk}(\dv_k) - \Theta(\dv_k)\mathrm{rk}(\dv)\\
& \\
& = 0 ~\quad \quad
(\mathrm{by}~~ \eqref{eq-slope-equal} )\\
\end{array}
\]
Since $M_k$ are $\mu_{\Theta, \mathrm{rk}}$-stable, the isotropy group of $x$ of $G$
is isomorphic to $\prod_{k=1}^m \mathrm{GL}(L_k)$. Hence, from the above computations,
it follows that the isotropy group acts trivially on the fibre over $x$.
Therefore, by Kempf's descent criterion, we get a line bundle $\lambda_\U(\dv)$ on the
moduli space $\msc(\dv)$.
Moreover, a global section $\Theta_{\gamma'}(\mathbb{M})$ being a $G$-invariant section
of $\lambda_\U(\mathbb{M})$, it descends to a section $\Theta_{\gamma'}(\dv)$ of
$\lambda_\U(\dv)$ (cf. \cite[Proposition 7.5]{AK07}).
Since $\lambda_\U(\mathbb{M})$ is a $G$-linearized line bundle which is used to
construct the moduli space $\msc(\dv)$, it follows that the descended line bundle
$\lambda_\U(\dv)$ is ample on $\msc(\dv)$. Moreover, we have
$$
H^0(\rsc, \lambda_\U(\mathbb{M}))^G \cong H^0(\msc(\dv), \lambda_\U(\dv))
$$
(cf. \cite[Proposition 7.6]{AK07}).
\end{proof}
Using \cite[Theorem 1]{DW1}, for sufficiently large dimensional vector spaces $\U$
satisfying \eqref{eq-theta-1}, we get finitely many maps $\gamma_0, \dots, \gamma_m$
as in the proof of Proposition \ref{det-prop-1} such that the corresponding map
$$
\Theta_{\gamma} \colon \ms(\dv) \ra \mathbb{P}^m
$$
is a closed scheme-theoretic embedding.
Let $\msc(\dv)^\mathrm{ff}$ be the closed subscheme of $\msc(\dv)$ corresponding
to $\rsc^\mathrm{ff}$. Then, we have the following result which follows from the
similar arguments as in \cite[Theorem 7.8]{AK07}.
\begin{proposition}\label{det-prop-2}
For any dimension vector $\dv$, we can find finite-dimensional vector spaces
$\U = \{U_{0_w}, U_{1_w}\}_{w\in (\QAl)_0}$ satisfying \eqref{eq-theta-1}
and finitely many maps $\gamma'_0, \dots, \gamma'_m$ as in Proposition
\ref{det-prop-1} such that the map
$$
\Theta_{\gamma'} \colon \msc(\dv)^\mathrm{ff} \ra \mathbb{P}^m
$$
given by $[M]\mapsto (\Theta_{\gamma'_0}(M) : \cdots : \Theta_{\gamma'_N}(M))$ is
scheme-theoretic closed embedding.
\end{proposition}
We notice that this embedding $\Theta_{\gamma'}$ is precisely the restriction of
$\Theta_{\gamma}$ (cf. Proposition \ref{det-prop-1}).
Let $M_{\mathrm{fil}}^\mathrm{JH}(\dv)^\mathrm{ff} :=
M_{\mathrm{fil}}^\mathrm{JH}(\dv) \cap \msc(\dv)^\mathrm{ff}$
(respectively, $\msf(\dv)^\mathrm{ff} := \msf(\dv) \cap \msc(\dv)^\mathrm{ff}$) . Then $M_{\mathrm{fil}}^\mathrm{JH}(\dv)^\mathrm{ff}$ and $\msf(\dv)^\mathrm{ff}$ are
locally closed subschemes of $\msc(\dv)$.
\begin{cor}\label{det-cor-2}
For any dimension vector $\dv$, we can find finite dimensional vector spaces
$\U = \{U_{0_w}, U_{1_w}\}_{w\in (\QAl)_0}$
satisfying \eqref{eq-theta-1} and finitely many maps
$\tilde{\gamma_0}, \dots, \tilde{\gamma_m}$ as in
Corollary \ref{det-cor-1} such that the map
$$
\Theta_{\tilde{\gamma}} \colon M_{\mathrm{fil}}^\mathrm{JH}(\dv)^\mathrm{ff} \ra \mathbb{P}^m
$$
given by
$[M]\mapsto (\Theta_{\tilde{\gamma_0}}(M) : \cdots : \Theta_{\tilde{\gamma_m}}(M))$
is a scheme-theoretic locally closed embedding.
\end{cor}
\begin{proof}
There is an ample line bundle $\tilde{\lambda}_U(\dv) = \eta^*\lambda_U(\dv)$ on the
moduli space $M_{\mathrm{fil}}^\mathrm{JH}(\dv)$, and for $\tilde{\gamma_i} = \kappa(\gamma_i)$,
we have
$$
\eta^*\Theta_{\gamma_i} = \Theta_{\tilde{\gamma_i}}
$$
Now using Corollary \ref{det-cor-1}, the result follows by combining the embedding
$\eta \colon M_{\mathrm{fil}}^\mathrm{JH}(\dv)^\mathrm{ff} \ra \msc(\dv)^\mathrm{ff}$
and $$\Theta_\gamma \colon \msc(\dv)^\mathrm{ff} \ra \mathbb{P}^m \, .$$
\end{proof}
In particular if we choose suitable slope function we get following result.
\begin{cor}\label{det-cor-3}
For any dimension vector $\dv$ and a slope function $\mu_s$ as in Example
\ref{ex:sink slope}, we can find finite dimensional vector spaces
$\U = \{U_{0_w}, U_{1_w}\}_{w\in (\QAl)_0}$ satisfying \eqref{eq-theta-1} and finitely
many maps $\tilde{\gamma_0}, \dots, \tilde{\gamma_m}$ as in Corollary \ref{det-cor-1}
such that the map
$$
\Theta_{\tilde{\gamma}} \colon \msf(\dv)^\mathrm{ff} \ra \mathbb{P}^m
$$
given by $[M]\mapsto (\Theta_{\tilde{\gamma_0}}(M) : \cdots : \Theta_{\tilde{\gamma_m}}(M))$
is a scheme-theoretic locally closed embedding.
\end{cor}
\begin{remark}\rm{
For the components which are not faithful, we can have the similar statements as in
the Proposition \ref{det-prop-2} and the Corollary \ref{det-cor-2}, \ref{det-cor-3}
by further introducing extra relations. More precisely, the projective modules in
presentations \eqref{eq-det-1} and \eqref{eq-det-2} will change up to extra relations.
}
\end{remark}
\subsection*{Acknowledgement} The first-named author would like to acknowledge the
support of SERB-DST (India) grant number YSS/2015/001182 and MTR/2018/000475. The
second name author would like to acknowledge the support of DAE and DST INSPIRE grant number
IFA 13-MA-25. We also thank HRI and IIT Gandhinagar for providing excellent infrastructure.
| 156,846
|
TITLE: Compact spaces in which any closed set can be partitioned into finitely many closed sets whose clopen subsets extend to the whole space
QUESTION [6 upvotes]: Let $X $ be a compact topological space (not necessarilly Hausdorff). I am looking for a charactrization for the following property:
Property: If $C $ is a closed subset of $X $, then there are pairwise disjoint closed subsets $C_1$,...,$C_n $ of $X $ such that $C=C_1\cup\dots\cup C_n $, and each $C_i $ has the property that if $A $ is a clopen subest of $C_i $, then there exists a clopen subset $B $ of $X $ with $A=C_i\cap B$?
Any comment is very welcome.
REPLY [3 votes]: I don't see any reason to believe there is a simpler necessary and sufficient condition in general. When $X$ is Hausdorff there is one, though: your condition is equivalent to $X$ being totally disconnected.
To prove this, first suppose $X$ is compact Hausdorff and totally disconnected. Then clopen sets separate points of $X$ (see Any two points in a Stone space can be disconnected by clopen sets). It follows by compactness that clopen sets also separate closed sets (the proof is the same as the proof that a compact Hausdorff space is normal, just using clopen sets instead of open sets everywhere). So in particular, if $C\subseteq X$ is closed and $A$ is clopen in $C$, then $A$ and $C\setminus A$ can be separated by clopen subsets of $X$ (since they are disjoint and closed in $X$). This means that your condition holds, with $n=1$.
Conversely, suppose $X$ is compact Hausdorff and not totally disconnected. Your property is inherited by closed subspaces, so we may replace $X$ with one of its nontrivial connected components. So, we assume $X$ is compact Hausdorff and connected and has more than one point, and must show it does not satisfy your condition.
To prove this, pick an infinite discrete subset $D\subset X$ (see Every infinite Hausdorff space has an infinite discrete subspace) and consider $C=\overline{D}$. Suppose a decomposition $C=C_1\cup\dots\cup C_n$ with your property existed. Note that each point of $D$ is isolated in $C$, and some $C_i$ must contain infinitely many points of $D$. In particular, some $C_i$ must be disconnected, so there is a nontrivial clopen subset $A\subset C_i$. But $X$ is connected, so it has no nontrivial clopen subsets, and so $A$ cannot be $B\cap C_i$ for any clopen $B\subseteq X$.
Of course, total disconnectedness is not necessary in the non-Hausdorff case (I don't know whether it is sufficient). Besides trivial examples like the indiscrete topology, there is also the cofinite topology on any set. More generally, as Es.Ro commented, a sufficient condition is that every closed subset of $X$ has only finitely many connected components, since then you can take the connected components as the $C_i$.
| 150,853
|
I.
3 comments:
That poster is awesome. I have saved it. In the future when you rise to power, it shall come in handy...
Add one vote.
Okay, I am voting for Von. I will look forward to that board game. All I have is playing backgammon online nowadays.
| 119,752
|
1.0L Stainless Steel Coffee Kettle Hight Quality Drip Kettle Pot - Silver
Note: For multiple item orders, the processing time will be based on the item with the longest processing time. Dispatched in: Item ships within 5-10 business days.
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- Color: SILVER
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Material: 304 Stainless Steel
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5.Teapot with strainer makes tea together ,more convenient to drink
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| 3,646
|
: Gracie's, IllustrationFriday
"Entangled" by Isaac
Labels: IllustrationFriday, Isaac's
Yikes!
Saturday, November 07, 2009
Book_1<<
Labels: FanArt, Group, of note
Monday, November 02, 2009
"Skinny" by Lily
Labels: IllustrationFriday, Lily's
"Skinny" by Isaac
Labels: IllustrationFriday, Isaac's
| 256,228
|
TITLE: Does a linear differential equation of a matrix variable have a closed form solution?
QUESTION [1 upvotes]: Let $X(t), A, B, C$ be matrices and $A, B, C$ are constant matrices. Does the following linear differential equation have a closed form solution?
$$
\frac{d X(t)}{dt} = A + BXC.
$$
Thank you!
REPLY [0 votes]: This expands user254433's answer. First, note that
$$
\frac{\mathrm{d}}{\mathrm{d}t}\left[\sum_{k\geq0}\frac{t^{k}}{k!}B^{k}MC^{k}\right]=B\left(\sum_{k\geq1}\frac{t^{k-1}}{\left(k-1\right)!}B^{k-1}MC^{k-1}\right)C=B\left(\sum_{k\geq0}\frac{t^{k}}{k!}B^{k}MC^{k}\right)C.
$$
Now, suppose $B$ and $C$ are nonsingular. Motivated by the above,
take
$$
X(t)=\sum_{k\geq0}\frac{t^{k}}{k!}B^{k}\left(X(0)+B^{-1}AC^{-1}\right)C^{k}-B^{-1}AC^{-1}
$$
as a solution. You can verify that the above is a solution directly:
$$
X^{\prime}(t)=B\left[\sum_{k\geq0}\frac{t^{k}}{k!}B^{k}\left(X(0)+B^{-1}AC^{-1}\right)C^{k}\right]C=A+BX(t)C.
$$
| 48,132
|
? Like a virgin, sort of | Main | Scout Walker Kama Sutra ?
August 30, 2003
Prowling the icy wasteland
I found this at Classical Values. Mine also demonstrates an eerie accuracy.
Posted by Ghost of a flea at August 30, 2003 11:22 AM
Looking like a scared bunny who turned and fled, even though he was actually an angry rabbit who was going to fight in another fight, away from the first fight, Scrutineer raced forth and snarled:
"I'm seriously going to brutalize you so heinously, you will not be able to see straight!!"
Posted by: Michael at August 30, 2003 12:28 PM
I'm going to reduce you to ash, and sell you as spice!"
Posted by: Ray at August 30, 2003 08:38 PM
| 225,697
|
Guest post by Richie Robertson, Director of University of the West Indies Seismic Research Centre
La Soufriere volcano looms largely in the consciousness of most Vincentians. Although it occupies the northern third of the island and is often shrouded in clouds, it does not take very much to command our attention. There have been numerous instances in the past when the occasional smell of sulphur, thunderstorms over the volcano and the spurious observation of someone visiting the summit crater, have led to rumours of an eruption. It is, therefore, not surprising that the most recent eruption of the volcano still remains fresh in the minds of those of us who experienced it.
St Vincent in eruption, April 1979. Photo credit – Steve Sparks.
My own path to becoming a volcanologist and current leader of the regional volcano monitoring organization in the English-speaking Eastern Caribbean began with the 1979 eruption. I…
View original post 567 more words
| 395,991
|
Recovery- She will be mine, oh yes. She will be mine.
Motivation- Get confident. Get hairy. Get women.
Recovery- She will be mine, oh yes. She will be mine.
Motivation- Get confident. Get hairy. Get women.
First workout as part of the New Strength Cycle Template Bench Workoutz.
Warmup– Jump around and stuff
Strength
Conditioning
*Sub 30 double unders or 10 burpees for the row if equipment is unavailable
Motivation- Use the forest as your weight room
Warmup- Follow your dreams. But first, do this.
Dynamic- EMOM (Every Minute On the Minute) for 8 minutes- Flat Bench- 3 explosive reps @ 50-60% 1RM
Volume (3-0-3 tempo: 3 seconds down, 3 seconds up, 60-90 seconds b/t supersets):
Warmup– Let this be your warmup, young grasshopper
Strength–
Conditioning- “Mila” (If you’ve already completed this workout, increase the working weight and try to beat your previous time)
Motivation- Dom. Gym. Curls. Pump. Need I say more? This one’s an oldie but a goodie.
Warmup- Keep it secret, keep it swole.
Dynamic- EMOM for 8 minutes (EMOM = Every Minute On the Minute), 3 Explosive Deadlifts@ ~50% 1RM
Volume (Rest periods should be 60-90 seconds b/t supersets)
Last week of this bodybuilding micro-cycle so pack on the weight, get the MEGAPUMPFEELS© and celebrate the birth of this great country.
Motivation- What is freedom? See below:
Recovery
Motivation
Warmup
Dynamic
Volume
Dynamic
Volume.
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TITLE: Abelian $2$-groups
QUESTION [2 upvotes]: Is every abelian group $A$ where every element has order two isomorphic to a direct product of cyclic groups of order two, $A\cong C_2\times C_2\times\ldots$?
I ask because I used this "fact" in one of my old answers here (which is relevant to some work I am doing), and have just realised that this is not obvious, and so perhaps not true.
Am I perhaps just not seeing something which I thought was obvious at the time? Or is there something more subtle going on?
(Note that there is no assumption that $A$ is finitely generated.)
REPLY [8 votes]: An abelian group is a $\mathbb{Z}$-module. An abelian group of exponent dividing $n$ is a $\mathbb{Z}/n\mathbb{Z}$ module. In your case, $A$ is a $\mathbb{Z}/2\mathbb{Z}$-module, so a vector space over the field $\mathbb{Z}/2\mathbb{Z}$. By standard axioms such as the axiom of choice, $A$ has a basis, and so is the direct sum of one dimensional subspaces. In other words, $A$ is the restricted direct product or direct sum of copies of $C_2$.
$\mathbb{Z}/n\mathbb{Z}$ is a self-injective ring if $n$ is nonzero, and this often gives you nice decompositions. See "DSC" group.
$A$ need not be a direct product. For instance if $A$ is countably infinite, then it is not a direct product of copies of $C_2$, since finitely many $C_2$s produces finite cardinality, and infinitely many $C_2$s produces at least a continuum cardinality.
| 68,018
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After the success of some Hogwarts-inspired pieces previously, Penneys has launched a new Harry Potter range – and we love it! As always, Penneys (or Primark if you’re in the UK) has absolutely come up trumps. With everything from cushions and bed linen to Christmas decorations and clothes, you (and the kids) can get set for some serious Hogwarts style.
Sign up for our free Newsletter stuffed full of ideas, competitions and offers.
PS Did we mention it’s free?
Many of the items in the new Harry Potter range hit stores this month, with more coming in September and October. Start checking your local store now, as these won’t last long (remember the drama with the Beauty & the Beast Chip purse…?).
You might also enjoy [WATCH] Goodbye Christopher Robin Trailer
Get Cosy with Hogwarts Style
What house would the Sorting Hat have put you in? Are you a Hufflepuff, Ravenclaw, Slytherin or Gryffindor at heart? Pick your favourite Hogwarts house and decorate accordingly with these house-inspired cushions (€5/£4 each) and co-ordinating throw blankets (€8/£6 each). There’s also a white Hedgwig-shaped cushion (€8/£6) to cuddle up with.
These would be perfect for kids’ rooms, and are a really quick and easy way to theme the bedroom decor.
Enchanting Harry Potter Style
Deck yourself out from head to toe! There are loads of fun bits in the Harry Potter range to choose from, including hoodies and sweatshirts, t-shirts, pyjamas, onesies and dressing gowns designed to look like Hogwarts robes or printed with the Marauder’s Map. If kitting yourself out for bed isn’t enough, you can also get themed bed linen!
Don’t forget to accessorise – take your pick from a Hermione’s fantastic time turner necklace, pendant chokers, earrings, pin badges, slippers, socks, trainers and lots more.
Clockwise from top left: burgundy velour glasses sweatshirt (€12/£10), Hufflepuff makeup bag, available in each house colour (€5/£4), kids pyjamas (€11/£8), time turner pendant necklace (€4/£3), pack of 5 Harry Potter socks (€8/£6), Hufflepuff hoodie, available in each house colour (€14/£12); quidditch pyjama vest top (€6/£4.50); boot trainer (€14/£12); lace up pumps (€11/£8).
You might also enjoy Our August Picks for New Family Movies
Add a Touch of Magic to the Kitchen
Your morning coffee has never looked so good! We are completely besotted with the range of mugs coming into stores, particularly these Keeper and Catch mugs (€8/£6) for quidditch fans, as well as a black cauldron-shaped mug (€7/£6) for a really magical cuppa.
Complete the look in the kitchen with hanging message chalkboards (€5/£4 each), a Hogwarts-design pin board (€10/£8), and a set of four drinks coasters representing each of the Hogwarts houses (€5/£4).
Quirky and Cool Harry Potter Style
If quirky is your style, then these are for you. These would also make a great way to set the scene for a Harry Potter-themed birthday or party.
Clockwise from top left: Hogwarts bunting (€4.50/£3); hanging arrows sign (€8/£6); Harry Potter face mug (€7/£5); Harry Potter mirror (€6/£5); Polyjuice potion mug with straw (€5/£4); potion bottle lights (€10/£8).
Have A Happy Hogwarts Christmas
True fans will love the Christmas baubles that are due in stores from the end of September.
Choose from a single Hogwarts crest decoration (€6/£5), a set of four different coloured house baubles (€5/£4), or a festive gold bauble decoration (€6/£5).
Can your Christmas tree compete with the grandeur and wonders of Hogwarts?
Calling All Future Wizards!
Even the littlest family member can get in on the action with baby items from the new Harry Potter range.
We love this three-pack of bandana-style bibs (€4/£3) in neutral colours, while the matching baby vests (€7/£5) and Future Wizard babygrow and hat set (€8/£6.50) would make great gifts for a new bundle.
You might also enjoy Hot Toys for Christmas 2017 – Top Predictions
If You Only Buy One Thing…
This Hogwarts letter cushion (€8/£6) is it. It is perfect, and unusual enough to add some style to any room in the house.
I don’t know about you, but I read the Harry Potter series and hoped (in vain, clearly) that a magical Hogwarts letter might appear. My daughter, now reading the books for the first time, is equally besotted and listed ‘Hogwarts professor’ as her dream occupation at school recently. I think I may need to get her one of these cushions too!
What’s your favourite piece from the Harry Potter range at Penneys? Leave a comment below and let us know – we’d love to hear from you!
| 278,428
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This is a small mountain located in the middle of Estes Park. Its summit is 8900 feet and can be accessed via for a fee via a vintage cable car during warm months of the year. There is even a gift shop & snack bar on top. Note, there are a bunch of mountain lions known to frequent these slopes. Watch your kids and smaller climbing partners! On its slopes, you can find a bunch of small crags, some of which have published routes on them, like the southern crags of Dihedral Rock, Shield Rock, The Fin, The Thimble, The Needle, and The Thumb. Apparently, some of these crags may be on private property. There are other crags on this mountain which have probably have been climbed by wandering souls. While this area can hardly be considered a destination for climbers, inclement weather, limited time, kids, when your kids are competing at the Hunter Jumper competitions, and short approaches might coax a wander in the area. There are routes here which vary from 1-2 pitches, 0 to 2 stars, 5.0 to 5.13, gear to bolts, all on gneiss and granite outcrops. The north, west, & east faces do see a bit of moisture, so you may find healthy stretches of lichen on these faces. You may find some guided groups in the area due to the ease of access, low-end routes, and general lack of competition for routes here. Close-in parking is quite limited (probably 8 cars maximum). Consider bringing a trash bag to help pick up debris left by partiers in the area.
In Estes Park, drive Mary's Lake Rd or CO Hwy 7 to Peak View Drive towards the south side of the mountain. Drive uphill on Curry (dirt road). Do not block the gate when you park! Consider driving compact vehicles.
| 186,797
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BRIGHTON, Mich. (WLNS) -- This week's snow storm created a much needed boost in business for local winter sports locations.
It's been non-stop at Mt. Brighton Ski Resort since 9 a.m. Saturday.
"It's been great. the phones have been ringing off the hook," said operations manager Rob Bruhn.
He tells 6 News Saturday is on track to be one of the best Saturdays of the season.
The great conditions and a good crowd allowed the resort to open all six of its lifts. "This is the first weekend this year we've had all six lifts going. With all the fresh snow, it's enabled us to get the whole mountain operating."
An estimated 2,000 skiers and snowboarders packed the slopes throughout the day management expected more throughout the night.
The large crowd surprised some customers like Soorim Yoo. "There's never really that many people around," said Yoo. "We couldn't even park in the spots, so we ended up parking way back."
"The skiing conditions are fantastic tonight, it's not too cold, not too windy, the snow is great," said skier Melinda Mooradian.
The hill won't close until 2:00 am Sunday which is how long some plan to stay like Yoo.
"I hear it's supposed to get warm tomorrow, so we don't really want to miss out. We're going to get as much done as we can," he said.
It's that kind of spirit has Bruhn in a good mood. "Last year was probably the worst in 50 years for us. Not just here, but all over the country, so when we get snow storms like this and have weekends like this, it brings a smile to everyone's face," said Bruhn.
He added that this weekend's ticket sales will help make up for the slow start to the season caused by the mild winter conditions, but it won't erase their deficit completely.
| 302,205
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\begin{document}
\maketitle
\begin{abstract}
Sensor sources submit updates to a monitor through an unslotted, uncoordinated, unreliable multiple access collision channel. The channel is unreliable; a collision-free transmission is received successfully at the monitor with some transmission success probability. For an infinite-user model in which the sensors collectively transmit updates as a Poisson process and each update has an independent exponential transmission time,
a stochastic hybrid system (SHS) approach is used to derive the average age of information (AoI) as a function of the offered load and the transmission success probability. The analysis is then extended to evaluate the individual age of a selected source. When the number of sources and update transmission rate grow large in fixed proportion, the limiting asymptotic individual age is shown to provide an accurate individual age approximation for a small number of sources.
\end{abstract}
\section{Introduction}
Consider a collection of sensors that transmit updates to a central monitor. In many applications, complexity and energy considerations dictate that the sensors be transmit-only devices that blindly send update measurements without regard to the activity of other sensors \cite{Blaszczyszyn2008,Zhang2017FHMMF}.
Because the transmit-only sources cannot coordinate, the transmissions are subject to collisions and the system operation is necessarily unslotted.
Since the timeliness may be important, this work examines the age of information (AoI) of these sensor updates. When the newest received update has time stamp $u(t)$, the age process is $\age(t)= t-u(t)$ \cite{Kaul-YG-Infocom2012} and the average age is $\limty{t} \E{\age(t)}$.
We note there has been growing interest in the AoI of sources sharing a communication facility, starting with multiple sources submitting updates through queues \cite{Huang-Modiano-isit2015,Kadota-UBSM-Allerton2016,Kaul-Yates-isit2018priority,Yates-Kaul-IT2018, Najm-Telatar-aoi2018,maatouk2019age}. In addition, AoI has been analyzed for multiple users sharing a slotted system with various levels of system coordination, including round-robin and Aloha-like contention \cite{Kaul-Yates-isit2017}, scheduled access \cite{Jiang-KZZN-isit2018,Hsu-isit2018,farazi2019fundamental,kosta2019age,maatouk2020optimality}, CSMA \cite{maatouk2019minimizing}, and random access with source-optimized contention policies \cite{chen2019age}. However, age of information (AoI) in transmit-only sensor updates has not been studied. The graphical method of age analysis introduced in \cite{Kaul-YG-Infocom2012} and then employed in e.g. \cite{Kam-KE-isit2013random,Kam-KE-isit2014diversity,Kam-KNE-IT2016diversity,Costa-CE-IT2016management,Costa-CE-isit2014,Champati-AG-aoi2018,Inoue-MTT-arxiv2017,Kam-KNWE-isit2016deadline,Feng-Yang-aoi2018}
has not enabled age analysis of the collision channel.
\subsection{System Model}
In the collision channel, a transmission is collision-free if all other transmitters are idle during that transmission. If an update suffers a collision, it is not received by the monitor. In addition, the communication channel is unreliable; a collision-free update will suffer an error and fail to be received by the monitor with probability $\Pe$.
A key advantage of an unslotted system is that the transmission times can have arbitrary durations \cite{Gallager1985multiaccess}. To avoid a combinatorial explosion of the state space, we assume the transmission times of the updates are modeled as independent exponential $(\mu)$ random variables. Furthermore, the collection of sensors in aggregate initiate update transmissions as a rate $\lambda$ Poisson point process. This is consistent with the ``infinite user'' model of historical importance in the analysis of the maximum stable throughput of collision resolution protocols \cite{abramson1970aloha,roberts1975aloha,gallager1978conflict, tsybakov1980random,Gallager1985multiaccess}.
\subsection{Paper Summary}
For the collection of uncoordinated sensors, we consider two types of age metrics. The {\em system age} is defined as the age of the most recent update received from any sensor in the system. For the system age, an update from any sensor reduces the age at the monitor. This is in contrast to the {\em individual age} of a selected sensor among $N$ sensors. Poisson arrivals of transmitted updates and exponential update transmission times enable the method of stochastic hybrid systems (SHS) for age analysis.
Section~\ref{sec:SHS}, provides a short introduction to the SHS method and then uses SHS to analyze the system age in Section~\ref{sec:analysis}.
Using the probability of correct detection $\Pc=1-P_e$, the system age analysis is extended to evaluate the individual age in Section~\ref{sec:indiv}. The individual age, in the limit of a large number of users and proportional system service rate, is shown to converge to simple function of the offered load, that approximates the individual age even for a small number of sources. The paper concludes with a discussion of open issues in Section~\ref{sec:conclusion}.
\section{Average System Age}
\subsection{SHS Background}\label{sec:SHS}
A stochastic hybrid system (SHS) \cite{Hespanha-2006modelling} has state $[q(t),\xv(t)]$ such that $\xv(t)\in\R^{1\times n}$ and $q(t)\in\Qcal=\set{0,\ldots,M}$ is a continuous-time
Markov chain.
For AoI analysis, $q(t)$ describes the discrete state of a network while the age vector $\xv(t)$ describes the continuous-time evolution of a collection of age-related processes.
The SHS approach was introduced in \cite{Yates-Kaul-IT2018}, where
it was shown that age tracking can be implemented as a simplified SHS with non-negative linear reset maps in which the continuous state is a piecewise linear process
\cite{Vermes-1980,Davis-1984,Deville-DDZ-siam2016moment}. For finite-state systems,
this led to a set of age balance equations and simple conditions \cite[Theorem~4]{Yates-Kaul-IT2018} under which $\E{\xv(t)}$ converges to a fixed point.
A description of this simplified SHS for AoI analysis now follows.
In the graph representation of the Markov chain $q(t)$, each state $q\in\Qcal$ is a node and each transition $l$ is a directed edge $(q_l,q'_l)$ with transition rate $\laml$ from state $q_l$ to $q'_l$.
Associated with each transition $l$, is transition reset mapping $\Amat_l\in\set{0,1}^{n\times n}$ that can induce a discontinuous jump $\xv'=\xv\Amat_l$ in the continuous state $\xv(t)$.
Unlike an ordinary continuous-time Markov chain, the SHS Markov chain may include self-transitions in which the discrete state is unchanged because a reset occurs in the continuous state. Furthermore, for a given pair of states $i,j\in\Qcal$, there may be multiple transitions $l$ and $l'$ in which $q(t)$ jumps from $i$ to $j$ but the transition maps $\Amat_l$ and $\Amat_{l'}$ are different.
For each state $\qbar$, we denote the respective sets of incoming and outgoing transitions by
\begin{align}\eqnlabel{Lcalqbar}
\Lcal'_{\qbar}\!=\!\set{l\in\Lcal: q'_l=\qbar},\ \
\Lcal_{\qbar}\!=\!\set{l\in\Lcal: q_l=\qbar}.
\end{align}
Assuming the Markov chain $q(t)$ is ergodic, the discrete state Markov chain $q(t)$ has stationary probabilities
$\bar{\piv}=\rvec{\pibar_0&\cdots&\pibar_M}$ satisfying
\begin{align}
\bar{\pi}_{\qbar}\sum_{l\in\Lcal_{\qbar}}\laml&=\sum_{l\in\Lcal'_{\qbar}}\laml\bar{\pi}_{\ql},\quad \qbar\in\Qcal,
\eqnlabel{AOI-SHS-pi}
\end{align}
and the normalization constraint $\sum_{\qbar\in\Qcal}\bar{\pi}_\qbar=1$.
The next theorem provides a way to derive the limiting average age vector $\E{\xv}=\limty{t}\E{\xv(t)}$.
\begin{theorem}\thmlabel{AOI-SHS}
\cite[Theorem~4]{Yates-Kaul-IT2018}
If the discrete-state Markov chain $q(t)$ is ergodic with stationary distribution $\bar{\piv}>0$ and there exists a non-negative vector $\vvbar=\rvec{\vvbar_0&\cdots\vvbar_M}$
such that
\begin{align}
\bar{\vv}_{\qbar}\sum_{l\in\Lcal_{\qbar}}\laml &=\onev[]\bar{\pi}_{\qbar}+ \sum_{l\in\Lcal'_{\qbar}}\laml \bar{\vv}_{\ql}\Amat_l,\quad \qbar\in\Qcal,\eqnlabel{AOI-SHS-v}
\end{align}
then
the average age vector is
$\E{\xv}=
\sum_{\qbar\in\Qcal} \vvbar_{\qbar}$.
\end{theorem}
In the next section, \Thmref{AOI-SHS} is employed to find the average age
for uncoordinated unslotted updating.
\begin{figure}[t]
\centering
\begin{tikzpicture}[->, >=stealth', auto, semithick, node distance=2cm]
\tikzstyle{every state}=[fill=none,draw=black,thick,text=black,scale=0.9]
\node[state] (0) {$0$};
\node[state] (1)[right of=0] {$1$};
\node[state] (2) [right of=1] {$2$};
\node (H) [right of=2] {$\cdots$};
\node[state] (M) [right of=H] {$M$};
\path
(0) edge[bend left=10,above] node{\small $\lambda$} (1)
(1) edge[bend left=10,below] node{\small $\Pc\mu$} (0)
(1) edge[bend left=60,below] node{\small $\Pe\mu$} (0)
(1) edge[bend left=10,above] node{\small $\lambda$} (2)
(2) edge[bend left=10,below] node{\small $2\mu$} (1)
(2) edge[bend left=10,above] node{\small $\lambda$} (H)
(H) edge[bend left=10,below] node{\small $3\mu$} (2)
(H) edge[bend left=10,above] node{\small $\lambda$} (M)
(M) edge[bend left=10,below] node{\small $M\mu$} (H);
\end{tikzpicture}
\caption{SHS Markov chain for the system age over an unslotted collision channel.}
\label{fig:unslotted-MC}
\vspace{-5mm}
\end{figure}
\subsection{SHS Analysis of the System Age}\label{sec:analysis}
For an SHS age model of the unslotted collision channel, the discrete state Markov chain for $q(t)$ is shown in Figure~\ref{fig:unslotted-MC} and the set of SHS transitions is given in Table~\ref{tab:overall-MC}.
The discrete state $q(t)\in\set{0,1,2,\ldots}$ is the number of active transmitters.
In the idle state $0$, the start of a transmission causes the system to jump to state $1$. This update is successfully delivered if it completes service before another update begins transmission. Otherwise,
a jump to state $2$ begins a collision period in which transmitted updates suffer collisions and are unsuccessful. In states $k\ge 2$, there are $k$ updates being transmitted in a $k$-way collision. A collision period ends when the system returns to the idle state.
\begin{table}[t]
\caption{SHS transitions for tracking the overall age in the Markov chain of Fig.~\ref{fig:unslotted-MC}.}\label{tab:overall-MC}
\begin{displaymath}\arraycolsep=2pt
\begin{array}{cccccc}
l & q_l\to q'_l & \laml & \xv\Amat_l & \Amat_l & \vv_{\ql}\Amat_l\\ \hline
1 & 0\to 1 &\lambda & \rvec{\xz&x_2}&\smvec{0 & 0\\0 & 1}&\rvec{\vz&v_{02}}\\
2 & 1\to 0 & \Pc\mu & \rvec{x_1&x_1}&\smvec{1 & 1\\ 0& 0} &\rvec{v_{11}&v_{11}}\\
3 & 1\to 0 & \Pe\mu & \rvec{x_2&x_2}&\smvec{0& 0\\1 & 1} &\rvec{v_{12}&v_{12}}\\
4 & 1\to 2 & \lambda & \rvec{x_2&x_2}&\smvec{0& 0\\1 & 1} &\rvec{v_{12}&v_{12}}\\
5 & 2\to 1 & 2\mu & \rvec{x_1&x_2}&\Imat&\rvec{v_{21}&v_{22}}\\
6 & 2\to 3 & \lambda & \rvec{x_1&x_2}&\Imat&\rvec{v_{21}&v_{22}}\\
7 & 3\to2 & 3\mu & \rvec{x_1&x_2}&\Imat&\rvec{v_{31}&v_{32}}\\
\vdots &\vdots&\vdots&\vdots&\vdots&\vdots\\
\vdots & M\!\to\! M\!-\!1 &M\mu & \rvec{x_1&x_2}&\Imat&\rvec{v_{M1}&v_{M2}}
\end{array}
\end{displaymath}
\vspace{-5mm}
\end{table}
The
age state is
$\xv(t)=\rvec{x_1(t)& x_2(t)}$ where $x_2(t)$ is the age at the monitor and $x_1(t)$ is what the age at the monitor would become if an update in service were to complete transmission at time $t$. Our objective is to calculate the average age at the monitor $\age=\limty{t}\E{x_2(t)}$. In each state $q(t) = \qbar$, the continuous state evolves according to
$\dot{\xv}(t)=\onev[]=\rvec{1&1}$.
With respect to AoI, an age reduction in $x_2(t)$ occurs only when a collision-free update is delivered successfully.
In particular, in transition $l=1$, the system goes from idle to having a single update in service. In this case, the mapping $\xv'=\xv\Amat_1=\rvec{0 & x_2}$ resets $x_1$ to $x_1=0$, the age of the fresh update that just began transmission. On the other hand, $x'_2=x_2$ is unchanged because it tracks the age at the monitor. In state $1$, the transition $l=2$ corresponds to the update being transmitted collision-free and also being successfully received. In this transition, $\xv'=\xv\Amat_2=\rvec{x_1 & x_1}$ resets $x_2$ to $x_2'=x_1$,
the age of the update that was just successfully received. By contrast, transition $l=3$ corresponds to the update being transmitted collision-free but it fails to be received. In this transition, $\xv'=\xv\Amat_3=\rvec{x_2 & x_2}$ leaves the age $x_2$ at the monitor unchanged. This transition also resets $x_1$ to $x_1'=x_2$, which destroys the ability of the update in transmission to reduce the age at the monitor. Similarly, $\Amat_4=\Amat_3$ because in transition $l=4$, a second update collides with an update in a transmission. Since this collision guarantees that neither update in transmission is successfully received, this transition also sets $x_1'=x_2$,
While state $1$ has exactly one update being in service, this update may or may not be collision-free. This information is encoded in the continuous state $\xv(t)$. Specifically, $x_1(t)<x_2(t)$ when there is single update with age $x_1(t)$ in the middle of a collision-free transmission at time $t$; otherwise $x_1(t)=x_2(t)$.
In particular, the transition $l=4$ into state $2$ initiates a collision period in which $x_1(t)=x_2(t)$.
This condition is preserved throughout the collision period, including when the system transitions through state $1$ and back to the idle state $0$.
A consequence of the Poisson update arrival process is that the number of updates being simultaneously transmitted can be arbitrarily large. However, in order to apply \Thmref{AOI-SHS}, the state space is truncated so that the largest collision has $M$ updates. The average age in the truncated system is $\age_M$. The average system age with an infinite user population is $\age=\limty{M}\age_M$.
To employ \Thmref{AOI-SHS}, observe first that \eqnref{AOI-SHS-pi} implies
$\lambda\pibar_0=\mu\pibar_1$ and for $k=1,\ldots,M-1$,
\begin{align}
(\lambda+k\mu)\pibar_{k}&=\lambda\pibar_{k-1}+(k+1)\mu\pibar_{k+1}.
\end{align}
Solving for $\pibar_k$, $k=1,\ldots,M$, in terms of $\rho=\lambda/\mu$ and enforcing the normalization constraint yields
\begin{align}\eqnlabel{pibar-all}
\pibar_0 =\bigl(\sum_{j=0}^M \rho^j/j!\bigr)^{\!-1},\quad
\pibar_{k}=\frac{\rho^k}{k!}\pibar_0.
\end{align}
From \eqnref{AOI-SHS-v}, we have for $\qbar\in\set{0,1,2,M}$ that
\begin{subequations}\eqnlabel{SHS-vector}
\begin{align}
\lambda\vvmbar{0}&=\onev[]\pibar_{0}+\Pc\mu\vvmbar{1}\Amat_2+\Pe\mu\vvmbar{1}\Amat_3,\\
(\lambda+\mu)\vvmbar{1}&=\onev[]\pibar_1+\lambda\vvmbar{0}\Amat_1+2\mu\vvmbar{2},\\
(\lambda+2\mu)\vvmbar{2}&=\onev[]\pibar_{2}+\lambda\vvmbar{1}\Amat_4+3\mu\vvmbar{3},\\
M\mu\vvmbar{M}&=\onev[]\pibar_{M}+\lambda\vvmbar{M-1},
\shortintertext{and for $\qbar=k\in\set{3,\ldots,M-1}$,}
(\lambda+k\mu)\vvmbar{k}&=\onev[]\pibar_{k}+\lambda\vvmbar{k-1} +(k+1)\mu\vvmbar{k+1}.
\end{align}
\end{subequations}
\begin{figure}
\centering
\includegraphics{alohaplot-eps-converted-to}
\caption{Average system age $\age$ in \Thmref{collision-age} as a function of offered load $\rho$; time is normalized so that $\mu=1$. Dashed lines show the lower bound $\age_1(\rho)/(\mu P_c)$.}\label{fig:ageplot}
\vspace{-5mm}
\end{figure}
Solving \eqnref{SHS-vector} for $\vvmbar{0},\ldots,\vvmbar{M}$, the average system age in the truncated system is \begin{align}
\age_M&=\E{x_2}=\limty{M}\sum_{k=0}^M \vbar_{k2}.
\end{align}
The average system age is then $\age=\limty{M}\age_M$. These steps can be found in the Appendix. For $j=1,2,\ldots$, we adopt the shorthand notation
\begin{align}\eqnlabel{beta-gamma-defn}
\beta_j\equiv\sum_{i=j}^\infty\frac{\rho^i}{i!}e^{-\rho},\qquad
\gamma_j
\equiv \sum_{k=0}^\infty
\frac{j!}{(j+k)!}\rho^k,
\end{align}
in order to state the following claim.\footnote{Note that $\beta_j=\prob{K\ge j}$ and $\gamma_j=\beta_j/\prob{K=j}$ for a Poisson $(\rho)$ random variable $K$, While it is possible to state \Thmref{collision-age} with $\gamma_j$ replaced by $\beta_j/\prob{K=j}$, this ratio of quantities that both go to zero as $j\to\infty$ can induce numerical stability issues in the calculation of $\age$.}
\begin{theorem}\thmlabel{collision-age}
Poisson updates through a collision channel achieve the average system age
\begin{align*}
\age=\frac{(1\!+\!\rho)e^\rho}{\mu P_c \rho }\!+\!\frac{\beta_1}{\mu}
\!+\!\frac{(3\!+\!\rho)\beta_2}{2\mu}\!+\!\frac{\rho(1\!+\!\rho)\beta_2\gamma_3}{6\mu}
\!+\!\sum_{j=3}^\infty \frac{\beta_j\gamma_j}{j\mu}.
\end{align*}
\end{theorem}
\begin{figure}
\centering
\includegraphics{alohavslotted-eps-converted-to}
\caption{Average system age $\age$ in \Thmref{collision-age} of the unslotted system
vs.~the average system age $\slotage$ of the corresponding slotted Aloha system as a function of offered load $\rho$.}\label{fig:slotted}
\vspace{-5mm}
\end{figure}
With time normalized so that $\mu=1$, Figure~\ref{fig:ageplot} depicts the system age in \Thmref{collision-age} as a function of the offered load $\rho$ for probability of correct reception $\Pc\in\set{0.5, 0.8,1}$. For all $\Pc$, the age becomes high when $\rho$ approaches zero or when $\rho$ becomes large and the system has too many collisions. For $\Pc=1$, the average age happens to be minimized at $\rho=\rho^*=0.5195$, achieving the minimum age of $\age^*=5.513$. As $\Pc$ decreases, the optimal offered load increases slightly. For example, when $\Pc=0.5$, the optimal load is $\rho^*=0.5625$; this achieves an average age of $\age^*=10.40$. We see from Figure~\ref{fig:ageplot} that the average age is not particularly sensitive to variations in $\rho$ near $\rho^*$.
We further observe that all terms in $\age$ are non-negative. With the definition
\begin{align}
\age_1(\rho)\equiv \paren{1+\frac{1}{\rho}}e^\rho, \eqnlabel{asymptotic-age}
\end{align}
the average system age satisfies the lower bound
\begin{align}
\age \ge \frac{\age_1(\rho)}{\mu P_c}
\end{align}
This simple lower bound, depicted in Figure~\ref{fig:ageplot} with dashed lines, is tight for small $\rho$.
It is also instructive to compare the system age of the unslotted and slotted systems. Consider the corresponding infinite-user slotted system. In each unit time slot, the number of fresh transmitted updates is a Poisson random variable $K$ with $\E{K}=\rho$. A fresh update is successfully transmitted in each time slot with probability $P_s=\prob{K=1}=\rho e^{-\rho}$. The average system age is \cite[Equation~(23)]{Kaul-Yates-isit2017}
\begin{align}
\slotage=\frac{1}{2}+\frac{1}{P_s}=\frac{1}{2}+\frac{e^\rho}{\rho}.
\end{align}
Figure~\ref{fig:slotted} compares average system age in the slotted and unslotted systems. We see that the age penalty for unslotted operation is negligible when the offered load $\rho$ is small. However, when the offered load is large, the age penalty becomes large because of the long collision periods induced by unslotted operation.
\section{Individual Age Analysis}\label{sec:indiv}
\begin{figure}[t]
\centering
\includegraphics{nalohaplot20norm-eps-converted-to}
\caption{Average individual age for $N=20$ sources. Time is normalized so that $\mu=20$. The figure compares $\age_{1|20}$ and $\age_1(\rho)$ against a simulation of $N=20$ on/off sources with aggregate update arrival rate $20\rho$. At each $\rho$,
$\color{red}\bullet$ marks the time-average individual ages of each of the $20$ sources while $\times$ marks the average age averaged over the $20$ sources.}
\label{fig:naloha20}
\end{figure}
In practice, the number of sources $N$ will be finite and it is desirable to characterize the age process of an individual source. Fortunately, the infinite user model of \Thmref{collision-age} can be employed to evaluate the individual age for one of $N$ sources by reinterpreting $P_c$, the probability of correct detection of a collision-free update, as the probability that the collision-free update reduces the age of a selected user. Specifically, suppose the aggregate updating rate $\lambda$ in the infinite user model is from $N$ independent sources, each offering updates as a Poisson process of rate $\lambda/N$. In this case, a transmitted update belongs to a source $i$ with probability $1/N$. Hence,
\Thmref{collision-age} can be employed with an update that is transmitted collision-free as belonging to source $i$ (and thus offering an age reduction for source $i$) with probability $P_c=1/N$. This yields the individual age
\begin{align}\eqnlabel{indiv-age}
\age_{1|N}=\frac{N(1+\rho)e^\rho}{\mu\rho }&+\frac{\beta_1}{\mu}
+\frac{(3+\rho)\beta_2}{2\mu}\nn
&+\frac{\rho(1+\rho)\beta_2\gamma_3}{6\mu}
+\sum_{j=3}^\infty \frac{\beta_j\gamma_j}{j\mu}.
\end{align}
For fixed service rate $\mu$ and fixed offered load $\rho$, \eqnref{indiv-age} implies that the individual age grows linearly with the number of users $N$. This is not surprising since the system bandwidth, as embodied in the fixed service rate $\mu$, is shared among $N$ sources. However, to provide good age performance as $N$ becomes large, the system needs bandwidth to grow in proportion to $N$. In this case, we assume the system has $N$ sources, each offering updates at rate $\lambda_0$ but the system bandwidth grows with $N$ so that the service rate of a transmission is $\mu=N\mu_0$. The normalized offered load remains fixed at $\rho=(N\lambda_0)/(N\mu_0)=\lambda_0/\mu_0$. A transmitted update belongs to the selected source with probability $P_c=1/N$. We also assume time is normalized so that $\mu_0=1$. Under these conditions, we observe as $N\to\infty$ that
\begin{align}
\age_{1|N}\to \age_1(\rho).
\end{align}
Here we can interpret $\age_1(\rho)$ as the individual age on a collision channel in the limit of the number of sources becoming large and the transmission time of an update approaching zero. In this asymptotic limit, the individual average age is minimized at $\rho=(\sqrt{5}-1)/2=0.618$.\footnote{It can be shown that $\rho=0.618$ also maximizes the probability the system is transmitting a collision-free update.}
\begin{figure}[t]
\centering
\includegraphics{nalohaplot100norm-eps-converted-to}
\caption{Average individual age for $N=100$ sources. Time is normalized so that $\mu=100$. The figure compares $\age_{1|100}$ and $\age_1(\rho)$ against a simulation of $N=100$ on/off sources with aggregate update arrival rate $100\rho$. At each $\rho$, $\color{red}\bullet$ marks the time-average individual ages of each of the $100$ sources while $\times$ marks the average age averaged over the $100$ sources.}\label{fig:naloha100}
\end{figure}
We will see this individual age model is somewhat pessimistic in that the Poisson update process of source $i$ can generate self-colliding updates; i.e., a source $i$ update can collide with time-overlapping updates also from source $i$. In practice, each source transmits one update at a time and never has a self-collision. In this sense, $\age_{1|N}$
and $\age_1(\rho)$
are approximations for the individual average age.
To evaluate these approximations, we simulate a system with $N$ independent on/off sources. Each source is either transmitting an update of exponential duration with expected value $1/\mu$, or being silent for an exponential period with expected length $1/\lambda_0-1/\mu$. By this construction, the two-state update process of each source offers updates at the longterm rate of $\lambda_0=\lambda/N$ updates per unit time.
As $N$ becomes large, we expect the aggregate update process to be reasonably approximated by a rate $N\lambda_0=\lambda$ Poisson process. We also expect each source to obtain average individual age that is approximated by $\age_{1|N}$.
Under these conditions, Figures~\ref{fig:naloha20} and~\ref{fig:naloha100} compare $\age_{1|N}$ and $\age_1(\rho)$ against the simulated time-average ages experienced by each of the $N$ on/off sources, each generating $50,000$ updates. Time is normalized so that $\mu=N$ and the average update transmission time is $1/\mu=1/N$. The aggregate offered load is $\rho=(N\lambda_0)/N=\lambda_0$.
In Figure~\ref{fig:naloha20} with $N=20$ sources, $\age_{1|N}$, which is derived from the infinite user model of \Thmref{collision-age}, is pessimistic in slightly (by 2-3\%) overestimating the average age received by a source. The asymptotic approximation, $\age_1(\rho)$, which discards terms of $\age_{1|N}$ that become negligible as $\mu=N$ becomes large, is observed to be an even better age approximation in the finite user system. In Figure~\ref{fig:naloha100} with $N=100$ sources, we see that that with more sources, the approximation $\age_1(\rho)$ becomes an increasingly accurate approximation to the average individual age.
\section{Conclusion}\label{sec:conclusion}
For uncoordinated transmit-only sensors, this work provides an exact analysis for the system age. The uncoordinated transmit-only system works well as long as the normalized offered load is near $\rho^*=0.6$. When these networks have a nontrivial number of sources, $\age_1(\rho)$ is a useful approximation for the individual age in a system with offered load $\rho$.
From $\age_1(\rho)$, we see that the individual age penalty is substantial (on the order of $10\times$) if the offered load is, say, $\rho^*/10$ or $10\rho^*$. Moreover, we saw in the comparison with the slotted system that age in the unslotted system is particularly sensitive to overloading the system. Configuring the network of transmit-only sources for the proper offered load would be an issue at time of deployment. On the other hand, adaptive configuration may also be possible if the sources have access to some minimal feedback.
In addition, there remain a number of open questions about how additional coordination mechanisms, such as collision detection and/or avoidance, and state-dependent updating policies, contribute to reducing AoI.
\appendix
\section*{Proof of \Thmref{collision-age}}
The mappings $\Amat_l$ induce $x_1(t)=x_2(t)$ in all states $k\neq 1$. This implies \eqnref{SHS-vector} has a solution such that $\vvmbar{k}=\rvec{\vbar_{k} &\vbar_{k}}=\vbar_{k}\onev[]$ for all $k\neq 1$. Only $\vvmbar{1}=\rvec{\vbar_{11} & \vbar_{12}}$ has distinct non-identical components. In terms of $\vbar_{11},\vbar_{12}$ and $\vbar_k$, $k\neq1$, \eqnref{SHS-vector} becomes
\begin{subequations}\eqnlabel{SHS-scalar}
\begin{align}
\rho\vbar_{0}&=\muinv \pibar_{0}+\Pc\vbar_{11}+\Pe\vbar_{12},\eqnlabel{vbar0}\\
(1+\rho)\vbar_{11} &=\muinv\pibar_1+2\vbar_{2},\eqnlabel{vbar11}\\
(1+\rho)\vbar_{12}&=\muinv \pibar_1+\rho\vbar_0+2\vbar_{2},\eqnlabel{vbar12}\\
(2+\rho)\vbar_{2}&=\muinv \pibar_{2}+\rho\vbar_{12}\!+\!3\vbar_{3},\!
\eqnlabel{vbar2}\\
M\vbar_{M}&=\muinv \pibar_M+\rho\vbar_{M-1}.\eqnlabel{vbarM}
\shortintertext{and for $3\le k\le M-1$,}
\rho_k\vbar_{k}&=\muinv \pibar_k + \rho\vbar_{k-1} +(k+1)\vbar_{k+1}\eqnlabel{vbark3plus}.
\end{align}
\end{subequations}
The average age in \Thmref{AOI-SHS} becomes
\begin{align}\eqnlabel{ageM}
\age_M = \vbar_{0}+\vbar_{12} +\vbar_2+\sum_{j=3}^M\vbar_j.
\end{align}
In the limit of large $M$, we obtain the limiting average age $\age=\limty{M}\age_M$.
Equations \eqnref{vbarM} and \eqnref{vbark3plus} admit the
solution
\begin{align}
\vbar_{k}=\frac{\rho}{k} \vbar_{k-1}+\frac{\beta_{k|M}}{k\mu}, \qquad 3 \le k\le M,\eqnlabel{vbark-step}
\end{align}
where
\begin{align}\eqnlabel{betaM-defn}
\beta_{k|M}=\sum_{i=k}^M\pibar_i =\pibar_0\sum_{i=k}^M\rho^i/i!
\end{align}
is the stationary probability of the system being in a collision of $k$ or more updates.
Now we observe that it follows from \eqnref{vbark-step} that for $3\le l\le M$,
\begin{align}\eqnlabel{vbarl}
\vbar_l =\sum_{j=3}^l \frac{(j-1)!}{\mu l!}\rho^{l-j}\beta_{j|M} +\frac{2\rho^{l-2}}{l!} \vbar_2.
\end{align}
Defining $V_{3:M}=\sum_{l=3}^M \vbar_l$, it then follows from \eqnref{vbarl} and reordering of the sums over $l$ and $j$ that
\begin{align}
\!\!\!V_{3:M}
&\!=\!\sum_{j=3}^M\!\frac{\beta_{j|M}}{j\mu}\sum_{l=j}^M\!\frac{j!}{l!}\rho^{l-j}\!+\!\frac{\rho\vbar_2}{3}\sum_{m=0}^{M-3} \frac{\rho^m}{(m\!+\!3)!}.
\eqnlabel{S3M-v2}
\end{align}
Defining
$\gamma_{j|M}=\sum_{k=0}^{M-j}\frac{j!}{(k+j)!}\rho^k$,
the index shift $k=l-j$ in \eqnref{S3M-v2} yields
\begin{align}
V_{3:M}=\sum_{j=3}^M\frac{\beta_{j|M}}{j\mu}\gamma_{j|M}+\frac{\rho}{3}\gamma_{3|M}\vbar_2.\eqnlabel{S3M-v3}
\end{align}
Applying \eqnref{vbark-step} with $k=3$ to \eqnref{vbar2} yields
\begin{align}
\vbar_2&= \frac{\pibar_{2}+\beta_{3|M}}{2\mu}+\frac{\rho}{2}\vbar_{12}=\frac{\beta_{2|M}}{2\mu}+\frac{\rho}{2}\vbar_{12}\eqnlabel{vbar2-v2}.
\end{align}
From \eqnref{vbar2-v2} and the identity $\pibar_1+\beta_{2|M}=\beta_{1|M}$, it follows from \eqnref{vbar11} and \eqnref{vbar12} that
\begin{align}
\vbar_{11}&=\frac{\rho^2\vbar_0}{1+\rho}+\frac{\beta_{1|M}}{\mu},\quad
\vbar_{12}=\rho\vbar_0+\frac{\beta_{1|M}}{\mu}.\eqnlabel{vbar1-v2}
\end{align}
From \eqnref{vbar1-v2} and the identity $\Pc+\Pe=1$, it follows from \eqnref{vbar0} that
\begin{align}
\vbar_0=\frac{1+\rho}{\mu\rho\Pc}.
\eqnlabel{vbar0-v2}
\end{align}
It then follows from \eqnref{vbar2-v2} and \eqnref{vbar1-v2} that
\begin{align}\eqnlabel{vbar2-v3}
\vbar_2&=\frac{\rho^2}{2}\vbar_0 +\frac{\beta_{2|M}+\rho\beta_{1|M}}{2\mu}.
\end{align}
Applying \eqnref{S3M-v3}, \eqnref{vbar1-v2}, and \eqnref{vbar2-v3} to
\eqnref{ageM}
and observing that $\limty{M}\beta_{j|M}=\beta_j$ and
$\limty{M}\gamma_{j|M}=\gamma_j$, the claim follows.
\newpage
\bibliographystyle{unsrt}
\bibliography{AOI-2019-01,collision}
\end{document}
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Of Course, IVF is only ever presented like this picture.Boo hiss to the nasty Catholics who object.
They never mention the take-home baby rate is something like 15%. And we never mention the hundreds of children who die in order that one might live. A bit like me wandering into my local school, killing all the children but the cutest one, and then making sure nobody hears about it, get myself and hubby and child into some arty soft-focus shots, and ensuring that anyone who objects is painted as a religious nutcase/killjoy.IVF doesn't even cure infertility, it merely bypasses it.
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\begin{document}
\begin{center}
\font\titlerm=cmr10 scaled\magstep4
\font\titlei=cmmi10 scaled\magstep4
\font\titleis=cmmi7 scaled\magstep4
\centerline{\titlerm Green's Function For Linear Differential}
\vspace{0.5cm}
\centerline{\titlerm Operators In One Variable }
\vspace{1.5cm}
\noindent{{
Adel Kassaian{$^*$}\footnote{E-mail:
a.kassaian@gmail.com},
}}\\
\end{center}
\begin{abstract}
General formula for causal Green's function of linear differential
operator of given degree in one variable, $
({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0} P_{k}(x) {\rd^{k}_{x}}) $, is
given according to coefficient functions of differential operator as
a series of integrals. The solution also provides analytic formula
for fundamental solutions of corresponding homogenous linear
differential equation, $\big({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0}
P_{k}(x) {\rd^{k}_{x}}\big)\,y(x)=0$, as series of integrals.
Furthermore, multiplicative property of causal Green's functions is
shown and by which explicit formulas for causal Green's functions of
some classes of decomposable linear differential operators are
given. A method to find Green's function of general linear
differential operator of given degree in one variable with arbitrary
boundary condition according to coefficient functions of
differential operator is demonstrated.
\end{abstract}
\section{Converting initial value problem for ordinary linear differential equation into Volterra's integral equation}
Initial value problem for linear differential equation of degree $n$
in one variable, \be\big({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0} P_{k}(x)
{\rd^{k}_{x}}\big)\,y(x)=g(x),\label{yak}\ee can be converted to
Volterra's integral equation of second kind. Following the reference
\cite{ik}, the key relation in the procedure is the identity;
\be
\int_{a}^{x}dz_1
\int_{a}^{z_{1}} dz_{2} \cdots\int_{a}^{z_{r-1}}\,\,dz_{r}\,\,
F(z_{r})= \int_{a}^{x} dz {(x-z)^{r-1}\over (r-1)!}F(z), \nn\ee
which can be easily proved for arbitrary function $F(x)$ using
integration by part and induction. For initial condition
$\rd^{\,i}_{x}\,y(a)=c_i$ for $ i=0,1,..,n-1$, putting $u(x)=
\rd_{x}^{\,n}\,y(x)$ in the above relation and setting $r=n-k$, for
$k=0,...,n-1$ we have;
\be \rd_{x}^{\,k}\,y(x)-\sum_{i=k}^{n-1}
c_{i}{(x-a)^{i-k}\over(i-k)!}=\int_{a}^{x} dz {(x-z)^{n-k-1}\over
(n-k-1)!}\,u(z).\label{head} \ee
Inserting $\rd_{x}^{\,n}\,y(x)=u(x)$ and
$\rd_{x}^{\,k}\,y(x)=\sum_{i=k}^{n-1}
c_{i}{(x-a)^{i-k}\over(i-k)!}+\int_{a}^{x} dz {(x-z)^{n-k-1}\over
(n-k-1)!}\,u(z)$ for $k=1,...,n-1$ into the left hand side of
equation (\ref{yak}) we get; $
\big({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0} P_{k}(x)
{\rd^{k}_{x}}\big)\,y(x)=u(x)+\sum_{k=0}^{n-1}\sum_{i=k}^{n-1}
c_{i}P_{k}(x){(x-a)^{i-k}\over(i-k)!}+\int_{a}^{x} dz
\sum^{n-1}_{k=0}P_{k}(x){(x-z)^{i-k}\over (i-k)!}\,u(z)$. Therefore
we have the following Volterra's equation for $u(x)$;
\be u(x)+\int_{a}^{x} dz\,
K(x,z) \,u(z)=g(x)+S(x),\label{volttera} \ee
where
$K(x,z)=\big(\sum_{k=0}^{n-1}P_{k}(x){{(x-z)^{n-k-1}}\over(n-k-1)!}\big)$
and $S(x)=-\sum_{k=0}^{n-1}\sum_{i=k}^{n-1}
c_{i}P_{k}(x){(x-a)^{i-k}\over(i-k)!}\,=\,
-\sum_{i=0}^{n-1}\sum^{i}_{k=0}
c_{i}{{P_{k}(x)(x-a)^{i-k}}\over(i-k)!}$. By setting $k=0$ in
equation (\ref{head}), the solution $y(x)$ is given by;
\be y(x)=D(x)+\int_{a}^{x} dz {(x-z)^{n-1}\over (n-1)!} u(z).\ee
where $D(x)=\sum_{i=0}^{n-1} c_{i}{(x-a)^{i}\over i !}$.
\section{Casual Green's function for linear differential
operators in one variable}
\label{tecneq}
For equation (\ref{yak}) with initial condition
$\rd^{\,i}_{x}y(a)=0$ for $ i=0,1,..,n-1$, the corresponding
Volterra's equation is given by;
\be u(x)+\int_{a}^{x} dz\,
K(x,z) \,u(z)=g(x),\nn \ee
where
$K(x,z)=\big(\sum_{k=0}^{n-1}P_{k}(x){{(x-z)^{n-k-1}}\over(n-k-1)!}\big)$
and $y(x)=\int_{a}^{x} dz {(x-z)^{n-1}\over (n-1)!} u(z)$.For
$g(x)\in L^2[a,b]$, the condition $(\int^{b}_{a}\int^{b}_{a} dx\,
dy\, |K(x,y)|^2)<\infty $ is sufficient condition for existence of
unique solution in $L^2[a,b]$, given by iteration (e.g. see
\cite{it}). Clearly this conditions can be satisfied if $P_{i}(x)$
(\,$i=0,1,...,n-1$) and $g(x)$ functions are taken to be continuous
on $[a,b]$. Therefore we can state the following theorem;
\begin{theorem}\label{FinalGreen's}{The Green's function for inhomogeneous linear differential equation
$\big({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0} P_{k}(x)\,{\rd^{k}_{x}}
\big)\,y(x)=g(x),$
where $P_{i}(x)$ \,(\,$i=0,1,...,n-1$) and $g(x)$ are in $\mc[a,b]$,
with the boundary condition; $\rd^{\,i}_{x}y(a)=0$ for $i=0,1,..,n-1$, is given
by;
\be G(x,y)=\theta(x-y)
\Big({(x-y)^{n-1}\over(n-1)!}+\int^{x}_{y}dz\,{(x-z)^{n-1}\over(n-1)!}\,R(z,y)\Big),\label{maryam}\ee
where \bea R(x,y)=h(x,y)+\sum_{r=2}^{\infty}\int_{y}^{x}dz_1
\int_{y}^{z_{1}} dz_{2} \cdots\int_{y}^{z_{r-2}}\,\,dz_{r-1}\,
h(x,z_1)\,h(z_{1}, z_{2})\nn\\\cdots h(z_{r-1}, y),\eea and
$\displaystyle\,\,
h(x,y)=-\sum_{k=0}^{n-1}P_{k}(x){{(x-y)^{n-k-1}}\over(n-k-1)!}.$ The
solution to inhomogeneous linear differential equation (\ref{yak}})
for $x\in[a,b]$ is then given by \,\,$ y(x)=\int_{a}^{\infty} dz
\,G(x,z) \,g(z).$ \\
\end{theorem}
{\bf{Proof}}. In order to prove (\ref{maryam}) is Green's function
of (\ref{yak}) it is enough to prove that for the two variables
function; \be
T(x,y)=\Big({(x-y)^{n-1}\over(n-1)!}+\int^{x}_{y}dz\,{(x-z)^{n-1}\over(n-1)!}\,R(z,y)\Big),\label{hasan}\ee
we have $\big({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0} P_{k}(x)
{\rd^{k}_{x}} \big)\,T(x,y)=0$ and $(\rd^{\,i}_{x} T(x,y))|_{x=y}=0$
for $i=0,1,..,n-2$ and $(\rd^{\,n-1}_{x} T(x,y))|_{x=y}=1$ [e.g. see
\cite{ij}]. This can be easily done by noting; \be \rd^{\,i}_{x}
T(x,y)=\Big({(x-y)^{n-i-1}\over(n-i-1)!}+\int^{x}_{y}dz\,{(x-z)^{n-i-1}\over(n-i-1)!}\,R(z,y)\Big)
\hspace{0.5cm} i=0,1,..,n-1\label{magic}\ee \be \rd^{\,n}_{x}
T(x,y)= R(x,y), \nn\ee and therefore, \bea
({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0} P_{k}(x) {\rd^{k}_{x}}
)\,\,T(x,y)=&&R(x,y)+\sum^{n-1}_{k=0}
{ P_{k}(x)(x-y)^{n-k-1}\over(n-k-1)!}\nn\\&&+\sum^{n-1}_{k=0}P_{k}(x)\int^{x}_{y}dz\,{(x-z)^{n-k-1}\over(n-k-1)!}R(z,y)\nn\\
=&& R(x,y)-h(x,y)-\int^{x}_{y} dz\, h(x,z)\,R(z,y) \nn\\=&&
R(x,y)-h(x,y)-\big(R(x,y)-h(x,y)\big)=0.\nn\eea
In the last line we used $\int^{x}_{y} dz\,
h(x,z)\,R(z,y)=\big(R(x,y)-h(x,y)\big)$, which comes from definition
of $R(x,y)$. By using (\ref{magic}) we have $(\rd^{\,i}_{x}
T(x,y))|_{x=y}=0$ for $i=0,1,..,n-2$ and $(\rd^{\,n-1}_{x}
T(x,y))|_{x=y}=1$.\\
The Green's function for (\ref{yak}) with mentioned boundary
condition is called causal solution which by method of variation of
parameters is given by;
\be G(x,y)= \big(\sum_{i=1}^{n}{W_{i}(y)\, u_{i}(x) \over W(y)}
\big) \theta(x-y), \label{korner}\ee
where $u_{1}(x), u_{2}(x),...,u_{n}(x)$ are fundamental solutions of
corresponding homogeneous differential equation;
\,$({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0} P_{k}(x) {\rd^{k}_{x}}
)\,u_{i}(x)=0$. $W(y)$ is the
Wronskian and $W_{i}(y)$ is the Wronskian with its $i^{\,th}$ column
in determinant is replace by $(0,0,..,0,1)$. Comparing this result
with (\ref{maryam}) we have the identity;
\be \sum_{i=1}^{n}{W_{i}(y)\, u_{i}(x) \over W(y)}
\,=\,{(x-y)^{n-1}\over(n-1)!}+\int^{x}_{y}dz\,{(x-z)^{n-1}\over(n-1)!}\,R(z,y).\label{kasai}\ee
For linear differential operator of first degree ($n$=$1$), like
$\rd_{x}-P(x)$, the causal Green's function using [theorem
\ref{FinalGreen's}] is equal to; \bea(\rd_{x}-P(x))^{-1}&&=
\theta(x-y)\Big(1+\sum_{k=1}^{\infty}\int_{y}^{x}dz_1
\cdots\int_{y}^{z_{n-2}}dz_{k-1}\int_{y}^{z_{k-1}}dz_{k}\,
P(z_1)\cdots P(z_{k-1})P(z_{k})\Big)\nn\\&&=
\theta(x-y)\Big(1+\sum_{k=1}^{\infty}{1\over
k!}\int_{y}^{x}\cdots\int_{y}^{x}\int_{y}^{x}dz_1 \cdots
dz_{k-1}dz_{k}\, P(z_1) \cdots P(z_{k-1})P(z_{k})\Big)\nn
\\&&= \theta(x-y)e^{\int^{x}_{y}dz P(z)}.\nn\eea For linear
differential operator of degree two in the form of;
$(\rd^{2}_{x}-P(x))$, the causal Green's function by using [theorem
\ref{FinalGreen's}] is given by $T_{s}(x,y)\theta(x-y)$ where; \bea
T_{s}(x,y)=\Big\{(x-y)+\sum_{k=1}^{\infty}\Big(\int_{y}^{x}dz_1
\cdots\int_{y}^{z_{k-2}}dz_{k-1}\int_{y}^{z_{k-1}}dz_{k}\nn\\\,
(x-z_1)P(z_1)(z_1-z_2)P(z_2)(z_2-z_3)\cdots
(z_{k-1}-z_{k})P(z_{k})(z_{k}-y)\Big)\Big\}.\label{shahla}\eea For
example
$(\rd^{2}_{x}-x)^{-1}=\theta(x-y)\,(x-y)+\theta(x-y)\int^{x}_{y}dz(x-z)z(z-y)
+\theta(x-y)\int^{x}_{y} dt\int^{t}_{y} dz\, \,\big((x-t)t
(t-z)z(z-y)\big)+
\cdots = \theta(x-y) \Big(\big(x - y\big)+ \big({x^4 \over 12} - {(x^3 y) \over 6}
+ {(x y^3) \over 6} - {y^4 \over 12}\big)+ \big({x^7 \over 504}
- {(x^6 y) \over 180}+ {(
x^4 y^3) \over 72} -
{(x^3 y^4) \over 72} + {(x y^6) \over 180} -
{y^7 \over 504}\big)+\cdots \Big),
$ which is consistent with solution
$\displaystyle(\rd^{\,2}_{x}-x)^{-1}=\theta(x-y)\big({-\mathrm{Ai}(x)\mathrm{Bi}(y)
+ \mathrm{Ai}(y)\mathrm{Bi}(x)\over{\mathrm{Ai}(y)\mathrm{Bi}'(y)
- \mathrm{Ai}'(y)\mathrm{Bi}(y)}}\big)$ derived by
(\ref{korner}).\\
It can be seen from (\ref{maryam}) that if $P_{i}(x)$
(\,$i=0,1,...,n-1$) functions are smooth on
$[a,b]$ then $T(x,y)$, given by (\ref{hasan}), is smooth function on
$[a,b]\times[a,b]$, in which case we state the following theorem;
\begin{theorem}\label{adel}{If $T_{1}(x,y)\theta(x-y)$ and $T_{2}(x,y)\theta(x-y)$ are
causal Green's functions for linear differential operators
$\mathcal{O}_{1}(x,\rd_{x})=\big({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0}
P_{k}(x) {\rd^{k}_{x}} \big)$ and
$\mathcal{O}_{2}(x,\rd_{x})=\big({\rd^{\,m}_{x}}+\sum^{m-1}_{k=0}
q_{k}(x) {\rd^{k}_{x}} \big)$ respectively (assuming $P_{i}(x)$'s
and $q_{i}(x)$'s functions are in ${\mc}^{\infty}[a,b]$) then
$T_{3}(x,y)\theta(x-y)$ where, \be T_{3}(x,y)=\int_{y}^{x} dz\,
T_{2}(x,z)\, T_{1}(z,y), \ee is the causal Green's function for
linear differential operator
$\mathcal{O}_{3}(x,\rd_{x})=\mathcal{O}_{1}(x,\rd_{x}).\mathcal{O}_{2}(x,\rd_{x})$}
\end{theorem}
{\bf{Proof.}} By assumption;
$\mathcal{O}_{1}(x,\rd_{x})\,T_{1}(x,y)=0$ and $(\rd^{\,i}_{x}
T_{1}(x,y))|_{x=y}=0$ for $i=0,1,..,n-2$ and $(\rd^{\,n-1}_{x}
T_{1}(x,y))|_{x=y}=1$ also
$\mathcal{O}_{2}(x,\rd_{x})\,T_{2}(x,y)=0$ and $(\rd^{\,i}_{x}
T_{2}(x,y))|_{x=y}=0$ for $i=0,1,..,m-2$ and $(\rd^{\,m-1}_{x}
T_{2}(x,y))|_{x=y}=1$, therefore we have; \be \rd^{\,i}_{x}
\,T_{3}(x,y)=\int_{y}^{x} dz\,(\rd^{\,i}_{x}( T_{2}(x,z)))\,
T_{1}(z,y), \hspace{0.5cm} i=0,1,..,m-1\hspace{0.5cm}\label{kant}\ee
\be \rd^{\,m}_{x} T_{3}(x,y)= T_{1}(x,y)+\int_{y}^{x}
dz\,(\rd^{\,m}_{x}( T_{2}(x,z)))\, T_{1}(z,y), \label{hapar}\ee \bea
\rd^{\,i}_{x} T_{3}(x,y)=
\rd^{\,i-m}_{x}(T_{1}(x,y))+\rd^{\,i-m}_{x}\big(\int_{y}^{x}
dz\,&(\rd^{\,m}_{x}( T_{2}(x,z)))\, T_{1}(z,y)\big),
\nn\\&i=m+1,..,m+n-1.\label{hava}\eea Concentrating on the second
term in (\ref{hava}), we have for $k=1,2,..,n-1$;
\bea \rd^{\,k}_{x}\big(&\int_{y}^{x}
dz\,(\rd^{\,m}_{x}(T_{2}(x,z)))\, T_{1}(z,y) \big)=
\Big\{\sum^{k-1}_{j=0}\rd_{x}^{j}\Big((\rd^{\,m+k-1-j}_{x}
T_{2}(x,z))|_{z=x} \nn\\& T_{1}(x,y) \Big)\Big\}+\big(\int_{y}^{x}
dz\,(\rd^{\,m+k}_{x}( T_{2}(x,z)))\, T_{1}(z,y)
\big)\nn\\&=\{\sum^{k-1}_{j=0}\sum^{j}_{r=0}{j \choose
r}\Big((\rd^{\,m+k-1-j+r}_{x}
T_{2}(x,z))|_{z=x}\,\rd_{x}^{j-r}T_{1}(x,y) \Big)\}\nn\\
&+\big(\int_{y}^{x} dz\,(\rd^{\,m+k}_{x}( T_{2}(x,z)))\, T_{1}(z,y)
\big)\label{shadi}\eea
From (\ref{kant}),(\ref{hapar}), (\ref{hava}) and (\ref{shadi}) we
have $(\rd^{\,i}_{x} T_{3}(x,y))|_{x=y}=0$ for
$i=0,1,...,m+n-3,m+n-2$ and $(\rd^{\,m+n-1}_{x}
T_{3}(x,y))|_{x=y}=1$. On the other hand; \bea
\mathcal{O}_{2}(x,\rd_{x})
T_{3}(x,y)&=&\big({\rd^{\,m}_{x}}+\sum^{m-1}_{k=0} q_{k}(x)
{\rd^{\,k}_{x}} \big) \int_{y}^{x} dz\, T_{2}(x,z)\,
T_{1}(z,y)\nn\\&=&\rd_{x}\big(\int_{y}^{x} dz\, \rd^{\,m-1}_{x}
T_{2}(x,z)\, T_{1}(z,y)\big)\nn\\&&+ \int_{y}^{x}
dz\,\big(\sum^{m-1}_{k=0} q_{k}(x) {\rd^{\,k}_{x}} T_{2}(x,z)\big)\,
T_{1}(z,y)\nn\\&=& T_{1}(x,y)+\int^{y}_{x}
dz\,\mathcal{O}_{2}(x,\rd_{x}) T_{2}(x,z)
T_{1}(z,y)\nn\\&=&T_{1}(x,y).\nn\eea Therefore
$\mathcal{O}_{1}(x,\rd_{x}).
\mathcal{O}_{2}(x,\rd_{x})T_{3}(x,y)=\mathcal{O}_{1}(x,\rd_{x})T_{1}(x,y)=0.$
\\
The following corollary comes as a consequence;
\begin{corollary}\label{lema786} {Causal Green's function for
differential operator,\be {\mathcal
O}(x,\rd_{x})=\big(\rd_{x}-p_{1}(x)\big)\,\big(\rd_{x}-p_{2}(x)\big)
\,\cdots\big(\rd_{x}-p_{n}(x)\big),
\label{formallimit2}\hspace{1cm}\ee where $p_{\,i}(x)\in
{\mc}^{\infty}[a,b]$ (for $i=1,...,n$)is given by; \bea
G(x,y)=\theta(x-y)&\int_{y}^{x}dz_1 \int_{y}^{z_{1}}dz_{2}
\cdots\int_{y}^{z_{r-2}}dz_{n-1}\big(\,{e^{\int^{x}_{z_{1}}dt_{n}
\,p_{n}(t_{n})}}\nn\\&{e^{\int^{z_{1}}_{z_{2}}dt_{n-1}
\,p_{n-1}(t_{n-1})}}\cdots{e^{\int^{z_{n-1}}_{y} dt_{1}
\,p_{1}(t_{1})}}\big)\label{iraj1}\eea}
\end{corollary}
For example for differential operator
$\mathcal{O}(x,\rd_{x})=\rd^{2}_{x}+3x\rd_{x}+(2x^{2}+2)$, since
$(\rd_{x}+x)(\rd_{x}+2 x)=\rd^{2}_{x}+3x\rd_{x}+(2x^{2}+2)$, by
using result (\ref{iraj1}) one gets $G(x,y) =\sqrt{\pi\over2}\ \,
e^{\,y^{2}-{x^{2}\over 2}}\, \{\mbox{Erf}
({{x\over\sqrt2}})-(\mbox{Erf}({{y\over\sqrt2}}))\}\,\,\theta(x-y)$.\\
\begin{corollary}\label{hasti} {\, Causal Green's function for linear differential operator ; \be{\mathcal
O}(x,\rd_{x})=\sum_{k=0}^{n}\alpha_{k}{\rd^{\,k}_{x}}\, ,\ee where
$\alpha_{k}\in \mc$ and $\alpha_{n}\ne 0$, is given by;
\be G(x,y)\,\,=\,{\theta(x-y)\over\alpha_{n}}\int_{y}^{x}dz_1
\,\int_{y}^{z_1}\,dz_2 \cdots \int_{y}^{z_{n-2}}dz_{n-1}\,\,
e^{\big( \beta_{1}(x-z_1)+\beta_{2}(z_1-z_2)\cdots
+\beta_{n}(z_{n-1}-y)\big)},\ee where $\beta_{1}$, $\beta_{2}\cdots
\beta_{n}$ are $n$ complex roots of equation
$\displaystyle\sum_{i=0}^{n}\alpha_{i} {X}^{\,i}=0$}.
\end{corollary}
{\bf Proof}. Differential operator ${\mathcal
O}(x,\rd_{x})=\sum_{i=0}^{n}\alpha_{i}{\rd^{\,i}_{x}}$, according to {\em{Fundamental theorem of
algebra}},
can be written as, $\,
\sum_{i=0}^{n}\alpha_{i}\rd_x^{{\,i}}=\alpha_n
(\rd_x-\beta_1)(\rd_x-\beta_2)\cdots(\rd_x-\beta_n)$.
Therefore by using (\ref{iraj1}) the result is proved. \\
For example $ (\rd^{2}_{x}-\omega^2)^{-1} = \theta(x-y)\int_{y}^{x}
dz_{1}
e^{(\omega(x-z_1)-\omega(z_1-y))}={{\sinh\omega(x-y)}\over\omega}\theta(x-y)$\,\,\
and also $ (\rd^{3}_{x}-i
\alpha\rd^{2}_{x}-\omega^2\rd_{x}+i\alpha\omega^2)^{-1} =
\theta(x-y)\int_{y}^{x} dz_{1}(\int_{y}^{z_1} dz_{2}
e^{(\omega(x-z_1)-\omega(z_1-z_2)+i \alpha(z_2-y))}) $ $
\hspace{0.5cm}=\theta(x-y)\big({{e^{\,\omega(x-y)}-e^{i\alpha(x-y)}}\over{\alpha^2+\omega^2}}
-{\sinh[\omega(x-y)]\over{i\,\alpha\,\omega+\omega^2}}\big). $\\
Lets consider a differential operator in form of \be
\mathcal{O}(x,\rd_x)=-\rd_{x}^{2}+v(x).\label{hamed}\ee By
decomposing it into two firs degree differential operators;
$-(\rd_{x}^{2}-v(x))=-(\rd_{x}-p(x))(\rd_{x}-q(x))$, we have
consequently $q(x)=-p(x)$ and $ p(x)^2-\rd_{x}p(x)=v(x)$. Therefore
according to (\ref{iraj1}) the causal the Green's function is given
by; \be \big(-\rd_{x}^{2}+v(x)\big)^{-1}=-\theta(x-y)\int^{x}_{y} dz
e^{(-\int^{x}_{z}dt p(t)+\int^{z}_{y}dt' p(t'))},\label{sarvin}\ee
where $p(x)$ is solution for first order nonlinear differential
equation $ p \,(x)^2-\rd_{x} p\, (x)=v(x)$. This is just Riccati
equation, thus the answers to $ p \,(x)^2-\rd_{x} p\, (x)=v(x)$ are
given by solutions of homogenous differential equation $
(-\rd_{x}^{2}+v(x))u_{1,2}(x)=0 $ where $p(x)=-({u_{1,2}'\over
u_{1,2}})$. Inserting $p(x)=-({u_{1}'\over u_{1}})$ into solution
(\ref{sarvin}), we have; $
\big(-\rd_{x}^{2}+v(x)\big)^{-1}=-\theta(x-y)u_{1}(x)u_{1}(y)\int^{x}_{y}
dz ({1\over {u(z)}^2}).$ Considering the relation $ u_{2}(z)=
u_{1}(z)\int dz {1\over {u_{1}(z)}^{\,2}}$ (valid for homogenous
differential equation $ (-\rd_{x}^{2}+v(x))u_{1,2}(x)=0 $) the
solution (\ref{sarvin}) becomes the standard solution,
$\big(-\rd_{x}^{2}+v(x)\big)^{-1}=-\theta(x-y)\Big(u_{2}(x)u_{1}(y)-u_{1}(x)u_{2}(y)\Big).$
\\
Considering [theorem \ref{adel}] one can introduce the following
infinite non-abelian group of operators on a subspace of
${\mc}^{\infty}[a,b]$. We call it "Lalescu Group";
\begin{itemize}{\item{\bf Lalescu Group.}} {\em The Group of
differential operators of the form;
$\big({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0} P_{k}(x) {\rd^{k}_{x}}\big)$
of all finite order, $n\geq 0$, where $P_{k}(x)\in
{\mc}^{\infty}[a,b]$ (for k=0,1,..,n-1) and their corresponding
causal Green's functions $G(x,y)=T(x,y)\theta(x-y)$ (given by
(\ref{maryam})), on subspace of ${\mc}^{\infty}[a,b]$ consisting of
functions which themselves and their derivatives to all orders are
zero at $x=a$, creates non-abelian group with operators
multiplication.}
\end{itemize}
Beside all differential operators
$\mathcal{O}(x,\rd_{x})=\big({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0}
P_{k}(x) {\rd^{k}_{x}}\big)$ and their causal Green's functions
$G(x,y)=T(x,y)\theta(x-y)$, the group also contains
integro-differential operators and their inverses, coming from
mixing these two groups of operators. For example
$\mathcal{O}_{1}(x,\rd_{x}).T_{2}(x,y)\theta(x-y)$, acting on
$\phi(x)$ in the function space as
$\mathcal{O}_{1}(x,\rd_{x})(\int_{a}^{x}\, dz \,T_{2}(x,z)\phi(z))$,
and its inverse $\mathcal{O}_{2}(x,\rd_{x}).T_{1}(x,y)\theta(x-y)$
(where $\mathcal{O}^{-1}_{1}(x,\rd_{x})=T_{1}(x,y)\theta(x-y)$ and
$\mathcal{O}^{-1}_{2}(x,\rd_{x})=T_{2}(x,y)\theta(x-y)$).
\section{Green's function for linear differential
operators in one variable with other boundary conditions}
\label{tecneq2}
In previous part the causal Green's function was derived according
to coefficient functions of linear differential operator, here in
this part it is shown that Green's function of general linear
differential operator, for other boundary conditions on $[a,b]$
(e.g. Sturm-Liouville problem), can also be derived according to
coefficient functions of differential operator. First we note that
homogeneous linear differential equation of degree $n$ in one
variable, $\big({\rd^{\,n}_{x}}+\sum^{n-1}_{k=0} P_{k}(x)
{\rd^{k}_{x}}\big)\,u(x)=0,$ for initial condition $\rd^{\,i}_{x}
\,u(a)=c_{i}$ for $ i=0,1,..,n-1$ can be converted to Volterra's
integral equation of second kind as; $\big(\mu(x)+\int_{a}^{x} dz\,
K(x,z) \,\mu(z)\big)=S(x),$ where $
K(x,z)=\big(\sum_{k=0}^{n-1}P_{k}(x){{(x-z)^{n-k-1}}\over(n-k-1)!}\big)$,
$u(x)=D(x)+\int_{a}^{x} dz {(x-z)^{n-1}\over (n-1)!} \mu(z)$, $\,
D(x)=\sum_{k=0}^{n-1}{c_{k}{(x-a)^{k}}\over k!}$ and $ S(x)=
-\sum_{i=0}^{n-1}\sum^{i}_{k=0}
c_{i}{{P_{k}(x)(x-a)^{i-k}}\over(i-k)!}$. Therefore we state the
following theorem;
\begin{theorem}{The Solution of homogeneous linear differential equation
$\big({\rd^{\,n}_{x}}$ $+\sum^{n-1}_{k=0} P_{k}(x)\,{\rd^{k}_{x}}
\big)\,u(x)=0,$ where $P_{i}(x)$ \,(\,$i=0,1,$ $...,n-1$) are in $\mc[a,b]$,
with initial condition; $\rd^{\,i}_{x}$ $u(a)= c_{i}$ for $i=0,1,$ $..,n-1$, is given
by;
\be u(x)=D(x)+\int^{x}_{a}dz\,T(x,z) S(z),\label{el}\ee
where $\displaystyle\, D(x)=\sum_{k=0}^{n-1}{c_{k}{(x-a)^{k}}\over
k!}$,\,$\,\,\,\displaystyle S(x)= -\sum_{i=0}^{n-1}\sum^{i}_{k=0}
c_{i}{{P_{k}(x)(x-a)^{i-k}}\over(i-k)!}$ and $T(x,y)$ is given by
(\ref{hasan}).}
\end{theorem}
By using the above theorem for solutions $u_{r}(x)$ ($r=0,...,n-1$)
with initial conditions $\rd_{x}^{\,i}u_{r}(a)=\delta_{r,i}$ for
$i=0,...,n-1$, one can find $n$ linearly independent solutions;
\begin{itemize}{\item{\bf Fundamental solutions of homogenous linear ordinary differential equation.\vspace{0.2cm}}} {\em
$\displaystyle u_{r}(x)={{(x-a)^{r}}\over r!}+\int^{x}_{a}dz\,T(x,z)
(-\sum_{k=0}^{r} {{P_{k}(z)(z-a)^{r-k}}\over(r-k)!})$ for
$r=0,...,n-1$ are $n$ linearly independent solutions of
$\vspace{0.2cm}\displaystyle \big({\rd^{\,n}_{x}}$
$+\sum^{n-1}_{k=0} P_{k}(x)\,{\rd^{k}_{x}}
\big)\,u(x)=0$. The Wornskian is given by Abel's identity as;
$\displaystyle W(u_0,u_1,...u_{n-1})=e^{-\int^{x}_{a} dz
P_{n-1}(z)}$.}
\end{itemize}
It is easy to check that for boundary condition; $\rd^{\,i}_{x}u(b)=
c_{i}$ for $i=0,1,..,n-1$ the answer for homogenous differential
equation in $\mc[a,b]$ is given by (\ref{el}) in which $a$ is
replaced by $b$. One can derive solutions of homogenous differential
equation, with specific boundary conditions on either $a$ or $b$, by
using (\ref{el}). By substituting solutions of homogenous
differential equation with suitable boundary conditions (derived by
(\ref{el})) in to the expressions for Green's functions given by
method of variation of parameters, one can derive the Green's
function according to coefficient functions of differential
operator. For example the Green's function for Sturm-Lioville
problem; \be (\rd_{x}^{2}-P(x)) y(x)= g(x), \hspace{1cm}
\text{B.C.}\hspace{0.6cm}y(a)=y(b)=0, \ee by method of variation of
parameters is given by; $G(x,y)=(W(y))^{-1}$ $\big(u_{1}(x)u_2(y)$
$\theta(y-x)$ $+u_{2}(x)u_{1}(y)\,$ $\theta(x-y)\big)$, where
$u_{1}(x)$ and $u_{2}(x)$ are answers of corresponding homogenous
differential equation, $W(y)$ is the Wronskian and we have
$u_{1}(a)=u_{2}(b)=0$. If we take B.C. $u_{1}(a)=0$ and $\rd_x
u_{1}(a)=1$ then according to (\ref{el}); we have
$u_{1}(x)=(x-a)+\int^{x}_{a}dz T_{s}(x,z)P(z)(z-a)=T_{s}(x,a)$ (The
function $T_{s}(x,y)$ is given by (\ref{shahla})). On the other hand
by taking B.C. $u_{2}(b)=0$ and $\rd_x u_{2}(b)=1$ we have
$u_{2}(x)=T_{s}(x,b)$. The Wronskian of $u_{1}(x)$ and $u_{2}(x)$ is
constant and can be calculated easily at point $x=a$ as;
$W(y)=-T_{s}(a,b)$ therefore; \be G(x,y)=(-T_{s}(a,b))^{-1}{
\Big(T_{s}(x,a)T_{s}(y,b)\theta(y-x)+(x\leftrightarrow y) \Big)
}.\ee
\section{Conclusion }\label{disc}
Causal Green's function for general linear differential operator in
one variable was given by [theorem \ref{FinalGreen's}] as series of
integrals. Multiplicative property of causal Green's functions is
shown by [theorem \ref{adel}]. For differential operators which are
equal to multiplications of first order linear differential
operators, explicit formula (\ref{iraj1}), was given for causal
Green's functions. An infinite non-abelian group of operators on a
subspace of ${\mc}^{\infty}[a,b]$ is introduced. Analytic formula
for fundamental solutions of homogenous linear differential equation
in one variable was given via equation (\ref{kasai}) and for
specific boundary condition via equation (\ref{el}) as series of
integrals. By using equation (\ref{el}) and the method of variation
of parameters a way to derive Green's functions with arbitrary
boundary conditions in one variable according to coefficient
functions of differential operators was given.
\section*{Acknowledgements} I acknowledge {\em Farhang Loran} for
his helps and useful discussions. I dedicate this paper to memory of
my father {\em Iraj Kassaian}.
| 72,507
|
\begin{document}
\maketitle
\begin{abstract}
We consider the inverse problem of determining an unknown vectorial source current distribution associated with the homogeneous Maxwell system. We propose a novel non-iterative reconstruction method for solving the aforementioned inverse problem from far-field measurements. The method is based on recovering the Fourier coefficients of the unknown source. A key ingredient of the method is to establish the relationship between the Fourier coefficients and the multi-frequency far-field data. Uniqueness and stability results are established for the proposed reconstruction method. Numerical experiments are presented to illustrate the effectiveness and efficiency of the method.
\medskip
\medskip
\noindent{\bf Keywords:}~~inverse source problem, Maxwell's system, Fourier expansion, multi-frequency, far-field
\noindent{\bf 2010 Mathematics Subject Classification:}~~35R30, 35P25, 78A46
\end{abstract}
\section{Introduction}\label{sect:1}
The inverse source problem is concerned with the reconstruction of an unknown/inaccess-ible active source from the measurement of the radiating field induced by the source. The inverse source problem arises in many important applications including acoustic tomography \cite{Anastasio, ClaKli, Klibanov2013, Liu2015}, medical imaging\cite{Ammari2002, Arridge1999,Fokas2004} and detection of pollution for the environment\cite{Badia2002}. In this paper, we are mainly concerned with the inverse source problem for wave propagation in the time-harmonic regime. In the last decades, many theoretical and numerical studies have been done in dealing with the inverse source problem for wave scattering.
The uniqueness and stability results can be found in \cite{Bao2017, Isa2 }. Several numerical reconstruction methods have also been proposed and developed in the literature. For a fixed frequency, we refer the reader to \cite{Ammari2002,Badia2013,He1998}.
However, with only one single frequency, the inverse source problem lacks of stability and it leads to severe ill-posedness.
In order to improve the resolution, multi-frequency measurements should be employed in the reconstruction \cite{Bao2017,Eller2009, Valdivia2012}.
The goal of this paper is to develop a novel numerical scheme for reconstructing an electric current source associated with the time-harmonic Maxwell system. Due to the existence of non-radiating sources \cite{Bleistein1977, Marengo2004}, the vectorial current sources cannot be uniquely determined from surface measurements. Albanese and Monk \cite{1Mon} showed that surface currents and dipole sources have a unique solution, but it is not valid for volume currents. Valdivia\cite{Valdivia2012} showed that
the volume currents could be uniquely identified if the current density is divergence free. Following the spirit of our earlier work \cite{WangGuo17,Wang17} by three of the authors of using Fourier method for inverse acoustic source problem, we develop a Fourier method for the reconstruction of a volume current associated with the time-harmonic Maxwell system. The extension from the scalar Hemholtz equation to the vectorial Maxwell system involves much subtle and technical analysis. First, we establish the one-to-one correspondence between the Fourier coefficients and the far-field data, so that the Fourier coefficients can be directly calculated. Second, the proposed method is stable and robust to measurement noise. This is rigorously verified by establishing the corresponding stability estimates. Finally, compared to near-field Fourier method, our method is easy to implement with cheaper computational costs.
The rest of the paper is organized as follows. Section 2 describes the mathematical setup of the inverse source problem of our study. The theoretical uniqueness and stability results of proposed Fourier method are given in Section 3 and Section 4, respectively. Section 5 presents several numerical examples to illustrate the effectiveness and efficiency of the proposed method.
\section{Problem formulation}
Consider the following time-harmonic Maxwell system in $\mathbb{R}^3$,
\begin{equation}\label{eq: Maxwell}
\left\{
\begin{aligned}
&\nabla\times \bm E-\mathrm{i}\omega \mu_0 \bm H=0, \\
&\nabla\times\bm H+\mathrm{i}\omega\varepsilon_0\bm E=\bm J,
\end{aligned}
\right.
\end{equation}
with the Silver-M\"uller radiation condition
\begin{equation*}
\lim_{|\bm x| \rightarrow +\infty} |\bm x|\left(\sqrt{\mu_0}\bm H\times \hat{\bm x}-\sqrt{\varepsilon_0}\bm E\right)=0,
\end{equation*}
where $\hat{\bm x}=\bm x/ |\bm x|$ and $\bm x=(x_1, x_2, x_3)^{\top}\in \mathbb{R}^3$.
Throughout the rest of the paper, we use non-bold and bold fonts to signify scalar and vectorial quantities, respectively. In \eqref{eq: Maxwell}, $\bm E$ denotes the electric filed, $ \bm H $ denotes the magnetic filed, $ \bm J $ is an electric current density, $\omega$ denotes the frequency, $\varepsilon_0$ denotes the electric permittivity and $\mu_0$ denotes the magnetic permeability of the isotropic homogeneous background medium.
By eliminating $\bm H$ or $\bm E$ in \eqref{eq: Maxwell}, we obtain
\begin{equation*}
\nabla\times\nabla\times \bm E-k^2 \bm E=\mathrm{i}\omega \mu_0 \bm J,
\end{equation*}
and
\begin{equation*}
\nabla\times\nabla\times \bm H -k^2 \bm H= \nabla\times\bm J,
\end{equation*}
where $ k:=\omega\sqrt{\mu_0\varepsilon_0} $.
With the help of the vectorial Green function \cite{Tai94}, the radiated field can be written as
\begin{equation}\label{eq:a1}
\displaystyle \bm E(\bm x)= \mathrm{i}\omega \mu_0 \left(\bm I+\frac{1}{k^2} \nabla \nabla\cdot\right)\int_{\mathbb{R}^3} \Phi(\bm x,\bm y)\, \bm J(\bm y) \,\mathrm{d}\bm y,
\end{equation}
and
\begin{equation}\label{eq:a2}
\displaystyle \bm H(\bm x)= \nabla\times \int_{\mathbb{R}^3} \Phi(\bm x,\bm y)\, \bm J(\bm y) \,\mathrm{d}\bm y,
\end{equation}
respectively, where $\bm I$ is the $3\times 3$ identity matrix and
\begin{equation*}
\displaystyle\Phi(\bm x,\bm y)=\frac{\mathrm{e}^{\mathrm{i}k|\bm x-\bm y|}}{4\pi|\bm x-\bm y|}, \quad \bm x\neq\bm y,
\end{equation*}
is the fundamental solution to the Helmholtz equation.
The radiating fields $\bm E, \bm H $ to the Maxwell system have the following asymptotic expansion \cite{Colton2013}
\begin{equation*}
\begin{aligned}
\bm E(\bm x)=\frac{e^{\mathrm{i}k|\bm x|}}{|\bm x|} \left\{ \bm E_{\infty}(\hat{\bm x})+ \mathcal{O}\left(\frac{1}{|\bm x|}\right) \right\},\quad |\bm x|\rightarrow +\infty,\\
\bm H(\bm x)=\frac{e^{\mathrm{i}k|\bm x|}}{|\bm x|} \left\{ \bm H_{\infty}(\hat{\bm x})+ \mathcal{O}\left(\frac{1}{|\bm x|}\right) \right\},\quad |\bm x|\rightarrow +\infty,
\end{aligned}
\end{equation*}
and by using the integral representations \eqref{eq:a1} and \eqref{eq:a2}, we have
\begin{align}
&\label{eq:E_far}\bm E_{\infty}(\hat{\bm x})= \frac{\mathrm{i}\omega \mu_0}{4\pi} \left(\bm I-\hat{\bm x} \hat{\bm x}^{\top} \right)\int_{\mathbb{R}^3} e^{-\mathrm{i}k\hat{\bm x}\cdot \bm y} \, \bm J(\bm y) \,\mathrm{d}\bm y,\\
&\label{eq:H_far} \bm H_{\infty}(\hat{\bm x})=\frac{\mathrm{i}k}{4\pi} \hat{\bm x}\times\int_{\mathbb{R}^3} e^{-\mathrm{i}k\hat{\bm x}\cdot \bm y} \, \bm J(\bm y) \,\mathrm{d}\bm y.
\end{align}
In what follows, we always assume that the electromagnetic source is a volume current that is supported in $D$.
As mentioned earlier, there exists non-radiating sources that produce no radiating field outside $D$. Hence, without any a prior knowledge, one can only recover the radiating part of the current density distribution. In order to formulate the uniqueness result, we assume that the current density distribution $\bm J$ only consists of radiating source, which is independent of the wavenumber $k$ and of the form
\begin{equation*}
\bm J\in \left(L^2(\mathbb{R}^3)\right)^3, \quad \mathrm{supp}\ \bm J\subset D,
\end{equation*}
where $D$ is a cube. Furthermore, the current density distribution $\bm J$ satisfies the transverse electric (TE) and transverse magnetic (TM) decomposition; that is, the source can be expressed in the form
\begin{equation}\label{eq:density}
\bm J=\bm p f+ \bm p\times\nabla g,
\end{equation}
where $f\in L^2(D)$ and $ g\in H^1(D)$. We also refer to \cite{Lindell88} for more details on the TE/TM decomposition.
Here, $\bm p$ is the polarization direction which is assumed to be known and yields the following admissible set
\begin{equation}\label{eq:p}
\mathbb{P}:=\left\{ \bm p \in \mathbb{S}^2 \mid \bm p\times \bm l \neq \bm 0, \quad \forall \ \bm l\in \mathbb{Z}^3\backslash\{\bm 0\} \right\}.
\end{equation}
From \eqref{eq:E_far} and \eqref{eq:H_far}, it is clear that
\begin{equation*}
\begin{aligned}
&\bm E_{\infty}(-\hat{\bm x})=-\overline{\bm E_{\infty}(\hat{\bm x})},
&\bm H_{\infty}(-\hat{\bm x})=\overline{\bm H_{\infty}(\hat{\bm x})},
\end{aligned}
\end{equation*}
where and also in what follows, the overbar stands for the complex conjugate in this paper. Therefore, for our inverse problem, the measurements of the far-field data could be from an upper hemisphere $\mathbb{S}_{+}^2$, say $x_3\geq0$. Figure \ref{fig:geometry} provides a schematic illustration of the geometric setting of the measurements. With the above discussion, the inverse source problem of the current study can be stated as follows,
\begin{ip}
{\rm Given a fixed polarization direction $\bm p\in \mathbb{R}^3$ and a finite number of wavenumbers $\{k\}$, we intend to recover the electromagnetic source $\bm J$ defined in \eqref{eq:density} from the electric far-field data $\{\bm E_\infty(\hat{\bm x}_k;k,\bm p)\}$ or the magnetic far-field data $\{\bm H_\infty(\hat{\bm x}_k;k,\bm p)\}$, where $\hat{\bm x}_k$ depends on the wavenumber $k$ and $\hat{\bm x}_k\in \mathbb{S}_{+}^2$.}
\end{ip}
\begin{figure}
\centering
{\includegraphics[width=0.6\textwidth]
{images/geometry}}
\caption{\label{fig:geometry} The schematic illustration of the inverse electromagnetic source problem by the far-field measurements with $x_3\geq0$.}
\end{figure}
\section{Uniqueness}
Prior to our discussion, we introduce some notations and relevant Sobolev spaces.
Without loss of generality, we let
\begin{equation*}
D=\left(-\frac{a}{2},\ \frac{a}{2}\right)^3, \quad a \in \mathbb{R}_+.
\end{equation*}
Introduce the Fourier basis functions that are defined by
\begin{equation}\label{eq:Fourier_basic}
\displaystyle \phi_{\bm l}(\bm x)=\exp\left(\mathrm{i}\frac{2\pi}{a} \bm l \cdot \bm x \right),\quad \bm l\in \mathbb{Z}^3, \ \bm x\in \mathbb{R}^3.
\end{equation}
By using the Fourier series expansion, the scalar functions $f\in L^2(D)$ and $g\in H^1(D)$ can be written as
\begin{equation*}
f=\sum_{\bm l \in \mathbb{Z}^3} \hat{f}_{\bm l}\, \phi_{\bm l}, \quad
g=\sum_{\bm l \in \mathbb{Z}^3\backslash \{\bm 0\}} \hat{g}_{\bm l}\, \phi_{\bm l},
\end{equation*}
where the Fourier coefficients are given by
\begin{align}
&\label{eq:f_hat0}\hat{f}_{\bm l}=\frac{1}{a^3} \int_{D}f(\bm x)\overline{\phi_{\bm l}(\bm x)}\,\mathrm{d}\bm x,\\
&\label{eq:g_hat0}\hat{g}_{\bm l}=\frac{1}{a^3} \int_{D}g(\bm x)\overline{\phi_{\bm l}(\bm x)}\,\mathrm{d}\bm x.
\end{align}
Therefore the Fourier expansion of the current density $\bm J$ is
\begin{equation}\label{eq:J_expansion}
\bm J=\bm p f+\bm p\times \nabla g=\bm p \sum_{\bm l \in \mathbb{Z}^3} \hat{f}_{\bm l}\, \phi_{\bm l} + \frac{2\pi \mathrm{i}}{a} \sum_{\bm l \in \mathbb{Z}^3\backslash \{\bm 0\}} (\bm p\times\bm l)\ \hat{g}_{\bm l}\, \phi_{\bm l}.
\end{equation}
The proposed reconstruction scheme in the current article is based on determining the Fourier coefficients $\hat{f}_{\bm l}$ and $\hat{g}_{\bm l}$ of the current density by using the corresponding electric or magnetic far-field data. For the subsequent use, we introduce the Sobolev spaces with $\sigma>0$
\begin{equation*}
(H_{\bm p}^\sigma(D))^3:=\left\{ \bm p f+\bm p\times \nabla g \mid f \in H^\sigma(D), \, g\in H^{\sigma+1}(D), \, \bm p\in \mathbb{S}^2
\right\},
\end{equation*}
equipped with the norm
\begin{equation*}
\|\bm G\|_{\bm p , \sigma}=\left(\sum_{{\bm l}\in \mathbb{Z}^3}\left( 1+|{\bm l}|^2 \right)^{\sigma}|\hat{f}_{\bm l}|^2 + \frac{4\pi^2}{a^2} \sum_{{\bm l}\in \mathbb{Z}^3\backslash \{\bm 0\}}\left( 1+|{\bm l}|^2 \right)^{\sigma}|\bm p\times \bm l|^2 |\hat{g}_{\bm l}|^2
\right)^{1/2}.
\end{equation*}
In addition, the wavenumber cannot be zero in \eqref{eq:E_far} and \eqref{eq:H_far}. Following \cite{WangGuo17}, we introduce the following definition of wavenumbers.
\begin{defn}[Admissible wavenumbers]\label{def:31}
Let $\lambda$ be a sufficiently small positive constant and the admissible wavenumbers can be defined by
\begin{equation}\label{eq:wavenumber}
k_{\bm l}:=\left\{
\begin{aligned}
& \frac{2\pi}{a}|\bm l|, \quad \bm l \in \mathbb{Z}^3\backslash\{\bm 0\}, \\
& \frac{2\pi}{a}\lambda, \quad \bm l =\bm 0.
\end{aligned}
\right.
\end{equation}
Correspondingly, the observation direction is given by
\begin{equation}\label{eq:x_hat}
\hat{\bm x}_{\bm l}:=\left\{
\begin{aligned}
& \hat{\bm l}, \quad \quad \bm l \in \mathbb{Z}^3\backslash\{\bm 0\}, \\
& (1,0,0), \quad \bm l =\bm 0.
\end{aligned}
\right.
\end{equation}
\end{defn}
By virtue of Definition~\ref{def:31}, the Fourier basis functions defined in \eqref{eq:Fourier_basic} could be written as
\begin{equation*}
\displaystyle \phi_{\bm l}(\bm x)= \exp\left(\mathrm{i}k_{\bm l}\, \hat{\bm l} \cdot \bm x \right),\quad \bm l\in \mathbb{Z}^3, \ \bm x\in \mathbb{R}^3.
\end{equation*}
Next we state the uniqueness result.
\begin{theorem}
Let $ k_{\bm l}$ and $ \hat{x}_{\bm l}$ be defined in \eqref{eq:wavenumber} and \eqref{eq:x_hat}, then the Fourier coefficients $\{\hat{f}_{\bm l}\}$ and $\{\hat{g}_{\bm l}\}$ in \eqref{eq:f_hat0} and \eqref{eq:g_hat0} could be uniquely determined by $ \{\bm E_{\infty}(\hat{\bm x}_{\bm l};k_{\bm l},\bm p)\}$ or $ \{\bm H_{\infty}(\hat{\bm x}_{\bm l};k_{\bm l},\bm p)\}$, where $ \bm l \in \mathbb{Z}^3 $.
\end{theorem}
\begin{proof}[\bf Proof.]
Let $\bm J$ be the electromagnetic source that produces the electric far-field data $\{\bm E_{\infty}(\hat{\bm x}_{\bm l};k_{\bm l})\}_{\bm l \in \mathbb{Z}^3}$ and the magnetic far-field data $\{\bm H_{\infty}(\hat{\bm x}_{\bm l};k_{\bm l})\}_{\bm l \in \mathbb{Z}^3}$ on $\mathbb{S}^2$.
First, we consider the recovery of $\bm J$ by the magnetic far-field data.
For every $\bm l\in \mathbb{Z}^3\backslash\{\bm 0\}$, using \eqref{eq:H_far} and \eqref{eq:J_expansion}, we have
\begin{equation}\label{eq:H_far1}
\begin{aligned}
&\bm H_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})\\
&=\frac{\mathrm{i}k_{\bm l}}{4\pi} \hat{\bm x}_{\bm l}
\times \int_D \left(\bm p \hat{f}_{\bm 0} \mathrm{e}^{-\mathrm{i}k_{\bm l}\hat{\bm x}_{\bm l}\cdot \bm y}
+ \sum_{\tilde{\bm l} \in \mathbb{Z}^3\backslash \{\bm 0\}}\left(\bm p \hat{f}_{\tilde{\bm l}}+ \frac{2\pi \mathrm{i}}{a} (\bm p\times\tilde{\bm l})\hat{g}_{\tilde{\bm l} }\right)\mathrm{e}^{\mathrm{i} (k_{\tilde{\bm l}}\hat{\tilde{\bm l}}-k_{\bm l}\hat{\bm x}_{\bm l})\cdot \bm y}\right) \mathrm{d}\bm y\\
&=\frac{\mathrm{i}k_{\bm l} a^3}{4\pi} \left( \hat{\bm x}_{\bm l}
\times \bm p \hat{f}_{\bm l} + \frac{2\pi \mathrm{i}}{a} \hat{\bm x}_{\bm l}
\times (\bm p\times \bm l) \hat{g}_{\bm l} \right).
\end{aligned}
\end{equation}
From \eqref{eq:p} and \eqref{eq:x_hat}, we see that $\{ \hat{\bm x}_{\bm l}, \bm p\times \hat{\bm x}_{\bm l}, \hat{\bm x}_{\bm l}\times (\bm p\times \hat{\bm x}_{\bm l}) \}$
forms an orthogonal basis of $\mathbb{R}^{3}$. Multiplying $ \hat{\bm x}_{\bm l} \times \bm p$ on the both sides of \eqref{eq:H_far1}, and using the orthogonality, we obtain
\begin{equation}\label{eq:Hf_hat}
\hat{f}_{\bm l}=\frac{4\pi \hat{\bm x}_{\bm l}\times \bm p \cdot\bm H_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})}{\mathrm{i}k_{\bm l} a^3
|\hat{\bm x}_{\bm l} \times \bm p|^2}.
\end{equation}
Similarly, multiplying $ \hat{\bm x}_{\bm l}\times (\bm p\times \bm l)$ on the both sides of \eqref{eq:H_far1}, we have
\begin{equation}\label{eq:Hg_hat}
\hat{g}_{\bm l}=-\frac{2 \hat{\bm x}_{\bm l}\times (\bm p\times \bm l) \cdot\bm H_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})}{k_{\bm l} a^2
|\hat{\bm x}_{\bm l} \times (\bm p\times\bm l)|^2}.
\end{equation}
For $\bm l=\bm 0$, we have
\begin{equation*}
\begin{aligned}
&\bm H_{\infty}(\hat{\bm x}_{\bm 0}; k_{\bm 0})\\
&=\frac{\mathrm{i}k_{\bm 0}}{4\pi} \hat{\bm x}_{\bm 0}
\times \int_D \left( \bm p \hat{f}_{\bm 0} \mathrm{e}^{-\mathrm{i}k_{\bm 0}\hat{\bm x}_{\bm 0}\cdot \bm y}+ \sum_{\bm l \in \mathbb{Z}^3\backslash \{\bm 0\}}\left(\bm p \hat{f}_{\bm l}+ \frac{2\pi \mathrm{i}}{a} (\bm p\times\bm l)\hat{g}_{\bm l}\right)\mathrm{e}^{\mathrm{i}(k_{\bm l}\hat{\bm l}-k_{\bm 0}\hat{\bm x}_{\bm 0})\cdot \bm y} \right)\mathrm{d}\bm y.
\end{aligned}
\end{equation*}
Multiplying $ \hat{\bm x}_{\bm 0} \times \bm p$ on the both side of the last equation, and also using the orthogonal property, we obtain
\begin{equation*}
\begin{aligned}
&\hat{\bm x}_{\bm 0} \times \bm p \cdot \bm H_{\infty}(\hat{\bm x}_{\bm 0}; k_{\bm 0})\\
&=\frac{\mathrm{i}k_{\bm 0}}{4\pi} |\hat{\bm x}_{\bm 0}
\times \bm p|^2 \left( a^3\hat{f}_{\bm 0} \frac{\sin \lambda \pi}{\lambda \pi}+ \sum_{\bm l \in \mathbb{Z}^3\backslash \{\bm 0\}}\hat{f}_{\bm l} \int_{D}
\mathrm{e}^{\mathrm{i}(k_{\bm l}\hat{\bm l}-k_{\bm 0}\hat{\bm x}_{\bm 0})\cdot \bm y} \, \mathrm{d}\bm y \right).
\end{aligned}
\end{equation*}
Thus,
\begin{equation}\label{eq:Hf0_hat}
\hat{f}_{\bm 0}= \frac{\lambda \pi}{a^3 \sin \lambda \pi} \left( \frac{4\pi \hat{\bm x}_{\bm 0}\times \bm p \cdot\bm H_{\infty}(\hat{\bm x}_{\bm 0}; k_{\bm 0})}{\mathrm{i}k_{\bm 0}
|\hat{\bm x}_{\bm 0} \times \bm p|^2}- \sum_{\bm l \in \mathbb{Z}^3\backslash \{\bm 0\}}\hat{f}_{\bm l} \int_{D}
\mathrm{e}^{\mathrm{i}(k_{\bm l}\hat{\bm l}-k_{\bm 0}\hat{\bm x}_{\bm 0})\cdot \bm y} \, \mathrm{d}\bm y \right).
\end{equation}
Next, we consider the recovery of $\bm J$ by the electric far-field data.
For every $\bm l\in \mathbb{Z}^3\backslash\{\bm 0\}$, using \eqref{eq:E_far} and \eqref{eq:J_expansion}, we have
\begin{equation}\label{eq:E_far1}
\begin{aligned}
\bm E_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})
=\frac{\mathrm{i}\omega \mu_0 a^3}{4\pi}\left( \bm I -\hat{\bm x}_{\bm l} \hat{\bm x}_{\bm l} ^{\top} \right) \left( \bm p \hat{f}_{\bm l} + \frac{2\pi \mathrm{i}}{a} (\bm p\times \bm l) \hat{g}_{\bm l} \right).
\end{aligned}
\end{equation}
Through straightforward calculations, one can verify that
\begin{equation*}
\hat{\bm x}_{\bm l} \times(\hat{\bm x}_{\bm l} \times \bm A)=-\left( \bm I -\hat{\bm x}_{\bm l} \hat{\bm x}_{\bm l} ^{\top} \right)\bm A, \quad \forall\bm A\in \mathbb{R}^3.
\end{equation*}
Combining the last two equations, one can show that
\begin{equation}\label{eq:E_far2}
\begin{aligned}
&\bm E_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})
=\frac{\mathrm{i}\omega \mu_0 a^3}{4\pi} \left( -\hat{\bm x}_{\bm l} \times(\hat{\bm x}_{\bm l} \times\bm p) \hat{f}_{\bm l} + \frac{2\pi \mathrm{i}}{a} ( -\hat{\bm x}_{\bm l} \times(\hat{\bm x}_{\bm l} \times(\bm p \times \bm l)) \hat{g}_{\bm l} \right).
\end{aligned}
\end{equation}
Multiplying $ \bm p $ on the both sides of \eqref{eq:E_far2}, and using the orthogonality, we obtain
\begin{equation*}
\begin{aligned}
&\bm p\cdot \bm E_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})\\
&=\frac{\mathrm{i}\omega \mu_0 a^3}{4\pi}\left( -\bm p\cdot\hat{\bm x}_{\bm l} \times(\hat{\bm x}_{\bm l} \times\bm p) \hat{f}_{\bm l} + \frac{2\pi \mathrm{i}}{a} ( -\bm p\cdot\hat{\bm x}_{\bm l} \times(\hat{\bm x}_{\bm l} \times(\bm p \times \bm l)) \hat{g}_{\bm l} \right)\\
&=\frac{\mathrm{i}\omega \mu_0 a^3}{4\pi}\left( (\hat{\bm x}_{\bm l}\times \bm p) \cdot(\hat{\bm x}_{\bm l} \times\bm p) \hat{f}_{\bm l} + \frac{2\pi \mathrm{i}}{a} (\hat{\bm x}_{\bm l}\times \bm p) \cdot(\hat{\bm x}_{\bm l} \times(\bm p \times \bm l)) \hat{g}_{\bm l} \right)\\
&=\frac{\mathrm{i}\omega \mu_0 a^3}{4\pi}\left| \hat{\bm x}_{\bm l}\times \bm p \right|^2 \hat{f}_{\bm l}.
\end{aligned}
\end{equation*}
Thus,
\begin{equation}\label{eq:Ef_hat}
\hat{f}_{\bm l}=\frac{4\pi \bm p \cdot\bm E_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})}{\mathrm{i}\omega \mu_0 a^3
|\hat{\bm x}_{\bm l} \times \bm p|^2}.
\end{equation}
Similarly, multiplying $ \bm p\times \bm l$ on the both sides of \eqref{eq:E_far2}, we obtain
\begin{equation}\label{eq:Eg_hat}
\hat{g}_{\bm l}=-\frac{2 (\bm p\times \bm l) \cdot\bm E_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})}{\omega\mu_0 a^2
|\hat{\bm x}_{\bm l} \times (\bm p\times\bm l)|^2}.
\end{equation}
For $\bm l=\bm 0$, we have
\begin{equation*}
\begin{aligned}
&\bm E_{\infty}(\hat{\bm x}_{\bm 0}; k_{\bm 0})\\
&=\frac{\mathrm{i}\omega \mu_0}{4\pi} \left( \bm I -\hat{\bm x}_{\bm 0} \hat{\bm x}_{\bm 0} ^{\top} \right) \int_D \left( \bm p \hat{f}_{\bm 0} \mathrm{e}^{-\mathrm{i}k_{\bm 0}\hat{\bm x}_{\bm 0}\cdot \bm y}+ \sum_{\bm l \in \mathbb{Z}^3\backslash \{\bm 0\}}\left(\bm p \hat{f}_{\bm l}+ \frac{2\pi \mathrm{i}}{a} (\bm p\times\bm l)\hat{g}_{\bm l}\right)\mathrm{e}^{\mathrm{i}(k_{\bm l}\hat{\bm l}-k_{\bm 0}\hat{\bm x}_{\bm 0})\cdot \bm y} \right)\mathrm{d}\bm y.
\end{aligned}
\end{equation*}
Multiplying $ \bm p$ on the both sides of the last equation, and also using the orthogonality, we obtain
\begin{equation*}
\begin{aligned}
& \bm p \cdot \bm E_{\infty}(\hat{\bm x}_{\bm 0}; k_{\bm 0})
=\frac{\mathrm{i}\omega \mu_0}{4\pi} |\hat{\bm x}_{\bm 0}
\times \bm p|^2 \left( a^3\hat{f}_{\bm 0} \frac{\sin \lambda \pi}{\lambda \pi}+ \sum_{\bm l \in \mathbb{Z}^3\backslash \{\bm 0\}}\hat{f}_{\bm l} \int_{D}
\mathrm{e}^{\mathrm{i}(k_{\bm l}\hat{\bm l}-k_{\bm 0}\hat{\bm x}_{\bm 0})\cdot \bm y} \, \mathrm{d}\bm y \right).
\end{aligned}
\end{equation*}
Thus,
\begin{equation*}
\hat{f}_{\bm 0}= \frac{\lambda \pi}{a^3 \sin \lambda \pi} \left( \frac{4\pi \bm p \cdot\bm E_{\infty}(\hat{\bm x}_{\bm 0}; k_{\bm 0})}{\mathrm{i} \omega \mu_0
|\hat{\bm x}_{\bm 0} \times \bm p|^2}- \sum_{\bm l \in \mathbb{Z}^3\backslash \{\bm 0\}}\hat{f}_{\bm l} \int_{D}
\mathrm{e}^{\mathrm{i}(k_{\bm l}\hat{\bm l}-k_{\bm 0}\hat{\bm x}_{\bm 0})\cdot \bm y} \, \mathrm{d}\bm y \right).
\end{equation*}
The proof is complete.
\end{proof}
In practical computations, we have to truncate the infinite series by a finite order $N \in \mathbb{N}$ to approximate $\bm J$ by
\begin{equation}\label{eq:J_N}
\bm J_N=\bm p \hat{f}_{\bm 0}+ \sum_{1\leq|\bm l|_{\infty}\leq N} \left(\bm p \hat{f}_{\bm l} + \frac{2\pi \mathrm{i}}{a}(\bm p\times\bm l) \,\hat{g}_{\bm l}\right) \phi_{\bm l},
\end{equation}
where $\hat{f}_{\bm 0}$ could be represented by magnetic far-field
\begin{equation}\label{eq:Hf0_hat}
\hat{f}_{\bm 0}\approx \frac{\lambda \pi}{a^3\sin \lambda \pi} \left( \frac{4\pi \hat{\bm x}_{\bm 0}\times \bm p \cdot\bm H_{\infty}(\hat{\bm x}_{\bm 0}; k_{\bm 0})}{\mathrm{i}k_{\bm 0}
|\hat{\bm x}_{\bm 0} \times \bm p|^2}- \sum_{1\leq|\bm l|_{\infty}\leq N} \hat{f}_{\bm l} \int_{D}
\mathrm{e}^{\mathrm{i}(k_{\bm l}\hat{\bm l}-k_{\bm 0}\hat{\bm x}_{\bm 0})\cdot \bm y} \, \mathrm{d}\bm y \right),
\end{equation}
or electric far-field
\begin{equation}\label{eq:Ef0_hat}
\hat{f}_{\bm 0}\approx \frac{\lambda \pi}{a^3 \sin \lambda \pi} \left( \frac{4\pi \bm p \cdot\bm E_{\infty}(\hat{\bm x}_{\bm 0}; k_{\bm 0})}{\mathrm{i} \omega \mu_0
|\hat{\bm x}_{\bm 0} \times \bm p|^2}- \sum_{1\leq|\bm l|_{\infty}\leq N}\hat{f}_{\bm l} \int_{D}
\mathrm{e}^{\mathrm{i}(k_{\bm l}\hat{\bm l}-k_{\bm 0}\hat{\bm x}_{\bm 0})\cdot \bm y} \, \mathrm{d}\bm y \right).
\end{equation}
\section{Stability}
In this section, we derive the stability estimates of recovering the Fourier coefficients of the electric current source by using the far-field data. We only consider the stability of using the magnetic far-field data, and the case with the electric far-field data can be treated in a similar manner. In what follows, we introduce $H_{\infty}^{\delta}(\hat{\bm x}_{\bm l}; k_{\bm l})$ such that
\begin{equation*}
|\bm H_{\infty}^{\delta}(\hat{\bm x}_{\bm l}; k_{\bm l})-\bm H_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})|\leq \delta |\bm H_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})|,
\end{equation*}
where $\delta>0$. We first present two auxiliary results.
\begin{theorem}
For $ \bm{l} \in \mathbb{Z}^{3},\, |\bm{l}|_{\infty} \leq N$, we have
\begin{align}
\label{eq:f_cefficient} & |\hat{f}^{\delta}_{\bm{l}}- \hat{f}_{\bm{l}}| \leq C_1 \delta, \quad \quad 1\leq |\bm{l}|_{\infty} \leq N, \\
\label{eq:g_cefficient} & |\hat{g}^{\delta}_{\bm{l}}- \hat{g}_{\bm{l}}| \leq C_2 \delta, \quad \quad 1\leq |\bm{l}|_{\infty} \leq N, \\
\label{eq:f_cefficient0} & |\hat{f}^{\delta}_{\bm 0}- \hat{f}_{\bm 0}| \leq C_3\delta + C_4 \lambda N \delta+ C_5\frac{\lambda}{\sqrt{N}},
\end{align}
where constants $C_1, C_2, C_3, C_4$ and $C_5$ depend on $ f, g , a $ and $\lambda$.
\end {theorem}
\begin{proof}[\bf Proof.]
For $ \bm {l}\in \mathbb{Z}^3\backslash\{\bm 0\} $, from Schwarz inequality and \eqref{eq:Hf_hat}, we have
\begin{align*}
|\hat{f}^{ \delta}_{\bm {l}}- \hat{f}_{\bm {l}} |
&= \left|\frac{4\pi \hat{\bm x }_{\bm l}\times \bm p }{\mathrm{i}k_{\bm l} a^3
|\hat{\bm x}_{\bm l}\times \bm p |^2}\cdot \left(\bm H_{\infty}^{\delta}(\hat{\bm x}_{\bm l}; k_{\bm l})-\bm H_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})\right)\right|\\
&\leq \frac{4\pi }{\mathrm{i}k_{\bm l} a^3
|\hat{\bm x}_{\bm l}\times \bm p |}\delta\left|\bm H_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})\right|\\
&\leq \frac{\delta}{ a^3
|\hat{\bm x}_{\bm l}\times \bm p |}\left| \hat{\bm x}\times\int_{D} e^{-\mathrm{i}k\hat{\bm x}_{\bm l}\cdot \bm y} \, \bm J(y) \,\mathrm{d}\bm y \right|\\
&\leq \frac{\delta}{ a^3
|\hat{\bm x}_{\bm l}\times \bm p |} |\hat{\bm x}\times \bm p| \left|\int_{D} e^{-\mathrm{i}k\hat{\bm x}_{\bm l}\cdot \bm y} \ ( f(\bm y)+|\nabla g(\bm y)|) \,\mathrm{d}\bm y \right|\\
&\leq \frac{\delta}{ a^3} \left(\int_{D} \left|e^{-\mathrm{i}k\hat{\bm x}_{\bm l}\cdot \bm y}\right|^2 \,\mathrm{d}\bm y\right)^{1/2} \left( \|f\|_{L^2(D)}+\|\nabla g\|_{L^2(D)}\right) \\
&\leq C_1 \delta
\end{align*}
where $ C_1 =(\|f\|_{L^2(D)}+\| g\|_{H^1(D)} )/{ a^{3/2}}$ and
it leads to estimate \eqref{eq:f_cefficient}.
Correspondingly, from \eqref{eq:Hg_hat}, we have
\begin{align*}
|\hat{g}^{ \delta}_{\bm {l}}- \hat{g}_{\bm {l}} |
&=\left|-\frac{2 \hat{\bm x}_{\bm l}\times (\bm p\times \bm l) }{k_{\bm l} a^2 |\hat{\bm x}_{\bm l} \times (\bm p\times\bm l)|^2} \cdot \left(\bm H_{\infty}^{\delta}(\hat{\bm x}_{\bm l}; k_{\bm l})-\bm H_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})\right)\right| \\
&\leq \frac{2 }{k_{\bm l} a^2
|\hat{\bm x}_{\bm l}\times (\bm p\times\bm l) |}\delta \left|\bm H_{\infty}(\hat{\bm x}_{\bm l}; k_{\bm l})\right|\\
&\leq \frac{\delta}{ 2\pi |\bm l|a^2
|\hat{\bm x}_{\bm l}\times \bm p |}\left| \hat{\bm x}\times\int_{D} e^{-\mathrm{i}k\hat{\bm x}_{\bm l}\cdot \bm y} \, \bm J(y) \,\mathrm{d}\bm y \right|\\
&\leq \frac{ \|f\|_{L^2(D)}+\| g\|_{H^1(D)} }{2\pi |\bm l| a^{1/2}}\ \delta\\
&\leq C_2 \delta,
\end{align*}
where $C_2= (\|f\|_{L^2(D)}+\| g\|_{H^1(D)})/(2\pi a^{1/2})$ and it verifies \eqref{eq:g_cefficient}.
For $ \bm {l}= \{\bm 0\} $, from Schwarz inequality and \eqref{eq:Hf0_hat}, we have
\begin{align*}
|\hat{f}_{\bm 0}^{\delta}-\hat{f}_{\bm 0}|
\leq & \frac{\lambda \pi}{a^3 \sin \lambda \pi} \left|\frac{4\pi \hat{\bm x }_{\bm 0}\times \bm p }{\mathrm{i}k_{\bm 0}
|\hat{\bm x}_{\bm 0}\times \bm p |^2}\cdot \left(\bm H_{\infty}^{\delta}(\hat{\bm x}_{\bm 0}; k_{\bm 0})-\bm H_{\infty}(\hat{\bm x}_{\bm 0}; k_{\bm 0})\right)\right|\\
& +\underbrace{\frac{\lambda \pi}{ a^3 \sin \lambda \pi}\sum_{1\leq|\bm {l}|_{\infty}\leq N}\left|\left(\hat{f}^{\delta}_{\bm {l}}- \hat{f}_{\bm {l}}\right) \int_{D}
\mathrm{e}^{\mathrm{i}(k_{\bm l}\hat{\bm l}-k_{\bm 0}\hat{\bm x}_{\bm 0})\cdot \bm y} \, \mathrm{d}\bm y \right|}_{I_1}\\
&+\underbrace{\frac{\lambda \pi}{a^3 \sin \lambda \pi} \sum_{|\bm {l}|_{\infty}\geq N}\left|\hat{f}_{\bm {l}} \int_{D}
\mathrm{e}^{\mathrm{i}(k_{\bm l}\hat{\bm l}-k_{\bm 0}\hat{\bm x}_{\bm 0})\cdot \bm y} \, \mathrm{d}\bm y \right|}_{I_2}\\
\triangleq & \ C_3 \delta+ I_{1}+I_{2}.
\end{align*}
where $ \displaystyle C_3={ \lambda \pi(\|f\|_{L^2(D)}+\| g\|_{H^1(D)})}/{(a^{9/2} \sin \lambda \pi)}$.
Define $\bm l=(l_1, l_2, l_3)\in\mathbb{ Z}^3$, from \eqref{eq:wavenumber} and \eqref{eq:x_hat}, we find that
\begin{equation*}
\int_{D} \mathrm{e}^{\mathrm{i}(k_{\bm l}\hat{\bm l}-k_{\bm 0}\hat{\bm x}_{\bm 0})\cdot \bm y} \, \mathrm{d}\bm y =\left\{
\begin{aligned}
& \frac{a^3 \sin \, (l_1-\lambda)\pi}{(l_1-\lambda)\pi}, &|\bm l|=|l_1|, \\
& 0, &|\bm l|\neq |l_1|,
\end{aligned}
\right.
\end{equation*}
which together with \eqref{eq:f_cefficient} gives
\begin{align*}
I_1 \leq &\frac{\lambda \pi}{a^3 \sin \lambda \pi}\sum_{1\leq|\bm {l}|_{\infty}\leq N}\left|\hat{f}^{\delta}_{\bm {l}}- \hat{f}_{\bm {l}}\right| \left|\frac{a^3\sin (l_1-\lambda)\pi}{(l_1-\lambda)\pi} \right| \\
\leq & \frac{\lambda \pi}{\sin \lambda \pi} 2 \sum_{j=1}^{N}\left( C_1 \delta \frac{ \sin \lambda\pi}{(j-\lambda)\pi}\right)\\
\leq & C_4 \lambda N \delta,
\end{align*}
where $C_4=2 C_1$.
On the other hand, one can deduce that
\begin{align*}
I_2 \leq &\frac{\lambda \pi}{a^3 \sin \lambda \pi}\sum_{|\bm {l}|_{\infty} > N}| \hat{f}_{\bm {l}}| \left|\frac{a^3 \sin (l_1-\lambda)\pi}{(l_1-\lambda)\pi} \right| \\
\leq & \frac{\lambda \pi}{\mathrm{sin}\,\lambda \pi}\left(\sum_{|\bm {l}|_{\infty} > N} | \hat{f}_{\bm {l}}|^2\right)^{1/2} \left( \sum_{|\bm {l}|_{\infty} > N}\left|\frac{\sin (l_1-\lambda)\pi}{(l_1-\lambda)\pi} \right|^2\right)^{1/2} \\
\leq & \frac{\lambda \pi}{\mathrm{sin}\,\lambda \pi}\frac{1}{a^3}\left \|f \right\|_{L^2(D)}\left( 2\sum_{j=N+1}^\infty \left|\frac{ \mathrm{sin}\,\lambda\pi}{(j-\lambda)\pi}\right |^2 \right)^{1/2}\\
\leq &\frac{2\lambda}{a^3\sqrt{N}}\|f\|_{L^2(D)} \\
= & C_5 \frac{\lambda}{\sqrt{N}},
\end{align*}
where $ C_5=2\|f\|_{L^2(D)}/a^3 $. Finally, we obtain
\begin{equation*}
|\hat{f}^{\delta}_{\bm 0}- \hat{f}_{\bm 0}|
\leq C_3\delta + C_4 \lambda N \delta+ C_5\frac{\lambda}{\sqrt{N}}.
\end{equation*}
The proof is complete.
\end{proof}
\begin{lemma}\label{a}\cite{Wang17}
Let $\bm J$ be a vector function in $(H_{\bm p}^{\sigma}(D))^3 $ and $0\leq\mu\leq\sigma$, then the following estimate holds
\begin{equation*}
\|\bm J_N-\bm J\|_{\bm p, \mu}\leq N^{\mu-\sigma}\|\bm J \|_{\bm p,\sigma}, \quad 0\leq\mu\leq\sigma.
\end{equation*}
\end{lemma}
The stability result is contained in the following theorem.
\begin{theorem}\label{A-2.3}
Let $\bm J \in (H_{\bm p}^{\sigma}(D))^3 $ and $0\leq\mu\leq\sigma$, then the following estimate holds
\begin{equation*}
\| \bm J_{N}^{\delta}- \bm J \|_{\bm p, \mu}
\leq C_6\delta + C_6 \lambda N \delta+ C_6\frac{\lambda}{\sqrt{N}}+ C_7 N^{\mu+3/2}\delta+ C_8 N^{\mu+5/2}\delta+N^{\mu-\sigma}\|\bm J \|_{\bm p,\sigma},
\end{equation*}
where $C_6, C_7, C_8$ depend only on $f, g, a $ and $\lambda$.
\end{theorem}
\begin{proof}[\bf Proof.]
It is readily seen that
\begin{equation}\label{3.4}
\begin{aligned}
&\| \bm J_{N}^{ \delta}- \bm J_{N}\|_{\bm p, \mu}\\
\leq &\left(\sum_{|\bm l|_{\infty}=0}^N \left( 1+|{\bm l}|^2 \right)^{\mu}|\hat{f}_{\bm l}^{\delta}-\hat{f}_{\bm l}|^2
+ \frac{4\pi^2}{a^2} \sum_{|\bm l|_{\infty}=1}^N\left( 1+|{\bm l}|^2 \right)^{\mu}|\bm p\times \bm l|^2 |\hat{g}_{\bm l}^{\delta}-\hat{g}_{\bm l}|^2
\right)^{1/2}\\
\leq & |\hat{f}_{\bm 0}^{\delta}-\hat{f}_{\bm 0}| + \left(\sum_{|\bm l|_{\infty}=1}^N \left( 1+|{\bm l}|^2 \right)^{\mu}|\hat{f}_{\bm l}^{\delta}-\hat{f}_{\bm l}|^2\right)^{1/2}\\
&+ \left(\frac{4\pi^2 }{a^2} \sum_{|\bm l|_{\infty}=1}^N\left( 1+|{\bm l}|^2 \right)^{\mu}|\bm p\times \bm l|^2 |\hat{g}_{\bm l}^{\delta}-\hat{g}_{\bm l}|^2 \right)^{1/2}\\
\leq & \left(C_3\delta + C_4 \lambda N \delta + C_5\frac{\lambda}{\sqrt{N}} \right) + C_1\delta \left(\sum_{|\bm l|_{\infty}=1}^N \left( 1+|{\bm l}|^2 \right)^{\mu}\right)^{1/2}\\
&+\frac{2\pi}{a} C_2\delta\left(\sum_{|\bm {l}|_{\infty}=1 }^N \left( 1+|\bm {l}|^2\right)^{\mu}\left|\bm l \right|^2 \right)^{1/2}\\
\leq & C_6\delta + C_6 \lambda N \delta+ C_6\frac{\lambda}{\sqrt{N}}+ C_7 N^{\mu+3/2}\delta+ C_8 N^{\mu+5/2}\delta,
\end{aligned}
\end{equation}
where $C_6=\max\, \{C_3, C_4, C_5\}$.
Hence, from \eqref{3.4} and Lemma \ref{a} , we obtain
\begin{equation*}
\| \bm J_{N}^{ \delta}- \bm J \|_{\bm p, \mu}
\leq C_6\delta + C_6 \lambda N \delta+ C_6\frac{\lambda}{\sqrt{N}}+ C_7 N^{\mu+3/2}\delta+ C_8 N^{\mu+5/2}\delta+N^{\mu-\sigma}\|\bm J \|_{\bm p,\sigma},
\end{equation*}
which completes the proof.
\end{proof}
\begin{rem}\label{rem:N}
If one takes $ N=\tau \delta^{-\frac{1}{\sigma+5/2}}$ with $\tau\geq 1$ in Theorem \ref{A-2.3} , we have
\begin{equation*}
\begin{aligned}
\| \bm J_{N}^{\delta}-\bm J \|_{\bm p, \mu}
\leq & C_6\delta +C_6\lambda\tau\delta^{\frac{\sigma+3/2}{\sigma+5/2}} +\frac{C_6\lambda}{ \sqrt{\tau}}\delta^{\frac{1}{2\sigma+5}}
+ C_7 \tau^{\mu+3/2}\delta^{\frac{1+\sigma-\mu}{\sigma+5/2}}\\
&+ C_8 \tau^{\mu+5/2}\delta^{\frac{\sigma-\mu}{\sigma+5/2}} +\tau^{\mu-\sigma}\delta^{\frac{\sigma-\mu}{\sigma+5/2}}\|\bm J \|_{\bm p, \sigma}, \quad 0\leq \mu\leq \sigma.
\end{aligned}
\end{equation*}
\end{rem}
\section{Numerical examples }
In this section, we carry out a series of numerical experiments to illustrate that the proposed Fourier reconstruction method is effective and efficient.
First, we briefly describe some parameters setting of our numerical experiments. Let $D=[-0.5, 0.5]^3$, namely, $a=1$. Assume that the wave propagates in the vacuum space, where $\mu_0=4\pi\times10^{-7}$ and $\varepsilon_0=8.8541\times10^{-12}$.
Synthetic electromagnetic far-field data are generated by solving the direct problem of
\eqref{eq: Maxwell} by using the quadratic finite elements on a truncated spherical domain enclosed
by a PML layer. The mesh of the forward solver is successively refined till
the relative error of the successive measured electromagnetic wave data is below $0.1\%$.
To show the stability of our proposed method, we also add some random noise to the synthetic far-field data by considering
\begin{equation*}
\begin{aligned}
&\bm E_{\infty}^ \delta:=\bm E_{\infty} +\delta r_1 |\bm E_{\infty} |_{\infty}\mathrm{e}^{\rm{i}\pi r_2},\\
&\bm H_{\infty}^ \delta:=\bm H_{\infty} +\delta r_1 |\bm H_{\infty} |_{\infty}\mathrm{e}^{\rm{i}\pi r_2},
\end{aligned}
\end{equation*}
where $r_1 $ and $ r_2$ are two uniform random numbers, both ranging from $-1$ to $1$, and $\delta>0$ represents the noise level.
From Remark \ref{rem:N}, the truncation $N$ is given by
\begin{equation}\label{eq:N}
N(\delta):=[3\delta^{-2/7}]+1,
\end{equation}
where $[X]$ denotes the largest integer that is smaller than $X+1$.
Next, we specify details of obtaining the artificial multi-frequency electromagnetic far-field data. Let
\begin{equation*}
\mathbb{L}_{N}:= \{\bm l \in \mathbb{Z}^{3}\mid 1 \leq |\bm{l}|_\infty\leq N \},
\end{equation*}
then the wavenumber set is given by
\begin{equation*}
\mathbb{K}_N:= \left\{2\pi|\bm{l}|:\bm{l}\in \mathbb{L}_{N} \right \} \cup \{2\pi\lambda\}, \quad \lambda=10^{-3},
\end{equation*}
and the observation directions are given by
\begin{equation*}
\mathbb{X}_N:= \left\{\frac{\bm l}{|\bm l|}:\bm{l}\in \mathbb{L}_{N} \right \} \cup \{(1,0,0)\}.
\end{equation*}
Thus, every wavenumber and observation direction can be denoted by $k_{j}\in \mathbb{K}_{N}$ and $ \hat{\bm x}_j \in \mathbb{X}_N $, respectively, where $\, j=1,2,\cdots,(2N+1)^{3}$.
Correspondingly, the frequency $\omega_j$ is chosen as $\omega_j=k_j/\sqrt{\mu_0 \varepsilon_0}$. With the admissible wavenumbers defined earlier, the artificial electromagnetic far-field data with noise can be written as
\begin{equation*}
\left \{\left(\bm E_\infty^\delta(\hat{x}_{j};k_{j}),\bm H_\infty^\delta(\hat{x}_{j};k_{j})\right ) : \hat{\bm x}_{j}\in \mathbb{K}_{N}, k_{j}\in \mathbb{K}_{N},\, j=1,2, \cdots,(2N+1)^{3}\right \}.
\end{equation*}
Finally, we specify details of the numerical inversion via the Fourier method. We reconstruct the electric current source $\bm J(\bm x),\, \bm x\in D$ by the truncated Fourier expansion $\bm J_N^{\delta}(\bm x),\, \bm x\in D$,
where
\begin{equation*}
\bm J=\begin{bmatrix}
J_{1}\\
J_{2} \\
J_{3}
\end{bmatrix}, \quad
\bm J_N^{\delta}=\begin{bmatrix}
J_{1}^{N}\\
J_{2}^{N} \\
J_{3}^{N}
\end{bmatrix}.
\end{equation*}
Given the noisy far-field data defined above,
if we use the electric far-field data $\{\bm E_\infty^\delta(\hat{x}_{j};k_{j})\}$, then
the Fourier coefficients $\hat{f}_{\bm l}, \hat{g}_{\bm l}, 1\leq |\bm l|_\infty \leq N $ and $\hat{f}_{\bm 0}$ are computed by \eqref{eq:Ef_hat}, \eqref{eq:Eg_hat} and \eqref{eq:Ef0_hat}, respectively. If we use the magnetic far-field data $\{\bm H_\infty^\delta(\hat{x}_{j};k_{j})\}$, then
the Fourier coefficients $\hat{f}_{\bm l}, \hat{g}_{\bm l}, 1\leq |\bm l|_\infty \leq N $ and $\hat{f}_{\bm 0}$ are computed by \eqref{eq:Hf_hat}, \eqref{eq:Hg_hat} and \eqref{eq:Hf0_hat}, respectively.
Divide the domain $D$ into a mesh with a uniform grid of size $ 50\times 50\times 50 $. The approximated Fourier series $\bm J_N^{\delta}(\bm z)$ are computed at the mesh nodes $\bm z_j,\, j=1,2, \cdots, 50^3$ by \eqref{eq:J_N}. The relative error is defined as
\begin{equation*}
\mathrm{relative \ error}=\frac{\|\bm J-\bm J_N^{\delta}\|_{L^2(D)}}
{\|\bm J\|_{L^2(D)}}.
\end{equation*}
Unless specified otherwise, we use the magnetic far-field data to reconstruct the electric current source.
Based on the above discussion, we formulate the reconstruction scheme by the Fourier method in Algorithm S as follows.
\begin{table}[htp]
\centering
\begin{tabular}{cp{.8\textwidth}}
\toprule
\multicolumn{2}{l}{{\bf Algorithm S:}\quad Fourier method for reconstructing the electromagnetic source }\\
\midrule
{\bf Step 1} & Choose the parameters $\lambda$, $N$, the wavenumber set $\mathbb{K}_N$ and observation direction set $\mathbb{X}_N$. \\
{\bf Step 2} & Collect the measured electric far-field data $\bm E_\infty^\delta(\hat{x}_{j};k_{j})$ or the magnetic far-field data $\bm H_\infty^\delta(\hat{x}_{j};k_{j})$ for $\hat{x}_{j}\in \mathbb{X}_N$ and $k_{j}\in \mathbb{K}_N $.\\
{\bf Step 3} & Compute the Fourier coefficients $\hat{f}_{\bm 0}$, $\hat{f}_{\bm l}$ and $\hat{g}_{\bm l}$ for $1\leq |\bm l|_{\infty}\leq N$. \\
{\bf Step 4} & Select a sampling mesh $\mathcal{T}_h$ in a region $D$. For each sampling point $z_j\in \mathcal{T}_h $, calculate the imaging functional $\bm J_N$ defined in $\eqref{eq:J_N}$, then $\bm J_N$ is the reconstruction of $\bm J$. \\
\bottomrule
\end{tabular}
\end{table}
\begin{example}\label{example1}
In this example, we numerically estimate the stability of the proposed method. We consider the following smooth source function
\begin{equation*}
\bm J=\bm p\times \nabla g,
\end{equation*}
where
\begin{equation*}
\begin{aligned}
&\bm p=\frac{1}{4}\left(\sqrt{5}, -2, \sqrt{7} \right),\\
&g(x_1, x_2, x_3)=10\left(x_1^2+x_2^2\right)\exp\left(-50\left(x_1^2+x_2^2+x_3^2\right)\right).
\end{aligned}
\end{equation*}
\end{example}
\begin{figure}
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J1_1_exact}}\hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J1_2_exact}} \hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J1_3_exact}} \hfill\\
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J1_1_recover}}\hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J1_2_recover}} \hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J1_3_recover}} \hfill
\caption{\label{fig:2} Contour plots of the exact and reconstructed source function of Example \ref{example1} at the plane $x_3=0$, where $\delta=2\%$. (a) $J_1$, (b) $J_2$, (c) $J_3$, (d) $J_1^{10}$, (e) $J_2^{10}$, (f) $ J_3^{10}$.}
\end{figure}
\begin{table}[h]
\center
\caption{ The relative errors of the reconstructions with different noise levels $\delta$. }\label{tab1}
\begin{tabular}{lcccccc}
\toprule
& $ \delta $ & 2\% & 5\% & 10\% &20\% \\
\midrule
& $ N(\delta) $ & 10 & 8 & 6 &5 \\
& Relative error & 0.10\% &2.10\% & 4.26\% & 8.94\% \\
& Time (second) & 78 & 40 & 12 &11 \\
\bottomrule
\end{tabular}
\end{table}
Figure \ref{fig:2} shows the comparison between the exact and the reconstructed source function at the plane $x_3=0$ with the additional noise $\delta=2\%$. We observe that
the reconstructions are very close to the exact one. To exhibit the accuracy quantitatively, we list the relative errors in $L^2$ in table \ref{tab1}. Meanwhile, table \ref{tab1} illustrates that the stability and CPU time increase as the truncation order $N(\delta)$ increases.
\begin{example}\label{example2}
In this example, we use the electric far-field data to recover the source. We aim to recover a smooth source as follows
\begin{equation*}
\bm J=\bm p f + \bm p\times \nabla g;
\end{equation*}
where
\begin{equation*}
\begin{aligned}
&\bm p= \frac{1}{3}\left(\sqrt{5}, -1, \sqrt{3}\right),\\
& f(x_1, x_2, x_3) =3\exp\left(-80\left((x_1-0.15)^2+(x_2-0.15)^2+x_3^2\right)\right),\\
&g(x_1, x_2, x_3) =0.3\exp\left(-40\left(x_1^2+x_2^2+x_3^2\right)\right).
\end{aligned}
\end{equation*}
\end{example}
\begin{figure}
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J2_1_exact}}\hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J2_2_exact}} \hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J2_3_exact}} \hfill\\
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J2_1_recover}}\hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J2_2_recover}} \hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J2_3_recover}} \hfill
\caption{\label{fig:3} Iso-surface plots of the exact and the reconstructed vectorial source function of Example \ref{example2}, where the red color denotes the iso-surface level being $10$ and the green color denotes iso-surface level being $ -10$. (a) $J_1$, (b) $J_2$, (c) $J_3$, (d) $J_1^{10}$, (e) $J_2^{10}$, (f) $ J_3^{10}$.}
\end{figure}
Figure \ref{fig:3} presents the iso-surface plots of the exact source and the reconstruction with noise $2\%$ , which demonstrate clearly that our proposed method performance nicely.
\begin{example}\label{example3}
In this example, we consider a discontinuous source function. For simplicity, the source function is given by
\begin{equation*}
\bm J=\bm p f,
\end{equation*}
where
\begin{equation*}
\begin{aligned}
&\bm p=\frac{1}{\sqrt{6}}\left(1, \sqrt{2}, \sqrt{3}\right),\\
& \displaystyle f(x_1, x_2, x_3) =
\begin{cases}
&\displaystyle 1, \quad \mathrm{if } \ (x_1+0.25)^2+x_2^2+x_3^2\leq 0.15^2,\medskip\\
& \displaystyle\frac{1}{2}, \quad \mathrm{if}\ 0.1\leq x_1\leq 0.4, -0.15\leq x_2\leq 0.15,-0.15\leq x_3\leq 0.15,\medskip\\
& \displaystyle 0, \quad \mathrm{elsewhere}.
\end{cases}
\end{aligned}
\end{equation*}
\end{example}
\begin{figure}
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J3_1_exact}}\hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J3_1_N5_recover}} \hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J3_1_N10_recover}} \hfill\\
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J3_1_N15_recover}}\hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J3_1_N20_recover}} \hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/J3_1_N25_recover}} \hfill
\caption{\label{fig:4} Contour plots of the exact and the reconstructed vector source function in Example \ref{example3} at the plane $x_3=0$. (a) exact $J_1$, (b) $J_1^5$, (c) $J_1^{10}$, (d) $J_1^{15}$, (e) $J_1^{20}$, (f) $J_1^{25}$.}
\end{figure}
\begin{figure}
\centering
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/Gibbs_5}}\hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/Gibbs_15}} \hfill
\hfill\subfigure[]{\includegraphics[width=0.32\textwidth]
{images/Gibbs_25}} \hfill\\
\caption{\label{fig:5} Gibbs phenomenon of the reconstructed source $J_1^N$ for different $N$ with $x_2=x_3=0$. (a) $N=5$, (b) $N=15$, (c) $N=25$.}
\end{figure}
Figure \ref{fig:4} shows the contour plots of the exact source and the reconstructions with different truncation order, $N=5, 10, 15, 20, 25$. It is clear that the resolution of the reconstructed results increase as the truncation order $N$ increases. Figure \ref{fig:5} shows the Gibbs phenomenon of the reconstructions over the line $x_2=x_3=0$ with the truncation order $N=5, 15 , 25$, respectively.
\section*{Acknowledgment}
The work of M. Song was supported by the NSFC grant under No. 11671113.
The work of Y. Guo was supported by the NSF grants of China under 11601107, 11671111 and 41474102.
The work of H. Liu was supported by the FRG and startup grants from Hong Kong Baptist University, Hong Kong RGC General Research Funds, 12302415 and 12302017.
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Mac software company Techspansion has decided to discontinue development of its applications and close its doors. The news was posted this morning on the front page of the company’s site, where company founder, Tyler Loch, said that he would be shutting down the company for “personal reasons.”
“I’ve decided (after much soul-searching) to close Techspansion, and take care of the things (and people) in my life that I’ve been neglecting, along with my physical, mental and emotional health,” Loch told Macworld in an e-mail.
Techspansion is best known for creating software for media conversion. It produced VisualHub, an application that lets you convert video files between many different formats; AudialHub, a similar program for audio files; and iSquint, an application that converts video into iPod-friendly formats.
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\begin{document}
\date{}
\maketitle
\textbf{Abstract}:
\small{We prove a Donsker and a Glivenko--Cantelli theorem for sequences of random discrete measures generalizing empirical measures. Those two results hold under standard conditions upon bracketing numbers of the indexing class of functions. As a byproduct, we derive a posterior consistency and a Bernstein--von Mises theorem for the Dirichlet process prior, under the topology of total variation, when the observation space is countable. We also obtain new information about the Durst--Dudley--Borisov theorem}.
\section{Introduction}
In this article we shall adopt the generic notation (for $r\in [1,\infty])$
\begin{align*}
\ell^r:=&\aoo \bp\in \RRR^{\NNN},\; \mmi \bp\mmi_r<\infty\aff\text{, where}\\
\mmi \bp\mmi_r^r:=&\sli_{i\in \NNN} \mid p_i\mid^r,\;\text{for } 1\le r<\infty,\text{ and where }\\
\mmi \bp \mmi_{\infty}:=&\sup_{i\in \NNN} \mid p_i\mid\text{, writing }\bp=(p_i)_{i\in \NNN}.
\end{align*}
Let $\po \mfX,\AAA_{\mfX}\pf$ be a measurable space. We shall use the notation
\beq Q(f):=\ili_{\mfX} f dQ\label{notation generique de l'integrale Q(f)}\eeq
for a given signed measure $Q$ in $\AAA_{\mfX}$ with finite total variation, and for each $f\in L^1(Q)$.
Any $\mfX$-valued sequence $\my=(y_i)_{i\in \NNN}$ combined with an element $\bp\in \ell^1$ defines a signed discrete measure on $\po \mathfrak{X},\AAA_{\mfX}\pf$ - with finite total variation - through the following formula:
\beq P_{\my,\bp}:=\sli_{i\in \NNN}p_i\dd_{y_i}.\label{Prbpmy}\eeq
Now substitute $\my$ by a $\AAA_{\mfX}^{\otimes \NNN}$ measurable sequence $\mY=(Y_i)_{i\in\NNN}$, and $\bp$ by a $\ell^1$-valued Borel random variable $\mbeta=(\beta_i)_{i\in \NNN}$ (both of them on a probability space $\wap$). Then the composition map $P_{\mY,\mbeta}$ defines random signed measure in the following sense: for any specified bounded Borel function $f$, the map
\beq P_{\mY,\mbeta}(f):\;\;\;\omega\rar P_{\mY_n(\omega),\mbeta_n(\omega)}(f)\label{Prn}\eeq
is Borel from $\po \Omega, \AAA\pf$ to $\RRR$. In the sequel we shall continue to adopt the same convention (\ref{notation generique de l'integrale Q(f)}) for $P(f)$ when $P$ is a random or non random measure, and we shall extend it - when meaningful - to functions $f$ that are not necessarily bounded.\lb
In \cite{Varron14Donsker}, Varron started the investigation on how well known results in empirical processes theory (see, e.g., \cite{Dudley,Vander} for monographs on the subject) could be carried over sequences of random signed measures of the form $P_{\mY_n,\mbeta_n}$
where, for each $n$, the sequence $(Y_{i,n})_{i \in \NNN}$ is independent and identically distributed \textit{given} $\mbeta_n$.
He showed that the uniform entropy numbers and the Koltchinskii--Pollard uniform entropy integral - two crucial notions in empirical processes theory - both adapt very well to that wider class of random measures, which not only encompasses the empirical measure, but also discrete nonparametric Bayesian priors. The latter notion of uniform entropy integral can be briefly defined as follows for a class $\FF$ of real Borel functions on $\po \mathfrak{X},\AAA_{\mfX}\pf$:
\begin{align} \nono J(\dd,\FF):=&\ili_{0}^{\dd}\sqrt{\log\poo \sup_{Q\;probab.} N\po \e\mmi F\mmi_{Q,2},\FF,\norm_{Q,2}\pf\pff}d\e,\;\; \dd\in (0,\infty].
\end{align}
Here $\norm_{Q,2}$ stands for the $L^2(Q)$ norm, $N(\e,\FF,\norm_{Q,2})$ denotes the minimal number of $\norm_{Q,2}$ balls with radius $\e$ needed to cover $\FF$, and $F$ stands for the minimal measurable envelope of the class $\FF$ - see, e.g., \cite[p. 85]{Vander}. Note that $F$ can be simply taken as
$$F(y):=\sup\aoo\mid f(y)\mid, f\in \FF\aff,\; y\in \mfX,$$
when $\FF$ is countable or pointwise measurable - see \S\ref{subsection definition de G_n} below.
When $J(\infty,\FF)$ is finite, Varron proved a Donsker theorem under natural asymptotic conditions upon $(\mbeta_n,\mY_n)_{n\geq 1}$. Those two asymptotic theorems (see \cite[Theorems 1 and 2]{Varron14Donsker}) involve processes of the form
\beq G_n(f):=\sli_{i\in \NNN} \beta_{i,n}\cooo f(Y_{i,n})-\EEE\poo f(Y_{i,n})\mid \beta_{i,n}\pff\cfff,\; f\in \FF,\label{definition de Gn}\eeq
indexed by a class $\FF$ of real Borel functions. A rigorous definition of $G_n(\cdot)$ is not immediate and is therefore voluntarily postponed to \S \ref{subsection definition de G_n}.\lb
While the uniform entropy has been celebrated as a very useful condition to prove that a class $\FF$ is Donsker or Glivenko--Cantelli, another condition turned out to be very fruitful as well: bracketing entropy. The bracket $\llbracket f^-,f^+\rrbracket$ between two Borel functions $f^-$ and $f^+$ is defined as the set of Borel functions $f$ fulfilling $f^-\prec f\prec f^+$, the symbol $\prec$ standing for the everywhere pointwise comparison between real functions on $\mfX$.
Denoting by $N_{[]}(\e,\FF,\norm_{Q,2})$ the minimal number of brackets with $\norm_{Q,2}$ diameter less than $\e$ needed to cover $\FF$, the $Q$ bracketing entropy of $\FF$ is defined as
\beq J_{[]}(\dd,\FF,Q):=\ili_{0}^{\dd}\sqrt{\log N_{[]}(\e,\FF,\norm_{Q,2})}d\e,\;\; \dd\in (0,\infty]\label{definition entropy crochet}.\eeq
A naturally arising question is then: \textit{does bracketing entropy adapt with the same efficiency to sequences of random measures such as in (\ref{Prn})?} The answer provided in the present article is: \textit{yes, but to a lesser extent}. More restrictions upon the weights are needed. First the $\beta_{i,n}$ have to be non negative, since the idea of bracketing relies on the comparison principle
$$f^-\prec f \prec f^+\Rightarrow Q(f^-)\le Q(f)\le Q(f^+), $$
when $Q$ is a non negative measure. Second, when looking for a Donsker theorem, $\mmi \mbeta_n\mmi_{\infty}$ has to tend to zero fast enough to counterbalance a the possible growth of $\mmi \mbeta_n\mmi_1$. The amount of compensation is directly linked to the moments of $F(Y_{1,n}),\; n\in \NNN^*$.\lb
Those two conditions were not required under the assumption that $J(\infty,\FF)$ is finite (see \cite[Theorems 1 and 2]{Varron14Donsker}). This difference can be explained by the fact that the use of the Koltchinskii--Pollard entropy is intimately linked to that of symmetrization, namely the study of
$$G_n^0(f):=\sli_{i\in \NNN}\e_i\beta_{i,n}f(Y_{i,n}),\; f\in \FF,$$
where the $\e_i$ are symmetric Bernoulli (or Rademacher) random variables, independent of $(\mY_n,\mbeta_n)$. By subgaussianity of Rademacher processes, the $G_n^0(\cdot)$ inherit several properties of infinite dimensional Gaussian analysis. In particular, Hilbert spaces take a predominant role. This explains why the results in \cite{Varron14Donsker} hold under conditions upon $\mmi\mbeta_n\mmi_2$ and $\mmi\mbeta_n\mmi_4$. On the other hand, bracketing methods do not rely on subgaussianity, but on a form of Bernstein's inequality. The latter is a tradeoff between subgaussian and subexponential tails for sums of independent random variables that are uniformly bounded. This roughly explains why $\mmi \mbeta_n\mmi_{\infty}$ - and its conjugate norm $\mmi \mbeta_n\mmi_1$ - needs to be controlled. Such a difference of extent between bracketing and uniform entropy was not visible on the empirical process for the following simple reason: when taking $\beta_{i,n}\equiv n^{-1/2}$ for $i\le n$ and $\beta_{i,n}\equiv 0$ otherwise, one has $\mmi \mbeta_n\mmi_{\infty}\equiv\mmi \mbeta_n\mmi_1^{-1}=n^{-1/2}$. This equality makes the counterbalance between those two norms hardly visible in the proof of the bracketing Donsker theorem.\lb
Various interesting classes admit a finite bracketing entropy - see, e.g., \cite[Chapter 2.7]{Vander}. In addition, several examples of posterior distributions in (discrete) Bayesian nonparametrics have the form $\mP_{\mY_n,\mbeta_n}$, or at least exhibit a predominant term that can be expressed as such - see \cite[Section 3]{Varron14Donsker}. Hence our main results present an interesting range of applications, which we will here illustrate through two examples. The first one takes place in the framework of frequentist asymptotic analysis of nonparametric Bayesian priors: for a countable observation space, we prove a posterior consistency and a Bernstein--von Mises theorem for the Dirichlet process prior, under the topology of total variation (see \S \ref{subsection posterior analysis}). Along the proof, we also revisit the Durst--Dudley--Borisov theorem and we obtain additional information about this phenomenon. Our second example of application is a Donsker theorem - under a bracketing condition - for a specific form of local empirical measures (see \S \ref{subsection mesure empirique locale}).
The remainder of this article is organized as follows: in \S \ref{subsection definitions} we give a careful description of the mathematical framework. Then our two main results are stated in \S \ref{section: results}. Applications follow in \S \ref{section: applications}. The proofs of those results are then written in \S \ref{section proofs}. Finally, the Appendix is dedicated to a minor proof.
\section{The mathematical framework}\label{subsection definitions}
In order to properly state our main results, we first need to carefully define their underlying probabilistic framework. This section may be skipped at first reading.
\subsection{The underlying probability space}\label{subsection: definition de wap}
Empirical processes carry over some lacks of measurability that are usually tackled by using outer expectations - see, e.g., \cite[Chapter 1.2]{Vander}. In order to make use of Fubini's theorem - which is in general untrue for outer expectations - mathematical rigor imposes to define the underlying probability space $\wap$ as a suitable product space. First, for fixed $n\geq 1$, consider a Markov transition kernel from $\ell^1$ to $\mfX$, i.e., a family $\{\mathbf{P}_{n,\bp},\;\bp\in \ell^1\}$ of probability measures for which the maps $\bp\rar \mP_{n,\bp}(A),\; A\in \AAA_{\mfX}$ are measurable from $\po \ell^1,Bor(\ell^1)\pf$ to $\po [0,1],Bor([0,1])\pf$. Also consider a probability measure $Q_n$ on $\po \ell^1,Bor(\ell^1)\pf$ and define:
\begin{align*}
&\tilde{\Omega}:= \ell^1\times\mfX^\NNN,\text{ endowed with its product }\sig\text{-algebra}\\
&\tilde{\AAA}:=Bor(\ell^1)\otimes\AAA_{\mfX}^{\otimes \NNN}
\text{, with probability law defined through the generic formula: }\\
&\PPP_n\poo \ao(\bp,\my)\in \tilde{\Omega}, \bp \in A, \forall j\in \{ 1,\ldots, k\} , y_i\in B_j\af\pff:= \ili_{\bp \in A}\prolijk \mP_{n,\bp}(B_j) dQ_n(\bp).
\end{align*}
Then define $\Omega:=\tilde{\Omega}^{\NNN^*}$, $\AAA:=\tilde{\AAA}^{\NNN^*}$,
$\PPP:=\bigotimes_{n\geq 1} \PPP_n$ on $\AAA$ and define the $\mY_n$ and $\mbeta_n$ as coordinate maps on $\Omega$~:
\begin{align*}
\mbeta_n(\bp_1,\my_1,\bp_2,\my_2,\ldots):=\bp_n,\text{ and }
\mY_n(\bp_1,\my_1,\bp_2,\my_2,\ldots):=\my_n.
\end{align*}
Note that, for fixed $n$ and $\bp\in \ell^1$, $\mP_{n,\bp}^{\otimes \NNN}$ is the law of $\mY_n$ given $\mbeta_n=\bp$. We shall denote by $\mP_n$ the law of $Y_{1,n}$.\lb
To simplify the notations we now adopt the following convention: each time a map $\mathfrak{h}$ is defined on a probability space, the symbol $\EEE^*(\mathfrak{h})$ will denote the outer expectation with respect to that probability space. We shall adopt the same convention for outer probabilities $\PPP^*$.
\subsection{Definition of $G_n$}\label{subsection definition de G_n}
From now on, and throughout all this article, we shall make the assumption that $\mP_n(F)<\infty$ for all $n\geq 1$.
We also assume that $\FF$ is \textit{pointwise measurable} with countable separant $\FF_0$ in the following sense: for any $f\in \FF$, there exists $(f_m)_{m\geq 1}\in \FF_0^{\NNN^*}$ such that $f_m(y)\rar f(y)$ for each $y\in \mfX$. Such a very standard assumption will be useful to tackle annoying measurability issues. \lb
Because the symbol $\Sig_{i\in \NNN}$ in (\ref{definition de Gn}) is ambiguous, we need to give a rigorous definition of the processes that will be involved in this article. Our definition differs from that used in \cite{Varron14Donsker} for two reasons. The first (minor) one is to cover the case where the $\mmi \mbeta_n\mmi_1$ are not deterministically equal to 1. The second one is for technical purposes: in our proofs, we shall truncate the $f\in \FF$ from above using thresholds that depend upon the weights.
First note that, for any bounded function $f$ and any Borel map $T$ from $\ell^1$ to $\RRR^+$, the map
$$\Phi_{f,T}:\; (\my,\bp)\rar \sli_{i\in \NNN}p_i\poo f\indic_{\{F\le T(\bp)\}}(y_i)-\mP_{n,\bp}\po f\indic_{\{F\le T(\bp)\}}(y_i)\pf\pff$$
is properly defined (through the limits in $\RRR$ of partial sums) and Borel from $\mfX^{\NNN}\times \ell^1$ to $\RRR$. One can hence define a random variable $G_n^T(f)$ by composition $G_n^T(f):=\Phi_{f,T}\circ(\mY_n,\mbeta_n)$.
We will say that a map $\psi$ from $\FF$ to $\RRR$ is \textit{$\FF_0$-separable} whenever we have $\mmi \psi\mmi_{\FF}=\mmi \psi \mmi_{\FF_0}$. We shall also denote by $\BB(\FF,\FF_0)$ the space of all bounded $\FF_0$-separable functions and by $\AAA_{\norm_{\FF}}$ is the $\sig$-algebra spanned by the $\norm_{\FF}$-balls.
\begin{lem}
For any choice of $T$ as above, the map
$$\Phi_{\FF,T}:\; (\my,\bp)\rar \ao f\rar \Phi_{f,T}(\my,\bp)\af$$
is measurable from $\mfX^{\NNN}\times \ell^1$ to $(\BB(\FF,\FF_0),\AAA_{\norm_{\FF}})$.
\end{lem}
\textbf{Proof}:
Fix $T$. Let us first prove that $\Phi_{\FF,T}$ takes its values in $\BB(\FF,\FF_0)$. Fix $\my\in \mfX^{\otimes \NNN}$ and $\bp\in \ell^1$. Since, for all $k\in \NNN$:
$$\sup_{f\in \FF}\Mid\sli_{i\geq k+1}p_i \coo f\indic_{\{F\le T(\bp)\}}(y_i)-\mP_{n,\bp}\po f\indic_{\{F\le T(\bp)\}}\pf\cff\Mid \le 2T(\bp)\sli_{i\geq k+1}\mid p_i\mid,$$
and since $\po \BB(\FF,\FF_0),\norm_{\FF}\pf$ is a Banach space, it is sufficient to prove that each trajectory $f\rar f\indic_{\{F\le T(\bp)\}}(y_i),\;i\in \NNN,$ and $f\rar \mP_{n,\bp}\po f\indic_{\{F\le T(\bp)\}}\pf$ is $\FF_0$-separable. To see this, take $f\in \FF$ and consider $(f_m)_{m\geq 1}\in \FF_0^{\NNN}$ such that $f_m\rar f_0$ pointwise. Thus $\mP_{n,\bp}(f_m\indic_{\{F\le T(\bp)\}})\rar \mP_{n,\bp}\po f\indic_{\{F\le T(\bp)\}}\pf$ by the dominated convergence theorem. Now since $\Phi_{\FF,T}$ takes its values in $\BB(\FF,\FF_0)$, the following equality holds for any $(\my,\bp)\in\mfX^{\NNN}\times \ell^1$:
\beq \sup_{f\in \FF}\mid \Phi_{f,T}(\my,\bp)\mid=\sup_{f\in \FF_0}\mid \Phi_{f,T}(\my,\bp)\mid,\eeq
and then the measurability of each $\Phi_{f,T},\; f\in \FF_0$ ensures that of $\Phi^T_{\FF}$ with respect to $\AAA_{\norm_{\FF}}$. $\Box$\lb
Now denote by $\tilde{\EE}_{\FF,\FF_0}$ the space of all measurable maps from $\po \Omega,\AAA\pf$ to $\po \BB(\FF,\FF_0),\AAA_{\norm_{\FF}}\pf$.
The preceding lemma gives the opportunity to define the processes $G_n$ on any class of functions $$\FF_M:=\ao f\indic_{\{F\le M\}},\;f\in \FF\af$$ since (confounding $M$ with a constant function on $\ell^1$) the composition map
$$G_n^M:=\Phi_{\FF,M}\circ(\mY_n,\mbeta_n)$$
belongs to $\tilde{\EE}_{\FF,\FF_0}$. Denote by $\EE_{\FF,\FF_0}$ the quotient space of $\tilde{\EE}_{\FF,\FF_0}$ with respect to the equivalence class
$$G\sim G'\Leftrightarrow \mmi G-G'\mmi_{\FF}=0,\; \PPP\text{-a.s.} ,$$
and endow $\EE_{\FF,\FF_0}$ with the compatible distance
\beq d(G,G'):=\EEE\poo \arctan \po \mmi G-G'\mmi_{\FF}\pf\pff,\label{definition de la distance L_0 processus}\eeq
which is that of $\norm_{\FF}$-convergence in probability.
The following lemma defines $G_n$ as a suitable limit of the $G_n^M$ when $M\rar \infty$.
\begin{lem}\label{lem: definition complete de Gn}
For fixed $n\geq 1$, the sequence $(G_n^M)_{M\geq 1}$ is Cauchy in the complete metric space $\po\EE_{\FF,\FF_0},d\pf$. It hence converges to a limit which we take as the definition of $G_n$. Moreover, for any sequence $(T_k)$ of Borel thresholding maps fulfilling $T_k(\mbeta_n)\cvproba \infty$ as $\kif$, we have
$d(G_n^{T_k},G_n)\rar 0$ as $\kif$.
\end{lem}
\textbf{Proof}:
For integers $M,M'$ we have, writing $f^{M,M'}:=f\indic_{\{M<F\le M'\}}$
\begin{align}
\nono &d\poo G_n^M,G_n^{M'}\pff \\
\nono = &\EEE\poooo \arctan \pooo \sup_{f\in \FF}\Mid \Phi_{f^{M,M'}}(\mY_n,\mbeta_n)\Mid \pfff\pffff \\
\le & \EEE\poooo \arctan \poooo \sli_{i\in \NNN}\mid \beta_{i,n}\mid\cooo F^{M,M'}(Y_{i,n})+\EEE\poo F^{M,M'}(Y_{i,n})\Mid \mbeta_n\pff\cfff\pffff\pffff.\label{pnop}
\end{align}
Using Fatou's lemma for conditional expectations and the concavity of $\arctan$ on $\RRR^+$ we have, almost surely:
\begin{align}
\nono&\EEE\poooo \arctan \pooo \sli_{i\in \NNN}\mid \beta_{i,n}\mid\cooo F^{M,M'}(Y_{i,n})+\EEE\poo F^{M,M'}(Y_{i,n})\Mid \mbeta_n\pff\cfff\pfff\Mid \mbeta_n\pffff\\
\nono\le&\arctan\pooo2\sli_{i\in \NNN} \mid \beta_{i,n}\mid \EEE\poo F^{M,M'}(Y_{i,n})\mid \mbeta_n\pff\pfff\\
\label{jil}=&\arctan\pooo2\mmi \mbeta_{n}\mmi_1\EEE\poo F^{M,M'}(Y_{1,n})\mid \mbeta_n\pff\pfff\\
\le &\arctan\pooo2\mmi \mbeta_{n}\mmi_1\EEE\poo F\indic_{\{F>M\}}(Y_{1,n})\mid \mbeta_n\pff\pfff,\label{bepplap}
\end{align}
where $(\ref{jil})$ comes from the fact that the law $\mY_n$ given $\mbeta_n=\bp$ is $\mP_{n,\bp}^{\otimes \NNN}$.
It hence suffices to prove that the right hand side (RHS) of (\ref{bepplap}) tends to $0$ in probability as $M\rar \infty$. This is true since $\mP_n(F)<\infty$. Now to prove the last statement of Lemma \ref{lem: definition complete de Gn}, formally replace $M$ by $T_k(\bp)$ in the preceding calculus and let $M'\rar\infty$ to obtain, using Fatou's lemma for conditional expectations:
\beq
d\po G_n^{T_k},G_n\pf
\le \EEE\poooo \arctan\pooo 2\mmi \mbeta_{n}\mmi_1\EEE\poo F\indic_{\{F>T_k(\mbeta_n)\}}(Y_{1,n})\mid \mbeta_n\pff\pfff\pffff\label{inegalite entre les esperance condtionnelles pour le reste de cauchy},\eeq
which tends to $0$ as $\kif$, by assumption upon $(T_k(\mbeta_n))_{k\geq 1}$ and since $\mP_n(F)<\infty$.
$\Box$
\subsection{Definition of the limit processes}
One of our results is a Donsker theorem (see Theorem \ref{T2} below). The limit processes are mixtures of $\FF$-indexed Brownian bridges, for which a rigorous definition is not immediate due to the non separability of $\ell^{\infty}(\FF)$. We shall use the definition of \cite{Varron14Donsker}. However, since the possibility to condition upon the weights was not made perfectly clear in \cite{Varron14Donsker}, we feel the need to give more precisions in its reminder. First fix $n\geq 1$.
For $p\geq 1$, $\mf:=(f_1,\ldots,f_p)\in \FF^p$ and $\bp\in\ell^1$, write $\mQnp^\mf$ for the centered Gaussian distribution on $\RRR^p$ with covariance matrix
$$\Sig_{n,\bp}^\mf:=\coo \mPnp\poo (f_j-\mPnp(f_j))(f_{j'}-\mPnp(f_{j'}))\pff\cff_{(j,j')\in \{1,\ldots, p\}^2}.$$
Now consider $\Omega':=\RRR^{\FF}$, endowed with its product Borel $\sig$-algebra $\AAA'$, i.e the $\sig$-algebra spanned by the $\pi$-system
\begin{align}
\nono &\AAA'_0:=\aoo C_{A_{f_1},\ldots,A_{f_k}},\: k\geq 1,\; \mf\in \FF^k,\; A_{f_j}\text{ Borel for each }j\in \{1,\ldots,k\}\aff\text{, with}\\
\nono &C_{A_{f_1},\ldots,A_{f_k}}:=\aoo \psi \in \Omega',\; \forall j\in \{1,\ldots,k\},\; \psi(f_j)\in A_{f_j}\aff.
\end{align}
For fixed $\bp\in \ell^1$, Kolmogorov's extension theorem (see, e.g., \cite[p. 115, Theorem 6.16]{Kallenberg}) ensures the existence of a unique probability measure $\PPP'_{n,\bp}$ on $\po \Omega',\AAA'\pf$ which is compatible with the system of marginals $\ao\mQnp^{\mf},\;\mf\in\FF^p,\; p\geq 1\af$, namely $$\PPP'_{n,\bp}\po C_{A_{f_1},\ldots,A_{f_k}}\pf=\mQnp^\mf\po A_{f_1}\times\ldots\times A_{f_k}\pf,$$
for all elements of $\AAA'_0$. Now since $\{\mP_{n,\bp},\;\bp\in \ell^1\}$ defines a transition kernel, then so does the family $\{\mQnp^\mf,\;\bp \in \ell^1\}$ for any specified $\mf$: it is a simple consequence of the fact that $\bp \rar \Sig_{n,\bp}^{\mf}$ is Borel. Consequently the map $\bp \rar \PPP'_{n,\bp}(C)$ is measurable for any specified $C\in \AAA'_0$. That measurability property is then immediately extended to all $C\in \AAA$ by Dynkin's $\pi$-$\lab$ theorem - see, e.g., \cite[p. 2, Theorem 1.1]{Kallenberg}. As a consequence, the family $\{\PPP'_{n,\bp},\;\bp\in \ell^1\}$ defines a transition kernel on $\po \Omega',\AAA'\pf$. Finally, we take $\Omega'':=\ell^1\times \Omega'$, endowed with $Bor(\ell^1)\otimes \AAA'$, and with probability $\PPP''_n$ defined by
$$\PPP'_n(A\times C):=\ili_{\bp \in A}\PPP'_{n,\bp}(C)dQ_n(\bp).$$
We then take $\AAA''_n$ as the completion of $Bor(\ell^1)\otimes \AAA'$ with respect to $\PPP''_n$ and we define $W_n$ as the coordinate map $(\bp,\psi)\rar \psi$ on $\po \Omega'',\AAA''_n,\PPP_n''\pf$. \begin{lem}
Let $n\geq 1$ be an integer. Assume that, $\PPP$-almost surely:
$$J_{[]}\poo\infty,\FF,\norm_{\mP_{n,\mbeta_n},2}\pff<\infty.$$
Then $W_n$ is $\PPP''_n$-almost surely bounded.
\end{lem}
\textbf{Proof:} Write $\mbeta'_n$ as the canonical map $(\bp,\psi)\rar \bp$ on $\po \Omega'',\AAA''_n,\PPP''_n\pf$ and note that $\mbeta'_n$ and $\mbeta_n$ are equal in law. For any finite subclass $\{f_1,\ldots,f_p\}\subset \FF$ we have, by conditioning on $\mbeta_n'$ and then using Dudley's chaining theorem (see, e.g., \cite[p. 101, Corollary 2.2.8]{Vander})
\begin{align*}
\EEE\pooo \max_{j\le p}\mid W_n(f_j)\mid\Mid \mbeta'_n\pfff\le &\mathfrak{C}_0
\ili_{0}^{+\infty}\sqrt{\log N\po \e,\FF,\norm_{\mP_{n,\mbeta'_n},2}\pf}d\e\\
\le &\mathfrak{C}_0
\ili_{0}^{+\infty}\sqrt{\log N_{[]}\po \e,\FF,\norm_{\mP_{n,\mbeta'_n},2}\pf}d\e\\
=&
\mathfrak{C}_0J_{[]}\poo\infty,\FF,\mP_{n,\mbeta'_n}\pff,\;\PPP''_n\text{-almost surely},
\end{align*}
where $\mathfrak{C}_0$ is a universal constant. It follows by Fatou's lemma for conditional expectations that
$$\EEE\pooo \sup_{f\in \FF_0}\mid W_n(f)\mid \Mid \mbeta'_n\pfff\le \mathfrak{C}_0J_{[]}\poo\infty,\FF,\mP_{n,\mbeta'_n}\pff,\;\PPP''_n\text{-almost surely}.$$
Now using the same arguments as those used to obtain \cite[p. 2314, assertion (49)]{Varron14Donsker}, one can show that
$$\PPP''_n\poo \ao \omega\in \Omega'',\; W_n(\omega)\text{ is not }\FF_0\text{ separable}\af\pff=0,$$
which concludes the proof.$\Box$ \lb
\section{Results}\label{section: results}
Before stating our two main results, let us briefly mention that the maps $\bp\rar N_{[]}\po\e,\FF,\norm_{\mP_{n,\bp},r}\pf$ and $\bp\rar J_{[]}\po \dd,\FF,\mP_{n,\bp}\pf$ are properly measurable for fixed $\e$ and $\dd$. This is proved in \S
\ref{sous section preuve de la mesurabilite}.
\subsection{A Glivenko--Cantelli theorem}
Our first result is a Glivenko--Cantelli theorem. Recall that $\mP_n$ is the law of $Y_{1,n}$. We shall denote by $\ell^{1,+}:=\ell^1\cap [0,\infty[^{\NNN}$ the set of non negative summable sequences.
\begin{theo}\label{T1}
Assume that
\beq \lim_{M\rar \infty}\; \lsn \mP_n\poo F\indic_{\{F\geq M\}}\pff=0\label{condition envelope GC},\eeq
and that, for any $\e>0$:
\beq \pooo N_{[]}\po\e,\FF,\norm_{\mP_{n,\mbeta_n},1}\pf\pfff_{n\geq 1}\text{ is bounded in probability}.\label{condition bracket GC}\eeq
Also assume that $\mbeta_n\in \ell^{1,+}$ is almost surely for all $n$, and that
\beq \po \mmi \mbeta_n\mmi_1\pf\suite\text{ is bounded in probability.}\label{condition GC mbeta_n borne en proba}\eeq Then, under the condition $\mmi \mbeta_n\mmi_2\cvproba 0$ we have
$\mmi G_n\mmi_{\FF}\cvproba 0.$
\end{theo}
\textbf{Remark}: It is important to compare the assumptions of Theorem \ref{T1} to
those of Dudley's bracketing Glivenko--Cantelli theorem \cite[p. 122, Theorem 2.4.1]{Vander} for the empirical measure
$P_{\mY_n,\mbeta^{Emp}_n}$, where $\beta_{i,n}^{Emp}:\equiv n^{-1}$ for $i\le n$ and is identically null otherwise, and where $\mY_n$ is constant in $n$ - hence with constant law $\mP_n=\mP_0$. A few easy arguments then show that, in this special case, those two theorems exactly coincide: first, for $\mbeta_n=\mbeta^{Emp}_n$ the convergence in probability is equivalent to an almost sure convergence by Pollard's reverse martingale argument (see, e.g., \cite[p. 124, Lemma 2.4.5]{Vander}). Second, (\ref{condition envelope GC})+(\ref{condition bracket GC}) is here equivalent to the finiteness of $N_{[]}\po \e,\FF,\norm_{\mP_0,1}\pf$ for each $\e>0$.
\subsection{A Donsker theorem}
For a sequence $Z_n$ of maps from $\Omega$ to $\RRR$ we shall write
$${\lsn}^{\PPP^*}\; Z_n:=\inf\aoo M\in \RRR,\; \limn \PPP^*(Z_n\geq M)=0\aff,$$
with the convention $\inf_{\emptyset}=+\infty$, and we shall simply write $\lsn^{\PPP}\;Z_n$ when the maps $Z_n$ are measurable.
Our second result is a Donsker theorem. \begin{theo}\label{T2}
Assume that
\begin{align}
&\mmi \mbeta_n\mmi_2\cvproba 1 \label{ condition Donsker norme L2 egale a 1},\\
&\mmi\mbeta_n\mmi_{\infty}\cvproba 0,\label{condition Donsker linfty tend vers 0}
\end{align}
and that, for some $p\in [2,\infty[$
\begin{align}
&\mmi\mbeta_n\mmi_1 \times \mmi \mbeta_n\mmi_{\infty}^{p-1}\text{ is bounded in probability},\label{condition Donsker produit l1linfty borne}\\
&\lim_{\dd\rar 0}\; {\lsn}^{\PPP} J_{[]}(\dd,\FF,\mP_{n,\mbeta_n})=0,\label{condition Donsker sur l'entropie crochet}\\
&\lim_{M\rar \infty}\; \lsn \mP_n\po F^p\indic_{\{F>M\}}\pf=0\label{ condition envelope Donsker}.
\end{align}
Also assume that there exists a semimetric $\rho$ that makes $\FF$ totally bounded, and fulfilling
\beq \lim_{\dd\rar 0}\;{\lsn}^{\PPP^*}\mathop{\sup_{(f_1-f_2)\in \FF^2,\;}}_{\rho(f_1,f_2)<\dd}\mP_{n,\mbeta_n}\poo (f_1-f_2)^2\pff=0\label{ condition sur la semimetrique rho}.\eeq
Then
\beq d_{BL}\poo G_n,W_{n}\pff:=\sup_{B\in BL1}\Mid \EEE^*\poo B\po G_n\pf\pff-\EEE^{*}\poo B\po W_n\pf\pff\Mid\rar 0,\label{convergence BL1}\eeq
where $BL1$ is the set of all 1-Lipschitz functions on $\po \ell^{\infty}(\FF),\norm_{\FF}\pf$ that are bounded by 1.\lb
Moreover, if $\FF$ is uniformly bounded, then (\ref{convergence BL1}) holds without assuming (\ref{condition Donsker produit l1linfty borne}) nor (\ref{ condition envelope Donsker}).
\end{theo}
\textbf{Remarks}:
We chose to state Theorem \ref{T2} under the most general assumptions that our methodology can afford. In order to give more substance to those conditions, it seems convenient to discuss on the place of Theorem \ref{T2} in the existing literature on Donsker theorems for empirical processes. \begin{enumerate}
\item When $\mbeta_n$ is the vector of rescaled empirical weights ($\beta_{i,n}\equiv n^{-1/2}$ for $i\le n$ and $\beta_{i,n}\equiv0$ otherwise), and when $\mP_n=\mP_0$ is constant in $n$, then $\mP_{\mY_n,\mbeta_n}$ is a sequence of empirical processes. Noting that - for $p=2$ - the $\mbeta_n$ obviously satisfy conditions (\ref{ condition Donsker norme L2 egale a 1}), (\ref{condition Donsker linfty tend vers 0}) and (\ref{condition Donsker produit l1linfty borne}) one can immediately conclude that - in this setup - Theorem \ref{T2} exactly coincides with Ossiander's bracketing Donsker theorem \cite[Theorem 3.1]{Ossiander87}. Andersen \textit{et al.} \cite{AndersenGOZ88} did also prove a Donsker theorem under more general conditions, where the finiteness of $J_{[]}(\infty,\FF,\mP_0)$ is relaxed a to more abstract assumption, involving majorizing measures on $\norm_{\mP_0,2}$ balls and "weak $\norm_{\mP_0,2}$" brackets. This possible extension of Theorem \ref{T2} is beyond the scope of the present article and may deserve future investigations.
\item Let us now relax the assumption that $\mP_n$ is constant in $n$. In that case the $G_n$ fall into the framework of triangular arrays of empirical processes with varying baseline measures, which were studied by Sheehy and Wellner \cite[Section 3]{SheehyW92}. These authors did prove a Donsker result for $G_n$ indexed by classes fulfilling $J(\infty,\FF)<\infty$, under the envelope condition (\ref{ condition envelope Donsker}), and assuming that $\mP_n$ converges to a limit $\mP_0$ in the following sense - see their Corollary 3.1:
\begin{align}
\nono &\sup_{(f_1,f_2)\in \FF^2}\max\aoo\mid\mP_n\po(f_1-f_2)^2\pf-\mP_0\po(f_1-f_2)^2\pf\mid,\; \mid\mP_n(f_1)-\mP_0(f_1)\mid,\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mid\mP_n(f_1^2)-\mP_0(f_1^2)\mid\aff\rar 0\label{SW}.\end{align}
It is then clear that our Theorem 2 puts forward an analogue of their result, replacing their assumption $J(\infty,\FF)<\infty$ by the bracketing condition (\ref{condition Donsker sur l'entropie crochet}). To see this, just note that (\ref{ condition sur la semimetrique rho}) is satisfied under (\ref{SW}), by choosing $\rho(f_1,f_2):=\mmi f_1-f_2\mmi_{\mP_0,2}$.
\item Let us now discuss on assumption (\ref{condition Donsker sur l'entropie crochet}), which might be the most cumbersome to verify for applications. If $J_{[]}(\infty,\FF,\mP_0)<\infty$, a simple way to check (\ref{condition Donsker sur l'entropie crochet}) - by direct comparison of bracketing numbers - is to prove that
\beq\sup_{(f_1,f_2)\in \FF^2}\frac{\mP_{n,\mbeta_n}\po (f_1-f_2)^2\pf}{\mP_0\po (f_1-f_2)^2\pf}\label{condition de domination}\text{ is bounded in probability.}\eeq
Such a sufficient condition is quite restrictive and seems far from necessary, but its verification is sometimes very simple to perform. This is for example the case for our application to the local empirical process at fixed point - see \S \ref{subsection mesure empirique locale}. \item We conclude this series of remarks by pointing out that, whereas involving random weights, Theorem 2 has almost no connections with Donsker theorems for bootstrap empirical measures. For more details see \cite[Remark 2.2]{Varron14Donsker}.
\end{enumerate}
\section{Applications}\label{section: applications}
\subsection{Posterior analysis of the Dirichlet process prior under the discrete total variation}\label{subsection posterior analysis}
Assume (in this subsection only) that $\mfX$ is infinite countable. The class $\FF$ of all indicator functions of subsets of $\mfX$:
\beq\FF:=\aoo \indic_{C},\: C\subset \mfX \aff\label{definition classe de tous les sous ensembles}\eeq
is rich enough to define the discrete total variation between two measures on $\mfX$, since
$$\mmi Q-Q'\mmi_{\FF}=\sup_{C\subset \mfX}\mid Q(C)-Q'(C)\mid=:\mmi Q-Q'\mmi_{Tot. var.}.$$
Clearly, $\FF$ is too large to satisfy $J(\infty,\FF)<\infty$. It was however shown in the celebrated Durst--Dudley--Borisov theorem that $\FF$ may have a finite bracketing entropy $J_{[]}(\infty,\FF,Q)$ under a simple necessary and sufficient criterion upon $Q$, namely
\begin{tabbing}
$ \hskip 0pt $ \= $(DDB(Q))\;:$ $ \hskip 20pt $ \= $\sli_{y\in \mfX}\sqrt{Q(\{y\})} <\infty$.
\end{tabbing}
\begin{theo}[Durst--Dudley--Borisov, 1981]\label{theo Durst}
For the class $\FF$ defined in (\ref{definition classe de tous les sous ensembles}) we have
$$J_{[]}(\infty,\FF,Q)<\infty\;\;\Leftrightarrow\;\; \po DDB(Q)\pf.$$
\end{theo}
We shall combine Theorem \ref{T1} with a refinement of Theorem \ref{theo Durst} - see Lemma \ref{lem des sommes de queues} - to prove both a posterior consistency and a Bernstein--von Mises theorem for the Dirichlet process prior, under the \textit{discrete} total variation. To properly state it, we need to introduce some more notations. From now on we shall denote by $DP(\alp,M)$ a Dirichlet process with mean probability measure $\alp$ on $\mfX$ and concentration parameter $M>0$. A possible representation of $DP(\alp,M)$ is that of Sethuraman \cite{Sethuraman94}:
\beq DP(\alp,M)=_{law}Pr_{\mY,\mbeta},\label{representation de Sethuraman}\eeq
where $\mY\leadsto \alp^{\otimes \NNN}$, and $\beta_i:=V_i\;\Pi_{j\le i-1}(1-V_j)$
with $(V_i)_{i\in \NNN}\leadsto Beta(1,M)^{\otimes \NNN}$ being independent of $\mY$.
Now consider the nonparametric Bayesian model where the prior $Pr$ has distribution $DP(\alp,M)$ and where the sample $\XXN$ has conditional law $\mP^{\otimes n}$ given $Pr=\mP$. In this model it is well known (see \cite{Ferguson73}) that a natural expression of the posterior distribution of $Pr$ given $\XXN=\xxn$ is $Post_n\xxn:=DP(\alp_{\xxn},M+n)$, where
\begin{align}
\nono \alp_{\xxn}:=&\tht_n\alp+(1-\tht_n)\mP_{\xxn}\text{, with }
\nono \tht_n:=\frac{M}{M+n},\text{ and } \mP_{\xxn}:=\frac{1}{n}\sliin \dd_{x_i}.
\end{align}
We shall take advantage of that explicit representation to prove the following two results.
\begin{coro}[Posterior consistency]\label{ coro:posterior}
Take $\FF$ as in (\ref{definition classe de tous les sous ensembles}).
Let $\mP_0$ be a probability measure on countable $\mfX$, and let $(X_n)_{n \geq 1}\leadsto \mP_0^{\NNN^*}$. Then for almost every sequence $(x_n)_{n\geq 1}$ we have
$$\Mmi Post_n\xxn - \mP_{\xxn}\Mmi_{Tot. Var.}\cvloi 0.$$
\end{coro}
\begin{coro}[Bernstein--von Mises]\label{coro: posterior et bernstein}
Assume in addition $(DDB(\mP_0))$ and $(DDB(\alp))$. Then for almost every sequence $(x_n)_{n\geq 1}$ we have
\beq \nono\sqrt{n} \pooo Post_n\xxn - \mP_{\xxn}\pfff\cvloi \GGG_{\mP_0},\text{ in }\ell^{\infty}(\FF).\eeq
As a consequence, for almost every sequence $(x_n)_{n\geq 1}$ we have
$$\sqrt{n}\Mmi Post_n\xxn - \mP_{\xxn}\Mmi_{Tot. Var.}\cvloi \mmi\GGG_{\mP_0}\mmi_{\FF}.$$
\end{coro}
The corresponding proofs are written in \S \ref{sous section preuve Bernstein}.
\subsection{A Donsker theorem for local empirical measures under a bracketing condition}\label{subsection mesure empirique locale}
Assume in this subsection that $\mfX=\RRR^d$.
The local empirical process indexed by functions, introduced by Einmahl and Mason \cite{EinmahlM97} has been intensively investigated during the last decades, due to its connections with several smoothing nonparametric methods. One of its particular forms can be written as follows:
$$T_{n,h_n}(f):=\frac{1}{\sqrt{nh_n^d}}\sliin \coooo f\poo h_n^{-1}\po Z_i-z\pf\pff-\EEE\pooo f\poo h_n^{-1}\po Z_i-z\pf\pff\pfff\cffff,$$
where $(h_n)_{n\geq 1}$ is a deterministic non negative sequence tending to $0$, and where $(Z_n)_{n\geq 1 }$ is an i.i.d. sequence. Implicit in the results of Einmahl and Mason \cite[Theorem 1.1]{EinmahlM97} is the following Donsker theorem.
\begin{theo}[from Einmahl and Mason, 1997]\label{ theo de Einmahl et Mason}
Let $(h_n)\suite$ be a non random sequence of non negative numbers such that $h_n\rar 0$ and $nh_n^d\rar \infty$.
Assume that $J(\infty,\FF)<\infty$. Assume that the support $S$ of $F$ is bounded, and that $Z_1$ admits a version of Lebesgue density $\mf$ on a neighborhood of $z$ that is continuous at $z$ and such that $\mf(z)>0$. Also assume that, taking $\mP_0$ as the uniform distribution on $S$, we have $\mP_0(F^2)<\infty$. Then we have the following weak convergence \beq \nono \frac{1}{\sqrt{\lab(S)\mf(z)}}T_{n,h_n}(\cdot)\cvloi \WW_{\mP_0}(\cdot),\text{ in }\ell^{\infty}(\FF), \eeq
where $\WW_{\mP_0}(\cdot)$ denotes the $L^2(\mP_0)$-isonormal Gaussian process indexed by $\FF$ (or $\mP_0$-Brownian motion).
\end{theo}
Their proof heavily relies on a representation of their own \cite[Proposition 3.1]{EinmahlM97}:
$$T_{n,h_n}(f):=_{law}\frac{1}{\sqrt{nh_n^d}}\times\cooo \sliin b_{i,n}\po f(Y_{i,n})-\mP_n(f)\pf\cfff+R_n(f),$$
as processes indexed by $\FF$, where:
\begin{itemize}
\item The $(b_{i,n})_{i\le n}$ are i.i.d Bernoulli with parameter $a_n:=\PPP\po h_n^{-1}(Y_1-z)\in S\pf$;
\item The $(Y_{i,n})_{i\le n}$ are i.i.d with law
\beq \mP_n:=\PPP\po Y_1\in \cdot\mid h_n^{-1}(Y_1-z)\in S\pf,\label{definition Pn local}\eeq
with $(b_{i,n})_{i\le n}\indep (Y_{i,n})_{i\le n}$;
\item The term
\beq R_n(f):= \frac{\sliin (b_{i,n}-a_n)}{\sqrt{nh_n^d}}\mP_n(f)\label{definition de R_n(f)}\eeq
plays the asymptotic role of a correcting drift between the Brownian bridge $\GGG_{\mP_0}$ and the Brownian motion $\WW_{\mP_0}$.\end{itemize}
The following corollary of Theorem \ref{T2} is a Donsker theorem for $T_{n,h_n}$ under the condition $J_{[]}(\infty,\FF,\mP_0)<\infty$. Its proof is written in \S \ref{sous section preuve de Donsker local}.
\begin{coro}\label{coro Donsker local}
Theorem \ref{ theo de Einmahl et Mason} still holds if assumption $J(\infty,\FF)<\infty$ is replaced by $J_{[]}(\infty,\FF,\mP_0)<\infty.$
\end{coro}
\section{Proofs}\label{section proofs}
\subsection{Proof of Theorem \ref{T1}}
The proof is divided in two lemmas.
\begin{lem}\label{ lem: GC sous un seuil arbitraire}
Take $M>0$. Under the assumptions of Theorem \ref{T1} we have
$\mmi G_n^M\mmi_{\FF}\cvproba 0$ as $\nif$.
\end{lem}
\textbf{Proof }: Fix $M>0,\; \e>0$, and choose - by (\ref{condition bracket GC}) and (\ref{condition GC mbeta_n borne en proba}) - an integer $N$ for which, $\PPP(\mbeta_n\in \SSS_n)>1-\e$ for all $n\geq 1$, with
$$\SSS_n:= \aoo \bp \in \ell^{1,+},\;\mmi \bp\mmi_1\le N\text{ and } N_{[]}\po \e,\FF,\norm_{\mP_{n,\bp},1}\pf\le N\aff.$$
Now fix $n\geq 1$, and $\bp \in \SSS_n$ and denote by $Br(n,\bp)=\ao(f_j^-,f_j^+),\;j=1,\ldots,N\af$ a covering bracket of $\FF$ with $\max_{j\le N}\mmi f_j^+-f_j^-\mmi_{\mP_{n,\bp},1}\le \e/N$. Using the same comparison argument as in \cite[p. 122]{Vander} we have, for $(f^-,f^+)\in Br(n,\bp)$, $f\in \llbracket f^-,f^+\rrbracket$ and $\my\in \mfX^\NNN$ (recall (\ref{Prbpmy})):
\begin{align}
\nono &\coo P_{\my,\bp}\poo f^-\indic_{\{F\le M\}}-\mP_{n,\bp}\po f^-\indic_{\{F\le M\}}\pf\pff \cff- N\mP_{n,\bp}( f^+ -f^-)\\
\nono \le & P_{\my,\bp}\poo f\indic_{\{F\le M\}}-\mP_{n,\bp}\po f\indic_{\{F\le M\}}\pf\pff\\
\le& \coo P_{\my,\bp}\poo f^+\indic_{\{F\le M\}}-\mP_{n,\bp}\po f^+\indic_{\{F\le M\}}\pf\pff\cff + N\mP_{n,\bp}( f^+ -f^-),\label{comparison 1}
\end{align}
from where, writing $B_{n,\bp}$ for the set of functions $f$ that are a side of a bracket (hence $\sharp B_{n,\bp}\le 2N$, where "$\sharp$" stands for "cardinal"):
\begin{align}
\nono &\sup_{f\in \FF}\Mid P_{\my,\bp}\poo f\indic_{\{F\le M\}}-\mP_{n,\bp}\po f\indic_{\{F\le M\}}\pf\pff\Mid\\
\nono \le& \max_{f\in B_{n,\bp}}\Mid P_{\my,\bp}\poo f\indic_{\{F\le M\}}-\mP_{n,\bp}\po f\indic_{\{F\le M\}}\pf\pff\Mid+\e.\end{align}
Note that the condition $\bp\in \ell^{1,+}$ is crucial to obtain (\ref{comparison 1}).
Now formally replacing $\my$ by an i.i.d sequence $(Y_i)_{i\in \NNN}$ having distribution $\mP_{n,\bp}$ we obtain, for $\bp\in\SSS_n$:
\beq \EEE\poooo \sup_{f\in \FF}\Mid \sli_{i\in \NNN}p_i\po f(Y_i)-\mP_{n,\bp}(f)\pf\Mid \pffff\le \Delta_n(\bp)+\e,\text{ where }\label{ha}\eeq
\begin{align}
\nono\Delta^2_n(\bp):=&\EEE^2\poooo \sli_{f\in B_{n,\bp}}\Mid \sli_{i\in \NNN}p_i\poo f(Y_i)-\mP_{n,\bp}(f)\pff\Mid\pffff\\
\nono\le & \EEE\poooo \pooo\sli_{f\in B_{n,\bp}}\Mid \sli_{i\in \NNN}p_i\poo f(Y_i)-\mP_{n,\bp}(f)\pff\Mid\pfff^2\pffff\\
\nono\le & (2N)^2\max_{f\in B_{n,\bp}}\;\EEE\poooo\Mid \sli_{i\in \NNN}p_i\poo f(Y_i)-\mP_{n,\bp}(f)\pff\Mid^2\pffff\\
\nono\le & (2N)^2 \max_{f\in B_{n,\bp}}\;\sli_{i\in \NNN}p_i^2\Var\poo f(Y_1)\pff\\ \le &(2NM \mmi \bp\mmi_2)^2.\label{haha}
\end{align}
Combining (\ref{ha}) and (\ref{haha}) yields, almost surely
$$\EEE\pooo \mmi G_n^M\mmi_\FF\Mid \mbeta_n\pfff\indic_{\SSS_n}(\mbeta_n)\le 2NM \mmi \mbeta_n\mmi_2+\e.$$
This concludes the proof, since $\mmi\mbeta_n\mmi_2\cvproba 0$ by assumption and since $\PPP\po \mbeta_n\notin \SSS_n\pf<\e$. $\Box$\lb
With Lemma \ref{ lem: GC sous un seuil arbitraire} at hand, the proof of Theorem \ref{T1} will be concluded as follows:
\begin{lem}\label{lem: deseuillage pour GC}
We have $$\lim_{M\rar \infty}\lsn d(G_n^M,G_n)=0.$$
\end{lem}
\textbf{Proof}: In view of (\ref{condition GC mbeta_n borne en proba}) it is sufficient to show that
\beq \nono \forall \e>0,\;\mathop{\overline{\lim}}_{M\rar \infty}\lsn\PPP\pooo \EEE\poo F\indic_{\{F>M\}}\po Y_{1,n}\pf\Mid \mbeta_n\pff>\e\pfff\le \e.\eeq
This is immediate by (\ref{condition envelope GC}) combined with Markov's inequality.
$\Box$
\subsection{Proof of Theorem \ref{T2}}
By (\ref{ condition Donsker norme L2 egale a 1}) we can assume without loss of generality that $\mmi \mbeta_n\mmi_2\equiv 1$ for all $n$.
First note that (\ref{condition Donsker sur l'entropie crochet}) immediately implies
\beq \forall \dd>0, \pooo N_{[]}\po \dd,\FF,\norm_{\mP_{n,\mbeta_n},2}\pf\pfff_{n\geq 1} \text{is bounded in probability}\label{condition Donsker sur mfa(delta)}.\eeq
The proof of Theorem \ref{T2} follows the same directions as in that of Theorem 2 in \cite{Varron14Donsker}. The only crucial point that changes is that of proving the following asymptotic equicontinuity condition
\beq \lim_{\dd\rar 0} \; {\lsn}^{\PPP^*} \sup_{(f_1,f_2)\in \FF^2,\; \rho(f_1,f_2)< \dd}\mid G_n(f_1)-G_n(f_2)\mid =0,\label{equicontinuite cruciale}\eeq
which would be the only missing ingredient to complete the proof of Theorem 2.
Proving (\ref{equicontinuite cruciale}) will be achieved by conditioning upon $\mbeta_n$ and using the following chaining argument. It is an extension of usual chaining arguments for the bracketing entropy \cite[p. 286, Lemma 19.34]{VanderAsymptotic} to unbalanced empirical measures. Due to the fact that infinitely many weights are involved, only uniformly bounded classes of functions are treated here for simplicity. This will be largely sufficient for our purposes.
\begin{lem}\label{lem: chaining pour des empiriques desequilibrees}
Let $\bp\in \ell^{1,+}$ such that $\mmi \bp\mmi_2=1$ let $Q$ be a probability measure and let $\GG$ be a uniformly bounded pointwise measurable class of functions with countable separant $\GG_0$. Let $\dd\in (0,\infty]$ be such that
\begin{align}
&\sup_{g\in \GG}\mmi g\mmi_{Q,2}\le \dd\text{ and }\label{bep variance}\\
&\sup_{g\in \GG,\; y\in \mfX}\;\mid g(y)\mid\le \mmi \bp\mmi_{\infty}^{-1}\mfa(\dd,Q),\text{ where}\label{bep norme infinie}\\
\nono &\mfa(\dd,Q):=\dd/\sqrt{\log N_{[]}\po \dd,\GG,\norm_{Q,2}\pf}.
\end{align}
Then for any i.i.d sequence $(Y_i)_{i\in \NNN}$ with distribution $Q$, we have
\beq \EEE\pooo \sup_{g\in \GG}\Mid \sli_{i\in \NNN}p_i\poo g(Y_i)-Q(g)\pff\Mid\pfff\le \mathfrak{C}_1
J_{[]}\po \dd,\GG,Q\pf,\label{ assertion du chaining unbalanced}\eeq
where $\mathfrak{C}_1$ is a universal constant.
\end{lem}
\textbf{Proof}:
We shall use the notations
\begin{align*}
\Delta^2(\GG,Q):=&\sup_{g\in \GG}Q(g^2),\;\text{and }\Gam(\GG):=\sup_{g\in \GG,\; y\in \mfX} \mid g(y)\mid.
\end{align*}
Given a finite class of functions $\tilde{\GG}$ and given $m\geq 1$ we have, by combining Lemmas 2.2.9 and 2.2.10 in \cite[p. 102]{Vander}:
\begin{align}
\nono &\EEE\poooo \max_{g\in \tilde{\GG}} \Mid \sli_{i=0}^m p_i \poo g(Y_i)-Q(g)\pff\Mid\pffff\\
\nono \le & 24 \coooo\sqrt{\sli_{i=0}^m p_i^2\Delta^2(\tilde{\GG},Q)\log(1+\sharp \tilde{\GG})}+ \max_{i\le m}\mid p_i\mid \Gam(\tilde{\GG})\log(1+\sharp \tilde{\GG})\cffff,\\
\le &24 \coooo \mmi \bp\mmi_2\Delta(\tilde{\GG},Q)\sqrt{\log(1+\sharp \tilde{\GG})}+\mmi \bp\mmi_{\infty}\Gam(\tilde{\GG})\log(1+\sharp \tilde{\GG})\cffff,\label{papou}
\end{align}
where the possible choice of factor 24 was actually shown in \cite[p. 285, Lemma 19.33]{VanderAsymptotic}. Since (\ref{papou}) does not depend upon $m$ we then have, as soon as $\Gam(\tilde{\GG})<\infty$ (and recalling that $\mmi \bp\mmi_2=1$):
\begin{align}
&\nono\EEE\poooo \max_{g\in \tilde{\GG}} \Mid \sli_{i\in \NNN} p_i \poo g(Y_i)-Q(g)\pff\Mid\pffff \\
=&\lim_{m\rar \infty} \EEE\poooo \max_{g\in \tilde{\GG}} \Mid \sli_{i=0}^m p_i \poo g(Y_i)-Q(g)\pff\Mid\pffff\label{pa}\\ \le &24 \cooo \Delta(\tilde{\GG},Q)\sqrt{\log(1+\sharp \tilde{\GG})}+\mmi \bp\mmi_{\infty}\Gam(\tilde{\GG})\log(1+\sharp \tilde{\GG})\cfff, \label{inegalite maximale}
\end{align}
where $(\ref{pa})$ is an application of the dominated convergence theorem, since all the involved random variables are bounded by $2\Gam(\tilde{\GG})$. Now with $(\ref{inegalite maximale})$ at hand, the remainder of the proof is as follows: a careful look at all the arguments of the proof of \cite[p. 286, Lemma 19.34]{VanderAsymptotic} - noting that their truncating argument is not needed here - shows that the latter are still true with the systematic formal change of $\sqrt{n}$ by $\mmi \bp \mmi_{\infty}^{-1}$. $\Box$\lb
We can now start our proof of (\ref{equicontinuite cruciale}). First fix $\e>0$. Using (\ref{condition Donsker sur l'entropie crochet}) and (\ref{ condition sur la semimetrique rho}) there exist $\dd_1,\dd_2>0$ and $n_0$ such that for all $n\geq n_0$ we have $1-\e\le \PPP_*\po \mbeta_n\in \SSS'_n\pf$, where $\SSS'_n$ is the set of all $\bp\in \ell^{1,+}$ satisfying the following conditions:
\begin{align}
&2\sqrt{2}\mathfrak{C}_1J_{[]}\poo \frac{\dd_1}{2},\FF,\mP_{n,\bp}\pff\le \e\label{ fil1}\\
\nono &\sup_{f\in \FF_{\dd_2}}\mmi f\mmi_{\mP_{n,\bp},2}<\dd_1,\text{ where }\\
\nono &\FF_{\dd_2}:= \aoo f_1-f_2,\; (f_1,f_2)\in \FF^2,\; \rho(f_1,f_2)<\dd_2\aff,
\end{align}
and where $\mathfrak{C}_1$ denotes the universal constant in (\ref{ assertion du chaining unbalanced}).
Now fix $\bp$, write
\begin{align*}
T(\bp):=& \mmi \bp\mmi_{\infty}^{-1}\mfa(\dd_1,\mP_{n,\bp})\indic_{\{\mmi \bp\mmi_{\infty}>0\}},\text{ and define}\\
\FF_{\bp,\dd_1}:= &\aoo (f_1-f_2)\indic_{\{F\le T(\bp)\}},\; (f_1,f_2)\in \FF^2,\; \mmi f_1-f_2\mmi_{\mP_{n,\bp},2}<\dd_1\aff.
\end{align*}
Next, apply Lemma \ref{lem: chaining pour des empiriques desequilibrees} for fixed $\bp \in \SSS'_n$ to obtain (noticing that $\FF_{\bp,\dd_1}$ satisfies (\ref{bep variance}) and (\ref{bep norme infinie}) for the choice of $Q:=\mP_{n,\bp}$ and $\dd:=\dd_1$)
\begin{align}
\nono \EEE\poooo \sup_{f\in \FF_{\bp,\dd_1}} \Mid \sli_{i\in \NNN} p_i\poo f (Y_i)-\mP_{n,\bp}\po f\pf\pff\Mid\pffff\le &\mathfrak{C}_1J_{[]}\poo \dd_1,\FF_{\bp,\dd_1},\mP_{n,\bp}\pff\,\\
\le &2\sqrt{2}\mathfrak{C}_1J_{[]}\poo \frac{\dd_1}{2},\FF,\mP_{n,\bp}\pff, \label{mpap}
\end{align}
where the $Y_i$ are i.i.d with law $\mP_{n,\bp}$ and where (\ref{mpap}) is a consequence of (\ref{ fil1}) and standard comparisons of entropy numbers. Now since the latter inequality is valid for all $\bp \in \SSS'_n$ we have
\begin{align}
\EEE\pooo \mmi G_n^T\mmi_{\FF_{\mbeta_n,\dd_1}}\Mid \mbeta_n\pfff\indic_{\SSS'_n}\po \mbeta_n\pf\le \e\indic_{\SSS'_n}\po \mbeta_n\pf,\text{ almost surely.}\label{oh}
\end{align}
Note that the measurability $\mmi G_n^T\mmi_{\FF_{\mbeta_n,\dd_1}}$ is not immediate at all, but can be proved using the same arguments as in \cite[proof of Proposition 4.2]{Varron14Donsker}.
In view of (\ref{oh}), and since $\FF_{\dd_2}\subset \FF_{\bp,\dd_1}$ for $\bp\in \SSS'_n$, the proof of (\ref{equicontinuite cruciale}) will be completed if we prove the following lemma.
\begin{lem}
We have
$d(G_n^T,G_n)\rar 0$ as $\nif$.
\end{lem}
\textbf{Proof}: From (\ref{inegalite entre les esperance condtionnelles pour le reste de cauchy}) we have (noting that $\mmi \beta_n\mmi_2\equiv1$ implies $\mmi\beta_n\mmi_{\infty}>0$ a.s.)
\begin{align}
\nono &d\po G_n^{T},G_n\pf\\
\le &\EEE\poooo \arctan\pooo 2\mmi \mbeta_{n}\mmi_1\EEE\poo F\indic_{\{F>T(\mbeta_n)\}}(Y_{1,n})\mid \mbeta_n\pff\pfff\pffff \label{hao}\\
\nono =&\EEE\poooo \arctan\pooo 2\mmi \mbeta_{n}\mmi_1\EEE\poo F\indic_{\{F>T(\mbeta_n)\}}(Y_{1,n})\mid \mbeta_n\pff\pfff\pffff\\
\nono \le & \EEE\poooo \arctan\pooo 2\frac{\mmi \mbeta_{n}\mmi_1}{T(\mbeta_n)^{p-1}}\EEE\poo F^p\indic_{\{F>T(\mbeta_n)\}}(Y_{1,n})\mid \mbeta_n\pff\pfff\pffff\\
\nono = & \EEE\poooo \arctan\pooo 2\frac{\mmi \mbeta_{n}\mmi_1\times \mmi \mbeta_n\mmi_{\infty}^{p-1}}{\mfa(\dd_1,\mP_{n,\mbeta_n})^{p-1}}\EEE\poo F^p\indic_{\{F>T(\mbeta_n)\}}(Y_{1,n})\mid \mbeta_n\pff\pfff\pffff.
\end{align}
Since, by (\ref{condition Donsker produit l1linfty borne}) and (\ref{condition Donsker sur mfa(delta)}), the sequence $\mmi \mbeta_{n}\mmi_1\times \po\mmi \mbeta_n\mmi_{\infty}/\mfa(\dd_1,\mP_{n,\mbeta_n})\pf^{p-1}$ is bounded in probability, it only remains to prove that
\beq E_n:=\EEE\pooo F^p\indic_{\{F>T(\mbeta_n)\}}(Y_{1,n})\mid \mbeta_n\pfff\cvproba 0\label{gt}.\eeq
To prove this, fix $\e>0$ and choose $M$ large enough so that $$\EEE\poo F^p\indic_{\{F>M\}}(Y_{1,n})\pff\le \e^2,$$
for all $n\geq 1$, which is possible by (\ref{ condition envelope Donsker}). Next apply Markov's inequality to $E_n$ on the set $\{ T(\mbeta_n)> M\}$ and then note that $\PPP\po T(\mbeta_n)\le M\pf\rar 0$ by (\ref{condition Donsker linfty tend vers 0}) and (\ref{condition Donsker sur mfa(delta)}). To conclude the proof, let us now consider the isolated case where $(\ref{condition Donsker produit l1linfty borne})$ and $(\ref{ condition envelope Donsker})$ are removed from the set of assumptions of Theorem \ref{T2}, but $\FF$ is uniformly bounded, i.e., $F\le M$ for some constant $M>0$. Then a look at (\ref{hao}) immediately yields the claim, noticing that $T(\mbeta_n)\cvproba \infty$. $\Box$
\subsection{Proof of Corollary \ref{ coro:posterior}}\label{subsection preuve posterior}With (\ref{representation de Sethuraman}) in mind, let us define
\beq A:=\aoo (x_n)_{n\geq 1},\; \Mmi \alp_{\xxn}-\mP_0\Mmi_{\FF} \rar 0\aff.\label{definition de A}\eeq
The class of indicators of subsets of a countable set is universally Glivenko--Cantelli - see, e.g., \cite[p. 217, Remark 6.4.3]{Dudley}. Therefore, since $\tht_n\rar 0$, the triangle inequality entails $\mP_0^{\NNN^*}(A)=1$. Now take an arbitrary sequence $(x_n)_{n\geq 1}\in A$. We shall apply Theorem \ref{T1} to the sequence $Post_n\xxn$. In this setup we have $\mP_{n,\bp}=\mP_n=\alp_{\xxn}$ for all $\bp \in \ell^1$, and $$\beta_{i,n}:=V_{i,n}\proli_{j=0}^{i-1}(1-V_{j,n}),\; i\in \NNN,\; n\geq 1,$$
with $(V_{i,n})_{i\in \NNN}\leadsto Beta(1,M+n)^{\otimes \NNN}$. To prove (\ref{condition bracket GC}) let us first remark that if $\llbracket f^-,f^+\rrbracket$ is a bracket between two indicator functions fulfilling $\mmi f^+-f^-\mmi_{\mP_{\xxn},1}\le \e$ then $\mmi f^+-f^-\mmi_{\alp_{\xxn},1}\le \tht_n+(1-\tht_n)\e$. Moreover, since the pointwise supremum/infimum of a set of indicator functions is itself an indicator function, any covering of $\FF$ by brackets can be converted into another covering with the same number of brackets, each of one between two indicator functions. Hence, since $\tht_n\rar 0$, we conclude that it is sufficient to prove that $N_{[]}(\e,\FF,\mP_{\xxn})$ is a bounded sequence for fixed $\e>0$. This is done as follows: let us first choose a finite set $C_0\subset \mfX$ such that $\mP_0(C_0)>1-\e$. Then by definition of $A$ one has $\mP_{\xxn}(C_0)>1-\e$ for all large enough $n$.
\beq\forall C\subset \mfX, C\cap C_0\subset C\subset \po C\cap C_0\pf \cup C_0^c\label{inclusion}.\eeq
Hence the the finite collection
$$ \aoo \llbracket \mathds{1}_{C},\mathds{1}_{C\cup C_0^c}\rrbracket,\; C\subset C_0\aff$$
defines a covering of $2^{\sharp C_0}$ brackets having $\norm_{\mP_{\xxn},1}$ diameters less than $\e$. This proves that
$N_{[]}(\e,\FF,\mP_{\xxn})\le 2^{\sharp C_0}$ for all large $n$, and hence proves (\ref{condition bracket GC}). Now
conditions (\ref{condition envelope GC}) and (\ref{condition GC mbeta_n borne en proba}) are immediate since $\FF$ is uniformly bounded and $\mmi \mbeta_n\mmi_1\equiv 1$ - see, e.g. \cite[p. 112]{HjortLivre}. Finally, standard calculus on beta distributions shows that
$\EEE\po \mmi\mbeta_n\mmi_2^2\pf\sim n^{-1}$, from where one can apply Theorem \ref{T1} and conclude the proof.
\subsection{Proof of Corollary \ref{coro: posterior et bernstein}}\label{sous section preuve Bernstein}
We shall now assume without loss of generality that the support of $\mP_0$ is infinite.
\subsubsection{Two preliminary results}
Theorem \ref{theo Durst} states that the finiteness of $\Sigma_{y\in \mfX}\sqrt{\mP_0(\{y\})}$ is equivalent to that of $J_{[]}(\infty,\FF,\mP_0)$. Our next lemma goes one step further: it shows that it is possible to control the magnitude of $J_{[]}(\dd,\FF,\mP_0)$, for small $\dd>0$, by "tail" sums of the $\sqrt{\mP_0(\{y\})}$.
\begin{lem}\label{lem des sommes de queues}
Define, for $k\in \NNN:$
\beq \mathbf{j}_{\mP_0}(k):=\min\aoo J\in \NNN, \;\sli_{y\in \mfX\;:\; \pl\le 16^{-J} }\pl\le 4^{-k}\aff\label{definition de j(k)}.\eeq
Then, for all $p\geq 1$ we have, for a universal constant $\mathfrak{C}_2$
$$J_{[]}\po 2^{-(p-1)},\FF,\mP_0\pf\le \mathfrak{C}_2 \sqrt{\sli_{y\in \mfX}\sqrt{\mP_0(\{y\})}}\times \sqrt{\sli_{y: \; \pl\le 16^{-\mathbf{j}_{\mP_0}(p)+1}}\sqrt{\mP_0(\{y\})}}.$$
Moreover if the support of $\mP_0$ is infinite we have $\mathbf{j}_{\mP_0}(p)\rar \infty$ as $p\rar \infty$.
\end{lem}
\textbf{Proof }: The very last statement is obvious. We shall now write $\mathbf{j}(\cdot)$ instead of $\mathbf{j}_{\mP_0}(\cdot)$ for concision. The proof consists in enriching the arguments of Dudley \cite[p. 245-246]{Dudley} with additional analytical precisions. We shall hence borrow his notations. First, for $j\in \NNN$ write
\beq \nono A_j:=\aoo y\in \mfX,\; 16^{-j-1}<\mP_0(\{y\})\le 16^{-j}\aff,\text{ and }r_j:=\sharp A_j\label{definition de Aj}.\eeq Now define the following maps on $\NNN$
\begin{align}
\nono&m(\cdot):\; k\rar \sli_{j=0}^{\mathbf{j}(k)}r_j=\sharp \bculi_{j=0}^{\mathbf{j}(k)}A_j,\\
\nono &k(\cdot):\; J\rar \min\aoo p\geq 1,\:4^{-p}< \sli_{y:\; \pl\le 16^{-J}}\pl\aff,\\
\nono &\kappa(\cdot):\; k\rar \min\aoo \kappa \in \NNN,\; \mathbf{j}(\kappa)=\mathbf{j}(k)\aff.
\end{align}
For consistency of notations in the following calculus, we shall also define $k(-1):=0$. Note that, writing $\KK$ for the range of $\kappa(\cdot)$, the map $\mathbf{j}(\cdot)$ is one to one on $\KK$.\lb
Fix $k\geq 1$. Similarly as in Dudley \cite[p. 245-246]{Dudley} we see that, for fixed $k\geq 1$, one can use the same arguments as for (\ref{inclusion}), with the formal replacement of $\e$ by $4^{-k}$ and $ C_0$ by
$$C_k:= \bculi_{j=0}^{\mathbf{j}(k)}A_j,$$
which satisfies $\mP_0(C_k)\geq 1-4^{-k}$ by (\ref{definition de j(k)}). This implies
\beq \forall k\geq 1,\; N_{[]}\po 2^{-k},\FF,\norm_{\mP_0,2}\pf \le 2^{m(k)}.\label{borne crochet mk}\eeq
Now, for any $p\geq 1$, by monotonicity of the involved functions:
\begin{align*}
J_{[]}\po 2^{-(p-1)},\FF,\mP_0\pf
\le& \sli_{k\geq p}\sqrt{\log N_{[]}\po 2^{-k},\FF,\norm_{\mP_0,2}\pf}\po 2^{-(k-1)}-2^{-k}\pf\\
\le& \sqrt{\log (2)}\sli_{k\geq p}\frac{\sqrt{m(k)}}{2^k} \text{ by }(\ref{borne crochet mk}).
\end{align*}
Next, fix $p\geq 1$ and write\begin{align*}
\sli_{k\geq p}\frac{\sqrt{m(k)}}{2^k}\le& \sli_{k\geq p}\sli_{j=0}^{\mathbf{j}(k)}\sqrt{r_j}\;2^{-k}\\
\le &\sli_{k\geq p}\sli_{j=0}^{\mathbf{j}(k)}2\sqrt{\sli_{y\in A_j}\sqrt{\pl}}\;2^{j-k}\text{, since }r_j\le \sum_{y\in A_j}4^{j+1}\sqrt{\pl}\\
=&2\sli_{j\geq 0}\sqrt{\sli_{y\in A_j}\sqrt{\pl}}\mathop{\sli_{k:\;k\geq p,}}_{\mathbf{j}(k)\geq j}2^{j-k}\\
\le& 2\sqrt{\sli_{j\geq 0}\sli_{y\in A_j}\sqrt{\pl}}\times\sqrt{\sli_{j\geq 0}\po \mathop{\sli_{k:\;k\geq p,}}_{\mathbf{j}(k)\geq j}2^{j-k}\pf^2},\text{ using Cauchy--Schwartz}\\
=& 2\sqrt{\sli_{y\in \mfX}\sqrt{\pl}}\times \sqrt{\sli_{j\geq 0}\po \mathop{\sli_{k\geq p,}}_{\mathbf{j}(k)\geq j}2^{j-k}\pf^2}.
\end{align*}
Now we have
\begin{align*}
&\sli_{j\geq 0}\po \mathop{\sli_{k:\;k\geq p,}}_{\mathbf{j}(k)\geq j}2^{j-k}\pf^2\\
\le &4\sli_{j\geq 0}4^{j-k(j-1)}\wedge 4^{j-p},\text{ since } \mathbf{j}(k)\geq j\text{ implies } k\geq k(j-1)\\
=& 4\sli_{k\geq 0}\;\sli_{j: \;\mathbf{j}(k-1)\le j-1<\mathbf{j}(k)}4^{j-k}\wedge 4^{j-p} \text{, since }k(j)=k \text{ for }\mathbf{j}(k-1)\le j< \mathbf{j}(k)\\
=&4\sli_{k\le p} \;\sli_{j:\;\mathbf{j}(k-1)\le j-1< \mathbf{j}(k)}4^{j-p}+4\sli_{k\geq p+1} \;\sli_{j:\;\mathbf{j}(k-1)\le j-1< \mathbf{j}(k)}4^{j-k}\\
\le &8\coo4^{\mathbf{j}(p)-p}+\sli_{k\geq p+1}4^{\mathbf{j}(k)-k}\cff\\
=& 8\sli_{k\geq p}4^{\mathbf{j}(k)-k}\\
\le &8\sli_{k\geq p}4^{\mathbf{j}(k)-k+\kappa(k)}\times \sli_{\ell\;:\;\pl\le 16^{-\mathbf{j}(k)+1}}\pl,\text{ by }(\ref{definition de j(k)})\text{ and since }\mathbf{j}(k)=\mathbf{j}(\kappa(k))\\
= &8\sli_{k\geq p}4^{\mathbf{j}(k)-k+\kappa(k)}\times \sli_{j\geq \mathbf{j}(k)-1}\;\sli_{\ell\;\in A_j}\pl\\
\le & 8\sli_{j\geq 0}\poo \sli_{y\in A_j}\pl\pff\poo\sli_{k:\;k\geq p,\; \mathbf{j}(k)-1\le j} 4^{\mathbf{j}(k)-k+\kappa(k)}\pff.
\end{align*}
Now notice that, when $\mathbf{j}(p)>j+1$ the set of indices $\{k\geq p,\; \mathbf{j}(k)-1\le j\}$ is empty, from where\begin{align*}
&\sli_{j\geq 0} \poo\sli_{y\in A_j}\pl\pff \poo \sli_{k:\;k\geq p,\; \mathbf{j}(k)-1\le j} 4^{\mathbf{j}(k)-k+\kappa(k)}\pff\\
\le & \sli_{j\geq \mathbf{j}(p)-1}\poo \sli_{y\in A_j}\pl\pff \poo \sli_{k:\; \mathbf{j}(k)-1\le j} 4^{\mathbf{j}(k)+\kappa(k)-k}\pff\\
\le& \sli_{j\geq \mathbf{j}(p)-1} \poo \sli_{y\in A_j}\pl \pff \poo \sli_{k'\in \KK,\; \mathbf{j}(k')\le j+1}4^{\mathbf{j}(k')+k'}\sli_{k:\; \kappa(k)=k'}4^{-k}\pff\\
\le&4\sli_{j\geq \mathbf{j}(p)-1} \poo \sli_{y\in A_j}\pl\pff \poo\sli_{k'\in \KK,\; \mathbf{j}(k')\le j+1}4^{\mathbf{j}(k')}\pff\text{, since }\kappa(k)=k'\text{ implies } k\geq k'\\
\le &32\sli_{j\geq \mathbf{j}(p)-1} \sli_{y\in A_j}\pl 4^j,\text{ since }\kappa(\cdot)\text{ is one to one on }\KK\\
\le &32 \sli_{j\geq \mathbf{j}(p)-1}\sli_{y\in A_j}\sqrt{\pl},\text{ since }y\in A_j\text{ implies }\pl 4^j\le \sqrt{\pl}\\
=&32\sli_{y: \; \pl\le 16^{-\mathbf{j}(p)+1}}\sqrt{\pl}.\end{align*}
This concludes the proof.$\Box$\lb
Our second preliminary result is as follows.
\begin{lem}\label{lem application du multiplier CLT}
Write
$$I_{\e}:=\ao y\in \mfX,\; \pl \le \e\af,\e\in \QQQ^+.$$
Then for $\mP_0^{\otimes \NNN^*}$-almost any sequence $(x_n)_{n\geq 1}$ we have:
\beq \forall \e\in \QQQ^+,\; \limn \sli_{y\in I_\e}\sqrt{\mP_{\xxn}(\{y\})}= \sli_{y\in I_\e}\sqrt{\mP_0(\{y\})}.\eeq
\end{lem}
\textbf{Proof:} Since the class $\FF$ is $\mP_0$-Donsker and admits a square integrable envelope ($F\equiv1$), the conditional multiplier Donsker theorem applies for a suitable i.i.d. standard normal sequence $(\xi_n)_{n\geq 1}$ - see, e.g., \cite[p. 183, Theorem 2.9.7]{Vander}. Hence for $\mP_0^{\NNN^*}$-almost every sequence $(x_n)_{n\geq 1}$ we have - recalling that $\WW_{\mP_0}$ stands for the $L^2(\mP_0)$-isonormal Gaussian process indexed by $\FF$:
\begin{align}
\pooo \WW_{\xxn}(f)\pfff_{f\in \FF} \cvloi \poo \WW_{\mP_0}(f)\pff_{f\in \FF},\label{conditional CLT}\text{ where }\\
\nono\WW_{\xxn}(f):=\frac{1}{\sqrt{n}}\sliin \xi_if(x_i),\; f\in \FF.
\end{align}
Here the weak convergence holds in the sense of Hoffman-J\H{o}rgensen holds taking the underlying probability space as the canonical product space for $(\xi_n)_{n\geq 1}$. Moreover the involved processes are Gaussian, hence weak convergence implies convergence of first moments of absolute suprema. As a consequence, for such a sequence $(x_n)_{n\geq 1}$ fulfilling (\ref{conditional CLT}) we have, for all $\e\in \QQQ^+$
$$\EEE\poooo2\sup_{A\subset I_\e }\frac{1}{\sqrt{n}}\Mid \sliin \xi_i\mathds{1}_{A}(x_i)\Mid- \frac{1}{\sqrt{n}}\Mid \sliin \xi_i\mathds{1}_{I_\e}(x_i)\Mid\pffff\rar \EEE\poooo2\sup_{A\subset I_\e } \Mid \WW_{\mP_0}\po \mathds{1}_A\pf\Mid-\Mid \WW_{\mP_0}\po\mathds{1}_{I_\e}\pf\Mid\pffff.$$
Finally, by the standard equality
$$\sup_{A\subset I_{\e}}\Mid \sli_{y\in A}g(y)\Mid =\frac{1}{2}\poo \sli_{y\in I_{\e}}\mid g(y)\mid+\Mid \sli_{y\in I_{\e}}g(y)\Mid\pff,$$
we have (with $g(y):=n^{-1/2}\sliin \xi_i\mathds{1}_{\{y\}}(x_i)$)
\begin{align*}
&\EEE\poooo2\sup_{A\subset I_\e }\frac{1}{\sqrt{n}}\Mid \sliin \xi_i\mathds{1}_{A}(x_i)\Mid- \frac{1}{\sqrt{n}}\Mid \sliin \xi_i\mathds{1}_{I_\e}(x_i)\Mid\pffff\\
=&
\EEE\poooo \sli_{y\in I_\e}\Mid \frac{1}{\sqrt{n}} \sliin \xi_i\mathds{1}_{\{y\}}(x_i)\Mid
\pffff\\
=&\sli_{y\in I_{\e}}\sqrt{\mP_{\xxn}(\{y\})},
\end{align*}
and similarly (with now $g(y):=\WW_{\mP_0}(\mathds{1}_{\{y\}})$)
\begin{align*}
\EEE\pooo 2\sup_{A\subset I_\e } \Mid \WW_{\mP_0}\po \mathds{1}_A\pf\Mid-\Mid \WW_{\mP_0}\po\mathds{1}_{I_\e}\pf\Mid\pfff=\sli_{y\in I_\e}\sqrt{\mP_0(\{y\})}\,
\end{align*}
which concludes the proof.$\Box$
\subsubsection{Use of Theorem \ref{T2} }
Recall that $A$ was defined in (\ref{definition de A}) and has probability one.
Let us consider the set
\begin{align*}
B:=& \aooo (x_n)_{n\geq 1},\; \forall \e\in \QQQ^+,\; \lsn \sli_{y:\; \alp_{\xxn}(\{y\})\le \e}\sqrt{\alp_{\xxn}(\{y\})}\le \sli_{y\in I_{2\e}}\sqrt{\mP_0(\{y\})}\afff.
\end{align*}
We have, for any $n\geq 1$ and $\e\in \QQQ^+$ (using $\sqrt{a+b}\le \sqrt{a}+\sqrt{b}$)
\begin{align*}
&\sli_{y:\; \alp_{\xxn}(\{y\})\le \e}\sqrt{\alp_{\xxn}(\{y\})}\\
\le& \sqrt{\tht_n}\sli_{y:\; \alp_{\xxn}(\{y\})\le \e}\sqrt{\alp(\{y\})}+\sqrt{1-\tht_n}\sli_{y:\; \alp_{\xxn}(\{y\})\le \e}\sqrt{\mP_{\xxn}(\{y\})}.
\end{align*}
Now if $(x_n)_{n\geq 1}$ belongs to $A$ and since $\FF$ induces the total variation distance we have, for all $n$ large enough
$$\ao y\in \mfX,\; \alp_{\xxn}(\{y\})\le \e\af\subset \ao y\in \mfX,\; \mP_0(\{y\})\le 2\e\af=I_{2\e}.$$
We hence conclude that $\mP_0^{\NNN^*}(A\cap B)=1$ - recalling that $\tht_n\rar 0$ and $(DDB(\alp))$ holds. Let us now consider a sequence $(x_n)_{n\geq 1}\in A\cap B$. Similarly as in \S \ref{subsection preuve posterior}, we shall prove Corollary \ref{coro: posterior et bernstein} by verifying all the assumptions of Theorem \ref{T2}, for the choice of $\mPnp=\mP_n:=\mP_{\xxn}$.
Because the class $\FF$ is uniformly bounded by $1$, the conditions upon
$$\beta_{i,n}:=\sqrt{n}V_{i,n}\proli_{j=0}^{i-1}(1-V_{j,n}),\; i\in \NNN,$$
that we need to check are (\ref{ condition Donsker norme L2 egale a 1}) and $(\ref{condition Donsker linfty tend vers 0})$, or equivalently
$$\mmi \mbeta_n\mmi_2\cvproba 1,\text{ and }\mmi \mbeta_n\mmi_4\cvproba 0.$$
These are respectively proved by direct computations of expectations and variances. It now remains to verify (\ref{condition Donsker sur l'entropie crochet}) and (\ref{ condition sur la semimetrique rho}).
By definition of $A$ and since $\tht_n\rar0$, the sequence $\mP_n$ obviously fulfills (\ref{SW}) and therefore satisfies (\ref{ condition sur la semimetrique rho}). Now in view of Lemma \ref{lem des sommes de queues}, assertion (\ref{condition Donsker sur l'entropie crochet}) will be proved if we show that
\beq\lim_{p\rar \infty}\; \lsn \sli_{y: \;\alp_{\xxn}(\{y\})\le 16^{-\mathbf{j}_{\mP_n}(p)+1}}\sqrt{\mP_n(\{y\})}=0.\label{a prouver}\eeq
\begin{lem}\label{lem1}
Take $(x_n)_{n\geq 1}\in A$. For any $p\geq 1$ we have $\mathbf{j}_{\mP_n}(p)\geq \mathbf{j}_{\mP_0}(p)$ for all large enough $n$.
\end{lem}
\textbf{Proof}: Fix $p\geq 1$. By definition of $\mathbf{j}_{\mP_0}$ we have
$$\sli_{y:\; \mP_0(\{y\})\le 16^{-\mathbf{j}_{\mP_0}(p)+1}}\mP_0(\{y\})> 4^{-p}.
$$
Now since $(x_n)_{n\geq 1}\in A$, we have, for all $n$ large enough:
$$\sli_{y:\; \mP_0(\{y\})\le 16^{-\mathbf{j}_{\mP_0}(p)+1}}\mP_n(\{y\})> 4^{-p},
$$
whence $\mathbf{j}_{\mP_0}(p)-1\le \mathbf{j}_{\mP_n}(p)-1$ by definition of $\mathbf{j}_{\mP_n}$. $\Box$\lb
Now applying Lemma \ref{lem1} we have, writing $\e(p):=16^{-\mathbf{j}_{\mP_0}(p)+1}$:
\begin{align*}
&\lim_{p\rar \infty} \lsn\;\; \sli_{y:\; \;\alp_{\xxn}(\{y\})\le 16^{-\mathbf{j}_{\mP_n}(p)+1}}\sqrt{\mP_n(\{y\})}\\
\le &\lim_{p\rar \infty} \lsn \;\;\sli_{y:\; \;\alp_{\xxn}(\{y\})\le \e(p)}\sqrt{\mP_n(\{y\})}\\
\le &\lim_{p\rar \infty} \lsn \;\;\sli_{y\in I_{2\e(p)}}\sqrt{\mP_0(\{y\})},\text{ since }(x_n)_{n\geq 1}\in B\\
=&0,
\end{align*}
by $(DDB(\mP_0))$ together with $\lim \mathbf{j}_{\mP_0}(p)\rar \infty$. This proves (\ref{a prouver}) and we can now apply Theorem \ref{T2} to obtain
$$d_{BL}\poooo\sqrt{n}\poo Post_n\xxn-\alp_{\xxn}\pff,\GGG_{\alp_{\xxn}}\pffff\rar 0.$$
But since $(x_n)_{n\geq 1}\in A$ the sequence $\mP_n:=\alp_{\xxn}$ satisfies (\ref{SW}) from where (see \cite[Remark 2.2]{Varron14Donsker}):
$$\GGG_{\alp_{\xxn}}\cvloi \GGG_{\mP_0}\text{, in }\ell^{\infty}(\FF),$$
which concludes the proof of Corollary \ref{coro: posterior et bernstein}. $\Box$
\subsection{Proof of Corollary \ref{coro Donsker local}}\label{sous section preuve de Donsker local}
Recall that $\mP_0$ denotes here the uniform distribution on $S$ and that $\mP_n$ has been defined in (\ref{definition Pn local}). Let $\VV$ be a neighborhood of $z$ on which $Z_1$ admits the density $\mf$. Since $S$ is bounded and $h_n\rar 0$ we have $z+h_nS\subset \VV$ for $n$ large enough. We may assume without loss of generality that this is the case for all $n\geq 1$.
\begin{lem}\label{lem convergence du PE randomise pour le PE local}
We have (taking here the convention $0/0=0$)
$$\poooo\sliin \frac{b_{i,n}}{\sqrt{\sliin b_{i,n}^2}}\poo f(Y_{i,n})-\mP_n(f)\pff\pffff_{f\in \FF}\cvloi \GGG_{\mP_0},$$
where $\GGG_{\mP_0}$ denotes the $\mP_0$ Brownian bridge.
\end{lem}
\textbf{Proof}:
Write $$\beta_{i,n}:=\frac{b_{i,n}}{\sqrt{\sliin b_{i,n}^2}},\text{ for }i=1,\ldots,n. $$ Since $\mf$ is continuous at $z$ we have
\begin{align}
& a_n\sim \lab(S)\mf(z)h_n^d\label{an equivalent de f(z)hn^dlab(S)},\text{ from where }na_n\rar \infty\text{ and }a_n\rar 0.
\end{align}
This property ensures that the sequence $\mbeta_n$ satisfies (\ref{ condition Donsker norme L2 egale a 1}), (\ref{condition Donsker linfty tend vers 0}) and (\ref{condition Donsker produit l1linfty borne}) of Theorem 2 - taking $p:=2$ and recalling that $b_{i,n}\equiv b_{i,n}^2$. In order to verify (\ref{condition Donsker sur l'entropie crochet}) and (\ref{condition Donsker sur mfa(delta)}) we will now prove (\ref{condition de domination}), noting here that $\mP_{n,\bp}:=\mP_n$ for all $\bp\in \ell^{1,+}$. The usual change of variable $u=h_n^{-1}(v-z)$ in the next integrals gives, for an arbitrary non negative function $g$ with support included in $S$
\begin{align}
\nono \mP_n(g)=& \frac{1}{a_n}\ili_{z+h_nS}g\poo h_n^{-1}(v-z)\pff \mf(v) dv\\
\nono =&\frac{h_n^d}{a_n}\ili_{S}g(u)\mf(z+h_nu)du\\
\nono \le & \sup_{u\in S}\mf(z+h_n u)\;\frac{h_n^d}{a_n}\ili_{S} g(u)du\\
\nono = & \sup_{u\in S}\mf(z+h_n u)\;\frac{h_n^d}{a_n}\lab(S)\mP_0(g).
\end{align}
This proves (\ref{condition de domination}) by applying that inequality to elements of the form $(f_1-f_2)^2$, $(f_1,f_2)\in \FF^2$ and recalling (\ref{an equivalent de f(z)hn^dlab(S)}) together with the continuity of $\mf$ at $z$. This also proves (\ref{ condition envelope Donsker}), taking $g:=F^2\indic_{\{F> M\}}$. Let us now verify (\ref{ condition sur la semimetrique rho}) by proving (\ref{SW}).
Using a calculus similar as above we have, for an arbitrary function $g\prec (2F)^2\vee (2F)$ \begin{align}
&\nono \mf(z)\lab(S)\Mid \mP_n(g)-\mP_0(g)\Mid\\
\nono =& \Mid \frac{\mf(z)\lab(S)h_n^d}{a_n} \ili_S g(u)\mf(z+h_nu) du-\ili_{S} g(u)\mf(z) du\Mid\\
\nono \le&\frac{\mf(z)\lab(S)h_n^d}{a_n}\times \Mid \ili_S g(u)\mf(z+h_nu) du-\mf(z)\ili_{S} g(u)du \Mid\\
\nono &\;\; +\Mid \frac{\mf(z)\lab(S)h_n^d}{a_n}-1\Mid \times \mf(z)\ili_{S} \mid g (u)\mid du \\
\nono \le & \frac{\mf(z)\lab(S)h_n^d}{a_n}\times \sup_{v\in S} \mid \mf(z+h_nv)-\mf(z)\mid\times \ili_S (2F)^2\vee (2F) du\\
&\;\; + \Mid \frac{\mf(z)\lab(S)h_n^d}{a_n}-1\Mid \times \mf(z)\ili_{S} (2F)^2\vee (2F) du\label{ borne 2},
\end{align}
which tends to zero independently of $g\prec (2F)^2\vee (2F)$. This proves $(\ref{SW})$ and concludes the proof of Lemma \ref{lem convergence du PE randomise pour le PE local}.$\Box$ \lb
Let us now continue the proof of Corollary \ref{coro Donsker local}. First, note that
we have
\begin{align}
&\sliin b_{i,n}^2\sim\lab(S)\mf(z)nh_n^d \text{ in probability, from where}\\
& \poooo\frac{1}{\sqrt{\mf(z)\lab(S)nh_n^d}}\sliin \beta_{i,n}\poo f(Y_{i,n})-\EEE\po f(Y_{i,n})\pf\pff\pffff_{f\in \FF}\cvloi \GGG_{\mP_0},
\end{align}
and hence that sequence of processes is asymptotically tight (see, e.g., \cite[p. 20, Definition 1.3.7]{Vander}). Now elementary probability calculus shows that
\beq\frac{\sliin (b_{i,n}-a_n)}{\sqrt{\mf(z)\lab(S)nh_n^d}}\cvloi Z,\label{normalite asymptotique des bin}\eeq
where $Z$ is standard normal. Moreover, since $\mP_n$ satisfies (\ref{SW}) and since
$f\rar \mP_0(f)$ is continuous with respect to $\norm_{\mP_0,2}$, which makes $\FF$ totally bounded, the (deterministic) sequence $\mP_n(\cdot)$ is relatively compact in $\ell^{\infty}(\FF)$. This, combined with (\ref{normalite asymptotique des bin}), implies that the sequence $R_n(\cdot)$ - defined in (\ref{definition de R_n(f)}) - is asymptotically tight, and hence so is $T_{n,h_n}(\cdot)$ by summation. It will hence be proved to converge to $\WW_{\mP_0}$ if we prove finite marginal convergences. This is done by elementary analysis of characteristic functions, using the change of variable $u=h_n^{-1}(v-z)$ in the integrals. We omit details. $\Box$
\section{Appendix: a minor proof}\label{sous section preuve de la mesurabilite}
In this section we prove the measurability properties claimed in \S \ref{section: results}.
\begin{lem}\label{lem de mesurabilite de brackets}
For fixed $r\geq 1$ and $n\geq 1$, the map $(\e,\bp)\rar N_{[]}\poo \e,\FF,\norm_{\mP_{n,\bp},r}\pff$ is Borel from $]0,\infty[\times \ell^1$ to $\RRR^+$.
As a consequence, the maps
$$\bp \rar J_{[]}\po \dd,\FF,\mP_{n,\bp}\pf,\; \dd>0,$$
are Borel.
\end{lem}
\textbf{Proof}: Fix $r\geq 1$ and $n\geq 1$. Any bracket is closed for the the pointwise topology, i.e., the topology spanned by the evaluation maps $\ao \{f\rar f(y)\},\; y\in \mfX\af$. Hence so is any finite union of brackets that covers $\FF_0$. Since $\FF$ is included in the closure of $\FF_0$ for the pointwise topology, we deduce that
\beq \forall (\e,\bp)\in ]0,\infty[\times \ell^1,\; N_{[]}\poo\e,\FF,\norm_{\mP_{n,\bp},r}\pff=N_{[]}\poo\e,\FF_0,\norm_{\mP_{n,\bp},r}\pff.\label{egalite bracket FF et FF0}\eeq
Now the proof of Lemma \ref{lem de mesurabilite de brackets} boils down to proving the measurability of
$$H:\; (\e,\bp)\rar N_{[]}\poo\e,\FF_0,\norm_{\mP_{n,\bp},r}\pff.$$
This is done by noting that, for any $K\in \NNN$, the set
\beq B_K:=\aoo (f_j^-,f_j^+)_{j=1,\ldots,K}\in {(\FF_0^2)}^K,\; \FF_0\subset \bculi_{j=1}^K \llbracket f_j^-,f_j^+\rrbracket\aff\nono \eeq
is countable, and that
\begin{align*}
H(\e,\bp)>K\;\;\Leftrightarrow\;\; \forall (f_j^-,f_j^+)_{j=1,\ldots,K}\in B_K,\; \exists j\in \{ 1,\ldots,K\},\; \mmi f_j^+-f_j^-\mmi_{\mP_{n,\bp},r}>\e,
\end{align*}
which yields the claimed result, since for fixed Borel non negative $g$, the map $\bp\rar \mmi g\mmi_{\mP_{n,\bp},r}$ is Borel (recall that $\{\mP_{n,\bp},\;\bp\in \ell^1\}$ is regular).$\Box$
\bibliographystyle{plain}
\bibliography{biblioAM-new,biblioNZ-new}
\end{document}
| 113,650
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\begin{document}
\maketitle
\begin{abstract}
A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads $q$ bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter $q$ is called the locality of the tester.
LTCs were initially studied as important components of PCPs, and since then the topic has evolved on its own. High rate LTCs could be useful in practice: before attempting to decode a received word, one can save time by first quickly testing if it is close to the code.
An outstanding open question has been whether there exist ``$c^3$-LTCs'', namely LTCs with \textbf{c}onstant rate, \textbf{c}onstant distance, and \textbf{c}onstant locality.
In this work we construct such codes based on a new two-dimensional complex which we call a left-right Cayley complex. This is essentially a graph which, in addition to vertices and edges, also has squares. Our codes can be viewed as a two-dimensional version of (the one-dimensional) expander codes, where the codewords are functions on the squares rather than on the edges.
\end{abstract}
\section{Introduction}
A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads $q$ bits (randomly - but not necessarily uniformly - chosen) from a given word, and rejects words with probability proportional to their distance from the code. The parameter $q$ is called the locality of the tester.
A random code has, with high probability, constant rate and distance, but locality that is proportional to the length. This is true even for random LDPC codes \cite{Ben-SassonHR05}, and a priori the mere existence of codes with constant locality is not obvious. The first LTCs appear implicitly in works on program checking \cite{BLR} and on probabilistically checkable proofs (PCPs) \cite{BFL,LFKN, BFLS,AS,ALMSS}. A formal definition of an LTC appeared simultaneously in several places \cite{BFLS,RuSu96,FriedlS13, Arora-thesis} (see \cite{Goldreich2010} for a detailed history).
Spielman, in his PhD thesis \cite{Spielman96}, discusses the possibility of having an error-correcting code that is locally testable (he uses the term `checkable code') and explains its potential applicability: {\em ``A checker would be able to read only a constant number of bits of a received signal and then estimate the chance that a decoder will be able to correct the errors, then the checker can instantly request a retransmission of that block, before the decoder has wasted its time trying to decode the message.
Unfortunately all known codes with local-checkers have rate approaching zero."}
Goldreich and Sudan \cite{GolSud06} initiated a systematic study of LTCs as objects of interest in their own right. Over the years better and better LTCs were constructed \cite{PoliSpiel94, GolSud06, BenSassonSuVaWi03, BGHSV, BenSasson-Sudan05, Din07, KMRS17, GKORS18}, but, nevertheless, experts went back and forth on whether ``$c^3$-LTCs'' (namely, LTCs with {\bf c}onstant rate, {\bf c}onstant distance, and {\bf c}onstant locality) are likely to exist, compare \cite{Goldreich-LTCsurveyOG} with \cite[Section 3.3.2]{Goldreich2010}.
We construct the first such family of LTCs,
\begin{theorem}\label{thm:main}
For all $0<r<1$, there exist $\delta,\kappa>0$ and $q\in \mathbb{N}$ and a polynomial-time construction of an infinite family of error-correcting codes $\set{C_n}$ with rate $r$ and distance $\delta$, such that for all $n$, $C_n$ is $\kappa$-locally testable with $q$ queries.
Namely, every code $C_n$ comes with a randomized local tester that reads at most $q$ bits from a given word $w$ and then accepts or rejects, such that
\begin{itemize}
\item For all $w\in C_n$, $\Pr[\hbox{accept}]=1.$
\item For all $w\not\in C_n$, $\Pr[\hbox{reject}] \geq \kappa\cdot \dist(w,C_n)$.
\end{itemize}
\end{theorem}
We remark that \cite{KMRS17,GKORS18} have shown (see \cite[Section 1.2]{GKORS18}) how to take an LTC with rate arbitrarily close to $1$ and with constant distance, and construct a new LTC with rate and distance approaching the Gilbert-Varshamov bound, and only a constant overhead in the locality $q$. So the theorem above holds for all $r,\delta>0$ that satisfy $r+h(\delta)<1$ where $h(\cdot)$ is the binary entropy function.
\subsection*{Expander codes, one dimension up}
The celebrated expander-codes of Sipser and Spielman \cite{SipserSp96} are a family of error-correcting codes constructed from a single base code $C_0\subseteq \bits^d$ and a family of $d$-regular expander graphs $G_n=(V_n,E_n)$ such that the code corresponding to $G_n$ consists of functions on $E_n$ such that for every vertex in $V_n$, the local view from the neighboring edges (assuming some arbitrary fixed ordering) is itself in the base code $C_0$,
\[ C = \sett{f:E_n\to\bits}{\forall v\in V_n, f|_{edges(v)}\in C_0}.
\]
Similarly, our codes will also be defined via a fixed base-code and an infinite family of expander graphs. Our graphs will have, in addition to vertices and edges, also two-dimensional faces, called squares, where each square touches four edges and four vertices.
Our codewords are functions {\em on the squares} such that for every edge, the bits on the neighboring squares form a codeword in the base code. It is natural to view our code as a Tanner code \cite{Tanner81} with bits on the squares and constraints on the edges; whereas the expander-codes have bits on the edges and constraints on the vertices.
Inspecting our code on the set of squares neighboring a fixed vertex, we see an intermediate code, whose constraints come from the edges neighboring that vertex.
We thus have three codes for the three dimensions of links: the base code $C_1$ at the link of an edge, the intermediate code $C_0$ at the link of a vertex, and the global code $C$ at the link of the empty face which is the set of all squares.
\paragraph{Left-Right Cayley Complex}
Let us describe our construction of a graph-with-squares, namely a square complex (for a more formal description see Definition \ref{def:LRC}). Let $G$ be a finite group with two sets of generators $A,B$. We define the left-right Cayley complex
$X = Cay^2(A, G,B)$ as follows
\begin{itemize}
\item The vertices are $X(0)=G$.
\item The edges are $X(1) = X^A(1)\sqcup X^B(1)$ where
\[X^A(1) = \sett{\set{g,ag}}{g\in G, a\in A}, \qquad X^B(1) = \sett{\set{g,gb}}{g\in
G, b\in B}. \]
\end{itemize}
The fact that with $A$ we multiply on the left, and with $B$ we multiply on the right, gives a local commutativity which generates many four-cycles, namely, squares.
Indeed for every $a,g,b$ the graph has a cycle of length $4$ with alternating $A$ and $B$ edges, given by the walk $g,gb,agb,ag,g$. We place a square for each of these four-cycles.
\begin{itemize}
\item The squares are a set of the following four-cycles in the graph,
\[ X(2) = \sett{(g,gb,agb,ag,g)}{g\in G,a\in A,b\in B}.\]
We denote by $[a,g,b]$ the square containing the edges $\set{g,ag}$ and $\set{g,gb}$. By changing the `root' of the square we get $[a,g,b]=[a^{-1},ag,b] = [a^{-1},ab,b^{-1}]= [a,gb,b^{-1}]$.
\end{itemize}
\paragraph{The Code}
Fix a left-right Cayley complex $X=Cay^2(A,G,B)$, and fix a pair of base codes $C_A\subseteq \bits^A$ and $C_B\subseteq \bits^B$ (assuming $|A|=|B|=d$ we can take both to be isomorphic to some $C_1\subseteq\bits^d$). Our code is defined to be
\[
C[A,G,B,C_A,C_B] = \sett{f:X(2)\to \bits}{ \forall a,g,b,\; f([\cdot,g,b])\in C_A,\hbox{ and } f([a,g,\cdot]) \in C_B}.
\]
Observe that for a codeword $f\in C$ and a fixed vertex $g\in G$, the restriction of $f$ to the squares touching $g$ is $f([\cdot,g,\cdot])$. It is not difficult to check that this word necessarily belongs to the tensor code $C_A\otimes C_B$, see Lemma \ref{lem:tensor}. Thus, by putting the constraints around each edge, we get an intermediate code on the squares touching a vertex, which turns out to be a tensor code! Tensor codes have non-trivial dependencies among the constraints defining them. This often implies local testability of tensor codes \cite{BenSasson-Sudan-tensors, DSW06, BVidweakly}, and turns out important for showing that our code can be locally tested by the following simple test:\\
\textbf{Local test:} Choose a random vertex $g$, and accept iff $f([\cdot,g,\cdot])\in C_A\otimes C_B$.\\
\noindent We discuss below the type of local to global propagation that goes into proving that this test works.
Finally, to complete our construction of locally testable codes and to prove Theorem \ref{thm:main}, we describe in Section \ref{sec:LRCC} an explicit construction of a family of groups and pairs of generating sets which give good left-right Cayley complexes, and in Section \ref{sec:ins} a matching choice of base codes $C_A,C_B$.
\paragraph{Propagation from local to global}
Sipser and Spielman proved distance of their expander codes \cite{SipserSp96} through propagation: expansion of the underlying graph is used to ``lift'' the distance of the base code to the distance of the global code.
In our codes distance is shown via a similar argument.
More interestingly, local testability of our codes is also shown through expansion.
We show that if a received word violates only a small amount of constraints, then locally it can be corrected, as long as the intermediate code $C_A\otimes C_B$ is itself {\em robustly locally testable}. We describe an iterative decoding algorithm (Algorithm \ref{alg}) and prove that it converges thanks to sufficient expansion of certain edge-to-edge random walks on our square complex.
Conceptually, local local-testability (of the intermediate code $C_A\otimes C_B$), implies global local-testability (of the entire code), through expansion.
The existence of many dependencies among the constraints defining our codes is the point where our codes most clearly differ from expander codes: in expander codes one can have a single violated constraint that does not propagate, and leads to a word that is far from the code but no tester can detect it, as proven in \cite{Ben-SassonHR05}.
\subsection*{Locally Testable Codes: historical background and techniques}
As mentioned earlier, the study of LTCs arose naturally in works on program checking and PCPs. The Hadamard code was the first code proven to be locally testable in the work of Blum, Luby, and Rubinfeld on linearity testing \cite{BLR}. The low (logarithmic) rate of this code was quickly improved to polynomial rate by moving from linear functions (codewords of the Hadamard code) to low degree polynomial functions (codewords of the Reed Muller code). Subsequent works studied ``low degree tests'' which are in fact proofs that the Reed-Muller code is locally testable. These works were crucial for progress leading up to the proof of the PCP theorem. More on the relation between PCPs and LTCs, as well as the historical development, can be found in Goldreich's survey \cite{Goldreich2010}.
A systematic study of LTCs was initiated by Goldreich and Sudan in \cite{GolSud06}, and a sequence of works constructed both LTCs and PCPs with improved parameters \cite{GolSud06,BenSassonSuVaWi03, BGHSV,Ben-SassonS08,Din07}, achieving constant locality and distance, but rate $1/\poly\log n$. Some experts believed that low rate is inherently needed and some attempts to prove upper bounds on the rate have been made \cite{Ben-SassonGKSV10, DinurK11,ben2012towards, BabaiSS05}, although these lower bounds are in rather restrictive models.
This, perhaps, has triggered works from the other end of the spectrum \cite{KMRS17,GKORS18} which focused on constructing error correcting codes with constant rate and distance, that are locally testable with smallest possible locality. These works achieve constant rate and quasi-poly-logarithmic distance and locality.
In terms of techniques, many of the earlier constructions of LTCs have two notable features. Firstly, they are based on the properties of low degree polynomials, and secondly, they come hand in hand with a PCP constructions, so that both share the same composition-recursion structure.
The gap amplification technique \cite{Din07} of the first author is a construction of both a PCP and an LTC that relies on expander graphs and concatenation and departs from the domain of low degree polynomials. Meir \cite{Meir08} gave a tensor-code-based construction of LTCs that is neither related to low degree functions nor to PCPs altogether. Further works \cite{KMRS17,GKORS18} also construct LTCs without any PCP counterpart.
A feature shared by all previous constructions of LTCs with mildly high rate is their recursive nature. One first constructs codes with weaker properties and then enhances them by concatenation, possibly with different iterations. The overall composed structure of the code is somewhat complicated and begs for a more direct ``one-shot'' construction.
A path leading towards a one-shot construction seemed to open up with the connection to high dimensional expanders.
\subsection*{High Dimensional Expansion}
The current paper is mainly elementary and almost self-contained (with the exception of Section \ref{sec:LRCC} which uses the existence of some Ramanujan Cayley graphs with specific properties and can be taken as a black box). But it came up as a result of a much longer and intensive journey. Some interesting open problems were left aside along the way. It is, therefore, worthwhile to give the story here.
The journey started by the first and fourth authors during a year-long program at the IIAS (Israeli Institute
of Advanced Studies) on high dimensional expanders in 2017: the hope was to use the Ramanujan complexes (a la \cite{LSV1,LSV2}) to construct LTCs as high-dimensional versions of expander-codes over Ramanujan graphs as explained above.
Although expander codes are typically not locally testable \cite{Ben-SassonHR05} the hope was that higher dimensional versions would be.
This optimistic belief was inspired by local to global behavior of certain high dimensional complexes that was uncovered already by Garland in his seminal work \cite{Garland}.
In that paper, Garland proved a conjecture of Serre, that the cohomology of co-compact lattices in high-rank simple $p$-adic groups vanishes. Equivalently, if $X$ is a finite simplicial quotient of a Bruhat-Tits building of dimension at least two, its cohomology vanishes. The proof of Garland is ``local-to-global'': he showed that if the links of relevant cells have a spectral gap, then so does the global Laplacian of $X$. Namely, if $X$ is locally an expander, then it is also globally so. (For a purely combinatorial treatment and generalizations - see \cite{Oppenheim18}). The global spectral gap implies the vanishing of the cohomology.
This ``local to global'' approach is a high-dimensional phenomenon that does not hold for graphs! In graphs, the local structure does not reveal any information about the global expansion. To illustrate this, the reader may recall the LPS-Ramanujan graphs \cite{LPS} which are (p+1)-regular expander graphs with large girth. One can easily get (p+1)-regular graphs with large girth (and hence locally isomorphic to the LPS ones) which are far from being expanders. In contrast, the Garland method shows that local expansion implies global expansion in the high dimensional case.
The local to global approach was also the key ingredient, in \cite{KKL14,EvraK16} where Gromov's overlapping problem was solved using the Ramanjaun complexes.
At this point there was already some interest from the theoretical computer science community. The fact that high dimensional expansion is related to property testing in computer science was observed for the first time in \cite{KaufmanL14}. The first author and Kaufman proved that high dimensional expansion implies an efficient agreement-test \cite{DK17}, which is related to both PCPs and LTCs. Anari et al \cite{anari2019log} resolved a conjecture regarding convergence of certain Markov chains by analyzing the global random walk through local analysis at the links.
Inspired by all this, the idea was to construct LTC codes by using the local-to-global behavior of the Ramanujan complexes in an analog to the way \cite{SipserSp96} used Ramanujan graphs for LDPC codes. For simplicity, we will describe it from now on only in dimension $2$, but one can do the same in higher dimensions.
\remove{As described above, we choose a base code for the highest links, say repeats the above: the SS code can be thought of as being the subspace of the functions on the edges of a Ramanujan graph whose "local view" at the link of every vertex is in a "small code". If the small code has a good rate and distance the Ramanujan property of the graph enables the propagation of these properties for the "big code". In fact, propagating the rate does not even need the spectral gap.}
The original idea was as follows: fix a large prime $p$ and take an infinite family of Ramanujan complexes $X$, quotients of the Bruhat-Tits building of $G=SL(3,\mathbb{Q}_p)$. The complex $X$ is a $2$-dimensional complex, the link of every edge of it is in one-to-one correspondence with the projective line $\mathbb{P}^1$ over $\F_p$ and the link of every vertex is the graph of lines versus points of the projective plane over $\F_p$. One can define a base code (``the small code'') $C_1$ on $\mathbb{P}^1$ to be a "projective" variant of the Reed-Solomon code. This code induces a "big code" $C$ as a subspace of the $\F_p$ functions on $X(2)$- the $2$-dimensional cells of $X$- whose local views at every edge are in the base code of the edge. The goal was then to propagate the rate, distance, and local-testability of Reed-Solomon codes from the small code $C_1$ to the big code $C$.
This turned out to be easier to say than to do.
At some point, we were hoping to use $p$-adic uniformization.
Recall the work of Mumford \cite{Mumford} who used the combinatorial structure of one such Ramanujan complex to prove a result on algebraic surfaces appearing as locally symmetric quotients of $SU(2,1)$. We were hoping to go in the opposite direction and to use the theory of algebraic surfaces to study our combinatorial objects. The theory of $p$-adic uniformization was developed in depth by Varshavsky in his thesis \cite{varshavsky1998p} (written under the supervision of the 3rd author of the current paper). This is an opportunity to thank Yakov Varshavsky who gave upon our request a semester-long course describing this work. While we eventually are not using this, we were fortunate to be exposed to an amazing chapter of deep mathematics.
Propagating local testability from the small code to the big code when these are defined over a high dimensional expander is possible. This was proved in \cite{DiksteinDFH18} with the hope that it would serve our original plan. For our codes to fit, the intermediate code, $C_0$ - the one that is defined on the link of a vertex through the small Reed-Solomon codes $C_1$ on the edges - needed to be itself locally-testable. Unfortunately we failed to prove that $C_0$ is locally testable. Here the problem is very concrete: Find $C_1$ inside $\F_p^{\mathbb{P}^1}$ such that the induced intermediate code $C_0$ on the link of a vertex is locally testable. Here, the link of a vertex is nothing but the lines versus points graph of the projective plane.
One can generalize this challenge to get such a code also on higher dimensional spherical buildings.
This is interesting also in higher dimensions: are such spherical codes locally testable?\\
We, therefore, changed direction and replaced $G=SL(3,\mathbb{Q}_p)$ by a product $G=SL(2,\mathbb{Q}_p)\times SL(2,\mathbb{Q}_q)$. This time the quotients obtained from congruence lattices in $G$ give rise to square complexes. These complexes were shown long ago to be Ramanujan cubical complexes \cite{JL} and the dynamic of walks along them was studied in \cite{mozes135zero}.
This time the local intermediate code look like tensor codes (since the link of every vertex is the {\em complete} bipartite graph) and there are plenty of tensor codes that are locally testable as mentioned above. A subtle obstacle arose at this point which does not exist in the graph codes of \cite{SipserSp96}: one needs to name the squares in such a way that the function defined on the link of an edge $\set{u,v}$ will be in or out the code independently if we look at it from the vertex $u$ or the vertex $v$. It might be that this challenge can be overcome, but at that point, we realized that by changing from these square complexes to the left-right Cayley complexes as defined above, this problem is easily fixed. Moreover, it became also easier to argue about the rate- making the whole paper much simpler than we expected!
As explained, our long journey left a number of unsettled issues. We believe they are interesting in their
own right (and in all dimensions) even if not needed anymore for the concrete goal of locally testable codes.
The left-right Cayley complexes seem objects that are worth studying for their own sake. It is actually somewhat surprising that in spite of over 100 years of studying Cayley graphs, these objects, as far as we know, have never been studied before. An immediate curiosity is whether there are higher-dimensional analogs or whether a group ``has only two sides'' and hence these exist only in dimension $2$.
Anyway, it seems that this paper solves one problem but opens many others.
\section*{Acknowledgements}
We wish to thank Prahladh Harsha and Avi Wigderson for many interesting discussions along the way of this project. We also wish to thank Tali Kaufman for her influential role in connecting LTCs and high dimensional expansion.
This work was presented by the first author on October 6, 2021
at the Simon's Institute for the Theory of Computing \cite{breakthrough-talk} as part of the lecture series on breakthroughs in computer science, and at the Institute for Advanced Study in Princeton on October 25-26, 2021 \cite{IAStalk}. It was also presented by the 4th author on October 27, 2021 at the Simon's HDX21 workshop \cite{SimonsHDX-talk}. The authors are very grateful to these institutions and for the remarks of the audience which improved the exposition of the paper.
\section{Preliminaries}\label{sec:prelim}
\subsection{Expander Graphs}
A $d$-regular graph $G$ is said to be a $\lambda$-one-sided expander if it has eigenvalues $d=\lambda_1 \ge \lambda_2 \ge ... \ge \lambda_n \geq -d$ which satisfy $\lambda_i \leq \lambda\cdot d$ for all $i>1$.
The following is a standard lemma by Alon and Chung,
\begin{lemma}[{\cite{alon1988explicit}}]\label{lemma:AC}
Let $G=(V,E)$ be a $d$-regular $\lambda$ one-sided expander. Let $T\subseteq
V$ be such that the graph induced on $T$, denoted $G(T)$, has average degree at least $\delta d$. Then $|T| \ge (\delta - \lambda )\cdot |V|$, and the number of edges in $G(T)$ is at least $(\delta - \lambda )\delta \cdot|E|$.
\end{lemma}
This lemma holds in more general situations where instead of a $d$-regular graph we have a weighted Markov operator as long as it has a basis of eigenvectors. Let $\D$ be any probability distribution over a finite set $V$, and define an inner product by
\[
\iprod{\cdot,\cdot}_\D:\R^V\times \R^V\to \R, \qquad
\iprod{f,f'}_\D = \E_{x\sim \D}[f(x)f'(x)].
\]
Denote by $\one\in\R^V$ the constant $1$ function.
\begin{lemma}\label{lemma:AC-general}
Let $M:\R^V\to \R^V$ be a symmetric Markov operator such that $M \one
= \one$, and such that for all $f$ with $\iprod{f, \one}_\D=0$, $\iprod{f,Mf}_\D\leq
\lambda\iprod{f,f}_\D$. Let $f=\ind_T$ be the indicator of a set $T\subseteq
V$, so that $\iprod{f,f}_\D=\Pr_\D[T]$. If $\iprod{f,Mf}_\D\geq \delta \cdot \iprod{f,f}_\D$ then $\Pr_\D[T] \ge \delta - \lambda $, and $\iprod{f,Mf}_\D \geq \delta(\delta-\lambda)$.
\end{lemma}
\begin{proof}
Denote $p = \Pr_\D[T]$.
We can write $f = p
\one + f_\perp$ with $\iprod{f_\perp,\one}_\D=0$. We get
\[\delta \cdot p \leq \iprod{f,Mf}_\D = \iprod{p
\one + f_\perp,M(p
\one + f_\perp)}_\D =
p^2 + \lambda\iprod{f_\perp,f_\perp}_\D \leq
p^2 + \lambda p .
\] which, when rearranging, gives the lemma.
\end{proof}
\subsection{Error Correcting Codes}
A linear code $C\subset \bits^n$ is an $\bits$-linear subspace of $\bits^n$.
The block-length of the code is $n$. The rate and distance of the code are the relative dimension of the code and relative Hamming weight of the smallest weight non-zero codeword, respectively, namely,
\[
\Rate(C) = \frac 1 n \dim(C) \qquad \mbox{ and } \qquad
\dist(C) = \frac 1 n \min_{w\in C-\set{0}} |\sett{i\in [n]}{w_i \neq 0}|.
\]
We recall the definition of locally testable codes from \cite{GolSud06}. The definition given here is that of a ``strong'' LTC, and implies all other definitions of locally testable codes. See \cite[Chapter 13]{goldreich2017introduction}.
\begin{definition}[Locally Testable Code (LTC)]\label{def:LTC}
For $\kappa>0$ and $q\in \mathbb{N}$ we say that an error-correcting code $C\subseteq \bits^n$ is {\em $\kappa$-locally testable with $q$ queries} if there is a distribution over a collection of $q$-element subsets $S\subset [n]$ such that each subset $S$ is associated with a set $V_S\subset \bits^S$ of allowed local views, and such that, denoting by $f|_S$ the restriction of $f$ to the set $S$, the following hold.
\begin{itemize}
\item If $f\in C$ then for every $S$, $f|_S\in V_S$.
\item For every $f\in\bits^n$,
\[ \kappa\cdot \dist(f,C) \leq \Pr_S [ f|_S\not\in V_S].
\]
\end{itemize}
\end{definition}
\begin{definition}[Tensor Code]
Let $n_1,n_2\in\N$ and let $C_i \subset \set{f:[n_i]\to\bits}$ for $i=1,2$
be two linear codes. Define the tensor code $C=C_1\otimes C_2$ by
\[ C = \sett{M:[n_1]\times[n_2]\to \bits}{\forall i\in [n_1],j\in [n_2],
M(i,\cdot)\in C_2, M(\cdot,j)\in C_1}.
\]
\end{definition}
It is easy to check that $\dim(C_1 \otimes C_2)= \dim(C_1) \cdot \dim(C_2)$, and that $\dist(C_1 \otimes C_2) = \dist(C_1)\dist(C_2)$.
We view the elements of $C$ as $n_1$-by-$n_2$ matrices and write $w(i,\cdot)\in \bits^{n_2}$ for the $i$-th row of $w$, and similarly $w(\cdot,j)\in\bits^{n_1}$ is the $j$-th column of $w$.
A natural {\em test} for whether a given matrix $f\in \bits^{n_1\times n_2}$ is in $C_1\otimes C_2$ is as follows:
Randomly choose a row or a column, and check whether the restriction of $f$ to that column (or row) is in $C_1$ (or $C_2$).
The quality of the test is measured by the relation between the rejection probability and the distance of $f$ from the tensor code. Formally, this is captured by the notion of robust-testability.
\begin{definition}[Robust testability of tensor codes]
Fix $C_i \subseteq \bits^{n_i}$ linear error correcting codes for $i=1,2$. For $f:[n_1]\times[n_2]\to\bits$, let
\[
\dcol(f) = \dist(f, C_1\otimes \bits^{n_2}), \qquad
\drow(f) = \dist(f, \bits^{n_1}\otimes C_2).
\]
and
\[ \rho(f) = (\dcol(f) + \drow(f))/2.\]
The robust testability of $C_1\otimes C_2$ is defined to be
\[ \rho = \min_{f\not\in C_1\otimes C_2} \frac{\rho(f)}{\dist(f,C_1\otimes C_2)} ,
\]
and we say that $C_1\otimes C_2$ is $\rho$-robustly testable.
\end{definition}
The robust testability of tensor codes was first studied in \cite{BenSasson-Sudan-tensors}, where it was shown that for any code $C$ with sufficiently high distance, the $d$-dimensional tensor code $C^{\otimes d}$ is robustly testable for all $d\geq 3$. The requirement $d\geq 3$ was puzzling because the tensor of Reed Solomon codes is known \cite{PoliSpiel94} to be robustly testable even for $d=2$ and this was considered the prototype for locally testable codes. Surprisingly, Paul Valiant discovered \cite{PValiant05} that there are codes $C$ for which $C\otimes C$ is {\em not} robustly testable, see also \cite{GoldreichM07}. Quickly after that \cite{DSW06} formulated a notion of smooth codes, broadened later to `weakly smooth' in \cite{BVidweakly}, and showed that the tensor product of a smooth code and any other code is in fact robustly testable. To define smooth codes recall the definition of LDPC codes,
\begin{definition}[LDPC code]\label{def:ldpc}
Let $c,d,n\in \mathbb{N}$. A $(c,d,n)$-LDPC code is given by a $(c,d)$-regular bipartite graph $([n],[m],E)$ (called a factor graph) with $n$ left vertices and $m=nc/d$ right vertices, called parity checks, such
that all right vertices have degree $d$ and all left vertices have degree
$c$. The code is defined to be
\[ C = \sett{w:[n]\to\bits}{\forall j\in [m], \sum_{i:ij\in E}w(i) =0 \mod 2}.
\]
\end{definition}
\begin{definition}[Smooth code]\label{def:smooth}
Let $c,d,n\in \mathbb{N},\alpha,\beta,\delta>0$. A $(c,d,n)$ LDPC code $C\subset
\bits^n$ is $(\alpha,\beta,\delta)$-smooth if for every $Y\subseteq [n]$
with $|Y|\leq \alpha \cdot m$ there is some $X\subseteq [n]$ with $|X|\leq
\beta\cdot n$ such that the code $C(\bar Y)|_{\bar X}$ has distance at least
$\delta$.
Here the code $C(\bar{Y})|_{\bar X}$ is the code obtained by removing
the constraints in $Y$ and then removing the coordinates in $X$.
\end{definition}
In Section \ref{sec:ins} we show that random low density parity check codes (LDPC) are smooth.
\paragraph{Agreement-Testability}
A related testing notion focuses on the agreement between pairs of overlapping local views. We think of the following situation,
\begin{itemize}
\item For each column we are given a codeword of $C_1$, and these are aggregated into $w_1 \in C_1\otimes \bits^{n_2}$.
\item For each row we are given a codeword of $C_2$, and these are aggregated into $w_2 \in \bits^{n_1}\otimes C_2$.
\item We check ``agreement'', namely, pick a random pair of row $i$ and column $j$, and check whether they agree on their intersection, i.e. whether
\[ w_1(i,j) \stackrel ? = w_2(i,j).\]
\end{itemize}
Testability is defined to be the ratio between the amount of pairwise disagreement to the distance from the code. Formally,
\begin{definition}[agreement-testability]\label{def:atest}
Let $\kappa>0$. Let $C_i \subset \set{f:[n_i]\to\bits}$ for $i=1,2$. We say that $C_1\otimes C_2$ is $\kappa$-agreement-testable if for every $w_1 \in C_1\otimes \bits^{n_2}$, $w_2\in \bits^{n_1}\otimes C_2$, there exists $w\in C_1\otimes C_2$ such that
\begin{equation*}
\kappa \cdot (\Pr_i[w_1(i,\cdot)\neq w(i,\cdot)] + \Pr_j[w_2(\cdot,j)\neq w(\cdot,j)])
\leq \Pr_{i\in [n_1],j\in[n_2]}[w_1(i,j)\neq w_2(i,j)] .
\end{equation*}
\end{definition}
In words, given a word $w_1$ whose rows are in $C_1$, and given a word $w_2$ whose columns are in $C_2$, we say that $C_1\otimes C_2$ is $\kappa$-agreement-testable if the amount of disagreement between $w_1$ and $w_2$ is an upper-bound for the fraction of rows or columns one needs to change in order to get to the closest word $w\in C_1\otimes C_2$, times $\kappa$.
It is well known (see for example \cite{DH09}) that agreement testability is equivalent to robust testability:
\begin{lemma}\label{lem:robagr}
Let $C_i \subseteq\bits^{n_i}$, and assume $\delta_i=\dist(C_i)$ for $i=1,2$. \begin{itemize}
\item If $C_1\otimes C_2$ is $\rho$-robustly testable then it is $\kappa$-agreement testable, for $\kappa^{-1} = \frac 1 {2\delta_1\rho} + \frac {1+1/(2\rho)} {\delta_2} $.
\item If $C_1\otimes C_2$ is $\kappa$-agreement testable, then it is $\rho$-robustly testable for $\rho = \frac\kappa{2(\kappa+1)} $.
\end{itemize}
\end{lemma}
We prove this lemma in Appendix~\ref{app:robagr}.
\section{The Left-Right Cayley Complex}
We describe a new construction of a Cayley graph that in addition to vertices and edges also has two-dimensional faces, called squares. Each square contains four edges that constitute a four-cycle.
\begin{definition}[Left-Right Cayley Complex]\label{def:LRC}
Let $G$ be a finite group with two symmetric sets of generators $A,B$, namely, each is
closed under taking inverses. We assume that the identity element of $G$ is neither in $A$ nor in $B$. Define the {\em Left-Right Cayley Complex} $X = Cay^2(A, G,B)$ as follows
\begin{itemize}
\item The vertices are $X(0)=G$.
\item The edges are $X(1) = X^A(1)\sqcup X^B(1)$ where
\[X^A(1) = \sett{\set{g,ag}}{g\in G, a\in A}, \qquad X^B(1) = \sett{\set{g,gb}}{g\in
G, b\in B}. \]
\item The squares are $X(2) = A\times G\times B / \sim$ where for every $a\in A,b\in B,g\in G$,
\[ (a,g,b)\sim (a^{-1},ag,b)\sim (a^{-1},agb,b^{-1})\sim (a,gb,b^{-1}),\]
and denote the equivalence class of $(a,g,b)$ by $[a,g,b]$, so \[[a,g,b]=\set{(a,g,b),(a^{-1},ag,b),(a^{-1},agb,b^{-1}), (a,gb,b^{-1})}.\]
\end{itemize}
\end{definition}
The graph $(X(0),X^A(1))$ is none other than the Cayley graph $Cay(G,A)$. Similary $(X(0),X^B(1))$ is the Cayley graph $Cay(G,B)$. The fact that with $A$ we multiply on the left, and with $B$ we multiply on the right, gives a local commutativity which generates many four-cycles, namely, squares.
\begin{remark}
Given a group $G$ and a set of generators $A$, the Cayley graph $Cay^{left}(G,A)$ with left-multiplication edges is isomorphic to the Cayley graph $Cay^{right}(G,A)$ with right multiplication edges via the map $g\mapsto g^{-1}$. The left-multiplication edge $\set{g,ag}$ maps to the right multiplication edge $\set{g^{-1}, g^{-1}a^{-1}}$.
This justifies talking about a Cayley graph without specifying left or right multiplication.
\end{remark}
\begin{remark}
The product of two graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ is a square complex $X=G_1\times G_2$ defined as follows.
\begin{itemize}
\item The vertices are $X(0)= V_1\times V_2$. \item The edges are $X(1)=E_1\times V_2\;\sqcup \;V_1\times E_2$, where an edge $(\set{u,u'},v)\in E_1\times V_2$ connects $(u,v)$ with $(u',v)$, and similarly an edge $(u,\set{v,v'})\in V_1\times E_2$ connects $(u,v)$ with $(u,v')$.
\item The squares $X(2)$ are identified with $E_1\times E_2$, so that the square corresponding to the pair of edges $e_1=\set{u,u'}\in E_1$ and $e_2=\set{v,v'}\in E_2$ is the four-cycle
$(u,v)\to (u,v')\to (u',v')\to (u',v)\to (u,v)$.
\end{itemize}
The left-right Cayley complex is the quotient of the Cartesian product of $G_A = (G,X^A(1))$ and $G_B=(G,X^B(1))$ obtained by identifying the vertex $(g,g')$ with $(gh,h^{-1}g')$ for all $h\in G$. One can check that the map $(g,g')\mapsto gg'$ gives a homomorphism from $G_A\times G_B$ to $Cay^2(A,G,B)$.
\end{remark}
\begin{remark}
Left-right Cayley complexes are examples of two-dimensional cubical complexes. Cubical complexes are well-studied, and in particular there are constructions of Ramanujan cubical complexes \cite{JL} with bounded degree and any dimension, whose walk dynamics was studied in \cite{mozes135zero}. The left-right Cayley complexes have an additional matching labels feature that other complexes are not known to have.
\end{remark}
\begin{definition}[Links]\label{def:bij}
For each $g\in G$, the link of $g$ is $X_g = \sett{[a,g,b]}{a\in A,b\in B}$. There is a natural map $(a,b)\mapsto [a,g,b]$.
For every edge $e=\set{g,ag}$, the link of $e$ is denoted $X_e = \sett{[a,g,b]}{b\in B}$. Similarly if $e=\set{g,gb}$ we let $X_e = \sett{[a,g,b]}{a\in A}$.
\end{definition}
\begin{definition}\label{def:TNC}
A left-right Cayley complex satisfies the total no-conjugacy condition if
\[\forall a\in A,b\in B,g\in G,\qquad g^{-1}ag\neq b.\label{eq:nc} \tag{TNC}\]
\end{definition}
Here are a few easy properties of left-right Cayley complexes.
\begin{claim}\label{claim:NC}
Assuming \eqref{eq:nc}, each vertex has exactly $|A|+|B|$ distinct neighbors; and each square contains exactly four distinct vertices; and the map $(a,b)\mapsto [a,g,b]$ is a bijection from $A\times B$ to $X_g$ for each $g\in G$.
\end{claim}
\begin{proof}
Let $a\neq a'\in A$ and $b\neq b'\in B$. Clearly $ag\neq a'g$ and $gb\neq gb'$. If $ag = gb$ then $g^{-1}ag = b$ which contradicts \eqref{eq:nc}. So $g$ has $|A|+|B|$ distinct neighbors.
Next we show that each square $[a,g,b]\in X(2)$ must have four distinct vertices. Clearly $g\neq ag$ and $g\neq gb$, and we already saw that $ag\neq gb$. Now, if $g = agb$ we would contradict \eqref{eq:nc} because it implies $g^{-1}a^{-1}g=b$ making $a^{-1}\in A$ and $b\in B$ conjugates.
Finally, let us see that the map $(a,b)\mapsto [a,g,b]$ is a bijection between $A\times B$ and $X_g$ for all $g$. Assume that $[a,g,b]=[a',g,b']$ for some $(a,b), (a',b')\in A\times B$. This implies that $(a',g,b')\in \set{(a,g,b),(a^{-1},ag,b),(a^{-1},agb,b^{-1}), (a,gb,b^{-1})}$. We have seen that $g\neq ag,gb,agb$ so this implies that $(a,g,b)=(a',g,b')$ which means that $(a,b)=(a',b')$.
\end{proof}
\begin{remark}\label{rem:sizes}
It follows that assuming \eqref{eq:nc} \[|X(1)| = \frac{\card A + \card B}2\cdot |G|\quad \ve\quad |X(2)| = \frac{|A||B|}4 \cdot |G|.\]
\end{remark}
\remove{\begin{claim}\label{claim:expansion}
Assume $Cay(G,A),Cay(G,B)$ are $\lambda$-expanders. Then the $1$-skeleton of $X$, namely the graph $(X(0),X(1))$, is also a $\lambda$-expander.
\end{claim}
\begin{proof}
The graph $(X(0),X(1))$ is a $(|A|+|B|)$-regular graph whose edges are the union of the edges two $\lambda$-expanders on the same vertex set. Let $M_0$ be the Markov operator describing the random walk on $(X(0),X(1))$, and let $M_A,M_B$ be the operators of the random walks on $Cay(G,A)$ and $Cay(G,B)$ respectively. Denote $\alpha = \frac{|A|}{|A|+|B|}$ and $\beta = \frac{|B|}{|A|+|B|}=1-\alpha$, then $M_0 = \alpha M_A + \beta M_B$. Clearly $M_A\one = M_B\one = M_0\one = \one$.
Fix $f:X(0)\to\R$ with $\iprod{f,\one}=0$. We have \[ \iprod{f,M_0 f} = \alpha\iprod{f,M_A f} + \beta\iprod{f,M_B f} \leq \alpha\cdot \lambda \cdot \iprod{f,f}+\beta\cdot \lambda \cdot \iprod{f,f}= \lambda \cdot \iprod{f,f}.\]
\end{proof}
}
It will be natural to consider a weighted version of the $1$-skeleton of $X$, where the weight is distributed evenly between the $A$ and the $B$ edges. When $|A|=|B|$ this is the usual unweighted graph.
\begin{definition}\label{def:distr}
Let $\D_1$ be the distribution over $X(1)$ given by selecting with probability half a uniform edge in $X^A(1)$, and with probability half a uniform edge in $X^B(1)$. (In case $A,B$ have equal size $\D_1$ is the uniform distribution over $X(1)$.)
We define an inner product on functions over $X(1)$ $f,f':X(1)\to\R$ by
\begin{equation}\label{def:iprod}
\iprod{f,f'}_{\D_1} = \E_{e\sim \D_1}[f(e)f'(e)] = \frac 1 2 \E_{e\in X^A(1)}[f(e)f'(e)] + \frac 1 2 \E_{e\in X^B(1)}[f(e)f'(e)].
\end{equation}
This will be the only inner product we consider for functions over $X(1)$ so we sometimes omit the subscript and simply write $\iprod{f,f'}=\iprod{f,f'}_{\D_1}$. As usual we let $\norm f = \iprod{f,f}$.
\end{definition}
\paragraph{Parallel Random Walk}
In addition to the standard random walk on the $1$-skeleton of $X$, we will be interested in a random walk on the edges called
the parallel walk, which takes an edge $e$ to a random edge $e'$ that is ``parallel'' to it.
\begin{definition}[Labels]
For each $s \in A\cup B$ let $[s] = \set{s,s^{-1}}$. Let $\tilde A = \sett{[a]}{a\in A}$ and let $\tilde B = \sett{[b]}{b\in B}$. The label of an edge $\set{g,ag}$ is defined to be $\set{a,a^{-1}}$, and this is independent of the presentation of the edge as $\set{g,ag}$ or $\set{(ag),a^{-1}(ag)}$. Similarly, the label of an edge $\set{g,gb}$ is defined to be $\set{b,b^{-1}}$.
Let $\tilde A \cup \tilde B$ denote the set of labels of the edges in the complex.
For any $\sigma\in \tilde A \cup \tilde B$, denote by $X^\sigma(1)$ the set of edges labelled $\sigma$.
\end{definition}
\begin{claim}\label{claim:par}
If $\sigma=\set{c,c^{-1}} \in \tilde A \cup \tilde B$ and $c\neq c^{-1}$, then $X^\sigma(1)$ has size $|G|$, otherwise it has size $|G|/2$.
\end{claim}
\begin{proof}
We shall prove the claim for $\sigma=\set{a,a^{-1}} \in \tilde A$, the claim for $\sigma \in \tilde B$ is proven analogously.
Observe that every vertex $g$ participates in two edges labelled $\sigma=\set{a,a^{-1}}$, namely $\set{g,ag}$ and $\set{g,a^{-1}g}$. Since every edge is counted twice, from each of its two endpoints, we get that $|G| = |X^\sigma(1)|$. \footnote{Note also that when $a\neq a^{-1}$ the operator $\mpar_{[a]}$ on $X^\sigma(1)$ is isomorphic to the standard random walk on $Cay(G,B)$, and similarly if $b\neq b^{-1}$ then $\mpar_{[b]}$ is isomorphic to $Cay(G,A)$.}
In case $a=a^{-1}$each vertex participates in only a single edge labelled $[a]$, but still every edge has two endpoints so after accounting for the double counting we get $|X^\sigma(1)|=|G|/2$.
\end{proof}
Define a Markov operator $\mpar_\sigma:\R^{X^\sigma(1)}\to \R^{X^\sigma(1)}$ on the space of functions on $X^\sigma(1)$ by setting, for any $f:X^\sigma(1)\to\R$,
\[\mpar_{[a]}f(\set{g,ag}) = \E_b f(\set{gb,agb}),\qquad\mpar_{[b]}f(\set{g,gb}) = \E_a f(\set{ag,agb}).\]
We define a Markov operator $\mpar:\R^{X^\sigma(1)}\to \R^{X^\sigma(1)}$ on the space of functions on the entire set of edges $X(1)$ by letting, for any $f:X(1)\to\R$,
\begin{equation}\label{eq:split}
\mpar f = \sum_\sigma \mpar_\sigma (f|_{X^\sigma(1)}).
\end{equation}
\begin{definition}[Parallel Random Walk]
We define a random walk on the set of edges $X(1)$ as follows. Starting from an edge $e$, choose uniformly a square containing $e$ and then move to the unique edge $e'\neq e$ with the same label as $e$. (If \eqref{eq:nc} doesn't hold the square might not contain an edge $e'\neq e$ with the same label, in which case the walk will stay in place).
\end{definition}
The Markov operator corresponding to this walk is exactly $\mpar$, because starting at an edge $e=\set{g,ag}$, a random square containing $e$ is $[a,g,b]$ for a uniformly chosen $b\in B$, and then the only other $[a]$-labeled edge in this square is the edge $e'=\set{gb,agb}$.
\begin{lemma}\label{lem:par}
Assume both $Cay(G,A)$ and $Cay(G,B)$ are $\lambda$-expanders. Suppose $R\subseteq X(1)$ and assume $f=\one_R:X(1)\to\R$ satisfies $\iprod{f,\mpar f} \geq c\cdot \iprod{f,f}$. Then there exists some $\sigma\in \tilde A\cup \tilde B$ such that
\[ |R\cap X^\sigma(1)| \geq (c-\lambda)|G|/2.
\]
\end{lemma}
\begin{proof}
We expand $\iprod{f,\mpar f}$ according to \eqref{eq:split}, and get
\[\iprod{f,\mpar f} = \E_\sigma \E_{e\in X^\sigma(1)}[f(e)\mpar_\sigma f(e)],\]
where the expectation over $\sigma$ is obtained by choosing, with probability half, a random label in $\tilde A$, and with probability half, a random label in $\tilde B$.
Clearly also
\[\iprod{f,f} = \E_{e\sim \D_1}[f(e)^2] = \E_\sigma \E_{e\in X^\sigma(1)}[f(e)^2].\]
Plugging these into the inequality $\iprod{f,\mpar f} - c\cdot \iprod{f,f}\geq 0$ we get
\[ \E_\sigma \E_{e\in X^\sigma(1)} \left[ {f(e)\mpar_\sigma f(e)} -c\cdot {f(e)^2} \right] \geq 0
\] so there must be at least one $\sigma$ for which \begin{equation}
\E_{e\in X^\sigma(1)}[ f(e) \mpar_{\sigma} f(e) ] \geq c \cdot\E_{e\in X^\sigma(1)}
{[f(e)^2]}.
\end{equation}
Fix, say, $\sigma = [a]$ and define $h_a:G\to\R$ by $h_a(g) = f(\set{g,ag})$. (The case $\sigma=[b]$ is analogous and omitted). Now,
\begin{multline}
c \cdot \iprod{h_a,h_a}
= c\cdot \E_g[h_a(g)^2]
= c\cdot \E_{g}[f(\set{g,ag})^2]
= c\cdot \E_{e\in X^{[a]}(1)}[f(e)^2]
\leq \E_{e\in X^{[a]}(1)}[f(e)\mpar_{\sigma} f(e)]
\\ = \E_{g\in G}\left[f(\set{g,ag})\E_{b\in B}[f(\set{gb,agb})] \right]
= \E_{g\in G}\left[h_a(g) \E_{b\in B}h_a(gb)\right]
= \iprod{h_a,M_B h_a},
\end{multline}
where $M_B$ is the random walk operator on $Cay(G,B)$. We relied here on the fact that choosing a uniform edge in $X^{[a]}(1)$ can be done by choosing a uniform $g\in G$ and looking at $\set{g,ag}$. Observe now that $h_a$ indicates the set $T = \sett{g\in G}{f(\set{g,ag})\neq 0}$, so by Lemma~\ref{lemma:AC-general} applied on the graph $Cay(G,B)$ with the operator $M_B$ we deduce that
$|T| \geq (c-\lambda)|G| $. Since every non-zero value for $f$ can cause at most two non-zero values in $h_a$, we get that $|R\cap X^\sigma(1)|=|f^{-1}(1)\cap X^\sigma(1)|\ge |h_a^{-1}(1)|/2 = |T|/2
\geq (c-\lambda)\cdot |G|/2$.
\end{proof}
\section{Error Correcting Code on a Left-Right Cayley Complex}
Let $G, A, B$ and $X=Cay^2(G,A,B)$ as in the previous section. Recall that for any vertex $g \in X(0)$ (resp. any edge $e \in X(1)$) we denote by $X_g \subset X(2)$ (resp. $X_e \subset X(2)$) the set of squares in $X$ containing the vertex $g$ (resp. the edge $e$). Let $C_A\subset \bits^A$ and let $C_B\subset \bits^B$ be two fixed linear error correcting codes with rates $\rho_A = \Rate(C_A),\rho_B = \Rate(C_B)$ and distances $\delta_A = \dist(C_A),\delta_B = \dist(C_B)$, respectively.
Define the code $C=C[G,A,B,C_A,C_B]$ as follows.
For an edge $e=\set{g,ag}\in X^A(1)$ we define a local code
\[ C_e = \sett{f:X_e\to\bits}{f([a,g,\cdot]) \in C_B}.
\]
Similarly, for an edge $e=\set{g,gb}\in X^B(1)$ we define a local code
\[ C_e = \sett{f:X_e\to\bits}{f([\cdot,g,b]) \in C_A}.
\]
Note that this definition appears to depend on the choice of $g\in e$ but it does not.
Finally, we define a global code
\[ C = \sett{f:X(2)\to \bits}{\forall e\in X(1), f|_{X_e}\in C_e}.
\]
\inote{talk here about the "intermediate code", not done yet.}
For each vertex $g \in X(0)$, define the local tensor code around the vertex $g$ to be
\[ C_g = \sett{f:X_g\to \bits}{f([\cdot,g,\cdot]) \in C_A\otimes C_B}.
\]
\begin{lemma}[$C$ is a lifted tensor-code]\label{lem:tensor}
\[
C = \sett{f:X(2)\to \bits}{\forall g\in X(2), f|_{X_g}\in C_g}.
\]
\end{lemma}
\begin{proof} Immediate from the fact that $f([\cdot,g,\cdot]) \in C_A\otimes C_B$ for any $g\in X(0)$ if and only if $f([a,g,\cdot]) \in C_B$ and $f([\cdot,g,b]) \in C_A$ for any $g\in X(0)$, $a\in A$ and $b \in B$.
\end{proof}
Observe that for the local code at each vertex to be a tensor code, we must make sure that around every $A$ edge we have {\em the same} code $C_A$, and similarly for $B$. If we choose different base codes at different edges we might still get a code with rate and distance, but local testability will probably fail, because we lose the local tensor structure. This is in contrast to the case of expander codes where the local base code can be chosen arbitrarily and differently at each vertex.
\subsection{Properties of The Code}
\inote{need to name the code as well}
We now look at the rate, distance and local testability of the code $C = C[G,A,B,C_A,C_B]$.
Recall $\rho_A = \Rate(C_A)$, $\rho_B = \Rate(C_B)$ and $\delta_A = \dist(C_A)$, $\delta_B = \dist(C_B)$
\begin{lemma}[Rate]\label{lem:rate}
The rate of the code $C$ is bounded from below by
\[
\Rate(C) \geq 2(\rho_A+\rho_B) - 3.
\]
\end{lemma}
\begin{proof}
For each $e\in X^A(1)$, $codim(C_e) = codim(C_B) = |B|\cdot(1-\rho_B)$. Similarly
for each $e\in X^B(1)$, $codim(C_e) = codim(C_A) = |A|\cdot(1-\rho_A)$. The
number of linearly independent constraints on $f\in C$ is at most
\[
|X^A(1)|\cdot |B| (1-\rho_B) + |X^B(1)|\cdot |A| (1-\rho_A) = |G||A||B|(1 - \frac {\rho_A+\rho_B}2)
\]
On the other hand, the dimension of the ambient space is the number of squares $|X(2)| = |G||A||B|/4$, see Remark \ref{rem:sizes}.
Subtracting the number of constraints from the number of bits we get a lower bound on the dimension of the code,
\[
dim(C)\geq \frac{1}{4} |G||A||B|(1 - (4 - 2(\rho_A+\rho_B))) = \frac{1}{4} |G||A||B|(2(\rho_A+\rho_B)-3).
\]
\end{proof}
In fact, we can do a little better. Recall that a {\em vertex cover} of a graph is a set of vertices that touch all of the edges. For example, if the graph is bipartite, then it has a vertex cover whose size is half the size of the graph.
\begin{lemma}[Rate - better bound]
Suppose the underlying graph of $X$ has a vertex cover of size $\nu|G|$. Then the rate of the code is at least $4\nu\rho_A\rho_B +1-4\nu$.
In particular, if the graph is bipartite, then $\nu=\frac 1 2$ and we get that
\[
\Rate(C) \geq 2 \rho_A \rho_B - 1.
\]
\end{lemma}
It is interesting to mention that in the expander codes of Tanner \cite{Tanner81}, (whose distance and decoding was later analyzed in \cite{SipserSp96}), if the local code $C_0$ has rate $\rho_0$ then the global rate is shown to be at least $2\rho_0-1$. In our code the rate of the local code is $\Rate(C_g) = \Rate(C_A\otimes C_B) = \rho_A\rho_B$, and in case the graph is bipartite, we get the same bound of $2(\rho_A\rho_B)-1$ on the rate of the global code.
\begin{proof}
Let $V^*\subset G$ be a vertex cover, namely, a set of vertices that touches every edge. Then $f\in C$ if and only if for every $g\in V^*$, $f|_{X_g}\in C_g$. The reason is that every edge $e$ touches some $g\in V^*$ and the constraint $f|_{X_e}\in C_e$ is implied by $f|_{X_g}\in C_g$.
Since $C_g$ is isomorphic to $C_A\otimes C_B$ it has $|A|\cdot|B|(1 -\rho_A\rho_A)$ linearly independent constraints. The dimension of the code is at least
\begin{multline}
\dim(C) \geq |G||A||B|\frac 1 4 - |V^*|\cdot |A|\cdot|B|(1 -\rho_A\rho_A) \\
\geq \frac 1 4 |G||A||B|\cdot(1 - 4\nu(1-\rho_A\rho_B))
= \frac 1 4 |G||A||B|\cdot (4\nu\rho_A\rho_B +1-4\nu).
\end{multline}
\end{proof}
\begin{lemma}[Distance] \label{lem:distance}
Suppose that both $Cay(G,A)$ and $Cay(G,B)$ are $\lambda$-expanders for $\lambda<1$.
Then the distance of the code $C$ is bounded from below by
\[
\dist(C) \geq \delta_A \delta_B \cdot \left( \min(\delta_A,\delta_B) - \lambda \right).
\]
\end{lemma}
\begin{proof}
Let $0\neq f\in C$.
Let $g_0\in X(0)$ be some vertex such that $w_{g_0} = f|_{X_{g_0}} \neq 0$ (if they are all zero then $f=0$). Observe that since $0\neq w_{g_0}\in C_A\otimes C_B$ then $w_{g_0}$ has at least $\delta_A|A|$ non zero columns
and at least $\delta_B|B|$ non-zero rows. Let $A_1\subset A$ be the labels of these columns, and fix $a_1\in A_1$. We first show that
\begin{equation}\label{eq:rows}
\Pr_{g,b}[f([a_1,g,b])\neq 0] \ge \delta_B(\delta_B-\lambda).
\end{equation}
To prove \eqref{eq:rows} consider the graph $Cay(G,B)$ whose vertices are $X(0)$ and the edges are $X^B(1)$, and define a function $f_{a_1}:X^B(1) \to\bits$ by $f_{a_1}(\set{g,gb}) = f([a_1,g,b])$, observing that $f_{a_1}$ is well defined because for $g' = gb$,
\[
f_{a_1}(\set{g,g'}) = f_{a_1}(\set{g,gb})=f([a_1,g,b])=f([a_1,g',b^{-1}])= f_{a_1}(\set{g',g'b^{-1}}) = f_{a_1}(\set{g',g}).
\]
Since $f_{a_1}\neq 0$, it must have large weight because it belongs to the expander code defined on $Cay(G,B)$ with local code $C_B$. More elaborately, for every vertex $g$ that touches an edge where $ f_{a_1}\neq 0$, there must be at least $\delta_B|B|$ non-zero edges touching $g$. By Lemma \ref{lemma:AC} we get at least $\delta_B(\delta_B-\lambda)|X^B(1)|$ edges on which $f_{a_1}\neq 0$, which proves \eqref{eq:rows}.
For every $a\in A_1$, the weight of $f_{a}$ is at least $\delta_B(\delta_B-\lambda)$, so if we choose a random $a\in A$ and then a random edge in $X^B(1)$, the probability that $a\in A_1$ is at least $\delta_A$, and conditioned on this, the probability that $f_{a}(e)\neq 0$ is at least $\delta_B(\delta_B-\lambda)$, so altogether
\[ \Pr_{a,g,b}[f([a,g,b])\neq 0] \geq \Pr_a[a\in A_1]\cdot \Pr_{g,b}[f_{a}(\set{g,gb})\neq 0\;|\; a\in A_1] \geq \delta_A \delta_B(\delta_B-\lambda).
\]
Symmetrically, the weight of $f$ is also at least $\delta_B\delta_A(\delta_A-\lambda)$, and the lemma follows.
\end{proof}
\begin{theorem}[Local Testability]\label{thm:mainltc}
Suppose $X=Cay^2(A,G,B)$ is a left-right Cayley complex such that both $Cay(G,A)$ and $Cay(G,B)$ are $\lambda$-expanders, and such that \eqref{eq:nc} holds. Assume $C_A\subset \bits^A$ and $C_B\subset \bits^B$ are error correcting codes with relative distances $\delta_A,\delta_B>0$ respectively and such that $C_A\otimes C_B$ is $\kappa_0$-agreement-testable.
If
\begin{equation}\label{eq:c}
c=\frac{ \kappa_0}{8+\kappa_0} \cdot \min(\delta_A,\delta_B) >\lambda
\end{equation}
then $C = C[G,A,B,C_A,C_B]$
is $\kappa$-locally testable with $|A|\cdot|B|$ queries, for $\kappa^{-1} = \max(4(1+|A|+|B|), \frac{2(|A|+|B|)}{c-\lambda})$. Namely, for every $f:X(2)\to\bits$
\[ \kappa \cdot \dist(f,C) \leq \Pr_{g\in X(0)}[f|_{X_g}\not \in C_g].
\]
\end{theorem}
In words, given some potential codeword $f$, each vertex $g$ is associated with a local test that reads $f$ at all of the $|A|\cdot|B|$ squares touching $g$ and checks that these vaues form a codeword in the base code $C_g$. The theorem says that the distance of $f$ to the code is upper bounded by a constant multiple of the fraction of violated local tests.
We prove the theorem in the next section, by describing an iterative correction algorithm that finds a codeword close to $f$ if the probability that the test rejects is not too large.
\subsection{Local Self-Correction Algorithm}
In this section we describe a local self-correction algorithm that starts with a given string $f:X(2)\to\bits$ and either finds a codeword $f_0\in C$ or gives up. We denote \[\vrej(f) = \Pr_{g}(f|_{X_g}\not\in C_{g}),\]
the fraction of rejecting local tests. We will show that if $\rho(f)\leq \rho_0$ for some constant $\rho_0>0$, then the algorithm finds $f_0\in C$ such that $\dist(f_0,f)\leq O(\rho(f))$.
For each vertex $g$, let $\L_g\in C_g$ be a closest codeword to $f|_{X_g}$ (breaking ties arbitrarily).
We focus on the collection of local views $\set{\L_g}$ and whether the local views of neighboring vertices agree on the common squares.
\begin{definition}
Given a collection of local views $\L=\sett{\L_g\in C_g}{g\in G}$, we define the disagreement of the collection to be
\begin{equation}\label{eq:Eps}
\disagr(\L) = \Pr_{e=\set{g,g'}\in X(1)}[\L_g|_{X_{e}} \neq \L_{g'}|_{X_{e}}]. \end{equation}
\end{definition}
\inote{would params improve if $\D_1$ were used here?}
\RestyleAlgo{ruled}
\begin{algorithm}\label{alg}
\begin{enumerate}
\item (Initialization:) For each vertex $g$, let $\L_g\in C_g$ be a closest codeword to $f|_{X_g}$ (breaking ties arbitrarily). \[ \L_g = argmin_{w\in C_g} \dist(w,f|_{X_g}).\]
\item (Main loop:) If there is a vertex $g$ and a choice $w\in C_g$ that reduces $\disagr(\L)$ then replace $\L_g$ by $w$ and repeat.
\item (End:) If $\disagr(\L)>0$ output ``far''. Otherwise, $\disagr(\L)=0$, define $f_0:X(2)\to\bits$ by choosing for each square $s\in X(2)$ an arbitrary vertex $g\in s$ and setting $f_0(s) = \L_g(s)$. Output $f_0$.
\end{enumerate}
\caption{Iterative decoding algorithm. (input: $f:X(2)\to\bits$)}
\end{algorithm}
Observe that $\disagr(\L)|X(1)|$ is a non-negative integer, and this value decreases by at least $1$ every step of the algorithm, so the algorithm must halt.
\begin{proposition}\label{prop:completeness}
If the algorithm outputs $f_0$ then $f_0\in C$ and
\[ \dist(f,C)\leq \dist(f,f_0) \leq 4(1+|A|+|B|)\cdot \rho(f).
\]
\end{proposition}
Let $\L^1 = \set{\L^1_g}$ be the collection of local views initially defined in step $1$ of the algorithm, and let $\L=\set{\L_g}$ be the final collection, at the end of the algorithm.
\begin{proposition}\label{prop:soundness}
If the algorithm outputs ``far'' then $\disagr(\L)\geq \eps_0 = \frac{c-\lambda}{|A|+|B|}$, where $c=\frac{ \kappa_0}{8+\kappa_0} \cdot \min(\delta_A,\delta_B)$ is defined in \eqref{eq:c}.
\end{proposition}
We will show that this immediately means that $\rho(f)\geq \disagr(\L)/2\geq \frac{c-\lambda}{2(|A|+|B|)}$ and this in turn means that $\dist(f,C)\leq 1 \leq \frac{2(|A|+|B|)}{c-\lambda}\cdot \rho(f),$ which will prove the theorem.
\begin{proof}[Proof of Theorem~\ref{thm:mainltc}]
Given $f:X(2)\to\bits$, run the algorithm above. The output is either a function $f_0$, which by Proposition~\ref{prop:completeness}, satisfies $\dist(f,C)\leq \dist(f,f_0) \leq 4(1+|A|+|B|)\cdot \rho(f)$; or the output is ``far'', in which case $\disagr(\L)\geq\eps_0$ by Proposition~\ref{prop:soundness}. We observe that
\begin{equation}\label{eq:rho}
\disagr(\L^1) \leq 2\rho(f).
\end{equation}
The reason is that for each edge $\set{g,g'}$ that contributes to $\disagr$ either $f|_{\T g} \neq \L^1_g$
or $f|_{\T {g'}} \neq \L^1_{g'}$, otherwise \[\L^1_g|_{\T{gg'}} = (f|_{\T {g}})|_{\T{gg'}}
= f|_{\T {gg'}} = (f|_{\T {g'}})|_{\T{gg'}}= \L^1_{g'}|_{\T{gg'}}.\]
So the process of selecting an edge uniformly and then a random endpoint of it will lead to a rejecting vertex with probability at least $\disagr(\L^1)/2$, proving \eqref{eq:rho}. Now $\rho(f) \geq \disagr(\L^1)/2 \geq \disagr(\L)/2 \geq \eps_0/2 = \frac{c-\lambda}{2(|A|+|B|)}$, so we can write
\[\dist(f,C)\leq 1\leq \frac{2(|A|+|B|)}{c-\lambda}\cdot\rho(f).\]
All in all we get,
\[ \dist(f,C) \leq \max(4(1+|A|+|B|),\frac{2(|A|+|B|)}{(c-\lambda)})\cdot \rho(f) = \kappa \cdot \Pr_{g}(f|_{X_g}\not\in C_{g})
\] as needed.
\end{proof}
\begin{remark}
Algorithm \ref{alg} is clearly also a decoding algorithm in the standard sense: if we know that the given word $f$ is close enough to the code, then the regular structure of the tester implies that it will be rejected with probability proportional to $\dist(f,C)$. The analysis herein shows that for small enough (constant) distance, the algorithm will then find the nearest codeword.
The difficulty in our analysis is to show the same without any a priori guarantee on the distance of $f$ from the code.
\end{remark}
We now turn to prove the two propositions.
\begin{proof}[Proof of Proposition \ref{prop:completeness}]
By assumption, $\disagr(\L)=0$. We first observe that the value of $f_0(s)$ does not depend on the choice of $g\in s$ because $\disagr(\L)=0$ implies that $\L_{g}(s) = \L_{g'}(s)$ for any
$g,g'\in s$. (Suppose $g_1,g_2\in s$ disagree. If they are adjacent this
means that $\L_{g_1}$ disagrees with $\L_{g_2}$ contradicting $\disagr(\L)=0$.
If they are non-adjacent, they have a common neighbor which cannot agree with both of them). It follows that $f_0\in C$, because for each $g$, $f_0|_{X_g}=\L_g\in C_g$.
To bound $\dist(f,f_0)$, let \[
V_0 = \sett{g\in X(0)}{f|_{\T {g}}\neq \L^1_g},\qquad
V_1 = \sett{g\in X(0)}{\L^1_g\neq \L_g}.\]
So $V_0$ is the set of vertices whose local view doesn't perfectly satisfy the constraints of the code, and $V_1$ is the set of vertices $g$ for which $\L_g$ at the end of the algorithm differs from its initial value.
Observe that $g\in V_0$ iff $f|_{\T g}\not\in C_g$, so by definition,
\begin{equation}\label{eq:V0}
|V_0| = \vrej(f)\cdot |X(0)|.
\end{equation}
Any square $s$ that does not touch $V_0\cup V_1$ must have for every $g\in s$
\[f_0 (s) = \L_g(s) = \L_g^0(s) = f(s),\]
where the second equality is because $g\not \in V_1$ and the third is because $g\not \in V_0$.
We bound $|V_1|$ by the number of iterations of the algorithm, which is at most $|V_1|\leq \disagr(\L^1) \cdot |X(1)|$. We recall from \eqref{eq:rho} that $
\disagr(\L^1) \leq 2\rho(f)$.
Thus, we have,
\begin{equation}\label{eq:V1}
|V_1|\leq \disagr(\L^1) \cdot |X(1)| \leq 2\vrej(f)\cdot \frac {|A|+|B|}2 |X(0)|.
\end{equation}
Altogether, since every vertex touches $|A||B|$ squares, and since $|X(2)| = |A||B||X(0)|/4$, and using \eqref{eq:V0} and \eqref{eq:V1}, we get
\[ \dist(f, f_0) \leq \frac {|A||B| \cdot |V_0\cup V_1| }{|X(2)|} =
\frac {4 \cdot |V_0\cup V_1| }{|X(0)|} \leq
4(1+ {|A|+|B|})\vrej(f) .
\]
\end{proof}
The interesting part of the proof is to show that if $\disagr(\L)>0$ after
the algorithm ends, then $\disagr(\L) > \eps_0 = \frac {c-\lambda}{|A|+|B|} $.
\begin{proof}[Proof of Proposition~\ref{prop:soundness}]
Let
\[R = \sett{e=\set{g,g'}\in X(1)}{\L_g|_{\T{e}}\neq \L_{g'}|_{\T{e}}}\] be the set of ``dispute'' edges.
The rest of the proof is aimed towards showing $\disagr(\L)\geq \eps_0$ or equivalently, since $\disagr(\L) = |R|/|X(1)|$, that
\begin{equation}\label{eq:R} |R| \geq \frac{c-\lambda}{|A|+|B|}\cdot|X(1)| =\frac{c-\lambda}{2} \cdot |G|.
\end{equation}
First some more notations.
For an edge $\set{g,ag}\in X^A(1)$ let
\[\gpar(\set{g,ag}) = \sett{\set{gb,agb}\in X^A(1)}{b\in B}\] and similarly
for an edge $\set{g,gb}\in X^B(1)$, \[\gpar(\set{g,gb}) = \sett{\set{ag,agb}\in
X^B(1)}{a\in A}.\]
For a vertex $g$, let
\[ \ga(g) = \sett{\set{g,ag}}{a\in A},\qquad \gb(g) = \sett{\set{g,gb}}{b\in
B}. \]
We now make two claims on the local structure of $R$. The first is due to
the local distance, and the second is due to the local testability of tensor
codes.
\begin{claim}\label{claim:s1}
Suppose $\set{g,ag}\in R$, then
\[ |R\cap \gb(g)| + |R\cap \gb(ag)| + |R\cap\gpar{\set{g,ag}}|\geq \delta_B|B|.\]
Similarly, suppose $\set{g,gb}\in R$, then
\[ |R\cap \ga(g)| + |R\cap \ga(gb)| + |R\cap\gpar{\set{g,gb}}|\geq \delta_A|A|.\]
\end{claim}
\begin{proof}Let $e=\set{g,ag}\in R$, so $\L_g|_{X_e}\neq \L_{ag}|_{X_e}$. Since $\L_g|_{X_e},\L_{ag}|_{X_e}\in C_e$, these are two distinct codewords of $C_e$, and must disagree on at least $\delta_B|B|$ squares. Let $[a,g,b]$ be such a square, and look at the three edges of the square that are not $e$: $\set{g,gb},\set{gb,agb}$ and $\set{agb,ag}$. At least one of the three edges must be in $R$, because $\L_g,\L_{gb},\L_{agb},\L_{ag}$ cannot all agree on the value of $[a,g,b]$ without contradicting $\L_g([a,g,b]) \neq \L_{ag}([a,g,b])$. This implies the first part of the claim, and the second part is proven similarly.
\end{proof}
Recall that we assume $C_A\otimes C_B$ is agreement-testable, as per Definition \ref{def:atest}. \begin{claim}\label{claim:s2}
Assume $C_A\otimes C_B$ is $\kappa_0$-agreement-testable. For every $g\in G$,
\begin{equation}\label{eq:ltc}\Pr_a[\set{g,ag}\in R]+\Pr_b[\set{g,gb}\in R]\leq \kappa_0^{-1}\cdot \Pr_{a\in A,b\in B}[\set{ag,agb}\in R\hbox{ or }\set{gb,agb}\in R
].
\end{equation}
\end{claim}
\begin{proof}
Define $w_0,w_1,w_2:A\times B\to\bits$ as follows. First, let $w_0(a,b) = \L_g([a,g,b])$.
Next, let
$w_1(a,b) = \L_{ag}([a^{-1},ag,b])$. Similarly let $w_2(a,b)=\L_{gb}([a,gb,b^{-1}])$.
In words, the $a$th row of $w_1$ comes from the ``opinion'' of $\L_{ag}$,
and the $b$th column of $w_2$ comes from the ``opinion'' of $\L_{gb}$. Observe
that $w_0\in C_A\otimes C_B$, $w_1\in \bits^A\otimes
C_B$, and $w_2 \in C_A\otimes \bits^B$.
Now observe that $w_1(a,\cdot)\neq w_0(a,\cdot)$ iff $\set{g,ag}\in R$, and
$w_2(\cdot,b) \neq w_0(\cdot,b)$ iff $\set{g,gb}\in R$. Finally, $w_1(a,b)\neq
w_2(a,b)$ implies that the event on the RHS of \eqref{eq:ltc} holds, namely, $\set{ag,agb}\in
R\hbox{ or } \set{gb,agb}\in R$.
By the $\kappa_0$-agreement-testability of $C_A\otimes C_B$, there is a word $w\in C_A\otimes C_B$ such that
\[ \Pr_a[w(a,\cdot)\neq w_1(a,\cdot)] + \Pr_b[w(\cdot,b)\neq w_2(\cdot,b)] \leq \kappa_0^{-1}\cdot \Pr_{a,b}[w_1(a,b)\neq w_2(a,b)].
\]
Since the iterative algorithm has terminated, we know that
\[
\Pr_a[w_0(a,\cdot)\neq w_1(a,\cdot)] + \Pr_b[w_0(\cdot,b)\neq w_2(\cdot,b)] \leq \Pr_a[w(a,\cdot)\neq w_1(a,\cdot)] + \Pr_b[w(\cdot,b)\neq w_2(\cdot,b)]
\]
otherwise the algorithm would have flipped from $\L_g=w_0$ to $\L_g=w$. Combining the inequalities the claim follows,
\begin{align*}
\Pr_a[\set{g,ag}\in R]+\Pr_b[\set{g,gb}\in R] &=
\Pr_a[w_0(a,\cdot)\neq w_1(a,\cdot)] + \Pr_b[w_0(\cdot,b)\neq w_2(\cdot,b)] \\
&\leq
\Pr_a[w(a,\cdot)\neq w_1(a,\cdot)] + \Pr_b[w(\cdot,b)\neq w_2(\cdot,b)] \\
&\leq \kappa_0^{-1}\cdot \Pr_{a,b}[w_1(a,b)\neq w_2(a,b)]\\
&\leq \kappa_0^{-1}\cdot \Pr_{a\in A,b\in B}[\set{ag,agb}\in R\hbox{ or }\set{gb,agb}\in R].
\end{align*}
\end{proof}
Let $M_0 = \frac 1 2 M_A+\frac 1 2 M_B$. Clearly for any $f:X(0)\to\R$ such that $E[f]=0$, $\iprod{f,M_0 f} = \frac 1 2 \iprod{f,M_A f}+\frac 1 2 \iprod{f,M_B f} \leq \lambda\iprod{f,f}$.
Recall the distribution $\D_1$ over $X(1)$ from Definition~\ref{def:distr} and the corresponding inner product $\iprod{\cdot,\cdot}_{\D_1}$.
Define $\Down:\R^{X(1)}\to\R^{X(0)}$, $\Up:\R^{X(0)}\to\R^{X(1)}$ to be down and up operators, moving us from functions on edges to functions on vertices and vice versa. Namely,
\[\forall f_1\in \R^{X(1)},\qquad \Down f_1(g) = \E_{e\sim\D_1|g}[f_1(e)]= \frac 1 2 \E_{a\in A}[f_1(\set{g,ag})]+ \frac 1 2 \E_{b\in B}[f_1(\set{g,gb})]\] and \[\forall f_0\in \R^{X(0)},\qquad \Up f_0(\set{g_1,g_2}) = \E_{g\in \set{g_1,g_2}} [f_0(g)]=\frac 1 2(f_0(g_1)+f_0(g_2)).
\]
Note that these are averaging operators so they never increase norms, e.g. $\norm{\Down f} \leq \norm f$ for all $f$.
\begin{claim}
Let $M=\Up M_0\Down:\R^{X(1)}\to \R^{X(1)}$. Then $M$ has second largest eigenvalue at most $\lambda$.
\end{claim}
\begin{proof}
We rely on the fact that $\D_1$ can be described by first choosing a uniform vertex $g$ and then a random edge containing $g$ such that with probability half we choose an $A$ edge and with probability half a $B$ edge.
For any $f_1:X(1)\to\R$ and $f_0:X(0)\to\R$ we have
\[ \iprod{\Down f_1,f_0} = \E_{g}[\E_{e\sim\D_1|g} [f_1(e)]\cdot f_0(g)] = \E_{e\sim \D_1} [f_1(e)\E_{g\in e} [f_0(g)]] = \iprod{f_1,\Up f_0}_{\D_1}.
\] Now, if $\iprod{f_1,\one}=0$ then $\iprod{\Down f_1,\one}=0$, so
\[ \iprod{f_1,Mf_1}=\iprod{f_1,\Up M_0 \Down f_1} = \iprod{\Down f_1,M_0 \Down f_1} \leq \lambda\iprod{\Down f_1,\Down f_1}\leq \lambda\iprod{f_1,f_1}.
\]
\end{proof}
The following lemma is based on Claims \ref{claim:s1} and \ref{claim:s2}.
\begin{lemma}\label{lemma:stay} Fix $\gamma = \frac{\kappa_0}{8+\kappa_0}$. Let $M = \Up M_0 \Down$ and let $f=\ind_R:X(1)\to\R$ be the indiator function of the edge set $R$. Then
\[ \iprod{f,(\gamma\mpar +(1-\gamma) M)f}_{\D_1} \geq \gamma \cdot\min(\delta_A, \delta_B) \cdot \iprod{f,f}_{\D_1}.
\]
\end{lemma}
\def\EBA{\mathsf{E}_{BA}}
\def\EAB{\mathsf{E}_{AB}}
\def\Et{\mathsf{E}_{3}}
\begin{proof}
We give a combinatorial interpretation to $\gamma\mpar +(1-\gamma) M$ by observing that for a fixed $e\in X(1)$, $(\gamma\mpar +(1-\gamma) M)f(e)$ is the probability that $e'\in R$ in the following random process.
\begin{enumerate}
\item Start from an edge $e\in X(1)$.
\item\label{step:par} With probability $\gamma$, output a uniformly random edge $e'\in \gpar(e)$ and halt. With probability $1-\gamma$ continue.
\item\label{step:down} Choose at random one of the endpoints of the edge, $g_1 \in e$.
\item With probability $\frac 1 2$ let $g_2 = a_1g_1$ for a random $a_1\in A$, and with probability $\frac 1 2$ let $g_2= g_1b_1$ for a random $b_1\in B$.
\item With probability $\frac 1 2$ let $e'=\set{g_2, a_2g_2}$ for a random $a_2\in A$, and with probability $\frac 1 2$ let $g_2= g_2b_2$ for a random $b_2\in B$. Output $e'$.
\end{enumerate}
We will prove the lemma by showing that for every $e\in R$,
\begin{equation}\label{eq:stay}
(\gamma\mpar +(1-\gamma) M)f(e)\geq \gamma\cdot\min(\delta_A,\delta_B).
\end{equation}
So fix some $e \in R$, and for convenience assume $e = \set{g,ag}$ for some $g\in G,a\in A$ (if $e=\set{g,gb}$ the argument is symmetric). Let
\[r_0=|R\cap \gpar(e)|,\quad r_1 =|R\cap \gb(g)|,\quad r_2=|R\cap \gb(ag)|.\]
By Claim \ref{claim:s1}, $r_0+r_1+r_2 \geq \delta_B|B|$. With probability $\gamma$ step \ref{step:par} outputs a random $e'\in \gpar(e)$, and the probability it is in $R$ is $r_0/|B|$.
\begin{equation}\label{eq:step2}
\Pr[e'\in R] = \gamma \cdot r_0/|B| + (1-\gamma)\cdot \Pr[e'\in R\,|\,\hbox{the process entered step 3}]
\end{equation}
\inote{Alex, I think it is ok and not $r_0/|A|$}
Assume we entered step \ref{step:down}. Due to Claim \ref{claim:s2},
\begin{equation}\label{eq:edge}
\Pr_{a,b}[\set{ag_1,ag_1b}\in R \hbox{ or }\set{g_1b,ag_1b}\in R] \geq \kappa_0^{-1}\cdot r_i/|B|
\end{equation}
where $i\in\set{1,2}$ depending on whether $g_1=g$ or $g_1=ag$ as chosen in step \ref{step:down}.
What is the probability that $e'$ is one of the edges $\set{ag_1,ag_1b}$ and $\set{g_1b,ag_1b}$ considered in the LHS of \eqref{eq:edge}? This happens exactly if in steps $4$ and $5$ we will walk in alternating colors ($A,B$ or $B,A$). Let $\EAB$ be the event that in step $4$ we choose an $A$-edge, i.e. $g_2=a_1g_1$ for some $a_1\in A$ and then in step $5$ we set $e'$ to be a $B$-edge, i.e. $e' = \set{a_1g_1,a_1g_1b_2}$ for some $b_2\in B$. Similarly let $\EBA$ be the event that $g_2=g_1b_1$ and $e' = \set{g_1b_1,a_2g_1b_1}$. Clearly \[ \Pr[\EAB] = \Pr[\EBA] = \frac 1 4.
\]
Now,
\begin{equation}\label{eq:ab1}
\Pr[e'\in R\ve \EAB] = \frac 1 4\cdot \Pr_{a_1,b_2}[\set{a_1g_1,a_1g_1b_2}\in R],
\end{equation}
and
\begin{equation}\label{eq:ab2}
\Pr[e'\in R\ve \EBA] = \frac 1 4\cdot \Pr_{a_2,b_1}[\set{g_1b_1,a_2g_1b_1}\in R].
\end{equation}
where the probability is taken over the randomness of the random process above conditioned on having entered step $3$. Since $\EAB$ and $\EBA$ are disjoint events,
\begin{align*}
\Pr[e'\in R] &\geq
\Pr[e'\in R \ve \EAB]+\Pr[e'\in R \ve \EBA]\\
&\geq \frac 1 4 \cdot (\Pr_{a,b}[\set{a g_1,ag_1b}\in R]+\Pr_{a,b}[\set{g_1b,ag_1b}\in R])\\
&\geq \frac 1 4 \cdot
\Pr_{a,b}[\set{ag_1,ag_1b}\in R \hbox{ or }\set{g_1b,ag_1b}\in R]\\
&\geq \frac 1 4 \kappa_0\cdot r_i/|B| = \frac {r_i\kappa_0} {4|B|}
\end{align*}
where in the last inequality we have used \eqref{eq:edge}.
We conclude that if in step $3$ we choose $g_1=g$, then $\Pr[e'\in R]\geq\frac {r_1\kappa_0} {4|B|}$, whereas if in step $3$ we choose $g_1=ag$, then $\Pr[e'\in R]\geq\frac {r_2\kappa_0} {4|B|}$.
Altogether, recalling \eqref{eq:step2},
\[ \Pr[e'\in R] \ge \gamma\cdot \frac{r_0}{|B|} + (1-\gamma)\cdot \frac{\kappa_0}{ 4|B|}(r_1+r_2)/2.
\]
Plugging in $\gamma = \frac {\kappa_0} {8+\kappa_0}$ we get $1-\gamma ={8\gamma/\kappa_0}$, and recalling that $r_0+r_1+r_2\geq \delta_B|B|$,
\[\Pr[e'\in R] \ge \gamma (r_0+r_1+r_2)/|B| \geq \gamma \delta_B.
\]
We have seen that if $e=\set{g,ag}$ for some $a,g$ is in $R$, then $e'\in R$ with probability at least $\gamma \delta_B$.
Symmetrically, if $e=\set{g,gb}$ for some $g,b$ is in $R$ then we would get that $e'\in R$ with probability at least $\gamma \delta_A$. Together this proves \eqref{eq:stay} and completes the proof of the lemma.
\end{proof}
Recall from \eqref{eq:c} that $c = \frac{ \kappa_0}{8+\kappa_0} \cdot \min(\delta_A,\delta_B)$. By Lemma \ref{lemma:stay}, $\iprod{f,(\gamma\mpar + (1-\gamma)M)f} \geq c\cdot\iprod{f,f}$ so either
\begin{equation}\label{eq:parlarge}
\iprod{f,Mf}\geq c\iprod{f,f}
\end{equation}
or
\begin{equation}\label{eq:Mlarge}
\iprod{f,\mpar f}\geq c\iprod{f,f}.
\end{equation}
If \eqref{eq:parlarge} holds, then by Lemma \ref{lemma:AC-general}, applied with the operator $M$ whose vertex set is $X(1)$ is endowed with the distribution $\D_1$, we get $\Pr_{\D_1}[R] \geq c - \lambda$ which means that $|R|\geq \frac{|G|}{2}\cdot\min(|A|,|B|)(c-\lambda)$.
Otherwise, assume that \eqref{eq:Mlarge} holds.
By Lemma~\ref{lem:par} there exists some $\sigma\in \tilde A\cup \tilde B$ such that, $|R\cap X^\sigma(1)| \geq |G| (c-\lambda)/2$.
This completes the proof of Proposition~\ref{prop:soundness} showing that if $\disagr(\L)>0$ then $\disagr(\L)> \frac{2(c-\lambda)}{|A|+|B|}$.
\end{proof}
\section{A Concrete Construction}\label{sec:ins}
In the previous section we have described a code scheme: Given a left-right Cayley complex $Cay^2(A,G,B)$ together with two base codes $C_A\subseteq \bits^A$ and $C_B\subseteq\bits^B$, we get an error-correcting code $C[G,A,B,C_A,C_B]$.
In this section we prove our main theorem by showing how to find an infinite family of left-right Cayley complexes and base codes that yield locally testable codes.
\begin{theorem*}[Restatement of Theorem \ref{thm:main}]
For all $0<r<1$, there exist $\delta,\kappa>0$, $q\in \mathbb{N}$ and an explicit construction of an infinite family of error-correcting codes $\set{C_n}_n$, such that for each $n$, $\Rate(C_n) \geq r$, $\dist(C_n) \geq \delta$ and $C_n$ is $\kappa$-locally testable with $q$ queries.
\end{theorem*}
The proof of the theorem relies on the following two lemmas.
\begin{lemma}[Good base code]\label{lem:GBC}
For all $0<r_0<1$, there exist $\delta_0, \kappa_0 > 0$ and $d_0, D_0 \in\mathbb{N}$, such that for every integer $D>D_0$ that is divisible by $d_0$, there exists a linear error correcting code $C_0\subseteq\bits^D$ with rate at least $r_0$, distance at least $\delta_0$, and such that the tensor code $C_0 \otimes C_0$ is $\kappa_0$-agreement testable.
\end{lemma}
\begin{lemma}[Good left-right Cayley complexes]\label{lem:Cay}
Let $d_0,D_0 \in \mathbb{N}$.
Let $q$ be any odd prime power such that $q \geq \max \set{2d_0^2, D_0}$ and define $D = d_0 \cdot \lfloor \frac{q+1}{d_0} \rfloor$.
Then there exist an explicit construction of an infinite family of finite groups $G_i = PSL_2(q^i)$, with two symmetric generating subsets $A_i,B_i \subset G_i$, such that for each $i$, both $A_i$ and $B_i$ are of size $D$ hence divisible by $d_0$, $A_i$ and $B_i$ satisfy condition \eqref{eq:nc}, and the Cayley graphs $\mbox{Cay}(G_i,A_i)$ and $\mbox{Cay}(G_i,B_i)$ are $\lambda$-expanders where $ \lambda \leq 4 D^{-1/2}$.
\end{lemma}
We prove Lemma \ref{lem:GBC} in Subsection \ref{subsec:GBC} by showing that random LDPC codes are smooth.
We prove Lemma \ref{lem:Cay} in Section \ref{sec:LRCC} using the known constructions of Ramanujan graphs by Lubotzky, Samuels and Vishne \cite{LSV2} and Morgenstern \cite{morgenstern1994existence}.
Let us now deduce Theorem \ref{thm:main} from Lemmas \ref{lem:GBC} and \ref{lem:Cay}.
\begin{proof}[Proof of Theorem \ref{thm:main}]
Fix $0<r<1$ and set $r_0 = \frac{{r+3}}{4}$ so that $r=4r_0-3$.
By Lemma \ref{lem:GBC}, given $r_0$, there exist $\delta_0, \kappa_0 > 0$ and $d_0,D_0 \in \mathbb{N}$, such that for any $D>D_0$ divisible by $d_0$, there exists a code $C_0 \subset \bits^D$ with $\Rate(C_0) \geq r_0$, $\dist(C_0) \geq \delta_0$ and such that $C_0\otimes C_0$ is $\kappa_0$-agreement-testable.
Define $q_0 = \max \lbrace 2D_0,\; 2d_0^2,\; 32\left(\frac{\kappa_0 + 8}{\kappa_0 \delta_0} \right)^2 \rbrace$.
For any $q \geq q_0$ odd prime power denote $D= d_0 \cdot \lfloor \frac{q+1}{d_0} \rfloor$.
Note that $q+1 \geq D \geq q + 1 - d_0 > q - \sqrt{q} > \frac{1}{2} q$.
In particular $D > \frac{1}{2} q_0$, hence $4D^{-1/2} < \sqrt{ \frac{32}{q_0} } \leq \frac{\kappa_0 \delta_0}{8 + \kappa_0}$, which also implies $4D^{-1/2} < \delta_0$.
By Lemma \ref{lem:Cay} there exists an explicit construction of an infinite family of groups $G_i = PSL_2(q^i)$ together with generating sets $A_i,B_i$ such that for each $i\in\mathbb{N}$, $|A_i|=|B_i|=D $, conditions \eqref{eq:nc} holds, and both $Cay(G_i,A_i)$ and $Cay(G_i,B_i)$ are $\lambda = 4D^{-1/2}$ expanders.
In particular, from our choice of $D$, equation \eqref{eq:c} holds and $\lambda < \delta_0$.
By Lemma \ref{lem:GBC} there exists a code $C_0$ of length
$D$, with rate at least $r_0$, distance at least $\delta_0$ and such that the tensor code $C_0 \otimes C_0$ is $\kappa_0$-agreement testable. Since $D$ is a constant we can, theoretically, enumerate over all possible codes in search of a good one.
Define our family of global codes to be $C_i = C[G_i,A_i,B_i,C_0,C_0]$, $i\in \mathbb{N}$, and by the above choices it has the following parameters:
\begin{itemize}
\item Block-length $\frac{1}{4}|G_i|D^2$, where $|G_i| = \frac{1}{2}(q^{3i} - q^i)$.
\item Distance at least $\delta = \delta_0^2(\delta_0 - 4D^{-1/2}) > 0$, by Lemma \ref{lem:distance},
\item Rate at least $r = 4r_0-3 > 0$, by Lemma \ref{lem:rate},
\item It is $\kappa$-locally-testable with $D^2$ queries, by Theorem \ref{thm:mainltc}, for
\begin{equation}
\kappa = \min \left\lbrace \frac{1}{4+8D} \;,\; \frac{1}{4D} \left( \frac{\delta_0 \kappa_0}{8+\kappa_0} - 4D^{-1/2} \right) \right\rbrace .
\end{equation}
\end{itemize}
\end{proof}
\subsection{Good Base Codes} \label{subsec:GBC}
In this section we prove Lemma \ref{lem:GBC} by relying on the notion of smooth codes from \cite{DSW06}, which was consequently broadened to weakly-smooth codes in \cite{BVidweakly}. These works showed that the tensor product of a smooth code and any other code is robustly testable and therefore, by Lemma \ref{lem:robagr}, also agreement-testable.
\inote{replace with def of BV below?}
\begin{definition}[Smooth Code]
Let $c,d,n\in \mathbb{N},\alpha,\beta,\delta>0$. A $(c,d,n)$ LDPC code $C\subset
\bits^n$ is $(\alpha,\beta,\delta)$-smooth if for every $Y_0\subseteq Y$
with $|Y_0|\leq \alpha|Y|$ there is some $X_0\subseteq X$ with $|X_0|\leq
\beta|X|$ such that the code $C(\bar Y_0)|_{\bar X_0}$ has distance at least
$\delta$.
Here the code $C(\bar{Y_0})|_{\bar X_0}$ is the code obtained by removing
the constraints in $Y_0$ and then removing the coordinates in $X_0$.
\end{definition}
\subsubsection{Random LDPC Codes}
We will next show that random LDPC codes are smooth.
Random LDPC codes, see Definition \ref{def:ldpc}, were famously introduced by Gallager in his PhD thesis \cite{Gallager63}.
Given a $(c,d,n)$-LDPC code, let $m=nc/d$. By counting constraints it is easy to see that the dimension of an LDPC code is at least $n-m = n(1-\frac c d)$, so the rate is at least $1-\frac c d$.
Spielman described in his thesis \cite{Spielman96} the following expansion property,
\begin{definition}
A $(c,d)$-regular bipartite graph $([n],[m],E)$ is a $(\delta,\gamma)$-expander if every set of left vertices $A\subset [n]$ whose size is at most $\delta n$, has at least $c|A|(1-\gamma)$ neighbors.
\end{definition}
An LDPC code whose factor graph is a $(\delta,\gamma)$ expander immediately has distance at least $\delta$, as long as $\gamma<\frac 1 2$ \cite{Spielman96}.
A random $(c,d,n)$-code is given by selecting a random $(c,d)$-regular bipartite graph, which in turn is done by taking a random matching between the $nc$ ``half-edges'' on the left and the $md$ ``half-edges'' on the right, where we assume that $nc/d$ is an integer.
\begin{claim}[Claim 6.4 in \cite{Ben-SassonHR05}\footnote{A $(\delta,\gamma)$-expander here is called a $(c(1-\gamma),\delta)$-left-expander in \cite{Ben-SassonHR05}.}]\label{claim:cdexp-exist}
For every $c>2$, $d$, any constant $\gamma > \frac {1}c$, and sufficiently large $D$ such that $Dc/d$ is an integer, a random $(c,d)$-regular bipartite graph with $D$ left vertices is with high probability a $(\delta,\gamma)$-expander for any $\delta$ satisfying
\[
\delta\leq \left( 2e^{1+c(1-\gamma)}(d-d\gamma)^{c\gamma}\right)^{-\frac{1}{c\gamma-1}}. \qed
\]
\end{claim}
\begin{remark}
It can be extracted from the proof of Claim 6.4 in \cite{Ben-SassonHR05} that the first $D$ for which such a $(c,d)$-regular $(\delta,\gamma)$-expander exists, denote it by $D_0$, is upper bounded by a function of $c,d$ and $\gamma$.
More explicitly, if $c=4$ and $\gamma = \frac{5}{12}$, then
\[
D_0 \leq 2^{42} d^{15}.
\]
\end{remark}
The proof is similar Gallager's proof \cite{Gallager63} that a random LDPC code has constant distance with high probability.
Tensors of these codes are robustly testable,
\begin{theorem}[Robust testability of expander codes]\label{thm:tensors}
Let $C$ be a $(c,d,D)$-code whose factor graph is a $(c,d)$-regular $(\delta,\gamma)$-expander. Let $C'$ be any linear code with distance $\delta'$. Then $C \otimes C'$ is $\rho$-robustly testable for
\begin{itemize}
\item $\rho \ge \frac{\delta\delta' \cdot (\frac{1}{6} - \gamma)}{2d}$ when $\gamma<1/6$ \cite{DSW06}, and
\item $\rho \ge \frac{\delta\delta' \cdot }{d^{\log_{0.5 + \gamma}0.05}}$ for all $\gamma<1/2$ \cite{BVidweakly}.
\end{itemize}
\end{theorem}
Finally, we can prove Lemma \ref{lem:GBC}, which we restate for convenience,
\begin{lemma*}[Restatement of Lemma \ref{lem:GBC}]
For all $0<r_0<1$, there exist $\delta_0, \kappa_0 > 0$ and $d_0, D_0 \in\mathbb{N}$, such that for every integer $D>D_0$ that is divisible by $d_0$, there exists a linear error correcting code $C_0\subseteq\bits^D$ with rate at least $r_0$, distance at least $\delta_0$, and such that the tensor code $C_0 \otimes C_0$ is $\kappa_0$-agreement testable.
\end{lemma*}
\begin{proof}
We fix $\gamma=0.15<1/6$ and set $c_0=7$ so that $\gamma>1/c_0$. We choose $d_0 = \lceil\frac 7 {1-r_0} \rceil$ such that $\frac {c_0} {d_0} \leq 1- r_0$. Claim \ref{claim:cdexp-exist} guarantees existence of $\delta_0>0$ and $D_0$ such that for all $D>D_0$ divisible by $d_0$, a random $(c_0,d_0)$-regular bipartite graph with $D$ left vertices is with high probability a $(\delta_0,\gamma)$-expander.
For each such bipartite graph, we take $C_0$ to be the corresponding $(c_0,d_0,D)$-LDPC code. This code has rate at least $r_0$, distance at least $\delta_0$, and by taking $C' = C_0$ in Theorem \ref{thm:tensors}, we get that $C_0 \otimes C_0$ is robustly testable with $\rho=\Omega(\delta_0^2/d_0) = \Omega(\delta_0^2(1-r_0))$.
By Claim \ref{claim:rtoag} these codes are $\kappa_0$-agreement-testable for
$\kappa_0 = \Omega(\delta_0^3(1-r_0))$.
\end{proof}
We remark that the divisibility condition on $D$ is not really necessary. For $D$ not divisible by $d_0$ one can redistribute at most $d_0$ extra edges so that the graph is slightly irregular. The resulting graph is still a $(\delta,\gamma)$-expander, and one can also prove smoothness, mutatis mutandis, with a negligible change to the parameters.
\remove{
\subsubsection{Tensors of Reed-Solomon Codes}
\def\rs{\textsf{RS}}
The Reed-Solomon code is a linear code whose codewords are the pointwise evaluation of uni-variate polynomials $p\in \F_q[x]$ of degree at most $d$,
\[ \rs_d = \sett{f:\F_q\to\F_q}{\exists a_0,\ldots,a_d\in \F_q, f(x) = \sum_{i=0}^d a_i x^i}.\]
The rate of this code is $r=\frac {d+1} q$ and the distance is $\delta = \frac {q-d}q = 1-r+\frac 1 q$.
Tensor powers of this code are smooth. We focus on the $3$-wise tensor because it will fit nicely in our construction (the two-wise tensor is smooth but the robustness isn't enough).
Given $r>0$, let $r'$ be such that $r'^3>r$ and letting $d=r'q$ take $\bc = (\rs_d)^{\otimes 3}$ for any large enough $q$.
The distance of the code $\bc$ is $\delta^3$ and the rate is at least $r$. The block-length of this code is $D=q^3$, and the code is defined by constraints whose length is $d+2 \leq q=\sqrt[3]D$.
We will show that this code is weakly smooth. Let
\[\rs_d^\perp = \sett{ g:\F_q\to\F_q}{ \forall f\in \rs_d, \; \sum_{x\in \F_q} f(x)g(x)=0, \; wt(g)=d+2}\]
denote the set of constraints of length $d+2$ that define $\rs_d$. For a set $A\subset \F_q$ let $\rs^\perp_d(A) = \sett{g\in \rs^\perp_d }{ supp(g)\cap A=\phi }$ be the set of constraints that avoid touching $A$. The following is true because every $d+2$ points on a line are constrained in a Reed-Solomon code.
\begin{fact}\label{fact}
Let $A\subset \F_q$. Let
\[C' = (\rs_d^\perp(A))^\perp
= \sett{f:\F_q\to\F_q}{\forall g\in \rs^\perp_d(A),\; \sum_{x\in F_q} f(x)g(x)=0 }.\]
Then $C'|_{\F_q\setminus A} = \rs_d|_{\F_q\setminus A}$.
\end{fact}
\begin{lemma} The code $\bc$ is $(\frac{\delta^3}{432},\frac\delta 6,\frac{\delta^3}{64},d+2)$-weakly-smooth.
\end{lemma}
\begin{proof}
A line is a set of points obtained by fixing two coordinates, e.g. $(x,y,\cdot)$. A plane is a set of points obtained by fixing one coordinate, e.g. $(x,\cdot,\cdot)$.
Fix a set of points $A_0\subset [q]^3$ such that $\alpha_0 = |A_0|/q^3 \leq \frac{\delta^3}{432}$. We will show that there exists $A_1,A_2$ such that $|A_1\cup A_2|\leq \frac \delta 6\cdot q^3$ and for $A=A_0\cup A_1\cup A_2$ the following holds. Let
\[ constr_{\leq d+2}(A_0) = \sett{g\in \bc^\perp}{wt(g)\leq d+2, supp(g)\cap A_0=\phi}
\]
and let $C'= constr_{\leq d+2}(A_0)^\perp$. We will show that every non zero word $w\in C'|_{\F_q^3\setminus A}$ has weight at least $\delta^3/64 q^3$.
We say that a line $(x,y,\cdot)$ (or $(x,\cdot,z)$ or $(\cdot,y,z)$) is good if it contains at most $\delta q/4$ points from $A_0$. We say that a plane is good if it contains at most $\delta/8$ fraction of bad lines.
Let
\begin{align*}
A_1 &= \sett{(x,y,z)}{(x,y,\cdot)\hbox{ or }(x,\cdot,z)\hbox{ or }(\cdot,y,z)\hbox{ is a bad line}}\\
A_2 &= \sett{(x,y,z)}{(x,\cdot,\cdot)\hbox{ or }(\cdot,y,\cdot)\hbox{ or }(\cdot,\cdot,z)\hbox{ is a bad plane}}
\end{align*}
By an averaging argument,
\begin{align*}
|A_1| &\leq \frac{4\alpha_0}{\delta}\cdot 3q^2\cdot q = \frac{12\alpha_0}{\delta}\cdot q^3\\
|A_2| &\leq \frac{12\alpha_0}{\delta^2}\cdot 3q\cdot q^2 = \frac{36\alpha_0}{\delta^2}\cdot q^3.
\end{align*}
We choose $\alpha_0=\delta^3/432$ so that $|A_1|,|A_2| \leq \frac \delta {12} q^3$.
Let us call a point $(x,y,z)$ ``non-zero'' if it is not in $A = A_0\cup A_1\cup A_2$ and $w(x,y,z)\neq 0$.
\begin{claim}
If $(x_0,y_0,z_0)$ is non-zero then $(x_0,y_0,\cdot)$ has at least $\delta q/4$ non-zero points, and similarly for $(x_0,\cdot,z_0)$ and $(\cdot,\cdot,z_0)$. This implies that the weight of $w$ is at least $(\delta q/4)^3$.
\end{claim}
\begin{proof}
There are at most $\delta/12$ fraction of bad planes, so there are at most $\frac \delta 4 q$ bad planes of the form $(\cdot,y,\cdot)$ (similarly $(x,\cdot,\cdot)$, $(\cdot,\cdot,z)$).
By the assumption, $(x_0,\cdot,\cdot)$ is a good plane, otherwise $(x_0,y_0,z_0)\in A$. By definition, there must be at most $\delta/8$ fraction of bad lines contained in this plane, so at most $\delta q/4$ lines of the form $(x_0,y,\cdot)$ are bad. This means that on the line $(x_0,y_0,\cdot)$ at least $q(1-\frac{3\delta}{4})$ points are not in $A$, because at most $\delta/4$ of the points are in $A_0$, at most another $\delta/4$ of the points are in $A_1$, and at most $\delta/4$ of the points are in $A_2$. Using Fact \ref{fact} above, $w$ restricted to this set of points must have at least $q(\delta-3\delta/4)$ non-zero points.
We conclude that for every non-zero point, each of the three lines passing through it must have at least $\delta q/4$ non-zero points. From here it is immediate that the weight of $w$ is at least $(\delta q/4)^3 = \frac {\delta^3} {64}\cdot q^3$.
\end{proof}
\end{proof}
}
\section{Good Left-Right Cayley Complexes} \label{sec:LRCC}
In the previous section we showed how to construct good locally testable codes on good left-right Cayley complexes provided the latter exists.
To finish the proof of the main result of the paper, we should show that such complexes indeed exist and to give explicit construction.
Namely, in this section we prove Lemma \ref{lem:Cay}.
More generally, we show that for every $\lambda > 0$, there exist $k_1, k_2 \in \mathbb{N}$ and an infinite family of finite groups $G_i$, with two symmetric subsets of generators $A_i, B_i$, such that for each $i$, $|A_i| = k_1$ and $|B_i| = k_2$, the two sets $A_i$ and $B_i$ satisfies \eqref{eq:nc}, and the second largest eigenvalues of the adjecancy matrices of $\mbox{Cay}(G_i,A_i)$ and $\mbox{Cay}(G_i,B_i)$, denoted $\lambda(\mbox{Cay}(G_i,A_i))$ and $\lambda(\mbox{Cay}(G_i,B_i))$, are bounded from above by $\lambda$.
Moreover, we can take $\lambda = \Theta(k_1^{-1/2}) = \Theta(k_2^{-1/2})$, making both Cayley graphs quasi-Ramanujan.
There are a number of ways in the literature to find Cayley graphs with small $\lambda(\mbox{Cay}(G,S))$.
There are even various methods to give different sets of generators for the same group (see \cite{Lub10}, \cite{LSV2}).
The difficulty is to ensure that condition \eqref{eq:nc} is satisfied.
We will show two (actually three) ways to do so.
In all of our constructions, the elements in the sets $B_i$ will be of order $2$, while all the elements in $A_i$ will be of order greater then $2$.
This ensures that \eqref{eq:nc} is automatically satisfied.
\subsection{The Morgenstern Generators, $q=2^\ell$}
In \cite{morgenstern1994existence}, Morgenstern presented for every prime power $q$, infinitely many groups $G_i = PGL_2(q^i)$ or $G_i = PSL_2(q^i)$ each with a symmetric set $B_i$ of $q+1$ generators such that $\mbox{Cay}(G_i,B_i)$ are Ramanujan, i.e., $\lambda(\mbox{Cay}(G_i,B_i))\leq\frac{2\sqrt{q}}{q+1}$.
The case of $q$ even, i.e., $q = 2^{\ell}$, is special in two ways.
First of all, here $PGL_2(q^i) = PSL_2(q^i)$, so this is always a simple group.
But more importantly, in this case all the elements of $B_i$ are of order $2$ (see Remark \ref{rem:B_i}).
Assume $q$ is even from now on.
Morgenstern constructed an explicit arithmetic lattice $\Gamma$ in the group $PSL_2(\mathbb{F}_q((t)))$ which is isomorphic to the free product $\langle b_0\rangle*\ldots*\langle b_q\rangle$, where $B=\{b_0,\ldots,b_q\}$ is a set of elements of order $2$ (see \cite[Section 5]{morgenstern1994existence}).
The above mentioned Cayley graphs $\mbox{Cay}(G_i,B_i)$ are identified as quotients of this $\Gamma$ by normal congruence subgroups, where $B_i = \phi_i(B)$ is the image of $B$ under an epimorphism $\phi_i\,:\,\Gamma\rightarrow G_i$.
Note that by \cite{morgenstern1994existence} these Cayley graphs are all Ramanujan.
Let us now show how to get another symmetric set of generators $A_{i}$ for $G_{i}=PSL_{2}(q^{i})$ with $\lambda(\mbox{Cay}(G_{i},A_{i}))$ small, and such that $A_i$ and $B_i$ satisfy \eqref{eq:nc}.
Let $\Lambda$ be the index $2$ subgroup of $\Gamma$ - the kernel of the homomorphism $\phi\,:\,\Gamma\rightarrow C_2$ (= the cyclic group of order $2$) where $\phi$ sends each $b_j$ to the unique non-trivial element of $C_2$.
One can see easily that $\Lambda$ is exactly the subgroup of all elements of $\Gamma$ of even length w.r.t. $B$.
It is generated by the set $A=\{b_t b_s \;|\; b_t, b_s \in B,\; t\ne s\}$ which is of size $k_1 = q^2+q$.
We claim
\begin{claim} \label{Claim:Morgenstern-A}
(i) For $i > 1$, the image $A_i = \phi_i(A)$ of $A$ in $G_i$ generates $G_i = PSL_2(q^i)$.
(ii) $\lambda(\mbox{Cay}(G_i,A_i)) < \frac{3q-1}{q^2+q} < \frac{3\sqrt{k_1-1}}{k_1}$.
(iii) For $i > 4$, the images of the elements of $A$ in $G_i$ are distinct from one another, and each element in $A_i$ has order $>2$.
\end{claim}
\begin{proof}
(i) Since $\Lambda=\langle A\rangle$ is of index two in $\Gamma$ then $\langle A_{i}\rangle$ is of index at most two in $G_{i}$.
But $G_i = PSL_2(q^i)$ is simple, hence it has no index $2$ subgroup (a subgroup of index $2$ must be normal), which implies $\langle A_i \rangle = G_i$ .
(ii) Let $T_{B}$ and $T_{A}$ be the (non-normalized) adjacency matrices of $\mbox{Cay}(G_{i},B_{i})$ and $\mbox{Cay}(G_{i},A_{i})$, respectively.
Note that $T_{B}^{2}=T_{A}+(q+1)I$.
Hence if $\mu$ is an eigenvalue of $T_{A}$, then $\mu=\lambda^{2}-(q+1)$ for some eigenvalue $\lambda$ of $T_{B}$.
Since $\mbox{Cay}(G_{i},B_{i})$ is Ramanujan, $|\lambda|=q+1$ or $|\lambda|\leq2\sqrt{q}$.
Therefore $\mu=q^{2}+q$ or $\mu\leq(2\sqrt{q})^{2}-(q+1)=3q-1$.
(iii) It suffices to show that each reduced word which is a product of length at most $4$ in $B$ is not in the kernel of $\phi_i$, which is equivalent to the girth of $\mbox{Cay}(G_i,B_i)$ being greater than $4$.
By \cite[Theorem 5.13 (3)]{morgenstern1994existence} the girth of $\mbox{Cay}(G_i,B_i)$ is at least $\frac{2}{3}\log_q|G_i|\geq i$, which completes the proof.
\end{proof}
Thus, given $\lambda>0$ by taking $q$ large enough so that $\frac{3\sqrt{q^{2}+q-1}}{q^{2}+q}<\lambda$, we get the desired $\lambda$-expanding left-right Cayley complexes with $k_{1}=q^{2}+q$ and $k_{2}=q+1$.
We can do slightly better. Note that $\Lambda$ above, being a normal subgroup of a free product of finite groups, with trivial intersection with each factor is a free group (see Section 34 in \cite{Kurosh}).
In fact, by the Reidemeister-Schreier algorithm applied to the transversal set $\{1,b_{0}\}$ of $\Lambda$ in $\Gamma$ (or by inspection) one can see that $\Lambda$ is a free group on the $q$ generators $\{b_{0}b_{j}\;:\;j=1,\ldots,q\}$.
As $(b_{0}b_{j})^{-1}=b_{j}b_{0}$ we deduce that $A'=\{b_{0}b_{j},\,b_{j}b_{0}\;:\;j=1,\ldots,q\}$ is a symmetric set of generators of $\Lambda$.
We can now look at the image $A'_i = \phi_i(A')$ under the epimorphism $\phi_i\,:\,\Gamma \rightarrow G_i$.
Arguing similarly to the proof of Claim \ref{Claim:Morgenstern-A} (i), $A'_i$ generates $G_i$, and by the proof of (iii) above, the images are all different.
Finally:
\begin{claim}\label{Claim:Morgenstern-A'}
$\lambda(\mbox{Cay}(G_{i},A_{i}'))<\frac{3\sqrt{2q-1}}{2q}$.
\end{claim}
\begin{proof}
Let $V_{i}=\{f\,:\,G_{i}\rightarrow \mathbb{C}\}$ and for any element $s\in G_{i}$, define the $s$-adjecancy $T_{s}\,:\,V_{i}\rightarrow V_{i}$, $T_{s}f(g)=f(gs)$, and for any multiset $S$ of $G_{i}$, define the $S$-adjecancy operator $T_{S}\,:\,V_{i}\rightarrow V_{i}$, $T_{S}=\sum_{s\in S}T_{s}$.
Note that for any two multisets $S,S'$ of $G_{i}$, $T_{S\cup S'}=T_{S}+T_{S'}-T_{S\cap S'}$ and $T_{S}T_{S'}=T_{SS'}$, where $SS'=\{ss'\,:\,s\in S,s'\in S'\}$ counted with multiplicities.
Therefore $T_{A_{i}'}=T_{b}T_{B_{i}}+T_{B_{i}}T_{b}-2I$, where $b=\phi_{i}(b_{0})$.
Let $f\in V_{i}$ be such that $f\perp1_{G_{i}}$, i.e. $\sum_{g\in G_{i}}f(g)=0$.
Note that for any $s\in G_{i}$, then $T_{s}f\perp1_{G_{i}}$ and $\|T_{s}f\|=\|f\|$.
By \cite[Theorem 5.11]{morgenstern1994existence}, we have $\|T_{B_{i}}f\|\leq2\sqrt{q}\|f\|$ for any $f\perp1_{G_{i}}$.
Then
\[
\|T_{A_{i}'}f\|\leq\|T_{b}T_{B_{i}}f\|+\|T_{B_{i}}T_{b}f\|+2\|f\|\leq\|T_{B_{i}}f\|+\|T_{B_{i}}(T_{b}f)\|+2\|f\|
\]
\[
\leq\left(2\sqrt{q}-1\right)\|f\|+\left(2\sqrt{q}-1\right)\|f\|+2\|f\|=4\sqrt{q}\|f\|\leq3\sqrt{2q-1}\|f\|
\]
which completes the proof.
\end{proof}
So this time we have a family of $\lambda$-expanders left-right Cayley complexes with $k_{1}=2q$ and $k_{2}=q+1$, for any $\lambda>\frac{3\sqrt{2q-1}}{2q}$.
\begin{remark} \label{rem:B_i}
Everything said above is explicit.
In fact the generator set $B_i$ of $PSL_2(q^i)$ are given explicitly in \cite[equation (21)]{morgenstern1994existence}.
Assume $i$ is even.
Let $\textbf{i} \in \mathbb{F}_{q^i}$ be such that $\textbf{i} \not \in \mathbb{F}_q$ and $\epsilon = \textbf{i}^2 + \textbf{i} \in \mathbb{F}_q$.
Let $x\in\mathbb{F}_{q^i}$ be such that $1,x,\ldots,x^{e_i-1}$ form a basis for $\mathbb{F}_{q^i}$ over $\mathbb{F}_q$.
Then the $q+1$ elements of $B_i$ are
\begin{equation} \label{eq:B_i-Morgenstern}
\phi_{i}(b_j)=\left(\begin{array}{cc} 1 & \gamma_j + \delta_j\textbf{i}\\ x(\gamma_j + \delta_j + \delta_j\textbf{i}) & 1 \end{array}\right),\qquad j=0,\ldots,q,
\end{equation}
where $(\gamma_j,\delta_j)\in\mathbb{F}_q^2$ are the $q+1$ solutions in $\mathbb{F}_q$ for $\gamma^2 + \gamma\delta + \delta^2 \epsilon = 1$.
One indeed sees that each of the elements of $B_i$ is of order $2$.
\end{remark}
We will pass now to a different construction, which will give us Cayley graphs of $G_{i}$ w.r.t. $A_{i}$ and $B_{i}$ of the same size:
$|A_{i}|=|B_{i}|=q+1$, and both are Ramanujan.
\subsection{The LSV Generators, $q$ odd}
In \cite{LSV2}, Lubotzky, Samuels and Vishne constructed Ramanujan complexes, based on an arithmetic lattice $\Gamma$, discovered
by Cartwright and Steger \cite{cartwright1998family}, which acts simply transitively on the Bruhat-Tits building of $PGL_{d}(\mathbb{F}_{q}((t)))$.
The special case $d=2$ gave some new Ramanujan graphs.
These Ramanujan graphs were highlighted in \cite{kaufman2011edge}, as edge-transitive Ramanujan graphs which have been used there to construct symmetric LDPC codes.
The arithmetic group $\Gamma$, acting simply transitive on the Bruhat-Tits tree of $PGL_{2}(\mathbb{F}_{q}((t)))$ ($q$ any odd prime power) is obtained there as a the group generated by the $q+1$ conjugates of a specific element $b$, conjugated by the non-split torus $T$ of order $q+1$ in $PGL_{2}(\mathbb{F}_{q})$.
This is a symmetric set of generators $A$ for $\Gamma$ which generates a free group on $\frac{q+1}{2}$ generators.
We will present below a different choice for $b$, this time $b'$ - an element of order $2$, whose conjugation under $T$ forms a symmetric
set $B$ of size $q+1$ and generate a group $\Gamma'$ which also acts simply transitive on the Bruhat-Tits tree.
Moreover, $\Gamma$ and $\Gamma'$ are both finite index subgroups of an arithmetic group $G(R)$ - to be defined below.
In \cite{LSV2} (see also \cite{kaufman2011edge}) it was shown that $G(R)$ has infinitely many finite congruence quotients $G_{i}$, under the maps $\phi_{i}\,:\,G(R)\rightarrow G_{i}$, for which $\mbox{Cay}(G_{i},\phi_{i}(A))$ are Ramanujan $(q+1)$-regular graphs.
We will observe below that the same holds for $\mbox{Cay}(G_{i},\phi_{i}(B))$.
For $i$ large enough (see Claim \ref{Claim:LSV-generators-order}) the elements of $\phi_{i}(A)$ are of order $>2$ while $\phi_{i}(B)$ contains only elements of order $2$.
Hence we will get two-sided Cayley square complexes with $k_{1}=k_{2}=q+1$ and $\lambda\leq\frac{2\sqrt{q}}{q+1}$.
By choosing $q$ large enough, they will be $\lambda$-expanders for arbitrarly small $\lambda>0$.
Now, in more details: Let $0\ne\epsilon\in \mathbb{F}_{q}$ be a non-square element, let $R=\mathbb{F}_{q}[y,\frac{1}{y},\frac{1}{1+y}]$ be the subring of $\mathbb{F}_{q}(y)$, generated by $y$, $\frac{1}{y}$ and $\frac{1}{1+y}$, and let $A(R)$ be the quaternion $R$-algebra,
\begin{equation}
A(R)=R+R\alpha+Rz+R\alpha z\qquad:\qquad\alpha^{2}=\epsilon,\quad z^{2}=1+y,\quad z\alpha=-\alpha z.\label{eq:A(R)}
\end{equation}
\begin{remark}
We note that our choice of basis for $A(R)$, $\{1,\alpha,z,\alpha z\}$, is based on \cite{kaufman2011edge}, while \cite{LSV2} used a different basis for $A(R)$, $\{\xi,\xi^{q},\xi z,\xi^{q}z\}$, where $\{\xi,\xi^{q}\}$ forms an $\mathbb{F}_{q}$-basis for $\mathbb{F}_{q^{2}}=\mathbb{F}_{q}[\alpha]$.
The change of bases does not affect any of the following constructions.
\end{remark}
For any ring $D$, denote by $D^{*}$ its group of units.
Note that an element of $r(y)\in R$ belongs to $R^{*}$ if and only if it is of the form $r(y)=cy^{n}(1+y)^{m}$ , $c\in\mathbb{F}_{q}^{*}$, $n,m\in\mathbb{Z}$, and that an element $a=a_{1}+a_{2}\alpha+a_{3}z+a_{4}\alpha z\in A(R)$ belongs to $A(R)^{*}$ if and only if its norm $N(a):=a_{1}^{2}-\epsilon a_{2}^{2}-(1+y)a_{3}^{2}-\epsilon(1+y)a_{4}^{2}\in R$ belongs to $R^{*}$.
Note also that $R$ is the center of $A(R)$ and $R^{*}$ is the center of $A(R)^{*}$.
Then the principal arithmetic group $G(R)$ is defined to be
\[
G(R)=A(R)^{*}/R^{*}=\left\{ a\in A(R)\,:\,N(a)\in R^{*}\right\} /R^{*}.
\]
The Cartwright--Steger arithmetic lattice $\Gamma$, and the second arithmetic lattice $\Gamma'$, are defined to be the subgroups of $G(R)$, generated by the symmetric sets of size $q+1$, $A$ and $B$, which are the sets of $T$ conjugates of the elemenets, $b$ and $b'$, respectively, where $T=\mathbb{F}_{q}[\alpha]^{*}/\mathbb{F}_{q}^{*}\leq G(R)$ is a non-split torus of order $q+1$, $b=\left(1-\frac{1}{1+y}z\right)R^{*}\in G(R)$ and $b'=\alpha b=\left(\alpha-\frac{1}{1+y}\alpha z\right)R^{*}\in G(R)$, namely,
\[
\Gamma=\langle A\rangle\leq G(R),\quad A=\left\{ tbt^{-1}\,:\,t\in T\right\} ,\quad \Gamma'=\langle B\rangle\leq G(R),\quad B=\left\{ tb't^{-1}\,:\,t\in T\right\} ,
\]
\[
T=\mathbb{F}_{q}[\alpha]^{*}/\mathbb{F}_{q}^{*},\qquad b=\left(1-\frac{1}{1+y}z\right)R^{*},\qquad b'=\alpha b=\left(\alpha-\frac{1}{1+y}\alpha z\right)R^{*}.
\]
Note that $b$ and $b'$ belongs to $G(R)$, since $N\left(1-\frac{1}{1+y}z\right)=1-(1+y)\frac{1}{(1+y)^{2}}=\frac{y}{1+y}\in R^{*}$ and $N\left(\alpha-\frac{1}{1+y}\alpha z\right)=N(\alpha)\cdot N\left(1-\frac{1}{1+y}z\right)=-\epsilon\cdot\frac{y}{1+y}\in R^{*}$.
\begin{claim} \label{Claim:LSV-generators-order}
(i) Every element of $A$ is of infinite order, while every element of $B$ is of order $2$.
(ii) For $i > 2$, every element of $A_{i}=\phi_{i}(A)$ is of order $>2$, while every element of $B_{i}=\phi_{i}(B)$ is of order $2$.
\end{claim}
\begin{proof}
(i) The claim about the elements of $A$ follows from \cite[Corollary 5.4]{LSV2}.
For the claim about the elements of $B$, since they are all conjugate of one another, it suffice to show $b'^{2}=1$, or equivalently, $\left(\alpha-\frac{1}{1+y}\alpha z\right)^{2}\in R^{*}$.
This follows from the following computations,
\[
\left(\alpha-\frac{1}{1+y}\alpha z\right)^{2}=\alpha^{2}-\frac{1}{1+y}\alpha\alpha z-\frac{1}{1+y}\alpha z\alpha+\frac{1}{(1+y)^{2}}\alpha z\alpha z=*,
\]
and by equation \ref{eq:A(R)}, as $\alpha z=-z\alpha$, $\alpha^{2}=\epsilon$ and $z^{2}=1+y$, we get
\[
*=\alpha^{2}-\frac{1}{(1+y)^{2}}\alpha^{2}z^{2}=\epsilon-\frac{\epsilon}{1+y}=\epsilon\frac{y}{1+y}\in R^{*}.
\]
(ii) This follows from an injectivity radius argument for congruence subgroups, see for instance \cite{lubotzky2007moore}.
\end{proof}
Let $\mathcal{B}$ be the Bruhat-Tits tree of $PGL_{2}(\mathbb{F}_{q}((t)))$, which is a $(q+1)$-regular infinite tree.
By \cite[Section 3]{LSV2}, $\Gamma$, $\Gamma'$ and $G(R)$ are subgroups of $PGL_{2}(\mathbb{F}_{q}((t)))$, hence acts on $\mathcal{B}$.
In the notation of \cite{LSV2}, let $v_{0}=[L_{0}]$ be the fundamental vertex in $\mathcal{B}$, and let $\Omega$ be the set of its neighbors.
\begin{claim} \label{Claim:LSV-generators-Cayley-Ramanujan}
(i) For each set $X=A$ or $X=B$, the map $g\leftrightarrow g.v_{0}$ is a bijection between $X$ and $\Omega$.
(ii) The subgroups, $\Gamma$ and $\Gamma'$, acts simply transitively on the Bruhat-Tits tree.
(iii) Both subgroups, $\Gamma$ and $\Gamma'$, are normal in $G(R)$ and of index $2(q+1)$.
(iv) If $\phi\,:\,G(R)\rightarrow PSL_{2}(q^{e})$ is an epimorphism, then both subsets, $\phi(A)$ and $\phi(B)$, are symmetric set of generators for $PSL_{2}(q^{e})$.
(v) If $\phi\,:\,G(R)\rightarrow PSL_{2}(q^{e})$ is an epimorphism whose kernel is a congruence subgroup $G(R,\phi)$ of $G(R)$, then both Cayley graphs, $\mbox{Cay}(PSL_{2}(q^{e}),\phi(A))$ and $\mbox{Cay}(PSL_{2}(q^{e}),\phi(B))$, are Ramanujan $(q+1)$-regular graphs.
\end{claim}
\begin{proof}
(i) The claim for $A$ is \cite[Proposition 4.3]{LSV2}.
The claim for $B$ follows from the claim for $A$ and the identity $tb't^{-1}.v_{0}=t\alpha bt^{-1}.v_{0}=(t\alpha)b(t\alpha)^{-1}(t\alpha t^{-1}).v_{0}$.
Now, $\alpha\in T$ and $T$ fixes $v_{0}$, so $\{tb't^{-1}.v_{0}|t\in T\}=\{tbt^{-1}.v_{0}|t\in T\}$.
(ii) The transitivity claim for $\Gamma$ is \cite[Proposition 4.5]{LSV2}, which relies solely on the validity of claim (i) for the generating set $A$ of $\Gamma$, hence the same proof works also for $\Gamma'$.
Moreover, the same proof can actually show that for any $n\in\mathbb{N}$, for any vertex $v$ of distance $n$ from $v_{0}$, there exists a reduced word $g=s_{1}\cdots s_{n}\in\Gamma$ (resp. $\Gamma'$), $s_{1},\ldots,s_{n}\in A$ (resp. $B$), such that $g.v_{0}=v$.
This proves that the action is also simple since the number of vertices of distance $n$ is equal the number of reduced words of length $n$, for any $n\in\mathbb{N}$.
(iii) The claim for $\Gamma$ follows from \cite[Propositions 4.9 and 3.5]{LSV2}, and the same proof also works for $\Gamma'$.
The fact that the index is $2(q+1)$ follows also from the fact that $\Gamma'$ acts simply transitively on the Bruhat-Tits tree by (ii).
Hence the index of $\Gamma'$ in $G(R)$ is equal to the order of the stabilizer of $v_{0}$ in $G(R)$, which by \cite[Proposition 3.5]{LSV2}, is of size $2(q+1)$.
(iv) By (iii) both images, $\phi(\Gamma)$ and $\phi(\Gamma')$, are normal subgroups of index $\leq2(q+1)$ in $PSL_{2}(q^{e})$, and since $PSL_{2}(q^{e})$ is a simple group of size $\geq\frac{1}{2}(q+1)q(q-1)>2(q+1)$, we get that $\phi(\Gamma)=PSL_{2}(q^{e})=\phi(\Gamma')$.
(v) The claim for $\mbox{Cay}(PSL_{2}(q^{e}),\phi(A))$ is \cite[Theorem 7.1]{LSV2}, and the same proof holds also for $\mbox{Cay}(PSL_{2}(q^{e}),\phi(B))$.
Another way to see this is to observe that both graphs are isomorphic to $G(R,\phi)\backslash \mathcal{B}$ and in particular they are isomorphic, so if one is Ramanujan so is the other.
\end{proof}
\subsection{Proof of Lemma \ref{lem:Cay} and Degree Reduction}
First we use the LSV generators constructed in the previous subsection to prove the following Lemma.
\begin{claim}\label{claim:LSV-graphs}
For any odd prime power $q$ there exist an explicit construction of an infinite family of finite groups $G_i = PSL_2(q^i)$, with two symmetric generating subsets $A_i,B_i$ of $G_i$, such that for each $i$, $|A_i|=|B_i|=q+1$, condition \eqref{eq:nc} holds for $A_i$ and $B_i$, and the Cayley graphs $\mbox{Cay}(G_i,A_i)$ and $\mbox{Cay}(G_i,B_i)$ are Ramanujan, in particular they are $\lambda$-expanders with $ \lambda \leq 2 (q+1)^{-1/2}$.
\end{claim}
\begin{proof}
From Claim \ref{Claim:LSV-generators-Cayley-Ramanujan} we get that for any $i$, there exists two symmetric generating subsets $A_i$ and $B_i$ of the finite group $G_i = PSL_2(q^i)$, both sets are of size $q+1$, and the Cayley graphs $\mbox{Cay}(G_i,A_i)$ and $\mbox{Cay}(G_i,B_i)$ are both Ramanujan.
By Claim \ref{Claim:LSV-generators-order}, for any $i > 2$, the two sets $A_i$ and $B_i$ satisfy condition \eqref{eq:nc}.
\end{proof}
Next we prove the following degree reduction trick, which allows us to start with a $\lambda$-expander Cayley graph, and to remove a few elements from the generating set with only negligible effect on $\lambda$.
\begin{claim} \label{claim:degree-reduction}
(i) Let $G$ be a finite group, let $S' \subset S$ be two symmetric subset of $G$, and denote $\lambda = \lambda(\mbox{Cay}(G,S))$ and $\lambda' = \lambda(\mbox{Cay}(G,S'))$ the normalized second largest eigenvalues of the corresponding Cayley graphs.
Then
\[
\lambda' \leq \lambda + \frac{|S \setminus S'|}{|S'|}.
\]
(ii) In particular, if $c \leq \lambda \cdot |S|^{1/2} \leq C$, where $0<c<C$, and $|S\setminus S'| \leq \frac{1}{2} \cdot c \cdot |S|^{1/2}$, then
\[
\lambda' \leq 2 \lambda \leq 2C\cdot |S'|^{1/2}.
\]
\end{claim}
\begin{proof}
(i) Let $M = M_A$ and $M' = M_{A'}$ be the adjacency matrices of $\mbox{Cay}(G,S)$ and $\mbox{Cay}(G,S')$, respectively.
Since $S$ (resp. $S'$) generates $G$, the largest eigenvalue of $M$ (resp. $M'$), which is $|S|$ (resp. $|S'|$), has multiplicity one, and its eigenvector is the constant function $1_G$.
By the Courant-Fischer Formula we get that
\[
\lambda \cdot |S| = \max_{0 \ne v \perp 1_G} \frac{v^tMv}{v^tv} \qquad \mbox{and} \qquad \lambda' \cdot |S'| = \max_{0 \ne v \perp 1_G} \frac{v^tM'v}{v^tv}.
\]
Now the matrix $M-M'$ can be considered as the adjacency matrix of the set $S\setminus S'$, which by the Perron-Frobenius Theorem, all of its eigenvalues are bounded in absolute value by $|S \setminus S'|$, and by the Courant-Fischer Formula $|S \setminus S'| = \max_{0 \ne v} \frac{v^t(M' - M)v}{v^tv} $.
Therefore we get that
\[
\lambda' \cdot |S'| = \max_{0 \ne v \perp 1_G} \frac{v^tM'v}{v^tv}
\leq \max_{0 \ne v \perp 1_G} \frac{v^tMv}{v^tv} + \max_{0 \ne v \perp 1_G} \frac{v^t(M' - M)v}{v^tv}
\leq \lambda \cdot |S| + |S \setminus S'|,
\]
and after dividing by $|S'|$ we get the claim.
(ii) follows from (i) together with the fact that
\[
|S \setminus S'| \leq \frac{1}{2} \cdot c \cdot |S|^{1/2} \leq \frac{\lambda |S|}{1 + \lambda} \quad \Rightarrow \quad \frac{|S\setminus S'|}{|S|} \leq \lambda.
\]
\end{proof}
Finally we combine the above two Claims to prove Lemma \ref{lem:Cay}.
\begin{lemma*}[Restatement of Lemma \ref{lem:Cay}]
Let $d_0,D_0 \in \mathbb{N}$.
Let $q$ be any odd prime power such that $q \geq \max \set{2d_0^2, D_0}$ and define $D = d_0 \cdot \lfloor \frac{q+1}{d_0} \rfloor$.
Then there exist an explicit construction of an infinite family of finite groups $G_i = PSL_2(q^i)$, with two symmetric generating subsets $A_i,B_i \subset G_i$, such that for each $i$, both $A_i$ and $B_i$ are of size $D$ hence divisible by $d_0$, $A_i$ and $B_i$ satisfy condition \eqref{eq:nc}, and the Cayley graphs $\mbox{Cay}(G_i,A_i)$ and $\mbox{Cay}(G_i,B_i)$ are $\lambda$-expanders where $ \lambda \leq 4 D^{-1/2}$.
\end{lemma*}
\begin{proof}[Proof of Lemma \ref{lem:Cay}]
First note that $D$ is by definition the largest integer $\leq q+1$ which is divisible by $d_0$, and that $q+1-D \leq d_0 \leq \frac{1}{2} \sqrt{D}$.
By Claim \ref{claim:LSV-graphs}, for each $i$, there exist $\tilde A_i,\tilde B_i$ two symmetric generating subset of $G_i = PSL_2(q^i)$, such that $\tilde A_i,\tilde B_i$ are both of size $q+1$, they satisfy \eqref{eq:nc} and such that the corresponding Cayley graphs are Ramanujan, i.e. $\lambda$-expandrs for $\lambda\leq \frac{2\sqrt{q}}{q+1}\leq 2(q+1)^{-1/2}$.
Let $A_i \subset \tilde A_i$ and $B_i \subset \tilde B_i$ be any two symmetric subsets of size $D$.
Since $\tilde A_i$ and $\tilde B_i$ satisfy \eqref{eq:nc}, any subsets of them must also satisfy \eqref{eq:nc}.
By Claim \ref{claim:degree-reduction}, we get that for $G= G_i$, $S= \tilde A_i$ or $\tilde B_i$, and $S' = A_i$ or $B_i$, respectively, we get that
\[
\lambda(\mbox{Cay}(G,S')) \leq 2 \lambda(\mbox{Cay}(G,S)) \leq 4 D^{-1/2},
\]
which completes the proof of the Lemma.
\end{proof}
\newcommand{\etalchar}[1]{$^{#1}$}
| 6,700
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Detectives in Harrogate are investigating after an elderly woman was targeted by a bogus caller in Cundall.
The victim, 74-year-old woman, lost a large amount of cash when she was tricked into letting a man claiming to be a workman into her home.
At around 10.30am on Monday 7 February 2011, the suspect called at the victim’s house on Swale Avenue and tricked his way in by claiming he needed to test the water.
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For further advice about guarding against distraction burglaries visit the North Yorkshire Police website
| 373,223
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\begin{document}
\begin{frontmatter}
\title{Density of periodic points, invariant measures and almost equicontinuous points of Cellular Automata }
\author[M]{Pierre Tisseur}
\ead{pierre@xaravve.trentu.ca}
\address[M]{Department of Mathematics, Trent University,
Peterborough, Ontario, Canada}
\date{}
\maketitle
\begin{abstract}
Revisiting the notion of $\mu$-almost equicontinuous cellular automata introduced by R. Gilman, we show that
the sequence of image measures of a shift ergodic measure $\mu$ by iterations of a $\mu$-almost equicontinuous
cellular automata $F$, converges in
Cesaro mean to an invariant measure $\mu_c$.
If the initial measure $\mu$ is a Bernouilli measure, we prove that the Cesaro mean limit
measure $\mu_c$ is shift mixing.
Therefore we also show that for any shift ergodic and $F$-invariant measure $\mu$, the existence of
$\mu$-almost equicontinuous
points implies that the set of periodic points is
dense in the topological support $S(\mu )$ of the invariant measure $\mu$.
Finally we give a non trivial example of a
couple ($\mu$-equicontinuous cellular automata $F$, shift ergodic and $F$-invariant measure $\mu$) which has no
equicontinuous point in $S(\mu )$.
\end{abstract}
\begin{keyword}
Cellular automata, ergodic theory, discrete dynamical systems
\MSC 37B15, 37A35,37A25
\end{keyword}
\end{frontmatter}
\section{Introduction}
A one-dimensional cellular automaton (CA) is a discrete mathematical idealization of a space-time
physical system. The space, called configuration space, is the set of
doubly infinite sequences of elements of a finite set $A$. The discrete time is
represented by the action of a cellular automaton $F$ on this space.
Using extensive computer simulations, Wolfram \cite{Wo86} has proposed a first empirical (visual)
classification of one dimensional cellular automata.
In \cite{GI87} Gilman propose a formal and measurable classification by roughly dividing the
set of cellular automata in two parts, those with {\it almost equicontinuous
points or equicontinuous points} and those with {\it almost expansive points} (partition in order and disorder).
The Gilman's classes are defined thanks to
a Bernoulli measure $\mu$ which correspond to the Wolfram's simulations that
use random entry.
The measure does not need to be invariant, so the
classification can be apply to any cellular automata.
In \cite{Ku94}, Kurka, introduce a topological classification based on the
equicontinuity, sensitivity and expansiveness properties.
If a cellular automaton has equicontinuous points, then there exist finite configurations that
stop the propagation of the perturbations on the one dimensional lattice. If a cellular automaton
has $\mu$-almost equicontinuous
points then the probability that a perturbation move to infinity is equal to zero (see \cite{GI87}).
Remark that the class of cellular automata with almost equicontinuous points contains the
topological class of CA with equicontinuous points.
In this paper we consider the definitions of Gilman ($\mu$ expansiveness and $\mu$ equicontinuity)
in the more general case of shift ergodic measures. We show that under this condition, if a cellular automaton $F$ has
$\mu$-equicontinuous points then the sequence $(\mu\circ F^{-n})$ converges in Cesaro mean
to an invariant measure $\mu_c$. We prove that $F$ has still $\mu_c$-equicontinuous points and
if the initial shift ergodic measure is a Bernouilli measure, then $\mu_c$ is
a shift mixing measure. Remark that the convergence in Cesaro mean of CA with
equicontinuous points in $S(\mu )$ (the topological support of a measure $\mu$), had been done by
Blanchard an Tisseur in \cite{BLTI}.
In \cite{GI87} Gilman gives an example of a $\mu$-equicontinuous CA that has no equicontinuous points.
The invariant measure $\mu_c$, (limit by Cesaro mean of $(\mu\circ F^{n})$)
that we can construct (using ou results) for this particular automaton still has $\mu_c$-equicontinuous
points, but the restriction of this CA to the topological support of $\mu_c$ has equicontinuous points.
Using Cesaro mean of images measures, we describe a cellular automaton with a non trivial dynamic
which keep the sensitiveness property (no equicontinuous points)
if we restrict its action to the topological support $S(\mu_c )$ of the
invariant measure $\mu_c$.
This example use a "counter'' dynamic and is defined
thanks to a composition of 5 cellular automata acting on different shifts.
In \cite{BK} Boyle and Kitchen have shown that closing cellular automata
have always a dense set of periodic points. The expansive CA and some
cellular automata with equicontinuous points belong to this large class.
Here, we prove that if $\mu$ is a shift ergodic measure and $F$ is a $\mu$-invariant cellular automaton
with $\mu$-equicontinuous points then the set of $F$-periodic points is dense in the
the topological support $S(\mu )$.
This result extends a previous result on the density of periodic points of
surjective with equicontinuous points, cellular automata acting on a mixing subshift of finite type
(see \cite{BLTI}).
Finally, even if $\mu$-equicontinuity appears to have a more complex dynamic that equicontinuity for almost all the points,
we show that the measurable entropy $h_\mu (F)$ of any cellular automaton $F$
with $\mu$-almost equicontinuous points is equal to zero if the measure $\mu$ is shift ergodic and $F$ invariant.
\section{Definitions and preliminary results}
\subsection{Symbolics systems and cellular automata}
Let $A$ be a finite set or alphabet. Denote by $A^*$ the set of all
concatenations of letters in $A$. These concatenations are called words. The
length of a word $u\in A^*$ is denoted by $\vert u\vert$.
The set of bi-infinite sequences
$x=(x_i)_{i\in\ZZ}$ is denoted by $A^\ZZ$. A point $x\in A^\ZZ$ is called a
configuration. For $i\le j$ in $\ZZ$ we denote by $x(i,j)$ the word $x_i\ldots
x_j$ and by $x(p,\infty )$ the infinite sequence $(v_i)_{i\in\NN}$ such that for
all $i\in\NN$ one has $v_i=x_{p+i-1}$. We endow $A^\ZZ$ with the product
topology. The shift $\sigma \colon A^\ZZ\to A^\ZZ$ is defined by :
$\sigma (x)=(x_{i+1})_{i\in \ZZ}$. For each integer $t$ and each word $u$, we
call cylinder the set $[u]_t=\{x\in A^\ZZ : x_t=u_1\ldots ;x_{t+\vert
u\vert}=u_{\vert u\vert}\}$. For this topology $A^\ZZ$ is a compact metric
space. A metric compatible with this topology can be defined by the distance
$d(x,y)=2^{-i}$ where $i=\min\{\vert j\vert \,\mbox{ such that } x(j)\ne y(j)\}$.
The dynamical system $(A^\ZZ ,\sigma )$ is called the full shift. A subshift $X$
is a closed shift-invariant subset $X$ of $A^\ZZ$ endowed with the shift
$\sigma$.
If $\alpha =\{A_1,\ldots ,\,
A_n\}$ and $\beta =\{B_1,\ldots ,\, B_m\}$ are two partitions denote
by $\alpha \vee \beta$ the partition $\{A_i\cap A_j\, i=1\ldots n, \,\,
j=1,\ldots ,\, m\}$.
Consider a probability measure $\mu$ on the Borel sigma-algebra
$\mathcal{B}$ of $A^\ZZ$. If $\mu$ is $\sigma$-invariant then the topological
support of $\mu$ (which is the smallest closed subset of mesure 1) is a subshift denoted by $S(\mu )$.
The metric entropy $h_\mu (T)$ of a transformation $T$
is an isomorphism invariant between two $\mu$-preserving
transformations. Put $H_\mu (\alpha ) = \sum_{A\in\alpha}\mu (A)\log \mu (A)$.
The entropy of the partition $\alpha$ is defined as $h_\mu (\alpha ) =
\lim_{n\to\infty}1/nH_\mu (\vee_{i=0}^{n-1}T^{-i}\alpha )$ and the entropy of $(X,T,\mu )$ as
$\sup_\alpha h_\mu (\alpha )$.
A cellular automaton (CA) is a continuous self-map $F$ on
$A^\ZZ$ commuting with the shift. The Curtis-Hedlund-Lyndon theorem
states that for every cellular automaton $F$ there exist an integer $r$ and a
block map $f$ from $A^{2r+1}$ to $A$ such that: $F(x)_i=f(x_{i-r},\ldots ,x_i
,\ldots ,x_{i+r}).$
The integer $r$ is called the radius of the cellular
automaton. If the block map of a cellular automaton is such that
$F(x)_i=f(x_{i},\ldots ,\ldots ,x_{i+r})$,
the cellular automaton is called
one-sided and can be extended a map on a two-sided shift $A^\ZZ$ or a map on a
one-sided shift $A^\NN$. If $X$ is a subshift of $A^\ZZ$ and one has
$F(X)\subset X$, the restriction of $F$ to $X$ determines a dynamical system
$(X,F)$; it is
called a cellular automaton on $X$.
For example, given any shift invariant measure we can consider the restriction of the
cellular automaton $(F, A^\ZZ)$ to $(F, S(\mu ))$.
A closed subset of $Y\subset A^\ZZ$ (not necessarily shift-invariant) such that
$F(Y)\subset Y$ is said $F$-invariant.
\subsection{Almost equicontinuous points of cellular automata}
In \cite{GI87} Gilman shows that for a Bernoulli measure $\mu$ it is possible to divide the
cellular automata set in three following classes: The class of CA where there exists equicontinuous points,
the class of CA with $\mu$-almost equicontinuous points but without equicontinuous point and the class of
almost expansive CA.
In this section we recall the topological and measurable definitions for
the expansiveness and equicontinuous classes
of cellular automata $F$ of radius $r$ acting on set $A^\ZZ$ where $A$ is a finite set.
Since all the Gilman proof use the shift ergodicity of the Bernoulli measure, we extend the initial definitions
to any shift ergodic measure.
For any real $\epsilon >0$, define $D(x,\epsilon )$ the set of the point $y$ such that for all
$i\in\NN$ one has $d(F^i(x),F^i(y))\le\epsilon$ and for all positive integer $n$ write
$B_n(x)=D(x,2^{-n})$.
Let $\mu$ be any probability measure on $A^\ZZ$ and for any $x\in A^\ZZ$ define the cylinder set
$C_n(x)$ as the set of $y$ such $y_i=x_i$ with $-n\le i\le n$. We use these two types of set to rewrite
the following definitions.
\begin{defis} Equicontinuity
$\mbox{ }$
-A point $x\in A^\ZZ$ is called an equicontinuous point if for all positive integer $n$ there exist another
positive integer $m$ such that $B_n(x)\supset C_n(x)$.
-A cellular automata is an almost equi\-continuous CA if there exists equi\-conti\-nuous points.
-A cellular automaton is equicontinuous is all the points $x\in A^\ZZ$ are equicontinuous points.
-A point $x$ is called a $\mu$-almost equicontinuous if $\mu (B_n(x))>0$ for all $n\ge r$.
-If $\mu$ is a shift ergodic measure, a cellular automaton $F$ is $\mu$-almost equicontinuous if there exists some
$\mu$-almost equicontinuous point $x$.
\end{defis}
\begin{defis} Expansiveness
$\mbox{ }$
-A Cellular automaton is positively expansive if there exists a positive integer $n$
such that for all $x\in A^\ZZ$ one has $B_n(x)=\{x\}$.
-A cellular automaton $F$ is almost expansive if there exists a positive integer $n$
such that for all $x\in A^\ZZ$,
$\mu (B_n(x))=0$ where $\mu$ is a shift ergodic measure.
-A cellular automaton is sensitive if for all points $x\in A^\ZZ$ and each
integer $m>0$ one has $C_m(x)\subsetneq B_r(x)$.
\end{defis}
\begin{rem}
If $x$ is an equicontinuous point for a CA $F$ which belong to the topological
support $S(\mu )$ of some probability measure $\mu$, then $x$ is also a $\mu$-equicontinuous point
for $F$.
If the measure $\mu$ is shift ergodic, then $F$ is a $\mu$-equicontinuous CA.
If there exists a point $x$ and integer $m>0$ such that $C_m(x)\subset B_r(x)$ then $x(-m,m)$
is called a blocking word.
In \cite{GI87} Gilman prove the partition of the space of CA into two classes ($\mu$-equicontinuous
and $\mu$-equicontinuous classes) with
respect to a Bernoullli measure $\mu$ .
The partition in $\mu$-equicontinuous and $\mu$-expansive CA (see \cite{GI87}) is still true
for any shift ergodic measure $\mu$.
\end{rem}
The following result appears in \cite{GI87} (in a slightly different form and for Bernoulli measures).
\begin{pro}\label{pro1}\cite{GI87}
Let $\mu$ be a shift-ergodic measure and $F$ a cellular automaton with
radius $r$. The following are properties are equivalent:
\hskip .3 true cm (i) There exist a point $x\in A^\ZZ$ such that
$\mu (B_r(x))>0$.
\hskip .3 cm (ii) The set of $\mu$-equicontinuous points has measure 1 for $F$.
\hskip .3 cm (iii) Almost all points $x$ verify that
for any integer $m\ge 0$ one has
$$
\lim_{n\to\infty}\left(\mu \left(C_n(x)\cap B_m(x)\right)/
(\mu (C_n(x))\right)=1.
$$
\end{pro}
\begin{ques}
Is it possible that there exist some non $\mu$-equicontinuous point $x$ such that there exist an
integer $m\ge r$ with $\mu (B_m(x))>0$?
\end{ques}
The next topological result is due to Gilman (see \cite{GI88}).
\begin{pro}\label{pro2}\cite{GI88}
If there exist a point $x$ and an integer $m\neq 0$ such that $B_n(x)\cap \sigma^{m}B_n(x)\neq\emptyset$
with $n\ge r$ then the common
sequence {\small $(F^i(y)(-n,n))_{i\in\NN}$} of all points $y\in B_n(x)$ is ultimatly periodic.
\end{pro}
{\it Proof:}
Since all the elements of $B_n(x)$ share the same sequence $(F^i(x)(-n,n))_{i\in\NN}$ and
the orbit under $F$ of each shift periodic point is ultimatly periodic, it
is sufficient to show that $B_n(x)$ contains a shift periodic element.
Take an integer $n$ greater than the radius $r$ of $F$ and
pick two points $z_1=z_1^-x(-n,n)z_1^+$ and $z_2=z_2^-x(-n,n)z_2^+$ in $B_n(x)$.
As the automaton depends on a local rule of radius $r$, each point
$z_{(i,j)}=z_i^-x(-n,n)z_j^+$ belongs to $B_n(x)$ for all
$(i,j)\in \{1,2\}\times\{1,2\}$.
Pick a point $y_1=y_1^-x(-n,n)y_1^+=y_1^{-*}x(-n+m,n+m)y_1^{+*}$ in $B_n(x)\cap\sigma^{m}B_n(x)$ and
remark that the point $y_2=y_1^{-*}x(-n,n)y_1^{+}$ belongs to $B_n(x)$.
Suppose without loosing generalities that $m>0$ and define the word $w=x(-n,-n+m)$.
By construction we get $y_2(-n,-n+2m)=ww$ and since $y_2\in \cup_{i=0}^2\sigma^{-i}B_n(x)$
we obtain that $\sigma^{-m}y_2\in B_n(x)$.
Repeating the same process, we obtain for all
positive integer $k$ a point $y_k\in \cup_{i=0}^{k-1}\sigma^{-i}B_n(x)$ such that $y_k(-n,-n+km)=w^k$.
As $B_n(x)$ is a closed and compact set,
we can conclude by saying that the $\sigma$ periodic point
$y=\lim_{i\to\infty}\sigma^{-im}y_{2i}$ belong to $B_n(x)$.
\hfill$\Box$
In \cite{GI88} Gilman state the following result using the ergodic properties
of any Bernoulli measure $\mu$.
\begin{pro}\cite{GI88}\label{pro3}
Let $\mu$ be a shift ergodic measure. If a cellular automaton $F$ has a $\mu$-equicontinuous point,
then for all $\epsilon >0$ there exists a $F$-invariant closed set $Y$
such that $\mu (Y)>1-\epsilon$ and the restriction of $F$ to $Y$ is equicontinuous.
\end{pro}
{\it Squetch of the Proof:}
Let $x$ be a $\mu$-equicontinuous point and $m$ a positive integer greater than
the radius $r$ of the cellular automaton $F$. Since $\mu (B_m(x))>0$, and
$\mu$ is a shift ergodic measure, there exists a set $S$ of measure 1 such that
all points $y$ in $S$ intersect infinitively often the set $B_n(x)$.
From Proposition \ref{pro2} for each positive integer $k$ the sequences
$(F^n(y)(-k,k))_{n\in\NN}$
are ultimatly periodic.
Let $Y_{(p(k))}$ be the set of point $y$ such that each sequence $(F^n(y)(-k,k))_{n\in\NN}$ are
periodic of period $p$ and preperiod $p(k)$.
As the measure $\mu$ is shift ergodic, for each real $\epsilon >0$ there exists a map
$p_\epsilon\colon \NN\to\NN$ such that for all $n\in\NN$ we have $\mu (Y_{(p_\epsilon (n))})>1-\epsilon\times 2^{-n}$.
The set $Y=\lim_{n\to\infty}\cap_{i=1}^nY_{(p(i))}$ is closed since it is an intersection of
the closed sets
$Y_{(p(i)}$ and $\mu (Y)> 1-\epsilon$.
Each point $y\in Y$ is clearly an equicontinuous point.
\hfill$\Box$
\section{Main results}
\subsection{Measure entropy and density of the set of periodic points}
\begin{pro}\label{pro4}
If a cellular automaton $F$ have some $\mu$-equicontinuous points with $\mu$
a $F$-invariant and shift-ergodic measure then its measure entropy $h_\mu (F)$ is equal to zero.
\end{pro}
{\it Proof:}
Let $\alpha_p$ be the partition of $A^\ZZ$ by the $2p+1$ central coordinates. Two points $x$ and
$y$ belong to the same element of $\alpha_p$ if and only if $x(-p,p)=y(-p,p)$. Let
$\alpha_n^p(x)$ be the element of the partition $\alpha_p\cap F^{-1}\alpha_p\ldots F^{-n+1}\alpha_p$
which contains $x$.
Clearly for all $n\in\NN$ we have $\alpha^n_p(x)\supset B_p(x)$.
From Proposition 1, there exist a set of points $Z$ with measure 1 such that
if $y\in Z$ then $\mu (B_{m}(y))>0$ for all integer $m\ge 0$.
This implies that for almost all $y$ and positive integer $p$, we have
$\lim_{n\to\infty}\frac{-\log \mu (\alpha_p^n(y))}{n}\le \lim_{n\to\infty}\frac{-\log \mu (B_p(x))}{n}=0$.
Using the Shannon theorem which tell that
$$
h_\mu (F,\alpha_p )=\int_{A^\ZZ} \lim_{n\to\infty}\frac{-\log \mu (\alpha_p^n(y))}{n}d\mu (y),
$$
we can conclude that $h_\mu (F)=\sup_{\alpha_p}h_\mu (F,\alpha_p)=0$.
\hfill$\Box$
\begin{rem}
From remark 1, all cellular automata $F$ which have equicontinuous points
in the topological support of a shift ergodic measure $\mu$, verify : $h_\mu (F)=0$.
\end{rem}
\begin{pro}\label{pro5}
Let $\mu$ be a shift ergodic and cellular automaton $F$, invariant measure .
If $F$ has $\mu$-almost equicontinuous points then the set of $F$-periodic points is dense in
the topological support $S(\mu )$.
\end{pro}
{\it Proof:}
We are going to show that
for any point $z\in S(\mu )$ and positive integer $p$ we can construct a $\sigma$ and $F$ periodic point
$\bar{w}=^\infty w^\infty$ such that there exists an integer $r\le k\le \vert w\vert$
with $w(k, 2p+1+k)=z(-p,p)$. We recall that $r$ is the radius of $F$.
Since there is a $\mu$-almost equicontinuous point $x\in S(\mu)$ then $\mu (B_r(x))>0$.
Since $\mu$ is shift ergodic, there exist positive integers $i$ and $j$ such that
$\mu \big(C_p(z)$ $\cap\sigma^{-(i+p)}B_r(x)\cap\sigma^{j+p}B_r(x)\big )>0$.
To simplify, write $S=C_p(z)$ $\cap\sigma^{-(i+p)}B_r(x)\cap\sigma^{j+p}B_r(x)$ and pick a point $y\in S$.
From Proposition \ref{pro2}, there exist a shift periodic point $\overline{w}\in S$ such that
$\overline{w}(-r-i-p,j+p)=w=y(-r-i-p,j+p)$.
From the Poincar\'e recurrence theorem, there exists an integer $m$ such that
$\mu\left( S\cap F^{-m}S\right)>0$. This implies that there exist a point $y\in S$ such that
$F^m(y)(-r-i-p,r+j+p)=y(-r-i-p,r+j+p)$.
But from the definition of $B_r(x)$, all the points $y\in S$ share the same sequence
$\left(F^i(y)(-r-i-p,j+p)\right)$ $=\left(F^{i}(\overline{w})(-r-i-p,j+p)\right)_{i\in\NN}$.
Since the common $\sigma$ period of $\overline{w}$ is $r+i+j+2p+1$ we obtain that
$F^m(\overline{w})=\overline{w}$.
Arguing that the sequence of images by a cellular automaton of any shift periodic point is
ultimatly periodic we can assert that $(F^i(\overline{w}))$ is a periodic sequence and conclude.
\hfill$\Box$
\subsection{Invariant measures as limit of Cesaro means}
Proposition \ref{pro2} also allows us to prove a Cesaro mean convergence result.
\begin{thm}\label{thm1}
Let $\mu$ be a shift-ergodic measure. If a cellular automaton $F$ has some
$\mu$-almost equicontinuous points then $(\mu\circ F^{-n})$ converges vaguely in C\'esaro mean under $F$
to an invariant measure $\mu_c$.
\end{thm}
{\it Proof}
To show that the sequence of measure $(\frac{1}{n}\sum_{i=0}^{n-1}\mu\circ F^{-i})_{n\in\NN}=(\mu_n)_{n\in\NN}$
converge vaguely in measure we need to show
that for all $x\in S(\mu)$ and $m\in\NN$ the sequence $\left(\mu_n (C_m(x))\right)_{n\in\NN}$ converges.
Since there exist a point $z$ and an integer $m>0$ with
$\mu (B_m(z))>0$ where $\mu$ a shift ergodic measure, we get
$\lim_{n\to\infty}\mu (\cup_{i=-n}^{n}\sigma^{-i}B_m(z))=1$.
Using the same arguments than in Proposition \ref{pro3} we can assert that there exists a set $Y_{({\bf P}_\epsilon (k))}$
of measure greater than $1-\epsilon$ such that all the sequences
$(F^n(y)(-k,k))_{n\in\NN}$ are eventually periodic with preperiod $pp_\epsilon (k)$ and
period $p_\epsilon (k)$ if $y\in Y_{({\bf P}_\epsilon (k),k)}$ and
${\bf P}_\epsilon (k)=(pp_\epsilon (k),p_\epsilon (k))$.
Hence for all $x\in A^\ZZ$
$$
\begin{array}{ll}
\mu_n (C_m(x)\cap Y_{({\bf P}_\epsilon (k))})=&\frac{1}{n}\sum_{i=0}^{pp_\epsilon (k)-1}\mu \left
( F^{-i} \left
(C_m(x)\right )\cap Y_{({\bf P}_\epsilon (k))}\right )\cr
&+\frac{1}{n}\sum_{i=pp_\epsilon (k)}^{n-1}\mu \left ( F^{-i}
\left (C_m(x)\right )\cap Y_{({\bf P}_\epsilon (k))}\right ).
\end{array}
$$
The first term tends to $0$; using periodicity one gets
$$
\lim_{n\to\infty}\mu_n (C_m(x)\cap Y_{({\bf P}_\epsilon (k))})=\frac{1}{p_\epsilon (k)}
\sum_{i=0}^{p_\epsilon (k)-1}\mu
\left ( F^{-(i+pp_\epsilon (k))} (C_m(x) \cap Y_{({\bf P}_\epsilon (k))}\right ).
$$
Clearly if $k\ge m$ we have
$\lim_{\epsilon\to 0}\mu_n(C_m(x)\cap Y_{({\bf P}_\epsilon (k))})=\mu_n(C_m(x))$.
The convergence is uniform with respect to $\epsilon$ since for all $x$ and $m\in\NN$
$$
\left\vert\mu_n (C_m(x)\cap Y_{({\bf P}_\epsilon (k))} )-\mu_n (C_m(x))\right\vert \le \frac{n\epsilon}{n}
=\epsilon .
$$
Consequently, letting $\epsilon$ going to $0$ and assuming that $k\ge m$, we get the result by inversing the limits
$$
\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\mu\circ F^{-i}(C_m(x))
=\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\lim_{\epsilon\to 0}
\mu\circ F^{-i}(C_m(x)\cap Y_{({\bf P}_\epsilon (k))} )
$$
$$
=\lim_{\epsilon\to 0}\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}
\mu\circ F^{-i}(C_m(x)\cap Y_{({\bf P}_\epsilon (k))} )
$$
$$
=\lim_{\epsilon\to 0}\frac{1}{p_\epsilon (k)}\sum_{i=0}^{p_\epsilon (k)-1}\mu
\left ( F^{-(i+pp_\epsilon (k))} (C_m(x)
) \cap Y_{({\bf P}_\epsilon (k))}\right )=\mu_c(C_m(x)).
$$
We denote by $\mu_c$ the Cesaro mean limit of $(\mu\circ F^{n})_{n\in\NN}$.
\hfill$\Box$
\bigskip
In the following Proposition and Corrolary, we suppose that $\mu_c$ is a
Probability measure on $A^\ZZ$ which came from the Cesaro mean of $(\mu\circ F^{-n})_{n\in\NN}$.
Remark that the three results of this subsection, remain true for cellular automata with equicontinuous points in $S(\mu )$.
\begin{pro}\label{pro7}
If $\mu$ is a Bernouilli measure and $F$ a $\mu$-equicontinuous CA, then
the Cesaro mean measure $\mu_c$ is a shift mixing measure.
\end{pro}
{\it Proof}
It is enough to show that for any points $x$ and $y$ and positive integer $m$ and
$n$ we have $\lim_{t\to\infty}\mu_c(C_m(x)\cap \sigma^{-t}C_n(y))=\mu_c(C_m(x))\times\mu_c(C_n(y))$.
From the proof of Theorem 3, we get for any $k\ge \max\{m,n\}$
$$
\mu_c(C_m(x))=
\lim_{\epsilon\to 0}\frac{1}{p_\epsilon (k)}\sum_{i=0}^{p_\epsilon (k)-1}\mu
\left ( F^{-(i+pp_\epsilon (k))} (C_m(x)
) \cap Y_{({\bf P}_\epsilon (k))}\right )
$$
and
$$
\mu_c(C_n(y))=
\lim_{\epsilon\to 0}\frac{1}{p_\epsilon (k)}\sum_{i=0}^{p_\epsilon (k)-1}\mu
\left ( F^{-(i+pp_\epsilon (k))} (C_n(y)
) \cap Y_{({\bf P}_\epsilon (k))}\right ).
$$
Remark that for all $y\in Y_{({\bf P}_\epsilon (k),k)}$, the sequence
$\left( F^{pp_\epsilon (k)+n}(y)(-k,k)\right)_{n\in\NN}$ is periodic.
Since $\mu$ is a Bernouilli measure it follows that
when $t\ge 2k+1+2pp_\epsilon (k)\times r$ we obtain for all $i\in\NN$
$$
\mu \left ( F^{-i} (C_m(x))\cap Y_{({\bf P}_\epsilon (k))}\cap F^{-i}(\sigma^{-t}C_n(y))\cap \sigma^{-t}
Y_{({\bf P}_\epsilon (k))}\right )
$$
$$
=\mu \left ( F^{-i} (C_m(x))\cap Y_{({\bf P}_\epsilon (k))}\right )\times
\mu \left(F^{-i}(\sigma^{-t}C_n(y))\cap \sigma^{-t}
Y_{({\bf P}_\epsilon (k))}\right ).
$$
We write $t_\epsilon =2k+1+2pp_\epsilon (k)\times r$.
Remark that for all $t\in\NN$ and $\epsilon >0$, we have $\mu (Y_{({\bf P}_\epsilon (k))}\cap
\sigma^{-t}Y_{({\bf P}_\epsilon (k)))}>1-2\epsilon$.
Let $A=\liminf_{t\to\infty}\mu_c (C_m(x)\cap \sigma^{-t}C_n(y))=$
\small
$$
\liminf_{t\to\infty}\lim_{n\to\infty}\lim_{\epsilon\to 0}\frac{1}{n}\sum_{i=0}^{n-1}
\mu \left ( F^{-i} (C_m(x))\cap \sigma^{-t}C_n(y))\cap Y_{({\bf P}_\epsilon (k))}
\cap \sigma^{-t}
Y_{({\bf P}_\epsilon (k))}\right ).
$$
\normalsize
Using similar arguments of those arising in the proof of Theorem 3, for the
convergence with respect to $n$ and the uniform convergence
with respect to $\epsilon$, we can write that $A$ is equal to
\small
$$
\liminf_{t\to\infty}\lim_{\epsilon\to 0}\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}
\mu \left ( F^{-i} (C_m(x))\cap \sigma^{-t}C_n(y))\cap Y_{({\bf P}_\epsilon (k))}
\cap \sigma^{-t}
Y_{({\bf P}_\epsilon (k))}\right ).
$$
Using the uniform convergence with respect to $\epsilon$, we can invert the limitinf
with respect to $t$ and the limit with respect to $\epsilon$ and obtain that $A$ is equal to
$$
\lim_{\epsilon\to 0}\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}
\mu \left ( F^{-i} (C_m(x))\cap \sigma^{-t_\epsilon}C_n(y))\cap Y_{({\bf P}_\epsilon (k))}
\cap \sigma^{-t_\epsilon}
Y_{({\bf P}_\epsilon (k))}\right )
$$
$$
=\lim_{\epsilon\to 0}\frac{1}{p_\epsilon (k)}\sum_{i=0}^{p_\epsilon (k)-1}
\mu \left ( F^{-i+pp_\epsilon (k)} \left(C_m(x))\cap \sigma^{-t_\epsilon}C_n(y)\right)
\cap Y_{({\bf P}_\epsilon (k))}\cap \sigma^{-t_\epsilon}
Y_{({\bf P}_\epsilon (k))}\right ).
$$
Using the independance of the finite sets
$\{F^{-i+pp_\epsilon (k)} \left(C_m(x))\right)\cap Y_{({\bf P}_\epsilon (k))}\, i\in\NN\}$ and
the image by $\sigma^{-t_\epsilon}$ of
$\{F^{-i+pp_\epsilon (k)} \left(C_n(y))\right)\cap Y_{({\bf P}_\epsilon (k))}\, i\in\NN\}$
with respect to the measure $\mu$ it follows that
$
A=\mu_c (C_m(x))\times \mu_c (C_n(y)).$
If we substitute $\limsup$ instead of $\liminf$ in $A$ we obtain the same result,
so we can conclude that $\lim_{t\to\infty}\mu_c(C_m(x)\cap \sigma^{-t}C_n(y))=\mu_c(C_m(x))\times\mu_c(C_n(y))$.
\normalsize
\hfill$\Box$
\begin{rem}\label{remmesure}
If the initial measure $\mu$ is the direct product of a Bernouilli measure and some
atomic measure on a point of the type $\overline{a}=\ldots aaaa\ldots$ which consists in the repetition of the
same letter $a$ in the alphabet $A$, we obtain that the
limit measure $\mu_c$ is shift mixing
using the same arguments than in Theorem \ref{thm1}.
\end{rem}
\begin{pro}\label{pro8}
If $(S (\mu ) ,F)$ is a $\mu$-almost equicontinuous cellular automaton and
the Cesaro mean limit measure $\mu_c=\frac{1}{n}\sum \mu\circ F^{-i}$ is shift mixing, then
$\mu_c$ is still $\mu_c$-equi\-con\-tinuous for $(S(\mu_c ), F)$.
\end{pro}
{\it Proof}
Since $\mu_c$ is a shift ergodic measure,
it remains to show that there exist a point $z$ and an integer $m>0$ such that $\mu_c (B_m (z))>0$.
If $F$ is $\mu$ equicontinuous then there exist a point $x$ and a positive integer $m$ such that
$\mu (B_m(x))>0$.
As $\mu$ is a shift ergodic measure, there exists an integer $t>0$ such that
$B_m(x)\cap \sigma^{-t}B_m(x)\neq\emptyset$. From Proposition 3, there exist positive integers
$pp>0$ and $p>0$ such that $(F^{n+pp}(x)(-m,m))_{n\in\NN}$ is periodic of period $p$.
Let $z=F^{pp}(x)$, we can conclude arguing that
$$
\mu_c(B_m(z))\ge \lim_{p\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} \mu (F^{-i}(B_m(z)\cap B_m(x)),
$$
$$
\mu_c(B_m(z))\ge \frac{1}{p}\sum_{i=0}^{p-1} \mu (F^{-i-pp}(B_m(z)\cap B_m(x))
$$
and
$$
\mu_c(B_m(z))\ge \frac{1}{p} \mu (F^{-pp}(B_m(z)\cap B_m(x))=\frac{1}{p}\mu (B_m(x))>0.
$$
\hfill$\Box$
\subsection{Example of $\mu$-equi\-con\-tinuous CA
without equi\-continuous points}
When we study and compare some measurable (given an invariant measure $\mu$) and topological properties of
the dynamic of cellular automata, it is
natural to consider the dynamical system $(F,S(\mu ))$ which is the restriction to the topological
support $S(\mu )$ of the action of a cellular automaton $F$.
The automaton $F$ is a surjective map $S(\mu )\to S(\mu)$ and from Proposition
\ref{pro5} the set of periodic points in $S(\mu )$ is dense when $\mu$ is also shift ergodic
and $F$ has $\mu$-equicontinuous points.
In the case of basic examples of $\mu$-equicontinuous CA (see the example given in \cite{GI87}), the systems $(F,S(\mu ))$ always contains
equicontinuous points even if $(F, A^\ZZ)$ may not have these kind of points.
Roughly, if $\mu$ is a shift ergodic measure, to have $\mu$-equicontinuous points without
equicontinuous points requires that there exist some `perturbations' that can move to
infinity but the probability that these perturbations move to infinity is equal to zero.
One way to get theses properties for an automaton $(F,S(\mu ))$, is that $F$ generate
permanently `propagating structures'' of different sizes. The ``length of life'' of the
``propagating structures'' depends on their size.
This is roughly the dynamic of the following cellular automaton $F_e$.
\subsubsection{Definition of the cellular automaton $F_e$}
The automaton $F_e$ we consider act on $X=X_0\times X_1\times X_2$ where $X_0=\{0,1\}^\ZZ$,
$X_1=\{E_0,E_1,E_2,E_3,0,R,L\}^\ZZ$ and $X_2=\{0,1\}^\ZZ$.
We define $F_e$ as the composition of 5 other cellular automata
$F_e=F_3\circ F_2\circ F_1\circ F_p^{X_0}\circ F_p^{X_1}$.
To simplify we write $\hat{E}=\{E_0,E_1,E_2,E_3\}$ and $\overline{E}=\{0,L,R\}$.
In the following we denote by
$x=(x^0,x^1,x^2)$ any point $x\in X$ where
$x^i\in X_i$ with $0\le j\le 2$. The letter in position $i$ in the restriction of $x\in X$ to $X_j$ is
denoted by $x_i^j$. We denote by ${\bf 1}_{S}(x)$ the map which is equal to one if $x\in S$ and zero
otherwise.
The automaton $F_1$ is the identity on $X_0\times X_1$ and its restriction to $X_2$ came from the
following block map $f_1$ of radius 3
$$
f_1(x_{i-3}^2,\ldots ,x_{i}^2,\ldots ,x_{i+3}^2)=
{\bf 1}_{\{1\}}(x_{i-3}^2)\times{\bf 1}_{\{1\}}(x_{i-2}^2)\times{\bf 1}_{\{1\}}(x_{i-1}^2).
$$
The automaton $F_2$ is still the identity on $X_0\times X_1$ but its action on $X_2$ depends on $X_1$.
The block map $f_2$ is defined by:
$$
f_2\left(
\begin{array}{l}
x_{i-2}^1,\ldots , x_i^1, \ldots , x_{i+2}^1\cr
x_{i-2}^2,\ldots , x_i^2, \ldots , x_{i+2}^2\cr
\end{array}
\right)
=
\left(
\begin{array}{c}
x_i^1\cr
x_i^2\vee_{j=0}^2{\bf 1}_{\{E_0\}}(x_{i-j}^1)
\end{array}
\right )
$$
where $\vee_{i=0}^{2} {\bf 1}_{\{E_0\}}x_{i-j}^1$ is equal to 1 when at least one $x_{i-j}^1$ is equal to 1.
The automaton $F_3$ is the identity map on $X_0\times X_2$
and is defined thanks to a local rule $f_3$ on $X_1$.
The block map $f_3$ is defined by the following rules :
$$
\begin{array}{ll}
f_3(x_{i-11}^1,\ldots x_i^1,\ldots x_{i+11}^1)&=R \mbox{ if }
x_{i-10}^1,\ldots x_i^1,\ldots x_{i+k}^1=R0^{11+k}\cr
&\mbox{ where }m=\min\{10, \min \{k\; | x_{i+k}\in \hat{E}\}\}\cr
&=L \mbox{ if } x_{i-k}^1,\ldots x_i^1,\ldots x_{i+10}^1=0^{11+k}L\cr
&\mbox{ where }m=\min\{10, \min \{k\; | x_{i-k}\in \hat{E}\}\}\cr
&=R \mbox{ if }\exists\, 0\le k,j\le 9\mbox{ such that }\cr
& x_{i-j-k}^1, \ldots , x_i^1,\ldots , x_{i+10}^1=E^*0^{k}L0^{j+10}\cr
&\mbox{ with } j+2k+1=10\mbox{ and }E^*\in\hat{E}.\cr
&=L \mbox{ if }\exists\, 0\le k,j\le 9\mbox{ such that }\cr
& x_{i-10}^1, \ldots , x_i^1,\ldots , x_{i+j+k}^1=0^{10+j}R0^{k}E^*\cr
&\mbox{ with } j+2k+1=10 \mbox{ and }E^*\in\hat{E} .\cr
\end{array}
$$
Consider now the case where the central coordinate $x_i$ is an element of $\hat{E}$.
$$
\begin{array}{ll}
\mbox{For each } i\in\{0,1,2,3\}\cr
f_3(x_{i-10}^1,\ldots ,E_{i},\ldots x_{i+10}^1)&=E_{i+1} \mbox{ if } x_{i-k}^1, \ldots, x_i^1=R0^{k-1}E_i
\mbox{ with } 0\le k\le 9 \cr
&\mbox{ }\mbox{ }\mbox{ }\mbox{ where the addition `$i+1$' is made modulo 4.}
\end{array}
$$
For all the other cases where the central coordinate $x_i$ is an element $E^*$ in
$\hat{E}$
we have $f_3(x_{i-10}^1,\ldots ,E^*,\ldots x_{i+10}^1)=E^*$.
In all the other cases that have not been discribed above we have
$f_3(x_{i-10}^1,\ldots $ $,x_i^1,\ldots x_{i+10}^1)=0$.
The automaton $F_p^{X_0}$ is the identity on $X_2$ and its action on $X_0\times X_1$ came from the following
local rule:
\small
$$
f_p^{X_0}\left(
\begin{array}{ll}
&x_{i-10}^0,... , x_{i}^0, ... , x_{i+10}^0\cr
&x_{i-10}^1, ... , x_{i}^1, ... ,x_{i+10}^1\cr
\end{array}
\right)
\hskip -.1 cm = \hskip -.1 cm
\left(
\begin{array}{c}
f_p^{(X_0,0)} \cr
{\bf 1}_{0}(x_{i-1}^0)x_{i}^1 +{\bf 1}_{1}(x_{i-1}^0)\left[ {\bf 1}_{\overline{E}}(x_i^1)x_i^1\vee
{\bf 1}_{\hat{E}}(x_i^1) E_1 \right]\cr
\end{array}
\right)
$$
\normalsize
where
$$
\begin{array}{ll}
f_p^{(X_0,0)}&=1\mbox{ if } x_{i-1}^1\in\hat{E}\cr
&=1 \mbox{ if } x_i^1\notin\hat{E}\; ; \, x_{i-n}^1,\ldots x_{i+m}^1=0^{m+n+1} \mbox{ where }m=\min \cr
&\{10, \min \{k\; | x_{i-k}\in \hat{E}\}\} \mbox{ and } n=\min \{10, \min \{k\; | x_{i+k}\in \hat{E}\}\}\cr
&\mbox{ and } \exists\; 0\le l\le \lceil \frac{m-1}{2}\rceil \mbox{ such that } x_{i-l}^0=1\cr
&=0 \mbox{ otherwise.}
\end{array}
$$
\medskip
The automaton $F_p^{X_1}$ is the identity on $X_0$ and $X_2$ and its
action on $X_1$ is given by the local rule $f_p^{X_1}$:
$$
f_p^{X_1}(x_{i-152}^1, ... ,x_i^1, ... ,x_{i+152}^1)=
{\bf 1}_{\overline{E}}(x_i^1) x_i^1+
{\bf 1}_{\hat{E}}(x_i^1) x_i^1\times\prod_{j=-152}^{152}{\bf 1}_{\overline{E}}(x_{i+j}^1) .
$$
\subsubsection{The dynamic of $F_e$}
In this subsection, we describe the global dynamic of $F_e$ by showing the
contribution of each of the 5 cellular automata.
{\sc The dynamic of $F_1$}
\medskip
The action of $F_1$ on $X_2$ is only the shift of consecutive sequences (or ``trains'') of letters ``1'' of one
coordinate to the right and the destruction of the two last letters ``1'' at the left side of this {\it train}.
Action of $F_1$ on a finite configuration of $X_1$:
$$
01111110000 \hskip .2 cm\raisebox{.4 cm}{$F_1$}\hskip -.55 cm\mapsto
00001111000\hskip .2 cm\raisebox{.4 cm}{$F_1$}\hskip -.55 cm\mapsto 00000001100
\hskip .2 cm\raisebox{.4 cm}{$F_1$}\hskip -.55 cm\mapsto 0000000000
$$
Remark that a {\it train }of 1 with a length $2k+1$ will move of $k$ coordinates to
the right before collapsing.
\medskip
{\sc The dynamic of $F_2$}
The action of the cellular automaton $F_2$ is to 'create' a sequence of three letters `1' in
$X_2$ when there is a letter $E_0$ in $x_i^1$.
$$
\begin{array}{ll}
\mbox{Example:}&
\binom{E_0}{0}\binom{*}{0}\binom{*}{0}\hskip .2 cm\raisebox{.4 cm}{$f_2$}\hskip -.55 cm\mapsto
\binom{E_0}{1}\binom{*}{1}\binom{*}{1}\hskip .2 cm\raisebox{.4 cm}{$f_2$}\hskip -.55 cm\mapsto
\binom{E_0}{1}\binom{*}{1}\binom{*}{1}.
\end{array}
$$
The symbol * replace any letter in $\{0,L,R\}$.
{\sc Action of $F_2\circ F_1$}
If $F_e^i(x^1)=E_0$ for $0\le i\le n$ then there is at least a {\it train of 1} of length $n+3$ moving
to the left of at least $\lceil \frac{n+3}{2}\rceil$ coordinates.
Action of $F_2\circ F_1$ (* is any letter in $\hat{E}\cup \overline{E}$)
$$
\begin{array}{lllr}
&&^\infty 0E_0**********\ldots &x^0\cr
&&^\infty 0000000000\cdots & x^1\cr
&(F_2\circ F_1)^n&\downarrow \cr
&&^\infty 0E_0**********\ldots &F^n (x)^0\cr
&&\underbrace{^\infty 01111\cdots 1}_{n+3 \mbox{ times}}000\cdots &F^n(x)^1
\end{array}
$$
\medskip
{\sc The dynamic of $F_p^{X_1}$}
\medskip
The cellular automaton $F_p^{X_1}$ is a projection of $X$ to
$X'=X_0\times X'_1\times X_2$ where $X'_1$ is the subshift of finite type where there
is at least $l_m=152$ letters in $\overline{E}$ between 2 letters in $\hat{E}$.
Since in $X'_1$ there no 2 consecutive letters in $\hat{E}$, it induce a special kind of dynamic ({\it counter dynamic}) due to
the action of $F_3$ on $X'_1$.
In the following (except in a part of the proof of Proposition \ref{pro10}) we will consider the action of $F_e$
on $X'$ rather than $X$.
\medskip
{\sc The `counter' dynamic of $F_3$ on $X'_1$}
Under the action of $F_3$, a letter $R$ surrounded by 10 letters $0$ in the right and left coordinates moves
of 10 coordinates to the right. A letter $L$ also surrounded by 10 letters $0$ moves of 10 coordinates to the left.
When a letter $R$ surrounded by only letters $0$ and one letter $E_i\in\hat{E}$, reaches a position situated
at less than 10 coordinates to the right side of the letter
$E_i$, the letter $E_i$ changes in $E_{i+1}$, (the addition is made modulo 4) and the letter
$L$ becomes a letter $R$ and start to move to the left side.
When a letter $L$ surrounded by only letters $0$ and one letter $E_i\in\hat{E}$ at a coordinate situated at
less than 10 coordinates to the left, the letter $R$ becomes a letter $L$ without modify the letter $E_i$.
Now consider a pattern of the form $E^**^lE^*$ where $*\in\overline{E}$ and $E^*\in\hat{E}$.
First remark that for any points $x$ which contains this pattern, the evolution under $F_3$ of the subpattern
$*^lE^*$ depends only on the initial
$E^**^lE^*$. Since the block map $f_3$ applied on a word $u\in \{\hat{E}\cup \overline{E}\}^{23}$ gives a letter $0$ when appears more than
one letter $R$ or $L$ in $u$,
it is clear that after $\lceil\frac{l}{10}\rceil$ iterations it remains at most only one letter $L$ or $R$ in
any pattern $E^**^lE^*$ in $F^{\lceil\frac{l}{10}\rceil}(x)$.
We call {\it counter} of size $i+j+1$, any finite configuration of the form $E^*0^i*0^jE^*$ and
{\it void counter} any pattern of the form $E^*0^kE^*$ where $k\in\NN$, $E^*\in\hat{E}$ and $*\in\{R,L\}$.
The action of $F_3$ on the $l+1$ last coordinates of {\it counters} of size $l$ is a rotation.
The letter $R$ move to the right and change in $L$ in the neighborhood of the last letter $E_i$ which
change in $E_{i+1}$. Then the $L$ return to the left side and change in $R$ in the neighborhood of the first $E^*$.
The period of a {\it counter} of size $l$ is approximatly equal to $\frac{l}{5}$
(between $\lfloor\frac{l}{5}\rfloor$ and $\lceil\frac{l}{5}\rceil$).
This period corresponds to the number of iterations needed for the commuting letters $L,R$ to go and return
in the neighborhood (less than 10 coordinates) of the first $E^*\in\hat{E}$.
Let's see a typical evolution of a {\it counter}.
\small
$$
^\infty\hskip -.1 cm *E^*0R0^{140}000000000RLE_3 \hskip .2 cm\raisebox{.4 cm}{$F_3^{14}$}\hskip -.65 cm\mapsto
^\infty\hskip -.1 cm *E^*000^{140}R0000000000E_3 \hskip .3 cm\raisebox{.1 cm}{$F_3$} \hskip -.55 cm\raisebox{-.3 cm}{$\hookleftarrow$}
$$
$$
\hookrightarrow
^\infty \hskip -.1 cm *E^*000^{140}0000000000RE_3
\raisebox{.4 cm}{$F_3$}\hskip -.50 cm\mapsto
^\infty \hskip -.1 cm *E^*000^{140}R0000000000E_0 \hskip .2 cm\raisebox{.4 cm}{$F_3^{14}$} \hskip -.55 cm\raisebox{-.1 cm}{$\hookleftarrow$}
$$
$$
\hskip -.7 cm \hookrightarrow
^\infty \hskip -.1 cm *E^*0R0^{140}0000000000E_0 \hskip .2 cm\raisebox{.4 cm}{$F_3$}\hskip -.55 cm\mapsto
^\infty \hskip -.1 cm *E^*00000000L0^{140}0E_0 \hskip .2 cm\raisebox{.4 cm}{$F_3^{14}$} \hskip -.55 cm\raisebox{-.1 cm}{$\hookleftarrow$}
$$
$$
\hskip -1 cm \hookrightarrow
^\infty \hskip -.1 cm *E^*000^{140}000000000RE_0 \hskip .2 cm\raisebox{.4 cm}{$F_3$}\hskip -.55 cm\mapsto
^\infty \hskip -.1 cm *E^*000^{140}L000000000E_1
$$
\normalsize
Considering the additional action of $F_3\circ F_2\circ F_1$ and the set $X'$, the set of {\it counters}
split in two subsets :
The {\it void counters}
(which represent the patterns of the form $E^*0^kE_i$ with $i\in\{1,2,3\}$) and the {\it counters} whose
patterns is of the form $E^*0^kE_0$.
In the second case, under the action of $F_3\circ F_2$, the last letter $E_0$ generate a continuous
flow of letters $1$ moving to the right side.
In the following section we will see that under the action of $F_p^{X_0}$, the {\it void counters} of the type
$E^*0^kE_0$ will disapear after
less than $\lceil\frac{k}{5}\rceil$ iterations.
Remark that since a finite configuration ($E^*0^kE_0$) generate a continuous flow
(or an infinite length {\it train}) of letters 1,
there would exist equicontinuous points for $(F_e, X)$ (and for ($F_e, S(\mu_c)$) see section 3.3.2) without the action of $F_p^{X_0}$.
{\sc Action $F_p^{X_0}$ and $F_p^{X_0}\circ F_3$} on $^\infty 0^\infty\times X'_1$
Consider a point $x\in ^\infty 0^\infty\times X'_1\times X_2$.
If $x_i^1=E^*$ ($E^*\in\hat{E}$ and $x\in X'_1$) then $F_p^{X_0}(x_{i+1}^0)=1$.
If there is no letter $R$ or $L$ in the word $x_{i}^1,\ldots x_{i+k}^1$, letters $1$ will appear in
position $i+1$ to $i+k$ in $x^0$ at the next iteration.
Then consider some cylinder in $X'_1$ of the form
$[E^{(*,1)}0^j*0^kE^{(*,2)}]_i\cap X_1'=[E^{(*,1)}0^j*0^kE^{(*,2)}]_i^{X'_1}$
($i,j,k\in\NN$; $*\in\{R,L\}$; $E^{(*,\binom{1}{2})}\in\hat{E}$).
Since the letter $R$ and $L$ move two times faster than the letters 1 in $X_0$
we obtain for
each $x\in ^\infty 0^\infty \times [E^{(*,1)}0^j*0^kE^{(*,2)}]_i^{X'_1}\times X_2$ that
$y^0_{i+t_1}\ldots y^0_{i+t_2}=0^{\lceil\frac{t_1}{2}\rceil}$
when $y=F_p^{X_0}\circ F_3(x)$, $t_1=\lceil\frac{2(j+k+1)}{3}\rceil$ and $t_2=j+k+1$.
It follows that in this case, $F_p^{X_0}$ does not modify the last letter $E^{(*,2)}$.
So $F_p^{X_0}$ does not affect the {\it counters} of the type $E^{(*,1)}0^i*0^jE^{(*,2)}$ in
$^\infty 0^\infty \times X'_1\times X_2$ where $*\in\{R,L\}$.
Nevertheless, if $x\in ^\infty 0^\infty \times [E^{(*,1)}0^lE^{(*,2)}]_i^{X_1}\times X_2$, after
$\lceil\frac{l}{5}\rceil$ iterations,
a letter $1$ will appear in position $i+l+1$ in $x^0$ and at the next iteration the final letter $E^{*,2}$ will
be fixed to $E_1$.
Hence after a while ($\lceil\frac{l}{5}\rceil$) all the patterns $E^*0^lE^{(*,2)}$ with $E^{(*,2)}\neq E_1$
will disapear.
In the more general case ($x\in X'$), the action of $F_p^{X_0}$ on $X'_1$ will be the identity after a while
(which means after the iterations of $F_e$).
Notice that the dynamic of the restriction of $F_p^{X_0}$ to $^\infty0^\infty\times X'_1$ is enough
for the results of
subsection 3.3.3 .
Typical actions of $F_p^{X_0}$ on $^\infty 0^\infty \times X_1$ with a {\it void counter} of size $10n$:
\scriptsize
$$
\left( F_p^{X_0}\right) ^n\left(^\infty\binom{0}{0}\binom{0}{E_2}\binom{0}{0}\binom{\cdots}
{\cdots}\binom{0}{0}\binom{0}{E_0}\right) =
\left(F_p^{X_0}\right)^{n-1}\left(^\infty\binom{0}{0}\binom{0}{E_2}\binom{1}{0}\binom{0}{0}\binom{\cdots}{\cdots}\binom{0}{E_0}\right)
$$
$$
=F_p^{X_0}\left(^\infty\binom{0}{0}\binom{0}{E_2}\binom{1}{0}\binom{1}{0}\binom{\cdots}{\cdots}
\binom{1}{0}\binom{0}{E_0}\right)
=^\infty\binom{0}{0}\binom{0}{E_2}\binom{1}{0}\binom{1}{0}\binom{\cdots}{\cdots}\binom{1}{0}\binom{0}{E_1}.
$$
\normalsize
{\sc Action of $F_3\circ F_2\circ F_1$ on $X'_1\circ X_2$}
Consider the {\it counter} of size $l$ : $\mathcal{C}_l$ of the form $E^{(*,1)}0^j*0^kE^{(*,2)}$
($j+k+1=l$, $*\in\{R,L\}$ and $E^{(*,1)},E^{(*,1)}\in\hat{E}$).
Under the action of $F_3\circ F_2\circ F_1$, the last letter $E^{(*,2)}$ stay in the state
$E_0$ during a period of $\frac{l}{5}\le P_l\le \frac{l}{5}+1$.
Taking into consideration the action of $F_2\circ F_1$, the rotation of the
{\it counter} $\mathcal{C}_{l}$ will generate periodically a {\it train of 1} of length
$P_l+3$ (see action of $F_2\circ F_3$).
During $3P_l$ iterations, the {\it counter} $\mathcal{C}_l$ is in the non emmitting phase
(the last letter is in $\{E_i\vert \, 1\le i\le 3\}$).
{\sc Action of $F_e$ on {counters}}
Since a {\it counter} $\mathcal{C}_l$ generate a {\it train of 1} with an approximatly length $\frac{l}{5}$,
it can influence some patterns situated at
$(\frac{l}{5}+3+(\frac{l}{5}+3)/2=\frac{3l}{10}+9/2)$ coordinates to the right of the right extremity of the {\it counter}.
Recall that a {\it train of 1} looses 2 elements and moves of 1 coordinate in one iteration.
Remark that using some concatenation process (with other {\it counters}), the {\it train of 1} generated
by a {\it counter} $\mathcal{C}_l$ may produce perturbations further than $\frac{3l}{10}+\frac{9}{2}$
coordinates. Roughly, a {\it train of 1} of length $t$ looses $2l$ elements when it cross a second {\it counter} of size
$l$ to the right side and can gain $\frac{l}{5}+3$ elements (with a good synchronization) thanks to this second counter.
The Figure 1 represents a typical action of $F_e$ on
5 {\it counters} $\mathcal{C}_{2100}$, $\mathcal{C}_{304}$ and three $\mathcal{C}_{152}$.
The first {\it counter} of size 152 is a {\it void counter} which never generate any sequence of 1
in $X_1$. To the left side there is one large {\it counter} of size 3000 which is in a non emmitting
state (last letter not an $E_0$).
For simplification, we do not specify the states of the {\it counters} and their evolution because
the interesting part of their dynamic can be deduce from the evolution of the {\it train of 1} in $X_1$.
Remark that the non `emmitting period' of the {\it counters} last 3 times more than the `emmitting' one.
\begin{rem}
Recall that 152 is the minimum size of a {\it counter} thanks to the action of $F_p^{X_1}$.
This minimum size is required to simplify the proof of Proposition \ref{noneq} by
using quantitative arguments on flows
rather than study the complex dynamic of concatenations of {\it train of 1}.
\end{rem}
\begin{figure}[t]
\centering
\includegraphics[width=1\textwidth,height=0.5\textheight]{aut3}
\caption{An illustration of the dynamic of $F_e$ on 5 {\it counters} and the resulting dynamic of
{\it train of 1} in $X_2$. Remark that each line represents $30$ iterations of $F_e$.
The black horizontal lines represent the
{\it trains of 1} and blank ones, the sequences of $0$.
The extremity of the counters are delimited by arrows.}
\label{fig1}
\end{figure}
\subsubsection{The topological and measurable properties of $F_e$}
\begin{pro}\label{pro10}
There exist a shift mixing and $F_e$-invariant measure $\mu_c$ such that
the cellular automaton $F_e$ has $\mu_c$-equicontinuous points.
\end{pro}
{\it Proof}
Let $\mu_1$ be the uniform Bernouilli measure on $\hat{E}\cup\overline{E}$ and
$\mu_I=\delta_{^\infty 0^\infty}\times\mu_1\times \delta_{^\infty 0^\infty}$ where
$\delta_{^\infty 0^\infty}$ is the atomic measure on the fix points
$^\infty 0^\infty$ in $X_0$ and $X_2$. The measure $\mu_c$ we consider is
simply the Cesaro limit of some converging subsequence $(\mu_I\circ F^{-n})_{n\in\NN}$.
Since $\mu_I$ is the direct product of a Bernouilli measure and atomic measures on shift
invariant points, Theorem \ref{thm1}, Proposition \ref{pro7} and Remark \ref{remmesure} tell us that
$(\mu_I\circ F^{-n})_{n\in\NN}$ converges to a shift mixing measure if
there exists some $\mu_I$-equicontinuous points.
Furthemore from Proposition \ref{pro8}, if there exists $\mu_I$-equicontinuous points,
there also exists $\mu_c$-equicontinuous points. So to prove that
$F_e$ is
a $\mu_c$-equicontinuous CA we only need to show that $F_e$ contains a $\mu_I$-equicontinous point.
To complete the proof we are going to show that there exists a point $x$ and an integer $m\ge r$ such that
$\mu_I(B_m(x))>0$.
For each $l\in\NN$, denote by $\mathcal{C}_l[i]$ the union of all the sets
$^\infty 0^\infty\times [U]_i^{X'_1}\times ^\infty 0^\infty$
where $U=[E^{(*,1)}0^j*0^kE^{(*,2)}]_i^{X'1_1}$ is a cylinder in $X_1$, * replace one letter in $\{L,R\}$,
$E^{(*,\binom{1}{2})}$ are letters in $\hat{E}$ and the positive integers verify $j+k+1=l$.
Let $\overline{\mathcal{C}_k[i]}$ be the union of sets
($^\infty 0^\infty\times[E^*0^kE^*]_{i}^{X'_1}\times ^\infty 0^\infty$) where $E^*$ replace any letter in $\hat{E}$.
We call respectively {\it counters} in position $i$ and {\it void counters} in positions $i$ the sets
$\mathcal{C}_l[i]$ and $\overline{\mathcal{C}_l[i]}$.
Next we denote by $\mathcal{C}^*_l[i]$ the union of all the sets
$^\infty 0^\infty\times [E^{(*,1)}*^lE^{(*,2)}]_i\times ^\infty 0^\infty$ where $*\in\overline{E}$ and
$E^{(*,\binom{1}{2})}\in \hat{E}$. From section 3.3.2, each element in $\mathcal{C}^*_l[i]$ will
enter in a {\it counter} $\mathcal{C}_l[i]$ or $\overline{\mathcal{C}_l[i]}$ after less than $\frac{l}{5}$
iterations. Remark that the iterations of each element $y$ in a {\it precounters} $\mathcal{C}^*_l[i]$ will never
generate a {\it train of 1} from its coordinate $i+l+1$ longer than $\frac{l}{5}+3$ because
the letter $E^*$ in position $i+l+1$ never stay in the emmitting state $E_0$ for more than
$\frac{l}{5}$ iterations.
Consider $x_0=(^\infty 0^\infty , ^\infty 0^\infty,^\infty 0^\infty)$ and for each $k\in\NN$, pick
a point $x_k\in[0^{k}]_{(-k-1-r)}\cap B_r(x_0)$.
In the following, we will prove that there exist integers $k>0$ such that
$\mu_I(B_r(x_0))=\mu_I (B_r(x_k))>0$ by showing that
$$
\mu_I([0^{k}]_{(-k-1-r)}\cap B_r(x_k)^\complement)<\mu_I ([0^{k}]_{(-k-1-r)}).
$$
The set $[0^{k}]_{(-k-1-r)}\cap B_r(x_k)^\complement$ is the set of points that contains
{\it counters} in the left side of $[0^{k}]_{(-k-1-r)}$ that are able to generate
{\it trains of 1} which
move to the right and cross completly the $k$ coordinates betwen $-k-r-1$ to $r$
(the {\it trains} of ``1'' enter in the central
coordinates ($[-r,r]$)).
Now, consider the map $S(p)$ which gives the minimum size of the {\it counter} $\mathcal{C}^*_p[-p-l-k-r]$
in order that it produce {\it trains of 1} that move further than the coordinate $-r$.
Recall that $l_m=152$ is the minimum size of the {\it counters} due to the projection of $F_p^{X_1}$.
Clearly the set$[0^{k}]_{(-k-1-r)}\cap B_r(x_k)^\complement$
is a subset of
$\mathbb{S}_k=\left\{\cup_{i=S(0)}^\infty\mathcal{C}^*_i[-i-k-1-r]
\cup_{p=l_m}^\infty\{\cup_{j=S(p)}^\infty \mathcal{C}^*_j[-j-p-k-1-r]\}\right\}
\cap[0^{k}]_{(-k-1-r)}$.
Remark that we do not try to identified the cases where appears between
the coordinates $-p-r-1$ to $-r-1$ {\it precounters} $\mathcal{C}^*_j[-j-p-k-1-r]\}$
big enought to influence the coordinates $[-r,r]$.
In order to find a good upper bound for $\mu_I(\mathbb{S}_k)$ we are going
to show that $(S(p))_{p\in\NN}$ is a strictly increasing sequence.
Let $T_l=\frac{l}{5}+3$ be the original length of the {\it trains of 1} generated by a {\it precounter}
$\mathcal{C}^*_j[-j-p-k-1-r]$. When the left extremity of the {\it train of 1} arrive in
position $-k-1-r$
it has lost $2p$ elements and gain some other ones due to possible concatenations
with other {\it precounters} situated between the coordinates $-p-k-1-r$ and $-k-1-r$.
Clearly, the maximum gains in terms of concatenation of letters 1 arrives when there is
only one {\it precounter} of size $p-1$ between the coordinates $-p-k-1-r$ and $-k-1-r$.
In this case the remaining {\it train of 1} called $\mathcal{T}$ which came from $\mathcal{C}^*_j[-j-p-k-1-r]$
would have a length of $T_l-2p+\frac{p-1}{5}+3$. Since the right front of this {\it train} is
at coordinate $k-1-r+\frac{p}{5}+3$, it needs to cross $k-(\frac{p}{5}+3)$ coordinates
to influence the central coordinates $(-r,r)$.
As $\mathcal{T}$ has a size of $T_l-2p+\frac{p-1}{5}+3$,
it can cross at most
$(T_l-2p+\frac{p}{5}+3)/2=\frac{l}{10}+3-\frac{11p}{10}$ coordinates and
since it has a right extremity in position $k-1-r+\frac{p}{5}+3$, it follows that $S(p)\ge 10(k+p)$.
Under the action of $F_p^{X_1}$, a letter $E^*\in\hat{E}$ will be change in $0$ is there is another
element of $\hat{E}$ situated in a neighborhood of $l_m$ coordinates. This implies that
for all integer $l\ge l_m$ and $i\in\NN$ we have $\mu_I(\mathcal{C}^{*}_l[i])=(\hat{q})^2\times
(\overline{q})^{2l_m+l}$ where $\overline{q}=\mu_I(\cup_{E^*\in\overline{E}}[E^*]_i)$ and
$\hat{q}=\cup_{*\in\hat{E}}[*]_i$ for some integer $i$.
We set $q_l=\mu_I(\mathcal{C}^{*}_l[i])$ and $q^*=[0^{k}]_{(-k-1-r)}$
and we remark that if $l\ge l_m$, for all $n\in\NN$ we have
$\mu_I(\mathcal{C}_{l+n}[i])=q_l\times \overline{q}^n$.
Since $\mathbb{S}_k=$
\footnotesize
$$
\left\{\cup_{i=S(0)}^\infty\mathcal{C}^*_i[-i-k-1-r]
\cup_{p=l_m}^\infty\{\cup_{j=S(p)}^\infty \mathcal{C}^*_j[-j-p-k-1-r]\}\right\}
\cap [0^{k}]_{(-k-1-r)}
$$
\normalsize
we obtain that
$$
\mu_I(\mathbb{S}_k)\le q^*\left(q_{S(0)}\sum_{i=0}^\infty \overline{q}^i+q_{S(l_m)}\sum_{j=0}^\infty
\overline{q}^j(\sum_{i=0}^\infty \overline{q}^i) \right)
$$
$$
=q^*q_{S(0)}\left(\frac{1}{1-q}+\overline{q}^{10l_m}(\frac{1}{1-\overline{q}})^2\right).
$$
Since $S(0)=10k$, there will be an integer $k\ge 0$ such that
$$
q_{S(0)}\left(\frac{1}{1-\overline{q}}+\overline{q}^{10l_m}(\frac{1}{1-\overline{q}})^2\right)<0
$$
which prove that
$\mu_I([0^{k}]_{(-k-1-r)}\cap B_r(x_k)^\complement)
<\mu_I ([0^{k}]_{(-k-1-r)})=q^*$.
Then we can conclude arguing that there exists an integer $k$ such that
$$
\mu_I(B_r(x_k))=\mu_I(x_0)>0.
$$
\hfill$\Box$
\begin{rem}
It is possible to give a simpler proof of Proposition \ref{pro10} using only the limit measure $\mu_c$ but
in this case we can not show that $(\mu_I\circ F^{-n})_{n\in\NN}$ is a converging sequence.
We conjecture that the limit in Cesaro mean of the sequence $(\mu_u\circ F_e^{-n})_{n\in\NN}$
where $\mu_u$ is measure of the uniform measure
on $X$ will be identical to the measure $\mu_c$ defined in the Proof of Proposition \ref{pro10}.
In the following $\mu_c$ will always denote the limit in Cesaro mean
of the sequence $(\mu_I\circ F_e^{-n})_{n\in\NN}$.
\end{rem}
\begin{pro}\label{noneq}
The cellular automaton $(F_e,S(\mu_c ))$ is sensitive (has no equi\-conti\-nuous point in the
topological support $S(\mu_c )$).
\end{pro}
{\it Proof}
Suppose that there exist an equicontinuous point.
We need to show that there exists a point $x\in S(\mu_c)$ and an integer $m$ such that
$C_m(x)\subset B_r(x)$.
First remark that if for some $x$ there exist integers $i>0$, such that
$F^i(x)^2_0=0$, then for all $\epsilon =2^{-m}$ there exists $y\in C_m(x)$ such that
$F^i(y)^2=1$.
This implies that if there exists $x\in A^\ZZ$ and
$m>0$ such that $C_m(x)\subset B_r(x)$, then for all $y\in C_m(x)$ and $i\in\NN$,
one has $F^i(y)^2=1$ (condition (*)).
Hence, if there exists an equicontinuous point, there exists a finite configuration
that produce a continuous and permanent `flow of 1'.
Let's try to construct such a configuration.
Recall that thanks to the action of $F^{X_0}_p$ (see section 3.3.2), after a while ($k$ iterations),
there is no {\it void counter} (patterns
of the type $E^*0^kE_0$ in $X'_1$ with $E^*\in\hat{E}$) able to generate a continuous flow of letters `1'.
It follows that $S(\mu_c)\cap X$ does not contains any cylinder of the type $[E^*0^kE_0]_i$ ($i\in \ZZ, k\in \NN$).
From section 3.3.2, after a while (a period less than $\frac{l}{10}$) there is at most one
letter in $\{R,L\}$ between two occurences of letters in $\hat{E}$.
Since we search for a finite configuration that produce a permanent `flow of 1', we can only consider
the {\it counters } $\mathcal{C}$ and {\it void counter} $\overline{\mathcal{C}}$ and
as we are going to use a quantitative arguments on flow of 1, we will only consider the
real {\it counters} $\mathcal{C}$.
If we suppose that there exists an equicontinuous point, then there is a finite sequence of $k$
consecutive counters
$\mathcal{C}_{l_k}, \mathcal{C}_{l_{k-1}}, \ldots \mathcal{C}_{l_0}$ that generate a continuous
flow of letters 1
(there exists a point $x\in \mathcal{C}_{l_k}, \mathcal{C}_{l_{k-1}}, \ldots \mathcal{C}_{l_0}$ such that
$\forall n\in\NN\; F_e^n(x)(-r,r)=1^{2r+1}$).
First remark (see the analysis of the $F_3$ dynamic in subsection 3.3.2) that a {\it train} of 1 generated by a
{\it counter} $\mathcal{C}_{l_k}$ will reach the second emittor of
the neighbour {\it counter} $\mathcal{C}_{l_0}$ if for all $1\le i\le k$, one has
$l_{i-1}\le \frac{3l_i}{10}+\frac{9}{2}$.
Recall that each {\it counter} $\mathcal{C}_{l_i}$ generate a {\it train} of 1, which last at most
$P_i$ iterations at the coordinate $0$ and do not influence the central coordinates
during at least $4P_i$ iterations.
Remark that in the case of {\it void counter} of type $[E^*0^kE_1]$ ($E^*$ is an element of $\hat{E}$),
no {\it train of `1'} are generated.
Were are going to show that there exist integers $k>0$ such that
$F^{i+k}(x)^2$ $\neq 1$ when $0\le i\le 4P_k$ and $\mathcal{C}_{l_k}, \mathcal{C}_{l_{k-1}}\ldots
\mathcal{C}_{l_0}$ is the finite sequence of {\it counters} which appears in the left coordinates of $x$.
In order to do that, we will show that, considering
all the {\it trains} generated by the $k$ counters, it always remains some holes,
during a period of $3P_k$ where $P_k$ is the period of the first and largest {\it counter} $\mathcal{C}_{l_k}$.
Without loosing generalities, we can suppose that the train of 1 generated by the first counter of size $l_k$
arrive in coordinate 0 at $t=0$ and last at most $P_k$ iterations (we do not know a priori
the global dynamic). For time $t=P_k+1$ to
$4\times P_k$, there is no train of 1 due to this first {\it counter} that pass through the
central coordinate.
The {\it train} of 1 generated by the second counter from the left : $\mathcal{C}_{l_{k-1}}$ last at
most $P_{k-1}$ iterations and its effect stops for a period of $3P_{k-1}$ in the interval
time $t=P_k+1$ to $t=4P_k$.
Clearly, between $t=P_k+1$ and $t=4P_k$, if $P_{k-1}$ is small enought, there is at least one interval
of length at least $3P_{k-1}$ that will be not affected by the two
first {\it counters} if $3P_k-(2\times 3+1)(P_{k-1})\ge 0$. This interval is minimum when the {\it train }
of 1 generated by the second {\it counter} pass exactly in the middle of the interval
$[P_{k}+1, 4P_k]$ between 2 {\it trains} of the first {\it counter}.
The condition $3P_k-(2\times 3+1)(P_{k-1})\ge 0$ is equivalent to $P_{k-1}\le \frac{3P_k}{7}$
and since $l_{k-1}\le \frac{3l_k}{10}+\frac{9}{2}$, $P_k=\frac{l_k}{5}+3$, then $l_k$
must be greater than $\frac{39\times 35}{9}\approx 152$.
Since, the propagation of the {\it trains} of 1 from one counter to the other,
requires that $l_{i-1}\le \frac{3l_i}{10}+\frac{9}{2}$ and thanks to the automaton $F^{X_1}_p$, all the {\it counters}
have a size $l\ge 152$,
the condition $P_{i-1}\le \frac{3P_i}{7}$ is true for all $1\le i\le k$ and repeating $k$ times the
first process there will remain intervals of length $3P_0$ between $P_k+1$ and $4P_k$ that will
be not affected by any of the {\it trains } of 1 generated by the $k$ {\it counters}.
Since for any sequence of {\it counters} $\mathcal{C}_{l_k}, \mathcal{C}_{l_{k-1}}, \ldots \mathcal{C}_{l_0}$
and points $y$ that contains such counters,
there exists an integer $n\in\NN$ such that $F_e^n(y)\neq 1$, we can conclude.
\hfill$\Box$
\begin{rem}
Clearly, The cellular automaton ($F_e$, $X$) has no equicontinuous point too.
\end{rem}
\begin{figure}[t]
\centering
\includegraphics[width=1\textwidth,height=0.3\textheight]{flow2}
\caption{The flow of 1 generated by a sequence of $k$ {\it counters} }
\label{fig2}
\end{figure}
\subsection*{Questions}
- Is it possible to construct an infinite sequence that generate a continuous flow for the
particular automaton $F_e$?
-In general, what are the conditions on the counters of automata similar to $F_e$ in order that they can not produce
equicontinuous points?
-Is it possible to find a $\mu$-equicontinuous CA without equicontinuous point for a $F$-invariant measure
$\mu$ whose topological support $S(\mu)$ is a subshift of finite type ?
-Is there an example similar to the one we have presented in the case of an only two states
cellular automaton?
-The dynamic of the example $F_e$ given in this paper (propagation of trains) seems to appear in different
simulations of one dimensional cellular automata (even in the two states case like the class studied by
Wolfram, see \cite{Wo86}).
How common are the CA with the same properties?
| 18,387
|
TITLE: Complexity of quantum simulation
QUESTION [4 upvotes]: Richard Feynman showed that Quantum simulation on a Turing machine will have an exponential slowdown. If that is so, does this put quantum simulation outside of P (complexity class)? I thought quantum simulation was polynomially possible on quantum computers, but there is still no proof that BQP is strictly bigger than P. So either quantum simulation has not been shown to lie outside of P, or it is not in BQP. I can't seem to find the answer.
REPLY [7 votes]: Feynman never showed that quantum simulation has an exponential cost. He probably mentioned that it seemed to be the case.
In complexity theory, it's surprisingly hard to prove lower bounds on the cost of solving a problem or to prove separations between complexity classes. The behavior of computer programs is wildly diverse, and eliminating all of them as candidates is pretty tricky. For example, complexity theorists trying to resolve P-vs-NP have proved lower bounds... on how hard the proof will be.
Classical vs quantum, i.e. BPP vs BQP, is one of the many examples of a complexity class pair in the "conjectured to be different, but no one knows how to prove it" bucket.
As an example of why this is tricky, suppose you conjectured that quantum computers have a space advantage over classical computers. That BQP $\not\subset$ PSPACE. You make all kinds of arguments about the number of amplitudes being exponentially large, etc, etc. Except we know that, actually, BQP $\subset$ PSPACE. You can compute a single output amplitude without using much space by iterating over all the possible paths contributing to it. Use that to iterate over the outputs, and return one probabilistically.
Proving there's nothing that does for time what path integrals did for space is hard.
| 45,877
|
Girl, Wash Your Face
We're so excited that it's time for our first book for our Online Book Club! February's book is Girl, Wash Your Face by Rachel Hollis and we can't wait to share a few of our favorite parts with you! If you've read it, we hope you'll share what you love about it. And if this book is on your To-Read list, we hope this gives you a good idea of what you have to look forward to!
| 155,954
|
TITLE: Coefficient of Taylor expansion of rational function?
QUESTION [0 upvotes]: We know that given a rational number of the from $p/q$ if we expand it in the decimal base that is an expansion in power of 10 and we know that coefficient is repeating after finite step. In other words, we can say that we need only finitely many coefficients to describe the rest.
Let $F(z)$ be a rational function that is $F(z)=\frac{p(z)}{q(z)}$. I take the Taylor series expansion of $F(z)$ at the point away from the pole (maybe in a local coordinate around the point). Say
$$F(z)\mid_z=a \sum t_k (z-a)^k $$
My question is there any finite condition on the coefficient of $t_k$ ?
That is we need only finitely many $t_k$'s to determine the rest? If so what will be the proof?
REPLY [2 votes]: For simplicity of notation, we can consider $a = 0$. Then we have $$\frac {P(z)}{Q(z)} = \sum_{n=0}^\infty c_nz^n$$
or $$P(z) = \sum_{n=0}^\infty c_nz^nQ(z)$$
if $Q(z) = \sum_{n=0}^k b_nz^n$, the coefficient of $z^n$ on the RH side for sufficiently high $n$ will be given by $$\sum_{j=0}^k b_jc_{n-j}$$
When $n$ is greater than the degree of $P$, this will be $0$. So we can solve (assuming $b_0 \ne 0$) to get $$c_n = -\sum_{j-1}^k \frac{b_j}{b_0}c_{n-j}$$
So, yes, rational functions have a recursion formula for their Taylor series coefficients, which holds for $n$ greater than the degrees of both $P$ and $Q$.
| 97,956
|
\begin{document}
\title {Note on the best approximation in $L^1$ metric}
\author{ Alexey Solyanik }
\date{21 June 2016}
\dedicatory{To Yuriy Kryakin}
\address{Caribbean Sea, St. Marten, France}
\email{transbunker@gmail.com}
\newtheorem{remark}{Remark}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{example}{Example}
\maketitle
\vspace{1 cm}
\section{Introduction}
In this note we present one of the approaches to find the best (or good) approximation of the given function by trigonometric polynomials in $L^1$ metric.
The method is as follows. First we try to represent the given function as the sum of Bernoulli kernels. For this purpose we use Fourier Analysis. Then we apply well-known Favard Theorem \cite{favard_1} concerning best approximation of Bernoulli kernels in $L^1$ metric.
This method is not new and actually goes back to the second paper of J. Favard \cite{favard_2} where decomposition of the given function as the infinite sum of shifted Bernoulli kernel (convolution) was applied for good (and some times the best) approximation of smooth function.
The new ingredient is that we decompose a given function not only to the sum of shifted \textit{one} Bernoulli kernel but also to the series of \textit{all } even (or odd) Bernoulli kernels.
We remind that for $r=1,2, \dots $
\begin{equation}
\label{intr_ber_r}
\mathcal{B}_r(\theta) =\sum_{k\neq 0} \frac{e^{ik\theta}}{(ik)^r}
\end{equation}
are the \textit{Bernoulli kernels}.
If $m\geq 2$ this series converge absolutely on $\mathbb{T}=\mathbb{R}/2\pi \mathbb{Z}$ and hence its sum is a continued function. For $m=1$ this series converge everywhere but to the discontinued function with a jump equal $2\pi$ at the point $0$.
We note that our definition of Bernoulli kernels is different by the constant factor $2$ with the usual one (see e.g. \cite{devore}, p. 150) since we use $2\pi$ instead of $\pi$ in the definitions of norm, convolution and Fourier coefficients. Thus for $f\in L^1$
$$
\|f\|_{L^1}=\frac{1}{2\pi}\int_0^{2\pi}|f(\varphi)|d\varphi
$$
and
$$
\widehat{f}(k)=\frac{1}{2\pi}\int_0^{2\pi}f(\varphi)e^{-ik\varphi} d\varphi
$$ We refer to \cite{katz} for background theory of Fourier series and for the notations.
In the seminal paper \cite{favard_1} Jean Favard found the exact approximation of Bernoulli kernels by the trigonometric polynomials in $L^1$ metric:
\begin{equation}
\label{intr_fav_best}
E_{n-1}(\mathcal{B}_r)_{L^1}= \frac{K_r}{n^r}
\end{equation}
where
\begin{equation}
\label{intr_fav_const}
K_r=\frac{4}{\pi}\sum_{k\in 4\mathbb{Z}+1} \frac{1}{k^{r+1}}
\end{equation}
and trigonometric polynomial $\tau_{n-1}^r (\theta)$ of best approximation has explicit formula.
Here and later in the sequel
$$
E_m(f)_X=\inf_{t\in T_m} \| f-t\|_X
$$
where $X$ is $L^1$ or $L^\infty$ and $T_m=\lbrace t(\varphi) : t(\varphi)=\sum_{|k|\leq m }c_k e^{ik\varphi} \rbrace$ is the space of trigonometric polynomials (with complex coefficients $c_k$) of the degree less or equal $m$.
Favard Theorem (\ref{intr_fav_best}) was a corner stone of many approximation theorems (see e.g. \cite{favard_1}, \cite{favard_2}, \cite{favard_3}, \cite{nik}, \cite{tem}, \cite{kry} or chapter 7 of the book \cite{devore}). In \cite{nik} S. M. Nikolsky applied (\ref{intr_fav_best}) to obtain estimates of approximation of Lipschitz functions by algebraic polynomials on the interval $[-1,1]$ and observed that these estimates depended on the position of the point $x$ on the interval (see e.g. chapter 8 of \cite{devore} or Section \ref{_sec_alg_appr} of this note for the precise definitions).
The starting point of this article was a question posed to the author by Y. V. Kryakin ----- to improve constants in the Nikolsky type inequality
\begin{equation}
\label{intr_alg_nik}
|f(x)-P_{n}(f,x)| \leq \frac{C_1}{n}
\sqrt{1-x^2}+ \frac{C_2}{n^2}\sqrt{x^2}
\end{equation}
After standard substitution $x=\cos\theta$ this problem at once reduced to the trigonometric case and as usual, the main ingredient of the proof is good (or best) approximation of the corresponding kernels $\mathcal{K}_1(\theta)=\mathcal{B}_1(\theta)\cos\theta $ and $\mathcal{K}_2(\theta)= \mathcal{B}_1(\theta)\sin\theta$ in $L^1$-metric by trigonometric polynomials.
Trigonometric polynomial interpolating kernel $\mathcal{K} $ at the equidistant points (nodes) is the desired polynomial of the best approximation. In order to prove this one have to show that the difference between kernel and interpolating polynomial change sign at every node and only at these nodes. Unlike Bernoulli kernels the kernels $\mathcal{K}_1$ and $\mathcal{K}_2$ are not an algebraic polynomials, but quasi-polynomials. This fact not allow to apply standard arguments --- quasi-polynomials does not vanishing after many differentiation.
We gave three different proofs of this important property which at once implies the desired estimates (Theorem \ref{theo_quasi_bern}). First one is based on the modification of the \textit{Sturm sequence} in the way of N. Tschebotarow \cite{tsch} . Second one is based on the application of \textit{Sturmian arguments} of zero set analysis (see e.g. \cite{gal} p. 188).
Unfortunately both proofs was not too short.
The third proof, which is based on the decomposition of kernel to the series of Bernoulli kernels is the subject of present note and this approach seems to be new.
\section{Favard constants}
Constants $K_r$ in (\ref{intr_fav_best}) usually called the \textit{Favard constants} (see e.g. \cite{devore} , p.149) and from (\ref{intr_fav_const}) $K_r=\frac{4}{\pi}S(r+1)$, where
$$
S(r)=\sum_{k\in 4\mathbb{Z}+1} \frac{1}{k^{r}}=1+\frac{(-1)^r}{3^r}+\frac{1}{5^r}+\frac{(-1)^r}{7^r}+\frac{1}{9^r}+\frac{(-1)^r}{11^r} + \cdots
$$
Hence
\begin{equation}
\label{intr_fav_osc}
K_2<K_4<\cdots < \frac{4}{\pi} < \cdots K_3<K_1
\end{equation}
We also claim that for odd $r$
\begin{equation}
\label{intr_fav_const_odd}
K_{r}=\frac{4}{\pi}\sum_{k=0}^\infty \frac{1}{(2k+1)^{r+1}}
\end{equation}
and for even $r$
\begin{equation}
\label{intr_fav_const_even}
K_{r}=\frac{4}{\pi}\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^{r+1}}
\end{equation}
If we (following \cite{el}) compose the generating function for $K_r$ (where we define for convenience $K_0=1$)
\begin{equation}
\label{intr_bern_gen}
K(z)=\sum_{r=0}^\infty K_rz^r=\sum_{r=0}^\infty\frac{4}{\pi} S(r+1)z^r=\frac{4}{\pi }\sum_{r=0}^\infty \sum_{m\in 4\mathbb{Z}+1}\frac{z^r}{m^{r+1}}
\end{equation}
and then change order of sums, which is possible for $|z|<1$ we get
\begin{equation}
\label{intr_sum}
K(z)=\frac{4}{\pi }\sum_{m\in 4\mathbb{Z}+1}\frac{1}{m-z}=\frac{4}{\pi }(\frac{1}{1-z}-\frac{1}{3+z}+ \frac{1}{5-z}-\frac{1}{7+z}+\frac{1}{9-z}-\dots
\end{equation}
Thus, according to the well-known formulas for $\tan z$ and $\sec z$ (see e. g. \cite{aar} formulas (1.2.7) and (1.2.8))
\begin{equation}
\label{fav_const_tan}
\frac{\pi}{2}\tan\frac{\pi z}{2}=\frac{1}{1-z}- \frac{1}{1+z}+ \frac{1}{3-z}-\frac{1}{3+z}+ \frac{1}{5-z}-\frac{1}{5+z}+\dots
\end{equation}
\begin{equation}
\label{fav_const_sec}
\frac{\pi}{2}\sec\frac{\pi z}{2}=\frac{1}{1-z}+\frac{1}{1+z}- \frac{1}{3-z}-\frac{1}{3+z}+ \frac{1}{5-z}+\frac{1}{5+z}-\dots
\end{equation}
we have
\begin{equation}
\label{intr_final_formula_tan_cosec}
K(z)=\sum_{r=0}^\infty K_rz^r=\sum_{r=0}^\infty K_{2r+1}z^{2r+1}+\sum_{r=0}^\infty K_{2r} z^{2r}=\tan \frac{\pi z}{2}+\sec \frac{\pi z}{2}
\end{equation}
Develop right side in Taylor series near $z=0$ (\cite{as}, p. 75)
\begin{equation}
\label{intr_sec}
\sec z=1+ \frac{z^2}{2}+\frac{5z^4}{24}+\frac{61z^6}{720} +\dots = \sum_{n=0}^\infty \frac{(-1)^n E_{2n}}{(2n)!}z^{2n}
\end{equation}
and
\begin{equation}
\label{intr_tan}
\tan z= z+ \frac{z^3}{3}+\frac{2z^5}{15}+\frac{17z^7}{315}+ \dots=\sum_{n=1}^\infty \frac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}
\end{equation}
where $E_n$ and $B_n$ are Euler and Bernoulli numbers respectively (\cite{as}, p. 804)
\begin{equation}
\label{intr_eul_num}
E_0= 1, ~~~ E_2=-1,~~~ E_4=5, ~~~
\end{equation}
\begin{equation}
\label{intr_eul_num}
B_0= 1, ~~~ B_1=-\frac{1}{2},~~~ B_2=\frac{1}{6},~~~B_4=-\frac{1}{30}
\end{equation}
Now comparing coefficients near $z^n$ in both sides of ({\ref{intr_final_formula_tan_cosec}) we can obtain exact values of $K_r$
\begin{equation}
\label{intr_first_favar}
K_1=\frac{\pi}{2},~~~ K_2=\frac{\pi^2}{8},~~~K_3=\frac{\pi^3}{24}, ~~~K_4=\frac{5\pi^4}{384},~~~K_5=\frac{\pi^5}{240},~~~K_6=\frac{61\pi^6}{46080},\dots
\end{equation}
Since coefficients in Taylor expansion of functions $\sec z$ and $\tan z$ near $z=0$ has only rational numbers, we can conclude that all $K_r$ are $\pi^r$-rational.
\section{Best approximation of Steklov kernels}
To explain the method we start with some well known result of the best approximation in $L^1$ metric (see \cite{kry} ). The only advantage of our approach is that it is a little bit shorter.
Define
$$
\chi_h(\theta)=h^{-1}~~~\text{for}~~~\vert \theta \vert \leq \pi h~~~ \text{and} ~~~0 ~~~\text{otherwise on}~~~\mathbb{T}
$$
and
$$
\chi^m_h(\theta)=(\chi_h\ast\chi_h\ast \dots \chi_h) (\theta)~~~ (m ~~~\text{times})
$$
Then it is easy to see that
\begin{equation}
\label{chi}
\chi^m_h(\theta)=1+(2\pi h )^{-m}\sum_{p=0}^m(-1)^{m-p} \binom{m} p \mathcal{B}_m(\theta+(2p-m)\pi h)
\end{equation}
or
\begin{equation}
\label{chi_1}
\chi^m_h(\theta)=1+(2\pi h)^{-m}\Delta_{\pi h}^m \mathcal{B}_m(\theta)
\end{equation}
where $\Delta_h^m$ is the central difference.
In order to prove (\ref{chi}) we claim that
\begin{equation}
\label{chi_f}
\widehat{\chi_h}(k)=\frac{\sin \pi hk}{\pi h k}
\end{equation}
and hence
\begin{equation}
\label{chi_f_m}
\widehat{\chi_h^m}(k)=\left(\frac{\sin \pi hk}{\pi h k}\right)^m=(2\pi h)^{-m} (e^{i\pi h k }- e^ {-i \pi h k})^m\widehat{\mathcal{B}_m}(k)
\end{equation}
for $k\neq 0$. Hence for $k\neq 0$ the Fourier coefficients of the left side and of the right side of (\ref{chi}) are equal.
But for $k=0$ obviously $\widehat{\chi_h^m}(0)=1$ and for the sum $\Sigma$ with Bernoulli kernel in the right side of (\ref{chi}) we always have $\widehat{\Sigma}(0)=0$, since $\widehat{\mathcal{B}_m}(0)=0$.
Thus for all $k\in \mathbb{Z}$ Fourier coefficients of left and right sides of (\ref{chi}) coincides and hence the functions are equal.
The representation (\ref{chi}) and Favard identity (\ref{intr_fav_best}) immediately implies
\begin{equation}
\label{best_chi}
E_{n-1}(\chi_h^m)_{L^1}\leq \frac{K_m}{(\pi h n)^m}
\end{equation}
which is \textit{a good,} but not \textit{the best} estimate for \textit{all combinations }of $h$ and $n$.
Meanwhile, for some combinations of $n$ and $h$ it is the best one and to show this we use duality.
The fundamental E. Helly (or Hahn-Banach) Theorem
\begin{equation}
\label{intr_dual_helly}
(X /Y)^*\cong Y^\bot
\end{equation}
implies that
\begin{equation}
\label{intr_dual_trig}
E_{n-1}(f)_{L^1}=\sup_{\sigma\in T^\bot_{n-1}, \|\sigma\|_\infty=1} \int_0^{2\pi} f (\varphi)\overline{ \sigma(\varphi)}d\varphi/ 2\pi
\end{equation}
Here $T^\bot_{n-1}$ is ''annihilator space'' for $T_{n-1}$. i.e. space of functionals, which equal zero on each element from $T_{n-1}$, i. e. they annihilate polynomials.
Or in other words, if $\langle\cdot , \sigma \rangle$ is a functional on the space $L^1$ generated by $\sigma\in L^\infty$, then
$$
T^\bot_{n-1} =\lbrace \langle\cdot , \sigma \rangle \in (L^1)^* : \ker \langle\cdot , \sigma \rangle =T_{n-1}\rbrace
$$
Elementary Fourier analysis tell us that $\sigma\in T^\bot_{n-1}$ if and only if $\hat{\sigma}(k)=0$ for all $|k|\leq n-1$. Hence
\begin{equation}
\label{intr_dual_ber}
E_{n-1}(\chi_h^m)_{L^1}=\sup_{\sigma\in T^\bot_{n-1}, \|\sigma\|_\infty=1} \int_0^{2\pi}\chi_h^m(\varphi)\overline{ \sigma(\varphi)}d\varphi/ 2\pi
\end{equation}
and from representation (\ref{chi})
\begin{equation}
\label{intr_dual_ber_TT}
E_{n-1}(\chi_h^m)_{L^1}=
\end{equation}
$$
(2\pi h)^{-m}\sup_{\sigma\in T^\bot_{n-1}, \|\sigma\|_\infty=1} \sum_{p=0}^m(-1)^{m-p} \binom{m} p \int_0^{2\pi} \mathcal{B}_m(\varphi)\overline{ \sigma(\varphi-(2p-m)\pi h)}d\varphi/ 2\pi
$$
Define
\begin{equation}
\label{intr_dual_sin}
s_n(\varphi)=s(n\varphi)=\text{sgn} \sin (n \varphi)=\frac{2}{\pi }\sum_{k\in 2\mathbb{Z}+1} \frac{e^{ikn\varphi}}{ik}
\end{equation}
and
\begin{equation}
\label{intr_dual_cos}
c_n(\varphi)=c(n\varphi)=\text{sgn} \cos (n \varphi)= \text{sgn} \sin (n \varphi +\pi/2)=\frac{2}{\pi }\sum_{k\in 2\mathbb{Z}+1} (-1)^{\frac{k-1}{2}}\frac{e^{ikn\varphi}}{k}
\end{equation}
Then obviously $c_n( \varphi) \in T^\bot_{n-1}$ and has a unit $L^\infty$ norm. Remark, that since $\chi_h^m$ is an even function for all $m$, we have to choose even $\sigma(\varphi)$ in (\ref{intr_dual_ber_TT}) to get estimate from below.
Let from now $m$ be an \textit{odd} number, let say $m=2r+1$ and $h=\frac{2k-1}{2n}$ for $k=1,2,\dots n$. Note, that for these values of $h$
$$
c_n(\varphi-(2p-m)\pi h)=(-1)^{p-r-k}\text{sign}\cos(n\varphi-\pi/2 )=(-1)^{p-r-k}s(n\varphi)
$$
Define $\sigma(\varphi)=(-1)^k c_n(\varphi )$. Then from (\ref{intr_dual_ber_TT})
$$
E_{n-1}(\chi_h^m)_{L^1}\geq (-1)^k (2\pi h)^{-m} \sum_{p=0}^m(-1)^{m-p+p-r-k} \binom{m} p \int_0^{2\pi} \mathcal{B}_m(\varphi)\overline{s(n\varphi)}d\varphi/ 2\pi =
$$
$$
(-1)^{r}(\pi h)^{-m}\int_0^{2\pi} \mathcal{B}_m(\varphi)\overline{s(n\varphi)}d\varphi/ 2\pi =(-1)^{r}(\pi h)^{-m}\frac{2i}{\pi }\sum_{k\in 2\mathbb{Z}+1} \frac{\widehat{\mathcal{B}_m}(kn)}{k}=
$$
$$
(-1)^{r}(\pi hn)^{-m}\frac{2i}{\pi }\sum_{k\in 2\mathbb{Z}+1} \frac{1}{k(ik)^{2r+1}}=(-1)^r(\pi hn)^{-m}\frac{2}{\pi }\sum_{k\in 2\mathbb{Z}+1} \frac{(-1)^r}{k^{2r+2}}=
$$
$$
(\pi hn)^{-m}\frac{4}{\pi }\sum_{k=0}^\infty \frac{1}{(2k+1)^{2r+2}}=\frac{K_m}{(\pi h n)^m}
$$
which is an opposite inequality to (\ref{best_chi}) and hence we have an equality for odd $m$ and $h\in \mathcal{I}_n$, where
$$
\mathcal{I}_n=\left\lbrace \frac{1}{2n}, \frac{3}{2n}, \frac{5}{2n}, \dots \frac{2n-1}{2n}\right\rbrace
$$
Let now $m$ be an \textit{even} number, let say $m=2r$ and again $h\in \mathcal{I}_n$. Note, that for even $m$
$$
c_n(\varphi-(2p-m)\pi h)=(-1)^{p-r}\text{sign}\cos(n\varphi)=(-1)^{p-r}c(n\varphi)
$$
Hence, if we take $\sigma(\varphi)=c_n(\varphi)$ in (\ref{intr_dual_ber_TT}) we get
$$
E_{n-1}(\chi_h^m)_{L^1}\geq (2\pi h)^{-m} \sum_{p=0}^m(-1)^{m-p} \binom{m} p \int_0^{2\pi} \mathcal{B}_m(\varphi)(-1)^{p-r}\overline{c(n\varphi)}d\varphi/ 2\pi =
$$
$$
(-1)^r(\pi h)^{-m}\int_0^{2\pi} \mathcal{B}_m(\varphi)\overline{c(n\varphi)}d\varphi/ 2\pi =(-1)^r(\pi h)^{-m}\frac{2}{\pi }\sum_{k\in 2\mathbb{Z}+1} (-1)^{\frac{k-1}{2} }\frac{\widehat{\mathcal{B}_m}(kn)}{k}=
$$
$$
(\pi hn)^{-m}\frac{2}{\pi }\sum_{k\in 2\mathbb{Z}+1} \frac{(-1)^{\frac{k-1}{2}}}{k^{2r+1}}=(\pi hn)^{-m}\frac{4}{\pi}\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^{2r+1}}=\frac{K_m}{(\pi h n)^m}
$$
where we use (\ref{intr_fav_const_even}), since $m=2r$. This is an opposite inequality to (\ref{intr_fav_best}) and hence we have an equality for even $m$ and $h\in \mathcal{I}_n$.
We claim also that $\chi_h^m(\varphi)=0$ for $|\varphi|>\pi mh$. Hence if $\pi/2n\geq \pi mh$ we can take $\sigma(\varphi)=c_n(\varphi)$ to obtain
$$
\int_0^{2\pi}\chi_h^m(\varphi)\overline{ \sigma(\varphi)}d\varphi/ 2\pi=\int_0^{2\pi}\chi_h^m(\varphi)d\varphi/ 2\pi=1
$$
This observation implies
\begin{equation}
\label{best_chi_1}
E_{n-1}(\chi_h^m)_{L^1}= 1
\end{equation}
for all $h$ such that $|h|\leq \frac{1}{2mn}$.
If we compare (\ref{best_chi}) with (\ref{best_chi_1}) for $m=1$ we see that right side of (\ref{best_chi}) equal $1$ for $h=1/2n$.
For $m\geq 2$ and $h=\frac{1}{2mn}$ right side of (\ref{best_chi}) equal $(2m/ \pi)^m K_m > 1$ which indicate that our estimates are rude for these values of $h$.
Let we summarize all our observations together. Let numbers $n$ and $m$ are fixed. Then
\begin{equation}
\label{best_chi_first}
\text{for} ~~~ 0< h \leq\frac{1}{2mn} ~~~ \text{we have}~~~ E_{n-1}(\chi_h^m)_{L^1}= 1
\end{equation}
\begin{equation}
\label{best_chi_all}
\text{for} ~~~ \frac{1}{2mn} \leq h \leq 1~~~ \text{we have}~~~ E_{n-1}(\chi_h^m)_{L^1}\leq \frac{K_m}{(\pi h n)^m}
\end{equation}
\begin{equation}
\label{best_chi_odd}
\text{for} ~~~~h\in \left\lbrace \frac{1}{2n}, \frac{3}{2n}, \frac{5}{2n}, \dots \frac{2n-1}{2n}\right\rbrace~~~ \text{we have}~~~ E_{n-1}(\chi_h^m)_{L^1}= \frac{K_m}{(\pi h n)^m}
\end{equation}
\section{Best approximation of quasi-Bernoulli kernels}
Our second example are kernels $\mathcal{K}_1(\theta)=\mathcal{B}_1(\theta)\cos\theta$ and $\mathcal{K}_2(\theta)=\mathcal{B}_1(\theta)\sin\theta$, which play an important role in questions of approximation of smooth functions by \textit{algebraic } polynomials on the segment $[-1,1]$. This is the subject of the last section and actually the starting point of this note. We call kernels $\mathcal{K}$ as well as more general kernels, the \textit{quasi-Bernoulli }kernels since obvious analogy with quasi-polynomials.
\begin{theorem}
\label{theo_quasi_bern} Let $\mathcal{K}_1(\theta)=\mathcal{B}_1(\theta)\cos\theta$ and $\mathcal{K}_2(\theta)=\mathcal{B}_1(\theta)\sin\theta$. Then for all natural $n\geq 2$
\begin{equation}
\label{quasi_ber_ba_1}
E_{n-1} (\mathcal{K}_1)_{L^1}= \tan\frac{\pi}{2n}
\end{equation}
\begin{equation}
\label{quasi_ber_ba_2}
E_{n-1} (\mathcal{K}_2)_{L^1}= \sec\frac{\pi}{2n}-1
\end{equation}
\end{theorem}
\begin{proof}
To find the best approximation of these kernels by trigonometric polynomials we use formulas which seems unknown before --
\begin{equation}
\label{intr_golden_formula_1}
\mathcal{K}_1(\theta)=\frac{1}{2}\sin \theta+\sum_{r=0}^\infty((-1)^r\mathcal{B}_{2r+1}(\theta)-2\sin \theta))
\end{equation}
and
\begin{equation}
\label{intr_golden_formula_2}
\mathcal{K}_2(\theta)=1+\frac{1}{2}\cos \theta + \sum_{r=1}^\infty((-1)^{r+1}\mathcal{B}_{2r}(\theta)+2\cos\theta)
\end{equation}
To prove these formulas one have only to compute Fourier coefficients of both sides which is a routine work (see also (\ref{kry_1_coef}), (\ref{kry_1_coef_1}), (\ref{kry_2_coef}), (\ref{kry_2_coef_1}))
Since, obviously for $r\geq 1$ we have
$|(-1)^r \mathcal{B}_{2r+1}(\theta) -2\sin\theta | \leq 6 \cdot 2^{-(2r+1)}$ and $|(-1)^{r +1}\mathcal{B}_{2r}(\theta) +2\cos \theta | \leq 6 \cdot 2^{-2r}$, these series converges absolutely and hence uniformly on $\mathbb{T}$.
The corresponding trigonometric polynomials of best approximation of $\mathcal{K}_1$ and $\mathcal{K}_2$ in $L^1$ metric are
$$
\tau_{n-1}(\mathcal{K}_1)(\theta) =\frac{1}{2}\sin \theta+\sum_{r=0}^\infty((-1)^r\tau_{n-1}(\mathcal{B}_{2r+1})(\theta) -2\sin \theta)
$$
and
$$
\tau_{n-1}(\mathcal{K}_2)(\theta) =1+\frac{1}{2}\cos \theta+\sum_{r=1}^\infty((-1)^{r+1}\tau_{n-1}(\mathcal{B}_{2r})(\theta) +2\cos \theta)
$$
where series converges absolutely, since
$$
|(-1)^r\tau_{n-1}(\mathcal{B}_{2r+1})(\theta) -2\sin \theta|\leq E_{n-1}(\mathcal{B}_{2r+1})_{L^1}+|(-1)^r \mathcal{B}_{2r+1}(\theta) -2\sin\theta | \leq
$$
$$
\frac{K_{2r+1}}{n^{2r+1}}+ 6 \cdot 2^{-(2r+1)}
$$
and
$$
|(-1)^{r+1}\tau_{n-1}(\mathcal{B}_{2r})(\theta) +2\cos \theta|\leq E_{n-1}(\mathcal{B}_{2r})_{L^1}+|(-1)^{r+1} \mathcal{B}_{2r}(\theta) +2\cos\theta | \leq
$$
$$
\frac{K_{2r}}{n^{2r}}+ 6 \cdot 2^{-2r}
$$
Now we immediately get the desired upper estimates for $E_{n-1}(\mathcal{K} )$ from the Favard identities (\ref{intr_fav_best}) and formula (\ref{intr_final_formula_tan_cosec}).
In order to prove opposite inequality we again use duality. Define
\begin{equation}
e_n(\mathcal{K}_1)=\frac{1}{2\pi}\int_0^{2\pi} \mathcal{K}_1(\varphi)\text{sign}\sin n\varphi d\varphi
\end{equation}
and
\begin{equation}
\label{kry_2_mat}
e_n(\mathcal{K}_2)=\frac{1}{2\pi}\int_0^{2\pi} \mathcal{K}_2(\varphi)(-\text{sign}\cos n\varphi) d\varphi
\end{equation}
Then by the duality arguments we have $E_{n-1}(\mathcal{K} ) \geq e_n(\mathcal{K})$.
For $n\geq 2$ in the same manner as in the preceding section, we obtain
$$
e_n(\mathcal{K}_1)=\frac{2}{\pi}\sum_{k\in 2\mathbb{Z}+1} \frac{\widehat{\mathcal{K}_1}(-kn)}{ik}=\frac{2}{\pi}\sum_{k\in 2\mathbb{Z}+1} \frac{-kn}{ik((kn)^2-1)i}
=
$$
$$
\frac{2}{\pi}\sum_{k\in 2\mathbb{Z}+1} \frac{n}{((kn)^2-1)}=\frac{4}{\pi n}\sum_{k=0}^\infty \frac{1}{((2k+1)^2-n^{-2})}
$$
In order to calculate the last sum we can use (\ref{fav_const_tan})
\begin{equation}
\label{kry_1_formula}
\sum_{k=0}^\infty \frac{1}{(2k+1)^2-z^2}=\frac{\pi}{4z}\tan\frac{\pi z}{2}
\end{equation}
Hence
\begin{equation}
\label{kry_1_finito}
e_n(\mathcal{K}_1)=\tan\frac{\pi}{2n}
\end{equation}
Remark, that
\begin{equation}
\label{kry_1_asympt}
e_n(\mathcal{K}_1)\asymp \frac{\pi}{2n}=\frac{K_1}{n} ~~~ \text{for} ~~~~~~ n\to \infty
\end{equation}
and
\begin{equation}
\label{kry_2_asympt_l}
e_n(\mathcal{K}_1) >\frac{\pi}{2n} +\frac{\pi^3}{24n^3}=\frac{K_1}{n}+\frac{\pi^3}{24n^3}
\end{equation}
where $K_1$ is the first Favard constant (see e.g. \cite{as}, p. 75).
Analogously for $n\geq 2$
\begin{equation}
\label{kry_2_mat_calc}
e_n(\mathcal{K}_2)=\frac{2}{\pi}\sum_{k\in 2\mathbb{Z}+1}(-1)^\frac{k+1}{2} \frac{\widehat{\mathcal{K}_2}(-kn)}{k}=\frac{2}{\pi }\sum_{k\in 2\mathbb{Z}+1}(-1)^\frac{k-1}{2} \frac{n^{-2}}{k(k^2-n^{-2})}
=
\end{equation}
$$
\frac{2}{\pi}\sum_{k\in 2\mathbb{Z}+1} (-1)^\frac{k-1}{2}(\frac{1/2}{k-n^{-1}}+\frac{1/2}{k+n^{-1}}-\frac{1}{k})=
$$
$$
=\frac{2}{\pi}(\frac{1}{1-n^{-1}}+\frac{1}{1+n^{-1}}-\frac{1}{3-n^{-1}}-\frac{1}{3+n^{-1}}+\frac{1}{5-n^{-1}}+\frac{1}{5+n^{-1}}-\dots -2\frac{\pi}{4})
$$
Now we used (\ref{fav_const_sec}) to evaluate the last sum and obtain
\begin{equation}
\label{kry_2_finito}
e_n(\mathcal{K}_2)=\sec\frac{\pi}{2n}-1
\end{equation}
Remark, that
\begin{equation}
\label{kry_1_asympt}
e_n(\mathcal{K}_2)\asymp \frac{\pi^2}{8n^2} =\frac{K_2}{n^2}~~~ \text{for} ~~~~~~ n\to \infty
\end{equation}
and
\begin{equation}
\label{kry_2_asympt_l}
e_n(\mathcal{K}_2) >\frac{\pi^2}{8n^2} +\frac{5\pi^4}{384 n^4 }=\frac{K_2}{n^2}+\frac{5\pi^4}{384 n^4 }
\end{equation}
where $K_2$ is the second Favard constant (see e.g. \cite{as}, p. 75).
\end{proof}
\section{Bernoulli series}
From previous two sections we see that both representations has the form
\begin{equation}
\label{repr_gen}
\mathcal{K}=T+\sum_{r=1}^\infty \mathcal{B}_r\ast \mu_r
\end{equation}
with some trivial distribution $T$ ( trigonometric polynomial ) and non-trivial distributions $\lbrace \mu_r \rbrace$. We remind that distributions is a continues linear functionals over $C^\infty (\mathbb{T)}$. The typical examples are measures (for instance the Dirac measure $\langle f, \delta_\theta \rangle=f(\theta)$ with $\widehat{\delta}(k)= e^{-ik\theta}$), derivative $\mu_r=\partial^r$ with $\widehat{\mu_r}=(ik)^r$ or absolutely continues measures $\mu_r=g_r(\theta)d\theta$. See ( \cite{ed} p. 52) for definitions and explanations.
Indeed, representation (\ref{chi}) has the form (\ref{repr_gen}) with $T_0(\theta)=1$ and all $\mu_r$ equal $0$ except $r=m$, where
$$
\mu_m=(2 \pi h )^{-m} (\delta_{\pi h} -\delta_{-\pi h})^m
$$
On the other hand representation (\ref{intr_golden_formula_1}) has the form (\ref{repr_gen}) with $T(\theta)=\frac{1}{2}\sin \theta$ and $\mu_{2r}=0$ and $\mu_{2r+1}=(-1)^r\delta_0 -2\sin \theta d\theta$.
We remark that if we choose $\mu_r=\partial^r$ and $\mu_k=0$ for $k\neq r$ then the representation (\ref{repr_gen}) (as a functional over space $W^r$ ) is the familiar Euler-Maclauren formula (in the periodic case) with $\mathcal{K}=\delta_0$ and $\langle f, T\rangle =\widehat{f}(0)$.
We also claim, that any function $\mathcal{K}\in L^1$ has representation (\ref{repr_gen}) if we choose $\mu_r(\theta)=\widehat{K}(-r)(-ir)^re^{-ir\theta}+\widehat{K}(r)(ir)^re^{ir\theta}$ and $T(\theta)=\widehat{\mathcal{K}}(0)$, but we mainly interested in such representations where distributions are measures with non trivial discrete part.
In this section we show that expansion (\ref{repr_gen}) with measures with non trivial discrete part hold for quite general class of functions.
Next proposition in this direction corresponds to the expansion (\ref{repr_gen}) with $\mu_r=c_r(\delta-D_N(\theta)d\theta)$, where $\delta$ is the Dirac measure and $D_N (\theta)$ is the Dirichlet kernel.
\begin{lemma}
\label{gen_g}
Let $\mathcal{K}(\theta)\in L^1(\mathbb{T})$ and there exist natural number $N$ such that for $k\in \mathbb{Z}$ and $|k|\geq N+1$
\begin{equation}
\label{gen_g_f}
\widehat{\mathcal{K}}(k)=g(k)
\end{equation}
where $g(z)$ is an analytic function in $\lbrace z: ~~|z|>N\rbrace$ and has a zero at infinity.
Then
\begin{equation}
\label{gen_g_b}
\mathcal{K}(\theta)=T_N(\theta) + \sum_{m=1}^\infty c_m( \mathcal{B}_m(\theta)-S_N(\mathcal{B}_m,\theta))
\end{equation}
where
\begin{equation}
\label{gen_g_co}
c_m=\frac{1}{2\pi i}\int_{C} (i\zeta)^mg(\zeta)\frac{d\zeta}{\zeta}
\end{equation}
and integral is over some contour $C$ around closed disc $\lbrace z : |z|\leq N\rbrace$.
Here $S_N(\mathcal{B}_m ,\theta)$ is the $N$-th partial sum of Fourier series of the Bernoulli kernel. i. e.
$$
S_N(\mathcal{B}_{2r+1} ,\theta)=(-1)^{r}2\sum_{k=1}^N \frac{\sin k \theta}{k^{2r+1}}
$$
$$
S_N(\mathcal{B}_{2r} ,\theta)=(-1)^{r}2\sum_{k=1}^N \frac{\cos k \theta}{k^{2r}}
$$
and
$$
T_N(\theta)=\sum_{|k|\leq N} \widehat{\mathcal{K}}(k)e^{ik\theta}
$$
\end{lemma}
\begin{proof}
Since $g(z)$ is analytic outside of the closed disc $\lbrace z: |z|\leq N \rbrace$ and has a zero at infinity, it can be expanded in $\lbrace z : |z|> N\rbrace $ into the Laurent series
$$
g(z)=\sum_{m=1}^\infty \frac{c_m}{(iz)^m}
$$
with $c_m$ defined by (\ref{gen_g_co}) (see e.g. \cite{ww}, p. 100).
Denote $\mathcal{S}(\theta)$ the right side of (\ref{gen_g_b}) and claim that $|\mathcal{B}_m(\theta)-S_N(\mathcal{B}_m, \theta)|\leq 6\cdot (N+1)^{-m}$ on $\mathbb{T}$ and from (\ref{gen_g_co}) follows that $|c_m|\leq M (N+0.5)^m$. Hence series (\ref{gen_g_b}) converge absolutely and we can change sum and integral. This implies that for $|k|\leq N$
\begin{equation}
\label{gen_g_proof_k_f_N}
\widehat{\mathcal{S}}(k)=\widehat{T_N}(k)=\widehat{\mathcal{K}}(k)
\end{equation}
and for $|k|> N$
\begin{equation}
\label{gen_g_proof_k_f}
\widehat{\mathcal{S}}(k)=\sum_{m=1}^\infty c_m \widehat{\mathcal{B}_m} (k)=\sum_{m=1}^\infty \frac{c_m}{(ik)^m}=g(k)=\widehat{\mathcal{K}}(k)
\end{equation}
Thus all Fourier coefficients of $\mathcal{S}(\theta)$ and $\mathcal{K}(\theta)$ coincides and hence they are equal.
\end{proof}
Now we can apply this lemma to kernels, which arises in the questions of approximations of functions with bounded derivative of high order by algebraic polynomials on the segment $[-1,1]$.
Let
\begin{equation}
\label{bern_ser_quasi_ber}
\mathcal{K}(\theta)=\mathcal{B}_1 (\theta) t_1(\theta)+\mathcal{B}_2(\theta) t_2(\theta)+ \dots \mathcal{B}_m(\theta) t_m(\theta)
\end{equation}
where $\mathcal{B}_k(\theta) $ are Bernoulli kernels and $t_k(\theta)$ are (complex) trigonometric polynomials. Then we call $\mathcal{K}(\theta)$ the \textit{quasi-Bernoulli} kernel
For example $\mathcal{K}_1(\theta)=\mathcal{B}_1(\theta) \cos \theta$ and $\mathcal{K}_2(\theta)=\mathcal{B}_1(\theta) \sin \theta$ are quasi-Bernoulli kernels. We see that for $k\neq\pm 1$
\begin{equation}
\label{kry_1_coef}
\widehat{\mathcal{K}}_1(k) =\frac{1}{2}(\mathcal{B}_1(\theta) e^{i\theta}+\mathcal{B}_1(\theta)e^{-i \theta})\hat{}(k)=\frac{1}{2}(\widehat{\mathcal{B}_1}(k-1)+\widehat{\mathcal{B}_1}(k+1))=\frac{k}{(k^2-1) i}
\end{equation}
and
\begin{equation}
\label{kry_1_coef_1}
\widehat{\mathcal{K}}_1(\pm 1) =\pm \frac{1}{4i}, ~~~~~~\widehat{\mathcal{K}}_1(0) =0
\end{equation}
Analogously for $k\neq\pm 1$
\begin{equation}
\label{kry_2_coef}
\widehat{\mathcal{K}}_2(k) =\frac{1}{2i}(\mathcal{B}_1(\theta) e^{i\theta}-\mathcal{B}_1(\theta)e^{-i \theta})\hat{}(k)=\frac{1}{2i}(\widehat{\mathcal{B}_1}(k-1)-\widehat{\mathcal{B}_1}(k+1))=-\frac{1}{(k^2-1) }
\end{equation}
and
\begin{equation}
\label{kry_2_coef_1}
\widehat{\mathcal{K}}_2(\pm 1) = \frac{1}{4}, ~~~~~\widehat{\mathcal{K}}_2(0) =1
\end{equation}
Hence $\widehat{\mathcal{K}}_1(k) =g(k)$ for $|k|>1$ with
$$
g(z)=\frac{z}{i(z^2-1)}
$$
The function $g(z)$ is analytic in $|z|>1$ and obviously have a zero at infinity. Thus we can apply Lemma \ref{gen_g}. To calculate the coefficients we can use the residue theorem (\cite{ww}, p. 112) and get
\begin{equation}
\label{gen_g_co_k_1}
c_m=\frac{i^m}{2\pi i}\int_{C} \frac{\zeta^{m}}{i(\zeta^2-1)}d\zeta=\frac{i^{m-1}}{2\pi i}\int_{C} \zeta^{m}\frac{1}{2}( \frac{1}{\zeta -1} - \frac{1}{\zeta +1} )d\zeta=
\end{equation}
$$
i^{m-1}\frac{1}{2}(1-(-1)^m)
$$
Hence for $r=0, 1, 2, \dots$
\begin{equation}
\label{gen_g_co_k_1_2r}
c_{2r}=0 ~~~~~\text{and}~~~~~c_{2r+1}=(-1)^{r}
\end{equation}
We see that according to Lemma \ref{gen_g}
$$
\mathcal{K}_1(\theta)=\sum_{|k|\leq 1} \widehat{\mathcal{K}}(k)e^{ik\theta}+\sum_{r=0}^\infty c_{2r+1}( \mathcal{B}_{2r+1}(\theta)-S_1(\mathcal{B}_{2r+1},\theta))=
$$
$$
\frac{1}{2}\sin \theta +\sum_{r=0}^\infty (-1)^r( \mathcal{B}_{2r+1}(\theta)-(-1)^{r}2\sin\theta)
$$
which coincide with (\ref{intr_golden_formula_1})
Analogously we can obtain (\ref{intr_golden_formula_2}). Moreover we can use the same approach to calculate coefficients in representation for any quasi-Bernoulli kernel defined by the formula (\ref{bern_ser_quasi_ber}), since $g(z)$ in this case is always rational function and has all poles inside of the disc with radius bigger that maximum degree of polynomials $t_m(\theta)$.
\section{Approximation of Lipschitz functions by algebraic polynomials}
\label{_sec_alg_appr}
In \cite{favard_3} J. Favard posed and partially solved the problem of the best approximation of Lipschitz's functions by algebraic polynomials on the segment $\mathbb{I}=[-1,1]$.
Denote $W^1=W^1(\mathbb{I})$ the space of all Lipschitz functions, or in other words absolutely continues functions with finite norm $\|f\|_{W^1}=\|f'\|_{L^\infty}$. We prefer use the same symbols for the algebraic case, since it is clear from the context of the exposition which case we consider. Thus we define
$$
E_m(f)_X=\inf_{p\in P_m} \| f-p\|_X
$$
where $X$ is $L^1(\mathbb{I})$ or $L^\infty(\mathbb{I} ) $ and $P_m=\lbrace p(x) : p(x)=\sum_{k=0}^m p_k x^k \rbrace$ is the space of algebraic polynomials of the degree less or equal $m$. We also define
$$
E_m(X)=\sup_{f: \| f\|_X \leq 1} E_m(f)_X
$$
The result of J. Favard can be expressed in the form
$$
\frac{1}{n}<E_{n-1}(W^1)<\frac{K_1}{n}
$$
This result was improved in \cite{nik} by S. M. Nikolsky .
\begin{equation}
\label{alg_nik}
E_{n-1}(W^1)= \frac{K_1}{n}-\epsilon_n
\end{equation}
with
$$
\epsilon_n>0,~~~~~~ \epsilon_n=O(\frac{1}{n\log n})
$$
The exact value (formula) for $E_n(W^1)$ seems still unknown.
Since algebraic case reducing to the trigonometric case by the substitution $x=\cos \varphi$, the main questions concentrate around best constants in the inequalities and how constants depends of the point position on the interval. This dependence was first observed by S. M. Nikolsky in the same paper, were was noticed that for the algebraic Favard means $U_{n-1}(f,x)$ which are an algebraic polynomial of degree less or equal $n-1$
\begin{equation}
\label{intr_alg_nik_log}
|f(x)-U_{n-1}(f,x)| \leq \frac{K_1}{n}\sqrt{1-x^2}+ C\frac{\log n}{n^2}\sqrt{x^2}
\end{equation}
for all functions $f\in W^1$. Here $K_1$ is the first Favard constant. This is immediately corollary of Favard Theorem in trigonometric case, but S. M. Nikolsky also proved that for \textit{Favard means} $\log n$ factor can not be removed.
The last result indicate that in the algebraic case, instead of trigonometric, the constant near the term $1/n$ depends on the position of the point $x$ on the interval $[-1,1]$ and the reminder term (if exist) disappear near point $0$ and near point $1$ dominate.
Then many papers was dedicated to remove $\log$ factor using some linear methods of approximation by algebraic polynomials and to generalise Nikolsky's type inequality to high derivatives. The factor was removed and theorems was generalised, but constants, which appeared in the proofs was given with no fine estimates.
V. N. Temlyakov in \cite{tem} improved constants in the Nikolsky's type inequality (\ref{intr_alg_nik_log}) but, of course, for different approximation polynomials and, of course, with a little bigger factor near $\sqrt{1-x^2}$. But the factor near $\sqrt{x^2}$ (reminder term) have fine and simple expression.
\begin{equation}
\label{alg_tem}
|f(x)-P_{n}(f,x)| \leq \frac{K_1}{n}
\sqrt{1-x^2}+ \frac{2 K_2}{n^2}\sqrt{x^2}
\end{equation}
Here $K_1$ and $K_2$ are first and second Favard constants.
In this note we found, in some sense, optimal factors near $\sqrt{1-x^2} $ and $\sqrt{x^2}$ in (\ref{alg_tem}).
\begin{theorem}
\label{theo_main}
Let $f(x)$ be a function defined on $\mathbb{I}$ and $\|f\|_{W^1}\leq 1$.
Then for every $n\geq 2$ there exist an algebraic polynomial $P_{n}(f,x)$ of degree less or equal $n$, such that
\begin{equation}
\label{alg_est_t_s}
|f(x)-P_{n}(f,x)| \leq T(n)
\sqrt{1-x^2}+ S(n) \sqrt{x^2}
\end{equation}
where
$$
T(n)=\tan\frac{\pi}{2n}
$$
and
$$
S(n)=\sec\frac{\pi}{2n}-1
$$.
\end{theorem}
Before proof some remarks are in order.
Asymptotically for big $n$
\begin{equation}
\label{intr_tem_T}
T(n)=\frac{K_1}{n}+ O(n^{-3})
\end{equation}
and
\begin{equation}
\label{intr_tem_S}
S(n)=\frac{K_2}{n^2}+O(n^{-4})
\end{equation}
Thus, factor near $\sqrt{1-x^2}$ is asymptotically the same as in Temlyakov's Theorem, but the factor near $\sqrt{x^2}$ is $2$ times better.
We also note that the first factor $T(n)$ can not be equal or less than factor in Temlyakov's Theorem with the optimal second factor.
These estimate are sharp in the sense that factors near $\sqrt{1-x^2}$ and $\sqrt{x^2}$ can not be improved \textit{simultaneously} too much. The exact sense of the last remark will be a subject of forthcoming note.
\begin{proof}
First we transform the question to the trigonometric case by substitution $x=\cos \theta$. This is a standard approach since \cite{favard_3} and \cite{nik} (see also \cite{bust}). After this the original problem of approximation by algebraic polynomials looks as follows.
Let function $g\in N^1(\mathbb{T})$, i. e. $g(\theta)$ is periodic with period $2\pi$, even, absolutely continues function, such that $g'(\theta)=h(\theta)\sin \theta$, where $h(\theta)\in L^\infty$. Define $\|g\|_{N^1}=\|h\|_{L^\infty}$. We shall approximate $g(\theta)$ by trigonometric polynomials of degree less or equal $n-1$ and since in $N^1(\mathbb{T})$ the norm is not translation invariant, we expect that the desired estimate have to depend of the position of the point $\theta$ on $\mathbb{T}$.
As usual, we use the representation (see e.g. \cite{devore}, p. 211 and \cite{tem})
\begin{equation}
\label{ker_repr}
g(\theta)=\int_0^{2\pi} \mathcal{B}_1(\theta -\varphi) h(\varphi)\sin\varphi d\varphi/2\pi=\sin \theta (h\ast \mathcal{K}_1)(\theta)-\cos \theta (h\ast \mathcal{K}_2)(\theta)
\end{equation}
where
\begin{equation}
\label{kry_1}
\mathcal{K}_1(\theta)=\mathcal{B}_1(\theta)\cos\theta
\end{equation}
and
\begin{equation}
\label{kry_1}
\mathcal{K}_2(\theta)=\mathcal{B}_1(\theta)\sin \theta
\end{equation}
Let $\tau_{n-1}(\mathcal{K}_1)$ and $\tau_{n-1}(\mathcal{K}_2)$ be a trigonometric polynomials of best approximation of $\mathcal{K}_1$ and $\mathcal{K}_2$ in $L^1$ metric.
Then
\begin{equation}
\label{ker_repr}
g(\theta)-(h\ast \tau_{n-1}(\mathcal{K}_1))(\theta) \sin\theta + (h\ast \tau_{n-1}(\mathcal{K}_2)(\theta)\cos\theta =
\end{equation}
$$
= h\ast (\mathcal{K}_1-\tau_{n-1}(\mathcal{K}_1))(\theta)\sin \theta -h\ast (\mathcal{K}_2-\tau_{n-1}(\mathcal{K}_2))(\theta)\cos \theta
$$
and hence
\begin{equation}
\label{ker_est}
| g(\theta)-t_n(g)(\theta)|\leq E_{n-1} (\mathcal{K}_1) |\sin\theta| +E_{n-1} (\mathcal{K}_2) |\cos\theta|
\end{equation}
for trigonometric polynomial
\begin{equation}
\label{ker_pol}
t_n(g)(\theta)=(h\ast \tau_{n-1}(\mathcal{K}_1))(\theta) \sin\theta - (h\ast \tau_{n-1}(\mathcal{K}_2)(\theta)\cos\theta
\end{equation}
of the degree $n$.
This is standard arguments. The problem is how to find polynomials of best approximations of $\mathcal{K}$ kernels and how to calculate the values of these best approximations.
We apply Theorem \ref{theo_quasi_bern}, which implies that
\begin{equation}
\label{ker_est}
| g(\theta)-t_n(g)(\theta)|\leq T(n)|\sin\theta| +S(n)|\cos\theta|
\end{equation}
and after substitution $x=\cos \theta$ we get (\ref{alg_est_t_s}) where $P_n(f,x)$ is a corresponding algebraic polynomial in the Tschebicheff basis.
\end{proof}
\begin{remark} In the book \cite{bust} Temlyakov's estimate (\ref{alg_tem}) (Theorem 2.3.2 on the page 14) printed with the wrong factor near $\sqrt{x^2}$ --- instead $2K_2/n^2$ author put $K_2/n^2$, probably since of misprint. But we shall to stress out, that estimate with factor $K_2/n^2$ near $\sqrt{x^2}$ is actually wrong. This will be a subject of forthcoming note.
\end{remark}
| 36,816
|
TITLE: Decorated permutations and subset permutations
QUESTION [7 upvotes]: Decorated permutations are defined as permutations where the fix-points come in two colors (say $\overline{\cdot}$ and $\underline{\cdot}$). For example, the 16 decorated permutations of length 3 are
$$\overline{1}\overline{2}\overline{3},\underline{1}\overline{2}\overline{3},\overline{1}\underline{2}\overline{3},\overline{1}\overline{2}\underline{3},\underline{1}\underline{2}\overline{3},\underline{1}\overline{2}\underline{3},\overline{1}\underline{2}\underline{3},\underline{1}\underline{2}\underline{3},$$
$$\overline{1}32,\underline{1}32,\hspace{10pt}
21\overline{3},21\underline{3},\hspace{10pt}
231,\hspace{10pt}
312,\hspace{10pt}
3\overline{2}1,3\underline{2}1\ .$$
They play an important role in the context of the positive Grassmannian and are in bijection with many other combinatorial objects such as bounded affine permutations, Grassmann necklaces, Le-diagrams, positroids and equivalence classes of plabic graphcs, see, for example, here or here. These combinatorial objects have certainly been studied in detail in recent years.
Their total number of clearly $\sum_{k=0}^n F(n,k)2^k$, where $F(n,k)$ denotes the number of permutations of length $n$ with $k$ fixed-points.
When looking for these on the OEIS, I saw that neither of the above names was found there, whilst subset permutations are found there as A000522. These are, given a set with $n$ elements, obtained by choosing $k$ elements among them, and then permute these $k$ elements, giving in total $\sum_{i=0}^{n} {n \choose k}k! = \sum_{k=0}^{n} n!/k!\ $.
For example, for $n=3$, we obtain the following 16 elements:
$$-,1,2,3,12,21,13,31,23,32,123,132,213,231,312,321.$$
I claim (and quickly prove below) that subset permutations also belong to the list of combinatorial objects in bijection with the above.
My questions are now
Have the later already appeared in this context?
and
Is the equality $\sum_{k=0}^n F(n,k)2^k = \sum_{k=0}^{n} n!/k!$ otherwise directly implied?
Construction of the bijection:
Take a subset permutation. This is, take a set of $k$ element in $\{1,\ldots,n\}$ together with a permutation of these.
Consider this as a permutation of length $n$ with the remaining $n-k$ elements being fixed-points.
Now label these $n-k$ fixed-points by $\underline{\cdot}$ and the additional fixed-points from the permutation of the $k$ elements by $\overline{\cdot}$.
Than this is a bijection from subset permutations to decorated permutations.
For example, $143768$ as a subset permutation of $\{1,\ldots,8\}$ is mapped to $\overline{1}\underline{2}43\underline{5}76\overline{8}$.
Also, the preimages of the 16 decorated permutations at the beginning are
$$123,23,13,12,3,2,1,-,132,32,213,21,231,312,321,31.$$
REPLY [3 votes]: Max's answer is neat in using $e^{2x}\frac{e^{-x}}{1-x}=\frac{e^x}{1-x}$. We offer one approach without generating functions. To this end, let $f_n:=\sum_{k=0}^n\frac1{k!}$ and $g_n:=\sum_{k=0}^n\frac{2^k}{k!}\sum_{j=0}^{n-k}\frac{(-1)^j}{j!}$. Assume empty sums to be $0$. Clearly, $f_{n+1}-f_n=\frac1{(n+1)!}$. On the other hand, we have
\begin{align} g_{n+1}-g_n
&=\sum_{k=0}^{n+1}\frac{2^k}{k!}\sum_{j=0}^{n+1-k}\frac{(-1)^j}{j!}
-\sum_{k=0}^{n+1}\frac{2^k}{k!}\sum_{j=0}^{n-k}\frac{(-1)^j}{j!} \\
&=\sum_{k=0}^{n+1}\frac{2^k}{k!}\frac{(-1)^{n+1-k}}{(n+1-k)!}
=\frac{(-1)^{n+1}}{(n+1)!}\sum_{k=0}^{n+1}\binom{n+1}k(-2)^k \\
&=\frac{(-1)^{n+1}}{(n+1)!}(-1)^{n+1}=\frac1{(n+1)!}.
\end{align}
Therefore, $f_{n+1}-f_n=g_{n+1}-g_n$. Since $f_0=g_0=1$, it follows $f_n=g_n$ and $n!f_n=n!g_n$. Obviously, $n!f_n=\sum_{k=0}^n\frac{n!}{k!}$ and
$$n!g_n=\sum_{k=0}^n2^k\frac{n!}{k!}\sum_{j=0}^{n-k}\frac{(-1)^j}{j!}=\sum_{k=0}^n2^k\binom{n}k\cdot !(n-k)=\sum_{k=0}^nF(n,k)2^k.$$
| 123,752
|
TITLE: Number of trials required to get value in a certain range
QUESTION [0 upvotes]: We do $M$ trials. In each trial the result is a uniform RV in $[0,1]$. What is the minimum no of tosses needed to be $90\%$ sure of getting a value in range $[0.8, 0.9]$.
I figure the answer is $9$ since each independent trial has probability of hitting this range as $0.1$. Can someone please confirm?
Thanks.
REPLY [0 votes]: As requested, here is my comment in answer form:
The problem with the method in the post is that you can't simply add the probabilities, there are overlaps. Clearest example is coin tosses: How many tosses do you need to have at least a 90% chance of getting at least one H? If you simply added you'd think 2 tosses was enough to guarantee an H (as each slot has a probability of $\frac 12$). But this is wrong, as the probability of TT is $\frac 14$. Better to work from the other direction. The only way NOT to get at least one H in n tosses is to throw n T in a row. The probability of that is $(\frac 12)^n$, hence the probability that you get at least one H is 1 - $(\frac 12)^n$. A little calculation shows that 3 tosses is insufficient, but 4 tosses gives you a 93.75% chance.
Approaching the original problem the same way, we ask for the probability that each of n trials misses the specified subinterval. That probability is clearly $(\frac {9}{10})^n$. Hence the probability that at least one of your trials lands in the right interval is 1 - $(\frac {9}{10})^n$. We want the smallest n for which this is at least 90%. The calculation takes a bit longer than it did for the coins; in the end we see that 22 trials just makes it, coming in at about 90.15%.
| 6,775
|
TITLE: How many ways can you get a sum $300$ from $100$ times choosing a number from $(0, 7, 11, 18)$
QUESTION [0 upvotes]: How many ways can you get a sum $300$ from $100$ times choosing a number from $(0,7,11,18)$? Sets are ordered, for example, if sum is $18$, and you can choose 3 times, the answer is:
$(0, 0, 18), (0, 18, 0), (18, 0, 0), (7, 11, 0), (7, 0, 11), (0, 7, 11), (0, 11, 7), (11, 0, 7), (11, 7, 0)$. The number is very big (checked with python). What have I done so far: there are only $4$ ways to get $300$ with unordered sets from $(7, 11):$ $300 = 2 \cdot 7 + 26 \cdot 11 = 13 \cdot 7 + 19 \cdot 11 = 24 \cdot 7 + 12 \cdot 11 = 35 \cdot 7 + 5 \cdot 11$
REPLY [1 votes]: @JimmyK4542 gave you a beneficial hint ,so i want you give you another approach that may facilitate your work.You said that i do not know generating fucntions ,but i believe that do not need to be master over generating functions to understand my suggested method.
We said that each places in the tuple can be seen like a distinguishable boxes and we want to put some numbers into these boxes such that each box can contain one number and the summation of the numbers in the boxes is equal to $300$.Whats more , it is given that there are $100$ "boxes".
Now , we know that each box can take either $0$ or $7$ or $11$ or $18$. Then ,lets show these numbers as exponential of a variable such as $\mathbf{\text{x}}$. Moreover , as you realize , we said that "each box can take either $0$ or $7$ or $11$ or $18$" ,so if you remember the basic counting rules , we use $\mathbf{\text{"+"}}$ when there is $\mathbf{\text{"or"}}$ connector.By using these informations , we can write the generating function form of a "box" by $$(x^0 +x^7 +x^{11} +x^{18})$$
As you mentioned in question ,we want to find all possible arragements such that $(18,0,7,...,11),(0,11,..,18,...,0)$ etc. Now , because of there are $100$ "boxes".I should work over $$(x^0 +x^7 +x^{11} +x^{18})^{100}$$ to find the coefficent of $x^{300}$ in the expansion of $(x^0 +x^7 +x^{11} +x^{18})^{100}$.Because ,as ou remember from binomial theorem , when we find the coefficient of an variable $x^n$ ,we choose one element form each paranteses to construct $x^n$. The coefficient of $x^n$ give use the number of all possible ways to construct $x^n$.
When we expand $(x^0 +x^7 +x^{11} +x^{18})^{100}$ , we select only one variable from each paranteses to contruct $x^{300}$. Do you get the idea ? It is so easy even for those who do not generating functions.
When we come to how to find the coefficients in these expansions. It can be found by hands , when you learn generating functions ,you will se some techniques , but i recommend you to use softwares to save time. When you click the link , you will see the calculation. As you see , the result is $$84,821,934,421,635,014,089,177,660,022,757,410,400$$
| 208,708
|
\begin{document}
\maketitle
\begin{abstract}
Motivated by applications in intelligent highway systems, the
paper studies the problem of guiding mobile agents in a
one-dimensional formation to their desired relative positions.
Only coarse information is used which is communicated from a
guidance system that monitors in real time the agents' motions.
The desired relative positions are defined by the given distance
constraints between the agents under which the overall formation
is rigid in shape and thus admits locally a unique realization. It
is shown that even when the guidance system can only transmit at
most four bits of information to each agent, it is still possible
to design control laws to guide the agents to their desired
positions. We further delineate the thin set of initial conditions
for which the proposed control law may fail using the example of a
three-agent formation.
Tools from non-smooth analysis are utilized for the convergence
analysis.
\end{abstract}
\section{Introduction} \label{se:intro}
In recent years, various ideas have been proposed to realize
intelligent highway systems to reduce traffic congestions and
improve safety levels. It is envisioned that navigation,
communication and automatic driver assistance systems are critical
components \cite{BrVa97,Bi05,Sh07}. A great deal of monitoring and
controlling capabilities have been implemented through roadside
infrastructures, such as cameras, sensors, and control and
communication stations. Such systems can work together to monitor
in real time the situations on highways and at the same time
guide vehicles to move in a coordinated fashion, e.g. to keep
appropriate distances from the vehicles in front of and behind
each individual vehicle. In intelligent highway systems, the
guiding commands are expected to be simple and formatted as short
digital messages to scale with the number of vehicles and also to
avoid conflict with the automatic driver assistance systems
installed within the vehicles. Similar guided formation control
problems also arise when navigating mobile robots or docking
autonomous vehicles \cite{TuCrTrMoBoDo06}.
Motivated by this problem of guiding platoons of vehicles on
highways, we study in this paper the problem of controlling a
one-dimensional multi-agent formation using only \emph{coarsely}
quantized information. The formation to be considered are rigid
under inter-agent distance constraints and thus its shape is
uniquely determined locally. Most of the existing work on
controlling rigid formations of mobile agents, e.g.
\cite{AnYuFiHe08,CaMoCDC07,KrBrFr08}, assumes that there is no
communication bandwidth constraints and thus real-valued control
signals are utilized. The idea of quantized control through
digital communication channels has been applied to consensus
problems, e.g.\ \cite{KaBaSr07,FrCaFaZa09} and references therein,
and more recently to formation control problems \cite{DiJo10}. The
uniform quantizer and logarithmic quantizer \cite{NaFaZaEv07} are
among the most popular choices for designing such controllers with
quantized information. Moreover, the paper \cite{CePeFr10} has
discussed Krasowskii solutions and hysteretic quantizers in
connection with continuous-time average consensus algorithms under
quantized measurements.
The problem studied in this paper distinguishes itself from the
existing work in that it explores the limit of the least bandwidth
for controlling a one-dimensional rigid formation by using a
quantizer in its simplest form with only two quantization levels.
As a result, for each agent in the rigid formation, at most four
bits of bandwidth is needed for the communication with the
navigation controller. The corresponding continuous-time model
describing the behavior of the overall multi-agent formation is,
however, non-smooth and thus an appropriate notion of solution
\cite{CePe07} has to be defined first. We use both the Lyapunov
approach and trajectory-based approach to prove convergence since
the former provides a succinct view about the dynamic behavior
while the latter leads to insight into the set of initial
positions for which the proposed controller may fail. We also
discuss some situations when different assumptions about the
quantization scheme are made and indicate those scenarios in which
the formation control problem with quantized information can be
challenging to solve.
The rest of the paper is organized as follows. We first formulate
the one-dimensional guided formation control problem with coarsely
quantized information in section \ref{se:formulation}. Then in
section \ref{se:analysis}, we provide the convergence analysis
results first using the Lyapunov method and then the
trajectory-based method.
Simulation results are presented in section \ref{se:simulation} to
validate the theoretical analysis. We make concluding remarks in
section \ref{se:conclusion}.
\section{Problem formulation} \label{se:formulation}
The one-dimensional guided formation that we are interested in
consists of $n$ mobile agents. We consider the case when the
formation is rigid \cite{AnYuFiHe08}; to be more specific, if we
align the given one-dimensional space with the $x$-axis in the
plane and label the agents along the positive direction of the
$x$-axis by $1, \ldots, n$, then the geometric shape of the
formation is specified by the given pairwise distance constraints
$|x_i-x_{i+1}| = d_i$, $i = 1, \ldots, n-1$, where $d_i>0$ are
desired distances. Although the guidance system can monitor the
motion of the agents in real time, we require that it can only
broadcast to the mobile agents quantized guidance information
through digital channels. In fact, we explore the limit for the
bit constraint by utilizing the quantizer that only has two
quantization levels and consequently its output only takes up one
bit of bandwidth. The quantizer that is under consideration takes
the form of the following sign function: For any $z\in \R$,
\[
\textrm{sgn}(z)=\left\{ \ba{ll}
+1 & z\ge 0\\
-1 & z<0\;. \ea\right.
\]
Each agent, modeled by a kinematic point, then moves according to
the following rules utilizing the coarsely quantized information:
\be\label{quantized.n.agent.system} \ba{rcl}
\dot x_1 &=& -k_1 \textrm{sgn} (x_1-x_2) \textrm{sgn} (|x_1-x_2|-d_1)\\
\dot x_i &=& \textrm{sgn}(x_{i-1}-x_i) \textrm{sgn}(|x_{i-1}-x_i|-d_{i-1})-\\
&& k_i \textrm{sgn}(x_i-x_{i+1}) \textrm{sgn}(|x_i-x_{i+1}|-d_{i}),\\ &&\hfill i=2,\ldots,n-1\\
\dot x_n &=& \textrm{sgn}(x_{n-1}-x_n)
\textrm{sgn}(|x_{n-1}-x_n|-d_{n-1}) \ea \ee
where $x_i\in \R$ is the position of agent $i$ in the
one-dimensional space aligned with the $x$-axis, and $k_i>0$ are
gains to be designed. Note that since each agent is governed by at
most two distance constraints, as is clear from
(\ref{quantized.n.agent.system}), a bandwidth of four bits is
sufficient for the communication between the guidance system and
the agents $2, \ldots, n-1$ and the required bandwidths for the
guidance signals for agents $1$ and $n$ are both 2 bits. Hence, in
total only $4n-2$ bits of bandwidth is used.
The main goal of this paper is to demonstrate under this extreme
situation of using coarsely quantized information, the formation
still exhibits satisfying convergence properties under the
proposed maneuvering rules. Towards this end, we introduce the
variables of relative positions among the agents \be\label{z}
z_i\dfb x_i-x_{i+1}\;,\quad i=1,2,\ldots,n-1\;. \ee Let us express
the system in the $z$-coordinates to obtain
\be\label{quantized.n.agent.system.z} \ba{rcl}
\dot z_1 &=& -(k_1+1) \textrm{sgn}(z_1) \textrm{sgn}(|z_1|-d_1)\\[2mm] && + k_2 \textrm{sgn}(z_2) \textrm{sgn}(|z_2|-d_2)\\[2mm]
\dot z_i &=& \textrm{sgn}(z_{i-1}) \textrm{sgn}(|z_{i-1}|-d_{i-1}) \\[2mm]&& -(k_i+1) \textrm{sgn}(z_i) \textrm{sgn}(|z_i|-d_{i})\\[2mm]
&& +k_{i+1} \textrm{sgn}(z_{i+1}) \textrm{sgn}(|z_{i+1}|-d_{i+1}),\\[2mm] && \hfill i=2,\ldots,n-2\\[2mm]
\dot z_{n-1} &=& \textrm{sgn}(z_{n-2})
\textrm{sgn}(|z_{n-2}|-d_{n-2})\\ [2mm] &&-(k_{n-1}+1)
\textrm{sgn}(z_{n-1}) \textrm{sgn}(|z_{n-1}|-d_{n-1})\;. \ea \ee
To study the dynamics of the system above, we need to first
specify what we mean by the solutions of the system. Since the
vector field $f(z)$ on the right-hand side is discontinuous, we
consider Krasowskii solutions, namely solutions to the
differential inclusion $\dot z\in {\cal K}(f(z))$, where
\[
{\cal K}(f(z))=\bigintersect_{\delta
>0}{\overline {\rm co}}\, (f (B(z,\delta)))\;,
\]
$\overline {\rm co}$ denotes the involutive closure of a set, and
$B(z,\delta)$ is the ball centered at $z$ and of the radius
$\delta$. The need to consider these solutions becomes evident in
the analysis in the next section. Since the right-hand side of
(\ref{quantized.n.agent.system}) is also discontinuous, its
solutions are to be intended in the Krasowskii sense as well. Then
we can infer conclusions on the behavior of
(\ref{quantized.n.agent.system}) provided that each solution $x$
of (\ref{quantized.n.agent.system}) is such that $z$ defined in
(\ref{z}) is a Krasowskii solution of
(\ref{quantized.n.agent.system.z}). This is actually the case by
\cite{BP-SSS:87}, Theorem 1, point 5), and it is the condition
under which we consider (\ref{quantized.n.agent.system.z}). It
turns out that the $z$-system (\ref{quantized.n.agent.system.z})
is easier to work with for the convergence analysis that we
present in detail in the next section.
\section{Convergence analysis} \label{se:analysis}
In this section, after identifying the equilibria of the system,
we present two different approaches for convergence analysis. The
first is based on a Lyapunov-like function and the second examines
the vector field in the neighborhood of the system's trajectories.
\subsection{Equilibria of the system}
We start the analysis of system (\ref{quantized.n.agent.system.z})
by looking at the discontinuity points of the system. A
discontinuity point is a point at which the vector field on the
right-hand side of the equations above is discontinuous. Hence,
the set ${\cal D}$ of all the discontinuity points is:
\[
{\cal D}=\{z\in \R^{n-1}: \Pi_{i=1}^{n-1} z_i (|z_i|-d_{i})=0\}\;.
\]
It is of interest to characterize the set of equilibria:
\begin{proposition}\label{lemma.equilibria}
Let $k_1+1>k_2$, $k_i>k_{i+1}$ for $i=2,\ldots,n-2$, and
$k_{n-1}>0$. The set of equilibria, i.e.\ the set of points for
which $\mathbf{0}\in {\cal K}(f(z))$ with $f(z)$ being the vector
field on the right-hand side of
(\ref{quantized.n.agent.system.z}), is given by
\[
{\cal E}=\{z\in \R^{n-1}: \sum_{i=1}^{n-1} |z_i|
||z_i|-d_{i}|=0\}\;.
\]
\end{proposition}
The proof of this proposition relies on the following lemma.
\begin{lemma} \label{lemma.claim}
For $i\in \{2,\ldots, n-2\}$, if $|z_j|\,||z_j|-d_{j}|=0$ for
$j=1,2,\ldots, i-1$, and $\mathbf{0}\in {\cal K}(f(z))$, then
$|z_{i}|\, ||z_{i}|-d_{i}|=0$.
\end{lemma}
\medskip
\textit{Proof:} Suppose by contradiction that $|z_{i}|\,
||z_{i}|-d_{i}|\ne 0$. Observe that $z$ belongs to a discontinuity
surface where in particular $|z_{i-1}|\, ||z_{i-1}|-d_{i-1}|=0$.
This implies that in a neighborhood of this point, the state space
is partitioned into different regions where $f(z)$ is equal to
constant vectors. In view of (\ref{quantized.n.agent.system.z}),
the component $i$ of these vectors is equal to one of the
following values: $1-(k_i+1)+k_{i+1}$, $1-(k_i+1)-k_{i+1}$,
$-1-(k_i+1)+k_{i+1}$, $-1-(k_i+1)-k_{i+1}$, if $\textrm{sgn}(z_i)
\textrm{sgn}(|z_i|-d_{i})=1$, or $1+(k_i+1)+k_{i+1}$,
$1+(k_i+1)-k_{i+1}$, $-1+(k_i+1)+k_{i+1}$, $-1+(k_i+1)-k_{i+1}$,
if $\textrm{sgn}(z_i) \textrm{sgn}(|z_i|-d_{i})=-1$. Any $v\in
{\cal K}(f(z))$ is such that its component $i$ belongs to (a
subinterval of) the interval
$[-1-(k_i+1)-k_{i+1},1-(k_i+1)+k_{i+1}]$ if $\textrm{sgn}(z_i)
\textrm{sgn}(|z_i|-d_{i})=1$ (respectively, to the interval
$[-1+(k_i+1)-k_{i+1},1+(k_i+1)+k_{i+1}]$ if $\textrm{sgn}(z_i)
\textrm{sgn}(|z_i|-d_{i})=-1$). In both cases, if $k_i>k_{i+1}$,
then the interval does not contain $0$ and this is a
contradiction. This ends the proof of the lemma. \hfill $\square$
\textit{Proof of Proposition \ref{lemma.equilibria}:} First we
show that if $\mathbf{0}\in {\cal K}(f(z))$, then $z\in {\cal E}$.
As a first step, we observe that $\mathbf{0}\in {\cal K}(f(z))$
implies $|z_1| ||z_1|-d_{1}|=0$. In fact, suppose by contradiction
that the latter is not true. This implies that at the point $z$
for which $\mathbf{0}\in {\cal K}(f(z))$, any $v\in {\cal
K}(f(z))$ is such that the first component takes values in the
interval $[-(k_1+1)-k_2, -(k_1+1)+k_2]$, or in the interval
$[(k_1+1)-k_2, (k_1+1)+k_2]$. In both cases, if $k_1+1>k_2$, then
$0$ does not belong to the interval and this contradicts that
$\mathbf{0}\in {\cal K}(f(z))$. Hence, $|z_1| ||z_1|-d_{1}|=0$.
This and Lemma \ref{lemma.claim} show that $|z_{i}|\,
||z_{i}|-d_{i}|=0$ for $i=1,2,\ldots, n-2$. To prove that also
$|z_{n-1}|\, ||z_{n-1}|-d_{n-1}|=0$, consider the last equation of
(\ref{quantized.n.agent.system.z}), and again suppose by
contradiction that $|z_{n-1}|\, ||z_{n-1}|-d_{n-1}|\ne 0$. Then
the last component of $v\in {\cal K}(f(z))$ belongs to a
subinterval of $[-1-(k_{n-1}+1), 1-(k_{n-1}+1)]$ or to a
subinterval of $[-1+(k_{n-1}+1), 1+(k_{n-1}+1)]$. If $k_{n-1}>0$,
then neither of these intervals contain $0$ and this is again a
contradiction. This concludes the first part of the proof, namely
that $\mathbf{0}\in {\cal K}(f(z))$ implies $z\in {\cal E}$.
Now we let $z\in {\cal E}$ and prove that $\mathbf{0}\in {\cal
K}(f(z))$. By definition, if $z\in {\cal E}$, then $z$ lies at the
intersection of $n-1$ planes, which partition $\R^n$ into
$\nu\stackrel{\cdot}{=} 2^{n-1}$ regions, on each one of which
$f(z)$ is equal to a different constant vector. Any $v \in {\cal
K}(f(z))$ is the convex combination of these $\nu$ vectors, which
we call $v^{(1)}, \ldots, v^{(\nu)}$. We construct $v \in {\cal
K}(f(z))$ such that $v=\mathbf{0}$. We observe first that, the
component $1$ of the vectors $v^{(i)}$'s can take on four possible
values, namely $(k_1+1)+k_2$, $(k_1+1)-k_2$, $-(k_1+1)+k_2$,
$-(k_1+1)-k_2$, and that there are exactly $\frac{\nu}{4}$ (we are
assuming that $n\ge 3$, as the case $n=2$ is simpler and we omit
the details) vectors among $v^{(1)},\ldots,v^{(\nu)}$ whose first
component is equal to $(k_1+1)+k_2$, $\frac{\nu}{4}$ whose first
component is equal to $(k_1+1)-k_2$ and so on. As a consequence,
if $\lambda_i=\frac{1}{\nu}$ for all $i=1,2,\ldots,\nu$, then
$\sum_{j=1}^\nu \lambda_j v_1^{(j)}=0$.\\
Similarly, the component $i$, with $i=2,\ldots, n-2$, can take on
eight possible values ($1+(k_i+1)+k_{i+1}$,
$1+(k_i+1)-k_{i+1}$,$\ldots, -1-(k_i+1)-k_{i+1}$ -- see the
expression of $\dot z_i$ in (\ref{quantized.n.agent.system.z}))
and as before, the set $v^{(1)},\ldots,v^{(\nu)}$ can be
partitioned into $\frac{\nu}{8}$ sets, and each vector in a set
has the component $i$ equal to one and only one of the eight
possible values. Moreover, these values are such that
$\sum_{j=1}^\nu
\lambda_j v_i^{(j)}=0$. \\
Finally, if $i=n-1$, the set $v^{(1)},\ldots,v^{(\nu)}$ can be
partitioned into four sets, and each vector in a set has the last
component equal to one and only one of the four possible values
$1+(k_{n-1}+1)$, $1-(k_{n-1}+1)$, $-1+(k_{n-1}+1)$,
$-1-(k_{n-1}+1)$. Hence, $\sum_{j=1}^\nu \lambda_j
v_{n-1}^{(j)}=0$. Let now $v\in {\cal K}(f(z))$ be such that
$v=\sum_{i=1}^{\nu} \lambda_i v^{(i)}$, with
$\lambda_i=\frac{1}{\nu}$ for all $i$. Since $\sum_{j=1}^\nu
\lambda_j v_i^{(j)}=0$ for all $i=1,2,\ldots, n-1$, then
$v=\mathbf{0}$ and this proves that for all $z\in {\cal E}$, we
have $\mathbf{0}\in {\cal K}(f(z))$. This completes the proof.
\hfill $\square$
Next, we show that the equilibrium set $\scr{E}$ is attractive.
\subsection{Lyapunov function based analysis}
Now we are in a position to present the main convergence result.
\begin{theorem}
If \be\label{gains} k_1\ge k_2 \;,\; k_i\ge k_{i+1}+1\;,\;
i=2,\ldots,n-2\;,\; k_{n-1}\ge 1\;, \ee then all the Krasowskii
solutions to (\ref{quantized.n.agent.system.z}) converge to (a
subset of) the equilibria set ${\cal E}$.
\end{theorem}
\textit{Proof:} Let
\[
V(z)=\frac{1}{4}\sum_{i=1}^{n-1} (z_i^2-d_i^2)^2
\]
be a smooth non-negative function. We want to study the expression
taken by $\nabla V(z) f(x)$, where $f(z)$ is the vector field on
the right-hand side of (\ref{quantized.n.agent.system.z}). We
obtain:
\begin{eqnarray*}
&&\nabla_{z_i} V(z) \dot z_i \\
&= & \left\{ \ba{ll}
z_1 (z_1^2-d_1^2) [-(k_1+1)\textrm{sgn}(z_1) \textrm{sgn}(|z_1|-d_1)\\+ k_2 \textrm{sgn}(z_2) \textrm{sgn}(|z_2|-d_2)] \\ \qquad i=1\\
\qquad \\
z_i(z_i^2-d_i^2)[-(k_i+1) \textrm{sgn}(z_i) \textrm{sgn}(|z_i|-d_{i}) \\ + \textrm{sgn}(z_{i-1}) \textrm{sgn}(|z_{i-1}|-d_{i-1}) \\
+k_{i+1} \textrm{sgn}(z_{i+1}) \textrm{sgn}(|z_{i+1}|-d_{i+1})] \\ \qquad i=2,\ldots, n-2\\
\qquad \\
z_{n-1}(z_{n-1}^2-d_{n-1}^2)[\textrm{sgn}(z_{n-2})
\textrm{sgn}(|z_{n-2}|-d_{n-2})
& \\
-(k_{n-1}+1) \textrm{sgn}(z_{n-1}) \textrm{sgn}(|z_{n-1}|-d_{n-1})] \\
\qquad i=n-1 \ea \right.
\end{eqnarray*}
If $z\not \in {\cal D}$, i.e.\ if $z$ is not a point of
discontinuity for $f(z)$, then:
\begin{eqnarray*}
&&\nabla_{z_i} V(z) \dot z_i \\
&\le& \left\{ \ba{ll}
-(k_1+1-k_2) |z_1|\,|z_1^2-d_1^2|& i=1\\[2mm]
-(k_i-k_{i+1}) |z_i|\,|z_i^2-d_i^2| & i=2,\ldots, n-2\\[2mm]
-k_{n-1} |z_{n-1}|\,|z_{n-1}^2-d_{n-1}^2| & i=n-1 \ea \right.
\end{eqnarray*}
where we have exploited the fact that
$\textrm{sgn}(z_i^2-d_i^2)=\textrm{sgn}(|z_i|-d_i)$. Hence, if
(\ref{gains}) holds, then
\[
\nabla V(z) f(z)\le -\sum_{i=1}^{n-1}|z_i|\,|z_i^2-d_i^2|<0\;.
\]
If $z\in {\cal D}$, we look at the set
\[
\dot{\overline V}(z)=\{a\in \R: \exists v\in {\cal K}(f(z))\,\,\,
{\rm s.t.}\,\, \, a=\nabla V(z)\cdot v\}\;.
\]
We distinguish two cases, namely (i) $z\in {\cal E}\subseteq {\cal
D}$ and (ii) $z\in {\cal D}\setminus {\cal E}$. In case (i),
$\nabla V(z)=\mathbf{0}^T$, and therefore, $\dot{\overline
V}(z)=\{0\}$. In case (ii), there must exist at least one agent
such that $|z_i|\,|z_i^2-d_i^2|=0$ and at least one agent such
that $|z_j|\,|z_j^2-d_j^2|\ne 0$. Let ${\cal I}_1(z)$
(respectively, ${\cal I}_2(z)$) be the set of indices
corresponding to agents for which $|z_i|\,|z_i^2-d_i^2|=0$
($|z_j|\,|z_j^2-d_j^2|\ne 0$).
Clearly, ${\cal I}_1(z)\cup {\cal I}_2(z)=\{1,2,\ldots,n-1\}$.\\
Since $\nabla_{z_i} V(z)=z_i(z_i^2-d_i^2)=0$ if $i\in {\cal
I}_1(z)$, then
\[\ba{rcl}
\nabla V(z) \cdot v &=& \dst\sum_{i=1}^{n-1} z_i(z_i^2-d_i^2)
v_i\\
&=& \dst\sum_{i\in {\cal I}_2(z)} z_i(z_i^2-d_i^2) v_i\;. \ea
\]
Let $i\in {\cal I}_2(z)$ and $v\in {\cal K}(f(z))$. In view of
(\ref{quantized.n.agent.system}), for $i=1,2,\ldots, n-1$, it
holds:
\begin{eqnarray*}
&&v_i\in \{\mu\in \R:
\mu=(2\lambda_1-1)\tilde k_{i+1} \\
&& \qquad \qquad - (k_i+1) \textrm{sgn}(z_i) \textrm{sgn}(|z_i|-d_{i}),\,
\lambda_i\in [0,1]\}\;,
\end{eqnarray*}
with
\[
\tilde k_{i+1} =\left\{ \ba{ll}
k_{2} & i=1\\
1+k_{i+1} & i=2,\ldots, n-2\\
1 & i=n-1\;. \ea \right.
\]
Then
\[\ba{rcl}
\nabla V(z) \cdot v &=& \dst\sum_{i\in {\cal I}_2(z)}
z_i(z_i^2-d_i^2)
v_i\\
&\le& \dst\sum_{i\in {\cal I}_2(z)} -(k_i+1) |z_i| |z_i^2-d_{i}^2|
\\&& + \tilde k_{i+1} |z_i| |z_i^2-d_{i}^2| |2\lambda_j-1| \;. \ea
\]
By (\ref{gains}), $k_i+1-\tilde k_{i+1}\ge 1$ for all $i$, and
therefore, if $z\in {\cal D}\setminus {\cal E}$, then
\[\ba{rcl}
\nabla V(z) \cdot v &\le & -\dst\sum_{i\in {\cal I}_2(z)}
|z_i||z_i^2-d_i^2|<0 \;, \ea
\]
for all $v\in {\cal K}(f(z))$. This shows that for all $z\in {\cal
D}\setminus {\cal E}$, either $\max \dot{\overline V}(z)< 0$ or
$\dot{\overline V}(z)=\emptyset$. In summary, for all $z\in
\R^{n-1}$, either $\max \dot{\overline V}(z)\le 0$ or
$\dot{\overline V}(z)=\emptyset$, and $0\in \dot{\overline V}(z)$
if and only if $z\in {\cal E}$.
It is known (Lemma 1 in \cite{BaCe99}) that if $\varphi(t)$ is a
solution of the differential inclusion $\dot z\in {\cal K}(f(z))$,
then $\frac{d}{dt} V(\varphi(t))$ exists almost everywhere and
$\frac{d}{dt} V(\varphi(t))\in \dot{\overline V}(\varphi(t))$. We
conclude that $V(\varphi(t))$ is non-increasing. Let $z_0\in S$,
with $S\subset \R^{n-1}$ a compact and strongly invariant set for
(\ref{quantized.n.agent.system.z}). For any $z_0$, such a set
exists and includes the point $(d_1,d_2, \ldots, d_{n-1})\in {\cal
E}$ (hence $S\cap {\cal E}\ne \emptyset$), by definition of $V(z)$
and because $V(z)$ is non-increasing along the solutions of
(\ref{quantized.n.agent.system.z}). Since $\max\dot{\overline
V}(z)\le 0$ or $\dot{\overline V}(z)=\emptyset$ for all $z\in
\R^{n-1}$, then by the LaSalle invariance principle for
differential inclusions \cite{BaCe99, Co08}, any solution
$\varphi(t)$ to the differential inclusion converges to the
largest weakly invariant set in $S\cap \overline{{\cal E}}=S\cap
{\cal E}$ (${\cal E}$ is closed). Since the choice (\ref{gains})
yields that the gains $k_i$'s satisfy the condition in Lemma
\ref{lemma.equilibria}, ${\cal E}$ is the set of equilibria of
(\ref{quantized.n.agent.system.z}) (and therefore it is weakly
invariant) and since $S\cap {\cal E}\ne \emptyset$, we conclude
that any solution $\varphi(t)$ converges to the set of points
$S\cap {\cal E}$. \hfill $\square$
Since the equilibrium set $\scr{E}$ contains those points for
which two agents coincide with each other, it is of interest to
characterize those initial conditions under which the asymptotic
positions of some of the agents become coincident. In the next
subsection, we use a three-agent formation as an example to show
how such analysis can be carried out.
\subsection{Trajectory based analysis}
\noindent We specialize the rigid formation examined before to the
case $n=3$. Letting $k_1=k_2=1$, the one-dimensional rigid
formation becomes:
\be\label{quantized.3.agent.system} \ba{rcr}
\dot x_1 &=& -\textrm{sgn}(x_1-x_2) \textrm{sgn}(|x_1-x_2|-d_1)\\
\dot x_2 &=& \textrm{sgn}(x_1-x_2) \textrm{sgn}(|x_1-x_2|-d_1)-\\
&& \textrm{sgn}(x_2-x_3) \textrm{sgn}(|x_2-x_3|-d_2)\\
\dot x_3 &=& \textrm{sgn}(x_2-x_3) \textrm{sgn}(|x_2-x_3|-d_2)\;.
\ea \ee
Let us express the system in the coordinates $z_1$, $z_2$, so as
to obtain:
\be\label{quantized.3.agent.system.z} \ba{rcr}
\dot z_1 &=& -2 \textrm{sgn}(z_1) \textrm{sgn}(|z_1|-d_1)+\textrm{sgn}(z_2) \textrm{sgn}(|z_2|-d_2)\\
\dot z_2 &=& \textrm{sgn}(z_1) \textrm{sgn}(|z_1|-d_1)-
2\textrm{sgn}(z_2) \textrm{sgn}(|z_2|-d_2)\;. \ea \ee
We study the solutions of the system above. In what follows, it is
useful to distinguish between two sets of points:
\begin{eqnarray*}
{\cal E}_1&=&\{z\in \R^2: |z_i|=d_i,\;i=1,2\}, \\ {\cal
E}_2&=&\{z\in \R^2: |z_i|=d_i\;\mbox{or}\; |z_i|=0,\;i=1,2\}\;.
\end{eqnarray*}
Clearly, ${\cal E}_1\subset {\cal E}_2$. We now prove that all the
solutions converge to the desired set ${\cal E}_1$ except for
solutions which originates on the $z_1$- or the $z_2$-axis:
\begin{theorem}\label{p1}
All Krasowskii solutions of (\ref{quantized.3.agent.system.z})
converge in finite time to the set ${\cal E}_2$. In particular,
the solutions which converge to the points $\{(d_1, 0), (0, d_2),
(-d_1, 0), (0, -d_2)\}$ must originate from the set of points
$\{z: z_1\cdot z_2=0, z\ne 0\}$. Moreover, the only solution which
converges to $(0,0)$ is the trivial solution which originates from
$(0,0)$.
\end{theorem}
\textit{Proof:} Because of the symmetry of $f(z)$, it suffices to
study the solutions which originate in the first quadrant only. In
the first quadrant we distinguish four regions: (i) ${\cal
R}_1=\{z\in \R^2: z_i\ge d_i, i=1,2\}$, (ii) ${\cal R}_2=\{z\in
\R^2: 0\le z_1<d_1,\; z_2\ge d_2\}$, (iii) ${\cal R}_3=\{z\in
\R^2: 0\le z_i<d_i,\;i=1,2\}$, (iv) ${\cal R}_4=\{z\in \R^2:
z_1\ge d_1,\; 0\le z_2< d_2\}$. Now we examine
the solutions originating in these regions.\\
(i) $z(0)\in {\cal R}_1$. If both $z_1(0)> d_1$ and $z_2(0)> d_2$,
then the system equations become
\[
\dot z_1 = -1\;,\quad \dot z_2 = -1
\]
and the solution satisfies $z_2(t)=z_1(t)+z_2(0)-z_1(0)$. In other
words, the solution evolves along the line of slop $+1$ and
intercept $z_2(0)-z_1(0)$. If $z_2(0)-z_1(0)=d_2-d_1$, then the
solution $z(t)$ converges to the point $z=(d_1,d_2)$ in finite
time. In particular $z(t_1)=(d_1,d_2)$ with
$t_1=z_1(0)-d_1=z_2(0)-d_2$. If $z_2(0)-z_1(0)>d_2-d_1$, then
$z(t)$ converges in finite time to the semi-axis $\{z:z_1=d_1,\;
z_2>d_2\}$. This is a set of points at which $f(z)$ is
discontinuous, since for $z_1\ge d_1$, $f(z)=(-1, -1)$, and for
$z_1< d_1$, $f(z)=(3, -3)$. Since at these points $F(z)={\rm
co}\{(-1, -1), (3, -3)\}$,\footnote{Here ${\rm
co}\{v_1,\ldots,v_m\}$ denotes the smallest closed convex set
which contains $v_1,\ldots,v_m$.} and vectors in $F(z)$ intersect
the tangent space at the semi-axis in those points, a sliding
mode along the semi-axis must occur. Since $\dot z(t)\in F(z(t))$,
we conclude that the sliding mode must satisfy the equations
\[
\dot z_1(t)=0,\; \dot z_2(t)=-\frac{3}{2}\;,
\]
and therefore, after a finite time, the solution converges to the
point $(d_1,d_2)$. On the other hand, if $z_2(0)-z_1(0)<d_2-d_1$,
then the solution reaches the ray $\{z:z_1>d_1,\; z_2=d_2\}$.
Similar considerations as before can show that a sliding mode
occurs along the ray and that it satisfies the equations
\[
\dot z_1(t)=-\frac{3}{2},\; \dot z_2(t)=0\;,
\]
and again convergence in finite time to $(d_1,d_2)$ is inferred.
Finally we examine the case $z(0)=(d_1,d_2)$. At the point
$(d_1,d_2)$,
\[
F(d_1,d_2)={\rm co}\{(-1, -1), (3, -3), (1, -1), (1,1)\}\;,
\]
i.e. $\mathbf{0}\in F(d_1,d_2)$ and $(d_1,d_2)$ is an equilibrium
point. Similarly as before, one shows that the solution which
originates from $(d_1,d_2)$ must stay in $(d_1,d_2)$.\\
(ii) $z(0)\in {\cal R}_2$. If $z_1(0)> 0$ and $z_2(0)>d_2$, then
the map $f(z)$ is equal to the vector $(3,-3)$ and the solution
$z(t)$ satisfies $z_2(t)=-z_1(t)+z_1(0)+z_2(0)$. If
$z_1(0)+z_2(0)=d_1+d_2$, then $z(t)$ converges to $(d_1,d_2)$,
while if $z_1(0)+z_2(0)=d_1+d_2$, it first converges to the
semi-axis $\{z:z_1=d_1,\; z_2>d_2\}$, and then it slides towards
$(d_1,d_2)$. When $z_1(0)+z_2(0)=d_1+d_2$, the solution reaches
the segment $\{z:0<z_1<d_1,\; z_2=d_2\}$. On this segment,
$F(z)={\rm co}\{(3, -3), (1, 1)\}$, and since this intersects the
tangent space at the segment, a sliding mode occurs. The sliding
mode obeys the equations
\[
\dot z_1(t)=\frac{3}{2},\; \dot z_2(t)=0\;,
\]
which show that the state reaches $(d_1,d_2)$.\\
If $z_1(0)=0$ and $z_2(0)>d_2$, then the initial condition lies on
another discontinuity surface of $f(z)$. Observe that, for those
points such that $-d_1<z_1\le 0 $ and $z_2> d_2$, $f(z)=(-1, -1)$.
Hence, $F(z)={\rm co}\{(-1, -1), (3, -3)\}$ intersects the tangent
space at the semi-axis in those points, and the solutions can
slide along the semi-axis until they reach the point $(0,d_2)$ and
stop, or can enter the region ${\cal R}_2\setminus
\{z:z_1=0,z_2>d_2\}$, and then converge to $(d_1,d_2)$, or they
can enter the region $\{z:
-d_1<z_1<0, z_2>d_2\}$ and converge to the point $(-d_1,d_2)$.\\
The point $(0,d_2)$ is an equilibrium, and if $z(0)=(0,d_2)$,
solutions stay at the equilibrium.\\
We review the remaining cases succinctly, as they are
qualitatively similar to the cases
examined above. \\
(iii) $z(0)\in {\cal R}_3$. If $z_i(0)> 0$ for $i=1,2$, then the
solutions converge to $(d_1,d_2)$ possibly sliding along the
segments $\{z:0<z_1\le d_1, z_2=d_2\}$ or $\{z:z_1=d_1, 0<z_2\le
d_2\}$. If $z_1(0)=0$ and $z_2(0)>0$, then the solution can
converge to the points $(-d_1, d_2)$, $(0, d_2)$ or $(d_1,d_2)$.
If $z_1(0)>0$ and $z_2(0)=0$, then the solutions can converge to
$(d_1,d_2)$, $(d_1,0)$ or $(d_1,-d_2)$. Finally, if $z_i(0)=0$ for
$i=1,2$, the solutions can converge to any of the points in ${\cal
E}_2$. In particular,
a possible solution is the one which remains in $(0,0)$.\\
(iv) $z(0)\in {\cal R}_4$. Solutions which start from initial
conditions such that $z_1(0)>d_1$ and $z_2(0)>0$ converge to
$(d_1, d_2)$. If $z_1(0)=d_1$ and $z_2(0)>0$, then the solution
converge to $(d_1, d_2)$ possibly sliding on the segment
$\{z:z_1=d_1, 0<z_2<d_2\}$. If $z_1(0)>d_1$ and $z_2(0)=0$, the
solutions can converge to one of the three possible points: $(d_1,
-d_2)$, $(d_1, 0)$, $(d_1, d_2)$. \hfill $\square$
A few comments are in order:
\begin{itemize}
\item Sliding modes arise naturally for those situations in which, for
instance, the state reaches the semi-axis $\{z:z_1>d_1,
z_2=d_2\}$. This forces us to consider Krasowskii solutions rather
than Carath\'eodory solutions. On the other hand, the set of
Krasowskii solutions may be too large in some cases, as it is
evident for instance for those solutions which start on the $z_1$-
or $z_2$-axis.
\item The occurrence of sliding modes are not acceptable in practice as
they would require fast information transmission. A mechanism to
prevent sliding modes in the system
(\ref{quantized.3.agent.system}) can be introduced following
\cite{CePeFr10}.
\end{itemize}
\section{Simulation results} \label{se:simulation}
In this section, we present simulation results for the guided
formation control with coarsely quantized information. We consider
a formation consisting of 6 agents, labeled by $1, \ldots, 6$. The
distance constraints are $|x_i-x_{i+1} |=1$, $i= 1, \ldots, 5$.
The initial positions of agents 1 to 6 are 0, 0.5, 1, 2, 4 and 5
respectively. Then the shape of the initial formation is shown in
figure \ref{fig1}.
\begin{figure}[ht]
\centerline{\scalebox{.5}{\includegraphics{initial.eps}}}
\caption{The initial shape of the 6-agent formation.}
\label{fig1}
\end{figure}
We choose $k_1 =6$, $k_2 = 5$, $k_3=4$, $k_4=3$ and $k_5=2$ and
simulate the agents' motion under the control laws
(\ref{quantized.n.agent.system}). In figure \ref{fig2}, we show
the shape of the final formation.
\begin{figure}[ht]
\centerline{\scalebox{.5}{\includegraphics{final.eps}}}
\caption{The final shape of the 6-agent formation.}
\label{fig2}
\end{figure}
To see how the shape evolves with time, we present the curve of
the Lyapunov function $V(z) = \frac{1}{4}\sum_{i=1}^5
(z_i^2-d_i^2)^2$ in figure \ref{fig3}.
\begin{figure}[ht]
\centerline{\scalebox{.5}{\includegraphics{Lyapunov.eps}}}
\caption{The curve of the Lyapunov function $V$.}
\label{fig3}
\end{figure}
Since our analysis has been carried out using Krasowskii
solutions, when we further look into the dynamics of $z$, it is
clear that the sliding mode may still happen when the Krasowskii
solution converges. But this effect due to the system's
non-smoothness is within an acceptable level as shown in figure
\ref{fig4} which presents the curve of $z_1$.
\begin{figure}[ht]
\centerline{\scalebox{.5}{\includegraphics{z1.eps}}}
\caption{The dynamics of $z_1$.}
\label{fig4}
\end{figure}
\section{Concluding remarks} \label{se:conclusion}
In this paper, we have studied the problem of controlling a
one-dimensional guided formation using coarsely quantized
information. It has been shown that even when the guidance system
adopts quantizers that return only the one-bit sign information
about the quantized signal, the formation can still converge to
the desired equilibrium under the proposed control law.
The point model we have used throughout the analysis is a
simplified description of vehicle dynamics. When more detailed
models are taken into consideration, we need to deal with
collision avoidance and other practical issues as well. So it is
of great interest to continue to study the same problem with more
sophisticated vehicle models and more physical constraints from
the applications.
| 201,608
|
\begin{document}
\begin{abstract}
We study groups of germs of complex diffeomorphisms having a
property called {\it irreducibility}. The notion is motivated by the
similar property of the fundamental group of the complement of an
irreducible hypersurface in the complex projective space. Natural
examples of such groups of germ maps are given by holonomy groups
and monodromy groups of integrable systems (foliations) under
certain conditions. We prove some finiteness results for these
groups extending previous results in \cite{CL}. Applications are
given to the framework of germs of holomorphic foliations. We prove
the existence of first integrals under certain irreducibility or
more general conditions on the tangent cone of the foliation after
a punctual blow-up.
\end{abstract}
\maketitle
\newtheorem{theorem}{Theorem}
\renewcommand*{\thetheorem}{\Alph{theorem}}
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Corollary}{Corollary}[section]
\newtheorem{Proposition}{Proposition}[section]
\newtheorem{Lemma}{Lemma}[section]
\newtheorem{Claim}{Claim}[section]
\newtheorem{Definition}{Definition}[section]
\newtheorem{Example}{Example}[section]
\newtheorem{Remark}{Remark}[section]
\newtheorem*{cltheorem}{Theorem}
\newtheorem{Question}{Question}[section]
\newcommand\virt{\rm{virt}}
\newcommand\SO{\rm{SO}}
\newcommand\G{\varGamma}
\newcommand\Om{\Omega}
\newcommand\Kbar{{K\kern-1.7ex\raise1.15ex\hbox to 1.4ex{\hrulefill}}}
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\def\be{{\beta}}
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\def\rv{\right\vert}
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\def\vol{\rm{{Vol}}}
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\def\Dom{\rm{{Dom}}}
\def\supp{\rm{{supp}}}
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\def\Exp{\rm{{Exp}}}
\def\Hom{\rm{{Hom}}}
\def\codim{\rm{{codim}}}
\def\cotg{\rm{{cotg}}}
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\def\VIP{\rm{{VIP}}}
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\def\Ker{\rm{{Ker}}}
\def\Sat{\rm{{Sat}}}
\def\diag{\rm{{diag}}}
\def\rank{\rm{{rank}}}
\def\Sing{\rm{{Sing}}}
\def\sing{\rm{{sing}}}
\def\hot{\rm{{h.o.t.}}}
\def\Fol{\rm{{Fol}}}
\def\grad{\rm{{grad}}}
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\def\Aut{\rm{{Aut}}}
\def\Sep{\rm{{Sep}}}
\def\Res{\rm{{Res}}}
\def\ord{\rm{{ord}}}
\def\h.o.t.{\rm{{h.o.t.}}}
\def\Hol{\mbox{Hol}}
\def\Diff{\mbox{Diff}}
\def\SL{\rm{{SL}}}
\tableofcontents
\section{Introduction}
Let $C\subset \mathbb P^2$ be an algebraic curve of degree $\nu$ in
the complex projective plane. By a classical theorem of Zariski and
Fulton-Deligne (\cite{Deligne,Fulton}), if $C\subset \mathbb P^2$ is
irreducible and smooth or with only normal double crossings
singularities, then the fundamental group of the complement $\mathbb
P^2 \setminus C$ is finite abelian, isomorphic to $\mathbb Z / \nu
\mathbb Z$. Indeed, in Deligne's paper it is observed that, in the
irreducible case, the fundamental group of the complement $\mathbb
P^2\setminus C$ is isomorphic to the local fundamental group
$\pi_1(V\setminus C)$ where $V$ is a suitable arbitrarily small
neighborhood of $C$ in $\mathbb P^2$. As a consequence, the group
$\pi_1(\mathbb P^2 \setminus C)$ has the following irreducibility
property: given two simple loops $\alpha, \beta $ in $\mathbb
P^2\setminus C$, there is a loop $\delta \subset \mathbb P^2
\setminus C$ that conjugates the classes $[\alpha], [\beta] \in
\pi_1(\mathbb P^2\setminus C)$. For this it is not necessary to
assume that the curve is smooth or has only double points as
singularities, just the irreducibility of the curve is required. We
shall use this as a motivation for the definition of an {\it
irreducibility} property for a group of germs of analytic
diffeomorphisms.
Expand a germ of a complex diffeomorphism $f$ at the origin $0 \in
\mathbb C$ as $f(z)= e^{2 \pi i \lambda} z + a_{k+1} z^{k+1} + ...$.
The {\it multiplier} $f^\prime(0)=e^{2 \pi i \lambda}$ does not
depend on the coordinate system. We shall say that the germ $f \in
\Diff(\mathbb C,0)$ is {\it flat} or {\it tangent to the identity}
if $f^\prime(0) =1$.
Using these ideas and some features from arithmetics, in \cite{CL}
the authors prove a finiteness theorem for groups of germs of
complex diffeomorphisms in one variable. This reads as follows:
\begin{Theorem}[\cite{CL}, Theorem~1
page 221] \label{Theorem:CL} Let $G$ be a subgroup of $\Diff(\mathbb
C,0)$ generated by elements $f_1,...,f_{\nu+1}$. Assume that:
\begin{enumerate}
\item $f_k(z)= \mu z + \cdots$ with $\mu=e^{2 i\pi/ \nu+1}$ and
$f_{\nu+1} \circ \cdots \circ f_1=\id$.
\item the $f_k$ are pairwise conjugate in $G$.
\item $\nu+1 = p^s$ with $p $ prime and $s\in \mathbb N^*$.
\end{enumerate}
Then $G$ is a finite group; in particular $G$ is conjugate to a
finite group of rotations fixing $0$.
\end{Theorem}
Here we have $\mathbb N=\{0,1,2,...\}$ and $\mathbb N ^*= \mathbb N
\setminus \{0\}=\{1,2,...\}$. Notice that if $\nu+1=2$ so
that $G=<f_1,f_2>$ where $f_1\circ f_2=\id$. This implies that
$G=<f_1>$ is cyclic. Since $f_1$ and $(f_1)^{-1}$ are conjugate in
$G$ we have $f_1=(f_1)^{-1}$ and therefore $(f_1)^2=\id$. Thus $G$
is finite of order $\leq 2$.
As a consequence of Theorem~\ref{Theorem:CL} the same authors obtain a theorem
(cf. \cite{CL} Theorem~0 page 218) about the existence of a
holomorphic first integral for a germ of holomorphic foliation of
codimension one singular at the origin $0 \in\mathbb C^n, n \geq 2$
provided that: (i) the foliation is not dicritic; (ii) the tangent
cone of the foliation is irreducible of degree $\nu +1=p^s$ where
$p$ is a prime number and $s\in \mathbb N^*$.
It is our aim in this paper to study further the classification of
groups of germs of complex diffeomorphisms exhibiting this
irreducibility property and possible applications of this to the
study of holomorphic foliations singularities. We shall then study
groups for which the irreducibility holds but with a number of
generators not necessarily of the form $p^s$ for $p$ prime.
For simplicity we consider the group $\Diff(\mathbb C,0)$ of germs of
complex diffeomorphisms fixing the origin $0 \in \co$. We shall
start with a definition:
\begin{Definition}[irreducible group]
\label{Definition:irreduciblegroup}
{\rm A subgroup $G \subset \Diff(\mathbb C,0)$ is {\it irreducible} if it admits a
finite set of generators
$f_1,f_2,\ldots,f_{\nu +1}\in G$ such that:
\begin{enumerate}
\item[(a)] $f_1\circ f_2\circ\ldots\circ f_{\nu+1}=\id$.
\item[(b)] $f_i$ and $f_j$ are conjugate in $G$ for all $i,j$.
\end{enumerate}}
\end{Definition}
The above maps $f_j$ will be called {\it basic generators} of the
group $G$. By definition an irreducible group is finitely generated.
The above definition does not exclude the possibility that we have a
repetition of basic generators, ie., $f_i=f_j$. Note also that an
irreducible abelian group is finite cyclic: indeed, since the group
is abelian we have $f_i=f_j$, for all $i,j$. Therefore the group is
generated by an element $f_1$ of finite order because $f_1^{\nu +1}
=\id$.
Let us give some examples illustrating our above definition:
\begin{Example}
{\rm Every cyclic subgroup {\it of finite order} $G\subset
\Diff(\mathbb C,0)$ is irreducible. In particular every finite
subgroup $G \subset \Diff(\mathbb C,0)$ is irreducible. Indeed, such
a group is analytically conjugate to a cyclic group of rational
rotations say, $G=<z\mapsto e^{2 \pi i /\nu} z>$ for some $\nu
\in\mathbb N^*$. On the other hand a cyclic group is irreducible
only if it has finite order. If $G\subset \Diff(\mathbb C,0)$ is
abelian and irreducible then $G$ is also finite. Indeed, all
generators $f_1,...,f_{\nu+1}$ in the definition of irreducible
group are conjugate so $f_1=\ldots=f_{\nu+1}$ and then
$f_1^{\nu+1}=\id$. This shows that $G$ is finite. Later on we shall
give examples of (nonabelian) irreducible groups (see
\S~\ref{section:examples}).}
\end{Example}
A (more geometrical) example is given below:
\begin{Example}
{\rm Let $C\subset \mathbb P^2$ be an algebraic curve of degree
$\nu$. By a classical theorem of Zariski and Fulton-Deligne, if
$C\subset \mathbb P^2$ is irreducible and smooth or with only
normal double crossings singularities, then the fundamental group of
the complement $\mathbb P^2 \setminus C$ is finite abelian,
isomorphic to $\mathbb Z / \nu \mathbb Z$. If this the case, then
any representation $\vr\colon \pi_1(\mathbb P^2\setminus C) \to
\mathbb \Diff(\mathbb C,0)$ has as image an irreducible group. For
instance, assume that we have a closed rational one-form $\Omega$ on
$\mathbb P^2$ with polar divisor $(\Omega)_\infty =C$ irreducible.
Then the monodromy of $\Omega$ gives an irreducible subgroup of
$\Diff(\mathbb C,0)$. }
\end{Example}
Now we extend Theorem~\ref{Theorem:CL} for the case some of
generators has a multiplier which is a root of the unity of order
$p^s$ for a prime $p$ and natural number $s \geq 0$, {\sl but
without further restrictions on the number of generators}. This
includes the case where some of the basic generators is tangent to
the identity.
\begin{theorem}
\label{Theorem:2irreducibledim1} Let $G\subset \Diff(\mathbb C,0)$
be an irreducible group. If the multiplier of some basic generator
is a root of the unit of order $p^s$, where $p$ is prime and
$s\in\mathbb{N}$, then $G$ is a finite group.
\end{theorem}
A straightforward consequence of
Theorem~\ref{Theorem:2irreducibledim1} is that if some basic
generator is tangent to the identity ( ie., $s=0$) then $G$ is
trivial. Indeed, a finite subgroup $G\subset \Diff(\mathbb C,0)$ is
analytically linearizable.
The fact that the basic generators are conjugate in the group $G$, not
only in $\Diff(\mathbb C,0)$ (condition (b) in
Definition~\ref{Definition:irreduciblegroup} above), is essential
(cf. Example~\ref{Example:essential}).
Let us consider a germ of codimension one holomorphic foliation with
a singularity at the origin $0 \in \mathbb C^n, n \geq 2$. This is
given by an integrable germ of a holomorphic one-form $\omega$ at
$0$. The integrability condition writes $\omega \wedge d \omega=0$
and we may assume that $codim \sing(\omega)\geq 2$. In the case
where $codim \sing(\omega) \geq 3$ a theorem of Malgrange
(\cite{malgrange}) assures the existence of a holomorphic first
integral, ie., a germ of a holomorphic function $f \in \mathcal O_n$
such that $ df \wedge \omega=0$. Denote by $\pi \colon \tilde
{\mathbb C^n} \to \mathbb C^n$ the blow-up at the origin of $\mathbb
C^n$. The {\it tangent cone} of $\fa$ is defined as the intersection
$C(\fa)$ of the singular set of the lifted foliation $\tilde
\fa=\pi^*(\fa)$ with the exceptional divisor $E=\pi^{-1}(0)\cong
\mathbb P^{n-1}$, ie., $C(\fa) = \sing(\tilde \fa) \cap E$
(\cite{Cerveau-Mattei}). This is a (not necessarily irreducible)
algebraic subset of $\mathbb P^{-1}$ of codimension $\geq 1$. Assume
now that $\fa$ is non dicritical, ie., that $E$ is invariant by
$\tilde \fa$. In this case we have a leaf $\tilde L= E\setminus
C(\fa)$ of $\tilde \fa$. This leaf has a holonomy group called {\it
projective holonomy} of the foliation $\fa$. It can be viewed as
follows: take a point $\tilde q \in \tilde L$ and a transverse disc
$\Sigma$ to $\tilde L$ with center at $\tilde q$. Then consider the
representation of the fundamental group $\pi(\tilde L,\tilde q)$ in
the group $\Diff(\Sigma, \tilde q)$ of germs of complex
diffeomorphisms of $\Sigma$ fixing $\tilde q$, obtained as the image
of the holonomy maps of the loops $\gamma$ in $\tilde L$ based at
$\tilde q$. The projective holonomy group is the image $H(\fa)$ of
this representation, that can may be also viewed as a subgroup of
$\Diff(\mathbb C,0)$ under an identification $(\Sigma, \tilde q)=
(\mathbb C,0)$. Given a codimension $\geq 2$ component
$\Lambda_2\subset C(\fa)$ there is an isomorphism between
fundamental groups $\pi_1(E) \cong \pi_1(E \setminus \Lambda_2)$
because the real codimension of $\Lambda_2$ in $E$ is $\geq 4$. Thus
if $C_1(\fa)\subset C(\fa)$ denotes the union of all codimension
one irreducible components of $C(\fa)$ we have $\pi_1(E\setminus
C_1(\fa))\cong \pi_1(E\setminus C(\fa))=\pi_1(\tilde L)$. Given a
small simple loop $\gamma \subset E \setminus C(\fa)$ around an
irreducible component of codimension one of $C(\fa)$ (such loops
generate the fundamental group $\pi_1(\mathbb P^2 \setminus C(\fa))$
\cite{Deligne}) we shall call the corresponding holonomy map a {\it
basic holonomy generator} of the projective holonomy of $\fa$.
Assume now that the codimension one component of $C(\fa)$ is
irreducible. The projective holonomy group is the representation of
an group which by Deligne's theorem mentioned in the beginning, has
the irreducibility property. Thus $H(\fa)\subset \Diff(\mathbb C,0)$
is an irreducible group, having as basic generators in
Definition~\ref{Definition:irreduciblegroup}, the above introduced
basic holonomy generators of the projective holonomy. Notice that we
do know whether $ C(\fa)$ is smooth or has only ordinary points, so
we cannot apply Deligne's results and assure that $H(\fa)$ is
finite. The finiteness of $H(\fa)$ is assured under some conditions
as in the case studied in \cite{CL}. As an application of
Theorem~\ref{Theorem:2irreducibledim1} we may extend Theorem~0 in
\cite{CL} as follows:
\begin{theorem}
\label{Theorem:firstintegral1} Let $\fa$ be a non-dicritic germ of
codimension one holomorphic foliation with a singularity at the
origin $ 0 \in \mathbb C^n, n \geq 3$. Suppose that:
\begin{enumerate}
\item The codimension one component of the
tangent cone $ C(\fa)$ of $\fa$ is irreducible.
\item There is a basic generator of the projective holonomy with
multiplier $1$ or a root of the unity of order $p^s$ for some prime
$p$ and $s\in\mathbb{N}^*$.
\end{enumerate}
Then $\fa$ admits a holomorphic first integral.
\end{theorem}
We observe that if $ C(\fa)$ is irreducible of codimension one and
degree $p^s$ for some prime $p$ and $s \in \mathbb N^*$ then
condition (2) above is automatically satisfied (\cite{CL} pages
219-220).
Given a codimension one foliation germ $\fa$ at $0 \in \mathbb C^n,
n \geq 3$ we consider maps $\tau \colon (\mathbb C^2,0) \to (\mathbb
C^n,0)$ {\it in general position} with respect to $\fa$ in the sense
of \cite{Ma-Mo} (see also \cite{C-LN-S} and \cite{Cerveau-Mattei}).
We shall call the inverse image $\tau^*(\fa)$ of $\fa$ by such a map
$\tau$, a {\it generic plane section} of $\fa$. In \cite{CL} it is
observed that for a nondicritical foliation germ, the irreducibility
of the tangent cone implies that the generic plane sections admit a
reduction by one single blow-up. In this sense we can extend
Theorem~\ref{Theorem:firstintegral1} above as follows:
\begin{theorem}
\label{Theorem:firstintegralreducible} Let $\fa$ be a non-dicritic
germ of codimension one holomorphic foliation with a singularity at
the origin $ 0 \in \mathbb C^n, n \geq 3$. Suppose that:
\begin{enumerate}
\item The codimension one component of the
tangent cone $ C(\fa)$ of $\fa$ is a normal crossings divisor.
\item Each codimension one irreducible component $\Lambda_j$ of $ C(\fa)$
has degree $p_j^{s_j}$ for a prime $p_j$ and $s_j \in \mathbb N\cup
\{0\}$.
\item A generic plane section $\tau^*(\fa)$ of $\fa$, can be reduced
with a single blow-up $\pi\colon \widetilde {\mathbb C^2} \to
\mathbb C^2$.
\end{enumerate}
Then $\fa$ admits a holomorphic first integral.
\end{theorem}
\section{Irreducible groups with an element tangent to the identity}
In this section we consider irreducible groups having a generator
with multiplier $1$, ie., of the form $f_k(z) = z + a_{k+1} z^{k+1}
+ \cdots$. Since any two generators are conjugate by hypothesis,
this implies that all elements in $G$ are tangent to the identity,
ie., $G$ is {\it flat}. We shall then start to prove
Theorem~\ref{Theorem:2irreducibledim1}.
\begin{proof}[Proof of Theorem~\ref{Theorem:2irreducibledim1}: flat case]
There exists a finite set of basic generators
$f_1,f_2,\ldots,f_{\nu +1}\in G$ such that:
\begin{enumerate}
\item[(a)] $f_1\circ f_2\circ\ldots\circ f_{\nu+1}=\id$.
\item[(b)] $f_i$ and $f_j$ are conjugate in $G$ for all $i,j$.
\end{enumerate}
From (b) we have that all generators have the same multiplier.
Indeed, for $i\neq j$ there exists $g\in G $ of the form
$$g(z)=\alpha z+h.o.t.,\;\;\alpha\neq0$$
such that $$f_i\circ g=g\circ f_j$$ hence we obtain
$$f_i^\prime(0)\cdot\alpha=f_j^\prime(0)\cdot\alpha$$ then
$$f_i^\prime(0)=f_j^\prime(0),\;\mbox{for all }i,j.$$
Since some generator is tangent to identity we have
$f^\prime_1(0)=\ldots=f^\prime_{\nu+1}(0)=1$. Then we have
$$f_i(z)=z+h.o.t.,\mbox{ for all }1\leq i\leq {\nu+1}.$$
Consequently, every element $g\in G$ is of the form
$$g(z)=z+h.o.t.$$
In this case, we will show that
\begin{equation}\label{identity1}
f_1=f_2=\ldots=f_{\nu+1}=\id.
\end{equation}
Consequently $G$ is trivial.
The idea is to use Taylor series expansion and create an algorithm using derivation.
\vglue.2in
Indeed, for $i\neq j$ by (b) there is $g\in G$ such that
\begin{equation}\label{dercomp01}
f_i\circ g=g\circ f_j.
\end{equation}
Derivating (\ref{dercomp01}) we have
\begin{equation}\label{dercomp11}
f_i^\prime(g)\cdot g^\prime=g^\prime(f_j)\cdot f_j^\prime.
\end{equation}
Also, derivating (\ref{dercomp11}) have
\begin{equation}\label{dercomp21}
f_i^{\prime\prime}(g)\cdot (g^\prime)^2+f_i^\prime(g)\cdot
g^{\prime\prime}=g^{\prime\prime}(f_j)\cdot
(f_j^\prime)^2+g^\prime(f_j)\cdot f_j^{\prime\prime}
\end{equation}
as $g(0)=f_i(0)=f_j(0)=0$ and
$g^\prime(0)=f^\prime_i(0)=f_j^\prime(0)=1$ we have in
(\ref{dercomp21})
$$f_i^{\prime\prime}(0)+g^{\prime\prime}(0)=g^{\prime\prime}(0)+f_j^{\prime\prime}(0)$$
therefore $$f^{\prime\prime}_i(0)=f_j^{\prime\prime}(0).$$
So there is $a_2=\dfrac{f^{\prime\prime}_i(0)}{2!}\in\co$ for all $i$ such that
$$f_i(z)=z+a_2z^2+h.o.t.,\mbox{ for all }1\leq i\leq \nu+1.$$
Then by (a) we have that $a_2=0$ therefore
$$f^{\prime\prime}_i(0)=f_j^{\prime\prime}(0)=0.$$
With the objective of obtaining a similar result for higher order
derivatives we have to verify the following statement, variant of a
well-known result of Leibniz:
\begin{Claim}\label{Leibnitzgeral1}
For any $\varphi$, $\psi$ diffeomorphisms and $n\in\na$, $n\geq3$ we have that
\begin{equation}\label{hipind1}
(\varphi\circ\psi)^{(n)}=\varphi^{(n)}(\psi)\cdot (\psi^\prime)^n+\displaystyle\sum^{n-1}_{k=2}\varphi^{(k)}(\psi)\cdot R_k(\psi^\prime,\psi^{\prime\prime},\ldots,\psi^{(n+1-k)})+\varphi^\prime(\psi)\cdot \psi^{(n)}
\end{equation}
where $R_k\in \mathbb C[x_1,...,x_{n+1-k}]$ are homogeneous polynomials of degree $k$ in $\co^{n+1-k}$ which has null coefficient in the monomial $x_1^k$.
\end{Claim}
\begin{proof}[Proof of Claim~\ref{Leibnitzgeral1}]
In fact, let us show this by induction on $n$, for $n=3$ we have
$$(\varphi\circ\psi)^{(3)}=\varphi^{\prime\prime\prime}(\psi)\cdot (\psi^\prime)^3+3\varphi^{\prime\prime}(\psi)\cdot \psi^\prime \cdot \psi^{\prime\prime}+\varphi^\prime(\psi)\cdot \psi^{\prime\prime\prime}$$
therefore equality is valid. Let us assume that equality is
satisfied for $n$ by showing that it is valid for $n+1$. By the
hypothesis of induction (\ref{hipind1}) is valid. Derivating
(\ref{hipind1}) we have
\begin{equation}\label{derhipind1}
\begin{array}{c}
(\varphi\circ\psi)^{(n+1)}=\varphi^{(n+1)}(\psi)\cdot
(\psi^\prime)^{n+1}+n\varphi^{(n)}(\psi)\cdot (\psi^\prime)^{n-1}
\cdot \psi^{\prime\prime}+\\
\\\displaystyle\sum^{n-1}_{k=2}\left[\varphi^{(k+1)}(\psi)\cdot\psi^\prime\cdot
R_k(\psi^\prime,\psi^{\prime\prime},\ldots,\psi^{(n+1-k)})+\varphi^{(k)}(\psi)\cdot
\left(R_k(\psi^\prime,\psi^{\prime\prime},\ldots,\psi^{(n+1-k)})\right)^\prime
\right]\\ \\+\varphi^{\prime\prime}(\psi)\cdot \psi^\prime\cdot
\psi^{(n)}+\varphi^\prime(\psi)\cdot \psi^{(n+1)}.
\end{array}
\end{equation}
It is not difficult to see that
$\left(R_k(\psi^\prime,\psi^{\prime\prime},
\ldots,\psi^{(n+1-k)})\right)^\prime$ is a homogeneous polynomial
degree $k$ in $\co^{n+2-k}$ which has null coefficient in the
monomial $x_1^k$. Denoting
$$\tilde{R}_n(\psi^\prime,\psi^{\prime\prime})=n(\psi^\prime)^{n-1}\cdot
\psi^{\prime\prime}+\psi^\prime\cdot R_{n-1}(\psi^\prime,\psi^{\prime\prime})$$
$$\tilde{R}_l(\psi^\prime,\psi^{\prime\prime},\ldots,\psi^{(n+2-l)})=\psi^\prime\cdot R_{l-1}(\psi^\prime,\psi^{\prime\prime},\ldots,\psi^{(n+2-l)})+\left(R_{l}(\psi^\prime,\psi^{\prime\prime},\ldots,\psi^{(n+1-l)})\right)^\prime,\;\mbox{ for }2<l<n.$$
and
$$ \tilde{R}_2(\psi^\prime,\psi^{\prime\prime},\ldots,\psi^{(n)})=\psi^\prime\cdot\psi^{(n)}+\left(R_{2}(\psi^\prime,\psi^{\prime\prime},\ldots,\psi^{(n-1)})\right)^\prime$$
we can re-write (\ref{derhipind1}) as
$$ (\varphi\circ\psi)^{(n+1)}=\varphi^{(n+1)}(\psi)\cdot
(\psi^\prime)^{n+1}+\displaystyle\sum^{n}_{l=2}\varphi^{(l)}(\psi)
\cdot
\tilde{R}_l(\psi^\prime,\psi^{\prime\prime},\ldots,\psi^{(n+2-l)})+
\varphi^\prime(\psi)\cdot \psi^{(n+1)}
$$
where $\tilde{R}_l$ are homogeneous polynomials of degree $l$ in
$\co^{n + 2-l}$ which has null coefficient in the monomial $x_1^l$.
This proves Claim~\ref{Leibnitzgeral1}.
\end{proof}
Our next step is:
\begin{Claim}\label{null1}
\begin{equation}\label{iterednull1}
f_i^{(n)}(0)=f_j^{(n)}(0)=0
\end{equation}
for all $n\in\mathbb{N}$, $n\geq3$.
\end{Claim}
\begin{proof}[Proof of Claim~\ref{null1}]
As before, we shall use induction on $n$ to prove the claim.
For $n=3$, derivating (\ref{dercomp21}) we have to
\begin{equation}\label{dercomp3p1}
f_i^{\prime\prime\prime}(g)\cdot (g^\prime)^3+3f_i^{\prime\prime}(g)
\cdot g^\prime\cdot g^{\prime\prime}+f_i^\prime(g)\cdot
g^{\prime\prime\prime}=g^{\prime\prime\prime}(f_j)\cdot
(f_j^\prime)^3+ 3g^{\prime\prime}(f_j)\cdot f^\prime_j\cdot
f^{\prime\prime}_j+g^\prime(f_j)\cdot f^{\prime\prime\prime}_j
\end{equation}
as $g(0)=f_i(0)=f_j(0)=0$,
$g^\prime(0)=f^\prime_i(0)=f_j^\prime(0)=1$ and
$f^{\prime\prime}_i(0)=f_j^{\prime\prime}(0)=0$ we have in
(\ref{dercomp3p1})
$$f_i^{\prime\prime\prime}(0)+g^{\prime\prime\prime}(0)=g^{\prime\prime\prime}(0)+f_j^{\prime\prime\prime}(0)$$therefore $$f^{\prime\prime\prime}_i(0)=f_j^{\prime\prime\prime}(0)$$
hence there is $a_3=\dfrac{f^{\prime\prime\prime}_i(0)}{3!}\in\co$
for all $i$ such that
$$f_i(z)=z+a_3z^3+h.o.t.,\mbox{ for all }1\leq i\leq \nu+1.$$
Then by (a) we have that $a_3=0$ therefore
$$f^{\prime\prime\prime}_i(0)=f_j^{\prime\prime\prime}(0)=0.$$
Suppose the statement is satisfied for $ 3\leq l <n$ by showing that
it is valid for $n$. Now derivating $n$-times (\ref{dercomp01}) and
using Claim~\ref{Leibnitzgeral1} we have to
\begin{equation}\label{dercompnp1}
\begin{array}{c}
f_i^{(n)}(g)\cdot (g^\prime)^n+\displaystyle\sum^{n-1}_{l=2}f_i^{(l)}(g)\cdot R_l(g^\prime,g^{\prime\prime},\ldots,g^{(n+1-l)})+f_i^\prime(g)\cdot g^{(n)}\\
\\
=g^{(n)}(f_j)\cdot (f_j^\prime)^n+\displaystyle\sum^{n-1}_{l=2}g^{(l)}(f_j)\cdot \tilde{R}_l(f_j^\prime,f_j^{\prime\prime},\ldots,f_j^{(n+1-l)})+g^\prime(f_j)\cdot f_j^{(n)}
\end{array}
\end{equation}
where $R_l$ and $\tilde{R}_l$ are homogeneous polynomials of degree
$l$ in $\co^{n+1-l}$ which has null coefficient in the monomial
$x_1^l$. By the induction hypothesis (\ref{iterednull1}) it is valid
for all $3\leq l <n$ so
$$f^{(l)}_i(0)=f_j^{(l)}(0)=0,\;\mbox{for all }3\leq l<n,$$
$g(0)=f_i(0)=f_j(0)=0$, $g^\prime(0)=f^\prime_i(0)=f_j^\prime(0)=1$
and $f^{\prime\prime}_i(0)=f_j^{\prime\prime}(0)=0$ we have in (\ref{dercompnp1})
$$f_i^{(n)}(0)+g^{(n)}(0)=g^{(n)}(0)+f_j^{(n)}(0)$$
therefore $$f^{(n)}_i(0)=f_j^{(n)}(0)$$hence there is
$a_n=\dfrac{f^{(n)}_i(0)}{n!}\in\co$ for all $i$ such that
$$f_i(z)=z+a_nz^n+\ldots,\mbox{ for all }1\leq i\leq \nu+1.$$
So by (a) we have that $a_n=0$ therefore
$$f^{(n)}_i(0)=f_j^{(n)}(0)=0.$$ This ends the proof of Claim~\ref{null1}.
\end{proof}
By Claim~\ref{null1} we have equality (\ref{identity1}) and
therefore $G=\{id\}$. This proves
Theorem~\ref{Theorem:2irreducibledim1} in case some generator is
tangent to the identity.
\end{proof}
\section{Irreducible groups with a primitive root of the unity}
Recall that if $f_1,...,f_{\nu+1}$ is a set of generators of an
irreducible group $G$ as in
definition~\ref{Definition:irreduciblegroup} then we have
$f_k^\prime(0)=\mu$ for all $k$, with $\mu ^{\nu+1}=1$. Thus the
multiplier of any generator is a root of the unity. Its order is a
divisor of $\nu+1$ but it is not necessarily of the form $p^s$ for a
prime $p$ (see examples~\ref{Contraexample1} and
\ref{Example:more}). In Theorem~\ref{Theorem:2irreducibledim1} we
assume that $\mu$ has order $p^s$. The proof of this theorem goes
partially as the one given in \cite{CL} for
Theorem~\ref{Theorem:CL}. Nevertheless, we shall make use of our
preceding case in Theorem~\ref{Theorem:2irreducibledim1} and of some
techniques borrowed from \cite{CL}.
\begin{proof}[Proof of Theorem~\ref{Theorem:2irreducibledim1}: remaining case]
Suppose that the multiplier $\mu\neq 1$ of some (and therefore of
each) basic generator is a root of the unit of order $p^s$ for
some prime $p$ and some $s \in \mathbb N^*$. By the above remark
$p^s$ divides $\nu+1$. We will perform an adaptation of the
arguments in the proof of \cite[Theorem 1]{CL} in order to conclude
that the group is finite. The idea is to show by a formal algorithm
that $G$ is formally linearizable. More precisely, let us prove
that:
\begin{Claim} If the group $G$ is linear (by a conjugation) to order $k$, that is,
$$f_i(z)=\mu z+t_iz^{k+1}+h.o.t,\;\;\mbox{where }t_i\in\co$$ for all $i$
then either the $t_i=0$ for all $i$ (and the group is linearized to
order $k+1$) or $\nu+1$ does not divide $k$ and
$t_1=t_2=\ldots=t_{\nu+1}$.
\end{Claim}
In the latter case, we can linearize $G$ to order $k+1$ by
conjugating by
$$z\mapsto z+\dfrac{t_1}{\mu-\mu^{k+1}}z^{k+1}.$$
This produces a convergent algorithm in the Krull topology
producing a complex conjugate diffeomorphism $G$ to the group
generated by the rational rotation $R(z)=\mu z$.
\begin{proof}[proof of the claim]
Suppose that $\nu+1$ divides $k$, we define the following
application
$$\begin{array}{c c c c} \varphi:&G&\to& \co \\
&az+bz^{k+1}+h.o.t.&\mapsto&\dfrac{b}{a} \end{array}$$
which defines a morphism from $(G,\circ)$ into $(\co,+)$: indeed
given $f,g\in G$ we write
$$f(z)=az+bz^{k+1}+h.o.t.\;\;\mbox{and}\;\;g(z)=cz+dz^{k+1}+h.o.t.$$
and how $f\circ g\in G$ is written
$$f\circ g(z)=acz+(ad+bc^{k+1})z^{k+1}+h.o.t.$$
so we have
$$\varphi(f\circ g)=\dfrac{ad+bc^{k+1}}{ac}.$$
Since $\nu+1$ divides $k$ and $c=\mu^n$ for some integer $n$ then
we have $c^{k+1}=c$ therefore
$$\varphi(f\circ g)=\dfrac{ad+bc}{ac}=\dfrac{b}{a}+\dfrac{d}{c}=\varphi(f)+\varphi(g).$$
Now from (a) we have
$$\begin{array}{c} \varphi(f_1\circ f_2\circ\ldots\circ f_{\nu+1})=\varphi(\id)=0\\ \\
\varphi(f_1)+\varphi(f_2)+\ldots+\varphi(f_{\nu+1})=0\\ \\
\dfrac{t_1}{\mu}+\dfrac{t_2}{\mu}+\ldots+\dfrac{t_{\nu+1}}{\mu}=0.
\end{array}$$
Now for $i\neq j$ by (b) we have $h\in G$ such that
$$f_i\circ h=h\circ f_j$$
applying the morphism we have
$$\begin{array}{c} \varphi(f_i\circ h)=\varphi(h\circ f_j)\\ \\
\varphi(f_i)+\varphi(h)=\varphi(h)+\varphi(f_j)\\ \\
\varphi(f_i)=\varphi(f_j)\\
\\
\dfrac{t_i}{\mu}=\dfrac{t_j}{\mu}
\\
\\
t_i=t_j.
\end{array}$$
Therefore $t_1=t_2=\ldots=t_{\nu+1}=0$.
Suppose that $\nu+1$ does not divide $k$, we define the following
application $\psi\colon G\to \Aff(\co), \,
\psi(az+bz^{k+1}+h.o.t.)=\dfrac{az+b}{a^{k+1}}$, which defines a
morphism from $(G,\circ)$ into $(\Aff(\co),\circ)$: indeed given
$f,g\in G$ we write
$$f(z)=az+bz^{k+1}+h.o.t.\;\;\mbox{and}\;\;g(z)=cz+dz^{k+1}+h.o.t.$$
and since $f\circ g\in G$ is written as
$$f\circ g(z)=acz+(ad+bc^{k+1})z^{k+1}+h.o.t.$$
so we have
$$\psi(f\circ g)=\dfrac{acz+(ad+bc^{k+1})}{(ac)^{k+1}}$$
on the other hand
$$\psi(f)\circ\psi(g)=\dfrac{a\left(\dfrac{cz+d}{c^{k+1}}\right)+b}{a^{k+1}}=
\dfrac{acz+(a+bc^{k+1})}{(ac)^{k+1}}=\psi(f\circ g).$$
Denote by $G_0$ the image of $G$ by $\psi$. Therefore $G_0$ is an
affine group generated by the transformations
$g_1,g_2,\ldots,g_{\nu+1}$, pairwise conjugate in $G_0$, where
$$g_i(z)=\dfrac{\mu z+t_i}{\mu^{k+1}}.$$
We now apply the following lemma whose proof is found in \cite{CL}
(page 222):
\begin{Lemma} Let $\eta$ be a $l$-th root of the unit, $l>1$,
$\beta_1,\beta_2,\ldots,\beta_{r+1}\in\co$ and $\Gamma$
an affine group generated by the transformations
$h_i(z)=\eta z+\beta_i$, $i=1,2,\ldots,r+1$.
Then the $h_i$'s are pairwise conjugate in $\Gamma$ if and only if
either $l$ has two distinct prime divisors or $l=q^m$, for some
prime $q$ and some $m\in\mathbb{N}^*$ and
$\beta_1=\beta_2=\ldots=\beta_{r+1}$.
\end{Lemma}
Taking $\eta=\dfrac{1}{\mu^k}$, $l=p^s$, $r=\nu$ and
$\beta_i=\dfrac{t_i}{\mu^{k+1}}$ ($i=1,...,\nu+1$) by the previous
lemma we have $p^s=q^m$, $q$ prime, $m\in\mathbb{N}^*$ and
$$\dfrac{t_1}{\mu^{k+1}}=\dfrac{t_2}{\mu^{k+1}}=\ldots=\dfrac{t_{\nu+1}}{\mu^{k+1}}.$$
Therefore $q=p$, $m=s$ and
$$t_1=t_2=\ldots=t_{\nu+1}.$$
This proves the claim. \end{proof}
The proof of Theorem~\ref{Theorem:2irreducibledim1} is now complete.
\end{proof}
As for the case of groups of germs of real analytic diffeomorphisms
we promptly obtain:
\begin{Corollary}
Let $f_1,f_2,\ldots,f_{\nu +1}\in \Diff^w(\rr,0)$ be germs of real
analytic diffeomorphisms and let $G\subset \Diff^w(\rr,0)$ the group
generated by them. Assume that:
\begin{enumerate}
\item[(a)] $f_1\circ f_2\circ\ldots\circ f_{\nu+1}=\id$.
\item[(b)] $f_i$ and $f_j$ are conjugate in $G$ for all $i,j$.
\item[(c)] Some $f_j$ has a multiplier of order $p^s$ for
some prime $p$ and $s \in \mathbb N$.
\end{enumerate}
Then $G$ is finite of order $\leq 2$. If $s=0$ then $G$ is trivial.
\end{Corollary}
\begin{proof}
It is enough to consider the group $G^\mathbb C\subset \Diff(\mathbb
C,0)$ generated by the complexification of the maps
$f_j,\;j=1,...,\nu+1$. The group satisfies the hypotheses of
Theorem~\ref{Theorem:2irreducibledim1} and therefore it is finite.
Since the complexification defines an injective morphism $G \to
G^{\mathbb C}$ the group $G$ must finite. Since $G$ is analytically
linearizable, the conclusion follows from the fact that a finite
order germ of real analytic diffeomorphism has order one or two.
\end{proof}
\section{Examples and counterexamples}
\label{section:examples} Now we give some examples illustrating our
results and the necessity of the hypotheses we make. The first two
examples show that the condition on the order of the multiplier of a
generator cannot be dropped in
Theorem~\ref{Theorem:2irreducibledim1}.
\begin{Example}\label{Contraexample1}
{\rm Consider $G\subset \Diff(\mathbb C,0)$ the subgroup generated
by the maps
$$f_1(z)=f_2(z)=f_3(z)=f_4(z)=\dfrac{z}{a}, f_5(z)=\dfrac{z}{a+z}\;\;
\mbox{ and }\;\;f_6(z)=\dfrac{z}{a-a^5z}$$where $a^6=1$ so that
$a=\dfrac{1}{1-a}$. We claim that $G$ is irreducible and not
abelian. The first condition is satisfied
$$\begin{array}{r c l}f_1\circ f_2\circ f_3\circ
f_4\circ f_5\circ f_6(z)&=&f_1\circ f_2\circ f_3\circ
f_4\left(\dfrac{z}{a^2}\right)=\dfrac{z}{a^6}=z.
\end{array}$$
To check the second condition take $g_1(z)=f_1\circ f_5\circ
f_1^4(z)=\dfrac{z}{1+az}$ then $g_1\in G$ and note that
$$f_5\circ g_1(z)=\dfrac{z}{a+(1+a^2)z}$$
$$g_1\circ f_1(z)=\dfrac{z}{a+az}.$$ Since
$a=\dfrac{1}{1-a}$ we have $1+a^2=a$ therefore $f_5\circ
g_1=g_1\circ f_1$. Similarly, since $a^3=-1$ and $1+a^2=a$ we have
$f_1\circ g_2=g_2\circ f_6$. Finally, because $1+a-a^5=1+a+a^2$ we
have $f_5\circ g_3=g_3\circ f_6$. Consequently the $f_j$ are
pairwise conjugate in the group $G$. Finally, we observe that $G$ is
not abelian and in particular, it is not analytically linearizable:
indeed, if $g$ linearizes $G$, then $g\circ f_1\circ g^{-1}= f_1$. }
\end{Example}
Other examples of irreducible groups which are not finite are
briefly listed below:
\begin{Example}
\label{Example:more}{\rm (1) Let $G_{10}\subset \Diff(\mathbb C,0)$
be the subgroup finitely generated by the maps
$$f_1(z)=\cdots=f_8(z)=\dfrac{z}{a}, f_9(z)=\dfrac{z}{a+z}\;\;
\mbox{ and }\;\;f_{10}(z)=\dfrac{z}{a-a^9z}$$where $a^{10}=1$ so
that $a^3+a=\dfrac{1}{1-a}$.
(2) Consider $G_{12}\subset \Diff(\mathbb C,0)$ the subgroup
finitely generated by the maps
$$f_1(z)=\cdots=f_{10}(z)=\dfrac{z}{a},
f_{11}(z)=\dfrac{z}{a+z}\;\;\mbox{ and }\;\;
f_{12}(z)=\dfrac{z}{a-a^{11}z}$$where $a^{12}=1$ so that
$a=\dfrac{1}{1-a}$.
(3) Take $G_{12}^\prime \subset \Diff(\mathbb C,0)$ as the subgroup
finitely generated by the maps
$$f_1(z)=\cdots=f_{10}(z)=\dfrac{z}{a},
f_{11}(z)=\dfrac{z}{a+z}\;\;\mbox{ and }\;\;
f_{12}(z)=\dfrac{z}{a-a^{11}z}$$where $a^{12}=1$ so that
$a^3+a^2=\dfrac{1}{1-a}$.
(4) Let $G_{14}\subset \Diff(\mathbb C,0)$ be the subgroup finitely
generated by the maps
$$f_1(z)=\cdots=f_{12}(z)=\dfrac{z}{a},
f_{13}(z)=\dfrac{z}{a+z}\;\;\mbox{ and }\;\;
f_{14}(z)=\dfrac{z}{a-a^{13}z}$$where $a^{14}=1$ so that
$a^5+a^3+a=\dfrac{1}{1-a}$.
(5) Consider $G_{18}\subset \Diff(\mathbb C,0)$ the subgroup
finitely generated by the applications
$$f_1(z)=\cdots=f_{16}(z)=\dfrac{z}{a},
f_{17}(z)=\dfrac{z}{a+z}\;\;\mbox{ and }\;\;
f_{18}(z)=\dfrac{z}{a-a^{17}z}$$where $a^{18}=1$ so that
$a=\dfrac{1}{1-a}$.
(6) Put $G_{18}^\prime \subset \Diff(\mathbb C,0)$ as the subgroup
finitely generated by the applications
$$f_1(z)=\cdots=f_{16}(z)=\dfrac{z}{a},
f_{17}(z)=\dfrac{z}{a+z}\;\;\mbox{ and }\;\;
f_{18}(z)=\dfrac{z}{a-a^{17}z}$$where $a^{18}=1$ so that
$a^5+a^4+a^3=\dfrac{1}{1-a}$.
}
\end{Example}
We now show that condition (b) in the definition of irreducibility
(Definition~\ref{Definition:irreduciblegroup}) is essential in
Theorem~\ref{Theorem:2irreducibledim1}.
\begin{Example}
\label{Example:essential} {\rm Indeed, given $p \in \mathbb N$
consider $G=<f,f^{-1}>\subset\Diff(\co,0)$ where
$f(z)=\frac{z}{(1-z^p)^{1/p}}$, $f^{-1}(z)=\frac{z}{(1+z^p)^{1/p}}$.
Take $h(z)=e^{\pi i/p}z$. Note that $f\circ f^{-1}=\id$ and
$f(h(z))=\frac{e^{\pi i/p}z}{(1+z^p)^{1/p}}= e^{\pi
i/p}\frac{z}{(1+z^p)^{1/p}}=h(f^{-1}(z))$. Thus $f$ and $f^{-1}$ are
conjugate in $\Diff(\mathbb C,0)$, but clearly not in $G$. Note
also that $f^{(n)}(z)=\frac{z}{(1-nz^p)^{1/p}}\neq z$ for all
$n\in\mathbb{N}$. Therefore $G$ it is not finite. }
\end{Example}
\section{Germs of foliations: proof of
Theorems~\ref{Theorem:firstintegral1} and
~\ref{Theorem:firstintegralreducible}}
We consider a germ $\fa$ of a codimension one holomorphic foliation
singular at the origin $0 \in \mathbb C^n, n \geq 3$. We consider
the blow-up $\pi \colon \widetilde {\mathbb C^n} \to \mathbb C^n$ of
$\mathbb C^n$ at the origin $0 \in \mathbb C^n$ and denote by
$\tilde \fa$ the lifted foliation $\tilde \fa= \pi^*(\fa)$. We also
denote by $E=\pi^{-1}(0)\cong \mathbb P^{n-1}$ the exceptional
divisor of the blow-up.
\begin{proof}[Proof of Theorem~\ref{Theorem:firstintegral1}]
The proof goes as the proof of Theorem~0 in \cite{CL}. The key point
is then the following fact:
\begin{Claim}
The projective holonomy $H(\fa)$ is a finite group.
\end{Claim}
This is proved as an immediate consequence of the irreducibility of
the tangent cone, the fact that the projective holonomy is generated
by the basic holonomy generators, and
Theorem~\ref{Theorem:2irreducibledim1}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Theorem:firstintegralreducible}]
Again the proof has steps in common with the proof of Theorem~0 in
\cite{CL}. As already mentioned (see the paragraph before
Theorem~\ref{Theorem:firstintegralreducible}), it is observed in
\cite{CL} that the irreducibility of the tangent cone implies that a
generic plane section of $\fa$ can be reduced with a single blow-up.
We have this same property by hypothesis now. The main point is
then, again, the following fact:
\begin{Claim}
The projective holonomy $H(\fa)$ is a finite group.
\end{Claim}
\begin{proof}[proof of the claim]
Let $L(\fa)=E\setminus C(\fa)$ be the leaf of $\tilde \fa$ in the
blow-up $\widetilde{\mathbb C^n}$. We shall prove that the image
$H(\fa)$ of any holonomy representation $\Hol\colon \pi_1(L(\fa))\to
\Diff(\mathbb C,0)$ is a finite group. Notice that the codimension
$\geq 3$ singularities of $\tilde \fa$ do not affect the fundamental
group of $L(\fa)$. Therefore $L(\fa)$ has the homotopy type of the
complement of a normal crossing divisor in $\mathbb P^n$. By
Deligne's theorem this fundamental group is abelian. Moreover, this
group is generated by simple loops around the codimension one
irreducible components of $ C(\fa)$. This proves that the group
$\pi_1(L(\fa))$ is the direct product of the groups
$\pi_1(E\setminus \Lambda_j), j=1,...,r$ where $\Lambda_j\subset E,
j=1,...,r$ are the codimension one irreducible components of
$C(\fa)\subset E$. This implies that any holonomy representation
$\Hol\colon \pi_1(L(\fa)) \to \Diff(\mathbb C,0)$ has as image an
abelian group $H(\fa)\subset \Diff(\mathbb C,0)$ which is a direct
product of groups $H_j=\Hol(\pi_1(E\setminus \Lambda_j))\subset
\Diff(\mathbb C,0)$. Each group $H_j\subset \Diff(\mathbb C,0)$ is
irreducible and is finite because of the hypothesis on the degree of
the components $\Lambda_j$ and of the conclusion of
Theorem~\ref{Theorem:2irreducibledim1}. Thus the group $H(\fa)$ is
abelian as well.
\end{proof}
Once we have proved that the projective holonomy is finite and
knowing that the generic hyperplane sections of the foliation can be
reduced with a single blow-up we can proceed as in \cite{CL} and
conclude.
\end{proof}
\section{Foliations in projective spaces and local examples}
We turn our attention to
the case of foliations in the projective space $\mathbb P^n$ of
dimension $n \ge 2$. It is well-known that a codimension one
holomorphic foliation (with singularities) on a complex projective
space $\mathbb P^n$ can be defined in homogeneous coordinates
$(z_1,...,z_{n+1})$ in $\co^{n+1}$ by a differential 1-form
$$
\om = \sum_{i=1}^{n+1} a_i(z)dz_i \qquad z \in \co^{n+1} $$ where
the $a_i(z)$ are homogeneous polynomials of the same degree (without
common factors and) satisfying $\sum\limits_{i=1}^{n+1} z_i\,a_i(z)
= 0$,
and the integrability
condition $\om \wedge d\om = 0$. The singular set of $\om$ is
$S(\om) := \{z \in \co^{n+1}; a_j(z) = 0\,,\,\, j=1,\ldots,n+1\}$.
We denote by $\widehat\fa(\om)$ the codimension one singular
holomorphic foliation on $\co^{n+1}$ defined by $\om$ with singular
set $\sing (\om)$. If $\pi\colon \co^{n+1}\backslash\{0\} \to
\mathbb P^n$ denotes the canonical projection, this induces a
foliation $\fa = \fa(\om)$ on $\mathbb P^n$ with singular set
$\sing\,\fa = \pi(S(\om))$ and $\pi^*(\fa) = \widehat\fa(\om)$. As
we have remarked above there is no loss of generality if we assume
that $codim\,\sing\,\fa \ge 2$. Before going into our main example,
we must recall the notion of Kupka singularity:
\begin{Definition}[Kupka singularity] {\rm
Let $\omega=\sum_{j=1}^na_jdx_j$ be a holomorphic integrable 1-form
on a neighborhood $\U$ of $0\in \co^n$. We assume that $n\ge3$ and
that $\sing(\omega)=\{p\in \U\mid \omega(p)=0\}$ is a codimension
$\geq 2$ analytic subset. A singularity $q\in\sing(\omega)$ is a
{\it Kupka singularity\/} if $d\omega(q)\ne0$. We denote by
$K(\omega)$ the subset of Kupka singularities.}
\end{Definition}
The main properties of the Kupka singularities are stated in
\cite{Omegar}, \cite{Kupka}. These singularities are of local
product type, by a foliation in dimension two.
\begin{Example}
\label{Example:Kupka}{\rm Let $k \in \mathbb N, k\geq 2$ be given.
Let also $a,\;b,\;c,\;\alpha,\;\beta,\;\gamma\in\co^*$ be fixed. We
consider the integrable homogeneous 1-form $\Omega$ in $\co^4$ with
coordinates $(x,y,z,w)$ by:
$$\begin{array}{l l
l}\Omega&=&\left[-x^2(ay^{k-1}+bz^{k-1}+cw^{k-1})+\alpha
y^{k+1}+\beta z^{k+1}+\gamma
w^{k+1}\right]dx\\&&\\&&+xy^{k-2}(ax^2-\alpha
y^2)dy+xz^{k-2}(bx^2-\beta z^2)dz+xw^{n-2}(cx^2-\gamma
w^2)dw\end{array}$$
Notice that $\Omega ({\vec R})=0$ for the radial vector field ${\vec
R}=x\frac{\partial}{\partial x} + y \frac{\partial}{\partial y}+ z
\frac{\partial}{\partial z} + w\frac{\partial}{\partial w}$. Thus
$\Omega$ defines a codimension one holomorphic foliation $\fa$ of
degree $k$ in $\mathbb P^3$. This foliation leaves invariant the
hyperplane of infinity $H=\pi(x=0)$, where
$\pi:\co^4\setminus\{0\}\to \mathbb P^3$ is the canonical
projection.
\begin{Claim}
We have $\sing(\fa)\cap H =S$ is irreducible and smooth. Moreover,
all singularities of $\fa$ in $H$ are of Kupka type.
\end{Claim}
\begin{proof}[proof of the claim]
First of all we note that $$H\cap \sing(\fa)=\{[0:y:z:w];\;\alpha
y^{k+1}+\beta z^{k+1} +\gamma w^{k+1}=0\}$$ is a smooth irreducible
hypersurface of degree $k+1$. Furthermore
$$d\left(\dfrac{\Omega}{F}\right)=0$$where $F$ is
the homogeneous polynomial of degree $k+2$ given by
$$F=x^2\left[x\left(a\dfrac{y^{k-1}}{k-1}+
b\dfrac{z^{k-1}}{k-1}+c\dfrac{w^{k-1}}{k-1}\right)
-\alpha\dfrac{y^{k+1}}{k+1}-\beta\dfrac{z^{k+1}}{k+1}
-\gamma\dfrac{w^{k+1}}{k+1}+x^{k+1}\right].$$We can write $F=xQ$,
where $Q$ is the polynomial of degree $k$ given by
$$Q=x^2\left(a\dfrac{y^{k-1}}{k-1}+b\dfrac{z^{k-1}}{k-1}
+c\dfrac{w^{k-1}}{k-1}\right)-\alpha\dfrac{y^{k+1}}{k+1}
-\beta\dfrac{z^{k+1}}{k+1}-\gamma\dfrac{w^{k+1}}{k+1}+x^{k+1}.$$
Thus
$$\Omega=x^{k+2} d\left(\dfrac{Q}{x^{k+1}}\right)=-(k+1)Qdx+xdQ.$$
Let now $p=(x,y,z,w) \in \mathbb C^4$ be such that
$\Omega(p)=d\Omega (p)=0$. From $\Omega(p)=0$ we have $Q(p)=x(p)=0$.
From $\Omega=-(k+1)Qdx+xdQ$ we get $d\Omega=-(k+2) dQ \wedge dx$.
Thus $d\Omega(p)=0$ implies $dQ(p)\wedge dx(p)=0$. From the
expression of $Q$ above, taking into account that $x(p)=0$, we have
$dQ(p)\wedge dx(p)=0 \implies y(p)=z(p)=w(p)=0$. Since
$(x,y,z,w)(p)\ne 0$ in homogeneous coordinates, we have that every
singularity in $S$ is of Kupka type.
A final remark is that $\dfrac{Q}{x^{k+1}}$ is a first meromorphic
integral of $\fa$.
\end{proof}
}
\end{Example}
The above example allows us to construct local examples of the
situation we deal with in Theorem~\ref{Theorem:firstintegral1}.
\begin{Example}
{\rm Example~\ref{Example:Kupka}, originally placed in the
projective space $\mathbb P^3$ may be used in order to produce a
local example, illustrating our
Theorem~\ref{Theorem:firstintegralreducible}. Indeed, the idea is to
find a germ of foliation at $0 \in \mathbb C^3$ that produces after
a blow-up at the origin, a foliation having an invariant exceptional
divisor which will be bimeromorphically mapped into the hyperplane
at infinity $H\subset \mathbb P^3$ of the above example. For this
consider the one-form $\Omega$ given in $\co^3$ by
$$\Omega=\left[-x^2(y^2+z^2)+y^4+z^4\right]dx+xy(x^2-y^2)dy+xz(x^2-z^2)dz.$$
Using the relations
$$x=\dfrac{1}{X},\;y=\dfrac{Y}{X^2},\;z=\dfrac{Z}{X^2}$$
we have the pull-back one-form
$$\begin{array}{r c l} \tilde{\Omega}&=&\left[-\dfrac{1}{X^2}\left( \dfrac{Y^2}{X^4}+\dfrac{Z^2}{X^4}\right)+\dfrac{Y^4}{X^8}+\dfrac{Z^4}{X^8}\right]d\left(\dfrac{1}{X}\right) +\dfrac{1}{X}\dfrac{Y}{X^2}\left(\dfrac{1}{X^2}-\dfrac{Y^2}{X^4}\right)d\left(\dfrac{Y}{X^2}\right)
\\&&\\&&+\dfrac{1}{X}\dfrac{Z}{X^2}\left(\dfrac{1}{X^2}-\dfrac{Z^2}{X^4}\right)d\left(\dfrac{Z}{X^2}\right)\\&&\\
&=&
\left[\dfrac{X^2(Y^2+Z^2)-Y^4-Z^4}{X^{10}}\right]dX+\dfrac{Y(X^2-Y^2)}{X^{10}}\left(XdY-2YdX\right)+\dfrac{Z(X^2-Z^2)}{X^{10}}\left(XdZ-2ZdX\right)
\end{array} $$
and then
$$\begin{array}{r c l} X^{10}\tilde{\Omega}
&=&
\left[-X^2(Y^2+Z^2)+Y^4+Z^4\right]dX+XY(X^2-Y^2)dY+XZ(X^2-Z^2)dZ.\end{array}$$
Thus by construction
$\alpha(X,Y,Z)=\left[-X^2(Y^2+Z^2)+Y^4+Z^4\right]dX+XY(X^2-Y^2)dY+XZ(X^2-Z^2)dZ$
is an integrable one-form in $\mathbb C^3$, it admits the following
first integral
\[
f=\frac{1}{X^4} \big[X^2 \big(\frac{Y^2 + Z^2}{2}\big) -
\frac{Y^4}{4} - \frac{Z^4} {4}\big].
\]
We then consider the birational change of coordinates
$(x,y,z)=(1/x,y,z)=(X,Y,Z)$ which gives
\[
f(x,y,z)=x^2(y^2+z^2)/2 -x^4y^4/4-x^4z^4/4
\]
and the one-form
\[
\omega=[(y^2 + z^2) + x^2 ( y^4 + z^4)]dx + xy(1- x^2 y^2)dy + zx (1
- x^2 z^2)dz.
\]
Then, finally, $\omega$ is an integrable one-form in a neighborhood
of the origin $0 \in \mathbb C^3$. The corresponding foliation
$\fa_\omega: \omega=0$ has a tangent cone given by the degree $3$
homogeneous equation $(P_3=0)\subset \mathbb P^2$ for
$P_3=x(y^2+z^2)+xy^2+xz^2=2x(y^2+z^2)$. Therefore $C(\fa_\omega)$ is
a normal crossings divisor with components given by three projective
lines, induced by $x=0, y + iz=0, y - iz=0$. Because of the Kupka
type of the singularities at infinity of the starting foliation
$\fa_\Omega: \Omega=0$, we have that a generic plane section
$\tau^*(\fa_\omega)$ of $\fa_\omega$, can be reduced with a single
blow-up $\pi\colon \widetilde {\mathbb C^2} \to \mathbb C^2$,
verifying then condition (3) in
Theorem~\ref{Theorem:firstintegralreducible}.
}
\end{Example}
\subsection{Formal case}
Theorem~\ref{Theorem:2irreducibledim1} also holds in the formal
framework. Indeed, the proofs are {\it ipsis litteris} the same.
Denote by $\widehat\Diff(\mathbb C,0)$ the group of formal complex
diffeomorphims fixing the origin $0 \in \mathbb C$. An element $\hat
f\in \widehat\Diff(\mathbb C,0)$ is a formal power series $\hat
f(z)=\mu z + \sum\limits_{j=1}^\infty a_j z^j\in \mathbb C[[z]]$
with $\mu \ne 0$.
We may then state:
\begin{Theorem}
\label{Theorem:2irreducibledim1formal} Let $\hat G\subset \widehat
\Diff(\mathbb C,0)$ be an irreducible group, ie., a finitely
generated subgroup with generators $\hat f_1,...,\hat f_{\nu+1}$
pairwise conjugate in $\hat G$ and such that $\hat f_{\nu+1}\circ
\cdots \circ \hat f_1=\id$. If the multiplier of any basic generator
is a $p^s$-root of the unit, where $p$ is prime and
$s\in\mathbb{N}$, then $\hat G$ is a finite group, trivial in the
first case $s=0$.
\end{Theorem}
\bibliographystyle{amsalpha}
| 192,846
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This whole Google/Verizon net neutrality debate has reached critical mass and is quickly entering the “beating a dead horse” territory, but AT&T threw a bit more gasoline on the fire last Friday with its “Wireless is Different” post on the company’s Public Policy Blog.
As the title suggests, the post asserts that wireless internet is different from hard-wired broadband internet. This is not to be confused with Wi-Fi, which is a short range wireless-ification of hard-wired broadband via a router. There are too many vaguely similar terms being thrown around, so let’s use “mobile internet” and “home internet” to describe what AT&T, Verizon, and Google are talking about.
To recap the debate, Google and Verizon think that home internet from the likes of Comcast or Time Warner or FiOS or whatever you use at your house should be left alone. Why? Because the pipeline into your house is relatively big and fast and can, in a very simple sense, pretty much handle whatever you and everyone around you are trying to download without buckling much.
The same can’t be said for mobile internet. If you’ve ever been to a big concert or sporting event or any place where thousands of people are trying to use their iPhones at the same time, AT&T’s network buckles pretty hard. It happens to other networks too—Verizon, Sprint, T-Mobile, etc.—but it’s more prevalent on AT&T because so many people have iPhones. Some would also argue that AT&T’s network and overall coverage isn’t as good as its competitors, which may or may not be true. Watch what happens to Verizon if and when it gets an iPhone, though.
Anyway.
What Verizon and AT&T want, and what Google’s given up on fighting against, is to differentiate the idea of home and mobile internet service. The hands-off approach that works for home internet doesn’t work for mobile internet since, according to AT&T at least, there’s “insatiable demand” for mobile data that must be delivered over “wireless networks of finite and shared resources.” The pipe is much smaller, in other words.
So if you’re AT&T or Verizon, how do you deal with insatiable demand over a finite pipeline? Sell fewer phones and sign fewer people up for two-year contracts until you can deliver those 3G speeds you’ve been talking about in all your ads? No way, you’ve got a bottom line to protect! You sell as many phones and activate as many new customers as you can and then selectively throttle their connections when your network starts to buckle.
Or as AT&T puts it, “In order to provide consumers with the high quality wireless broadband services that they demand, wireless carriers must be able to dynamically manage traffic and operate their networks in an environment free from burdensome, arbitrary and unnecessary regulations.”
That plan sounds semi-reasonable given the notion that mobile companies aren’t going to artificially restrict the number of handsets available on their networks. The problem—and what consumers and the press are going ape about—is that “an environment free from burdensome, arbitrary and unnecessary regulations” leaves the door open for the likes of AT&T, Verizon, Sprint, T-Mobile, and any other mobile operator to go beyond simply throttling traffic once their networks start to choke.
Imagine if Google, who owns YouTube and has plenty of money, went to all the mobile operators and said, “Hey, we’ll give you guys each a million bucks a month if you don’t throttle traffic to YouTube, Google, Gmail, or any of our other products.” The next time you’re at the Red Sox game, all the sites that Google owns would load right up while Discount Doug’s Better Than Gmail E-mail Service doesn’t load up or takes a long time to load up because everyone at Fenway is on their phones at the same time.
Eventually people would be less and less likely to use Discount Doug’s Better Than Gmail E-mail Service because they’d say, “Even though Doug’s really is better, Gmail works much faster on my phone so I’m sticking with Google.”
Doug goes out of business because he couldn’t afford to pay for priority traffic and the next Doug that comes along doesn’t even bother trying to create a better e-mail system or search engine or video sharing site at all. What’s the point? The ability for small players to innovate and compete drops dramatically when the big guys can pay for their own express lane.
But this stuff only pertains to mobile internet access, so why worry too much about it? Because mobile internet will eventually replace home internet, just like home broadband internet replaced dial-up. It’s happening already as more and more people use their phones as their primary internet devices and 4G mobile networks slowly begin to blanket the country.
That’s where Google really gets caught up in the middle of all this. Its position is that home internet should be open, equal, and indiscriminate, which is friendly to consumers and innovators alike. However, the company is positioning its own strategy as “mobile first.”
In other words: Google sees the internet’s future, and the future is mobile.
While most people halfway expect to get jerked around by their mobile company in the interest of profits, Google’s supposed “Don’t Be Evil” mantra and its newfound coziness with the carriers thanks to the Android platform has a lot of people feeling a bit betrayed.
More on Techland: Google and Verizon Push For Open Internet, Net Neutrality Still In Effect
| 93,031
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\begin{document}
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\title{On the exactness for polynomial optimization strengthened with Fritz John conditions}
\author{
Ngoc Hoang Anh Mai\footremember{1}{CNRS; LAAS; 7 avenue du Colonel Roche, F-31400 Toulouse; France.}
}
\maketitle
\begin{abstract}
We utilize the same technique as in [arXiv:2205.04254 (2022)] to provide some representations of polynomials non-negative on a basic semi-algebraic set, defined by polynomial inequalities, under more general conditions.
Based on each representation, we obtain semidefinite programs which return a sequence of values that finitely converges to the optimal value of a given polynomial optimization problem under generic assumption.
Consequently, we can compute exactly the minimal value of any polynomial over a basic convex semi-algebraic set which is defined by the inequalities of concave polynomials.
\end{abstract}
\keywords{sum-of-squares; Nichtnegativstellensatz; gradient ideal; Fritz John conditions; polynomial optimization}
\tableofcontents
\section{Introduction}
The study of representations of a polynomial $f$ non-negative on a basic semi-algebraic set $S$ (also called Nichtnegativstellens\"atz) is a central interest in real algebraic geometry with influential applications in polynomial optimization.
We refer the readers to \cite{hilbert1888darstellung,putinar1993positive,
lasserre2001global,krivine1964anneaux,scheiderer2000sums,
scheiderer2003sums,scheiderer2006sums,schmudgen1991thek,
demmel2007representations,nie2014optimality,
marshall2006representations,marshall2009representations}
for extensive discussions of various aspects of this subject.
Accordingly, it is important, difficult and challenging to provide Nichtnegativstellens\"atz, which have the same forms as Putinar's and Schm\"udgen's Positivstellensatz, in cases where: (i) polynomial $f$ is non-negative with infinitely many zeros on basic semi-algebraic sets $S$ or (ii) the Karush–Kuhn–Tucker conditions do not hold for $\min_x \{f(x)\,:\,x\in S\}$ at any zero of $f$ on $S$.
Initially developed by Demmel, Nie and Powers in their celebrated work \cite{demmel2007representations}, the techniques in \cite{mai2022exact} enable us to deal with above cases, and hence obtain exact semidefinite programs for a polynomial optimization problem in these cases.
Roughly speaking, we utilize a variety $V$, generated by polynomials in the Fritz John conditions for $\min_x \{f(x)\,:\,x\in S\}$, whose intersection with $S$ contains the zeros of $f$.
It turns out that $f$ has finitely many values on $V$, so has nice representations without denominator involving quadratic module and preordering associated with $S\cap V$ under a generic condition.
Following the same research line as in \cite{mai2022exact} our main results in this paper based on this approach state some Nichtnegativstellens\"atz under more general conditions.
In particular, $f$ still has the representation involving preordering when $f$ has finitely many values on the intersection $V\cap S$ (instead of only $V$ as above) under the more general condition.
Let $\R[x]$ denote the ring of polynomials with real coefficients in the vector of variables $x$.
Given $f,g_1,\dots,g_m\in\R[x]$, consider polynomial optimization problem:
\begin{equation}\label{eq:pop}
f^\star:=\inf\limits_{x\in S(g)} f(x)\,,
\end{equation}
where $S(g)$ is the basic semi-algebraic set associated with $g=(g_1,\dots,g_m)$ defined by
\begin{equation}
S(g):=\{x\in\R^n\,:\,g_j(x)\ge 0\,,\,j=1,\dots,m\}\,.
\end{equation}
Given $p\in\R[x]$, we denote by $\nabla p$ the gradient of $p$, i.e., $\nabla p=(\frac{\partial p}{\partial x_1},\dots,\frac{\partial p}{\partial x_n})$.
We state in the following lemma the Fritz John conditions, which are equivalent to but different from the ones in \cite[Lemma 1]{mai2022exact}:
\begin{lemma}\label{lem:FJ}
Let $f,g_1,\dots,g_m\in\R[x]$. If $u$ is a local minimizer for problem \eqref{eq:pop}, then the Fritz John conditions hold for problem \eqref{eq:pop} at $u$, i.e., there exists a real vector $(\lambda_0,\dots,\lambda_m)$ such that
\begin{equation}
\begin{array}{rl}
& \lambda_0^2 \nabla f(u)=\sum_{j=1}^m \lambda_j^2 \nabla g_j(u)\,,\\
& \lambda_j^2 g_j(u) =0\,,\,j=1,\dots,m\,,\\
& \sum_{j=1}^m \lambda_j^2=1\,.
\end{array}
\end{equation}
\end{lemma}
Denote by $\Sigma^2[x]$ the cone of sum of squares of polynomials in $\R[x]$.
Given $g_1,\dots,g_m\in\R[x]$, let $Q(g)[x]$ be the quadratic module associated with $g=(g_1,\dots,g_m)$ defined by
\begin{equation}
Q(g)[x]:=\Sigma^2[x] +\sum_{j=1}^m g_j\Sigma^2[x]\,.
\end{equation}
We say that $S(g)$ satisfies the Archimedean condition if there exists $R>0$ such that $R-x_1^2-\dots-x_n^2\in Q(g)[x]$.
Given $g_1,\dots,g_m\in\R[x]$, with $g=(g_1,\dots,g_m)$, let $\Pi g$ be the vector of products of $g_1,\dots,g_m$ defined by
\begin{equation}\label{eq:prod.g}
\Pi g:=(g^\alpha)_{\alpha\in\{0,1\}^m\backslash \{0\}}\,,
\end{equation}
where $\alpha=(\alpha_1,\dots,\alpha_m)$ and $g^\alpha:=g_1^{\alpha_1}\dots g_m^{\alpha_m}$.
We call $Q(\Pi g)[x]$ the preordering generated by $g$, denoted by $P(g)[x]$.
Obviously, if $m=1$, it holds that $P(g)[x]=Q(g)[x]$.
Given $h_1,\dots,h_l$, let $V(h)$ be the variety generated by $h=(h_1,\dots,h_l)$ defined by
\begin{equation}
V(h):=\{x\in\R^n\,:\,h_j(x)=0\,,\,j=1,\dots,l\}\,.
\end{equation}
and let $I(h)[x]$ be the ideal generated by $h$ defined by
\begin{equation}
I(h)[x]:= \sum_{j=1}^l h_j \R[x]\,.
\end{equation}
Given $g_1,\dots,g_m\in\R[x]$, let $\varphi^g:\R^{n}\to \R^{(n+m)\times m}$ be a function associated with $g=(g_1,\dots,g_m)$ defined by
\begin{equation}
\varphi^g(x)=
\begin{bmatrix}
\nabla g_1(x)& \dots& \nabla g_m(x)\\
g_1(x)&\dots&0\\
.&\dots&.\\
0&\dots&g_m(x)
\end{bmatrix}\,.
\end{equation}
Given a real matrix $A$, we denote by $\rank(A)$ the largest number of columns of $A$ which are not linearly dependent over $\R$.
We say that a set $\Omega$ is finite if its cardinal number is a non-negative integer.
Let $C(g)$ be the set of critical points associated with $g$ defined by
\begin{equation*}
C(g):=\{x\in\R^n\,:\,\rank(\varphi^g(x))< m\}.
\end{equation*}
In the following theorem, we recall the main representation in \cite[Theorem 1]{mai2022exact} :
\begin{theorem}\label{theo:rep.old}
Let $f,g_1,\dots,g_m\in\R[x]$.
Assume that $f$ is non-negative on $S(g)$ with $g:=(g_1,\dots,g_m)$ and $f(C(g))$ is finite.
Then there exists $q\in P(g)[x,\bar \lambda]$ such that $f-q$ vanishes on $ V(h_\text{FJ})$, where $\bar\lambda:=(\lambda_0,\dots,\lambda_m)$ and
\begin{equation}\label{eq:.polyFJ}
h_\text{FJ}:=(\lambda_0\nabla f-\sum_{j=1}^m \lambda_j \nabla g_j,\lambda_1g_1,\dots,\lambda_mg_m,1-\sum_{j=0}^m\lambda_j^2)\,.
\end{equation}
Moreover, if $S(g)$ satisfies the Archimedean condition, we can take $q\in Q(g)[x,\bar \lambda]$.
\end{theorem}
We state the first main result in the following theorem:
\begin{theorem}\label{theo:rep}
Let $f,g_1,\dots,g_m\in\R[x]$.
Assume that $f$ is non-negative on $S(g)$ with $g:=(g_1,\dots,g_m)$ and $f(C(g)\cap S(g))$ is finite.
Then there exists $q\in P(g)[x,\bar \lambda]$ such that $f-q$ vanishes on $(S(g)\times\R^{m+1})\cap V(h_\text{FJ})$, where $\bar\lambda:=(\lambda_0,\dots,\lambda_m)$ and $h_\text{FJ}$ is defined as in \eqref{eq:.polyFJ}.
Moreover, if $S(g)$ satisfies the Archimedean condition, we can take $q\in Q(g)[x,\bar \lambda]$.
\end{theorem}
The proof of Theorem \ref{theo:rep} is postponed to Section \ref{sec:proof}.
Given a real matrix $A$, we denote by $\rank^+(A)$ the largest number of columns of $A$ whose convex hull over $\R$ has no zeros.
Let $C^+(g)$ be the set of critical points associated with $g$ defined by
\begin{equation*}
C^+(g):=\{x\in \R^n\,:\,\rank^+(\varphi^g(x))< m\}.
\end{equation*}
We state the second main result in the following theorem:
\begin{theorem}\label{theo:rep.plus}
Let $f,g_1,\dots,g_m\in\R[x]$.
Assume that $f$ is non-negative on $S(g)$ with $g:=(g_1,\dots,g_m)$ and $f(C^+(g))$ is finite.
Then there exists $q\in P(g)[x,\bar \lambda]$ such that $f-q$ vanishes on $V(h_\text{FJ}^+)$, where $\bar\lambda:=(\lambda_0,\dots,\lambda_m)$ and
\begin{equation}\label{eq:.polyFJ.plus}
h_\text{FJ}^+:=(\lambda_0^2\nabla f-\sum_{j=1}^m \lambda_j^2 \nabla g_j,\lambda_1^2g_1,\dots,\lambda_m^2g_m,1-\sum_{j=0}^m\lambda_j^2)\,.
\end{equation}
Moreover, if $S(g)$ satisfies the Archimedean condition, we can take $q\in Q(g)[x,\bar \lambda]$.
\end{theorem}
\begin{proof}
To prove Theorem \ref{theo:rep.plus}, we do similarly to the proof of \cite[Theorem 1]{mai2022exact} by replacing $C(g)$ and $h_\text{FJ}$ in \cite[Section 2.2]{mai2022exact} with $C^+(g)$ and $h_\text{FJ}^+$, respectively.
Note that the equality $C^+(g)=\pi(V(h_\text{FJ}^+)\cap \{\lambda_0=0\})$ follows thanks to the following equivalences:
\begin{equation}\label{eq:equi.Cplus}
\begin{array}{rl}
& x\in C^+(g)\\
\Leftrightarrow & \rank^+(\varphi^g(x)) < m\\
\Leftrightarrow & \exists \lambda\in\R^{m}\,:\, \sum_{j=1}^m\lambda_j^2=1\,,\, \sum_{j=1}^m \lambda_j^2 \nabla g_j( x)=0\,,\, \lambda_j^2 g_j =0\\
\Leftrightarrow & \exists \bar \lambda\in\R^{m+1}\,:\,
\sum_{j=0}^m\lambda_j^2=1\,,\,\lambda_0=0\,,\,\lambda_0^2\nabla f(x)=\sum_{j=1}^m \lambda_j^2 \nabla g_j(x)\,,\, \lambda_j^2 g_j =0\\
\Leftrightarrow & \exists \bar \lambda\in\R^{m+1}\,:\,(x,\bar\lambda)\in V(h_\text{FJ}^+)\cap \{\lambda_0=0\}\\
\Leftrightarrow& x\in\pi(V(h_\text{FJ}^+)\cap \{\lambda_0=0\})\,.
\end{array}
\end{equation}
Moreover, the Lagrangian function becomes
$L(x,\bar \lambda) = f(x)+\sum_{j=1}^m \left(\frac{\lambda_j}{\lambda_0}\right)^2 g_j (x)$.
\end{proof}
We state the third main result in the following theorem:
\begin{theorem}\label{theo:rep.plus2}
Let $f,g_1,\dots,g_m\in\R[x]$.
Assume that $f$ is non-negative on $S(g)$ with $g:=(g_1,\dots,g_m)$ and $f(C^+(g)\cap S(g))$ is finite.
Then there exists $q\in P(g)[x,\bar \lambda]$ such that $f-q$ vanishes on $S(g)\cap V(h_\text{FJ}^+)$, where $\bar\lambda:=(\lambda_0,\dots,\lambda_m)$ and $h_\text{FJ}^+$ is defined as in \eqref{eq:.polyFJ.plus}.
Moreover, if $S(g)$ satisfies the Archimedean condition, we can take $q\in Q(g)[x,\bar \lambda]$.
\end{theorem}
\begin{proof}
The proof is processed similarly to the one of Theorem \ref{theo:rep} by replacing $C(g)$ and $h_\text{FJ}$ in \cite[Section 2.2]{mai2022exact} with $C^+(g)$ and $h_\text{FJ}^+$, respectively.
\end{proof}
\begin{remark}
Since $C(g)\cap S(g)$, $C^+(g)$ and $C^+(g)\cap S(g)$ are subsets of $C(g)$, the assumptions of Theorems \ref{theo:rep}, \ref{theo:rep.plus} and \ref{theo:rep.plus2} that $f(C(g)\cap S(g))$, $f(C^+(g))$ and $f(C^+(g)\cap S(g))$ are finite hold generically thanks to \cite[Theorem 2]{mai2022exact}, respectively.
\end{remark}
The paper is organized as follows:
Section \ref{sec:proof} is mostly to prove Theorem \ref{theo:rep}.
We give in Section \ref{sec:examples} some illustrated examples for Theorems \ref{theo:rep} and \ref{theo:rep.plus}.
Section \ref{sec:application} is to present the application of our results in computing exactly the optimal value for a polynomial optimization problem.
\section{Proof}
\label{sec:proof}
We use the same notation as in \cite{mai2022exact} throughout this section.
We generalize \cite[Lemma 10]{mai2022exact} in the following lemma:
\begin{lemma}\label{lem:quadra}
Let $f\in\R[x]$, let $W$ be a complex variety generated by finitely many polynomials in $\R[x]$, and let $A$ be a semi-algebraic subset of $\R^n$.
Assume that $f(A\cap W)$ is finite.
Then there exist a finite sequence of subsets $W_1,\dots,W_r$ such that the following conditions hold:
\begin{enumerate}
\item $W_1,\dots,W_r$ are pairwise disjoint complex varieties generated by finitely many polynomials in $\R[x]$;
\item for $j=1,\dots,r$, $W_j\subset W$ and $f$ is constant on $W_j$;
\item $(W_1\cup\dots\cup W_r)\cap A=W\cap A$.
\end{enumerate}
\end{lemma}
The proof of Lemma \ref{lem:quadra} is similar to the one of \cite[Lemma 10]{mai2022exact} since we only need to replace $\R^n$ in the proof of \cite[Lemma 10]{mai2022exact} with $A$.
The following lemma is based on \cite[Lemma 3.3]{demmel2007representations}:
\begin{lemma}\label{lem:constant}
Let $f,g_1,\dots,g_m\in\R[x]$.
Assume that $f(C(g))$ with $g:=(g_1,\dots,g_m)$ is finite.
Let $h_\text{FJ}$ be defined as in \eqref{eq:.polyFJ}.
Let $W$ be a semi-algebraically path connected component of $(S(g)\times\R^{m+1})\cap V(h_\text{FJ})$. Then $f$ is constant on $W$.
\end{lemma}
\begin{proof}
The proof is similar to the one of \cite[Lemma 13]{mai2022exact}.
When proving that $\tau\mapsto f(x(\tau))$ is constant on $[a_j,b_j]$, we note that \begin{equation}
f(S(g)\cap C(g))=f(S(g)\cap \pi(V(h_\text{FJ})\cap \{\lambda_0=0\}))
\end{equation}
is finite since $f(S(g)\cap\pi(V(h_\text{FJ})\cap \{\lambda_0=0\}))\supset f(x([\tau_1,\tau_2]))$, which is due to the fact that
\begin{equation}
\begin{array}{rl}
S(g)\cap\pi(V(h_\text{FJ})\cap \{\lambda_0=0\})=&\pi((S(g)\times\R^{m+1})\cap V(h_\text{FJ})\cap \{\lambda_0=0\})\\
\supset& \pi(W\cap \{\lambda_0=0\}) \supset x([\tau_1,\tau_2])\,.
\end{array}
\end{equation}
\end{proof}
\subsubsection*{Proof of Theorem \ref{theo:rep}}
\begin{proof}
Using \cite[Lemma 8]{mai2022exact}, we decompose $(S(g)\times\R^{m+1})\cap V(h_\text{FJ})$ into semi-algebraically path connected components:
$Z_1,\dots,Z_s$.
Lemma \ref{lem:constant} shows that $f$ is constant on $Z_i$.
Thus $f((S(g)\times\R^{m+1})\cap V(h_\text{FJ}))$ is finite.
Observe that $(S(g)\times\R^{m+1})\cap V(h_\text{FJ})=(S(g)\times\R^{m+1})\cap V_\C(h_\text{FJ})$.
By using Lemma \ref{lem:quadra}, we obtain a finite sequence of subsets $W_{1},\dots,W_{r}$ such that the following conditions hold:
\begin{itemize}
\item $W_{1},\dots,W_{r}$ are pairwise disjoint complex varieties generated by finitely many polynomials in $\R[x,\bar\lambda]$;
\item for $j=1,\dots,r$, $W_{j}\subset V_\C(h_\text{FJ})$ and $f$ is constant on $W_{j}$;
\item $(W_{1}\cup\dots\cup W_{r})\cap (S(g)\times\R^{m+1})=V_\C(h_\text{FJ})\cap (S(g)\times\R^{m+1})=(S(g)\times\R^{m+1})\cap V(h_\text{FJ})$.
\end{itemize}
Let $D$ be the union of $W_{1},\dots,W_{r}$.
Let $b=1-\lambda_0^2-\dots-\lambda_m^2$.
From this, \cite[Lemma 11]{mai2022exact} yields that there exists $p\in P(g,b)[x,\bar\lambda]$ such that $f-p$ vanishes on $D\cap\R^{n+m+1}\supset (S(g)\times\R^{m+1})\cap V(h_\text{FJ})$.
We write
\begin{equation}\label{eq:preo.rep.b}
p=\sum_{\alpha\in\{0,1\}^m}\sigma_\alpha g^\alpha+b \sum_{\beta\in\{0,1\}^m}\psi_\beta g^\beta\,,
\end{equation}
for some $\sigma_\alpha,\psi_\beta\in\Sigma^2[x,\bar{\lambda}]$.
Let
$q=\sum_{\alpha\in\{0,1\}^m}\sigma_\alpha g^\alpha\in P(g)[x,\bar\lambda]$.
Since $b=0$ on $V(h_\text{FJ})$, it holds that $f=p=q$ on $(S(g)\times\R^{m+1})\cap V(h_\text{FJ})$.
Assume that $S(g)$ satisfies the Archimedean condition. Then there exists $R>0$ such that $g_{m+1}=R-x_1^2-\dots-x_n^2\in Q(g)[x]$.
It implies that $S(g,b)$ with $b=1-\lambda_0^2-\dots-\lambda_m^2$ satisfies the Archimedean condition.
This is due to the fact that
\begin{equation}\label{eq:large.ball}
(R+1)-x_1^2-\dots-x_n^2-\lambda_0^2-\dots-\lambda_m^2=b+g_{m+1}\in Q(g,b)[x,\bar\lambda]\,.
\end{equation}
From this, \cite[Lemma 11]{mai2022exact} shows that there exists $p\in Q(g,b)[x,\bar\lambda]$ such that $f-p$ vanishes on $D\cap\R^{n+m+1}\supset (S(g)\times\R^{m+1})\cap V(h_\text{FJ})$.
We write
\begin{equation}\label{eq:quadra.rep.b}
p=\sigma_0+\sum_{j=1}^m\sigma_j g_j+b\sigma_{m+1}\,,
\end{equation}
for some $\sigma_j\in\Sigma^2[x,\bar{\lambda}]$.
Let
$q=\sigma_0+\sum_{j=1}^m\sigma_j g_j\in Q(g)[x,\bar\lambda]$.
Since $b=0$ on $V(h_\text{FJ})$, $f=p=q$ on $(S(g)\times\R^{m+1})\cap V(h_\text{FJ})$.
This completes the proof.
\end{proof}
\section{Illustrated examples}
\label{sec:examples}
In this section, we illustrate our Nichtnegativstellens\"atz stated in Theorems \ref{theo:rep}, \ref{theo:rep.plus2} and \ref{theo:rep.plus} with several explicit examples.
The following lemma shows a case where $S(g)$ is convex and $C^+(g)$ is empty:
\begin{lemma}\label{lem:convex.nonemptyinter}
Let $g=(g_1,\dots,g_m)$ with $g_j\in\R[x]$.
Assume that each $g_j$ is concave and $S(g)$ has nonempty interior.
Then $S(g)$ is convex and $C^+(g)=\emptyset$.
\end{lemma}
\begin{proof}
It is a simple matter to prove that $S(g)$ is convex.
Assume by contradiction that there is $x\in C^+(g)\cap S(g)$.
Since $x\in C^+(g)$, by \eqref{eq:equi.Cplus}, there exists $\lambda\in\R^{m}\backslash\{0\}$ such that $\sum_{j=1}^m \lambda_j^2 \nabla g_j( x)=0$ and $\lambda_j^2 g_j =0$.
Set $G(x)=\sum_{j=1}^m \lambda_j^2 g_j( x)$.
Then $G$ is concave since all $g_j$ is concave.
In addition, $\nabla G(x)=0$ yields that $G(x)=0$ is the maximal value of $G$.
Let $a$ be in the interior of $S(g)$.
Then $0\ge G(a)=\sum_{j=1}^m \lambda_j^2 g_j(a)$ implies that $\lambda_1=\dots=\lambda_m=0$ since all $g_j(a)$ are positive.
This contradicts $\lambda\ne 0$, and hence it holds that $C^+(g)=\emptyset$.
\end{proof}
The following lemma is a consequence of Lemma \ref{lem:convex.nonemptyinter} and Theorem \ref{theo:rep.plus}:
\begin{lemma}\label{lem:rep.convex.nomempinter}
Let $f,g_1,\dots,g_m\in\R[x]$.
Assume that $f$ is non-negative on $S(g)$ with $g:=(g_1,\dots,g_m)$, each $g_j$ is concave and $S(g)$ has nonempty interior.
Then $S(g)$ is convex and there exists $q\in P(g)[x,\bar \lambda]$ such that $f-q$ vanishes on $V(h_\text{FJ}^+)$, where $\bar\lambda:=(\lambda_0,\dots,\lambda_m)$ and $h_\text{FJ}^+$ is defined as in \eqref{eq:.polyFJ.plus}.
Moreover, if $S(g)$ satisfies the Archimedean condition, we can take $q\in Q(g)[x,\bar \lambda]$.
\end{lemma}
To illustrate the representations in Lemma \ref{lem:rep.convex.nomempinter}, see \cite[Examples 6, 7 and 8]{mai2022exact}.
Note that if $C(g)=\emptyset$, then so is $C^+(g)$ since $C^+(g)\subset C(g)$.
Contrary to Lemma \ref{lem:rep.convex.nomempinter}, the following example shows the representation of any polynomial non-negative on a non-convex semi-algebraic set $S(g)$:
\begin{example}\label{exam:infinite.critical}
Consider the problem \eqref{eq:pop} with $n=2$, $m=3$ and $g=(g_1,g_2,g_3)=(x_1+1,1-x_2^2,1-(x_1-1)^2-x_2^2)$.
Then $S(g)$ is non-convex since it contains all points in the hypercube $[-1,1]^2$ but not in the open ball of center $(1,0)$ with unit radius.
The condition $\rank^+(\varphi^g(x))< m$ can be expressed as
\begin{equation}
\begin{cases}
\exists \lambda\in \R_+^3\backslash\{0\}\,:\,\sum_{j=1}^3\lambda_j\nabla g_j(x)=\lambda_1\begin{bmatrix}
1\\
0
\end{bmatrix}+\lambda_2\begin{bmatrix}
0\\
-2x_2
\end{bmatrix}+\lambda_3\begin{bmatrix}
2(x_1-1)\\
-2x_2
\end{bmatrix}=0\,,\\
\lambda_1g_1(x)=\lambda_1(x_1+1)=0\,,\,\lambda_2g_2(x)=\lambda_2(1-x_2^2)=0\,,\\
\lambda_3g_3(x)=\lambda_3(1-(x_1-1)^2-x_2^2)=0\,.
\end{cases}
\end{equation}
It implies that $\lambda_1=0$ or $x_1=-1$.
If $\lambda_1=0$, $2\lambda_3(x_1-1)=0$ which gives $\lambda_3=0$ or $x_1=1$.
If $\lambda_1=\lambda_3=0$, $\lambda_2\ne 0$ which implies $-2x_2=0$ and $1-x_2^2=0$, hence this is impossible.
If $\lambda_1=0$ and $x_1=1$, then $-2(\lambda_2+\lambda_3)x_2=0$ and $\lambda_2(1-x_2^2)=\lambda_3(1-x_2^2)=0$.
If $\lambda_1=0$, $x_1=1$ and $x_2=0$, then $\lambda_2=\lambda_3=0$ which contradicts $\lambda\ne 0$.
If $\lambda_1=0$, $x_1=1$ and $\lambda_2+\lambda_3=0$, then $\lambda_2=\lambda_3=0$ (since $\lambda_j\ge 0$) which also contradicts $\lambda\ne 0$.
Thus we get $x_1=-1$.
Then $\lambda_1-4\lambda_3=0$, $-2(\lambda_2+\lambda_3)x_2=0$, $\lambda_2(1-x_2^2)=0$ and $\lambda_3(-3-x_2^2)=0$.
It implies $\lambda_3=0$, so $\lambda_1=0$, $\lambda_2x_2=0$ and $\lambda_2(1-x_2^2)=0$.
Since $\lambda_1=\lambda_3=0$, we obtain $\lambda_2>0$ which gives $x_2=0=1-x_2^2$.
This is impossible, hence $C^+(g)=\emptyset$ which implies that $f(C^+(g))=\emptyset$.
By Theorem \ref{theo:rep.plus}, if $f$ is non-negative on $S(g)$, there exists $q\in P(g)[x,\bar \lambda]$ and $f-q$ vanishes on $V(h_\text{FJ}^+)$.
\end{example}
In the following example, we reconsider \cite[Example 10]{mai2022exact} which the previous result (Theorem \ref{theo:rep.old}) is inapplicable to but the the new one (Theorem \ref{theo:rep}) is applicable to:
\begin{example}\label{exam:counterexample}
Consider the problem \eqref{eq:pop} with $n=2$, $m=2$, $f=x_1+x_2$ and $g=(-x_1^2,-x_2^2)$.
Then we get $f^\star=0$, $S(g)=\{(0,0)\}$ and $(0,0)$ is the unique global minimizer for this problem.
We get $C(g)=\{(t,0),(0,t)\,:\,t\in\R\}$ which gives that $f(C(g))=\R$ is infinite.
However $C(g)\cap S(g)=\{(0,0)\}$ implies $f(C(g)\cap S(g))=\{0\}$ is finite.
Let $h_\text{FJ}$ be as in \eqref{eq:.polyFJ}.
Then
\begin{equation}
V(h_\text{FJ})=\{(t,0,0,0,\pm 1),(0,t,0,\pm 1,0),(0,0,0,\cos{t},\sin{t})\,:\,t\in\R\}\,.
\end{equation}
By Theorem \ref{theo:rep}, there exists $q\in P(g)[x,\bar \lambda]$ such that $f-q$ vanishes on
\begin{equation}
(S(g)\times\R^{m+1})\cap V(h_\text{FJ})=\{(0,0,0,\cos{t},\sin{t})\,:\,t\in\R\}\,.
\end{equation}
It is not hard to take $q=0$.
\end{example}
The following lemma shows a case where $C^+(g)$ is a singleton:
\begin{example}\label{exam:nonkkt}
Consider the problem \eqref{eq:pop} with $n=2$, $m=3$, $f=x_1-1$ and $g=(g_1,g_2,g_3)=(x_1,x_2,(x_1-1)^3-x_2)$.
It is easy to check that $S(g)$ is non-convex and $f^\star=0$.
Moreover, the point $(1,0)$ is the unique global minimizer for this problem and Karush--Kuhn--Tucker conditions do not hold for \eqref{eq:pop} at this point.
The condition $\rank^+(\varphi^g(x))< m$ can be expressed as
\begin{equation}
\begin{cases}
\exists \lambda\in \R_+^3\backslash\{0\}\,:\,\sum_{j=1}^3\lambda_j\nabla g_j(x)=\lambda_1\begin{bmatrix}
1\\
0
\end{bmatrix}+\lambda_2\begin{bmatrix}
0\\
1
\end{bmatrix}+\lambda_3\begin{bmatrix}
3(x_1-1)^2\\
-1
\end{bmatrix}=0\,,\\
\lambda_1g_1(x)=\lambda_1 x_1=0\,,\,\lambda_2g_2(x)=\lambda_2 x_2=0\,,\,\lambda_3g_3(x)=\lambda_3 ((x_1-1)^3-x_2)=0\,.
\end{cases}
\end{equation}
It implies that $\lambda_3=\lambda_2$ so $\lambda_1+3\lambda_2(x_1-1)^2=0$.
It follows that $\lambda_1=0$ and $\lambda_2(x_1-1)=0$.
If $\lambda_2=0$, then $\lambda=0$, hence this is impossible.
Thus we get $\lambda_2>0$ which gives $x_1=1$ and $x_2=0$.
Thus $C^+(g)=\{(1,0)\}$ is a singleton then so is $f(C^+(g))$.
From this, Theorem \ref{theo:rep.plus} shows that there exists $q\in P(g)[x,\bar \lambda]$ with $\bar\lambda=(\lambda_0,\lambda_1)$ such that $f-q$ vanishes on $V(h_\text{FJ}^+)$ with $h_\text{FJ}^+$ defined as in \eqref{eq:.polyFJ.plus}, namely
\begin{equation}
\begin{array}{rl}
h_\text{FJ}^+=&(\lambda_0^2\begin{bmatrix}
1\\
0
\end{bmatrix}-\lambda_1^2\begin{bmatrix}
1\\
0
\end{bmatrix}-\lambda_2^2\begin{bmatrix}
0\\
1
\end{bmatrix}-\lambda_3^2\begin{bmatrix}
3(x_1-1)^2\\
-1
\end{bmatrix},\\\\
&\lambda_1^2 x_1,\lambda_2^2x_2,\lambda_3^2 ((x_1-1)^3-x_2),1-\sum_{j=0}^3\lambda_j^2)\,.
\end{array}
\end{equation}
Let $(x,\bar\lambda)\in V(h_\text{FJ}^+)$.
We get $\lambda_3^2=\lambda_2^2$, so $\lambda_0^2-\lambda_1^2-3\lambda_2^2(x_1-1)^2=0$ and $\lambda_1^2x_1=\lambda_2^2x_2=\lambda_2^2 ((x_1-1)^3-x_2)=0$.
If $\lambda_2=0$, then $\lambda_0^2=\lambda_1^2=\frac{1}{2}$ and $x_1=0$.
If $\lambda_2\ne 0$, then $x_2=0$, $x_1=1$ and $\lambda_1=\lambda_0=0$ which gives $\lambda_2^2=\lambda_3^2=\frac{1}{2}$.
Thus we obtain
\begin{equation}
V(h_\text{FJ}^+)=\{(0,t,\frac{1}{\sqrt{2}}r_1,\frac{1}{\sqrt{2}}r_2,0,0),(1,0,0,0,\frac{1}{\sqrt{2}}r_1,\frac{1}{\sqrt{2}}r_2)\,:\,r_j=\pm 1\,,\,t\in\R\}\,.
\end{equation}
It is not hart to check that $q=(x_1-1)^3=g_2+g_3\in Q(g)[x,\bar \lambda]\subset P(g)[x,\bar \lambda]$ and $f-q$ vanishes on $V(h_\text{FJ}^+)$.
\end{example}
\begin{remark}
From Lemma \ref{lem:convex.nonemptyinter} and Examples \ref{exam:infinite.critical}, \ref{exam:nonkkt}, if $C^+(g)$ is finite, then there are cases where $S(g)$ is convex, and there are cases where $S(g)$ is non-convex.
This implies that the finiteness of $C^+(g)$ does not depend on the convexity of $S(g)$.
Thus it is still open to find all explicit cases of $g$ where $C^+(g)$ is finite.
\end{remark}
Next we show in the following example that Theorems \ref{theo:rep}, \ref{theo:rep.plus2} and \ref{theo:rep.plus} are inapplicable to \cite[Example 4]{mai2022exact}:
\begin{example}\label{exam:infinite.critical2}
Consider the problem \eqref{eq:pop} with $n=2$, $m=1$, $f=x_1+x_2^2$ and $g=(g_1)=(-x_1^2)$.
Then the point $(0,0)$ is the unique global minimizer.
The condition $\rank^+(\varphi^g(x))< m$ can be expressed as
\begin{equation}
\exists \lambda_1\in \R_+\backslash\{0\}\,:\,\lambda_1\nabla g_1(x)=\lambda_1\begin{bmatrix}
-2x_1\\
0
\end{bmatrix}=0\,,\,\lambda_1g_1(x)=-\lambda_1 x_1^2=0\,.
\end{equation}
It is equivalent to $x_1=0$.
Thus $C(g)=C^+(g)=S(g)=\{(0,t)\,:\,t\in\R\}$ is infinite which implies that
\begin{equation}
f(C(g))=f(C^+(g))=f(C(g)\cap S(g))=f(C^+(g)\cap S(g))=\{t^2\,:\,t\in\R\}=[0,\infty)
\end{equation}
is infinite.
\end{example}
\section{Exact polynomial optimization}
\label{sec:application}
\subsection{Moment-SOS relaxations}
In this subsection we recall some preliminaries of the Moment-SOS relaxations originally developed by Lasserre in \cite{lasserre2001global}.
Given $d\in\N$, let $\N^n_d:=\{\alpha\in\N^n\,:\,\sum_{j=1}^n \alpha_j\le d\}$.
Given $d\in\N$, we denote by $v_d$ the vector of monomials in $x$ of degree at most $d$, i.e., $v_d=(x^\alpha)_{\alpha\in\N^n_d}$ with $x^\alpha:=x_1^{\alpha_1}\dots x_n^{\alpha_n}$.
For each $p\in\R[x]_d$, we write $p=c(p)^\top v_d=\sum_{\alpha\in\N^n_d}p_\alpha x^\alpha$, where $c(p)$ is denoted by the vector of coefficient of $p$, i.e., $c(p)=(p_\alpha)_{\alpha\in\N^n_d}$ with $p_\alpha\in\R$.
Given $A\in\R^{r\times r}$ being symmetric, we say that $A$ is positive semidefinite, denoted by $A\succeq 0$, if every eigenvalue of $A$ is non-negative.
Given $y=(y_\alpha)_{\alpha\in\N^n}\subset \R$, let $L_y:\R[x]\to\R$ be the Riesz linear functional defined by $L_y(p)=\sum_{\alpha\in\N^n} p_\alpha y_\alpha$ for every $p\in\R[x]$.
Given $d\in\N$, $p\in\R[x]$ and $y=(y_\alpha)_{\alpha\in\N^n}\subset \R$, let $M_d(y)$ be the moment matrix of order $d$ defined by $(y_{\alpha+\beta})_{\alpha,\beta\in\N^n_d}$ and let $M_d(py)$ be the localizing matrix of order $d$ associated with $p$ defined by $(\sum_{\gamma\in\N^n}p_\gamma y_{\alpha+\beta+\gamma})_{\alpha,\beta\in\N^n_d}$.
Given $g_1,\dots,g_m\in\R[x]$, let $Q_d(g)[x]$ be the truncated quadratic module of order $d$ associated with $g=(g_1,\dots,g_m)$ defined by
\begin{equation}
Q_d(g)[x]=\{\sigma_0+\sum_{j=1}^m\sigma_jg_j\,:\,\sigma_j\in\Sigma^2[x]\,,\,\deg(\sigma_0)\le 2d\,,\,\deg(\sigma_jg_j)\le 2d\}\,.
\end{equation}
Given $h_1,\dots,h_l\in\R[x]$, let $I_d(h)$ be the truncated ideal of order $d$ associated with $h=(h_1,\dots,h_l)$ defined by
\begin{equation}
I_d(h)[x]=\{\sum_{j=1}^l\psi_jh_j\,:\,\psi_j\in\R[x]\,,\,\deg(\psi_jh_j)\le 2d\}\,.
\end{equation}
Given $k\in\N$ and $f,g_1,\dots,g_m,h_1,\dots,h_l\in\R[x]$, consider the following primal-dual semidefinite programs associated with $f$, $g=(g_1,\dots,g_m)$ and $h=(h_1,\dots,h_l)$:
\begin{equation}\label{eq:mom.relax}
\begin{array}{rl}
\tau_k(f,g,h):=\inf\limits_y& L_y(f)\\
\text{s.t} &M_k(y)\succeq 0\,,\,M_{k-d_j}(g_jy)\succeq 0\,,\,j=1,\dots,m\,,\\
&M_{k-r_t}(h_ty)=0\,,\,t=1,\dots,l\,,\,y_0=1\,,
\end{array}
\end{equation}
\begin{equation}\label{eq:sos.relax}
\begin{array}{rl}
\rho_k(f,g,h):=\sup\limits_{\xi,G_j,u_t} & \xi\\
\text{s.t} & G_j\succeq 0\,,\\
&f-\xi=v_k^\top G_0v_k+\sum_{j=1}^m g_jv_{k-d_j}^\top G_jv_{k-d_j}\\
&\qquad\qquad+\sum_{t=1}^l h_tu_t^\top v_{2r_t}\,,\\
\end{array}
\end{equation}
where $d_j=\lceil \deg(g_j)/2\rceil$ and $r_t=\lceil \deg(h_t)/2\rceil$.
Using \cite[Lemma 15]{mai2022exact}, we obtain
\begin{equation}\label{eq:equi.sos}
\rho_k(f,g,h):=\sup_{\xi\in\R}\{ \xi\,:\,f-\xi\in Q_k(g)[x]+I_k(h)[x]\}\,.
\end{equation}
Primal-dual semidefinite programs \eqref{eq:mom.relax}-\eqref{eq:sos.relax} is known as the Moment-SOS relaxations of order $k$ for problem
\begin{equation}\label{eq:pop.equality}
\bar f^\star:=\inf\limits_{x\in S(g)\cap V(h)} f(x)\,.
\end{equation}
We state in the following lemma some recent results involving the Moment-SOS relaxations:
\begin{lemma}\label{lem:mom.sos}
Let $f,g_1,\dots,g_m,h_1,\dots,h_l\in\R[x]$.
Let $\bar f^\star$ be as in \eqref{eq:pop.equality} with $g=(g_1,\dots,g_m)$ and $h=(h_1,\dots,h_l)$.
Then the following statements hold:
\begin{enumerate}
\item For every $k\in\N$, $\tau_k(f,g,h)\le \tau_{k+1}(f,g,h)$ and $\rho_k(f,g,h)\le \rho_{k+1}(f,g,h)$.
\item For every $k\in\N$, $\rho_k(f,g,h)\le \tau_{k}(f,g,h)\le \bar f^\star$.
\item If $S(g)\cap V(h)$ has non-empty interior, for $k\in\N$ sufficient large, the Slater condition holds for the Moment relaxation \eqref{eq:mom.relax} of order $k$.
\item If $S(g)\cap V(h)$ satisfies the Archimedean condition, $\rho_k(f,g,h)\to \bar f^\star$ as $k\to \infty$.
\item If there exists $R>0$ such that $g_m+h_l=R-x_1^2-\dots-x_n^2$, for $k\in\N$ sufficient large, the Slater condition holds for the SOS relaxation \eqref{eq:sos.relax} of order $k$.
\item If there exists $q\in Q(g)[x]$ such that $f-\bar f^\star-q$ vanishes on $V(h)$, then there exists $k\in\N$ such that $\rho_k(f,g,h)=\bar f^\star$.
\item If there exists $q\in P(g)[x]$ such that $f-\bar f^\star-q$ vanishes on $V(h)\cap S(g)$, then there exists $k\in\N$ such that $\rho_k(f,\Pi g,h)=\bar f^\star$.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof of the first six statements can be found in \cite[Lemma 16]{mai2022exact}.
The final statement is proved similarly to the fifth statement.
It is sketched as follows:
Let $u=f-\bar f^\star-q$.
By assumption,we get $u=0$ on $S(g)\cap V(h)$.
From this, Krivine--Stengle's Nichtnegativstellens\"atz \cite{krivine1964anneaux} say that there exist a positive integer $r$ and $w \in P(g)[x]$ such that $u^{2r} + w \in I(h)$.
Let $c=\frac{1}{2r}$.
Then it holds that $1+t+ct^{2r}\in\Sigma^2[t]$.
Thus for all $\varepsilon>0$, we have
\begin{equation}
f-\bar f^\star+\varepsilon=q + \varepsilon(1+\frac{u}{\varepsilon}+c\left(\frac{u}{\varepsilon}\right)^{2r})-c\varepsilon^{1-2r}(u^{2r} + w) +c\varepsilon^{1-2r}\sigma\in P(g)[x]+I(h)[x]\,.
\end{equation}
Moreover, the degree of the right hand side has an upper bound independent from $\varepsilon$.
This implies that there exists $k\in\N$ such that for all $\varepsilon>0$, $f-\bar f^\star+\varepsilon \in P_k(g)[x]+I_k(h)[x]$.
Then we for all $\varepsilon>0$, $\bar f^\star-\varepsilon$ is a feasible solution of \eqref{eq:equi.sos} of the value $\rho_k(f,\Pi g,h)$.
It gives $\rho_k(f,\Pi g,h)\ge \bar f^\star-\varepsilon$, for all $\varepsilon>0$, and, in consequence, we get $\rho_k(f,\Pi g,h)\ge \bar f^\star$.
Using the third statement, we obtain that $\rho_k(f,\Pi g,h)= \bar f^\star$, yielding the final statement.
\end{proof}
\begin{remark}
In Lemma \ref{lem:mom.sos}, if $m\ge 2$ and there exists $q\in Q(g)[x]$ such that $f-\bar f^\star-q$ vanishes on $V(h)\cap S(g)$, we are not sure that $\rho_k(f, g,h)=\bar f^\star$ for some $k\in\N$.
\end{remark}
As a consequence of Lemma \ref{lem:FJ}, the following lemma is obtained:
\begin{lemma}\label{lem:equi.prob}
Let $f,g_1,\dots,g_m\in\R[x]$.
Let $f^\star$ be as in problem \eqref{eq:pop} with $g=(g_1,\dots,g_m)$.
Let $h_\text{FJ}^+$ be as in \eqref{eq:.polyFJ.plus}.
If problem \eqref{eq:pop} has a global minimizer, then it holds that
\begin{equation}\label{eq:eqi.POP}
\begin{array}{rl}
f^\star:=\min\limits_{x,\bar\lambda}& f(x)\\
\text{s.t.}& x\in S(g)\,,\,(x,\bar\lambda)\in V(h_\text{FJ}^+)\,.
\end{array}
\end{equation}
\end{lemma}
We present in the following two theorems the main application of Theorems \ref{theo:rep.old}, \ref{theo:rep}, \ref{theo:rep.plus} and \ref{theo:rep.plus2} to polynomial optimization:
\begin{theorem}\label{theo:pop}
Let $f,g_1,\dots,g_m\in\R[x]$.
Let $f^\star$ be as in problem \eqref{eq:pop} with $g=(g_1,\dots,g_m)$.
Let $h_\text{FJ}$ be as in \eqref{eq:.polyFJ} and $h_\text{FJ}^+$ be as in \eqref{eq:.polyFJ.plus}.
Assume that problem \eqref{eq:pop} has at least one global minimizer.
Let $\Pi g$ be as in \eqref{eq:prod.g}.
Then the following statements hold:
\begin{enumerate}
\item If $f(C(g)\cap S(g))$ is finite, there exists $k\in\N$ such that $\rho_k(f,\Pi g,h_\text{FJ})=f^\star$.
\item If $f(C^+(g)\cap S(g))$ is finite, there exists $k\in\N$ such that $\rho_k(f,\Pi g,h_\text{FJ}^+)=f^\star$.
\end{enumerate}
\end{theorem}
\begin{remark}
Since $C^+(g)\cap S(g)\subset C^+(g)$, the second statement of Theorem \ref{theo:pop} holds if $f(C^+(g))$ is finite.
\end{remark}
The proof of Theorem \ref{theo:pop}, which relies on Theorems \ref{theo:rep}, \ref{theo:rep.plus2} and the final statement of Lemma \ref{lem:mom.sos}, is similar to the one of \cite[Theorem 3]{mai2022exact}.
\begin{theorem}\label{theo:pop2}
Let $f,g_1,\dots,g_m\in\R[x]$.
Let $f^\star$ be as in problem \eqref{eq:pop} with $g=(g_1,\dots,g_m)$.
Let $h_\text{FJ}$ be as in \eqref{eq:.polyFJ} and $h_\text{FJ}^+$ be as in \eqref{eq:.polyFJ.plus}.
Assume that problem \eqref{eq:pop} has at least one global minimizer and $S(g)$ satisfies the Archimedean condition.
Then the following statements hold:
\begin{enumerate}
\item If $f(C(g))$ is finite, there exists $k\in\N$ such that $\rho_k(f,g,h_\text{FJ})=f^\star$.
\item If $f(C^+(g))$ is finite, there exists $k\in\N$ such that $\rho_k(f,g,h_\text{FJ}^+)=f^\star$.
\end{enumerate}
Furthermore, if these exists $R>0$ such that $g_m=R-x_1^2-\dots-x_n^2$, for $k\in\N$ sufficient large, the Slater condition holds for the SOS relaxation \eqref{eq:sos.relax} of order $k$ with $h=h_\text{FJ}$ or $h=h_\text{FJ}^+$.
\end{theorem}
The proof of Theorem \ref{theo:pop2}, which relies on Theorems \ref{theo:rep.old} and \ref{theo:rep.plus} together with the fifth and sixth statements of Lemma \ref{lem:mom.sos}, is similar to the one of \cite[Theorem 3]{mai2022exact}.
Combining Lemma \ref{lem:rep.convex.nomempinter} and the sixth statement of Lemma \ref{lem:mom.sos}, we obtain the following corollary:
\begin{corollary}\label{coro:ball.box.simplex}
Let $f,g_1,\dots,g_m\in\R[x]$.
Let $f^\star$ be as in problem \eqref{eq:pop} with $g=(g_1,\dots,g_m)$.
Let $h_\text{FJ}^+$ be as in \eqref{eq:.polyFJ.plus}.
Assume that problem \eqref{eq:pop} has at least one global minimizer, each $g_j$ is concave and $S(g)$ has nonempty interior.
Then $S(g)$ is convex and there exists $k\in\N$ such that $\rho_k(f,\Pi g,h_\text{FJ}^+)=f^\star$ with $\Pi g$ being defined as in \eqref{eq:prod.g}.
Moreover, if $S(g)$ satisfies the Archimedean condition, there exists $k\in\N$ such that $\rho_k(f,g,h_\text{FJ}^+)=f^\star$.
\end{corollary}
\subsection{Numerical examples}
\label{sec:num}
In this subsection we report numerical results obtained by solving the SOS relaxations of the values $\rho_k(f,g,0)$ and $\rho_k(f,g,h_\text{FJ})$ for problem \eqref{eq:pop}, where $h_\text{FJ}$ is defined as in \eqref{eq:.polyFJ}.
The first one is the standard semidefinite program in \cite{lasserre2001global} while the second one is the semidefinite program modeled by our method in this paper.
We indicate the data of each semidefinite program, namely ``value", ``time" and ``size"
correspond to the numerical value of the optimal value of the semidefinite program, the running time in seconds to obtain this numerical value and the size of the semidefinite program, respectively.
Here the size of the semidefinite program includes the largest matrix size, the number of affine constraints, the number of scalar variables and the number of matrix variables.
The experiments are performed in Julia 1.3.1 with the softwares TSSOS \cite{wang2021tssos} and Mosek 9.1 \cite{mosek2010mosek}.
We use a desktop computer with an Intel(R) Core(TM) i7-8665U CPU @ 1.9GHz $\times$ 8 and 31.2 GB of RAM.
Our test problem is taken from Example \ref{exam:nonkkt}, namely $n=2$, $m=3$, $f=x_1-1$ and $g=(g_1,g_2,g_3)=(x_1,x_2,(x_1-1)^3-x_2)$.
It is clear that $f^\star=0$ which is attained at the unique global minimizer $(1,0)$ for problem \eqref{eq:pop}.
Moreover, the Karush--Kuhn--Tucker conditions do not hold at this minimizer.
By using \cite[Proposition 3.4]{nie2014optimality}, we get $\rho_k(f,g,0)< f^\star$ for all $k\in\N$.
As shown in Example \ref{exam:nonkkt}, there exists $q\in Q(g)[x,\bar \lambda]$ and $f-q$ vanishes on $V(h_\text{FJ}^+)$.
From this, the sixth statement of Theorem \ref{lem:mom.sos} yields that $\rho_d(f,g,h_\text{FJ}^+)=f^\star$ for some $d\in\N$.
Thus we obtain $\rho_k(f,g,0)<f^\star=\rho_d(f,g,h_\text{FJ}^+)$, for all $k\in\N$.
Since $h_\text{FJ}^+$ has sign symmetry at $\bar \lambda$, we use TSSOS to exploit this structure when computing $\rho_k(f,g,h_\text{FJ}^+)$.
We display the numerical results in Table \ref{tab:bench}.
\begin{table}
\footnotesize
\caption{\footnotesize Lower bounds on $f^\star=0$}
\label{tab:bench}
\begin{center}
\begin{tabular}{ cccc }
\hline
\multicolumn{4}{c}{$\rho_k(f,g,h_\text{FJ}^+)$} \\
\hline
$k$& value & time & size\\
$3$ &$-0.38154$ & 0.09& (21,186,216,19)\\
$4$&$-0.01199$ & 0.38& (42,476,1056,54)\\
$5$ &$-0.00822$ & 2.07& (84,1110,4393,100)\\
\hline
\multicolumn{4}{c}{$\rho_k(f,g,0)$}\\
\hline
$k$&value & time & size \\
$13$&$-0.02419$& 1.27 & (105,378,1,4)\\
$14$&$-0.02254$& 2.08 & (120,435,1,4)\\
$15$&$-0.02254$& 3.13 & (136,496,1,4)\\
\hline
\multicolumn{4}{c}{$\rho_k(f,(g,b),0)$} \\
\hline
$k$&value & time & size \\
$13$&$-0.01490$& 1.68 & (105,378,1,5)\\
$14$&$-0.01305$& 2.43 & (120,435,1,5)\\
$15$&$-0.01267$& 3.03 & (136,496,1,5)\\
\end{tabular}
\end{center}
\end{table}
It shows that the numerical value $-0.00822$ of $\rho_5(f,g,h_\text{FJ}^+)$ is the best lower bound on $f^\star$.
It takes around $2$ seconds to obtain this numerical value.
It is worth pointing out that the standard SOS relaxations of the values $\rho_k(f,g,0)$ and $\rho_k(f,(g,b),0)$ cannot reach the bound $-0.00822$ in less than $3$ seconds in spite of using the additional ball constraint $b=2-x_1^2-x_2^2$.
Another observation is that the size of the SOS relaxations of the value $\rho_k(f,g,h_\text{FJ}^+)$ grows faster than the ones of the values $\rho_k(f,g,0)$ and $\rho_k(f,(g,b),0)$ when $k$ increases and even when the sparsity is exploited.
\paragraph{Acknowledgements.}
The author was supported by the MESRI funding from EDMITT.
\bibliographystyle{abbrv}
\input{GradientIdeals.bbl}
\end{document}
| 159,229
|
11 types of Uber and Grab driver you'll meet in Southeast Asia
It is always a lot easier to use ride-hailing services like Uber and Grab when going out on the weekends or when heading somewhere nearby.
For starters, you don’t have to worry about finding parking lots, or you don’t have feel stress in a traffic jam, ‘cos that’s the driver’s job. All you need to do is sit back and relax comfortably inside the car.
But of course, there are times when your drivers will attempt to communicate and be nice with you in order to get on your good sides, and this is also because it’s part of their job.
As much as we love our Uber and Grab drivers, sometimes, certain drives can go a little bit overboard with their friendliness (or unfriendliness).
See if you’re familiar with these drivers below.
1. The chatterbox
This kind of driver is usually very friendly and helpful. They can talk about anything from A to Z, and let’s just say, throughout the journey, their mouths are never shut. The shy and quiet passengers might find it hard to deal with this kind of driver, but they usually have no ill intention, just overly friendly and talkative.
2. The silent type
Contrary to the chatterbox, this driver is a quiet and shy type, and wouldn’t spout a word throughout the journey. The better ones would at least turn on the radio, while others entertain you with the sound of car engine. Even if you try to talk to them, at most, they would reply only in a single, short, sentence. So take a hint – they don’t want to talk!
3. The flirt
Similar to the chatterbox, this driver talks with a motive. They usually started off with common questions like “What’s your job?” or “Where are you from?”, and then move on to more personal questions like “Are you married?” or “Are you seeing anyone?”. So if you received those kinds of questions, it is obvious that the driver is into you, so be careful!
4. The newbie
This driver is always lost, and sometimes he or she would in turn ask you how the app works. When you get out of the car, they would also ask if you’re using cash or card, even though it’s already indicated in the app. So from there, you can assume that the driver is new, though they would usually admit it themselves.
5. The salesman
Also similar to the chatterbox and the flirt, this driver would try to gain your affection but with a different motive. They would start off by telling you their own stories and why they become Grab or Uber drivers, and then start to tell you about their other jobs with the hope that you would support them. These drivers are usually credit-card or insurance salesman.
6. The flatterer
This driver would endlessly throw compliments at you, not because they’re into you or anything, but because they want you to give them a five-star rating. They would constantly remind you to give them five stars even though their service might not be the best.
7. The whiner
Another talkative driver who can’t stop complaining. The driver would usually complain about probably other riders, drivers or the management of the apps. It’s quite a useful insight as you get to hear how the drivers gain from the service and what are their losses.
8. The absent mind
This driver is always lost too, but not because he or she is new, but it is because they are always not paying attention to their surroundings. Sometimes, they’re in daze, sometimes it’s because they’re too busy telling you stories or talking on the phones, and other times is because they forgot that they have a passenger at the back of their seat.
9. The GPS denier
The know-it-all! This driver knows every single corner, short-cuts, and long-cuts of every road. Usually it’s because they are former cab drivers or maybe because their main or former jobs require them to drive on the street a lot. So, they would keep telling you that they’re going the right way, even though your GPS shows another road – and most of the time, they are right!
10. The curious man
This driver is like an investigator. He or she likes to hear feedbacks from the customers. Instead of complaining, they want to know what you think of the service so that they can use as knowledge or to improve themselves.
11. The Grab/Uber hater
Since both Grab and Uber offer similar service, some drivers tend to get a little competitive. They would bash each other and claim that their service are always the better ones, and if you think otherwise, they would refute you like there’s no tomorrow!
| 257,876
|
- Yield
- 24 cookies
Ingredients
- ½ cup butter
- 1 cup sugar
- ¼ teaspoon baking soda
- ¼ teaspoon cream of tarter
- 1 egg
- ½ teaspoon vanilla
- 1½ cup all-purpose flour
- 2 tablespoons sugar
- 1 teaspoon ground cinnamon
Preparation
- In a mixing bowl beat the butter with an electric mixer for 30 seconds. Add the 1 cup sugar, baking soda, and cream of tarter. Beat till combined, scraping sides of bowl. Beat in the egg and vanilla till combined. Beat in as much of the flour as you can with the mixer. Combine the 2 tablespoon sugar and the cinnamon. Shape the dough into 1-inch balls. Roll balls in the sugar-cinnamon mix and place on the cookie sheet. With the bottom of a glass gently tap down the cookies. Bake at 375 oven for 10 to 11 minutes or till edges are golden.
Related Video
Nutritional Info
- Calories102
- Carbohydrates15 g(5%)
- Fat4 g(6%)
- Protein1 g(2%)
- Saturated Fat3 g(13%)
- Sodium16 mg(1%)
- Polyunsaturated Fat0 g
- Fiber0 g(1%)
- Monounsaturated Fat1 g
- Cholesterol17 mg(6%)
| 91,576
|
TITLE: Can I fix a point in Minkowski space to give it a vector space structure?
QUESTION [7 upvotes]: I looked up the term Minkowski space on Wikipedia. It said
There is an alternative definition of Minkowski space as an affine space which views Minkowski space as a homogenous space of the Poincaré group with the Lorentz group as the stabilizer.
In their book Metric Affine Geometry, Snapper and Troyer state on page 59:
It cannot be stressed enough that the affine space $X$ is not a vector space. Its points cannot be added and there is no way to multiply by scalars. No point in $X$ is preferred; they all play the same role. In particular, there is no point in $X$ which makes a better origin for a vector space than any other point.
The situation changes radically if we choose a point $c$ in $X$ and keep it fixed. It is now possible to make $X$ into a left vector space over $k$ by using the one-to-one mapping $f$ from $X$ onto $V$ defined by $f(x) = \overrightarrow{c,x}$ for each $x \in X$. All we do is carry the vector space structure of $V$ over to $X$ by means of the mapping $f$.
So here's my question:
As I understand it, it makes sense to think of Minkowski space as an affine space since the basic principle of Special Relativity is that no point in $X$ is a preferred reference frame. But does that mean it is then impossible to "fix" a point in $X$ as Snapper and Troyer say can be done? In other words, is there any physical meaning to the idea of fixing a point in the affine space or is that impossible according to SR?
Obviously I am trying to use a mathematician's idea to interpret what can be done physically with Minkowski space.
REPLY [6 votes]: Fixing a point is more or less like fixing a coordinate system on your affine space. Then you can identify $X$ with $V$ as stated in the book, where the fixed point $c\in X$ is mapped to the origin of $V$. In other words, fixing a point $c$ in $X$ is like glueing a copy of $V$ onto $X$ in such a way that $O\in V$ overlaps with $c\in X$. As far as the Lorentz group is considered then the coordinates (i.e. the component of the glued copy of $V$ onto $X$) really behave like vectors, but this is no longer the case under more general transformations (consider for instance translations, or the action of the ray inversion from the conformal group).
| 122,722
|
El fraude electoral ha quedado demostrado
- Collection
- Creator
-
- Creation Date
- between 1984 and 2009
- Geographic
-
- Topic
-
Format
View formats within this collection
-
- Language
- Spanish
- Digital Object Made Available By
Special Collections & Archives, UC San Diego, La Jolla, 92093-0175 ()
Unknown (M.
| 362,037
|
Return of the Nephilim
by Chuck Missler
Koinonia House: 1997. (audio only)
UFOs, Stonehenge, the Book of Revelations – what more can you ask for? Missler explains that Genesis chapter 6 means that when fallen angels were having sex with women, their offspring were monstrous ‘people’ called Nephilim.
He suggests that these Nephilim, or perhaps their spirits (demons?) are reappearing today, this time as UFOs and aliens. He also suggests that they have done so for thousands of years and were involved in building Stonehenge, the Great Pyramid and other monuments. Heathen legends which talk of intercourse between gods and humans are further evidence. CIA and other government cover-ups indicate our government is aware of the situation and is on the wrong side.
Although he does a good job talking about the Nephilim, he does not convincingly tie them to UFO appearances. He gives too much credence to wackos who claim alien abduction. He says that every single abductee has dabbled in the occult, and yet he takes them at face value when they claim to have had an embryo implanted. I always wonder what the babies are like when they’re born…?
| 69,216
|
A Procrastinator's Guide to Year-End Fundraising.
Reader Comments (2)
(please delete this comment after responding).
Sarah Sheard sheard at 3MilSys.com
| 356,917
|
Release information, detailed
5.6.2.43 / GUI 1.8.4
System
Updated: Production status condition for personalized token order from manufacturer.
Updated: Production status flags for token manufacturer.
Updated: TokenTemplateManager will sort certificate template items in ascending name order at select.
Updated: Database context.
Configuration
Added config.js parameters "startupLanguage" and "languageList" for possibility to override allowed languages and startup language (intended for English only)
GUI
Added:
\tas allowed character in type "text" policy
Added: About dialog with information of current Net iD Portal versions (API and GUI).
Added: Presentation of additional token serial numbers in search result.
Changed: Presentation of Token > Model. Will show ProfileLabel from server if available (instead of local token model name).
Changed: Small behavior change for config autoSwitchOrganizationTheme.
Updated: More formatting possibilities in string table for texts used in task information.
Fixed — API
Fixed: Issue with Column name when using UpdateTokenStatus against TimerService-Monitor module.
Fixed: Issue with External permission with EndEntity objects when using TimerService-Monitor module.
Fixed: Issue with Null exceptions when selecting from 'hist_tkns' database table.
Fixed: Issue with Permission when using SelfView privilege only and together with dynamic token attribute structure.
Fixed: Issue with Publish history tokens when expired when using TimerService.
Fixed: Issue with Select multiple card orders when send to manufacturer.
Fixed: Issue with Synchronize history tokens when upgrade database.
Fixed: Issue with opening tasks with status 'Wait'.
Fixed — GUI
Fixed: Administration selection by category for whitelists.
Fixed: Auto-activate OTP for bootstrap.
Fixed: Behavior for config 'logonOneTimePasswordWithUser'
Fixed: Default color scheme for SecMaker
Fixed: Policy-valid-marker for new-password-field.
Fixed: Removed Microsoft special UTF-8 tags in begining of some files
Fixed: Local task action 'PluginCloseToken'.
| 189,887
|
TITLE: Why is the solution of Clairaut's Differential Equation valid?
QUESTION [3 upvotes]: Clairaut's Differential Equation is:
$$y=xy^\prime+f(y^\prime)$$
where $f$ is supposed to be continuously differentiable.
Every proof for the solution of this equation that I have seen starts by differentiating both sides of this equation; however, this makes $y^{\prime\prime}$ appear.
Why can we assume that $y$ has a second derivative?
REPLY [1 votes]: The equation $y_0=px_0+f(p)$ need not have a solution $p$ for all initial values $(x_0,y_0)$. In these points there is no solution of the Clairaut differential equation.
Assume now that the initial point $(x_0,y_0)$ is well-behaved, that is a $p_0$ with $y_0=x_0p_0+f(p_0)$ exists and $x_0+f'(p_0)\ne 0$. Then by the implicit function theorem for the function
$$
F(x,y,p)=px+f(p)-y
$$
at $(x,y)\approx(x_0,y_0)$ the equation $$F=0\iff y=xp+f(p)$$ is solvable for $p\approx p_0$, and the local solution $p=g(x,y)$ is as smooth as $f$. That is, as $f\in C^k$, with $k\ge 1$, giving $F\in C^k$, so is $g\in C^k$ (locally).
The solution of $y'=g(x,y)$ now in consequence is $C^{k+1}$, so at least $C^2$, as long as it stays in the neighborhood of $(x_0,y_0)$.
Note that where the linear solutions $y=cx+f(c)$ meet the singular solution of $x+f'(y')=0$, which can be parametrized as $$x(p)=-f'(p), ~~ y(p)=f(p)-f'(p)p,$$ one can switch the branches of solutions and at that point the second derivative jumps.
| 129,183
|
\chapter{On special quadratic birational transformations whose base locus has dimension at most three}\label{chapter: transformations whose base locus has dimension at most three}
In this chapter we continue the study of special quadratic birational transformations
$\varphi:\PP^n\dashrightarrow\sS:=\overline{\varphi(\PP^n)}\subseteq\PP^{N}$
started in Chapter \ref{sec: transformations into a hypersurface},
by reinterpreting techniques and well-known results on
special Cremona transformations.
While in Chapter \ref{sec: transformations into a hypersurface} we required that $\sS$ was a hypersurface,
here we allow more freedom in the choice of $\sS$,
but we only treat the case in which
the dimension of the base
locus $\B$ is $r=\dim(\B)\leq3$.
Note that for every closed subscheme $X\subset\PP^{n-1}$
cut out by the quadrics containing it, we can
consider $\PP^{n-1}$ as a hyperplane in $\PP^n$
and hence $X$ as a subscheme of $\PP^n$. So
the linear system $|\I_{X,\PP^n}(2)|$ of all quadrics in $\PP^n$
containing $X$ defines a quadratic rational map
$\psi:\PP^n\dashrightarrow\PP^N$
($N=h^0(\I_{X,\PP^n}(2))-1=n+h^0(\I_{X,\PP^{n-1}}(2))$),
which is birational onto the image
and whose inverse is defined by linear forms,
i.e. $\psi$ is of type $(2,1)$.
Conversely, every birational transformation
$\psi:\PP^n\dashrightarrow\overline{\psi(\PP^n)}\subseteq\PP^N$
of type $(2,1)$ whose image is nondegenerate, normal and linearly normal
arise in this way.
From this it follows that there are many (special) quadratic transformations.
However,
when the image $\sS$ of the
transformation $\varphi$
is sufficiently regular,
by straightforward generalization of Proposition \ref{prop: dimension formula},
we obtain
strong numerical and geometric restrictions on the base locus $\B$.
For example, as soon as $\sS$ is not too much singular,
the secant variety $\Sec(\B)\subset\PP^n$ has to be a hypersurface and $\B$
has to be a $QEL$-variety of type $\delta=\delta(\B)=2\dim(\B)+2-n$; in particular
$n\leq 2\dim(\B)+2$ and $\Sec(\B)$ is a hyperplane if and only if
$\varphi$ is of type $(2,1)$.
So the classification of
transformations $\varphi$ of type $(2,1)$ whose base locus
has dimension $\leq 3$
essentially follows
from classification results on
$QEL$-manifold:
Proposition \ref{prop: LQEL with delta=r and delta=r-1},
Theorem \ref{prop: classification CC-varieties} and
\cite[Theorems~4.10 and 7.1]{ciliberto-mella-russo}.
When $\varphi$ is of type $(2,d)$ with $d\geq2$, then $\Sec(\B)$
is a nonlinear hypersurface
and it is not so easy to exhibit examples.
The most difficult cases of this kind
are those for which $n=2r+2$ i.e. $\delta=0$.
In order to classify these transformations,
we proceed as in Propositions
\ref{prop: 2-fold in P6} and \ref{prop: 3-fold in P8 - S nonlinear}
(see also \cite{mella-russo-baselocusleq3}).
That is, we first determine the Hilbert
polynomial of $\B$ in Lemmas \ref{lemma: r=2 B nondegenerate} and
\ref{lemma: r=3 B nondegenerate},
by using
the usual Castelnuovo's argument, Castelnuovo's bound and
some refinement of Castelnuovo's bound
(see Chap. \ref{cap: castelnuovo theory});
consequently we deduce
Propositions \ref{prop: r=2 B nondegenerate}
and \ref{prop: r=3 B nondegenerate} by applying the classification
of smooth varieties of low degree:
\cite{ionescu-smallinvariants},
\cite{ionescu-smallinvariantsII},
\cite{ionescu-smallinvariantsIII},
\cite{fania-livorni-nine},
\cite{fania-livorni-ten},
\cite{besana-biancofiore-deg11},
\cite{ionescu-degsmallrespectcodim}.
We also apply the
double point formula in Lemmas:
\ref{lemma: double point formula r=2},
\ref{lemma: double point formula},
\ref{lemma: quadric fibration},
\ref{lemma: scroll over surface} and
\ref{lemma: scroll over curve},
in order to obtain additional informations on
$d$ and $\Delta=\deg(\sS)$.
We summarize our classification
results in Table \ref{tabella: all cases 3-fold}.
In particular, we provide
an answer to a question left open
in the recent preprint \cite{alzati-sierra}.
\section{Notation and general results}\label{sec: notation}
Throughout the chapter we work over $\CC$ and keep the following setting.
\begin{assumption}\label{assumption: base}
Let $\varphi:\PP^n\dashrightarrow\sS:=\overline{\varphi(\PP^n)}\subseteq\PP^{n+a}$ be
a quadratic birational transformation
with smooth connected base locus $\B$
and with $\sS$ nondegenerate,
linearly normal and factorial.
\end{assumption}
Recall that we can resolve the
indeterminacies of
$\varphi$ with the diagram
\begin{equation}
\xymatrix{ & \widetilde{\PP^n} \ar[dl]_{\pi} \ar[dr]^{\pi'}\\ \PP^n\ar@{-->}[rr]^{\varphi}& & \sS }
\end{equation}
where $\pi:\widetilde{\PP^n}=\Bl_{\B}(\PP^n)\rightarrow\PP^n$ is the blow-up of
$\PP^n$ along $\B$ and
$\pi'=\varphi\circ\pi:\widetilde{\PP^n}\rightarrow\sS$.
Denote by $\B'$ the base locus of $\varphi^{-1}$,
$E$ the exceptional divisor of $\pi$,
$E'=\pi'^{-1}(\B')$,
$H=\pi^{\ast}(H_{\PP^n})$,
$H'={\pi'}^{\ast}(H_{\sS})$, and note that, since
$\pi'|_{\widetilde{\PP^n}\setminus E'}:\widetilde{\PP^n}\setminus E'\rightarrow \sS\setminus\B'$
is an isomorphism,
we have $(\sing(\sS))_{\mathrm{red}}\subseteq (\B')_{\mathrm{red}}$.
We also put
$r=\dim(\B)$,
$r'=\dim(\B')$,
$\lambda=\deg(\B)$,
$g=g(\B)$ the sectional genus of $\B$,
$c_j=c_j(\T_{\B})\cdot H_{\B}^{r-j}$ (resp. $s_j=s_j(\N_{\B,\PP^n})\cdot H_{\B}^{r-j}$)
the degree of the $j$-th Chern class (resp. Segre class) of $\B$,
$\Delta=\deg(\sS)$,
$c=c(\sS)$ the \emph{coindex} of $\sS$
(the last of which is defined
by $-K_{\reg(\sS)}\sim (n+1-c)H_{\reg(\sS)}$,
whenever $\Pic(\sS)=\ZZ\langle H_{\sS}\rangle$).
\begin{assumption}\label{assumption: liftable}
We suppose that there exists a rational map
$\widehat{\varphi}:\PP^{n+a}\dashrightarrow\PP^n$
defined by a sublinear system of $|\O_{\PP^{n+a}}(d)|$ and having
base locus $\widehat{\B}$
such that $\varphi^{-1}=\widehat{\varphi}|_{\sS}$ and $\B'=\widehat{\B}\cap\sS$.
We then will say that $\varphi^{-1}$ is \emph{liftable}\footnote{If $a\geq 2$
and $\psi:\PP^n\dashrightarrow\mathbf{Z}:=\overline{\psi(\PP^n)}\subset\PP^{n+a}$ is
a birational transformation with $\mathbf{Z}$ factorial,
from \cite{mella-polastri} it follows that there exists a Cremona transformation
$\widetilde{\psi}:\PP^{n+a}\dashrightarrow\PP^{n+a}$
such that $\overline{\widetilde{\psi}(\mathbf{Z})} \simeq\PP^n\subset\PP^{n+a}$ and
$\psi^{-1}=\widetilde{\psi}|_{\mathbf{Z}}$; in particular,
if $\varpi$ denotes the linear projection
of $\PP^{n+a}$ onto $\overline{\widetilde{\psi}(\mathbf{Z})}$, we have
$\psi^{-1}=(\varpi\circ\widetilde{\psi})|_{\mathbf{Z}}$. But this
in general does not ensure the liftability of $\psi^{-1}$, because
we only have that $\mathrm{Bs}(\psi^{-1})\subseteq \mathrm{Bs}(\varpi\circ\widetilde{\psi}) \cap \mathbf{Z}$.}
and that $\varphi$ is \emph{of type} $(2,d)$.
\end{assumption}
The above assumption
yields the relations:
\begin{equation}\label{eq: lift}
\begin{array}{ll}
H' \sim 2H-E, & H \sim dH'-E', \\
E'\sim (2d-1)H-dE, & E \sim (2d-1)H'-2E' ,
\end{array}
\end{equation}
and hence also
$ \Pic(\widetilde{\PP^n})\simeq \ZZ\langle H \rangle\oplus \ZZ\langle E \rangle
\simeq \ZZ\langle H'\rangle\oplus \ZZ\langle E'\rangle $.
Note that, by the proofs of
\cite[Proposition~1.3 and 2.1(a)]{ein-shepherdbarron} and
by factoriality of $\sS$, we obtain that $E'$
is a reduced and irreducible divisor.
Moreover we have
$\Pic(\sS)\simeq \Pic(\sS\setminus\B')\simeq \Pic(\widetilde{\PP^n}\setminus E')
\simeq \ZZ\langle H'\rangle\simeq \ZZ\langle H_{\sS}\rangle$.
Finally, we require the following:
\begin{assumption}\label{assumption: ipotesi}
$(\sing(\sS))_{\mathrm{red}}\neq (\B')_{\mathrm{red}}$.
\end{assumption}
Now we point out that, just as in Proposition \ref{prop: dimension formula},
since $E'$ is irreducible,
by Assumption \ref{assumption: ipotesi} and \cite[Theorem~1.1]{ein-shepherdbarron},
we deduce that
$\pi'|_V:V\rightarrow U$ coincides with the blow-up of $U$ along $Z$,
where $U=\reg(\sS)\setminus\sing((\B')_{\mathrm{red}})$,
$V=\pi'^{-1}(U)$ and $Z=U\cap (\B')_{\mathrm{red}}$.
It follows that
$K_{\widetilde{\PP^n}} \sim (-n-1)H+(n-r-1)E \sim (c-n-1)H'+(n-r'-1)E'$,
from which, together with (\ref{eq: lift}), we obtain
$2r+3-n=n-r'-1$ and
$c=\left( 1-2d\right) r+dn-3d+2$.
One can also easily see that,
for the general point
$x\in\Sec(\B)\setminus \B$,
$\overline{\varphi^{-1}\left(\varphi\left(x\right)\right)}$
is a linear space of dimension $n-r'-1$ and
$\overline{\varphi^{-1}\left(\varphi\left(x\right)\right)}\cap \B$
is a quadric hypersurface, which coincides with
the entry locus $\Sigma_{x}(\B)$ of $\B$ with respect to $x$.
So we can generalize Proposition \ref{prop: dimension formula},
obtaining one of the main results useful for purposes of this chapter:
\begin{proposition}\label{prop: B is QEL}
$\Sec(\B)\subset\PP^n$ is a hypersurface of degree $2d-1$ and
$\B$ is a $QEL$-variety of type $\delta=2r+2-n$.
\end{proposition}
In many cases, $\B$ has a much stronger property of being $QEL$-variety.
Recall that a subscheme $X\subset\PP^n$ is said to have the $K_2$ property if
$X$ is cut out by quadratic forms $F_0,\ldots,F_N$
such that the Koszul relations among the $F_i$ are generated by linear syzygies.
We have the following fact (see \cite{vermeire} and \cite{alzati-syz}):
\begin{fact}\label{fact: K2 property}
Let $X\subset\PP^n$ be
a smooth variety
cut out by quadratic forms $F_0,\ldots,F_N$
satisfying $K_2$ property and let
$F=[F_0,\ldots,F_N]:\PP^n\dashrightarrow\PP^N$ be the
induced rational map. Then
for every $x\in\PP^n\setminus X$,
$\overline{F^{-1}\left(F\left(x\right)\right)}$ is
a linear space of dimension
$n+1-
\mathrm{rank}\left(\left({\partial F_i}/{\partial x_j}(x)\right)_{i,j}\right)$;
moreover,
$\dim(\overline{F^{-1}\left(F\left(x\right)\right)})>0$
if and only if $x\in\Sec(X)\setminus X$ and in this case
$\overline{F^{-1}\left(F\left(x\right)\right)}\cap X$ is a quadric hypersurface,
which coincides with
the entry locus $\Sigma_{x}(X)$ of $X$ with respect to $x$.
\end{fact}
We have a simple sufficient
condition for the $K_2$ property (see \cite{saint-donat}, \cite{green-lazarsfeld-1988} and \cite[Proposition~2]{alzati-russo-subhomaloidal}):
\begin{fact}\label{fact: test K2}
Let $X\subset\PP^n$ be a smooth linearly normal variety and suppose $h^1(\O_X)=0$
if $\dim(X)\geq2$. Putting $\lambda=\deg(X)$ and $s=\mathrm{codim}_{\PP^n}(X)$ we have:
\begin{itemize}
\item if $\lambda\leq 2s+1$, then $X$ is arithmetically Cohen-Macaulay;
\item if $\lambda\leq 2s$, then the homogeneous ideal of $X$
is generated by quadratic forms;
\item if $\lambda\leq2s-1$, then the syzygies of the generators
of the homogeneous ideal of $X$ are generated by the linear ones.
\end{itemize}
\end{fact}
\begin{remark}\label{remark: samuel conjecture}
Let $\psi:\PP^n\dashrightarrow\mathbf{Z}:=\overline{\psi(\PP^n)}\subseteq\PP^{n+a}$ be
a birational transformation ($n\geq3$).
We point out that, from Grothendieck's Theorem on parafactoriality (Samuel's Conjecture)
\cite[\Rmnum{11} Corollaire~3.14]{sga2} it follows that $\mathbf{Z}$ is factorial
whenever it is a local complete intersection
with $\dim(\sing(\mathbf{Z}))<\dim(\mathbf{Z})-3$.
Of course, every complete intersection in a smooth variety
is a local complete intersection.
Moreover, $\psi^{-1}$ is liftable whenever
$\Pic(\mathbf{Z})=\ZZ\langle H_{\mathbf{Z}}\rangle$
and $\mathbf{Z}$ is factorial and projectively normal.
So, from \cite{larsen-coomology} and \cite[\Rmnum{4} Corollary~3.2]{hartshorne-ample},
$\psi^{-1}$ is liftable whenever
$\mathbf{Z}$ is either
smooth and projectively normal with $n\geq a+2$
or a factorial
complete intersection.
\end{remark}
\section{Numerical restrictions}
Proposition \ref{prop: B is QEL} already provides
a restriction on the invariants of the transformation $\varphi$;
here we give further restrictions of this kind.
\begin{proposition}\label{prop: hilbert polynomial}
Let $\epsilon=0$ if $\langle \B \rangle =\PP^n$ and let $\epsilon=1$ otherwise.
\begin{itemize}
\item If $r=1$ we have:
\begin{eqnarray*}
\lambda &=& \frac{n^2-n+2\,\epsilon-2\,a-2}{2} , \\
g &=& \frac{n^2-3\,n+4\,\epsilon-2\,a-2}{2} .
\end{eqnarray*}
\item If $r=2$ we have:
\begin{eqnarray*}
\chi(\O_{\B}) &=& \frac{2\,a-n^2+5\,n+2\,g-6\,\epsilon+4}{4} , \\
\lambda &=& \frac{n^2-n+2\,g+2\,\epsilon-2\,a-4}{4} . \\
\end{eqnarray*}
\item If $r=3$ we have:
\begin{eqnarray*}
\chi(\O_{\B}) &=& \frac{4\lambda-n^2+3n-2g-4\epsilon+2a+6}{2} .
\end{eqnarray*}
\end{itemize}
\end{proposition}
\begin{proof}
By Proposition \ref{prop: B is QEL} we have
$h^0(\PP^n,\I_{\B}(1))=\epsilon$.
Since $\sS$ is normal and linearly normal,
we have $h^0(\PP^n,\I_{\B}(2))=n+1+a$ (see Lemma \ref{prop: cohomology I2B}).
Moreover,
since $n\leq 2r+2$ (being $\delta\geq0$),
proceeding as in Lemma \ref{prop: cohomology twisted ideal}
(or applying
\cite[Proposition~1.8]{mella-russo-baselocusleq3}),
we obtain
$h^j(\PP^n,\I_{\B}(k))=0$ for every $j,k\geq1$.
So we obtain
$\chi(\O_{\B}(1))=n+1-\epsilon$ and
$\chi(\O_{\B}(2))= (n+1)(n+2)/2 - (n+1+a)$.
\end{proof}
\begin{proposition}\label{prop: segre and chern classes} \hspace{1pt}
\begin{itemize}
\item If $r=1$ we have:
\begin{eqnarray*}
{c}_{1} &=& 2-2\,g , \\
{s}_{1} &=& -n\,\lambda-\lambda-2\,g+2 , \\
d &=& \frac{ 2\,\lambda-{2}^{n} }{ 2\,n\,\lambda-2\,\lambda-{2}^{n+1}-4\,g+4 } , \\
\Delta &=& -n\,\lambda+\lambda+{2}^{n}+2\,g-2 .
\end{eqnarray*}
\item If $r=2$ we have:
\begin{eqnarray*}
c_1 &=& \lambda-2\,g+2 , \\
c_2 &=& -\frac{{n}^{2}\,\lambda}{2}+\frac{3\,n\,\lambda}{2}+{2}^{n}+2\,g\,n-2\,n-2\,g-\Delta+2 , \\
s_1 &=& -n\,\lambda-2\,g+2 , \\
s_2 &=& 2\,n\,\lambda+{2}^{n}+4\,g\,n-4\,n-\Delta , \\
d\,\Delta &=& -n\,\lambda+2\,\lambda+{2}^{n-1}+2\,g-2 .
\end{eqnarray*}
\item If $r=3$ we have:
\begin{eqnarray*}
c_1 &=& 2\,\lambda-2\,g+2 , \\
c_2 &=& -\frac{{n}^{2}\,\lambda}{2}+\frac{5\,n\,\lambda}{2}-\lambda+{2}^{n-1}+2\,g\,n-2\,n-6\,g-d\,\Delta+6 , \\
c_3 &=& \frac{{n}^{3}\,\lambda}{3}-2\,{n}^{2}\,\lambda+\frac{11\,n\,\lambda}{3}-2\,\lambda-n\,{2}^{n-1}+3\,{2}^{n-1}-g\,{n}^{2}+{n}^{2}+3\,g\,n+ \\ && + d\,\Delta\,n-3\,n-4\,g-d\,\Delta-\Delta+4 , \\
s_1 &=& -n\,\lambda+\lambda-2\,g+2 , \\
s_2 &=& 2\,n\,\lambda-2\,\lambda+{2}^{n-1}+4\,g\,n-4\,n-4\,g-d\,\Delta+4 , \\
s_3 &=& \frac{2\,{n}^{3}\,\lambda}{3}-4\,{n}^{2}\,\lambda+\frac{10\,n\,\lambda}{3}-n\,{2}^{n}+{2}^{n}-4\,g\,{n}^{2}+4\,{n}^{2}+4\,g\,n+2\,d\,\Delta\,n-4\,n-\Delta .
\end{eqnarray*}
\end{itemize}
\end{proposition}
\begin{proof}
See also \cite{crauder-katz-1989} and \cite{crauder-katz-1991}.
By \cite[page~291]{crauder-katz-1989} we see that
\begin{displaymath}
H^j\cdot E^{n-j}=
\left\{
\begin{array}{ll}
1, & \mbox{if } j= n ; \\
0, & \mbox{if } r+1\leq j \leq n-1 ; \\
(-1)^{n-j-1} s_{r-j}, & \mbox{if } j \leq r .
\end{array}
\right.
\end{displaymath}
Since $H'=2H-E$ and $H=dH'-E'$ we have
\begin{eqnarray}
\Delta &=& {H'}^n=(2H-E)^n, \\
d \Delta &=& d{H'}^{n}={H'}^{n-1}\cdot(dH'-E')=(2H-E)^{n-1}\cdot H.
\end{eqnarray}
From the exact sequence
$0\rightarrow\mathcal{T}_{\B}\rightarrow \mathcal{T}_{\PP^n}|_{\B}\rightarrow\mathcal{N}_{\B,\PP^n}\rightarrow0$
we get:
\begin{eqnarray}
s_1 &=& - \lambda \left( n+1\right) + {c}_{1} ,\\
s_2 &=& \lambda \begin{pmatrix}n+2\cr 2\end{pmatrix}-{c}_{1} \left( n+1\right) +{c}_{2} ,\\
s_3 &=& -\lambda \begin{pmatrix}n+3\cr 3\end{pmatrix}+{c}_{1} \begin{pmatrix}n+2\cr 2\end{pmatrix}-{c}_{2} \left( n+1\right) +{c}_{3} ,\\
& \vdots & \nonumber
\end{eqnarray}
Moreover $c_1=-K_{\B}\cdot H_{\B}^{r-1}$
and it can be expressed as a function of $\lambda$ and $g$.
Thus we found $r+3$
independent equations on the $2r+5$ variables:
$c_1,\ldots,c_r,s_1,\ldots,s_r,d,\Delta,\lambda,g,n$.
\end{proof}
\begin{remark}
Proposition \ref{prop: segre and chern classes}
holds under less restrictive assumptions, as shown in the above proof.
Here we treat the special case:
let $\psi:\PP^8\dashrightarrow\mathbf{Z}:=\overline{\psi(\PP^8)}\subseteq\PP^{8+a}$ be a quadratic
rational map whose base locus
is a smooth irreducible $3$-dimensional variety $X$.
Without any other restriction on $\psi$, denoting
with $\pi:\Bl_X(\PP^8)\rightarrow \PP^8$
the blow-up of $\PP^8$ along $X$ and with $s_i(X)=s_i(\N_{X,\PP^8})$, we have
\begin{equation}\label{eq: grado mappa razionale}
\deg(\psi)\deg(\mathbf{Z}) = (2\pi^{\ast}(H_{\PP^8})-E_X)^8 =
-s_3(X)-16s_2(X)-112s_1(X)-448\deg(X)+256.
\end{equation}
Moreover, if $\psi$ is birational with liftable inverse and
$\dim(\sing(\mathbf{Z}))\leq 6$,
we also have
\begin{equation}\label{eq: sollevabile}
d \deg(\mathbf{Z}) =(2\pi^{\ast}(H_{\PP^8})-E_X)^7\cdot \pi^{\ast}(H_{\PP^8}) = -s_2(X) -14 s_1(X) -84 \deg(X) +128,
\end{equation}
where $d$ denotes the degree of the linear system defining $\psi^{-1}$.
\end{remark}
Proposition \ref{prop: double point formula}
is a translation of the well-known
\emph{double point formula} (see for example \cite{peters-simonis} and \cite{laksov}),
taking into account Proposition \ref{prop: B is QEL}.
\begin{proposition}\label{prop: double point formula}
If $\delta=0$
then
$$
2(2d-1)=
\lambda^2 -
\sum_{j=0}^{r}\begin{pmatrix} 2r+1 \cr j \end{pmatrix} s_{r-j}(\T_{\B})\cdot H_{\B}^{j}.
$$
\end{proposition}
\section{Case of dimension 1}\label{sec: dim 1}
Lemma \ref{lemma: numerical 1-fold}
directly follows from
Propositions \ref{prop: hilbert polynomial}
and \ref{prop: segre and chern classes}.
\begin{lemma}\label{lemma: numerical 1-fold}
If $r=1$, then
one of the following cases holds:
\begin{enumerate}[(A)]
\item $n=3$, $a=1$, $\lambda=2$, $g=0$, $d=1$, $\Delta=2$;
\item $n=4$, $a=0$, $\lambda=5$, $g=1$, $d=3$, $\Delta=1$;
\item $n=4$, $a=1$, $\lambda=4$, $g=0$, $d=2$, $\Delta=2$;
\item\label{case: escluso 1-fold} $n=4$, $a=2$, $\lambda=4$, $g=1$, $d=1$, $\Delta=4$;
\item $n=4$, $a=3$, $\lambda=3$, $g=0$, $d=1$, $\Delta=5$.
\end{enumerate}
\end{lemma}
\begin{proposition}\label{prop: possibili casi 1-fold}
If $r=1$, then
one of the following cases holds:
\begin{enumerate}[(I)]
\item $n=3$, $a=1$, $\B$ is a conic;
\item $n=4$, $a=0$, $\B$ is an elliptic curve of degree $5$;
\item $n=4$, $a=1$, $\B$ is the rational normal quartic curve;
\item $n=4$, $a=3$, $\B$ is the twisted cubic curve.
\end{enumerate}
\end{proposition}
\begin{proof}
From Lemma \ref{lemma: numerical 1-fold}
it remains only to exclude case (\ref{case: escluso 1-fold}).
In this case $\B$ is a complete intersection of two quadrics in $\PP^3$
and also
it is an $OADP$-curve.
This is absurd because the only $OADP$-curve is the twisted cubic curve.
\end{proof}
\section{Case of dimension 2}\label{sec: dim 2}
Proposition \ref{prop: possibili casi 2-fold} follows from
Proposition \ref{prop: LQEL with delta=r and delta=r-1} and
\cite[Theorem~4.10]{ciliberto-mella-russo}.
\begin{proposition}\label{prop: possibili casi 2-fold}
If $r=2$, then either $n=6$, $d\geq2$, $\langle \B \rangle = \PP^6$, or
one of the following cases holds:
\begin{enumerate}[(I)]
\setcounter{enumi}{4}
\item\label{case 2-fold a} $n=4$, $d=1$, $\delta=2$,
$\B=\PP^1\times\PP^1\subset\PP^3\subset\PP^4$;
\item\label{case 2-fold b} $n=5$, $d=1$, $\delta=1$,
$\B$ is a hyperplane section of $\PP^1\times\PP^2\subset\PP^5$;
\item\label{case 2-fold c} $n=5$, $d=2$, $\delta=1$,
$\B=\nu_2(\PP^2)\subset\PP^5$ is the Veronese surface;
\item\label{case 2-fold d} $n=6$, $d=1$, $\delta=0$, $\B\subset\PP^5$ is an $OADP$-surface, i.e. $\B$ is as in one of the following cases:
\begin{enumerate}[($\ref{case 2-fold d}_1$)]
\item\label{case 2-fold d1} $\PP_{\PP^1}(\O(1)\oplus\O(3))$ or
$\PP_{\PP^1}(\O(2)\oplus\O(2))$;
\item\label{case 2-fold d2} del Pezzo surface of degree $5$
(hence the blow-up of
$\PP^2$ at $4$ points
$p_1,\ldots,p_4$ and $|H_{\B}|=|3H_{\PP^2}-p_1-\cdots-p_4|$).
\end{enumerate}
\end{enumerate}
\end{proposition}
\begin{lemma}\label{lemma: r=2 B nondegenerate}
If $r=2$, $n=6$ and $\langle \B \rangle = \PP^6$,
then one of the following cases holds:
\begin{enumerate}[(A)]
\item \label{case 2-fold a=0 lambda=7}
$a=0$, $\lambda=7$, $g=1$, $\chi(\O_{\B})=0$;
\item \label{case 2-fold a leq 3}
$0\leq a \leq 3$, $\lambda=8-a$, $g=3-a$, $\chi(\O_{\B})=1$.
\end{enumerate}
\end{lemma}
\begin{proof}
By Proposition \ref{prop: hilbert polynomial}
it follows that
$g=2\lambda+a-13$ and $\chi(\O_{\B})=\lambda+a-7$.
By Lemma \ref{prop: castelnuovo argument} and using that $g\geq0$
(proceeding as in Proposition \ref{prop: 2-fold in P6}),
we obtain $(13-a)/2 \leq \lambda \leq 8-a$.
\end{proof}
\begin{lemma}\label{lemma: double point formula r=2}
If $r=2$, $n=6$ and $\langle \B \rangle = \PP^6$, then
one of the following cases holds:
\begin{itemize}
\item $a=0$, $d=4$, $\Delta=1$;
\item $a=1$, $d=3$, $\Delta=2$;
\item $a=2$, $d=2$, $\Delta=4$;
\item $a=3$, $d=2$, $\Delta=5$.
\end{itemize}
\end{lemma}
\begin{proof}
We have $s_1(\T_{\B})\cdot H_{\B} = -c_1 $ and
$ s_2(\T_{\B}) = c_1^2-c_2=12\chi(\O_{\B}) -2c_2 $.
So, by Proposition \ref{prop: double point formula}, we obtain
\begin{equation}
2(2d-1) = \lambda^2-10\lambda-12\chi(\O_{\B})+2c_2+5c_1 .
\end{equation}
Now, by Propositions \ref{prop: hilbert polynomial} and
\ref{prop: segre and chern classes},
we obtain
\begin{equation}
d\Delta = 2a+4,\quad
\Delta = (g^2+(-2a-4)g-16d+a^2-4a+75)/8 ,
\end{equation}
and then we conclude by Lemma \ref{lemma: r=2 B nondegenerate}.
\end{proof}
\begin{proposition}\label{prop: r=2 B nondegenerate}
If $r=2$, $n=6$ and $\langle \B\rangle=\PP^6$
then one of the following cases holds:
\begin{enumerate}[(I)]
\setcounter{enumi}{8}
\item $a=0$, $\lambda=7$, $g=1$, $\B$ is an elliptic scroll $\PP_{C}(\E)$ with $e(\E)=-1$;
\item $a=0$, $\lambda=8$, $g=3$, $\B$ is the blow-up of $\PP^2$ at $8$ points $p_1\ldots,p_8$, $|H_{\B}|=|4H_{\PP^2}-p_1-\cdots-p_8|$;
\item $a=1$, $\lambda=7$, $g=2$, $\B$ is the blow-up of $\PP^2$ at $6$ points $p_0\ldots,p_5$, $|H_{\B}|=|4H_{\PP^2}-2p_0-p_1-\cdots-p_5|$;
\item $a=2$, $\lambda=6$, $g=1$, $\B$ is the blow-up of $\PP^2$ at $3$ points $p_1,p_2,p_3$, $|H_{\B}|=|3H_{\PP^2}-p_1-p_2-p_3|$;
\item $a=3$, $\lambda=5$, $g=0$, $\B$ is a rational normal scroll.
\end{enumerate}
\end{proposition}
\begin{proof}
For $a=0$, $a=1$ and $a\in\{2,3\}$
the statement follows, respectively, from \cite{crauder-katz-1989},
Proposition \ref{prop: 2-fold in P6} and \cite{ionescu-smallinvariants}.
\end{proof}
\section{Case of dimension 3}\label{sec: dim 3}
Proposition \ref{prop: possibili casi C1} follows from:
Proposition \ref{prop: LQEL with delta=r and delta=r-1},
\cite{fujita-3-fold},
Theorems \ref{prop: classification CC-varieties} and
\ref{prop: classification del pezzo varieties}
and
\cite{ciliberto-mella-russo}.
\begin{proposition}\label{prop: possibili casi C1}
If $r=3$, then either $n=8$, $d\geq2$, $\langle \B \rangle = \PP^8$, or
one of the following cases holds:
\begin{enumerate}[(I)]
\setcounter{enumi}{13}
\item\label{case C1 a} $n=5$, $d=1$, $\delta=3$, $\B=Q^3\subset\PP^4\subset\PP^5$ is a quadric;
\item\label{case C1 b} $n=6$, $d=1$, $\delta=2$, $\B=\PP^1\times\PP^2\subset\PP^5\subset\PP^6$;
\item\label{case C1 c} $n=7$, $d=1$, $\delta=1$, $\B\subset\PP^6$ is as
in one of the following cases:
\begin{enumerate}[($\ref{case C1 c}_1$)]
\item\label{case C1 c1} $\PP_{\PP^1}(\O(1)\oplus\O(1)\oplus\O(2))$;
\item\label{case C1 c2} linear section of $\GG(1,4)\subset\PP^9$;
\end{enumerate}
\item\label{case C1 d} $n=7$, $d=2$, $\delta=1$, $\B$ is a hyperplane section of $\PP^2\times\PP^2\subset\PP^8$;
\item\label{case C1 e} $n=8$, $d=1$, $\delta=0$, $\B\subset\PP^7$ is an $OADP$-variety,
i.e. $\B$ is as in one of the following cases:
\begin{enumerate}[($\ref{case C1 e}_1$)]
\item\label{case C1 e1} $\PP_{\PP^1}(\O(1)\oplus\O(1)\oplus\O(3))$ or
$\PP_{\PP^1}(\O(1)\oplus\O(2)\oplus\O(2))$;
\item\label{case C1 e2} Edge variety of degree $6$ (i.e. $\PP^1\times\PP^1\times\PP^1$) or
Edge variety of degree $7$;
\item\label{case C1 e3} $\PP_{\PP^2}(\E)$, where $\E$
is a vector bundle with $c_1(\E)=4$ and $c_2(\E)=8$, given as an extension by
the following exact sequence
$0\rightarrow\O_{\PP^2}\rightarrow\E\rightarrow
\I_{\{p_1,\ldots,p_8\},\PP^2}(4)\rightarrow0$.
\end{enumerate}
\end{enumerate}
\end{proposition}
In the following we denote by
$\Lambda\subsetneq C\subsetneq S\subsetneq \B$
a sequence of general linear sections of $\B$.
\begin{lemma}\label{lemma: r=3 B nondegenerate}
If $r=3$, $n=8$ and $\langle \B \rangle = \PP^8$,
then one of the following cases holds:
\begin{enumerate}[(A)]
\item \label{a=0,lambda=13}
$a=0$, $\lambda=13$, $g=8$, $K_S\cdot H_S=1$, $K_S^2=-1$;
\item \label{a=1,lambda=12}
$a=1$, $\lambda=12$, $g=7$, $K_S\cdot H_S=0$, $K_S^2=0$;
\item \label{a geq2}
$0\leq a\leq6$, $\lambda=12-a$, $g=6-a$, $K_S\cdot H_S=-2-a$.
\end{enumerate}
\end{lemma}
\begin{proof}
Firstly we note that,
from the exact sequence
$0\rightarrow\mathcal{T}_{S}\rightarrow\mathcal{T}_{\B}|_{S}\rightarrow\O_{S}(1)\rightarrow0$,
we deduce
$c_2=c_2(S)+c_1(S)=12\chi(\O_{S})-K_{S}^2-K_{S}\cdot H_{S}$
and hence
\begin{equation}
K_S^2=14\lambda+12\chi(\O_S)-12g+d\Delta-116 = -22\lambda+12g+d\Delta-12a+184.
\end{equation}
Secondly we note that (see Lemma \ref{prop: castelnuovo argument}), putting
$h_{\Lambda}(2):=h^0(\PP^5,\O(2))-h^0(\PP^5,\I_{\Lambda}(2))$,
we have
\begin{equation}\label{hilbert-function}
\mathrm{min}\{\lambda,11\} \leq h_{\Lambda}(2)\leq
21-h^0(\PP^8,\I_{\B}(2))=12-a.
\end{equation}
Now we establish the following:
\begin{claim}\label{claim: KsHs<0}
If $K_S\cdot H_S\leq0$ and $K_S\nsim 0$, then $\lambda=12-a$ and $g=6-a$.
\end{claim}
\begin{proof}[Proof of the Claim]
Similarly to Case \ref{case: KS nsim 0}, we obtain that
$P_{\B}(-1)=0$ and $P_{\B}(0)=1-q$, where
$q:=h^1(S,\O_S)=h^1(\B,\O_{\B})$; in particular $g=-5q-a+6$ and $\lambda=-3q-a+12$.
Since $g\geq0$ we have $5q\leq 6-a$ and
the possibilities are:
if $a\leq1$ then $q\leq1$; if $a\geq 2$ then $q=0$.
If $(a,q)=(0,1)$ then $(g,\lambda)=(1,9)$
and the case is excluded by
\cite[Theorem~12.3]{fujita-polarizedvarieties}\footnote{Note that
$\B$ cannot be a scroll over a curve
(this follows from (\ref{eq: relation scroll}) and (\ref{eq: second relation scroll}) below
and also it follows from \cite[Proposition~3.2(i)]{mella-russo-baselocusleq3}).};
if $(a,q)=(1,1)$ then $(g,\lambda)=(0,8)$
and the case is excluded by
\cite[Theorem~12.1]{fujita-polarizedvarieties}.
Thus we have $q=0$ and hence
$g=6-a$ and $\lambda=12-a$;
in particular we have $a\leq 6$.
\end{proof}
Now we discuss the cases according
to the value of $a$.
\begin{case}[$a=0$]
It is clear that $\varphi$ must be of type $(2,5)$ and hence
$K_S^2=-22\lambda+12g+189$.
By Claim \ref{claim: KsHs<0}, if $K_S\cdot H_S=2g-2-\lambda<0$,
we fall into case (\ref{a geq2}).
So we suppose that $K_S\cdot H_S\geq0$, namely that $g\geq\lambda/2+1$.
From Castelnuovo's bound it follows that
$\lambda\geq12$ and if $\lambda=12$ then
$K_S\cdot H_S=0$, $g=7$ and hence $K_S^2=9$.
Since this is impossible by Claim \ref{claim: KsHs<0}, we conclude that
$\lambda\geq 13$.
Now by (\ref{hilbert-function}) it follows that
$11\leq h_{\Lambda}(2)\leq12$, but
if $h_{\Lambda}(2)=11$
from Castelnuovo Lemma (Proposition \ref{prop: castelnuovo lemma})
we obtain a contradiction.
Thus we have $h_{\Lambda}(2)=12$ and
$h^0(\PP^5,\I_{\Lambda}(2))=h^0(\PP^8,\I_{\B}(2))=9$.
So from Theorem \ref{prop: extension castelnuovo lemma by fano harris} we deduce
that $\lambda\leq 14$ and furthermore,
by the refinement of Castelnuovo's bound contained in
Theorem \ref{prop: refinement castelnuovo bound by ciliberto},
we obtain $g\leq 2\lambda-18$.
In summary we have the following possibilities:
\begin{enumerate}[(i)]
\item\label{case: T1} $\lambda=13$, $g=8$, $K_S\cdot H_S=1$, $\chi(\O_S)=2$, $K_S^2=-1$;
\item\label{case: T2} $\lambda=14$, $g=8$, $K_S\cdot H_S=0$, $\chi(\O_S)=-1$, $K_S^2=-23$;
\item\label{case: T3} $\lambda=14$, $g=9$, $K_S\cdot H_S=2$, $\chi(\O_S)=1$, $K_S^2=-11$;
\item\label{case: T4} $\lambda=14$, $g=10$, $K_S\cdot H_S=4$, $\chi(\O_S)=3$, $K_S^2=1$.
\end{enumerate}
Case (\ref{case: T1}) coincides with case (\ref{a=0,lambda=13}).
Case (\ref{case: T2}) is excluded by Claim \ref{claim: KsHs<0}.
In the circumstances of case (\ref{case: T3}), we have $h^1(S,\O_S)=h^2(S,\O_S)=h^0(S,K_S)$.
If $h^1(S,\O_S)>0$, since
$(K_{\B}+4H_{\B})\cdot K_S=K_S^2+3 K_S\cdot H_S=-5<0$,
we see that $K_{\B}+4H_{\B}$ is not nef and then
we obtain a contradiction by \cite{ionescu-adjunction}.
If $h^1(S,\O_S)=0$, then we also have $h^1(\B,\O_{\B})=h^2(\B,\O_{\B})=0$ and hence
$\chi(\O_{\B})=1-h^3(\B,\O_{\B})\leq 1$, against the fact that $\chi(\O_{\B})=2\lambda-g-17=2$.
Thus case (\ref{case: T3}) does not occur.
Finally, in the circumstances of case (\ref{case: T4}), note
that $h^0(S,K_S)=2+h^1(S,\O_S)\geq2$ and we write
$|K_S|=|M|+F$, where $|M|$ is the mobile part of the linear system $|K_S|$
and $F$ is the fixed part. If $M_1=M$
is a general member of $|M|$, there exists $M_2\in|M|$
having no common irreducible components with $M_1$ and
so $M^2=M_1\cdot M_2=\sum_{p}\left(M_1\cdot M_2\right)_{p}\geq0$;
furthermore, by using Bertini Theorem, we see that
$\sing(M_1)$ consists of points $p$ such that
the intersection multiplicity $\left(M_1\cdot M_2\right)_{p}$ of $M_1$ and $M_2$ in $p$ is at least $2$.
By definition, we also have $M\cdot F\geq0$ and so we deduce
$2p_a(M)-2=M\cdot (M+K_S)= 2 M^2+ M\cdot F\geq 0$, from which
$p_a(M)\geq 1$ and $p_a(M)=2$ if $F=0$.
On the other hand, we have $M\cdot H_S\leq K_S\cdot H_S=4$ and,
since $S$ is cut out by quadrics,
$M$ does not contain planar curves of degree $\geq3$.
If $M\cdot H_S=4$, then $F=0$,
$M^2=1$ and $M$ is a (possibly disconnected) smooth curve;
since $p_a(M)=2$, $M$ is actually disconnected
and so it is a disjoint union of twisted cubics, conics and lines.
But then we obtain the contradiction that
$p_a(M)=1-\#\{\mbox{connected components of }M\}<0$.
If $M\cdot H_S\leq3$, then $M$ must be either
a twisted cubic or a union of conics and lines. In all these cases
we again obtain the contradiction that
$p_a(M)=1-\#\{\mbox{connected components of }M\}\leq 0$.
Thus case (\ref{case: T4}) does not occur.
\end{case}
\begin{case}[$a=1$] By Proposition \ref{prop: 3-fold in P8 - S nonlinear}
we fall into case (\ref{a=1,lambda=12}) or (\ref{a geq2}).
\end{case}
\begin{case}[$a\geq2$] By (\ref{hilbert-function}) it follows that
$\lambda\leq 10$ and by Castelnuovo's bound it follows that
$K_S\cdot H_S\leq -4<0$. Thus, by Claim \ref{claim: KsHs<0}
we fall into case (\ref{a geq2}).
\end{case}
\end{proof}
Now we apply the double point formula
(Proposition \ref{prop: double point formula})
in order to obtain additional numerical restrictions
under the hypothesis of Lemma \ref{lemma: r=3 B nondegenerate}.
\begin{lemma}\label{lemma: double point formula}
If $r=3$, $n=8$ and $\langle \B\rangle=\PP^8$, then
$$
K_{\B}^3=\lambda^2+23\lambda-24g-(7d+1)\Delta-4d+36a-226 .
$$
\end{lemma}
\begin{proof}
We have (see \cite[App. A, Exercise~6.7]{hartshorne-ag}):
\begin{eqnarray*}
s_1(\T_{\B})\cdot H_{\B}^2 &=& -c_1(\B)\cdot H_{\B}^2=K_{\B}\cdot H_{\B}^2 , \\
s_2(\T_{\B})\cdot H_{\B} &=& c_1(\B)^2\cdot H_{\B}-c_2(\B)\cdot H_{\B}
= K_{\B}^2\cdot H_{\B}-c_2(\B)\cdot H_{\B} \\
&=& 3K_{\B}\cdot H_{\B}^2-2H_{\B}^3-2c_2(\B)\cdot H_{\B}+12\left(\chi(\O_{\B}(H_{\B}))-\chi(\O_{\B})\right), \\
s_3(\T_{\B}) &=& -c_1(\B)^3+2c_1(\B)\cdot c_2(\B)-c_3(\B)
=K_{\B}^3+48\chi(\O_{\B})-c_3(\B).
\end{eqnarray*}
Hence, applying the double point formula and using
the relations
$\chi(\O_{\B})=2\lambda-g+a-17$, $\chi(\O_{\B}(H_{\B}))=9$, we obtain:
\begin{eqnarray*}
4d-2 &=& 2\,\deg(\Sec(\B))\\
&=& \deg(\B)^2-s_3(\T_{\B})-7\,s_2(\T_{\B})\cdot H_{\B}-21\,s_1(\T_{\B})\cdot H_{\B}^2-35\,H_{\B}^3 \\
&=& \deg(\B)^2-21\,\deg(\B)-42\,K_{\B}\cdot H_{\B}^2+14\,c_2(\B)\cdot H_{\B}-K_{\B}^3 \\
&& +c_3(\B)-84\,\chi(\O_{\B}(H_{\B}))+36\,\chi(\O_{\B}) \\
&=& -K_{\B}^3+\lambda^2+23\lambda-24g-(7d+1)\Delta+36a-228.
\end{eqnarray*}
\end{proof}
\begin{lemma}\label{lemma: quadric fibration}
If $r=3$, $n=8$, $\langle \B\rangle=\PP^8$ and
$\B$ is a quadric fibration over a curve,
then one of the following cases holds:
\begin{itemize}
\item $a=3$, $\lambda=9$, $g=3$, $d=3$, $\Delta=5$;
\item $a=4$, $\lambda=8$, $g=2$, $d=2$, $\Delta=10$.
\end{itemize}
\end{lemma}
\begin{proof}
Denote by $\beta:(\B,H_{\B})\rightarrow (Y,H_Y)$
the projection over the curve $Y$
such that $\beta^{\ast}(H_Y)=K_{\B}+2H_{\B}$.
We have
\begin{eqnarray*}
0&=&\beta^{\ast}(H_Y)^2\cdot H_{\B}
= K_{\B}^2\cdot H_{\B}+4K_{\B}\cdot H_{\B}^2+4H_{\B}^3, \\
0&=& \beta^{\ast}(H_Y)^3= K_{\B}^3+6K_{\B}^2\cdot H_{\B}+12 K_{\B}\cdot H_{\B}^2+8 H_{\B}^3, \\
\chi(\O_{\B}(H_{\B})) &=& \frac{1}{12} K_{\B}^2\cdot H_{\B}-\frac{1}{4}K_{\B}\cdot H_{\B}^2+\frac{1}{6}H_{\B}^3+\frac{1}{12}c_2(\B)\cdot H_{\B}+\chi(\O_{\B}), \\
\end{eqnarray*}
from which it follows that
\begin{eqnarray}
K_{\B}^3 &=& -8\lambda+24g-24, \\
c_2(\B)\cdot H_{\B}
&=& -36\lambda+26g-12a+298.
\end{eqnarray}
Hence, by Lemma \ref{lemma: double point formula} and
Proposition \ref{prop: segre and chern classes},
we obtain
\begin{eqnarray}
d\Delta&=& 23\lambda-16g+12a-180 , \\
\Delta+4d&=&\lambda^2-130\lambda+64g-48a+1058 .
\end{eqnarray}
Now the conclusion follows from Lemma
\ref{lemma: r=3 B nondegenerate},
by observing that
the case $a=6$ cannot occur. In fact,
if $a=6$, by \cite{ionescu-smallinvariants}
it follows that $\B$ is a rational normal scroll
and by a direct calculation (or by Lemma \ref{lemma: scroll over curve}) we see that
$d=2$ and $\Delta=14$.
\end{proof}
\begin{lemma}\label{lemma: scroll over surface}
If $r=3$, $n=8$, $\langle \B\rangle=\PP^8$ and
$\B$ is a scroll over a smooth surface $Y$,
then we have:
\begin{eqnarray*}
c_2\left(Y\right) &=& \left(\left(7d-1\right)\lambda^2+\left(177-679d\right)\lambda+\left(292d-92\right)g-28d^2 \right. \\
&& \left. +\left(5554-252a\right)d+36a-1474\right)/\left(2d+2\right), \\
\Delta &=& \left(\lambda^2-107\lambda+48g-4d-36a+878\right)/\left(d+1\right) .
\end{eqnarray*}
\end{lemma}
\begin{proof}
Similarly to Lemma \ref{lemma: quadric fibration},
denote by $\beta:(\B,H_{\B})\rightarrow (Y,H_Y)$
the projection over the surface $Y$ such that
$\beta^{\ast}(H_Y)=K_{\B}+2H_{\B}$. Since $\beta^{\ast}(H_Y)^3=0$ we obtain
\begin{eqnarray*}
K_{\B}^3 &=&-8H_{\B}^3-12K_{\B}\cdot H_{\B}^2-6K_{\B}^2\cdot H_{\B} \\
&=& -30K_{\B}\cdot H_{\B}^2+4H_{\B}^3+6c_2(\B)\cdot H_{\B}-72\chi(\O_{\B}(H_{\B}))+72\chi(\O_{\B}) \\
&=& 130\lambda-72g-6d\Delta+72a-1104.
\end{eqnarray*}
Now we conclude comparing the last formula
with Lemma \ref{lemma: double point formula} and
using the relation
\begin{equation}
70\lambda-44g+(7d-1)\Delta-596=c_3(\B)=c_1(\PP^1)c_2(Y)=2c_2(Y).
\end{equation}
\end{proof}
\begin{lemma}\label{lemma: scroll over curve}
If $r=3$, $n=8$, $\langle \B\rangle=\PP^8$ and
$\B$ is a scroll over a smooth curve,
then we have:
$a=6$, $\lambda=6$, $g=0$, $d=2$, $\Delta=14$.
\end{lemma}
\begin{proof}
We have a projection $\beta:(\B,H_{\B})\rightarrow (Y,H_Y)$
over a curve $Y$ such that
$\beta^{\ast}(H_Y)=K_{\B}+3H_{\B}$. By expanding the expressions
$\beta^{\ast}(H_Y)^2\cdot H_{\B}=0$ and $\beta^{\ast}(H_Y)^3=0$ we obtain
$K_{\B}^2\cdot H_{\B}=3\lambda-12g+12$ and
$K_{\B}^3=54(g-1)$,
and hence by Lemma \ref{lemma: double point formula} we get
\begin{equation}\label{eq: relation scroll}
\lambda^2+23\lambda-78g-(7d+1)\Delta-4d+36a-172 = 0.
\end{equation}
Also, by expanding the expression $\chi(\O_{\B}(H_{\B}))=9$ we obtain
$c_2=-35\lambda+30g-12a+294 $
and hence by Proposition \ref{prop: segre and chern classes} we get
\begin{equation}\label{eq: second relation scroll}
22\lambda-20g-d\Delta+12a-176 = 0.
\end{equation}
Now the conclusion follows from Lemma \ref{lemma: r=3 B nondegenerate}.
\end{proof}
Finally we conclude our discussion about classification with the following:
\begin{proposition}\label{prop: r=3 B nondegenerate}
If $r=3$, $n=8$ and $\langle \B\rangle=\PP^8$,
then one of the following cases holds:
\begin{enumerate}[(I)]
\setcounter{enumi}{18}
\item $a=0$, $\lambda=12$, $g=6$, $\B$ is a scroll $\PP_{Y}(\E)$ over a rational surface $Y$ with $K_Y^2=5$, $c_2(\E)=8$ and $c_1^2(\E)=20$;
\item $a=0$, $\lambda=13$, $g=8$, $\B$ is obtained as the blow-up of a Fano variety $X$ at a point $p\in X$, $|H_{\B}|=|H_{X}-p|$;
\item\label{case: cubic hypersurface} $a=1$, $\lambda=11$, $g=5$, $\B$ is the blow-up of $Q^3$ at $5$ points $p_1,\ldots,p_5$, $|H_{\B}|=|2H_{Q^3}-p_1-\cdots-p_5|$;
\item $a=1$, $\lambda=11$, $g=5$, $\B$ is a scroll over $\PP_{\PP^1}(\O\oplus\O(-1))$;
\item $a=1$, $\lambda=12$, $g=7$, $\B$ is a linear section of $S^{10}\subset\PP^{15}$;
\item\label{case: scroll over Q2 or quadric fibration} $a=2$, $\lambda=10$, $g=4$, $\B$ is a scroll over $Q^2$;
\item $a=3$, $\lambda=9$, $g=3$, $\B$ is a scroll over $\PP^2$ or a quadric fibration over $\PP^1$;
\item $a=4$, $\lambda=8$, $g=2$, $\B$ is a hyperplane section of $\PP^1\times Q^3$;
\item $a=6$, $\lambda=6$, $g=0$, $\B$ is a rational normal scroll.
\end{enumerate}
\end{proposition}
\begin{proof}
For $a=6$ the statement follows from \cite{ionescu-smallinvariants}.
The case with $a=5$ is excluded by \cite{ionescu-smallinvariants} and Example
\ref{example: a=5}.
For $a=4$ the statement follows from \cite{ionescu-smallinvariantsIII}.
For $a\in\{2,3\}$,
by \cite{fania-livorni-nine},
\cite{fania-livorni-ten} and
\cite{ionescu-smallinvariantsII}
it follows that the abstract structure of
$\B$ is as asserted, or $a=2$ and $\B$ is a quadric
fibration over $\PP^1$; the last case is
excluded by
Lemma \ref{lemma: quadric fibration}.
For $a=1$ the statement is just Proposition \ref{prop: 3-fold in P8}.
Now we treat the cases with $a=0$.
\begin{case}[$a=0, \lambda=12$]
Since $\deg(\B)\leq 2\mathrm{codim}_{\PP^8}(\B)+2$, it follows that
$(\B,H_{\B})$ must be as in one of the cases
(a),\ldots,(h) of \cite[Theorem~1]{ionescu-degsmallrespectcodim}.
Cases (a), (d), (e), (g), (h) are of course impossible
and case (c) is excluded by Lemma \ref{lemma: quadric fibration}.
If $\B$ is as in case (b),
by Lemma \ref{lemma: scroll over curve}
we obtain that
$\B$ is a scroll over a birationally ruled surface
(hence over a rational surface since $q=0$).
Now suppose that $(\B,H_{\B})$ is as in case (f).
Thus there is a reduction $(X,H_X)$ as in one of the cases:
\begin{enumerate}[(f1)]
\item\label{case1} $X=\PP^3$, $H_X\in|\O(3)|$;
\item\label{case2} $X=Q^3$, $H_X\in|\O(2)|$;
\item\label{case3} $X$ is a $\PP^2$-bundle over a smooth curve such that
$\O_X(H_X)$ induces $\O(2)$ on each fiber.
\end{enumerate}
By definition of reduction we have $X\subset\PP^{N}$,
where $N=8+s$,
$\deg(X)=\lambda+s=12+s$ and
$s$ is the number of points blown up on $X$ to get $\B$.
Case (f\ref{case1}) and (f\ref{case2})
are impossible because
they force $\lambda$ to be
respectively $16$ and $11$.
In case (f\ref{case3}),
we have a projection $\beta:(X,H_{X})\rightarrow (Y,H_Y)$
over a curve $Y$ such that
$\beta^{\ast}(H_Y)=2K_{X}+3H_{X}$.
Hence we get
\begin{displaymath}
K_X H_X^2= (2K_X+3H_X)^2\cdot H_X/12 -K_X^2\cdot H_X/3 - 3H_X^3/4 = -K_X^2\cdot H_X/3 - 3H_X^3/4 ,
\end{displaymath}
from which we deduce that
\begin{eqnarray*}
0&=&(2K_X+3H_X)^3
= 8K_X^3+36K_X^2\cdot H_X+54K_X\cdot H_X^2+27H_X^3 \\
&=& 8K_X^3+18K_X^2\cdot H_X-27 H_X^3/2 \\
&=& 8( K_{\B}^3 - 8s )+18K_X^2\cdot H_X-27 (\deg(\B)+s)/2 \\
&=& 18 K_X^2\cdot H_X-155s/2-210.
\end{eqnarray*}
Since $s\leq 12$ (see \cite[Lemma~8.1]{besana-biancofiore-numerical}),
we conclude that case (f)
does not occur.
Thus, $\B=\PP_{Y}(\E)$ is a scroll over a surface $Y$;
moreover,
by Lemma \ref{lemma: scroll over surface} and
\cite[Theorem~11.1.2]{beltrametti-sommese},
we obtain $K_Y^2=5$, $c_2(\E)=K_Y^2-K_S^2=8$ and $c_1^2(\E)=\lambda+c_2(\E)=20$.
\end{case}
\begin{case}[$a=0, \lambda=13$]
The proof is located in \cite[page~16]{mella-russo-baselocusleq3},
but we sketch it for the reader's convenience.
By Lemma \ref{lemma: r=3 B nondegenerate} we know that
$\chi(\O_S)=2$ and $K_S$ is an exceptional curve of the first kind.
Thus, if we blow-down the divisor $K_S$, we obtain
a $K3$-surface.
By using adjunction theory
(see for instance \cite{beltrametti-sommese} or Ionescu's papers cited in the
bibliography)
and by Lemmas \ref{lemma: quadric fibration},
\ref{lemma: scroll over surface} and \ref{lemma: scroll over curve} it follows that
the adjunction map $\phi_{|K_{\B}+2H_{\B}|}$ is a generically finite morphism;
moreover, since $(K_{\B}+2H_{\B})\cdot K_S=0$, we see that
$\phi_{|K_{\B}+2H_{\B}|}$ is not a finite morphism.
So, we deduce that
there is a $(\PP^2,\O_{\PP^2}(-1))$ inside $\B$
and, after the blow-down of this divisor, we get a smooth Fano $3$-fold
$X\subset\PP^9$ of sectional genus $8$ and degree $14$.
\end{case}
\end{proof}
\section{Examples}\label{sec: examples}
As in \S \ref{sec: examples hypersurface},
in order to verify the calculations in the following examples,
we suggest the use of \cite{macaulay2} or \cite{sagemath}.
\begin{example}[$r=1,2,3; n=3,4,5; a=1; d=1$]\label{example: 1}
As already said in \S \ref{sec: type 2-1},
if $Q\subset\PP^{n-1}\subset\PP^n$ is a smooth quadric, then
the linear system $|\I_{Q,\PP^n}(2)|$ defines a birational transformation
$\psi:\PP^n\dashrightarrow\sS\subset\PP^{n+1}$ of type $(2,1)$ whose image
is a smooth quadric.
\end{example}
\begin{example}[$r=1; n=4; a=0; d=3$]\label{example: 2}
See also \cite{crauder-katz-1989}. If $X\subset\PP^4$
is a nondegenerate curve of genus $1$ and degree $5$, then $X$ is
the scheme-theoretic intersection of the quadrics (of rank $3$) containing $X$ and
$|\I_{X,\PP^4}(2)|$ defines a Cremona transformation
$\PP^4\dashrightarrow\PP^4$ of type $(2,3)$.
\end{example}
\begin{example}[$r=1,2,3; n=4,5,7; a=1,0,1; d=2$]\label{example: 3}
As already said in Example \ref{example: d=2 Delta=2},
if $X\subset\PP^n$
is either $\nu_2(\PP^2)\subset\PP^5$ or $\PP^2\times\PP^2\subset\PP^8$,
then $|\I_{X,\PP^n}(2)|$ defines
a birational transformation
$\psi:\PP^n\dashrightarrow\PP^n$ of type $(2,2)$.
The restriction of $\psi$
to a general hyperplane is a birational transformation
$\PP^{n-1}\dashrightarrow\sS\subset\PP^n$ of type $(2,2)$ and
$\sS$ is a smooth quadric.
\end{example}
\begin{example}[$r=1; n=4; a=2; d=1$ - not satisfying \ref{assumption: ipotesi}]\label{example: B2=singSred}
This essentially gives an example for case (\ref{case: escluso 1-fold}) of Lemma \ref{lemma: numerical 1-fold}.
We have a special birational transformation
$\psi:\PP^4\dashrightarrow\sS\subset\PP^6$ of type $(2,1)$
with base locus $X$, image $\sS$
and base locus of the inverse $Y$, as follows:
\begin{eqnarray*}
X &=& V(x_0x_1-x_2^2-x_3^2,-x_0^2-x_1^2+x_2x_3,x_4), \\
\sS &= & V(y_2y_3-y_4^2-y_5^2-y_0y_6,y_2^2+y_3^2-y_4y_5+y_1y_6), \\
P_{\sS}(t) &=& (4t^4+24t^3+56t^2+60t+24)/4!, \\
\sing(\sS) &=& V(y_6,y_5^2,y_4y_5,y_3y_5,y_2y_5,y_4^2,y_3y_4,y_2y_4,2y_1y_4+y_0y_5, \\
&& y_0y_4+2y_1y_5,y_3^2,y_2y_3,y_2^2,y_1y_2+2y_0y_3,2y_0y_2+y_1y_3), \\
P_{\sing(\sS)}(t) &=& t + 5, \\
(\sing(\sS))_{\mathrm{red}} &=& V(y_6,y_5,y_4,y_3,y_2), \\
Y=(Y)_{\mathrm{red}}&=&(\sing(\sS))_{\mathrm{red}}= V(y_6,y_5,y_4,y_3,y_2).
\end{eqnarray*}
\end{example}
\begin{example}[$r=1,2,3; n=4,5,6; a=3; d=1$]\label{example: 5}
See also \cite{russo-simis} and \cite{semple}.
If $X=\PP^1\times\PP^2\subset\PP^5\subset\PP^{6}$,
then $|\I_{X,\PP^6}(2)|$ defines
a birational transformation
$\psi:\PP^{6}\dashrightarrow \sS\subset\PP^{9}$ of type $(2,1)$
whose base locus is $X$ and whose image is $\sS=\GG(1,4)$.
Restricting $\psi$ to a general $\PP^5\subset\PP^{6}$
(resp. $\PP^4\subset\PP^{6}$)
we obtain a birational transformation
$\PP^5\dashrightarrow\sS\subset\PP^{8}$
(resp. $\PP^4\dashrightarrow\sS\subset\PP^{7}$)
whose image is a smooth linear section of $\GG(1,4)\subset\PP^{9}$.
\end{example}
\begin{example}[$r=2; n=6; a=0; d=4$]\label{example: 6}
See also \cite{crauder-katz-1989} and \cite{hulek-katz-schreyer}.
Let $Z=\{p_1,\ldots,p_8\}$ be
a set of $8$ points in $\PP^2$
such that
no $4$ of the $p_i$ are collinear and no $7$ of the $p_i$ lie on a conic and
consider the blow-up $X=\Bl_Z(\PP^2)$
embedded in $\PP^6$ by $|4H_{\PP^2}-p_1-\cdots-p_8|$.
Then the homogeneous ideal of $X$ is
generated by quadrics and $|\I_{X,\PP^6}(2)|$
defines a Cremona transformation $\PP^6\dashrightarrow\PP^6$ of type $(2,4)$.
The same happens when
$X\subset\PP^6$ is a septic elliptic scroll with $e=-1$.
\end{example}
\begin{example}[$r=2; n=6; a=1; d=3$]\label{example: 7}
As already said in Examples \ref{example: d=3 Delta=2} and \ref{example: d=3 Delta=2 continuing},
if $X\subset\PP^6$ is a general hyperplane section of an
Edge variety of dimension $3$ and degree $7$ in $\PP^7$,
then $|\I_{X,\PP^6}(2)|$ defines a birational transformation
$\psi:\PP^6\dashrightarrow\sS\subset\PP^7$ of type $(2,3)$ whose
image is a rank $6$ quadric.
\end{example}
\begin{example}[$r=2; n=6;a=2;d=2$]\label{example: 8}
If $X\subset\PP^6$
is the blow-up of $\PP^2$ at $3$ general points $p_1,p_2,p_3$ with
$|H_{X}|=|3H_{\PP^2}-p_1-p_2-p_3|$, then
$\Sec(X)$ is a cubic hypersurface.
By Fact \ref{fact: K2 property} and \ref{fact: test K2} we deduce
that $|\I_{X,\PP^6}(2)|$ defines a birational transformation
$\psi:\PP^6\dashrightarrow\sS\subset\PP^8$ and its type is $(2,2)$.
The image $\sS$ is a complete intersection of two quadrics, $\dim(\sing(\sS))=1$ and
the base locus of the inverse is $\PP^2\times\PP^2\subset\PP^8$.
Alternatively, we can obtain the transformation
$\psi:\PP^6\dashrightarrow\sS\subset\PP^8$
by restriction to a general $\PP^6\subset\PP^8$
of the special Cremona transformation $\PP^8\dashrightarrow\PP^8$ of type $(2,2)$.
\end{example}
\begin{example}[$r=2; n=6; a=3; d=2$]\label{example: 9}
See also \cite{russo-simis} and \cite{semple}.
If $X=\PP_{\PP^1}(\O(1)\oplus\O(4))$ or $X=\PP_{\PP^1}(\O(2)\oplus\O(3))$, then
$|\I_{X,\PP^6}(2)|$ defines a birational
transformations
$\psi:\PP^6\dashrightarrow\sS\subset\PP^9$
of type $(2,2)$ whose base locus is $X$ and whose image is
$\sS=\GG(1,4)$.
\end{example}
\begin{example}[$r=2,3;n=6,7;a=5; d=1$]\label{example: 10}
See also \cite[\Rmnum{3} Theorem~3.8]{zak-tangent}.
If $X=\GG(1,4)\subset\PP^9\subset\PP^{10}$, then
$|\I_{X,\PP^{10}}(2)|$ defines a birational transformation
$\psi:\PP^{10}\dashrightarrow \sS\subset\PP^{15}$ of type $(2,1)$
whose base locus is $X$ and whose image is the spinorial variety $\sS=S^{10}\subset\PP^{15}$.
Restricting $\psi$
to a general
$\PP^7\subset\PP^{10}$
(resp. $\PP^6\subset\PP^{10}$)
we obtain a special birational transformation
$\PP^7\dashrightarrow\sS\subset\PP^{12}$
(resp. $\PP^6\dashrightarrow\sS\subset\PP^{11}$)
whose dimension of the base locus is
$r=3$ (resp. $r=2$)
and whose image is a linear section of $S^{10}\subset\PP^{15}$.
In the first case $\sS=\overline{\psi(\PP^7)}$ is smooth while in the second case
the singular locus of $\sS=\overline{\psi(\PP^6)}$ consists of
$5$ lines, image of the $5$ Segre $3$-folds
containing del Pezzo surface of degree $5$
and spanned by its pencils of conics.
\end{example}
\begin{example}[$r=2,3; n=6,7; a=6; d=1$]\label{example: 11}
See also \cite{russo-simis}, \cite{semple} and \cite[\Rmnum{3} Theorem~3.8]{zak-tangent}.
We have a birational transformation
$\psi:\PP^{8}\dashrightarrow\GG(1,5)\subset\PP^{14}$ of type $(2,1)$
whose base locus
is $\PP^1\times\PP^3\subset\PP^7\subset\PP^{8}$ and whose image is $\GG(1,5)$.
Restricting $\psi$
to a general $\PP^7\subset\PP^{8}$
we obtain a birational transformation
$\PP^7\dashrightarrow\sS\subset\PP^{13}$
whose base locus $X$ is
a rational normal scroll
and whose image $\sS$ is a smooth linear section of
$\GG(1,5)\subset\PP^{14}$.
Restricting $\psi$
to a general $\PP^6\subset\PP^{8}$
we obtain a birational transformation
$\psi=\psi|_{\PP^6}:\PP^6\dashrightarrow\sS\subset\PP^{12}$
whose base locus $X$ is
a rational normal scroll
(hence either $X=\PP_{\PP^1}(\O(1)\oplus\O(3))$
or $X=\PP_{\PP^1}(\O(2)\oplus\O(2))$)
and whose image $\sS$ is a singular linear section of
$\GG(1,5)\subset\PP^{14}$. In this case,
we denote by $Y\subset\sS$ the base locus of the inverse of $\psi$ and by
$F=(F_0,\ldots,F_5):\PP^5\dashrightarrow\PP^5$
the restriction of $\psi$ to $\PP^5=\Sec(X)$.
We have
\begin{eqnarray*}
Y&=&\overline{\psi(\PP^5)}=\overline{F(\PP^5)}=\GG(1,3)\subset\PP^5\subset\PP^{12} , \\
J_4&:=&\left\{x=[x_0,...,x_5]\in\PP^5\setminus X:
\rk\left(\left({\partial F_i}/{\partial x_j}(x)\right)_{i,j}\right)\leq 4 \right\}_{\mathrm{red}}\\
&=& \left\{x=[x_0,...,x_5]\in\PP^5\setminus X: \dim\left(\overline{F^{-1}\left(F(x)\right)}\right)\geq2 \right\}_{\mathrm{red}}\mbox{ and }\dim\left(J_4\right) = 3,\\
\overline{\psi\left(J_4\right)} &=& \left(\sing\left(\sS\right)\right)_{\mathrm{red}} =\PP_{\PP^1}(\O(2)) \subset Y. \\
\end{eqnarray*}
\end{example}
\begin{example}\label{example: parzialecremona}
We have a good candidate for the restriction to a general $\PP^6\subset\PP^8$
of a quadratic Cremona transformation
of $\PP^8$ whose base locus is a $3$-fold of
degree $12$ and sectional genus $6$.
In fact, there exists a smooth nondegenerate curve
$C\subset\PP^6$ of degree $12$, genus $6$,
and having homogeneous ideal generated
by $9$ quadrics.
Moreover, if
$\psi:\PP^6\dashrightarrow\mathbf{Z}\subset\PP^{8}$ is
the rational map
defined by $|\I_{C,\PP^6}(2)|$, then the image $\mathbf{Z}$ has degree $14$ and
so $\psi$ is birational. We also have that the homogeneous ideal of $\mathbf{Z}$
is generated by quintics and sextics.
The curve $C\subset\PP^6$
can be constructed as follows:
let $Y\subset\PP^5$ be a del Pezzo surface of degree $5$. Consider
the Veronese embedding $\nu_2:\PP^5\rightarrow\PP^{20}$ and let $Y'=\nu_2(Y)$.
$Y'$ is a surface of degree $20$, sectional genus $6$ and its linear span is a
$\PP^{15}\subset\PP^{20}$.
Thus, via a sequence of $8$ inner projections, we obtain
a surface $S\subset\PP^7$ of degree $12$ and sectional genus $6$.
Now we take a smooth hyperplane section of $S$.
This is what the following Macaulay2 code does.
{\footnotesize
\begin{verbatim}
i1 : ringP2=QQ[z_0..z_2];
i2 : ringP5=QQ[t_0..t_5];
i3 : g=map(ringP2,ringP5,gens image basis(3,intersect(ideal(z_0,z_1),ideal(z_1,z_2),
ideal(z_2,z_0),ideal(z_0-z_1,z_2-z_0))));
i4 : ringP20=QQ[t_0..t_20];
i5 : v=map(ringP5,ringP20,gens (ideal vars ringP5)^2);
i6 : ringP15=ringP20/(ideal image basis(1,saturate kernel(g*v)));
i7 : p=map(ringP20,ringP15,vars ringP20);
i8 : f=g*v*p;
i9 : L=ideal image basis(1,intersect(preimage(f,ideal(z_0,z_1-z_2)),
preimage(f,ideal(z_1,z_2-z_0)),preimage(f,ideal(z_2,z_0-z_1)),
preimage(f,ideal(z_0,z_1+z_2)),preimage(f,ideal(z_1,z_2+z_0)),
preimage(f,ideal(z_2,z_0+z_1)),preimage(f,ideal(z_0-z_1,z_2+z_0)),
preimage(f,ideal(z_0+z_1,z_2-z_0))));
i10 : ringP7=QQ[x_0..x_7];
i11 : h=f*map(ringP15,ringP7,gens(L));
i12 : idealS=saturate kernel h;
i13 : H=sub(idealS, {x_7=>x_0+x_1+x_2+x_3+x_4+x_5+x_6});
i14 : ringP6=QQ[x_0..x_6];
i15 : idealC=saturate sub(H,ringP6);
i16 : C=Proj(ringP6/idealC);
i17 : dim singularLocus C
o17 = -infinity
i18 : dim C
o18 = 1
i19 : degree C
o19 = 12
i20 : genus C
o20 = 6
i21 : numgens idealC
o21 = 9
i22 : ringP8=QQ[y_0..y_8];
i23 : psi=map(ringP6,ringP8,gens idealC);
i24 : Z=Proj(coimage(psi));
i25 : dim Z
o25 = 6
i26 : degree Z
o26 = 14
i27 : DegreeOfpsi=(-5*degree(C)+2*genus(C)+62)/degree(Z)
o27 = 1
\end{verbatim}
}
\end{example}
\begin{example}[$r=3;n=8;a=0;d=5$]\label{example: 12}
See also \cite{hulek-katz-schreyer}. If $\mathcal{X}\subset\PP^9$ is a
general $3$-dimensional
linear section of $\GG(1,5)\subset \PP^{14}$,
$p\in \mathcal{X}$ is a general point and
$X\subset\PP^8$ is the image of $\mathcal{X}$ under the projection from $p$,
then the homogeneous ideal of $X$ is generated by quadrics
and $|\I_{X,\PP^8}(2)|$ defines a Cremona transformation $\PP^8\dashrightarrow\PP^8$
of type $(2,5)$.
\end{example}
\begin{example}[$r=3;n=8;a=1;d=3$]\label{example: 13}
As already said in Example \ref{example: d=3 Delta=3},
if $X\subset\PP^8$
is the blow-up
of the smooth quadric $Q^3\subset\PP^4$
at $5$ general points $p_1,\ldots,p_5$ with
$|H_{X}|=|2H_{Q^3}-p_1-\cdots-p_5|$, then
$|\I_{X,\PP^8}(2)|$ defines a birational transformation
$\psi:\PP^8\dashrightarrow\sS\subset\PP^9$ of type $(2,3)$ whose
image is a cubic hypersurface with singular locus of dimension $3$.
\end{example}
\begin{example}[$r=3;n=8;a=1;d=4$ - incomplete]\label{example: 14}
By \cite{alzati-fania-ruled} (see also \cite{besana-fania-flamini-f1}) there exists
a smooth irreducible nondegenerate linearly normal $3$-dimensional variety
$X\subset\PP^8$ with
$h^1(X,\O_X)=0$,
degree $\lambda=11$, sectional genus $g=5$, having the structure of a
scroll $\PP_{\FF^1}(\E)$ with $c_1(\E)=3C_0+5f$ and $c_2(\E)=10$
and hence having degrees of the Segre classes
$s_1(X)=-85$, $s_2(X)=386$, $s_3(X)=-1330$.
Now, by Fact \ref{fact: test K2}, $X\subset\PP^8$ is
arithmetically Cohen-Macaulay
and by Riemann-Roch Theorem, denoting with $C$ a general curve section of $X$,
we obtain
\begin{equation}\label{eq: riemann-rock}
h^0(\PP^8,\I_X(2))=h^0(\PP^6,\I_C(2))
= h^0(\PP^6,\O_{\PP^6}(2))-h^0(C,\O_C(2))
=28-(2\lambda+1-g),
\end{equation}
hence $h^0(\PP^8,\I_X(2))=10$.
If the homogeneous ideal of $X$ is generated
by quadratic forms\footnote{One says that
a subvariety $X\subset\PP^n$ is $2$-regular (in the sense of Castelnuovo-Mumford)
if $h^j(\PP^n,\I_{X}(2-j))=0$ for all $j>0$.
If $X$ is $2$-regular,
then its homogeneous ideal is generated
by forms of degrees $\leq2$ (see \cite{mumford-curves}).
Now, if $X\subset\PP^8$ is a scroll over $\FF_1$ as above, we have
$h^j(\PP^8,\I_{X}(2-j))=0$ for $j>0$ and $j\neq 4$,
but unfortunately we also have
$h^4(\PP^8,\I_{X}(-2))=h^3(X,\O_{X}(-2))=-P_{X}(-2)=5$.}
or at least if
$X=V(H^0(\I_X(2)))$,
the linear system $|\I_X(2)|$ defines a rational map
$\psi:\PP^8\dashrightarrow\sS=\overline{\psi(\PP^8)}\subset\PP^{9}$
whose base locus is
$X$ and whose image $\sS$ is nondegenerate.
Now, by (\ref{eq: grado mappa razionale}) we deduce $\deg(\psi)\deg(\sS)=2$,
from which
$\deg(\psi)=1$ and $\deg(\sS)=2$.
We have a good candidate for the restriction of $\psi$ to a general $\PP^6\subset\PP^8$.
In fact, there exists a smooth irreducible nondegenerate linearly normal curve
$C\subset\PP^6$ of degree $\lambda=11$, genus $g=5$,
and having homogeneous ideal generated
by the $10$ quadrics:
\begin{equation}\label{eq: 10 quadrics generating C}
\begin{array}{c}
x_5^2-x_4 x_6 , \
-x_3 x_6+x_2 x_6+x_4 x_5-x_0 x_1 , \
x_3 x_5-x_2 x_6 , \
x_2 x_5-x_1 x_6 , \\
x_3 x_4-x_1 x_6 , \
x_2 x_4-x_1 x_5 , \
x_2 x_6-x_1 x_6+x_1 x_5-x_4^2-x_3^2+x_2 x_3+x_0^2 , \\
x_0 x_6-x_1 x_5+x_1 x_3 , \
x_0 x_6-x_1 x_5+x_2^2 , \
x_0 x_5-x_1 x_4+x_1 x_2 .
\end{array}
\end{equation}
The quadrics (\ref{eq: 10 quadrics generating C}) give a rational map
$\psi':\mathbb{P}^6\dashrightarrow\sS'\subset\mathbb{P}^9$ and since
$\deg(\psi')\deg(\sS')=-5\lambda+2g+62=17$, we have that
$\psi'$ is birational. Moreover,
the homogeneous ideal of $\sS'$ is generated
by $6$ quartics and a rank $6$ quadric defined by:
$$
y_3^2-y_3y_4+y_2y_5+y_0y_7-y_0y_8 .
$$
\end{example}
\begin{example}[$r=3;n=8;a=1;d=4$]\label{example: 15}
As already said in Example \ref{example: d=4 Delta=2},
if $X\subset\PP^8$ is
a general linear $3$-dimensional section
of the spinorial variety $S^{10}\subset\PP^{15}$, then
$|\I_{X,\PP^8}(2)|$ defines a
birational transformation
$\psi:\PP^8\dashrightarrow\sS\subset\PP^9$
of type $(2,4)$
whose image is a smooth quadric.
\end{example}
\begin{example}[$r=3;n=8;a=2;d=3$]\label{example: 16}
By \cite{fania-livorni-ten} (see also \cite{besana-fania-threefolds}) there exists
a smooth irreducible nondegenerate linearly normal $3$-dimensional variety
$X\subset\PP^8$ with $h^1(X,\O_X)=0$,
degree $\lambda=10$, sectional genus $g=4$,
having the structure of a scroll
$\PP_{Q^2}(\E)$ with $c_1(\E)=\O_Q(3,3)$ and $c_2(\E)=8$
and hence having degrees of the Segre classes
$s_1(X)=-76$, $s_2(X)=340$, $s_3(X)=-1156$.
By Fact \ref{fact: test K2}, $X\subset\PP^8$ is
arithmetically Cohen-Macaulay and its homogeneous ideal is generated by quadratic forms.
So by (\ref{eq: riemann-rock}) we have $h^0(\PP^8,\I_X(2))=11$ and
the linear system $|\I_X(2)|$ defines a rational map
$\psi:\PP^8\dashrightarrow\sS\subset\PP^{10}$
whose base locus is
$X$ and whose image $\sS$ is nondegenerate.
By (\ref{eq: grado mappa razionale}) it follows that $\deg(\psi)\deg(\sS)=4$
and hence $\deg(\psi)=1$ and $\deg(\sS)=4$.
\end{example}
\begin{example}[$r=3;n=8;a=3;d=2$]\label{example: 17}
By \cite{fania-livorni-nine} (see also \cite{besana-fania-threefolds}) there exists
a smooth irreducible nondegenerate linearly normal $3$-dimensional variety
$X\subset\PP^8$ with $h^1(X,\O_X)=0$,
degree $\lambda=9$, sectional genus $g=3$,
having the structure of a scroll $\PP_{\PP^2}(\E)$ with $c_1(\E)=4$ and $c_2(\E)=7$
and hence having degrees of the Segre classes
$s_1(X)=-67$, $s_2(X)=294$, $s_3(X)=-984$.
By Fact \ref{fact: test K2}, $X\subset\PP^8$ is
arithmetically Cohen-Macaulay and its homogeneous ideal is generated by quadratic forms.
So by (\ref{eq: riemann-rock}) we have $h^0(\PP^8,\I_X(2))=12$ and
the linear system $|\I_X(2)|$ defines a rational map
$\psi:\PP^8\dashrightarrow\sS\subset\PP^{11}$
whose base locus is
$X$ and whose image $\sS$ is nondegenerate.
By (\ref{eq: grado mappa razionale}) it follows that
$\deg(\psi)\deg(\sS)=8$
and in particular
$\deg(\psi)\neq 0$ i.e.
$\psi:\PP^8\dashrightarrow\sS$ is
generically quasi-finite.
Again by Fact \ref{fact: test K2} and Fact \ref{fact: K2 property} it follows
that $\psi$ is birational and hence $\deg(\sS)=8$.
Now we find the equations for such an $X$.
Consider the set of $7$ points in $\PP^2$,
$$T:=\{[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[0,1,1],[1,1,1]\}.$$
The homogeneous ideal of $T$ is generated by $3$ cubics; we have
$\dim |\I_{T,\PP^2}(4)|=7$ and let $v=v_{|\I_{T,\PP^2}(4)|}:\PP^2\dashrightarrow\PP^7$.
The image $S=\overline{v(\PP^2)}\subset\PP^7$ is a smooth nondegenerate surface,
of degree $9$,
sectional genus $3$ and with homogeneous ideal generated by $12$ quadrics.
These $12$ quadrics define a
special quadratic birational map $\psi':\PP^7\dashrightarrow\PP^{11}$
whose image is a complete intersection of $4$ quadrics and whose inverse can be defined
by quadrics.
We then consider the $8$ quadrics defining
$\psi'^{-1}$ and the $4$ quadrics defining $\overline{\psi'(\PP^7)}$.
These $12$ quadrics give a Cremona transformation of $\PP^{11}$.
Explicitly, there is a
Cremona transformation $\phi:\PP^{11}\dashrightarrow\PP^{11}$ defined
by:
\begin{equation}
\begin{array}{c}
x_6x_{10}-x_5x_{11} , \
x_1x_{10}-x_4x_{10}+x_3x_{11} , \
x_6x_8-x_2x_{11} , \
x_5x_8-x_2x_{10} , \\
x_3x_8-x_0x_{10} , \
x_1x_8-x_4x_8+x_0x_{11} , \
x_6x_7-x_1x_9+x_4x_9-x_4x_{11} , \
x_5x_7+x_3x_9-x_4x_{10} , \\
x_2x_7-x_4x_8+x_0x_9 , \
x_1x_5-x_4x_5+x_3x_6 , \
x_2x_3-x_0x_5 , \
x_1x_2-x_2x_4+x_0x_6 .
\end{array}
\end{equation}
The inverse of $\phi$ is defined by:
\begin{equation}
\begin{array}{c}
-y_5y_{10}+y_4y_{11} , \
y_5y_9-y_8y_9+y_6y_{10}-y_1y_{11}+y_7y_{11} , \
y_2y_{10}+y_3y_{11} , \
y_4y_9-y_1y_{10} , \\
-y_8y_9+y_6y_{10}+y_7y_{11} , \
y_3y_9+y_0y_{10} , \
y_2y_9-y_0y_{11} , \
y_4y_6+y_5y_7-y_1y_8 , \\
y_2y_4+y_3y_5 , \
-y_3y_6+y_2y_7-y_0y_8 , \
y_1y_3+y_0y_4 , \
y_1y_2-y_0y_5 .
\end{array}
\end{equation}
The base locus of $\phi$ (resp. $\phi^{-1}$) is a variety
of dimension $6$,
degree $9$,
sectional genus $3$,
and the support of the singular locus is a plane $\PP^2\subset\PP^{11}$.
In particular, by restricting the above Cremona transformation
to a general $\PP^8\subset\PP^{11}$, we obtain an explicit example
of special quadratic birational transformation
$\psi:\PP^8\dashrightarrow\mathbf{S}\subset\PP^{11}$;
that is we obtain explicit equations
for a $3$-fold scroll $X\subset\PP^8$ over $\PP^2$, of degree
$9$ and sectional genus $3$. (For example, by restricting to the subspace
$V(x_0+x_1+x_2+x_5+x_8-x_{11}, x_1+x_2+x_3+x_4+x_7-x_{10}, x_0+x_3+x_4+x_5+x_6-x_9)$,
everything works fine and the image is a complete intersection
with singular locus of dimension $3$.)
The following Macaulay2 code computes the maps $\phi$ and $\phi^{-1}$.
{\footnotesize
\begin{verbatim}
i1 : installPackage("AdjointIdeal");
i2 : installPackage("Parametrization");
i3 : ringP2=QQ[z_0..z_2];
i4 : points=intersect(ideal(z_0,z_1),ideal(z_0,z_2),ideal(z_1,z_2),ideal(z_0,z_1-z_2),
ideal(z_1,z_0-z_2),ideal(z_2,z_0-z_1),ideal(z_0-z_1,z_0-z_2));
i5 : parametr=gens(image(basis(4,points)));
i6 : ringP7=QQ[t_0..t_7];
i7 : idealS=saturate(kernel(map(ringP2,ringP7,parametr)));
i8 : -- You could work with the surface S,
-- but we prefer to work with a sectional curve for efficiency reasons.
ringP6=QQ[t_0..t_6];
i9 : idealC=sub(sub(idealS, {t_7=>0}),ringP6);
i10 : ImInv=invertBirationalMap(ideal(ringP6),gens(idealC));
i11 : ringP11=QQ[x_0..x_11];
i12 : idealX=sub(ideal(image(basis(2,saturate(ideal(ImInv#0)+ImInv#1)))),vars(ringP11));
i13 : X=Proj(ringP11/idealX);
i14 : ImInv2=invertBirationalMap(ideal(ringP11),gens(idealX));
i15 : ringP11'=QQ[y_0..y_11];
i16 : Phi=map(ringP11,ringP11',gens(idealX));
i17 : InversePhi=map(ringP11',ringP11,sub(transpose(ImInv2#0),vars(ringP11')));
\end{verbatim}
}
\end{example}
\begin{example}[$r=3;n=8;a=3;d=3$]\label{example: 17nuovo}
By \cite{fania-livorni-nine} (see also \cite{besana-fania-threefolds}) there exists
a smooth irreducible nondegenerate linearly normal $3$-dimensional variety
$X\subset\PP^8$ with $h^1(X,\O_X)=0$,
degree $\lambda=9$, sectional genus $g=3$,
having the structure
of a quadric fibration over $\PP^1$
and hence having degrees of the Segre classes
$s_1(X)=-67$, $s_2(X)=295$, $s_3(X)=-997$.
By Fact \ref{fact: test K2}, $X\subset\PP^8$ is
arithmetically Cohen-Macaulay and its homogeneous ideal is generated by quadratic forms.
So by (\ref{eq: riemann-rock}) we have $h^0(\PP^8,\I_X(2))=12$ and
the linear system $|\I_X(2)|$ defines a rational map
$\psi:\PP^8\dashrightarrow\sS\subset\PP^{11}$
whose base locus is
$X$ and whose image $\sS$ is nondegenerate.
By (\ref{eq: grado mappa razionale}) it follows that
$\deg(\psi)\deg(\sS)=5$
and hence $\deg(\psi)=1$ and $\deg(\sS)=5$.
Now we find the equations for such an $X$.
Consider the rational normal scroll
$S(1,1,1,2)=\PP_{\PP^1}(\O(1)\oplus\O(1)\oplus\O(1)\oplus\O(2))\subset\PP^8$,
which is defined by the $2\times2$ minors of the matrix:
\begin{displaymath}
\left(\begin{array}{ccccc} x_0 & x_2 & x_4 & x_6 & x_7 \\ x_1 & x_3 & x_5 & x_7 & x_8 \end{array}\right)
\end{displaymath}
Intersect $S(1,1,1,2)$ with the quadric $Q\subset\PP^8$ defined by:
\begin{displaymath}
x_3^2+x_3x_4+x_0x_5+x_1x_5+x_2x_5+x_3x_5+x_1x_6+x_1x_7+x_6x_7+x_7^2+x_1x_8+x_7x_8 .
\end{displaymath}
We have $S(1,1,1,2)\cap Q=X\cup P$, where
$P$ is the linear variety $V(x_8,x_7,x_5,x_3,x_1)$, while $X\subset\PP^8$
is a nondegenerate smooth variety of degree $9$, sectional genus $3$
and having homogeneous ideal generated by $12$ quadrics.
These $12$ quadrics give the birational map
$\psi:\PP^8\dashrightarrow\sS\subset\PP^{11}$ defined by:
\begin{equation}
\begin{array}{c}
x_7^2-x_6x_8 , \
x_5x_7-x_4x_8 , \
x_3x_7-x_2x_8 , \
x_1x_7-x_0x_8 , \\
x_5x_6-x_4x_7 , \
x_3x_6-x_2x_7 , \
x_1x_6-x_0x_7 , \
x_3x_4-x_2x_5 , \
x_1x_4-x_0x_5 , \\
x_3^2+x_0x_5+x_1x_5+2x_2x_5+x_3x_5+x_0x_7+x_6x_7+x_0x_8+x_1x_8+x_6x_8+x_7x_8 , \\
x_2x_3+x_0x_4+2x_2x_4+x_0x_5+x_2x_5+x_0x_6+x_6^2+x_0x_7+x_6x_7+x_0x_8+x_6x_8 , \\
x_1x_2-x_0x_3 .
\end{array}
\end{equation}
The image $\sS$
is defined by the quadrics:
\begin{equation}
\begin{array}{c}
y_6y_7-y_5y_8+y_4y_{11} , \
y_3y_7-y_2y_8+y_1y_{11} , \
y_3y_5-y_2y_6+y_0y_{11} , \\
y_3y_4-y_1y_6+y_0y_8 , \
y_2y_4-y_1y_5+y_0y_7 .
\end{array}
\end{equation}
It coincides with the cone over $\GG(1,4)\subset V(y_9,y_{10})\simeq\PP^9\subset\PP^{11}$.
\end{example}
\begin{example}[$r=3;n=8;a=4;d=2$]\label{example: 18}
Consider the composition
$$
f:\PP^1\times\PP^3\longrightarrow\PP^1\times Q^3\subset \PP^1\times\PP^4\longrightarrow\PP^9,
$$
where the first map is the identity of $\PP^1$
multiplied by
$[z_0,z_1,z_2,z_3]\mapsto [z_0^2,z_0z_1,z_0z_2,z_0z_3,z_1^2+z_2^2+z_3^2]$,
and the last map is
$([t_0,t_1],[y_0,\ldots,y_4])\mapsto [t_0y_0,\ldots,t_0y_4,t_1y_0,\ldots,t_1y_4]
=[x_0,\ldots,x_9]$.
In the equations defining
$\overline{f(\PP^1\times\PP^3)}\subset\PP^{9}$,
by replacing
$x_9$ with $x_0$, we obtain the quadrics:
\begin{equation}\label{equazioni-di-B-a=4}
\begin{array}{c}
-x_0x_3 + x_4x_8, \
-x_0x_2 + x_4x_7, \
x_3x_7 - x_2x_8, \
-x_0x_5 + x_6^2 + x_7^2 + x_8^2, \
-x_0x_1 + x_4x_6, \\
x_3x_6 - x_1x_8, \
x_2x_6 - x_1x_7, \
-x_0^2 + x_1x_6 + x_2x_7 + x_3x_8, \
-x_0^2 + x_4x_5, \
x_3x_5 - x_0x_8, \\
x_2x_5 - x_0x_7, \
x_1x_5 - x_0x_6, \
x_1^2 + x_2^2 + x_3^2 - x_0x_4.
\end{array}
\end{equation}
Denoting with $I$ the ideal generated by the
quadrics (\ref{equazioni-di-B-a=4}) and $X=V(I)$,
we have that $I$ is saturated
and $X$ is smooth.
The linear system $|\I_{X,\PP^8}(2)|$ defines
a birational transformation
$\psi:\PP^8\dashrightarrow\sS\subset \PP^{12}$
whose base locus is $X$ and whose image is the variety $\sS$
with homogeneous ideal generated by:
\begin{equation}
\begin{array}{c}
y_6y_9-y_5y_{10}+y_2y_{11}, \
y_6y_8-y_4y_{10}+y_1y_{11}, \
y_5y_8-y_4y_9+y_0y_{11}, \
y_2y_8-y_1y_9+y_0y_{10}, \\
y_2y_4-y_1y_5+y_0y_6, \
y_2^2+y_5^2+y_6^2+y_7^2-y_7y_8+y_0y_9+y_1y_{10}+y_4y_{11}-y_3y_{12}.
\end{array}
\end{equation}
We have $\deg(\sS)=10$ and $\dim(\sing(\sS))=3$.
The inverse of $\psi:\PP^8\dashrightarrow\sS$
is defined by:
\begin{equation}
\begin{array}{c}
-y_7y_8+y_0y_9+y_1y_{10}+y_4y_{11}, \
y_0y_5+y_1y_6-y_4y_7-y_{11}y_{12}, \
y_0y_2-y_4y_6-y_1y_7-y_{10}y_{12}, \\
-y_1y_2-y_4y_5-y_0y_7-y_9y_{12}, \
-y_0^2-y_1^2-y_4^2-y_8y_{12}, \
-y_3y_8-y_9^2-y_{10}^2-y_{11}^2, \\
-y_3y_4-y_5y_9-y_6y_{10}-y_7y_{11}, \
-y_1y_3-y_2y_9-y_7y_{10}+y_6y_{11}, \
-y_0y_3-y_7y_9+y_2y_{10}+y_5y_{11}.
\end{array}
\end{equation}
Note that $\sS\subset\PP^{12}$
is the intersection of a quadric hypersurface in $\PP^{12}$
with the cone over $\GG(1,4)\subset\PP^9\subset\PP^{12}$.
\end{example}
\begin{example}[$r=3;n=8;a=5$ - with non liftable inverse]\label{example: a=5}
If $X\subset\PP^8$ is the blow-up of $\PP^3$ at a point $p$ with
$|H_{X}|=|2H_{\PP^3}-p|$, then (modulo a change of coordinates)
the homogeneous ideal of $X$ is generated by the quadrics:
\begin{equation}
\begin{array}{c}
x_6x_7-x_5x_8,\
x_3x_7-x_2x_8,\
x_5x_6-x_4x_8,\
x_2x_6-x_1x_8,\
x_5^2-x_4x_7,\
x_3x_5-x_1x_8,\
x_2x_5-x_1x_7,\\
x_3x_4-x_1x_6,\
x_2x_4-x_1x_5,\
x_2x_3-x_0x_8,\
x_1x_3-x_0x_6,\
x_2^2-x_0x_7,\
x_1x_2-x_0x_5,\
x_1^2-x_0x_4 .
\end{array}
\end{equation}
The linear system $|\I_{X,\PP^8}(2)|$ defines a birational transformation
$\psi:\PP^{8}\dashrightarrow\PP^{13}$ whose base locus is $X$ and whose
image is the variety $\sS$ with homogeneous ideal generated by:
\begin{equation}
\begin{array}{c}
y_8y_{10}-y_7y_{12}-y_3y_{13}+y_5y_{13},\
y_8y_9+y_6y_{10}-y_7y_{11}-y_3y_{12}+y_1y_{13},\
y_6y_9-y_5y_{11}+y_1y_{12},\\
y_6y_7-y_5y_8-y_4y_{10}+y_2y_{12}-y_0y_{13},\
y_3y_6-y_5y_6+y_1y_8+y_4y_9-y_2y_{11}+y_0y_{12},\\
y_3y_4-y_2y_6+y_0y_8,\
y_3^2y_5-y_3y_5^2+y_1y_3y_7-y_2y_3y_9+y_2y_5y_9-y_0y_7y_9-y_1y_2y_{10}+y_0y_5y_{10}.
\end{array}
\end{equation}
We have $\deg(\sS)=19$, $\dim(\sing(\sS))=4$ and
the degrees of Segre classes of $X$ are: $s_1=-49$, $s_2=201$, $s_3=-627$. So,
by (\ref{eq: sollevabile}), we deduce that
the inverse of
$\psi:\PP^8\dashrightarrow\sS$ is not liftable;
however, a representative of the equivalence class of $\psi^{-1}$ is defined by:
\begin{equation}
\begin{array}{c}
y_{12}^2-y_{11}y_{13},\
y_8y_{12}-y_6y_{13},\
y_8y_{11}-y_6y_{12},\
-y_6y_{10}+y_7y_{11}+y_3y_{12}-y_5y_{12},\
y_8^2-y_4y_{13},\\
y_6y_8-y_4y_{12},\
y_3y_8-y_2y_{12}+y_0y_{13},\
y_6^2-y_4y_{11},\
y_5y_6-y_1y_8-y_4y_9.
\end{array}
\end{equation}
We also point out that
$\Sec(X)$ has dimension
$6$ and degree $6$ (against Proposition \ref{prop: B is QEL}).
\end{example}
\begin{example}[$r=3;n=8;a=6;d=2$]\label{example: 20}
See also \cite{russo-simis} and \cite{semple}.
If
$X=\PP_{\PP^1}(\O(1)\oplus\O(1)\oplus\O(4))$ or
$X=\PP_{\PP^1}(\O(1)\oplus\O(2)\oplus\O(3))$ or
$X=\PP_{\PP^1}(\O(2)\oplus\O(2)\oplus\O(2))$, then
the linear system
$|\I_{X,\PP^8}(2)|$ defines a birational transformation
$\PP^8\dashrightarrow\sS\subset\PP^{14}$ of type $(2,2)$
whose base locus is $X$ and whose image is $\sS=\GG(1,5)$.
\end{example}
\begin{example}[$r=3; n=8; a=7; d=1$]\label{example: oadpDegree8}
See also \cite[Example~2.7]{ciliberto-mella-russo} and \cite{ionescu-smallinvariantsIII}.
Let $Z=\{p_1,\ldots,p_8\}\subset\PP^2$ be such that
no $4$ of the $p_i$ are collinear and no $7$ of the $p_i$ lie on a conic and
consider the scroll $\PP_{\PP^2}(\mathcal{E})\subset\PP^7$ associated
to the very ample vector bundle $\mathcal{E}$ of rank $2$,
given as an extension by the following exact sequence
$0\rightarrow\O_{\PP^2}\rightarrow\E\rightarrow
\I_{Z,\PP^2}(4)\rightarrow0.$
The homogeneous ideal of $X\subset\PP^7$ is
generated by $7$ quadrics and so
the linear system $|\I_{X,\PP^8}(2)|$ defines a birational transformation
$\psi:\PP^8\dashrightarrow\sS\subset\PP^{15}$ of type $(2,1)$.
Since we
have
$c_1(X)=12$,
$c_2(X)=15$,
$c_3(X)=6$, we deduce
$s_1(\mathcal{N}_{X,\PP^8})=-60$,
$s_2(\mathcal{N}_{X,\PP^8})=267$,
$s_3(\mathcal{N}_{X,\PP^8})=-909$,
and hence $\deg(\sS)=29$, by (\ref{eq: grado mappa razionale}).
The base locus of the inverse of
$\psi$ is $\psi(\PP^7)\simeq \PP^6\subset\sS\subset\PP^{15}$.
We also observe that the restriction of $\psi|_{\PP^7}:\PP^7\dashrightarrow\PP^6$
to a general hyperplane $H\simeq\PP^6\subset\PP^7$
gives rise to a transformation as in Example \ref{example: 6}.
\end{example}
\begin{example}[$r=3; n=8; a=8,9; d=1$]\label{example: edge}
If $X\subset\PP^7\subset\PP^8$ is a $3$-dimensional
Edge variety of degree $7$ (resp. degree $6$), then
$|\I_{X,\PP^8}(2)|$ defines a birational transformation
$\PP^8\dashrightarrow\sS\subset\PP^{16}$
(resp. $\PP^8\dashrightarrow\sS\subset\PP^{17}$) of type $(2,1)$
whose base locus is $X$ and
whose degree of the image is $\deg(\sS)=33$ (resp. $\deg(\sS)=38$).
For memory overflow problems,
we were not able to calculate the scheme $\sing(\sS)$;
however, it is easy to obtain that
$1\leq \dim(\sing(\sS))<\dim(Y)=6$
and $\dim(\sing(Y))=1$, where $Y$ denotes the base locus of the inverse.
\end{example}
\begin{example}[$r=3; n=8; a=10; d=1$]\label{example: oadp10}
See also \cite{russo-simis}, \cite{semple} and \cite[\Rmnum{3} Theorem~3.8]{zak-tangent}.
We have a birational transformation
$\PP^{10}\dashrightarrow\GG(1,6)\subset\PP^{20}$ of type $(2,1)$
whose base locus
is $\PP^1\times\PP^4\subset\PP^9\subset\PP^{10}$ and whose image is $\GG(1,6)$.
Restricting it
to a general $\PP^8\subset\PP^{10}$
we obtain a birational transformation
$\psi:\PP^8\dashrightarrow\sS\subset\PP^{18}$
whose base locus $X$ is
a rational normal scroll
(hence either $X=\PP_{\PP^1}(\O(1)\oplus\O(1)\oplus\O(3))$
or $X=\PP_{\PP^1}(\O(1)\oplus\O(2)\oplus\O(2))$)
and whose image $\sS$ is a linear section of
$\GG(1,6)\subset\PP^{20}$.
We denote by $Y\subset\sS$ the base locus of the inverse of $\psi$ and by
$F=(F_0,\ldots,F_9):\PP^7\dashrightarrow\PP^9$
the restriction of $\psi$ to $\PP^7=\Sec(X)$.
We have
\begin{eqnarray*}
Y&=&\overline{\psi(\PP^7)}=\overline{F(\PP^7)}=\GG(1,4)\subset\PP^9\subset\PP^{18} , \\
J_6&:=&\left\{x=[x_0,...,x_7]\in\PP^7\setminus X:
\rk\left(\left({\partial F_i}/{\partial x_j}(x)\right)_{i,j}\right)\leq 6 \right\}_{\mathrm{red}}\\
&=& \left\{x=[x_0,...,x_7]\in\PP^7\setminus X: \dim\left(\overline{F^{-1}\left(F(x)\right)}\right)\geq2 \right\}_{\mathrm{red}}\mbox{ and }\dim\left(J_6\right) = 5,\\
\overline{\psi\left(J_6\right)} &=& \left(\sing\left(\sS\right)\right)_{\mathrm{red}} \subset Y\mbox{ and }\dim\left(\overline{\psi\left(J_6\right)}\right) = 3. \\
\end{eqnarray*}
\end{example}
\section{Summary results}\label{sec: table}
\begin{theorem}\label{theorem: classification}
Table \ref{tabella: all cases 3-fold}
classifies all special quadratic transformations $\varphi$
as in \S \ref{sec: notation} and with $r\leq3$.
\end{theorem}
As a consequence,
we generalize Corollary \ref{prop: classification type 2-3 into cubic}.
\begin{corollary}\label{corollary: coindex 2}
Let $\varphi:\PP^n\dashrightarrow\sS\subseteq\PP^{n+a}$ be
as in \S \ref{sec: notation}. If $\varphi$ is of type $(2,3)$ and
$\sS$ has coindex $c=2$,
then $n=8$, $r=3$ and one of the following cases holds:
\begin{itemize}
\item $\Delta=3$, $a=1$, $\lambda=11$, $g=5$, $\B$ is the blow-up of $Q^3$ at $5$ points;
\item $\Delta=4$, $a=2$, $\lambda=10$, $g=4$, $\B$ is a scroll over $Q^2$;
\item $\Delta=5$, $a=3$, $\lambda=9$, $g=3$, $\B$ is a quadric fibration over $\PP^1$.
\end{itemize}
\end{corollary}
\begin{proof}
We have that $\B\subset\PP^n$ is a $QEL$-variety of type
$\delta=(r-d-c+2)/d=(r-3)/3$ and $n=((2d-1)r+3d+c-2)/d=(5r+9)/3$.
From Divisibility Theorem (Theorem \ref{prop: divisibility theorem}), we deduce
$(r,n,\delta)\in\{(3,8,0),(6,13,1),(9,18,2)\}$ and
from the classification
of $CC$-manifolds (Theorem \ref{prop: classification CC-varieties}),
we obtain $(r,n,\delta)=(3,8,0)$. Now we apply the results in \S \ref{sec: dim 3}.
\end{proof}
In the same fashion, one can prove the following:
\begin{proposition}
Let $\varphi$ be as in \S \ref{sec: notation} and of type $(2,1)$. If
$c=2$, then $r\geq1$ and $\B$ is $\PP^1\times\PP^2\subset\PP^5$
or one of its linear sections.
If $c=3$, then $r\geq2$ and $\B$ is either
$\PP^1\times\PP^3\subset\PP^7$
or $\GG(1,4)\subset\PP^9$
or one of their linear sections.
If $c=4$, then $r\geq3$ and $\B$ is either an $OADP$ $3$-fold in $\PP^7$
or $\PP^1\times\PP^4\subset\PP^{9}$
or one of its hyperplane sections.
\end{proposition}
\begin{remark}
Imitating the proof of Proposition \ref{prop: invariants d=2 Delta=3}
(resp. Proposition \ref{prop: invariants d=2 Delta=4}),
one can also compute all possible Hilbert polynomials and
Hilbert schemes of lines through a general point
of the base locus of a
special quadratic birational
transformation
of type $(2,2)$ into
a complete intersection of two quadrics
(resp. three quadrics).
\end{remark}
In Table \ref{tabella: all cases 3-fold} we use the following shortcuts:
\begin{description}
\item[$\exists^{\ast}$] flags cases for which
is known a transformation $\varphi$ with base locus $\B$ as required,
but we do not know if the image $\sS$
satisfies all the assumptions in \S \ref{sec: notation};
\item[$\exists^{\ast\ast}$] flags cases for which
is known that there is a smooth irreducible variety $X\subset\PP^n$
such that, if $X=V(H^0(\I_X(2)))$, then the linear system $|\I_X(2)|$ defines
a birational transformation
$\varphi:\PP^n\dashrightarrow\sS=\overline{\varphi(\PP^n)}\subset\PP^{n+a}$
as stated;
\item[$?$] flags cases for which we do not know if there exists at least
an abstract variety
$\B$ having the structure and the invariants required;
\item[$\exists$] flags cases for which everything works fine.
\end{description}
\begin{table}
\centering
\tabcolsep=5.4pt
\begin{tabular}{|c||c|c|c|c|c|c|c|c||ll|}
\hline
$r$ & $n$ & $a$ & $\lambda$ & $g$ & Abstract structure of $\B$ & $d$ & $\Delta$ & $c$ & \multicolumn{2}{|c|}{Existence} \\
\hline
\hline
\multirow{4}{*}{$1$} & $3$ & $1$ & $2$ & $0$ & $\nu_2(\PP^1)\subset\PP^2$ & $1$ & $2$ & $1$ & $\exists$ & Ex. \ref{example: 1} \\
\cline{2-11}
& $4$ & $0$ & $5$ & $1$ & Elliptic curve & $3$ & $1$ & $0$ & $\exists$ & Ex. \ref{example: 2} \\
\cline{2-11}
& $4$ & $1$ & $4$ & $0$ & $\nu_4(\PP^1)\subset\PP^4$ & $2$ & $2$ & $1$ & $\exists$ & Ex. \ref{example: 3} \\
\cline{2-11}
& $4$ & $3$ & $3$ & $0$ & $\nu_3(\PP^1)\subset\PP^3$ & $1$ & $5$ & $2$ & $\exists$ & Ex. \ref{example: 5} \\
\hline
\hline
\multirow{14}{*}{$2$} & $4$ & $1$ & $2$ & $0$ & $\PP^1\times\PP^1\subset\PP^3$ & $1$ & $2$ & $1$ & $\exists$ & Ex. \ref{example: 1} \\
\cline{2-11}
& $5$ & $0$ & $4$ & $0$ & $\nu_2(\PP^2)\subset\PP^5$ & $2$ & $1$ & $0$ & $\exists$ & Ex. \ref{example: 3} \\
\cline{2-11}
& $5$ & $3$ & $3$ & $0$ & Hyperplane section of $\PP^1\times\PP^2\subset\PP^5$ & $1$ & $5$ & $2$ & $\exists$ & Ex. \ref{example: 5} \\
\cline{2-11}
& $6$ & $0$ & $7$ & $1$ & Elliptic scroll $\PP_{C}(\E)$ with $e(\E)=-1$ & $4$ & $1$ & $0$& $\exists$ & Ex. \ref{example: 6} \\
\cline{2-11}
& $6$ & $0$ & $8$ & $3$ & \begin{tabular}{c} Blow-up of $\PP^2$ at $8$ points $p_1,\ldots,p_8$,\\ $|H_{\B}|=|4H_{\PP^2}-p_1-\cdots-p_8|$ \end{tabular} & $4$ & $1$ & $0$&$\exists$ & Ex. \ref{example: 6} \\
\cline{2-11}
& $6$ & $1$ & $7$ & $2$ & \begin{tabular}{c} Blow-up of $\PP^2$ at $6$ points $p_0,\ldots,p_5$,\\ $|H_{\B}|=|4H_{\PP^2}-2p_0-p_1-\cdots-p_5|$ \end{tabular} & $3$ & $2$ & $1$& $\exists$ & Ex. \ref{example: 7} \\
\cline{2-11}
& $6$ & $2$ & $6$ & $1$ & \begin{tabular}{c} Blow-up of $\PP^2$ at $3$ points $p_1,p_2,p_3$,\\ $|H_{\B}|=|3H_{\PP^2}-p_1-p_2-p_3|$ \end{tabular} & $2$ & $4$ & $2$& $\exists$ & Ex. \ref{example: 8} \\
\cline{2-11}
& $6$ & $3$ & $5$ & $0$ & $\PP_{\PP^1}(\O(1)\oplus\O(4))$ or $\PP_{\PP^1}(\O(2)\oplus\O(3))$ & $2$ & $5$ & $2$ & $\exists$ & Ex. \ref{example: 9} \\
\cline{2-11}
& $6$ & $5$ & $5$ & $1$ & \begin{tabular}{c} Blow-up of $\PP^2$ at $4$ points $p_1\ldots,p_4$,\\ $|H_{\B}|=|3H_{\PP^2}-p_1-\cdots-p_4|$ \end{tabular} & $1$ & $12$ & $3$ & $\exists$ & Ex. \ref{example: 10} \\
\cline{2-11}
& $6$ & $6$ & $4$ & $0$ & $\PP_{\PP^1}(\O(1)\oplus\O(3))$ or $\PP_{\PP^1}(\O(2)\oplus\O(2))$ & $1$ & $14$ & $3$ & $\exists$ & Ex. \ref{example: 11} \\
\hline
\hline
\multirow{23}{*}{$3$} & $5$ & $1$ & $2$ & $0$ & $Q^3\subset\PP^4$ & $1$ & $2$ & $1$&$\exists$ & Ex. \ref{example: 1} \\
\cline{2-11}
& $6$ & $3$ & $3$ & $0$ & $\PP^1\times\PP^2\subset\PP^5$ & $1$ & $5$ & $2$& $\exists$ & Ex. \ref{example: 5} \\
\cline{2-11}
& $7$ & $1$ & $6$ & $1$ & Hyperplane section of $\PP^2\times\PP^2\subset\PP^8$ & $2$ & $2$ & $1$& $\exists$ & Ex. \ref{example: 3} \\
\cline{2-11}
& $7$ & $5$ & $5$ & $1$ & Linear section of $\GG(1,4)\subset\PP^9$ & $1$ & $12$ &$3$ & $\exists$ & Ex. \ref{example: 10} \\
\cline{2-11}
& $7$ & $6$ & $4$ & $0$ & $\PP_{\PP^1}(\O(1)\oplus\O(1)\oplus\O(2))$ & $1$ & $14$ &$3$ & $\exists$ & Ex. \ref{example: 11} \\
\cline{2-11}
& $8$ & $0$ & $12$ & $6$ & \begin{tabular}{c} Scroll $\PP_{Y}(\E)$, $Y$ rational surface, \\ $K_Y^2=5$, $c_2(\E)=8$, $c_1^2(\E)=20$ \end{tabular} & $5$ & $1$ & $0$ & $?$ & Ex. \ref{example: parzialecremona} \\
\cline{2-11}
& $8$ & $0$ & $13$ & $8$ & \begin{tabular}{c} Variety obtained as the projection \\ of a Fano variety $X$ from a point $p\in X$ \end{tabular} & $5$ & $1$ & $0$ & $\exists$ & Ex. \ref{example: 12} \\
\cline{2-11}
& $8$ & $1$ & $11$ & $5$ & \begin{tabular}{c} Blow-up of $Q^3$ at $5$ points $p_1,\ldots,p_5$, \\ $|H_{\B}|=|2H_{Q^3}-p_1-\cdots-p_5|$ \end{tabular} & $3$ & $3$ & $2$ & $\exists$ & Ex. \ref{example: 13} \\
\cline{2-11}
& $8$ & $1$ & $11$ & $5$ & Scroll over $\PP_{\PP^1}(\O\oplus\O(-1))$ & $4$ & $2$& $1$ & $\exists^{\ast\ast}$ & Ex. \ref{example: 14} \\
\cline{2-11}
& $8$ & $1$ & $12$ & $7$ & Linear section of $S^{10}\subset\PP^{15}$ & $4$ & $2$& $1$ &$\exists$ & Ex. \ref{example: 15} \\
\cline{2-11}
& $8$ & $2$ & $10$ & $4$ & Scroll over $Q^2$ & $3$ & $4$ & $2$ & $\exists^{\ast}$ & Ex. \ref{example: 16} \\
\cline{2-11}
& $8$ & $3$ & $9$ & $3$ & Scroll over $\PP^2$ & $2$ & $8$ & $3$ & $\exists$ & Ex. \ref{example: 17} \\
\cline{2-11}
& $8$ & $3$ & $9$ & $3$ & Quadric fibration over $\PP^1$ & $3$ & $5$ & $2$ & $\exists^{\ast}$ & Ex. \ref{example: 17nuovo} \\
\cline{2-11}
& $8$ &$4$ & $8$ & $2$ & Hyperplane section of $\PP^1\times Q^3$ & $2$ & $10$ & $3$ & $\exists^{\ast}$ & Ex. \ref{example: 18} \\
\cline{2-11}
& $8$ &$6$ & $6$ & $0$ & Rational normal scroll & $2$ & $14$ & $3$ & $\exists$ & Ex. \ref{example: 20} \\
\cline{2-11}
& $8$ & $7$ & $8$ & $3$ & \begin{tabular}{c} $\PP_{\PP^2}(\E)$, where $0\rightarrow\O_{\PP^2}\rightarrow$ \\$\rightarrow\E\rightarrow \I_{\{p_1,\ldots,p_8\},\PP^2}(4)\rightarrow0$ \end{tabular} & $1$ & $29$ & $4$ & $\exists^{\ast}$ & Ex. \ref{example: oadpDegree8} \\
\cline{2-11}
& $8$ & $8$ & $7$ & $2$ & Edge variety & $1$ & $33$ & $4$ & $\exists^{\ast}$ & Ex. \ref{example: edge} \\
\cline{2-11}
& $8$ & $9$ & $6$ & $1$ & $\PP^1\times\PP^1\times\PP^1\subset\PP^7$ & $1$ & $38$ & $4$ & $\exists^{\ast}$ & Ex. \ref{example: edge} \\
\cline{2-11}
& $8$ & $10$ & $5$ & $0$ & Rational normal scroll & $1$ & $42$ & $4$ & $\exists$ & Ex. \ref{example: oadp10} \\
\hline
\end{tabular}
\caption{All transformations $\varphi$ as in \S \ref{sec: notation} and with $r\leq3$}
\label{tabella: all cases 3-fold}
\end{table}
| 68,042
|
TITLE: If $k \neq i \Rightarrow \sum_{j=1}^n a_{kj}A_{ij}=0$
QUESTION [1 upvotes]: The $(i,j)$ cofactor $A_{ij}$ is defined in terms of the minor by $A_{ij} = (-1)^{i+j}\det(M_{ij})$
Corollary. If $k \neq i \Rightarrow \sum_{j=1}^n a_{kj}A_{ij}=0$.
Now proof in my lecture notes is this. Let's $A$ be matrix such that $\forall j[a_{ij} = a_{kj}] $
Then $\det A = \sum_{j=1}^n a_{ij}A_{ij}= \sum_{j=1}^n a_{kj}A_{ij}$ from here $\Rightarrow$ $\det A=0$.
Now my question is what if $A$ is not a matrix such that $\forall j[a_{ij} = a_{kj}] $? If this doesn't hold for this $A$ than how corollary is true?
REPLY [0 votes]: What the author means is this: copy the elements of $A$ to a new matrix $B$. Then, replace the $i$-row of $B$ by the $k$-th row of $A$. Now we have $b_{ij}=a_{kj}$ (because the $i$-th row of $B$ is the $k$-row of $A$) and $B_{ij}=A_{ij}$ (because $A$ and $B$ differ only in the $i$-rows). Therefore, by Laplace expansion along the $i$-th row,
$$
\det(B)=\sum_{j=1}^nb_{ij}B_{ij}=\sum_{j=1}^na_{kj}A_{ij}.
$$
However, $B$ has two identical rows (both the $i$-th and $k$-th rows of $B$ are equal to the $k$-th row of $A$), $\det(B)$ must be zero. Therefore $\sum_{j=1}^na_{kj}A_{ij}=0$.
| 189,253
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