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TITLE: Symmetric function and product of two functions
QUESTION [1 upvotes]: Question: [amended referring to the comments below]
In reading up on a nice result among generative mechanisms for outcomes that are described by power law distributions, I came across this paper, where the central claim is that given a function $f(x, y)$ that governs the link formation probability between two nodes, one can derive the conditions under which a scale-free network structure emerges.
The authors assert that $f(x,y)$ is a function that is symmetric with respect to its arguments and consider the case when $f(x, y) = g(x)h(y)$. Based on this, they conclude that $g(x) \equiv h(x)$? I am trying to trace the arguments to support this assertion.
Attempt:
I started by assuming that $g(x) \neq h(x)$ for any $x$. So, let $g(x) = \lambda h(x), \lambda \in \mathbb{R}$. Now, $f(x, y) = \lambda h(x)h(y)$ and $f(y, x) = \lambda h(y) h(x)$. This means that $f(x, y) = f(y, x)$ irrespective of $\lambda$, i.e., we are unable to conclude that the original assumption of unequal $g$ and $h$ is true unless $\lambda = 1$.
Any other approach to prove the question?
REPLY [2 votes]: This is not true at all! For example, the function $f(x,y) \equiv 2$ is symmetric, obviously, and can also be written as $f(x,y) = g(x)h(y)$ where $g(x) \equiv 2$ and $h(x) \equiv 1$.
It is true that $g(x)h(y) = g(y)h(x)$ for all $x,y$ by symmetry. Only if, we furthermore suppose there is some $X$ such that $g(X) = h(X) \neq 0$, then for all $Y$, $g(Y) = h(Y)$ by cancellation. However, this may not always happen, as the example above shows. | 136,736 |
Full original article can be found on USA Today by Dalvin Brown. or aluminum.
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TITLE: Finite morphism $f:X \to \mathbb{P}_k^n$ is surjective?
QUESTION [1 upvotes]: Let $X$ be an $n$-dimensional projective $k$-scheme and let $f: X \to \mathbb{P}_k^n$ be a finite morphism.
Is $f$ necessarily surjective?
If not, then what else do we need to impose such that $f$ is surjective?
What I tried:
If $f$ is dominant, then $f(X)$ is dense and closed and hence equal to $\mathbb{P}_k^n$. Now $f$ is dominant if and only if $f(X)$ contains the generic point of $\mathbb{P}_k^n$.
Since $f$ is finite, $f(X)$ is closed and thus by the irreducibility of $\mathbb{P}_k^n$
$$
f(X) = \mathbb{P}_k^n \quad \Leftrightarrow \quad \operatorname{dim}f(X) = \operatorname{dim}\mathbb{P}_k^n = n
$$
Now if finite morphism do (to some extent) preserve dimensions, then this could be an approach.
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Since $f\vert X_0:X_0\to Y$ is finite and surjective we have $n=\operatorname {dim}(X_0)=\operatorname {dim} (Y)$.
But now the inclusion of irreducible varieties $Y\subset \mathbb P^n_k$ both of dimension $n$ forces $f(X_0)=Y=\mathbb P^n_k$, so that a fortiori $f(X)=\mathbb P^n_k$, i.e. $f$ is surjective as desired. | 157,165 |
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Who's Who in Hospitality
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\begin{document}
\maketitle
\begin{abstract}
In this paper we introduce and study the concept of cyclic
subgroup commutativity degree of a finite group $G$. This quantity
measures the probability of two random cyclic subgroups of $G$
commuting. Explicit formulas are obtained for some particular
classes of groups. A criterion for a finite group to be an Iwasawa group
is also presented.
\end{abstract}
\noindent{\bf MSC (2010):} Primary 20D60, 20P05; Secondary 20D30,
20F16, 20F18.
\noindent{\bf Key words:} cyclic subgroup commutativity degree,
subgroup commu\-ta\-ti\-vity degree, poset of cyclic subgroups,
subgroup lattice.
\section{Introduction}
In the last years there has been a growing interest in the use of
probability in finite group theory. One of the most important
aspects which have been studied is the probability that two
elements of a finite group $G$ commute. It is called the {\it
commutativity degree} of $G$ and has been investigated in many
papers, such as \cite{2,3} and \cite{5}--\cite{9}. Inspired by
this concept, in \cite{16} we introduced a similar notion for the
subgroups of $G$, called the {\it subgroup commutativity degree}
of $G$. This quantity is defined by
$$\barr{lcl} sd(G)&=&\frax1{|L(G)|^2}\,\left|\{(H,K)\in L(G)^2\mid
HK=KH\}\right|=\vsp &=&\frax1{|L(G)|^2}\,\left|\{(H,K)\in
L(G)^2\mid HK\in L(G)\}\right|\earr\0(1)$$ (where $L(G)$ denotes
the subgroup lattice of $G$) and it measures the probability that
two subgroups of $G$ commute, or equivalently the probability that
the product of two subgroups of $G$ be a subgroup of $G$ (recall
also the natural generalization of $sd(G)$, namely the {\it
relative subgroup commutativity degree} of a subgroup of $G$,
introduced and studied in \cite{17}).
\bigskip
Another two probabilistic notions on $L(G)$ have been investigated
in \cite{21} and \cite{18}: the {\it normality degree} and the
{\it cyclicity degree} of $G$. They are defined by
$$ndeg(G)=\dd\frac{|N(G)|}{|L(G)|}\hspace{1mm}\mbox{ and }\hspace{1mm} cdeg(G)=\dd\frac{|L_1(G)|}{|L(G)|}\,,$$where $N(G)$
and $L_1(G)$ denote the normal subgroup lattice and the poset of
cyclic subgroups of $G$, and measure the probability of a random
subgroup of $G$ to be normal or cyclic, respectively.
\bigskip
Clearly, in the definition of $sd(G)$ we may restrict to one of
the above remarkable subsets of $L(G)$. In the case of $N(G)$
nothing can be said, since normal subgroups commute with all
subgroups of $G$. By taking $L_1(G)$ instead of $L(G)$ in (1) a
new significant quantity is obtained, namely
$$\barr{lcl} csd(G)&=&\frax1{|L_1(G)|^2}\,\left|\{(H,K)\in L_1(G)^2\mid
HK=KH\}\right|=\vsp &=&\frax1{|L_1(G)|^2}\,\left|\{(H,K)\in
L_1(G)^2\mid HK\in L(G)\}\right|.\earr$$This measures the
probability that two cyclic subgroups of $G$ commute and will be
called the {\it cyclic subgroup commutativity degree} of $G$. Its
study is the purpose of the current paper.
\bigskip
The paper is organized as follows. Some basic properties and
results on cyclic subgroup commutativity degree are presented in
Section 2. Section 3 deals with cyclic subgroup commutativity
degrees for some special classes of finite groups: $P$-groups,
dihedral groups and $p$-groups possessing a cyclic maximal
subgroup. As an application, in Section 4 we give a criterion
for a finite group to be an Iwasawa group. In the final section some
further research directions and a list of open problems are indicated.
\bigskip
Most of our notation is standard and will usually not be repeated
here. Elementary notions and results on groups can be found in
\cite{4,14}. For subgroup lattice concepts we refer the reader to
\cite{13,15,20}.
\section{Basic properties of cyclic subgroup\\ commutativity degree}
Let $G$ be a finite group. First of all, we remark that the cyclic
subgroup commutativity degree $csd(G)$ satisfies the following
relation $$0<csd(G)\le 1.$$Moreover, by consequence (9) on page
202 of \cite{13}, the permutability of a subgroup $H\in L_1(G)$
with all cyclic subgroups of $G$ is equivalent with the
permutability of $H$ with all subgroups of $G$. This shows that
$$csd(G)=1\Longleftrightarrow sd(G)=1$$and therefore the finite
groups $G$ satisfying $csd(G)=1$ are in fact the Iwasawa groups,
i.e. the nilpotent modular groups (see [13, Exercise 3, p. 87]).
Notice that a complete description of these groups is given by a
well-known Iwasawa's result (see Theorem 2.4.13 of \cite{13}). In
particular, we infer that $csd(G)=1$ for all Dedekind groups $G$.
\bigskip
Given $H\in L_1(G)$, we will denote by $C_1(H)$ the set consisting
of all cyclic subgroups of $G$ commuting with $H$, that is
$$C_1(H)=\{K\in L_1(G)\mid HK=KH\}.$$Then
$$csd(G)=\frax1{|L_1(G)|^2}\dd\sum_{H\in L_1(G)}|C_1(H)|,\0(2)$$which
leads to a precise expression of $csd(G)$ for finite groups $G$
whose cyclic subgroup structure is known.
\bigskip\noindent{\bf Example 2.1.} The alternating group $A_4$ has
eight cyclic subgroups, namely: the trivial subgroup $H_1$, three
subgroups $H_i\cong\mathbb{Z}_2$, $i=2,3,4$, and four subgroups
$H_i\cong\mathbb{Z}_3$, $i=5,6,7,8$. We can easily see that
$|C_1(H_1)|=8$, $|C_1(H_i)|=4$ for $i=\ov{2,4}$, and
$|C_1(H_i)|=5$ for $i=\ov{5,8}$. Hence
$$csd(A_4)=\frax{1}{64}\left(8+3\cdot4+4\cdot5\right)=\frax{5}{8}\,.$$
\bigskip
Clearly, we have $L(H)\cup\left(N(G)\cap L_1(G)\right)\subseteq
C_1(H)$, $\forall\hspace{1mm} H\in L_1(G)$, implying that
$$csd(G)\geq\frax1{|L_1(G)|^2}\dd\sum_{H\in L_1(G)}|L(H)\cup\left(N(G)\cap
L_1(G)\right)|.$$By this inequality some lower bounds for $csd(G)$
can be inferred, namely
$$csd(G)\geq\frax1{|L_1(G)|^2}\dd\sum_{H\in L_1(G)}|N(G)\cap
L_1(G)|=\frax{|N(G)\cap L_1(G)|}{|L_1(G)|}$$and
$$csd(G)\geq\frax1{|L_1(G)|^2}\dd\sum_{H\in
L_1(G)}|L(H)|\geq\frax{2|L_1(G)|-1}{|L_1(G)|^2}\,,$$since
$|L(H)|\geq 2$\, for every non-trivial cyclic subgroup $H$ of $G$.
Another lower bound for $csd(G)$ follows by the simple remark that
for every subgroup $M$ of $G$ we have
$$\{(H,K)\in L_1(G)^2\mid HK=KH\}\supseteq\{(H,K)\in L_1(M)^2\mid
HK=KH\}.$$Thus
$$csd(G)\geq\left(\frax{|L_1(M)|}{|L_1(G)|}\right)^2 csd(M).$$In
particular, if $M$ is abelian, then $csd(M)=1$ and so
$$csd(G)\geq\left(\frax{|L_1(M)|}{|L_1(G)|}\right)^2.$$
Assume next that $G$ and $G'$ are two finite groups. If $G\cong
G'$, then $csd(G)=csd(G')$. The same thing cannot be said in the
case when $G$ and $G'$ are only lattice-isomorphic, as shows the
following elementary example.
\bigskip\noindent{\bf Example 2.2.} It is well-known that the subgroup lattices of
$G=\mathbb{Z}_3\times\mathbb{Z}_3$ and $G'=S_3$ are isomorphic. On
the other hand, we have $csd(G)=1$ because $G$ is abelian, but
$csd(G')\neq 1$ because $G'$ is not nilpotent (more precisely, we
can easily check that $csd(G')=19/25$).
\bigskip
By a direct calculation, one obtains
$$csd(S_3\times\mathbb{Z}_3)=\frax{85}{121}\ne\frax{19}{25}=csd(S_3)csd(\mathbb{Z}_3)$$and
consequently in general we don't have $csd(G\times
G')=csd(G)csd(G')$. A sufficient condition in order to this
equality holds is that $G$ and $G'$ be of coprime orders. This
remark can naturally be extended to arbitrary finite direct
products.
\bigskip\noindent{\bf Proposition 2.3.} {\it Let
$(G_i)_{i=\overline{1,k}}$ be a family of finite groups having
coprime orders. Then
$$csd(\xmare{i=1}k G_i)=\prod_{i=1}^k csd(G_i).\0(3)$$}
\bigskip
The following immediate consequence of Proposition 2.3 shows that
computing the cyclic subgroup commutativity degree of a finite
nilpotent group is reduced to finite $p$-groups.
\bigskip\noindent{\bf Corollary 2.4.} {\it If $G$ is a finite
nilpotent group and $(G_i)_{i=\ov{1,k}}$ are the Sylow subgroups
of $G$, then $$csd(G)=\prod_{i=1}^k csd(G_i).$$}
\bigskip\noindent{\bf Remark 2.5.} The condition in the hypothesis of
Proposition 2.3 is not necessary to obtain the equality (3). For
example, we have
$$csd(S_3\times\mathbb{Z}_2)=\frax{19}{25}=csd(S_3)csd(\mathbb{Z}_2),$$even
if the groups $S_3$ and $\mathbb{Z}_2$ are not of coprime orders.
\section{Cyclic subgroup commutativity degrees\\ for some classes of finite groups}
In this section we will compute explicitly the cyclic subgroup
commutativity degree of several semidirect products for which we
are able to describe the cyclic subgroup structure.
\bigskip
\bigskip
\noindent{\bf 3.1. The cyclic subgroup commutativity degree of
finite $P$-groups}
\bigskip
First of all, we recall the notion of $P$-group, according to
\cite{13}. Let $p$ be a prime, $n\geq 2$ be a cardinal number and
$G$ be a group. We say that $G$ belongs to the class $P(n,p)$ if
it is either elementary abelian of order $p^n$, or a semidirect
product of an elementary abelian normal subgroup $M$ of order
$p^{n-1}$ by a group of prime order $q\neq p$ which induces a
nontrivial power automorphism on $M$. The group $G$ is called a
$P$-$group$ if $G\in P(n,p)$ for some prime $p$ and some cardinal
number $n\geq 2$. It is well-known that the class $P(n,2)$
consists only of the elementary abelian group of order $2^n$.
Also, for $p>2$ the class $P(n,p)$ contains the elementary abelian
group of order $p^n$ and, for every prime divisor $q$ of $p-1$,
exactly one non-abelian $P$-group with elements of order $q$.
Moreover, the order of this group is $p^{n-1}q$ if $n$ is finite.
The most important property of the groups in a class $P(n,p)$ is
that they are all lattice-isomorphic (see Theorem 2.2.3 of
\cite{13}).
\bigskip
In the following, we will focus on finite non-abelian $P$-groups.
So, assume that $p>2$ and $n\in\mathbb{N}$ are fixed, and take a
divisor $q$ of $p-1$. The non-abelian group of order $p^{n-1}q$ in
the class $P(n,p)$ will be denoted by $G_{n,p}$. By Remarks 2.2.1
of \cite{13}, it is of type
$$G_{n,p}=M\langle x\rangle,$$ where $M\cong \mathbb{Z}_p^{n-1}$ (i.e. the direct product of $n-1$ copies
of $\mathbb{Z}_p$), $o(x)=q$ and there exists an integer $r$ such
that $x^{-1}yx=y^r$, for all $y\in M$. Notice that we have
$$N(G_{n,p})=L(M)\cup\{G_{n,p}\}.$$The set $L_1(G_{n,p})$ has been described in \cite{15}: it consists of the trivial subgroup
1, of the subgroups of order $p$ in $M$ and of the subgroups of
type $\langle yx\rangle$ with $y\in M$. Then
$$|L_1(G_{n,p})|=1+\frax{p^{n-1}-1}{p-1}+p^{n-1}=2+p+p^2+...+p^{n-1}.$$On the other hand, we
have $$C_1(H)=L_1(G_{n,p}), \mbox{ for all } H\leq M,$$ and
$$C_1(\langle yx\rangle)=L_1(M)\cup\{\langle yx\rangle\}, \mbox{ for
all } y\in M.$$In this way, an explicit value of $csd(G_{n,p})$ is
obtained by using (2).
\bigskip\noindent{\bf Theorem 3.1.1.} {\it The cyclic subgroup commutativity degree
of the $P$-group $G_{n,p}$ is given by the following equality:
$$csd(G_{n,p}){=}\frax{\left(2{+}p{+}p^2{+}...{+}p^{n-2}\right)\hspace{-0,5mm}\left(2{+}p{+}p^2{+}...{+}p^{n-1}\right)\hspace{-0,5mm}{+}p^{n-1}\hspace{-0,5mm}\left(3{+}p{+}p^2{+}...{+}p^{n-2}\right)}{\left(2{+}p{+}p^2{+}...{+}p^{n-1}\right)^2}\,.$$}
\smallskip
We observe that for $p=3$, $q=2$ and $n=2$ we have $G_{2,3}\cong
S_3$, and hence $csd(S_3)=\frax{19}{25}$ can be also computed by
the above formula. The following consequence of Theorem 3.1.1 is
immediate, too.
\bigskip\noindent{\bf Corollary 3.1.2.} {\it $\dd\lim_{n\to\infty}csd(G_{n,p})=\frax{2}{p}\,.$}
\bigskip
\bigskip
\noindent{\bf 3.2. The cyclic subgroup commutativity degree of
finite dihedral groups}
\bigskip
The dihedral group $D_{2m}$ $(m\ge 2)$ is the symmetry group of a
regular polygon with $m$ sides and it has the order $2m$. The most
convenient abstract description of $D_{2m}$ is obtained by using
its generators: a rotation $x$ of order $m$ and a reflection $y$
of order $2$. Under these notations, we have
$$D_{2m}=\langle x,y\mid x^m=y^2=1,\ yxy=x^{-1}\rangle.$$It is well-known that for every divisor $r$ or $m$, $D_{2m}$ possesses a subgroup isomorphic to $\mathbb{Z}_r$, namely $H^r_0=\langle x^{\frac{m}{r}}\rangle$, and $\frac{m}{r}$ subgroups isomorphic to $D_{2r}$, namely $H^r_i=\langle x^{\frac{m}{r}},x^{i-1}y\rangle,$ $i=1,2,...,\frac{m}{r}\hspace{1mm}.$ The
normal subgroups of $D_{2m}$ are
$$N(D_{2m})=\left\{\barr{lll}
L(H^m_0)\cup\{G\},& m\equiv 1 \hspace{1mm}({\rm mod}\hspace{1mm} 2)\\
\\
L(H^m_0)\cup\{G, H^{\frac{m}{2}}_1, H^{\frac{m}{2}}_2\},& m\equiv
0 \hspace{1mm}({\rm mod}\hspace{1mm} 2),\earr\right.$$while the
cyclic subgroups of $D_{2m}$ are
$$L_1(D_{2m})=L(H^m_0)\cup\{H^1_i\mid i=1,2,...,m\}.$$It follows
that $$|L_1(D_{2m})|=\tau(m)+m,$$where $\tau(m)$ denotes the
number of divisors of $m$. Clearly, we have
$$|C_1(H)|=\tau(m)+m, \mbox{ for all } H\in L(H^m_0).$$On the
other hand, it is easy to see that
$$C_1(H^1_i)=\left\{\barr{lll}
L(H^m_0)\cup\{H^1_i\},& m\equiv 1 \hspace{1mm}({\rm mod}\hspace{1mm} 2)\\
\\
L(H^m_0)\cup\{H^1_i, H^1_{i+\frac{m}{2}}\},& m\equiv 0
\hspace{1mm}({\rm mod}\hspace{1mm} 2)\earr\right.$$and therefore
$$|C_1(H^1_i)|=\left\{\barr{lll}
\tau(m)+1,& m\equiv 1 \hspace{1mm}({\rm mod}\hspace{1mm} 2)\\
\\
\tau(m)+2,& m\equiv 0 \hspace{1mm}({\rm mod}\hspace{1mm}
2),\earr\right.$$for all $i=1,2,...,m$. Then (2) leads to the
following result.
\bigskip\noindent{\bf Theorem 3.2.1.} {\it The cyclic subgroup commutativity degree
of the dihedral group $D_{2m}$ is given by the following equality:
$$csd(D_{2m})=\left\{\barr{lll}
\frax{\tau(m)(\tau(m)+m)+m(\tau(m)+1)}{(\tau(m)+m)^2}\,,& m\equiv 1 \hspace{1mm}({\rm mod}\hspace{1mm} 2)\\
\\
\frax{\tau(m)(\tau(m)+m)+m(\tau(m)+2)}{(\tau(m)+m)^2}\,,& m\equiv
0 \hspace{1mm}({\rm mod}\hspace{1mm} 2)\,.\earr\right.$$}
\smallskip
The cyclic subgroup commutativity degree of the dihedral group
$D_{2^n}$ is obtained directly from Theorem 3.2.1.
\bigskip\noindent{\bf Corollary 3.2.2.} {\it We have
$$csd(D_{2^n})=\frax{n^2+(n+1)2^n}{(n+2^{n-1})^2}$$and in particular
$$csd(D_8)=\frax{41}{49}\,.$$}
\bigskip
We are also able to compute the limit of $csd(D_{2^n})$ when
$n\to\infty$.
\bigskip\noindent{\bf Corollary 3.2.3.} {\it $\dd\lim_{n\to\infty}csd(D_{2^n})=0.$}
\bigskip
\newpage
\bigskip
\noindent{\bf 3.3. The subgroup commutativity degree of finite
$p$-groups\\ possessing a cyclic maximal subgroup}
\bigskip
Let $p$ be a prime, $n\ge 3$ be an integer and denote by $\calg$
the class consisting of all finite $p$-groups of order $p^n$
having a maximal subgroup which is cyclic. Obviously, $\calg$
contains finite abelian $p$-groups of type
$\mathbb{Z}_p\times\mathbb{Z}_{p^{n-1}}$ whose cyclic subgroup
commutativity degree is $1$, but some finite non-abelian $p$-groups
belong to $\calg$, too. They are exhaustively described in Theorem
4.1, \cite{14}, II: a non-abelian group is contained in $\calg$ \io
it is isomorphic to $M(p^n)$ when $p$ is odd, or to one of the
following groups
\begin{itemize}
\item[--] $M(2^n)\ (n\ge 4),$
\item[--] the dihedral group $D_{2^n}$,
\item[--] the generalized quaternion group
$$Q_{2^n}=\langle x,y\mid x^{2^{n-1}}=y^4=1,\ yxy^{-1}=x^{2^{n-1}-1}\rangle,$$
\item[--] the quasi-dihedral group
$$S_{2^n}=\langle x,y\mid x^{2^{n-1}}=y^2=1,\ y^{-1}xy=x^{2^{n-2}-1}\rangle\,\, (n\ge 4)$$
\end{itemize}
when $p=2$.
\bigskip
In the following the cyclic subgroup commutativity degrees of the
above $p$-groups will be determined. As we observed in Section 2,
we have $$csd(M(p^n))=1.$$Because $csd(D_{2^n})$ has been obtained
in 3.2, we need to focus only on computing $csd(Q_{2^n})$ and
$csd(S_{2^n})$.
\bigskip\noindent{\bf Theorem 3.3.1.} {\it The cyclic subgroup commutativity degree
of the generalized quaternion group $Q_{2^n}$ is
$$csd(Q_{2^n})=\frax{n^2+(n+1)2^{n-1}}{(n+2^{n-2})^2}\,.$$In particular, we have
$$csd(Q_{16})=\frax{7}{8}\,.$$}
\smallskip
\noindent{\bf Proof.} Under the above notation, it is easy to see
that $L_1(Q_{2^n})$ consists of all subgroups contained in
$\langle x\rangle$ and of all subgroups of type $\langle
x^ky\rangle$, $k=0,1,...,2^{n-2}-1$. Moreover, we have
$$|C_1(H)|=|L_1(Q_{2^n})|=n+2^{n-2},
\forall\hspace{1mm}H\leq\langle x\rangle.$$We also remark that
$$\langle x^{k_1}y\rangle\langle x^{k_2}y\rangle=\langle x^{k_2}y\rangle\langle
x^{k_1}y\rangle\Longleftrightarrow k_1=k_2 \mbox{ or }
|k_1-k_2|=2^{n-3}.$$This leads to $$|C_1(\langle
x^ky\rangle)|=n+2, \,\forall\hspace{1mm}k=0,1,...,2^{n-2}-1.$$One
obtains \bigskip
$$\hspace{-30mm}csd(Q_{2^n})=\frax{1}{(n+2^{n-2})^2}\dd\sum_{H\in L_1(Q_{2^n})}|C_1(H)|=$$
$$\hspace{27mm}=\frax{1}{(n+2^{n-2})^2}\left[\dd\sum_{^{H\in
L_1(Q_{2^n})}_{\hspace{3mm}H\leq\langle
x\rangle}}|C_1(H)|+\dd\sum_{k=0}^{2^{n-2}-1}|C_1(\langle
x^ky\rangle)|\right]=$$
$$\hspace{7mm}=\frax{1}{(n+2^{n-2})^2}\left[n(n+2^{n-2})+(n+2)2^{n-2}\right]=$$
$$\hspace{-37,5mm}=\frax{n^2+(n+1)2^{n-1}}{(n+2^{n-2})^2}\,,$$as desired.
\hfill\rule{1,5mm}{1,5mm}
\smallskip
\bigskip\noindent{\bf Corollary 3.3.2.} {\it $\dd\lim_{n\to\infty}csd(Q_{2^n})=0.$}
\bigskip
The same type of reasoning will be used to calculate
$csd(S_{2^n})$.
\bigskip\noindent{\bf Theorem 3.3.3.} {\it The cyclic subgroup commutativity degree
of the quasi-dihedral group $S_{2^n}$ is
$$csd(S_{2^n})=\frax{n^2+3n\cdot2^{n-2}+5\cdot2^{n-3}}{(n+3\cdot2^{n-3})^2}\,.$$In particular, we have
$$csd(S_{16})=\frax{37}{50}\,.$$}
\smallskip
\noindent{\bf Proof.} It is a simple exercise to check that the
poset $L_1(S_{2^n})$ of cyclic subgroups of $S_{2^n}$ consists of
$$L(\langle x\rangle)\cup\{\langle x^{2k}y\rangle\mid k=0,1,...,2^{n-2}-1\}\cup\{\langle x^{2k+1}y\rangle\mid
k=0,1,...,2^{n-3}-1\}.$$Again, we have
$$|C_1(H)|=|L_1(S_{2^n})|=n+3\cdot2^{n-3},
\forall\hspace{1mm}H\leq\langle x\rangle.$$In order to study the
commutativity of the other two types of subgroups of $S_{2^n}$,
the following remarks are essential:
\begin{itemize}
\item[--] $\langle x^{2k_1}y\rangle\langle x^{2k_2}y\rangle=\langle x^{2k_2}y\rangle\langle x^{2k_1}y\rangle{\Longleftrightarrow} 2^{n-3}\mid k_1-k_2$;
\item[--] $\langle x^{2k_1+1}y\rangle\langle x^{2k_2+1}y\rangle=\langle x^{2k_2+1}y\rangle\langle x^{2k_1+1}y\rangle{\Longleftrightarrow} 2^{n-3}\mid k_1-k_2{\Longleftrightarrow} k_1=k_2$;
\item[--] $\langle x^{2k_1}y\rangle\langle x^{2k_2+1}y\rangle\neq\langle x^{2k_2+1}y\rangle\langle x^{2k_1}y\rangle, \forall\hspace{1mm} k_1,k_2$.
\end{itemize}We infer that
$$|C_1(\langle x^{2k}y\rangle)|=n+2, \forall\hspace{1mm}k=0,1,...,2^{n-2}-1$$and
$$|C_1(\langle x^{2k+1}y\rangle)|=n+1, \forall\hspace{1mm}k=0,1,...,2^{n-3}-1.$$Hence
\bigskip
$$\hspace{-42mm}csd(S_{2^n})=\frax{1}{(n+3\cdot2^{n-3})^2}\dd\sum_{H\in L_1(S_{2^n})}|C_1(H)|=$$
$$=\frax{1}{(n+3\cdot2^{n-3})^2}\left[\dd\sum_{^{H\in
L_1(S_{2^n})}_{\hspace{3mm}H\leq\langle
x\rangle}}\hspace{-3mm}|C_1(H)|+\hspace{-3mm}\dd\sum_{k=0}^{2^{n-2}-1}|C_1(\langle
x^{2k}y\rangle)|+\hspace{-3mm}\dd\sum_{k=0}^{2^{n-3}-1}|C_1(\langle
x^{2k+1}y\rangle)|\right]{=}$$
$$\hspace{22mm}=\frax{1}{(n+3\cdot2^{n-3})^2}\left[n(n+3\cdot2^{n-3})+(n+2)2^{n-2}+(n+1)2^{n-3}\right]=$$
$$\hspace{-43mm}=\frax{n^2+3n\cdot2^{n-2}+5\cdot2^{n-3}}{(n+3\cdot2^{n-3})^2}\,,$$completing
the proof.
\hfill\rule{1,5mm}{1,5mm}
\bigskip\noindent{\bf Corollary 3.3.4.} {\it $\dd\lim_{n\to\infty}csd(S_{2^n})=0.$}
\section{A criterion for a finite group to be Iwasawa}
A famous result by Gustafson \cite{3} concerning the commutativity degree states that if $d(G)>5/8$ then $G$ is abelian,
and we have $d(G)=5/8$ if and only if $G/Z(G)\cong\mathbb{Z}_2\times\mathbb{Z}_2$. In this section a similar problem
is studied for the cyclic subgroup commutativity degree, namely: \textit{is there a constant $c\in(0,1)$ such that if $csd(G)>c$ then
$G$ is Iwasawa}?
\bigskip
The answer to this problem is negative, as shows the following theorem.
\bigskip\noindent{\bf Theorem 4.1.} {\it The cyclic subgroup commutativity degree
of the non-Iwasawa group $\mathbb{Z}_{2^n}\times Q_8$, $n\geq 2$, tends to $1$ when $n$ tends to infinity.}
\bigskip
\noindent{\bf Proof.} Let $a=(a_1,a_2)\in \mathbb{Z}_{2^n}\times Q_8$. Then $o(a)=2^k$ if and only if either $o(a_1)=2^k$ and $o(a_2)\leq 2^k$ or $o(a_1)<2^k$ and $o(a_2)=2^k$.
We infer that $\mathbb{Z}_{2^n}\times Q_8$ has one element of order $1$, $3$ elements of order $2$, $28$ elements of order $4$, and $2^{k+2}$ elements of order $2^k$, $\forall\, k=3,4,...,n$.
These generate one cyclic subgroup of order $1$, $3$ cyclic subgroups of order $2$, $14$ cyclic subgroups of order $4$, and $8$ cyclic subgroups of order $2^k$, $\forall\, k=3,4,...,n$.
Consequently, $$|L_1(\mathbb{Z}_{2^n}\times Q_8)|=1+3+14+8(n-2)=8n+2.$$Then $$\barr{lcl} csd(\mathbb{Z}_{2^n}\times Q_8)&=&\frax1{(8n+2)^2}\,\left|\{(H,K)\in L_1(\mathbb{Z}_{2^n}\times Q_8)^2\mid
HK=KH\}\right|=\vsp &=&1-\frax1{(8n+2)^2}\,\left|\{(H,K)\in
L_1(\mathbb{Z}_{2^n}\times Q_8)^2\mid HK\neq KH\}\right|.\earr$$One the other hand, by Theorem 2.15 of \cite{1} we know that $\mathbb{Z}_{2^n}\times Q_8$ has $24(n+2)$ pairs of subgroups which do not permute. This implies that $$csd(\mathbb{Z}_{2^n}\times Q_8)\geq 1-\frax{24(n+2)}{(8n+2)^2}$$and so $\dd\lim_{n\to\infty}csd(\mathbb{Z}_{2^n}\times Q_8)=1$, completing the proof.
\hfill\rule{1,5mm}{1,5mm}
\bigskip\noindent{\bf Corollary 4.2.} {\it There is no constant $c\in(0,1)$ such that if $csd(G)>c$ then
$G$ is Iwasawa.}
\bigskip
However, we can get a positive answer to the above problem if we replace the condition "$csd(G)>c$" by the stronger condition "$csd^*(G)>c$",
where $$csd^*(G)={\rm min}\{csd(S) \mid S \mbox{ section of } G\}.$$This was suggested by the fact that a $p$-group is modular if and only
if each of its sections of order $p^3$ does. Moreover, if a $p$-group is not modular then it contains a section isomorphic to $D_8$ or to $E(p^3)$,
the non-abelian group of order $p^3$ and exponent $p$ for $p>2$ (see Lemma 2.3.3 of \cite{13}).
\bigskip\noindent{\bf Lemma 4.3.} {\it Let $G$ be a finite $p$-group such that $csd^*(G)>41/49$. Then $G$ is modular, and consequently an Iwasawa group.}
\bigskip
\noindent{\bf Proof.} Assume that $G$ is not modular. Then there is a section $S$ of $G$ such that $S\cong D_8$ or $S\cong E(p^3)$ for $p>2$. We can easily check that
$$csd(E(p^3))=\frax{p^3+5p^2+4p+4}{(p^2+p+2)^2}<\frax{41}{49}=csd(D_8).$$Therefore $csd(S)\leq 41/49$, contradicting our assumption.
\hfill\rule{1,5mm}{1,5mm}
\bigskip\noindent{\bf Lemma 4.4.} {\it Let $G$ be a finite group such that $csd^*(G)>19/25$. Then $G$ is nilpotent.}
\bigskip
\noindent{\bf Proof.} We will show by induction on $|G|$ that if $G$ is not nilpotent then $csd^*(G)\leq 19/25$, i.e. there is a section $S$ of $G$ with $csd(S)\leq 19/25$.
For $|G|=6$ we have $G\cong S_3$ and the desired conclusion follows by taking $S=G$. Assume now that it is true for all non-nilpotent
groups of order $<|G|$. We distinguish the following two cases.
If $G$ contains a proper non-nilpotent subgroup $H$, then $H$ has a section $S$ with $csd(S)\leq 19/25$ by the inductive hypothesis and we are done since
$S$ is also a section of $G$.
If all proper subgroups of $G$ are nilpotent, then $G$ is a Schmidt group. By \cite{12} (see also \cite{10}) it follows that $G$ is a solvable group of order $p^mq^n$
(where $p$ and $q$ are different primes) with a unique Sylow $p$-subgroup $P$ and a cyclic Sylow $q$-subgroup $Q$, and hence $G$ is a semidirect product of $P$ by $Q$.
Moreover, we have:
\begin{itemize}
\item[-] if $Q=\langle y\rangle$ then $y^q\in Z(G)$;
\item[-] $Z(G)=\Phi(G)=\Phi(P)\times\langle y^q\rangle$, $G'=P$, $P'=(G')'=\Phi(P)$;
\item[-] $|P/P'|=p^r$, where $r$ is the order of $p$ modulo $q$;
\item[-] if $P$ is abelian, then $P$ is an elementary abelian $p$-group of order $p^r$ and $P$ is a minimal normal subgroup of $G$;
\item[-] if $P$ is non-abelian, then $Z(P)=P'=\Phi(P)$ and $|P/Z(P)|=p^r$.
\end{itemize}We infer that $S=G/Z(G)$ is also a Schmidt group of order $p^rq$ which can be written as a semidirect product of an elementary abelian $p$-group $P_1$ of order $p^r$ by a cyclic group $Q_1$ of order $q$ (note that $S_3$ and $A_4$ are examples of such groups). Then $L_1(S)=L_1(P_1)\cup \{Q_1^x \mid x\in S\}$ and $$|L_1(S)|=\frax{p^r-1}{p-1}+1+p^r=\frax{p^{r+1}+p-2}{p-1}\,.$$One obtains: $$csd(S)=\frax{5p+4}{(p+2)^2}\hspace{3mm} \mbox{ for } r=1\0(a)$$and $$csd(S)=\frax{p^{2r}+3p^{r+2}-4p^{r+1}-p^r+p^2-4p+4}{(p^{r+1}+p-2)^2}\hspace{3mm} \mbox{ for } r\geq 2.\0(b)$$In both cases ($a$) and ($b$) we can easily check that $$csd(S)\leq\frax{19}{25}\,,$$as desired.
\hfill\rule{1,5mm}{1,5mm}
\bigskip
We are now able to prove the main result of this section.
\bigskip\noindent{\bf Theorem 4.5.} {\it Let $G$ be a finite group such that $csd^*(G)>41/49$. Then $G$ is an Iwasawa group. Moreover, we have
$csd^*(G)=41/49$ if and only if $G\cong G'\times G''$, where $G'$ is a $2$-group with $csd^*(G')=41/49$ and $G''$ is an Iwasawa group of odd order.}
\bigskip
\noindent{\bf Proof.} Since $csd^*(G)>41/49>19/25$, Lemma 4.4 implies that $G$ is nilpotent. Then it can be written as
$$G=\xmare{i=1}k G_i\,,\0(4)$$where $G_i$ is a Sylow $p_i$-subgroup of $G$, $i=1,2,...,k$. For each $i$ we have $$csd^*(G_i)\geq csd^*(G)>\frax{41}{49}\,,$$and therefore
$G_i$ is an Iwasawa group by Lemma 4.3. Consequently, $G$ is also an Iwasawa group.
Suppose now that $csd^*(G)=41/49$. Then $G$ is nilpotent by Lemma 4.4, and therefore it has a direct decomposition of type $(4)$, where we can assume $p_1<p_2<...<p_k$. Remark that
$p_1=2$. Indeed, if $p_1>2$ then all $p_i$'s are odd, which implies that $G_i$ cannot have sections isomorphic with $D_8$, $\forall\, i=1,2,...,k$. On the other hand, $G_i$ cannot also have sections isomorphic with $E(p_i^3)$ because $csd^*(G_i)\geq csd^*(G)=41/49$. Thus $G_i$ is Iwasawa, $\forall\, i=1,2,...,k$, and the same thing can be said about $G$, a contradiction. Hence $p_1=2$ and we are done by taking $$G'=G_1 \mbox{ and } G''=\xmare{i=2}k G_i.$$
Conversely, since $G'$ and $G''$ are of coprime orders, every section $S$ of $G\cong G'\times G''$ is of type $S\cong S'\times S''$, where $S'$ and $S''$ are sections of $G'$ and $G''$, respectively. Then $$csd(S)=csd(S')csd(S'')=csd(S')$$because $G''$ is Iwasawa. This shows that $$csd^*(G)=csd^*(G')=\frax{41}{49}\,,$$completing the proof.
\hfill\rule{1,5mm}{1,5mm}
\bigskip
We end this section by noting that the problem of finding the structure of $2$-groups $G'$ with $csd^*(G')=41/49$ remains open.
\section{Conclusions and further research}
Similarly with our previous concepts of \textit{subgroup commutativity
degree}, \textit{normality degree} or \textit{cyclicity degree} of
a finite group, the \textit{cyclic subgroup commutativity degree}
can also constitute a significant aspect of probabilistic finite
group theory. Clearly, the study started in this paper can
successfully be extended to other classes of finite groups and all
problems on $sd(G)$, $ndeg(G)$, $cdeg(G)$ (see e.g. \cite{16}-\cite{21})
can be investigated for $csd(G)$, too. On the other hand, the connections
between the above concepts seem to be very inte\-resting. These will surely
constitute the subject of some further research.
\smallskip
Finally, we formulate several specific open problems on cyclic
subgroup commutativity degrees.
\bigskip\noindent{\bf Problem 5.1.} Compute explicitly the cyclic
subgroup commutativity degree of ${\rm ZM}(m,n,r)$ (see
\cite{18}), or, more generally, the cyclic subgroup
commutativity degree of an arbitrary metacyclic group.
\bigskip\noindent{\bf Problem 5.2.} Let $G$ be a finite group. Study the properties
of the map $csd: L(G)\longrightarrow [0,1]$, $H\mapsto csd(H)$. Is
it true that for every $H,K\in L(G)$, we have $H\subseteq
K\Longrightarrow csd(H)\geq csd(K)$?
\bigskip\noindent{\bf Problem 5.3.} For many finite groups $G$, the commutativity
of $x,y\in G$ is strongly connected with the commutativity of
$\langle x\rangle, \langle y\rangle\in L_1(G)$. Can be extended
this to a connection between $d(G)$ and $csd(G)$?
\bigskip\noindent{\bf Problem 5.4.} Does exist finite groups $G$
such that $csd(G)=sd(G)\neq 1$? | 187,159 |
TITLE: Prove that a "tensor product" principal $G$-bundle coincides with a "pullback" via topos morphism
QUESTION [1 upvotes]: From Moerdijk, Classifying spaces and classifying topoi, page 22. Consider a right $G$-set $S$ with the discrete topology. Let $E$ be a principal $G$-bundle over the topological space $X$. One can construct another principal $G$-bundle $S\otimes_G E$ as the quotient of $S\times E$ by the equivalence relation $(s\cdot g,e)\sim(s, g\cdot e)$. Moreover, one can construct another right $G$-set substituting $E$ by the set $\tilde G$, which is $G$ with the right multiplication action (but which, being $G$, supports also left multiplication and so the relation is well defined). So consider an arbitrary topos morphism $f:Sh(X)\longrightarrow \mathcal BG$, where the domain is the topos of sheaves on $X$ and the codomain is the topos of right $G$-sets.
The aim is to prove that, for any $S$,
$$(S\otimes_G\tilde G)\otimes_G f^*(\tilde G)\cong f^*(S \otimes_G\tilde G);$$
from this it will follow that, since $S\otimes_G\tilde G\cong S$, then
$$S\otimes_G f^*(G)\cong f^*(S \otimes_G\tilde G)\cong f^*(S)$$
and so the functors $-\otimes_G f^*(G)$ and $f^*$ coincide.
Can someone give me a hint in order to prove the key isomorphism? I tried with universal properties, but I find some problems in dealing with maps from $G$-sets to sheaves, which are not very natural unless they are actually $f^*$...
Thank you in advance.
REPLY [2 votes]: It's easier to just prove directly that $f^{\ast}(-)$ coincides with $(-) \otimes_G f^{\ast}(G)$. The point is that both of these functors are cocontinuous in their argument, and right $G$-sets is (freely) generated under colimits by $G$. So to show that they agree it suffices to show that they agree at $G$. Plugging in $G$ gives $f^{\ast}(G)$ and $G \otimes_G f^{\ast}(G) \cong f^{\ast}(G)$ respectively. | 185,164 |
TITLE: Convert this equation to find nth instead
QUESTION [2 upvotes]: I have this equation
a(n) = 2^(n - 1) n
for the series
1,4,12,32,80,192,448,.....
So when n = 4, a(n) = 32
What I am looking for is to get n for a(n), but a(n) is not always in the series
Example
Get n for 50?
50 is between a(4) and a(5)
In this case it will be the smaller one
even if we are getting n for 447, which is between a(6) and a(7), but clearly it is much closer to a(7), I will require n =6
I will be using the equation in a computer program, so if the convertion equasion returns a fraction/decimal, that is fine, as converting it into integer will do the trick.
So basically, I need the above equation converted into
n = .....
Sorry, I don't know what tag best suits the question, please amend if you don't mind.
REPLY [0 votes]: Unless you need a large range of $n$ values, a simple solution is to tabulate $n2^{n-1}$ and find the desired number by dichotomic search.
The values that fit in a $32$ bits unsigned are
$$\begin{align}n&\to n2^n&\text{bits}\\
0 &\to 0\\
1 &\to 1 &\to 1 \\
2 &\to 4 &\to 3 \\
3 &\to 12 &\to 5 \\
4 &\to 32 &\to 6 \\
5 &\to 80 &\to 8 \\
6 &\to 192 &\to 9 \\
7 &\to 448 &\to 10 \\
8 &\to 1024 &\to 11 \\
9 &\to 2304 &\to 13 \\
10 &\to 5120 &\to 14 \\
11 &\to 11264 &\to 15 \\
12 &\to 24576 &\to 16 \\
13 &\to 53248 &\to 17 \\
14 &\to 114688 &\to 18 \\
15 &\to 245760 &\to 19 \\
16 &\to 524288 &\to 20 \\
17 &\to 1114112 &\to 22 \\
18 &\to 2359296 &\to 23 \\
19 &\to 4980736 &\to 24 \\
20 &\to 10485760 &\to 25 \\
21 &\to 22020096 &\to 26 \\
22 &\to 46137344 &\to 27 \\
23 &\to 96468992 &\to 28 \\
24 &\to 201326592 &\to 29 \\
25 &\to 419430400 &\to 30 \\
26 &\to 872415232 &\to 31 \\
27 &\to 1811939328 &\to 32 \\\end{align}$$
CAUTION: the $\text{bits}$ column is not correct.
You can even do better by relying on the number of significant bits of your number, which tell you quickly where in the table you will land. For a given number of bits, you have the choice between only two values.
Use an array indexed with the number of bits of the value and in every slot store the values of $n2^n$ that achieves at least this length, and corresponding $n$ (some of the $n$ will be repeated). | 151,961 |
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Discover Glass (Glassblowing Workshop)
What to expect
- Get a brief taste of glassblowing as you unleash your creative side during this beginners workshop
- Learn the basics of a new artform with a talented craftsman to assist you
- Customize your own paperweight or bowl to take home as a souvenir
Book nowSelect a time slot to book (212) 858 9559
About this Activity
Working side-by-side with a talented artist, get your first taste of the wonderful world of glass blowing and create your own paperweight or double-walled bowl to take home! This class is designed to introduce you to the basics of glassblowing in a relaxed environment without requiring a significant time commitment. You will be able to customize your creation by choosing the colors and shape of your one-of-a-kind glass souvenir. Come feel the heat and unlock your creative spirit you'll be dying to come back for more!
Key Highlights
Cancellation Policy
Cancellations must be made at least 24 hours in advance of your scheduled start time. If a cancellation is made within 24 hours of your scheduled start time, a $50 fee will be charged to your account. If a cancellation is made after the equipment has been turned on, you will be charged for a 3-hour slot (currently $135 or $165 depending on work station). If you are late to your start time, you will be charged for the entire time that was scheduled. Staying later than your scheduled time is only possible if there is no one scheduled after you and will be charged the hourly rental rate. | 25,859 |
subsubsection \<open>Preliminary: Register machine sums are Diophantine\<close>
theory Register_Machine_Sums imports Diophantine_Relations
"../Register_Machine/RegisterMachineSimulation"
begin
fun sum_polynomial :: "(nat \<Rightarrow> polynomial) \<Rightarrow> nat list \<Rightarrow> polynomial" where
"sum_polynomial f [] = Const 0" |
"sum_polynomial f (i#idxs) = f i [+] sum_polynomial f idxs"
lemma sum_polynomial_eval:
"peval (sum_polynomial f idxs) a = (\<Sum>k=0..<length idxs. peval (f (idxs!k)) a)"
proof (induction "idxs" rule: List.rev_induct)
case Nil
then show ?case by auto
next
case (snoc x xs)
moreover have suc: "peval (sum_polynomial f (xs @ [x])) a = peval (sum_polynomial f (x # xs)) a"
by (induction xs, auto)
moreover have list_property: "xa < length xs \<Longrightarrow> (xs ! xa) = (xs @ [x]) ! xa" for xa
by (simp add: nth_append)
ultimately show ?case by auto
qed
definition sum_program :: "program \<Rightarrow> (nat \<Rightarrow> polynomial) \<Rightarrow> polynomial"
("[\<Sum>_] _" [100, 100] 100) where
"[\<Sum>p] f \<equiv> sum_polynomial f [0..<length p]"
lemma sum_program_push: "m = length ns \<Longrightarrow> length l = length p \<Longrightarrow>
peval ([\<Sum>p] (\<lambda>k. if g k then map (\<lambda>x. push_param x m) l ! k else h k)) (push_list a ns)
= peval ([\<Sum>p] (\<lambda>k. if g k then l ! k else h k)) a"
unfolding sum_program_def apply (induction p, auto)
oops
definition sum_radd_polynomial :: "program \<Rightarrow> register \<Rightarrow> (nat \<Rightarrow> polynomial) \<Rightarrow> polynomial"
("[\<Sum>R+] _ _ _") where
"[\<Sum>R+] p l f \<equiv> [\<Sum>p] (\<lambda>k. if isadd (p!k) \<and> l = modifies (p!k) then f k else Const 0)"
lemma sum_radd_polynomial_eval[defs]:
assumes "length p > 0"
shows "peval ([\<Sum>R+] p l f) a = (\<Sum>R+ p l (\<lambda>x. peval (f x) a))"
proof -
have 1: "x \<le> length p - Suc 0 \<Longrightarrow> x < length p" for x using assms by linarith
have 2: "x \<le> length p - Suc 0 \<Longrightarrow> peval (f ([0..<length p] ! x)) a = peval (f x) a" for x
using assms
by (metis diff_Suc_less less_imp_diff_less less_le_not_le nat_neq_iff nth_upt plus_nat.add_0)
show ?thesis
unfolding sum_radd_polynomial_def sum_program_def sum_radd.simps sum_polynomial_eval
by (auto, rule sum.cong, auto simp: 1 2)
qed
definition sum_rsub_polynomial :: "program \<Rightarrow> register \<Rightarrow> (nat \<Rightarrow> polynomial) \<Rightarrow> polynomial"
("[\<Sum>R-] _ _ _") where
"[\<Sum>R-] p l f \<equiv> [\<Sum>p] (\<lambda>k. if issub (p!k) \<and> l = modifies (p!k) then f k else Const 0)"
lemma sum_rsub_polynomial_eval[defs]:
assumes "length p > 0"
shows "peval ([\<Sum>R-] p l f) a = (\<Sum>R- p l (\<lambda>x. peval (f x) a))"
proof -
have 1: "x \<le> length p - Suc 0 \<Longrightarrow> x < length p" for x using assms by linarith
have 2: "x \<le> length p - Suc 0 \<Longrightarrow> peval (f ([0..<length p] ! x)) a = peval (f x) a" for x
using assms
by (metis diff_Suc_less less_imp_diff_less less_le_not_le nat_neq_iff nth_upt plus_nat.add_0)
show ?thesis
unfolding sum_rsub_polynomial_def sum_program_def sum_rsub.simps sum_polynomial_eval
by (auto, rule sum.cong, auto simp: 1 2)
qed
definition sum_sadd_polynomial :: "program \<Rightarrow> state \<Rightarrow> (nat \<Rightarrow> polynomial) \<Rightarrow> polynomial"
("[\<Sum>S+] _ _ _") where
"[\<Sum>S+] p d f \<equiv> [\<Sum>p] (\<lambda>k. if isadd (p!k) \<and> d = goes_to (p!k) then f k else Const 0)"
lemma sum_sadd_polynomial_eval[defs]:
assumes "length p > 0"
shows "peval ([\<Sum>S+] p d f) a = (\<Sum>S+ p d (\<lambda>x. peval (f x) a))"
proof -
have 1: "x \<le> length p - Suc 0 \<Longrightarrow> x < length p" for x using assms by linarith
have 2: "x \<le> length p - Suc 0 \<Longrightarrow> peval (f ([0..<length p] ! x)) a = peval (f x) a" for x
using assms
by (metis diff_Suc_less less_imp_diff_less less_le_not_le nat_neq_iff nth_upt plus_nat.add_0)
show ?thesis
unfolding sum_sadd_polynomial_def sum_program_def sum_sadd.simps sum_polynomial_eval
by (auto, rule sum.cong, auto simp: 1 2)
qed
definition sum_ssub_nzero_polynomial :: "program \<Rightarrow> state \<Rightarrow> (nat \<Rightarrow> polynomial) \<Rightarrow> polynomial"
("[\<Sum>S-] _ _ _") where
"[\<Sum>S-] p d f \<equiv> [\<Sum>p] (\<lambda>k. if issub (p!k) \<and> d = goes_to (p!k) then f k else Const 0)"
lemma sum_ssub_nzero_polynomial_eval[defs]:
assumes "length p > 0"
shows "peval ([\<Sum>S-] p d f) a = (\<Sum>S- p d (\<lambda>x. peval (f x) a))"
proof -
have 1: "x \<le> length p - Suc 0 \<Longrightarrow> x < length p" for x using assms by linarith
have 2: "x \<le> length p - Suc 0 \<Longrightarrow> peval (f ([0..<length p] ! x)) a = peval (f x) a" for x
using assms
by (metis diff_Suc_less less_imp_diff_less less_le_not_le nat_neq_iff nth_upt plus_nat.add_0)
show ?thesis
unfolding sum_ssub_nzero_polynomial_def sum_program_def sum_ssub_nzero.simps sum_polynomial_eval
by (auto, rule sum.cong, auto simp: 1 2)
qed
definition sum_ssub_zero_polynomial :: "program \<Rightarrow> state \<Rightarrow> (nat \<Rightarrow> polynomial) \<Rightarrow> polynomial"
("[\<Sum>S0] _ _ _") where
"[\<Sum>S0] p d f \<equiv> [\<Sum>p] (\<lambda>k. if issub (p!k) \<and> d = goes_to_alt (p!k) then f k else Const 0)"
lemma sum_ssub_zero_polynomial_eval[defs]:
assumes "length p > 0"
shows "peval ([\<Sum>S0] p d f) a = (\<Sum>S0 p d (\<lambda>x. peval (f x) a))"
proof -
have 1: "x \<le> length p - Suc 0 \<Longrightarrow> x < length p" for x using assms by linarith
have 2: "x \<le> length p - Suc 0 \<Longrightarrow> peval (f ([0..<length p] ! x)) a = peval (f x) a" for x
using assms
by (metis diff_Suc_less less_imp_diff_less less_le_not_le nat_neq_iff nth_upt plus_nat.add_0)
show ?thesis
unfolding sum_ssub_zero_polynomial_def sum_program_def sum_ssub_zero.simps sum_polynomial_eval
by (auto, rule sum.cong, auto simp: 1 2)
qed
end | 130,247 |
Program and Relationship Manager
- Employer
- Northeastern University
- Location
- Massachusetts, United States
- Salary
- Salary Not specified
- Posted date
- Oct 1, 2021
View more
- Position Type
- Faculty Positions, Business & Management, Management
- Employment Level
- Administrative
- Employment Type
- Full Time
Program and Relationship:
Direct and support implementation of large, complex projects that often require multiple resources and functional integration to coordinate activities of the team and ensure customer satisfaction. Manage digital programs within the EDGE team's portfolio to deliver on Northeastern's values around experiential learning.
Qualifications:
- Demonstrated understanding of Project Management processes, strategies and methods.
- Must have strong people skills and be able to influence project contributors across departments and organizations without having a direct oversight role.
- Experience working in a highly collaborative environment and promoting a teamwork mentality.
- High level of initiative and drive with a strong sense of urgency and accountability.
- Ability to work independently, but also as a team member when the situation warrants.
- Strong organizational, leadership, operations, and time management skills
- Ability to solve problems, analyze learner survey and performance data and synthesize and summarize information.
- Demonstrated strong business writing, internet research and project management/report writing skills.
- Builds positive relationships across departments, and shows a commitment to collegiality, mutual respect, supportiveness and inclusiveness. Is comfortable working in cross-functional teams and understands how to manage various stakeholders. Acts in a fair, flexible, honest and ethical manner at all times.
- Values and demonstrates consistently excellent communication skills. Is able to communicate and present to multiple levels of audiences including leadership, external partners, and students.
- Consistently looks for a better way to do things in order to create organizational efficiency and effectiveness. Acts as a good steward of the university's resources.
- Knowledge and skills required are normally obtained through a Bachelor's Degree plus 5 to 7 years of related project and/or operations management experience. Master's Degree preferred.-a1970483e7f1d94bb0f202fa403368e9
Get job alerts
Create a job alert and receive personalised job recommendations straight to your inbox.Create alert | 139,953 |
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What Makes Me Feel Optimistic In A World Full Of Bad News
If you read my blog you know I’m capable of catastrophic Irish morbidity. Case in point. When my daughters strike an aloof attitude toward me
If you read my blog you know I’m capable of catastrophic Irish morbidity. Case in point. When my daughters strike an aloof attitude toward me
We Collearys are not a religious people. Henry’s a recovering Catholic. I was raised Mormon and love them dearly, but simply could not commit to been Professor McGonegle. I’ve been Medusa. And now I’ve been Makeup-Artist-to-the-Stars, Gilda. I blame my myriad incarnations on my daughter Bridget, because she loves
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Showing items 1-23 of 23. | 57,503 |
\begin{document}
\title[A survey on the Non-inner Automorphism Conjecture]{A survey on the Non-inner Automorphism Conjecture}
\author{Renu Joshi and Siddhartha Sarkar}
\address{Department of Mathematics\\
Indian Institute of Science Education and Research Bhopal\\
Bhopal Bypass Road, Bhauri \\
Bhopal 462 066, Madhya Pradesh\\
India}
\email{[email protected], [email protected]}
\keywords{Finite $p$-groups; Non-inner automorphisms}
\subjclass{
Primary 20D15
}
\begin{abstract}
In this survey article, we try to summarize the known results towards the long-standing non-inner automorphism conjecture, which states that every finite non-abelian $p$-group has a non-inner automorphism of order $p$.
\end{abstract}
\maketitle
\section{Introduction}
\label{introsec}
\noindent We begin with some motivation towards the problem. A group $G$ will always assumed to be finite, and the notations used are listed below. For an abelian group, every non-trivial automorphism is non-inner. Also, while $G$ is a non-abelian $p$-group it has its center $1 \neq Z(G) \subsetneq G$ and $G/Z(G) \cong {\mathrm{Inn}}(G)$, which implies it always has an inner automorphism of order $p$.
\smallskip
\noindent First note that for an abelian $p$-group $G$ other than $G \cong C_p$, its automorphism group ${\mathrm{Aut}}(G)$ always contain a non-inner automorphism: to see this first notice
\begin{eqnarray*}
{\mathrm{Aut}}(C_{p^n}) & \cong & C_{p^{n-1}(p-1)} ~~~~{\mathrm{if}}~~~ p \neq 2 \\
& \cong & C_2 \times C_{2^{n-2}} ~~~{\mathrm{if}}~~~ p = 2
\end{eqnarray*}
\noindent So if, $G \cong C_{p^{e_1}} \times C_{p^{e_2}} \times \dotsc \times C_{p^{e_n}}$ with $1 \leq e_1 \leq e_2 \leq \dotsc \leq e_n$, the conclusion is clear from the easy embedding
\[
{\mathrm{Aut}}(C_{p^{e_1}}) \times {\mathrm{Aut}}(C_{p^{e_2}}) \times \dotsc \times {\mathrm{Aut}}(C_{p^{e_n}}) \hookrightarrow {\mathrm{Aut}}(G)
\]
unless $e_n = 1$. So while $e_1 = \dotsc = e_n = 1$, we have ${\mathrm{Aut}}(G) \cong GL(n,p)$ and hence its order is divisible by $p^N$, where $N = {n \choose 2} \geq 1$.
\smallskip
\noindent In 1973, Berkovich \cite[4.13]{kou} posed the following conjecture :
\smallskip
\noindent {\bf Non-inner automorphism conjecture (NIAC):} Prove that every finite non-abelian $p$-group admits an automorphism of order $p$, which is not an inner one.
\smallskip
\noindent This is one of the "simple to state," and notoriously hard problem in group theory and so far has known to be affirmative for a broader class of finite $p$-groups.
\bigskip
\noindent {\bf Notations}
$\bullet$ $C_n$ denote the cyclic group of order $n$,
$\bullet$ ${\mathrm{exp}}(G) := {\mathrm{l.c.m.}} \{ |x| ~:~ x \in G \}$ is exponent of a group $G$,
$\bullet$ $[H, K] = \langle [x, y] = x^{-1} y^{-1} xy ~:~ x \in H, y \in K \rangle$, where $H, K \leq G$,
$\bullet$ $\gamma_1(G) = G, \gamma_{i+1}(G) = [\gamma_i(G), G]$ for $i \geq 1$,
$\bullet$ $Z(G)$ is the center of $G$,
$\bullet$ $cl(G)$ is the nilpotency class of $G$,
$\bullet$ For $x \in G$, $x^G$ denote the conjugacy class of $G$ that contain $x$,
$\bullet$ $d(G)$ denote the minimum number of generators of a group $G$,
$\bullet$ For a finite $p$-group,
\smallskip
\hspace*{.08in} (i) $\mho_i(G) = \langle x^{p^i} ~:~ x \in G \rangle, ~ i \geq 0$, \\
\hspace*{.22in} (ii) $\Phi(G) = \mho_1(G) [G, G]$ is the Frattini subgroup of $G$.
\bigskip
\section{\bf Cohomologically trivial modules and NIAC for regular $p$-groups}
\label{cohomology}
\smallskip
\noindent Let $A$ be a $G$-module, then $A$ is called {\bf cohomologically trivial} if the Tate cohomology $H^n(H,A) = 0$ for all integers $n$ and for all subgroups $H \leq G$.
\smallskip
\noindent Around 1966, the following result was proved independently by Gasch\"{u}tz \cite{gas1} and Uchida \cite{uch} :
\begin{theorem} If $G$ and $A$ are finite $p$-groups, then $A$ is cohomologically trivial if $H^n(G,A) = 0$ only for one integer $n$.
\end{theorem}
\noindent Until this point, a weaker version of this problem was also known as Tannaka's conjecture, which was pursued by several mathematicians, including Tannaka and Nakayama (see \cite{uch}). One of the remarkable consequence of the above Thm. is (see \cite{gas2}) :
\begin{theorem}\label{gas2}
Every finite $p$-group $G \neq C_p$ admits a non-inner automorphism of $p$-power order.
\end{theorem}
\noindent However, this is far away from concluding anything about NIAC.
\smallskip
\noindent A finite $p$-group is called {\bf regular} if for every $x, y \in G$ we have
\[
x^p y^p \equiv (xy)^p \hspace*{.1in} {\mathrm{mod}}~~ \mho_1(\gamma_2(\langle x, y \rangle))
\]
\noindent Regularity is commonly known as a generalization of abelian property among finite $p$-groups (see \cite[Thm.2.10]{fer} for more details).
\smallskip
\noindent For a finite $p$-group $G$ and $1 \neq N \unlhd G$, set $Q := G/N$. Equip $A := Z(N)$ a (right) $Q$-module structure given via conjugation; i.e.,
\[
a^{Ng} := g^{-1} ag = a^g \hspace*{.2in} (a \in A, g \in G)
\]
A {\bf crossed homomorphism} $f : Q \rightarrow A$ is a map that satisfy
\[
f(q_1 q_2) = f(q_1)^{q_2} f(q_2) \hspace*{.2in} (q_1, q_2 \in Q)
\]
Clearly, a crossed homomorphism maps identity of $Q$ to the identity of $A$. Moreover, the set $Z^1(Q,A)$ of all crossed homomorphisms form an abelian $p$-group with respect to componentwise addition in which trivial map is the identity element. If $f \in Z^1(Q,A)$, then the map $g \mapsto gf(gN)$ is an automorphism of $G$ that fixes $N$ and $G/N$ elementwise \cite[Satz.I.17.1]{hup}.
\smallskip
\begin{lemma}\label{crossed-hom-elementary}
Let $G$ be a regular $p$-group. Equip $Z(\Phi(G))$ with a $G/{\Phi(G)}$-module structure via conjugation. Then $Z^1(G/{\Phi(G)}, Z(\Phi(G)))$ is elementary abelian.
\end{lemma}
\smallskip
\noindent {\bf Proof :} For $f \in Z^1(G/{\Phi(G)}, Z(\Phi(G)))$ and $g \in G$, it is enough to show that $f(g \Phi(G))^p = 1$. So assume $g \not\in \Phi(G)$ and we have $f(g^p \Phi(G)) = 1$. Applying the definition of crossed homomorphism we get
\[
f(g \Phi(G))^{g^{p-1}} \dotsc f(g \Phi(G))^g f(g \Phi(G)) = 1
\]
Denoting $w = f(g \Phi(G))$ we have $(gw)^p = g^p$. It is now enough to show that $(gw)^p = g^p w^p$ : we set $H := \langle g, w \rangle$. Then $\gamma_2(H) = \langle [g,w]^x ~:~ x \in G \rangle$. Since $g^p \in \Phi(G)$, we have $[g^p, w] = 1$. Using regularity property of $G$ (see \cite[Lem.2.13]{fer}) we have $[g,w]^p = 1$. Since $H$ is generated by elements of order $\leq p$ we have ${\mathrm{exp}}(H) \leq p$ (see \cite[Lem.2.11]{fer}). This implies our claim from the definition of regularity. \QED
\smallskip
\noindent In 1980, Schmid \cite{sch} proved the following result :
\begin{theorem}\label{schmid-thm}
Let $G$ be a regular $p$-group and $1 \neq N \unlhd G$. Let $Q = G/N$ and consider the $Q$-module structure on $Z(N)$ via conjugation. If $Q$ is not cyclic, then $H^n(Q, Z(N)) \neq 0$ for all $n$.
\end{theorem}
\noindent As a consequence, it follows that while $G$ is regular, then following the methods of Thm.\ref{gas2}, non-inner automorphisms of $G$ of order $p$ are constructible, thereby confirming NIAC for regular $p$-groups. This now brings up a mysterious connection of NIAC with the following problem posed by Schmid \cite[Problem 17.2]{kou}: Does there exist finite $p$-group $G$ so that
\[
H^1 \left( {\frac{G}{\Phi(G)}}, Z(\Phi(G)) \right) = 0
\]
\noindent These are also called non-Schmid (NS-) groups, and Abdollahi \cite{abd1} confirmed their existence. Later in \cite{gho1}, it was shown that NS-groups could not have non-inner automorphisms that fixes $\Phi(G)$ elementwise. One can then ask :
\smallskip
\noindent {\bf Question 1 :} Give a complete classification of NS-groups.
\smallskip
\noindent Now let us come back to NIAC for regular $p$-group. We elaborate the proof of Schmid \cite{sch}, which is slightly easier to understand using Thm.\ref{non-frat-NIAC}.
\smallskip
\begin{theorem}
Let $p$ be any prime and $G$ be a finite regular $p$-group other than $C_p$. Then $G$ admits a non-inner automorphism of order $p$ that fixes $\Phi(G)$ elementwise.
\end{theorem}
\smallskip
\noindent {\bf Proof.} We may assume $G$ is non-abelian. Then $G/{\Phi(G)}$ is not cyclic and from Thm.\ref{schmid-thm}, we have $H^1(G/{\Phi(G)},Z(\Phi(G))) \neq 0$.
Since, ${\mathrm{exp}}(H^1(G/{\Phi(G)},Z(\Phi(G))))$ divides $|G/{\Phi(G)}|$ (see \cite[Satz.I.16.19]{hup}) we have $H^1(G/{\Phi(G)},Z(\Phi(G)))$ is a non-trivial abelian $p$-group. Now
let
\[
C_{\mathrm{Aut}(G)} \left( \Phi(G) ; \frac{G}{\Phi(G)} \right) := \Big\{ \alpha \in {\mathrm{Aut}}(G) ~:~ [\Phi(G), \alpha] = 1, [G, \alpha] \leq \Phi(G) \Big\},
\]
i.e., the set of automorphisms that fixes $\Phi(G)$ and $G/{\Phi(G)}$ elementwise. Using the definitions of $1$-cocycles and $1$-coboundaries we have the natural isomorphisms
\[
Z^1 \left( \frac{G}{\Phi(G)}, Z(\Phi(G)) \right) \cong C_{\mathrm{Aut}(G)} \left( \Phi(G) ; \frac{G}{\Phi(G)} \right), ~~~ I(Z(\Phi(G))) \cong B^1 \left( \frac{G}{\Phi(G)},Z(\Phi(G) \right),
\]
where $I(Z(\Phi(G)))$ denotes the group of all inner automorphism of $G$ induced by $Z(\Phi(G))$. Using Thm.\ref{non-frat-NIAC} we assume $C_{G}(Z(\Phi(G))) = \Phi(G)$. Then we have (see the proof of \cite[Satz.17.1(c)]{hup})
\[
I(Z(\Phi(G))) \cong B^1 \left( \frac{G}{\Phi(G)},Z(\Phi(G) \right) \cong {\mathrm{Inn}}(G) \bigcap C_{\mathrm{Aut}(G)} \left( \Phi(G) ; \frac{G}{\Phi(G)} \right)
\]
Suppose that $C_{\mathrm{Aut}(G)} \left( \Phi(G) ; \frac{G}{\Phi(G)} \right)\subseteq {\mathrm{Inn}}(G)$, then $B^1 \left( \frac{G}{\Phi(G)},Z(\Phi(G) \right) \cong Z^1 \left( \frac{G}{\Phi(G)}, Z(\Phi(G)) \right)$, contradicting $H^1(G/{\Phi(G)},Z(\Phi(G))) \neq 0$. This implies $C_{\mathrm{Aut}(G)} \left( \Phi(G) ; \frac{G}{\Phi(G)} \right)\nsubseteq {\mathrm{Inn}}(G)$. Now using Lem.\ref{crossed-hom-elementary}, we have $Z^1(G/{\Phi(G)},Z(\Phi(G)))$ is elementary abelian and hence $G$ contain a non-inner automorphism of order $p$ that fixes $\Phi(G)$ and $G/{\Phi(G)}$ elementwise. \QED
\bigskip
\section{\bf NIAC for groups with small class}
\bigskip
\noindent The first attempt to solve NIAC for finite $p$-groups of class $2$ was due to Liebeck \cite{lie}. Liebeck proved that :
\begin{theorem}\label{liebeck-class2-odd-p}
For every odd prime $p$, a finite $p$-group of class $2$ admits a non-inner automorphism of order $p$ that fixes $\Phi(G)$ elementwise.
\end{theorem}
\noindent From now on we will call a finite $p$-group as {\bf NIAC$(+)$-group} if it contain a non-inner automorphism of order $p$ that fixes $\Phi(G)$ elementwise. If $G$ satisfy NIAC but not NIAC$(+)$, we will call it a {\bf NIAC$(-)$-group} otherwise. Liebeck \cite{lie} also constructed an example of a $2$-group of order $128$ that is not NIAC$(+)$.
\smallskip
\noindent Before we go further, it is a good point to mention the work of Deaconescu and Silberberg in 2002 \cite{dsi}, which encompass a large part of finite $p$-groups that satisfy NIAC.
\begin{theorem}\label{non-frat-NIAC}
Every finite $p$-group $G$ with the condition $C_{G}(Z(\Phi(G))) \neq \Phi(G)$ satisfy NIAC.
\end{theorem}
\noindent The proof of this Thm. uses certain reductions towards the case of R\'{e}dei $p$-groups \cite[Aufgabe 22, Pg.309]{hup} and establish the proof for these groups. A finite $p$-group $G$ that satisfy $C_{G}(Z(\Phi(G))) = \Phi(G)$ are called {\bf strongly Frattinian}. This reduces to verify NIAC for strongly Frattinian groups.
\smallskip
\noindent In 2006, Abdollahi \cite{abd2} completed the class $2$ case using \ref{non-frat-NIAC} pointing out the following revised version of Liebeck's result for class $2$ and $p=2$. This shows that the example of Liebeck \cite{lie} of order $128$ is indeed a NIAC$(-)$-group.
\begin{theorem}
Every finite $2$-group of class $2$ admits a non-inner automorphism of order $2$ that fixes either $\Phi(G)$ or $\Omega_1(Z(G))$ elementwise.
\end{theorem}
\noindent This is further extended in 2013 by Abdollahi et. al. for class $3$ \cite{agw} and leave the following question open :
\bigskip
\noindent {\bf Question 1 :} Verify NIAC for finite $p$-groups of class $cl(G) \geq 4$.
\bigskip
\noindent We want to point out that the methods given in \cite{abd2} are quite strong, and can provide a reasonably compact proof of Thm.\ref{liebeck-class2-odd-p} compared to \cite{lie}. This we will outline now :
\begin{proposition}
Let $G$ be a finite non-abelian $p$-group. Let $G$ contain an element $z \in Z(G) \setminus [G,G]$ of order $p$. Then $G$ has a non-inner automorphism of order $p$ that fixes $\Phi(G)$ elementwise.
\end{proposition}
\smallskip
\noindent {\bf Proof.} Consider a maximal subgroup $M \leq G$ with $z \in G$. Define $\alpha : G \rightarrow G$ by $(g^i m)^{\alpha} = g^i z^i m$ with $m \in M$ and $0 \leq i \leq p-1$. Then we have
\[
g^i m_1 g^j m_2 = g^{i+j} m_1 [m_1, g^j] m_2 = g^s m^{\prime}
\]
where $m^{\prime} = g^{i+j-s} m_1 [m_1, g^j] m_2 \in M, i+j \equiv s$ mod $p$ for some $0 \leq s \leq p-1$. Hence
\[
(g^i m_1 g^j m_2)^{\alpha} = g^s z^s m^{\prime} = (g^i m_1)^{\alpha} (g^j m_2)^{\alpha}
\]
and $\alpha$ is a homomorphism. As $\alpha$ fixes $M$ elementwise it has image a subgroup larger than $M$, showing $\alpha$ is surjective. Since $G$ is finite, it must be injective as well. \QED
\smallskip
\begin{proposition} Let $G$ be a finite nilpotent group of class $2$ such that $[G,G] = \langle [a,b] \rangle$ for some $a, b \in G$. Then $G = H C_G(H)$ with $H = \langle a, b \rangle$.
\end{proposition}
\smallskip
\noindent {\bf Proof :} Remark 2.2, \cite{abd2}. \QED
\smallskip
\noindent We first prove the case of $2$-generated finite $p$-group $G$ with class $2$. Using this, we will subsequently prove the main theorem by reducing to the $d(G) = 2$ case.
\smallskip
\begin{theorem}
Let $p$ be an odd prime and $G$ be a finite $p$-group of class $2$ with $G = \langle a,b \rangle$. Then $G$ has a non-inner automorphism of order $p$ that fixes $\Phi(G)$ elementwise.
\end{theorem}
\smallskip
\noindent {\bf Proof.} Suppose the order of $[a, b]$ be $p^n$ where $n \geq 1$. Since $G$ has class $2$, we have $[G, G] = \langle [a, b] \rangle$. The condition $|[a,b]| = p^n$ implies that $a^{p^n}, b^{p^n} \in Z(G)$ and $a^{p^{n-1}}, b^{p^{n-1}} \not\in Z(G)$. We will now prove that $Z(G) = \langle a^{p^n}, b^{p^n}, [a,b] \rangle$.
\smallskip
\noindent Let $g \in Z(G)$. Using $[G, G] = \langle [a, b] \rangle$, write $g = a^i b^j [a, b]^t$ for some non-negative integers $i,j,t$. Now we have
\[
1 = [a, g] = [a, b]^j
\]
This shows $p^n \mid j$. Similarly $p^n \mid i$. This shows $Z(G) = \langle a^{p^n}, b^{p^n}, [a,b] \rangle$.
\bigskip
\noindent If $n = 1$, we have $Z(G) = \Phi(G)$. Then $C_G(Z(\Phi(G))) = C_G(Z(G)) = G \neq \Phi(G)$ unless $G$ is cyclic of order $p$. Using Thm.\ref{non-frat-NIAC} we now assume that $n \geq 2$.
\smallskip
\noindent We first prove that $Z(G)$ is cyclic : if not, then for any $z \in \Omega_1(Z(G)) \setminus [G,G]$ we may construct a non-inner automorphism by Prop.3.4. Note that the elements $a[G,G], b[G,G]$ have order $p^n$ in $G/[G,G]$.
\smallskip
\noindent We now show that we may assume that the order of $b$ in $G$ is $p^n$ : since $Z(G)$ is cyclic, without loss of generality assume that $b^{p^n} \in \langle a^{p^n} \rangle$ and write $b^{p^n} = a^{p^n j}$. Since $p$ is odd, we have,
\[
(a^{-j}b)^{p^n} = a^{-p^n j} b^{p^n} [b, a^{-j}]^{p^n \choose 2} = 1
\]
If $(a^{-j}b)^{p^{n-1}} \in Z(G)$, we have
\[
1 = [a, (a^{-j}b)^{p^{n-1}}] = [a, a^{-j}b]^{p^{n-1}} = [a, b]^{p^{n-1}}
\]
which contradicts $|[a, b]| = p^n$. As $G = \langle a, a^{-j}b \rangle$, replacing $b$ by $a^{-j}b$ proves our assertion. Now we have $|b| = p^n$ and $b^{p^{n-1}} \not\in Z(G)$. Then using lemma 1 of \cite{lie} we may construct the non-inner automorphism $\sigma$ of order $p$ defined by $a^{\sigma} = ab^{p^{n-1}}, b^{\sigma} = b$ that fixes $[G, G]$ elementwise. Now $\Phi(G) = \langle a^p, b^p, [a,b] \rangle$ and $(a^p)^{\sigma} = (ab^{p^{n-1}})^p = a^p b^{p^n} [b^{p^{n-1}}, a]^{p \choose 2} = a^p$, which shows $\sigma$ fixes $\Phi(G)$ elementwise. This is a contradiction. \QED
\bigskip
\noindent {\bf Proof of \ref{liebeck-class2-odd-p}.} Suppose that the assertion is not true. Then using part (a), Thm.1 of \cite{lie} we have $[G, G]$ is cyclic. Let $[G, G] = \langle [a,b] \rangle$ with $|[a, b]| = p^n$ and consider the subgroup $H = \langle a, b \rangle$. Then $G = HC_G(H)$ and by previous Thm. we have $\varphi \in {\mathrm{Aut}}(H) \setminus {\mathrm{Inn}}(H)$ of order $p$ that fixes $\Phi(H)$ elementwise.
\smallskip
\noindent From above calculations $Z(H) = \langle a^{p^n}, b^{p^n}, [a,b] \rangle \subseteq \langle a^p, b^p, [a,b] \rangle = \Phi(H)$. Hence $\varphi$ fixes $Z(H)$ as well.
\smallskip
\noindent Now define $\psi : G \rightarrow G$ as $(hk)^{\psi} = h^{\varphi} k$ for $h \in H, k \in C_G(H)$. If we have two expression $g = h_1 k_1 = h_2 k_2$ for $h_i \in H, k_i \in C_G(H)$ of $g \in G$, then $h^{-1}_2 h_1 = k_2 k^{-1}_1 \in H \cap K \subseteq Z(H)$. Hence $(h^{-1}_2 h_1)^{\varphi} = h^{-1}_2 h_1 = k_2 k^{-1}_1$. Hence $h_1^{\varphi} k_1 = h_2^{\varphi} k_2$. Thus $\psi$ is well defined. Since $[H, C_G(H)] = 1$, the map $\psi$ is a homomorphism. It is clearly surjective and hence an automorphism of $G$. Since $\varphi$ has order $p$, the order of $\psi$ is also $p$. We need to show that $\psi$ is non-inner. Here we notice that $\psi \vert_H = \varphi$.
\smallskip
\noindent Let $x \in G$ so that $g^{\psi} = x^{-1} gx$ for every $g \in G$. Write $x = uv$ with $u \in H, v \in C_G(H)$. Then for any $h \in H$ we have
\[
h^{\psi} = h^{\varphi} = v^{-1} u^{-1} huv = u^{-1} hu
\]
which shows $\varphi$ is an inner automorphism of $H$, a contradiction. Final step is to check that $\psi$ fixes $\Phi(G)$ elementwise. By hypothesis, it fixes $\Phi(H) = H^p[H,H] \supseteq [H,H] = [G,G]$ elementwise. We need to show it fixes $G^p$ elementwise. Let $g = hk \in G$ with $h \in H, k \in C_G(H)$. Then $g^p = h^p k^p$. But $\varphi$ fixes $h^p$. Hence $(g^p)^{\psi} = (h^p)^{\varphi} k^p = h^p k^p = g^p$. This concludes the proof. \QED
\bigskip
\section{\bf NIAC for groups with small co-class}
\bigskip
\noindent For a finite $p$-group $G$ of order $p^n$ and $cl(G)=l$, it's {\bf co-class} $c$ is defined to be $c:=n-l$ which must be at least $1$. In the classification program for finite $p$-groups (yet to be complete), the attempts through the co-class are much more successful. The groups with fixed co-class are much more richer and very much similar to their counterparts in pro-$p$-groups with fixed co-class. See \cite{lmc} for more details and the references therein.
\smallskip
\noindent The first attempt to solve NIAC through co-class was due to Abdollahi in 2010 \cite{abd3}. This can be made through the following fundamental observation :
\begin{theorem}\label{bounding-coclass} \cite[Thm.2.5]{abd3}
Let $G$ be a finite non-abelian $p$-group of co-class $c$ such that $G$ has no non-inner automorphism of order $p$ leaving $\Phi(G)$ elementwise fixed. Then
\[
d(Z(G)) \left( d(G) + 1 \right) \leq c+1.
\]
\end{theorem}
\noindent The $p$-groups $G$ of co-class $1$ satisfy $d(Z(G)) = 1$ and $d(G) = 2$ which solve NIAC for them. This result also shows how the co-class controls the growth of the minimum number of generators of $G$ and $Z(G)$ for NIAC$(-)$-groups.
\smallskip
\noindent Fouladi and Orfi \cite{for} first attempted to prove NIAC for co-class $2$ for odd primes. Here they assumed the case for $|G| \leq p^6$ using \cite{bpi} for $p \geq 5$ and the case for $|G| \leq 3^6$ was checked through GAP \cite{gap}. Later in 2014, Abdollahi and four other authors \cite{aggrw} completed the case for co-class $2$ without using GAP or any classification of low order groups.
\smallskip
\noindent The final work in this direction was made by Ruscitti and two other authors in 2017 \cite{rly} for $p$-groups of co-class $3$ while $p \neq 3$. The proofs of their work depends on various complicated reductions and extensive use of derivations. Their work leave the following questions open :
\bigskip
\noindent {\bf Question 2 :} Verify NIAC for finite $3$-groups of co-class $3$.
\bigskip
\noindent {\bf Question 3 :} Verify NIAC for finite $p$-groups of co-class $c \geq 4$.
\bigskip
\section{\bf NIAC for other families of $p$-groups}
\bigskip
\noindent A finite $p$-group is said to be {\bf powerful} if $\gamma_2(G) \subseteq \mho_1(G)$ while $p \geq 3$ and $\gamma_2(G) \subseteq \mho_2(G)$ while $p=2$. In 2010, Abdollahi \cite{abd3} proved that if $G$ is a non-abelian $p$-group with $G/{Z(G)}$ is powerful then $G$ admits an non-inner automorphism which either fixes $\Phi(G)$ or $\Omega_1(Z(G))$ elementwise. This settles the case for powerful $p$-groups since for any normal subgroup $N$ of a finite $p$-group $G$ we have $\mho_i(G/N) = \mho_i(G)N/N$ \cite[Thm.2.4]{fer}.
\smallskip
\noindent For a finite $p$-group $G$ and a proper non-trivial normal subgroup $N$ of $G$, we call $(G,N)$ a {\bf Camina pair} if $xN \subseteq x^G$ for every $x \in G-N$. In 2013, Ghoraishi \cite{gho} proved that for an odd prime $p$, a finite $p$-group $G$ is a NIAC$(+)$-group if $(G,Z(G))$ is a Camina pair. In case $p=2$ and $(G,Z(G))$ is a Camina pair the group $G$ admits a non-inner automorphism of order either $2$ or $4$ that fixes $\Phi(G)$ elementwise \cite{agh}. This also leave the following question open:
\bigskip
\noindent {\bf Question 4 :} Let $G$ be a $2$-group with $(G,Z(G))$ a Camina pair. Does $G$ contain a non-inner automorphism of order $2$ that fixes either $\Phi(G)$ or $\Omega_1(Z(G))$ elementwise?
\bigskip
\noindent Let us turn to the case of strongly Frattinian groups. In 2009, Shabani-Attar \cite{att} verified NIAC for a subclass of strongly Frattinian groups. In fact, the following result was proved :
\smallskip
\begin{theorem}
Let $G$ be a finite non-abelian $p$-group satisfying one of the following conditions:
\begin{enumerate}
\item $rank(\gamma_2(G)\cap Z(G)) \neq rank(Z(G))$
\item $\frac{Z_2(G)}{Z(G)}$ is cyclic
\item $G$ is strongly Frattinian and $\frac{Z_2(G)\cap Z(\Phi(G))}{Z(G)}$ is not elementary abelian of rank $d(G) rank(Z(G))$,
\end{enumerate}
then $G$ has a non-inner central automorphism of order $p$ which fixes $\Phi(G)$ elementwise.
\end{theorem}
\smallskip
\noindent For a finite $p$-group $G$, let $Z^{\ast}_2(G)$ denote the pre-image of $\Omega_1(Z_2(G)/{Z(G)})$ in $G$. In 2014, Ghoraishi \cite{gho2} proved that if $G$ fails to satisfy the condition
\begin{equation}\label{star-central}
Z^*_2(G)\leq C_G(Z^*_2(G))=\Phi(G),
\end{equation}
then $G$ is a NIAC$(+)$-group. In fact, the condition (\ref{star-central}) implies $G$ is strongly Frattinian (see Thm.\ref{non-frat-NIAC}), and there are infinitely many groups which are strongly Frattinian but does not satisfy (\ref{star-central}). So this improves the requirement of verifying NIAC for groups with (\ref{star-central}).
\smallskip
\noindent In 2013, Jamali and Viseh \cite{jvi} proved that if $G$ is a finite $p$-group with $\gamma_2(G)$ cyclic, then $G$ is either NIAC$(+)$ or it admits an non-inner automorphism that fixes $Z(G)$ elementwise.
\smallskip
\noindent In 2017, Abdollahi and Ghoraishi {\cite{agh2}} confirmed NIAC for $2$-generated finite $p$-groups with abelian Frattini subgroup.
\smallskip
\noindent Recently Fouladi and Orfi \cite{for2} proved that an odd prime $p$, a finite non-abelian $p$-group with $|\frac{Z_3(G)}{Z(G)}|\leq p^{d(G)+1}$, has non-inner automorphism of order $p$. This leave the following questions open:
\bigskip
\noindent {\bf Question 5:} Verify NIAC for finite $p$-groups with $\lvert \frac{Z_3(G)}{Z(G)} \rvert > p^{d(G)+1}$ for odd prime $p$.
\bibliographystyle{plain}
\bibliography{surveyfinal}
\end{document} | 43,168 |
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TITLE: What is a natural number?
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\begin{document}
\title{Characteristic cycles
and the conductor
of direct image}
\author{Takeshi Saito}
\maketitle
\begin{abstract}
We prove the functoriality
for proper push-forward
of the characteristic cycles
of constructible complexes
by morphisms of
smooth projective schemes over
a perfect field,
under the assumption that
the direct image of the
singular support has the
dimension at most
that of the target
of the morphism.
The functoriality
is deduced from a conductor
formula which is a special case
for morphisms to curves.
The conductor formula in the
constant coefficient case gives
the geometric case of
a formula conjectured by Bloch.
\end{abstract}
Let $k$ be a perfect field
and $\Lambda$ be a finite field
of characteristic invertible in $k$.
For a constructible complex
${\cal F}$ of $\Lambda$-modules
on a smooth scheme $X$ over $k$,
the characteristic cycle
$CC{\cal F}$ is defined in \cite[Definition 5.10]{CC}
as a cycle supported on
the singular support $SS{\cal F}$
defined by Beilinson in \cite{Be}
as a closed conical subset of the cotangent bundle $T^*X$.
We study the functoriality
of characteristic cycles
for proper push-forward.
Let $f\colon X\to Y$
be a morphism
of smooth projective
schemes over $k$.
Then, we prove in
Theorem \ref{thmf*}
the equality
\begin{equation*}
CCRf_*{\cal F}=
f_!CC{\cal F}
\leqno{\rm (\ref{eqcnf})}
\end{equation*}
conjectured in \cite[Conjecture 1]{prop}
under the assumption
$\dim f_\circ SS{\cal F}\leqq \dim Y$
for the direct image
$f_\circ SS{\cal F}\subset T^*Y$.
The precise definitions
will be given in Subsection \ref{ssdc}.
We can slightly weaken the assumption, as is seen in Theorem
\ref{thmf*}.
The formula (\ref{eqcnf})
is an algebraic analogue of \cite[Proposition 9.4.2]{KSc} where
functorial properties of characteristic cycles are studied in a transcendental context.
In the case where $Y={\rm Spec}\ k$,
the equality (\ref{eqcnf}) is
the index formula
\begin{equation*}
\chi(X_{\bar k},{\cal F})
= (CC{\cal F},T^*_XX)_{T^*X}
\leqno{\rm (\ref{eqind})}
\end{equation*}
computing the Euler-Poincar\'e characteristic as an intersection number
proved in
\cite[Theorem 7.13]{CC}.
We deduce the functoriality
(\ref{eqcnf})
from the index formula
(\ref{eqind})
in Subsection \ref{ssccd} as follows.
By taking a projective embedding
of $Y$ and a good pencil,
we reduce it to the case
where $Y$ is
a projective smooth curve.
By the index formula
(\ref{eqind}) applied to a general fiber,
the equality (\ref{eqcnf})
is equivalent to a conductor formula
\begin{equation*}
-a_yRf_*{\cal F}=
(CC{\cal F},df)_{T^*X,X_y}
\leqno{\rm (\ref{eqcf})}
\end{equation*}
proved in
Theorem \ref{thmay},
where the left hand side
denotes the Artin conductor
at a closed point $y\in Y$
of the direct image.
In the case where ${\cal F}$
is the constant sheaf $\Lambda$,
the right hand side equals
the localized self-intersection product
defined in \cite{Bl}
and the formula (\ref{eqcf})
specializes to the geometric case,
Corollary \ref{corBl}, of
the conductor formula conjectured
in \cite{Bl} by Bloch.
Further the index formula implies
that we have an equality (\ref{eqsum})
for the sums over $y\in Y$
of the both sides
in (\ref{eqcf}).
To deduce (\ref{eqcf}) from (\ref{eqsum})
for the sums,
it suffices to show the existence of
a covering of $Y$
\'etale at a fixed point $y$
killing the contributions of the
other points.
For the vanishing of
the left hand side,
the local acyclicity of
$f\colon X\to Y$ relative to ${\cal F}$
is a sufficient condition.
The $SS{\cal F}$-transversality of
$f\colon X\to Y$
defined in Definition \ref{dftrans}
and studied in Subsection \ref{ssFtr}
after some preliminaries in
Subsection \ref{ssCtr}
is a stronger condition
and is a sufficient
condition for the vanishing of
the right hand side.
Thus, the proof of (\ref{eqcf})
is reduced to showing
variants of the stable reduction theorem
on the existence of ramified covering of $Y$
such that
the base change of
$f\colon X\to Y$ is
locally acyclic relatively to
a modification ${\cal F}'$ of the pull-back
and is $SS{\cal F}'$-transversal.
We show that
$f\colon X\to Y$ is
locally acyclic relatively to
a modification of a perverse sheaf
${\cal F}$ if
the inertia action on
the nearby cycles complex
$R\Psi{\cal F}$ is trivial
in Proposition \ref{prS}.2.
This is rather a direct consequence
of the relation of
the direct image by the open
immersion of the generic fiber
with the nearby cycles complex.
As we work with torsion coefficients,
the condition is satisfied over
a ramified covering of $Y$.
Further,
we show that
the local acyclicity of
$f\colon X\to Y$ relatively to ${\cal F}$
implies
the existence of a ramified covering
such that the
base change of
$f\colon X\to Y$ is
$SS{\cal F}'$-transversal
for the pull-back ${\cal F}'$
of ${\cal F}$
in Corollary \ref{corV}
of Theorem \ref{thmV2}.
Theorem \ref{thmV2} is deduced
from a weaker version
Proposition \ref{prV2}
which is proved by using
the alteration \cite[Theorem 8.2]{dJ}.
In Proposition \ref{prV2},
the ramified covering may inseparable,
while it is generically \'etale
in Theorem \ref{thmV2}.
Theorem \ref{thmV2}
is proved by an argument similar to
that in the proof of
\cite[Proposition 3.2]{TF}
by using a consequence of
the stable reduction theorem
\cite[Theorem 1.5]{Tem}.
We also prove an index formula
Proposition \ref{prvan}
for vanishing cycles complex.
The author thanks A.\ Beilinson
for the remark that
Theorem \ref{thmay}
implies the geometric case
of the conductor formula
conjectured by Bloch in \cite{Bl}
and for showing
the proof of Lemma \ref{lmch0}
in the characteristic zero case.
The author thanks H.\ Haoyu
for discussion on the subject
of Subsection \ref{ssiv}
and thanks H.\ Kato
for pointing out an error
in the proof of Proposition \ref{prvan}
in an earlier version.
The research was supported
by JSPS Grants-in-Aid
for Scientific Research
(A) 26247002.
\tableofcontents
\section{Local acyclicity and transversality}\label{str}
\subsection{Nearby cycles and local acyclicity}\label{ssS}
We fix some conventions on
perverse sheaves.
Let $X$ be a noetherian
scheme and let $\Lambda$ be
a finite field of characteristic $\ell$
invertible on $X$.
We say that a complex ${\cal F}$ of
$\Lambda$-modules
on the \'etale site of $X$
is constructible
if the cohomology sheaf
${\cal H}^q{\cal F}$ is
constructible for every integer $q$
and
${\cal H}^q{\cal F}=0$
except for finitely many $q$.
Let $D^b_c(X,\Lambda)$
denote the category of constructible
complexes of $\Lambda$-modules.
First we recall the case
where $X$ is a scheme of finite
type over a field $k$.
Let $\Lambda$ be
a finite field of characteristic $\ell$
invertible in $k$.
Then, the $t$-structure
$(^{\rm p}\!D^{\leqq 0},\,
^{\rm p}\!D^{\geqq 0})$
on $D^b_c(X,\Lambda)$
relative to the middle perversity
is defined in \cite[2.2.10]{BBD}
and the perverse sheaves
on $X$ form an abelian
subcategory
${\rm Perv}(X,\Lambda)
=\,
^{\rm p}\!D^{\leqq0}
\cap\,
^{\rm p}\!D^{\geqq0}.$
Next, we consider the case
where $X$ is a scheme of
finite type over
the spectrum $S$ of
a discrete valuation ring as in \cite[4.6]{au}.
Let $s$ and $\eta$ denote the
closed point and the generic point
of $S$ respectively
and let $i\colon X_s\to X$
and $j\colon X_\eta\to X$
be the closed immersion and the open immersion
of the fibers.
Let $\Lambda$ be
a finite field of characteristic $\ell$
invertible on $S$.
Then, we consider
the $t$-structure on
$D^b_c(X,\Lambda)$
obtained by gluing
(\cite[1.4.10]{BBD})
the $t$-structure
$(^{\rm p}\!D^{\leqq 0},\,
^{\rm p}\!D^{\geqq 0})$ on
$D^b_c(X_s,\Lambda)$ and
the $t$-structure
$(^{\rm p}\!D^{\leqq -1},\,
^{\rm p}\!D^{\geqq -1})$ on
$D^b_c(X_\eta,\Lambda)$.
In particular,
a constructible complex
${\cal F}\in D^b_c(X,\Lambda)$
is contained in $^{\rm p}\!D^{\leqq 0}$
if and only if we have
$i^*{\cal F}
\in\, ^{\rm p}\!D^{\leqq 0}$ and
$j^*{\cal F}\in\, ^{\rm p}\!D^{\leqq -1}$.
Note that if the $t$-structure
on $D^b_c(X_\eta,\Lambda)$
where $X_\eta$ is regarded
as a scheme over $\eta$ is
$(^{\rm p}\!D^{\leqq 0},\,
^{\rm p}\!D^{\geqq 0})$,
then that
on $D^b_c(X_\eta,\Lambda)$
where $X_\eta$ is regarded
as a scheme over $S$ is
$(^{\rm p}\!D^{\leqq -1},\,
^{\rm p}\!D^{\geqq -1})$.
To distinguish them,
we call the former
the $t$-structure on
$X_\eta$ over $\eta$
and the latter
the $t$-structure on
$X_\eta$ over $S$.
We use the same
terminology for perverse sheaves on
$X_\eta$.
The functors $j_!,Rj_*\colon
D^b_c(X_\eta,\Lambda)
\to D^b_c(X,\Lambda)$
are $t$-exact with respect to
the $t$-structure on
$X_\eta$ over $S$.
This follows from \cite[Th\'eor\`eme 3.1]{XIV}
by the argument in
\cite[4.6 (a)]{au}.
Let ${\cal F}
\in {\rm Perv}(X_\eta,\Lambda)$
be a perverse sheaf on
$X_\eta$ over $S$.
Then the intermediate extension
$j_{!*}{\cal F}\in
{\rm Perv}(X,\Lambda)$
is defined as the image
$$j_{!*}{\cal F}
={\rm Im}(j_!{\cal F}
\to Rj_*{\cal F}).$$
We have ${}^{\rm p}\!{\cal H}^qi^*
Rj_*{\cal F}=0$
for $q\neq 0,-1$ and
the morphism
$j_{!*}{\cal F}
\to Rj_*{\cal F}$
induces an isomorphism
\begin{equation}
i^*j_{!*}{\cal F}
\to{}^{\rm p}\!{\cal H}^{-1}
i^*Rj_*{\cal F}.
\label{eqH0}
\end{equation}
This is deduced similarly as
\cite[(4.1.12.1)]{BBD}
from a consequence \cite[(4.1.11.1)]{BBD}
of the $t$-exactness of
the functors $j_!$ and $Rj_*$.
Assume that $S$ is strictly local.
Let $\bar \eta$ be a geometric point
above $\eta$
and let
$\bar j\colon X_{\bar \eta}
\to X_\eta$
denote the canonical morphisms.
Then, the nearby cycles
functor
$$R\Psi= i^*R(j\bar j)_*\bar j^*
\colon D^b_c(X_\eta,\Lambda)
\to D^b_c(X_s,\Lambda)$$
is $t$-exact
with respect to
the $t$-structure
on $X_\eta$ over $\eta$
\cite[Corollaire 4.5]{au}.
\begin{lm}\label{lmS}
Let $S={\rm Spec}\ {\cal O}_K$ be the spectrum
of a strictly local discrete valuation ring
and let $s$ and $\eta$ denote
the closed and the generic point of $S$
respectively.
Let $X$ be a scheme of finite type over $S$,
and let $i\colon X_s\to X$
and $j\colon X_\eta\to X$
denote the immersions.
Let ${\cal F}$ be a perverse sheaf
of $\Lambda$-modules on $X_\eta$
over $\eta$.
Then the morphism
$i^*Rj_*{\cal F}\to R\Psi {\cal F}$
induces an isomorphism
\begin{equation}
{}^{\rm p}\!{\cal H}^0i^*
Rj_*{\cal F}
\to (R\Psi {\cal F})^I
\label{eqI}
\end{equation}
to the inertia fixed part
as a perverse sheaf on
$X_s$.
\end{lm}
\proof{
Let $P\subset I$ denote the
wild inertia subgroup.
Then, since the functor
taking the $P$-invariant parts
is an exact functor,
we have an isomorphism
$i^*Rj_*{\cal F}\to
R\Gamma(I/P,(R\Psi {\cal F})^P)$.
Since the profinite group
$I/P$ is cyclic, the assertion follows.
\qed}
\medskip
We study the local acyclicity
of a morphism to
the spectrum of a discrete valuation
ring with respect to a perverse sheaf.
\begin{pr}\label{prS}
Let $S={\rm Spec}\ {\cal O}_K$ be the spectrum
of a discrete valuation ring
and let $s$ and $\eta$ denote
the closed and the generic point of $S$
respectively.
Let $X$ be a scheme of finite type over $S$,
and let $i\colon X_s\to X$
and $j\colon X_\eta\to X$
denote the immersions.
{\rm 1.}
Let ${\cal G}$ be
a perverse sheaf of $\Lambda$-modules on $X$.
If $X\to S$ is locally acyclic relatively
to ${\cal G}$,
then ${\cal G}$ has no non-zero
subquotient supported
on the closed fiber
and is isomorphic to $j_{!*}j^*{\cal G}$.
{\rm 2.}
For a perverse sheaf ${\cal F}$
of $\Lambda$-modules on $X_\eta$
over $S$,
the following conditions are
equivalent:
{\rm (1)}
The morphism
$X\to S$ is locally acyclic relatively to
$j_{!*}{\cal F}$.
{\rm (2)}
Let $\bar s$ be a geometric point
above the closed point $s\in S$
and let $\bar i\colon X_{\bar s}\to X$
denote the canonical morphism.
Then, the canonical morphism
\begin{equation}
\bar i^*j_{!*}{\cal F}\to R\Psi {\cal F}
\label{eqij}
\end{equation}
is an isomorphism.
{\rm (3)}
The inertia group $I$ of $K$
acts trivially
on the nearby cycles complex $R\Psi{\cal F}$.
{\rm (4)}
The formation of
$j_{!*}{\cal F}$ commutes with the pull-back by
faithfully flat morphisms
$S'\to S$ of the spectra
of discrete valuation rings.
\end{pr}
\proof{
1.
The local acyclicity
is equivalent to
the vanishing
$R\Phi {\cal G}=0$.
Since the shifted vanishing cycles functor
$R\Phi[-1]\colon
D^b_c(X,\Lambda)\to
D^b_c(X_s,\Lambda)$
is $t$-exact
\cite[Corollaire 4.6]{au},
it is reduced to the case
where ${\cal G}$ is
a simple perverse sheaf.
If ${\cal G}$ is supported
on the closed fiber,
we have $R\Phi{\cal G}[-1]
={\cal G}$
and the assertion follows.
2.
(1)$\Leftrightarrow$(2):
The condition (2) is equivalent to
that for every geometric point
$x$ of $X_s$,
the canonical morphism
$j_{!*}{\cal F}_x\to
R\Gamma(X_{(x)}\times_{S_{(s)}}\bar \eta,{\cal F})$ is an isomorphism.
(2)$\Leftrightarrow$(3):
By (\ref{eqH0}) and
(\ref{eqI}),
the morphism (\ref{eqij})
induces an isomorphism
$\bar i^*j_{!*}{\cal F}\to
(R\Psi {\cal F})^I$.
(2)$\Rightarrow$(4):
Since the formation of nearby cycles complex
$R\Psi{\cal F}$
commutes with base change
\cite[Proposition 3.7]{TF},
the isomorphism (\ref{eqij})
implies the condition (4).
(4)$\Rightarrow$(2):
There exists a finite extension $K'$ of $K$
such that
the inertia action $I'\subset I$
on $R\Psi{\cal F}$
is trivial, since $\Lambda$ is a finite field.
Let $j'\colon X_{K'}\to X_{S'}$
denote the base change
of the open immersion $j$
by $S'={\rm Spec}\, {\cal O}_{K'}\to S$,
let $i'\colon X_{\bar s}\to X_{S'}$
denote the canonical morphism
and let ${\cal F}'$ denote the
pull-back of ${\cal F}$ on
$X_{K'}$.
We factorize the morphism (\ref{eqij})
as the composition of
$\bar i^*j_{!*}{\cal F}\to
\bar i^{\prime*}j'_{!*}{\cal F}'\to R\Psi {\cal F}.$
By (3)$\Rightarrow$(2) already proven,
the second arrow
is an isomorphism.
The condition (4) implies
that the first arrow is an isomorphism.
Hence the composition (\ref{eqij})
is an isomorphism.
\qed}\medskip
Finally, we consider the
case where
$X$ is a scheme of
finite type over
a regular
noetherian connected
scheme $S$ of dimension $1$.
Let $\Lambda$ be
a finite field of characteristic $\ell$
invertible on $S$.
Then
the $t$-structure
$(^{\rm p}\!D^{\leqq 0},\,
^{\rm p}\!D^{\geqq 0})$ on
$D^b_c(X,\Lambda)$ is defined
as the intersection of
the inverse images of
the $t$-structures
$(^{\rm p}\!D^{\leqq 0},\,
^{\rm p}\!D^{\geqq 0})$ on
$D^b_c(X\times_SS_s,\Lambda)$
for the base changes by
the localizations $S_s\to S$
at closed points $s\in S$.
If $Y=S$ is a smooth curve over a field
$k$ and if $f\colon X\to Y$
is a morphism of schemes
of finite type over $k$,
the $t$-structure on
$D^b_c(X,\Lambda)$ defined
above is the same as
that defined by considering
$X$ as a scheme of finite type over $k$.
\begin{cor}\label{corS}
Let $S$ be a regular
noetherian scheme of dimension $1$.
Let $X$ be a scheme of finite
type over $S$
and ${\cal F}$ be
a perverse sheaf of
$\Lambda$-modules on $X$.
Let $V\subset S$ be a dense
open subscheme such
that the base change
$X_V\to V$ is universally locally acyclic relatively to the
restriction ${\cal F}_V$ of ${\cal F}$.
Then, there exists a
finite faithfully flat and generically \'etale morphism
$S'\to S$ of regular schemes
such that the base change
$X'\to S'$ is locally acyclic relatively to $j'_{!*}{\cal F}_{V'}$
where ${\cal F}_{V'}$
denotes the pull-back of ${\cal F}$
on $V'=V\times_SS'$
and $j'\colon X'_{V'}\to X'$
denote the base change.
\end{cor}
\proof{
By Proposition \ref{prS}.2
(1)$\Rightarrow$(4)
and weak approximation,
it suffices to consider locally
on a neighborhood of each point
of the complement $S\sm V$.
Since the coefficient field $\Lambda$
is finite, the assertion follows from
Proposition \ref{prS}.2 (3)$\Rightarrow$(1).
\qed}
\subsection{$C$-transversality}
\label{ssCtr}
We introduce some terminology on
proper intersection.
\begin{lm}\label{lmpi}
Let $f\colon C\to X$
and $h\colon W\to X$
be morphisms of schemes
of finite type over a field $k$.
Assume that $C$ is irreducible
of dimension $n$ and
that $h$ is locally of complete
intersection of relative
virtual dimension $d$.
Then every irreducible component
of $h^*C=C\times_XW$
is of dimension $\geqq n+d$.
\end{lm}
\proof{
Since the assertion is local on $W$,
we may decompose
$h=gi$ as the composition of
a smooth morphism $g$ with
a regular immersion of codimension
$c$.
Since the assertion is clear for
$g$, we may assume that
$h=i$ is a regular immersion.
Then, it follows from
\cite[Proposition (5.1.7)]{EGA4}.
\qed}
\begin{df}\label{dfpi}
Let $f\colon C\to X$
and $h\colon W\to X$
be morphisms of schemes
of finite type over a field $k$.
Assume that every irreducible component
of $C$ is of dimension $n$ and
that $h$ is locally of complete
intersection of relative
virtual dimension $d$.
We say that $h\colon W\to X$
meets $f\colon C\to X$ {\em properly}
if $h^*C=C\times_XW$
is of dimension $n+d$.
\end{df}
By Lemma \ref{lmpi},
the condition that
$h^*C=C\times_XW$
is of dimension $n+d$
is equivalent to the condition that
every irreducible component
of $h^*C=C\times_XW$
is of dimension $n+d$.
\begin{lm}\label{lmpi2}
Let $f\colon C\to X$
be a morphism of schemes
of finite type over a field $k$.
Assume that $X$ is equidimensional of dimension $m$
and that
$C$ is equidimensional of dimension $n\geqq m$.
We consider the following conditions:
{\rm (1)}
Every morphism $h\colon W\to X$
locally of complete intersection
meets $C$ properly.
{\rm (2)}
For every closed point $x$ of $X$,
the fiber $C\times_Xx$
is of the dimension $n-m$.
{\rm 1.}
We have {\rm (2)}$\Rightarrow${\rm (1)}.
Assume that the condition {\rm (2)}
is satisfied
and let $h\colon W\to X$
be a morphism
locally of complete intersection
of relative virtual dimension $d$
of schemes of
finite type over $k$.
Then $C\times_XW$ is
equidimensional of dimension $n+d$
and the morphism
$C\times_XW\to W$
satisfies the condition {\rm (2)}
and hence {\rm (1)}.
{\rm 2.}
If $X$ is regular,
we have
{\rm (1)}$\Rightarrow${\rm (2)}.
{\rm 3.}
Assume that $X={\mathbf P}$ is a projective space
and let $c$ be an integer.
Then, the linear subspaces
$V\subset {\mathbf P}$ of
codimension $c$
such that the immersion
$V\to {\mathbf P}$ meets $C$
properly form
a dense open subset
of the Grassmannian variety
${\mathbf G}$.
\end{lm}
\proof{
1.
Assume that the condition (2)
is satisfied and
let $h\colon W\to X$
be a morphism
locally of complete intersection
of relative dimension $d$.
Then, we have
$\dim C\times_XW\leqq \dim W+n-m
=n+d$.
Hence, $C\times_XW$
is equidimensional of dimension
$n+d$ by Lemma \ref{lmpi}.
The rest is clear.
2.
If $X$ is regular
and $x$ is a closed point,
the closed immersion
$i\colon x\to X$ is
a regular immersion of
codimension $m$ and hence
the condition (1) implies that
$\dim C\times_Xx=n-m$.
3.
Let ${\mathbf V}\subset
{\mathbf P}\times {\mathbf G}$
be the universal family
of linear subspaces of codimension
$c$
and we consider the cartesian diagram
$$\xymatrix{
&C_{\mathbf V}\ar[r]\ar[d]^{
\hspace{5mm}\square}\ar[ld]
&C\ar[d]\\
{\mathbf G}
&{\mathbf V}\ar[l]\ar[r]
&{\mathbf P}.
}$$
Then, since
the projection
${\mathbf V}\to {\mathbf P}$
is smooth of relative dimension
$\dim {\mathbf G}-c$,
we have
$\dim C_{\mathbf V}=
\dim {\mathbf G}+n-c$.
Hence
the open subset
of ${\mathbf G}$
consisting of $V$
such that
$\dim C\times_{\mathbf P}V\leqq n-c$
is dense.
\qed}
\medskip
Recall that a closed subset
$C$ of a vector bundle
$E$ on a scheme $X$ is
said to be {\em conical} if
it is stable under the action
of the multiplicative group.
For a closed conical subset
$C\subset E$,
the intersection $B=C\cap X$
with the $0$-section
identified with a closed subset
of $X$ is called the base of $C$.
We say that a morphism
$f\colon X\to Y$ of
noetherian schemes is finite
(resp. proper)
on a closed subset $Z\subset X$
if its restriction $Z\to Y$
is finite (resp. proper)
with respect to a
closed subscheme structure of $Z\subset X$.
\begin{df}\label{dfpt}
Let $f\colon X\to Y$ be
a morphism
of smooth schemes
over a field $k$ and
let $C\subset T^*X$ be a
closed conical subset.
{\rm 1.
(\cite[1.2]{Be})}
We say that $f\colon X\to Y$
is $C$-{\em transversal}
if the inverse image of
$C$ by the canonical morphism
$X\times_YT^*Y\to T^*X$
is a subset of the $0$-section
$X\times_YT^*_YY
\subset X\times_YT^*Y$.
{\rm 2.}
Assume that every irreducible component
of $X$ is of dimension $n$
and that
every irreducible component
of $C$ is of dimension $n$.
Assume that every irreducible component
of $Y$ is of dimension $m\leqq n$.
We say that $f\colon X\to Y$
is {\em properly} $C$-transversal
if $f\colon X\to Y$
is $C$-transversal and if
for every closed point $y$ of
$Y$,
the fiber
$C\times_Yy$
is of dimension $n-m$.
\end{df}
\begin{df}\label{dfpth}
Let $h\colon W\to X$ be
a morphism
of smooth schemes
over a field $k$ and
let $C\subset T^*X$ be a
closed conical subset.
Let $K\subset
W\times_XT^*X$ be the inverse image
of the $0$-section
$T^*_WW\subset T^*W$
by the canonical morphism
$W\times_XT^*X\to T^*W$.
{\rm 1.
(\cite[1.2]{Be})}
We say that $h\colon W\to X$
is $C$-{\em transversal}
if the intersection
$(W\times_XC)\cap K
\subset W\times_XT^*X$
is a subset of the $0$-section
$W\times_XT^*_XX$.
If $h\colon W\to X$ is
$C$-transversal,
a closed conical subset
$h^\circ C\subset T^*W$
is defined to be the image
of $h^*C=W\times_XC$
by $W\times_YT^*X\to T^*W$.
{\rm 2.
(\cite[Definition 7.1]{CC})}
Assume that every irreducible component
of $X$ is of dimension $n$
and that
every irreducible component
of $C$ is of dimension $n$.
Assume that every irreducible component
of $W$ is of dimension $m$.
We say that $h\colon W\to X$
is {\em properly} $C$-transversal
if $h\colon W\to X$
is $C$-transversal and if
$h\colon W\to X$
meets $C\to X$ properly.
\end{df}
If $h\colon W\to X$ is $C$-transversal,
the morphism
$W\times_XT^*X\to T^*W$
is finite on
$h^*C=W\times_XC$
and hence
$h^\circ C\subset T^*W$
is a closed subset
by \cite[Lemma 1.2 (ii)]{Be}.
For a morphism
$r\colon X\to Y$
of smooth schemes
proper on the base
$B=C\cap T^*_XX\subset X$
of a closed conical subset
$C\subset T^*X$,
the closed conical subset
$r_\circ C\subset T^*Y$
is defined to be
the image by the projection
$X\times_YT^*Y\to T^*Y$
of the inverse image of $C$
by the canonical morphism
$X\times_YT^*Y\to T^*X$.
\begin{lm}\label{lmtrbc}
Let $f\colon X\to Y$ be
a {\em smooth} morphism
of smooth schemes
over a field $k$ and
let $C\subset T^*X$ be a
closed conical subset.
Let
$$\begin{CD}
X@<h<< W\\
@VfVV
\hspace{-10mm}
\square
\hspace{7mm}
@VVgV\\
Y@<i<<Z
\end{CD}$$
be a cartesian diagram
of smooth schemes over
$k$.
{\rm 1.}
Assume that
$f\colon X\to Y$ is
$C$-transversal
(resp.\ properly $C$-transversal).
Then,
$h\colon W\to X$ is
$C$-transversal
(resp.\ properly $C$-transversal)
and
$g\colon W\to Z$ is
$h^\circ C$-transversal
(resp.\ properly $h^\circ C$-transversal).
{\rm 2.}
Assume that
$f\colon X\to Y$ is proper
on the base of $C$.
Then,
$i\colon Z\to Y$ is
$f_\circ C$-transversal
if and only if
$h\colon W\to X$ is
$C$-transversal.
If these equivalent conditions
are satisfied, we have
$i^\circ f_\circ C=
g_\circ h^\circ C$.
\end{lm}
\proof{
1.
The assertion for the transversality
is proved in \cite[Lemma 3.9.2]{CC}.
The proper transversality
of $h\colon W\to X$ follows from
the transversality
and Lemma \ref{lmpi2} applied
to $C\to Y$ and $Z\to Y$.
The proper $h^\circ C$-transversality
of $g\colon W\to Z$ follows
from that
$h^*C\to h^\circ C$ is finite.
2.
We consider the commutative diagram
$$\begin{CD}
T^*X@<<<
W\times_XT^*X@>{dh}>> T^*W\\
@AAA
\hspace{-15mm}
\square
\hspace{16mm}
@AAA
@AAA
\\
X\times_YT^*Y@<<<
W\times_YT^*Y@>{g^*(di)}>>
W\times_ZT^*Z\\
@VVV
\hspace{-15mm}
\square
\hspace{16mm}
@VV{g_*}V
\hspace{-17mm}
\square
\hspace{14mm}
@VVV
\\
T^*Y@<<<
Z\times_YT^*Y@>{di}>> T^*Z
\end{CD}$$
with cartesian squares
indicated by $\square$.
The upper vertical arrows
are injections.
Since $dh$ induces an isomorphism
$W\times_XT^*X/Y\to T^*W/Z$
for the relative cotangent bundles
and $f\colon X\to Y$ is smooth,
the upper right square is
also cartesian.
Let $K$ and $K'$
be the inverse image
of the $0$-sections
by $dh\colon W\times_XT^*X\to T^*W$
and
$di\colon Z\times_YT^*Y\to T^*Z$
respectively.
Since the upper right square is cartesian,
$K$ is identified with
the inverse image of
the $0$-section by
$g^*(di)\colon
W\times_YT^*Y
\to W\times_ZT^*Z$
which equals
$g_*^{-1}(K')
\subset W\times_YT^*Y$.
Since the lower left square
is cartesian,
the pull-back
$Z\times_Yf_\circ C$
is the image $g_*(C')$ of
$C'=(W\times_X C)
\cap
(W\times_YT^*Y)$.
Hence the condition that
$(Z\times_Yf_\circ C) \cap K'
=g_*(C')\cap K'
=g_*(C'\cap g_*^{-1}(K'))$
is a subset
of the $0$-section
is equivalent to the condition
that $(W\times_X C)
\cap K=C'\cap g_*^{-1}(K')$ is a subset
of the $0$-section.
If these conditions are satisfied,
the equality
$i^\circ f_\circ C=
g_\circ h^\circ C$
follows from the cartesian diagram.
\qed}
\begin{lm}\label{lmLeg}
Let ${\mathbf P}$
be a projective space of
dimension $n$ and
let $C\subset T^*{\mathbf P}$
be a closed conical subset.
{\rm 1.}
Let ${\mathbf P}^\vee$
be the dual projective space,
let
$Q\subset {\mathbf P}
\times {\mathbf P}^\vee$
be the universal family of
hyperplanes
and let
\begin{equation}
\begin{CD}
{\mathbf P}
@<p<< Q
@>{p^\vee}>>
{\mathbf P}^\vee
\end{CD}
\label{eqpv}
\end{equation}
be the projections.
Let $C^\vee
=p^\vee_\circ p^\circ C$
be the Legendre transform.
Let $V\subset {\mathbf P}$
be a linear subspace
and let $V^\vee\subset {\mathbf P}^\vee$
be the dual subspace.
Then the immersion
$V\to {\mathbf P}$
is $C$-transversal if and only if
$V^\vee\to {\mathbf P}^\vee$
is $C^\vee$-transversal.
{\rm 2.}
Assume that
every irreducible component of
$C$ is of dimension $n
=\dim {\mathbf P}$ and
let $0\leqq c\leqq n$ be an integer.
Then, the linear subspaces
$V\subset {\mathbf P}$ of
codimension $c$
such that the immersion
$V\to {\mathbf P}$ is
properly $C$-transversal form
a dense open subset
of the Grassmannian variety
${\mathbf G}$.
\end{lm}
\proof{
1.
The $C$-transversality of
$V\to {\mathbf P}$ means
${\mathbf P}(T^*_V{\mathbf P})
\cap {\mathbf P}(C)=\varnothing$
and similarly for
the $C^\vee$-transversality of
$V^\vee\to {\mathbf P}^\vee$.
Then, the assertion follows
from
${\mathbf P}(T^*_V{\mathbf P})
={\mathbf P}(T^*_{V^\vee}{\mathbf P}^\vee)$
and ${\mathbf P}(C)=
{\mathbf P}(C^\vee)$
under the identification
${\mathbf P}(T^*{\mathbf P})
=Q
={\mathbf P}(T^*{\mathbf P}^\vee)$.
2.
Since the condition is an open condition
on $V$, it suffices to show the existence.
By induction on $c$,
it is reduced to the case $c=1$.
By 1,
the hyperplanes $H$ such that
the immersion $H\to {\mathbf P}$
is $C$-transversal
is parametrized by
the complement of
the image $p^\vee({\mathbf P}(C))
\subsetneqq {\mathbf P}^\vee$.
Hence, the assertion follows
from this and Lemma \ref{lmpi2}.3.
\qed}
\subsection{$SS{\cal F}$-transversality}
\label{ssFtr}
For the definitions and
basic properties
of the singular support
of a constructible complex
on a smooth scheme over a field,
we refer to \cite{Be}
and \cite{CC}.
Let $k$ be a field and
let $\Lambda$ be a finite field
of characteristic $\ell$ invertible in $k$.
Let $X$ be a smooth scheme over $k$
such that every irreducible
component is of dimension $n$
and let
${\cal F}$ be a constructible complex on $X$.
The singular support
$SS{\cal F}$ is defined in \cite{Be}
as a closed conical subset of
the cotangent bundle $T^*X$.
By \cite[Theorem 1.3 (ii)]{Be},
every irreducible component $C_a$
of the singular support
$$SS{\cal F}=C=\bigcup_aC_a$$
is of dimension $n=\dim X$.
Further if $k$ is perfect,
the characteristic cycle
$$CC{\cal F}=\sum_am_aC_a$$
is defined as a linear combination
with ${\mathbf Z}$-coefficients
in \cite[Definition 5.10]{CC}.
It is characterized by the
Milnor formula
\begin{equation}
-\dim {\rm tot}
\phi_u({\cal F},f)=
(CC{\cal F},df)_{T^*U,u}
\label{eqMil}
\end{equation}
for morphisms $f\colon U\to Y$
to smooth curves $Y$
defined on an \'etale neighborhood $U$
of an isolated characteristic point $u$.
For more detail on the notation,
we refer to \cite[Section 5.2]{CC}.
\begin{lm}\label{lmh}
Let $h\colon W\to X$
be a morphism of smooth
schemes over a field $k$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$
and let $C$ denote the singular
support $SS{\cal F}$.
If $h\colon W\to X$
is properly $C$-transversal,
we have
$$SSh^*{\cal F}=
h^\circ SS{\cal F}.$$
\end{lm}
\proof{
By \cite[Theorem 1.4 (iii)]{Be},
we may assume that $k$
is perfect.
Suppose $\dim W=\dim X+d$.
If ${\cal F}$ is a perverse sheaf on $X$,
then $h^*{\cal F}[d]$ is a perverse sheaf on $W$
by the assumption that
$h\colon W\to X$
is $C$-transversal and by
\cite[Lemma 8.6.5]{CC}.
Hence by \cite[Theorem 1.4 (ii)]{Be},
we may assume that
${\cal F}$ is a perverse sheaf.
By \cite[Proposition 5.14.2]{CC},
we have $CC{\cal F}\geqq 0$ and
the support of $CC{\cal F}$
equals the singular support
$SS{\cal F}$.
Also
we have $(-1)^dCCh^*{\cal F}\geqq 0$ and
the support of $CCh^*{\cal F}$
equals the singular support
$SSh^*{\cal F}$.
By the assumption that
$h\colon W\to X$
is properly $C$-transversal and
by \cite[Theorem 7.6]{CC},
we have $CCh^*{\cal F}
=h^!CC{\cal F}$.
Hence by the positivity
\cite[Proposition 7.1 (a)]{Ful}, the singular support
$SSh^*{\cal F}$
equals the support $h^\circ SS{\cal F}$
of $h^!CC{\cal F}$.
\qed}
\begin{lm}\label{lmtrans}
Let $k$ be a field and
$f\colon X\to Y$ be
a morphism of schemes
of finite type over $k$.
Assume that
$Y$ is smooth over $k$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$.
Let
$$\begin{CD}
X@>i>> P\\
@V{i'}VV @VVgV\\
P'@>{g'}>>Y
\end{CD}$$
be a commutative
diagram of schemes over $k$
such that
$i$ and $i'$ are closed immersions and
the schemes $P$ and $P'$
are smooth over $k$.
Let
$C=SSi_*{\cal F}\subset
X\times_PT^*P\subset T^*P$
and
$C'=SSi'_*{\cal F}\subset
X\times_{P'}T^*P'\subset T^*P'$
denote the singular supports
of the direct images.
Then,
$P\to Y$ is $C$-transversal
(resp.\ properly $C$-transversal)
if and only if
$P'\to Y$ is $C'$-transversal
(resp.\ properly $C'$-transversal).
\end{lm}
\proof{
By factorizing $P\to Y$
as the composition of
the graph $P\to P\times Y$
and the projection
$P\times Y$,
we may assume that
$P\to Y$ is smooth.
Similarly,
we may assume that
$P'\to Y$ is smooth.
By considering the morphism
$(i, i')\colon X\to
P\times_YP'$,
we may assume that
there exists a smooth
morphism
$P'\to P$ compatible
with the immersions of $X$
and the morphisms to $Y$.
Since the assertion is \'etale local
on $P$, we may assume
that there exists
a section $s\colon P\to P'$
compatible with the immersions of $X$
and the morphisms to $Y$.
Then, we have
$C'=s_\circ C$
and the assertion follows
from \cite[Lemma 3.8]{CC}.
\qed}
\medskip
Lemma \ref{lmtrans}
allows us to make the
following definition.
\begin{df}\label{dftrans}
Let $k$ be a field and
$f\colon X\to Y$ be
a morphism of schemes
of finite type over $k$.
Assume that
$Y$ is smooth over $k$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$.
We say that $f\colon X\to Y$
is $SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal)
if locally on $X$
there exist a closed immersion
$i\colon X\to P$
to a smooth scheme $P$ over $k$
and a morphism
$g\colon P\to Y$ over
$k$ such that $f=g\circ i$
and that
$g\colon P\to Y$
is $C$-transversal
(resp.\ properly
$C$-transversal)
for $C=SSi_*{\cal F}$.
\end{df}
In Definition \ref{dftrans},
we obtain an equivalent
condition by requiring
that $g$ is {\em smooth}.
Let $f\colon X\to Y$ be
a morphism of schemes
of finite type over a field $k$
such that
$Y$ is smooth over $k$
and let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$.
For an open subset
$U\subset X$,
we say $f\colon X\to Y$
is $SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal)
on $U$
if the restriction
$U\to Y$ of $f$
is $SS{\cal F}_U$-transversal
(resp.\ properly
$SS{\cal F}_U$-transversal)
for the restriction
${\cal F}_U$ of ${\cal F}$ on $U$.
Similarly, for an open subset
$V\subset Y$,
we say $f\colon X\to Y$
is $SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal)
on $V$
if the base change
$X\times_YV\to V$ of $f$
is $SS{\cal F}_{X\times_YV}$-transversal
(resp.\ properly
$SS{\cal F}_{X\times_YV}$-transversal)
for the restriction
${\cal F}_{X\times_YV}$ of ${\cal F}$ on $X\times_YV$.
\begin{lm}\label{lmtr}
Let $f\colon X\to Y$ be
a morphism of schemes
of finite type over a field $k$.
Assume that
$Y$ is smooth over $k$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$.
{\rm 1.}
Assume that $f\colon X\to Y$ is smooth
and that ${\cal F}$ is locally constant.
Then, $f\colon X\to Y$ is
properly $SS{\cal F}$-transversal.
{\rm 2.}
Assume that
$f\colon X\to Y$ is
$SS{\cal F}$-transversal.
Or more weakly,
suppose that
there exists a quasi-finite faithfully
flat morphism
$Y'\to Y$ of smooth schemes
over $k$ such that
the base change
$f'\colon X'\to Y'$ is $SS{\cal F}'$-transversal
for the pull-back ${\cal F}'$
of ${\cal F}$ on $X'=X\times_YY'$.
Then,
$f\colon X\to Y$ is
universally locally acyclic
relatively to
${\cal F}$.
{\rm 3.}
The following conditions are equivalent:
{\rm (1)}
$f\colon X\to Y$ is
$SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal).
{\rm (2)}
For every integer $q$
and for every constituant ${\cal G}$
of the perverse sheaf $^{\rm p}\!{\cal H}^q{\cal F}$,
the morphism
$f\colon X\to Y$ is
$SS{\cal G}$-transversal
(resp.\ properly
$SS{\cal G}$-transversal).
{\rm 4.}
Let $k'$ be an extension of $k$.
Then $f\colon X\to Y$ is
$SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal)
if and only if
the base change $f'\colon X'\to Y'$
by ${\rm Spec}\ k'\to {\rm Spec}\ k$ is
$SS{\cal F}'$-transversal
(resp.\ properly
$SS{\cal F}'$-transversal)
for the pull-back ${\cal F}'$
on $X'$ of ${\cal F}$.
\end{lm}
\proof{
1.
If ${\cal F}$ is locally constant,
then the singular support
$SS{\cal F}$ is a subset of the $0$-section
$T^*_XX$.
Hence the assertion follows.
Since the remaining
assertions 2-4 are local on $X$,
we may take a closed
immersion $i\colon X\to P$
to a smooth scheme $P$ over $k$
such that $f$ is the composition
of $i$ with a morphism
$P\to Y$ over $k$.
Replacing $X$ and ${\cal F}$
by $P$ and $i_*{\cal F}$,
we may assume that $X$
is smooth over $k$.
Set $C=SS{\cal F}$.
2.
If $f\colon X\to Y$
is $C$-transversal,
the morphism
$f\colon X\to Y$ is
universally locally acyclic
relatively to
${\cal F}$
by the definition of
singular support.
Thus under the weaker assumption,
the morphism
$f'\colon X'\to Y'$ is universally
locally acyclic with respect
to the pull-back ${\cal F}'$.
Since $Y'\to Y$ is quasi-finite
and faithfully flat,
the morphism
$f\colon X\to Y$ itself is universally
locally acyclic with respect
to ${\cal F}$.
3.
By \cite[Theorem 1.4 (ii)]{Be},
the singular support
$SS{\cal F}$ equals the union
of $SS{\cal G}$
for the constituants ${\cal G}$
of the perverse sheaves
$^{\rm p}\!{\cal H}^q{\cal F}$
for integers $q$.
Hence the assertion follows.
4.
By \cite[Theorem 1.4 (iii)]{Be},
the construction of the singular support
commutes with change of
base fields.
Hence the assertion follows.
\qed}
\begin{lm}\label{lmtrV}
Let $f\colon X\to Y$ be
a morphism of schemes
of finite type over a field $k$.
Assume that
$Y$ is smooth over $k$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$.
Assume that $f\colon X\to Y$ is
$SS{\cal F}$-transversal.
{\rm 1.}
Assume that ${\cal F}$
is a perverse sheaf.
Let $V\subset Y$
be a dense open subscheme
and
$j\colon X_V=X\times_YV\to V$
be the open immersion.
Then, there is a unique isomorphism
${\cal F}\to j_{!*}j^*{\cal F}$.
{\rm 2.}
There exists
a dense open subscheme
$V\subset Y$ such that
the base change $f\colon X\to V$ is
properly $SS{\cal F}$-transversal
on $V$.
\end{lm}
\proof{
1.
By \cite[Corollaire 1.4.25]{BBD},
it suffices to show
that for every constituant of ${\cal F}$,
its restriction on $X_V$ is non-trivial.
Let ${\cal G}$ be a constituant of ${\cal F}$.
By Lemma \ref{lmtr}.3 and 2,
the morphism
$f\colon X\to Y$ is
locally acyclic relatively to ${\cal G}$.
Let $x$ be a geometric point of $X$
such that ${\cal G}_x\neq 0$
and let
$y\to f(x)$ be a specialization
for a geometric point $y$ of $V$.
Then, since the canonical morphism
${\cal G}_x\to
R\Gamma(X_{(x)}\times_{Y_{(f(x))}}y,{\cal G})$
is an isomorphism,
the restriction of ${\cal G}$ on $X_V$
is non-trivial. Thus the assertion is proved.
2.
As in the proof of Lemma \ref{lmtr},
we may assume that $X$
is smooth over $k$.
Set $C=SS{\cal F}$.
There exists a dense open subset
$V\subset Y$
such that for every irreducible
component $C_a$ with
the reduced scheme structure
of $C=\bigcup_aC_a$,
the base change
$C_a\times_YV\to V$ is flat.
\qed}
\begin{lm}\label{lmtrb}
Let $f\colon X\to Y$ be
a morphism of schemes
of finite type over a field $k$.
Assume that
$Y$ is smooth over $k$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$.
Let $Y'\to Y$ be a morphism
of smooth schemes over $k$
and let
$$\begin{CD}
X@<h<< X'\\
@VfVV
\hspace{-10mm}
\square
\hspace{7mm}
@VV{f'}V\\
Y@<<< Y',
\end{CD}$$
be a cartesian diagram.
Let ${\cal F}'$ denote the
pull-back of ${\cal F}$ on $X'$.
{\rm 1.}
We consider the following conditions:
{\rm (1)}
$f\colon X\to Y$ is
$SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal).
{\rm (2)}
$f'\colon X'\to Y'$ is
$SS{\cal F}'$-transversal
(resp.\ properly
$SS{\cal F}'$-transversal).
Then, we have
{\rm (1)}$\Rightarrow${\rm (2)}.
Conversely,
if $Y'\to Y$ is \'etale surjective,
we have
{\rm (2)}$\Rightarrow${\rm (1)}.
{\rm 2.}
Assume that $f\colon X\to Y$ is
$SS{\cal F}$-transversal, that
${\cal F}$ is a perverse sheaf on $X$
and that $\dim Y'=\dim Y+d$.
Then ${\cal F}'[d]$ is
a perverse sheaf on $X'$.
{\rm 3.}
Assume that $f\colon X\to Y$ is smooth
and is properly $SS{\cal F}$-transversal.
Then, we have
$SS{\cal F}'=h^\circ SS{\cal F}$.
Further if $k$ is perfect,
we have
$CC{\cal F}'=h^! CC{\cal F}$.
\end{lm}
\proof{
Since the assertions are local on $X$,
we may take a closed
immersion $i\colon X\to P$
to a smooth scheme $P$ over $Y$.
As in the proof of Lemma \ref{lmtr},
we may assume that $f\colon
X\to Y$ is smooth.
Set $C=SS{\cal F}$.
1.
Assume that $f\colon X\to Y$
is $C$-transversal.
The pair $(h,f')$ of morphisms
is $C$-transversal by
Lemma \ref{lmtrbc}.1.
Hence,
${\cal F}'=h^*{\cal F}$
is micro-supported on $h^\circ C$
by \cite[Lemma 4.2.4]{CC}
and we have
an inclusion
$SS{\cal F}'\subset h^\circ C$
and $f'$ is $SS{\cal F}'$-transversal.
Thus the implication
(1)$\Rightarrow$(2) is proved
for the $C$-transversality.
The assertion on
the proper $C$-transversality
follows from this
and Lemma \ref{lmtrbc}.1.
Since the formation of
singular support is \'etale local,
we have
(2)$\Rightarrow$(1)
if $Y'\to Y$ is \'etale surjective.
{\rm 2.}
Since $h\colon X'\to X$
is $C$-transversal by Lemma \ref{lmtrbc}.1,
the assertion follows from
\cite[Lemma 8.6.5]{CC}.
{\rm 3.}
Since $h\colon X'\to X$
is properly $C$-transversal by Lemma \ref{lmtrbc}.1,
the assertion for
$SS{\cal F}'$
(resp.\ for $CC{\cal F}'$)
follows from
Lemma \ref{lmh}
(resp.\ \cite[Theorem 7.6]{CC}).
\qed}
\medskip
Lemma \ref{lmtrb}.3 is
closely related to the subject studied in
\cite{EH}.
\begin{lm}\label{lmtrfun}
Let $f\colon X\to Y$ be
a morphism of schemes
of finite type over a field $k$.
Assume that
$Y$ is smooth over $k$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$.
{\rm 1.}
Let $g\colon Y\to Z$
be a smooth morphism
of smooth schemes over $k$.
If $f\colon X\to Y$ is
$SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal),
then the composition
$gf\colon X\to Z$ is
$SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal).
{\rm 2.}
Let
$h\colon W\to X$ be a smooth
morphism
of schemes of finite type over $k$.
If $f\colon X\to Y$ is
$SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal),
then the composition
$fh\colon W\to Y$ is
$SSh^*{\cal F}$-transversal
(resp.\ properly
$SSh^*{\cal F}$-transversal).
{\rm 3.}
Let $$
\xymatrix{
X\ar[r]^r\ar[rd]_f&
X'\ar[d]^{f'}\\
&Y}$$
be a commutative diagram
of morphisms
of schemes of finite type over $k$.
Assume that
$r\colon X\to X'$ is {\rm proper}
on the support of ${\cal F}$
and that $f\colon X\to Y$ is
quasi-projective.
If $f\colon X\to Y$ is
$SS{\cal F}$-transversal,
then $f'\colon X'\to Y$ is
$SS\, Rr_*{\cal F}$-transversal.
\end{lm}
\proof{
1.
As in the proof of Lemma \ref{lmtr},
we may assume that $X$
is smooth over $k$.
Set $C=SS{\cal F}$.
Since $g\colon Y\to Z$
is smooth,
the $C$-transversality of
$f$ implies that of $gf$
by \cite[Lemma 3.6.3]{CC}.
The assertion on
the proper $C$-transversality
follows from this and
the smoothness of $g$.
2.
Since the question is \'etale local
on $W$,
we may assume that
there exists a cartesian diagram
$$\xymatrix{
W\ar[r]^h\ar[d]^{
\hspace{5mm}\square}&
X\ar[r]^f\ar[d]^i&Y\\
Q\ar[r]&P\ar[ru]
}$$
of morphisms of schemes over $k$
such that
the vertical arrows are
closed immersions
and the horizontal arrow
$Q\to P$
is a smooth morphism of smooth schemes over $k$.
By replacing $X$ and ${\cal F}$
by $P$ and $i_*{\cal F}$,
we may assume that $X$ is smooth.
Since
$W\times_XT^*X\to T^*W$
is an injection and
$SSh^*{\cal F}
=h^\circ SS{\cal F}$
by Lemma \ref{lmh},
the assertion follows.
3.
Since the assertion is
local on $X'$,
we may assume that
$X'$ and $Y$ are affine
and hence $X$ is quasi-projective.
By taking
a closed immersion $i'\colon
X'\to P'$ to an affine space
and by factorizing
$X'\to Y$
as the composition of
the immersion
$(i',f')\colon X'\to P'\times Y$
and the projection
$P'\times Y\to Y$,
we may assume that
$X'$ is smooth.
Similarly, we take
an open subscheme $P$
of a projective
space and a closed immersion
$i\colon X\to P$.
Then, by factorizing
$X\to X'$
as the composition of
the immersion
$(i,r)\colon X\to P\times X'$
and the projection
$P\times X'\to X'$,
we may also assume that
$X$ is smooth,
by \cite[Lemma 3.8
(2)$\Rightarrow$(1)]{CC}.
By \cite[Lemma 2.2 (ii)]{Be},
we have $SS\, Rr_*{\cal F}
\subset r_\circ SS{\cal F}$.
Hence the assertion follows
from \cite[Lemma 3.8 (2)$\Rightarrow$(1)]{CC}.
\qed}
\medskip
We give two methods to
establish $SS{\cal F}$-transversality.
\begin{lm}\label{lmtrZ}
Let $Y\to S$ be
a smooth morphism
of smooth schemes
of finite type over a field $k$
and let $f\colon X\to Y$ be
a morphism of schemes
of finite type over a field $k$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$.
Assume that the composition
$X\to S$ is {\em properly}
$SS{\cal F}$-transversal.
{\rm 1.}
Assume that $k$ is perfect.
Then, the following conditions
are equivalent:
{\rm (1)}
$f\colon X\to Y$ is $SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal).
{\rm (2)}
For every closed point $s\in S$,
the fiber
$f_s\colon X_s\to Y_s$ is
$SS{\cal F}_s$-transversal
(resp.\ properly
$SS{\cal F}_s$-transversal)
for the pull-back
${\cal F}_s$
of ${\cal F}$ on
$X_s=X\times_Ss$.
{\rm 2.}
Assume that
${\cal F}$ is a perverse sheaf on $X$
and that
$f\colon X\to Y$ is
locally acyclic relatively to
${\cal F}$.
If there exists a closed subset
$Z\subset X$
{\em quasi-finite} over $S$ such that
$f\colon X\to Y$
is $SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal)
on the complement
of $Z$, then
$f\colon X\to Y$
is $SS{\cal F}$-transversal
(resp.\ properly
$SS{\cal F}$-transversal)
on $X$.
\end{lm}
\proof{
1.
The implication (1)$\Rightarrow$(2)
is a special case of
Lemma \ref{lmtrb}.1.
We show
(2)$\Rightarrow$(1).
Since the question is local on $X$,
we may assume that $f\colon X\to Y$
is smooth.
Let $T^*X/S$ and
$T^*Y/S$ denote
the relative cotangent bundles
and let
$C=SS{\cal F}$.
By the assumption that
$X\to S$ is $C$-transversal,
the canonical surjection
$T^*X\to T^*X/S$
is finite on $C$
by \cite[Lemma 1.2 (ii)]{Be}.
Hence its image
$\bar C
\subset T^*X/S$
is a closed conical subset
and $C\to \bar C$ is finite.
The morphism
$X\to Y$ is $C$-transversal
if and only if
the inverse image of $\bar C
\subset T^*X/S$
by the canonical injection
$X\times_YT^*Y/S\to T^*X/S$
is a subset of the $0$-section.
This is equivalent to that
for every closed point
$s\in S$ and
the closed immersion
$i_s\colon X_s\to X$,
the morphism
$f_s\colon X_s\to Y_s$
is $i_s^\circ C$-transversal.
Further, under the
assumption that
$f\colon X\to Y$ is $C$-transversal,
this is properly $C$-transversal
if and only if
$f_s\colon X_s\to Y_s$
is properly $i_s^\circ C$-transversal
for every closed point
$s\in S$.
By the assumption that
$X\to S$ is properly $SS{\cal F}$-transversal and by
Lemma \ref{lmtrb}.3,
we have
$SS{\cal F}_s=
i_s^\circ SS{\cal F}=
i_s^\circ C$
for every closed point
$s\in S$.
Hence the assertion is proved.
2.
By Lemma \ref{lmtr}.4,
we may assume that $k$
is perfect.
By 1 and Lemma \ref{lmtrb}.2, we may assume
that $S$ is a point
and further that
$S={\rm Spec}\ k$.
As in the proof of Lemma \ref{lmtrb},
we may assume that $f\colon X\to Y$
is smooth of relative dimension $d$.
Let $u\in Z$.
By replacing $X$
by a neighborhood of $u$,
we may assume $Z=\{u\}$.
Set $C=SS{\cal F},\ v=f(u)\in Y$
and regard
$X\times_YT^*Y$ as a closed subscheme
of $T^*X$.
We show that
$f\colon X\to Y$ is $C$-transversal,
assuming that $f\colon X\to Y$
is locally acyclic relatively to
${\cal F}$.
Namely, we show that
the intersection $C'
=C\cap (X\times_YT^*Y)$ is
a subset of the $0$-section
$X\times_YT^*_YY$.
By the assumption that
$f\colon X\to Y$
is $C$-transversal outside $u$,
the intersection $C'
=C\cap (X\times_YT^*Y)$ is
a subset of the union
$(X\times_YT^*_YY)
\cup (u\times_YT^*Y)$ with
the fiber at $u$.
Let $\omega
\in u\times_YT^*Y
=v\times_YT^*Y$
be a non-zero element.
After shrinking $Y$
to a neighborhood of $v=f(u)$
if necessary,
we take a smooth morphism
$Y\to {\mathbf A}^1
={\rm Spec}\ k[t]$
such that $dt(v)=\omega$.
Then, by
\cite[Lemma 3.6.3]{CC},
the point
$u$ is at most an isolated $C$-characteristic point
of the composition
$g\colon X\to Y\to {\mathbf A}^1$.
Since ${\cal F}$ is a perverse sheaf,
the characteristic cycle
$CC{\cal F}$ is an effective
cycle and its support equals
$C=SS{\cal F}$
by
\cite[Proposition 5.14]{CC}.
Let $dg$ denote the section of
$X\times_YT^*Y\subset T^*X$
defined by the function
$g^*(t)$.
Since the composition
$X\to Y\to {\mathbf A}^1$
is locally acyclic relatively to
${\cal F}$ by
\cite[Corollaire 5.2.7]{app},
we have
$(CC{\cal F},dg)_{T^*X,u}=0$
by the Milnor formula (\ref{eqMil}).
Therefore by the positivity
\cite[Proposition 7.1 (a)]{Ful}, the intersection
$SS{\cal F}\cap dg
=C'\cap dg$
is empty and
hence $\omega\notin C'$.
Since $\omega$ is any non-zero
element of $u\times_YT^*Y$,
we conclude that
$C'\cap
(u\times_YT^*Y)\subset 0$
and that
$f\colon X\to Y$
is $C$-transversal.
Assume further that
$f\colon X\to Y$
is properly $C$-transversal
outside $u$.
Since
$f\colon X\to Y$
is $C$-transversal,
the morphism
$T^*X\to T^*X/Y$
to the relative cotangent bundle
is finite on $C$
by \cite[Lemma 1.2 (ii)]{Be}
and the image
$\bar C\subset T^*X/Y$
of $C$ is a closed conical subset.
It is sufficient to
show that for every point
$y\in Y$,
the fiber $\bar C\times_Yy$
is of dimension $\leqq d$.
For $y\neq f(u)$,
this follows from the assumption.
Assume $y=f(u)$.
Then, every irreducible component of
$\bar C\times_Yy$
is either
a closure of an irreducible
component of
$\bar C\times_Yy
\cap (X\times_Yy\sm \{u\})$
or a subset of
the fiber
$T^*_u(X\times_Yy)$.
Since $\dim
T^*_u(X\times_Yy)=d$,
the assertion is proved.
\qed}
\begin{lm}\label{lmtrqf}
Let
$$\begin{CD}
W@>h>> X@>f>> Y\\
@A{j'}AA
\hspace{-10mm}
\square
\hspace{7mm}
@AAjA\\
U'@>{h_U}>> U
\end{CD}$$
be a cartesian diagram of schemes
of finite type over a field $k$.
Assume that $Y$ is smooth over $k$
and that $j$ is an open immersion.
Let ${\cal F}_U$
be a perverse sheaf
of $\Lambda$-modules on $U$.
Let ${\cal F}=j_{!*}{\cal F}_U$
and let ${\cal F}'$ be
a perverse sheaf on $W$
such that the restriction
${\cal F}'_{U'}$ on $U'$
is isomorphic to the pull-back
$h_U^*{\cal F}_U$.
If one of the following conditions
{\rm (1)} and {\rm (2)} below
is satisfied and if
$f\circ h\colon W\to Y$
is $SS{\cal F}'$-transversal,
then $f\colon X\to Y$
is $SS{\cal F}$-transversal.
{\rm (1)}
The morphism
$h\colon W\to X$
is proper, surjective and generically finite and
the composition
$W\to Y$ is quasi-projective.
The scheme $U$ is smooth of dimension $d$
over $k$
and there exists a locally constant sheaf
${\cal G}$ of $\Lambda$-modules
on $U$
such that ${\cal F}_U={\cal G}[d]$.
{\rm (2)}
The morphism
$h$ is quasi-finite
and faithfully flat.
For every constituant
${\cal G}$ of
the perverse sheaf ${\cal F}_U$,
the pull-back $h_U^*{\cal G}$ is
also a perverse sheaf on $U'$.
\end{lm}
\proof{
Assume that (1) is satisfied.
Since $h$ is surjective,
the canonical morphism
${\cal G}\to h_{U*}h_U^*{\cal G}$
is an injection.
Hence the constituants of
the perverse sheaf
${\cal F}_U={\cal G}[d]$ are constituants of
$^{\rm p}\! {\cal H}^0Rh_{U*}
{\cal F}'_{U'}
=j^*{}^{\rm p}\! {\cal H}^0Rh_*{\cal F}'$.
Consequently, the constituants
of ${\cal F}=j_{!*}{\cal F}_U$
are constituants of
$^{\rm p}\! {\cal H}^0Rh_*{\cal F}'$.
Since $W\to Y$ is quasi-projecitive,
the assertion follows
from Lemma \ref{lmtrfun}.3
and Lemma \ref{lmtr}.3.
Assume that (2) is satisfied.
By Lemma \ref{lmtr}.3
and the perversity assumption on
the pull-backs,
we may assume that ${\cal F}_U$
is a simple perverse sheaf.
Then, by \cite[Th\'eor\`eme 4.3.1 (ii)]{BBD},
there exists a locally closed
immersion $i\colon V\to U$
of a scheme $V$ smooth of dimension $d$
over $k$ and a locally constant sheaf
${\cal G}$ on $V$ such
that ${\cal F}_U=i_{!*}{\cal G}[d]$.
By replacing $X$ and $U$
by the closure of the image of $V\to X$
and $V$,
we may assume that $V=U$.
Since the assertion is \'etale local
on $X$ by Lemma \ref{lmtrb}.1,
we may assume that
$h$ is finite
and that $X, W$ and $Y$ are affine.
Then the assertion follows
from the case (1).
\qed}
\subsection{Alteration and transversality}\label{ssat}
Let $f\colon X\to Y$
be a morphism of smooth
schemes over a field $k$
and let $D\subset Y$
be a divisor smooth over $k$.
In this article,
we say that $f\colon X\to Y$
is {\em semi-stable}
relatively to $D$
if \'etale locally on $X$
and on $Y$,
there exists a cartesian diagram
$$\begin{CD}
X@>f>>Y@<<<D\\
@VVV
\hspace{-10mm}
\square
\hspace{7mm}
@VVV
\hspace{-10mm}
\square
\hspace{7mm}
@VVV\\
{\mathbf A}^n
@>>>
{\mathbf A}^1@<<<0
\end{CD}$$
where
the lower left horizontal arrow
${\mathbf A}^n={\rm Spec}\, k[t_1,\ldots,t_n]\to
{\mathbf A}^1
={\rm Spec}\, k[t]$
is defined by $t\mapsto
t_1\cdots t_n$
and the lower right horizontal arrow is
the inclusion of the origin
$0\in {\mathbf A}^1$.
A semi-stable morphism
$f\colon X\to Y$
is flat and
the base change
$f_V\colon X\times_YV\to V=Y\sm D$
is smooth.
We recall statements on
the existence of alteration.
\begin{lm}\label{lmdJ}
Let $k$ be a perfect field
and let
$f\colon X\to Y$
be a dominant separated morphism
of integral schemes
of finite type over $k$.
{\rm 1.}
There exists a commutative diagram
\begin{equation}
\begin{CD}
X@<<< W\\
@VfVV@VV{g}V\\
Y@<<<Y'
\end{CD}
\label{eqdJ}
\end{equation}
of integral
schemes of finite type over $k$
satisfying the following condition:
The bottom horizontal arrow
$Y'\to Y$
is dominant
and is the composition $gh$
of an \'etale morphism $g$
and a finite flat radicial morphism $h$.
The schemes $W$ and $Y'$
are smooth over $k$
and the morphism $g\colon W\to Y'$
is quasi-projective and smooth.
The induced morphism
$W\to X\times_YY'$
is proper surjective
and generically finite.
{\rm 2.}
Let $\xi\in Y$ be a point
such that the local ring
${\cal O}_{Y,\xi}$
is a discrete valuation ring.
Then,
there exists a commutative diagram
{\rm (\ref{eqdJ})}
of integral
schemes of finite type over $k$
satisfying the following condition:
The bottom horizontal arrow
$Y'\to Y$ is quasi-finite
and flat and its image is an open neighborhood
of $\xi$.
The schemes $W$ and $Y'$
are smooth over $k$,
the closure $D'\subset Y'$
of the inverse image
of $\xi$ is a divisor smooth over $k$ and
the morphism $g\colon W\to Y'$
is quasi-projective
and is semi-stable relatively to $D'$.
The induced morphism
$W\to X\times_YY'$
is proper surjective
and generically finite.
\end{lm}
\proof{
1. Let $\eta$ be the generic point
of $Y$.
Then, it suffices to apply
\cite[Theorem 4.1]{dJ} to
the generic fiber $X\times_Y\eta$.
2.
Let $S={\rm Spec}\
{\cal O}_{Y,\xi}$ be the localization
at $\xi$.
Then, it suffices to apply
\cite[Theorem 8.2]{dJ} to
the base change $X\times_YS\to S$.
\qed}
\medskip
We prove an analogue
of the generic local acyclicity theorem
\cite[Th\'eor\`eme 2.13]{TF}.
\begin{pr}\label{prV1}
Let $f\colon X\to Y$ be
a morphism of schemes
of finite type over a {\em perfect} field $k$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$.
Then, there exists a cartesian diagram
\begin{equation}
\begin{CD}
X@<<< X'\\
@VfVV
\hspace{-10mm}
\square
\hspace{7mm}
@VV{f'}V\\
Y@<<<Y'
\end{CD}
\label{eqdJ2}
\end{equation}
of schemes of finite type over $k$
satisfying the following conditions:
The scheme $Y'$ is smooth over $k$
and the morphism $Y'\to Y$
is dominant and
is the composition $gh$
of an \'etale morphism $g$
and a finite flat radicial morphism $h$.
The morphism
$f'\colon X'\to Y'$
is properly $SS{\cal F}'$-transversal
for the pull-back
${\cal F}'$ of ${\cal F}$
on $X'$.
\end{pr}
\proof{
We may assume that
${\cal F}$ is a simple
perverse sheaf
by Lemma \ref{lmtr}.3
and Lemma \ref{lmtrb}.
Hence, we may
assume that
there exist a locally closed immersion
$j\colon Z\to X$ of
a smooth irreducible
scheme of dimension $d$
and
a simple locally constant
sheaf ${\cal G}$
of $\Lambda$-modules
such that $j_{!*}{\cal G}[d]={\cal F}$
by \cite[Th\'eor\`eme 4.3.1 (ii)]{BBD}.
By replacing $X$ by the closure
of $j(Z)$,
we may assume that $j\colon
Z\to X$ is an open immersion.
It suffices to consider the
case where
$Z\to Y$ is dominant
since the assertion is clear
if otherwise.
Let $Z_1\to Z$ be
a finite \'etale covering
such that the pull-back of
${\cal G}$ is constant
and let $X_1$ be the normalization
of $X$ in $Z_1$.
Applying Lemma \ref{lmdJ}.1
to $X_1\to Y$,
we obtain a commutative diagram
\begin{equation}
\begin{CD}
X@<r<<W\\
@VfVV@VVV\\
Y@<<<Y'
\end{CD}
\label{eqWXY}
\end{equation}
of schemes over $k$
satisfying the following conditions:
The scheme $Y'$ is
smooth and the morphism $Y'\to Y$
is dominant and is the composition $gh$
of an \'etale morphism $g$
and a finite flat radicial surjective
morphism $h$.
The morphism
$W\to Y'$ is quasi-projective
and smooth.
The induced morphism
$r'\colon W\to X'=X\times_YY'$
is proper surjective and
generically finite.
The pull-back ${\cal G}'_W$
of ${\cal G}$ on $W\times_XZ$
is a constant sheaf.
We consider the cartesian diagram
$$\begin{CD}
Z@<<< Z'@<<< W\times_XZ\\
@VjVV
\hspace{-10mm}
\square
\hspace{7mm}
@V{j'}VV
\hspace{-15mm}
\square
\hspace{12mm}
@VV{j_W}V\\
X@<<< X'@<{r'}<< W
\end{CD}$$
and let ${\cal G}'$
be the pull-back of ${\cal G}$
on $Z'$.
Since the finite radicial surjective morphism $h$ is universally
a homeomorphism, we have
${\cal F}'=j'_{!*}{\cal G}'[d]$.
Since
${\cal G}'_W$
is a constant sheaf on $W\times_XZ$
and $W$ is smooth over $k$,
the intermediate extension
$j_{W!*}{\cal G}'_W[d]$
is constant.
The smooth morphism
$W\to Y'$ is properly
$SSj_{W!*}{\cal G}'_W[d]$-transversal
by Lemma \ref{lmtr}.1.
Since $W\to X'$ is proper
and $W\to Y'$ is quasi-projective,
the morphism $X'\to Y'$ is
$SS{\cal F}'$-transversal by
the case (1) in Lemma \ref{lmtrqf}.
After shrinking $Y'$,
the morphism $X'\to Y'$ is
properly $SS{\cal F}'$-transversal
by Lemma \ref{lmtrV}.2.
\qed}
\begin{cor}\label{corFr}
Let $f\colon X\to Y$ and ${\cal F}$
be as in Proposition {\rm \ref{prV1}}
and assume that $k$
is of characteristic $p>0$.
Then, there exist a
dense open subscheme
$V\subset Y$ smooth over $k$
and an iteration $\tilde V\to V$ of Frobenius
such that the base change
$X\times_Y\tilde V\to \tilde V$
is $SS\tilde {\cal F}$-transversal
for the pull-back $\tilde {\cal F}$
on $\tilde X_V=X\times_Y\tilde V$.
\end{cor}
\proof{
After shrinking $Y'$
in the conclusion of
Proposition \ref{prV1},
we may assume that
$Y'\to Y$
is the composition $jgh$
of an open immersion $j
\colon V\to Y$,
a finite surjective radical morphism
$g$ and an \'etale surjective
morphism $h$.
By Lemma \ref{lmtrb}.1,
we may assume that
$Y'\to Y$ is $jg$.
Thus, the assertion follows.
\qed}
\medskip
We show an analogue of
the stable reduction theorem.
\begin{pr}\label{prV2}
Let
\begin{equation}
\begin{CD}
X@<\supset<< U\\
@VfVV
\hspace{-10mm}
\square
\hspace{7mm}
@VV{f_V}V\\
Y@<\supset<< V
\end{CD}
\label{eqXUV2}
\end{equation}
be
a cartesian diagram of schemes
of finite type over a {\em perfect} field $k$.
Assume that $Y$ is normal
and that $V$ is
a dense open subset of $Y$
smooth over $k$.
Let ${\cal F}_U$
be a perverse sheaf
of $\Lambda$-modules on $U$
such that
$f_V\colon U\to V$
is $SS{\cal F}_U$-transversal.
Then, there exists a cartesian diagram
\begin{equation}
\begin{CD}
X@<<< X'@<{j'}<<U'&=U\times_XX'\\
@VfVV
\hspace{-10mm}
\square
\hspace{7mm}
@VV{f'}V
\hspace{-10mm}
\square
\hspace{7mm}
@VVV\\
Y@<g<<Y'@<\supset<<V'&=V\times_YY'
\end{CD}
\label{eqV2}
\end{equation}
of schemes of finite type over $k$
satisfying the following conditions:
The scheme $Y'$ is smooth over
$k$ and $V'\subset Y'$ is
the complement of
a divisor $D'\subset Y'$
smooth over $k$.
The morphism $g\colon Y'\to Y$
is quasi-finite flat and
the complement $Y\sm g(Y')$
is of codimension $\geqq 2$ in $Y$.
The pull-back
${\cal F}'_{U'}$ of ${\cal F}_U$
is a perverse sheaf on $U'$
and
for ${\cal F}'=j'_{!*}{\cal F}'_{U'}$ on $X'$,
the morphism
$f'\colon X'\to Y'$
is $SS{\cal F}'$-transversal.
\end{pr}
\medskip
First, we prove a basic case.
\begin{lm}\label{lmst}
Let $X$ and $Y$ be smooth schemes
over a field $k$.
Let $D\subset Y$
be a divisor smooth over $k$
and $V=Y\sm D$ be the complement.
Let $f\colon X\to Y$ be a morphism
over $k$ semi-stable
relatively to $D$.
Assume that $\dim X=n$.
For a cartesian diagram {\rm (\ref{eqV2})}
such that $Y'\to Y$
is a quasi-finite flat morphism
of smooth schemes over $k$, let ${\cal F}'$ be the
perverse sheaf
${\cal F}'=j'_{!*}\Lambda_{U'}[n]$ on $X'$.
{\rm 1.}
Assume that $\dim Y=1$.
Let $Y'\to Y$
be a flat morphism of smooth curves over $k$
such that for every
$y'\in Y'\sm V'$,
the action of the
inertia group $I_{y'}$
on $R\Psi_{y'}\Lambda_{U'}$
is trivial.
Then the morphism
$X'\to Y'$ is properly
$SS{\cal F}'$-transversal.
{\rm 2.}
There exists a quasi-finite faithfully flat morphism
$Y'\to Y$ of smooth schemes over $k$
satisfying the following conditions:
The open subscheme $V'\subset Y'$
is the complement of
a divisor $D'$ smooth over $k$
and the morphism
$X'\to Y'$ is properly
$SS{\cal F}'$-transversal.
\end{lm}
\proof{
1.
Since the question is \'etale local,
we may assume that
$Y={\mathbf A}^1_k=
{\rm Spec}\ k[t]$,
that
$X=X_n=
{\mathbf A}^n_k={\rm Spec}\ k[t_1,\ldots,t_n]$
and that the morphism
$f\colon X\to Y$
is defined by $t\mapsto
t_1\cdots t_n$.
We prove the assertion by induction on $n$.
If $n=1$, then
$f\colon X\to Y$ is \'etale and
${\cal F}'$ is constant.
Hence the assertion follows in this case
by Lemma \ref{lmtr}.1.
Assume $n>1$.
Outside the closed point
$u\in X$ defined by $t_1=\cdots=t_n=0$,
locally there exists a smooth morphism
$X=X_n\to X_{n-1}$ over $Y$.
Hence, the induction hypothesis implies
the assertion on the complement
$X\sm \{u\}$
by Lemma \ref{lmtrfun}.2.
Thus,
the morphism $f'\colon X'\to Y'$
is properly $SS{\cal F}'$-transversal outside
the inverse image of $u$.
By Proposition \ref{prS}.2
(3)$\Rightarrow$(1),
the morphism $f'\colon X'\to Y'$
is locally acyclic relatively to
${\cal F}'$.
Hence by Lemma \ref{lmtrZ}.2,
the morphism $f'\colon X'\to Y'$
is properly $SS{\cal F}'$-transversal
on $X'$.
2.
It follows from 1 by Lemma \ref{lmtrb}.1 and Lemma \ref{lmtrV}.1.
\qed}
\proof[Proof of Proposition {\rm \ref{prV2}}]{
The proof is similar to
that of Proposition \ref{prV1}.
By Lemma \ref{lmtrb}
and Lemma \ref{lmtrV},
it suffices to show the
assertion on a neighborhood
of each point $\xi\in Y$
of codimension $1$
not contained in $V$.
Thus, we may assume
that the closure $D$ of
$\xi$ is a divisor smooth over $k$
and that
$V=Y\sm D$.
We may assume that
${\cal F}_U$ is a simple
perverse sheaf
by Lemma \ref{lmtr}.3
and Lemma \ref{lmtrb}.
Hence, similarly as
in the proof of Proposition \ref{prV1},
we may
assume that
there exist a dense open immersion
$j\colon Z\to U$ of
a smooth irreducible
scheme of dimension $d$
and
a simple locally constant
sheaf ${\cal G}$
of $\Lambda$-modules
such that ${\cal F}_U=
j_{!*}{\cal G}[d]$.
Further, we may assume
that $Z\to Y$ is dominant.
Taking a finite \'etale
covering trivializing ${\cal G}$
and applying Lemma \ref{lmdJ}.2
as in the proof of
Proposition \ref{prV1},
we obtain a commutative diagram
\begin{equation}
\begin{CD}
X@<r<<W_1\\
@VfVV@VVV\\
Y@<g<<Y_1
\end{CD}
\label{eqWXY2}
\end{equation}
of schemes over $k$
satisfying the following conditions:
The scheme $Y_1$ is
smooth over $k$, the morphism $g_1\colon
Y_1\to Y$
is quasi-finite and flat
and $Y\sm g_1(Y_1)$ is
of codimension $\geqq 2$ in $Y$.
The inverse image $V\times_YY_1$
is the complement of a divisor
$D_1$ smooth over $k$
and the morphism
$W_1\to Y_1$ is quasi-projective
and is semi-stable relatively to $D_1$.
The induced morphism
$r_1\colon W_1\to X_1
=X\times_YY_1$
is proper surjective and
generically finite.
The pull-back ${\cal G}'_1$
of ${\cal G}$ on $W_1\times_XZ$
is a constant sheaf.
By Lemma \ref{lmst}.2 applied
to the semi-stable morphism
$W_1\to Y_1$, we obtain
a quasi-finite faithfully flat
morphism $Y'\to Y_1$
of smooth schemes
satisfying the condition loc.\ cit.
We consider the cartesian diagram
$$\begin{CD}
Z@<<< Z'@<<< W'\times_XZ\\
@V{j_Z}VV
\hspace{-18mm}
\square
\hspace{15mm}
@V{j_{Z'}}VV
\hspace{-20mm}
\square
\hspace{16mm}
@VV{j_W}V\\
X@<<< X'@<{r'}<< W'\\
@. =X\times_YY'@. =W_1\times_{Y_1}Y'
\end{CD}$$
and let ${\cal G}'$ and
${\cal G}'_{W'}$
denote the pull-backs of
${\cal G}$ on $Z'$
and on $W'\times_XZ$ respectively.
Since
${\cal G}'_{W'}$
is a constant sheaf on $W'\times_XZ$,
the morphism $W'\to Y'$
is $SSj_{W!*}{\cal G}'_{W'}[d]$-transversal
by Lemma \ref{lmst}.2.
The pull-back ${\cal F}'_{U'}$
is a perverse sheaf by Lemma \ref{lmtrb}.
The perverse sheaf
${\cal F}'=j'_{!*}{\cal F}'_{U'}$
is canonically identified
with $j_{Z'!*}{\cal G}'[d]$.
Since $r'\colon W'\to X'$
is proper surjective
and generically finite
and since
$r'\colon W'\to Y'$ is quasi-projective,
the morphism $X'\to Y'$ is
$SS{\cal F}'$-transversal
by the case (1) in Lemma \ref{lmtrqf}.
\qed}
\begin{cor}\label{corla}
Let the cartesian diagram
{\rm (\ref{eqXUV2})}
and a perverse sheaf ${\cal F}_U$
on $U=X\times_YV$ be
as in Proposition {\rm \ref{prV2}}.
Assume further that
$Y$ is smooth over $k$ and
that $V$ is the complement
of a divisor $D\subset Y$
smooth over $k$.
Then,
there exist a cartesian diagram
{\rm (\ref{eqV2})}
satisfying the following conditions:
The scheme $Y'$ is smooth over
$k$ and $V'\subset Y'$ is
the complement of
a divisor $D'\subset Y'$
smooth over $k$.
The morphism $g\colon Y'\to Y$
is quasi-finite flat,
the morphism
$D'\to D$ is dominant and
the morphism $V'\to V$ is {\em \'etale}.
The pull-back
${\cal F}'_{U'}$ of ${\cal F}_U$
is a perverse sheaf on $U'$
and the morphism
$f'\colon X'\to Y'$
is universally locally acyclic
relatively to ${\cal F}'=j'_{!*}{\cal F}'$.
\end{cor}
\proof{
Let $V'\subset Y'$ be
as in the conclusion of
Proposition \ref{prV2}.
Let $\bar Y''$ be
the normalization of $Y$
in the separable closure of
$k(Y)$ in $k(Y')$.
Then, there exists
a dense open subset
$Y''\subset \bar Y''$ smooth over $k$
of the image of $Y'\to \bar Y''$
such that
$g''\colon Y''\to Y$ is flat,
that $V''=V\times_YY''$
is the complement of
a divisor $D''$ smooth over $k$,
that $D''\to D$ is dominant,
and that $V''\to V$ is {\em \'etale}.
Since $Y'\times_{\bar Y''}Y''\to
Y''$ is finite surjective
radicial,
the cartesian diagram
(\ref{eqV2}) defined
by $Y''\to Y$ in place of
$Y'\to Y$ satisfies the conditions.
\qed}
\begin{cor}\label{corla1}
Let $X\to Y$ be a morphism
of schemes of finite type
over a field $k$
and assume that
$Y$ is smooth of dimension $1$.
Then, for a constructible complex
${\cal F}$ of $\Lambda$-modules
on $X$,
the following conditions are
equivalent:
{\rm (1)}
$X\to Y$ is locally acyclic
relatively to ${\cal F}$.
{\rm (2)}
$X\to Y$ is universally locally acyclic
relatively to ${\cal F}$.
{\rm (3)}
There exists a finite faithfully
flat morphism
$Y'\to Y$ of smooth curves
over $k$ such that
the base change
$X'\to Y'$ is $SS{\cal F}'$-transversal
for the base change ${\cal F}'$
of ${\cal F}$ on $X'$.
\end{cor}
The equivalence (1)$\Leftrightarrow$(2) is proved in \cite{Or}.
\proof{
We show (1)$\Rightarrow$(3).
Since the nearby cycles
functor is $t$-exact,
we may assume that ${\cal F}$
is a perverse sheaf.
Then, the assertion
follows from Propositions \ref{prV1},
\ref{prV2}
and \ref{prS}.
The implication
(3)$\Rightarrow$(2) is proved in Lemma \ref{lmtr}.2.
The implication
(2)$\Rightarrow$(1) is trivial.
\qed}
\medskip
The following example shows that
taking a covering $Y'\to Y$ in condition (3)
is necessary.
\begin{ex}\label{ex}
{\rm
Let $k$ be a field of characteristic $p>2$.
Let $X={\mathbf A}^1\times {\mathbf P}^1$
and $j\colon U={\mathbf A}^1\times {\mathbf A}^1
={\rm Spec}\ k[x,y]\to X$
be the open immersion.
Let ${\cal G}$ be the locally constant sheaf
of $\Lambda$-modules of
rank $1$ on $U$
defined by the Artin-Schreier covering
$t^p-t=xy$ and by a non-trivial character
${\mathbf F}_p\to \Lambda^\times$.
Then,
the second projection
${\rm pr}_2\colon X\to Y={\mathbf P}^1$
is locally acyclic relatively to ${\cal F}=j_!{\cal G}$
\cite[Th\'eor\`eme 2.4.4]{KL}.
On the other hand,
the singular support $C=SS{\cal F}$
is the union of the zero-section
$T^*_XX$ and the conormal bundles
$T^*_{X_\infty}X$ of the fiber
$X_\infty={\rm pr}_2^{-1}(\infty)$
and
$T^*_{(0,\infty)}X$ of the point $(0,\infty)$.
Hence
the projection
${\rm pr}_2\colon X\to Y={\mathbf P}^1$
is not $C$-transversal.
Let $Y'={\mathbf P}^1\to Y={\mathbf P}^1$
be the Frobenius.
Then, for the pull-back ${\cal F}'$
of ${\cal F}$
on $X'=X\times_YY'$,
the singular support $C'=SS{\cal F}'$
is the union of the zero-section
$T^*_{X'}X'$ and the image
of the pull-back
$X'_\infty\times_{{\mathbf A}^1}
T^*{\mathbf A}^1\to T^*X'$ with respect to the
first projection on the fiber
$X'_\infty={\rm pr}_2^{-1}(\infty)$ at infinity.
Consequently, the projection
${\rm pr}_2\colon X'\to Y'={\mathbf P}^1$
is $C'$-transversal.
Let $Y''\to Y'$ be a flat morphism
of smooth curves over $k$ and
let ${\cal F}''$ be the pull-back
of ${\cal F}'$ on
$X''=X'\times_{Y'}Y''$.
Then, the morphism
$h\colon
X''\to X'$ is properly $C'$-transversal
and hence
$SS{\cal F}''
=h^\circ SS{\cal F}'$
is the union of
$T^*_{X''}X''$ and
$X''_\infty\times_{{\mathbf A}^1}
T^*{\mathbf A}^1\subset T^*X''$
by Lemma {\rm \ref{lmh}}.
}
\end{ex}
\subsection{Potential transversality}\label{sst}
We prove a refinement of the analogue of
the stable reduction theorem,
using the following consequence
of the stable reduction theorem
for curves.
\begin{lm}\label{lmTem}
Let
$$
\begin{CD}
U@>{\subset}>>X\\
@V{f_V}VV
\hspace{-10mm}
\square
\hspace{7mm}
@VVfV\\
V@>{\subset}>>
Y@>g>>S
\end{CD}
$$
be a cartesian diagram of morphisms of
smooth schemes of finite type
over a {\em perfect} field $k$
satisfying the following conditions:
The morphism
$f\colon X\to Y$ is flat
and the morphisms
$g\colon Y\to S$
and $f_V\colon U\to V$ are
smooth of relative dimension $1$.
The horizontal arrows are
open immersions and
the open subset $V\subset Y$ is the
complement of
a divisor $D\subset Y$
smooth over $k$ and
quasi-finite and flat over $S$.
Then, there exists a commutative
diagram
$$
\begin{CD}
X'@>{f'}>>Y'@>{g'}>>S'\\
@VVV@VVV@VVV\\
X@>f>>Y@>g>>S
\end{CD}
$$
of smooth schemes over $k$
satisfying the following conditions:
The morphisms
$S'\to S$, $Y'\to Y\times_SS'$
and $X'\to X\times_YY'$ are
quasi-finite flat and dominant.
The morphisms $g'\colon Y'\to S'$
and $f'\colon X'\to Y'$ are smooth of relative
dimension $1$
and that $V'=V\times_YY'
\subset Y'$ is the
complement of
a divisor $D'\subset Y'$
smooth over $k$
and quasi-finite and flat over $S'$.
The morphism
$V'\to V\times_SS'$
is {\em \'etale} and the morphism
$U'=X'\times_{Y'}V' \to U\times_VV'$
is an isomorphism.
The quasi-finite morphisms
$D'\to D$ and
$X'\times_{Y'}D'\to
X\times_YD'$ are dominant.
\end{lm}
\proof{
Let $\bar \eta$
be a geometric point of
$S$ defined by an algebraic
closure of the function
field of an irreducible
component.
Then, it suffices to apply \cite[Theorem 1.5]{Tem}
to the base change of
$X\to Y\to S$ by $\bar \eta\to S$.
\qed}
\begin{thm}\label{thmV2}
Let
$$
\begin{CD}
U@>{\subset}>>X\\
@VVV
\hspace{-10mm}
\square
\hspace{7mm}
@VVfV\\
V@>{\subset}>>
Y@>>>S
\end{CD}
$$
be a cartesian
diagram of morphisms of
schemes of finite type
over a {\em perfect} field $k$.
Assume that $Y$ and $S$ are
smooth over $k$,
that
$Y\to S$ is smooth of relative
dimension $1$
and that $V\subset Y$ is the
complement of
a divisor $D$ smooth over $k$ and
quasi-finite and flat over $S$.
Let ${\cal F}_U$ be a
perverse sheaf
of $\Lambda$-modules
on $U=X\times_YV$
such that
$U\to V$ is $SS{\cal F}_U$-transversal.
Then, there exists a commutative
diagram
\begin{equation}
\begin{CD}
V'@>\subset>>Y'@>>> S' \\
@VVV\hspace{-10mm}
\square
\hspace{7mm}
@VVV@VVV\\
V@>\subset>>Y@>>> S
\end{CD}
\label{eqYS}
\end{equation}
of smooth schemes over $k$
satisfying the following
conditions {\rm (1)}
and {\rm (2):}
{\rm (1)}
The morphisms
$S'\to S$ and $Y'\to Y\times_SS'$
are quasi-finite flat and dominant.
The horizontal arrow
$Y'\to S'$ is smooth of relative
dimension $1$.
The left square is cartesian and
$V'\subset Y'$
is the complement
of a divisor $D'\subset Y'$
smooth over $k$ and
quasi-finite and flat over $S'$.
The induced morphism
$V'\to V\times_SS'$
is {\em \'etale}
and
$D'\to D$ is {\em dominant}.
{\rm (2)}
Let
\begin{equation}
\begin{CD}
U'@>{j'}>>X'@>{f'}>> Y'\\
@VVV
\hspace{-10mm}
\square
\hspace{7mm}
@VVV
\hspace{-10mm}
\square
\hspace{7mm}
@VVV\\
U@>>>X@>f>> Y'\\
\end{CD}
\label{eqXX'}
\end{equation}
be a cartesian diagram
and let ${\cal F}'_{U'}$
denote the pull-back of
${\cal F}_U$ on $U'$.
Then the morphism
$f'\colon X'\to Y'$
is $SS{\cal F}'$-transversal
for ${\cal F}'
=j'_{!*}{\cal F}'_{U'}$.
\end{thm}
\proof{
Since the assertion is local on $X$,
we may assume that
there exists a closed immersion
$i\colon X\to {\mathbf A}^n_Y$
for an integer $n\geqq 0$.
By replacing $X$ and ${\cal F}_U$
by ${\mathbf A}^n_Y$
and ${i|_U}_*{\cal F}_U$
on ${\mathbf A}^n_V$,
we may assume that
$X$ is an open subscheme
of ${\mathbf A}^n_Y$.
We prove the assertion by
induction on $n$.
Assume $n=0$ and hence
$X\to Y$ is an open immersion.
Since the open immersion
$U\to V$ is $SS{\cal F}_U$-transversal,
the singular support
$SS{\cal F}_U$
is a subset of the $0$-section
$T^*_UU$ by Lemma \cite[Lemma 3.6.3]{CC}.
Hence ${\cal F}_U$ is
locally constant by \cite[Lemma 2.1(iii)]{Be}.
Let $U_1\to U$ be a finite \'etale covering
such that the pull-back
of ${\cal F}_U$ is constant.
Let $Y_1$ be the normalization of
$Y$ in $U_1$.
There exists a
quasi-finite flat and dominant
morphism $S'\to S$
of smooth scheme
such that
the normalization
$Y'$ of $Y_1\times_SS'$
is smooth over $S'$
and that
$V'\subset Y'$ is the complement
of a divisor $D'$ \'etale over $S'$.
After shrinking $S'$,
we may assume that $Y'\to Y\times_SS'$ is flat.
After shrinking $Y'$ keeping
$D'$ dominant over $D$,
we may assume that
$V'\to V\times_SS'$ \'etale.
Then, the condition (1) is satisfied.
Since ${\cal F}'$ on $X'
\subset Y'$ is constant,
the condition (2) is also satisfied
by Lemma \ref{lmtr}.1.
Assume that $n\geqq 1$
and that the assertion holds
for $n-1$.
For the proof of
the induction step,
we first show the following weaker assertion.
\begin{cl}\label{lmind}
Let
$X\subset {\mathbf A}^n_Y\to
{\mathbf A}^1_Y$
be a projection
and assume that its restriction
$U\subset {\mathbf A}^n_V
\to {\mathbf A}^1_V$
is $SS{\cal F}_U$-transversal.
Then, there exist a commutative
diagram {\rm (\ref{eqYS})}
satisfying the condition {\rm (1)}
and
an open subset
$W'\subset {\mathbf A}^1_{Y'}$
satisfying the following condition:
{\rm (2$'$)}
The intersection
$W'\cap {\mathbf A}^1_{D'}$
is dense in ${\mathbf A}^1_{D'}$.
For the cartesian diagram
{\rm (\ref{eqXX'})}
and for the pull-back ${\cal F}'_{U'}$
of ${\cal F}$ on $U'$
and ${\cal F}'=j'_{!*}{\cal F}'_{U'}$
on $X'$,
the morphism
$f'\colon X'\to Y'$
is $SS{\cal F}'$-transversal on
the inverse image
$X'\times_{{\mathbf A}^1_{Y'}}W'
\subset X'$.
\end{cl}
\proof[Proof of Claim]{
By the induction hypothesis
applied to
$X\subset {\mathbf A}^n_Y\to {\mathbf A}^1_Y
\to {\mathbf A}^1_S$,
there exists a commutative diagram
$$\begin{CD}
Y_1@>>> S_1\\
@VVV@VVV\\
{\mathbf A}^1_Y
@>>> {\mathbf A}^1_S
\end{CD}$$
satisfying the conditions
{\rm (1)} and {\rm (2)} in Theorem
\ref{thmV2}.
We consider the cartesian diagram
$$
\begin{CD}
U_1@>{j_1}>>
X_1@>>>Y_1\\
@VVV
\hspace{-10mm}
\square
\hspace{7mm}
@VVV
\hspace{-10mm}
\square
\hspace{7mm}
@VVV\\
U@>>>X
@>>>{\mathbf A}^1_Y
\end{CD}$$
and let ${\cal F}_{U_1}$
be the pull-back of ${\cal F}_U$.
Then,
for ${\cal F}_1=
j_{1!*}{\cal F}_{U_1}$
on $X_1$,
the morphism
$X_1\to Y_1$
is $SS{\cal F}_1$-transversal.
The inverse image
$V_1=V\times_YY_1\subset
Y_1$
is the complement of
a divisor $D_1\subset Y_1$
smooth over $k$
and quasi-finite and flat over $S_1$.
The quasi-finite morphism
$V_1\to V\times_SS_1$ is {\em \'etale}
and the quasi-finite morphism
$D_1\to {\mathbf A}^1_D$
is dominant.
Since the morphism
$S_1\to {\mathbf A}^1_S$
is quasi-finite and flat,
there exists a
quasi-finite, flat
and dominant morphism
$S'\to S$
of smooth schemes over $k$
such that
the normalization
$S'_1$ of $S_1\times_SS'$
is smooth over $S'$
and that
the induced morphism
$S'_1\to S_1$
is also quasi-finite, flat
and dominant.
After shrinking $S'_1$ if necessary,
we may assume that
the morphism
$Y_1\times_{S_1}S'_1
\to
{\mathbf A}^1_Y\times_{
{\mathbf A}^1_S}S'_1$
of smooth curves over
$S'_1$ is flat.
Hence, by replacing $S,Y,S_1$ and
$Y_1$ by $S',Y\times_SS',
S'_1$ and $Y_1\times_{S_1}S'_1$,
we may assume that
$S_1\to S$ is smooth
of relative dimension 1.
We consider the commutative
diagram
\begin{equation}
\begin{CD}
V_1@>>> Y_1@>>> S_1\\
@VVV
\hspace{-10mm}
\square
\hspace{7mm}
@VVV@VVV\\
V@>>> Y@>>> S
\end{CD}
\label{eqVW1}
\end{equation}
where the left square is cartesian.
Since $V_1\to
V\times_SS_1$ is \'etale,
the left vertical arrow
$V_1\to V$
is also smooth of relative dimension 1.
The middle vertical arrow
$Y_1\to Y$ is flat.
Hence, by Lemma \ref{lmTem}
applied to (\ref{eqVW1}),
there exists a commutative diagram
$$\begin{CD}
Y'_1@>>>Y'@>>> S'\\
@VVV@VVV@VVV\\
Y_1@>>>Y @>>> S
\end{CD}$$
of smooth schemes over $k$
satisfying the following conditions:
The morphisms
$S'\to S$,
$Y'\to Y\times_SS'$
and
$Y'_1\to Y_1\times_YY'$
are quasi-finite flat
and dominant.
The morphisms
$Y'\to S'$
and $Y'_1\to Y'$
are smooth of relative dimension 1.
The inverse image
$V'=V\times_YY'$
is the complement
$Y'\sm D'$ of
a divisor $D'\subset Y'$ smooth over $k$ and
quasi-finite and flat over $S'$.
The morphism
$V'\to
V\times_SS'$ is {\em \'etale} and
the morphism
$Y'_1\times_{Y'}V'\to
V_1\times_VV'$
is an isomorphism.
The quasi-finite morphisms
$D'\to D$ and $Y'_1\times_{Y'}D'
\to Y_1\times_YD'$
are dominant.
Thus the condition (1)
in Theorem \ref{thmV2}
is satisfied.
The composition
$Y'_1\to Y_1\times_YY'\to
{\mathbf A}^1_{Y'}$
is quasi-finite and flat.
We consider the cartesian diagram
$$\begin{CD}
X'' @>>> Y'_1\\
@VVV
\hspace{-10mm}
\square
\hspace{7mm}
@VVV\\
X'@>>>{\mathbf A}^1_{Y'}
@>>> Y'
\end{CD}$$
and the pull-back
${\cal F}''$ of ${\cal F}_1$
on $X''$.
Then, since
$X_1\to Y_1$
is $SS{\cal F}_1$-transversal,
the morphism
$X''\to Y'_1$
is $SS{\cal F}''$-transversal
by Lemma \ref{lmtrb}.1.
Since $Y'_1\to Y'$ is smooth,
the composition
$X''\to Y'$
is also $SS{\cal F}''$-transversal
by Lemma \ref{lmtrfun}.1.
By Lemma \ref{lmtrb}.2 and
Lemma \ref{lmtr}.3,
the pull-back on $U''=U'\times_{X'}X''$ of
every constituant ${\cal G}$ of ${\cal F}'_{U'}$
is a perverse sheaf.
Since $Y'_1\to {\mathbf A}^1_{Y'}$
is quasi-finite and flat,
the morphism $f'\colon X'\to Y'$ is
$SS{\cal F}'$-transversal
on the image of $X''$
by the case (2) in Lemma \ref{lmtrqf}.
Let $W'\subset {\mathbf A}^1_{Y'}$
be the image of $Y'_1$.
The image of $X''\to X'$ equals
the inverse image
$X'\times_{{\mathbf A}^1_{Y'}}W'$.
Hence,
$f'\colon X'\to Y'$ is
$SS{\cal F}'$-transversal
on $X'\times_{{\mathbf A}^1_{Y'}}W'
\subset X'$.
Since
$Y'_1\times_{Y'}D'
\to Y_1\times_YD'$
and $D_1\to {\mathbf A}^1_D$
are dominant,
the intersection
$W'\cap {\mathbf A}^1_{D'}$
is dense in ${\mathbf A}^1_{D'}$.
Thus the condition (2$'$)
in Claim is also satisfied.
\qed}
\smallskip
To complete the proof of
the induction step,
we use the following elementary
lemma.
\begin{lm}\label{lmVC}
Let $X$ be an open subset
of a vector space $V$
of dimension $n$ over an infinite field $k$
regarded as a smooth scheme over $k$.
Let $C\subset T^*X$
be a closed conical subset of
dimension $\leqq n$.
Then, there exists an isomorphism
$V\to {\mathbf A}^n$
of vector spaces over $k$
such that the compositions
$X\to V\to {\mathbf A}^n
\to {\mathbf A}^1$
with the projections ${\rm pr}_i,
i=1,\ldots,n$
have at most isolated $C$-characteristic points.
\end{lm}
\proof{
Identify the cotangent bundle
$T^*X$ with the product $X\times V^\vee$
with the dual
and let ${\mathbf P}(C)
\subset
{\mathbf P}(T^*X)
=
X\times {\mathbf P}(V^\vee)$
be the projectivization.
Then, by the assumption
$\dim C\leqq n$,
the projection
${\mathbf P}(C)
\to {\mathbf P}(V^\vee)$
is generically finite.
By the assumption that $k$ is infinite,
there exists a basis
$p_1,\ldots,p_n$ of $V^\vee$
such that the fibers
of ${\mathbf P}(C)
\to {\mathbf P}(V^\vee)$
at
$\bar p_1,\ldots,\bar p_n\in
{\mathbf P}(V^\vee)$
are finite.
Then, the product of
$p_1,\ldots,p_n\colon
V\to {\mathbf A}^1$
satisfies the condition.
\qed}
\medskip
Set $C_U=SS{\cal F}_U
\subset T^*U$.
By the assumption that
$U\subset{\mathbf A}^n_V\to V$ is
$SS{\cal F}_U$-transversal,
the morphism
$T^*U\to T^*U/V$
to the relative cotangent bundle
is finite on $C_U$
by \cite[Lemma 1.2 (ii)]{Be}.
The image $\bar C_U\subset T^*U/V$
of $C_U$
and its closure $\bar C\subset T^*X/Y$
are closed conical subsets.
Since every irreducible
component of $C_U$
is of dimension $\dim X$,
every irreducible
component of $\bar C_U$
is also of dimension $\dim X$.
Hence,
for the generic point of
each irreducible component of $Y$,
the fiber of $\bar C_U$ is
of dimension $\leqq n=\dim X-\dim Y$.
Consequently,
for the generic point of
each irreducible component of
$D\subset Y$,
the fiber of $\bar C$ is
also of dimension $\leqq n$.
By Lemma \ref{lmVC}
applied to the fibers of
the generic points of
irreducible components of $D$,
after replacing $S$
by a dense open subset,
there exists a coordinate
of ${\mathbf A}^n_Y\supset X$
such that,
for each $i=1,\ldots,n$,
there exist
a dense open subset $W_i
\subset {\mathbf A}^1_D$ and
an open neighborhood $X_i
\subset X$
of the inverse image
$W_i\times_{{\mathbf A}^1_Y}X$
by the $i$-th projection ${\rm pr}_i$
satisfying the following
condition:
The inverse image of
$\bar C\subset T^*X/Y$
by the morphism
$X\times_{{\mathbf A}^1_Y}
T^*{\mathbf A}^1_Y/Y
\to T^*X/Y$
of the relative cotangent bundles
induced by ${\rm pr}_i$
is a subset of the $0$-section on $X_i$.
Then, the restriction
$U\to {\mathbf A}^1_V$
of ${\rm pr}_i$ is
$SS{\cal F}_U$-transversal
on $X_i\cap U$.
By Claim
applied to the restriction
$X_i\to {\mathbf A}^1_Y$ of
${\rm pr}_i$,
there exist a commutative
diagram {\rm (\ref{eqYS})}
satisfying the condition {\rm (1)}
and for each $i=1,\ldots, n$
a dense open subset
$W'_i\subset {\mathbf A}^1_{Y'}$
satisfying the condition (2$'$).
Hence $X'\to Y'$
is $SS{\cal F}'$-transversal
on the union
$W'=\bigcup_{i=1}^n
{\rm pr}_i^{-1}W'_i
\subset X'
\subset{\mathbf A}^n_{Y'}$
of the inverse images
by the projections.
Since $X'\to Y'$
is $SS{\cal F}'$-transversal
on $U'$,
it is $SS{\cal F}'$-transversal
on $W'\cup U'$.
By shrinking $S'$ if necessary,
we may assume that
$Z'=X'\sm (W'\cup U')
=\prod_{i=1}^n
( {\mathbf A}^1_{D'}
\sm ( {\mathbf A}^1_{D'}\cap W'_i))
\subset {\mathbf A}^n_{D'}$
is quasi-finite over $S'$.
By Corollary \ref{corla},
there exists a
cartesian diagram
$$\begin{CD}
U''@>{j''}>>X''@>{f''}>>Y''@<<<V''\\
@VVV\hspace{-10mm}
\square
\hspace{7mm}
@VVV\hspace{-10mm}
\square
\hspace{7mm}
@VVV\hspace{-10mm}
\square
\hspace{7mm}
@VVV\\
U'@>{j'}>>X'@>{f'}>>Y'@<<<V'
\end{CD}$$
of smooth schemes over $k$
satisfying the following condition:
The morphism
$V''\to V'$ is \'etale
and
$V''\subset Y''$
is the complement of
a divisor $D''\subset Y''$ smooth over $k$.
The morphism
$D''\to D'$ is dominant.
For the pull-back
${\cal F}''_{U''}$ of ${\cal F}'_{U'}$
on $U''$
and ${\cal F}''=j''_{!*}{\cal F}''$,
the morphism
$f''\colon X''\to Y''$
is universally locally acyclic relatively
to ${\cal F}''$.
By Lemma \ref{lmtrb}.1
and Lemma \ref{lmtrV}.1,
${\cal F}''$ is the pull-back
of ${\cal F}'$ outside the
inverse image $Z''$ of $Z'$
and $f''\colon X''\to Y''$
is $SS{\cal F}''$-transversal
outside the inverse image
$Z''$.
Let $S''\to S'$ be a quasi-finite
flat dominant morphism
of smooth schemes over $k$
such that
the normalization $Y'''$ of
$Y''\times_{S'}S''$
is smooth over $S''$
of relative dimension
$1$ and
that $V'''=V''\times_{Y''}Y'''$
is the complement of
a divisor $D'''$ smooth over $k$.
Since $V''\to V'$ is \'etale,
the morphism $V''\to S'$ is smooth
and $V'''\to V''\times_{S'}S''$
is an isomorphism.
Hence the morphism
$V'''\to V'\times_{S'}S''$
is \'etale.
The morphism $D'''\to D'$ is dominant.
We consider the commutative diagram
$$\begin{CD}
U'''@>{j'''}>>X'''@>{f'''}>>Y'''@>>>S''\\
@VVV\hspace{-10mm}
\square
\hspace{7mm}
@VVV\hspace{-10mm}
\square
\hspace{7mm}
@VVV@VVV\\
U'@>{j'}>>X'@>{f'}>>Y'@>>>S'
\end{CD}$$
where the left and middle
squares are cartesian.
Then for the pull-back
${\cal F}'''$ of ${\cal F}''$
on $X'''$, the morphism
$f'''\colon X'''\to Y'''$
is universally locally acyclic
and is $SS{\cal F}'''$-transversal
outside the inverse image
$Z'''$ of $Z''$
quasi-finite over $S''$.
Shrinking $S''$,
we may further assume that
$X'''\to S''$
is properly $SS{\cal F}'''$-transversal
by Lemma \ref{lmtrV}.2.
Then, the morphism
$f'''\colon X'''\to Y'''$ is
$SS{\cal F}'''$-transversal
by Lemma \ref{lmtrZ}.2.
Further we have an isomorphism
${\cal F}'''
=j'''_{!*}j^{\prime\prime\prime*}
{\cal F}'''$
by Lemma \ref{lmtrV}.1.
Thus, the commutative diagram
$$\begin{CD}
V'''@>\subset>>Y'''@>>>S''\\
@VVV\hspace{-10mm}
\square
\hspace{7mm}
@VVV@VVV\\
V@>\subset>>Y@>>>S
\end{CD}$$
satisfies the required conditions.
\qed}
\begin{cor}\label{corV}
Let $f\colon X\to Y$
be a morphism of scheme
of finite type over a perfect field
$k$.
Assume that $Y$ is smooth of
dimension $1$.
Let ${\cal F}$ be a constructible
complex of $\Lambda$-modules
on $X$.
Assume that
$f\colon X\to Y$ is
locally acyclic relatively to ${\cal F}$ and
that there exists a dense
open subset $V\subset Y$
such that
$f\colon X\to Y$
is $SS{\cal F}$-transversal on $V$.
Then, there exists
a cartesian diagram
$$
\begin{CD}
X@<<< X'\\
@VfVV\hspace{-10mm}
\square
\hspace{7mm}
@VV{f'}V\\
Y@<<< Y'
\end{CD}$$
of morphisms of schemes
of finite type over $k$
satisfying the following condition:
The morphism
$Y'\to Y$ is a
finite generically {\em \'etale} morphism
of smooth curves.
For the pull-back ${\cal F}'$ of
${\cal F}$ on $X'$,
the morphism $f'\colon X'\to Y'$
is $SS{\cal F}'$-transversal.
\end{cor}
\proof{
Since the shifted vanishing cycles functor
$R\Phi[-1]$ is $t$-exact,
we may assume that
${\cal F}$ is a perverse sheaf.
Then the assertion follows
from the case
where $S={\rm Spec}\ k$
in Theorem \ref{thmV2},
Proposition \ref{prS}.1
and weak approximation.
\qed}
\section{Characteristic cycles and the direct image}\label{spp}
\subsection{Direct image
of a cycle}\label{ssdc}
To state the compatibility with
push-forward, we fix some terminology
and notations.
Recall that a morphism
$f\colon X\to Y$ of
noetherian schemes is said to be proper
on a closed subset $Z\subset X$
if its restriction $Z\to Y$
is proper with respect to a
closed subscheme structure of $Z\subset X$.
Let $f\colon X\to Y$ be
a morphism of smooth schemes over
a field $k$ and we
consider the diagram
\begin{equation}
\begin{CD}
T^*X@<<< X\times_YT^*Y
@>>> T^*Y
\end{CD}
\label{eqJD}
\end{equation}
as an algebraic correspondence
from $T^*X$ to $T^*Y$.
Assume that
every irreducible component
of $X$ (resp.\ of $Y$)
is of dimension $n$ (resp.\ $m$).
Let $B\subset X$
be a closed subset
on which $f\colon X\to Y$
is proper
and let $C\subset T^*X$
be a closed subset of
$B\times_XT^*X$.
Then, the closed subset
$f_\circ C\subset
T^*Y$
is defined as the image
by the right arrow
in (\ref{eqJD})
of the inverse image of
$C$ by the left arrow.
It is a closed subset by
the assumption that
$f$ is proper on $B$.
The composition of
the Gysin map
\cite[6.6]{Ful}
for the first arrow and
the push-forward map
for the second arrow
defines a morphism
\begin{equation}
\begin{CD}
f_!\colon {\rm CH}_n(C)
@>>>
{\rm CH}_m(f_\circ C)
\end{CD}
\label{eqCH}
\end{equation}
since $\dim T^*X
-\dim X\times_YT^*Y
=n-m$.
If every irreducible component
of $C$
(resp.\ $f_\circ C$) is of dimension
$\leqq n$
(resp.\ $\leqq m$),
the morphism (\ref{eqCH})
defines
a morphism
\begin{equation}
\begin{CD}
f_!\colon Z_n(C)
@>>>
Z_m(f_\circ C)
\end{CD}
\end{equation}
of free abelian groups of cycles.
\begin{lm}\label{lmCHtr}
Let $$
\xymatrix{
X\ar[r]^g
\ar[rd]_f &
X'\ar[d]^{f'}\\
&Y}$$ be
a commutative diagram of
morphisms of smooth schemes over $k$.
Assume that
every irreducible component
of $X$ (resp.\ of $X'$ and $Y$)
is of dimension $n$ (resp.\
$n'$ and $m$).
Let $B\subset X$
be a closed subset
on which $f\colon X\to Y$
is proper
and let $C\subset T^*X$
be a closed subset of
$B\times_XT^*X$.
Then, the diagram
$$
\xymatrix{
{\rm CH}_n(C)\ar[r]^{g_!}
\ar[rd]_{f_!} &
{\rm CH}_{n'}(g_\circ C)
\ar[d]^{f'_!}\\
&{\rm CH}_m(f_\circ C)}$$
is commutative.
\end{lm}
\proof{
We consider the diagram
\begin{equation*}
\begin{CD}
T^*X@<<< X\times_{X'}T^*X'
@>>> T^*X'@.\\
@. @AAA
\hspace{-20mm}
\square
\hspace{17mm}
@AAA\\
@.X\times_YT^*Y
@>>>X'\times_YT^*Y
@>>> T^*Y
\end{CD}
\end{equation*}
with cartesian square.
After decomposing the
right vertical arrow into the composition
of a smooth morphism and a regular immersion,
it suffices to apply \cite[Theorem 6.2 (a)]{Ful}.
\qed}
\begin{lm}\label{lmptrA}
Let $f\colon X\to Y$
be a smooth morphism of smooth irreducible schemes
over a perfect field $k$.
Assume that $X$
(resp.\ of $Y$) is
of dimension $n$
(resp.\ $m$).
Let $C=\bigcup_aC_a\subset T^*X$
be a closed conical subset such
that every irreducible component
$C_a$ is of dimension $n$
and that $f\colon X\to Y$ is
properly $C$-transversal and
is proper on the base $B=C\cap T^*_XX\subset X$.
Let $A=\sum_am_aC_a$ be a linear combination.
Let $y\in Y$ be a closed point,
let $A_y=i_y^!A$ be the pull-back
{\rm \cite[Definition 7.1]{CC}} by
the closed immersion $i_y\colon X_y\to X$
of the fiber
and let
$(A_y,T^*_{X_y}X_y)_{T^*X_y}$
denote the intersection number.
Then, we have
\begin{equation}
f_! A=
(-1)^m
(A_y,T^*_{X_y}X_y)_{T^*X_y}
\cdot
[T^*_YY]
\label{eqAy0}
\end{equation}
in $Z_m(T^*_YY)$.
\end{lm}
\proof{
Since the closed immersion
$i_y\colon X_y\to X$
is properly $C$-transversal by Lemma \ref{lmtrbc}.1,
the pull-back
$A_y=i_y^!A$
is defined.
Further by the assumption that
$f\colon X\to Y$
is $C$-transversal,
we have an inclusion
$f_\circ C\subset T^*_YY$
and $f_!A$
is defined as an element of
${\rm CH}_m(f_\circ C)
=Z_m(f_\circ C)\subset Z_m(T^*_YY)$.
Hence it suffices to
show that the
coefficient of $T^*_YY$
in $f_!A$ equals the intersection number
$(-1)^m(A_y,T^*_{X_y}X_y)_{T^*X_y}$.
We consider the cartesian diagram
$$\begin{CD}
T^*X@<<< X\times_YT^*Y@>>> T^*Y\\
@AAA
\hspace{-20mm}
\square
\hspace{15mm}
@AAA
\hspace{-20mm}
\square
\hspace{15mm}
@AAA\\
X_y\times_XT^*X@<<< X_y\times_YT^*Y@>>> y\times_YT^*Y\\
@VVV
\hspace{-20mm}
\square
\hspace{15mm}
@VVV
\hspace{-20mm}
\square
\hspace{15mm}
@VVV\\
T^*X_y@<0<< X_y@>>> y.
\end{CD}$$
We regard the four sides of the exterior square of the diagram
as algebraic correspondences.
The coefficient of $f_!A$
is the image of $A$ by the composition
via the upper right corner.
It equals the composition
via the upper right corner
by \cite[Theorem 6.2 (a)]{Ful}
applied to the upper right and the lower left squares.
Since the definition of $i_y^!A$ in
\cite[Definition 7.1]{CC} involves
the sign $(-1)^{\dim X-\dim X_y}=(-1)^m,$
the assertion follows.
\qed}
\medskip
We study the case where $Y$
is a smooth curve
and $\dim f_\circ C=1$.
Let $f\colon X\to Y$
be a morphism of smooth schemes
over $k$.
Assume that
every irreducible
component of $X$
(resp.\ of $Y$) is
of dimension $n$
(resp.\ $1$).
Let $C\subset T^*X$
be a closed conical subset
such that every irreducible
component $C_a$ of
$C=\bigcup_aC_a$
is of dimension $n$ and that $f\colon X\to Y$
is proper on the base
$B=C\cap T^*_XX\subset X$.
Let $V\subset Y$ be a dense open subscheme
such that the base change
$f_V\colon X_V\to V$
is properly $C_V$-transversal
for the restriction
$C_V$ of $C$ on $X_V$.
Let $y\in Y\sm V$ be a closed point
on the boundary and
let $t$ be a uniformizer at $y$
and let $df$ denote the section
of $T^*X$ defined on a neighborhood
of the fiber $X_y$ by the pull-back $f^*dt$.
Then, on a neighborhood of
$X_y$,
the intersection $C\cap df
\subset T^*X$ is supported on the
inverse image of the intersection
$B\cap X_y$.
Hence for a linear combination
$A=\sum_am_aC_a$,
the intersection product
\begin{equation}
(A,df)_{T^*X,X_y}
\label{eqAdf}
\end{equation}
supported on the fiber $X_y$
is defined as an element of
${\rm CH}_0(B\cap X_y)$.
Since $C$ is conical,
the intersection product
$(A,df)_{T^*X,X_y}$
does not depend on the choice of $t$.
Thus the intersection number
also denoted $(A,df)_{T^*X,X_y}$
is defined as its image by
the degree mapping
${\rm CH}_0(B\cap X_y)
\to
{\rm CH}_0(y)={\mathbf Z}$.
\begin{lm}\label{lmd1A}
Let $f\colon X\to Y$
be a morphism of smooth irreducible schemes
over a perfect field $k$.
Assume that $X$
(resp.\ of $Y$) is
of dimension $n$
(resp.\ $1$).
Let $C=\bigcup_aC_a\subset T^*X$
be a closed conical subset
as in Lemma {\rm \ref{lmptrA}}.
{\rm 1.}
The following conditions
are equivalent:
{\rm (1)}
$\dim f_\circ C\leqq 1$.
{\rm (2)}
There exists a dense open subscheme
$V\subset Y$
such that the base change
$f_V\colon X_V\to V$
is $C_V$-transversal
for the restriction
$C_V$ of $C$ on $X_V$.
{\rm (3)}
There exists a dense open subscheme
$V\subset Y$
such that the base change
$f_V\colon X_V\to V$
is properly $C_V$-transversal
for the restriction
$C_V$ of $C$ on $X_V$.
{\rm 2.}
Let $V\subset Y$ be a dense open subscheme
satisfying the condition {\rm (3)}
above.
Let $A=\sum_am_aC_a$
be a linear combination,
let $v\in V$ be a closed point
and define the intersection number
$(A_v,T^*_{X_v}X_v)_{T^*X_v}$
as in Lemma {\rm \ref{lmptrA}}.
Then, we have
\begin{equation}
f_! A=
-
(A_v,T^*_{X_v}X_v)_{T^*X_v}\cdot
[T^*_YY]+
\sum_{y\in Y\sm V}
(A,df)_{T^*X,X_y}\cdot [T^*_yY]
\label{eqAy}
\end{equation}
in $Z_1(f_\circ C)$.
\end{lm}
\proof{
1.
Since $f_\circ C$ is a closed conical subset
of the line bundle $T^*Y$,
the condition (1) is equivalent
to the existence of a dense open subset
$V\subset Y$ such that
$f_\circ C\subset T^*_YY
\cup\bigcup_{y\in Y\sm V}T^*_yY$.
This is equivalent to the condition (2).
The equivalence
(2)$\Leftrightarrow$(3) follows
from Lemma \ref{lmtrV}.2.
2.
It suffices to compare the coefficients
of the $0$-section
$T^*_YY$
and of the fibers
$T^*_yY$ respectively.
For those of $T^*_YY$,
it is proved in Lemma \ref{lmptrA}.
For those of $T^*_yY$,
it follows from
the projection formula
\cite[Theorem 6.2 (a)]{Ful}
applied to
the cartesian square in the diagram
$$
\begin{CD}
T^*X@<<< X\times_YT^*Y@>>> T^*Y\\
@.@A{df}AA
\hspace{-15mm}
\square
\hspace{10mm}
@AA{dt}A\\
@. X@>>>Y.
\end{CD}$$
\qed}
\medskip
\begin{lm}\label{lmcE}
Let $X$ be a scheme of
finite type of dimension $d$
over a field $k$
and let $E$ be a vector bundle
on $X$ associated
to a locally free ${\cal O}_X$-module
${\cal E}$ of rank $n$.
Let $s\colon X\to E$
be a section,
$0\colon X\to E$ be the zero section
and
$Z=Z(s)
=0(X)\cap s(X)\subset X$ be the zero locus
of $s$.
Let ${\cal K}=[{\cal O}_X
\overset s\to {\cal E}]$
be the complex of ${\cal O}_X$-modules
where ${\cal E}$ is put on
degree $0$ and let
${c_n}^X_Z({\cal K})$ be the localized Chern
class defined in {\rm \cite[Section 1]{Bl}}.
Then, we have
$$(0(X),s(X))_E
=
{c_n}^X_Z({\cal K})\cap [X]$$
in ${\rm CH}_{d-n}(Z)$.
\end{lm}
\proof{
We may assume that $X$ is integral
and $Z\subsetneqq X$.
By taking the blow-up at $Z$
and by \cite[Proposition 2.3.1.6]{KS},
we may assume that $Z$ is a Cartier divisor $D
\subset X$.
Then, we have an exact sequence of
$0\to {\cal L}\to {\cal E}\to {\cal F}\to 0$
of locally free ${\cal O}_X$-modules
where ${\cal L}$ and ${\cal F}$ are
of rank 1 and $n-1$ respectively
and $s\in \Gamma(X,{\cal E})$
is defined by
$s\in \Gamma(X,{\cal L})$.
Then, the right hand side
equals $c_{n-1}({\cal F})\cap [D]$
by \cite[Proposition (1.1) (iii)]{Bl}.
The left hand side also equals
$c_{n-1}({\cal F})\cap [D]$
by the excess intersection formula
\cite[Theorem 6.3]{Ful}.
\qed}
\medskip
We define the specialization of
a cycle.
Let $f\colon X\to Y$
be a smooth morphism
of smooth schemes over
a perfect field $k$ and
assume that $X$ (resp.\ $Y$)
is equidimensional of dimension
$n+1$ (resp.\ 1).
Let $y\in Y$ be a closed point,
$V=Y\sm \{y\}$
be the complement
and $U=X\times_YV$
be the inverse image.
Let $C\subset T^*U$
be a closed conical subset
equidimensional of dimension
$n+1$
and assume that
$U\to V$ is properly $C$-transversal.
We define its specialization
$${\rm sp}_yC
\subset T^*X_y$$
as follows.
By the assumption that
$U\to V$ is properly $C$-transversal
and \cite[Lemma 3.1]{CC},
the morphism
$T^*U\to T^*U/V$
to the relative cotangent bundle
is finite on $C$.
Hence its image
$C'\subset T^*U/V$
is a closed conical subset.
Let $\bar C'\subset T^*X/Y$
be the closure and
define
${\rm sp}_yC
\subset T^*X_y$
to be the fiber
$\bar C'\times_Yy
\subset T^*X/Y\times_Yy
=T^*X_y$.
The specialization
${\rm sp}_yC\subset T^*X_y$
is a closed conical subset
equidimensional of
dimension $n$.
For a linear combination
$A=\sum_am_aC_a$
of irreducible components
of $C=\bigcup_aC_a$,
we define its specialization
$${\rm sp}_yA
\in Z_n({\rm sp}_yC)$$
as follows.
First, we define
$A'\in Z_{n+1}(C')$
as the push-forward of
$A$ by the morphism
$T^*U\to T^*U/V$ finite on $C$.
Let $\bar A'\in Z_{n+1}(\bar C')$
be the unique element
extending $A'\in Z_{n+1}(C')$.
Then, we define
${\rm sp}_yA
\in Z_n({\rm sp}_yC)$
to be the {\em minus} of
the pull-back of $A'$
by the Gysin map
for the immersion
$i_y\colon X_y\to X$.
If $X\to Y$ is proper,
for a closed point $v\in V$
and the closed immersion
$i_v\colon X_v\to X$,
we have
\begin{equation}
({\rm sp}_yA,
T^*_{X_y}X_y)_{T^*X_y}
=
(i_v^!A,
T^*_{X_v}X_v)_{T^*X_v}
\label{eqspiv}
\end{equation}
since the definition of $i_v^!A$ in
\cite[Definition 7.1]{CC} involves
the sign $(-1)^{\dim X-\dim X_v}=-1$.
\begin{lm}\label{lmsp}
Let $f\colon X\to Y$
be a smooth morphism
of smooth schemes over
a perfect field $k$ and
assume that $X$ (resp.\ $Y$)
is equidimensional of dimension
$n+1$ (resp.\ $1$).
Let $y\in Y$ be a closed point,
$i_y\colon X_y\to X$
be the closed immersion
of the fiber,
$V=Y\sm \{y\}$
be the complement
and $U=X\times_YV$
be the inverse image.
Let $C\subset T^*X$
be a closed conical subset
equidimensional of dimension
$n+1$ such that
$f\colon X\to Y$ is properly $C$-transversal.
{\rm 1.}
For the restriction $C_U$
of $C$ on $U$, we have
\begin{equation}
{\rm sp}_yC_U=
i_y^\circ C.
\label{eqspC}
\end{equation}
{\rm 2.}
For a linear combination
$A=\sum_am_aC_a$
of irreducible components
of $C=\bigcup_aC_a$
and its restriction $A_U$ on $U$,
we have
\begin{equation}
{\rm sp}_yA_U=
i_y^!A.
\label{eqspA}
\end{equation}
\end{lm}
\proof{
1.
By the assumption that
$f\colon X\to Y$ is properly
$C$-transversal
and \cite[Lemma 3.1]{CC},
the morphism
$T^*X\to T^*X/Y$
to the relative cotangent bundle
is finite on $C$
and hence its image
$C'\subset T^*X/Y$
is a closed conical subset.
Further $C'$ with reduced scheme
structure is flat over $Y$.
Hence it equals the closure
of the restriction $C'_U$
and we obtain (\ref{eqspC}).
2.
We consider the cartesian diagram
$$\begin{CD}
T^*X@>>> T^*X/Y\\
@AAA\hspace{-17mm}
\square\hspace{12mm} @AAA\\
X_y\times_XT^*X@>>> T^*X_y.
\end{CD}$$
The right hand side is
the minus of the image of
$A$ by the push-forward
and the pull-back via
upper right.
The left hand side is
the minus of the image of
$A$ by the pull-back and
the push forward
via lower left.
Hence the assertion follows from
the projection formula
\cite[Theorem 6.2 (a)]{Ful}.
\qed}
\subsection{Characteristic cycle of
the direct image}\label{ssccd}
Let $k$ be a field and
let $\Lambda$ be a finite field
of characteristic $\ell$ invertible in $k$.
Let $X$ be a smooth scheme over $k$
such that every irreducible
component is of dimension $n$.
Let ${\cal F}$ be a constructible complex of $\Lambda$-modules
on $X$
and $C=SS{\cal F}$ be the singular support.
Then, every irreducible
component $C_a$ of $C=
\bigcup_aC_a$
has the same dimension
as $X$
\cite[Theorem 1.3 (ii)]{Be} and
the base $B=C\cap T^*_XX
\subset T^*_XX=X$
defined as the intersection
with the $0$-sections
equals the support of ${\cal F}$
\cite[Lemma 2.1 (i)]{Be}.
Let $f\colon X\to Y$
be a morphism of smooth schemes
over $k$, proper on the
support of ${\cal F}$.
Then, we have an inclusion
\begin{equation}
SSRf_*{\cal F}\subset f_\circ SS{\cal F}
\end{equation}
by
\cite[Lemma 2.2 (ii)]{Be}.
We restate a conjecture
from \cite[Conjecture 1]{prop}.
\begin{cn}\label{cnf*}
Let $f\colon X\to Y$
be a morphism
of smooth schemes
over a perfect field $k$.
Assume that
every irreducible
component of $X$
(resp.\ of $Y$) is
of dimension $n$
(resp.\ $m$).
Let ${\cal F}$ be a constructible complex of $\Lambda$-modules
on $X$
and $C=SS{\cal F}$ be the singular support.
Assume that $f$ is proper on
the support of ${\cal F}$.
Then, we have
\begin{equation}
CCRf_*{\cal F}=
f_!CC{\cal F}
\label{eqcnf}
\end{equation}
in ${\rm CH}_m(f_\circ SS{\cal F})$.
\end{cn}
If $\dim f_\circ SS{\cal F}\leqq m$,
the equality (\ref{eqcnf})
is an equality as cycles
in ${\rm CH}_m(f_\circ SS{\cal F})
=Z_m(f_\circ SS{\cal F})$
without rational equivalence.
A weaker version of Conjecture \ref{cnf*}
is proved in the case $k$ is finite
and $X$ and $Y$ are projective
in \cite{UYZ}
using $\varepsilon$-factors.
If $Y={\rm Spec}\ k$,
the equality (\ref{eqcnf}) means
the index formula
\begin{equation}
\chi(X_{\bar k},{\cal F})
= (CC{\cal F},T^*_XX)_{T^*X}
\label{eqind}
\end{equation}
where the right hand side denotes
the intersection number.
Further if $X$ is projective,
the equality (\ref{eqind})
is proved in
\cite[Theorem 7.13]{CC}.
\begin{lm}\label{lmfun}
Let $f\colon X\to Y$
be a morphism of smooth schemes
over $k$
and let ${\cal F}$
be a constructible
complex of $\Lambda$-modules.
Assume that $f\colon X\to Y$
is proper on the
support of ${\cal F}$.
Assume that
every irreducible
component of $X$
(resp.\ of $Y$) is
of dimension $n$
(resp.\ $m$).
{\rm 1.}
Let
$$
\xymatrix{
X\ar[r]^g
\ar[rd]_f &
X'\ar[d]^{f'}\\
&Y}$$ be
a commutative diagram of
morphisms of smooth schemes over $k$.
Then, we have
\begin{equation}
f_!CC{\cal F}=
f'_!(g_!CC{\cal F})
\label{eqfg}
\end{equation}
in ${\rm CH}_m(f_\circ SS{\cal F})$.
{\rm 2.}
Assume that one of the following conditions
{\rm (1)} and {\rm (2)} is
satisfied:
{\rm (1)} $f\colon X\to Y$
is an immersion.
{\rm (2)}
$f\colon X\to Y$ is
quasi-projective and
$SS{\cal F}$-transversal.
\noindent
Then,
we have $\dim (f_\circ SS{\cal F})
\leqq m=\dim Y$
and
\begin{equation*}
CCRf_*{\cal F}=
f_!CC{\cal F}
\leqno{(\ref{eqcnf})}
\end{equation*}
in $Z_m(f_\circ SS{\cal F})$.
\end{lm}
\proof{
1.
It follows from
Lemma \ref{lmCHtr}.
2.
The case (1) is proved in
\cite[Lemma 5.13.2]{CC}.
We show the case (2).
Since $f_\circ SS{\cal F}$
is a subset of the $0$-section $T^*_YY$,
we have $\dim (f_\circ SS{\cal F})
\leqq m=\dim Y$.
We may assume that $Y$ is connected and affine
and hence $X$ is quasi-projective.
Let $X\to P$ be an immersion
to a projective space
and factorize $X\to Y$
as the composition of an immersion
$X\to Y\times P$
and the projection $Y\times P\to Y$.
Then, by 1 and the case (1),
we may assume
that $f\colon X\to Y$ is projective and smooth.
By the assumption
that $f\colon X\to Y$ is
$SS{\cal F}$-transversal,
it is locally acyclic relatively
to ${\cal F}$ by Lemma \ref{lmtr}.1.
Since $f\colon X\to Y$ is proper,
the direct image
$Rf_*{\cal F}$ is locally constant
by \cite[5.2.4]{app}.
By Lemma \ref{lmtrV}.2,
there exists a dense
open subscheme $V\subset Y$
such that
$f\colon X\to Y$
is properly $SS{\cal F}$-transversal
on $V$.
By \cite[Lemma 5.11.1]{CC}
and Lemma \ref{lmptrA},
it suffices to show
the equality
\begin{equation}
{\rm rank}\ Rf_*{\cal F}
= (i_y^!CC{\cal F},T^*_{X_y}X_y)
_{T^*X_y}
\label{eqptr}
\end{equation}
for a closed point $y\in V$.
Since
${\rm rank}\ Rf_*{\cal F}
=\chi(X_{\bar y},i_y^*{\cal F})$,
the equality
(\ref{eqptr}) follows from the compatibility
$CCi_y^*{\cal F}=
i_y^!CC{\cal F}$
with the pull-back \cite[Theorem 7.6]{CC}
and
the index formula \cite[Theorem 7.13]{CC}.
\qed}
\medskip
We consider the case where $Y$
is a smooth curve
and $\dim f_\circ SS{\cal F}\leqq 1$.
We recall the definition
of the Artin conductor
and the description of
the characteristic cycle
of a sheaf on a curve.
Let $Y$ be a smooth irreducible curve over
a perfect field $k$
and let ${\cal G}$ be
a constructible complex of
$\Lambda$-modules on $Y$.
Let $V\subset Y$ be a dense open subscheme
such that the restriction
${\cal G}_V$ is locally constant
i.\ e.\
the cohomology sheaf
${\cal H}^q{\cal G}_V$ is locally constant
for every integer $q$.
For a closed point $y\in Y$,
the Artin conductor
$a_y{\cal G}$ is defined by
\begin{equation}
a_y{\cal G}
=
{\rm rank}\
{\cal G}_V
-{\rm rank}\
{\cal G}_{\bar y}
+
{\rm Sw}_y{\cal G}.
\label{eqay}
\end{equation}
Here
$\bar y$ denotes a geometric point above $y$
and ${\rm Sw}_y$
denotes the alternating sum of the Swan conductor.
The characteristic cycle is given by
\begin{equation}
CC{\cal G}
=-\Bigr({\rm rank}\ {\cal G}_V
\cdot [T^*_YY]
+\sum_{y\in Y\sm V}a_y{\cal G}\cdot
[T^*_yY]\Bigr)
\label{eqccY}
\end{equation}
by \cite[Lemma 5.11.3]{CC}.
Here
$T^*_yY$ is the fiber of $y$.
Let $f\colon X\to Y$
be a morphism
of smooth schemes
over a perfect field $k$
and $y\in Y$ be a closed point.
Assume that $\dim Y=1$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$.
Assume that
$f\colon X\to Y$ is proper on
the support of ${\cal F}$.
Under the assumption
$\dim f_\circ SS{\cal F}\leqq 1$,
the equality
$CCRf_*{\cal F}=
f_!CC{\cal F}$
(\ref{eqcnf})
in $Z_1(f_\circ SS{\cal F})$
is equivalent to
the equality
\begin{equation}
-a_yRf_*{\cal F}=
(CC{\cal F},df)_{T^*X,X_y}.
\label{eqcf}
\end{equation}
for every closed point
$y\subset Y\sm V$
by (\ref{eqccY}),
Lemma \ref{lmfun}.2 (2)
and Lemma \ref{lmd1A}.2,
where the right hand side is defined
as in (\ref{eqAdf}).
\begin{thm}\label{thmay}
Let $f\colon X\to Y$
be a quasi-projective morphism
of smooth schemes
over a perfect field $k$
and $y\in Y$ be a closed point.
Assume that $\dim Y=1$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$.
Assume that
$f\colon X\to Y$ is proper on
the support of ${\cal F}$
and is properly $SS{\cal F}$-transversal
on a dense open subscheme
$V\subset Y$.
Then, we have
\begin{equation*}
-a_yRf_*{\cal F}=
(CC{\cal F},df)_{T^*X,X_y}.
\leqno{(\ref{eqcf})}
\end{equation*}
\end{thm}
\proof{
We may assume that $k$
is algebraically closed.
By the same argument as in the proof
of Lemma \ref{lmfun}.2,
we may assume that
$f\colon X=Y\times P\to Y$
is the projection
for a projective space $P$.
By Lemma \ref{lmd1A}.1
and by replacing $Y$
by a projective smooth
curve over $k$
containing $Y$ as a dense
open subscheme,
we may assume that $Y$
is projective
and smooth.
By Lemma \ref{lmfun}
applied to $X\to Y\to {\rm Spec}\ k$,
we obtain
$$(f_!CC{\cal F},T^*_YY)_{T^*Y}
=
(CC{\cal F},T^*_XX)_{T^*X}.$$
By the index formula \cite[Theorem 7.13]{CC},
we have
$$
(CCRf_*{\cal F},T^*_YY)_{T^*Y}
=\chi(Y_{\bar k},Rf_*{\cal F})
=\chi(X_{\bar k},{\cal F})
=(CC{\cal F},T^*_XX)_{T^*X}.
$$
Thus, we have
$$
(CCRf_*{\cal F}-
f_!CC{\cal F},T^*_YY)_{T^*Y}
=0.$$
Since the coefficients of
$T^*_YY$ in
$CCRf_*{\cal F}$
and
$f_!CC{\cal F}$ are equal
by (\ref{eqAy0}), (\ref{eqccY})
and the index formula \cite[Theorem 7.13]{CC},
we obtain
\begin{equation}
\sum_{y\in Y\sm V}
-a_yRf_*{\cal F}=
\sum_{y\in Y\sm V}
(CC{\cal F},df)_{T^*X,X_y}.
\label{eqsum}
\end{equation}
By d\'evissage using
Lemma \ref{lmtr}.3
and \cite[Lemma 5.13.1]{CC}, we may assume
that ${\cal F}$ is a perverse sheaf.
Set $\widetilde V=V\cup \{y\}$
and $Z=Y\sm \widetilde V$.
By Corollary \ref{corS}, Corollary \ref{corV}
and weak approximation,
there exists a faithfully flat
finite morphism
$Y'\to Y$ of projective smooth curves
{\em \'etale at} $y$
satisfying the following condition:
Let
$$\begin{CD}
X@<<< X'@<{\tilde j'}<<
X'_{\widetilde V'}\\
@VfVV\hspace{-10mm}
\square
\hspace{6mm}
@V{f'}VV
\hspace{-13mm}
\square
\hspace{8mm}
@VVV\\
Y@<<< Y'@<<<\widetilde V'&=
Y'\times_Y\widetilde V
\end{CD}$$
be a cartesian diagram
and set
${\cal F}'=\tilde j'_{!*}
{\cal F}'_{\widetilde V'}$
for the pull-back
${\cal F}'_{\widetilde V'}$
of ${\cal F}$ on $X'_{\widetilde V'}$.
Then on $Y'_0=Y'\sm Y'\times_Yy$, the morphism
$f'\colon X'\to Y'$
is $SS{\cal F}'$-transversal and hence is
universally locally acyclic relatively to
${\cal F}'$.
For each $y'\in Z'=
Z\times_YY'$,
we have
$a_{y'}Rf'_*{\cal F}'=
(CC{\cal F}',df')_{T^*X',y'}=0$.
Since $Y'\to Y$ is \'etale at $y$,
for each $y'\in Y'\times_Yy$,
we have
$a_yRf_*{\cal F}=
a_{y'}Rf'_*{\cal F}'$
and
$(CC{\cal F},df)_{T^*X,X_y}=
(CC{\cal F}',df')_{T^*X',y'}$.
Thus,
by applying (\ref{eqsum}) to
$f'\colon X'\to Y'$ and ${\cal F}'$,
we obtain
$$-[Y':Y]\cdot
a_yRf_*{\cal F}=
[Y':Y]\cdot
(CC{\cal F},df)_{T^*X,y}$$
and hence (\ref{eqcf}).
\qed}
\begin{cor}[{\rm cf.\ \cite[Conjecture]{Bl}}]\label{corBl}
Let $f\colon X\to Y$
be a projective flat morphism
of smooth schemes
over a perfect field $k$.
Assume that
$\dim X=n$,
$\dim Y=1$
and that there exists
a dense open subscheme
$V\subset Y$
such that the base change
$f_V\colon X\times_YV\to V$
is smooth.
Then, for a closed point $y\in Y$,
we have
\begin{equation}
-a_yRf_*\Lambda
=
(-1)^n{c_n}^X_{X_y}(\Omega^1_{X/Y})\cap [X].
\label{eqBl}
\end{equation}
\end{cor}
\proof{
Applying Theorem \ref{thmay}
to the constant sheaf ${\cal F}=\Lambda$
and $CC\Lambda=
(-1)^n[T^*_XX]$,
we obtain
$-a_yRf_*\Lambda
=
(-1)^n
(T^*_XX,df)_{T^*X,X_y}$.
By applying Lemma \ref{lmcE} to the right hand side
and $[f^*\Omega^1_{Y/k}
\to \Omega^1_{X/k}]$,
we obtain (\ref{eqBl}).
\qed}
\begin{thm}\label{thmf*}
Let $f\colon X\to Y$
be a morphism
of smooth schemes
over a perfect field $k$.
Let ${\cal F}$
be a constructible complex
of $\Lambda$-modules on $X$
and $C=SS{\cal F}$ be
the singular support.
Assume that $Y$ is projective,
that $f\colon X\to Y$ is quasi-projective and
is proper on the support
of ${\cal F}$ and that we have an inequalty
\begin{equation}
\dim f_\circ C
\leqq \dim Y=m.
\label{eqdim}
\end{equation}
Then, we have
\begin{equation}
CCRf_*{\cal F}=
f_!CC{\cal F}
\label{eqCCf}
\end{equation}
in $Z_m(f_\circ SS{\cal F})$.
\end{thm}
\proof{
We may assume that
$k$ is algebraically closed.
Since $X$ is quasi-projective,
there exists a locally closed
immersion $i\colon X\to P$
to a projective space $P$.
By decomposing $f$
as the composition of
the immersion
$(i,f)\colon X\to P\times Y$
and the second projection
$P\times Y\to Y$,
we may assume that $f$
is projective and smooth
by Lemma \ref{lmfun}.
Set $C=f_\circ SS{\cal F}
\subset T^*Y$.
We have
$SSRf_*{\cal F}
\subset
f_\circ SS{\cal F}=C$.
By the assumption, we have
$\dim C\leqq m$.
By the index formula
\cite[Theorem 7.13]{CC} and
Theorem \ref{thmay},
the equality (\ref{eqCCf}) is proved for
$Y$ of dimension $\leqq 1$.
We show the general case
by reducing to the case $\dim Y=1$.
We take a closed immersion
of $Y$ to a projective space
$i\colon Y\to {\mathbf P}$.
We use the notations
${\mathbf P}
\overset p\gets Q
\overset{p^\vee}\to
{\mathbf P}^\vee$ in (\ref{eqpv})
and let $p_X\colon
X\times_{\mathbf P}Q\to X$
be the projection.
After replacing the immersion $i$
by the composition with a Veronese
embedding if necessary,
we may assume that
the restriction to
${\mathbf P}(i_\circ C)
\subset Q=
{\mathbf P}(T^*{\mathbf P})$
of the projection
$p^\vee \colon Q\to {\mathbf P}^\vee$
is generically radicial
by the assumption $\dim C\leqq m=\dim Y$
and by \cite[Corollary 3.21]{CC}.
Let $C^\vee=p^\vee_\circ p_Y^\circ C
\subset T^*{\mathbf P}^\vee$ and
let $D$ denote the image
$p^\vee({\mathbf P}(i_\circ C))
\subset {\mathbf P}^\vee$.
By Lemma \ref{lmLeg},
Lemma \ref{lmpi2}.3
and the Bertini theorem,
there exists a line
$L\subset {\mathbf P}^\vee$
satisfying the following conditions:
The immersion
$h\colon L\to{\mathbf P}^\vee$
is properly $C^\vee$-transversal.
The morphism
$h\colon L\to{\mathbf P}^\vee$ meets
$p_X^\circ SS{\cal F}$
properly.
The axis $A_L$ of $L$
meets $Y$ transversely and
$L$ meets $D$ transversely.
Since $A_L$ meets $Y$ transversely,
the blow-up $Y'$ of $Y$ at
$Y\cap A_L$ is smooth.
We consider the cartesian diagram
\begin{equation}
\begin{CD}
X@<{p_X}<<X\times_{\mathbf P}Q
@<{h_X}<<X'\\
@VfVV
\hspace{-15mm}
\square
\hspace{10mm}
@V{\tilde f}VV
\hspace{-13mm}
\square
\hspace{8mm}
@VV{f'}V\\
Y@<{p_Y}<<Y\times_{\mathbf P}Q
@<{h_Y}<<Y'\\
@.@V{p^\vee}VV
\hspace{-13mm}
\square
\hspace{8mm}
@VV{p_L}V\\
@.{\mathbf P}^\vee @<h<<L
\end{CD}
\label{eqL}
\end{equation}
of projective smooth schemes over $k$.
The equality
(\ref{eqCCf})
is equivalent to
\begin{equation*}
p^\vee_! p_Y^!CCRf_*{\cal F}=
p^\vee_! p_Y^!f_!CC{\cal F}.
\end{equation*}
It suffices to compare the
coefficients of
$C_a^\vee=p^\vee_\circ p_Y^\circ C_a$
for each irreducible component
of $C=\bigcup_aC_a$ of
dimension $m=\dim Y$.
Hence, this is further equivalent to
\begin{equation}
h^!p^\vee_! p_Y^!CCRf_*{\cal F}=
h^!p^\vee_! p_Y^!f_!CC{\cal F}
\label{eqLCCf}
\end{equation}
since ${\mathbf P}(i_\circ C)
\to D$ is generically radicial,
$h\colon L\to {\mathbf P}^\vee$
is properly $C^\vee$-transversal
and
$L$ meets $D$ transversely.
Let $\pi_X\colon X'\to X$
denote the composition
$p_X\circ h_X$ of the top line
in (\ref{eqL}).
We show that
the equality (\ref{eqLCCf})
is equivalent to
\begin{equation}
CCR(p_Lf')_*\pi_X^*{\cal F}=
(p_Lf')_!CC\pi_X^*{\cal F}.
\label{eqCCf'}
\end{equation}
First, we compare the left hand sides.
By \cite[Corollary 7.12]{CC}
applied to $i_*Rf_*{\cal F}$ on
${\mathbf P}$,
the left hand side of (\ref{eqLCCf}) equals
$h^!CCRp^\vee_* p_Y^*Rf_*{\cal F}$.
Since $SSRp^\vee_* p_Y^*Rf_*{\cal F}
\subset C^\vee$
and since $h\colon L\to {\mathbf P}^\vee$
is properly $C^\vee$-transversal,
the left hand side further equals
$CCh^*Rp^\vee_* p_Y^*Rf_*{\cal F}$
by \cite[Theorem 7.6]{CC}.
By proper base change theorem,
this is equal to the left hand side
$CCR(p_Lf')_*\pi_X^*{\cal F}$
of (\ref{eqCCf'}).
Next, we compare the right hand sides.
The right hand side of
(\ref{eqLCCf}) is equal to
$(p_L f')_!\pi_X^!CC{\cal F}$
by the projection formula
\cite[Theorem 6.2 (a)]{Ful}.
Since $h\colon L\to {\mathbf P}^\vee$
is $C^\vee$-transversal
and $C^\vee=p^\vee_\circ p_Y^\circ C
=(p^\vee \tilde f)_\circ p_X^\circ
SS{\cal F}$,
the immersion
$h_X\colon X'\to X\times_{\mathbf P}Q$
is
$p_X^\circ SS{\cal F}$-transversal
by Lemma \ref{lmtrbc}.2.
Further since $h\colon L\to {\mathbf P}^\vee$
meets $p_X^\circ SS{\cal F}$ properly,
the immersion
$h_X\colon X'\to X\times_{\mathbf P}Q$
is properly
$p_X^\circ SS{\cal F}$-transversal.
Since $p_X$ is smooth, the composition
$\pi_X=p_X\circ h_X$
is properly
$SS{\cal F}$-transversal.
Thus by \cite[Theorem 7.6]{CC},
it further equals
to the right hand side
$(p_Lf')_!CC\pi_X^*{\cal F}$
of (\ref{eqCCf'}).
We show the equality (\ref{eqCCf'})
by applying Theorem \ref{thmay}
to complete the proof.
Since ${\mathbf P}(i_\circ C)
\subset
Y\times_{\mathbf P}Q
={\mathbf P}(Y\times_{\mathbf P}T^*{\mathbf P})$
is the complement of the largest
open subset where
$p^\vee\colon
Y\times_{\mathbf P}Q
\to {\mathbf P}^\vee$
is $p_Y^\circ C$-transversal
and since $p_Y^\circ C=
p_Y^\circ f_\circ SS{\cal F}=
\tilde f_\circ p_X^\circ SS{\cal F}
=
\tilde f_\circ SSp_X^*{\cal F}$,
the composition
$p^\vee\tilde f\colon
X\times_{\mathbf P}Q
\to {\mathbf P}^\vee$
is $SSp_X^*{\cal F}$-transversal
on the complement $
{\mathbf P}^\vee\sm D$
by \cite[Lemma 3.8 (1)$\Rightarrow$(2)]{CC}.
By Lemma \ref{lmtrbc},
the morphism $p_Lf'\colon X'\to L$
is $SS\pi_X^*{\cal F}$-transversal
on the dense open subset
$L\sm L\cap D$.
Hence the equality (\ref{eqCCf'})
follows from Theorem \ref{thmay}
applied to $\pi_X^*{\cal F}$
and the equality (\ref{eqCCf})
is proved.
\qed}
\medskip
In the case of characteristic $0$,
we recover the classical result
as in \cite[Proposition 9.4.2]{KSc},
in a slightly weaker form.
Let $X$ be a smooth scheme
equidimensional of dimension $n$ over a field $k$
and let $\omega_X\in \Omega^2(T^*X)$ denote
the canonical symplectic form
on the cotangent bundle $T^*X$.
Let $C\subset T^*X$ be a closed conical subset.
We say that $C$ is
{\em isotropic} if the restriction of
$\omega_X$ on $C$ is $0$.
We say that $C$ is
{\em Lagrangean} if it is isotropic and
if $C$ is equidimensional of dimension $n$.
\begin{lm}\label{lmch0}
Let $k$ be a field of characteristic $0$
and let $f\colon X\to Y$ be a morphism
of smooth schemes over $k$.
Assume that $X$ (resp.\ $Y$) is
equidimensional of dimension $n$
(resp.\ $m$).
Let $C\subset T^*X$ be a closed conical subset.
If $C\subset T^*X$ is isotropic,
then $f_\circ C\subset T^*Y$ is also isotropic.
\end{lm}
The author learned the following proof
from Beilinson.
\proof{
Let $T^*_\Gamma(X\times Y)
\subset T^*(X\times Y)$
be the normal bundle
of the graph $\Gamma\subset X\times Y$
of $f\colon X\to Y$
and let $p_2\colon T^*(X\times Y)=
T^*X\times T^*Y\to T^*Y$
be the projection.
The direct image $f_\circ C\subset T^*Y$
equals the image by $p_2$ of the intersection
$C_1=T^*_\Gamma(X\times Y)
\cap (C\times T^*Y)$.
Since the normal bundle
$T^*_\Gamma(X\times Y)
\subset T^*(X\times Y)$
is isotropic
and since
$\omega_{X\times Y}$ equals
the sum $p_1^*\omega_X+p_2^*\omega_Y$
of the pull-backs by projections,
the assumption that
$C\subset T^*X$ is isotropic implies
that the restriction of
$p_2^*\omega_Y$ on $C_1$ is 0.
Since $k$ is of characteristic $0$,
for each irreducible component
$C'$ of $f_\circ C\subset T^*Y$,
there exists a closed subset
$C'_1\subset C_1$
generically \'etale over $C'$.
Hence the assertion follows.
\qed}
\begin{pr}\label{prch0}
Let $k$ be a field of characteristic $0$
and let $X$ be a smooth schemes over $k$.
Let ${\cal F}$ be a constructible
complex of $\Lambda$-modules on $X$.
{\rm 1.}
The singular support
$SS{\cal F}$ is Lagrangean.
{\rm 2.}
Let $f\colon X\to Y$ be a morphism
of smooth schemes over $k$.
Assume that $f$ is proper on
the support of ${\cal F}$.
Then, the inequality
{\rm (\ref{eqdim})} holds.
Further if $f\colon X\to Y$ is quasi-projective,
the equality {\rm (\ref{eqCCf})} holds.
\end{pr}
\proof{
1.
We may assume that $X$
is equidimensional of dimension $n$.
Since the singular support $SS{\cal F}$
is equidimensional of dimension $n$
\cite[Theorem 1.3 (ii)]{Be},
it suffices to show that
$SS{\cal F}$
is isotropic.
By devissage,
we may assume that
there exist a locally closed immersion
$i\colon V\to X$ of smooth scheme,
a locally constant sheaf
${\cal G}$ on $V$
and
${\cal F}=i_!{\cal G}$.
Since the resolution of singularity
is known in characteristic $0$,
the immersion $i$ is decomposed
by an open immersion
$j\colon V\to W$ and a
proper morphism $h\colon W\to X$
such that $W$ is smooth
and $V$ is the complement of
a divisor with simple normal crossings.
Thus, by the inclusion $SS{\cal F}
=SS Rh_*j_!{\cal G}
\subset h_\circ SS j_!{\cal G}$
and Lemma \ref{lmch0},
it is reduced to the case
where $i=j$ is an open immersion
of the complement of
a divisor with simple normal crossings.
Since $k$ is of characteristic $0$,
this case is proved in
\cite[Proposition 4.11]{CC}.
2.
By 1 and Lemma \ref{lmch0},
the direct image
$f_\circ SS{\cal F}$ is
isotropic. Hence
the inequality
$\dim f_\circ SS{\cal F}\leqq
\dim Y$ (\ref{eqdim}) holds.
We show the equality
$CCRf_*{\cal F}
=f_!CC{\cal F}$ (\ref{eqCCf}).
Similarly as in the proof of Theorem \ref{thmf*},
we may assume that $Y$ is affine
and $f\colon X=P\times Y\to Y$
is the projection for a projective smooth scheme
$P$ over $k$.
By resolution of singularity,
we may assume that $Y$ is projective and smooth.
Then since
the inequality (\ref{eqdim}) holds,
we may apply Theorem \ref{thmf*}.
\qed}
\subsection{Index formula for vanishing cycles}\label{ssiv}
We prepare some notation
to formulate an index formula
for vanishing cycle complex.
Let $f\colon X\to Y$
be a smooth morphism
of smooth schemes
over a perfect field $k$.
Assume that $X$
(resp.\ $Y$) is
equidimensional of
dimension $n+1$ (resp.\ $1$).
Let ${\cal F}$ be a constructible
complex of $\Lambda$-modules
on $X$.
Let $y\in Y$ be a closed point
and $i_y\colon X_y\to X$
be the closed immersion
of the fiber.
Assume that $f\colon X\to Y$
is properly $SS{\cal F}$-transversal
on the complement $X\sm X_y$
of the fiber $X_y=f^{-1}(y)$.
Then, the specialization
\begin{equation}
{\rm sp}_ySS{\cal F}
\subset
T^*X_y
\label{eqspSS}
\end{equation}
is defined as a closed
conical subset
equidimensional of dimension $n$.
Further, the specialization
\begin{equation}
{\rm sp}_yCC{\cal F}
\in
Z_n({\rm sp}_ySS{\cal F})
\label{eqspCC}
\end{equation}
is defined as a cycle.
\begin{lm}\label{lmspPs}
Let $f\colon X\to Y$
be a smooth morphism
of smooth schemes over a field $k$
and assume $\dim Y=1$.
Let ${\cal F}$ be a constructible
complex of $\Lambda$-modules
on $X$ and assume
that $f\colon X\to Y$
is properly $SS{\cal F}$-transversal.
Let $y\in Y$ be a closed point.
Then,
we have
\begin{equation}
SSR\Psi_y{\cal F}
={\rm sp}_ySS{\cal F}.
\label{eqSSsp}
\end{equation}
Further if $k$ is perfect, we have
\begin{equation}
CCR\Psi_y{\cal F}
={\rm sp}_yCC{\cal F}.
\label{eqCCsp}
\end{equation}
\end{lm}
\proof{
Let $i_y\colon X_y\to X$
denote the closed immersion
of the fiber.
Then, by the assumption that
$f\colon X\to Y$
is properly $SS{\cal F}$-transversal,
we have
${\rm sp}_ySS{\cal F}
=i_y^\circ SS{\cal F}$
and
${\rm sp}_yCC{\cal F}
=i_y^! CC{\cal F}$.
Recall that the definitions
of ${\rm sp}_y$ and $i_y^!$
both involve the minus sign.
Since $f\colon X\to Y$ is
locally acyclic relatively to
${\cal F}$ by Lemma \ref{lmtr}.2,
the canonical morphism
$i_y^*{\cal F}
\to R\Psi_y{\cal F}$
is an isomorphism.
Hence the equalities
(\ref{eqSSsp}) and (\ref{eqCCsp})
follow from Lemma \ref{lmh}
and \cite[Theorem 7.6]{CC}
respectively.
\qed}
\medskip
The following example
shows that the inclusion
$SSR\Psi{\cal F}
\subset {\rm sp}_ySS{\cal F}$
does not hold in general.
\begin{ex}\label{ex2}
{\rm
Let $k$ be a field of characteristic $p>2$.
Let $X={\mathbf A}^1\times {\mathbf P}^1$
and $j\colon U={\mathbf A}^1\times {\mathbf A}^1
={\rm Spec}\ k[x,y]\to X$
be the open immersion.
Let ${\cal G}$ be the locally constant sheaf
of $\Lambda$-modules of
rank $1$ on $U$
defined by the Artin-Schreier covering
$t^p-t=x^py^2$ and by a non-trivial character
${\mathbf F}_p\to \Lambda^\times$.
Then, the nearby cycles complex
$R\Psi_\infty{\cal F}$
is acyclic except at
the closed point $(0,\infty)$
or at degree $1$
and
$\dim R^1\Psi{\cal F}_{(0,\infty)}
=1.$
Hence, the singular support
$SSR\Psi_\infty{\cal F}$
equals the fiber
$T^*_{(0,\infty)}X_\infty$
at the closed point
and is not a subset of
the zero-section
${\rm sp}_\infty SS{\cal F}=
T^*_{X_\infty}X_\infty$.
}
\end{ex}
Let $Z\subset X_y$
be a closed subset.
Assume that $f\colon X\to Y$
is properly $SS{\cal F}$-transversal
on the complement of $Z$.
Then,
on the complement
$X_y\sm Z$,
we have
${\rm sp}_yCC{\cal F}
=
i_y^!CC{\cal F}
=
CCi_y^*{\cal F}$
by Lemma \ref{lmsp}
and the compatibility with
the pull-back \cite[Theorem 7.6]{CC}.
Thus, the difference
\begin{equation}
\delta_y
CC{\cal F}
=
{\rm sp}_yCC{\cal F}
-
CCi_y^*{\cal F}
\label{eqdel}
\end{equation}
is defined as a cycle in
$Z_n\bigl(Z\times_X(
{\rm sp}_ySS{\cal F}
\cup
SSi_y^*{\cal F})\bigr)$
supported on $Z$.
If $Z$ is proper over $Y$,
the intersection number
$(\delta_y SS{\cal F},T^*_{X_y}X_y)_
{T^*X_y}$ is
defined.
\begin{pr}\label{prvan}
Let $f\colon X\to Y$
be a smooth morphism of smooth schemes
over a perfect field $k$.
Assume that $X$
(resp.\ $Y$) is
equidimensional of
dimension $n+1$ (resp.\ $1$).
Let ${\cal F}$ be a constructible
complex of $\Lambda$-modules
on $X$.
Let $y\in Y$ be a closed point
and let $Z\subset X_y$
be a closed subset.
Assume that $f\colon X\to Y$
is properly $SS{\cal F}$-transversal
on the complement of $Z$
and that either of the following
conditions {\rm (1)}
and {\rm (2)} is satisfied:
{\rm (1)}
$f\colon X\to Y$ is projective.
{\rm (2)}
$\dim Z=0$.
\noindent
Then,
for the vanishing cycles complex
$R\Phi_y{\cal F}$, we have
\begin{equation}
\chi(Z_{\bar k},
R\Phi_y{\cal F})
=
(\delta_y CC{\cal F},T^*_{X_y}X_y)_
{T^*X_y}.
\label{eqvan}
\end{equation}
\end{pr}
\proof{
We may assume that
$k$ is algebraically closed.
We show the case (1).
Let $v\in Y$ be a closed point
different from $y$
and let $i_v\colon X_v\to X$
be the closed immersion.
Then, since the projective
morphism $f\colon X\to Y$
is locally acyclic relative to ${\cal F}$
outside $Z$ by Lemma \ref{lmtr}.2,
the left hand side of
(\ref{eqvan})
equals
\begin{equation}
\chi(Z,
R\Phi_y{\cal F})
=
\chi(X_y,
R\Psi_y{\cal F})
-
\chi(X_y,
i_y^*{\cal F})
=
\chi(X_v,
i_v^*{\cal F})
-
\chi(X_y,
i_y^*{\cal F})
\label{eqchv}
\end{equation}
The right hand side of (\ref{eqvan})
\begin{equation*}
(\delta_y CC{\cal F},T^*_{X_y}X_y)_
{T^*X_y}
=
({\rm sp}_y CC{\cal F},T^*_{X_y}X_y)_
{T^*X_y}
-
(CCi_y^*{\cal F},
T^*_{X_y}X_y)_
{T^*X_y}
\end{equation*}
equals
\begin{equation}
(i_v^! CC{\cal F},T^*_{X_v}X_v)_
{T^*X_v}
-
(CCi_y^*{\cal F},
T^*_{X_y}X_y)_
{T^*X_y}
\label{eqdelv}
\end{equation}
by (\ref{eqspiv}).
Since $i_v\colon X_v\to X$
is properly $SS{\cal F}$-transversal
by Lemma \ref{lmtrbc},
the right hand side of (\ref{eqchv})
equals (\ref{eqdelv})
by the compatibility with
the pull-back \cite[Theorem 7.6]{CC}
and the index formula
\cite[Theorem 7.13]{CC}.
Thus the equality
(\ref{eqvan}) is proved.
We show the case (2).
Since the formation of nearby cycles
complex commutes with base change
by \cite[Proposition 3.7]{TF},
we may assume that
the action of the inertia group $I_y$
on $R\Psi_y{\cal F}$
is trivial.
Since the vanishing cycles functor
is $t$-exact by
\cite[Corollaire 4.6]{au},
we may assume that ${\cal F}$
is a simple perverse sheaf.
First, we consider the case
${\cal F}$ is supported
on the closed fiber $X_y$.
By the assumption that
$f\colon X\to Y$
is properly $SS{\cal F}$-transversal
on the complement of $Z$,
the morphism
$f\colon X\to Y$
is locally acyclic relatively to
${\cal F}$ on the complement of $Z$.
Thus ${\cal F}$ is supported on $Z$
and the assertion follows in this case.
We may assume that
the restriction ${\cal F}|_{X_\eta}$
on the generic fiber is non-trivial.
Then, by Proposition \ref{prS}.2,
the morphism
$f\colon X\to Y$
is locally acyclic relatively to
${\cal F}$.
Hence by Lemma \ref{lmtrZ}.2,
the morphism $f\colon X\to Y$
is properly $SS{\cal F}$-transversal
and the assertion follows
from Lemma \ref{lmspPs}.
\qed}
\medskip
In the case (2) $\dim Z=0$,
Proposition \ref{prvan}
means
$CCR\Phi_y{\cal F}=
\delta_yCC{\cal F}$.
However,
Examples \ref{ex} and \ref{ex2}
show that one cannot expect
to have
$CCR\Psi_y{\cal F}=
{\rm sp}_yCC{\cal F}$ or
equivalently
$CCR\Phi_y{\cal F}=
\delta_yCC{\cal F}$
in general. | 48,928 |
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\begin{document}
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\numberwithin{equation}{section}
\thispagestyle{empty}
\title{\Large {\bf
Two interacting Ising chains\\[2mm]
in relative motion
}}
\author{{H.J.~Hilhorst}\\[5mm]
{\small Laboratoire de Physique Th\'eorique, B\^atiment 210}\\[-1mm]
{\small Universit\'e Paris-Sud and CNRS,
91405 Orsay Cedex, France}\\}
\maketitle
\begin{small}
\begin{abstract}
\noindent
We consider
two parallel cyclic Ising chains counter-rotating
at a relative velocity $v$,
the motion actually being a succession of discrete steps.
There is an in-chain interaction
between nearest-neighbor spins and a
cross-chain interaction
between instantaneously opposite spins.
For velocities $v > 0$ the system, subject to
a suitable markovian dynamics at a temperature $T$,
can reach only a nonequilibrium steady state (NESS).
This system was introduced
by Hucht et al., who showed that for $v=\infty$
it undergoes a para- to ferromagnetic
transition, essentially due to the fact that each chain exerts an
effective field on the other one.
The present study of the $v=\infty$ case
determines the consequences of the fluctuations of this effective field
when the system size N is finite.
We show that whereas to leading order the system obeys
detailed balancing with respect to an
effective time-independent Hamiltonian, the higher order
finite-size corrections violate detailed balancing.
Expressions are given to various orders in
$N^{-1}$ for the interaction free energy between the chains,
the spontaneous magnetization, the in-chain and cross-chain
spin-spin correlations, and the spontaneous magnetization.
It is shown how finite-size
scaling functions may be derived explicitly.
This study was motivated by
recent work on a two-lane traffic problem
in which a similar phase transition was found.\\
\noindent
{{\bf Keywords:} kinetic Ising model, nonequilibrium stationary state,
phase transition}
\end{abstract}
\end{small}
\vspace{12mm}
\noindent LPT Orsay 11/03
\section{Introduction}
\label{secintroduction}
Recently Hucht \cite{Hucht09} (see also \cite{Kadauetal08}),
motivated by the phenomenon of magnetic friction,
formulated a nonequilibrium steady state (NESS) Ising model of a new type.
It consists of two parallel linear Ising chains having a relative
velocity $v$.
In addition to a nearest-neighbor interaction in each chain,
any pair of spins facing each other
on the two chains has an instantaneous interaction.
In the version of the model easiest to study, each chain is finite and
periodic; we will therefore speak of
cyclic counter-rotating Ising chains (CRIC).
The model, subject to suitable temperature dependent
Markovian dynamics,
was shown \cite{Hucht09} at velocity $v=\infty$
to have a para- to ferromagnetic phase transition
which in the limit of infinitely long chains may be
understood in terms of an equivalent equilibrium model.
The CRIC seems to us to be of the same fundamental importance
as Glauber's
\cite{Glauber63} original kinetic Ising model.
First, it is of interest in its own right as a new member
of the class of NESS.
Second, its interest is enhanced in the wider context of recent work on Ising
models that in one way or another are driven,
dissipate energy, or have some novel type of coupling;
such work has appeared
in a variety of contexts \cite{DemeryDean10,Pleimlingetal10,Pradosetal10}.
In particular, the present CRIC was extended to a Potts version by
Igl\'oi {\it et al.} \cite{Igloietal11}, who find remarkable
nonequilibrium phase transitions.
In this paper we contribute further to the study of the CRIC.
We focus on finite chains and on how to derive
known and new properties from the master equation that defines the model.
\vspace{3mm}
Hucht's solution \cite{Hucht09} is based on showing that
at $v=\infty$ the stationary state dynamics of
the CRIC is actually that of an equilibrium Ising chain
in an effective magnetic field $H_0$,
this field being zero above the transition
temperature and nonzero below. This equivalence
is valid in the limit where the chain length $N$ tends to infinity.
In this work we show
that it is possible to formulate this problem as an expansion
in powers of $N^{-1/2}$. To lowest order we recover the equivalent
equilibrium system found in reference \cite{Hucht09}.
To higher orders fluctuations of the field $H_0$ come into play and
appear as finite-size effects.
The finite $N$ case is of interest, first of all, on the level of principles,
and secondly, for the analysis of finite size effects in simulations
as were carried out in \cite{Hucht09} and by ourselves.
We expect, furthermore, that our approach will help prepare the way
for future work on the $v<\infty$ case, which is considerably harder.
The effective transition rates satisfy detailed balancing
to leading order in the large-$N$ expansion \cite{Hucht09};
our analysis reveals, however, that
to higher orders in $N^{-1/2}$ the
detailed balancing (DB) symmetry of the effective rates is broken.
The stationary state distribution may be found explicitly,
at least to the lowest DB-violating order.
Knowing this state one can calculate all desired NESS properties.
\vspace{5mm}
In section \ref{secdefmodel} of this paper
we define the rules of the markovian dynamics for general relative
velocity $v$ and then specialize to $v=\infty$. These dynamical
equations are the starting point for all that follows.
In section \ref{secdetbal} we discuss the DB violation
that occurs in higher orders of $N^{-1}$.
In section \ref{seczero} we consider the stationary state to zeroth
order, as was already done by Hucht \cite{Hucht09}.
In sections \ref{secexpansion} and \ref{secfirst}
we show how $N^{-1}$ can be introduced
as an expansion parameter and we define a `leading order', composed of
the zeroth order and a first-order correction.
In section \ref{secsecond} we show how for the stationary state
distribution an expansion may be found in powers of $N^{-1}$. We
present the explicit result to next-to-leading order.
In section \ref{secaverages} we calculate for various quantities of
physical interest their stationary state
averages to successive orders in the expansion.
In section \ref{sectraffic} we briefly discuss the relation of the
present model to a two-lane road traffic model studied earlier.
In section \ref{secconclusion} we conclude.
\section{Counter-rotating Ising chains}
\label{secdefmodel}
\subsection{A stochastic dynamical system}
\label{secmodel}
\begin{figure}
\begin{center}
\scalebox{.55}
{\includegraphics{ladder1h.eps}}
\end{center}
\caption{\small Ladder of spins with an intrachain nearest-neighbor
interaction $J_1=J$.
The two chains constituting the ladder
have a relative velocity $v$, the motion taking place in
discrete steps of one lattice unit.
There is an interchain nearest-neighbor
interaction $J_2=\eta J$ between each pair of spins facing each other
at any instant in opposite chains.}
\label{figladder}
\end{figure}
We consider Ising spins on the ladder lattice shown in figure \ref{figladder}.
The spins in the upper chain are denoted by $r_j$, those in the lower chain by
$s_i$, where the integers $j$ and $i$ are site indices.
There is a nearest-neighbor interaction
$J_1=J$ inside each chain
and an interaction $J_2=\eta J$ between each pair of spins facing each other
in opposite chains. We take $J>0$ and $\eta$ of arbitrary sign.
The feature \cite{Kadauetal08} and \cite{Hucht09}
that distinguishes this model from the standard Ising model on a
ladder lattice,
is that the two chains move with respect to one another at a speed $v>0$.
This will mean the following:
the time axis is discretized in intervals
of duration $\tau=a_0/v$ (where $a_0$ is the lattice spacing)
and at the end of each interval the upper
chain is shifted one lattice spacing $a_0$
to the right with respect to the lower one.
The Hamiltonian ${\cal H}(t)$ of this system
is therefore time-dependent and given by
\beq
{\cal H}(t) = -J\sum_j \left[ r_jr_{j+1} + s_js_{j+1} \right]
-\eta J \sum_j r_{j}s_{\lfloor j+vt/a_0 \rfloor}\,,
\label{Hamiltonian}
\eeq
where $\lfloor x \rfloor$ is the largest integer less than or
equal to $x$.
We will consider cyclic boundary conditions
\footnote{In connection with the traffic problem
open boundary conditions
are certainly also worthy of consideration. These have however
the inconvenience of breaking the translational symmetry.}.
In this case the chains become counter-rotating
loops of length say $N$;
the site indices $i$, $j$, and
$\lfloor j+vt/a_0 \rfloor$ must then be interpreted modulo $N$.
Employing
the shorthand notation $r=\{r_j|j=1,2,\ldots,N\}$
and $s=\{s_j|j=1,2,\ldots,N\}$, we may indicate
a spin configuration of the system by $(r,s)$.
We associate with ${\cal H}(t)$ a stochastic
time evolution of $(r,s)$. Its precise definition requires that
we exercise some caution.
We will first define it as a Monte Carlo procedure and then write down
the master equation and pass to analytic considerations.
Single-spin reversals are attempted
at uniformly distributed random instants of time
at a rate of $1/\tau_0$ per site
\footnote{We may scale time such that $\tau_0=1$.}.
Each attempt
is governed by transition probabilities.
Since there are $2N$ sites, there are $2N$ different single-spin flips
by which a state $(r,s)$ may be entered or exited.
Given that a reversal attempt takes place, let
$(2N)^{-1}\Wr_j(r;s;t)$ and $(2N)^{-1}\Ws_j(s;r;t)$ be the probabilities that
$r_j$ and $s_j$ are flipped, respectively.
The reversal attempt will remain unsuccessful with the
complementary probability
\beq
1-A_{\rm acc} = 1-(2N)^{-1}\sum_{j=1}^N
\left[ \Wr_j(r;s;t) + \Ws_j(s;r;t) \right],
\label{defAacc}
\eeq
where $A_{\rm acc}$ is what is usually called the `acceptance
probability'.
We now specify the $\Wr_j$ and $\Wr_j$ in such a way that
at any time $t$ the system strives to attain the canonical equilibrium
at a given temperature $T$ with respect to the
instantaneous Hamiltonian ${\cal H}(t)$.
The choice is not unique.
We choose
\bea
\Wr_j(r;s;t)&=&\tfrac{1}{4}
\big[ 1-\tfrac{1}{2}r_j(r_{j-1}+r_{j+1})\tanh 2K \big]
\big[ 1-r_js_i\tanh\eta K \big], \nonumber\\[2mm]
\Ws_i(s;r;t)&=&\tfrac{1}{4}
\big[ 1-\tfrac{1}{2}s_i(s_{i-1}+s_{i+1})\tanh 2K \big]
\big[ 1-s_ir_j\tanh\eta K \big],
\label{defw}
\eea
where we have set $K=J/T$
(with $T$ measured in units of Boltzmann's constant)
and where in {\it both\,} equations $i$ and $j$ are related by
\beq
i = \lfloor j+vt/a_0 \rfloor\!\!\mod N.
\label{relijt}
\eeq
Equation (\ref{defw}) is different both from the heat bath (or: Glauber)
and from the Metropolis transition probabilities.
We will refer to it as the ``factorizing rate''.
The factor
\beq
\wGj(r)=\tfrac{1}{2}\big[ 1-\tfrac{1}{2}r_j(r_{j-1}+r_{j+1})\tanh 2K \big]
\label{defwG}
\eeq
represents the Glauber transition probability.
The $\Wr_j$ and $\Ws_j$ define an easy-to-simulate Markov chain
\footnote{No confusion should arise with the two
legs of the ladder lattice, to which we refer also as `chains'.}
with time-dependent transition probabilities.
\footnote{The reversal attempts, that is, the steps of the Markov
chain, are Poisson distributed on the time axis.
This makes it possible at any time to probabilistically
connect the elapsed time $t$ to the
number of spin reversal attempts $n$. In the large $t$ limit
of course $n\simeq t/\tau_0$.}
In the special case
$v=0$ the Hamiltonian ${\cal H}(t)$ reduces to the equilibrium
Hamiltonian of the ladder lattice. For $v$ arbitrary but $\eta=0$
it reduces to the equilibrium Hamiltonian of two decoupled chains. In
both special cases the dynamics is standard and obeys detailed balancing.
In the general case, since
the Hamiltonian is time-dependent, the system will not
reach equilibrium but instead enter a NESS.
Actually, for generic $v$,
because of the periodic discrete shifts, the NESS is a
$\tau$-periodic function of time;
NESS averages are naturally defined to include an average over this period.
In the limiting case
$v=\infty$ we have $\tau=0$ and this complication disappears.
The infinite velocity
NESS is the subject of our interest in the remaining sections.
It is a problem that depends only on the two parameters $K$ and $\eta$.
We note finally that as compared to ours, there is an extra prefactor
\beq
\frac{2+(1-r_{j-1}r_{j+1})\tanh 2K}{1+\tanh 2K}(1+\ee^{-2\eta K})
\label{prefactor}
\eeq
in Hucht's expression for the transition probability $\Wr_j(r;s;t)$,
and an analogous prefactor for $\Ws_i(s;r;t)$.
These factors may easily be carried along in the calculation.
\subsection{The limit $v\to\infty$}
\label{secvinfty}
Let $P(r,s;n)$ be the probability distribution on the configurations
$(r,s)$ after $n$ spin reversal attempts.
We will now write down the formal evolution equation for
$P(r,s;n)$ for the case of $v=\infty$,
where important simplifications occur.
When $v=\infty$ there is no relation between the indices $i$ and $j$
and hence the chain has transition probabilities
$w_j(r;s)$ given by the average of (\ref{defw})
on all $i$, which is now considered as an independent variable.
We denote this average by $w_j(r,s)$ and thus have
\begin{subequations}\label{defwrsj}
\bea
w_j(r;s) &=& \frac{1}{N}\sum_{i=1}^N \Wr_j(r;s;t) \nonumber\\[2mm]
&=& \tfrac{1}{4}\big[ 1-\tfrac{1}{2}r_j(r_{j-1}+r_{j+1})\tanh 2K \big]
\big[ 1-r_j\mu(s)\tanh\eta K\big] \nonumber\\[2mm]
&=& \wGj(r)\times\tfrac{1}{2}\big[ 1-r_j\mu(s)\tanh\eta K\big],
\label{defwrj}\\[2mm]
w_j(s;r) &=& \frac{1}{N}\sum_{i=1}^N \Ws_j(s;r;t) \nonumber\\[2mm]
&=&\wGj(s)\times\tfrac{1}{2}\big[ 1-s_j\mu(r)\tanh\eta K\big],
\label{defwsj}
\eea
\end{subequations}
where
\beq
\mu(s)=\frac{1}{N}\sum_{i=1}^Ns_i, \qquad \mu(r)=\frac{1}{N}\sum_{i=1}^Nr_i\,.
\label{defmus}
\eeq
We will write $r^j$ for the configuration
obtained from $r$ by reversing $r_j$
(that is, by carrying out the replacement $r_j \mapsto -r_j$),
and similarly define $s^j$.
Summing on all $2N$ flips by which it is possible to enter or to exit
$(r,s)$ we find that the evolution of $P(r,s;n)$ is described by the
master equation
\bea
P(r,s;n+1) &=& \frac{1}{2N}\sum_{j=1}^N \left[\, w_j(r^j;s)P(r^j,s;n)
+ w_j(s^j;r)P(r,s^j;n) \right.\nonumber\\[2mm]
&& + \left. \big( 1-w_j(r;s) \big)P(r,s;n)
+ \big( 1-w_j(s;r) \big)P(r,s;n) \,\right],
\nonumber\\
&&
{}\label{defMchain}
\eea
where the second line corresponds to the probability of an unsuccessful spin
reversal attempt. In vector notation equation
(\ref{defMchain}) may be written
\beq
P(n+1) = ({\bf 1} + {\cal W}) P(n),
\label{defW}
\eeq
where $P(n)$ is the $2^{2N}$-dimensional vector of elements
$P(r,s;n)$, the symbol ${\bf 1}$ denotes the unit matrix, and
${\cal W}$ is a matrix composed of entries $w_j$ for
which comparison of (\ref{defMchain})
and (\ref{defW}) yields
\bea
{\cal W}(r,s;r',s')&=&\delta_{r'r^j}\delta_{s's}w_j(r^j,s)
+\delta_{r'r}\delta_{s's^j}w_j(r,s^j) \nonumber\\[2mm]
&&-\,\delta_{r'r}\delta_{s's}\sum_{j=1}^N \left[ w_j(r;s)+w_j(s;r) \right].
\label{relwW}
\eea
The discrete-time master equation,
(\ref{defMchain}) together with the
Poisson statistics of the reversal attempts on the time axis,
fully defines the CRIC for $v=\infty$.
This equation may be studied analytically, as is
the purpose of this work, or may be implemented in a Monte Carlo
simulation.
\section{Detailed balancing and its violation}
\label{secdetbal}
Henceforth we consider the case $v=\infty$.
Our purpose is now to find the
stationary state distribution $P_{\rm st}(r,s)$ of the evolution
equation (\ref{defMchain}).
This distribution
is the solution of $P(r,s;n)=P(r,s;n+1)=P_{\rm st}(r,s)$, which means
\beq
0={\cal W}P_{\rm st}\,.
\label{defstst}
\eeq
Combining equations (\ref{defstst}) and (\ref{defMchain})
yields the $v=\infty$ stationary state equation
\bea
0 &=& \sum_{j=1}^N \left[w_j(r^j;s)\Pst(r^j,s)
+ w_j(s^j;r)\Pst(r,s^j) \right. \nonumber\\
&&\phantom{XXXX}
- \left. w_j(r;s)\Pst(r,s) - w_j(s;r)\Pst(r,s)
\right].
\label{defPst}
\eea
If the transition probabilities
satisfy the condition
of detailed balancing, the solution of (\ref{defPst}) is easily constructed;
in case of the contrary, there are no general methods.
We examine therefore first the question of whether equation (\ref{defMchain})
satisfies detailed balancing.
A Markov chain satisfies detailed balancing (DB)
if and only if
its transition probabilities are such that
any loop in configuration space is traversed with equal probability in
either direction.
To show that the transition probabilities $w_j$
fail to obey DB
we consider an elementary
loop of four single-spin flips,
\beq
(r,s) \mapsto (r^j,s) \mapsto (r^j,s^j) \mapsto (r,s^j) \mapsto (r,s).
\label{defloop}
\eeq
Given the system is in $(r,s)$, we denote by $p_+(\eta)$ and $p_-(\eta)$
the probability that in the next four attempts it goes
through this loop in forward and in backward direction, respectively.
That is,
\bea
p_+(\eta) &=& w_j(r;s)w_j(s;r^j)w_j(r^j;s^j)w_j(s^j;r), \nonumber\\[2mm]
p_-(\eta) &=& w_j(s;r)w_j(r;s^j)w_j(s^j;r^j)w_j(r^j;s).
\label{xppmeta}
\eea
For $\eta=0$ the two chains are decoupled, and as discussed below
equation (\ref{relijt}), each of them separately satisfies DB;
it is easy indeed to verify explicitly that $p_+(0)=p_-(0)\equiv p(0)$.
For general $\eta$ we may
work out the difference $p_+(\eta)-p_-(\eta)$ with the aid of (\ref{defwrj}),
(\ref{defmus}), and the relations
\beq
\mu(r^j) = \mu(r)-\frac{2r_j}{N}\,, \qquad \mu(s^j) = \mu(s)-\frac{2s_j}{N}\,,
\label{relmurjmur}
\eeq
which yields
\bea
p_+(\eta)-p_-(\eta) &=& 4N^{-1} p(0)\,\tanh^2\eta K\,
[r_j\mu(s)-s_j\mu(r)] \nonumber\\[2mm]
&& \times \big\{ [r_j\mu(r)-s_j\mu(s)]+2N^{-1}\tanh\eta K \big\}.
\label{xdiffppmeta}
\eea
This shows that DB is violated in the general case of
nonzero coupling ($\eta\neq 0$)
between the chains. It becomes valid again only asymptotically in
the limit $N\to\infty$.
We therefore cannot hope to rely on any general methods to construct
$P_{\rm st}(r,s)$ for finite $N$.
Indeed, writing out the stationary state equation
(\ref{defstst}) fully explicitly for $N=3,4$
(only $N=2$ is trivial) has confirmed the nontriviality of the
stationary state but has not provided us with any useful insight.
\section{Stationary state $P_{\rm st}(r,s)$ to zeroth order}
\label{seczero}
The limit $N\to\infty$ was considered by Hucht \cite{Hucht09,Kadauetal08},
and we briefly recall the results.
One may suppose that in this limit $\mu(r)$ and $\mu(s)$
have vanishing fluctuations around an
as yet unknown common average to be called $m_0(K,\eta)$.
We will denote the $N\to\infty$ limit of $w_j$ by $w_{j,0}$\,.
It then follows from (\ref{defwrj}) that
\beq
w_{j,0}(r) = \wGj(r) \times \tfrac{1}{2}[1-r_jm_0\tanh\eta K].
\label{defwrj0}
\eeq
With the transition probabilities (\ref{defwrj0}) the $r$- and the
$s$-chain decouple. Moreover, the expression for these $w_{j,0}$
is such that the spin dynamics satisfies
DB with respect to the pair of uncoupled
nearest-neighbor Ising Hamiltonians in a field,
\beq
{\cal H}_0(r,s)/{T} =
- K\sum_{j=1}^{N} \big[ r_jr_{j+1} + s_js_{j+1} \big]
- H_0\sum_{j=1}^{N} \big[ r_j + s_j \big]\,.
\label{defH0}
\eeq
where $H_0$ is defined in terms of $m_0$ by
\beq
\tanh H_0 = m_0\tanh\eta K
\label{deffield}
\eeq
and where $K$ and $H_0$ both include a
factor $1/T$.
The quantity ${\cal H}_0(r,s)$ is an effective time-independent Hamiltonian.
Let $m(K,z)$ denote the magnetization per spin of the one-dimensional (1D)
Ising chain with coupling $K$ in a field that we will for convenience
denote by $z$. This quantity is well-known and given by
\beq
m(K,z)=\frac{\sinh z}{\sqrt{ \sinh^2 z + {\rm e}^{-4K} }}\,.
\label{xmKz}
\eeq
Consistency requires that
\beq
m_0=m(K,H_0).
\label{relconst}
\eeq
Upon combining (\ref{deffield}) with (\ref{relconst})
one obtains an equation for $H_0$ [or equivalently $m_0$].
The solution $H_0$ is a function of the two system parameters $K$
and $\eta$ and given by
\beq
\tanh H_0(K,\eta)=\left\{
\begin{array}{ll}
\left( \dfrac{ \tanh^2\eta K-{\rm e}^{-4K} }
{ 1-{\rm e}^{-4K} } \right)^{\half}, \phantom{XX}& K>\Kc, \\[5mm]
0, & K\leq\Kc\,,
\end{array}
\right.
\label{solnH0}
\eeq
in which there appears a critical coupling $\Kc = J/\Tc$ that is the
solution of
\footnote{Equation (\ref{xKc}) may be rewritten as
$\sinh(2J_1/\Tc)\sinh(2J_2/\Tc)=1$, which shows, as
was also noticed in reference \cite{Hucht09}, that $\Tc$ is exactly
(but accidentally) equal to the critical
temperature of Onsager's square Ising model with horizontal and
vertical couplings $J_1$ and $J_2$.}
\beq
\tanh \eta \Kc = \ee^{-2\Kc}.
\label{xKc}
\eeq
The magnetization $m_0(K,\eta)$ follows directly from (\ref{deffield})
and (\ref{solnH0}). For $T\to\Tc^-$ it vanishes
as $m_0 \propto (\Tc-T)^{\beta}$ with a classical exponent $\beta=\frac{1}{2}$.
For later use it is worthwhile to notice that also
$H_0(T)\propto (T-\Tc)^{1/2}$ when $T\leq\Tc$\,.
The DB property found below equation (\ref{defwrj0})
now allows us to conclude that for $N\to\infty$
the stationary state distribution $P_{{\rm st},0}(r,s)$
is the Boltzmann distribution corresponding to (\ref{defH0}), that is,
\beq
P_{{\rm st},0}(r,s) = {\cal N}_0\, \ee^{-{\cal H}_0(r,s)/T}
\label{xstst0}
\eeq
where ${\cal N}_0$ is the normalization.
In reference \cite{Hucht09} several
system properties were calculated in this $N\to\infty$
limit by averaging with respect to $P_{{\rm st},0}(r,s)$.
\section{Expansion procedure for $P_{\rm st}(r,s)$}
\label{secexpansion}
As has become clear in section \ref{secdetbal},
the inverse system size $1/N$ is a measure of the degree of DB
violation.
In the present case this will lead us to attempt
to find the finite $N$ stationary state by expanding around
the known $N=\infty$ solution (\ref{xstst0}), which will
play the role of the zeroth order result.
At the basis of the expansion is the hypothesis, to be confirmed
self-con\-sistent\-ly, that the fluctuations
$\delta\mu$ of the chain magnetizations, defined by
\beq
\delta\mu(r)=\mu(r)-m_0\,, \qquad \delta\mu(s)=\mu(s)-m_0\,.
\label{defmurs}
\eeq
are of order $N^{-1/2}$.
\vspace{3mm}
A naive attempt to set up the expansion
would be to notice that
the transition probability (\ref{defwrj})
can be written as a sum of its average and a correction,
$w_j(r;s) = w_{j,0}(r) + \bar{w}_{j}(r;s)$,
where $w_{j,0}(r)$ is given by (\ref{defwrj0}) and
$\bar{w}_{j}(r;s) = \wGj(r) \times(-\tfrac{1}{2}r_j) \delta\mu(s) \tanh\eta K$
is of order $N^{-1/2}$. One might then
think that there exists a corresponding expansion
$P_{\rm st}(r,s)=P_{{\rm st},0}(r,s)[1+\ldots]$.
However, the dot terms turn out to be of order ${\cal O}(1)$ as $N\to\infty$,
which is a sign that this is not the right way to expand.
The reason for this failure is that $\Pst$ is the
exponential of the extensive quantity ${\cal H}_0$;
one should therefore ask first if this exponential contains
any corrections of less divergent order in $N$ before attempting to
multiply it by a series of type $[1+\ldots]$.
In the next section we describe how the expansion can be set up
successfully.\\
Knowing how to calculate higher order corrections to the
stationary state distribution, although
certainly of diminishing practical interest,
has a definite theoretical merit.
What we will find in the end
is that in fact to first order in the expansion detailed balancing continues
to hold, but with respect to a Hamiltonian ${\cal H}^{(1)}(r,s)$
that acquires a first order correction.
In section \ref{secfirst} we present the solution, to be denoted as
$P^{(1)}_{\rm st}(r,s)$, of the
stationary state to first order. In section \ref{secsecond}
we will show how higher orders can be calculated and
find that from the second order on DB violation appears.
Section \ref{secsecond}
also provides the demonstration of the correctness of the expansion.
\section{Stationary state $P_{\rm st}(r,s)$ to first order}
\label{secfirst}
We use the upper index `$(1)$' to indicate any quantity correct
up to first order in the expansion.
We will prove that the correct expansion takes the form
\beq
\Pst(r,s) = \Pstupone(r,s)\left[ 1 + q_1(r,s) + q_2(r,s) + \ldots\right],
\label{seriestwo}
\eeq
where
the $q_k$ $(k=1,2,\ldots)$,
that we will show how to determine later, are of
of order ${\cal O}(N^{-k/2})$ and
where $\Pstupone(r,s)$, which includes a first order correction
to the zeroth order result, is explicitly given by
\begin{subequations}\label{deffirstorder}
\beq
\Pstupone(r,s) = {\cal N}^{(1)} \exp
\left( -\frac{{\cal H}^{(1)}(r,s)}{T} \right),
\label{deffirstP}
\eeq
\beq
\frac{{\cal H}^{(1)}(r,s)}{T} = \frac{{\cal H}_0(r,s)}{T}
-g_0 N \delta\mu(r)\delta\mu(s),
\label{deffirstH}
\eeq
\beq
g_0 = \cosh^2H_0 \tanh\eta K,
\label{deffirstg}
\eeq
\end{subequations}
in which ${\cal N}^{(1)}$ is the appropriate normalization.
The second term on the RHS of (\ref{deffirstH})
is a correction to the zeroth order effective Hamiltonian.
It is ${\cal O}(1)$ for $N\to\infty$ and, since it is proportional to
$g_0$, it vanishes as expected when $\eta=0$.
In order to demonstrate (\ref{seriestwo})-(\ref{deffirstorder})
we split ${\cal W}$ according to
\beq
{\cal W} = \calWupone + \sum_{k=2}^\infty {\cal W}_k\,,
\label{splittwo}
\eeq
where we take for $\calWupone$ the matrix with the factorizing
transition probabilities
{\it that ensure detailed balancing with respect to\,} ${\cal
H}^{(1)}$,
and in which the ${\cal W}_k$ will be defined shortly.
Expression (\ref{deffirstorder}) for Hamiltonian ${\cal H}^{(1)}$ shows that a
spin $r_j$ is subject to a total field $H_0 + g_0\delta\mu(s)$.
Hence by analogy to (\ref{defwrj0})
the transition probabilities that enter $\calWupone$ are
\beq
\wupone_j(r;s) = \wGj(r) \times
\tfrac{1}{2}[ 1 - r_j\tanh\{H_0+g_0\delta\mu(s)\} ].
\label{xwupone}
\eeq
We then have by construction that
\beq
\calWupone\Pstupone =0,
\label{sseqfirstorder}
\eeq
which is the combined zeroth and first order result.
It may be obtained in explicit form
from (\ref{defPst}) by the substitutions
$w_j \mapsto w_j^{(1)}$ and $\Pst \mapsto \Pstupone$.
A remark on terminology is in place at this point.
Since the zeroth and first order will often be combined,
we will refer to equation (\ref{sseqfirstorder}) as describing the
`leading order'.
The terms $q_1$, $q_2$, \ldots in the
series (\ref{seriestwo}) will be referred to as `higher order' corrections.
\section{Stationary state to higher orders}
\label{secsecond}
The validity of the expansion procedure
of this section hinges on our being
able to show that the corrections take effectively
the form of the series of $q_k$ in (\ref{seriestwo}), where the terms
are proportional to increasing powers of $N^{-1/2}$.
\subsection{The perturbation series for $P_{\rm st}(r,s)$}
\label{secdefseries}
In order to show that the higher order corrections to $\Pst$ can be
expressed as the series of equation (\ref{seriestwo}),
we must first define the ${\cal W}_k$\, in equation
(\ref{splittwo}). Let us define $\delta w_j(r;s)$ by
\beq
w_j(r;s) = \wupone_j(r;s) + \delta w_j(r;s).
\label{defdeltaw}
\eeq
Starting from (\ref{defdeltaw}) we employ the explicit expressions
(\ref{defwrj}) and (\ref{xwupone}) for $w_j$ and $\wupone_j$,
respectively, perform a straightforward Taylor expansion in
$\delta\mu$, and still use (\ref{deffield}) to eliminate $m_0$ in
favor of $H_0$. This leads to
\bea
\delta w_j(r;s) &=& \wGj(r) \times \Big[ \tfrac{1}{2}[1-r_j\mu(s)\tanh\eta K]
- \tfrac{1}{2}[1-r_j\tanh\{H_0+g_0\delta\mu(s)\}] \Big]
\nonumber\\[2mm]
&=& \wGj(r) \times (-\tfrac{1}{2}r_j) \Big[ \{m_0+\delta\mu(s)\}\tanh\eta K
- \tanh\{H_0+g_0\delta\mu(s)\} \Big]
\nonumber\\[2mm]
&=& \wGj(r) \times (-\tfrac{1}{2}r_j) \sum_{k=2}^\infty a_k\,\delta\mu^k(s)
\nonumber\\
&=& \sum_{k=2}^\infty w_{j,k}(r;s),
\label{defwjk}
\eea
where the last equality, supposed to hold term by term in $k$,
defines $w_{j,k}$ and shows that it is of order $N^{-k/2}$.
In the third line of (\ref{defwjk})
the vanishing of the term linear in $\delta\mu$ has of course been
pre-arranged.
The first two coefficients $a_k$ in that line are given by
\bea
a_2 &=& g_0^2 (1-\tanh^2 H_0)\tanh H_0\,, \nonumber\\[2mm]
a_3 &=& \tfrac{1}{3} g_0^3 (1-\tanh^2 H_0)(1-3\tanh^2 H_0).
\label{xak}
\eea
It becomes clear now that there is a qualitative difference between the
high temperature regime $T\geq \Tc$ where we have
$H_0=0, a_2=0$, and
\beq
a_3=\frac{1}{3}\tanh^3\eta K, \qquad T\geq\Tc\,,
\label{xa3}
\eeq
and the low
temperature regime $T<\Tc$ where $H_0>0, a_2>0$.
We define the matrices ${\cal W}_k$ in expansion
(\ref{splittwo}) in terms of the $w_{j,k}$ by analogy to (\ref{relwW}).
Hence for $T\geq\Tc$ we have that ${\cal W}_2=0$.
\subsection{The higher order equations}
\label{sechigher}
The leading order equation (\ref{sseqfirstorder}) being satisfied,
we now turn to the higher orders. Substitution of (\ref{splittwo}) in
(\ref{defstst}) and use of (\ref{sseqfirstorder}) leads to an
expansion of which the first term is
\begin{subequations}\label{sseqnextorder}
\beq
\calWupone\Pstupone q_1 + {\cal W}_2\Pstupone = 0, \qquad T<\Tc\,.
\label{ssnextlow}
\eeq
In the high temperature phase the fact that
${\cal W}_2=0$ implies that $q_1$=0 and therefore
(\ref{ssnextlow}) is replaced by the next term in the expansion,
\beq
\calWupone\Pstupone q_2 + {\cal W}_3\Pstupone = 0, \qquad T\geq\Tc\,.
\label{ssnexthigh}
\eeq
\end{subequations}
Either will be referred to as the `next-to-leading order' equation.
One obtains
all higher-order equations in explicit form by inserting
in the full stationary state equation
(\ref{defPst}) the expansions
(\ref{seriestwo}) for $\Pst(r,s)$ and (\ref{defdeltaw})-(\ref{defwjk})
for $w_j(r;s)$.
\subsection{Equation for $T<\Tc$}
\label{seclowT}
By the procedure indicated above we obtain
for the next-to-leading order equation (\ref{ssnextlow})
the explicit form
\bea
0 &=& \sum_j \Big[ \wjtwo(r^j;s)\Pstupone(r^j,s) - \wjtwo(r;s)\Pstupone(r,s)
\nonumber\\
&& {}\phantom{XX} + \wjtwo(s^j;r)\Pstupone(r,s^j) - \wjtwo(s;r)\Pstupone(r,s)
\nonumber\\[2mm]
&& {}\phantom{XX} + \wupone_j(r^j;s)\Pstupone(r^j,s)q_1(r^j,s)
- \wupone_j(r;s) \Pstupone(r,s)q_1(r,s)
\nonumber\\[2mm]
&& {}\phantom{XX} + \wupone_j(s^j;r)\Pstupone(r,s^j)q_1(r,s^j)
- \wupone_j(s;r) \Pstupone(r,s)q_1(r,s) \Big].
\nonumber\\[2mm]
&& {}
\label{ststeqn3}
\eea
We wish to divide (\ref{ststeqn3})
by $\Pstupone(r,s)$ and therefore have to compute
\beq
\frac{ \Pstupone(r^j,s) }{ \Pstupone(r,s) } \equiv \ee^{-2R_j(r;s)}.
\label{defR}
\eeq
We easily find
\bea
2R_j(r;s) &=& [ \calHupone(r^j,s) - \calHupone(r,s) ]/T
\nonumber\\[2mm]
&=& [ {\cal H}_0(r^j,s) - {\cal H}_0(r,s) ]/T
-g_0N\delta\mu(s)[\delta\mu(r^j)-\delta\mu(r)]
\nonumber\\[2mm]
&=& -2Kr_j(r_{j-1}+r_{j+1}) -2r_j\{H_0+g_0\delta\mu(s)\},
\label{xR}
\eea
where we used (\ref{deffirstorder}) and (\ref{defH0}).
Detailed balancing says that
\beq
\wupone_j(r^j;s)\Pstupone(r^j,s)(r^j,s) = \wupone_j(r;s) \Pstupone(r,s).
\label{detbalW1}
\eeq
Using (\ref{xR}) in the first two lines
and (\ref{detbalW1})
in the last two lines of (\ref{ststeqn3}) we obtain
\bea
0 &=& \sum_j \Big[ \wjtwo(r^j;s)\ee^{-2R_j(r;s)} - \wjtwo(r;s)
\nonumber\\
&& {}\phantom{XX} + \wjtwo(s^j;r)\ee^{-2R_j(s;r)} - \wjtwo(s;r)
\nonumber\\[2mm]
&& {}\phantom{XX} + \wupone_j(r;s) \{q_1(r^j,s)-q_1(r,s)\}
\nonumber\\[2mm]
&& {}\phantom{XX} + \wupone_j(r;s) \{q_1(r,s^j)-q_1(r,s)\} \Big].
\label{ststeqn4}
\eea
The expression in the first line of (\ref{ststeqn4}) may be rewritten as
\beq
\begin{split}
\wjtwo(r^j;s)&\ee^{-2R_j(r;s)} - \wjtwo(r;s)\\[2mm]
& = \wGj(r) \times \tfrac{1}{2}[1-r_j\tanh H_0]\times
4r_j\delta\mu^2(s) g_0^2\tanh H_0\,,
\end{split}
\label{x1}
\eeq
of which the first two factors on the RHS
are again exactly $w_{j,0}$\,.
In (\ref{ststeqn4})
$\wupone_j$ is of order $N^0$ but contains corrections of higher order in
$N^{-1/2}$. In (\ref{ststeqn4}),
to leading order in $N^{-1/2}$, we may therefore replace it by its
$N\to\infty$ limit, that is, by $w_{j,0}$ defined by (\ref{defwrj0}).
When we substitute (\ref{x1}) in (\ref{ststeqn4})
and apply to $\wupone_j$ the $N\to\infty$ limit, we obtain the final form of the
equations for the next-to-leading order correction to the stationary state,
\bea
0 &=& \sum_j \Big[
\wjzero(r) \times 4r_j g_0^2\tanh H_0\,\delta\mu^2(s)
\nonumber\\
&& {}\phantom{XX} + \wjzero(s) \times 4s_j g_0^2\tanh H_0\,\delta\mu^2(r)
\nonumber\\[2mm]
&& {}\phantom{XX} + \wjzero(r)\{ q_1(r^j,s) - q_1(r,s) \}
\nonumber\\[2mm]
&& {}\phantom{XX} + \wjzero(s)\{ q_1(r,s^j) - q_1(r,s) \}
\Big].
\label{ststeqn5a}
\eea
\subsection{Equation for $T \geq \Tc$}
\label{sechighT}
For $T \geq \Tc$ we have $a_2=0$ whence $q_1=0$.
Equation (\ref{ssnexthigh}), when rendered explicit,
leads to expressions that are identical
to successively (\ref{ststeqn3}), (\ref{ststeqn4}), and (\ref{ststeqn5a})
apart from the substitutions
$q_1\mapsto q_2$ and $\wjtwo\mapsto w_{j,3}$.
In this case $\wjzero(r)=\tfrac{1}{2}\wGj(r)$ where
$\wGj(r)$ is given by (\ref{defwG}), and $\delta\mu=\mu$ since $H_0=m_0=0$.
Hence instead of (\ref{ststeqn5a}) we get
\bea
0 &=& \sum_j \Big[
\wGj(r) \times 4r_j a_3 \mu^3(s)
\nonumber\\
&& {}\phantom{XX} + \wGj(s) \times 4s_j a_3 \mu^3(r)
\nonumber\\[2mm]
&& {}\phantom{XX} + \wGj(r)\{ q_2(r^j,s) - q_2(r,s) \}
\nonumber\\[2mm]
&& {}\phantom{XX} + \wGj(s)\{ q_2(r,s^j) - q_2(r,s) \}
\Big].
\label{ststeqn5b}
\eea
Finding the solutions of (\ref{ststeqn5a}) and (\ref{ststeqn5b})
will be the subject of the next two subsections.
We will first consider the easier case of $T\geq\Tc$ and then the
case $T<\Tc$.
\subsection{Solution for $T \geq \Tc$}
\label{secsolhigh}
We start with the high temperature phase, where equation
(\ref{ststeqn5b}) applies.
Detailed balancing would be satisfied if the expression under the sum on $j$
were zero, that is, if we had
\bea
q_2(r^j,s) - q_2(r,s) &=& -a_3 r_j \mu^3(s),
\nonumber\\[2mm]
q_2(r,s^j) - q_2(r,s) &=& -a_3 s_j \mu^3(r).
\label{detbal5b}
\eea
It can easily be shown that it is impossible to satisfy these equations.
However, they suggest that we look for a solution $q_2$ of the form
\beq
q_2(r,s) = NC_2a_3[ \mu(r)\mu^3(s) + \mu(s)\mu^3(r) ]
\label{tryb}
\eeq
where only the constant $C_2$ is still adjustable.
The difference $q_2(r^j,s) - q_2(r,s)$ is easy to calculate, but we are
interested only in its leading order.
This leads to
\begin{subequations}\label{difftryb}
\bea
q_2(r^j,s) - q_2(r,s) &=& -2C_2 a_3 r_j [ \mu^3(s) + 3\mu(s)\mu^2(r) ]
+ {\cal O}(N^{-2}),\phantom{XXX}
\label{difftryb1}\\[2mm]
q_2(r,s^j) - q_2(r,s) &=& -2C_2 a_3 s_j [ \mu^3(r) + 3\mu(r)\mu^2(s) ]
+ {\cal O}(N^{-2}).\phantom{XXX}
\label{difftryb2}
\eea
\end{subequations}
It should be noted that whereas (\ref{tryb}) is of order $N^{-1}$,
the differences (\ref{difftryb}) are of order $N^{-3/2}$.
We now need
\beq
\sum_j \wGj(r) \{ q_2(r^j,s) - q_2(r,s) \} =
-2C_2a_3 \left( \sum_j \wGj(r)r_j \right) [ \mu^3(s) + 3\mu(s)\mu^2(r) ].
\label{sumwGj}
\eeq
With the aid of the explicit expression for $\wGj(r)$ one evaluates easily
\beq
\sum_j \wGj(r)r_j = \tfrac{1}{4}(1-\gamma)N\mu(r).
\label{sumb}
\eeq
we see that the equation is
satisfied for $C_2=\tfrac{1}{8}$. Hence from (\ref{tryb}) we get
\beq
q_2(r,s) = \tfrac{1}{24}N (\tanh^3\eta K) [ \mu(r)\mu^3(s) + \mu(s)\mu^3(r) ].
\label{q2final}
\eeq
This is of order $N^{-1}$.
\vspace{5mm}
\subsection{Solution for $T <\Tc$}
\label{secsollow}
In the low-temperature regime equation (\ref{ststeqn5a}) applies.
In order to solve this equation we now postulate
\beq
q_1(r,s) = NC_1b_2 [ \delta\mu^3(r) + \delta\mu^3(s) ]
\label{trya}
\eeq
where $C_1$ is an adjustable constant and
\beq
b_2 = 4a_2/(1-\tanh^2H_0) = 4g_0^2\tanh H_0\,.
\label{defb2}
\eeq
Expression (\ref{trya}) is of order $N^{-1/2}$.
Instead of (\ref{difftryb}) we now have the difference
\beq
q_1(r^j,s) - q_1(r,s) = -6C_1b_2 r_j \delta\mu^2(r) + {\cal O}(N^{-3/2}).
\label{difftrya}
\eeq
which is of order $N^{-1}$.
The first two lines of (\ref{ststeqn5a}) require that we evaluate
\beq
\sum_j \wjzero(r)r_j = \tfrac{1}{4} \sum_j
[ 1-\tfrac{1}{2}\gamma r_j( r_{j-1}-r_{j+1} )][ 1-r_j\tanh H_0 ]r_j
\label{suma0}
\eeq
Unlike the sum in (\ref{sumwGj}),
this is not a sum of zero-average random terms.
It will produce a result of order $N$, which we may replace by its average.
This yields
\bea
\sum_j \wjzero(r)r_j &=& \tfrac{1}{4}N[ (1-\gamma)m_0 -
(1-\gamma a_H) \tanh H_0 ]
\nonumber\\
&\equiv& NG,
\label{suma}
\eea
where the last equality defines $G$ and where $a_H$
is the nearest neighbor spin-spin correlation $\la r_jr_{j+1} \ra$
of a 1D Ising chain in a field as described by ${\cal H}_0$
[equation (\ref{defH0})].
Expression (\ref{suma}), contrary to its $T\geq \Tc$ counterpart
(\ref{sumb}), has no spin dependence and is therefore
equal for the $r$- and $s$- spins.
The first two lines of (\ref{ststeqn5a}), to be denoted $S_1$, become
\beq
S_1 = 4 NG g_0^2 \tanh H_0\, \big[ \delta\mu^2(s) + \delta\mu^2(r) \big].
\label{xS1}
\eeq
We use (\ref{difftrya}) to write the last two lines of (\ref{ststeqn5a}) as
\bea
S_2 &=& -6C_1b_2 \left[
\Big( \sum_j\wjzero(r)r_j \Big) \delta\mu^2(r)
+ \Big( \sum_j\wjzero(s)s_j \Big) \delta\mu^2(s)
\right] \nonumber\\[2mm]
&=& -6 NG C_1b_2\,\big[ \delta\mu^2(r) + \delta\mu^2(s) \big].
\label{xS2}
\eea
The stationary state equation (\ref{ststeqn5a})
may the be written as $S_1+S_2=0$
and we see that it is satisfied for $C_1=\tfrac{1}{6}$.
\beq
q_1(r,s) = \tfrac{2}{3}N g_0^2 \tanh H_0\, \big[ \delta\mu^3(r) +
\delta\mu^3(s) \big].
\label{q1final}
\eeq
\subsection{Section summary}
\label{sec}
We have studied in the preceding subsections the large-$N$ expansion
of the stationary state distribution $P_{\rm st}(r,s)$
of the infinite velocity CRIC defined in section
\ref{secdefmodel}.
We have shown, for $T<\Tc$ and $T\geq\Tc$ separately,
the existence of a series of correction terms $q_k$
that multiplies the leading order result $\Pstupone$ in
(\ref{seriestwo}),
which itself is again composed of a zeroth and a first order contribution.
This expansion also furnishes the necessary proof that
the prefactor $\Pstupone$
represents indeed the `leading order' behavior.
We have determined explicitly
the first nonzero correction term in this series:
$q_1$ for $T\geq\Tc$ and $q_2$ for $T<\Tc$.
When looking ahead beyond this leading order correction,
it appears that the $q_k$ (for $k\geq 2$ when $T<\Tc$
and for $k \geq 3$ when $T\geq\Tc$) involve not only $\delta\mu(r)$
and $\delta\mu(s)$, but also energy fluctuations such as
$N^{-1}\sum_j(r_jr_{j+1} - a_H)$, if not longer-range correlations.
Therefore, even though
on the basis of the results of this section
one might be tempted to postulate a general solution of the simple type
$\Pst(r,s) = \Pstupone(r,s)Q(\delta\mu(r),\delta\mu(s))$,
it is unlikely that the true $P_{\rm st}(r,s)$ is of this form.
\section{Stationary state averages}
\label{secaverages}
Stationary state averages of observables $A(r,s)$
are averages with respect to $\Pst(r,s)$, so that using (\ref{seriestwo})
and (\ref{deffirstP}) we have
\bea\label{defaverage}
\la A\ra &=& \frac{ \sum_{r,s}A(r,s)\ee^{-{\calHupone}(r,s)/T}
[ 1 + q_1(r,s) + q_2(r,s) + \ldots ] }
{ \sum_{r,s}\ee^{-{\calHupone}(r,s)/T}
[ 1 + q_1(r,s) + q_2(r,s) + \ldots ] }
\nonumber\\[2mm]
&=& \la A \raupone +[\la Aq_\ell\raupone -
\la A\raupone \la q_\ell\raupone] +\ldots,
\eea
where $\la \ldots \raupone$ indicates an average with weight
$\Pstupone(r,s)$
[equation (\ref{deffirstorder})],
the second line results from a straightforward expansion, and
\beq
\ell = \left\{
\begin{array}{ll}
2,\phantom{XXX} & T \geq \Tc\,, \\[2mm]
1, & T < \Tc\,,
\end{array}
\right.
\eeq
for the lowest order nonzero terms in the expansion.
Although the $q_k$ are accompanied by increasing powers of
$N^{-1/2}$, the order in $N^{-1/2}$ of each of the terms in the series
(\ref{defaverage}) must be analyzed for each observable $A$
separately.
\subsection{Integral representation of the partition function}
\label{secintrepr}
The denominator in the first line of (\ref{defaverage})
is a normalization factor to which we may refer
(although slightly improperly) as the partition function $Z$.
In order to find expressions for the averages $\la \ldots \raupone$
in the second line of (\ref{defaverage}),
we begin by evaluating $Z$ to leading order,
\beq
\Zstupone(K,H_0,g_0) \equiv \sum_{r,s} \ee^{-{\calHupone}(r,s)/T},
\label{defZeff}
\eeq
with $\calHupone$ given by (\ref{deffirstH}) in
which one should substitute (\ref{defH0}) and (\ref{defmurs}).
To this order (\ref{defZeff}) is a true partition function, {\it viz.}
the trace of a Boltzmann factor.
The notation $\Zstupone(K,H_0,g_0)$ is meant to indicate that we wish
to consider this quantity as a function of three independent parameters,
ignoring for the moment expression (\ref{deffirstg}) for $g_0$\,.
The $r$- and $s$-spins in (\ref{defZeff})
may be decoupled by the integral representation
\bea
\Zstupone &=& \frac{N}{\pi g_0}
\int_{-\infty}^{\infty}\!\dd x \int_{-\infty}^{\infty}\!\dd y\,\,
\ee^{ -g_0^{-1}{N}(x^2+y^2) }
\nonumber \\[2mm]
&& \times \left[ \ee^{-(x+{\rm i}y)Nm_0 }
\sum_r \ee^{K\sum_j r_j r_{j+1} + (H_0+x+{\rm i}y)\sum_j r_j} \right]
\nonumber\\[2mm]
&& \times \left[ \ee^{-(x-{\rm i}y)Nm_0 }
\sum_s \ee^{K\sum_j s_j s_{j+1} + (H_0+x-{\rm i}y)\sum_j s_j} \right]
\nonumber\\[2mm]
\label{intreprZeff}
\eea
in which $m_0=m(K,H_0)$ follows from (\ref{deffield}) and (\ref{solnH0}).
The two factors in brackets in (\ref{intreprZeff})
are seen to be the partition functions $\zeta(K,H_0 + x \pm {\rm i}y)$
of independent standard
Ising chains in magnetic fields $H_0 + x \pm {\rm i}y$. Hence
\beq
\Zstupone = \frac{N}{\pi g_0}
\int_{-\infty}^{\infty}\!\dd x \int_{-\infty}^{\infty}\!\dd y\,
\ee^{ -g_0^{-1}N(x^2+y^2) -2xNm_0 }
\big| \zeta(K,H_0+x+{\rm i}y) \big|^2.
\label{intZeff}
\eeq
We recall that
\beq
\zeta(K,B) \equiv \lambda_+^N + \lambda_-^N\,,
\label{xZeff}
\eeq
where
\beq
\lambda_\pm(K,B) = \ee^K \left[ \cosh B \pm \sqrt{ \sinh^2 B + \ee^{-4K} }
\right].
\label{eigtransf}
\eeq
are the transfer matrix eigenvalues.
\subsection{Stationary point and fluctuations}
\label{secstatpoint}
The $x$ and $y$ integrals in (\ref{intZeff}) are easily evaluated by
the saddle point meyhod,
In the limit of large $N$,
we may neglect in (\ref{xZeff})
the exponentially small corrections due to $\lambda_-$
and get from (\ref{intZeff})
\beq
\Zstupone \simeq \frac{N}{\pi g_0}
\int_{-\infty}^{\infty}\!\dd x \int_{-\infty}^{\infty}\!\dd y\,\,
\ee^{ -N {\cal F}(x,y) },
\label{xZeff2}
\eeq
where
\beq
{\cal F}(x,y)=
g_0^{-1} (x^2+y^2) +2xm_0
- \log\big|\lambda_+(K,H_0+x+{\rm i}y)\big|^2
\label{defcalF}
\eeq
Let $(x^*,y^*)$ denote the stationary point of the integration in
(\ref{xZeff2}).
The stationary point equations ${\cal F}_x={\cal F}_{y} =0$
can be expressed as
\beq
g_0^{-1}(x^*\pm{\rm i}y^*) = m(K,H_0+x^*\mp{\rm i}y^*) - m_0\,,
\label{statpoint}
\eeq
with the magnetization
$m(K,B)=\lambda_+^{-1}(K,B){\partial\log\lambda_+(K,B)}/{\partial B}$
given by (\ref{xmKz}).
For reasons of symmetry the stationary point
must have $y^*=0$.
This reduces (\ref{statpoint}) to the single real equation
\beq
g_0^{-1}x^* = m(K,H_0+x^*) - m(K,H_0),
\label{statpointreal}
\eeq
where we used that $m_0=m(K,H_0)$ [equation (\ref{relconst})].
Equation (\ref{statpointreal})
has for all $H_0$ the obvious solution $x^*=0$.
We investigate the stability of the stationary point $(x^*,y^*)$
by calculating the matrix of second derivatives,
\bea
&&{\cal F}_{xx}^* = 2[g_0^{-1} - \chi(K,H_0)], \qquad
{\cal F}_{yy}^* = 2[g_0^{-1} + \chi(K,H_0)], \nonumber\\[2mm]
&&{\cal F}_{xy}^* = {\cal F}^*_{yx} = 0,
\label{calFsecond}
\eea
where the asterisk indicates evaluation in the stationary point
and where $\chi(K,B)=\partial m(K,B)/\partial B$ is the magnetic
susceptibility. We obtain
the eigenvalues ${\cal F}_{xx}^*$ and ${\cal F}_{yy}^*$ explicitly
by substituting in (\ref{calFsecond}) for $g_0$
the expressions (\ref{deffirstg}) and for
$\chi$ the expression
\beq
\chi(K,B) =
\frac{\ee^{-4K}\cosh B}{ \left( \sinh^2 B + \ee^{-4K} \right)^{3/2} }\,,
\label{defchiKB}
\eeq
where (\ref{xmKz}) has been used. This yields
\beq
{\cal F}_{xx,yy}^* =
\left\{
\begin{array}{ll}
\dfrac{ 2(\ee^{-2K} \mp \tanh\eta K) }{\ee^{-2K}\tanh\eta K}, & T>\Tc\,, \\[4mm]
\dfrac{ 2 (1-\tanh^2\eta K) (\tanh^2\eta K \mp {\ee}^{-4K}) }
{ (1-{\ee}^{-4K}) \tanh^3\eta K }\,, & T<\Tc\,,
\end{array}
\right.
\label{xchilow}
\eeq
in which the upper (lower) sign refers to the $xx$ (to the $yy$) derivative.
It can be seen that ${\cal F}_{yy}^*$ is positive for all
temperatures, but that ${\cal F}_{xx}^*$\,,
which is positive in both the high and
the low-temperature phase, vanishes as $T\to\Tc$.
Hence for all $T\neq\Tc$ the stability is ensured by the quadratic terms
in the expansion of ${\cal F}(x,y)$ around the stationary point.
\subsection{Free energy}
We are now in a position to calculate various physical quantities of interest.
The first one will be the
{\it interaction\,} free energy per spin between the two
chains which (divided by $T$)
will be called $F_{\rm int}$. It will turn out to have an expansion
\beq
F_{\rm int} = F_{\rm int}^{(0)} + N^{-1}f_{\rm int} +\ldots
\label{formFint}
\eeq
To show this we pursue the calculation of $\Zstupone$
begun in (\ref{xZeff2}). We there substitute the expansion
\beq
{\cal F}(x,y) = {\cal F}^* + \tfrac{1}{2}{\cal F}_{xx}^* x^2
+ \tfrac{1}{2}{\cal F}_{yy}^* y^2 + \ldots\,.
\label{expcalF}
\eeq
We can then carry out the integrations in (\ref{xZeff2}) by the saddle point
method and find that only the quadratic terms in
(\ref{expcalF}) contribute.
The result has the form
\beq
\Zstupone \simeq \ee^{-N{\cal F}^* - f_{\rm int}}
\left[1+{\cal O}(N^{-1})\right]
\label{resZeff}
\eeq
where
\beq
N{\cal F}^*=N{\cal F}(0,0)=-2N\log\lambda_+(K,H_0)
\label{xcalFstar}
\eeq
and
\beq
f_{\rm int}(K,\eta) = \tfrac{1}{2}\log\left[ 1-g_0^2\chi^2(K,H_0) \right].
\label{xfint}
\eeq
Here ${\cal F}^*$ is the free energy (divided by $T$)
of two independent Ising chains in an effective
field $H_0$.
Since $H_0$ is proportional to the coupling $\eta K$
between the chains,
the field dependent
part of ${\cal F}^*$ actually represents the bulk interaction free energy
$NF^{(0)}_{\rm int}$ between the chains, that is,
\beq
NF^{(0)}_{\rm int}(K,\eta) =
\left\{
\begin{array}{ll}
0, & T\geq\Tc\,,\\[2mm]
-2N\log\left( \dfrac{\lambda_+(K,H_0)}{\lambda_+(K,0)} \right), & T<\Tc\,;
\end{array}
\right.
\label{xxFint}
\eeq
and furthermore $f_{\rm int}(K,\eta)$
is a residual interaction free energy between them which remains of
order $N^0$ as $N\to\infty$.
The energy that one drives from it has a cusp singularity and hence
the exponent $\alpha=0$ \cite{Hucht09}.
Beyond this leading order result
we obtain $f_{\rm int}$ explicitly in terms of the two system parameters
$K$ and $\eta$ by substituting in (\ref{xfint}) the expressions for
$g_0$ and $\chi$ given in (\ref{deffirstg}) and (\ref{defchiKB}),
respectively, and (when $T<\Tc$) eliminating $H_0$. The result is that
\bea
f_{\rm int}(K,\eta) &=& \left\{
\begin{array}{ll}
\tfrac{1}{2}\log\left( 1-{\ee}^{4K}\tanh^2\eta K \right), & T >\Tc\,,\\[2mm]
\tfrac{1}{2}\log\left( 1-{\ee}^{-8K}\tanh^{-4}\eta K \right), & T <\Tc\,.
\end{array}
\right.
\label{xxfint}
\eea
In view of (\ref{xxFint})
we see that $F_{\rm int}$ has a linear cusp at $T=\Tc$, and
(\ref{xxfint}) shows that
$f_{\rm int}$ diverges logarithmically for
$T\to\Tc$. In spite of this weak divergence,
the finite size correction $f_{\rm int}$
to the interaction free energy $F_{\rm int}$
also conforms the classical specific heat exponent
$\alpha=0$.
\subsection{Finite size scaling of the free energy near $\Tc$}
\label{fsscaling}
We will show how our approach allows for
finding the finite size scaling functions.
By the way of an example we consider the singular part of the
free energy. For $T\to\Tc$ the quantity $f_{\rm int}$ diverges
due to the second order derivative ${\cal F}_{xx}^*$ becoming zero.
In order
for the integral (\ref{xZeff2}) combined with (\ref{expcalF})
to converge at $T=\Tc$, we have to include higher order
terms in the expansion (\ref{expcalF}).
We will write
\beq
{\cal F}(x,y) = {\cal F}^*
+ \tfrac{1}{2}{\cal F}_{xx}^* x^2
+ \tfrac{1}{2}{\cal F}_{yy}^* y^2
+ \tfrac{1}{6}{\cal F}_{xxx}^* x^3
+ \tfrac{1}{24}{\cal F}_{xxxx}^* x^4 + \ldots
\label{expcalFTc}
\eeq
and will argue below
that near $\Tc$ the terms not exhibited explicitly in this
series are of higher order
\footnote{Terms with an odd number of $y$ derivations vanish by symmetry.}.
In order to find the coefficients in (\ref{expcalFTc}
we perform a
straightforward derivation of (\ref{defcalF}) and set $x^*=y^*=0$.
We then define
\beq
\epsilon=\frac{T-\Tc}{\Tc}=-\frac{K-\Kc}{\Kc}
\label{defeps}
\eeq
which, in the vicinity of $\Tc$\,, leads to
\beq
H_0 = \Bc\,\epsilon^{1/2} + {\cal O}(\epsilon)
\label{xKHepsilon}
\eeq
where from (\ref{solnH0}) we have
\beq
\Bc^2 = \left\{
\begin{array}{ll}
0, & T>\Tc\,,\\[2mm]
2\ee^{-2\Kc}
\big( \eta + 1/\sinh 2\Kc \big)\Kc\,,
& T<\Tc\,.
\end{array}
\right.
\label{defBc}
\eeq
When using (\ref{xKHepsilon}) in the coefficients found above
we obtain
\bea
{\cal F}^*_{xx} &=& a_{\pm}\epsilon + {\cal O}(\epsilon^2),
\nonumber\\[2mm]
{\cal F}^*_{yy} &=& 4\ee^{2\Kc} + {\cal O}(\epsilon),
\nonumber\\[2mm]
{\cal F}^*_{xxx} &=& b_\pm(-\epsilon)^{1/2} + {\cal O}(\epsilon^{3/2}),
\nonumber\\[2mm]
{\cal F}^*_{xxxx} &=& c +{\cal O}(\epsilon),
\label{coefficients}
\eea
where
\bea
a_\pm &=& \left\{
\begin{array}{r}
2\\[2mm]
4
\end{array}
\right\} \big( \ee^{4\Kc}-1 \big)
\big( \eta + 1/\sinh 2\Kc \big)\Kc\,,
\qquad
\begin{array}{l}
T>\Tc\,,\\[2mm]
T<\Tc\,.
\end{array}
\nonumber\\[2mm]
b^2_\pm &=& \left\{
\begin{array}{ll}
0, & T>\Tc\,,
\nonumber\\[2mm]
4 \big( 3\ee^{4\Kc}-1 \big)
\big( \eta + 1/\sinh 2\Kc \big)\Kc\,,\quad{}
& T<\Tc\,.
\end{array}
\right.\\[2mm]
c &=& 6\ee^{2\Kc},
\label{defabc}
\eea
We substitute the explicit expressions (\ref{coefficients}) in
(\ref{expcalFTc}) and use that expansion in the integral
(\ref{xZeff2}). When we introduce the scaled variables of integration
$u$ and $v$ defined by
\beq
x=N^{-1/4}u, \qquad y=N^{-1/2}v,
\label{scaledvar}
\eeq
as well as the scaling variable
\beq
\tau=\epsilon N^{1/2},
\label{deftau}
\eeq
the factor $N$ disappears from the exponential. After carrying out
the Gaussian integration on $v$ we get
\beq
\Zstupone \simeq \ee^{-N{\cal F}^*} \times
\frac{N^{1/4}\ee^{\Kc}}{\sqrt{2\pi}}
\,{\cal Z}(\tau),
\label{xZKc}
\eeq
valid in the scaling limit $N\to\infty$, $T\to\Tc$ with $\tau$ fixed,
and where ${\cal Z}$ is the scaling function
\beq
{\cal Z}(\tau) = \int_{-\infty}^{\infty}\!\dd u\,
\exp\left[-\tfrac{1}{2} a_\pm|\tau| u^2
-\tfrac{1}{6} b_\pm(-\tau)^{1/2}u^3
-\tfrac{1}{24} cu^4 \right].
\label{defcalZ}
\eeq
It is of a type that occurs
standardly in problems with mean field type critical
behavior; they have been studied recently by Gr\"uneberg and Hucht
\cite{GrunebergHucht04}.
It has the limiting behavior
\beq
{\cal Z}(\tau) \simeq
\left\{
\begin{array}{ll}
{\cal Z}(0) \equiv
\int_{-\infty}^\infty\!\dd u\,\ee^{-cu^4/24}, & \tau\to 0, \\[4mm]
\left( \dfrac{2\pi}{a_\pm}|\tau| \right)^{1/2}\,,& \tau\to\pm\infty.
\end{array}
\right.
\label{xcalZlimit}
\eeq
Upon combining (\ref{resZeff}) and (\ref{xZKc}) we find that
\bea
f_{\rm int}(K,\eta) = -\tfrac{1}{4}\log N
-\log {\cal Z}(tN^{1/2})
-\tfrac{1}{2}\log\left( \dfrac{\ee^{2\Kc}}{2\pi} \right) + \ldots,
\label{scalingfint}
\eea
again valid in the scaling limit,
and where the dots stand for terms that vanish as $N\to\infty$.
It follows, in particular, that equation (\ref{xxfint}) may
now be completed by
\beq
f_{\rm int}(\Kc,\eta) \simeq \tfrac{1}{4}\log N + \log{\cal Z}(0) + \ldots,
\qquad T=\Tc\,, \quad N\to\infty,
\label{xxfintTc}
\eeq
where the dots stand for terms that vanish as $N\to\infty$.
\subsection{Susceptibilities}
\label{secsusceptibilities}
Of primary interest are the correlations between the fluctuations of
the magnetizations in the two chains. We set as before $\delta\mu=\mu-m_0$.
The general expression that we
will study here is
\bea
\chi_{k\ell}&\equiv& \la \delta\mu^k(r)\delta\mu^\ell(s)\ra
\nonumber\\[2mm]
&=&\la \delta\mu^k(r)\delta\mu^\ell(s)\raupone + \ldots\,
\label{defchimn}
\eea
where the dots in the last line, obtained according to (\ref{defaverage}),
represent higher order terms.
Special cases that we will consider
are the cross-chain susceptibility $\chi_{\rm int}$
and the single-chain susceptibility $\chi_{\rm sin}$, defined as
\begin{subequations}\label{chispecial}
\bea
\chi_{\rm int} &=& N\chi_{11}=N\la\delta\mu(r)\delta\mu)(s)\ra,
\label{chispecialint}\\[2mm]
\chi_{\rm sin} &=& N\chi_{20}=N\la\delta\mu^2(r)\ra,
\label{chispecialsin}
\eea
\end{subequations}
in which, of course, the latter is also equal to $\chi_{02}$ by symmetry.
\subsubsection{Cross-susceptibility}
\label{seccrosssusc}
We first consider the correlations between the
fluctuating magnetizations of the two chains.
The cross-susceptibility $\chi_{\rm int}$
is the quantity most characteristic of these correlations.
From equations (\ref{deffirstH}) and (\ref{defZeff}) it is clear
that $\chi_{\rm int}=\partial\log\Zstupone/\partial g_0$
where the derivative has to be evaluated at fixed $K$ and $H_0$,
considering $g_0$ as an independent parameter in (\ref{intreprZeff}).
Doing the calculation for $\Zstupone$ given by (\ref{resZeff}),
(\ref{xcalFstar}), and (\ref{xfint}), we observe that
${\cal F}^*={\cal F}(0,0)$ is independent of $g_0$ so that
\bea
\chi_{\rm int}(K,\eta) =\frac{\partial f_{\rm int}}{\partial g_0}
&=& \frac{g_0\chi^2}{1-g_0^2\chi^2} \nonumber\\[2mm]
&=& \left\{
\begin{array}{ll}
\dfrac{ \tanh\eta K }{ \ee^{-4K}-\tanh^2\eta K } & T>\Tc\,,\\[4mm]
\dfrac{ {\rm e}^{-8K} (1-\tanh^2\eta K) }
{ (\tanh\eta K) (1-{\rm e}^{-4K})
(\tanh^4\eta K-{\rm e}^{-8K})} & T<\Tc\,.
\end{array}
\right.
\label{reschiint}
\eea
For $T\to\Tc$ this quantity diverges as
$|T-\Tc|^{-\gamma_{\rm int}}$
with
$\gamma_{\rm int}=1$.
It is a signal that at $T=\Tc$ this correlation scales with another
power of $N$. A scaling function for $\chi_{\rm int}$ may be derived from
the one for $f_{\rm int}$\,, but
we will not try to be exhaustive.
Since at speed $v=\infty$ all index pairs $(i,j)$ are equivalent,
the correlations between the $r$- and the $s$-spins are given by
\beq
\langle r_i s_j \rangle -m_0^2 =N^{-1}\chi_{\rm int}(K,\eta).
\label{xrisjav}
\eeq
\subsubsection{Single-chain chain susceptibility}
\label{secsinglesusc}
The single-chain susceptibilities $\chi_{\rm sin}$
is defined in equation (\ref{chispecial}).
Let us now consider the general expression (\ref{defchimn})
for $\chi_{k\ell}$\,, for which the appropriate
approach differs slightly from that of the preceding subsection.
One may generate insertions $\delta\mu^k(r$ [or $\delta\mu^\ell(s)$]
in the integral (\ref{intZeff}) by
passing from $x$ and $y$ to the two
independent variables $z=(x+{\rm i}y)$ and $\bz=(x-{\rm i}y)$ and
letting $N^{-k}\partial^k/\partial z^k$
[or $N^{-\ell}\partial^\ell/\partial \bz^\ell$]
act on $\ee^{-2zNm_0}Z(K,H_0+z)$ [or on $\ee^{-2\bz Nm_0}Z(K,H_0+\bz)$].
We find, using (\ref{xZeff}) and neglecting again the effect of $\lambda_-$
which is exponentially small in $N$,
\beq
N^{-k}\frac{\partial^k}{\partial z^k}\,\,
\big[ \ee^{-zNm_0}Z(K,H_0+z) \big] = J_k(z)\,Z(K,H_0+z),
\label{partialkz}
\eeq
in which
\bea
J_0(z)&=& 1, \nonumber\\[2mm]
J_1(z)&=& \tm-m_0\,, \nonumber\\[2mm]
J_2(z) &=& (\tm-m_0)^2+N^{-1}\tchi, \nonumber\\[2mm]
J_3(z) &=& (\tm-m_0)^3 +3N^{-1}(\tm-m_0)\tchi +N^{-2}\tchi', \nonumber\\[2mm]
J_4(z) &=& (\tm-m_0)^4 + 6N^{-1}(\tm-m_0)^2\tchi + 4N^{-2}(\tm-m_0)\tchi'
\nonumber\\[2mm]
& & +3N^{-2}\tchi^2 + N^{-3}\tchi^{\prime\prime},
\label{defJkz}
\eea
where, in this formula,
we abbreviated $\tm=m(K,H_0+z)$ and $\tchi=\chi(K,H_0+z)$
[see equations (\ref{xmKz}) and (\ref{defchiKB})]
in order to emphasize the $z$ dependence of these quantities,
and where the
primes on $\tchi$ stand for differentiations with respect to $H_0$.
Equations (\ref{partialkz}) and (\ref{defJkz}) of course have
counterparts obtained by
letting $r\mapsto s$, $k\mapsto\ell$ and $z\mapsto\bar{z}$.
When (\ref{partialkz}) is substituted in (\ref{defchimn}) we obtain
\beq
\chi_{k\ell} = \la J_k(z)J_{\ell}(\bar{z})\raupone + \ldots,
\label{xchimnJ}
\eeq
where the dots stand for higher-than-leading order terms in the
$N^{-1}$ expansion.
By virtue of equations (\ref{xchimnJ}) and (\ref{defJkz})
it follows that
\bea
\chi_{20} &=& \la J_2(z)\raG \nonumber\\[2mm]
&=& \la \big( m(K,H_0+z)-m_0 \big)^2 \ra
+ N^{-1}\la\chi(K,H_0+z)\ra
\label{x2chi20}
\eea
We now expand $m$ and $\chi$ for small $z$
anticipating that upon integration
with weight $\exp(-N{\cal F})$ each factor $z^2$
will, to leading order,
produce a factor $N^{-1}$. After multiplication by $N$ this yields
\beq
\chi_{\rm sin}(K,\eta)
= N\chi^2(K,H_0)\la z^2\raG + \chi(K,H_0) +{\cal O}(N^{-1}).
\label{x2chi20bis}
\eeq
Anticipating again that each factor $z$ or $\bz$ will produce a
factor $N^{-1/2}$, we see that all terms exhibited explicitly on the
right hand sides in (\ref{xavJ}) are of order $N^{-1}$.
We have replaced the averages $\la\ldots\raupone$, which are with
respect to $\exp(-N{\cal F}(x,y)$,
by averages $\la\ldots\raG$ in which ${\cal F}(x,y)$ of equation (\ref{defcalF})
is replaced with the Gaussian terms in its expansion, shown in (\ref{expcalF}).
Upon using in (\ref{x2chi20}) the explicit evaluations
\bea
\la z^2 \raG &=& \la x^2 \raG - \la y^2 \raG \nonumber\\[2mm]
&=& \frac{1}{N}\left( \frac{1}{{\cal F}_{xx}^*} - \frac{1}{{\cal F}_{yy}^*}
\right) =
\frac{g_0^2 \chi}{N(1-g_0^2\chi^2)}\,, \qquad T\neq\Tc\,.
\label{xavzz}
\eea
we arrive at
\beq
\chi_{\rm sin}(K,\eta)
= \frac{\chi}{1-g_0^2\chi^2}\,, \qquad T\neq \Tc\,,
\label{x3chi20}
\eeq
valid in the limit $N\to\infty$.
Hence the in-chain susceptibility $\chi_{\rm sin}$
is equal to the susceptibility of the 1D Ising model enhanced
by a factor $(1-g_0^2\chi^2)^{-1}$ due to
the presence of the other chain.
Using expressions (\ref{deffirstg}) and (\ref{defchiKB})
for $g_0$ and $\chi$, respectively,
we may render (\ref{x3chi20})
explicit in terms of $K$ and $\eta$ and get
\beq
\chi_{\rm sin}(K,\eta) =
\left\{
\begin{array}{ll}
\dfrac{ \ee^{-2K} }{ \ee^{-4K}-\tanh^2\eta K }\,,
\phantom{XXX} & T>\Tc\,,
\\[4mm]
\dfrac{ {\rm e}^{-4K} (\tanh\eta K) (1-\tanh^2\eta K) }
{ (1-{\rm e}^{-4K}) (\tanh^4\eta K-{\rm e}^{-8K})}\,, & T<\Tc\,.
\end{array}
\right.
\label{x3chi20bis}
\eeq
For $T\to\Tc$ the susceptibility $\chi_{\rm sin}$
diverge as $(T-\Tc)^{-\gamma}$ with, again, the classical critical
exponent $\gamma=1$.
For $\eta=0$ (whence $\Tc=0$) the first one of
equations (\ref{x3chi20bis}) reduces to
the standard susceptibility of the zero field 1D Ising chain.
In agreement with the symmetry of the problem, $\chi_{\rm int}$ is odd
and $\chi_{\rm sin}$ is even in $\eta$.
Both above and below $\Tc$ one easily verifies that
in agreement with Schwarz's inequality
we have $\chi_{\rm int}/\chi_{\rm sin}\leq 1$.
\subsection{Spontaneous magnetization}
\label{secspontmagn}
For $T\geq\Tc$ symmetry dictates that the magnetization
$\la\mu(r)\ra$ and $\la\mu(s)\ra$ are zero to all orders.
However, for $T<\Tc$ the magnetization
$\mu(r)=N^{-1}\sum_{j=1}^Nr_j$ has, to leading order, a Gaussian
probability distribution of width $N^{-1/2}$ around $m_0(K,H_0)$.
As a consequence
$\la\delta\mu(r)\ra$ vanishes to order $N^{-1/2}$.
However, to order $N^{-1}$ there appear
nonzero corrections terms to $\la\mu(r)\ra$.
As an application of equation (\ref{defaverage}) we calculate in this
subsection these correction terms.
Upon using (\ref{defaverage}) for the spacial case $A=\delta\mu(r)$
and inserting in it the explicit expression
(\ref{q1final}) for $q_1$ we obtain
\beq
\la\delta\mu(r)\ra = \la\delta\mu(r)\raupone + \tfrac{2}{3}Ng_0^2\tanh H_0
\left[ \la J_4(z)\raupone \,+\,
\la J_1(z)J_3(\bz)\raupone \right].
\label{xmurav}
\eeq
When substituting (\ref{defJkz}) in the second term of
(\ref{xmurav}) we see that we need
\bea
\la J_4(z)\raupone &=& \chi^4\la z^4\raG +6N^{-1}\chi^3\la z^2\raG
+3N^{-2}\chi^2 + {\cal O}(N^{-5/2}),
\nonumber\\[2mm]
\la J_1(z)J_3(\bz)\raupone &=& \chi^4\la z\bz^3\raG
+3N^{-1}\chi^3\la z\bz\raG + {\cal O}(N^{-5/2}).
\label{xavJ}
\eea
We have replaced the averages $\la\ldots\raupone$
by averages $\la\ldots\raG$ for the same reasons as in the preceding
subsection. Taking into account again that each factor $z$or $\bar{z}$
brings in a power $N^{-1/2}$, we see that all terms explicitly
exhibited on the right hand sides of equations (\ref{xavJ})
are of the same order in $N$, namely
${\cal O}(N^{-2})$.
The Gaussian averages are easily calculated and we are led to
\beq
\la J_4(z)\raupone \,+\, \la J_1(z)J_3(\bz)\raupone =
\frac{ 3\chi^3(\chi+g_0) }{ N^2(1-g_0^2\chi^2)^2 } + {\cal O}(N^{-5/2}).
\label{finJav}
\eeq
We should now evaluate the first term on the right hand side of
(\ref{xmurav}), namely
\beq
\la\delta\mu(r)\raupone = \la J_1(z)\raupone = \chi\la z\raupone.
\label{zavfull1}
\eeq
The Gaussian average $\la z\raG$
vanishes on account of symmetry. However, when
the third order terms in the Taylor expansion (\ref{expcalF})
of ${\cal F}(x,y)$
are kept and we expand these
we get after a straightforward
calculation that we will not reproduce here,
\bea
\la\delta\mu(r)\raupone &=& \
\tfrac{1}{3}N\chi\chi'\big[ \la x^4\raG - 3\la x^2y^2\raG \big]
\nonumber\\[2mm]
&=& N\chi\chi'\la x^2\raG\big[ \la x^2\raG -
\la y^2\raG \big] +{\cal O}(N^{-2})
\nonumber\\[2mm]
&=&\frac{g_0^3\chi^2\chi'}{2N(1-g_0\chi)^2(1+g_0\chi)} + {\cal O}(N^{-2}).
\label{zavfull2}
\eea
The final result for $\la\delta\mu(r)\ra$ is obtained by substitution
of (\ref{zavfull2}) and (\ref{finJav}) in (\ref{xmurav}).
We see that $\la\delta\mu(r)\ra$ has two contributions of order $
N^{-1}$. The contribution $\la\delta\mu(r)\raupone$ comes from the
effective leading order Hamiltonian $\calHupone$.
The second contribution accompanies the violation of detailed
balancing symmetry and is therefore essentially a non-thermodynamic effect.
\subsection{Pair correlation function}
\label{secpaircorrelation}
It is of interest to study the pair correlation
\beq
g_N(\ell)\equiv\la r_jr_{j+\ell}\ra
\label{defgNell}
\eeq
in a single chain. To that end we
consider again expansion (\ref{defaverage}),
now with $A=r_jr_{j+\ell}$. Its first term may be written
\beq
\gupone_N(\ell) = {\Zstell}/{\Zstupone}
\label{defgNell1}
\eeq
where $\Zstell$ is given by (\ref{intreprZeff}) but with an
insertion $r_jr_{j+\ell}$ in the sum on $r$. Equivalently, $\Zstupone$
is given by the
same integral as (\ref{xZeff2}) but with an insertion
$\tgupone_N(\ell;K,H_0+z)$, this quantity being
the pair correlation of the 1D Ising chain
in a field $H_0+x+{\rm i}y$.
Evaluation by means of the standard transfer matrix method yields
\beq
\tgupone_N(\ell;K,H_0+z) = m^2(K,H_0+z)
+\,\frac{\ee^{-4K}\,\tLambda^\ell(K,H_0+z)}{\sinh^2(H_0+z)+\ee^{-4K}}\,,
\label{xtgamma}
\eeq
well-known in the case $z=0$, in which we defined
$\tLambda = {\lambda_-}/{\lambda_+}$\,,
where the tilde serves as a reminder of the $z$ dependence,
and where contributions exponentially small in $N$ have again been neglected.
In order to obtain the desired physical correlation function
$g_N(\ell)$ of this system we now have to average (\ref{xtgamma})
with an appropriately normalized weight
$\exp\big[ -N{\cal F}(x,y) \big]$.
We will consider this quantity in the high-temperature regime $T>\Tc$
where $H_0=0$.
Knowing that $z$ is of order $N^{-1/2}$ we expand
(\ref{xtgamma}) for small $z$, which gives
\bea
\tgupone_N(\ell;K,H_0+z) &=& \nonumber\\[2mm]
\ee^{4K}z^2 &+& (\tanh^\ell K)(1-\ee^{4K}z^2)
\exp\left( -(\ee^{-4K}+\ee^{2K}\ell)z^2 \right) + {\cal O}(N^{-2}).\\
&&
\label{xtgammaexp}
\eea
To leading order the average on $z$ may be carried out with the weight
$\exp\big[ -N{\cal F}(x,y) \big]$ in which the expansion ${\cal F}$
is limited to its quadratic terms.
Straightforward calculation yields
\beq
g_N(\ell)= \tanh^\ell K
+ (1-\tanh^\ell K)\frac{g_0^2\chi^3}{1-g_0^2\chi^2}\,N^{-1} + {\cal O}(N^{-2}),
\qquad T>\Tc\,,
\label{resgNell2}
\eeq
valid for $N\to\infty$ at fixed $\ell$,
where as before $\chi$ stands for the susceptibility $\chi(K,0)=\ee^{2K}$
of the 1D Ising chain and where $g_0=\tanh\eta K$.
In the scaling limit $\ell,N\to\infty$ with a fixed ratio one obtains
\beq
g_N(\ell) \simeq
(\tanh^\ell K) \phi(\ell N^{-1}) + \frac{g_0^2\chi^3}{1-g_0^2\chi^2}\,N^{-1},
\quad T>\Tc\,,\quad \ell,N\to\infty,
\label{resgNell1}
\eeq
in which each of the two terms is valid
up to corrections of relative order $ N^{-1}$
and in which $\phi$ is the scaling function defined by
\beq
\phi^2(x)=\frac{1-g_0^2\chi^2}
{1-g_0^2\chi^2 +2g_0^2\chi^2 x}\,.
\label{defphi}
\eeq
We observe the noncommutativity
\beq
\lim_{N\to\infty} \sum_{\ell=-N/2+1}^{N/2}g_N(\ell)
\neq \sum_{\ell=-\infty}^{\infty}\lim_{N\to\infty}g_N(\ell).
\label{noncomm}
\eeq
The right hand side of this inequality is equal to $\chi(K,0)$
whereas the right hand side is equal to
$\chi(K,0)+\chi_{\rm int}(K,\eta)$.
We conclude by noting that the pair correlation function may also be studied
to higher order in $N^{-1}$ in the low-temperature regime. For $T<\Tc$
the fluctuations of the magnetic field $z$ are asymmetric and greater
care is required. We will not include such a calculation here.
\section{Traffic model}
\label{sectraffic}
Motivated by an interest very different from that of references
\cite{Hucht09,Kadauetal08}
we recently introduced a new traffic model describing
vehicles that may overtake each other
on a road with two opposite lanes \cite{ARHS10}.
That work shows the appearance of a phase transition when the traffic
intensity, supposed equal on the two lanes, attains a critical value.
Above the critical intensity the symmetry between the two traffic lanes is
broken: one lane has dense and slow, the other one dilute
and fast traffic.
The study of reference \cite{ARHS10}
invoked a mean-field-type assumption that
couples the velocity of a vehicle in a given lane to the
{\it average\,} of the vehicle velocities in the opposite lane.
This assumption was justified by the argument that
a vehicle in one lane
encounters, in the course of time, all vehicles in the opposite lane.
Although there is no one-to-one correspondence between the two models,
they share essentially the same features, as may be seen as
follows.
For $J_2<0$ the two chains of the CRIC studied here
have opposite spontaneous
magnetizations; up-spins may then be regarded as the vehicles of the
traffic problem; they will be denser in one chain (traffic lane) than in
the other.
The CRIC is more amenable to analysis than the traffic model.
It was shown analytically \cite{Hucht09,Kadauetal08}
that the CRIC phase transition
disappears when $v$ is finite.
Our simulations \cite{ARHS11} of the traffic model have shown,
nevertheless, that this problem is
close to the critical point $v=\infty$. This explains the
critical-point-like phenomena
that we observed, namely
fluctuations that last longer than the simulation time.
\section{Conclusion}
\label{secconclusion}
We have considered in this paper the nonequilibrium steady state (NESS)
of a model consisting of two counter-rotating interacting Ising
chains introduced by Kadau {\it et al.} \cite{Kadauetal08} and by
Hucht \cite{Hucht09}. The model is
related to a road traffic model studied earlier by ourselves \cite{ARHS10}.
Its dynamics is governed by a master equation parametrized by two
interaction constants $J/T$ and $\eta$.
The model has a phase transition, known to be of mean field type,
at a critical temperature $T=\Tc$\,.
Starting from the master equation
we have shown that in the limiting case
of a relative velocity $v=\infty$ of the two chains,
the stationary state distribution $P_{\rm st}$ may be studied
in an expansion in powers of the inverse system size $N^{-1}$.
Knowing this distribution we have calculated, also as
expansions in $N^{-1}$, of averages of physical interest:
the interaction free energy between the chains, the in-chain and
cross-chain susceptibilities, the correlation function (for $T>\Tc$),
and the spontaneous magnetization (for $T<\Tc$).
We have shown how
near criticality scaling functions may be explicitly calculated.
Whereas to leading order the force exerted by
one chain on the other is that of
an effective magnetic field $H_0$,
the $N^{-1}$ expansion requires that we take into account the
fluctuations of this field around its average.
It then appears that to leading order the dynamics obeys detailed
balancing with respect to an effective Hamiltonian, as was found by
Hucht \cite{Hucht09}, but that to higher order in the expansion the
detailed balancing is violated.
In this work we have addressed many different, albeit interrelated,
aspects of the finite-size CRIC.
We have not tried to be exhaustive and have not considered, for example,
energy dissipation.
Similarly, the parallel problem with open boundary conditions
has been left aside.
We hope that the results of this work will be helpful in guiding the study,
which we believe to be worthwhile,
of the finite-velocity ($v<\infty$) version of the model.
\section*{Acknowledgments}
The author thanks C\'ecile Appert-Rolland and Gr\'egory Schehr
(Orsay, France)
and Samyr J\'acobe (UFRN, Natal, Brazil) for
discussions on and around the subject of this work.
\appendix | 205,908 |
: EVP’
Podcast: The Great Debate
| 323,135 |
So a few things are happening in my life right now. I don't really get into too much detail because I don't think I'm comfortable enough sharing really intimate details of my life. Although I do divulge more in personal emails and letters.
Here's a small life update...
♥ My oldest brother got remarried at the registry office yesterday! They will have a ceremony later on in the year.
♥ My new nephew Micah will be born towards the end of this month! So I will be an aunty for the third time!
♥ Jason's car got side swiped (by a truck we presume) while it was parked at the train station. It's not worth fixing as it's really old but we were wanting to get a new car by mid year. Instead we have to get one now (and I will be able to drive it at least since we will get an automatic - looking at a Mazda 3)
♥ I really need a new job because I'm getting no hours at my current job and it's making it difficult to do anything.
♥ Trying not to get all the negative things get me down so I'm holding onto God even more (and I feel it's a test of faith at the moment).
♥ Heartened by all the encouraging thoughts and messages I receive.
And I'm finally making items for my shop again. Concentrating on making things keeps me focused.
I haven't decided how I will use these. On a headband but maybe a thicker elastic one or a hard headband?
A few with my new fabric.
A new lot with coloured skinny elastic headbands.
I hope you like them. I might have a shop coupon code soon once I start putting more things back into the shop.
Have a wonderful rest of week.
20 comments:
Keep your chin up girl! Congrats to your brother and on becoming an aunt again! So exciting.
Love the new items for your shop, especially the ones in the first picture.
My boyfriend has a Mazda 3! It's snazzy :)
Hi Gracie!! First of all, very sorry to hear about the truck and the job. It sucks either way, but I hope something good comes out of it all.
Secondly, your yummy etsy items look fab and I am going to check out more as soon as I am done typing this.
And last but not least... your Wonderland Party looks amazing!! I love that first photo and your dress is divine. You look gorgeous as always, and you did such an amazing job on the decorations. Loving all the wonderful little details!!
xoxo Diana
Sweet Gracie, I really hope that things get easier for you soon! They always do, but sometimes it's hard to be patient. And you should know how much I love my valentine's headband I ordered from you. :)
Crossing my fingers that good luck finds you soon doll!
those are darling!
life is hard sometimes but I hope you keep chugging along :D good luck with finding a fantastic job soon, I'm sure it is out there somewhere.
Oh I wish I could give you a big hug but am sending virtual positive thoughts instead. Hope all gets better soon x
Glad Jason was ok after that happened.
Love those headbands so pretty xx
your new head bands are so so pretty!!! and definitely keep your head up, you can handle anything!!
Beautiful lady, I hope you find a dream job and feel brighter soon. Your etsy shop is definitely a good distraction and fun passion job to keep you going.
Hope things will start to turn around for the better soon! :))
Loving the headbands you have for your etsy store. You should make some on hard headbands! I'll totally get one! ;)
xx
my fingers and toes are crossed for you that everything is a ok!
NEW CAR SMELL!!!!!!!!!!!!! thats something to look forward to :)
love all of the new pieces. keep thinking positive thoughts, i hope you find a job soon! so sorry to hear about the truck tho! but congrats to your brother!! xoxo jcd :: cornflake dreams
your new headbands are soo cute! love the fabrics!! wow, soo many things going on for u!! good luck on the job hunt!!
Oh I am so sorry about your job, I hope you will find a great one!
Beautiful headbands, I love them!! Lots of things happening with you, but troubles in life just makes you stronger.. And you can look forward to those happy things, like being an auntie again :)
Good luck with the new old car, and I hope you are able to find a new job quickly.
And congrats to your brother and on your new nephew!
Congratulations to your brother!
That sucks about Jason's car. Did they leave a note or anything? I guess the upside is, you'll have a car you can drive soon!
Love the new stuff for your shop!
x Jasmine
love all of your new Etsy headbands - they are so cute! hang on with the job love - i've been unemployed for some time and recently picked up freelance work. it's really helped - maybe you could look into it, xoxo. | 68,519 |
Be bold, be brave, but above all, be true to yourself.
Photo: Charlton Brown / Dirk Lindner
Jo Maudsley and Chris Pask, directors of Charlton Brown, discuss this year’s trends in interior design and examine what a professional interior designer can bring to a project.
Q: Can you tell us about some of your projects?
We work across all scales, including masterplans, individual homes and apartments, as well as interior design and bespoke furniture. Much of our work focuses on high-end, residential projects where contemporary meets traditional – carefully restoring historic properties to celebrate their original features while transforming them into homes for 21st-century family life. We deliver contemporary interiors and new-build homes, including country homes and often work to Passivhaus standards. We are passionate about creating homes that are tailored to our clients and their spaces. Every project we work on is different, and our design response is always as unique as the story, brief, budget and timeframe we are working with.
Q: What inspires your interior designers?
Our Hampstead-based practice offers a range of architectural and interior design services, from feasibility and design through to planning and delivery. We see interior design as an extension of our sensitive, bespoke approach to architecture. Informed by its context and the history of the homes we work in, we draw upon forms, shapes and colours to create the perfect, timeless fit. We are also increasingly expanding our furniture, fixtures, and equipment side of the business, developing unique furniture collections for our designed spaces. Furniture is not only a fundamental part of a carefully considered interior, but also of a welcoming, restful home.
Q: Why employ a professional interior designer?
Interior designers bring experience, and knowledge of what works and what to avoid. This can be invaluable when it comes to decision making and tips for creating successful spaces.
We make sure the process is driven by you, through a bespoke approach that encompasses all aspects of an interior, without the hassle. It will also ensure you’ve considered everything. You will have a clear idea of your timeline and how your design will sit within budget from the start, making sure you get the most for your investment, without overspending.
Q: What are the main interior design and decorating trends this season?
There are, without doubt, certain surface finishes, internal fittings and features that we’ve noticed cropping up recently. It can be fun designing an interior scheme around a particular colour or material: however, we often find our clients’ homes themselves provide richer inspiration. In fact, there is a noticeable trend towards personal interior design: brave and increasingly design-savvy clients are opting to reject safe design and material choices – and are getting striking results in return. Our designers ensure you’re making the most of your home’s potential. We can also weave in pieces, designs and valuables that mean something to you.
Q: Which colours are especially popular?
The Victorian and Georgian (and sometimes much older or younger) houses we work with have their own style and a common approach is to decide which elements should look like they’ve been there forever, and which shouldn’t. Building a palette of finishes and colours in the same way as the Georgians or Victorians is a great way to create striking interiors that feel timeless. We are often asked if it’s possible to use bold colour without resulting in garish and uncomfortable interiors and the answer is “yes!”
We’ve spent an unhealthy amount of time looking at the colour theories of designers and decorators of the past and we’ve come to learn our ancestors were every bit as individual as we are. We are not constrained by the cost and technological factors that limited the range of colours available to them, but we can benefit by learning from the huge amount of care that went into pairing colours, building palettes and matching finishes.
Q: Is grey still trendy?
When used correctly, grey can bring masses of character to a space. Rooms that receive direct natural light can be painted in grey to make the most of the movement of shadows across the room. That being said, there are some wonderful colours in the tertiary range that are relatively neutral, like grey, while adding so much more to a palette than grey does, through their subtle hues. Again, we can learn a lot from our ancestors who were masters of tertiary colours like olive, slate and russet. A sophisticated palette for a library or dining room can be built by tempering rich reds with shades of slate and citrine.
Q: Which furniture is in fashion?
A diverse range of furniture is very popular. A combination of furniture styles can often be used in the same spaces, for example, a mix of very contemporary furniture with 20th-century design classics, together with antiques and pieces which show the patina of age can work well. This provides a space with character and homeliness – a feeling of pieces collected over time (even if this is not the case).
Q: Do you have any tips for those considering redecorating?
Start with a brief, so you know exactly what you’re looking for and your budget. Creating a mood board from Pinterest or Instagram can help you to keep your vision in mind while shopping for styles and items and is a great way to find inspiration for your look or to inform your designer. Don’t feel like you need to follow current trends – these can often date your spaces and you can fall out of love with them quickly. Successful design makes the most of your favourite pieces, colours and textures and is what will make your home unique to you.
For more information, visit charltonbrown.com or email [email protected]. | 86,345 |
Requirements:
- Secure storage of valuable equipment
- Highly dense storage
- Free up space while storing the same amount of goods
- Integration of the storage lift in the university building with separate retrieval option for students and employees
Solution:
- Storage shuttle installed with two access openings over two floors: one on the lower floor (library) for students and one on the upper floor (offices) for employees
- Items secured by assigning individual access codes
- Same amount of goods stored on a footprint 5% the size of the original
- Freed-up space has been used to create new offices
Scope of delivery:
- 1 Shuttle XP 250 system | 156,624 |
New Sanctions Risk Plunging the People of Mali Further into Humanitarian Crisis, Warn 13 NGOs
- population of Mali already faces.
The United States has also underlined its support for ECOWAS, while France - in its first weeks of its Presidency of the Council of the European Union - has suspended flights to Mali.
The 13 organisations.
Mali, ECOWAS and the members of the international community supporting these sanctions must monitor their impact, and unequivocally commit to applying humanitarian exemptions in line with existing guidelines - taking all necessary measures to limit the impact of these measures on civilians.
Franck Vannetelle, the International Rescue Committee’s Country Director in Mali, says:
.”
The full list of signatories includes:
● International Rescue Committee
● Action Against Hunger
● CARE
● CECI
● Danish Refugee Council
● Mercy Corps
● Norwegian Church Aid
● Norwegian Refugee Council
● Oxfam
● Plan International
● Terre des Hommes
● World Vision | 131,849 |
by Ian S. Palmer
Join bet365 NOW to BET ON SOCCER online!
Manchester United returns home to Old Trafford this Saturday, January 23 when they host Southampton. Both clubs haven’t lived up to their potential as of yet this season and it appears realistically that the league title will be out of their reach. On the bright side though, one or both of the teams could land a top-four spot and a berth in the 2016/17 European Champions League, with United having the edge in that scenario. United is coming off of a big 1-0 win at Anfield over Liverpool last weekend while Southampton took down West Bromwich Albion at home by a score of 3-0.
Manchester United vs Southampton – bet365 Soccer BETTING LINE:
- bet365 currently lists Man United as the favorites at 5/6 with Southampton 3/1 and a draw at 5/2
Manchester United vs Southampton – Head to Head:
- The teams met earlier this season with Man United winning a 3-2 thriller in Southampton. United won 2-1 on the south coast last season with Southampton winning 1-0 at Old Trafford. It was Southampton’s only win in their last 14 meetings in all competitions and they have just four victories in the past 25 encounters. Overall, they’ve met 117 times with United winning, 61, losing 27 and drawing 29.
Man United has just two wins in their past nine league contests and enters the weekend in fifth place with 37 points from 10 wins, seven draws and five losses. The team has 28 goals for and 20 against. They have five wins at home along with for draws and a loss with 12 goals scored and just four against. United has seven clean sheets at Old Trafford and has failed to score there on four occasions. Overall, they have 11 clean sheets and have been shut out seven times. They have two wins, a loss and three draws in their past six home outings.
Southampton occupies 10th place with 30 points from eight wins, six draws and eight losses with 31 goals scored and 24 against. The Saints have just two wins on the road with four draws and four defeats. They’ve scored just nine away goals and conceded 11. They have three clean sheets on the road and have been shut out four times. Overall, they have eight clean sheets and have failed to score seven times. Southampton has a win, a draw and four defeats in their past half dozen road fixtures. United heads into the game just a pair of points behind fourth place while the Saints are eight back of United.
Southampton has won two games in a row, but struggle away from home. They’ve lost four straight road trips and will need to stop the bleeding if they have any chance of a top-four finish. It looks like United will be without forward Ashley Young who was injured last week at Liverpool. They’ll also be without the services of defenders Marcos Rojo, Michael Carrick, Phil Jones, Bastian Schweinsteiger, Antonio Valencia and Luke Shaw. However, midfielder Adnan Januzaj should be back in the squad after returning from a loan spell.
Southampton is expected to be missing Gaston Ramirez and Florin Gardos while Charlie Austin is expected to make his Saints debut after the 26-year-old striker was recently signed from Championship League team Queen’s Park Rangers in the transfer window. Austin had 18 goals in the Premier League last campaign and cost Southampton four million pounds. | 347,961 |
First Night: General Information
SAFETY FIRST
With the assistance of the St. Louis Metropolitan Police Department, Grand Center will be staffed with police officers and security stationed throughout the festival footprint. In the interest of safety, please do not bring coolers or glass containers with you. Purses, backpacks and large bags will be subject to a visual search.
LOST AND FOUND
If you have become disconnected from your child or a family member, go to the Command Center in the Grand Center, Inc. offices located at 3526 Washington, Suite 300. Volunteers and police will be enlisted to help re-connect you with your family. If you have lost or found a personal item of value, please turn it in or leave your contact information at the Command Center.
FIRST AID
For minor injuries, cuts or scrapes, visit the Command Center in the Grand Center, Inc. offices located at 3526 Washington, Suite 200. An EMS unit will also be located there in case of a medical emergency.
RESTROOMS
All cultural institutions, the churches and most of the buildings have public restrooms. Portable facilities will be available on Olive, on the West side of Grand and on Grand in front of 634 N. Grand.
SERVICES FOR PERSONS WITH DISABILITIES
Handicap accessible parking is available at each of the parking lots and garages. All Metro buses are wheelchair accessible.
All cultural institutions are wheelchair accessible.
Several of our performances will be interpreted for the hearing impaired. Please check the printed program guide the night of the event for the specific performances. Also, we have indicated the performances that will be interpreted in the program descriptions on the web site.
Service animals are welcome.
INFORMATION AND ADMISSION BUTTONS
Maps and program schedules are available where Admission Buttons are purchased: First Night Souvenir and Button Tent, in front of 500 N. Grand Blvd. and in the Saint Louis University Busch Student Center.
SOUVENIR SHOP
From First Night commemorative t-shirts to hundreds of glo-in-the-dark items, you’ll want to check out our fun and colorful souvenir shop at the Busch Student Center and at the First Night Souvenir and Button Tent at 500 N. Grand Blvd.
HOW TO DRESS
Wear a warm coat, hat and gloves. While most of the First Night events are indoors, we want you to experience as many of them as you can so that means walking from venue to venue as well as enjoying all of the visual sights and surprises we have prepared for you! And, you won’t want to miss the opening and closing ceremonies and the spectacular fireworks… so dress warm.
FOOD AND DRINK
It’s a winter food festival! “Warm up” with a cup of hot chocolate or fresh, brewed coffee as you stroll and find “comfort” with many of your favorite snacks, treats and sweets located throughout the festival.
Several neighborhood establishments will be open to the public: Best Steakhouse, The Bistro in Grand Center, BaiKu, City Diner at the Fox, Dooley’s, Field House, Lucha, Triumph Grill and Vito’s Sicilian Restaurant. We also have several food and beverage trucks available along Olive and Washington Blvd.
Food and drink are not permitted inside the theaters, concert halls and churches.
First Night is a non-alcoholic event.
IN THE EVENT OF SEVERE WEATHER
First Night will go on regardless of temperature, rain, sleet or snow!
A FEW MORE THINGS YOU SHOULD KNOW
Food, drinks and smoking will not be permitted inside the theaters, concert halls or churches.
Stroller space is limited, however, we will make every effort to accommodate your needs.
First Night is not responsible for lost or stolen items.
First Night is an alcohol free event.
Support Grand Center Inc.
Your gift helps to support the arts, preserve a legacy and transform a neighborhood into a vibrant community.. | 368,660 |
TITLE: Partial differential equation $[\partial_t + v(x)\partial_x - \rho(x)] D(t,x) = 0$
QUESTION [3 upvotes]: I'm stuck with the following problem I found (without a proof) in the Peskin and Schroeder textbook on quantum field theory (the differential equation mentioned below is equivalent to the Callan-Symanzik equation describing the evolution of the n-point correlation functions under variation of the energy scale at which the theory is defined):
Show that the solution of the equation
$$
[\partial_t + v(x)\partial_x - \rho(x)] D(t,x) = 0
$$
has the form
$$
D(t,x) = D(0,X_t(x)) \exp\left(\int_0^t d t' \rho(X_{t'}(x)) \right),
$$
where $X_t(x)$ is the solution of the equation
$$
\partial_t X_t(x) = - v(X_t(x))
$$
with the initial condition
$$
X_0(x) = x.
$$
Let's compute e.g.
$$
\partial_t \left[D(0,X_t(x)) \exp\left(\int_0^t d t' \rho(X_{t'}(x)) \right) \right]= \left[\partial_t D(0,X_t(x))\right] \exp\left(\int_0^t d t' \rho(X_{t'}(x)) \right) + D(0,X_t(x)) \rho(X_t(x)) \exp\left(\int_0^t d t' \rho(X_{t'}(x)) \right)
$$
I don't see any possible simplifications with other terms.
I wonder how to show that the solution of the mentioned differential equation is indeed of the above form.
REPLY [4 votes]: This is a simple transport type equation. We shall solve the PDE using the methods of characteristics. Suppose we could rewrite the PDE as follows
\begin{align}
\partial_t D + \nu(x) \partial_x D- \rho(x)D =&\ \partial_t D+ X'(t)\partial_x D -\rho(x)D \\
=&\ \frac{d}{dt}D(t, X(t)) -\rho(x)D = 0
\end{align}
where $X'(t) = \nu(X)$. Solving for the characteristic $X$ leads to the solution
\begin{align}
\frac{X'}{\nu(X)} = 1 \ \ \Rightarrow \ \ \int \frac{dX}{\nu(X)} = t+ C
\end{align}
which really depends on $\nu(X)$ (nevertheless it exists). Hence let us assume that we found $X(t)$. Back to the PDE, we have
\begin{align}
\frac{d}{dt}D(t, X(t)) = \rho(X(t)) D(t, X(t)) \ \ \Rightarrow& \ \ \frac{d}{dt}\log|D(t, X(t))| =\rho(X(t))\\
\Rightarrow&\ \ \ \log|D(t, X(t))| -\log|D(0, X(0))| = \int^t_0 dt' \rho(X(t'))\\
\Rightarrow&\ \ D(t, X(t)) = D(0, X(0))\exp\left( \int^t_0 dt' \rho(X(t'))\right).
\end{align}
We are almost done, but not done. If we impose the initial condition that $X(0) = x$, then we have a unique curve that pasts through the point $(0, x)$ which will dictate the values of $D(t, x)$ when $(t, X)$ lies on the trace of $(t, X(t))$. Hence, we shall use the notation $X= X_t(x)$ to indicate the curve. Thus, your solution becomes
\begin{align}
D(t, X) = D(0,x)\exp\left( \int^t_0 dt' \rho(X_t(x)) \right).
\end{align} | 35,380 |
I have a big problem with my GV in that the engine is only running on 3 out of 4 cylinders! Mechanic says the valves are bent - at least one of them, maybe more. He hasn't opened the engine head up to see but he has replaced the timing chain. The tensioner failed and the chain broke, I'm £800 deep just for that replacement along with compression diagnostics etc.
He says that to open up the engine head and fix the bent valves will cost an extra £2500, bringing the cost up to £3200!! I think the car is worth £2-3k on a good day so I feel like I could just pay the £800 and cut my losses. Maybe try to sell it as a project for someone who is more of a piston head than me!
It had to be towed in to the garage over a month ago because there was no compression in the engine. He found the problem was the chain and said I might get lucky if the valves were not bent when the new chain started the engine. They are, hence the big price. It's done 90,000 miles and I've had it for 18 months, bought for 4k. It came with a warranty but they sent an inspector to the garage who said the fault is from wear and tear on the chain (because the tensioner failed). I said how could I account for that? There were no signs. They still won't cover any costs and the warranty maxes out at £1k anyway! Not worth the paper its written on really.
Anyway, sorry for the long read... anyone got experience of these fixes or sales in this condition?
Thanks
| 56,833 |
Air pollution is one of the largest problems facing the world today. What are its causes and what measures can be proposed to solve this problem?
Global warming has become a serious concern for many countries, one of the widely faced issues relates to that of air pollution. This essay will discuss the reasons of air pollution which largely involves increasing number of vehicles and effects of industrialization. It will also provide some plausible solutions to such problems including improved public transportation and enforcement of industry laws. To begin with, air pollution have become an increasingly hazardous problem over the past few years. Although, there are several factors which contributes to this problem, foremost is the massive increase in number of private motor vehicles emitting smoke into the air. One of the main reasons behind this trend is inefficiency of public transportation system which is driving force for public to purchase and drive their own vehicles. For example, one will be left with no other option but to purchase one’s own car when one does not find a solution to commute between office and home through public conveyance. Moreover, in addition to that powerplants being operated on fossil fuels emit high pollutants into the air. Research suggests that more than 50% of the air pollution is caused by heavy industries. Therefore, private vehicles and industries are considered as one of the major contributors in air pollution. Nevertheless, there are also numerous solutions to these problems. However, an improved public commute system can play an important role in reduction of private transport means such as cars and motor bikes. For instance, if one is able to reach a destination in less time and cost as compared to going on a private motor vehicle, one will prefer public transport over private and hence, there will be drastic reduction in private cars. Consequently, emission of carbon dioxide will also be reduced. Furthermore, strict environmental laws can be enforced on industries by regulatory bodies. To illustrate, a sizeable plantation can be made mandatory for all industries depending on their size so that effected greenhouse environment can be compensated in terms of increased number of trees. To conclude, in today’s fast paced and industrialized world, more and more air is being polluted. This essay discussed how air pollution is often caused by private vehicles & powerplants and also provided with conceivable solutions. | 358,101 |
Monday Apr 25, 2022
Niche Real Estate Agents, Home Upgrade Ideas, and Leawood vs Prairie Village Kansas
In this podcast episode, we're going to be talking about niche real estate agents, home upgrade ideas, and being a real estate agent in Leawood vs being a real estate agent in Prairie Village KS.
We'll give you our thoughts on each of these topics and let you know what we think is the best option for you.
So whether you're just starting out in the world of real estate or you're looking to upgrade your home, make sure to listen to this podcast episode!
To leave or reply to comments, please download free Podbean or
To leave or reply to comments,
please download free Podbean App. | 87,385 |
This is a short sleeve pullover, with a round yoke in the lace section, which has raglan shaping in the sleeve and body once the lace section is complete. Instructions are fully written, however a chart sample is given for both the lace in the yoke, and the (FPtr, dc) pattern used in the sleeve and body, to assist crocheters who prefer charted instructions.
Designer | 87,108 |
. Once you’re back online, your new files and edits will sync to OneDrive automatically. You can view and edit your files, as well as collaborate in real-time and share files/folders easily and directly from your desktop — without using a web browser. Also, everyone who has access to these files will always have the latest version of the file at hand.
Figure 1: Note: With OneDrive you can easily sync files to your desktop.
More from my site
Contact Sales
- +852 2207 9988
- [email protected]
Customer Service
- +852 2353 1445
- [email protected]
| 224,440 |
\begin{document}
\selectlanguage{american}
\global\long\def\1{\mbox{1\hspace{-1mm}I}}
\selectlanguage{english}
\title{CONTROL PROBLEM ON SPACE OF RANDOM VARIABLES AND MASTER EQUATION }
\author{Alain Bensoussan\\
International Center for Decision and Risk Analysis\\
Jindal School of Management, University of Texas at Dallas\thanks{Also with the College of Science and Engineering, Systems Engineering
and Engineering Management, City University Hong Kong. Research supported
by the National Science Foundation under grant DMS-1303775 and the
Research Grants Council of the Hong Kong Special Administrative Region
(CityU 500113). } \\
Sheung Chi Phillip Yam\\
Department of Statistics, The Chinese University of Hong Kong\thanks{Research supported by The Hong Kong RGC GRF 14301015 with the project
title: Advance in Mean Field Theory.}}
\maketitle
\begin{abstract}
We study in this paper a control problem in a space of random variables.
We show that its Hamilton Jacobi Bellman equation is related to the
Master equation in Mean field theory. P.L. Lions in (\cite{PLL}),(\cite{PL2}
introduced the Hilbert space of square integrable random variables
as a natural space for writing the Master equation which appears in
the mean field theory. W. Gangbo and A. \'{S}wi\k{e}ch \cite{GAS}
considered this type of equation in the space of probability measures
equipped with the Wasserstein metric and use the concept of Wasserstein
gradient. We compare the two approaches and provide some extension
of the results of Gangbo and \'{S}wi\k{e}ch.
\end{abstract}
\section{INTRODUCTION }
We study first an abstract control problem where the state is in a
Hilbert space. We then show how this model applies when the Hilbert
space is the space of square integrable random variables, and for
certain forms of the cost functions. We see that it applies directly
to the solution of the Master equation in Mean Field games theory.
We compare our results with those of W. Gangbo and A. \'{S}wi\k{e}ch
\cite{GAS} and show that the approach of the Hilbert space of square
integrable random variables simplifies greatly the development.
\section{AN ABSTRACT CONTROL PROBLEM}
\subsection{\label{sub:SETTING-OF-THE}SETTING OF THE PROBLEM}
We begin by defining an abstract control problem, without describing
the application. We consider a Hilbert space $\mathcal{H}$, whose
elements are denoted by $X.$ We identify $\mathcal{H}$ with its
dual. The scalar product is denoted by $((,))$ and the norm by ||.||.
We then consider functionals $\mathcal{F}(X)$ and $\mathcal{F}_{T}(X)$
which are continuously differentiable on $\mathcal{H}$. The gradients
$D_{X}\mathcal{F}(X)$ and $D_{X}\mathcal{F}_{T}(X)$ are Lipschitz
continuous
\begin{align}
||D_{X}\mathcal{F}(X_{1})-D_{X}\mathcal{F}(X_{2})|| & \leq c||X_{1}-X_{2}||\label{eq:2-1}\\
||D_{X}\mathcal{F}_{T}(X_{1})-D_{X}\mathcal{F}_{T}(X_{2})|| & \leq c||X_{1}-X_{2}||\nonumber
\end{align}
To simplify notation, we shall also assume that
\begin{equation}
||D_{X}\mathcal{F}(0)||,\:||D_{X}\mathcal{F}_{T}(0)||\leq c\label{eq:2-1-1}
\end{equation}
So we have
\begin{equation}
||D_{X}\mathcal{F}(X)||\leq c(1+||X||)\label{eq:2-11-1}
\end{equation}
and
\begin{equation}
|\mathcal{F}(X)|\leq C(1+||X||^{2}),\;\label{eq:2-12-4}
\end{equation}
where we denote by $C$ a generic constant. The same estimates hold
also for $\mathcal{F}_{T}(X).$
A control is a function $v(s)$ which belongs to $L^{2}(0,T;\mathcal{H}).$
We associate to a control $v(.)$ the state $X(s)$ satisfying
\begin{align}
\frac{dX}{ds} & =v(s)\label{eq:2-2}\\
X(t) & =X\nonumber
\end{align}
We may write it as $X_{Xt}(s)$ to emphasize the initial conditions
and even $X_{Xt}(s;v(.))$ to emphasize the dependence in the control.
The function $X(s)$ belongs to the Sobolev space $H^{1}(t,T;\mathcal{H}).$
We then define the cost functional
\begin{equation}
J_{Xt}(v(.))=\frac{\lambda}{2}\int_{t}^{T}||v(s)||^{2}ds+\int_{t}^{T}\mathcal{F}(X(s))ds+\mathcal{F}_{T}(X(T))\label{eq:2-3}
\end{equation}
and the value function
\begin{equation}
V(X,t)=\inf_{v(.)}\,J_{Xt}(v(.))\label{eq:2-4}
\end{equation}
\subsection{BELLMAN EQUATION}
We want to show the following
\begin{thm}
\label{theo2-1} We assume (\ref{eq:2-1}), (\ref{eq:2-1-1}) and
\begin{equation}
\lambda>cT(1+T)\label{eq:2-5}
\end{equation}
The value function (\ref{eq:2-4}) is $C^{1}$ and satisfies the growth
conditions
\begin{align}
|V(X,t)| & \leq C(1+||X||^{2})\label{eq:2-6}\\
||D_{X}V(X,t)|| & \leq C(1+||X||),\:|\frac{\partial V(X,t)}{\partial t}|\leq C(1+||X||^{2})\nonumber
\end{align}
where $C$ is a generic constant. Moreover $D_{X}V(X,t)$ and $\dfrac{\partial V(X,t)}{\partial t}$
are Lipschitz continuous, more precisely
\begin{align}
||D_{X}V(X^{1},t^{1})-D_{X}V(X^{2},t^{2})|| & \leq C||X^{1}-X^{2}||+C|t^{1}-t^{2}|(1+||X^{1}||+||X^{2}||)\label{eq:2-7}\\
|\dfrac{\partial V(X^{1},t^{1})}{\partial t}-\dfrac{\partial V(X^{2},t^{2})}{\partial t}| & \leq C||X^{1}-X^{2}||(1+||X^{1}||+||X^{2}||)+C|t^{1}-t^{2}|(1+||X^{1}||^{2}+||X^{2}||^{2})\nonumber
\end{align}
It is the unique solution, satisfying conditions (\ref{eq:2-6}) and
(\ref{eq:2-7}) of Bellman equation
\end{thm}
\begin{align}
\dfrac{\partial V}{\partial t}-\frac{1}{2\lambda}||D_{X}V||^{2}+\mathcal{F}(X) & =0\nonumber \\
V(X,T)=\mathcal{F}_{T}(X)\label{eq:2-8}
\end{align}
The control problem (\ref{eq:2-2}), (\ref{eq:2-3}) has a unique
solution.
\begin{proof}
We begin by studying the properties of the cost functional $J_{Xt}(v(.)).$
We first claim that $J_{Xt}(v(.))$ is Gâteaux differentiable in the
space $L^{2}(t,T;\mathcal{H})$, for $X,t$ fixed. Define $X_{v}(s)$
by
\[
\frac{dX_{v}(s)}{ds}=v(s),\:X_{v}(t)=X
\]
and $Z_{v}(s)$ by
\[
-\frac{dZ_{v}(s)}{ds}=D_{X}\mathcal{F}(X_{v}(s)),\:Z_{v}(t)=D_{X}\mathcal{F}_{T}(X_{v}(T))
\]
then we can prove easily that
\begin{equation}
\frac{d}{d\mu}J_{Xt}(v(.)+\mu\tilde{v}(.))|_{\mu=0}=\int_{t}^{T}((\lambda v(s)+Z_{v}(s),\tilde{v}(s)))ds\label{eq:2-9}
\end{equation}
Let us prove that the functional $J_{Xt}(v(.))$ is strictly convex.
Let $v_{1}(.)$ and $v_{2}(.)$ in $L^{2}(t,T;\mathcal{H}).$ We write
\[
J_{Xt}(\theta v_{1}(.)+(1-\theta)v_{2}(.))=J_{Xt}(v_{1}(.)+(1-\theta)(v_{2}(.)-v_{1}(.)))
\]
\[
=J_{Xt}(v_{1}(.))+\int_{0}^{1}\frac{d}{d\mu}J_{Xt}(v_{1}(.)+\mu(1-\theta)(v_{2}(.)-v_{1}(.)))\,d\mu
\]
From formula (\ref{eq:2-9}) we have also
\[
\frac{d}{d\mu}J_{Xt}(v(.)+\mu\theta\tilde{v}(.))=\theta\int_{t}^{T}((\lambda(v(s)+\mu\theta\tilde{v}(.))+Z_{v+\mu\theta\tilde{v}}(s),\tilde{v}(s)))ds
\]
Therefore
\[
\int_{0}^{1}\frac{d}{d\mu}J_{Xt}(v_{1}(.)+\mu(1-\theta)(v_{2}(.)-v_{1}(.)))\,d\mu=(1-\theta)\int_{0}^{1}d\mu\left\{ \int_{t}^{T}((\lambda(v_{1}(s)+\mu(1-\theta)(v_{2}(s)-v_{1}(s)))\right.
\]
\[
+\left.Z_{v_{1}+\mu(1-\theta)(v_{2}-v_{1})}(s),\,v_{2}(s)-v_{1}(s)\,))ds\right\}
\]
Similarly we write
\[
J_{Xt}(\theta v_{1}(.)+(1-\theta)v_{2}(.))=J_{Xt}(v_{2}(.)+\theta(v_{1}(.)-v_{2}(.)))
\]
\[
=J_{Xt}(v_{2}(.))+\int_{0}^{1}\frac{d}{d\mu}J_{Xt}(v_{2}(.)+\mu\theta(v_{1}(.)-v_{2}(.)))\,d\mu
\]
and
\begin{align*}
\int_{0}^{1}\frac{d}{d\mu}J_{Xt}(v_{2}(.)+\mu\theta(v_{1}(.)-v_{2}(.)))\,d\mu & =\theta\int_{0}^{1}d\mu\left\{ \int_{t}^{T}((\lambda(v_{2}(s)+\mu\theta(v_{1}(s)-v_{2}(s)))\right.+\\
+ & \left.Z_{v_{2}+\mu\theta(v_{1}-v_{2})}(s),\,v_{1}(s)-v_{2}(s)\,))ds\right\}
\end{align*}
We shall set $Z_{1}(s)=Z_{v_{1}+\mu(1-\theta)(v_{2}-v_{1})}(s)$
and $Z_{2}(s)=Z_{v_{2}+\mu\theta(v_{1}-v_{2})}(s).$ Combining formulas,
we can write
\begin{equation}
J_{Xt}(\theta v_{1}(.)+(1-\theta)v_{2}(.))=\theta J_{Xt}(v_{1}(.))+(1-\theta)J_{Xt}(v_{2}(.))+\label{eq:2-10}
\end{equation}
\[
+\theta(1-\theta)\left[-\frac{\lambda}{2}\int_{t}^{T}||v_{1}(s)-v_{2}(s)||^{2}ds+\int_{0}^{1}d\mu\int_{t}^{T}((Z_{1}(s)-Z_{2}(s),v_{2}(s)-v_{1}(s)\,))ds\right]
\]
Let $X_{1}(s)$ and $X_{2}(s)$ denote the states corresponding to
the controls $v_{1}(.)+\mu(1-\theta)(v_{2}(.)-v_{1}(.))$ and $v_{2}(.)+\mu\theta(v_{1}(.)-v_{2}(.))$.
One checks easily that
\[
X_{1}(s)-X_{2}(s)=(1-\mu)\int_{t}^{s}(v_{1}(\sigma)-v_{2}(\sigma))d\sigma
\]
and from the definition of $Z_{1}(.)$, $Z_{2}(.)$ we obtain
\[
||Z_{1}(s)-Z_{2}(s)||\leq c[||X_{1}(T)-X_{2}(T)||+\int_{s}^{T}||X_{1}(\sigma)-X_{2}(\sigma)||d\sigma]
\]
and combining formulas, we can assert
\[
||Z_{1}(s)-Z_{2}(s)||\leq c(1-\mu)(1+T)\int_{t}^{T}||v_{1}(\sigma)-v_{2}(\sigma)||d\sigma
\]
Going back to (\ref{eq:2-10}) we obtain easily
\begin{equation}
J_{Xt}(\theta v_{1}(.)+(1-\theta)v_{2}(.))\leq\theta J_{Xt}(v_{1}(.))+(1-\theta)J_{Xt}(v_{2}(.))+\label{eq:2-10-1}
\end{equation}
\[
-\frac{\theta(1-\theta)}{2}(\lambda-cT(1+T))\int_{t}^{T}||v_{1}(s)-v_{2}(s)||^{2}ds
\]
and from the assumption (\ref{eq:2-5}) we obtain immediately that
$J_{Xt}(v(.))$ is strictly convex. Next we write
\[
\mathcal{F}(X(s))-\mathcal{F}(X)=\int_{0}^{1}((D_{X}\mathcal{F}(X+\theta\int_{t}^{s}v(\sigma)d\sigma),\int_{t}^{s}v(\sigma)d\sigma))
\]
so , using (\ref{eq:2-11-1}) we obtain
\[
|\mathcal{F}(X(s))-\mathcal{F}(X)|\leq c(1+||X||)||\int_{t}^{s}v(\sigma)d\sigma||+\frac{c}{2}||\int_{t}^{s}v(\sigma)d\sigma||^{2}
\]
\[
\leq\frac{c^{2}(1+||X||)^{2}}{2\delta}+\frac{c+\delta}{2}||\int_{t}^{s}v(\sigma)d\sigma||^{2}
\]
for any $\delta>0.$ Using (\ref{eq:2-12-4}) we can assert that
\[
|\mathcal{F}(X(s))|\leq C_{\delta}(1+||X||^{2})+\frac{c+\delta}{2}T\int_{t}^{T}||v(\sigma)||^{2}d\sigma
\]
A similar estimate holds for $\mathcal{F}_{T}(X(T)).$ Therefore,
collecting results, we obtain
\begin{align*}
|\int_{t}^{T}\mathcal{F}(X(s))ds+\mathcal{F}_{T}(X(T))| & \leq C_{\delta}(1+||X||^{2})(1+T)\\
+ & \frac{c+\delta}{2}T(1+T)\int_{t}^{T}||v(s)||^{2}ds
\end{align*}
It follows that
\begin{align}
J_{Xt}(v(.)) & \geq\frac{\lambda-(c+\delta)T(1+T)}{2}\int_{t}^{T}||v(s)||^{2}ds-C_{\delta}(1+||X||^{2})(1+T)\label{eq:2-12-2}
\end{align}
Since $\lambda-cT(1+T)>0,$we can find $\delta>0$ sufficiently small
so that $\lambda-(c+\delta)T(1+T)>0.$ This implies that $J_{Xt}(v(.))\rightarrow+\infty$
as $\int_{t}^{T}||v(s)||^{2}ds\rightarrow+\infty.$ This property
and the strict convexity imply that the functional $J_{Xt}(v(.))$
has a minimum which is unique. The Gâteaux derivative must vanish
at this minimum denoted by $u(.).$ The corresponding state is denoted
by $Y(.).$ From formula (\ref{eq:2-9}) we obtain also the existence
of a solution of the two-point boundary value problem
\begin{align}
\frac{dY}{ds} & =-\frac{Z(s)}{\lambda},\quad-\frac{dZ}{ds}=D_{X}\mathcal{F}(Y(s))\label{eq:2-11}\\
Y(t) & =X,\qquad Z(T)=D_{X}\mathcal{F}_{T}(Y(T))\nonumber
\end{align}
and the optimal control $u(.)$ is given by the formula
\begin{equation}
u(s)=-\frac{Z(s)}{\lambda}\label{eq:2-12}
\end{equation}
In fact, the system (\ref{eq:2-11}) can be studied directly, and
we can show directly that it has one and only one solution. We notice
that it is a 2nd order differential equation, since
\begin{align}
\frac{d^{2}Y}{ds^{2}} & =\frac{1}{\lambda}D_{X}\mathcal{F}(Y(s))\label{eq:2-12-1}\\
Y(t)=X & \quad\frac{dY}{ds}(T)=-\frac{1}{\lambda}D_{X}\mathcal{F}_{T}(Y(T))\nonumber
\end{align}
We can write also (\ref{eq:2-12-1}) as an integral equation
\begin{equation}
Y(s)=X-\frac{s-t}{\lambda}D_{X}\mathcal{F}_{T}(Y(T))-\frac{1}{\lambda}\int_{t}^{T}D_{X}\mathcal{F}(Y(\sigma))(s\wedge\sigma-t)d\sigma\label{eq:2-13}
\end{equation}
and we can view this equation as a fixed point equation in the space
$C^{0}([t,T];\mathcal{H}),$ namely $Y(.)=\mathcal{K}(Y(.)),$ where
$\mathcal{K}$ is defined by the right hand side of (\ref{eq:2-13}).
One can show that $\mathcal{K}$ is a contraction, hence $Y(.)$ is
uniquely defined. Note also, that if we have a solution of (\ref{eq:2-11})
and if $u(.)$ is defined by (\ref{eq:2-12}) the control $u(.)$
satisfies the necessary condition of optimality for the functional
$J_{Xt}(v(.)).$ Since this functional is convex, the necessary condition
of optimality is also sufficient and thus $u(.)$ is optimal. The
value function is thus defined by the formula
\begin{equation}
V(X,t)=\frac{1}{2\lambda}\int_{t}^{T}||Z(s)||^{2}ds+\int_{t}^{T}\mathcal{F}(Y(s))ds+\mathcal{F}_{T}(Y(T))\label{eq:2-14}
\end{equation}
We now study the properties of the value function. We begin with the
first property (\ref{eq:2-6}). Using (\ref{eq:2-12-2}) we obtain
\[
V(X,t)\geq-C(1+||X||^{2})
\]
On the other hand, we have
\[
V(X,t)\leq J_{Xt}(0)=(T-t)\mathcal{F}(X)+\mathcal{F}_{T}(X)\leq C(1+||X||^{2})
\]
and the first estimate (\ref{eq:2-6}) is obtained.
We proceed in getting estimates for the solution $Y(.)$ of (\ref{eq:2-13}).
We write
\[
||Y(.)||=\sup_{t\leq s\leq T}||Y(s)||
\]
Using easy majorations, we obtain
\begin{align}
||Y(.)|| & \leq\frac{||X||\lambda+cT(T+1)}{\lambda-cT(T+1)}\label{eq:2-15}\\
||Z(.)|| & \leq\frac{\lambda(1+T)c(1+||X||)}{\lambda-cT(T+1)}\nonumber \\
||u(.)|| & \leq\frac{(1+T)c(1+||X||)}{\lambda-cT(T+1)}\nonumber
\end{align}
We then study how these functions depend on the pair $X,t.$ We recall
that $Y(s)=Y_{Xt}(s).$ Let us consider two points $X_{1},t_{1}$
and $X_{2},t_{2}$ and denote $Y_{1}(s)=Y_{X_{1}t_{1}}(s),$ $Y_{2}(s)=Y_{X_{2}t_{2}}(s)$.
To fix ideas we assume $t_{1}<t_{2}.$ For $s>t_{2}$ we have
\begin{align*}
Y_{1}(s)-Y_{2}(s) & =X_{1}-X_{2}-\frac{1}{\lambda}(D_{X}\mathcal{F}_{T}(Y_{1}(T))-D_{X}\mathcal{F}_{T}(Y_{2}(T)))(s-t_{2})-\frac{1}{\lambda}D_{X}\mathcal{F}_{T}(Y_{1}(T))(t_{2}-t_{1})\\
-\frac{1}{\lambda} & \int_{t_{2}}^{T}(D_{X}\mathcal{F}(Y_{1}(\sigma))-D_{X}\mathcal{F}(Y_{2}(\sigma)))(s\wedge\sigma-t_{2})d\sigma-\frac{1}{\lambda}\int_{t_{1}}^{t_{2}}D_{X}\mathcal{F}(Y_{1}(\sigma))(s\wedge\sigma-t_{1})d\sigma
\end{align*}
From which we obtain
\begin{align*}
\sup_{t_{2}\leq s\leq T}||Y_{1}(s)-Y_{2}(s)|| & \leq||X_{1}-X_{2}||+\frac{c}{\lambda}T(1+T)\sup_{t_{2}\leq s\leq T}||Y_{1}(s)-Y_{2}(s)||+\\
+ & \frac{t_{2}-t_{1}}{\lambda}[||D_{X}\mathcal{F}_{T}(Y_{1}(T))||+\int_{t_{1}}^{T}||D_{X}\mathcal{F}(Y_{1}(s))||ds]
\end{align*}
Using the properties of $D_{X}\mathcal{F}$ and $D_{X}\mathcal{F}_{T}$
and (\ref{eq:2-15}) we can assert that
\[
\sup_{t_{2}\leq s\leq T}||Y_{1}(s)-Y_{2}(s)||\leq\frac{\lambda}{\lambda-cT(T+1)}\left(||X_{1}-X_{2}||+(t_{2}-t_{1})(1+T)c\frac{1+||X_{1}||}{\lambda-cT(T+1)}\right)
\]
More globally we can write
\begin{equation}
\sup_{\max(t_{1},t_{2})\leq s\leq T}||Y_{X_{1}t_{1}}(s)-Y_{X_{2}t_{2}}(s)||\leq\frac{\lambda}{\lambda-cT(T+1)}\left(||X_{1}-X_{2}||+|t_{2}-t_{1}|(1+T)c\frac{1+\,\max(||X_{1}||,||X_{2}||)}{\lambda-cT(T+1)}\right)\label{eq:2-16}
\end{equation}
In particular
\begin{equation}
\sup_{t\leq s\leq T}||Y_{X_{1}t}(s)-Y_{X_{2}t}(s)||\leq\frac{\lambda||X_{1}-X_{2}||}{\lambda-cT(T+1)}\label{eq:2-17}
\end{equation}
Recalling that from the system (\ref{eq:2-11}) we have
\[
Z(s)=\int_{s}^{T}D_{X}\mathcal{F}(Y(\sigma))d\sigma+D_{X}\mathcal{F}_{T}(Y(T))
\]
and noting $Z(s)=Z_{Xt}(s)$ we deduce from (\ref{eq:2-17}) that
\begin{equation}
\sup_{t\leq s\leq T}||Z_{X_{1}t}(s)-Z_{X_{2}t}(s)||\leq\frac{c(T+1)\lambda||X_{1}-X_{2}||}{\lambda-cT(T+1)}\label{eq:2-18}
\end{equation}
We next write
\[
J_{X_{1}t}(u_{1}(.))-J_{X_{2}t}(u_{1}(.))\leq V(X_{1},t)-V(X_{2},t)\leq J_{X_{1}t}(u_{2}(.))-J_{X_{2}t}(u_{2}(.))
\]
where $u_{1}(.)$ and $u_{2}(.)$ are the optimal controls for the
problems with initial conditions $(X_{1},t)$ and $(X_{2},t),$ respectively.
Denoting by $Y_{X_{1}t}(s)$ and $Y_{X_{2}t}(s)$ the optimal states
and by $Y_{X_{1}t}(s;u_{2}(.)),\;Y_{X_{2}t}(s;u_{1}(.))$ the trajectories
( not optimal) when the control $u_{2}(.)$ is used with the initial
conditions $(X_{1},t)$ and when the control $u_{1}(.)$ is used with
the initial conditions $(X_{2},t),$ we have
\begin{align*}
Y_{X_{1}t}(s;u_{2}(.))-Y_{X_{2}t}(s) & =Y_{X_{1}t}(s)-Y_{X_{2}t}(s;u_{1}(.))=X_{1}-X_{2}
\end{align*}
Therefore
\[
V(X_{1},t)-V(X_{2},t)\leq\int_{t}^{T}(\mathcal{F}(Y_{X_{2}t}(s)+X_{1}-X_{2})-\mathcal{F}(Y_{X_{2}t}(s)))ds+\mathcal{F}_{T}(Y_{X_{2}t}(T)+X_{1}-X_{2})-\mathcal{F}_{T}(Y_{X_{2}t}(T))
\]
and by techniques already used it follows
\[
V(X_{1},t)-V(X_{2},t)\leq((\int_{t}^{T}D_{X}\mathcal{F}(Y_{X_{2}t}(s))ds+D_{X}\mathcal{F}_{T}(Y_{X_{2}t}(T)),X_{1}-X_{2}))+\frac{c}{2}(1+T)||X_{1}-X_{2}||^{2}
\]
which is in fact
\begin{equation}
V(X_{1},t)-V(X_{2},t)\leq((Z_{X_{2}t}(t),X_{1}-X_{2}))+\frac{c}{2}(1+T)||X_{1}-X_{2}||^{2}\label{eq:2-19}
\end{equation}
By interchanging the roles of $X_{1}$ and $X_{2}$ we also obtain
\begin{equation}
V(X_{1},t)-V(X_{2},t)\geq((Z_{X_{1}t}(t),X_{1}-X_{2}))-\frac{c}{2}(1+T)||X_{1}-X_{2}||^{2}\label{eq:2-20}
\end{equation}
Using the estimate (\ref{eq:2-18}) we can also write
\begin{equation}
V(X_{1},t)-V(X_{2},t)\geq((Z_{X_{2}t}(t),X_{1}-X_{2}))-c(T+1)[\frac{\lambda}{\lambda-cT(T+1)}+\frac{1}{2}]||X_{1}-X_{2}||^{2}\label{eq:2-21}
\end{equation}
Combining (\ref{eq:2-19}) and (\ref{eq:2-21}) we immediately get
\begin{equation}
|V(X_{1},t)-V(X_{2},t)-((Z_{X_{2}t}(t),X_{1}-X_{2}))|\leq c(T+1)[\frac{\lambda}{\lambda-cT(T+1)}+\frac{1}{2}]||X_{1}-X_{2}||^{2}\label{eq:2-22}
\end{equation}
This shows immediately that $V(X,t)$ is differentiable in $X$ and
that
\begin{equation}
D_{X}V(X,t)=Z(t)=-\lambda u(t)\label{eq:2-23}
\end{equation}
From the 2nd estimate (\ref{eq:2-15}) we immediately obtain the 2nd
estimate (\ref{eq:2-6}). We continue with the derivative in $t.$
We first write the optimality principle
\begin{equation}
V(X,t)=\frac{\lambda}{2}\int_{t}^{t+\epsilon}||u(s)||^{2}ds+\int_{t}^{t+\epsilon}\mathcal{F}(Y(s))ds+V(Y(t+\epsilon),t+\epsilon)\label{eq:2-24}
\end{equation}
which is a simple consequence of the definition of the value function
and of the existence of an optimal control. From (\ref{eq:2-22})
we can write
\[
V(X_{2},t)-V(X_{1},t)-((Z_{X_{2}t}(t),X_{2}-X_{1}))\leq C||X_{1}-X_{2}||^{2}
\]
where $C$ is the constant appearing in the right hand side of (\ref{eq:2-22}).
We apply with $X_{2}=Y(t+\epsilon)$ , $X_{1}=X,$$t=t+\epsilon.$
We note that $Z_{Y(t+\epsilon),t+\epsilon}(t+\epsilon)=Z_{Xt}(t+\epsilon)=-\lambda u(t+\epsilon),$since
$u(s)$ for $t+\epsilon<s<t$ is optimal for the problem starting
with initial conditions $Y(t+\epsilon),\,t+\epsilon.$ Therefore
\[
V(Y(t+\epsilon),t+\epsilon)-V(X,t+\epsilon)\leq-\lambda((u(t+\epsilon),\int_{t}^{t+\epsilon}u(s)ds))+C||\int_{t}^{t+\epsilon}u(s)ds||^{2}
\]
Using this inequality in (\ref{eq:2-24}) yields
\[
V(X,t)-V(X,t+\epsilon)\leq\frac{\lambda}{2}\int_{t}^{t+\epsilon}||u(s)||^{2}ds+\int_{t}^{t+\epsilon}\mathcal{F}(Y(s))ds-\lambda((u(t+\epsilon),\int_{t}^{t+\epsilon}u(s)ds))+C||\int_{t}^{t+\epsilon}u(s)ds||^{2}
\]
from which we obtain
\begin{equation}
\liminf_{\epsilon\rightarrow0}\frac{V(X,t+\epsilon)-V(X,t)}{\epsilon}\geq\frac{\lambda}{2}||u(t)||^{2}-\mathcal{F}(X)\label{eq:2-25}
\end{equation}
Next we have
\[
V(X,t+\epsilon)\leq\frac{\lambda}{2}\int_{t+\epsilon}^{T}||u(s)||^{2}ds+\int_{t+\epsilon}^{T}\mathcal{F}(Y(s)-\int_{t}^{t+\epsilon}u(\sigma)d\sigma)ds+\mathcal{F}_{T}(Y(T)-\int_{t}^{t+\epsilon}u(\sigma)d\sigma)
\]
therefore
\begin{align*}
V(X,t+\epsilon)-V(X,t) & \leq-\frac{\lambda}{2}\int_{t}^{t+\epsilon}||u(s)||^{2}ds-\int_{t}^{t+\epsilon}\mathcal{F}(Y(s))ds+\\
+ & \int_{t+\epsilon}^{T}(\mathcal{F}(Y(s)-\int_{t}^{t+\epsilon}u(\sigma)d\sigma)-\mathcal{F}(Y(s)))ds+\\
+ & \mathcal{F}_{T}(Y(T)-\int_{t}^{t+\epsilon}u(\sigma)d\sigma)-\mathcal{F}_{T}(Y(T))
\end{align*}
and using assumptions on $\mathcal{F}$, $\mathcal{F}_{T}$ it follows
that
\[
V(X,t+\epsilon)-V(X,t)\leq-\frac{\lambda}{2}\int_{t}^{t+\epsilon}||u(s)||^{2}ds-\int_{t}^{t+\epsilon}\mathcal{F}(Y(s))ds+
\]
\[
-((\int_{t+\epsilon}^{T}D_{X}\mathcal{F}(Y(s))ds+D_{X}\mathcal{F}_{T}(Y(T)),\int_{t}^{t+\epsilon}u(\sigma)d\sigma))+\frac{c}{2}(1+T)||\int_{t}^{t+\epsilon}u(\sigma)d\sigma||^{2}
\]
which means
\begin{align*}
V(X,t+\epsilon)-V(X,t) & \leq-\frac{\lambda}{2}\int_{t}^{t+\epsilon}||u(s)||^{2}ds-\int_{t}^{t+\epsilon}\mathcal{F}(Y(s))ds+\\
+ & \lambda((u(t+\epsilon),\int_{t}^{t+\epsilon}u(\sigma)d\sigma))+\frac{c}{2}(1+T)||\int_{t}^{t+\epsilon}u(\sigma)d\sigma||^{2}
\end{align*}
We then obtain
\begin{equation}
\limsup_{\epsilon\rightarrow0}\frac{V(X,t+\epsilon)-V(X,t)}{\epsilon}\leq\frac{\lambda}{2}||u(t)||^{2}-\mathcal{F}(X)\label{eq:2-26}
\end{equation}
and comparing with (\ref{eq:2-25}) we obtain immediately that $V(X,t)$
is differentiable in $t$ and the derivative is given by
\begin{equation}
\frac{\partial V}{\partial t}(X,t)=\frac{\lambda}{2}||u(t)||^{2}-\mathcal{F}(X)\label{eq:2-27}
\end{equation}
Recalling (\ref{eq:2-23}) we see immediately that $V(X,t)$ is solution
of the HJB equation (\ref{eq:2-8}). The 2nd estimate (\ref{eq:2-6})
is an immediate consequence of the equation and the estimate on $||D_{X}V(X,t)||.$
We next turn to check the addtional estimates (\ref{eq:2-7}). We
have
\[
D_{X}V(X_{1},t_{1})-D_{X}V(X_{2},t_{2})=Z_{X_{1}t_{1}}(t_{1})-Z_{X_{2}t_{2}}(t_{2})
\]
We assume $t_{1}<t_{2}$ then we can write
\begin{align}
Z_{X_{1}t_{1}}(t_{1})-Z_{X_{2}t_{2}}(t_{2}) & =\int_{t_{1}}^{t_{2}}(D_{X}\mathcal{F}(Y_{X_{1}t_{1}}(s))-D_{X}\mathcal{F}(Y_{X_{2}t_{2}}(s)))ds+\nonumber \\
+ & D_{X}\mathcal{F}_{T}(Y_{X_{1}t_{1}}(T))-D_{X}\mathcal{F}_{T}(Y_{X_{2}t_{2}}(T))\label{eq:2-28}
\end{align}
Using previously used majorations, we can check
\begin{equation}
||Z_{X_{1}t_{1}}(t_{1})-Z_{X_{2}t_{2}}(t_{2})||\leq\frac{\lambda c(T+1)}{\lambda-cT(T+1)}\left(||X_{1}-X_{2}||+|t_{2}-t_{1}|(1+T)c\frac{1+\,\max(||X_{1}||,||X_{2}||)}{\lambda-cT(T+1)}\right)\label{eq:2-29}
\end{equation}
and the first estimate (\ref{eq:2-7}) follows immediately. The 2nd
estimate (\ref{eq:2-7}) is a direct consequence of the HJB equation
and of the first estimate (\ref{eq:2-7}). So the value function has
the regularity indicated in the statement and satisfies the HJB equation.
Let us show that such a solution is necessarily unique. This is a
consequence of the verification property. Indeed consider any control
$v(.)\in L^{2}(t,T;\mathcal{H})$ and the state $X(s)$ solution of
(\ref{eq:2-2}). Let $V(x,t)$ be a solution of the HJB equation which
is $C^{1}$ and satisifies (\ref{eq:2-6}), (\ref{eq:2-7}). Then
the function $V(X(s),s)$ is differentiable and
\begin{align*}
\frac{d}{ds}V(X(s),s) & =\frac{\partial V}{\partial s}(X(s),s)+((D_{X}V(X(s),s),v(s)))=\\
= & -\mathcal{F}(X(s))+\frac{1}{2\lambda}||D_{X}V(X(s),s)||^{2}+((D_{X}V(X(s),s),v(s)))\\
\geq & -\mathcal{F}(X(s))-\frac{\lambda}{2}||v(s)||^{2}
\end{align*}
from which we get immediately by inegration $V(X,t)\leq J_{Xt}(v(.)).$
Now if we consider the equation
\begin{equation}
\dfrac{d\hat{X}(s)}{ds}=-\frac{1}{\lambda}D_{X}V(\hat{X}(s),s),\;\hat{X}(t)=X\label{eq:2-30}
\end{equation}
it has a unique solution, since $D_{X}V(X,s)$ is uniformly Lipschitz
in $X.$ If we set $\hat{v}(s)=-\dfrac{1}{\lambda}D_{X}V(\hat{X}(s),s),$
we see easily that $V(X,t)=J_{Xt}(\hat{v}(.))$. So $V(X,t)$ coincides
with the value function, and thus we have only one possible solution.
This completes the proof of the theorem. $\blacksquare$
\end{proof}
\section{THE MASTER EQUATION }
\subsection{FURTHER REGULARITY ASSUMPTIONS. }
We now assume that
\begin{equation}
\mathcal{F},\:\mathcal{F}_{T}\,\text{are}\:C^{2}\label{eq:3-1}
\end{equation}
The operators $D_{X}^{2}\mathcal{F}(X)$,$D_{X}^{2}\mathcal{F}_{T}(X)$
belong to $\mathcal{L}(\mathcal{H};\mathcal{H}).$ According to the
assumptions (\ref{eq:2-1}) we can assert that
\begin{equation}
||D_{X}^{2}\mathcal{F}(X)||,\:||D_{X}^{2}\mathcal{F}_{T}(X)||\leq c\label{eq:3-2}
\end{equation}
where the norm of the operators is the norm of $\mathcal{L}(\mathcal{H};\mathcal{H}).$
Recalling the equation (\ref{eq:2-13}) for $Y(s),$ we differentiate
formally with respect to $X$ to obtain
\begin{align}
D_{X}Y(s) & =I-\frac{s-t}{\lambda}D_{X}^{2}\mathcal{F}_{T}(Y(T))D_{X}Y(T)\label{eq:3-3}\\
-\frac{1}{\lambda} & \int_{t}^{T}D_{X}^{2}\mathcal{F}(Y(\sigma))D_{X}Y(\sigma)(s\wedge\sigma-t)d\sigma\nonumber
\end{align}
so, $D_{X}Y(.)$ appears as the solution of a linear equation, and
we see easily that it has one and only one solution verifying
\begin{equation}
\sup_{t\leq s\leq T}||D_{X}Y(s)||\leq\frac{\lambda}{\lambda-cT(T+1)}\label{eq:3-4}
\end{equation}
It is then easy to check that $D_{X}Y(s)$ is indeed the gradient
of $Y_{Xt}(s)$ with respect to $X$, and the estimate (\ref{eq:3-4})
is coherent with (\ref{eq:2-7}). Since $D_{X}V(X,t)=Z(t)=Z_{Xt}(t)$
with
\[
Z(t)=\int_{t}^{T}D_{X}\mathcal{F}(Y(s))ds+D_{X}\mathcal{F}_{T}(Y(T))
\]
we can differentiate to obtain
\begin{equation}
D_{X}^{2}V(X,t)=\int_{t}^{T}D_{X}^{2}\mathcal{F}(Y(s))D_{X}Y(s)ds+D_{X}^{2}\mathcal{F}_{T}(Y(T))D_{X}Y(T)\label{eq:3-5}
\end{equation}
and
\begin{equation}
||D_{X}^{2}V(X,t)||\leq\frac{\lambda c(T+1)}{\lambda-cT(T+1)}\label{eq:3-6}
\end{equation}
which is coherent with (\ref{eq:2-18}).
\subsection{MASTER EQUATION}
We obtain the Master equation, by simply differentiating the HJB equation
(\ref{eq:2-8}) with respect to $X.$ We set $\mathcal{U}(X,t)=D_{X}V(X,t)$.
We know from (\ref{eq:2-15}) that
\begin{equation}
||\mathcal{U}(X,t)||\leq\frac{\lambda(1+T)c(1+||X||)}{\lambda-cT(T+1)}\label{eq:3-7}
\end{equation}
The function $\mathcal{U}(X,t)$ maps $\mathcal{H}\times(0,T)$ into
$\mathcal{H}.$ From (\ref{eq:3-6}) we see that it is differentiable
in $X,$ with $D_{X}\mathcal{U}(X,t):\mathcal{H}\times(0,T)\rightarrow\mathcal{L}(\mathcal{H};\mathcal{H})$
and
\begin{equation}
||D_{X}\mathcal{U}(X,t)||\leq\frac{\lambda c(1+T)}{\lambda-cT(T+1)}\label{eq:3-8}
\end{equation}
From the HJB equation we see that $\mathcal{U}(X,t)$ is differentiable
in $t$ and satisfies the equation
\begin{align}
\frac{\partial\mathcal{U}}{\partial t}-\frac{1}{\lambda}D_{X}\mathcal{U}(X,t)\,\mathcal{U}(X,t)+D_{X}\mathcal{F}(X) & =0\label{eq:3-9}\\
\mathcal{U}(X,T)=D_{X}\mathcal{F}_{T}(X)\nonumber
\end{align}
We have the
\begin{prop}
\label{prop2} We make the assumptions of Theorem\ref{theo2-1} and
(\ref{eq:3-1}). Then equation (\ref{eq:3-9}) has one and only one
solution satisfying the estimates (\ref{eq:3-7}), (\ref{eq:3-8}).\end{prop}
\begin{proof}
We have only to prove uniqueness. Noting that
\[
D_{X}\mathcal{U}(X,t)\,\mathcal{U}(X,t)=\frac{1}{2}D_{X}||\mathcal{U}(X,t)||^{2}
\]
we see immediately from the equation that $\mathcal{U}(X,t)$ is
a gradient. So $\mathcal{U}(X,t)=D_{X}\tilde{V}(X,t).$ Therefore
(\ref{eq:3-9}) reads
\begin{align*}
D_{X}(\frac{\partial\tilde{V}}{\partial t}-\frac{1}{2\lambda}||D_{X}\tilde{V}||^{2}+\mathcal{F}(X)) & =0\\
D_{X}\tilde{V}(X,T)=D_{X}\mathcal{F}_{T}(X)
\end{align*}
We thus can write
\begin{align*}
\frac{\partial\tilde{V}}{\partial t}-\frac{1}{2\lambda}||D_{X}\tilde{V}||^{2}+\mathcal{F}(X) & =f(t)\\
\tilde{V}(X,T)=D_{X}\mathcal{F}_{T}(X)+h
\end{align*}
where $f(t)$ is purely function of $t$ and $h$ is a constant. If
we introduce the function $\varphi(t)$ solution of
\[
\frac{\partial\varphi}{\partial t}=f(t),\quad\varphi(T)=h
\]
the function $\tilde{V(}X,t)-\varphi(t)$ is solution of the HJB
equation (\ref{eq:2-8}) and satisfies the regularity properties of
Theorem \ref{theo2-1}. From the uniqueness of the solution of the
HJB equation we have $\tilde{V(}X,t)-\varphi(t)=V(X,t)$ the value
function, hence $\mathcal{U}(X,t)==D_{X}V(X,t),$ which proves the
uniqueness.$\blacksquare$
\end{proof}
\section{FUNCTIONALS ON PROBABILITY MEASURES}
\subsection{GENERAL COMMENTS}
If we have a functional on probability measures, the idea , introduced
by P.L. Lions \cite{PLL}, \cite{PL2} is to consider it as a functional
on random variables, whose probability laws are the probability measures.
Nevertheless, it is possible to work with the space of probability
measures directly, which is a metric space. The key issue is to define
the concept of gradient. For the space of probability measures, it
is the Wasserstein gradient. We shall see that, in fact, it is equivalent
to the gradient in the sence of the Hilbert space of random variables.
\subsection{WASSERSTEIN GRADIENT}
We consider the space $\mathcal{P}_{2}(R^{n})$ of probability measures
on $R^{n}$ , with second order moments, equipped with the Wasserstein
metric $W_{2}(\mu,\nu),$defined by
\begin{equation}
W_{2}^{2}(\mu,\nu)=\inf_{\gamma\in\Gamma(\mu,\nu)}\int_{R^{n}\times R^{n}}|\xi-\eta|^{2}\gamma(d\xi,d\eta)\label{eq:4-1}
\end{equation}
where $\Gamma(\mu,\nu)$ denotes the set of joint probability measures
on $R^{n}\times R^{n}$ such that the marginals are $\mu$ and $\nu$
respectively. It is useful to consider a probability space $\Omega,$$\mathcal{A},P$
and random variables in $\mathcal{H=}$$L^{2}(\Omega,\mathcal{A},P;R^{n}).$
We then can write $\mu=\mathcal{L}_{X}$ and
\[
W_{2}^{2}(\mu,\nu)=\inf_{\begin{array}{c}
X,Y\in\mathcal{H}\\
\mathcal{L}_{X}=\mu\\
\mathcal{L}_{Y}=\nu
\end{array}}E|X-Y|^{2}
\]
When the probability law has a density with respect to Lebesgue measure,
say $m(x)$ belonging to $L^{1}(R^{n})$ and positive, we replace
the law by its density. Note that $\int|x|^{2}m(x)dx$ $<+\infty.$
We call $L_{m}^{2}(R^{n};R^{n})$ the space of functions $f:\,R^{n}\rightarrow R^{n}$
such that $\int_{R^{n}}|f(x)|^{2}m(x)dx<+\infty.$ We consider functionals
$F(\mu)$ on $\mathcal{P}_{2}(R^{n}).$ If $\mu$ has a density $m$
we write $F(m).$ If $m\in L^{2}(R^{n})$, we say that $F(m)$ has
a Gâteaux differential at $m$, denoted by $\dfrac{\partial F(m)}{\partial m}(x)$
if we have
\begin{equation}
\lim_{\theta\rightarrow0}\dfrac{F(m+\theta\mu)-F(m)}{\theta}=\int_{R^{n}}\dfrac{\partial F(m)}{\partial m}(x)\mu(x)dx,\,\forall\mu\in L^{2}(R^{n})\label{eq:4-2}
\end{equation}
and $\dfrac{\partial F(m)}{\partial m}(x)\in L^{2}(R^{n}).$ For probability
densities, we shall extend this concept as follows. We say that $\dfrac{\partial F(m)}{\partial m}(x)\in L_{m}^{1}(R^{n})$
is the \emph{functional derivative} of $F$ at $m$ if for any sequence
of probability densities $m_{\epsilon}$ in $\mathcal{P}_{2}(R^{n})$
such that $W_{2}(m_{\epsilon},m)\rightarrow0$ then $\dfrac{\partial F(m)}{\partial m}(.)\in$$L_{m_{\epsilon}}^{1}(R^{n})$
and
\begin{equation}
\frac{F(m_{\epsilon})-F(m)-\int_{R^{n}}\dfrac{\partial F(m)}{\partial m}(x)(m_{\epsilon}(x)-m(x))dx}{W_{2}(m_{\epsilon},m)}\rightarrow0,\:\text{as}\:\epsilon\rightarrow0\label{eq:4-22}
\end{equation}
The function $\dfrac{\partial F(m)}{\partial m}(.)$ is called the
\emph{functional derivative }of $F(m)$ at point $m.$ Let us see
the connection with the concept of Wasserstein gradient on the metric
space $\mathcal{P}_{2}(R^{n})$ . We shall simply give the definition
and the expression of the gradient. For a detailed theory, we refer
to Otto \cite{OTT}, Ambrosio- Gigli- Savaré \cite{AGS}, Benamou-Brenier
\cite{BEB}, Brenier \cite{BRE}, Jordan-Kinderlehrer-Otto \cite{JKO},
Otto \cite{OTT}, Villani \cite{VIL}.
The first concept is that of optimal transport map, also called Brenier's
map. Given a probability $\nu\:\in\mathcal{P}_{2}(R^{n}),$the Monge
problem
\[
\inf_{\{T(.)|\,T(.)m=\nu\}}\int_{R^{n}}|x-T(x)|^{2}m(x)dx
\]
has a unique solution which is a gradient $T(x)=D\Phi(x).$ The notation
$T(.)m=\nu$ means that $\nu$ is the image of the probability whose
density is $m.$ The optimal solution is the Brenier's map. It is
noted $T_{m}^{\nu}$. We do not necessarily assume that $\nu$ has
a density. The following property holds
\begin{equation}
W_{2}^{2}(m,\nu)=\int_{R^{n}}|x-D\Phi(x)|^{2}m(x)dx\label{eq:4-3}
\end{equation}
This motivates the definition of tangent space $\mathcal{T}(m)$ of
the metric space $\mathcal{P}_{2}(R^{n})$ at point $m$ as
\[
\mathcal{T}(m)=\overline{\{D\Phi|\Phi\in C_{c}^{\infty}(R^{n})\}}
\]
We next consider curves on $\mathcal{P}_{2}(R^{n}),$ defined by densities
$m(t)\equiv m(t)(x)=m(x,t)$. The evolution of $m(t)$ is defined
by a velocity vector field $v(t)\equiv v(t)(x)=v(x,t)$ if $m(x,t)$
is the solution of the continuity equation
\begin{align}
\frac{\partial m}{\partial t}+\text{div }(v(x,t)m(x,t)) & =0\label{eq:4-4}\\
m(x,0)=m(x)\nonumber
\end{align}
We can interpret this equation in the sense of distributions, and
it is sufficient to assume that $\int_{0}^{T}\int_{R^{n}}|v(x,t)|^{2}m(x,t)dxdt<+\infty,\,\forall T<+\infty.$
This evolution model has a broad sprectrum and turns out to be equivalent
to the property that $m(t)$ is absolutely continuous in the sense
\[
W_{2}(m(s),m(t))\leq\int_{s}^{t}\rho(\sigma)d\sigma,\,\forall s<t
\]
with $\rho(.)$ locally $L^{2}.$ Now, for a given absolutely continuous
curve $m(t),$ the corresponding velocity field is not necessarily
unique. We can define the velocity field with minimum norm, i.e. $\hat{v}(x,t)$
solution of
\begin{equation}
\inf\left\{ \int_{0}^{T}\int_{R^{n}}|v(x,t)|^{2}m(x,t)dxdt\,\left|\begin{array}{cc}
\frac{\partial m}{\partial t}+\text{div }(v(x,t)m(x,t)) & =0\end{array}\right.\right\} \label{eq:4-5}
\end{equation}
The Euler equation for this minimization problem is
\[
\int_{0}^{T}\int_{R^{n}}\hat{v}(x,t).v(x,t)m(x,t)dt=0,\:\forall v(x,t)|\text{div }(v(x,t)m(x,t))=0\,\text{a.e. }
\]
which implies immediately that $\hat{v}(t)\in\mathcal{T}(m(t))$ a.e.
$t.$ Consequently, to a given absolutely continuous curve $m(t)$
we can associate a unique velocity field $\hat{v}(t)$ in the tangent
space $\mathcal{T}(m(t))$ a.e. $t.$ It is called the \uline{tangent
vector field }to the curve $m(t).$ It can be expressed by the following
formula
\begin{equation}
\hat{v}(x,t)=\lim_{\epsilon\rightarrow0}\frac{T_{m(t)}^{m(t+\epsilon)}(x)-x}{\epsilon}\label{eq:4-6}
\end{equation}
the limit being understood in $L_{m(t)}^{2}(R^{n};R^{n}).$ The function
$T_{m(t)}^{m(t+\epsilon)}(x)$ is uniquely defined. Since by (4.3)
, $||T_{m(t)}^{m(t+\epsilon)}(x)-x||_{L_{m(t)}^{2}}=W_{2}(m(t),m(t+\epsilon)),$
we see that. for any absolutely continuous curve
\begin{equation}
W_{2}(m(t),m(t+\epsilon))\leq C(t)\epsilon\label{eq:4-60}
\end{equation}
In the definition of the functional derivative, see (\ref{eq:4-22})
we can write
\begin{equation}
\frac{F(m_{\epsilon})-F(m)-\int_{R^{n}}\dfrac{\partial F(m)}{\partial m}(x)(m_{\epsilon}(x)-m(x))dx}{\epsilon}\rightarrow0,\:\text{as}\:\epsilon\rightarrow0\label{eq:4-22-1}
\end{equation}
provided the sequence $m_{\epsilon}$ is absolutely continuous.
Suppose that we consider the curve corrresponding to a gradient $D\Phi(x)$
where $\Phi(x)$ is smooth with compact support, i.e the curve $m(t)$
is defined by
\begin{align}
\frac{\partial m}{\partial t}+\text{div }(D\Phi(x)m(x,t)) & =0\label{eq:4-7}\\
m(x,0)=m(x)\nonumber
\end{align}
Since it is a gradient, $D\Phi(x)$ has minimal norm and we can claim
from (\ref{eq:4-6}) that
\begin{equation}
D\Phi(x)=\lim_{\epsilon\rightarrow0}\frac{T_{m}^{m(\epsilon)}(x)-x}{\epsilon}\:\text{in}\:L_{m}^{2}(R^{n};R^{n})\label{eq:4-8}
\end{equation}
We consider now a functional $F(m)$ on $\mathcal{P}_{2}(R^{n})$,
and limit ourselves to densities. We say that $F(m)$ is differentiable
at $m$ if there exists a function $\Gamma(x,m)$ belonging to the
tangent space $\mathcal{T}(m)$ with the property
\begin{equation}
\frac{F(m(\epsilon))-F(m)-\int_{R^{n}}\Gamma(x,m).(T_{m}^{m(\epsilon)}(x)-x)m(x)dx}{W_{2}(m,m(\epsilon))}\rightarrow0,\,\text{as }\epsilon\rightarrow0\label{eq:4-9}
\end{equation}
We recall that , see ( \ref{eq:4-3}) $W_{2}(m,m(\epsilon))=||T_{m}^{m(\epsilon)}(x)-x||_{L_{m}^{2}}.$
The function $\Gamma(x,m)$ is called the Wasserstein gradient and
denoted $\nabla F_{m}(m)(x).$ If we apply this property to the map
$m(t)$ defined by (\ref{eq:4-7}), this is equivalent to
\[
\dfrac{F(m(\epsilon))-F(m)}{\epsilon}\rightarrow\int_{R^{n}}\Gamma(x,m).D\Phi(x)m(x)dx
\]
From the continuity equation (\ref{eq:4-7}), using the regularity
of $\Phi,$ we can state that
\[
\dfrac{m(x,\epsilon)-m(x)}{\epsilon}\rightarrow-\text{div }(D\Phi(x)m(x)),\:\text{as }\epsilon\rightarrow0,\:\text{in the sense of distributions}
\]
If $F(m)$ has a functional derivative we obtain
\[
\dfrac{F(m(\epsilon))-F(m)}{\epsilon}\rightarrow-\int_{R^{n}}\frac{\partial F(m)}{\partial m}(x)\text{div }(D\Phi(x)m(x))dx
\]
Therefore we obtain
\begin{align*}
\int_{R^{n}}\Gamma(x,m).D\Phi(x)m(x)dx & =-\int_{R^{n}}\frac{\partial F(m)}{\partial m}(x)\text{div }(D\Phi(x)m(x))dx\\
& =\int_{R^{n}}D\frac{\partial F(m)}{\partial m}(x).D\Phi(x)m(x)dx
\end{align*}
If we assume that $D\frac{\partial F(m)}{\partial m}(x)\in L_{m}^{2}(R^{n};R^{n}),$
we can replace $D\Phi(x)$ by any element of $\mathcal{T}(m)$. Since
$\Gamma(x,m)$ and $D\frac{\partial F(m)}{\partial m}(x)$ belong
to $\mathcal{T}(m),$ it follows that
\begin{equation}
\nabla F_{m}(m)(x)=D\frac{\partial F(m)}{\partial m}(x)\label{eq:4-10}
\end{equation}
So the Wasserstein gradient is simply the gradient of the functional
derivative.
\begin{rem}
\label{rem3-1} The concept of functional derivative, defined in (\ref{eq:4-22})
uses a sequence of probability densities $m_{\epsilon}\rightarrow m,$
so it is not equivalent to the concept of Gâteaux differential in
the space $L^{2}(R^{n}),$ which requires to remove the assumptions
of positivity and $\int_{R^{n}}m(x)dx=1.$ We will develop the differences
in examples in which explicit formulas are available, see section
\ref{sec:QUADRATIC-CASE}.
\end{rem}
\subsection{\label{sub:GRADIENT-IN-THE}GRADIENT IN THE HILBERT SPACE $\mathcal{H}$.}
The functional $F(m)$ can now be written as a functional $\mathcal{F}(X)$
on $\mathcal{H}$ , with $m=\mathcal{L}_{X}.$ We assume that random
variables with densities form a dense subspace of $\mathcal{H}.$
Consider a random variable $Y\in\mathcal{H}$ and let $\pi(x,y)$
be the joint probability density on $R^{n}\times R^{n}$ of the pair
$(X,Y).$ So $m(x)=\int_{R^{n}}\pi(x,y)dy.$ Consider then the random
variable $X+tY.$ Its probability density is given by
\[
m(x,t)=\int_{R^{n}}\pi(x-ty,y)dy
\]
and it sastisfies the continuity equation
\[
\frac{\partial m}{\partial t}=-\text{div }(\int_{R^{n}}\pi(x-ty,y)ydy)
\]
We have $\mathcal{F}(X+tY)=F(m(t)).$ Next
\[
\lim_{t\rightarrow0}\frac{\mathcal{F}(X+tY)-\mathcal{F}(X)}{t}=((D_{X}\mathcal{F}(X),Y))
\]
and
\begin{align*}
\lim_{t\rightarrow}\frac{F(m(t))-F(m)}{t} & =-\int_{R^{n}}\frac{\partial F(m)}{\partial m}(x)\text{div }(\int_{R^{n}}\pi(x,y)ydy)dx\\
& =\int_{R^{n}}D\frac{\partial F(m)}{\partial m}(x).(\int_{R^{n}}\pi(x,y)ydy)dx\\
& =((D\frac{\partial F(m)}{\partial m}(X),Y))
\end{align*}
Thus necessarily
\begin{equation}
D\frac{\partial F(m)}{\partial m}(X)=\nabla F_{m}(m)(X)=D_{X}\mathcal{F}(X)\label{eq:4-11}
\end{equation}
So, the gradient in $\mathcal{H}$ reduces to the Wasserstein gradient,
in which the argument is replaced with the random variable. In the
sequel, we will use the gradient in $\mathcal{H}$.
\section{MEAN FIELD TYPE CONTROL PROBLEM}
\subsection{PRELIMINARIES}
Consider a function $f(x,m)$ defined on $R^{n}\times\mathcal{P}_{2}(R^{n})$.
As usual we consider only $m$ which are densities of probability
measures, and use also the notation $f(x,\mathcal{L}_{X}).$ We then
define $\mathcal{F}(X)=Ef(X,\mathcal{L}_{X}).$ This implies
\begin{equation}
\mathcal{F}(X)=\Phi(m)=\int_{R^{n}}f(x,m)m(x)dx\label{eq:5-1}
\end{equation}
We next consider the functional derivative
\begin{equation}
\frac{\partial\Phi(m)}{\partial m}(x)=F(x,m)=f(x,m)+\int_{R^{n}}\frac{\partial f}{\partial m}(\xi,m)(x)m(\xi)d\xi\label{eq:5-2}
\end{equation}
and we have
\begin{equation}
D_{X}\mathcal{F}(X)=D_{x}F(X,\mathcal{L}_{X})\label{eq:5-3}
\end{equation}
We make the assumptions
\begin{equation}
|D_{x}F(x,m)|\leq\dfrac{c}{2}(1+|x|+(\int|\xi|^{2}m(\xi)d\xi)^{\frac{1}{2}})\label{eq:5-4}
\end{equation}
\begin{equation}
|D_{x}F(x_{1},m_{1})-D_{x}F(x_{2},m_{2})|\leq\dfrac{c}{2}(|x_{1}-x_{2}|+W_{2}(m_{1},m_{2}))\label{eq:5-5}
\end{equation}
which implies immediately the properties (\ref{eq:2-1}), (\ref{eq:2-1-1}).
\subsection{EXAMPLES}
We consider first quadratic functionals. We use the notation $\bar{x}=\int_{R^{n}}xm(x)dx.$
We then consider
\begin{equation}
f(x,m)=\frac{1}{2}(x-S\bar{x})^{*}\bar{Q}(x-S\bar{x})+\frac{1}{2}x^{*}Qx\label{eq:5-53}
\end{equation}
then assuming that $\int_{R^{n}}m(x)dx=1,$ i.e. $m$ is a probability
density we have
\begin{equation}
F(x,m)=\frac{1}{2}x^{*}(Q+\bar{Q})x+\frac{1}{2}\bar{x}^{*}S^{*}\bar{Q}S\bar{x}-\bar{x}^{*}(\bar{Q}S+S^{*}\bar{Q}-S^{*}\bar{Q}S)x\label{eq:5-54}
\end{equation}
\begin{equation}
D_{x}F(x,m)=(Q+\bar{Q})x-(\bar{Q}S+S^{*}\bar{Q}-S^{*}\bar{Q}S)\bar{x}\label{eq:5-55}
\end{equation}
We see easily that assumptions (\ref{eq:5-4}),(\ref{eq:5-5}) are
satisfied.
We can give an additonal example
\begin{equation}
f(x,m)=\frac{1}{2}\int_{R^{n}}K(x,\xi)m(\xi)d\xi\label{eq:5-6}
\end{equation}
with $K(x,\xi)=K(\xi,x)$ and
\begin{equation}
|K(x_{1},\xi_{1})-K(x_{2},\xi_{2})|\leq C(1+|x_{1}|+|x_{2}|+|\xi_{1}|+|\xi_{2}|)(|x_{1}-x_{2}|+|\xi_{1}-\xi_{2}|)\label{eq:5-67}
\end{equation}
\begin{align}
|D_{x}K(x_{1},\xi_{1})-D_{x}K(x_{2},\xi_{2})| & \leq\dfrac{c}{2}(|x_{1}-x_{2}|+|\xi_{1}-\xi_{2}|)\label{eq:5-7}\\
|D_{x}K(0,0)| & \leq\dfrac{c}{2}\nonumber
\end{align}
We have
\[
\frac{\partial f}{\partial m}(\xi,m)(x)=\frac{1}{2}K(\xi,x)=\frac{1}{2}K(x,\xi)
\]
hence $\int_{R^{n}}\frac{\partial f}{\partial m}(\xi,m)(x)m(\xi)d\xi=f(x,m)$
which implies
\begin{equation}
F(x,m)=2f(x,m)=\int_{R^{n}}K(x,\xi)m(\xi)d\xi\label{eq:5-7-1}
\end{equation}
We thus have
\begin{align*}
|D_{x}F(x,m)|| & \leq\int_{R^{n}}|D_{x}K(x,\xi)|m(\xi)d\xi\leq\\
& \leq\dfrac{c}{2}(1+|x|+\int|\xi|m(\xi)d\xi)\leq\\
& \leq\dfrac{c}{2}(1+|x|+(\int|\xi|^{2}m(\xi)d\xi)^{\frac{1}{2}})
\end{align*}
If we take 2 densities $m_{1},m_{2},$ we may consider 2 random variables
$\Xi_{1},\Xi_{2}$ with the probabilities $m_{1},m_{2}.$ Therefore
\begin{align*}
|D_{x}F(x_{1},m_{1})-D_{x}F(x_{2},m_{2})| & \leq|\int D_{x}(K(x_{1},\xi)-K(x_{2},\xi))m_{1}(\xi)d\xi||+\\
+ & |E\:D_{x}(K(x_{2},\Xi_{1})-K(x_{2},\Xi_{2}))|\leq\\
\dfrac{c}{2}|x_{1}-x_{2}| & +\dfrac{c}{2}\sqrt{E|\Xi_{1}-\Xi_{2}|^{2}}
\end{align*}
and since $\Xi_{1},\Xi_{2}$ are arbitrary, with marginals $m_{1},m_{2}$we
can write (\ref{eq:5-5}). In the sequel we also consider a functional
$h(x,m)$ with exactly the same properties as $f$ and write
\begin{align}
F_{T}(x,m) & =h(x,m)+\int_{R^{n}}\frac{\partial h}{\partial m}(\xi,m)(x)m(\xi)d\xi\label{eq:5-8}\\
\mathcal{F}_{T}(X) & =\int_{R^{n}}h(x,m)m(x)dx,\;D_{X}\mathcal{F}_{T}(X)=D_{x}F_{T}(X,\mathcal{L}_{X})\nonumber
\end{align}
\subsection{\label{sub:MEAN-FIELD-TYPE}MEAN FIELD TYPE CONTROL PROBLEM}
We can formulate the following mean field type control problem. Let
us consider a dynamical system in $R^{n}$
\begin{align}
\frac{dx}{ds} & =v(x(s),s)\label{eq:5-9}\\
x(t) & =\xi\nonumber
\end{align}
where $v(x,s)$ is a feedback to be optimized. The initial condition
is a random variable with probability density $m(x).$The Fokker-Planck
equation of the evolution of the density is
\begin{align}
\frac{\partial m}{\partial s}+\text{div}(v(x)m) & =0\label{eq:5-10}\\
m(x,t)=m(x)\nonumber
\end{align}
We denote the solution by $m_{v(.)}(x,s)$. Similarly we call the
solution of (\ref{eq:5-9}) $x(s;v(.)).$ We then consider the cost
functional
\begin{align}
J_{m,t}(v(.)) & =\frac{\lambda}{2}\int_{t}^{T}\int_{R^{n}}|v(x,s)|^{2}m_{v(.)}(x,s)dxds+\int_{t}^{T}\int_{R^{n}}m_{v(.)}(x,s)f(x,m_{v(.)}(s))dxds+\label{eq:5-11}\\
& +\int_{R^{n}}m_{v(.)}(x,T)h(x,m_{v(.)}(T))dx\nonumber
\end{align}
which is equivalent to the expression
\begin{equation}
J_{m,t}(v(.))=\frac{\lambda}{2}\int_{t}^{T}E|v(x(s;v(.)))|^{2}ds+\int_{t}^{T}Ef(x(s;v(.)),\mathcal{L}_{x(s)})ds+Eh(x(T;v(.)),\mathcal{L}_{x(T)})\label{eq:5-12}
\end{equation}
This is a standard mean field type control problem, not a mean field
game. In \cite{BFY} we have associated to it a coupled system of
HJB and FP equations, see p. 18, which reads here
\begin{align}
-\frac{\partial u}{\partial s}+\frac{1}{2\lambda}|Du|^{2} & =F(x,m(s))\label{eq:5-13}\\
u(x,T) & =F_{T}(x,m(s))\nonumber \\
\frac{\partial m}{\partial s}-\frac{1}{\lambda}\text{div} & (Du\,m)=0\nonumber \\
m(x,t) & =m(x)\nonumber
\end{align}
This system expresses a necessary condition of optimality. The function
$u(x,t)$ is not a value function, but an adjoint variable to the
optimal state, which is $m(x,s)$. The optimal feedback is given by
\begin{equation}
\hat{v}(x,s)=-\frac{1}{\lambda}Du(x,s)\label{eq:5-14}
\end{equation}
We proceed formally, although we shall be able to give an explicit
solution of this system. If $\hat{v}(x,s)$ is the optimal feedback,
then the value function $V(m,t)=J_{m,t}(\hat{v}(.))$ is given by
\begin{align}
V(m,t) & =\frac{1}{2\lambda}\int_{t}^{T}\int_{R^{n}}m(x,s)|Du(x,s)|^{2}dxds+\int_{t}^{T}\int_{R^{n}}m(x,s)f(x,m(s))dxds+\label{eq:5-15}\\
+ & \int_{R^{n}}m(x,T)h(x,m(T))dx\nonumber
\end{align}
The value function is solution of Bellman equation, see \cite{BFY1},
\cite{LPI}, written formally (it will be justified later)
\begin{align}
\frac{\partial V}{\partial t}-\frac{1}{2\lambda}\int_{R^{n}}|D_{\xi}\frac{\partial V(m,t)}{\partial m}(\xi)|^{2}m(\xi)d\xi+\int_{R^{n}}f(\xi,m)m(\xi)d\xi & =0\label{eq:5-16}\\
V(m,T)=\int_{R^{n}}h(\xi,m)m(\xi)d\xi\nonumber
\end{align}
\subsection{SCALAR MASTER EQUATION}
We derive the master equation, by considering the function
\[
U(x,m,t)=\frac{\partial V(m,t)}{\partial m}(x)
\]
and we note that
\[
\frac{\partial U}{\partial m}(x,m,t)(\xi)=\frac{\partial^{2}V(m,t)}{\partial m^{2}}(x,\xi)
\]
therefore the function is symmetric in $x,\xi$ which means
\[
\frac{\partial U}{\partial m}(x,m,t)(\xi)=\frac{\partial U}{\partial m}(\xi,m,t)(x)
\]
By differentiating (\ref{eq:5-16}) in $m$, and using the symmetry
property, we obtain the equation
\begin{align}
\frac{\partial U}{\partial t}-\frac{1}{\lambda}\int_{R^{n}}D_{\xi}\frac{\partial U}{\partial m}(x,m,t)(\xi).D_{\xi}U(\xi,m,t)m(\xi)d\xi-\label{eq:5-17}\\
-\frac{1}{2\lambda}|D_{x}U(x,m,t)|^{2}+F(x,m)=0\nonumber \\
U(x,m,T)=F_{T}(x,m)\nonumber
\end{align}
This function allows to uncouple the system of HJB-FP equations, given
in (\ref{eq:5-13}). Indeed, we first solve the FP equation, replacing
$u$ by $U,$ i.e.
\begin{align}
\frac{\partial m}{\partial s}-\frac{1}{\lambda}\text{div}(DU\,m) & =0\label{eq:5-18}\\
m(x,t) & =m(x)\nonumber
\end{align}
then $u(x,s)=U(x,m(s),s)$ is solution of the HJB equation (\ref{eq:5-13}),
as easily checked. In particular , we have
\begin{equation}
u(x,t)=U(x,m,t)\label{eq:5-19}
\end{equation}
\subsection{VECTOR MASTER EQUATION}
We next consider $\mathcal{U}(x,m,t)=D_{x}U(x,m,t).$ Differentiating
(\ref{eq:5-17}) we can write
\begin{equation}
\frac{\partial\mathcal{U}}{\partial t}-\frac{1}{\lambda}\int_{R^{n}}D_{\xi}\frac{\partial\mathcal{U}}{\partial m}(x,m,t)(\xi)\,\mathcal{U}(\xi,m,t)m(\xi)d\xi-\frac{1}{\lambda}D_{x}\mathcal{U}(x,m,t)\,\mathcal{U}(x,m,t)+D_{x}F(x.m)=0\label{eq:5-20}
\end{equation}
\[
\mathcal{U}(x,m,T)=D_{x}F_{T}(x.m)
\]
\section{CONTROL PROBLEM IN THE SPACE $\mathcal{H}$.}
\subsection{FORMULATION}
If we set
\begin{align}
\mathcal{F}(X) & =Ef(X,\mathcal{L}_{X})=\int f(x,m)m(x)dx\label{eq:6-1}\\
\mathcal{F}_{T}(X) & =Eh(X,\mathcal{L}_{X})=\int h(x,m)m(x)dx\nonumber
\end{align}
\begin{align}
F(x,m) & =f(x,m)+\int\frac{\partial f(\xi,m)}{\partial m}(x)m(\xi)d\xi\label{eq:6-2}\\
F_{T}(x,m) & =h(x,m)+\int\frac{\partial h(\xi,m)}{\partial m}(x)m(\xi)d\xi\nonumber
\end{align}
We have
\begin{align}
D_{X}\mathcal{F}(X) & =D_{x}F(X,\mathcal{L}_{X})\label{eq:6-3}\\
D_{X}\mathcal{F}_{T}(X) & =D_{x}F_{T}(X,\mathcal{L}_{X})\nonumber
\end{align}
We assume that
\begin{align}
|D_{x}F(x_{1},m_{1})-D_{x}F(x_{2},m_{2})| & \leq\frac{c}{2}(|x_{1}-x_{2}|+W_{2}(m_{1},m_{2}))\label{eq:6-4}\\
|D_{x}F_{T}(x_{1},m_{1})-D_{x}F_{T}(x_{2},m_{2})| & \leq\frac{c}{2}(|x_{1}-x_{2}|+W_{2}(m_{1},m_{2}))\nonumber
\end{align}
\begin{align}
|D_{x}F(x,m)| & \leq\dfrac{c}{2}(1+|x|+\sqrt{\int_{R^{n}}|\xi|^{2}m(\xi)d\xi})\label{eq:6-4-1}\\
|D_{x}F_{T}(x,m)| & \leq\dfrac{c}{2}(1+|x|+\sqrt{\int_{R^{n}}|\xi|^{2}m(\xi)d\xi})\nonumber
\end{align}
It follows that
\begin{align*}
||D_{X}\mathcal{F}(X_{1})-D_{X}\mathcal{F}(X_{2})|| & \leq||D_{x}F(X_{1},\mathcal{L}_{X_{1}})-D_{x}F(X_{2},\mathcal{L}_{X_{1}})||+\\
+ & ||D_{x}F(X_{2},\mathcal{L}_{X_{1}})-D_{x}F(X_{2},\mathcal{L}_{X_{2}})||
\end{align*}
\[
\leq c||X_{1}-X_{2}||
\]
and similar estimate for $\mathcal{F}_{T}.$Therefore the set up
of section \ref{sub:SETTING-OF-THE} is satisfied. We can reinterpret
the problem (\ref{eq:5-9}), (\ref{eq:5-12}) or (\ref{eq:5-10}),
(\ref{eq:5-11}) as (\ref{eq:2-2}), (\ref{eq:2-3}) which has been
completely solved in Theorem \ref{theo2-1}. We shall study the solution
of the abstract setting. Of course, the initial state $X$ has probability
law $\mathcal{L}_{X}=m.$
\subsection{\label{sub:INTERPRETATION-OF-THE}INTERPRETATION OF THE SOLUTION }
The key point of the proof of Theorem \ref{theo2-1} is the study
of the system (\ref{eq:2-11}) which has one and only one solution.
We proceed formally. Consider the HJB-FP system (\ref{eq:5-13}).
The initial conditions are the pair $(m,t),$ so we can write the
solution as $u_{m,t}(x,s)$, $m_{m,t}(x,s).$ We introduce the differential
equation
\begin{align}
\frac{dy}{ds} & =-\frac{1}{\lambda}Du(y(s),s)\label{eq:6-5}\\
y(t) & =x\nonumber
\end{align}
The solution ( if it exists) can be written $y_{xmt}(s).$Now let
us set $z_{xmt}(s)=Du_{mt}(y_{xmt}(s),s).$ Differentiating the HJB
equation (\ref{eq:5-13}) and computing the derivative $\dfrac{dz}{ds}$
we obtain
\begin{align}
-\dfrac{dz}{ds} & =D_{x}F(y(s),m(s))\label{eq:6-6}\\
z(T) & =D_{x}F_{T}(y(T),m(T))\nonumber
\end{align}
Now, from the definition of $m(s)$ solution of the FP equation, we
can write
\begin{equation}
m(s)=y_{mt}(s)(.)(m)\label{eq:6-7}
\end{equation}
in which we have used the notation $y_{mt}(s)(x)=y_{mt}(x,s)=y_{xmt}(s)$
and $y_{mt}(s)(.)(m)$ means the image measure of $m$ by the map
$y_{mt}(s)(.).$ So we can write the system (\ref{eq:6-5}), (\ref{eq:6-6})
as
\begin{align}
\frac{d^{2}y}{ds^{2}} & =\frac{1}{\lambda}D_{x}F(y(s),y(s)(.)(m))\label{eq:6-8}\\
y(t)=x & \;\frac{dy}{ds}(T)=-\frac{1}{\lambda}D_{x}F_{T}(y(T),y(T)(.)(m))\nonumber
\end{align}
This is also written in integral form
\begin{align}
y(s) & =x-\frac{s-t}{\lambda}D_{x}F_{T}(y(T),y(T)(.)(m))\label{eq:6-9}\\
- & \frac{1}{\lambda}\int_{t}^{T}D_{x}F(y(\sigma),y(\sigma)(.)(m))(s\wedge\sigma-t)d\sigma\nonumber
\end{align}
Now if we take $y_{X,\mathcal{L}_{X},t}(s),$ then $y(s)(.)(\mathcal{L}_{X})=\mathcal{L}_{y(s)}$
. Writing $y_{X,\mathcal{L}_{X},t}(s)=Y(s)$ to emphasize that we
are dealing with a random variable, we can write (\ref{eq:6-9}) as
\begin{align}
Y(s) & =X-\frac{s-t}{\lambda}D_{x}F_{T}(Y(T),\mathcal{L}_{Y(T)})-\label{eq:6-10}\\
- & \frac{1}{\lambda}\int_{t}^{T}D_{x}F(Y(\sigma),\mathcal{L}_{Y(\sigma)})(s\wedge\sigma-t)d\sigma\nonumber
\end{align}
which is nothing else than (\ref{eq:2-3}) recalling the values of
$D_{X}\mathcal{F}(X),\:D_{X}\mathcal{F}_{T}(X)$, cf (\ref{eq:6-3}).
We know from Theorem \ref{theo2-1}that (\ref{eq:6-10}) has one and
only one solution in $C^{0}([t,T];\mathcal{H})$ and in fact in $C^{2}([t,T];\mathcal{H}).$
This result, of course, does not allow to go from (\ref{eq:6-10})
to (\ref{eq:6-9}), but it easy to mimic the proof. We state the result
in the following
\begin{prop}
\label{prop3}We assume (\ref{eq:6-4}),(\ref{eq:6-4-1}) and condition
(\ref{eq:2-5}). For given $m,t$ there exists one and only one solution
$y_{mt}(x,s)$ of (\ref{eq:6-9}) in the space $C(t,T;L_{m}^{2}(R^{n};R^{n})).$ \end{prop}
\begin{proof}
We use a fixed point argument. We define a map from $C(t,T;L_{m}^{2}(R^{n};R^{n}))$
to itself. Let $z(x,s)$ a function in $C(t,T;L_{m}^{2}(R^{n};R^{n})).$
We define
\begin{align*}
\zeta(x,s) & =x-\frac{s-t}{\lambda}D_{x}F_{T}(z(x,T),z(T)(.)(m))-\\
- & \frac{1}{\lambda}\int_{t}^{T}D_{x}F(z(x,\sigma),z(\sigma)(.)(m))(s\wedge\sigma-t)d\sigma
\end{align*}
We have
\begin{align*}
|\zeta(x,s)| & \leq|x|+\dfrac{T}{\lambda}\dfrac{c}{2}(1+|z(x,T)|+(\int_{R^{n}}|z(\xi,T)|^{2}m(\xi)d\xi)^{1/2})+\\
+ & \dfrac{cT}{2\lambda}\int_{t}^{T}(1+|z(x,\sigma)|+(\int_{R^{n}}|z(\xi,\sigma)|^{2}m(\xi)d\xi)^{1/2})d\sigma
\end{align*}
hence , from norm properties
\begin{align*}
\sqrt{\int_{R^{n}}|\zeta(x,s)|^{2}m(x)dx} & \leq\sqrt{\int_{R^{n}}|x|^{2}m(x)dx}+\dfrac{T}{\lambda}\dfrac{c}{2}(1+2(\int_{R^{n}}|z(\xi,T)|^{2}m(\xi)d\xi)^{1/2})+\\
+ & \dfrac{cT}{2\lambda}\int_{t}^{T}(1+2(\int_{R^{n}}|z(\xi,\sigma)|^{2}m(\xi)d\xi)^{1/2})d\sigma
\end{align*}
and we conclude easily that $\zeta$ belongs to $C(t,T;L_{m}^{2}(R^{n};R^{n})).$
We set $\zeta=\mathcal{T}$(z). Using the assumptions and similar
estimates, one checks that $\mathcal{T}$ is a contraction. We prove
indeed that
\begin{equation}
||\mathcal{T}(z_{1})-\mathcal{T}(z_{2})||_{C(t,T;L_{m}^{2})}\leq(1-\dfrac{cT(T+1)}{\lambda})||z_{1}-z_{2}||_{C(t,T;L_{m}^{2})}\label{eq:6-11}
\end{equation}
$\blacksquare$
\end{proof}
It follows immediately that the solution $y_{xmt}(s)=y_{mt}(x,s)$
satisfies the estimate
\begin{equation}
||y_{mt}(.)||_{C(t,T;L_{m}^{2})}\leq\frac{\lambda\sqrt{\int_{R^{n}}|x|^{2}m(x)dx}+cT(T+1)}{\lambda-cT(T+1)}\label{eq:6-12}
\end{equation}
Since $Y_{Xt}(s)=y_{X,\mathcal{L}_{X},t}(s)$ we deduce the first
estimate (\ref{eq:2-15}). We consider next
\begin{align}
z_{xmt}(s) & =z_{mt}(x,s)=D_{x}F_{T}(y(x,T),y(T)(.)(m))+\label{eq:6-12-1}\\
+ & \int_{s}^{T}D_{x}F(y(x,\sigma),y(\sigma)(.)(m))d\sigma\nonumber
\end{align}
and from the assumption (\ref{eq:6-4-1}) we obtain easily
\[
||z_{mt}(.)||_{C(t,T;L_{m}^{2})}\leq c(1+T)(1+||y_{mt}(.)||_{C(t,T;L_{m}^{2})})
\]
hence
\begin{equation}
||z_{mt}(.)||_{C(t,T;L_{m}^{2})}\leq\lambda c(1+T)\frac{\sqrt{\int_{R^{n}}|x|^{2}m(x)dx}+1}{\lambda-cT(T+1)}\label{eq:6-13}
\end{equation}
Clearly $Z(s)=Z_{Xt}(s)=z_{X,\mathcal{L}_{X},t}(s)$, see (\ref{eq:2-11}),
and we recover the 2nd estimate (\ref{eq:2-15}).
We can give more properies on $y_{xmt}(s).$ We write first
\begin{align*}
y_{mt}(x_{1},s)-y_{mt}(x_{2},s) & =x_{1}-x_{2}-\frac{s-t}{\lambda}(D_{x}F_{T}(y_{mt}(x_{1},T),y_{mt}(T)(.)(m))-D_{x}F_{T}(y_{mt}(x_{2},T),y_{mt}(T)(.)(m)))-\\
- & \frac{1}{\lambda}\int_{t}^{T}(D_{x}F(y_{mt}(x_{1},\sigma),y_{mt}(\sigma)(.)(m))-D_{x}F(y_{mt}(x_{2},\sigma),y_{mt}(\sigma)(.)(m)))(s\wedge\sigma-t)d\sigma
\end{align*}
From (\ref{eq:6-4}) we obtain easily
\begin{equation}
\sup_{t<s<T}|y_{mt}(x_{1},s)-y_{mt}(x_{2},s)\leq\frac{\lambda|x_{1}-x_{2}|}{\lambda-cT(T+1)}\label{eq:6-14}
\end{equation}
Also
\begin{equation}
\sup_{t<s<T}|y_{mt}(x,s)|\leq\lambda\frac{|x|+\dfrac{Tc(1+T)(1+\sqrt{\int_{R^{n}}|\xi|^{2}m(\xi)d\xi})}{\lambda-cT(T+1)}}{\lambda-cT(T+1)}\label{eq:6-15}
\end{equation}
A similar estimate holds for $\sup_{t<s<T}|z_{mt}(x,s)|$.
\section{BELLMAN EQUATION AND MASTER EQUATION}
\subsection{THE VALUE FUNCTION }
The value function of the control problem in $\mathcal{H}$ is given
by
\[
V(X,t)=\frac{1}{2\lambda}\int_{t}^{T}||Z(s)||^{2}ds+\int_{t}^{T}\mathcal{F}(Y(s))ds+\mathcal{F}_{T}(Y(T))
\]
in which $Y(s)=$$y_{X,\mathcal{L}_{X},t}(s)$ and $Z(s)=z_{X,\mathcal{L}_{X},t}(s).$
From this representation and the definition of $\mathcal{F}$ and
$\mathcal{F}_{T}$ we can assert that $V(X,t)$ depends only on $\mathcal{L}_{X}$
and thus can be written $V(m,t)$ with
\begin{align}
V(m,t) & =\frac{1}{2\lambda}\int_{t}^{T}\int_{R^{n}}|z_{xmt}(s)|^{2}m(x)dxds+\int_{t}^{T}\int_{R^{n}}f(y_{xmt}(s),y_{mt}(s)(.)(m))m(x)dxds+\label{eq:7-1}\\
& +\int_{R^{n}}h(y_{xmt}(T),y_{mt}(T)(.)(m))m(x)dx\nonumber
\end{align}
From (\ref{eq:6-13}) we have
\[
\int_{t}^{T}\int_{R^{n}}|z_{xmt}(s)|^{2}m(x)dxds\leq\dfrac{T\lambda^{2}c^{2}(1+T)^{2}(1+\int_{R^{n}}|x|^{2}m(x)dx)}{(\lambda-cT(T+1))^{2}}
\]
and $|\mathcal{F}(Y(s))|\leq C(1+||Y(s)||^{2}),$ therefore
\[
|\int_{R^{n}}f(y_{xmt}(s),y_{mt}(s)(.)(m))m(x)dx|\leq C(1+\int|y_{xmt}(s)|^{2}m(x)dx)
\]
and from the estimate (\ref{eq:6-12}) we obtain
\[
|\int_{t}^{T}\int_{R^{n}}f(y_{xmt}(s),y_{mt}(s)(.)(m))m(x)dxds|\leq CT[1+\dfrac{\lambda^{2}\int_{R^{n}}|x|^{2}m(x)dx+T^{2}(T+1)^{2}}{(\lambda-cT(T+1))^{2}}]
\]
and the third term in the right hand side of $(\ref{eq:7-1}$) satisfies
a similar estimate. We thus have obtained
\begin{equation}
|V(m,t)|\leq C(1+\int_{R^{n}}|x|^{2}m(x)dx)\label{eq:7-2}
\end{equation}
which is, of course, equivalent to the 1st estimate (\ref{eq:2-6}).
We turn now to $U(x,m,t)=$$\dfrac{\partial V(m,t)}{\partial m}(x)$.
We have seen formally in (\ref{eq:5-19}) that $U(x,m,t)=u(x,t)=u_{mt}(x,t).$
We need to prove it. We begin by giving a solution to the system HJB-FP
equations (\ref{eq:5-13}). We have the
\begin{lem}
\label{lem4} We make the assumptions of Proposition \ref{prop3}.
We can give an explicit formula to the system (\ref{eq:5-13}). We
have
\begin{align}
u_{mt}(x,t) & =\dfrac{1}{2\lambda}\int_{t}^{T}|z_{xmt}(s)|^{2}ds+\int_{t}^{T}F(y_{xmt}(s),y_{mt}(s)(.)(m))ds+\label{eq:7-3}\\
& +F_{T}(y_{xmt}(T),y_{mt}(T)(.)(m))\nonumber
\end{align}
and $m_{mt}(s)=y_{mt}(s)(.)(m).$\end{lem}
\begin{proof}
Indeed, if we look at $F(x,m(s))$ and $F_{T}(x,m(T))$ in which $m(.)$
is frozen, the HJB equation appears as a standard one for a deterministic
control problem. This problem is simply
\begin{align*}
\dfrac{dx}{ds} & =v(s)\\
x(t) & =x
\end{align*}
\[
J_{xt}(v(.))=\frac{\lambda}{2}\int_{t}^{T}|v(s)|^{2}ds+\int_{t}^{T}F(x(s),m(s))ds+F_{T}(x(T),m(T))
\]
in which the function $m(s)$ is frozen, but not arbitrary. It is
the function solution of the FP equation, in the system (\ref{eq:5-13})
If we write the necessary conditions of optimality, one checks easily
that in view of the specific value of $m(s),$the optimal state is
$y_{xmt}(s)$ and the optimal control is $-\dfrac{1}{\lambda}z_{xmt}(s).$
In plugging these values in the cost function, we obtain formula (\ref{eq:7-3}).
$\blacksquare$
\end{proof}
We may assume that
\begin{align}
|F(x,m)| & ,|F_{T}(x,m)|\leq C(1+|x|^{2}+\int|\xi|^{2}m(\xi)d\xi)\label{eq:7-4}
\end{align}
We shall also assume that
\begin{equation}
|\frac{\partial F(x,m)}{\partial m}(\xi)|,\:|\frac{\partial F_{T}(x,m)}{\partial m}(\xi)|\leq C(1+|x|^{2}+|\xi|^{2}+\int|\eta|^{2}m(\eta)d\eta)\label{eq:7-41}
\end{equation}
\begin{equation}
|D_{x}D_{\xi}\frac{\partial F(x,m)}{\partial m}(\xi)|\leq C\label{eq:7-42}
\end{equation}
We also make an assumption which simplifies proofs, but which can
be overcome, with technical difficulties.
\begin{align}
\int_{R^{n}}(F(x,m_{1})-F(x,m_{2})(m_{1}(x)-m_{2}(x))dx & \geq0\label{eq:7-43}\\
\int_{R^{n}}(F_{T}(x,m_{1})-F_{T}(x,m_{2})(m_{1}(x)-m_{2}(x))dx & \geq0\nonumber
\end{align}
This assumption allows to obtain the following interesting in itself
result
\begin{prop}
\label{prop5-1} We assume (\ref{eq:7-43}). Then considering the
system of HJB-FP equations (\ref{eq:5-13}) with initial conditions
$m_{1}(x)$ and $m_{2}(x)$ and calling $u_{1}(x,s),m_{1}(x,s)$,
respectively $u_{2}(x,s),m_{2}(x,s)$ the solutions, we have the property
\begin{equation}
\int_{R^{n}}(u_{1}(x,t)-u_{2}(x,t))(m_{1}(x)-m_{2}(x))dx\geq0\label{eq:7-44}
\end{equation}
\end{prop}
\begin{proof}
From the system HJB-FP we can write
\begin{align*}
-\frac{\partial}{\partial s}(u_{1}-u_{2})+\frac{1}{2\lambda}|Du_{1}|^{2}-\frac{1}{2\lambda}|Du_{2}|^{2} & =F(x,m_{1}(s))-F(x,m_{2}(s))\\
u_{1}(x,T)-u_{2}(x,T) & =F_{T}(x,m_{1}(T))-F_{T}(x,m_{2}(T))
\end{align*}
\begin{align*}
\frac{\partial}{\partial s}(m_{1}-m_{2}) & =\frac{1}{\lambda}\text{div}(Du_{1}m_{1}-Du_{2}m_{2})\\
m_{1}(x,t) & -m_{2}(x,t)=m_{1}(x)-m_{2}(x)
\end{align*}
then a simple calculation shows that
\begin{align*}
\frac{d}{ds}\int_{R^{n}}(u_{1}(x,s)-u_{2}(x,s))(m_{1}(x,s)-m_{2}(x,s))dx & =-\int_{R^{n}}(F(x,m_{1}(s))-F(x,m_{2}(s)))(m_{1}(x,s)-m_{2}(x,s))dx\\
- & \frac{1}{2\lambda}\int_{R^{n}}(m_{1}(x,s)+m_{2}(x,s))|Du_{1}(x,s)-Du_{2}(x,s)|^{2}dx
\end{align*}
and the result follows immediately, recalling that $m_{1},m_{2}$
are positive and using the assumption (\ref{eq:7-43}). $\blacksquare$
\end{proof}
We now state the
\begin{prop}
\label{prop6} We make the assumptions of Proposition \ref{prop3}
and (\ref{eq:7-4}), (\ref{eq:7-41}), (\ref{eq:7-42}), (\ref{eq:7-43}).
We then have
\begin{equation}
U(x,m,t)=\frac{\partial V}{\partial m}(m,t)(x)=u_{mt}(x,t)\label{eq:7-45}
\end{equation}
Moreover, we have the estimate
\begin{equation}
|U(x,m,t)|\leq C(1+|x|^{2}+\int_{R^{n}}|\xi|^{2}m(\xi)d\xi)\label{eq:7-450}
\end{equation}
\end{prop}
\begin{proof}
We recall the definition of the value function $V(m,t),$ see section
\ref{sub:MEAN-FIELD-TYPE}, and formulas (\ref{eq:5-15}) and (\ref{eq:7-3}).
Let $m_{1}(x)$ be some probability density and the functions $u_{1}(x,s)=u_{m_{1}t}(x,s),$
$m_{1}(x,s)=m_{m_{1}t}(x,s)$ solutions of the system HJB-FP (\ref{eq:5-13}).
The feedback $\hat{v}_{1}(x,s)=-\dfrac{1}{\lambda}Du_{1}(x,s)$ is
optimal for the control problem (\ref{eq:5-9}), (\ref{eq:5-10}),
(\ref{eq:5-11}). The corresponding optimal trajectory, starting from
a deterministic value $x$ is $y_{xm_{1}t}(s).$ The probability density
$m_{\hat{v}_{1}}(x,s)$ corresponding to the feedback $\hat{v}_{1}(x,s)$
is the image of $m_{1}$ by the map $x\rightarrow y_{xm_{1}t}(s)$,
so we can write
\[
m_{1}(s)=m_{\hat{v}_{1}}(s)=y_{m_{1}t}(s)(.)(m_{1})
\]
We now consider another initial probability density $m_{2}(x)$ and
the same feedback $\hat{v}_{1}.$ Namely we compute $J_{m_{2}t}(\hat{v}_{1}(.)).$
The probability density at time $s,$ with initial condition at time
$t$ equal to $m_{2}$ and feedback $\hat{v}_{1}$ is $y_{m_{1}t}(s)(.)(m_{2})$
denoted $m_{12}(s)=m_{12}(x,s).$ It is solution of the FP equation
\begin{align*}
\frac{\partial m_{12}}{\partial s}-\dfrac{1}{\lambda}\text{div}(Du_{1}(x)m_{12}) & =0\\
m_{12}(x,t)=m_{2}(x)
\end{align*}
We can then write
\begin{align*}
J_{m_{2}t}(\hat{v}_{1}(.)) & =\frac{1}{2\lambda}\int_{t}^{T}\int_{R^{n}}|Du_{1}(x,s)|^{2}m_{12}(x,s)dxds+\int_{t}^{T}\int_{R^{n}}m_{12}(x,s)f(x,m_{12}(s))dxds+\\
+ & \int_{R^{n}}m_{12}(x,T)h(x,m_{12}(T))dx
\end{align*}
Therefore we have the inequality
\begin{align}
V(m_{2},t)-V(m_{1},t) & \leq J_{m_{2}t}(\hat{v}_{1}(.))-J_{m_{1}t}(\hat{v}_{1}(.))\label{eq:7-46}\\
= & \frac{1}{2\lambda}\int_{t}^{T}\int_{R^{n}}|Du_{1}(x,s)|^{2}(m_{12}(x,s)-m_{1}(x,s))dxds+\nonumber \\
+ & \int_{t}^{T}\int_{R^{n}}(m_{12}(x,s)f(x,m_{12}(s))-m_{1}(x,s)f(x,m_{1}(s)))dxds+\nonumber \\
+ & \int_{R^{n}}(m_{12}(x,T)h(x,m_{12}(T))-m_{1}(x,T)h(x,m_{1}(T)))dx\nonumber
\end{align}
We note that
\begin{align*}
\frac{\partial(m_{12}-m_{1})}{\partial s}-\dfrac{1}{\lambda}\text{div}(Du_{1}(x)(m_{12}-m_{1})) & =0\\
m_{12}(x,t)-m_{1}(x,t)=m_{2}(x)-m_{1}(x)
\end{align*}
\begin{align*}
-\frac{\partial}{\partial s}u_{1}+\frac{1}{2\lambda}|Du_{1}|^{2} & =F(x,m_{1}(s))\\
u_{1}(x,T) & =F_{T}(x,m_{1}(T))
\end{align*}
hence, as easily seen
\begin{align*}
\int_{R^{n}}u_{1}(x,t)(m_{2}(x)-m_{1}(x))dx & =\frac{1}{2\lambda}\int_{t}^{T}\int_{R^{n}}|Du_{1}(x,s)|^{2}(m_{12}(x,s)-m_{1}(x,s))dxds+
\end{align*}
\[
+\int_{t}^{T}\int_{R^{n}}F(x,m_{1}(s))(m_{12}(x,s)-m_{1}(x,s))dxds+\int_{R^{n}}F_{T}(x,m_{1}(T))(m_{12}(x,T)-m_{1}(x,T))dx
\]
Combining with (\ref{eq:7-46}) we can write
\[
V(m_{2},t)-V(m_{1},t)\leq\int_{R^{n}}u_{1}(x,t)(m_{2}(x)-m_{1}(x))dx+
\]
\[
+\int_{t}^{T}\int_{R^{n}}[m_{12}(x,s)f(x,m_{12}(s))-m_{1}(x,s)f(x,m_{1}(s))-F(x,m_{1}(s))(m_{12}(x,s)-m_{1}(x,s))]dxds+
\]
\[
\int_{R^{n}}[m_{12}(x,T)h(x,m_{12}(T))-m_{1}(x,T)f(x,m_{1}(s))-F_{T}(x,m_{1}(T))(m_{12}(x,T)-m_{1}(x,T))]dx
\]
Recalling that $F(x,m)$ is the functional derivative of $\int_{R^{n}}f(x,m)m(x)dx,$
we can write the above inequality as follows
\begin{equation}
V(m_{2},t)-V(m_{1},t)\leq\int_{R^{n}}u_{1}(x,t)(m_{2}(x)-m_{1}(x))dx+\label{eq:7-47}
\end{equation}
\[
+\int_{t}^{T}\int_{R^{n}}\int_{R^{n}}\int_{0}^{1}\int_{0}^{1}\theta\frac{\partial F}{\partial m}(x,m_{1}(s)+\theta\mu(m_{12}(s)-m_{1}(s))(\xi)(m_{12}(x,s)-m_{1}(x,s))(m_{12}(\xi,s)-m_{1}(\xi,s))dxd\xi dsd\theta d\mu+
\]
\[
+\int_{R^{n}}\int_{R^{n}}\int_{0}^{1}\int_{0}^{1}\theta\frac{\partial F}{\partial m}(x,m_{1}(T)+\theta\mu(m_{12}(T)-m_{1}(T))(\xi)(m_{12}(x,T)-m_{1}(x,T))(m_{12}(\xi,T)-m_{1}(\xi,T))dxd\xi d\theta d\mu
\]
Recalling that $m_{12}(s)=y_{m_{1}t}(s)(.)(m_{2})$ and $m_{1}(s)=y_{m_{1}t}(s)(.)(m_{1})$,
we can write for a test function $\varphi(x,\xi)$
\[
\int_{R^{n}}\int_{R^{n}}\varphi(x,\xi)(m_{12}(x,s)-m_{1}(x,s))(m_{12}(\xi,s)-m_{1}(\xi,s))dxd\xi=
\]
\[
\int_{R^{n}}\int_{R^{n}}\varphi(y_{xm_{1}t}(s),y_{\xi m_{1}t}(s))(m_{2}(x)-m_{1}(x))(m_{2}(\xi)-m_{1}(\xi))dxd\xi
\]
We introduce a pair of random variables $X_{1},X_{2}$whose marginals
are $m_{1},m_{2}.$We then introduce an independent copy $Y_{1},Y_{2}.$
It is easy to convince oneself that we have the relation
\[
\int_{R^{n}}\int_{R^{n}}\varphi(y_{xm_{1}t}(s),y_{\xi m_{1}t}(s))(m_{2}(x)-m_{1}(x))(m_{2}(\xi)-m_{1}(\xi))dxd\xi=
\]
\begin{align*}
\int_{0}^{1}\int_{0}^{1}E\,D_{\xi}D_{x}\varphi(y_{X_{1}m_{1}t}(s)+\theta(y_{X_{2}m_{1}t}(s)-y_{X_{1}m_{1}t}(s)),y_{Y_{1}m_{1}t}(s)+\mu(y_{Y_{2}m_{1}t}(s)-y_{Y_{1}m_{1}t}(s)))\\
(y_{X_{2}m_{1}t}(s)-y_{X_{1}m_{1}t}(s))(y_{Y_{2}m_{1}t}(s)-y_{Y_{1}m_{1}t}(s))d\theta d\mu
\end{align*}
If we have $||D_{\xi}D_{x}\varphi(x,\xi)||\leq C,$then we get
\[
|\int_{R^{n}}\int_{R^{n}}\varphi(y_{xm_{1}t}(s),y_{\xi m_{1}t}(s))(m_{2}(x)-m_{1}(x))(m_{2}(\xi)-m_{1}(\xi))dxd\xi|\leq|
\]
\[
CE|y_{X_{2}m_{1}t}(s)-y_{X_{1}m_{1}t}(s)||y_{Y_{2}m_{1}t}(s)-y_{Y_{1}m_{1}t}(s)
\]
and from the independence property
\[
\leq C(E|y_{X_{2}m_{1}t}(s)-y_{X_{1}m_{1}t}(s)|)^{2}\leq CE|y_{X_{2}m_{1}t}(s)-y_{X_{1}m_{1}t}(s)|^{2}
\]
Using property (\ref{eq:6-14}) we obtain also
\[
|\int_{R^{n}}\int_{R^{n}}\varphi(y_{xm_{1}t}(s),y_{\xi m_{1}t}(s))(m_{2}(x)-m_{1}(x))(m_{2}(\xi)-m_{1}(\xi))dxd\xi|\leq CE|X_{2}-X_{1}|^{2}
\]
and since$X_{1},X_{2}$ have an arbitrary correlation, this implies
also
\[
|\int_{R^{n}}\int_{R^{n}}\varphi(y_{xm_{1}t}(s),y_{\xi m_{1}t}(s))(m_{2}(x)-m_{1}(x))(m_{2}(\xi)-m_{1}(\xi))dxd\xi|\leq CW_{2}^{2}(m_{1},m_{2})
\]
We may apply this result with $\varphi(x,$$\xi)=\frac{\partial F}{\partial m}(x,m_{1}(s)+\theta\mu(m_{12}(s)-m_{1}(s))(\xi).$
Thanks to assumption (\ref{eq:7-42}) the same result carries over.
Therefore we conclude easily the estimate
\begin{equation}
V(m_{2},t)-V(m_{1},t)\leq\int_{R^{n}}u_{1}(x,t)(m_{2}(x)-m_{1}(x))dx+CW_{2}^{2}(m_{1},m_{2})\label{eq:7-48}
\end{equation}
Interchanging the role of $m_{1},m_{2}$, we have also
\begin{align*}
V(m_{1},t)-V(m_{2},t) & \leq\int_{R^{n}}u_{2}(x,t)(m_{1}(x)-m_{2}(x))dx+CW_{2}^{2}(m_{1},m_{2})\\
\leq & \int_{R^{n}}u_{1}(x,t)(m_{1}(x)-m_{2}(x))dx+\int_{R^{n}}(u_{2}(x,t)-u(x_{1},t)(m_{1}(x)-m_{2}(x))dx+CW_{2}^{2}(m_{1},m_{2})
\end{align*}
and from Proposition \ref{prop5-1} and assumption (\ref{eq:7-43})
the 2nd integral is negative, which implies
\[
V(m_{1},t)-V(m_{2},t)\leq\int_{R^{n}}u_{1}(x,t)(m_{1}(x)-m_{2}(x))dx+CW_{2}^{2}(m_{1},m_{2})
\]
or
\[
V(m_{2},t)-V(m_{1},t)\geq\int_{R^{n}}u_{1}(x,t)(m_{2}(x)-m_{1}(x))dx-CW_{2}^{2}(m_{1},m_{2})
\]
and comparing with (\ref{eq:7-48}) we can assert
\begin{equation}
|V(m_{2},t)-V(m_{1},t)-\int_{R^{n}}u_{1}(x,t)(m_{2}(x)-m_{1}(x))dx|\leq CW_{2}^{2}(m_{1},m_{2})\label{eq:7-49}
\end{equation}
Now we have
\begin{equation}
|u_{mt}(x,t)|\leq C(1+|x|^{2}+\int_{R^{n}}|\xi|^{2}m(\xi)d\xi\label{eq:7-50}
\end{equation}
So for any curve $m_{\epsilon}\in\mathcal{P}_{2}$, $u_{mt}(x,t)\in L_{m_{\epsilon}}^{1}$.
From the estimate (\ref{eq:7-49}) we get immediately the result (\ref{eq:7-45}).
The proof has been completed. $\blacksquare$
\end{proof}
\subsection{OBTAINING BELLMAN EQUATION}
We have seen in section \ref{sub:INTERPRETATION-OF-THE}that
\[
z_{xmt}(s)=Du_{mt}(y_{xmt}(s),s)
\]
and thus
\begin{equation}
D_{x}U(x,m,t)=z_{xmt}(t)\label{eq:7-8}
\end{equation}
Therefore from the estimate (\ref{eq:6-15}) we can assert that
\begin{equation}
|D_{x}U(x,m,t)|\leq C[1+|x|+\sqrt{\int|\xi|^{2}m(\xi)d\xi}]\label{eq:7-9}
\end{equation}
In particular, we can see that $D_{x}U(x,m,t)$ belongs to $L_{m}^{2}(R^{n};R^{n}).$
But then, recalling the correspondance $V(X,t)=V(m,t)|_{m=\mathcal{L}_{X}},$we
can write
\begin{equation}
D_{X}V(X,t)=D_{x}U(X,\mathcal{L}_{X},t)\label{eq:7-10}
\end{equation}
and
\[
||D_{X}V(X,t)||^{2}=\int_{R^{n}}|D_{x}\frac{\partial V(m,t)}{\partial m}(x)|^{2}m(x)dx
\]
If we look at the Bellman equation in the Hilbert space $\mathcal{H}$,
see (\ref{eq:2-8}) we obtain exactly (\ref{eq:5-16}). So we can
state
\begin{prop}
\label{prop5} We make the assumptions of Proposition \ref{prop6}.
The value function $V(m,t)$ of the problem (\ref{eq:5-9}),(\ref{eq:5-12})
or equivalently (\ref{eq:5-10}),(\ref{eq:5-11}) satisfies the estimates
(\ref{eq:7-2}), (\ref{eq:7-450}) (with $U(x,m,t)=\dfrac{\partial V(m,t)}{\partial m}(x)$
) and (\ref{eq:7-9}). It is the unique solution, satisfying these
estimates, of the Bellman equation (\ref{eq:5-16}). Moreover, we
have the explicit formula (\ref{eq:7-1}) with $y_{xmt}(s)$ being
the unique solution of (\ref{eq:6-9}) and $z_{xmt}(s)=-\lambda\dfrac{dy_{xmt}(s)}{ds}$
\end{prop}
\subsection{OBTAINING THE SCALAR MASTER EQUATION}
We can derive the scalar master equation from the probabilistic master
equation (\ref{eq:3-9}), which we write as follows
\begin{align}
\frac{\partial\mathcal{U}}{\partial t}-\frac{1}{2\lambda}D_{X}||\mathcal{U}(X,t)||^{2}+D_{X}\mathcal{F}(X) & =0\label{eq:3-9-1}\\
\mathcal{U}(X,T)=D_{X}\mathcal{F}_{T}(X)\nonumber
\end{align}
We know that $\mathcal{U}(X,t)=D_{x}U(X,\mathcal{L}_{X},t)$ and
$U(x,m,t)=u_{mt}(x,t)$ and $D_{x}u_{mt}(x,t)=z_{xmt}(t).$ Therefore
$||\mathcal{U}(X,t)||^{2}=\int_{R^{n}}|D_{\xi}U(\xi,m,t)|^{2}m(\xi)d\xi.$
Since this functional of $m$ only has a derivative in the Hilbert
space $\mathcal{H}$ it can be written as follows
\[
D_{X}||\mathcal{U}(X,t)||^{2}=D_{x}\frac{\partial}{\partial m}\,(\int_{R^{n}}|D_{\xi}U(\xi,m,t)|^{2}m(\xi)d\xi)(X)
\]
Recalling that $D_{X}\mathcal{F}(X)=D_{x}F(X,m),$ $D_{X}\mathcal{F}_{T}(X)=D_{x}F_{T}(X,m),$
we see that (\ref{eq:3-9-1}) can be wriiten as follows
\begin{align*}
D_{x}[\frac{\partial U(X,m,t)}{\partial t}-\frac{1}{2\lambda}\frac{\partial}{\partial m}\,(\int_{R^{n}}|D_{\xi}U(\xi,m,t)|^{2}m(\xi)d\xi)(X)+F(X,m)] & =0\\
D_{x}U(X,m,T)=D_{x}F_{T}(X,m)
\end{align*}
This leads to
\begin{align*}
\frac{\partial U(x,m,t)}{\partial t}-\frac{1}{2\lambda}\frac{\partial}{\partial m}\,(\int_{R^{n}}|D_{\xi}U(\xi,m,t)|^{2}m(\xi)d\xi)(x)+F(x,m) & =0\\
U(x,m,T)=F_{T}(x,m)
\end{align*}
which we can write as (\ref{eq:5-17}), taking account of the symmetry
property $\dfrac{\partial}{\partial m}U(x,m,t)(\xi)=\dfrac{\partial}{\partial m}U(\xi,m,t)(x)$.
$\blacksquare$
This proof is not fully rigorous. It assumes implicitly the existence
of $\dfrac{\partial}{\partial m}U(x,m,t)(\xi),$ which is the 2nd
derivative of the function $V(m,t).$ To study it rigorously and give
an implicit formula for $\dfrac{\partial}{\partial m}U(x,m,t)(\xi)$
, one can use the system of HJB-FP equations (\ref{eq:5-13}) and
write the solution as $u_{mt}(x,s),m_{mt}(x,s)$ to emphasize the
initial conditions $m,t.$ We then consider the functional derivatives
$\dfrac{\partial u_{mt}(x,s)}{\partial m}(\xi)$, $\dfrac{\partial m_{mt}(x,s)}{\partial m}(\xi)$
and differentiate formally the system of HJB-FP equations. To simplify
notation, we take a test function $\tilde{m}(\xi)$ and consider
\[
\int_{R^{n}}\dfrac{\partial u_{mt}(x,s)}{\partial m}(\xi)\tilde{m}(\xi)\,d\xi,\:\int_{R^{n}}\dfrac{\partial m_{mt}(x,s)}{\partial m}(\xi)\tilde{m}(\xi)\,d\xi
\]
which we note $\tilde{u}_{mt;\tilde{m}}(x,s),\:$ $\tilde{m}_{mt;\tilde{m}}(x,s)$
and to simplify further $\tilde{u}(x,s),$ $\tilde{m}(x,s).$ In particular
, $\tilde{u}(x,t)=\int\dfrac{\partial}{\partial m}U(x,m,t)(\xi)\tilde{m}(\xi)\,d\xi$.
The pair $\tilde{u}(x,s),\tilde{m}(x,s)$ is solution of a system
of linear P.D.E. as follows
\begin{align}
-\dfrac{\partial\tilde{u}}{\partial s}+\frac{1}{\lambda}D\tilde{u}.Du & =\int\frac{\partial F}{\partial m}(x,m(s))(\xi)\tilde{m}(\xi,s)d\xi\label{eq:7-11}\\
\tilde{u}(x,T) & =\int\frac{\partial F_{T}}{\partial m}(x,m(T))(\xi)\tilde{m}(\xi,T)d\xi\nonumber
\end{align}
\begin{align*}
\dfrac{\partial\tilde{m}}{\partial s}-\frac{1}{\lambda}\text{div}(D\tilde{u}\,m+Du\,\tilde{m}) & =0\\
\tilde{m}(x,t)=\tilde{m}(x)
\end{align*}
This system is obtained by linearization of the system (\ref{eq:5-13}).
The functions $u(x,s),$$m(x,s)$ are solutions of the system (\ref{eq:5-13}).
We can write also
\begin{equation}
\tilde{u}(x,s)=\int_{s}^{T}\int_{R^{n}}\frac{\partial F}{\partial m}(y_{xmt}(\sigma),y_{mt}(\sigma)(.)(m))(\xi)\tilde{m}(\xi,\sigma)d\xi d\sigma+\int_{R^{n}}\frac{\partial F_{T}}{\partial m}(y_{xmt}(T),y_{mt}(T)(.)(m))(\xi)\tilde{m}(\xi,T)d\xi\label{eq:7-12}
\end{equation}
We can then study (\ref{eq:7-12}) as a fixed point equation in the
function $\tilde{u}(x,s).$
\section{\label{sec:QUADRATIC-CASE}QUADRATIC CASE}
\subsection{ASSUMPTIONS }
We consider the quadratic case, (\ref{eq:5-53}). We also take
\begin{equation}
h(x,m)=\frac{1}{2}(x-S_{T}\bar{x})^{*}\bar{Q_{T}}(x-S_{T}\bar{x})+\frac{1}{2}x^{*}Q_{T}x\label{eq:5-13-1}
\end{equation}
In the space $\mathcal{H}$ we have
\begin{align}
\mathcal{F}(X) & =\frac{1}{2}EX^{*}(Q+\bar{Q})X+\frac{1}{2}EX^{*}(S^{*}\bar{Q}S-\bar{Q}S-S^{*}\bar{Q})EX\nonumber \\
\mathcal{F}_{T}(X) & =\frac{1}{2}EX^{*}(Q_{T}+\bar{Q})X+\frac{1}{2}EX^{*}(S^{*}\bar{Q}S-\bar{Q}S-S^{*}\bar{Q})EX\label{eq:8-1}
\end{align}
We can write $\mathcal{F}(X)=Ef(X,\mathcal{L}_{X})$ with
\begin{equation}
f(x,m)=\frac{1}{2}(x-S\bar{x})^{*}\bar{Q}(x-S\bar{x})+\frac{1}{2}x^{*}Qx\label{eq:8-2}
\end{equation}
in which $\bar{x}=\int xm(x)dx$, assuming the probability law of
$X$ has a density, $m.$ So we can also write
\begin{align*}
\mathcal{F}(X) & =\Phi(m)=\dfrac{1}{2}\int_{R^{n}}x*(Q+\bar{Q)}xm(x)dx+\dfrac{1}{2}\int_{R^{n}}x^{*}m(x)dx\,(S^{*}\bar{Q}S-\bar{Q}S-S^{*}\bar{Q})\,\int_{R^{n}}xm(x)dx\\
& =\int_{R^{n}}f(x,m)m(x)dx
\end{align*}
If we take $m\in L^{2}(R^{n})\cap L^{1}(R^{n})$ not necessarily
a probability density, then we have to introduce $m_{1}=\int_{R^{n}}m(x)dx$
and write
\begin{equation}
\Phi(m)=\int_{R^{n}}f(x,m)m(x)dx=\label{eq:8-3}
\end{equation}
\[
=\dfrac{1}{2}\int_{R^{n}}x*(Q+\bar{Q)}xm(x)dx+\dfrac{1}{2}\bar{x}^{*}S^{*}\bar{Q}S\bar{x}\,m_{1}-\dfrac{1}{2}\bar{x}^{*}(\bar{Q}S+S^{*}\bar{Q})\bar{x}
\]
We have noted $F(x,m)=\dfrac{\partial\Phi(m)}{\partial m}(x).$ Then
as a Gâteaux differential we have
\begin{equation}
F(x,m)=\dfrac{1}{2}x*(Q+\bar{Q)}x+\bar{x}^{*}(S^{*}\bar{Q}S\,m_{1}-\bar{Q}S-S^{*}\bar{Q})x+\dfrac{1}{2}\bar{x}^{*}S^{*}\bar{Q}S\bar{x}\label{eq:8-4}
\end{equation}
We note that
\begin{equation}
D_{X}\mathcal{F}(X)=(Q+\bar{Q})X+(S^{*}\bar{Q}S-\bar{Q}S-S^{*}\bar{Q})EX\label{eq:8-5}
\end{equation}
So the equality $D_{X}\mathcal{F}(X)=D_{x}F(X,m)$ is true only when
$m_{1}=1.$ It is important to keep in mind that when we work with
Gâteaux differentials, we have to make calculations with the term
$m_{1},$even though that eventually, when applied to $m=$ probability
density, we shall have $m_{1}=1.$ To understand further this point,
let us compute the 2nd derivative. We have
\begin{equation}
\frac{\partial F}{\partial m}(x,m)(\xi)=\bar{x}^{*}S^{*}\bar{Q}S(x+\xi)+\xi^{*}(S^{*}\bar{Q}S\,m_{1}-\bar{Q}S-S^{*}\bar{Q})x\label{eq:8-6}
\end{equation}
We see that this formula is symmetric in $x,\xi$ as needed. Without
the term $m_{1}$ in (\ref{eq:8-4}) this will not be true. We have
\[
D_{x}^{2}F(x,m)=Q+\bar{Q},\;D_{x}D_{\xi}\frac{\partial F}{\partial m}(x,m)(\xi)=(S^{*}\bar{Q}S\,m_{1}-\bar{Q}S-S^{*}\bar{Q})
\]
and , see \cite{BFY2}
\begin{align*}
D_{X}^{2}\mathcal{F}(X)Z & =D_{x}^{2}F(X,\mathcal{L}_{X})Z+E_{Y\tilde{Z}}D_{x}D_{y}\frac{\partial F}{\partial m}(X,\mathcal{L}_{X})(Y)\tilde{Z}=\\
& =(Q+\bar{Q})Z+(S^{*}\bar{Q}S-\bar{Q}S-S^{*}\bar{Q})EZ
\end{align*}
which is exactly what we obtain by differentiating (\ref{eq:8-5})
in the Hilbert space.
\subsection{BELLMAN EQUATION}
Bellman equation ( \ref{eq:5-16}) writes
\begin{equation}
\frac{\partial V}{\partial t}-\frac{1}{2\lambda}\int_{R^{n}}|D_{\xi}\frac{\partial V(m,t)}{\partial m}(\xi)|^{2}m(\xi)d\xi+\label{eq:8-7}
\end{equation}
\[
+\dfrac{1}{2}\int_{R^{n}}\xi*(Q+\bar{Q)}\xi m(\xi)d\xi+\dfrac{1}{2}\bar{x}^{*}S^{*}\bar{Q}S\bar{x}\,m_{1}-\dfrac{1}{2}\bar{x}^{*}(\bar{Q}S+S^{*}\bar{Q})\bar{x}=0
\]
\[
V(m,T)=\dfrac{1}{2}\int_{R^{n}}\xi*(Q_{T}+\bar{Q_{T}})\xi m(\xi)d\xi+\dfrac{1}{2}\bar{x}^{*}S_{T}^{*}\bar{Q}_{T}S_{T}\bar{x}\,m_{1}-\dfrac{1}{2}\bar{x}^{*}(\bar{Q}_{T}S_{T}+S_{T}^{*}\bar{Q}_{T})\bar{x}
\]
The solution is
\begin{equation}
V(m,t)=\dfrac{1}{2}\int_{R^{n}}\xi*P(t)\xi m(\xi)d\xi+\dfrac{1}{2}\bar{x}^{*}\Sigma(t;m_{1})\bar{x}\label{eq:8-8}
\end{equation}
with
\begin{align}
\frac{dP}{dt}-\frac{P^{2}}{\lambda}+Q+\bar{Q} & =0\label{eq:8-9}\\
P(T)=Q_{T}+\bar{Q_{T}}\nonumber
\end{align}
\begin{align}
\frac{d\Sigma}{dt}-\frac{1}{\lambda}(\Sigma P+P\Sigma)-\frac{1}{\lambda}\Sigma^{2}m_{1}+S^{*}\bar{Q}S\,m_{1}-(\bar{Q}S+S^{*}\bar{Q}) & =0\label{eq:8-10}\\
\Sigma(T;m_{1})=S_{T}^{*}\bar{Q}_{T}S_{T}\,m_{1}-(\bar{Q}_{T}S_{T}+S_{T}^{*}\bar{Q}_{T})\nonumber
\end{align}
\subsection{MASTER EQUATION}
The scalar Master equation (\ref{eq:5-17}) reads
\begin{align}
\frac{\partial U}{\partial t}-\frac{1}{\lambda}\int_{R^{n}}D_{\xi}\frac{\partial U}{\partial m}(x,m,t)(\xi).D_{\xi}U(\xi,m,t)m(\xi)d\xi-\frac{1}{2\lambda}|D_{x}U(x,m,t)|^{2}\label{eq:5-17-1}\\
+\dfrac{1}{2}x*(Q+\bar{Q)}x+\bar{x}^{*}(S^{*}\bar{Q}S\,m_{1}-\bar{Q}S-S^{*}\bar{Q})x+\dfrac{1}{2}\bar{x}^{*}S^{*}\bar{Q}S\bar{x}=0\nonumber \\
U(x,m,T)=\dfrac{1}{2}x*(Q_{T}+\bar{Q}_{T})x+\bar{x}^{*}(S_{T}^{*}\bar{Q_{T}}S_{T}\,m_{1}-\bar{Q_{T}}S_{T}-S_{T}^{*}\bar{Q}_{T})x+\dfrac{1}{2}\bar{x}^{*}S_{T}^{*}\bar{Q_{T}}S_{T}\bar{x}\nonumber
\end{align}
Its solution is
\begin{equation}
U(x,m,t)=\frac{\partial V(m,t)}{\partial m}(x)=\frac{1}{2}x^{*}P(t)x+\bar{x}^{*}\Sigma(t;m_{1})x+\frac{1}{2}\bar{x}^{*}\frac{\partial\Sigma(t;m_{1})}{\partial m_{1}}\bar{x}\label{eq:5-18-1}
\end{equation}
We have
\begin{equation}
D_{x}U(x,m,t)=P(t)x+\Sigma(t;m_{1})\bar{x}\label{eq:5-19-1}
\end{equation}
\[
\frac{\partial U}{\partial m}(x,m,t)(\xi)=\bar{x}^{*}\frac{\partial\Sigma(t;m_{1})}{\partial m_{1}}(x+\xi)+\xi^{*}\Sigma(t;m_{1})x+\frac{1}{2}\bar{x}^{*}\frac{\partial^{2}\Sigma(t;m_{1})}{\partial m_{1}^{2}}\bar{x}
\]
\[
D_{\xi}\frac{\partial U}{\partial m}(x,m,t)(\xi)=\frac{\partial\Sigma(t;m_{1})}{\partial m_{1}}\bar{x}+\Sigma(t;m_{1})x
\]
We note that $\Gamma(t;m_{1})=\dfrac{\partial\Sigma(t;m_{1})}{\partial m_{1}}$
satisfies the equation
\begin{align}
\frac{d\Gamma}{dt}-\frac{1}{\lambda}(\Gamma(P+\Sigma m_{1})+(P+\Sigma m_{1})\Gamma)-\frac{1}{\lambda^{2}}\Sigma^{2}+S^{*}\bar{Q}S & =0\label{eq:5-19-2}\\
\Gamma(T;m_{1})=S_{T}^{*}\bar{Q}_{T}S_{T}\nonumber
\end{align}
and we check easily that the function $U(x,m,t)$ defined by (\ref{eq:5-18-1})
is solution of the scalar master equation (\ref{eq:5-17-1}).
We turn to the vector master equation (\ref{eq:5-20}) which reads
\begin{align}
\frac{\partial\mathcal{U}}{\partial t}-\frac{1}{\lambda}\int_{R^{n}}D_{\xi}\frac{\partial\mathcal{U}}{\partial m}(x,m,t)(\xi)\,\mathcal{U}(\xi,m,t)m(\xi)d\xi-\frac{1}{\lambda}D_{x}\mathcal{U}(x,m,t)\,\mathcal{U}(x,m,t)+\label{eq:5-20-1}\\
+(Q+\bar{Q})x+(S^{*}\bar{Q}Sm_{1}-\bar{Q}S-S^{*}\bar{Q})\bar{x}=0\nonumber
\end{align}
\[
\mathcal{U}(x,m,T)=(Q_{T}+\bar{Q_{T}})x+(S_{T}^{*}\bar{Q_{T}}S_{T}m_{1}-\bar{Q}_{T}S_{T}-S_{T}^{*}\bar{Q_{T}})\bar{x}
\]
whose solution is
\begin{equation}
\mathcal{U}(x,m,t)=D_{x}U(x,m,t)=P(t)x+\Sigma(t;m_{1})\bar{x}\label{eq:5-21}
\end{equation}
This statement is easily verified.
\subsection{SYSTEM OF HJB-FP EQUATIONS}
We now look at the system (\ref{eq:5-13}) which reads
\begin{align}
-\frac{\partial u}{\partial s}+\frac{1}{2\lambda}|Du|^{2} & =\dfrac{1}{2}x*(Q+\bar{Q)}x+\bar{x}^{*}(s)(S^{*}\bar{Q}S\,m_{1}(s)-\bar{Q}S-S^{*}\bar{Q})x+\dfrac{1}{2}\bar{x}^{*}(s)S^{*}\bar{Q}S\bar{x}(s)\label{eq:5-13-2}\\
u(x,T) & =\dfrac{1}{2}x*(Q_{T}+\bar{Q}_{T})x+\bar{x}^{*}(T)(S_{T}^{*}\bar{Q}_{T}S_{T}\,m_{1}(T)-\bar{Q}_{T}S_{T}-S_{T}^{*}\bar{Q}_{T})x+\dfrac{1}{2}\bar{x}^{*}(T)S_{T}^{*}\bar{Q}_{T}S_{T}\bar{x}(T)\nonumber \\
\frac{\partial m}{\partial s}-\frac{1}{\lambda}\text{div} & (Du\,m)=0\nonumber \\
m(x,t) & =m(x)\nonumber
\end{align}
It is immediate to see that $m_{1}(s)=m_{1}.$The function $\bar{x}(s)$
represents the mean $\int_{R^{n}}\xi m(\xi,s)d\xi.$We do not need
to obtain the full probability $m(x,s).$The mean is sufficient. One
can check the formula
\begin{equation}
u(x,s)=\frac{1}{2}x^{*}P(s)x+x^{*}\Sigma(s;m_{1})\bar{x}(s)+\frac{1}{2}\bar{x}(s)^{*}\frac{\partial\Sigma(s;m_{1})}{\partial m_{1}}\bar{x}(s)\label{eq:5-14-1}
\end{equation}
In particular $u(x,t)=U(x,m,t)$ given by (\ref{eq:5-18-1}). Also
$u(x,s)=U(x,m(s),s).$ Note that $\bar{x}(s)$ evolves as follows
\begin{align}
\frac{d\bar{x}}{ds}+\frac{1}{\lambda}(P(s)+m_{1}\Sigma(s;m_{1}))\bar{x}(s) & =0\label{eq:5-15-1}\\
\bar{x}(t)=\bar{x}\nonumber
\end{align}
We have seen that $U(x,m,t)$ is differentiable in $m$ with
\[
\frac{\partial U}{\partial m}(x,m,t)(\xi)=\bar{x}^{*}\frac{\partial\Sigma(t;m_{1})}{\partial m_{1}}(x+\xi)+\xi^{*}\Sigma(t;m_{1})x+\frac{1}{2}\bar{x}^{*}\frac{\partial^{2}\Sigma(t;m_{1})}{\partial m_{1}^{2}}\bar{x}
\]
If we consider a test function $\tilde{m}(\xi)$ and define
\[
\tilde{u}(x,t)=\tilde{u}_{mt;\tilde{m}}(x,t)=\lim_{\theta\rightarrow0}\frac{u_{m+\theta\tilde{m},t}(x,t)-u_{mt}(x,t)}{\theta}
\]
we get
\begin{equation}
\tilde{u}(x,t)=x^{*}\Sigma(t;m_{1})\tilde{\bar{x}}+\bar{x}^{*}\frac{\partial\Sigma(t;m_{1})}{\partial m_{1}}\tilde{\bar{x}}+\tilde{m}_{1}(\bar{x}^{*}\frac{\partial\Sigma(t;m_{1})}{\partial m_{1}}x+\frac{1}{2}\bar{x}^{*}\frac{\partial^{2}\Sigma(t;m_{1})}{\partial m_{1}^{2}}\bar{x})\label{eq:5-16-1}
\end{equation}
We can also compute
\[
\tilde{u}(x,s)=\tilde{u}_{mt;\tilde{m}}(x,s)=\lim_{\theta\rightarrow0}\frac{u_{m+\theta\tilde{m},t}(x,s)-u_{mt}(x,s)}{\theta}
\]
We have
\begin{equation}
\tilde{u}(x,s)=x^{*}(\Sigma(s;m_{1})\tilde{\bar{x}}(s)+\frac{\partial\Sigma(s;m_{1})}{\partial m_{1}}\bar{x}(s)\tilde{m}_{1})+\bar{x}^{*}(s)\frac{\partial\Sigma(s;m_{1})}{\partial m_{1}}\tilde{\bar{x}}(s)+\tilde{m}_{1}\frac{1}{2}\bar{x}^{*}(s)\frac{\partial^{2}\Sigma(s;m_{1})}{\partial m_{1}^{2}}\bar{x}(s)\label{eq:8-1-1}
\end{equation}
in which
\begin{equation}
\frac{d\tilde{\bar{x}}(s)}{ds}+\frac{1}{\lambda}(P(s)+m_{1}\Sigma(s;m_{1}))\tilde{\bar{x}}(s)+\frac{1}{\lambda}\tilde{m}_{1}(\Sigma(s;m_{1})+m_{1}\frac{\partial\Sigma(s;m_{1})}{\partial m_{1}})\bar{x}(s)=0\label{eq:8-2-1}
\end{equation}
and where $\tilde{m}_{1}=\int_{R^{n}}\tilde{m}(\xi)d\xi.$ The function
$\tilde{u}(x,s)$ is the solution of the linearized equation (\ref{eq:7-11}),
namely
\begin{align}
-\dfrac{\partial\tilde{u}}{\partial s}+\frac{1}{\lambda}D\tilde{u}.(P(s)x+\Sigma(s;m_{1})\bar{x}(s)) & =x^{*}\left(S^{*}\bar{Q}S\,\bar{x}(s)\tilde{m}_{1}+(S^{*}\bar{Q}S\,m_{1}-\bar{Q}S-S^{*}\bar{Q})\tilde{\bar{x}}(s)\right)+\label{eq:8-3-1}\\
+ & \bar{x}(s)^{*}S^{*}\bar{Q}S\tilde{\bar{x}}(s)\nonumber
\end{align}
\begin{align*}
\tilde{u}(x,T) & =x^{*}\left(S^{*}\bar{Q}S\,\bar{x}(T)\tilde{m}_{1}+(S^{*}\bar{Q}S\,m_{1}-\bar{Q}S-S^{*}\bar{Q})\tilde{\bar{x}}(T)\right)+\\
+ & \bar{x}(T)^{*}S^{*}\bar{Q}S\tilde{\bar{x}}(T)
\end{align*}
which can be checked by direct calculation.
\subsection{STATE EQUATION }
We consider equation (\ref{eq:6-9}) in the quadratic case. Since
we know the function $u_{mt}(x,s)$ see (\ref{eq:5-14-1}) the best
is to use the fact that $y_{xmt}(s)$ is the solution of
\begin{align*}
\frac{dy}{ds} & =-\frac{1}{\lambda}Du(y(s),s)\\
y(t) & =x
\end{align*}
We get the explicit solution
\begin{equation}
y_{xmt}(s)=\exp-\frac{1}{\lambda}\int_{t}^{s}P(\sigma)d\sigma\,x-\frac{1}{\lambda}\int_{t}^{s}(\exp-\frac{1}{\lambda}\int_{\sigma}^{s}P(\tau)d\tau\,\Sigma(\sigma)\exp-\frac{1}{\lambda}\int_{t}^{\sigma}(\Sigma(\tau)+P(\tau))d\tau)d\sigma\,\bar{x}]\label{eq:8-4-1}
\end{equation}
\subsection{FORMULATION IN THE HILBERT SPACE }
We can formulate Bellman equation and the Master equation in the Hilbert
space $\mathcal{H}$ . We have first Bellman equation
\begin{equation}
\frac{\partial V}{\partial t}-\frac{1}{2\lambda}||D_{X}V||^{2}+\frac{1}{2}EX^{*}(Q+\bar{Q})X+\frac{1}{2}EX^{*}(S^{*}\bar{Q}S-\bar{Q}S-S^{*}\bar{Q})EX=0\label{eq:8-5-1}
\end{equation}
\[
V(X,T)=\frac{1}{2}EX^{*}(Q_{T}+\bar{Q}_{T})X+\frac{1}{2}EX^{*}(S_{T}^{*}\bar{Q}_{T}S_{T}-\bar{Q}_{T}S_{T}-S_{T}^{*}\bar{Q}_{T})EX
\]
whose solution is
\begin{equation}
V(X,t)=\frac{1}{2}EX^{*}P(t)X+\frac{1}{2}EX^{*}\Sigma(t)EX\label{eq:8-6-1}
\end{equation}
with $\Sigma(t)=\Sigma(t;1).$ The Master equation reads
\begin{equation}
\frac{\partial\mathcal{U}}{\partial t}-\frac{1}{\lambda}D_{X}\mathcal{U}(X,t)\,\mathcal{U}(X,t)+(Q+\bar{Q})X+(S^{*}\bar{Q}S-\bar{Q}S-S^{*}\bar{Q})EX=0\label{eq:8-7-1}
\end{equation}
\[
\mathcal{U}(X,T)=(Q_{T}+\bar{Q}_{T})X+(S_{T}^{*}\bar{Q}_{T}S_{T}-\bar{Q}_{T}S_{T}-S_{T}^{*}\bar{Q}_{T})EX
\]
whose solution is $\mathcal{U}(X,t)=P(t)X+\Sigma(t)EX.$ Note that
$D_{X}\mathcal{U}(X,t)Z=P(t)Z+\Sigma(t)EZ$ . The state equation is
the solution of
\begin{equation}
\frac{dY}{ds}=-\frac{1}{\lambda}(P(s)Y(s)+\Sigma(s)EY(s))\label{eq:8-8-1}
\end{equation}
\[
Y(t)=X
\]
hence the formula
\begin{align}
Y(s) & =\exp-\frac{1}{\lambda}\int_{t}^{s}P(\sigma)d\sigma X-\label{eq:8-9-1}\\
- & \int_{t}^{s}(\exp-\frac{1}{\lambda}\int_{\sigma}^{s}P(\tau)d\tau\,\Sigma(\sigma)\,\exp-\frac{1}{\lambda}\int_{t}^{\sigma}(P(\tau)+\Sigma(\tau))d\tau)d\sigma)EX\nonumber
\end{align} | 100,522 |
,000
Idaman Residence, Kuala Lumpur
iCondo Directory
See more pictures and read information on Idaman Residence in the iCondo Directory.
7 picture(s)
Property Details
- QuikPro No:UP1128444
- Property Type:Condominium
- Built-Up:1,700 sq. ft.
- Rental Price:RM 7,000
- Bedrooms:2
- Bathrooms:2
- Occupancy:Vacant
- Furnishing:Fully Furnished
- Posted Date:11/01/2012
- Facilities:Barbecue Area, Playground, Swimming Pool, 24hr Security | 76,151 |
TITLE: cokernel pairs left adjoint to equalizers
QUESTION [4 upvotes]: Given a category $ C $ which has both cokernel pairs and equalizers, how can I see that the functor $ C^\downarrow\longrightarrow C^{\downarrow\downarrow} $, which takes every arrow in $ C $ to its cokernel pair has as right adjoint the functor in the converse direction, which assigns to each pair of parallel arrows its equalizing arrow?
There are so many arrows involved, that all my attempts end up in confusion.
REPLY [1 votes]: Consider a bigger category $\mathcal D$ which disjointly contains $C^\downarrow$ and $C^{\downarrow\downarrow}$, and additionally it contains the arrow $g$ of $C$ as an arrow from $f$ in $C^\downarrow$ to $(u,v)$ in $C^{\downarrow\downarrow}$ whenever $fgu=fgv$.
All compositions are defined straightforwardly.
Now, verify that in $\mathcal D$, the reflection of an object $f\,\in C^\downarrow$ in the full subcategory $C^{\downarrow\downarrow}$ will be just the cokernel pair of $f$ (whenever it exists) and the coreflection of an object $(u,v)\,\in C^{\downarrow\downarrow}$ will be just the equalizer of $(u,v)$.
Finally, conclude the adjunction. (See also my answer on this.) | 95,987 |
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TITLE: what ordinals were used in proving unique factorization over $\mathbb{Z}$
QUESTION [0 upvotes]: I don't fully understand this other question, but there's a clear relationship between logic and number theory
the strength of saying "each sentence of true arithmetic has a recursive proof"
Here's a statement: every integer $n \in \mathbb{Z}\backslash\{0\}$ has a unique prime factorization which could be thought of as defining a tree structure on the integers.
For a theoretical computer scientist this is just like any other tree, which can be iterated through or breadth-first search or DFS, etc. The fact that the nodes are integers is almost immaterial.
All I know is that certain number theory statements could be be proven with first order logic and others with second order logic, but I doubt anyone details which logic structures were actually used.
Even more basic, does the Euclidean algorithm define a recursive structure on pairs of integers or sequence of integers?
REPLY [2 votes]: Let me first address the specific issue of how statements in number theory correspond to recursive "infinitary tree proofs."
Well, first of all, the point is that every true statement about natural numbers has a recursive proof. So asking whether something like the (statement of the effectiveness of the) Euclidean algorithm has a recursive proof, will always receive the answer "yes." Here's an example of a recursive proof of the Euclidean algorithm's correctness and termination:
For each specific natural number $n$, we can prove using PA that the Euclidean algorithm "works" on $n$ (or that $n=0$). Moreover, we may effectively find the first proof (in some reasonable indexing) of this fact, $\pi_n$.
The function $f$ taking $n$ to $\pi_n$ - "search until you find a proof that EA works for $n$" - is recursive; and so constitutes a "level-one" recursive proof. That is, we only need one use of the recursive $\omega$-rule.
The overall structure of this proof is a tree: it has a root, branching $\omega$-many times, and at each branch we have a usual first-order proof. (Note that this tree really doesn't come from the details of the algorithm at all, just the fact that PA can prove each specific instance of it working; the complexity of the algorithm is not necessarily related to the complexity of the recursive proof.)
Note that indeed, this picture applies to any true $\Pi^0_1$ sentence (in particular, it works regardless of whether that sentence is provable in PA alone). More complicated statements require more uses of the recursive $\omega$-rule, leading to more complicated trees of rank up to $\omega^2$ (I believe that bound is sharp).
EDIT: By a small tweak, we can in fact show that any true $\Sigma^0_3$ sentence $\exists x\forall y\exists z\varphi(x, y, z)$ (with $\varphi$ having only bounded quantifiers) has a recursive proof of height 1, as follows:
Begin the proof by guessing correctly the right value of $x$. We don't have a good way of knowing what $x$ should be, but this is just one bit of information - so "coding it in" to the recursive proof doesn't break anything. (This is exactly analogous to how the set $\{x:$ the Riemann hypothesis is true and $x=0\}$ is computable, even though we don't know if it is $\emptyset$ or $\{0\}$.)
Now, consider a function $f$ which - on a given $y$ - searches for some $z$ such that $\varphi(x, y, z)$ holds together with a proof ("certificate") of this fact. Since such a $z$ always exists, and whether $\varphi(x, y, z)$ holds can be effectively tested since $\varphi$ has only bounded quantifiers, a proof that such a $z$ works also exists, and moreover the function $f$ can be taken to be recursive.
But then this $f$ (basically) constitutes a recursive proof.
So it's at the $\Pi^0_3$ level that things first can become interesting - the idea being that we somehow need to "guess correctly" at the existential quantifier in the middle not once but infinitely many times, one for each possible value of the outermost "$\forall$;" so the "guess $x$ correctly" trick above doesn't work. Moreover, telling whether a given guess works is a $\Pi^0_1$ (noneffective, that is) question, so even if by luck we happen to have guessed the right witness, we can't obviously verify its correctness, and that's necessary for a recursive proof.
So this is the subtlety of the Shoenfield completeness theorme: the naive idea of just "bootstrapping" up the $\Pi^0_1$ idea (which does work if we replace the recursive $\omega$-rule with the full $\omega$-rule, and then every true sentence has a proof of finite height!) doesn't work, and something more complicated has to be done.
Now what do ordinals have to do with all this? Well, the linked question is about the difficulty of proving "Every true statement has a recursive proof," and specifically asks how much transfinite induction is needed to prove this (and the stronger statement that only height $\omega^2$ is needed). Well, in order to formalize transfinite induction principles in PA, we need some way to talk about ordinals (e.g. how do you write "$\epsilon_0$ is well-founded" in the language of arithmetic alone?), so this is what the "appropriate coding of ordinals" is about. And if you look at Franzen's paper, you'll see how this shows up in the statement and proof of the result.
This, of course, leaves off an important question: why are transfinite induction principles the right metric to use to gauge how hard it is to prove something? This is a general theme in proof theory (and elsewhere), that transfinite induction principles help you prove statements about the existence (or non-existence) of well-founded labelled trees with certain properties; e.g. here it's about the existence of well-founded labelled trees which correspond to recursive proofs, while in ordinal analysis it's about the nonexistence of finite labelled trees corresponding to proofs of $0=1$ from specific families of axioms.
In particular, the analysis of the Euclidean algorithm case is provable in PA alone (and indeed much less), so no ordinal strength at all is needed to show that that statement has a recursive proof (and indeed, I believe the vast majority of statements in basic number theory will have recursive proofs provably in PA alone, although I'm not an expert here so don't quote me on that). | 22,048 |
TITLE: what are the p-adic division algebras?
QUESTION [23 upvotes]: Is there a classification of division algebras over $\mathbb{Q}_p$? There are field extensions of $\mathbb{Q}_p$, but are there any others? In particular, I want to know if they are all commutative.
When I posed this question commuting algebra of an irreducible representation yesterday, I thought that the important point was the irreducibility of the representation, but now I realize that it might be the $p$-adic property that is important.
REPLY [29 votes]: There are hundreds of people who can answer this question better than I do here, but here is at least a partial answer. I hope that I do not misrepresent the truth too egregiously. (And I welcome corrections from people who are more deeply informed than I am.)
The (isomorphism classes of) finite-dimensional central division algebras over a field $K$ are classified by a gadget called the Brauer Group of $K$, and it is a basic fact of Local Class Field Theory that the BG of $\mathbb Q_p$ is isomorphic to $\mathbb Q/\mathbb Z$. The element of the BG corresponding to a central division algebra $H$ is called the invariant of $H$. So the answer to your question is, yes, there are plenty of division algebras over $\mathbb Q_p$, even ones whose center is $\mathbb Q_p$ itself. (That’s what one means when one speaks of a “central division algebra” over $\mathbb Q_p$.)
Here's a simple description of a cda of invariant $1/n$: let $U$ be the unramified extension of $\mathbb Q_p$ of degree $n$. I’m going to make an $n$-dimensional vector space over $U$ into a noncommutative ring. Let the basis be the set $\{1,\pi,\pi^2,\cdots,\pi^{n-1}\}$, where $\pi$ simply behaves like an $n$-th root of $p$: that is, $\pi^n=p$. And the question is how $\pi$ commutes with scalars from $U$. Here it is, $\pi u=u^\sigma\pi$, where by $u^\sigma$ I mean the image of $u$ under the Frobenius automorphism $\sigma$ of $U$ over $\mathbb Q_p$. Remember that the Galois group of $U$ over $\mathbb Q_p$ is cyclic, and $\sigma$ is a generator, satisfying the property that for $z$ in the ring of local integers of $U$, we have the relation $z^\sigma\equiv z^p\bmod{(p)}$.
If you’re really interested in this, I recommend that you try this out in the smallest case, $p=n=2$, and calculate the reciprocal of, for instance, $1+\omega+\pi$, where $\omega$ is a primitive cube root of $1$ over $\mathbb Q_2$, and $\pi^2=2$. | 58,360 |
TITLE: set of immersions on a manifold is an open set in the set of all mappings to $\mathbb{R}^{2m+1}$
QUESTION [1 upvotes]: I'm studying a proof of Takens' Delay Embedding Theorem. A key fact used is that the set of immersions is open in the set of all mappings (mappings from an $m$-dimensional manifold M to $\mathbb{R}^{2m+1}$).
I don't know how dumb this question is, but in what sense is it open? In which topology do we have this openness of our set of interest? I'm trying to develop an intuition on the set of such mappings, so I ask this question.
REPLY [2 votes]: It's a good question. Usually for mapping spaces, one uses the compact-open topology. It has as a subbase consisting of sets $C(K, U) = \left\{f: X \to Y \;|\; f(K) \subseteq U \right\}$, where $K$ varies over compact $K \subseteq X$ and open $U \subseteq Y$. This topology enjoys many natural properties (especially when the spaces are Hausdorff) that are itemized on the wiki page.
Exactly what constitutes a map $f:X \to Y$ depends on the category that you're interested in. It's likely that you're considering smooth maps, given the (differential-topology) tag. | 144,450 |
TITLE: How to choose $\alpha$ such that the improper integral with respect to $\alpha$ is finite?
QUESTION [1 upvotes]: Give and example of a function $f$ and a choice of $\alpha$ that is strictly increasing such that
$$\int^\infty_0 f\,dx=+\infty \quad \text{and} \quad \int^\infty_0 f\,d\alpha<\infty.$$
REPLY [1 votes]: Let $f(x)=\chi_{[0,1)}(x)x+\chi_{[1,\infty)}(x)x^{-1}$ and $\alpha(x)=\displaystyle\int_{0}^{x}f(t)dt$, then $\alpha$ is strictly increasing and that $\alpha'(x)=f(x)$, so
\begin{align*}
\int_{0}^{\infty}f(x)d\alpha(x)&=\int_{0}^{\infty}(f(x))^{2}dx\\
&=\int_{0}^{1}x^{2}dx+\int_{1}^{\infty}\dfrac{1}{x^{2}}dx\\
&<\infty,
\end{align*}
but
\begin{align*}
\int_{0}^{\infty}f(x)dx>\int_{1}^{\infty}\dfrac{1}{x}dx=\infty.
\end{align*} | 26,916 |
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January 9, 2007
As annual meeting closes, boards in all 3 divisions decide -- or punt -- on a range of legislation. 8, 2007
A federal judge has issued an injunction ordering the State University of New York at New Paltz to reinstate two student government leaders who were suspended because of allegations (which they deny) that they harassed a university administrator.
January 8, 2007
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The simple, little economic trilogy of scarcity, choice, and cost may well be one of the most useful concept gadgets in your tool bag of decision making. Keep the tool handy and use it a lot.
One of the most interesting aspects of our present consumer culture is our fixation with the printed price tag. What does this item cost? Or, this week we are rolling back all the prices, or, the cost of this item is reduced until Friday. The cost to be considered in the purchase of a product or service is presented to be on the bar code or price tag. But the money you tender to the cashier may not be the cost at all. The retailer, however, has even narrowed the logic of your decision to purchase down to the common denominator of do I have enough money to buy this?
In your lifetime the culture has even erased the criterion of whether you have enough money or not in your pocket to make the purchase by supplying you with a convenient piece of plastic that nullifies the logic and allows you to make the purchase, regardless. You walk away with your purchase fully confident that you did the really smart thing because you simply could not have lived without the item another day at that irresistible price. I actually know folks who will drive all the way across town to purchase their gasoline if the price is one cent per gallon cheaper than the competition. They spend three dollars to get across town to save twenty-four cents at the pump. What was the real cost?
I remember listening to a reporter’s interview with one of the early astronauts. The reporter’s question was, how do you feel, sitting in a spacecraft that has been built by the cheapest bidder? Our thinking regarding price tags or bids sometimes does not consider the real cost.
In the discipline of economics, cost is viewed through a different set of glasses. Cost is not determined by how much money you spent or how many plastic swipes you transacted. Cost is inexorably linked to scarcity and choice. In fact, scarcity, choice, and cost are at the very heart of the study of economics. When a good or service is scarce it means that the choice of one alternative requires that the next highest alternative be given up. The very existence of alternative uses forces us to make choices. The opportunity cost of any choice is the value of the next best alternative that was lost or forgone in making that choice. The true cost of the alternative that you did not choose may or may not have anything to do with money as represented on a bar code or printed price tag.
It would be easy to give lots of illustrations of foregone opportunity cost: the university student who gives up four years of earned income and pays thousands and thousands of dollars to attend university classes he hopes will better his position for the balance of his life; the girl who gives up marrying her childhood sweetheart in lieu of a musical career in London; the farmer who decides to continue farming the old family acreage instead of selling the property to the big box store.
But I don’t have to go farther to find true-to-life examples than to consider some of my friends from the organization that I love so much: Project C.U.R.E. We have over 16,000 volunteers within the United States at our various warehouse operations and collection cities. Every one of those sixteen thousand individuals has a personal story of scarcity, choice, and cost. The stories are packed with drama and passion because they each represent personal desires and values. None has to come and volunteer. But they faithfully come by the scores . . . but at a cost. What was the value of the alternative that was lost when they decided to forego that next highest alternative and choose to volunteer their time and efforts for no money payment?
In 1999, an operating room nurse named Barb Youngberg came with another nurse friend to check out Project C.U.R.E. They had heard at a hospital staff meeting about the humanitarian organization that was functioning out of the huge Continental Airport hangers at the old Denver Stapleton airport. They were intrigued by what they saw and the idea that their efforts were saving lives all around the world. Barb and her friend made a decision to start sorting medical supplies one night a month for Project C.U.R.E.
As Project C.U.R.E. had to move around the city from one donated location to another, Barb followed and began donating more and more of her time. In 2005, she retired from her hospital job but kept volunteering. In 2007, the doctors discovered that Barb had congestive heart failure. To complicate the diagnostic procedures, they found she was allergic to the dye required for the tests. She knew exactly what the surgery entailed because she had scrubbed down and assisted in many such surgeries in the past.
Just hours after the successful diagnosis, the doctors performed open heart surgery that also included the insertion of a pacemaker. As she lay in the recovery room and then spent two and a half weeks in rehabilitation, she knew she had to make a huge decision. She was now realizing that the years of life she had left were a very scarce commodity. What would she do with those years? She had to make a decision. Would she take it easy on some leisure cruises, or relax and stay around home . . . or what?
Barb made her decision. Just as soon as she was able, she returned to Project C.U.R.E. She had decided to give the best of her life for the rest of her life helping other people be better off. Project C.U.R.E. had just moved into its permanent location near Park Meadows, in Centennial. Barb volunteered to take over the entire bio-med tech area and build it into an efficient and successful department. She already possessed the knowledge of the various pieces of equipment. Over the years of her career she had accumulated the valuable experience of working with the medical machines and pieces of technical equipment. Those were her talents. Now she could manage the inventory and lots of other people to help her build and run a successful department.
When the pieces of bio-med-tech equipment, like x-ray machines, defibrillators, e.k.g. machines, ultrasound machines, or anesthesia machines arrive on Project C.U.R.E.’s docks, Barb Youngberg knows exactly what to do to get those valuable, life-saving items ready to be shipped to the targeted hospitals and clinics in over 130 different countries working with Project C.U.R.E. around the world.
Utilizing the economic trilogy of scarcity, choice, and cost can help us make better decisions throughout our lives. In Barb Youngberg’s case, realizing she faced the scarcity of years, and knowing she had to make a choice, she gave up the options of cruises and leisure in lieu of helping to save countless lives of kids and adults around the world. We salute her for her nearly fourteen years of voluntary service at Project C.U.R.E. and her determination to make good choices.
Next Week: A weakness of the economic trilogy
(Research ideas from Dr. Jackson’s new writing project on Cultural Economics)
© Dr. James W. Jackson
Permissions granted by Winston-Crown Publishing House | 302,450 |
Don Williams - We've Got A Good Fire Goin' Lyrics
There's a storm rollin' over the hill
And the willow trees are blowin'
I'm standin' here starin' out the window
Safe and warm
I feel her put her arms around me
Oh, I've got a good woman
And we've got a good fire goin'.
We've got a feast on the supper table
And bread for breakin'
A blessing from the Lord
For makin' me such a fortunate man
The light of my life in the candle
Her face a glowin'
Oh, I've got a good woman
'.
[Instrumental]'.
Oh, I've got a good woman
And we've got a good fire goin'...
Other Lyrics by Artist
- Don Williams - I'll Be There If You Ever Want Me
- Don Williams - Diamonds To Dust
- Don Williams - Donald And June
- Don Williams - Back In My Younger Days
- Don Williams - Just 'Cause I'm In Love With You
- Don Williams - Come A Little Closer
- Don Williams - Darlin' That's What Your Love Does
- Don Williams - Lord Have Mercy On A Country Boy
- Don Williams - True Love
- Don Williams - Then It's Love
- Don Williams - She's A Heart Full
- Don Williams - Desperately
- Don Williams - We've Got A Good Fire Goin'
- Don Williams - What's The Score
- Don Williams - I Wouldn't Be A Man
- Don Williams - I'll Never Be In Love Again
- Don Williams - Jamaica Farewell
- Don Williams - Loving You's Like Coming Home
Rand Lyrics
Don Williams We've Got A Good Fire Goin' Comments
Great backup by the Gatlin Brothers.
One of his best and I love it so much
jimreev es
Not funny this fire is really hot. Help me baby. Great song. Great Singer. This I dedicate to someone. Help me please!!! Begging!!! Puppy dog eyes. Lol
Thank you for loving me so much..my dear...
Don Williams was a great great musician..... MISRIP
Beautiful song by one of Greatest Legendary in country music.
Let the fire of love burn us to give the warmth of it....
Never ending flame...in deed...
And you do feel it on your side too..my love.......Thank you dear Don. W...R.I.P. dear...you do light our flames of love ...with your beautiful songs...
Iam from Pacific island... I love your songs and as a gentleman feel satisfy.. Love you Don... RIP
Um homem de arrasar quarteirão belo mas está feliz dando continuidade a sua vida no mundo espiritual 🙏🙏🙏🙏🙏🙏
We do have a good fire going. LOML
Hmmm.....Love You ..Love You...Love You....Nothing seems to put off the Fire of our Love dear....I wish we could keep the rest of the world around us warm too....
You live forever in our hearts.Always remembered. Love you D.W.
Love enflames the wick of life...
I love this song and I love you Devon salmond
Anyone listening in 2019?
Um luxo iloveyou um brindaco 🍹♥️ as boas vibrações positivas que nos embalam na paz infinitamente maior
Iloveyou sempre 👍🌷💋❤️
RIP, Don.
Dave Logins wrote this, just in case you wanted to know.
Let the rain fall til morning.
WHY don't 'they' give writer credits? I'd really like to know. (Nevermind, thanks Eddie!)
Don must have had a fair share of enemies, I wonder why. I keep seeing a significant number of dislikes in all his songs and I can't help wondering what is there to dislike
Shame this ballad spent 2 weeks at #1 on Radio & Records country charts . but only #3 on Billboard. "One Good Well, "Back In My Younger Days", "True Love", and "I've Been Loved By The Best " are all #1 country songs on all charts!! So there!!
rip don ❤️
He is gifted artist and fortunate man . His voice is flowing in my heart and I feel I've a good fire going.
this is simply amazing ...full of passion.
Conscience and passion, voices of the soul. The earth, the sound of perseverance. RIP gentle giant
Thanks for sharing great song loved it
The perfect song for that cold rainy night when you are stuck indoors with your woman. Just the two of you.
Love you Don. You are one of the greatest.
il chantais tres bien mais son caractere raciste ne nous permet pas de le considerer.
This is my favorite songs of Don Williams. Very relaxing song.
yeah me got me a good man
All time great
I so love this song
Great song love all Don Williams songs Thank you 🍀🍀🍀
This song warms me inside out...I pray and hope one day I find a man who feels like this about me and I feel the same🙏🏽
I'd like to get my hands on the 30 people who didn't like this song!
J'adore
What a good voice and good story behind his music
With wifey
So lovely. :)
And the taste of life's sweet wine !
This or I Recall a Gypsy Woman Like you say one of the sexiest song that you'd play in front of your Grandmother.
God had seen fit to endow Don Williams with the power to turn all love around us into music, with his unique singing voice..just pure magic at work
I've got a good woman and we've got a good fire going!
Just when you thought Don Williams had recorded every good song that existed, he came out with this one. Don Williams "The Gentle Giant" has never recorded a Bad song in fact ALL of his songs are absolutely Beautiful. I am so glad that I got to see him in concert.
This was the song that we chose for our wedding dance 29 years ago. Still married to the same wonderful man. Thank you Mr. Williams for a wonderful song. Rest in peace.
Beautiful song. Rest In Peace Don and thank you for your music.
The rain of the entire music community is falling at the loss of Sir Don! He was one of God's gifts to those who find peace in the simplicity and honesty of uncompromised lyrics.
RIP Don your song will be remembered and loved .
RIP legend, i am mourning
RIP king of country .. your music will live forever.
RIP Don. Thank you for so much you gave us, especially that voice.
Gentle Giant,Rest in Peace,lay on the cold ground and rise live gently among the serene angels where you will feel eternal bliss and sing the Lord's praise forever.Wait for us on the pearly gates !!Rest In Peace Legend Rest in Peace
RIP , Don Williams❤️
I'm a metalhead by nature but I cannot deny the genius that was Don Williams. Rest In Peace, Gentle Giant.
My favorite Don Williams song. RIP. Your music was beautiful.
RIP Dr. Don the doctor of songs
Rest in peace.
This song simply grounds me.... I could listen to it over and over. Innocent and seductive all at the same time. You will be certainly missed!
Thank you, Don. Thank you for meaning so much to me all of my life. I will never ever replace the feelings your music provides me. Glad to have seen you in person twice... Godspeed my friend, may you deliver a gorgeous concert to the angels.
Wish i could have seen him,it was a life goal of mine!May he rest in peace.....butoH LOrd!Please Have mercy on a country boy
I always relaxed with such music, may he have a long life,Oh don
I just love Don Williams.
no after party at my wedding no friends just a private wedding m]for me and sam
This song reminds me of my campus days. I could listen to this album on a rainy evening and the memories of that feeling are indescribable....this song and Phil Collins 'Everyday'
I've always loved this man. Don, you are the BEST
RIP Gentle Giant,the legacy lives on
Thanks Don for giving me some joy may your soul rest in peace
Oh Don, wish the rain and the fire were going here with me!
Whenever i listen to this song, i just feel good. Nice song!
listening to this song is better than a valium
Haha...thats true!!
my dad dedicated this song to my mom the first day he seen her I remember he would play this song an sing it after she passed away I was only 16 I miss u mom rip an I miss u dad rip its only been 7 months you've been gone but my heart hurts everyday you guys aren't here
It's a pity that no-one has mentioned the song writer Dave Loggins - he's also supplying the backing vocals on this recording. Great song - beautifully presented
Eddie Edwards Did you ever stop to think maybe nobody was aware of it?
If people weren’t aware of it why would you blame them for not bringing it up?
Very good, I love to hear it.
bello, romantico e geniale.....
Don Williams - We've Got A Good Fire Going
No one better.a man,his guitar on stage.sawhim in Huntsville alabama.best concert I ever seen
Just superb
A Dave Loggins song with Don Williams singing lead and Dave singing back up. One of the best!
a song full of so much memory. he's a genius performer and I can't think of a single bad song he ever did....but this one is special.
brilliant singer and a great song writer, he had it all
THANKS FOR THE UPLOAD
great tune love it!!
What an example of a musician and a man.
"...a feast on the suppertable..." This song, the composition, arrangement, performance is tops. Sophisticated, timeless, poetic even.
Loon sez: It's hard from me to imagine him doing anything else but singing the tunes he does.... born to the craft. I'm known to rock but before 'rock 'n' roll', I grew up with this kind of American music. And now and then, nothing else will do. Right now, for example.
agreed.what a great tune.and performer.and writers.technicians.sounds fantastic!!!!
cette chanson me rappel mon voyage a okondja a l est du GABON
cool....
wow the great Ever
i like za song
Don Williams...The Best Ever.
Magnificient Don Williams does it again
true love song
What a song for where our life is right now. November evening, 23 degrees outside, a good woman next to me, and a good fire going. Life is good. Scoat
My loooooove is over flowing...
Don williams it was of the 80s the best man singer of country.I love him so much.
When I'm happy when l am sad or just want to be alone or need company all I need is to hear don williams n all s right in my world.love u and bless u.
Menka Khan there's no easy listening like don
My hero in country music. The legend Don Williams RIP always remembered.
Gotta dedicate this one to the best dance partner ever! 32 years now. Love ya Deb W. !!
The "Gentle Giant" never gets old...love his voice.
Ekeson ndukwe odum
Hé lit my life like a candle
M'y love is overfloying
This song is so good
This song is so good | 264,275 |
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\begin{document}
\title{Space of initial conditions for a cubic Hamiltonian system}
\author{Thomas Kecker}
\date{}
\maketitle
\begin{abstract}
\noindent
In this paper we perform the analysis that leads to the space of initial conditions for the Hamiltonian system $q' = p^2 + zq + \alpha$, $p' = -q^2 - zp - \beta$, studied by the author in a previous article \cite{kecker1}. By compactifying the phase space of the system from $\mathbb{C}^2$ to $\mathbb{CP}^2$ three base points arise in the standard coordinate charts covering the complex projective space. Each of these is removed by a sequence of three blow-ups, a construction to regularise the system at these points. The resulting space, where the exceptional curves introduced after the first and second blow-up are removed, is the so-called Okamoto's space of initial conditions for this system which, at every point, defines a regular initial value problem in some coordinate chart of the space. The solutions in these coordinates will be compared to the solutions in the original variables.
\end{abstract}
\section{Introduction}
When studying the solutions of a differential equation in the complex plane a natural question to ask is what types of singularities can occur by analytic continuation of a local analytic solution, which exists by Cauchy's local existence and uniqueness theorem around every point where the equation is defined as a regular initial value problem. Points where the equation is itself singular are called fixed singularities of the equation. All other singularities of solutions, which arise somewhat spontaneously, i.e.\ they cannot be read off from the equation itself, are called movable singularities, as in fact their position varies with the initial conditions prescribed for the equation. Some differential equations are very special in this respect as the only movable singularities that can occur by analytic continuation of a local analytic solution are poles: an equation of this type is said to have the Painlev\'e property. In the class of second-order ordinary differential equations the six Painlev\'e equations stand out as nonlinear equations with this property. The six Painlev\'e equations are well-studied by many authors and have a rich mathematical structure of families of rational and special function solutions, B\"acklund transformations, and relations to integrable systems and isomonodromic deformation problems of associated linear systems. Another feature of the Painlev\'e equations was studied by Okamoto \cite{okamoto}: the so-called space of initial conditions. This space is constructed to deal with the equations at the points at infinity of the phase space, in particular at the movable singularities of the solutions. It is obtained by extending the phase space of the variables $(y,y')$ of the system to a compact space including the points where $y$ or $y'$ (or both) become infinite, e.g. $\mathbb{CP}^2$. Okamoto \cite{okamoto} constructed the space of initial conditions for all six Painlev\'e equations. Starting from so-called Hirzebruch surfaces (certain compact rational surfaces), each equation re-written in local coordinates on the surface can be regularised by a sequence of $8$ {\it blow-ups} (one needs $9$ blow-ups if starting from $\mathbb{CP}^2$). He showed that at every point of the resulting space, the equation written in some local coordinate chart in a neighbourhood of this point, forms a regular initial value problem.
All six Painlev\'e equations can be written in the form of equivalent Hamiltonian systems. Sakai \cite{sakai} has studied the spaces of initial conditions for all six Painlev\'e equations in Hamiltonian form from a geometric viewpoint based on the symmetries of the underlying rational surfaces. The space of initial conditions for the Painlev\'e equations in Hamiltonian form was also studied in \cite{matano}. Joshi et.\ al.\ study the asymptotic behaviour of the solutions of the first Painlev\'e equation \cite{joshi1}, second Painlev\'e equation \cite{joshi2} and fourth Painlev\'e equation \cite{joshi3} via the space of initial conditions in the limit when the independent variable goes to infinity.
In this article we construct the space of initial conditions for the cubic Hamiltonian system (\ref{hamsys}) below, which was previously studied in \cite{kecker1} and is fact related to the Painlev\'e IV equation, but different to the standard Hamiltonian system considered e.g. in \cite{sakai} or \cite{matano}. Although the space of initial conditions is isomorphic to the one obtained there, the blow-up calculations in local coordinates are somewhat easier because of a $3$-fold symmetry in the singularity structure of the solutions: all singularities are simple poles with residues given by third roots of unity $1,\omega,\bar{\omega}$, where $\omega = \frac{-1+i\sqrt{3}}{2}$. After compactifying the phase space of the system to $\mathbb{CP}^2$ one finds three points where the system is indeterminate, the so-called {\it base points}, each of which can be resolved by a sequence of three blow-ups. Thus, the total number of blow-ups needed to regularise the system is also $9$ as in the case of Okamoto. However, the calculations to resolve each of these three base points are essentially the same up to factors of $\omega$ and $\bar{\omega}$. Each blow-up extends the phase space by introducing an additional line, called the exceptional curve. We show that the exceptional curves arising from the first two blow-ups are repellors of the dynamic system, meaning that the only singularities that can arise occur when the solution passes through the exceptional curve after the third blow-up. On this line the system of equations forms a regular initial value problem, which is a manifestation of the Painlev\'e property of this system. Thus the space of initial conditions, formed by the extended compact phase space after the three blow-ups for each base point, with the exceptional curves after the first and second blow-up removed, has the property that at each point there exist local coordinates such that the system of equations has a local analytic solution. We will see how the solutions of the system after the last blow-up are related to the solutions in the original variables.
The regular systems obtained after the third blow-up of each base point were also used in \cite{kecker1} for an alternative proof of the Painlev\'e property for the Hamiltonian system (\ref{hamsys}). The solutions of system (\ref{hamsys}) were studied further by Steinmetz \cite{steinmetz} by the re-scaling method, in particular their asymptotic behaviour in sectors of the complex plane and the distribution of poles.
\section{A cubic Hamiltonian system}
We consider the following Hamiltonian system, introduced in \cite{kecker1}, with Hamiltonian cubic in $p$ and $q$,
\begin{equation*}
H(z,p,q) = \frac{1}{3} \left( p^3 + q^3 \right) + zpq + \alpha p + \beta q,
\end{equation*}
the Hamiltonian equations being
\begin{equation}
\label{hamsys}
q' = \frac{\partial H}{\partial p} = p^2 + z q + \alpha, \quad p' = -\frac{\partial H}{\partial q} = -q^2 - z p - \beta.
\end{equation}
This system is related to the fourth Painlev\'e equation in the following way. Namely, the combination $w = p + q - z$ satisfies the equation
\begin{equation}
\label{p4scal}
2ww'' = w'^2 - w^4 -4zw^3 -(2\alpha+2\beta+3z^2)w^2 - (1-\alpha+\beta)^2,
\end{equation}
which becomes $P_{I\!V}$ after a simple rescaling of variables. Furthermore, the combinations $w = \rho p + \bar{\rho}q - z$, $\rho \in \{\omega,\bar{\omega}\}$, satisfy the same equation (\ref{p4scal}) but with the parameters $\alpha$ and $\beta$ replaced by $\rho \alpha$, $\bar{\rho} \beta$. Hence, by linear combination of solutions of equation (\ref{p4scal}) with different parameters, the solutions of system (\ref{hamsys}) can be expressend completely in terms of the fourth Painlev\'e transcendents.
Although it is already granted by the connection with the Painlev\'e IV equation that the system (\ref{hamsys}) has the Painlev\'e property, in \cite{kecker1} an alternative proof of this statement was given. At any singularity $z_\ast$ of system (\ref{hamsys}), its solution is represented by a Laurent series, convergent in a punctured neighbourhood of $z_\ast$, of the form
\begin{equation}
\label{poleexpansion}
\begin{aligned}
q(z) =& \frac{-\rho}{z-z_\ast} + \frac{\rho z_\ast}{2} + \left( \rho \left(1+\frac{z_\ast^2}{4} \right) - \frac{\alpha}{3} + \frac{2}{3} \bar{\rho} \beta \right) (z-z_\ast) + h (z-z_\ast)^2 + \sum_{n=3}^\infty c_n (z-z_\ast)^n \\
p(z) =& \frac{\bar{\rho}}{z-z_\ast} + \frac{\bar{\rho} z_\ast}{2} + \left( \bar{\rho} \left( 1 - \frac{z_\ast^2}{4} \right) - \frac{2}{3} \rho \alpha + \frac{\beta}{3} \right) (z-z_\ast)+ k(z-z_\ast)^2 + \sum_{n=3}^\infty d_n (z-z_\ast)^n,
\end{aligned}
\end{equation}
$\rho \in \{1,\omega,\bar{\omega} \}$, having simple poles with residues given in terms of the third roots of unity. Here, $h$ and $k$ are complex parameters, coupled by the linear relation
\begin{equation*}
\rho h - k = \left( \frac{5}{4} \bar{\rho} - \frac{\alpha}{2} \rho + \frac{\beta}{2} \right) z_\ast,
\end{equation*}
so there is essentially only one free parameter. Fixing this parameter is similar to prescribing initial conditions for the system of equations, and we will see how this is done at the end of this article.
The $3$-fold structure of the solutions of the system will be important in the following: it will allow us to construct the space of initial conditions for the system in a symmetric manner. The proof in \cite{kecker1} of the Painlev\'e property of system (\ref{hamsys}) relies on the fact that at any simple pole of the system (\ref{hamsys}) the function
\begin{equation*}
W(z) = H(z,p(z),q(z)) + \frac{p(z)^2}{q(z)}
\end{equation*}
remains finite. This in turn relies on the fact that $W$ satisfies the first-order linear differential equation
\begin{equation*}
W' + 3 \frac{p}{q^2} W = \beta \frac{p}{q} + 2 \alpha \left( \frac{p}{q} \right)^2 + 3 \left( \frac{p}{q} \right)^3,
\end{equation*}
and Lemma 6 in \cite{kecker2}, showing that the coefficients in this equation, i.e. $\frac{p}{q^2}$ and powers of $\frac{p}{q}$, remain bounded at a singularity. The auxiliary function $W$ will also play an important role below when we are showing that certain points at infinity in the space of initial conditions cannot be reached by analytic continuation of a solution. Due to the nature of the blow-up computations the expressions we are dealing with become somewhat long and we have used \textsc{Mathematica} to perform the symbolic calculations.
\section{Constructing the space of initial conditions}
At the movable singularities of a solution, the system of equations itself is well-defined and nothing can be said in general about the nature of the solution in a neighbourhood of this point just from the structure of the equation. To obtain some information on how the solution behaves in the vicinity of a movable singularity one has to include the points at infinity of the phase space of the system as the solution will be unbounded in this space. Thus the first step in constructing the space of initial conditions is to extend the system of equations in the variables $(q,p) \in \mathbb{C}^2$ to a compact space which includes the points where either $p$ or $q$ (or both) are infinite. In general any rational surface can serve as compactification but in the following we compactify the phase space of the Hamiltonian system to $\mathbb{CP}^2$. To this end we express the system of equations in the three standard coordinate charts of complex projective space, denoted by $(p,q)$, $(u_1,u_2)$ and $(v_1,v_2)$, where
\begin{equation*}
[1:q:p] = [u_1:1:u_2] = [v_1:v_2:1],
\end{equation*}
which together cover $\mathbb{CP}^2$. The sets of points $u_1 = 0$ in the variable $(u_1,u_2)$ and $v_1 = 0$ in the variables $(v_1,v_2)$ represent the line at infinity of $\mathbb{CP}^2$, denoted by $L$ in the following. In these two coordinate charts, the system of equations becomes
\begin{equation}
\label{usystem}
u_1' = -\alpha u_1^2 - z u_1 - u_2^2, \quad u_2' = -\beta u_1 - \gamma u_1 u_2 - 2 z u_2 - \frac{u_2^2+1}{u_1}
\end{equation}
and, respectively,
\begin{equation}
\label{vsystem}
v_1' = \beta v_1^2 + z v_1 + v_2^2, \quad v_2' = \alpha v_1 + \beta v_1 v_2 + 2 z v_2 + \frac{v_2^3+1}{v_1}.
\end{equation}
We see that at the points $(u_1,u_2), (v_1,v_2) \in \{(0,-1), (0,-\omega), (0,-\bar{\omega}) \}$ the right hand side of the second equation in (\ref{usystem}) or (\ref{vsystem}) becomes indeterminate, i.e.\ is of the form $\frac{0}{0}$ and nothing can be said about the behaviour of the solutions. These are the {\it base points} of the system extended on $\mathbb{CP}^2$. However, the pairs of coordinates $(u_1,u_2) = (v_1,v_2) = (0,-1)$, $(u_1,u_2) = (0,-\omega), (v_1,v_2) = (0,-\bar{\omega})$ and $(u_1,u_2) = (0,-\bar{\omega})$, $(v_1,v_2) = (0,-\omega)$ each describe the same point in $\mathbb{CP}^2$, so there are only three base points for our system. The following Lemma shows that, apart from these three base points, the line at infinity cannot be reached by analytic continuation of a solution in the complex plane.
\begin{lemma}
\label{lemL}
Let $\Gamma$ be a rectifiable path in the complex plane with endpoint $z_\ast$ such that analytic continuation of a solution of the system in the variables $(u_1,u_2)$ or $(v_1,v_2)$ along $\Gamma$ leads to a point $P \in L$, represented by the coordinates $(u_1,u_2) = (0,c)$ or $(v_1,v_2) = (0,c)$, $c \in \mathbb{C}$, respectively. Then we have $c \in \{-1,-\omega,-\bar{\omega}\}$.
\end{lemma}
\begin{proof}
Let $W_{(u)}(z)$ and $W_{(v)}(z)$ denote the functions obtained by re-writing the auxiliary function $W(z)$ in the variables $(u_1,u_2)$ and $(v_1,v_2)$, respectively. We perform the analysis for $W_{(u)}$, the case for $W_{(v)}$ being similar,
\begin{equation*}
W_{(u)}(z) = \frac{1+3 \beta u_1^2+3 z u_1 u_2+3 \alpha u_1^2 u_2+3 u_1^2 u_2^2+u_2^3}{3 u_1^3}.
\end{equation*}
Note that for any point $P \in L$, not one of the three base points, the values of $W_{(u)}$ and $W_{(v)}$ are infinite. At the base points themselves $W_{(u)}$ and $W_{(v)}$ are of the indeterminate form $\frac{0}{0}$. Now consider the logarithmic derivative
\begin{equation*}
\frac{d}{d z} \log(W_{(u)}(z)) = \frac{W_{(u)}'(z)}{W_{(u)}(z)} = -\frac{3 u_1 \left(u_2+2 \beta u_1^2 u_2+3 z u_1 u_2^2+\alpha u_1^2 u_2^2+u_2^4\right)}{1+3 \beta u_1^2+3 z u_1 u_2+3 \alpha u_1^2 u_2+3 u_1^2 u_2^2+u_2^3}.
\end{equation*}
Suppose that $P$ has coordinates $(0,c)$ where $c \notin \{-1,-\omega,-\bar{\omega}\}$. In some neighbourhood $U$ of $P$ the logarithmic derivative is bounded, say by some number $M$. Let $z_0 \in \Gamma$ denote a point on the curve such that $W_{(u)}(z_0) \neq 0$ and $(u_1(z),u_2(z)) \in U$ for all $z \in \Gamma_{z_0}$, the part of $\Gamma$ following $z_0$. Integrating along the path $\Gamma_{z_0}$ shows that
\begin{equation*}
|\log(W_{(u)}(z_\ast))| = \Big| \log(W_{(u)}(z_0)) + \int_{z_0}^{z_\ast} \frac{d}{d z} \log(W_{(u)}(z)) dz \Big| \leq |\log(W_{(u)}(z_0))| + \int_{z_0}^{z_\ast} M |dz| < \infty.
\end{equation*}
On the other hand, since $(u_1(z_\ast),u_2(z_\ast)) = P$, we would have $W_{(u)}(z_\ast)$ infinite, contradicting the above. Hence we must have $P=(0,-1)$, $P=(0,-\omega)$ or $P=(0,-\bar{\omega})$.
\end{proof}
We will now describe the procedure of blowing up the surface at the points $(u_1,u_2) = (0,-\rho)$, where $\rho \in \{1,\omega,\bar{\omega}\}$. The analysis for all three points is similar, differing only in various factors of $\omega$ and $\bar{\omega}$. The blow-up at a point $P=(p_1,p_2) \in \mathbb{C}^2$ in the coordinates $(u_1,u_2)$ is defined by the following construction,
\begin{equation*}
\text{Bl}_P = \{ ((u_1,u_2),[\zeta_1,\zeta_2]) \in \mathbb{C}^2 \times \mathbb{CP}^1 : (u_1-p_1) \zeta_2 = (u_2-p_2) \zeta_1 \},
\end{equation*}
where $[\zeta_1:\zeta_2]$ are homogeneous coordinates on $\mathbb{CP}^1$. Note that a blow-up is a local operation on the space when seen as follows. We define the projection $\pi: \text{Bl}_P \to \mathbb{C}^2$ by $((u_1,u_2),[\zeta_1,\zeta_2]) \mapsto (u_1,u_2)$. For any point $Q \neq P$, the pre-image $\pi^{-1}(Q)$ consists of a single point whereas the pre-image of $P$ itself is $\pi^{-1}(P) = P \times \mathbb{CP}^1$. This is called the exceptional curve in $\text{Bl}_P$. So we see that by a blow-up at $P$ the point itself becomes inflated to a sphere whereas away from $P$ the local geometry of the space does not change. In the coordinates $(u_1,u_2)$ the blow-up is performed by introducing two new coordinate charts, denoted by $(u_{1,1},u_{1,2})$ and $(u_{2,1},u_{2,2})$, respectively. The first new coordinate chart is given by
\begin{equation*}
u_{1,1} = \frac{\zeta_1}{\zeta_2} = \frac{u_1 - p_1}{u_2 - p_2}, \quad u_{1,2} = u_2 - p_2,
\end{equation*}
and covers the part of $\text{Bl}_P$ where $\zeta_2 \neq 0$, whereas the second chart is
\begin{equation*}
u_{2,1} = u_1 - p_1, \quad u_{2,2} = \frac{\zeta_2}{\zeta_1} = \frac{u_2 - p_2}{u_1 - p_1},
\end{equation*}
covering the part of $\text{Bl}_P$ where $\zeta_1 \neq 0$. The exceptional curves introduced by blowing up each base point will be denoted $L_1^{(\rho)}$, $\rho \in \{1,\omega,\bar{\omega}\}$, and are parametrised by $(u_{1,1},u_{1,2}) = (c,0)$ in the first chart and $(u_{2,1},u_{2,2}) = (0,c)$ in the second chart, $c \in \mathbb{C}$. When looking for new base points after performing the blow-up, we only need to look on the exceptional curve. Re-written in the variables after the blow-up, the system of equations in the first chart becomes
\begin{equation*}
\begin{aligned}
u_{1,1}' &= \frac{2\bar{\rho} - 2 \rho z u_{1,1}}{u_{1,2}} + (\beta - \rho \alpha)u_{1,1}^2 + z u_{1,1} - \rho, \\
u_{1,2}' &= (\rho \alpha-\beta) u_{1,1} u_{1,2} - \alpha u_{1,1} u_{1,2}^2 + 2z(\rho - u_{1,2}) - \frac{u_{1,2}^2 - 3\rho u_{1,2} + 3\bar{\rho}}{u_{1,1}},
\end{aligned}
\end{equation*}
where it is indeterminate at the point $(u_{1,1},u_{1,2}) = \left( \frac{\rho}{z},0 \right)$. In the second chart,
\begin{equation*}
\begin{aligned}
u_{2,1}' &= -\bar{\rho} - z u_{2,1} - \alpha u_{2,1}^2 + 2\rho u_{2,1} u_{2,2} - u_{2,1}^2 u_{2,2}^2 \\
u_{2,2}' &= \rho \alpha - \beta - zu_{2,2} + \rho u_{2,2}^2 + \frac{2\rho z - 2\bar{\rho} u_{2,2}}{u_{2,1}},
\end{aligned}
\end{equation*}
with indeterminacy at $(u_{2,1},u_{2,2}) = ( 0, \bar{\rho}z )$. Since $u_{1,1} = u_{2,2}^{-1}$ we see that the base points of these systems in fact correspond to the same point on the exceptional curve, so there is only one new base point. Also note that the location of the base point becomes $z$ dependent. We will now show that a solution cannot pass through the exceptional curve except at the base points.
\begin{lemma}
\label{lemL1}
Let $\Gamma$ be a rectifiable path in the complex plane with end point $z_\ast$ such that analytic continuation of a solution along $\Gamma$ leads to a point $P$ on one of the exceptional curves $L_1^{(\rho)}$, $\rho \in \{1, \omega, \bar{\omega} \}$. Let $P$ have coordinates $(u_{1,1},u_{1,2}) = (c^{-1},0)$ in the first chart and $(u_{2,1},u_{2,2}) = (0,c)$, $c \in \mathbb{C}$, in the second chart, respectively. Then we must have $c = \bar{\rho} z_\ast$.
\end{lemma}
\begin{proof}
The proof runs along the same lines as Lemma \ref{lemL}, by considering the auxiliary function $W$ re-written in the variables $(u_{1,1},u_{1,2})$, denoted by $W_1$, and in the variables $(u_{2,1},u_{2,2})$, denoted $W_2$. Again, we will only perform the analysis for $W_1$, the case for $W_2$ being similar. We have
\begin{equation*}
W_1(z) = u_{1,1}^{-3} u_{1,2}^{-2} \cdot P_1(z,u_{1,1},u_{1,2}),
\end{equation*}
where
\begin{equation*}
P_1(z,u_{1,1},u_{1,2}) = \bar{\rho} - z \rho u_{1,1} - \rho u_{1,2} + z u_{1,1} u_{1,2}+ \bar{\rho} \left(1 - \bar{\rho} \alpha + \rho \beta \right) u_{1,1}^2 u_{1,2} + \frac{1}{3} u_{1,2}^2+ (\alpha - 2 \rho) u_{1,1}^2 u_{1,2}^2 + u_{1,1}^2 u_{1,2}^3.
\end{equation*}
On the exceptional curve $(u_{1,1},u_{1,2}) = (c^{-1},0)$, $c \in \mathbb{C}$, $W_1$ is infinite apart from the point with $c = \bar{\rho} z$, where it is of the indeterminate form $\frac{0}{0}$. On the other hand, the logarithmic derivative of $W_1(z)$ is
\begin{equation*}
\begin{aligned}
\frac{W_1'}{W_1} = & P_1(z,u_{1,1},u_{1,2})^{-1} \cdot 3 u_{1,1} u_{1,2} \left(3 - 3 z \bar{\rho} u_{1,1} - 6 \bar{\rho} u_{1,2} + 6 z \rho u_{1,1} u_{1,2} + (2 \rho \beta -\bar{\rho} \alpha) u_{1,1}^2 u_{1,2} + 4 \rho u_{1,2}^2 - 3 z u_{1,1} u_{1,2}^2 \right. \\ & \left. + 2 (\rho \alpha - \beta) u_{1,1}^2 u_{1,2}^2 - u_{1,2}^3 - \alpha u_{1,1}^2 u_{1,2}^3 \right),
\end{aligned}
\end{equation*}
which is bounded in a neighbourhood of any point $(u_{1,1},u_{1,2}) = (c^{-1},0)$, $c \neq \bar{\rho} z$. By a similar integral estimate as in Lemma \ref{lemL}, we obtain a contradiction that $|W_1(z_\ast)| < \infty$. Hence we must have $c = \bar{\rho} z_\ast$.
\end{proof}
We will now perform the second blow-up, with the computations carried out for the base point $(0,\bar{\rho} z)$ in the variables $(u_{2,1},u_{2,2})$. We introduce two new coordinate charts,
\begin{equation*}
\hat{u}_{1,1} = \frac{u_{2,1}}{\hat{u}_{2,2} - \bar{\rho}z}, \quad \hat{u}_{1,2} = u_{2,2} - \bar{\rho} z,
\end{equation*}
and
\begin{equation*}
\hat{u}_{2,1} = u_{2,1}, \quad \hat{u}_{2,2} = \frac{u_{2,2} - \bar{\rho} z}{u_{2,1}}.
\end{equation*}
In these coordinates the system of equations takes the following form,
\begin{equation*}
\begin{aligned}
\hat{u}_{1,1}' = & \frac{\bar{\rho} + (\bar{\rho} + \beta - \rho \alpha) \hat{u}_{1,1}}{\hat{u}_{1,2}} + \rho \hat{u}_{1,1} \hat{u}_{1,2} -(\rho z^2 + \alpha) \hat{u}_{1,1}^2 \hat{u}_{1,2} - 2 \bar{\rho} z \hat{u}_{1,1}^2 \hat{u}_{1,2}^2 - \hat{u}_{1,1}^2 \hat{u}_{1,2}^3 \\
\hat{u}_{1,2}' = & \rho \alpha - \beta - \bar{\rho} + z \hat{u}_{1,2} + \rho \hat{u}_{1,2}^2 - \frac{2\bar{\rho}}{\hat{u}_{1,1}} \\
\hat{u}_{2,1}' = & -\rho^2 + z \hat{u}_{2,1} - (\rho z^2 + \alpha) \hat{u}_{2,1}^2 + 2\rho \hat{u}_{2,1}^2 \hat{u}_{2,2} - 2z\bar{\rho} \hat{u}_{2,1}^3 \hat{u}_{2,2} - \hat{u}_{2,1}^4 \hat{u}_{2,2}^2 \\
\hat{u}_{2,2}' = & \frac{\rho \alpha - \beta - \bar{\rho} - \bar{\rho} \hat{u}_{2,2}}{\hat{u}_{2,1}} + (\rho z^2 + \alpha) \hat{u}_{2,1} \hat{u}_{2,2} - \rho \hat{u}_{2,1} \hat{u}_{2,2}^2 + 2 z \bar{\rho} \hat{u}_{2,1}^2 \hat{u}_{2,2}^2 + \hat{u}_{2,1}^3 \hat{u}_{2,2}^3.
\end{aligned}
\end{equation*}
Still, after the second blow-up the indeterminacy in the system of equations prevails, namely in the first chart at the coordinates $(\hat{u}_{1,1},\hat{u}_{1,2}) = \left((\bar{\rho} \alpha - \rho \beta - 1)^{-1},0\right)$ and at $(\hat{u}_{2,1},\hat{u}_{2,2}) = (0,\bar{\rho} \alpha -\rho \beta - 1)$ in the second coordinate chart, these representing the same point. The exceptional curves introduced by the second blow-ups will be denoted by $L_2^{(1)}$, $L_2^{(\omega)}$, $L_2^{(\bar{\omega})}$. Similar to Lemmas \ref{lemL} and \ref{lemL1}, the next Lemma shows that the exceptional curve cannot be reached by analytic continuation of a solution except at the base points.
\begin{lemma}
\label{lemL2}
Let $\Gamma$ be a rectifiable path in the complex plane with endpoint $z_\ast$ such that analytic continuation of a solution along $\Gamma$ leads to a point $P$ on one of the exceptional curves $L_2^{(\rho)}$, $\rho \in \{1,\omega,\bar{\omega}\}$. Let $P$ have coordinates $(\hat{u}_{1,1},\hat{u}_{1,2}) = (c^{-1},0)$ and $(\hat{u}_{2,1},\hat{u}_{2,2}) = (0,c)$. Then we must have $c= \bar{\rho} \alpha - \rho \beta -1$.
\end{lemma}
\begin{proof}
Again we consider the auxiliary function $W$, re-written in the variables $(\hat{u}_{1,1},\hat{u}_{1,2})$ and $(\hat{u}_{2,1},\hat{u}_{2,2})$, denoted $\hat{W}_1$ and $\hat{W}_2$, respectively. Again we only consider the case $\hat{W}_1$,
\begin{equation*}
\hat{W}_1 = \hat{u}_{1,1}^{-2} \hat{u}_{1,2}^{-1} \cdot \hat{P}_1(z,\hat{u}_{1,1},\hat{u}_{1,2}),
\end{equation*}
where
\begin{equation*}
\begin{aligned}
\hat{P}_1 = & \Big( \bar{\rho}+ (\beta - \rho \alpha + \bar{\rho}) \hat{u}_{1,1} - z \hat{u}_{1,1} \hat{u}_{1,2} + \left( (\bar{\rho} \alpha - 2) z + z^3/3 \right) \hat{u}_{1,1}^2 \hat{u}_{1,2} - \rho \hat{u}_{1,1} \hat{u}_{1,2}^2+ ( \alpha - 2\rho + \rho z^2) \hat{u}_{1,1}^2 \hat{u}_{1,2}^2 \\ & + z^2 \rho \hat{u}_{1,1}^3 \hat{u}_{1,2}^2+ z \bar{\rho} \hat{u}_{1,1}^2 \hat{u}_{1,2}^3+2 z \bar{\rho} \hat{u}_{1,1}^3 \hat{u}_{1,2}^3+ \frac{1}{3} \hat{u}_{1,1}^2 \hat{u}_{1,2}^4+ \hat{u}_{1,1}^3 \hat{u}_{1,2}^4 \Big).
\end{aligned}
\end{equation*}
We note that on the exceptional curve $(\hat{u}_{1,1},\hat{u}_{1,2}) = (c^{-1},0)$, $c \in \mathbb{C}$, $\hat{W}_1$ is infinite, apart from at the base point $c = \bar{\rho} \alpha - \rho \beta -1$, where it is of the indeterminate form $\frac{0}{0}$. The logarithmic derivative of $\hat{W}_1$ is given by
\begin{equation*}
\begin{aligned}
\frac{\hat{W}_1'}{\hat{W}_1} = & \hat{P}_1(z,\hat{u}_{1,1},\hat{u}_{1,2})^{-1} \cdot \hat{u}_{1,1} \hat{u}_{1,2} \left(3 + (2 \rho \beta - \bar{\rho} \alpha) \hat{u}_{1,1} - 6 z \rho \hat{u}_{1,1} \hat{u}_{1,2} - ( 2z(\bar{\rho} \beta - \alpha) - \rho z^3) \hat{u}_{1,1}^2 \hat{u}_{1,2} - 6 \bar{\rho} \hat{u}_{1,1} \hat{u}_{1,2}^2 \right. \\ & \left. + (2 \rho \alpha - 2 \beta + 6 \bar{\rho} z^2) \hat{u}_{1,1}^2 \hat{u}_{1,2}^2 - ( \rho \alpha z^2 + \bar{\rho} z^4) \hat{u}_{1,1}^3 \hat{u}_{1,2}^2 + 9 z \hat{u}_{1,1}^2 \hat{u}_{1,2}^3 - (2 \bar{\rho} \alpha z + 4z^3) \hat{u}_{1,1}^3 \hat{u}_{1,2}^3 + 4 \rho \hat{u}_{1,1}^2 \hat{u}_{1,2}^4 \right. \\ & \left. - (\alpha + 6z^2) \hat{u}_{1,1}^3 \hat{u}_{1,2}^4 - 4 z \bar{\rho} \hat{u}_{1,1}^3 \hat{u}_{1,2}^5 - \hat{u}_{1,1}^3 \hat{u}_{1,2}^6\right).
\end{aligned}
\end{equation*}
Again, in a neighbourhood of any point on the exceptional curve other than the base point, the logartihmic derivative of $\hat{W}_1$ is bounded. By an integral estimate similar to the one in Lemmas \ref{lemL} and \ref{lemL1} it follows, by analytic continuation along $\Gamma$, that $|\hat{W}_1(z_\ast)| < \infty$, in contradiction to the fact that $\hat{W}_1$ is infinite there. Hence the solution must run into the base point $(u_{1,1},u_{1,2}) = (( \bar{\rho} \alpha - \rho \beta -1)^{-1},0)$.
\end{proof}
We will now show that one further blow-up for each base point will resolve the indeterminacy in the system of equations so that one obtains a regular initial value problem. We will perform the blow-up in the variables $(\hat{u}_{2,1},\hat{u}_{2,2})$. For this we again introduce two new coordinate charts,
\begin{equation*}
\tilde{u}_{1,1} = \frac{\hat{u}_{2,1}}{\hat{u}_{2,2} + 1 - \bar{\rho} \alpha + \rho \beta}, \quad \tilde{u}_{1,2} = \hat{u}_{2,2} + 1 - \bar{\rho} \alpha + \rho \beta,
\end{equation*}
and
\begin{equation*}
\tilde{u}_{2,1} = \hat{u}_{2,1}, \quad \tilde{u}_{2,2} = \frac{\hat{u}_{2,2} + 1 - \bar{\rho} \alpha + \rho \beta}{\hat{u}_{2,1}}.
\end{equation*}
The equations after the third blow-up read, in the variables $(\tilde{u}_{1,1},\tilde{u}_{1,2})$,
\begin{equation*}
\begin{aligned}
\tilde{u}_{1,1}' = & z \tilde{u}_{1,1}+ \rho \left(1 + z^2 + \beta \rho \right) \left(1 - \bar{\rho} \alpha + \rho \beta \right) \tilde{u}_{1,1}^2 + 2\left( \alpha - 2 \rho - z^2 \rho - 2 \beta \bar{\rho}\right) \tilde{u}_{1,1}^2 \tilde{u}_{1,2}+3 \rho \tilde{u}_{1,1}^2 \tilde{u}_{1,2}^2 \\ & -2z \bar{\rho} (1- \bar{\rho} \alpha + \rho \beta)^2 \tilde{u}_{1,1}^3 \tilde{u}_{1,2}+ 6z \bar{\rho} \left(1 - \bar{\rho} \alpha + \rho \beta \right) \tilde{u}_{1,1}^3 \tilde{u}_{1,2}^2-4 z \bar{\rho} \tilde{u}_{1,1}^3 \tilde{u}_{1,2}^3 +(1-\bar{\rho} \alpha + \rho \beta)^3 \tilde{u}_{1,1}^4 \tilde{u}_{1,2}^2 \\ & -4 (1- \bar{\rho} \alpha + \rho \beta)^2 \tilde{u}_{1,1}^4 \tilde{u}_{1,2}^3+ 5\left(1 - \bar{\rho} \alpha + \rho \beta \right) \tilde{u}_{1,1}^4 \tilde{u}_{1,2}^4-2 \tilde{u}_{1,1}^4 \tilde{u}_{1,2}^5 \\
\tilde{u}_{1,2}' = & -\frac{\bar{\rho}}{\tilde{u}_{1,1}} -\rho \left(1 + z^2+ \rho \beta \right) \left( 1 - \bar{\rho} \alpha + \rho \beta \right) \tilde{u}_{1,1} \tilde{u}_{1,2}+\left(-\alpha +2 \rho +z^2 \rho +2 \beta \bar{\rho}\right) \tilde{u}_{1,1} \tilde{u}_{1,2}^2-\rho \tilde{u}_{1,1} \tilde{u}_{1,2}^3 \\ & + 2z \bar{\rho} (1 - \bar{\rho} \alpha + \rho \beta)^2 \tilde{u}_{1,1}^2 \tilde{u}_{1,2}^2 -4z\bar{\rho} \left(1 - \bar{\rho} \alpha + \rho \beta \right) \tilde{u}_{1,1}^2 \tilde{u}_{1,2}^3+2 z \bar{\rho} \tilde{u}_{1,1}^2 \tilde{u}_{1,2}^4 -(1-\bar{\rho} \alpha + \rho \beta)^3 \tilde{u}_{1,1}^3 \tilde{u}_{1,2}^3 \\ & + 3(1- \bar{\rho} \alpha + \rho \beta)^2 \tilde{u}_{1,1}^3 \tilde{u}_{1,2}^4 -3 \left(1 -\bar{\rho} \alpha + \rho \beta \right) \tilde{u}_{1,1}^3 \tilde{u}_{1,2}^5+ \tilde{u}_{1,1}^3 \tilde{u}_{1,2}^6,
\end{aligned}
\end{equation*}
and in the variables $(\tilde{u}_{2,1},\tilde{u}_{2,2})$,
\begin{equation}
\label{finalsys}
\begin{aligned}
\tilde{u}_{2,1}' = & -\bar{\rho}+z \tilde{u}_{2,1}+\left(\alpha -2 \rho -z^2 \rho -2 \beta \bar{\rho}\right) \tilde{u}_{2,1}^2+\left(1 - \bar{\rho} \alpha + \rho \beta \right) \tilde{u}_{2,1}^5 \tilde{u}_{2,2}-\tilde{u}_{2,1}^6 \tilde{u}_{2,2}^2+ 2z \bar{\rho} \left(1- \bar{\rho} \alpha + \rho \beta \right) \tilde{u}_{2,1}^3 \\ & + 2 \rho \tilde{u}_{2,1}^3 \tilde{u}_{2,2} -(1-\bar{\rho} \alpha + \rho \beta)^2 \tilde{u}_{2,1}^4 - 2 z \bar{\rho} \tilde{u}_{2,1}^4 \tilde{u}_{2,2} \\
\tilde{u}_{2,2}' = & -\rho (1 + z^2 + \rho \beta)(1-\bar{\rho}\alpha+\rho \beta) -z \tilde{u}_{2,2} -5 \left(1 - \bar{\rho} \alpha + \rho \beta \right) \tilde{u}_{2,1}^4 \tilde{u}_{2,2}^2+2 \tilde{u}_{2,1}^5 \tilde{u}_{2,2}^3 + 2z \bar{\rho} (1 - \bar{\rho} \alpha + \rho \beta)^2 \tilde{u}_{2,1} \\ & + \left(-2 \alpha +4 \rho +2 z^2 \rho +4 \beta \rho ^2\right) \tilde{u}_{2,1} \tilde{u}_{2,2} - (1-\bar{\rho} \alpha + \rho \beta)^3 \tilde{u}_{2,1}^2 - 6z \bar{\rho} (1 -\bar{\rho} \alpha + \rho \beta) \tilde{u}_{2,1}^2 \tilde{u}_{2,2} -3 \rho \tilde{u}_{2,1}^2 \tilde{u}_{2,2}^2 \\ & +4(1-\bar{\rho} \alpha + \rho \beta)^2 \tilde{u}_{2,1}^3 \tilde{u}_{2,2}+4z \bar{\rho} \tilde{u}_{2,1}^3 \tilde{u}_{2,2}^2.
\end{aligned}
\end{equation}
From these equation we see that on the exceptional curves $L_3^{(1)}$, $L_3^{(\omega)}$ and $L_3^{(\bar{\omega})}$, introduced by the third blow-ups of each base point, the system of equations becomes a regular initial value problem in the variables $(\tilde{u}_{2,1},\tilde{u}_{2,2})$. If we denote by $\mathcal{S}$ the compact space obtained by the three blow-ups at each base point, covered by all the coordinate systems introduced in the process, the system describes a regular intial value problem on the space
\begin{equation*}
\mathcal{I} = \mathcal{S} \setminus \left( L \cup L_1^{(1)} \cup L_1^{(\omega)} \cup L_1^{(\bar{\omega})} \cup L_2^{(1)} \cup L_2^{(\omega)} \cup L_2^{(\bar{\omega})} \right),
\end{equation*}
this is the space of initial conditions. The changes of variables introduced by the three blow-ups amount to the following relationship to the original variables $(p,q)$,
\begin{equation*}
\tilde{u}_{1,1}(z) = \frac{1}{q(z) r(z)}, \quad \tilde{u}_{1,2}(z) = r(z), \quad \tilde{u}_{2,1}(z) = \frac{1}{q(z)}, \quad \tilde{u}_{2,2}(z) = q(z) r(z),
\end{equation*}
where $r(z) = 1 - \bar{\rho} \alpha + \rho \beta - \bar{\rho} z q(z) + \rho q(z)^2 + q(z) p(z)$. These bi-rational relations can be inverted easily to yield
\begin{equation*}
\begin{aligned}
q(z) &= \frac{1}{\tilde{u}_{2,1}(z)} \\
p(z) &= -\frac{\rho}{\tilde{u}_{2,1}(z)} + \bar{\rho} z - (1-\bar{\rho} \alpha + \rho \beta ) \tilde{u}_{2,1} + \tilde{u}_{2,1}^2 \tilde{u}_{2,2}.
\end{aligned}
\end{equation*}
We can seek local analytic solutions of the final system (\ref{finalsys}), with initial conditions $(\tilde{u}_{2,1}(z_\ast),\tilde{u}_{2,2}(z_\ast)) = (0,c)$ on the exceptional curve, in the form of power series
\begin{equation*}
\tilde{u}_{2,1}(z) = \sum_{n=1}^\infty a_n(z-z_\ast)^n, \quad \tilde{u}_{2,2}(z) = c + \sum_{n=1}^\infty b_n(z-z_\ast)^n,
\end{equation*}
where all coefficients $a_n$, $b_n$, $n=1,2,3,\dots$, can be obtained recursively. One finds
\begin{equation*}
a_1 = -\bar{\rho}, \quad a_2 = -\frac{z_\ast \bar{\rho}}{2}, \quad a_3 = \frac{\rho \alpha - 2\beta}{3} - \bar{\rho} \left( 1 + \frac{z_\ast^2}{2} \right), \quad a_4 = -\frac{c \rho}{2} + \left(\frac{5\alpha \rho}{6} - \frac{7\beta}{6} - \frac{15\bar{\rho}}{8} \right) z_*-\frac{3}{8} \bar{\rho } z_*^3, \quad \cdots
\end{equation*}
\begin{equation*}
\begin{aligned}
b_1 &= \alpha - \beta^2 - \rho + \alpha \beta \rho - 2 \beta \bar{\rho} - c z_\ast + (\alpha - \bar{\rho} \beta - \rho) z_\ast^2, \\ b_2 &= c \left(-\frac{5}{2} -2 \beta \rho + \alpha \bar{\rho} \right) +\frac{1}{2} \left(5 \alpha -\beta ^2-3 \rho +3 \alpha \beta \rho -2 \alpha ^2 \bar{\rho }-4 \beta \bar{\rho }\right) z_\ast-\frac{c z_\ast^2}{2}, \quad \cdots
\end{aligned}
\end{equation*}
Thus a solution in the variables $(\tilde{u}_{2,1},\tilde{u}_{2,2})$, locally analytic in the neighbourhood of a point on one of the exceptional curves $L_3^{(\rho)}$, $\rho \in\{1,\omega,\bar{\omega}\}$, where $\tilde{u}_{2,1}=0$, becomes a simple pole in the original variables $(p,q)$, corresponding to the expansions (\ref{poleexpansion}). The parameters $h$ and $k$ in the expansions (\ref{poleexpansion}) are determined in terms of $\tilde{u}_{2,2}(z_\ast)=c$, the position of the initial point on the exceptional curve $L_3^{(\rho)}$, by the expressions
\begin{equation}
\label{hkrelation}
h = \frac{c}{2} + \left(-\frac{\alpha}{2} + \frac{7 \rho}{8} + \frac{\beta \bar{\rho}}{2} \right) z_\ast, \quad k = \frac{c \rho}{2} - \frac{3 \bar{\rho} z_\ast}{8}.
\end{equation}
Although the space $\mathcal{I}$ itself is not compact, Lemmas \ref{lemL}, \ref{lemL1} and \ref{lemL2} show that a solution, when analytically continued along some path in the complex plane cannot pass through the line at infinity $L$ or any of the exceptional curves $L_1^{(\rho)}$, $L_2^{(\rho)}$, $\rho \in \{1,\omega,\bar{\omega}\}$. We have thus shown the following theorem by which we conclude this article.
\begin{thm}
Let $(p(z),q(z))$ be a local analytic solution of the system (\ref{hamsys}) in a neighbourhood of a point $z_0 \in \mathbb{C}$. Let $\Gamma$ be a rectifiable path from $z_0$ to some point $z_\ast$ such that $(p(z),q(z))$ can be analytically continued along $\Gamma$ up to, but not including the point $z_\ast$. Then, in some coordinate chart of $\mathcal{I}$, the solution, re-written in these coordinates, can be analytically continued to $z_\ast$ leading to a point $P \in L_3^{(\rho)}$, $\rho \in \{1,\omega,\bar{\omega} \}$, not covered by the original coordinate chart $(p,q)$. The local analytic solution about $P$ corresponds to a simple pole of the form (\ref{poleexpansion}) in the variables $(p,q)$ with the parameters $h$ and $k$ fixed by the location of $P$ via the expressions (\ref{hkrelation}).
\end{thm}
\bibliographystyle{plain} | 78,121 |
\date{October 27, 2002}
\chapter{Just-Infinite Branch Groups}\label{chapter:just-infinite}
\begin{definition}
A group $G$ is \emdef{just-infinite} if it is infinite but all of
its proper quotients are finite, i.e., if all of its nontrivial normal
subgroups have finite index.
\end{definition}
The following simple criterion from~\cite{grigorchuk:jibg} characterizes
the just-infinite branch groups acting on a tree.
\begin{theorem}\label{theorem:jicriterion}
Let $G$ be a branch group acting on a tree and let $(L_i,H_i)_{i\in\N}$
be a corresponding branch structure. The following three conditions are equivalent
\begin{enumerate}
\item $G$ is just-infinite.
\item the abelianization $H_i^{ab}$ of $H_i$ is finite, for each $i\in\N$.
\item the commutator subgroup $H_i'$ of $H_i$ has finite index in $G$,
for each $i\in\N$.
\end{enumerate}
\end{theorem}
The statement is a corollary of the fact that $H_i'$, being characteristic
in the normal subgroup $H_n$, is a normal subgroup of $G$, for each
$i\in\N$, and the following useful lemma that says that weakly branch
groups satisfy the following property:
\begin{lemma}
Let $G$ be a weakly branch group acting on a tree and let $(L_i,H_i)_{i\in\N}$
be a corresponding branch structure. Then every non-trivial normal
subgroup $N$ of $G$ contains the commutator subgroup $H_n'$, for some $n$
depending on $N$.
\end{lemma}
\begin{proof}
Let $g$ be an element in $G \setminus \Stab_G(\LL_1)$ and let $N=\langle g
\rangle^G$ be its normal closure in G. Then $g=ha$ for some $h \in
\Stab_{\Aut(\tree)}(\LL_1)$ with decomposition $h =(h_1,\dots,h_{m_1})$
and $a$ a nontrivial rooted automorphism of $\tree$. Without loss of
generality we may assume that $1^a=m_1$.
For arbitrary elements $\xi,\nu \in L_1$, we define $f,t\in H_1$ by
$f=(\xi,1,\dots,1)$ and $t=(\nu,1,\dots,1)$ and calculate
\begin{gather*}
[g,f] = (\xi,1,\dots,1,(\xi^{-1})^{h_1}), \\
[[g,f],t] = ([\xi,\nu],1,\dots,1).
\end{gather*}
Since $[[g,f],t]$ is always in $N=\langle g \rangle^G$, we obtain
$L_1'\times 1 \times\dots\times 1 \preceq N$ and, by the spherical
transitivity of $G$, it follows that
\[ L_1' \times L_1' \times \dots \times L_1'= H_1' \preceq N. \]
Thus any normal subgroup of $G$ that contains $g$ also contains $H_1'$.
Similarly, if $g$ is an element in $\Stab_G(\LL_n) \setminus
\Stab_G(\LL_{n+1})$ and $N$ is the normal closure $N= \langle g^G
\rangle$, then $N$ contains $H_{n+1}'$.
\end{proof}
In particular, the above results imply that all finitely generated
torsion weakly branch groups are just-infinite branch groups.
The study of just-infinite groups is motivated by their minimality. More
precisely, we have the following
\begin{theorem}\cite{grigorchuk:jibg}
Every finitely generated infinite group has a just-infinite
quotient.
\end{theorem}
Therefore, if $\mathcal C$ is a class of groups closed for taking
quotients and it contains a finitely generated infinite group, then it
contains a finitely generated just-infinite group.
Note that there are non-finitely generated groups that do not have
just-infinite quotients, for example, the additive group of rational
numbers $\Q$.
It is known (see~\cite{wilson:jig}) that a just-infinite group with
non-trivial Baer radical is a finite extension of a free abelian group of
finite rank (recall that the \emdef{Baer radical} of the group $G$ is the
subgroup of $G$ generated by the cyclic subnormal subgroups of $G$).
Moreover, the only just-infinite group with non-trivial center is the
infinite cyclic group $\Z$. Therefore, an abelian group has just-infinite
quotient if and only if it can be mapped onto $\Z$. In particular, no
abelian torsion group has just-infinite quotients. The last fact is in a
sharp contrast with the fact that there are large classes of centerless,
torsion, just-infinite, branch groups, for instance \GG\ groups with
finite directed part $B$ (see Chapter~\ref{chapter:torsion}) and many
\GGS\ groups.
\begin{definition}
A group $G$ is \emdef{hereditarily just-infinite} if it is
residually finite and all of its non-trivial normal subgroups are
just-infinite.
\end{definition}
We mention that our definition of hereditarily just-infinite group differs
from the one given in~\cite{wilson:segal} in that we require residual
finiteness. Note that all non-trivial normal subgroups of a group $G$ are
just infinite if and only if all subgroups of finite index in $G$ are just
infinite. This is true since every subgroup of finite index in $G$
contains a normal subgroup of $G$ of finite index.
Examples of hereditarily just-infinite groups are the infinite cyclic
group $\Z$, the infinite dihedral group $D_\infty$ and the projective
special linear groups $\PSL(n,\Z)$, for $n\geq 3$. However, the whole
class is far from well understood and described.
The following result from~\cite{grigorchuk:jibg}, which modifies the
result of \JWilson\ from~\cite{wilson:jig} (see also~\cite{wilson:segal}),
strongly motivates the study of the branch groups.
\begin{theorem}[Trichotomy of just-infinite groups]\label{theorem:tri}
Let $G$ be a finitely generated just-infinite group. Then exactly one of the
following holds:
\begin{enumerate}
\item\label{theorem:tri:1} $G$ is a branch group.
\item\label{theorem:tri:2} $G$ has a normal subgroup $H$ of finite index of the form
\[H = L^{(1)} \times \dots \times L^{(k)}=L^k,\]
where the factors are copies of a group $L$, the conjugations by
the elements in $G$ transitively permute the factors of $H$ and $L$ has
exactly one of the following two properties:
\begin{enumerate}
\item\label{theorem:tri:2a} $L$ is hereditary just-infinite (in case
$G$ is residually finite).
\item\label{theorem:tri:2b} $L$ is simple (in case $G$ is not
residually finite).
\end{enumerate}
\end{enumerate}
\end{theorem}
The proof of this theorem presented in~\cite{grigorchuk:jibg} uses only
the statement from~\cite{wilson:jig} that every subnormal subgroup in a
just-infinite group with trivial Bear radical has a near complement (but
this is probably one of the most important facts from Wilson's theory).
The proof actually works for any (not necessarily finitely generated)
just-infinite group with trivial Baer radical, for instance just-infinite
groups which are not virtually cyclic.
The results of Wilson in~\cite{wilson:jig} (see also~\cite{wilson:segal})
combined with the above trichotomy result show that the following
characterization of just-infinite branch groups is possible. Define an
equivalence relation on the set of subnormal subgroups of a group $G$ by,
$H \sim K$ if the intersection $H \cap K$ has a finite index booth in $H$
and in $K$. The set of equivalence classes of subnormal subgroups, ordered
by the order induced by inclusion, forms a Boolean lattice, which,
following \JWilson, is called the \emdef{structure lattice} of $G$.
\begin{theorem}
Let $G$ be a just-infinite group. Then $G$ is branch group if and
only if it has infinite structure lattice.
Moreover, in such a case, the structure lattice is isomorphic to the
lattice of closed and open subsets of the Cantor set.
\end{theorem}
On the intuitive level, the just-infinite groups should be
considered as ``small'' groups in contrast to, say, the free or
non-elementary hyperbolic groups, which are ``large'' groups.
There is a rigorous approach to the concept of largeness in
groups. Namely, following Pride (\cite{pride:large}), we say that
a group $G$ is \emdef{larger} than a group $H$, and we denote $G
\succeq H$, if $H$ has a subgroup of finite index that is a
homomorphic image of a subgroup of $G$ of finite index. The groups
$G$ and $H$ are \emdef{equally large} (or \emdef{Pride
equivalent}) if $G \succeq H$ and $H \succeq G$. The set of
equivalence classes of equally large groups is partially ordered
by $\succeq$ and the class of finite groups is the obvious
smallest element.
We denote the class of groups equally large to $G$ by $[G]$. A group $G$
is called \emdef{minimal} if the only class below $[G]$ is the class of
finite groups $[1]$. The \emdef{height} of a group $G$ is the height of
the class $[G]$ in the ordering, i.e. the length of a maximal chain
between $[1]$ and $[G]$. Therefore, the minimal groups are the groups of
height 1. Such groups are called \emdef{atomic} in~\cite{neumann:pride}
\begin{theorem}[\cite{grigorchuk-w:minimality}]\label{thm:minimal}
The first Grigorchuk group $\Gg$ and the Gupta-Sidki $p$-groups are
minimal.
\end{theorem}
A number of questions about just-infinite groups was asked
in~\cite{pride:large,e-pride:large}. Positive answer to Problem~5
from~\cite{pride:large} (Problem~4' in~\cite{e-pride:large}) that
asks if there exist finitely generated just-infinite groups that
do not satisfy the ascending chain condition on subnormal
subgroups was provided in~\cite{grigorchuk:gdegree}. Later,
\PNeumann\ constructed in~\cite{neumann:pride} more examples of
finitely generated just-infinite regular branch groups answering
the same question (and also some other quastions) raised by
\MEdjvet\ and \SPride\ in~\cite{e-pride:large}. In particular,
\PNeumann\ provided negative answer to the question if every
finitely generated minimal group is finite-by-$D_2$-by-finite
(here $D_2$ denotes the class of groups in which every nontrivial
subnormal subgroup has finite index). Negative answer to this last
question also follows form Theorem~\ref{thm:minimal} above.
The question of possible heights of finitely generated just-infinite
groups is an interesting one. All hereditarily just-infinite and all
infinite simple groups are minimal. It is plausible that the Grigorchuk
2-groups from~\cite{grigorchuk:gdegree} that are defined by non-periodic
sequences have infinite height (see Question~\ref{question:height}). | 165,449 |
\begin{document}
\maketitle
\begin{abstract}
We aim to characterize the category of injective $*$-homomorphisms
between commutative C*-subalgebras of a given C*-algebra. We reduce
this problem to finding a weakly terminal commutative
subalgebra, and solve it for various C*-algebras, including
all commutative ones and all type I von Neumann algebras.
This answers a natural generalization of the Mackey--Piron programme: which
lattices are those of closed subspaces of Hilbert space?
We also discuss how the categorification differs from the original
question.
\end{abstract}
\section{Introduction}
The collection $\C(A)$ of commutative C*-subalgebras of a fixed
C*-algebra $A$ can be made into a category under various choices of
morphisms. Two natural ones are inclusions and injective $*$-homomorphisms,
resulting in categories $\Cs(A)$ and $\Cm(A)$, respectively. The goal
of this article is to characterize these categories.
Categories based on $\C(A)$ are interesting for a number of reasons.
A first motivation to study such categories is the hope
that they could lead to a noncommutative extension of Gelfand duality.
It is known that $\Cs(A)$ determines $A$ as a partial
C*-algebra~\cite{vdbergheunen:colim}. Equivalently, $\Cs(A)$
determines precisely the quasi-Jordan structure of $A$~\cite{hamhalter:pseudojordan,hamhalterturilova:pseudojordan}.
Thus, $\C(A)$ in itself is already an interesting invariant of
$A$. Moreover, structures based on $\C(A)$ circumvent obstructions to
a noncommutative Gelfand duality that afflict many
other candidates~\cite{vdbergheunen:nogo}.
Indeed, the forthcoming article~\cite{heunenreyes:awstar} shows that
for C*-algebras $A$ with enough projections, adding a little more
structure to $\C(A)$ fully determines the algebra structure of
$A$. To get a full noncommutative Gelfand
duality for such algebras, it then suffices to characterize the
structures based on $\C(A)$ that arise this way; an important step is
obviously to characterize categories of the form $\C(A)$.
Second, there is a physical perspective on $\C(A)$. The underlying
idea, due to Bohr, is that one can only empirically access a quantum mechanical
system, whose observables are modeled by a (noncommutative)
C*-algebra, through its classical subsystems, as modeled by commutative C*-subalgebras~\cite{heunenetal:bohrification}.
Categories based on $\C(A)$ are of paramount importance in the
recent use of topos theory in research in foundations of physics based
on this idea, that proposes a new form of quantum logic~\cite{doeringisham:topos,heunenetal:topos}.
Knowing which categories are of the form $\C(A)$ also characterizes
which toposes are of the form studied in that programme. This should
increase insight into the intrinsic structure of such toposes, and
hence shed light on the foundations of quantum physics such toposes
aim to describe logically.
Third, more generally, a characterization of $\C(A)$ satisfactorily addresses a
general theme in research in foundations of quantum
mechanics. For example, it answers (a categorification of)
the Mackey--Piron programme. This programme asks: which
orthomodular lattices are those of closed subspaces of Hilbert
space? (See~\cite{piron:foundations,soler:orthomodular,kalmbach:measures}.)
A characterization of $\C(A)$ provides an answer, because choosing a
commutative C*-subalgebra of the matrix algebra $M_n(\field{C})$
amounts to choosing an orthonormal subset and hence a closed subspace
of $\field{C}^n$, and an
appropriate generalization to infinite dimension holds as well (see
also~\cite{heunen:complementarity}).
Similarly, a characterization of $\C(A)$ has consequences in the study
of test spaces. These are defined as collections of orthogonal
subsets of a Hilbert space satisfying some conditions, and have been
proposed as axioms for operational quantum mechanics. One of the major
questions there is again which test spaces arise from propositions on Hilbert
spaces~\cite{wilce:testspaces}.
Our main result is to reduce characterizing $\Cm(A)$ to
finding a weakly terminal commutative subalgebra of $A$.
This is closely related to analyzing all maximal abelian subalgebras
(masas). Explicating the structure of masas of C*-algebras in general is a
hard problem, and not much seems to be known systematically outside of
the case of type I factors;
see~\cite{bures:masas,sinclairsmith:masas}. Fortunately, finding
a weakly terminal commutative subalgebra is generally easier than
finding all masas. We prove that the following classes of C*-algebras $A$
possess weakly terminal commutative subalgebras, and therefore find a
full characterization of $\Cm(A)$ for:
\begin{itemize}
\item type I von Neumann algebras, including all
finite-dimensional C*-algebras;
\item commutative C*-algebras.
\end{itemize}
Hopefully future investigations will establish weakly terminal commutative
subalgebras for larger classes of C*-algebras.
The strategy behind our characterization is as follows. The key
insight is to recognize $\Cm(D)$ for a commutative C*-algebra $D$ as
the Grothendieck construction of an action of a monoid $M$ on a
partially ordered set $P$. We characterize such so-called
amalgamations. Next, we use known results to characterize the
partially ordered set $P=\Cs(D)$, consisting of
partitions of the Gelfand spectrum of $D$. Then, we show that $\Cm(A)$ is
equivalent to $\Cm(D)$ for a weakly terminal object $D$ in $\Cm(A)$.
Finally, we establish such a weakly terminal object $D$ for the
various types of C*-algebras $A$ mentioned, finishing the characterization.
This last step is the only one limiting our characterization to
C*-algebras $A$ with weakly terminal commutative subalgebras.
Summarizing:
\begin{enumerate}[(1)]
\item show that a C*-algebra $A$ has a weakly terminal commutative
subalgebra $D$;
\item show that $\Cm(A)$ is equivalent to $\Cm(D)$;
\item show that $\Cm(D)$ is equivalent to $P(X) \rtimes S(X)$, with $X$
the spectrum of $D$;
\item characterize $P(X) \rtimes S(X)$ in terms of $P(X)$ and $S(X)$;
\item a characterization of $P(X)$ exists;
\item in the cases in question, $X$, and hence $S(X)$, is easy to characterize.
\end{enumerate}
The paper is structured as follows. We start with
Section~\ref{sec:preliminaries}, that introduces the poset $\Cs(A)$ and the
category $\Cm(A)$ and discusses their basic properties and motivation. A more
in-depth analysis of the relationship between the two, again depending
on the Grothendieck construction, is made later, in
Section~\ref{sec:inclusionsinjections}. Our main results
are presented in between. To aid intuition, we first cover the finite-dimensional
case, and only then refine to the subtleties of the
infinite-dimensional case. Section~\ref{sec:groupamalgamations}
characterizes amalgamations of groups and posets, which is then used
in Section~\ref{sec:finite} to establish the characterization in the
finite-dimensional case. Then, Section~\ref{sec:monoidamalgamations}
refines the earlier analysis to characterize amalgamations of monoids
and posets. This is used in Section~\ref{sec:infinite} to establish
the characterization in the infinite-dimensional case.
Appendix~\ref{sec:inversesemigroups} records some intermediate results
of independent interest. In particular, it discusses an alternative
way to investigate the relationship between $\Cm(A)$ and $\Cs(A)$.
\section{Motivation}\label{sec:preliminaries}
\begin{definition}
Write $\C(A)$, or simply $\C$, for the collection of nonzero commutative
C*-subalgebras $C$ of a C*-algebra $A$. Here, we do not require
C*-algebras to have a unit. This set of objects can be made
into a category by various choices of morphisms, such as:
\begin{itemize}
\item inclusions $C \hookrightarrow C'$, given by $c \mapsto c$,
yielding a (posetal) category $\Cs(A)$;
\item injective $*$-morphisms $C \rightarrowtail C'$, giving a
(left-cancellative) category $\Cm(A)$.
\end{itemize}
\end{definition}
These two categories are interesting for two related reasons. First, they
form a major ingredient in an attack on a noncommutative
extension of Gelfand
duality~\cite{vdbergheunen:colim,vdbergheunen:nogo,heunenreyes:awstar}. Essentially, one
could think of them as invariants of a C*-algebra.
Second, they play an important role in the recent use of topos theory
in the foundations of quantum physics. From this perspective, one
could think of them as encoding the logic of a quantum-mechanical
system whose observables are modeled by the C*-algebra $A$.
We will discuss these two perspectives in turn, but first we consider
functoriality of the construction $A \mapsto \C(A)$.
Section~\ref{sec:inclusionsinjections} below discusses the
relationship between the two choices of morphisms, $\Cm(A)$ or
$\Cs(A)$ in more detail.
\subsection*{Functoriality}
The assignment $A \mapsto \Cs(A)$ extends to a functor: given a
$*$-homomorphism $\varphi \colon A \to B$, direct images $C \mapsto \varphi(C)$ form
a morphism $\Cs(A) \to \Cs(B)$ of posets, for if $C \subseteq C'$,
then $\varphi(C) \subseteq \varphi(C')$. Well-definedness relies on
the following fundamental fact, that we record as a lemma for future reference.
\begin{lemma}\label{cstarimage}
The set-theoretic image of a C*-algebra under a $*$-homomorphism is
again a C*-algebra.
\end{lemma}
\begin{proof}
See~\cite[Theorem~4.1.9]{kadisonringrose:operatoralgebras}.
\end{proof}
The assignment $A \mapsto \Cm(A)$ has to be adapted to be made
functorial. Either we only consider injective $*$-homomorphisms $A
\rightarrowtail B$, or we restrict the target category $\Cm(A)$ as follows.
\begin{lemma}\label{Cmfunctorial}
There is a functor $\Cat{Cstar} \to \Cat{Cat}$, sending $A$ to the
subcategory of $\Cm(A)$ with morphisms those $i \colon C \rightarrow
C'$ satisfying
\begin{equation*}\label{injectionextends}
i^{-1}(I \cap C') = I \cap C
\end{equation*}
for all closed (two-sided) ideals $I$ of $A$.
\end{lemma}
\begin{proof}
Let $\varphi \colon A \to B$ be a $*$-homomorphism, and let $i$ be as
in the statement of the lemma. Then $i$ induces a well-defined
injective $*$-homomorphism $\varphi(C) \to
\varphi(C')$ precisely when $\varphi(c_1)=\varphi(c_2) \Longleftrightarrow
\varphi(i(c_1)) = \varphi(i(c_2))$. Since $\varphi$ and $i$ are linear, this
comes down to $\varphi(c)=0 \Longleftrightarrow \varphi(i(c))=0$, \ie
$\ker(\varphi) \cap C = \ker(\varphi \after i)$. This becomes $I
\cap C = i^{-1}(I \cap C')$ for $I=\ker(\varphi)$, and is therefore satisfied.
\end{proof}
Notice that $*$-homomorphisms satisfying the condition of the previous
lemma are automatically injective, as is seen by taking $I=\{0\}$.
Notice also that when $A$ is a topologically simple algebra, such as a
C*-algebra of $n$-by-$n$ complex matrices, then the subcategory of the
previous lemma is actually the whole category $\Cm(A)$.
\subsection*{Invariants}
Let us temporarily consider von Neumann algebras $A$ and their von
Neumann subalgebras $\V(A)$, giving categories $\Vs$ and $\Vm$. We
will show that $\Vs$ contains exactly the same information as the
lattice $\Proj(A)$ of projections of $A$. This lattice has been
studied in depth, so from the point of view of (new) invariants of $A$, the
category $\Vm$ is more interesting.
See also Remark~\ref{invariants} below. By extension, $\Cm$ is
possibly more interesting as an invariant than $\Cs$, because
$\C(A)$ and $\V(A)$ coincide for finite-dimensional C*-algebras $A$.
Denote the category of von Neumann algebras and unital normal
$*$-homo\-mor\-phisms by $\Cat{Neumann}$, and write $\Cat{cNeumann}$ for
the full subcategory of commutative algebras. Denote the category of
orthomodular lattices and lattice morphisms preserving the
orthocomplement by $\Cat{Ortho}$. The functor $\Proj \colon
\Cat{Neumann} \to \Cat{Ortho}$ takes $A$ to $\{p \in A \mid
p^2=p=p^*\}$ under the ordering $p \leq q$ iff $pq=p$. On morphisms
$f \colon A \to B$ it acts as $p \mapsto f(p)$. Denote the essential
image of $\Proj$ by $\cat{D}$; traditional quantum logic is the study
of this subcategory of $\Cat{Ortho}$~\cite{redei:quantumlogic}.
Denote by $\Cat{Poset}[\Cat{cNeumann}]$ the category whose objects are
sets of commutative von Neumann algebras $C$, partially ordered by
inclusion (\ie $C \leq C'$ iff $C \subseteq C'$), and whose morphisms
are monotonic functions. We may regard
$\Vs$ as a functor $\Cat{Neumann} \to
\Cat{Poset}[\Cat{cNeumann}]$. Denote the essential image of $\Vs$ by
$\cat{C}$; this is a subcategory of $\Cat{Poset}[\Cat{cNeumann}]$.
We now define two new functors, $F \colon \cat{C} \to \cat{D}$ and $G
\colon \cat{D} \to \cat{C}$. The functor $F$ acts on an object $\Vs(A)$ as
follows. For each $C \in \Vs(A)$, we know that $\Proj(C)$ is a Boolean
algebra~\cite[4.16]{redei:quantumlogic}. Because additionally the
hypothesis of Kalmbach's Bundle lemma, recalled below, is
satisfied, these Boolean algebras unite into an orthomodular lattice
$F(\Vs(A))$. This assignment extends naturally to morphisms.
\begin{lemma}
Let $\{B_i\}$ be a family of Boolean algebras such that $\vee_i=\vee_j$, $\neg_i=\neg_j$,
and $0_i=0_j$ on intersections $B_i \cap B_j$. If $\leq$ on
$\bigcup_i B_i$ is transitive and makes it into a lattice,
then $\bigcup_i B_i$ is an orthomodular lattice.
\end{lemma}
\begin{proof}
See~\cite[1.4.22]{kalmbach:orthomodularlattices}.
\end{proof}
The functor $G$ acts on the projection lattice $L$ of a von Neumann
algebra as follows. Consider all complete Boolean sublattices $B$ of $L$ as a
poset under inclusion. For each $B$, the continuous functions on its
Stone spectrum form a commutative von Neumann algebra. Thus we obtain
an object $G(L)$ in $\cat{C}$, and this assignment extends naturally to morphisms.
\begin{theorem}\label{VsProj}
The objects $\Proj$ and $\Vs$ of $\Cat{Neumann}/\Cat{Cat}$ are equivalent.
\[\xymatrix@R-2ex{
& \Cat{Neumann} \ar_-{\Vs}[dl] \ar^-{\Proj}[dr] \\
\cat{C} \ar@<.25ex>^-{F}[rr] \ar@<-.33ex>@{}|-{\simeq}[rr]
&& \cat{D} \ar@<1ex>^-{G}[ll]
}\]
\end{theorem}
\begin{proof}
Follows directly from the definitions and the previous lemma.
\end{proof}
Indeed, both $\Vs(A)$ and $\Proj(A)$ capture the Jordan algebra structure of $A$~\cite{hardingdoering:jordan}.
Returning to the setting of C*-algebras, notice that the previous theorem
fails, because there are C*-algebras without any nontrivial
projections. But every C*-algebra has many commutative
C*-subalgebras: every self-adjoint element generates one, and every element of a
C*-algebra is a linear combination of self-adjoint
elements. For C*-algebras, $\Cs(A)$ captures precisely the
pseudo-Jordan algebra structure of $A$~\cite{hamhalter:pseudojordan,hamhalterturilova:pseudojordan}.
In this regard, it is also worth remarking that the functor $\Cs \colon\Cat{Cstar} \to \Cat{Poset}[\Cat{cCstar}]$
factors through the category of partial C*-algebras~\cite{vdbergheunen:colim}.
\subsection*{Toposes in foundations of physics}
The main theorem in the application of topos theory to foundations of
quantum physics is the following. The canonical functor $C \mapsto C$ is an
internal (possibly nonunital)
C*-algebra~\cite[Theorem~6.4.8]{heunenetal:bohrification}. It holds in
both toposes $\Cat{Set}^{\Cs}$ and $\Cat{Set}^{\Cm}$ because
of the fundamental Lemma~\ref{cstarimage} above. Categorically, $\Cm$
is a more natural choice than $\Cs$. As argued above, this choice is
also more interesting from an algebraic point of view.
But to characterize a presheaf category is the same as characterizing
the category it is based on, by Morita equivalence; see also
Section~\ref{sec:inclusionsinjections} and
Appendix~\ref{sec:inversesemigroups} below. Thus, our main results
also characterize toposes of the form $\Cat{Set}^{\Cm}$.
For a more or less practical account of the above
folklore knowledge we refer to~\cite{bunge:presheaves}.
\section{Poset-group-amalgamations}\label{sec:groupamalgamations}
This section recalls the Grothendieck construction, focusing on the
special case of an action of a group on a poset. We will call the
resulting categories poset-group-amalgamations. The goal of this
section is to characterize such categories. This is interesting in its
own right, but even more so because we will see that $\Cm$ is of
this form in Section~\ref{sec:infinite}. For that reason, we
prefer a practical characterization. Therefore, we will not pursue the
highest possible level of generality: the discussion in this
section is in elementary terms, spelling out what is probably folklore
knowledge. In particular, the characterization in this section can
be extended to poset-category-amalgamations, and perhaps even to a
characterization of Grothendieck constructions of arbitrary indexed
categories, but we will not pursue this here.
We will meet the Grothendieck construct, also called the
category of elements, again in Section~\ref{sec:inclusionsinjections},
where it is discussed more abstractly. The main idea in this section
is to separate out symmetries into a monoid action, leaving just the
partial order.
\begin{definition}
An \emph{action} of a monoid $M$ on a category $\cat{C}$ is
a functor $F \colon M \to \Cat{Cat}(\cat{C},\cat{C})$. Write $mx$
for the action of $Fm$ on an object $x$ of $\cat{C}$, and $mf$ for
the action of $Fm$ on a morphism $f$ of $\cat{C}$.
\end{definition}
\begin{definition}
If a monoid $M$ acts on a category $\cat{C}$, then we can perform
the \emph{Grothendieck construction}: we can make a new
category $\cat{C} \rtimes M$ whose objects are those
of $\cat{C}$, and whose morphisms $x \to y$ are pairs $(m,f)$ such
that $\dom(f) = x$ and $\cod(f)=my$. Composition and identities are
inherited from $M$ and $\cat{C}$. Explicitly, $\id[x] = (1,\id[x])$,
and $(n,g) \circ (m,f) = (mn,(mg)f)$.
\end{definition}
If the category $\cat{C}$ in the previous definition is
a partially ordered set $P$, then $P \rtimes M$ has as objects $p \in P$, and
morphisms $p \to q$ are $m \in M$ such that $p \leq mq$, with unit
and composition from $M\op$.
An illustrative example to keep in mind is the following. Let $M$ be
the group of unitary $n$-by-$n$ matrices. Let $P$ be the lattice of
subspaces of $\field{C}^n$, ordered by inclusion. Then $M$
acts on $P$ by $UV = \{ U(v) \mid v \in V \}$ for $U \in M$ and $V \in
P$. Morphisms in $P \rtimes M$ between subspaces $V \subseteq
\field{C}^n$ and $W \subseteq \field{C}^n$ are unitary matrices $U$
such that $U^{-1}(v) \in W$ for all $v \in V$.
This section characterizes categories of the form $P \rtimes G$ for an
action of a group $G$ on a poset $P$ with a least
element. Our characterization will rely on weakly initial objects to
recover $P$ from $P \rtimes G$. Categorically, this is trivial, but as
we will see in Sections~\ref{sec:finite} and~\ref{sec:infinite}, it is a very
important step in our application.
An object $0$ is \emph{weakly initial} when for any
object $x$ there exists a (not necessarily unique) morphism $0 \to
x$. If a category $\cat{A}$ has a weak initial object $0$, we can
regard the endohomset monoid $\cat{A}(0,0)$ as a one-object category.
Recall that a \emph{retraction} of a functor is a left-inverse.
\begin{lemma}
If a category $\cat{A}$ has a weak initial object $0$ and a faithful
retraction $F$ of the inclusion $\cat{A}(0,0) \hookrightarrow \cat{A}$, then its
objects are preordered by
\[
x \leq y \iff \exists f \in \cat{A}(x,y) .\, F(f)=1.
\]
\end{lemma}
\begin{proof}
Clearly $\leq$ is reflexive, because $F(\id[x])=1$. It is also
transitive, for if $x \leq y$ and $y \leq z$, then there are $f
\colon x \to y$ and $g \colon y \to z$ with $F(f)=1=F(g)$, so that
$g \after f \colon x \to z$ satisfies $F(g \after f) = F(g) \after
F(f) = 1 \after 1 = 1$ and $x \leq z$.
\end{proof}
Thus we can recover the group $G$ from $\cat{A}=P \rtimes G$ by
looking at $\cat{A}(0,0)$. We can also recover the poset $P$ from
$\cat{A}$ by the previous lemma. What is left is to reconstruct the action
of $G$ on $P$ given just $\cat{A}$. For $m \in G$ and $p \in P$, we
can access the object $mq$ through the morphisms $m \colon p \to q$ in
$\cat{A}$. There is always at least one such morphism, namely $m
\colon mq \to q$, because trivially $mq \leq mq$. In fact, this is
always an isomorphism. We will now use this fact to recover the action of
$G$ on $P$ from $\cat{A}$.
\begin{definition}
A category $\cat{A}$ is called a \emph{poset-group-amalgamation}
when there exist a partial order $P$ and a group $G$ such that:
\begin{enumerate}
\item[(A1)] there is a weak initial object $0$, unique up to isomorphism;
\item[(A2)] there is a faithful retraction $F$ of the inclusion
$\cat{A}(0,0) \hookrightarrow \cat{A}$;
\item[(A3)] there is an isomorphism $\alpha \colon \cat{A}(0,0) \to G\op$ of
monoids;
\item[(A4)] there is an equivalence $\smash{\xymatrix@1{(\cat{A},\leq)
\ar@<1ex>|-{\beta}[r] & P \ar@<.5ex>|-{\beta'}[l]}}$ of preorders;
\item[(A5)] for each object $x$ there is an isomorphism $f \colon x
\to \beta'(\beta(x))$ with $\alpha F(f)=1$;
\item[(A6)] for each $y$ and $m$ there is an isomorphism $f \colon x \to y$ with $\alpha F(f)=m$.
\end{enumerate}
\end{definition}
\begin{example}
If $P$ is a partial order with least element, and $G$ is a
group acting on $P$, then $P \rtimes G$ satisfies (A1)--(A6).
\end{example}
\begin{proof}
The least element $0$ of $P$ is a weak initial object, satisfying
(A1). Conditions (A2)--(A4) are satisfied by definition, and (A5) is vacuous.
To verify (A6) for $q \in P$ and $m \in G$, notice that $mq \leq
mq$, so $f=1\colon mq \to mq$ is an isomorphism with $\alpha F(f)=1$.
\end{proof}
\begin{lemma}\label{recoverGaction}
If $\cat{A}$ satisfies (A1)--(A6), then it induces an action of $G$
on $P$ given by $mp = \beta(x)$ if $f \colon x \to
\beta'(p)$ is an isomorphism with $\alpha(F(f))=m$.
\end{lemma}
\begin{proof}
First, notice that for any $p \in P$ and $m \in G$ there exists an
isomorphism $f \colon x \to \beta'(p)$ with $\alpha(F(f))=m$ by
(A6). If there is another isomorphism $f' \colon x' \to \beta'(p)$
with $\alpha(F(f'))=m$, then their composition gives $x \cong x'$,
and therefore $\beta(x) \cong \beta(x')$. But because $P$ is a
partial order, this means $\beta(x)=\beta(x')$. Thus the action is
well-defined on objects.
To see that it is well-defined on morphisms, suppose that $p \leq
q$. Then there is a morphism $f \colon \beta'(p) \to \beta'(q)$ with
$F(f)=1$. For any $m \colon 0 \to 0$, axiom (A6) provides
isomorphisms $f_p \colon x_p \to \beta'(p)$ and $f_q \colon x_q \to
\beta'(q)$ with $\alpha(F(f_p)) = m = \alpha(F(f_q))$. Then
$f=f_q^{-1} f f_p \colon x_p \to x_q$ is an isomorphism satisfying
$\alpha F(f) = m m^{-1} = 1$. So $mp \leq mq$.
Next, we verify that this assignment is functorial $G \to
\Cat{Cat}(P,P)$. Clearly $\id[\beta'(p)]$ is an isomorphism
$x \to \beta'(p)$ with $F(\id[\beta'(p)])=1$. Therefore $1p=\beta(\beta'(p))=p$.
Finally, for $m_2,m_1 \in M$ and $p \in P$, we have $m_1p=\beta(x_1)$ where $f_1
\colon x_1 \to \beta'(p)$ is an isomorphism with $\alpha(F(f_1))=m_1$. So
$m_2(m_1p) = \beta(x_2)$ where $f_2 \colon x_2 \to
\beta'(\beta(x_1))$ is an isomorphism with $\alpha(F(f_2))=m_2$. By (A5),
there is an isomorphism $h \colon x_1 \to \beta'(\beta(x_1))$ with
$F(h)=1$. So $f = f_1 h^{-1} f_2$ is an isomorphism $x_2 \to
\beta'(p)$ with $\alpha(F(f)) = m_2 m_1$. Thus
$(m_2m_1)p=\beta(x_2)=m_2(m_1p)$.
\end{proof}
\begin{theorem}\label{charPG}
If $\cat{A}$ satisfies (A1)--(A6), then there is an equivalence
$\cat{A} \to P \rtimes G$ given by $x \mapsto \beta(x)$ on objects
and $f \mapsto \alpha(F(f))$ on morphisms.
\end{theorem}
\begin{proof}
First we verify that the assignment of the statement is
well-defined, \ie that $\alpha(F(f))$ is indeed a morphism of
$P \rtimes G$. Given $f \colon x \to y$, we need to show
that $\beta(x) \leq \alpha(F(f)) \cdot \beta(y)$. Unfolding the
definition of action, this means finding an isomorphism $k
\colon x' \to \beta'(\beta(y))$ with $\alpha(F(k))=\alpha(F(f))$ and
$\beta(x) \leq \beta(x')$. Unfolding the definition of the preorder,
the latter means finding a morphism $h' \colon \beta'(\beta(x)) \to
\beta'(\beta(x'))$ with $F(h')=1$. By (A5), it suffices to find $h
\colon x \to x'$ with $F(h)=1$ instead. But (A6) provides an
isomorphism $k \colon x' \to \beta'(\beta(y))$ with
$\alpha(F(k))=\alpha(F(f))$. By (A5) again, there exists an
isomorphism $l \colon y \to \beta'(\beta(y))$ with
$\alpha(F(l))=1$. Finally, we can take $h = k^{-1} l f \colon x \to
x'$. This morphism indeed satisfies $\alpha(F(h)) = \alpha(F(f)) \cdot \alpha(F(l))
\cdot \alpha(F(k))^{-1} = \alpha(F(k)) \cdot \alpha(F(k))^{-1} = 1$.
Functoriality follows directly from the previous lemma, so indeed we
have a well-defined functor $\cat{A} \to P \rtimes G$. Moreover, our
functor is essentially surjective because $\beta$ is an equivalence,
and it is faithful because $F$ is faithful.
Finally, to prove fullness, let $m \colon \beta(x) \to \beta(y)$ be a
morphism in $P \rtimes G$. This means that $\beta(x) \leq
m \beta(y)$, which unfolds to: there are a morphism $f \colon
x \to z$ and an isomorphism $h \colon z \to \beta'(\beta(y))$ in
$\cat{A}$ with $\alpha(F(f))=1$ and $\alpha(F(h))=m$. By (A5), this is
equivalent to the existence of a morphism $f \colon x \to z$ with
$\alpha(F(f))=1$ and an isomorphism $h \colon z \to y$ in $\cat{A}$ with
$\alpha(F(h))=m$. Now take $k=hf \colon x \to y$ in $\cat{A}$. Then
\[
\alpha(F(k)) = \alpha(F(hf)) = \alpha(F(f)) \cdot \alpha(F(h)) = 1 \cdot m = m.
\]
Hence our functor is full, and we conclude that it is (half
of) an equivalence.
\end{proof}
\section{The finite-dimensional case}\label{sec:finite}
This section uses poset-group-amalgamations to completely
characterize the category $\Cm(A)$ for finite-dimensional C*-algebras
$A$. En passant, we will also characterize the poset $\Cs(C)$ for
commutative finite-dimensional C*-algebras $C$.
\subsection*{Finite partition lattices}
We start with identifying the appropriate poset $P$.
Recall that a \emph{partition} $p$ of $\{1,\ldots,n\}$ is a family of
disjoint subsets $p_1,\ldots,p_k$ of $\{1,\ldots,n\}$ whose union is
$\{1,\ldots,n\}$. Partitions are ordered by \emph{refinement}: $p \leq
q$ whenever each $p_i$ is contained in a $q_j$. Ordered this way, the
partitions of $\{1,\ldots,n\}$ form a lattice, called the
\emph{partition lattice}, that we denote by $P(n)$.
It is known precisely when a lattice is (isomorphic to) the partition
lattice $P(n)$. We recall such a characterization below, but first we
briefly have to recall some terminology.
Recall that a lattice is \emph{semimodular} if $a \vee b$ covers $b$
whenever $a$ covers $a \wedge b$. A lattice is \emph{geometric} when
it is atomic and semimodular. An element $x$ in a lattice is
\emph{modular} when $a \vee (x \wedge y) = (a \vee x) \wedge y$ for
all $a \leq y$.
The \emph{M{\"o}bius function} of a finite lattice is the unique function
$\mu \colon L \to \field{Z}$ satisfying $\sum_{y \leq x} \mu(x) =
\delta_{0,x}$. It can be defined recursively by $\mu(0)=1$ and $\mu(x)
= -\sum_{y \leq x} \mu(x)$ for $x>0$;
see~\cite{blasssagan:mobius}. The \emph{characteristic
polynomial} of a finite lattice $L$ is $\sum_{x \in L} \mu(x) \cdot
\lambda^{\dim(1)-\dim(x)}$.
\begin{theorem}\label{yoon}
A lattice $L$ is isomorphic to $P(n+1)$ if and only if:
\begin{enumerate}
\item[(P1)] it is geometric;
\item[(P2)] if $\rk(x)=\rk(y)$, then $\mathop{\uparrow} x \cong \mathop{\uparrow} y$;
\item[(P3)] it has a modular coatom;
\item[(P4)] its characteristic polynomial is $(\lambda-1) \cdots (\lambda-n)$.
\end{enumerate}
\end{theorem}
\begin{proof}
See~\cite{yoon:partitionlattices}.
\end{proof}
This immediately extends to a characterization of $\Cs(A)$ for
finite-dimensional commutative C*-algebras $A$.
\begin{corollary}
A lattice $L$ is isomorphic to $\Cs(A)\op$ for a commutative
C*-algebra $A$ of dimension $n+1$ if and only if it satisfies (P1)--(P4).
\end{corollary}
\begin{proof}
The lattice $\Cs(A)$ is that of subobjects of $A$ in the category of
finite-dimensional commutative C*-algebras and unital
$*$-homomorphisms. Recall that a \emph{subobject} is an equivalence
class of monomorphisms into a given object, where two monics are identified
when they factor through one another by an isomorphism. The dual
notion is a \emph{quotient}: an equivalence class of epimorphisms out of a
given object. By Gelfand duality, $\Cs(A)$ is isomorphic to the opposite
of the lattice of quotients of the discrete topological space
$\Spec(A)$ with $n+1$ points. But the latter is precisely $P(n+1)\op$.
\end{proof}
\subsection*{Symmetric group actions}
The appropriate group to consider is the symmetric group $S(n)$ of all
permutations $\pi$ of $\{1,\ldots,n\}$. The group $S(n)$ acts on
$P(n)$. Explicitly,
$\pi p = (\pi p_1,\ldots,\pi p_k)$ for $p=(p_1,\ldots,p_k) \in P(n)$
and $\pi \in S(n)$, where $\pi p_l = \{ \pi(i) \mid i \in \pi_l
\}$. The following lemma might be considered the main insight of this
article.
\begin{lemma}\label{lem:charcommfinite}
If $A$ is a commutative C*-algebra of dimension $n$, then there is
an isomorphism $\Cm(A)\op \cong P(n) \rtimes S(n)$ of categories.
\end{lemma}
\begin{proof}
We may assume that $A=\mathbb{C}^n$. Objects $C$ of $\Cm(A)$ then
are of the form $C=\{(x_1,\ldots,x_n) \in \mathbb{C}^n \mid \forall
k \forall i,j \in p_k \colon x_i = x_j \}$ for some partition
$p=(p_1,\ldots,p_l)$ of $\{1,\ldots,n\}$. But these are precisely
the objects of $P(n)$, and hence of $P(n) \rtimes S(n)$.
If $f \colon C' \to C$ is a morphism of $\Cm(A)$, \ie an injective
$*$-homomorphism, then $f(C') \subseteq C$ is a C*-subalgebra. Say $C'
= \{ x \in \mathbb{C}^n \mid \forall k \forall i,j \in p'_k \colon
x_i = x_j \}$ for a partition $p'=(p'_1,\ldots,p'_{l'})$. Then we
see that $f$ must be induced by an injective function
$\{1,\ldots,n\} \to \{1,\ldots,n\}$, which we can extend to a
permutation $\pi \in S(n)$. Then $C' \to C$ means that $\pi p' \leq
p$. But this is precisely a morphism in $(P(n) \rtimes S(n))\op$.
\end{proof}
\subsection*{Terminal subalgebras}
A maximal abelian subalgebra $D$ of a C*-algebra $A$ is a maximal
element in $\Cs(A)$. If $A$ is finite-dimensional, such $D$ are unique
up to conjugation with a unitary.
The prime example is the following:
if $A$ is the C*-algebra $M_n(\mathbb{C})$ of $n$-by-$n$ complex
matrices, then maximal abelian subalgebras $D$ are precisely the
subalgebras consisting of all matrices that are diagonal in some fixed basis.
In finite dimension, maximal elements of $\Cs(A)$ are the same as
terminal objects of $\Cm(A)$. For the following
lemma, weakly terminal objects of $\Cs(A)$ are in fact enough. Recall
that an object $D$ is weakly terminal when every object $C$ allows a
morphism $C \to D$.
\begin{lemma}\label{terminal}
If $\Cm(A)$ has a weak terminal object $D$, then there is an
equivalence $\Cm(A) \simeq \Cm(D)$ of categories.
\end{lemma}
\begin{proof}
Clearly the inclusion $\Cm(D) \hookrightarrow \Cm(A)$ is a full and
faithful functor, so it suffices to prove that it is essentially
surjective. Let $C \in \Cm(A)$. Then there exists an injective
$*$-homomorphism $f \colon C \to D$ because $D$ is weakly
terminal. Hence $C \cong f(C) \in \Cm(D)$.
\end{proof}
\subsection*{The characterization}
We can now bring all the pieces together.
\begin{theorem}\label{charffactor}
For a category $\cat{A}$, the following are equivalent:
\begin{itemize}
\item the category $\cat{A}$ is equivalent to $\Cm(M_n(\field{C}))\op$;
\item the category $\cat{A}$ is equivalent to $P(n) \rtimes S(n)$;
\item the following hold: \begin{itemize}
\item $\cat{A}$ satisfies (A1)--(A6);
\item $(\cat{A}, \leq)$ satisfies (P1)--(P4) for $n-1$;
\item $\cat{A}(0,0)\op$ is isomorphic to the symmetric group on $n$ elements.
\end{itemize}
\end{itemize}
\end{theorem}
\begin{proof}
Combine the previous two lemmas with Theorem~\ref{charPG} and
Theorem~\ref{yoon}.
\end{proof}
We can actually do better than
factors $A=M_n(\field{C})$: the next theorem characterizes $\Cm(A)$ for any
finite-dimensional C*-algebra $A$.
\begin{lemma}\label{directsums}
If $\Cm(A_i)$ has a weak terminal object $D_i$ for each $i$ in a set
$I$, then the C*-direct sum $\bigoplus_{i \in I} D_i$ is a weak terminal object in
$\Cm(\bigoplus_{i \in I} A_i)$.
\end{lemma}
\begin{proof}
Let $C \in \C(\bigoplus_{i \in I} A_i)$.
Then $C$ is contained in the commutative subalgebra $\bigoplus_{i
\in I} \pi_i(C)$ of $\bigoplus_{i \in I} A_i$. Because each $D_i$
is weakly terminal, there exist morphisms $f_i \colon \pi_i(C) \to
D_i$. Therefore $\bigoplus_{i \in I} f_i $ is a morphism
$\bigoplus_{i \in I} \pi_i(C) \to \bigoplus_{i \in I} D_i$,
and thus the latter is weakly terminal in $\Cm(\bigoplus_{i \in I} A_i)$.
\end{proof}
\begin{theorem}\label{charfinite}
A category $\cat{A}$ is equivalent to $\Cm(A)\op$ for a
finite-dimensional C*-algebra $A$ if and only if there are
$n_1,\ldots,n_k \in \field{N}$ such that:
\begin{itemize}
\item $\cat{A}$ satisfies (A1)--(A5) and (A6');
\item $(\cat{A},\leq)$ satisfies (P1)--(P4) for $(\sum_{i=1}^k n_i)-1$;
\item $\cat{A}(0,0)\op$ is isomorphic to the symmetric group on
$\sum_{i=1}^k n_i$ elements;
\item $\sum_{i=1}^k n_i^2 = \dim(A)$.
\end{itemize}
\end{theorem}
\begin{proof}
Every finite-dimensional C*-algebra $A$ is of the form $\bigoplus_{i=1}^k
M_{n_i}(\field{C})$ with $n=\sum_{i=1}^k
n_i^2$~\cite[Theorem~III.1.1]{davidson:cstar}. By
Lemmas~\ref{terminal} and~\ref{directsums}, we have
\[
\Cm(A)
\simeq \Cm(\bigoplus_{i=1}^k \field{C}^{n_i})
\cong \Cm(\field{C}^{(\sum_{i=1}^k n_i)}).
\]
So by Lemma~\ref{lem:charcommfinite}, $\Cm(A)\op \simeq P(m) \rtimes S(m)$ for
$m=\sum_{i=1}^k n_i$. Now the
statement follows from Theorem~\ref{charffactor}.
\end{proof}
\section{Poset-monoid-amalgamations}\label{sec:monoidamalgamations}
The main idea of our characterization of $\Cm(A)$ for
finite-dimensional C*-algebras $A$ holds unabated in the
infinite-dimensional case. However, the technical implementation of
the idea needs some adapting. For example, the appropriate monoid is
no longer a group. Therefore, we will have to refine axiom (A6) into
(A6') and (A7') as follows. We re-list the other axioms for convenience.
\begin{definition}
A category $\cat{A}$ is called a \emph{poset-monoid-amalgamation}
when there exist a partial order $P$ and a monoid $M$ such that:
\begin{enumerate}
\item[(A1')] there is a weak initial object $0$, unique up to isomorphism;
\item[(A2')] there is a faithful retraction $F$ of the inclusion
$\cat{A}(0,0) \hookrightarrow \cat{A}$;
\item[(A3')] there is an isomorphism $\alpha \colon \cat{A}(0,0) \to M\op$ of
monoids;
\item[(A4')] there is an equivalence $\smash{\xymatrix@1{(\cat{A},\leq)
\ar@<1ex>|-{\beta}[r] & P \ar@<.5ex>|-{\beta'}[l]}}$ of preorders;
\item[(A5')] for each object $x$ there is an isomorphism $f \colon x
\to \beta'(\beta(x))$ with $\alpha F(f)=1$;
\item[(A6')] for each object $y$ and $m \colon 0 \to 0$,
there is $f \colon x \to y$ such that $F(f)=m$, that is universal
in the sense that $f'=fg$ with
$F(g)=1$ for any $f' \colon x' \to y$ with $F(f')=m$;
\item[(A7')] if $F(f)=m_2m_1$ for a morphism $f$, then $f=f_1f_2$ with $F(f_i)=m_i$.
\end{enumerate}
\end{definition}
The idea behind axiom (A6') is that in $P \rtimes M$, we can access
the object $mq$ through the morphisms $m \colon p \to q$. There is
always at least one such morphism, namely $m \colon mq \to q$, because
trivially $mq \leq mq$. This might not be an isomorphism, but it still
has the universal property that all other morphisms $m \colon p \to
q$ factor through it.
We can rephrase this universality as follows: for each object $y$ of
$P \rtimes M$ and $m \in M$, there is a maximal element of the set $\{ f
\colon x \to y \mid \alpha(F(f))=m\}$, preordered by $f \leq g$ iff $f=hg$ for
some morphism $h$ satisfying $\alpha(F(h))=1$.
\[\xymatrix{
x \ar^-{f}[r] & y \\
z \ar_-{g}[ur] \ar@{-->}^-{h}[u]
}\]
Also, notice the swap in (A7'). It is caused by the contravariance in the
composition of $P \rtimes M$ and (A3'), and is not a mistake, as the
following example shows.
\begin{example}
If $P$ is a partial order with least element, and $M$ is a
monoid acting on $P$, then $P \rtimes M$ is a poset-monoid-amalgamation.
\end{example}
\begin{proof}
The least element $0$ of $P$ is a weak initial object, satisfying
(A1). Conditions (A2)--(A4) are satisfied by definition, and (A5) is vacuous.
To verify (A6) for $q \in P$ and $m \in M$, notice that $mq \leq
mq$, and if $p \leq mq$, then certainly $p \leq 1mq$. Finally, (A7)
means that if $p \leq m_2 m_1 r$, we should be able to provide $q$
such that $p \leq m_2 q$ and $q \leq m_1 r$; taking $q=m_1 r$
suffices.
\end{proof}
\begin{lemma}\label{recoverMaction}
If $\cat{A}$ satisfies (A1')--(A7'), then it induces an
action of $M$ on $P$ given by $pm = \beta(x)$ if $f \colon x \to
\beta'(p)$ is a maximal element with $\alpha(F(f))=m$.
\end{lemma}
\begin{proof}
First, notice that for any $p \in P$ and $m \in M$ there exists a
maximal $f \colon x \to \beta'(p)$ with $\alpha(F(f))=m$ by
(A6'). If there is another maximal $f' \colon x' \to \beta'(p)$ with
$\alpha(F(f'))=m$, then there are morphisms $g \colon x \to x'$ and
$g' \colon x' \to x$ with $F(g)=1=F(g')$. Hence $F(gg')=1=F(g'g)$,
and because $F$ is faithful, $g$ is an isomorphism with $g'$ as
inverse. So $x \cong x'$, and therefore
$\beta(x)\cong\beta(x')$. But because $P$ is a partial order, this
means $\beta(x)=\beta(x')$. Thus the action is well-defined on objects.
To see that it is well-defined on morphisms, suppose that $p \leq
q$. Then there is a morphism $f \colon \beta'(p) \to \beta'(q)$ with
$F(f)=1$. For any $m \colon 0 \to 0$, we can find maximal $f_p
\colon x_p \to \beta'(p)$ with $F(f_p)=m$, and maximal $f_q \colon
x_q \to \beta'(q)$ with $F(f_q)=m$. Now $ff_p \colon x_p \to
\beta'(q)$ has $F(ff_p)=m$. Because $f_q$ is a maximal such
morphism, $ff_p$ factors through $f_q$. That is, there is $h \colon
x_p \to x_q$ with $f_qh=ff_p$ and $F(h)=1$. So $mp \leq mq$.
Next, we verify that this assignment is functorial $M \to
\Cat{Cat}(P,P)$. Clearly $\id[\beta'(p)]$ is maximal among morphisms
$f \colon x \to \beta'(p)$ with $F(f)=1$. Therefore $1p=\beta(\beta'(p))=p$.
For $m_2,m_1 \in M$ and $p \in P$, we have $m_1p=\beta(x_1)$ where $f_1
\colon x_1 \to \beta'(p)$ is maximal such that $\alpha(F(f_1))=m_1$. Therefore
$m_2(m_1p) = \beta(x_2)$, where the morphism $f_2 \colon x_2 \to
\beta'(\beta(x_1))$ is maximal such that $\alpha(F(f_2))=m_2$. By axiom (A5'),
there is an isomorphism $h \colon x_1 \to \beta'(\beta(x_1))$ with
$F(h)=1$. This gives $f =f_1h^{-1}f_2 \colon x_2
\to \beta'(p)$ with $\alpha(F(f))=\alpha(F(f_2)) \cdot
\alpha(F(h))^{-1} \cdot \alpha(F(f_1)) = m_2m_1$. We will now show
that $f$ is universal with this property. If $g \colon y \to
\beta'(p)$ also has $\alpha(F(g))=m_2m_1$, then (A7') provides $g_2
\colon y \to z$ and $g_1 \colon z \to \beta'(p)$ with $g=g_1g_2$ and
$\alpha(F(g_i))=m_i$.
\[\xymatrix@C-2ex@R-3ex{
x_2 \ar^-{f_2}[rr] \ar@(u,u)|-{f}[rrrrrr]
&& \beta'(\beta(x_1)) \ar^-{h^{-1}}[rr]
&& x_1 \ar^-{f_1}[rr]
&& \beta'(p) \\
&&& z \ar@{-->}^-{hk}[ul] \ar@{-->}^-{k}[ur] \ar|-{g_1}[urrr] \\
y \ar@{-->}_-{l}[uu] \ar|-{g_2}[urrr] \ar@(r,d)|-{g}[uurrrrrr]
}\]
By maximality of $f_1$, there exists $k$ with $g_1=f_1k$ and
$\alpha(F(k))=1$. And by maximality of $f_2$, there is exists $l$ with
$hkg_2=f_2l$ and $\alpha(F(l))=1$. Hence
\[
g=g_1g_2=f_1kg_2=f_1h^{-1}hkg_2=f_1h^{-1}f_2l=fl.
\]
So $f$ is
maximal with $F(f)=m_2m_1$. Thus $(m_2m_1)p=\beta(x_2)=m_2(m_1p)$.
\end{proof}
\begin{theorem}\label{charPM}
If $\cat{A}$ satisfies (A1')--(A7'), then there is an equivalence
$\cat{A} \to P \rtimes M$ given by $x \mapsto \beta(x)$ on objects
and $f \mapsto \alpha(F(f))$ on morphisms.
\end{theorem}
\begin{proof}
First, it follows from (A6') that the assignment of the statement is
well-defined, \ie that $\alpha(F(f))$ is indeed a morphism of
$P \rtimes M$. Indeed, if $f \colon x \to y$, then we need to show
that $\beta(x) \leq \alpha(F(f)) \cdot \beta(y)$. Unfolding the
definition of the action, this means we need to find a maximal $k
\colon x' \to \beta'(\beta(y))$ with $F(f)=F(k)$, such that
$\beta(x) \leq \beta(x')$. Unfolding the definition of the preorder,
this means we need to find a morphism $h' \colon \beta'(\beta(x)) \to
\beta'(\beta(x'))$ with $F(h')=1$. By (A5'), it suffices to find $h
\colon x \to x'$ with $F(h)=1$ instead. But by (A6'), there exists
maximal $k \colon x' \to \beta'(\beta(y))$ with $F(k)=F(f)$. By its
maximality, there exists $h \colon x \to x'$ with $F(h)=1$ and
$f=kh$. In particular, $\beta(x) \leq \beta(x')$.
Functoriality follows directly from the previous lemma, so indeed we
have a well-defined functor $\cat{A} \to P \rtimes M$. Moreover, our
functor is essentially surjective because $\beta$ is an equivalence,
and it is faithful because $F$ is faithful.
Finally, to prove fullness, let $m \colon \beta(x) \to \beta(y)$ be a
morphism in $P \rtimes M$. This means that $\beta(x) \leq
\beta(y) m$, which unfolds to: there are morphisms $f \colon
x \to z$ and $h \colon z \to \beta'(\beta(y))$ with $F(f)=1$ and $h$
maximal with $\alpha(F(h))=m$. By (A5'), this is
equivalent to the existence of a morphism $f \colon x \to z$ with
$F(f)=1$ and a morphism $h \colon z \to y$ maximal with
$\alpha(F(h))=m$. Now take $k=hf \colon x \to y$. Then
\[
\alpha(F(k)) = \alpha(F(hf)) = \alpha(F(f))\alpha(F(h)) = 1 \cdot m
= m.
\]
Hence our functor is full, and we conclude that it is (half
of) an equivalence.
\end{proof}
\section{The infinite-dimensional case}\label{sec:infinite}
To adapt Theorem~\ref{charffactor} to the infinite-dimensional case,
we have to make three more adaptations. First, the poset $P$ now becomes
a lattice of partitions of a (locally) compact Hausdorff
space. Second, the symmetric group gets replaced by symmetric monoids
on (locally) compact Hausdorff spaces. Third, we have to be more
careful about maximal abelian subalgebras.
\subsection*{Infinite partition lattices}
For arbitrary (locally) compact Hausdorff spaces, it is more
convenient to talk about equivalence relations then partitions.
An equivalence relation $\sim$ on a (locally) compact Hausdorff space
$X$ is called \emph{closed} when the set $\{x \in X \mid \exists u
\in U .\, x \sim u \}$ is closed for every closed $U \subseteq
X$. Closed equivalence relations on $X$ are also called
\emph{partitions}, and form a partial order $P(X)$ under \emph{refinement}:
\[
\mathop{\sim} \leq \mathop{\approx}
\quad \iff \quad
\big(\forall x,y \in X .\, x \sim y \implies x \approx y \big).
\]
Notice that quotients of a (locally) compact Hausdorff space by an
equivalence relation are again (locally) compact Hausdorff if and only if the equivalence relation is closed.
Fortunately, a characterization of $P(X)$ is known, and is due to
Firby~\cite{firby:compactifications1,firby:compactifications2}.
This also gives a characterization of $\Cs(A)$ for
commutative C*-algebras $A$. As in Section~\ref{sec:finite}, we first
briefly recall the necessary terminology.
An element $b$ of a lattice is called \emph{bounding} when (i) it is zero
or an atom; or (ii) it covers an atom and dominates exactly three
atoms; or (iii) for distinct atoms $p,q$ there exists an atom $r \leq
b$ such that there are exactly three atoms less than $r \vee p$ and
exactly three atoms less than $r \vee q$. A collection of atoms of a
lattice with at least four elements is called \emph{single} when it is a
maximal collection of atoms of which the join of any two dominates
exactly three atoms (not necessarily in the collection).
A collection $B$ of nonzero bounding elements of a lattice is called a
\emph{1-point} when (i) its atoms form a single collection; and (ii) if
$a$ is bounding and $a \geq b \in B$, then $a \in B$; and (iii) any $a
\in B$ dominates an atom $p \in B$.
\begin{theorem}\label{charPX}
A lattice $L$ with at least four elements is isomorphic to $P(X)$
for a compact Hausdorff space $X$ if and only if:
\begin{enumerate}
\item[(P1')] $L$ is complete and atomic;
\item[(P2')] the intersection of any two 1-points contains exactly one atom, \\
and any atom belongs to exactly two 1-points;
\item[(P3')] for bounding $a,b \in L$ that are contained in a 1-point,
\begin{align*}
\quad\qquad & \{ p \in \mathrm{Atoms}(L) \mid p \leq a \vee b \} \\ & =
\{ p \in \mathrm{Atoms}(L) \mid \text{ if } x \text{ is a 1-point with } p
\in x \text{ then } a \in x \text{ or } b \in x \};
\end{align*}
for bounding $a,b \in L$ that are not contained in a 1-point,
\[
\;
\{ p \in \mathrm{Atoms}(L) \mid p \leq a \vee b \} =
\{ p \in \mathrm{Atoms}(L) \mid p \leq a \text{ or } p \leq b \};
\]
\item[(P4')] for 1-points $x\neq y$ there are bounding $a,b$ with $a
\not\in x$, $b \not\in y$, and $a \vee b=1$;
\item[(P5')] joins of nests of bounding elements are bounding;
\item[(P6')] for nonzero $a \in L$, the collection $B$ of bounding
elements equal to or covered by $a$ is the unique one satisfying:
\begin{itemize}
\item $\bigvee B = a$;
\item no 1-point contains two members of $B$;
\item if $c$ is bounding, $b_1 \in B$, and no 1-point contains
$b_1$ and $c$, then there is a bounding $b \geq c$ such that (i)
there is no 1-point containing both $b$ and $b_1$, and (ii)
whenever there is a 1-point containing both $b$ and $b_2 \in B$,
then $b \geq b_2$;
\end{itemize}
\item[(P7')] any collection of nonzero bounding elements that is not
contained in a 1-point has a finite subcollection that is not contained in a 1-point;
\end{enumerate}
and $X$ is (homeomorphic to) the set of 1-points of
$L$, where a subset is closed if it is a singleton 1-point or it
is the set of 1-points containing a fixed bounding element.
\end{theorem}
\begin{proof}
See~\cite{firby:compactifications2}.
\end{proof}
\begin{remark}
The axiom responsible for compactness of $X$ is (P7'). The previous
theorem holds for locally compact Hausdorff spaces $X$ when we replace (P7') by
\begin{itemize}
\item[(P7'')] every 1-point contains a bounding $b$ such that
$\{ l \in L \mid l \geq b \}$ satisfies (P7').
\end{itemize}
Indeed, because (P1')--(P6') already guarantee Hausdorffness, we may
take local compactness to mean that every point has a compact
neighbourhood that is closed. And closed sets correspond to sets of
1-points containing a fixed bounding element.
\end{remark}
As before, this directly leads to a characterization of $\Cs(A)$ for
commutative C*-algebras $A$.
\begin{corollary}
A lattice $L$ is isomorphic to $\Cs(A)\op$ for a commutative
C*-algebra $A$ of dimension at least three if and only if it
satisfies (P1')--(P6') and (P7''). The C*-algebra $A$ is unital if and
only if $L$ additionally satisfies (P7').
\end{corollary}
\begin{proof}
The lattice $\Cs(A)$ is that of subobjects of $A$ in the category of commutative
(unital) C*-algebras and (unital) nondegenarate
$*$-homomorphisms. Recall that a \emph{subobject} is an equivalence class of
monomorphisms into a given object, where two monics are identified
when they factor through one another by an isomorphism. The dual
notion is a \emph{quotient}: an equivalence class of epimorphisms out of a
given object. By Gelfand duality, $\Cs(A)$ is isomorphic to the opposite
of the lattice of quotients of $X=\Spec(A)$.
But the latter is precisely $P(X)\op$, because categorical
quotients in the category of (locally) compact Hausdorff spaces are
quotient spaces.
\end{proof}
\subsection*{Symmetric monoid actions}
We write $S(X)$ for the monoid of continuous functions
$f \colon X \twoheadrightarrow X$ with dense image on a locally
compact Hausdorff space $X$, called the \emph{symmetric monoid} on $X$.
\begin{proposition}\label{SXaction}
For any locally compact Hausdorff space $X$, the monoid $S(X)$ acts
on $P(X)$ by
\[
(f \mathop{\sim}) \;=\; (f \times f)^{-1}(\mathop{\sim}).
\]
\end{proposition}
\begin{proof}
First of all, notice that $f\mathop{\sim}$ is reflexive, symmetric and
transitive, so indeed is a well-defined equivalence relation on $X$, which is closed
because $f$ is continuous. Concretely, $x (f\mathop{\sim}) y$ if and only if $f(x) \sim
f(y)$. Moreover, clearly $\id\mathop{\sim} = \mathop{\sim}$, and $g(f\mathop{\sim}) =
(gf)\mathop{\sim}$, so the above is a genuine action.
\end{proof}
As before, this directly leads to a characterization of $\Cm(A)$ for
commutative C*-algebras $A$.
\begin{lemma}\label{charCmcomm}
If $A=C(X)$ for a locally compact Hausdorff space $X$, there is an isomorphism
$\Cm(A)\op \cong P(X) \rtimes S(X)$ of categories.
\end{lemma}
\begin{proof}
By definition, objects $C$ of $\Cm(A)$ are subobjects of $C(X)$ in the category of
commutative C*-algebras. By Gelfand duality, these correspond to quotients of $X$ in the
category of locally compact Hausdorff spaces. But these, in turn, correspond to closed
equivalence relations on $X$, establishing a bijection between the objects of $\Cm(A)$
and $P(X)$.
A morphism $C \to C'$ in $\Cm(A)$ corresponds through Gelfand duality to an
epimorphism $g \colon Y' \twoheadrightarrow Y$ between the corresponding spectra.
Writing the quotients as $Y=X/\mathop{\sim}$ and $Y'=X/\mathop{\approx}$ for closed
equivalence relations $\sim$ and $\approx$, we find that $g$
corresponds to a continuous $f \colon X \to X$ with dense
image respecting equivalence:
\[
x \approx y \implies f(x) \sim f(y).
\]
But this just means $\mathop{\approx} \leq (\mathop{\sim} f)$.
In other words, this is precisely a morphism $f \colon \mathop{\approx} \to
\mathop{\sim}$ in $P(X) \rtimes S(X)$.
\end{proof}
\subsection*{Weakly terminal subalgebras}
In the infinite-dimensional case, it is no longer true that all
maximal abelian subalgebras of a C*-algebra $A$ are
isomorphic. However, it suffices if there exists a commutative
subalgebra into which all others embed by an injective
$*$-homomorphism. To be
precise, a commutative subalgebra $D$ of a C*-algebra $A$ is
\emph{weakly terminal} when every $C \in \C(A)$ allows an injective
$*$-homomorphisms $C \to D$ (that is not necessarily an inclusion, and
not necessarily unique). Equivalently, every masa is isomorphic to a
subalgebra of $D$. Weakly terminal commutative subalgebras $D$ are
maximal up to isomorphism, in the sense that if $D$ can be extended to
a larger commutative C*-subalgebra $E$, then $D \cong E$. This does
not imply that $D=E$, \ie that $D$ is maximal. For a counterexample,
take $A=E=C([0,1])$ and $D=\{f \in E \mid f \mbox{ constant on
}[0,\tfrac{1}{2}]\}$. Then $D \subsetneq E$, but $D \cong
C([\tfrac{1}{2},1]) \cong E$.
\begin{lemma}\label{separablemasas}
If $A=B(H)$ for an infinite-dimensional separable Hilbert space $H$,
then $\Cm(A)$ has a weak terminal object, $*$-isomorphic to
$L^\infty(0,1) \oplus \ell^\infty(\field{N})$.
\end{lemma}
\begin{proof}
Let $C \in \Cm(A)$. By Zorn's lemma, $C$ is a C*-subalgebra of a
maximal element of $\Cs(A)$.
A maximal element in $\Cs(A)$ for a von Neumann algebra $A$ is itself
a von Neumann algebra, because it must equal its weak
closure. It is known that maximal abelian von Neumann subalgebras of
$A=B(H)$ for an infinite-dimensional separable Hilbert space $H$ are
unitarily equivalent to one of the following: $L^\infty(0,1)$,
$\ell^\infty(\{0,\ldots,n\})$ for $n \in \field{N}$,
$\ell^\infty(\field{N})$, $L^\infty(0,1) \oplus
\ell^\infty(\{0,\ldots,n\})$ for $n \in \field{N}$, or $L^\infty(0,1)
\oplus \ell^\infty(\field{N})$
(see~\cite[Theorem~9.4.1]{kadisonringrose:operatoralgebras}).
Each of these allows an injective $*$-homomorphism into the latter
one, $D=L^\infty(0,1) \oplus \ell^\infty(\field{N})$.
Therefore, there exists a morphism $C \to D$ in $\Cm(A)$ for each
$C$ in $\Cm(A)$, so that $D$ is weakly terminal in $\Cm(A)$.
\end{proof}
If $\dim(H)$ is uncountable, the situation becomes a bit more involved. A
complete classification of (maximal) abelian subalgebras of $B(H)$ is
known~\cite{segal:decomposition}, and we will use this to establish a
weakly terminal commutative subalgebra.
\begin{lemma}\label{masas}
If $A=B(H)$ for an infinite-dimensional Hilbert space $H$,
then $\Cm(A)$ has a weak terminal object, $*$-isomorphic to
$\bigoplus_{\alpha,\beta \leq \dim(H)} L^\infty\big( (0,1)^\alpha
\big)$, where $\alpha,\beta$ are cardinals, and $(0,1)^\alpha$ is
the product measure space of Lebesgue unit intervals.
\end{lemma}
\begin{proof}
Maximal abelian subalgebras $C$ of $B(H)$ are isomorphic to a direct
sum of $L^\infty\big( (0,1)^\alpha\big)$ ranging over cardinal
numbers $\alpha$~\cite{segal:decomposition}.
We must show that $D = \bigoplus_{\alpha,\beta \leq
\dim(H)} L^\infty\big( (0,1)^\alpha \big)$ embeds as a subalgebra of $B(H)$
that embeds any such $C$.
A commutative algebra $L^\infty\big( (0,1)^\alpha \big)$ acts on the
Hilbert space $L^2\big( (0,1)^\alpha \big)$.
Observe that $L^2(0,1)$ is separable: the periodic functions $x
\mapsto \exp(2\pi i n x)$ form an orthonormal basis.
Hence $\dim\big( L^2\big( (0,1)^\alpha \big)\big) = \max(\alpha,
\aleph_0)$ unless $\alpha=0$, in which case the dimension
vanishes. Therefore $\dim \big( L^2\big( (0,1)^\alpha \big)
\big) \leq \dim(H)$ if and only if $\alpha \leq \dim(H)$.
Because $H$ is infinite-dimensional, we have the equation $\dim(H) =
\dim(H)^3$ of cardinal numbers.
Thus any maximal abelian subalgebra $C$ embeds into $D$, and $D$
itself embeds as a maximal abelian subalgebra of $B(H)$.
\end{proof}
The following infinite-dimensional analogue of Lemma~\ref{directsums}
will allow us to deduce from the previous lemma that arbitrary type I
von Neumann algebras possess weakly terminal commutative
subalgebras. (For direct integrals,
see~\cite[Chapter~14]{kadisonringrose:operatoralgebras}.)
\begin{lemma}\label{directintegrals}
Let $(M,\mu)$ be a measure space, and for each $t \in M$ let $A_t$
be a von Neumann algebra. If $\Cm(A_t)$ has a weakly terminal
object $D_t$ for almost every $t$, then the direct integral
$\int_M^\oplus D_t \mathrm{d}\mu(t)$ is a weak terminal object in
$\Cm(\int_M^\oplus A_t \mathrm{d} \mu(t))$.
\end{lemma}
\begin{proof}
Let $C \in \Cm(\int_M^\oplus A_t \mathrm{d}\mu(t))$.
Supposing $A_t$ acts on a Hilbert space $H_t$, then $C$ is contained
in $\int_M^\oplus C_t \mathrm{d}\mu(t)$, where $C_t$ is the von
Neumann subalgebra of $B(H_t)$ generated by $\{a_t \mid a \in C\}$.
But because almost every $D_t$ is weakly terminal, this in turn
embeds into $\int_M^\oplus D_t \mathrm{d}\mu(t)$, which is therefore
weakly terminal.
\end{proof}
\begin{corollary}\label{typeiterminal}
Every type I von Neumann algebra $A$ possesses a weakly terminal
commutative subalgebra $D$. More precisely: if $A=\int_M^\oplus A_t
\mathrm{d}\mu(t)$ for a measure space $(M,\mu)$ and type I factors
$A_t$ acting on Hilbert spaces $H_t$, then $D$ is $*$-isomorphic to
$\Spec\big(\int_M^\oplus \bigoplus_{\alpha,\beta\leq \dim(H_t)} L^\infty((0,1)^\alpha)\big)$.
\end{corollary}
\begin{proof}
Every type I von Neumann algebra is a direct integral of type I
factors~\cite[Section~14.2]{kadisonringrose:operatoralgebras}. Since
the latter have weakly terminal commutative
subalgebras by Lemma~\ref{masas}, we can deduce that the original
algebra has a weakly terminal commutative subalgebra by
Lemma~\ref{directintegrals}.
\end{proof}
Much less is known about the structure of (maximal) abelian
subalgebras of von Neumann algebras of type II and III; see~\cite{bures:masas,sinclairsmith:masas}.
The results of~\cite{tomiyama:masas} indicate that the previous lemma
might extend to show that $\Cm(A)$ has a weak terminal object for
\emph{any} von Neumann algebra $A$.
It would also be interesting to see if the previous corollary implies
that type I C*-algebras have weakly terminal commutative subalgebras.
\subsection*{The characterization}
We now arrive at our main
result: the next theorem characterizes $\Cm$ for infinite-dimensional
type I von Neumann algebras.
\begin{theorem}\label{chartypei}
For a category $\cat{A}$, and an infinite-dimensional type I von Neumann algebra
$A=\int_M^\oplus B(H_t)\mathrm{d}\mu(t)$ for a measure space $(M,\mu)$
and Hilbert spaces $H_t$, the following are equivalent:
\begin{itemize}
\item the category $\cat{A}$ is equivalent to $\Cm(A)\op$;
\item the category $\cat{A}$ is equivalent to $P(X) \rtimes S(X)$,
where $X$ is the topological space $\Spec\big(\int_M^\oplus
\bigoplus_{\alpha,\beta\leq \dim(H_t)}
L^\infty((0,1)^\alpha)\big)$;
\item the following hold:
\begin{itemize}
\item $\cat{A}$ satisfies (A1')--(A7');
\item $(\cat{A}, \leq)$ satisfies (P1')--(P6'),(P7''), giving a topological
space $X$;
\item $\cat{A}(0,0)\op$ is isomorphic to the monoid $S(X)$;
\item $X$ is homeomorphic to $\Spec\big(\int_M^\oplus
\bigoplus_{\alpha,\beta\leq \dim(H_t)}
L^\infty((0,1)^\alpha)\big)$.
\end{itemize}
\end{itemize}
When $A=B(H)$ for an infinite-dimensional Hilbert space $H$, the
space $X$ simplifies to $\bigsqcup_{\alpha,\beta\leq \dim(H)}
\Spec\big(L^\infty\big((0,1)^\alpha\big)\big)$. When $H$ is
separable, $X$ further simplifies to $\Spec(L^\infty(0,1)) \sqcup
\Spec(\ell^\infty(\field{N}))$.
\end{theorem}
\begin{proof}
Combine the previous four lemmas with Theorems~\ref{charPM}
and~\ref{charPX}. For the last condition, remember that
Gelfand duality turns products of C*-algebras into coproducts of
compact Hausdorff spaces.
\end{proof}
The Gelfand spectrum of $\ell^\infty(\field{N})$ is the
Stone-{\v C}ech compactification of the discrete topology of
$\field{N}$. In other words, $\Spec(\ell^\infty(\field{N}))$
consists of the ultrafilters on $\field{N}$.
A topological space is homeomorphic to $\Spec(L^\infty(0,1))$ if and
only if it is compact, Hausdorff, totally disconnected, and its clopen
subsets are isomorphic to the Boolean algebra of (Borel) measurable subsets of
the interval $(0,1)$ modulo (Lebesgue) negligible ones.
Notice that both spaces are compact, so we could have used (P7')
instead of (P7'') in the previous theorem for the case $A=B(H)$ with
$H$ separable.
\section{Inclusions versus injections}\label{sec:inclusionsinjections}
This section compares $\Cs$ to $\Cm$. We will show for
C*-algebras $A$ and $B$ that:
\begin{itemize}
\item if $\Cm(A)$ and $\Cm(B)$ are isomorphic, $\Cs(A)$ and
$\Cs(B)$ are isomorphic;
\item if $\Cm(A)$ and $\Cm(B)$ are equivalent, $\Cs(A)$ and
$\Cs(B)$ are Morita-equivalent.
\end{itemize}
For any category $\cat{C}$, recall that the category $\int_{\cat{C}}
P$ of elements of a presheaf $P \in \PSh(\cat{C})$ is defined as
follows. Objects are pairs $(C,x)$ of $C \in \cat{C}$ and $x \in
P(C)$. A morphism $(C,x) \to (D,y)$ is a morphism $f \colon C \to D$
in $\cat{C}$ satisfying $x=P(f)(y)$.
\begin{lemma}
For any $P \in \PSh(\cat{C})$, the toposes $\PSh(\cat{C})/P$
and $\PSh(\int_{\cat{C}} P)$ are equivalent.
\end{lemma}
\begin{proof}
See~\cite[Exercise~III.8(a)]{maclanemoerdijk:sheaves}; we write out
a proof for the sake of explicitness. Define a functor $F \colon
\PSh(\cat{C})/P \to \PSh(\int_{\cat{C}} P)$ by
\begin{align*}
F\big(Q \stackrel{\alpha}{\Rightarrow} P\big)(C,x) & = \alpha_C^{-1}(x), \\
F\big(Q \stackrel{\alpha}{\Rightarrow} P\big)\big( (C,x) \stackrel{f}{\to}
(D,y) \big) & = Q(f), \\
F(Q \stackrel{\beta}{\Rightarrow} Q')_{(C,x)} & = \beta_C.
\end{align*}
Define a functor $G \colon \PSh(\int_{\cat{C}} P) \to \PSh(\cat{C})/P$
by $G(R)=(Q \stackrel{\alpha}{\Rightarrow} P)$ where
\begin{align*}
Q(C) & = \coprod_{x \in P(C)} R(C,x), \\
Q(C \stackrel{f}{\to} D) & = R\big( (C,P(f)(y)) \stackrel{f}{\to} (D,y) \big), \\
\alpha_C(\kappa_x(r)) & = x,
\end{align*}
where $\kappa_x \colon R(C,x) \to \coprod_{x \in P(C)} R(C,x)$ is
the coproduct injection. The functor $G$ acts on morphisms as
\[
G(R \stackrel{\beta}{\Rightarrow} R')_C = \coprod_{x \in P(C)} \beta_{(C,x)}.
\]
Then one finds that $GF(Q \stackrel{\alpha}{\Rightarrow} P) = (Q
\stackrel{\alpha}{\Rightarrow} P)$, and $FG(R) = \hat{R}$, where
\begin{align*}
\hat{R}(C,x) & = \{ x \} \times R(C,x), \\
\hat{R}\big( (C,x) \stackrel{f}{\to} (D,y) \big) & = \id \times R
\big( (C,P(f)(y)) \stackrel{f}{\to} (D,y) \big).
\end{align*}
Thus there is a natural isomorphism $R \cong \hat{R}$, and
$F$ and $G$ form an equivalence.
\end{proof}
\begin{definition}
Define a presheaf $\Aut \in \PSh(\Cm)$ by
\begin{align*}
\Aut(C) & = \{ i \colon C \stackrel{\cong}{\to} C' \mid C' \in \C
\}, \\
\Aut\big( C \stackrel{k}{\rightarrowtail} D \big) \big( j \colon D
\stackrel{\cong}{\to} D' \big) & = j \big|_{k(C)} \after k
\colon C \stackrel{\cong}{\to} j(k(C)) .
\end{align*}
\end{definition}
Notice that $\Aut(C)$ contains the automorphism group
of $C$. Also, any automorphism of $A$ induces an element of
$\Aut(C)$.
The category $\int_{\Cm}\Aut$ of elements of $\Aut$ unfolds explicitly
to the following. Objects are pairs $(C,i)$ of
$C \in \C$ and a $*$-isomorphism $i \colon C \smash{\stackrel{\cong}{\to}}
C'$. A morphism $(C,i) \to (D,j)$ is an injective $*$-homomorphism $k
\colon C \rightarrowtail D$ such that $i = j \after k$.
\begin{lemma}
The categories $\Cs$ and $\int_{\Cm}\Aut$ are equivalent.
\end{lemma}
\begin{proof}
Define a functor $F \colon \Cs \to \int_{\Cm}\Aut$ by
$F(C)=(C, \id[C])$ on objects and $F(C \subseteq D) =
(C \hookrightarrow D)$ on morphisms. Define a functor $G \colon
\int_{\Cm}\Aut \to \Cs$ by $G(C,i) = i(C) = \cod(i)$ on objects and
$G\big(k \colon (C,i) \to (D,j)\big) = (i(C) \subseteq j(D))$ on
morphisms. Then $GF(C)=C$, and $FG(C,i) = (i(C), \id[i(C)]) \cong
(C,i)$, so that $F$ and $G$ implement an equivalence.
\end{proof}
\begin{theorem}\label{CsCm}
The toposes $\PSh(\Cs)$ and $\PSh(\Cm)/\Aut$ are equivalent.
\end{theorem}
\begin{proof}
Combining the previous two lemmas, the equivalence is implemented
explicitly by the functor $F \colon \PSh(\Cm)/\Aut \to
\PSh(\Cs)$ defined by
\begin{align*}
F\big(P \stackrel{\alpha}{\Rightarrow} \Aut \big)(C) & =
\alpha_C^{-1}(\id[C]) \\
F\big(P \stackrel{\alpha}{\Rightarrow} \Aut \big)(C \subseteq D) & =
P(C \hookrightarrow D)
\end{align*}
and the functor $G \colon \PSh(\Cs) \to \PSh(\Cm)/\Aut$
defined by $G(R) = \big( P \stackrel{\alpha}{\Rightarrow} \Aut \big)$,
\begin{align*}
P(C) & = \coprod_{i \colon C \stackrel{\cong}{\to} C'} R(i(C)), \\
P\big(C \stackrel{k}{\rightarrowtail} D \big) & = \coprod_{j \colon D
\stackrel{\cong}{\to} D'} R\big(j(k(C)) \subseteq j(D)\big), \\
\alpha_C(\kappa_i(r)) & = i.
\end{align*}
This proves the theorem.
\end{proof}
Hence the topos $\PSh(\Cm)$ is an \emph{{\'e}tendue}: the unique natural
transformation from $\Aut$ to the terminal presheaf is (objectwise)
epic, and the slice topos $\PSh(\Cm)/\Aut$ is (equivalent to) a
localic topos.
\begin{lemma}\label{slices}
If $F \colon \cat{C} \to \cat{D}$ is (half of) an equivalence, $X$
is any object of $\cat{C}$ and $Y \cong F(X)$, then the slice
categories $\cat{C}/X$ and $\cat{D}/Y$ are equivalent.
\end{lemma}
\begin{proof}
Let $G \colon \cat{D} \to \cat{C}$ be the other half of the given
equivalence, and pick an isomorphism $i \colon Y \to F(X)$. Define a
functor $H \colon \cat{C}/X \to \cat{D}/Y$ by $H(a \colon A \to X) =
(i \after Fa \colon FA \to Y)$ and $H(f \colon a \to b) = Ff$. Define a
functor $K \colon \cat{D}/Y \to \cat{C}/X$ by $K(a \colon A \to Y) =
(\eta_X^{-1} \after Gi \after Ga \colon GA \to X)$ and $K(f \colon a
\to b) = Gf$. By naturality of $\eta^{-1}$ we then have $KH(a) \cong
a$. And because $G\varepsilon = \eta^{-1}$ we also have $HK(a) \cong a$.
\end{proof}
\begin{lemma}\label{catvsPSh}
If the categories $\cat{C}$ and $\cat{D}$ are equivalent, then
the toposes $\PSh(\cat{C})$ and $\PSh(\Cat{D})$ are equivalent.
\end{lemma}
\begin{proof}
Given functors $F \colon \cat{C} \to \cat{D}$ and $G \colon
\cat{D} \to \cat{C}$ that form an equivalence, one directy verifies
that $(-) \after G\colon\PSh(\cat{C}) \to \PSh(\cat{D})$ and $(-)
\after F \colon\PSh(\cat{D}) \to \PSh(\cat{C})$ also form an equivalence.
\end{proof}
\begin{theorem}\label{CmCs}
If $\Cm(A)$ and $\Cm(B)$ are equivalent categories, then $\Cs(A)$
and $\Cs(B)$ are Morita-equivalent posets, \ie the toposes $\PSh(\Cs(A))$ and
$\PSh(\Cs(B))$ are equivalent.
\end{theorem}
\begin{proof}
If $\Cm(A) \simeq \Cm(B)$, then $\PSh(\Cm(A)) \simeq \PSh(\Cm(B))$ by
Lemma~\ref{catvsPSh}. Moreover, the object $\Aut_B$ is (isomorphic
to) the image of the object $\Aut_A$ under this equivalence.
Hence
\[
\PSh(\Cs(A)) \simeq \PSh(\Cm(A))/\Aut_A \simeq \PSh(\Cm(B))/\Aut_B
\simeq \PSh(\Cs(B))
\]
by Theorem~\ref{CsCm}.
\end{proof}
\begin{remark}\label{invariants}
Hence $\Cm(A)$ is an invariant of the topos $\PSh(\Cs(A))$ as well as of the C*-algebra
$A$. It is not a complete invariant for the latter, however, as shown by Lemma~\ref{masas}.
For example, $\Cm(M_n(\field{C})) \simeq \Cm(\field{C}^n)$, but $\Cs(M_n(\field{C}))
\not\cong \Cs(\field{C}^n)$, and certainly $M_n(\field{C}) \not\cong
\field{C}^n$.
We have relied heavily on equivalences of categories, and indeed a
logical formula holds in the topos $\PSh(\cat{C})$ if and only
if it holds in $\PSh(\cat{D})$ for equivalent categories
$\cat{C}$ and $\cat{D}$. Therefore one might argue that $\Cm$ has
too many morphisms, as compared to $\Cs$, for toposes based on
it to have internal logics that are interesting from the point of
view of foundations of quantum mechanics. Instead of equivalences, one
could consider isomorphisms of categories. This also resembles
the original Mackey--Piron question more closely. After all, an equivalence of
partial orders is automatically an isomorphism. The following
theorem shows that $\Cm(A)$ is a weaker invariant of $A$ than
$\Cs(A)$, in this sense.
\end{remark}
\begin{theorem}
If $\Cm(A)$ and $\Cm(B)$ are isomorphic categories, then $\Cs(A)$
and $\Cs(B)$ are isomorphic posets.
\end{theorem}
\begin{proof}
Let $K \colon \Cm(A) \to \Cm(B)$ be an isomorphism. Suppose that
$C,D \in \Cm(A)$ satisfy $C \subseteq D$. Consider the subcategory
$\Cm(D)$ of $\Cm(A)$. On the one hand, by Lemma~\ref{charCmcomm}
it is isomorphic to $P(X) \rtimes S(X)$ for $X=\Spec(D)$, and therefore
has a faithful retraction $F_A$ of the inclusion $\Cm(D) \to
\Cm(D)(0,0)$ by Theorem~\ref{charPM}. On the other hand, $K$ maps it
to $\Cm(K(D))$, which is isomorphic to $P(Y) \rtimes S(Y)$ for
$Y=\Spec(K(D))$, and therefore similarly has a retraction
$F_B$. Moreover, we have $KF_A = F_BK$. Now, by
Theorem~\ref{charPM}, inclusions in $\Cm$ are characterized among
all morphisms $f$ by $F(f)=1$. Hence $F_B(K(C \hookrightarrow D)) =
KF_A(C \hookrightarrow D)=K(1)=1$, and therefore $K(C) \subseteq K(D)$.
\end{proof}
It remains open whether existence of an isomorphism $\Cs(A)
\cong \Cs(B)$ implies existence of an isomorphism $\Cm(A) \cong
\Cm(B)$. This question can be reduced as follows, at least in finite
dimension, because every
injective *-morphism factors uniquely as a $*$-isomorphism followed by
an inclusion. Write $\Ci(A)$ for the category with $\C(A)$ for objects
and $*$-isomorphisms as morphisms.
Supposing an isomorphism $F \colon \Cs(A) \to \Cs(B)$,
we have $\Cm(A) \cong \Cm(B)$ if and only if there is an isomorphism $G \colon
\Ci(A) \to \Ci(B)$ that coincides with $F$ on objects. Now, in case $A$ is
(isomorphic to) $M_n(\field{C})$, (so is $B$, and) if $C,D \in
\C(A)$ are isomorphic then so are $F(C)$ and $F(D)$: if $C \cong
D$, then $\dim(C)=\dim(D)$, so $\dim(F(C))=\dim(F(D))$ because $F$
preserves maximal chains, and hence
$F(C) \cong F(D)$. However, it is not clear whether this behaviour is
functorial, \ie extends to a functor $G$, or generalizes to infinite dimension.
\appendix
\section{Inverse semigroups and {\'e}tendues}\label{sec:inversesemigroups}
The direct proof of Theorem~\ref{CsCm} follows from~\cite[A.1.1.7]{johnstone:elephant},
but it can also be arrived at through a detour via inverse semigroups, based
on results due to Funk~\cite{funk:semigroupsandtoposes}. This appendix
describes the latter intermediate results, which might be of
independent interest. Fix a unital C*-algebra $A$.
\begin{definition}
Define a set $T$ with functions $T \times T \stackrel{\cdot}{\to} T$
and $T \stackrel{*}{\to} T$ by:
\begin{align*}
& T = \left\{ C \stackrel{i}{\rightarrowtail} A
\mid C \in \C, \; i \text{ is an injective
$*$-homomorphism} \right\},\\
& (C' \stackrel{i'}{\rightarrowtail} A) \cdot (C
\stackrel{i}{\rightarrowtail} A) =
(i^{-1}(C') \stackrel{i' \after i}{\rightarrowtail} A), \\
& (C \stackrel{i}{\rightarrowtail} A)^* = (i(C)
\stackrel{i^{-1}}{\rightarrowtail} A).
\end{align*}
The multiplication is well-defined, because the inverse image of a
*-algebra under a $*$-homomorphism is again a *-algebra, and the inverse
image of a closed set is again a closed set, so that $i^{-1}(C)$ is
indeed a commutative C*-algebra. The operation * is well-defined
because of Lemma~\ref{cstarimage}; and on the image,
$i^{-1}$ is a well-defined injective $*$-homomorphism. One can
verify that together, these data form an inverse semigroup; that is,
multiplication is associative, and $i^*$ is the unique element with
$ii^*i=i$ and $i^*ii^*=i^*$.
\end{definition}
\begin{lemma}\label{domcodT}
For $(C \stackrel{i}{\rightarrowtail} A) \in T$, we have $i^*i = (C
\hookrightarrow A)$ and $ii^* = (i(C) \hookrightarrow A)$.
\end{lemma}
\begin{proof}
For the former claim:
\[
(C \stackrel{i}{\rightarrowtail} A)^* \cdot (C \stackrel{i}{\rightarrowtail} A)
= (i(C) \stackrel{i^{-1}}{\rightarrowtail} A) \cdot (C \stackrel{i}{\rightarrowtail} A)
= (i^{-1}(i(C)) \stackrel{i^{-1} \after i}{\rightarrowtail} A)
= (C \hookrightarrow A).
\]
For the latter claim:
\begin{align*}
(C \stackrel{i}{\rightarrowtail} A) \cdot (C \stackrel{i}{\rightarrowtail} A)^*
& = (C \stackrel{i}{\rightarrowtail} A) \cdot (i(C) \stackrel{i^{-1}}{\rightarrowtail} A) \\
& = ((i^{-1})^{-1}(C) \stackrel{i \after i^{-1}}{\rightarrowtail} A)
= (i(C) \hookrightarrow A).
\end{align*}
This proves the lemma.
\end{proof}
\begin{definition}
For any inverse semigroup $T$, one can define the groupoid $G(T)$ whose
objects are the idempotents of $T$, i.e. the elements $e \in T$ with $e^2=e$. A morphism
$e \to f$ is an element $t \in T$ satisfying $e=t^*t$ and $tt^*=f$.
\end{definition}
\begin{proposition}\label{GTA}
The groupoids $G(T)$ and $\Ci$ are isomorphic.
\end{proposition}
\begin{proof}
An element $(C \stackrel{i}{\rightarrowtail} A)$ of $T$ is idempotent when $i^{-1}(C)=C$
and $i^2=i$ on $C$. That is, the objects of $G(T)$ are the
inclusions $(C \hookrightarrow A)$ of commutative C*-subalgebras; we
can identify them with $\C$.
A morphism $C \to C'$ in $G(T)$ is an element $(D \stackrel{j}{\rightarrowtail} A)$ of $T$
such that $(C \hookrightarrow A) = j^*j = (D \hookrightarrow A)$
and $(C' \hookrightarrow A) = jj^* = (j(D) \hookrightarrow A)$, \ie
$C=D$ and $C'=j(D)$. That is, a morphism $C \to C'$ is an injective
$*$-homomorphism $j \colon C \rightarrowtail C'$ that satisfies
$j(D)=C'$, \ie that is also surjective. In other words, a morphism
$C \to C'$ is a $*$-isomorphism $C \to C'$.
\end{proof}
\begin{definition}
For any inverse semigroup $T$, one can define a partial order on the
set $E(T)=\{ e \in T \mid e^2=e \}$ of idempotents by $e \leq f$ iff $e=fe$.
\end{definition}
In fact, $G(T)$ is an ordered groupoid, with $G(T)_0 = E(T)$.
\begin{proposition}\label{ETA}
The posets $E(T)$ and $\Cs$ are isomorphic.
\end{proposition}
\begin{proof}
As with $G(T)$, objects of $E(T)$ can be identified with $\C$. Moreover,
there is an arrow $C \to C'$ if and only if
\[
(C \hookrightarrow A) = (C' \hookrightarrow A) \cdot (C
\hookrightarrow A) = (C \cap C' \hookrightarrow A),
\]
\ie when $C \cap C'=C$. That is, there is an arrow $C \to C'$ iff
$C \subseteq C'$.
\end{proof}
Also, $G(T)$ is always a subcategory of the following category $L(T)$.
\begin{definition}
For any inverse semigroup $T$, one can define the left-cancellative
category $L(T)$ whose objects are the idempotents of $T$. A morphism
$e \to f$ is an element $t \in T$ satisfying $e=t^*t$ and $t=ft$.
\end{definition}
\begin{proposition}\label{LTA}
The categories $L(T)$ and $\Cm$ are isomorphic.
\end{proposition}
\begin{proof}
As with $G(T)$, objects of $L(T)$ can be identified with $\C$.
A morphism $C \to C'$ in $L(T)$ is an element
$(j \colon D \rightarrowtail A)$ of $T$ such that $(C
\hookrightarrow A) = j^*j = (D \hookrightarrow A)$ and
\[
(D \stackrel{j}{\rightarrowtail} A) = (C' \hookrightarrow A) \cdot
(D \stackrel{j}{\rightarrowtail} A) = (j^{-1}(C')
\stackrel{j}{\rightarrowtail} A).
\]
That is, a morphism $C \to C'$ is an injective $*$-homomorphism $j \colon
C \rightarrowtail A$ such that $C=j^{-1}(C')$.
Hence we can identify morphisms $C \to C'$ with injective
$*$-homomorphisms $j \colon C \rightarrowtail C'$.
\end{proof}
Every ordered groupoid $G$ has a classifying topos $\B(G)$. We now
describe the topos $\B(G(T))$ explicitly, unfolding the definitions
on~\cite[page~487]{funk:semigroupsandtoposes}.
For a presheaf $P \colon \Cs\op \to \Cat{Set}$, define another
presheaf $P^* \colon \Cs\op \to \Cat{Set}$ by
\[
P^*(C) = \{ (j,x) \mid j \in \Ci(A)(C,C'), x \in
P(C') \}.
\]
On a morphism $C \subseteq D$, the presheaf $P^* \colon P^*(D) \to P^*(C)$ acts as
\[
(k \colon D' \stackrel{\cong}{\to} D, y \in P(D'))
\longmapsto
\big( k\big|_{C} \colon C \stackrel{\cong}{\to} k(C), \,P(k(C) \subseteq
D')(y) \big).
\]
An object of $\B(G(T))$ is a pair $(P,\theta)$ of a presheaf $P \colon
\Cs\op \to \Cat{Set}$ and a natural transformation $\theta \colon P^*
\Rightarrow P$. A morphism $(P,\theta) \to (Q,\xi)$ is a natural
transformation $\alpha \colon P \Rightarrow Q$ satisfying $\alpha \after
\theta = \xi \after \alpha^*$, where the natural transformation $\alpha^*
\colon P^* \Rightarrow Q^*$ is defined by $\alpha^*_C(j,x) = (j,\alpha_C(x))$.
\begin{lemma}\label{CmB}
The toposes $\PSh(\Cm)$ and $\B(G(T))$ are equivalent.
\end{lemma}
\begin{proof}
Combine Proposition~\ref{LTA} with
\cite[Proposition~1.12]{funk:semigroupsandtoposes}.
Explicitly, $(P, \theta)$ in $\B(G(T))$ gets mapped to $F \colon
\Cm(A)\op \to \Cat{Set}$ defined by $F(C)=P(C)$ and
\[
F(k \colon C \rightarrowtail D)(y) = \theta_C( k \colon C
\stackrel{\cong}{\to} k(C), \, P(k(C) \subseteq D)(y) ).
\]
Conversely, $F$ in $\PSh(\Cm)$ gets mapped to
$(P,\theta)$, where
\begin{align*}
P(C) & = F(C), \\
P(C \subseteq D) & = F(C \hookrightarrow D), \\
\theta_C (j \colon C \stackrel{\cong}{\to} C', x \in F(C')) & = F(C'
\stackrel{j^{-1}}{\to} C \subseteq D) (x).
\end{align*}
\end{proof}
There is a canonical object $\mathbf{S}=(S,\theta)$ in $\B(G(T))$,
defined as follows.
\begin{align*}
S(C) & = \{ i \colon C \rightarrowtail A \}, \\
S(C \subseteq D)(j \colon D \rightarrowtail A) & = (j\big|_C \colon
C \rightarrowtail A).
\end{align*}
In this case $S^*$ becomes
\begin{align*}
S^*(C) & = \{(j,i) \mid j \colon C \stackrel{\cong}{\to} C', i
\colon C' \rightarrowtail A \}, \\
S^*(C \subseteq D)(j,i) & = (j \mid_C \colon C \stackrel{\cong}{\to}
j(C),\, i\big|_{j(C)} \colon j(C) \rightarrowtail A ).
\end{align*}
Hence we can define a natural transformation $\theta \colon S^*
\Rightarrow S$ by
\[
\theta_C(j,i) = i \after j.
\]
The equivalence of the previous lemma maps $\mathbf{S}$ in $\B(G(T))$
to $\mathbf{D}$ in $\PSh(\Cm)$:
\begin{align*}
\mathbf{D}(C) & = \{ i \colon C \rightarrowtail A \}, \\
\mathbf{D}(k \colon C \rightarrowtail D)(j \colon D
\rightarrowtail A) & = (j \after k \colon C \rightarrowtail A).
\end{align*}
Technically, the topos $\B(G(T))$ is an {\'e}tendue: the
unique morphism from some object $\mathbf{S}$ to the terminal object
is epic, and the slice topos $\B(G(T))/\mathbf{S}$ is (equivalent
to) a localic topos. The following lemma makes the latter equivalence
explicit.
\begin{lemma}
The toposes $\B(G(T))/\mathbf{S}$ and $\PSh(\Cs)$ are equivalent.
\end{lemma}
\begin{proof}
Combine Proposition~\ref{ETA} with equation (1) in \cite[page
488]{funk:semigroupsandtoposes}.
\end{proof}
Combining the previous two lemmas, we find:
\begin{theorem}
The toposes $\PSh(\Cm)/\mathbf{D}$ and $\PSh(\Cs)$ equivalent.
\qed
\end{theorem}
In our specific application, we have more information and it is
helpful to reformulate things slightly. By Lemma~\ref{cstarimage},
giving an injective $*$-homomorphism $i \colon C \rightarrowtail A$ is
the same as giving a $*$-isomorphism $C \cong C'$ for
some $C' \in \C$ (by taking $C'=i(C)$). Hence $\mathbf{S}$ is
isomorphic to the object
$\mathbf{Aut}=(\mathrm{Aut},\theta)$ in $\B(G(T))$ with
$\theta_C(j,i)=i \after j$. This leads to Theorem~\ref{CsCm}.
\bibliographystyle{plain}
\bibliography{cccc}
\end{document} | 216,104 |
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\begin{document}
~
\vspace{5mm}
\begin{center}
{{\Large \bf Toward Quantization of Inhomogeneous Field Theory }}
\\[17mm]
Jeongwon Ho$^1$, ~~O-Kab Kwon$^2$, ~~Sang-A Park$^3$, ~~Sang-Heon Yi$^1$ \\[2mm]
{\it $^1$Center for Quantum Spacetime, Sogang University, Seoul 04107, Republic of Korea} \\
{\it $^2$Department of Physics,~Institute of Basic Science, Sungkyunkwan University, Suwon 16419, Korea} \\
{\it $^3$Department of Physics, Yonsei University, Seoul 03722, Korea}
\\[2mm]
{\it [email protected], [email protected], [email protected], [email protected]}
\end{center}
\vspace{15mm}
\begin{abstract}
\noindent
We explore the quantization of a $(1+1)$-dimensional inhomogeneous scalar field theory in which Poincar\'{e} symmetry is explicitly broken. We show the classical equivalence between an inhomogeneous scalar field theory and a scalar field theory on curved spacetime background.
This implies that a hidden connection may exist among some inhomogeneous field theories, which corresponds to general covariance in field theory on curved spacetime.
Based on the classical equivalence, we propose how to quantize a specific field theory with broken Poincar\'{e} symmetry inspired by standard field theoretic approaches, canonical and algebraic methods, on curved spacetime.
Consequently, we show that the Unruh effect can be realized in inhomogeneous field theory and propose that it may be tested by a condensed matter experiment.
We suggest that an algebraic approach is appropriate for the quantization of a generic inhomogeneous field theory.
\end{abstract}
\thispagestyle{empty}
\newpage
\section{Introduction}
There are growing interests in physical systems described by particles or fields with non-constant mass, related to the condensed matter physics, neutron physics, cosmology and so on. For instance, in the study of electronic and transport properties of some materials, the Schr\"{o}dinger equation with effectively position-dependent mass is an important and useful tool~\cite{Bastard:1988,Roos:1983,Filho:2011}. In nuclear physics and neutron star physics, this equation has also interesing applications~\cite{Ring:1980,Chamel:2005nd}. In the perspective of field theory, the so-called waterfall field in a hybrid model of inflation~\cite{Linde:1993cn} can be regarded effectively as an incarnation of a time-dependent mass. Recently, there have been some renewed interests in the supersymmetric extension of the position-dependent mass and couplings~\cite{Bak:2003jk,DHoker:2006qeo,Kim:2008dj,Kim:2009wv,Kim:2018qle,Kim:2019kns,Arav:2020obl,Kim:2020jrs,Kwon:2021flc}. In these examples, some quantum nature has been explored in the non-relativistic quantum mechanics setup and some classical aspects in field theories have been analyzed. Nevertheless, the formalism and conceptual understanding of non-constant mass in field theories are a bit deterred partly because of the lack of sufficient symmetries or conservation properties. Furthermore, analytic solutions to equations of motion are rare. Accordingly, the quantum aspects of field theory with non-constant mass raises various issues and awaits further explorations.
In this paper, we consider a position-dependent mass and couplings in the context of bosonic field theories. We will call this kind of field theory as {\it inhomogeneous field theory} (IFT). Since the position dependence of mass and couplings breaks the Poincar\'{e} symmetry explicitly, the usual Wigner representation of the symmetry on fields would be unavailable. In other words, scalar, vector or tensor field distinction or spin representation of fields are obscured in IFT, unlike in the Poincar\'{e} symmetric case. Despite this, we may simply take the standpoint that our IFT is derived from the usual field theory with constant mass and couplings by elevating those to be position-dependent. Therefore, it would be legitimate to adopt the usual terminology and to designate a scalar field in IFT, for instance. This is an obvious abuse of terminology, but wouldn't lead to any hurdles or glitches.
On the other hand, a frame dependence might be a real trouble in quantizing IFT. One of the related hurdles in the quantization is the absence of Poincar\'{e} invariant vacuum, which will require a careful treatment. This situation reminds us {\it field theory on curved spacetime} (FTCS), in which case one encounters similar issues like the absence of the preferred vacuum. In the case of FTCS, there has been a large development to overcome difficulties with preserving general covariance, which is eventually evolved to a subject known as the algebraic quantum field theory~\cite{Haag:1992hx,Wald:1995yp,Yngvason:2004uh,Halvorson:2006wj,Hollands:2009bke,Benini:2013fia,Hollands:2014eia,Khavkine:2014mta,Fredenhagen:2014lda}. See also~\cite{Witten:2018zxz,Witten:2021jzq,Dedushenko:2022zwd}. Based on the similar difficulties in IFT to those in FTCS, one may guess that it is very tempting to adopt the algebraic formulation for IFT. In particular, the van Hove model, which may be thought to be a kind of IFT, has been a good example for such a formulation~\cite{Fewster:2019ixc}.
However, in general, it doesn't seem to be straightforward to materialize this expectation.
In this work, we would like to make this anticipation into a concrete working example in the case of a $(1+1)$-dimensional scalar IFT. Toward quantization of IFT, we suggest a new direction inspired by an algebraic quantization process in FTCS. One of our main points is to show that {\it inhomogeneous quantum field theory} (IQFT) is equivalent to {\it quantum field theory on curved spacetime} (QFTCS). We also propose a ``generalized stress tensor'' in IQFT, which is designed to be conserved. Based on this, we show the existence of the Unruh-like effect in IQFT.
The paper is organized as follows. In section 2, we show the classical equivalence between $(1+1)$-dimensional IFT and $(1+1)$-dimensional FTCS. As an example, we focus on a free scalar field. The implication of general covariance in conjunction with the equivalence is explored. In some examples, the limiting behaviors of a position-dependent mass are inspected and their geometric interpretation is given. In section 3, based on the classical equivalence, we present our main proposal for the quantization of our model in IFT. Following the canonical quantization in curved spacetime, we quantize IFT by the canonical quantization. In section 4, the Hadamard method is summarized which is a very useful algebraic construction superseding the canonical quantization. By applying this method to IFT, we suggest some interesting quantum aspects including the Unruh-like effect in IFT. In section 5, we summarize our results and provide some future directions. In Appendix \ref{AppA} and \ref{AppB}, we collect some formulae useful for the main text.
\section{Classical Equivalence }
In conventional relativistic quantum field theories with constant mass and coupling parameters in the Minkowski spacetime, the Poincar\'{e} symmetry is essential in the canonical quantization. On the contrary, the Poincar\'{e} symmetry is explicitly broken in IFT whose mass and coupling parameters depend on spatial coordinates. Therefore, it isn't guaranteed to apply the canonical quantization method to IFT. In this paper, we would like to propose a quantization method for IFT, based on a classical equivalence between IFT and FTCS in the $(1+1)$-dimensional spacetime. Before presenting our proposal for the quantization, we show the classical equivalence of IFT and FTCS in this section.
And we also discuss interesting implication of general covariance in FTCS in the context of IFT.
Then we investigate limiting behaviors of a position-dependent mass in terms of background metric in FTCS through our equivalence.
\subsection{$(1+1)$-dimensional IFT and FTCS}
As a simple example of IFTs, let us consider a $(1+1)$-dimensional scalar field theory with a position dependent mass $m(x)$, couplings $g_{n}(x)$'s, and a source $J(x)$, whose action is given by
\begin{equation} \label{IFT0}
S_{{\rm IFT}} =\int d^2 x \mathcal{L}_{{\rm IFT}} = \int d^2 x\left( -\frac{1}{2}\partial _{\mu}\phi \partial^{\mu}\phi - \frac{1}{2}m^{2}(x)\phi^{2} - \sum_{n=3}g_{n}(x)\phi^{n} +J(x)\phi\right)\,,
\end{equation}
where the Poincar\'{e} symmetry is partially broken. Using the remaining symmetry, its supersymmetric extension with $J(x)=0$ was done in \cite{Kwon:2021flc}. Attempting to quantize this theory, we encounter various difficulties due to the broken Poincar\'{e} symmetry. Recalling that any parameters in field theory may be promoted to the background values of certain fields in an enlarged field theory or string theory\footnote{This is an old folklore realized in various cases. For instance, axion field is a field elevation of the original constant $\theta$ angle parameter. The usefulness of space-dependent parameters in supersymmetric field theory is emphasized in~\cite{Seiberg:1993vc}.}, we may regard $m(x)$, $g_{n}(x)$'s, and $J(x)$ as vacuum expectation values of those fields. For instance, we might embed the above action into a higher dimensional theory with the Poincar\'{e} symmetry. In this way, we may perform a quantization of IFT from the Poincar\'{e} invariant theory. However, it doesn't seem to be straightforward to realize this embedding.
Instead, we explore another way for a quantization of the above IFT. For this purpose, we consider a $(1+1)$-dimensional scalar FTCS, whose action is given by
\begin{equation} \label{GraLag0}
S_{{\rm FTCS}} =\int d^2 x \sqrt{-g} \mathcal{L}_{{\rm FTCS}} = \int d^{2}x \sqrt{-g}\bigg[ -\frac{1}{2}\nabla _{\mu}\phi \nabla^{\mu}\phi - \frac{1}{2}m_0^{2} \phi^{2} - \sum_{\ell=1}f_{\ell}({\cal R})\phi^{\ell} \bigg]\,,
\end{equation}
where $m_0$ is a constant, ${\cal R}$ denotes the scalar curvature of the background metric, and $f_{\ell}$'s are some functions of ${\cal R}$.
The action \eqref{GraLag0} enjoys general covariance. In the following, we take the above metric, $g_{\mu\nu}$, as non-dynamical. Now, let us take a $(1+1)$-dimensional metric in the form of
\begin{equation} \label{Ourmet0}
ds^{2} = e^{2\omega(x)}(-dt^{2} + dx^{2})\,,
\end{equation}
which is a generic conformal form of the metric in $(1+1)$ dimensions. The absence of the time coordinate in $\omega$ is matched to the time independence of coupling parameters in IFT. See the action \eqref{IFT0}. Some properties of this metric are given in Appendix~\ref{AppA}. In particular, one may note that $\sqrt{-g} = e^{2 \omega (x)}$ and ${\cal R} = -2 \omega'' e^{-2\omega}$, where ${}'$ denotes the derivative with respect to $x$.
Inserting the above metric in the FTCS action~\eqref{GraLag0}, one can see that the action is converted to the IFT action in \eqref{IFT0} with the parameter matching as follows:
\begin{align} \label{matching}
&m^{2}(x) = \sqrt{-g} \Big(m^{2}_{0} +2 f_{2}(\mathcal{R})\Big)\,,
\nonumber\\
&
g_{n}(x) = \sqrt{-g} f_{n} (\mathcal{R})\,, \nonumber\\
&J(x) = -\sqrt{-g} f_{1}(\mathcal{R})\,.
\end{align}
For any functions $f_{\ell}(\mathcal{R})$'s in \eqref{GraLag0}, we can convert a $(1+1)$-dimensional scalar FTCS to a scalar IFT at the action level, as far as $m^{2}(x)\ge0$.
The equations of motion of the scalar fields are completely identical in those two actions with the conversion rule~\eqref{matching}. Thus we can say that the same physics is described by two differently-looking languages, IFT and FTCS, {\it i.e.}, a kind of dual description.
For instance, a scalar IFT with a position-dependent mass and a scalar FTCS with constant mass describe the same physics. We will call such a connection between two theories as the {\it classical equivalence} and denote:
\begin{equation}
\boxed{\text{IFT $\iff$ FTCS}}\,.
\end{equation}
It is crucial in this equivalence that the kinetic term in \eqref{GraLag0} in $(1+1)$ dimensions is always independent of the metric for the conformal form in \eqref{Ourmet0}. The basic concepts and methods in the above can be extended to other field theories including fermionic ones in $(1+1)$ dimensions. However, note that in general the classical equivalence is not straightforwardly extended to higher dimensional theories.
To illustrate our approach concretely, we focus on the following simplest quadratic action on curved spacetime, which is given by\footnote{This is not the conformally coupled case in $(1+1)$ dimensions.}
\begin{equation} \label{GraLag2}
S_{{\rm FTCS}} = \int d^{2}x \sqrt{-g} \Big[-\frac{1}{2}\nabla _{\mu}\phi \nabla^{\mu}\phi -\frac{1}{2}m^{2}_{0}\phi^{2} - \frac{\xi}{2} \mathcal{R} \phi^{2}\Big]\,,
\end{equation}
where $\xi$ is a constant. The stress tensor for this action is given by
\begin{align} \label{}
T_{\mu\nu}
&= \nabla_{\mu}\phi\nabla_{\nu}\phi -\frac{1}{2}g_{\mu\nu}\Big[(\nabla\phi)^2 + m_{0}^2\, \phi^2\Big] + \xi \Big(- \nabla_{\mu}\nabla_{\nu} + g_{\mu\nu}\nabla^{2} \Big)\phi^{2}\,.
\end{align}
Just like in the general case, one can see that the above FTCS action is converted to the IFT action on the Minkowski spacetime in the form of
\begin{equation} \label{IFT2}
S_{{\rm IFT}} = \int d^{2}x \Big[ -\frac{1}{2}\partial _{\mu}\phi \partial^{\mu}\phi -\frac{1}{2}m^{2}(x)\, \phi^{2}\Big]\,, \qquad
\end{equation}
where the squared mass function is given by
\begin{equation} \label{Matrel}
m^2(x;m_0,\xi) = \sqrt{-g} (m_0^2 + \xi \mathcal{R})=e^{2\omega(x)}m_0^2-2\omega''\xi \,.
\end{equation}
When we consider the inverse conversion from IFT to FTCS, we could take any form of the $f_{2}({\cal R})$ function in~\eqref{matching} for a given mass function $m(x)$. This means that we have freedom in the conversion process and these should be taken into account in the equivalence. Though a minimal choice is to take $\xi =0$, we consider a two-parameter conversion to set the scalar IFT action~\eqref{IFT2} to the form of \eqref{GraLag2}. This choice in \eqref{Matrel} contrasts to taking $m(x)$ as the vacuum expectation value of a certain scalar field and reveals some interesting features in our equivalence, especially in the expression of ``stress tensor'' in IFT, $T_{\mu\nu}^{\rm IFT}$.
From the stress tensor in the scalar FTCS, we read the ``stress tensor'' in the scalar IFT as
\begin{align} \label{Tclassical}
T_{\mu\nu}^{\rm IFT} & = \partial_{\mu}\phi\partial_{\nu}\phi -\frac{1}{2}\eta_{\mu\nu}\Big[(\partial\phi)^{2} + m^{2}_{0}e^{2\omega}\phi^{2}\Big] +\xi\Big[-\partial_{\mu} \partial_{\nu} + \Gamma^{\alpha}_{\mu\nu,\, {\rm IFT}} \partial_{\alpha} + \eta_{\mu\nu} \partial^{2} \Big]\phi^{2}\,,
\end{align}
where $\Gamma^{\alpha}_{\mu\nu,\, {\rm IFT}}$ denotes
\begin{equation} \label{}
\Gamma^{\alpha}_{\mu\nu,\, {\rm IFT}} \equiv \omega'(x) \Big[2 \delta^{(x}_{\mu}\delta^{t)}_{\nu} \delta^{\alpha}_{t}+ (\delta^{x}_{\mu} \delta^{x}_{\nu} + \delta^{t}_{\mu}\delta^{t}_{\nu})\delta^{x}_{\alpha}\Big] \,.
\end{equation}
Here, $\omega'(x)$ should read from the chosen conversion rule in~\eqref{Matrel}. We would like to emphasize that our ``stress tensor'' satisfies the ``conservation law'' derived from the counter part in FTCS, which takes a very unusual form from the viewpoint of IFT, {\it i.e.},
\begin{align}
\nabla^{\mu}_{{\rm IFT}}T_{\mu\nu}^{\rm IFT} \equiv \eta^{\alpha\beta} \big[ \partial_{\alpha}T_{\beta\nu}^{\rm IFT} - \Gamma^{\rho}_{\alpha\beta,\, {\rm IFT}}~ T_{\rho\nu}^{\rm IFT} - \Gamma^{\rho}_{\alpha\nu,\, {\rm IFT}}~ T_{\beta\rho}^{\rm IFT} \big] =0\,,
\end{align}
where the indices are raised and lowered by $\eta_{\mu\nu}$ in IFT. Indeed, one can check explicitly in our case that
\begin{align} \label{}
\nabla^{\rm IFT}_{\mu} T^{\mu}_{~\,t, {\rm IFT} } &=\partial_{\mu} T^{\mu}_{~\,t, {\rm IFT} } =0\,, \nonumber \\
\nabla^{\rm IFT}_{\mu} T^{\mu}_{~\,x, {\rm IFT} } &= \partial_{\mu} T^{\mu}_{~\,x, {\rm IFT} } +\omega'(x) \Big[m^{2}_{0}e^{2\omega} -\xi\partial^2 \Big]\phi^{2} =0\,.
\end{align}
The first equation corresponds to the energy conservation which comes from the $t$-translation symmetry in our scalar IFT. The second equation obviously reveals the absence of the $x$-translation symmetry in IFT, which implies that the conserved stress tensor cannot be introduced in the usual sense~\cite{Kwon:2021flc}. In other words, there is no conserved quantity coming from the second equation. Nevertheless we can interpret the above quantity as the conserved ``stress tensor'' originated from the scalar FTCS.
To see legitimacy of our construction of the ``stress tensor'' and to verify the energy conservation, one may return to the canonical Hamiltonian.
In our specific position dependent IFT in~\eqref{IFT2}, the canonical Hamiltonian (density) can be introduced as
\begin{equation} \label{}
{\cal H}_{\rm can} = \frac{1}{2}\Big[\dot{\phi}^{2} + (\phi')^{2} + m^{2}(x) \phi^{2}\Big]\,,\qquad \dot{}\equiv\frac{\partial}{\partial t}\,,
\end{equation}
which corresponds to the time translation generator. To see the matching of this to the ``stress tensor'', let us check that
the $tt$-component of our ``stress tensor'' is the same with ${\cal H}_{\rm can}$ up to the total derivative term,
\begin{equation} \label{}
T^{tt}_{\rm IFT}
= {\cal H}_{\rm can} - \Big[ \xi\Big ((\phi^{2})' -\omega'\phi^{2}\Big)\Big]' \,,
\end{equation}
where one should recall that the indices are raised by $\eta_{\mu\nu}$ in \eqref{Tclassical}.
If the fall-off condition of the scalar field $\phi$ is taken appropriately, the energy defined by the Hamiltonian would be the same as that obtained from the $tt$-component of our ``stress tensor''.
As discussed previously, the classical equivalence provides us a new way to explore the quantization of the IFT with a position dependent mass. Before going ahead, we present some notable aspects of our model and its correspondence.
\subsection{Implications of general covariance}
As we have already discussed, general covariance is manifest in the quadratic FTCS action \eqref{GraLag2}. Though the equivalence of \eqref{GraLag2} and \eqref{IFT2} for the metric \eqref{Ourmet0} is shown, the meaning of general covariance is obscured in the IFT action point of view since the measures of the action integral may be changed by general coordinate transformations. In this subsection, we try to figure out the implication of the general covariance of FTCS in the context of IFT in association with the classical equivalence.
As we showed in the previous subsection, the FTCS action \eqref{GraLag2} is converted to the IFT action \eqref{IFT2} as
\begin{align}\label{IFT=g1}
\int d^2 x\, \sqrt{-g} \mathcal{L}_{{\rm FTCS}} (g_{\mu\nu}, m_0, \xi)= \int d^2 x\, \mathcal{L}_{{\rm IFT }} (m(x)),
\end{align}
where $m^2(x) = \sqrt{-g} (m_0^2 + \xi \mathcal{R})$ and the metric is given in \eqref{Ourmet0}. We would like to ask how the general covariance in the left-hand side is encoded in the right-hand side of \eqref{IFT=g1}.
Performing a general coordinate transformation $(t,x) \,\, \longrightarrow \,\, (T(x^\mu), X(x^\mu))$ for the FTCS action in the left-hand side of \eqref{IFT=g1}, we obtain
\begin{align}\label{GC}
\int d^2 x\, \sqrt{-g} \mathcal{L}_{{\rm FTCS}} (g_{\mu\nu}, m_0, \xi)= \int d^2 X \sqrt{-\bar{g}}\, \mathcal{L}_{{\rm FTCS}} (\bar{g}_{\mu\nu}, m_0, \xi)\,,
\end{align}
where $\bar{g}_{\mu\nu} (X)\equiv \frac{\partial x^\rho}{ \partial X^\mu} \frac{\partial x^\sigma}{ \partial X^\nu} g_{\rho\sigma}(x)$. Under this transformation, the integration region could be changed.
Now we convert the transformed FTCS action to the IFT action:
\begin{align}\label{IFT=g2}
\int d^2 X \sqrt{-\bar{g}}\, \mathcal{L}_{{\rm FTCS}} (\bar{g}_{\mu\nu}, m_0, \xi)= \int d^2 X \mathcal{L}_{{\rm IFT }} (\bar{m}(\mathbf{X})),
\end{align}
where the mass function in the right-hand side is given by
\begin{align}\label{mpX}
\bar{m}^2 (\mathbf{X}) = \sqrt{-\bar{g}} (m_0^2 + \xi \bar{\mathcal{R}}).
\end{align}
Compared to the position-dependent mass function $m(x)$, the mass function $\bar{m}(\mathbf{X})$ in the scalar IFT converted from the transformed scalar FTCS can depend on both coordinates $\mathbf{X}=(T,X)$. That is, two seemingly different IFT's with $m(x)$ and $\bar{m}(\mathbf{X})$ are connected each other. This result is quite surprising from the view point of IFT, while it is natural within our equivalence. Based on this observation, we would like to interpret this relation between two IFT's as the existence of a hidden connection in IFT.
\begin{figure}
\begin{center}
\includegraphics{SP/fig-geo.pdf}
\caption{The geometry under consideration via \eqref{Rmetric1} and \eqref{RinMin1} is presented. The Minkowski spacetime \eqref{RinMin1} is covered by $(T,X)$ without any restrictions in the coordinates. On the other hand, the Rindler spacetime \eqref{Rmetric1} covers only the right or left Rindler wedge(region I or II) in $(T,X)$-coordinates, while $-\infty<t,x<\infty$. }
\label{fig:geo}
\end{center}
\end{figure}
In order to show this phenomenon explicitly, we consider the Rindler spacetime and a coordinate transformation in the side of FTCS. First, we take the metric in the FTCS action as
\begin{align} \label{Rmetric1}
ds^{2} = e^{2bx}(-dt^{2} + dx^{2})\,,
\end{align}
where $b>0$ corresponds to a constant proper acceleration, related to a static observer in the Rindler spacetime.
Then the corresponding IFT action by our equivalence becomes
\begin{align} \label{SIFT1}
S_{{\rm IFT}} = \int d^{2}x \Big[ -\frac{1}{2}\partial _{\mu}\phi \partial^{\mu}\phi -\frac{1}{2}m^{2}(x)\, \phi^{2}\Big]\,,\qquad m^{2}(x) = m^{2}_{0}\, e^{2bx}\,.
\end{align}
Second, let us carry out the coordinate transformation in the FTCS side as $(t,x)\to (T,X)$,
\begin{align}\label{CoordTr}
T = b^{-1} e^{b x} \sinh (b t),\qquad X = b^{-1} e^{b x} \cosh (b t)\,.
\end{align}
This brings the Rindler metric \eqref{Rmetric1} to the Minkowski metric,
\begin{align} \label{RinMin1}
ds^2 = - dT^2 + dX^2
\end{align}
with the range of coordinates
$X > |T|$. See Fig.~\ref{fig:geo}. This is the so-called right Rindler wedge of the Minkowski spacetime. This right Rindler wedge spacetime is a globally hyperbolic but geodesically incomplete spacetime. To avoid the encounter with the spacetime boundary within a finite proper time/affine parameter, one usually extend the spacetime to be geodesically complete. In this case, the extended one is the Minkowski spacetime with $-\infty<T,X<\infty$.
This is natural in the context of general covariance in FTCS, which is displayed in the left side of Fig.~\ref{fig:rel}.
\begin{figure}[h]
\begin{center}
\includegraphics{SP/fig-equiv.pdf}
\caption{FTCS's and IFT's are presented on left-side and right-side columns, respectively. Our classical equivalence relates FTCS and IFT in the first row. In the left-hand side, the first and the second theories are connected by the coordinate transformation~\eqref{CoordTr}. On the other hand, the theories in the second row are trivially identical to each other as they are nothing more than changing dummy variables on the same region. So the relation $\textcolor{red}{(*)}$ between two different IFT's is deduced. In the left-hand side, the first and the third theories are connected by general covariance which includes the extension of the range of domains. IFT's with different mass functions in the same range can be related by a hidden relation $\textcolor{red}{(**)}$.
}
\label{fig:rel}
\end{center}
\end{figure}
Now, we convert the transformed FTCS action to the corresponding IFT action by our equivalence, to see the effect of the coordinate transformation in the IFT point of view. Adopting the equivalence \eqref{IFT=g2}, we obtain the converted action with the full range of coordinates, $T$ and $X$,
\begin{align} \label{SIFT2}
S_{{\rm IFT}} = \int d^{2}X \Big[ -\frac{1}{2} \bar \partial _{\mu}\phi \bar \partial^{\mu}\phi -\frac{1}{2}\bar{m}^{2} (X)\, \phi^{2}\Big]\,, \qquad \bar{m}^{2}(X) = m_0^{2}\,,
\end{align}
where $\bar\partial_{\mu} \equiv \frac{\partial}{\partial X^\mu}$.
Superficially, two IFT actions in \eqref{SIFT1} and \eqref{SIFT2} look different.
Nevertheless, any physical quantities derived from $S_{\rm IFT}[m_0^2e^{2bx};\phi(t,x)]$ and $S_{\rm IFT}[m_0^2;\phi(T,X)]$ should be related by a definite connecting rule, according to our equivalence and general covariance of FTCS.
This leads to an unexpected consequence from the perspective in IFT, which is a definite relation\footnote{From the perspective of IFT, this relation between the exponential mass function and the constant mass can be understood as a Wyel rescaling of the flat metric $\eta$ as $\eta \rightarrow e^{2\omega(x)}\eta\,.$} between the scalar IFT with the exponential mass function and the scalar IFT with constant mass, as displayed in the right side of Fig.~\ref{fig:rel}.
\subsection{Limiting behaviors of mass function}
In this subsection, we investigate limiting behaviors of the mass function $m(x)$ in terms of background metric in FTCS through our equivalence. In the conventional field theory with a constant mass, $m_0^2$ should be taken as a nonnegative value for a nontachyonic behavior. To avoid unnecessary intricacy we focus on $m^2(x)\geq0$. Under this condition, limiting behaviors of mass function such as $m^{2}(x)\rightarrow0$ and $m^2(x)\rightarrow\infty$ would be interesting. In the case of $m^2(x)=0$, scaling symmetry appears in IFT. This symmetry corresponds to the Weyl symmetry in FTCS.
Now, we provide a geometrical interpretation of massless and infinitely-massive points of the mass function $m(x)$ in the scalar IFT for appropriate $x$-values.
First of all, the massless condition ($m^{2}(x_{*})=0$) can be read from (\ref{Ourmet0}) and (\ref{Matrel}) as
\begin{equation} \label{massless}
m^{2}(x) \Big|_{x=x_{*}}= m^{2}_{0}\, e^{2\omega(x)} -2\xi \omega''(x)\Big|_{x=x_{*}}=0\,.
\end{equation}
There are two possibilities to achieve this condition, as far as $m_{0}\xi \neq 0$. One possibility is that neither $e^{2\omega(x)}$ nor $\omega''(x)$ vanishes at $x=x_{*}$.
In this case, $\omega(x)$ should behave as $\omega''(x) \sim e^{2\omega(x)}$, to satisfy the massless condition. By solving (\ref{massless}), we see that $\omega(x)$ near the massless point $x=x_*$ behaves as
\begin{equation} \label{localdS}
e^{\omega (x) } \sim \frac{1}{\cosh \gamma (x+c)}\,, \qquad R\sim 2\gamma^{2}\,, \qquad m(x) \sim \frac{\#}{\cosh \gamma (x+c)}\,,
\end{equation}
or
\begin{equation} \label{localAdS}
e^{\omega (x) } \sim \frac{1}{\sinh \gamma (x+c)}\,, \qquad R\sim -2\gamma^{2}\,, \qquad m(x) \sim \frac{\#}{\sinh \gamma (x+c)}\,,
\end{equation}
where $\gamma$, $c$ and $\#$ are some constants. The massless point is given by $|x_*|\rightarrow \infty$. Without loss of generality, we set $c=0$ translating the $x$-variable in our metric. Since the local curvature is constant in these cases, one can identify these special cases as locally dS or flat, and locally AdS spaces.
The other possibility is that both $e^{2\omega(x)}$ and $\omega''(x)$ independently vanish at the massless point.
Note that this point corresponds to (at least) a coordinate singularity since our metric becomes degenerate at this point as can be seen from our metric in (\ref{Ourmet0}). Recalling that the Ricci scalar for our metric is given by ${\cal R} = -2\omega''(x) e^{-2\omega(x)}$,
we see that the Ricci scalar divergence at the massless point is determined by the scaling behaviors of $e^{2\omega(x)}$ and $\omega''(x)$. If the scaling behaviors at the massless point are given by
\begin{equation} \label{SndPos}
e^{-2\omega(x)}\sim \frac{1}{\epsilon^{2}}\,, \qquad \omega''(x) \sim \epsilon^{2-\alpha}\,, \qquad 0<\alpha<2\,, \quad \epsilon \rightarrow 0\,,
\end{equation}
the Ricci scalar diverges at the massless point and thus this corresponds to a curvature singularity\footnote{This second possibility will be studied in the subsequent work~\cite{Kwon:2022}.}.
The above observations tell us that the massless point should correspond to either a coordinate singularity or a curvature singularity. Our following examples are not included in the case of (\ref{SndPos}) and we restrict our attention to the first possibility.
In fact, one can show that the coordinate singularity in our cases corresponds to a Killing horizon.
This is shown by using a time-like Killing vector, $K=\frac{\partial}{\partial t}$, which always exists for our metric in Eq.~(\ref{Ourmet0}) and whose normalization is taken as $K^{2}\rightarrow -1$ when $e^{2\omega}\rightarrow 1$. In the case of asymptotically AdS or dS spacetime, the normalization of $K$ should be taken appropriately\footnote{The AdS and dS cases are presented in the following.}.
Then, from the definition of a Killing horizon ${\cal H}$, one can see that it is specified by the property $K^{2}|_{\cal H}=0$ which corresponds to $e^{2\omega}=0$.
Even in the locally dS (\ref{localdS}) and the locally AdS (\ref{localAdS}) spacetimes, which correspond to the static patch of dS spacetime and the Rinder patch of AdS spacetime, respectively, there are Killing horizons specified by $e^{2\omega}=0$ . This tells us that the coordinate singularity given by $e^{2\omega}=0$ always corresponds to a Killing horizon in our case.
On the contrary, it is rather simple to describe
the infinitely massive point $m^{2}(x)\rightarrow \infty$
in the corresponding FTCS. The mass formula in~(\ref{Matrel}) tells us that $e^{2\omega}$ or $\omega''$ should diverge at that point. Just like in the massless point case, we focus on the scaling behaviors of $e^{2\omega}$ and $\omega''$ as
\begin{equation} \label{}
e^{2\omega} \sim \frac{1}{\varepsilon^{2}}\,, \qquad \omega'' \sim \varepsilon^{\beta -2}\,, \qquad 0 < \beta \,, \quad \varepsilon \rightarrow 0\,,
\end{equation}
which means that $\omega''$ diverges slowly than $e^{2\omega}$. In this case, the Ricci scalar vanishes at the infinitely massive point. Then, the infinitely massive point is realized as the flat background metric on the corresponding FTCS.
In the following we present various examples which provide some concrete realization of scaling behaviors of $e^{2\omega}$ and $\omega''$. As a first example, we consider the Rindler spacetime which is described by $\omega(x) = bx$. See (\ref{SIFT1}).
In this case, the massless point is given by $x\rightarrow -\infty$ which corresponds to the Rindler horizon, while the infinitely massive point is given by $x\rightarrow \infty$ which corresponds to the future and past null infinities of the Rindler spacetime.
This example is consistent with our scaling arguments in the above.
In the next we show several non-flat examples.
In the case of dS$_{2}$, the Ricci scalar is given by $\mathcal{R}= 2/\ell^{2}$ with the de Sitter radius $\ell$. One can see that
\begin{equation} \label{}
m^{2}(x) = \Big( m^{2}_{0} + \frac{2\xi}{\ell^{2}} \Big)\frac{1}{\cosh^{2}(x/\ell)}\,,
\end{equation}
which leads to positive $m(x)$ within our choice $\xi>0$. The massless condition $m(x)\rightarrow0$ is achieved by $x\rightarrow\pm\infty$ which are de Sitter horizons in our metric.
In the case of AdS$_{2}$, the Ricci scalar is given by $\mathcal{R}= -2/\ell^{2}$ with the anti-de Sitter radius $\ell$. Here we take the range of $x$ as $(0,+\infty)$. In this case, $x=0$ corresponds to the boundary of AdS$_{2}$ and
\begin{equation} \label{}
m^{2}(x) = \Big( m^{2}_{0} - \frac{2\xi}{\ell^{2}} \Big)\frac{1}{\sinh^{2}(x/\ell)}\,,
\end{equation}
where $m(x)$ always becomes positive under our choice $\xi < 0$ and
\begin{equation} \label{}
m(x) \rightarrow \infty \quad \text{as} \quad x\rightarrow 0\,, \qquad m(x) \rightarrow 0 \quad \text{as} \quad x\rightarrow \infty\,.
\end{equation}
Indeed the massless point and the infinitely massive point correspond to the Killing horizon and the flat region, respectively.
Let us consider $(1+1)$-dimensional black hole backgrounds, whose metric is taken as~\cite{Mandal:1991tz,Giddings:1992ff}
\begin{equation} \label{}
ds^{2} = -\frac{dudv}{ 1 - \lambda^{2}uv}\,.
\end{equation}
Under the coordinate transformation
\begin{align}
u = -\frac{1}{b}e^{-b(t-x)}\,, \qquad v = \frac{1}{b}e^{b(t+x)} \,,
\end{align}
we can set it as
\begin{equation} \label{MSWmetric}
ds^{2} = \frac{-dt^2+dx^2}{\frac{\lambda^{2}}{b^2} + e^{-2bx} }\,.
\end{equation}
Then the mass function in IFT is given by
\begin{equation} \label{}
m^{2}(x) = \Big( m^{2}_{0} + \frac{4\xi\lambda^2e^{-2bx}}{\frac{\lambda^2}{b^2}+e^{-2bx}} \Big)\frac{1}{\frac{\lambda^2}{b^2}+e^{-2bx}}\,,
\end{equation}
which is positive for $\xi >0$ and
\begin{equation}
m(x) \longrightarrow
\left\{ \begin{array}{lll}
m_{0}/\lambda& \text{as} & x\rightarrow +\infty \\
~ 0& \text{as} & x\rightarrow -\infty
\end{array} \right.\,. \label{}
\end{equation}
So the massless point corresponds to the black hole horizon and the infinitely massive point does not exist. The limit of $\lambda\rightarrow1$ is nothing but the Witten's $(1+1)$-dimensional black hole~\cite{Witten:1991yr}. Note also that in the limit of $\lambda\rightarrow0$, the metric becomes the Rindler spacetime.
In all the above examples, massless points and infinitely massive points in IFT correspond to Killing horizons and flat regions in FTCS, respectively. Note that the scaling argument is valid for all the above examples.
\section{Quantization}\label{QIFT}
In the previous section, we have discussed the classical equivalence between the scalar IFT and the scalar FTCS in $(1+1)$ dimensions. Based on this equivalence, we would like to give a proposal for the quantization of IFT.
Our proposal includes the canonical quantization for the specific IFT, which is read from the canonical quantization procedure in the FTCS (See Appendix~\ref{AppB}). This proposal tells us how to quantize the specific IFT. As an example, we consider the quantization of the specific IFT in which the mass function is given in an exponential form.
Yet, it has been known that the canonical quantization of FTCS is insufficient to provide a comprehensive general framework, at least conceptually. A more adequate and conceptually favorable framework is based on an algebraic construction, which is called an algebraic formulation of QFTCS. Our proposal actually is made on this framework. Some aspects in this framework are given in the next section.
\subsection{Proposal}
The quantization of IFT is not well established in a comprehensive way contrary to the conventional homogeneous field theory which possesses the Poincar\'{e} symmetry. Concretely speaking, the Poincar\'{e} invariant vacuum does not exist because the position-dependent mass and couplings break the Poincar\'{e} symmetry explicitly. This situation is somewhat similar to the case of FTCS in which the preferred vacuum state cannot be selected because of the absence of the Poincar\'{e} symmetry in a generic curved spacetime background. This naturally motivates us to adopt similar quantization method in the FTCS to the IFT. Triggered by this motivation and elevating the classical equivalence to the quantum level, we arrive at the following proposal to the quantization of IFT in $(1+1)$ dimensions:
\begin{equation}
\boxed{\text{IQFT $\iff$ QFTCS}}\,.
\end{equation}
One implication of this proposal is that the canonical quantization in FTCS is transcribed to that of IFT. As a concrete example, we will consider the field theory on the Rindler spacetime \eqref{Rmetric1} and the corresponding IFT with exponential mass function in the next subsection.
This proposal is not just about canonical quantization, but has broader implications.
As is well-known, the difficulty in introducing the preferred vacuum state on curved spacetime leads to an algebraic formulation of QFTCS.
Briefly speaking, the algebraic formulation starts from appropriate algebraic relations among quantum fields (local algebra) with some appropriate properties or axioms: isotony, covariance, locality, and causality, existence of dynamics. And then, algebraic states are introduced as normalized positive linear functionals on the field algebra. Through the so-called Gelfand-Naimark-Segal (GNS) construction, we can construct a relevant Hilbert space from the algebraic states. Especially, the free field case can be formulated in a rigorous way. See~\cite{Haag:1992hx,Wald:1995yp,Yngvason:2004uh,Halvorson:2006wj,Hollands:2009bke,Benini:2013fia,Hollands:2014eia,Khavkine:2014mta,Fredenhagen:2014lda} for reviews on algebraic formulation of QFTCS.
Our proposal incorporates these aspects of QFTCS into the quantum equivalence between the scalar IQFT and the scalar QFTCS.
We would like to emphasize that the classical equivalence in the previous section does not automatically warrant its quantum version. As is well-known, the ordering ambiguity in the operator elevation of classical variables results in the trivial example of inequivalent quantum theories with the classical equivalence. Therefore, our proposal should be taken as one possible way to quantize IFT and may be tested only by experiments.
Of course, this quantum equivalence should be taken with some caution, since a generic IFT cannot be realized even classically by a FTCS in $(1+1)$ dimensions. Therefore, the above quantum equivalence should be taken only when the classical equivalence holds. Though our proposal, which is applicable only in the form of metric \eqref{Ourmet0}, is not completely generic, it leads to a concrete way to compute the quantum effects in the IFT. In the following, we present a canonical quantization method in a specific IFT model before giving an algebraic Hadamard approach in the next section.
\subsection{IQFT with an exponential mass function}
According to our proposal, the quantization of IFT with an exponential mass function is achieved by the canonical quantization on the Rindler spacetime. First we present a brief summary of this procedure. The Klein-Gordon equation of the IFT with the mass function $m(x)=m_0e^{bx}$ is given by
\begin{equation}
\left(\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2} +m_0^2e^{2bx} \right)\phi(t,x)=0\,,
\end{equation}
which is the same form for the scalar field in the Rindler spacetime. This result is a trivial consequence from our equivalence. Thus we can apply all the results in the Rindler spacetime to the IFT with the exponential mass function. In particular, the mode solution of the equation of motion on the Rindler spacetime is given by~\cite{Fulling:1972md,Takagi:1986kn,Fulling:1989nb}
\begin{align}
&u_{\Omega}(t,x)=\frac{1}{\sqrt{2\Omega}}\theta(\rho)h_{\Omega}(\rho)e^{-i\Omega \eta}\,,
\\
&\eta\equiv bt\,,\quad \rho\equiv b^{-1}e^{bx}\,,\label{rhoeta}
\end{align}
where $\Omega$ is a positive energy eigenvalue of the mode, $\theta$ is the step function, and $h$ is given by the modified Bessel function of second kind as
\begin{equation}
h_{\Omega}(\rho)=R^*\sqrt{\frac{2}{\pi}}\frac{K_{i\Omega}(m_0\rho)}{|\Gamma(i\Omega)|}\,,\qquad R^*\equiv \left(\frac{m_0}{2b}\right)^{2i\Omega} \left(\frac{\Gamma(-i\Omega)}{|\Gamma(i\Omega)|}\right)^2\,.
\end{equation}
These mode solutions can be used for the IFT.
As is well-known, it is straightforward to perform the canonical quantization in the Rindler spacetime. A brief explanation of the canonical quantization on curved spacetime is given in Appendix \ref{AppB}. The equal time commutator
\begin{align}
&[ \phi(t,x), \pi (t, y) ] = i\delta (x-y)\,
\end{align}
gives the commutation relation $[b_{\Omega},b^{\dagger}_{\Omega'}]=\delta(\Omega-\Omega')$, where $b_{\Omega}$ and $b^{\dagger}_{\Omega}$ are the annihilation and the creation operators, respectively. And then the scalar field is expanded as:
\begin{equation}
\phi(t,x)=\int_{0}^{\infty}d\Omega\,\Big(b_{\Omega}\,u_{\Omega}(t,x)+b_{\Omega}^{\dagger}\,u_{\Omega}^*(t,x)\Big)\,.
\end{equation}
The Rindler vacuum, $|0\rangle_{\rm R}$, is defined by
\begin{equation}
b_{\Omega}|0\rangle_{\rm R}=0\,,
\end{equation}
from which we can construct the Fock space with creation operators, $b^{\dagger}_{\Omega}$.
All these results in QFTCS are transcribed to IQFT setup. According to our quantum equivalence, the Rindler vacuum, $|0\rangle_{\rm R}$, and the annihilation/creation operators, $b_{\Omega}/b_{\Omega}^{\dagger}$, are identified with those of IQFT.
So the Fock spaces of QFTCS and IQFT are identical and so does the two-point functions. Since we are considering free field theories, the two-point functions of both sides determine any $n$-point functions. Therefore all $n$-point functions of two theories are the same, which is the meaning of our quantum equivalence. To emphasize that the quantization is done in the context of the scalar IFT, we denote the vacuum of the scalar IQFT with an exponential mass function as $|\underline{0}\rangle_{\rm IFT}$.
In the next section, we explore further aspects of the quantum equivalence focusing on the Unruh effect.
\section{Quantum Aspects of IQFT}
In this section, we present a quantization of IFT based on the algebraic method which supersedes the canonical quantization considered in the previous section. One feature of the algebraic approach is that it democratically treats the pure and the mixed states in the canonical quantization.
From our proposal that the $(1+1)$-dimensional IQFT is equivalent to the $(1+1)$ dimensional QFTCS, it is natural to anticipate that the quantization and renormalization methods in QFTCS should be carried over to IQFT in a rather straightforward manner. After a brief review on the Hadamard method in QFTCS, we apply this to IQFT to see some quantum effects including the Unruh effect.
\subsection{Hadamard renormalization}
In the algebraic formulation of QFTCS, a physically important class of quantum states are given by Gaussian Hadamard states which may serve as substitute of the preferred vacuum state. Hadamard state is defined as an algebraic state satisfying the Hadamard condition which is motivated by some reasonable physical considerations. The condition includes that the short distance singularity structure of the $n$-point functions of the Hadamard state on curved spacetime should be given by that of the $n$-point functions of the vacuum state in the Minkowski spacetime, the ultra-high energy mode of quantum fields resides essentially in the ground state, and the singular structure of the $n$-point functions should be of positive frequency type~\cite{Fulling:1978ht,Fulling:1981cf,Kay:1988mu,Radzikowski:1996pa}.
Contrary to the ordinary vacuum state, the Hadamard state is not unique for a given background spacetime but forms a class in general. In the case of a Gaussian Hadamard state\footnote{A Gaussian state, which is also called a quasi-free state, is defined by the condition that the connected $n$-point functions of the state vanish, or any $n$-point functions can be obtained from $1$- and $2$-point functions.}, one can obtain a Fock space representation of the algebra of quantum fields and can identify the Gaussian Hadamard state with the Fock space vacuum. In this way, one can see that some well-known vacua belong to the Hadamard class. The Hadamard method encompasses the usual Fock space canonical quantization and implements appropriately relevant requirements such as general covariance of stress tensor, while it connects unitarily inequivalent representations of the algebras of observables.~\cite{Kay:1988mu,Wald:1995yp,Hollands:2014eia}.
Under this scheme, the Gaussian Hadamard state, $\omega_{\rm H}$ is defined by the renormalized two point function of scalar field $\phi$ as\footnote{The normalization condition of an algebraic state $\omega$, is $\omega(\mathbf{1})=1$, where $\mathbf{1}$ is an identity element in the field algebra. }
\begin{equation} \label{HadamardFTCS}
\omega_{\rm H}\big(\phi (\mathbf{x}) \phi (\mathbf{x}')\big) = F(\mathbf{x},\mathbf{x}') - H(\mathbf{x},\mathbf{x}') \,,
\end{equation}
where $F(\mathbf{x},\mathbf{x}')$ is an unrenormalized two point function known as the Hadamard function, and the function $H(\mathbf{x},\mathbf{x}')$ is so called as the Hadamard parametrix~\cite{Hadamard:1952}. In the following, $\omega_{\rm H}(\phi (\mathbf{x}) \phi (\mathbf{x}'))$ is also denoted by $\langle \phi (\mathbf{x}) \phi (\mathbf{x}')\rangle_{\rm H} $. Here, $H(\mathbf{x},\mathbf{x}')$ is a local covariant function of the half of squared geodesic length, $\sigma(\mathbf{x},\mathbf{x}')$ between two points $\mathbf{x}$ and $\mathbf{x}'$, written in terms of the metric and the curvature. The Hadamard function $F(\mathbf{x},\mathbf{x}')$ is symmetric and satisfies
\begin{equation} \label{kgeq}
(-\Box +m^{2} + \xi {\cal R}) F(\mathbf{x},\mathbf{x}') = \delta (\mathbf{x}-\mathbf{x}')\,,
\end{equation}
which can also be represented by a real part of the `positive frequency' $2$-point Wightman function, $\textrm{Re}~ G^{+}(\mathbf{x},\mathbf{x}')$. It is known that $H(\mathbf{x},\mathbf{x}')$ takes the same form for any Hadamard states. Then, the Hadamard renormalization is achieved by subtracting the singular part $H(\mathbf{x},\mathbf{x}')$ from the function $F(\mathbf{x},\mathbf{x}')$.
Explicitly, the Hadamard parametrix $H$ is given by
\begin{align} \label{}
H(\mathbf{x},\mathbf{x}') &= \alpha_{D} \frac{U(\mathbf{x},\mathbf{x}')}{\sigma^{\frac{D}{2}-1}(\mathbf{x},\mathbf{x}')} + \beta_{D} V(\mathbf{x},\mathbf{x}') \ln \left(\mu^2\, \sigma(\mathbf{x},\mathbf{x}')\right) \quad\text{for even $D$}\,,\nonumber \\
H(\mathbf{x},\mathbf{x}') &= \alpha_{D} \frac{U(\mathbf{x},\mathbf{x}')}{\sigma^{\frac{D}{2}-1}(\mathbf{x},\mathbf{x}')} \quad\hskip4.8cm \text{for odd $D$}\,,
\end{align}
where $\alpha_{D}, \beta_{D}$ are numerical constants depending on the dimension $D$ and $\mu$ is a certain mass scale introduced from the dimensional reason. Symmetric bi-scalars $U(\mathbf{x},\mathbf{x}')$ and $V(\mathbf{x},\mathbf{x}')$, which are regular for $\mathbf{x}' \rightarrow \mathbf{x}$, are universal geometrical objects independent of any Hadamard states. They can be expanded in terms of $\sigma (\mathbf{x}, \mathbf{x}')$ as
\begin{equation} \label{}
U(\mathbf{x},\mathbf{x}') = \sum_{n=0}^{D/2 - 2} U_n (\mathbf{x},\mathbf{x}') \sigma^n (\mathbf{x}, \mathbf{x}') , \qquad V(\mathbf{x},\mathbf{x}') = \sum_{n=0}^{+ \infty} V_n (\mathbf{x},\mathbf{x}') \sigma^n (\mathbf{x}, \mathbf{x}') .
\end{equation}
Here, $U_n (\mathbf{x},\mathbf{x}')$ and $V_n (\mathbf{x},\mathbf{x}')$ can be completely determined by recursion relations and boundary conditions, which are obtained by comparing the power of $\sigma$ on the both sides of the equation, $(-\Box +m^{2} + \xi {\cal R}) H(\mathbf{x},\mathbf{x}') = \delta (\mathbf{x}-\mathbf{x}') $ ~\cite{Wald:1977up}.
Concretely, in two dimensions $\alpha_2=0$ and the Hadamard parametrix becomes
\begin{equation}
H(\mathbf{x},\mathbf{x}') = \frac{V(\mathbf{x},\mathbf{x}')}{2\pi} \ln \left(\mu^2\, \sigma(\mathbf{x},\mathbf{x}') \right)\,,
\end{equation}
where the bi-scalar $V(\mathbf{x},\mathbf{x}') $ is given by~\cite{Decanini:2005eg}
\begin{equation}
V(\mathbf{x},\mathbf{x}') = -1 - \frac{1}{24} \mathcal{R} g_{\mu \nu} \nabla^{\mu} \sigma \nabla^{\nu} \sigma - \frac{1}{2} \left( m_0^2 + \xi - \frac{1}{6} \right) \mathcal{R} \sigma + {\cal O}(\sigma^{3/2}).
\end{equation}
Since the renormalized stress tensor enters in various kinds of semi-classically improved energy conditions, it has been one of important topics in QFTCS.
Based on the above procedure, one can obtain the renormalized stress tensor by acting an appropriate differential bi-vector operator, ${\cal T}_{\mu\nu'}$ on the renormalized $2$-point function as
\begin{equation} \label{StressTensor}
\langle T_{\mu\nu} (\mathbf{x}) \rangle_{\rm H} = \lim_{\mathbf{x}'\rightarrow \mathbf{x}} {\cal T}_{\mu\nu'} \langle \phi (\mathbf{x}) \phi (\mathbf{x}')\rangle_{\rm H}\,.
\end{equation}
For instance, in our two dimensional case~\eqref{GraLag2}, the differential bi-vector is given by
\begin{align} \label{DiffBiV}
{\cal T}_{\mu\nu'} = & (1-2\xi) \partial_{\mu}\partial_{\nu'} +\Big(2\xi- \frac{1}{2}\Big)g_{\mu\nu'} g^{\alpha\beta'}\partial_{\alpha}\partial_{\beta'} -\frac{1}{2}g_{\mu\nu'}m_0^{2} \nonumber \\
& -2\xi\delta^{\mu'}_{\mu}\partial_{\mu'}\partial_{\nu'} + 2\xi g_{\mu\nu'}g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta} +\xi \Big(\mathcal{R}_{\mu\nu'} - \frac{1}{2} g_{\mu\nu'}\mathcal{R} \Big)\,.
\end{align}
Though there still remain some ambiguities in this subtraction method, one can construct an essentially unique stress tensor under Wald's axiom. The ambiguous terms in stress tensor are written in geometrical quantities~\cite{Wald:1995yp,Moretti:2001qh}.
Sometimes, the above procedure has been known as a `point-splitting method'~\cite{Birrell:1982ix}, while it is now regarded as more reliable results on a firm mathematical ground. We will apply this well-established prescription in QFTCS to IQFT in the following subsection.
Some comments are in order.
The stress tensor in QFTCS is expected to satisfy some natural axioms~\cite{Wald:1978pj}.
For instance, the quantum expectation value of the stress tensor should be local, covariant, covariantly-conserved, and etc.
Now, it is widely believed that the so-called Hadamard renormalization is well suited to this purpose. For instance, a global Hadamard state consistent with the Hadamard renormalization would lead to
\begin{equation} \label{Hadamard}
\langle T_{\mu\nu} (\mathbf{x}) \rangle_{\rm H} = \frac{\partial y^{\alpha}}{\partial x^{\mu}} \frac{\partial y^{\beta}}{\partial x^{\nu}} ~ \langle T_{\alpha\beta}(\mathbf{y}) \rangle_{\rm H} \,,
\end{equation}
where it may be noted that the vacuum states in each coordinate $\mathbf{x}$ and $\mathbf{y}$ do not need to be realized on the same Fock space vacuum in general.
Note that the Minkowski vacuum is a global Hadamard state on the whole Minkowski space, while the Rindler vacuum is not a global Hadamard one on the whole Minkowski space since it diverges on the Rindler horizon. However, the Rindler vacuum is a Hadamard state on the Rindler wedge. To proceed, let us denote the Poincar\'{e} invariant Minkowski vacuum $|0\rangle_{\rm M}$ and the Rindler vacuum $|0\rangle_{\rm R}$ on each Fock spaces. As is well-known, the Minkowski vacuum is realized as a mixed state on the Rindler wedge.
To avoid cluttering the discussion, we repeat the expression in \eqref{Hadamard} in terms of the algebraic state $\omega_{\rm M}$ which is a Gaussian Hadamard state in the Minkowski spacetime as
\begin{equation} \label{omegaRel}
\omega_{\rm M}(T^{\rm M}_{\mu\nu}(\mathbf{x})) = \frac{\partial y^{\alpha}}{\partial x^{\mu}} \frac{\partial y^{\beta}}{\partial x^{\nu}} ~ \omega_{\rm M}( T^{\rm R}_{\alpha\beta}(\mathbf{y}) )\,,
\end{equation}
where $T^{\rm M}_{\mu\nu}(\mathbf{x})$ denotes the stress tensor on the Minkowski spacetime and $T^{\rm R}_{\alpha\beta}(\mathbf{y})$ does from the Rindler one.
In the Fock space representation of this state on the Minkowski spacetime is give by
\begin{equation} \label{}
\omega_{\rm M}(T^{\rm M}_{\mu\nu}(\mathbf{x}))={}_{\rm M}\langle 0| T^{\rm M}_{\mu\nu} (\mathbf{x})|0 \rangle_{\rm M}\,,
\end{equation}
which is represented by a mixed state on the Rindler patch. In the standard convention, we take the Minkowski vacuum energy to be zero which means that $\omega_{\rm M}(T^{\rm M}_{\mu\nu})=0$.
Now we can consider the Gaussian Hadamard state on the right Rindler wedge, $\underline{\omega}_{\rm R}$ which is not a global Hadamard state on the Minkowski spacetime, since it diverges on the Rindler horizon. Just like the algebraic state $\omega_{\rm M}$, in the Fock space representation of $\underline{\omega}_{\rm R}$ on the Rindler spacetime, $\underline{\omega}_{\rm R}$ is given by the Rindler vacuum as
\begin{equation} \label{}
\underline{\omega}_{\rm R}(T^{\rm R}_{\mu\nu}(\mathbf{y}))=\langle T_{\mu\nu}^{\rm R}(\mathbf{y}) \rangle_{\rm H}^{\rm R} ={}_{\rm R}\langle 0| T^{\rm R}_{\mu\nu} (\mathbf{y})|0 \rangle_{\rm R}<0\,.
\end{equation}
which is consistent with $\omega_{\rm M}(T^{\rm R}_{\mu\nu})=0$. This is also consistent with the Unruh effect.
Another way to understand the Unruh effect in terms of stress tensor is to consider a normal ordering prescription. Let us define a normal ordering of stress tensor operator in the Minkowski spacetime by the subtraction of its vacuum expectation value as
\begin{equation} \label{}
:T^{\rm M}_{\mu\nu}(\mathbf{x}) : \;\equiv T^{\rm M}_{\mu\nu}(\mathbf{x}) - {}_{\rm M} \langle 0| T^{\rm M}_{\mu\nu}(\mathbf{x}) | 0\rangle_{\rm M}~ {\bf 1}\,,
\end{equation}
where ${\bf 1}$ denotes the identity operator. Our choice of $\omega_{\rm M}(T^{\rm M}_{\mu\nu})=0$ means $:T^{\rm M}_{\mu\nu}(\mathbf{x}) :=\;T^{\rm M}_{\mu\nu}(\mathbf{x})$.
By taking the same definition of a normal ordering in the Rindler spacetime,
\begin{equation} \label{}
:T^{\rm R}_{\mu\nu}(\mathbf{y}) : \; = T^{\rm R}_{\mu\nu}(\mathbf{y}) - {}_{\rm R} \langle 0| T^{\rm R}_{\mu\nu}(\mathbf{y}) | 0\rangle_{\rm R}~ {\bf 1}\,,
\end{equation}
one can see that
\begin{equation} \label{}
\omega_{\rm M}( :T^{\rm R}_{\mu\nu}(\mathbf{y}) : ) = \omega_{\rm M}( T^{\rm R}_{\mu\nu}(\mathbf{y})) - {}_{\rm R} \langle 0| T^{\rm R}_{\mu\nu}(\mathbf{y}) | 0\rangle_{\rm R}= - {}_{\rm R} \langle 0| T^{\rm R}_{\mu\nu}(\mathbf{y}) | 0\rangle_{\rm R} ~ > ~ 0\,,
\end{equation}
where we used $\omega_{\rm M}(T^{\rm R}_{\mu\nu})=0$. This is another way of explanations for the Unruh effect.
\subsection{Interpretation in IQFT}
In this subsection, we focus on the IQFT with the exponential mass function, which corresponds to the quantum field theory on the Rindler spacetime background~\eqref{Rmetric1}. According to our quantum equivalence, all the construction of previous subsection can be transcribed to the scalar IQFT. Especially, we consider the stress tensor construction in the scalar IQFT.
The procedure of the construction goes as follows. The vacuum expectation value of stress tensor in QFTCS is given by (recall that $V(\mathbf{x},\mathbf{x}')=-1$ in the flat case)
\begin{equation} \label{}
\langle T_{\mu\nu}\rangle_{\rm H} = \lim_{\mathbf{x}'\rightarrow \mathbf{x}} {\cal T}_{\mu\nu'} \bigg[ F(\mathbf{x},\mathbf{x}') + \frac{1}{4\pi}\ln \left( \mu^{2}\sigma(\mathbf{x},\mathbf{x}') \right) \bigg]\,.
\end{equation}
We are interested in the counterpart in the scalar IQFT of the Unruh effect in the scalar QFTCS by setting $\xi=0$ in \eqref{GraLag2}.
To simplify the description, we take the limit of $m_0\rightarrow0$. Because of the infra-red divergence, the $(1+1)$-dimensional scalar theory with $m_0=0$ is not well-defined in the strict sense~\cite{Coleman:1973ci}. However, in ~\cite{Takagi:1986kn}, the limit of $m_0\rightarrow0$ is carefully taken into account to see the Unruh effect in the massless case. In the context of our quantum equivalence, all the consequences of QFTCS including the results and the process of the limit $m_0\rightarrow0$ are carried over to IQFT.
The Hadamard function in the limit of $m_0\rightarrow0$ is given by~\cite{Dowker:1978aza,Moretti:1995fa}
\begin{align} \label{}
F(\mathbf{x},\mathbf{x}') &= \frac{1}{2}\,{}_{\rm R}\langle0|\{\phi(\mathbf{x}),\phi(\mathbf{x}')\}|0\rangle_{\rm R} \nonumber \\
&= \frac{1}{4\pi}\ln \frac{1}{\rho\rho' |\alpha^{2} -(\eta-\eta')^{2}|}\,, \qquad \cosh \alpha \equiv 1+ \frac{(\rho-\rho')^{2}}{2\rho\rho'}\,,
\end{align}
where $\rho$ and $\eta$ have been introduced in~\eqref{rhoeta}. In the IFT coordinates the Hadamard function becomes
\begin{align}
F_{\rm IFT}(\mathbf{x},\mathbf{x}')&= \frac{1}{2}\,{}_{\rm IFT}\langle\underline{0}|\{\phi(\mathbf{x}),\phi(\mathbf{x}')\}|\underline{0}\rangle_{\rm IFT} \nonumber \\
&=-\frac{b}{4\pi}(x+x')-\frac{1}{4\pi}\ln \Big|(x-x')^2-(t-t')^2 \Big|\,,
\end{align}
which can be interpreted as the Hadamard function in the IFT with the exponential mass function.
By using the explicit form of the squared geodesic distance in this case,
\begin{equation}
2\sigma = \Big| (\rho-\rho')^{2}-\rho\rho'\Big(2\sinh\frac{\eta-\eta'}{2}\Big)^{2} \Big|=\frac{4}{b^2}e^{b(x+x')}\Big|\sinh^2\frac{b}{2}(x-x')-\sinh^2\frac{b}{2}(t-t') \Big|\,,
\end{equation}
one can obtain the following expanded expression in the limit of $\mathbf{x}\rightarrow\mathbf{x}'$,
\begin{equation} \label{}
F_{\rm IFT}(\mathbf{x},\mathbf{x}') + \frac{1}{4\pi}\ln 2\sigma
=\frac{b^2}{48\pi}\big[(x-x')^2+(t-t')^2\big]+\cdots \,,
\end{equation}
where the renormalization scale $\mu$ is removed because it does not affect our result.
From our quantum equivalence, we anticipate that the Unruh-like effect in the scalar IQFT with the exponential mass function. To see this effect, we apply the Hadamard method borrowed from \eqref{StressTensor} to our case, resulting in
\begin{equation} \label{}
\langle T_{\mu\nu}^{\rm IFT} (\mathbf{x}) \rangle_{\rm H}^{\rm IFT} \equiv \lim_{\mathbf{x}'\rightarrow \mathbf{x}} {\cal T}^{\rm IFT}_{\mu\nu'} \langle \phi (\mathbf{x}) \phi (\mathbf{x}')\rangle_{\rm H}^{\rm IFT}\,,
\end{equation}
where the Hadamard state in IFT is defined in the same way with FTCS in~\eqref{HadamardFTCS}.
And the differential bi-vector,
\begin{align} \label{}
{\cal T}_{\mu\nu'}^{\rm IFT} = & \partial_{\mu}\partial_{\nu'} -\frac{1}{2} \eta_{\mu\nu'} \eta^{\alpha\beta'}\partial_{\alpha}\partial_{\beta'} -\frac{1}{2}\eta_{\mu\nu'}m^{2}(x) \,,
\end{align}
comes from the classical expression of the ``stress tensor'' in \eqref{Tclassical} with $\xi=0$. The straightforward computation in IQFT leads to the vacuum expectation value of ``stress tensor'' in the form of
\begin{equation} \label{NegE1}
\langle T_{tt}^{\rm IFT}\rangle_{\rm H}^{\rm IFT}=\langle T_{xx}^{\rm IFT}\rangle_{\rm H}^{\rm IFT}=-\frac{b^2}{24\pi}\,,\quad \langle T_{\rm IFT}{}^{\mu}_{~\mu} \rangle_{\rm H}^{\rm IFT} =0\,.
\end{equation}
This result is the counterpart of the well-known result~\cite{Unruh:1976db,Takagi:1986kn,Crispino:2007eb,Fulling:2018lez} for the Unruh effects for minimally coupled massless scalar field in $(1+1)$-dimensional Rindler spacetime background,
\begin{equation} \label{NegE2}
\langle T_{\eta\eta}^{\rm R}\rangle_{\rm H}^{\rm R} = -\frac{1}{24\pi}\,, \qquad \langle T_{\rho\rho}^{\rm R}\rangle_{\rm H}^{\rm R} = -\frac{1}{24\pi}\frac{1}{\rho^{2}}\,, \qquad \langle T_{\rm R}{}^{\mu}_{~\mu} \rangle_{\rm H}^{\rm R} =0\,.
\end{equation}
Indeed by the coordinate transformation in~\eqref{rhoeta}, \eqref{NegE2} is covariantly transformed to \eqref{NegE1}.
Note that the Rindler vacuum energy is negative relative to the Minkowski vacuum energy. So it is not strange that the energy of the vacuum in the IQFT with the exponential mass function is negative in \eqref{NegE1}.
All the above results can be translated to the Unruh-like effect in IQFT. Therefore we anticipate an experimental verification of the Unruh effect in the setup of IQFT. There are many attempts to capture the Unruh effect by using hydrodynamical analog of the Schwarzschild metric~\cite{Unruh:1981,Chen:1998kp} or in high energy experiments~\cite{Lynch:2019hmk} which have many technical hurdles to overcome. If one can engineer a $(1+1)$-dimensional condensed matter system realizing the scalar IFT with an exponential mass function, it would be easier to verify the Unruh effect experimentally.
\section{Conclusion}
IFT does not have the Poincar\'{e} symmetry and so it disallows the conventional quantization method. In order to overcome such difficulties, we suggested a kind of dual description, which allows a quantization of IFT. As a first step toward quantization of IFT, the classical equivalence is established between a scalar IFT and a scalar FTCS through the explicit expressions of actions and the equations of motion in $(1+1)$-dimensions.
Along this line of reasoning, we proposed a generalized ``stress tensor'' in IFT motivated from the counterpart in FTCS. This ``stress tensor'' is conserved by using a covariant derivative newly introduced in IFT.
Since we focused on a free scalar IFT, only the meaningful parameter function in the action is the mass function. Thus some details about limiting behaviors of mass functions in IFT are explored.
We have shown that the massless point of the mass function in IFT corresponds to the horizon of the background spacetime in FTCS.
In this regard, we expect that physical properties on the horizon would be related to those of massless point of $(1+1)$-dimensional scalar IFT.
As is well-known, FTCS enjoys general covariance. When the classical equivalence is combined with this general covariance, a new connection among IFT's is obtained. This connection is given by the procedure of ``equivalence --- general covariance --- equivalence''. See Fig.~\ref{fig:rel}.
Based on the classical equivalence, we proposed a quantum equivalence of QFTCS and IQFT. As an example we have studied IQFT with an exponential mass function which is shown to be equivalent with quantum field theory on the Rindler spacetime.
Especially, we have identified the Unruh-like effect in IQFT.
Along this line, it is natural to consider other patch of the Minkowski spacetime shown in region III in Fig.~\ref{fig:geo}. We can take a coordinate transformation in this patch as
\begin{align}
T-X=\frac{1}{b}e^{b(\bar{t}-\bar{x})}\,,\qquad T+X=\frac{1}{b}e^{b(\bar{t}+\bar{x})}\,.
\end{align}
In this case the mass function depends on the time-like coordinate $\bar{t}$: $m^2(\bar{t})=m_0^2e^{2b\bar{t}}$. See Fig.~\ref{fig:geo}. Therefore two differently looking IFTs with mass functions, $m(x)$ and $m(\bar{t})$, are related by general covariance. It would be interesting to study the relation between two IFTs at quantum level.
There are many open issues related to our work to be pursued. We expect that our equivalence for scalar field theories can be extended to other field theories: fermion, gauge, tensor, and higher spin ones. Thus the extension to supersymmetric field theory would be possible. The extension to a higher dimensional case is another important future direction. It would also be interesting to explore finite temperature effects in IQFT. Another interesting direction is to include interactions in IFT. In conjunction with condensed matter physics, it is desirable to study non-relativistic limit of IQFT. One interesting subject is to implement IQFT in the context of AdS/CFT correspondence. Since the ``connection path'' of IFT's given by ``equivalence --- general covariance --- equivalence'' may not be unique, the (in)dependence of the path from the view point of IFT needs to be studied further.
From a more provisional point of view, it would be interesting to identify the type of von Neumann algebra factors for a local algebra of IFT, which is related to a local algebra in QFTCS in our setup~\cite{Chandrasekaran:2022cip}.
According to our quantum equivalence, one can ask how to realize the information loss problem in the view point of IQFT. That may be related to understand the Hawking radiation in the context of IQFT. In this regard, we guess that the CGHS model~\cite{Callan:1992rs} becomes a good test ground for this physics.
\vskip 1cm
\section*{Acknowledgments}
We appreciate conversations and discussions with Dongsu Bak, Seungjoon Hyun, Chanju Kim, Kyung Kiu Kim, Wontae Kim, Yoonbai Kim, Miok Park, and Driba D. Tolla.
This work was supported by the National Research Foundation of Korea(NRF) grant with grant number NRF-2019R1F1A1056815(O.K.), NRF-2020R1A2C1014371(O.K. and J.H.), \\ NRF-2020R1C1C1012330(S.-A.P.),
NRF-2021R1A2C1003644(S.-H.Y.) and supported by Basic Science Research Program through the NRF funded by the Ministry of Education 2020R1A6A1A0304787(S.-H.Y. and J.H.).
\newpage
\begin{center} {\Large \bf Appendix}
\end{center}
\begin{appendix}
\section{Isometries in $(1+1)$-dimensional Background}\label{AppA}
In this Appendix, we summarize some formulae used in the main text. For the two-dimensional metric,
\begin{align}
ds^2=e^{2\omega(x)}(-dt^2+dx^2)\,,
\end{align}
the non-vanishing Christoffel symbols are given by
\begin{equation} \label{}
\Gamma^{t}_{tx}= \Gamma^{t}_{xt}= \Gamma^{x}_{xx} = \Gamma^{x}_{tt}= \omega'(x)\,,
\end{equation}
where ${}'$ denotes the differentiation with respect to $x$.
In this geometry, the Ricci tensor and the curvature scalar are given by
\begin{align}
R_{\mu\nu} = -g_{\mu\nu}~ \omega''\,, \qquad R=-2e^{-2\omega}\omega''\,.
\end{align}
Now we show that the spacetime described by the above metric admits only a time-like Killing vector, excepting dS, AdS, and Minkowski spacetimes which have three Killing vectors.
Killing condition on this background $\nabla_{(\mu} \xi_{\nu)}=0$ becomes
\begin{equation} \label{}
\xi^{x} = C(t) e^{-\omega(x)}\,, \qquad (\xi^{x})^{\Large\boldsymbol{\cdot}}-(\xi^{t})' =0\,, \qquad (\xi^{t})^{\Large\boldsymbol{\cdot}} + \omega' \xi^{x}= (\xi^{t})^{\Large\boldsymbol{\cdot}} - (\xi^{x})' =0\,,
\end{equation}
where $\Large\boldsymbol{\cdot}$ denotes the differentiation with respect to $t$. This condition leads to
\begin{equation} \label{}
\frac{\ddot{C}}{C} = \frac{(e^{-2\omega})''}{e^{-2\omega}} = A_{0} = const. \quad \text{if} \quad C\neq 0\,.
\end{equation}
When $C=0$, one obtains $\xi^{\mu} = (1,0)$ or $\xi = \partial_{t}$, up to normalization. In the case of $C\neq 0$,
we can solve the differential equation $ \frac{(e^{-2\omega})''}{e^{-2\omega}} = - A_{0}$, which leads to
\begin{equation} \label{}
e^{-\omega} =
\left\{ \begin{array}{lll }
D_{c}\cosh Bx + D_{s}\sinh Bx \,, & \text{when} & A_{0} = B^{2} > 0 \\
D\cos B(x-x_{0})\,, & \text{when} & A_{0} = - B^{2} < 0 \\
D_{1}x + D_{2}\,, & \text{when} & A_{0} = 0
\end{array} \right. \,,
\end{equation}
where $D_{c/s}, D, D_{1/2}$, and $x_{0}$ are integration constants. One can check that upper two cases ($A_{0}$ is positive or negative) correspond to dS$_{2}$ and AdS$_{2}$, respectively. While the last one with $D_{1}=0$ corresponds to the Minkowski spacetime. If $D_{1}\neq 0$, then there is a singularity.
This computation tells us that there is no other Killing vector except for $\xi = \partial_{t}$ ($C=0$ case) for a generic non-singular metric.
As an example, let us consider the case of $A_{0} =B^{2}$ with $D_{c}=0$. The independent Killing vectors up to normalization are obtained as
\begin{equation} \label{}
\xi = \xi^{t}\partial_{t} + \xi^{x}\partial_{\xi} =
\left\{ \begin{array}{l}
\cosh Bx\sinh Bt~ \partial_{t} + \sinh Bx \cosh Bt~ \partial_{x} \\
\cosh Bx\cosh Bt~ \partial_{t} + \sinh Bx \sinh Bt~ \partial_{x} \\
\partial_{t}
\end{array} \right. \,.
\end{equation}
One can check that these three Killing vectors form a $SO(2,1)$ algebra. Indeed, performing the coordinate transformation $r=r_{H}\coth Bx$, we obtain the Rindler wedge of AdS$_{2}$ geometry.
In the case of $A_{0} =B^{2}$ with $D_{s}=0$, one obtains
\begin{equation} \label{}
\xi = \xi^{t}\partial_{t} + \xi^{x}\partial_{\xi} =
\left\{ \begin{array}{l}
\sinh Bx\sinh Bt~ \partial_{t} + \cosh Bx \cosh Bt~ \partial_{x} \\
\sinh Bx\cosh Bt~ \partial_{t} + \cosh Bx \sinh Bt~ \partial_{x} \\
\partial_{t}
\end{array} \right. \,,
\end{equation}
which corresponds to the Killing vectors in the static path of dS$_{2}$ spacetime.
\vskip1cm
\section{Canonical Quantization in Curved Spacetime} \label{AppB}
In this Appendix, we review the quantization procedure in a $(1+1)$-dimensional curved background~\cite{Birrell:1982ix,Jacobson:2003vx}. We consider the quadratic action \eqref{GraLag2} for the real scalar field with the coupling $\xi$ on a background geometry. The equation of motion of the model \eqref{GraLag2} is given by
\begin{align}\label{EOM1}
\left(-\Box + m_0^2 + \xi \mathcal{R} \right) \phi = 0, \qquad \Box \equiv \frac{1}{\sqrt{-g}} \partial_{\mu}\left(\sqrt{-g} g^{\mu\nu} \partial_{\nu}\right).
\end{align}
The canonical momentum for the field $\phi$ at a constant time $t$ is read as
\begin{align}
\pi (t,x) \equiv \frac{\delta S_{{\rm FTCS}}}{\delta \partial_{t} \phi (t,x)} = \sqrt{h} \, n^\mu \partial_{\mu} \phi(t,x),
\end{align}
where $n^\mu = \frac{g^{\mu 0}}{\sqrt{-g^{00}}}$ is the unit normal vector to the hypersurface $\Sigma$, and $h$ is the determinant of the induced spatial metric on the surface. In order to quantize the field $\phi (t,x)$, one has to promote the fields, $\phi (t,x)$ and $\pi (t,x)$, to Hermitian operators and require the commutation relation at a fixed time $t$,
\begin{align}\label{ETCR1}
&[ \phi(t,x), \pi (t, y) ] = i\delta (x-y)\,,
\nonumber\\
&[ \phi(t,x), \phi (t, y) ] = [ \pi(t,x), \pi (t, y) ]=0.
\end{align}
The commutation relations in \eqref{ETCR1} are the same forms of those in the Minkowski spacetime. Following the quantization procedure in the Minkowski spacetime, we define an inner product
\begin{align}\label{brckt1}
\langle \phi_1,\, \phi_2 \rangle = \int_\Sigma d \Sigma^\mu J_\mu,
\end{align}
where $d \Sigma^\mu \equiv n^\mu d \Sigma$ and $J_\mu \equiv i \left(\phi_1^* \partial_{\mu} \phi_2 - \partial_{\mu} \phi_1^* \phi_2 \right)$ is a current satisfying the on-shell relation $\nabla^\mu J_\mu = 0$. This bracket is called the Klein-Gordon inner product, and it does not depend on the choice of the spacelike hypersurface $\Sigma$ in the case that the fields decay sufficiently fast at spatial infinity. That is, if we consider another hypersurface $\Sigma'$ at a different time $t'$, we have the relation \begin{align}
\int_\Sigma d \Sigma^\mu J_\mu - \int_{\Sigma'} d \Sigma^\mu J_\mu = \int_{M} d^2 x \sqrt{-g} \, \nabla^\mu J_\mu = 0,
\end{align}
where $M$ is the manifold bounded by the hypersurfaces, $\Sigma$ and $\Sigma'$. This independence of the hypersurfaces realizes the time-independence of the inner product in the Minkowski spacetime. For complex functions $f$ and $g$ satisfying the equation of motion \eqref{EOM1}, the inner product satisfies the following relations,
\begin{align}\label{brckt2}
\langle f,\, g\rangle^* = - \langle f^*,\, g^* \rangle = \langle g,\, f\rangle,
\end{align}
which implies $\langle f,\, f^*\rangle =0$.
In analogy with the quantization of $\phi$ in Minkowski spacetime, we define the annihilation operator related to the function $f$ in terms of the inner product,
\begin{align}
a(f) = \langle f,\, \phi \rangle,
\end{align}
which is independent of the hypersurface $\Sigma$. Using the properties of the inner product in \eqref{brckt2} and the Hermiticity of the field operator $\phi$, we obtain the Hermitian conjugate of $a(f)$, $a^\dagger(f) = - a (f^*)$. Using these relations and the commutation relations in \eqref{ETCR1}, we obtain the following commutation relations
\begin{align}\label{numop1}
[a(f), a^\dagger (g)] = \langle f,\, g\rangle, \quad [a(f), a (g)] = -\langle f,\, g^*\rangle, \quad [a^\dagger (f), a^\dagger (g)] =- \langle f^*,\, g\rangle.
\end{align}
When the complex solution $f$ satisfies $\langle f,\, f\rangle = 1$, by setting $g = f$ in \eqref{numop1} one can easily see that the relations in \eqref{numop1} are nothing but commutation relations of number operators in harmonic oscillator.
Therefore in order to quantize the scalar field $\phi$ in a curved background we have to fine a complete orthonormal basis of solutions to \eqref{EOM1} satisfying the inner product relations
\begin{align}
\langle u_i,\, u_j\rangle = \delta_{ij}, \quad \langle u^*_i,\, u_j\rangle = 0, \quad \langle u_i^*,\, u_j^*\rangle = - \delta_{ij},
\end{align}
where corresponding annihilation and creation operators are denoted by $a_i$ and $a_i^\dagger$, respectively.
Then one can expand the scalar field $\phi$ as
\begin{align}
\phi(t,x) =\sum_{i} \left( a_i u_i + a_i^\dagger u_i^*\right)
\end{align}
with the commutation relations $[a_i,\, a_j^\dagger ] = \delta_{ij}$. The Fock space can be constructed by these operators.
\end{appendix} | 147,439 |
Residential Property-Related Affordability is important to both lending institutions as well as their mortgage lending clients, as shifts in affordability have implications in terms of the level of financial pressure that they can exert on households.
Therefore, whereas a dramatic improvement in home and home-related affordability from late-2008 to 2012 was instrumental in lowering levels of bad debt and distressed home selling, early signs of mild affordability deterioration in 2014 begin to pose financial "limitations" on the Household Sector.
ESTIMATES OF HOME BUYING AFFORDABILITY
The December SARB Quarterly Bulletin enabled us to update our own 2 housing affordability indices for the 2nd quarter of 2014, using the SARB Average Employee Remuneration Index, the FNB House Price Index, and a Prime Rate time series. As at the 2nd quarter of 2014, we saw a further slight deterioration in these measures of affordability..
Nevertheless, the affordability levels remain far improved on the highs of "in-affordability" experienced back around 2007/8. The cumulative decline (improvement) in the 2 affordability indices since their 2007 peak levels are -31.7% in the case of the Average Price/Income Ratio Index and -50.4% in the Instalment/Income Ratio Index.
2015 PROMISES FURTHER MILD AFFORDABILITY DETERIORATION
While we do not yet have the updated Reserve Bank Average Employee Remuneration Index for the 3rd quarter, the evidence available suggests that not too much happened on the Residential Affordability front in the latter half of 2014.
The StatsSA Average Non-Farm Employee Earnings data pointed to something of a normalization in Average Employee Earnings growth to 6.6% year-on-year in the 3rd Quarter, after a dip to 4.8% in the 2nd Quarter. This is virtually in line with 6.4% year-on-year house price inflation in the 3rd quarter.
However, a mild deterioration in the Bond Repayment/Average Remuneration Ratio may have taken place due to a further 25 basis point interest rate increase in July.
Looking forward into 2015, we believe that certain key factors could cause further mild home and home-related affordability deterioration.
Over the past 3 years, we have seen a broad improvement in residential demand relative to supply, reflected in both FNB's Valuers' Market Strength Index which continues to rise, as well as in a broad declining trend in the average time of properties on the market prior to sale, as per the FNB Estate Agent Survey.
While this improving market balance, and mounting residential supply shortages, are expected to bring about a resurgence in residential building completions in 2015, we don't expect that this will be in time to prevent slightly higher house price inflation this year. As such, we expect a further acceleration in average house price growth into the 8-9% range, and do not believe that Average Employee Remuneration will follow suit.
The net result is expected to be some further moderate increase (deterioration) in both of the above mentioned affordability ratios.
In addition, our interest rate forecast is for a further 75 basis points' worth of interest rate hiking during 2015, with the Reserve Bank continually signaling its intention of "normalizing" interest rates from abnormally low levels.
However, the slump in oil prices and Global food prices increases the possibility of rate hikes being shifted outwards, so we would place a bigger probability of stronger house price inflation causing affordability deteriorations in 2015 as opposed to interest rate hiking.
THE HOUSING-RELATED AFFORDABILITY PICTURE "SLID SOMEWHAT" TOO IN 2014
Considering other affordability measures strongly related to housing, 2014 saw a broad deterioration in this group too. Unsurprisingly the "Municipal Rates, Tariffs, Maintenance and Repairs/Average Employee Remuneration" Index has been on a gradual broad rising trend through the entire 2008-14 period.
This affordability measure rose (deteriorated) slightly during the 2nd quarter of 2014, by +0.4% on the previous quarter, and it is now +4.22% above its level at the beginning of 2008, having been driven higher mostly by high inflation in the area of electricity tariffs.
In addition, real house price levels (house prices adjusted for consumer price inflation) rose slightly in 2014 as a whole, indicating that housing lost some "price competitiveness" to the competing consumer goods and services last year.
Finally, there is the matter of credit affordability, which is a function of how much credit is outstanding, the level of disposable income, and of course the prevailing level of interest rates. The best measure of the affordability of Household Sector credit is the Household Debt-Service Ratio (The cost of servicing the household sector debt burden, expressed as a percentage of Household Sector Disposable Income).
The SARB's "interest only" version of this ratio, like the 2 housing affordability indices, after ending its downward trend in 2013, began to rise late in 2013, from a revised 8.5% in the 3rd quarter of that year to 9.1% by the 3rd quarter of 2014, lifted in part by the SARB's January and July interest rate hikes..
As yet, the deterioration has been small, and residential-related affordability remains vastly improved from 2007/8.
Looking into 2015, we expect a further slow affordability deterioration, based on our projection of slightly higher house price growth in the region of 8-9%, and the expectation that average employee remuneration growth will not quite keep pace. | 123,880 |
TITLE: Functional equation $f(x+y)=f(x)f(y)$ for complex-valued $f$
QUESTION [4 upvotes]: It is well known that the only continuous functions $f: \mathbb{R} \to \mathbb{R}$ which satisfy the equation
$$f(x+y)=f(x)f(y)$$
are the exponential functions $f(x)=a^x$.
I am trying to prove a similar result when we allow $f$ to take complex values, i.e. if $f$ is a continuous function from $\mathbb{R}$ to $\mathbb{C}$ which satisfies $f(x+y)=f(x)f(y)$ for all $x,y \in \mathbb{R}$, then there exists some $z\in \mathbb{C}$ such that $f(x)=e^{zx}$. However, I am totally stuck after attempting a few different methods.
In real analysis, the steps towards proving this result is: (1) Use induction to show that $f(x)=a^x$ is true for all $x\in \mathbb{Z}$; (2) Use $f(1)=(f(\frac{1}{n}))^n$ to show that $f(\frac{1}{n})=(f(1))^{\frac{1}{n}}$, hence $f(x)=a^x$ is true for all $x\in \mathbb{Q}$; and (3) Use continuity to extend this to all $x\in \mathbb{R}$. However, when we allow $f$ to take value in $\mathbb{C}$, it seems that we're stuck at step (2), since every complex number has n distinct complex n-th roots and we do not know which one to choose for $f(\frac{1}{n})$. We do have the additional assumption of continuity, but I don't quite see how it can be employed in this step.
Any help will be appreciated.
REPLY [4 votes]: First we find all continuous maps $f:%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow S^{1}$ such that $f(x+y)=f(x)f(y)$ for all $x,y.$ Then we replace the range by $\mathbb C$.
Note that $f(0)=1$. Fix a positive integer $N$. By a standard
argument in Complex Analysis there exists a unique continuous function $%
h_{N}:[-N,N]\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ such that $f(x)=e^{ih_{N}(x)}$ $(\left\vert x\right\vert \leq N)$ and $%
h_{N}(0)=1$. It follows easily that $h_{N}^{\prime }s$ define a continuous
function $h:%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ such that $h(0)=0$ and $f(x)=e^{ih(x)}$ for all real numbers $x$. Note
that $e^{i[h(a+b)-h(a)-h(b)]}=1$ so $h(a+b)-h(a)-h(b)=2n\pi $ for some
integer $n$. By continuity of $h$ we conclude that $n$ does not depend on $a$
and $b$. Since $h(0)=0$ we conclude that $h$ is additive.
Since $h$ is additive and continuous there is a real number $a$ such that $%
h(x)=ax$ for all $x$. Hence $f(x)=e^{iax}$. Now consider the second case.
Since $f(0)=f^{2}(0)$ either $f(0)=0$ or $f(0)=1$. If $f(x)=0$ for some $x$
then $f(x+y)=f(x)f(y)=0$ for all $y$ which gives $f\equiv 0$. If this is not
the case then $f(0)=1$ and $f$ never vanishes. Let $g(x)=\frac{f(x)}{%
\left\vert f(x)\right\vert }$. The first part can be applied to $g$ and we
get $f(x)=e^{iax}\left\vert f(x)\right\vert $. Also $\log \left\vert
f(x)\right\vert $ is an additive continuous function on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
,$ so $\left\vert f(x)\right\vert =e^{bx}$ for some real number $b$. We now
have $f(x)=e^{(b+ia)x}$. | 203,009 |
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Closes the current SDF file, if any, and exits the program. | 417,765 |
Ep 104. [Private Call] Bringing accountability to my business (this will help you)
What do you do when you’re stuck? How do you bring priority and purpose into your life? I was on a private accountability call with my operations guy, Evan Karadimov, and as we started chatting I knew I had to hit record. I know that some of the pain points that Evan has been feeling lately are going to resonate with you. | 79,528 |
Apple has canceled Bastards, a drama series led by Richard Gere and based on the Israeli-based series called Nevelot (via Variety).
Bastards
Mr. Gere was set to star in Bastards, with Howard Gordon and Warren Leight serving as showrunners and writers. Sources at Apple cite creative differences between Apple, Mr. Gordon, and Mr. Leight.
Gere would have starred as one of two elderly Vietnam vets and best friends who find their monotonous lives upended when a woman they both loved fifty years ago is killed by a car.
Further Reading:
[New Lego Star Wars Battles Arrives in 2020]
[French Police Defeat Retadup Botnet Infecting 850,000 Computers]
Can’t help but think the title wasn’t as family friendly as Apple would have wanted. Problem with giving creative freedom to someone who seems to want to be Hollywood’s Bono, you might get disputes. But somehow, I think someone picked the wrong title. Call it a Cook-hunch. | 126,594 |
TITLE: Discrete math question about surjective, injective function and domain, range
QUESTION [0 upvotes]: I'm a first year computer science student and I'm learning discrete math by myself (teacher unavailable) due to the quarantine and I dont understand these two little questions :
1) Lets say we have a function $f : X \to Y$ that has an inverse function. How do I find the function $I$($x$) = $f^{-1}(f(x))$ and how can I find the domain and range of $I(x)$ ? This one is very confusing and I love an good explanation for it.
2) Prove that if $f$ and $g$ are both surjective, then $g \circ f$ is surjective. I think that I have to prove that its image is equal to its codomain, but I have no idea how to do this.
Thanks. Your help is very appreciated.
REPLY [1 votes]: That is not always easy to find the inverse function when you have general sets $X$ and $Y$.
However, if $X$ and $Y$ are subsets of $\mathbb R$, for instance, you can find $f^{-1}$ with the following idea: Solve the equation $f(y)=x$ for $y$, then $y$ will give you the formula for the inverse.
For instance, consider the function $f: (0, 1)\longrightarrow \mathbb R$ given by $$f(x)= \frac{x}{1+x}.$$ Then $f$ is bijective. Let us find $f^{-1}$. Well:
$$f(y)=x\Leftrightarrow \frac{y}{1+y}=x\Leftrightarrow y= x(1+y)\Leftrightarrow y-xy=x\Leftrightarrow y(1-x)=x\Leftrightarrow y= \frac{x}{1-x}.$$ The inverse will be given by $$f^{-1}(x)=\frac{x}{1-x}.$$ Let us check how this works:
$$f(f^{-1}(x))= \frac{f^{-1}(x)}{1+f^{-1}(x)}=\frac{x/(1-x)}{1+(x/(1-x)}=\frac{x/(1-x)}{(1-x+x)/(1-x)}= \frac{x/(1-x)}{1/(1-x)}=x.$$ On the other hand:
$$f^{-1}(f(x))= \frac{f(x)}{1-f(x)}=\frac{x/(1+x)}{1-(x/(1+x))}=\frac{x/(1+x)}{(1+x-x)/(1+x)}=\frac{x/(1+x)}{1/(1+x)}=x.$$
In general, if $f:X\longrightarrow Y$ is bijective then $f^{-1}$ will have $Y$ as domain, and $X$ as image.
Finally, if $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are surjective then $g\circ f: X\longrightarrow Z$ is surjective. In fact, given $z\in Z$ there exists $y\in Y$ such that $g(y)=z$ since $g$ is surjective. Once $f$ is surjective, there exists $x\in X$ such that $f(x)=y$. Hence, the element $x\in X$ is such that $$(g\circ f)(x)=g(f(x))=g(y)=z,$$
that is, every element of $Z$ is in the imagem of $g\circ f$, this show $Z\subset \textrm{Im}(g\circ f)$. Since, $\textrm{Im}(g\circ f)$ is always a subset of $Z$, it follows $Z=\textrm{Im}(g\circ f)$, therefore $g\circ f$ is surjective. | 25,096 |
TITLE: Showing that a function is pointwise convergent
QUESTION [0 upvotes]: Here is how I answered the question. Could you please say what is not correct if it's incorrect?
Taking $ϵ = 1$ in the definition of the uniform convergence, we find that there exists $N ∈ N $such that $|f_{n}(x) − f(x)| < 1$ for all $x ∈ A $ if $p > q$. Choose some $n > N$. Then, since $f_n$ is bounded, there is a constant $Mn ≥ 0$ such that $|f_n(x)| ≤ Mn$ for all $x ∈ A$. It follows that $|f(x)| ≤ |f(x) − f_n(x)| + |f_n(x)| < 1 + Mn$ for all $x ∈ A$, meaning that $f$ is bounded on $A $(by $1 + Mn$).
If $ 0 < x ≤ 1$, then $fn(x) = 0$ for all $ n ≥ 1/x$, so $ f_n(x) → 0$ as $n → ∞$; and if $x = 0$, then $f_n(x) = 0$ for all $n$, so $f_n(x) → 0 $ also. It follows that $f_n → 0$ pointwise on $[0, 1]$. This is the case even though $||f_n|| = n → ∞ $ as $n → ∞$. Thus, a pointwise convergent sequence of functions need not be bounded, even if it converges to zero. As the boundaries are not set within the exclusive set of range values that satisfies the condition of $p,q>0 $ thus the convergence of the range uniformly sets on the interval of the $[0.01, 1] $
REPLY [1 votes]: Remark: above you have written that $f_n(x) = 0$ for all $n \geq 1/x$. But
$f_n(x) = n^p x \exp(-n^q x) > 0$ for all $x > 0$. To prove the uniform convergence it is sufficient to use the following equivalence:
$$ f_{n} \rightarrow f \; (\mbox{uniformly}) \iff \| f_n - f \|_{\infty} \rightarrow 0 \; \mbox{when } n \rightarrow + \infty. $$ In your case,
we have $\|f_n\|_{\infty} = n^{p - q} e^{-1}$ which goes to zero when $n \rightarrow + \infty$. So, if $q > p$ we have uniform convergence. On the other hand, if $p > q$, then $n^{p - q} \rightarrow + \infty$ as $n \rightarrow + \infty$ (If $p = q$, $\|f_n\|_{\infty} = \frac{1}{e}$). Hence, in this case, using the above equivalence, we conclude that the convergence cannot be uniform. | 113,449 |
\begin{document}
\maketitle
\begin{abstract}
The aim of this paper is to prove the existence and uniqueness of solutions of the following $q$-Cauchy problem of second order linear $q$-difference problem associated with the Rubin's $q$-difference operator $\partial_q$ in a neighborhood of zero
\begin{equation}
\left\{
\begin{array}{cc}
q\,a_0(x)\,\partial_q^2y(qx)\,+\,a_1(x)\,\partial_qy(x)\,+\,a_2(x)y(x)&\;=\;b(x),\quad \hbox{if $y$ is odd;}\\
q\,a_0(x)\partial_q^2y(qx)\,+\,q\,a_1(x)\partial_qy(qx)\,+\,a_2(x)y(x)&\;=\;b(x),\quad \hbox{if $y$ is even,}
\end{array}
\right.
\end{equation}
with the initial conditions
\begin{equation}
\partial_{q}^{i-1}y(0)= b_{i};\quad b_{i} \in{\mathbb{C}},\; i=1,2
\end{equation}
where $a_i$, $i=0,1,2$, and $b$ are defined, continuous at zero and bounded on an interval $I$ containing zero such that $a_0(x)\neq 0$ for all $x\in I$.
Then, as application of the main results, we study the second order homogenous linear $q$-difference equations as well as the $q$-Wronskian associated with the Rubin's $q$-difference operator $\partial_q$. Finally, we construct a fundamental set of solutions for the second order linear homogeneous $q$-difference equations in the cases when the coefficients are constants and $a_1(x)=0$ for all $x\in I$.\\
\keywords{Rubin's $q$-difference operator $\partial_q$; $q$-difference equations; q-initial value problems; $q$-Wronskian associated with the Rubin's $q$-difference operator $\partial_q$.}
\end{abstract}
\section{Introduction}
Studies on $q$-difference equations appeared already at the beginning of the last century in intensive works especially by F.H. Jackson$~{\cite{N}}$, R.D. Carmichael$~{\cite{R.D}}$, T.E. Mason$~{\cite{T.E}}$, C.R. Adams$~{\cite{C.R}}$ and other authors such us Poincare, Picard, Ramanujan. Apart from this old history of $q$-difference equations, the subject received a considerable interest of many mathematicians and from many aspects, theoretical and practical.
Since years eighties$~{\cite{Hahn}}$, an intensive and somewhat surprising interest in the subject reappeared in many areas of mathematics and applications including mainly new $q$-difference calculus and $q$-orthogonal polynomials, $q$-combinatorics, $q$-arithmetics, $q$-integrable systems.
The present article is devoted for developing the theory of the second order linear $q$-difference equations associated with the Rubin's $q$-difference operator $\partial_q$ in a neighborhood of zero, studying the $q$-wronskian associated with the Rubin’s $q$-difference operator $\partial_q$ and to showing how it plays a central role in solving the second order linear $q$-difference equations. As application, we study the second order linear homogeneous $q$-difference equations associated with the Rubin’s $q$-difference operator $\partial_q$ when the coefficients are constants. As M.H. Annaby and Z.S. Mansour$~{\cite{j6}}$, we establish some results associated with the Rubin’s $q$-difference operator $\partial_q$.
We mention that, in this paper, we will follow all the so-mentioned works which, for a convergence argument, they imposed that the parameter $q\in]0,1[$ satisfies the condition
$$\frac{\ln{(1-q)}}{\ln(q)}\in 2\,\mathbb{Z}.$$
This paper is organized as follows: in Section 2, we recall some necessary fundamental concepts of quantum analysis. Section 3 is devoted to prove the existence and uniqueness of solutions of the $q$-Cauchy problem of second order linear $q$-difference equations in a neighborhood of zero. In Section 4, we study the second order linear $q$-difference equations associated with the Rubin’s $q$-difference operator $\partial_q$. In Section 5, we introduce the $q$-Wronskian associated with the Rubin’s $q$-difference operator $\partial_q$ and we establish some of its properties. In Section 6, we construct a fundamental set of solutions for the second order linear homogeneous $q$-difference equations when the coefficients are constants. Finally, in Section 7, we purpose some examples.
\section{Notations and preliminaries}
In this section, we introduce some necessary fundamental concepts of quantum analysis which will be used in this paper. For this purpose, we refer the reader to the book by G. Gasper and M. Rahman $~{\cite{L}}$, for the definitions, notations and properties of the $q$-shifted factorials and the $q$-hypergeometric functions.
Throughout this paper, we assume $q \in]0, 1[$ and we denote
$${\mathbb R}_{q}\; =\; \{\pm q^k , k\in \mathbb Z\},\quad {\mathbb R}_{q,+} \;=\; \{+ q^k , k\in \mathbb Z\}\quad \text{and}\quad \widetilde{{\mathbb R}}_{q}\;=\; {\mathbb R}_{q}\, \bigcup\, \{0\}.$$
\subsection{Basic symbols}
For a complex number $a$, the $q$-shifted factorials are defined by:
$${(a,q)}_{0}\;:=\;1;\quad{(a,q)}_{n}\;:=\;\displaystyle\prod_{k=0}^{n-1}(1-a{q}^{k}),\; n\geq 1\quad \text{and}\quad {(a,q)}_{\infty}\;=\;\prod_{k=0}^{\infty}(1-a{q}^{k}).$$
We also denote
$$ {[n]}_{q}\;=\;\frac{1-q^n}{1-q}\quad\text{and}\quad{[n]_{q}}!\;=\;\frac{(q,q)_{n}}{(1-q)^n},\quad n\in\mathbb{N}.$$
If we change $q$ by $q^{-1}$, we obtain
$$[n]_{q^{-1}}!\;=\;q^{-\frac{n(n-1)}{2}}\, [n]_{q},\quad n\in\mathbb{N}.$$
The Gauss $q$-binomial coefficient $\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]_{q}$ is defined by (see$~{\cite{L}}$)
$${\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]_{q}}\;=\;\D{{\frac {[n]_{q}!}{[n-k]_{q}![k]_{q}!}}} \;=\;\frac{(q,q)_{n}}{(q,q)_{k}\,(q,q)_{n-k}},\quad n\geq k\geq 0.$$
Moreover, the $q$-binomial theorem is given by
$$(-a,q)_n\;=\;\D\sum_{k=0}^{n}{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]_{q}}q^{\frac{k(k-1)}{2}}a^k,\quad a\in\mathbb{C}.$$
\subsection{Operators and elementary $q$-special functions}
Let $\mathcal{A} \subseteq \mathbb{R}$ be a $q$-geometric set containing zero and satisfying for every $x\in\mathcal{A}$, $\pm q^{\pm1}x\in\mathcal{A}$. A $q$-difference equation is an equation that contains $q$-derivatives of a function defined on $\mathcal{A}$. Let $f$ be a function, real or complex valued, defined on a $q$-geometric set $\mathcal{A}$. The $q$-difference operator $D_q$, the Jackson $q$-derivative is defined by
\begin{equation}\label{ddd}
D_{q}f(x)\;:=\;\frac{f(x)-f(qx)}{(1-q)x},\quad \text{for all}\;x\in\mathcal{A}\backslash\{0\}.
\end{equation}
In the $q$-derivative, as $q\rightarrow 1$, the $q$-derivative is reduced to the classical derivative.\\
The $q$-derivative at zero is defined by
\begin{equation}
D_qf(0)\;:=\;\lim_{n\rightarrow +\infty}\dfrac{f(q^nx)-f(0)}{q^nx},\quad \text{for all}\;x\in\mathcal{A},
\end{equation}
if the limit exists and does not depend on $x$.\\
We recall that for a function $f(x)$ defined on $\mathcal{A}$ and $n\in\mathbb{N}$, we have
\begin{equation}\label{lm}
\displaystyle D_{q}^{n}f(x)\;=\;{\frac {(-1)^{n}q^{-\frac{n(n-1)}{2}}}{(1-q)^{n}x^{n}}}\,\sum _{k=0}^{n}(-1)^{k}\,{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]_{q}}\,q^{\frac{k(k-1)}{2}}f(q^{n-k}x).
\end{equation}
The Rubin's $q$-difference operator $\partial_q$ is defined in $~{\cite{&,3+}}$ by
\begin{equation}\label{/}
\partial_{q}f(x)\;=\;\left\{
\begin{array}{ll}
\D\dfrac{f(q^{-1}x)+f(-q^{-1}x)-f(qx)+f(-qx)-2f(-x)}{2(1-q)x}&,\; \hbox{$x\neq0$;} \\
\D\lim_{x\rightarrow0}\partial_{q}f(x)&,\; \hbox{$x=0$.}
\end{array}
\right.
\end{equation}
It is straightforward to prove that if a function $f$ is differentiable at a point $z$, then
$$\D{\lim_{q\rightarrow 1^{-}}\partial_{q}f(x)\,=\, f\,'(x)}.$$
A repeated application of the Rubin's $q$-difference operator $n$ times is denoted by:
$$\partial_{q}^0f\;=\;f\quad{\text{and}}\quad \partial_{q}^{n+1}f\;=\;\partial_{q}(\partial_{q}^{n}f),\quad n\in\mathbb{N}.$$
We define the $q$-shift operators by:
\begin{eqnarray}
(\Lambda_qf)(x) \;=\,& f(qx)\quad &\text{and}\quad (\Lambda_q^{-1} f)(x)\;=\; f(xq^{-1}),\label{Lam}\\
\partial_q\,\Lambda_qf\;=\,& q\Lambda_q\partial_q \quad &\text{and}\quad
\partial_q\,\Lambda_q^{-1}f\;=\; q^{-1}\,\Lambda_q^{-1}\,\partial_q.\label{Lam1}
\end{eqnarray}
We mention that $\partial_q$ is closely related to the Jackson’s $q$-derivative operator $D_q$ and use relation $~(\ref{lm})$, we can easily prove the following result:
\begin{prop}{$:$}\label{code}
Let $f$ be a function defined on $\mathbb{R}_q$. Then for all $n\in\mathbb{N}$ we have:
\begin{enumerate}
\item \quad \begin{equation}\label{seconde}
\partial_{q}^{2n}f\;=\;q^{-n(n+1)}(D_{q}^{2n}f_{e})o\Lambda_{q}^{-n}+q^{-n^2}(D_{q}^{2n}f_{o})o\Lambda_{q}^{-n},\end{equation}
\item \quad \begin{equation}
\partial_{q}^{2n+1}f\;=\;q^{-(n+1)^2}(D_{q}^{2n+1}f_{e})o\Lambda_{q}^{-(n+1)}+q^{-n(n+1)}(D_{q}^{2n+1}f_{o})o\Lambda_{q}^{-n},\end{equation}
\end{enumerate}
where $f_{e}$ and $f_{o}$ are, respectively, the even and the odd parts of $f$ and $\Lambda_{q}^{-n}$ is the function given by $\Lambda_q^{-n}(x)\,=\,q^{-n}x$.
\end{prop}
A right inverse to the $q$-derivative, the $q$-integration is given by Jackson as
\begin{equation}
\int_{0}^{x}f(t)d_{q}t \; := \; x\,(1-q)\,\displaystyle\sum_{n=0}^{+\infty}q^{n}f(q^{n}\,x),\quad \text{for all}\;x\in\mathcal{A},
\end{equation}
provided that the series converges. In general,
\begin{equation}
\D\int_{a}^{b}f(t)d_{q}t \; :=\; \int_{0}^{b}f(t)d_{q}t\, -\, \int_{0}^{a}f(t)d_{q}t, \quad \text{for all}\;a,b\in\mathcal{A}.
\end{equation}
The $q$-integration for a function is defined in $~{\cite{HW}}$ by the formulas
\begin{align}
\D \int_{0}^{+\infty}f(t)d_{q}t &\;=\; (1-q)\,\displaystyle\sum_{n=-\infty}^{+\infty}\,q^{n}\,f(q^{n}),\label{a29} \\
\D \int_{-\infty}^{0}f(t)d_{q}t &\;=\; (1-q)\,\displaystyle\sum_{n=-\infty}^{+\infty}\,q^{n}\,f(-q^{n}),\label{a29*} \\
\D\int_{-\infty}^{+\infty}f(t)d_{q}t &\;=\; (1-q)\,\D\sum_{n=-\infty}^{+\infty}\,q^{n}\,\left[f(q^{n})\,+\,f(-q^{n})\right],
\end{align}
provided that the series converges.
\begin{rem}{$:$}
Note that when $f$ is continuous on $[0,a]$, it can be shown that
$$\D\lim_{q\rightarrow1}\int_{0}^{a}f(t)d_{q}t \;=\; \int_{0}^{a}f(t)dt.$$
\end{rem}
\begin{defn}
A function $f$ which is defined on $\mathcal{A}$, $0\in\mathcal{A}$, is said to be $q$-regular at zero if
\begin{equation}
\lim_{n\rightarrow \infty}f(xq^n)\;=\;f(0),\quad \text{for all}\;x\in\mathcal{A}.
\end{equation}
\end{defn}
Through the remainder of the paper, we deal only with functions that are $q$-regular at zero.
The following results hold by direct computation.
\begin{lemma}\label{m"}
Let $f$ be a function defined on $\mathcal{A}$. Then, if $\D\int_{0}^{x}f(t)\,d_{q}t$ exists, we have:
\begin{enumerate}
\item \quad for all integer $n$,\quad $\D\int_{-\infty}^{+\infty}f(q^nt)d_{q}t\;=\;q^{-n}\,\D\int_{-\infty}^{+\infty}f(t)d_{q}t.$
\item \quad if $f$ is odd, then
\begin{equation}\label{mr'}
\partial_{q}\D\int_{0}^{x}\,f(t)d_{q}t \;=\; f(x),
\end{equation}
and
\begin{equation}\label{mr}
\D\int_{0}^{x}\,\partial_{q}f(t)d_{q}t \;=\; f(x)\,-\, \lim_{n\rightarrow \infty}f(xq^n).
\end{equation}
\item \quad if $f$ is even, then
\begin{equation}\label{mr1'}
\partial_{q}\D\int_{0}^{x}\,f(t)d_{q}t \;=\; f(q^{-1}x),
\end{equation}
and
\begin{equation}\label{mr1}
\D\int_{0}^{x}\,\partial_{q}f(t)d_{q}t \;=\; f(q^{-1}x)\,-\,\lim_{n\rightarrow \infty}f(xq^{n-1}).
\end{equation}
\end{enumerate}
\end{lemma}
\begin{prop}{\bf[The rule of $q$-integration by parts]}
Let $f$ and $g$ be two functions defined on $[-a,a]$, for all $a>0$. Then, if $\D\int_{-a}^{a}\partial_qf(t)\,g(t)\,d_{q}t$ exists, the rule of $q$-integration by parts is given by:
\begin{equation}
\D\int_{-a}^{a}\partial_qf(t)\,g(t)\,d_{q}t\;=\;2\left[f_e(q^{-1}a)g_o(a)\,+\,f_o(a)g_e(q^{-1}a)\right]\,-\,\D\int_{-a}^{a}f(t)\,\partial_qg(t)\,d_{q}t,
\end{equation}
where $f_e$, $f_o$ are the even and the odd parts of $f$ respectively and $g_e$, $g_o$ are the even and the odd parts of $g$ respectively.
\end{prop}
\begin{corollary}
Let $f$ and $g$ be two functions defined on $\mathbb{R}_q$. Then, if $\D\int_{-\infty}^{+\infty}\partial_qf(t)\,g(t)\,d_{q}t$ exists, the rule of $q$-integration by parts is given by:
\begin{equation}
\D\int_{-\infty}^{+\infty}\partial_qf(t)\,g(t)\,d_{q}t\;=\;-\,\D\int_{-\infty}^{+\infty}f(t)\,\partial_qg(t)\,d_{q}t.
\end{equation}
\end{corollary}
\begin{defn}
Let $f$ be a function defined on $\mathcal{A}$. We say that $f$ is $q$-integrable on $\mathcal{A}$ if and only if $\D \int_{0}^{x}f(t)d_{q}t$ exists for all $x\in\mathcal{A}$.
\end{defn}
The $q$-trigonometric functions $q$-cosine and $q$-sine (see$~{\cite{&,3+}}$) are defined, respectively, on $\mathbb{C}$ by:
\begin{equation}\label{cos}
\cos(x,q^2)\;:=\;\D\sum_{n=0}^{+\infty}\,(-1)^n\,b_{2n}(x,q^2)\end{equation}
and
\begin{equation}\label{sin}
\sin(x,q^2)\;:=\;\D\sum_{n=0}^{+\infty}\,(-1)^n\,b_{2n+1}(x,q^2)
\end{equation}
where
\begin{equation}\label{mas}
b_n(x,q^2)\;=\;q^{[\frac{n}{2}]([\frac{n}{2}]+1)}\dfrac{x^{n}}{[n]_q!},\quad n\in\mathbb{N}.
\end{equation}
These two functions induce a $\partial_q$-adapted $q$-analogue exponential function as
\begin{equation}\label{exp3}
e(x,q^2)\;:=\;\cos(-ix,q^2)\,+\,i\sin(-ix,q^2)\;=\;\D\sum_{n=0}^{+\infty}b_{n}(x,q^2).
\end{equation}
Remark that $e(x,q^2)$ is absolutely convergent for all $x$ in the complex plane since both of its component functions are. Moreover, $\D\lim_{q\rightarrow1^{-}}e(x,q^2)\,=\,e^x$ (exponential function) point-wise and uniformly on compacts. The following results hold by direct computation.
\begin{lemma}\label{sit}
\begin{enumerate}
\item For all $x\in\mathbb{C}$ and $\lambda\in\mathbb{C}$, we have
\begin{equation}
\partial_q\cos(\lambda x,q^2)\;=\;-\,\lambda\,\sin(\lambda x,q^2),\quad \partial_q\sin(\lambda x,q^2)\;=\;\lambda\,\cos(\lambda x,q^2)
\end{equation}
and
\begin{equation}
\partial_q e(\lambda x,q^2)\;=\;\lambda\,e(\lambda x,q^2).
\end{equation}
\item For all function $f$ defined on $\mathcal{A}$, we have
\begin{equation}\label{PIM}
\partial_{q}f(x)\;=\;\D\dfrac{f_e(q^{-1}x)-f_e(x)}{(1-q)x}\,+\,\D\dfrac{f_o(x)-f_o(qx)}{(1-q)x},\quad x\in\mathcal{A}\backslash\{0\}\end{equation}
Here, $f_e$ and $f_o$ are the even and the odd parts of $f$, respectively.
\item For two functions $f$ and $g$ defined on $\mathcal{A}$, we have \\
-- \quad if $f$ is even and $g$ is odd then
\begin{equation}\label{marwa1}
\partial_{q}(fg)(x)\;=\;f(x)\partial_{q}g(x)+q\,g(qx)\partial_{q}f(qx)\;=\;q\,g(x)\partial_{q}f(qx)+f(qx)\partial_{q}g(x).
\end{equation}
-- \quad if $f$ and $g$ are even then
\begin{equation}\label{marwa2}
\partial_{q}(fg)(x)\;=\;g(q^{-1}x)\partial_{q}f(x)+f(x)\partial_{q}g(x).
\end{equation}
-- \quad if $f$ and $g$ are odd then
\begin{equation}\label{marwa3}
\partial_{q}(fg)(x)\;=\;q^{-1}\,g(q^{-1}x)\partial_{q}f(q^{-1}x)+q^{-1}f(x)\partial_{q}g(q^{-1}x).
\end{equation}
\end{enumerate}
\end{lemma}
\section{$q$-Initial Value Problems in a Neighborhood of Zero}
In this section, we prove the existence and uniqueness of solutions of the $q$-Cauchy problem of second order $q$-difference equations in a neighborhood of zero.
In the sequel $X$ is a Banach space with norm $\|\cdot\|$ and $I\subseteq\mathbb{R}$ is an interval containing zero.
\begin{defn}
Let $S$ and $R$ be defined, respectively, by
$$S(y_0,\beta)\,:=\,\{y\in\mathbb{X}\,:\,\|y-y_0\|\,\leq\,\beta\},$$
and
$$R\,:=\,\{(x,y)\in I \times\mathbb{X}\,:\,|x|\leq \alpha,\,\|y-y_0\|\,\leq\,\beta\},$$
where $y_0\in\mathbb{X}$ and $\alpha$, $\beta$ are fixed positive real numbers.
\end{defn}
By a $q$-initial value problem ($q$-IVP) in a neighborhood of zero we mean the problem of finding continuous functions at zero
satisfying system \begin{equation}\label{IP1}
\partial_{q}{y(x)}\;=\;f(x,y(x)),\quad y(0)\;=\;y_0,\quad {x}\in I.\end{equation}
\begin{theorem}
Let $f:R\rightarrow {\mathbb{X}}$ be a continuous function at $(0,y_0)\in R$ , and $\phi$ be a function defined on $I$. Then $\phi$ is a solution of the
$q$-IVP $~(\ref{IP1})$ if, and only if,
\begin{enumerate}
\item For all $x\in I$, $(x,\phi(x))\in R$.
\item $\phi$ is continuous at zero.
\item For all $x\in I$,
\begin{equation}\label{IP5}
\int_{0}^{x}f(t,\phi(t))d_qt\;=\;\left\{
\begin{array}{cc}
\phi(x)\,-\,y_0,&\quad \hbox{if $y$ is odd;}\\
\phi(q^{-1}x)\,-\,y_0,&\quad \hbox{if $y$ is even,}
\end{array}
\right.
\end{equation}
\end{enumerate}
\end{theorem}
\begin{dem}
Let $\phi$ be a solution of the $q$-IVP $~(\ref{IP1})$. Then
\begin{equation}\label{IP2}
\partial_{q}{\phi(x)}\;=\;f(x,\phi(x)),\quad \phi(0)\;=\;y_0,\quad {x}\in I.\end{equation}
which implies $(x,\phi(x))\in R$ for all $x \in I$. Also, since $\phi$ is $q$-differentiable on $I$, then it is continuous at zero. Finally, integrating both sides of $~(\ref{IP2})$ from 0 to $x$ and using Lemma$~\ref{m"}$ we get
$$\int_{0}^{x}f(t,\phi(t))d_qt\;=\;\left\{
\begin{array}{cc}
\phi(x)\,-\,\phi(0),&\quad \hbox{if $y$ is odd;}\\
\phi(q^{-1}x)\,-\,\phi(0),&\quad \hbox{if $y$ is even,}
\end{array}
\right.$$
Consequently,
$$\int_{0}^{x}f(t,\phi(t))d_qt\;=\;\left\{
\begin{array}{cc}
\phi(x)\,-\,y_0,&\quad \hbox{if $y$ is odd;}\\
\phi(q^{-1}x)\,-\,y_0,&\quad \hbox{if $y$ is even,}
\end{array}
\right.$$
Conversely, assume the items (1), (2) and (3) are satisfied, then $\phi$ is ordinary differentiable at zero. Consequently, it is $q$-differentiable on $I$ with $\partial_{q}{\phi(x)}\;=\;f(x,\phi(x))$ and $\phi(0)\;=\;y_0$. Therefore, $\phi$ is a solution of the $q$-IVP $~(\ref{IP1})$.
\hfill$\blacksquare$
\end{dem}
To prove the existence and uniqueness of the solution of the $q$-IVP $~(\ref{IP1})$, we need some preliminary results:
\begin{defn}
Let ${\displaystyle (\mathbb{X},d)}$ be a complete metric space. Then a map ${\displaystyle T\colon \mathbb{X}\to \mathbb{X}}$ is called a contraction mapping on ${\displaystyle \mathbb{X}}$ if there exists $0\leq\rho<1$ such that
$${\displaystyle d(T(x),T(y))\leq \rho\,d(x,y)}$$
for all ${\displaystyle x,y}\in {\displaystyle \mathbb{X}}$.
\end{defn}
\begin{theorem}{\bf(Banach's fixed point theorem).}\label{BPT}\;Let ${\displaystyle (\mathbb{X},d)}$ be a non-empty complete metric space with a contraction mapping ${\displaystyle T\colon \mathbb{X}\to \mathbb{X}}$. Then $T$ has a unique fixed-point $x^* \in \mathbb{X}$ (i.e. $T(x^*) = x^*$). Furthermore, for any $x\in \mathbb{X}$ and $n\geq1$ the iterative sequence $\{T^n(x)\}$ converges to $x^*$.
\end{theorem}
\begin{theorem}\label{th1}
Let $f:R\rightarrow {\mathbb{X}}$ be a continuous function at $(0,y_0)\in R$ and satisfies the Lipschtiz condition with respect to $y$ in $R$, that is, there exists a positive constant $L$ such that:
\begin{equation}
\big\| f(x,y_{1})-f(x,y_{2})\big\| \leq{L}\big\|y_{1}-y_{2} \big\|, \quad\textit{for all } (x,y_{1}), (x,y_{2})\in{R}.
\end{equation}
Then the $q$-IVP $~(\ref{IP1})$ has a unique solution on $[-h,h]$, where
$$h\;:=\;\min\{\alpha,\frac{\beta}{L\beta\,+\,M},\frac{\rho}{L}\}\quad \text{with}\quad M\,=\,\sup_{(x,y)\in{R}}\|f(x,y)\|<+\infty,\quad 0\leq\rho<1.$$
\end{theorem}
\begin{dem}
We prove the theorem for $x\in[0,h]$ and the proof for $x \in [-h,0]$ is similar. Define the operator $T$ by
\begin{equation}\label{IP3}
T\,y(x)\;=\;y_0\,+\,\int_{0}^{x}f(t,y(t))d_qt.
\end{equation}
Let $\mathcal{C}([0,h])$ be the space of all continuous functions at zero and bounded on the interval $[0,h]$ with the supremum norm such that for $y\in\mathcal{C}([0,h])$, we have
$$\|y\|_{\infty}\;=\;\displaystyle\sup_{x\in[0,h] }\,\|y(x)\|.$$
This space is complete. Let
$$\widetilde{S}(y_0,\beta)\,:=\,\{y\in\mathcal{C}([0,h])\,:\,\|y\,-\,y_0\|_{\infty}\,\leq\,\beta\},\quad \beta >0.$$
As $\widetilde{S}(y_0,\beta)$ is a closed subset of the complete space $\mathcal{C}([0,h])$, it is also complete metric space.
First, we prove that $T:\, \widetilde{S}(y_0,\beta)\rightarrow \widetilde{S}(y_0,\beta)$. Let $\phi\in \widetilde{S}(y_0,\beta)$,
\begin{align*}
\big\|T\,\phi(x)\,-\,y_0\big\|_\infty&\;=\;\displaystyle\sup_{x\in[0,h] }\,\big\| \int_{0}^{x}f(t,\phi(t))d_qt\big\|\\
&\;=\;\displaystyle\sup_{x\in[0,h] }\,\big\| \int_{0}^{x}\left(f(t,\phi(t))\,-\,f(t,y_0)\,+\,f(t,y_0)\right)d_qt\big\|\\
&\;\leq\;\displaystyle\sup_{x\in[0,h] }\, \int_{0}^{x}\big\|f(t,\phi(t))\,-\,f(t,y_0)\,+\,f(t,y_0)\big\|d_qt\\
&\;\leq\; \displaystyle\sup_{x\in[0,h] }\,\int_{0}^{x}\big\|f(t,\phi(t))\,-\,f(t,y_0)\big\|\,+\,\big\|f(t,y_0)\big\|d_qt\\
&\;\leq\;\displaystyle\sup_{x\in[0,h] }\, \int_{0}^{x}\left(L\,\big\|\phi(t)\,-\,y_0\big\|\,+\,M\right)d_qt\\
&\;\leq\;\left(L\,\big\|\phi\,-\,y_0\big\|_\infty\,+\,M\right)\,\int_{0}^{x}d_qt\\
&\;\leq\; \left(L\,\beta\,+\,M\right)\,\int_{0}^{x}d_qt\\
&\;\leq\; \left(L\,\beta\,+\,M\right)\,x\\
&\;\leq\; \left(L\,\beta\,+\,M\right)\,h.\end{align*}
Then, using the fact that $h\leq \frac{\beta}{L\beta\,+\,M}$, we obtain
$$\|T\,\phi\,-\,y_0\|_\infty\leq \beta,\quad \text{i.e.},\quad T\,\phi\in \widetilde{S}(y_0,\beta).$$
Next, we prove that $T$ is a contraction mapping. Assume that $\phi_1,\phi_2\in \widetilde{S}(y_0,\beta)$, then
\begin{align*}
\big\|T\,\phi_1(x)\,-\,T\,\phi_2(x) \big\|&\;=\;\big\| \int_{0}^{x}\left(f(t,\phi_1(t))\,-\,f(t,\phi_2(t))\right)d_qt\big\|\\
&\;\leq\; \int_{0}^{x}\big\|f(t,\phi_1(t))\,-\,f(t,\phi_2(t))\big\|d_qt\\
&\;\leq\; \int_{0}^{x}L\,\big\|\phi_1(t)\,-\,\phi_2(t)\big\|d_qt\\
&\;\leq\;L\,\big\|\phi_1\,-\,\phi_2\big\|_{\infty} \,\int_{0}^{x}d_qt\\
&\;\leq\; L\,\big\|\phi_1\,-\,\phi_2\big\|_{\infty}\,x\\
&\;\leq\;L\,\big\|\phi_1\,-\,\phi_2\big\|_{\infty} \,h.
\end{align*}
But, the fact that
$$h\leq \frac{\rho}{L}\quad \text{with}\quad 0\leq\rho<1$$
gives
$$ \|T\,\phi_1(x)\,-\,T\,\phi_2(x) \|\;\leq\;\rho\,\|\phi_1\,-\,\phi_2\|_{\infty}.$$
Then $T$ is a contraction mapping. By Banach’s fixed point theorem$~\ref{BPT}$, $T$ has a unique fixed point in $\widetilde{S}(y_0,\beta)$ and then the $q$-IVP $~(\ref{IP1})$ has a unique solution in $\widetilde{S}(y_0,\beta)$.
\hfill$\blacksquare$
\end{dem}
\begin{theorem}\label{th2}
Let the functions $f_{i}(x,y_{1},y_{2})$, $i=1,2$, be defined on $I \times\prod_{i=1}^{2} S_{i}(b_i, \beta_i)$, such that the following conditions are satisfied:
\begin{description}
\item[(i)] for $y_{i}\in S_{i}(b_i,\beta_i)$, $i=1,2$, $f_{i}(x,y_{1},y_{2})$ are continuous at zero,
\item[(ii)] there is a positive constant $L$ such that, for $x\in I$, $y_{i}, \tilde{y}_{i}\in S_{i}(b_i,\beta_i)$, $i=1,2,$ the following Lipschitz condition is satisfied:
$$\big\| f_{i}(x,y_{1},y_{2})-f_{i}(x, \tilde{y}_{1},\tilde{y}_{2})\big\| \leq L \sum _{i=1}^{2}\|y_{i}-\tilde{y}_{i} \|.$$
\end{description}
Then there exists a unique solution of the $q$-initial value problem,
\begin{equation}\label{IVP4}
\partial_{q}y_{i}(x)=f_{i}\bigl(x,y_{1}(x),y_{2}(x) \bigr),\quad y_{i}(0)=b_i\in\mathbb{X},\; i =1,2,\; x \in I.
\end{equation}
\end{theorem}
\begin{dem}
Let $y_0=(b_{1},b_2)^{T}$ and $\beta=(\beta_{1},\beta_{2})^{T}$, where $(\cdot ,\cdot)^{T}$ stands for vector transpose. The function $f:I\times \prod_{i=1}^{2}S_{i}(b_i,\beta_i)\rightarrow{\mathbb{X}}\times{\mathbb{X}}$ is defined by
$$f(x,y_{1},y_{2})\;=\; (f_{1}(x,y_{1},y_{2}),f_{2}(x, y_{1}, y_{2}) )^{T}.$$
It is easy to show that system $~(\ref{IVP4})$ is equivalent to the $q$-IVP $~(\ref{IP1})$.\\
Since each $f_i$ is continuous at zero, $f$ is continuous at zero. We claim that $f$ satisfies the Lipschitz condition. Indeed for $y=(y_1,y_2)$ and $\tilde{y}=(\tilde{y}_1,\tilde{y}_2)$ in $\prod_{i=1}^{2}S_{i}(b_i,\beta_i)$
\begin{align*}
\big\| f(x,y)-f(x,\tilde{y}) \big\| &\;=\; \big\| f(x,y_{1},y_{2})-f(x, \tilde{y}_{1}, \tilde{y}_{2}) \big\| \\ &\;=\;\sum_{i=1}^{2} \big\| f_{i}(x,y_{1},y_{2})-f_{i}(x, \tilde{y}_{1},\tilde {y}_{2}) \big\| \\
&\;\leq\; L\,\sum_{i=1}^{2}\|y_{i}- \tilde{y}_{i}\|\\
&\;=\; L\,\|y-\tilde{y}\|.\end{align*}
Applying Theorem$~\ref{th1}$, there exists $h>0$ such that $~(\ref{IP1})$ has a unique solution on $[-h,h]$. Hence, the $q$-IVP $~(\ref{IVP4})$ has a unique solution on $[-h,h]$.
\hfill$\blacksquare$
\end{dem}
\begin{corollary}\label{cor1}
Let $f(x,y_{1},y_{2})$ be a function defined on $I\times\D\prod_{i=1}^{2} S_{i}(b_i,\beta_i)$ such that the following conditions are satisfied:
\begin{description}
\item[(i)] for any values of $y_{i}\in S_{i}(b_i,\beta_i)$, $i=1,2$, $f$ is continuous at zero,
\item[(ii)] $f$ satisfies the following Lipschitz condition
$$\big\| f(x,y_{1},y_{2})-f(x,\tilde{y}_{1}, \tilde{y}_{2}) \big\| \leq L\,\sum_{i=1}^{2} \big\|y_{i} -\tilde{y}_{i}\big\|,$$\end{description}
where $L>0$, $y_{i},\tilde{y}_{i}\in S_{i}(b_i,\beta_i)$, $i=1,2$ and $x \in I$. Then
\begin{equation}\label{IP5}
\partial_{q}^{2}y(qx)\;=\;\left\{
\begin{array}{cc}
f\bigl(x,y(x),\partial_{q}y(x)\bigr),\quad \hbox{if $y$ is odd;}\\
f\bigl(x,y(x),\partial_{q}y(qx)\bigr),\quad \hbox{if $y$ is even,}
\end{array}
\right.
\end{equation}
with the initial conditions
\begin{equation}
\partial_{q}^{i-1}y(0)=b_i,\quad i=1,2
\end{equation}
has a unique solution on $[-h,h]$.
\end{corollary}
\begin{dem}
Consider equation $~(\ref{IP5})$. It is equivalent to $~(\ref{IVP4})$, where $\{\phi_{i}(x)\}_{i=1}^{2}$ is a solution of $~(\ref{IVP4})$ if and only if $\phi_{1}(x)$ is a solution of $~(\ref{IP5})$. Here,
\begin{align*} f_{i}(x,y_{1},y_{2})\;=\;\left \{ \textstyle\begin{array}{l@{\quad}l} y_{2},& i=1, \\ f (x,y_{1},y_{2}),& i=2. \end{array}\displaystyle \right . \end{align*}
Hence, by Theorem$~\ref{th2}$, there exists $h>0$ such that system $~(\ref{IVP4})$ has a unique solution on $[-h,h]$.
\hfill$\blacksquare$
\end{dem}
The following corollary gives us the sufficient conditions for the existence and uniqueness of the solutions of the $q$-Cauchy problem $~(\ref{IP5})$.
\begin{corollary}\label{cor11}
Assume the functions $a_{j}(x):I\rightarrow \mathbb{C}$, $j=0,1,2$, and $b(x)\,:\,I \rightarrow \mathbb{X}$ satisfy the following conditions:
\begin{description}
\item[(i)] $a_{j}(x)$, $j=0,1,2$, and $b(x)$ are continuous at zero with $a_{0}(x)\neq0$ for all $x \in I$,
\item[(ii)] $\dfrac{a_{j}(x)}{a_{0}(x)}$ is bounded on $I$, $j=1,2$. Then
\begin{equation}\label{orderff'}
\left\{
\begin{array}{cc}
q\,a_0(x)\,\partial_q^2y(qx)\,+\,a_1(x)\,\partial_qy(x)\,+\,a_2(x)y(x)&\;=\;b(x),\quad \hbox{if $y$ is odd;}\\
q\,a_0(x)\partial_q^2y(qx)\,+\,q\,a_1(x)\partial_qy(qx)\,+\,a_2(x)y(x)&\;=\;b(x),\quad \hbox{if $y$ is even,}
\end{array}
\right.
\end{equation}
with the initial conditions
\begin{equation}\label{abs2}
\partial_{q}^{i-1}y(0)=b_i;\quad b_i\in\mathbb{C},\; i=1,2
\end{equation}
\end{description}
has a unique solution on subinterval $J\subseteq I$ containing zero.
\end{corollary}
\begin{dem}
Dividing $~(\ref{orderff'})$ by $a_0(x)$, we get
\begin{equation}\label{IP6}
\left\{
\begin{array}{cc}
\partial_{q}^{2}y(qx)\;=\; A_{1}(x)\partial_{q}y(x)+A_{2}(x)y(x)+B(x),\quad \hbox{if $y$ is odd;}\\
\partial_{q}^{2}y(qx)\;=\; q\,A_{1}(x)\partial_{q}y(qx)+A_{2}(x)y(x)+B(x),\quad \hbox{if $y$ is even,}
\end{array}
\right.
\end{equation}
where $A_{j}(x)=-q^{-1}\,\dfrac{a_{j}(x)}{a_{0}(x)}$ and $B(x)=\dfrac{b(x)}{a_{0}(x)}$. Since $A_{j}(x)$ and $B(x)$ are continuous at zero, the function $f(x,y_{1},y_{2})$, defined by
$$\left\{
\begin{array}{cc}
f(x,y_{1},y_{2})\;=\; A_{1}(x)\partial_{q}y(x)+A_{2}(x)y(x)+B(x),\quad \hbox{if $y$ is odd;}\\
f(x,y_{1},y_{2})\;=\; q\,A_{1}(x)\partial_{q}y(qx)+A_{2}(x)y(x)+B(x),\quad \hbox{if $y$ is even,}
\end{array}
\right.$$
is continuous at zero. Furthermore, $A_{j}(x)$ is bounded on $I$. Consequently, there is $L>0$ such that $|A_{j}(x)|\leq L$ for all $x \in I$. We can see that $f$ satisfies the Lipschitz condition with Lipschitz constant $L$. Thus, $f(x,y_{1},y_{2})$ satisfies the conditions of Corollary $~\ref{cor1}$. Hence, there exists a unique solution of $~(\ref{IP6})$ on $J$.
\hfill$\blacksquare$
\end{dem}
\section{Second Order Homogeneous Linear $q$-Difference Equations Associated with the Rubin's $q$-Difference Operator $\partial_q$}
Consider the second order non-homogeneous $q$-difference equation associated with the Rubin's $q$-difference operator $\partial_q$ in a neighborhood of zero
\begin{equation}
\left\{
\begin{array}{cc}
q\,a_0(x)\,\partial_q^2y(qx)\,+\,a_1(x)\,\partial_qy(x)\,+\,a_2(x)y(x)&\;=\;b(x),\quad \hbox{if $y$ is odd;}\\
q\,a_0(x)\partial_q^2y(qx)\,+\,q\,a_1(x)\partial_qy(qx)\,+\,a_2(x)y(x)&\;=\;b(x),\quad \hbox{if $y$ is even,}
\end{array}
\right.
\end{equation}
with the initial conditions
\begin{equation}
\partial_{q}^{i-1}y(0)= b_{i};\quad b_{i} \in{\mathbb{C}},\; i=1,2
\end{equation}
where $a_i$, $i=0,1,2$, and $b$ are defined, continuous at zero and bounded on an interval $I$ containing zero such that $a_0(x)\neq 0$ for all $x\in I$. In this section, we shall study the second order homogeneous linear $q$-difference problem associated with the Rubin's $q$-difference operator $\partial_q$ of the form
\begin{equation}\label{order}
\left\{
\begin{array}{cc}
q\,a_0(x)\,\partial_q^2y(qx)\,+\,a_1(x)\,\partial_qy(x)\,+\,a_2(x)y(x)&\;=\;0,\quad \hbox{if $y$ is odd;}\\
q\,a_0(x)\partial_q^2y(qx)\,+\,q\,a_1(x)\partial_qy(qx)\,+\,a_2(x)y(x)&\;=\;0,\quad \hbox{if $y$ is even.}
\end{array}
\right.
\end{equation}
The following result summarizes some properties of $~(\ref{order})$ which we can state at once.
\begin{prop}\label{pr}
Let $x$ in a subinterval $J$ of $I$ which contains zero. Then
\begin{enumerate}
\item If $\phi_1$ and $\phi_2$ are two solutions of $~(\ref{order})$ and have the same parity, then
$$\phi(x)\;=\;c_1\phi_1(x)\,+\,c_2\phi_2(x),$$
where $c_1$ and $c_2$ are constants, is also a solution of $~(\ref{order})$.
\item If $\phi$ is a solution of $~(\ref{order})$ such that $\partial_q^{\,i-1}\phi(0)\,=\,0$, $i=1,2$, then
$$\phi(x)\;=\;0.$$
\end{enumerate}
\end{prop}
\begin{dem}
\begin{enumerate}
\item Since $\phi_1$ and $\phi_2$ are two solutions of Equation $~(\ref{order})$ and have the same parity, we distinguish two cases:
\begin{description}
\item[--] If $\phi_1$ and $\phi_2$ are odd, then, $\phi(x)\,=\,c_1\phi_1(x)\,+\,c_2\phi_2(x)$ is an odd function. Therefore, using the basic rules for $q$-differentiation, we get
\begin{align*}
q\,a_0(x)\,&\partial_q^{\,2}\phi(qx)\,+\,a_1(x)\,\partial_q\phi(x)\,+\,a_2(x)\,\phi(x)\\
\,=\;&q\,a_0(x)\partial_q^{\,2}\left[c_1\,\phi_1(qx)+c_2\,\phi_2(qx)\right]\,+\,a_1(x)\partial_q\left[c_1\,\phi_1(x)+c_2\,\phi_2(x)\right]\\
&\,+\;a_2(x)[c_1\,\phi_1(x)+c_2\,\phi_2(x)]\\
\,=\;&q\,a_0(x)\left[c_1\,\partial_q^{\,2}\phi_1(qx)+c_2\,\partial_q^2\phi_2(qx)\right]+a_1(x)\left[c_1\,\partial_q\phi_1(x)+c_2\,\partial_q\phi_2(x)\right]\\
&\,+\;a_2(x)[c_1\,\phi_1(x)+c_2\,\phi_2(x)]\\
\,=\;&c_1\,\left[q\,a_0(x)\partial_q^{\,2}\phi_1(qx)\,+\,a_1(x)\partial_q\phi_1(x)\,+\,a_2(x)\phi_1(x)\right]\\
&\,+\;c_2\,\left[q\,a_0(x)\partial_q^{\,2}\phi_2(qx)\,+\,a_1(x)\partial_q\phi_2(x)\,+\,a_2(x)\phi_2(x)\right]\\
\,=\;&0.
\end{align*}
\item[--] If $\phi_1$ and $\phi_2$ are even, then, $\phi(x)\,=\,c_1\,\phi_1(x)\,+\,c_2\,\phi_2(x)$ is an even function. Therefore, using the basic rules for $q$-differentiation, we get
\begin{align*}
q\,a_0(x)\,&\partial_q^{\,2}\phi(qx)\,+\,q\,a_1(x)\,\partial_q\phi(qx)\,+\,a_2(x)\,\phi(x)\\
\,=\;&q\,a_0(x)\partial_q^{\,2}\left[c_1\,\phi_1(qx)+c_2\,\phi_2(qx)\right]\,+\,q\,a_1(x)\partial_q\left[c_1\,\phi_1(qx)+c_2\,\phi_2(qx)\right]\\
&\,+\;a_2(x)[c_1\,\phi_1(x)+c_2\,\phi_2(x)]\\
\,=\;&q\,a_0(x)\left[c_1\,\partial_q^{\,2}\phi_1(qx)+c_2\,\partial_q^2\phi_2(qx)\right]+q\,a_1(x)\left[c_1\,\partial_q\phi_1(qx)+c_2\,\partial_q\phi_2(qx)\right]\\
&\,+\;a_2(x)[c_1\,\phi_1(x)+c_2\,\phi_2(x)]\\
\,=\;&c_1\,\left[q\,a_0(x)\partial_q^{\,2}\phi_1(qx)\,+\,q\,a_1(x)\partial_q\phi_1(qx)\,+\,a_2(x)\phi_1(x)\right]\\
&\,+\;c_2\,\left[q\,a_0(x)\partial_q^{\,2}\phi_2(qx)\,+\,q\,a_1(x)\partial_q\phi_2(qx)\,+\,a_2(x)\phi_2(x)\right]\\
\,=\;&0.
\end{align*}
\end{description}
Thus, $\phi(x)$ is also a solution of Equation $~(\ref{order})$.
\item The function $\phi_0(x)$ which is identically zero in $I$ clearly satisfies $~(\ref{order})$ and the initial conditions
$$\partial_q^{\,i-1}\phi_0(0)\;=\;0,\quad i=1,2.$$
Thus $\phi(x)$ and $\phi_0(x)$ satisfy the same initial conditions at zero and therefore, by Corollary $~\ref{cor11}$, there exists a unique solution of $~(\ref{order})$. Then, we have
$$\phi(x)\;=\;\phi_0(x)\;=\;0,$$
\end{enumerate}
for all $x$ in $I$.\hfill$\blacksquare$
\end{dem}
\begin{defn}
A set of two linearly independent solutions of $~(\ref{order})$ is called a fundamental set of it.
\end{defn}
The existence of fundamental sets of $~(\ref{order})$ is established in the following lemma.
\begin{lemma}\label{lem}
Let $b_{ij}$, $i,j=1,2$, be any real or complex numbers. For each $j=1,2$, let $\phi_j$ be the solution of $~(\ref{order})$ which satisfies the initial conditions
\begin{equation}
\partial_q^{\,i-1}\phi_j(0)\;=\;b_{ij},\quad i=1,2.
\end{equation}
Then a necessary and sufficient condition that $\{\phi_1,\phi_2\}$ is a fundamental set of $~(\ref{order})$ is that $\det (b_{ij})\neq 0$.
\end{lemma}
\begin{dem}
\underline{Necessary:} Let $\{\phi_1,\phi_2\}$ is a fundamental set but suppose that $\det (b_{ij}) = 0$. Then there are numbers $\alpha_j$, $j=1,2$, not all zero, such that
\begin{equation}\label{xx}
\left\{
\begin{array}{ll}
\alpha_1b_{11}\,+\,\alpha_2b_{12}&\;=\;0\\
\alpha_1b_{21}\,+\,\alpha_2b_{22}&\;=\;0.
\end{array}
\right.\end{equation}
Now define $\phi(x)\;=\;\alpha_1\phi_{1}(x)\,+\,\alpha_2\phi_{2}(x)$, for all $x$ in a subinterval $J$ of $I$ which contains zero.
By Proposition $~\ref{pr}$, $\phi$ is a solution of $~(\ref{order})$ and, by $~(\ref{xx})$, we have
$$\phi(0)\;=\;\partial_q\phi(0)\;=\;0.$$
Hence, by Proposition $~\ref{pr}$, $\phi(x)\,=\,0$. But, since the $\alpha_j$, $j=1,2$, are not all zero, this contradicts the linear independence of the $\phi_j(x)$, and so we must have $\det (b_{ij})\neq 0$.\\
\underline{Sufficient:} Let $\det (b_{ij})\neq 0$. Then we have to show that the relation
\begin{equation}\label{rr}\alpha_1\phi_{1}(x)\,+\,\alpha_2\phi_{2}(x)\;=\;0\end{equation}
is possible for all $x$ in $J$ only when $\alpha_1\,=\,\alpha_2\,=\,0$. Differentiating $(i-1)$ times $~(\ref{rr})$, $i=1,2$, and putting $x=0$, we obtain the equations $~(\ref{xx})$. Since $\det (b_{ij})\neq 0$, $~(\ref{xx})$ implies that $\alpha_1\,=\,\alpha_2\,=\,0$, as required.
\hfill$\blacksquare$ \end{dem}
\section{The $q$-Wronskian Associated with the Rubin's $q$-Difference Operator $\partial_q$}
To determine if two solutions of the second order $q$-difference equations associated with the Rubin's $q$-difference operator $\partial_q$ form a fundamental set, we introduce a $q$-analogue of the Wronskian.
\begin{defn}\label{wq}
Let $y_1$ and $y_2$ be two $q$-differentiable functions defined on a $q$-geometric set $\mathcal{A}$. The $q$-Wronskian associated with the Rubin's $q$-difference operator $\partial_q$ of the functions $y_1$ and $y_2$ which will be denoted by $W_q(y_1,y_2)$ is defined by:
\begin{description}
\item[--] If $y_1$ is even and $y_2$ is odd, we have
\begin{equation}\label{ff1}
W_q(y_1,y_2)(x)\;=\;{\displaystyle \begin{vmatrix}y_{1}(x)&y_{2}(x)\\q\,\partial_qy_{1}(qx)&\partial_qy_{2}(x) \end{vmatrix}}\;=\;y_{1}(x)\partial_qy_{2}(x)\,-q\,\,y_{2}(x)\partial_qy_{1}(qx).
\end{equation}
\item[--] If $y_1$ and $y_2$ are odd, we have
\begin{equation}\label{ff2}
W_q(y_1,y_2)(x)\;=\;{\displaystyle \begin{vmatrix}y_{1}(x)&y_{2}(x)\\\partial_qy_{1}(x)&\partial_qy_{2}(x) \end{vmatrix}}\;=\;y_{1}(x)\partial_qy_{2}(x)\,-\,y_{2}(x)\partial_qy_{1}(x).
\end{equation}
\item[--] If $y_1$ and $y_2$ are even, we have
\begin{equation}\label{ff3}
W_q(y_1,y_2)(x)\;=\;{\displaystyle \begin{vmatrix}y_{1}(x)&y_{2}(x)\\q\,\partial_qy_{1}(qx)&q\,\partial_qy_{2}(qx) \end{vmatrix}}
\;=\;q\,y_{1}(x)\partial_qy_{2}(qx)\,-\,q\,y_{2}(x)\partial_qy_{1}(qx).
\end{equation}
\end{description}
It is easy to see that, if $q$ tends to $1^{-}$, the $q$-Wronskian tends to the ordinary Wronskian
\begin{equation}\label{w}
W(y_1,y_2)(x)\;=\;{\displaystyle \begin{vmatrix}y_{1}(x)&y_{2}(x)\\y_{1}\,'(x)&y_{2}\,'(x) \end{vmatrix}}\;=\;y_{1}(x)y_{2}'(x)\,-\,y_{2}(x)y_{1}'(x).
\end{equation}
\end{defn}
\begin{corollary}
Let $y_1$ and $y_2$ be two differentiable functions defined on a $q$-geometric set $\mathcal{A}$.
The $q$-Wronskian which is defined by $~(\ref{wq})$ can be rewritten in the form
\begin{equation}\label{n2}
W_q(y_1,y_2)(x)\;=\;\dfrac{y_{2}(x)y_{1}(qx)\,-\,y_{1}(x)y_{2}(qx)}{(1-q)x},\quad x\in\mathcal{A}\backslash\{0\}.
\end{equation}
\end{corollary}
\begin{prop}\label{pl}
Let $y_1, y_2$ be two solutions of $~(\ref{order})$ defined on $\mathcal{A}$. Then, for all $x\in\mathcal{A}\backslash\{0\}$, we have
\begin{description}
\item[--] If $y_1$ and $y_2$ have opposite parity, then
\begin{equation}\label{pl1}
\partial_qW_q(y_1,y_2)(x)\;=\;y_1(x)\,\partial_q^2\,y_2(x)\,-\,y_2(x)\,\partial_q^2\,y_1(x).
\end{equation}
\item[--] If $y_1$ and $y_2$ have the same parity, then
\begin{equation}\label{pl1'}
\partial_qW_q(y_1,y_2)(x)\;=\;q\,y_1(qx)\,\partial_q^2\,y_{2}(qx)\,-\,q\,y_2(qx)\,\partial_q^2\,y_{1}(qx).
\end{equation}
\end{description}
\end{prop}
\begin{dem}
To prove $~(\ref{pl1})$, we involve several cases such as:
\begin{description}
\item[--] If $y_1$ is an even function and $y_2$ is an odd function, the $q$-Wronskian is given by $~(\ref{ff1})$. Using the fact that $\partial_qy_1$ is odd and $\partial_qy_2$ is even, we get by the help of $~(\ref{marwa2})$ and $~(\ref{marwa3})$ as follows
\begin{align*}
\partial_qW_q(y_1,y_2)(x)&\;=\;\partial_q\left[y_{1}(x)\partial_qy_{2}(x)\right]-q\,\partial_q\left[y_{2}(x)\partial_qy_{1}(qx)\right]\\
&\;=\;\partial_q\,y_2(q^{-1}x)\,\partial_qy_1(x)\,+\,y_1(x)\,\partial_q^2y_2(x)\\
&\qquad\;-\,q\,\left[q^{-1}\,\partial_q\,y_1(x)\,\partial_qy_2(q^{-1}x)\,+\,q^{-1}\,y_2(x)\,\partial_q^2y_1(x)\right]\\
&\;=\;y_1(x)\,\partial_q^2y_2(x)\,-\,y_2(x)\,\partial_q^2y_1(x).
\end{align*}
\item[--] If $y_1$ and $y_2$ are odd, the $q$-Wronskian is given by $~(\ref{ff2})$. Using the fact that $\partial_qy_1$ and $\partial_qy_2$ are even, we get by the help of $~(\ref{marwa1})$ as follows
\begin{align*}
\partial_qW_q(y_1,y_2)(x)&\;=\;\partial_q\left[y_{1}(x)\partial_qy_{2}(x)\right]\,-\,\partial_q\left[y_{2}(x)\partial_qy_{1}(x)\right]\\
&\;=\;q\,y_1(qx)\,\partial_q^2y_{2}(qx)\,-\,q\,y_2(qx)\,\partial_q^2y_{1}(qx).
\end{align*}
\item[--] If $y_1$ and $y_2$ are even, the $q$-Wronskian is given by $~(\ref{ff3})$. Using the fact that $\partial_qy_1$ and $\partial_qy_2$ are odd, we get by the help of $~(\ref{marwa1})$ as follows
\begin{align*}
\partial_qW_q(y_1,y_2)(x)&\;=\;q\,\partial_q\left[y_{1}(x)\partial_qy_{2}(qx)\right]\,-\,q\,\partial_q\left[y_{2}(x)\partial_qy_{1}(qx)\right]\\
&\;=\;q\,y_1(qx)\,\partial_q^2y_{2}(qx)\,-\,q\,y_2(qx)\,\partial_q^2y_{1}(qx).
\end{align*}
\end{description}
This completes the proof.
\hfill$\blacksquare$ \end{dem}
\begin{theorem}{\bf($q$-analogue of Abel's theorem)}\label{Abel}
Let $I$ be an interval containing zero. If $y_1, y_2$ are solutions of $~(\ref{order})$ in a subinterval $J$ of $I$, $J=[-h,h]$, $h > 0$, then their $q$-Wronskian associated with the Rubin's $q$-difference operator $\partial_q$ satisfies the linear first order $q$-difference equation as follows:
\begin{description}
\item[--] If $y_1$ and $y_2$ have opposite parity, then
\begin{equation}\label{Wqs1}
\partial_q\,W_q(y_1,y_2)(x)\;=\;-q^{-1}\,E(q^{-1}x)\,W_q(y_1,y_2)(q^{-1}x);
\end{equation}
\item[--] If $y_1$ and $y_2$ have the same parity, then
\begin{equation}\label{Wqs2}
\partial_q\,W_q(y_1,y_2)(x)\;=\;-E(x)\,W_q(y_1,y_2)(x);
\end{equation}
\end{description}
where
\begin{equation}\label{R(x)}
E(x)\;=\;\dfrac{a_1(x)\,+\,x(1-q)\,{a_2(x)}}{a_0(x)},
\end{equation}
for all $x\in J\backslash\{0\}$.
\end{theorem}
\begin{dem}
To prove $~(\ref{Wqs1})$ and $~(\ref{Wqs2})$, we involve several cases according to the parity of $y_1$ and $y_2$. Then, we have
\begin{description}
\item[--] If $y_1$ is an even function and $y_2$ is an odd function, from $~(\ref{order})$, we obtain
$$\partial_q^2y_{1}(x)\;=\;-\,\dfrac{q\,a_1(q^{-1}x)\partial_qy_1(x)+a_2(q^{-1}x)y_1(q^{-1}x)}{q\,a_0(q^{-1}x)}$$
and
$$\partial_q^2y_{2}(x)\;=\;-\dfrac{a_1(q^{-1}x)\partial_qy_2(q^{-1}x)+a_2(q^{-1}x)y_2(q^{-1}x)}{q\,a_0(q^{-1}x)}.$$
It follows from $~(\ref{pl1})$ that
\begin{align*}
\partial_qW_q(y_1,y_2)(x)\;=&\;y_1(x)\,\partial_q^2y_{2}(x)\,-\,y_2(x)\,\partial_q^2y_{1}(x)\\
\;=&\;-q^{-1}\,\dfrac{a_1(q^{-1}x)}{a_0(q^{-1}x)}\left(y_1(x)\,\partial_qy_2(q^{-1}x)\,-\,q\,y_2(x)\,\partial_qy_1(x)\right)\\
&\;-\;q^{-1}\,\dfrac{a_2(q^{-1}x)}{a_0(q^{-1}x)}\left(y_1(x)\,y_2(q^{-1}x)\,-\,y_2(x)\,y_1(q^{-1}x)\right).
\end{align*}
Hence, using $~(\ref{ff1})$ and $~(\ref{n2})$, we obtain
$$\partial_qW_q(y_1,y_2)(x)\;=\;-\,q^{-1}\,\left[\dfrac{a_1(q^{-1}x)\,+\,q^{-1}x(1-q)\,{a_2(q^{-1}x)}}{a_0(q^{-1}x)}\right]\,W_q(y_1,y_2)(q^{-1}x).$$
From $~(\ref{R(x)})$, we conclude that
$$\partial_qW_q(y_1,y_2)(x)\;=\;-\,q^{-1}\,E(q^{-1}x)\,W_q(y_1,y_2)(q^{-1}x),\quad x\in J\backslash\{0\}.$$
\item[--] If $y_1$ and $y_2$ are odd, from $~(\ref{order})$, we obtain
$$\partial_q^2y_{1}(qx)\;=\;-\dfrac{a_1(x)\partial_qy_1(x)+a_2(x)y_1(x)}{q\,a_0(x)}$$
and
$$\partial_q^2y_{2}(qx)\;=\;-\dfrac{a_1(x)\partial_qy_2(x)+a_2(x)y_2(x)}{q\,a_0(x)}.$$
It follows from $~(\ref{pl1'})$ that
\begin{align*}
\partial_qW_q(y_1,y_2)(x)\;=&\;q\,y_1(qx)\,\partial_q^2y_{2}(qx)\,-\,q\,y_2(qx)\,\partial_q^2y_{1}(qx)\\
\;=&\;-\dfrac{a_1(x)}{a_0(x)}\left(y_1(qx)\,\partial_qy_2(x)\,-\,y_2(qx)\,\partial_qy_1(x)\right)\\
&\;-\;\dfrac{a_2(x)}{a_0(x)}\left(y_1(qx)\,y_2(x)\,-\,y_2(qx)\,y_1(x)\right).
\end{align*}
Hence, using $~(\ref{ff2})$ and $~(\ref{n2})$, we obtain
$$\partial_qW_q(y_1,y_2)(x)\;=\;-\,\left[\dfrac{a_1(x)\,+\,(1-q)\,x\,{a_2(x)}}{a_0(x)}\right]\,W_q(y_1,y_2)(x).$$
From $~(\ref{R(x)})$, we conclude that
$$\partial_qW_q(y_1,y_2)(x)\;=\;-\,E(x)\,W_q(y_1,y_2)(x),\quad x\in J\backslash\{0\}.$$
\item[--] If $y_1$ and $y_2$ are even, from $~(\ref{order})$, we obtain
$$\partial_q^2y_{1}(qx)\;=\;-\dfrac{q\,a_1(x)\partial_qy_1(qx)+a_2(x)y_1(x)}{q\,a_0(x)}$$
and
$$\partial_q^2y_{2}(qx)\;=\;-\dfrac{q\,a_1(x)\partial_qy_2(qx)+a_2(x)y_2(x)}{q\,a_0(x)}.$$
It follows from $~(\ref{pl1'})$ that
\begin{align*}
\partial_qW_q(y_1,y_2)(x)\;=&\;q\,y_1(qx)\,\partial_q^2y_{2}(qx)\,-\,q\,y_2(qx)\,\partial_q^2y_{1}(qx)\\
\;=&\;-\dfrac{a_1(x)}{a_0(x)}\left(q\,y_1(qx)\,\partial_qy_2(qx)\,-\,q\,y_2(qx)\,\partial_qy_1(qx)\right)\\
&\;-\;\dfrac{a_2(x)}{a_0(x)}\left(y_1(qx)\,y_2(x)\,-\,y_2(qx)\,y_1(x)\right).
\end{align*}
Hence, using $~(\ref{ff3})$ and $~(\ref{n2})$, we obtain
$$\partial_qW_q(y_1,y_2)(x)\;=\;-\,\left[\dfrac{a_1(x)\,+\,(1-q)\,x\,{a_2(x)}}{a_0(x)}\right]\,W_q(y_1,y_2)(x).$$
From $~(\ref{R(x)})$, we conclude that
$$\partial_qW_q(y_1,y_2)(x)\;=\;-\,E(x)\,W_q(y_1,y_2)(x),\quad x\in J\backslash\{0\}.$$
\end{description}
This finishes the proof of the theorem.
\hfill$\blacksquare$ \end{dem}
The following theorem gives a $q$-type Liouville’s formula for the $q$-Wronskian associated with the Rubin’s $q$-difference operator $\partial_q$.
\begin{theorem}{\bf($q$-Liouville’s formula)}\label{li}
Suppose that $x(1\,-\,q)\,E(x)\neq -1$ for all $x$ in a subinterval $J$ of $I$ which contains zero. Then the $q$-Wronskian of any set of solutions $\{y_1,y_2\}$ of $~(\ref{order})$ is given by
\begin{equation}\label{Liou}
W_q(y_1,y_2)(x)\;=\;\dfrac{W_q(y_1,y_2)(0)}{\D\prod_{k=0}^{+\infty}\left(1\,+\,x(1\,-\,q)q^k\,E(xq^k)\right)},\quad x\in J
\end{equation}
where $E(x)$ is defined by $~(\ref{R(x)})$.
\end{theorem}
\begin{dem}
Suppose that $x(1\,-\,q)\,E(x)\neq -1$ for all $x$ in a subinterval $J$ of $I$ which contains zero. Here we distinguish two cases:
\begin{description}
\item[--] In case $y_1$ and $y_2$ have opposite parity, the $q$-Wronskian $~(\ref{ff1})$ is even. Using Theorem$~\ref{Abel}$, the $q$-Wronskian $W_q(y_1,y_2)(x)$ satisfies the linear first order $q$-difference equation $~(\ref{Wqs1})$. Then, we get
$$W_q(y_1,y_2)(x)\;=\;(1\,+\,q^{-1}\,x(1\,-\,q)\,E(q^{-1}x))\,W_q(y_1,y_2)(q^{-1}x),\quad x\in J\backslash\{0\}.$$
Replacing $x$ by $qx$ in the previous equation, we obtain
$$W_q(y_1,y_2)(qx)\;=\;(1\,+\,x(1\,-\,q)\,E(x))\,W_q(y_1,y_2)(x),\quad x\in J\backslash\{0\}.$$
\item[--] In case $y_1$ and $y_2$ have the same parity, the $q$-Wronskian is odd. Using Theorem $~\ref{Abel}$, the $q$-Wronskian $W_q(y_1,y_2)(x)$ satisfies the linear first order $q$-difference equation $~(\ref{Wqs2})$. Then, we get
$$W_q(y_1,y_2)(qx)\;=\;(1\,+\,x(1\,-\,q)\,E(x))\,W_q(y_1,y_2)(x),\quad x\in J\backslash\{0\},$$
\end{description}
Hence, under the assumption $1\,+\,x(1\,-\,q)\,E(x)\,\neq\, 0$, we deduce for all solutions $y_1$ and $y_2$ that
$$W_q(y_1,y_2)(x)\;=\;\dfrac{W_q(y_1,y_2)(qx)}{1\,+\,x(1\,-\,q)\,E(x)}.$$
Now we use induction on $n$ to see that we have for $n\in\mathbb{N} $,
$$W_q(y_1,y_2)(x)\;=\;\dfrac{W_q(y_1,y_2)(xq^n)}{\D\prod_{k=0}^{n-1}\left(1\,+\,x(1\,-\,q)q^k\,E(xq^k)\right)}.$$
Since all functions $\dfrac{a_j}{a_0}$, $1\leq j\leq n$, are continuous at zero, then $\D\sum_{k=0}^{+\infty}q^k\,|E(xq^k)|$ is convergent. Consequently,
$$\D\prod_{k=0}^{n-1}\left(1\,+\,x(1\,-\,q)q^k\,E(xq^k)\right)\quad \text{converges for every $x\in J$}.$$
Letting $n \rightarrow \infty$ and noting that $0 < q < 1$, one gets $q^n$ tends to 0. Thus, using the continuity of $W_q(y_1,y_2)(x)$ at zero, $~(\ref{Liou})$ follows.
\hfill$\blacksquare$ \end{dem}
\begin{corollary}\label{bb5}
Let $\{y_1,y_2\}$ be a set of solutions of $~(\ref{order})$ in some subinterval $J$ of $I$ which contains zero. Then $W_q(y_1,y_2)(x)$ is either never zero in $I$ if and only if $\{y_1,y_2\}$ is a fundamental set of $~(\ref{order})$.
\begin{dem}
From Lemma $~\ref{lem}$, the functions $\{y_1,y_2\}$ form a fundamental set of $~(\ref{order})$ if and only if $W_q(0)\,\neq\,0$. Hence, the result is a direct consequence of Theorem $~\ref{li}$.
\hfill$\blacksquare$ \end{dem}
\end{corollary}
\section{Homogeneous Equations with constant coefficients}
We start now to study the second order homogeneous linear $q$-difference equations associated with the Rubin's $q$-difference operator that contain constant coefficients only:
\begin{equation}\label{order"}
a\,\partial_{q}^{2}y(x)\,+\,b\,y(x)=\,0,\quad a \neq 0,\end{equation}
with the initial conditions
\begin{equation}\label{vb}
y_1(0)\,=\,1,\quad \partial_qy_1(0)\,=\,0\quad \text{and} \quad y_2(0)\,=\,0,\quad \partial_qy_2(0)\,=\,1,\end{equation}
respectively, where $a$ and $b$ are constants. In the following result, we will solve the equation $~(\ref{order"})$.
\begin{prop}\label{fgf'}
Let $x\in\mathbb{R}_q$. Then, the $q$-difference equation $~(\ref{order"})$ have two distinct solutions
$$y_1(x)\;=\;\cos(\sqrt{\dfrac{b}{a}}x,q^2)\quad\text{and}\quad y_2(x)\;=\;\sqrt{\dfrac{a}{b}}\sin(\sqrt{\dfrac{b}{a}}x,q^2),$$
respectively, where $a$ and $b$ are constants, $a \neq 0$.
\end{prop}
\begin{dem}
First, it easy to see that, if the solution $y(x)$ of the $q$-difference problem $~(\ref{order"})$ is an analytic function at the origin, it can be developed in entire powers series. Then we have
\begin{equation}
y(x)\;=\;\D\sum_{n=0}^{\infty}a_n\,x^n,\quad x\in\mathbb{R}_q.
\end{equation}
Here we distinguish two cases:
\begin{description}
\item[--] If $n\,=\,2p$, $p\in\mathbb{N}$, the solution $y_1$ of $~(\ref{order"})$ is even. Then, we have
\begin{equation}\label{fg1}
\partial_q^2y_1(x)\;=\;\D\sum_{p\geq0}^{}\,q^{-2(p+1)}\dfrac{1-q^{2p+2}}{1-q}\,\dfrac{1-q^{2p+1}}{1-q}\,a_{2p+2}\,x^{2p},\quad p\in\mathbb{N}.
\end{equation}
Loading $~(\ref{fg1})$ in $~(\ref{order"})$ and equating the coefficients of $x^{2p}$, one obtains
$$a\,q^{-2(p+1)}\,\dfrac{1-q^{2p+2}}{1-q}\,\dfrac{1-q^{2p+1}}{1-q}\,a_{2p+2}\;=\;-b\,a_{2p},\quad p\in\mathbb{N}.$$
Hence, We get the following recurrence relation for the coefficients:
$$a_{2p}\;=\;-\,\dfrac{b}{a}\,q^{2p}\,\dfrac{(1-q)^2}{(1-q^{2p})\,(1-q^{2p-1})}\,a_{2p-2},\quad p\geq 1.$$
So, by induction on $p$ and the fact that $ a_{0}\,=\,1$, we obtain:
$$a_{2p}\;=\;(-1)^p\,q^{p(p+1)}\,\left(\dfrac{b}{a}\right)^p\,\dfrac{(1-q)^{2p}}{(q,q)_{2p}}\;=\;(-1)^p\,\dfrac{q^{p(p+1)}}{[2p]_q!}\,\,\left(\sqrt{\dfrac{b}{a}}\right)^{2p}\quad p\in\mathbb{N}.$$
From the definitions $~(\ref{cos})$, we get
$$y_1(x)\;=\;\cos(\sqrt{\dfrac{b}{a}}x,q^2).$$
\item[--] If $n\,=\,2p+1$, $p\in\mathbb{N}$, the solution $y_2$ of $~(\ref{order"})$ is odd. Then, we have
\begin{equation}\label{fg4}
\partial_{q}^{2}y_2(x)\;=\;\D\sum_{p\geq0}^{}\,q^{-2(p+1)}\dfrac{1-q^{2p+3}}{1-q}\,\dfrac{1-q^{2p+2}}{1-q}\,a_{2p+3}\,x^{2p+1},\quad p\in\mathbb{N}.
\end{equation}
Loading $~(\ref{fg4})$ in $~(\ref{order"})$ and equating the coefficients of $x^{2n+1}$, one obtains
$$a\,q^{-2(p+1)}\,\dfrac{1-q^{2p+3}}{1-q}\,\dfrac{1-q^{2p+2}}{1-q}\,a_{2p+3}\;=\;-b\,a_{2p+1},\quad p\in\mathbb{N}.$$
Hence, we get the following recurrence relation for the coefficients:
$$a_{2p+1}\;=\;-\,\dfrac{b}{a}\,q^{2p}\,\dfrac{(1-q)^2}{(1-q^{2p+1})\,(1-q^{2p})}\,a_{2p-1},\quad p\geq 1.$$
So, by induction on $p$ and the fact that $ a_{1}\,=\,1$, we obtain:
\begin{align*}
a_{2p+1}\;=&\;(-1)^p\,q^{p(p+1)}\,\left(\dfrac{b}{a}\right)^p\,\dfrac{(1-q)^{2p+1}}{(q,q)_{2p+1}}\\
\;=&\;\sqrt{\dfrac{a}{b}}\,(-1)^p\,\dfrac{q^{p(p+1)}}{[2p+1]_q!}\,\left(\sqrt{\dfrac{b}{a}}\right)^{2p+1},\quad p\in\mathbb{N}.\end{align*}
From the definitions $~(\ref{sin})$, we get
$$y_2(x)\;=\;\sqrt{\dfrac{a}{b}}\sin(\sqrt{\dfrac{b}{a}}x,q^2).$$
\end{description}
Then, the functions $y_1(x)$ and $y_2(x)$, $x\in\mathbb{R}_q$, are solutions of $~(\ref{order"})$. This completes the proof.
\hfill$\blacksquare$
\end{dem}
\begin{corollary}\label{nb}
Let $x\in\mathbb{R}_q$. Then Equation $~(\ref{order"})$ can be written another way:
\begin{equation}\label{eex1'}
\left\{
\begin{array}{cc}
a\,\partial_q^2y(qx)\,-\,b(1-q)\,x\,\partial_qy(qx)\,+\,b\,y(x)\;=\;0,\quad \hbox{if $y$ is odd;}\\
a\,\partial_q^2y(qx)\,-\,b\,q\,(1-q)\,x\,\partial_qy(qx)\,+\,b\,y(x)\;=\;0,\quad \hbox{if $y$ is even,}
\end{array}
\right.
\end{equation}
where $a$ and $b$ are constants, $a \neq 0$.
\end{corollary}
\begin{dem}
Two cases arise according to $y$ is even or odd. Then we have
\begin{description}
\item[--] If $y$ is odd, the $\partial_qy$ would be an even function. Detailing the Rubin’s $q$-difference operator $\partial_q$ in $~(\ref{order"})$, the equation becomes
\begin{equation}\label{eex41}
a\,\partial_qy(q^{-1}x)\,-\,a\,\partial_qy(x)\,+\,b\,(1-q)\,x\,y(x)\;=\;0.\end{equation}
Applying the $\partial_q\,\Lambda_q$ derivative in the previous equation and using $~(\ref{marwa3})$, one gets
$$a\,\partial_q^2y(x)\,-\,a\,q\,\partial_q^2y(qx)\,+\,b\,(1-q)\,y(x)\,+\,b\,q\,(1-q)\,x\,\partial_qy(x)\;=\;0.$$
Loading $~(\ref{order"})$ in the last equation and next dividing the lefthand by $(-q)$, we obtain
$$a\,\partial_q^2y(qx)\,-\,b\,(1-q)\,x\,\partial_qy(x)\,+\,b\,y(x)\;=\;0.$$
\item[--] If $y$ is even, the $\partial_qy$ would be an odd function. Detailing the Rubin’s $q$-difference operator $\partial_q$ in $~(\ref{order"})$, the equation reads
\begin{equation}\label{eex4}
a\,\partial_qy(x)\,-\,a\,\partial_qy(qx)\,+\,b\,(1-q)\,x\,y(x)\;=\;0.\end{equation}
Applying the $\partial_q$ derivative in the previous equation and using $~(\ref{marwa1})$, one gets
$$a\,\partial_q^2y(x)\,-\,a\,q\,\partial_q^2y(qx)\,+\,b\,(1-q)y(x)\,+\,b\,q^2\,(1-q)\,x\,\partial_qy(qx)\;=\;0.$$
Loading $~(\ref{order"})$ in the last equation and next dividing the lefthand by $(-q)$, we obtain
$$a\,\partial_q^2y(qx)\,-\,b\,q\,(1-q)\,x\,\partial_qy(qx)\,+\,b\,y(x)\;=\;0.$$
\end{description}
Thus the result hold.
\hfill$\blacksquare$
\end{dem}
Let us now calculate the $q$-Wronskian of the solutions of $~(\ref{order"})$ in the following result:
\begin{prop}\label{nb1}
Let $x\in \mathbb{R}_q$. Then the $q$-Wronskian $W_q(x)$ of the solutions of the $q$-difference equation $~(\ref{order"})$ subject to the initial conditions $~(\ref{vb})$ is given by
\begin{equation}
W_q(x)\;=\;1,\quad x\in\mathbb{R}_q.
\end{equation}
Moreover, the set $\{y_1, y_2\}$ form a fundamental set of $~(\ref{order"})$.
\end{prop}
\begin{dem}
Comparing $~(\ref{eex1'})$ with $~(\ref{order})$, we get
$$a_0(x)\,=\,aq^{-1},\quad a_1(x)\,=\,-b\,x\,(1-q)\quad \text{and}\quad a_2(x)\,=\,b,$$
where $a$, $b$ are constants and $a\neq0$. Using the formula $~(\ref{R(x)})$, we obtain
$$E(x)\;=\;0,\quad x\in\mathbb{R}_q.$$
So, from the $q$-Liouville’s formula $~(\ref{Liou})$, we deduce that
$$W_q(x)\;=\;W_q(0),\quad x\in\mathbb{R}_q.$$
But, it follows from Proposition $~\ref{fgf'}$ that the solutions of the $q$-difference equation $~(\ref{order"})$ are the functions
$$y_1(x)\;=\;\cos(\sqrt{\dfrac{b}{a}}x,q^2)\quad\text{and}\quad y_2(x)\;=\;\sqrt{\dfrac{a}{b}}\sin(\sqrt{\dfrac{b}{a}}x,q^2),$$
with the initial conditions $~(\ref{vb})$, where $a$ and $b$ are constants, $a \neq 0$.
Then, by $~(\ref{ff1})$, we get
\begin{align*}
W_q(0)&\;=\;W_q(\cos(\sqrt{\dfrac{b}{a}}x,q^2),\sqrt{\dfrac{a}{b}}\sin(\sqrt{\dfrac{b}{a}}x,q^2))\mid_{x=0}\\
&\;=\;\left(\cos(\sqrt{\dfrac{b}{a}}x,q^2)\cos(\sqrt{\dfrac{b}{a}}x,q^2)
\,+\,q\,\sin(\sqrt{\dfrac{b}{a}}x,q^2)\sin(q\,\sqrt{\dfrac{b}{a}}x,q^2)\right)\mid_{x=0}\\
&\;=\;1.
\end{align*}
Consequently, $W_q(x)\,=\,1$, for all $x\in \mathbb{R}_q$. Moreover, by $~(\ref{bb5})$, the set $\{y_1, y_2\}$ form a fundamental set of $~(\ref{order"})$ and the result is proved.
\hfill$\blacksquare$
\end{dem}
\section{Examples}
\begin{ex} We purpose to solve the following $q$-difference equation associated with the Rubin’s $q$-difference operator
\begin{equation}\label{fg}
\partial_q^2y(x)\,+\,y(x)\,=\,0,\quad x\in\mathbb{R}_q
\end{equation}
with the initial conditions
$$y_1(0)\,=\,1,\quad \partial_qy_1(0)\,=\,0\quad \text{and} \quad y_2(0)\,=\,0,\quad \partial_qy_2(0)\,=\,1,$$
respectively.\\
By Proposition$~\ref{fgf'}$, the solutions of $~(\ref{fg})$ are the functions $\cos(x,q^2)$, $\sin(x,q^2)$, $x\in\mathbb{R}_q$, respectively.\\
From Proposition $~\ref{nb1}$, the $q$-Wronskian $W_q(x)$ of the solutions of the $q$-difference equation $~(\ref{fg})$ is given by
$$W_q(x)\;=\;1,\quad x\in\mathbb{R}_q.$$
consequently, the set $\{\cos(x,q^2)$, $\sin(x,q^2)\}$ form a fundamental set of $~(\ref{fg})$.\end{ex}
\begin{ex} We define a pair of basic $q$-trigonometric functions by studying the solutions of the
second order q-difference equation associated with the Rubin’s $q$-difference operator
\begin{equation}\label{fg0}
q\,\partial_q^2y(x)\,+\,y(x)\,=\,0,\quad x\in\mathbb{R}_q
\end{equation}
with the initial conditions
$$y_1(0)\,=\,1,\quad \partial_qy_1(0)\,=\,0\quad \text{and} \quad y_2(0)\,=\,0,\quad \partial_qy_2(0)\,=\,1,$$
respectively.\\
By Proposition$~\ref{fgf'}$, the functions $\cos(q^{-1/2}x,q^2)$, $q^{1/2}\,\sin(q^{-1/2}x,q^2)$, $x\in\mathbb{R}_q$, are solutions of $~(\ref{fg0})$, respectively.\\
From Proposition $~\ref{nb1}$, the $q$-Wronskian $W_q(x)$ of the solutions of the $q$-difference equation $~(\ref{fg0})$ is given by
$$W_q(x)\;=\;1,\quad x\in\mathbb{R}_q.$$
consequently, the set $\{\cos(q^{-1/2}x,q^2)$, $q^{1/2}\,\sin(q^{-1/2}x,q^2)$ form a fundamental set of $~(\ref{fg0})$.\end{ex} | 23,625 |
\begin{document}
\title[]{On conjugacy of Cartan subalgebras
in extended affine Lie algebras}
\author{V. Chernousov}
\address{Department of Mathematics, University of Alberta,
Edmonton, Alberta T6G 2G1, Canada}
\thanks{ V. Chernousov was partially supported by the Canada Research
Chairs Program and an NSERC research grant} \email{[email protected]}
\author{E. Neher}
\address{Department of Mathematics and Statistics, University of Ottawa,
Ottawa, Ontario K1N 6N5, Canada}
\thanks{E.~Neher was partially supported by a Discovery grant from NSERC}
\email{[email protected]}
\author{A. Pianzola}
\address{Department of Mathematics, University of Alberta,
Edmonton, Alberta T6G 2G1, Canada.
\newline
\indent Centro de Altos Estudios en Ciencia Exactas, Avenida de Mayo 866, (1084) Buenos Aires, Argentina.}
\thanks{A. Pianzola wishes to thank NSERC and CONICET for their
continuous support}\email{[email protected]}
\author{U. Yahorau}
\address{Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada\newline \indent
present address: Department of Mathematics and Statistics, University of
Ottawa, Ottawa, Ontario K1N 6N5, Canada} \email{[email protected]}
\begin{abstract} That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of the most elegant results in mathematics. The definition of the root system is done by fixing a Cartan subalgebra of the given Lie algebra. The remarkable fact is that (up to isomorphism) this construction is independent of the choice of the Cartan subalgebra. The modern way of establishing this fact is by showing that all Cartan subalgebras are conjugate.
For symmetrizable Kac-Moody Lie algebras, with the appropriate definition
of Cartan subalgebra, conjugacy has been established by Peterson and Kac.
An immediate consequence of this result is that the root systems and
generalized Cartan matrices are invariants of the Kac-Moody Lie algebras.
The purpose of this paper is to establish conjugacy of Cartan subalgebras
for extended affine Lie algebras; a natural class of Lie algebras that
generalizes the finite-dimensional simple Lie algebra and affine Kac-Moody
Lie algebras.
\end{abstract}
\maketitle
\section*{Introduction}
Let $\gg$ be a finite-dimensional split simple Lie algebra over a field $k$
of characteristic $0$, and let $\bG$ be the simply connected
Chevalley-Demazure algebraic group associated to $\gg$. Chevalley's theorem
(\cite[VIII, \S3.3, Cor de la Prop.~10]{bou:Lie78}) asserts that all split
Cartan subalgebras $\h$ of $\gg$ are conjugate under the adjoint action of
$\bG(k)$ on $\gg.$ This is one of the central results of classical Lie
theory. One of its immediate consequences is that the corresponding root
system is an invariant of the Lie algebra (i.e., it does not depend on the
choice of Cartan subalgebra).
We now look at the analogous question in the infinite dimensional set up as it relates to extended affine Lie algebras (EALAs for short). We assume henceforth that $k$ is algebraically closed, but the reader should keep in mind that our results are more akin to the setting of Chevalley's theorem for general $k$ than to conjugacy of Cartan subalgebras in finite-dimensional simple Lie algebras over algebraically closed fields.
The role of $(\gg, \h)$ is now played by a pair $(E,H)$ consisting of a Lie
algebra $E$ and a ``Cartan subalgebra" $H$. There are other Cartan
subalgebras, and the question is whether they are conjugate and, if so, under
the action of which group.
The first example is that of untwisted affine Kac-Moody Lie algebras. Let $R
= k[t^{\pm 1}]$. Then
\begin{equation}\label{KacMoody}
E = \gg \ot_k R \oplus kc \oplus kd
\end{equation}
and \begin{equation}\label{cartan} H = \mathfrak{h} \ot 1 \oplus kc \oplus
kd.
\end{equation}
The relevant information is as follows. The $k$-Lie algebra $\gg \ot_k R
\oplus kc$ is a central extension (in fact the universal central extension)
of the $k$-Lie algebra $\gg \ot_k R$. The derivation $d$ of $\gg \ot_k R$
corresponds to the degree derivation $t d/dt$ acting on $R$. Finally
$\mathfrak{h}$ is a fixed Cartan subalgebra of $\gg.$ The nature of $H$ is
that it is abelian, it acts $k$-diagonalizably on $E$, and it is maximal with
respect to these properties. Correspondingly, these algebras are called MADs
(Maximal Abelian Diagonalizable) subalgebras. A celebrated theorem of
Peterson and Kac \cite{PK} states that all MADs of $E$ are conjugate (under
the action of a group that they construct which is the analogue of the simply
connected group in the finite-dimensional case). Similar results hold for the
twisted affine Lie algebras. These algebras are of the form
$$ E = L \oplus kc \oplus kd. $$
The Lie algebra $L$ is a loop algebra $L = L(\gg, \si)$ for some finite
order automorphism $\si$ of $\gg$ (see \ref{ssec:rev-lt} below for details).
If $\si$ is the identity, we are in the untwisted case. The ring $R$ can be
recovered as the centroid of $L$.
Extended affine Lie algebras can be thought of as multi-variable
generalizations of finite-dimensional simple Lie algebras and affine
Kac-Moody algebras. For example, taking $R=k[t_1^{\pm 1}, \ldots, t_n^{\pm
1}]$ in \eqref{KacMoody} and increasing $kc$ and $kd$ correspondingly leads
to toroidal algebras, an important class of examples of EALAs. But as is
already the case for affine Kac-Moody algebras, there are many interesting
examples where $\g\ot_k R$ is replaced by a more general algebra, a so-called
Lie torus (see \ref{def:lietor}).
In the EALA set up, the Lie algebras $\gg$ as above are the case of nullity
$n = 0$, while the affine Lie algebras are the case of nullity $n = 1$. In
higher nullity $n$ we have $R = k[t_1^{\pm 1},\ldots,t_\ell^{\pm 1}]$ for
some $\ell \leq n,$ where again $R$ is the centroid of the centreless core
$E_{cc}$ of the given EALA. Most of our work will concentrate in the case
when $\ell = n$. In this situation $E_{cc}$ is finitely generated as a module
over the centroid $R$ (called the {\it fgc condition} in EALA theory). We
hasten to add that the non-fgc algebras are fully understood and classified
(see \ref{lietor-prop} below), but it is presently not known if our conjugacy
theorem holds in this case. The crucial result about the fgc case is that
$E_{cc}$ is necessarily a multiloop algebra, hence a twisted form of $\gg
\ot_k R$ for some (unique) $\gg$. This allows methods from Galois cohomology
to be used in the study of the algebras under consideration (all of this,
with suitable references, will be explained in the main text).
Part of the properties of an EALA $(E,H)$ is a root space decomposition:
$E=\bigoplus_{\al \in \Psi} E_\al$ with $E_0 = H$. The ``root system" $\Psi$
is an example of an extended affine root system.
The main question, of course, is whether $\Psi$ is an invariant of $E$. In
other words, if $H'$ is a subalgebra of $E$ for which the pair $(E, H')$ is
given an EALA structure, is the resulting root system $\Psi'$ isomorphic (in
the sense of [extended affine] root systems) to $\Psi$? That this is true
follows immediately from the main result of our paper.
\begin{theorem}[Theorem {\ref{main-res}}]\label{main} Let $(E,H)$ be an extended affine Lie algebra of fgc type. Assume $E$ admits the second structure $(E,H')$ of an extended affine Lie algebra.
Then $H$ and $H'$ are conjugate, i.e., there exists a $k$-linear automorphism
$f$ of the Lie algebra $E$ such that $f(H)=H'$.
\end{theorem}
The main idea of the proof is as follows. Just as for the affine algebras, an
EALA $E$ can be written in the form $E = L \oplus C \oplus D$. Unlike the
affine case, starting with $L$ (which is a multiloop algebra given our fgc
assumption), one can construct an infinite number of $E's$. The exact nature
of all possible $C$ and $D$, and what the resulting Lie algebra structure is,
has been described in works by one of the authors (Neher). For the reader's
convenience we will recall this construction below. By the main result of
\cite{CGP} one knows that conjugacy holds for $L$. The challenge, which is
far from trivial, is to ``lift" this conjugacy to $E$. It worth noting that
\cite{PK} proceeds to some extend in the opposite direction. They establish
conjugacy ``upstairs", i.e. for $E$, and use this to obtain conjugacy
``downstairs", i.e. for $L$. It is also worth emphasizing that in the affine
case, the most important and useful result is conjugacy upstairs. The same
consideration applies to EALAs.
Built into the EALA definition is the existence of a certain ideal, the
so-called core $E_c$ of an EALA $(E,H)$. For example, for $E$ as in
\eqref{KacMoody} we have $E_c = \g \ot_k R \oplus kc$, while in the
realization $E=L \oplus C \oplus D$ of above the core is $E_c = L \oplus C$.
An important step in our proof of Theorem~\ref{main} is to show in
Corollary~\ref{cores are the same} that the cores of two EALA structures on
$E$ are the same, not only isomorphic. It then follows immediately that the
core $E_c$ of an EALA $(E,H)$ is stable under automorphisms of $E$
(Proposition~\ref{automorphism gives another EALA}). These new structural
results are true for any, not necessarily fgc EALA.
A priori, it is not clear at all that conjucacy at the level of the
centreless core can be ``lifted" to the EALA. As a rehearsal to get insight
into the difficulties that this question poses it is natural to look at the
case of EALAs of nullity 1, which are precisely the affine Kac-Moody Lie
algebras. This is the content of \cite{CGPY}. The positive answer on nullity
1 motivated us to try to tackle the general case, which resulted in the
present work. It is worth mentioning that the methods needed to establish the
general case are far more delicate than those used in \cite{CGPY}.
\medskip
{\bf Notation}: We suppose throughout that $k$ is a field of characteristic
$0$. Starting with section \S\ref{sec:subEALA} we assume that $k$ is
algebraically closed.
For convenience $\ot = \ot_k.$
\section{Some general results}
Some of the key results needed later to establish our main theorem are true
and easier to prove in a more general setting. This is the purpose of this
section.
Throughout $L$ will denote a Lie algebra over $k$.
\subsection{Cohomology}\label{cohom} Let $V$ be an $L$-module. We denote by $\rmZ^2(L, V)$ the $k$-space of $2$-cocycles of $L$ with coefficients in $V$. Its elements consist of alternating maps $\si\co L \times L \to V$ satisfying the cocycle condition $(l_i \in L)$
\begin{equation}\label{cohomo1}
\begin{split}
& l_1 \cdot \si(l_2, l_3) + l_2 \cdot \si(l_3, l_1) + l_3 \cdot \si(l_1, l_2)
\\ &\qquad = \si([l_1, l_2], l_3) + \si([l_2, l_3], l_1) + \si([l_3, l_1], l_2) \end{split} \end{equation}
Given such a $2$-cocycle $\si$, the vector space $L \oplus V$ becomes a Lie
algebra with respect to the product
\[
[l_1 + v_1, \, l_2 + v_2] = [l_1, l_2]_L + \big( l_1 \cdot v_2 - l_2 \cdot v_1 + \si(l_1, l_2) \big)
\]
We will denote this Lie algebra by $L \oplus_\si V$. Note that the projection
onto the first factor $\pr_L \co L \oplus_\si V \to L$ is an epimorphism of
Lie algebras whose kernel is the abelian ideal $V$. Note that $L$ is not
necessarily a subalgebra of $L \oplus_\si V$.
A special case of this construction is the situation when $V$ is a trivial
$L$-module. In this case a $2$-cocycle will be called a {\em central
$2$-cocycle\/}. Note that all terms on the left hand side of \eqref{cohomo1}
vanish. For a central $2$-cocycle, $V$ is a central ideal of $L \oplus_\si V$
and $\pr_L \co L \oplus_\si V \to L$ is a central extension.
\subsection{Invariant bilinear forms}\label{gen:ibf} A bilinear form $\be \co L \times L \to k$ is {\em invariant} if $\be([l_1, l_2], l_3) = \be(l_1, [l_2, l_3])$ holds for all $l_i \in L$.
Let $\g$ be a finite-dimensional split simple Lie algebra with Killing form
$\ka$. Let $R\in \kalg$. For any linear form $\vphi \co R \to k$, i.e., an
element of $R^*$, we obtain an invariant bilinear form $\inpr$ of the Lie
algebra $\g \ot_k R$ by $(x \ot r \mid y \ot s) = \ka(x,y) \, \vphi(rs)$. We
mention that every invariant bilinear form of $\g \ot_k R$ is obtained in
this way for a unique $\vphi \in R^*$ (see Cor. 6.2 of \cite{NPPS}).
\subsection{Central $2$-cocycles and invariant bilinear forms} \label{cen-ibf}
Assume our Lie algebra $L$ comes equipped with an invariant bilinear form
$\inpr$. We denote by $\Der_k(L)$ the Lie algebra of derivations of $L$ and
by $\SDer(L)$ the subalgebra of skew derivations, i.e., those derivations $d$
satisfying $\big(d (l) \mid l\big) = 0$ for all $l\in L$. Let $D$ be a
subalgebra of $\SDer (L)$ and denote by $D^*= \Hom_k(D, k)$ its dual space.
It is well-known and easy to check that then $\si_D \co L \times L \to D^*$
defined by
\begin{equation}\label{eala-cons0}
\si_D\big( l_1, \, l_2)\, (d) = \big( d(l_1) \mid l_2)
\end{equation}
is a central $2$-cocycle. We have not included the dependence of $\si_D$ on
$\inpr$ in our notation since later on the bilinear form $\inpr$ will be
unique up to a scalar and hence the cocycles defined by different forms also
differ only by a scalar, see Remark~\ref{rem:scalar}.
\subsection{A general construction of Lie algebras}\label{gen-data} We consider the following data:
\begin{enumerate}[(i)]
\item Two Lie algebras $L$ and $D$;
\item an action of $D$ on $L$ by derivations of $L$, written as $d\cdot
l$ or sometimes also as $d(l)$ for $d\in D$, $l\in L$ (thus $[d_1,
d_2] \cdot l = d_1 \cdot (d_2 \cdot l) - d_2\cdot (d_1 \cdot l)$
and $d\cdot [l_1, l_2] = [d\cdot l_1, l_2] + [l_1, d \cdot l_2]$ for
$d,d_i \in D$ and $l, l_i \in L$);
\item a vector space $V$ which is a $D$-module and which will also be
considered as a trivial $L$-module;
\item a central $2$-cocycle $\si \co L \times L \to V$ and a $2$-cocycle
$\ta \co D \times D \to V$.
\end{enumerate}
Given these data, we define a product on \[ E= L \oplus V \oplus D\] by
($v_i \in V$, $l_i \in L$, and $d_i \in D$)
\begin{equation} \label{n:gencons3} \begin{split}
[ l_1 + v_1 + d_1, \, l_2 + v_2 + d_2 ]
& =
\big( [l_1, l_2]_L + d_1 \cdot l_2 - d_2 \cdot l_1 \big)
\\&\quad +
\big(\si(l_1, l_2) +d_1 \cdot v_2 - d_2 \cdot v_1
+ \ta(d_1, d_2) \big)
\\&\quad
+ [d_1, d_2]_D.
\end{split} \end{equation}
Here $[.,.]_L$ and $[.,.]_D$ are the Lie algebra products of $L$ and $D$
respectively. To avoid any possible confusion we will sometimes denote the
product of $E$ by $[.,.]_E$.
\begin{proposition}\label{gen-constr}
The algebra $E$ defined in \eqref{n:gencons3} is a Lie algebra.\end{proposition}
We will henceforth denote this Lie algebra $(L, \si,\tau)$.
\begin{proof} The product is evidently alternating. For $e_i \in E$ let $J(e_1, e_2, e_3) = \big[ [e_1,e_2]\, e_3 \big] + \big[ [e_2,e_3]\, e_1 \big] + \big[ [e_3,e_1]\, e_2 \big]$ for $e_i \in E$. That $J(E,E,E)=0$ follows from tri-linearity of $J$ and the following special cases:
$J(D,D,D)=0$ since $D$ is a Lie algebra and $\ta$ is a $2$-cocycle; $J(D,D,L)
= 0$ since $L$ is a $D$-module; $J(D,D,V) = 0$ since $V$ is a $D$-module;
$J(D,V,V)=0 = J(D,L,V)$ since all terms vanish by definition
\eqref{n:gencons3}; $J(D,L,L)=0$ since $D$ acts on $L$ by derivations;
$J(L\oplus V, L \oplus V, L \oplus V)=0$ since $L\oplus_\si V$ is a Lie
algebra by \ref{cohom}.
\end{proof}
We will later use this construction for different data. For example, it is
the standard construction of an EALA as reviewed in \S\ref{sec:review}.
One of the central themes of this paper is to extend automorphisms from the
Lie algebra $L$ to the Lie algebra $E=(L, \si,\tau)$. Recall that the
elementary automorphism group $\EAut(M)$ of a Lie $k$-algebra $M$ is by
definition the subgroup of $\Aut_k(M)$ generated by the automorphisms $\exp
(\ad_M x)$ for $\ad_M x$ a nilpotent derivation. Clearly, any elementary
automorphism is $\Ctd_k(M)$-linear, where here and below $\Ctd_k$ denotes the
centroid of a $k$-algebra.\footnote{We recall that for an arbitrary
$k$-algebra $A$, $\Ctd_k(A) = \{ \chi \in \End_k(A) : \chi(ab) = \chi(a)b =
a\chi(b) \, \forall \, a,b \in A \}$. The space $A$ is naturally a left
$\Ctd_k(A)$-module via $\chi \cdot a = \chi(a).$ If $\Ctd_k(A)$ is
commutative, for example if $A$ is perfect, the above action endows $A$ with
an algebra structure over $\Ctd_k(A)$. The reader may refer to \cite{bn} for
general facts about centroids.}
\begin{proposition}\label{li-elem} Every elementary automorphism $f$ of $L$ lifts to an elementary automorphism $\tilde f$ of $E=(L,\si,\ta)$ with the following properties: \begin{enumerate}[\rm (i)]
\item $\tilde f(L) \subset L \oplus V$; the $L$-component of $\tl f|_L$ is $f$, i.e., $\pr_L \circ \tl f|_L =f$.
\item $\tl f(V) \subset V.$ In fact $\tl f|_V = \Id_V$.
\item For $d\in D$ the $D$-component of $\tl f (d)\in E$ is $d$, i.e., $\tilde{f}(d) = d + x_{f,d}$ for some $x_{f,d} \in L \oplus V$.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $x\in L$ and denote by $\ad_L x$ and $\ad_E x$ the corresponding inner derivation of $L$ and $E$ respectively. We let $e=l + v + d \in E$ be an arbitrary element of $E$ with the obvious notation. Then
\[ (\ad_E x)(e) = \big([x,l]_L - d \cdot x \big) + \si(x,l) \in L \oplus V.\]
Putting $e_1 = [x,l] -d \cdot x$, an easy induction shows that
\[ (\ad_E x)^n(e) = (\ad_L x)^{n-1}(e_1) + \si\big( x, (\ad_L x)^{n-2}(e_1)\big) \in L \oplus V, \quad n\ge 2.\]
In particular, if $\ad_L x$ is nilpotent then so is $\ad_E x$. Assuming this to be the case, it is immediate from the product formula \eqref{n:gencons3} that (i)--(iii) hold for $\tl f = \exp (\ad_E x)$.
\end{proof}
\section{Review: Lie tori and EALAs} \label{sec:review}
\subsection{Lie tori} \label{def:lietor}
In this paper the term ``root system'' means a finite, not necessarily
reduced root system $\Delta$ in the usual sense, except that we will assume
$0 \in \Delta$, as for example in \cite{AABGP}. We denote by $\Delta\ind = \{
0 \} \cup \{ \alpha\in \Delta: \frac{1}{2} \alpha \not\in \Delta\}$ the
subsystem of indivisible roots and by $\euQ(\Delta)=\Span_\Z(\Delta)$ the
root lattice of $\Delta$. To avoid some degeneracies we will always assume
that $\Delta\ne \{0\}$.
\smallskip Let $\Delta$ be a finite irreducible root system, and let $\La$ be an abelian
group. A \textit{Lie torus of type $(\Delta,\La)$\/} is a Lie algebra $L$
satisfying the following conditions (LT1) -- (LT4). \smallskip
\begin{itemize}
\item[(LT1)] (a) $L$ is graded by $\euQ(\Delta) \oplus \La$. We write this
grading as $L = \bigoplus_{\alpha \in \euQ(\Delta), \la \in \La}
L_\alpha^\la$ and thus have $[L_\alpha^\la, L_\beta^\mu] \subset L^{\la
+ \mu}_{\alpha + \beta}$. It is convenient to define \[ L_\alpha = \ts
\bigoplus_{\la \in \La} L_\alpha^\la \quad \hbox{and}\quad L^\la =
\bigoplus_{\alpha \in \euQ(\Delta)} L_\alpha^\la.\]
(b) We further assume that $\supp_{\euQ(\Delta)} L = \{ \alpha \in
\euQ(\Delta); L_\alpha \ne 0\} = \Delta$, so that $L =
\bigoplus_{\alpha \in \Delta} L_\alpha$.
\item[(LT2)] (a) If $L_\alpha^\la \ne 0$ and $\alpha \ne 0$, then there
exist $e_\alpha^\la \in L_\alpha^\la$ and $f_\alpha^\la \in
L_{-\alpha}^{-\la}$ such
that
\[ L_\alpha^\la = k e_\alpha^\la, \quad L_{-\alpha}^{-\la} = k f_\alpha^\la,
\]
and \[ [[e_\alpha^\la, f_\alpha^\la],\, x_\beta] = \lan \beta,
\alpha\ch\ran x_\beta \]
for all $\beta \in \Delta$ and $x_\beta \in L_\beta .$\footnote{Here and
elsewhere $\alpha\ch$ denotes the coroot corresponding to $\alpha$ in the
sense of \cite{Bbk}.}
(b) $L_\alpha^0 \ne 0$ for all $0 \ne \alpha \in \Delta\ind.$ \smallskip
\item[(LT3)] As a Lie algebra, $L$ is generated by $\bigcup_{0\ne \alpha
\in \Delta} L_\alpha$. \smallskip
\item[(LT4)] As an abelian group, $\La$ is generated by $\supp_\La L = \{
\la \in \La : L^\la \ne 0\}$.
\end{itemize} \smallskip
We define the {\em nullity} of a Lie torus $L$ of type $(\Delta,\La)$ as the
rank of $\La$ and the {\em root-grading type\/} as the type of $\Delta$. We
will say that $L$ is a \emph{Lie torus} (without qualifiers) if $L$ is a Lie
torus of type $(\Delta,\La)$ for some pair $(\Delta,\La)$. A Lie torus is
called \textit{centreless\/} if its centre $\scZ(L) = \{0\}$. If $L$ is an
arbitrary Lie torus, its centre $\scZ(L)$ is contained in $L_0$ from which
it easily follows that $L/\scZ(L)$ is in a natural way a centreless Lie torus
of the same type as $L$ and nullity (see \cite[Lemma~1.4]{y:lie}).
An obvious example of a Lie torus of type $(\Delta,\Z^n)$ is the Lie
$k$-algebra $\g \ot R$ where $\g$ is a finite-dimensional split simple Lie
algebra of type $\Delta$ and $R=k[t_1^{\pm 1}, \ldots, t_n^{\pm 1}]$ is the
Laurent polynomial ring in $n$-variables with coefficients in $k$ equipped
with the natural $\Z^n$-grading. Another important example, studied in
\cite{bgk}, is $\lsl_l(k_q)$ for $k_q$ a quantum torus.
Lie tori have been classified, see \cite{Al} for a recent survey of the many
papers involved in this classification. Some more background on Lie tori
is contained in the papers \cite{abfp2,n:persp, n:eala-summ}.
\subsection{Some known properties of centreless Lie tori}\label{lietor-prop}
We review the properties of Lie tori used in our present work. This is not a
comprehensive survey. The reader can find more information in
\cite{abfp2,n:persp,n:eala-summ}. We assume that $L$ is a centreless Lie
torus of type $(\Delta,\La)$ and nullity $n$. \smallskip
For $e^\la_\alpha$ and $f_\alpha^\la$ as in (LT2) we put $h_\alpha^\la =
[e_\alpha^\la, f_\alpha^\la]\in L_0^0$ and observe that $(e_\alpha^\la,
h_\alpha^\la, f_\alpha^\la)$ is an $\lsl_2$-triple. Then \begin{equation}
\label{def:h}
\frh = \Span_k \{ h^\la_\alpha\} = L_0^0\end{equation} is a toral
\footnote{A subalgebra $T$ of a Lie algebra $L$ is toral, sometimes also
called $\ad$-diagonalizable, if $L = \bigoplus_{\al\in T^*} L_\al(T)$ for
$L_\al(T) = \{ l \in L : [t,l] = \al(t)l \hbox{ for all $t\in T$}\}$. In this
case $\{ \ad t : t\in T\}$ is a commuting family of $\ad$-diagonalizable
endomorphisms. Conversely, if $\{ \ad t : t\in T\}$ is a commuting family of
$\ad$-diagonalizable endomorphisms and $T$ is a finite-dimensional
subalgebra, then $T$ is a toral.} subalgebra of $L$ whose root
spaces are the $L_\alpha$, $\alpha \in \Delta$.
Up to scalars, $L$ has a unique nondegenerate symmetric
bilinear form $\inpr$ which is $\La$-graded in the sense that $(L^\la \mid
L^\mu) = 0$ if $\la + \mu \ne 0$, \cite{NPPS, y:lie}. Since the subspaces
$L_\alpha$ are the root spaces of the toral subalgebra $\frh$ we also know
$(L_\alpha \mid L_\ta) = 0$ if $\alpha + \ta \ne 0$. \smallskip
The centroid $\Ctd_k(L)$ of $L$ is isomorphic to the group ring $k[\Xi]$ for
a subgroup $\Xi$ of $\La$, the so-called {\em central grading
group}.\footnote{In \cite{n:persp} the central grading group is denoted by
$\Ga.$ We will reserve this notation for the Galois group of an extension
$S/R$ which is prominently used later in our work.} Hence $\Ctd_k(L)$ is a
Laurent polynomial ring in $\nu$ variables, $0 \le \nu \le n$,
(\cite[7]{n:tori}, \cite[Prop.~3.13]{bn}). (All possibilities for $\nu$ do in
fact occur.) We can thus write $\Ctd_k(L)= \bigoplus_{\xi \in \Xi} k
\chi^\xi$, where the $\chi^\xi$ satisfy the multiplication rule $\chi^\xi
\chi^\de = \chi^{\xi + \de}$ and act on $L$ as endomorphisms of $\La$-degree
$\xi$. \smallskip
$L$ is a prime Lie algebra, whence $\Ctd_k(L)$ acts without torsion on $L$
(\cite[Prop.~4.1]{Al}, \cite[7]{n:tori}). As a $\Ctd_k(L)$-module, $L$ is
free. If $L$ is fgc, namely finitely generated as a module over its centroid,
then $L$ is a multiloop algebra \cite{abfp2}. \smallskip
If $L$ is not fgc, equivalently $\nu < n$, one knows (\cite[Th.~7]{n:tori})
that $L$ has root-grading type ${\rm A}$. Lie tori with this root-grading
type are classified in \cite{bgk,bgkn,y1}. It follows from this
classification together with \cite[4.9]{ny} that $L\simeq \lsl_l(k_q)$ for
$k_q$ a quantum torus in $n$ variables and $q=(q_{ij})$ an $n\times n$
quantum matrix with at least one $q_{ij}$ not a root of unity.
\smallskip
Any $\theta \in \Hom_\Z(\La, k)$ induces a so-called {\em degree derivation}
$\pa_\theta$ of $L$ defined by $\pa_\theta (l^\la) = \theta(\la) l^\la$ for
$l^\la \in L^\la$. We put $\euD = \{ \pa_\theta: \theta \in \Hom_\Z(\La, k)
\}$ and note that $\theta \mapsto \pa_\theta$ is a vector space isomorphism
from $\Hom_\Z(\La, k)$ to $\euD$, whence $\euD\simeq k^n$. We define $\ev_\la
\in \euD^*$ by $\ev_\la(\pa_\theta) = \theta(\la)$. One knows
(\cite[8]{n:tori}) that $\euD$ induces the $\La$-grading of $L$ in the sense
that $L^\la= \{ l \in L : \pa_\theta(l) = \ev_\la(\pa_\theta) l \hbox{ for
all } \theta \in \Hom_\Z(\La, k)\}$ holds for all $\la \in \La$.\smallskip
If $\chi \in \Ctd_k(L)$ then $\chi d \in \Der_k(L)$ for any derivation $d\in
\Der_k(L)$. We call
\begin{equation}\label{Volodya1} \CDer_k(L) := \Ctd_k(L) \euD = \ts \bigoplus_{\ga \in
\Xi} \chi^\xi \euD\end{equation}
the {\em centroidal derivations\/} of $L$. Since
\begin{equation}\label{derbracket}[
\chi^\xi \pa_\theta, \, \chi^\de \pa_\psi ] = \chi^{\xi + \de}( \theta(\de)
\pa_\psi - \psi(\xi) \pa_\theta)
\end{equation}
it follows that $\CDer(L)$
is a
$\Xi$-graded subalgebra of $\Der_k(L)$, a generalized Witt algebra. Note
that $\euD$ is a toral subalgebra of $\CDer_k(L)$ whose root spaces are the
$\chi^\xi \euD = \{ d\in \CDer(L): [t, d] = \ev_\xi(t) d \hbox{ for all }
t\in \euD\}$. One also knows (\cite[9]{n:tori}) that
\begin{equation} \label{lietor-prop1}
\Der_k(L) = \IDer(L) \rtimes \CDer_k(L) \quad (\hbox{semidirect product}).
\end{equation}
For the construction of EALAs, the $\Xi$-graded subalgebra $\SCDer_k(L)$ of
{\em skew-centroidal derivations\/} is important:
\begin{align*}
\SCDer_k(L) &= \{ d\in \CDer_k(L) : (d \cdot l \mid l) = 0 \hbox{ for all } l\in L\}
\\ &= \ts \bigoplus_{\xi \in \Xi } \SCDer_k(L)^\xi, \\
\SCDer_k(L)^\xi &= \chi^\xi \{ \pa_\theta : \theta(\xi) = 0 \}.
\end{align*}
Note $\SCDer_k(L)^0 = \euD$ and $[ \SCDer_k(L)^\xi, \, \SCDer_k(L)^{-\xi}]
=0$, whence
\[\SCDer_k(L) = \euD \ltimes \ts\big( \bigoplus_{\xi \ne 0} \SCDer(L)^\xi \big) \quad
(\hbox{semidirect product}).\footnote{The left-hand side depends a priori
on the choice of invariant bilinear form on $L$, while the right-hand side does not.
This is as it should be given that the non-degenerate invariant bilinear form is unique up to non-zero scalar.} \]
\subsection{Extended affine Lie algebras (EALAs)} \label{def:eala}
An \textit{extended affine Lie algebra\/} or EALA
for short, is a triple $\big(E,H, \inpr\big)$ (but see Remark~\ref{comp:def})
consisting of a Lie algebra $E$ over $k$, a subalgebra $H$ of $E$ and a
nondegenerate symmetric invariant bilinear form $\inpr$ satisfying the axioms
(EA1) -- (EA5) below.
\begin{itemize}
\item[(EA1)] $H$ is a nontrivial finite-dimensional toral and
self-centra\-li\-zing subalgebra of $E$.
\end{itemize}
Thus $E = \ts\bigoplus_{\al \in H^*} E_\al $ for $E_\al = \{ e\in E: [h,e] =
\al(h)e \hbox{ for all } h\in H\}$ and $E_0 = H$. We denote
by $\Psi=\{\al \in H^*: E_\al \ne 0\}$ the set of roots of $(E,H)$ --
note that $0 \in \Psi$! Because the restriction of $\inpr$ to $ H \times H $ is nondegenerate,
one can in the usual way transfer this bilinear form to $H^*$ and
then introduce anisotropic roots $\Psi\an= \{ \al \in \Psi : (\al \mid
\al) \ne 0\}$ and isotropic (= null) roots $\Psi^0 = \{ \al\in \Psi : (\al
\mid \al) = 0\}$. The \textit{core of $\big(E,H, \inpr \big)$} is by definition the
subalgebra generated by $\bigcup_{\al \in \Psi\an} E_\al$. It will be henceforth denoted by $E_c.$\smallskip
\begin{itemize}
\item[(EA2)] For every $\al \in \Psi\an$ and $x_\al \in E_\al$, the
operator $\ad x_\al$ is locally nilpotent on $E$. \smallskip
\item[(EA3)] $\Psi\an$ is connected in the sense that for any decomposition
$\Psi\an = \Psi_1 \cup \Psi_2$ with $\Psi_1 \neq \emptyset$ and $\Psi_2
\neq \emptyset$ we have $(\Psi_1 \mid \Psi_2) \neq 0$.
\smallskip
\item[(EA4)] The centralizer of the core $E_c$ of $E$ is contained in
$E_c$, i.e., $\{e \in E : [e, E_c] =0 \} \subset E_c$. \smallskip
\item[(EA5)] The subgroup $\La = \Span_\Z(\Psi^0) \subset H^*$ generated by
$\Psi^0$ in $(H^*,+)$ is a free abelian group of finite rank.
\end{itemize}
The rank of $\La$ is called the \textit{nullity} of $\big(E,H, \inpr \big)$.
Some references for EALAs are \cite{AABGP, bgk, bgkn, Ne4, n:persp,
n:eala-summ}. It is immediate that any finite-dimensional split simple Lie
algebra is an EALA of nullity $0$. The converse is also true,
\cite[Prop.~5.3.24]{n:eala-summ}. It is also known that any affine Kac-Moody
algebra is an EALA -- in fact, by \cite{abgp}, the affine Kac-Moody algebras
are precisely the EALAs of nullity $1$. The core $E_c$ of an EALA is in fact
an ideal.
\begin{remark}\label{comp:def}
In \cite{Ne4, n:persp, n:eala-summ} an EALA is defined as a pair $(E,H)$
consisting of a Lie algebra $E$ and a subalgebra $H\subset E$ satisfying the
axioms (EA1) -- (EA5) of \ref{def:eala} as well as
\begin{enumerate}
\item[(EA0)] $E$ has an invariant nondegenerate symmetric bilinear form
$\inpr$.
\end{enumerate}
As we will see in Corollary~\ref{cor:ff} below the choice of the invariant
bilinear form is not important. To be precise, the sets of isotropic and
anisotropic roots, which a priori depend on the form $\inpr$, are actually
independent of the choice of $\inpr$. In other words, two EALAs of the form
$\big(E,H, \inpr\big)$ and $\big(E,H, \inpr' \big)$ have the same $\Psi$
(this is obvious), $\Psi\an$ and $\Psi^0$, and hence also the same core $E_c$
and centreless core $E_{cc}= E_c/Z(E_c)$. The role of $\inpr$ is to show that
$\Psi$ is an extended affine root system (EARS) \cite{AABGP}\footnote{EARS
can be defined without invariant forms \cite[Prop.~5.4, \S5.3]{LN}} and to
pair the dimensions between the homogeneous spaces $C^{\la}$ and $D^{-\la}$,
introduced in \ref{eala-cons}. In fact, as indicated in \cite[\S6]{n:persp},
it is natural to consider more general EALA structure in which the existence
of an invariant form is replaced by the requirement that the set of roots of
$(E,H)$ has a specific structure without changing much the structure of
EALAs.
\end{remark}
\subsection{Isomorphisms of EALAs}\label{def-isom} An {\it isomorphism} between EALAs $\big(E,H, \inpr\big)$ and $\big(E',H', \allowbreak \inpr'\big)$ is a Lie algebra isomorphism $f \co E \to E'$ that maps $H$ onto $H'$. Any such map induces an isomorphism between the corresponding EARS.
We point out that no assumption is made about the compatibility of the
bilinear forms with the given Lie algebra isomorphism $f \co E \to E'$ . In
particular, $f$ is not assumed to be an isometry up to scalar as in
\cite{AF:isotopy}. There is a good reason for not making this assumption.
While the form is unique on the core $E_c$ up to a scalar, there are many
ways to extend it from $E_c$ to an invariant form on $E$ without changing the
algebra structure. This can already be seen at the example of an affine
Kac-Moody Lie algebra $E$ with the standard choice of $H$ for which there
exists an infinite number of invariant bilinear forms $\inpr$ on $E$ which
are not scalar multiple of each other and such that $\big(E,H, \inpr\big)$ is
an EALA. The isometry up to scalar condition will render all these EALAs
non-isomorphic. Removing this condition yields the equivalence (up to Lie
algebra isomorphism) between the affine Kac-Moody Lie algebras and EALAs of
nullity one (see above).
\subsection{Roots}\label{rev:root} The set $\Psi$ of roots of an EALA $E$ has special properties: It is a so-called
extended affine root system in the sense of \cite[Ch.~I]{AABGP}. We will not
need the precise definition of an extended affine root system or the more
general affine reflection system in this paper and therefore refer the
interested reader to \cite{AABGP} or the surveys \cite[\S2, \S3]{n:persp} and
\cite[\S5.3]{n:eala-summ}. But we need to recall the structure of $\Psi$ as
an affine reflection system: There exists an irreducible root system $\Delta
\subset H^*$, an embedding $\Delta\ind \subset \Psi$ and a family
$(\La_\alpha : \alpha \in \Delta)$ of subsets $\La_\alpha \subset \La$ such
that \begin{equation} \label{root1}
\Span_k(\Psi) = \Span_k(\Delta) \oplus \Span_k(\La) \quad \hbox{and} \quad
\Psi = \ts \bigcup_{\alpha \in \Delta} ( \alpha + \La_\alpha).
\end{equation}
Using this (non-unique) decomposition of $\Psi$, we write any $\psi \in \Psi$
as $\psi = \al + \la$ with $\al \in \Delta$ and $\la \in \La_\alpha \subset
\La$ and define $(E_c)_\al^\la = E_c \cap E_\psi$. Then $E_c =
\bigoplus_{\alpha \in \Delta, \la \in \La} (E_c)_\alpha^\la$ is a Lie torus
of type $(\Delta,\La)$. Hence $E_{cc} = E_c/\scZ(E_c)$ is a centreless Lie
torus, called the {\em centreless core of $E_c$}.
\subsection{Construction of EALAs} \label{eala-cons} To construct an EALA one reverses the
process described in \ref{rev:root}. We will use data $(L,\sigma_D,\ta)$
described below. Some background material can be found in \cite[\S6]{n:persp}
and \cite[\S5.5]{n:eala-summ}:
\begin{itemize}
\item $L$ is a centreless Lie torus of type $(\Delta,\La)$. We fix a
$\La$-graded invariant nondegenerate symmetric bilinear form $\inpr$
and let $\Xi$ be the central grading group of $L$.
\item $D=\bigoplus_{\xi \in \Xi} D^\xi$ is a graded subalgebra of $
\SCDer_k(L)$ such that the evaluation map $\ev_{D^0} : \La \to D^{0\,
*}$, $\la \to \ev_\la \mid_{D^0}$ is injective. Since $(L^\la \mid
L^\mu) =0$ if $\la + \mu \ne 0$ and since $D^\xi(L^\la) \subset L^{\xi
+ \la}$ it follows that the central cocycle $\si_D$ of
\eqref{eala-cons0} has values in the graded dual $D^{\gr *}=:C$ of
$D$. Recall $C=\bigoplus_{\xi \in \Xi} C^\xi$ with $C^\xi =
(D^{-\xi})^* \subset D^*$. We also note that the contragredient action
of $D$ on $D^*$ leaves $C$ invariant. In the following we will always
use this $D$-action on $C$. In particular, $d\in D^0$ acts on $C^\xi$
by the scalar $-\ev_\la(d)$.
\item $\ta : D\times D \to C$ is an \textit{affine cocycle\/} defined to
be a $2$-cocycle satisfying for all $d,d_i \in D$ and $d^0 \in D^0$
\begin{align*}
\ta(d^0, d) &= 0, \quad \hbox{and} \quad
\ta(d_1, d_2)(d_3) = \ta(d_2, d_3)(d_1).
\end{align*}
\end{itemize}
It is important to point out that there do exist non-trivial affine cocycles,
see \cite[Rem.~3.71]{bgk}.
\noindent The data $(L,\sigma_D,\ta)$ as above satisfy all the axioms of our
general construction \ref{gen-data} and hence, by \ref{gen-constr}, is a Lie
algebra with respect to the product \eqref{n:gencons3}.\footnote{Strictly
speaking we should write $\EA(L,D, \inpr_L, \ta)$. The effect that different
choice of forms has on the resulting EALA is explained in Remark
\ref{rem:scalar}.} We will denote this Lie algebra by $E$. By construction we
have the decomposition
\begin{equation}\label{eala-cons1} E = L \oplus C \oplus D. \end{equation} Note that $E$ has the toral subalgebra \[H=\frh \oplus C^0\oplus D^0\]
for $\frh$ as in \ref{lietor-prop}. The symmetric bilinear form $\inpr$ on
$E$, defined by
\[ \big( l_1 + c_1 + d_1 \mid l_2 + c_2 + d_2\big)
= (l_1 \mid l_2)_L + c_1(d_2) + c_2(d_1), \]
is nondegenerate and invariant. Here $\inpr_L$ is of course our fixed chosen
invariant bilinear form of the Lie torus $L$. We have now indicated part of
the following result.
\begin{theorem}[{\cite[Th.~6]{Ne4}}]\label{n:mainconst} {\rm (a)} The triple
$\big(E,H, \inpr \big)$ constructed above is an extended affine Lie
algebra,\footnote{See Remark \ref{comp:def} above.} denoted $\EA(L,D,\ta)$.
Its core is $L \oplus D^{\gr\, *}$ and its centreless core is $L$. \smallskip
{\rm (b)} Conversely, let $\big(E,H, \inpr \big)$ be an extended affine Lie
algebra, and let $L=E_c/Z(E_c)$ be its centreless core. Then there exists a
subalgebra $D\subset \SCDer_k(L)$ and an affine cocycle $\ta$ satisfying the
conditions in {\rm \ref{eala-cons}} such that $\big(E,H, \inpr \big) \simeq
\EA(L, \inpr_L, D,\ta)$ for some $\Lambda$-graded invariant nondegenerate
bilinear form $\inpr_L$ on $L.$
\end{theorem}
\begin{remark}\label{rem:scalar} As mentioned in \ref{lietor-prop}, invariant $\La$-graded bilinear forms on $L$ are unique up to a scalar. Changing the form on $L$ by the scalar $s\in k$, will result in multiplying the central cocycle $L \times L \to C$ by $s$. Including for a moment the bilinear form $\inpr$ on $L$ in the notation, the map
$\Id_L \oplus s\Id_C\oplus \Id_D$ is an isomorphism from $\EA(L,\inpr_L,
D,\ta)$ to $\EA(L, s\inpr_L, D, s\ta)$. \end{remark}
\section{Invariance of the core}\label{equality of cores}
In this section $\big(E,H, \inpr \big)$ is an EALA whose centreless core
$E_{cc} = E_c/Z(E_c)$ is an arbitrary Lie torus $L$, hence not necessarily
fgc. We decompose $E$ in the form
$$E = L \oplus C \oplus D$$
as described in the previous section. We have a canonical map
$\overline{\phantom{0}} \co E_c \to E_c /Z(E_c)=L.$
We start by proving a result of independent interest on the structure of
ideals of the Lie algebra $E$.
\begin{proposition}
\label{prop:ideal-stru} Let $I$ be an ideal of the Lie algebra $E$. Then either $I \subset C= Z(E_c)$ or $E_c \subset I$.
\end{proposition}
Since $L$ is centreless, the centre of $E_c$ is $C$. We note that it is
immediate that $C\ideal E$.
\begin{proof}
We assume that $I \not \subset C$ and set $I_c = I \cap E_c$ and $I_{cc} =
\overline{I_c}.$ We will proceed in several steps using without further
comments the notation introduced in \S\ref{sec:review}.
(I) $I_{cc} \ne \{0\}$: Let $ e= x + c + d \in I$ where $x\in L, c \in C$
and $d\in D$. For any $l \in L$ we then get $[e, l]_E = (\ad_L x + d)(l) +
\si_D(x,l) \in I$, whence $(\ad_L x + d) (l) \in I_{cc}$. If for all $e\in I$
the corresponding derivation $\ad_L x + d = 0$ it follows that $x=0=d$ since
$L$ is centreless. But then $I \subset C$ which we excluded. Therefore some
$e\in I$ has $\ad_L x + d \ne 0$, hence $0 \ne (\ad_L x + d)(l)\in I_{cc}$
for some $l\in L$.\smallbreak
(II) $d \cdot x \in I_{cc}$ for all $d\in D$ and $x\in I_{cc}$: There exists
$c\in C$ such that $ x + c\in I_c$. Hence $[d, x + c]_E = d \cdot x + d\cdot
c\in I_c$ since $I_c$ is an ideal of $E$. Therefore $d \cdot x \in I_{cc}$.
(III) $I_{cc}=L$: Since the $\La$-grading of $L$ is induced by the action of
$D^0 \subset D$ on $L$, it follows from (II) that $I_{cc}$ is a $\La$-graded ideal.
By \cite[Lemma 4.4]{Yo}, $L$ is a $\Lambda$-graded simple. Hence $I_{cc} = L$.
(IV) $E_c \subset I$: Let $c\in C$ be arbitrary. Since $E_c$ is perfect,
there exist $l_i, l_i'\in L$ such that $c= \sum_i [l_i, l_i']_E$. By (III)
there exist $c_i \in C$ such that $ l_i + c_i\in I_c$. Then $[l_i, l_i']_E =
[l_i + c_i, l'_i]_E \in I_c$ implies $c\in I_c$ which together with (III)
forces $E_c \subset I$.
\end{proof}
\begin{corollary}\label{cores are the same}
Let $\big(E,H, \inpr \big)$ and $(E,H', \inpr' \big)$ be two extended affine
Lie algebra structures on $E$ with cores $E_c$ and $E'_c$ respectively. Then
$E_c = E'_c$.
\end{corollary}
For special types of EALAs, namely those for which the the root system $\De$
in \eqref{root1} is reduced, Corollary~\ref{cores are the same} is proven in
\cite[Th.~5.1]{maribel} with a completely different method.
\begin{proof} Since $E'_c$ is an ideal of $E$, Proposition~\ref{prop:ideal-stru} says that either $E'_c \subset Z(E_c)$ or $E_c \subset E'_c$. In the first case $E'_c$ is abelian, a contradiction to the assumption that anisotropic roots exist. Hence $E_c \subset E'_c$. By symmetry, also $E'_c \subset E_c$. \end{proof}
\begin{corollary}\label{cor:ff}
Let $(E,H,\inpr)$ and $(E,H,\inpr')$ be two EALAs. We distinguish the notation of {\rm \ref{def:eala}} for $(E,H,\inpr')$ by $'$. \sm
{\rm (a)} $\Psi=\Psi'$, $\Psi^0 = \Psi^{\prime\, 0}$, $\Psi\an =
{\Psi'\,}\an.$ \sm
{\rm (b)} There exists $0\ne a\in k$ such that $\inpr|_{E'_c\times E'_c} = a
\inpr|_{E_c \times E_c}$.
\end{corollary}
\begin{proof} (a) The equality $\Psi=\Psi'$ is obvious since $\Psi$ is the set of roots of $H$. By Corollary~\ref{cores are the same}, we have $E_c = {E'}_c$. The algebra $E_c$ is a Lie torus whose root-grading by a finite irreducible root system $\Delta$ is induced by $H_c = H \cap E_c$. Let $\pi \co H^* \to H_c^*$ be the canonical restriction map. The structure of the root spaces of $E$, see for example \cite[6.9]{n:persp}, shows that $\Psi^0 = \pi^{-1}(\{0\})$ whence $\Psi^0={\Psi'}^{0}$.
(b) Because $E_c$ is perfect, the centre of $E_c$ equals the radical of
$\inpr|_{E_c \times E_c}$. Indeed, let $z\in E_c$. Then, using that $\inpr$
is nondegenerate and invariant and that $E_c$ is perfect we have $z\in Z(E_c)
\iff 0=([z,E_c]\mid E) = (z\mid [E_c, E]) = (z\mid E_c) \iff$ $ z$ lies in
the radical of the restriction of $\inpr$ to $E_c.$ Now (b) follows from the
fact that invariant bilinear forms on $E_{cc}$ are unique up to a scalar.
\end{proof}
As a consequence, when no explicit use of the form is being made, we will
denote EALAs as couples $(E,H)$.
As an application of Corollary~\ref{cores are the same} we can now prove
\begin{proposition}\label{automorphism gives another EALA}
The core $E_c$ of an EALA $(E,H)$ is stable under automorphisms of the
algebra $E$, i.e., $f (E_c)=E_c$ for any $f \in \Aut_k(E)$.
\end{proposition}
\begin{proof}
Let $f\in {\rm Aut}_{k}(E)$. Denote $H'=f(H)$. Let $\inpr'$ be the bilinear
form on $E$ given by
$$
(x\,|\,y)'=\big( f^{-1}(x)\,|\,f^{-1}(y)\big).
$$
Clearly, $\big(E, H', \inpr' \big)$ is another EALA-structure on the Lie
algebra $E$. Therefore, by Corollary~\ref{cores are the same}, we have that
the core $E'_c$ of $\big(E,H', \inpr' \big)$ is equal to $E_c$. It remains to
show that $E'_c=f(E_c)$.
Let $\alpha\in \Psi$ be a root with respect to $H$. There exists a unique
element $t_{\alpha}$ in $H$ such that $(t_{\alpha}\,|\, h)=\alpha (h)$ for
all $h\in H$. Recall that $\alpha$ is anisotropic if
$(t_{\alpha}\,|\,t_{\alpha})\neq 0$ and that $E_c$ is generated (as an ideal)
by $\cup_{\alpha \in \Psi\an} E_{\alpha}$. Let $\Psi'$ be the set of roots of
$(E,H')$. The mapping $^t{f_{|H}^{-1}}: H^*\rightarrow H'^*$ satisfies
$^t{f_{|H}^{-1}}(\Psi)=\Psi'$. Notice that
$f(t_{\alpha})=t_{(^tf)^{-1}(\alpha)}$. We next have
$(t_{(^tf)^{-1}(\alpha)}\,|\,t_{(^tf)^{-1}(\alpha)})'=(f(t_\alpha)\,|\,f(t_{\alpha}))'=
(t_{\alpha}\,|\,t_{\alpha})$. Therefore, $ ^t{f^{-1}}(\Psi\an) =(\Psi')\an$,
$f(E_{\alpha})=E'_{^t{f^{-1}}}(\alpha)$, and this implies $f(E_c)=E'_c=E_c$.
\end{proof}
By Proposition~\ref{automorphism gives another EALA} we have a well-defined
restriction map
\[
\res_c \co \Aut_k(E) \longrightarrow \Aut_k (E_c).
\]
Since $L$ is centreless, the centre of $E_c$ is $C$. It is left invariant by
any automorphism of $E_c$. Hence $\overline{\phantom{0}} \co E_c \to L$
induces a natural group homomorphism \[ \overline{\res} \co \Aut_{k}(E_c)\to
\Aut_{k}(L). \] Composing the two group homomorphisms yields
\begin{equation}\label{def:rescc}
\rescc := \overline{\res} \circ \res_c \co \Aut_{k}(E)\to \Aut_{k}(L).
\end{equation}
We can easily determine the kernel of $\rescc$. For its description we recall
that a $k$-linear map $\psi \co D \to C$ is called a {\em derivation\/} if
$\psi([d_1, d_2]) = d_1 \cdot \psi(d_2) - d_2 \cdot \psi (d_1)$ holds for all
$d_i \in D$. We denote by $\Der_k(D,C)$ the $k$-vector space of derivations
from $D$ to $C$.
\begin{proposition}
\label{res-ker}
{\rm (a)} $\overline{\res}$ is injective.
{\rm (b)} The kernel of\/ $\rescc$ consists of the maps $f$ of the form
\begin{equation} \label{res-ker1} f( l + c + d) = l + \big( c + \psi(d)\big) + d, \qquad
\psi \in \Der_k(D,C). \end{equation}
In particular, $\Ker (\rescc)$ is a vector group isomorphic to
$\Der_k(D,C)$.
\end{proposition}
\begin{proof}
(a) is immediate from the fact that $ L \oplus C = [L,L]_E$. It implies that $\Ker (\rescc )= \Ker (\res_c)$. Let $f\in \Ker (\res_c)$. Then there exist linear maps $f_{CD} \in \Hom_k(D,C)$, $f_{LD} \in \Hom_k(D,L)$ and $f_D\in \End_k(D)$ such that $f(d) = f_{LD}(d) + f_{CD}(d) + f_D(d)$ holds for all $d\in D$. For $l\in L$ we then get $d\cdot l = f([d,l]) = [f_{LD}(d) + f_{CD}(d) + f_D(d), l] = \big( \ad_L f_{LD}(d) + f_D(d)\big)(l)$, i.e., $d = \ad_L f_{LD}(d) + f_D(d)$. Since $D \cap \IDer L = \{0\}$ it follows that $f_{LD} = 0$ and $f_D = \Id_D$. One then sees that $f_{CD}$ is a derivation by applying $f$ to a product $[d_1, d_2]_E$. That conversely any map of the form \eqref{res-ker1} is an automorphism, is a straightforward verification.
\end{proof}
Our next goal is to study in detail the image of $\rescc$. From
Proposition~\ref{li-elem} we know
\[ \EAut(L) \subset \rescc\big( \Aut_k(E) \big). \]
For the Conjugacy Theorem \ref{main-res} it is necessary to know that a
bigger group of automorphisms of $L$ lies in the image of $\rescc$. We will
do this in Theorem~\ref{litwi}. Its proof requires some preparations to which
the next two sections are devoted.
\section{Fgc EALAs as subalgebras of untwisted EALAs}\label{sec:subEALA}
We remind the reader that from now on $k$ is assumed to be algebraically
closed. In this section we will describe how to embed an fgc EALA into an
untwisted EALA. Here, we say that an EALA $E$ is {\em fgc\/} if its
centreless core is so, and we say that $E$ is {\em untwisted} if its
centreless core $E_{cc},$ as a Lie torus, is of the form $E_{cc} = \g \ot R$
for some finite-dimensional simple Lie algebra $\g$ over $k$ and Laurent
polynomial ring $R$ in finitely many variables.\sm
\subsection{Multiloop algebras}\label{ssec:rev-lt}
In order to realize an fgc EALA as a subalgebra of an untwisted EALA, we need
some preparation starting with a review of fgc Lie tori which by \cite{abfp2}
are multiloop algebras $L=L(\g,\bs)$. They are constructed as follows: $\g$
is a simple finite-dimensional Lie algebra and $\bs = (\si_1, \ldots,
\si_n)$ is a family of commuting finite order automorphisms. We will denote
the order of $\si_i$ by $m_i$. We fix once and for all a compatible set
$(\ze_\ell)_{\ell \in \Bbb{N}}$ of primitive $\ell$-th roots of unity, i.e.
$\zeta_{n\ell}^n = \zeta_{\ell}$ for $n \in \N$.
The second ingredient are two rings, \begin{equation*}
R = k[t_1^{\pm 1}, \ldots, t_n^{\pm 1}] \quad \hbox{and} \quad S = k [
t_1^{\pm \frac{1}{m_1}}, \ldots, t_n^{\pm \frac{1}{m_n}}].
\end{equation*}
For convenience we set $z_i = t_i^\frac{1}{m_i}.$ Thus $z_i^{m_i} = t_i$ and
$S = k [ z_1^{\pm{1}}, \ldots, z_n^{\pm 1}]$.
Let $\Lambda = \Z^n$. For $\la = (\la_1, \cdots , \la_n) \in \Lambda$ let
\[z^\la = z_1^{\la_1}\cdots z_n^{\la_n} := t_1^\frac{\la_1}{m_1}\cdots t_n^\frac{\la_n}{m_n}.\]
The $k$-algebra $S$ has a natural $\Lambda$-grading by declaring that $z^\la$
is of degree $\la.$ Then $R$ is a graded subalgebra of $S$ whose homogeneous
components have degree belonging to the subgroup
\[ \Xi= m_1\ZZ \oplus \cdots \oplus m_n\ZZ \subset \Lambda. \]
Note that $\Xi \simeq \ZZ^n.$
We set $\overline{\La} = \Lambda/\Xi$ and let $\overline{\phantom{0}} :
\Lambda \to \overline{\La}$ denote the canonical map. After the natural
identification of $\Xi$ with $\Z^n$, this is nothing but the canonical map $
\overline{\phantom{0}} : \ZZ^n \to \ZZ/m_1\ZZ \oplus \cdots \oplus
\ZZ/m_n\ZZ.$
The automorphisms $\si_i$ can be simultaneously diagonalized. For $\bar{\la}
= (\overline{\la_1}, \cdots , \overline{\la_n}) \in \overline{\Lambda}$ we
set
\[ \g^{\bar{\la}} = \{ x\in \g : \si_i(x) = \ze_{m_i}^{\bar \la_i} x, \; 1 \le i \le n\}\]
then $\g = \ts \bigoplus_{\bar{\la} \in \overline{\La}} \g^{\bar{\la}}$.
Note that $\g \ot S$ is a centreless $\Lambda$-graded Lie algebra with
homogeneous subspaces $(\g \ot S)^\la = \g \ot S^\la$. By definition, the
multiloop algebra $L(\g, \bs)$ is the graded subalgebra of $\g \otimes S$
given by
\begin{equation} \label{ssec:rev-lt1}
L = L(\g, \bs) = \ts \bigoplus_{\la \in \overline{\La}} \, \g^{\bar{\la}} \ot z^{\la} \subset \g \ot S. \end{equation}
Note that the $\La$-grading of $L$ is given by $L^\la = L \cap (\g \ot S)^\la
= \g^{\bar{\la}} \ot z^\la$. The grading group of $L$ is
\[ \La_L := \Span \{ \la \in \La : L^\la \ne 0 \} = \Span \{ \la \in \La: \g^{\bar{\la}} \ne 0 \} \subset \La.\]
We shall later see that in the cases we are interested in, namely those
related to the realization of Lie tori and EALAs, we always have $\La_L =
\La.$
\subsection{The EALA construction with $L(\g,\bs)$ as centreless core}\label{ssec:msl}
From now on we consider an EALA $E$ whose centreless core is fgc. By
\cite[Prop.~3.2.5 and Th.~3.1]{abfp2} one then knows that $E_{cc}$ is a
multiloop algebra $L(\g, \bs)$ with $\g$ simple and $\bs$ as above. The
(admittedly delicate) choice of $\bs$ is such that the $\La$-grading of
$L(\g, \bs)$ yields the $\La$-grading of the Lie torus $E_{cc}$. With such a
choice $\g^0$ is simple.
By \cite{bn, GP1} the ring $R$ is canonically isomorphic to the centroid
$\Ctd_k(L)$ of the Lie algebra $L = L(\g, \bs).$ More precisely, for $r\in R$
let $\chi_r \in \End(L)$ be the homothety $l \mapsto rl$. Then the centroid
$\Ctd_k(L)$ of $L$ is $\{ \chi_ r : r\in R\}$ and the map $r \mapsto \chi_r$
is a $k$-algebra isomorphism $R \to \Ctd_k(L).$ We will henceforth identify
these two rings without further mention and view $L$ naturally as an $R$-Lie
algebra.
Let $\eps \in S^*$ be the linear form defined by $\eps(z^\la) = \de_{\la,
{\bf 0}}$. We will also view $\eps$ as a symmetric bilinear form on $S$
defined by $\eps(s_1, s_2) = \eps(s_1 s_2)$ for $s_i \in S$. We denote by
$\ka$ the Killing form of $\g$ and define a bilinear form $\inpr_S$ on $\g
\ot S$ by
\[ (x_1 \ot s_1 \mid x_2 \ot s_2)_S = \ka(x_1, x_2) \, \eps(s_1 s_2),\]
i.e., $\inpr_S = \ka \ot \eps$. The bilinear form $\inpr_S$ is invariant,
nondegenerate and symmetric. By \cite[Cor.~7.4]{NPPS}, the restriction
$\inpr_L$ of $\inpr_S$ to the subalgebra $L(\g,\bs)$ has the same properties
and is up to a scalar the only such bilinear form.
Since $S$ is $\La$-graded, every $\theta \in \Hom_\ZZ(\La, k)$ gives rise to
a derivation $\pa_\theta$ of $S$, defined by $\pa_\theta(z^\la) =
\theta(\la)z^\la$ for $\la\in \La$. We get a subalgebra $\euD_S = \{
\pa_\theta : \theta \in \Hom_\ZZ( \La, k) \}$ of degree $0$ derivations of
$S$. The map $\theta \mapsto \pa_\theta$ is a vector space isomorphism. It is
well-known, cf.\ \eqref{Volodya1} and \eqref{derbracket}, that $\Der_k(S) =
S \euD_S$. It follows that $\Der_k(S)$ is a $\La$-graded Lie algebra with
homogeneous subspace $(\Der_k(S))^\la = S^\la \euD$. The analogous facts hold
for the $\Xi$-graded algebra $R$, i.e., putting $\euD_R = \{ \pa_\xi : \xi
\in \Hom_\ZZ(\Xi, k)\}$ the Lie algebra $\Der_k(R) = R \euD_R$ is
$\Xi$-graded with $\Der_k(R)^\xi = R^\xi \euD$. But we can identify $\euD_S$
with $\euD_R$ and then denote $\euD = \euD_S = \euD_R$ since the restriction
map $\Hom_\ZZ(\La, k)\to \Hom_\ZZ(\Xi, k)$ is an isomorphism of vector spaces
(this because $\La/\Xi = \Gamma$ is a finite group and $k$ is torsion-free).
Hence $\Der_k(R) = R \euD \subset \Der_k(S) = S \euD$. Observe that the
embedding $\Der_k(R) \subset \Der_k(S)$ preserves the degrees of the
derivations.\footnote{Since $S$ is an \'etale covering of $R$, in fact even
Galois, every $k$-linear derivation $\delta \in \Der_k(R)$ uniquely extends
to a derivation $\hat \delta$ of $S$. Under our inclusion $\Der_k(R) \subset
\Der_k(S)$ we have $\delta=\hat \delta$.}
One easily verifies that $z^\la \pa_\theta$ is skew-symmetric with respect to
the bilinear form $\eps$ of $S$ if and only if $\theta(\la) = 0$. The
analogous fact holds for $R$:
\begin{align*} \SDer_k(R) &= \{ \delta \in \Der_k(R) : \delta \hbox{ is skew-symmetric} \} \\ & = \ts \bigoplus_{\xi \in \Xi} z^\xi \{ \pa_\theta : \theta \in \Hom_\ZZ(\Xi, k), \theta (\xi) = 0 \} \subset \SDer_k (S).
\end{align*}
We now consider derivations of $\g \ot S$ and of $L$. It is well-known that
the map $\delta \mapsto \Id_\g \ot \delta$ identifies $\Der_k(S)$ with the
subalgebra $\CDer_k(\g \ot S)$ of centroidal derivations of $\g \ot S$; it
maps $\SDer_k(S)$ onto $\SCDer_k(\g \ot S)$. Analogously, $\Der_k(R)\to
\CDer(L)$, $\delta \mapsto (\Id_\g \ot \delta)|_{L}$ is an isomorphism of Lie
algebras \cite{P}.\footnote{Note that in the expression $\Id_\g \ot \delta$
the element $\delta \in \Der_k(R)$ is viewed as an element of $\Der_k(S)$
under the inclusion $\Der_k(R) \subset \Der_k(S)$ described above.} One can
check that under this isomorphism $\SDer_k(R)$ is mapped onto $\SCDer_k(L)$.
The embedding $ \SDer_k(R) \subset \SDer_k(S)$ of above then gives rise to an
embedding
\begin{equation}
\label{ssec:msl1} \SCDer_k(L) \subset \SCDer_k(\g \ot S).
\end{equation}
To construct an EALA $E$ with $E_{cc } = L$ we follow \ref{eala-cons} and
take a graded subalgebra $D\subset \SCDer_k (L) \simeq \SDer_k (R)$ such that
the evaluation map $\ev \co \La \to {D^0}^*$ is injective. This then provides
us with the central cocycle $\si_D \co L \times L \to C=D^{\gr *}$. Using
Theorem~\ref{n:mainconst} it follows that $ \EA(L,D,\ta)$ is an EALA with
centreless core $L$ for any affine cocycle $\ta \co D \times D \to C$ and,
conversely, any EALA $E$ with $E_{cc} \simeq L$ is isomorphic to $\EA(L,
D,\ta)$ for appropriate choices of $D$ and $\ta$.
\begin{example}[\bf untwisted EALA] Let $\si_i = \Id$ for all $i.$ Then $S=R$, $L=\g \ot S = \g \ot R.$ Using the invariant bilinear form $\inpr_S$ on $\g \ot S$ described above we observe that for any $D\subset \SCDer(L)$ as above and affine cocycle $\ta$ we have an EALA $\EA(\g \ot S, D, \ta)$. Any EALA isomorphic to such an EALA will be called {\em untwisted}.
\end{example}
\begin{remark}\label{scaling}Note that if we replace $\inpr_S$ by $s \inpr_S$ for some $s \in k^\times$, then, as explained in Remark \ref{rem:scalar}, the resulting EALA is $\EA(\g \ot S, D, s\ta),$ which is again an untwisted EALA.
\end{remark}
By taking into account that the invariant bilinear form $\inpr_L$ on $L =
L(\g, \bs)$ is by assumption the restriction of $\inpr_S$ to $L,$ the
following lemma is immediate from the above.
\begin{lemma}\label{subeals-split}
Let $E=\EA(L,D,\ta)=L \oplus C \oplus D$ be an EALA with centreless core
$L=L(\g,\bs)$ as in \eqref{ssec:rev-lt1}. Assume, without loss of generality,
that the invariant bilinear form $\inpr_E$ of $E$ is such that its
restriction to $L$ is the form $\inpr_L$ above. By means of
\eqref{ssec:msl1} view $D$ as a subalgebra of $\SCDer(\g \ot S)$. Then
\begin{equation} \label{subeals-split1} E_S = \EA(\g \ot S, D, \ta) = \g \ot
S \oplus C \oplus D \end{equation} is an untwisted EALA containing $E$ as a
subalgebra.
\end{lemma}
\begin{remark}\label{clarification} That there is no loss of generality on the choice of $\inpr_E$ follows from Remark \ref{scaling}. Indeed, scaling a given form to produce $\inpr_L$ when restricted to L will result in replacing $\EA(\g \ot S, D, \ta)$ by $\EA(\g \ot S, D, s\ta).$ The relevant conclusion that $E$ is a subalgebra of an untwisted EALA remains valid.
\end{remark}
The following lemma will be useful later.
\begin{lemma}\label{lift-fix} Let $E=L \oplus C \oplus D$ be an EALA with centreless core an fgc Lie torus.
{\rm (a)} Let $g\in \Aut_k(L)$. Then the endomorphism $f_g$ of $E$ defined
by
\begin{equation} \label{lift-fix1}
f_g (l \oplus c \oplus d) = g(l) \oplus c \oplus d \end{equation}
is an automorphism of $E$ if and only if $g \circ d = d \circ g$ holds for
all $d\in D$. \sm
{\rm (b)} The map $g \mapsto f_g$ is an isomorphism between the groups
\begin{align*}
\Aut_D (L) = \{ g\in \Aut_k(L) : g \circ d = d \circ g \hbox{ for all } d\in D\} \end{align*}
and $\{f \in \Aut_k(E) : f (L) = L, f|_{C\oplus D} = \Id \}$. In particular,
for any $g$ in
\begin{equation} \label{lift-fix2}
\{ g\in \Aut_R(L) : g (L^\la) = L^\la \hbox{ for all } \la \in \La \} \subset \Aut_D(L)\end{equation}
the map $f_g$ of \eqref{lift-fix1} is an automorphism of $E$.
\end{lemma}
\begin{proof}
(a) It is immediate from \eqref{lift-fix1} and the multiplication rules \eqref{n:gencons3} that $f_g$ is an automorphism of $E$ if and only if \begin{enumerate}
\item[(i)] $ g \circ d = d \circ g $ holds for all $d\in D$ and
\item[(ii)] $\si\big( g(l_1), \, g(l_2)\big) = \si(l_1, l_2)$ holds for all $l_i \in L$.
\end{enumerate}
To show that the second condition is implied by the first, recall that $\si$
is defined by \eqref{eala-cons0}, whence (ii) is equivalent to $\big( (d
\circ g)(l_1) \mid g(l_2) \big) = \big( d(l_1) \mid l_2)$. Because of (i)
this holds as soon as $g$ is orthogonal with respect to $\inpr$. But this is
exactly what \cite[Cor.~7.4]{NPPS} says.
The first part of (b) is immediate. Any automorphism stabilizing the
homogeneous spaces $L^\la$ commutes with $\euD$ viewed as a subset of
$\SCDer(L)$. If it is also $R$-linear it commutes with all of $\SCDer(L)$ and
so in particular with the subalgebra $D\subset \SCDer(L)$. \end{proof}
\section{Lifting automorphisms in the untwisted case} \label{sec:lift-split}
In this section we assume that $E$ is an extended affine Lie algebra whose
centreless core $E_{cc}$ is untwisted in the sense that $E_{cc} = L = \g \ot
R.$ In other words $L = L(\g, \text{\bf \rm Id}).$ In particular $R = S$ and
$t_i = z_i.$
\subsection{Notation.} We let $\bG$ and $\widetilde{\bG}$ be the adjoint and simply connected algebraic $k$--groups corresponding to $\g.$ Recall that we have a central isogeny
\begin{equation}\label{isogeny}
1 \to \bmu \to \wti{\G} \to \bG \to 1
\end{equation}
where ${\bmu}$ is either $\bmu_m$ or $\bmu_2 \times \bmu_2.$
The algebraic $k$--group of automorphisms of $\g$ will be denoted by
$\bAut(\g).$ For any (associative commutative unital) $k$-algebra $K$ by
definition $\bAut(\g)(K)$ is the (abstract) group $\Aut_K(\g \otimes K)$ of
automorphisms of the $K$--Lie algebra $\g \otimes K.$
Recall that we have a split exact sequence of $k$--groups (see \cite{SGA3}
Exp. XXIV Th\'eor\`eme 1.3 and Proposition 7.3.1)
\begin{equation}\label{splitexact}
1 \to \bG \to \bAut(\g) \to \bOut(\g) \to 1
\end{equation}
where $\bOut(\g)$ is the finite constant $k$--group $\Out(\g)$ corresponding
to the group of symmetries of the Dynkin diagram of $\g$.\footnote{The group
$\bOut(\g)$ is denoted by $\bAut\big(\text{\rm Dyn}(\g)\big)$ in
\cite{SGA3}.}
There is no canonical splitting of the above exact sequence. A splitting is
obtained (see \cite{SGA3}) once we fix a base of the root system of a Killing
couple of $\wti{\bG}$ or $\bG.$ Accordingly, we henceforth fix a maximal
(split) torus $\widetilde{\mathbf T}\subset \widetilde{\dG}$. Let $\Sigma =
\Sigma(\widetilde{\dG},\widetilde{\mathbf T})$ be the root system of
$\widetilde{\dG}$ relative to $\widetilde{\mathbf T}$. We fix a Borel
subgroup $\widetilde{\mathbf T}\subset \widetilde{\mathbf B}\subset
\widetilde{\dG}$. It determines a system of simple roots
$\{\alpha_1,\ldots,\alpha_\ell \}$. Fix a Chevalley basis $\lbrace
H_{\alpha_1},\ldots H_{\alpha_\ell},\ X_{\alpha},\ \alpha\in\Sigma\rbrace $
of $\dg$ corresponding to the pair $(\widetilde{\mathbf
T},\widetilde{\mathbf B})$. The Killing couple $(\widetilde{\bB},
\widetilde{\bT})$ induces a Killing couple $(\bB,\bT)$ of $\bG.$
In what follows we need to consider the $R$-groups obtained by the base the
change $R/k$ of all of the algebraic $k$--groups described above. Note that
$\bAut(\g)_R = \bAut(\g \otimes R).$ Since no confusion will arise we will
omit the use of the subindex $R$ (so that for example (\ref{isogeny}) and
(\ref{splitexact}) should now be thought as an exact sequence of group
schemes over $R$).
\begin{theorem}\label{lifting theorem} The group
$\Aut_{R}(\g\otimes R)$ is in the image of the map $\rescc$ of
\eqref{def:rescc}, i.e., every $R$-linear automorphism of $\g \otimes R$ can
be lifted to an automorphism of $E$.
\end{theorem}
\begin{proof}
By (\ref{splitexact}) we have
\begin{equation} \label{lift-thm1}
\Aut_{R}(\g\otimes R)=\G(R)\rtimes \Out(\g). \end{equation}
We will proceed in 3 steps: \begin{enumerate}[(1)]
\item Lifting of automorphisms in the image of $\widetilde{\G}(R)$ in
$\G(R)$.
\item Lifting of automorphisms in $\G(R)$.
\item Lifting of the elements of $\Out(\g)$.
\end{enumerate}
To make our proof more accessible we start by recalling the main ingredients
of the construction of $E$, see \ref{gen-data} and \ref{eala-cons}.
\begin{itemize}
\item[(a)] Up to a scalar in $k$, the Lie algebra $\g \ot R$ has a unique nondegenerate invariant bilinear form $\inpr_R$, namely $(x\ot r \mid x'\ot r')_R = \ka(x,x')\, \ep(rr')$ where $\ka$ is the Killing form of $\g$, $x,x'\in \g$ and $\eps \in R^*$ is given by $\eps(\sum_{\la \in \La} a_\la t^\la) = a_{\bf 0}$. Recall that $t^\la = t_1^{\la_1} \cdots t_n^{\la_n}$ for $\la = (\la_1, \ldots, \la_n)\in \La = \Z^n$.
\item[(b)] The Lie algebra $D$ is a $\La$-graded Lie algebra of skew-centroidal derivations of $R$ acting on $\g \ot R$ by $\Id \ot d$ for $d\in D$. Every homogeneous $d\in D$, say of degree $\la$, can be uniquely written as $d=t^\la \pa_\theta$ for some additive map $\theta \co \La \to k$, where
$\pa_\theta(t^\mu) = \theta(\mu) t^\mu$ for $\mu \in \La$.
\item[(c)] The Lie algebra $E$ is constructed using the general construction \ref{gen-data} with $L=\g \ot R$, $D$ as above, $V=C=D^{\gr*}$ with the canonical $D$-action on $L$ and $C$, the central $2$-cocycle of \eqref{eala-cons0} using the bilinear form $\inpr_R$ of (a) above, and some $2$-cocycle $\ta\co D \times D \to C$.
\end{itemize}
In our proofs of steps 1 and 2 we will embed $E$ as a subalgebra of a Lie
algebra $\wti E$ and use the following general result.
\begin{lemma} \label{embed-gen}
Assume that $R$ is a subring of a commutative associative ring $\wti R$. We
put $\wti L = \g \ot \wti R$.
{\rm (a)} Assume that \begin{itemize}
\item[\rm (i)] the action of $D$ on $R$ extends to an action of $D$ on $\wti
R$ by derivations.
\item[\rm (ii)] $\wti \si \co \wti L \times \wti L \to C$ is a central
$2$-cocycle such that $\wti \si |_{L \times L } = \si$.
\end{itemize}
Then $D$ acts on $\wti L$ by $d (x \ot s) = x \ot d(s)$ for $d\in D$, $x\in \g$ and $s\in \widetilde{R}$. The data $(\wti L, \wti \si, \ta)$ satisfy the conditions of the construction {\rm \ref{gen-data}}, hence define a Lie algebra $\wti E = \wti L \oplus C \oplus D$. It contains $E$ as a subalgebra.
{\rm (b)} Let $\tilde f \in \Aut(\wti E)$ satisfy $\tilde f (L \oplus C) = L
\oplus C$. Then $\tilde f (E) = E$.
\end{lemma}
\begin{proof} The easy proof of (a) will be left to the reader. In (b) it remains to show that $\tilde f(D) \subset L \oplus C \oplus D$. We fix $d\in D$. We then know $\tilde f(d) = \tilde l + \tilde c + \tilde d$ for appropriate $\tilde l \in \wti L$, $\tilde c \in C$ and $\tilde d \in D$. We claim that $\tilde l \in L$. For arbitrary $l \in L$ we have $d \cdot l= [d,l]_E = [d,l]_{\wti E}$ where $[.,.]_E$ and $[.,.]_{\wti E}$ are the products of $E$ and $\wti E$ respectively. Hence $\tilde f\big( d \cdot l\big) = [\tilde f(d), \, \tilde f(l)]_{\wti E} = [\tilde l + \tilde c + \tilde d, \, \tilde f(l)]_{\wti E}$. Denoting by $(\cdot)_{\wti L}$ the $\wti L$-component of elements of $\wti E$, it follows that
\[
\tilde f\big( d(l)\big)_{\wti L} = [\tilde l, \, \tilde f(l)_{\wti L}]_{\wti E} + [ \tilde d, \tilde f(l)_{\wti L}]_{\wti E}.
\] By assumption for all $x\in L$, $\tilde f(x)_{\wti L} \in L$. It follows that the last term in the displayed equation and the left hand side lie in $L$. Since $C$ is the centre of $\wti L \oplus C$ we know $\tilde f(C) = C$ whence $(\pr_L \circ \tilde f)(L) = L$ for $\pr_L \co L \oplus C \to L$ the canonical projection. The displayed equation above therefore implies $[\tilde l, L]_{\wti L} \subset L$.
We will prove that this in turn forces $\tilde l \in L$. Indeed, let $\{r_i :
i\in I\}$ be a $k$-basis of $R$ and extend it to a $k$-basis of $\wti R$, say
by $\{s_j : j\in J\}$. Thus $\tilde l = \sum_i x_i \ot r_i + \sum_j y_j \ot
s_j$ for suitable $x_i$, $y_j \in \g$. For every $z\in \g$ we then have
$[\tilde l , z\ot 1] = \sum_i [x_i, z] \ot r_i + \sum_j [y_j , z] \ot s_j \in
\g \ot R$. Hence $[y_j, z] = 0$ for all $j\in J$. Since this holds for all
$z\in \g$, we get $y_j = 0$ for all $j\in J$ proving $\tilde l \in \g \ot R$.
\end{proof}
After these preliminaries we can now start the proof of Theorem~\ref{lifting
theorem} proper. In what follows we view $R$ as a subring of the iterated
Laurent series field $F= k((t_1))((t_2)) \cdots ((t_n)).$\footnote{The field
$F$ is more natural to use than the function field $K = k(t_1, \cdots,
t_n)$. The extensions of forms and derivations of $R$ are easier to see on
$F$ than $K$. There is also a much more important reason: The absolute Galois
group of $F$ coincides with the algebraic fundamental group of $R.$ This fact
is essential in \cite{GP}.}\sm
{\it{Step 1. Lifting of automorphisms of $\bG(R)$ coming from
$\widetilde{\G}(R)$.}}
We will follow the strategy suggested by Lemma~\ref{embed-gen} and construct
a Lie algebra $\wti E = (\g\ot F) \oplus C \oplus D$ containing $E=(\g \ot R)
\oplus C \oplus D$ as subalgebra, and then show that if $g\in {\wti \dG}(R)
\subset {\wti \dG}(F),$ then $\Ad g \in \Aut_F(\g \otimes F)$ can be lifted
to an automorphism of $\wti E$ that stabilizes $E$ and whose image under
$\rescc$ is precisely $\Ad g \in \Aut_{R}(\dg \ot R)$.
The following lemma implies that the conditions of Lemma~\ref{embed-gen}(a) are satisfied.
\begin{lemma}\label{const-f}
{\rm (a)} The linear form $\eps\in R^*$ extends to a linear form $\wti \eps$ of $F$.
{\rm (b)} The bilinear form $\inpr^{\wti{}}$ defined by $(x\ot f\mid x'\ot f')^{\wti{}} = \ka(x,x') \, \wti \eps (ff')$ for $x,x'\in \g$, $f,f'\in F$, is an invariant symmetric bilinear form extending the bilinear form $\inpr$ of $\g \ot R$.
{\rm (c)} Every derivation $d\in D$ extends to a derivation $\tilde d$ of $F$ such that \begin{enumerate}
\item[\rm (i)] $\tilde d$ is skew symmetric with respect to the bilinear form $\inpr^{\wti{}}$,
\item[\rm (ii)] $d \mapsto \tilde d$ is an embedding of $D$ into $\Der_k (F)$.
\item[\rm (iii)] Every $d\in D$ acts on $\wti L = \g \ot F$ by the derivation $\Id\ot \tilde d$ which is skew-symmetric with respect to the bilinear form $\inpr^{\wti{}}$.
\end{enumerate}
{\rm (d)} Let $\si_D \co \wti L \times \wti L \to D^*$ be the central $2$-cocycle of \eqref{eala-cons0} with respect to the action of $D$ on $\widetilde{L}$ defined in {\rm (c)}. Let $\pr \co D^* \to C$ be any projection of $D^*$ onto $C$ whose restriction to $C \subset D^*$ is the identity map. Then $\wti \si = \pr \circ \si_D \co \wti L \times \wti L \to C=D^{\gr*}$ is a central $2$-cocycle such that $\wti \si |_{L \times L} $ is the central $2$-cocycle appearing in the construction of $E$.
\end{lemma}
\begin{proof} An {\it arbitrary} $k$--derivation of $R$ extends to a $k$--derivation of $F$. To see this use the fact that $\Der_k(R)$ is a free $R$--module admitting the degree derivations $\pa_i = t_i \pa/\pa t_i$ as a basis. It is thus sufficient to show that the $\pa_i$ extend to $k$--derivations of $F$, but this is easy to see.
The rest of the proof is straightforward and will be left to the reader.
\end{proof}
We can now apply Lemma~\ref{embed-gen}(a) and get a Lie algebra $\wti E =
\wti L \oplus C \oplus D,$ with $\wti L = \g \ot F,$ containing $E= L \oplus
C \oplus D$ as a subalgebra.
Since $\ad X_\al$, $\al \in \Sigma$, is a nilpotent derivation, $\exp( \ad f
X_\al)$ is an elementary automorphism of $\g \ot F$ for all $f \in F$. It is
well-known that, since $F$ is a field, the group $\widetilde{\dG}(F)$ is
generated by root subgroups $U_{\alpha} = \{ x_{\alpha}(f)\mid f \in F\}$,
$\alpha\in\Sigma$ and that $\Ad x_\al(f) = \exp( \ad f X_\al)$. By
Proposition~\ref{li-elem}, $\Ad x_\al(f)$ lifts to an automorphism of $\wti
E$ which maps $\g \ot F $ to $(\g \ot F) \oplus C$ and such that its $(\g \ot
F)$-component is $\Ad x_\al(f)$. Consequently, for any $g\in
\widetilde{\dG}(F)$ there is an automorphism $\tilde f_{g}\in
\Aut_{k}(\widetilde{E})$ such that $(\pr_{\g\otimes F_n}\circ \tilde
f_g)|(\g\otimes F)=\Ad g \in \Aut_{F}(\g\otimes F)$. Moreover, again by
Proposition~\ref{li-elem}, $\tilde f_g(C) = C$, whence $\tilde f_g(L \oplus C
) = L \oplus C$ whenever $g \in \widetilde{\bf G}(R)$. Therefore, by
Lemma~\ref{embed-gen}, we get $\tilde f_g(E) = E$. This finishes the proof of
{\it Step 1}.
\medskip
{\it{Step 2. Lifting automorphisms from $\G(R).$ }}
We begin with a preliminary simple observation.
\begin{lemma}\label{newimage} There exist an integer $m > 0$ such that the algebra $\widetilde{R}=k[t_1^{\pm \frac{1}{m}},\ldots, t_n^{\pm \frac{1}{m}}]$ has the following property: All the elements of $\bG(R)$, when viewed as elements of $\bG(\wti{R})$, belong to the image of $\wti{\bG}(\wti{R})$ in $\bG(\wti{R}).$
\end{lemma}
\begin{proof} Recall that $H^1(R, \bmu_m) \simeq R^\times/{(R^\times)}^m$. Let $m$ be the order of $\bmu(k)$ (if $\bmu = \bmu_2 \times \bmu_2$ we can take $m = 2$ instead of $m = 4.)$ Consider the exact sequence
$$\wti{\bG}(R) \to \bG(R) \to H^1(R,\bmu)$$
resulting from (\ref{isogeny}). Let $g \in \bG(R)$ and consider its image
$[g] \in H^1(R, \bmu).$ Then $g$ is in the image of $\wti{\bG}(R)$ if and
only if $[g] = 1.$ It is clear that the image of $[g]$ in $H^1(\wti{R},
\bmu)$ is trivial. The lemma follows.
\end{proof}
Let $\wti{R}$ be as in the previous lemma, and let $\wti L = \g \ot \wti{R}$.
By Lemma~\ref{embed-gen}(a) we have a Lie algebra $\wti E = \wti L \oplus C
\oplus D$ containing $E= L \oplus C \oplus D$ as a subalgebra.
Let $g\in\G(R) \subset \bG(\wti{R})$. To avoid any possible confusion when
$g$ is viewed as an element of $\bG(\wti{R})$ we denote it by $\wti{g}$. By
{\it{Step 1}} there is a lifting $\tilde f_g\in \Aut_k(\wti E)$ of $\Ad
\wti{g} \in \Aut_{\wti{R}} (\g \ot \wti{R})$. To establish this we used that
$\tilde f_g (\wti{L} \oplus C) = \wti{L} \oplus C$. But since $g \in \bG(R)$ and $\tilde{f}_g$ lifts $\Ad \wti{g}$ we conclude that $\tilde f_g (L \oplus C) = L \oplus C$. We can thus apply Lemma~\ref{embed-gen}(b) and conclude $\tilde f(E) = E$. Hence $\tilde f|_E$ is the desired lift of $\Ad g\in \Aut_{R}(\g \otimes R)$. This completes the proof of {\it Step 2}.
{\it{Step 3. Lifting automorphisms from $\Out(\g)$}.}
Let $g$ be a diagram automorphism of $\g$, or more generally any automorphism
of $\g$. We identify $g$ with $g\ot \Id_R$ and note that $g$ is an $R$-linear
automorphism of $\g \ot R$ preserving the $\La$-grading. Hence
Lemma~\ref{lift-fix}(b) shows that $g$ lifts to an automorphism of $E$. This
completes the proof of Theorem~\ref{lifting theorem}.
\end{proof}
\section{Lifting automorphisms in the fgc case}
In this section we will consider an EALA $E$ whose centreless core $L$ is an
fgc Lie torus. If $R = k[t_1^{\pm 1}, \ldots, t_n^{\pm 1}]$ is the centroid
of $L$ (see the second paragraph of \ref{ssec:msl}), we will show that any
$R$-linear automorphism of $L$ lifts to an automorphism of $E$. Although our
method of proof is inspired by general Galois descent considerations, we will
give a self-contained presentation (with some hints for the expert readers
regarding the Galois formalism) . \sm
Throughout we will use the notation and definitions of \S\ref{sec:subEALA}.
Thus $L=L(\g, \bs)$ is a multiloop Lie torus with $\bs=(\si_1, \ldots,
\si_n)$ consisting of commuting automorphisms $\si_i\in \Aut_k(\g)$ of order
$m_i$. The crucial point here is that the subalgebras $L\subset \g\ot S$ and
$E\subset E_S$ are the fixed point subalgebras under actions of $\Ga =
\ZZ/m_1\ZZ \oplus \cdots \oplus \ZZ/m_n\ZZ$ on $\g \ot S$ and $E_S$
respectively. In this section we will write the group operation of $\Ga$ as
multiplication.
Indeed, let $\ga_i$ be the image of $(0, \ldots, 0,1,0, \ldots , 0)\in
\ZZ^n$ in $\Ga$. Then $\ga_i$ can be viewed as an automorphism of $S$ via
$\gamma_i \cdot z^\la=\zeta_{m_i}^{ \la_i}z^\la$ for $\la \in \La = \Z^n$.
This defines in a natural way an action of $\Ga$ as automorphisms of $S$.
Clearly $R=S^\Ga$. The group $\Ga$ also acts on $\g$ by letting $\ga_i$ act
on $\g$ via $\si_i^{-1}$. The two actions of $\Ga$ combine to the tensor
product action of $\Ga$ on $\g \ot S$. Note that $\Ga$ acts on $\g \ot S$ as
automorphisms. The subalgebra $L\subset \g \ot S$ is the fixed point
subalgebra under this action.\footnote{In fact, $S/R$ is a Galois extension
with Galois group $\Ga$. The action of $\Ga$ on $\g \ot S$ is the twisted
action of $\Ga$ given by the loop cocycle $\eta(\bs)$ mapping $\ga_i\in \Ga$
to $\si_i^{-1} \ot \Id_S \in \Aut_S(\g\ot S)$.}
By construction every $\ga \in \Ga$ acts on $\g \ot S$ by an $R$-linear
automorphism preserving the $\La$-grading of $\g \ot S$. Identifying (with
any risk of confusion) $\ga \in \Ga$ with this automorphism, the inclusion
\eqref{lift-fix2} applied to $E_S = \mathfrak{g} \ot S \oplus C \oplus D$
says that $\ga$ extends to an automorphism $f_{\ga} \in \Aut_k(E_S)$ given by
\eqref{lift-fix1}. Moreover, the group homomorphism $\ga\mapsto f_\ga$
defines an action of $\Ga$ on $E_S$ by automorphisms. By construction, $E$ is
the fixed point subalgebra of $E_S$ under this action. To summarize,
\[ L=(\g \ot S)^\Ga \quad \hbox{and} \quad E=(E_S)^\Ga.\]
The action of $\Ga$ on $\g \ot S$ gives rise to an action of $\Ga$ on the
automorphism group $\Aut_k(\g \ot S)$ by conjugation: $\ga \cdot g = \ga
\circ g \circ \ga^{-1}$ for $g\in \Aut_k(\g \ot S)$ and $\ga \in \Gamma.$
Similarly, $\Ga$ acts on $\Aut_k(E_S)$ by conjugation. The first part of the
following theorem shows that these two actions are compatible with the
restriction map $\rescc$ of \eqref{def:rescc}.
\begin{theorem} \label{litwi}
{\rm (a)} The restriction map $\rescc \co \Aut_k(E_S) \to \Aut_k(\g \ot S)$
is $\Ga$-equivariant. Its kernel is fixed pointwise under the action of
$\Ga$. \sm
{\rm (b)} The canonical map
\[ \Aut_k(E_S)^\Ga \to \Ima (\rescc) ^\Ga \]
induced by $\rescc$ is surjective. \sm
{\rm (c)} Every $R$-linear automorphism $g$ of $L$ lifts to an automorphism
$f_g$ of $E$, i.e., $\rescc({f_g}) = g$.
\end{theorem}
\begin{proof} (a) Let $\ga\in \Ga$ and view $\ga$ as an automorphism of $\g \ot S$.
By construction $\rescc(f_\ga) = \ga$. Since $\rescc$ is a group
homomorphism, for any $f\in \Aut_k(E_S)$ we get $\rescc(\ga \cdot f) =
\rescc(f_\ga \circ f \circ f_\ga^{-1}) = \ga \circ \rescc(f) \circ \ga^{-1} =
\ga \cdot \rescc(f).$ We have determined the kernel of $\rescc$ in
Proposition~\ref{res-ker}. The description in loc.\ cit.\ together with the
definition of the lift $f_\ga$ in \eqref{lift-fix1} implies the last
statement of (a).
(b) By \cite[I\S5.5, Prop.~38]{serre}, the exact sequence of $\Gamma$-modules $1 \to \Ker(\rescc) \to \Aut_k(E_S) \to \Ima(\rescc)\to 1$ gives rise to the long exact cohomology sequence
\[ 1 \to \Ker(\rescc) \to \Aut_k(E_S)^\Ga \to \Image(\rescc)^\Ga \to H^1\big( \Ga, \Ker(\rescc)\big) \to \cdots
\]
of pointed sets. Since $\Ker(\rescc)$ is a torsion-free abelian group and $\Ga$ is finite, we have $H^1\big( \Ga, \Ker(\rescc)\big)= 1.$ Now (b) follows.
(c) Every automorphism $g\in \Aut_k(\g \ot S)^\Ga$ leaves $(\g \ot S)^\Ga =
L$ invariant and in this way gives rise to an automorphism $\rho_L(g) \in
\Aut_k(L)$. Similarly we have a group homomorphism $\rho_E \co
\Aut_k(E_S)^\Ga \to \Aut_k(E)$. Since $\rescc \co \Aut_k(E_S) \to \Aut_k(\g
\ot S)$ is $\Ga$-equivariant, it preserves the $\Ga$-fixed points. We thus
get the following commutative diagram where $\overline{\res}_{c,E} \co
\Aut_k(E) \to \Aut_k(L)$ is the map \eqref{def:rescc}:
\begin{equation} \label{litwi1} \vcenter{
\xymatrix{
\Aut_k(E_S)^\Ga \ar[r]^{\rescc} \ar[d]_{\rho_E} & \Aut_k(\g \ot S)^\Ga \ar[d]^{\rho_L} \\
\Aut_k(E) \ar[r]^{\overline{\res}_{c,E}} & \Aut_k(L)
}}\end{equation}
We will prove (c) by restricting the diagram \eqref{litwi1} to subgroups.
Observe that $\rho_L$ maps $\Aut_S(\g \ot S)^\Ga$ to $\Aut_R(L)$. In fact,
we claim
\[ \rho_L \co \Aut_S(\g \ot S)^\Ga \to \Aut_R(L) \quad \hbox{is an isomorphism.}
\]
This can be proven as a particular case of a general Galois descent result of
affine group schemes. That said, due to the concrete nature of the algebras
involved it is easy to give a direct proof (which we now do). The Lie
algebra $L$ is an $S/R$-form of $\g \ot S$. Indeed, the $S$-linear Lie
algebra homomorphism
\[ \theta \co L \ot_R S \to \g \ot_k S, \quad \textstyle \sum_i x_i \ot s_i \ot s \mapsto \sum_i x_i \ot s_i s\]
where $\sum_i x_i \ot s_i \in L$, $s\in S$, is an isomorphism. This can be
checked directly \cite[Lem.~5.7]{abp2.5}, or derived from the fact that $L$
is given by the Galois descent described in the last footnote. It follows
that $L \subset \g\ot S$ is a spanning set of the $S$-module $\g \ot S$,
which implies that $\rho_L$ is injective. For the proof of surjectivity, we
associate to $g\in \Aut_R(L)$ the automorphisms $g \ot \Id_S\in \Aut_S(L \ot
S)$ and $\tilde g = \theta \circ (g \ot \Id_S) \circ \theta^{-1} \in
\Aut_S(\g \ot S)$. We contend that $\tilde g \in \Aut_S(\g\ot S)^\Ga$, i.e.,
$\ga \circ \tilde g \circ \ga^{-1}= \tilde g$ holds for all $\ga \in \Ga$.
Since both sides are $S$-linear, it suffices to prove this equality by
applying both sides to $l\in L$. Since $\theta(l\ot 1) = l$ we get $(\theta
\circ (g \ot \Id) \circ \theta^{-1})(l) = \big( \theta \circ (g \ot
\Id)\big)(l\ot 1) = \theta^{-1}(g(l) \ot 1) = g(l)$ and since $\ga$ fixes $L
\subset \g \ot S$ pointwise the invariance of $\tilde g $ follows. It is
immediate that $\rho_L(\tilde g) = g$. \sm
By Theorem~\ref{lifting theorem}, every $S$-linear automorphism of $\g \ot S$
lifts to an automorphism of $E_S$, in other words $\Aut_S(\g \ot S) \subset
\Image(\rescc)$. Using (b) this implies that the canonical map $\rescc^{-1}
\big( \Aut_S(\g \ot S)^\Ga\big) \to \Aut_S(\g \ot S)^\Ga$ is surjective. By
restricting the diagram \eqref{litwi1} we now get the commutative diagram
\[ \xymatrix{
\rescc^{-1} \big( \Aut_S(\g \ot S)^\Ga\big) \ar@{->>}[r] \ar[d]_{\rho_E}
& \Aut_S(\g \ot S)^\Ga \ar[d]_\simeq^{\rho_L}\\
\overline{\res}_{c,E}^{-1}\big( \Aut_R(L)\big) \ar[r] &\Aut_R(L)
}\]
which implies that the bottom horizontal map is surjective and thus finishes
the proof.
\end{proof}
\section{The conjugacy theorem}\label{sec:conju}
In this section we will prove the main result of our paper: Theorem
\ref{main} asserting the conjugacy of Cartan subalgebras of a Lie algebra $E$
which give rise to fgc EALA structures on a Lie algebra $E$
(Theorem~\ref{main-res}). Assume therefore that $H$ and $H'$ are subalgebras
of $E$ such that $(E,H)$ and $(E,H')$ are fgc EALAs.\footnote{We have seen
that the core, in particular the fgc assumption, is independent of the
chosen invariant bilinear form.} The strategy of our proof is as follows:
(a) Show that the canonical images $H_{cc}$ and $H'_{cc}$ of $H$ and $H'$
respectively in the centreless core $E_{cc}$ are conjugate by an automorphism
of $E_{cc}$ that can be lifted to $E.$
This allows us to assume $H_{cc} = H'_{cc}$. Then we prove that
(b) Two Cartan subalgebras $H$ and $H'$ of $E$ with $H_{cc} = H'_{cc}$ are
conjugate in $\Aut_k(E)$.
It turns out that part (b) can be proven for all EALAs, not only for fgc
EALAs. In view of later applications we therefore start with part (b), which
is the theorem below.
\begin{theorem}
\label{main-non-fgc}
Let $(E,H)$ and $(E,H')$ be two EALA structures on the Lie algebra $E$. We put $H_c = H \cap E_c$, $H_{cc} = \overline{H_c} \subset E_{cc}$ and use $'$ to denote the analogous data for $(E,H')$ keeping in mind that $E_c = E'_c$ by Corollary~{\rm \ref{cores are the same}}. Assume $H_{cc} = H'_{cc}$. Then
\sm
{\rm (a)} $H_c = H'_c.$
{\rm (b)} There exists $f\in \Ker(\rescc)\subset \Aut_k(E)$ such that $f(H) = H'$.
\end{theorem}
\begin{proof} (a)
Let $x\in H_c$. Since $H_{cc}=H'_{cc}$ there exists $y\in H'_c$ such that
$\overline{x}=\overline{y}\in E_{cc}$. Then $c=x-y\in C =Z(E_c)$, so that the
elements $x$ and $y$ commute. Being elements of $H_c$ and $H'_c$, both $\ad_E
x$ and $\ad_E y$ are $k$-diagonalizable endomorphisms of $E$. It follows
that $\ad_E c$ is also $k$-diagonalizable.
We now note that it follows from $[C,D]_E\subset C$ and $[C,E_c]_E=0$ that
any eigenvector of $\ad_E c$ with a nonzero $D$-component necessarily
commutes with $c$. Therefore $c\in Z(E)\subset H'_c.$ Thus $x=y+c\in H'_c$,
and therefore $H_c\subset H'_c$. Thus $H'_c = H_c$ by symmetry finishing the
proof of (a). \sm
Since the proof of (b) is much more involved, we have divided it into a
series of lemmas (Lemma~\ref{almost-n} -- Lemma~\ref{deri}). The reader will
find the proof of (b) after the proof of Lemma~\ref{deri}.
Because $H'_c=H_c=H_{cc}\oplus C^0$ we have decompositions $H=H_{cc}\oplus
C^0\oplus D^0$ and $H'=H_{cc}\oplus C^0\oplus D'^0$ for a (non-unique)
subspace $D'^0 \subset E$. Our immediate goal is restrict the possibilities
for $D'^0$.
\begin{lemma}
\label{almost-n}
$D'^0 \subset H_{cc} \oplus C \oplus D^0$.
\end{lemma}
\begin{proof}
Let $d'^0 \in D'^0$, say $d'^0 = l' + c + d$ with obvious notation. Since $[d'^0, h]_E = 0$ for $h\in H'_{cc} = H_{cc}$ we get
$ 0 = [l' + c + d, h]_E = \big( [l',h]_L + d(h) \big) + \si(l',h) = [l', h]_L $
because $\CDer(L)^0(H_{cc}) = 0$ and therefore $d(h) = \si(l',h) = 0$. Thus $l'\in C_L(H_{cc}) = L_0$.
We have two Lie tori structures on $L$, the second one is denoted by $L'$;
the $L'$-structure has a $\Lambda '$-grading $L'=\oplus _{\lambda'\in
\Lambda'}L^{\lambda'}$, induced by $D'^0.$ Similarly, $L=\oplus_{\lambda\in
\Lambda} L^{\lambda}$ is induced by $D^0$. Since $H_{cc} = H'_{cc}$ the
identity map of $L$ is an isotopy (see \cite[Theorem 7.2]{Al}). Thus
\[
L^{\lambda}_{\alpha}=L'^{\, \phi_{\Lambda}(\lambda)+\phi_s(\alpha)}_{\phi_r(\alpha)}.
\]
The nature of the maps $\phi$ is given in {\it loc.\ cit.} All that is
relevant to us is the fact that for all $\la, \al$ there exist appropriate
$\al', \la'$ such that $L_{\alpha}^{\lambda} = {L'}_{\alpha'}^{\lambda'}$.
Since $D'^0$ induces the $\La$-grading of $L$, we have for $l ^\la \in L
^\la$ that
\[ k l ^\la \ni [d'^0, l ^\la]_E = [l' + c + d, l ^\la]_E =
\big( [l',l ^\la]_L + d(l ^\la) \big) + \si(l',l ^\la).\]
Thus $0 = \si(l',l ^\la)(\tilde d) = (\tilde d (l') \mid l ^\la)$ for all
$\tilde d \in D$ and all $l ^\la$. By the nondegeneracy of $\inpr$ on $L$ we
get $\tilde d(l') = 0$ for all $\tilde d \in D$. As $D^0 \subset D$ induces
the $\La$-grading of $L$ this forces $l'\in L^0$, whence $l'\in L^0_0 =
H_{cc}$. But then $[l', l ^\la]_L \in k l ^\la$ so that the equation above
implies $d(l ^\la) \in k l ^\la$. We can write $d=\sum_{\ga \in \Ga} r^\ga
d^{0\ga}$ for some $r^\ga \in R^\ga$ and $d^{0\ga} \in D^0$. Since $r^\ga
d^{0\ga} (l ^\la) \in L ^{\la + \ga}$ we get $r^\ga d^{0\ga}(l ^\la)= 0$ for
all $\ga \ne 0$. But $R$ acts without torsion on $L$, so $r^\ga = 0$ or
$d^{0\ga} = 0$ for $\ga \ne 0$, and $d\in D^0$ follows.
\end{proof}
We keep the above notation and set $C^{\neq \mu } =\oplus _{\la\neq
\mu}C^{\la}$.
\begin{lemma} \label{lem-psi}
There exists a subspace $V\subset H'$ such that
\begin{enumerate}[\rm (a)]
\item $H'=H_c\oplus V$, $V\subset C^{\neq 0}\oplus D^0$, and
\item $V$ is the graph of some linear map $\xi\in \Hom(D^0, C^{\neq 0})$.
\end{enumerate}
\end{lemma}
\begin{proof} (a)
By the already proven part (a) of Theorem~\ref{main-non-fgc} we have
$H'=H'_c\oplus D'^0=H_c\oplus D'^0$ and by Lemma~\ref{almost-n}, $D'^0\subset
H_{cc}\oplus C\oplus D^0$. We decompose
\begin{equation} \label{nona1}
H_{cc}\oplus C\oplus D^0=(H_{cc}\oplus C^0)\oplus (C^{\neq 0}\oplus D^0).
\end{equation}
Let $p:H_{cc}\oplus C\oplus D^0\to C^{\neq 0}\oplus D^0$ be the projection
with kernel $H_{cc} \oplus C^0$ and put $V=p(D'^0)$. Since $D'^0\cap
(H_{cc}\oplus C^0)\subset D'^0 \cap E_c=0$, we see that $p|_{D'^0} \co D'^0
\to V $ is a vector space isomorphism. Note also that $V \subset H'$. Indeed,
every $v\in V$ is of the form $v=p(d'^0)$ for some $d'^0\in D'^0$, whence
$d'^0=h+c^0+v$ for unique $c^0 \in C^0, h\in H_{cc}$. Since $h,c^0\in H'$ it
follows that $v=d'^0-c^0-h \in H'$. Moreover the inclusion $V\subset C^{\neq
0}\oplus D^0$ implies $V\cap (H_{cc}\oplus C^0)=0$ by \eqref{nona1}. By a
dimension argument we now get $H'=H_c\oplus V$.
(b) The multiplication rule \eqref{derbracket} together with the fact that
the $\La$-grading of $D$ is induced by $D^0$ shows $[D,D] = \bigoplus_{\la
\ne 0} D^\la$. Hence, using \eqref{n:gencons3} and the perfectness of $E_c$,
we have $E=[E,E] \oplus D^0$ and then $D^0 \simeq E/[E,E] \simeq D'^0$. In
particular, $\dim (V) = \dim (D'^0) = \dim (D^0)$. Note also that $V \cap
C^{\neq 0} = \{0\}$. Indeed, let $v=p(d'^0)$ for some $d'^0 = h_{cc} + c^0 +
c^{\neq 0} + d^0$ (obvious notation). Then $p(d'^0) = c^{\ne 0} + d^0\in
C^{\neq 0}$ forces $d^0=0$, whence $d'^0 \in E_c$. But then $d'^0 = 0$
because $E_c \cap D'^0 = \{0\}$. Therefore $v=p(d'^0) = 0$. It now follows
that the projection $p_1 \co C^{\neq 0} \oplus D^0 \to D^0$ with kernel
$C^{\neq 0}$ is injective on $V$. By reasons of dimensions $p_1 |_V \co V \to
D^0$ is a vector space isomorphism. Its inverse followed by the projection
onto $C^{\neq 0}$ is the map $\xi$ whose graph is $V$.
\end{proof}
\begin{lemma}
\label{wei}
{\rm (a)} The weights of the toral subalgebra $V$ of $C \oplus D$ are the
linear forms $\ev'_\mu \in V^*$ for $\mu \in \supp (C) = \supp (D) \subset
\La$, defined by
\[ \ev'_\mu(\xi(d^0) + d^0) = \ev_\mu (d^0)\]
for $d^0 \in D^0$ and $\xi$ as in Lemma~{\rm \ref{lem-psi}}.
{\rm (b)} There exists a unique linear map $\psi_\mu\co D^\mu \to C^{\neq
\mu}$ such that the $\ev'_\mu$-weight space of $C\oplus D$ is given by
\begin{equation} \label{wei0}
(C\oplus D)_{\ev'_\mu} = C^\mu \oplus \{ \psi_\mu (d^\mu) + d^\mu : d^\mu \in D^\mu\}.\end{equation}
{\rm (c)} We have $\psi_0 = \xi$. \end{lemma}
\begin{proof}
(a) Since $V \subset H'\cap (C \oplus D)$ the space $V$ is indeed a toral
subalgebra of $C\oplus D$. We write the elements of $V$ in the form $\xi(d^0)
+ d^0$. Since $\ta(D^0, D) = 0$ we then have the following multiplication
rule for the action of $V$ on $C \oplus D$:
\begin{equation}
\label{wei1} [\xi(d^0)+d^0, \, c+d]_E= (d^0\cdot c-d\cdot \xi(d^0)) + [d^0,d]_D.
\end{equation}
It follows that $C^\mu$ is contained in $(C \oplus D)_{\ev'_\mu}$. Moreover,
for any eigenvector $c+d$ of $\ad V$ with $d\ne 0$ the $D$-component $d$ is
an eigenvector of the toral subalgebra $D^0$ of $D$, whence $d\in D^\mu$ for
some $\mu\in \supp D$ and thus $c+d \in (C\oplus D)_{\ev'_\mu}$.
(b) By \eqref{wei1} we have $c+d \in (C \oplus D)_{\ev'_\mu}$ with $d\ne 0$
if and only if $d=d^\mu$ and
\begin{align*}
\ev_\mu(d^0)\, (c + d^\mu) &= \ev'_\mu( \xi(d^0) + d^0)\, (c + d^\mu)
= [\xi(d^0) + d^0, \, c + d^\mu]_E \\
&= \big( d^0 \cdot c - d^\mu \cdot \xi(d^0)\big) + \ev_\mu(d^0) d^\mu
\end{align*}
holds for all $d^0 \in D^0$. Thus $\ev_\mu(d^0)c=d^0\cdot c - d^\mu\cdot
\xi(d^0)$. Writing $c$ in the form $c=\sum_{\lambda\in \Lambda} c^{\lambda}$
with $c^\la \in C^\la$ and comparing homogeneous components we get
$\ev_\mu(d^0)c^\la= \ev_\la( d^0)c^\la-(d^{\mu}\cdot \xi(d^0))^\la$ for every
$\lambda\in \Lambda$, whence
\begin{equation}\label{equation}
(d^{\mu}\cdot \xi(d^0))^{\lambda}=\ev_{\lambda-\mu}(d^0)c^{\lambda}.
\end{equation}
Since $C^\mu \subset (C \oplus D)_{\ev'_\mu}$ we can assume $c^\mu = 0$. But
for $\lambda\neq \mu$ there exists $d^0\in D^0$ such that
$\ev_{\lambda-\mu}(d^0)\neq 0$ and then \eqref{equation} uniquely determines
$c^\la$. Thus $c + d = \psi_\mu(d^\mu) + d^\mu$ for a unique $\psi_\mu(d^\mu)
\in C^{\neq \mu}$. That $\psi_\mu$ is linear now follows from uniqueness.
(c) We have $C^0 \oplus V \subset (C \oplus D)_{\ev'_0}$ by definition of the
$\ev'_0$-weight space. Moreover, by \eqref{wei0} and Lemma~\ref{lem-psi}(b),
$\dim (C^0 \oplus V) = 2\dim D^0 = \dim (C \oplus D)_{\ev'_0}$, whence $C^0
\oplus V = (C \oplus D)_{\ev'_0}$. But then $\psi_0 = \xi $ follows from
Lemma~\ref{lem-psi}(b) and the uniqueness of $\psi_0$.
\end{proof}
\begin{lemma}\label{deri}
Let $\psi \co D \to C$ be the unique linear map satisfying $\psi|_{D^\mu} =
\psi_\mu$ with $\psi_\mu$ as in Lemma~{\rm \ref{wei}}. Then $\psi$ is a
derivation, i.e., for $d^\la \in D^\la$ and $d^\mu \in D^\mu$ we have
\begin{equation} \label{wei4}
\psi_{\la + \mu}([d^\la, d^\mu]_D)
= d^\la \cdot \psi_\mu(d^\mu) - d^\mu \cdot \psi_\la (d^\la).\end{equation}
\end{lemma}
\begin{proof} The multiplication in $C \oplus D$ yields
\[
[ \psi_\la (d^\la) + d^\la, \, \psi_\mu(d^\mu) + d^\mu]_{C \oplus D}
= \big( \tau(d^\la, d^\mu) + d^\la \cdot \psi_\mu(d^\mu) - d^\mu \cdot \psi_\la (d^\la) \big) + [d^\la, d^\mu]_D. \]
Since $\ta(d^\la, d^\mu) \in C^{\la + \mu}$ the $C^{\neq (\la +
\mu)}$-component of this element is
\begin{equation} \label{wei5}
[ \psi_\la (d^\la) + d^\la, \, \psi_\mu(d^\mu) + d^\mu]_{C^{\neq (\la + \mu)}}
= d^\la \cdot \psi_\mu(d^\mu) - d^\mu \cdot \psi_\la (d^\la).
\end{equation}
But because $\psi_\la(d^\la) + d^\la \in (C\oplus D)_{\ev'_\la}$ and
$\psi_\mu(d^\mu) + d^\mu \in (C\oplus D)_{\ev'_\mu}$ we also know
\[ [\psi_\la (d^\la) + d^\la, \, \psi_\mu(d^\mu) + d^\mu]_{C\oplus D}\in (C\oplus D)_{\ev'_{\la + \mu}}.\]
By \eqref{wei0} there are therefore two cases to be considered, $[d^\la,
d^\mu]_D \ne 0$ and $[d^\la, d^\mu]_D = 0$.
{\em Case $[d^{\lambda},d^{\mu}]_D\neq 0$}: In this case
\[[ \psi_\la (d^\la) + d^\la, \, \psi_\mu(d^\mu) + d^\mu]_{C \oplus D} = \psi_{\la + \mu}([d^\la, d^\mu]_D) + [d^\la, d^\mu]_D\]
with $C^{\neq(\la + \mu)}$-component equal to $\psi_{\la + \mu}([d^\la,
d^\mu]_D)$ so that \eqref{wei4} follows by comparison with \eqref{wei5}.
{\em Case $[d^{\lambda},d^{\mu}]_D =0$}: In this case \eqref{wei4} becomes
\[ d^\la \cdot \psi_\mu (d^\mu) = d^\mu \cdot \psi_\la(d^\la)\]
with both sides being contained in $C^{\ne (\la + \mu)}$. We prove this
equality by comparing the $C^{\rho}$-component of both sides for some $\rho
\ne \la + \mu$. By \eqref{equation}
\begin{align*}
\ev_{(\rho-\la)-\mu}(d^0)\,\psi(d^{\mu})^{ \rho-\la}
&= d^{\mu}\cdot (\xi (d^0)^{(\rho-\la)-\mu}) \quad \mbox{and} \\
\ev_{(\rho-\mu)-\lambda}(d^0)\, \psi(d^{\lambda})^{\rho-\mu} &=
d^{\lambda}\cdot (\xi (d^0)^{(\rho-\mu)-\lambda}).
\end{align*}
Hence, choosing $ d^0\in D^0$ such that $\ev_{\rho-\lambda-\mu}(d^0)\neq 0$,
setting $e=\ev_{\rho-\lambda-\mu}(d^0)^{-1}$ and using $[d^\la, d^\mu]_D = 0$
we have
\[ \begin{array}{lll}
d^{\lambda}\cdot \psi(d^{\mu})^{\rho-\lambda}
& = & d^{\lambda}\cdot (e\, d^{\mu}\cdot \xi (d^0)^{\rho-\lambda-\mu})
= e\,d^{\mu}\cdot(d^{\lambda}\cdot \xi (d^0)^{\rho-\lambda-\mu}) \\
& = &
e\,d^{\mu}\cdot(\ev_{\rho-\lambda-\mu}(d^0)
\psi(d^{\lambda})^{\rho-\mu}) = d^{\mu}\cdot \psi (d^{\lambda})^{\rho-\lambda}.
\end{array}\]
This finishes the proof of \eqref{wei4}. \end{proof}
{\em End of the proof of Theorem}~\ref{main-non-fgc}(b): It follows from
Lemma~\ref{deri} that the map $f$ defined by \eqref{res-ker1} lies in
$\Ker(\rescc)$. This map fixes $L \oplus C$ pointwise and maps $D^0$ to
$(\psi + \Id)(D^0) = V$. Thus $f(H) = H'$ in view of Lemma~\ref{lem-psi}.
\end{proof}
We can now prove the main result of this paper: Conjugacy of Cartan
subalgebras of a Lie algebra $E$ which give rise to fgc EALA structures on
$E$.
\begin{theorem}\label{main-res} Let $(E,H)$ be an EALA whose centreless core $E_{cc}$ is fgc, and let $(E,H')$ be a second EALA structure. Then there exists an automorphism $f$ of the {\em Lie algebra\/} $E$ such that $f(H)=H'$.
\end{theorem}
\begin{proof} Using the notation of Theorem~\ref{main-non-fgc}, we know that $(E_{cc}, H_{cc}$) and $(E_{cc}, H'_{cc})$ are fgc Lie tori. Both subalgebras $H_{cc}$ and $H'_{cc}$ are MADs of $L=E_{cc}$ (\cite[Cor.~5.5]{Al}). We can now apply \cite{CGP}: Both $H$ and $H'$ are Borel-Mostow MADs in the sense of \cite[\S13.1]{CGP} and satisfy the conditions of the general Conjugacy Theorem \cite[Thm.~12.1]{CGP}. Hence there exists
$g\in \Aut_{R}(L)$ such that $g(H'_{cc})=H_{cc}$.\footnote{Even though it is
not needed for this work, we remind the reader that $g$ can be chosen in the
image of a natural map $\widetilde{\mathfrak{G}}(R)\to {\rm Aut}_{R}(L)$
where $\widetilde{\mathfrak{G}}$ is a simple simply connected group scheme
over $R$ with Lie algebra $L$.} According to Theorem~\ref{litwi}(c), $g\in
\Aut_{R}(L)\subset \Aut_{k}(L)$ can be lifted to an automorphism, say $f_g$,
of $E$. So replacing the second structure $(E,H')$ by $(E, f(H'))$ we may
assume without loss of generality that $H_{cc}=H'_{cc}$.\footnote{We leave to
the reader to check that $(E, f_g(H'))$ has a natural EALA structure. For
example if $\inpr'$ was the invariant bilinear form of $(E,H')$ then on $(E,
\phi(H')$ we use $(\inpr' \circ(f^{-1}\times f^{-1})$.} An application of
Theorem~\ref{main-non-fgc} now finishes the proof.
\end{proof}
\begin{remarks} (a) We point out that conjugacy does not hold for all maximal $\ad$-diagonaliz\-able subalgebras of an EALA $(E,H)$, see \cite{yaho}.
\sm
(b) In the setting of Theorem~\ref{main-res} let $\Psi$ and $\Psi'$ be the
root systems of the EALA structures $(E,H)$ and $(E,H')$ respectively, cf.\
axiom (EA1) of the Definition~\ref{def:eala}. The dual map of the isomorphism
$f|_H$ is an isomorphism $\Psi' \to \Psi$, namely an isomorphism $H^{\prime
*} \to H^*$ sending $\Psi'$ to $\Psi$ and ${\Psi '}{}\re$ to $\Psi\re$. The
root system $\Psi$ and $\Psi'$ are extended affine root systems and are thus
given in terms of finite irreducible, but possibly non-reduced root systems
$\dot \Psi$ and $\dot \Psi'$ (\cite{AABGP}, or \cite{LN} where $\dot \Psi$
and $\dot \Psi'$ are called quotient root systems). It follows from
\cite[4.1]{LN} that isomorphic extended affine root systems have isomorphic
quotient root systems. Thus $\dot \Psi' \simeq \dot \Psi$. We thus recover
\cite[Prop.~6.1(i)]{Al} where this was proven by a different method. \sm
(c) We can be more precise about the automorphism $f$ needed for conjugacy in
Theorem~ \ref{main-res}. Namely, let $\res_D: \Aut_k(E)\rightarrow
\Aut_k(E/E_c)\simeq \Aut_k(D)$ be the canonical map. Then the conjugating
automorphism $f$ can be chosen in the normal subgroup
\[ G = \Ker (\res_D)\cap \rescc^{-1}\big( \Aut_R(E_{cc}) \big)\]
of $\Aut_k(E)$. Indeed, the automorphism $f$ of the proof of
Theorem~\ref{main-res} has the form $f=f'\circ f_g$ where $f'\in \Ker
(\rescc)$ and thus $f'\in G$ by Proposition~\ref{res-ker}(b). Moreover, $f_g$
is a certain lift of $g\in \Aut_R(E_{cc})$. That $\res_D(f_g)=1$ follows from
the proof of Theorem~\ref{lifting theorem}, Proposition~\ref{li-elem}(iii)
and Lemma~\ref{lift-fix}.
\end{remarks} | 59,799 |
Those of us who were the first in our families to go to college face some particular challenges. The biggest challenge is not having a trusted mentor to guide our decisions.
Source: First in the Family A first in the family student: “Everyone was telling me to go in there and get a business degree. I don’t know what the reason was but, yeah, sure, I’ll do it. I think that was another reason I struggled in early college. Why do I need to know this? As soon as I shifted over to political science I was loving it.”.
We see in the BUZZ Today quote how the absence of a trusted mentor impacts success in college. Notice this student was taking the advice of “everyone”. Business was what others steered him to with little regard for what fit him. In fact, what “everyone” was saying was what they thought about him and about business. Probably the “everyone” didn’t know very much about either one.
Those that come from a family where college education was the experience of earlier family generations have a real advantage. Not only do the family elders have the experience of college, they have two equally important things to bring to the first in the family college student.
Knowing how to register for classes and where to look for scholarships are important, but there even more important things. A mother or father with a college degree know their son or daughter. As parents who have, for eighteen years, spent almost every day with their child, they know that child. They know their likes and dislikes. They know their habits and above all they know what motivates them. Without motivation all is lost. So the educated parent can act as a mentor and avoid the “everyone” advice in the BUZZ Today quote.
But even more important, the parent is a trusted adviser. The son or daughter knows the parent after nearly two decades in the relationship. It is that trust which can help the student avoid the risk of losing motivation because the courses simply do not fit who they are.
In the end, every first in the family college student needs a trusted source of advice and guidance. A good place to start is with the book Your Future is Calling. | 208,179 |
Kittie have shared a new live video for their song “Charlotte.” The footage is from the band’s recently released live film “Kittie: Live At The London Music Hall.” That effort features the band’s October 27, 2017 20th anniversary show at the London Music Hall in London, ON. A number of former members joined the group onstage that night including: Ivy Vujic, Jennifer Arroyo, Jeff Phillips, Fallon Bowman, and Tanya Candler. | 371,993 |
All Results for Rosinko
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Try this: With soundex on the surname | 323,719 |
Posts: 4889
Registered: 07-02
Platinum Shell said:
Hey Mark IV, I think I have a pretty good idea for another separate module.
Basically, when DooMGuy's health reaches critical levels, he starts to slow down. I tested and found that it seems to work best when his turbo is set to 60-70 when he's bleeding. It would not only add more depth to the game, but would add more challenge, and serve as a reminder for the player to get more health. Plus, you've also made a few sprites for the DooMGuy when bleeding (along with your other good spritework), so going into chasecam and seeing his slowed speed along with him bleeding and limping would actually be pretty neat to see. (ignore my fascination with dark things).
Combine this with the health regeneration module and it would make a good combo. Again, just a suggestion.
Posts: 805
Registered: 08-11
Mr. Chris said:
An earlier version (I think 0.9 or 0.10) had the low health slowdown but it can impede gameplay, especially when you need to make a jump that only a full speed running can bypass.
Posts: 400
Registered: 12-09
Posts: 2682
Registered: 07-02
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Registered: 07-03
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Registered: 11-11
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Posts: 409
Registered: 12-11
Last edited by HavoX on 12-11-11 at 14:20
Posts: 1647
Registered: 04-03
Mr. Chris said:
Congratulations for winning this year's Cacoward for Best Gameplay Mod.
Posts: 32
Registered: 11-11
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Registered: 12-11
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Registered: 12-09
Posts: 9
Registered: 12-11
webcider said:
...my netbook can't keep up with all the gore is it possible to make the gore a bit lighter?
In the first post, Sergeant_Mark_IV said:
- If your framerate is low, type "gl_use_models 0" to hide the bloodpool 3D models. This helps a lot on old computers.
Last edited by HavoX on 12-14-11 at 21:21
HavoX said:
The SSG and Rocket Launcher are already powerful enough as they are. Why fix what isn't broken?
webcider said:
just for your information i can't get this to work at all in the console
GL_USE_Model "0"
webcider said:
I got one minor issue the fix that is decribed, doesn't work with GZdoom and thats the one i am using. Should i change source port to something lighter perhaps?? I really like the lightning the OpenGL gives makes it kind of nice in dark places to be able to spot armor and health :D
Last edited by HavoX on 12-16-11 at 21:31
Posts: 308
Registered: 01-11
__________________
Chris's uploads - Link
Brutal Video Vault
Brutal Tumblr
Last edited by Glaice on 12-19-11 at 06:47
johnfulgor said:
4)The removal of a bug in the Shotgun reload animation. It happend to me that afer a sequence of continous shots (keeping pressed the fire button) the reloading sequence was played every two shots.
Posts: 289
Registered: 02-09
webcider said:
just for your information i can't get this to work at all in the console
GL_USE_Model "0"
Bloodite Krypto said:
That command doesn't exist, apparently.
Posts: 52
Registered: 05-11
> | 69,796 |
Fares based on your travel habits? Welcome to the future.
It’s common knowledge that passengers pay different prices for their seats. Someone who bought well in advance is likely to have paid a different airfare than a passenger who purchased their ticket at the last minute, while those who set flight alerts have a good chance of scoring a deal.
But according to one software company, airlines may soon begin charging passengers different prices based on who they are.
Revenue management software company PROS — which works with more than 80 international airlines — said that select airlines have already begun implementing “dynamic pricing” structures on their websites.
“2018 will be a very phenomenal year in terms of traction,” John McBride, director of product management for PROS, told Travel Weekly. “Based on our backlog of projects, there will be a handful of large carriers that move toward dynamic pricing science.”
Dynamic or “surge” pricing is an economic strategy wherein companies price services based on demand (think Uber charging more for rides in the rain or at rush hour). And although airlines have already implemented this type of fare — for example, making flights around the holidays more expensive — using new technology, they can tailor fares to specific passengers.
Airline websites will be able to identify customers by their IP addresses and mine data for their flying history. The revenue management system would then create a person-specific fare based off criteria like loyalty status or business/leisure traveler. Loyal customers and leisure travelers would likely pay less, while those who are willing to pay more — like business travelers with a company credit card — would likely see higher prices.
However, before airlines implement a “pure” version of dynamic pricing, reports Travel Weekly, they have to move away from the legacy distribution system put in place after 1978 deregulation. Airline pricing has a limited number of fare classes, each with their own price points and restrictions. Airlines rotate which fare classes are available through their sale cycles. While current customers may believe their fare is unique, they’ve actually fallen into a specific fare class.
If airlines switch to dynamic pricing, each customer on the plane could, theoretically, pay a completely different price. As George Hobica notes in USA Today, dynamic pricing could make finding a good price on airfare even more confusing than it already is. | 138,526 |
In brief view, we shall examine the following:
– Sex and Gender
– Gender stereotype and roles
– The origins of gender stereotype
Gender: WHAT DOES IT REALLY MEAN?
Gender refers to the socially constructed categorisation of individuals into masculine and feminine. Unlike gender which is a social constraint, SEX on the other hand refers to the categorisation of an individual on the basis of the genetic materials they produce during sexual reproduction or intercourse. Masculine and feminine are gender categories, while male and female are sex categories. Thus, sexes are biological categorisation of individuals based on their reproductive responsibilities and capabilities and gender are social categorisation of males and females into defined roles and responsibilities. Further differentiation of gender and sex can be broken thus
Sex:
Men have the XY chromosomes while women have the XX chromosomes
Women menstruate, men do not
Women have vaginas, men have penises
Women develop breasts, men do not.
Men have testicles, women do not.
Gender:
In most places in the world, men do not do babysitting; it is presumed a female responsibility.
In many places, women wear high heels, long nail, and skirts, men do not.
In Saudi Arabia, women are not allowed to drive and in many other places they are not allowed to participate in politics and certain trade.
THE CONCEPT OF SEX: What defines an individual as a male or female or neither?
Members of species of living organisms across all domains are divided into 2 or more categories and it is based on the complimentary materials they are capable of reproducing during sexual reproduction. Typically, species of living organisms have the male and female sexes. The female sex is defined as the individual who produces the larger gamete, that is, the one which is capable of bearing offspring. Sex is primarily based on the reproductive capability of individual specie.
In plant kingdoms, species are mostly hermaphrodite that is, they bear both the male and female reproductive capabilities. In other cases some individual species bear single gametes the case may be. However, in the animal kingdom, sexes differ in broadways across species. In mammals for example in humans, sex is determined by the X and Y chromosomes (XX for females, XY for males), thus making the sex in human a dichotomous one. All individuals in the human species have at least one X chromosome. The Y chromosome is shorter than the X.
GENDER STEREOTYPE AND ROLES
Gender roles and stereotype is more or less a cultural dictum of modalities, etiquettes, instructions, mannerism, responsibilities, etc. which men and women are expected of. Without gender, there cannot be any gender role and stereotype. Gender stereotype are laid down social manuscripts which men and women are required to correspond with. They are culturally defined social standards of relations between male and female. What is obtained as a gender acceptable norm in one society may not be obtained in the another, however, gender stereotype and roles are universal and are culturally distinct from the other and most times very similar.
In Saudi Arabia, where women are not allowed to drive, engage in active politics or be ambitions, which are culturally induced, is not obtainable in Western societies where such perceptions are frown at. Thus, gender stereotypes and roles are culturally induced standards of behaviour and relations for the sexes. If individuals put up behaviours that do not confirm to these perceived standards of their gender, the social consequences maybe unpleasant. For example, in Africa where nail painting is a feminine fashion, a man who paints his nails may get a backlash and negative label from those around him.
Generally, gender stereotype cuts across cultures and domains. In a study by Williams and Best (1982) across 30 countries, stereotyping of females and males are pervasive. Men are generally believed to be more dominant, independent, aggressive and achievement-oriented while females are believed to be nurturing, affiliative, sensitive, gentle, etc. In another study by Williams and Best (1989), men and women who lived in highly developed countries perceived themselves as being more similar than their counterparts in less developed countries. A simple explanation for this is that women are more educated and independent in developed countries than their counterparts in less developed countries. Also, respondents in Christian societies were more likely to perceive similarities between the sexes than those from Muslim countries.
One may wonder: HOW and WHERE do these gender stereotypes come from?
Gender role learning and subsequently stereotype are rooted in socialization. Socialization is simply the process where the norms, beliefs, culture, perceptions, sentiments, etc of a social enclave are passed onto their members. The primary agents of socialization of gender role and subsequently, gender stereotype are: FAMILY, EDUCATION, PEER GROUP AND THE MASS MEDIA. Each of these agents services the commonly perceived gender roles and images in the minds of the members of a given society, thus setting standards and expectations of what is generally perceived as suitable for male and female. Other agents of socialization are religious organizations, social gatherings, workplace, etc.
Repeated exposure to cues from these agents of socialization elicits the individual’s perception that what he or she defines as the right or suitable gender attitude is natural, but unknown to him or her, he or she is simply a product of social conditioning. Thus, the socially accepted standards of behaviours and roles for a gender is nothing more than indoctrinated perception from social construction as a result of our repetitive interaction with the above agents of socialization. Thus, gender role can be defined as set of behavioural and social norms that are considered as generally appropriate for a man or woman. A typical example in Africa is that it is inappropriate for a woman to make passes at a man, specifically, to chase after a man. The people in these societies adhere to this and see such conception as a natural standard for gender interpersonal relationships, but these are individuals unknown to them are merely acting on a false sense of natural dictum rather than socialized sentiments.
Stereotypes generally come with negativity. The negative stereotype that emanates from gender leads to sexism- the discrimination of a person based on their gender. Women are mostly are on the receiving end of the negativity of gender. These stereotypes have evolved from the old type typical backlash of women as being less smart, less competent, less responsible, and less creative than men to the modern form of sexism which merely deals on the denial of sexism. Although in less develop countries in Africa and the middle East in particular, some these old type of sexism persists, like women being refused into political participation, jobs, violence and abuse. In places like Pakistan, Saudi Arabia and elsewhere, women are objectified as sex objects and household pets. The recognition for their talents, capabilities and roles outside the home and marriage is often frowning at violently opposed. | 414,323 |
Single Family Detached
- 3 beds
- 1.5 baths
- $2,200 price
- 219062159 mls id
- Half Baths: 1
- Square Feet: 1,125 sqft
- Year Built: 1954
- Lot Size: 0.23 acres
- Neighborhood: Beverly Hills Vlg
- Status: Active
Description and previous landlord. Security Deposit 3300. Cleaning fee $150. No pets. 13 months minimum lease.
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Wonderful solid ranch in a nice quiet Beverly Hills neighborhood. Great location close to shopping. Birmingham Schools. Wood floors throughout, newer windows, cove ceilings, huge Florida room, tiled kitchen floor, granite counters,(kitchen updated in 2014), huge basement with high ceilings, new...
MLS ID 219067453
31746 E Sheridan Drive Beverly Hills, MI
3 beds, 2 baths, 1,587 sqft, $2,400
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Listing Agent: Mary Kay Buckley
Updated: 15th July, 2019 6:50 PM. | 328,393 |
TITLE: Proving identity through combinatorial model
QUESTION [3 upvotes]: How to prove this identity using combinatorial argument?
$\displaystyle\sum_{r=0}^{n}\binom{n}{r}\binom{p}{r+s}\binom{q+r}{m+n} = \sum_{r=0}^{n}\binom{n}{r}\binom{q}{r+m}\binom{p+r}{n+s}$
Only thing I could think of was to change $\binom{n}{r}$ to $\binom{n}{n-r}$ and then we can probably think of choosing $(n-r+r+s+m+n) = (n+s+m+n)$ people or something but its dead end for me. Please help. Also it seems if I can figure one side, other might trivially follow. Please don't ignore the problem.
REPLY [6 votes]: Suppose you have a fictional country where there are two big parties. The $\color{red}{red}$ and the $\color{blue}{blue}$ parties. Initially, there are $\color{red}{q}$ and $\color{blue}{p}$ of each party. Suppose you want to pick a political committee (senate?) consisting of $\color{red}{(n+m)}$ and $\color{blue}{(n+s)}$ members. Assume, also, that you have a pool of $n$ candidates that can turn red or blue, they are somewhat neutral and can represent either party.
By law you have to choose at least $\color{red}{m}$ from the original red party and at least $\color{blue}{s}$ from the original blue party.
There are two ways this elections will happen. Either first the red party chooses or the blue party chooses.
For the LHS suppose that first the $\color{blue}{blue}$ party chooses. Assume that $r+s$ is the number of people selected from the original party. Then we need to choose $n-r$ from the pool. This we can do in $\binom{p}{r+s}\binom{n}{n-r}$ ways. Then, the rest of the pool ($r$ of them) will become $\color{red}{red},$ so in the other election you have to choose from $q+r$ candidates, giving you $\binom{q+r}{n+m}$ ways. This gives us a total of $\binom{p}{r+s}\binom{n}{n-r}\binom{q+r}{n+m}$ ways to do this election.
The RHS is the same, but the $\color{red}{red}$ people vote first. | 27,026 |
The AP Tour
All Time Low, The Matches, Sonny, Forever the Sickest Kids
House of Blues, Orlando, Fl
April 18, 2008
by Jen Cray
Alternative Press Magazine's third annual tour has reached an All Time Low. If this is what's popular among American teenagers, I fear for the upcoming generation of bands who were weaned on such drivel.
At just $13 a ticket for six bands, this tour was in the very least a good bargain for fans. It was no surprise that the House of Blues show in Orlando sold out.
Brightly colored ADD kids, lost somewhere between Saturday Night Fever and Hot Hot Heat, go by the moniker Forever the Sickest Kids. The cinematic Tristar Pictures theme music played as they stepped out onto the stage, setting the tone for an evening heavily reliant on the dance pop of the Eighties. The wildly happy and energetic bunch didn't offer much more than just a dance party, but -- lucky for them -- that and a pretty face was all this crowd seemed to have come for. They danced about on-stage while tossing out Top 40-ready nuggets like "Hey Brittany" and "Believe Me I'm Lying" and the audience seemed set to eat them up with a spoon. My prediction is that these guys are going to become familiar faces in the upcoming months.
When Sonny Moore left From First to Last in 2007 to pursue a solo career, I didn't pay much notice. Now that I've seen what the little man has been working up in the studio on his own, I care even less. The indiscriminate noise that this discombobulated band made blended hip hop, emo, and club music in an indigestible mixture. The only redeeming elements of this "band" were drummer Sean Friday and guitarist Chris Null. The rest, Moore in particular, were like a bad imitation of Ugly Kid Joe... only worse. Even climbing his way from one end of the audience to the other couldn't save this sad display.
If we're grading on a curve, The Matches aced the course for best band on the bill. Dapper in dress, looking much like Orlando Bloom dressed up like Ducky from Pretty In Pink, vocalist/guitarist Shawn Harris was a captivating front man who didn't have to constantly reach for the audience or run in circles around his band mates to hold the crowd's attention. The music was not quite so effortless. Where once they were just another pop punk band with emo leanings, this California four piece are reaching for mainstream pop land in their latest music.
They're the eldest bunch, having been a band since 1997, but a youthful air and lighthearted modern A.F.I.-like approach to their genre make them a perfect fit on the AP Tour. I won't go so far as to call myself a "fan," but I will say that their 30 minutes on-stage were the bit that I paid the most attention to. At one point they were joined by All Time Low singer Alex Gaskarth, which garnered all kinds of Teen Beat heartthrob screams.
The band that I had otherwise only known as "the kids with the weird haircuts who always get photographed in their underwear," are this year's Fall Out Boy or Panic! At the Disco. They write the same simple emo pop songs that pushed those bands into the stratosphere of celebrity. They've got pre-teen crush-worthy looks, and a non-threatening stage presence that wins over the parents in the crowd as well as their kids. Sure, they've got the words "cock" and "balls" written on their amps in duct tape, but this can be shrugged off as the same sort of goofy humor that made Blink 182 huge.
Blink 182 is the Nirvana to the generation born in the late Eighties. When their "classic" songs were played before All Time Low's set, it became a massive sing-a-long and my rationalization of it is this: in the late Nineties when Blink blew up, they were the antithesis to popular music. Those were the days when 'N Sync, Backstreet Boys, and Britney Spears reigned. Blink was punk in the eyes of the young, when compared to manufactured pop. It makes perfect sense that those fans who grew up watching them went on to start bands of their own and cite them as a major influence.
All of this rolled around in my head as I tried to listen to All Time Low with an open mind. "Six Feet Under the Stars," "Dear Maria," these are the exact same songs that Fall Out Boy wrote when they called them "Sugar, We're Goin' Down" and "Dance, Dance." They're catchy no matter how high the listener's resistance to them. It doesn't make them good, but at times I find my head nodding along because -- dammit -- I'm not made of stone! A good melody can overtake anyone if you forget to hate it.
As co-headliners, All Time Low share top billing on this tour with The Rocket Summer.
| 61,037 |
TITLE: What is the definition of a qubit and a copy/clone of a qubit?
QUESTION [2 upvotes]: A qubit with state $|\psi \rangle =\alpha|0\rangle + \beta|1\rangle$ is defined as : if we have infinite copies of $|\psi \rangle$ and measure them all in the basis $\{|0\rangle,|1\rangle\}$ then $|\alpha|^2$ percent of them would measure $|0\rangle$ after measurement and $|\beta|^2$ percent of them would measure $|1\rangle$. So definition of a qubit with a particular known state depends on the definition of copies of a qubit ( clone/copy ).
We define two qubits A and B being copy/clone of each other if, same percent cent of qubits give same measurement results if we take infinite copies of qubit A and qubit B separately and measure them in $\{|0\rangle,|1\rangle\}$ basis. So definition of being a clone/copy is dependent on its own definition. Thus even the definition of a qubit with a specific known state appears recursive to me. What am I missing ?
REPLY [2 votes]: A qubit is simply not defined the way you define it, neither are clones. This is mostly, because you describe a state by referring to a state. It's not the problem that you are actually referring to copies of the state, you are referring to copies of the state, which is yet to be defined.
Let's go to what I have learned as the Ludwig school of thought about states and experiments, which is an operational definition (and therefore probably closest to the kind of definition you are searching).
We have to start out with a physical experiment. How can I describe this? Well, in an experiment, you have a stage where you prepare your system. After the system is prepared (e.g. creating a beam of polarized light with a laser beam and a polarization filter for instance), you measure it. If you repeat your measurement, you'll get a probability distribution over your measurement outcomes.
Now the key thing to realize is that the preparation procedure describes the state of a system. In other words, a state is an abstract description of how to actually create e.g. a particle with certain prefixed properties. A state is something like a class in object oriented programming and it must not(!) be confused with an instance of that class. In your example, a qubit might be described by the procedure of how to prepare a photon in a superposition state of up and down-polarized light ($|0\rangle,1\rangle$) according to some parameters ($\alpha,\beta$). The single photon is not(!) a state, it is only an instance of the state.
In everyday language, we often confuse the concept of a state and its instances, because it rarely matters, if we know how to interpret the probability distribution produced by a state (which, if we see the state as a preparation procedure, becomes the asymptotic empirical distribution of the instances of this state), but to me it seems crucial to highlight the problem with your definition of states.
Note that until now I have not even mentioned measurements - that is, because they are not important for the definition of states, they form something like its dual part. Now, if you want to compare two states, because, say, you have two preparation procedures and want to determine whether the abstract state behaves the same way, then you need to consider measurements and you would say that two states are equivalent if their empirical distributions according to an informationally complete measurement are the same in the asymptotics of large numbers of instances of the state. [As an aside, note that your measurement procedure above is NOT informationally complete: You do not retrieve the phase information, you can only compare $|\alpha|^2$ and $|\beta|^2$. To be able to say that two states are equivalent, you will have to know the relative phase between $\alpha$ and $\beta$]. It seems to me that what I call "equivalent states" are what you think of as "copied states".
But then, what are cloned states? In fact, we would not define a "cloned state", but a "cloning machine". A cloning machine takes an instance of the state as an input and outputs the same instance and an instance of another state that is defined by this procedure. If the corresponding states are the same, then we consider the machine a "cloning machine". In other words: We have two states, state A which we start out with and another one, state B, with the preparation procedure: "prepare an instance of A, send it to the cloning machine and take the outcome". Now, if these two states are equivalent, then we would say that state B is a clone of state A and the machine was a cloning machine. If we could find a cloning machine that produces a clone for whatever input state I define, then this would be a universal cloning machine and as you very well know, this is forbidden by quantum mechanics.
Now, I said above that this is what I came to know as the "Ludwig school" of thought (an operational approach to quantum mechanics, which was mostly developed by the German physicist Günther Ludwig and his students).
A different definition would be, if you consider the statistical interpretation of quantum mechanics. Then you'd never have a single instance of a state, your "state" would rather be a statistical ensemble of many such instances - at least that's what I think you do.
Another approach would be to just see the "state" as an abstract description of what I called an "instance of a state". It is just not defined operationally. You'd interpret Born's rule as saying: Take a state, measure it. If you perform this experiment more often, then the empirical distribution of the outcomes would asymptotically be given by Born's rule. It does not(!) say: Take copies of the state. It says: Do the experiment all over. If you think about it, then you'll see that this is pretty close to what I described above, because this actually means "prepare the state and measure it". | 192,218 |
81SMEXIZELDAMANMon 19th Feb 2018 Good way to not buy games a lot is not be on nintendolife.....where they talk about new games all the time xD Just play the ones you have instead Lord Head Admin of SonyLife ♥♥♥Videogames are lame♥♥♥
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LONGLEY, WILLIAM PRESTON
LONGLEY, WILLIAM PRESTON (1851–1878). William Preston (Bill) Longley, outlaw, son of Campbell and Sarah Longley, was born in Austin County, Texas, on October 6, 1851. By April 1853 his family had moved to Evergreen, in what was then Washington County, where Longley went to school and worked on the family farm. Tales of Longley's criminal career are a mixture of actual facts and his boasts, but it is known that at the end of the Civil War a rebellious Longley took up with other young men and terrorized newly-freed slaves. On December 20, 1868, Longley, Johnson McKeown, and James Gilmore intercepted three ex-slaves from Bell County; this incident resulted in the death of Green Evans. Longley would later claim that after this he worked as a cowboy in Karnes County, and then killed a soldier as he rode through Yorktown, but there is no corroboration for these stories. He also claimed that he rode with bandit Cullen M. Baker in northeast Texas, but this is unlikely. In 1869–70, he and his brother-in-law, John W. Wilson, were terrorizing residents of south central Texas, and it was alleged that in February 1870, in Bastrop County, they killed a black man named Brice. In March the military authorities offered a $1,000 reward for them. They were also accused of killing a black woman. After Wilson's death in Brazos County, Longley traveled north, later claiming that he killed a traildriver named Rector, fought Indians, killed a horse thief named McClelland, and killed a soldier at Leavenworth, Kansas, for insulting the virtue of Texas women. None of these claims have been corroborated. At Cheyenne, Wyoming Territory, Longley joined a gold-mining expedition into the Wind River Mountains, but was stranded when the United States Army stopped the group. In June 1870 he enlisted in the United States cavalry and promptly deserted. He was captured, court-martialed, and sentenced to two years' confinement at Camp Stambaugh, Wyoming Territory. After about six months he was released back to his unit, where he remained until he again deserted on June 8, 1872. Longley claimed that he lived and rode with Chief Washakie and his Shoshone Indians, which is questionable, and then returned to Texas via Parkerville, Kansas, where he claimed he killed a Charlie Stuart, of whom there is no record. He returned to Texas and Bell County, where his parents had moved, and claimed that he worked as a cowboy in Comanche County and what was then Brown County, allegedly killing a black man and engaging in a gunfight at the Santa Anna Mountains in Coleman County.
In July 1873 Longley was arrested by Mason county sheriff J. J. Finney in Kerr County and taken to Austin so that Finney could be paid a reward. When the reward was not paid, Finney was supposedly paid off by a Longley relative and Longley was released. In late 1874 Longley and his brother James Stockton Longley rode from Bell County to the Lee County home of their uncle, Caleb Longley, who implored Longley to kill a Wilson Anderson for allegedly killing his son. On March 31, 1875, Longley shotgunned Anderson to death while Anderson was plowing a field, and the two brothers fled north to the Indian Territory. They returned to Bell County in July, where James turned himself in; James was later acquitted of any part in Anderson's murder. In November 1875 Longley killed George Thomas in McLennan County, then rode south to Uvalde County, where, in January 1876, he killed William (Lou) Shroyer in a running gunfight. By February 1876 Longley was in Delta County, Texas, sharecropping for the Reverend William R. Lay. A dispute with a local man over a girl led to Longley's arrest. He burned himself out of the Delta County jail and, on June 13, 1876, killed the Reverend Lay while Lay was milking a cow. On June 6, 1877, Longley was captured in DeSoto Parish, Louisiana, by Nacogdoches county sheriff Milton Mast; Longley was returned to Lee County to stand trial for the murder of Wilson Anderson. Longley promptly began writing letters to a local newspaper about his "adventures," claiming that he had killed thirty-two men. On September 5, 1877, he was found guilty of murder and sentenced to hang. He was held in the Galveston County jail until the Court of Appeals affirmed his conviction in March 1878. Longley was baptized into the Catholic Church. On October 11, 1878, before a crowd of thousands in Giddings, Texas, Longley was executed by Lee county sheriff James Madison Brown. Just before his execution, Longley claimed that he had only killed eight men. Rumors persisted that Longley's hanging had been a hoax and that he had gone to South America, and a claim was made in 1988 that he had later reappeared and died in Louisiana. Between 1992 and 1994 an effort was made to find his body in the Giddings Cemetery, but to no avail. There is also some evidence that his body may have been returned to Bell County after his execution.
BIBLIOGRAPHY:
Ed Ellsworth Bartholomew, Wild Bill Longley: A Texas Hard-Case (Houston: Frontier Press of Texas, 1953). Frontier Times, June 1926, June 1927. Henry Clay Fuller, The Adventures of Bill Longley (Nacogdoches, Texas: Baker, n.d.). Galveston Daily News, September 16,ick Miller, "Longley, William Preston," accessed October 17, 2017,.
Uploaded on June 15, 2010. Modified on September 13, 2017. Published by the Texas State Historical Association. | 326,871 |
\begin{document}
\title[Blow up for mass critical NLH ]{On the blow up phenomenon for the mass critical focusing Hartree equation with inverse-square potential}
\author[Y. Chen]{Yu Chen}
\address
{China Academy Of Engineering Physics, \ Beijing,\ China,\
100088,}\email{[email protected]}
\author[C. Lu]{Chao Lu}
\address
{China Academy Of Engineering Physics, \ Beijing,\ China,\
100088,}\email{[email protected]}
\author[J. Lu]{Jing Lu}
\address
{College of Science, China Agricultural University, \ Beijing,\ China,\
100193,}
\email{[email protected]}
\begin{abstract} In this paper, we consider the dynamics of the solution to the mass critical focusing Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$. The main difficulties are the equation is \emph{not} space-translation invariant and the nonlinearity is non-local. We first prove that if the mass of the initial data is less than that of ground states, then the solution will be global. Although we don't know whether the ground state is unique, we can verify all the ground states have the same, minimal mass threshold. Then at the minimal mass threshold, we can construct the finite-time blow up solution, which is a pseudo-conformal transformation of the ground state, up to the symmetries of the equation. Finally, we establish an mass concentration phenomenon of the finite-time blow up solution to the equation.
\end{abstract}
\maketitle
\section{Introduction}
We study the following mass critical focusing Hartree equation with inverse-square potential in $d\ge 3$,
\begin{equation}\label{problem-eq: NLS-H}
\left\{
\aligned
(i\partial_{t}-\Delta + \frac{a}{|x|^{2}}) u = (|\cdot|^{-2}\ast |u|^{2})u\\
u|_{t=0}=u_{0}\in H^{1}(\mathbb R^{d})
\endaligned
\right.
\end{equation}
where $u:\mathbb R\times\mathbb R^{d}\to \mathbb C$ is a complex valued function, $\Delta = \sum_{k=1}^{d}\frac{\partial^{2}}{\partial x_{k}^{2}}$ is the Laplace operator and $-\left(\frac{d-2}{2}\right)^{2}<a<0$. Note that $\La = -\Delta + \frac{a}{|x|^{2}}$ for convenience.
{Solutions to \eqref{problem-eq: NLS-H} conserve the \emph{mass} and \emph{energy}, defined respectively by
\begin{eqnarray}
\nonumber
M(u) &=& \frac{1}{2}\int_{\mathbb R^{d}} |u|^{2}=M(u_{0}), \\
E(u) &=& H(u) - L_{V}(u)=E(u_{0}),
\end{eqnarray}
where
\[
\aligned
H(u) &= \frac{1}{2}\int_{\mathbb R^{d}}\left(|\nabla u(x)|^{2} + \frac{a}{|x|^{2}} |u(x)|^{2}\right)dx, \\
L_{V}(u)& = \frac{1}{4} \iint_{\mathbb R^{d} \times \mathbb R^{d}} \frac{|u(x)|^{2}|u(y)|^{2}}{|x-y|^{2}}\;dx dy.
\endaligned
\]
For the nonlinear Hartree equation with inverse-square potential,
\begin{equation}\label{nlsha}
(i\partial_t-\Delta+\frac{a}{|x|^2}) u = (|\cdot|^{-\gamma}\ast |u|^{2})u,~~0<\gamma<d.
\end{equation}
When $a=0$, \eqref{nlsha} reduces to the `free' nonlinear Hartree equation:
\begin{equation}\label{nls0'}
(i\partial_t-\Delta) u = (|\cdot|^{-\gamma}\ast |u|^{2})u,~~0<\gamma<d.
\end{equation}
Like \eqref{nls0'}, the equation \eqref{nlsha}
enjoys the scaling symmetry
\begin{equation}\label{scaling}
u(t,x) \mapsto u^\lambda(t,x) : = \lambda^{\frac{d+2-\gamma}{2}} u(\lambda^2t, \lambda x).
\end{equation}
This symmetry identifies $\dot H_x^{s_{c}}(\R^d)$ as the scaling-critical space of initial data, where $s_{c}=\frac{\gamma}{2}-1$.
The \emph{mass-critical} problem corresponds to $s_c=0$ (or $\gamma=2$), in which case $M(u)\equiv M(u^\lambda)$. The \emph{energy-critical} problem corresponds to $s_c=1$ (or $\gamma=4$), in which case $E(u)\equiv E(u^\lambda)$. In this paper, we just consider the mass-critical case.
Recently, more and more scientists have been devoted to studying the behavior of the blow-up solution to the dispersive equations, such as the classical nonlinear Schr\"odinger equations and Hartree equations. In the context of the focusing mass-critical nonlinear Schr\"odinger equations $(NLS)$, the characterization
of the minimal mass blowup solutions begins with F. Merle \cite{Mer}, where he showed that if
an $H_x^1$-solution with minimal mass blows up at finite time, then up to symmetries of the equation,
it must be the pseudoconformal ground state. The proof, which was later simplified by
Hmidi and Keraani \cite{HmiKer} relies heavily on the finiteness of the blowup time.
For the mass-critical $(NLS)$, Merle and Tsutsumi \cite{MerTsu} further showed that there must be one point with the same mass focused as the ground state(the ground state is unique) as the time goes, if the solution's initial data is in $H^{1}$ and it blows up in finite time. But for the normal mass critical blow-up solution whose initial data is in $L^{2}$, \cite{Bourgain98} has showed that there is at least one point where the mass concentrates and the speed is parabolic in $d=2$. In particular, we have
Later, \cite{BV2007}and \cite{Ker2006} extended this result to $d=1$ and $d\ge3$. For the focusing mass-critical free
nonlinear Hartree equations,
Miao, Xu and Zhao \cite{MXZ-6} adapted Keraani¡¯s argument \cite{HmiKer}
and showed that any finite time blowup solution with ground state mass and $H_x^1$ initial data must
be the pseudoconformal ground state up to symmetries of the equation.
About the characterization of the minimal mass
blowup solution blowing up at infinite time, Killip, Li, Visan and Zhang \cite{KLVZ} first solved the problem for the focusing mass-critical nonlinear Schr\"odinger equations under the spherically
symmetric assumption.
Later \cite{LiZhang} give the
characterization of the minimal mass
blowup solution blowing up at infinite time for the focusing mass-critical Hartree equation and they showed that any global solution with ground state mass which is spherically symmetric and which
does not scatter must be the solitary wave $e^{it}Q$ up to symmetries.
For other results about the dynamics of the classical Hartree equations, the reader can refer to
\cite{ G,GMX}, \cite{LMZ}, \cite{MXZ-1,MXZ-3,MXZ-2,MXZ-4,MXZ-5}, \cite{MXZ-9,MXZ-7,MXZ-8} and other references.
The Laplace operator with inverse-square potential $\La$ is the limiting form of $-\Delta+a|x|^{-2-\varepsilon}$, which can't be researched by Kato's distrubance methods. So \cite{KMVZZ} utilized Mikhalin Multiplier theorem to establish the equivalence norm theorem between $\La$--Sobolev norm and $\Delta$--Sobolev norm.
For the defocusing nonlinear Schr\"odinger equation with inverse-square potential $(NLS_a)$,
\cite{ZZ-Sca} used the Strichartz estimate and the equivalence norm theorem in \cite{BPSTZ} to establish the interacted Morawetz estimate in order to get the $H^{1}$
scattering theory with energy subcritical case. Furthermore, for the energy critical $(NLS_a)$, \cite{KMVZ-EnrCri} obtained the $\dot H^{1}$ scattering theory in $d=3$. But note that the range of $a$ need be restricted because of the restriction of the target in the equivalence norm theorem. For the focusing $(NLS_a)$: In the energy subcritical case, \cite{KMVZ-focus} established the threshold of the blow-up and scatter if the $H^{1}$ initial satisfies $M^{1-s_{c}}E^{s_{c}}(u_{0}) < M^{1-s_{c}}E^{s_{c}}(Q_{\min\{a, 0\}})$.
For the energy critical case, \cite{CsoGen} established the rigidity argument of the minimal mass blow-up solution with the initial data in $H^{1}$. Compared to the classical Schr\"odinger equation, the rigidity description of Schr\"odinger equation with inverse-square potential can remove the effect of translation. \cite{Bens-Dinh} described the mass concentration phenomenon of the blow-up solution with $H^{1}$ initial data.
So far, there are few results about the dynamics of the solution to the mass critical focusing Hartree equation with inverse-square potential. Inspired by the above works, we consider the dynamics of the solution to the mass critical focusing Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$. The main difficulties are the equation \eqref{problem-eq: NLS-H} is \emph{not} space-translation invariant and the nonlinearity is non-local.
We first prove that if the mass of the initial data is less than that of ground states, then the solution will be global. Although we don't know the ground state is unique, we can verify all the ground states have the same, minimal mass threshold. Then at the minimal mass threshold, we can construct the finite-time blow up solution, which is a pseudo-conformal transformation of the ground state, up to the symmetries of the equation. Finally, we establish an mass concentration phenomenon of the finite-time blow up solution to the equation.
Before we show the main result, we utilize the variational characterization to gain the following important proposition.
\begin{proposition}[Ground State]\label{ground state}
Functional
\[
J(u) : = \frac{M(u)H(u)}{L_{V}(u)}, \qquad u\in H^{1}(\mathbb R^{d}\setminus \{0\})
\]
can gain the minimal value when $J_{\min}$, and the minimal point $W$ has the form like $W(x)=e^{i\theta}mQ (nx)$, where $m, n>0$, $\theta\in\mathbb R$, and $Q\neq0$ is the non-negative non-empty radial solution of the equation
\begin{equation}\label{eq:ground-state}
(-\Delta + a |x|^{-2} )Q + Q = (|\cdot|^{-2}\ast |Q|^{2}) Q,
\end{equation}
where $-(\tfrac{d-2}2)^{2}<a<0$.
if $Q\ge0$ is a non-negative non-empty radial solution of the equation \eqref{eq:ground-state}, and $J(Q)=J_{\min}$, we say $Q$ is a {\bf Ground state}. We define that the set of all ground state is called $\mathcal G$. All ground state has the same mass, which is defined as $M_{gs}$.
\end{proposition}
Our main result in this paper is as follows:
\begin{theorem}\label{thm:main}Suppose that $d\ge3$, $-(\tfrac{d-2}2)^{2}<a<0$, then
\begin{enumerate}[$(1)$]
\item If $M(u_{0})<M_{gs}$, then the solution $u(t)$ of the equation \eqref{problem-eq: NLS-H} is global.
\item If $M(u_{0})= M_{gs}$ and the solution $u(t,x)\in C([0,T), H^1(\mathbb{R}^d))$ blows up in finite time $T>0$, i.e. $\lim_{t\to T^\ast}H(u(t))=\infty$, then we have
\[
u\in
\left\{
e^{i\frac{|\cdot|^{2}}{4(T^{\ast}-t)}} e^{i\theta} { \lambda}^{\frac{d}{2}} Q(\lambda \cdot)
:
\theta\in\mathbb R,
\lambda>0,
Q\in\mathcal G
\right\}.
\]
\item In particular, let $u$ be the solution to \eqref{problem-eq: NLS-H} which blows up in finite time $T>0$, and the function $\lambda(t)$ satisfy $\lim_{t\to T^\ast}\lambda(t)\sqrt{H(u(t))}=\infty$ then there exists a function $x:[0, T^\ast)\to\mathbb R^{d}$, such that
\[
\lim_{t\to T^{\ast}}\frac{1}{2}\int_{|x-x(t)|\le \lambda(t)} |u(t, x)|^{2}dx \ge M_{gs}.
\]
\end{enumerate}
\end{theorem}
\begin{remark}
We require $a\le0$ here, because the variational description is invalid when $a>0$. Without the minimal point in corresponding minimal problem, we can't confirm the result. But we can utilize the ways in \cite{Bens-Dinh} to extend the result to the radial case under the condition $a>0$. \end{remark}
In this chapter, we show some preparation and the theory on the local well-posedness in section $2$. In section $3$, we give the variational characterization and prove the first part of Theorem \ref{thm:main}, that is to say, solution does not blow up if its mass is small enough. In section $4$, we establish the rigid portrays and profile decomposition to describe the blow-up phenomenon in finite time. In section $5$, we give the second part of the proof of theorem \ref{thm:main}--the rigid portrays of the minimal mass blow-up solution in finite time and the third one -- the mass critical phenomenon which is not lower than one of the ground state.
\section{Preliminaries}
In this section, we will show some important tools of harmonic analysis and give the local well-posedness result.
\vskip1em
\subsection{Harmonic analysis adapted to $\La$} In this section, we describe some harmonic analysis tools adapted to the operator $\La$. The primary reference for this section is \cite{KMVZZ1}.
Recall that by the sharp Hardy inequality, one has
\begin{equation}\label{iso}
\|\sqrt{\La}\, f\|_{L_x^2}^2 \sim \|\nabla f\|_{L_x^2}^2\textrm{for} a>-(\tfrac{d-2}2)^{2}.
\end{equation}
Thus, the operator $\La$ is positive for $a> -(\frac{d-2}2)^2$. To state the estimates below, it is useful to introduce the parameter
\begin{equation}\label{rho}
\rho:=\tfrac{d-2}2-\bigr[\bigl(\tfrac{d-2}2\bigr)^2+a\bigr]^{\frac12}.
\end{equation}
We first give the following result concerning equivalence of Sobolev spaces was established in \cite{KMVZZ1}; it plays an important role throughout this paper.
\begin{lemma}[Equivalence of Sobolev spaces, \cite{KMVZZ1}]\label{pro:equivsobolev} Let $d\geq 3$, $a> -(\frac{d-2}{2})^2$, and $0<s<2$. If $1<p<\infty$ satisfies $\frac{s+\rho}{d}<\frac{1}{p}< \min\{1,\frac{d-\rho}{d}\}$, then
\[
\||\nabla|^s f \|_{L_x^p}\lesssim_{d,p,s} \|(\La)^{\frac{s}{2}} f\|_{L_x^p}\textrm{for all} f\in C_c^\infty(\R^d\backslash\{0\}).
\]
If $\max\{\frac{s}{d},\frac{\rho}{d}\}<\frac{1}{p}<\min\{1,\frac{d-\rho}{d}\}$, then
\[
\|(\La)^{\frac{s}{2}} f\|_{L_x^p}\lesssim_{d,p,s} \||\nabla|^s f\|_{L_x^p} \textrm{for all} f\in C_c^\infty(\R^d\backslash\{0\}).
\]
\end{lemma}
Next, we recall some fractional calculus estimates due to Christ and Weinstein \cite{CW}. Combining these estimates with Lemma~\ref{pro:equivsobolev}, we can deduce analogous statements for powers of $\La$ (with suitably restricted sets of exponents).
\begin{lemma}[Fractional calculus]\text{ }
\begin{itemize}
\item[(i)] Let $s\geq 0$ and $1<r,r_j,q_j<\infty$ satisfy $\tfrac{1}{r}=\tfrac{1}{r_j}+\tfrac{1}{q_j}$ for $j=1,2$. Then
\[
\| |\nabla|^s(fg) \|_{L_x^r} \lesssim \|f\|_{L_x^{r_1}} \||\nabla|^s g\|_{L_x^{q_1}} + \| |\nabla|^s f\|_{L_x^{r_2}} \| g\|_{L_x^{q_2}}.
\]
\item[(ii)] Let $G\in C^1(\C)$ and $s\in (0,1]$, and let $1<r_1\leq \infty$ and $1<r,r_2<\infty$ satisfy $\tfrac{1}{r}=\tfrac{1}{r_1}+\tfrac{1}{r_2}$. Then
\[
\| |\nabla|^s G(u)\|_{L_x^r} \lesssim \|G'(u)\|_{L_x^{r_1}} \|u\|_{L_x^{r_2}}.
\]
\end{itemize}
\end{lemma}
Strichartz estimates for the propagator $e^{-it\La}$ were proved in \cite{BPSTZ}. Combining these with the Christ--Kiselev lemma \cite{CK}, we arrive at the following:
\begin{proposition}[Strichartz, \cite{BPSTZ}] Fix $a>-(\tfrac{d-2}{2})^2$. The solution $u$ to
\[
(i\partial_t-\La)u = F
\]
on an interval $I\ni t_0$ obeys
\[
\|u\|_{L_t^q L_x^r(I\times\R^d)} \lesssim \|u(t_0)\|_{L_x^2(\R^d)} + \|F\|_{L_t^{\tilde q'} L_x^{\tilde r'}(I\times\R^d)}
\]
for any $2\leq q,\tilde q\leq\infty$ with $\frac{2}{q}+\frac{d}{r}=\frac{2}{\tilde q}+\frac{d}{\tilde r}= \frac{d}2$ and $(q,\tilde q)\neq (2,2)$. \end{proposition}
We call such pairs $(q,r)$ and $(\tilde{q},\tilde{r})$ \emph{admissible} pairs.
\subsection{Several useful inequalities}
\begin{lemma}[Hardy Inequality \cite{Cav, ZZ-Sca}]
Supposed that $\alpha>0$ , $1< p <\infty$, $\alpha p<d$. then there exists the constant $C>0$, such that
\[
\left\| \frac{u}{|\cdot|^{\alpha}}\right\|_{L^{p}(\mathbb R^{d})} \le C\||\nabla|^{\alpha}u\|_{L^{p}(\mathbb R^{d})}.
\]
If $1\le p<\infty$, $0\le \alpha \le 1$ and $sp<d$, we have
\[
\left(\frac{d-sp}{p}\right)^{s}
\left\| \frac{u}{|\cdot|^{\alpha}}\right\|_{L^{p}(\mathbb R^{d})}
\le
\| u\|_{L^{p}(\mathbb R^{d})}^{1-s}
\| \nabla u\|_{L^{p}(\mathbb R^{d})}^{s}.
\]
Specially, we have
\[
\frac{d-2}{2}\left\| \frac{u}{|\cdot|}\right\|_{L^{2}(\mathbb R^{d})} \le \| \nabla u\|_{L^{2}(\mathbb R^{d})}.
\]
\end{lemma}
\begin{lemma}[Hardy-Littlewood-Sobolev Inequality,\cite{LL}]
If $1< p, q< \infty$, $0<\alpha<d$ and $\frac{1}{p}+\frac{1}{q}+\frac{\alpha}{d}=2$, we have
\begin{equation}\label{Hardy-Littlewood-Sobolev }
\left|
\iint_{\mathbb R^{d}\times \mathbb R^{d}}\frac{f(x)g(y)}{|x-y|^{\alpha}} dxdy
\right|
\lesssim \|f\|_{L^{p}(\mathbb R^{d})}\|g\|_{L^{q}(\mathbb R^{d})}
\end{equation}
\end{lemma}
\begin{lemma}
[Riesz Rearrangement Inequality,\cite{LL}]\label{riesz}
We denote that $\;f^{\ast}$ is the radial non-increase symmetrical rearrangement of the function $f$, that is to say, denote $f^{\ast}$ as the rearrangement of $f$. Then we have
\begin{equation}\label{eq:rearrangement}
\left|\iint_{\mathbb R^{d}\times\mathbb R^{d}} f(x)g(y)h(x-y)dxdy\right|
\le
\large\left| \iint_{\mathbb R^{d}\times\mathbb R^{d}} f^{\ast}(x)g^{\ast}(y)h^{\ast}(x-y)dxdy \large\right|
\end{equation}
\end{lemma}
\subsection{The local wellposedness theory}\label{S:LWP}
We next discuss the local theory for \eqref{problem-eq: NLS-H}.
We begin by making our notion of solution precise.
\begin{definition}[Solution]\label{def:soln} Let $t_0\in\R$ and $u_0\in H_a^1(\R^d)$. Let $I$ be an interval containing $t_0$. We call $u:I\times\R^d\to\C$ a \emph{solution} to
\[
(i\partial_t - \mathcal{L})u = \mu |u|^\alpha u,\quad u(t_0)=u_0
\]
if it belongs to $C_t H_a^1(K\times\R^d)\cap S_a^1(K)$ for any compact $K\subset I$ and obeys the Duhamel formula
\begin{equation}\label{duhamel}
u(t) = e^{-i(t-t_0)\mathcal{L}}u_0-i\mu\int_{t_0}^t e^{-i(t-s)\mathcal{L}}\bigl(|u|^\alpha u\bigr)(s)\,ds
\end{equation}
for all $t\in I$. We call $I$ the \emph{lifespan} of $u$. We call $u$ a \emph{maximal-lifespan solution} if it cannot be extended to a strictly larger interval. We call $u$ \emph{global} if $I=\R$.
\end{definition}
\begin{theorem}
[The local wellposedness]
Supposed that $d\ge3$, $a>-(\tfrac{d-2}2)^{2}$. Then there exists $T=T(\|u_{0}\|_{H^{1}(\mathbb R^{d})})>0$, such that there exists a unique solution $u(t,x)$ of the equation \eqref{problem-eq: NLS-H} satisfying
$$u\in C([0, T);H^{1}(\mathbb R^{d}))\bigcap_{(q, r)\in \Lambda_{0}} L^{q}((0, T), W_{a}^{1, r}(\mathbb R^{d})). $$
\end{theorem}
\begin{proof}
The proofs follow along standard lines using the contraction mapping principle.
Because of the equivalent norm theorem and the Hardy-Littlewood-Sobolev inequality, we need take target carefully. Take $ 0<s<1$, and take $0<\varepsilon\ll1$ which satisfies that $\frac{1+\rho}{d}<\frac{1}{2}-\varepsilon$, $1-s-d\varepsilon>0$. Denote
\begin{gather*}
(\tfrac{1}{q'}, \tfrac{1}{r'}) = (1-\tfrac{d\varepsilon}{2}, \tfrac{1}{2}+\varepsilon)\\
(\tfrac{1}{q_{1}}, \tfrac{1}{r_{1}}, \tfrac{1}{{\tilde r_{1}}}) = (\tfrac{d\varepsilon}{2}, \tfrac{1}{2}-\varepsilon, \tfrac{1}{2}-\varepsilon)\\
(\tfrac{1}{q_{2}}, \tfrac{1}{r_{2}}, \tfrac{1}{{\tilde r_{2}}})
= (\tfrac{1-s-d\varepsilon}{2}, \tfrac{1}{2}+\tfrac{s-1}{d}+\varepsilon, \tfrac{1}{2}-\tfrac{1}{d}+\varepsilon).
\end{gather*}
On one hand, the section of $\varepsilon>0$ guarantees the validity of the Hardy-Littlewood-Sobolev inequality when $\min\{\tilde r_{1}, \tilde r_{2}, \tilde r_{3}\}>r'$. On the other hand, it ensures the condition in which the Sobolev equivalent norm
$ \||\nabla|f\|_{L^{{ r_1}}}\lesssim\|(\La)^{\frac{1}2} f\|_{L^{{ r_1}}}$
and
$ \||\nabla|^{s}f\|_{L^{{ r_2}}}\lesssim\|(\La)^{\frac{s}2} f\|_{L^{{ r_2}}}$
is valid, which is $\frac{1+\rho}{d}<\frac{1}{r_{1}}, \frac{s+\rho}{d}<\frac{1}{r_{2}}$.
Denote the time interval is $I=[0, T]$. Therefore, we have a nonlinear estimate: For $\sigma\in\{0, 1\}$, we have
\begin{align*}
& \Big\|
(|\cdot|^{-2}\ast|u|^{2})u
-
(|\cdot|^{-2}\ast|v|^{2})v\big]
\Big\|_{L^{q'}_{t}(I, \dot W_{a}^{\sigma, r'})}
\\
\lesssim & \Big\|
(|\cdot|^{-2}\ast|u|^{2})u
-
(|\cdot|^{-2}\ast|v|^{2})v\big]
\Big\|_{L^{q'}_{t}(I, \dot W^{\sigma, r'})}
\\
\le & \Big\|(|\cdot|^{-2}\ast(u\overline{(u-v)}))u\Big\|_{L^{q'}_{t}(I, \dot W^{\sigma, r'})}
+
\Big\| ( |\cdot|^{-2}\ast(({u-v})\overline{v}))u\Big\|_{L^{q'}_{t}(I, \dot W^{\sigma, r'})}
\\& +
\Big\| (|\cdot|^{-2}\ast|v|^{2})(u-v)\Big\|_{L^{q'}_{t}(I, \dot W^{\sigma, r'})}
\\\triangleq &I_1+I_2+I_3
\end{align*}
We only estimate $I_2$, since the estimates of $I_1$ and $I_3$ are similar. Using the equivalent norm theorem, Fractional derivative law for space and H\"older inequality for time, we can obtain
\begin{align*}
I_2
\lesssim\; &
T^{s}
\|u-v\|_{L_{t}^{q_{1}}(I, \dot W^{\sigma, \tilde r_{1}})}
\|u\|_{L_{t}^{q_{2}}(I, \dot W^{0, \tilde r_{2}})}
\|v\|_{L_{t}^{q_{2}}(I, \dot W^{0, \tilde r_{2}})}
\\ & +
T^{s}
\|u\|_{L_{t}^{q_{1}}(I, \dot W^{\sigma, \tilde r_{1}})}
\|u-v\|_{L_{t}^{q_{2}}(I, \dot W^{0, \tilde r_{2}})}
\|v\|_{L_{t}^{q_{2}}(I, \dot W^{0, \tilde r_{2}})}
\\ & +
T^{s}
\|v\|_{L_{t}^{q_{1}}(I, \dot W^{\sigma, \tilde r_{1}})}
\|u\|_{L_{t}^{q_{2}}(I, \dot W^{0, \tilde r_{2}})}
\|u-v\|_{L_{t}^{q_{2}}(I, \dot W^{0, \tilde r_{2}})}
\\
\lesssim\; &
T^{s}
\|u-v\|_{L_{t}^{q_{1}}(I, \dot W^{\sigma, r_{1}})}
\|u\|_{L_{t}^{q_{2}}(I, \dot W^{s, r_{2}})}
\|v\|_{L_{t}^{q_{2}}(I, \dot W^{s, r_{2}})}
\\ & +
T^{s}
\|u\|_{L_{t}^{q_{1}}(I, \dot W^{\sigma, r_{1}})}
\|u-v\|_{L_{t}^{q_{2}}(I, \dot W^{s, r_{2}})}
\|v\|_{L_{t}^{q_{2}}(I, \dot W^{s, r_{2}})}
\\ & +
T^{s}
\|v\|_{L_{t}^{q_{1}}(I, \dot W^{\sigma, r_{1}})}
\|u\|_{L_{t}^{q_{2}}(I, \dot W^{s, r_{2}})}
\|u-v\|_{L_{t}^{q_{2}}(I, \dot W^{s, r_{2}})}
\\
\lesssim\; &
T^{s}
\|u-v\|_{L_{t}^{q_{1}}(I, \dot W_a^{\sigma, r_{1}})}
\|u\|_{L_{t}^{q_{2}}(I, \dot W_a^{s, r_{2}})}
\|v\|_{L_{t}^{q_{2}}(I, \dot W_a^{s, r_{2}})}
\\ & +
T^{s}
\|u\|_{L_{t}^{q_{1}}(I, \dot W_a^{\sigma, r_{1}})}
\|u-v\|_{L_{t}^{q_{2}}(I, \dot W_a^{s, r_{2}})}
\|v\|_{L_{t}^{q_{2}}(I, \dot W_a^{s, r_{2}})}
\\ & +
T^{s}
\|v\|_{L_{t}^{q_{1}}(I, \dot W_a^{\sigma, r_{1}})}
\|u\|_{L_{t}^{q_{2}}(I, \dot W_a^{s, r_{2}})}
\|u-v\|_{L_{t}^{q_{2}}(I, \dot W_a^{s, r_{2}})}
\end{align*}
If we define the norm $X(I)$ as
\[
\|u\|_{X(I)}:= \| u\|_{L_t^{\infty}(I, W_{a}^{1, 2})} + \| u\|_{L_t^{q_1}(I, W_{a}^{1, r_1})} + \| u\|_{L_t^{q_2}(I, W_{a}^{1, r_2})}
\]
Noting $0<s<1$,
we can get
\begin{equation*}
\Big\| \bigl[ |\cdot|^{-2}\ast(({u-v})\overline{v})u\Big\|_{L^{q'}_{t}(I, \dot W^{\sigma, r'})}
\lesssim
T^{s} \|u-v\|_{X(I)}(\|u\|_{X(I)}^{2}+ \|v\|_{X(I)}^{2})
\end{equation*}
Thus, we have
\begin{equation}\label{eq:estimate-nonlinear-term-Hartree-inverse}
\Big\|
(|\cdot|^{-2}\ast|u|^{2})u
-
(|\cdot|^{-2}\ast|v|^{2})v\big]
\Big\|_{L^{q'}_{t}(I, \dot W_{a}^{\sigma, r'})}
\lesssim
T^{s} \|u-v\|_{X(I)}(\|u\|_{X(I)}^{2}+ \|v\|_{X(I)}^{2})
\end{equation}
Define the operator as
\[
\mathcal T u = e^{it\La}u_0-i\int_{0}^te^{i(t-s)\La}\bigl( (|\cdot|^{-2}\ast |u|^{2})u\bigr)(s)\, ds
\]
By Stricartz estimate and the nonlinear estimate \eqref{eq:estimate-nonlinear-term-Hartree-inverse}, we get
\begin{equation}
\aligned
\|(\mathcal Tu -\mathcal T v)\|_{X(I)}&\le C T^{s} \|u-v\|_{X(I)}(\|u\|_{X(I)}^{2}+ \|v\|_{X(I)}^{2})\\
\|(\mathcal Tu)\|_{X(I)}&\le C\|u_{0}\|_{H_{a}^{1}} + C T^{s} \|u\|_{X(I)}^{3}
\endaligned
\end{equation}
Denote the space as
\[
\mathcal S(I):= \{u\in C(I, H_{a}^{1}): \|u\|_{X(I)}\le 2C\|u_{0}\|_{H_{a}^{1}}\}
\]
Then, the operator $\mathcal T$ is the contraction mapping in $(\mathcal S(I), \|\cdot\|_{X(I)})$ if $T=T(\|u_{0}\|_{H_{a}^{1}})$ is small enough. Furthermore, there is unique solution of the equation \eqref{problem-eq: NLS-H} in $C(I, H_{a}^{1})\cap X(I)$.
Note that the equivalence between $H_{a}^{1}$ and $H^{1}$, use the Strichartz estimate again, we gain that
$$u\in\bigcap_{(q, r)\in \Lambda_{0}} L^{q}(I, W_{a}^{1, r}(\mathbb R^{d})), $$
which complete the proof of the local wellposedness.
\end{proof}
\begin{remark}
$T=T(\|u_{0}\|_{H^{1}(\mathbb R^{d})})>0$ means that, if the maximal life interval of solution $u$ is $[0, T^{\ast})$, $T^{\ast}<\infty$, we have $\lim_{t\to T^{\ast}} \|u(t)\|_{H_{a}^{1}}=\infty$. Combining with the mass conservation, the solution blows-up in finite time means that $\lim_{t\to T^{\ast}} H(u(t))=\infty$.
\end{remark}
\section{Variational Characterization and Global Well-posedness}
In this section, we are in the position to prove the global well-posedness result. We will show the variational characterization Proposition \ref{ground state} which is related to the optimal Gargliardo-Nirenberg inequality, then we use the inequality to obtain our global well-posedness result.
Before proving the proposition, we give two simple lemmas which will be used later. First, we show a primary embedded lemma.
\begin{lemma}\label{lem:sobolev-Lv}
If
\[
\lim_{n\to\infty} \| u_{n} - u\|_{L^{\frac{2d}{d-1}}(\mathbb R^{d})}=0,
\]
we have
\[
\lim_{n\to\infty}L_{V}(u_{n} - u)= 0,
\quad
\text{ and }
\lim_{n\to\infty} L_{V}(u_{n}) = L_{V}(u).
\]
\end{lemma}
\begin{proof}
By Hardy-Littlewood-Sobolev inequality, we gain
\begin{equation}\label{formula:Hardy-L-S in fact}
\aligned
\left| \iint_{\mathbb R^{d}\times R^{d}} \frac{f_{1}(x)f_{2}(x)f_{3}(y)f_{4}(y)}{|x-y|^{2}}dxdy \right|
& \lesssim
\|f_{1}f_{2}\|_{L^{\frac{d}{d-1}}(\mathbb R^{d})}\cdot
\|f_{3}f_{4}\|_{L^{\frac{d}{d-1}}(\mathbb R^{d})}\\
&\le \prod_{k=1}^{4}\|f_{k}\|_{L^{\frac{2d}{d-1}}(\mathbb R^{d})}
\endaligned
\end{equation}
Therefore,
\[
L_{V}(u_{n}- u) \lesssim \|u_{n}- u \|_{L^{\frac{2d}{d-1}}(\mathbb R^{d})}^{4}\to 0,
\quad
\text{ when }
n\to\infty.
\]
Note that
\[
|u(x)|^{2} |u(y)|^{2} - |v(x)|^{2} |v(y)|^{2}
=
(|u(x)|^{2}-|v(x)|^{2}) |u(y)|^{2} + |v(x)|^{2} ( |u(y)|^{2}-|v(y)|^{2} )
\]
By the inequality \eqref{formula:Hardy-L-S in fact}, we get
\[
\aligned
\left|L_{V}(u)-L_{V}(v)\right|
\lesssim
&\||u|^{2}-|v|^{2}\|_{L^{\frac{d}{d-1}}(\mathbb R^{d})}
\left(\| u\|^{2}_{L^{\frac{2d}{d-1}}(\mathbb R^{d})} +\|v\|^{2}_{L^{\frac{2d}{d-1}}(\mathbb R^{d})}\right)
\\
\lesssim
&
\|u-v\|_{L^{\frac{2d}{d-1}}(\mathbb R^{d})}
\left(
\|u\|^{3}_{L^{\frac{2d}{d-1}}(\mathbb R^{d})}+ \|u-v\|^{3}_{L^{\frac{2d}{d-1}}(\mathbb R^{d})}
\right)
\endaligned
\]
It means that if
\[
\lim_{n\to\infty} \| u_{n} - u\|_{L^{\frac{2d}{d-1}}(\mathbb R^{d})}=0,
\]
we have
\[
\lim_{n\to\infty} L_{V}(u_{n}) = L_{V}(u).
\]
\end{proof}
Next we give the Schwartz symmetrical rearrangement argument about the functional $J$.
\begin{lemma}\label{lemma:rearrangement}
Assume $u\in H^{1}(\mathbb R^{d})$ is a non-radial function, denote $u^{\ast}$ as the Schwartz symmetrical rearrangement of $u$, then $u^{\ast}\ge0$,
\[
J(u^{\ast}) < J(u).
\]
\end{lemma}
\begin{proof}
By the classical Schwartz symmetrical rearrangement argument, we know that $u^{\ast}$ satisfies
\[
\gathered
M(u)=M(u^{\ast}), \\
\|\nabla u^{*}\|_{L^{2}(\mathbb R^{d})} \le \|\nabla u\|_{L^{2}(\mathbb R^{d})}. \\
\endgathered
\]
By Lemma \ref{riesz}, we get
\[
L_{V}(u)\le L_{V}(u^{\ast}).
\]
Since $u$ is nonradial, then we have $u\neq0$ and
\[
\int_{\mathbb R^{d}}\frac{|u|^{2}}{|x|^{2}} < \int_{\mathbb R^{d}}\frac{|u^{*}|^{2}}{|x|^{2}}. \\
\]
Therefore
\[
J(u^{\ast})<J(u)
\]
holds.
\end{proof}
Now we will prove Proposition \ref{ground state}. By solving a minimization problem, the minimum is attained at the ground state of the corresponding stationary equation.
\begin{proof}[The proof of Proposition $\ref{ground state}$]
We need to show the minimum can be attained first.
Suppose that the non-zero function sequence $\{u_{n}\}$ is the minimal sequence of the functional $J$, that is to say,
\[
\lim_{n\to\infty} J(u_{n}) = \inf\left\{J(u) : u\in H^{1}(\mathbb R^{d}\setminus \{0\})\right\},
\]
By Lemma \ref{lemma:rearrangement}, without loss of generality, we can assume $u_{n}$ is non-negative radial.
Note that for any $u\in H^{1}(\mathbb R^{d}\setminus \{0\})$, $\mu , \nu >0$, we have
\begin{equation}\label{eq:scaling4MHLJ}
\left\{
\gathered
M(\mu u(\nu \cdot)) = \mu ^{2}\nu ^{-d}M(u), \\
H(\mu u(\nu \cdot)) = \mu ^{2}\nu ^{2-d}H(u), \\
L_{V}(\mu u(\nu \cdot)) = \mu ^{4}\nu ^{2-2d}L_{V}(u), \\
J(\mu u(\nu \cdot)) = J(u). \\
\endgathered
\right.
\end{equation}
Denote
\[
v_{n}(x) = \frac{\left(M(u_{n})\right)^{\frac{d-2}{4}}}{\left(H(u_{n})\right)^{\frac{d}{4}}} u_{n}\left(\left(\frac{M(u_{n})}{H(u_{n})}\right)^{\frac{1}{2}}x\right)
\]
Then, $v_{n}$ is non-negative radial, and
\[
\aligned
M(v_{n})=H(v_{n})\equiv1, \quad J(v_{n}) = J(u_{n}), \\
\lim_{n\to\infty} J(v_{n})= \inf\left\{J(u) : 0\ne u\in H^{1}(\mathbb R^{d})\right\}.
\endaligned
\]
Note that $v_{n}$ is bounded in $H^{1}_{\text{rad}}(\mathbb R^{d})$ and
\[
H^{1}_{\text{rad}}(\mathbb R^{d})\hookrightarrow\hookrightarrow L^{\frac{2d}{d-1}}(\mathbb R^{d}),
\]
then there exist a subsequence $v_{n_{k}}$ and $v^{\ast}\in H^{1}_{\text{rad}}(\mathbb R^{d})$, such that as $k\to\infty$, we have $v_{n_{k}}\rightharpoonup v^{\ast}$ in $H^{1}(\mathbb R^{d})$ and $v_{n_{k}}\to v^{\ast}$ in $L^{\frac{2d}{d-1}}(\mathbb R^{d})$.
By the weak low semi-continuity of the functional $M$ and $H$, we obtain
\[
M(v^{\ast})\le 1, H(v^{\ast})\le 1.
\]
Since $\|v_{n_{k}} - v^{\ast}\|_{L^{\frac{2d}{d-1}}(\mathbb R^{d})}\to 0$, by Lemma \ref{lem:sobolev-Lv}, we have
\[
L_{V}(v^{\ast} ) =\lim_{k\to\infty} L_{V}(v_{n_{k}}).
\]
Therefore,
\[
\aligned
J(v^{\ast})
&= \frac{M(v^{\ast}) H(v^{\ast})}{L_{V}(v^{\ast})} \le \lim_{k\to\infty}\frac{1}{ L_{V}(v_{n_{k}})}
\\
&= \lim_{k\to\infty}J(v_{n_{k}})
= \inf\left\{J(u) : 0\ne u\in H^{1}(\mathbb R^{d})\right\}.
\endaligned
\]
Thus we proved that the minimum can be attained.
\vskip1em
Next, consider the variational derivatives of $M$, $H$, $L_{V}$: fix $u\neq 0$, for any $\varphi\in H^{1}(\mathbb R^{d})$,
\begin{align}\label{formula: variational derivative}
\frac{d}{d\epsilon}\Big|_{\epsilon=0}M(u+\epsilon\varphi) =& \Re \int_{\mathbb R^d} u\bar\varphi, \\
\frac{d}{d\epsilon}\Big|_{\epsilon=0}H(u+\epsilon\varphi) =& \Re \int_{\mathbb R^d} \left(-\Delta + \frac{ a}{|x|^{2}}\right)u\cdot \bar\varphi, \\
\frac{d}{d\epsilon}\Big|_{\epsilon=0}L_{V}(u+\epsilon\varphi) =& \Re\iint\frac{|u(x)|^{2}u(y)\bar\varphi(y)}{|x-y|^{2}}dxdy,
\end{align}
If the functional $J$ attains the minimum at $W$, then we have for any $\varphi\in H^{1}(\mathbb R^{d})$,
\[
\aligned
0
= &\frac{d}{d\epsilon}\Big|_{\epsilon=0}J(W+\epsilon\varphi)
=\frac{d}{d\epsilon}\Big|_{\epsilon=0}\frac{M(W+\epsilon\varphi)H(W+\epsilon\varphi)}{L_{V}(W+\epsilon\varphi)}\\
=&\frac{\displaystyle \left(\Re\int_{\mathbb R^d} W\bar\varphi\right)\cdot H(W)}{L_{V}(W)}
+\frac{\displaystyle M(W)\cdot \Re \int_{\mathbb R^d} \left(-\Delta + \frac{ a}{|x|^{2}}\right)W\cdot \bar\varphi }{L_{V}(W)}\\
&-\frac{\displaystyle M(W)H(W)\cdot \Re\int_{\mathbb R^d}\frac{|W(x)|^{2} W(y)\bar\varphi(y)}{|x-y|^{2}}dxdy}{(L_{V}(W))^{2}}.
\endaligned
\]
It means that
\begin{equation}\label{element-ground-state}
\aligned
L_{V}(W)H(W)\cdot W+ L_{V}(W) M(W) \cdot \left(-\Delta W+\frac{a}{|x|^{2}}W\right)
\\-M(W)H(W)\cdot (|\cdot|^{-2}\ast W^{2})W=0,
\endaligned
\end{equation}
i.e.
\[
(-\Delta +a|x|^{-2})W+\alpha W= \beta(|\cdot|^{-2}\ast |W|^{2})W,
\]
where
\[
\alpha= \frac{H(W)}{M(W)}, \quad \beta= \frac{H(W)}{L_{V}(W)}.
\]
By a direct calculation, we know
\begin{align*}
(-\Delta +a|\cdot|^{-2})\bigl[\mu u(\nu\cdot )\bigr](x) =& \mu\nu^{2} (-\Delta +a|\cdot|^{-2})[u](\nu\cdot ), \\
\bigl[(|\cdot|^{-2}\ast |\mu W(\nu \cdot)|^{2})\mu W(\nu \cdot)\bigr](x) =& \mu^{3}\nu^{2-d}[(|\cdot|^{-2}\ast |W|^{2})W](\nu\cdot )
\end{align*}
Therefore, $Q$ is the solution of \eqref{eq:ground-state} using the scaling $W(x)=\alpha^{\frac{d}{4}}\beta^{-\frac{1}{2}}Q(\sqrt\alpha x)$.
\vskip1em
Next we prove that if $W$ is the minimal element, then $W$ is radial and there exists a constant $\theta\in\mathbb R$ such that $W= e^{i\theta}|W|$.
If $W$ is non-radial, then by Lemma \ref{lemma:rearrangement}, $J(W^{\ast}) \linebreak[3]< J(W)$, which is contradict to the minimality of $W$. So $W$ must be radial.
Since $J(|W|)\le J(W)$, $|W|$ is also a minimal element. Suppose that $W(x)= e^{i\theta(x)}|W|(x)$, where $\theta(x)$ is a real-valued function, then
\begin{equation}\label{eq:iii}
\aligned
|\nabla W(x)|^{2}
& = \Bigl|e^{i\theta(x)}|W|(x)\cdot i\nabla\theta(x)+ e^{i\theta(x)}(\nabla|W|)(x)\Bigr|^{2} \\
& = |W(x)|^{2}|\nabla\theta(x)|^{2} + \Bigl|\nabla|W|(x)\Bigr|^{2}
\endaligned
\end{equation}
By the minimality of $J(W)$, $J(W)=J(|W|)$. But by $M(W) = M(|W|)$ and $L_{V}(W) = L_{V}(|W|)$, we have $H(W)=H(|W|)$.
So $\nabla\theta(x)\equiv0$ in \eqref{eq:iii}, thus $\theta(x)\equiv\text{constant}$.
Therefore, $W(x)=e^{i\theta}mQ (nx)$, where $m, n>0$, $\theta\in\mathbb R$, and $Q\neq0$ is the non-negative non-zero radial solution of \eqref{eq:ground-state}.
\vskip1em
Finally, we prove that all ground states have the same mass.
For $\lambda\in(0, \infty)$,
\begin{align*}
&M(\lambda^{\alpha}Q(\lambda^{\beta}\cdot))+ E(\lambda^{\alpha}Q(\lambda^{\beta}\cdot))
\\&=
\lambda^{2\alpha-\beta d}M(Q) + \lambda^{2\alpha + 2\beta - \beta d}H(Q) - \lambda^{4\alpha + 2\beta -2\beta d}L_{V}(Q)
\end{align*}
Using the chain rules and variational derivatives \eqref{formula: variational derivative}, then letting $\lambda=1$ in the left side, we can obtain
\[
\aligned
&\Re\int_{\mathbb R^{d}} \left( -\Delta + \frac{ a}{|x|^{2}}+1 - |\cdot|^{-2}\ast|Q|^{2} \right) Q\cdot\
\overline{\left( \frac{d}{d\lambda}\Big|_{\lambda=1} \lambda^{\alpha}Q(\lambda^{\beta}x) \right) } dx \\
=&(2\alpha-\beta d)M(Q) + (2\alpha + 2\beta - \beta d)H(Q) - (4\alpha + 2\beta -2\beta d)L_{V}(Q),
\endaligned
\]
Since $Q$ satisfies \eqref{eq:ground-state}, we have
\[
(2\alpha-\beta d)M(Q) + (2\alpha + 2\beta - \beta d)H(Q) - (4\alpha + 2\beta -2\beta d)L_{V}(Q)\equiv0, \quad\forall \alpha, \beta.
\]
A simple calculation yields
\[
\aligned
\text{ let }&\ \alpha=d, \beta=2, & \text{ we get }& \phantom{-} 4H(Q)-4L_{V}(Q) =0;\\
\text{ let }&\ \alpha=d-2, \beta=2, & \text{ we get }& -4M(Q)+4L_{V}(Q) =0;\\
\text{ let }&\ \alpha=2d-2, \beta=4, &\text{ we get }& -4M(Q)+4H(Q) =0;
\endaligned
\]
So
\begin{equation}\label{Q}
M(Q) = H(Q) = L_{V}(Q) = J(Q)=\inf\{ J(u): u\in H^{1}(\mathbb R^{d}\backslash\{0\})\}=:M_{gs}
\end{equation}
holds.
\end{proof}
Using the above proposition, we can directly obtain the following corollary.
\begin{corollary}
[Gagliardo-Nirenberg inequality]
\begin{equation}\label{GN-inequality}
L_{V}(u)\le \frac{M(u)H(u)}{M_{gs}}, \qquad \forall u\in H^{1}(\mathbb R^{d}).
\end{equation}
The equality holds if and only if $u\in H^{1}(\mathbb R^{d})$ is a minimal element of functional $J(u)$, that is to say $u\in\mathcal G$, or $u=0$ .
\end{corollary}
Applying the Gagliardo-Nirenberg inequality directly, we can prove that the solution of the equation \eqref{problem-eq: NLS-H} is global if its mass is less than the mass of the ground state.
\begin{theorem}
If $u_{0}\in H^{1}(\mathbb R^{d})$ and satisfies $M(u_{0})<M_{gs}$, then the solution of the equation \eqref{problem-eq: NLS-H} is global.
\end{theorem}
\begin{proof}
In order to prove the solution is global, we only need to verify $M(u_{0})<M_{gs}$, since it means that $H(u(t))$ is uniformly bounded in time.
Using Gagliardo-Nirenberg inequality \eqref{GN-inequality} to $L_{V}(u)$ yields
\[
\aligned
E(u(t))&=E(u)=H(u) - L_{V}(u)\\
&\ge H(u(t)) - \frac{M(u(t))}{M_{gs}}H(u(t))\\
&=\left(1-\frac{M(u(t))}{M_{gs}}\right)H(u(t))
\endaligned
\]
then by $M(u_{0})<M_{gs}$ we have
\[
\left(1-\frac{M(u_0)}{M_{gs}}\right)H(u(t))\le E(u_{0}),
\]
where we used conservation of the energy and the mass.
So $H(u(t))$ is uniformly bounded in time.
\end{proof}
\section{\textbf{Rigidity argument} and profile decomposition}
We are devoted to describing the dynamics of the blow-up solution in this section.
At first, we show several key propositions and lemmas.
\begin{proposition}\label{static-rigidity}
If $u\in H^{1}(\mathbb R^{d})$ satisfies£º $M(u)=M_{gs}$ and $E(u)= 0$, then there exist $\theta\in\mathbb R$, $\lambda>0$ and $Q\in\mathcal G$, such that
\[
u(x) = e^{i\theta}\lambda^{\frac{d}{2}} Q(\lambda x).
\]
\end{proposition}
\begin{proof}
By Gagliardo-Nirenberg inequality, $E(u)=0$ means that $u$ is the minimal element of $J(u)$. By Proposition \ref{ground state}, there exist $m, n>0$, $\theta\in\mathbb R$, $Q\in\mathcal G$, such that
\[
u = e^{i\theta} m Q(n\cdot)
\]
Owing to $M(u) = M_{gs}=M(Q)$, thus $m=n^{\frac{d}{2}}$, there exist $\theta\in\mathbb R$, $\lambda>0$, $Q\in\mathcal G$ such that
\[
u(x) = e^{i\theta}\lambda^{\frac{d}{2}} Q(\lambda x).
\]
\end{proof}
\begin{proposition}
[Linear profile decomposition]\label{profile decomposition}
Suppose that $\{v_{n}\}$ is bounded in $H^{1}(\mathbb R^{d})$. Then there exists an subsequence,
which is still denoted as $\{v_{n}\}$ such that
\begin{equation}\label{profile}
v_{n} = \sum_{j=1}^{J}V^{j}(\cdot - x_{n}^{j}) + \omega_{n}^{J}, \quad\forall J\in\mathbb N,
\end{equation}
where $\{V^{j}\}_{j=1}^{\infty}\subset H^{1}(\mathbb R^{d})$, $x_{n}^{j}\in\mathbb R^{d}$, $1\le j, n \in\mathbb N$ and the following orthogonality conditions holds:
\begin{enumerate}[\quad$(a). $]
\item If $k\neq j$, we have $|x_n^k - x_n^j|\to \infty$, when $n\to\infty$;
\item
\begin{align*}
M(v_n) =& \sum_{j=1}^{J} M(V^j) + M(\omega_n^J) + o_n(1), \\
\|\nabla v_n\|_{L^{2}(\mathbb R^{d})}^{2} =& \sum_{j=1}^{J} \|\nabla V^j\|_{L^{2}(\mathbb R^{d})}^{2} + \|\nabla \omega_n^J\|_{L^{2}(\mathbb R^{d})}^{2} + o_n(1), \\
H(v_n) =& \sum_{j=1}^{J} H(V^j(\cdot - x_{n}^{j})) + H(\omega_n^J) + o_n(1), \\
\lim_{J\to\infty}\limsup_{n\to\infty} \|\omega_n^J\|_{L^p(\mathbb R^d)}=&0, \text{ µ± } 2<p<2^{*};\\
L_{V}(v_{n}) = & \sum_{j=1}^{J} L_{V}(V^{j}) + \varepsilon_{n, J}, \text{ ÇÒ } \lim_{J\to\infty}\limsup_{n\to\infty} \varepsilon_{n, J}=0.
\end{align*}
where $2^{*}=\frac{2d}{d-2}$.
\end{enumerate}
\end{proposition}
The proof of this proposition is standard
except we may deal with the difficulties which the potential term brings
to confirming the orthogonality structure and
showing the orthogonality result of $L_{V}$. Here we omit the proof, the reader can refer to \cite{Bens-Dinh, KMVZ-EnrCri} for details.
Now we establish the following propositions which plays an important role in the classification of minimal mass blow-up solution.
\begin{proposition}
\label{dynamic-rigidity}
Assume the sequence $\{u_{n}\}$ satisfies
\[
M(u_{n})\equiv M_{gs},~
0<\limsup_{n\to\infty}H(u_{n})<\infty,~
\limsup_{n\to\infty}E(u_{n})\le 0,
\]
Then there exist a subsequence (still denoted as $u_{n}$, $\theta\in\mathbb R$, $\lambda>0$ and $Q\in\mathcal G$ such that
\[
\lim_{n\to\infty}\| u_{n} - e^{i\theta}\, \lambda^{\frac{d}{2}} Q(\lambda\cdot)\|_{H^{1}(\mathbb R^{d})} = 0.
\]
\end{proposition}
\begin{proof}
For any function $u\ne0$ and $0< M(u) < M_{gs}$, we have
\begin{align*}
E(u) & = H(u)-L_{V}(u)\ge H(u)\left(1-\frac{M(u)}{M_{gs}}\right) \\
& \cong_{a, d}\|\nabla u\|_{L^{2}(\mathbb R^{d})}^{2} \left(1-\frac{M(u)}{M_{gs}}\right)> 0,
\end{align*}
so we have
\[
\inf_{y\in\mathbb R^{d}}E(u(\cdot +y))> 0.
\]
By the profile decomposition, there exist a subsequence (still denoted as $\{u_{n}\}_{n=1}^{\infty}$), such that
\[
\begin{aligned}
u_{n}(x)=&\sum_{j=1}^{J}V^{j}(x-x_{n}^{j})+\omega_{n}^{J}(x), \\
H(u_{n})=&\sum_{j=1}^{J} H(V^j(\cdot - x_{n}^{j}))+H(\omega_{n}^{J})+o_{n}(1), \\
L_{V}(u_{n}) = &\sum_{j=1}^{J} L_{V}(V^j) +\epsilon_{n, J},
\quad
\lim_{J\to\infty}\limsup_{n\to\infty} |\epsilon_{n, J}|=0
\end{aligned}
\]
Since $M(u)$ and $L_{V}(u)$ are translation invariant, we have
\[
\aligned
\limsup_{n\to+\infty}E(u_{n})
&\ge
\limsup_{n\to+\infty}\left( \sum_{j=1}^{J} E(V^{j}(\cdot - x_{n}^{j})) + H(\omega_{n}^{J}) - \epsilon_{n, J}\right)\\
&\ge
\limsup_{n\to+\infty}\left( \sum_{j=1}^{J} \inf_{y\in\mathbb R^{d}}E(V^{j}(\cdot -y)) + H(\omega_{n}^{J}) - \epsilon_{n, J}\right)\\
&\ge
\sum_{j=1}^{J} \inf_{y\in\mathbb R^{d}}E(V^{j}(\cdot -y)) - \liminf_{n\to+\infty}\epsilon_{n, J} \\
\endaligned
\]
Since $\lim_{J\to\infty}\limsup_{n\to\infty} |\epsilon_{n, J}|=0$, we know
\[
\limsup_{n\to+\infty}E(u_{n})
\ge
\sum_{j=1}^{\infty} \inf_{y\in\mathbb R^{d}}E(V^{j}(\cdot -y)).
\]
By the profile decomposition, we know $0\le \sum_{j}M(V^{j})\le M_{gs}$, so for any $j\in \mathbb N$, $\inf_{y\in\mathbb R^{d}}E(V^{j}(\cdot -y))\ge 0.$ Furthermore, by $\limsup_{n\to\infty}E(u_{n})\le0$ we have $M(V^{j}) = 0$ or $M(V^{j})=M_{gs}$.
Owing to $0\le \sum_{j}M(V^{j})\le M_{gs}$, we just need to consider two cases: $V^{j}\equiv0 , \forall j\ge 1$; or $M(V^{1})=M_{gs}$, $V^{j}\equiv0 , \forall j\ge 2$.
For the first case, the profile decomposition yields
\[
\lim_{n\to\infty} L_{V}(u_{n}) = \lim_{n\to\infty}(0 + \epsilon_{n, J})=0
\]
So
\[
\limsup_{n\to\infty}E(u_{n}) = \limsup_{n\to\infty} H(u_{n}),
\]
which is contradicted with the condition $0<\limsup_{n\to\infty}H(u_{n})<\infty, \limsup_{n\to\infty}E(u_{n})\le 0$.
So we only consider the second case, i.e. $u_{n}(x)= V(x-x_{n})+r_{n}(x)$ satisfies
\[
\begin{gathered}
r_{n}(x+ x_{n})\rightharpoonup 0 \text{ ÔÚ } L^{2}(\mathbb R^{d}), \dot H^{1}(\mathbb R^{d}), H^{1}(\mathbb R^{d})\text{ ÖÐ };\\
M(u_{n}) = M(V) + M(r_{n}) + o_{n}(1);\\
\|\nabla u_{n}\|_{ L^{2}(\mathbb R^{d})}^{2} = \|\nabla V\|_{ L^{2}(\mathbb R^{d})}^{2}+\|\nabla r_{n}\|_{ L^{2}(\mathbb R^{d})}^{2}+o_{n}(1);\\
H(u_{n}) = H(V(\cdot - x_{n})) + H(r_{n}) + o_{n}(1);\\
\limsup_{n\to\infty}\|r_{n}\|_{p}\to 0, \quad 2<p<2^{*};\\
L_{V}(u_{n}) = L_{V}(V) + o_{n}(1).
\end{gathered}
\]
and $M(V)=M_{gs}$.
In order to complete the proof, we firstly show $\{x_{n}\}$ be bounded. Otherwise, there exists a subsequence $\{x_{n_{k}}\}_{k=1}^{\infty}$ such that
\[
\lim_{k\to\infty} x_{n_{k}} = \infty .
\]
Note that the orthogonality conclusion of the profile decomposition tells us that
\[
\int\frac{|u_{n}|^{2}}{|x|^{2}}dx= \int\frac{|V(x-x_{n})|^{2}}{|x|^{2}}dx+\int\frac{|r_{n}(x)|^{2}}{|x|^{2}}dx+o_{n}(1).
\]
On one hand, For any $\varphi\in C_{c}(\mathbb R^{d})$, we have,
\[
\aligned
\int\frac{|V(x-x_{n_k})|^{2}}{|x|^{2}}dx
\le &2\int\Bigg{(}\frac{|(V-\varphi)(x-x_{n_k})|^{2}}{|x|^{2}}+\frac{|\varphi(x-x_{n_k})|^{2}}{|x|^{2}}\Bigg{)}dx\\
\lesssim &\|V-\varphi\|^{2}_{\dot{H}^{1}(\mathbb R^{d})}+\int\frac{|\varphi(x)|^{2}}{|x+x_{n_k}|^{2}}dx\\
& \longrightarrow \|V-\varphi\|^{2}_{\dot{H}^{1}(\mathbb R^{d})} ,
\quad
\text{ when } k\to\infty.
\endaligned
\]
where we used the Hardy inequality.
By the density, we get
\[
\lim_{k\to\infty}\int\frac{|V(x-x_{n_k})|^{2}}{|x|^{2}}dx = 0.
\]
On the other hand, note that $M(V)=M_{gs}$ and using Gargliardo-Nirenberg inequality, we conclude that $\inf_{y\in\mathbb R^{d}}E(V(\cdot + y))\ge 0$. Then
\[
\begin{aligned}
0\ge &\limsup_{n\to\infty}E(u_{n})=\limsup_{n\to\infty}(H(V(\cdot-x_{n}))+H(r_{n}))-L_{V}(V)\\
\ge&\limsup_{n\to\infty}H(r_{n})\ge 0
\end{aligned}
\]
Thus,
\begin{equation}\label{}
\|\nabla r_{n}\|_{L^{2}(\mathbb R^{d})}\cong H(r_{n})\to 0,~~\textrm{as}~~n\to\infty.
\end{equation}
By Hardy inequality, we have
\[
\lim_{n\to\infty}\int_{\mathbb R^{d}} \frac{|r_{n}(x)|^{2}}{|x|^{2}} dx= 0.
\]
So there exists a subsequence $\{x_{n_{k}}\}_{k=1}^{\infty}$, such that
\[
\lim_{k\to\infty}\int_{\mathbb R^{d}} \frac{|u_{n_{k}}(x)|^{2}}{|x|^{2}} dx= 0.
\]
then
\[
\begin{aligned}
0\ge&\limsup_{n\to\infty}E(u_{n_{k}})\ge\limsup_{k\to\infty}E(u_{n})=\limsup_{n\to\infty}H(u_{n_k})-L_{V}(V)\\
=&\limsup_{n\to\infty}\frac{1}{2}\|\nabla u_{n_k}\|_{2}^{2}-L_{V}(V)
\ge\frac{1}{2}\|\nabla V\|_{2}^{2}-L_{V}(V) > E(V)=0.
\end{aligned}
\]
which is a contradiction. Therefore, $\{x_{n}\}_{n=1}^{\infty}\subset\mathbb R^{d}$ must be bounded.
By
\[
\lim_{n\to\infty} M(r_{n})=0, \quad
\lim_{n\to\infty} H(r_{n})=0
\]
we know
\[
\lim_{n\to\infty}\| u_{n}- V(\cdot -x_{n})\|_{H^{1}(\mathbb R^{d})}=\lim_{n\to\infty}\| r_{n}\|_{H^{1}(\mathbb R^{d})}=0.
\]
Since $\{x_{n}\}_{n=1}^{\infty}\subset\mathbb R^{d}$ is bounded, then there exists $x_0$ such that \linebreak[2] $\lim_{n\to\infty}x_{n}=x_{0}\in\mathbb R^{d}$ up to a subsequence, so
\[
\lim_{n\to\infty}\| u_{n}- V(\cdot -x_{0})\|_{H^{1}(\mathbb R^{d})}=0.
\]
Utilizing Gagliardo-Nirenberg inequality and $M(V)=M_{gs}$ again, we have
\[
\begin{aligned}
0\ge\limsup_{n\to\infty}E(u_{n})= \limsup_{n\to\infty}(H(u_{n})-L_{V}(u_{n})) = H(V(\cdot -x_{0}))- L_{V}(V)\ge 0\
\end{aligned}
\]
Then $M(V(\cdot -x_{0}))=M_{gs}$, $E(V(\cdot -x_{0}))=0$, by Propositions \ref{static-rigidity}, we get
\[
V(\cdot -x_{0})=e^{i\theta}\, \lambda^{\frac{d}{2}} Q(\lambda\cdot), \quad \text{ ÆäÖÐ} Q\in\mathcal G, \theta\in\mathbb R,
\]
Therefore,
\[
\lim_{n\to\infty}\| u_{n}- e^{i\theta}\, \lambda^{\frac{d}{2}} Q(\lambda\cdot)\|_{H^{1}(\mathbb R^{d})}=0,
\]
which completes the proof of the proposition.
\end{proof}
\section{The description of blow-up solution in finite time}
In this section, we consider the dynamics of blow-up solution.
We first prove the second part of Theorem \ref{thm:main} to describe he minimal mass blow-up solution in finite time.
\begin{theorem}\label{thm:critical-mass-blowup}
If $M(u_{0})= M_{gs}$ and the solution $u$ blows up in finite time,i.e., there exists $0<T^{\ast}<\infty$ such that $\lim_{t\to T^{\ast}}H(u(t))=\infty$, then
\[
u\in
\left\{
e^{i\frac{|\cdot|^{2}}{4(T^{\ast}-t)}} e^{i\theta} { \lambda}^{\frac{d}{2}} Q(\lambda \cdot)
:
\theta\in\mathbb R,
\lambda>0,
Q\in\mathcal G
\right\}.
\]
\end{theorem}
Denote the space $\Sigma$ as
\[
\Sigma:=\{u\in H^{1}(\mathbb R^{d}):xu\in L^{2}(\mathbb R^{d})\}.
\]
And for $u(t)\in \Sigma$, define the function
\[
\Gamma(t):=\int_{\mathbb R^{d}}|x|^{2}|u(t, x)|^{2}dx.
\]
We now give the virial identities for \eqref{problem-eq: NLS-H} without proof.
\begin{lemma}\label{lemma:Virial Identity}
Suppose $u$ is the solution of Hartree equation with inverse-square potential \eqref{problem-eq: NLS-H} in time interval $[0, T), T>0$ satisfying $u(t)\in \Sigma$ for all $t\in[0, T)$. Then, for any $t\in[0, T)$, we have the following identities:
\begin{align} \label{eq:Virial Identity-I}
\Gamma'(t)=&-4 \mathrm{Im}\int_{\mathbb R^{d}}\bar{u}(t, x)(\nabla u(t, x)\cdot x)dx, \\
\label{eq:Virial Identity-II}
\Gamma''(t)=&16E(u(t)).
\end{align}
\end{lemma}
\begin{lemma}
\label{lemma: Virial estimate}
Suppose $u\in H^{1}(\mathbb R^{d})$ satisfies $M(u)=M_{gs}$. Then for any function $\theta\in C^{\infty}_{c}(\mathbb R^{d})$, we have
\[
\left|
\int_{\mathbb R^{d}}\nabla\theta\cdot\Im(\bar{u}\nabla u)
\right|
\le
\sqrt{2E(u)}\left(\int_{\mathbb R^{d}}|\nabla \theta|^{2}|u|^{2}\right)^{1/2}.
\]
\end{lemma}
\begin{proof}
For any two functions $\theta$ and $s\in\mathbb C$, a direct computation $\nabla (u e^{is\theta}) = e^{is\theta}\left(\nabla u + is\nabla\theta\cdot u\right) $ yields,
\[
|\nabla (u e^{is\theta})|^{2} = |\nabla u|^{2} + |s|^{2}|\nabla\theta|^{2} |u|^{2} + 2\Im (\nabla u\cdot\overline{s\nabla\theta u}).
\]
So for any function $\theta\in C^{\infty}_{c}(\mathbb R^{d})$ and $s\in\mathbb R$, we have
\begin{equation}\label{eq: rotated energy identity}
E(u e^{is\theta}) = E(u) + s\cdot \int_{\mathbb R^{d}}\nabla\theta\cdot\Im(\bar{ u}\nabla u) + s^{2}\cdot\frac{1}{2}\int_{\mathbb R^{d}}|\nabla\theta|^{2} |u|^{2}.
\end{equation}
Note that $M(u e^{is\theta}) = M(u)=M_{gs}$, and by Gagliardo-Nirenberg inequality, we have $ E(u e^{is\theta}) \ge 0$. So
\[
\left|
\int_{\mathbb R^{d}}\nabla\theta\cdot\Im(\bar{u}\nabla u)
\right|^{2}
-
4\cdot E(u)\cdot \frac{1}{2}\int_{\mathbb R^{d}}|\nabla\theta|^{2} |u|^{2}
\le
0.
\]
which completes the proof of the lemma.
\end{proof}
Now let us come to prove Theorem \ref{thm:critical-mass-blowup}.
\begin{proof}[The proof of Theorem \ref{thm:critical-mass-blowup}]
Suppose $u$ is the solution to the equation (\ref{problem-eq: NLS-H}) and satisfies
\[
M(u_{0})=M_{gs},
\quad
\lim\limits_{t\to T^{\ast}}H(u(t))=+\infty.
\]
For any time sequence $\{t_{n}\}_{n=1}^{\infty}\subset[0, T^{\ast})$ such that $\lim_{n\to\infty}t_{n}=T^{\ast}$, define
\[
v_{n}(x) = \left(\frac{1}{H(u(t_{n}))}\right)^{-\frac{d}{4}}
u\left(t_{n}, \left(\frac{1}{H(u(t_{n}))}\right) ^{-\frac{1}{2}}x\right),
\]
by conservation of energy, we have
\[
\aligned
M(v_{n})& = M(u(t_{n}))\equiv M(u_{0}) = M_{gs}, \\
H(v_{n})& = \frac{1}{H(u(t_{n}))} H(u(t_{n})) \equiv1, \\
E(v_{n})& = \frac{1}{H(u(t_{n}))} E(u(t_{n}))=\frac{1}{H(u(t_{n}))} E(u_{0})\to0,
\text{ when } n\to\infty.
\endaligned
\]
By Proposition \ref{dynamic-rigidity}, there exist a subsequence $\{v_{n_{k}}\}$ and $Q\in\mathcal{G}, \theta\in\mathbb{R}, \lambda>0$, such that
\begin{equation}\label{H1}
\lim_{k\to\infty}\|v_{n_{k}}-e^{i\theta}\lambda^{\frac{d}{2}} Q(\lambda\cdot)\|_{H^{1}(\mathbb{R}^{d})}=0.
\end{equation}
Therefore,
\[
\aligned
&\quad \limsup_{k\to\infty} \int_{\mathbb R^{d}}
\bigl|
|v_{n_{k}}|^{2}-|\lambda^{\frac{d}{2}} Q(\lambda\cdot)|^{2}
\bigr|
\\
& =
\limsup_{k\to\infty} \int_{\mathbb R^{d}}
\bigl|
(|v_{n_{k}}| -|\lambda^{\frac{d}{2}} Q(\lambda\cdot)|)
(|v_{n_{k}}| +|\lambda^{\frac{d}{2}} Q(\lambda\cdot)|)
\bigr|
\\
& \le
\limsup_{k\to\infty} \int_{\mathbb R^{d}}
|v_{n_{k}} - e^{i\theta}\lambda^{\frac{d}{2}} Q(\lambda\cdot)|
(|v_{n_{k}}| +|\lambda^{\frac{d}{2}} Q(\lambda\cdot)|)
\\
& \le
\limsup_{k\to\infty}
\|v_{n_{k}} - e^{i\theta}\lambda^{\frac{d}{2}} Q(\lambda\cdot)\|_{L^{2}(\mathbb R^{d})}
\left(
\|v_{n_{k}}\|_{L^{2}(\mathbb R^{d})} +\|\lambda^{\frac{d}{2}} Q(\lambda\cdot)\|_{L^{2}(\mathbb R^{d})}
\right)
\\
& \le
\limsup_{k\to\infty}
\|v_{n_{k}} - e^{i\theta}\lambda^{\frac{d}{2}} Q(\lambda\cdot)\|_{H^{1}(\mathbb R^{d})}
\cdot
(\sqrt{2M_{gs}} + \sqrt{2M_{gs}})
\\
& =0.
\endaligned
\]
So
\begin{align}\label{L1con}
|v_{n_{k}}|^{2}\to |Q|^{2}~~ \textrm{in} ~~L^{1}(\mathbb R^{d}), ~\textrm{as}~~ k\to\infty.
\end{align}
By the definition of $v_{n}$, for any $\varphi\in\mathcal S(\mathbb R^{d})$ we have
\begin{align*}
&\quad
\left|\int_{\mathbb{R}^{d}}|u(t_{n_{k}})(x)|^{2}\varphi(x)dx-2M_{gs}\varphi(0)\right|
\\
= &
\left|
\int_{\mathbb{R}^{d}}
|v_{n_{k}}(x)|^{2}\varphi\bigl(\sqrt{H(u(t_{n_{k}}))}x\bigr)dx
-
\varphi(0)
\int_{\mathbb{R}^{d}}
|\lambda^{\frac{d}{2}} Q(\lambda\cdot)|^{2}dx\right|
\\
\le &
\left|
\int_{\mathbb{R}^{d}}
\left(
|v_{n_{k}}(x)|^{2}- |\lambda^{\frac{d}{2}} Q(\lambda\cdot)(x)|^{2}
\right)
\varphi\bigl(\sqrt{H(u(t_{n_{k}}))}x\bigr)dx
\right|
\\
&
+
\left|
\int_{\mathbb{R}^{d}}
|\lambda^{\frac{d}{2}} Q(\lambda\cdot)|^{2}
\left(
\varphi\bigl(\sqrt{H(u(t_{n_{k}}))}x\bigr) - \varphi(0)
\right)
dx\right|
\\
\le &
\|\varphi\|_{L^{\infty}(\mathbb R^{d})} \left\|v_{n_{k}}|^{2}-|\lambda^{\frac{d}{2}} Q(\lambda\cdot)|^{2}\right\|_{L^{1}(\mathbb R^{d})}dx\\
&+\int_{\mathbb{R}^{d}}|\lambda^{\frac{d}{2}} Q(\lambda\cdot)(x)|^{2}|\varphi(\sqrt{H(u(t_{n_{k}}))} x)-\varphi(0)|dx.
\end{align*}
Using \eqref{L1con}, the fact that $\lim_{n\to\infty} H(u(t_{n})) = \infty$ and Lebesgue control convergence theorem, we have for any $\varphi\in\mathcal S(\mathbb R^{d})$,
\[
\lim_{k\to\infty}\left|\int_{\mathbb{R}^{d}}|u(t_{n_{k}})(x)|^{2}\varphi(x)dx-2M_{gs}\varphi(0)\right| =0.
\]
In the sense of the distribution $\mathcal S'(\mathbb R^{d})$, we have
\begin{equation}\label{eq:Schwartz-limit-to-delta}
|u(t_{n_{k}})|^{2}\to 2 M_{gs}\delta,
\quad
\text{ as } k\to\infty.
\end{equation}
For any $R>0$, define $\phi_{R}(x)=R^{2}\phi(x/R)$, where $\phi\in C^{\infty}_{c}(\mathbb{R}^{d})$ is non-negative, radial and there exists a constant $C>0$, such that
\[
\gathered
\phi(x)=|x|^{2}, \quad \text{if} |x|\le 1;\\
|\nabla\phi(x)|^{2}\le C\phi(x), \quad \forall x\in\mathbb R^{d}. \\
\endgathered
\]
For any $t\in[0, T^{\ast})$, define
\[
\Gamma_{R}(t)=\int_{\mathbb{R}^{d}}\phi_{R}(x)|u(t, x)|^{2}dx.
\]
Similar to the proof of \eqref{eq:Virial Identity-I}, we have
\[
\aligned
\Gamma'_{R}(t)
&=2\Re\int\phi_{R}\bar u(t) \partial_{t}u(t)\\
&=-2\Re\int\phi_{R}\cdot i\bar u(t) \left[\Delta u(t)-\frac{a}{|x|^{2}}u(t)+(|\cdot|^{-2}\ast|u(t)|^{2})u(t)\right]\\
&=-2\int\nabla\phi_{R}(x)\cdot \Im\Bigl(\bar u(t) \nabla u(t) \Bigr).
\endaligned
\]
Since $M(u)=M_{gs}$ , using Lemma \ref{lemma: Virial estimate} and $|\nabla\phi_{R}|^{2}\le C\phi_{R}$, we get
\[
|\Gamma'_{R}(t)|\le 2\sqrt{2E(u(t))}(\int|\nabla\phi_{R}|^{2}|u(t)|^{2})^{1/2}\lesssim \sqrt{E(u_{0})}\sqrt{\Gamma_{R}(t)},
\]
then
\[
\left| \frac{d}{dt} \sqrt{\Gamma_{R}(t)} \right|\lesssim 1,
\qquad
\forall t\in[0, T^{\ast}).
\]
By the mean value theorem, we have
\[
\left|\sqrt{\Gamma_{R}(t)}-\sqrt{\Gamma_{R}(t_{n_k})}\right|\lesssim |t-t_{n_k}|.
\]
Note that the slow increasing limit formula \eqref{eq:Schwartz-limit-to-delta} means
\[
\lim_{k\to\infty}
\Gamma_{R}(t_{n_k})
=
\lim_{k\to\infty}
\int_{\mathbb{R}^{d}}|u(t_{n_k})(x)|^{2}\phi_{R}(x)dx=2 M_{gs}\phi_{R}(0)=0.
\]
So for any $t\in [0, T^{\ast})$ and any $R>0$,
\[
\Gamma_{R}(t)\lesssim (T^{\ast}-t)^{2}.
\]
Then let $R\to\infty$, we know for any $t\in[0, T^{\ast})$,
\begin{equation}\label{eq:Virial-estimate-in-situation}
u(t)\in\Sigma\quad \text{and} \quad 0\le\Gamma(t)\lesssim (T^{\ast}-t)^{2}.
\end{equation}
By Lemma \ref{lemma:Virial Identity}, for any $t\in [0, T^{\ast})$, $\Gamma$ is $ C^{2}$ in $[0, T^{\ast})$, and
\begin{equation}\label{eq:Virial-second}
\Gamma''(t)=16E(u(t))=16E(u_{0}).
\end{equation}
Combining \eqref{eq:Virial-estimate-in-situation} with \eqref{eq:Virial-second}, we know that for any $t\in[0, T^{\ast})$,
\[
\Gamma(t)=8E(u_{0})(T^{\ast}-t)^{2}.
\]
Then
\[
\Gamma'(t)=-16E(u_{0})(T^{\ast}-t).
\]
Utilizing the definition of $\Gamma$ again, the lemma \ref{lemma:Virial Identity}
\[
\aligned
\Gamma(t)&=\int|x|^{2}|u(t,x)|^{2}, \\
\Gamma'(t)&=-4\int x\cdot \Im (\bar{u}(t,x)\nabla u(t,x)),
\endaligned
\]
and identity \eqref{eq: rotated energy identity} to calculate
\[
\aligned
E(ue^{-i\frac{|\cdot|^{2}}{4(T^{\ast}-t)}})
=&E(u)-\frac{1}{2(T^{\ast}-t)}\int x\cdot \Im (\bar{u}\nabla u)+\frac{1}{8(T^{\ast}-t)^{2}}\int|x|^{2}|u|^{2}\\
=&E(u)+ \frac{1}{8(T^{\ast}-t)}\Gamma'(t)+\frac{1}{8(T^{\ast}-t)^{2}}\Gamma(t)\\
=&E(u)+ \frac{1}{8(T^{\ast}-t)}\Bigl(-16E(u)(T^{\ast}-t)\Bigr)+\frac{1}{8(T^{\ast-t})^{2}}8E(u)(T^{\ast}-t)^{2}\\
=&0.
\endaligned
\]
We can get $M(ue^{-i\frac{|\cdot|^{2}}{4(T^{\ast}-t)}})=M_{gs}$ and $E(ue^{-i\frac{|\cdot|^{2}}{4(T^{\ast}-t)}})=0$, then by Proposition \ref{static-rigidity}, there exist $\tilde\lambda>0$, $\tilde \theta\in\mathbb{R}$, $\tilde{Q}\in\mathcal G$ such that
\[
ue^{-i\frac{|\cdot|^{2}}{4(T^{\ast}-t)}} = e^{i\tilde\theta} {\tilde \lambda}^{\frac{d}{2}}\tilde Q(\tilde\lambda \cdot).
\]
that is to say
\[
u\in
\left\{
e^{i\frac{|\cdot|^{2}}{4(T^{\ast}-t)}} e^{i\theta} { \lambda}^{\frac{d}{2}} Q(\lambda \cdot)
:
\theta\in\mathbb R,
\lambda>0,
Q\in\mathcal G
\right\}.
\]
\end{proof}
At the end of this section, we show a mass concentration phenomenon to prove the third part of Theorem \ref{thm:main}: If the solution of the equation \eqref{problem-eq: NLS-H} blows up in finite time, then we have the following concentration of mass:
\begin{theorem}\label{thm:mass-concentration}
Suppose $u$ is a blow-up solution of Hartree equation \eqref{problem-eq: NLS-H})with inverse-square potential and it blows up at finite time $T^{\ast}>0$.
If $\lambda(t)>0$ and satisfies $\lim_{t\to T^{\ast}}\lambda(t)\sqrt{H(u(t))}= +\infty$,
then there exists a function $x:[0, T^{\ast})\to\mathbb R^{d}$, such that
\[
\liminf_{t\to T^{\ast}}
\frac{1}{2}\int_{|x-x(t)|\le\lambda(t)}|u|^{2}(x)dx\ge M_{gs}.
\]
\end{theorem}
Before proving the theorem, we prove a vital proposition first.
\begin{proposition}\label{prop:masss-concentratio-profile}
Suppose $\{u_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{d})$ and satisfies
\[
0<\limsup_{n\to\infty}H(u_{n}), \limsup_{n\to\infty}L_{V}(u_{n}) <\infty,
\]
Then up to a subsequence, there exists $\{x_{n}\}\subset \mathbb{R}^{d}$, such that
\[
u_{n}(\cdot+x_{n})\rightharpoonup V
\]
in $H^{1}$, and
\begin{equation}
M(V)\ge\frac{\limsup\limits_{n\to\infty}L_{V}(u_{n})}{\limsup\limits_{n\to\infty}H(u_{n})}M_{gs}.
\end{equation}
\end{proposition}
\begin{proof}
Firstly, we can choose a subsequence $\{u_{n_{k}}\}_{k=1}^{\infty}$, such that
\[
\limsup_{k\to\infty}H(u_{n_{k}})= \limsup_{n\to\infty}H(u_{n}), \limsup_{k\to\infty}L_{V}(u_{n_{k}}) =\limsup_{n\to\infty}L_{V}(u_{n}),
\]
Without loss of generality, we can assume the upper limit unchanges.
By the profile decomposition, up to a subsequence, we have
\[
\begin{aligned}
u_{n}=&\sum_{j=1}^{J}V^{j}(\cdot-x_{n}^{j})+\omega^{J}_{n}, \\
M(u_{n})=&\sum_{j=1}^{J}M(V^{j})+M(\omega_{n}^{J})+o_{n}(1). \\
H(u_{n})=&\sum_{j=1}^{J}H(V^{j}(\cdot - x_{n}^{j}))+H(\omega_{n}^{J})+o_{n}(1). \\
L_{V}(u_{n})=&\sum_{j=1}^{J}L_{V}(V^{j})+\epsilon_{n, J},
\qquad
\lim_{J\to\infty}\limsup_{n\to\infty} |\epsilon_{n, J}| = 0. \\
\end{aligned}
\]
Then
\begin{equation}\label{eq:concentration}
\begin{aligned}
\limsup_{n\to\infty}L_{V}(u_{n})
\le
&
\sum_{j=1}^{\infty}L_{V}(V^{j})
\le
\sum_{j=1}^{\infty}\frac{M(V^{j})}{M_{gs}}\liminf_{n\to\infty}H(V^{j}(\cdot-x_{n}^{j}))\\
\le&
\frac{\sup_{j}(M(V^{j}))}{M_{gs}}\lim_{J\to\infty}\left(\sum_{j=1}^{J}\liminf_{n\to\infty}H(V^{j}(\cdot-x_{n}^{j}))\right)\\
\le&\frac{\sup_{j}(M(V^{j}))}{M_{gs}}\lim_{J\to\infty}\limsup_{n\to\infty}H(u_{n})\\
= &\frac{\sup_{j}(M(V^{j}))}{M_{gs}}\limsup_{n\to\infty}H(u_{n})\\
\end{aligned}
\end{equation}
Note that $\{u_{n}\}$ is bounded in $H^{1}(\mathbb R^{d})$ ,
\[
\begin{aligned}
\sum_{j=1}^{\infty}M(V^{j})\le\limsup_{n\to\infty}M(u_{n})<\infty .
\end{aligned}
\]
So there exists $j_{0}$, such that
\[
M(V^{j_{0}}) = \sup_{j}(M(V^{j})).
\]
At the same time, we also have
\[
u_{n}(\cdot + x_{n}^{j_{0}}) \rightharpoonup V^{j_{0}}, ~~\textrm{in}~~H^{1}(\mathbb R^{d}).
\]
$V^{j_{0}}$ is just $V $ which is needed.
\end{proof}
\begin{proof}[The proof of Theorem \ref{thm:mass-concentration}]
Choose a time sequence $\{t_{n}\}_{n=1}^{\infty}$ to satisfy
\[
\{t_{n}\}_{n=1}^{\infty}\subset[0, T^{\ast}),
\quad
\lim_{n\to\infty} t_{n} = T^{\ast}.
\]
Denote that
\begin{equation}\label{scaling-data-un}
v_{n}(x)=\left(\frac{M_{gs}}{H(u(t_{n}))}\right)^{\frac{d}{4}}u\left(t_{n}, \left(\frac{M_{gs}}{H(u(t_{n}))}\right)^{\frac{1}{2}}x\right).
\end{equation}
By simple scaling analysis, we have
\[
\begin{aligned}
M(v_{n})=&M(u(t_{n}))\equiv M(u_{0}), \\
H(v_{n})=&\frac{M_{gs}}{H(u(t_n))}\cdot H(u(t_n))\equiv M_{gs}, \\
E(v_{n})=&\frac{M_{gs}}{H(u(t_n))}\cdot E(u(t_n))=\frac{M_{gs}}{H(u(t_n))}\cdot E(u_{0})\to 0,
~~\text{as }~~ n\to\infty. \\
\end{aligned}
\]
By the definition of $E(v_n)$, we have
\[
\lim_{n\to\infty} H(v_{n}) = \lim_{n\to\infty} L_{V}(v_{n})=M_{gs}\in (0, \infty).
\]
Using Proposition \ref{prop:masss-concentratio-profile}, there exists a subsequence $\{v_{n_{k}}\}_{k=1}^{\infty}$ of $\{v_{n}\}_{n=1}^{\infty}$ such that
\begin{equation}\label{eq:weak-limit}
v_{n_{k}}(\cdot+x_{k})\rightharpoonup V,~~\textrm{in}~~H^{1}(\mathbb R^{d}), L^{2}(\mathbb R^{d}), \dot H^{1}(\mathbb R^{d}), \textrm{as}~k\to\infty
\end{equation}
and
\[
M(V)\ge \frac{\lim\limits_{n\to\infty}L_{V}(v_{n})}{\lim\limits_{n\to\infty}H(v_{n})}M_{gs} =M_{gs}.
\]
By the weak convergence \eqref{eq:weak-limit}, for any $R>0$, we have
\[
\liminf_{k\to\infty}\int_{|x|\le R}|v_{n_{k}}(x+x_{k})|^{2}dx\ge\int_{|x|\le R}|V(x)|^{2}dx.
\]
By \eqref{scaling-data-un}, we have
\[
\begin{aligned}
\liminf_{k\to\infty}
\int_{|x-x_{n_{k}}|\le R}
\left(\frac{M_{gs}}{H(u(t_{n_k}))}\right)^{\frac{d}{2}}\left| u\left(t_{n_k}, \left(\frac{M_{gs}}{H(u(t_{n_k}))}\right)^{\frac{1}{2}}x\right)\right|^{2}dx
\ge&\int_{|x|\le R}|V(x)|^{2}dx,
\\
\liminf_{k\to\infty}
\int_{\Bigl|x-\bigl(\frac{M_{gs}}{H(u(t_{n_k}))}\bigr)^{\frac{1}{2}}x_{k}\Bigr|
\le \left(\frac{M_{gs}}{H(u(t_{n_k}))}\right)^{\frac{1}{2}}R}
|u(t_{n_k}, x)|^{2}dx
\ge&\int_{|x|\le R}|V(x)|^{2}dx,
\end{aligned}
\]
Let $x(t_{n_{k}}) =\bigl(\frac{M_{gs}}{H(u(t_{n_k}))}\bigr)^{\frac{1}{2}}x_{k} $,
\[
\liminf_{k\to\infty}
\int_{|x- x(t_{n_{k}})|
\le \left(\frac{M_{gs}}{H(u(t_{n_k}))}\right)^{\frac{1}{2}}R}
|u(t_{n_k}, x)|^{2}dx
\ge\int_{|x|\le R}|V(x)|^{2}dx.
\]
Since $\lim_{t\to T^{\ast}}\lambda(t)H(u(t))^{\frac{1}{2}}=\infty$, then for any $R>0$, if $k$ is large enough, there must be
\[
\lambda(t_{n_{k}}) \ge \left(\frac{M_{gs}}{H(u(t_{n_k}))}\right)^{\frac{1}{2}}R.
\]
So
\[
\liminf_{k\to\infty}
\int_{|x- x(t_{n_{k}})|
\le \lambda(t_{n_{k}}) }
|u(t_{n_k}, x)|^{2}dx
\ge\int_{|x|\le R}|V(x)|^{2}dx
\]
By the arbitrariness of $R>0$, we have
\[
\liminf_{k\to\infty}
\int_{|x- x(t_{n_{k}})|
\le \lambda(t_{n_{k}}) }
|u(t_{n_k}, x)|^{2}dx
\ge\int_{\mathbb R^{d}}|V(x)|^{2}dx = 2M_{gs}.
\]
By the arbitrariness of $\{t_{n}\}_{n=1}^{\infty}$ , there exists $x(t)$ in $[0, T^{\ast})$, such that
\[
\liminf_{k\to\infty}
\int_{|x- x(t)|
\le \lambda(t) }
|u(t, x)|^{2}dx
\ge2M_{gs},
\]
which completes the proof of the theorem.
\end{proof} | 100,077 |
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\begin{document}
\newcommand{\TITLE}{Common Divisors of Elliptic Divisibility
Sequences over Function Fields}
\newcommand{\TITLERUNNING}{Common Divisors of Elliptic Divisibility Sequences}
\newcommand{\AUTHOR}{Joseph H. Silverman}
\newcommand{\DATE}{January 2004}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{example}{Example}
\newtheorem{question}{Question}
\newtheorem*{acknowledgement}{Acknowledgements}
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\begin{list}
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\def\Gcal{{\mathcal G}}
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\def\Ical{{\mathcal I}}
\def\Jcal{{\mathcal J}}
\def\Kcal{{\mathcal K}}
\def\Lcal{{\mathcal L}}
\def\Mcal{{\mathcal M}}
\def\Ncal{{\mathcal N}}
\def\Ocal{{\mathcal O}}
\def\Pcal{{\mathcal P}}
\def\Qcal{{\mathcal Q}}
\def\Rcal{{\mathcal R}}
\def\Scal{{\mathcal S}}
\def\Tcal{{\mathcal T}}
\def\Ucal{{\mathcal U}}
\def\Vcal{{\mathcal V}}
\def\Wcal{{\mathcal W}}
\def\Xcal{{\mathcal X}}
\def\Ycal{{\mathcal Y}}
\def\Zcal{{\mathcal Z}}
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\title[\TITLERUNNING]{\TITLE}
\date{\DATE}
\author{\AUTHOR}
\address{Mathematics Department, Box 1917, Brown University,
Providence, RI 02912 USA}
\email{[email protected]}
\subjclass{Primary: 11D61; Secondary: 11G35}
\keywords{divisibility sequence, elliptic curve, common divisor}
\begin{abstract}
Let $E/k(T)$ be an elliptic curve defined over a rational function
field of characteristic zero. Fix a Weierstrass equation for~$E$. For
points $R\in E(k(T))$, write $x_R=A_R^{\vphantom{2}}/D_R^2$ with
relatively prime polynomials $A_R(T),D_R(T)\in k[T]$. The sequence
$\left\{D_{nR}\right\}_{n\ge1}$ is called the \emph{elliptic
divisibility sequence of~$R$}. \par Let $P,Q\in E(k(T))$ be
independent points. We conjecture that
\[
\deg \bigl(\gcd(D_{nP},D_{mQ})\bigr)~\text{is bounded for $m,n\ge1$,}
\]
and that
\[
\gcd(D_{nP},D_{nQ})
= \gcd(D_{P},D_{Q})~\text{for infinitely many $n\ge1$.}
\]
We prove these conjectures in the case that $j(E)\in k$. More
generally, we prove analogous statements with~$k(T)$ replaced by the
function field of any curve and with~$P$ and~$Q$ allowed to lie on
different elliptic curves. If instead~$k$ is a finite field of
characteristic~$p$ and again assuming that $j(E)\in k$, we show that
$\deg \bigl(\gcd(D_{nP},D_{nQ})\bigr)$ is as large as $n+O(\sqrt{n})$
for infinitely many~$n\not\equiv0\pmod{p}$.
\end{abstract}
\maketitle
\section*{Introduction}
A \textit{divisibility sequence} is a sequence~$\{d_n\}_{n\ge1}$ of
positive integers with the property that
\[
m|n \Longrightarrow d_m|d_n.
\]
Classical examples include sequences of the form $a^n-1$ and various
other linear recurrence sequences such as the Fibonacci sequence.
See~\cite{BPvdP} for a complete classification of linear recurrence
divisibility sequences.
\par
Bugeaud, Corvaja and Zannier have shown that independent divisibility
sequences of this type have only limited common factors. For example,
they prove that if $a,b\in\ZZ$ are multiplicatively independent integers,
then for every $\e>0$ there is a constant $c=c(a,b,\e)$ so that
\begin{equation}
\label{equation:BCZ}
\log\gcd(a^n-1,b^n-1) \le \e n + c
\qquad\text{for all $n\ge1$.}
\end{equation}
(This result is proven in~\cite{BCZ}. See also~\cite{CZ1,CZ2} for
more general results.)
\par
It is natural to consider the case of function fields.
For multiplicatively independent polynomials $a,b\in k[T]$ with
coefficients in a field~$k$ of characteristic~$0$,
Ailon and Rudnick~\cite{AR} prove the strong result that there
is a constant $c=c(a,b)$ so that
\begin{equation}
\label{equation:AR}
\deg\gcd(a^n-1,b^n-1) \le c
\qquad\text{for all $n\ge1$,}
\end{equation}
and that
\begin{equation}
\label{equation:AR1}
\gcd(a^n-1,b^n-1) =\gcd(a-1,b-1)
\qquad\text{for infinitely many $n\ge1$,}
\end{equation}
Somewhat surprisingly, if~$a(T)$ and~$b(T)$ have coefficients in a
finite field, then neither~\eqref{equation:AR}, nor even the weaker
statement~\eqref{equation:BCZ}, is true, even if the set of
allowable~$n$'s is restricted in various reasonable ways.
(See~\cite{SilvermanGCDoverFF}.)
\par
A divisibility sequence of the form $a^n-1$ comes from a rank~1
subgroup of the multiplicative group~$\GG_m$. It is interesting to
consider divisibility sequences coming from other algebraic groups,
for example from elliptic curves. The classical definition of an
elliptic divisibility sequence~\cite{Ward1,Ward2} uses the nonlinear
relation satisfied by division polynomials, but we will use an
alternative definition\footnote{The alternative definition
of elliptic divisibility sequence that we use,
which is based directly on elliptic curves, gives a slightly different
collection of divisibility sequences than is given by the classical
non-linear recurrence formula. See~\cite[\S10.3]{EPS} and~\cite{SHIP}.}
that has the dual advantages of being more natural
and more easily generalized to other algebraic groups. (See
\cite{Durst,EGW,EPS,EW1,EW2,SHIP} for additional material on elliptic
divisibility sequences and~\cite{SilvermanGCDinFGgps} for a
discussion of general algebraic divisibility sequences and their
relation to Vojta's conjecture .)
\par
Let~$E/\QQ$ be an elliptic curve given by a (minimal) Weierstrass
equation
\begin{equation}
\label{equation:WE}
y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.
\end{equation}
Any nonzero rational point $P\in E(\QQ)$ can be written in
the form
\[
P = (x_{P},y_{P})
= \left(\frac{A_{P}}{D_{P}^2},\frac{B_{P}}{D_{P}^3}\right)
\quad\text{with $\gcd(A_P,D_P)=\gcd(B_P,D_P)=1$.}
\]
Assume now that~$P\in E(\QQ)$ is a nontorsion point. The
\textit{elliptic divisibility sequence} associated to~$E/\QQ$ and~$P$
is the sequence of denominators of the multiples of~$P$:
\[
\left\{ D_{nP} : n=1,2,3,\ldots\right\}
\]
The elliptic analogue of~\eqref{equation:BCZ} is part~(a) of the following
conjecture, while part~(b) gives an elliptic analogue of
a conjecture of Ailon and Rudnick~\cite{AR}.
\begin{conjecture}
\label{conjecture:intro:GCDonECoverQ}
With notation as above, let $P,Q\in E(\QQ)$ be independent nontorsion
points.
\begin{parts}
\Part{(a)}
For every $\e>0$ there is a constant $c=c(E/\QQ,P,Q,\e)$ so that
\[
\log\gcd(D_{nP},D_{nQ}) \le \e n^2 + c
\qquad\text{for all $n\ge1$.}
\]
\Part{(b)}
There is an equality
\[
\gcd(D_{nP},D_{nQ}) = \gcd(D_P,D_Q)
\qquad\text{for infinitely many $n\ge1$.}
\]
\end{parts}
\end{conjecture}
\begin{remark}
Siegel's theorem~\cite[Theorem~IX.3.1]{AEC} implies that
\[
\log D_{nP}\gg\ll n^2,
\]
so the $n^2$ appearing in
Conjecture~\ref{conjecture:intro:GCDonECoverQ}(a) is the natural quantity
to expect. See also~\cite{SilvermanGCDinFGgps} for a proof that
Vojta's conjecture~\cite{Vojta} implies
Conjecture~\ref{conjecture:intro:GCDonECoverQ}(a).
\end{remark}
Following the lead of Ailon and Rudnick, we replace~$\QQ$ with the
rational function field~$k(T)$ and replace~$\ZZ$ with the ring of
polynomials~$k[T]$. Then we can look at an elliptic curve $E/k(T)$
given by a (minimal) Weierstrass equation~\eqref{equation:WE} and we
can write the $x$-coordinate of a point $P\in E(k(T))$ in the form
\[
x_P=\frac{A_P}{D_P^2}
\quad\text{with $A_P(T),D_P(T)\in k[T]$ satisfying
$\gcd(A_P,D_P)=1$.}
\]
This leads to a conjectural elliptic analogue of~\eqref{equation:AR}
and~\eqref{equation:AR1}.
\begin{conjecture}
\label{conjecture:intro:GCDonECoverFF}
Let~$k$ be an algebraically closed field of characteristic~$0$, let
$E/k(T)$ be an elliptic curve, and let $P,Q\in E(k(T))$ be independent
points. Then there is a constant $c=c(E,P,Q)$ so that
\[
\deg\gcd(D_{nP},D_{nQ}) \le c
\qquad\text{for all $n\ge1$.}
\]
Further, there is an equality
\[
\gcd(D_{nP},D_{nQ}) = \gcd(D_{P},D_{Q})
\qquad\text{for infinitely many $n\ge1$.}
\]
\end{conjecture}
Our principal result in this paper is a proof of
Conjecture~\ref{conjecture:intro:GCDonECoverFF} in the case that~$E$
has constant $j$-invariant. We note that even in this special case,
the proof requires nontrivial tools such as Raynaud's
theorem~\cite{Raynaud1,Raynaud2} bounding torsion points on
subvarieties of abelian varieties.
\begin{theorem}
\label{theorem:intro:maintheorem}
Conjecture~\ref{conjecture:intro:GCDonECoverFF} is true for elliptic
curves with constant $j$-invariant, i.e., with $j(E)\in
k$.\end{theorem}
We actually prove something more general than
Theorem~\ref{theorem:intro:maintheorem}. First, we replace~$k(T)$
by the function field of an arbitrary algebraic curve. Second, we
allow different integer multipliers for~$P$ and~$Q$. Third, we
allow the points~$P$ and~$Q$ to lie on different elliptic curves. For
the complete statement, see Conjecture~\ref{conjecture:ECoverFF} and
Theorem~\ref{theorem:ECoverFFconstantj}. This added generality does
not significantly lengthen the proof and makes parts of the argument
more transparent.
We also consider the case that~$E$ is an elliptic curve over a field
$\FF_q(T)$ of characteristic~$p$. In this case, nothing like
Conjecture~\ref{conjecture:intro:GCDonECoverFF} is true, even with the
natural restriction that~$n$ be prime to~$p$. We prove
(Theorem~\ref{theorem:gcdonECoverFFconstantj}) that if~$E/\FF_q(T)$ has
constant~$j$ invariant and $P,Q\in E(\FF_q(T))$, then
\[
\deg\gcd(D_{nP},D_{nQ}) \ge n + O(\sqrt{n})
\]
for infinitely many~$n$ satisfying $p\notdivide n$. We conjecture that
the same is true for all~$E/\FF_q(T)$.
\begin{acknowledgement}
The author would like to thank Gary Walsh for rekindling his interest
in the arithmetic properties of divisibility sequences and for bringing
to his attention the articles~\cite{AR} and~\cite{BCZ}.
\end{acknowledgement}
\section{Preliminaries}
\label{section:preliminaries}
In this section we set some notation, recall a deep theorem,
and prove two basic estimates that will be required for our main results.
We begin with notation.
\begin{notation}
\item[$k$]
an algebraically closed field of characteristic zero.
\item[$C/k$]
a smooth projective curve.
\item[$K$]
the function field of $C$.
\item[$E/K$]
an elliptic curve.
\item[$\Ecal/C$]
a minimal smooth projective elliptic surface $\Ecal\to C$ with generic
fiber~$E$.
\item[$\s_P$]
the section $\s_P:C\to\Ecal$ corresponding to a point $P\in E(K)$.
\item[$\Obar$]
the ``zero divisor'' $\Obar=\s_O(C)\in\Div(\Ecal)$ corresponding to
the point $O\in E(K)$.
\end{notation}
We recall that~$E/K$ is said to \textit{split over~$K/k$} if it
is~$K$-isomorphic to an elliptic curve defined over~$k$, and
that~$E/K$ is said to be \textit{constant over~$k$} if $j(E)\in k$.
Clearly split curves are constant, while a constant curve can always
be split over a finite extension of~$K$.
\par
We observe that if a Weierstrass equation is chosen for~$E/K$ and if
$P=(x_P,y_P)\in E(K)$, then the pullback divisor $\s_P^*(\Obar)$ is,
roughly, one half the polar divisor of~$x_P$. The following elementary
result shows the stability of $\s_{mP}^*(\Obar)$ at at a fixed point
of~$C$ for multiples~$mP$ of~$P$. This is well known, but for
completeness and since it is false when the residue characteristic is
positive, we include a proof.
\begin{lemma}
\label{lemma:stabilityofvanishing}
With notation as above, let $\g\in C$ and let $P\in E(K)$ be a
nontorsion point.
\begin{parts}
\Part{(a)}
If $\ord_\g \s_{P}^*(\Obar)\ge 1$, then
\[
\ord_\g\s_{mP}^*(\Obar) = \ord_\g\s_{P}^*(\Obar)
\qquad\text{for all $m\ne 0$.}
\]
\Part{(b)}
There is an integer $m'=m'(E/K,P,\g)$ so that
\[
\ord_\g\s_{mP}^*(\Obar) \in \{0,m'\}
\qquad\text{for all $m\ne 0$.}
\]
In particular, $\ord_\g \s_{mP}^*(\Obar)$ is bounded independently of~$m$.
\end{parts}
\end{lemma}
\begin{proof} (a)
Let $[m]:\Ecal\to\Ecal$ be the multiplication-by-$m$ map. Notice that
\[
[m]^*\Obar = \Obar + D_m,
\]
where the divisor $D_m\in\Div(\Ecal)$ is the divisor of nonzero
$m$-torsion points. For example, if the fiber~$\Ecal_\g$ is nonsingular,
then the intersection of~$D_m$ with~$\Ecal_\g$ consists of the nonzero
$m$-torsion points of the elliptic curve~$\Ecal_\g$. It is thus clear
that at least on the nonsingular fibers, the divisors~$\Obar$ and~$D_m$ do
not intersect. (This is where we are using the characteristic zero
assumption. More generally, it is enough to assume that~$m$ is
relatively prime to the residue characteristic.) However, even if
the fiber~$\Ecal_\g$ is singular, the map~$[m]:\Ecal\to\Ecal$ is \'etale
in a neighborhood at the zero point~$O_\g$ of~$\Ecal_\g$,
so $\Obar\cap D_m=\emptyset$.
\par
We have
\[
\s_{mP}^*(\Obar)
= \s_{P}^*\bigl([m]^*(\Obar)\bigr)
= \s_{P}^*(\Obar) + \s_{P}^*(D_m).
\]
The assumption that $\ord_\g\s_{P}^*(\Obar)\ge1$ is equivalent to the
statement that $\s_{P}(\g)=O_\g\in\Ecal_\g$.
Since $\Obar\cap D_m=\emptyset$ from above, it follows that
the support of $\s_{P}^*(D_m)$ does
not contain~$\g$, so
\[
\ord_\g \s_{mP}^*(\Obar)
= \ord_\g\s_{P}^*(\Obar) + \ord_\g\s_{P}^*(D_m)
= \ord_\g\s_{P}^*(\Obar).
\]
This completes the proof of~(a). \par In order to prove~(b), we may
suppose without loss of generality that there exists some $m\ne0$ such
that $\ord_\g \s_{mP}^*(\Obar)\ge1$, since otherwise we may take
$m'=0$. Suppose that \text{$\ord_\g \s_{m_1P}^*(\Obar)\ge1$} and
\text{$\ord_\g \s_{m_2P}^*(\Obar)\ge1$}. Then applying~(a), first to~$m_1P$
with $m=m_2$ and second to~$m_2P$ with $m=m_1$, we find that
\[
\ord_\g \s_{m_1P}^*(\Obar) =\ord_\g \s_{m_2m_1P}^*(\Obar)
=\ord_\g \s_{m_2P}^*(\Obar) .
\]
\end{proof}
\begin{remark}
Lemma~\ref{lemma:stabilityofvanishing} readily generalizes to algebraic
groups. For the group~$\GG_m/k(\PP^1)$, the proof is especially
transparent and helps to illustrate the general case, so we recall it
here. Let $R(T)\in k(T)$ be a rational function, say $R(T)=A(T)/B(T)$,
and suppose that $\ord_\g(R(T)-1)=e\ge 1$. This implies that $B(\g)\ne0$
and that
\[
A(T)-B(T) = (T-\g)^e C(T)
\qquad\text{for some $C(T)\in k[T]$.}
\]
Then
\begin{align*}
A(T)^{m}-B(T)^{m}
&=\prod_{\z\in\bfmu_m} \bigl(A(T)-\z B(T)\bigr)\\
&=\prod_{\z\in\bfmu_m} \bigl((T-\g)^e C(T)+ (1-\z)B(T)\bigr).
\end{align*}
Hence
\[
\left.\frac{A(T)^{m}-B(T)^{m}}{A(T)-B(T)}\right|_{T=\g}
= \prod_{\substack{\z\in\bfmu_m\\\z\ne1\\}} (1-\z)B(\g)
\ne 0,
\]
which proves that
$\ord_\g\bigl(A(T)^{m}-B(T)^{m}\bigr)=\ord_\g\bigl(A(T)-B(T)\bigr)$.
\end{remark}
We will also need the following elementary result, which says that a
$\Kbar$-isogeny mapping even one $K$-rational nontorsion point to a
$K$-rational point is necessarily itself defined over~$K$.
\begin{lemma}
\label{lemma:isogenyofrationalpt}
Let~$K$ be a field of characteristic~$0$, let~$E_1/K$ and~$E_2/K$ be
elliptic curves, and let $G:E_2\to E_1$ be an isogeny defined
over~$\Kbar$. Suppose that there is a $K$-rational point $P\in
E_2(K)$ so that the image~$G(P)$ is also $K$-rational, i.e., $G(P)\in
E_1(K)$. Then either~$P$ has finite order or else~$G$ is defined over~$K$.
\end{lemma}
\begin{proof}
For each $s\in\Gal(\Kbar/K)$, define an isogeny~$g_s$ by
\[
g_s:E_2\longrightarrow E_1,\qquad
g_s(Q) = G^s(Q) - G(Q).
\]
The assumption on the point~$P$ implies that
\[
G(P) = \bigl(G(P)\bigr)^s = G^s(P^s) = G^s(P),
\]
so we see that $P\in\ker(g_s)$ for all $s\in\Gal(\Kbar/K)$.
Let $d_s=\deg(g_s)$. Applying the dual isogeny, it follows that
$P\in E_2[d_s]$ for all $s\in\Gal(\Kbar/K)$.
Hence either~$P$ is a torsion point, or else $d_s=0$
for all $s\in\Gal(\Kbar/K)$.
But
\[
d_s=0 \iff g_s=[0] \iff G^s=G,
\]
so
\[
\text{$d_s=0$ for all $s\in\Gal(\Kbar/K)$}
\iff
\text{$G$ is defined over~$K$.}
\]
This completes the proof that either~$P$ is a torsion point or else~$G$
is defined over~$K$.
\end{proof}
We conclude this section by recalling a famous result of
Raynaud. We will apply Raynaud's theorem to a curve embedding in an
abelian surface.
\begin{theorem}[Raynaud's Theorem]
\label{theorem:raynaud}
Let $k$ be a field of characteristic zero, let~$A/k$ be an abelian
variety, and let $V\subset A$ be a subvariety. Then the Zariski
closure of $V\cap A_\tors$ is equal to a finite union of translates of
abelian subvarieties of~$A$ by torsion points.
\end{theorem}
\begin{proof}
See~\cite{Raynaud2} for the general case. For the case that~$V$ is a
curve, which is the case that we will need, see~\cite{Raynaud1}.
\end{proof}
\section{Common Divisors on Elliptic Curves over
Characteristic~$0$ Function Fields}
We continue with the notation set in Section~\ref{section:preliminaries}.
For any two \underbar{effective} divisors $D_1,D_2\in\Div(C)$, we define
the \textit{greatest common divisor} in the usual way as
\[
\GCD(D_1,D_2)
= \sum_{\g\in C} \min\bigl\{\ord_\g(D_1),\ord_\g(D_2)\bigr\}\cdot(\g)
\in \Div(C).
\]
(Here~$\ord_\g(D)$ is the coefficient of~$\g$ in the divisor~$D$.)
\begin{definition}
Let~$E/K$ be an elliptic curve over a function field as above, and let
$P,Q\in E(K)$ be points, not both zero. Then the
\textit{{\upshape(}elliptic{\upshape)} greatest common divisor} of~$P$
and~$Q$ is
\[
\GCD(P,Q) = \GCD\bigl(\s_P^*(\Obar),\s_Q^*(\Obar)\bigr).
\]
\end{definition}
\begin{remark}
As noted earlier, the divisor $\s_P^*(\Obar)$ is, roughly, one half
the polar divisor of~$x_P$. Thus the elliptic~$\GCD$ is a natural
generalization of the definition given in the introduction.
\end{remark}
More generally, we can work with points on different curves.
\begin{definition}
Let~$E_1/K$ and~$E_2/K$ be elliptic curves as above, and let $P_1\in
E_1(K)$ and $P_2\in E_2(K)$ be points, not both zero.
The \textit{{\upshape(}elliptic{\upshape)} greatest common divisor}
of~$P_1$ and~$P_2$ is the divisor
\[
\GCD(P_1,P_2) = \GCD\bigl(\s_{P_1}^*(\Obar_{\Ecal_1}),
\s_{P_2}^*(\Obar_{\Ecal_2})\bigr)\in\Div(C).
\]
\end{definition}
In order to prove boundedness of~$\GCD(P_1,P_2)$, we need~$P_1$ and~$P_2$
to be independent in some appropriate sense, which prompts the following
definition.
\begin{definition}
\label{definition:independence}
We say that~$P_1$ and~$P_2$ are \textit{dependent} if there are
isogenies $F:E_1\to E_1$ and $G:E_2\to E_1$, not both zero, so that
$F(P_1)=G(P_2)$; otherwise we say that~$P_1$ and~$P_2$ are
\textit{independent}. If~$E_1$ and~$E_2$ are defined over a field~$K$,
we say that~$P_1$ and~$P_2$ are \textit{$K$-dependent} if the
isogenies~$F$ and~$G$ can be defined over~$K$.
\end{definition}
\begin{remark}
We observe that independence is an equivalence
relation, since if $F(P_1)=G(P_2)$, then $(\hat G\circ F)(P_1)=(\hat
G\circ G)(P_2)$, where~$\hat G$ is the dual isogeny to~$G$. We also
note that a torsion point can never be part of an independent pair,
since if (say) $NP_1=0$, then we can take $F=[N]$ and $G=0$ to show that~$P_1$
and any~$P_2$ are dependent.
\end{remark}
\begin{conjecture}
\label{conjecture:ECoverFF}
Let~$K$ be a characteristic zero function field as above, let~$E_1/K$
and~$E_2/K$ be elliptic curves, and let $P_1\in E(K)$ and
$P_2\in E_2(K)$ be $K$-independent points. (See
Definition~\ref{definition:independence}.)
\begin{parts}
\Part{(a)}
There is a constant
$c=c(K,E_1,E_2,P_1,P_2)$ so that
\[
\deg\GCD(n_1P_1,n_2P_2) \le c
\qquad\text{for all $n_1,n_2\ge1$.}
\]
\Part{(b)}
Further, there is an equality
\[
\GCD(nP_1,nP_2) = \GCD(P_1,P_2)
\qquad\text{for infinitely many $n\ge1$.}
\]
\end{parts}
\end{conjecture}
We prove Conjecture~\ref{conjecture:ECoverFF} in the case that~$E_1$
and~$E_2$ have constant $j$-invariant. We note that even this
``special case'' is far from trivial, since it relies on Raynaud's
Theorem (Theorem~\ref{theorem:raynaud}). For~(b), we prove a stronger
positive density result.
\begin{theorem}
\label{theorem:ECoverFFconstantj}
Let~$K$ be a characteristic zero function field as above, let~$E_1/K$
and~$E_2/K$ be elliptic curves, and let $P_1\in E(K)$ and $P_2\in
E_2(K)$ be $K$-independent points. Assume further that the elliptic
curves $E_1/K$ and~$E_2/K$ both have constant $j$-invariant, i.e.,
\text{$j(E_1),j(E_2)\in k$}.
\begin{parts}
\Part{(a)}
There is a constant
$c=c(K,E_1,E_2,P_1,P_2)$ so that
\[
\deg\GCD(n_1P_1,n_2P_2) \le c
\qquad\text{for all $n_1,n_2\ge1$.}
\]
\Part{(b)}
The set
\[
\bigl\{n\ge1 : \GCD(nP_1,nP_2) = \GCD(P_1,P_2) \bigr\}
\]
has positive density.
\end{parts}
\end{theorem}
\begin{proof}
The fact that~$E_1$ and~$E_2$ have constant $j$-invariants means that
they split over some finite extension of~$K$. Taking a common
splitting field, there is a finite cover $C'\to C$ and elliptic
curves~$E_1',E_2'/k$ so that
\[
\Ecal_i\times_CC' \cong_{/k}
E_i'\times_kC'.
\]
(N.B., $E_i'$ is defined over the constant field~$k$.)
We thus get commutative diagrams
\[
\begin{CD}
E_i' \times_k C' @>>> \Ecal_i \\
@VVV @VVV \\
C' @>f>> C \\
\end{CD}
\qquad\text{for $i=1,2$.}
\]
Each point $P_i\in E_i(K)$ gives a section $\s_{P_i}:C\to\Ecal_i$,
which in turn lifts to a unique section
\[
\t_{P_i}\times 1 :C'\to E_i'\times_kC'.
\]
In other words, each point $P_i\in E_i(K)$ gives a unique morphism
\text{$\t_{P_i}:C'\to E_i'$} so that the following diagram commutes:
\begin{equation}
\label{cd:sectionsplits}
\begin{CD}
E_i'\times_k C' @>>> \Ecal_i \\
@A\t_{P_i}\times 1AA @AA\s_{P_i}A \\
C' @>f>> C \\
\end{CD}
\end{equation}
\par
We now fix two $K$-independent points $P_1\in E_1(K)$ and
$P_2\in E_2(K)$ and define a morphism
\[
\f = \t_{P_1}\times\t_{P_2} : C' \longrightarrow E_1'\times_k E_2'.
\]
Suppose that a point $\g\in C$ is in the support of $\GCD(n_1P_1,n_2P_2)$
for some $n_1,n_2\ge1$. This means that
\[
\g\in\Support(\s_{n_1P_1}^*(\Obar_{\Ecal_1}))
\qquad\text{and}\qquad
\g\in\Support(\s_{n_2P_2}^*(\Obar_{\Ecal_2})).
\]
Tracing around the commutative diagrams, this means that
for every point $\g'\in f^{-1}(\g)\in C'$,
\[
\t_{n_1P_1}(\g')=O_1
\qquad\text{and}\qquad
\t_{n_2P_2}(\g')=O_2,
\]
where~$O_i\in E_i'(k)$ is the zero point.
Equivalently, $\t_{P_1}(\g')\in E_1'[n_1]$ and $\t_{P_2}(\g')\in E_2'[n_2]$,
so in particular,~$\t_{P_1}(\g')$ and~$\t_{P_2}(\g')$ are
torsion points of~$E_1'$ and~$E_2'$, respectively.
Hence $\f(\g')=\bigl(\t_{P_1}(\g'),\t_{P_2}(\g')\bigr)$ is a torsion point
of the abelian surface~\text{$E_1'\times E_2'$}.
\par
To recapitulate, we have proven that
\begin{multline}
\label{equation:supportGCDistorsion}
\g\in\Support(\GCD(n_1P_1,n_2P_2)) \\
\Longrightarrow
\f(\g')\in E_1'[n_1]\times E_2'[n_2]
\subset (E_1'\times E_2')_\tors \\
\quad\text{for all $\g'\in f^{-1}(\g)$.}
\end{multline}
To ease notation, we let
\[
A=E_1'\times E_2' \qquad\text{and}\qquad V = \f(C')\subset A.
\]
There are several cases to consider:
\subsection*{Case I: $\t_{P_1}$ and $\t_{P_2}$ are both constant maps}
\hfill\break From diagram~\eqref{cd:sectionsplits}, the assumption
that~$\t_{P_i}$ is constant and nonzero implies that the divisor
$\s_{P_i}^*(\Obar_{\Ecal_i})$ is supported on the set of ramification
points~$\Rcal_f$ of the map \text{$f:C'\to C$}.
More generally, the independence
assumption implies that~$P_i$ is nontorsion, so $nP_i\ne O_i$ for all
$n\ge1$. Hence $\s_{nP_i}^*(\Obar)$ is supported on~$\Rcal_f$.
Further, Lemma~\ref{lemma:stabilityofvanishing}(b) says that for any
particular point~$\g\in\Rcal_f$, the multiplicity
$\ord_\g\s_{nP_i}^*(\Obar_{\Ecal_i})$ is bounded independently
of~$n$. Therefore
\[
\text{$\deg\s_{nP_i}^*(\Obar_{\Ecal_i})$ is bounded for all $n\ge1$.}
\]
Thus~$E_i(K)$ contains infinitely many points of bounded degree, or
what amounts to the same thing, of bounded height.
(See~\cite[III~\S4]{ATAEC}, where $h(P)=2\deg\s_P^*(\Obar)+O(1)$.) It
follows from~\cite[Theorem~III.5.4]{ATAEC} that $\Ecal_i\to C$ splits
as a product over~$k$.\footnote{For Case~I, it is also possible to
give a more elementary proof that~$\Ecal_i$ splits using explicit
Weierstrass equations and considering different types of twists.}
\par
Thus Case~I leads to the conclusion that both~$E_1$ and~$E_2$
are $K$-isomorphic to elliptic curves defined over~$k$, so we may
replace them with curves that are defined over~$k$. Then
$\Ecal_i=E_i\times_kC$, and any point~$Q_i\in E_i(K)$ is associated to a
$k$-morphism $\t_{Q_i}:C\to E_i$. Our assumption that~$\t_{P_i}$ is
constant is equivalent to saying that $P_i\in E_i(k)$, so as long
as $nP_i\ne O_i$, we have
\[
\Support\left(\s_{nP_i}^*(\Obar_{\Ecal_i})\right) =
\bigl(\{nP_i\}\times C'\bigr) \cap \bigl(\{O_i\}\times C'\bigr)
= \emptyset.
\]
Hence the assumption that~$P_1$ and~$P_2$ are nontorsion points leads,
in Case~I, to the conclusion that
\[
\GCD(n_1P_1,n_2P_2) = 0\qquad\text{for all $n_1,n_2\ge1$.}
\]
Thus Case~I gives a strong form of both~(a) and~(b).
\subsection*{Case II: $\t_{P_1}$ or $\t_{P_2}$ is nonconstant,
and $V\cap A_\tors$ is infinite}
\hfill\break
The assumption that one of~$\t_{P_1}$ or~$\t_{P_2}$ is nonconstant implies
that~$V=\f(C')$ is an irreducible curve, and then Raynaud's
Theorem~\ref{theorem:raynaud} tells us that $V\cap A_\tors$ can only
be infinite if it is contained in the translate of an elliptic curve
(abelian subvariety of~$A$) by a torsion point of~$A$. Thus there is
an elliptic curve $W\subset A$ and a torsion point $t\in A$ so that
$V=W+t$. Let~$N$ be the order of the point~$t$. Then composing with
the multiplication-by-$N$ map yields
\[
[N]\circ\f = [N]\circ(\t_{P_1}\times\t_{P_2}) = \t_{NP_1}\times\t_{NP_2},
\]
and since~$W$ is an elliptic curve, we see that~$[N]\circ\f$ maps~$C'$
onto \text{$NV=N(W+t)=NW=W$}. Hence we get a commutative diagram
\[
\begin{matrix}
& & C' \\
& \overset{\t_{NP_1}}{\swarrow}
& \phantom{XX}\big\downarrow{\scriptstyle [N]\circ\f}
& \overset{\t_{NP_2}}{\searrow} \\[1ex]
E_1' & \overset{\pi_1}{\longleftarrow} & W
& \overset{\pi_2}{\longrightarrow} & E_2' \\
\end{matrix}
\]
where~$\pi_1$ and~$\pi_2$ are the projections
\text{$\pi_i:E_1'\times E_2'\to E_i'$}.
\par
Let $d_2=\deg(\pi_2)$. Since~$W$ is an elliptic curve, there is a
dual isogeny $\hat\pi_2:E_2'\to W$ with the property that
$\hat\pi_2\circ\pi_2=[d_2]$. We compute
\begin{align}
[d_2]\circ \t_{NP_1}
&= [d_2]\circ \pi_1 \circ [N]\circ\f \notag \\
&= \pi_1 \circ [d_2] \circ [N]\circ\f \notag \\
&= \pi_1 \circ \hat\pi_2 \circ \pi_2 \circ [N]\circ\f \notag \\
&= \pi_1 \circ \hat\pi_2 \circ \t_{NP_2}
\label{equation:P2P1relation}
\end{align}
\par
Let $G':E_2'\to E_1'$ be the isogeny
\[
G' = \pi_1\circ\hat\pi_2\circ[N] \in \Hom_k(E_2',E_1').
\]
Recall that $K'=k(C')$ is the extension of~$K$ over which~$E_1$ and~$E_2$
become isomorphic to~$E_1'$ and~$E_2'$, respectively. Thus~$G'$ induces
an isogeny
\[
G:E_2\to E_1\quad\text{defined over $K'$,}
\]
but \textit{a priori}, there is no reason that~$G$ need be defined over~$K$.
However, the relation~\eqref{equation:P2P1relation} gives a commutative
diagram
\[
\begin{CD}
E_2' \times_k C' @>G'\times1>> E_1'\times_k C' \\
@A\t_{P_2}\times1AA @AA\t_{[d_1N]P_1}\times1A \\
C' @= C' \\
\end{CD}
\]
which is equivalent to the equality
\begin{equation}
\label{equation:Gmapsrationalpts}
G(P_2) = [d_1N](P_1)
\end{equation}
of points in~$E_1(K)$.
\par
The curves~$E_1$ and~$E_2$ and the points~$P_1$ and~$P_2$ are rational
over~$K$ by assumption, hence the same is true of the multiple
$[d_1N](P_1)$ of~$P_1$. Thus~\eqref{equation:Gmapsrationalpts} says
that the isogeny~$G$ maps at least one $K$-rational point of~$E_2$ to a
$K$-rational point of~$E_1$. Further, the independence assumption
on~$P_1$ and~$P_2$ ensures that they are not torsion points. Hence
Lemma~\ref{lemma:isogenyofrationalpt} tells us that~$G$ is indeed
defined over~$K$. Then~\eqref{equation:Gmapsrationalpts} contradicts
the $K$-independence of~$P_1$ and~$P_2$, which shows that Case~II
cannot occur.
\subsection*{Case III: $\t_{P_1}$ or $\t_{P_2}$ is nonconstant,
and $V\cap A_\tors$ is finite}
\hfill\break
The assumption that one of~$\t_{P_1}$ or~$\t_{P_2}$ is nonconstant implies
that the map~$\f$ is nonconstant, and hence that $\f:C'\to V$ is
finite-to-one. We showed earlier~\eqref{equation:supportGCDistorsion} that
\[
\Support(\GCD(n_1P_1,n_2P_2)) \subset f\bigl(\f^{-1}(V\cap A_\tors)\bigr),
\]
so the assumption that $V\cap A_\tors$ is finite implies that
$\GCD(n_1P_1,n_2P_2)$ is supported on a finite set of points that is
\textit{independent of~$n_1$ and~$n_2$}. Since
Lemma~\ref{lemma:stabilityofvanishing}(b) tells us that for any
particular point~$\g\in C$, the order of~$\GCD(n_1P_1,n_2P_2)$ at~$\g$ is
bounded independently of~$n_1$ and~$n_2$, this shows that
$\deg\GCD(n_1P_1,n_2P_2)$ is bounded, which completes the proof of~(a).
\par
In order to prove~(b), we return to~\eqref{equation:supportGCDistorsion},
which actually provides the more accurate information that
\[
\Support(\GCD(nP_1,nP_2)) \subset f\bigl(\f^{-1}(V\cap A[n])\bigr).
\]
Since $V\cap A_\tors$ is finite by assumption, we can find an integer~$N$
so that $V\cap A_\tors$ is contained in~$A[N]$. It follows that
\[
V\cap A[n] = V\cap A[\gcd(n,N)]
\qquad\text{for all $n\ge1$,}
\]
and hence in particular that
\[
V\cap A[n] = V\cap \{0\}
\qquad\text{for all $n$ with $\gcd(n,N)=1$.}
\]
Hence
\begin{multline*}
\Support(\GCD(nP_1,nP_2)) = \Support(\GCD(P_1,P_2))\\
\qquad\text{for all $n$ with $\gcd(n,N)=1$.}
\end{multline*}
On the other hand, Lemma~\ref{lemma:stabilityofvanishing}(a) tells us
that the multiplicities of $\GCD(nP_1,nP_2)$ and $\GCD(P_1,P_2)$ are
the same at every point in the support of the latter. Therefore
\[
\GCD(nP_1,nP_2)=\GCD(P_1,P_2)
\qquad\text{for all $n$ with $\gcd(n,N)=1$,}
\]
which completes the proof of~(b), and with it the proof of
Theorem~\ref{theorem:ECoverFFconstantj}.
\end{proof}
\begin{remark}
One can easily formulate other variants of the common divisor problem
on algebraic groups. For example, let $a(T)\in\CC[T]$ be a nonconstant
polynomial, let $E/\CC(T)$ be an elliptic curve, and let $P\in E(\CC(T))$
be a nontorsion point. Then it is plausible to guess that there is
a constant $c=c(a,E,P)$ so that
\[
\deg\gcd(D_{nP},a^m-1)\le c
\qquad\text{for all $n_1,n_2\ge1$.}
\]
If~$E$ has constant $j$-invariant, one can probably prove that this is
true using a generalization of Raynaud's theorem to semiabelian
varieties~\cite{C-L,DP}. The situation over~$\QQ$ is somewhat more
complicated due to the different growth rates of~$D_{nP}$ and~$a^m$,
but Vojta's conjecture applied to the blowup of \text{$E\times\GG_m$}
at $(0,1)$ implies that for every $\e>0$ there exists a proper Zariski
closed subset $Z=Z(a,E,P,\e)$ of \text{$E\times\GG_m$} so that
\begin{multline*}
\log\gcd(D_{nP},a^m-1)\le \e\max\{n^2,m\} \\
\text{provided that $(nP,a^m)\notin Z$.}
\end{multline*}
See~\cite{SilvermanGCDinFGgps} for details.
\end{remark}
\section{Common Divisors on Elliptic Curves over
Characteristic~$p$ Function Fields}
We continue with the notation set in Section~\ref{section:preliminaries},
except that rather than working over a field of characteristic~0,
we work instead over a finite field~$k=\FF_q$.
\par
Let $a(T),b(T)\in k[T]$ be multiplicatively independent polynomials.
As noted in the introduction, Ailon and Rudnick~\cite{AR} prove that
\text{$\gcd(a(T)^n-1,b(T)^n-1)$} is bounded for $n\ge1$ when~$k$ is a
field of characteristic zero, but the author~\cite{SilvermanGCDoverFF}
has shown that there is no analogous bound when~$k$ has
characteristic~$p$, even if the exponent~$n$ is subject to some
reasonable restrictions such as~$p\notdivide n$.
\par
It is natural to ask for a result similar to~\cite{SilvermanGCDoverFF}
for elliptic curves over~$\FF_q(T)$, as given in the following
conjecture.
\begin{conjecture}
\label{conjecture:gcdonECoverFF}
Let~$\FF_q$ be a finite field of characteristic~$p$, let~$E/\FF_q(T)$
be an elliptic curve, and let~$P,Q\in E(\FF_q(T))$ be nontorsion
points. Then there is a constant $c=c(q,E,P,Q)>0$ so that
\begin{equation}
\label{equation:gcdonECoverFF}
\deg\GCD(nP,nQ) \ge cn
\qquad\text{for infinitely many $n\ge1$ with $p\notdivide n$.}
\end{equation}
\end{conjecture}
\begin{remark}
It is tempting to conjecture a lower bound of the form~$cn^2$, since
the only obvious upper bound comes from $\deg D_{nP}\gg\ll n^2$, but
there is really no evidence either for or against the stronger bound.
\end{remark}
\begin{remark}
It is easy to prove~\eqref{equation:gcdonECoverFF} if one
allows~$p$ to divide~$n$. To see this, factor $[p]=\f\circ\fhat:E\to
E$, where $\f:E^{(p)}\to E$ is the Frobenius map and~$\fhat$ its dual.
Let~$\Obar$ denote, as usual, the zero divisor on a model of~$E$
over~$\PP^1$, and let~$\Obar'$ similarly denote the zero divisor on a
model of~$E^{(p)}$. Then $\f^*(\Obar)=p\Obar'$ and
$\fhat^*(\Obar')=\Obar+D$ for some effective divisor~$D$, which allows
us to estimate
\begin{align*}
\deg\GCD(p^iP,p^iQ)
&=\deg\GCD\left(\s_P^*\circ\left.{\fhat^i}\right.^*\circ{\f^i}^*(\Obar),
\s_Q^*\circ\left.{\fhat^i}\right.^*\circ{\f^i}^*(\Obar)\right) \\
&= p^i\deg\GCD\left(\s_P^*\circ\left.{\fhat^i}\right.^*(\Obar'),
\s_Q^*\circ\left.{\fhat^i}\right.^*(\Obar')\right), \\
&\ge p^i\deg\GCD\left(\s_P^*(\Obar),\s_Q^*(\Obar)\right).
\end{align*}
Hence
\[
\deg\GCD(nP,nQ) \ge n\cdot\deg\GCD(P,Q)
\quad\text{for all $n=p^i$, $i=1,2,3,\ldots$.}
\]
\end{remark}
We prove a strong form of Conjecture~\ref{conjecture:gcdonECoverFF}
for elliptic curves with constant $j$-invariant.
\begin{theorem}
\label{theorem:gcdonECoverFFconstantj}
Let~$\FF_q$ be a finite field of characteristic~$p\ge5$,
let $E/\FF_q(T)$ be an elliptic curve, let~$P,Q\in E(\FF_q(T))$ be
nontorsion points, and suppose that $j(E)\in\FF_q$. Then
\begin{multline*}
\deg\GCD(nP,nQ) \ge n + O(\sqrt{n})\\
\text{for infinitely many $n\ge1$ with $p\notdivide n$,}
\end{multline*}
where the big-$O$ constant depends only on~$E/\FF_q(T)$.
\end{theorem}
\begin{proof}
For the moment, we take~$E/\FF_q(T)$ to be any elliptic curve, not
necessarily with constant~$j$ invariant, and we fix a (minimal)
Weierstrass equation for~$E$. For each integer $N\ge1$, let
\[
S_{q,N} = \{\pi\in\FF_q[T] : \text{$\pi$ is monic, irreducible, and
$\deg\pi=N$}\}.
\]
Given any $\pi\in S_{q,N}$, we reduce~$E$ modulo~$\pi$ to obtain
an elliptic curve $\Etilde_\pi$ defined over the finite field
$\FF_\pi=\FF_q[T]/(\pi)$. The residue fields~$\FF_\pi\cong\FF_{q^N}$
associated to the various~$\pi$ are all isomorphic, but the elliptic
curves~$\Etilde_\pi$ for different primes need not (and generally will
not) be isomorphic. The Hasse estimate~\cite[V.1.1]{AEC} says that
\begin{equation}
\label{equation:hasseestimate}
n_\pi(E) =\#\Etilde_\pi(\FF_\pi)=q^N + 1 - a_\pi(E)
\quad\text{with $|a_\pi(E)|\le2q^{N/2}$.}
\end{equation}
\par
Suppose now that $j(E)\in\FF_q$. For simplicity, we assume that
$j(E)\ne0,1728$. The other two cases, which can be handled similarly,
will be left for the reader. This means that there is an elliptic
curve $E'/\FF_q$ so that~$E$ is a quadratic twist of~$E'$.
More prosaically, if~$E'$ is given by a Weierstrass equation
$y^2=x^3+ax+b$ with $a,b,\in\FF_q^*$, then~$E$ has a Weierstrass
equation~(cf.~\cite[X~\S5]{AEC})
\[
E : y^2=x^3+\d^2ax+\d^3b
\qquad\text{for some squarefree $\d\in\FF_q[T]$.}
\]
Replacing~$a,b$ by~$r^2a,r^3b$ and~$\d(T)$ by $r^{-1}\d(T)$ for an
appropriate~$r\in\FF_q^*$, we may assume that~$\d(T)$ is monic.
For now, we assume that~$\d(T)\ne1$, so~$E$ is a nontrivial
twist of~$E'$.
\par
For any~$\pi\in S_{q,N}$ with $\pi\notdivide\d$, the
curve~$\Etilde_\pi/\FF_\pi$ is isomorphic over~$\FF_\pi$ to
either~$E'/\FF_{q^N}$ or to the unique quadratic twist
of~$E'/\FF_{q^N}$. More precisely,~$\Etilde_\pi/\FF_\pi$ is
isomorphic over~$\FF_\pi$ to~$E'/\FF_{q^N}$ if~$\d$ is a square
in~$\FF_\pi$ and it is isomorphic to the twist if~$\d$ is not
a square in~$\FF_\pi$. It follows from this and from the standard proof
of~\eqref{equation:hasseestimate} using the action of Galois on
the Tate module~\cite[V.1.1]{AEC} that
\[
a_\pi(E) = \LS{\d}{\pi}a_N(E'),
\]
where $a_N(E')=q^N+1-\#E'(\FF_{q^N})$ and where $\LS{\d}{\pi}$ is the
Legendre symbol. We divide the set of primes~$S_{q,N}$ into two
subsets,
\begin{align*}
S_{q,N}^+(\d) &= \left\{ \pi\in S_{q,N} : \LS{\d}{\pi}=+1 \right\}, \\
S_{q,N}^-(\d) &= \left\{ \pi\in S_{q,N} : \LS{\d}{\pi}=-1 \right\}.
\end{align*}
Then
\[
n_\pi(E) = \begin{cases}
q^N + 1 - a_N(E')&\text{for all $\pi\in S_{q,N}^+$,} \\
q^N + 1 + a_N(E')&\text{for all $\pi\in S_{q,N}^-$.} \\
\end{cases}
\]
\par
For a fixed~$\d$, the (quadratic) reciprocity law
for~$\FF_q[T]$~\cite[Theorem~3.5]{RosenFF} says that
\[
\LS{\d}{\pi} = (-1)^{\frac{q-1}{2}\cdot N\cdot\deg(\d)}\LS{\pi}{\d}.
\]
Notice that the power of~$-1$ depends only on the degree of~$\pi$. It
follows that half the possible congruence classes for~$\pi$
modulo~$\d$ yield~$\LS{\d}{\pi}=+1$ and the other half
yield~$\LS{\d}{\pi}=-1$. Now Dirichlet's theorem
\cite[Theorem~4.8]{RosenFF} implies that
\[
\#S_{q,N}^+ = \frac{q^N}{2N} + O\left(\frac{q^{N/2}}{N}\right)
\qquad\text{and}\qquad
\#S_{q,N}^- = \frac{q^N}{2N} + O\left(\frac{q^{N/2}}{N}\right).
\]
\par
Let $n=q^N+1-a_N(E')$. Then~$n=n_\pi(E)$ for every $\pi\in S_{q,N}^+$,
so~$n$ annihilates~$\Etilde_\pi(\FF_\pi)$, and hence $D_{nP}$ is
divisible by all of these primes. Since the same is true of~$nQ$, we
obtain the lower bound
\begin{multline*}
\deg\GCD(nP,nQ)
\ge \sum_{\pi\in S_{q,N}^+} \deg(\pi) \\
=\#S_{q,N}^+\cdot N
= \frac{1}{2}q^N + O(q^{N/2})
= n + O(n^{1/2}).
\end{multline*}
Similarly, if $n=q^N+1-a_N(E')$, then the same argument using
the primes $\pi\in S_{q,N}^-$ yields the same lower bound.
\par
To recapitulate, we have proven that
\begin{multline}
\deg\GCD(nP,nQ)\ge n+O(n^{1/2}) \\
\text{for all $n=q^N+1\pm a_N(E')$ with $N=1,2,3,\ldots$,}
\label{equation:FFlowerbound}
\end{multline}
where we may take either sign. This estimate is exactly the lower
bound that we are trying to prove, subject to the additional constraint
that we want~$n$ to be relatively prime to~$p$. However, it is clear
that at least one of the numbers \text{$q^N+1+a_N(E')$}
and \text{$q^N+1-a_N(E')$} is prime to~$p$, since otherwise~$p$ would
divide their sum, and hence~$p=2$, contrary to assumption.
Therefore~\eqref{equation:FFlowerbound} holds for infinitely many values
of~$n$ with~$p\notdivide n$, which completes the proof of
Theorem~\ref{theorem:gcdonECoverFFconstantj}.
\par
It remains to consider that case that~$E$ is a trivial twist of~$E'$,
i.e., the case that~$E$ is~$\FF_q(T)$-isomorphic to a cruve defined
over~$\FF_q$. But then $E(\FF_q(T))=E'(\FF_q(T))=E'(\FF_q)$, since a
nonconstant point in~$E'(\FF_q(T))$ would correspond to a nonconstant
morphism $\PP^1\to E'$. But the group~$E'(\FF_q)$ is finite,
so~$E(\FF_q(T))$ has no nontorsion points and the statement of the
theorem is vacuously true.
\end{proof}
\begin{remark}
We continue with the notation from the proof of
Theorem~\ref{theorem:gcdonECoverFFconstantj}. It is well known that
$a_N(E')=\a^N+\b^N$, where~$\a$ and~$\b$ are the complex roots
of $X^2-a_1(E')X+q$. Thus for any particular~$E'$, one may find
more precise information about the values of~$n$ being used in the
statement of the theorem.
\end{remark} | 198,237 |
TITLE: sequential $US$-space
QUESTION [1 upvotes]: A topological space is called a US-space provided that
each convergent sequence has a unique limit.
the notion of strongly KC-spaces, that is, those spaces
in which every countably compact subset is closed.
If $(X,\tau )$ is a sequential $US$-space, then $X $ is strongly $ KC$.
P r o o f . Let $A$ be a countably compact subset of $X$. If $A $ is not closed, since
$(X,\tau )$ is a sequential space, there is some $x \in \operatorname{cl}(A) - A$ and a sequence $ \{x_{n} \}_{n \in \omega} ⊂$
A convergent to $x$. Since $A$ is countably compact, $ \{x_{n} : n ∈ \omega \}$ must have an
accumulation point $y$ in $A$ and so $ \{ x_{n} : n ∈ \omega \} ∪ \{x \} $is not closed in $X$. Again since
$X$ is sequential, it follows that there is some sequence $ \{x_{n_{k}} : k ∈ \omega \} ⊂ \{x_{n} : n ∈ \omega \}$
which converges to $x′$ and $x′ \not\in \{x_{n} \} ∪ \{x\}$. Then $ \{ x_{n_{k}} : k ∈ \omega \} $ must also converge
to $x$, contradicting the definition of$ US$-space. Therefore, $X $is strongly $KC$.
Why $ \{ x_{n} : n ∈ \omega \} ∪ \{x \} $is not closed in $X$?
REPLY [3 votes]: Let $S=\{x_n:n\in\omega\}\cup\{x\}$. I don't immediately see why $S$ is not closed, but we don't need this to make the argument work. Suppose that $S$ is closed; $y$ is an accumulation point of $\{x_n:n\in\omega\}$, so $y\in\operatorname{cl}\{x_n:n\in\omega\}\subseteq\operatorname{cl}S$. And $y\ne x$ (since $x\notin A$), so $y=x_m$ for some $m\in\omega$.
Let $S'=\{x_n:n>m\}\cup\{x\}$; clearly $S'$ is not closed, since $x_m\in(\operatorname{cl}S')\setminus S$, and $\langle x_n:n>m\rangle$ still converges to $x$. Thus, there is some $x'\in(\operatorname{cl}S')\setminus S'$ and some subsequence $\langle x_{n_k}:k\in\omega\rangle$ of $\langle x_n:n>m\rangle$ converging to $x'$, and the proof can be completed as before. | 205,517 |
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\chapter[Unstable $G$-global homotopy theory]{Unstable \texorpdfstring{\for{toc}{$G$}\except{toc}{$\bm G$}}{G}-global homotopy theory}\label{chapter:unstable}
In this chapter we will introduce several models of \emph{unstable $G$-global homotopy theory}, generalizing Schwede's unstable global homotopy theory \cite[Chapter~1]{schwede-book}. These models are already geared towards the study of $G$-global algebraic $K$-theory, and in particular, while we will be ultimately interested in \emph{stable} $G$-global homotopy theory and in the theory of $G$-global infinite loop spaces, the comparisons proven here will be instrumental in establishing results on the latter.
\section{Equivariant homotopy theory for monoids}\label{section:equiv-models}
\index{G-equivariant homotopy theory@$G$-equivariant homotopy theory}
Let $G$ be a discrete group. In unstable $G$-equivariant homotopy theory one is usually interested in $G$-spaces or $G$-simplicial sets up to so-called \emph{(genuine) $G$-weak equivalences},\index{G-weak-equivalence@$G$-weak equivalence|textbf}\index{genuine G-weak equivalence@genuine $G$-weak equivalence|seeonly{$G$-weak equivalence}} i.e.~$G$-equivariant maps that induce weak homotopy equivalences on $H$-fixed points for all subgroups $H\subset G$. More generally, one can consider the \emph{$\mathcal F$-weak equivalences}\index{F-weak equivalence@$\mathcal F$-weak equivalence|textbf} for any collection $\mathcal F$\nomenclature[aF]{$\mathcal F$}{generic collection of subgroups} of subgroups of $G$, i.e.~those maps that induce weak homotopy equivalences on $H$-fixed points for all $H\in\mathcal F$. If $\mathcal F=\mathcal A\ell\ell$\nomenclature[aAll]{$\mathcal A\ell\ell$}{collection of all subgroups} is the collection of all subgroups, this recovers the previous notion; at the other extreme, if $\mathcal F$ consists only of the trivial subgroup, then the $\mathcal F$-weak equivalences are precisely the underlying weak homotopy equivalences. We will at several points encounter \emph{proper equivariant homotopy theory}, where one considers the class $\mathcal Fin$\nomenclature[aFin]{$\mathcal Fin$}{collection of finite subgroups} of \emph{finite} subgroups of $G$; of course, $\mathcal Fin=\mathcal A\ell\ell$ if $G$ is finite, but for infinite $G$ these differ.
\index{G-equivariant homotopy theory@$G$-equivariant homotopy theory!proper|seeonly{proper $G$-equivariant homotopy theory}}
\index{proper G-equivariant homotopy theory@proper $G$-equivariant homotopy theory|textbf}
Our first approach to unstable $G$-global homotopy theory will rely on a generalization of this to actions of \emph{simplicial monoids}, and this section is devoted to generalizing several basic results from unstable equivariant homotopy theory to this context. In particular, we will construct equivariant model structures, prove a version of Elmendorf's Theorem, and discuss functoriality with respect to monoid homomorphisms.
\subsection{Equivariant model structures} There are several approaches to the construction of the usual equivariant model structures for group actions, for example the criteria of Dwyer and Kan \cite{dwyer-kan-equivariant} or Stephan \cite{cellular}. In this subsection, we will use the work of Dwyer and Kan to prove:
\begin{prop}\label{prop:equiv-model-structure}
Let $M$ be a simplicial monoid and let $\mathcal F$ be a collection of finite subgroups of $M_0$. Then the category $\cat{$\bm M$-SSet}$ of $M$-objects in $\cat{SSet}$ admits a unique model structure such that a map $f\colon X\to Y$ is a weak equivalence or fibration if and only if $f^H\colon X^H\to Y^H$ is a weak homotopy equivalence or Kan fibration, respectively, for all $H\in\mathcal F$.
We call this the \emph{$\mathcal F$-model structure}\index{F-model structure@$\mathcal F$-model structure|textbf}\index{F-model structure@$\mathcal F$-model structure!injective|seeonly{injective $\mathcal F$-model structure}} (or, if $\mathcal F$ is clear from the context, the \emph{$M$-equivariant model structure},\index{M-equivariant model structure@$M$-equivariant model structure|seeonly {$\mathcal F$-model structure}} or even simply the \emph{equivariant model structure})\index{equivariant model structure|seeonly{$\mathcal F$-equivariant model structure}} and its weak equivalences the \emph{$\mathcal F$-weak equivalences}\index{F-weak equivalence@$\mathcal F$-weak equivalence|textbf} (or simply \emph{$M$-weak equivalences};\index{M-weak equivalence@$M$-weak equivalence|seeonly {$\mathcal F$-weak equivalence}} \emph{equivariant weak equivalences}\index{equivariant weak equivalence|seeonly{$\mathcal F$-weak equivalence}}). It is simplicial, combinatorial, and proper. A possible set of generating cofibrations is given by
\begin{equation*}
I=\{ M/H\times\del\Delta^n\hookrightarrow M/H\times\Delta^n : n\ge0,\text{ }H\in\mathcal F\},
\end{equation*}
and a possible set of generating acyclic cofibrations is given by
\begin{equation*}
J=\{ M/H\times\Lambda^n_k\hookrightarrow M/H\times\Delta^n : 0\le k\le n,\text{ }H\in\mathcal F\}.
\end{equation*}
Moreover, filtered colimits are homotopical in $\cat{$\bm M$-SSet}$.
\end{prop}
The finiteness condition on the subgroups $H$ is not necessary for the existence of the model structure, but it guarantees that filtered colimits are fully homotopical, which will simplify several arguments later. It is crucial for our argument that we only test weak equivalences and fibrations with respect to discrete sub\emph{groups}.
The proof of Proposition~\ref{prop:equiv-model-structure} will be given below. However, we already note:
\begin{lemma}\label{lemma:equivariant-weak-equivalences-prod-homotopical}
The weak equivalences of the above model structure are closed under finite products and small (i.e.~set-indexed) coproducts.
\begin{proof}
As fixed points commute with products and coproducts, this is immediate from the corresponding statement for ordinary simplicial sets.
\end{proof}
\end{lemma}
In order to conveniently formulate the result of Dwyer and Kan, we introduce the following notion:
\begin{defi}\label{defi:cellular-family}\index{cellular family|textbf}
Let $\mathscr D$ be a category enriched and tensored over $\cat{SSet}$. We call a family $(\Phi_i)_{i\in I}$ (for some set $I$) of enriched functors $\mathscr D\to\cat{SSet}$ \emph{cellular} if the following conditions are satisfied:
\begin{enumerate}
\item Each $\Phi_i$ preserves filtered colimits.
\item Each $\Phi_i$ is corepresentable in the enriched sense.
\item For each $i,j\in I$, $n\ge 0$, and some (hence any) $X_i$ corepresenting $\Phi_i$, the functor $\Phi_j$ sends pushouts along $\incl\otimes X_i\colon\del\Delta^n\otimes X_i\to\Delta^n\otimes X_i$ to homotopy pushouts in $\cat{SSet}$ (with respect to the weak homotopy equivalences).\label{item:cf-gluing}
\end{enumerate}
\end{defi}
By Proposition~\ref{prop:U-sharp-closure} (or an easy direct argument), $\Phi_j$ more generally sends pushouts along $f\otimes X_i\colon K\otimes X_i\to L\otimes X_i$ for any cofibration $f\colon K\to L$ of simplicial sets to homotopy pushouts. Thus, the above conditions immediately imply that the corepresenting elements $\{X_i:i\in I\}$ form a \emph{set of orbits} in the sense of \cite[2.1]{dwyer-kan-equivariant}. Our terminology is instead motivated by \cite[Proposition~2.6]{cellular}.
\begin{thm}[Dwyer \& Kan]\label{thm:dk-equivariant}
Let $\mathscr D$ be a complete and cocomplete category that is in addition enriched, tensored, and cotensored over $\cat{SSet}$. Let $(\Phi_i)_{i\in I}$ be a cellular family, and fix for each $i\in I$ an $X_i\in\mathscr D$ corepresenting $\Phi_i$.
Then there is a unique model structure on $\mathscr D$ in which a map $f$ is a weak equivalence or fibration if and only if $\Phi_i(f)$ is a weak equivalence or fibration, respectively, in $\cat{SSet}$ for each $i\in I$. This model structure is simplicial and moreover cofibrantly generated with generating cofibrations
\begin{equation*}
\{\incl\otimes X_i\colon \del\Delta^n\otimes X_i\to\Delta^n\otimes X_i : n\ge 0,i\in I\}
\end{equation*}
and generating acyclic cofibrations
\begin{equation*}
\{\incl\otimes X_i\colon \Lambda^n_k\otimes X_i\to \Delta^n\otimes X_i : 0\le k\le n,i\in I\}.
\end{equation*}
Finally, filtered colimits in $\mathscr D$ are homotopical.
\begin{proof}
\cite[Theorem~2.2]{dwyer-kan-equivariant} and its proof show that the model structure exists and that it is cofibrantly generated by the above sets of (acyclic) cofibrations. The final statement follows immediately from the corresponding statement for $\cat{SSet}$ as each $\Phi_i$ preserves filtered colimits.
\end{proof}
\end{thm}
Next, we want to establish some additional properties of this model structure:
\begin{prop}\label{prop:dk-equivariant-addendum}
In the above situation, $\mathscr D$ is a proper model category. A commutative square in $\mathscr D$ is a homotopy pushout if and only if its image under $\Phi_i$ is a homotopy pushout in $\cat{SSet}$ for each $i\in I$.
\end{prop}
Here we use the term \emph{homotopy pushout}\index{homotopy pushout|textbf} for the dual of what Bousfield and Friedlander called a \emph{homotopy fibre square} in \cite[Appendix~A.2]{bousfield-friedlander}.
To prove the proposition, it will be convenient to realize the above model structure on $\mathscr D$ as a \emph{transferred model structure}:
\begin{defi}\index{transferred model structure|textbf}
Let $\mathscr D$ be a complete and cocomplete category, let $\mathscr C$ be a model category, and let $F\colon\mathscr C\rightleftarrows\mathscr D :\!U$ be an (ordinary) adjunction. The \emph{model structure transferred along $F\dashv U$} on $\mathscr D$ is the (unique if it exists) model structure where a morphism $f$ is a weak equivalence or fibration if and only if $Uf$ is.
\end{defi}
Transferred model structures will play a role at several points in this monograph and we recall some basic facts about them in Appendix~\ref{appendix:transfer}.
\begin{constr}\nomenclature[aPhi1]{$\Phi$}{right adjoint in Elmendorf's Theorem}
Fix for each $i\in I$ an $X_i\in\mathscr D$ corepresenting $\Phi_i$, and let $\cat{O}_{\Phi_\bullet}\subset\mathscr D$ be the full simplicial subcategory spanned by the $X_i$'s. We define the enriched functor $\Phi\colon\mathscr D\to\FUN(\textbf{O}_{\Phi_\bullet}^\op,\cat{SSet})$ as the composition
\begin{equation*}
\mathscr D\xrightarrow{\textup{enriched Yoneda}}\FUN(\mathscr D^\op,\cat{SSet})\xrightarrow{\textup{restriction}}\FUN(\textbf{O}_{\Phi_\bullet}^\op, \cat{SSet}),
\end{equation*}
where $\FUN$\nomenclature[aFun2]{$\FUN$}{enriched category of enriched functors} denotes the enriched category of simplicially enriched functors.
In other words, $\Phi(X)(Y)=\Maps_{\mathscr D}(Y,X)$ with the obvious functoriality. In particular, if $Y\in\textbf{O}_{\Phi_\bullet}$ corepresents $\Phi_i$, then we have an enriched isomorphism
\begin{equation}\label{eq:Phi-level}
\ev_Y\circ \Phi\cong \Phi_i.
\end{equation}
\end{constr}
\begin{constr}
It is well-known---see e.g.~\cite[Theorem~4.51]{enriched-presheaves} together with \cite[Theorem~3.73${}^\op$]{enriched-presheaves} for a statement in much greater generality---that for any essentially small simplicial category $T$, any cocomplete category $\mathscr D$ enriched and tensored over $\cat{SSet}$, and any simplicially enriched functor $F\colon T\to\mathscr D$, there exists an induced simplicial adjunction $\FUN(T^\op,\cat{SSet})\rightleftarrows\mathscr D$ with right adjoint $R$ given by $R(Y)(t)=\Maps(F(t),Y)$ for all $t\in T$, $Y\in\mathscr D$ with the obvious functoriality in each variable.
The left adjoint $L$ can be computed by the simplicially enriched coend
\begin{equation*}
L(X)=\int^{t\in T} X(t)\otimes F(t)
\end{equation*}
for any enriched presheaf $X$, together with the evident functoriality.
\end{constr}
In particular, applying this to the inclusion of $\textbf{O}_{\Phi_\bullet}$ yields:
\begin{cor}\nomenclature[aLambda]{$\Lambda$}{left adjoint in Elemendorf's Theorem}
The simplicial functor $\Phi$ has a simplicial left adjoint $\Lambda$.\qed
\end{cor}
By $(\ref{eq:Phi-level})$, the model structure from Theorem~\ref{thm:dk-equivariant} is transferred along $\Lambda\dashv\Phi$ from the projective model structure, allowing us to use the general results from Appendix~\ref{appendix:transfer}.
\begin{proof}[Proof of Proposition~\ref{prop:dk-equivariant-addendum}]
Right properness of the model structure on $\mathscr D$ is immediate from Lemma~\ref{lemma:transferred-properties}-$(\ref{item:tpr-proper})$.
For left properness and the characterization of homotopy pushouts, we observe that homotopy pushouts in $\FUN(\cat{O}_{\Phi_\bullet}^\op,\cat{SSet})$ can be checked levelwise, so that it is enough by Lemma~\ref{lemma:U-pushout-preserve-reflect} that $\Phi$ sends pushouts along cofibrations to homotopy pushouts. By Proposition~\ref{prop:U-sharp-closure} it suffices to check this for a set of generating cofibrations, which is then an instance of Condition~$(\ref{item:cf-gluing})$ of Definition~\ref{defi:cellular-family}.
\end{proof}
We now want to apply this to construct the $M$-equivariant model structure, for which we use the following well-known observation, cf.~e.g.~\cite[1.2]{dwyer-kan-equivariant} for the case of topological groups acting on spaces or \cite[Example~2.14]{cellular} for discrete groups acting on simplicial sets.
\begin{lemma}\label{lemma:fixed-points-cellular}
Let $M$ be a simplicial monoid and let $\mathcal F$ be a collection of finite subgroups of $M_0$. For any $H\in\mathcal F$, the enriched functor $(\blank)^H\colon\cat{$\bm M$-SSet}\to\cat{SSet}$ is corepresented by $M/H$ via evaluation at the class of $1\in M$. It preserves filtered colimits and pushouts along underlying cofibrations. In particular, the family $\big((\blank)^H\big)_{H\in\mathcal F}$ is cellular.
\begin{proof}
The corepresentability statement is obvious. As limits and colimits in $\cat{$\bm M$-SSet}$ are created in $\cat{SSet}$, filtered colimits commute with all finite limits, hence in particular with fixed points with respect to finite groups. Similarly, one reduces the statement about pushouts to the corresponding statement in $\cat{Set}$, which is easy.
As each of the maps $\del\Delta^n\times M/H\hookrightarrow\Delta^n\times M/H$ is in particular an underlying cofibration, the above immediately implies that the family of fixed point functors is cellular, finishing the proof.
\end{proof}
\end{lemma}
\begin{proof}[Proof of Proposition~\ref{prop:equiv-model-structure}]
By the previous lemma, we may apply Theorem~\ref{thm:dk-equivariant} and Proposition~\ref{prop:dk-equivariant-addendum}, so it only remains to show that this model category is combinatorial. But we know it is cofibrantly generated, and as an ordinary category $\cat{$\bm M$-SSet}$ is just the category of enriched functors of the category $BM$ with one object and endomorphism space $M$ into $\cat{SSet}$,\nomenclature[aBM]{$BM$}{(enriched) category with one object and hom set/mapping space $M$} hence locally presentable.
\end{proof}
\subsection{Elmendorf's Theorem}
The classical Elmendorf Theorem\index{Elmendorf's Theorem!classical|textbf} \cite{elmendorf} explains how $G$-equivariant homotopy theory (with respect to a fixed topological or simplicial group $G$) can be modelled in terms of fixed point data. Dwyer and Kan \cite[Theorem~3.1]{dwyer-kan-equivariant} provided a generalization of this to the above context:
\begin{thm}[Dwyer \& Kan]
If $(\Phi_i)_{i\in I}$ is any cellular family on $\mathscr D$, then the simplicial adjunction
\begin{equation*}
\Lambda\colon\FUN(\cat{O}_{\Phi_\bullet}^\op,\cat{SSet})\rightleftarrows\mathscr D :\!\Phi
\end{equation*}
is a Quillen equivalence for the projective model structure on the source.\qed
\end{thm}
If $M$ is a simplicial monoid and $\mathcal F$ is a collection of finite subgroups of $M_0$, then we write $\cat{O}_{\mathcal F}$ (or simply $\cat{O}_M$ if $\mathcal F$ is clear from the context)\nomenclature[aOFOM]{$\cat{O}_{\mathcal F}$ (also $\cat{O}_{M}$)}{(simplicially enriched) orbit category} for the full subcategory of $\cat{$\bm M$-SSet}$ spanned by the $M/H$ for $H\in\mathcal F$. The above then specializes to:
\begin{cor}\label{cor:elmendorf}\index{Elmendorf's Theorem!for monoids|textbf}
The simplicial adjunction
\begin{equation*}
\Lambda\colon\FUN(\cat{O}_{\mathcal F}^\op,\cat{SSet})\rightleftarrows\cat{$\bm M$-SSet}_{\textup{$\mathcal F$-equivariant}} :\!\Phi
\end{equation*}
is a Quillen equivalence for any simplicial monoid $M$ and any collection $\mathcal F$ of finite subgroups of $M_0$. Here $\Phi(X)(M/H)=\Maps_M(M/H,X)\cong X^H$ with the evident functoriality.\qed
\end{cor}
\subsection{Injective model structures}
For a group $G$, it is an easy observation that the cofibrations of the $\mathcal A\ell\ell$-model structure on $\cat{$\bm G$-SSet}$ are precisely the underlying cofibrations. In the case of a general collection $\mathcal F$, this will of course no longer be true---for example, if $\mathcal F$ is closed under subconjugates, then we can explicitly characterize the cofibrations as those injections such that all simplices not in the image have isotropy in $\mathcal F$, see e.g.~\cite[Proposition~2.16]{cellular}.
However, it is well-known that there is still a model structure with the same weak equivalences and whose cofibrations are the underlying cofibrations called the \emph{mixed} or \emph{injective $\mathcal F$-model structure}, see e.g.~\cite[Proposition~1.3]{shipley-mixed} for a pointed version. We will now construct an analogue of this in our situation, which will use:
\begin{cor}\label{cor:homotopy-pushout-M-SSet}
Pushouts in $\cat{$\bm M$-SSet}$ along underlying cofibrations are homotopy pushouts (for any collection $\mathcal F$ of finite subgroups of $M_0$).
\begin{proof}
By Lemma~\ref{lemma:fixed-points-cellular}, each $(\blank)^H$ sends such a pushout to a pushout again. As taking fixed points moreover obviously preserves underlying cofibrations, this is then a homotopy pushout in $\cat{SSet}$, so the claim follows from Proposition~\ref{prop:dk-equivariant-addendum}.
\end{proof}
\end{cor}
\begin{cor}\label{cor:equivariant-injective-model-structure}\index{injective F-model structure@injective $\mathcal F$-model structure|textbf}\index{equivariant injective model structure|seeonly {injective $\mathcal F$-model structure}}
Let $M$ be any simplicial monoid and let $\mathcal F$ be a collection of finite subgroups of $M_0$. Then there is a unique model structure on $\cat{$\bm{M}$-SSet}$ whose weak equvialences are the $\mathcal F$-weak equivalences and whose cofibrations are the injective cofibrations (i.e.~levelwise injections). We call this the \emph{injective $\mathcal F$-model structure} (or \emph{equivariant injective model structure} if $\mathcal F$ is clear from the context). It is combinatorial, simplicial, proper, and filtered colimits in it are homotopical.
\begin{proof}
As an ordinary category, $\cat{$\bm{M}$-SSet}$ is just a category of enriched functors into $\cat{SSet}$, and hence the usual injective model structure (which has weak equivalences the underlying non-equivariant weak homotopy equivalences) on it exists and is combinatorial. On the other hand, the $\mathcal F$-equivariant model structure is combinatorial, and its weak equivalences are stable under filtered colimits as well as pushouts along underlying cofibrations by the previous corollary.
Thus, we can apply Corollary~\ref{cor:mix-model-structures} to combine the cofibrations of the injective model structure with the $\mathcal F$-weak equivalences, yielding the desired model structure and proving that it is combinatorial, proper, and that filtered colimits in it are homotopical. It only remains to prove that it is simplicial, which means verifying the Pushout Product Axiom.\index{Pushout Product Axiom!for simplicial model categories} So let $i\colon K\to L$ be a cofibration of simplicial sets and let $f\colon X\to Y$ be an underlying cofibration of $M$-simplicial sets. Because $\cat{SSet}$ is a simplicial model category, we immediately see that the pushout product map\nomenclature[zo]{$\ppo$}{pushout product}
\begin{equation*}
\begin{tikzcd}
K\times X\arrow[d, "i\times X"']\arrow[r, "K\times f"] & K\times Y\arrow[d]\arrow[rdd, "i\times Y", bend left=15pt]\\
L\times X\arrow[r]\arrow[rrd, "L\times f"', bend right=10pt] & (K\times Y)\amalg_{K\times X}(L\times X)\arrow[rd, "i\ppo f" description, dashed]\\
& & L\times Y
\end{tikzcd}
\end{equation*}
is again an underlying cofibration. It only remains to prove that this is a weak equivalence provided that either $i$ or $f$ is. For this we observe that the equivariant weak equivalences are stable under finite products by Lemma~\ref{lemma:equivariant-weak-equivalences-prod-homotopical}; moreover, a weak homotopy equivalence between simplicial sets with trivial $M$-action is already an equivariant weak equivalence. Hence, if $i$ is an acyclic cofibrations of simplicial sets, then the cofibration $i\times X$ is actually acyclic in the equivariant injective model structure, and so is $i\times Y$. Moreover, $K\times Y\to(K\times Y)\amalg_{K\times X}(L\times X)$ is also an acyclic cofibration as the pushout of an acyclic cofibration. It follows by $2$-out-of-$3$ that also $i\ppo f$ is an equivariant weak equivalence. The argument for the case that $f$ is an acyclic cofibration is analogous, and this finishes the proof.
\end{proof}
\end{cor}
\subsection{Functoriality} We will now explain how the above model structures for different monoids or collections of subgroups relate to each other.
\subsubsection{Change of monoid}\index{functoriality in homomorphisms!for M-SSet@for $\cat{$\bm{M}$-SSet}$|(}
If $\alpha\colon H\to G$ is any group homomorphism, then $\alpha^*$\nomenclature[aalphaaupperstar]{$\alpha^*$}{restriction along $\alpha$} obviously preserves cofibrations, fibrations, and weak equivalences of the $\mathcal A\ell\ell$-model structures. It follows immediately that the simplicial adjunctions $\alpha_!\dashv\alpha^*$\nomenclature[aalphalowershriek]{{$\alpha_{"!}$}}{left adjoint to restriction} and $\alpha^*\dashv\alpha_*$\nomenclature[aalphalowerstar]{{$\alpha_*$}}{right adjoint to restriction} are Quillen adjunctions. For monoids and general $\mathcal F$ one instead has to distinguish between the usual $\mathcal F$-model structure and the injective one:
\begin{lemma}\label{lemma:alpha-shriek-projective}
Let $\alpha\colon M\to N$ be any monoid homomorphism, let $\mathcal F$ be a collection of finite subgroups of $M_0$, and let $\mathcal F'$ be a collection of finite subgroups of $N_0$ such that $\alpha(H)\in\mathcal F'$ for all $H\in\mathcal F$. Then $\alpha^*$ sends $\mathcal F'$-weak equivalences to $\mathcal F$-weak equivalences and it is part of a simplicial Quillen adjunction
\begin{equation*}
\alpha_!\colon\cat{$\bm M$-SSet}_{\textup{$\mathcal F$-equivariant}}\rightleftarrows\cat{$\bm N$-SSet}_{\textup{$\mathcal F'$-equivariant}} :\!\alpha^*.
\end{equation*}
\begin{proof}
If $f$ is any morphism in $\cat{$\bm N$-SSet}$ and $H\subset M_0$ is any subgroup, then $(\alpha^*f)^H=f^{\alpha(H)}$. Thus, the claim follows immediately from the definition of the weak equivalences and fibrations of the equivariant model structures.
\end{proof}
\end{lemma}
\begin{lemma}\label{lemma:alpha-star-injective}
In the situation of Lemma~\ref{lemma:alpha-shriek-projective}, also
\begin{equation*}
\alpha^*\colon\cat{$\bm N$-SSet}_{\textup{$\mathcal F'$-equivariant injective}}\rightleftarrows\cat{$\bm M$-SSet}_{\textup{$\mathcal F$-equivariant injective}} :\!\alpha_*.
\end{equation*}
is a simplicial Quillen adjunction.
\begin{proof}
We have seen in the previous lemma that $\alpha^*$ is homotopical. Moreover, it obviously preserves injective cofibrations.
\end{proof}
\end{lemma}
The questions when $\alpha_!$ is left Quillen for the injective model structures or when $\alpha_*$ is right Quillen for the usual model structures are more complicated. The following propositions will cover the cases of interest to us:
\begin{prop}\label{prop:alpha-shriek-injective}
Let $\alpha\colon H\to G$ be an \emph{injective} homomorphism of discrete groups and let $M$ be any simplicial monoid. Let $\mathcal F$ be any collection of finite subgroups of $M_0\times H$ and let $\mathcal F'$ be a collection of finite subgroups of $M_0\times G$ such that the following holds: for any $K\in\mathcal F'$, $g\in G$ also $(M\times\alpha)^{-1}(gKg^{-1})\in\mathcal F$. Then
\begin{equation*}
\alpha_!\colon\cat{$\bm{(M\times H)}$-SSet}_{\textup{$\mathcal F$-equiv.~inj.}}\rightleftarrows\cat{$\bm{(M\times G)}$-SSet}_{\textup{$\mathcal F'$-equiv.~inj.}} :\!\alpha^*=(M\times\alpha)^*
\end{equation*}
is a simplical Quillen adjunction; in particular, $\alpha_!$ is homotopical.
\begin{proof}
While we have formulated the result above in the way we later want to apply it, it will be more convenient for the proof to switch the order in which we write the actions, i.e.~to work with $(H\times M)$- and $(G\times M)$-simplicial sets.
We may assume without loss of generality that $H$ is a subgroup of $G$ and $\alpha$ is its inclusion. Then $G\times_H\blank$ (with the $M$-action pulled through via enriched functoriality) is a model for $\alpha_!$; if $X$ is any $H$-simplicial set, then the $n$-simplices of $G\times_HX$ are of the form $[g,x]$ with $g\in G$ and $x\in X_n$ where $[g,x]=[g',x']$ if and only if there exists an $h\in H$ with $g'=gh$ and $x=h.x'$.
Now let $K\subset G\times M_0$ be any subgroup. We set $S\mathrel{:=}\{g\in G: g^{-1}Kg\subset H\times M_0\}$ and observe that this is a right $H$-subset of $G$. We fix representatives $(s_i)_{i\in I}$ of the orbits. If now $X$ is any $(H\times M)$-simplicial set, then we define
\begin{equation*}
\iota\colon\coprod_{i\in I} X^{s_i^{-1}Ks_i} \to G\times_HX
\end{equation*}
as the map that is given on the $i$-th summand by $x\mapsto[s_i,x]$.
The following splitting ought to be well-known:
\begin{claim*}
The map $\iota$ is natural in $X$ (with respect to the evident functoriality on the left hand side) and it defines an isomorphism onto $(G\times_HX)^K$.
\begin{proof}
The naturality part is obvious. Moreover, it is clear from the choice of the $s_i$ as well as the above description of the equivalence relation that $\iota$ is injective, so that it only remains to prove that its image equals $(G\times_H X)^K$.
Indeed, assume $[g,x]$ is a $K$-fixed $n$-simplex. Then in particular $k_1g\in gH$ for any $k=(k_1,k_2)\in K$ by the above description of the equivalence relation, hence $g^{-1}k_1g\in H$ which is equivalent to $g^{-1}kg\in H\times M_0$. Letting $k$ vary, we conclude that $g\in S$, and after changing the representative if necessary we may assume that $g=s_i$ for some $i\in I$. But then
\begin{align*}
[s_i,x]&=k.[s_i,x]=[k_1s_i, k_2.x]=[s_i (s_i^{-1}k_1s_i), k_2.x]\\
&=[s_i, (s_i^{-1}k_1s_i, k_2).x]=[s_i,(s_i^{-1}ks_i).x]
\end{align*}
for any $k\in K$, and hence $x\in X^{s_i^{-1}Ks_i}$ as $H$ acts faithfully on $G$. Thus, $\im\iota$ contains all $K$-fixed points. Conversely, going through the above equation backwards shows that $[s_i,x]$ is $K$-fixed for any $(s_i^{-1}Ks_i)$-fixed $x$, i.e.~also $\im\iota\subset(G\times_HX)^K$.
\end{proof}
\end{claim*}
In particular, for $K=1$ this recovers the fact that non-equivariantly $G\times_HX$ is given as disjoint union of copies of $X$; we immediately conclude that $\alpha_!$ preserves injective cofibrations. On the other hand, if $K\in\mathcal F'$, then we conclude from the claim that for any morphism $f$ in $\cat{$\bm{(H\times M)}$-SSet}$ the map $(G\times_Hf)^K$ is conjugate to $\coprod_{i\in I} f^{s_i^{-1}Ks_i}$ for some $s_i\in G$ with $s_i^{-1}Ks_i\subset H\times M_0$ for all $i\in I$. Then by assumptions on $\mathcal F$ already $s_i^{-1}Ks_i\in\mathcal F$, so that each $f^{s_i^{-1}Ks_i}$ is a weak homotopy equivalence whenever $f$ is a $\mathcal F$-weak equivalence. As coproducts of simplicial sets are fully homotopical, we conclude that $(G\times_Hf)^K$ is a weak homotopy equivalence, and letting $K$ vary this shows $G\times_Hf$ is an $\mathcal F'$-weak equivalence.
\end{proof}
\end{prop}
\begin{prop}\label{prop:alpha-lower-star-homotopical}
In the situation of Proposition~\ref{prop:alpha-shriek-injective}, also
\begin{equation*}
\alpha^*\colon\cat{$\bm{(M\times G)}$-SSet}_{\textup{$\mathcal F'$-equivariant}}\rightleftarrows\cat{$\bm{(M\times H)}$-SSet}_{\textup{$\mathcal F$-equivariant}} :\!\alpha_*.
\end{equation*}
is a simplicial Quillen adjunction. Moreover, if the index $(G:\im\alpha)$ is finite, then $\alpha_*$ is fully homotopical.
\begin{proof}
We may again assume that $\alpha$ is the inclusion of a subgroup, so that $\alpha_*$ can be modelled as usual by $\Maps^H(G,\blank)$.
Let $K\subset M_0\times G$ be any subgroup, and let $K_2$ be its projection to $G$. We pick a system of representatives $(g_i)_{i\in I}$ of $H\backslash G/K_2$, and we let $L_i=(M\times H)\cap (g_iKg^{-1}_i)$. Similarly to the previous proposition, one checks that we have an isomorphism
$\coprod_{i\in I} (M\times H)/L_i \to (M\times G)/K$ given on summand $i$ by $[m,h]\mapsto[m,hg_i]$. Together with the canonical isomorphism $\Maps^H(G,\blank)^K\cong\Maps^{M\times H}((M\times G)/K,\blank)$ induced by the projection, this shows that for any $(M\times H)$-equivariant map $f\colon X\to Y$ the map $\alpha_*(f)^K$ is conjugate to $\prod_{i\in I}f^{L_i}$. If $K\in\mathcal F'$, then the assumptions guarantee that $L_i\in\mathcal F$, so $\alpha_*$ is obviously right Quillen. If in addition $(G:H)<\infty$, then $H\backslash G$ is finite, and hence so is $I$. As finite products in $\cat{SSet}$ are homotopical, so is $\alpha_*$ in this case.
\end{proof}
\end{prop}
\begin{rk}\index{graph subgroup|textbf}
Let $A,B$ be groups. We recall that a \emph{graph subgroup} $C\subset A\times B$ is a subgroup of the form $\{(a,\phi(a)) : a\in A'\}$ for some subgroup $A'\subset A$ and some group homomorphism $\phi\colon A'\to B$; note that this is not symmetric in $A$ and $B$. Both $A'$ and $\phi$ are uniquely determined by $C$, and we write $C\mathrel{=:}\Gamma_{A',\phi}$.\nomenclature[aGammaHphi]{$\Gamma_{H,\phi}$}{graph subgroup $\{(h,\phi(h)):h\in H\}$} A subgroup $C\subset A\times B$ is a graph subgroup if and only if $C\cap (1\times B)=1$.
If $A$ and $B$ are monoids, then we can define its graph subgroups as the graph subgroups of the maximal subgroup $\core(A\times B)$\nomenclature[acore]{$\core$}{maximal subgroup of a monoid} of $A\times B$. If $A'\subset\core(A)$ and $\phi\colon A'\to B$ is a homomorphism, then we will abbreviate $(\blank)^\phi\mathrel{:=}(\blank)^{\Gamma_{A',\phi}}$.\nomenclature[aphi]{$(\blank)^\phi$}{fixed points for the graph subgroup corresponding to $\phi$}
\end{rk}
\begin{ex}
Let $\mathcal E$ be any collection of finite subgroups of $M_0$ closed under taking subconjugates. Then the assumptions of the previous two propositions are in particular satisfied if we take $\mathcal F=\mathcal G_{\mathcal E,H}$\nomenclature[aG1]{$\mathcal G_{A,B}$}{graph subgroups of $A\times B$ for homomorphisms $A\to B$}\nomenclature[aG2]{$\mathcal G_{\mathcal E,B}$}{graph subgroups $\Gamma_{H,\phi}\subset A\times B$ with $H\in\mathcal E$} to be the collection of those graph subgroups $\Gamma_{K,\phi}$ of $M_0\times H$ with $K\in\mathcal E$, and similarly $\mathcal F'=\mathcal G_{\mathcal E,G}$.
\end{ex}
Let us consider a general homomorphism $\alpha\colon H\to G$ now. Then $(M\times\alpha)^*$ is right Quillen with respect to the $\mathcal G_{\mathcal E,H}$- and $\mathcal G_{\mathcal E,G}$-model structures for $\mathcal E$ as above, so Ken Brown's Lemma implies that $\alpha_!$ preserves weak equivalences between cofibrant objects. On the other hand, Proposition~\ref{prop:alpha-shriek-injective} says that $\alpha_!$ is fully homotopical if $\alpha$ is injective. The following proposition interpolates between these two results:
\begin{prop}\label{prop:free-quotient-general}
Let $\mathcal E$ be a collection of finite subgroups of $M_0$ closed under subconjugates, let $\alpha\colon H\to G$ be a homomorphism, and let $f\colon X\to Y$ be a $\mathcal G_{\mathcal E,H}$-weak equivalence in $\cat{$\bm{(M\times H)}$-SSet}$ such that $\ker(\alpha)$ acts freely on both $X$ and $Y$. Then $\alpha_!(f)$ is a $\mathcal G_{\mathcal E,G}$-weak equivalence.
\begin{proof}
By Proposition~\ref{prop:alpha-shriek-injective} we may assume without loss of generality that $\alpha$ is the quotient map $H\to H/\ker(\alpha)$, so that the functor $\alpha_!$ can be modelled by quotiening out the action of the normal subgroup $K\mathrel{:=}\ker(\alpha)$.
The following splitting follows from a simple calculation similar to the above arguments, which we omit. It can also be obtained from the discrete special case of \cite[Lemma~A.1]{hausmann-equivariant} by adding disjoint basepoints:
\begin{claim*}
Let $L\subset M_0$ be any subgroup and let $\phi\colon L\to H/K$ be any homomorphism. Then we have for any $(L\times H)$-simplicial set $Z$ on which $K$ acts freely a natural isomorphism
\begin{equation*}
\coprod_{[\psi\colon L\to H]} Z^{\psi}/(\centralizer_H(\im\psi)\cap K) \xrightarrow\cong (Z/K)^\phi
\end{equation*}
given on each summand by $[z]\mapsto[z]$. Here the coproduct runs over $K$-conjugacy classes of homomorphisms lifting $\phi$, and $\centralizer_H$ denotes the centralizer in $H$.\nomenclature[aCH]{$\centralizer_H(K)$}{centralizer of $K$ in $H$}\qed
\end{claim*}
We can now prove the proposition. Let $L\in\mathcal E$ and let $\phi\colon L\to H$. In order to show that $(f/K)^\phi$ is a weak homotopy equivalence it suffices by the claim that $f^\psi/(\centralizer_H(\im\psi)\cap K)$ be a weak homotopy equivalence for all $\psi\colon L\to H$ lifting $\phi$. But indeed, as $K$ acts freely on $X$ and $Y$, so does $\centralizer_H(\im\psi)\cap K$; in particular, it also acts freely on $X^\psi$ and $Y^\psi$. The claim follows as $f^\psi$ is a weak homotopy equivalence by assumption and since free quotients preserve weak homotopy equivalences of simplicial sets (for example by the special case $M=1$ of the above discussion).
\end{proof}
\end{prop}
\index{functoriality in homomorphisms!for M-SSet@for $\cat{$\bm{M}$-SSet}$|)}
\subsubsection{Change of subgroups} We now turn to the special case that the monoid $M$ is fixed (i.e.~$\alpha=\id_M$), but the collection $\mathcal F$ is allowed to vary. For this we will use the notion of \emph{quasi-localizations}, which we recall in Appendix~\ref{appendix:quasi-loc}; in particular, we will employ the notation $\mathscr C^\infty_W$ (or simply $\mathscr C^\infty$) introduced there for `the' quasi-localization of a category $\mathscr C$ at a class $W$ of maps.
\begin{prop}\label{prop:change-of-family-sset}
Let $M$ be a simplicial monoid and let $\mathcal F,\mathcal F'$ be collections of finite subgroups of $M_0$ such that $\mathcal F'\subset\mathcal F$. Then the identity descends to a quasi-localization
\begin{equation}\label{eq:to-smaller-family}
\cat{$\bm M$-SSet}_{\textup{$\mathcal F$-weak equivalences}}^\infty\to\cat{$\bm M$-SSet}_{\textup{$\mathcal F'$-weak equivalences}}^\infty
\end{equation}
at the $\mathcal F'$-weak equivalences, and this functor admits both a left adjoint $\lambda$\nomenclature[alambda]{$\lambda$}{left adjoint to forgetful functor along change of subgroups} as well as a right adjoint $\rho$.\nomenclature[arho1]{$\rho$}{right adjoint to forgetful functor along change of subgroups} Both $\lambda$ and $\rho$ are fully faithful.
\begin{proof}
The identity obviously descends to the quasi-localization $(\ref{eq:to-smaller-family})$. It then only remains to construct the desired adjoints, as they will automatically be fully faithful as adjoints of quasi-localizations, see e.g.~\cite[Proposition~7.1.17]{cisinski-book}.
But indeed, Lemma~\ref{lemma:alpha-shriek-projective} specializes to yield a Quillen adjunction
\begin{equation*}
\id\colon\cat{$\bm M$-SSet}_{\textup{$\mathcal F'$-equivariant}}\rightleftarrows\cat{$\bm M$-SSet}_{\textup{$\mathcal F$-equivariant}} :\!\id
\end{equation*}
so that the left derived functor $\textbf{L}\id$ in the sense of Theorem~\ref{thm:derived-adjunction} defines the desired left adjoint. To construct the right adjoint, we observe that while
\begin{equation*}
\id\colon\cat{$\bm M$-SSet}_{\textup{$\mathcal F$-equivariant}}\rightleftarrows\cat{$\bm M$-SSet}_{\textup{$\mathcal F'$-equivariant}} :\!\id
\end{equation*}
is typically \emph{not} a Quillen adjunction with respect to the usual model structures, it becomes one if we use Corollary~\ref{cor:enlarge-generating-cof} to enlarge the cofibrations on the right hand side to contain all generating cofibrations of the $\mathcal F$-model structure (which we are allowed to do by Corollary~\ref{cor:homotopy-pushout-M-SSet}), or alternatively that it is a Quillen adjunction for the corresponding injective model structures by Lemma~\ref{lemma:alpha-star-injective}.
\end{proof}
\end{prop}
\begin{rk}\label{rk:change-of-family-sset-explicit}
By the above proof, $\lambda$ can be modelled by taking a cofibrant replacement with respect to the $\mathcal F'$-model structure.
\end{rk}
\section[$G$-global homotopy theory via monoid actions]{\texorpdfstring{\except{toc}{$\bm G$}\for{toc}{$G$}}{G}-global homotopy theory via monoid actions}\label{sec:equivariant-models}
\subsection{The universal finite group}
Schwede \cite{schwede-orbi} proved that unstable global homotopy theory with respect to all compact Lie groups can be modelled by spaces with the action of a certain topological monoid $\mathcal L$, that he calls the \emph{universal compact Lie group},\index{universal compact Lie group} and which we will recall in Section~\ref{sec:global-vs-g-global}. For unstable global homotopy theory with respect to \emph{finite} groups, we will instead be interested in a certain discrete analogue $\mathcal M$, which (under the name $M$) also plays a central role in Schwede's approach \cite{schwede-k-theory} to global algebraic $K$-theory.
\begin{defi}
We write $\omega=\{0,1,2,\dots\}$,\nomenclature[aomega]{$\omega$}{set of non-negative integers} and we denote by $\mathcal M$\nomenclature[aM]{$\mathcal M$}{`universal finite group,' monoid of self-injections of $\omega$} the monoid (under composition) of all injections $\omega\to\omega$.
\end{defi}
Analogously to the Lie group situation \cite[Definition~1.6]{schwede-orbi}, when we model a `global space' by an $\mathcal M$-simplicial set $X$, we do not expect the fixed point spaces $X^H$ for all finite $H\subset\mathcal M$ to carry homotopical information, but only those for certain so-called \emph{universal} $H$. To define these, we first need the following terminology, cf.~\cite[Definition~2.16]{schwede-k-theory}:
\begin{defi}\label{defi:set-universe}
Let $H$ be any finite group. A countable $H$-set $\mathcal U$ is called a \emph{complete $H$-set universe}\index{complete H-set universe@complete $H$-set universe|textbf} if the following equivalent conditions hold:
\begin{enumerate}
\item Every finite $H$-set embeds $H$-equivariantly into $\mathcal U$.
\item Every countable $H$-set embeds $H$-equivariantly into $\mathcal U$.
\item There exists an $H$-equivariant isomorphism
\begin{equation*}
\mathcal U\cong\coprod_{i=0}^\infty\coprod_{\substack{{\scriptstyle K\subset H}\\{\scriptstyle\text{subgroup}}}} H/K.
\end{equation*}\label{item:su-concrete-example}
\item Every subgroup $K\subset H$ occurs as stabilizer of infinitely many distinct elements of $\mathcal U$.
\end{enumerate}
\end{defi}
The proof that the above conditions are indeed equivalent is easy and we omit it. For all of these statements except for the second one this also appears without proof as \cite[Proposition~2.17 and Example~2.18]{schwede-k-theory}.
The following lemmas are similarly straightforward to prove from the definitions, and they also appear without proof as part of \cite[Proposition~2.17]{schwede-k-theory}.
\begin{lemma}\label{lemma:supersets-of-universe}
Let $\mathcal U\subset \mathcal V$ be $H$-sets, assume $\mathcal U$ is a complete $H$-set universe and $\mathcal V$ is countable. Then also $\mathcal V$ is a complete $H$-set universe.\qed
\end{lemma}
\begin{lemma}\label{lemma:restriction-universe}
Let $\mathcal U$ be a complete $H$-set universe and let $\alpha\colon K\to H$ be an injective group homomorphism. Then $\alpha^*\mathcal U$ (i.e.~$\mathcal U$ with $K$-action given by $k.x=\alpha(k).x$) is a complete $K$-set universe.\qed
\end{lemma}
\begin{defi}
A finite subgroup $H\subset\mathcal M$ is called \emph{universal}\index{universal subgroup!for M@for $\mathcal M$|textbf} if the restriction of the tautological $\mathcal M$-action on $\omega$ to $H$ makes $\omega$ into a complete $H$-set universe.
\end{defi}
Lemma~\ref{lemma:restriction-universe} immediately implies:
\begin{cor}\label{cor:subgroup-universal-subgroup-universal}
Let $K\subset H\subset\mathcal M$ be subgroups and assume that $H$ is universal. Then also $K$ is universal.\qed
\end{cor}
\begin{lemma}\label{lemma:uniqueness-of-universal-homom}
Let $H$ be any finite group. Then there exists an injective monoid homomorphism $i\colon H\to\mathcal M$ with universal image. Moreover, if $j\colon H\to\mathcal M$ is another such homomorphism, then there exists a $\phi\in\core\mathcal M$ such that
\begin{equation}\label{eq:phi-def-condition}
j(h)=\phi i(h)\phi^{-1}
\end{equation}
for all $h\in H$.
\begin{proof}
This is similar to \cite[Proposition~1.5]{schwede-orbi}: first, we observe that there exists a complete $H$-set universe, for example
\begin{equation*}
\mathcal U\mathrel{:=}\coprod_{i=0}^\infty\coprod_{\substack{{\scriptstyle K\subset H}\\{\scriptstyle\text{subgroup}}}} H/K.
\end{equation*}
As this is countable, we can pick a bijection of sets $\omega\cong\mathcal U$, which yields an $H$-action on $\omega$ turning it into a complete $H$-set universe. The $H$-action amounts to a homomorphism $H\to\Sigma_\omega\subset\mathcal M$ which is injective as the action is faithful, providing the desired homomorphism $i$.
If $j$ is another such homomorphism, then both $i^*\omega$ and $j^*\omega$ are complete $H$-set universe, and hence there exists an $H$-equivariant isomorphism $\phi\colon i^*\omega\to j^*\omega$, see part $(\ref{item:su-concrete-example})$ of Definition~\ref{defi:set-universe}. The $H$-equivariance of $\phi$ then precisely means that $\phi(i(h)(x))=j(h)(\phi(x))$ for all $h\in H$ and $x\in\omega$, i.e.~$\phi i(h)\phi^{-1}=j(h)$ as desired.
\end{proof}
\end{lemma}
\subsection{Global homotopy theory}\label{subsec:global}\index{global homotopy theory|(}
Before we introduce models of unstable $G$-global homotopy theory based on the above monoid $\mathcal M$ in the next subsection, let us first consider the ordinary global situation in order to present some of the main ideas without being overly technical.
Heuristically, we would like to think of a `global space' $X$ as having for each \emph{abstract} finite group $H$ a fixed point space $X^H$ and for each abstract group homomorphism $f\colon H\to K$ a suitably functorial restriction map $f^*\colon X^K\to X^H$. One way to make this heuristic rigorous is given by Schwede's orbispace model, see \cite[Theorem~2.12]{schwede-cat}---in fact, it turns out that there is also some additional $2$-functoriality. The above lemma already tells us that we can assign to an $\mathcal M$-simplicial set $X$ for each abstract finite group $H$ an essentially unique fixed point space $X^H$ as follows: we pick an injective group homomorphism $i\colon H\to\mathcal M$ with universal image and set $X^H\mathrel{:=}X^{i(H)}$. This is indeed independent of the group homomorphism $i$ up to isomorphism: namely, if $j$ is another such group homomorphism, the lemma provides us with a $\phi$ such that $j(h)=\phi i(h)\phi^{-1}$ for all $h\in H$, and $\phi.\blank\colon X\to X$ clearly restricts to an isomorphism $X^{i(H)}\to X^{j(H)}$.
However, there is an issue here---namely, the element $\phi$ (or more precisely, its action) is not canonical (in particular, it is not clear how to define $1$-functoriality):
\begin{ex}\label{ex:too-many-endomorphisms}
Let us consider the special case $H=1$, so that there is in particular only one homomorphism $i=j\colon H\to\mathcal M$. Then \emph{any} $\phi\in\core\mathcal M$ satisfies the condition $(\ref{eq:phi-def-condition})$ and hence produces an endomorphism $\phi.\blank$ of $X=X^{\{1\}}$.
Looking at the orbispace model, we should expect all such endomorphisms of $X^{\{1\}}$ to be homotopically trivial. However, for $X=\mathcal M$ any $\phi\not=1$ gives us a non-trivial endomorphism.
\end{ex}
This suggests that $\mathcal M$-simplicial sets with respect to maps inducing weak homotopy equivalences on fixed points for universal subgroups are not yet a model of unstable global homotopy theory. In order to solve the issue raised in the example, we will enhance $\mathcal M$ to a simplicial (or categorical) monoid in particular trivializing the above action. This uses:
\begin{constr}
Let $X$ be any set. We write $EX$ for the (small) category with objects $X$ and precisely one morphism $x\to y$ for each $x,y\in X$, which we denote by $(y,x)$. We extend $E$ to a functor $\cat{Set}\to\cat{Cat}$ in the obvious way.\nomenclature[aE]{$E$}{right adjoint to $\Ob\colon\cat{Cat}\to\cat{Set}$; right adjoint to $\ev_0\colon\cat{SSet}\to\cat{Set}$}
We will moreover also write $EX$ for the simplicial set given in degree $n$ by
\begin{equation*}
(EX)_n=X^{\times (1+n)}\cong\Maps(\{0,\dots,n\},X)
\end{equation*}
with structure maps via restriction and with the evident functoriality in $X$. We remark that the simplicial set $EX$ is indeed canonically isomorphic to the nerve of the category $EX$, justifying the clash of notation. In fact, it will be useful at several points to switch between viewing $EX$ as a category or as a simplicial set.
\end{constr}
\begin{rk}
It is clear that the category $EX$ is a groupoid and that it is contractible for $X\not=\varnothing$. In particular, the simplicial set $EX$ is a Kan complex, again contractible unless $X=\varnothing$.
\end{rk}
The functor $E\colon\cat{Set}\to\cat{Cat}$ is right adjoint to the functor $\Ob\colon\cat{Cat}\to\cat{Set}$; likewise $E\colon\cat{Set}\to\cat{SSet}$ is right adjoint to the functor sending a simplicial set to its set of zero simplices. In particular, $E$ preserves products, so $E\mathcal M$\nomenclature[aEM]{$E\mathcal M$}{simplicial (or categorical) monoid obtained from $\mathcal M$} is canonically a simplicial monoid. As it is contractible, any two translations $u.\blank,v.\blank$ for $u,v\in\mathcal M$ are homotopic on any $E\mathcal M$-object $X$; in fact, there is unique edge $(v,u)$ from $u$ to $v$ in $E\mathcal M$, and acting with this gives an explicit homotopy $u.\blank\Rightarrow v.\blank$.
We conclude that $E\mathcal M$ avoids the issue detailed in Example~\ref{ex:too-many-endomorphisms}. Indeed, Theorems~\ref{thm:script-I-global-model-structure} and~\ref{thm:strict-global-I-model-structure} together with Theorem~\ref{thm:L-vs-I} will show that $\cat{$\bm{E\mathcal M}$-SSet}$ is a model of global homotopy theory in the sense of \cite{schwede-book} with respect to finite groups. Maybe somewhat surprisingly, the main result of this subsection (Theorem~\ref{thm:m-vs-em}) will be that the same homotopy theory can be modelled by $\mathcal M$-simplicial sets with respect to a slightly intricate notion of weak equivalence.
\begin{ex}\label{ex:global-classifying space}
Let $H$ be any finite group and let $A$ be a countable faithful $H$-set. The set $\Inj(A,\omega)$ of injections $A\to\omega$ \nomenclature[aInj]{$\Inj(A,B)$}{set of injective maps $A\to B$} has a natural $\mathcal M$-action via postcomposition and a commuting $H$-action via precomposition, inducing an $E\mathcal M$-action on $E\Inj(A,\omega)/H$.
Note that $H$ acts freely from the right on $\Inj(A,\omega)$ and hence on the contractible simplicial set $E\Inj(A,\omega)$ as injections of sets are monomorphisms. In particular, ignoring the $E\mathcal M$-action, $E\Inj(A,\omega)/H$ is just an Eilenberg-Mac Lane space of type $K(H,1)$. However, the $E\mathcal M$-simplicial set $E\Inj(A,\omega)/H$ contains interesting additional equivariant information compared to an ordinary $K(H,1)$ equipped with trivial $E\mathcal M$-action, and we call it `the' \emph{global classifying space} of $H$.
\end{ex}
The basis for our comparison between $\cat{$\bm{E\mathcal M}$-SSet}$ and $\cat{$\bm{\mathcal M}$-SSet}$ will be the following model categories provided by Proposition~\ref{prop:equiv-model-structure}:
\begin{cor}\label{cor:m-model-structure}
The category $\cat{$\bm{\mathcal M}$-SSet}$ admits a unique model structure such that a map $f\colon X\to Y$ is a weak equivalence or fibration if and only if the map $f^H$ is a weak homotopy equivalence or Kan fibration, respectively, for each universal $H\subset\mathcal M$. We call this the \emph{universal model structure}\index{universal model structure|textbf} and its weak equivalences the \emph{universal weak equivalences}.\index{universal weak equivalence|textbf} It is simplicial, combinatorial, and proper. A possible set of generating cofibrations is given by
\begin{equation*}
I=\{ \mathcal M/H\times\del\Delta^n\hookrightarrow \mathcal M/H\times\Delta^n : n\ge0,\text{ $H\subset\mathcal M$ universal}\}
\end{equation*}
and a possible set of generating acyclic cofibrations is given by
\begin{equation*}
J=\{ \mathcal M/H\times\Lambda^n_k\hookrightarrow \mathcal M/H\times\Delta^n : 0\le k\le n,\text{ $H\subset\mathcal M$ universal}\}.
\end{equation*}
Moreover, filtered colimits in $\cat{$\bm{\mathcal M}$-SSet}$ are homotopical. \qed
\end{cor}
As already mentioned above, this will not yet model global equivariant homotopy theory, so we reserve the names `global model structure' and `global weak equivalences' for a different model structure.
\begin{cor}\label{cor:EM-SSet-model-structure}
The category $\cat{$\bm{E\mathcal M}$-SSet}$ of $E\mathcal M$-simplicial sets admits a unique model structure such that a map $f\colon X\to Y$ is a weak equivalence or fibration if and only if it is so when considered as a map in $\cat{$\bm{\mathcal M}$-SSet}$. We call this the \emph{global model structure}\index{global model structure|seealso{$G$-global model structure}}\index{global model structure!on EM-SSet@on $\cat{$\bm{E\mathcal M}$-SSet}$|textbf} and its weak equivalences the \emph{global weak equivalences}.\index{global weak equivalence|seealso{$G$-global weak equivalence}}\index{global weak equivalence!in EM-SSet@in $\cat{$\bm{E\mathcal M}$-SSet}$|textbf} It is simplicial, proper, and combinatorial with possible set of generating cofibrations
\begin{equation*}
I=\{(E\mathcal M)/H\times\del\Delta^n\hookrightarrow (E\mathcal M)/H\times\Delta^n : n\ge0,\text{ $H\subset\mathcal M$ universal}\}
\end{equation*}
and possible set of generating acyclic cofibrations
\begin{equation*}
J=\{(E\mathcal M)/H\times\Lambda^n_k\hookrightarrow (E\mathcal M)/H\times\Delta^n : 0\le k\le n,\text{ $H\subset\mathcal M$ universal}\}.
\end{equation*}
Moreover, filtered colimits in $\cat{$\bm{E\mathcal M}$-SSet}$ are homotopical. \qed
\end{cor}
\begin{constr}
The forgetful functor $\cat{$\bm{E\mathcal M}$-SSet}\to\cat{$\bm{\mathcal M}$-SSet}$ admits both a simplicial left and a simplicial right adjoint. While they exist for abstract reasons (e.g.~as simplicially enriched Kan extensions), they are also easy to make explicit:
Let $X$ be any $\mathcal M$-simplicial set. We write $E\mathcal M\times_{\mathcal M}X$\nomenclature[aEMM]{$E\mathcal M\times_{\mathcal M}\blank$}{left adjoint to forgetful functor $\cat{$\bm{E\mathcal M}$-SSet}\to\cat{$\bm{\mathcal M}$-SSet}$} for the following $E\mathcal M$-simplicial set: as a simplicial set, this is the quotient of $E\mathcal M\times X$ under the equivalence relation generated in degree $n$ by $(u_0v,\dots,u_nv;x)\sim(u_0,\dots,u_n;v.x)$ for all $u_0,\dots,u_n,v\in\mathcal M$ and $x\in X_n$. As usual, we denote the class of $(u_0,\dots,u_n;x)$ by $[u_0,\dots,u_n;x]$.
The $E\mathcal M$-action on $E\mathcal M\times_{\mathcal M}X$ is induced by the obvious $E\mathcal M$-action on the first factor. If $f\colon X\to Y$ is any $\mathcal M$-equivariant map, then $E\mathcal M\times_{\mathcal M}f$ is induced by $E\mathcal M\times f$, and similarly for higher cells $\Delta^n\times X\to Y$. We omit the easy verification that is well-defined.
We have a natural $\mathcal M$-equivariant map $\eta\colon X\to\forget(E\mathcal M\times_{\mathcal M}X)$ given in degree $n$ by sending an $n$-simplex $x$ to the class $[1,\dots,1;x]$. Moreover, if $Y$ is an $E\mathcal M$-simplicial set, we have an $E\mathcal M$-equivariant map $\epsilon\colon E\mathcal M\times_{\mathcal M}(\forget Y)\to Y$ giving in degree $n$ by acting, i.e.~$[u_0,\dots,u_n;y]\mapsto(u_0,\dots,u_n).y$. We leave the easy verification to the reader that these are well-defined, enriched natural, and define unit and counit, respectively, of a simplicial adjunction $E\mathcal M\times_{\mathcal M}\blank\dashv\forget$.
Similarly, the forgetful functor has a simplicial right adjoint $\Maps^{\mathcal M}(E\mathcal M,\blank)$\nomenclature[amapsMEM]{$\Maps^{\mathcal M}(E\mathcal M,\blank)$}{right adjoint to forgetful functor $\cat{$\bm{E\mathcal M}$-SSet}\to\cat{$\bm{\mathcal M}$-SSet}$} (the simplicial set of $\mathcal M$-equivariant maps), with $E\mathcal M$-action induced by the right $E\mathcal M$-action on itself via postcomposition.
\end{constr}
By definition of the model structures we immediately get:
\begin{cor}\label{cor:forget-right-Quillen}
The simplicial adjunction
\begin{equation}\label{eq:em-times-forget}
E\mathcal M\times_{\mathcal M}\blank\colon \cat{$\bm{\mathcal M}$-SSet}\rightleftarrows\cat{$\bm{E\mathcal M}$-SSet}:\!\forget
\end{equation}
is a Quillen adjunction and the right adjoint creates weak equivalences.\qed
\end{cor}
As already suggested by our heuristic above, this is not a Quillen equivalence; more precisely, $\forget^\infty$ is not essentially surjective:
\begin{ex}
If $X$ is any $E\mathcal M$-simplicial set, then all $u\in\mathcal M$ act on $X$ by weak homotopy equivalences, and hence the same will be true for any $\mathcal M$-simplicial set $Y$ weakly equivalent to $\forget X$. On the other hand, $\mathcal M$ considered as a discrete $\mathcal M$-simplicial set with $\mathcal M$-action by postcomposition does not satisfy this, and hence can't lie in the essential image.
\end{ex}
In the example we have only looked at the underlying non-equivariant homotopy type of a given $\mathcal M$-simplicial set. However, in order to have a sufficient criterion for the essential image of the forgetful functor, we should better take equivariant information into account. While the translation maps will usually not be $\mathcal M$-equivariant (related to the fact that $\mathcal M$ is highly non-commutative), we can look at those parts of the action that we can still expect to be preserved:
\begin{defi}\label{defi:semistable}
An $\mathcal M$-simplicial set $X$ is called \emph{semistable}\index{semistable!M-simplicial set@$\mathcal M$-simplicial set|textbf}\index{semistable|seealso{$G$-semistable}} if for each universal subgroup $H\subset\mathcal M$ and each $u\in\mathcal M$ centralizing $H$ the translation map
\begin{equation*}
u.\blank\colon X\to X
\end{equation*}
is an $H$-equivariant weak equivalence.
\end{defi}
We observe that this is equivalent to demanding that for any such $u\in\mathcal M$ and $H\subset\mathcal M$ the restriction of $u.\blank$ to $X^H\to X^H$ is a weak homotopy equivalence (this uses Corollary~\ref{cor:subgroup-universal-subgroup-universal}).
\begin{ex}\label{ex:em-semistable}
Strengthening the previous example, any $E\mathcal M$-simplicial set $X$ is in fact semistable when viewed as an $\mathcal M$-simplicial set: namely, if $u\in\mathcal M$ centralizes an (arbitrary) subgroup $H\subset\mathcal M$, then $(u,1)$ provides an $H$-equivariant homotopy between $u.\blank$ and the identity.
\end{ex}
\begin{rk}
The term `semistable' refers to Schwede's characterization \cite[Theorem~4.1 and Lemma~2.3-(iii)]{schwede-semistable} of semistable symmetric spectra, i.e.~symmetric spectra whose na\"ive homotopy groups agree with their true homotopy groups,\index{semistable!symmetric spectrum} as those spectra for which a certain canonical $\mathcal M$-action on the na\"ive homotopy groups is given by isomorphisms, also cf.~\cite[Corollary~3.32 and Proposition~3.16]{hausmann-equivariant} for a similar characterization in the equivariant case due to Hausmann.
Both Schwede and Hausmann prove that in the respective situation the action on na\"ive homotopy groups is actually trivial, i.e.~all elements of $\mathcal M$ act by the identity. Likewise, it will follow from Theorem \ref{thm:m-vs-em} together with the argument from Example~\ref{ex:em-semistable} above that for a semistable $\mathcal M$-simplicial set the translation $u.\blank$, $u\in\mathcal M$ centralizing some universal subgroup $H\subset\mathcal M$, is in fact the identity in the $H$-equivariant homotopy category.
\end{rk}
Obviously, semistability is invariant under universal weak equivalences, so it is a necessary condition to lie in the essential image of $\forget$ by the above example. As the main result of this section, we will show that it is also sufficient, and moreover the above is everything that prevents $\forget^\infty$ from being an equivalence:
\begin{thm}\label{thm:m-vs-em}
The adjunction $(\ref{eq:em-times-forget})$ induces a Bousfield localization
\begin{equation*}
E\mathcal M\times_{\mathcal M}^{\textbf{\textup L}}\blank\colon \cat{$\bm{\mathcal M}$-SSet}^\infty\rightleftarrows\cat{$\bm{E\mathcal M}$-SSet}^\infty:\!\forget^\infty;
\end{equation*}
in particular, $\forget^\infty$ is fully faithful. Moreover, its essential image consists precisely of the semistable $\mathcal M$-simplicial sets.
\end{thm}
Here we have tacitly identified the objects of $\cat{$\bm{\mathcal M}$-SSet}^\infty$ with those of $\cat{$\bm{\mathcal M}$-SSet}$, cf.~Remark~\ref{rk:associated-quasi}.
\begin{rk}
The theorem in particular tells us that the forgetful functor identifies $\cat{$\bm{E\mathcal M}$-SSet}^\infty$ with the full subcategory of $\cat{$\bm{\mathcal M}$-SSet}^\infty$ spanned by the semistable objects. In view of Proposition~\ref{prop:localization-subcategory}, the latter is canonically identified with the quasi-localization of semistable $\mathcal M$-simplicial sets at the universal weak equivalences, so we do not have to be careful to distinguish them. Put differently: semistable $\mathcal M$-simplicial sets with respect to the universal weak equivalences are a model of unstable global homotopy theory.
\end{rk}
Our proof of the theorem will proceed indirectly via the alternative models provided by the Elmendorf Theorem for monoids (Corollary~\ref{cor:elmendorf}): namely, we will exhibit the \emph{global orbit category} $\textbf{O}_{E\mathcal M}$\index{global orbit category|textbf}\index{global orbit category|seealso{$G$-global orbit category}} as an explicit simplicial localization\index{simplicial localization} of $\textbf{O}_{\mathcal M}$ in the sense of Definition~\ref{defi:simplicial-localization}, and then deduce the theorem from the universal property of simplicial localizations (or more precisely its model categorical manifestation Theorem~\ref{thm:simpl-localization-model-cat}).
To do so, let us begin by understanding these categories a bit better:
\index{global orbit category|(}
\begin{lemma}\label{lemma:group-hom-associated}
Let $H,K\subset\mathcal M$, and let $u_0,\dots,u_n\in\mathcal M$ such that $[u_0,\dots,u_n]$ is $H$-fixed in $(E\mathcal M/K)_n$. Then there exists for any $h\in H$ a unique $\sigma(h)\in K$ such that $hu_i=u_i\sigma(h)$ for all $i$. Moreover, $\sigma\colon H\to K$ is a homomorphism.
Conversely, whenever such a map $\sigma$ exists, $[u_0,\dots,u_n]$ is $H$-fixed in $(E\mathcal M)/K$.
\begin{proof}
As $[u_0,\dots,u_n]$ is $H$-fixed, $(u_0,\dots,u_n)\sim (hu_0,\dots,hu_n)$ for any $h\in H$,
so by definition there indeed exists some $\sigma(h)\in K$ such that $hu_i=u_i\sigma(h)$ for all $i$; moreover, $\sigma(h)$ is unique as $K$ acts freely from the right on $\mathcal M$.
To check that $\sigma$ is a group homomorphism, let $h_1,h_2\in H$ arbitrary. Then $h_1h_2u=h_1u\sigma(h_2)=u\sigma(h_1)\sigma(h_2)$, hence $\sigma(h_1h_2)=\sigma(h_1)\sigma(h_2)$ by uniqueness.
The proof of the converse is trivial.
\end{proof}
\end{lemma}
\begin{constr}
By definition, $\textbf{O}_{E\mathcal M}\subset\cat{$\bm{E\mathcal M}$-SSet}$ is the full simplicial subcategory spanned by the $E\mathcal M/H$ for universal $H\subset\mathcal M$. We have seen in Lemma~\ref{lemma:fixed-points-cellular} that the simplicial set $\Maps_{\textbf{O}_{E\mathcal M}}(E\mathcal M/H, E\mathcal M/K)$ is isomorphic to $(E\mathcal M/K)^H$ via evaluation at $[1]\in E\mathcal M/H$. On $0$-simplices, this gives $\Maps_{\textbf{O}_{E\mathcal M}}(E\mathcal M/H, E\mathcal M/K)_0\cong (\mathcal M/K)^H$; an inverse is then given by sending $u\in(\mathcal M/K)^H$ to $\blank\cdot u\colon [v_0,\dots,v_n]\mapsto[v_0u,\dots,v_nu]$.
More generally, an $n$-simplex of the mapping space $\Maps_{\textbf{O}_{E\mathcal M}}(E\mathcal M/H, E\mathcal M/K)$ can be represented by an $(n+1)$-tuple $(u_0,\dots,u_n)$ such that $[u_0,\dots,u_n]\in E\mathcal M/K$ is $H$-fixed, which by Lemma~\ref{lemma:group-hom-associated} is equivalent to the existence of a group homomorphism $\sigma\colon H\to K$ such that $hu_i=u_i\sigma(h)$ for all $i=0,\dots,n$. Two tuples represent the same morphism iff they become equal in $E\mathcal M/K$, i.e.~iff they only differ by right multiplication with some $k\in K$.
Moreover, one immediately sees by direct inspection, that if the $n$-simplex $f\in\Maps_{\textbf{O}_{E\mathcal M}}(E\mathcal M/H,E\mathcal M/K)_n$ is represented by $(u_0,\dots,u_n)$ and the $n$-simplex $f'\in\Maps_{\textbf{O}_{E\mathcal M}}(E\mathcal M/K,E\mathcal M/L)_n$ is represented by $(u_0',\dots,u_n')$, then their composition $f'f$ is represented by $(u_0u_0',\dots,u_nu_n')$ (note the different order!).
Likewise, $\textbf{O}_{\mathcal M}$ is the full (simplicial) subcategory of $\cat{$\bm{\mathcal M}$-SSet}$ spanned by the $\mathcal M/H$, and we have an isomorphism $\Hom(\mathcal M/H,\mathcal M/K)\cong (\mathcal M/K)^H$ via evaluation at $[1]$; composition is again induced from multiplication in $\mathcal M$.
We now define a functor $i\colon\textbf{O}_{\mathcal M}\to\textbf{O}_{E\mathcal M}$ as follows: an object $\mathcal M/H$ is sent to $E\mathcal M/H$. On morphism spaces, $i$ is given as the composition
\begin{align*}
\Maps_{\textbf{O}_{\mathcal M}}(\mathcal M/H,\mathcal M/K)&\cong (\mathcal M/K)^H\cong\Maps_{\textbf{O}_{E\mathcal M}}(E\mathcal M/H,E\mathcal M/K)_0\\ &\hookrightarrow\Maps_{\textbf{O}_{E\mathcal M}}(E\mathcal M/H,E\mathcal M/K),
\end{align*}
which then just sends the morphism represented by $u$ to the morphism represented by the same element. As an upshot of the above discussion, this is indeed functorial.
\end{constr}
\begin{defi}\index{centralizing morphism|textbf}\index{centralizing morphism|seealso{$G$-centralizing morphism}}
Let $H\subset\mathcal M$ be universal. A map $f\colon \mathcal M/H\to \mathcal M/H$ is called \emph{centralizing} if there exists a $u\in\mathcal M$ centralizing $H$ such that $f$ is given by right multiplication by $u$. Analogously, we define centralizing morphisms in $\textbf{O}_{E\mathcal M}$.
\end{defi}
The following will be the main ingredient to the proof of Theorem~\ref{thm:m-vs-em}:
\begin{prop}\label{prop:em-quasi-localization}\index{global orbit category!as a simplicial localization}
The functor $i\colon\textbf{\textup O}_{\mathcal M}\to \textbf{\textup O}_{E\mathcal M}$ is a simplicial localization at the centralizing morphisms.
\begin{proof}
By construction, $i$ induces an isomorphism onto the underlying category $\und\textbf{O}_{E\mathcal M}=(\textbf{O}_{E\mathcal M})_0$\nomenclature[au]{$\und$}{underlying category (of $0$-simplices) of a simplicially enriched category} of $\textbf{O}_{E\mathcal M}$. Thus, it is enough to prove that $\und\textbf{O}_{E\mathcal M}\hookrightarrow\textbf{O}_{E\mathcal M}$ is a quasi-localization at the centralizing morphisms, for which we will verify the assumptions of Proposition~\ref{prop:enrichment-vs-localization}, i.e.~that $\cat{O}_{E\mathcal M}$ has fibrant mapping spaces, that the centralizing morphisms are homotopy equivalences, and that the functors
\begin{equation}\label{eq:emql-iterated-degeneracy}
s^*\colon\big((\textbf{O}_{E\mathcal M})_0,W\big)\to \big((\textbf{O}_{E\mathcal M})_n,s^*W\big)
\end{equation}
induce equivalences on quasi-localizations, where $W$ denotes the class of centralizing morphisms and $s$ is the unique map $[n]\to[0]$ in $\Delta$.
For the first claim we will show that all mapping spaces in $\cat{O}_{E\mathcal M}$ are actually even nerves of groupoids. Indeed, it suffices to prove that each $E\mathcal M/H$ is, for which it is then in turn enough to observe that $E\mathcal M$ is the nerve of a groupoid by construction and that we can form the quotient by $H$ already in the category of groupoids as the nerve preserves quotients by \emph{free} group actions.
For the second claim we note that if $u$ centralizes $H$, then $\blank\cdot u$ is even homotopic to the identity via the edge $[1,u]$ in $(E\mathcal M/H)^H\cong\Maps_{{\textbf O}_{E\mathcal M}}(E\mathcal M/H,E\mathcal M/H)$.
Finally, for the third claim it is by Corollary~\ref{cor:homotopy-equivalence} enough to prove that $(\ref{eq:emql-iterated-degeneracy})$ is a homotopy equivalence in the sense of Definition~\ref{defi:homotopy-equivalence}. For this we will show that $i\colon[0]\to[n],0\mapsto 0$ induces a homotopy inverse. Indeed, $i^*$ is obviously homotopical and moreover $i^*s^*=(si)^*=\id$ by functoriality. It remains to prove that $s^*i^*$ is homotopic to the identity of $(\textbf{O}_{E\mathcal M})_n$.
We begin by picking for each universal $H\subset\mathcal M$ an $H$-equivariant isomorphism $\omega\amalg\omega\cong\omega$, where $H$ acts on each of the three copies of $\omega$ in the tautological way; such an isomorphism indeed exists as both sides are complete $H$-set universes. Restricting to the two copies of $\omega$ then gives injections $\alpha_H,\beta_H\in\mathcal M$ such that:
\begin{enumerate}
\item $\alpha_H$ and $\beta_H$ centralize $H$\label{item:centralizing}
\item $\omega=\im(\alpha_H)\sqcup\im(\beta_H)$, i.e.~the images of $\alpha_H$ and $\beta_H$ partition $\omega$.\nomenclature[zsqcup]{$\sqcup$}{internal disjoint union of sets}\label{item:disjoint-cover}
\end{enumerate}
We now define $f\colon (\textbf{O}_{E\mathcal M})_n\to(\textbf{O}_{E\mathcal M})_n$ as follows: $f$ is the identity on objects. A morphism $(E\mathcal M)/H\to (E\mathcal M)/K$ represented by $(u_0,\dots,u_n)\in\mathcal M^{n+1}$ is sent to the morphism represented by $(v_0,\dots,v_n)$ where $v_i$ satisfies
\begin{equation}\label{eq:O-M-vs-O-EM-f-defining}
v_i\alpha_K = \alpha_H u_i\qquad\text{and}\qquad v_i\beta_K=\beta_Hu_0.
\end{equation}
We first observe that there is indeed a unique such $v_i$ as $\alpha_K$ and $\beta_K$ are injections whose images form a partition of $\omega$. Moreover, this is an injection as $\alpha_H$ and $\beta_H$ have disjoint image and as both $\alpha_H u_i$ and $\beta_Hu_0$ are injective.
Next, we show that $(v_0,\dots,v_n)$ indeed defines a morphism, i.e.~it represents an $H$-fixed simplex of $(E\mathcal M)/K$. Indeed, as $(u_0,\dots,u_n)$ represents an $H$-fixed simplex, there is a (unique) group homomorphism $\sigma\colon H\to K$ such that $hu_i=u_i\sigma(h)$ for all $h\in H$. But then
\begin{equation*}
hv_i\alpha_K = h\alpha_H u_i=\alpha_H hu_i=\alpha_H u_i\sigma(h)=v_i\alpha_K\sigma(h)=v_i\sigma(h)\alpha_K,
\end{equation*}
where we have used Condition~$(\ref{item:centralizing})$ twice as well as the definition of $v_i$. Analogously one shows $hv_i\beta_K= v_i\sigma(h)\beta_K$; as the images of $\alpha_K$ and $\beta_K$ together cover $\omega$, we conclude that $hv_i=v_i\sigma(h)$ for all $i$ and all $h$, and hence $[v_0,\dots,v_n]\in (E\mathcal M)/K$ is $H$-fixed as desired.
Moreover, this is independent of the choice of representative: if we pick any other representative $(u_0',\dots,u_n')$, then there is some $k\in K$ such that $u_i'=u_ik$ for all $i$, and thus the associated $v_i'$ satisfy
\begin{equation*}
v_i'\alpha_K=\alpha_H u_i'=\alpha_H u_i k=v_i\alpha_K k = v_ik\alpha_K,
\end{equation*}
where we have used the definitions of $v_i$ and $v_i'$ as well as $(\ref{item:centralizing})$. Analogously one shows $v_i'\beta_K= v_ik\beta_K$; as before we conclude that $v_i'=v_ik$, so that $(v_0,\dots,v_n)$ and $(v_0',\dots,v_n')$ represent the same morphism.
With this established, one easily checks that $f$ is a functor $(\textbf{O}_{E\mathcal M})_n\to(\textbf{O}_{E\mathcal M})_n$. As our notion of homotopy equivalence does not require the intermediate functors to be homotopical (although $f$ actually is by a computation analogous to the above), it only remains to prove that $f$ is homotopic to both the identity and $s^*i^*$. For this we observe that we have by the defining equation $(\ref{eq:O-M-vs-O-EM-f-defining})$ natural transformations
\begin{equation*}
\id\Leftarrow f\Rightarrow s^*i^*
\end{equation*}
where the left hand transformation is given on $(E\mathcal M)/H$ by the morphism corresponding to $s^*[\alpha_H]=[\alpha_H,\dots,\alpha_H]$ while the right hand transformation corresponds to $s^*[\beta_H]$, and these are levelwise weak equivalences by Condition~$(\ref{item:centralizing})$ above.
\end{proof}
\end{prop}
\begin{cor}
In the Quillen adjunction
\begin{equation*}
(i^\op)_!\colon \FUN(\textbf{\textup O}_{\mathcal M}^\op,\cat{SSet})\rightleftarrows
\FUN(\textbf{\textup O}_{E\mathcal M}^\op,\cat{SSet}) :\!(i^\op)^*,
\end{equation*}
the right adjoint is homotopical and the induced functor between associated quasi-categories is fully faithful with essential image precisely those simplicial pre\-sheaves on $\textbf{\textup O}_{\mathcal M}$ that invert the centralizing morphisms.
\begin{proof}
It is clear, that $(i^\op)^*$ is homotopical. By the previous proposition, $i\colon\textbf{\textup O}_{\mathcal M}\to\textbf{\textup O}_{E\mathcal M}$ is a simplicial localization at the centralizing morphisms, and hence $i^\op$ is a simplicial localization at their opposites. As both source and target of $i^\op$ are small and locally fibrant, the claim now follows from Theorem~\ref{thm:simpl-localization-model-cat}.\index{global orbit category|)}
\end{proof}
\end{cor}
\begin{proof}[Proof of Theorem~\ref{thm:m-vs-em}]
We already know from Corollary~\ref{cor:forget-right-Quillen} that $E\mathcal M\times_{\mathcal M}(\blank)\dashv\forget$ is a Quillen adjunction with homotopical right adjoint. It therefore suffices to prove that $\forget^\infty$ is fully faithful with essential image the semistable $\mathcal M$-simplicial sets.
\begin{claim*}
The diagram
\begin{equation}\label{diag:forget-vs-restr}
\begin{tikzcd}[column sep=large]
\cat{$\bm{E\mathcal M}$-SSet} \arrow[r, "\forget"]\arrow[d, "\Phi"'] & \cat{$\bm{\mathcal M}$-SSet}\arrow[d, "\Phi"]\\
\FUN(\textbf{\textup O}_{E\mathcal M}^\op, \cat{SSet})\arrow[r, "(i^\op)^*"'] &[4em] \FUN(\textbf{\textup O}_{\mathcal M}^\op, \cat{SSet})
\end{tikzcd}
\end{equation}
of homotopical functors commutes up to natural isomorphism.
\begin{proof}
An explicit choice of such an isomorphism $\tau$ is given as follows: if $X$ is any $E\mathcal M$-simplicial set and $H\subset\mathcal M$ is universal, then $\tau_X(\mathcal M/H)$ is the composition
\begin{align*}
\Phi(X)(i(\mathcal M/H))=\Phi(X)(E\mathcal M/H)&\xrightarrow{\ev_{[1]}} X^H=(\forget X)^H\\ &\xrightarrow{(\ev_{[1]})^{-1}}\Phi(\forget X)(\mathcal M/H).
\end{align*}
To see that this is well-defined, let $H,K\subset\mathcal M$ universal, and let $u\in\mathcal M$ define an $K$-fixed point of $\mathcal M/H$. Then we have commutative diagrams
\begin{equation}\label{diag:Phi-vs-fixed-points-E}
\begin{tikzcd}
\Phi(X)(E\mathcal M/H)=\Maps^{E\mathcal M}(E\mathcal M/H, X)\arrow[d, "(\blank\cdot u)^*"']\arrow[r, "\ev_{[1]}", "\cong"'] & X^H\arrow[d, "u.\blank"]\\
\Phi(X)(E\mathcal M/K)=\Maps^{E\mathcal M}(E\mathcal M/K, X)\arrow[r, "\ev_{[1]}", "\cong"'] & X^K
\end{tikzcd}
\end{equation}
and, for each $Y\in\cat{$\bm{\mathcal M}$-SSet}$,
\begin{equation}\label{diag:Phi-vs-fixed-points}
\begin{tikzcd}
\Phi(Y)(\mathcal M/H)=\Maps^{\mathcal M}(\mathcal M/H, Y)\arrow[d, "(\blank\cdot u)^*"']\arrow[r, "\ev_{[1]}", "\cong"'] & Y^H\arrow[d, "u.\blank"]\\
\Phi(Y)(\mathcal M/K)=\Maps^{\mathcal M}(\mathcal M/K, Y)\arrow[r, "\ev_{[1]}", "\cong"'] & Y^K.
\end{tikzcd}
\end{equation}
Taking $Y=\forget X$ in $(\ref{diag:Phi-vs-fixed-points})$, these two together then show that $\tau_X$ is natural (and hence defines a morphism in $\FUN(\textbf{\textup O}_{\mathcal M}^\op, \cat{SSet})$). It is then obvious that $\tau$ is natural (say, in the unenriched sense), as it is levelwise given by a composition of natural transformations.
\end{proof}
\end{claim*}
We can now deduce the theorem: the vertical functors in $(\ref{diag:forget-vs-restr})$ induce equivalences on quasi-localizations by Corollary~\ref{cor:elmendorf}, and by the previous corollary the bottom horizontal arrow is fully faithful with essential image those pre\-sheaves that invert centralizing isomorphisms. Thus, $\forget^\infty$ is fully faithful with essential image those $\mathcal M$-simplicial sets $X$ such that $\Phi(X)$ inverts centralizing isomorphisms. Taking $Y=X$ and a $u\in\mathcal M$ centralizing $H=K$ in $(\ref{diag:Phi-vs-fixed-points})$, we see that $\Phi(X)$ inverts centralizing morphisms if and only if $X$ is semistable, finishing the proof.
\end{proof}
\index{global homotopy theory|)}
\subsection[$G$-global model structures]{\texorpdfstring{$\bm G$}{G}-global model structures}
Let us fix some (possibly infinite) discrete group $G$. We now want to extend the above discussion to yield a \emph{$G$-global model structure} on the category $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$ of simplicial sets with a $G$-action and a commuting $E\mathcal M$-action, which we can equivalently think of as simplicial sets with an action of $E\mathcal M\times G$, or as the category of $G$-objects in $\cat{$\bm{E\mathcal M}$-SSet}$.
For $G=1$ this will recover the previous model; however, as soon as $G$ contains torsion, the weak equivalences will be strictly finer than the underlying global weak equivalences. In particular, we will show later in Theorem~\ref{thm:G-global-vs-proper-sset} that the weak equivalences are fine enough that one can recover proper $G$-equivariant homotopy theory as a Bousfield localization.
\begin{defi}\index{universal graph subgroup|textbf}
A graph subgroup $\Gamma=\Gamma_{H,\phi}$ of $\mathcal M\times G$ is called \emph{universal} if the corresponding subgroup $H\subset\mathcal M$ is universal. A graph subgroup $\Gamma\subset(E\mathcal M\times G)_0$ is universal if it is universal as a subgroup of $\mathcal M\times G$.
\end{defi}
\begin{ex}\nomenclature[aphiG]{$\blank\times_\phi G$}{quotient of $\blank\times G$ by graph subgroup}\label{ex:EM-fixed-points-corep}
Let $H\subset\mathcal M$ be a subgroup and let $\phi\colon H\to G$ be a group homomorphism. We write
\begin{equation*}
\mathcal M\times_\phi G\mathrel{:=} (\mathcal M\times G)/\Gamma_{H,\phi}
\qquad\text{and}\qquad
E\mathcal M\times_\phi G\mathrel{:=} (E\mathcal M\times G)/\Gamma_{H,\phi}
\end{equation*}
and call them \emph{$G$-global classifying spaces}.
It follows immediately from the constructions that $\mathcal M\times_\phi G$ corepresents $(\blank)^\phi$ on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ and that $E\mathcal M\times_\phi G$ corepresents $(\blank)^\phi$ on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$.
\end{ex}
As in the global situation we then conclude from Proposition~\ref{prop:equiv-model-structure}:
\begin{cor}
There exists a unique model structure on the category $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ of $\mathcal M$-$G$-simplicial sets in which a morphism $f$ is a weak equivalence or fibration if and only if $f^\phi$ is a weak homotopy equivalence or Kan fibration, respectively, for each universal $H\subset\mathcal M$ and each homomorphism $\phi\colon H\to G$.
We call this the \emph{$G$-universal model structure}\index{G-universal model structure@$G$-universal model structure|textbf}\index{universal model structure|seealso{$G$-universal model structure}} and its weak equivalences the \emph{$G$-universal weak equivalences}.\index{G-universal weak equivalence@$G$-universal weak equivalence|textbf}\index{universal weak equivalence|seealso{$G$-universal weak equivalence}} It is simplicial, combinatorial, proper, and filtered colimits in it are homotopical. A possible set of generating cofibrations is given by
\begin{equation*}
\{(\mathcal M\times_\phi G)\times(\del\Delta^n\hookrightarrow\Delta^n) : n\ge 0,H\subset\mathcal M\text{ universal},\phi\colon H\to G\text{ homomorphism}\}
\end{equation*}
and a possible set of generating acyclic cofibrations by
\begin{equation*}
\pushQED{\qed}
\{(\mathcal M\times_\phi G)\times(\Lambda^n_k\hookrightarrow\Delta^n) : 0\le k\le n,H\subset\mathcal M\text{ universal},\phi\colon H\to G\}.\qedhere
\popQED
\end{equation*}
\end{cor}
\begin{cor}\label{cor:EM-G-model-structure}
There is a unique model structure on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$ in which a map $f$ is a weak equivalence or fibration if and only if it is so in $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$.
We call this the \emph{$G$-global model structure}\index{G-global model structure@$G$-global model structure!on EM-G-SSet@on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$|textbf} and its weak equivalences the \emph{$G$-global weak equivalences}.\index{G-global weak equivalence@$G$-global weak equivalence!in EM-SSet@in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$|textbf} It is simplicial, combinatorial, proper, and filtered colimits in it are homotopical. A possible set of generating cofibrations is given by
\begin{equation*}
\{(E\mathcal M\times_\phi G)\times(\del\Delta^n\hookrightarrow\Delta^n) : n\ge 0,H\subset\mathcal M\text{ universal},\phi\colon H\to G\text{ homomorphism}\}
\end{equation*}
and a possible set of generating acyclic cofibrations by
\begin{equation*}
\pushQED{\qed}
\{(E\mathcal M\times_\phi G)\times(\Lambda^n_k\hookrightarrow\Delta^n) : 0\le k\le n,H\subset\mathcal M\text{ universal},\phi\colon H\to G\}.\qedhere
\popQED
\end{equation*}
\end{cor}
\begin{ex}\label{ex:G-globally-contractible}
As a concrete instance of Example~\ref{ex:EM-fixed-points-corep}, we can consider the case where $G$ itself is a universal subgroup of $\mathcal M$ (in particular finite), and $\phi$ is the identity. In this case, $E\mathcal M\times_\id G\cong E\mathcal M$ with $G$ acting via precomposition.
We claim that this is $G$-globally weakly contractible in the sense that the unique map to the terminal object is a $G$-global weak equivalence. Indeed, $(E\mathcal M)^\phi\cong E(\mathcal M^\phi)$, so it suffices that $\mathcal M^\phi\not=\varnothing$, i.e.~that there exists an $H$-equivariant injection $\phi^*\omega\to\omega$. This follows immediately from the universality of $H$.
More generally, the same argument shows that $E\Inj(A,\omega)$ is $G$-globally weakly contractible for any countable faithful $G$-set $A$.
\end{ex}
Analogously to the global situation, we will now show that the forgetful functor $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}\to\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ is fully faithful on associated quasi-categories and characterize its essential image.
\begin{defi}\label{defi:G-semistable}
An $\mathcal M$-$G$-simplicial set $X$ is called \emph{$G$-semistable}\index{G-semistable@$G$-semistable!M-G-simplicial set@$\mathcal M$-$G$-simplicial set|textbf} if the $(H\times G)$-equivariant map $u.\blank\colon X\to X$ is a $\mathcal G_{H,G}$-weak equivalence for any universal $H\subset\mathcal M$ and any $u\in\mathcal M$ centralizing $H$, i.e.~for any homomorphism $\phi\colon H\to G$ the induced map $u.\blank\colon X^\phi\to X^\phi$ is a weak homotopy equivalence.
\end{defi}
\subsubsection{Combinatorics of the $G$-global orbit category}\index{G-global orbit category@$G$-global orbit category|(}
As before, our comparison will proceed indirectly via the respective orbit categories. The mapping spaces in $\cat{O}_{E\mathcal M\times G}$ are again given by fixed points of quotients, and this section is devoted to clarifying their structure. In fact we will do all of this in greater generality now as we will need the additional flexibility later. For this let us fix a group $K$ together with a (finitely or infinitely) countable faithful $K$-set $A$ and a $G$-$K$-biset $X$.
We begin with the following trivial observation:
\begin{lemma}\label{lemma:emg-basic-general}
Let $(u_0,\dots,u_n;x),(v_0,\dots,v_n;y)\in\Inj(A,\omega)^{n+1}\times X$.
\begin{enumerate}
\item These represent the same $n$-simplex of $E\Inj(A,\omega)\times_KX$ (where `${\times_K}\mskip-1.25\thinmuskip$' means that we divide out the diagonal right $K$-action) if and only if there exists some $k\in K$ such that $y=x.k$ and $v_i=u_i.k$ for all $i=0,\dots,n$.\label{item:emgbg-def}
\item Assume these indeed represent the same $n$-simplex and that there is some $i$ such that $u_i=v_i$. Then $x=y$ and $u_j=v_j$ for \emph{all} $j=0,\dots,n$.\label{item:emgbg-unique}
\item Assume $x=y$. Then these represent the same $n$-simplex if and only if there is a $k\in\stabilizer_K(x)$ (where $\stabilizer_K$ denotes the stabilizer)\nomenclature[aStabHx]{$\stabilizer_H(x)$}{stabilizer of $x$ with respect to a given action of $H$} such that $v_i=u_i.k$ for $i=0,\dots,n$.\label{item:emgbg-stab}
\end{enumerate}
\begin{proof}
The first statement holds by definition and the third one is obviously a special case of this. Finally, the second statement follows from the first by freeness of the right $K$-action on $\Inj(A,\omega)$.
\end{proof}
\end{lemma}
We can now characterize the $\phi$-fixed points for any universal $H\subset\mathcal M$ and any homomorphism $\phi\colon H\to G$, generalizing Lemma~\ref{lemma:group-hom-associated}:
\begin{lemma}\label{lemma:emg-fixed-point-characterization-general}
Let $(u_0,\dots,u_n;x)\in\Inj(A,\omega)\times X$ such that $[u_0,\dots,u_n;x]\in (E\Inj(A,\omega)\times_KX)^\phi$. Then there exists for any $h\in H$ a unique $\sigma(h)\in K$ with
\begin{equation}\label{emg-gen:sigma-defining}
hu_i=u_i.\sigma(h)\qquad\text{for all $i=0,\dots,n$},
\end{equation}
and this satisfies
\begin{equation}\label{emg-gen:sigma-vs-phi}
x.\sigma(h)=\phi(h).x.
\end{equation}
The converse holds: whenever there exists a set map $\sigma\colon H\to K$ satisfying $(\ref{emg-gen:sigma-defining})$ and $(\ref{emg-gen:sigma-vs-phi})$ for all $h\in H$, then $[u_0,\dots,u_n;x]$ is a $\phi$-fixed point. Moreover, $\sigma$ is automatically a group homomorphism in this case.
\begin{proof}
That $(u_0,\dots,u_n;x)$ is $\phi$-fixed means by definition that for each $h\in H$
\begin{equation*}
(u_0,\dots,u_n;x)\sim(h,\phi(h)).(u_0,\dots,u_n;x)=(hu_0,\dots,hu_n;\phi(h).x)
\end{equation*}
which again means by definition that there exists a $\sigma(h)\in K$ such that $hu_i=u_i.\sigma(h)$ and moreover $\phi(h).x=x.\sigma(h)$. This $\sigma(h)$ is already uniquely characterized by the first property (for $i=0$) as $K$ acts freely from the right on $\Inj(A,\omega)$, proving the first half of the proposition.
Conversely, if such a $\sigma$ exists, then $[u_0,\dots,u_n;x]$ is clearly $\phi$-fixed. Moreover,
\begin{equation*}
u_0.\sigma(hh')=hh'u_0=hu_0.\sigma(h')=u_0.\sigma(h)\sigma(h'),
\end{equation*}
and hence $\sigma(hh')=\sigma(h)\sigma(h')$ by freeness of the right $K$-action.
\end{proof}
\end{lemma}
In the situation of Lemma~\ref{lemma:emg-fixed-point-characterization-general} we write $\sigma_{(u_0,\dots,u_n)}$\nomenclature[asigmau0un]{$\sigma_{(u_0,\dots,u_n)}$}{homomorphism witnessing that $[u_0,\dots,u_n]$ is a fixed point} for the unique (homomorphism) $H\to K$ satisfying $(\ref{emg-gen:sigma-defining})$. One can show that the lemma provides a complete characterization of the homomorphisms arising this way, which we will only need on the level of vertices:
\begin{cor}\label{cor:emg-equiv-group-hom-realization-general}
Let $\sigma\colon H\to K$ be any group homomorphism. Then there exists $u\in\Inj(A,\omega)$ such that $hu=u.\sigma(h)$ for all $h\in H$. Moreover, if $x\in X$ satisfies $(\ref{emg-gen:sigma-vs-phi})$, then $[u;x]$ is a $\phi$-fixed point of $E\Inj(A,\omega)\times_KX$.
\begin{proof}
When equipped with the tautological $H$-action, $\omega$ is a complete $H$-set universe; on the other hand, $\sigma^*A$ is a countable $H$-set by assumption, so that there exists an $H$-equivariant injection $u\colon\sigma^*A\to\omega$. The $H$-equivariance of $u$ directly translates to $hu=u.\sigma(h)$, and Lemma~\ref{lemma:emg-fixed-point-characterization-general} then proves that $[u;x]$ is $\phi$-fixed.
\end{proof}
\end{cor}
\subsubsection{The comparison} We can now describe the orbit category $\textbf{O}_{E\mathcal M\times G}$ (with respect to the universal graph subgroups) in more concrete terms:
\begin{rk}
The objects of $\textbf{O}_{E\mathcal M\times G}$ are the $E\mathcal M$-$G$-simplicial sets of the form $E\mathcal M\times_\phi G$ where $H\subset\mathcal M$ is universal and $\phi\colon H\to G$ is a homomorphism. If $K\subset\mathcal M$ is another universal subgroup and $\psi\colon K\to G$ a homomorphism, then Lemma~\ref{lemma:emg-fixed-point-characterization-general} tells us that any $n$-simplex of $\Maps_{\textbf{O}_{E\mathcal M\times G}}(E\mathcal M\times_\phi G, E\mathcal M\times_\psi G)$ can be represented by a tuple $(u_0,\dots,u_n;g)\in \mathcal M^{1+n}\times G$ such that there exists a (necessarily unique) group homomorphism $\sigma\colon H\to K$ satisfying
\begin{equation}\label{eq:fixed-point-relation}
hu_i=u_i\sigma(h) \qquad\text{and}\qquad \phi(h)g=g\psi(\sigma(h))
\end{equation}
for all $i=0,\dots,n$ and $h\in H$. Another such tuple $(u_0',\dots,u_n';g')$ represents the same morphism if and only if there exists a $k\in K$ such that $u_i'=u_ik$ for all $i=0,\dots,n$ and $g'=g\psi(k)$.
If $L\subset\mathcal M$ is another universal subgroup, $\theta\colon L\to G$ a group homomorphism, and if $(u_0',\dots,u_n';g')$ represents a morphism $E\mathcal M\times_\psi G\to E\mathcal M\times_\theta G$, then the composition $(u_0',\dots,u_n';g')(u_0,\dots,u_n;g)$ is represented by $(u_0u_0',\dots,u_nu_n';gg')$.
Similarly, objects of $\textbf{O}_{\mathcal M\times G}$ are the $\mathcal M$-$G$-sets $\mathcal M\times_\phi G$ with $\phi$ as above, and maps can be represented by pairs $(u;g)$ with $u\in\mathcal M$ and $g\in G$ satisfying analogous conditions to the above. Compositions are once more given by multiplication in $\mathcal M$ and $G$. In particular, we again get a functor $i\colon \textbf{O}_{\mathcal M\times G}\to\textbf{O}_{E\mathcal M\times G}$, sending $\mathcal M\times_\phi G$ to $E\mathcal M\times_\phi G$ and a morphism $\mathcal M\times_\phi G\to\mathcal M\times_\psi G$ represented by $(u;g)$ to the morphism $E\mathcal M\times_\phi G\to E\mathcal M\times_\psi G$ represented by the same pair $(u;g)$.
\end{rk}
\begin{defi}\index{G-centralizing morphism@$G$-centralizing morphism|textbf}
A morphism $f\colon\mathcal M\times_\phi G\to \mathcal M\times_\phi G$ in $\textbf{O}_{\mathcal M\times G}$ is called \emph{$G$-centralizing} if there exists a $u\in\mathcal M$ centralizing $H$ such that $f$ is represented by $(u;1)$. Analogously, we define $G$-centralizing morphisms in $\textbf{O}_{E\mathcal M\times G}$.
\end{defi}
\begin{prop}\index{G-global orbit category@$G$-global orbit category!as a simplicial localization}
The above functor $i\colon\textbf{\textup O}_{\mathcal M\times G}\to\textbf{\textup O}_{E\mathcal M\times G}$ is a simplicial localization at the $G$-centralizing morphisms.
\begin{proof}
Let us write $W\subset(\textbf{\textup O}_{E\mathcal M\times G})_0$ for the subcategory of $G$-centralizing morphisms. As in the purely global setting (Proposition~\ref{prop:em-quasi-localization}), this consists of homotopy equivalences and $\cat{O}_{E\mathcal M\times G}$ has fibrant mapping spaces, so that it is enough to prove that for each $n\ge 0$ the homotopical functor
\begin{equation*}
s^*\colon ((\textbf{\textup O}_{E\mathcal M\times G})_0, W)\to ((\textbf{\textup O}_{E\mathcal M\times G})_n, s^*W)
\end{equation*}
induced by the unique map $s\colon[n]\to[0]$ is a homotopy equivalence. As before we have a strict left inverse given by restriction along $i\colon [0]\to[n],0\mapsto 0$ and it suffices to construct a zig-zag of levelwise weak equivalences between $s^*i^*$ and the identity.
For this we recall from the proof of Proposition~\ref{prop:em-quasi-localization} that we can choose for each universal subgroup $H\subset\mathcal M$ injections $\alpha_H,\beta_H\in\mathcal M$ centralizing $H$ and such that $\omega=\im(\alpha_H)\sqcup\im(\beta_H)$.
Now let $H,K\subset\mathcal M$ be universal and let $\phi\colon H\to G$ and $\psi\colon K\to G$ be group homomorphisms. Then any morphism $E\mathcal M\times_\phi G\to E\mathcal M\times_\psi G$ in $(\textbf{\textup O}_{E\mathcal M\times G})_n$ can be represented by a tuple $(u_0,\dots,u_n;g)$ such that there exists a group homomorphism $\sigma\colon H\to K$ satisfying the relations $(\ref{eq:fixed-point-relation})$. We recall from the proof of the ordinary global case that there exists for each $i=0,\dots,n$ a unique $v_i$ such that
\begin{equation*}
v_i\alpha_K=\alpha_Hu_i\qquad\text{and}\qquad v_i\beta_K=\beta_Hu_0.
\end{equation*}
We claim that $(v_0,\dots,v_n;g)$ again defines a morphism, i.e.~its class in $E\mathcal M\times_\psi G$ is $\phi$-fixed. Indeed, we have seen in the global case that $hv_i=v_i\sigma(h)$, hence
\begin{align*}
(h,\phi(h)).(v_0,\dots,v_n;g)&=(hv_0,\dots,hv_n;\phi(h)g)\\ &=(v_0\sigma(h),\dots,v_n\sigma(h);g\psi(\sigma(h)))\sim (v_0,\dots,v_n;g)
\end{align*}
as desired. Similarly, one uses the argument from the non-equivariant case to show that the morphism represented by $(v_0,\dots,v_n;g)$ does not depend on the chosen representative $(u_0,\dots,u_n;g)$.
We now define a functor $f\colon(\textbf{\textup O}_{E\mathcal M\times G})_n\to(\textbf{\textup O}_{E\mathcal M\times G})_n$ as follows: $f$ is the identity on objects and on morphisms given by the above construction. Using that the above is independent of choices one easily checks that $f$ is indeed a functor. As before, we have by construction natural transformations $\id\Leftarrow f\Rightarrow s^*i^*$,
where the left hand transformation is given on $E\mathcal M\times_\phi G$ by $s^*[\alpha_H;1]=[\alpha_H,\dots,\alpha_H;1]$ and the right hand one by $s^*[\beta_H;1]$. As $\alpha_H$ and $\beta_H$ centralize $H$ by definition, these are weak equivalences, finishing the proof.
\end{proof}
\end{prop}
By the same arguments as in the ordinary global setting we deduce:
\begin{thm}\label{thm:em-vs-m-equiv}
The adjunction
\begin{equation*}
E\mathcal M\times_{\mathcal M}^{\textbf{\textup L}}\blank\colon\cat{$\bm{\mathcal M}$-$\bm G$-SSet}_{\textup{$G$-universal}}^\infty\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\infty :\!\forget^\infty
\end{equation*}
is a Bousfield localization;
in particular, $\forget^\infty$ is fully faithful. Moreover, its essential image consists precisely of the $G$-semistable $\mathcal M$-$G$-simplicial sets.\index{G-global orbit category@$G$-global orbit category|)}\qed
\end{thm}
\subsubsection{Additional model structures}
The following corollary lifts the above comparison of quasi-categories to the level of model categories:
\begin{cor}\label{cor:em-vs-m-equiv-model-cat}\index{G-global model structure@$G$-global model structure!on M-G-SSet@on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$|textbf}
There is a unique model structure on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ in which a map $f\colon X\to Y$ is a weak equivalence iff $E\mathcal M\times_{\mathcal M}^{\textbf{\textup L}}f$ is an isomorphism in $\Ho(\cat{$\bm{E\mathcal M}$-$\bm G$-SSet})$, and a cofibration iff it is so in the $G$-universal model structure on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$.
An object $X\in\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ is fibrant in this model structure if and only if it is fibrant in the $G$-universal model structure and moreover $G$-semistable in the sense of Definition~\ref{defi:G-semistable}.
We call this the \emph{$G$-global model structure} and its weak equivalences the \emph{$G$-global weak equivalences}.\index{G-global weak equivalence@$G$-global weak equivalence!in M-G-SSet@in $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$|textbf} It is combinatorial with generating cofibrations
\begin{equation*}
\{(\mathcal M\times_\phi G)\times(\del\Delta^n\hookrightarrow\Delta^n) : n\ge 0, H\subset\mathcal M\text{ universal},\phi\colon H\to G\text{ homomorphism}\},
\end{equation*}
simplicial, left proper, and filtered colimits in it are homotopical.
Finally, the simplicial adjunction
\begin{equation*}
E\mathcal M\times_\mathcal M\blank\colon \cat{$\bm{\mathcal M}$-$\bm G$-SSet}_{\textup{$G$-global}} \rightleftarrows \cat{$\bm{E\mathcal M}$-$\bm G$-SSet} :\!\forget
\end{equation*}
is a Quillen equivalence with homotopical right adjoint.
\begin{proof}
Theorem~\ref{thm:em-vs-m-equiv} allows us to invoke Lurie's localization criterion (Theorem~\ref{thm:lurie-localization-criterion}) which proves all of the above claims except for the part about filtered colimits, which is in turn an instance of Lemma~\ref{lemma:filtered-still-homotopical}.
\end{proof}
\end{cor}
As a special case of Corollary~\ref{cor:equivariant-injective-model-structure}, the $G$-global weak equivalences of $E\mathcal M$-$G$-simplicial sets are part of an injective model structure.\index{injective G-global model structure@injective $G$-global model structure!on EM-G-SSet@on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$} We will now prove the analogue of this for $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$:
\begin{thm}\label{thm:G-M-injective-semistable-model-structure}\index{G-global model structure@$G$-global model structure!injective|seeonly{injective $G$-global model structure}}\index{injective G-global model structure@injective $G$-global model structure!on M-G-SSet@on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$|textbf}
There exists a unique model structure on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ whose cofibrations are the underlying cofibrations and whose weak equivalences are the $G$-global weak equivalences. We call this the \emph{injective $G$-global model structure}. It is combinatorial, simplicial, left proper, and filtered colimits in it are homotopical.
Finally, the simplicial adjunction
\begin{equation*}
\forget\colon\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\textup{inj.~$G$-global}}\rightleftarrows\cat{$\bm{\mathcal M}$-$\bm G$-SSet}_{\textup{inj.~$G$-global}} :\!\Maps^{\mathcal M}(E\mathcal M,\blank)
\end{equation*}
is a Quillen equivalence with homotopical left adjoint.
\begin{proof}
We first claim that the $G$-global weak equivalences are stable under pushout along \emph{injective} cofibrations. For this we let $f\colon A\to B$ be a $G$-global weak equivalence and $i\colon A\to C$ an injective cofibration. Applying the factorization axiom in the $G$-global model structure, we can factor $f=pk$ where $k$ is an acyclic cofibration and $p$ is an acyclic fibration. As the $G$-global and $G$-universal model structures on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ have the same cofibrations, they also have the same acyclic fibrations; in particular, $p$ is a $G$-universal weak equivalence.
We now consider the iterated pushout
\begin{equation*}
\begin{tikzcd}
A \arrow[rr, bend left=25pt, "f"]\arrow[r, "k"']\arrow[d, "i"'] & X\arrow[r, "p\vphantom{k}"']\arrow[d, "j"] & B\arrow[d]\\
C \arrow[r, "\ell"'] & Y \arrow[r, "q\vphantom{k}"'] & D.
\end{tikzcd}
\end{equation*}
Then $\ell$ is an acyclic cofibration in the $G$-global model structure as a pushout of an acyclic cofibration. Moreover, $j$ is an injective cofibration as a pushout of an injective cofibration, so $q$ is a $G$-universal (and hence in particular $G$-global) weak equivalence by left properness of the equivariant injective model structure. The claim follows as $q\ell$ is a pushout of $f$ along $i$.
We therefore conclude from Corollary~\ref{cor:mix-model-structures} that the model structure exists and that it is combinatorial and left proper. Moreover, Lemma~\ref{lemma:filtered-still-homotopical} shows that filtered colimits in it are still homotopical, so it only remains to verify the Pushout Product Axiom for the simplicial tensoring.
For this we can argue precisely as in the proof of Corollary~\ref{cor:equivariant-injective-model-structure} once we show that for any simplicial set $K$ the functor $K\times\blank$ preserves $G$-global weak equivalences, and that for any $\mathcal M$-$G$-simplicial set $X$ the functor $\blank\times X$ sends weak equivalences of simplicial sets to $G$-global weak equivalences.
For the second statement it is actually clear that $\blank\times X$ sends weak equivalences even to $G$-universal weak equivalences. For the first statement it is similarly clear that $K\times\blank$ preserves $G$-universal weak equivalences, but it also preserves acyclic cofibrations in the usual $G$-global model structure as the latter is simplicial. The claim again follows as any $G$-global weak equivalence can be factored as a $G$-global acyclic cofibration followed by a $G$-universal weak equivalence.
\end{proof}
\end{thm}
\begin{rk}
With a bit of (combinatorial) work one can show that $\forget\dashv\Maps^{\mathcal M}(E\mathcal M,\blank)$ is also a Quillen equivalence for the usual $G$-global model structures. On the other hand, I do not know whether $E\mathcal M\times_{\mathcal M}\blank$ preserves injective cofibrations or $G$-global weak equivalences in general, so it is not clear whether $E\mathcal M\times_{\mathcal M}\blank\dashv\forget$ is also a Quillen equivalence for the injective $G$-global model structures.
\end{rk}
\subsection{An explicit \texorpdfstring{$\bm G$}{G}-semistable replacement}\label{sec:g-semistable-replacement}\index{G-semistable@$G$-semistable!M-G-simplicial set@$\mathcal M$-$G$-simplicial set!replacement|(}
If one wants to check if a morphism $f\colon X\to Y$ in $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ is a $G$-global weak equivalence straight from the definition, one runs into trouble as soon as at least one of $X$ or $Y$ is not cofibrant because computing $E\mathcal M\times_{\mathcal M}^{\cat{L}}f$ then involves cofibrant replacements, and the ones provided by the small object argument are completely intractable.
On the other hand, the \emph{$G$-universal} weak equivalences are much easier to understand, so one could instead try to take $G$-semistable replacements of $X$ and $Y$ and then apply the following general observation about Bousfield localizations:
\begin{lemma}\label{lemma:semistable-between-semistable}
A morphism $f\colon X\to Y$ of $G$-semistable $\mathcal M$-$G$-simplicial sets is a $G$-global weak equivalence if and only if it is a $G$-universal weak equivalence.\qed
\end{lemma}
However, this leaves us with the problem of finding (functorial) $G$-semistable replacements. While the $G$-global model structure asserts that these exist, they are again completely inexplicit. We could also try to construct them by means of $E\mathcal M\times_{\mathcal M}^{\textbf L}\blank$, but then we would of course be back where we started.
In this subsection we will remedy this situation by constructing explicit $G$-semistable replacements, yielding a characterization of the $G$-global weak equivalences. This characterization will become crucial later (see e.g.~Theorem~\ref{thm:tame-M-sset-vs-EM-sset}), and in particular I do not know how to prove the respective statements `by hand.'
\begin{rk}
Before we come to the construction, let us think about how it should look like. The simplicial set $E\mathcal M$ is (weakly) contractible, and one can conclude from this, see e.g.~\cite[Proposition~4.2.4.4]{htt}, that $\cat{$\bm{E\mathcal M}$-SSet}$ models ordinary non-equivariant homotopy theory when equipped with the \emph{underlying} weak equivalences. More precisely, with respect to these weak equivalences, the homotopical functors
\begin{equation*}
\const\colon\cat{SSet}\to\cat{$\bm{E\mathcal M}$-SSet}\qquad\text{and}\qquad
\forget\colon\cat{$\bm{E\mathcal M}$-SSet}\to\cat{SSet}
\end{equation*}
induce mutually quasi-inverse equivalences on associated quasi-categories. It follows that the composition
\begin{equation*}
\cat{$\bm{\mathcal M}$-SSet}^\infty_{\textup{$G$-universal}}\xrightarrow{E\mathcal M\times_{\mathcal M}^{\textbf L}\blank}\cat{$\bm{E\mathcal M}$-SSet}^\infty_{\textup{$G$-global}}\xrightarrow{\forget}\cat{SSet}^\infty
\end{equation*}
is equivalent to taking homotopy colimits over $\mathcal M$, also cf.~\cite[Definition~3.2]{I-vs-M-1-cat}.
\end{rk}
\subsubsection{Action categories}
The remark suggests that we might be lucky and succeed in constructing the replacement by means of a suitable equivariant enhancement of one of the standard constructions of homotopy colimits. This will indeed work for the model of what is usually called the \emph{action groupoid} (although it won't be a groupoid in our case), which we now recall:
\begin{constr}\nomenclature[aM]{$(\blank)\hq\mathcal M$}{$G$-global action category/bar construction}
Let $X$ be any $\mathcal M$-set. We write $X\hq\mathcal M$ for the \emph{action category},\index{action category|textbf} i.e.~the category with set of objects $X$ and for any $x\in X$ and $u\in\mathcal M$ a morphism $u\colon x\to u.x$; we emphasize that this means that if $u\not=v$ with $u.x=v.x$ then $u$ and $v$ define two distinct morphisms $x\to u.x=v.x$. The composition in $X\hq\mathcal M$ is given by multiplication in $\mathcal M$.
\end{constr}
The $\mathcal M$-action on $X$ immediately gives an $\mathcal M$-action on $\Ob(X\hq\mathcal M)$; however, it is not entirely clear how to extend this to morphisms. For an invertible element $\alpha\in\core\mathcal M$, the condition that $\alpha.f$ for $f\colon x\to y$ should be a morphism $\alpha.x\to \alpha.y$ naturally leads to the guess $\alpha.f=\alpha f\alpha^{-1}$. While general elements of $\mathcal M$ are not invertible, there is still a notion of \emph{conjugation}. This is made precise by the following construction, which is implicit in \cite[proof of Lemma~5.2]{schwede-semistable} (which Schwede attributes to Strickland) and also appeared in an earlier version of \cite{schwede-k-theory}:
\begin{constr}\label{constr:M-conjugation}\index{conjugation on M@conjugation on $\mathcal M$|textbf}
Let $\alpha\in\mathcal M$. We define for any $u\in\mathcal M$ the \emph{conjugation} $c_\alpha(u)$\nomenclature[acalpha]{$c_\alpha$}{conjugation by $\alpha\in\mathcal M$} of $u$ by $\alpha$ via
\begin{equation*}
c_\alpha(u)(x)=\begin{cases}
\alpha u(y) & \text{if }x=\alpha(y)\\
x & \text{if }x\notin\im \alpha.
\end{cases}
\end{equation*}
We remark that this is well-defined (as $\alpha$ is injective), and one easily checks that this is again injective, so that we get a map $c_\alpha\colon\mathcal M\to\mathcal M$.
\end{constr}
Put differently, $c_\alpha(u)$ is the unique element of $\mathcal M$ such that
\begin{equation*}
c_\alpha(u)\alpha=\alpha u\qquad\text{and}\qquad c_\alpha(u)(x)=x\text{ for $x\notin\im \alpha$}.
\end{equation*}
The first condition justifies the name `conjugation,' and it means in particular that $c_\alpha(u)=\alpha u\alpha^{-1}$ for invertible $\alpha$.
For a group $G$, conjugation by a fixed element $g$ defines an endomorphism of $G$, and letting $g$ vary this yields an action of $G$ on itself. The analogous statement holds for the above construction:
\begin{lemma}\label{lemma:conjugation-homomorphism}
For any $\alpha\in\mathcal M$, the map $c_\alpha\colon\mathcal M\to\mathcal M$ is a monoid homomorphism. Moreover, for varying $\alpha$ this defines an action of $\mathcal M$ on itself, i.e.~$c_1=\id_{\mathcal M}$ and $c_\alpha\circ c_\beta=c_{\alpha\beta}$ for all $\alpha,\beta\in\mathcal M$.
\begin{proof}
We will only prove the first statement, the calculations for the other claims being similar. For this let $u,v\in\mathcal M$. Then
\begin{equation*}
c_\alpha(uv)\alpha=\alpha uv= c_\alpha(u)\alpha v=c_\alpha(u)c_\alpha(v)\alpha.
\end{equation*}
On the other hand, if $x\notin\im(\alpha)$, then $c_\alpha(uv)(x)=x$ and $c_\alpha(v)(x)=x\notin\im\alpha$, hence also $(c_\alpha(u)c_\alpha(v))(x)=x$. We conclude that $c_\alpha(uv)=c_\alpha(u)c_\alpha(v)$. Moreover, clearly $c_\alpha(1)=1$, so that $c_\alpha$ is indeed a monoid homomorphism.
\end{proof}
\end{lemma}
\begin{constr}
We define a $\mathcal M$-action on $X\hq\mathcal M$ as follows: the action of $\mathcal M$ on objects is the one on $X$, and on morphisms $\alpha\in\mathcal M$ acts by sending
\begin{equation*}
x\xrightarrow{u}u.x\qquad\text{to}\qquad \alpha.x\xrightarrow{c_\alpha(u)}\alpha.(u.x);
\end{equation*}
note that this is indeed a morphism as $c_\alpha(u).(\alpha.x)=(c_\alpha(u)\alpha).x=(\alpha u).x=\alpha.(u.x)$ by construction of $c_\alpha$. By the previous lemma, this is then an endofunctor of $X\hq\mathcal M$, and for varying $\alpha$ this yields an $\mathcal M$-action.
If $f\colon X\to Y$ is a map of $\mathcal M$-sets, then we write $f\hq\mathcal M$ for the functor that is given on objects by $f$ and that sends a morphism
\begin{equation*}
x\xrightarrow{u}u.x\qquad\text{to}\qquad f(x)\xrightarrow{u}u.f(x)=f(u.x).
\end{equation*}
One easily checks that this is well-defined, functorial in $f$, and that $f\hq\mathcal M$ is $\mathcal M$-equivariant. Postcomposing with the nerve we therefore get a functor $\cat{$\bm{\mathcal M}$-Set}\to\cat{$\bm{\mathcal M}$-SSet}$ that we denote by $(\blank)\hq\mathcal M$ again. If $G$ is any group, we moreover get an induced functor $\cat{$\bm{\mathcal M}$-$\bm G$-Set}\to\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ by pulling through the $G$-action (again denoted by the same symbol).
If $X$ is any $\mathcal M$-set, then there is a unique functor from $X$ (viewed as a discrete category) to $X\hq\mathcal M$ that is the identity on objects. This then yields a natural transformation $\pi\colon\discr\Rightarrow(\blank)\hq\mathcal M$ from the functor that sends an $\mathcal M$-$G$-set $X$ to the discrete simplicial set $X$ with the induced action.
\end{constr}
\begin{constr}
Let $X$ be any $\mathcal M$-$G$-simplicial set. Applying the above construction levelwise yields a bisimplicial set $X\#\mathcal M$ with $(X\#\mathcal M)_{n,\bullet}=X_n\hq\mathcal M$, and this receives a map from the bisimplicial set $\Discr X$ with $(\Discr X)_{n,\bullet}=\discr X_n$. This yields a functor $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}\to\cat{$\bm{\mathcal M}$-$\bm G$-BiSSet}$\nomenclature[aBiSSet]{$\cat{BiSSet}$}{category of bisimplicial sets} receiving a natural transformation $\Pi\colon\Discr\Rightarrow\#$.
Taking diagonals, we then obtain a functor $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}\to\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$, that we again denote by $(\blank)\hq\mathcal M$, together with a natural transformation $\id\Rightarrow(\blank)\hq\mathcal M$, that we again denote by $\pi$. We remark that on $\mathcal M$-$G$-sets (viewed as discrete simplicial sets) this recovers the previous construction.
\end{constr}
\subsubsection{The detection result}
Now we are ready to state the main result of this subsection:
\begin{thm}\label{thm:hq-M-semistable-replacement}
\begin{enumerate}
\item For any $X\in\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$, the map $\pi_X\colon X\to X\hq\mathcal M$ is a $G$-global weak equivalence, and $X\hq\mathcal M$ is $G$-semistable.
\item For any $f\colon X\to Y$ in $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ the following are equivalent:
\begin{enumerate}
\item $f$ is a $G$-global weak equivalence.
\item $f\hq\mathcal M$ is a $G$-global weak equivalence.
\item $f\hq\mathcal M$ is a $G$-universal weak equivalence.
\end{enumerate}
\end{enumerate}
\end{thm}
The proof of the theorem will occupy the rest of this subsection.
\begin{rk}
We defined the bisimplicial set $X\#\mathcal M$ in terms of the action category construction. We can also look at this bisimplicial set `from the other side,' which recovers the bar construction (just as in the usual construction of non-equivariant homotopy quotients):
If $Y$ is any $\mathcal M$-$G$-set, then $(Y\hq\mathcal M)_m$ consists by definition of the $m$-chains
\begin{equation*}
y\xrightarrow{u_1}u_1.y\xrightarrow{u_2}(u_2u_1).y\xrightarrow{u_3}\cdots\xrightarrow{u_m}(u_m\cdots u_1).y
\end{equation*}
of morphisms in $Y\hq\mathcal M$. Such a chain is obviously uniquely described by the source $y\in Y$ together with the injections $u_m,\dots, u_2,u_1\in\mathcal M$, which yields a bijection $(Y\hq\mathcal M)_m\cong \mathcal M^m\times Y$. This bijection becomes $\mathcal M$-$G$-equivariant, when we let $G$ act via its action on $X$ and $\mathcal M$ via its action on $X$ and the conjugation action on each of the $\mathcal M$-factors.
The assignment $Y\mapsto\mathcal M^m\times Y$ becomes a functor in $Y$ in the obvious way, and with respect to this the above bijection is clearly natural in $\mathcal M$-equivariant maps. Applying this levelwise, we therefore get a natural isomorphism
\begin{equation}\label{eq:x-hash-m-other-levels}
(X\#\mathcal M)_{\bullet,m}\cong\mathcal M^m\times X
\end{equation}
of $\mathcal M$-$G$-simplicial sets. While we will not need this below, we remark that unravelling the definitions, one can work out that under the isomorphism $(\ref{eq:x-hash-m-other-levels})$ the simplicial structure maps of $X\#\mathcal M$ indeed correspond to those of the usual bar construction.
\end{rk}
By construction and the previous remark, we understand the bisimplicial set $X\#\mathcal M$ in both its simplicial directions individually. In non-equivariant simplicial homotopy theory, the \emph{Diagonal Lemma} then often allows to leverage this to prove statements about the diagonal ($X\hq\mathcal M$ in our case). Luckily, this immediately carries over to our situation:
\begin{lemma}\label{lemma:equivariant-diagonal}\index{Diagonal Lemma!equivariant|textbf}\index{Diagonal Lemma}
Let $M$ be a monoid, let $\mathcal F$ be a collection of subgroups of $M$, and let $f\colon X\to Y$ be a map of $M$-bisimplicial sets. Assume that for each $n\ge 0$ the map $f_{n,\bullet}\colon X_{n,\bullet}\to Y_{n,\bullet}$ is a $\mathcal F$-weak equivalence, or that for each $m\ge 0$ the map $f_{\bullet,m}\colon X_{\bullet,m}\to Y_{\bullet,m}$ is. Then also $\diag f\colon\diag X\to\diag Y$ is a $\mathcal F$-weak equivalence.
\begin{proof}
By symmetry it suffices to consider the first case. If $H\in\mathcal F$ is any subgroup, then $(f_{n,\bullet})^H$ literally agrees with $(f^H)_{n,\bullet}$ (if we take the usual construction of fixed points), and likewise $(\diag f)^H=\diag(f^H)$. Thus, the claim follows immediately from the usual Diagonal Lemma, see e.g.~\cite[Proposition~IV.1.7]{goerss}.
\end{proof}
\end{lemma}
Let us draw some non-trivial consequences from this:
\begin{cor}\label{cor:X-hq-M-homotopical}
The above functor $(\blank)\hq\mathcal M$ preserves $G$-universal weak equivalences of $\mathcal M$-$G$-simplicial sets.
\begin{proof}
By the previous lemma (applied to the universal graph subgroups of the monoid $\mathcal M\times G$), it suffices that each $(\blank\#\mathcal M)_{\bullet,m}$ does, which follows from the isomorphism~$(\ref{eq:x-hash-m-other-levels})$ together with Lemma~\ref{lemma:equivariant-weak-equivalences-prod-homotopical}.
\end{proof}
\end{cor}
\begin{lemma}\label{lemma:X-hq-M-semistable}
Let $X$ be any $\mathcal M$-$G$-simplicial set. Then $X\hq\mathcal M$ is $G$-semistable.
\begin{proof}
Let $H\subset\mathcal M$ be any subgroup and let $\alpha\in\mathcal M$ centralize $H$. We will show that $\alpha.\blank\colon X\hq\mathcal M\to X\hq\mathcal M$ is even an $(H\times G)$-weak equivalence.
By Lemma~\ref{lemma:equivariant-diagonal} we reduce this to the case that $X$ is a \emph{$\mathcal M$-$G$-set}, in which case $X\hq\mathcal M$ is the nerve of the action category. We then observe that the maps $\alpha\colon x\to \alpha.x$ assemble into a natural transformation $a\colon\id\Rightarrow(\alpha.\blank)$ by virtue of the identity $c_\alpha(u)\alpha=\alpha u$. This natural transformation is obviously $G$-equivariant, but it is also $H$-equivariant: if $h\in H$ and $x\in X$ are arbitrary, then $h.a_x$ is by definition the map $h\alpha h^{-1}\colon h.x\to h.(\alpha.x)$; as $h$ commutes with $\alpha$, this agrees with $a_{h.x}$ as desired. Upon taking nerves, $a$ therefore induces an $(H\times G)$-equvariant homotopy between the identity and $\alpha.\blank$; in particular, $\alpha.\blank$ is an $(H\times G)$-weak equivalence.
\end{proof}
\end{lemma}
The following two statements are again easily deduced from the isomorphism $(\ref{eq:x-hash-m-other-levels})$, and we omit their proofs.
\begin{cor}\label{cor:hq-M-homotopy-po}
The functor $(\blank)\hq\mathcal M\colon\cat{$\bm{\mathcal M}$-$\bm G$-SSet}\to\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ is cocontinuous and it preserves injective cofibrations.\qed
\end{cor}
\begin{cor}\label{cor:hq-M-tensors}
The functor $(\blank)\hq\mathcal M\colon\cat{$\bm{\mathcal M}$-$\bm G$-SSet}\to\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ has a natural simplicial enrichment with respect to which it preserves tensors. The natural transformation $\pi$ is simplicially enriched.\qed
\end{cor}
The only remaining input that we need to prove the theorem is an explicit computation of $(\blank)\hq\mathcal M$ on the $\mathcal M$-$G$-sets $\mathcal M\times_\phi G$ for universal $H\subset\mathcal M$ and any group homomorphism $\phi\colon H\to G$. We will actually do this in greater generality, which will become important later in the proof of Theorem~\ref{thm:tame-M-sset-vs-EM-sset}:
\begin{thm}\label{thm:hq-M-computation}
Let $H\subset\mathcal M$ be universal, let $A$ be a countable faithful $H$-set, and let $X$ be any $G$-$H$-biset. If we consider $\Inj(A,\omega)$ as an $\mathcal M$-$H$-biset in the obvious way, then the unique functor
$\Inj(A,\omega)\hq\mathcal M \to E\Inj(A,\omega)$ that is the identity on objects induces a $G$-universal weak equivalence
\begin{equation}\label{eq:hq-m-comparison}
\big(\Inj(A,\omega)\times_HX\big)\hq\mathcal M\cong
\big(\Inj(A,\omega)\hq\mathcal M\big)\times_HX
\xrightarrow{\sim} \big(E\Inj(A,\omega)\big)\times_HX.
\end{equation}
Here the unlabelled isomorphism comes from the cocontinuity of $(\blank)\hq\mathcal M$.
\end{thm}
\begin{rk}
The theorem of course also holds for $A$ uncountable (as then both sides are just empty). However, some lemmas we will appeal to only hold for countable $A$ and this is also the only case we will be interested in. Therefore, we have decided to state the theorem in the above form.
\end{rk}
The proof of the theorem needs some preparations and will be given at the end of this subsection. Before that, let us already use it to deduce the detection result. This will involve a standard `cell induction'\index{cell induction|textbf} argument; as similar lines of reasoning will appear again later, we formalize this part once and for all:
\begin{lemma}\label{lemma:saturated-objects}
Let $\mathscr C$ be a cocomplete category, let $I$ be a class of morphisms and let $\mathscr S$ be a class of objects in $\mathscr C$ such that the following conditions are satisfied:
\begin{enumerate}
\item $\mathscr S$ contains the initial object $\varnothing$\label{item:so-empty}.
\item If
\begin{equation}\label{diag:pushout-generating-cof}
\begin{tikzcd}
A\arrow[d]\arrow[r, "i"] & B\arrow[d]\\
C \arrow[r] & D
\end{tikzcd}
\end{equation}
is a pushout with $C\in\mathscr S$ and $i\in I$, then $D\in\mathscr S$.\label{item:so-pushout-closure}
\item $\mathscr S$ is closed under filtered colimits.\label{item:so-closure-filtered}
\end{enumerate}
Then $\mathscr S$ contains all $I$-cell complexes. If in addition also
\begin{enumerate}
\setcounter{enumi}{3}
\item $\mathscr S$ is closed under retracts,\label{item:so-retract-closure}
\end{enumerate}
then $\mathscr S$ contains all retracts of $I$-cell complexes. In particular, if $\mathscr C$ is a cofibrantly generated model category and $I$ a set of generating cofibrations, then $\mathscr S$ contains all cofibrant objects.
\begin{proof}
It is enough to prove the first statement. For this let $X$ be an $I$-cell complex, i.e.~there exists an ordinal $\alpha$ and a functor $X_\bullet\colon\{\beta<\alpha\}\to\mathscr C$ such that the following conditions hold:
\begin{enumerate}
\item[(A)] $X_0=\varnothing$
\item[(B)] For each $\beta$ with $\beta+1<\alpha$, the map $X_\beta\to X_{\beta+1}$ can be written as a pushout of some generating cofibration $i\in I$.
\item[(C)] If $\beta<\alpha$ is a limit ordinal, then the maps $X_\gamma\to X_\beta$ for $\gamma<\beta$ exhibit $X_\beta$ as a (filtered) colimit.
\end{enumerate}
If $\alpha$ is a limit ordinal, we extend this to $\{\beta\le\alpha\}$ via $X_\alpha\mathrel{:=}\colim_{\beta<\alpha}X_\beta$ together with the obvious structure maps; if $\alpha$ is a successor ordinal instead, we replace $\alpha$ by its predecessor. In both cases we obtain a functor $X_\bullet\colon\{\beta\le\alpha\}\to\mathscr C$ satisfying conditions (A)--(C) above (with `$<$' replaced by `$\le$') and such that $X_\alpha=X$. We will prove by transfinite induction that $X_\beta\in\mathscr S$ for all $\beta\le\alpha$ which will then imply the claim.
By Conditions $(\ref{item:so-empty})$ and (A) we see that $X_0=\varnothing\in\mathscr S$. Now assume $\beta>0$, and we know the claim for all $\gamma<\beta$. If $\beta$ is a successor ordinal, $\beta=\gamma+1$, then Conditions $(\ref{item:so-pushout-closure})$ and (B) together with the induction hypothesis imply that $X_\beta\in\mathscr S$. On the other hand, if $\beta$ is a limit ordinal, then the maps $X_\gamma\to X_\beta$ for $\gamma<\beta$ express $X_\beta$ as a filtered colimit of elements of $\mathscr S$ by Condition (C) together with the induction hypothesis. Thus, Condition~$(\ref{item:so-closure-filtered})$ immediately implies the claim, finishing the proof.
\end{proof}
\end{lemma}
\begin{cor}\label{cor:saturated-trafo}
Let $\mathscr C$ be a cocomplete category and let $\mathscr D$ be a left proper model category such that filtered colimits in it are homotopical. Let $I$ be any collection of morphisms in $\mathscr C$, and let $F,G\colon\mathscr C\to\mathscr D$ be functors together with a natural transformation $\tau\colon F\Rightarrow G$. Assume the following:
\begin{enumerate}
\item $\tau_\varnothing$ is a weak equivalence and for every map $(X\to Y)\in I$ both $\tau_X$ and $\tau_Y$ are weak equivalences. \label{item:st-base}
\item Both $F$ and $G$ send pushouts along maps $i\in I$ to homotopy pushouts in $\mathscr D$.\label{item:st-pushout}
\item $F$ and $G$ preserve filtered colimits up to weak equivalence in the sense that for any filtered category $J$ and any diagram $X\colon J\to\mathscr C$ the canonical maps $\colim_J F\circ X\to F(\colim_JX)$ and $\colim_J G\circ X\to G(\colim_JX)$ are weak equivalences.\label{item:st-filtered}
\end{enumerate}
Then $\tau$ is a weak equivalence on all retracts of $I$-cell complexes. In particular, if $\mathscr C$ is a cofibrantly generated model category and $I$ is a set of generating cofibrations, then $\tau_X$ is a weak equivalence for all cofibrant $X$.
\begin{proof}
Let $\mathscr S$ be the class of objects $X\in\mathscr C$ such that $\tau_X$ is a weak equivalence. It will suffice to verify the conditions of the previous lemma. Indeed, Condition $(\ref{item:so-empty})$ of the lemma is in immediate consequence of our Condition~$(\ref{item:st-base})$.
In order to verify Condition~$(\ref{item:so-pushout-closure})$ of the lemma, we consider a pushout as in $(\ref{diag:pushout-generating-cof})$. By naturality, this induces a commutative cube
\begin{equation*}
\begin{tikzcd}[row sep=small, column sep=small]
& GA\arrow[dd] \arrow[rr, "Gi"] && GB\arrow[dd]\\
FA\arrow[dd]\arrow[ur] \arrow[rr, "Fi" near end, crossing over] && FB\arrow[ur]\\
& GC\arrow[rr] && GD\\
FC\arrow[ur]\arrow[rr] && FD\arrow[from=uu, crossing over]\arrow[ur]
\end{tikzcd}
\end{equation*}
with all the diagonal maps coming from $\tau$. The assumption $C\in\mathscr S$ implies that the front-to-back map at the lower left corner is a weak equivalence, and our Condition $(\ref{item:st-base})$ tells us that the two top front-to-back maps are weak equivalences. Moreover, both front and back square are homotopy pushouts by Condition~$(\ref{item:st-pushout})$. We conclude that also the lower right front-to-back map is a weak equivalence, just as desired.
To verify Condition~$(\ref{item:so-closure-filtered})$ of the lemma, we observe that for any filtered category $J$ and any $X_\bullet\colon J\to\mathscr C$ the map $\tau_{\colim_{j\in J} X_j}$
fits into a commutative diagram
\begin{equation*}
\begin{tikzcd}[column sep=huge]
\colim_{j\in J} F(X_j)\arrow[d]\arrow[r, "{\colim_{j\in J}\tau_{X_j}}"] & \colim_{j\in J} G(X_j)\arrow[d]\\
F(\colim_{j\in J} X_j) \arrow[r, "{\tau_{\colim_{j\in J} X_j}}"'] & G\big(\colim_{j\in J} X_j\big)
\end{tikzcd}
\end{equation*}
where the vertical maps are the canonical comparison maps and hence weak equivalences by our Condition~($\ref{item:st-filtered}$). If now all $X_i$ lie in $\mathscr S$, then the top horizontal map is a filtered colimit of weak equivalences and hence a weak equivalence by assumption, proving Condition $(\ref{item:so-closure-filtered})$ of the lemma.
Finally, Condition~($\ref{item:so-retract-closure}$) for $\mathscr S$ is automatic as weak equivalences in any model category are closed under retracts. This finishes the proof.
\end{proof}
\end{cor}
\begin{proof}[Proof of Theorem~\ref{thm:hq-M-semistable-replacement}]
We will only prove the first statement; the second one will then follow formally from this (cf.~Lemma~\ref{lemma:semistable-between-semistable}.)
We already know by Lemma~\ref{lemma:X-hq-M-semistable} that $X\hq\mathcal M$ is $G$-semistable for any $X\in\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$. It therefore only remains to prove that $\pi_X\colon X\to X\hq\mathcal M$ is a $G$-global weak equivalence. Since we have seen in Corollary~\ref{cor:X-hq-M-homotopical} that $(\blank)\hq\mathcal M$ preserves $G$-universal weak equivalences, it suffices to prove this for cofibrant $X$, and Corollary~\ref{cor:hq-M-homotopy-po} together with Corollary~\ref{cor:saturated-trafo} reduces this further to the case that $X$ is the source or target of one of the standard generating cofibrations, i.e.~$X=(\mathcal M\times_\phi G)\times\del\Delta^n$ or $X=(\mathcal M\times_\phi G)\times\Delta^n$ for some $n\ge 0$. By Corollary~\ref{cor:hq-M-tensors} we are then further reduced to $X=\mathcal M\times_\phi G$.
In this case we observe that the unit $\eta$ of the adjunction $E\mathcal M\times_{\mathcal M}(\blank)\dashv\forget$, evaluated at $\mathcal M\times_\phi G$ can be factored as
\begin{equation*}
\mathcal M\times_\phi G\xrightarrow{\pi} (\mathcal M\times_\phi G)\hq\mathcal M\xrightarrow{(\ref{eq:hq-m-comparison})}E\mathcal M\times_\phi G\cong\forget\big(E\mathcal M\times_{\mathcal M}(\mathcal M\times_\phi G)\big),
\end{equation*}
where the middle arrow comes from applying Theorem~\ref{thm:hq-M-computation} for $A=\omega$ and $X=G$ with its left regular $G$-action and $H$ acting from the right via $\phi$, and the final isomorphism uses that the left adjoint $E\mathcal M\times_{\mathcal M}\blank$ is cocontinuous. The above composition is a $G$-global weak equivalence by Corollary~\ref{cor:em-vs-m-equiv-model-cat}, and the middle arrow is even a $G$-universal weak equivalence by Theorem~\ref{thm:hq-M-computation}. Therefore, $2$-out-of-$3$ implies that $\pi$ is a $G$-global weak equivalence as desired. By the above reduction, this completes the proof.\index{G-semistable@$G$-semistable!M-G-simplicial set@$\mathcal M$-$G$-simplicial set!replacement|)}
\end{proof}
It remains to prove that the map $(\ref{eq:hq-m-comparison})$ is a $G$-universal weak equivalence.
\begin{ex}
As the proof of the theorem will be quite technical, let us begin with something much easier that will nevertheless give an idea of the general argument: we will show that the map in question is a non-equivariant weak equivalence for $A$ finite and $H=1$, which amounts to saying that $\Inj(A,\omega)\hq\mathcal M$ is weakly contractible as a simplicial set. This argument also appears as \cite[Example~3.3]{I-vs-M-1-cat}.
By construction, the category $\Inj(A,\omega)\hq\mathcal M$ has objects the injections $i\colon A\to\omega$ and it has for every $u\in\mathcal M$ and $i\in\Inj(A,\omega)$ a map $u\colon i\to ui$. If now $i,j\in\Inj(A,\omega)$ are arbitrary, then we can pick a bijection $u\in\mathcal M$ with $ui=j$, i.e.~$i$ and $j$ are isomorphic in $\Inj(A,\omega)\hq\mathcal M$. It follows that for our favourite $i\in\Inj(A,\omega)$ the induced functor $B\End(i)\to \Inj(A,\omega)\hq\mathcal M$ is an equivalence of categories. Therefore it suffices that the monoid $\End(i)$ has weakly contractible classifying space. However, $\End(i)$ consists precisely of those $u\in\mathcal M$ that fix $\im(i)$ pointwise. Picking a bijection $\omega\cong \omega\setminus\im(i)$ therefore yields an isomorphism of monoids $\mathcal M\cong\End(i)$. Since $\mathcal M$ has weakly contractible classifying space by \cite[Lemma~5.2]{schwede-semistable} (whose proof Schwede attributes to Strickland), this finishes the proof.
\end{ex}
\subsubsection{Equivariant analysis of action categories}
Fix a universal subgroup $K\subset\mathcal M$ and a homomorphism $\phi\colon K\to G$. By definition, the simplicial set $(\Inj(A,\omega)\times_HX)\hq\mathcal M$ is the nerve of the category of the same name, and as the nerve is a right adjoint, this is compatible with $\phi$-fixed points. In the following, we want to understand these fixed point categories better and in particular describe them as disjoint unions of monoids in analogy with the above example.
However, in Theorem~\ref{thm:hq-M-computation} we allow $A$ to be infinite (and $A=\omega$ is the case we actually used in the proof of Theorem~\ref{thm:hq-M-semistable-replacement}). In this case, there are `too many isomorphism classes' in $(\Inj(A,\omega)\hq\mathcal M)\times_HX$: for example, not all objects in $\mathcal M\hq\mathcal M$ are isomorphic, though they all receive a map from $1\in\mathcal M$. To salvage this situation we introduce the full subcategory $C_K\subset(\Inj(A,\omega)\times_HX)\hq\mathcal M$ spanned by those $[u,x]$ for which $\im(u)^c\subset\omega$ contains a complete $K$-set universe (this is independent of the choice of representative as $\im(u)=\im(u.h)$ for all $h\in H$).
\begin{lemma}\label{lemma:c-K-whe}
The inclusion $C_K\hookrightarrow(\Inj(A,\omega)\hq\mathcal M)\times_HX$ induces a $(K\times G)$-homotopy equivalence on nerves.
\begin{proof}
Let $\alpha\in\mathcal M$ be $K$-equivariant such that $\im(\alpha)^c$ contains a complete $K$-set universe. Then $\alpha.\blank$ is $(K\times G)$-equivariant and it takes all of $(\Inj(A,\omega)\hq\mathcal M)\times_HX$ to $C_K$. We claim that this is a $(K\times G)$-homotopy inverse to the inclusion. Indeed, the proof of Lemma~\ref{lemma:X-hq-M-semistable} shows that the maps $\alpha\colon x\to\alpha.x$ define a natural transformation from the identity to $\alpha.\blank$ as endofunctors of $(\Inj(A,\omega)\hq\mathcal M)\times_HX$, showing that $\alpha.\blank$ is a right $(K\times G)$-homotopy inverse. However, as $C_K$ is a full subcategory, these also define such a transformation for the other composite, proving that $\alpha.\blank$ is also a left $(K\times G)$-homotopy inverse.
\end{proof}
\end{lemma}
The next lemma in particularly tells us that $C_K$ avoids the aforementioned issue. For its proof we need the following notation:
\begin{defi}
Let $A,B$ be sets, let $A=A_1\sqcup A_2$ be any partition, and let $f_i\colon A_i\to B$ ($i=1,2$) be any maps of sets. Then we write $f_1+f_2$ for the unique map $A\to B$ that agrees on $A_1$ with $f_1$ and on $A_2$ with $f_2$.
\end{defi}
By slight abuse, we will also apply the above when $f_1$ and $f_2$ are maps into subsets of $B$. Obviously, $f_1+f_2$ will be injective if and only if $f_1$ and $f_2$ are injections with disjoint image.
\begin{lemma}\label{lemma:map-implies-isomorphic}
Let $p,q\in C_K^\phi$ and fix a representative $(u,x)\in\Inj(A,\omega)\times X$ of $p$. Then the following are equivalent:
\begin{enumerate}
\item There exists a map $f\colon p\to q$ in $C_K^\phi$.
\item There exists a representative of $q$ of the form $(v,x)$ such that in addition $\sigma_u=\sigma_v$ (see the discussion after Lemma~\ref{lemma:emg-fixed-point-characterization-general}).
\item There exists an isomorphism $f'\colon p\to q$ in $C_K^\phi$.
\end{enumerate}
\begin{proof}
Obviously $(3)\Rightarrow(1)$; we will prove that also $(1)\Rightarrow(2)$ and $(2)\Rightarrow(3)$.
Assume $f\colon p\to q$ is any morphism in $C_K^\phi$. Then $q=f.p$, so that $(fu,x)$ is a representative of $q$. We claim that this has the desired property, i.e.~$v\mathrel{:=}fu$ satisfies $\sigma_v=\sigma_u$. But indeed, as $f$ is $\phi$-fixed, it has to be $K$-equivariant by definition of the action. Then
\begin{equation*}
v.\sigma_v(k)=kv=kfu=fku=fu.\sigma_u(k)=v.\sigma_u(k),
\end{equation*}
and hence $\sigma_v(k)=\sigma_u(k)$ as $H$ acts freely on $\Inj(A,\omega)$. This proves $(1)\Rightarrow(2)$.
For the proof of $(2)\Rightarrow(3)$, we observe that $\im(u)$ and $\im(v)$ are both $K$-subsets of $\omega$: indeed, $k.u(a)=(ku)(a)=u(\sigma_u(k).a)$ for all $a\in A$, and similarly for $v$. Thus, $\omega\setminus\im(u)$ and $\omega\setminus\im(v)$ are $K$-sets in their own right; as they are obviously countable and moreover contain a complete $K$-set universe each by definition of $C_K$, they are themselves complete $K$-set universes. It follows that there exists a $K$-equivariant bijection $f'_1\colon\omega\setminus\im(u)\cong\omega\setminus\im(v)$.
On the other hand, $f'_0=vu^{-1}\colon\im(u)\to\im(v)$ is also $K$-equivariant because
\begin{equation*}
f'_0(k.u(a))=f'_0(u(\sigma_u(k).a))=v(\sigma_u(k).a)=v(\sigma_v(k).a)=k.v(x)=k.f'_0(u(a))
\end{equation*}
for all $k\in K$, $a\in A$. As it is moreover obviously bijective, we conclude that $f'\mathrel{:=}f'_0+f'_1$ defines an isomorphism $[u,x]\to[v,x]$ in $C_K^\phi$ as desired.
\end{proof}
\end{lemma}
We fix for each isomorphism class of $C_K^\phi$ a representative $p\in C_K^\phi$ and for each such $p$ in turn a representative $(u,x)\in\Inj(A,\omega)\times X$. Let us write $\mathscr I$ for the set of all these. The above lemma then implies:
\begin{cor}\label{cor:c-K-phi-decomposition}
The tautological functor
\begin{equation*}
\Phi\colon\coprod_{(u,x)\in\mathscr I}B\End_{C_K^\phi}([u,x])\to C_K^\phi
\end{equation*}
is an equivalence of categories.\qed
\end{cor}
Let us now study these monoids more closely:
\begin{prop}\label{prop:tau-group-completion}
Let $(u,x)\in\Inj(A,\omega)\times X$ such that $[u,x]\in C_K^\phi$.
\begin{enumerate}
\item Let $f\colon[u,x]\to[u,x]$ be any map in $C_K^\phi$. Then there exists a unique $\tau(f)\in H$ such that $fu=u.\tau(f)$. Moreover, $\tau(f)$ centralizes $\im\sigma_u$ and stabilizes $x\in X$. (We caution the reader that $\tau$ depends on the chosen representative.)
\item The assignment $f\mapsto\tau(f)$ defines a monoid homomorphism
\begin{equation*}
\tau\colon\End_{C_K^\phi}([u,x])\to\centralizer_H(\im\sigma_u)\cap\stabilizer_H(x).
\end{equation*}
\item The homomorphism $\tau$ induces a weak equivalence on classifying spaces.
\end{enumerate}
\end{prop}
For the proof of the proposition we will need:
\begin{lemma}\label{lemma:M-K-classifying}
The monoid $\mathcal M^K$ of $K$-equivariant injections $\omega\to\omega$ has weakly contractible classifying space.
\begin{proof}
This is an equivariant version of~\cite[proof of Lemma~5.2]{schwede-semistable}. As $K$ is universal, we can pick a $K$-equivariant bijection $\omega\cong\omega\amalg\omega$ which yields two $K$-equivariant injections $\alpha,\beta\in\mathcal M$ whose images partition $\omega$. Then the conjugation homomorphism $c_\alpha\colon\mathcal M\to\mathcal M$ satisfies
\begin{equation*}
c_\alpha(u)\alpha=\alpha u\qquad\text{and}\qquad c_\alpha(u)\beta=\beta
\end{equation*}
for all $u\in\mathcal M$. The first equality proves that $\alpha$ defines a natural transformation from the identity to $B(c_\alpha)\colon B\mathcal M\to B\mathcal M$ (also cf.~the proof of Lemma~\ref{lemma:X-hq-M-semistable}) while the second one shows that $\beta$ defines a natural transformation from the constant functor to it. Upon taking nerves, we therefore get a zig-zag of homotopies between the identity and a constant map, proving the claim.
\end{proof}
\end{lemma}
\begin{proof}[Proof of Proposition~\ref{prop:tau-group-completion}]
For the first statement we observe that there exists at most one such $\tau(f)$ by freeness of the action. On the other hand, $f$ being a morphism in particular means that $f.[u,x]=[u,x]$. Plugging in the definition of the action and of the equivalence relation we divided out, this means that there exists $\tau(f)\in H$ with $(fu,x)=(u.\tau(f),x.\tau(f))$. Thus, it only remains to show that $\tau(f)$ centralizes $\im(\sigma_u)$. Indeed, if $k\in K$ is arbitrary, then on the one hand
\begin{equation*}
kfu=fku=fu.\sigma_u(k)=u.(\tau(f)\sigma_u(k))
\end{equation*}
(where we have used that $f$ is $K$-equivariant since it is a morphism in $C_K^\phi$), and on the other hand
\begin{equation*}
kfu=ku.\tau(f)=u.(\sigma_u(k)\tau(f)).
\end{equation*}
Thus, $u.(\tau(f)\sigma_u(k))=u.(\sigma_u(k)\tau(f))$, whence indeed $\tau(f)\sigma_u(k)=\sigma_u(k)\tau(f)$ by the freeness of the right $H$-action. This finishes the proof of $(1)$.
For the second statement, we observe that $ff'u=fu.\tau(f')=u.(\tau(f)\tau(f'))$ and hence $\tau(ff')=\tau(f)\tau(f')$ by the above uniqueness statement. Analogously, $1u=u=u.1$ shows $\tau(1)=1$.
For the final statement, we will first prove:
\begin{claim*} The assignment
\begin{align*}
\textup{T}\colon\End_{C_K^\phi}([u,x]) &\to \Inj(\omega\setminus\im u,\omega\setminus\im u)^K\times\big(\centralizer_H(\im\sigma_u)\cap\stabilizer_H(x)\big)\\
f &\mapsto \big(f|_{\omega\setminus\im(u)}\colon\omega\setminus\im(u)\to\omega\setminus\im(u), \tau(f)\big)
\end{align*}
defines an isomorphism of monoids.
\begin{proof}
This is well-defined by the first part and since the injection $f$ restricts to a self-bijection of $\im(u)$ by the above, so that it also has to preserve $\omega\setminus\im(u)$. It is then obvious (using the second part of the proposition) that $\textup{T}$ is a monoid homomorphism.
We now claim that it is actually an isomorphism of monoids. For injectivity, it suffices to observe that if $\tau(f)=\tau(f')$, then $fu=u\tau(f)=u\tau(f')=f'u$.
For surjectivity, we let $f_1\colon\omega\setminus\im(u)\to\omega\setminus\im(u)$ be any $K$-equivariant injection and we let $t\in\centralizer_H(\im\sigma_u)\cap\stabilizer_H(x)$. Then there is a unique map $f_0\colon\im(u)\to\im(u)$ with $f_0u=u.t$ and this is automatically injective (in fact, even bijective). We claim that it is also $K$-equivariant. Indeed, if $k\in K$ is arbitrary, then
\begin{equation*}
k.f_0(u(x))=ku(t.x)=u(\sigma_u(k)t.x)=u(t\sigma_u(k).x)=f_0(u(\sigma_u(k).x))=f_0(k.u(x))
\end{equation*}
where we have used that $t$ commutes with $\sigma_u(k)$. Thus, $f\mathrel{:=}f_0+f_1$ defines a $K$-equivariant injection $\omega\to\omega$ and we claim that this is an endomorphism of $[u,x]$ in $C_K$, hence the desired preimage of $(f_1,t)$. Indeed, $f.[u,x]$ is represented by
\begin{equation*}
(fu,x)=(u.t,x)\sim (u,x.t^{-1})=(u,x),
\end{equation*}
where we have used that $t$ and hence also $t^{-1}$ stabilizes $x$.
\end{proof}
\end{claim*}
To prove that $\tau$ induces a weak homotopy equivalence on classifying spaces, we now observe that the induced map factors as
\begin{align*}
\nerve\big(B\End([u,x])\big)&\xrightarrow{\nerve(B\textup{T})}\nerve\big(B(\Inj(\omega\setminus\im u,\omega\setminus\im u)^K\times L)\big)\\
&\cong\nerve\big(B(\Inj(\omega\setminus\im u,\omega\setminus\im u)^K)\big)\times\nerve(BL)\xrightarrow{\pr}\nerve(BL),
\end{align*}
where $L\mathrel{:=}\centralizer_H(\im\sigma_u)\cap\stabilizer_H(x)$. The first map is an isomorphism by the previous statement, so it suffices that $\Inj(\omega\setminus\im u,\omega\setminus\im u)^K$ has trivial classifying space. But indeed, as in the proof of Lemma~\ref{lemma:map-implies-isomorphic} we see that $\omega\setminus\im u$ is a complete $K$-set universe, so that $\Inj(\omega\setminus\im u,\omega\setminus\im u)^K\cong\mathcal M^K$ as monoids. Thus, the claim follows from Lemma~\ref{lemma:M-K-classifying}.
\end{proof}
\subsubsection{Equivariant analysis of quotient categories}
We recall that the simplicial set $E\Inj(A,\omega)\times X$ is canonically and equivariantly isomorphic to the nerve of the groupoid of the same name. On the other hand, the right $H$-action on $E\Inj(A,\omega)\times X$ is free, and as the nerve preserves quotients by \emph{free} group actions, the simplicial set $E\Inj(A,\omega)\times_H X$ is again canonically identified with the nerve of the corresponding quotient of categories, which we denote by the same name.
In the following we want to devise a description of this category and its fixed points analogous to the above results. For this we first observe that while hom sets in quotient categories are in general hard to describe, the situation is easier here because this particular quotient is preserved by the nerve: namely, any morphism $p\to q$ (for $p,q\in \Inj(A,\omega)\times_KX$) is represented by a triple $(u,v;x)$ with $u,v\in\Inj(A,\omega)$, $x\in X$ such that $[u;x]=q$ and $[v,x]=p$; moreover, a triple $(u',v';x')$ represents the same morphism if and only if there exists an $h\in H$ with $u'=u.h$, $v'=v.h$, and $x'=x.h$. The following lemma gives a more concrete description once we have fixed representatives of $p$ and $q$:
\begin{lemma}\label{lemma:emg-equiv-hom-sets-general}
Let $(u,x),(v,y)\in\Inj(A,\omega)\times X$. Then the assignment
\begin{align*}
\{h\in H: y.h=x\} &\to\Hom_{E\Inj(A,\omega)\times_HX}([u;x],[v;y])\\
h&\mapsto [v.h,u;x]=[v,u.h^{-1},y]
\end{align*}
is well-defined and bijective. In particular, the assignment
\begin{equation}\label{eq:emgehs-endo}
\begin{aligned}
\stabilizer_H(x)&\to\End_{E\Inj(A,\omega)\times_HX}([u;x])\\
h&\mapsto [u.h,u;x]
\end{aligned}
\end{equation}
is bijective; in fact, this even defines an isomorphism of groups.
\begin{proof}
Let us denote the above map by $\alpha$. We first observe that for any $h$ on the left hand side the representative $(v.h,u;x)$ differs from $(v,u.h^{-1},y)$ only by right multiplication by $h$, so that the two given definitions of $\alpha(h)$ indeed agree. In particular, they define an edge from $[u;x]$ to $[v;y]$, proving that $\alpha$ is well-defined.
Lemma~\ref{lemma:emg-basic-general}-$(\ref{item:emgbg-unique})$ already implies that $\alpha$ is injective. For surjectivity, we pick an edge on the right hand side and let $(a,b;c)$ be a representative. By definition $[b;c]=[u;x]$, so after acting suitably from the right by $H$ on $(a,b;c)$, we may assume $b=u$ and $c=x$, i.e.~our chosen representative takes the form $(a,u;x)$. But by assumption this represents an edge to $[v;y]$ and hence $[a;x]=[v;y]$, i.e.~there exists an $h\in H$ such that $a=v.h$ and $x=y.h$, which then provides the desired preimage.
Specializing to $(v;y)=(u;x)$ shows that $(\ref{eq:emgehs-endo})$ is bijective. The calculation
\begin{equation*}
[u.hh',u;x]=[u.hh',u.h';x][u.h',u;x]=[u.h,u;x][u.h',u;x]
\end{equation*}
(where the final equality uses that $x.(h')^{-1}=x$) then shows that it is in fact an isomorphism of groups.
\end{proof}
\end{lemma}
Similarly, we can describe the hom sets in the fixed point categories:
\begin{lemma}\label{lemma:emg-equiv-hom-sets-general-phi}
Let $(u,x),(v,y)\in\Inj(A,\omega)\times X$ such that $[u;x],[v;y]$ are $\phi$-fixed points of $E\Inj(A,\omega)\times_HX$. Then
\begin{align*}
\{h\in H: y.h=x, h\sigma_u(k)h^{-1}=\sigma_v(k)\text{ $\forall k\in K$}\} &\to
\Hom_{(E\Inj(A,\omega)\times_HX)^\phi}([u;x],[v;y])\\
h&\mapsto [v.h,u;x]=[v,u.h^{-1},y].
\end{align*}
is well-defined and bijective. In particular, this yields a bijection
\begin{align*}
\stabilizer_H(x)\cap\centralizer_H(\im\sigma_u)&\to\End_{(E\Inj(A,\omega)\times_HX)^\phi}([u;x])\\
h&\mapsto [u.h,u;x].
\end{align*}
This map is in fact even an isomorphism of groups.
\begin{proof}
By the previous lemma we are reduced to proving that $[v.h,u;x]$ is $\phi$-fixed if and only if $\sigma_v(k)=h\sigma_u(k)h^{-1}$ for all $k\in K$. Indeed,
\begin{align*}
(k,\phi(k)).(v.h,u;x)&=(kv.h,k.u;\phi(k).x)=(v.\sigma_v(k)h,u.\sigma_u(k);x.\sigma_u(k))\\
&\sim\big(v.(\sigma_v(k)h\sigma_u(k)^{-1}),u;x\big)
\end{align*}
where we have applied Lemma~\ref{lemma:emg-fixed-point-characterization-general} to $(u;x)$. By freeness of the right $H$-action this represents the same element as $(v.h,u;x)$ if and only if $\sigma_v(k)h\sigma_u(k)^{-1}=h$, which is obviously equivalent to the above condition.
\end{proof}
\end{lemma}
\begin{prop}\label{prop:EInj-decomposition}
The functor
\begin{equation*}
\Psi\colon\coprod_{(u,x)\in\mathscr I}B\big(\centralizer_H(\im\sigma_u)\cap\stabilizer_H(x)\big)\to \big(E\Inj(A,\omega)\times_HX\big)^\phi
\end{equation*}
given on the $(u,x)$-summand by sending $t\in\centralizer_H(\im\sigma_u)\cap\stabilizer_H(x)$ to the morphism $[u.t,u;x]$ is an equivalence of groupoids.
\begin{proof}
Lemma \ref{lemma:emg-equiv-hom-sets-general-phi} implies that this indeed lands in the $\phi$-fixed points and that each of the maps
\begin{equation*}
\centralizer_H(\im\sigma_u)\cap\stabilizer_H(x)\to\End_{\big(E\Inj(A,\omega)\times_HX\big)^\phi}([u;x])
\end{equation*}
is a group isomorphism; in particular, $\Psi$ is a functor. As both source and target of $\Psi$ are groupoids, it suffices then to show that $\mathscr I$ also is a system of (representatives of) representatives of the isomorphism classes on the right hand side.
To see that $\mathscr I$ hits every isomorphism class at most once, assume $(u,x),(v,y)\in\mathscr I$ represent isomorphic elements in $(E\Inj(A,\omega)\times_HX)^\phi$. Lemma~\ref{lemma:emg-equiv-hom-sets-general-phi} implies that there exists an $h\in H$ with $y.h=x$ and $h^{-1}\sigma_v(k)h=\sigma_u(k)$ for all $k\in K$. Then $(w,x)\mathrel{:=}(v.h,y.h)$ represents the same element as $(v,y)$ in both $C_K^\phi$ as well as $(E\Inj(A,\omega)\times_HX)^\phi$, and one easily checks that $\sigma_w=\sigma_u$. Thus, $[u,x]\cong[w,x]=[v,y]$ in $C_K^\phi$ by Lemma~\ref{lemma:map-implies-isomorphic}, and hence $(u,x)=(v,y)$ by definition of $\mathscr I$.
But $\mathscr I$ also covers the isomorphism classes of $(E\Inj(A,\omega)\times_HX)^\phi$: if $(v;y)$ represents an arbitrary element of it and $\alpha\in\mathcal M$ is again $K$-equivariant with $\omega\setminus\im\alpha$ a complete $K$-set universe, then one easily checks that $[\alpha v,v;y]$ is $\phi$-fixed, so that it witnesses $[v;y]\cong[\alpha v;y]$. In other words, we may assume that $v$ misses a complete $K$-set universe. But in this case it defines an element of $C_K$, which is then by definition isomorphic in $C_K$ to some $[u,x]$ with $(u,x)\in\mathscr I$, and hence also in $(E\Inj(A,\omega)\times_HX)^\phi$ by functoriality.
\end{proof}
\end{prop}
\subsubsection{Quotient vs.~action categories} Putting all of the above results together, we finally get:
\begin{proof}[Proof of Theorem~\ref{thm:hq-M-computation}]
As before let $K\subset\mathcal M$ be a universal subgroup, and let $\phi\colon K\to G$ be any group homomorphism. We have to show that $(\ref{eq:hq-m-comparison})$ induces a weak homotopy equivalence on $\phi$-fixed points.
To this end, we consider the diagram of categories and functors
\begin{equation*}
\begin{tikzcd}[column sep=-15pt]
\smash{\coprod\limits_{(u,x)\in\mathscr I}}B\End_{C_K^\phi}([u,x]) \arrow[rr, "\coprod B\tau"]\arrow[d, "\Phi"'] && \smash{\coprod\limits_{(u,x)\in\mathscr I}}B\big(\centralizer_H(\im\sigma_u)\cap\stabilizer_H(x)\big)\arrow[d, "\Psi"]\\
C_K^\phi\arrow[r, hook] & \big((\Inj(A,\omega)\times_HX)\hq\mathcal M)^\phi\arrow[r] & (E\Inj(A,\omega)\times_HX)^\phi
\end{tikzcd}
\end{equation*}
where the unlabelled arrow is induced by the map in question (viewed as a functor). The top path through this diagram sends $f\colon[u,x]\to[u,x]$ to $[u.\tau(f),u;x]$ while the lower one sends it to $[fu,u;x]$. As $u.\tau(f)=fu$ by definition of $\tau$, these two agree, i.e.~the above diagram commutes.
We now observe that the top map is a weak homotopy equivalence by Proposition~\ref{prop:tau-group-completion}, that the vertical maps are equivalences by Corollary~\ref{cor:c-K-phi-decomposition} and Proposition~\ref{prop:EInj-decomposition}, respectively, and that the lower left inclusion is a weak homotopy equivalence by Lemma~\ref{lemma:c-K-whe}. The claim now follows by $2$-out-of-$3$.
\end{proof}
\subsection{Functoriality}
\index{functoriality in homomorphisms!for EM-G-SSet@for $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$|(}
We will now study various change-of-group functors for the above models of $G$-global homotopy theory. We begin with the versions for $E\mathcal M$-actions, where Lemma~\ref{lemma:alpha-shriek-projective} and Lemma~\ref{lemma:alpha-star-injective}, respectively, specialize to:
\begin{cor}
Let $\alpha\colon H\to G$ be any group homomorphism. Then
\begin{equation*}
\alpha_!\colon\cat{$\bm{E\mathcal M}$-$\bm{H}$-SSet}_{\textup{$H$-global}}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm{G}$-SSet}_{\textup{$G$-global}} :\!\alpha^*
\end{equation*}
is a simplicial Quillen adjunction with homotopical right adjoint.\qed
\end{cor}
\begin{cor}
Let $\alpha\colon H\to G$ be any group homomorphism. Then
\begin{equation*}
\alpha^*\colon\cat{$\bm{E\mathcal M}$-$\bm{G}$-SSet}_{\textup{$G$-global injective}}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm{H}$-SSet}_{\textup{$H$-global injective}} :\!\alpha_*
\end{equation*}
is a simplicial Quillen adjunction.\qed
\end{cor}
On the other hand, Propositions~\ref{prop:alpha-shriek-injective} and \ref{prop:alpha-lower-star-homotopical} imply:
\begin{cor}\label{cor:alpha-shriek-injective-EM}
Let $\alpha\colon H\to G$ be an \emph{injective} group homomorphism. Then
\begin{equation*}
\alpha_!\colon\cat{$\bm{E\mathcal M}$-$\bm H$-SSet}_{\textup{$H$-global injective}}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\textup{$H$-global injective}} :\!\alpha^*
\end{equation*}
is a simplicial Quillen adjunction. In particular, $\alpha_!$ is homotopical.\qed
\end{cor}
\begin{cor}\label{cor:alpha-lower-star-injective-EM}
Let $\alpha\colon H\to G$ be an \emph{injective} group homomorphism. Then
\begin{equation*}
\alpha^*\colon\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\textup{$G$-global}}\rightleftarrows \cat{$\bm{E\mathcal M}$-$\bm H$-SSet}_{\textup{$H$-global}}:\!\alpha_*
\end{equation*}
is a simplicial Quillen adjunction. If $(G:\im\alpha)<\infty$, then $\alpha_*$ is homotopical.\qed
\end{cor}
Finally, Proposition~\ref{prop:free-quotient-general} specializes to:
\begin{cor}\label{cor:free-quotient-EM}
Let $\alpha\colon H\to G$ be any group homomorphism. Then
\begin{equation*}
\alpha_!\colon\cat{$\bm{E\mathcal M}$-$\bm H$-SSet}\to\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}
\end{equation*}
preserves weak equivalences between objects with free $\ker(\alpha)$-action.\index{functoriality in homomorphisms!for EM-G-SSet@for $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$|)}\qed
\end{cor}
The case of $\mathcal M$-actions needs slightly more work:
\index{functoriality in homomorphisms!for M-G-SSet@for $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$|(}
\begin{cor}\label{cor:alpha-shriek-projective-M}
Let $\alpha\colon H\to G$ be any group homomorphism. Then
\begin{equation*}
\alpha_!\colon\cat{$\bm{\mathcal M}$-$\bm{H}$-SSet}_{\textup{$H$-global}}\rightleftarrows\cat{$\bm{\mathcal M}$-$\bm{G}$-SSet}_{\textup{$G$-global}} :\!\alpha^*
\end{equation*}
is a simplicial Quillen adjunction.
\begin{proof}
This follows as before for the $H$-universal and $G$-universal model structure, respectively. Thus, it suffices by Proposition~\ref{prop:cofibrations-fibrant-qa} that $\alpha^*$ sends $G$-semistable objects to $H$-semistable ones, which is obvious from the definition.
\end{proof}
\end{cor}
\begin{prop}\label{prop:alpha-star-EM-homotopical}
Let $\alpha\colon H\to G$ be any group homomorphism. Then
\begin{equation*}
\alpha^*\colon\cat{$\bm{\mathcal M}$-$\bm{G}$-SSet}_{\textup{$G$-global injective}}\rightleftarrows\cat{$\bm{\mathcal M}$-$\bm{H}$-SSet}_{\textup{$H$-global injective}} :\!\alpha_*
\end{equation*}
is a simplicial Quillen adjunction. In particular, $\alpha^*$ is homotopical.
\begin{proof}
It is clear that $\alpha^*$ preserves injective cofibrations, so it only remains to show that it is homotopical. However, while $\alpha^*$ commutes with $E\mathcal M\times_{\mathcal M}\blank$, it is not clear a priori that it is also suitably compatible with $E\mathcal M\times_{\mathcal M}^{\textbf{L}}\blank$ since $\alpha^*$ usually does not preserve cofibrant objects of the universal model structures.
Instead, we consider the commutative diagram
\begin{equation*}
\begin{tikzcd}
\cat{$\bm{\mathcal M}$-$\bm G$-SSet}_{\textup{$G$-global}}\arrow[r, "\blank\hq\mathcal M"]\arrow[d, "\alpha^*"'] & \cat{$\bm{\mathcal M}$-$\bm G$-SSet}_{\textup{$G$-universal}}\arrow[d,"\alpha^*"]\\
\cat{$\bm{\mathcal M}$-$\bm H$-SSet}_{\textup{$H$-global}}\arrow[r, "\blank\hq\mathcal M"'] & \cat{$\bm{\mathcal M}$-$\bm H$-SSet}_{\textup{$H$-universal}}.
\end{tikzcd}
\end{equation*}
Then the horizontal arrows create weak equivalences by Theorem~\ref{thm:hq-M-semistable-replacement}, while the right vertical arrow is clearly homotopical. Thus, also the left hand vertical arrow is homotopical.
\end{proof}
\end{prop}
\begin{cor}
Let $\alpha\colon H\to G$ be an \emph{injective} group homomorphism. Then
\begin{equation*}
\alpha_!\colon\cat{$\bm{\mathcal M}$-$\bm H$-SSet}_{\textup{$H$-global injective}}\rightleftarrows\cat{$\bm{\mathcal M}$-$\bm G$-SSet}_{\textup{$H$-global injective}} :\!\alpha^*
\end{equation*}
is a simplicial Quillen adjunction. In particular, $\alpha_!$ is homotopical.
\begin{proof}
By Proposition~\ref{prop:alpha-shriek-injective}, $\alpha_!$ preserves injective cofibrations and it sends $H$-universal weak equivalences to $G$-universal ones. On the other hand, any $H$-global weak equivalence factors as an $H$-global acyclic cofibration followed by an $H$-universal weak equivalence. Since $\alpha_!$ sends the former to $G$-global weak equivalences by Corollary~\ref{cor:alpha-shriek-projective-M}, $2$-out-of-$3$ implies that $\alpha_!$ is also homotopical.\index{functoriality in homomorphisms!for M-G-SSet@for $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$|)}
\end{proof}
\end{cor}
\subsection{\texorpdfstring{$\bm{G}$}{G}-global homotopy theory vs.~\texorpdfstring{$\bm G$}{G}-equivariant homotopy theory}\label{sec:g-global-vs-g-em}\index{proper G-equivariant homotopy theory@proper $G$-equivariant homotopy theory!vs G-global homotopy theory@vs.~$G$-global homotopy theory|(}
As promised, we will now explain how to exhibit classical (proper) $G$-equivariant homotopy theory as a Bousfield localization of our models of $G$-global homotopy theory. In fact, we will construct for both models a chain of four adjoint functors, in particular yielding two Bousfield localizations each. Before we can do any of this however, we need to understand a particular case of the `change of family' adjunction from Proposition~\ref{prop:change-of-family-sset} better:
\begin{defi}\label{defi:class-E}
We define $\mathcal E$ (for `equivariant')\nomenclature[aE]{$\mathcal E$}{collection of graph subgroups $\Gamma_{H,\phi}\subset\mathcal M\times G$ with $H$ universal and $\phi$ injective} as the collection of those graph subgroups $\Gamma_{H,\phi}$ of $\mathcal M\times G$ such that $H$ is universal and $\phi$ is \emph{injective}.
\end{defi}
Below we will need a characterization of the essential image of the left adjoint
\begin{equation}\label{eq:lambda-injective}
\lambda\colon\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\textup{$\mathcal E$-w.e.}}^\infty\to\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\textup{$G$-global w.e.}}^\infty
\end{equation}
of the localization functor. Let us give some intuition for this: on the left hand side, we only remember the fixed points for injective $\phi\colon H\to G$. If now $X$ is any $E\mathcal M$-$G$-simplicial set, then $\lambda(X)^\phi$ and $X^\phi$ are weakly equivalent by abstract nonsense. On the other hand, if $\psi\colon H\to G$ is not necessarily injective, then $\lambda(X)^\psi$ should be somehow computable from the fixed points for injective homomorphisms. A natural guess is that $\lambda(X)^\psi$ be weakly equivalent to $X^{\bar\psi}$ where $\bar\psi\colon H/\ker(\psi)\to G$ is the induced homomorphism (and we have secretly identified $H/\ker(\psi)$ with a universal subgroup of $\mathcal M$ isomorphic to it), and this indeed turns out to be true. However, we of course do not want $\lambda(X)^\psi$ and $\lambda(X)^{\bar\psi}$ to be merely abstractly weakly equivalent, but instead we want some explicit and suitably coherent way to identify them. The following definition turns this vague heuristic into a rigorous notion:
\begin{defi}\label{defi:ker-oblivious}\index{kernel oblivious|textbf}\index{kernel oblivious|(}
An $E\mathcal M$-$G$-simplicial set $X$ is called \emph{kernel oblivious} if the following holds: for any universal $H,H'\subset\mathcal M$, any surjective group homomorphism $\alpha\colon H\to H'$, any arbitrary group homomorphism $\phi\colon H'\to G$, and any $u\in\mathcal M$ such that $hu=u\alpha(h)$ for all $h\in H$, the map
\begin{equation*}
u.\blank\colon X^\phi\to X^{\phi\alpha}
\end{equation*}
is a weak homotopy equivalence of simplicial sets.
\end{defi}
\begin{ex}\label{ex:triv-oblivious}
Any $E\mathcal M$-$G$-simplicial set with trivial $E\mathcal M$-action is kernel oblivious: in fact, in this case $X^\phi=X^{\phi\alpha}$, and $u.\blank$ is just the identity.
\end{ex}
\begin{thm}\label{thm:im-lambda}
The following are equivalent for an $E\mathcal M$-$G$-simplicial set $X$:
\begin{enumerate}
\item $X$ is kernel oblivious.\label{item:il-oblivious}
\item $X$ lies in the essential image of $(\ref{eq:lambda-injective})$.\label{item:il-ess-im}
\item $X$ is \emph{$G$-globally} equivalent to an $\mathcal E$-cofibrant $X'\in\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$.\label{item:il-some-cof}
\item In any cofibrant replacement $\pi\colon X'\to X$ in the $\mathcal E$-model structure on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$, $\pi$ is actually a $G$-global weak equivalence.\label{item:il-any-cof}
\end{enumerate}
\end{thm}
The proof will require some preparations.
\begin{lemma}\label{lemma:I-between-ker-oblivious}
Let $f\colon X\to Y$ be an $\mathcal E$-weak equivalence in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$ such that $X$ and $Y$ are kernel oblivious. Then $f$ is a $G$-global weak equivalence.
\begin{proof}
Let $H\subset\mathcal M$ be universal and let $\phi\colon H\to G$ be any group homomorphism. We have to show that $f^\phi$ is a weak homotopy equivalence.
For this we choose a universal subgroup $H'\subset\mathcal M$ together with an isomorphism $H'\cong H/\ker\phi$, which gives rise to a surjective homomorphism $\alpha\colon H\to H'$ with $\ker(\alpha)=\ker(\phi)$. By the universal property of quotients, there then exists a unique $\bar\phi\colon H'\to G$ with $\bar\phi\alpha=\phi$; moreover, $\bar\phi$ is injective.
We now appeal to Corollary~\ref{cor:emg-equiv-group-hom-realization-general} to find a $u\in\mathcal M$ such that $hu=u\alpha(h)$ for all $h\in H$, yielding a commutative diagram
\begin{equation*}
\begin{tikzcd}
X^{\bar\phi}\arrow[d, "f^{\bar\phi}"']\arrow[r, "u.\blank"] & X^{\bar\phi\alpha}=X^\phi\arrow[d, "f^\phi", shift left=13pt]\\
Y^{\bar\phi}\arrow[r, "u.\blank"'] & Y^{\bar\phi\alpha}=Y^\phi.
\end{tikzcd}
\end{equation*}
The horizontal maps are weak equivalences as $X$ and $Y$ are kernel oblivious, and the left hand vertical map is a weak equivalence because $f$ is an $\mathcal E$-weak equivalence. Thus, also the right hand map is a weak equivalence as desired.
\end{proof}
\end{lemma}
\begin{prop}\label{prop:EM-inj-discrete}
Let $K\subset\mathcal M$ be any subgroup and let $\psi\colon K\to G$ be an \emph{injective} homomorphism. Then the projection $E\mathcal M\times_\psi G\to G/\im\psi$ is a $G$-global weak equivalence (where the right hand side carries the trivial $E\mathcal M$-action).
\begin{proof}
The map in question is conjugate to the image of the unique map $p\colon E\mathcal M\to *$ under $\psi_!\colon\cat{$\bm{E\mathcal M}$-$\bm{K}$-SSet}\to\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$, where $K$ acts on $E\mathcal M$ from the right via $k.(u_0,\dots,u_n)=(u_0k^{-1},\dots,u_nk^{-1})$.
As $\psi_!$ is homotopical (Corollary~\ref{cor:alpha-shriek-injective-EM}), it is then enough to show that $p$ is a $K$-global weak equivalence, which is just an instance of Example~\ref{ex:G-globally-contractible}.
\end{proof}
\end{prop}
\begin{cor}
Let $K\subset\mathcal M$ be universal, let $\psi\colon K\to G$ be injective, and let $L$ be any simplicial set. Then $(E\mathcal M\times_\psi G)\times L$ is kernel oblivious.
\begin{proof}
The kernel oblivious $E\mathcal M$-$G$-simplicial sets are obviously closed under $G$-global weak equivalences. Thus, the claim follows from the previous proposition together with Example~\ref{ex:triv-oblivious}.
\end{proof}
\end{cor}
\begin{proof}[Proof of Theorem~\ref{thm:im-lambda}]
The implications $(\ref{item:il-any-cof})\Rightarrow(\ref{item:il-ess-im})$ and $(\ref{item:il-ess-im})\Rightarrow(\ref{item:il-some-cof})$ follow immediately from Remark~\ref{rk:change-of-family-sset-explicit}. For the remaining implications we will use:
\begin{claim*}
If $X$ is cofibrant in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\mathcal E}$, then $X$ is kernel oblivious.
\begin{proof}
Let $\alpha,\phi,u$ as in Definition~\ref{defi:ker-oblivious}. It suffices to verify that the natural transformation $u.\blank\colon (\blank)^\phi\Rightarrow(\blank)^{\phi\alpha}$ satisfies the assumptions of Corollary~\ref{cor:saturated-trafo}. But indeed, Condition~$(\ref{item:st-base})$ is an instance of the previous corollary, while all the remaining conditions are part of Lemma~\ref{lemma:fixed-points-cellular}.
\end{proof}
\end{claim*}
The implication $(\ref{item:il-some-cof})\Rightarrow(\ref{item:il-oblivious})$ now follows immediately from the claim. On the other hand, if $X$ is kernel oblivious, and $\pi\colon X'\to X$ is any $\mathcal E$-cofibrant replacement, then also $X'$ is kernel oblivious by the claim, so that $\pi$ is a $G$-global weak equivalence by Lemma~\ref{lemma:I-between-ker-oblivious}. This shows $(\ref{item:il-oblivious})\Rightarrow(\ref{item:il-any-cof})$, finishing the proof.\index{kernel oblivious|)}
\end{proof}
We now consider the functor $\triv_{E\mathcal M}\colon\cat{$\bm G$-SSet}\to\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$ that equips a $G$-simplicial set with the trivial $E\mathcal M$-action. The equality of functors
\begin{equation}\label{eq:triv-fixed-points}
(\blank)^\phi\circ\triv_{E\mathcal M}=(\blank)^{\im\phi}
\end{equation}
(for any $H\subset\mathcal M$ and any homomorphism $\phi\colon H\to G$) shows that this is homotopical with respect to the proper weak equivalences on the source and the $G$-global or $\mathcal E$-weak equivalences on the target.
\begin{cor}\label{cor:triv-vs-lambda-sset}
The diagram
\begin{equation}\label{diag:triv-lambda-triv-sset}
\begin{tikzcd}[column sep=small]
& \cat{$\bm G$-SSet}_{\textup{proper}}^\infty\arrow[dl, "\triv_{E\mathcal M}^\infty"', bend right=10pt]\arrow[dr, "\triv_{E\mathcal M}^\infty", bend left=10pt]\\
\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\mathcal E}^\infty\arrow[rr, "\lambda"'] &&\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\textup{$G$-global}}^\infty
\end{tikzcd}
\end{equation}
commutes up to canonical equivalence.
\begin{proof}
This is is obviously true if we replace $\lambda$ by its right adjoint; in particular, there is a natural transformation filling the above, induced by the unit $\eta$ of $\lambda\dashv\textup{localization}$. To see that this is in fact an equivalence, it suffices (as $\lambda$ is fully faithful) that the right hand arrow lands in the essential image of $\lambda$. But by Theorem~\ref{thm:im-lambda} this is equivalent to demanding that $\triv_{E\mathcal M}X$ be kernel oblivious for every $G$-simplicial set $X$, which is just an instance of Example~\ref{ex:triv-oblivious}.
\end{proof}
\end{cor}
We can now prove the comparison between $G$-global and proper $G$-equivariant homotopy theory:
\begin{thm}\label{thm:G-global-vs-proper-sset}
The functor $\triv_{E\mathcal M}\colon\cat{$\bm G$-SSet}_{\textup{proper}}\to\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\textup{$G$-global}}$\nomenclature[atriv]{$\triv$}{embedding of proper $G$-equivariant into $G$-global homotopy theory via trivial actions} descends to a fully faithful functor on associated quasi-categories with essential image the kernel oblivious $E\mathcal M$-$G$-simplicial sets.
This induced functor admits both a left adjoint $\textbf{\textup L}(E\mathcal M\backslash\blank)$\nomenclature[aLEM]{$\textbf{\textup L}(E\mathcal M\backslash\blank)$}{left adjoint to $\triv$} as well as a right adjoint $(\blank)^{\textbf{\textup R}E\mathcal M}$.\nomenclature[aREM]{$(\blank)^{\textbf{\textup R}E\mathcal M}$}{right adjoint to $\triv$} Moreover, $(\blank)^{\textbf{\textup R}E\mathcal M}$
is a quasi-localization at the $\mathcal E$-weak equivalences, and it in turn admits another right adjoint $\mathcal R$,\nomenclature[aR]{$\mathcal R$}{right adjoint to right adjoint to usual embedding from proper $G$-equivariant into $G$-global homotopy theory ($\triv$ or $\const$)} which is again fully faithful.
\end{thm}
In particular, Example~\ref{ex:triv-oblivious} accounts for all kernel oblivious $E\mathcal M$-$G$-simplicial sets up to homotopy.
\begin{proof}
We first observe that the adjunctions
\begin{equation}\label{eq:triv-EM-G-global}
E\mathcal M\backslash\blank\colon\cat{$\bm{E\mathcal M}$-$\bm{G}$-SSet}_{\textup{$G$-global}}\rightleftarrows
\cat{$\bm{G}$-SSet}_{\textup{proper}} :\!\triv_{E\mathcal M}
\end{equation}
and
\begin{equation}\label{eq:triv-EM-I}
E\mathcal M\backslash\blank\colon\cat{$\bm{E\mathcal M}$-$\bm{G}$-SSet}_{\mathcal E}\rightleftarrows
\cat{$\bm{G}$-SSet}_{\textup{proper}} :\!\triv_{E\mathcal M}
\end{equation}
are Quillen adjunctions with homotopical right adjoints by the equality $(\ref{eq:triv-fixed-points})$, and in particular $(\ref{eq:triv-EM-G-global})$ induces the desired left adjoint of $\triv_{E\mathcal M}^\infty$.
To construct the right adjoint of $\triv_{E\mathcal M}$ it suffices to observe that while
\begin{equation*}
\triv_{E\mathcal M}\colon\cat{$\bm G$-SSet}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet} :\!(\blank)^{E\mathcal M}
\end{equation*}
is not a Quillen adjunction with respect to the above model structures, it becomes one once we use Corollary~\ref{cor:enlarge-generating-cof} to enlarge the generating cofibrations of the $G$-global model structure as to contain all $G/H\times\del\Delta^n\hookrightarrow G/H\times\Delta^n$ for $H\subset G$ finite and $n\ge 0$ (which we are allowed to do by Corollary~\ref{cor:homotopy-pushout-M-SSet}); in particular, $\triv_{E\mathcal M}$ is left Quillen with respect to the injective $G$-global model structure on the target.
To prove the remaining statements we will need:
\begin{claim*}
The Quillen adjunction $(\ref{eq:triv-EM-I})$ is a Quillen equivalence. In particular,
\begin{equation*}
\triv_{E\mathcal M}^\infty\colon\cat{$\bm G$-SSet}_{\textup{proper}}^\infty\to\cat{$\bm{E\mathcal M}$-$\bm{G}$-SSet}_{\textup{$\mathcal E$-w.e.}}^\infty
\end{equation*}
is an equivalence of quasi-categories.
\begin{proof}
It suffices to prove the first statement. The equality $(\ref{eq:triv-fixed-points})$ shows that the right adjoint preserves and reflects weak equivalences. It is therefore enough that the ordinary unit $\eta\colon X\to\triv_{E\mathcal M}(E\mathcal M\backslash X)$, which is given by the projection map, is an $\mathcal E$-weak equivalence for any $\mathcal E$-cofibrant $E\mathcal M$-$G$-simplicial set $X$. By the above, both $E\mathcal M\backslash\blank$ as well as $\triv_{E\mathcal M}$ are left Quillen (after suitably enlarging the cofibrations in the target), and they moreover commute with tensoring with simplicial sets. By Corollary~\ref{cor:saturated-trafo} it therefore suffices that $\eta$ is a weak equivalence for every $E\mathcal M\times_\psi G$ with $K\subset\mathcal M$ universal and $\psi\colon K\to G$ injective.
An easy calculation shows that the projection $E\mathcal M\times G\to G$ descends to an isomorphism $E\mathcal M\backslash(E\mathcal M\times_\psi G)\to G/\im\psi$, so we want to show that the projection $E\mathcal M\times_\psi G\to G/\im\psi$ is an $\mathcal E$-weak equivalence. But this is actually even a $G$-global weak equivalence by Proposition~\ref{prop:EM-inj-discrete}, finishing the proof of the claim.
\end{proof}
\end{claim*}
We now contemplate the diagram $(\ref{diag:triv-lambda-triv-sset})$ from Corollary~\ref{cor:triv-vs-lambda-sset}. By the above claim together with Proposition~\ref{prop:change-of-family-sset} we then conclude that
\begin{equation*}
\triv_{E\mathcal M}^\infty\colon\cat{$\bm G$-SSet}_{\textup{proper}}^\infty\to\cat{$\bm{E\mathcal M}$-$\bm{G}$-SSet}_{\textup{$G$-global}}^\infty
\end{equation*}
is fully faithful, and by Theorem~\ref{thm:im-lambda} its essential image then consists precisely of the kernel oblivious $E\mathcal M$-$G$-simplicial sets. Moreover, we deduce by uniqueness of adjoints that its right adjoint $(\blank)^{\textbf{R}E\mathcal M}$ is canonically equivalent to the composite
\begin{equation}\label{eq:EM-derived-fixpoints-alternative}
\begin{aligned}
\cat{$\bm{E\mathcal M}$-$\bm{G}$-SSet}_{\textup{$G$-global}}^\infty&\xrightarrow{\textup{localization}}
\cat{$\bm{E\mathcal M}$-$\bm{G}$-SSet}_{\textup{$\mathcal E$-w.e.}}^\infty\\
&\xrightarrow{(\triv_{E\mathcal M}^\infty)^{-1}}\cat{$\bm G$-SSet}_{\textup{proper}}
\end{aligned}
\end{equation}
(where the right hand arrow denotes any chosen quasi-inverse) and hence indeed a quasi-localization at the $\mathcal E$-weak equivalences. Finally, $(\ref{eq:EM-derived-fixpoints-alternative})$ has a fully faithful right adjoint by Proposition~\ref{prop:change-of-family-sset}, given explicitly by
\begin{equation*}
\cat{$\bm G$-SSet}_{\textup{proper}}^\infty\xrightarrow{\triv_{E\mathcal M}^\infty}
\cat{$\bm{E\mathcal M}$-$\bm{G}$-SSet}_{\textup{$\mathcal E$-w.e.}}^\infty\xrightarrow{\rho}\cat{$\bm{E\mathcal M}$-$\bm{G}$-SSet}_{\textup{$G$-global}}^\infty
\end{equation*}
which is then also right adjoint to $(\blank)^{\textbf{R}E\mathcal M}$, finishing the proof.
\end{proof}
\begin{rk}
In summary, we in particular have two Bousfield localizations
\begin{equation*}
\textbf{L}(E\mathcal M\backslash\blank)\dashv\triv_{E\mathcal M}\qquad\text{and}\qquad(\blank)^{\textbf{R}E\mathcal M}\dashv\mathcal R.
\end{equation*}
In the ordinary global setting one is mostly interested in the analogue of the adjunction $\triv_{E\mathcal M}\dashv(\blank)^{\textbf{R}E\mathcal M}$ (cf.~Remark~\ref{rk:R-vs-R}) which is a `wrong way' (i.e.~right) Bousfield localization.
\end{rk}
\begin{warn}
Using the Quillen equivalence $(\ref{eq:triv-EM-I})$, we can give another description of the composition $(\ref{eq:EM-derived-fixpoints-alternative})$ and hence of the right adjoint $(\blank)^{\textbf{R}E\mathcal M}$ of $\triv_{E\mathcal M}^\infty$: namely, this can be computed by taking a cofibrant replacement \emph{with respect to the $\mathcal E$-model structure} and then dividing out the left $E\mathcal M$-action.
In contrast to this, the \emph{left} adjoint of $\triv_{E\mathcal M}^\infty$ is computed by taking a cofibrant replacement \emph{with respect to the $G$-global model structure} and then dividing out the action. These two functors are \emph{not} equivalent even for $G=1$: namely, let $H\subset\mathcal M$ be any non-trivial universal subgroup and consider the projection $p\colon E\mathcal M\to E\mathcal M/H$. As $E\mathcal M$ and $E\mathcal M/H$ are both cofibrant in $\cat{$\bm{E\mathcal M}$-SSet}$, we can calculate the value of $\textbf{L}(E\mathcal M\backslash\blank)$ at $p$ simply by $E\mathcal M\backslash p\colon E\mathcal M\backslash E\mathcal M\to E\mathcal M\backslash E\mathcal M/H$, which is a map between terminal objects, hence in particular an equivalence.
On the other hand, $p$ is not an $\mathcal E$-weak equivalence (i.e.~underlying weak equivalence) as $E\mathcal M/H$ is a $K(H,1)$ while $E\mathcal M$ is contractible. But $(\blank)^{\textbf RE\mathcal M}$ is a quasi-localization at the $\mathcal E$-weak equivalences, and these are saturated as they are part of a model structure. Thus $p^{\textbf{R}E\mathcal M}$ is not an equivalence, and in particular it cannot be conjugate to $\textbf{L}(E\mathcal M\backslash\blank)(p)$.
\end{warn}
The above descriptions of the adjoints are not really suitable for computations. However, for finite $G$ we can give easier constructions of $(\blank)^{\textbf{R}E\mathcal M}$ and $\mathcal R$:
\begin{prop}\label{prop:G-finite-simple}
Assume $G$ is finite and choose an injective homomorphism $i\colon G\to\mathcal M$ with universal image, inducing $(i,\id)\colon G\to E\mathcal M\times G$. Then
\begin{equation}\label{eq:G-global-vs-G-finite}
(i,\id)^*\colon\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\textup{$G$-global}}\rightleftarrows\cat{$\bm G$-SSet} :\!(i,\id)_*
\end{equation}
is a Quillen adjunction with homotopical left adjoint, and there are preferred equivalences
\begin{equation*}
\big((i,\id)^*\big)^\infty\simeq(\blank)^{\textbf{\textup R}E\mathcal M}\qquad \text{and}\qquad\textbf{\textup R}(i,\id)_*\simeq\mathcal R.
\end{equation*}
\begin{proof}
It is obvious from the definition that $(i,\id)^*$ sends $\mathcal E$-weak equivalences (and hence in particular $G$-global weak equivalences) to $G$-weak equivalences. Moreover, it preserves cofibrations as the cofibrations on the right hand side are just the underlying cofibrations. We conclude that $(\ref{eq:G-global-vs-G-finite})$ is a Quillen adjunction and that $(i,\id)^*$ descends to $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\mathcal E}^\infty\to\cat{$\bm G$-SSet}^\infty$.
On the other hand, by Theorem~\ref{thm:G-global-vs-proper-sset} also $(\blank)^{\cat{R}E\mathcal M}$ descends accordingly and the resulting functor is quasi-inverse to the one induced by $\triv_{E\mathcal M}$.
The equality $(i,\id)^*\circ\triv_{E\mathcal M}=\id_{\cat{$\bm G$-SSet}}$ of homotopical functors then also exhibits the functor induced by $(i,\id)^*$ on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\mathcal E}^\infty$ as left quasi-inverse to the one induced by $\triv_{E\mathcal M}$ which provides the first equivalence. The second one is then immediate from the uniqueness of adjoints.
\end{proof}
\end{prop}
Finally, let us consider the analogues for $\mathcal M$-actions:
\begin{cor}
The homotopical functor
\begin{equation*}
\triv_{\mathcal M}\colon \cat{$\bm G$-SSet}_{\textup{proper}}\to\cat{$\bm{\mathcal M}$-$\bm G$-SSet}_{\textup{$G$-global}}
\end{equation*}
descends to a fully faithful functor on associated quasi-categories. This induced functor admits both a left adjoint $\textbf{\textup L}(\mathcal M\backslash\blank)$\nomenclature[aLM]{$\textbf{\textup L}(\mathcal M\backslash\blank)$}{left adjoint to $\triv$} as well as a right adjoint $(\blank)^{\textbf{\textup R}\mathcal M}$.\nomenclature[aRM]{$(\blank)^{\textbf{\textup R}\mathcal M}$}{right adjoint to $\triv$} The latter is a quasi-localization at those $f$ such that $E\mathcal M\times_{\mathcal M}^{\textbf{\textup L}}f$ is an $\mathcal E$-weak equivalence, and it in turn admits another right adjoint $\mathcal R$ which is again fully faithful.
Moreover, the diagram
\begin{equation}\label{diag:triv-M-vs-EM}
\begin{tikzcd}[column sep=small]
& \cat{$\bm G$-SSet}_{\textup{proper}}^\infty\arrow[dl, "\triv_{E\mathcal M}^\infty"', bend right=10pt]\arrow[dr, "\triv_{\mathcal M}^\infty", bend left=10pt]\\
\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\infty_{\textup{$G$-global}}\arrow[rr, "\forget^\infty"'] & & \cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\infty_{\textup{$G$-global}}
\end{tikzcd}
\end{equation}
commutes up to canonical equivalence.
\end{cor}
It follows formally that the forgetful functor is also compatible with the remaining functors in the two adjoint chains constructed above, and likewise for its own adjoints $E\mathcal M\times^{\cat{L}}_{\mathcal M}\blank$ and $\cat{R}\Maps^{\mathcal M}(E\mathcal M,\blank)$.
\begin{proof}
We obviously have a Quillen adjunction
\begin{equation*}
\triv_{\mathcal M}\colon\cat{$\bm G$-SSet}_{\textup{proper}}\rightleftarrows\cat{$\bm{\mathcal M}$-$\bm G$-SSet}_{\textup{injective $G$-global}} :\!(\blank)^{\mathcal M},
\end{equation*}
justifying the above description of the right adjoint; in fact, as in the proof of Theorem~\ref{thm:G-global-vs-proper-sset}, it would have been enough to enlarge the generating cofibrations by the maps $G/H\times\del\Delta^n\hookrightarrow G/H\times\Delta^n$ for $H$ finite and $n\ge0$.
For the left adjoint, we now want to prove that also
\begin{equation*}
\mathcal M\backslash\blank\colon\cat{$\bm{\mathcal M}$-$\bm G$-SSet}_{\textup{$G$-global}}\rightleftarrows\cat{$\bm G$-SSet}_{\textup{proper}} :\!\triv_{\mathcal M}
\end{equation*}
is a Quillen adjunction. For this we observe that this is true for the $G$-universal model structure on the left hand side (as $\triv_{\mathcal M}$ is then obviously right Quillen); in particular $\mathcal M\backslash\blank$ preserves cofibrations and $\triv_{\mathcal M}$ sends fibrant $G$-simplicial sets to $G$-universally fibrant $\mathcal M$-$G$-simplicial sets. As this adjunction has an obvious simplicial enrichment, it therefore suffices by Proposition~\ref{prop:cofibrations-fibrant-qa} and the characterization of the fibrant objects provided in Corollary~\ref{cor:em-vs-m-equiv-model-cat} that $\triv_{\mathcal M}$ has image in the $G$-semistable $\mathcal M$-$G$-simplicial sets, which is in fact obvious from the definition.
To prove that $(\ref{diag:triv-M-vs-EM})$ commutes up to canonical equivalence, it suffices to observe that the evident diagram of homotopical functors inducing it actually commutes on the nose. All the remaining statements then follow formally from the commutativity of $(\ref{diag:triv-M-vs-EM})$ as before.\index{proper G-equivariant homotopy theory@proper $G$-equivariant homotopy theory!vs G-global homotopy theory@vs.~$G$-global homotopy theory|)}
\end{proof}
\section{Tameness}\label{sec:tame}
In this section we will be concerned with the notion of \emph{tameness} for $\mathcal M$-actions and $E\mathcal M$-actions, and we will in particular show that the models from the previous sections have tame analogues, that still model unstable $G$-global homotopy theory.
Our reason to study tameness here is twofold: firstly, the categories of tame $\mathcal M$- and $E\mathcal M$-simplicial sets carry interesting symmetric monoidal structures, which we will study in Chapter~\ref{chapter:coherent}, and which play a central role in the construction of global algebraic $K$-theory \cite{schwede-k-theory}; secondly, tame $\mathcal M$- and $E\mathcal M$-simplicial sets are intimately connected to the diagram spaces we will consider in the next section, and the theory developed here will be crucial in establishing those models.
\subsection{A reminder on tame \texorpdfstring{$\bm{\mathcal M}$}{M}-actions}
We begin with the notion of tame $\mathcal M$-actions, which first appeared (for actions on abelian groups) as \cite[Definition~1.4]{schwede-semistable}, and which were then further studied (for actions on sets and simplicial sets) in \cite{I-vs-M-1-cat}, see in particular~\cite[Definition~2.2 and Definition~3.1]{I-vs-M-1-cat}.
\index{support!for M-actions@for $\mathcal M$-actions|(}
\begin{defi}\label{defi:M-set-tame-support}
Let $A\subset\omega$ be any finite set. We write $\mathcal M_A\subset\mathcal M$ for the submonoid of those $u\in\mathcal M$ that fix $A$ pointwise, i.e.~$u(a)=a$ for all $a\in A$.
Let $X$ be any $\mathcal M$-set. An element $x\in X$ is said to be \emph{supported on $A$} if $u.x=x$ for all $u\in\mathcal M_A$; we write $X_{[A]}\subset X$ for the subset of those elements that are supported on $A$, i.e.~$X_{[A]}=X^{\mathcal M_A}$.\nomenclature[aA]{$(\blank)_{[A]}$}{(simplicial) subset of elements supported on $A$}
We call $x$ \emph{finitely supported} if it is supported on some finite set, and we write
\begin{equation*}
X^\tau\mathrel{:=}\bigcup_{A\subset\omega\textup{ finite}} X_{[A]}
\end{equation*}
for the subset of all finitely supported elements. We call $X$ \emph{tame} if $X=X^\tau$.\index{tame!M-action@$\mathcal M$-action|textbf}\nomenclature[atau]{$(\blank)^\tau$}{(simplicial) subset of finitely supported elements; subcategory of tame objects}
\end{defi}
On $\mathcal M$-simplicial sets everything can be extended levelwise:
\begin{defi}\label{defi:tame-M-simplicial-set}
Let $X$ be an $\mathcal M$-simplicial set. An $n$-simplex $x$ is \emph{supported} on the finite set $A\subset\omega$ if it is supported on $A$ as an element of the $\mathcal M$-set $X_n$ of $n$-simplices of $X$. Analogously, $x$ is said to be \emph{finitely supported} if it is so as an element of $X_n$. We define $X^\tau$ via $(X^\tau)_n=(X_n)^\tau$, i.e.~as the family of all finitely supported simplices. We call $X$ \emph{tame} if $X^\tau=X$.\index{tame!M-action@$\mathcal M$-action|textbf}
\end{defi}
\subsubsection{Basic properties} Let us record some basic facts about the above notions. All of these can be found explicitly in \cite{I-vs-M-1-cat} for $\mathcal M$-sets and are easily extended to $\mathcal M$-simplicial sets (for which they also appear implicitly in \emph{op.~cit.}).
\begin{lemma}\label{lemma:support-vs-morphism-M}\label{lemma:first-M-basic}
\begin{enumerate}
\item If $f\colon X\to Y$ is a map of $\mathcal M$-sets and $A\subset\omega$ is finite, then $f$ restricts to $f_{[A]}\colon X_{[A]}\to Y_{[A]}$, hence in particular to $f^\tau\colon X^\tau\to Y^\tau$.
\item If $X$ is any $\mathcal M$-simplicial set and $A\subset\omega$ is finite, then $X_{[A]}$ is a simplicial subset of $X$. In particular, $X^\tau$ is a simplicial subset.
\item If $f\colon X\to Y$ is a map of $\mathcal M$-simplicial sets and $A\subset\omega$ is finite, then $f$ restricts to $f_{[A]}\colon X_{[A]}\to Y_{[A]}$. In particular, it restricts to $f^\tau\colon X^\tau\to Y^\tau$.
\end{enumerate}
\begin{proof}
The first statement is a trivial calculation which we omit; this also appears without proof in \cite[discussion before Lemma~2.6]{I-vs-M-1-cat}. The second statement follows by applying the first one to the structure maps, and the third one follows then by applying the first one levelwise.
\end{proof}
\end{lemma}
\begin{rk}\label{rk:tame-M-colimits}
The above lemma provides us with a functor $(\blank)^\tau\colon\cat{$\bm{\mathcal M}$-SSet}\to\cat{$\bm{\mathcal M}$-SSet}^\tau$ right adjoint to the inclusion of the full subcategory of tame $\mathcal M$-simplicial sets. It follows formally that $\cat{$\bm{\mathcal M}$-SSet}^\tau$ is complete and cocomplete, with colimits created in $\cat{SSet}$, also see~\cite[Lemma~2.6]{I-vs-M-1-cat}.
In fact, it is trival to check that tame $\mathcal M$-simplicial sets are preserved by \emph{finite} products and passing to $\mathcal M$-subsets, so also finite limits in $\cat{$\bm{\mathcal M}$-SSet}^\tau$ are created in $\cat{SSet}$, also cf.~\cite[Example~4.11]{schwede-k-theory}.
\end{rk}
\begin{defi}\index{support!for M-actions@for $\mathcal M$-actions|textbf}
Let $X$ be any $\mathcal M$-set and let $x\in X$ be finitely supported. Then the \emph{support} $\supp(x)$\nomenclature[asupp]{$\supp$}{support (with respect to $\mathcal M$- or $E\mathcal M$-action)} is the intersection of all finite sets $A\subset\omega$ on which $x$ is supported.
\end{defi}
\begin{lemma}
In the above situation, $x$ is supported on $\supp(x)$.
\begin{proof}
This is immediate from \cite[Proposition~2.3]{I-vs-M-1-cat}.
\end{proof}
\end{lemma}
\begin{ex}\label{ex:M-Inj-support}
Let $A$ be a finite set. Then the $\mathcal M$-set $\Inj(A,\omega)$ is tame and the support of an injection $i\colon A\to\omega$ is simply its image, also see~\cite[Example~2.9]{I-vs-M-1-cat}: namely, it is clear from the definition that any injection $i\colon A\to\omega$ is supported on its image; on the other hand, if $B\not\supset\im(i)$ is any finite set, then we pick an $a\in\im(i)\setminus B$ together with an injection $u\in\mathcal M_B$ such that $a\notin\im(u)$. Then $a\notin\im(ui)$, so $u.i\not=i$, and hence $i$ cannot be supported on $B$.
On the other hand, if $A$ is countably infinite, then a similar computation shows that $\Inj(A,\omega)$ is not tame, and in fact even $\Inj(A,\omega)^\tau=\varnothing$. In particular, $\mathcal M$ itself is not tame.
\end{ex}
\begin{lemma}\label{lemma:support-vs-action-M}
Let $u\in\mathcal M$ and let $X$ be any $\mathcal M$-simplicial set. Then $\supp(u.x)=u(\supp(x))$ for any finitely supported simplex $x$. In particular, $u.\blank\colon X\to X$ restricts to $X_{[A]}\to X_{[u(A)]}$ for any finite $A\subset\omega$, and $X^\tau$ is an $\mathcal M$-simplicial subset of $X$.
\begin{proof}
The first statement is \cite[Proposition~2.5-(ii)]{I-vs-M-1-cat}, which immediately implies the second one. The final statement then in turn follows from the second one, also cf.~\cite[discussion after Proposition~2.5-(ii)]{I-vs-M-1-cat}.
\end{proof}
\end{lemma}
\begin{lemma}\label{lemma:support-agree-M}\label{lemma:last-M-basic}
Let $X\in\cat{$\bm{\mathcal M}$-SSet}$, $u,u'\in\mathcal M$, and let $x\in X_n$. Assume that $x$ is supported on some finite set $A\subset\omega$ such that $u|_A=u'|_A$. Then $u.x=u'.x$.
\begin{proof}
This is \cite[Proposition~2.5-(i)]{I-vs-M-1-cat}, applied to the $\mathcal M$-set $X_n$.\index{support!for M-actions@for $\mathcal M$-actions|)}
\end{proof}
\end{lemma}
\subsubsection{The structure of tame \texorpdfstring{$\mathcal M$-$G$}{M-G}-simplicial sets}
\cite[Theorem~2.11]{I-vs-M-1-cat}, which we recall below, describes how tame $\mathcal M$-sets decompose into some standard pieces. As a consequence of this we will prove:
\begin{thm}\label{thm:structure-tame-M-G-SSet}
Let us define
\begin{equation}\label{eq:definition-I-tame}
\begin{aligned}
I_{\textup{tame}}\mathrel{:=}&\{(\Inj(A,\omega)\times_{\Sigma_A}X)\times\del\Delta^n\hookrightarrow(\Inj(A,\omega)\times_{\Sigma_A}X)\times\Delta^n :\\
&\hphantom{\{}\text{$A\subset\omega$ finite, $X$ a $G$-$\Sigma_A$-biset, $n\ge 0$}\}.
\end{aligned}
\end{equation}
Then the $I_{\textup{tame}}$-cell complexes in $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ are precisely the tame $\mathcal M$-$G$-simplicial sets.
\end{thm}
If $X$ is a tame $\mathcal M$-set, let us write $s_n(X)$ for the subset of those $x\in X$ with $\supp(x)=\{1,\dots,n\}$. Lemma~\ref{lemma:support-agree-M} provides for any $u\in\Inj(\{1,\dots,n\},\omega)$ a well-defined map $s_n(X)\to X$ obtained by acting with any extension of $u$ to an $\bar{u}\in\mathcal M$, and Lemma~\ref{lemma:support-vs-action-M} shows that this restricts to a $\Sigma_n$-action on $s_n(X)$.
\begin{thm}\label{thm:structure-tame-m-set}
Let $X$ be any tame $\mathcal M$-set. Then the map
\begin{equation}\label{eq:tame-decomposition}
\coprod_{n=0}^\infty\Inj(\{1,\dots,n\},\omega)\times_{\Sigma_n}s_n(X)\to X
\end{equation}
induced by the above construction is well-defined and an isomorphism of $\mathcal M$-sets.
\begin{proof}
See \cite[Theorem~2.11]{I-vs-M-1-cat}.
\end{proof}
\end{thm}
\begin{cor}\label{cor:structure-tame-m-g-set}
Let $X$ be any tame $\mathcal M$-$G$-set. Then each $s_n(X)$ is a $G$-subset of $X$, and the map $(\ref{eq:tame-decomposition})$ is an isomorphism of $\mathcal M$-$G$-sets.
\begin{proof}
In order to prove that $s_n(X)$ is closed under the action of $G$, we have to show that $\supp(g.x)=\supp(x)$ for all $g\in G$ and $x\in X$. The inclusion `$\subset$' is an instance of Lemma~\ref{lemma:support-vs-morphism-M} because the $G$-action commutes with the $\mathcal M$-action. The inclusion `$\supset$' then follows by applying the same argument to $g^{-1}$ and $g.x$, or by simply observing that \emph{injective} $\mathcal M$-equivariant maps strictly preserve supports.
Again using that the $\mathcal M$-action commutes with the $G$-action, we see that $(\ref{eq:tame-decomposition})$ is $G$-equivariant, hence an isomorphism of $\mathcal M$-$G$-sets by the previous theorem.
\end{proof}
\end{cor}
The only remaining ingredient for the proof of Theorem~\ref{thm:structure-tame-M-G-SSet} is the following:
\begin{lemma}\label{lemma:tame-complement}
Let $X$ be a tame $\mathcal M$-$G$-set and let $Y\subset X$ be an $\mathcal M$-$G$-subset. Then also $X\setminus Y$ is an $\mathcal M$-$G$-subset.
\end{lemma}
We caution the reader that the above is not true in general for non-tame actions---for example, the subset $Y\subset\mathcal M$ of \emph{non-surjective} maps is closed under the left regular $\mathcal M$-action, but its complement is not.
\begin{proof}
The set $X\setminus Y$ is obviously closed under the $G$-action. The fact that it is moreover closed under the $\mathcal M$-action appeared in a previous version of \cite{I-vs-M-1-cat}; let us give the argument for completeness. Assume $x\in X\setminus Y$ and let $u\in\mathcal M$ such that $u.x\in Y$. By tameness, there exists a finite set $A$ on which $x$ is supported. We now pick any invertible $v\in\mathcal M$ such that $v|_A=u|_A$. By Lemma~\ref{lemma:support-agree-M} we then have $v.x=u.x\in Y$ and hence also $x=v^{-1}.(v.x)\in Y$, which is a contradiction.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:structure-tame-M-G-SSet}]
Obviously all sources and targets of maps in $I_{\textup{tame}}$ are tame. As the tame $\mathcal M$-$G$-simplicial sets are closed under all colimits, we see that all $I_{\textup{tame}}$-cell complexes are tame (cf.~Lemma~\ref{lemma:saturated-objects}).
Conversely, let $X$ be any tame $\mathcal M$-$G$-simplicial set; we consider the usual skeleton filtration $\varnothing=X^{(-1)}\subset X^{(0)}\subset X^{(1)}\subset\cdots$ of $X$. It suffices to prove that each $X^{(n-1)}\to X^{(n)}$ is a relative $I_{\textup{tame}}$-cell complex.
For this we contemplate the (a priori non-equivariant) pushout
\begin{equation*}
\begin{tikzcd}
X_n^{\textup{nondeg}}\times\del\Delta^n\arrow[d]\arrow[r, hook] &X_n^{\textup{nondeg}}\times\Delta^n\arrow[d]\\
X^{(n-1)}\arrow[r,hook]&X^{(n)}
\end{tikzcd}
\end{equation*}
where $X_n^{\textup{nondeg}}$ denotes the subset of nondegenerate $n$-simplices. The \emph{degenerate} simplices obviously form an $\mathcal M$-$G$-subset of $X_n$, and hence so do the nondegenerate ones by the previous lemma. With respect to this action, all maps in the above square are obviously $\mathcal M$-$G$-equivariant so that this is a pushout in $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$. But applying Corollary~\ref{cor:structure-tame-m-g-set} to $X_n^{\textup{nondeg}}$ expresses the top horizontal map as a coproduct of maps in $I_{\textup{tame}}$, finishing the proof.
\end{proof}
\subsection{Tame \texorpdfstring{$\bm{E\mathcal M}$}{EM}-actions}
Next, we will introduce and study analogues of tameness and support for $E\mathcal M$-simplicial sets.
\begin{defi}\label{defi:tame-EM-simplicial-set}
Let $A\subset\omega$ be finite and let $X$ be an $E\mathcal M$-simplicial set. An $n$-simplex $x$ of $X$ is said to be \emph{supported on $A$} if $E(\mathcal M_A)$ acts trivially on $x$, i.e.~the composition
\begin{equation*}
E(\mathcal M_A)\times\Delta^n\hookrightarrow E\mathcal M\times\Delta^n\xrightarrow{E\mathcal M\times x} E\mathcal M\times X\xrightarrow{\textup{act}} X
\end{equation*}
agrees with
\begin{equation*}
E(\mathcal M_A)\times\Delta^n\xrightarrow{\pr} \Delta^n\xrightarrow{x} X.
\end{equation*}
The simplex $x$ is \emph{finitely supported} if it is supported on some finite set $A$.
We write $X_{[A]}$ for the family of simplices supported on a finite set $A\subset\omega$ and $X^\tau\mathrel{:=}\bigcup_{A\subset\omega\textup{ finite}} X_{[A]}$ for the family of all finitely supported simplices. We call $X$ \emph{tame} if $X=X^\tau$, i.e.~if all its simplices are finitely supported.\index{support!for EM-actions@for $E\mathcal M$-actions}
\index{tame!EM-actions@$E\mathcal M$-actions|textbf}
\end{defi}
\begin{ex}\label{ex:EM-Inj-support}
Let $A$ be a finite set. Then $E\Inj(A,\omega)$ is tame, and the support of an $n$-simplex $(i_0,\dots,i_n)$ is given by $B\mathrel{:=}\im(i_0)\cup\cdots\im(i_n)$: namely, to prove that $(i_0,\dots,i_n)$ is supported on $B$ we have to show that
\begin{equation*}
(u_0,\dots,u_m).f^*(i_0,\dots,i_n)=f^*(i_0,\dots,i_n)
\end{equation*}
for any $f\colon [m]\to[n]$ in $\Delta$ and any $u_0,\dots,u_m\in\mathcal M_B$. But plugging in the definition, the left hand side evaluates to $(u_0i_{f(0)},\dots,u_mi_{f(m)})$ while the right hand side is given by $(i_{f(0)},\dots,i_{f(m)})$, so the claim is obvious. Conversely, one argues as in Example~\ref{ex:M-Inj-support} that $(i_0,\dots,i_n)$ is not supported on any finite $C\not\supset B$, or one simply notes that if $(i_0,\dots,i_n)$ is supported on $C$, then so are the individual injections $i_0,\dots,i_n$ as elements of the $\mathcal M$-set $\Inj(A,\omega)$.
Again, one similarly shows that $(E\mathcal M)^\tau=\varnothing$; in particular, $E\mathcal M$ is not tame.
\end{ex}
\begin{lemma}\label{lemma:EM-support-preserve}
Let $f\colon X\to Y$ be a map of $E\mathcal M$-simplicial sets, and let $A\subset\omega$ be any finite set. Then $X_{[A]}$ and $Y_{[A]}$ are simplicial subsets of $X$ and $Y$, respectively, and $f$ restricts to $f_{[A]}\colon X_{[A]}\to Y_{[A]}$.
\begin{proof}
The first statement is clear from the definition.
For the second statement we have to show that $(u_0,\dots,u_m).g^*f(x)=g^*f(x)$ for any $g\colon[m]\to[n]$ in $\Delta$ and any $u_0,\dots,u_m\in\mathcal M_A$. But using that $f$ is simplicial and $E\mathcal M$-equivariant, the left hand side equals $f((u_0,\dots,u_m).g^*x)$, while the right hand side equals $f(g^*x)$, so the claim follows from the definitions.
\end{proof}
\end{lemma}
It is a straight-forward but somewhat lengthy endeavor to also verify the analogues of the other basic properties of tame $\mathcal M$-actions established above in the world of $E\mathcal M$-simplicial sets. We will not do this at this point as they have shorter proofs once we know the following theorem, that is also of independent interest:
\begin{thm}\label{thm:support-EM-vs-M}\index{support!these notions agree|textbf}
Let $X$ be an $E\mathcal M$-simplicial set, let $n\ge 0$, and let $A\subset\omega$ be any finite set. Then $x\in X_n$ is supported on $A$ in the sense of Definition~\ref{defi:tame-EM-simplicial-set} if and only if it is supported on $A$ as a simplex of the underlying $\mathcal M$-simplicial set of $X$ (see Definition~\ref{defi:tame-M-simplicial-set}). In other words, $X_{[A]}=(\forget X)_{[A]}$ as simplicial sets, where $\forget$ denotes the forgetful functor $\cat{$\bm{E\mathcal M}$-SSet}\to\cat{$\bm{\mathcal M}$-SSet}$.
Moreover, the subfunctor $(\blank)_{[A]}\colon\cat{$\bm{E\mathcal M}$-SSet}\to\cat{SSet}$ of the forgetful functor is corepresented in the enriched sense by $E\Inj(A,\omega)$ via evaluation at the $0$-simplex given by the inclusion $A\hookrightarrow\omega$.
\end{thm}
The proof requires some combinatorial preparations:
\begin{prop}
Let $A\subset\omega$ be any finite set, and let $u_0,\dots,u_n\in\mathcal M$. Then there exists a $\chi\in\mathcal M_A$ such that $\im(u_0\chi)\cup\cdots\cup\im(u_n\chi)$ has infinite complement in $\omega$.
\begin{proof}
We will construct strictly increasing chains $B_0\subsetneq B_1\subsetneq\cdots$ and $C_0\subsetneq C_1\subsetneq\cdots$ of finite subsets of $\omega\setminus A$ and $\omega$, respectively, such that for all $j\ge 0$
\begin{equation}\label{eq:induction-hypothesis-B-C}
C_j\cap\bigcup_{i=0}^nu_i(B_j)=\varnothing.
\end{equation}
Let us first show how this yields the proof of the claim: we set $B_\infty\mathrel{:=}\bigcup_{j=0}^\infty B_j$ and $C_\infty\mathrel{:=}\bigcup_{j=0}^\infty C_j$. Then both of these are infinite sets, and moreover each $u_i(B_\infty)$ misses $C_\infty$ by Condition $(\ref{eq:induction-hypothesis-B-C})$ and since the unions are increasing. As $B_\infty$ is infinite, we find an injection $c\colon\omega\setminus A\to\omega$ with image $B_\infty$. We claim that $\chi\mathrel{:=}c+\id_A$ has the desired properties: indeed, this is again an injection as $B_\infty\cap A=\varnothing$, and it is the identity on $A$ by construction. On the other hand,
\begin{equation*}
\bigcup_{i=0}^n \im(u_i\chi)=\underbrace{\bigcup_{i=0}^n u_i(A)}_{{}\mathrel{=:}A'}\cup\underbrace{\bigcup_{i=0}^n u_i(B_\infty)}_{{}\mathrel{=:}B'}
\end{equation*}
and $B'$ has infinite complement in $\omega$ (namely, at least $C_\infty$) whereas $A'$ is even finite; we conclude that also their union has infinite complement as desired.
It therefore only remains to construct the chains $B_0\subsetneq\cdots$ and $C_0\subsetneq\cdots$, for which we will proceed by induction. We begin by setting $B_0=C_0=\varnothing$ which obviously has all the required properties. Now assume we've already constructed the finite sets $B_j$ and $C_j$ satisfying $(\ref{eq:induction-hypothesis-B-C})$.
The set $\left(A\cup B_j\cup\bigcup_{i=0}^n u_i^{-1}(C_j)\right)$ is finite as $A,B_j,C_j$ are finite and each $u_i$ is injective, so we can pick a $b\in\omega$ not contained in it. We now set $B_{j+1}\mathrel{:=}B_j\cup\{b\}$, which is obviously finite and a proper superset of $B_j$ by construction. We moreover observe that $C_j$ misses all $u_i(B_{j+1})$ as it misses $u_i(B_j)$ by the induction hypothesis and moreover $u_i(b)\notin C_j$ for any $i$ by construction.
By the same argument we can pick $c\in\omega\setminus\left(C_j\cup \bigcup_{i=0}^n u_i(B_{j+1})\right)$ and define $C_{j+1}\mathrel{:=}C_j\cup\{c\}$; this is obviously again finite and a proper superset of $C_j$. We claim that Condition $(\ref{eq:induction-hypothesis-B-C})$ is satisfied for $B_{j+1}$ and $C_{j+1}$. Indeed, we have already seen that $C_j$ misses all of $u_i(B_{j+1})$. On the other hand, also $c\notin u_i(B_{j+1})$ for each $i$ by construction, verifying the condition. This finishes the inductive construction and hence the proof of the proposition.
\end{proof}
\end{prop}
\begin{prop}
Let $A\subset\omega$ be finite, and let $(u_0,\dots,u_n), (v_0,\dots,v_n)\in\mathcal M^{1+n}$ such that $u_i|_A=v_i|_A$ for $i=0,\dots,n$. Then $[u_0,\dots,u_n]=[v_0,\dots,v_n]$ in $\mathcal M^{1+n}/\mathcal M_A$.
\begin{proof}
Applying the above to the $2n+2$ injections $u_0,\dots,u_n,v_0,\dots,v_n$, we may assume without loss of generality that
\begin{equation*}
B\mathrel{:=}\omega\setminus\left(\bigcup_{i=0}^n\im u_i\cup\bigcup_{i=0}^n\im v_i\right)
\end{equation*}
is infinite. We can therefore choose an injection $\phi\colon\omega\setminus A\to\omega$ with image in $B$, and we moreover pick a bijection $\omega\setminus A\cong (\omega\setminus A)\amalg(\omega\setminus A)$, yielding two injections $j_1,j_2\colon\omega\setminus A\to\omega\setminus A$ whose images partition $\omega\setminus A$. We now define for $i=0,\dots,n$
\begin{equation*}
w_i(x)\mathrel{:=}\begin{cases}
u_i(x) & \textup{if }x\in A\\
u_i(y) & \textup{if }x = j_1(y)\textup{ for some $y\in\omega\setminus A$}\\
\phi(y) & \textup{if }x = j_2(y)\textup{ for some $y\in\omega\setminus A$.}
\end{cases}
\end{equation*}
This is indeed well-defined as $\omega$ is the disjoint union $A\sqcup \im(j_1)\sqcup\im(j_2)$ and since $j_1$ and $j_2$ are injective. We now claim that $w_i$ is injective (and hence an element of $\mathcal M$): indeed, assume $w_i(x)=w_i(x')$ for $x\not=x'$. Since $\im(u_i)$ is disjoint from $\im\phi$ and since $\phi$ is injective, we conclude that $x,x'\notin\im j_2$. As moreover $w_i|_A=u_i|_A$ and $w_ij_1=u_i|_{\omega\setminus A}$ are injective, we can assume without loss of generality that $x\in A$ and $x'=j_1(y')$ for some $y'\in\omega\setminus A$. But then $u_i(x)=w_i(x)=w_i(x')=u_i(y')$, which contradicts the injectivity of $u_i$ as $y'\notin A$ and hence in particular $y'\not=x$.
We now observe that by construction $w_i(\incl_A+j_1)=u_i$ and $w_i(\incl_A+j_2)=u_i|_A+\phi$. On the other hand, $\incl_A+j_1$ and $\incl_A+j_2$ are obviously injections fixing $A$ pointwise, so that they witness the equalities
\begin{equation*}
[u_0,\dots,u_n]=[w_0,\dots,w_n]=[u_0|_A+\phi,\dots,u_n|_A+\phi]
\end{equation*}
in $\mathcal M^{1+n}/\mathcal M_A$. Analogously, one shows that $[v_0,\dots,v_n]=[v_0|_A+\phi,\dots,v_n|_A+\phi]$, and as $v_i|_A=u_i|_A$ by assumption, this further agrees with $[u_0|_A+\phi,\dots,u_n|_A+\phi]$, finishing the proof.
\end{proof}
\end{prop}
\begin{cor}\label{cor:support-agree-EM}
Let $(u_0,\dots,u_n),(u_0',\dots,u_n')\in (E\mathcal M)_n$, let $X$ be an $E\mathcal M$-simplicial set, and let $x\in X_n$. Assume that $x$ is supported \emph{as an element of the $\mathcal M$-set $X_n$} on some finite set $A\subset\omega$ such that $u_i|_A=u_i'|_A$ for $i=0,\dots,n$.
Then $(u_0,\dots,u_n).x = (u_0',\dots,u_n').x$.
\begin{proof}
We begin with the special case that there exists an $\alpha\in\mathcal M_A$ such that $u_i'=u_i\alpha$ for all $i$. In this case
\begin{equation*}
(u_0',\dots,u_n').x=(u_0\alpha,\dots,u_n\alpha).x=(u_0,\dots,u_n).\alpha.x=(u_0,\dots,u_n).x
\end{equation*}
as desired, where the last step uses that $x$ is fixed by $\alpha\in\mathcal M_A$.
We conclude that $\mathcal M^{1+n}\to X_n$, $(u_0,\dots,u_n)\mapsto (u_0,\dots,u_n).x$ descends to $\mathcal M^{1+n}/\mathcal M_A$; the claim therefore follows from the previous proposition.
\end{proof}
\end{cor}
\begin{proof}[Proof of Theorem~\ref{thm:support-EM-vs-M}]\index{support!these notions agree}
Let $\chi\colon E\Inj(A,\omega)\times\Delta^n\to X$ be any $E\mathcal M$-equivariant map. We claim that the image of the $n$-simplex $(i,\dots,i;\id_{[n]})$, where $i$ denotes the inclusion $A\hookrightarrow\omega$, is supported on $A$. Indeed, $(i,\dots,i;\id_{[n]})$ is supported on $A$ by the argument from Example~\ref{ex:EM-Inj-support}, hence so is its image by Lemma~\ref{lemma:EM-support-preserve}.
On the other hand, let $x\in X_n$ be supported on $A$ \emph{with respect to the underlying $\mathcal M$-action}. We define $\chi_m\colon (E\Inj(A,\omega)\times\Delta^n)_m\to X_m$ as follows: we send a tuple $(u_0,\dots,u_m;f)$, where the $u_i$ are injections $A\to\omega$ and $f\colon[m]\to[n]$ is any map in $\Delta$, to $(\bar{u}_0,\dots,\bar{u}_m).f^*x$ where each $\bar{u}_i$ is an extension of $u_i$ to all of $\omega$, i.e.~to an element of $\mathcal M$. Such extensions can certainly be chosen as $A$ is finite, and we claim that this is in fact independent of the choice of extension: indeed, as $x$ is fixed by $\mathcal M_A$, so is $f^*x$, and hence this follows from the previous corollary. With this established, it is trivial to prove that the $\chi_m$ assemble into a simplicial map $E\Inj(A,\omega)\times\Delta^n\to X$ and that this is $E\mathcal M$-equivariant. Moreover, a possible extension of $i\colon A\hookrightarrow\omega$ is given by the identity of $\omega$ and hence we see that $\chi_m(i,\dots,i;\id_{[n]})=x$.
We conclude that if $x\in X_n$ is supported on $A$ with respect to the underlying $\mathcal M$-action, then it is obtained by evaluating some $E\mathcal M$-equviariant $\chi\colon E\Inj(A,\omega)\times\Delta^n\to X$ at the canonical element (by the second paragraph), and hence it is actually supported on $A$ with respect to the $E\mathcal M$-action (by the first one). As the converse holds for trivial reasons, we conclude that the two notions of `being supported on $A$' indeed agree. It then follows from the above that evaluation at the canonical element defines a surjection
\begin{equation}\label{eq:corepr-supported-on-A}
\Maps(E\Inj(A,\omega),X)\to X_{[A]}.
\end{equation}
As we have seen in Lemma~\ref{lemma:EM-support-preserve}, $X\mapsto X_{[A]}$ defines a subfunctor of the forgetful functor $\cat{$\bm{E\mathcal M}$-SSet}\to\cat{SSet}$, and with respect to this $(\ref{eq:corepr-supported-on-A})$ is obviously natural. To finish the proof of the claimed corepresentability result, it therefore suffices that $(\ref{eq:corepr-supported-on-A})$ is also injective. For this we let $\chi,\chi'\colon E\Inj(A,\omega)\times\Delta^n\to X$ with $\chi(i,\dots,i;\id_{[n]})=\chi'(i,\dots,i;\id_{[n]})$. Then we have for any $(u_0,\dots,u_m)\in\mathcal M^{1+m}$ and $f\colon[m]\to[n]$ in $\Delta$
\begin{align*}
\chi(u_0,\dots,u_m;f)&=(\bar{u}_0,\dots,\bar{u}_m).\chi(i,\dots,i;f)=(\bar{u}_0,\dots,\bar{u}_m).f^*\chi(i,\dots,i;\id_{[n]})\\
&=(\bar{u}_0,\dots,\bar{u}_m).f^*\chi'(i,\dots,i;\id_{[n]})=\chi'(u_0,\dots,u_m;f)
\end{align*}
where again $\bar{u}_i$ is any extension of $u_i$ to an element of $\mathcal M$. This finishes the proof of corepresentability and hence of the theorem.
\end{proof}
The above theorem has the following computational consequence:
\begin{cor}\label{cor:E-Inj-corepr}
For any simplicial set $K$, any finite set $A$, and any $G$-$\Sigma_A$-biset $X$, the map
\begin{equation*}
E\mathcal M\times_{\mathcal M}\big((\Inj(A,\omega)\times_{\Sigma_A}X)\times K\big)\to(E\Inj(A,\omega)\times_{\Sigma_A}X)\times K
\end{equation*}
adjunct to the product of the inclusion of the $0$-simplices with the identity of $K$ is an isomorphism of $E\mathcal M$-$G$-simplicial sets.
\begin{proof}
As $E\mathcal M\times_{\mathcal M}\blank$ is a simplicial left adjoint, we are reduced to proving that the corresponding map $E\mathcal M\times_{\mathcal M}\Inj(A,\omega)\to E\Inj(A,\omega)$ is an isomorphism (it is obviously left-$\mathcal M$-right-$\Sigma_A$-equivariant). For this we observe that by the above theorem $E\Inj(A,\omega)$ corepresents $(\blank)_{[A]}$ by evaluating at the inclusion $\iota\colon A\hookrightarrow\omega$. On the other hand, it is obvious that $\Inj(A,\omega)$ corepresents $(\blank)_{[A]}\colon\cat{$\bm{\mathcal M}$-SSet}\to\cat{SSet}$ by evaluating at the same element, also see \cite[Example~2.9]{I-vs-M-1-cat}. By adjointness, $E\mathcal M\times_{\mathcal M}\Inj(A,\omega)$ therefore corepresents $(\blank)_{[A]}\circ\forget$ via evaluating at $[1;\iota]$. By the previous theorem this agrees with $(\blank)_{[A]}$, and as the above map sends $[1;\iota]$ to $\iota$, this completes the proof.
\end{proof}
\end{cor}
We now very easily prove the $E\mathcal M$-analogue of Lemma~\ref{lemma:support-vs-action-M}:
\begin{lemma}\label{lemma:support-vs-action-EM}
Let $(u_0,\dots,u_n)\in (E\mathcal M)_n$, let $A\subset\omega$ be finite, and let $X$ be any $E\mathcal M$-simplicial set. Then the composition
\begin{equation}\label{eq:EM-support-restricted-action}
\Delta^n\times X_{[A]}\xrightarrow{(u_0,\dots,u_n)\times\incl}E\mathcal M\times X\xrightarrow{\textup{act}} X
\end{equation}
has image in $X_{[u_0(A)\cup\cdots\cup u_n(A)]}$; in particular, $X^\tau$ is an $E\mathcal M$-simplicial subset of $X$.
\begin{proof}
It suffices to prove the first statement. For this we observe that any simplex in the image of $(\ref{eq:EM-support-restricted-action})$ can be written as $\big(f^*(u_0,\dots,u_n)\big).x$ for some $m$-simplex $x$ of $X_{[A]}$ and some $f\colon[m]\to[n]$ in $\Delta$. We now calculate for any $v\in\mathcal M$
\begin{equation}\label{eq:double-action}
v.(f^*(u_0,\dots,u_n).x)=v.(u_{f(0)},\dots,u_{f(m)}).x=(vu_{f(0)},\dots,vu_{f(n)}).x.
\end{equation}
If now $v$ is the identity on $u_0(A)\cup\cdots\cup u_n(A)$, then $vu_{f(i)}$ and $u_{f(i)}$ agree on $A$ for all $i=0,\dots,m$. Hence, if $x$ is supported on $A$, then
\begin{equation*}
(vu_{f(0)},\dots, vu_{f(m)}).x=(u_{f(0)},\dots,u_{f(m)}).x=f^*(u_0,\dots,u_n).x
\end{equation*}
by Corollary~\ref{cor:support-agree-EM}. Together with $(\ref{eq:double-action})$ this precisely yields the claim.
\end{proof}
\end{lemma}
\begin{cor}\label{cor:EM-tau-colim-lim}
The full simplicial subcategory $\cat{$\bm{E\mathcal M}$-SSet}^\tau\subset\cat{$\bm{E\mathcal M}$-SSet}$ is complete and cocomplete, and it is closed under all small colimits and finite limits.
\begin{proof}
As in Remark~\ref{rk:tame-M-colimits}, the previous lemma provides a right adjoint $(\blank)^\tau$ to the inclusion $\cat{$\bm{E\mathcal M}$-SSet}^\tau\hookrightarrow\cat{$\bm{E\mathcal M}$-SSet}$, which shows that $\cat{$\bm{E\mathcal M}$-SSet}^\tau$ is complete and cocomplete with colimits created in $\cat{$\bm{E\mathcal M}$-SSet}$. As the forgetful functor $\cat{$\bm{E\mathcal M}$-SSet}\to\cat{$\bm{\mathcal M}$-SSet}$ creates limits and preserves and reflects tameness by Theorem~\ref{thm:support-EM-vs-M}, the identification of finite limits follows from Remark~\ref{rk:tame-M-colimits}.
\end{proof}
\end{cor}
\begin{rk}
Of course, we could have just turned Theorem~\ref{thm:support-EM-vs-M} into the definition instead---however, this does not buy as anything, as we then instead would have had to do all of the above work in order to prove Corollary~\ref{cor:support-agree-EM} (or equivalently the corepresentability statement), which will be crucial below. Moreover, defining support by means of the mere $\mathcal M$-action is `evil' from a homotopical viewpoint, as we \emph{a priori} throw away a lot of higher information: for example, if $u\in\mathcal M$ and $x$ is a vertex of an $E\mathcal M$-simplicial set, then we can think of the edge $(u,1).(s^*x)$, where $s\colon[1]\to[0]$ is the unique morphism in $\Delta$, as providing a natural comparison between $x$ and $u.x$. From the homotopical viewpoint we should then always be interested in this edge itself and not only in its endpoints.
\end{rk}
A hindrance to understanding the $G$-global weak equivalences of general $\mathcal M$-simplicial sets was that it is not clear whether $E\mathcal M\times_{\mathcal M}\blank$ is fully homotopical. Using the above theory, we can prove the following comparison result, which in particular tells us that this issue goes away when restricting to tame actions:
\begin{thm}\label{thm:tame-M-sset-vs-EM-sset}
The simplicial adjunction $E\mathcal M\times_{\mathcal M}\blank\dashv\forget$ restricts to
\begin{equation}\label{eq:EM-adjunction-tame}
E\mathcal M\times_{\mathcal M}\blank\colon\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau :\!\forget.
\end{equation}
Both functors in $(\ref{eq:EM-adjunction-tame})$ preserve and reflect $G$-global weak equivalences, and they descend to mutually inverse equivalences on associated quasi-categories.
\begin{proof}
The forgetful functor obviously restricts to the full subcategories of tame objects. To see that also $E\mathcal M\times_{\mathcal M}\blank$ restricts accordingly, we appeal to Theorem~\ref{thm:structure-tame-M-G-SSet}: as the tame $E\mathcal M$-$G$-simplicial sets are closed under all colimits, we are reduced (Lemma~\ref{lemma:saturated-objects}) to showing that $E\mathcal M\times_{\mathcal M}\big((\Inj(A,\omega)\times_{\Sigma_A}X)\times K\big)$ is tame for every $G$-$\Sigma_A$-biset $X$ and any simplicial set $K$, which follows easily from Corollary~\ref{cor:E-Inj-corepr}.
The forgetful functor creates weak equivalences (even without the tameness assumption) as it is homotopical and part of a Quillen equivalence by Corollary~\ref{cor:em-vs-m-equiv-model-cat}.
Let us now prove that the unit $\eta\colon Y\to\forget E\mathcal M\times_{\mathcal M}Y$ is a $G$-global weak equivalence for every $Y\in\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau$. We caution the reader that this is not just a formal consequence of Corollary~\ref{cor:em-vs-m-equiv-model-cat} because we are \emph{not} deriving $E\mathcal M\times_{\mathcal M}\blank$ in any way here. Instead, we will use Theorem~\ref{thm:hq-M-semistable-replacement} together with the full generality of Theorem~\ref{thm:hq-M-computation}:
Both $E\mathcal M\times_{\mathcal M}\blank$ as well as $\forget$ are left adjoints and hence cocontinuous. As both functors preserve tensors, the composition $\forget(E\mathcal M\times_{\mathcal M}\blank)$ sends the maps in $I_{\textup{tame}}$ to underlying cofibrations. Corollary~\ref{cor:saturated-trafo} therefore reduces this to the case that $Y=(\Inj(A,\omega)\times_{\Sigma_A}X)\times K$ for $H,A,X$ as above and $K$ any simplicial set. Again using that both $E\mathcal M\times_{\mathcal M}\blank$ and $\forget$ preserve tensors (and that the above is a simplicial adjunction) we reduce further to the case that $Y=\Inj(A,\omega)\times_{\Sigma_A}X$. After postcomposing with the isomorphism from Corollary~\ref{cor:E-Inj-corepr}, the unit simply becomes the inclusion of the $0$-simplices, and this factors as
\begin{equation*}
\Inj(A,\omega)\times_{\Sigma_A}X\xrightarrow{\pi}\big(\Inj(A,\omega)\times_{\Sigma_A}X\big)\hq\mathcal M\to E\Inj(A,\omega)\times_{\Sigma_A}X,
\end{equation*}
where the right hand map is the $G$-universal weak equivalence from Theorem~\ref{thm:hq-M-computation}. As the left hand map is moreover a $G$-global weak equivalence by Theorem~\ref{thm:hq-M-semistable-replacement}, thus so is the unit.
Now let $f\colon X\to Y$ be any map in $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau$. In the naturality square
\begin{equation*}
\begin{tikzcd}[column sep=1in]
X\arrow[d, "\eta"']\arrow[r, "f"] & Y\arrow[d,"\eta"]\\
\forget E\mathcal M\times_{\mathcal M} X\arrow[r, "\forget E\mathcal M\times_{\mathcal M}f"'] & \forget E\mathcal M\times_{\mathcal M}Y
\end{tikzcd}
\end{equation*}
both vertical maps are $G$-global weak equivalences by the above. Thus, if $f$ is a $G$-global weak equivalence, then so is the lower horizontal map. But $\forget$ reflects these, hence also $E\mathcal M\times_{\mathcal M}f$ is a $G$-global weak equivalence, proving that $E\mathcal M\times_{\mathcal M}\blank$ is homotopical. Conversely, if $E\mathcal M\times_{\mathcal M}f$ is a $G$-global weak equivalence, then so is $\forget E\mathcal M\times_{\mathcal M}f$ and hence also $f$ by the above square, i.e.~$E\mathcal M\times_{\mathcal M}\blank$ also reflects $G$-global weak equivalences.
Finally, if $X\in\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ is arbitrary, then the triangle identity for adjunctions shows that $\forget\epsilon_X\colon\forget E\mathcal M\times_{\mathcal M}(\forget X)\to\forget X$ is right inverse to $\eta_{\forget X}$, hence a $G$-global weak equivalence. As $\forget$ reflects these, we conclude that also $\epsilon$ is levelwise a $G$-global weak equivalence, proving that the functors in $(\ref{eq:EM-adjunction-tame})$ induce mutually inverse equivalences of quasi-categories.
\end{proof}
\end{thm}
\subsection{The Taming of the Shrew} As promised, we can now finally prove that also tame $E\mathcal M$-$G$-simplicial sets and tame $\mathcal M$-$G$-simplicial sets are models of $G$-global homotopy theory. At this point, we will only consider them as categories with weak equivalences whereas suitable $G$-global model structures are the subject of Subsection~\ref{subsec:tame-model-structures}.
\begin{thm}\label{thm:shrew}
The inclusions
\begin{equation*}
\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau\hookrightarrow\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}
\qquad\text{and}\qquad
\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau\hookrightarrow\cat{$\bm{\mathcal M}$-$\bm G$-SSet}
\end{equation*}
are homotopy equivalences with respect to the $G$-global weak equivalences.
\end{thm}
For the proof we will need:
\begin{prop}
Let $H\subset\mathcal M$ be a universal subgroup, let $A\subset\omega$ be a faithful $H$-subset, and let $\phi\colon H\to G$ be any homomorphism. Then:
\begin{enumerate}
\item The restriction $r\colon E\mathcal M\times_\phi G\to E\Inj(A,\omega)\times_\phi G$ is a $G$-global weak equivalence in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$.
\item The restriction $r\colon \mathcal M\times_\phi G\to \Inj(A,\omega)\times_\phi G$ is a $G$-global weak equivalence in $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$.
\end{enumerate}
\begin{proof}
For the first statement, we note that the restriction $E\mathcal M\to E\Inj(A,\omega)$ is an $H$-global weak equivalence (with respect to $H$ acting on $\mathcal M$ and $A$ via its tautological action on $\omega$) by Example~\ref{ex:G-globally-contractible}. On the other hand, the $H$-action on both sides is free, so Corollary~\ref{cor:free-quotient-EM} implies that its image under $\phi_!$ is a $G$-global weak equivalence. But this is clearly conjugate to $r$, finishing the proof of the first statement.
For the second statement, we note that the inclusion $\mathcal M\times_\phi G\hookrightarrow E\mathcal M\times_\phi G$ is a $G$-global weak equivalence by Theorem~\ref{thm:em-vs-m-equiv}, and so is $\Inj(A,\omega)\times_\phi G\hookrightarrow E\Inj(A,\omega)\times_\phi G$ by the proof of Theorem~\ref{thm:tame-M-sset-vs-EM-sset}. The claim therefore follows from the first statement together with $2$-out-of-$3$.
\end{proof}
\end{prop}
\begin{cor}\label{cor:injective-almost-tame}
Let $H\subset\mathcal M$ be a universal subgroup, and let $A\subset\omega$ be a finite faithful $H$-subset.
\begin{enumerate}
\item Let $X$ be fibrant in the injective $G$-global model structure on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$. Then the $H$-action on $X$ restricts to $X_{[A]}$, and $X_{[A]}\hookrightarrow X$ is a $\mathcal G_{H,G}$-weak equivalence. In particular, $X^\tau\hookrightarrow X$ is a $G$-global weak equivalence.
\item Let $Y$ be fibrant in the injective $G$-global model structure on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$. Then the $H$-action on $Y$ restricts to $Y_{[A]}$, and $Y_{[A]}\hookrightarrow Y$ is a $\mathcal G_{H,G}$-weak equivalence. In particular, $Y^\tau\hookrightarrow Y$ is a $G$-universal weak equivalence.
\end{enumerate}
\begin{proof}
We will only prove the claims for $E\mathcal M$-actions, the proof of the other ones being analogous.
Let $\phi\colon H\to G$ be any homomorphism. Using Theorem~\ref{thm:support-EM-vs-M}, we see that the inclusion $X_{[A]}^\phi\hookrightarrow X^\phi$ agrees up to conjugation by isomorphisms with
\begin{equation}\label{eq:inclusion-corep}
r^*\colon\Maps^{E\mathcal M\times G}(E\Inj(A,\omega)\times_\phi G,X)\to\Maps^{E\mathcal M\times G}(E\mathcal M\times_\phi G,X),
\end{equation}
where $r$ is as in the previous proposition. As $r$ is a $G$-global weak equivalence between injectively cofibrant $E\mathcal M$-$G$-simplicial sets, and since the injective $G$-global model structure is simplicial by Corollary~\ref{cor:equivariant-injective-model-structure}, $(\ref{eq:inclusion-corep})$ is a weak homotopy equivalence, proving the first statement.
For the second statement we consider the commutative diagram
\begin{equation*}
\begin{tikzcd}
\colim_{A} X_{[A]}\arrow[r]\arrow[d] & \colim_{A}X\arrow[d]\\
X^\tau\arrow[r,hook] & X,
\end{tikzcd}
\end{equation*}
where the colimits run over the filtered poset of finite faithful $H$-subsets $A\subset\omega$, and all maps are induced by the inclusions. The right hand vertical arrow is an isomorphism as the indexing category is filtered, and so is the left hand vertical map since a simplex supported on some finite set $B$ is also supported on the finite faithful $H$-subset $HB\cup F$, where $F\subset\omega$ is any chosen free $H$-orbit (which exists by universality). The claim therefore follows from the first statement as $\mathcal G_{H,G}$-weak equivalences are closed under filtered colimits.
\end{proof}
\end{cor}
\begin{proof}[Proof of Theorem~\ref{thm:shrew}]
Again, we will only prove the first statement. For this, we factor the inclusion through the full subcategory $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^{w\tau}$ of those $E\mathcal M$-$G$-simplicial sets $X$ for which $X^\tau\hookrightarrow X$ is a $G$-global weak equivalence. It suffices to prove that both intermediate inclusions are homotopy equivalences.
Indeed, a homotopy inverse to $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^{\tau}\hookrightarrow\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^{w\tau}$ is obviously given by $(\blank)^\tau$, whereas the previous corollary implies that taking functorial fibrant replacements in the injective $G$-global model structure provides a homotopy inverse to the remaining inclusion.
\end{proof}
\section[$G$-global homotopy theory via diagram spaces]{\texorpdfstring{\except{toc}{$\bm G$}\for{toc}{$G$}}{G}-global homotopy theory via diagram spaces}
We will now consider models of unstable $G$-global homotopy theory in terms of more general `diagram spaces,' i.e.~functors from suitable indexing categories to $\cat{SSet}$. These models will in particular be useful in Chapter~\ref{chapter:stable} to connect unstable and stable $G$-global homotopy theory.
\begin{defi}
We write $I$\nomenclature[aI1]{$I$}{category of finite sets and injections} for the category of finite sets and injections, and we write $\mathcal I$\nomenclature[aI2]{$\mathcal I$}{simplicially enriched category built from $I$} for the simplicial category obtained by applying $E$ to the hom sets. An \emph{$I$-simplicial set} is a functor $I\to\cat{SSet}$, and we write $\cat{$\bm I$-SSet}$ for the simplicially enriched functor category $\FUN(I,\cat{SSet})$. An $\emph{$\mathcal I$-simplicial set}$ is a simplicially enriched functor $\mathcal I\to\cat{SSet}$, and we write $\cat{$\bm{\mathcal I}$-SSet}\mathrel{:=}\FUN(\mathcal I,\cat{SSet})$.
\end{defi}
In the literature, the category $I$ is also denoted by $\mathbb I$ \cite{lind} and unfortunately also by $\mathcal I$ \cite{sagave-schlichtkrull}. Sagave and Schlichtkrull proved that $\cat{$\bm I$-SSet}$ models ordinary homotopy theory, see~\cite[Theorem~3.3]{sagave-schlichtkrull}.
We can also view $I$ as a discrete analogue of the topological category $L$ used in Schwede's model of unstable global homotopy theory in terms $L$-spaces, see~\cite[Sections~1.1--1.2]{schwede-book}. In an earlier version, Schwede also sketched that $I$-spaces model global homotopy theory (with respect to finite groups), also cf.~\cite[Section~6.1]{hausmann-global}, for which we will give a full proof as Theorem~\ref{thm:L-vs-I}.
\subsection{Model structures} Next, we will introduce $G$-global model structures on $G$-$I$- and $G$-$\mathcal I$-simplicial sets (i.e.~$G$-objects in $\cat{$\bm I$-SSet}$ or $\cat{$\bm{\mathcal I}$-SSet}$, respectively), and prove that they are equivalent to the models from the previous sections. As usual in this context, we begin by constructing a suitable level model structure that we will then later Bousfield localize at the desired weak equivalences:
\index{G-global model structure@$G$-global model structure!strict level|seeonly{strict level model structure}}
\index{strict level model structure!on G-I-SSet@on $\cat{$\bm G$-$\bm I$-SSet}$|textbf}
\begin{prop}\label{prop:strict-model-structure}
There is a unique model structure on $\cat{$\bm{G}$-$\bm{I}$-SSet}$ in which a map $f\colon X\to Y$ is a weak equivalence or fibration if and only if $f(A)\colon X(A)\to Y(A)$ is a $\mathcal G_{\Sigma_A,G}$-weak equivalence or fibration, respectively, for every finite set $A$, i.e.~for each finite group $H$ acting faithfully on $A$ and each homomorphism $\phi\colon H\to G$ the induced map $X(A)^\phi\to Y(A)^\phi$ is a weak equivalence or fibration, respectively.
We call this the \emph{strict level model structure} and its weak equivalences the \emph{strict level weak equivalences}\index{strict level weak equivalence!in G-I-SSet@in $\cat{$\bm G$-$\bm{I}$-SSet}$|textbf}. It is proper, simplicial, combinatorial, and filtered colimits in it are homotopical. A possible set of generating cofibrations is given by the maps
\begin{equation*}
(I(A,\blank)\times_\phi G)\times\del\Delta^n\hookrightarrow(I(A,\blank)\times_\phi G)\times\Delta^n
\end{equation*}
for $n\ge 0$ and $H$, $A$, and $\phi$ as above, and a set of generating acyclic cofibrations is likewise given by the maps
\begin{equation*}
(I(A,\blank)\times_\phi G)\times\Lambda^n_k\hookrightarrow(I(A,\blank)\times_\phi G)\times\Delta^n
\end{equation*}
with $0\le k\le n$.
\index{strict level model structure!on G-II-SSet@on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$|textbf}
The analogous model structure on $\cat{$\bm{G}$-$\bm{\mathcal I}$-SSet}$ exists and has the same properties; we again call it the \emph{strict level model structure} and its weak equivalences the \emph{strict level weak equivalences}.\index{strict level weak equivalence!in G-II-SSet@in $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$|textbf} A possible set of generating cofibrations is given by the maps
\begin{equation*}
(\mathcal I(A,\blank)\times_\phi G)\times\del\Delta^n\hookrightarrow(\mathcal I(A,\blank)\times_\phi G)\times\Delta^n
\end{equation*}
for $n\ge 0$ and $H$, $A$, and $\phi$ as above, and a set of generating acyclic cofibrations is likewise given by the maps
\begin{equation*}
(\mathcal I(A,\blank)\times_\phi G)\times\Lambda^n_k\hookrightarrow(\mathcal I(A,\blank)\times_\phi G)\times\Delta^n
\end{equation*}
with $0\le k\le n$.
Finally, the forgetful functor is part of a simplicial Quillen adjunction\nomenclature[aII]{$\mathcal I\times_I\blank$}{left adjoint to forgetful functor $\cat{$\bm{\mathcal I}$-SSet}\to\cat{$\bm I$-SSet}$}
\begin{equation}\label{eq:I-vs-script-I-restricted}
\mathcal I\times_{I}\blank\colon\cat{$\bm{G}$-$\bm{I}$-SSet}_{\textup{strict level}}\rightleftarrows\cat{$\bm{G}$-$\bm{\mathcal I}$-SSet}_{\textup{strict level}} :\!\forget.
\end{equation}
\end{prop}
\begin{rk}
To be entirely precise, the above generating (acyclic) cofibrations of course do not form a set since there are too many finite groups $H$ and finite faithful $H$-sets $A$. However, this issue is easily solved by restricting to a set of finite groups $H$ and sets of finite $H$-sets such that these cover all isomorphism classes, and we will tacitly do so. Similar caveats apply to several of the other model structures considered below.
\end{rk}
\subsubsection{Generalized projective model structures}
We will obtain these model structures as an instance of a more general construction from \cite{schwede-book} of `generalized projective model structures' for suitable index categories. For this we will need the following terminology:
\begin{defi}\index{dimension function|seealso{generalized projective model structure, for categories with dimension function}}
\index{dimension function|textbf}
Let $\mathscr C$ be a complete and cocomplete closed symmetric monoidal category. We say that a $\mathscr C$-enriched category $\mathscr I$ \emph{has a dimension function} if there exists a function $\dim\colon\Ob(\mathscr I)\to\mathbb N$ such that
\begin{enumerate}
\item $\Hom(d,e)$ is initital in $\mathscr C$ whenever $\dim(e)<\dim(d)$.
\item If $\dim(d)=\dim(e)$, then $d\cong e$.
\end{enumerate}
\end{defi}
\begin{ex}
Both $I$ and $\mathcal I$ have dimension functions; a canonical choice is the function sending a finite set $A$ to its cardinality $|A|$. Moreover, if $G$ is any discrete group, then composing with the projection $I\times BG\to I$ or $\mathcal I\times BG\to\mathcal I$ yields a dimension function on $I\times BG$ or $\mathcal I\times BG$, respectively.
\end{ex}
\begin{prop}\label{prop:generalized-projective-dim}\index{generalized projective model structure!for categories with dimension function|textbf}
Let $\mathscr I$ be a $\mathscr C$-enriched category with dimension function $\dim$, and assume we are given for each $A\in\mathscr I$ a model structure on the category $\End(A)\textup{--}\mathscr C\mathrel{:=}\FUN(B\End(A),\mathscr C)$ of enriched functors such that the following `consistency condition' holds: if $\dim(A)\le\dim(B)$, then any pushout of $\Hom(A,B)\otimes_{\End(A)}i$ is a weak equivalence in $\End(B)\textup{--}\mathscr C$ for any acyclic cofibration $i$ in $\End(A)\textup{--}\mathscr C$.
Then there exists a unique model structure on $\FUN(\mathscr I,\mathscr C)$ such that a map $f\colon X\to Y$ is a weak equivalence or fibration if and only if $f(A)\colon X(A)\to Y(A)$ is a weak equivalence or fibration, respectively, in $\End(A)\textup{--}\mathscr C$ for each $A\in\mathscr I$.
Moreover, if each $\End(A)\textup{--}\mathscr C$ is cofibrantly generated with set of generating cofibrations $I_A$ and set of generating acyclic cofibrations $J_A$, then the resulting model structure is cofibrantly generated with set of generating cofibrations
\begin{equation*}
\{\Hom(A,\blank)\otimes_{\End(A)}i\colon A\in \mathscr I,i\in I_A\},
\end{equation*}
and generating acyclic cofibrations
\begin{equation*}
\{\Hom(A,\blank)\otimes_{\End(A)}j\colon A\in \mathscr I,j\in J_A\}.
\end{equation*}
\begin{proof}
See \cite[Proposition~C.23]{schwede-book}.
\end{proof}
\end{prop}
Here, $\Hom(A,\blank)\otimes_{\End(A)}\blank$ is the left adjoint of the functor $\FUN(\mathscr I,\cat{SSet})\to\End(A)\textup{--}\mathscr C$ given by evaluation at $A$. For $\mathscr C=\cat{SSet}$, we have a concrete model of $\Hom(A,B)\otimes_{\End(A)}X$ as the \emph{balanced product} $\Hom(A,B)\times_{\End(A)}X$, i.e.~the quotient of the ordinary product by the equivalence relation generated in each simplicial degree by $(f\sigma,x)\sim (f,X(\sigma)(x))$.
We can also chacterize the cofibrations of the above model structure in analogy with the usual characterization in Reedy model structures. For this we need the following notion from \cite[Construction C.13~and Definition~C.15]{schwede-book}:
\begin{constr}\label{constr:latching-general}
\index{latching object|textbf}\index{latching map|textbf}
For any $m\ge 0$, we define $\mathscr I^{{}\le m}\subset\mathscr I$ as the full subcategory of those objects $A$ satisfying $\dim(A)\le m$. Then restriction along $i_m\colon\mathscr I^{{}\le m}\hookrightarrow\mathscr I$ admits a left adjoint $i_{m!}\colon\Fun(\mathscr I^{{}\le m},\mathscr C)\to\Fun(\mathscr I,\mathscr C)$ via enriched left Kan extension, and we write $\epsilon_m$ for the counit of the adjunction $i_{m!}\dashv i_m^*$. We now define the \emph{$A^{\text{th}}$ latching object} of $X\colon\mathscr I\to\mathscr C$ via $L_A(X)\mathrel{:=}(i_{(\dim(A)-1)!}i_{\dim(A)-1}^*X)(A)$ and the \emph{$A^{\text{th}}$ latching map} $\ell_A\colon L_A(X)\to X(A)$ as $\epsilon_{\dim(A)-1}(A)$.
\end{constr}
\begin{rk}\index{latching category|textbf}\index{latching object|textbf}\index{latching map|textbf}
If $\mathscr I$ is an ordinary category with dimension function, then the \emph{latching category} $\del(\mathscr I\downarrow A)$ is defined for $A\in\mathscr I$ as the full subcategory of the slice $\mathscr I\downarrow A$ on all objects $B\to A$ that are not isomorphisms. In analogy with the usual terminology in Reedy categories, we can then define the \emph{$A^{\text{th}}$ latching object}\nomenclature[aLA]{$L_A$}{$A^{\text{th}}$ latching object} as
\begin{equation*}
L_A(X)\mathrel{:=}\colim\limits_{B\to A\in \del(\mathscr I\downarrow A)} X(B) \in\End(A)\textup{--}\mathscr C,
\end{equation*}
and the maps $X(B)\to X(A)$ induced by the given maps $B\to A$ assemble via the universal property of colimits into a map $\ell_A\colon L_A(X)\to X(A)$ that we again call the \emph{$A^{\text{th}}$ latching map}.\nomenclature[alA]{$\ell_A$}{$A^{\text{th}}$ latching map} Note that this is indeed a special case of the above general definition by the usual pointwise formula for left Kan extension.
\end{rk}
\begin{ex}
While we view $BG\times I$ as a simplicially enriched category, the category $\FUN(BG\times I,\mathscr C)$ of enriched functors is isomorphic to the usual functor category, so the previous remark applies to this setting to express the latching objects as colimits over the latching categories.
However, we can give an even simpler description in this case: namely, the evident inclusion of the poset $\{B\subsetneq A\}$ into $\del(BG\times I\downarrow A)$ is an equivalence for any $A\in I$, in particular cofinal. Thus, we can describe latching object for $X\in\cat{$\bm G$-$\bm I$-SSet}$ as $\colim_{B\subsetneq A}X(B)$ with the induced $(\Sigma_A\times G)$-action, and the latching map is again induced by the inclusions $B\hookrightarrow A$.
\end{ex}
\begin{prop}\label{prop:characterization-cofibration-general}\index{generalized projective model structure!for categories with dimension function!cofibrations|textbf}
A map $f\colon X\to Y$ is a cofibration in the model structure of Proposition~\ref{prop:generalized-projective-dim} if and only if the map
\begin{equation*}
X(A)\amalg_{L_A(X)}L_A(Y)\xrightarrow{(f(A),\ell_A)} Y(A)
\end{equation*}
(where the pushout is taken over the maps $\ell_A\colon L_A(X)\to X(A)$ and $L_A(f)$) is a cofibration in the given model structure on $\End(A)\textup{--}\mathscr C$ for all $A\in\mathscr I$.
\begin{proof}
This is part of \cite[Proposition~C.23]{schwede-book}
\end{proof}
\end{prop}
\subsubsection{Strict level model structures} Using the above criterion we now get:
\begin{proof}[Proof of Proposition~\ref{prop:strict-model-structure}]
We will only construct the model structure on $\cat{$\bm{G}$-$\bm{I}$-SSet}$, the construction for \cat{$\bm{G}$-$\bm{\mathcal I}$-SSet} being analogous. For this we want to appeal to Proposition~\ref{prop:generalized-projective-dim} (for $\mathscr C=\cat{SSet}$, $\mathscr I=BG\times I$), so we have to check the consistency condition. To this end we claim that in the above situation, the functor $(BG\times I)(A,B)\times_{G\times\Sigma_A}\blank\cong I(A,B)\times_{\Sigma_A}\blank$ sends acyclic cofibrations of the usual $\mathcal G_{\Sigma_A,G}$-model structure to acyclic cofibrations in the \emph{injective} $\mathcal G_{\Sigma_B,G}$-model structure. But indeed, by cocontinuity it suffices to check this on generating acyclic cofibrations, where this is obvious.
This already shows that the model structure on $\cat{$\bm G$-$\bm I$-SSet}$ exists, and that it is cofibrantly generated with generating (acyclic) cofibrations as claimed above. As $\cat{$\bm G$-$\bm I$-SSet}$ is locally presentable, we conclude that it is in fact combinatorial.
The category $\cat{$\bm G$-$\bm I$-SSet}$ is enriched, tensored, and cotensored over $\cat{SSet}$ in the obvious way, and as the $\mathcal G_{\Sigma_A,G}$-model structures are simplicial and since pullbacks, cotensors, and (acyclic) fibrations are defined levelwise, also $\cat{$\bm G$-$\bm I$-SSet}$ is simplicial. Similarly one proves right properness and the preservation of weak equivalences under filtered colimits.
Moreover, the forgetful functor admits a simplicial left adjoint (via simplicially left Kan extension along $I\to\mathcal I$), which we denote by $\mathcal I\times_I\blank$; explicitly, we can arrange that $\mathcal I\times_I(I(A,\blank)\times K)=\mathcal I(A,\blank)\times K$ for all $A\in I$ and $K\in\cat{$\bm G$-SSet}$ with the evident functoriality, and this in turn describes $\mathcal I\times_I\blank$ up to canonical (simplicial) isomorphism. It is then obvious from the definition that $\forget$ is right Quillen, so that $(\ref{eq:I-vs-script-I-restricted})$ is a Quillen adjunction.
It only remains to establish left properness, for which we observe that any of the above generating cofibrations is a levelwise cofibration, and hence so is any cofibration of the strict level model structure. The claim therefore follows from Corollary~\ref{cor:homotopy-pushout-M-SSet}.
\end{proof}
Proposition~\ref{prop:characterization-cofibration-general} specializes to:
\begin{cor}\label{cor:characterization-i-cof}
A map $f\colon X\to Y$ in $\cat{$\bm G$-$\bm I$-SSet}$ is a cofibration in the strict level model structure if and only if for each finite set $A$ the map
\begin{equation}\label{eq:latching-map}
X(A)\amalg_{L_A(X)}L_A(Y)\xrightarrow{(f(A),\ell_A)} Y(A)
\end{equation}
is a cofibration in the $\mathcal{G}_{\Sigma_A,G}$-model structure on $\cat{$\bm{(\Sigma_A\times G)}$-SSet}$.
In particular, $X$ is cofibrant in the strict level model structure if and only if for each finite set $A$ the latching map $\colim_{B\subsetneq A} X(B)\to X(A)$ is a cofibration in the above model category.\qedhere\qed
\end{cor}
\begin{ex}
If $G=1$, the $\mathcal G_{\Sigma_A,G}$-cofibrations are precisely the underlying cofibrations of simplicial sets. In this case, the strict level cofibrations on $\cat{$\bm I$-SSet}$ have been considered non-equivariantly under the name \emph{flat cofibrations} \cite[Definition~3.9]{sagave-schlichtkrull}.\index{flat cofibration!in I-SSet@in $\cat{$\bm I$-SSet}$}
\end{ex}
\subsubsection{$G$-global weak equivalences} Before we can introduce the $G$-global weak equivalences for the above models, we need some preparations.
\begin{constr}\label{constr:i-extension}\nomenclature[aI1z]{$\overline{I}$}{extension of $I$ to all sets}\nomenclature[aI2z]{$\overline{\mathcal I}$}{extension of $\mathcal I$ to all sets}
We write $\overline{I}$ for the category of all sets and injections, and we write $\overline{\mathcal I}$ for the simplicial category obtained by applying $E$ to each hom set. Then $I$ and $\mathcal I$ are full (simplicial) subcategories of $\overline{I}$ and $\overline{\mathcal I}$, respectively. We will now explain how to extend any $I$- or $\mathcal I$-simplicial set to $\overline{I}$ or $\overline{\mathcal I}$, respectively:
In the case of $X\colon I\to\cat{SSet}$ we define
\begin{equation*}
\overline{X}(A)=\colim\limits_{B\subset A\textup{ finite}} X(B)
\end{equation*}
for any set $A$. If $i\colon A\to A'$ is any injection, we define the structure map $\overline X(A)\to\overline X(A')$ as the map induced via the universal property of the above colimit by the family
\begin{equation*}
X(B)\xrightarrow{X(i|_B)} X(i(B))\to \colim\limits_{B'\subset A'\textup{ finite}} X(B')=\overline{X}(A')
\end{equation*}
for all finite $B\subset A$, where the unlabelled arrow is the structure map of the colimit for the term indexed by $i(B)\subset A'$. We omit the trivial verification that this functorial.
In the case of a simplicially enriched $X\colon\mathcal I\to\cat{SSet}$ we define the extension analogously on objects and morphisms. If now $(i_0,\dots,i_n)$ is a general $n$-cell of $\overline{\mathcal I}(A,A')$, then we define $\overline X(i_0,\dots,i_n)$ as the composition
\begin{equation*}
\Delta^n\times\colim\limits_{B\subset A\textup{ finite}} X(B)\cong
\colim\limits_{B\subset A\textup{ finite}}\Delta^n\times X(B)\to
\colim\limits_{B'\subset A'\textup{ finite}} X(B')
\end{equation*}
where the isomorphism is the canonical one and the unlabelled arrow is induced by $X(i_0|_B,\dots, i_n|_B)\colon\Delta^n\times X(B)\to X(i_0(B)\cup i_1(B)\cup\cdots\cup i_n(B))$ for all finite $B\subset A$. We omit the easy verification that this defines a simplicially enriched functor $\overline{\mathcal I}\to\cat{SSet}$.
One moreover easily checks that these become simplicially enriched functors by sending $F\colon\Delta^n\times X\to Y$ to the transformation given on a set $A$ by
\begin{equation*}
\Delta^n\times\colim\limits_{B\subset A\textup{ finite}} X(B)\cong
\colim\limits_{B\subset A\textup{ finite}}\Delta^n\times X(B)\xrightarrow{\colim F(B)}
\colim\limits_{B\subset A\textup{ finite}} Y(B),
\end{equation*}
and that with respect to this the structure maps
\begin{equation*}
X(A)\to \colim\limits_{B\subset A\textup{ finite}} X(B)=\overline X(A)
\end{equation*}
for finite $A$ define simplicially enriched natural isomorphisms; accordingly, we will from now on no longer distinguish notationally between the extension and the original object.
\end{constr}
\begin{rk}
It is not hard to check that the above is a model for the (simplicially enriched) left Kan extension; however, we will at several points make use of the above explicit description.
\end{rk}
\begin{rk}
Simply by functoriality, the above construction lifts to provide simplicially enriched extension functors
\begin{equation*}
\cat{$\bm G$-$\bm I$-SSet}\to\cat{$\bm G$-$\bm{\overline{I}}$-SSet}\qquad\text{and}\qquad
\cat{$\bm G$-$\bm{\mathcal I}$-SSet}\to\cat{$\bm G$-$\bm{\overline{\mathcal I}}$-SSet}
\end{equation*}
for any group $G$.
\end{rk}
For later use we record:
\begin{lemma}\label{lemma:evaluation-h-universe}
Let $\mathcal U$ be a complete $H$-set universe, let $A$ be any $H$-set, and let $i\colon\mathcal U\to A$ be an $H$-equivariant injection. Then $X(i)\colon X(\mathcal U)\to X(A)$ is an $(H\times G)$-weak equivalence for any $X\in\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$.
\end{lemma}
The proof will rely on the following easy observation:
\begin{lemma}\label{lemma:extension-filtered-colimit}
If $X$ is a $G$-$I$- or $G$-$\mathcal I$-simplicial set, then its extension preserves filtered colimits, i.e.~if $J$ is a small filtered category and $A_\bullet\colon J\to \overline{I}$ any functor, then the canonical map
\begin{equation*}
\colim_j X(A_j)\to X\big(\colim_j A_j\big)
\end{equation*}
is an isomorphism. (Here it does not matter whether we form the colimit on the right hand side in $\cat{Set}$ or in $\overline{I}$).
\begin{proof}
Unravelling the definition, the left hand side is given by the double colimit $\colim_j\colim_{B\subset A_j\textup{ finite}} X(B)$. Write $\overline{A}\mathrel{:=}\colim_{j\in J}A_j$; we define a map in the other direction as follows: a finite subset $B\subset\overline{A}$ is contained in the image of some structure map $i\colon A_j\to\overline{A}$, and we send $X(B)$ via $X(i^{-1})$ to $X(i^{-1}(B))$ in the $(j, i^{-1}(B))$-term on the left hand side. We omit the easy verification that this is well-defined and inverse to the above map.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lemma:evaluation-h-universe}]
Let us first assume that $A$ is countable. As $\mathcal U$ is a complete $H$-set universe by assumption, we can therefore find an $H$-equivariant injection $j\colon A\to\mathcal U$. Then $X(j)$ is an $(H\times G)$-homotopy inverse to $X(i)$, as exhibited by the $(H\times G)$-equivariant homotopies $X(\id_A, ij)$ and $X(\id_{\mathcal U}, ji)$, finishing the proof of the special case.
In the general case we now observe that the map in question factors as
\begin{equation*}
X(\mathcal U)\cong\colim\limits_{i(\mathcal U)\subset B\subset A\textup{ countable $H$-set}} X(\mathcal U)\xrightarrow{\sim}
\colim\limits_{i(\mathcal U)\subset B\subset A\textup{ countable $H$-set}} X(B) \cong X(A),
\end{equation*}
where the left hand map is induced by the inclusion of the term indexed by $i(\mathcal U)$ (which is an isomorphism because $\{i(\mathcal U)\subset B\subset A\}$ is a filtered poset, so that it has connected nerve), the second map uses the above special case levelwise, and the final isomorphism comes from the previous lemma. The claim follows immediately.
\end{proof}
\end{lemma}
\begin{constr}\nomenclature[aevomega]{$\ev_\omega$}{functor $\cat{$\bm I$-SSet}\to\cat{$\bm{\mathcal M}$-SSet}$ or $\cat{$\bm{\mathcal I}$-SSet}\to\cat{$\bm{E\mathcal M}$-SSet}$ given by evaluating at $\omega$}
By applying Construction~\ref{constr:i-extension} and then restricting along the inclusion $B\mathcal M\hookrightarrow\overline{I}$ or $B(E\mathcal M)\to\overline{\mathcal I}$, respectively, sending the unique object to $\omega$, we get functors
\begin{equation*}
\ev_\omega\colon \cat{$\bm{G}$-$\bm{I}$-SSet}\to\cat{$\bm{\mathcal M}$-$\bm{G}$-SSet}
\qquad\text{and}\qquad
\ev_\omega\colon \cat{$\bm{G}$-$\bm{\mathcal I}$-SSet}\to\cat{$\bm{E\mathcal M}$-$\bm{G}$-SSet}.
\end{equation*}
\end{constr}
\begin{defi}\index{G-global weak equivalence@$G$-global weak equivalence!in G-II-SSet@in $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$|textbf}\index{G-global weak equivalence@$G$-global weak equivalence!in G-I-SSet@in $\cat{$\bm G$-$\bm{I}$-SSet}$|textbf}
A map $f\colon X\to Y$ in $\cat{$\bm G$-$\bm I$-SSet}$ or $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ is called a \emph{$G$-global weak equivalence} if $f(\omega)=\ev_\omega f$ is a $G$-global weak equivalence in $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ or $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$, respectively.
\end{defi}
\begin{lemma}\label{lemma:strict-are-global}
Let $f\colon X\to Y$ be a strict level weak equivalence in $\cat{$\bm{G}$-$\bm{\mathcal I}$-SSet}$ or $\cat{$\bm{G}$-$\bm{I}$-SSet}$. Then $f$ is also a $G$-global weak equivalence.
\begin{proof}
It suffices to consider the second case. Let $H\subset\mathcal M$ be universal; we will show that $X(\omega)\to Y(\omega)$ is a $\mathcal G_{H,G}$-weak equivalence.
Pick a free $H$-orbit $F$ inside $\omega$ (which exists by universality). We then observe that we have a commutative diagram
\begin{equation*}
\begin{tikzcd}
\colim\limits_{F\subset A\subset\omega\textup{ finite $H$-set}} X(A)\arrow[d, "\colim f(A)"'] \arrow[r] & \colim\limits_{A\subset\omega\textup{ finite}} X(A)\arrow[d, "\colim f(A)"]\\
\colim\limits_{F\subset A\subset\omega\textup{ finite $H$-set}} Y(A)\arrow[r] & \colim\limits_{A\subset\omega\textup{ finite}} Y(A)
\end{tikzcd}
\end{equation*}
where the horizontal maps are induced from the inclusion of filtered posets $\{F\subset A\subset\omega\textup{ finite $H$-set}\}\hookrightarrow\{A\subset\omega\textup{ finite}\}$. This is is cofinal: any finite subset $A\subset\omega$ is contained in the finite $H$-set $F\cup HA$ which is an element of the left hand side.
Thus, the horizontal maps are isomorphisms, and it therefore suffices that the left hand vertical map is a $\mathcal G_{H,G}$ weak equivalence. But on this side, $H$ simply acts on each term of the colimit by functoriality, and each $f(A)$ is a $\mathcal G_{H,G}$-weak equivalence with respect to this action as $A$ is in particular faithful. The claim follows as the $\mathcal G_{H,G}$-weak equivalences are closed under filtered colimits.
\end{proof}
\end{lemma}
In order to later characterize the fibrant objects in the $G$-global model structure, we introduce:
\begin{defi}
A $G$-$I$-simplicial set (or $G$-$\mathcal I$-simplicial set) $X$ is called \emph{static},\index{static|textbf} if for all finite faithful $H$-sets $A$ and each $H$-equivariant injection $i\colon A\to B$ into another finite $H$-set, the induced map $X(i)\colon X(A)\to X(B)$ is a $\mathcal G_{H,G}$-weak equivalence.
\end{defi}
\begin{lemma}
Let $f\colon X\to Y$ be a map of static $G$-$I$-simplicial sets or $G$-$\mathcal I$-simplicial sets. Then $f$ is a $G$-global equivalence if and only if $f$ is a strict level weak equivalence.
\begin{proof}
Again it suffices to consider the first case.
The implication `$\Leftarrow$' holds without any assumptions by Lemma~\ref{lemma:strict-are-global}. For the remaining implication we first observe:
\begin{claim*}
Both $X(\omega)$ and $Y(\omega)$ are $G$-semistable.
\begin{proof}
It suffices to prove the first statement. Let $H\subset\mathcal M$ be universal, let $u\in\mathcal M$ centralize $H$, and pick a finite faithful $H$-subset $A\subset\omega$. We now consider the commutative diagram
\begin{equation*}
\begin{tikzcd}
X(A)\arrow[d, "X(u|_A)"'] \arrow[r] & \colim\limits_{A\subset B\subset\omega\textup{ finite $H$-set}} X(B) \arrow[r, "\cong"] & X(\omega)\arrow[d, "u.\blank"]\\
X(u(A)) \arrow[r] & \colim\limits_{u(A)\subset B\subset\omega\textup{ finite $H$-set}} X(B) \arrow[r, "\cong"'] & X(\omega)
\end{tikzcd}
\end{equation*}
where the isomorphisms on the right come from cofinality again, and the left hand horizontal maps are structure maps of the respective colimits, hence $\mathcal G_{H,G}$-weak equivalences as all transition maps are. As the left hand vertical map is an isomorphism for trivial reasons, we conclude that also the right hand vertical map is a $\mathcal G_{H,G}$-weak equivalence, i.e.~$X(\omega)$ is semistable as desired.
\end{proof}
\end{claim*}
Let $H$ be a finite group and $A$ a finite faithful $H$-set; we have to show that $f(A)$ is a $\mathcal G_{H,G}$-weak equivalence, for which we may assume without loss of generality that $H$ is a universal subgroup of $\mathcal M$ and $A$ an $H$-subset of $\omega$. But the same argument as above then shows that $f(A)$ agrees up to conjugation by $\mathcal G_{H,G}$-weak equivalences with $f(\omega)$. The latter is a $G$-global weak equivalence between $G$-semistable $\mathcal M$-$G$-simplicial sets by the above claim, hence a $G$-universal weak equivalence. Thus, also $f(A)$ is a $\mathcal G_{H,G}$-weak equivalence by $2$-out-of-$3$ as desired.
\end{proof}
\end{lemma}
\subsubsection{Connection to tame actions}
\index{tame!M-action@$\mathcal M$-action!tame M-simplicial sets vs flat I-simplicial sets@tame $\mathcal M$-simplicial sets vs.~flat $I$-simplicial sets|(}
In order to construct the $G$-global model structures on $\cat{$\bm G$-$\bm I$-SSet}$ and $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ and to compare them to our previous models, we will exploit a close connection between them on the pointset level. Namely, Sagave and Schwede showed that the $1$-category of tame $\mathcal M$-simplicial sets is equivalent to the full subcategory of $\cat{$\bm I$-SSet}$ spanned by the flat\index{flat!I-simplicial set@$I$-simplicial set} (i.e.~globally cofibrant) objects. In order to state their precise comparison, we need:
\begin{constr}\nomenclature[zbullet]{$(\blank)_\bullet$}{right adjoint to $\ev_\omega$, \textit{see also} $(\blank)_{[A]}$}
Let $X$ be any $\mathcal M$-simplicial set. We write $X_\bullet$ for the $I$-simplicial set with $(X_\bullet)(A)=X_{[A]}$ for each finite $A\subset\omega$; an injection $j\colon A\to B$ acts by extending it to an injection $\bar{\jmath}\in\mathcal M$ and then using the $\mathcal M$-action (this is well-defined by Lemma~\ref{lemma:support-vs-action-M} together with Lemma~\ref{lemma:support-agree-M}). As any finite set is isomorphic to a subset of $\omega$, there is an essentially unique way to extend this to a functor $I\to\cat{SSet}$, and we fix any such extension.
This becomes a simplicial functor by the enriched functoriality of the individual $X_{[A]}$; in particular, we get an induced functor $(\blank)_\bullet\colon\cat{$\bm{\mathcal M}$-$\bm G$-SSet}\to\cat{$\bm G$-$\bm I$-SSet}$ for any group $G$.
The inclusions $X_{[A]}\hookrightarrow X$ assemble into an enriched natural map $\epsilon\colon X_\bullet(\omega)\to X$ for any $\mathcal M$-$G$-simplicial set $X$. Moreover, if $Y$ is a $G$-$I$-simplicial set, then the structure map $Y(A)\to Y(\omega)$ for any finite $A\subset\omega$ factors through $Y(\omega)_{[A]}$, and for varying $A$ these assemble into an enriched natural map $Y\to Y(\omega)_\bullet$.
\end{constr}
The above construction is a `coordinate free' version of \cite[Construction~5.5]{I-vs-M-1-cat} applied in each simplicial degree (and with $G$-actions pulled through), also cf.~\cite[discussion before Corollary~5.7]{I-vs-M-1-cat}. In particular,~\cite[Proposition~5.6]{I-vs-M-1-cat} implies, also cf.~\cite[Corollary~5.7]{I-vs-M-1-cat}:
\begin{lemma}\label{lemma:M-SSet-vs-I-SSet-sagave-schwede}
The above defines a simplicial adjunction
\begin{equation}\label{eq:evaluation-support-adjunction}
\ev_\omega\colon\cat{$\bm G$-$\bm I$-SSet}\rightleftarrows\cat{$\bm{\mathcal M}$-$\bm G$-SSet} :\!(\blank)_\bullet
\end{equation}
where the left adjoint has image in $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau$, and for any $\mathcal M$-$G$-simplicial set $X$ the counit $X_\bullet(\omega)\to X$ factors through an isomorphism onto $X^\tau$. In particular, $(\ref{eq:evaluation-support-adjunction})$ restricts to a Bousfield localization $\cat{$\bm G$-$\bm I$-SSet}\rightleftarrows\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau$.
Finally, the right adjoint has essential image the flat $G$-$I$-simplicial sets, i.e.~the unit $\eta\colon Y\to Y(\omega)_\bullet$ is an isomorphism if and only if $Y$ is cofibrant in $\cat{$\bm I$-SSet}$.\index{tame!M-action@$\mathcal M$-action!tame M-simplicial sets vs flat I-simplicial sets@tame $\mathcal M$-simplicial sets vs.~flat $I$-simplicial sets|)}\qed
\end{lemma}
Using Theorem~\ref{thm:support-EM-vs-M}, we will now give an analogous comparison between $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ and $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$.
\index{tame!EM-action@$E\mathcal M$-action!tame EM-simplicial sets vs flat II-simplicial sets@tame $E\mathcal M$-simplicial sets vs.~flat $\mathcal I$-simplicial sets|(}
\begin{constr}
Let $X$ be any $E\mathcal M$-simplicial set. We write $X_\bullet$ for the $\mathcal I$-simplicial set with $(X_\bullet)(A)=X_{[A]}$ for every finite $A\subset\omega$; an $(n+1)$-tuple of injections $j_0,\dots,j_n\colon A\to B$ acts by extending each of them to an injection $\bar{\jmath}_k\colon\omega\to\omega$ and then using the $E\mathcal M$-action (which is well-defined by Corollary~\ref{cor:support-agree-EM} together with Lemma~\ref{lemma:support-vs-action-EM}). Again we fix an extension to all of $\mathcal I$.
This becomes a simplicial functor by enriched functoriality of the individual $X_{[A]}$; in particular, if $G$ is any group we then again get an induced simplicial functor $(\blank)_\bullet\colon\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}\to\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$.
We define $\epsilon\colon X_\bullet(\omega)\to X$ as the map induced by the inclusions $X_{[A]}\hookrightarrow X$. Moreover, if $Y\in\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$, then $Y(A)\to Y(\omega)$ factors through $Y(\omega)_{[A]}$ by definition of the action, and one easily checks that these assemble into $\eta\colon Y\to Y(\omega)_\bullet$.
\end{constr}
\begin{rk}
By Theorem~\ref{thm:support-EM-vs-M}, the diagram
\begin{equation}\label{eq:forget-vs-blank-bullet}
\begin{tikzcd}
\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}\arrow[r, "(\blank)_\bullet"]\arrow[d, "\forget"'] & \cat{$\bm G$-$\bm{\mathcal I}$-SSet}\arrow[d, "\forget"]\\
\cat{$\bm{\mathcal M}$-$\bm G$-SSet}\arrow[r, "(\blank)_\bullet"'] & \cat{$\bm G$-$\bm{I}$-SSet}
\end{tikzcd}
\end{equation}
commutes strictly, and the same can be arranged for $\ev_\omega$ instead of $(\blank)_\bullet$ by construction. Under these identifications, also the unit and counit are preserved, i.e.~$\forget(\eta_Y)=\eta_{\forget Y}$ and $\forget(\epsilon_X)=\epsilon_{\forget X}$.
\end{rk}
As the notions of tameness in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$ and $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ agree by another application of Theorem~\ref{thm:support-EM-vs-M}, we conclude from Lemma~\ref{lemma:M-SSet-vs-I-SSet-sagave-schwede}:
\begin{lemma}\label{lemma:evaluation-support-adjunction-E}
The above yields a simplicially enriched adjunction
\begin{equation}\label{eq:evaluation-support-adjunction-E}
\ev_\omega\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet} :\!(\blank)_\bullet
\end{equation}
where the left adjoint has image in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$. Moreover,
for any $E\mathcal M$-$G$-simplicial set $X$ the counit $(X_\bullet)(\omega)\to X$ factors through an isomorphism onto $X^\tau$, so that $(\ref{eq:evaluation-support-adjunction-E})$ restricts to a Bousfield localization $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$.
Finally, the essential image of $(\blank)_\bullet$ consists precisely of those $G$-$\mathcal I$-simplicial sets that are flat,\index{flat!II-simplicial set@$\mathcal I$-simplicial set} i.e.~whose underlying $I$-simplicial sets are globally cofibrant.\index{tame!EM-action@$E\mathcal M$-action!tame EM-simplicial sets vs flat II-simplicial sets@tame $E\mathcal M$-simplicial sets vs.~flat $\mathcal I$-simplicial sets|)}\qed
\end{lemma}
As an upshot of this, we can now very easily prove the following alternative description of the $G$-global weak equivalences of $G$-$I$-simplical sets, that will become useful at several points later:
\begin{thm}\label{thm:G-global-we-I-characterization}
\index{G-global weak equivalence@$G$-global weak equivalence!in G-I-SSet@in $\cat{$\bm G$-$\bm{I}$-SSet}$}
The following are equivalent for a map $f$ in $\cat{$\bm G$-$\bm I$-SSet}$:
\begin{enumerate}
\item $f$ is a $G$-global weak equivalence in $\cat{$\bm G$-$\bm{I}$-SSet}$.
\item $f(\omega)$ is a $G$-global weak equivalence in $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$.
\item $\mathcal I\times_If$ is a $G$-global weak equivalence in $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$.\label{item:I-underived}
\item $E\mathcal M\times_{\mathcal M} f(\omega)$ is a $G$-global weak equivalence in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$.\label{item:EM-underived}
\end{enumerate}
\end{thm}
We emphasize that the above functors are \emph{not} derived in any way.
\begin{proof}
The equivalence $(1)\Leftrightarrow(2)$ holds by definition, and $(2)\Leftrightarrow(4)$ is an instance of Theorem~\ref{thm:tame-M-sset-vs-EM-sset}. It therefore only remains to show that $(3)\Leftrightarrow(4)$, for which we observe that the total mate of $(\ref{eq:forget-vs-blank-bullet})$ provides a natural isomorphism filling
\begin{equation*}
\begin{tikzcd}
\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau &\arrow[l, "\ev_\omega"'] \cat{$\bm G$-$\bm{\mathcal I}$-SSet}\\
\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau\arrow[u, "E\mathcal M\times_{\mathcal M}\blank"] &\arrow[l, "\ev_\omega"] \cat{$\bm G$-$\bm{I}$-SSet}.\arrow[u, "\mathcal I\times_I\blank"']
\end{tikzcd}
\end{equation*}
The claim then follows immediately from the definitions.
\end{proof}
\subsubsection{$G$-global model structures} Using the above as well as our knowledge about tame $\mathcal M$- and $E\mathcal M$-actions we can now prove:
\begin{thm}\label{thm:script-I-global-model-structure}\index{G-global model structure@$G$-global model structure!on G-II-SSet@on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$|textbf}
There is a unique model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ whose weak equivalences are the $G$-global weak equivalences and with cofibrations those of the strict level model structure. We call this model structure the \emph{$G$-global model structure}. It is proper, combinatorial, simplicial, and filtered colimits in it are homotopical. Moreover, the fibrant objects of this model structure are precisely the strictly level fibrant static $G$-$\mathcal I$-simplicial sets.
Finally, the simplicial adjunction $(\ref{eq:evaluation-support-adjunction-E})$ is a Quillen equivalence with respect to the $G$-global injective model structure on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$.
\begin{proof}
By Lemma~\ref{lemma:strict-are-global}, $\ev_\omega$ sends strict level weak equivalences to $G$-global weak equivalences, and it clearly sends generating cofibrations to injective cofibrations. In particular, the simplicial adjunction
\begin{equation*}
\ev_\omega\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{strict level}}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\textup{injective $G$-global}} :\!(\blank)_\bullet
\end{equation*}
is a Quillen adjunction. We now want to apply Lurie's localization criterion (Theorem~\ref{thm:lurie-localization-criterion}) to this, for which we have to show that $\cat{R}(\blank)_\bullet$ is fully faithful with essential image the static $G$-$\mathcal I$-simplicial sets.
Indeed, Corollary~\ref{cor:injective-almost-tame} shows that $\cat{R}(\blank)_\bullet$ restricts accordingly, and that the counit $X_\bullet(\omega)\to X$ is a $G$-global weak equivalence for any injectively fibrant $X$. On the other hand, for any $G$-$\mathcal I$-simplicial set $Y$, the map $\eta(\omega)\colon Y(\omega)\to Y(\omega)_\bullet(\omega)$ is a one-sided inverse of $\epsilon_{Y(\omega)}$, hence an isomorphism by Lemma~\ref{lemma:evaluation-support-adjunction-E} as $Y(\omega)$ is tame. Since $(\blank)_\bullet$ clearly preserves $G$-global weak equivalences in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^{w\tau}$ (i.e.~the subcategory of those $E\mathcal M$-$G$-simplicial sets $X$ for which $X^\tau\hookrightarrow X$ is a $G$-global weak equivalence), and as this contains both the injectively fibrant objects by Corollary~\ref{cor:injective-almost-tame} as well as the tame $E\mathcal M$-$G$-simplicial set $Y(\omega)$ for trivial reasons, we conclude that the derived unit $\eta_Y$, represented by the composition $Y\to Y(\omega)_\bullet\to Z_\bullet$ for some injectively fibrant replacement $Y(\omega)\to Z$, is a $G$-global weak equivalence. Thus, if $Y$ is static, then the derived unit is a $G$-global weak equivalence between static $G$-$\mathcal I$-simplicial sets, hence a strict level weak equivalence by Lemma~\ref{lemma:strict-are-global}. This completes the proof of the claim.
Lurie's criterion then shows that the desired model structure exists, and that it is left proper, combinatorial, simplicial, has the fibrant objects described above, and that $(\ref{eq:evaluation-support-adjunction-E})$ becomes a Quillen equivalence for this model structure. Moreover, Lemma~\ref{lemma:filtered-still-homotopical} shows that filtered colimits in this model structure are homotopical.
Finally, we consider a pullback square
\begin{equation*}
\begin{tikzcd}
P\arrow[r, "f"]\arrow[d, "p"'] & X\arrow[d, "q"]\\
Y\arrow[r, "g"'] & Z
\end{tikzcd}
\end{equation*}
in $\cat{$\bm{G}$-$\bm{\mathcal I}$-SSet}$ such that $q$ is a strict level fibration and $g$ is a $G$-global weak equivalence. We will show that also $f$ is a $G$-global weak equivalence, which will in particular imply right properness of the $G$-global model structure.
As finite limits in $\cat{SSet}$ commute with filtered colimits, and as limits commute with each other, we get for any universal subgroup $H\subset\mathcal M$ and any group homomorphism $\phi\colon H\to G$ a pullback square
\begin{equation*}
\begin{tikzcd}[column sep=large]
P(\omega)^\phi\arrow[r, "f(\omega)^\phi"]\arrow[d, "p(\omega)^\phi"'] & X(\omega)^\phi\arrow[d, "q(\omega)^\phi"]\\
Y(\omega)^\phi\arrow[r, "g(\omega)^\phi"'] & Z(\omega)^\phi
\end{tikzcd}
\end{equation*}
in $\cat{SSet}$. The map $g(\omega)^\phi$ is a weak equivalence by definition, and we have to show that also $f(\omega)^\phi$ is. For this it suffices by right properness of $\cat{SSet}$ that $q(\omega)^\phi$ is a Kan fibration. But as before, after picking a free $H$-orbit $F\subset\omega$, it can be identified with the filtered colimit
\begin{equation*}
\colim\limits_{F\subset A\subset\omega\textup{ finite $H$-set}} q(A)^\phi
\end{equation*}
of Kan fibrations, and hence is itself a Kan fibration as desired.
\end{proof}
\end{thm}
\begin{thm}\label{thm:strict-global-I-model-structure}
There is a unique model structure on $\cat{$\bm G$-$\bm I$-SSet}$ whose cofibrations are the strict level cofibrations and whose weak equivalences are the $G$-global weak equivalences. We call this the \emph{$G$-global model structure}.\index{G-global model structure@$G$-global model structure!on G-I-SSet@on $\cat{$\bm G$-$\bm I$-SSet}$|textbf} It is left proper, combinatorial, simplicial, and filtered colimits in it are homotopical. Moreover, its fibrant objects are precisely the static strictly fibrant ones.
Finally, the simplicial adjunctions
\begin{align}
\ev_\omega\colon\cat{$\bm G$-$\bm I$-SSet}_{\textup{$G$-global}}&\rightleftarrows\cat{$\bm{\mathcal M}$-$\bm G$-SSet}_{\textup{injective $G$-global}} :\!(\blank)_\bullet\nonumber\\
\label{eq:I-vs-script-I}
\mathcal I\times_I\blank\colon\cat{$\bm G$-$\bm I$-SSet}_{\textup{$G$-global}}&\rightleftarrows\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global}}:\!\forget
\end{align}
are Quillen equivalences, and both functors in $(\ref{eq:I-vs-script-I})$ are fully homotopical.
\begin{proof}
All statements except for those about the adjunction $(\ref{eq:I-vs-script-I})$ are proven just as in the previous theorem.
For the remaining statements, we observe that $\mathcal I\times_I\blank$ preserves cofibrations by Proposition~\ref{prop:strict-model-structure} while it is homotopical by Theorem~\ref{thm:G-global-we-I-characterization}; in particular it is left Quillen. On the other hand, the forgetful functor is clearly homotopical. We then consider the diagram
\begin{equation*}
\begin{tikzcd}
\cat{$\bm G$-$\bm{\mathcal I}$-SSet}\arrow[d, "\ev_\omega"']\arrow[r, "\forget"] &[1em] \cat{$\bm G$-$\bm{I}$-SSet}\arrow[d, "\ev_\omega"]\\
\cat{$\bm{E\mathcal M}$-$\bm G$-SSet} \arrow[r, "\forget"'] & \cat{$\bm{\mathcal M}$-$\bm G$-SSet}
\end{tikzcd}
\end{equation*}
of homotopical functors (with respect to the $G$-global weak equivalences everywhere), which commutes up to canonical isomorphism. By the above, the vertical maps induce equivalences on associated quasi-categories, and so does the lower horizontal map by Corollary~\ref{cor:em-vs-m-equiv-model-cat}. The claim follows by $2$-out-of-$3$.
\end{proof}
\end{thm}
\subsubsection{Further model structures}
For later use we record the existence of \emph{positive $G$-global model structures}, which can be constructed in precisely the same way as above; we leave the details to the reader.
\begin{thm}\label{thm:positive-G-global-script-I}\index{positive G-global model structure@positive $G$-global model structure!on G-II-SSet@on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$|textbf}
\index{G-global model structure@$G$-global model structure!positive|seeonly{positive $G$-global model structure}}
There is a unique cofibrantly generated model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ with weak equivalences the $G$-global weak equivalences and generating cofibrations the maps
\begin{equation*}
\mathcal I(A,\blank)\times_\phi G\times\del\Delta^n\hookrightarrow\mathcal I(A,\blank)\times_\phi G\times\Delta^n
\end{equation*}
for $n\ge 0$, finite groups $H$, homomorphisms $\phi\colon H\to G$ and \emph{non-empty} finite faithful $H$-sets $A$. We call this the \emph{positive $G$-global model structure}. It is combinatorial, simplicial, proper, and filtered colimits in it are homotopical. Moreover, a $G$-$\mathcal I$-simplicial set $X$ is fibrant if and only if $X(A)$ is fibrant in the $\mathcal G_{\Sigma_A,G}$-equivariant model structure for all \emph{non-empty} $A$ and $X$ is \emph{positively static}\index{static!positive|seeonly{positively static}}\index{positively static|textbf} in the sense that $X(i)\colon X(A)\to X(B)$ is a $\mathcal G_{\Sigma_A,G}$-weak equivalence for any injection $i\colon A\to B$ of \emph{non-empty} finite sets.
Finally, the identity adjunction $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{positive $G$-global}}\rightleftarrows\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global}}$ is a Quillen equivalence.\qed
\end{thm}
\begin{thm}\label{thm:positive-global-I-model-structure}
There is a unique cofibrantly generated model structure on $\cat{$\bm G$-$\bm I$-SSet}$ with weak equivalences the $G$-global weak equivalences and generating cofibrations the maps
\begin{equation*}
I(A,\blank)\times_\phi G\times\del\Delta^n\hookrightarrow I(A,\blank)\times_\phi G\times\Delta^n
\end{equation*}
for $n\ge 0$, finite groups $H$, homomorphisms $\phi\colon H\to G$ and \emph{non-empty} finite faithful $H$-sets $A$. We call this the \emph{positive $G$-global model structure}.\index{positive G-global model structure@positive $G$-global model structure!on G-I-SSet@on $\cat{$\bm G$-$\bm I$-SSet}$|textbf} It is left proper, combinatorial, simplicial, and filtered colimits in it are homotopical. Moreover, its fibrant objects are precisely the positively static strictly fibrant ones. Finally, the simplicial adjunctions
\begin{align*}
\mathcal I\times_I\blank\colon\cat{$\bm G$-$\bm{I}$-SSet}_{\textup{positive $G$-global}}&\rightleftarrows\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{positive $G$-global}} :\!\forget\\
\id\colon\cat{$\bm G$-$\bm{I}$-SSet}_{\textup{positive $G$-global}}&\rightleftarrows\cat{$\bm G$-$\bm{I}$-SSet}_{\textup{$G$-global}} :\!\id
\end{align*}
are Quillen equivalences.\qed
\end{thm}
\begin{rk}
Again, there are suitable \emph{positive level model structures}\index{positive level model structure} in the background of which the above are Bousfield localizations.
\end{rk}
\begin{lemma}\label{lemma:pos-cof-script-I}
Let $f\colon X\to Y$ be a cofibration in either of the $G$-global positive model structures. Then $f(\varnothing)\colon X(\varnothing)\to Y(\varnothing)$ is an isomorphism.
\begin{proof}
The class of such maps is obviously closed under retracts, pushouts, and transfinite compositions. Thus, it suffices to verify the claim for each generating cofibration $i$. But in this case both source and target are obviously empty in degree $\varnothing$, in particular $i(\varnothing)$ is an isomorphism.
\end{proof}
\end{lemma}
Finally, we come to injective model structures:
\begin{thm}\label{thm:script-I-vs-EM-injective}
\index{injective G-global model structure@injective $G$-global model structure!on G-II-SSet@on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$|textbf}
There exists a unique model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ whose cofibrations are the injective cofibrations and whose weak equivalences are the $G$-global weak equivalences. We call this the \emph{injective $G$-global model structure}. It is combinatorial, proper, simplicial, and filtered colimits in it are homotopical.
\end{thm}
For the proof we will need:
\begin{lemma}\label{lemma:G-global-I-injective-pushout}
The $G$-global weak equivalences in $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ and $\cat{$\bm G$-$\bm{I}$-SSet}$ are stable under pushout along injective cofibrations, and a commutative square in either of these is a homotopy pushout if and only if its image under $\ev_\omega$ is.
\begin{proof}
As the left adjoint functor $\ev_\omega$ preserves injective cofibrations and creates $G$-global weak equivalences, this is simply an instance of Lemma~\ref{lemma:U-pushout-preserve-reflect} together with the existence of the injective $G$-global model structures on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$ and $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$ (Theorem~\ref{thm:G-M-injective-semistable-model-structure} and Corollary~\ref{cor:equivariant-injective-model-structure}, respectively).
\end{proof}
\end{lemma}
\begin{proof}[Proof of Theorem~\ref{thm:script-I-vs-EM-injective}]
As observed in the proof of Proposition~\ref{prop:strict-model-structure}, the cofibrations of the $G$-global model structure are in particular injective cofibrations. On the other hand, pushouts along injective cofibrations preserve $G$-global weak equivalences by Lemma~\ref{lemma:G-global-I-injective-pushout}. Corollary~\ref{cor:mix-model-structures} therefore shows that the model structure exists, that it is combinatorial and proper, and that filtered colimits in it are homotopical.
It only remains to verify the Pushout Product Axiom for the simplicial tensoring, which in turn follows immediately for the Pushout Product Axiom for $\cat{SSet}$ and for the injective model structure on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$.
\end{proof}
\begin{thm}\label{thm:I-vs-M-injective}
\index{injective G-global model structure@injective $G$-global model structure!on G-I-SSet@on $\cat{$\bm G$-$\bm{I}$-SSet}$|textbf}
There exists a unique model structure on $\cat{$\bm G$-$\bm{I}$-SSet}$ whose cofibrations are the injective cofibrations and whose weak equivalences are the $G$-global weak equivalences. We call this the \emph{injective $G$-global model structure}. It is combinatorial, left proper, simplicial, and filtered colimits in it are homotopical. Moreover, the forgetful functor is part of a simplicial Quillen equivalence
\begin{equation}\label{eq:forget-lQ-injective-I-SSet}
\forget\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{inj.~$G$-global}}\rightleftarrows\cat{$\bm G$-$\bm I$-SSet}_{\textup{inj.~$G$-global}} :\Maps_I(\mathcal I,\blank).
\end{equation}
\begin{proof}
The first part is analogous to the proof of the previous theorem.
For the final statement, we observe that the forgetful functor admits a simplicial right adjoint given by simplicially enriched right Kan extension along $I\to\mathcal I$, which we denote by $\Maps_I(\mathcal I,\blank)$.\nomenclature[amapsII]{$\Maps_I(\mathcal I,\blank)$}{right adjoint to forgetful functor $\cat{$\bm{\mathcal I}$-SSet}\to\cat{$\bm{I}$-SSet}$}
As the forgetful functor preserves weak equivalences and injective cofibrations for trivial reasons, it is left Quillen. Thus, $(\ref{eq:forget-lQ-injective-I-SSet})$ is a Quillen equivalence by Theorem~\ref{thm:strict-global-I-model-structure}.
\end{proof}
\end{thm}
\subsection{Functoriality}\index{functoriality in homomorphisms!for G-II-SSet@for $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$|(}
We will now discuss change of group functors for the above models.
\begin{lemma}\label{lemma:alpha-shriek-projective-script-I}
Let $\alpha\colon H\to G$ be any group homomorphism. Then
\begin{equation}\label{eq:alpha-shriek-projective-script-I}
\alpha_!\colon\cat{$\bm H$-$\bm{\mathcal I}$-SSet}_{\textup{$H$-global}}\rightleftarrows\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global}} :\!\alpha^*
\end{equation}
is a simplicial Quillen adjunction with fully homotopical right adjoint, and likewise for the positive model structures on either side.
\begin{proof}
For the first statement, one immediately checks that $\alpha^*$ is right Quillen with respect to the strict level model structures. On the other hand, it obviously sends static $G$-$\mathcal I$-simplicial sets to static $H$-$\mathcal I$-simplicial sets, so that also the simplicial adjunction $(\ref{eq:alpha-shriek-projective-script-I})$ is a Quillen adjunction by Proposition~\ref{prop:cofibrations-fibrant-qa}.
To see that $\alpha^*$ is homotopical, we observe that in the commutative diagram
\begin{equation*}
\begin{tikzcd}
\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_\textup{$G$-global}\arrow[r, "\ev_\omega"]\arrow[d, "\alpha^*"'] & \cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_\textup{$G$-global}\arrow[d, "\alpha^*"]\\
\cat{$\bm H$-$\bm{\mathcal I}$-SSet}_\textup{$H$-global}\arrow[r, "\ev_\omega"'] & \cat{$\bm{E\mathcal M}$-$\bm H$-SSet}_\textup{$H$-global}
\end{tikzcd}
\end{equation*}
the horizontal arrows create weak equivalences by definition while the right hand vertical arrow obviously preserves weak equivalences; the claim follows immediately.
The proof for the positive model structures is analogous.
\end{proof}
\end{lemma}
\begin{cor}
Let $\alpha\colon H\to G$ be any group homomorphism. Then
\begin{equation*}
\alpha^*\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global injective}}\rightleftarrows\cat{$\bm H$-$\bm{\mathcal I}$-SSet}_{\textup{$H$-global injective}} :\!\alpha_*
\end{equation*}
is a simplicial Quillen adjunction.
\begin{proof}
It is obvious that $\alpha^*$ preserves injective cofibrations, and it is moreover homotopical by the previous lemma, hence left Quillen.
\end{proof}
\end{cor}
\begin{lemma}\label{lemma:alpha-shriek-injective-script-I}
Let $\alpha\colon H\to G$ be an \emph{injective} group homomorphism. Then
\begin{equation*}
\alpha_!\colon\cat{$\bm H$-$\bm{\mathcal I}$-SSet}_{\textup{$H$-global injective}}\rightleftarrows\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global injective}} :\!\alpha^*
\end{equation*}
is a simplicial Quillen adjunction. In particular, $\alpha_!$ is homotopical.
\begin{proof}
We may assume without loss of generality that $H$ is a subgroup of $G$ and that $\alpha$ is its inclusion, in which case $\alpha_!$ can be modelled by applying $G\times_H\blank$ levelwise. We immediately see that $\alpha_!$ preserves injective cofibrations. To finish the proof it suffices now to show that it is also homotopical, for which we consider
\begin{equation*}
\begin{tikzcd}
\cat{$\bm H$-$\bm{\mathcal I}$-SSet}\arrow[d, "\alpha_!"']\arrow[r, "\ev_\omega"] & \cat{$\bm{E\mathcal M}$-$\bm H$-SSet}\arrow[d, "\alpha_!"]\\
\cat{$\bm G$-$\bm{\mathcal I}$-SSet}\arrow[r, "\ev_\omega"'] & \cat{$\bm{E\mathcal M}$-$\bm G$-SSet}.
\end{tikzcd}
\end{equation*}
We claim that this commutes up to isomorphism. Indeed, as $\alpha_!$ is cocontinuous, there is a canonical $G$-equivariant isomorphism filling this, and one easily checks that this isomorphism is also $E\mathcal M$-equivariant.
But the horizontal arrows in the above diagram preserve and reflect weak equivalences by definition and the right hand arrow is homotopical by Corollary~\ref{cor:alpha-shriek-injective-EM}, so the claim follows immediately.
\end{proof}
\end{lemma}
\begin{lemma}\label{lemma:alpha-lower-star-injective-script-I}
Let $\alpha\colon H\to G$ be an \emph{injective} group homomorphism. Then
\begin{equation*}
\alpha^*\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global}}\rightleftarrows
\cat{$\bm H$-$\bm{\mathcal I}$-SSet}_{\textup{$H$-global}} :\!\alpha_*
\end{equation*}
is a simplicial Quillen adjunction, and likewise for the positive model structures. If $(G:\im\alpha)<\infty$, then $\alpha_*$ is homotopical.
\begin{proof}
Let us first show that this is a Quillen adjunction. We already know that $\alpha^*$ is homotopical, so it suffices to show that the above is a Quillen adjunction for the strict level model structures, which follows in turn by applying Proposition~\ref{prop:alpha-lower-star-homotopical} levelwise.
Finally, if $(G:\im\alpha)<\infty$, then $\alpha_*$ is non-equivariantly just given by a finite product. Using that filtered colimits commute with finite products in $\cat{SSet}$, one concludes similarly to the argument from the previous lemma that $\alpha_*$ commutes with $\ev_\omega$. The claim follows as $\alpha_*\colon\cat{$\bm{E\mathcal M}$-$\bm H$-SSet}\to\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$ is homotopical by Corollary~\ref{cor:alpha-lower-star-injective-EM}.
The proof for the positive model structures is again analogous.\index{functoriality in homomorphisms!for G-II-SSet@for $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$|)}
\end{proof}
\end{lemma}
As an application of the calculus just developed we can now prove:
\begin{prop}\label{prop:injective-is-really-static-script-I}
Let $X$ be fibrant in the injective $G$-global model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ and let $i\colon A\to B$ be a $G$-equivariant injection of (not necessarily finite) $G$-sets. Then $X(i)\colon X(A)\to X(B)$ is a proper $G$-equivariant weak equivalence with respect to the diagonal $G$-action.
\begin{proof}
Fix a finite subgroup $H\subset G$; we have to show that $X(i)^H\colon X(A)^H\to X(B)^H$ is a weak homotopy equivalence. By $2$-out-of-$3$ we may assume without loss of generality that $A=\varnothing$, and filtering $B$ by its finite $H$-subsets we may assume that $B$ itself is finite.
We now observe that $X$ is also fibrant in the $H$-global injective model structure by Lemma~\ref{lemma:alpha-shriek-injective-script-I}. On the other hand, by the Yoneda Lemma $X(i)^H$ agrees up to conjugation by isomorphisms with
\begin{equation*}
\Maps^H(p, X)\colon \Maps^H(*,X)\to\Maps^H(\mathcal I(B,\blank), X)
\end{equation*}
where $p\colon\mathcal I(B,\blank)\to *$ is the unique map, and $H$ acts on $\mathcal I(B,\blank)$ via $B$.
As the injective $H$-global model structure is simplicial and since all its objects are cofibrant, it therefore suffices that $p$ is an $H$-global weak equivalence, which by definition amounts to saying that $E\Inj(B,\omega)\to *$ is an $H$-global weak equivalence of $E\mathcal M$-$H$-simplicial sets. This is however just the content of Example~\ref{ex:G-globally-contractible}.
\end{proof}
\end{prop}
\index{functoriality in homomorphisms!for G-I-SSet@for $\cat{$\bm G$-$\bm I$-SSet}$|(}
All of the above functoriality properties have analogues for the models based on $I$-simplicial sets. Let us demonstrate this for a selection of these:
\begin{cor}\label{cor:alpha-shriek-left-Quillen-I-SSet}
Let $\alpha\colon H\to G$ be any group homomorphism. Then
\begin{equation*}
\alpha_!\colon\cat{$\bm H$-$\bm{I}$-SSet}_{\textup{$H$-global}}\rightleftarrows\cat{$\bm G$-$\bm{I}$-SSet}_{\textup{$G$-global}} :\!\alpha^*
\end{equation*}
is a simplicial Quillen adjunction with fully homotopical right adjoint, and likewise for the corresponding positive model structures on either side.
\begin{proof}
One proves as in Lemma~\ref{lemma:alpha-shriek-projective-script-I} that these are Quillen adjunctions. To prove that $\alpha^*$ is homotopical, we consider the commutative diagram
\begin{equation*}
\begin{tikzcd}
\cat{$\bm G$-$\bm{I}$-SSet}\arrow[r, "\mathcal I\times_I\blank"]\arrow[d, "\alpha^*"'] & \cat{$\bm G$-$\bm{\mathcal I}$-SSet}\arrow[d, "\alpha^*"]\\
\cat{$\bm H$-$\bm{I}$-SSet}\arrow[r, "\mathcal I\times_I\blank"'] & \cat{$\bm H$-$\bm{\mathcal I}$-SSet}.
\end{tikzcd}
\end{equation*}
The horizontal arrows preserve and reflect weak equivalences by Theorem~\ref{thm:G-global-we-I-characterization} while the right hand vertical arrow is homotopical by Lemma~\ref{lemma:alpha-shriek-projective-script-I}; the claim follows immediately.
\end{proof}
\end{cor}
\begin{cor}
Let $\alpha\colon H\to G$ be any group homomorphism. Then
\begin{equation*}
\alpha^*\colon\cat{$\bm G$-$\bm{I}$-SSet}_{\textup{$G$-global injective}}\rightleftarrows\cat{$\bm H$-$\bm{I}$-SSet}_{\textup{$H$-global injective}} :\!\alpha_*
\end{equation*}
is a simplicial Quillen adjunction.
\begin{proof}
By the previous corollary $\alpha^*$ is homotopical and it obviously preserves injective cofibrations, so it is left Quillen.
\end{proof}
\end{cor}
\begin{cor}
Let $\alpha\colon H\to G$ be an \emph{injective} group homomorphism. Then
\begin{equation*}
\alpha_!\colon\cat{$\bm H$-$\bm{I}$-SSet}_{\textup{$H$-global injective}}\rightleftarrows\cat{$\bm G$-$\bm{I}$-SSet}_{\textup{$G$-global injective}} :\!\alpha^*
\end{equation*}
is a simplicial Quillen adjunction. In particular, $\alpha_!$ is homotopical.
\begin{proof}
One proves analogously to Lemma~\ref{lemma:alpha-shriek-injective-script-I} that $\alpha_!$ preserves injective cofibrations. We now consider the commutative square on the left in
\begin{equation*}
\begin{tikzcd}
\cat{$\bm H$-$\bm{I}$-SSet} &\arrow[l, "\forget"'] \cat{$\bm H$-$\bm{\mathcal I}$-SSet}\\
\cat{$\bm G$-$\bm{I}$-SSet}\arrow[u, "\alpha^*"] &\arrow[l, "\forget"] \cat{$\bm G$-$\bm{\mathcal I}$-SSet}\arrow[u, "\alpha^*"']
\end{tikzcd}
\qquad
\begin{tikzcd}
\cat{$\bm H$-$\bm{I}$-SSet}\arrow[d, "\alpha_!"']\arrow[r, "\mathcal I\times_I\blank"] & \cat{$\bm H$-$\bm{\mathcal I}$-SSet}\arrow[d, "\alpha_!"]\\
\cat{$\bm G$-$\bm{I}$-SSet}\arrow[r, "\mathcal I\times_I\blank"'] & \cat{$\bm G$-$\bm{\mathcal I}$-SSet}.
\end{tikzcd}
\end{equation*}
Passing to total mates yields a canonical isomorphism filling the square on the right. But the horizontal arrows in this preserve and reflect weak equivalences by Theorem~\ref{thm:G-global-we-I-characterization} while the right hand vertical arrow is homotopical by Lemma~\ref{lemma:alpha-shriek-injective-script-I}. The claim follows immediately.
\end{proof}
\end{cor}
\begin{rk}
By direct computation, the (homotopical) restriction functors $\alpha^*$ for any homomorphism $\alpha\colon H\to G$ are compatible with $\ev_\omega\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}\to\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$, $\ev_\omega\colon\cat{$\bm G$-$\bm{I}$-SSet}\to\cat{$\bm{\mathcal M}$-$\bm G$-SSet}$, as well as all the forgetful functors.
It follows by abstract nonsense that $(\alpha^*)^\infty$ is actually compatible with all the equivalences of associated quasi-categories constructed above, and so are its adjoints $\cat{L}\alpha_!$ and $\cat{R}\alpha_*$.\index{functoriality in homomorphisms!for G-I-SSet@for $\cat{$\bm G$-$\bm I$-SSet}$|)}
\end{rk}
Using the characterization of the cofibrations given in Corollary~\ref{cor:characterization-i-cof} and the above functoriality properties, we can now prove:
\begin{thm}\label{thm:forget-left-quillen-global}
The simplicial adjunction
\begin{equation*}
\forget\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global}}\rightleftarrows\cat{$\bm G$-$\bm{I}$-SSet}_{\textup{$G$-global}} :\!\Maps_I(\mathcal I,\blank)
\end{equation*}
is a Quillen equivalence.
\begin{proof}
As the forgetful functor is homotopical and descends to an equivalence of associated quasi-categories (Theorem~\ref{thm:strict-global-I-model-structure}), it only remains to prove that it sends generating cofibrations to cofibrations.
\begin{claim*}
Let $A,B$ be finite sets and let $n\ge 0$. Then the latching map
\begin{equation*}
\colim_{C\subsetneq B} I(A,C)^{n+1}\to I(A,B)^{n+1}
\end{equation*}
is injective.
\begin{proof}
Let $(f_0,\dots,f_n)$ be a family of injections $A\to C$ and let $(f_0',\dots,f_n')$ be a family of injections $A\to C'$ for proper subsets $C,C'\subsetneq B$, such that both are sent to the same element of $I(A,B)^{n+1}$, i.e.~for each $a\in A$ and $i=0,\dots,n$
\begin{equation*}
C\ni f_i(a)=f_i'(a)\in C'.
\end{equation*}
We conclude that $f_i$ and $f_i'$ both factor through the same injection $f_i''\colon A\to C\cap C'$. But then obviously $(f_0,\dots,f_n)$ represents the same element of the colimit as $(f_0'',\dots,f_n'')$, and so does $(f_0',\dots,f_n')$, finishing the proof of the claim.
\end{proof}
\end{claim*}
Now let $A$ be a finite faithful $H$-set and let $B$ be any finite set. We can then view $I(A,B)^{n+1}$ as a $(\Sigma_B\times H)$-set. We claim that the isotropy group of \emph{any} $(f_0,\dots,f_n)\in I(A,B)^{n+1}$ is contained in $\mathcal G_{\Sigma_B,H}$. Indeed, this just amounts to saying that $H$ acts freely on $I(A,B)$ via its action on $A$, which is trivial to check.
We are now ready to finish the proof of the proposition: let $A$ be any finite faithful $H$-set, and let $B$ be any finite set. By the above claim, the latching map
\begin{equation}\label{eq:latching-base}
\ell_B\colon\colim_{C\subsetneq B} \mathcal I(A,C)\to\mathcal I(A,B)
\end{equation}
is injective, and the argument from the previous paragraph tells us in particular that any simplex not in the image has isotropy a graph subgroup of $\Sigma_B\times H$. This precisely means that $(\ref{eq:latching-base})$ is a cofibration for the $\mathcal G_{\Sigma_B,H}$-model structure on $\cat{$\bm{(\Sigma_B\times H)}$-SSet}$. We conclude from Corollary~\ref{cor:characterization-i-cof} that $\mathcal I(A,\blank)$ (with $H$ acting via $A$) is cofibrant in the $H$-global model structure on $\cat{$\bm{H}$-$\bm{I}$-SSet}$.
If now $\phi\colon H\to G$ is any homomorphism, then $\phi_!\colon\cat{$\bm H$-$\bm I$-SSet}\to\cat{$\bm G$-$\bm I$-SSet}$ is left Quillen for the $H$-global and $G$-global model structure, respectively, by Corollary~\ref{cor:alpha-shriek-left-Quillen-I-SSet}, so $\mathcal I(A,\blank)\times_\phi G\cong\phi_!\mathcal I(A,\blank)$ is cofibrant. As the $G$-global model structure is simplicial, we conclude that $\mathcal I(A,\blank)\times_\phi G\times\del\Delta^n\hookrightarrow\mathcal I(A,\blank)\times_\phi G\times\Delta^n$ is indeed a cofibration in $\cat{$\bm G$-$\bm I$-SSet}$ as desired.
\end{proof}
\end{thm}
\subsection{Another connection to monoid actions} We will now provide another comparison between $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ and $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$ in terms of a certain `reparametrization functor' appearing in the construction of global algebraic $K$-theory \cite[Constructions~3.3 and~8.2]{schwede-k-theory}.
\begin{constr}\label{constr:omega-bullet}\nomenclature[aomegabullet]{$(\blank)[\omega^\bullet]$}{reparametrization of $E\mathcal M$-action; $\mathcal I$-simplicial set built from an $E\mathcal M$-simplicial set this way}
We write $\overline{I}_\omega\subset\overline{I}$ for the full subcategory of countably infinite sets, and analogously $\overline{\mathcal I}_\omega\subset\overline{\mathcal I}$. Then the inclusion $B\mathcal M\to\overline I$ factors through an equivalence $i\colon B\mathcal M\to \overline{I}_\omega$. We pick once and for all a retraction $r$; this is then automatically quasi-inverse to $i$ and we fix $\tau\colon ir\cong\id$ with $\tau_\omega=\id_\omega$. The functor $r$ uniquely extends to a simplicially enriched functor $\overline{\mathcal I}_\omega\to BE\mathcal M$, which we denote by the same symbol; this is automatically a retraction of the inclusion $i$, and $\tau$ is a simplicially enriched isomorphism $ir\cong\id$.
For any $E\mathcal M$-$G$-simplicial set $X$, we now write $X[\blank]\mathrel{:=} X\circ r\colon\overline{\mathcal I}_\omega\to\cat{$\bm G$-SSet}$. We then define $X[\omega^\bullet]$ to be the following $G$-$\mathcal I$-simplicial set: if $A\not=\varnothing$, then $X[\omega^A]=X(r(\omega^A))$ as above. The $G$-action is as before, and for any injection $i\colon A\to B$ we take the structure map to be $X[i_!]\colon X[\omega^A]\to X[\omega^B]$, i.e.~it is given by applying $X\circ r$ to the `extension by zero map' $i_!\colon\omega^A\to\omega^B$ with
\begin{equation*}
i_!(f)(b)=\begin{cases}
f(a) & \text{if $b=i(a)$}\\
0 & \text{if $b\notin\im i$}.
\end{cases}
\end{equation*}
More generally, we let an $n$-simplex $(i_0,\dots,i_n)$ act by $X[i_{0!},\dots,i_{n!}]$. We remark that this means that as a $G$-simplicial set $X[\omega^A]=X$, and all of the above (higher) structure maps are given by acting with certain (inexplicit and mysterious) elements of $E\mathcal M$.
Finally, define $X[\omega^\varnothing]\mathrel{:=}X_{[\varnothing]}$. The structure maps $X[\omega^\varnothing]\to X=X[\omega^A]$ are given by the inclusions, and we choose all higher cells to be trivial. As all the remaining structure is given by acting with elements of $E\mathcal M$, this is easily seen to be functorial, yielding a $G$-$\mathcal I$-simplicial set.
This extends to a simplicially enriched functor $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}\to\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ by sending an $n$-simplex $f\colon\Delta^n\times X\to Y$ of the mapping space to the transformation given in non-empty degree by $f$ itself and in degree $\varnothing$ by $f_{[\varnothing]}$.
\end{constr}
\begin{rk}
There is an alternative `coordinate-free' perspective on the above construction, that we briefly sketch; for the arguments in this paper we will however only be interested in the above version of the construction.
If $A$ is any non-empty set, then $E\Inj(\omega^A,\omega)$ is left $E\mathcal M$-isomorphic to $E\mathcal M$ by precomposing with the isomorphism $\tau$; for $A=\varnothing$, $E\Inj(\omega^\varnothing,\omega)\cong E\omega$ corepresents the functor $(\blank)_{[\{0\}]}$.
It is then not hard to produce a natural map from $X[\omega^\bullet]$ to the $G$-$\mathcal I$-simplicial set sending $A\in \mathcal I$ to $\Maps^{E\mathcal M}(E\Inj(\omega^A,\omega),X)$ with the functoriality in $A$ is as above. This map is an isomorphism in all positive degrees and hence in particular a $G$-global weak equivalence. This `coordinate free' description is then analogous to \cite[construction after Proposition~3.5]{schwede-orbi}, also cf.~\cite[Section~8]{lind}.
\end{rk}
We can now state our comparison:
\begin{prop}\label{prop:omega-bullet-inverse}
The functors
\begin{equation*}
\ev_\omega\colon \cat{$\bm G$-$\bm{\mathcal I}$-SSet}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet} :\!(\blank)[\omega^\bullet]
\end{equation*}
are mutually inverse homotopy equivalences.
\end{prop}
For the proof we will need the following example of a complete $H$-set universe from \cite[Proposition~2.19]{schwede-k-theory}:
\begin{lemma}\label{lemma:exponential-universe}
Let $A$ be a finite $H$-set containing a free $H$-orbit. Then $\omega^A$ with left $H$-action via $(h.f)(a)=f(h^{-1}.a)$ is a complete $H$-set universe.\qed
\end{lemma}
\begin{rk}\label{rk:theta}
We can give an alternative description of $\ev_\omega\circ[\omega^\bullet]$ in non-empty degrees as follows: for any $A\not=\varnothing$ the isomorphism $\tau\colon\omega\to\omega^A$ from the construction of $(\blank)[\omega^\bullet]$ induces
\begin{equation}\label{eq:theta-isomorphism}
X(\omega^A)\xrightarrow{X(\tau^{-1})} X(\omega)=X(\omega)[\omega^A]
\end{equation}
and these are by definition compatible with all the relevant (higher) structure maps and moreover natural in $X$.
Using this, we define $\theta_X\colon X\to X(\omega)[\omega^\bullet]$ in degree $A\not=\varnothing$ as the composition
\begin{equation*}
X(A)\xrightarrow{X(e)} X(\omega^A)\xrightarrow{(\ref{eq:theta-isomorphism})} X(\omega)[\omega^A],
\end{equation*}
where $e\colon A\to\omega^A$ sends $a\in A$ to its characteristic function, i.e.~$e(a)(a)=1$ and $e(a)(b)=0$ otherwise.
In degree $\varnothing$, we define $\theta_X(\varnothing)\colon X\to X(\omega)_{[\varnothing]}$ via the unit of $\ev_\omega\dashv(\blank)_\bullet$. We omit the easy verification that $\theta_X$ is a map of $G$-$\mathcal I$-simplicial sets and natural in $X$.
\end{rk}
\begin{proof}[Proof of Proposition~\ref{prop:omega-bullet-inverse}]
We first show that $(\blank)[\omega^\bullet]$ is homotopical, for which we let $f$ be any $G$-global weak equivalence in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$. If now $H\subset\mathcal M$ is any universal subgroup, and $A$ is any finite $H$-set containing a free $H$-orbit (in particular $A\not=\varnothing$), then Lemma~\ref{lemma:exponential-universe} shows that $\omega^A$ is a complete $H$-set universe. We can therefore pick an $H$-equivariant isomorphism $\omega\cong\omega^A$, which shows that $f[\omega^A]$ agrees up to conjugation by $(H\times G)$-equivariant isomorphism with $f[\omega]=f$; in particular, $f[\omega^A]$ is a $\mathcal G_{H,G}$-weak equivalence. The argument from Lemma~\ref{lemma:strict-are-global} thus shows that $f[\omega^\bullet]$ is a $G$-global weak equivalence as desired.
Next, let us show that $\theta_X\colon X\to X(\omega)[\omega^\bullet]$ is a $G$-global weak equivalence for any $X\in\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$. As $\ev_\omega$ and $(\blank)[\omega^\bullet]$ are homotopical, we may assume without loss of generality that $X$ is static. We then let $A$ be any finite non-empty faithful $H$-set, and observe that $\theta_X(A)$ agrees by the usual cofinality argument up to $(H\times G)$-equivariant isomorphism with the composition
\begin{equation*}
X(A)\xrightarrow{X(e)} X(e(A))\to\colim\limits_{e(A)\subset B\subset\omega^A\text{ finite $H$-set}} X(B),
\end{equation*}
where the unlabelled arrow is the structure map. As the left hand map is an isomorphism and all transition maps of the colimit are $\mathcal G_{H,G}$-weak equivalences since $X$ is static, this is a $\mathcal G_{H,G}$-weak equivalence, so $\theta_X$ is a $G$-global weak equivalence as before. In particular, $\ev_\omega$ is right homotopy inverse to $(\blank)[\omega^\bullet]$.
To see that it is also left homotopy inverse, we consider the following zig-zag for each $E\mathcal M$-$G$-simplicial set $Y$:
\begin{equation*}
Y[\omega^\bullet](\omega)=\colim\limits_{A\subset\omega\textup{ finite}} Y[\omega^A]
\xrightarrow{\alpha} \colim\limits_{A\subset\omega\textup{ finite}}
Y[\omega^A\amalg\omega] \xleftarrow{\beta}
\colim\limits_{A\subset\omega\textup{ finite}} Y[\omega]\cong Y.
\end{equation*}
The transition maps on the two left hand colimits come from the extension by zero maps $\omega^A\to\omega^B$, and the transition maps of the remaining colimit are trivial. The group $G$ acts by its action on $Y$ everywhere. Moreover, $E\mathcal M$ acts on all the colimits analogously to Construction~\ref{constr:i-extension} (observe that this part of the action is trivial for the rightmost colimit), in addition on the middle colimit by its tautological action on the $\omega$-summand of $Y[\omega^A\amalg\omega]$, and finally by its given action on $Y[\omega]=Y$ for the final colimit. The maps $\alpha$ and $\beta$ are given in each degree by the inclusions $\omega^A\hookrightarrow\omega^A\amalg\omega\hookleftarrow\omega$. One immediately checks that they are well-defined and $E\mathcal M$-$G$-equivariant. Moreover, we can make the middle term into a functor in $Y$ in the obvious way and with respect to this the maps $\alpha$ and $\beta$ are clearly natural.
It only remains to prove that $\alpha$ and $\beta$ are $G$-global weak equivalences. We prove this for $\alpha$, the other argument being similar. For this let $H\subset\mathcal M$ be universal. We pick a free $H$-orbit $F$ inside $\omega$; by the same argument as before, $\alpha$ agrees up to conjugation by $(H\times G)$-equivariant isomorphisms with the map
\begin{equation}\label{eq:alpha-H-G-focus}
\colim\limits_{F\subset A\subset\omega\textup{ finite $H$-set}} Y[\omega^A]\to \colim\limits_{F\subset A\subset\omega\textup{ finite $H$-set}} Y[\omega^A\amalg\omega]
\end{equation}
still induced in each degree by the inclusion $i\colon\omega^A\hookrightarrow\omega^A\amalg\omega$. We claim that each of these maps is even an $(H\times G)$-equivariant homotopy equivalence. But indeed, as both $\omega^A$ and $\omega^A\amalg\omega$ are complete $H$-set universes by Lemma~\ref{lemma:exponential-universe}, there exists an $H$-equivariant injection $j\colon\omega^A\amalg\omega\to\omega^A$, and $Y[j]$ is $(H\times G)$-equivariantly homotopy inverse to $Y[i]$ as witnessed by the equivariant homotopies $Y[ij,\id]$ and $Y[ji,\id]$.
\end{proof}
\subsection{Comparison to $\bm G$-equivariant homotopy theory}\label{subsec:G-global-vs-G-equiv-script-I}\index{proper G-equivariant homotopy theory@proper $G$-equivariant homotopy theory!vs G-global homotopy theory@vs.~$G$-global homotopy theory|(}
We can now prove the analogues of the results of Section~\ref{sec:g-global-vs-g-em} for $G$-$\mathcal I$-simplicial sets:
\begin{cor}\label{cor:G-script-I-SSet-vs-G-SSet}
The homotopical functor
\begin{equation*}
\const\colon\cat{$\bm G$-SSet}_{\textup{proper}}\to\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global}}
\end{equation*}
induces a fully faithful functor on associated quasi-categories. This induced functor admits both a left adjoint $\textbf{\textup L}{\colim_{\mathcal I}}$ as well as a right adjoint $\textbf{\textup R}\ev_\varnothing$. The latter is a quasi-localization at those $f$ such that $f(\omega)$ is an $\mathcal E$-weak equivalence of $E\mathcal M$-$G$-simplicial sets (see Definition~\ref{defi:class-E}), and it in turn admits another right adjoint $\mathcal R$, which is again fully faithful.
Finally, the diagram
\begin{equation}\label{diag:triv-vs-const}
\begin{tikzcd}[column sep=small]
& \cat{$\bm G$-SSet}_{\textup{proper}}^\infty\arrow[dl, bend right=10pt, "\const^\infty"']\arrow[dr, bend left=10pt, "\triv_{E\mathcal M}^\infty"]\\
\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global}}^\infty\arrow[rr, "(\ev_\omega)^\infty"'] &&
\cat{$\bm G$-$\bm{E\mathcal M}$-SSet}_{\textup{$G$-global}}^\infty
\end{tikzcd}
\end{equation}
commutes up to canonical equivalence.
\begin{proof}
The adjunction
\begin{equation*}
\const\colon\cat{$\bm G$-SSet}_{\textup{proper}}\rightleftarrows\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global injective}} :\!\ev_\varnothing
\end{equation*}
is easily seen to be a Quillen adjunction, providing the above description of the right adjoint. Similarly, for the left adjoint we want to prove that the simplicial adjunction
\begin{equation}\label{eq:colim-vs-const-G-I}
\colim_{\mathcal I}\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global}}\rightleftarrows\cat{$\bm G$-SSet}_{\textup{proper}} :\!\const
\end{equation}
is a Quillen adjunction. For this we first observe that this is a Quillen adjunction when we equip the left hand side with the strict level model structure (as $\const$ is then obviously right Quillen). In particular, $\colim_{\mathcal I}$ preserves $G$-global (i.e.~strict level) cofibrations, and $\const$ sends fibrant $G$-simplicial sets to strictly fibrant $G$-$\mathcal I$-simplicial sets. It then suffices by Proposition~\ref{prop:cofibrations-fibrant-qa} and the characterization of the fibrant objects in $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global}}$ provided by Theorem~\ref{thm:script-I-global-model-structure} that $\const$ sends fibrant objects to static ones, which is immediate from the definitions.
In order to construct a canonical equivalence filling $(\ref{diag:triv-vs-const})$, it suffices to observe that the evident diagram of homotopical functors inducing it even commutes up to canonical isomorphism. The remaining statements then follow formally from Theorem~\ref{thm:G-global-vs-proper-sset} and commutativity of $(\ref{diag:triv-vs-const})$ as $(\ev_\omega)^\infty$ is an equivalence.
\end{proof}
\end{cor}
In the world of $G$-$\mathcal I$-simplicial sets there are rather explicit pointset models of the right adjoints $\textbf{R}\ev_\varnothing$ and $\mathcal R$, which we will introduce now. To this end, we define
\begin{equation}\label{eq:proper-set-universe}
\mathcal U_G\mathrel{:=}\coprod_{i=0}^\infty\coprod_{H\subset G,H\textup{ finite}} G/H.
\end{equation}
\nomenclature[aUG]{$\mathcal U_G$}{union of countably infinitely many copies of all transitive $G$-sets with finite isotropy up to isomorphism (replacement for complete set universe in the proper context)}
If $G$ is finite, the above is just a particular construction of a complete $G$-set universe. However, in general this need not be countable and it can even have uncountably many orbits (e.g.~for $G=\bigoplus_{\mathbb R}\mathbb Z/2\mathbb Z$). Despite these words of warning we always have:
\begin{lemma}\label{lemma:proper-set-universe}
Let $H$ be any finite group and let $\iota\colon H\to G$ be any injective group homomorphism. Then the $H$-set $\iota^*\mathcal U_G$ contains a complete $H$-set universe.
\begin{proof}
We may assume without loss of generality that $\iota$ is literally the inclusion of a finite subgroup $H$ of $G$. We then have an $H$-equivariant injection
\begin{equation*}
\coprod_{i=0}^\infty\coprod_{K\subset H} H/K\to\mathcal U_G
\end{equation*}
induced from $H\hookrightarrow G$ (and using that each subgroup $K\subset H$ is in particular a finite subgroup of $G$). As the left hand side is a complete $H$-set universe, this finishes the proof.
\end{proof}
\end{lemma}
By the universal property of enriched presheaves, the simplicial functor $\ev_{\mathcal U_G}$ given by evaluation at $\mathcal U_G$ has a simplicial right adjoint given explicitly by
\nomenclature[aR]{$R$}{pointset level model of $\mathcal R$ for $G$-$\mathcal I$-simplicial sets}
\begin{equation*}
R(X)(A)=\Maps(E\Inj(A,\mathcal U_G),X)
\end{equation*}
(where $\Maps$ denotes the simplicial set of all maps, with $G$ acting by conjugation) with the obvious functoriality in each variable. We can now prove:
\begin{prop}\label{prop:evaluation-proper-vs-evaluation-empty}
The simplicial adjunction
\begin{equation}\label{eq:evaluate-proper-universe}
\und_G\mathrel{:=}\ev_{\mathcal U_G}\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{$G$-global}}\rightleftarrows\cat{$\bm G$-SSet}_{\textup{proper}} :\!R
\end{equation}
\nomenclature[aug]{$\und_G$}{underlying proper $G$-equivariant space of a $G$-$\mathcal I$-simplicial set}
is a Quillen adjunction with homotopical left adjoint, and there are preferred equivalences
\begin{equation*}
\big(\ev_{\mathcal U_G}\big)^\infty\simeq\textbf{\textup{R}}\ev_\varnothing\qquad\text{and}\qquad \textbf{\textup R}R\simeq\mathcal R.
\end{equation*}
\begin{proof}
Let us call a map of $G$-$\mathcal I$-simplicial sets that becomes an $\mathcal E$-weak equivalences after evaluating at $\omega$ an \emph{$\mathcal E$-weak equivalence} again. Because of the canonical isomorphism $\ev_{\mathcal U_G}\circ\const\cong\id$ it then suffices that $(\ref{eq:evaluate-proper-universe})$ is a Quillen adjunction and that $\ev_{\mathcal U_G}$ is homotopical in $\mathcal E$-weak equivalences.
If $H\subset G$ is any finite subgroup, then we pick an injective homomorphism $\iota\colon H'\to G$ with image $H$ from a universal subgroup $H'\subset\mathcal M$. With this notation we then have for any $G$-$\mathcal I$-simplicial set $X$ an actual equality
\begin{equation*}
\big(X(\mathcal U_G)\big)^H=X\big(\iota^*(\mathcal U_G)\big)^\iota,
\end{equation*}
and analogously for morphisms. By the previous lemma there exists an $H'$-equi\-variant embedding $\omega\to \iota^*(\mathcal U_G)$ (with respect to the tautological action on the left hand side), and by Lemma~\ref{lemma:evaluation-h-universe} we conclude that the induced natural transformation $(\blank)^\iota\circ\ev_\omega\Rightarrow(\blank)^H\circ\ev_{\mathcal U_G}$ is a weak equivalence. Thus, $\ev_{\mathcal U_G}$ is homotopical in $\mathcal E$-weak equivalences (and hence in particular in $G$-global weak equivalences).
It only remains to prove that $\ev_{\mathcal U_G}$ sends the standard generating cofibrations to proper cofibrations. As the proper model structure is simplicial, we are reduced to showing that the $G$-simplicial set $E\Inj(A,\mathcal U_G)\times_\phi G$ is cofibrant in the proper model structure (i.e. has finite isotropy groups) for any finite group $H$, any finite (faithful) $H$-set $A$ and any group homomorphism $\phi\colon H\to G$. For this we let $(f_0,\dots,f_n;g)$ represent any $n$-simplex. If $g'$ fixes $[f_0,\dots,f_n;g]$, then in particular $g'g=g\phi(h)$ for some $h\in H$, and hence $g'=g\phi(h)g^{-1}$. As the right hand side can only take finitely many values for any fixed $g$, the claim follows.
\end{proof}
\end{prop}
\begin{rk}\label{rk:R-vs-R}
For $G=1$ the above adjunction is the $\mathcal I$-analogue of \cite[Remark~1.2.24 and Proposition~1.2.27]{schwede-book}. As we will show in Section~\ref{sec:global-vs-g-global}, there exists a zig-zag of homotopical functors
\begin{equation*}
\cat{$\bm{\mathcal I}$-SSet}\xrightarrow{\forget}\cat{$\bm I$-SSet}\xrightarrow{|\blank|}\cat{$\bm I$-Top}\xleftarrow{\text{forget}}\cat{$\bm{L}$-Top}
\end{equation*}
where $\cat{$\bm L$-Top}$ denotes Schwede's orthogonal spaces
(with respect to the global weak equivalences for the class of \emph{finite} groups), and all of these induce equivalences on associated quasi-categories. It is then not hard to check directly that under this identification the adjunction $\const^\infty\dashv\cat{R}\ev_\varnothing\dashv\mathcal R$ corresponds to the adjunction $\textbf{L}_1\dashv\textbf{R}\ev_0\dashv\textbf{R}R$ considered in \emph{loc.~cit.} for the trivial group.
In fact, there is also a completely abstract way to prove this: namely, in both cases the leftmost adjoint under consideration preserves the terminal object (in our case because we have explicitly constructed a further left adjoint, in Schwede's case by direct inspection). If we now have \emph{any} equivalence $\Phi$ between $\cat{$\bm L$-Top}^\infty$ and $\cat{$\bm{\mathcal I}$-SSet}^\infty$, then both ways through the diagram
\begin{equation*}
\begin{tikzcd}[column sep=small]
& \cat{SSet}^\infty\arrow[dl, "\const^\infty"', bend right=10pt]\arrow[dr, "\textbf{L}_1", bend left=10pt]\\
\cat{$\bm{\mathcal I}$-SSet}^\infty\arrow[rr, "\Phi"'] & & \cat{$\bm L$-Top}^\infty
\end{tikzcd}
\end{equation*}
are cocontinuous and send the terminal object to the terminal object. By the universal property of spaces \cite[Theorem~5.1.5.6]{htt}, there is therefore a contractible space of natural equivalences filling this, and as before $\Phi$ is then also compatible with the other adjoints.\index{proper G-equivariant homotopy theory@proper $G$-equivariant homotopy theory!vs G-global homotopy theory@vs.~$G$-global homotopy theory|)}
\end{rk}
\subsection{Model structures for tame actions}\label{subsec:tame-model-structures}
Above, we have used our understanding of tame $\mathcal M$- and $E\mathcal M$-actions to introduce the $G$-global model structures on $\cat{$\bm G$-$\bm I$-SSet}$ and $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$. Conversely, we will now use the results of this section to construct $G$-global model structures on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ and $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau$:
\begin{thm}\index{positive G-global model structure@positive $G$-global model structure!on EM-G-SSettau@on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$|textbf}\label{thm:pos-G-global-EM-tau}
There exists a unique model structure on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ in which a map $f$ is a weak equivalence, fibration, or cofibration if and only if $f_\bullet$ is a weak equivalence, fibration, or cofibration, respectively, in the positive $G$-global model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$. We call this the \emph{positive $G$-global model structure}. Its weak equivalences are precisely the $G$-global weak equivalences, and hence they are in particular closed under filtered colimits.
This model structure is proper, simplicial, and combinatorial with generating cofibrations
\begin{equation}\label{eq:EM-tau-generating-cof}
(E\Inj(A,\omega)\times_\phi G)\times\del\Delta^n\hookrightarrow(E\Inj(A,\omega)\times_\phi G)\times\Delta^n
\end{equation}
where $H$ runs through finite groups, $A\not=\varnothing$ is a finite faithful $H$-set, $\phi$ is a homomorphism $H\to G$, and $n\ge 0$. Finally, the simplicial adjunctions
\begin{align}\label{eq:em-tau-vs-em}
\incl\colon\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau&\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}_{\textup{injective $G$-global}} :\!(\blank)^\tau\\
\intertext{and}
\label{eq:em-tau-vs-script-I}
\ev_\omega\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}&\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau :\!(\blank)_\bullet
\end{align}
are Quillen equivalences.
\begin{proof}\begingroup\parskip=\the\parskip plus .76pt
\baselineskip=\the\baselineskip plus .1pt
The adjunction $\ev_\omega\dashv(\blank)_\bullet$ exhibits $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ as accessible Bousfield localization of the locally presentable category $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$, so it is itself locally presentable, cf.~\cite[Remark~5.5.1.6]{htt}.
Let us now show that the above defines a model structure on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$. Instead of using Crans' Transfer Criterion (Proposition~\ref{prop:transfer-criterion}), we can actually easily verify the model structure axioms directly here as the adjunction $\ev_\omega\dashv(\blank)_\bullet$ is already well-behaved $1$-categorically, also see \cite[Theorem~5.10 and Corollary~5.11]{I-vs-M-1-cat} where a similar argument is used to non-equivariantly relate $\cat{$\bm I$-SSet}$ and $\cat{$\bm{\mathcal M}$-SSet}^\tau$:
The $2$-out-of-$3$ property for weak equivalences as well as the closure under retracts for all three classes are obvious. Moreover, as $(\blank)_\bullet$ is fully faithful, the lifting axioms are inherited from the lifting axioms for $\cat{$\bm{G}$-$\bm{\mathcal I}$-SSet}$.
It only remains to verify the factorization axioms, for which we let $f\colon X\to Y$ be any map of tame $E\mathcal M$-$G$-simplicial sets. Then we can factor $f_\bullet$ as a cofibration $i\colon X_\bullet\to Z$ followed by an acyclic fibration $p\colon Z\to Y_\bullet$. But cofibrations in the positive $G$-global model structure are in particular cofibrations in $\cat{$\bm I$-SSet}$ (Theorem~\ref{thm:forget-left-quillen-global} together with Lemma~\ref{lemma:alpha-lower-star-injective-script-I}), so $Z$ is flat again and hence lies in the essential image of $(\blank)_\bullet$. We can therefore assume without loss of generality that $Z=Z'_\bullet$ for some $Z'\in\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$. By full faithfulness of $(\blank)_\bullet$ we can then write $i=i'_\bullet$, $p=p'_\bullet$, which yields the desired factorization $f=p'i'$. The remaining factorization axiom is proven analogously.
This completes the proof of the existence of the positive $G$-global model structure. By definition, $f\colon X\to Y$ is a weak equivalence if and only if $f_\bullet$ is, which in turn is equivalent by definition to $f_\bullet(\omega)$ being a $G$-global weak equivalence of $E\mathcal M$-$G$-simplicial sets. As $(\blank)_\bullet$ is fully faithful, $f_\bullet(\omega)$ is conjugate to $f$, which shows that the weak equivalences are precisely the $G$-global weak equivalences. In particular, they are closed under filtered colimits (which are created in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$).
The model structure is right proper since it is transferred from a right proper model stucture, see Lemma~\ref{lemma:transferred-properties}-$(\ref{item:tpr-proper})$. Moreover, $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ is tensored and cotensored over $\cat{SSet}$ with the tensoring given by the tensoring on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$ and the cotensoring given by applying $(\blank)^\tau$ to the usual cotensoring. As $(\ref{eq:em-tau-vs-script-I})$ is a simplicial adjunction, the positive $G$-global model structure on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ is then simplicial by Lemma~\ref{lemma:transferred-properties}-$(\ref{item:tpr-simplicial})$.
It is clear from the construction that $(\blank)_\bullet$ is right Quillen so that $(\ref{eq:em-tau-vs-script-I})$ is a Quillen adjunction. Moreover, both adjoints preserve and reflect weak equivalences by definition and the above characterization of the weak equivalences. To show that $(\ref{eq:em-tau-vs-script-I})$ is a Quillen equivalence it is therefore enough that the counit $X_\bullet(\omega)\to X$ be a $G$-global weak equivalence for each tame $E\mathcal M$-$G$-simplicial set $X$, but we already know that it is even an isomorphism.
Now let $I$ and $J$ be sets of generating cofibrations and generating acyclic cofibrations, respectively, of the positive $G$-global model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$. We claim that the positive $G$-global model structure on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ is cofibrantly generated (hence combinatorial) with generating cofibrations $I(\omega)\mathrel{:=}\{i(\omega) : i\in I\}$ and generating acyclic cofibrations $J(\omega)$. Indeed, $I(\omega)$ and $J(\omega)$ permit the small object argument as $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ is locally presentable, and they detect acyclic fibrations and fibrations, respectively, by adjointness. Taking $I$ to be the usual set of generating cofibrations and using the canonical isomorphism $\mathcal I(A,\blank)(\omega)\cong E\Inj(A,\omega)$ shows that $(\ref{eq:EM-tau-generating-cof})$ is a set of generating cofibrations.
Next, let us show that $(\ref{eq:em-tau-vs-em})$ is a simplicial Quillen adjunction. It is obvious that the left adjoint preserves tensors, so that this is indeed a simplicial adjunction. Moreover, $\incl$ sends the above generating cofibrations to injective cofibrations, and it is homotopical by the above characterization of the weak equivalences, hence left Quillen. Finally, the inclusion descends to an equivalence on homotopy categories by Theorem~\ref{thm:shrew}, i.e.~$(\ref{eq:em-tau-vs-em})$ is a Quillen equivalence.
It only remains to show that $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ is left proper. But indeed, we have seen that the weak equivalences are precisely the $G$-global weak equivalences, and since $(\ref{eq:em-tau-vs-em})$ is a Quillen adjunction, the cofibrations are in particular injective cofibrations. As pushouts in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ can be computed inside all of $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$, the claim therefore follows from the left properness of the injective $G$-global model structure on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$.
\endgroup
\end{proof}
\end{thm}
\begin{rk}
While we will be almost exclusively interested in the positive $G$-global model structure considered above, the analogous statement for the $G$-global model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ holds and can be proven in the same way. We call the resulting model structure the \emph{$G$-global model structure}\index{G-global model structure@$G$-global model structure!on EM-G-SSettau@on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$}; its generating cofibrations are again given by evaluating the usual generating cofibrations for the corresponding model structures on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$ at $\omega$.
\end{rk}
\begin{rk}\label{rk:support-em-tau-model-structure}
Similarly to Corollary~\ref{cor:injective-almost-tame}, if $X$ is fibrant in the $G$-global positive model structure, then all its simplices have `small support up to weak equivalence.' More precisely, let $H\subset\mathcal M$ be universal and let $A\subset\omega$ be a non-empty faithful $H$-set. Then the $H$-action on $X$ restricts to $X_{[A]}$, and the inclusion $X_{[A]}\hookrightarrow X$ is a $\mathcal G_{H,G}$-weak equivalence as $X_\bullet$ is fibrant in the positive $G$-global model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$, hence in particular positively static.
Of course, the analogous statement for the $G$-global model structure holds.
\end{rk}
For later use, we record two properties of the above cofibrations:
\begin{lemma}\label{lemma:g-global-pos-cof}
Let $f\colon X\to Y$ be a map in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$.
\begin{enumerate}
\item If $f$ is a cofibration in the $G$-global positive model structure, then $f$ restricts to an isomorphism $f_{[\varnothing]}\colon X_{[\varnothing]}\to Y_{[\varnothing]}$.
\item If $f$ is a $G$-global cofibration (for example, if $f$ is a positive $G$-global cofibration), then it is also a $\mathcal G_{H,G}$-cofibration for any subgroup $H\subset\mathcal M$, i.e.~$f$ is injective and $G$ acts freely on $Y$ outside the image of $f$.
\end{enumerate}
\begin{proof}
For the first statement we observe that $f_\bullet$ is a $G$-global positive cofibration by definition, so $f_{[\varnothing]}=f_\bullet(\varnothing)$ is an isomorphism by Lemma~\ref{lemma:pos-cof-script-I}.
For the second statement, it suffices to prove this for the generating cofibrations. As the $\mathcal G_{H,G}$-model structure is simplicial, it suffices further to show that $E\Inj(A,\omega)\times_\phi G$ is cofibrant in the $\mathcal G_{H,G}$-model structure, i.e.~that $G$ acts freely on it. This is immediate from Lemma~\ref{lemma:emg-basic-general}-(\ref{item:emgbg-unique}).
\end{proof}
\end{lemma}
While the above argument for the construction of the positive $G$-global model structure does not apply for the injective $G$-global model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$, we still have:
\begin{thm}
There is a unique model structure on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ with cofibrations the injective cofibrations and weak equivalences the $G$-global weak equivalences. We call this the \emph{injective $G$-global model structure}.\index{injective G-global model structure@injective $G$-global model structure!on EM-G-SSettau@on $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$|textbf} It is combinatorial, proper, simplicial, and filtered colimits in it are homotopical. Moreover, the simplicial adjunction
\begin{equation}\label{eq:ev-omega-quillen-tame}
\ev_\omega\colon\cat{$\bm G$-$\bm{\mathcal I}$-SSet}_{\textup{inj.~$G$-global}}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau_{\textup{inj.~$G$-global}} :\!(\blank)_\bullet
\end{equation}
is a Quillen equivalence.
\begin{proof}
Let $\hat I$ be a set of generating cofibrations for the injective $G$-global model structure on $\cat{$\bm G$-$\bm{\mathcal I}$-SSet}$, and define $I\mathrel{:=}\hat I(\omega)=\{i(\omega):i\in \hat I\}$.
\begin{claim*}
The injective cofibrations are precisely the retracts of relative $I$-cell complexes.
\begin{proof}
It is clear that any element of $I$ is an injective cofibration. As the latter are closed under pushouts, retracts, and transfinite composition, it suffices to show conversely that any injective cofibration is a retract of an $I$-cell complex.
But indeed, if $f$ is an injective cofibration, then so is $f_\bullet$ by direct inspection, so that it can be written as a retract of a relative $\hat I$-cell complex. As $\ev_\omega$ is a left adjoint, we conclude that $f_\bullet(\omega)$ is a retract of a relative $\hat I(\omega)=I$-cell complex. But by full faithfulness of $(\blank)_\bullet$, $f_\bullet(\omega)$ is conjugate to $f$, which completes the proof.
\end{proof}
\end{claim*}
We have seen in the proof of Theorem~\ref{thm:pos-G-global-EM-tau} that pushouts along injective cofibrations in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ preserve $G$-global weak equivalences, and that positive $G$-global cofibrations are in particular injective cofibrations. As the injective cofibrations are generated by the set $I$, Corollary~\ref{cor:enlarge-generating-cof} therefore shows that the model structure exists, that is combinatorial and proper, and that filtered colimits in it are homotopical.
To prove that the model structure is also simplicial, it suffices to observe that colimits and tensors can be computed in all of $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$ and that the latter is simplicial by Corollary~\ref{cor:equivariant-injective-model-structure}.
Finally, it is clear that $\ev_\omega$ preserves injective cofibrations and weak equivalences, so that $(\ref{eq:ev-omega-quillen-tame})$ is a Quillen adjunction, hence a Quillen equivalence by Theorem~\ref{thm:pos-G-global-EM-tau}.
\end{proof}
\end{thm}
Next, we come to an analogue of Theorem~\ref{thm:pos-G-global-EM-tau} for tame $\mathcal M$-actions, which can be proven in exactly the same way:
\begin{thm}\label{thm:M-G-tau-model-structure}
There exists a model structure on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau$ in which a map $f$ is a weak equivalence, fibration, or cofibration, if and only if $f_\bullet$ is a weak equivalence, fibration, or cofibration, respectively, in the positive $G$-global model structure on $\cat{$\bm G$-$\bm{I}$-SSet}$. We call this the \emph{positive $G$-global model structure}.\index{positive G-global model structure@positive $G$-global model structure!on M-G-SSettau@on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau$|textbf} Its weak equivalences are precisely the $G$-global weak equivalences.
This model structure is left proper, simplicial, combinatorial with generating cofibrations
\begin{equation*}
\begin{aligned}
&\{(\Inj(A,\omega)\times_\phi G)\times\del\Delta^n\hookrightarrow(\Inj(A,\omega)\times_\phi G)\times\Delta^n :{}\\
&\qquad\text{$H$ finite group, $A\not=\varnothing$ finite faithful $H$-set, $\phi\colon H\to G$ homomorphism}\},
\end{aligned}
\end{equation*}
and filtered colimits in it are homotopical. Finally, the simplicial adjunctions
\begin{align*}
\incl\colon\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau&\rightleftarrows\cat{$\bm{\mathcal M}$-$\bm G$-SSet}_{\textup{injective $G$-global}} :\!(\blank)^\tau\\
\ev_\omega\colon\cat{$\bm G$-$\bm{I}$-SSet}&\rightleftarrows\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau :\!(\blank)_\bullet
\end{align*}
are Quillen equivalences.\qed
\end{thm}
\begin{cor}
The simplicial adjunction
\begin{equation*}
E\mathcal M\times_{\mathcal M}\blank\colon\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau :\forget
\end{equation*}
is a Quillen equivalence, and both adjoints are homotopical.
\begin{proof}
We have seen in Theorem~\ref{thm:tame-M-sset-vs-EM-sset} that both adjoints are homotopical, and that they descend to equivalences on homotopy categories. It only remains to show that $E\mathcal M\times_{\mathcal M}\blank$ sends the above generating cofibrations to cofibrations, which is immediate from Corollary~\ref{cor:E-Inj-corepr}.
\end{proof}
\end{cor}
\begin{rk}
Again we get an analogous result for the $G$-global model structure\index{G-global model structure@$G$-global model structure!on M-G-SSettau@on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau$}. Moreover, one can construct an injective $G$-global model structure\index{injective G-global model structure@injective $G$-global model structure!on M-G-SSettau@on $\cat{$\bm{\mathcal M}$-$\bm G$-SSet}^\tau$} by an argument similar to the above. We leave the details to the interested reader.
\end{rk}
Finally, let us discuss functoriality for $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$:
\index{functoriality in homomorphisms!for EM-G-SSettau@for $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$|(}
\begin{lemma}\label{lemma:alpha-lower-shriek-tau}
Let $\alpha\colon H\to G$ be any group homomorphism. Then
\begin{equation*}
\alpha_!\colon\cat{$\bm{E\mathcal M}$-$\bm H$-SSet}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet} :\!\alpha^*
\end{equation*}
restricts to a Quillen adjunction
\begin{equation}\label{eq:alpha-shriek-tau}
\alpha_!\colon\cat{$\bm{E\mathcal M}$-$\bm H$-SSet}^\tau_{\textup{positive $H$-global}}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau_{\textup{positive $G$-global}} :\!\alpha^*.
\end{equation}
The right adjoint is fully homotopical, and if $\alpha$ is injective, so is the left adjoint.
\begin{proof}
It is clear that $\alpha^*$ preserves tameness, and so does $\alpha_!$ as the full subcategory $\cat{$\bm{E\mathcal M}$-SSet}^\tau\subset\cat{$\bm{E\mathcal M}$-SSet}$ is closed under colimits.
To see that $(\ref{eq:alpha-shriek-tau})$ is a Quillen adjunction with homotopical right adjoint, it suffices to observe that $\alpha^*$ commutes with $(\blank)_\bullet$ on the nose, so that the claim follows from Lemma~\ref{lemma:alpha-shriek-projective-script-I}. Finally, if $\alpha$ is injective, then $\alpha_!$ sends $H$-global weak equivalences to $G$-global weak equivalences by Corollary~\ref{cor:alpha-shriek-injective-EM}.
\end{proof}
\end{lemma}
Similarly, one deduces from Corollary~\ref{cor:free-quotient-EM}:
\begin{cor}\label{cor:free-quotient-tame}
In the above situation, $\alpha_!$ preserves $H$-global weak equivalences between objects with free $\ker(\alpha)$-action.\qed
\end{cor}
The situation for right adjoint is a bit more complicated: of course, $\alpha^*$ still has a right adjoint $\alpha_*$, and $\alpha^*\dashv\alpha_*$ is a Quillen adjunction for the injective model structures---however, $\alpha_*\colon\cat{$\bm{E\mathcal M}$-$\bm H$-SSet}^\tau\to\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ will usually not be given as restriction of $\alpha_*\colon\cat{$\bm{E\mathcal M}$-$\bm H$-SSet}\to\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}$ because the inclusion $\cat{$\bm{E\mathcal M}$-SSet}^\tau\hookrightarrow\cat{$\bm{E\mathcal M}$-SSet}$ does not preserve infinite limits.
However, we still have:
\begin{lemma}\label{lemma:alpha-lower-star-tame}
If $\alpha\colon H\to G$ is injective with $(G:\im\alpha)<\infty$, then
\begin{equation*}
\alpha^*\colon\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm H$-SSet}:\alpha_*
\end{equation*}
restricts to a Quillen adjunction
\begin{equation*}
\alpha^*\colon\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau_{\textup{positive $G$-global}}\rightleftarrows\cat{$\bm{E\mathcal M}$-$\bm H$-SSet}^\tau_{\textup{positive $H$-global}}:\alpha_*
\end{equation*}
in which both functors are fully homotopical.
\begin{proof}
We already know that $\alpha^*$ preserves tameness and is fully homotopical. To see that also $\alpha_*$ preserves tameness, we observe that as an $E\mathcal M$-simplicial set, $\alpha_*X$ is just a $(G:\im\alpha)$-fold product of copies of $X$, and that $\cat{$\bm{E\mathcal M}$-SSet}^\tau\subset\cat{$\bm{E\mathcal M}$-SSet}$ is closed under \emph{finite} limits by Corollary~\ref{cor:EM-tau-colim-lim}.
It only remains to show that $\alpha_*$ preserves fibrations as well as weak equivalences. But indeed, as $\alpha^*$ commutes with $\ev_\omega$, $\alpha_*$ commutes with $(\blank)_\bullet$ up to (canonical) isomorphism, so these follow from Lemma~\ref{lemma:alpha-lower-star-injective-script-I}.
\end{proof}
\end{lemma}
By abstract nonsense, (finite) products of fibrations in $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ are fibrations, and we have seen that also finite products of $G$-global weak equivalences are weak equivalences. If now $S$ is any finite set and $X$ is any $E\mathcal M$-$G$-simplicial set, then $X^{\times S}\mathrel{:=}\prod_{s\in S}X$ carries a natural $\Sigma_S$-action by permuting the factors, and this way $(\blank)^{\times S}$ lifts to $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau\to\cat{$\bm{E\mathcal M}$-$\bm{(G\times\Sigma_S)}$-SSet}^\tau$. We will later need the following (non-formal) strengthening of this observation:
\begin{cor}\label{cor:twisted-product}
The above lift of $(\blank)^{\times S}$ sends $G$-global weak equivalences or fibrations to $(G\times\Sigma_S)$-global weak equivalences or fibrations, respectively.
\begin{proof}
The claim is trivial if $S$ is empty. Otherwise, we pick $s_0\in S$, and we write $\Sigma_S^0\subset\Sigma_S$ for the subgroup of permutations fixing $s_0$, $p\colon G\times\Sigma_S^0\to G$ for the projection to the second factor, and $i\colon G\times\Sigma_S^0\hookrightarrow G\times\Sigma_S$ for the inclusion.
\begin{claim*}
The functor $(\blank)^{\times S}$ is isomorphic to $i_*\circ p^*$.
\begin{proof}
Fix for each $s\in S$ a permutation $\sigma_s\in\Sigma_S$ with $\sigma_s(s_0)=s$. Then we consider for $X\in \cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$ the natural map $\Maps^{G\times\Sigma_S^0}(G\times\Sigma_S, X)\to X^{\times S}$ given on the $s$-th factor by evaluating at $(1,\sigma_s^{-1})$. We omit the easy verification that this is $(G\times\Sigma_S)$-equivariant and an isomorphism.
\end{proof}
\end{claim*}
Thus, the statement follows from Lemmas~\ref{lemma:alpha-lower-shriek-tau} and~\ref{lemma:alpha-lower-star-tame}.\index{functoriality in homomorphisms!for EM-G-SSettau@for $\cat{$\bm{E\mathcal M}$-$\bm G$-SSet}^\tau$|)}
\end{proof}
\end{cor}
\begin{rk}
Again, there are similar functoriality properties for the remaining models. We leave the details to the interested reader.
\end{rk}
\section{Comparison to global homotopy theory}\label{sec:global-vs-g-global}
In this section we prove as promised that our theory generalizes Schwede's \emph{unstable global homotopy theory} with respect to finite groups. More precisely, we will give a chain of Quillen adjunctions between $\cat{$\bm I$-SSet}$ and Schwede's \emph{orthogonal spaces} that on associated quasi-categories exhibits the former as a (right Bousfield) localization with respect to an explicit class of `$\mathcal Fin$-global weak equivalences.'
\subsection{A reminder on orthogonal spaces}
Orthogonal spaces are based on a certain topological analogue $L$ of the categories $I$ and $\mathcal I$ considered above. Explicitly, the objects of $L$\nomenclature[aL3]{$L$}{topologically enriched category of finite dimensional real inner product spaces and linear isometric embeddings} are the finite dimensional real inner product spaces $V$, and as a set $L(V,W)$ is given by the linear isometric embeddings $V\to W$. For $V=W$, $L(V,W)$ carries the topology of the orthogonal group $O(V)$, and in general $L(V,W)$ is topologized as a Stiefel manifold; since we can completely black box the topology, we omit the details and refer the curious reader to \cite[discussion before Definition~1.1.1]{schwede-book} instead.
\begin{defi}\label{defi:orthogonal-space}\index{orthogonal space|textbf}
An \emph{orthogonal space} is a topologically enriched functor $L\to\cat{Top}$. We write $\cat{$\bm L$-Top}\mathrel{:=}\FUN(L,\cat{Top})$.
\end{defi}
Schwede \cite[Definition~1.1.1]{schwede-book} denotes the above category by `\textit{spc}' and he constructs a global model structure on it that we will recall now.
\subsubsection{The strict level model structure}
\index{global homotopy theory|(}
As before, we begin by constructing a suitable level model structure.
\begin{defi}
A map $f\colon X\to Y$ of orthogonal spaces is called a \emph{strict level weak equivalence}\index{strict level weak equivalence|textbf} or \emph{strict level fibration} if $f(V)$ is a weak equivalence or fibration, respectively, in the equivariant model structure on $\cat{$\bm{O(V)}$-Top}$ with respect to all closed subgroups for every $V\in L$, i.e.~$f(V)^H$ is a weak homotopy equivalence or Serre fibration, respectively, for every closed subgroup $H\subset O(V)$.
\end{defi}
Schwede \cite[Definition~1.1.8]{schwede-book} calls the above `strong level equivalences' and `strong level fibrations,' respectively.
If $H$ is a compact Lie group, then an \emph{orthogonal $H$-representation} is an object $V\in L$ together with a continuous homomorphism $\rho\colon H\to O(V)$; equivalently, we can view this as a pair of a finite dimensional real inner product space $V$ together with a continuous $H$-action by linear isometries. Restricting along $\rho$, $X(V)$ is then naturally an $H$-space for every orthogonal space $X$, and $f(V)\colon X(V)\to Y(V)$ is $H$-equivariant for any map $f\colon X\to Y$ of orthogonal spaces. The above condition can then be rephrased as saying that $f(V)$ should be a weak equivalence or fibration, respectively, in $\cat{$\bm H$-Top}$ for every compact Lie group $H$ and every orthogonal $H$-representation $V$, see~\cite[Lemmas~1.2.7 and~1.2.8]{schwede-book}.
\begin{prop}
There is a unique model structure on $\cat{$\bm L$-Top}$ with weak equivalences the strict level weak equivalences and fibrations the strict level fibrations. It is topological and cofibrantly generated with generating cofibrations
\begin{equation*}
\{L(V,\blank)/H\times\del D^n\hookrightarrow L(V,\blank)/H\times D^n : V\in L, H\subset O(V)\text{ closed},n\ge0\}.
\end{equation*}
\begin{proof}
This is \cite[Proposition~1.2.10]{schwede-book} and the discussion after it.
\end{proof}
\end{prop}
\subsubsection{Global weak equivalences}
The global weak equivalences of orthogonal spaces are slightly intricate to define due to some pointset topological issues. Intuitively speaking, however, they should again be created by `evaluating at $\mathbb R^\infty$,' analogously to the approach for $I$- and $\mathcal I$-simplicial sets:
\begin{constr}
We write $\mathcal L$ (the `universal compact Lie group')\nomenclature[aL4]{$\mathcal L$}{`universal compact Lie group,' topological monoid of linear isometric embeddings $\mathbb R^\infty\to\mathbb R^\infty$}\index{universal compact Lie group|textbf} for the topological monoid of linear isometric embeddings $\mathbb R^\infty\to\mathbb R^\infty$ under composition; here the scalar product on $\mathbb R^\infty=\mathbb R^{(\omega)}$ (the vector space of functions $\omega\to\mathbb R$ vanishing almost everywhere) is so that the canonical basis consisting of the characteristic functions of elements of $\omega$ is orthonormal. The topology on $\mathcal L$ is given as a subspace of the mapping space.
Let $X$ be an orthogonal space. Then \cite[Construction~3.2]{schwede-orbi} describes how $X$ yields a space $X(\mathbb R^\infty)$ with a continuous $\mathcal L$-action. Explicitly, we define
\begin{equation*}
X(\mathbb R^\infty)\mathrel{:=} \colim_{V\subset\mathbb R^\infty\text{ finite dimensional}} X(V),
\end{equation*}
where the structure maps of the colimit system are induced via $X$ from the inclusions. A continuous $\mathcal L$-action is given as follows: if $x$ is contained in the image of $X(V)\to X(\mathbb R^\infty)$, then $u.x$ is the image of $x$ under the composition
\begin{equation*}
X(V)\xrightarrow{X(u|_V\colon V\to u(V))} X(u(V))\to X(\mathbb R^\infty)
\end{equation*}
for any $u\in\mathcal L$, where the right hand map is the structure map of the colimit.
By functoriality of colimits this yields a functor $\ev_{\mathbb R^\infty}\colon\cat{$\bm L$-Top}\to\cat{$\bm{\mathcal L}$-Top}$.
\end{constr}
Unlike for simplicial sets, filtered (and even sequential) colimits of topological spaces do not preserve weak equivalences in general, which already suggests that $\ev_{\mathbb R^\infty}$ is not homotopically meaningful. \cite[Definition~1.1.2]{schwede-book} avoids this problem by defining the weak equivalences in terms of a `homotopy extension lifting property' instead. We will take a different approach following \cite{schwede-orbi} here:
Namely, non-equivariantly we could solve this issue by replacing the above colimit by a homotopy colimit, but this of course does not retain equivariant information: surely, whether $f\colon X\to Y$ is a global weak equivalence should not only depend on $f(\mathbb R^\infty)$ as a map of ordinary topological spaces, but it should take into account the $\mathcal L$-action and in particular suitable actions of all compact Lie groups. Concretely, see~\cite[Definitions~1.3, 1.4, and~1.6]{schwede-orbi}:
\begin{defi}
A compact subgroup $H\subset\mathcal L$ is called \emph{universal}\index{universal subgroup!for L@for $\mathcal L$|textbf} if it admits the structure of a Lie group and the tautological $H$-action on $\mathbb R^\infty$ makes the latter into a complete $H$-universe,\index{complete H-universe@complete $H$-universe} i.e.~any finite dimensional orthogonal $H$-representation embeds $H$-equivariantly linearly isometrically into $\mathbb R^\infty$.
A map $f\colon X\to Y$ of $\mathcal L$-spaces is a \emph{global weak equivalence}\index{global weak equivalence!in L1-Top@in $\cat{$\bm{\mathcal L}$-Top}$|textbf} if $f^H\colon X^H\to Y^H$ is a weak homotopy equivalence for every universal $H\subset\mathcal L$.
\end{defi}
\begin{rk}
Analogously to the situation for $\mathcal M$, any compact Lie group is isomorphic to a universal subgroup of $\mathcal L$, and any two such embeddings are conjugate \cite[Proposition~1.5]{schwede-orbi}.
\end{rk}
One way to calculate sequential homotopy colimits is by replacing the diagram in question by a sequence of closed embeddings. The same strategy works \emph{mutatis mutandis} in our situation:
\begin{defi}\label{defi:closed-orth-space}
An orthogonal space $X$ is called \emph{closed}\index{orthogonal space!closed|textbf} if for every map $\phi\colon V\to W$ in $L$ the induced map $X(\phi)\colon X(V)\to X(W)$ is a closed embedding.
\end{defi}
\begin{ex}\label{ex:cofibrant-closed}
Any orthogonal space that is cofibrant in the strict level model structure is closed, see~\cite[Proposition~1.2.11-(iii)]{schwede-book}.
\end{ex}
\begin{defi}
Let $f\colon X\to Y$ be a map of orthogonal spaces and let
\begin{equation*}
\begin{tikzcd}
\hat X\arrow[d,"\hat f"']\arrow[r, "\sim"] & X\arrow[d, "f"]\\
\hat Y\arrow[r, "\sim"'] & Y
\end{tikzcd}
\end{equation*}
be a commutative diagram such that the horizontal maps are strict level weak equivalences and $\hat X,\hat Y$ are closed. Then $f$ is called a \emph{global weak equivalence}\index{global weak equivalence!in L2-Top@in $\cat{$\bm L$-Top}$|textbf} if $\hat f(\mathbb R^\infty)$ is a global weak equivalence of $\mathcal L$-spaces.
\end{defi}
Note that we can always find such a square by just taking functorial cofibrant replacements in the strict level model structure. Moreover, \cite[Proposition~3.5]{schwede-orbi} together with \cite[Proposition~1.1.9-(i)]{schwede-book} shows that the above is independent of the choice of replacement and equivalent to Schwede's original definition.
We can now finally introduce the \emph{global model structure} on $\cat{$\bm L$-Top}$ \cite[Theorem~1.2.21]{schwede-book}:
\begin{thm}\label{thm:L-global-model-structure}\index{global model structure!on L-Top@on $\cat{$\bm L$-Top}$|textbf}\index{orthogonal space!global model structure|seeonly{global model structure, on $\cat{$\bm L$-Top}$}}
There is a unique model structure on $\cat{$\bm L$-Top}$ with the same cofibrations as the strict level model structure and with the global weak equivalences as weak equivalences. This model structure is topological, proper, and cofibrantly generated with generating cofibrations
\begin{equation*}
\{L(V,\blank)/H\times\del D^n\hookrightarrow L(V,\blank)/H\times D^n : V\in L, H\subset O(V)\text{ closed}, n\ge 0\}.
\end{equation*}
Moreover, an orthogonal space $X$ is fibrant if and only if it is static in the sense that $X(\phi)\colon X(V)\to X(W)$ is an $H$-equivariant weak equivalence for every compact Lie group $H$ and every $H$-equivariant linear isometric embedding $\phi\colon V\to W$ of faithful finite dimensional orthogonal $H$-representations.\qed
\end{thm}
\begin{rk}
Again, there is also a \emph{positive global model structure}\index{positive global model structure!on L-Top@on $\cat{$\bm L$-Top}$}\index{orthogonal space!positive global model structure|seeonly{positive global model structure, on $\cat{$\bm L$-Top}$}} where one restricts the generating cofibrations by demanding in addition that $V\not=0$, see~\cite[Proposition~1.2.23]{schwede-book}.
\end{rk}
By design, our models of global homotopy theory considered in the previous sections only see equivariant information with respect to finite groups, so we should not hope for them to be equivalent to the above model category. Instead, we consider the following coarser notion of weak equivalence:
\begin{defi}\label{defi:Fin-global-we}
A map of $\mathcal L$-spaces is called a \emph{$\mathcal Fin$-global weak equivalence}\index{Fin-global weak equivalence@$\mathcal Fin$-global weak equivalence|textbf}\index{global weak equivalence!in L2-Top@in $\cat{$\bm L$-Top}$|seealso{$\mathcal Fin$-global weak equivalence}} if it restricts to an $H$-equivariant weak equivalence for each \emph{finite} universal $H\subset\mathcal L$.
A map $f$ of orthogonal spaces is called a \emph{$\mathcal Fin$-global weak equivalence} if $\hat f(\mathbb R^\infty)$ is a $\mathcal Fin$-global weak equivalence in $\cat{$\bm{\mathcal L}$-Top}$ for some (hence any) replacement of $f$ up to strict level weak equivalence by a map $\hat f$ between closed orthogonal spaces.
\end{defi}
\begin{rk}\label{rk:Fin-global-model-structure}
As remarked without proof in \cite[Remark~3.11]{schwede-orbi}, there is a version for the global model structure on $\cat{$\bm L$-Top}$ which only sees representations of groups belonging to a given \emph{global family} $\mathcal F$, i.e.~a collection of compact Lie groups closed under isomorphisms and subquotients. For $\mathcal F=\mathcal Fin$ the family of finite groups this precisely recovers the above $\mathcal Fin$-global weak equivalences.
\end{rk}
\subsection{$\bm I$-spaces} The intermediate step in our comparison will be a global model structure on $\cat{$\bm I$-Top}$. For this the following terminology will be useful, see e.g.~\cite[Definition~A.28 and discussion after it]{schwede-book}:
\begin{defi}
Let $\mathscr C$ be a category enriched and tensored over $\cat{Top}$. Then a map $f\colon A\to B$ in $\mathscr C$ is an \emph{h-cofibration}\index{h-cofibration|textbf} if the natural map $\big(A\times[0,1]\big)\amalg_A B\to B\times[0,1]$ from the mapping cyclinder of $f$ admits a retraction.
\end{defi}
\begin{ex}
For $\mathscr C=\cat{Top}$ with the usual enrichment, the h-cofibrations are the classical (Hurewicz) cofibrations.
\end{ex}
\begin{lemma}
Let $G$ be a finite group and let
\begin{equation}\label{diag:pushout-along-h-cof}
\begin{tikzcd}
A\arrow[d]\arrow[r, "i"] & B\arrow[d]\\
C\arrow[r] & D
\end{tikzcd}
\end{equation}
be a pushout in $\cat{$\bm G$-Top}$ such that $i$ is an h-cofibration. Then $\Sing\colon\cat{$\bm G$-Top}\to\cat{$\bm G$-SSet}$ sends $(\ref{diag:pushout-along-h-cof})$ to a homotopy pushout.
\begin{proof}
Let $H\subset G$ be any subgroup. The functor $(\blank)^H$ preserves pushouts along closed embeddings by \cite[Proposition~B.1-(i)]{schwede-book} and it clearly preserves tensors; it easily follows that it preserves h-cofibrations, also cf.~\cite[Corollary~A.30-(ii)]{schwede-k-theory}. We conclude that $i^H$ is a Hurewicz cofibration and that the square on the left in
\begin{equation*}
\begin{tikzcd}
A^H\arrow[d]\arrow[r, "i^H"] & B^H\arrow[d]\\
C^H\arrow[r] & D^H
\end{tikzcd}
\qquad\qquad
\begin{tikzcd}[column sep=large]
(\Sing A)^H\arrow[d]\arrow[r,"(\Sing i)^H"] & (\Sing B)^H\arrow[d]\\
(\Sing C)^H\arrow[r]&(\Sing D)^H
\end{tikzcd}
\end{equation*}
is a pushout in $\cat{Top}$. It is well-known that $\Sing$ sends pushouts along Hurewicz cofibrations to homotopy pushouts (which also follows from Lemma~\ref{lemma:U-pushout-preserve-reflect} applied to $|\blank|$, using that the counit $\epsilon\colon |\Sing X|\to X$ is a weak homotopy equivalence for any $X\in\cat{Top}$), and as it moreover commutes with fixed points, we conclude that the square on the right is a homotopy pushout in $\cat{SSet}$.
The claim now follows as homotopy pushouts in $\cat{$\bm G$-SSet}$ can be checked on fixed points by Proposition~\ref{prop:dk-equivariant-addendum}.
\end{proof}
\end{lemma}
\begin{cor}\label{cor:h-cofibration-I-po}
Let $i\colon A\to B$ be any h-cofibration of $I$-spaces. Then $\Sing\colon\cat{$\bm I$-Top}\to\cat{$\bm I$-SSet}$ sends pushouts along $i$ to homotopy pushouts in the global model structure.
\begin{proof}
The evaluation functors $\ev_A\colon\cat{$\bm I$-Top}\to\cat{$\bm{\Sigma_A}$-Top}$ for $A\in I$ are cocontinuous and preserve tensors, so h-cofibrations of $I$-spaces are in particular levelwise h-cofibrations by \cite[Corollary~A.30-(ii)]{schwede-book}.
Thus, if we are given a pushout in $\cat{$\bm I$-Top}$ along $i$, then applying the previous lemma levelwise shows that its image under $\Sing$ is a levelwise homotopy pushout in the sense that evaluating at any $A\in I$ yields a homotopy pushout in $\cat{$\bm{\Sigma_A}$-SSet}$. As one easily concludes from the existence of the injective global model structure on $\cat{$\bm I$-SSet}$, such a levelwise homotopy pushout is in particular a homotopy pushout, finishing the proof.
\end{proof}
\end{cor}
\begin{prop}\label{prop:I-Top-global}
There is a unique model structure on $\cat{$\bm I$-Top}$ in which a map $f$ is a weak equivalence or fibration if and only if $\Sing f$ is a weak equivalence or fibration, respectively, in the global model structure on $\cat{$\bm I$-SSet}$. We call this the \emph{global model structure}\index{global model structure!on I-Top@on $\cat{$\bm I$-Top}$|textbf} and its weak equivalences the \emph{global weak equivalences}\index{global weak equivalence|in I-Top@in $\cat{$\bm I$-Top}$|textbf}. An $I$-space $X$ is fibrant in this model structure if and only if it is static\index{static} in the sense that $X(i)^H\colon X(A)^H\to X(B)^H$ is a weak homotopy equivalence for every finite group $H$ and every $H$-equivariant injection $i\colon A\to B$ of finite faithful $H$-sets.
The global model structure is topological, left proper, and cofibrantly generated with generating cofibrations
\begin{equation*}
I(A,\blank)/H\times\del D^n\hookrightarrow
I(A,\blank)/H\times D^n,
\end{equation*}
where $A$ and $H$ are as above and $n\ge 0$. Moreover, pushouts along h-cofibrations are homotopy pushouts.
Finally, the adjunction
\begin{equation}\label{eq:real-sing-I}
|\blank|\colon\cat{$\bm I$-SSet}_{\textup{global}}
\rightleftarrows\cat{$\bm I$-Top}_{\textup{global}} :\!\Sing
\end{equation}
is a Quillen equivalence in which both adjoints are homotopical.
\begin{proof}
The adjunction $|\blank|\colon\cat{SSet}\rightleftarrows\cat{Top} :\!\Sing$ is a Quillen equivalence in which both adjoints are homotopical. In particular, the unit $\eta\colon X\to\Sing|X|$ is a weak homotopy equivalence for any space $X$. As both adjoints commute with finite limits, it follows further that for any group $K$ acting on $X$, the unit $\eta_X$ induces weak homotopy equivalences on $K'$-fixed points for all finite $K'\subset K$, so that the unit of $(\ref{eq:real-sing-I})$ is a strict level weak equivalence for all $X\in\cat{$\bm I$-SSet}$.
Let us now construct the model structure, for which we will verify the conditions of Crans' Transfer Criterion (Proposition~\ref{prop:transfer-criterion}): we let $I_\Delta$ be the usual set of generating cofibrations, and we pick any set $J_\Delta$ of generating acyclic cofibrations of $\cat{$\bm I$-SSet}_{\textup{global}}$. As each cofibration is in particular a levelwise injection, we conclude that $|I_\Delta|$ and $|J_\Delta|$ consist of closed embeddings. As $\Sing$ preserves transfinite compositions along closed embeddings (see e.g.~\cite[Proposition~2.4.2]{hovey}), the local presentability of $\cat{$\bm I$-SSet}$ implies that $|I_\Delta|$ and $|J_\Delta|$ permit the small object argument.
It remains to show that any relative $|J_\Delta|$-cell complex is sent to a global weak equivalence under $\Sing$. Again using that $\Sing$ preserves transfinite compositions along closed embeddings, it suffices to show that pushouts of maps in $|J_\Delta|$ are global weak equivalences.
It is clear that the maps in $|I_\Delta|$ are h-cofibrations, hence so is the geometric realization of any cofibration by \cite[Corollary~A.30-(i)]{schwede-book} and in particular any map in $|J_\Delta|$. As the unit is a levelwise global (in fact, even strict level) weak equivalence, all maps in $|J_\Delta|$ are sent to global weak equivalences under $\Sing$, and hence so is any pushout of a map in $|J_\Delta|$ by Corollary~\ref{cor:h-cofibration-I-po}. This completes the proof of the existence of the model structure. Moreover, again using that the unit is a levelwise global weak equivalence and in addition that the right adjoint creates weak equivalences by definition, we immediately conclude that $(\ref{eq:real-sing-I})$ is a Quillen equivalence and that also $|\blank|$ is homotopical.
By Corollary~\ref{cor:h-cofibration-I-po}, $\Sing$ sends pushouts along h-cofibrations (hence in particular along cofibrations of the above model structure) to homotopy pushouts. Thus, Lemma~\ref{lemma:U-pushout-preserve-reflect} implies that $\cat{$\bm I$-Top}$ is left proper and that $\Sing$ creates homotopy pushouts. In particular, pushouts along h-cofibrations are homotopy pushouts.
As a functor category, $\cat{$\bm I$-Top}$ is enriched, tensored, and cotensored over $\cat{Top}$ in the obvious way. Restricting along the adjunction $|\blank|\colon\cat{SSet}\rightleftarrows\cat{Top}:\!\Sing$ therefore makes it into a category enriched, tensored, and cotensored over $\cat{SSet}$. With respect to this, $(\ref{eq:real-sing-I})$ is naturally a simplicial adjunction, so $\cat{$\bm I$-Top}$ is a simplicial model category by Lemma~\ref{lemma:transferred-properties}-$(\ref{item:tpr-simplicial})$. To see that it is topological, we then simply observe that it suffices to verify the Pushout Product Axiom for generating cofibrations and generating (acyclic) cofibrations, and that the usual ones for $\cat{Top}$ agree up to conjugation by isomorphisms with the images under geometric realization of the standard generating (acyclic) cofibrations of $\cat{SSet}$.
\end{proof}
\end{prop}
\begin{rk}\label{rk:I-Top-closed}
In analogy to Definition~\ref{defi:closed-orth-space}, let us call an $I$-space $X$ \emph{closed} if all structure maps $X(A)\to X(B)$ are closed embeddings. As $\Sing$ preserves sequential colimits along closed embeddings, it easily follows that $\Sing$ commutes with $\ev_\omega$ on the subcategory of \emph{closed} $I$-spaces. In particular, if $f\colon X\to Y$ is a map of closed $I$-spaces that is a \emph{weak equivalence at infinity}\index{weak equivalence at infinity} in the sense that $f(\omega)^H$ is a weak homotopy equivalence for each universal $H\subset\mathcal M$, then $f$ is already a $G$-global weak equivalence.
\end{rk}
\subsection{Proof of the comparison}
\index{orthogonal space!vs I-spaces@vs.~$I$-spaces|(}
In order to compare $I$-spaces to orthogonal spaces, we will relate their indexing categories:
\begin{constr}\label{constr:I-to-L}
We define a functor $\mathbb R^\bullet: I\to L$ as follows: a finite set $A$ is sent to the real vector space $\mathbb R^A$ of maps $A\to\mathbb R$, where the inner product on $\mathbb R^A$ is the unique one such that the characteristic functions of elements of $A$ form an orthonormal basis.
If $f\colon A\to B$ is an injection of finite sets, then $\mathbb R^f\colon\mathbb R^A\to\mathbb R^B$ is defined to be the unique $\mathbb R$-linear map sending the characteristic function of $a\in A$ to the characteristic function of $f(a)\in B$. It is clear that $\mathbb R^f$ is isometric, and that this makes $\mathbb R^\bullet$ into a well-defined functor.
\end{constr}
Restricting along $\mathbb R^\bullet$ yields a forgetful functor $\cat{$\bm L$-Top}\to\cat{$\bm I$-Top}$. By topologically enriched left Kan extension, this admits a topological left adjoint $L\times_I\blank$ satisfying $L\times_I I(A,\blank) = L(A,\blank)$ for any finite set $A$; the unit is then given on such corepresentables by $\mathbb R^\bullet\colon I(A,\blank)\to L(\mathbb R^A,\mathbb R^\bullet)= \forget L(\mathbb R^A,\blank)$.
\begin{rk}\label{rk:set-universe-vs-universe}
Let $H$ be a finite group and let $U$ be a countable set containing infinitely many free $H$-orbits (for example if $U$ is a complete $H$-set universe). Then the $\mathbb R$-linearization $\mathbb R^{(U)}$ contains infinitely many copies of the regular representation $\mathbb R^H$, so it is a complete $H$-universe in the usual sense. In particular, if $H\subset\mathcal M$ is universal, then the induced $H$-action on $\mathbb R^\infty=\mathbb R^{(\omega)}$ makes the latter into a complete $H$-universe.
\end{rk}
\begin{prop}\label{prop:L-vs-I-unit-corep}
The map $\mathbb R^\bullet\colon I(A,\blank)/H\times X\to {L}(\mathbb R^A,\mathbb R^\bullet)/H\times X$ is a global weak equivalence in $\cat{$\bm I$-Top}$ for any finite group $H$, any finite faithful $H$-set $A$, and any CW-complex $X$.
\begin{proof}
Taking products with $X$ obviously preserves strict level weak equivalences, and it preserves global acyclic cofibrations as $\cat{$\bm I$-Top}$ is topological. Thus, $\blank\times X$ is fully homotopical, and it suffices that $\mathbb R^\bullet\colon I(A,\blank)/H\to L(\mathbb R^A,\mathbb R^\bullet)/H$ is a global weak equivalence. For this we consider the commutative diagram
\begin{equation}\label{diag:semistable-replacements}
\begin{tikzcd}
& {|\mathcal I(A,\blank)|/H}\\
{I(A,\blank)/H}\arrow[ur, bend left=10pt]\arrow[r]\arrow[dr, bend right=10pt] & {\big(|\mathcal I(A,\blank)|\times {L}(\mathbb R^A,\mathbb R^\bullet)\big)/H}\arrow[u, "\pr"']\arrow[d, "\pr"]\\
& {{L}(\mathbb R^A,\mathbb R^\bullet)/H}
\end{tikzcd}
\end{equation}
where the maps from left to right are induced by the inclusion $I(A,\blank)\hookrightarrow|\mathcal I(A,\blank)|$ and by $\mathbb R^\bullet\colon I(A,\blank)\to {L}(\mathbb R^A,\mathbb R^\bullet)$. We begin by showing that the vertical arrows on the right are global weak equivalences.
It is clear that the $I$-space $|\mathcal I(A,\blank)|$ is closed, and so is ${L}(\mathbb R^A,\mathbb R^\bullet)$ by Example~\ref{ex:cofibrant-closed}. From this we can conclude by \cite[Proposition~B.13-(iii)]{schwede-book} that all the $I$-spaces on the right of $(\ref{diag:semistable-replacements})$ are closed, so that it suffices that the vertical maps are weak equivalences at infinity (see Remark~\ref{rk:I-Top-closed}).
Let $K\subset\mathcal M$ be a universal subgroup. Obviously, $|\mathcal I(A,\blank)|(\omega)\cong|E\Inj(A,\omega)|$ is $\mathcal A\ell\ell$-cofibrant in $\cat{$\bm{(K\times H)}$-Top}$, and so is ${L}(\mathbb R^A,\mathbb R^\bullet)(\omega)$ by \cite[Proposition~1.1.19-(ii)]{schwede-book} together with \cite[Proposition~A.5-(ii)]{schwede-orbi}. We claim that both are classifying spaces for $\mathcal G_{K,H}$ in the sense that their $T$-fixed points for $T\subset K\times H$ are contractible if $T\in\mathcal G_{K,H}$, and empty otherwise. Indeed, for ${L}(\mathbb R^A,\mathbb R^\bullet)(\omega)$ this is an instance of \cite[Proposition~1.1.26-(i)]{schwede-book}. On the other hand, if $K'\subset K$, $\phi\colon K'\to H$, then $|\mathcal I(A,\blank)|(\omega)^\phi\cong |E(\Inj(A,\omega)^\phi|$ is weakly contractible by Example~\ref{ex:G-globally-contractible}. Finally, $H$ acts freely on $\Inj(A,\omega)$, so $\mathcal I(A,\omega)^T$ has no vertices when $T\subset K\times H$ is not contained in $\mathcal G_{K,H}$, whence $|\mathcal I(A,\blank)|(\omega)^T$ has to be empty.
Thus, the projections
\begin{equation*}
|\mathcal I(A,\blank)|(\omega)\gets\big(|\mathcal I(A,\blank)|\times {L}(\mathbb R^A,\mathbb R^\bullet)\big)(\omega)\to{L}(\mathbb R^A,\mathbb R^\bullet)(\omega)
\end{equation*}
are weak equivalences of cofibrant objects in $\cat{$\bm{(K\times H)}$-Top}$ with respect to the $\mathcal A\ell\ell$-model structure. Applying Ken Brown's Lemma to the Quillen adjunction
\begin{equation*}
p_!\colon\cat{$\bm{(K\times H)}$-Top}_{\textup{$\mathcal A\ell\ell$}}\rightleftarrows\cat{$\bm{K}$-Top}_{\mathcal A\ell\ell} :\!p^*
\end{equation*}
where $p\colon K\times H\to K$ denotes the projection, therefore shows that after evaluation at $\omega$ the vertical maps in $(\ref{diag:semistable-replacements})$ become $K$-weak equivalences. The claim follows by letting $K$ vary.
To finish the proof, we observe now that $I(A,\blank)/H\to|\mathcal I(A,\blank)|/H$ agrees up to conjugation by isomorphisms with the image of
\begin{equation}\label{eq:semistabilization-simplicial}
I(A,\blank)/H\hookrightarrow\mathcal I(A,\blank)/H
\end{equation}
under geometric realization. The proposition follows as $(\ref{eq:semistabilization-simplicial})$ is a global weak equivalence by Theorem~\ref{thm:strict-global-I-model-structure} and since geometric realization is fully homotopical by Proposition~\ref{prop:I-Top-global}.
\end{proof}
\end{prop}
We define a monoid homomorphism $i\colon\mathcal M\to\mathcal L$ by sending $f\colon\omega\to\omega$ to $\mathbb R^f\colon\mathbb R^\infty\to\mathbb R^\infty$, i.e.~the unique linear isometry sending the $i$-th standard basis vector to the $f(i)$-th one.
\begin{lemma}
Let $X$ be an $\mathcal L$-space. Then $\Sing(i^*X)\in\cat{$\bm{\mathcal M}$-SSet}$ is semistable.
\begin{proof}
Let $H\subset\mathcal M$ be universal, and let $u\in\mathcal M$ centralize $H$. It suffices to show that $i(u).\blank\colon X\to X$ is an $i(H)$-equivariant weak equivalence.
Indeed, Remark~\ref{rk:set-universe-vs-universe} implies that $i(H)$ is a universal subgroup of $\mathcal L$, so $\mathcal L^{i(H)}$ is contractible by \cite[Proposition~1.1.26-(i)]{schwede-book} together with \cite[Proposition~A.10]{schwede-orbi}. In particular, there exists a path $\gamma\colon[0,1]\to \mathcal L^{i(H)}$ connecting $i(u)$ to the identity. The composition
\begin{equation*}
[0,1]\times X\xrightarrow{\gamma\times X} \mathcal L\times X\xrightarrow{\textup{action}} X
\end{equation*}
is then an $i(H)$-equivariant homotopy from $i(u).\blank$ to the identity, so $i(u).\blank$ is in particular an $i(H)$-equivariant (weak) homotopy equivalence.
\end{proof}
\end{lemma}
\begin{prop}\label{prop:Fin-global-we}\index{Fin-global weak equivalence@$\mathcal Fin$-global weak equivalence}
Let $f\colon X\to Y$ be a map of orthogonal spaces. Then the following are equivalent:
\begin{enumerate}
\item $f$ is a $\mathcal Fin$-global weak equivalence (Definition~\ref{defi:Fin-global-we}).
\item $\forget f$ is a global weak equivalence of $I$-spaces.
\end{enumerate}
Moreover, if $X$ and $Y$ are closed, then also the following statements are equivalent to the above:
\begin{enumerate}
\item[(3)] $\Sing\big(i^*f(\mathbb R^\infty)\big)$ is a universal weak equivalence of $\mathcal M$-simplicial sets.
\item[(4)] $\Sing\big(i^*f(\mathbb R^\infty)\big)$ is a global weak equivalence of $\mathcal M$-simplicial sets.
\end{enumerate}
\begin{proof}
Let us first assume that $X$ and $Y$ are closed; we will show that all of the above statements are equivalent.
For the equivalence $(1)\Leftrightarrow(3)$ we observe once more that $i\colon\mathcal M\to\mathcal L$ sends universal subgroups to universal subgroups. As any two abstractly isomorphic universal subgroups of $\mathcal L$ are conjugate \cite[Proposition~1.5]{schwede-orbi}, the claim now follows from the definitions.
The equivalence $(3)\Leftrightarrow(4)$ is immediate from the previous lemma. Finally, for $(2)\Leftrightarrow(4)$ it suffices by Remark~\ref{rk:I-Top-closed} to show that the diagram
\begin{equation*}
\begin{tikzcd}
\cat{$\bm L$-Top}\arrow[r, "\ev_{\mathbb R^\infty}"]\arrow[d, "\forget"'] & \cat{$\bm{\mathcal L}$-Top}\arrow[d, "i^*"]\\
\cat{$\bm I$-Top}\arrow[r, "\ev_\omega"'] & \cat{$\bm{\mathcal M}$-Top}
\end{tikzcd}
\end{equation*}
commutes up to natural isomorphism. This follows easily from the definitions once we observe that the map of posets $\{A\subset\omega\text{ finite}\}\to\{V\subset\mathbb R^\infty\text{ finite dimensional}\}$ sending $A$ to the image of the canonical map $\mathbb R^A\to\mathbb R^\infty$ is cofinal: if $v\in\mathbb R^\infty$, then there is only a finite set $S(v)$ of $i\in\omega$ such that $\pr_i(v)\not=0$ for the projection $\pr_i\colon\mathbb R^\infty\to\mathbb R$ to the $i$-th summand. Thus, if $V\subset\mathbb R^\infty$ is any finite dimensional subspace, and $v_1,\dots,v_n$ is a basis, then $S\mathrel{:=}S(v_1)\cup\cdots\cup S(v_n)$ is finite, and obviously $V$ is contained in the image of $\mathbb R^S\to\mathbb R^\infty$.
Now assume $X$ and $Y$ are not necessarily closed. If $j\colon A\to B$ is a strict level weak equivalence of orthogonal spaces, then $j$ is in particular a $\mathcal Fin$-global weak equivalence; moreover $\Sing(\forget(j))$ is obviously a strict level weak equivalence of $I$-simplicial sets, hence in particular a global weak equivalence. Thus, $(1)\Leftrightarrow(2)$ follows from the above special case together with $2$-out-of-$3$.
\end{proof}
\end{prop}
\begin{thm}\label{thm:L-vs-I}\index{orthogonal space!vs I-spaces@vs.~$I$-spaces|textbf}
The topologically enriched adjunction
\begin{equation}\label{eq:L-vs-I}
L\times_I\blank\colon \cat{$\bm I$-Top}\rightleftarrows\cat{$\bm L$-Top}:\forget
\end{equation}
is a Quillen adjunction. The induced adjunction of associated quasi-categories is a right Bousfield localization with respect to the $\mathcal Fin$-global weak equivalences.
\end{thm}
Non-equivariantly, this comparison was proven by Lind \cite[Theorem~6.2]{lind}.
\begin{proof}
Let us first show that $(\ref{eq:L-vs-I})$ is a Quillen adjunction. As it is a topologically enriched adjunction of topological model categories, it in particular becomes a simplicial adjunction of simplicial model categories when we restrict along the usual adjunction $\cat{SSet}\rightleftarrows\cat{Top}$. It therefore suffices (Proposition~\ref{prop:cofibrations-fibrant-qa}) to observe that $L\times_I\blank$ obviously preserves generating cofibrations, and that the forgetful functor preserves fibrant objects by the characterizations given in Theorem~\ref{thm:L-global-model-structure} and Proposition~\ref{prop:I-Top-global}, respectively.
By the previous proposition, the forgetful functor is homotopical and it precisely inverts the $\mathcal Fin$-global weak equivalences. It therefore only remains to show that the unit $X\to\forget(L\times_I X)$ is a global weak equivalence for any cofibrant $X\in\cat{$\bm I$-Top}$.
This will again be a cell induction argument (although we cannot literally apply Corollary~\ref{cor:saturated-trafo}): if $X$ is the source or target of one of the standard generating cofibrations of $\cat{$\bm I$-Top}$, then the claim is an instance of Proposition~\ref{prop:L-vs-I-unit-corep}. On the other hand, any pushout
\begin{equation*}
\begin{tikzcd}
I(A,\blank)/H\times\del D^{n-1}\arrow[r, hook]\arrow[d] & I(A,\blank)/H\times D^n\arrow[d]\\
X\arrow[r] & Y
\end{tikzcd}
\end{equation*}
along a generating cofibration is a homotopy pushout by left properness, and its image under $\forget\circ (L\times_I\blank)$ is a pushout along an h-cofibration by \cite[Corollary~A.30-(ii)]{schwede-book}, hence again a homotopy pushout by Proposition~\ref{prop:I-Top-global} above. We conclude that $\eta_Y$ is a global weak equivalence if $\eta_X$ is.
Using that transfinite compositions \emph{of closed embeddings} in $\cat{$\bm I$-Top}$ are homotopical (as they are preserved by $\Sing$), we see that $\eta_Z$ is a global weak equivalence for any cell complex $Z$ in the generating cofibrations. The claim follows as any cofibrant object of $\cat{$\bm I$-Top}$ is a retract of such a cell complex.
\end{proof}
\begin{cor}
The functor $\Sing\circ\forget:\cat{$\bm L$-Top}\to\cat{$\bm I$-SSet}$ preserves global fibrations and global weak equivalences, and it induces a quasi-localization $\cat{$\bm L$-Top}^\infty\to \cat{$\bm I$-SSet}^\infty$ at the $\mathcal Fin$-global weak equivalences.\qed
\end{cor}
\begin{rk}
It is not hard to show that $(\ref{eq:L-vs-I})$ is a Quillen adjunction with respect to the $\mathcal Fin$-global model structure on $\cat{$\bm L$-Top}$ mentioned in Remark~\ref{rk:Fin-global-model-structure}, hence a Quillen equivalence by the above theorem.
On the other hand, one can easily adapt the above proof to transfer the global model structure from $\cat{$\bm I$-Top}$ to $\cat{$\bm L$-Top}$. This way one obtains a model structure with the $\mathcal Fin$-global weak equivalences as weak equivalences, but slightly fewer cofibrations than the model structure sketched by Schwede.
\end{rk}
\index{global homotopy theory|)}
\index{orthogonal space!vs I-spaces@vs.~$I$-spaces|)} | 47,317 |
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Mediarock necesitaba un nuevo diseño de logotipo y lanzó un concurso de diseño en 99designs.
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You are here: Home >> Articles & Tutorials >> Nikon Coolpix S6000 Digital Camera - Reviews by Real People!
By DonalKeegan on Jul 27, 2011 |Product Reviews
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With a massive 14.2 mega-pixels image resolution, the Coolpix S6000 can produce high quality photographs that you can print as large as 20"x30".
Also featuring a 7x wide-angle Optical VR Zoom Nikkor ED lens, and a 2.7" Clear Color screen, you'll be able to take many fantastic photos, and view them very easily and clearly.
Not only are you able to take photographs with this Nikon, you can also record 720p HD videos and view them utilizing the HDMI output. Other technical features include a Smart Portrait System, Advanced Flash Control and EXPEED Image Processing.
To lessen the unwanted effects which come from camera shake, this digital camera has got 4 unique features for optimum image stabilization. These are the lens shift VR stabilization, high ISO capabilities, Motion Detection and advanced flash control.
With a sleek, contemporary, yet ultra slim design, this Nikon S6000 doesn't just look amazing, but also is filled with both user friendly settings, and other options enabling you more creative control over your photograph. Nikon's Smart Portrait System consists of face-detection technology to get the best exposure and focus for portraits or groups of people, Smile Timer to automatically take the picture as soon as it detects that your subject is smiling, a Blink Warning to inform you of the fact that your subject might have blinked, and even in-camera Red-Eye correction. In addition, one more handy feature is the Subject Tracking system which could come in handy especially when taking pictures of young children who could be moving about a lot.
Pros:
The majority of people were happy with the image quality, and that, at the end of the day, is the most important thing.
A noted strength were the camera's zoom capabilities.
There were also very good feedback on the video quality. Some other individuals enjoyed the fact that it comes with an easy to access button on the back of the camera which makes it quick and simple to change between camera and video mode.
Consumers enjoy the size and weight of the camera, combined with it being easy and straightforward to operate along with its general speed of the start up.
The battery life was considered very good, with one person stating that they took over 200 photographs with no warning to suggest that the battery would have to be charged up again.
Cons
One frequent complaint that people had was the fact that this camera doesn't include a dedicated battery charger, but just a cable to allow you to connect the entire camera up to a power supply to charge.
With regards to the camera build quality, some people mentioned that the covers for battery, memory card etc were lightweight and prone to break after extended use.
There were also unfavorable comments on the less than impressive Nikon software package that accompanies the camera.
Despite some negatives, the Coolpix S6000 is an attractive looking ultra-slim compact digital camera, crammed full of Nikon's imaginative and intelligent features. It's got enough going for it that it will not leave you feeling disappointed.
For more information go to our Nikon Coolpix S6000 Digital Camera page.
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WILLIAMSTOWN, Mass., November 28, 2016—A recent graduate of Williams College’s Center for Development Economics (CDE), Diala Issam Al Masri ’15, is one of three newly named Rhodes Scholars from Syria, Jordan, Lebanon, or Palestine. This is the first year that the Rhodes Trust is choosing scholars from that region, the result of a new partnership with the Saïd Foundation that was announced in June. Al Masri is the first CDE graduate to be awarded a Rhodes Scholarship.
The Rhodes Scholarship is among the oldest and most prestigious academic awards for college graduates. It provides two or three years of study at the University of Oxford in England. Al Masri was selected from among hundreds of applicants from the new Middle East region.
Al Masri heard the news she’d been selected for the Rhodes while celebrating Thanksgiving at the home of Williams professor Magnus Bernhardsson. “There were a few moments of disbelief, then pure, overwhelming joy,” she said. “I feel deeply grateful to everyone who got me here, and at the same time, a responsibility to live up to the trust placed in me.”
A native of Lebanon, Al Masri received a Fulbright Scholarship to further her studies at Williams, taking a path less traveled to the CDE. Typically, CDE students are already working as economists in their home countries, but Al Masri came to the CDE directly after completing her bachelor’s degree in political science/international affairs with a minor in economics in 2014 at the Lebanese American University in Beirut.
The CDE offers an intensive, one academic year master’s degree program designed for economists from low- and middle-income countries, to provide them with a thorough understanding of the development process, emphasizing analytical techniques helpful to policymakers. It is one of two master’s degree programs offered at Williams (the other is a program in the history of art).
Since receiving her master’s in policy economics from the CDE, Al Masri has remained at Williams, working first as a teaching assistant at the CDE for the Class of 2016, and now in the Economics Department as a research assistant for professors Peter Pedroni and Peter Montiel. Her research encompasses determinants of capital flows and the long-term relationship of finance and development.
Pedroni said that while working with Al Masri, it became quite clear that her talent and ability for research outpaced what one would normally expect from a research assistant.
“Diala is at this stage of her career without doubt and by no small margin the strongest student to come from the CDE program,” Pedroni noted in his letter to the Rhodes selection committee. “In terms of aptitude and technical ability, I would say Diala is far more comparable to the best of the Williams undergraduates who double major in economics and mathematics, many of whom go on to study among the top five economics Ph.D. programs in the U.S. But most importantly, Diala has developed a sense of research maturity that exceeds even the most talented of our top undergraduates.”
Al Masri says her plan is to pursue graduate work in economics at Oxford to further her studies of econometric techniques and continue in a career as an economist and an academic specializing in low- and middle-income countries and fragile states. She points to two specific programs at Oxford—the Institute for Global Economic Development (OxIGED) and the Institute for Economic Modeling (INET)—as programs she would be involved with in her studies.
Al Masri’s educational journey started in her hometown in a rural area in the Lebanese mountains. She recalls hearing many times how being a woman in a male-dominated society would impose limits on her, but instead she says it empowered her. As an undergraduate student in Lebanon, Al Masri was very involved with leadership and humanitarian organizations. She was one of six Lebanese students chosen to be part of the Middle East Partnership Initiative Student Leaders program, where she interacted with leaders from the Arab region.
She participated as a leader in the Global Classrooms UNA-USA Model United Nations and served as a trainer for Mercy Corps, a global humanitarian aid agency that helps people survive after conflict, crisis, or natural disaster. She helped implement a pilot project on women’s empowerment with the United Nations Development Programme and worked with the World Youth Alliance’s Middle East operation to support Syrian refugees who were arriving in her hometown.
Pedroni said that working with Al Masri is a pleasure not only because of her technical ability, but also because of her deep sense of intellectual devotion that she conveys for the subjects that she pursues. “I have no doubt that Diala will make good use of her intellectual talents and abilities for social contributions,” he said.
Al Masri is the 38th Williams student to be named a Rhodes Scholar since the program began in 1902. The most recent previous Williams recipient was Brian McGrail ’14. More on the scholarships and profiles of the three Middle East region recipients can be found on the Rhodes Trust | 130,707 |
TITLE: Confused on determining equilibria of a differential equation
QUESTION [0 upvotes]: So my final for calculus III is coming up, and I'm running through all the material from the past quarter. I am confused on one thing, though I'm sure it'll click as soon as someone helps me out, it's late and I'm confusing myself.
Question is: To find all equilibria as a function of the constant $a$
The given diffeq is $ x' = ax-x^2$
first of all, I'm assuming that the independent variable is time, right? $x' = dx/dt$?
Regardless, here's where I am:
$x' = g(x) = ax-x^2$
equilibria @ $g(x) = ax-x^2 = 0$
So when it says to "determine all equilibria as a function of the constant $a$", does this just mean that my answer can be: "Equillibria @ $a$ when $a=x$"?
Then, I can make a plot of $g(x)$ vs $x$, the curve of that plot will start at $(0,0)$, climb to a maximum, turn around, then hit $g(x) = 0$ at $(a, 0)$. And this is my Phase Plane, correct?
So by the stability criterion, this is a stable equilibrium, because $g'(x) < 0$.
So, a phase plane is a plot in which the x-intercepts tell me the equilibria points, and the derivative of $g(x)$ tells me the stability of each equilibia. Then what is the other plot called that shows families of solutions and identifies equilibria by horizontal asymptotes? In the case of the problem presented here, would I make one of those? By plotting $a$ vs $t$?
REPLY [0 votes]: Apparently there are 3 cases: $a<0$, $a=0$ and $a>0$.
You have to decide the stable point in each of the case and then describe the appearance of the solutions for different initial conditions....
I think that is what you are asked to do. | 64,950 |
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\begin{document}
\title{Central limit theorems for stationary random fields under weak dependence with application to ambit and mixed moving average fields \\
}
\author{Imma Valentina Curato, Robert Stelzer and Bennet Str{\"o}h\footnote{Ulm University, Institute of Mathematical Finance, Helmholtzstra\ss e 18, 89069 Ulm, Germany. Emails: [email protected], [email protected], [email protected]. }}
\maketitle
\textwidth=160mm \textheight=225mm \parindent=8mm \frenchspacing
\vspace{3mm}
\begin{abstract}
We obtain central limit theorems for stationary random fields employing a novel measure of dependence called $\theta$-lex weak dependence. We show that this dependence notion is more general than strong mixing, i.e., it applies to a broader class of models. Moreover, we discuss hereditary properties for $\theta$-lex and $\eta$-weak dependence and illustrate the possible applications of the weak dependence notions to the study of the asymptotic properties of stationary random fields.
Our general results apply to mixed moving average fields (MMAF in short) and ambit fields. We show general conditions such that MMAF and ambit fields, with the volatility field being an MMAF or a $p$-dependent random field, are weakly dependent. For all the models mentioned above, we give a complete characterization of their weak dependence coefficients and sufficient conditions to obtain the asymptotic normality of their sample moments. Finally, we give explicit computations of the weak dependence coefficients of MSTOU processes and analyze under which conditions the developed asymptotic theory applies to CARMA fields.
\end{abstract}
\noindent
{\it MSC 2020: primary 60G10, 60G57, 60G60,62M40;
secondary 62F10, 62M30.}
\\
\\
{\it Keywords: stationary random fields, weak dependence, central limit theorems, mixed moving average fields, CARMA fields, ambit fields.}
\section{Introduction}
Many modern statistical applications consider the modeling of phenomena evolving in time and/or space with either a countable or uncountable index set.
To this end, we can employ random fields on $\Z^m$ or $\R^m$ which are defined, for example, as solutions of recurrence equations, e.g., in \cite{DT2007}, or stochastic partial differential equations \cite{B2019b, CK2015, P2018}. Noticeable examples of the latter come from the class of ambit and mixed moving average fields.
The mixed moving average fields, MMAF in short, are defined as
\begin{equation}
\label{begin}
X_t=\int_{S} \int_{\R^m} f(A,t-s) \, \Lambda(dA,ds),\,\,\, t \in \R^m,
\end{equation}
where $S$ is a Polish space, $f$ a deterministic function called kernel and $\Lambda$ a L\'evy basis.
The above model encompasses Gaussian and non-Gaussian random fields by choosing the L\'evy basis $\Lambda$.
Ambit fields are defined by considering an additional multiplicative random function in the integrand (\ref{begin}) called volatility or intermittency field. However, an ambit field is typically defined without the variable $A$ in its kernel function. We refer the reader to \cite{BBV2018} for a comprehensive introduction to ambit fields which provide a rich class of spatio-temporal models on $\R \times \R^m$. Overall, MMAF and ambit fields are used in many applications throughout different disciplines, like geophysics \cite{H2002}, brain imaging \cite{JN2013}, physics \cite{BS2004}, biology \cite{BJJS2007,BNS2007}, economics and finance \cite{BBV2010,BBV2015,BM2017,NV2015,NV2017}.
The generality and flexibility of these models motivate an in-depth analysis of their asymptotic properties.
Central limit theorems for moving average fields, a sub-class of MMAF, are discussed in \cite{B2019} where the author shows the asymptotic normality of its sample mean and autocovariance. However, we do not pursue this approach because it is not directly applicable to the study of higher-order sample moments. Under strong mixing conditions, several central limit theorems for stationary random fields are available in the literature, see \cite{B1982, C1991, D1998, IL89, M1999, N1988}. In \cite{IL89}, we also find some of the first results related to the analysis of central limit theorems for autocovariance functions. Note that, in general, caution must be used when applying some of the classical strong mixing dependence notions to random fields. We refer to \cite{B1989} and \cite[Chapter 29]{B2007} for a thorough investigation on this point. The above said, for an MMAF on $\R$, i.e., a mixed moving average process, several difficulties already arise in showing that it is strongly mixing, see \cite{CS2018}. Usually, strong mixing is established by using a Markovian representation and showing geometric ergodicity of it. In turn, this often requires smoothness conditions on the driving random noise, and it is well-known that even autoregressive processes of order one are not strongly mixing when the distribution of the noise is not sufficiently regular, see \cite{A1984}. For $m \geq 2$, a Gaussian MMAF on $\R^m$ satisfying the conditions of \cite[Theorem $7$, pg. 73]{R1985} is strong mixing. However, for general driving L\'evy bases, no results in the literature can be found regarding the strong mixing of MMAF. Sharp central limit results for stationary random fields can also be obtained under the dependence notion of association (see \cite{BS2007, N1980} for a comprehensive introduction on this topic). However, in this case, central limit theorems for MMAF hold just under restrictive conditions on the kernel function $f$ in (\ref{begin}), see, e.g., \cite[Theorem $3.27$]{BS2007}. Moreover, association is inherited only under monotone functions, restricting the possible extension of its related asymptotic theory.
Concerning purely temporal ambit fields, i.e., L\'evy semistationary processes, in \cite{BCP2011,BPS2014,BHP2018} the authors obtain infill asymptotic results for this class of processes, that is, under the assumption that the number of observations in a given interval approaches to infinity. For ambit fields on $\R \times \R^m$ with $m \geq 1$ where $\Lambda$ is of Gaussian type and the volatility field is independent of $\Lambda$, the asymptotic behavior of the lattice power variation of the field is studied in \cite{P2014}. We notice that in the literature, there are no asymptotic results for partial sums of ambit fields when the number of observations approaches infinity without infill asymptotics.
We are interested in studying the asymptotic behavior of the partial sums (and of higher-order sample moments) of MMAF and ambit fields in general, i.e., without imposing regularity conditions on the driving L\'evy basis $\Lambda$ apart from moment conditions. To do so, we apply the $\eta$-weak dependence as defined in \cite{DMT2008} and a new notion of dependence called $\theta$-lex weak dependence. Although all the examples of our theory come from the model classes mentioned above, we want to emphasize that we present general central limit theorem results that apply to different stationary random fields.
To introduce the notion of $\theta$-lex weak dependence, let us start with a brief digression into the notions of $\eta$ and $\theta$-weak dependence for stochastic processes defined in \cite{DL1999} and \cite{DD2003}, respectively. $\eta$-weak dependence is typically associated with the study of non-causal processes, whereas $\theta$-weak dependence is related to the analysis of the causal ones. Central limit theorems for $\theta$-weakly dependent processes hold under weaker conditions compared to results for $\eta$-weakly dependent processes (different demands on the decay rate of the $\eta$ and $\theta$-coefficients as determined in \cite[Theorem 2.2]{DW2007} and \cite[Theorem 2]{DD2003}). We have that the definition of $\eta$ and $\theta$-weak dependence can be easily extended to the random field case by following \cite[Remark 2.1]{DDLLLP2008}. However, just for $\eta$-weakly dependent random fields, asymptotics of the partial sums of stationary random fields have been so far analyzed in \cite{DMT2008}. We aim to determine a central limit theorem that improves the results obtained in \cite{DMT2008}. We achieve this by defining the notion of $\theta$-lex-weak dependence, which is a modification of the original definition of $\theta$-weak dependence. We show that for $\theta$-lex-weakly dependent random fields, the sufficient conditions of a very powerful central limit theorem from Dedecker \cite{D1998} hold. Moreover, we obtain hereditary properties for $\theta$-lex and $\eta$-weakly dependent random fields, which allow us to easily extend the asymptotic results under weak dependence to the study of higher-order sample moments.
We then investigate the relationship between $\theta$-lex weak dependence and strong mixing. We prove that for random fields defined on $\Z^m$, $\theta$-lex weak dependence is a more general notion of dependence than $\alpha_{\infty,1}$-mixing as defined in Dedecker \cite{D1998}, i.e., it applies to a broader class of models. In the case of processes, we also show that $\theta$-lex weak dependence is a more general notion of dependence than $\alpha$-mixing as defined in \cite{B2007}, see for more details Section \ref{sec2-2}.
Let us now look at the class of MMAF. We distinguish in our theory between influenced and non-influenced MMAF, see Definition \ref{definition:influenced}. Influenced MMAF represent a possible extension of causal mixed moving average processes, see \cite[Section 3.2]{CS2018}, to random fields.
Hence, we show that influenced MMAF are $\theta$-lex-weakly dependent and that non-influenced MMAF are $\eta$-weakly dependent with coefficients computable in terms of the kernel function $f$ and the characteristic quadruplet of the L\'evy basis $\Lambda$. From this, we notice that in the case of influenced MMAF, the conditions ensuring asymptotic normality of the partial sums of $X$ are weaker-- in terms of the decay rate of the weak dependence coefficients-- in comparison with the one obtained for non-influenced MMAF. We then observe a parallel between our results and the one obtained for causal and non-causal mixed moving average processes \cite{CS2018}. Moreover, we exploit the hereditary properties of $\eta$ as well as $\theta$-lex-weak dependence and obtain conditions for the sample moments of order $p$ with $p \geq 1$ to be asymptotic normally distributed. Finally, we give explicit computations for mixed spatio-temporal Ornstein-Uhlenbeck processes \cite{NV2017}, also called MSTOU processes, and L\'evy-driven CARMA fields \cite{BM2017,P2018}. In particular, our calculations in the case of the MSTOU processes show that it is possible to determine the asymptotic normality of the generalized method of moments estimator, GMM in short, proposed in \cite{NV2017}.
At last, we apply our theory to ambit fields. We assume that the volatility field is an MMAF or a $p$-dependent random field which is independent of the L\'evy basis $\Lambda$. Under these assumptions, we show that homogeneous and stationary ambit fields are $\theta$-lex-weakly dependent and give sufficient conditions on the $\theta$-lex-coefficients to ensure asymptotic normality of the sample moments.
The paper is structured as follows. In Section \ref{sec2}, we introduce $\eta$-weak dependence and the novel $\theta$-lex-weak dependence. In Section \ref{sec2-2} we state central limit theorems for $\theta$-lex weakly dependent random fields in an ergodic, non-ergodic, and multivariate setting. Additionally, we analyze the relationship between $\theta$-lex weak dependence and strong mixing and provide some insight into possible functional extensions of the central limit theorem.
In Section \ref{sec3}, we discuss the weak dependence properties of MMAF. We first give a comprehensive introduction to L\'evy bases and its related integration theory, leading to the formal definition of an MMAF.
We discuss conditions on MMAF to be $\theta$-lex or $\eta$-weakly dependent and their related sample moment asymptotics.
In Section \ref{sec3-7}, we apply the developed theory to MSTOU processes and give explicit conditions assuring their sample moments' asymptotic normality under a Gamma distributed mean reversion parameter. We conclude Section \ref{sec3} by giving conditions under which the developed asymptotic theory can be applied to L\'evy-driven CARMA fields. In Section \ref{sec4}, we discuss weak dependence properties and related limit theorems for ambit fields. Section \ref{sec5} contains the detailed proofs of most of the results presented in the paper.
\section{Weak dependence and central limit theorems}
\label{sec2}
\subsection{Notations}
$\N_0$ denotes the set of non-negative integers, $\N$ the set of positive integers, and $\R^+$ the set of the non-negative real numbers. For $x\in \R^d$, $x^(j)$ denotes the $j$-th coordinate of $x$, $\norm{x}$ its Euclidean norm and we define $|x|=\norm{x}_\infty=\max_{j=1,\ldots,d}|x^{(j)}|$. For $d, k \geq 1$ and $F:\R^d\ra \R^k$, we define $\norm{F}_\infty=\sup_{t\in\R^d}\norm{F(t)}$. Let $A \in M_{n \times d}(\R)$, $A^{\prime}$ denote the transpose of the matrix $A$.
In the following Lipschitz continuous is understood to mean globally Lipschitz. For $u,n\in\N$, let $\mathcal{G}_u^*$ be the class of bounded functions from $(\R^n)^u$ to $\R$ and $\mathcal{G}_u$ be the class of bounded, Lipschitz continuous functions from $(\R^n)^u$ to $\R$ with respect to the distance $\sum_{i=1}^{u}\norm{x_i-y_i}$, where $x,y\in(\R^n)^u$. For $G\in\mathcal{G}_u$ we define
\begin{gather*}
Lip(G)=\sup_{x\neq y}\frac{|G(x)-G(y)|}{\norm{x_1-y_1}+\ldots+\norm{x_u-y_u}}.
\end{gather*}
We assume that all random elements in this paper are defined on a given complete probability space $(\Omega, \mathcal{F},P)$. $\norm{\cdot}_p$ for $p>0$ denotes throughout the $L^p$-norm of a random element. For a random field $X=(X_t)_{t\in\R^m}$ and a finite set $\Gamma\subset\R^m$ with $\Gamma=(i_1,\ldots,i_u)$, we define the vector $X_\Gamma=(X_{i_1},\ldots,X_{i_u})$. $A\subset B$ denotes a not necessarily proper subset $A$ of a set $B$, $|B|$ denotes the cardinality of $B$ and $dist(A,B)=\inf_{i\in A, j\in B} \norm{i-j}_\infty$ indicates the distance of two sets $A,B\subset\R^m$.
Hereafter, we often use the lexicographic order on $\R^m$. For distinct elements $y=(y_1,\ldots,y_m)\in\R^m$ and $z=(z_1,\ldots,z_m)\in\R^m$ we say $y<_{lex}z$ if and only if $y_1<z_1$ or $y_p<z_p$ for some $p\in\{2,\ldots,m\}$ and $y_q=z_q$ for $q=1,\ldots,p-1$. Moreover, $y\leq_{lex}z$ if $y<_{lex}z$ or $y=z$ holds. Finally, let us define the sets $V_t=\{s\in\R^m:s<_{lex}t\}\cup\{t\}$ and $V_t^h=V_t\cap \{s\in\R^m: \norm{t-s}_\infty\geq h \}$ for $h>0$. The definitions of the sets $V_t$ and $V_t^h$ are also used when referring to the lexicographic order on $\Z^m$.
\subsection{Weak dependence properties}
\label{sec2-1}
\begin{Definition}\label{thetaweaklydependent}
Let $X=(X_t)_{t\in\R^m}$ be an $\R^n$-valued random field. Then, $X$ is called $\theta$-lex-weakly dependent if
\begin{gather*}
\theta(h)=\sup_{u\in\N}\theta_{u}(h) \underset{h\ra\infty}{\longrightarrow} 0,
\end{gather*}
where
\begin{align*}
\theta_{u}(h)=\sup\bigg\{&\frac{|Cov(F(X_{\Gamma}),G(X_j))|}{\norm{F}_{\infty}Lip(G)},
F\in\mathcal{G}^*_u,G\in\mathcal{G}_1,j\in\R^m, \Gamma \subset V_j^h, |\Gamma|= u \bigg\}.
\end{align*}
We call $(\theta(h))_{h\in\R^+}$ the $\theta$-lex-coefficients.
\end{Definition}
\begin{Remark}
Our definition of $\theta$-lex-weak dependence differs from the $\theta$-weak dependence definition for random fields given in \cite[Remark 2.1]{DDLLLP2008}. In fact, instead of considering the covariance of two arbitrary finite-dimensional samples $X_\Gamma$ and $X_{\tilde{\Gamma}}$, for $\Gamma, \tilde{\Gamma} \subset \R^m$, we control the covariance of a finite-dimensional sample $X_\Gamma$ and an arbitrary one point sample $X_j$. Secondly, assuming that all points in the sampling set $\Gamma$ are lexicographically smaller than $j$, we provide order in the sampling scheme.
For $m=1$, i.e., in the process case, our definition of $\theta$-lex-weak dependence coincides with the definition of $\theta$-weak dependence given in \cite{DD2003}.
\end{Remark}
\begin{Definition}[{\cite[Definition 2.2 and Remark 2.1]{DDLLLP2008}}]\label{etaweaklydependent}\
Let $X=(X_t)_{t\in\R^m}$ be an $\R^n$-valued random field. Then, $X$ is called $\eta$-weakly dependent if
\begin{gather*}
\eta(h)=\sup_{u,v\in\N}\eta_{ u,v}(h) \underset{h\ra\infty}{\longrightarrow} 0,
\end{gather*}
where
\begin{align*}
\eta_{ u,v}(h) =\sup\bigg\{&\frac{|Cov(F(X_{\Gamma}),G(X_{\tilde\Gamma}))|}{u\norm{G}_{\infty}Lip(F)+v\norm{F}_{\infty}Lip(G)},
\\ &F\in\mathcal{G}_u,G\in\mathcal{G}_v, \Gamma,\tilde\Gamma\subset\R^m,|\Gamma|= u, |\tilde\Gamma|= v,dist(\Gamma,\tilde\Gamma)\geq h \bigg\}.
\end{align*}
We call $(\eta(h))_{h\in\R^+}$ the $\eta$-coefficients.
\end{Definition}
Let $(X_t)_{t\in\R^m}$ be $\theta$-lex- or $\eta$-weakly dependent and $h:\R^n\ra\R^k$ be an arbitrary Lipschitz function, then the field $(h(X_t))_{t\in\R^m}$ is also $\theta$-lex- or $\eta$-weakly dependent. The latter can be readily checked based on Definition \ref{thetaweaklydependent} and \ref{etaweaklydependent}.
In the next proposition, we give conditions for hereditary properties of functions that are only locally Lipschitz continuous. The proof of the result below is analogous to Proposition 3.2 \cite{CS2018}.
\begin{Proposition}\label{proposition:mmathetahereditary}
Let $X=(X_t)_{t\in\R^m}$ be an $\R^n$-valued stationary random field and assume that there exists a constant $C>0$ such that $E[\norm{X_0}^p]\leq C$, for $p>1$. Let $h:\R^n\ra\R^k$ be a function such that $h(0)=0, h(x)=(h_1(x),\ldots,h_k(x))$ and
\begin{gather*}
\norm{h(x)-h(y)}\leq c\norm{x-y}(1+\norm{x}^{a-1}+\norm{y}^{a-1}),
\end{gather*}
for $x,y\in\R^n$, $c>0$ and $1\leq a<p$. Define $Y=(Y_t)_{t\in\R^m}$ by $Y_t=h(X_t)$. If $X$ is $\theta$-lex or $\eta$-weakly dependent, then $Y$ is $\theta$-lex or $\eta$-weakly dependent respectively with coefficients
\begin{gather*}
\theta_Y(h)\leq\mathcal{C}\theta_X(h)^{\frac{p-a}{p-1}} \,\,\text{ or } \,\,
\eta_Y(h)\leq\mathcal{C}\eta_X(h)^{\frac{p-a}{p-1}}
\end{gather*}
for all $h>0$ and a constant $\mathcal{C}$ independent of $h$.
\end{Proposition}
\subsection{Mixing properties}
\label{sec2-2}
Let $\mathcal{M}$ and $\mathcal{V}$ be two sub-$\sigma$-algebras of $\mathcal{F}$. We define the strong mixing coefficient of Rosenblatt \cite{R1956}
\begin{align*}
\alpha(\mathcal{M},\mathcal{V})=\sup\{|P(M)P(V)-P(M\cap V)|,M\in\mathcal{M},V\in\mathcal{V}\}.
\end{align*}
A random field $X=(X_t)_{t\in\Z^m}$ is said to be $\alpha_{u,v}$-mixing for $u,v\in\N\cup\{\infty\}$ if
\begin{align*}
\alpha_{u,v}(h)=\sup\left\{ \alpha(\sigma(X_\Gamma),\sigma(X_{\tilde{\Gamma}})),\, \Gamma,\tilde{\Gamma}\subset\R^m,|\Gamma|\leq u, |\tilde{\Gamma}|\leq v, dist(\Gamma,\tilde{\Gamma})\geq h \right\}
\end{align*}
converges to zero as $h\rightarrow \infty$. Moreover, for $m=1$, the stochastic process $X$ is said to be $\alpha$-mixing if
\begin{align*}
\alpha(h)=\alpha\left( \sigma(\{X_s, s\leq0\}),\sigma(\{X_s, s\geq h\})\right)
\end{align*}
converges to zero as $h\rightarrow \infty$. Clearly, we have that $\alpha(h) \leq \alpha_{\infty,\infty}(h)$ for $m=1$. For a comprehensive discussion on the coefficients $\alpha_{u,v}(h)$, $\alpha(h)$ and their relation to other strong mixing coefficients we refer to \cite{B2007, BCS2020, D1998}.\\
The following proposition establishes a relationship between the $\theta$-lex-coefficients and the mixing coefficients $\alpha(h)$ and $\alpha_{\infty,1}(h)$.
\begin{Proposition}\label{proposition:thetavsalpha}
Let $X=(X_t)_{t\in\Z^m}$ be a stationary real-valued random field such that $E[\norm{X_0}^q]<\infty$ for some $q>1$. Then, for all $h\in\R^+$ and $m=1$, we have that
$$
\theta(h)\leq 2^{\frac{2q-1}{q}} (\alpha(h))^{\frac{q-1}{q}} \norm{X_0}_q.
$$
Moreover, for all $h\in\R^+$ and $m \geq 1$
$$\theta(h)\leq 2^{\frac{2q-1}{q}} (\alpha_{\infty,1}(h))^{\frac{q-1}{q}} \norm{X_0}_q.$$
\end{Proposition}
\begin{proof}
See Section \ref{sec5-1}.
\end{proof}
\begin{Remark}
If a stationary real-valued random field admits all finite moments, then Proposition \ref{proposition:thetavsalpha} ensures that $\theta(h)\leq C \alpha(h)$ for $m=1$ and $\theta(h)\leq C \alpha_{\infty,1}(h)$ for all $m\geq1$, where $C>0$ is a constant independent of $h$.
\end{Remark}
\begin{Proposition}\label{proposition:AR1notmixingbuttheta}
Let $(\xi_k)_{k\in\Z}$ be a sequence of independent random variables such that $\xi_k\sim Ber(\frac{1}{2})$ for all $k\in\Z$. Then, the stationary process $X_t=\sum_{j=0}^\infty 2^{-j-1}\xi_{t-j}$ for $t\in\Z$ is
$\theta$-lex-weakly dependent but neither $\alpha$-mixing nor $\alpha_{\infty,1}$-mixing.
\end{Proposition}
\begin{proof}
See Section \ref{sec5-1}.
\end{proof}
\begin{Remark}
Proposition \ref{proposition:thetavsalpha} shows that every stationary $\alpha$-mixing stochastic process and every stationary $\alpha_{\infty,1}$-mixing random field with finite $q$-th moments are $\theta$-lex-weakly dependent. On the other hand, Proposition \ref{proposition:AR1notmixingbuttheta} shows that there exists a stationary $\theta$-lex-weakly dependent process with finite variance, that is neither $\alpha$-mixing nor $\alpha_{\infty,1}$-mixing. Therefore, $\theta$-lex-weak dependence is a more general notion of dependence than $\alpha$- and $\alpha_{\infty,1}$-mixing.
\end{Remark}
We cite for completeness the results available in the literature regarding the relationship between $\eta$-weak dependence and $\alpha$-mixing. For integer-valued processes, the authors show in \cite[Proposition 1]{DFL2012} that $\eta$-weak dependence implies $\alpha$-mixing. Moreover, Andrews \cite{A1984} gives an example of an $\eta$-weakly dependent process that is not $\alpha$-mixing.
\subsection{Central limit theorems for $\theta$-lex-weakly dependent random fields}
\label{sec2-3}
In the theory of stochastic processes, one of the typical ways to prove central limit type results is to approximate the process of interest by a sequence of martingale differences. This approach was first introduced by Gordin \cite{G1969}. However, the latter does not apply to high-dimensional random fields as successfully as to processes. This unpleasant circumstance has been known among researchers for almost 40 years, as Bolthausen \cite{B1982} noted that martingale approximation appears a difficult concept to generalize to dimensions greater or equal than two.
For stationary random fields $X=(X_t)_{t \in \Z^m}$, Dedecker derived a central limit result in \cite{D1998} under the projective criterion
\begin{gather}\label{equation:l1criteria}
\sum_{k\in V_0^1}|X_kE[X_0|\mathcal{F}_{\Gamma(k)}]|\in L^1, \,\,\, \textrm{for $\mathcal{F}_{\Gamma(k)}=\sigma(X_k :k\in V_0^{|k|})$}.
\end{gather}
This condition is weaker than a martingale-type assumption and provides optimal results for mixing random fields.
Early use of such a projective criterion can be found in the central limit theorems for stationary processes derived in \cite{DM2002, PU2006}.
We show in this section that (\ref{equation:l1criteria}) is also fulfilled by appropriate $\theta$-lex-weakly dependent random fields.
In the following, by stationarity we mean stationarity in the strict sense. Let $\Gamma$ be a subset of $\Z^m$. We define $\partial\Gamma=\{i\in\Gamma: \exists j \notin \Gamma: \norm{i-j}_\infty=1\}$. Let $(D_n)_{n\in\N}$ be a sequence of finite subsets of $\Z^m$ such that
\begin{gather*}
\lim_{n\rightarrow\infty} |D_n|=\infty \text{ and }\lim_{n\rightarrow\infty} \frac{|\partial D_n|}{|D_n|}=0.
\end{gather*}
\begin{Theorem}\label{theorem:clt}
Let $X=(X_t)_{t\in\Z^m}$ be a stationary centered real-valued random field such that $E[|X_t|^{2+\delta}]<\infty$ for some $\delta>0$.
Additionally, assume that $\theta(h)\in \mathcal{O}(h^{-\alpha})$ with $\alpha>m(1+\frac{1}{\delta})$. Define
\begin{gather*}
\sigma^2=\sum_{k\in\Z^m}E[X_0X_k|\mathcal{I}],
\end{gather*}
where $\mathcal{I}$ is the $\sigma$-algebra of shift invariant sets as defined in \cite[Section 2]{D1998} (see \cite[Chapter 1]{K1985} for an introduction to ergodic theory). Then, $\sigma^2$ is finite, non-negative and
\begin{gather}\label{eq:clt}
\frac{1}{|D_n|^{\frac{1}{2}}}\sum_{j\in D_n}X_j\underset{n\ra\infty}{\xrightarrow{\makebox[2em][c]{d}}}\varepsilon \sigma,
\end{gather}
where $\varepsilon$ is a standard normally distributed random variable which is independent of $\sigma^2$.
\end{Theorem}
\begin{proof}
See Section \ref{sec5-1}.
\end{proof}
The multivariate extension of Theorem \ref{theorem:clt}, appearing below, is obtained by applying the Cram\'er-Wold device and noting that linear functions are Lipschitz.
\begin{Corollary}\label{corollary:ergodicclt}
Let $X=(X_t)_{t\in\Z^m}$ be a stationary ergodic centered $\R^n$-valued random field such that $E[\norm{X_t}^{2+\delta}]<\infty$ for some $\delta>0$.
Additionally, let us assume that $\theta(h)\in \mathcal{O}(h^{-\alpha})$ with $\alpha>m(1+\frac{1}{\delta})$. Then,
\begin{gather*}
\Sigma=\sum_{k\in\Z^m}E[X_0X_k']
\end{gather*}
is finite, positive definite and
\begin{gather*}
\frac{1}{|D_n|^{\frac{1}{2}}}\sum_{j\in D_n}X_j\underset{n\ra\infty}{\xrightarrow{\makebox[2em][c]{d}}}N(0,\Sigma),
\end{gather*}
where $N(0,\Sigma)$ denotes the multivariate normal distribution with mean $0$ and covariance matrix $\Sigma$.
\end{Corollary}
\begin{Remark}
Using Proposition \ref{proposition:thetavsalpha} the condition $\theta(h)\in \mathcal{O}(h^{-\alpha})$ with $\alpha>m(1+\frac{1}{\delta})$ in Theorem \ref{theorem:clt} can be replaced by $\alpha_{\infty,1}(h)\in \mathcal{O}(h^{-\beta})$ or $\alpha(h) \in \mathcal{O}(h^{-\beta})$ with $\beta>m(1+\frac{2}{\delta})$.\\
For real valued stochastic processes, the latter condition represents the sharpest one available for $\alpha$-mixing coefficients (see \cite[Theorem 25.70]{B2007}).
\end{Remark}
\begin{Remark}
It is natural to ask for conditions ensuring a functional extension of Theorem \ref{theorem:clt}. As a matter of fact, results of this kind are strongly related to the following $L^p$-projective criterion
\begin{gather}\label{equation:lpcriteria}
\sum_{k\in V_0}E[|X_kE[X_0|\mathcal{F}_{V_0^{|k|}}]|^p]<\infty,~ p\in[1,\infty],
\end{gather}
where $\mathcal{F}_{\Gamma}=\sigma(X_k,k\in\Gamma)$.\\
For $m=1$, Dedecker and Rio show in \cite[Theorem]{DR2000} that if (\ref{equation:lpcriteria}) holds for $p=1$, then a functional central limit theorem holds.\\
In the general case $m>1$, Dedecker proved in \cite[Theorem 1]{D2001} a functional central limit theorem if (\ref{equation:lpcriteria}) holds for $p>1$.\\
Since we can establish the connection between the $L^p$-projective criterion (\ref{equation:lpcriteria}) and the summability condition of the $\theta$-lex-coefficients of $X$ just for $p=1$, there is no functional extension of Theorem \ref{theorem:clt} readily obtainable, except for $m=1$ (see \cite[Remark 4.2]{CS2018}).
\end{Remark}
\section{Mixed moving average fields}
\label{sec3}
In this section we first introduce MMAF driven by a L\'evy basis. Then, we discuss weak dependence properties of such MMAF and derive sufficient conditions such that the asymptotic results of Section \ref{sec2-3} apply.
\subsection{Preliminaries}
\label{sec3-1}
Let $S$ denote a non-empty Polish space, $\mathcal{B}(S)$ the Borel $\sigma$-algebra on $S$, $\pi$ some probability measure on $(S,\mathcal{B}(S))$ and $\mathcal{B}_b(S \times \R^m)$ the bounded Borel sets of $S \times \R^m$.
\begin{Definition}
\label{basis}Consider a family $\Lambda=\{\Lambda(B), B \in \mathcal{B}_b(S\times\R^m) \}$ of $\R^d$-valued random variables. Then $\Lambda$ is called an $\R^d$-valued L\'evy basis or infinitely divisible independently scattered random measure on $S\times\R^m$ if
\begin{enumerate}[(i)]
\item the distribution of $\Lambda(B)$ is infinitely divisible (ID) for all $B\in \mathcal{B}_b(S\times\R^m)$,
\item for arbitrary $n\in \N$ and pairwise disjoint sets $B_1,\dots,B_n \in \mathcal{B}_b(S\times\R^m)$ the random variables $\Lambda(B_1),\ldots,\Lambda(B_n)$ are independent and
\item for any pairwise disjoint sets $B_1,B_2,\ldots \in \mathcal{B}_b(S\times\R^m)$ with $\bigcup_{n\in\N} B_n \in \mathcal{B}_b(S\times\R^m)$ we have, almost surely, $\Lambda (\bigcup_{n\in\N} B_n)= \sum_{n\in\N}\Lambda(B_n)$.
\end{enumerate}
\end{Definition}
In the following we will restrict ourselves to L\'evy bases which are homogeneous in space and time and factorisable, i.e. L\'evy bases with characteristic function
\begin{equation}\label{equation:fact}
\varphi_{\Lambda(B)}(u)=E\left[e^{\text{i}\langle u,\Lambda(B) \rangle}\right]=e^{\Phi(u)\Pi(B)}
\end{equation}
for all $u\in \R^d$ and $B\in\mathcal{B}_b(S\times\R^m)$, where $\Pi=\pi\times\lambda$ is the product measure of the probability measure $\pi$ on $S$ and the Lebesgue measure $\lambda$ on $\R^m$. Furthermore,
\begin{align}\label{equation:phi}
\Phi(u)=\text{i}\langle\gamma,u\rangle -\frac{1}{2} \langle u,\Sigma u\rangle +\int_{\R^d} \left(e^{\text{i}\langle u,x \rangle}-1-\text{i}\langle u,x\rangle \mathbb{1}_{[0,1]}(\norm{x})\right)\nu(dx)
\end{align}
is the cumulant transform of an ID distribution with characteristic triplet $(\gamma,\Sigma,\nu)$, where $\gamma \in \R^d$, $\Sigma\in M_{d\times d}(\R)$ is a symmetric positive-semidefinite matrix and $\nu$ is a L\'evy-measure on $\R^d$, i.e.
\begin{align*}
\nu(\{0\})=0 \quad \text{and}
\int_{\R^d}\left(1\wedge\norm{x}^2\right)\nu(dx)<\infty.
\end{align*}
The quadruplet $(\gamma, \Sigma,\nu,\pi)$ determines the distribution of the L\'evy basis completely and therefore it is called the characteristic quadruplet.
Following \cite{P2003}, it can be shown that a L\'evy basis has a L\'evy-It\^o decomposition.
\begin{Theorem}\label{theorem:levyitobasis}
Let $\{\Lambda(B), B\in \mathcal{B}_b(S\times\R^m)\}$ be an $\R^d$-valued L\'evy basis on $S\times\R^m$ with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$. Then, there exists a modification $\tilde{\Lambda}$ of $\Lambda$ which is also a L\'evy basis with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$ such that there exists an $\R^d$-valued L\'evy basis $\tilde{\Lambda}^G$ on $S\times \R^m$ with characteristic quadruplet $(0,\Sigma,0,\pi)$ and an independent Poisson random measure $\mu$ on $(\R^d\times S\times \R^m, \mathcal{B}(\R^d\times S\times \R^m))$ with intensity measure $\nu\times\pi\times\lambda$ such that
\begin{align}
\begin{split}\label{equation:levyito}
\tilde{\Lambda}(B)=\gamma(\pi\times\lambda)(B) + &\tilde{\Lambda}^G(B)+\int_{\norm{x}\leq1}\int_Bx(\mu(dx,dA,ds)-ds\pi(dA)\nu(dx))\\ &+\int_{\norm{x}>1}\int_B x \mu(dx,dA,ds)
\end{split}
\end{align}
for all $B\in \mathcal{B}_b(S\times\R^m)$.\\
If the L\'evy measure additionally fulfills $\int_{\norm{x}\leq1}\norm{x}\nu(dx)<\infty$, it holds that
\begin{align}\label{equation:levyitofinvar}
\tilde{\Lambda}(B)=\gamma_0(\pi\times\lambda)(B) +\tilde{\Lambda}^G(B)+\int_{\R^d}\int_B x \mu(dx,dA,ds)
\end{align}
for all $B\in \mathcal{B}_b(S\times\R^m)$ with
\begin{align}\label{equation:gammazero}
\gamma_0:=\gamma-\int_{\norm{x}\leq1}x\nu(dx).
\end{align}
Note that the integral with respect to $\mu$ exists $\omega$-wise as a Lebesgue integral.
\end{Theorem}
\begin{proof}
Analogous to \cite[Theorem 2.2]{BS2011}.
\end{proof}
We refer the reader to \cite[Section 2.1]{JS2003} for further details on the integration with respect to Poisson random measures. From now on we assume that any L\'evy basis has a decomposition (\ref{equation:levyito}).
Let us recall the following multivariate extension of \cite[Theorem 2.7]{RR1989}.
\begin{Theorem}\label{theorem:2}
Let $\Lambda=\{\Lambda(B), B\in \mathcal{B}_b(S\times\R^m)\}$ be an $\R^d$-valued L\'evy basis with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$, $f:S\times\R^m\rightarrow M_{n\times d}(\R)$ be a $\mathcal{B}(S\times\R^m)$-measurable function. Then $f$ is $\Lambda$-integrable in the sense of \cite{RR1989}, if and only if
\begin{gather}
\int_S\!\int_{\R^m}\!\! \Big\| f(A,s)\gamma\!+\!\! \!\int_{\R^d}\!\!\, f(A,s)x\left(\mathbb{1}_{[0,1]}(\|f(A,s)x\|)\!-\!\!\mathbb{1}_{[0,1]}(\|x\|)\right)\!\nu(dx)\Big\| ds \pi(dA)\!\!<\!\!\infty, \label{equation:intcond1} \\
\int_S\int_{\R^m}\|f(A,s)\Sigma f(A,s)'\|\ ds\pi(dA)<\infty \text{ and} \label{equation:intcond2}\\
\int_S\int_{\R^m}\int_{\R^d} \Big(1\wedge \|f(A,s)x\|^2 \Big) \nu(dx)ds\pi(dA)<\infty. \label{equation:intcond3}
\end{gather}
If $f$ is $\Lambda$-integrable, the distribution of the stochastic integral $\int_S\int_{\R^m}f(A,s)\Lambda(dA,ds)$ is ID with the characteristic triplet $(\gamma_{int},\Sigma_{int},\nu_{int})$ given by
\begin{gather*}
\gamma_{int}=\int_S\int_{\R^m} \Big(f(A,s)\gamma+\int_{\R^d}f(A,s)x\left(\mathbb{1}_{[0,1]}(\|f(A,s)x\|)-\mathbb{1}_{[0,1]}(\|x\|)\Big)\nu(dx)\right)ds \pi(dA), \\[-5pt]
\Sigma_{int}=\int_S\int_{\R^m}f(A,s)\Sigma f(A,s)'ds\pi(dA) \text{ and} \\
\nu_{int}(B)= \int_S\int_{\R^m}\int_{\R^d}\mathbb{1}_B(f(A,s)x)\nu(dx) ds\pi(dA)
\end{gather*}
for all Borel sets $B\subset \R^n\backslash \{0\}$.
\end{Theorem}
\begin{proof}
Analogous to \cite[Proposition 2.3]{BS2011}.
\end{proof}
Implicitly, we always assume that $\Sigma_{int}$ or $\nu_{int}$ are different from zero throughout the paper to rule out the deterministic case.
For $m=1$ it is known that the L\'evy-It\^o decomposition simplifies if the underlying L\'evy process $L_t=\Lambda(S\times (0,t])$ is of finite variation (if and only if $\Sigma=0$ and $\int_{|x|\leq1}|x|\nu(dx)<\infty$).
Extending this one-dimensional notion, we speak of the finite variation case whenever $\Sigma=0$ and $\int_{\norm{x}\leq1}\norm{x}\nu(dx)<\infty$.
\begin{Corollary}\label{corollary:1}
Let $\Lambda=\{\Lambda(B), B\in \mathcal{B}_b(S\times\R^m)\}$ be an $\R^d$-valued L\'evy basis with characteristic quadruplet $(\gamma,0,\nu,\pi)$ satisfying $\int_{\|x\|\leq1}\|x\|\nu(dx)<\infty$, and define $\gamma_0$ as in (\ref{equation:gammazero}), such that for $\Phi(u)$ in (\ref{equation:fact}) we have $\Phi(u)=\text{i}\langle\gamma_0,u\rangle +\int_{\R^d} \left(e^{\text{i}\langle u,x \rangle}-1\right) \nu(dx)$. Furthermore, let $f:S\times\R^m\rightarrow M_{n\times d}(\R)$ be a $\mathcal{B}(S\times\R^m)$-measurable function satisfying
\begin{gather}
\int_S\int_{\R^m}\|f(A,s) \gamma_0\|\ ds\pi(dA)<\infty \text{ and} \label{equation:intcondfinvar1}\\
\int_S\int_{\R^m}\int_{\R^d} \Big(1\wedge \|f(A,s)x\| \Big)\nu(dx)ds\pi(dA)<\infty.\label{equation:intcondfinvar2}
\end{gather}
Then,
\begin{gather*}
\int_S\int_{\R^m}f(A,s)\Lambda(dA,ds)=\int_S\int_{\R^m}f(A,s)\gamma_0 \ ds \pi(dA)+ \int_{\R^d}\int_{S}\int_{\R^m} f(A,s) x \mu(dx,dA,ds),
\end{gather*}
where the right hand side denotes an $\omega$-wise Lebesgue integral. Additionally, the distribution of the stochastic integral $\int_S\int_{\R^m}f(A,s)\Lambda(dA,ds)$ is ID with characteristic function
\begin{gather*}
E\left[e^{\text{i}\langle u,\int_S\int_{\R^m}f(A,s)\Lambda(dA,ds) \rangle}\right]=e^{\text{i}\langle u,\gamma_{int,0}\rangle+\int_{\R^d} \left(e^{\text{i}\langle u,x \rangle}-1\right) \nu_{int}(dx)}, \ u\in\R^d,
\end{gather*}
where
\begin{gather*}
\gamma_{int,0}=\int_S\int_{\R^m}f(A,s)\gamma_0\ ds \pi(dA), \, \, \textrm{and} \,\,
\nu_{int}(B)= \int_S\int_{\R^m}\int_{\R^d} \mathbb{1}_B(f(A,s)x)\nu(dx)ds\pi(dA).
\end{gather*}
\end{Corollary}
\subsection{The MMAF framework}
\label{sec3-2}
\begin{Definition}
Let $\Lambda=\{\Lambda(B), B\in \mathcal{B}_b(S\times\R^m)\}$ be an $\R^d$-valued L\'evy basis and let $f:S\times\R^m\rightarrow M_{n\times d}(\R)$ be a $\mathcal{B}(S\times\R^m)$-measurable function satisfying the conditions (\ref{equation:intcond1}), (\ref{equation:intcond2}) and (\ref{equation:intcond3}). Then, the stochastic integral
\begin{gather}\label{equation:MMAfield}
X_t:=\int_S\int_{\R^m}f(A,t-s)\Lambda(dA,ds)
\end{gather}
is stationary, well-defined for all $t\in\R^m$ and its distribution is ID. The random field $X$ is called an $\R^n$-valued mixed moving average field (MMAF) and $f$ its kernel function.
\end{Definition}
In the following result we give conditions ensuring finite moments of an MMAF and explicit formulas for the first- and second-order moments.
\begin{Proposition}\label{proposition:MMAexistencemoments}
Let $X$ be an $\R^n$-valued MMAF driven by an $\R^d$-valued L\'evy basis with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$ and with $\Lambda$-integrable kernel function $f:S\times\R^m\rightarrow M_{n\times d}(\R)$.
\begin{enumerate}[(i)]
\item If $\int_{\|x\|>1} \| x\|^r \,\nu(dx) < \infty \, \textrm{and} \, f\! \in\! L^r(S \times \R^m, \pi \otimes \lambda)$
for $r \in [2, \infty)$, then $E[\|X_t\|^r]<\infty$ for all $t\in\R^m$.
\item If $\int_{\|x\|>1} \| x\|^r \,\nu(dx) < \infty \, \textrm{and} \, f \!\in\! L^r(S \times \R^m, \pi \otimes \lambda)\cap L^2(S \times \R^m, \pi \otimes \lambda) $ for $r \in (0, 2)$, then $E[\|X_t\|^r]<\infty$ for all $t\in\R^m$.
\end{enumerate}
Consider the finite variation case, i.e. $\Sigma=0$ and $\int_{\norm{x}\leq1}\norm{x}\nu(dx)<\infty$, then the following holds:
\begin{enumerate}[(i)]
\item If $\int_{\|x\|>1} \| x\|^r \,\nu(dx) < \infty \, \textrm{and} \, f \!\in\! L^r(S \times \R^m, \pi \otimes \lambda)$
for $r \in [1, \infty)$, then $E[\|X_t\|^r]<\infty$.
\item If $\int_{\|x\|>1} \| x\|^r \,\nu(dx) < \infty \, \textrm{and} \, f \!\in\! L^r(S \times \R^m, \pi \otimes \lambda)\cap L^1(S \times \R^m, \pi \otimes \lambda) $ for $r \in (0, 1)$, then $E[\|X_t\|^r]<\infty$.
\end{enumerate}
\end{Proposition}
\begin{proof}
Analogous to \cite[Proposition 2.6]{CS2018}.
\end{proof}
\begin{Proposition}\label{proposition:MMAmoments}
Let $X$ be an $\R^n$-valued MMAF driven by an $\R^d$-valued L\'evy basis with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$ and with $\Lambda$-integrable kernel function $f:S\times\R^m\rightarrow M_{n\times d}(\R)$.
\begin{enumerate}[(i)]
\item If $\int_{\norm{x}>1}\norm{x}\nu(dx)<\infty$ and $f\in L^1(S\times\R^m,\pi\times\lambda)\cap L^2(S\times\R^m,\pi\times\lambda)$ the first moment of $X$ is given by
\begin{align*}
E[X_t]= \int_{S}\int_{\R^m}f(A,-s) \mu_\Lambda ds \pi(dA),
\end{align*}
where $\mu_\Lambda =\gamma+\int_{\norm{x}\geq1}x\nu(dx)$.
\item If $\int_{\R^d}\norm{x}^2\nu(dx)<\infty$ and $f\in L^2(S\times\R^m,\pi\times\lambda)$, then $X_t\in L^2$ and
\begin{align*}
Var(X_t)= \int_{S}\int_{\R^m}f(A,-s)\Sigma_\Lambda f(A,-s)'ds \pi(dA)\text{ and}\\
Cov(X_0,X_t)=\int_{S}\int_{\R^m}f(A,-s)\Sigma_\Lambda f(A,t-s)'ds \pi(dA),
\end{align*}
where $\Sigma_\Lambda =\Sigma+\int_{\R^d}xx'\nu(dx)$.
\item Consider the finite variation case, i.e. it holds that $\Sigma=0$ and $\int_{\norm{x}\leq1}\norm{x}\nu(dx)<\infty$. If $\int_{\norm{x}>1}\norm{x}\nu(dx)<\infty$ and $f\in L^1(S\times\R^m,\pi\times\lambda)$, the first moment of $X$ is given by
\begin{align*}
E[X_t]= \int_{S}\int_{\R^m}f(A,-s)\Big(\gamma_0+\int_{\R^d}x\nu(dx)\Big) ds \pi(dA),
\end{align*}
where $\gamma_0$ as defined in (\ref{equation:gammazero}).
\end{enumerate}
\end{Proposition}
\begin{proof}
Immediate from \cite[Section 25]{S2013} and Theorem \ref{theorem:2}.
\end{proof}
\subsection{Weak dependence properties of $(A,\Lambda)$-influenced MMAF}
\label{sec3-3}
Since there is no natural order on $\R^m$ for $m>1$, we cannot extend the definition of a natural filtration and, therefore, causal processes in a natural way to random fields. In the following, we will propose such an extension and prove $\theta$-lex-weak dependence for MMAF falling within this framework. Examples will be presented in Section \ref{sec3-7}.
\begin{Definition}\label{definition:influenced}
Let $X=(X_t)_{t\in\R^m}$ be a random field, $A=(A_t)_{t\in\R^m}\subset\R^m$ a family of Borel sets with Lebesgue measure strictly greater than zero and $M=\{M(B),B\in\mathcal{B}_b(S\times\R^m)\}$ an independently scattered random measure. Assume that $X_t$ is measurable with respect to $\sigma(M(B),B\in \mathcal{B}_b(S\times A_t))$. We then call $A$ the \emph{sphere of influence}, $M$ the \emph{influencer}, $(\sigma(M(B),B\in \mathcal{B}_b(S\times A_t)))_{t\in\R^m}$ the \emph{filtration of influence} and $X$ an \emph{$(A,M)$-influenced random field}. If $A$ is translation invariant, i.e. $A_t=t+A_0$, the sphere of influence is fully described by the set $A_0$ and we call $A_0$ the \emph{initial sphere of influence}.
\end{Definition}
Note that for $m=1$, the class of causal mixed moving average processes driven by a L\'evy basis $\Lambda$ equals the class of $(A,\Lambda)$-influenced mixed moving average processes driven by $\Lambda$ with $A_t=V_t$.
Let $A=(A_t)_{t\in\R^m}$ be a translation invariant sphere of influence with initial sphere of influence $A_0$. In this section we consider the filtration $(\mathcal{F}_{t})_{t\in\R^{m}}$ generated by $\Lambda$, i.e. the $\sigma$-algebra generated by the set of random variables $\{\Lambda(B): B\in\mathcal{B}(S\times A_t)\}$ with $t\in\R^{m}$.\\
Consider an MMAF $X$ that is adapted to $(\mathcal{F}_{t})_{t \in\R^{m}}$. Then, $X$ is $(A,\Lambda)$-influenced and can be written as
\begin{gather}\label{equation:influencedMMAfield}
\begin{gathered}
X_t=\int_S\int_{\R^m}f(A,t-s)
\Lambda(dA,ds)=\int_S\int_{A_t}f(A,t-s)\Lambda(dA,ds).
\end{gathered}
\end{gather}
Note that the translation invariance of $A$ is required to ensure stationarity of $X$.
In the following, we discuss under which assumptions an $(A,\Lambda)$-influenced MMAF is $\theta$-lex-weakly dependent. We start with a preliminary definition.
\begin{Definition}[{\cite[Definition 2.4.1]{BV2004}}]\label{definition:propercone}
$K\subset\R^m$ is called a closed convex proper cone if it satisfies the following properties
\begin{enumerate}[(i)]
\item $K+K\subset K$ (ensures convexity)
\item $\alpha K\subset K$ for all $\alpha\geq0$ (ensures that $K$ is a cone)
\item $K$ is closed
\item $K$ is pointed (i.e., if $x\in K$ and $-x\in K$, then $x=0$).
\end{enumerate}
\end{Definition}
We then apply a truncation technique to show that $X$ is $\theta$-lex-weakly dependent. Define $X_j$, $X_{\Gamma}$ as in Definition \ref{thetaweaklydependent} such that $j\in\R^m$ and $\Gamma\subset V_j^h$ (see Figure \ref{Plot1}). We truncate $X_j$ such that the truncation $\tilde{X}_j$ and $X_{\Gamma}$ become independent. From our construction, it will become clear that it is enough to find a truncation such that $\tilde{X}_j$ and $X_i$ are independent for the lexicographic greatest point $i\in V_j^h$. \\
For a given point $j$, we determine the truncation of $X_j$ by intersecting the integration set with $V_j^\psi$ for $\psi>0$ such that it does not intersect with $A_i$ (see Figure \ref{Plot2} and \ref{Plot3}). In the following, we will describe the choice of $\psi$. The figures illustrate the case $m=2$.
Let $i\in V_j^h$ be the lexicographic greatest point in $V_j^h$, i.e. $k\leq_{lex}i$ for all $k\in V_j^h$. In the following $dist(A,B)=\inf_{a\in A, b\in B}\norm{a-b}$ denotes the Euclidean distance of the sets $A$ and $B$. To ensure the existence of the above truncation, we assume that there exists an $\alpha\in\R^m\backslash\{0\}$ such that
\begin{gather}
\sup_{x\in A_0, x\neq0}\frac{\alpha'x}{\norm{x}}<0.\label{condition:scalarproduct}
\end{gather}
Intuitively, (\ref{condition:scalarproduct}) ensures that the initial sphere of influence $A_0$ can be covered by a closed convex proper cone.
Moreover, w.l.o.g. by applying a rotation to $A_0$, we can always assume to work with $A_0 \subset V_0$. The following remark discusses such transformation.
\begin{Remark}\label{remark:halfspaces}
Let $A_0$ be a subset of a half-space with Lebesgue measure strictly greater than zero such that $A_0 \nsubseteq V_0$.
Define the translation invariant sphere of influence $A=(A_t)_{t\in\R^m}$ by $A_t=(A_0+ t)_{t\in\R^m}$ and consider the $(A,\Lambda)$-influenced MMAF $X=(X_{t})_{t\in\R^m}$ of the form $X_{t}=\int_S\int_{A_0+t}f(A,t-s)\Lambda(dA,ds)$.
Note that if $A_0$ had Lebesgue measure zero, $X$ would be $0$ since the Lebesgue measure of $A_0$ is zero.
Define the hyperplane $D=\{x\in\R^m: \alpha'x=0\}$. Using the principal axis theorem we find an orthogonal matrix $O$ such that the axis of the first coordinate is orthogonal to the rotated hyperplane $OD$. Since $O$ is orthogonal, it holds that $|Det(D\varphi)(u)|=|Det(O)|=1$, where $D\varphi$ denotes the Jacobian matrix of the function $\varphi:u\mapsto Ou$. Additionally, for the rotated initial set $OA_0$ it holds that $OA_0\backslash V_0\subset \{0\}\times[0,\infty)^{m-1}$, such that $\lambda(\{0\}\times[0,\infty)^{m-1})=0$. By substitution for multiple variables we obtain for $\tilde{t}=O t$
\begin{align}\label{equation:MMArotateinitialsphere}
\begin{aligned}
X_{t}&=\!\!\int_S\!\int_{\R^m}f(\!A,t-s)\mathbb{1}_{A_0+t}(s)\Lambda(dA,ds)\\
&=\!\!\int_S\!\int_{\R^m}f(\!A,O^{-1}(Ot-Os))\mathbb{1}_{OA_0+Ot}(Os)\Lambda(dA,ds)\\
&=\!\!\int_S\!\int_{O\!A_0+\tilde{t}}\!f(\!A,O^{-1}\!(\tilde{t}-\tilde{s}))\Lambda(dA,d\tilde{s})\!
=\!\!\int_S\!\int_{O\!A_0\cap V_0+\tilde{t}}\!f(\!A,O^{-1}\!(\tilde{t}-\tilde{s}))\Lambda(dA,d\tilde{s})\\
&=\!\!\int_S\!\int_{V_{\tilde{t}}}\tilde{f}^{O}(\!A,\tilde{t}-\tilde{s})\Lambda(dA,d\tilde{s})=\tilde{X}_{\tilde{t}}
\end{aligned}
\end{align}
with $\tilde{f^O}(A,t-s)= f(A,O^{-1}(t-s))\mathbb{1}_{\{s\in OA_0+t \}}$.
\end{Remark}
Figure \ref{Plot4} shows the smallest closed convex proper cone covering $A_i$, which is called $K$. Note that all conditions can be formulated in terms of $A_0$ since the sphere of influence $A$ is translation invariant.
\begin{figure}[H]
\begin{minipage}[H]{3.5cm}
\center{
\includegraphics[width=4.5cm]{psinewconstruction1.pdf}\label{figurePlot1}\\
\caption{\small{Integration sets $A_j$ and $A_i$} of $X_j$ and $X_i$}\label{Plot1}}
\end{minipage}
\hspace{0.9cm}
\begin{minipage}[H]{3.5cm}
\vspace{0.19cm}
\centering
\includegraphics[width=4.5cm]{psinewconstruction2.pdf}\label{figurePlot2}\\
\caption{\small{$A_j$ and $A_i$ together with $V_j^\psi$}}
\label{Plot2}
\end{minipage}
\hspace{0.9cm}
\begin{minipage}[H]{3.5cm}
\vspace{0.6cm}
\centering
\includegraphics[width=4.5cm]{psinewconstruction3.pdf}\label{figurePlot3}\\
\caption{\small{Integration sets $A_i$ and $A_j \backslash V_j^{\psi}$ of $X_i$ and $\tilde{X}_j$}}
\label{Plot3}
\end{minipage}
\end{figure}
In order to choose $\psi$ we first define
\begin{gather}\label{equation:xi}
b=\sup_{\substack{x\in A_0\\ \norm{x}=1}}\frac{\alpha'x}{\norm{\alpha}} \text{ and }\tilde{K}=\left\{x\in\R^m: \frac{\alpha'x}{\norm{x}}\leq b \right\}.
\end{gather}
Due to (\ref{condition:scalarproduct}) (see Figure \ref{Plot4}), it holds $-1 \leq b<0$. For $x_1,x_2\in\tilde{K}$ it holds
\begin{gather*}
\frac{\alpha'(x_1+x_2)}{\norm{x_1+x_2}}= \frac{\alpha'x_1}{\norm{x_1+x_2}}+\frac{\alpha'x_2}{\norm{x_1+x_2}}\leq b \frac{\norm{x_1}+\norm{x_2}}{\norm{x_1+x_2}}\leq b
\end{gather*}
such that $\tilde{K}$ is a closed convex proper cone. It can be interpreted as the smallest equiangular closed convex proper cone that contains $A_0$.
Then, $\cos(\beta+\frac{\pi}{2})=b$ such that $\beta=\arcsin(-b)\in [0,\frac{\pi}{2})$ (see Figure \ref{Plot5}) and $dist(j,\tilde{K})\geq \sin(\beta)h=-bh$ (see Figure \ref{Plot6}). We choose $\psi$ as
\begin{gather}\label{equation:psi}
\psi(h)=\frac{-bh}{\sqrt{m}}.
\end{gather}
In particular we have $\psi(h)= \mathcal{O}(h)$.
Let $l\in V_j^h$ be an arbitrary point. From the given choice of $\psi$ and $i$ it holds $dist(l,j)\geq dist(i,j)$, $A_i\cap (A_j\backslash V_j^\psi) =\emptyset$, $A_i=i+A_0\subset i+ \tilde{K}$ and $A_l=l+A_0\subset l+ \tilde{K}$. Since $\tilde{K}$ is an equiangular closed convex proper cone we get $A_l\cap (A_j\backslash V_j^\psi) =\emptyset$.
\begin{figure}[H]
\begin{minipage}[H]{3.5cm}
\centering
\includegraphics[width=4.5cm]{psinewconstruction4.pdf}\label{figurePlot4}\\
\caption{\small{ Choice of $\alpha$ and $\beta$ }}\label{Plot4}
\end{minipage}
\hspace{0.9cm}
\begin{minipage}[H]{3.5cm}
\vspace{-0.04cm}
\centering
\includegraphics[width=4.5cm]{psinewconstruction5.pdf}\label{figurePlot5}\\
\caption{\small{ Choice of $\tilde{K}$ }} \label{Plot5}
\end{minipage}
\hspace{0.9cm}
\begin{minipage}[H]{3.5cm}
\centering
\includegraphics[width=4.5cm]{psinewconstruction6.pdf}\label{figurePlot6}\\
\caption{\small{ Construction of $\psi$ }}
\label{Plot6}
\end{minipage}
\end{figure}
The conditions below, which are expressed in terms of the kernel function $f$ and the characteristic quadruplet of the driving L\'evy basis, are sufficient to show that an $(A,\Lambda)$-influenced MMAF is $\theta$-lex-weakly dependent.
\begin{Proposition}\label{proposition:mmathetaweaklydep}
Let $\Lambda$ be an $\R^d$-valued L\'evy basis with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$ and $f:S\times\R^{m}\ra M_{n\times d}(\R)$ a $\mathcal{B}(S\times\R^m)$-measurable function. Consider the $(A,\Lambda)$-influenced MMAF
\begin{gather*}
X_{t}=\int_S\int_{A_{t}}f(A,t-s)\Lambda(dA,ds), \ t\in\R^{m}
\end{gather*}
with translation invariant sphere of influence $A$ such that (\ref{condition:scalarproduct}) holds.
\begin{enumerate}[(i)]
\item If $\int_{\norm{x}>1}\norm{x}^2\nu(dx)<\infty$, $\gamma+\int_{\norm{x}>1}x\nu(dx)=0$ and $f\in L^2(S\times\R^{m},\pi\otimes\lambda)$, then $X$ is $\theta$-lex-weakly dependent with $\theta$-lex-coefficients satisfying
\begin{gather}\label{equation:thetalexcoefficientsmmafield}
\theta_X(h)\leq 2 \Big(\int_S\int_{A_0\cap V_{0}^{\psi(h)}}\textup{tr}(f(A,-s)\Sigma_\Lambda f(A,-s)')ds\pi(dA)\Big)^{\frac{1}{2}}=:\hat{\theta}_X^{(i)}(h).
\end{gather}
\item If $\int_{\norm{x}>1}\norm{x}^2\nu(dx)<\infty$ and $f\in L^2(S\times\R^{m},\pi\otimes\lambda)\cap L^1(S\times\R^{m},\pi\otimes\lambda)$, then $X$ is $\theta$-lex-weakly dependent with $\theta$-lex-coefficients satisfying
\begin{gather}
\begin{aligned}
\theta_X(h)\leq 2& \bigg(\int_S\int_{A_0\cap V_{0}^{\psi(h)}}\textup{tr}(f(A,-s)\Sigma_\Lambda f(A,-s)')ds\pi(dA)\\
&+\Big\lVert\int_S\int_{A_0\cap V_{0}^{\psi(h)}} f(A,-s)\mu_\Lambda ds\pi(dA)\Big\rVert^2\bigg)^{\frac{1}{2}}=:\hat{\theta}_X^{(ii)}(h).\label{equation:thetalexcoefficientsmeanmmafield}
\end{aligned}
\end{gather}
\item If $\int_{\R^d}\norm{x}\nu(dx)<\infty$, $\Sigma=0$ and $f\in L^1(S\times\R^m,\pi\otimes\lambda)$ with $\gamma_0$ as in (\ref{equation:gammazero}), then $X$ is $\theta$-lex-weakly dependent with $\theta$-lex-coefficients satisfying
\begin{gather}
\begin{aligned}
\theta_X(h)\leq 2& \bigg(\int_S\int_{A_0\cap V_{0}^{\psi(h)}}\norm{f(A,-s)\gamma_0}ds\pi(dA)\\
+&\int_S\int_{A_0\cap V_{0}^{\psi(h)}} \int_{\R^d}\norm{f(A,-s)x} \nu(dx)ds\pi(dA)\bigg)=:\hat{\theta}_X^{(iii)}(h).\label{equation:thetalexcoefficientsfinitevariationmmafield}
\end{aligned}
\end{gather}
\item If $\int_{\norm{x}>1}\norm{x}\nu(dx)<\infty$ and $f\in L^1(S\times\R^m,\pi\otimes\lambda)\cap L^2(S\times\R^m,\pi\otimes\lambda)$, then $X$ is $\theta$-lex-weakly dependent with $\theta$-lex-coefficients satisfying
\begin{gather*}
\theta_X(h)\leq 2\bigg(\int_S\int_{A_0\cap V_0^{\psi(h)}}\textup{tr}(f(A,-s)\Sigma_{\Lambda_1} f(A,-s)')ds\pi(dA)\\
+\Big\lVert\int_S\int_{A_0\cap V_0^{\psi(h)}} f(A,-s)\gamma ds\pi(dA)\Big\rVert^2\bigg)^{\frac{1}{2}}\\
+2\int_S\int_{A_0\cap V_0^{\psi(h)}} \int_{\norm{x}>1}\norm{f(A,-s)x} \nu(dx)ds\pi(dA)=:\hat{\theta}_X^{(iv)}(h).
\end{gather*}
\end{enumerate}
The results above hold for all $h>0$ with $\psi$ as defined in $(\ref{equation:psi})$, $\Sigma_\Lambda =\Sigma+\int_{\R^d}xx'\nu(dx)$, $\Sigma_{\Lambda_1} =\Sigma+\int_{\norm{x}\leq1}xx'\nu(dx)$ and $\mu_\Lambda =\gamma-\int_{\norm{x}\geq1}x\nu(dx)$.
\end{Proposition}
\begin{proof}
See Section \ref{sec5-2}.
\end{proof}
In the next proposition, we consider a vector of a shifted real-valued $(A,\Lambda)$-influenced MMAF, and we show that it is $\theta$-lex weakly dependent.
This result is necessary to analyze, for example, the asymptotic behavior of the sample autocovariances. Define the set of possible shifts
\begin{gather}
S_k=\{(a,b)'\in\{0,\ldots,k\}\times\{-k,\ldots,k\}^{m-1} \},\ k\in\N_0\label{equation:shiftedmmafieldvector}
\end{gather}
and consider the enumeration $\{s_1,\ldots,s_{|S_k|}\}$ of $S_k$, where $|S_k|=(k+1)(2k+1)^{m-1}$.
Besides the hereditary properties from Proposition \ref{proposition:mmathetahereditary} we show that the field
\begin{gather}\label{equation:shiftedrandomfield}
Z_t=(X_t,X_{t+s_1},X_{t+s_2},\ldots,X_{t+s_{|S_k|}})
\end{gather}
inherits weak dependence properties.
\begin{Proposition}\label{proposition:vectorinfluencedweaklydep}
Let $\Lambda$ be an $\R^d$-valued L\'evy basis with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$ and $f:S\times\R^m\ra M_{1\times d}(\R)$ be a $\Lambda$-integrable, $\mathcal{B}(S\times\R^m)$-measurable function. Consider the $(A,\Lambda)$-influenced MMAF
\begin{gather*}
X_{t}=\int_S\int_{A_{t}}f(A,t-s)\Lambda(dA,ds), \ t\in\R^{m}
\end{gather*}
with translation invariant sphere of influence $A$ such that (\ref{condition:scalarproduct}) holds. Then
\begin{gather*}
Z_t:=\int_S\int_{A_t}g(A,t-s)\Lambda(dA,ds), \ t\in\R^m,
\end{gather*}
where $g(A,s)=(f(A,s),f(A,s-s_1),\ldots, f(A,s-s_{|S_k|}))'$ is a $\mathcal{B}(S\times\R^m)$-measurable function with values in $M_{(k+1)(2k+1)^{m-1}\times d}(\R)$ for $k\in\N_0$, is an $(A,\Lambda)$-influenced MMAF.\\
If $X$ additionally satisfies the conditions of Proposition \ref{proposition:mmathetaweaklydep} (i), (ii), (iii) or (iv), then $Z$ is $\theta$-lex-weakly dependent with coefficients respectively given by
\begin{gather}
\begin{aligned}\label{eq:mmainfluencedvectorweakly}
\theta_Z^{(i)}(h)\leq&\mathcal{D}\hat{\theta}_X^{(i)}(h-\psi^{-1}(k)),\qquad\qquad
\theta_Z^{(ii)}(h)\leq&\mathcal{D} \hat{\theta}_X^{(ii)}(h-\psi^{-1}(k)),\\
\theta_Z^{(iii)}(h)\leq&\mathcal{C} \hat{\theta}_X^{(iii)}(h-\psi^{-1}(k)),\text{ and}\qquad
\theta_Z^{(iv)}(h)\leq&\mathcal{C}\hat{\theta}_X^{(iv)}(h-\psi^{-1}(k)),
\end{aligned}
\end{gather}
where $\mathcal{D}=|S_k|^{m/2}$, $\mathcal{C}=|S_k|^{m}$ for $\psi(h)>k$ and $\hat{\theta}^{(\cdot)}(h)$ are defined as in Proposition \ref{proposition:mmathetaweaklydep}.
\end{Proposition}
\begin{proof}
See Section \ref{sec5-2}.
\end{proof}
\subsection{Sample moments of $(A,\Lambda)$-influenced MMAF}
\label{sec3-4}
Let us consider an $\R^n$-valued $(A,\Lambda)$-influenced MMAF
\begin{gather}
X=(X_u)_{u\in\Z^m} \text{ with } X_u=\int_S\int_{A_u}f(A,u-s)\Lambda(dA,ds),\label{equation:discreteinfluencedMMAfield}
\end{gather}
translation invariant sphere of influence $A$, and initial sphere of influence $A_0\subset V_0$ such that (\ref{condition:scalarproduct}) holds. We assume that we observe $X$ on the finite sampling sets $D_n\subset \Z^m$, such that
\begin{gather}\label{condition:samplingset}
\lim_{n\rightarrow\infty} |D_n|=\infty \text{ and }\lim_{n\rightarrow\infty} \frac{|D_n|}{|\partial D_n|}=0.
\end{gather}
We note that this includes in particular the equidistant sampling
\begin{gather}\label{equation:equidistantsample}
E_n=(0,n]^m\cap\Z^m\text{ such that }|E_n|=n^{m}, n\in\N.
\end{gather}
The sample mean of the random field $X$ is then defined as
\begin{gather}\label{equation:samplemean}
\frac{1}{|D_n|}\sum_{u\in D_n} X_{u}.
\end{gather}
If $\int_{\norm{x}>1}\norm{x}\nu(dx)<\infty$, we define the centered MMAF $\tilde{X}_u=X_u-E[X_u]$ and the sample autocovariance on $E_n$ at lag $k\in\N_0\times\Z^{m-1}$
\begin{gather}\label{equation:sampleautocovariance}
\frac{1}{|E_{n-\tilde{k}}|}\sum_{u\in E_{n-\tilde{k}}} \tilde{X}_{u}\tilde{X}_{u+k},\ k\in\N_0\times\Z^{m-1},
\end{gather}
where $\tilde{k}=|k|$. Let us start by analyzing the asymptotic properties of the sample mean (\ref{equation:samplemean}) for a centered $(A,\Lambda)$-influenced MMAF.
\begin{Theorem}\label{theorem:thetasamplemean}
Let $X=(X_u)_{u\in\Z^m}$ be an $(A,\Lambda)$-influenced MMAF as defined in (\ref{equation:discreteinfluencedMMAfield}) such that $\int_{\norm{x}>1}\norm{x}^{2+\delta}\nu(dx)<\infty$, $\gamma+\int_{\norm{x}>1}x\nu(dx)=0$ and $f\in L^2(S\times\R^{m},\pi\otimes\lambda)\cap L^{2+\delta}(S\times\R^{m},\pi\otimes\lambda)$ for some $\delta>0$. Assume that $X$ has $\theta$-lex-coefficients satisfying $\theta_X(h)=\mathcal{O}(h^{-\alpha})$, where $\alpha>m(1+\frac{1}{\delta})$. Then,
\begin{gather*}
\Sigma=\sum_{k\in\Z^m}E[X_0X_k']
\end{gather*}
is finite, positive semidefinite and
\begin{gather}\label{eq:mmathetaclt}
\frac{1}{|D_n|^{\frac{1}{2}}}\sum_{j\in D_n}X_j\underset{n\ra\infty}{\xrightarrow{\makebox[2em][c]{d}}}N(0,\Sigma).
\end{gather}
\end{Theorem}
\begin{proof}
By \cite[Theorem 3.6]{PV2017}, it follows that an MMAF is ergodic. Then, the result follows from Corollary \ref{corollary:ergodicclt}.
\end{proof}
In the theorem above, the initial sphere of influence $A_0$ must satisfy (\ref{condition:scalarproduct}). Additionally, we observe a trade-off between moment conditions on $X$ and the decay rate of the $\theta$-lex coefficients.
However, one can derive similar results for the sample mean of an MMAF by relaxing condition (\ref{condition:scalarproduct}) and exploiting the second order moment structure of an MMAF. On the other hand, the following technique does not carry over to higher-order moments.
\begin{Theorem}\label{theorem:thetasamplemeanspecial}
Let $X=(X_u)_{u\in\Z^m}$ be an $(A,\Lambda)$-influenced MMAF defined by
\begin{gather*}
X_u=\int_S\int_{A_u}f(A,u-s)\Lambda(dA,ds),
\end{gather*}
with translation invariant sphere of influence $A$ and initial sphere of influence $A_0\subset V_0$ such that $\gamma+\int_{\norm{x}>1}x\nu(dx)=0$ and $E[\norm{X_0}^{2}]<\infty$. Assume that $X$ has $\theta$-lex-coefficients satisfying $\theta_X(h)=\mathcal{O}(h^{-\alpha})$, where $\alpha>m$. Then,
\begin{gather*}
\Sigma=\sum_{k\in\Z^m}E[X_0X_k']
\end{gather*}
is finite, positive definite and
\begin{gather}\label{eq:mmathetacltspecial}
\frac{1}{|D_n|^{\frac{1}{2}}}\sum_{j\in D_n}X_j\underset{n\ra\infty}{\xrightarrow{\makebox[2em][c]{d}}}N(0,\Sigma).
\end{gather}
\end{Theorem}
\begin{proof}
See Section \ref{sec5-3}.
\end{proof}
To lighten notation, we assume in the following that $X$ is real-valued and centered, i.e., $E[X_0]=0$. In order to derive asymptotic properties for the distribution of (\ref{equation:sampleautocovariance}), we need to show weak dependence properties of the random field $Y=(Y_{j,k})_{j\in\Z^m}$ defined as
\begin{gather}\label{equation:sampleautocovy}
Y_{j,k}=X_jX_{j+k}-R(k),\ k\in\N_0\times\Z^{m-1},
\end{gather}
where
\begin{gather*}
R(k)=Cov(X_0,X_k)=E[X_0X_k']=\int_S\int_{A_0\cap A_k}\!\!\!\!f(A,-s)\Sigma_\Lambda f(A,k-s)'ds\pi(dA),
\end{gather*}
$k\in\N_0\times\Z^{m-1}$ with $\Sigma_\Lambda =\Sigma+\int_{\R^d}xx'\nu(dx)$ for an $(A,\Lambda)$-influenced MMAF $X$ with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$. The last equality follows from Proposition \ref{proposition:MMAmoments}.
\begin{Proposition}\label{proposition:deltavererbungtheta}
Let $X=(X_u)_{u\in\Z^m}$ be a real-valued $(A,\Lambda)$-influenced MMAF as defined in (\ref{equation:discreteinfluencedMMAfield}) such that $E[X_0]=0$ and $E[\norm{X_0}^{2+\delta}]<\infty$ for some $\delta>0$ with $\theta$-lex-coefficients $\theta_X$. Then, $(Y_{j,k})_{j\in\Z^m}$, $k\in\N_0\times\Z^{m-1}$ as defined in (\ref{equation:sampleautocovy}) is $\theta$-lex-weakly dependent with coefficients
\begin{gather*}
\theta_Y(h)\leq\mathcal{C}\left(\sqrt{2}\hat{\theta}^{(i)}_X\left(h-\psi^{-1}(|k|)\right)\right)^{\frac{\delta}{1+\delta}},
\end{gather*}
where $\mathcal{C}$ is a constant independent of $h$, $\psi$ as defined in $(\ref{equation:psi})$, and $\hat\theta_X^{(i)}(\cdot)$ is defined as in Proposition \ref{proposition:vectorinfluencedweaklydep}.
Furthermore, in the finite variation case and for $\hat\theta_X^{(iii)}(\cdot)$ defined as in Proposition \ref{proposition:vectorinfluencedweaklydep}, it holds
\begin{gather*}
\theta_Y(h)\leq\mathcal{C}\left(2\hat{\theta}^{(iii)}_X\left(h-\psi^{-1}(|k|)\right)\right)^{\frac{\delta}{1+\delta}}.
\end{gather*}
\end{Proposition}
\begin{proof}
Consider the 2-dimensional process $Z=(X_j,X_{j+k})_{j\in\Z^m}$ with $k\in\N_0\times\Z^{m-1}$. Proposition \ref{proposition:vectorinfluencedweaklydep} implies that $Z$ is $\theta$-lex-weakly dependent and from the proof we obtain
\begin{gather*}
\theta_Z(h)\leq\sqrt{2}\hat{\theta}^{(i)}_X(h-\psi^{-1}(|k|))\text{ for $\psi(h)>|k|$}.
\end{gather*}
Consider the function $h:\R^2\ra\R$ such that $h(x_1,x_2)=x_1x_2$. The function $h$ satisfies the assumptions of Proposition \ref{proposition:mmathetahereditary} for $p=2+\delta$, $c=1$ and $a=2$. Considering $h(Z)=X_jX_{j+k}$, we obtain the $\theta$-lex-coefficients of $(Y_{j,k})_{j\in\Z^m}$
\begin{gather*}
\theta_Y(h)\leq\mathcal{C}(\sqrt{2}\hat{\theta}^{(i)}_X(h-\psi^{-1}(|k|)))^{\frac{\delta}{1+\delta}} \text{ for $\psi(h)>|k|$.}
\end{gather*}
The coefficients for the finite variation case can be obtained from Proposition \ref{proposition:mmathetahereditary} and (\ref{eq:mmainfluencedvectorweakly}).
\end{proof}
The next corollary gives asymptotic properties of the sample autocovariances (\ref{equation:sampleautocovariance}) for $(A,\Lambda)$-influenced MMAF, i.e. we can give a distributional limit theorem for the process $(Y_{j,k})_{j\in\Z^m}$ by determining the asymptotic distribution of
\begin{gather*}
\frac{1}{|E_{n-\tilde{k}}|^{\frac{1}{2}}}\sum_{j\in E_{n-\tilde{k}}}Y_{j,k},\ k\in\N_0\times\Z^{m-1},
\end{gather*}
where $\tilde{k}=|k|$.
\begin{Corollary}\label{corollary:sampleautocovariancetheta}
Let $X=(X_u)_{u\in\Z^m}$ be a real-valued $(A,\Lambda)$-influenced MMAF as defined in (\ref{equation:discreteinfluencedMMAfield}) such that $E[X_0]=0$ and $E[\norm{X_0}^{4+\delta}]<\infty$ for some $\delta>0$. Let $\hat\theta_X^{(i)}$ be defined as in Proposition \ref{proposition:vectorinfluencedweaklydep}. If $\hat{\theta}_X^{(i)}(h)=\mathcal{O}(h^{-\alpha})$ for $\alpha>m\left(1+\frac{1}{\delta}\right)(\frac{3+\delta}{2+\delta})$, then
\begin{gather*}
\Sigma=\sum_{l\in\Z^m}Cov\left(\left(\begin{array}{c}
Y_{0,0}\\
\vdots \\
Y_{0,k}
\end{array}\right),
\left(\begin{array}{c}
Y_{l,0}\\
\vdots \\
Y_{l,k}
\end{array}\right)
\right)=\sum_{l\in\Z^m}Cov\left(\left(\begin{array}{c}
X_0X_0\\
\vdots \\
X_0X_k
\end{array}\right),
\left(\begin{array}{c}
X_lX_l\\
\vdots \\
X_lX_{l+k}
\end{array}\right)
\right)
\end{gather*}
is finite, positive semidefinite and
\begin{gather*}
\frac{1}{|E_{n-\tilde{k}}|^{\frac{1}{2}}}\sum_{j\in E_{n-\tilde{k}}}
\left(\begin{array}{c}
Y_{j,0}\\
\vdots \\
Y_{j,k}
\end{array}\right)
\underset{N\ra\infty}{\xrightarrow{\makebox[2em][c]{d}}}N\left(0,\Sigma\right),
\end{gather*}
where $\tilde{k}=|k|$.
\end{Corollary}
\begin{proof}
Analogous to Theorem \ref{theorem:thetasamplemean} we obtain the stated convergence using Proposition \ref{proposition:deltavererbungtheta}.
\end{proof}
\begin{Corollary}\label{corollary:samplemomentsofhigherorder}
Let $X=(X_u)_{u\in\Z^m}$ be a real-valued $(A,\Lambda)$-influenced MMAF as defined in (\ref{equation:discreteinfluencedMMAfield}) and $p\geq1$ such that $E[|X_0|^{2p+\delta}]<\infty$ for some $\delta>0$.
Let $\hat\theta_X^{(i)}$ be defined as in Proposition \ref{proposition:vectorinfluencedweaklydep}. If $\hat{\theta}_X^{(i)}(h)=\mathcal{O}(h^{-\alpha})$ for $\alpha>m\left(1+\frac{1}{\delta}\right)(\frac{2p-1+\delta}{p+\delta})$, then
\begin{gather*}
\Sigma=\sum_{k\in\Z^m}Cov(X_0^{p},{X_k^{p}})
\end{gather*}
is finite, positive semidefinite and
\begin{gather*}
\frac{1}{|E_{n}|^{\frac{1}{2}}}\sum_{j\in E_{n}}(X_j^p-E[X_0^p])\underset{N\ra\infty}{\xrightarrow{\makebox[2em][c]{d}}}N(0,\Sigma).
\end{gather*}
\end{Corollary}
\begin{Remark}\label{remark:gmmestimation}
The theory developed in this section is an essential step in showing the asymptotic normality of parametric estimators based on moment functions as the generalized method of moments (for a comprehensive introduction, see \cite{H2005}). The weak dependence properties and related central limit theorems analyzed in this section find application in the study of the GMM estimators presented in \cite[Section 6.1]{CS2018}, where the authors analyze parametric estimators of the supOU process.
\end{Remark}
\subsection{Weak dependence properties of non-influenced MMAF}
\label{sec3-5}
We now consider a general MMAF $X=(X_t)_{t\in\R^m}$ as defined in (\ref{equation:MMAfield}), i.e.
\begin{gather*}
X_t=\int_S\int_{\R^m} f(A,t-s)\Lambda(dA,ds),\ t\in\R^m,
\end{gather*}
and discuss under which assumptions a non-influenced MMAF is $\eta$-weakly dependent.
Note that we do not demand any additional assumption on the structure of $X$ as assumed in Section \ref{sec3-2} and \ref{sec3-3}.
\begin{Proposition}\label{proposition:mmaetaweaklydep}
Let $\Lambda$ be an $\R^d$-valued L\'evy basis with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$ and $f:S\times\R^{m}\ra M_{n\times d}(\R)$ a $\mathcal{B}(S\times\R^m)$-measurable function. Consider the MMAF $X=(X_t)_{t\in\R^m}$ with
\begin{gather*}
X_t=\int_S\int_{\R^m}f(A,t-s)\Lambda(dA,ds), \ t\in\R^m.
\end{gather*}
\begin{enumerate}[(i)]
\item If $\int_{\norm{x}>1}\norm{x}^2\nu(dx)<\infty$, $\gamma+\int_{\norm{x}>1}x\nu(dx)=0$ and $f\in L^2(S\times\R^{m},\pi\otimes\lambda)$, then $X$ is $\eta$-weakly dependent with $\eta$-coefficients satisfying
\begin{gather*}
\eta_X(h)\leq\Bigg(\int_S\int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c}\textup{tr}(f(A,-s)\Sigma_\Lambda f(A,-s)')ds\pi(dA)\Bigg)^{\frac{1}{2}}=\hat{\eta}_X^{(i)}(h).
\end{gather*}
\item If $\int_{\norm{x}>1}\norm{x}^2\nu(dx)<\infty$ and $f\in L^2(S\times\R^{m},\pi\otimes\lambda)\cap L^1(S\times\R^{m},\pi\otimes\lambda)$, then $X$ is $\eta$-weakly dependent with $\eta$-coefficients satisfying
\begin{align*}
\eta_X(h) & \leq
\bigg(\int_S\int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c}\textup{tr}(f(A,-s)\Sigma_\Lambda f(A,-s)')ds\pi(dA)\\
&+\Big\lVert\int_S\int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c} f(A,-s)\mu_\Lambda ds\pi(dA)\Big\rVert^2\bigg)^{\frac{1}{2}}=\hat{\eta}_X^{(ii)}(h).
\end{align*}
\item If $\int_{\R^d}\norm{x}\nu(dx)<\infty$, $\Sigma=0$ and $f\in L^1(S\times\R^m,\pi\otimes\lambda)$ with $\gamma_0$ as in (\ref{equation:gammazero}), then $X$ is $\eta$-weakly dependent with $\eta$-coefficients satisfying
\begin{align*}
\eta_X(h) &\leq
\int_S\int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c}\norm{f(A,-s)\gamma_0}ds\pi(dA)\\
&+\int_S\int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c} \int_{\R^d}\norm{f(A,-s)x} \nu(dx)ds\pi(dA)=\hat{\eta}_X^{(iii)}(h).
\end{align*}
\item If $\int_{\norm{x}>1}\norm{x}\nu(dx)<\infty$ and $f\in L^1(S\times\R^m,\pi\otimes\lambda)\cap L^2(S\times\R^m,\pi\otimes\lambda)$, then $X$ is $\eta$-weakly dependent with $\eta$-coefficients satisfying
\begin{align*}
\eta_X(h) &\leq
\bigg(\int_S\int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c}\textup{tr}(f(A,-s)\Sigma_{\Lambda_1} f(A,-s)')ds\pi(dA)\\
&+\Big\lVert\int_S\int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c} f(A,-s)\gamma ds\pi(dA)\Big\rVert^2\bigg)^{\frac{1}{2}}\\
&+\int_S\int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c} \int_{\norm{x}>1}\norm{f(A,-s)x} \nu(dx)ds\pi(dA)=\hat{\eta}_X^{(iv)}(h).
\end{align*}
\end{enumerate}
The results above hold for all $h>0$, where $\Sigma_\Lambda =\Sigma+\int_{\R^d}xx'\nu(dx)$, $\Sigma_{\Lambda_1} =\Sigma+\int_{\norm{x}\leq1}xx'\nu(dx)$ and $\mu_\Lambda =\gamma-\int_{\norm{x}\geq1}x\nu(dx)$.
\end{Proposition}
\begin{proof}
See Section \ref{sec5-4}.
\end{proof}
Analogous to Proposition \ref{proposition:vectorinfluencedweaklydep} we obtain the following result.
\begin{Proposition}\label{proposition:vectorgeneralmmaweaklydep}
Let $\Lambda$ be an $\R^d$-valued L\'evy basis with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$ and $f:S\times\R^m\ra M_{1\times d}(\R)$ be a $\Lambda$-integrable, $\mathcal{B}(S\times\R^m)$-measurable function. Consider the real-valued MMAF
\begin{gather*}
X_{t}=\int_S\int_{\R^m}f(A,t-s)\Lambda(dA,ds), \ t\in\R^{m}.
\end{gather*}
Then,
\begin{gather*}
Z_t:=\int_S\int_{\R^m}g(A,t-s)\Lambda(dA,ds), \ t\in\R^m,
\end{gather*}
is an MMAF, where $g(A,s)=(f(A,s),f(A,s-s_1),\ldots, f(A,s-s_{|S_k|}))'$ is a $\mathcal{B}(S\times\R^m)$-measurable function with values in $M_{(k+1)(2k+1)^{m-1}\times d}(\R)$ for $k\in\N_0$.\\
If $X$ additionally satisfies the conditions of Proposition \ref{proposition:mmaetaweaklydep} (i), (ii), (iii) or (iv), then $Z$ is $\eta$-weakly dependent with coefficients respectively given by
\begin{gather}
\begin{aligned}\label{eq:mmavectorweakly}
\eta_Z^{(i)}(h)\leq&\mathcal{D}\hat{\eta}_X^{(i)}(h-2k),\qquad\qquad
\eta_Z^{(ii)}(h)\leq&\mathcal{D} \hat{\eta}_X^{(ii)}(h-2k),\\
\eta_Z^{(iii)}(h)\leq&\mathcal{C} \hat{\eta}_X^{(iii)}(h-2k)\text{ and}\qquad
\eta_Z^{(iv)}(h)\leq&\mathcal{C}\hat{\eta}_X^{(iv)}(h-2k),
\end{aligned}
\end{gather}
where $\mathcal{D}=|S_k|^{m/2}$, $\mathcal{C}=|S_k|^{m}$ for $h>2k$, and $\hat{\eta}^{(\cdot)}(h)$ are defined as in Proposition \ref{proposition:mmaetaweaklydep}.
\end{Proposition}
\begin{proof}
Analogous to Proposition \ref{proposition:vectorinfluencedweaklydep}.
\end{proof}
\subsection{Sample moments of non-influenced MMAF}
\label{sec3-6}
Let us consider an $\R^n$-valued MMAF
\begin{gather}
X=(X_u)_{u\in\Z^m} \text{ with } X_u=\int_S\int_{\R^m}f(A,u-s)\Lambda(dA,ds).\label{equation:discretegeneralMMAfield}
\end{gather}
As in Section \ref{sec3-2} we assume that we observe $X$ on a sequence of finite sampling sets $D_n\subset\Z^m$, such that (\ref{condition:samplingset}) holds.
\begin{Theorem}\label{theorem:etasamplemean}
Let $(X_u)_{u\in\Z^m}$ be an MMAF as defined in (\ref{equation:discretegeneralMMAfield}) such that $E[X_0]=0$ and $E[\norm{X_0}^{2+\delta}]<\infty$ for some $\delta>0$. Assume that $X$ has $\eta$-coefficients satisfying $\eta_X(h)=\mathcal{O}(h^{-\beta})$, where $\beta>m\max\left(2,\left(1+\frac{1}{\delta}\right)\right)$.
Then,
\begin{gather}\label{equation:sigmaeta}
\Sigma=\sum_{u\in\Z^m}Cov(X_0,X_u)=\sum_{u\in\Z^m}E[X_0X_u']
\end{gather}
is finite, positive semidefinite and
\begin{gather}\label{equation:etaasympnorm}
\frac{1}{|D_n|^{\frac{1}{2}}}\sum_{u\in D_n}X_u\underset{n\ra\infty}{\xrightarrow{\makebox[2em][c]{d}}}N(0,\Sigma).
\end{gather}
\end{Theorem}
\begin{proof}
$X$ is $\lambda$-weakly dependent, see \cite[Definition 1]{DMT2008}. Then, \cite[Theorem 2]{DMT2008} implies the summability of $\sigma^2$ and the result stated in (\ref{equation:etaasympnorm}). The multivariate extension follows by using the Cram\'er-Wold device.
\end{proof}
\begin{Remark}
Theorem \ref{theorem:etasamplemean} can be formulated as a functional central limit theorem, following \cite[Theorem 3]{DMT2008}. For $t\in [0,1]^m$, where $[0,1]^m$ denotes the $m$-fold Cartesian product of $[0,1]$, we set $S_n(t)=\sum_{j\in tE_n}X_j$ with $E_n$ as defined in (\ref{equation:equidistantsample}) and the additional assumption that $S_n(t)=0$ if one coordinate of $t$ equals zero. The product $tE_n$ has to be understood coordinatewise. Then, under the assumptions of Theorem \ref{theorem:etasamplemean} it holds that
\begin{gather}
\frac{1}{n^{\frac{m}{2}}}S_n(t) \underset{n\ra\infty}{\xrightarrow{\mathcal{D}([0,1]^m)}}\sigma W(t),
\end{gather}
where $W=\{W(t), t\in[0,1]^m\}$ denotes a Brownian sheet, i.e., a centered Gaussian process such that $Cov(W(t_1,\ldots,t_m),W(s_1,\ldots,s_m)')=\prod_{i=1}^m t_i \wedge s_i$ for all $t_1,\ldots,t_m,s_1,\ldots,s_m\in[0,1]$, and $\underset{n\ra\infty}{\xrightarrow{\mathcal{D}([0,1]^m)}}$ denotes the convergence in the Skorokhod space (see e.g. \cite[Section 3]{BW1971} for a definition of the Skorokhod topology on $[0,1]^m$).
\end{Remark}
Analogous to Proposition \ref{proposition:deltavererbungtheta} we show the following result.
\begin{Proposition}\label{proposition:deltavererbungeta}
Let $(X_u)_{u\in\Z^m}$ be a real-valued MMAF as defined in (\ref{equation:discretegeneralMMAfield}) such that $E[X_0]=0$ and $E[\norm{X_0}^{2+\delta}]<\infty$ for some $\delta>0$.
Then, $(Y_{j,k})_{j\in\Z^m}$, $k\in\N_0\times\Z^{m-1}$ as defined in (\ref{equation:sampleautocovy}) is $\eta$-weakly dependent with coefficients
\begin{gather*}
\eta_Y(h)\leq\mathcal{C}(\sqrt{2}\hat{\eta}^{(i)}_X(h-2|k|))^{\frac{\delta}{1+\delta}},
\end{gather*}
where $\mathcal{C}$ is a constant independent of $h$ and $\hat\eta_X^{(i)}(\cdot)$ is defined as in Proposition \ref{proposition:vectorgeneralmmaweaklydep}.
Furthermore, in the finite variation case and for $\hat\eta_X^{(iii)}(\cdot)$ defined as in Proposition \ref{proposition:vectorgeneralmmaweaklydep}, it holds
\begin{gather*}
\eta_Y(h)\leq\mathcal{C}(2\hat{\eta}^{(iii)}_X(h-2|k|))^{\frac{\delta}{1+\delta}}.
\end{gather*}
\end{Proposition}
In the following, we give asymptotic properties of the sample autocovariances (\ref{equation:sampleautocovariance}).
\begin{Corollary}\label{corollary:sampleautocovarianceeta}
Let $(X_u)_{u\in\Z^m}$ be a real-valued MMAF as defined in (\ref{equation:discretegeneralMMAfield}) such that $\int_{\norm{x}>1}\norm{x}^{4+\delta}\nu(dx)<\infty$, $\gamma+\int_{\norm{x}>1}x\nu(dx)=0$ and $f:S\times\R^m\ra M_{1\times d}(\R)$ satisfies $f\in L^2(S\times\R^{m},\pi\otimes\lambda)\cap L^{4+\delta}(S\times\R^{m},\pi\otimes\lambda)$ for some $\delta>0$. If $\hat{\eta}^{(i)}_X(h)=\mathcal{O}(h^{-\beta})$, with $\hat\eta_X^{(i)}$ defined as in Proposition \ref{proposition:vectorgeneralmmaweaklydep}, and $\beta>m\max\left(2,\left(1+\frac{1}{\delta}\right)\right)(\frac{3+\delta}{2+\delta})$, then
\begin{gather*}
\Sigma=\sum_{l\in\Z^m}Cov\left(\left(\begin{array}{c}
Y_{0,0}\\
\vdots \\
Y_{0,k}
\end{array}\right),
\left(\begin{array}{c}
Y_{l,0}\\
\vdots \\
Y_{l,k}
\end{array}\right)
\right)=\sum_{l\in\Z^m}Cov\left(\left(\begin{array}{c}
X_0X_0\\
\vdots \\
X_0X_k
\end{array}\right),
\left(\begin{array}{c}
X_lX_l\\
\vdots \\
X_lX_{l+k}
\end{array}\right)
\right)
\end{gather*}
is finite, positive semidefinite and
\begin{gather*}
\frac{1}{|E_{n-\tilde{k}}|^{\frac{1}{2}}}\sum_{j\in E_{n-\tilde{k}}}
\left(\begin{array}{c}
Y_{j,0}\\
\vdots \\
Y_{j,k}
\end{array}\right)
\underset{N\ra\infty}{\xrightarrow{\makebox[2em][c]{d}}}N\left(0,\Sigma\right),
\end{gather*}
where $\tilde{k}=|k|$.
\end{Corollary}
\begin{proof}
Analogous to Theorem \ref{theorem:etasamplemean} we obtain the stated convergence using Proposition \ref{proposition:deltavererbungeta}.
\end{proof}
\begin{Remark}
Note that for $m=1$ Theorem \ref{theorem:etasamplemean} improves the only existing central limit theorem for MMA processes based on $\eta$-weak dependence (see \cite[Theorem 4.1]{CS2018}) by reducing the necessary decay of the $\eta$-coefficients from $\beta>4+\frac{2}{\delta}$ to $\beta>\max\left(2,\left(1+\frac{1}{\delta}\right)\right)$.
\end{Remark}
\begin{Remark}
Let $X$ be an $(A,\Lambda)$-influenced MMAF satisfying the conditions of Proposition \ref{proposition:mmathetaweaklydep} (i). Then $X$ is $\theta$-lex- and $\eta$-weakly dependent with the same weak dependence coefficients and both the asymptotic results in Section \ref{sec3-4} and \ref{sec3-6} can be applied. \\
Note that the asymptotic results in Section \ref{sec3-4} hold under weaker decay demands for the weak dependence coefficients compared to the results in Section \ref{sec3-6}.
\end{Remark}
\subsection{Example of $(A,\Lambda)$-influenced MMAF: MSTOU processes}
\label{sec3-7}
We apply the developed asymptotic theory to mixed spatio-temporal Ornstein-Uhlenbeck (MSTOU) processes. MSTOU processes were introduced in \cite{NV2017} and extend the spatio-temporal Ornstein-Uhlenbeck (STOU) processes (see \cite{BS2004,NV2015}) by additionally mixing the mean reversion parameter. This extension allows versatile modeling of short-range as well as long-range dependence structures in space-time.
In the following, we will treat the temporal and spatial domains separately. MSTOU processes are an example of $(A,\Lambda)$-influenced MMAF where the sphere of influence is a family of ambit sets, i.e. $A_t(x)\subset \R\times\R^m$ such that
\begin{gather}\label{equation:ambitset}
\begin{cases}
A_t(x)=A_0(0)+(t,x), \text{ (Translation invariant)}\\
A_s(x)\subset A_t(x),\\
A_t(x)\cap((t,\infty))\times\R^m=\emptyset.\text{ (Non-anticipative)}.
\end{cases}
\end{gather}
\begin{Proposition}
\label{mstou}
Let $\Lambda$ be a real-valued L\'evy basis on $(0,\infty)\times\R\times\R^m$ with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$ such that $\int_{|x|>1}x^2\nu(dx)<\infty$ and $f(\lambda)$ be the density function of $\pi$ (i.e. the mean reversion parameter $\lambda$) with respect to the Lebesgue measure. Furthermore, let $A=(A_t(x))_{(t,x)\in\R\times\R^m}$ be an ambit set. If
\begin{gather*}\label{equations:MSTOUcond1}
\int_0^\infty \int_{A_t(x)} \exp(-\lambda(t-s)) \,ds\, d\xi\, f(\lambda) \,d\lambda<\infty,
\end{gather*}
then the $(A,\Lambda)$-influenced MMAF
\begin{gather*}\label{equation:MSTOU}
Y_t(x)=\int_0^\infty \int_{A_t(x)}\exp(-\lambda(t-s)) \, \Lambda(d\lambda,ds,d\xi),\quad (t,x)\in\R\times\R^d
\end{gather*}
is well defined and we call $Y_t(x)$ a mixed spatio-temporal Ornstein-Uhlenbeck (MSTOU) process.
\end{Proposition}
\begin{proof}
Follows immediately from \cite[Corollary 1]{NV2017}.
\end{proof}
To calculate the assumptions under which the asymptotic results of Section \ref{sec3-3} hold, it becomes necessary to specify a family of ambit sets. In the following, we will consider c-class MSTOU processes, a sub-class of the $g$-class MSTOU processes as given in \cite[Definition 9]{NV2017}.
\begin{Definition}
Let $Y_t(x)$ be an MSTOU process as in Proposition \ref{mstou}. If, for a constant
$c>0$,
\begin{gather*}
A_t(x)=\{ (s,\xi): s\leq t, \norm{x-\xi}\leq c|t-s| \},
\end{gather*}
then $Y_t(x)$ is called a c-class MSTOU process.
A c-class MSTOU process is well defined if
\begin{gather}
\int_0^\infty \frac{1}{\lambda^{m+1}}f(\lambda) d\lambda<\infty.
\label{equations:gclassMSTOUcond1}
\end{gather}
\end{Definition}
The next theorem expresses the $\theta$-lex coefficients of c-class MSTOU processes in terms of the characteristic quadruplet of the driving L\'evy basis. We note that $A_0(0)$ is a closed convex proper cone with Lebesgue measure strictly greater than zero satisfying (\ref{condition:scalarproduct}). From (\ref{equation:psi}) it follows that $\psi(h)=\frac{1}{\sqrt{c^2+1}}\frac{h}{\sqrt{m+1}}$.
\begin{Theorem}
Let $(Y_t(x))_{(t,x)\in\R\times\R^m}$ be a c-class MSTOU process and $(\gamma,\Sigma,\nu,\pi)$ the characteristic quadruplet of its driving L\'evy basis. Moreover, let $f(\lambda)$ be the density of $\pi$ with respect to the Lebesgue measure.
\begin{enumerate}[(i)]
\item If $\int_{|x|>1} x^2\nu(dx)<\infty$ and $\gamma+\int_{|x|>1}x \,\nu(dx)=0$, then $Y_t(x)$ is $\theta$-lex-weakly dependent. Let $c\in(0,1]$, then for
\begin{alignat*}{2}
m&=1,\qquad &&\theta_Y(h)\leq\Bigg(2c\Sigma_\Lambda\int_0^\infty \frac{(2\lambda \psi(h)+1)}{\lambda^2}e^{-2\lambda \psi(h)}f(\lambda)d\lambda \Bigg)^{\frac{1}{2}},\\
\textrm{and for} & && \\
m& \geq 2,\qquad &&\theta_Y(h)\leq2\left(V_m(c)\Sigma_\Lambda \int_0^\infty
\frac{m!\sum_{k=0}^m \frac{1}{k!}(2\lambda \psi(h))^k }{(2\lambda)^{m+1}} e^{-2\lambda \psi(h)}
f(\lambda)d\lambda\right)^{\frac{1}{2}}.
\end{alignat*}
Let $c>1$, then for
\begin{alignat*}{2}
m&=1,\qquad &&\theta_Y(r) \leq 2 \Big( c\Sigma_\Lambda\int_0^\infty \frac{(2\lambda \frac{\psi(h)}{c}+1)}{2\lambda^2}e^{-2\lambda \frac{\psi(h)}{c}}ds f(\lambda)d\lambda\Big)^{\frac{1}{2}},\\
\textrm{and for} & && \\
m&\geq 2,\qquad &&\theta_Y(h)\leq2\Bigg(V_m(c)\Sigma_\Lambda \int_0^\infty \frac{m!\sum_{k=0}^m \frac{1}{k!}(2\frac{\lambda \psi(h)}{c})^k }{(2\lambda)^{m+1}} e^{-2\frac{\lambda \psi(h)}{c}}f(\lambda)d\lambda\Bigg)^{\frac{1}{2}}.
\end{alignat*}
\item If $\int_{\R} |x| \,\nu(dx)<\infty$, $\Sigma=0$ and $\gamma_0$ as defined in (\ref{equation:gammazero}), then $Y_t(x)$ is $\theta$-lex-weakly dependent. Let $c \in (0,1]$, then for
\begin{alignat*}{2}
m&\in\N,\qquad && \theta_Y(h)\leq2V_m(c) \gamma_{abs} \left(\int_0^\infty \frac{m!\sum_{k=0}^m \frac{1}{k!}(\lambda \psi(h))^k }{\lambda^{m+1}} e^{-\lambda \psi(h)}f(\lambda)d\lambda\right),
\end{alignat*}
whereas for $c>1$ and
\begin{alignat*}{2}
m&\in\N, \qquad && \theta_Y(h)\leq 2 V_m(c)\gamma_{abs}\Bigg(\int_0^\infty \frac{m!\sum_{k=0}^m \frac{1}{k!}(\frac{\lambda \psi(h)}{c})^k }{\lambda^{m+1}} e^{-\frac{\lambda \psi(h)}{c}}f(\lambda)d\lambda\Bigg).
\end{alignat*}
\end{enumerate}
The results above hold for all $h>0$, where $\gamma_{abs}=|\gamma_0|+\int_\R|x|\nu(dx)$, $V_m(c)=\frac{\left(\Gamma(\frac{1}{2})c\right)^m }{\Gamma(\frac{m}{2}+1)}$ denotes the volume of the $m$-dimensional ball with radius $c$, $\psi(h)=\frac{1}{\sqrt{c^2+1}}\frac{h}{\sqrt{m+1}}$ and $\Sigma_\Lambda =\Sigma+\int_{\R}x^2\nu(dx)$.
\end{Theorem}
\begin{proof}
\begin{enumerate}
\item [(i)] Let us consider the case $m=1$. From Proposition \ref{proposition:mmathetaweaklydep} we deduce
\begin{gather}\label{eq:MSTOUtemp}
\theta_Y(h)\leq 2 \Big( \Sigma_\Lambda\int_0^\infty \int_{A_0(0)\cap V_{0}^{\psi(h)}} \exp(2s\lambda) ds \, d\xi \, f(\lambda)d\lambda\Big)^{\frac{1}{2}}.
\end{gather}
As first step, one has to evaluate the truncated integration set $A_0(0)\cap V_{0}^{\psi(h)}$. Depending on the width of $A_0(0)$, we distinguish the two cases illustrated in the following figures. Figure \ref{Plotcases1} and \ref{Plotcases2} consider the case $c\in(0,1]$ and Figure \ref{Plotcases3} and \ref{Plotcases4} cover the case $c>1$.
\begin{figure}[H]
\begin{minipage}[H]{3cm}
\centering
\includegraphics[width=3.5cm]{ccases1.pdf}\label{figurePlotcases1}\\
\caption{\small{Integration set $A_0(0)$ with $(V_{(0,0)}^h)^c$ for $c=\frac{1}{\sqrt{2}}$ and $h=4\sqrt{3}$.}}\label{Plotcases1}
\end{minipage}
\hfill
\begin{minipage}[H]{3cm}
\centering
\includegraphics[width=3.5cm]{ccases2.pdf}\label{figurePlotcases2}\\
\caption{\small{Truncated set $A_0(0)\cap V_{(0,0)}^{\psi(h)}$ for $c=\frac{1}{\sqrt{2}}$ and $h=4\sqrt{3}$.}}
\label{Plotcases2}
\end{minipage}
\hfill
\begin{minipage}[H]{3cm}
\centering
\includegraphics[width=3.5cm]{ccases3.pdf}\label{figurePlotcases3}\\
\caption{\small{Integration set $A_0(0)$ with $(V_{(0,0)}^h)^c$ for $c=\sqrt{2}$ and $h=4\sqrt{6}$.}}
\label{Plotcases3}
\end{minipage}
\hfill
\begin{minipage}[H]{3cm}
\centering
\includegraphics[width=3.5cm]{ccases4.pdf}\label{figurePlotcases4}\\
\caption{\small{Truncated set $A_0(0)\cap V_{(0,0)}^{\psi(h)}$ for $c=\sqrt{2}$ and $h=4\sqrt{6}$.}}
\label{Plotcases4}
\end{minipage}
\end{figure}
Let $c\in (0,1]$, then (\ref{eq:MSTOUtemp}) is equal to
\begin{align*}
&2 \Big(\! \Sigma_\Lambda\!\!\int_0^\infty\!\! \int_{-\infty}^{-\psi(h)}\!\! \int_{\|\xi\|\leq cs} d\xi \, e^{2s\lambda} ds f(\lambda)d\lambda\Big)^{\frac{1}{2}}\!\! =2 \Big( \Sigma_\Lambda\int_0^\infty \!\!\int_{-\infty}^{-\psi(h)} (-2cs) e^{2s\lambda} ds f(\lambda)d\lambda\Big)^{\frac{1}{2}} \nonumber\\
&= \Big( 2c\Sigma_\Lambda\int_0^\infty \frac{(2\lambda \psi(h)+1)}{\lambda^2}e^{-2\lambda \psi(h)} f(\lambda)d\lambda\Big)^{\frac{1}{2}}.\nonumber
\end{align*}
The integral $\int_{\|\xi\|\leq cs} d\xi$ is the volume of an $m$-dimensional ball of radius $cs$, which for $m=1$ is equal to $-2cs$. For $c>1$ we can bound (\ref{eq:MSTOUtemp}) by
\begin{align*}
2 \Big( \Sigma_\Lambda\!\!\int_0^\infty\! \!\!\int_{-\infty}^{-\frac{\psi(h)}{c}}\!\!\!\! \!(-2cs) e^{2s\lambda} ds f(\lambda)d\lambda \Big)^{\frac{1}{2}}\!
=2 \Big( \!c\Sigma_\Lambda\!\!\int_0^\infty \!\! \frac{(2\lambda \frac{\psi(h)}{c}+1)}{2\lambda^2}e^{-2\lambda \frac{\psi(h)}{c}}ds f(\lambda)d\lambda\Big)^{\frac{1}{2}}.
\end{align*}
In a similar way, one can derive the $\theta$-lex coefficients for $m \geq 2$.
\item [(ii)] Analogous to (i).
\end{enumerate}
\end{proof}
We now give explicit computations of the $\theta$-lex-coefficients of a c-class MSTOU process in the case in which the mean reverting parameter $\lambda$ is gamma distributed. For a $Gamma(\alpha,\beta)$ distributed mean reversion parameter $\lambda$, i.e. $f(\lambda)=\frac{\beta^\alpha}{\Gamma(\alpha)}\lambda^{\alpha-1}e^{-\beta\lambda}$ $\mathbb{1}_{[0,\infty)}(\lambda)$, the c-class MSTOU process is well defined if $\alpha>m+1$ and $\beta>0$ due to condition (\ref{equations:gclassMSTOUcond1}).
\begin{Theorem}
Let $(Y_t(x))_{(t,x)\in\R\times\R^m}$ be a c-class MSTOU process and $(\gamma,\Sigma,\nu,\pi)$ the characteristic quadruplet of its driving L\'evy basis. Moreover, let the mean reversion parameter $\lambda$ be $Gamma(\alpha,\beta)$ distributed with $\alpha >m+1$ and $\beta>0$.
\begin{enumerate}[(i)]
\item If $\int_{|x|>1} x^2 \,\nu(dx)<\infty$ and $\gamma+\int_{|x|>1}x\nu(dx)=0$, then $Y_t(x)$ is $\theta$-lex-weakly dependent. Let $c\in[0,1]$, then for
\begin{alignat*}{2}
m&=1,\quad && \theta_Y(h)\leq2\left(\frac{c\Sigma_\Lambda \beta^\alpha}{2\Gamma(\alpha)} \left(\frac{\Gamma(\alpha-2)}{(2\psi(h)+\beta)^{\alpha-2}}+\frac{2\psi(h)\Gamma(\alpha-1)}{(2\psi(h)+\beta)^{\alpha-1}} \right)\right)^{\frac{1}{2}}, \\
\textrm{and for} & && \\
m& \geq 2,\quad &&\theta_Y(h)\leq2\left(V_m(c)\frac{m!\Sigma_\Lambda \beta^{\alpha} }{2^{m+1}} \sum_{k=0}^m \frac{(2\psi(h))^k}{k!(2\psi(h)+\beta)^{\alpha-m-1+k}}\frac{\Gamma(\alpha-m-1+k)}{\Gamma(\alpha)}\right)^{\frac{1}{2}}.
\end{alignat*}
Let $c>1$, then for
\begin{alignat*}{2}
m&\in\N,\ \ &&\theta_Y(h)\leq2\Bigg(V_m(c)\frac{m!\Sigma_\Lambda \beta^{\alpha} }{2^{m+1}} \sum_{k=0}^m \frac{\left(\frac{2\psi(h)}{c}\right)^k}{k!\left(\frac{2\psi(h)}{c}+\beta\right)^{\alpha-m-1+k}}\frac{\Gamma(\alpha-m-1+k)}{\Gamma(\alpha)}\Bigg)^{\frac{1}{2}}.
\end{alignat*}
The above implies that, in general, $\theta_Y(h)=\mathcal{O}(h^{\frac{(m+1)-\alpha}{2}})$.
\item If $\int_{\R} |x| \,\nu(dx)<\infty$, $\Sigma=0$ and $\gamma_0$ as defined in (\ref{equation:gammazero}), then $Y_t(x)$ is $\theta$-lex-weakly dependent. Let $c \in (0,1]$, then for
\begin{alignat*}{2}
m&\in\N,\qquad&& \theta_Y(h)\leq2V_m(c) m! \beta^{\alpha} \gamma_{abs} \sum_{k=0}^m \frac{\psi(h)^k}{k!(\psi(h)+\beta)^{\alpha-m-1+k}}\frac{\Gamma(\alpha-m-1+k)}{\Gamma(\alpha)},
\end{alignat*}
whereas for $c>1$ and
\begin{alignat*}{2}
m&\in\N,\qquad &&\theta_Y(h)\leq2V_m(c)d! \beta^{\alpha}\gamma_{abs} \sum_{k=0}^m \frac{\left(\frac{\psi(h)}{c}\right)^k}{k!\left(\frac{\psi(h)}{c}+\beta\right)^{\alpha-m-1+k}}\frac{\Gamma(\alpha-m-1+k)}{\Gamma(\alpha)},
\end{alignat*}
where $\gamma_{abs}=|\gamma_0|+\int_\R|x|\nu(dx)$, and $V_m(c)$ denotes the volume of the $m$-dimensional ball with radius $c$. the above implies that, in general, $\theta_Y(h)=\mathcal{O}(h^{(m+1)-\alpha})$.
\end{enumerate}
\end{Theorem}
This implies the following sufficient conditions for the asymptotic normality of the sample mean and the sample autocovariance function.
\begin{Corollary}\label{corollary:cltMSTOU}
Let $(Y_t(x))_{(t,x)\in\R\times\R^m}$ be a c-class MSTOU process and $(\gamma,\Sigma,\nu,\pi)$ the characteristic quadruplet of its driving L\'evy basis. Moreover, let the mean reversion parameter $\lambda$ be $Gamma(\alpha,\beta)$ distributed with $\alpha>m+1$ and $\beta>0$.
\begin{enumerate}
\item[(i)] If $\gamma+\int_{ |x|>1}x\nu(dx)=0$, $\int_{|x|>1} |x|^{2+\delta}\nu(dx)<\infty$ for some $\delta>0$ and $\alpha>(m+1) \left(3+\frac{2}{\delta} \right)$, then the sample mean of $Y_t(x)$ as defined in (\ref{equation:samplemean}) is asymptotically normal.
\item[(ii)] If $\gamma+\int_{ |x|>1}x\nu(dx)=0$, $\int_{|x|>1} |x|^{4+\delta}\nu(dx)<\infty$ for some $\delta>0$ and $\alpha>(m+1)\left(\frac{3+\delta}{2+\delta}\right)\left(3+\frac{2}{\delta}\right)$, then the sample autocovariances as defined in (\ref{equation:sampleautocovariance}) are asymptotically normal.
\end{enumerate}
\end{Corollary}
\begin{Corollary}\label{corollary:cltMSTOU2}
Let $(Y_t(x))_{(t,x)\in\R\times\R^m}$ be a c-class MSTOU process and $(\gamma,\Sigma,\nu,\pi)$ the characteristic quadruplet of its driving L\'evy basis. Moreover, let the mean reversion parameter $\lambda$ be $Gamma(\alpha,\beta)$ distributed such that $\alpha>m+1$ and $\beta>0$.
\begin{enumerate}
\item [(i)] If $\int_{\R} |x|\nu(dx)<\infty$, $\Sigma=0$, and $\alpha>(m+1) \left(2+\frac{2}{\delta} \right)$, then the sample mean of $Y_t(x)$ as defined in (\ref{equation:samplemean}) is asymptotically normal.
\item [(ii)]If $\int_{\R} |x|\nu(dx)<\infty$, $\Sigma=0$, $\int_{|x|>1} |x|^{4+\delta}\nu(dx)<\infty$ for some $\delta>0$ and $\alpha>(m+1)\left(\frac{3+\delta}{2+\delta}\right)\left(2+\frac{2}{\delta}\right)$, then the sample autocovariances as defined in (\ref{equation:sampleautocovariance}) are asymptotically normal.
\end{enumerate}
\end{Corollary}
\begin{Remark}
Since the c-class MSTOU processes satisfy the assumptions of Theorem \ref{theorem:thetasamplemeanspecial}, we can derive asymptotic normality of its sample mean under the weaker assumptions $E[Y_t(x)^{2}]<\infty$ and $\alpha>3(m+1)$.
\end{Remark}
We conclude with some remarks regarding the short and long range dependence of an MSTOU process.
\begin{Definition}
A stationary random field $Y=(Y_t(x))_{(t,x)\in\R\times\R^m}$ is said to have temporal short-range dependence if
\begin{gather*}
\int_0^{\infty}Cov(Y_t(x),Y_{t+\tau}(x))d\tau<\infty,
\end{gather*}
and temporal long-range dependence if the integral is infinite.\\
If $Cov(Y_t(x),Y_t(x+m_x))=C(|m_x|)$ for all $m_x\in\R^m$ and a positive definite function $C$ the random field $Y$ is called isotropic. Now, an isotropic random field is said to have spatial short-range dependence if
\begin{gather*}
\int_0^{\infty}C(r)dr<\infty.
\end{gather*}
\end{Definition}
We have that an MSTOU process is a stationary and isotropic random field, see Theorem 5 \cite{NV2017}.
By assuming a $Gamma(\alpha,\beta)$-distributed mean reversion parameter $\lambda$, we have the following results, as shown in Section 6 \cite{CS2018} and Section 3.3 \cite{NV2017}:
\begin{enumerate}[(i)]
\item For $m=0$, we have that $Y$ is a supOU process which is well-defined for $\alpha >1$ and $\beta >0$. Thus, we obtain a long-memory process for $1<\alpha \leq 2$ and a short memory one for $\alpha > 2$.
\item For $m=1$, $Y$ is well-defined if $\alpha>2$ and $\beta>0$. $Y$ exhibits temporal as well as spatial long-range dependence for $2<\alpha\leq3$. If $\alpha>3$ we observe temporal and spatial short-range dependence.
\item For $m=3$, $Y$ is well-defined if $\alpha >4$ and $\beta >0$. $Y$ exhibits temporal as well as spatial long-range dependence for $4<\alpha\leq 5$. If $\alpha>5$ we observe temporal and spatial short-range dependence.
\end{enumerate}
It is then easy to see that the assumptions in the Corollaries \ref{corollary:cltMSTOU} and \ref{corollary:cltMSTOU2} imply that we are in the realm of short-range dependence.
\begin{Remark}(GMM estimator)
For $m=0$, a consistent GMM estimator for the supOU process is defined in \cite{STW2015}. In \cite{CS2018}, the authors show asymptotic normality of the estimator and that if the underlying L\'evy process is of finite variation and all moments exist, then the GMM estimator is asymptotic normally distributed for $\alpha >2$.
For $m\geq 1$, a consistent GMM estimator for a c-class MSTOU process is introduced in \cite{NV2017}. The results in Corollaries \ref{corollary:cltMSTOU} and \ref{corollary:cltMSTOU2} should pave the way for an analysis of the asymptotic normality of the GMM estimator defined in \cite{NV2017} using arguments similar to \cite{CS2018}.
For example, when $m=1$, in the finite variation case and when all moments exist, we can apply our results to short-range dependent MSTOU processes with $\alpha>4$.
\end{Remark}
\subsection{Example of non-influenced MMAF: L\'evy-driven CARMA fields}
\label{sec3-8}
We conclude the section by showing that our developed asymptotic theory can be applied to the class of L\'evy-driven CARMA fields defined on $\R^m$.
CARMA (continuous autoregressive moving average) fields are an extension of the well-known CARMA processes (see, e.g., \cite{B2014} for a comprehensive introduction) and have been introduced in \cite{B2019b, BM2017,KP2019, P2018}.
In \cite{BM2017}, the authors define CARMA fields as isotropic random fields
\begin{gather}\label{equation:isotropicCARMA}
Y(t)=\int_{\R^m}g(t-s)dL(s),\quad t\in\R^m,
\end{gather}
where $g$ is a radially symmetric kernel and $L$ a real-valued L\'evy basis on $\R^m$.
When the L\'evy basis $L$ has a finite second-order structure, the CARMA fields generate a rich family of isotropic covariance functions on $\R^m$, which are not necessarily non-negative or monotone.
On the other hand, in \cite{P2018}, the author defines CARMA(p,q) fields based on a system of stochastic partial differential equations. For $0 \leq q <p$, the mild solution of the system is called a causal CARMA field and is given by
\begin{gather}\label{equation:causalCARMA}
Y(t)= b^T \int_{-\infty}^{t_1}\cdots \int_{-\infty}^{t_m} e^{A_1(t_1-s_1)}\cdots e^{A_m(t_m-s_m)}c~dL(s), \quad (t_1,\ldots,t_m)\in\R^m,
\end{gather}
where $A_1,\ldots,A_m$ are companion matrices, $L$ is a real-valued L\'evy basis on $\R^m$, $b=(b_0,\ldots,b_{p-1})^T \in \R^p$ with $b_q \neq 0$ and $b_i=0$ for $i >q$ and $c=(0,\ldots,0,1)^T\in\R^p$, see \cite[Definition 3.3]{P2018}.\\
In \cite{B2019b}, the author shows the existence of a mild solution for the CARMA stochastic partial differential equation, c.f. \cite[equation (1.7)]{B2019b}, in \cite[Theorem 5.3]{B2019b}. The causal CARMA fields presented in \cite{P2018} can be seen as a special case of the CARMA random fields defined in \cite{B2019b}. A more subtle relationship exists between the definition of CARMA field in \cite{B2019b} and \cite{BM2017} just when $m$ is odd, see \cite[Section 7]{B2019b}.
In general, our framework can be applied to the class of CARMA fields introduced in \cite{B2019b} and \cite{BM2017} when the assumption of the theorem below holds.
\begin{Theorem}\label{theorem:levydrivenexpdecaykernel}
Let $L$ be an $\R^d$-valued L\'evy basis with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$ such that $\int_{\norm{x}>1}\norm{x}^2\nu(dx)<\infty$ and $\gamma+\int_{\norm{x}>1}x\nu(dx)=0$. Let $g:\R^m\ra M_{n\times d}(\R)$ such that $g$ is exponentially bounded in norm, i.e. there exists $M,K\in\R^+$ such that
\begin{gather}\label{equation:expdecaykernel}
\norm{g(t)}^2\leq M e^{-K\norm{t}}, \text{ for all }t\in\R^m.
\end{gather}
Then, the moving average field $X_t=\int_{\R^m}g(t-s)L(ds)$, $t\in\R^m$ is an $\eta$-weakly dependent field with exponentially decaying $\eta$-coefficients.
\end{Theorem}
Due to the equivalence of norms, the result does not depend on a specific choice of norms.
\begin{proof}
See Section \ref{sec5}.
\end{proof}
\begin{Remark}
Since the kernels in (\ref{equation:isotropicCARMA}) and (\ref{equation:causalCARMA}) satisfy equation (\ref{equation:expdecaykernel}), for example, we can show that these fields are $\eta$-weakly dependent by applying Theorem \ref{theorem:levydrivenexpdecaykernel}.
\end{Remark}
\section{Ambit fields}
\label{sec4}
In the following, we will briefly introduce stationary ambit fields. We discuss weak dependence properties of such fields and give sufficient conditions for the applicability of the results in Section \ref{sec2-3}.
\subsection{The ambit framework}
\label{sec4-1}
Let $A_t(x)\subset\R\times\R^m$ for $(t,x)\in\R\times\R^m$ be an ambit set as defined in (\ref{equation:ambitset}). By $\mathcal{P}'$ we denote the usual predictable $\sigma$-algebra on $\R$, i.e. the $\sigma$-algebra generated by all left-continuous adapted processes. Then, a random field $X:\Omega\times\R\times\R^m\rightarrow\R$ is called predictable if it is measurable with respect to the $\sigma$-algebra $\mathcal{P}$ defined by $\mathcal{P}=\mathcal{P}' \otimes\mathcal{B}(\R^m)$.
\begin{Definition}\label{definition:ambitfield}
Let $\Lambda$ be a real-valued L\'evy basis on $\R\times\R^m$ with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$, $\sigma$ a predictable stationary random field on $\R\times\R^m$ independent of $\Lambda$.
Furthermore, let $l:\R^m\times\R\rightarrow\R$ be a measurable function and $A_t(x)$ an ambit set. We assume that $f(\xi,s)=\mathbb{1}_{A_t(x)}(\xi,s)l(\xi,s)\sigma_s(\xi)$ satisfies (\ref{equation:intcond1}), (\ref{equation:intcond2}) and (\ref{equation:intcond3}) almost surely.
Then, the random field
\begin{gather}\label{eq:ambitfield}
\begin{align}
\begin{aligned}
Y_t(x)=\int_{A_t(x)} l(x-\xi,t-s) \sigma_s(\xi) \Lambda(d\xi,ds),~ (t,x)\in\R\times\R^m,
\end{aligned}
\end{align}
\end{gather}
is called an ambit field and it is stationary (see p.~185 \cite{BBV2018}).
\end{Definition}
\begin{Remark}\label{remark:stochintegrand}
Ambit fields require us to define integrals with respect to L\'evy bases where the integrand is stochastic. Although the integration theory from Rajput and Rosinski just enables us to define stochastic integrals with respect to deterministic integrands \cite{RR1989}, one can extend this theory to stochastic integrands which are predictable and independent of the L\'evy basis. We can condition on the $\sigma$-algebra generated by the field $\sigma$ and use again the integration theory introduced in \cite{RR1989}. Then, such integrals are well defined if the kernel function satisfies the sufficient conditions (\ref{equation:intcond1}), (\ref{equation:intcond2}) and (\ref{equation:intcond3}) almost surely.
Allowing for dependence between the volatility field and the L\'evy basis demands the use of a different integration theory as presented in \cite[Section 1.2.1]{BBPV2014}, \cite[Proposition 39]{BBV2018}, \cite[Theorem 3.2]{BGP2013} and \cite{CK2015}.
\end{Remark}
We conclude this section by giving explicit formulas for the first and second moment of an ambit field.
\begin{Proposition}\label{proposition:ambitmoments}
Let $Y$ be an ambit field as defined in (\ref{eq:ambitfield}) driven by a real-valued L\'evy basis with characteristic quadruplet $(\gamma,\Sigma,\nu,\pi)$ and $\Lambda$-integrable kernel function $f(\xi,s)=\mathbb{1}_{A_t(x)}(\xi,s)l(\xi,s)\sigma_s(\xi)$, where $\sigma$ is predictable, stationary and independent of $\Lambda$.
\begin{enumerate}[(i)]
\item If $E[|Y_t(x)|]<\infty$, the first moment of $Y$ is given by
\begin{align*}
E[Y_t(x)]= \mu_\Lambda E[\sigma_t(x)] \int_{A_t(x)}l(x-\xi,t-s) d\xi\, ds,
\end{align*}
where $\mu_\Lambda =\gamma+\int_{|x|\geq1}x\nu(dx)$.
\item If $E[Y_t(x)^2]<\infty$, it holds
\begin{align*}
Var(Y_t(x))= &\Sigma_\Lambda E[\sigma_t(x)^2]\int_{A_t(x)} l(x-\xi,t-s)^2 d\xi\, ds \\
&+\mu_\Lambda ^2 \int_{A_t(x)}\int_{A_t(x)} l(x-\xi,t-s)l(x-\tilde{\xi},t-\tilde{s}) \rho(s,\tilde{s},\xi,\tilde{\xi}) d\xi\, ds\, d\tilde{\xi}\, d\tilde{s}\, \text{ and}\\
Cov(Y_t(x),Y_{\tilde{t}}(\tilde{x}))=&\Sigma_\Lambda E[\sigma_t(x)^2] \int_{A_t(x)\cap A_{\tilde{t}}(\tilde{x})} l(x-\xi,t-s)l(\tilde{x}-\xi,\tilde{t}-s) d\xi\, ds\, \\
&+\mu_\Lambda ^2 \int_{A_t(x)}\int_{A_{\tilde{t}}(\tilde{x})} l(x-\xi,t-s)l(\tilde{x}-\tilde{\xi},\tilde{t}-\tilde{s}) \rho(s,\tilde{s},\xi,\tilde{\xi}) d\xi\, ds\, d\tilde{\xi}\, d\tilde{s},
\end{align*}
where $\Sigma_\Lambda =\Sigma+\int_{\R}x^2\nu(dx)$ and $\rho(s,\tilde{s},\xi,\tilde{\xi})= E[\sigma_s(\xi)\sigma_{\tilde{s}}(\tilde{\xi})]-E[\sigma_s(\xi)]E[\sigma_{\tilde{s}}(\tilde{\xi})]$.
\end{enumerate}
\end{Proposition}
\begin{proof}
Immediate from \cite[Proposition 41]{BBV2018}.
\end{proof}
\subsection{Weak dependence properties of ambit fields}
\label{sec4-2}
Let us consider a stationary ambit field $Y=(Y_t(x))_{(t,x)\in\R\times\R^m}$ as defined in (\ref{eq:ambitfield}).
To analyze the covariance structure of $Y$, it becomes necessary to specify a model for $\sigma$.
In \cite{BBV2012} the authors proposed to model $\sigma$ by kernel-smoothing of a homogeneous L\'evy basis, i.e. a moving average random field
\begin{gather}
\label{sigma}
\sigma_t(x)=\int_{A_t^\sigma(x)}j(x-\xi,t-s) \Lambda^\sigma(d\xi,ds),
\end{gather}
where $\Lambda^\sigma$ is a real valued L\'evy basis independent of $\Lambda$ with characteristic quadruplet $(\mu_{\sigma},\Sigma_{\sigma},\nu_{\sigma},\pi_{\sigma})$, $A^\sigma=(A^\sigma_t(x))_{(t,x)\in\R\times\R^m}$ an ambit set as defined in (\ref{equation:ambitset}) and $j$ a real valued $\Lambda^\sigma$-integrable function. In the following, we extend this model and assume $\sigma$ to be an $(A^\sigma,\Lambda^\sigma)$-influenced MMAF, i.e.
\begin{gather}\label{equation:volatilityfield}
\sigma_t(x)=\int_S\int_{A_t^\sigma(x)}j(A,x-\xi,t-s) \Lambda^\sigma(dA,d\xi,ds).
\end{gather}
\begin{Proposition}\label{proposition:ambitthetaweaklydep}
Let $Y=(Y_t(x))_{(t,x)\in\R\times\R^m}$ be an ambit field as defined in (\ref{eq:ambitfield}) with $\sigma=(\sigma_t(x))_{(t,x)\in\R\times\R^m}$ being a predictable $(A^\sigma,\Lambda^\sigma)$-influenced MMAF as defined in (\ref{equation:volatilityfield}) and such that $A_0(0)$ and $A_0^\sigma(0)$ satisfy (\ref{condition:scalarproduct}), $j\in L^1(S\times\R\times\R^m, \pi\otimes\lambda) \cap L^2(S\times\R^m\times\R, \pi\otimes\lambda)$, where $\lambda$ indicates the Lebesgue measure on $\R^{m+1}$, and $\int_{|x|>1}|x|^2\nu_\sigma(dx)<\infty$.
\begin{enumerate}[(i)]
\item If $l\in L^2(\R\times\R^m)$, $\int_{|x|>1}|x|^2\nu(dx)<\infty$ and $\gamma + \int_{|x|>1}x\nu(dx)=0$, then $Y$ is $\theta$-lex-weakly dependent with $\theta$-lex-coefficients $\theta_Y(h)$ satisfying
\begin{gather}\label{equation:thetalexcoefficientsambitfield}
\begin{aligned}
\theta_Y(h)\leq&2 \Big(\Sigma_\Lambda E[\sigma_0(0)^2]
\int_{A_0(0) \cap V_{(0,0)}^{\psi(h)}}l(-\xi,-s)^2 d\xi ds\Big)^{\frac{1}{2}}\\
&+ 2\Bigg( \Bigg(\Sigma_{\Lambda^\sigma} \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^{\psi(h)}} j(A,-\xi,-s)^2 d\xi ds \pi(dA)\\
&+ \mu_{\Lambda^\sigma}^2 \Bigg(\!\int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^{\psi(h)}} j(A,-\xi,-s) d\xi ds \pi(dA)\Bigg)^2 \ \Bigg)\\
&\qquad\times \Sigma_\Lambda\!\! \int_{A_0(0) \backslash V_{(0,0)}^{\psi(h)}}l(-\xi ,-s)^2 d\xi ds\Bigg)^{\frac{1}{2}}.
\end{aligned}
\end{gather}
\item If $l\in L^1(\R\times\R^m)\cap L^2(\R\times\R^m)$ and $\int_{|x|>1}|x|^2\nu(dx)<\infty$, then $Y$ is $\theta$-lex-weakly dependent with $\theta$-lex-coefficients $\theta_Y(h)$ satisfying
\begin{align}
\theta_Y(h)\leq&
2 \Bigg(\Big(\Sigma_\Lambda E[\sigma_0(0)^2]
\int_{A_0(0) \cap V_{(0,0)}^{\psi(h)}}l(-\xi,-s)^2 d\xi ds \nonumber\\
&+\mu_\Lambda E[\sigma_0(0)^2] \Big(\int_{A_0(0) \cap V_{(0,0)}^{\psi(h)}}l(-\xi,-s) d\xi ds\Big)^2 \Bigg)^{\frac{1}{2}} \label{equation:thetalexcoefficientszeroambitfield}
\end{align}
\begin{align*}
\hspace{2cm} &+ 2\Bigg( \Bigg(\Sigma_{\Lambda^\sigma} \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^{\psi(h)}} j(A,-\xi,-s)^2 d\xi ds \pi(dA) \\&+\mu_{\Lambda^\sigma}^2\Bigg( \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^{\psi(h)}} j(A,-\xi,-s) d\xi ds \pi(dA)\Bigg)^2 \Bigg)\\ &\times\Bigg( \Sigma_\Lambda \int_{A_0(0) \backslash V_{(0,0)}^{\psi(h)}}l(-\xi ,-s)^2 d\xi ds\\
&\qquad+ \mu_\Lambda \Big(\int_{A_0(0) \backslash V_{(0,0)}^{\psi(h)}}l(-\xi ,-s) d\xi ds\Big)^2 \Bigg)\Bigg)^{\frac{1}{2}}.
\end{align*}
\item If $l\in L^1(\R\times\R^m)$, $\int_{\R}|x|\nu(dx)<\infty$ and $\Sigma=0$, then $Y$ is $\theta$-lex-weakly dependent with $\theta$-lex-coefficients $\theta_Y(h)$ satisfying
\begin{gather}\label{equation:thetalexcoefficientsfinitevariationambitfield}
\begin{aligned}
\theta_Y(h)\leq&2\Sigma_\sigma \Big(|\gamma_0|+\int_{\R}|x|\nu(dx)\Big) \Big(\int_{A_0(0) \cap V_{(0,0)}^{\psi(h)}}|l(-\xi,-s)| d\xi ds\Big)\\
&+2 \Bigg(\Sigma_{\Lambda^\sigma} \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^{\psi(h)}} j(A,-\xi,-s)^2 d\xi ds \pi(dA)\\
&+\mu_{\Lambda^\sigma}^2\Bigg( \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^{\psi(h)}} j(A,-\xi,-s) d\xi ds \pi(dA)\Bigg)^2 \Bigg)\\
&\times \Big(|\gamma_0|+\int_{\R}|x|\nu(dx)\Big)
\Big(\int_{A_0(0) \backslash V_{(0,0)}^{\psi(h)}}|l(-\xi,-s)| d\xi ds\Big).
\end{aligned}
\end{gather}
\end{enumerate}
The above results hold for all $h>0$, where $\psi(h)=\frac{-bh}{2\sqrt{m+1}}$ and $b$ are defined as in $(\ref{equation:xi})$, $\mu_\Lambda =\gamma+\int_{|x|\geq1}x\nu(dx)$, $\Sigma_\Lambda =\Sigma+\int_{\R}x^2\nu(dx)$, $\mu_{\Lambda^\sigma} =\gamma_\sigma+\int_{|x|\geq1}x\nu_\sigma(dx)$ and $\Sigma_{\Lambda^\sigma}=\Sigma_\sigma+\int_{\R}x^2\nu_\sigma(dx)$.
\end{Proposition}
\begin{proof}
See Section \ref{sec5-5}.
\end{proof}
We now analyze the case in which $\sigma$ is a $p$-dependent random field for $p\in\N$.
\begin{Proposition}\label{proposition:ambitmdepthetaweaklydep}
Let $Y=(Y_t(x))_{(t,x)\in\R\times\R^m}$ be an ambit field as defined in (\ref{eq:ambitfield}) with a predictable $p$-dependent stationary random field $\sigma_t(x)$ for $p\in\N$. Assume that $A_0(0)$ satisfies (\ref{condition:scalarproduct}). Additionally assume that $l\in L^2(\R^m\times\R)$, $\int_{|x|>1}|x|^2\nu(dx)<\infty$ and $\gamma + \int_{|x|>1}x\nu(dx)=0$. Then, for sufficiently big $h$, $Y$ is $\theta$-lex-weakly dependent with coefficients
\begin{gather}\label{equation:thetalexcoefficientsambitfieldiid}
\begin{gathered}
\theta_Y(h)\leq2 \Big(\Sigma_\Lambda E[\sigma_0(0)^2]\int_{A_0(0) \cap V_{(0,0)}^{\psi(h)}}l(-\xi,-s)^2 d\xi ds\Big)^{\frac{1}{2}},
\end{gathered}
\end{gather}
with $\psi(h)=\frac{-bh}{2\sqrt{m+1}}$ and $b$ as defined in $(\ref{equation:xi})$, $\Sigma_\Lambda =\Sigma+\int_{\R}x^2\nu(dx)$.
\end{Proposition}
\begin{proof}
See Section \ref{sec5-5}.
\end{proof}
\subsubsection{Volatility fields}
Let $\sigma$ be a $(A^\sigma,\Lambda^\sigma)$-influenced MMAF as defined in (\ref{sigma}), $j$ a non-negative kernel function and the following assumptions hold
\begin{equation*}
\textrm{(H)}: \left\{\begin{array}{l}
\textrm{The L\'evy basis $\Lambda^{\sigma}$ has generating quadruple $(\gamma_{\sigma},0, \nu_{\sigma},\pi_{\sigma})$ such that}\\
\int_{\R} |x| \nu_{\sigma}(dx) < \infty,\,\,\ \gamma_{\sigma}-\int_{|x|\leq 1} x \nu_{\sigma}(dx) \geq 0\,\,\,\textrm{and}\,\,\,\nu_{\sigma}(\R^{-})=0.
\end{array}\right.
\end{equation*}
Then, $\sigma$ has values in $\R^+$, and we call it volatility or intermittency field. Note that Assumption (H) implies that $\Lambda^\sigma$ satisfies the finite variation case and that this model is used in several applications of the ambit fields, see \cite{BBV2018}.
By assuming additionally that $j\in L^1(S\times\R\times\R^m, \pi\otimes\lambda) \cap L^2(S\times\R^m\times\R, \pi\otimes\lambda)$ and $\int_{|x|>1}|x|^2\nu_\sigma(dx)<\infty$, the results in Proposition \ref{proposition:ambitthetaweaklydep} (i) and (ii) hold.
On the other hand, the bound in Proposition \ref{proposition:ambitthetaweaklydep} (iii) can be tightened.
\begin{Corollary}\label{corollary:finitefiniteambitthetaweaklydep}
Let $Y=(Y_t(x))_{(t,x)\in\R\times\R^m}$ be an ambit field as defined in (\ref{eq:ambitfield}) with predictable volatility field $\sigma_t(x)$ being an $(A^\sigma,\Lambda^\sigma)$-influenced MMAF such that $A_0(0)$ and $A_0^\sigma(0)$ satisfy (\ref{condition:scalarproduct}), $j \in L^1(S\times\R\times\R^m, \pi\otimes\lambda)$, $l \in L^1(\R\times\R^m)$ and Assumption (H) holds. Let $\gamma_0$ with respect to $\Lambda$ and $\gamma_{0,\sigma}$ with respect to $\Lambda^\sigma$ be defined as in (\ref{equation:gammazero}). Then, $Y$ is $\theta$-lex-weakly dependent with coefficients
\begin{gather}\label{equation:thetalexcoefficientsfinitefinitevariationambitfield}
\begin{aligned}
\theta_Y(h)\leq&2 \Big(|\gamma_{0,\sigma}|+\int_{\R}|x|\nu_\sigma(dx)\Big)\Big(|\gamma_0|+\int_{\R}|x|\nu(dx)\Big) \\&\quad\times\Big(\int_S\int_{A_0(0)}|j(A,-\xi,-s)| d\xi ds\pi(dA)\Big)\Big( \int_{A_0(0) \cap V_{(0,0)}^{\psi(h)}}|l(-\xi,-s)| d\xi ds\Big)\\
&+2\Big(|\gamma_{0,\sigma}|+\int_{\R}|x|\nu_\sigma(dx)\Big)\Big(|\gamma_0|+\int_{\R}|x|\nu(dx)\Big)\\
&\quad\times\Big( \int_{A_0(0) \backslash V_{(0,0)}^{\psi(h)}}|l(-\xi,-s)| d\xi ds\Big) \Big(\int_S \int_{A_0(0) \cap V_{(0,0)}^{\psi(h)}}|j(A,-\xi,-s)| d\xi ds\pi(dA)\Big),
\end{aligned}
\end{gather}
for all $h>0$, with $\psi(h)=\frac{-bh}{2\sqrt{m+1}}$ and $b$ as defined in $(\ref{equation:xi})$.
\end{Corollary}
\begin{proof}
Analogous to Proposition \ref{proposition:ambitthetaweaklydep}.
\end{proof}
\subsection{Sample moments of ambit fields}
\label{sec4-3}
In this section, we study the asymptotic distribution of sample moments of $Y$. As in Section \ref{sec3-2} we assume that we observe $Y$ on a sequence of finite sampling sets $D_n\subset\Z\times\Z^m$, such that (\ref{condition:samplingset}) holds.
\begin{Theorem}\label{theorem:ambitthetasamplemean}
Let $Y=(Y_t(x))_{(t,x)\in\Z\times\Z^m}$ be an ambit field as defined in (\ref{eq:ambitfield}) such that $E[Y_t(x)]=0$, $E[|Y_t(x)|^{2+\delta}]<\infty$ for some $\delta>0$. Additionally assume that $Y$ is $\theta$-lex-weakly dependent with $\theta$-lex-coefficients satisfying $\theta_Y(h)=\mathcal{O}(h^{-\alpha})$ for $\alpha>m(1+\frac{1}{\delta})$ and defined $\mathcal{I}$ as in Theorem \ref{theorem:clt}. Then,
\begin{gather*}
\sigma^2=\sum_{(u_t,u_x)\in\Z\times\Z^m}E[Y_{0}(0)Y_{u_t}(u_x)|\mathcal{I}]
\end{gather*}
is finite, non-negative and
\begin{gather}\label{eq:ambitthetaclt}
\frac{1}{|D_n|^{\frac{1}{2}}}\sum_{(u_t,u_x)\in D_n}Y_{u_t}(u_x)\underset{n\ra\infty}{\xrightarrow{\makebox[2em][c]{d}}}\varepsilon \sigma,
\end{gather}
where $\varepsilon$ is a standard normally distributed random variable independent of $\sigma^2$.
\end{Theorem}
\begin{proof}
The result follows from Theorem \ref{theorem:clt}.
\end{proof}
\begin{Corollary}\label{corollary:ambitsamplemomentsofhigherorder}
Let $Y=(Y_t(x))_{(t,x)\in\Z\times\Z^m}$ be an ambit field as defined in (\ref{eq:ambitfield}) such that $E[|Y_t(x)|^{2p+\delta}]<\infty$ for $p\geq1$ and some $\delta>0$. Additionally, let us assume that $Y$ is $\theta$-lex-weakly dependent with $\theta$-lex-coefficients satisfying $\theta_Y(h)=\mathcal{O}(h^{-\alpha})$, for $\alpha>m\left(1+\frac{1}{\delta}\right)(\frac{2p-1+\delta}{p+\delta})$ and define $\mathcal{I}$ as in Theorem \ref{theorem:clt}. Then,
\begin{gather*}
\Sigma=\sum_{(u_t,u_x)\in\Z\times\Z^m}Cov(Y_0(0)^{p},Y_{u_t}(u_x)^{p} |\mathcal{I})
\end{gather*}
is finite, non-negative and
\begin{gather}\label{eq:ambitthetaclthighermoments}
\frac{1}{|D_n|^{\frac{1}{2}}}\sum_{(u_t,u_x)\in D_n}Y_{u_t}(u_x)^p-E[Y_{0}(0)^p]\underset{n\ra\infty}{\xrightarrow{\makebox[2em][c]{d}}}\varepsilon \sigma,
\end{gather}
where $\varepsilon$ is a standard normally distributed random variable which is independent of $\sigma^2$.
\end{Corollary}
\begin{proof}
Analogous to Corollary \ref{corollary:samplemomentsofhigherorder}.
\end{proof}
\begin{Remark}
Theorem \ref{theorem:ambitthetasamplemean} and Corollary \ref{corollary:ambitsamplemomentsofhigherorder} are important first steps to develop statistical inference for the class of ambit fields. However, we note that the limits in (\ref{eq:ambitthetaclt}) and (\ref{eq:ambitthetaclthighermoments}) are of mixed Gaussian type. Conditions that ensure the ergodicity of an ambit field with a deterministic kernel can be found in \cite[Theorem 3.6]{PV2017} whereas, for the case of a non-deterministic kernel, this remains an open problem.
\end{Remark}
\section{Proofs}
\label{sec5}
\subsection{Proofs of Section \ref{sec2-2} and \ref{sec2-3}}
\label{sec5-1}
We first extend some of the results obtained in \cite{DD2003} to random fields. This will enable us to connect the sufficient conditions given in \cite{D1998} with our definition of the $\theta$-lex-coefficients.
Let us define the space of bounded, Lipschitz continuous functions $\mathscr{L}_1=\{g:\R\rightarrow \R,$ bounded and Lipschitz continuous with $Lip(g)\leq1\}$. For a $\sigma$-algebra $\mathscr{M}$ and an $\R^n$-valued integrable random field $X=(X_t)_{t\in\Z^m}$ we define the mixingale-type measures of dependence
$$\gamma(\mathscr{M},X)=\norm{E[X|\mathscr{M}]-E[X]}_{1}, \quad \textrm{and} \quad \theta(\mathscr{M},X)=\sup_{g\in \mathscr{L}_1} \norm{E[g(X)|\mathscr{M}]-E[g(X)]}_{1}.$$
Using the above measures of dependence we define the following dependence coefficients
\begin{equation}
\label{eq:gammathetanonstat}
\gamma_h= \sup_{j\in\Z^m} \gamma\big(\mathcal{F}_{V_j^h},X_j\big), \quad \textrm{and} \quad \theta_h= \sup_{j\in\Z^m} \theta\big(\mathcal{F}_{V_j^h},X_j\big),
\end{equation}
for $h\in\N$. Obviously, it holds $\gamma(\mathscr{M},X)\leq2\norm{X}_1$ and $\gamma(\mathscr{M},X)\leq\theta(\mathscr{M},X)$ such that $\gamma_h\leq\theta_h$ for all $h\in\N$.
If $X$ is stationary we can write $\gamma_h$ and $\theta_h$ from (\ref{eq:gammathetanonstat}) as
\begin{equation}
\label{eq:gammathetastat}
\gamma_h= \gamma\big(\mathcal{F}_{V_0^h},X_0\big), \quad \text{and} \quad \theta_h= \theta\big(\mathcal{F}_{V_0^h},X_0\big),
\end{equation}
for $h\in\N$. First, we extend Proposition 2.3 from \cite{DDLLLP2008} and connect the $\theta$-lex-coefficients $\theta(h)$ from Definition \ref{thetaweaklydependent} with the mixingale-type coefficient $\theta_h$ defined above.
\begin{Lemma}\label{lemma:thetaiscondexp}
Let $X=(X_t)_{t\in\Z^m}$ be a real-valued random field. Then it holds that
\begin{gather*}
\theta(h)=\theta_h, \quad h\in\N.
\end{gather*}
\end{Lemma}
\begin{proof}
Fix $u,h\in\N$. We first show $\theta_u(h)\leq\theta_{h}$. Let $F\in \mathcal{G}^*_k$, $G\in \mathcal{G}_1$, $j\in\R^m$, $k\leq u$ and $\Gamma=\{i_1,\ldots,i_k\}$ with $i_1,\ldots,i_k\in V_j^h$. Now
\begin{align*}
&\left| Cov \left( \frac{F(X_\Gamma)}{\norm{F}_\infty}, \frac{G(X_j)}{ Lip(G)} \right) \right|
= \left| E\left[ \frac{F(X_\Gamma)}{\norm{F}_\infty}\frac{G(X_j)}{Lip(G)} - E\left[ \frac{F(X_\Gamma)}{\norm{F}_\infty}\right] E\left[\frac{G(X_j)}{Lip(G)}\right] \right]\right|\\
&= \left| E\left[E\left[\frac{F(X_\Gamma)}{\norm{F}_\infty}\frac{G(X_j)}{Lip(G)}\bigg| \mathcal{F}_{V_j^h} \right] - \frac{F(X_\Gamma)}{\norm{F}_\infty} E\left[\frac{G(X_j)}{Lip(G)}\right] \right]\right|\\
&\leq E\left[ \left|\frac{F(X_\Gamma)}{\norm{F}_\infty}\right| \left|E\left[ \frac{G(X_j)}{Lip(G)}\bigg| \mathcal{F}_{V_j^h} \right] - E\left[\frac{G(X_j)}{Lip(G)}\right]\right| \right]\\
&\leq \left\lVert E\left[ \frac{G(X_j)}{Lip(G)}\bigg| \mathcal{F}_{V_j^h} \right] - E\left[\frac{G(X_j)}{Lip(G)}\right] \right\rVert_1
= \theta(\mathcal{F}_{V_j^h},X_j)\leq \theta_{h}.
\end{align*}
Taking the supremum on the left hand side we obtain $\theta_u(h)\leq \theta_{h}$ and finally $\theta(h)\leq \theta_{h}$.\\
To prove the converse inequality, we first remark that by the martingale convergence theorem
\begin{gather}
\theta(\mathcal{F}_{V_j^h},X_j)=\lim_{k\rightarrow\infty} \theta( \mathcal{F}_{V_j^h\backslash V_j^k},X_j).\label{eq:convergencetailalgebra}
\end{gather}
Now, let $G\in\mathscr{L}_1$, i.e. $G\in\mathcal{G}_1$ with $Lip(G)\leq1$ and $j\in\R^m$. We first define $X_j^h(k)=\{ X_i:i\in V_j^h\backslash V_j^k \}$ and $F(X_j^h(k))=sign(E[G(X_j) | \mathcal{F}_{V_j^h\backslash V_j^k}]-E[g(X_j)])$ for $k>h$. Then $F\in\mathcal{G}^*_{u}$ for $u=|V_j^h\backslash V_j^k|\in\N$ with $\norm{F}_\infty=1$ and it holds
\begin{align*}
&E\left[ \left| E[G(X_j) | \mathcal{F}_{V_j^h\backslash V_j^k}]-E[G(X_j)]\right| \right]
=E\left[ \left(E[G(X_j) | \mathcal{F}_{V_j^h\backslash V_j^k}]-E[G(X_j)]\right)F\Big(X_j^h(k)\Big)\right]\\
&=E\left[ E\left[F\Big(X_j^h(k)\Big)G(X_j) | \mathcal{F}_{V_j^h\backslash V_j^k}\right]-E\left[F\Big(X_j^h(k)\Big)\right] E\left[G(X_j)\right] \right]\\
&=Cov\left(F\Big(X_j^h(k)\Big), G(X_j)\right)\leq \theta(h).
\end{align*}
Using (\ref{eq:convergencetailalgebra}) we can deduce the stated equality.
\end{proof}
We define $Q_X$ as the generalized inverse of the tail function $x\mapsto P(|X|>x)$ and $G_X$ as the inverse of $x\mapsto \int_0^xQ_{X}(u)du$.
\begin{Lemma}\label{lemma:finitesumclt}
Let $X=(X_t)_{t\in\Z^m}$ be a stationary centered real-valued random field such that $\norm{X_0}_2<\infty$ and assume that
\begin{gather}
\int_{0}^{\norm{X_0}_1}\tilde\theta(u) Q_{X_0}\circ G_{X_0}(u)du<\infty,\label{eq:summability}
\end{gather}
with $Q_X$ and $G_X$ as defined above and $\tilde\theta(u)= \sum_{k\in V_0} \mathbb{1}_{\left\{u<\theta_{|k|}\right\}}$. Then,
\begin{gather}
\sum_{k\in V_0}|E[X_kE_{|k|}[X_0]]|<\infty,\label{eq:finitesumclt}
\end{gather}
where $E_{|k|}[X_0]=E[X_0|F_{V_0^{|k|}}]$.
\end{Lemma}
\begin{proof}
First, let us observe that $X_k$ is $\mathcal{F}_{V_0^{|k|}}$ measurable, since $k\in V_0^{|k|}$. Then, we define $ \varepsilon_{k}=sign(E_{|k|}[X_0])$ such that
\begin{align*}
&\sum_{k\in V_0}|E[X_kE_{|k|}[X_0]]|
\leq \sum_{k\in V_0}E[|X_k||E_{|k|}[X_0]|]
= \sum_{k\in V_0}E[|X_k| \varepsilon_{k} E_{|k|}[X_0]]\\
&= \sum_{k\in V_0}E[ E_{|k|}[|X_k| \varepsilon_{k}X_0]]
=\sum_{k\in V_0} Cov(|X_k| \varepsilon_{k},X_0).
\end{align*}
We use Equation (4.2) of \cite[Proposition 1]{DD2003} to get
\begin{align*}
&\leq 2\sum_{k\in V_0} \int_{0}^{\gamma\Big(\mathcal{F}_{V_0^{|k|}},X_0\Big)/2} Q_{ \varepsilon_{k}|X_k|}\circ G_{X_0}(u)du\\
&=2 \int_{0}^{\norm{X_0}_1} \sum_{k\in V_0}\mathbb{1}_{\left\{u<\gamma\Big(\mathcal{F}_{V_0^{|k|}},X_0\Big)/2\right\}} Q_{X_k}\circ G_{X_0}(u)du\\
&\leq 2 \int_{0}^{\norm{X_0}_1} \sum_{k\in V_0}\mathbb{1}_{\left\{u< \theta\Big(\mathcal{F}_{V_0^{|k|}},X_0\Big)/2\right\}} Q_{X_k}\circ G_{X_0}(u)du\\
&\leq 2 \int_{0}^{\norm{X_0}_1} \sum_{k\in V_0}\mathbb{1}_{\left\{u< \theta_{|k|}/2 \right\}} Q_{X_k}\circ G_{X_0}(u)du.
\end{align*}
Now, let $\tilde\theta(u)= \sum_{k\in V_0} \mathbb{1}_{\left\{u<\theta_{|k|}\right\}}$ and note that $Q_{X_k}=Q_{X_0}$, such that
\begin{gather*}
\leq 2 \int_{0}^{\norm{X_0}_1}\tilde\theta(u) Q_{X_0}\circ G_{X_0}(u)du.
\end{gather*}
This shows that (\ref{eq:finitesumclt}) holds if (\ref{eq:summability}) is satisfied.
\end{proof}
We now derive sufficient criteria such that (\ref{eq:summability}) holds similar to \cite[Lemma 2]{DD2003}.
\begin{Lemma}\label{lemma:summability}
Let $X=(X_t)_{t\in\Z^m}$ be a stationary real-valued random field and $\theta_h$ defined as above. Then (\ref{eq:summability}) holds if $\norm{X}_r<\infty$ for some $r>p>1$ and $\sum_{h=0}^\infty (h+1)^{m(p-1)\frac{(r-1)}{(r-p)}-1} \theta_h<\infty$. In particular, for $p=2$ and $r=2+\delta$ with $\delta>0$ the above condition holds if $\theta_{h}\in \mathcal{O}(h^{-\alpha})$ for $\alpha>m(1+\frac{1}{\delta})$.
\end{Lemma}
\begin{proof}
As stated in \cite[Proof of Lemma 2]{DD2003} we note that $\int_{0}^{\norm{X}_1} Q_{X}^{r-1}\circ G_{X}(u)du=\int_0^1Q_X^r(u)d u=E[|X|^r]$. Applying H\"older's inequality with $q=\frac{r-1}{r-p}$ and $q'=\frac{r-1}{p-1}$ gives
\begin{align*}
&\left(\int_{0}^{\norm{X}_1}\tilde\theta(u)^{p-1} Q_{X}^{p-1}\circ G_{X}(u)du\right)^{(r-1)}\\
&\leq\left( \int_{0}^{\norm{X}_1}\tilde\theta(u)^{(p-1)\frac{(r-1)}{(r-p)}} du\right)^{(r-p)} \left(\int_{0}^{\norm{X}_1}Q_{X}^{r-1}\circ G_{X}(u)du\right)^{(p-1)}\\
&=\left( \int_{0}^{\norm{X}_1}\tilde\theta(u)^{(p-1)\frac{(r-1)}{(r-p)}} du\right)^{(r-p)} \norm{X}_r^{(rp-r)}.
\end{align*}
Let us note that $\theta_h$ as defined in (\ref{eq:gammathetastat}) is non-increasing. Then, for any function $f$ we have
\begin{align*}
f(\tilde\theta(u))&=f\left( \sum_{k\in V_0} \mathbb{1}_{\left\{u<\theta_{|k|}\right\}} \right) =\sum_{h=0}^\infty f\left( \sum_{k\in V_0} \mathbb{1}_{\left\{u<\theta_{|k|}\right\}} \right) \mathbb{1}_{\{ \theta_{h+1}\leq u<\theta_{h} \}}\\
&=\sum_{h=0}^\infty f\left( \sum_{k\in V_0:|k|\leq h} 1 \right) \mathbb{1}_{\{ \theta_{h+1}\leq u<\theta_{h} \}}.
\end{align*}
Note that $ \sum_{k\in V_0:|k|\leq h} 1=\sum_{i=0}^{m-1}h(2h+1)^i=\frac{1}{2}\left( (2h+1)^m-1\right)$ such that the above is equal to
\begin{gather}
\label{p1}
\sum_{h=0}^\infty f\left(\frac{1}{2}\Big((2h+1)^m-1\Big) \right) \mathbb{1}_{\{ \theta_{h+1}\leq u<\theta_{h} \}}.
\end{gather}
Let us assume that $f$ is monotonically increasing, sub-multiplicative and $f(0)=0$ such that $f\left(\frac{1}{2}\Big((2h+1)^m-1\Big) \right)\leq f(2^{m-1})f((h+1)^m)=\sum_{k=0}^h f(2^{m-1})\left(f\left((k+1)^m \right)-f\left(k^m \right)\right)$. Finally we can deduce that (\ref{p1}) is less than or equal to
\begin{gather*}
\sum_{h=0}^\infty f(2^{m-1}) f\left((h+1)^m \right) \mathbb{1}_{\{ \theta_{h+1}\leq u<\theta_{h} \}}
=f(2^{m-1})\sum_{h=0}^\infty \left( f\left((h+1)^m \right)- f\left(h^m\right)\right) \mathbb{1}_{\{ u<\theta_{h} \}}.
\end{gather*}
Applying the above result for $f(x)=x^{v}$ with $v=(p-1)\frac{(r-1)}{(r-p)}$ and noting that $(h+1)^{vm}-h^{vm}\leq vm (h+1)^{vm-1}$ for $vm\geq1$ and $(h+1)^{vm}-h^{vm}\leq vm~h^{vm-1}$ for $vm<1$ ($h>0$ by the mean value theorem), we get that for a constant $C=2^{v(m-1)}\theta_1>0$
\begin{align*}
&\Bigg( \int_{0}^{\norm{X}_1}(\tilde\theta(u))^{(p-1)\frac{(r-1)}{(r-p)}} du\Bigg)^{(r-p)}\\
&\leq \begin{cases}
\left(C+ \int_{0}^{\norm{X}_1}2^{v(m-1)} vm \sum_{h=0}^\infty (h+1)^{vm-1}\mathbb{1}_{\{ u<\theta_{h} \}} du\right)^{(r-p)}\!\!\!\!\!\!\!\!\!&,\text{ if $vm\geq1$}\\
\left(C+ \int_{0}^{\norm{X}_1}2^{v(m-1)} vm \sum_{h=1}^\infty h^{vm-1}\mathbb{1}_{\{ u<\theta_{h} \}} du\right)^{(r-p)}&, \text{ if $vm<1$}
\end{cases}\\
&=\begin{cases}
\left(C+2^{v(m-1)}vm \sum_{h=0}^\infty (h+1)^{vm-1} \theta_h\right)^{(r-p)}&,\text{ if $vm\geq1$}\\
\left(C+2^{v(m-1)}vm \sum_{h=1}^\infty h^{vm-1} \theta_h\right)^{(r-p)}&,\text{ if $vm<1$}
\end{cases}\\
&\leq \max\left(1,2^{r-p-1}\right)\left(C^{r-p}+\left((vm)2^{v(m-1)}\left(~ \sum_{h=0}^\infty (h+1)^{vm-1} \theta_h\right)\!\right)^{(r-p)}\right),
\end{align*}
which concludes the proof.
\end{proof}
Before we give the proof of Theorem \ref{theorem:clt}, we use Lemma \ref{lemma:thetaiscondexp} to show Proposition \ref{proposition:thetavsalpha} and Proposition \ref{proposition:AR1notmixingbuttheta}.
\begin{proof}[Proof of Proposition \ref{proposition:thetavsalpha}]
From Lemma \ref{lemma:thetaiscondexp}, \cite[Lemma 1]{DD2003}, \cite[Proposition 25.15 (I) (a)]{B2007}, and by applying H\"older's inequality for $q >1$ and noting that $\int_0^1Q_{X_0}^r(u)du=E[|X_0|^r]$, we obtain that
\begin{align*}
\theta(h)&=\theta_h=\theta\big(\mathcal{F}_{V_0^h},X_0\big)\leq2 \int_0^{2\alpha\big(\mathcal{F}_{V_0^h},\sigma(\{X_0\})\big)}Q_{X_0}(u)du \\[-10pt]
&= 2\!\int_0^{1}\!\!\mathbb{1}_{\left\{u\leq2\alpha\big(\mathcal{F}_{V_0^h},\sigma(\{X_0\})\big)\right\}}Q_{X_0}(u)du
\leq 2^{\frac{2q-1}{q}}\!\!\alpha\big(\mathcal{F}_{V_0^h},\sigma(\{X_0\})\big)^{\frac{q-1}{q}}\!\!\left( \int_0^{1}Q_{X_0}^q(u)du\right)^{\frac{1}{q}}.
\end{align*}
Thus, we have that $\theta(h)\leq 2^{\frac{2q-1}{q}}\alpha(h)^{\frac{q-1}{q}} \norm{X_0}_q$ for $m=1$, and $\theta(h)\leq 2^{\frac{2q-1}{q}}\alpha_{\infty,1}(h)^{\frac{q-1}{q}} \norm{X_0}_q$ for all $m\geq1$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{proposition:AR1notmixingbuttheta}]
Consider $\theta_{p,r}(h)$ and $\tilde\delta_{1,n}$ as defined in \cite[Definition 2.3 and Section 3.1.4]{DDLLLP2008}, respectively. Due to Lemma \ref{lemma:thetaiscondexp} it holds that $\theta(h)=\theta_h=\theta_{1,1}(h)\leq \theta_{1,\infty}(h)\leq \tilde\delta_{1,h}$, where the last inequality follows from \cite[Section 3.1.4]{DDLLLP2008}. We define $(\tilde\xi_{t})_{t\in\Z}$ as $\tilde\xi_{t}=\xi_t$ for $t>0$ and $\tilde\xi_{t}=\xi'_t$ for $t\leq0$, where $\xi'_t$ is an independent copy of $\xi_t$. For $\tilde{X}_t=\sum_{j=0}^\infty 2^{-j-1}\tilde\xi_{t-j}$, we have
\begin{align*}
\norm{X_h-\tilde{X}_h}_1\leq 2^{-h}=:\tilde\delta_{1,h}\underset{h\rightarrow\infty}{\longrightarrow}0.
\end{align*}
Thus, $X$ is $\theta$-lex-weakly dependent. To show that $X$ is neither $\alpha$- nor $\alpha_{\infty,1}$-mixing, we follow the idea given in \cite[Section 1.5]{DDLLLP2008}. The set $A=\{X_0\leq \frac{1}{2}\}$ belongs to the past $\sigma$-algebra $\sigma(\{X_s,s\leq0\})$, to the $\sigma$-algebra generated by $X_h$ and to the future $\sigma$-algebra $\sigma(\{X_s,s\geq h\})$. Hence for all $h$, $\alpha(h)\geq |P(A)P(A)-P(A)|=\frac{1}{4}$, and similarly $\alpha_{\infty,1}(h)\geq \frac{1}{4}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theorem:clt}]
In order to use \cite[Theorem 1]{D1998} we need to show that
\begin{gather*}
\sum_{k\in V_0}|E[X_kE_{|k|}[X_k]]|<\infty.
\end{gather*}
By Lemma \ref{lemma:finitesumclt} and Lemma \ref{lemma:summability} the result is proven if $\theta_{h}\in \mathcal{O}(h^{-\alpha})$ with $\alpha>m(1+\frac{1}{\delta})$.
Finally, since $X$ is stationary an application of Lemma \ref{lemma:thetaiscondexp} concludes.
\end{proof}
\subsection{Proofs of Section \ref{sec3-3}}
\label{sec5-2}
\begin{proof}[Proof of Proposition \ref{proposition:mmathetaweaklydep}]
\begin{enumerate}[(i)]
\item Let $t\in\R^{m}$, $\psi>0$. We restrict the MMAF $X$ to a finite support and define the truncated sequence
\begin{gather}\label{equation:truncatedtheta}
X_{t}^{(\psi)}=\int_S\int_{A_{t}\backslash V_{t}^\psi}f(A,t-s)\Lambda(dA,ds). \end{gather}
Note that the kernel function $f$ is square integrable such that (\ref{equation:intcond1}), (\ref{equation:intcond2}) and (\ref{equation:intcond3}) hold. Therefore, $f$ is $\Lambda$-integrable. Since $E[X_{t}X_{t}']<\infty$ for all $t\in\R^m$, by Proposition \ref{proposition:MMAexistencemoments} we can derive an upper bound of the expectation
\begin{align}
\begin{aligned}\label{eq:L1norminequality}
E\big[\norm{X_{t}-X_{t}^{(\psi)}}\big]&=E\bigg[ \Big\lVert\int_S\int_{A_t\cap V_{t}^\psi}f(A,t-s)\Lambda(dA,ds)\Big\rVert\bigg]\\
&\leq E\bigg[ \Big\lVert\int_S\int_{A_t\cap V_{t}^\psi}f(A,t-s)\Lambda(dA,ds)\Big\rVert^2\bigg]^{\frac{1}{2}}\\
& =\left(\sum_{\kappa=1}^nE\Bigg[ \bigg( \Big(\int_S\int_{A_t\cap V_{t}^\psi}f(A,t-s)\Lambda(dA,ds)\Big)^{(\kappa)}\bigg)^2\Bigg]\right)^{\frac{1}{2}}.
\end{aligned}
\end{align}
Using Proposition \ref{proposition:MMAmoments} and the translation invariance of $A_t$ and $V_t^\psi$, the above is equal to
\begin{gather*}
\Big(\int_S\int_{A_0 \cap V_{0}^\psi}\textup{tr}(f(A,-s)\Sigma_\Lambda f(A,-s)')ds\pi(dA)\Big)^{\frac{1}{2}}.
\end{gather*}
Let $u\in\N, h\in\R^{+}, \Gamma=\{i_1,\ldots,i_u\}\in(\R^m)^u$ and $j\in\R^m$ as in Definition \ref{thetaweaklydependent} such that $i_1,\ldots,i_u\in V_j^h$. Moreover, let $G\in\mathcal{G}_1$ and $F\in\mathcal{G}_u^*$. For $a\in\{1,\ldots,u\}$ define
\begin{gather*}
X_{i_a}=\int_S\int_{A_{i_a}}f(A,i_a-s)\Lambda(dA,ds), \quad \textrm{and} \quad
X_{j}^{(\psi)}=\int_S\int_{A_{j}\backslash V_{j}^\psi}f(A,j-s)\Lambda(dA,ds).
\end{gather*}
W.l.o.g. we assume that $i_a\leq_{lex}i_u$ for all $a\in\{1,\ldots,u\}$. If there exists a $\psi$ such that $A_{i_u}\cap A_{j}\backslash V_{j}^\psi=\emptyset$, then $A_{i_a}\cap A_{j}\backslash V_{j}^\psi=\emptyset$. \\
Now, $A$ is translation invariant with initial sphere of influence $A_0$. Furthermore, $A_0$ satisfies (\ref{condition:scalarproduct}). Then, for $\psi(h)$ as defined in (\ref{equation:psi}) it holds that $A_{i_u}\cap A_{j}\backslash V_{j}^\psi=\emptyset$.\\
From now on we set $\psi=\psi(h)$. We then get that $I_a=S\times A_{i_a}$ and $J=S\times A_{j}\backslash V_{j}^\psi$ are disjoint or have intersection on a set $S\times O$, where $O\subset\R^m$ and $\dim(O)<m$. Since $(\pi\times\lambda)(S\times O)=0$ and by the definition of a L\'evy basis, $X_{i_a}$ and $X_j^{(\psi)}$ are independent for all $a\in\{1,\ldots,u\}$. Finally, we get that $X_\Gamma$ and $X_j^{(\psi)}$ are independent and therefore also $F(X_\Gamma)$ and $G(X_j^{(\psi)})$. Now
\begin{align*}
& |Cov(F(X_\Gamma),G(X_j))|\\&\leq |Cov(F(X_\Gamma),G(X_j^{(\psi)}))|+|Cov(F(X_\Gamma),G(X_j)-G(X_j^{(\psi)}))|\\
&=|E[(G(X_j)-G(X_j^{(\psi)}))F(X_\Gamma)]-E[G(X_j)-G(X_j^{(\psi)})]E[F(X_\Gamma)]|\\
&\leq 2\norm{F}_\infty E\big[|G(X_j)-G(X_j^{(\psi)})|\big] \leq 2Lip(G) \norm{F}_\infty E[\norm{X_{j}-X_{j}^{(\psi)}}]\\
&\leq 2 Lip(G)\norm{F}_\infty \Big(\int_S\int_{A_0\cap V_{0}^\psi}\textup{tr}(f(A,-s)\Sigma_\Lambda f(A,-s)')ds\pi(dA)\Big)^{\frac{1}{2}},
\end{align*}
using (\ref{eq:L1norminequality}). Therefore, $X$ is $\theta$-lex weakly dependent with $\theta$-lex-coefficients
\begin{gather*}
\theta_X(h)\leq 2 \Big(\int_S\int_{A_0\cap V_{0}^\psi}\textup{tr}(f(A,-s)\Sigma_\Lambda f(A,-s)')ds\pi(dA)\Big)^{\frac{1}{2}},
\end{gather*}
which converge to zero as $h$ goes to infinity by applying the dominated convergence theorem.
\item Let $t\in\R^m$, $\psi>0$ and $X_t^{(\psi)}$ be defined as in (i). By applying Proposition \ref{proposition:MMAmoments}, we obtain
\begin{align*}
E\big[\norm{X_t-X_t^{(\psi)}}\big]&\leq\bigg(\int_S\int_{A_0\cap V_t^{\psi}}\textup{tr}(f(A,t-s)\Sigma_\Lambda f(A,t-s)')ds\pi(dA)\\ &\qquad+\Big\lVert\int_S\int_{A_0\cap V_t^{\psi}} f(A,t-s)\mu_\Lambda ds\pi(dA)\Big\rVert^2\bigg)^{\frac{1}{2}}.
\end{align*}
Finally, by proceeding as in the proof of (i), we obtain the stated bound for the $\theta$-lex-coefficients.
\item Since the kernel function $f$ is in $L^1$ the Equations (\ref{equation:intcondfinvar1}) and (\ref{equation:intcondfinvar2}) hold. Moreover, by Proposition \ref{proposition:MMAexistencemoments} $f$ is $\Lambda$-integrable and $E[X_{t}]<\infty$. Let $t\in\R^m$, $\psi>0$ and $X_t^{(\psi)}$ be defined as in (i). Then, we can derive with the help of Proposition \ref{proposition:MMAmoments} that
\begin{align*}
& E\big[\norm{X_{t}-X_{t}^{(\psi)}}\big]\\
&\leq \Big(\int_S\int_{A_0\cap V_{0}^\psi} \big\lVert f(A,-s)\gamma_0\big\rVert ds\pi(dA)+\int_S\int_{A_0 \cap V_{0}^\psi} \int_{\R^d} \big\lVert f(A,-s)y\big\rVert \nu(dy)ds\pi(dA) \Big),
\end{align*}
where we used that $E[\int_Ef(t)d\mu(t)]=\int_Ef(t)d\nu(t)$ for a Poisson random measure $\mu$ with corresponding intensity measure $\nu$ and an arbitrary set $E$.\\
Now for $F$, $G$, $X_\Gamma$, $X_j$ and $\psi=\psi(h)$ as described in the proof of (i) we get
\begin{align*}
|Cov(F(X_\Gamma),G(X_j))|
\leq2Lip(G)\norm{F}_\infty& \Big(\int_S\int_{A_0 \cap V_{0}^{\psi(h)}} \big\lVert f(A,-s)\gamma_0\big\rVert ds\pi(dA)\\&+\int_S\int_{A_0\cap V_{0}^{\psi(h)}} \int_{\R^d} \big\lVert f(A,-s)y\big\rVert \nu(dy)ds\pi(dA) \Big).
\end{align*}
Therefore $X$ is $\theta$-lex weakly dependent with $\theta$-lex-coefficients
\begin{align*}
\theta_X(h)\leq 2 \Big(&\int_S\int_{A_0 \cap V_{0}^{\psi(h)}} \big\lVert f(A,-s)\gamma_0\big\rVert ds\pi(dA)\\&+\int_S\int_{A_0\cap V_{0}^{\psi(h)}} \int_{\R^d} \big\lVert f(A,-s)y\big\rVert \nu(dy)ds\pi(dA) \Big),
\end{align*}
which converge to zero as $h$ goes to infinity by applying the dominated convergence theorem.
\item We use the notations described in (i). We have that $\Lambda$ is in distribution the sum of two $\R^d$-valued independent L\'evy bases $\Lambda_1$ and $\Lambda_2$ with characteristic quadruplets $(\gamma,\Sigma,\restr{\nu}{\norm{x}\leq1},\pi)$ and $(0,0,\restr{\nu}{\norm{x}>1},\pi)$, respectively. Since $f\in L^1\cap L^2$ we know that both integrals $X_t^{(\Lambda_1)}=\int_S\int_{\R^m}f(A,t-s)\Lambda_1(dA,ds)$ and $X_t^{(\Lambda_2)}=\int_S\int_{\R^m}f(A,t-s)\Lambda_2(dA,ds)$ exist. Additionally, it holds that $\int_S\int_{\R^m}f(A,t-s)\Lambda(dA,ds)=X_t^{(\Lambda_1)}+X_t^{(\Lambda_2)}$. By noting that
\begin{align*}
E\big[\norm{X_t-X_t^{(\psi)}}\big]&\leq E\big[\norm{X_t^{(\Lambda_1)}-(X_t^{(\Lambda_1)})^{(\psi)}}\big] +E\big[\norm{X_t^{(\Lambda_2)}-(X_t^{(\Lambda_2)})^{(\psi)}}\big]\\
&\leq E\bigg[ \Big\lVert X_t^{(\Lambda_1)}-(X_t^{(\Lambda_1)})^{(\psi)}\Big\rVert ^2\bigg]^{\frac{1}{2}}+E\bigg[\Big\lVert X_t^{(\Lambda_2)}-(X_t^{(\Lambda_2)})^{(\psi)}\Big\rVert\bigg]
\end{align*}
and following the proof of (ii) (for the first summand) and (iii) (for the second summand) we obtain the stated bound for the $\theta$-lex-coefficients.
\end{enumerate}
\end{proof}
\begin{proof}[Proof of Proposition \ref{proposition:vectorinfluencedweaklydep}]
In order to show that $Z$ is a well defined MMAF, we need to check that $g(A,s)$ is $\Lambda$-integrable as described in Theorem \ref{theorem:2}, i.e. $g(A,s)$ satisfies the conditions (\ref{equation:intcond1}), (\ref{equation:intcond2}) and (\ref{equation:intcond3}). For the sake of brevity we will consider in the following the norm $\norm{(x_1,\ldots,x_m)}=\norm{x_1}+\cdots+\norm{x_m}$ for $x_i\in\R$ for $i=1,\ldots,m$.\\
Let us start by showing that $g(A,s)$ is $\Lambda$-integrable for $k=1$, then
\begin{gather*}
g(A,s)=\big(f\big(A,s\big),f\big(A,s-s_1,\ldots, f\big(A,s-s_{|S_1|}\big)'\text{, where }|S_1|=2\cdot 3^{m-1}.
\end{gather*}
Note that for $x\in\R^d$
\begin{align*}
\mathbb{1}_{[0,1]}(\norm{g(A,s)x})&\leq\mathbb{1}_{[0,1]}(\norm{f(A,s)x}),\\
\mathbb{1}_{[0,1]}(\norm{g(A,s)x})&\leq\mathbb{1}_{[0,1]}(\norm{f(A,s-s_1)x}),\\
&\ldots \\
\mathbb{1}_{[0,1]}(\norm{g(A,s)x})&\leq\mathbb{1}_{[0,1]}(\norm{f(A,s-s_{|S_1|})x}),
\end{align*}
such that
{\fontsize{10}{4}
\begin{align*}
&\int_S\!\int_{\R^m} \Big\lVert g(A,s)\gamma\!+\!\! \int_{\R^d}\!g(A,s)x\left(\mathbb{1}_{[0,1]}(\norm{g(A,s)x})\!-\mathbb{1}_{[0,1]}(\norm{x})\right)\nu(dx)\Big\rVert ds \pi(dA) \\
&\leq \int_S\!\int_{\R^m} \Big\lVert f(A,s)\gamma\!+\!\! \int_{\R^d}\!f(A,s)x\left(\mathbb{1}_{[0,1]}(\norm{f(A,s)x})\!-\mathbb{1}_{[0,1]}(\norm{x})\right)\nu(dx)\Big\rVert ds \pi(dA)\\
&+\int_S\!\int_{\R^m} \Big\lVert f(A,s-s_1)\gamma+\!\! \int_{\R^d}\!f(A,s-s_1)x\left(\mathbb{1}_{[0,1]}(\norm{f(A,s-s_1)x})\!-\mathbb{1}_{[0,1]}(\norm{x})\right)\nu(dx)\Big\rVert ds \pi(dA)\\
&+\cdots+\!\!\int_S\!\int_{\R^m} \Big\lVert f\big(A,s-s_{|S_1|})\gamma\!+\!\! \int_{\R^d}\!f(A,s-s_{|S_1|})\\
&\qquad\qquad\times x\left(\mathbb{1}_{[0,1]}(\norm{f(A,s-s_{|S_1|})}\!-\mathbb{1}_{[0,1]}(\norm{x})\right)\!\nu(dx)\Big\rVert ds \pi(dA).
\end{align*}}
Since $f$ is $\Lambda$-integrable, we can conclude that the above expression is finite and (\ref{equation:intcond1}) holds. Now
\begin{gather*}
\int_S\!\int_{\R^m}\!\norm{g(A,s)\Sigma g(A,s)'} ds\pi(dA)\\
=\int_S\!\int_{\R^m}\!\!\norm{f(A,s)\Sigma f(A,s)'} ds\pi(dA) \!+\!\!\! \int_S\!\int_{\R^m}\!\!\norm{f(A,s-s_1)\Sigma f(A,s-s_1)'} ds\pi(dA)\\
+\ldots+\int_S\!\int_{\R^m}\!\!\norm{f(A,s-s_{|S_1|})\Sigma f(A,s-s_{|S_1|})'} ds\pi(dA),
\end{gather*}
and it is finite since $f$ is $\Lambda$-integrable and (\ref{equation:intcond2}) holds. Since $\left(\sum_{i=1}^na_i\right)^2\leq n \sum_{i=1}^na_i^2$, we have
\begin{gather*}
\norm{g(A,s)}^2\leq |S_1| \Big( \norm{f(A,s)}^2+\norm{f(A,s-s_1)}^2+\ldots+\norm{f(A,s-s_{|S_1|})}^2\Big),
\end{gather*}
and finally
\begin{align*}
&\int_S\int_{\R^m}\int_{\R^d} \Big(1\wedge \norm{g(A,s)x}^2 \Big)\nu(dx)ds\pi(dA)\\
&\leq |S_k| \bigg(\int_S\int_{\R^m}\int_{\R^d} \Big(1\wedge \norm{f(A,s)}^2 \Big)\nu(dx)ds\pi(dA)\\ &\qquad+\int_S\int_{\R^m}\int_{\R^d} \Big(1\wedge \norm{f(A,s-s_1)}^2 \Big)\nu(dx)ds\pi(dA)\\
&\qquad+\ldots+ \int_S\int_{\R^m}\int_{\R^d} \Big(1\wedge \norm{f(A,s-s_{|S_1|})}^2\Big)\nu(dx)ds\pi(dA)\bigg),
\end{align*}
that is finite since $f$ satisfies (\ref{equation:intcond3}). Thus, $g$ is $\Lambda$-integrable and $Z$ is an $(A,\Lambda)$-influenced MMAF. By induction, the above statement can be shown for each $k \in \N$.\\
Assume that $X$ satisfies the assumptions of Proposition \ref{proposition:mmathetaweaklydep} (i) and consider $\psi(h)$ as defined in (\ref{equation:psi}). Then,
\begin{align*}
\theta_Z^{(i)}(h)\leq&2\Bigg(\int_S\int_{A_0\cap V_{0}^{\psi(h)}}\textup{tr}\Big(g(A,-s)\Sigma_\Lambda g(A,-s)'\Big)ds\pi(dA)\Bigg)^{\frac{1}{2}}\\
=&2\Bigg(\int_S\int_{A_0\cap V_{0}^{\psi(h)}}\textup{tr}\Big(f(A,-s)\Sigma_\Lambda f(A,-s)'\Big)ds\pi(dA)\\
&~+\int_S\int_{A_0\cap V_{0}^{\psi(h)}}\textup{tr}\Big(f(A,s_1-s)\Sigma_\Lambda f(A,s_1-s)'\Big)ds\pi(dA)\\
&~+\ldots+\int_S\int_{A_0\cap V_{0}^{\psi(h)}}\textup{tr}\Big(f(A,s_{|S_k|}-s)\Sigma_\Lambda f(A,s_{|S_k|}-s\big)')ds\pi(dA)\Bigg)^{\frac{1}{2}}\\
\leq&2 |S_k|^{\frac{m}{2}} \Bigg(\int_S\int_{A_0\cap V_{0}^{\psi(h)-k}}\textup{tr}\Big(f\big(A,-s\big)\Sigma_\Lambda f\big(A,-s\big)'\Big)ds\pi(dA)\Bigg)^{\frac{1}{2}}\\
=&|S_k|^{\frac{m}{2}} \hat{\theta}_X^{(i)}(h-\psi^{-1}(k)).
\end{align*}
where $\psi^{-1}$ denotes the inverse of $\psi$, for all $\psi(h)>k$. Thus, $Z$ is a $(k+1)(2k+1)^{m-1}$-dimensional $\theta$-lex-weakly dependent MMAF. Similar calculations lead to the other statements in (\ref{eq:mmainfluencedvectorweakly}).
\end{proof}
\subsection{Proof of Section \ref{sec3-4}}
\label{sec5-3}
\begin{proof}[Proof of Theorem \ref{theorem:thetasamplemeanspecial}]
Let us first consider $X$ to be univariate. In order to use \cite[Theorem 1]{D1998} we need to show
\begin{gather}\label{equation:L1condition}
\sum_{k\in V_0}|X_kE_{|k|}[X_0]|\in L^1.
\end{gather} The H\"older inequality implies
\begin{gather*}
\norm{X_kE_{|k|}(X_0)}_{1}\leq \norm{X_k}_{2}\norm{E[X_0|\mathcal{F}_{V_0^{|k|}}]}_{2},
\end{gather*}
where $ \norm{X_k}_{2}<C$ for all $k$ and a constant $C$. Furthermore, we note that $\lVert E[X|\mathcal{F}] \rVert_{2}= \lVert E[E[X|\mathcal{H}]|\mathcal{F}] \rVert_{2}\leq \lVert E[X|\mathcal{H}]\rVert_{2}$ holds for a $\sigma$-algebra $\mathcal{H}$, a sub $\sigma$-algebra $\mathcal{F}$ and an $L^2$ random variable $X$, as the conditional expectation is the orthogonal projection in $L^2$. Now, using (\ref{equation:influencedMMAfield})
\begin{align*}
\norm{E[X_0|\mathcal{F}_{V_0^{|k|}}]}_{2}=& \left\lVert E\left[ \int_S\int_{V_0}\mathbb{1}_{A_0}(s) f(A,-s)\Lambda(dA,ds)\Big|\sigma(X_l:l\in V_0^{|k|} )\right]\right\rVert_{2}\\
\leq& \left\lVert E\left[ \int_S\int_{V_0}\mathbb{1}_{A_0}(s) f(A,-s)\Lambda(dA,ds)\Big|\sigma(\Lambda(B):B\in\mathcal{B}(V_0^{|k|}))\right]\right\rVert_{2}\\
=&\Bigg\lVert E\left[ \int_S\int_{V_0^{|k|}}\mathbb{1}_{A_0}(s) f(A,-s)\Lambda(dA,ds)\Big|\sigma(\Lambda(B):B\in\mathcal{B}(V_0^{|k|}))\right]\\
&~+E\left[\int_S\int_{V_0\backslash V_0^{|k|}}\mathbb{1}_{A_0}(s) f(A,-s)\Lambda(dA,ds)\Big|\sigma(\Lambda(B):B\in\mathcal{B}(V_0^{|k|}))\right]\Bigg\rVert_{2}.
\end{align*}
We note that $\int_S\int_{V_0^{|k|}}\mathbb{1}_{A_0}(s) f(A,-s)\Lambda(dA,ds)$ is measurable with respect to $\sigma(\Lambda(B):B\in\mathcal{B}(V_0^{|k|}))$. Since $\Lambda$ is a L\'evy basis (in particular independent for disjoint sets) we get that $\int_S\int_{V_0\backslash V_0^{|k|}}\mathbb{1}_{A_0}(s) f(A,-s)$ $\Lambda(dA,ds)$ is independent of $\sigma(\Lambda(B):B\in\mathcal{B}(V_0^{|k|}))$, such that the above equation is equal to
\begin{gather*}
\Bigg\lVert \int_S\int_{V_0^{|k|}}\mathbb{1}_{A_0}(s) f(A,-s)\Lambda(dA,ds)+E\left[\int_S\int_{V_0\backslash V_0^{|k|}}\mathbb{1}_{A_0}(s)f(A,-s)\Lambda(dA,ds)\right]\Bigg\rVert_{2}.
\end{gather*}
Since $\gamma+\int_{\norm{x}>1}x\nu(dx)=0$ the second summand is equal to zero and the above is equal to
\begin{align*}
&\Bigg\lVert \int_S\int_{V_0^{|k|}}\mathbb{1}_{A_0}(s) f(A,-s) f(A,-s)\Lambda(dA,ds)\Bigg\rVert_{2}\\
&=\Big(\int_S\int_{A_0\cap V_0^{|k|}}\textup{tr}(f(A,-s)\Sigma_\Lambda f(A,-s)')ds\pi(dA)\Big)^{\frac{1}{2}}
=\theta_X(|k|),
\end{align*}
using Proposition \ref{proposition:MMAmoments}.\\
The stated result then follows from \cite[Theorem 1]{D1998} using the dominated convergence theorem.
The Cram\'er-Wold device establishes the multivariate case straightforwardly.
\end{proof}
\subsection{Proofs of Section \ref{sec3-5}}
\label{sec5-4}
\begin{proof}[Proof of Proposition \ref{proposition:mmaetaweaklydep}]
\begin{enumerate}[(i)]
\item Let $t\in\R^m$ and $\psi>0$. We restrict the MMAF to a finite support, and define the sequence
\begin{align}\label{equation:truncatedeta}
\begin{aligned}
X_t^{(\psi)}&=\int_S\int_{\R^m}f(A,t-s)\mathbb{1}_{(-\psi,\psi)^m}(t-s)\Lambda(dA,ds)\\
&=\int_S\int_{(t-\psi,t+{\psi})^m}f(A,t-s)\Lambda(dA,ds).
\end{aligned}
\end{align}
Note that the kernel function $f$ is square integrable such that (\ref{equation:intcond1}), (\ref{equation:intcond2}) and (\ref{equation:intcond3}) hold. Therefore, $f$ is $\Lambda$-integrable. Since $E[X_tX_t']<\infty$ for all $t\in\R^m$, by Proposition \ref{proposition:MMAexistencemoments} we can derive an upper bound of the expectation
\begin{align}
\begin{aligned}\label{eq:L1norminequality2}
&E\big[\norm{X_t-X_t^{(\psi)}}\big]=E\bigg[ \Big\lVert\int_S\int_{\big((t-\psi,t+\psi)^m\big)^c}f(A,t-s)\Lambda(dA,ds)\Big\rVert\bigg]\\
&\qquad\leq E\bigg[ \Big\lVert\int_S\int_{\big((t-\psi,t+\psi)^m\big)^c}f(A,t-s)\Lambda(dA,ds)\Big\rVert^2\bigg]^{\frac{1}{2}}\\
&\qquad= \left(\sum_{\kappa=1}^nE\Bigg[ \bigg( \Big(\int_S\int_{\big((t-\psi,t+\psi)^m\big)^c}f(A,t-s)\Lambda(dA,ds)\Big)^{(\kappa)}\bigg)^2\Bigg]\right)^{\frac{1}{2}},
\end{aligned}
\end{align}
where $x^{(\kappa)}$ denotes the $\kappa$th coordinate of $x\in\R^n$.
Using Proposition \ref{proposition:MMAmoments} and the stationarity of $X$ this is equal to
\begin{gather*}
\Big(\int_S\int_{\big((\psi,\psi)^m\big)^c}\textup{tr}(f(A,-s)\Sigma_\Lambda f(A,-s)')ds\pi(dA)\Big)^{\frac{1}{2}}.
\end{gather*}
For $(u,v)\in\N\times\N$ let $F\in\mathcal{G}_u$, $G\in\mathcal{G}_v$, $h\in\R^{+}, \Gamma_i=\{i_1,\ldots,i_u\}\in(\R^m)^u$ and $\Gamma_j=\{j_1,\ldots,j_v\}\in(\R^m)^v$ as in Definition \ref{etaweaklydependent} such that $dist(\Gamma_i,\Gamma_j)\geq h$. For $a\in\{1,\ldots,u\}$ and $b\in\{1,\ldots,v\}$ define
\begin{align*}
X_{i_a}^{(\psi)}&=\int_S\int_{(i_a-{\psi},i_a+{\psi})^m}f(A,i_a-s)\,\Lambda(dA,ds) \text{ and }\\
X_{j_b}^{(\psi)}&=\int_S\int_{(j_b-{\psi},j_b+{\psi})^m}f(A,j_b-s)\,\Lambda(dA,ds).
\end{align*}
Now consider $a\in\{1,\ldots,u\}$ and $b\in\{1,\ldots,v\}$ such that $\inf_{1\leq x \leq u, 1\leq y \leq v}\norm{i_x-j_y}_\infty=\norm{i_a-j_b}_\infty$. Define the two sets $I_a=S\times (i_a-{\psi},i_a+{\psi})^m$ and $J_b=S\times (j_b-{\psi},j_b+{\psi})^m$. Let $\psi=\frac{h}{2}$ and $\norm{i_a-j_b}_\infty\geq h$. Then, it holds that $I_a$ and $J_b$ are disjoint as well as $I_{\tilde{a}}$ and $J_{\tilde{b}}$ for all $\tilde{a}=1,\ldots,u$ and $\tilde{b}=1,\ldots,v$. By the definition of a L\'evy basis $X_{i_a}^{(\psi)}$ and $X_{j_b}^{(\psi)}$ are independent for all $a\in\{1,\ldots,u\}$ and $b\in\{1,\ldots,v\}$. Finally, we get that $X_{\Gamma_i}^{(\psi)}$ and $X_{\Gamma_j}^{(\psi)}$ are independent and therefore also $F(X_{\Gamma_i}^{(\psi)})$ and $G(X_{\Gamma_j}^{(\psi)})$. Now,
\begin{align*}
&|Cov(F(X_{\Gamma_i}),G(X_{\Gamma_j}))|\\&\leq |Cov(F(X_{\Gamma_i})-F(X_{\Gamma_i}^{(\psi)}),G(X_{\Gamma_j}))|+|Cov(F(X_{\Gamma_i}^{(\psi)}),G(X_{\Gamma_j})-G(X_{\Gamma_i}^{(\psi)}))|\\
&= |E[(F(X_{\Gamma_i})-F(X_{\Gamma_i}^{(\psi)}))G(X_{\Gamma_j})]-E[F(X_{\Gamma_i})-F(X_{\Gamma_i}^{(\psi)})]E[G(X_{\Gamma_j})]|\\
&~+|E[(G(X_{\Gamma_j})-G(X_{\Gamma_j}^{(\psi)}))F(X_{\Gamma_i}^{(\psi)})]-E[G(X_{\Gamma_j})-G(X_{\Gamma_j}^{(\psi)})]E[F(X_{\Gamma_i}^{(\psi)})]|\\
&\leq 2\Big(\norm{G}_\infty E\big[|F(X_{\Gamma_i})-F(X_{\Gamma_i}^{(\psi)})|\big]+ \norm{F}_\infty E\big[|G(X_{\Gamma_j})-G(X_{\Gamma_i}^{(\psi)})|\big]\Big)\\
&\leq 2 \Big(\norm{G}_\infty Lip(F) \sum_{l=1}^u E[\norm{X_{i_l}-X_{i_l}^{(\psi)}}] +\norm{F}_\infty Lip(G) \sum_{k=1}^v E[\norm{X_{j_k}-X_{j_k}^{(\psi)}}] \Big)\\
&\leq 2 (u\norm{G}_\infty Lip(F)+v\norm{F}_\infty Lip(G))\\
&\qquad\times \Big(\int_S\int_{\big(\big(-\frac{{h}}{2},\frac{{h}}{2}\big)^m\big)^c}\textup{tr}(f(A,-s)\Sigma_\Lambda f(A,-s)')ds\pi(dA)\Big)^{\frac{1}{2}},
\end{align*}
using (\ref{eq:L1norminequality2}). Therefore, $X$ is $\eta$-weakly dependent with $\eta$-coefficients
\begin{gather*}
\eta_X(h)\leq 2 \Big(\int_S\int_{\big(\big(-\frac{{h}}{2},\frac{{h}}{2}\big)^m\big)^c}\textup{tr}(f(A,-s)\Sigma_\Lambda f(A,-s)')ds\pi(dA)\Big)^{\frac{1}{2}},
\end{gather*}
which converge to zero as $h$ goes to infinity by applying the dominated convergence theorem.
\item[(iii)] Since $f\in L^1$, (\ref{equation:intcondfinvar1}) and (\ref{equation:intcondfinvar2}) hold and $f$ is $\Lambda$-integrable. Let $t\in\R^m$, $\psi>0$ and $X_t^{(\psi)}$ be defined as in (i). Moreover, Proposition \ref{proposition:MMAexistencemoments} implies that $E[X_t]<\infty$. Then, using Proposition \ref{proposition:MMAmoments}
\begin{align*}
E\big[\norm{X_t-X_t^{(\psi)}}\big]
&\leq \Big(\int_S\int_{\big((t-\psi,t+{\psi})^m\big)^c} \big\lVert f(A,-s)\gamma_0\big\rVert ds\pi(dA)\\
&\qquad+\int_S\int_{\big((t-\psi,t+{\psi})^m\big)^c} \int_{\R^d} \big\lVert f(A,-s)x\big\rVert \nu(dx)ds\pi(dA) \Big).
\end{align*}
Now, for $F$, $G$, $X_{\Gamma_i}$ and $X_{\Gamma_j}$ and $\psi$ as described in (i), we get
\begin{align*}
|Cov(F(X_{\Gamma_i}),G(X_{\Gamma_j}))| &\leq
2 (u\norm{G}_\infty Lip(F)+v\norm{F}_\infty Lip(G))\\
&\quad \times\Bigg( \int_S\int_{\big(\big(-\frac{{h}}{2},\frac{{h}}{2}\big)^m\big)^c} \big\lVert f(A,-s)\gamma_0\big\rVert ds\pi(dA)\\
&\qquad\quad+\int_S\int_{\big(\big(-\frac{{h}}{2},\frac{{h}}{2}\big)^m\big)^c} \int_{\R^d} \big\lVert f(A,-s)x\big\rVert \nu(dx)ds\pi(dA)\Bigg).
\end{align*}
Therefore, $X$ is $\eta$ weakly dependent with $\eta$-coefficients
\begin{align*}
\eta_X(h)&\leq 2 \Bigg( \int_S\int_{\big(\big(-\frac{{h}}{2},\frac{{h}}{2}\big)^m\big)^c} \big\lVert f(A,-s)\gamma_0\big\rVert ds\pi(dA)\\
&\qquad +\int_S\int_{\big(\big(-\frac{{h}}{2},\frac{{h}}{2}\big)^m\big)^c} \int_{\R^d} \big\lVert f(A,-s)x\big\rVert \nu(dx)ds\pi(dA)\!\Bigg),
\end{align*}
which converge to zero as $h$ goes to infinity by applying the dominated convergence theorem.
\end{enumerate}
\noindent
Part (ii) and (iv) of the Proposition follow from the above results, analogously to Proposition \ref{proposition:mmathetaweaklydep} (ii) and (iv).
\end{proof}
\subsection{Proofs of Section \ref{sec3-8}}
\begin{proof}[Proof of Theorem \ref{theorem:levydrivenexpdecaykernel}]
Let $\norm{A}_F=\sqrt{tr(AA')}$ for $A\in M_{n\times d}(\R)$ denote the Frobenius norm and $\norm{x}_1=\sum_{\nu=1}^m|x^{(\nu)}|$ for $x\in\R^m$. From Proposition \ref{proposition:mmaetaweaklydep} it follows that $X$ is $\eta$-weakly dependent with $\eta$-coefficients
\begin{align*}
\eta_X(h)&=\Bigg(\int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c}\textup{tr}(g(-s)\Sigma_Lg(-s)')ds\Bigg)^{\frac{1}{2}}
=\Bigg(\int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c} \norm{g(-s)\Sigma^{\frac{1}{2}}}_F^2 ds\Bigg)^{\frac{1}{2}}\\
&\leq \Bigg(\norm{\Sigma^{\frac{1}{2}}}_F^2 \int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c} \norm{g(-s)}_F^2 ds\Bigg)^{\frac{1}{2}}
\leq \Bigg(\norm{\Sigma^{\frac{1}{2}}}_F^2M \int_{\left(\left(-\frac{h}{2},\frac{h}{2}\right)^m\right)^c} e^{-K\norm{s}_1}ds\Bigg)^{\frac{1}{2}}\\
&= \norm{\Sigma^{\frac{1}{2}}}_F M^{\frac{1}{2}}\Bigg(\int_{\R^m} e^{-K\norm{s}_1}ds-\int_{\left(-\frac{h}{2},\frac{h}{2}\right)^m} e^{-K\norm{s}_1}ds\Bigg)^{\frac{1}{2}}\\
&=\norm{\Sigma^{\frac{1}{2}}}_F M^{\frac{1}{2}}\Bigg(\left(\frac{1}{2K}\right)^m-\left(\frac{1}{2K}-\frac{e^{-\frac{K}{2}h}}{2K}\right)^m \Bigg)^{\frac{1}{2}}\\
&= \frac{\norm{\Sigma^{\frac{1}{2}}}_FM^{\frac{1}{2}}}{(2K)^\frac{m}{2}}\Bigg(1-\left(1-e^{-\frac{K}{2}h}\right)^m \Bigg)^{\frac{1}{2}}
\leq \frac{m\norm{\Sigma^{\frac{1}{2}}}_FM^{\frac{1}{2}}}{(2K)^\frac{m}{2}} e^{-\frac{K}{4}h},
\end{align*}
where the last inequality follows from Bernoulli's inequality.
\end{proof}
\subsection{Proofs of Section \ref{sec4-2}}
\label{sec5-5}
\begin{proof}[Proof of Proposition \ref{proposition:ambitthetaweaklydep}]
\begin{enumerate}[(i)]
\item Let $(t,x)\in\R\times\R^{m}$, $\psi>0$. We define the two truncated sequences
\begin{align*}
&\tilde{Y}_{t}^{(\psi)}(x)=\int_{A_t(x)\backslash V_{(t,x)}^\psi}l(x-\xi,t-s)\sigma_s(\xi)\Lambda(d\xi,ds), \,\,\, \text{and}\\
&Y_{t}^{(\psi)}(x)=\int_{A_{t}(x)\backslash V_{(t,x)}^\psi}l(x-\xi,t-s)\sigma_s^{(\psi)}(\xi)\Lambda(d\xi,ds),\,\, \text{where}\\
&\sigma_t^{(\psi)}(x)=\int_S\int_{A_{(t,x)}^\sigma(x)\backslash V_{(t,x)}^\psi}j(x-\xi,t-s) \Lambda^\sigma(dA,d\xi,ds).
\end{align*}
Since the kernel function $j$ is square integrable we have that (\ref{equation:intcond1}), (\ref{equation:intcond2}) and (\ref{equation:intcond3}) hold. Therefore, $j$ is $\Lambda^\sigma$-integrable and $\sigma$ is well-defined and stationary. Now, by Proposition \ref{proposition:MMAexistencemoments} it holds that $\sigma_t(x)\in L^2(\Omega)$. Since additionally $l\in L^2(\R^m\times\R)$ and $\sigma$ is stationary, it holds that $l\sigma\in L^2(\Omega\times\R^m\times\R)$. This implies that $l\sigma\in L^2(\R^m\times\R)$ almost surely. Then, $l\sigma$ satisfies (\ref{equation:intcond1}), (\ref{equation:intcond2}) and (\ref{equation:intcond3}) almost surely and the ambit field $Y$ is well-defined. Analogous to Proposition \ref{proposition:mmathetaweaklydep} and using Proposition \ref{proposition:ambitmoments}
\begin{align*}
&E\big[|Y_{t}(x)-Y_{t}^{(\psi)}(x)|\big]\leq E\big[|Y_{t}(x)-\tilde{Y}_{t}^{(\psi)}(x)|\big] + E\big[|\tilde{Y}_{t}^{(\psi)}(x)-Y_{t}^{(\psi)}(x)|\big] \\
&=E\bigg[ \Big|\int_{A_t(x)\cap V_{(t,x)}^\psi}l(x-\xi,t-s)\sigma_s(\xi)\Lambda(d\xi,ds)\Big|\bigg]\\
&\quad+E\bigg[ \Big|\int_{A_t(x)\backslash V_{(t,x)}^\psi}l(x-\xi,t-s)(\sigma_s(\xi)-\sigma_s^{(\psi)}(\xi))\Lambda(d\xi,ds)\Big|\bigg] \\
&\leq E\Bigg[ \bigg( \int_{A_t(x)\cap V_{(t,x)}^\psi}l(x-\xi,t-s)\sigma_s(\xi)\Lambda(d\xi,ds)\bigg)^2\Bigg]^{\frac{1}{2}}\\
&\quad+E\Bigg[ \bigg( \int_{A_t(x)\backslash V_{(t,x)}^\psi} l(x-\xi,t-s)(\sigma_s(\xi)-\sigma_s^{(\psi)}(\xi))\Lambda(d\xi,ds))\bigg)^2\Bigg]^{\frac{1}{2}}.
\end{align*}
Using Proposition \ref{proposition:ambitmoments} and the translation invariance of $A_t(x)$ and $V_{(t,x)}^\psi$, the above is equal to
\begin{align*}
&\Big(\Sigma_\Lambda E[\sigma_0(0)^2] \int_{A_0(0) \cap V_{(0,0)}^\psi}l(-\xi,-s)^2 d\xi ds\Big)^{\frac{1}{2}}\\
&\quad+ \Bigg( E\left[\left(\int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^\psi} j(A,-\xi,-s) \,\Lambda^\sigma(dA,d\xi,ds) \right)^2\right]\Sigma_\Lambda \, \int_{A_0(0) \backslash V_{(0,0)}^\psi} l(-\xi ,-s)^2 d\xi ds\,\Bigg)^{\frac{1}{2}}\\
&=\Big(\Sigma_\Lambda E[\sigma_0(0)^2] \int_{A_0(0) \cap V_{(0,0)}^\psi}l(-\xi,-s)^2 d\xi ds\Big)^{\frac{1}{2}}\\
&\quad+ \Bigg(\Bigg(\Sigma_{\Lambda^\sigma} \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^\psi} j(A,-\xi,-s)^2 d\xi ds \pi(dA)\\
&\quad+ \mu_{\Lambda^\sigma}^2\Bigg( \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^\psi} j(A,-\xi,-s) \, d\xi ds \pi(dA)\Bigg)^2 \Bigg)
\Sigma_\Lambda \, \int_{A_0(0) \backslash V_{(0,0)}^\psi} l(-\xi ,-s)^2 d\xi ds\Bigg)^{\frac{1}{2}}.
\end{align*}
Now let $G\in\mathcal{G}_1$ and $F\in\mathcal{G}^*_u$, i.e. $F,G$ are bounded and $G$ additionally Lipschitz-continuous for $u\in\N, h\in\R^{+}, \Gamma=\{(t_{i_1},x_{i_1}),\ldots,(t_{i_u},x_{i_u})\}\in(\R\times\R^m)^u$ and $(t_j,x_j)\in\R\times\R^m$ such that $(t_{i_1},x_{i_1}),\ldots,(t_{i_u},x_{i_u})\in V_{(t_j,x_j)}^h$. For $a\in\{1,\ldots,u\}$ define
\begin{align*}
&Y_{t_{i_a}}(x_{i_a})=\int_{A_{t_{i_a}}(x_{i_a})}l(x_{i_a}-\xi,t_{i_a}-s)\sigma_s(\xi) \Lambda(d \xi ,ds),\,\, \text{ and } \\
&Y_{t_j}^{(\psi)}(x_j)=\int_{A_{t_j}(x_j)\backslash V_{(t_j,x_j)}^{\psi(h)}}l(x_j-\xi,t_j-s)\sigma_s^{(\psi)}(\xi)\Lambda(d\xi,ds).
\end{align*}
W.l.o.g. we assume that $(t_{i_a},x_{i_a})\leq_{lex}(t_{i_u},x_{i_u})$ for all $a\in\{1,\ldots,u\}$. Since $A_0(0)\cup A_0^\sigma(0)$ satisfy (\ref{condition:scalarproduct}) we find analogous to (\ref{equation:psi}) a function $\psi(h)= \frac{-hb}{2\sqrt{m+1}}$, such that $A_{s_1}^\sigma(\xi_1)$ and $A_{s_2}^\sigma(\xi_2)\backslash V_{(s_2,\xi_2)}^{\psi(h)}$ are disjoint for all $(s_1,\xi_1)\in A_{(t_{i_u},x_{i_u})}$ and $(s_2,\xi_2)\in A_{(t_{j},x_{j})}\backslash V_{(t_j,x_j)}^{\psi(h)}$ or have intersection with zero Lebesgue measure. Then, by the definition of a L\'evy basis we get that $\sigma_{s_1}(\xi_1)$ and $\sigma_{s_2}^{\psi(h)}(\xi_2)$ are independent.
Furthermore, it holds that $A_{t_{i_u}}(x_{i_u})$ and $A_{t_{j}}(x_{j})\backslash V_{(t_j,x_j)}^{\psi(h)}$ are disjoint. We set $\psi=\psi(h)$ throughout. Finally, we get that $Y_{t_{i_a}}(x_{i_a})$ and $Y_{t_j}^{(\psi(h))}(x_j)$ are independent for all $a\in\{1,\ldots,u\}$ and therefore also $F(Y_\Gamma)$ and $G(Y_{t_j}^{(\psi(h))}(x_j))$. Now
\begin{align*}
&|Cov(F(Y_\Gamma),G(Y_{t_j}(x_j)))| \\
&\leq |Cov(F(X_\Gamma),G(Y_{t_j}^{(\psi(h))}(x_j)))|+|Cov(F(Y_\Gamma),G(Y_{t_j}(x_j))-G(Y_{t_j}^{(\psi(h))}(x_j)))|\\
&=|E[(G(Y_{t_j}(x_j))-G(Y_{t_j}^{(\psi(h))}(x_j))) F(X_\Gamma)]\\
&\quad -E[G(Y_{t_j}(x_j))-G(Y_{t_j}^{(\psi(h))}(x_j))]E[F(X_\Gamma)]|\\
&\leq 2\norm{F}_\infty E\big[|(G(Y_{t_j}(x_j))-G(Y_{t_j}^{(\psi(h))}(x_j)))|\big]\\
&\leq 2Lip(G) \norm{F}_\infty E[\norm{Y_{t_j}(x_j)-Y_{t_j}^{(\psi(h))}(x_j)}]\\
&\leq 2 Lip(G)\norm{F}_\infty\Bigg(\Big(\Sigma_\Lambda E[\sigma_0(0)^2] \int_{A_0(0) \cap V_{(0,0)}^{\psi(h)}}l(-\xi,-s)^2 d\xi ds\Big)^{\frac{1}{2}}\\
&+ \Bigg(\Bigg(\Sigma_{\Lambda^\sigma} \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^{\psi(h)}} j(A,-\xi,-s)^2 d\xi ds \pi(dA)\\
&+ \mu_{\Lambda^\sigma}^2\!\Bigg(\!\! \int_S\!\int_ {A_0^\sigma(0)\cap V_{(0,0)}^{\psi(h)}}\!\! j(A,-\xi,-s) d\xi ds \pi(dA)\Bigg)^2 \Bigg)
\Sigma_\Lambda \!\! \int_{A_0(0) \backslash V_{(0,0)}^{\psi(h)}}\!\!\!\!\!l(-\xi ,-s)^2 d\xi ds\Bigg)^{\frac{1}{2}}\Bigg),
\end{align*}
using the above inequality for $E\big[|Y_{t}(x)-Y_{t}^{(\psi(h))}(x)|\big]$. Therefore, $Y$ is $\theta$-lex weakly dependent with $\theta$-lex-coefficients
\begin{align*}
\theta_Y(h)&\leq 2\Bigg(\Big(\Sigma_\Lambda E[\sigma_0(0)^2] \int_{A_0(0) \cap V_{(0,0)}^{\psi(h)}}l(-\xi,-s)^2 d\xi ds\Big)^{\frac{1}{2}}\\
&\qquad+ \Bigg(\Bigg(\Sigma_{\Lambda^\sigma} \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^{\psi(h)}} j(A,-\xi,-s)^2 d\xi ds \pi(dA)\\
&\qquad+ \mu_{\Lambda^\sigma}^2\!\!\Bigg( \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^{\psi(h)}}\!\! j(A,-\xi,-s) d\xi ds \pi(dA)\Bigg)^2 \Bigg)\\
&\qquad\times\Sigma_\Lambda \int_{A_0(0) \backslash V_{(0,0)}^{\psi(h)}}\!\!\!\!\!l(-\xi ,-s)^2 d\xi ds\Bigg)^{\frac{1}{2}}\Bigg),
\end{align*}
which converges to zero as $h$ goes to infinity by applying the dominated convergence theorem.\\
\item Let $(t,x)\in\R\times\R^m$ and $\psi>0$, $Y_t^{(\psi)}(x)$ and $\tilde{Y}_t^{(\psi)}(x)$ be defined as in (i). By Proposition \ref{proposition:ambitmoments}
\begin{align*}
&E\big[|Y_{t}(x)-\tilde{Y}_{t}^{(\psi)}(x)|\big] + E\big[|\tilde{Y}_{t}^{(\psi)}(x)-Y_{t}^{(\psi)}(x)|\big]\\
&\leq\Bigg(\Big(\Sigma_\Lambda E[\sigma_0(0)^2]
\int_{A_t(x) \cap V_{(t,x)}^\psi}l(x-\xi,t-s)^2 d\xi ds \\
&\quad+\mu_\Lambda E[\sigma_0(0)^2] \Big(\int_{A_t(x) \cap V_{(t,x)}^\psi}l(x-\xi,t-s) d\xi ds\Big)^2 \Bigg)^{\frac{1}{2}}\\
&\quad+ \Bigg( \Bigg(\Sigma_{\Lambda^\sigma} \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^\psi} j(A,-\xi,-s)^2 d\xi ds \pi(dA)\\
&\quad+ \mu_{\Lambda^\sigma}^2\Bigg( \int_S\int_ {A_0^\sigma(0)\cap V_{(0,0)}^\psi} j(A,-\xi,-s) d\xi ds \pi(dA)\Bigg)^2 \Bigg)\\
&\qquad\times\Bigg( \Sigma_\Lambda \int_{A_t(x) \backslash V_{(t,x)}^\psi}\!\!\!\!\!\!\!l(x-\xi ,t-s)^2 d\xi ds
+ \mu_\Lambda \Big(\int_{A_t(x) \backslash V_{(t,x)}^\psi}\!\!\!\!\!\!\!l(x-\xi ,t-s) d\xi ds\Big)^2 \Bigg)\Bigg)^{\frac{1}{2}}.
\end{align*}
Finally, we can proceed as in the proof of part (i) to obtain the stated bound for the $\theta$-lex-coefficients.\\
\item Note that the kernel function $j$ is square integrable such that $\sigma$ is well defined and stationary. Now, by Proposition \ref{proposition:MMAexistencemoments} it holds that $\sigma\in L^1(\Omega)$. Since additionally $l\in L^1(\R^m\times\R)$ and $\sigma$ is stationary it holds that $l\sigma\in L^1(\Omega\times\R^m\times\R)$. This implies that $l\sigma\in L^1(\R^m\times\R)$ almost surely. Then $l\sigma$ satisfies (\ref{equation:intcondfinvar1}) and (\ref{equation:intcondfinvar2}) almost surely and the ambit field $Y$ is well defined. Let $(t,x)\in\R\times\R^m$ and $\psi>0$ be defined as in (i). By Proposition \ref{proposition:ambitmoments}
\begin{align*}
&E\big[|Y_{t}(x)-\tilde{Y}_{t}^{(\psi)}(x)|\big] + E\big[|\tilde{Y}_{t}^{(\psi)}(x)-Y_{t}^{(\psi)}(x)|\big]\\
&\leq E[|\sigma_0(0)|]\Big(|\gamma_0|+\int_{\R^d}|y|\nu(dy)\Big) \Big(\int_{A_0(0)\cap V_{(0,0)}^\psi} |l(-\xi,-s)| d\xi ds \Big)\\
&\quad+E[|\sigma_0(0)-\sigma_0^{(\psi)}(0)|]\Big(|\gamma_0|+\int_{\R^d}|y|\nu(dy)\Big) \Big(\int_{A_0(0)\backslash V_{(0,0)}^\psi} |l(-\xi,-s)| d\xi ds \Big) .
\end{align*}
Finally, we obtain a bound for the $\theta$-lex-coefficients by proceeding as in (i)
\end{enumerate}
\end{proof}
\begin{proof}[Proof of Proposition \ref{proposition:ambitmdepthetaweaklydep}]
Let $(t,x)\in\R\times\R^{m}$ and $\psi>0$ be defined as in Proposition \ref{proposition:ambitthetaweaklydep}. We define the truncated sequence
\begin{gather*}\label{equation:truncatedambitthetaiid}
Y_{t}^{(\psi)}(x)=\int_{A_t(x)\backslash V_{(t,x)}^\psi}l(x-\xi,t-s)\sigma_s(\xi)\Lambda(d\xi,ds).
\end{gather*}
Since $l\in L^2(\R^m\times\R)$, $\sigma\in L^2(\Omega)$ and $\sigma$ is stationary, it holds that $l\sigma\in L^2(\Omega\times\R^m\times\R)$. This implies $l\sigma\in L^2(\R^m\times\R)$ almost surely. Then $l\sigma$ satisfies (\ref{equation:intcond1}), (\ref{equation:intcond2}) and (\ref{equation:intcond3}) almost surely and the ambit field $Y$ is well defined. By Proposition \ref{proposition:ambitmoments}
\begin{align*}
E\big[|Y_{t}(x)-Y_{t}^{(\psi)}(x)|\big]&=E\bigg[ \Big|\int_{A_t(x)\cap V_{(t,x)}^\psi}l(x-\xi,t-s)\sigma_s(\xi)\Lambda(d\xi,ds)\Big|\bigg]\\
&\leq E\Bigg[ \bigg( \int_{A_t(x)\cap V_{(t,x)}^\psi}l(x-\xi,t-s)\sigma_s(\xi)\Lambda(d\xi,ds)\bigg)^2\Bigg]^{\frac{1}{2}}.
\end{align*}
Using the translation invariance of $A_t(x)$ and $V_{(t,x)}^\psi$ this is equal to
\begin{gather*}
\Big(\Sigma_\Lambda E[\sigma_0(0)] \int_{A_0(0) \cap V_{(0,0)}^\psi}l(-\xi,-s)^2 d\xi ds\Big)^{\frac{1}{2}}.
\end{gather*}
Define $\Gamma \in (\R\times\R^m)^u$, $(t_{j},x_{j})\in\R\times\R^m$ as in the proof of Proposition \ref{proposition:ambitthetaweaklydep}. Since $\sigma$ is $p$-dependent we get that $Y_\Gamma$ and $Y_{t_j}^{(\psi)}(x_j)$ are independent for a sufficiently big $h$. Then, for sufficiently big $h$, $Y$ is $\theta$-lex weakly dependent with $\theta$-lex-coefficients
\begin{gather*}
\theta_Y(h)\leq 2 \Big(\Sigma_\Lambda E[\sigma_0(0)^2] \int_{A_0(0) \cap V_{(0,0)}^{\psi(h)}}l(-\xi,-s)^2 d\xi ds\Big)^{\frac{1}{2}},
\end{gather*}
which converge to zero as $h$ goes to infinity by applying the dominated convergence theorem.
\end{proof}
\section*{Acknowledgements}
The authors are grateful to two anonymous referees for helpful comments, which considerably improved this work.
The third author was supported by the scholarship program of the Hanns-Seidel Foundation, funded by the Federal Ministry of Education and Research. | 6,658 |