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\noindent {\it Proof} . The fact that the operators within each set commute with each other follows from the commutativity
of the Pontryagin product. Now let us check that $P_{1,0}({\cal C})^{[d_1]}$ commutes with
$P_{0,1}({\cal C})^{[d_2]}$.
The composition $P_{0,1}({\cal C})^{[d_2]}\circ P_{1,0}({\cal C})^{[d_1]}$ acting on ${\Bbb C}H^*({\cal C}^{[N]})$
is given by the following correspondence ${\Bbb P}i$ from ${\cal C}^{[N]}$ to ${\cal C}^{[N+d_1-d_2]}$ equipped
with a class of dimension $N+d_1$:
$${\Bbb P}i=\{(D_1,E_1,D_2,E_2)\in{\cal C}^{[d_1]}\tildemes_S{\cal C}^{[N]}\tildemes_S{\cal C}^{[d_2]}\tildemes_S
{\cal C}^{[N+d_1-d_2]}\ |\ D_1+E_1=D_2+E_2\},$$
where the map $q:{\Bbb P}i\to{\cal C}^{[N]}\tildemes_S{\cal C}^{[N+d_1-d_2]}$ sends $(D_1,E_1,D_2,E_2)$ to
$(E_1,E_2)$. This correspondence is equipped with the natural intersection-product class of dimension
$N+d_1$
\betaegin{equation}\lambdabel{div-com-inter-class}
[{\cal C}^{[d_1]}\tildemes_S{\cal C}^{[N]}]\cdotot [{\cal C}^{[d_2]}\tildemes_S{\cal C}^{[N+d_1-d_2]}].
\operatorname{e}nd{equation}
On the other hand, the composition $P_{1,0}({\cal C})^{[d_1]}\circ P_{0,1}({\cal C})^{[d_2]}$ is given by the correspondence
$$
\betaegin{diagram}
& &{\cal C}^{[d_2]}\tildemes_S{\cal C}^{[N-d_2]}\tildemes_S{\cal C}^{[d_1]}& &\\
&\ldTo{\alpha_{d_2,N-d_2}p_{12}}& &\rdTo{\alpha_{N-d_2,d_1}p_{23}}&\\
{\cal C}^{[N]}& & & &{\cal C}^{[N+d_1-d_2]}
\operatorname{e}nd{diagram}
$$
The natural map
$${\cal C}^{[d_2]}\tildemes_S{\cal C}^{[N-d_2]}\tildemes_S{\cal C}^{[d_1]}\to{\Bbb P}i:(D_2,E',D_1)\mapsto
(D_1,D_2+E',D_2,D_1+E')$$
is an isomorphism onto an irreducible component ${\Bbb P}i_0\subset{\Bbb P}i$.
Other irreducible components ${\Bbb P}i_i\subset{\Bbb P}i$ are numbered by $i$, such that $1\le i\le\min(d_1,d_2)$.
Namely, ${\Bbb P}i_i$ is the image of the map
$${\cal C}^{[i]}\tildemes_S{\cal C}^{[d_2-i]}\tildemes_S{\cal C}^{[N-d_2+i]}\tildemes_S{\cal C}^{[d_1-i]}\to{\Bbb P}i:
(D_0,D'_2,E',D'_1)\mapsto (D_0+D'_1,D'_2+E', D_0+D'_2,D'_1+E').$$
Since the composition of this map with $q:{\Bbb P}i\to{\cal C}^{[N]}\tildemes_S{\cal C}^{[N+d_1-d_2]}$
factors through ${\cal C}^{[d_2-i]}\tildemes_S{\cal C}^{[N-d_2+i]}\tildemes_S{\cal C}^{[d_1-i]}$, we derive
that $q({\Bbb P}i_i)$ has dimension $\le N+d_1-i$. Therefore, the components ${\Bbb P}i_i$ with $i\ge 1$
give zero contribution to the composition $P_{0,1}({\cal C})^{[d_2]}\circ P_{1,0}({\cal C})^{[d_1]}$.
It remains to prove that $[{\Bbb P}i_0]$ appears in the intersection-product \operatorname{e}qref{div-com-inter-class}
with multiplicity $1$. To this end we can replace ${\cal C}^{[d_2]}$ by the cartesian product ${\cal C}^{d_2}$
and then use iteratively Lemma \ref{mult-lem2} (in the case $m=1$).
\operatorname{e}d
Let us denote by ${\cal D}_{t,u,{\Bbb Q}}={\Bbb Q}[t,u,\partial_t,\partial_u]$ the algebra of differential operators in two variables.
Let us also denote by
$${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]\subset{\cal D}_{t,u,{\Bbb Q}}$$
the subalgebra over ${\Bbb Z}$ generated by $t$, $\partial_u$ and by the divided powers of $u$ and $\partial_t$.
We will also consider the ${\Bbb Z}$-subalgebras ${\Bbb Z}[t,\partial_t^{[\betaullet]}]$ and ${\Bbb Z}[u^{[\betaullet]},\partial_u]$ in
this algebra.
\betaegin{cor}\lambdabel{Heis-cor}
Assume that we are given a point $p_0\in{\cal C}(S)$. Then
there is an action of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$
on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ such that
\betaegin{align*}
&t\mapsto P_{1,0}([p_0(S)])+\psi\cdotot P_{1,0}({\cal C}) \ \
(\text{Pontryagin product with }[p_0(S)]+\psi\cdotot[{\cal C}])
\\
&u^{[d]}\mapsto P_{1,0}({\cal C})^{[d]} \ \ (\text{Pontryagin product with } [{\cal C}^{[d]}]),\\
&\partial_t^{[d]}\mapsto P_{0,1}({\cal C})^{[d]},\\
&\partial_u\mapsto P_{0,1}([p_0(S)]),
\operatorname{e}nd{align*}
where $\psi=p_0^*K\in{\Bbb C}H^1(S)$.
\operatorname{e}nd{cor}
Here are some simple observations on actions of
${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$ (probably well known).
\betaegin{lem}\lambdabel{div-mod-lem}
Let $M$ be a ${\Bbb Z}[u^{[\betaullet]},\partial_u]$-module such that for every $x\in M$ one has
$\partial_u^nx=0$ for all $n\gg 0$ (resp., ${\Bbb Z}[t,\partial_t^{[\betaullet]}]$-module such that for every $x\in M$ one has
$\partial_t^{[n]}x=0$ for all $n\gg 0$).
Set $M_0=\{x\in M\ |\ \partial_ux=0\}$ (resp., $M_0=\{x\in M\ |\ \partial_t^{[n]}x=0 \text{ for all }n>0\}$).
Then the submodule $M_0[u^{[\betaullet]}]\subset M$ (resp., $M_0[t]\subset M$) consisting of elements
of the form $\sum a_i u^{[i]}$ (resp., $\sum a_i t^i$) with $a_i\in M_0$, coincides with the entire $M$.
\operatorname{e}nd{lem}
\noindent {\it Proof} . Assume first that $M$ is a module over ${\Bbb Z}[u^{[\betaullet]},\partial_u]$.
For $x\in M$ let $n$ be the minimal number such that $\partial_u^n(x)\in M_0[u^{[\betaullet]}]$.
Assume that $n>0$. Let
$$\partial_u^n(x)=a_0+a_1u+\ldots+a_ku^{[k]}.$$
where $a_i\in M_0$. Then
$$\partial_u(\partial_u^{n-1}(x)-a_0u-a_1u^{[2]}-\ldots-a_ku^{[k+1]})=0,$$
i.e., $\partial_u^{n-1}(x)-a_0u-a_1u^{[2]}-\ldots-a_ku^{[k+1]}\in M_0$. It follows that
$\partial_u^{n-1}(x)\in M_0[u^{[\betaullet]}]$ contradicting the choice of $n$.
Now, let $M$ be a module over ${\Bbb Z}[t,\partial_t^{[\betaullet]}]$. For $x\in M$ let $n$ be the minimal number
such that $\partial_t^{[k]}(x)\in M_0[t]$ for all $k\ge n$.
Assume that $n>0$ and let us lead this to contradiction.
Replacing $x$ by $\partial_t^{[n-1]}(x)$ we can reduce the proof to the case $n=1$.
In this case set for $k\ge 1$
$$f_k=\partial_t^{[k]}(x)=\sum_{i\ge 0}a_{k,i}t^i\in M_0[t].$$
Note that $f_k=0$ for $k\gg 0$, so we can form the finite sum
$$f=\sum_{k\ge 1}a_{k,0}t^k\in M_0[t].$$
Using the identities $\partial_t^{[m-k]}f_k={m\choose k}f_m$ one can easily check that
$\partial_t^{[k]}f=f_k$ for every $k\ge 1$. Hence, $x-f\in M_0$, i.e., $x\in M_0[t]$.
\operatorname{e}d
For every abelian group ${\Bbb G}a$ there is a natural structure of
a ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module on
${\Bbb G}a[t,u^{[\betaullet]}]$ such that $\partial_t^{[n]}(\gamma t^iu^{[j]})=\gamma{i\choose n}t^{i-n}u^{[j]}$.
Note that the operator $\partial_u$ on this module is surjective.
\betaegin{prop}\lambdabel{Heis-mod-prop}
Let $M$ be a ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module such that for every $x\in M$ one has
$\partial_t^{[n]}x=\partial_u^nx=0$ for all $n\gg 0$. Then
\noindent (i) $M\sigmameq M_0[t,u^{[\betaullet]}]$ as ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module, where
$$M_0=\{x\in M\ |\ \partial_ux=0, \partial_t^{[n]}x=0 \text{ for all }n>0\};$$
\noindent (ii) the operator $t:M\to M$ is injective, and the operator $\partial_u:M\to M$ is surjective.
\operatorname{e}nd{prop}
\noindent {\it Proof} . (i) First, it is easy to check that any submodule of an ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module of the form ${\Bbb G}a[t,u^{[\betaullet]}]$ itself
has the form ${\Bbb G}a'[t,u^{[\betaullet]}]$ for a subgroup ${\Bbb G}a'\subset{\Bbb G}a$. Indeed, this follows easily
from the fact that $\partial_t^{[m]}\partial_u^n(\gamma t^m u^{[n]})=\gamma$ for $\gamma\in{\Bbb G}a$.
Therefore, the natural morphism of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-modules
$M_0[t,u^{[\betaullet]}]\to M$ is injective (since $M_0$ embeds into $M$).
It remains to prove that this morphism is also surjective. Viewing $M$ as a
${\Bbb Z}[u^{[\betaullet]},\partial_u]$-module and applying Lemma \ref{div-mod-lem}, we derive that
$M=(\operatorname{ker} \partial_u)[u^{[\betaullet]}]$. Next, applying the same lemma to the ${\Bbb Z}[t,\partial_t^{[\betaullet]}]$-module
$\operatorname{ker}\partial_u$, we obtain $\operatorname{ker}\partial_u=M_0[t]$. Hence, $M=M_0[t,u^{[\betaullet]}]$.
\noindent
(ii) This follows from (i) since for a module of the form ${\Bbb G}a[t,u^{[\betaullet]}]$
injectivity of $t$ (resp., surjectivity of $\partial_u$) is clear.
\operatorname{e}d
In the case when $S$ is a point the following result is due to Collino~\cite{Col2}.
\betaegin{prop}\lambdabel{Col-prop} Let $i_{N}:{\cal C}^{[N-1]}\to{\cal C}^{[N]}$ be the closed embedding associated
with a point $p_0\in{\cal C}(S)$. Then the homomorphism $i_{N*}:{\Bbb C}H^*({\cal C}^{[N-1]})\to{\Bbb C}H^*({\cal C}^{[N]})$
(resp., $i_N^*:{\Bbb C}H^*({\cal C}^{[N]})\to{\Bbb C}H^*({\cal C}^{[N-1]})$) is injective (resp., surjective).
\operatorname{e}nd{prop}
\noindent {\it Proof} . Note that $i_{N*}=P_{1,0}([p_0(S)])$ and $i_N^*=P_{0,1}([p_0(S)])$. Now the
surjectivity of $P_{0,1}([p_0(S)])$ follows immediately from Corollary \ref{Heis-cor} and Proposition
\ref{Heis-mod-prop}(ii). To deal with injectivity of $P_{1,0}([p_0(S)])$ we
can modify the action of Corollary \ref{Heis-cor}(i) as follows:
\betaegin{align*}
&t\mapsto P_{1,0}(p_0(S)),\\
&u^{[d]}\mapsto P_{1,0}({\cal C})^{[d]},\\
&\partial_t^{[d]}\mapsto P_{0,1}({\cal C})^{[d]},\\
&\partial_u\mapsto P_{0,1}([p_0(S)])+\psi\cdotot P_{0,1}({\cal C}).
\operatorname{e}nd{align*}
It remains to apply Proposition \ref{Heis-mod-prop}(ii) to this action.
\operatorname{e}d
We also get the following corollary from Proposition \ref{Heis-mod-prop}.
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For every abelian group ${\Bbb G}a$ there is a natural structure of
a ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module on
${\Bbb G}a[t,u^{[\betaullet]}]$ such that $\partial_t^{[n]}(\gamma t^iu^{[j]})=\gamma{i\choose n}t^{i-n}u^{[j]}$.
Note that the operator $\partial_u$ on this module is surjective.
\betaegin{prop}\lambdabel{Heis-mod-prop}
Let $M$ be a ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module such that for every $x\in M$ one has
$\partial_t^{[n]}x=\partial_u^nx=0$ for all $n\gg 0$. Then
\noindent (i) $M\sigmameq M_0[t,u^{[\betaullet]}]$ as ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module, where
$$M_0=\{x\in M\ |\ \partial_ux=0, \partial_t^{[n]}x=0 \text{ for all }n>0\};$$
\noindent (ii) the operator $t:M\to M$ is injective, and the operator $\partial_u:M\to M$ is surjective.
\operatorname{e}nd{prop}
\noindent {\it Proof} . (i) First, it is easy to check that any submodule of an ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module of the form ${\Bbb G}a[t,u^{[\betaullet]}]$ itself
has the form ${\Bbb G}a'[t,u^{[\betaullet]}]$ for a subgroup ${\Bbb G}a'\subset{\Bbb G}a$. Indeed, this follows easily
from the fact that $\partial_t^{[m]}\partial_u^n(\gamma t^m u^{[n]})=\gamma$ for $\gamma\in{\Bbb G}a$.
Therefore, the natural morphism of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-modules
$M_0[t,u^{[\betaullet]}]\to M$ is injective (since $M_0$ embeds into $M$).
It remains to prove that this morphism is also surjective. Viewing $M$ as a
${\Bbb Z}[u^{[\betaullet]},\partial_u]$-module and applying Lemma \ref{div-mod-lem}, we derive that
$M=(\operatorname{ker} \partial_u)[u^{[\betaullet]}]$. Next, applying the same lemma to the ${\Bbb Z}[t,\partial_t^{[\betaullet]}]$-module
$\operatorname{ker}\partial_u$, we obtain $\operatorname{ker}\partial_u=M_0[t]$. Hence, $M=M_0[t,u^{[\betaullet]}]$.
\noindent
(ii) This follows from (i) since for a module of the form ${\Bbb G}a[t,u^{[\betaullet]}]$
injectivity of $t$ (resp., surjectivity of $\partial_u$) is clear.
\operatorname{e}d
In the case when $S$ is a point the following result is due to Collino~\cite{Col2}.
\betaegin{prop}\lambdabel{Col-prop} Let $i_{N}:{\cal C}^{[N-1]}\to{\cal C}^{[N]}$ be the closed embedding associated
with a point $p_0\in{\cal C}(S)$. Then the homomorphism $i_{N*}:{\Bbb C}H^*({\cal C}^{[N-1]})\to{\Bbb C}H^*({\cal C}^{[N]})$
(resp., $i_N^*:{\Bbb C}H^*({\cal C}^{[N]})\to{\Bbb C}H^*({\cal C}^{[N-1]})$) is injective (resp., surjective).
\operatorname{e}nd{prop}
\noindent {\it Proof} . Note that $i_{N*}=P_{1,0}([p_0(S)])$ and $i_N^*=P_{0,1}([p_0(S)])$. Now the
surjectivity of $P_{0,1}([p_0(S)])$ follows immediately from Corollary \ref{Heis-cor} and Proposition
\ref{Heis-mod-prop}(ii). To deal with injectivity of $P_{1,0}([p_0(S)])$ we
can modify the action of Corollary \ref{Heis-cor}(i) as follows:
\betaegin{align*}
&t\mapsto P_{1,0}(p_0(S)),\\
&u^{[d]}\mapsto P_{1,0}({\cal C})^{[d]},\\
&\partial_t^{[d]}\mapsto P_{0,1}({\cal C})^{[d]},\\
&\partial_u\mapsto P_{0,1}([p_0(S)])+\psi\cdotot P_{0,1}({\cal C}).
\operatorname{e}nd{align*}
It remains to apply Proposition \ref{Heis-mod-prop}(ii) to this action.
\operatorname{e}d
We also get the following corollary from Proposition \ref{Heis-mod-prop}.
\betaegin{cor}\lambdabel{module-cor}
For a point $p_0\in {\cal C}(S)$ consider the action of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$
on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ given by Corollary \ref{Heis-cor}. Then
there is an isomorphism of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module
$${\Bbb C}H^*({\cal C}^{[\betaullet]})\sigmameq K_{p_0}[t,u^{[\betaullet]}],$$
where
$$K_{p_0}=\{x\in{\Bbb C}H^*({\cal C}^{[\betaullet]})\ |\ P_{0,1}(p_0(S))x=0,
P_{0,1}({\cal C}))^{[d]}x=0 \text{ for all }d\ge 1\}.$$
\operatorname{e}nd{cor}
\betaegin{rems}
1. The isomorphism of the above corollary is compatible with the bigrading of ${\Bbb C}H^*(C^{[\betaullet]})$:
$t$ (resp., $u$) sends ${\Bbb C}H^i(C^{[N]})$ to ${\Bbb C}H^{i+1}(C^{[N+1]})$ (resp., ${\Bbb C}H^i(C^{[N+1]})$).
In the case $S=\operatorname{Spec}(k)$ the strong stability conjecture (see \cite{KV}, 2.13) is equivalent to the condition that $K_{p_0}\cap{\Bbb C}H^p(C^{[N]})$ is a torsion group for $N>2p$.
\noindent 2. In the case $S=\operatorname{Spec}(k)$ the decomposition of ${\Bbb C}H^*(C^{[\betaullet]})$ into the
direct summands of the form $K_{p_0}t^mu^{[n]}$ is the ${\Bbb Z}$-version of the well known motivic decomposition over ${\Bbb Q}$ obtained by using $\lambdambda$-operations (see \cite{dB1}).
\noindent 3. Using the divided powers of $P_{0,1}(C)$ we can avoid tensoring with ${\Bbb Q}$ in the
proof of Theorem \ref{curve-thm}. Instead one has to use the fact that $P_{0,1}(C)^{[d]}$
commutes with $P_{1,0}(a)$ for $a\in A_0(C)$, and hence the subgroup
$A_0(C)^{*n}\subset{\Bbb C}H_0(C^{[n]})$ is killed by all the operators $P_{0,1}(C)^{[d]}$.
\operatorname{e}nd{rems}
Using Corollary \ref{Heis-cor} we get an interesting $\operatorname{sl}_2$-action on the motive of ${\cal C}^{[N]}$.
In the case when $S$ is a point we obtain in this way a Lefschetz $\operatorname{sl}_2$-action on
${\Bbb C}H^*(C^{[N]})$.
\betaegin{thm}\lambdabel{Lefschetz-thm}
(i) Fix a point $p_0\in {\cal C}(S)$. Then for every $N\ge 0$ the operators
\betaegin{align*}
&e(x)=[{\cal R}]\cdotot x+\psi\cdotot P_{1,0}({\cal C})P_{0,1}([p_0(S)])(x),\\
&f=P_{1,0}({\cal C})P_{0,1}({\cal C}),\\
&h=P_{1,0}([p_0(S)])P_{0,1}({\cal C})-P_{1,0}({\cal C})P_{0,1}([p_0(S)])+\psi\cdotot P_{1,0}({\cal C})P_{0,1}({\cal C})
\operatorname{e}nd{align*}
(given by algebraic correspondences)
define compatible actions of the Lie algebra $\operatorname{sl}_2$ on ${\Bbb C}H^*({\cal C}^{[N]})$ and on
$H^*({\cal C}^{[N]},{\Bbb Q})$,
where ${\cal R}={\cal R}_N\subset{\cal C}^{[N]}$ is the divisor associated with $p_0$ (see section \ref{Jac-sec}).
\noindent
(ii) In the case when $S$ is a point (so we write ${\cal C}=C$)
the operator $h$ acts as $(i-N)\operatorname{id}$ on $H^i(C^{[N]},{\Bbb Q})$. In other words, the
action of $(e,f,h)$ on $H^*(C^{[N]},{\Bbb Q})$ is the Lefschetz action corresponding to the ample divisor
$R\subset C^{[N]}$ (associated with $p_0$).
\operatorname{e}nd{thm}
\noindent {\it Proof} . (i) Consider the action of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$
given by Corollary \ref{Heis-cor}. Then the operators $e=t\partial_u$, $f=u\partial_t$ and $h=t\partial_t-u\partial_u$
satisfy the relations of $\operatorname{sl}_2$.
By definition, we have
$$P_{1,0}([p_0])|_{{\cal C}^{[N-1]}}=i_{N*}, \ \ P_{0,1}([p_0])|_{{\cal C}^{[N]}}=i_{N}^*,$$
where $i_N:{\cal C}^{[N-1]}\to{\cal C}^{[N]}$ is
the embedding associated with $p_0$.
Hence, for $x\in{\Bbb C}H^*({\cal C}^{[N]})$ one has
$$P_{1,0}([p_0])P_{0,1}([p_0])(x)=i_{N*}i_N^*x=[{\cal R}]\cdotot x.$$
This implies our formula for $e$.
\noindent
(ii) It is easy to see that two $\operatorname{sl}_2$-triples $(e,f,h)$ and $(e',f',h')$ acting
on the same finite-dimensional space $V$ such that $e=e'$ and $[h,h']=0$ necessarily coincide
(i.e., $h=h'$ and $f=f'$). Indeed, consider the decomposition $V=\betaigoplus_{m,n} V_{m,n}$,
where $h$ (resp., $h'$) acts by $m\cdotot\operatorname{id}$ (resp., $n\cdotot\operatorname{id}$) on $V_{m,n}$.
Consider also the subspace $W=\operatorname{ker}(e)\subset V$ and the induced decomposition
$W=\betaigoplus_{m\ge 0,n\ge 0} W_{m,n}$, where $W_{m,n}=W\cap V_{m,n}$.
Viewing $V$ as a representation of $(e,f,h)$ we deduce that
$$e^iV\cap W=\betaigoplus_{m\ge i,n}W_{m,n}\text{ for all }i\ge 0.$$
On the other hand, using the triple $(e=e',f',h')$ we get
$$e^iV\cap W=\betaigoplus_{m,n\ge i}W_{m,n}\text{ for all }i\ge 0.$$
This immediately implies that $W_{m,n}=0$ for $m\neq n$, i.e., $h=h'$. It is well known that
this implies $f=f'$.
Now comparing the action of our operators $(e,f,h)$ on $H^*(C^{[N]},{\Bbb Q})$ with the Lefschetz action corresponding to $[R]$ and using the above observation we conclude that these two actions coincide.
\operatorname{e}d
\betaegin{rem} It is well known that the standard conjecture B of Lefschetz type for $X$ and
for a projective bundle over $X$ are equivalent (see \cite{Li}).
Also, if it is true for some variety $X$ then
it is true for an ample divisor in $X$. Thus, standard conjecture B for all $C^{[N]}$ and for the Jacobian
$J$ are equivalent.
In particular, from the above theorem we get a new proof of the standard conjecture B for $J$.
\operatorname{e}nd{rem}
Using our methods we easily recover the following result proved by S.~del Ba\~{n}o in
\cite{dB2}.
\betaegin{cor}\lambdabel{Lefschetz-cor}
The hard Lefschetz theorem for the operator $L$ of multiplication by the class
$[R]\in H^2(C^{[N]},{\Bbb Z})$ (associated with a point $p_0\in C$) holds over ${\Bbb Z}$, i.e., the map
$$L^i:H^{N-i}(C^{[N]},{\Bbb Z})\to H^{N+i}(C^{[N]},{\Bbb Z})$$
is an isomorphism for all $i\ge 0$.
\operatorname{e}nd{cor}
\noindent {\it Proof} . Since the action of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$ defined in Corollary \ref{Heis-cor} is given by algebraic correspondences, it can also be defined
for $H^*(C^{[N]},{\Bbb Z})$. The operator $L$ corresponds to the action of $e=t\partial_u$. As we have seen
in Theorem \ref{Lefschetz-thm}, the operator $h=t\partial_t-u\partial_u$ acts as $(i-N)\operatorname{id}$ on $H^i(C^{[N]},{\Bbb Z})$.
On the other hand, by Proposition \ref{Heis-mod-prop}(i) we have an isomorphism of $H^*(C^{[N]},{\Bbb Z})$
with a ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module of the form
${\Bbb G}a[t,u^{[\betaullet]}]$. For such a module we have
$h(\gamma t^mu^{[n]})=(m-n)\gamma t^mu^{[n]}$, and our assertion follows from the formula
$$e^{n-m}(\gamma t^mu^{[n]})=\gamma t^nu^{[m]} \text{ for }n\ge m.$$
\operatorname{e}d
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\noindent {\it Proof} . (i) Consider the action of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$
given by Corollary \ref{Heis-cor}. Then the operators $e=t\partial_u$, $f=u\partial_t$ and $h=t\partial_t-u\partial_u$
satisfy the relations of $\operatorname{sl}_2$.
By definition, we have
$$P_{1,0}([p_0])|_{{\cal C}^{[N-1]}}=i_{N*}, \ \ P_{0,1}([p_0])|_{{\cal C}^{[N]}}=i_{N}^*,$$
where $i_N:{\cal C}^{[N-1]}\to{\cal C}^{[N]}$ is
the embedding associated with $p_0$.
Hence, for $x\in{\Bbb C}H^*({\cal C}^{[N]})$ one has
$$P_{1,0}([p_0])P_{0,1}([p_0])(x)=i_{N*}i_N^*x=[{\cal R}]\cdotot x.$$
This implies our formula for $e$.
\noindent
(ii) It is easy to see that two $\operatorname{sl}_2$-triples $(e,f,h)$ and $(e',f',h')$ acting
on the same finite-dimensional space $V$ such that $e=e'$ and $[h,h']=0$ necessarily coincide
(i.e., $h=h'$ and $f=f'$). Indeed, consider the decomposition $V=\betaigoplus_{m,n} V_{m,n}$,
where $h$ (resp., $h'$) acts by $m\cdotot\operatorname{id}$ (resp., $n\cdotot\operatorname{id}$) on $V_{m,n}$.
Consider also the subspace $W=\operatorname{ker}(e)\subset V$ and the induced decomposition
$W=\betaigoplus_{m\ge 0,n\ge 0} W_{m,n}$, where $W_{m,n}=W\cap V_{m,n}$.
Viewing $V$ as a representation of $(e,f,h)$ we deduce that
$$e^iV\cap W=\betaigoplus_{m\ge i,n}W_{m,n}\text{ for all }i\ge 0.$$
On the other hand, using the triple $(e=e',f',h')$ we get
$$e^iV\cap W=\betaigoplus_{m,n\ge i}W_{m,n}\text{ for all }i\ge 0.$$
This immediately implies that $W_{m,n}=0$ for $m\neq n$, i.e., $h=h'$. It is well known that
this implies $f=f'$.
Now comparing the action of our operators $(e,f,h)$ on $H^*(C^{[N]},{\Bbb Q})$ with the Lefschetz action corresponding to $[R]$ and using the above observation we conclude that these two actions coincide.
\operatorname{e}d
\betaegin{rem} It is well known that the standard conjecture B of Lefschetz type for $X$ and
for a projective bundle over $X$ are equivalent (see \cite{Li}).
Also, if it is true for some variety $X$ then
it is true for an ample divisor in $X$. Thus, standard conjecture B for all $C^{[N]}$ and for the Jacobian
$J$ are equivalent.
In particular, from the above theorem we get a new proof of the standard conjecture B for $J$.
\operatorname{e}nd{rem}
Using our methods we easily recover the following result proved by S.~del Ba\~{n}o in
\cite{dB2}.
\betaegin{cor}\lambdabel{Lefschetz-cor}
The hard Lefschetz theorem for the operator $L$ of multiplication by the class
$[R]\in H^2(C^{[N]},{\Bbb Z})$ (associated with a point $p_0\in C$) holds over ${\Bbb Z}$, i.e., the map
$$L^i:H^{N-i}(C^{[N]},{\Bbb Z})\to H^{N+i}(C^{[N]},{\Bbb Z})$$
is an isomorphism for all $i\ge 0$.
\operatorname{e}nd{cor}
\noindent {\it Proof} . Since the action of ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$ defined in Corollary \ref{Heis-cor} is given by algebraic correspondences, it can also be defined
for $H^*(C^{[N]},{\Bbb Z})$. The operator $L$ corresponds to the action of $e=t\partial_u$. As we have seen
in Theorem \ref{Lefschetz-thm}, the operator $h=t\partial_t-u\partial_u$ acts as $(i-N)\operatorname{id}$ on $H^i(C^{[N]},{\Bbb Z})$.
On the other hand, by Proposition \ref{Heis-mod-prop}(i) we have an isomorphism of $H^*(C^{[N]},{\Bbb Z})$
with a ${\Bbb Z}[t,u^{[\betaullet]}, \partial_t^{[\betaullet]}, \partial_u]$-module of the form
${\Bbb G}a[t,u^{[\betaullet]}]$. For such a module we have
$h(\gamma t^mu^{[n]})=(m-n)\gamma t^mu^{[n]}$, and our assertion follows from the formula
$$e^{n-m}(\gamma t^mu^{[n]})=\gamma t^nu^{[m]} \text{ for }n\ge m.$$
\operatorname{e}d
\section{Tautological cycles}
\lambdabel{taut-sec}
Similarly to the case of cycles on the Jacobian considered in \cite{Bmain} and \cite{P-lie},
we are going to
define the subalgebra of tautological classes in ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ (resp., ${\Bbb C}H^*({\cal J})$).
We will show that the operators $(P_{m,k}(a))$ (resp., $T_k(m,a)$) act on this subalgebra by some differential operators.
We start with an abstract setup.
Let $R$ be a supercommutative ring,
$A$ a supercommutative $R$-algebra with a fixed even element ${\betaf a}_0\in A$,
${\cal D}(A,{\betaf a}_0)$ the corresponding Lie superalgebra (see the Introduction).
Let us also denote by ${\cal D}_+(A,{\betaf a}_0)\subset{\cal D}(A,{\betaf a}_0)$ the subalgebra generated by the operators
${\betaf P}_{m,k}(a)$ with $m\ge k$.
\betaegin{prop}\lambdabel{diff-op-prop} Let $B$ be a supercommutative $R$-algebra,
and let $x_m:A\to B$, $m\ge 0$, be a family of even $R$-linear maps.
\noindent
(i) Assume that ${\cal D}(A,{\betaf a}_0)$ (resp., ${\cal D}_+(A,{\betaf a}_0)$)
acts $R$-linearly on $B$ in such a way that
$${\betaf P}_{m,0}(a)(b)=x_m(a)\cdotot b$$
for all $m\ge 0$ and $a\in A$.
Assume also that
$${\betaf P}_{m,k}(a)(1)=0 \text{ for }k>0$$
(resp., for $m\ge k>0$).
Let us denote by ${\cal T} B\subset B$
the $R$-subalgebra generated by $(x_m(a))$.
Then ${\cal T} B$ is stable under the action of ${\cal D}(A,{\betaf a}_0)$ (resp., ${\cal D}_+(A,{\betaf a}_0)$).
Furthermore, if we view ${\cal T} B$ as the quotient of the superalgebra of polynomials
$R[x_m(a)\ |\ m\ge 0, a\in G]$,
where $G$ is some set of homogeneous generators of $A$ as an $R$-module,
then the action of this Lie algebra on ${\cal T} B$ is given by the formulas
\betaegin{align*}
&{\betaf P}_{m,k}(a)=\sum_{s\ge 0;k_1+\ldots+k_s=k, k_i\ge 1;
n_1,\ldots,n_s\ge 0; a_1,\ldots,a_s\in G}
(-1)^{k-s}\frac{k!}{s!}{n_1\choose k_1}\cdotot\ldots\cdotot{n_s\choose k_s}\tildemes\\
&x_{m-k+n_1+\ldots+n_s}(a a_s\ldots a_1\cdotot {\betaf a}_0^{k-s})\partial_{x_{n_1}(a_1)}\ldots\partial_{x_{n_s}(a_s)},
\operatorname{e}nd{align*}
where the case $s=0$ occurs only for $k=0$.
\noindent
(ii) Assume in addition that the above action of ${\cal D}(A,{\betaf a}_0)$ on $B$ extends to an action of
$\widetilde{U}_1(A,{\betaf a}_0)$, so that
$${\betaf P}_{m,0}(1)^{[d]}(b)=x_m(1)^{[d]}\cdotot b$$
for some elements $x_m(1)^{[d]}\in B$. Let $\widetilde{{\cal T}} B$ be the subalgebra generated by ${\cal T} B$ and
by all the elements $x_m(1)^{[d]}$. Then $\widetilde{{\cal T}} B$ is stable under the action of $\widetilde{U}_1(A,{\betaf a}_0)$,
and the action of ${\betaf P}_{m,k}(a)$ on it is given by the same formula as above (with $\partial_{x_m(1)}$ extended to the divided powers).
Similarly, if the action of ${\cal D}(A,{\betaf a}_0)$ on $B$ extends to an action of $\widetilde{U}_2(A,{\betaf a}_0)$
such that ${\betaf P}_{0,n}(1)^{[d]}(1)=0$ for $n\ge 1, d\ge 1$, then
${\cal T} B$ is stable under the action of $\widetilde{U}_2(A,{\betaf a}_0)$, and the action of
${\betaf P}_{0,n}(1)^{[d]}$ on ${\cal T} B$ is given by a differential operator of order $nd$.
\operatorname{e}nd{prop}
\noindent {\it Proof} . (i) For $k=0$ we have ${\betaf P}_{m,0}(a)=x_m(a)$ by assumption.
The general case follows by induction in $k$ using the (super)commutator formula
$$[{\betaf P}_{m,k}(a),x_n(a')]=[{\betaf P}_{m,k}(a),{\betaf P}_{n,0}(a')]=
\sum_{i\ge 1}(-1)^{i-1}i!\cdotot {k\choose i}{n\choose i}{\betaf P}_{m+n-i,k-i}(a\cdotot a'\cdotot {\betaf a}_0^{i-1})$$
together with the assumption that ${\betaf P}_{m,k}(a)(1)=0$ for $k>0$.
Indeed, it is straightforward to check that the similar commutation relation
holds for the differential operators in the right-hand side of the required formula.
\noindent (ii) Same proof as in (i) using the commutation relations in $\widetilde{U}_1(A,{\betaf a}_0)$ (resp.,
$\widetilde{U}_2(A,{\betaf a}_0)$).
\operatorname{e}d
Proposition \ref{diff-op-prop} can be applied to the algebra
$B={\Bbb C}H_*({\cal C}^{[\betaullet]})$ (equipped with Pontryagin product) and the operators
$(P_{m,k}(a))$ acting on it. This leads to a definition of the subalgebra of tautological classes.
\betaegin{defi} The {\it (big) algebra of tautological classes on ${\cal C}^{[\betaullet]}$}
$$\TCH^*({\cal C}^{[\betaullet]})\subset{\Bbb C}H^*({\cal C}^{[\betaullet]})$$
is the ${\Bbb C}H^*(S)$-subalgebra with respect to the Pontryagin product
generated by all the classes of the form
${\cal D}e_{n*}(a)\in{\Bbb C}H^*({\cal C}^{[n]})$, where ${\cal D}e_n:{\cal C}\to{\cal C}^{[n]}$ is the diagonal embedding,
$a\in{\Bbb C}H^*({\cal C})$, $n\ge 1$. Let us also denote by
$$\widetilde{\TCH}^*({\cal C}^{[\betaullet]})\subset{\Bbb C}H^*({\cal C}^{[\betaullet]})$$
the subalgebra generated by $\TCH^*({\cal C}^{[\betaullet]})$ along with all the classes
$\delta_n^{[d]}\in{\Bbb C}H^{(n-1)d}({\cal C}^{[nd]})$ (see section \ref{div-sec}).
Replacing Chow groups by cohomology we also define the
subalgebra of tautological classes ${\cal T}H^*({\cal C}^{[\betaullet]})\subset H^*({\cal C}^{[\betaullet]})$.
\operatorname{e}nd{defi}
We can also mimic the above definition of tautological classes in the case of the relative Jacobian
${\cal J}$ for a family of curves ${\cal C}/S$ equipped with a point $p_0\in{\cal C}(S)$.
\betaegin{defi} The {\it (big) algebra of tautological classes on ${\cal J}$}
$$\TCH^*({\cal J})\subset{\Bbb C}H^*({\cal J})$$
is the ${\Bbb C}H^*(S)$-subalgebra with respect to the Pontryagin product
generated by all the classes of the form $[n]_*(\iota_*a)$, where
$\iota=\sigma_1:{\cal C}\to{\cal J}$ is the embedding
associated with $p_0$, $a\in{\Bbb C}H^*({\cal C})$, $n\in{\Bbb Z}$.
Here we view ${\Bbb C}H^*({\cal J})$ as a ${\Bbb C}H^*(S)$-algebra via the homomorphism
$e_*:{\Bbb C}H^*(S)\to{\Bbb C}H^*({\cal J})$ associated with the neutral element $e\in{\cal J}(S)$.
Similarly, we define the subalgebra of tautological classes in cohomology.
\operatorname{e}nd{defi}
\betaegin{rem}
One can also consider smaller algebras of tautological classes by choosing a ${\Bbb C}H^*(S)$-subalgebra
${\Bbb A}A\subset{\Bbb C}H^*({\cal C})$ and considering only the classes ${\cal D}e_{n*}(a)$ with $a\in{\Bbb A}A$.
For example, one can take as ${\Bbb A}A$ the subalgebra generated by $[p_0(S)]\in{\Bbb C}H^1({\cal C})$,
or by the relative canonical class $K\in{\Bbb C}H^1({\cal C})$, or by some other divisor class. In \cite{P-lie} we
worked with the subalgebra generated by $\chi+[p_0]$, where $2\chi=K$ (in the case $S=\operatorname{Spec}(k)$).
\operatorname{e}nd{rem}
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\noindent {\it Proof} . (i) For $k=0$ we have ${\betaf P}_{m,0}(a)=x_m(a)$ by assumption.
The general case follows by induction in $k$ using the (super)commutator formula
$$[{\betaf P}_{m,k}(a),x_n(a')]=[{\betaf P}_{m,k}(a),{\betaf P}_{n,0}(a')]=
\sum_{i\ge 1}(-1)^{i-1}i!\cdotot {k\choose i}{n\choose i}{\betaf P}_{m+n-i,k-i}(a\cdotot a'\cdotot {\betaf a}_0^{i-1})$$
together with the assumption that ${\betaf P}_{m,k}(a)(1)=0$ for $k>0$.
Indeed, it is straightforward to check that the similar commutation relation
holds for the differential operators in the right-hand side of the required formula.
\noindent (ii) Same proof as in (i) using the commutation relations in $\widetilde{U}_1(A,{\betaf a}_0)$ (resp.,
$\widetilde{U}_2(A,{\betaf a}_0)$).
\operatorname{e}d
Proposition \ref{diff-op-prop} can be applied to the algebra
$B={\Bbb C}H_*({\cal C}^{[\betaullet]})$ (equipped with Pontryagin product) and the operators
$(P_{m,k}(a))$ acting on it. This leads to a definition of the subalgebra of tautological classes.
\betaegin{defi} The {\it (big) algebra of tautological classes on ${\cal C}^{[\betaullet]}$}
$$\TCH^*({\cal C}^{[\betaullet]})\subset{\Bbb C}H^*({\cal C}^{[\betaullet]})$$
is the ${\Bbb C}H^*(S)$-subalgebra with respect to the Pontryagin product
generated by all the classes of the form
${\cal D}e_{n*}(a)\in{\Bbb C}H^*({\cal C}^{[n]})$, where ${\cal D}e_n:{\cal C}\to{\cal C}^{[n]}$ is the diagonal embedding,
$a\in{\Bbb C}H^*({\cal C})$, $n\ge 1$. Let us also denote by
$$\widetilde{\TCH}^*({\cal C}^{[\betaullet]})\subset{\Bbb C}H^*({\cal C}^{[\betaullet]})$$
the subalgebra generated by $\TCH^*({\cal C}^{[\betaullet]})$ along with all the classes
$\delta_n^{[d]}\in{\Bbb C}H^{(n-1)d}({\cal C}^{[nd]})$ (see section \ref{div-sec}).
Replacing Chow groups by cohomology we also define the
subalgebra of tautological classes ${\cal T}H^*({\cal C}^{[\betaullet]})\subset H^*({\cal C}^{[\betaullet]})$.
\operatorname{e}nd{defi}
We can also mimic the above definition of tautological classes in the case of the relative Jacobian
${\cal J}$ for a family of curves ${\cal C}/S$ equipped with a point $p_0\in{\cal C}(S)$.
\betaegin{defi} The {\it (big) algebra of tautological classes on ${\cal J}$}
$$\TCH^*({\cal J})\subset{\Bbb C}H^*({\cal J})$$
is the ${\Bbb C}H^*(S)$-subalgebra with respect to the Pontryagin product
generated by all the classes of the form $[n]_*(\iota_*a)$, where
$\iota=\sigma_1:{\cal C}\to{\cal J}$ is the embedding
associated with $p_0$, $a\in{\Bbb C}H^*({\cal C})$, $n\in{\Bbb Z}$.
Here we view ${\Bbb C}H^*({\cal J})$ as a ${\Bbb C}H^*(S)$-algebra via the homomorphism
$e_*:{\Bbb C}H^*(S)\to{\Bbb C}H^*({\cal J})$ associated with the neutral element $e\in{\cal J}(S)$.
Similarly, we define the subalgebra of tautological classes in cohomology.
\operatorname{e}nd{defi}
\betaegin{rem}
One can also consider smaller algebras of tautological classes by choosing a ${\Bbb C}H^*(S)$-subalgebra
${\Bbb A}A\subset{\Bbb C}H^*({\cal C})$ and considering only the classes ${\cal D}e_{n*}(a)$ with $a\in{\Bbb A}A$.
For example, one can take as ${\Bbb A}A$ the subalgebra generated by $[p_0(S)]\in{\Bbb C}H^1({\cal C})$,
or by the relative canonical class $K\in{\Bbb C}H^1({\cal C})$, or by some other divisor class. In \cite{P-lie} we
worked with the subalgebra generated by $\chi+[p_0]$, where $2\chi=K$ (in the case $S=\operatorname{Spec}(k)$).
\operatorname{e}nd{rem}
\betaegin{thm}\lambdabel{taut-thm}
(i) The operators $(P_{m,k}(a))$ from the Introduction preserve the
subalgebra of tautological classes $\TCH^*({\cal C}^{[\betaullet]})\subset{\Bbb C}H^*({\cal C}^{[\betaullet]})$
(resp., $\widetilde{\TCH}^*({\cal C}^{[\betaullet]})$)
and act on it by the differential operators given in Proposition \ref{diff-op-prop} with
$x_n(a)={\cal D}e_{n*}(a)$.
\noindent
(ii) The operators $T_k(m,a)$ from section \ref{Jac-sec}
preserve the subalgebra of tautological classes
$\TCH^*({\cal J})\subset{\Bbb C}H^*({\cal J})$ and act on it by differential operators (with respect to the Pontryagin product), so that $T_k(m,a)$ acts by a differential operator of order $k$.
\noindent
(iii) The space of tautological classes with rational coefficients $\TCH^*({\cal J})_{{\Bbb Q}}\subset{\Bbb C}H^*({\cal J})_{{\Bbb Q}}$
is closed under the usual product and under the Fourier transform. It coincides with the
${\Bbb C}H^*(S)_{{\Bbb Q}}$-subalgebra
in ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$ with respect to the usual product, generated by the classes $\tau_k(a)$, $k\ge 0$,
$a\in{\Bbb C}H^*({\cal C})$ (see \operatorname{e}qref{tau-eq}).
\noindent
(iv) For every $N$ the map $\sigma_{N*}:{\Bbb C}H^*({\cal C}^{[N]})\to{\Bbb C}H^*({\cal J})$
(resp., $\sigma_N^*:{\Bbb C}H^*({\cal J})_{{\Bbb Q}}\to{\Bbb C}H^*({\cal C}^{[N]})_{{\Bbb Q}}$) sends
tautological classes to tautological classes (resp., with rational coefficients).
\noindent
(v) Similar statements hold for the cohomology.
\operatorname{e}nd{thm}
\noindent {\it Proof} . (i) This follows immediately from Proposition \ref{diff-op-prop}.
\noindent (ii) The operator $T_0(m,a)$ is simply the Pontryagin product with
$[m]_*\iota_*a\in{\Bbb C}H^*({\cal J})$.
The commutation relation of Theorem \ref{relations-thm} for $k'=0$ and $k\ge 1$ gives
\betaegin{align*}
&[T_k(m,a),T_0(m',a')]+\sum_{i\ge 1}\psi^i{k\choose i}m^{\operatorname{pr}ime i}T_{k-i}(m,a)T_0(m',a')=\\
&\sum_{i\ge 1}(-1)^{i-1}{k\choose i}m^{\operatorname{pr}ime i}T_{k-i}(m+m',aa'(K+2[p_0(S)])^{i-1})-
\psi^{k-1}m^{\operatorname{pr}ime k}p_0^*(a)T_0(m',a')\\
&-\sum_{i\ge 1}{k\choose i}m^{\operatorname{pr}ime i}\psi^{i-1}p_0^*(a')T_{k-i}(m,a).
\operatorname{e}nd{align*}
Since $T_k(m,a)(e_*x)=0$ this implies the assertion by induction in $k$.
\noindent
(iii) First of all, note that $g$-th Pontryagin power of $\sigma_{1*}({\cal C})$ is equal to $g![{\cal J}]$, so
$[{\cal J}]\in\TCH^*({\cal J})_{{\Bbb Q}}$. Next, let us check that $\TCH^*({\cal J})_{{\Bbb Q}}$ is closed under the
Fourier transform $F:{\Bbb C}H^*({\cal J})_{{\Bbb Q}}\to{\Bbb C}H^*({\cal J})_{{\Bbb Q}}$. We have seen in the proof of Lemma
\ref{four-lem} that
$$F([n]_*\iota_*a)=\sum_{k\ge 0}\frac{n^k}{k!}\tau_k(a).$$
Since $F(x*y)=F(x)\cdotot F(y)$, we derive the formula
\betaegin{equation}\lambdabel{F-prod-eq}
F(([n_1]_*\iota_*a_1)*\ldots*([n_s]_*\iota_*a_s))=
\sum_{k_1,\ldots,k_s}\frac{n_1^{k_1}\ldots n_s^{k_s}}{k_1!\ldots k_s!}\tau_{k_1}(a_1)\cdotot\ldots\cdotot
\tau_{k_s}(a_s).
\operatorname{e}nd{equation}
It remains to note that
$$\tau_{k_1}(a_1)\cdotot\ldots\tau_{k_s}(a_s)=T_{k_1}(0,a_1)\ldots T_{k_s}(0,a_s)([{\cal J}]),$$
hence, it is tautological by (ii). Thus, $\TCH^*({\cal J})_{{\Bbb Q}}$ is closed under the Fourier transform.
It follows that it is also closed under the usual product. Note that it contains all the classes
$\tau_k(a)=T_k(a)([{\cal J}])$. Now the fact that it is generated by these classes with respect to the usual product follows from \operatorname{e}qref{F-prod-eq}.
\noindent
(iv) Since the map $\sigma_*:{\Bbb C}H^*({\cal C}^{[\betaullet]})\to{\Bbb C}H^*({\cal J})$ respects the Pontryagin products,
the first assertion follows from the formula
$$\sigma_{n*}{\cal D}e_{n*}(a)=[n]_*\iota_*(a).$$
To check the assertion about the pull-backs we use the fact that the operators
$T_k(0,a)$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ are multiplications by the pull-backs of $\tau_k(a)$
(see \operatorname{e}qref{T-k-0-eq}).
Since $[{\cal C}^{[N]}]\in\TCH^0({\cal C}^{[N]})_{{\Bbb Q}}$ and the operators $T_k(0,a)$ preserve
$\TCH^*({\cal C}^{[\betaullet]})_{{\Bbb Q}}$, the result now follows from the fact that $\TCH^*({\cal J})_{{\Bbb Q}}$ is generated
by the classes $(\tau_k(a))$ with respect to the usual product (see part (iii)).
\noindent
(v) We can repeat the same proofs changing Chow groups to cohomology (and inserting
appropriate signs where needed).
\operatorname{e}d
\betaegin{rems}
1. Consider the situation of Proposition \ref{diff-op-prop}(i). Assume that the algebra $A$ is equipped with a nonnegative ${\Bbb Z}$-grading (compatible with the ${\Bbb Z}/2{\Bbb Z}$-grading) such that $\deltag({\betaf a}_0)=2$. Let us denote by ${\cal D}'(A,{\betaf a}_0)\subset{\cal D}(A,{\betaf a}_0)$ the subalgebra generated
by ${\betaf P}_{m,k}(a)$ with $m+k+\deltag(a)\ge 2$. Then the analogue of Proposition \ref{diff-op-prop}(i)
holds if we only have an action of the subalgebra ${\cal D}'(A,{\betaf a}_0)$. In particular, this can be applied to
the actions of the Lie algebra ${\cal HV}'({\Bbb Z})\subset{\cal HV}({\Bbb Z})$ (see the Introduction). In the case of the action of
${\cal HV}'({\Bbb Z})$ on ${\Bbb C}H^*(J)_{{\Bbb Q}}$ considered in \cite{P-lie} we recover the differential operators
obtained in {\it loc. cit.}.
\noindent
2. The subalgebra ${\cal D}_+({\Bbb C}H^*({\cal C}),K)\subset{\cal D}({\Bbb C}H^*({\cal C}),K)$
considered in Proposition \ref{diff-op-prop}
is closely related to the algebra generated by the operators $(T_k(m,a))$ (see section \ref{Jac-sec}).
We will study these algebras in a sequel to this paper.
\operatorname{e}nd{rems}
Let us give an example of the calculation involving tautological cycles and using the operators
introduced in this paper.
\betaegin{prop}\lambdabel{pull-back-prop}
(i) Consider the cycles $\tau_k({\cal C})\in{\Bbb C}H^{k-1}({\cal J})$ (see \operatorname{e}qref{tau-eq}).
Their pull-backs under the morphisms $\sigma_N:{\cal C}^{[N]}\to{\cal J}$ are given by
\betaegin{align*}
&\sigma_N^*\tau_k({\cal C})=(-1)^kN^k\psi^{k-1}[{\cal C}^{[N]}]+\\
&\sum_{i+n+m+p+l=k}(-1)^{n+l+m}\frac{k!}{(i+n)!(m+p)!l!}S(i+n,i)S(m+p,m)N^l\psi^{p+l}
{\cal D}e_{i*}(K^n)*[p_0]^{*m}*[{\cal C}^{[N-m-i]}]\\
&-\sum_{i+n+m+p+l+q=k, q\ge 1}(-1)^{n+l+m+q}\frac{k!q!}{(i+n+q)!(m+p)!l!}{i+q\choose i}
{m\choose q} S(i+n+q,i+q)S(m+p,m)N^l\tildemes\\
&\psi^{p+l+n+q-1}[p_0]^{*(m+i)}*[{\cal C}^{[N-m-i]}],
\operatorname{e}nd{align*}
where $[p_0]^{*m}$ denotes the $m$-th power with respect to the Pontryagin product,
$S(\cdotot,\cdotot)$ are the Stirling numbers of the second kind. Note that ${\cal D}e_0:{\cal C}\to S$
is just the projection to the base.
\noindent
(ii) For $k>g+\dim S+1$ one has $\tau_k({\cal C})=0$, and hence, $\sigma_N^*\tau_k({\cal C})=0$.
On the other hand, for $k>2g$ the class $\tau_k({\cal C})$ becomes zero in ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$.
\operatorname{e}nd{prop}
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\noindent
(iv) Since the map $\sigma_*:{\Bbb C}H^*({\cal C}^{[\betaullet]})\to{\Bbb C}H^*({\cal J})$ respects the Pontryagin products,
the first assertion follows from the formula
$$\sigma_{n*}{\cal D}e_{n*}(a)=[n]_*\iota_*(a).$$
To check the assertion about the pull-backs we use the fact that the operators
$T_k(0,a)$ on ${\Bbb C}H^*({\cal C}^{[\betaullet]})$ are multiplications by the pull-backs of $\tau_k(a)$
(see \operatorname{e}qref{T-k-0-eq}).
Since $[{\cal C}^{[N]}]\in\TCH^0({\cal C}^{[N]})_{{\Bbb Q}}$ and the operators $T_k(0,a)$ preserve
$\TCH^*({\cal C}^{[\betaullet]})_{{\Bbb Q}}$, the result now follows from the fact that $\TCH^*({\cal J})_{{\Bbb Q}}$ is generated
by the classes $(\tau_k(a))$ with respect to the usual product (see part (iii)).
\noindent
(v) We can repeat the same proofs changing Chow groups to cohomology (and inserting
appropriate signs where needed).
\operatorname{e}d
\betaegin{rems}
1. Consider the situation of Proposition \ref{diff-op-prop}(i). Assume that the algebra $A$ is equipped with a nonnegative ${\Bbb Z}$-grading (compatible with the ${\Bbb Z}/2{\Bbb Z}$-grading) such that $\deltag({\betaf a}_0)=2$. Let us denote by ${\cal D}'(A,{\betaf a}_0)\subset{\cal D}(A,{\betaf a}_0)$ the subalgebra generated
by ${\betaf P}_{m,k}(a)$ with $m+k+\deltag(a)\ge 2$. Then the analogue of Proposition \ref{diff-op-prop}(i)
holds if we only have an action of the subalgebra ${\cal D}'(A,{\betaf a}_0)$. In particular, this can be applied to
the actions of the Lie algebra ${\cal HV}'({\Bbb Z})\subset{\cal HV}({\Bbb Z})$ (see the Introduction). In the case of the action of
${\cal HV}'({\Bbb Z})$ on ${\Bbb C}H^*(J)_{{\Bbb Q}}$ considered in \cite{P-lie} we recover the differential operators
obtained in {\it loc. cit.}.
\noindent
2. The subalgebra ${\cal D}_+({\Bbb C}H^*({\cal C}),K)\subset{\cal D}({\Bbb C}H^*({\cal C}),K)$
considered in Proposition \ref{diff-op-prop}
is closely related to the algebra generated by the operators $(T_k(m,a))$ (see section \ref{Jac-sec}).
We will study these algebras in a sequel to this paper.
\operatorname{e}nd{rems}
Let us give an example of the calculation involving tautological cycles and using the operators
introduced in this paper.
\betaegin{prop}\lambdabel{pull-back-prop}
(i) Consider the cycles $\tau_k({\cal C})\in{\Bbb C}H^{k-1}({\cal J})$ (see \operatorname{e}qref{tau-eq}).
Their pull-backs under the morphisms $\sigma_N:{\cal C}^{[N]}\to{\cal J}$ are given by
\betaegin{align*}
&\sigma_N^*\tau_k({\cal C})=(-1)^kN^k\psi^{k-1}[{\cal C}^{[N]}]+\\
&\sum_{i+n+m+p+l=k}(-1)^{n+l+m}\frac{k!}{(i+n)!(m+p)!l!}S(i+n,i)S(m+p,m)N^l\psi^{p+l}
{\cal D}e_{i*}(K^n)*[p_0]^{*m}*[{\cal C}^{[N-m-i]}]\\
&-\sum_{i+n+m+p+l+q=k, q\ge 1}(-1)^{n+l+m+q}\frac{k!q!}{(i+n+q)!(m+p)!l!}{i+q\choose i}
{m\choose q} S(i+n+q,i+q)S(m+p,m)N^l\tildemes\\
&\psi^{p+l+n+q-1}[p_0]^{*(m+i)}*[{\cal C}^{[N-m-i]}],
\operatorname{e}nd{align*}
where $[p_0]^{*m}$ denotes the $m$-th power with respect to the Pontryagin product,
$S(\cdotot,\cdotot)$ are the Stirling numbers of the second kind. Note that ${\cal D}e_0:{\cal C}\to S$
is just the projection to the base.
\noindent
(ii) For $k>g+\dim S+1$ one has $\tau_k({\cal C})=0$, and hence, $\sigma_N^*\tau_k({\cal C})=0$.
On the other hand, for $k>2g$ the class $\tau_k({\cal C})$ becomes zero in ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$.
\operatorname{e}nd{prop}
\noindent {\it Proof} . (i) The idea is to use the formula
$$\sigma_N^*\tau_k({\cal C})=T_k(0,{\cal C})([{\cal C}^{[N]}])$$
(see \operatorname{e}qref{T-k-0-eq}). From Proposition \ref{T-P-prop}
we get the following expression for $T_k(0,{\cal C})$:
$$T_k(0,{\cal C})=(-1)^kP_{1,1}({\cal C})^k\psi^{k-1}+\sum_{i+n+j+l=k}(-1)^{n+j+l}\frac{k!}{(i+n)!j!l!}S(i+n,i)
P_{i,i}(K^n)P_{1,1}([p_0])^jP_{1,1}({\cal C})^l\psi^l.$$
Recall that $P_{1,1}({\cal C})x=Nx$ for $x\in{\Bbb C}H^*({\cal C}^{[N]})$.
Set $t=[p_0]\in{\Bbb C}H^1({\cal C})$, $u=[{\cal C}]\in{\Bbb C}H^0({\cal C})$.
It follows easily from Proposition \ref{diff-op-prop}(ii) that the subalgebra
${\Bbb C}H^*(S)[t,u^{[\betaullet]}]\subset\widetilde{\TCH}^*({\cal C}^{[\betaullet]})$ is preserved by
the operator $P_{1,1}([p_0])$, and its action is given by
$$P_{1,1}([p_0])|_{{\Bbb Z}[t,u^{[\betaullet]}]}=t(\partial_u-\psi \partial_t).$$
From this we deduce by induction in $j$ that
$$P_{1,1}([p_0])^j([{\cal C}^{[N]}])=P_{1,1}([p_0])^j(u^{[N]})=\sum_{m=0}^j(-\psi)^{j-m}S(j,m)t^mu^{[N-m]}$$
for $j\ge 0$.
Next, from Proposition \ref{diff-op-prop}(ii) we see that for an element $f\in{\Bbb C}H^*(S)[t,u^{[\betaullet]}]$
one has
$$P_{i,i}(K^n)f={\cal D}e_{i_*}(K^n)\partial_u^i f-
t^i\cdotot \sum_{q\ge 1}(-1)^q{i\choose q}\psi^{n+q-1}\partial^{i-q}_u\partial^q_t f.$$
From this we get formulas for $P_{i,i}(K^n)P_{1,1}([p_0])^j([{\cal C}^{[N]})$ for all $i,j,n$.
It remains to substitute them
into our expression for $T_k(0,{\cal C})([{\cal C}^{[N]}])$.
\noindent (ii) The first assertion is clear since $\tau_k({\cal C})\in{\Bbb C}H^{k-1}({\cal J})$. The second follows from the equality $\tau_k({\cal C})=T_k(0,{\cal C})[{\cal J}]$ together with the fact that
$T_k(0,{\cal C})=X_{0,k}$ is zero as an operator on ${\Bbb C}H^*({\cal J})_{{\Bbb Q}}$ for $k>2g$.
\operatorname{e}d
\betaegin{cor}\lambdabel{pull-back-cor} Assume that $S$ is a point.
\noindent
(i) For $k>1$ one has
\betaegin{align*}
&\sigma_N^*\tau_k(C)={\Bbb G}a_{e,k}*[C^{[N-k]}]
-\sum_{i+m=k-1}(-1)^m{k\choose m}{i+1\choose 2}{\cal D}e_{i*}(K)*[p_0]^{*m}*[C^{[N-k+1]}]\\
&-2\delta_{k,2}[p_0]*[C^{[N-1]}],
\operatorname{e}nd{align*}
where
$${\Bbb G}a_{e,k}=\sum_{i+m=k}(-1)^m{k\choose m}[{\cal D}e_i(C)]*[p_0]^{*m}\in{\Bbb C}H^{k-1}(C^{[k]})$$
\noindent
(ii) Assume that $K=(2g-2)[p_0]$. Then for $k>1$ one has
$$\sigma_N^*\tau_k(C)={\Bbb G}a_{e,k}*[C^{[N-k]}]-2g\delta_{k,2}[p_0]*[C^{[N-1]}].$$
Without this assumption one has
$$\sigma_N^*\tau_k(C)\sigmam_{a.e.}{\Bbb G}a_{e,k}*[C^{[N-k]}]-2g\delta_{k,2}[p_0]*[C^{[N-1]}],$$
where $\sigmam_{a.e.}$ denotes algebraic equivalence.
\noindent
(iii) Assume that $K=(2g-2)[p_0]$. Then one has ${\Bbb G}a_{e,k}=0$ for $k>g+1$.
Also, ${\Bbb G}a_{e,k}$ is a torsion class for $k\ge g/2+2$.
\noindent
(iv) One has ${\Bbb G}a_{e,k}\sigmam_{a.e.} 0$ for $k>g+1$.
Furthermore, if $C$ admits a morphism of degree $d\ge 1$ to ${\Bbb P}^1$
then for $k\ge d+1$ some multiple of ${\Bbb G}a_{e,k}$ is algebraically equivalent to zero.
\operatorname{e}nd{cor}
\noindent {\it Proof} . (i) This follows from Proposition \ref{pull-back-prop}(i).
\noindent (ii) This follows from (i) since ${\cal D}e_{i*}[p_0]=[p_0]^{*i}$.
\noindent (iii) Note that under these assumptions we have either $k>2$ or $g=0$. In either case
taking $N=k$ in (ii) we get $\sigma_k^*\tau_k(C)={\Bbb G}a_{e,k}$.
Now the first assertion follows from the trivial vanishing $\tau_k(C)=0$ for $k>g+1$.
The second assertion follows from vanishing of $\tau_k(C)$ in ${\Bbb C}H^*(J)_{{\Bbb Q}}$ for $k\ge g/2+2$ that
is checked as follows. Note that $\tau_k(C)/k!=p_{k-1}$, where $(p_n)$ are the tautological
classes considered in \cite{P-lie}. Now the required vanishing follows from Proposition 4.2 of \cite{P-lie}
(note that the condition $K=(2g-2)[p_0]$ implies the vanishing of all the classes $q_n$).
\noindent (iv) The first statement follows from $K\sigmam_{a.e.}(2g-2)[p_0]$ as in (iii). For the second statement we use the fact that for $k\ge d+1$ some multiple of
$\tau_k(C)$ is algebraically equivalent to zero in this case
by the result of Colombo and Van Geemen in \cite{CG}.
\operatorname{e}d
\betaegin{rem}
The {\it modified diagonal classes} ${\Bbb G}a_{e,k}\in{\Bbb C}H^{k-1}(C^{[k]})$ were introduced
by Gross and Schoen in \cite{GS}. Parts (iii) and (iv) of the above
corollary are related to Propositions 4.5 and 4.8 of {\it loc. cit.}. In the case when $C$ is hyperelliptic
and $p_0$ is stable under the hyperelliptic involution
Gross and Schoen show that ${\Bbb G}a_{e,3}=0$. By
Corollary \ref{pull-back-cor}(ii) this is equivalent to the fact that $\tau_3(C)=0$ in this case
(since $\sigma_N^*$ is injective for sufficiently large $N$).
\operatorname{e}nd{rem}
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Inventiones Math. 37 (1976), 215--228.
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\noindent
(ii) Assume that $K=(2g-2)[p_0]$. Then for $k>1$ one has
$$\sigma_N^*\tau_k(C)={\Bbb G}a_{e,k}*[C^{[N-k]}]-2g\delta_{k,2}[p_0]*[C^{[N-1]}].$$
Without this assumption one has
$$\sigma_N^*\tau_k(C)\sigmam_{a.e.}{\Bbb G}a_{e,k}*[C^{[N-k]}]-2g\delta_{k,2}[p_0]*[C^{[N-1]}],$$
where $\sigmam_{a.e.}$ denotes algebraic equivalence.
\noindent
(iii) Assume that $K=(2g-2)[p_0]$. Then one has ${\Bbb G}a_{e,k}=0$ for $k>g+1$.
Also, ${\Bbb G}a_{e,k}$ is a torsion class for $k\ge g/2+2$.
\noindent
(iv) One has ${\Bbb G}a_{e,k}\sigmam_{a.e.} 0$ for $k>g+1$.
Furthermore, if $C$ admits a morphism of degree $d\ge 1$ to ${\Bbb P}^1$
then for $k\ge d+1$ some multiple of ${\Bbb G}a_{e,k}$ is algebraically equivalent to zero.
\operatorname{e}nd{cor}
\noindent {\it Proof} . (i) This follows from Proposition \ref{pull-back-prop}(i).
\noindent (ii) This follows from (i) since ${\cal D}e_{i*}[p_0]=[p_0]^{*i}$.
\noindent (iii) Note that under these assumptions we have either $k>2$ or $g=0$. In either case
taking $N=k$ in (ii) we get $\sigma_k^*\tau_k(C)={\Bbb G}a_{e,k}$.
Now the first assertion follows from the trivial vanishing $\tau_k(C)=0$ for $k>g+1$.
The second assertion follows from vanishing of $\tau_k(C)$ in ${\Bbb C}H^*(J)_{{\Bbb Q}}$ for $k\ge g/2+2$ that
is checked as follows. Note that $\tau_k(C)/k!=p_{k-1}$, where $(p_n)$ are the tautological
classes considered in \cite{P-lie}. Now the required vanishing follows from Proposition 4.2 of \cite{P-lie}
(note that the condition $K=(2g-2)[p_0]$ implies the vanishing of all the classes $q_n$).
\noindent (iv) The first statement follows from $K\sigmam_{a.e.}(2g-2)[p_0]$ as in (iii). For the second statement we use the fact that for $k\ge d+1$ some multiple of
$\tau_k(C)$ is algebraically equivalent to zero in this case
by the result of Colombo and Van Geemen in \cite{CG}.
\operatorname{e}d
\betaegin{rem}
The {\it modified diagonal classes} ${\Bbb G}a_{e,k}\in{\Bbb C}H^{k-1}(C^{[k]})$ were introduced
by Gross and Schoen in \cite{GS}. Parts (iii) and (iv) of the above
corollary are related to Propositions 4.5 and 4.8 of {\it loc. cit.}. In the case when $C$ is hyperelliptic
and $p_0$ is stable under the hyperelliptic involution
Gross and Schoen show that ${\Bbb G}a_{e,3}=0$. By
Corollary \ref{pull-back-cor}(ii) this is equivalent to the fact that $\tau_3(C)=0$ in this case
(since $\sigma_N^*$ is injective for sufficiently large $N$).
\operatorname{e}nd{rem}
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532 (2001), 105--132.
\betaibitem{dB2} S.~del Ba\~{n}o, {\it On the Hodge theory of the symmetric powers of a curve},
Publ. Mat. 46 (2002), 17-25.
\betaibitem{B1} A.~Beauville, {\it Quelques remarques sur la
transformation de Fourier dans l'anneau de Chow d'une vari\'et\'e
ab\'elienne}, Algebraic Geometry (Tokyo/Kyoto 1982), Lecture Notes in Math.
1016, 238--260. Springer-Verlag, 1983.
\betaibitem{B2} A.~Beauville, {\it Sur l'anneau de Chow d'une vari\'et\'e
ab\'elienne}, Math. Ann. 273 (1986), 647--651.
\betaibitem{Bmain} A.~Beauville, {\it Algebraic cycles on Jacobian varieties},
Compositio Math. 140 (2004), 683--688.
\betaibitem{BV} A.~Beauville, C.~Voisin, {\it On the Chow ring of a K3 surface}, J.~Algebraic Geom. 13 (2004), 417--426.
\betaibitem{Bl} S.~Bloch, {\it Some elementary theorems about algebraic cycles on abelian varieties},
Inventiones Math. 37 (1976), 215--228.
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Illinois J. Math. 19 (1975), 567--583.
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Compositio Math. 88 (1993), 333--353.
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Mem. Amer. Math. Soc. 31 (1981), no. 243.
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\betaibitem{GS} B.~Gross, C. Schoen, {\it The modified diagonal cycle on the triple product of a pointed curve}, Annales de l'institut Fourier 45 (1995), 649--679.
\betaibitem{KV} S.~Kimura, A.~Vistoli, {\it Chow rings of infinite symmetric products},
Duke Math. J. 85 (1996), 411--430.
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Hilbert schemes of points on surfaces}, Invent. Math. 136 (1999), 157--207.
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\operatorname{e}nd{thebibliography}
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\begin{document}
\date{Version of \today}
\title[Finite dimensional approximation of forward prices]{Approximation of forward curve models in commodity markets with arbitrage-free
finite dimensional models}
\begin{abstract}
In this paper we show how to approximate a Heath-Jarrow-Morton dynamics for the forward prices in commodity markets with arbitrage-free models which have
a finite dimensional state space. Moreover, we recover a closed form representation of the forward price dynamics in the approximation models and derive the rate of convergence uniformly over an interval of time to maturity to the true dynamics under certain additional smoothness conditions. In the Markovian case we can strengthen the convergence to be uniform over time as well. Our results are based on the construction of a convenient
Riesz basis on the state space of the term structure dynamics.
\end{abstract}
\author[Benth]{Fred Espen Benth}
\address[Fred Espen Benth]{\\
Department of Mathematics \\
University of Oslo\\
P.O. Box 1053, Blindern\\
N--0316 Oslo, Norway}
\email[]{fredb\@@math.uio.no}
\urladdr{http://folk.uio.no/fredb/}
\author[Kr\"uhner]{Paul Kr\"uhner}
\address[Paul Kr\"uhner]{\\
Financial \& Actuarial Mathematics\\
Vienna University of Technology\\
Wiedner Hauptstr. 8/E105-1\\
AT-1040 Vienna, Austria}
\email[]{[email protected]}
\urladdr{https://fam.tuwien.ac.at/~paulkrue/}
\thanks{F.\ E.\ Benth acknowledges financial support from the project "Managing Weather Risk in Energy Markets (MAWREM)", funded by the ENERGIX program
of the Norwegian Research Council.}
\subjclass[2010]{91B24, 91G20}
\keywords{Energy markets, Heath-Jarrow-Morton, Non harmonic Fourier analysis, arbitrage free approximations}
\maketitle
\section{Introduction}
We develop arbitrage-free approximations to the forward term structure dynamics in commodity
markets. The approximating term structure models have finite dimensional state space, and therefore
tractable for further analysis and numerical simulation. We provide results on the convergence of the
approximating term structures and characterize the speed under reasonable smoothness properties
of the true term structure. Our results are based on the construction of a convenient Riesz basis on the state space of the term structure dynamics.
In the context of fixed-income markets, Heath, Jarrow and Morton~\cite{HJM} propose to model the entire term structure of interest rates. Filipovi\'c \cite{filipovic.01} reinterprets this approach in the so-called Musiela parametrisation, i.e., studying the so-called forward rates as solutions of first-order stochastic
partial differential equations. This class of stochastic partial differential equations is often referred to as
the Heath-Jarrow-Morton-Musiela (HJMM) dynamics. This highly successful method has been transferred to
other markets, including commodity and energy futures markets (see Clewlow and
Strickland~\cite{CS} and Benth,
Saltyte Benth and Koekebakker~\cite{BSBK-book}), where the term structure of forward and
futures prices are modelled by similar stochastic partial differential equations.
An important stream of research in interest rate modelling has been so-called finite dimensional realizations
of the solutions of the HJMM dynamics (see e.g., Bj\"ork and Svensson~\cite{BjorkSvensson}, Bj\"ork and Landen~\cite{BjorkLanden}, Filipovic and Teichmann~\cite{FT} and Tappe~\cite{Tappe2010}). Starting out with an equation for the forward rates driven by a
$d$-dimensional Wiener process, the question has been under what conditions on the volatility and drift
do we get solutions which belongs to a finite dimensional space, that is, when can the dynamics of the
whole curve be decomposed into a finite number of factors. This property has a close connection with
principal component analysis (see Carmona and Tehranchi~\cite{CT}), but is also convenient when
it comes to further analysis like estimation, simulation, pricing and portfolio management (see
Benth and Lempa~\cite{BL} for the latter).
In energy markets like power and gas, there is empirical and economical evidence for high-dimensional
noise. Moreover, the noise shows clear leptokurtic signs (see Benth, \v{S}altyt\.e Benth and Koekebakker~\cite{BSBK-book} and references therein). These empirical insights motivate the use of
infinite dimensional L\'evy processes driving the noise in the HJMM-dynamics modelling the forward term structure. We refer to Carmona and Tehranchi~\cite{CT} for a thorough analysis of HJMM-models with
infinite dimensional Gaussian noise in interest rate markets. Benth and Kr\"uhner~\cite{BK-stochastics}
introduced a convenient class of infinite dimensional L\'evy processes via subordination of Gaussian
processes in infinite dimensions. These models were used in analysing stochastic partial differential
equations with infinite dimensional L\'evy noise in Benth and Kr\"uhner~\cite{BK-coms}. Further,
pricing and hedging of derivatives in energy markets based on such models were studied in
Benth and Kr\"uhner~\cite{BK-siam}.
The present paper is motivated by the need of an arbitrage-free approximation of Heath, Jarrow, Morton style models -- using the Musiela parametrisation -- in electricity finance. Related research has been carried out by Henseler, Peters and Seydel \cite{Henseler.al.15} who construct a finite-dimensional affine model where a refined principle component analysis (PCA) method does yield an arbitrage free approximation of the term structure model.
Our main result Theorem~\ref{t:main statement} states that the arbitrage-free models for the underlying forward curve
process $f(t,x)$, $x\geq0$ being time to maturity and $t\geq 0$ is current time, can be approximated with processes of the form
$$
f_k(t,x) = S_k(t) + \sum_{n=-k}^k U_n(t)g_n(x) \,,
$$
where $S_k$ denotes the spot prices in the approximating model, $g_{-k},\dots,g_k$ are deterministic functions and $U_{-k},\dots,U_k$ are one-dimensional Ornstein Uhlenbeck type processes. Obviously, models of this type are much easier to handle in applications than general solutions for the HJMM equation. The
approximation $f_k$ is again a solution of an HJMM equation, and as such being an arbitrage-free model
for the forward term structure. We prove a uniform convergence in space of $f_k$ to the "real" forward
price curve $f$, pointwise in time. The convergence rate is of order $k^{-1}$ when the
forward curve $x\mapsto f(t,x)$ is twice continuously differentiable. Our approach
is an alternative to numerical approximations of the HJMM dynamics based on finite difference schemes or finite element
methods, where arbitrage-freeness of the approximating dynamics is not automatically ensured.
We refer to Barth~\cite{Barth} for an analysis of finite element methods
applied to
stochastic partial differential equations of the type we study.
We refine our results to the Markovian case, where the convergence is slightly strengthened to be uniform over time as well. Our approach goes via the explicit construction of
a Riesz basis for a subspace of the so-called Filipovi\'c space (see Filipovi\'c~\cite{filipovic.01}),
a separable Hilbert space of absolutely continuous functions on the positive real line with
(weak) derivative disappearing at a certain speed at infinity. The basis will be the functions $g_n$ in the
approximation $f_k$, and the subspace is defined by concentrating the functions in the Filipovi\'c space
to a finite time horizon $x\leq T$. This space was defined in Benth and Kr\"uhner~\cite{BK-coms},
and we extend the analysis here to accomodate the arbitrage-free finite dimensional approximation of
the HJMM-dynamics. We rest on properties of $C_0$-semigroups and stochastic integration with
respect to infinite dimensional L\'evy processes (see Peszat and Zabczyk~\cite{peszat.zabczyk.07})
in the analysis.
This paper is organised as follows. In Section~\ref{s:the model} we start with the mathematical formulation
of the HJMM dynamics for forward rates set in the Filipovi\'c space. The Riesz basis that will make the foundation for our approximation is defined and analysed in detail in Section~\ref{s:mathematical Preliminaries}. The arbitrage-free finite dimensional approximation to term structure modelling is
constructed in Section~\ref{s:approximation}, where we study convergence properties. The Markovian
case is analysed in the last Section~\ref{s:markovian}.
\section{The model of the forward price dynamics}
\label{s:the model}
Throughout this paper we use the Hilbert space
$$
H_\alpha := \left\{f\in AC(\mathbb R_+,\mathbb C): \int_0^\infty |f'(x)|^2e^{\alpha x} dx <\infty\right\}\,,
$$
where $AC(\mathbb R_+,\mathbb C)$ denotes the space of complex-valued absolutely continuous functions on $\mathbb R_+$. We endow
$H_{\alpha}$ with the scalar product $\langlef,g\rangle_\alpha:=f(0)\overline{g}(0) + \int_0^\infty f'(x)\overline{g}'(x) e^{\alpha x}dx$, and denote
the associated norm by $\|\cdot\|_{\alpha}$. Filipovi\'c~\cite[Section 5]{filipovic.01} shows that $(H_\alpha,\|\cdot\|_\alpha)$ is a separable Hilbert space\footnote{Note that
Filipovi\'c~\cite{filipovic.01} does not consider complex-valued functions. In our context, this minor extension is convenient, as will be clear later.}.
This space has been used in Filipovi\'c~\cite{filipovic.01} for term structure modelling of bonds and many mathematical properties have been derived therein.
We will frequently refer to $H_{\alpha}$ as the {\it Filipovi\'c space}.
We next introduce our dynamics for the term structure of forward prices in a commodity market. Denote by $f(t,x)$ the price at time $t$ of
a forward contract where time to delivery of the underlying commodity is $x\geq 0$. We treat $f$ as a stochastic process in time with values in
the Filipovi\'c space $H_{\alpha}$. More specifically, we assume that the process $\{f(t)\}_{t\geq 0}$ follows the HJM-Musiela model
which we formalize next.
On a complete filtered probability space $(\Omega,\{\mathcal{F}_t\}_{t\geq 0},\mathcal{F},P)$, where the filtration is assumed to be complete and right continuous,
we work with an $H_{\alpha}$-valued L\'evy process $\{L(t)\}_{t\geq 0}$ (cf.\ Peszat and Zabczyk~\cite[Theorem 4.27(i)]{peszat.zabczyk.07} for the construction of $H_\alpha$-valued L\'evy processes). We assume that $L$ has finite variance and mean equal to zero, and denote its covariance operator by $\mathcal Q$.
Let $f_0\in H_\alpha$ and $f$ be the solution of the stochastic partial differential equation (SPDE)
\begin{equation}
\label{e:HJMM-equation}
df(t) = \partial_x f(t) dt + \beta(t) dt + \Psi(t)dL(t),\quad t\geq 0, f(0)=f_0
\end{equation}
where $\beta\in L^1((\Omega\times\mathbb R_+,\mathcal P,P\otimes\lambda),H_\alpha)$, $\mathcal P$ being the predictable $\sigma$-field, and
$\Psi \in \mathcal{L}^2_{L}(H_\alpha):=\bigcup_{T>0}\mathcal{L}^2_{L,T}(H_\alpha) $ where the latter space is defined as in {Peszat and Zabczyk~\cite[page 113]{peszat.zabczyk.07}}. For $t\geq 0$, denote by $\mathcal{U}_t$ the shift semigroup on $H_{\alpha}$ defined by
$\mathcal{U}_t f=f(t+\cdot)$ for $f\in\mathcal H_{\alpha}$. It is shown in Filipovi\'c~\cite{filipovic.01} that $\{\mathcal{U}_t\}_{t\geq 0}$ is a $C_0$-semigroup on
$H_{\alpha}$, with generator $\partial_x$. Recall, that any $C_0$-semigroup admits the bound $\Vert\mathcal U_t\Vert_{\mathrm{op}}\leq Me^{wt}$ for some $w,M>0$ and any $t\geq 0$. Here, $\|\cdot\|_{\mathrm{op}}$
denotes the operator norm. In fact, in Filipovi\'c~\cite[Equation (5.10)]{filipovic.01} and Benth and Kr\"uhner~\cite[Lemma~3.4]{benth.kruehner.15}
it is shown that $\Vert\mathcal U_t\Vert_{\mathrm{op}}\leq C_{\mathcal{U}}$ for any $t\geq 0$ and a constant $C_{\mathcal{U}}:=\sqrt{2(1\wedge\alpha^{-1})}$. Thus $s\mapsto \mathcal U_{t-s}\beta(s)$ is Bochner-integrable and $s\mapsto \mathcal U_{t-s}\Psi(s)$ is integrable with respect to $L$. The unique mild solution of \eqref{e:HJMM-equation} is
\begin{equation}
\label{e:HJMM-equation-mild}
f(t)=\mathcal{U}_tf_0+\int_0^t\mathcal{U}_{t-s}\beta(s)\,ds+\int_0^t\mathcal{U}_{t-s}\Psi(s)\,dL(s)\,.
\end{equation}
If we model the forward price dynamics $f$ in a risk-neutral setting, the drift coefficient $\beta(t)$ will naturally be zero in order to ensure
the (local) martingale property of the process $t\mapsto f(t,\tau-t)$, where $\tau\geq t$ is the time of delivery of the forward. In this
case, the probability $P$ is to be interpreted as the equivalent martingale measure (also called the pricing measure). However, with
a non-zero drift, the forward model is stated under the market probability and $\beta$ can be related to the risk premium in the market.
In energy markets like power and gas, the forward contracts deliver over a period, and forward prices can be expressed
by integral operators on the Filipovi\'c space applied on $f$ (see Benth and Kr\"uhner~\cite{benth.kruehner.14,benth.kruehner.15} for more details).
The dynamics of $f$ can also be considered as a model for the forward rate in fixed-income theory, see Filipovi\'c~\cite{filipovic.01}. This is indeed the
traditional application area and point of analysis of the SPDE in \eqref{e:HJMM-equation}. Note, however, that the original no-arbitrage condition in the HJM approach for interest rate markets is different from the no-arbitrage condition used here. If $f$ is understood as the forward rate modelled
in the risk-neutral setting, there is a no-arbitrage relationship between the drift $\beta$, the volatility $\sigma$ and the covariance of the driving noise $L$. We refer to Carmona and Tehranchi~\cite{CT} for a detailed analysis.
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\section{A Riesz basis for the Filipovi\'c space}\label{s:mathematical Preliminaries}
In this section we introduce a Riesz basis for a suitable subspace of $H_\alpha$
defined in Benth and Kr\"uhner~\cite[Appendix A]{benth.kruehner.14} and present various of its
properties. Moreover, we give refined statements for this basis and also identify new properties.
We recall from Young~\cite{Young.80} that any Riesz basis $\{g_n\}_{n\in\mathbb{N}}$ on a separable Hilbert space can
be expressed by $g_n = \mathcal T e_n$ where $\{e_n\}_{n\in\mathbb N}$ is an orthonormal basis and $\mathcal T$ is a bounded invertible linear operator.
For further properties and definitions of Riesz bases, see Young~\cite{Young.80}.
In Section \ref{s:approximation} we want to employ the spectral method to an approximation of the SPDE in \eqref{e:HJMM-equation} involving the differential operator on the Filipovi\'c space $H_\alpha$. Thus, it would be convenient to have available the eigenvector basis for the differential operator. However, its eigenvectors do not seem to have nice basis properties. Instead, we propose to use a system of vectors which forms a Riesz basis which turns out to be almost an eigenvector system for the differential operator. This property will be made precise in Propositions~\ref{p:U and the riesz basis} and \ref{l:commutator of U and projectors}.
Finally, we will identify the convergence speed of the Riesz basis expansion.
Fix $\lambda>0$, $T>0$, and introduce
\begin{equation}
\mathrm{cut}:\mathbb R_+\rightarrow [0,T)\,,\qquad x\mapsto x-\max\{Tz: z\in\mathbb Z:Tz\leq x\}\,,
\end{equation}
and
\begin{equation}
\label{def-A-operator}
\mathcal A:L^2([0,T),\mathbb C)\rightarrow L^2(\mathbb R_+,\mathbb C)\,,\qquad
f\mapsto \left(x\mapsto e^{-\lambda x}f(\mathrm{cut}(x))\right)\,.
\end{equation}
Here, $L^2(A,\mathbb C)$ is the space of complex-valued square integrable functions on the Borel set $A\subset\mathbb R_+$ equipped with the
Lebesgue measure. The inner product of $L^2(A,\mathbb C)$ will be denoted $(\cdot,\cdot)_2$ and the corresponding norm $|\cdot|_2$. We remark that
the set $A$ will be clear from the context and thus not indicated in the notation for
norm and inner product.
We define
\begin{align}
g_*(x) &:= 1, \label{e:g-star-def}\\
g_n(x) &:= \frac{1}{\lambda_n\sqrt{T}}\left(\exp\left(\lambda_nx\right)-1\right)\,, \label{e:g-n-def}
\end{align}
where
\begin{equation}
\label{e:lambda-n-def}
\lambda_n:=\frac{2\pi i}{T}n-\lambda-\frac{\alpha}{2}\,,
\end{equation}
for any $n\in\mathbb Z$, $x\geq0$.
It is simple to verify that $g_n\in H_\alpha$ for any $n\in\mathbb Z$ and $g_*\in H_\alpha$.
As we will see, the system of vectors $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ forms a Riesz basis and we will use this to obtain arbitrage-free finite-dimensional approximations of the forward price dynamics \eqref{e:HJMM-equation}.
We start our analysis with some elementary properties of the operator $\mathcal A$ which have been proven in Benth and Kr\"uhner~\cite{benth.kruehner.14}.
\begin{lem}\label{l:stetige Einbettung}
$\mathcal A$ is a bounded linear operator and its range is closed in $L^2(\mathbb R_+,\mathbb C)$. Moreover,
$$ \frac{e^{-2T\lambda}}{1-e^{-2T\lambda}}| f|_2^2 \leq |\mathcal Af|_2^2 \leq \frac{1}{1-e^{-2T\lambda}}| f|_2^2$$
for any $f\in L^2([0,T),\mathbb C)$.
\end{lem}
\begin{proof}
This proof can be found in Benth and Kr\"uhner~\cite[Lemma A.1]{benth.kruehner.14}.
\end{proof}
In the following Proposition~\ref{p:Riesz basis on L2}, we calculate a Riesz basis of the space $\mathrm{ran}(\mathcal{A})$
and its biorthogonal system. The Riesz basis will be given as the image of an orthonormal basis of $L^2([0,T),\mathbb C)$. Consequently, its biorthogonal
system is given by the image of $(\mathcal A^{-1})^*$, which we calculate in the Lemma below:
\begin{lem}
\label{lem:dual_A}
The dual $(\mathcal A^{-1})^*$ of the inverse of $\mathcal A:L^2([0,T),\mathbb C)\rightarrow \mathrm{ran}(\mathcal{A})$
is given by
\begin{align*}
(\mathcal A^{-1})^*&:L^2([0,T),\mathbb C)\rightarrow \mathrm{ran}(\mathcal{A}) ,\\
(\mathcal A^{-1})^*f(x)&=(1-e^{-2\lambda T})e^{-\lambda x}\left( e^{2\lambda \mathrm{cut}(x)}f(\mathrm{cut}(x)) \right) \\
&= (1-e^{-2\lambda T})e^{2\lambda \mathrm{cut}(x)} \mathcal Af(x),\quad x\geq0\,.
\end{align*}
\end{lem}
\begin{proof}
Let $f,g\in L^2([0,T],\mathbb C)$
and define $h(x):=(1-e^{-2\lambda T})e^{2\lambda \mathrm{cut}(x)} \mathcal Af(x)$ for any $x\geq0$. Then we have
\begin{align*}
(h,\mathcal Ag)_2&= \int_0^\infty h(y) \overline{\mathcal Ag(y)} dy \\
&= (1-e^{-2\lambda T}) \sum_{n=0}^\infty \int_{nT}^{(n+1)T} e^{2\lambda(x-nT)} (e^{-\lambda x}f(x-nT))(e^{-\lambda x}\overline{g(x-nT)}) dx \\
&= (1-e^{-2\lambda T})\sum_{n=0}^\infty e^{-2\lambda nT} \int_{nT}^{(n+1)T} f(x-nT)\overline{g(x-nT)} dx \\
&= \int_0^T f(y)\overline{g(y)} dy\,.
\end{align*}
On the other hand,
\begin{align*}
((\mathcal A^{-1})^*f, \mathcal Ag)_2&=(f,g)_2= \int_0^Tf(y)\overline{g(y)} dy\,.
\end{align*}
Since $g$ is arbitrary, we have $h = (\mathcal A^{-1})^*f$ as claimed.
\end{proof}
Parts of the next proposition can be found in Benth and Kr\"uhner~\cite[Lemma A.3]{benth.kruehner.14}. In that paper there appears to be a gap in the proof which we have filled here.
\begin{prop}\label{p:Riesz basis on L2}
Define
$$ e_n(x) := \frac{1}{\sqrt{T}} \exp\left(\left(\frac{2\pi in}{T}-\lambda\right)x\right),\quad x\geq 0,n\in\mathbb Z.$$
Then $\{e_n\}_{n\in\mathbb Z}$ is a Riesz basis on the closed subspace $\mathrm{ran}(\mathcal{A})$ of $L^2(\mathbb R_+,\mathbb C)$ and
$$ F:=\{ f\in L^2(\mathbb R_+,\mathbb C): f(x)=0,x\in[0,T) \} $$
is a closed vector space compliment of $\mathrm{ran}(\mathcal{A})$. The continuous linear projector $\mathcal P_{\mathcal A}$ with range $\mathrm{ran}(\mathcal{A})$ and kernel $F$ has operator norm $\sqrt{\frac{1}{1-e^{-2\lambda T}}}$ and we have
$$ \mathcal P_{\mathcal A}f(x) = f(x),\quad x\in[0,T], f\in L^2(\mathbb R_+,\mathbb C).$$
The biorthogonal system $\{e_n\}^*_{n\in\mathbb Z}$ for the Riesz basis $\{e_n\}_{n\in\mathbb Z}$ is given by
$$ e_n^*(x) = \left(1-e^{-2\lambda T}\right)e^{2\lambda\mathrm{cut}(x)} e_n(x) $$
\end{prop}
\begin{proof}
Recall that the range of $\mathcal A$ is a closed subspace of $L^2(\mathbb R_+,\mathbb C)$ due to the lower bound given in Lemma \ref{l:stetige Einbettung}. Furthermore, $\{b_n\}_{n\in\mathbb Z}$ with
$$ b_n(x):= \frac{1}{\sqrt{T}}\exp\left(\frac{2\pi i n}{T}x\right),\quad n\in\mathbb Z,x\in[0,T)$$
is an orthonormal basis of $L^2([0,T],\mathbb C)$. Observe, that $e_n = \mathcal Ab_n$ and hence $\{e_n\}_{n\in\mathbb Z}$ becomes a Riesz basis of
$\mathrm{ran}(\mathcal{A})$.
Define the continuous linear operators
\begin{align*}
\mathcal M_{\lambda} &:L^2(\mathbb [0,T),\mathbb C)\rightarrow L^2([0,T),\mathbb C),\mathcal M_{\lambda}f(x):=e^{\lambda x}f(x),\\
\mathcal C &:L^2(\mathbb R_+,\mathbb C)\rightarrow L^2([0,T),\mathbb C),f\mapsto f\vert_{[0,T)}
\end{align*}
and $\mathcal P_{\mathcal A}:=\mathcal A\mathcal M_{\lambda}\mathcal C$. Observe, that $\mathcal M_{\lambda}\mathcal C\mathcal A$ is the identity operator on $L^2([0,T),\mathbb C)$ and hence $\mathcal P_{\mathcal A}^2=\mathcal P_{\mathcal A}$. Therefore, $\mathcal P_{\mathcal A}$ is a continuous linear projection with kernel $F$ and range $\mathrm{ran}(\mathcal{A})$.
Let $f\in L^2(\mathbb R_+,\mathbb C)$ be orthogonal to any element of the kernel of $\mathcal P_{\mathcal A}$. Then $f(x)=0$ Lebesgue-a.e.\ for any $x\geq T$. Hence, we have
\begin{align*}
|\mathcal P_{\mathcal A}f|_2^2 &= \sum_{n\in\mathbb N} \int_{nT}^{nT+T} (e^{-\lambda x} e^{\lambda(x-nT)})^2|f(x-nT)|^2dx \\
&= \sum_{n\in\mathbb N} e^{-2n\lambda T} \vert f\vert_2^2 \\
&= \frac{1}{1-e^{-2\lambda T}} \vert f\vert_2^2
\end{align*}
and it follows that $\Vert\mathcal P_{\mathcal A}\Vert_{\mathrm{op}} = \sqrt{\frac{1}{1-e^{-2\lambda T}}}$.
According to Lemma~\ref{lem:dual_A}, we have
\begin{align*}
e_n^*(x) &= (\mathcal{A}^{-1})^*b_n(x) \\
&= (1-e^{-2\lambda T})e^{-\lambda x}\left( e^{2\lambda \mathrm{cut}(x)}b_n(\mathrm{cut}(x)) \right) \\
&= \left(1-e^{-2\lambda T}\right)e^{2\lambda\mathrm{cut}(x)} e_n(x)\,,
\end{align*}
for any $n\in\mathbb Z$, $x\geq0$, as required.
\end{proof}
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Parts of the next proposition can be found in Benth and Kr\"uhner~\cite[Lemma A.3]{benth.kruehner.14}. In that paper there appears to be a gap in the proof which we have filled here.
\begin{prop}\label{p:Riesz basis on L2}
Define
$$ e_n(x) := \frac{1}{\sqrt{T}} \exp\left(\left(\frac{2\pi in}{T}-\lambda\right)x\right),\quad x\geq 0,n\in\mathbb Z.$$
Then $\{e_n\}_{n\in\mathbb Z}$ is a Riesz basis on the closed subspace $\mathrm{ran}(\mathcal{A})$ of $L^2(\mathbb R_+,\mathbb C)$ and
$$ F:=\{ f\in L^2(\mathbb R_+,\mathbb C): f(x)=0,x\in[0,T) \} $$
is a closed vector space compliment of $\mathrm{ran}(\mathcal{A})$. The continuous linear projector $\mathcal P_{\mathcal A}$ with range $\mathrm{ran}(\mathcal{A})$ and kernel $F$ has operator norm $\sqrt{\frac{1}{1-e^{-2\lambda T}}}$ and we have
$$ \mathcal P_{\mathcal A}f(x) = f(x),\quad x\in[0,T], f\in L^2(\mathbb R_+,\mathbb C).$$
The biorthogonal system $\{e_n\}^*_{n\in\mathbb Z}$ for the Riesz basis $\{e_n\}_{n\in\mathbb Z}$ is given by
$$ e_n^*(x) = \left(1-e^{-2\lambda T}\right)e^{2\lambda\mathrm{cut}(x)} e_n(x) $$
\end{prop}
\begin{proof}
Recall that the range of $\mathcal A$ is a closed subspace of $L^2(\mathbb R_+,\mathbb C)$ due to the lower bound given in Lemma \ref{l:stetige Einbettung}. Furthermore, $\{b_n\}_{n\in\mathbb Z}$ with
$$ b_n(x):= \frac{1}{\sqrt{T}}\exp\left(\frac{2\pi i n}{T}x\right),\quad n\in\mathbb Z,x\in[0,T)$$
is an orthonormal basis of $L^2([0,T],\mathbb C)$. Observe, that $e_n = \mathcal Ab_n$ and hence $\{e_n\}_{n\in\mathbb Z}$ becomes a Riesz basis of
$\mathrm{ran}(\mathcal{A})$.
Define the continuous linear operators
\begin{align*}
\mathcal M_{\lambda} &:L^2(\mathbb [0,T),\mathbb C)\rightarrow L^2([0,T),\mathbb C),\mathcal M_{\lambda}f(x):=e^{\lambda x}f(x),\\
\mathcal C &:L^2(\mathbb R_+,\mathbb C)\rightarrow L^2([0,T),\mathbb C),f\mapsto f\vert_{[0,T)}
\end{align*}
and $\mathcal P_{\mathcal A}:=\mathcal A\mathcal M_{\lambda}\mathcal C$. Observe, that $\mathcal M_{\lambda}\mathcal C\mathcal A$ is the identity operator on $L^2([0,T),\mathbb C)$ and hence $\mathcal P_{\mathcal A}^2=\mathcal P_{\mathcal A}$. Therefore, $\mathcal P_{\mathcal A}$ is a continuous linear projection with kernel $F$ and range $\mathrm{ran}(\mathcal{A})$.
Let $f\in L^2(\mathbb R_+,\mathbb C)$ be orthogonal to any element of the kernel of $\mathcal P_{\mathcal A}$. Then $f(x)=0$ Lebesgue-a.e.\ for any $x\geq T$. Hence, we have
\begin{align*}
|\mathcal P_{\mathcal A}f|_2^2 &= \sum_{n\in\mathbb N} \int_{nT}^{nT+T} (e^{-\lambda x} e^{\lambda(x-nT)})^2|f(x-nT)|^2dx \\
&= \sum_{n\in\mathbb N} e^{-2n\lambda T} \vert f\vert_2^2 \\
&= \frac{1}{1-e^{-2\lambda T}} \vert f\vert_2^2
\end{align*}
and it follows that $\Vert\mathcal P_{\mathcal A}\Vert_{\mathrm{op}} = \sqrt{\frac{1}{1-e^{-2\lambda T}}}$.
According to Lemma~\ref{lem:dual_A}, we have
\begin{align*}
e_n^*(x) &= (\mathcal{A}^{-1})^*b_n(x) \\
&= (1-e^{-2\lambda T})e^{-\lambda x}\left( e^{2\lambda \mathrm{cut}(x)}b_n(\mathrm{cut}(x)) \right) \\
&= \left(1-e^{-2\lambda T}\right)e^{2\lambda\mathrm{cut}(x)} e_n(x)\,,
\end{align*}
for any $n\in\mathbb Z$, $x\geq0$, as required.
\end{proof}
The statements collected in this section have been about the space $L^2(\mathbb R_+,\mathbb C)$ so far. However, we are actually interested in the space $H_\alpha$ which has a natural and simple isometry to $\mathbb C\times L^2(\mathbb R_+,\mathbb C)$. The next corollary will translate the $L^2(\mathbb R_+,\mathbb C)$-statements above to $H_\alpha$. Before stating it, we introduce a notation for later use:
Define
\begin{equation}
\label{def:Theta-op}
\mathcal Theta:H_\alpha\rightarrow \mathbb C\times L^2(\mathbb R_+,\mathbb C), f\mapsto (f(0), w_\alpha f')\,,
\end{equation}
where $w_\alpha(x):=e^{x\alpha/2}$ for $x\geq 0$. Then $\mathcal Theta$ is an isometry of Hilbert spaces. Its inverse is given by
\begin{equation}
\label{def:Theta-inv-op}
\mathcal Theta^{-1}:\mathbb C\times L^2(\mathbb R_+,\mathbb C)\rightarrow H_\alpha,(z,f)\mapsto z+\int_0^{(\cdot)} w_{\alpha}^{-1}(y)f(y)dy\,.
\end{equation}
We use these operators to prove:
\begin{cor}\label{k:Riesz basis on H}
The system $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ defined in \eqref{e:g-star-def}-\eqref{e:g-n-def} is a Riesz basis of a closed subspace $H_\alpha^{T}$ of $H_\alpha$.
Indeed, $H_\alpha^{T}$ is the space generated by
$\{g_*,\{g_n\}_{n\in\mathbb Z}\}$.
Moreover, there is a continuous linear projector $\Pi$ with range $H_\alpha^{T}$ and operator norm $\sqrt{\frac{1}{1-e^{-2\lambda T}}}$ such that
$$ \Pi h(x) = h(x), \quad h\in H_\alpha,x\in[0,T]. $$
Consequently, $\Pi\mathcal U_th(x) = \mathcal U_t\Pi h(x) = h(x+t)$ for any $t\in[0,T]$ and any $x\in[0,T-t]$.
The biorthogonal system $\{g_*^*,\{g_n^*\}_{n\in\mathbb Z}\}$ is given by
\begin{align*}
g_*^*(x) &= 1 \\
g_n^*(x) &= \int_0^x e^{-y\frac{\alpha}{2}} e_n^*(y) dy
\end{align*}
where $e_n^*$ is given in Proposition \ref{p:Riesz basis on L2} for any $n\in\mathbb Z$, $x\geq0$.
\end{cor}
\begin{proof}
Let $\{e_n\}_{n\in\mathbb Z}$ be the Riesz basis from Proposition \ref{p:Riesz basis on L2}, $V$ the linear vector space generated by $\{e_n\}_{n\in\mathbb Z}$
(which is in fact $\mathrm{ran}(\mathcal{A})$) and $\mathcal P_{\mathcal{A}}$ the projector from that proposition. Then $\{(1,0),\{(0,e_n)\}_{n\in\mathbb Z}\}$ is a Riesz basis of $\mathbb C\times V$. Furthermore, $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ is a Riesz basis of $\mathcal Theta^{-1}(\mathbb C\times V)$ because $g_*=\mathcal Theta^{-1}(1,0)$ and $g_n=\mathcal Theta^{-1}(0,e_n)$. Define $\Pi:=\mathcal Theta^{-1}(\mathrm{Id},\mathcal P_{\mathcal{A}})\mathcal Theta$. Then $\Pi$ is a linear projector with the same bound as $\mathcal P_{\mathcal A}$ where
$$
(\mathrm{Id},\mathcal P_{\mathcal A})(z,f):=(z,\mathcal P_{\mathcal A}f),\quad z\in\mathbb C,f\in L^2(\mathbb R_+,\mathbb C)\,.
$$
Let $h\in H_\alpha$. Observe that for any $x\in [0,T]$, $\mathrm{cut}(y)=y$ when $0\leq y\leq x$. We have from the definition of the various operators that
\begin{align*}
\Pi h(x)&=\mathcal Theta^{-1}(\mathrm{Id},\mathcal P_{\mathcal{A}})(h(0),\exp(\alpha\cdot/2)h') (x)\\
&=\mathcal Theta^{-1}\left((h(0),(\exp((\lambda+\alpha/2)\cdot)h')\vert_{[0,T)}(\mathrm{cut}(\cdot)\exp(-\lambda\cdot))\right)(x) \\
&=h(0)+\int_0^x e^{-(\lambda+\alpha/2)y} e^{(\lambda+\alpha/2)\mathrm{cut}(y)}
h'(\mathrm{cut}(y))\,dy \\
&=h(0)+\int_0^xh'(y)\,dy=h(x)\,.
\end{align*}
Hence, $\Pi h(x)=h(x)$ for any $x\in[0,T]$.
\end{proof}
We remark in passing that trivially $g_*^*=g_*$.
In the next proposition we compute the action of the shifting semigroup $\{\mathcal U_t\}_{t\geq0}$ on the Riesz basis of Corollary \ref{k:Riesz basis on H} and the dual semigroup on the biorthogonal system.
\begin{prop}\label{p:U and the riesz basis}
For the Riesz basis $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ in \eqref{e:g-star-def}-\eqref{e:g-n-def} and its biorthogonal system $\{g_*^*,\{g_n^*\}_{n\in\mathbb Z}\}$
derived in Corollary~\ref{k:Riesz basis on H}, it holds
\begin{enumerate}
\item $\mathcal U_t g_n = e^{\lambda_n t}g_n + g_n(t)g_*$ and
\item $\mathcal U^*_tg_n^* = e^{\overline{\lambda_n} t}g_n^*$,
\end{enumerate}
for any $n\in\mathbb Z$.
\end{prop}
\begin{proof}
Claim (1) follows from a straightforward computation.
For claim (2), we compute
\begin{align*}
\mathcal U_t^*g_n^* &= g_*\langle\mathcal U_t^*g_n^*,g_*\rangle_{\alpha} + \sum_{k\in\mathbb Z} g^*_k\langle\mathcal U_t^*g_n^*,g_k\rangle_{\alpha} \\
&= g_*\langle g_n^*,\mathcal U_tg_*\rangle_{\alpha} + \sum_{k\in\mathbb Z} g^*_k\langleg_n^*,\mathcal U_tg_k\rangle_{\alpha} \\
&= e^{\overline{\lambda_n}t}g_n^*
\end{align*}
for any $n\in\mathbb Z$, $t\geq 0$. Thus, the Proposition follows.
\end{proof}
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We remark in passing that trivially $g_*^*=g_*$.
In the next proposition we compute the action of the shifting semigroup $\{\mathcal U_t\}_{t\geq0}$ on the Riesz basis of Corollary \ref{k:Riesz basis on H} and the dual semigroup on the biorthogonal system.
\begin{prop}\label{p:U and the riesz basis}
For the Riesz basis $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ in \eqref{e:g-star-def}-\eqref{e:g-n-def} and its biorthogonal system $\{g_*^*,\{g_n^*\}_{n\in\mathbb Z}\}$
derived in Corollary~\ref{k:Riesz basis on H}, it holds
\begin{enumerate}
\item $\mathcal U_t g_n = e^{\lambda_n t}g_n + g_n(t)g_*$ and
\item $\mathcal U^*_tg_n^* = e^{\overline{\lambda_n} t}g_n^*$,
\end{enumerate}
for any $n\in\mathbb Z$.
\end{prop}
\begin{proof}
Claim (1) follows from a straightforward computation.
For claim (2), we compute
\begin{align*}
\mathcal U_t^*g_n^* &= g_*\langle\mathcal U_t^*g_n^*,g_*\rangle_{\alpha} + \sum_{k\in\mathbb Z} g^*_k\langle\mathcal U_t^*g_n^*,g_k\rangle_{\alpha} \\
&= g_*\langle g_n^*,\mathcal U_tg_*\rangle_{\alpha} + \sum_{k\in\mathbb Z} g^*_k\langleg_n^*,\mathcal U_tg_k\rangle_{\alpha} \\
&= e^{\overline{\lambda_n}t}g_n^*
\end{align*}
for any $n\in\mathbb Z$, $t\geq 0$. Thus, the Proposition follows.
\end{proof}
A certain Lie commutator plays a crucial role in comparing projected solutions to the SPDE~\eqref{e:HJMM-equation} with solutions to
the approximation. In the next proposition, we derive the essential results for convergence which will be used in the next
Section to analyse approximations of the SPDE~\eqref{e:HJMM-equation}.
\begin{prop}\label{l:commutator of U and projectors}
Let $k\in\mathbb N$, $t\geq0$, $H^{T}_\alpha$ be the closed subspace of $H_\alpha$ generated by the Riesz basis $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$
defined in \eqref{e:g-star-def}-\eqref{e:g-n-def} with biorthogonal system
$\{g_*^*,\{g_n^*\}_{n\in\mathbb Z}\}$ given in Corollary~\ref{k:Riesz basis on H}.
Define the projection
$$ \Pi_k:H^{T}_\alpha\rightarrow \text{span}\{g_*,g_{-k},\dots,g_k\},h\mapsto h(0)g_* + \sum_{n=-k}^k g_n\langleh,g_n^*\rangle_{\alpha}, $$
$c_{k,t}:=\sum_{\vert n\vert >k} g_n(t)g_n^*$, and the operator
$$ \mathcal C_{k,t}:H^{T}_\alpha\rightarrow \text{span}\{g_*\}, h\mapsto \langleh,c_{k,t}\rangle_{\alpha}g_*.$$
Then, $\|\Pi_k\|_{\text{op}}$ is bounded uniformly in $k$, $\Pi_kh\rightarrow h$, $\sup_{s\in[0,t]} \Vert\mathcal C_{k,s}h\Vert_\alpha\rightarrow 0$ for $k\rightarrow\infty$ and any $h\in H_\alpha^{T}$, and $[\Pi_k,\mathcal U_t] =\mathcal C_{k,t}$. Here, $[\Pi_k,\mathcal U_t]$ denotes the Lie commutator of $\Pi_k$ and $\mathcal U_t$, that is $[\Pi_k,\mathcal U_t]=\Pi_k\mathcal U_t-\mathcal U_t\Pi_k$.
Moreover, let $X$ be a stochastic process with values in $H_\alpha^{T}$
such that $X(t)=Y(t)+M(t)$ for some square integrable process $Y$ of finite variation and a square integrable martingale $M$. Then,
$$ \lim_{k\rightarrow\infty}\int_0^t\mathcal C_{k,t-s}dX(s)=0\,,$$
where the convergence is in $L^2(\Omega,H_\alpha)$.\footnote{$L^2(\Omega,H_\alpha)$ denotes the space of $H_{\alpha}$-valued random variables $Z$ with $\mathbb{E}[\Vert Z\Vert_\alpha^2]<\infty$.}
\end{prop}
\begin{proof}
Let $h\in H^T_\alpha$. Since $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ is a Riesz basis of $H^T_\alpha$ we have
$$
h = g_*\langleh,g_*\rangle_{\alpha} + \sum_{n\in\mathbb Z} g_n\langleh,g_n^*\rangle_{\alpha}\,,
$$
and hence we get $\Pi_kh\rightarrow h$ for $k\rightarrow \infty$.
We prove that the operator norm of $\Pi_k$ is uniformly bounded in $k\in\mathbb{N}$.
Recall from Corollary~\ref{k:Riesz basis on H} and \eqref{def:Theta-inv-op}
$g_n=\mathcal Theta^{-1}(0,\mathcal{A}b_n), n\in\mathbb{Z}$ and $g_*=\mathcal Theta^{-1}(1,0)$, where
$\mathcal{A}$ is defined in \eqref{def-A-operator} and $\{b_n\}_{n\in\mathbb{Z}}$ is
an orthonormal basis of $L^2([0,T],\mathbb{C})$. Without loss of generality, we assume
$h(0)=0$ for $h\in H_{\alpha}^T$, and find that
$$
\Pi_kh=\sum_{n=-k}^kg_n\langle h,g_n^*\mathrm{ran}gle_{\alpha}=\sum_{n=-k}^k
\mathcal{T}b_n(\mathcal{T}^{-1}h,b_n)_2=\mathcal {T}\sum_ {n=-k}^kb_n(\mathcal{T}^{-1}h,b_n)_2\,.
$$
Here, $\mathcal{T}f:=\mathcal Theta^{-1}(0,\mathcal{A}f)\in H_{\alpha}$ for
$f\in L^2([0,T],\mathbb{C})$, which is a bounded linear operator. Hence,
since $\sum_{n=-k}^kb_n(\mathcal{T}^{-1}h,b_n)_2$ is the projection of
$\mathcal{T}^{-1}h\in L^2([0,T],\mathbb{C})$ down to its first $2k+1$ coordinates,
$$
\|\Pi_kh\|_{\alpha}\leq|\mathcal{T}\|_{\text{op}}\left|\sum_{n=-k}^kb_n(\mathcal{T}^{-1}h,b_n)_2\right|_2\leq\|\mathcal{T}\|_{\text{op}}|\mathcal{T}^{-1}h|_2
$$
But since $\mathcal{T}^{-1}$ also is a bounded operator, it follows that
$\|\Pi_k\|_{\text{op}}\leq\|\mathcal{T}\|_{\text{op}}\|\mathcal{T}^{-1}\|_{\text{op}}$.
Benth and Kr\"uhner~\cite[Lemma 3.2]{benth.kruehner.14} yields that convergence in $H_\alpha$ implies local uniform convergence. Thus, as we know $h-\Pi_kh \rightarrow 0$,
it holds
$$ \sup_{s\in[0,t]} \vert h(s)-\Pi_{k}h(s)\vert \rightarrow 0\,, $$
for $k\rightarrow \infty$. Hence, we find
$$\sup_{s\in[0,t]}\left\vert\sum_{\vert n\vert>k}g_n(s)\langleh,g^*_n\rangle_{\alpha}\right\vert = \sup_{s\in[0,t]} \vert h(s)-\Pi_{k}h(s)\vert \rightarrow 0\,, $$
for $k\rightarrow\infty$. Therefore, $\sup_{s\in[0,t]}\Vert \mathcal C_{k,s}h\Vert_\alpha\rightarrow 0$ for $k\rightarrow\infty$.
Let $n\in\mathbb Z$. Then, by Proposition~\ref{p:U and the riesz basis}
\begin{align*}
[\Pi_k,\mathcal U_t]g_n &= \Pi_k(e^{\lambda_n t}g_n + g_n(t)g_*) - 1_{\{\vert n\vert\leq k\}}\mathcal U_tg_n \\
&= 1_{\{\vert n\vert\leq k\}}e^{\lambda_n t}g_n + g_n(t)g_* - 1_{\{\vert n\vert\leq k\}}(e^{\lambda_n t}g_n + g_n(t)g_*) \\
&= 1_{\{\vert n\vert>k\}}g_n(t)g_* \\
&= \mathcal C_{k,t}g_n
\end{align*}
for any $t\geq0$. Moreover,
\begin{align*}
[\Pi_k,\mathcal U_t]g_* = \Pi_kg_* - \mathcal U_tg_* = 0 = \mathcal C_{k,t}g_*.
\end{align*}
Let $\langle\langleM,M\rangle\rangle(t) = \int_0^t Q_sd\langleM,M\rangle(s)$ be the quadratic variation processes of the martingale $M$ given in
Peszat and
Zabczyk~\cite[Theorem 8.2]{peszat.zabczyk.07}\footnote{In Peszat and Zabczyk~\cite{peszat.zabczyk.07}, $\langle\langle\cdot,\cdot\rangle\rangle$ is called the operator angle bracket process, while $\langle\cdot,\cdot\rangle$ is the angle bracket process.}. Then, Peszat and Zabczyk~\cite[Theorem 8.7(ii)]{peszat.zabczyk.07} yields
\begin{align*}
\mathbb{E}\left(\Vert \int_0^t \mathcal C_{k,t-s}dM(s) \Vert_\alpha^2\right) &= \mathbb{E} \int_0^t \mathrm{Tr}(\mathcal C_{k,t-s}Q_s\mathcal C^*_{k,t-s})d\langleM,M\rangle(s)\,.
\end{align*}
Recall that for $h\in H_{\alpha}^T$, we find $\mathcal C_{k,t}h=\langleh,c_{k,t}\rangle_{\alpha}g_*$. Thus,
$$
\langleh,\mathcal C_{k,t}^*g_*\rangle_{\alpha}=\langle\mathcal C_{k,t}h,g_*\rangle_{\alpha}=\langleh,c_{k,t}\rangle_{\alpha}\,,
$$
which gives that $\mathcal C_{k,t}^*g_*=c_{k,t}$. For $g\in H_\alpha^T$ orthogonal to $g_*$ we have
$$ \langleh,\mathcal C_{k,t}^*g\rangle_\alpha =\langle\mathcal C_{k,t}h,g\rangle_{\alpha}=\langleh,c_{k,t}\rangle_{\alpha}\langleg_*,g\rangle_{\alpha}= 0 $$
for any $h\in H_\alpha^T$ and hence $\mathcal C_{k,t}^*g=0$. We get
\begin{align*}
\mathrm{Tr}(\mathcal C_{k,t-s}Q_s\mathcal C^*_{k,t-s}) &= \langle\mathcal C_{k,t-s}Q_s\mathcal C_{k,t-s}^*g_*,g_*\rangle_{\alpha} \\
&=\langleQ_sc_{k,t-s},c_{k,t-s}\rangle_{\alpha} \\
&\leq \Vert c_{k,t-s}\Vert_\alpha^2 \mathrm{Tr}(Q_s)\,.
\end{align*}
Hence,
\begin{align*}
\mathbb{E}\left(\left\Vert \int_0^t \mathcal C_{k,t-s}dM(s) \right\Vert_\alpha^2\right) &= \mathbb{E} \int_0^t \mathrm{Tr}(\mathcal C_{k,t-s}Q_s\mathcal C^*_{k,t-s})d\langleM,M\rangle(s) \\
&\leq \sup_{s\in[0,t]}\Vert c_{k,s}\Vert_\alpha^2 \mathbb{E}\left( \int_0^t\mathrm{Tr}(Q_s)d\langleM,M\rangle(s) \right) \\
&= \sup_{s\in[0,t]}\Vert c_{k,s}\Vert_\alpha^2 \mathbb{E}\left(\Vert M(t)-M(0)\Vert_\alpha^2\right) \\
&\rightarrow 0
\end{align*}
for $k\rightarrow\infty$. Similarily, we get
$$ \left\Vert \int_0^t \mathcal C_{k,t-s} dY(s) \right\Vert_\alpha^2 \leq \sup_{s\in[0,t]} \Vert c_{k,s}\Vert_\alpha^2\left(\int_0^t \Vert dY\Vert_\alpha(s)\right)^2 \rightarrow 0$$
as $k\rightarrow 0$, where $\Vert dY\Vert_\alpha$ denotes the total variation measure associated with $dY$ (see Dinculeanu~\cite[Definition \S 2.1]{dinculeanu.00}). The claim follows.
\end{proof}
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The projection operator $\Pi_k$ plays an important role in the arbitrage-free approximation of the forward term structure. For notational convenience,
we denote
\begin{equation}
\label{def-H-k-space}
H_{\alpha}^{T,k}:=\text{span}\{g_*,g_{-k},\dots,g_k\}\,,
\end{equation}
for any $k\in\mathbb N$. From the above considerations, we have that $\Pi_k$ projects the space $H_{\alpha}^T$ down to $H_{\alpha}^{T,k}$.
Our next aim is to identify the convergence speed of approximations in $H_{\alpha}^{T,k}$ of certain smooth
elements $f\in H_{\alpha}^T$, that is, how close is $\Pi_kf$ to $f$ in terms
of number of Riesz basis functions. We show a couple of technical results first.
\begin{cor}\label{k:distance to ONB}
Let $f\in H_\alpha^T$. Then, we have
\small{$$ \frac{e^{-2\lambda T}}{1-e^{-2\lambda T}} \left(|f(0)|^2 + \sum_{n\in\mathbb Z} \vert\langlef,g_n^*\rangle_{\alpha}\vert^2\right)\leq \Vert f\Vert_\alpha^2 \leq \frac{1}{1-e^{-2\lambda T}} \left(|f(0)|^2 + \sum_{n\in\mathbb Z} \vert\langlef,g_n^*\rangle_{\alpha}\vert^2\right)\,.$$}
\end{cor}
\begin{proof}
Corollary~\ref{k:Riesz basis on H} states that $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ is a Riesz basis of $H_\alpha^T$. Moreover, it is given by $g_*=\mathcal Theta^{-1}(1,0)$, $g_n=\mathcal Theta^{-1}(0,e_n)$ for any $n\in\mathbb Z$ where $\mathcal Theta$ is the isometry given in \eqref{def:Theta-inv-op} and $\{e_n\}_{n\in\mathbb Z}$ is the Riesz basis given in Proposition \ref{p:Riesz basis on L2}. Moreover, Lemma~\ref{l:stetige Einbettung} yields that $e_n=\mathcal Ab_n$ for any $n\in\mathbb Z$ where $\{b_n\}_{n\in\mathbb Z}$ is an orthonormal basis of
$L^2([0,T],\mathbb C)$ and $\Vert \mathcal A\Vert_{\mathrm{op}}^2\leq \frac{1}{1-e^{-2\lambda T}}$. Thus, we can construct a Hilbert space with orthonormal basis $\{b_*,\{b_n\}_{n\in\mathbb Z}\}$ and a bounded linear operator
$\mathcal B$ with $\Vert \mathcal B\Vert_{\mathrm{op}}^2\leq \frac{1}{1-e^{-2\lambda T}}$, such that $g_*=\mathcal Bb_*$, $g_n=\mathcal Bb_n$.
Thus, we have
\begin{align*}
\Vert f\Vert_\alpha^2 &= \Vert g_*\langlef,g_*\rangle_{\alpha} + \sum_{n\in\mathbb Z}g_n\langlef,g_n^*\rangle_{\alpha} \Vert_\alpha^2 \\
&= \Vert \mathcal Bb_*\langlef,g_*\rangle_{\alpha} + \sum_{n\in\mathbb Z}\mathcal Bb_n\langlef,g_n^*\rangle_{\alpha} \Vert_\alpha^2 \\
&\leq \frac{1}{1-e^{-2\lambda T}} \left(|\langlef,g_*\rangle_{\alpha}|^2 + \sum_{n\in\mathbb Z}|\langlef,g_n^*\rangle_{\alpha}|^2 \right) \\
\end{align*}
where $\{g_*,\{g^*_n\}_{n\in\mathbb Z}\}$ denotes the biorthogonal system to $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ given in Corollary~\ref{k:Riesz basis on H}.
The lower inequality simply uses the lower inequality of Lemma~\ref{l:stetige Einbettung} instead.
\end{proof}
The next technical result connects the inner product of elements in $H_{\alpha}^T$ with the
biorthogonal basis functions to a simple
Fourier-like integral on $[0,T]$:
\begin{cor}
\label{cor:alpha-2-inner-prod}
Assume $f\in H_{\alpha}^T$. Then, for any $n\in\mathbb Z$,
$$
\langlef,g_n^*\rangle_{\alpha}=(1-e^{-2\lambda T})^{-1}T^{-1/2}\int_0^Tf'(x)\exp\left((-\frac{2\pi i}{T}n-\lambda+\frac{\alpha}{2})x\right)\,dx
$$
\end{cor}
\begin{proof}
First, recall that $g_n^*=\mathcal Theta^*(0,e_n)$ for $n\in\mathbb Z$, where $\mathcal Theta$ is defined
in the \eqref{def:Theta-inv-op}. Thus,
\begin{align*}
\langlef,g_n^*\rangle&=\langlef,\mathcal Theta^*(0,e_n)\rangle_{\alpha} \\
&=(\mathcal Theta f,(0,e_n))_{\mathbb C\times L^2(\mathbb R_+)} \\
&=((f(0),e^{\alpha\cdot/2}f'),(0,e_n))_{\mathbb C\times L^2(\mathbb R_+)} \\
&=(e^{\alpha\cdot/2}f',e_n)_2\,.
\end{align*}
Note that $\exp(\alpha\cdot/2)f'$ and $e_n=\mathcal A b_n$ are elements of $\mathrm{ran}(\mathcal A)$.
If $h\in\mathrm{ran}(\mathcal{A})$, then there exists a $\hat{h}\in L^2([0,T],\mathbb C)$ such that
$h=\mathcal A \hat{h}$, or, $h(x)=\exp(-\lambda x)\hat{h}(\mathrm{cut}(x))$. Observe that
for $x\in[0,T]$, $\hat{h}(x)=\exp(\lambda x)h(x)$. Then, if $g\in\mathrm{ran}(\mathcal{A})$, we find
\begin{align*}
(h,g)_2&=\int_0^{\infty}h(x)\overline{g(x)}\,dx \\
&=\int_0^{\infty}e^{-2\lambda x}\hat{h}(\mathrm{cut}(x))\overline{\hat{g}(\mathrm{cut}(x)}\,dx \\
&=\sum_{n=0}^{\infty}e^{-2\lambda n T}\int_{nT}^{(n+1)T}e^{-2\lambda(x-nT)}\hat{h}(\mathrm{cut}(x))
\overline{\hat{g}(\mathrm{cut}(x))}\,dx \\
&=\sum_{n=0}^{\infty}e^{-2\lambda n T}\int_0^Te^{-2\lambda x}\hat{h}(x)\overline{\hat{g}(x)}\,dx \\
&=(1-e^{-2\lambda T})^{-1}\int_0^Th(x)\overline{g(x)}\,dx\,.
\end{align*}
Thus,
\begin{align*}
\langlef,g_n^*\rangle&=(1-e^{-2\lambda T})^{-1}\int_0^Te^{\alpha x/2}f'(x)\overline{e_n(x)}\,dx \\
&=(1-e^{-2\lambda T})^{-1}T^{-1/2}\int_0^Tf'(x)\exp\left((-\frac{2\pi i}{T}n-\lambda+\frac{\alpha}{2})x\right)\,dx
\end{align*}
Hence, the result follows.
\end{proof}
With this results at hand, we can prove a convergence rate of order $1/k$ for sufficiently smooth functions in $H_{\alpha}^T$.
\begin{prop}\label{p:approximation speed}
Assume $f\in H_\alpha^{T}$ is such that $f\vert_{[0,T]}$ is twice continuously differentiable.
Then, we have
$$
\left\Vert f - \Pi_kf\right\Vert^2_\alpha \leq\frac{C_1}{k}\,,
$$
for any $k\in\mathbb N$, where
$$
C_1=\frac{T\left\vert f'(T)e^{T(-\lambda+\alpha/2)}-f'(0)\right\vert^2+(\int_0^T \vert f''(x)\vert e^{x(-\lambda+\alpha/2)}\,dx)^2}{\pi^2(1-e^{-2\lambda T})^3}\,,
$$
and we recall the projection operator $\Pi_k$ from Proposition~\ref{l:commutator of U and projectors}.
\end{prop}
\begin{proof}
Corollary~\ref{k:distance to ONB} yields
\begin{align*}
\Vert f - \Pi_kf\Vert^2_\alpha &= \Vert \sum_{\vert n\vert>k} g_n\langlef,g_n^*\rangle_{\alpha}\Vert_\alpha^2
\leq C \sum_{\vert n\vert>k} \vert \langlef,g_n^*\rangle_{\alpha}\vert^2\,
\end{align*}
where $C:=(1-e^{-2\lambda T})^{-1}$. Define $h_n(x):=\exp(\xi_nx)$, $x\geq 0$, where we
denote $\xi_n=-\frac{2\pi i}{T}n-\lambda+\frac{\alpha}{2}$. Then, by Corollary~\ref{cor:alpha-2-inner-prod}
and integration-by-parts we find
\begin{align*}
\vert \langlef,g_n^*\rangle_{\alpha}\vert^2 &=C^2T^{-1}\left\vert\int_0^Tf'(x)h_n(x)dx\right\vert^2 \\
&= C^2T^{-1}\frac{1}{\vert\xi_n\vert^2}\left\vert f'(T)h_n(T)-f'(0)h_n(0)-\int_0^T f''(x)h_n(x)\,dx \right\vert^2 \\
& \leq \frac{2C^2}{T}\frac{1}{\vert\xi_n\vert^2}A_f\,,
\end{align*}
for any $n\in\mathbb Z\backslash\{0\}$, where the constant $A_f$ is
$$
A_f:=\left\vert f'(T)e^{T(-\lambda+\alpha/2)}-f'(0)\right\vert^2+(\int_0^T \vert f''(x)e^{x(\lambda-\alpha/2)}\, dx)^2\,.
$$
Moreover, we have
\begin{align*}
\sum_{\vert n\vert>k}\frac{1}{\vert\xi_n\vert^2}= 2 \sum_{n>k}\frac{1}{\vert\xi_n\vert^2}\leq \frac{T^2}{2\pi^2 k}.
\end{align*}
Putting the estimates together, we get
$$
\Vert f - \Pi_kf\Vert^2_\alpha \leq A_f\frac{C^3T}{\pi^2k}\,,
$$
as claimed.
\end{proof}
We can find a similar convergence rate for $c_{k,t}$, a result which becomes useful later:
\begin{lem}
\label{lemma:approximation speed_ckt}
Let $c_{k,t}$ be given as in Proposition~\ref{l:commutator of U and projectors}. Then,
$$
\Vert c_{k,t}\Vert_\alpha^2 \leq \frac{C_2}{k}\,,
$$
for any $k\in\mathbb N$, where $C_2=T/\pi^2(1-\exp(-2\lambda T))$.
\end{lem}
\begin{proof}
We appeal to Corollary~\ref{k:distance to ONB}, using $\{g_n^*\}_{n\in\mathbb Z}$ as the Riesz basis
with biorthogonal system $\{g_n\}_{n\in\mathbb Z}$, to find
\begin{align*}
\|c_{k,t}\|_{\alpha}^2&=\|\sum_{|n|>k}g_n(t)g_n^*\|_{\alpha}^2 \\
&\leq C\sum_{|n|>k}|g_n(t)|^2 \\
&=\frac{C}{T}\sum_{|n|>k}\frac{1}{\vert\lambda_n\vert^2}\left\vert e^{\lambda_n t}-1\right\vert^2 \\
&\leq\frac{2C}{T}(1+e^{-(2\lambda+\alpha)t})\sum_{|n|>k}\frac{1}{\vert\lambda_n\vert^2} \\
&\leq \frac{CT}{\pi^2}\frac1k\,,
\end{align*}
for $C=(1-\exp(-2\lambda T))^{-1}$. Hence, the result follows.
\end{proof}
With these results we are now in the position to study arbitrage-free approximations of the forward dynamics in \eqref{e:HJMM-equation}.
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\section{Arbitrage free approximation of forward term structure models}\label{s:approximation}
In this section we find an arbitrage-free approximation of a forward term structure model -- stated in the Heath-Jarrow-Morton-type setup -- which lives in a
finite dimensional state space. We furthermore derive the convergence speed of the approximation, and extend the results to account for
forward contracts delivering the underlying commodity over a period which is the case for electricity and gas.
Consider the SPDE \eqref{e:HJMM-equation} with a mild solution $f\in H_{\alpha}$ given by \eqref{e:HJMM-equation-mild}. We recall
from \eqref{e:g-star-def}-\eqref{e:g-n-def} and Corollary~\ref{k:Riesz basis on H} the
Riesz basis $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$ on the space $H_{\alpha}^T$ with the biorthogonal system
$\{g_*,\{g_n^*\}_{n\in\mathbb Z}\}$. Furthermore,
$\Pi$ is the projection of $H_{\alpha}$ on $H_{\alpha}^T$, while from \eqref{def-H-k-space} and
Proposition~\ref{p:U and the riesz basis} we have the projection $\Pi_k$ of $H_{\alpha}^T$
on $H_{\alpha}^{T,k}$ and the operator $\mathcal C_{k,t}$ for $k\in\mathbb N$, $t\geq 0$.
Let us define the continuous linear operator $\mathcal Lambda_k:H_{\alpha}\rightarrow H_{\alpha}^{T,k}$ by
\begin{equation}
\mathcal Lambda_k= \Pi_k\Pi
\end{equation}
for any $k\in\mathbb N$. The following theorem is one of the main results of the paper:
\begin{thm}\label{t:main statement}
For $k\in\mathbb N$, let $f_k$ be the mild solution of the SPDE
\begin{equation}
\label{e:approx-f-k}
df_k(t) = \partial_x f_k(t) dt + \mathcal Lambda_k\beta(t) dt + \mathcal Lambda_k\Psi(t) dL(t),\quad t\geq 0, f_k(0) = \mathcal Lambda_kf_0\,.
\end{equation}
Then, we have
\begin{enumerate}
\item $\mathbb{E}\left[\sup_{x\in[0,T-t]}\vert f_k(t,x) - f(t,x)\vert^2\right] \rightarrow 0$ for $k\rightarrow\infty$ and any $t\in [0,T]$,
\item $f_k$ takes values in the finite dimensional space $H_\alpha^{T,k}$, moreover, $f_k$ is a strong solution to the SPDE \eqref{e:approx-f-k}, i.e.\ $f_k\in\mathrm{dom}(\partial_x)$, {$t\mapsto \partial_xf_k(t)$} is $P$-a.s.\ Bochner-integrable and
$$ f_k(t) = f_k(0) + \int_0^t (\partial_xf_k(s)+\mathcal Lambda_k\beta(s))ds + \int_0^t \mathcal Lambda_k\Psi(s) dL(s)\,, $$
\item and,
\begin{align*}
f_k(t) &= S_k(t) + \sum_{n=-k}^k \left(e^{\lambda_n t} \langlef_k(0),g_n^*\rangle_{\alpha} + \int_0^t e^{\lambda_n (t-s)}dX_n(s)\right)g_n \,,
\end{align*}
where $S_k(t)= \delta_0(f_k(t))$ and $X_n(t) := \int_0^t \langle\Pi\beta(s)ds+\Pi\Psi(s)dL(s),g_n^*\rangle_{\alpha}$ for any $n\in\mathbb Z$, $t\geq0$.
\end{enumerate}
\end{thm}
\begin{proof}
(1) Define
$$f_\Pi(t) := \mathcal U_t\Pi f_0 + \int_0^t \mathcal U_{t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s))),\quad t\geq0.$$
Since $f_k$ is a mild solution, we have
\begin{align*}
f_k(t) &= \mathcal U_t\Pi_k\Pi f_0 + \int_0^t \mathcal U_{t-s}\Pi_k(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \\
&= \Pi_k\mathcal U_t\Pi f_0 + \int_0^t \Pi_k\mathcal U_{t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \\
&\hspace{0.3cm} - \mathcal C_{k,t}\Pi f_0 - \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \\
&= \Pi_k\left(\mathcal U_t\Pi f_0 + \int_0^t \mathcal U_{t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)))\right) \\
&\hspace{0.3cm} - \mathcal C_{k,t}\Pi f_0 - \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \\
&= \Pi_k(f_\Pi(t)) - \mathcal C_{k,t}\Pi f_0 - \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s))
\end{align*}
for any $t\geq0$. From Benth and Kr\"uhner~\cite[Lemma 3.2]{benth.kruehner.14} the sup-norm is dominated by the $H_{\alpha}$-norm. Thus, there is a constant $c>0$ such that
\begin{align*}
\mathbb{E}\left[\sup_{x\in[0,T-t]}\vert \Pi_k(f_\Pi(t,x)) - f_\Pi(t,x)\vert^2\right] & \leq c\mathbb{E}\left[ \Vert (\Pi_k-\mathcal I)f_\Pi(t) \Vert^2_\alpha \right]
\end{align*}
for any $t\geq 0$ where $\mathcal I$ denotes the identity operator on $H_\alpha$. The dominated convergence theorem yields that the right-hand side converges to $0$ for $k\rightarrow \infty$. Clearly, we have
$$ \sup_{x\in[0,T-t]}\vert \mathcal C_{k,t}f_\Pi(0,x) \vert \leq c\Vert \mathcal C_{k,t}f_\Pi(0) \Vert_\alpha \rightarrow 0\,, $$
for $k\rightarrow \infty$. Proposition~\ref{l:commutator of U and projectors} states that
$$ \mathbb{E}\left\Vert \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \right\Vert_\alpha^2 \rightarrow 0\,,$$
for $k\rightarrow 0$. Hence, we have
$$ \mathbb{E}\left(\sup_{x\in[0,T-t]}\vert f_k(t,x)-f_\Pi(t,x) \vert^2\right) \rightarrow 0\,, $$
for $k\rightarrow \infty$ and any $t\in [0,T]$. Since $f_\Pi(t,x) = f(t,x)$ for any $t\in[0,T]$, $x\in [0,T-t]$ the first part follows.
(2) Note first that $\partial_x g_n(x)=\exp(\lambda_n x)/\sqrt{T}=\lambda_ng_n(x)+g_*(x)/\sqrt{T}$, and hence
$\partial_x g_n\in H_{\alpha}^{T,k}$ whenever $|n|\leq k$. Thus, $H_{\alpha}^{T,k}$ is invariant under the
generator $\partial_x$, and its restriction to $H_{\alpha}^{T,k}$ is continuous and bounded. We find that $f_k$ takes values only in $H_\alpha^{T,k}$ because
\begin{align*}
f_k(t) &= \Pi_k\left(\mathcal U_t\Pi f_0 + \int_0^t \mathcal U_{t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)))\right) \\
&\hspace{0.3cm} - \mathcal C_{k,t}\Pi f_0 - \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds+\Pi\Psi(s)dL(s))\,,
\end{align*}
where all summands are clearly in $H_\alpha^{T,k}$.
(3) As $f_k(t)\in H_{\alpha}^{T,k}$, we have the representation
$$
f_k(t)=\langlef_k(t),g_*^*\rangle_{\alpha}g_*+\sum_{n=-k}^k \langlef_k(t),g_n^*\rangle_{\alpha}g_n\,.
$$
Since $g_*^*=1$, we find that $\langlef_k(t),g_*^*\rangle_{\alpha}=f_k(t,0)=\delta_0(f_k(t))$. Thus, from the mild solution of
\eqref{e:approx-f-k} we find, using Proposition \ref{p:U and the riesz basis}
\begin{align*}
f_k(t)&=S_k(t)+\sum_{n=-k}^k \left\langle\mathcal{U}_t f_k(0)+\int_0^t\mathcal U_{t-s}(\mathcal Lambda_k\beta(s)ds + \mathcal Lambda_k\Psi(s)dL(s)),g_n^*\right\rangle_{\alpha}g_n \\
&=S_k(t)+\sum_{n=-k}^{k}\langlef_k(0),\mathcal{U}_t^*g_n^*\rangle_{\alpha}g_n \\
&\qquad\qquad+\sum_{n=-k}^{k}\int_0^t\langle\mathcal Lambda_k\beta(s)ds + \mathcal Lambda_k\Psi(s)dL(s),\mathcal U_{t-s}^*g_n^*\rangle_{\alpha}g_n \\
&=S_k(t)+\sum_{n=-k}^{k}e^{\lambda_n t}\langlef_k(0),g_n^*\rangle_{\alpha}g_n \\
&\qquad\qquad+\sum_{n=-k}^k\int_0^t e^{\lambda_n(t-s)}\langle\mathcal Lambda_k\beta(s)ds + \mathcal Lambda_k\Psi(s)dL(s),g_n^*\rangle_{\alpha}g_n \,.
\end{align*}
Observe that for any $f\in H_{\alpha}$,
$$
\mathcal Lambda_k f=\Pi_k(\Pi f)=(\Pi f)(0)g_*+\sum_{m=-k}^k\langle\Pi f,g_m^*\rangle_{\alpha}g_m\,,
$$
and since $\{g_*,\{g_n\}_{n\in\mathbb Z}\}$, $\{g_*^*,\{g^*_n\}_{n\in\mathbb Z}\}$ are biorthogonal systems
\begin{align*}
\langle\mathcal Lambda_k f,g_n^*\rangle_{\alpha}&=(\Pi f)(0)\langleg_*,g_n^*\rangle_{\alpha}+\sum_{m=-k}^k\langle\Pi f,g_m^*\rangle_{\alpha}\langleg_m,g_n^*\rangle_{\alpha}
=\langle\Pi f,g_n^*\rangle_{\alpha}1_{\{\vert n\vert\leq k\}}\,.
\end{align*}
Hence, the claim follows.
\end{proof}
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Another view on Theorem \ref{t:main statement} is that all processes in the $k$-th approximation
of $f$ can be expressed in terms of the factor processes $X_*,X_{-k},\dots,X_k$, as stated below.
\begin{cor}\label{k:state variables}
Under the assumptions and notations of Theorem \ref{t:main statement}, we have for $k\in\mathbb N$,
\begin{align*}
f_k(t,x) &= S_k(t) + \sum_{n=-k}^k U_n(t) g_n(x)\,,
\end{align*}
for any $0\leq t<\infty$ and $x\geq 0$. Here,
\begin{align*}
S_k(t)&= S_k(0) + X_*(t) + \sum_{n=-k}^k \left( g_n(t)U_n(0) +\int_0^tg_n(t-s)dX_n(s)\right) \,,
\end{align*}
with,
\begin{align*}
X_n(t) &:= \left\langle \int_0^t(\Pi\beta(s)ds+\Pi\Psi(s)dL(s)),g_n^*\right\rangle_{\alpha}, \\
X_*(t) &:= \left\langle \int_0^t(\Pi\beta(s)ds+\Pi\Psi(s)dL(s)),g_*\right\rangle_{\alpha}, \\
U_n(t) &:= e^{\lambda_n t} \langlef_k(0),g_n^*\rangle + \int_0^t e^{\lambda_n (t-s)}dX_n(s) \end{align*}
for $n\in\{-k,\dots, k\}$.
\end{cor}
\begin{proof}
The first equation is a restatement of (3) in Theorem \ref{t:main statement}. Proposition \ref{p:U and the riesz basis} yields
$$ \langle\mathcal U_t h,g_*\rangle_{\alpha} = \langleh,g_*\rangle_{\alpha} + \sum_{n=-k}^k g_n(t)\langleh,g_n^*\rangle_{\alpha} $$
for any $h\in H_\alpha^{T,k}$ with $h=\langleh,g_*\rangle_{\alpha}g_*+\sum_{n=-k}^k \langleh,g_n^*\rangle_{\alpha}g_n$. Thus, since $g_*=1$ and $g_n(0)=0$ we have
\begin{align*}
S_k(t) &= f_k(t,0) \\
&=\langle f_k(t),g_*\rangle_{\alpha} \\
&= \langle \mathcal U_{t}f_k(0),g_*\rangle_{\alpha} + \int_0^t \langle\mathcal U_{t-s}(\mathcal Lambda_k\beta(s)\,ds+\mathcal Lambda_k\Psi(s)\,dL(s)),g_*\rangle_{\alpha} \\
&= \langlef_k(0),g_*\rangle_{\alpha}+\sum_{n=-k}^kg_n(t)\langlef_k(0),g_n^*\rangle_{\alpha} \\
&\qquad\qquad+\int_0^t\langle\mathcal Lambda_k\beta(s)\,ds+\mathcal Lambda_k\Psi(s)\,dL(s),g_*\rangle_{\alpha} \\
&\qquad\qquad+\sum_{n=-k}^k\int_0^tg_n(t-s)\langle\mathcal Lambda_k\beta(s)+\mathcal Lambda_k\Psi(s)\,dL(s),g_n^*\rangle_{\alpha}\,.
\end{align*}
As in the proof of Theorem~\ref{t:main statement}, we have $\langle\mathcal Lambda_k f,g_n^*\rangle_{\alpha}=\langle\Pi f,g_n^*\rangle_{\alpha}$ for any $f\in H_{\alpha}$.
Similarly, $\langle\mathcal Lambda_k f,g_*\rangle_{\alpha}=\langle\Pi f,g_*\rangle_{\alpha}$ for $n\in\mathbb Z$ with $\vert n\vert\leq k$. The result follows.
\end{proof}
The processes $S_k, U_{-k},\dots, U_k$ in Corollary~\ref{k:state variables} capture at any time $t$ the whole state of the market in the approximation model.
I.e., the spot price and the forward curve are simple functions of these state variables. As we will see in Corollary \ref{k:Forward prices} below, the forward prices
of contracts with delivery periods can be expressed in these state variables as well. Note that if we assume $\langle\Pi\beta,g_n^*\rangle$, $\langle\Pi\Psi,g_n^*\rangle$ are deterministic and constant, then $(X_{-k},\dots,X_k)$ is a $2k+1$-dimensional L\'evy process and $U_{-k},\dots,U_k$ are Ornstein-Uhlenbeck processes. This corresponds to the spot price model suggested in Benth, Kallsen and Meyer-Brandis~\cite{benth.et.al.05}.
From the proof of Corollary~\ref{k:state variables} we find that $S_k(0)=\langlef_k(0),g_*\rangle_{\alpha}$. But then
$$
S_k(0)=\langle\mathcal Lambda_k f_0,g_*\rangle_{\alpha}=\langle\Pi f_0,g_*\rangle_{\alpha}=(\Pi f_0)(0)=f_0(0)\,.
$$
Obviously, $f_0(0)$ is equal to today's spot price, so we obtain that the starting point of the process $S_k(t)$ in the approximation is today's spot price.
Furthermore, since we have $f_k(t,0)=S_k(t)$ because $g_n(0)=0$ for all $n\in\mathbb Z$, $S_k(t)$ is the approximative spot price dynamics associated
with $f_k(t)$. For $U_n(0)$, $n\in\mathbb Z$ invoking Corollary~\ref{cor:alpha-2-inner-prod} shows that
\begin{align*}
U_n(0)&=\langle\Pi f_0,g_n^*\rangle_{\alpha} \\
&=\frac1{\sqrt{T}(1-e^{-2\lambda T})}\int_0^{T}(\Pi f_0)'(y)
\exp((-\lambda+\alpha/2)x)\exp\left(\frac{2\pi i}{T}nx\right)\,dy\,.
\end{align*}
This is the Fourier transform of the initial forward curve $f_0$ (or, rather its derivative
scaled by an exponential function). In any case, both $S_k(0)$ and $U_n(0)$ are
given by (functionals of) the initial forward curve $f_0$.
Next, we would like to identify the convergence speed of our approximation, that is, the rate for the convergence in part (1) of Theorem \ref{t:main statement}.
\begin{prop}\label{p:convergence speed}
Assume that $x\mapsto f(t,x)$ is twice continuously differentiable and let
$f_k$ be the mild solution of the SPDE
$$ df_k(t) = \partial_x f_k(t) dt + \mathcal Lambda_k\beta(t) dt + \mathcal Lambda_k\Psi(t) dL(t),\quad t\geq 0, f_k(0) = \mathcal Lambda_kf_0\, .$$
Then, we have
$$
\mathbb{E}\left[\sup_{x\in[0,T-t]}\vert f_k(t,x) - f(t,x)\vert^2\right] \leq \frac{A(t)}{k}\,,
$$
for any $k>1$, where
\begin{align*}
A(t)&:=\frac{3T(1+\alpha^{-1})}{(1-e^{-2\lambda T})}\left\{\|\Pi f_0\|_{\alpha}^2+\int_0^T \mathbb{E}[\mathrm{Tr}(\Psi(s)Q\Psi^*(s))]ds+\left(\int_0^T \mathbb{E}\left[\Vert \beta(s)\Vert_\alpha\right]\,ds\right)^2 \right\} \\
&\qquad+\frac{3(1+\alpha^{-1})}{\pi^2(1-e^{-2\lambda T})^3}\left\{T\mathbb{E}\left[|\partial_xf_{\Pi}(t,T)
e^{T(-\lambda+\alpha/2)}-\partial_xf_{\Pi}(t,0)|^2\right] \right. \\
&\qquad\qquad\left.+\left(\int_0^T\mathbb{E}\left[|\partial^2_xf_{\Pi}(t,x)|\right]e^{x(-\lambda+\alpha/2)}\,dx\right)^2\right\}\,.
\end{align*}
\end{prop}
\begin{proof}
In the proof of Theorem~\ref{t:main statement} we have shown that
\begin{align*}
f_k(t) &= \Pi_k(f_\Pi(t)) - \mathcal C_{k,t}\Pi f_0 - \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s))\,,
\end{align*}
where $f_\Pi(t) := \mathcal U_t\Pi f_0 + \int_0^t \mathcal U_{t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)))$ for any $t\geq 0$. By Proposition~\ref{p:approximation speed} we have
$$
\left\Vert f_\Pi(t) - \Pi_k(f_\Pi(t))\right\Vert^2_\alpha \leq\frac{C_1(t)}{k}
$$
where $C_1(t)$ is a random variable defined by
$$
C_1(t)=\frac{T|\partial_xf_{\Pi}(t,T)e^{T(-\lambda+\alpha/2)}-\partial_xf_{\Pi}(t,0)|^2+(\int_0^T|\partial^2_xf_{\Pi}(t,x)|e^{x(-\lambda+\alpha/2)}\,dx)^2}{\pi^2(1-e^{-2\lambda T})^3}\,.
$$
Remark that from the proof of Theorem~\ref{t:main statement} we find for any $h\in H_{\alpha}^T$
\begin{align*}
\|\mathcal C_{k,t}h\|_{\alpha}^2&=\|\langleh,c_{k,t}\rangle_{\alpha}g_*\|_{\alpha}^2=|\langleh,c_{k,t}\rangle_{\alpha}|^2\leq \|h\|_{\alpha}^2\|c_{k,t}\|_{\alpha}^2\,,
\end{align*}
and therefore, from Lemma~\ref{lemma:approximation speed_ckt}
$$
\|\mathcal C_{k,t}h\|_{\alpha}^2\leq\|h\|_{\alpha}^2\frac{C_2}{k}\,,
$$
for the constant $C_2=T/\pi^2(1-e^{-2\lambda T})$.
Then, we have
\begin{align*}
\Vert f_k(t) - f_\Pi(t) \Vert_\alpha^2 &\leq 3\Vert \Pi_k(f_\Pi(t))-f_\Pi(t)\Vert_\alpha^2 + 3\Vert \mathcal C_{k,t}\Pi f_0\Vert_\alpha^2 \\
&\qquad\qquad+
3\Vert \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \Vert_\alpha^2 \\
&\leq \frac{3C_1(t)}{k} + \frac{3C_2}{k}\Vert \Pi f_0\Vert_\alpha^2 \\
&\qquad\qquad+ 3\Vert \int_0^t \mathcal C_{k,t-s}(\Pi\beta(s)ds + \Pi\Psi(s)dL(s)) \Vert_\alpha^2.
\end{align*}
By Lemma~3.2 in Benth and Kr\"uhner~\cite{benth.kruehner.14}, the supremum norm is bounded by the $H_{\alpha}$-norm with a constant
$c=\sqrt{1+\alpha^{-1}}$. Hence, taking expectations, yield
\begin{align*}
\mathbb{E}&\left[\sup_{x\in[0,T-t]}\vert f_k(t,x) - f(t,x) \vert^2 \right] \\
&\qquad \leq c^2\mathbb{E}\left[ \Vert f_k(t) - f_\Pi(t) \Vert_\alpha^2 \right] \\
&\qquad\leq \frac{3c^2}{k}\left(\mathbb{E}\left[C_1(t)\right]+ C_2\Vert \Pi f_0\Vert_\alpha^2\right) \\
&\qquad\qquad + \frac{3c^2}{k}C_2\left(\int_0^T \mathbb{E}[\mathrm{Tr}(\Psi(s)Q\Psi^*(s))]ds + \left(\int_0^T \mathbb{E}\left[\Vert \beta(s)\Vert_\alpha\right] ds\right)^2 \right) \,.
\end{align*}
The result follows.
\end{proof}
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In electricity and gas markets forward contracts deliver over a future period rather than at a fixed time. The holder of the forward contract receives a uniform stream of electricity
or gas over an agreed time period $[T_1,T_2]$. The forward prices of delivery period contracts can be derived from a "fixed-delivery time" forward curve model (see Benth et al.~\cite{BSBK-book})
by
\begin{equation}
\label{e:el-forward-def}
F(t,T_1,T_2) := \frac{1}{T_2-T_1}\int_{T_1}^{T_2} f(t,s-t)\,, ds
\end{equation}
where $f$ is given by the SPDE \eqref{e:HJMM-equation}. The following Corollary adapts Theorem~\ref{t:main statement} to the case of forward
contracts with delivery period.
\begin{cor}\label{k:Forward prices}
Assume the conditions of Theorem~\ref{t:main statement} and define
\begin{align*}
F_k(t,T_1,T_2) &:= \frac{1}{T_2-T_1}\int_{T_1}^{T_2} f_k(t,s-t) ds
\end{align*}
for any $0\leq t \leq T_1\leq T_2\leq T$. Then, we have
$$ F_k(t,T_1,T_2) \rightarrow F(t,T_1,T_2)$$
for $k\rightarrow\infty$ in $L^2(\Omega)$ where $F$ is given in \eqref{e:el-forward-def}. Furthermore,
$$
F_k(t,T_1,T_2) = S_k(t) + \sum_{n=-k}^k G_n(t,T_1,T_2) \left(e^{\lambda_n t} \langleg_n^*,f_k(0)\rangle_{\alpha}
+ \int_0^t e^{\lambda_n (t-s)}dX_n(s)\right)\,,
$$
for any $t\leq T_1\leq T_2\leq T$ where $S_k(t) = \delta_0(f_k(t))$,
$$G_n(t,T_1,T_2) = \frac{\exp(\lambda_n(T_2-t))-\exp(\lambda_n(T_1-t))-\lambda_n(T_2-T_1)}{\lambda_n^2\sqrt{T}(T_2-T_1)}$$
and $X_n(t):=\int_0^t\langle\Pi\beta(s)ds + \Pi\Psi(s)dL(s),g_n^*\rangle_{\alpha}$.
\end{cor}
\begin{proof}
Theorem \ref{t:main statement} yields uniform $L^2$ convergence of the integrands appearing in $F_k$ to the integrand appearing in $F$ and hence the convergence follows. The representation of $F_k$ follows immediately from part (3) of Theorem~\ref{t:main statement}.
\end{proof}
We remark in passing that temperature derivatives market (see e.g. Benth and \v{S}altyt\.{e} Benth~\cite{BSB-weather}) trades in forwards with a "delivery period" as well.
In this market, the forward is cash-settled against an index of the daily average temperature measured in a city over a given period.
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\section{Refinement to Markovian forward price models}\label{s:markovian}
In this Section we refine our analysis to Markovian forward price models, making the
additional assumption that the coefficients $\beta$ and $\Psi$ depend on the state of the forward curve.
More specifically, we assume that
\begin{align}
\beta(t) &= b(t,f(t)), \\
\Psi(t) &= \psi(t,f(t)),
\end{align}
where $b:\mathbb R_+\times H_\alpha\rightarrow H_\alpha$, $\psi:\mathbb R_+\times H_\alpha\rightarrow L(H_\alpha)$ are measurable Lipschitz-continuous functions of linear growth
in the sense
\begin{align}
\| b(t,f) - b(t,g)\|_{\alpha} &\leq C_b \| f-g\|_{\alpha}\,, \label{eq:lip-cond-b} \\
\Vert (\psi(t,f) - \psi(t,g))\mathcal Q^{1/2} \Vert_{\mathrm{HS}} &\leq C_\psi \| f-g\|_{\alpha}\,,
\label{eq:lip-cond-psi}
\end{align}
and
\begin{align}
\| b(t,f)\|_{\alpha} &\leq C_b(1+ \|f\|_{\alpha})\,, \label{eq:lingrowth-cond-b} \\
\Vert \psi(t,f)\mathcal Q^{1/2} \Vert_{\mathrm{HS}} &\leq C_\psi (1+ \|f\|_{\alpha})\,,
\label{eq:lingrowth-cond-psi}
\end{align}
for positive constants $C_b$, $C_\psi$. Under these conditions there exists a unique mild solution $f$ of
the semilinear SPDE
\begin{equation}
\label{eq:nonlinear_spde}
df(t) = (\partial_xf(t) + b(t,f(t))) dt + \psi(t,f(t-)) dL(t),\quad f(0) = f_0.
\end{equation}
We would like to note that semilinear SPDEs are treated in the book by Peszat and Zabczyk~\cite{peszat.zabczyk.07} and in Tappe~\cite{tappe.12}. Additionally, we assume that
\begin{align}
b(t,h) &= b(t,g), \label{eq:struct-cond-b}\\
\psi(t,h) &= \psi(t,g) \label{eq:struct-cond-psi}\,,
\end{align}
for any $h,g\in H_\alpha$ such that $h(x)=g(x)$ for any $x\in[0,T-t]$, i.e.\ the structure of the curve beyond our time horizon $T$ does not influence the
dynamics of the curve-valued process $f(t)$.
Before continuing our analysis of the arbitrage-free approximation in the Markovian case, we show a couple of useful lemmas. The first states a
version of Doob's $L^2$ inequality for Volterra-like Hilbert space-valued stochastic integrals with respect to the L\'evy process $L$, and is essentially collected from
Filipovi\'c, Tappe and Teichmann~\cite{FTT}.
\begin{lem}
\label{lem:doob}
Suppose that
$\Phi\in\mathcal L_L^2(H_{\alpha})$. Then,
\begin{align*}
\mathbb{E}\left[\sup_{s\in[0,t]} \Vert
\int_0^s\mathcal{U}_{s-r}\Phi(r)\,dL(r)
\Vert_{\alpha}^2\right] &\leq 4c_t^2 \int_0^t \mathbb{E}\left[\Vert\Phi(r)\mathcal Q^{1/2}\Vert_{\text{HS}}^2\right]\,dr\,,
\end{align*}
for $c_t>0$ being at most exponentially growing in $t$.
\end{lem}
\begin{proof}
Note first that due to Benth and Kr\"uhner~\cite[Lemma 3.5]{benth.kruehner.14} the $C_0$-semigroup $(\mathcal U_t)_{t\geq 0}$ is pseudo-contractive. Filipovi\'c, Tappe and Teichmann~\cite[Prop.~8.7]{FTT} state that there is a Hilbert space extension $H$
of $H_\alpha$ (i.e.\ $H$ is a Hilbert space and $H_\alpha$ is its
subspace and the norm of $H_\alpha$ equals the norm of $H$ restricted to
$H_\alpha$) and a $C_0$-group $(\mathcal V_t)_{t\in\mathbb R}$ on $H$ such that
$\mathcal V_t\vert_{H_\alpha} = \mathcal U_t$ for $t\geq 0$. Then, we have
\begin{align*}
\sup_{s\in[0,t]} \Vert\int_0^s\mathcal{U}_{s-r}\Phi(r)\,dL(r)\Vert_{\alpha}&\leq \sup_{s\in[0,t]}\Vert \mathcal V_{s-t}\Vert_{\mathrm{op}}
\Vert\int_0^s \mathcal U_{t-r}\Phi(r)\,dL(r) \Vert_{\alpha} \\
&\leq \sup_{s\in[0,t]}\Vert \mathcal V_{s}\Vert_{\mathrm{op}}
\sup_{s\in[0,t]}\Vert\int_0^s \mathcal U_{t-r}\Phi(r)\,dL(r) \Vert_{\alpha}\,.
\end{align*}
Thus, by Doob's maximal inequality, Thm.~2.2.7 in
Prevot and R\"ockner~\cite{PR}, we find
\begin{align*}
\mathbb{E}&\left[\sup_{s\in[0,t]} \Vert
\int_0^s\mathcal{U}_{s-r}\Phi(r)\,dL(r)
\Vert_{\alpha}^2\right] \\
&\qquad\qquad\leq \sup_{s\in[0,t]}\Vert \mathcal
V_{s}\Vert_{\mathrm{op}}^2\mathbb{E}\left[\sup_{s\in[0,t]} \Vert
\int_0^s\mathcal{U}_{t-r}\Phi(r)\,dL(r)\Vert_{\alpha}^2\right] \\
&\qquad\qquad\leq 4\sup_{s\in[0,t]}\Vert \mathcal
V_{s}\Vert_{\mathrm{op}}^2\mathbb{E}\left[\Vert\int_0^t\mathcal{U}_{t-r}\Phi(r)\,dL(r)\Vert_{\alpha}^2\right] \\
&\qquad\qquad=4\sup_{s\in[0,t]}\Vert \mathcal
V_{s}\Vert_{\mathrm{op}}^2\int_0^t\mathbb{E}\left[\|\mathcal U_{t-r}\Phi(r)\mathcal{Q}^{1/2}\Vert^2_{\text{HS}}\right]\,dr \\
&\qquad\qquad\leq 4\sup_{s\in[0,t]}\Vert \mathcal
V_{s}\Vert_{\mathrm{op}}^2\sup_{s\in[0,t]}\|\mathcal U_{s}\Vert_{\text{op}}^2\int_0^t\mathbb{E}\left[ \Vert\Phi(r)\mathcal{Q}^{1/2}\Vert^2_{\text{HS}}\right]\,dr
\end{align*}
This proves the Lemma by letting $c_t=\sup_{s\in[0,t]}\Vert \mathcal
V_{s}\Vert_{\mathrm{op}}\sup_{0\leq s\leq t}\Vert\mathcal{U}_{s}\Vert_{\text{op}}$
and recalling that any $C_0$-group is bounded in operator norm by
an exponentially increasing function in $t$. Hence, $c_t\leq c\exp(w t)$ for some
constants $c,w>0$.
\end{proof}
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We remark in passing that the above result holds for any pseudo-contractive semigroup
$\mathcal S_t$, $t\geq 0$.
The next lemma is a useful technical result on the distance between processes and the fixed point of an integral operator defined
via the mild solution of \eqref{eq:nonlinear_spde}. The lemma plays a crucial role in showing that certain arbitrage-free approximations of
\eqref{eq:nonlinear_spde} converge to the right limit.
\begin{lem}\label{l:fixpoint estimate}
For an $H_{\alpha}$-valued adapted and c\`adl\`ag stochastic process $h$, define
$$
V(h)(t) := \mathcal U_tf_0 + \int_0^t \mathcal U_{t-s}b(s,h(s))\,ds + \int_0^t \mathcal U_{t-s}\psi(s,h(s-))\,dL(s)\,,
$$
for any $t\geq 0$. Then, $V$ has a fixed point $\widehat{f}$ and it holds
$$
\mathbb{E}\left[\sup_{0\leq s\leq t}\| h(s)-\widehat{f}(s)\|^2_{\alpha} \right] \leq\frac{\pi^2}{6}\exp(4C_t)\mathbb{E}\left[\sup_{0\leq s\leq t}\| V(h)(s)-h(s)\|^2_{\alpha}\right] \,,
$$
for any $t\geq 0$ and any $H_{\alpha}$-valued adapted c\`adl\`ag stochastic processes $h$, with
$C_t$ being a positive constant depending on $t$.
\end{lem}
\begin{proof}
If $h$ is an adapted c\`adl\`ag $H_{\alpha}$-valued
stochastic process such that $\mathbb E[\int_0^t\|h(s)\|_{\alpha}^2\,ds]<\infty$, then from the linear growth assumption~\eqref{eq:lingrowth-cond-b}
on $b$ we find
\begin{align*}
\mathbb E[\int_0^t\|\mathcal U_{t-s}b(s,h(s))\|_{\alpha}\,ds] &\leq C_b e^{wt}(t+\mathbb E[\int_0^t\|h(s)\|_{\alpha}\,ds]) \\
&\leq C_b e^{wt}(t+\sqrt{t}\mathbb E[\int_0^t\|h(s)\|_{\alpha}^2\,ds]^{1/2})\\
&<\infty\,.
\end{align*}
Furthermore, from the linear growth condition~\eqref{eq:lingrowth-cond-psi}
on $\psi$
$$
\mathbb E[\int_0^t\|\mathcal{U}_{t-s}\psi(s,h(s))\|^2_{\alpha}\,ds]\leq 2C^2_{\psi}e^{2wt}\left(t+\mathbb E[\int_0^t\|h(s)\|^2_{\alpha}\,ds]\right)<\infty\,.
$$
Hence, $V(h)$ is well-defined, and it is an adapted c\`adl\`ag process. By a straightforward
estimation using again the linear growth of $b$ and $\psi$, we find similarly that
$$
\mathbb E[\int_0^t\|V(h)(s)\|_{\alpha}^2\,ds]\leq C_t\left(1+\mathbb E[\int_0^t\|h\|_{\alpha}^2\,ds]\right)<\infty\,,
$$
for some constant $C_t>0$
Therefore, $V$ maps into its own domain and, thus, can be iterated.
We note that by general theory, the SPDE
$$
df(t)=\partial_xf(t)\,dt+b(t,f(t))\,dt+\psi(t,f(t-))\,dL(t)
$$
has a unique mild solution $\widehat f$ which has a c\`adl\`ag modification, cf.\ Tappe~\cite[Theorem 4.5, Remark 4.6]{tappe.12}. By definition of mild solutions, we see that $\widehat f$
is a fix point for
$V$, i.e., $V(\widehat f)=\widehat f$.
Let $g,h$ be $H_{\alpha}$-valued adapted c\`adl\`ag stochastic processes and $t\geq 0$. Then, we have
\begin{align*}
\mathbb{E}&\left[\sup_{0\leq s\leq t}\|V(h)(s)-V(g)(s)\|^2_{\alpha}\right] \\
&\qquad\qquad\leq 2\mathbb{E}\left[\sup_{0\leq s\leq t}\|\int_0^s\mathcal U_{s-r}\left(b(r,h(r))-b(r,g(r))\right)\,dr\|_{\alpha}^2\right] \\
&\qquad\qquad\qquad+2\mathbb{E}\left[\sup_{0\leq s\leq t}\|\int_0^s\mathcal U_{s-r}\left(\psi(r,h(r-))-\psi(r,g(r-))\right)\,dL(r)\|^2_{\alpha}\right] \,.
\end{align*}
Consider the first term on the right hand side of the inequality. By the norm inequality for Bochner integrals and Lipschitz continuity of $b$ in
\eqref{eq:lip-cond-b}, we find
\begin{align*}
\mathbb{E}&\left[\sup_{0\leq s\leq t}\|\int_0^s\mathcal U_{s-r}\left(b(r,h(r))-b(r,g(r))\right)\,dr\|_{\alpha}^2\right] \\
&\qquad\qquad\leq\mathbb{E}\left[\sup_{0\leq s\leq t}\left(\int_0^s\|\mathcal U_{s-r}\|_{\text{op}}\|b(r,h(r))-b(r,g(r))\|_{\alpha}\,dr\right)^2\right] \\
&\qquad\qquad\leq t\mathbb{E}\left[\sup_{0\leq s\leq t}\int_0^s\|\mathcal U_{s-r}\|^2_{\text{op}}\|b(r,h(r))-b(r,g(r))\|^2_{\alpha}\,dr\right] \\
&\qquad\qquad\leq t^2\sup_{0\leq s\leq t}\|\mathcal{U}_s\|^2_{\text{op}}\mathbb{E}\left[\int_0^t\|b(r,h(r))-b(r,g(r))\|_{\alpha}^2\,dr\right] \\
&\qquad\qquad\leq t^2C_b^2\sup_{0\leq s\leq t}\|\mathcal U_s\|^2_{\text{op}}\int_0^t\mathbb{E}\left[\|h(r)-g(r)\|_{\alpha}^2\right]\,dr\,,
\end{align*}
where we have applied Cauchy-Schwartz' inequality. Recall that since
$\mathcal U_t$ is a pseudo-contractive semigroup, we find for some
$w>0$, it holds that $\sup_{0\leq s\leq t}\|\mathcal U_{s}\|_{\text{op}}^2\leq \exp(2w t)<\infty$.
For the second term, we find by appealing to Lemma~\ref{lem:doob} and the Lipschitz continuity in \eqref{eq:lip-cond-psi} of $\psi$,
\begin{align*}
\mathbb{E}&\left[\sup_{0\leq s\leq t}\|\int_0^s\mathcal U_{s-r}\left(\psi(r,h(r-))-\psi(r,g(r-))\right)\,dL(r)\|^2_{\alpha}\right] \\
&\qquad\qquad\leq 4c_t^2\int_0^t\mathbb{E}\left[\Vert(\psi(r,h(r))-\psi(r,g(r)))\mathcal Q^{1/2}\Vert^2_{\text{HS}}\right]\,dr \\
&\qquad\qquad\leq 4c_t^2C_{\psi}^2\int_0^t\mathbb{E}\left[\Vert h(r)-g(r)\Vert^2_{\alpha}\right]\,dr
\end{align*}
Here, the constant $c_t$ is from Lemma~\ref{lem:doob}. Denote by $C_t$ the constant
$$
C_t:=2C^2_b t^2 \sup_{s\in[0,t]}\Vert \mathcal U_s\Vert_{\mathrm{op}}+ 8c^2_tC_\psi^2t\,.
$$
Then, we have
\begin{align*}
\mathbb{E}&\left[\sup_{0\leq s\leq t}\| V^n(h)(s)-V^n(g)(s)\|^2_{\alpha}\right] \\
&\qquad\qquad\leq C_t\int_0^t \mathbb{E}\left[\|V^{n-1}(h)(s_1)-V^{n-1}(g)(s_1)\|^2_{\alpha}\right] \,ds_1\\
&\qquad\qquad\leq C_t^n \int_0^t\int_0^{s_1}\cdots\int_0^{s_{n-1}}\mathbb{E}\left[\|h(s_n)-g(s_n)\|_{\alpha}^2\right]ds_n\dots ds_1 \\
&\qquad\qquad\leq \frac{C_t^n}{n!}\mathbb{E}\left[\sup_{0\leq s\leq t}\|h(s)-g(s)\|^2_{\alpha} \right]\,,
\end{align*}
for any $n\in\mathbb N$. Denote by $L_{a}^2(\Omega,D([0,t],H_\alpha))$ the space of
$H_{\alpha}$-valued adapted c\`adl\`ag stochastic processes
$\{f(s)\}_{s\in[0,t]}$ for which $\mathbb{E}[\sup_{s\in[0,t]}\|f(s)\|_\alpha^2]<\infty$.
Equip this space with the norm $\|\cdot\|_t$ defined by
$$
\|f\|_t^2 := \mathbb{E}[\sup_{s\in[0,t]}\|f(s)\|_\alpha^2]
$$
for $f\in L_{a}^2(\Omega,D([0,t],H_\alpha))$. From the estimation above, we see
that $V$ operates on the normed space $L_{a}^2(\Omega,D([0,t],H_\alpha))$.
Moreover, $V^n$ is Lipschitz continuous with constant strictly less than $1$ for $n$ sufficiently large. Thus, by Banach's fixed point theorem there is at most one
fixed point for $V$. Hence, $\hat f$ is the unique fix point for $V$. Furthermore, we have
\begin{align*}
\mathbb{E}\left[\sup_{0\leq s\leq t}\|V^n(h)(s)-h(s)\|^2_{\alpha}\right]^{1/2}
& \leq \sum_{k=0}^{n-1} \mathbb{E}\left[\sup_{0\leq s\leq t}\|V^{k+1}(h)(s)-V^k(h)(s)\|^2_{\alpha}\right]^{1/2} \\
& \leq \mathbb{E}\left[\sup_{0\leq s\leq t}\| V(h)(s)-h(s)\|^2_{\alpha} \right]^{1/2}\sum_{k=0}^{n-1} \left(\frac{C_t^k}{k!}\right)^{1/2}\,.
\end{align*}
From Cauchy-Schwartz' inequality and we have that
\begin{align*}
\sum_{k=0}^{n-1} \left(\frac{C_t^k}{k!}\right)^{1/2}&=\sum_{k=0}^{n-1}(k+1)^{-1}\left(\frac{(k+1)^2C_t^k}{k!}\right)^{1/2} \\
&\leq\left(\sum_{k=0}^{n-1}\frac1{(k+1)^2}\right)^{1/2}\left(\sum_{k=0}^{n-1}\frac{(k+1)^2C_t^k}{k!}\right)^{1/2} \\
&\leq \frac{\pi}{\sqrt{6}}\left(\sum_{k=0}^{n-1}\frac{4^kC_t^k}{k!}\right)^{1/2} \\
&\leq \frac{\pi}{\sqrt{6}}\exp(2C_t)\,,
\end{align*}
where we have used the elementary inequality $k+1\leq 2^k$, $k\in\mathbb N$.
\end{proof}
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|
Let us define the Lipschitz continuous functions $b_\Pi:=\Pi\circ b$ and $\psi_\Pi:=\Pi\circ\psi$. Then,
Tappe~\cite[Theorem 4.5]{tappe.12} yields a mild solution $f_\Pi$ for the SPDE
\begin{equation}
df_\Pi(t) = (\partial_xf_\Pi(t) + b_\Pi(t,f_\Pi(t)))\, dt + \psi_\Pi(t,f_\Pi(t-))\,dL(t),\quad f_\Pi(0) = \Pi f_0\,.
\end{equation}
Furthermore, it will be convenient to use the notations
\begin{align}
b_k(t,h) := \mathcal Lambda_k(b(t,h)), \\
\psi_k(t,h) := \mathcal Lambda_k(\psi(t,h))
\end{align}
for any $h\in H_\alpha$, $t\geq0$.
In the proof of Theorem~\ref{t:main statement} we compared the solution $f$ to the projected solution $\Pi f$ which are essentially the same due to properties of $\Pi$. Then we compared $\Pi f$ to $f_\Pi$ which again had been essentially the same. Finally, we compared $\Pi_kf_\Pi$ to solutions of the projected SPDE where the difference was given by a certain Lie-commutator.
However, in the Markovian setting we want to change the dependencies of the coefficients as well, which complicates the proof of the approximation result.
\begin{thm}
\label{thm:main-markovian}
Denote by $\widehat f_k$ be the mild solution of the SPDE
$$
d\widehat f_k(t) = (\partial_x\widehat f_k(t) + b_k(t,\widehat f_k(t)))\,dt + \psi_k(t,\widehat f_k(t-))\,dL(t),\quad \widehat f_k(0) = \mathcal Lambda_k f_0, t\geq0\,.
$$
Then, $\widehat f_k\in H_{\alpha}^{T,k}$ is a strong solution, and we have
$$ \mathbb{E}\left[\sup_{t\in[0,T],x\in[0,T-t]} \vert \hat f_k(t,x)-f(t,x) \vert^2 \right] \rightarrow 0 $$
for $k\rightarrow \infty$.
\end{thm}
\begin{proof}
First we note that a unique mild solution $\widehat{f}_k$ of the SPDE exists due to Tappe~\cite[Theorem 4.5]{tappe.12}.
Define
$$
V_k(h)(t) := \mathcal U_t f_k(0) + \int_0^t \mathcal U_{t-s} (b_k(s,h(s))\,ds + \psi_k(s,h(s-))\,dL(s))\,,
$$
for any $k\in\mathbb N$, $t\geq 0$ and any adapted c\`adl\`ag stochastic process $h$ in $H_{\alpha}$.
Let $f_k$ be
defined as
\begin{align*}
f_k(t): &= \mathcal U_t f_k(0) + \int_0^t \mathcal U_{t-s} (b_k(s,f(s))\,ds + \psi_k(s,f(s))\,dL(s) \\
&= \mathcal U_t f_k(0) + \int_0^t \mathcal U_{t-s} (b_k(s,f_\Pi(s))\,ds + \psi_k(s,f_\Pi(s-))\,dL(s) \\
&= V_k(f_\Pi)(t)\,,
\end{align*}
for $f_k(0)=\mathcal Lambda_kf(0)$. Moreover, $\widehat f_k(t) = V_k(\widehat f_k)(t)$. By Lemma~\ref{l:fixpoint estimate}, it holds
\begin{align*}
\mathbb{E}\left[\sup_{0\leq s\leq t}\| f_\Pi(t) - \hat f_k(t) \|_\alpha^2\right]\leq\frac{\pi^2}{6}\exp(4C_t) \mathbb{E}\left[ \sup_{0\leq s\leq t}\| f_k(s)-f_\Pi(s)\|^ 2_\alpha\right]\,,
\end{align*}
for any $k\in \mathbb N$, $t\geq 0$ and $C_t$ given in the lemma (recall from Section~\ref{s:the model} that the operator norm of
the shift semigroup $\mathcal{U}_t$ is uniformly bounded by the constant $C_{\mathcal{U}}$).
By the definition of $f_k$ and $f_{\Pi}$ we find
\begin{align*}
\|f_k(s)-f_{\Pi}(s)\|_{\alpha}^2&\leq2\|\int_0^s\mathcal U_{s-r}(b_k(r,f_{\Pi}(r))-b_{\Pi}(r,f_{\Pi}(r)))\,dr\|_{\alpha}^2 \\
&\qquad+2\|\int_0^s\mathcal U_{s-r}(\psi_k(r,f_{\Pi}(r-))-\psi_{\Pi}(r,f_{\Pi}(r-)))\,dL(r)\|_{\alpha}^2\,.
\end{align*}
Consider the first term on the right-hand side of the inequality. By the norm inequality for Bochner integrals, Cauchy-Schwartz' inequality and
boundedness of the operator norm of
$\mathcal U_t$ we find (for $s\leq t$)
\begin{align*}
\|\int_0^s\mathcal U_{s-r}&(b_k(r,f_{\Pi}(r))-b_{\Pi}(r,f_{\Pi}(r)))\,dr\|_{\alpha}^2 \\
&\qquad\qquad\leq\left(\int_0^s\|\mathcal U_{s-r}(b_k(r,f_{\Pi}(r))-b_{\Pi}(r,f_{\Pi}(r)))\|_{\alpha}\,dr\right)^2 \\
&\qquad\qquad\leq t\int_0^t\|\mathcal U_{s-r}(b_k(r,f_{\Pi}(r))-b_{\Pi}(r,f_{\Pi}(r)))\|^2_{\alpha}\,dr \\
&\qquad\qquad\leq tC^2_{\mathcal U}\int_0^t\|b_k(r,f_{\Pi}(r))-b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\,dr \\
&\qquad\qquad\leq tC_{\mathcal U}^2\int_0^t\|(\Pi_k-\mathcal I)b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\,dr
\end{align*}
Here, $\mathcal I$ denotes the identity operator on $H_\alpha^T$. Hence, using
Lemma~\ref{lem:doob} and the fact that $\{\mathcal U\}_{t\geq 0}$ is pseudo-contractive,
\begin{align*}
\mathbb{E}&\left[ \sup_{0\leq s\leq t}\Vert f_k(s)-f_\Pi(s)\Vert^ 2_\alpha\right] \\
&\qquad\leq 2 tC_{\mathcal U}^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\right]\,dr \\
&\qquad\qquad + 2\mathbb{E}\left[\sup_{0\leq s\leq t}\Vert\int_0^s \mathcal U_{s-r}(\psi_k(r,f_\Pi(r-))-\psi_\Pi(r,f_\Pi(r-)))\,dL(r)\|_{\alpha}^2\right] \\
&\qquad\leq 2 tC_{\mathcal U}^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\right]\,dr \\
&\qquad\qquad + 8c_t^2\int_0^t\mathbb{E}\left[\|(\psi_k(r,f_\Pi(r))-\psi_\Pi(r,f_\Pi(r)))\mathcal Q^{1/2}\|_{\text{HS}}^2\right]\,dr \\
&\qquad\leq 2 tC_{\mathcal U}^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\right]\,dr \\
&\qquad\qquad + 8c_t^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)\psi_\Pi(r,f_\Pi(r))\mathcal Q^{1/2}\|_{\text{HS}}^2\right]\,dr \,.
\end{align*}
Denote by
\begin{align*}
K_t(k):&=2 tC_{\mathcal U}^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\right]\,dr \\
&\qquad+8c_t^2\int_0^t\mathbb{E}\left[\|(\Pi_k-\mathcal I)\psi_\Pi(r,f_\Pi(r))\mathcal Q^{1/2}\|_{\text{HS}}^2\right]\,dr \,,
\end{align*}
for $k\in\mathbb N$. By standard norm inequalities, we have
\begin{align*}
K_t(k):&=4 tC_{\mathcal U}^2(1+\|\Pi_k\|^2_{\text{op}})\int_0^t\mathbb{E}\left[\|b_{\Pi}(r,f_{\Pi}(r))\|_{\alpha}^2\right]\,dr \\
&\qquad+16c_t^2(1+\|\Pi_k\|^2_{\text{op}})\int_0^t\mathbb{E}\left[\|\psi_\Pi(r,f_\Pi(r))\|_{\text{op}}^2\right]\,dr \,,
\end{align*}
which is seen to be bounded uniformly in $k\in\mathbb N$ from
Proposition~\ref{l:commutator of U and projectors}.
Hence, we have $ K_{t}(k) \rightarrow 0$ for $k\rightarrow \infty$ and any $t\geq 0$ by the dominated convergence theorem because $(\Pi_k-\mathcal I)h\rightarrow 0$ for $k\rightarrow \infty$ and any $h\in H_\alpha^T$.
Thus, we find
$$
\mathbb{E}\left[\sup_{0\leq t\leq T}\Vert f_k(t) - \hat f_k(t) \Vert_\alpha^2\right] \rightarrow 0\,,
$$
for $k\rightarrow \infty$. Finally, $f_\Pi(t,x) = f(t,x)$ for any $t\in[0,T]$, $x\in[0,T-t]$. Moreover,
from Lemma~3.2 in Benth and Kr\"uhner~\cite{benth.kruehner.14} the sup-norm is dominated by the
$H_{\alpha}$-norm, and therefore we have
\begin{align*}
\mathbb{E}\left[\sup_{t\in[0,T],x\in[T-t]}\vert \hat f_k(t,x)- f(t,x)\vert^2\right] &\leq c \mathbb{E}\left[\sup_{0\leq t\leq T}\Vert \hat f_k(t)-f_\Pi(t) \Vert_\alpha^2\right]\rightarrow0\,,
\end{align*}
for $k\rightarrow \infty$.
The Proposition follows.
\end{proof}
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The philosophy in Thm.~\ref{thm:main-markovian} is to take $f(t)$ as the actual forward curve dynamics, and study
finite dimensional approximations $\widehat f_k(t)$ of it. By construction, $\widehat f_k$ solves a HJMM dynamics which yields
that the approximating forward curves become arbitrage-free. From the main theorem, the approximations
$\widehat f_k(t)$ converge uniformly to $f(t)$ for $x\in[0,T-t]$. As time $t$ progresses, the times to maturity $x\geq 0$
for which we obtain convergence shrink. The reason is that information of $f$ is transported
to the left in the dynamics of the SPDE. We recall that the approximation of $f$ is constructed by first
localizing $f$ to $x\in[0,T]$ for a fixed time horizon $T$ by the projection operator $\Pi$ down
to $H_{\alpha}^T$, and next creating finite-dimensional approximations of this.
Alternatively, we may use $f_{\Pi}(t)$ as our forward price model. Then, the finite dimensional
approximation $f_k(t)$ will converge uniformly over all $x\in[0,T]$. In practice, there will be a time
horizon for the futures market for which we have no information. For example, in liberalized power markets
like NordPool and EEX, there are no futures contracts traded with settlement beyond 6 years. Hence,
it is a delicate task to model the dynamics of the futures price curve beyond this horizon. The alternative is
then clearly to restrict the modelling perspective to the dynamics with the maturities confined in
$x\in[0,T]$.
Indeed, in such a context the
structural conditions \eqref{eq:struct-cond-b} and \eqref{eq:struct-cond-psi} will be trivially satisfied as
we restrict our model parameters in any case to the behaviour on $x\in[0,T]$.
We end our paper with a short discussion on a possible numerical implementation of $\widehat f_k(t)$, the
finite-dimensional approximation of $f(t)$. Since $\widehat f_k(t)\in H_{\alpha}^{T,k}$, we can express it as
$$
\widehat f_k(t)=\widehat f_{k,*}(t)+\sum_{n=-k}^kg_n\widehat f_{k,n}(t)\,,
$$
where $\widehat f_{k,*}(t)=\widehat f_k(t,0)g_*$ and $\widehat f_{k,n}(t)=\langle\widehat f_k(t),g_n^*\rangle_{\alpha}$ are $\mathbb C$-valued functions. For any $h\in H_{\alpha}^{T,k}$ it follows that $b_k(t,h)\in H_{\alpha}^{T,k}$. Define for $n=-k,\ldots,k$ the functions
\begin{align*}
\overline{b}_{k,n}&:\mathbb R_+\times \mathbb C^{2k+2}\rightarrow \mathbb C\,; \qquad (t,x_*,x_{-k},\ldots,x_k) \mapsto \left\langleb_k(t,x_*g_*+\sum_{j=-k}^kx_jg_j),g_n^*\right\rangle_{\alpha}\,, \\
\overline{b}_{k,*}&:\mathbb R_+\times \mathbb C^{2k+2}\rightarrow \mathbb C\,; \qquad (t,x_*,x_{-k},\ldots,x_k) \mapsto \left\langleb_*(t,x_*g_*+\sum_{j=-k}^kx_jg_j),g_n^*\right\rangle_{\alpha}\,.
\end{align*}
Furthermore, $\psi_k(t,h)\in L_{\text{HS}}(H_{\alpha},H_{\alpha}^{T,k})$. Thus, for any
$g\in H_{\alpha}$ we have that $\psi_k(t,h)(g)\in H_{\alpha}^{T,k}$. We define the mappings
\begin{align*}
\overline{\psi}_{k,n}&:\mathbb R_+\times\mathbb C^{2k+2}\rightarrow H^*_{\alpha};
(t,x_*,x_{-k},\ldots,x_k)\mapsto \left\langle\psi_k(t,x_*g_*+\sum_{j=-k}^kx_jg_j)(\cdot),g_n^*\right\rangle_{\alpha} \\
\overline{\psi}_{k,*}&:\mathbb R_+\times\mathbb C^{2k+2}\rightarrow H^*_{\alpha};
(t,x_*,x_{-k},\ldots,x_k)\mapsto \left\langle\psi_*(t,x_*g_*+\sum_{j=-k}^kx_jg_j)(\cdot),g_n^*\right\rangle_{\alpha}
\end{align*}
for $n=-k,\ldots,k$. Now, since $\partial_x g_*=0$ and $\partial_xg_n=\lambda_ng_n+g_*/\sqrt{T}$, we find
from the SPDE of $\widehat{f}_k$ the following $2k+2$ system of stochastic differential equations (after comparing terms with respect to the Riesz basis functions),
\begin{align*}
d\widehat{f}_{k,*}(t)&=\left(\frac1{\sqrt{T}}\sum_{n=-k}^k\widehat f_{k,n}(t)+
\overline{b}_{k,*}(t,\widehat{f}_{k,*}(t),\widehat{f}_{k,-k}(t),\ldots,\widehat{f}_{k,k}(t))\right)\,dt \\
&\qquad\qquad+
d\overline{\psi}_{k,*}(t,\widehat{f}_{k,*}(t-),\widehat{f}_{k,-k}(t-),\ldots,\widehat{f}_{k,k}(t-))(L(t)) \\
d\widehat{f}_{k,-k}(t)&=\left(\lambda_{-k}\widehat f_{k,-k}(t)+
\overline{b}_{k,-k}(t,\widehat{f}_{k,*}(t),\widehat{f}_{k,-k}(t),\ldots,\widehat{f}_{k,k}(t))\right)\,dt \\
&\qquad\qquad+
d\overline{\psi}_{k,-k}(t,\widehat{f}_{k,*}(t-),\widehat{f}_{k,-k}(t-),\ldots,\widehat{f}_{k,k}(t-))(L(t)) \\
\cdot & \cdots \\
\cdot & \cdots \\
d\widehat{f}_{k,k}(t)&=\left(\lambda_{k}\widehat f_{k,k}(t)+
\overline{b}_{k,k}(t,\widehat{f}_{k,*}(t),\widehat{f}_{k,-k}(t),\ldots,\widehat{f}_{k,k}(t))\right)\,dt \\
&\qquad\qquad+
d\overline{\psi}_{k,k}(t,\widehat{f}_{k,*}(t-),\widehat{f}_{k,-k}(t-),\ldots,\widehat{f}_{k,k}(t-))(L(t))
\end{align*}
In a compact matrix notation, defining $\mathbf{x}(t)=(x_1(t),x_2(t),\ldots,x_{2k+2}(t))'$ and
$$
A=\left[\begin{array}{ccccc} \frac1{\sqrt{T}} & \frac1{\sqrt{T}} & \frac1{\sqrt{T}} &\cdots & \frac1{\sqrt{T}} \\
0 & \lambda_{-k} & 0 & \cdots & 0 \\
0 & 0 & \lambda_{-k+1} & \cdots & 0 \\
\cdot & \cdot & \cdot & \cdots & \cdot \\
\cdot & \cdot & \cdot & \cdots & \cdot \\
0 & 0 & 0 & \cdots & \lambda_k \end{array}
\right]\,,
$$
we have the dynamics
$$
d\mathbf{x}(t)=(A\mathbf{x}(t)+\overline{\mathbf{b}}_k(t,\mathbf{x}(t)))\,dt+d\overline{\mathbf{\psi}}_k(t,\mathbf{x}(t-))(L(t))\,,
$$
with $\widehat{f}_{k,*}=x_1, \widehat{f}_{k,-k}=x_2,\ldots,\widehat{f}_{k,k}=x_k$.
Using for example an Euler approximation, we can derive an iterative numerical scheme for this
stochastic differential
equation in $\mathbb C^{2k+2}$. We refer to Kloeden and Platen~\cite{KP} for a detailed analysis
of numerical solution of stochastic differential equations driven by Wiener noise.
\end{document}
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\begin{document}
\bstctlcite{IEEEexample:BSTcontrol}
\title{In-memory Realization of In-situ \\ Few-shot Continual Learning with \\ a Dynamically Evolving Explicit Memory
}
\author{\IEEEauthorblockN{G. Karunaratne\IEEEauthorrefmark{1}\IEEEauthorrefmark{4},
M. Hersche\IEEEauthorrefmark{1}\IEEEauthorrefmark{4},
J. Langenegger\IEEEauthorrefmark{1}\IEEEauthorrefmark{4},
G. Cherubini\IEEEauthorrefmark{1},
M. Le Gallo\IEEEauthorrefmark{1},
U. Egger\IEEEauthorrefmark{1},\\
K. Brew\IEEEauthorrefmark{2},
S. Choi\IEEEauthorrefmark{2},
I. Ok\IEEEauthorrefmark{2},
C. Silvestre\IEEEauthorrefmark{2},
N. Li\IEEEauthorrefmark{2},
N. Saulnier\IEEEauthorrefmark{2},
V. Chan\IEEEauthorrefmark{2},
I. Ahsan\IEEEauthorrefmark{2},\\
V. Narayanan\IEEEauthorrefmark{3},
L. Benini\IEEEauthorrefmark{4},
A. Sebastian\IEEEauthorrefmark{1},
A. Rahimi\IEEEauthorrefmark{1}}\\
\IEEEauthorblockA{\IEEEauthorrefmark{1}IBM Research,
Z\"{u}rich, Switzerland
\IEEEauthorrefmark{2}IBM Research, Albany, NY, USA\\
\IEEEauthorrefmark{3}IBM T. J. Watson Research Center, NY, USA
\IEEEauthorrefmark{4}ETH Z\"{u}rich, Z\"{u}rich, Switzerland}\\}
\maketitle
\thispagestyle{firststyle}
\begin{abstract}
Continually learning new classes from few training examples without forgetting previous old classes demands a flexible architecture with an inevitably growing portion of storage, in which new examples and classes can be incrementally stored and efficiently retrieved. One viable architectural solution is to tightly couple a stationary deep neural network to a dynamically evolving explicit memory (EM). As the centerpiece of this architecture, we propose an EM unit that leverages energy-efficient in-memory compute (IMC) cores during the course of continual learning operations. We demonstrate for the first time how the EM unit can physically superpose multiple training examples, expand to accommodate unseen classes, and perform similarity search during inference, using operations on an IMC core based on phase-change memory (PCM). Specifically, the physical superposition of few encoded training examples is realized via in-situ progressive crystallization of PCM devices. The classification accuracy achieved on the IMC core remains within a range of 1.28\%--2.5\% compared to that of the state-of-the-art full-precision baseline software model on both the CIFAR-100 and miniImageNet datasets when continually learning 40 novel classes (from only five examples per class) on top of 60 old classes.
\end{abstract}
\begin{IEEEkeywords}
In-memory Computing, Continual Learning, Few-shot Learning, Hyperdimensional Computing, Non-volatile Memory Devices
\end{IEEEkeywords}
\section{Introduction}
Few-shot continual learning (FSCL), aka few-shot class-incremental learning~\cite{FSCIL_CVPR2020,shi_nips2021}, requires a learner to incrementally learn new classes from very few training examples, without forgetting the previously learned classes. The learner is exposed to a series of sessions, whereby each session introduces distinct unseen classes by providing only a few training examples per class. From these few examples, the learner should quickly and incrementally learn novel classes without forgetting the previously learned old classes. After learning novel classes in each session, the learner is evaluated on several query samples from all the classes, to which it was exposed so far (i.e., the union of the classes from the previous and the current sessions). FSCL is a very challenging research problem that could impose significant additional compute and memory costs on the learner.
Very recently, a low-cost solution was proposed that avoids expensive gradient-based computations for learning unseen classes during the course of FSCL~\cite{C-FSCIL_CVPR22}. This solution is inspired by a robust few-shot learner~\cite{kar_ncom_2021} that brings together deep neural networks with hyperdimensional computing~\cite{Kanerva2009,VSA03} to be able to represent raw images with high-dimensional holographic binary or bipolar vectors. Inspired by this powerful combination, the learner in~\cite{C-FSCIL_CVPR22} is composed of a \emph{frozen} controller and a \emph{dynamically evolving} explicit memory (EM) for FSCL. The controller is a deep convolutional network (including a final fully connected layer) that is interfaced with the EM unit to dynamically store or retrieve the acquired knowledge about the classes. The controller interacts with the EM through write and read operations using $d$-dimensional holographic vectors. Although the controller remains stationary, the EM dynamically updates its contents by new examples, and grows its size by storing new classes. Hence, it is desirable to have an EM unit with dynamically evolving contents that retains the acquired knowledge about already-seen classes in both compressed and nonvolatile manner.
\begin{figure}
\caption{\textbf{(a)}
\label{fig:phases}
\end{figure}
\section{Proposed Explicit Memory Unit for FSCL}
Fig.~\ref{fig:phases}(a)(b) illustrates the main stages of FSCL involved in~\cite{C-FSCIL_CVPR22}: a pretraining and metalearning stage, and an inference stage. The first stage is all done in software. The goal of this rather elaborate stage is to train the controller (here, a ResNet-12) to be able to generate $d$-dimensional\footnote{As seen later in Section~\ref{sec:setup}, $d$ is fixed to 256 so that the vectors occupy an entire column of a PCM crossbar array.} quasi-orthogonal real-valued vectors for different classes. By the end of this stage, the controller has learned how to assign 256-dimensional quasi-orthogonal, and thus dissimilar, vectors to novel classes in the EM, which allows the controller to remain stationary afterwards. This pretraining and metalearning stage is based on just the first session's (S1) training samples. Based on the datasets used in our experiments, as explained in Section~\ref{sec:dataset}, the first session contains 60 classes, each including 500 samples, out of the total 100 classes.
Next, the inference stage is composed of two phases, as shown in Fig.~\ref{fig:phases}(c): a continual learning phase of novel classes from very few training/support examples per class (in our results, 5 training examples, hence 5-shot), and a query evaluation phase in which we evaluate the accuracy over a batch of query examples containing a number of samples (100 in the datasets we used) per each class encountered so far. We quantize the controller’s output vector elements to 1-bit, hence the controller generates bipolar support vectors (of training examples) to be written, or accumulated, in the EM unit during the continual learning phase. For the query vectors (of query samples) during the evaluation phase, we quantize the controller’s output vector elements to 8-bit, so an analog input corresponding to the 8-bit query vector element is applied in each row of the crossbar array. This performs a similarity search between query vector and the class vectors. The resulting vector received via the columns of the crossbar array is used to classify the query sample. The class corresponding to the maximum element of the resulting vector is then taken as the predicted class.
\begin{figure}
\caption{\textbf{(a)}
\label{fig:arch}
\end{figure}
For the inference stage, the EM is implemented on an IMC core with a unit-cell array comprising PCM devices (see Fig.~\ref{fig:arch}). In every continual learning phase: (i) when the first example of a new class appears, the EM is expanded by choosing a fully reset column of unit-cells and, based on the sign of the bipolar support vector element, the unit-cell conductance is increased or decreased by the application of single SET pulses; ii) when an example from a previously seen class appears, a similar update is performed on the column of unit cells corresponding to that particular class.
We exploit the in-situ accumulation via progressive crystallization of PCM to realize physical superposition of few related support vectors on a single class vector. This means, as shown in Fig.~\ref{fig:arch}(b), starting from a fully reset pair of PCM devices, fine grained SET pulses are applied on either the positive or the negative device depending on whether the type of accumulation is incremental or decremental based on the input bipolar support vector element. This creates a multibit analog EM, whereby the size of the EM is set just by the number of classes, as opposed to the product of the number of classes and the number of training examples per class~\cite{kar_ncom_2021}. The nonvolatility of the resulting analog states preserves the acquired knowledge about already-seen classes. Finally, during the evaluation phase, the frequent similarity search between the 8-bit query vector (of a query image) and the analog class vectors are computed in-memory by exploiting Kirchhoff’s circuit laws, and the result is directly used for classification.
Moreover, given that the controller requires no parameter updates after the first stage of pretraining and metalearning, it can be treated as a deep neural network with stationary weights that can also benefit from an IMC-based implementation, as demonstrated for example in~\cite{joshi_ncom2020}.
\begin{figure}
\caption{Experimental Setup. On the right: the micrograph of the Hermes chip. The chip is accessed using a sequence of programming and MVM commands prepared and sent by the host computer via a FPGA-based interface.}
\label{fig:exp_setup}
\end{figure}
\begin{figure}
\caption{\textbf{(a)}
\label{fig:programming}
\end{figure}
\section{Experimental Setup}
\label{sec:setup}
The experiments are carried out on the Hermes chip with a 256x256 unit-cell array of PCM devices organized in a differential configuration \cite{Y2022khaddamJSSC} (see Fig.~\ref{fig:exp_setup}), accessed by a host computer via FPGA (Field Programmable Gate Array)-based interface. The bipolar support vectors are sent as SET pulses to the corresponding column of 256 unit cells of the initially fully RESET array. Based on the +1/-1 vector elements, the conductance of the positive/negative polarity devices are increased. The evolution of the conductance distribution of individual PCM devices as a function of the number of applied SET pulses on initially RESET devices is shown in Fig.~\ref{fig:programming}.
For similarity search in the evaluation phase, a 4-quadrant matrix-vector multiply (MVM) is performed between the 8-bit query vector and the set of analog class vectors stored in the unit-cell array.
\begin{figure}
\caption{A 2-D map of the array conductance (of the unit-cells with differential PCM devices) for the FSCL experiment on CIFAR-100 dataset. With each new session, more columns of the array are selected corresponding to the new classes in that session. And with each training example per class, the corresponding class vectors are updated with the application of SET pulses. The unit-cells activated for the update at each panel are highlighted in green. A close up view of the conductance evolution in a 10x10 region of the crossbar is shown in the bottom row.}
\label{fig:condmaps2}
\end{figure}
\begin{figure}
\caption{FSCL classification accuracy for IMC vs. FP32 software implementations for \textbf{(a)}
\label{fig:accuracy}
\end{figure}
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\section{Experimental Setup}
\label{sec:setup}
The experiments are carried out on the Hermes chip with a 256x256 unit-cell array of PCM devices organized in a differential configuration \cite{Y2022khaddamJSSC} (see Fig.~\ref{fig:exp_setup}), accessed by a host computer via FPGA (Field Programmable Gate Array)-based interface. The bipolar support vectors are sent as SET pulses to the corresponding column of 256 unit cells of the initially fully RESET array. Based on the +1/-1 vector elements, the conductance of the positive/negative polarity devices are increased. The evolution of the conductance distribution of individual PCM devices as a function of the number of applied SET pulses on initially RESET devices is shown in Fig.~\ref{fig:programming}.
For similarity search in the evaluation phase, a 4-quadrant matrix-vector multiply (MVM) is performed between the 8-bit query vector and the set of analog class vectors stored in the unit-cell array.
\begin{figure}
\caption{A 2-D map of the array conductance (of the unit-cells with differential PCM devices) for the FSCL experiment on CIFAR-100 dataset. With each new session, more columns of the array are selected corresponding to the new classes in that session. And with each training example per class, the corresponding class vectors are updated with the application of SET pulses. The unit-cells activated for the update at each panel are highlighted in green. A close up view of the conductance evolution in a 10x10 region of the crossbar is shown in the bottom row.}
\label{fig:condmaps2}
\end{figure}
\begin{figure}
\caption{FSCL classification accuracy for IMC vs. FP32 software implementations for \textbf{(a)}
\label{fig:accuracy}
\end{figure}
\section{Experimental Results}
\subsection{Datasets used for evaluation}
\label{sec:dataset}
For the accuracy evaluation, we use the CIFAR-100 and the miniImageNet dataset, restructured to comply with the FSCL setting~\cite{FSCIL_CVPR2020, shi_nips2021,C-FSCIL_CVPR22}.
Both datasets contain natural images of 100 classes in total, which are divided into a first session (S1) containing 60 classes with 500 training and 100 query examples per class, and eight novel sessions (S2--S9) with 5 novel classes introduced in each session containing 5 support examples and 100 query examples per class.
The updated class vectors occupy 256 rows and 60 columns (classes) on the array in S1, evolving to 100 columns (classes) in the last S9. The conductance evolution of unit cells of the crossbar array corresponding to CIFAR-100 is shown in Fig.~\ref{fig:condmaps2}.
\subsection{Classification accuracy}
The accuracy obtained for each dataset with our EM on the IMC hardware and various full-precision software baselines are illustrated in Fig.~\ref{fig:accuracy}. Compared to the full-precision software baseline in~\cite{C-FSCIL_CVPR22}, the accuracy degradation with our EM realization is at worst 2.5\% and at best 1.28\% across all sessions. This still makes the accuracy of our EM on the IMC hardware from 2.5\% to 11.01\% higher than the other best performing full-precision software methods reported in~\cite{FSCIL_CVPR2020,shi_nips2021}, considering all sessions across CIFAR-100 and miniImageNet datasets.
\subsection{Energy estimation}
The energy consumption during incremental class vector updates is estimated using the programming parameters, which include peak pulse current of 150\,uA, flat pulse duration 5\,ns, trailing edge pulse duration 40\,ns and source voltage 2.34\,V. These parameters yield an energy expense of 8.78\,pJ per PCM device during one programming cycle. Considering the vector dimension of 256, the total programming time and energy spent during an incremental update of one class vector are 11.5\,us and 2.25\,nJ, respectively. Given that 25 (5-shots from 5 classes) class vectors are updated during all subsequent sessions (S2--S9), the time and energy spent on updating the class vectors in these sessions are estimated to be 57.6\,us and 56.2\,nJ, respectively.
In comparison, the time and energy spent on similarity search of a single query (including digital to analog conversion, PCM read and analog to digital conversion) are estimated to be 520\,ns and 7.74\,nJ, respectively, during the last session. This leads to a total similarity search time and energy of 5.2\,ms and 77.3\,uJ respectively, as in this session we evaluate 10,000 queries (100 queries per class) in total, compared to just 25 updates of the class vectors. The limited number of updates and the application of SET pulses with low energy ensure that the durability of PCM is not significantly affected~\cite{Y2016tumaNatureNano,Y2020legalloJPD}.
\renewcommand{1}{1.3}
\begin{table}[!ht]
\centering
\caption{Comparison with the related works}\label{tab:results-perception-new}
\resizebox{\linewidth}{!}{
\begin{NiceTabular}{lcccc}
\toprule
& \begin{tabular}[c]{@{}c@{}}Kazemi\\ \textit{~et~al.}~\cite{kazemi_date2021}\end{tabular} & \begin{tabular}[c]{@{}c@{}}Li \textit{~et~al.}\\ \cite{li_vlsi2021,li_TED2021}\end{tabular} & \begin{tabular}[c]{@{}c@{}}Karunaratne\\ \textit{~et~al.}~\cite{kar_ncom_2021}\end{tabular} & This work \\
\cmidrule(r){1-1} \cmidrule(r){2-5}
Few-shot learning & \checkmark & \checkmark & \checkmark & \checkmark \\
Continual learning & & & & \checkmark \\
In-situ accumulation & & & & \checkmark \\
Holographic rep. in EM & & & \checkmark& \checkmark\\
Analog multibit EM & & & & \checkmark \\
Truly $\mathcal{O}$(1) search\tabularnote{Actual crossbar (no emulation based on individual memory devices) performing all dot product operations in parallel in-memory} & & \checkmark & & \checkmark \\
Query vector dim. ($d$) & 128--192 & 128 & 512 & 256\\
Number of classes & $\leq$20 & 32 & \textbf{$\leq$100} & \textbf{$\leq$100} \\
Similarity search energy\tabularnote{Core energy normalized to one class vector of length 256 ($d=256$)} & - & \textbf{17.5\,pJ} & 25.6\,pJ & 19.1\,pJ\\
Programming energy\tabularnote{per vector element} & - & 99\,pJ & 6240\,pJ & \textbf{8.78\,pJ}\\
Dataset(s) & Omniglot & Omniglot & Omniglot & \begin{tabular}[c]{@{}c@{}}miniImageNet\\ \& CIFAR100\end{tabular} \\
\bottomrule
\end{NiceTabular}
}
\label{tab:comparison}
\end{table}
\renewcommand{1}{1}
\subsection{Comparison}
We compare features of our work against the related works in Table~\ref{tab:comparison}. Our work shares few-shot learning capability with~\cite{kar_ncom_2021,kazemi_date2021,li_vlsi2021,li_TED2021}, holographic representation of vectors with~\cite{kar_ncom_2021}, and truly $\mathcal{O}(1)$ similarity search capability with~\cite{li_vlsi2021,li_TED2021}. However, our work is the first to demonstrate continual learning capability using the in-situ accumulation property. This makes our EM unit the fist multibit analog IMC core, whereas in \cite{li_vlsi2021,li_TED2021} storage and queries use binary vectors. Furthermore, our EM unit can handle up to 100 class vectors for the natural image datasets, making it the largest truly $\mathcal{O}(1)$ similarity search engine with analog states to date.
In Table~\ref{tab:comparison}, we also compare energy saving we gain by programming the support vectors using the in-situ accumulation instead of programming them from scratch in the novel bit lines. Programming with the in-situ accumulation is at least 4.7$\times$ energy efficient than the normal programming, because the in-situ accumulation requires a SET pulse of shorter duration. Our similarity search energy remains on par with other works.
\section{Conclusion}
We present a hardware based on an in-memory compute core consisting of PCM devices to implement the EM operations required in FSCL. We demonstrate for the first time how support vectors are accumulated in-situ using the progressive crystallization property of PCM. The proposed approach leads to physically superposed representations and enhanced energy savings, while retaining the accuracy within 2.5\% from the full-precision software baseline.
\tiny
\end{document}
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\begin{document}
\begin{center}
{\Large {\bf On the zeros of $\sum_{n=1}^\infty \frac{\lambda_\mathcal{P}(n)}{n^s}$}}
\\ Titus HILBERDINK and Eric SAIAS\\
Department of Mathematics, University of
Reading, Whiteknights,\\
PO Box 220, Reading RG6 6AX, UK\\
and\\
Sorbonne Universit\'{e}, LPSM, 4 Place Jussieu\\
F-75005 Paris, FRANCE
\\
\end{center}
\indent
\begin{abstract} Our main result is to answer a question of Michel Balazard by giving a Dirichlet series with only one zero in its half-plane of convergence. At the end of the paper we also give several sufficient conditions for the Generalized Riemann Hypothesis and ask if any of them are true.
\noindent
{\em 2010 AMS Mathematics Subject Classification}: 11M41, 11M06, 11M26.\newline
{\em Keywords and phrases}: zeros of Dirichlet series, completely multiplicative functions
\end{abstract}
In memory of Jean-Pierre Kahane\newline
\begin{center}
TABLE OF CONTENTS
\end{center}
\begin{description}
\item[1.] Introduction
\begin{description}
\item[1(a)] From Euler to Landau.
\item[1(b)] Zeros of Dirichlet series.
\item[1(c)] Generalization of $\lambda(n)$ to $\lambda_\mathcal{P}(n)$ and $\zeta(s)$ to $\zeta_\mathcal{P}(s)$.
\item[1(d)] Easy or known results on the set of zeros $Z_\mathcal{P}$.
\item[1(e)] New results on the multiset of zeros of Dirichlet series.
\item[1(f)] Zeros and abscissae of convergence for Dirichlet series with completely multiplicative coefficients.
\item[1(g)] Zeros of Helson's zeta functions.
\end{description}
\item[2.] Proof of Theorem 3.
\item[3.] Proof of Theorem 0.
\item[4.] Architecture of the proofs of Theorems 1 and 2.
\item[5.] Vocabulary, notations and results for Beurling primes.
\item[6.] Primes in short intervals.
\item[7.] Abscissae of convergence.
\item[8.] Proofs of Theorems 2 and 1.
\item[9.] Open questions related to GRH$\setminus$RH.
\end{description}
\pagebreak
\noindent
{\bf {\large 1. Introduction}\newline
1(a) From Euler to Landau.}\newline
Euler, in his paper \cite{E} of 1737, writes
\[ 1 - \frac{1}{2} - \frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\frac{1}{7}-\frac{1}{8}+\frac{1}{9}+\frac{1}{10}-\frac{1}{11}-\frac{1}{12}\quad\mbox{etc.} =0.\]
In modern words we define $\lambda(n)$, Liouville's function, to be the completely multiplicative function which is $-1$ at every prime. What Euler writes is then
\[ \sum_{n=1}^\infty \frac{\lambda(n)}{n}=0.\tag{1.1}\]
It is in this paper he uses the ``Euler'' product formula. He applies it first to the completely multiplicative function $\frac{1}{n}$ and obtains
\[ \prod_p \frac{1}{1-\frac{1}{p}} = \sum_{n=1}^\infty \frac{1}{n} = \infty.\tag{1.2}\]
Then he applies it a second time to the completely multiplicative function $\frac{\lambda(n)}{n}$ and obtains
\[ \sum_{n=1}^\infty \frac{\lambda(n)}{n} = \prod_p \frac{1}{1+\frac{1}{p}} = 0\tag{1.3}\]
due to (1.2), and (1.1) follows.
Let us now read formulas (1.2) and (1.3) with our modern eyes, with our definitions of infinite sums. Since the completely multiplicative function $\frac{1}{n}$ is positive, his proof of (1.2) is valid three centuries later. On the other hand the completely multiplicative function $\frac{\lambda(n)}{n}$ is not positive nor summable. Thus the first equality of (1.3) is not proved.
As a matter of fact, it is Riemann \cite{R} in 1859 for the first part, and de la Vall\'{e}e-Poussin \cite{dVP} and Hadamard \cite{H} 37 years later for the second part, by continuing Euler's $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$ as a meromorphic function in $\mathbb{C}$ and proving it does not vanish in the closed half-plane Re $s\ge 1$, who brought the tools to prove (1.1). More precisely, von Mangoldt \cite{vM} proved in 1897, just one year after de la Vall\'{e}e-Poussin and Hadamard, that
\[ \sum_{n=1}^\infty \frac{\mu(n)}{n}=0,\tag{1.4}\]
where $\mu(n)$ is the Mobius function. In 1907, Landau \cite{L} deduced (1.1) from (1.4). Thus 180 years separates Euler's claim and its proof! \newline
\noindent
{\bf 1(b) Zeros of Dirichlet series.}\newline
The series
\[ \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}\mbox{ vanishes only at $s=1$}.\tag{1.5}\]
(This result is part of Theorem 0 below, and we recall briefly its proof in section 3.)
In the situation of any Dirichlet series
\[ F(s)=\sum_{n=1}^\infty \frac{f(n)}{n^s},\]
we consider the zeros of $F(s)$ from a naive point of view. We denote by $Z(F)$ the set of complex numbers for which the above series converges and its sum is zero. In particular, we have $Z(F)\subset \{ s\in\mathbb{C}: \mathbb{R}e s\ge \sigma_c(F)\}$ where $\sigma_c(F)$ is the abscissa of convergence of $F$. Notice that the series may converge and have zero sum on the line $\sigma=\sigma_c(F)$. In other words, $Z(F)$ may contain points on this line.
\noindent
{\bf 1(c) Generalization of $\lambda(n)$ to $\lambda_\mathcal{P}(n)$ and $\zeta(s)$ to $\zeta_\mathcal{P}(s)$.}\newline
Let $\mathbb{P}$ denote the set of all primes and let $\mathcal{P}\subset\mathbb{P}$ be a subset. We define the generalized Liouville function associated to $\mathcal{P}$ as the completely multiplicative function defined on primes by
\[ \lambda_\mathcal{P}(p) = \left\{ \begin{array}{cl} -1 & \mbox{ if $p\in\mathcal{P}$}\\ 0 & \mbox{ if $p\not\in\mathcal{P}$}\end{array}\right. .\]
In this paper we study the set of zeros
\[ Z_\mathcal{P} = Z\biggl(\sum_{n=1}^\infty \frac{\lambda_\mathcal{P}(n)}{n^s}\biggr).\]
Let $\sigma_\mathcal{P}$ denote the abscissa of convergence of the series $\sum_{p\in\mathcal{P}} p^{-s}$. It is easy to see that
\[ \sigma_\mathcal{P}\le 1.\tag{1.6}\]
We generalize the usual $\zeta(s)$ by denoting
\[ \zeta_\mathcal{P}(s):= \prod_{p\in\mathcal{P}}\frac{1}{1-\frac{1}{p^s}}.\tag{1.7}\]
Of course $\lambda_\mathbb{P}=\lambda$ and $\zeta_\mathbb{P}=\zeta$. As in this particular important case $\mathcal{P}=\mathbb{P}$, the general zeta function $\zeta_\mathcal{P}(s)$ is a normally convergent Euler product in every fixed closed half-plane $\sigma\ge \max\{\sigma_\mathcal{P},0\}+\varepsilon$, for any fixed $\varepsilon>0$. If $\mathcal{P}$ is finite, (1.7) defines a non-vanishing meromorphic function in $\mathbb{C}$ whose multiset of poles is the union of $|\mathcal{P}|$ infinite arithmetic progressions of purely imaginary complex numbers. Except perhaps at $s=0$, all those poles are simple. At $s=0$, the multiplicity is $|\mathcal{P}|$.
Let us now study the case where $\mathcal{P}$ is infinite. Then
\[ \zeta_{\mathcal{P}}(s)\ne 0\quad\mbox{ for $\sigma>\sigma_{\mathcal{P}}$}.\tag{1.8}\]
Moreover, as in the basic usual case, if $\zeta_\mathcal{P}(s)$ has a meromorphic continuation in some open set across the line $\sigma=\sigma_\mathcal{P}$, we continue to denote this continuation by $\zeta_\mathcal{P}(s)$.\newline
Let $\mathcal{N}=\{ n\in\mathbb{N}: p|n \implies p\in \mathcal{P}\}$ (i.e. all the positive integers formed from the primes in $\mathcal{P}$) and let $\sigma_\mathcal{N}$ denote the abscissa of convergence of $\sum_{n\in\mathcal{N}}n^{-s}$. It is easy to prove that
\[ \sigma_\mathcal{N} = \left\{ \begin{array}{cl} -\infty & \mbox{ if $\mathcal{P}=\emptyset$}\\ \max\{ \sigma_\mathcal{P}, 0\} & \mbox{ if $\mathcal{P}\ne\emptyset$} \end{array} \right. .
\tag{1.9}\]
By introducing this complex parameter $s$ and generalizing to any set $\mathcal{P}$ of primes, we can interpret the two Euler formulas (1.2) and (1.3) by the absolutely convergent Euler products
\[ \sum_{n\in\mathcal{N}} \frac{1}{n^s} = \prod_{p\in\mathcal{P}} \frac{1}{1-\frac{1}{p^s}} = \zeta_\mathcal{P}(s), \quad (\sigma>\sigma_\mathcal{N})\]
and
\[ \sum_{n\in\mathbb{N}} \frac{\lambda_\mathcal{P}(n)}{n^s}=\sum_{n\in\mathcal{N}} \frac{\lambda(n)}{n^s} = \prod_{p\in\mathcal{P}} \frac{1}{1+\frac{1}{p^s}} = \frac{\zeta_\mathcal{P}(2s)}{\zeta_\mathcal{P}(s)}, \quad (\sigma>\sigma_\mathcal{N}).\tag{1.10}\]
\noindent
{\bf 1(d) Easy or known results on the set of zeros $Z_\mathcal{P}$.}\newline
\noindent
{\bf Theorem 0}\newline
{\em Let $\mathcal{P}$ and $\mathcal{P}^\prime$ be sets of primes.}
\begin{enumerate}
\item {\em If $\sigma_c(\sum_{p\in\mathcal{P}\mathbb{D}elta \mathcal{P}^\prime} p^{-s})\le 0$, then $Z_\mathcal{P} = Z_{\mathcal{P}^\prime}$;}
\item {\em $\overline{Z_\mathcal{P}} = Z_\mathcal{P}$;}
\item {\em if $\sigma_\mathcal{P}\le 0$, then $Z_\mathcal{P}=\emptyset$;}
\item {\em if $\sigma_\mathcal{P}>0$, then $\mathbb{R}e Z_\mathcal{P}\subset (0,\sigma_\mathcal{P}]\subset (0,1]$;}
\item $1\in Z_\mathcal{P}\Leftrightarrow \sum_{p\in\mathcal{P}}\frac{1}{p} =\infty$;
\item $Z_\mathbb{P}=\{1\}$;
\end{enumerate}
\noindent
{\bf Remark 1}\, Point (a) shows that the function $\mathcal{P}\to Z_\mathcal{P}$ is, in a way, locally constant.
\noindent
{\bf 1(e) New results on the multiset of zeros of Dirichlet series.}\newline
We know that the maximal open set where a Dirichlet series $F(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}$ is both convergent and holomorphic is $\mathbb{C}_{\sigma(F)}$ where $\mathbb{C}_\alpha = \{ s\in\mathbb{C}:$ Re $s>\alpha\}$.
Let $\alpha\in [-\infty,\infty)$. Thanks to Weierstrass (\cite{BG}, Theorem 3.3.1), we know that a necessary and sufficient condition for a multiset $Z$ of $\mathbb{C}_\alpha$ to be the multiset $Z(F)$ of some not identically zero holomorphic function $F$ in $\mathbb{C}_\alpha$, is for $Z$ to be locally finite in $\mathbb{C}_\alpha$.
Now for the same question where we ask $Z$ to be equal to the set $Z(F)\cap \mathbb{C}_\alpha$ for some Dirichlet series $F$ with $\sigma(F)=\alpha$, we are far from knowing the necessary and sufficient condition analogue to the Weierstrass theorem. It is even possible that it is impossible to give such a characterization.
In 2000, Balazard (unsolved problem 24 of \cite{MV}) asked the first question for this problem. He asked for an example of a Dirichlet series $F(s)$ for which $Z(F)\cap \mathbb{C}_{\sigma_c(F)}$ has only one element. Notice that under the Riemann Hypothesis, we have an example, namely the series in (1.5). For this series we then have $Z=\{1\}$ and
\[ \sigma_c=\frac{1}{2}. \tag{1.11}\]
This last formula (1.11) is classical under RH. Eighteen years later, we are able to provide an unconditional family of examples. More precisely we get the following result.\newline
\noindent
{\bf Theorem 1}\newline
{\em Let $a$ and $b$ be two real numbers such that
\[ 0<\max\Bigl\{ \frac{a}{2}, a-\frac{19}{40}\Bigr\}<b<a<1.\tag{1.12}\]
Then there exists a set $\mathcal{P}_{a,b}\subset\mathbb{P}$ such that $Z_{\mathcal{P}_{a,b}}=\{a\}$ with $a$ being a simple zero and $\sigma_c=b, \sigma_a=a$, where $\sigma_c$ and $\sigma_a$ are the abscissa of convergence and absolute convergence of $\sum_{n=1}^\infty \lambda_{\mathcal{P}_{a,b}}(n)n^{-s}$ respectively.
Under the Riemann Hypothesis, we can replace $(1.12)$ by $0<\frac{a}{2}<b<a<1$ or $(a,b)=(1,\frac{1}{2})$.}\newline
Let us remark that to provide an example of $\mathcal{P}$ with $|Z_\mathcal{P}|=1$, we were obliged to choose the zero real, because of the symmetry of $Z_\mathcal{P}$ about the real axis ($\overline{Z_\mathcal{P}}=Z_\mathcal{P}$).
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\noindent
{\bf 1(d) Easy or known results on the set of zeros $Z_\mathcal{P}$.}\newline
\noindent
{\bf Theorem 0}\newline
{\em Let $\mathcal{P}$ and $\mathcal{P}^\prime$ be sets of primes.}
\begin{enumerate}
\item {\em If $\sigma_c(\sum_{p\in\mathcal{P}\mathbb{D}elta \mathcal{P}^\prime} p^{-s})\le 0$, then $Z_\mathcal{P} = Z_{\mathcal{P}^\prime}$;}
\item {\em $\overline{Z_\mathcal{P}} = Z_\mathcal{P}$;}
\item {\em if $\sigma_\mathcal{P}\le 0$, then $Z_\mathcal{P}=\emptyset$;}
\item {\em if $\sigma_\mathcal{P}>0$, then $\mathbb{R}e Z_\mathcal{P}\subset (0,\sigma_\mathcal{P}]\subset (0,1]$;}
\item $1\in Z_\mathcal{P}\Leftrightarrow \sum_{p\in\mathcal{P}}\frac{1}{p} =\infty$;
\item $Z_\mathbb{P}=\{1\}$;
\end{enumerate}
\noindent
{\bf Remark 1}\, Point (a) shows that the function $\mathcal{P}\to Z_\mathcal{P}$ is, in a way, locally constant.
\noindent
{\bf 1(e) New results on the multiset of zeros of Dirichlet series.}\newline
We know that the maximal open set where a Dirichlet series $F(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}$ is both convergent and holomorphic is $\mathbb{C}_{\sigma(F)}$ where $\mathbb{C}_\alpha = \{ s\in\mathbb{C}:$ Re $s>\alpha\}$.
Let $\alpha\in [-\infty,\infty)$. Thanks to Weierstrass (\cite{BG}, Theorem 3.3.1), we know that a necessary and sufficient condition for a multiset $Z$ of $\mathbb{C}_\alpha$ to be the multiset $Z(F)$ of some not identically zero holomorphic function $F$ in $\mathbb{C}_\alpha$, is for $Z$ to be locally finite in $\mathbb{C}_\alpha$.
Now for the same question where we ask $Z$ to be equal to the set $Z(F)\cap \mathbb{C}_\alpha$ for some Dirichlet series $F$ with $\sigma(F)=\alpha$, we are far from knowing the necessary and sufficient condition analogue to the Weierstrass theorem. It is even possible that it is impossible to give such a characterization.
In 2000, Balazard (unsolved problem 24 of \cite{MV}) asked the first question for this problem. He asked for an example of a Dirichlet series $F(s)$ for which $Z(F)\cap \mathbb{C}_{\sigma_c(F)}$ has only one element. Notice that under the Riemann Hypothesis, we have an example, namely the series in (1.5). For this series we then have $Z=\{1\}$ and
\[ \sigma_c=\frac{1}{2}. \tag{1.11}\]
This last formula (1.11) is classical under RH. Eighteen years later, we are able to provide an unconditional family of examples. More precisely we get the following result.\newline
\noindent
{\bf Theorem 1}\newline
{\em Let $a$ and $b$ be two real numbers such that
\[ 0<\max\Bigl\{ \frac{a}{2}, a-\frac{19}{40}\Bigr\}<b<a<1.\tag{1.12}\]
Then there exists a set $\mathcal{P}_{a,b}\subset\mathbb{P}$ such that $Z_{\mathcal{P}_{a,b}}=\{a\}$ with $a$ being a simple zero and $\sigma_c=b, \sigma_a=a$, where $\sigma_c$ and $\sigma_a$ are the abscissa of convergence and absolute convergence of $\sum_{n=1}^\infty \lambda_{\mathcal{P}_{a,b}}(n)n^{-s}$ respectively.
Under the Riemann Hypothesis, we can replace $(1.12)$ by $0<\frac{a}{2}<b<a<1$ or $(a,b)=(1,\frac{1}{2})$.}\newline
Let us remark that to provide an example of $\mathcal{P}$ with $|Z_\mathcal{P}|=1$, we were obliged to choose the zero real, because of the symmetry of $Z_\mathcal{P}$ about the real axis ($\overline{Z_\mathcal{P}}=Z_\mathcal{P}$).
Before giving the answer to another question, let us begin with some obvious remarks. It is very easy to construct Dirichlet series with at least two zeros. The simple example is
\[ 1-\frac{1}{2^s}.\]
Now if we want to have at least two zeros with different real part, it is also easy: just choose
\[ \Bigl(1-\frac{1}{2^s}\Bigr)\Bigl(1-\frac{1}{2^{s-1}}\Bigr).\]
But now we ask the following question:
\[ \mbox{\em Find a Dirichlet series with completely multiplicative coefficients and two real zeros.}\tag{1.13}\]
As far as we know, (1.13) is an open question. Once again, some Dirichlet series $\frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)}$ with well chosen $\mathcal{P}$ give a family of examples which allow us to answer this question positively. \newline
\noindent
{\bf Theorem 2}\newline
{\em Let $a$ and $b$ be two real numbers such that
\[ 0<\max\Bigl\{ \frac{a}{2}, a-\frac{19}{40}\Bigr\}<b<a<1.\tag{1.14}\]
Then there exists a set $\mathcal{P}_{a,b}\subset\mathbb{P}$ such that $Z_{\mathcal{P}_{a,b}}\supset\{a,b\}$ with $a$ and $b$ being simple zeros and $\sigma_c\le \max\{\frac{3ab}{2(a+b)}, a-\frac{19}{40}\}, \sigma_a=a$, where $\sigma_c$ and $\sigma_a$ are the abscissae of simple and absolute convergence of $\sum_{n=1}^\infty \lambda_{\mathcal{P}_{a,b}}(n)n^{-s}$ respectively.
Under RH, we can replace $\max\{ \frac{a}{2}, a-\frac{19}{40}\}$ by $\frac{a}{2}$ in $(1.14)$, and $ \max\{\frac{3ab}{2(a+b)}, a-\frac{19}{40}\}$ by $\frac{3ab}{2(a+b)}$.}\newline
Notice that the conditions in Theorems 1 and 2 are the same. As a matter of fact, the proofs of the two results have similar structure, as will be seen in sections 4 and 8.
\newline
\noindent
{\bf 1(f) Zeros and abscissae of convergence for Dirichlet series with completely multiplicative coefficients}\newline
We continue with another question on the set of zeros of a Dirichlet series $F(s)=\sum_{n=1}^\infty\frac{f(n)}{n^s}$ with completely multiplicative coefficients $f(n)$. Denote the abscissa of convergence and absolute convergence by $\sigma_c$ and $\sigma_a$ respectively.
Let $V$ denote the set of values of the difference
\[ \mathbb{R}e \rho - \sigma_c\]
when $f$ varies through the completely multiplicative functions and $\rho\in Z(F)$. What can we say about the set $V$?
By Theorem 1, we know that $V\supset (0,\frac{19}{40})$ which is improved under the Riemann Hypothesis to $V\supset (0,\frac{1}{2})$. But, as a matter of fact, a little more is known unconditionally. Before telling our result, let us consider an analogous question for Dirichlet series with completely multiplicative coefficients.
Let $W$ denote the set of values of
\[ \sigma_a - \mathbb{R}e \rho\]
with $\rho$ as before. Now by Theorem 2, we have $W\supset (0,\frac{19}{40})$ which is improved under the Riemann Hypothesis to $W\supset (0,\frac{1}{2})$. But, in contrast to the case for $V$, we are not able to prove this unconditionally. \newline
\noindent
{\bf Theorem 3}\newline
{\em (i)\, $[0,\frac{1}{2}]\subset V\subset [0,1]$.\newline
(ii)\, There exists $v\in [\frac{1}{2},1]$ such that $V=[0,v]$ or $V=[0,v)$.\newline
(iii)\, $[0,\frac{19}{40})\cup \{\frac{1}{2}\}\subset W\subset [0,1]$, and under RH, we can replace (iii) by\newline
(iii$)^\prime$\, $[0,\frac{1}{2}]\subset W\subset [0,1]$.}\newline
Notice that these results cannot be extended to the case where $f$ is only multiplicative. To see this, consider $f$ defined by $f(1)=1$, $f(2) =-1$ and zero otherwise. Then $F(s)=1-2^{-s}$ vanishes at 0 and $\sigma_c(F) = \sigma_a(F)=-\infty$. \newline
\noindent
\noindent
{\bf 1(g) Zeros of Helson zeta functions}\newline
In \cite{S}, Seip studies the multiset of zeros of Helson zeta functions. For each completely multiplicative unimodular function $\chi$, Helson's zeta function $\zeta_\chi(s)$, is the meromorphic continuation (if any) of the Dirichlet series
\[ \sum_{n=1}^\infty \frac{\chi(n)}{n^s}.\]
We shall call a Helson zeta function {\em admissible} if it has a meromorphic continuation to $\mathbb{C}_{\frac{1}{2}}$.
Notice that by the Euler product, $\zeta_\chi(s)$ cannot vanish for $\sigma>1$. Moreover (\cite{S}, Theorem 2.1) an admissible Helson zeta function has at most one zero on the line $\sigma=1$ and, if it has one, it is simple. The function $\zeta(2s)/\zeta(s)$, which we already discussed, is an example where there is one.
Let us look now in the strip $S:=\{ s\in\mathbb{C}:1/2<$ Re$s<1\}$. Let $Z$ be a multiset belonging to $S$. We recall that by Weierstrass' Theorem (\cite{BG}, Theorem 3.3.1), a necessary condition for $Z$ to be the multiset of zeros of an admissible zeta function is:
\[ \mbox{\em $Z$ is locally finite in $\mathbb{C}_{\frac{1}{2}}$.}\tag{1.15}\]
Seip proves that under the Riemann Hypothsesis this necessary condition is also sufficient: if (1.15) is true, then there exists an admissible Helson zeta function whose multiset of zeros in $S$ is equal to $Z$.
The comparison of this result with ours is impressive. Under RH, Seip has found the necessary and sufficient condition. The main result here is only to make unconditional the (conditional on RH) 100 year-old result that there exists an example with a single zero in the half-plane of convergence!
We do not know if it is possible to use some of Seip's results and/or tools in \cite{S} to find new examples of sets of zeros of Dirichlet series with completely multiplicative coefficients themselves, and not of their possible meromorphic continuation.
\noindent
{\bf {\large 2. Proof of Theorem 3}}\newline
{\bf Step 1}\, We prove that $V\subset [0,1]$. Let $f$ be a completely multiplicative function. Let the Dirichlet series
\[ \sum_{n=1}^\infty \frac{f(n)}{n^s}\tag{2.1}\]
have abscissa of convergence and absolute convergence $\sigma_c$ and $\sigma_a$ respectively, and let $\rho$ be a zero of the series.
Then $\sigma_c\le \mathbb{R}e\rho\le \sigma_a$ for the series has to converge at $\rho$ and for $\sigma>\sigma_a$,
\[ \sum_{n=1}^\infty \frac{f(n)}{n^s} = \prod_p\frac{1}{1-\frac{f(p)}{p^s}}\ne 0. \]
Hence
\[ 0\le \mathbb{R}e\rho - \sigma_c\le \sigma_a-\sigma_c \le 1,\tag{2.2}\]
and $V\subset [0,1]$.\newline
\noindent
{\bf Step 2}\, We prove that $0\in V$. Define the completely multiplicative function $f$ on primes $p$ by
\[ f(p) = -\frac{1}{\log\log p}\qquad (p\ge 29)\]
and zero otherwise. As $29>e^e$, we have $-1\le f(p)\le 0$ for all $p$. Moreover, we have
\[ \sum_p \frac{f(p)}{p} = -\sum_{p\ge 29} \frac{1}{p\log\log p} = -\infty.\]
By applying Theorem 9 of \cite{KS1}, it follows that the Dirichlet series in (2.1) vanishes at $s=1$. To finish the proof, it suffices to verify that the abscissa of the series is 1.
Now
\[ \sum_{p\ge 29} \frac{1}{p^\sigma \log\log p} \asymp \log\log \frac{1}{\sigma -1} \quad\mbox{ $(1<\sigma<1+\frac{1}{100})$}.\]
As
\[ F(s) = \prod_{p\ge 29} \frac{1}{1+\frac{1}{p^s \log\log p}},\qquad (\sigma>1)\]
it follows that $\log F(\sigma) \asymp - \log\log\frac{1}{\sigma -1}$ and
\[ F(\sigma)\gg \frac{1}{( \log \frac{1}{\sigma-1})^2}.\]
As $F(1)=0$, $F(s)$ cannot be extended holomorphically to a neighbourhood of 1, and the result follows.\newline
\noindent
{\bf Remark 2}\, Here we have an example where $\sigma_c=\mathbb{R}e\rho = \sigma_a$.\newline
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Notice that these results cannot be extended to the case where $f$ is only multiplicative. To see this, consider $f$ defined by $f(1)=1$, $f(2) =-1$ and zero otherwise. Then $F(s)=1-2^{-s}$ vanishes at 0 and $\sigma_c(F) = \sigma_a(F)=-\infty$. \newline
\noindent
\noindent
{\bf 1(g) Zeros of Helson zeta functions}\newline
In \cite{S}, Seip studies the multiset of zeros of Helson zeta functions. For each completely multiplicative unimodular function $\chi$, Helson's zeta function $\zeta_\chi(s)$, is the meromorphic continuation (if any) of the Dirichlet series
\[ \sum_{n=1}^\infty \frac{\chi(n)}{n^s}.\]
We shall call a Helson zeta function {\em admissible} if it has a meromorphic continuation to $\mathbb{C}_{\frac{1}{2}}$.
Notice that by the Euler product, $\zeta_\chi(s)$ cannot vanish for $\sigma>1$. Moreover (\cite{S}, Theorem 2.1) an admissible Helson zeta function has at most one zero on the line $\sigma=1$ and, if it has one, it is simple. The function $\zeta(2s)/\zeta(s)$, which we already discussed, is an example where there is one.
Let us look now in the strip $S:=\{ s\in\mathbb{C}:1/2<$ Re$s<1\}$. Let $Z$ be a multiset belonging to $S$. We recall that by Weierstrass' Theorem (\cite{BG}, Theorem 3.3.1), a necessary condition for $Z$ to be the multiset of zeros of an admissible zeta function is:
\[ \mbox{\em $Z$ is locally finite in $\mathbb{C}_{\frac{1}{2}}$.}\tag{1.15}\]
Seip proves that under the Riemann Hypothsesis this necessary condition is also sufficient: if (1.15) is true, then there exists an admissible Helson zeta function whose multiset of zeros in $S$ is equal to $Z$.
The comparison of this result with ours is impressive. Under RH, Seip has found the necessary and sufficient condition. The main result here is only to make unconditional the (conditional on RH) 100 year-old result that there exists an example with a single zero in the half-plane of convergence!
We do not know if it is possible to use some of Seip's results and/or tools in \cite{S} to find new examples of sets of zeros of Dirichlet series with completely multiplicative coefficients themselves, and not of their possible meromorphic continuation.
\noindent
{\bf {\large 2. Proof of Theorem 3}}\newline
{\bf Step 1}\, We prove that $V\subset [0,1]$. Let $f$ be a completely multiplicative function. Let the Dirichlet series
\[ \sum_{n=1}^\infty \frac{f(n)}{n^s}\tag{2.1}\]
have abscissa of convergence and absolute convergence $\sigma_c$ and $\sigma_a$ respectively, and let $\rho$ be a zero of the series.
Then $\sigma_c\le \mathbb{R}e\rho\le \sigma_a$ for the series has to converge at $\rho$ and for $\sigma>\sigma_a$,
\[ \sum_{n=1}^\infty \frac{f(n)}{n^s} = \prod_p\frac{1}{1-\frac{f(p)}{p^s}}\ne 0. \]
Hence
\[ 0\le \mathbb{R}e\rho - \sigma_c\le \sigma_a-\sigma_c \le 1,\tag{2.2}\]
and $V\subset [0,1]$.\newline
\noindent
{\bf Step 2}\, We prove that $0\in V$. Define the completely multiplicative function $f$ on primes $p$ by
\[ f(p) = -\frac{1}{\log\log p}\qquad (p\ge 29)\]
and zero otherwise. As $29>e^e$, we have $-1\le f(p)\le 0$ for all $p$. Moreover, we have
\[ \sum_p \frac{f(p)}{p} = -\sum_{p\ge 29} \frac{1}{p\log\log p} = -\infty.\]
By applying Theorem 9 of \cite{KS1}, it follows that the Dirichlet series in (2.1) vanishes at $s=1$. To finish the proof, it suffices to verify that the abscissa of the series is 1.
Now
\[ \sum_{p\ge 29} \frac{1}{p^\sigma \log\log p} \asymp \log\log \frac{1}{\sigma -1} \quad\mbox{ $(1<\sigma<1+\frac{1}{100})$}.\]
As
\[ F(s) = \prod_{p\ge 29} \frac{1}{1+\frac{1}{p^s \log\log p}},\qquad (\sigma>1)\]
it follows that $\log F(\sigma) \asymp - \log\log\frac{1}{\sigma -1}$ and
\[ F(\sigma)\gg \frac{1}{( \log \frac{1}{\sigma-1})^2}.\]
As $F(1)=0$, $F(s)$ cannot be extended holomorphically to a neighbourhood of 1, and the result follows.\newline
\noindent
{\bf Remark 2}\, Here we have an example where $\sigma_c=\mathbb{R}e\rho = \sigma_a$.\newline
\noindent
{\bf Step 3}\, We prove the following: {\em if $f$ is completely multiplicative and $\beta^*+i\gamma$ is a zero of (2.1), then}
\[ (0,\beta^* - \sigma_c]\subset V.\]
To see this, let $f_1(n) = f(n)n^{-\sigma_c-i\gamma}$. The Dirichlet series for $f_1$ now has abscissa of convergence 0 and a real zero at some $\beta\ge 0$. We have to show that
\[ (0,\beta]\subset V.\tag{2.3}\]
Note that by (2.2), $\beta\le 1$. Let $\mathcal{P} = \{ p_1, p_2, \ldots \}$ where $p_k$ is an increasing sequence of prime numbers such that $p_k\asymp 2^k$ and let $\alpha\in (0,\beta)$.
Define a completely multiplicative function $g_\alpha$ by
\[ g_\alpha(p) = \left\{ \begin{array}{cl} p^\alpha & \mbox{ if $p\in\mathcal{P}$}\\ f_1(p) & \mbox{ if $p\not\in\mathcal{P}$} \end{array} \right. .\]
Let $\sigma >\alpha$. As $f_1(n)\ll n^\alpha$
\[ \sum_{p\in\mathcal{P}} \frac{|f_1(p)+g(p)|}{p^\sigma} \ll \sum_{p\in\mathcal{P}} \frac{1}{p^{\sigma-\alpha}}\asymp \sum_{k=1}^\infty \frac{1}{2^{(\sigma-\alpha)k}}<\infty.\]
It follows that the product
\[ H(s) = \prod_{p\in \mathcal{P}} \frac{1-\frac{f_1(p)}{p^s}}{1-\frac{g_\alpha(p)}{p^s}}\]
is absolutely convergent for $\sigma>\alpha$. Moreover, the Dirichlet series for $f_1$ and $g_\alpha$ are absolutely convergent for $\sigma>1$ and so, using Euler products, we have for $\sigma>1$
\[ \sum_{n=1}^\infty \frac{g_\alpha(n)}{n^s} = H(s) \sum_{n=1}^\infty \frac{f_1(n)}{n^s}.\]
It follows that the series on the left is actually convergent for $\sigma>\alpha$ and hence the above holds for $\sigma>\alpha$. Thus it is zero at $\beta$ and its abscissa of convergence is at most $\alpha$. However, $g_\alpha(p) = p^\alpha$ for $p\in\mathcal{P}$, so this abscissa is at least $\alpha$. Hence it equals $\alpha$. It follows that $\beta-\alpha\in V$. But $\alpha$ has been chosen arbitrarily in $(0,\beta)$, so (2.3) follows.\newline
\noindent
{\bf Step 4}\, $\frac{1}{2}\in V$. This follows immediately from the fact that the $L$-function associated to any non-principal Dirichlet character has zeros on the critical line and abscissa of convergence 0.\newline
These steps prove (i) and (ii) of Theorem 3. Now consider (iii).
By replacing $\mathbb{R}e\rho - \sigma_c$ with $\sigma_a-\mathbb{R}e\rho$ in (2.2) we obtain $W\subset [0,1]$.
Step 2 is also valid in the case of $W$ and shows $0\in W$. Next, the example in Step 4 (with $\sigma_a=1$) shows that $\frac{1}{2}\in W$.
By Theorem 2, we have $(0,\frac{19}{40})\subset W$, which concludes the proof of part (iii).
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf {\large 3. Proof of Theorem 0}}\newline
{\em Proof of} (a).\, Let $\rho = a+ib\in Z_{\mathcal{P}}$. Since for all $s$
\[ \sum_{n=1}^\infty \frac{\lambda_\emptyset(n)}{n^s}=1\ne 0,\]
we have $\mathcal{P}\ne\emptyset$.
Since $\lim_{n\to\infty} \frac{\lambda_\mathcal{P}(n)}{n^\rho}=0$, it follows that
\[ a+ib\in Z_{\mathcal{P}}\implies a>0.\tag{3.1}\]
Now we call a completely multiplicative function $h$ CMO if $\sum_{n=1}^\infty h(n)=0$. We suppose again that $\rho = a+ib\in Z_{\mathcal{P}}$. On one hand it means that $f(n):=\frac{\lambda_\mathcal{P}(n)}{n^\rho}$ is CMO. On the other hand, by (3.1), we have $a>0$. Thus for $g(n):=\frac{\lambda_{\mathcal{P}^\prime}(n)}{n^\rho}$, $g$ is completely multiplicative such that for all primes $p$, $|g(p)|<1$. As
$\sigma_c(\sum_{p\in\mathcal{P}\mathbb{D}elta \mathcal{P}^\prime} p^{-s})\le 0$, we have also $\sum_p |g(p)-f(p)|<\infty$. By using th\'{e}or\`{e}me 3 of \cite{KS1}, it follows that $g$ is CMO. In other words, $\rho\in Z_{\mathcal{P}^\prime}$, and (a) is proven.
Part (b) comes from the fact that $\lambda_\mathcal{P}$ is a real function.
Part (c) follows from the combination of $Z_\emptyset=\emptyset$ and part (a).
Part (d) follows from the combination of (3.1), (1.10), (1.9), (1.6) and (1.8).
Part (e) comes from th\'{e}or\`{e}me 9 of \cite{KS1}.
\noindent
{\em Proof of} (f).\, By (e), we know that $1\in Z_\mathbb{P}$.
To show that it contains no other points, let $\rho\in Z_\mathbb{P}$. Since the abscissa of convergence of $\sum_{n=1}^\infty\frac{\lambda(n)}{n^s}$ is at least $\frac{1}{2}$ due to the existence of Riemann zeros, it follows that Re $\rho\ge\frac{1}{2}$. Now, by Abel's Theorem
\[ 0=\sum_{n=1}^\infty\frac{\lambda(n)}{n^\rho} =\lim_{\varepsilon\to 0+}\sum_{n=1}^\infty\frac{\lambda(n)}{n^{\rho+\varepsilon}} = \lim_{\varepsilon\to 0+}\frac{\zeta(2\rho+2\varepsilon)}{\zeta(\rho+\varepsilon)}.\]
But $\zeta(2s)\ne 0$ for $\sigma\ge\frac{1}{2}$, so $\rho$ must be a pole of $\zeta$; i.e. $\rho =1$.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf {\large 4. Architecture of the proofs of Theorems 1 and 2}}\newline
Let $a\in (0,1)$. We begin by recalling the proof in \cite{KS2} of the existence of a set of primes $\mathcal{P}_a$ such that $a$ belongs to $Z_{\mathcal{P}_a}$. We know that $1$ is a zero of $F(s):=\frac{\zeta(2s)}{\zeta(s)}$. It follows that $a$ is a zero of $F(s/a)$. But
\[ F\Bigl(\frac{s}{a}\Bigr) = \frac{\zeta_{\mathbb{P}^{1/a}}(2s)}{\zeta_{\mathbb{P}^{1/a}}(s)},\]
where $\zeta_{\mathbb{P}^{1/a}}(s)$ is the zeta function associated to the set of Beurling primes $\mathbb{P}^{1/a} = \{p^{1/a}:p\in\mathbb{P}\}$. This set is sparse. It is the reason why the PNT allows us to approximate $\mathbb{P}^{1/a}$ by a subset $\mathcal{P}_a$ of $\mathbb{P}$ such that
\[ a\mbox{ is a zero of } \frac{\zeta_{\mathcal{P}_a}(2s)}{\zeta_{\mathcal{P}_a}(s)}.\tag{4.1}\]
We have
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_a}(n)}{n^s} = \frac{\zeta_{\mathcal{P}_a}(2s)}{\zeta_{\mathcal{P}_a}(s)}, \qquad (\sigma>a).\tag{4.2}\]
As for the usual case $a=1$ and $\mathcal{P}_1=\mathbb{P}$, we prove that we can deduce the wanted formula
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_a}(n)}{n^a} =0\tag{4.3}\]
from the combination of (4.1) and (4.2).
The architecture of the proofs of Theorems 1 and 2 is similar, but we need to introduce two new tools. Under RH, we recall that we know an example which answers Balazard's question. It is
\[ F(s) = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s} = \frac{\zeta(2s)}{\zeta(s)}\qquad \mbox{ for $\sigma>\frac{1}{2}$}\tag{4.4}\]
for which $s=1$ is the only (simple) zero of $F$ in the open half-plane where $\sigma>\frac{1}{2}$. But we do not know that RH is true. To get an unconditional example, we prove that it is possible to choose a set of $\mathcal{P}$ of primes such that (4.4) is replaced by
\[ F_\mathcal{P}(s) = \sum_{n=1}^\infty \frac{\lambda_\mathcal{P}(n)}{n^s} = \frac{\zeta_\mathcal{P}(2s)}{\zeta_\mathcal{P}(s)}\qquad \mbox{ for $\sigma>\sigma_c(F_\mathcal{P})$}\tag{4.5}\]
with $\sigma_c(F_\mathcal{P})<1$, and for which $s=1$ is always a simple zero.
More precisely, instead of working with the usual zeta function, we begin to work with Zhang's zeta function $\zeta_\mathcal{R}(s)$ (see \cite{Z} or \cite{DZ}) associated to an appropriate multiset $\mathcal{R}$ of generalized primes which share some of the properties of $\zeta(s)$ under RH:
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Part (b) comes from the fact that $\lambda_\mathcal{P}$ is a real function.
Part (c) follows from the combination of $Z_\emptyset=\emptyset$ and part (a).
Part (d) follows from the combination of (3.1), (1.10), (1.9), (1.6) and (1.8).
Part (e) comes from th\'{e}or\`{e}me 9 of \cite{KS1}.
\noindent
{\em Proof of} (f).\, By (e), we know that $1\in Z_\mathbb{P}$.
To show that it contains no other points, let $\rho\in Z_\mathbb{P}$. Since the abscissa of convergence of $\sum_{n=1}^\infty\frac{\lambda(n)}{n^s}$ is at least $\frac{1}{2}$ due to the existence of Riemann zeros, it follows that Re $\rho\ge\frac{1}{2}$. Now, by Abel's Theorem
\[ 0=\sum_{n=1}^\infty\frac{\lambda(n)}{n^\rho} =\lim_{\varepsilon\to 0+}\sum_{n=1}^\infty\frac{\lambda(n)}{n^{\rho+\varepsilon}} = \lim_{\varepsilon\to 0+}\frac{\zeta(2\rho+2\varepsilon)}{\zeta(\rho+\varepsilon)}.\]
But $\zeta(2s)\ne 0$ for $\sigma\ge\frac{1}{2}$, so $\rho$ must be a pole of $\zeta$; i.e. $\rho =1$.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf {\large 4. Architecture of the proofs of Theorems 1 and 2}}\newline
Let $a\in (0,1)$. We begin by recalling the proof in \cite{KS2} of the existence of a set of primes $\mathcal{P}_a$ such that $a$ belongs to $Z_{\mathcal{P}_a}$. We know that $1$ is a zero of $F(s):=\frac{\zeta(2s)}{\zeta(s)}$. It follows that $a$ is a zero of $F(s/a)$. But
\[ F\Bigl(\frac{s}{a}\Bigr) = \frac{\zeta_{\mathbb{P}^{1/a}}(2s)}{\zeta_{\mathbb{P}^{1/a}}(s)},\]
where $\zeta_{\mathbb{P}^{1/a}}(s)$ is the zeta function associated to the set of Beurling primes $\mathbb{P}^{1/a} = \{p^{1/a}:p\in\mathbb{P}\}$. This set is sparse. It is the reason why the PNT allows us to approximate $\mathbb{P}^{1/a}$ by a subset $\mathcal{P}_a$ of $\mathbb{P}$ such that
\[ a\mbox{ is a zero of } \frac{\zeta_{\mathcal{P}_a}(2s)}{\zeta_{\mathcal{P}_a}(s)}.\tag{4.1}\]
We have
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_a}(n)}{n^s} = \frac{\zeta_{\mathcal{P}_a}(2s)}{\zeta_{\mathcal{P}_a}(s)}, \qquad (\sigma>a).\tag{4.2}\]
As for the usual case $a=1$ and $\mathcal{P}_1=\mathbb{P}$, we prove that we can deduce the wanted formula
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_a}(n)}{n^a} =0\tag{4.3}\]
from the combination of (4.1) and (4.2).
The architecture of the proofs of Theorems 1 and 2 is similar, but we need to introduce two new tools. Under RH, we recall that we know an example which answers Balazard's question. It is
\[ F(s) = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s} = \frac{\zeta(2s)}{\zeta(s)}\qquad \mbox{ for $\sigma>\frac{1}{2}$}\tag{4.4}\]
for which $s=1$ is the only (simple) zero of $F$ in the open half-plane where $\sigma>\frac{1}{2}$. But we do not know that RH is true. To get an unconditional example, we prove that it is possible to choose a set of $\mathcal{P}$ of primes such that (4.4) is replaced by
\[ F_\mathcal{P}(s) = \sum_{n=1}^\infty \frac{\lambda_\mathcal{P}(n)}{n^s} = \frac{\zeta_\mathcal{P}(2s)}{\zeta_\mathcal{P}(s)}\qquad \mbox{ for $\sigma>\sigma_c(F_\mathcal{P})$}\tag{4.5}\]
with $\sigma_c(F_\mathcal{P})<1$, and for which $s=1$ is always a simple zero.
More precisely, instead of working with the usual zeta function, we begin to work with Zhang's zeta function $\zeta_\mathcal{R}(s)$ (see \cite{Z} or \cite{DZ}) associated to an appropriate multiset $\mathcal{R}$ of generalized primes which share some of the properties of $\zeta(s)$ under RH:
\begin{itemize}
\item $\zeta_\mathcal{R}(s) = \prod_{r\in \mathcal{R}} \frac{1}{1-\frac{1}{r^s}}$ is normally convergent in every half-plane $\sigma>1+\varepsilon$ with $\varepsilon>0$;
\item $\zeta_\mathcal{R}(s)$ has a non-vanishing meromorphic continuation of finite order to $\sigma>\frac{1}{2}$ with a unique simple pole at 1 with residue 1.
\end{itemize}
Now, more generally than in \cite{KS2}, we work here with the group of meromorphic functions in some non-empty open vertical half-plane generated by the function $\zeta_\mathcal{R}(\lambda s)$ where $\lambda>0$.
To prove Theorem 1 we use the function
\[ \frac{\zeta_\mathcal{R}(s/b)}{\zeta_\mathcal{R}(s/a)}\]
which has a simple zero at $a$ and a simple pole at $b$. To prove Theorem 2, we use the function
\[ \frac{1}{\zeta_\mathcal{R}(s/a)\zeta_\mathcal{R}(s/b)}\]
which has two simple zeros at $a$ and $b$. To finish off the proofs, we need to approximate these meromorphic functions by functions of the form
\[ \frac{\zeta_{\mathcal{P}_{a,b}}(2s)}{\zeta_{\mathcal{P}_{a,b}}(s)}\]
with $\mathcal{P}_{a,b}\subset\mathbb{P}$ such that this function has the same zeros and poles as described above and such that the formula
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_{a,b}}(n)}{n^s} = \frac{\zeta_{\mathcal{P}_{a,b}}(2s)}{\zeta_{\mathcal{P}_{a,b}}(s)}\]
is valid for $\sigma>b$ in the first case and for $s=a,b$ in the second.
To do this approximation, instead of using the PNT as in \cite{KS2}, we use the Baker, Harman and Pintz result on primes in short intervals
\[ \pi(x+y) - \pi(x) \asymp \frac{y}{\log x}\quad\mbox{ for $x^{\frac{21}{40}}\le y\le x$ and $x$ large enough.}\]
It is this number $\frac{21}{40}$ which explains the number $\frac{19}{40}$ in our results.
\newline
\noindent
{\bf {\large 5. Vocabulary, notations and results for Beurling primes}}\newline
In 1937, Beurling \cite{Be} (see also \cite{DZ}) had the idea to generalize the usual couple $(\mathbb{P}, \mathbb{N})$ formed by the usual sets $\mathbb{P}$ of primes and $\mathbb{N}$ of positive integers in the following way. He considers any multiset $\mathcal{P}$ of $(1,\infty)$ which is locally finite in $[1,\infty)$. The elements of $\mathcal{P}$ are called the {\em generalized} primes. He defines $\mathcal{N}$ to be the multiset of $[1,\infty)$ formed by the finite product of elements of $\mathcal{P}$ (the number 1 occurs as the product indexed by the subset $\emptyset$). We will talk of a discrete generalized prime system or just g-prime system for such a couple $(\mathcal{P}, \mathcal{N})$.
Let us mention that there are two natural generalizations of g-prime systems which we will not use directly here. We refer the interested reader to \cite{Hi}.
Let $(\mathcal{P}, \mathcal{N})$ be a g-prime system. Every element $n$ of the multiset $\mathcal{N}$ has a unique decomposition in generalized primes
\[ n=\prod_{p\in\mathcal{P}} p^{v_p(n)}.\]
We define the generalized M\"{o}bius function $\mu_\mathcal{P}$ on the multiset $\mathcal{N}$ by the formula
\[ \mu_\mathcal{P}(n) = \left\{ \begin{array}{cl} 0 & \mbox{ if $\exists p\in\mathcal{P}$ with $v_\mathcal{P}(n)\ge 2$}\\ (-1)^{\sum_{p\in\mathcal{P}}v_\mathcal{P}(n)} & \mbox{ if not}\end{array} \right. .\]
When $\mathcal{P}=\mathbb{P}$, $\mu_\mathcal{P}$ is the usual M\"{o}bius function.
Notice that for any sequence $(a_n)_{n\in\mathcal{N}}$ of complex numbers defined on $\mathcal{N}$, the function
\[ \sum_{n\in\mathcal{N}} \frac{a_n}{n^s}\]
is a generalized Dirichlet series. We shall use, without referencing anymore, the definitions and properties of generalized Dirichlet series. (See \cite{HR} for the theory of generalized Dirichlet series.)
Notice that in the introduction, we considered the particular example of a g-prime system $(\mathcal{P}, \mathcal{N})$ with $\mathcal{P}\subset\mathbb{P}$. The definition of $\zeta_\mathcal{P}(s)$ generalizes with no difficulty for any g-prime system. If the series $\sum_{n\in\mathcal{N}} n^{-s}$ has a finite abscissa of convergence $\sigma_c$, then
\[ \zeta_\mathcal{P}(s):= \sum_{n\in\mathcal{N}}\frac{1}{n^s} = \prod_{p\in\mathcal{P}} \frac{1}{1-\frac{1}{p^s}}\]
defines a non-vanishing holomorphic function for $\sigma>\sigma_c$. If $\zeta_\mathcal{P}(s)$ has a meromorphic continuation to some open set across the line $\sigma=\sigma_c$ we continue to write $\zeta_\mathcal{P}(s)$ for this continuation.
If $\mathcal{P},\mathcal{P}_1, \mathcal{P}_2$ are multisets of primes and $a>0$, we have
\[ \zeta_{\mathcal{P}_1\sqcup \mathcal{P}_2}(s) = \zeta_{\mathcal{P}_1} (s)\zeta_{\mathcal{P}_2}(s)\]
and
\[ \zeta_{\mathcal{P}^{1/a}}(s) = \zeta_\mathcal{P}(s/a).\]
Let $Q=\{q_1,q_2,\ldots \}$ be an infinite multiset of generalized primes, $I:Q\to (1,\infty)$ and $\sigma_0\ge 0$. We shall say that
\[ \prod_{q\in Q} \frac{1-\frac{1}{I(q)^s}}{1-\frac{1}{q^s}} \tag{5.1}\]
is a {\em simply absolutely convergent quotient of generalized Euler products in} $\sigma>\sigma_0$ if for all $s$ in the half-plane where $\sigma>\sigma_0$ the infinite product
\[ \prod_{k=1}^\infty \frac{1-\frac{1}{I(q_k)^s}}{1-\frac{1}{q_k^s}}\]
is absolutely convergent. Notice that the expression in (5.1) represents a non-vanishing holomorphic function in this half-plane.
We shall denote multisets of generalized integers by calligraphic letters ($\mathcal{A},\mathcal{B},\mathcal{C}, \ldots$) with the corresponding counting function by its capital equivalent.
eg for such a multiset $\mathcal{A}$, let
\[ A(x) = \sum_{\tiny \begin{array}{c} a \le x \\ a\in \mathcal{A}\end{array} } 1.\]
Moreover, if $c>0$, we write
\[ A^{1/c}(x): = | \{ a\in \mathcal{A}:a^{1/c}\le x\}| = A(x^c).\]
For $\mathbb{P}$ however, we shall keep the traditional notation $\pi(x)$ and $[x]$ for the counting funtions of the primes and natural numbers.\newline
\noindent
{\bf Definition.}\, Let $(\mathcal{R}, \mathcal{N})$ be a g-prime system. We say it is {\em good} if it satisfies the following properties:
\begin{align*}
N(x) & = x+O_\varepsilon(x^{1/2+\varepsilon})\tag{5.2}\\
R(x) & = \mbox{\rm{li}}(x)+O_\varepsilon(x^{1/2+\varepsilon})\tag{5.3}
\end{align*}
for all $\varepsilon>0$\footnote{Here li$(x)$ is the usual {\em logarithmic integral}, given by li$(x)=\int_2^x \frac{1}{\log y}\, dy$.}. As such, $\zeta_\mathcal{R}(s)$ has a non-vanishing meromorphic continuation to the half-plane $\sigma>\frac{1}{2}$, with exactly one (simple) pole at $1$ with residue 1.\newline
\noindent
{\bf Remark 3}\, Of course, under the Riemann Hypothesis, the basic g-prime system $(\mathbb{P}, \mathbb{N})$ is good.\newline
\noindent
{\bf Remark 4}\, Comparing to Zhang's work (\cite{Z} or \cite{DZ}) we changed conditions (5.2) and (5.3) a little.\newline
Now Zhang proved that, unconditionally, good systems exist (see \cite{DZ}, Theorem 17.11 and remark 17.12). \newline
\noindent
{\bf Theorem} (Zhang)\newline
{\em A good g-prime system exists.}\newline
\noindent
{\bf Lemma 5.1}\newline
{\em Let $(\cal{R}, \cal{N})$ be a good g-prime system Then for all $\varepsilon>0$, there exists $C>0$ such that }
\begin{align*}
|\zeta_{\cal{R}}(s)| & \le \exp \{ C(\log |t|)^{2(1-\sigma)+\varepsilon}\} \tag{$\frac{1}{2}+\varepsilon\le\sigma\le 1, |t|\ge 2$}\\
|\zeta^{-1}_{\cal{R}}(s)| &\le \exp\{ C(\log (|t|+2))^{2(1-\sigma)+\varepsilon}\} \tag{$\frac{1}{2}+\varepsilon\le\sigma\le 1, t\in\mathbb{R}$}
\end{align*}
This follows from Theorem 2.3 of \cite{HL}.\newline
\noindent
{\bf {\large 6. Primes in short intervals}}\newline
We begin with the usual primes.\newline
\noindent
{\bf Lemma 6.1} (Baker, Harman and Pintz)\newline
{\em We have, for $x^{\frac{21}{40}}\le y\le x$ and $x$ large enough
\[ \pi(x+y)-\pi(x)\ge \frac{y}{12\log x}.\]
Proof.}\, This is a consequence of Theorem 10.8 of \cite{Ha}.\newline\phantom{a}
$\Box$\newline
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Notice that in the introduction, we considered the particular example of a g-prime system $(\mathcal{P}, \mathcal{N})$ with $\mathcal{P}\subset\mathbb{P}$. The definition of $\zeta_\mathcal{P}(s)$ generalizes with no difficulty for any g-prime system. If the series $\sum_{n\in\mathcal{N}} n^{-s}$ has a finite abscissa of convergence $\sigma_c$, then
\[ \zeta_\mathcal{P}(s):= \sum_{n\in\mathcal{N}}\frac{1}{n^s} = \prod_{p\in\mathcal{P}} \frac{1}{1-\frac{1}{p^s}}\]
defines a non-vanishing holomorphic function for $\sigma>\sigma_c$. If $\zeta_\mathcal{P}(s)$ has a meromorphic continuation to some open set across the line $\sigma=\sigma_c$ we continue to write $\zeta_\mathcal{P}(s)$ for this continuation.
If $\mathcal{P},\mathcal{P}_1, \mathcal{P}_2$ are multisets of primes and $a>0$, we have
\[ \zeta_{\mathcal{P}_1\sqcup \mathcal{P}_2}(s) = \zeta_{\mathcal{P}_1} (s)\zeta_{\mathcal{P}_2}(s)\]
and
\[ \zeta_{\mathcal{P}^{1/a}}(s) = \zeta_\mathcal{P}(s/a).\]
Let $Q=\{q_1,q_2,\ldots \}$ be an infinite multiset of generalized primes, $I:Q\to (1,\infty)$ and $\sigma_0\ge 0$. We shall say that
\[ \prod_{q\in Q} \frac{1-\frac{1}{I(q)^s}}{1-\frac{1}{q^s}} \tag{5.1}\]
is a {\em simply absolutely convergent quotient of generalized Euler products in} $\sigma>\sigma_0$ if for all $s$ in the half-plane where $\sigma>\sigma_0$ the infinite product
\[ \prod_{k=1}^\infty \frac{1-\frac{1}{I(q_k)^s}}{1-\frac{1}{q_k^s}}\]
is absolutely convergent. Notice that the expression in (5.1) represents a non-vanishing holomorphic function in this half-plane.
We shall denote multisets of generalized integers by calligraphic letters ($\mathcal{A},\mathcal{B},\mathcal{C}, \ldots$) with the corresponding counting function by its capital equivalent.
eg for such a multiset $\mathcal{A}$, let
\[ A(x) = \sum_{\tiny \begin{array}{c} a \le x \\ a\in \mathcal{A}\end{array} } 1.\]
Moreover, if $c>0$, we write
\[ A^{1/c}(x): = | \{ a\in \mathcal{A}:a^{1/c}\le x\}| = A(x^c).\]
For $\mathbb{P}$ however, we shall keep the traditional notation $\pi(x)$ and $[x]$ for the counting funtions of the primes and natural numbers.\newline
\noindent
{\bf Definition.}\, Let $(\mathcal{R}, \mathcal{N})$ be a g-prime system. We say it is {\em good} if it satisfies the following properties:
\begin{align*}
N(x) & = x+O_\varepsilon(x^{1/2+\varepsilon})\tag{5.2}\\
R(x) & = \mbox{\rm{li}}(x)+O_\varepsilon(x^{1/2+\varepsilon})\tag{5.3}
\end{align*}
for all $\varepsilon>0$\footnote{Here li$(x)$ is the usual {\em logarithmic integral}, given by li$(x)=\int_2^x \frac{1}{\log y}\, dy$.}. As such, $\zeta_\mathcal{R}(s)$ has a non-vanishing meromorphic continuation to the half-plane $\sigma>\frac{1}{2}$, with exactly one (simple) pole at $1$ with residue 1.\newline
\noindent
{\bf Remark 3}\, Of course, under the Riemann Hypothesis, the basic g-prime system $(\mathbb{P}, \mathbb{N})$ is good.\newline
\noindent
{\bf Remark 4}\, Comparing to Zhang's work (\cite{Z} or \cite{DZ}) we changed conditions (5.2) and (5.3) a little.\newline
Now Zhang proved that, unconditionally, good systems exist (see \cite{DZ}, Theorem 17.11 and remark 17.12). \newline
\noindent
{\bf Theorem} (Zhang)\newline
{\em A good g-prime system exists.}\newline
\noindent
{\bf Lemma 5.1}\newline
{\em Let $(\cal{R}, \cal{N})$ be a good g-prime system Then for all $\varepsilon>0$, there exists $C>0$ such that }
\begin{align*}
|\zeta_{\cal{R}}(s)| & \le \exp \{ C(\log |t|)^{2(1-\sigma)+\varepsilon}\} \tag{$\frac{1}{2}+\varepsilon\le\sigma\le 1, |t|\ge 2$}\\
|\zeta^{-1}_{\cal{R}}(s)| &\le \exp\{ C(\log (|t|+2))^{2(1-\sigma)+\varepsilon}\} \tag{$\frac{1}{2}+\varepsilon\le\sigma\le 1, t\in\mathbb{R}$}
\end{align*}
This follows from Theorem 2.3 of \cite{HL}.\newline
\noindent
{\bf {\large 6. Primes in short intervals}}\newline
We begin with the usual primes.\newline
\noindent
{\bf Lemma 6.1} (Baker, Harman and Pintz)\newline
{\em We have, for $x^{\frac{21}{40}}\le y\le x$ and $x$ large enough
\[ \pi(x+y)-\pi(x)\ge \frac{y}{12\log x}.\]
Proof.}\, This is a consequence of Theorem 10.8 of \cite{Ha}.\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Lemma 6.2} \newline
{\em Let $(\mathcal{R},\mathcal{N})$ be a good g-prime system, $c\in (0,1)$, $h\in (0,1)$ and $\varepsilon>0$. Then
\[ R^{1/c}(x+x^h)-R^{1/c}(x) = \frac{x^{c+h-1}}{\log x} + O_\varepsilon(x^{\max\{\frac{c}{2}+\varepsilon,c+2h-2\}}).\]
Proof.}\, We have
\[ (x+x^h)^c = x^c\Bigl(1+\frac{1}{x^{1-h}}\Bigr)^c = x^c + cx^{c+h-1} + O(x^{c+2h-2}).\]
As $(\mathcal{R},\mathcal{N})$ be a good g-prime system, we have
\begin{align*}
R^{1/c}(x+x^h)-R^{1/c}(x) & = R((x+x^h)^c)-R(x^c) = \int_{x^c}^{(x+x^h)^c}\frac{1}{\log t}\, dt + O_\varepsilon(x^{c/2+\varepsilon})\\
& = \frac{x^{c+h-1}}{\log x} + O_\varepsilon(x^{\max\{\frac{c}{2}+\varepsilon,c+2h-2\}}).
\end{align*}
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Lemma 6.3} \newline
{\em Let $\mathcal{Q}$ and $\mathcal{Q}^*$ be two multisets of generalized primes. Let $\delta>0$ and $0<h\le 1$ be two real numbers such that
\[ Q(x)\ll x^\delta\tag{6.1}\]
\[ \lim_{x\to\infty} (Q^*(x)-Q(x))=\infty,\tag{6.2}\]
and for $x$ large enough,
\[ Q^*(x+x^h)-Q^*(x)\ge Q(x+x^h)-Q(x).\tag{6.3}\]
Then there exists an injection $I:\mathcal{Q}\to \mathcal{Q}^*$ such that
\[ \prod_{q\in \mathcal{Q}} \frac{1-\frac{1}{I(q)^s}}{1-\frac{1}{q^s}} = \frac{\zeta_{\mathcal{Q}}(s)}{\zeta_{I(\mathcal{Q})}(s)}\]
is a simply absolutely convergent quotient of generalized Euler products in $\sigma>\max\{\delta+h-1,0\}$.}
\newline
\noindent
{\em Proof.}\, Let $T_n$ be an increasing sequence of positive reals defined by $T_1=1$ and
\[ T_{n+1}=T_n+T_n^h \qquad (n\ge 1).\]
By (6.2) and (6.3) there exists a positive integer $n_0$ such that there is an injection $I_0:{\cal{Q}} \cap (1,T_{n_0}] \to {\cal{Q}}^* \cap (1,T_{n_0}]$ and for all $n\ge n_0$, there is also an injection
\[I_n: \mathcal{Q}\cap (T_n,T_{n+1}] \to \mathcal{Q}^*\cap (T_n,T_{n+1}].\]
As the intervals $(T_n,T_{n+1}]$ are disjoint, we get a global injection $I:\cal{Q} \to \cal{Q}^*$ such that $I(q)=q+O(q^h)$.
For fixed $s$ with $\sigma>0$, we have, uniformly in $q$,
\[ I(q)^s = q^s\Big(1+ O\Bigl(\frac{1}{q^{1-h}}\Bigr)\Bigr).\]
Thus
\[ \frac{1-\frac{1}{I(q)^s}}{1-\frac{1}{q^s}} = 1+ O\Bigl(\frac{1}{q^{\sigma+1-h}}\Bigr),\]
and using (6.1)
\[\log\biggl( \prod_{q\in \mathcal{Q}} \frac{1-\frac{1}{I(q)^s}}{1-\frac{1}{q^s}}\biggr) \ll \sum_{q\in\mathcal{Q}}\frac{1}{q^{\sigma+1-h}} = \sum_{n=0}^\infty\sum_{2^n<q\le 2^{n+1}}\frac{1}{q^{\sigma+1-h}} \ll \sum_{n=0}^\infty \frac{2^{\delta n}}{2^{(\sigma+1-h)n}} <\infty\]
if $\sigma>\delta+h-1$, as required.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf {\large 7. Abscissae of convergence}}\newline
Let $(\cal{R},\cal{N})$ be a good g-prime system and $c\in (\frac{1}{2},1)$. Let $\mathcal{Q}_c:=\mathcal{R}\cup \mathcal{R}^{1/c}$ and $\mathcal{M}_c$ the multiset of generalized integers associated to $\mathcal{Q}_c$.
\newline
\noindent
{\bf Lemma 7.1}\newline
{\em Let $(\cal{R},\cal{N})$ be a good g-prime system. Then}
\begin{align*}
\sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} &= \frac{x^{1-\alpha}}{1-\alpha} +
\left\{ \begin{array}{cl} O(x^{\frac{1}{2}-\alpha+\varepsilon}) & \mbox{ if $\alpha\in (0,\frac{1}{2}]$}\\ \zeta_{\mathcal{R}}(\alpha) + O(x^{\frac{1}{2}-\alpha+\varepsilon}) & \mbox{ if $\alpha\in (\frac{1}{2},1)$}\end{array} \right. \\
\sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} &= \zeta_{\mathcal{R}}(\alpha) - \frac{1}{(\alpha-1)x^{\alpha-1}} +O(x^{\frac{1}{2}-\alpha+\varepsilon}) \quad\mbox{ if $\alpha>1$.}
\end{align*}
{\em Proof.}\, These follow from writing
\[ \sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} = \int_{1-}^x \frac{1}{t^\alpha}\, dN(t) = \frac{N(x)}{x^\alpha}+\alpha\int_1^x \frac{N(t)}{t^{\alpha+1}}\, dt\]
and using the fact that $N(t)=t+O(t^{\frac{1}{2}+\varepsilon})$. Thus, for $\alpha\ne 1$,
\[ \sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} =x^{1-\alpha} + O(x^{\frac{1}{2}-\alpha+\varepsilon}) +
\frac{\alpha(x^{1-\alpha}-1)}{1-\alpha} +\alpha\int_1^x \frac{N(t)-t}{t^{\alpha+1}}\, dt.\]
If $\alpha\le\frac{1}{2}$, the integral is $O(x^{\frac{1}{2}-\alpha+\varepsilon})$ and the result follows. If $\alpha>\frac{1}{2}$, the integral is
\[ \alpha\int_1^\infty \frac{N(t)-t}{t^{\alpha+1}}\, dt -\alpha\int_x^\infty \frac{N(t)-t}{t^{\alpha+1}}\, dt = \alpha\int_1^\infty \frac{N(t)-t}{t^{\alpha+1}}\, dt+ O(x^{\frac{1}{2}-\alpha+\varepsilon}).\]
Now for $\alpha>1$, the integral on the right is just $\zeta_{\mathcal{R}}(\alpha) - \frac{\alpha}{\alpha-1}$. But by analytic continuation, this still holds for $\alpha\in (\frac{1}{2},1)$.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Lemma 7.2}\newline
{\em Let $(\cal{R},\cal{N})$ be a good g-prime system and $c\in (\frac{1}{2},1)$. Then for all fixed $\varepsilon>0$, we have
\[ M_c(x) = \zeta_\mathcal{R}\Bigl(\frac{1}{c}\Bigr)x + \zeta_{\mathcal{R}}(c)x^c
+O_\varepsilon(x^{\frac{3c}{2(1+c)}+\varepsilon}).\]
Proof.}\, Throughout this proof, $k$ and $\ell$ will implicitly be used to denote generalized integers of $\mathcal{N}$. We use Dirichlet's hyperbola method. Thus for every positive reals $x$ and $y$, we have
\begin{align*}
M_c(x) & = \sum_\ell \sum_{k\ell^{1/c}\le x} 1 = \sum_{\ell^{1/c}\le y} \sum_{k\le\frac{x}{\ell^{1/c}}} 1 + \sum_{k\le \frac{x}{y}} \sum_{\ell^{1/c}\le \frac{x}{k}} 1 - \sum_{\ell^{1/c}\le y} \sum_{k\le\frac{x}{y}} 1\\
& = S_1+S_2-S_3,
\end{align*}
where
\[ S_1 = \sum_{\ell^{1/c}\le y} N\Bigl(\frac{x}{\ell^{1/c}}\Bigr), \quad S_2=\sum_{k\le \frac{x}{y}} N\Bigl(\Bigl(\frac{x}{k}\Bigr)^c\Bigr) , \quad S_3 = N(y^c)N\Bigl(\frac{x}{y}\Bigr).\]
Now we use the formula $N(t)=t+O(t^{\frac{1}{2}+\varepsilon})$ to estimate these three sums, and at the end we optimize in $y$.
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\noindent
{\bf {\large 7. Abscissae of convergence}}\newline
Let $(\cal{R},\cal{N})$ be a good g-prime system and $c\in (\frac{1}{2},1)$. Let $\mathcal{Q}_c:=\mathcal{R}\cup \mathcal{R}^{1/c}$ and $\mathcal{M}_c$ the multiset of generalized integers associated to $\mathcal{Q}_c$.
\newline
\noindent
{\bf Lemma 7.1}\newline
{\em Let $(\cal{R},\cal{N})$ be a good g-prime system. Then}
\begin{align*}
\sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} &= \frac{x^{1-\alpha}}{1-\alpha} +
\left\{ \begin{array}{cl} O(x^{\frac{1}{2}-\alpha+\varepsilon}) & \mbox{ if $\alpha\in (0,\frac{1}{2}]$}\\ \zeta_{\mathcal{R}}(\alpha) + O(x^{\frac{1}{2}-\alpha+\varepsilon}) & \mbox{ if $\alpha\in (\frac{1}{2},1)$}\end{array} \right. \\
\sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} &= \zeta_{\mathcal{R}}(\alpha) - \frac{1}{(\alpha-1)x^{\alpha-1}} +O(x^{\frac{1}{2}-\alpha+\varepsilon}) \quad\mbox{ if $\alpha>1$.}
\end{align*}
{\em Proof.}\, These follow from writing
\[ \sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} = \int_{1-}^x \frac{1}{t^\alpha}\, dN(t) = \frac{N(x)}{x^\alpha}+\alpha\int_1^x \frac{N(t)}{t^{\alpha+1}}\, dt\]
and using the fact that $N(t)=t+O(t^{\frac{1}{2}+\varepsilon})$. Thus, for $\alpha\ne 1$,
\[ \sum_{\tiny \begin{array}{c} n \le x \\ n\in \mathcal{N}\end{array} } \frac{1}{n^\alpha} =x^{1-\alpha} + O(x^{\frac{1}{2}-\alpha+\varepsilon}) +
\frac{\alpha(x^{1-\alpha}-1)}{1-\alpha} +\alpha\int_1^x \frac{N(t)-t}{t^{\alpha+1}}\, dt.\]
If $\alpha\le\frac{1}{2}$, the integral is $O(x^{\frac{1}{2}-\alpha+\varepsilon})$ and the result follows. If $\alpha>\frac{1}{2}$, the integral is
\[ \alpha\int_1^\infty \frac{N(t)-t}{t^{\alpha+1}}\, dt -\alpha\int_x^\infty \frac{N(t)-t}{t^{\alpha+1}}\, dt = \alpha\int_1^\infty \frac{N(t)-t}{t^{\alpha+1}}\, dt+ O(x^{\frac{1}{2}-\alpha+\varepsilon}).\]
Now for $\alpha>1$, the integral on the right is just $\zeta_{\mathcal{R}}(\alpha) - \frac{\alpha}{\alpha-1}$. But by analytic continuation, this still holds for $\alpha\in (\frac{1}{2},1)$.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Lemma 7.2}\newline
{\em Let $(\cal{R},\cal{N})$ be a good g-prime system and $c\in (\frac{1}{2},1)$. Then for all fixed $\varepsilon>0$, we have
\[ M_c(x) = \zeta_\mathcal{R}\Bigl(\frac{1}{c}\Bigr)x + \zeta_{\mathcal{R}}(c)x^c
+O_\varepsilon(x^{\frac{3c}{2(1+c)}+\varepsilon}).\]
Proof.}\, Throughout this proof, $k$ and $\ell$ will implicitly be used to denote generalized integers of $\mathcal{N}$. We use Dirichlet's hyperbola method. Thus for every positive reals $x$ and $y$, we have
\begin{align*}
M_c(x) & = \sum_\ell \sum_{k\ell^{1/c}\le x} 1 = \sum_{\ell^{1/c}\le y} \sum_{k\le\frac{x}{\ell^{1/c}}} 1 + \sum_{k\le \frac{x}{y}} \sum_{\ell^{1/c}\le \frac{x}{k}} 1 - \sum_{\ell^{1/c}\le y} \sum_{k\le\frac{x}{y}} 1\\
& = S_1+S_2-S_3,
\end{align*}
where
\[ S_1 = \sum_{\ell^{1/c}\le y} N\Bigl(\frac{x}{\ell^{1/c}}\Bigr), \quad S_2=\sum_{k\le \frac{x}{y}} N\Bigl(\Bigl(\frac{x}{k}\Bigr)^c\Bigr) , \quad S_3 = N(y^c)N\Bigl(\frac{x}{y}\Bigr).\]
Now we use the formula $N(t)=t+O(t^{\frac{1}{2}+\varepsilon})$ to estimate these three sums, and at the end we optimize in $y$.
We have
\[ S_1 = x\sum_{\ell\le y^c} \frac{1}{\ell^{1/c}} +O\biggl(x^{\frac{1}{2}+\varepsilon}\sum_{\ell\le y^c} \frac{1}{\ell^{1/2c}}\biggr).\]
So, by Lemma 7.1, we obtain
\[ S_1 = x\biggl( \zeta_{\mathcal{R}}\Bigl(\frac{1}{c}\Bigr) - \frac{c}{(1-c)y^{1-c}}\biggr) + O\biggl(\frac{x}{y^{1-\frac{c}{2}-\varepsilon}}+x^{\frac{1}{2}+\varepsilon}y^{c-\frac{1}{2}}\biggr).\]
Next,
\[ S_2 = \sum_{k\le \frac{x}{y}} \Bigl(\frac{x}{k}\Bigr)^c + O\biggl( \sum_{k\le \frac{x}{y}} \Bigl(\frac{x}{k}\Bigr)^{\frac{c}{2}+\varepsilon} \biggr) = x^c\biggl( \frac{(x/y)^{1-c}}{1-c} + \zeta_{\mathcal{R}}(c) + O\Bigl(\Bigl(\frac{x}{y}\Bigr)^{\frac{1}{2}-c+\varepsilon}\Bigr)\biggr) + O\Bigl(x^{\frac{c}{2}+\varepsilon}\Bigl(\frac{x}{y}\Bigr)^{1-\frac{c}{2}}\Bigr).\]
Thus
\[ S_2 = \frac{x}{(1-c)y^{1-c}} + \zeta_{\mathcal{R}}(c)x^c +O\biggl(x^{\frac{1}{2}+\varepsilon}y^{c-\frac{1}{2}}+\frac{x^{1+\varepsilon}}{y^{1-\frac{c}{2}}}\biggr).\]
Finally,
\[S_3 = (y^c + O(y^{\frac{c}{2}+\varepsilon}))\Bigl(\frac{x}{y} + O\Bigl(\Bigl(\frac{x}{y}\Bigr)^{\frac{1}{2}+\varepsilon}\Bigr)\Bigr)=\frac{x}{y^{1-c}} + O\biggl(\frac{x}{y^{1-\frac{c}{2}-\varepsilon}}+x^{\frac{1}{2}+\varepsilon}y^{c-\frac{1}{2}}\biggr).\]
Combining these formulas, the different terms in $xy^{c-1}$ disappear and it follows that
\[ M_c(x) = x\zeta_{\mathcal{R}}\Bigl(\frac{1}{c}\Bigr)+ \zeta_{\mathcal{R}}(c)x^c + O\biggl(\frac{x^{1+\varepsilon}}{y^{1-c/2-\varepsilon}}+x^{\frac{1}{2}+\varepsilon}y^{c-\frac{1}{2}}\biggr).\]
Choosing $y=x^{\frac{1}{1+c}}$ gives the result.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Lemma 7.3}\newline
{\em Let $(\cal{R},\cal{N})$ be a good g-prime systems and $c\in (\frac{1}{2},1)$. Then for all fixed $\varepsilon>0$,}
\begin{align*}
(i)\quad \sum_{\tiny \begin{array}{c} m \le x \\ m\in \mathcal{M}_c\end{array} } \mu_{\mathcal{Q}_c}(m) & \ll x^{\frac{3c}{2(1+c)}+\varepsilon}\\
(ii)\quad \sum_{\tiny \begin{array}{c} k\ell^{1/c} \le x \\ k,\ell\in \mathcal{N}\end{array} } \mu_{\mathcal{R}}(k) & = \frac{x^c}{\zeta(c)} + O(x^{\frac{3c}{2(1+c)}+\varepsilon}).
\end{align*}
{\em Proof.}\, By Lemma 7.1, the abscissae of convergence of the series
\[ \sum_{m\in\mathcal{M}_c} \frac{\mu_{\mathcal{Q}_c}(m)}{m^s} \quad \mbox{and}\quad \sum_{k,l\in\mathcal{N}}\frac{\mu_{\mathcal{R}}(k)}{(kl^{1/c})^s}\]
are both at most 1. By the first effective Perron formula, it follows that for $n\ge 3$ and $T\ge 1$,
\begin{align*}
\sum_{\tiny \begin{array}{c} m \le x \\ m\in \mathcal{M}_c\end{array} } \mu_{\mathcal{Q}_c}(m) &= I_1+O(E) \\
\sum_{\tiny \begin{array}{c} k\ell^{1/c} \le x \\ k,\ell\in \mathcal{N}\end{array} } \mu_{\mathcal{R}}(k) &= I_{-1}+O(E)
\end{align*}
where
\begin{align*}
I_1 & = \frac{1}{2\pi i}\int_{1+\frac{1}{\log x} -iT}^{1+\frac{1}{\log x}+iT}\biggl( \sum_{m\in\mathcal{M}_c} \frac{\mu_{\mathcal{Q}_c}(m)}{m^s}\biggr) \frac{x^s}{s}\, ds = \frac{1}{2\pi i}\int_{1+\frac{1}{\log x} -iT}^{1+\frac{1}{\log x}+iT} \frac{x^s}{\zeta(s)\zeta(s/c)}\, \frac{ds}{s}\\
I_{-1} & = \frac{1}{2\pi i}\int_{1+\frac{1}{\log x} -iT}^{1+\frac{1}{\log x}+iT}\biggl( \sum_{k,l\in\mathcal{N}}\frac{\mu_{\mathcal{R}}(k)}{(kl^{1/c})^s} \biggr) \frac{x^s}{s}\, ds = \frac{1}{2\pi i}\int_{1+\frac{1}{\log x} -iT}^{1+\frac{1}{\log x}+iT} \frac{\zeta(s/c)x^s}{\zeta(s)}\, \frac{ds}{s},\mbox{ and}\\
E &= x\sum_{m\in\mathcal{M}_c}\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}.
\end{align*}
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Upper bound for $E$}\newline
We divide $E$ into $E=E_1+E_2+E_3$ where $E_1, E_2, E_3$ are as below.
\[ E_1:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ |m-x|\le \frac{x}{T}\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll M_c\Bigl(x+\frac{x}{T}\Bigr) - M_c\Bigl(x-\frac{x}{T}\Bigr)\ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x}{T}\]
by Lemma 7.2. Next $E_2 = E_{2,1}+E_{2,2}$ with
\[ E_{2,1}:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ x(1+\frac{1}{T})\le m\le 2x\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll \frac{x}{T}\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ x(1+\frac{1}{T})\le m\le 2x\end{array} } \frac{1}{m-x}.\]
The sum on the right is, using Lemma 7.2,
\begin{align*}
\int_{x(1+\frac{1}{T})}^{2x} \frac{dM_c(t)}{t-x} & = \frac{M_c(2x)}{x} + \int_{x(1+\frac{1}{T})}^{2x} \frac{M_c(t) - M_c(x(1+1/T))}{(t-x)^2}\, dt\\
& \ll 1+\frac{T}{x}x^{\frac{3c}{2(1+c)}+\varepsilon} + \int_{x(1+\frac{1}{T})}^{2x} \frac{dt}{t-x} =1+\frac{T}{x}x^{\frac{3c}{2(1+c)}+\varepsilon} + \log T.
\end{align*}
Thus
\[ E_{2,1} \ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x\log T}{T}.\]
We prove in the same way that
\[ E_{2,2}:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ \frac{x}{2}\le m \le x(1-\frac{1}{T})\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x\log T}{T}.\]
Furthermore,
\[ E_3:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ m<\frac{x}{2}\mbox{ or }m >2x\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll \frac{x\log x}{T}.\]
Putting together these bounds we obtain
\[ E \ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x\log (T+x)}{T}.\]
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\noindent
{\bf Upper bound for $E$}\newline
We divide $E$ into $E=E_1+E_2+E_3$ where $E_1, E_2, E_3$ are as below.
\[ E_1:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ |m-x|\le \frac{x}{T}\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll M_c\Bigl(x+\frac{x}{T}\Bigr) - M_c\Bigl(x-\frac{x}{T}\Bigr)\ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x}{T}\]
by Lemma 7.2. Next $E_2 = E_{2,1}+E_{2,2}$ with
\[ E_{2,1}:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ x(1+\frac{1}{T})\le m\le 2x\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll \frac{x}{T}\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ x(1+\frac{1}{T})\le m\le 2x\end{array} } \frac{1}{m-x}.\]
The sum on the right is, using Lemma 7.2,
\begin{align*}
\int_{x(1+\frac{1}{T})}^{2x} \frac{dM_c(t)}{t-x} & = \frac{M_c(2x)}{x} + \int_{x(1+\frac{1}{T})}^{2x} \frac{M_c(t) - M_c(x(1+1/T))}{(t-x)^2}\, dt\\
& \ll 1+\frac{T}{x}x^{\frac{3c}{2(1+c)}+\varepsilon} + \int_{x(1+\frac{1}{T})}^{2x} \frac{dt}{t-x} =1+\frac{T}{x}x^{\frac{3c}{2(1+c)}+\varepsilon} + \log T.
\end{align*}
Thus
\[ E_{2,1} \ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x\log T}{T}.\]
We prove in the same way that
\[ E_{2,2}:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ \frac{x}{2}\le m \le x(1-\frac{1}{T})\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x\log T}{T}.\]
Furthermore,
\[ E_3:=x\sum_{\tiny \begin{array}{c} m\in\mathcal{M}_c\\ m<\frac{x}{2}\mbox{ or }m >2x\end{array} }\frac{1}{m^{1+1/\log x}(1+T|\log(x/m)|)}\ll \frac{x\log x}{T}.\]
Putting together these bounds we obtain
\[ E \ll x^{\frac{3c}{2(1+c)}+\varepsilon} + \frac{x\log (T+x)}{T}.\]
\noindent
{\bf Upper bound for $I_1$}\newline
Using the fact that $(\mathcal{R},\mathcal{N})$ is good, the residue theorem and Lemma 5.1, we get
\begin{align*}
I_1 & = \frac{1}{2\pi i}\int_{\frac{1}{2}+\varepsilon -iT}^{\frac{1}{2}+\varepsilon +iT} \frac{x^s}{\zeta(s)\zeta(s/c)}\, \frac{ds}{s} + \frac{1}{2\pi i}\sum_{\gamma\in \{-1,1\} } \gamma \int_{\frac{1}{2}+\varepsilon +i\gamma T}^{1+\frac{1}{\log x} + i\gamma T} \frac{x^s}{\zeta(s)\zeta(s/c)}\, \frac{ds}{s} \\
& \ll x^{\frac{1}{2}+\varepsilon}T^\varepsilon + \frac{x}{T^{1-\varepsilon}}.
\end{align*}
{\bf Approximate formula for $I_{-1}$}\newline
This is almost the same calculation but now we pick up a residue at $c$ because of the pole of $\zeta(s/c)$. We have
\begin{align*}
I_{-1} & = \frac{x^c}{\zeta(c)} + \frac{1}{2\pi i}\int_{\frac{1}{2}+\varepsilon -iT}^{\frac{1}{2}+\varepsilon +iT} \frac{\zeta(s/c)x^s}{\zeta(s)}\, \frac{ds}{s} + \frac{1}{2\pi i}\sum_{\gamma\in \{-1,1\} } \gamma \int_{\frac{1}{2}+\varepsilon +i\gamma T}^{1+\frac{1}{\log x} + i\gamma T} \frac{\zeta(s/c)x^s}{\zeta(s)}\, \frac{ds}{s} \\
& = \frac{x^c}{\zeta(c)} + O(x^{\frac{1}{2}+\varepsilon}T^\varepsilon + \frac{x}{T^{1-\varepsilon}}).
\end{align*}
Finally, we get
\[ \sum_{\tiny \begin{array}{c} m \le x \\ m\in \mathcal{M}_c\end{array} } \mu_{\mathcal{Q}_c}(m) \ll x^{\frac{3c}{2(1+c)}+\varepsilon}T^\varepsilon + \frac{x}{T}(T^\varepsilon + \log x)\]
and
\[ \sum_{\tiny \begin{array}{c} k\ell^{1/c} \le x \\ k,\ell\in \mathcal{N}\end{array} } \mu_{\mathcal{R}}(k) = \frac{x^c}{\zeta(c)} + O\Bigl(x^{\frac{3c}{2(1+c)}+\varepsilon}T^\varepsilon + \frac{x}{T}(T^\varepsilon + \log x)\Bigr).\]
Choosing $T=x$ concludes the proof.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf Remark on the use of Perron's effective formula for Beurling primes}\newline
If we had $M_c(x)=[x]$, the counting function of the usual integers, then we would have had
\[ M_c\Bigl(x+\frac{x}{T}\Bigr) - M_c\Bigl(x-\frac{x}{T}\Bigr)\ll \frac{x}{T} + 1,\]
instead of the upper bound (7.1). But in the case of a general Beurling prime system, we do not always have
\[ N(x+h)-N(x)\ll h+1.\]
It is the reason why, in order to obtain an estimate for for generalized integers in short intervals, we needed first to compute the asymptotic development of the counting function $M_c(x)$ of Lemma 7.2.\newline
\pagebreak
\noindent
{\bf {\large 8. Proofs of Theorems 1 and 2}}\newline
{\bf Comments}\, The two proofs have similar structure. The theorems will follow easily from the two fundamental formulas (see (8.15) and (8.7)),
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_{a,b}}(n)}{n^s} = \frac{\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{R}}(s/a)}H(s),\quad (\sigma>b)\tag{8.1}\]
for Theorem 1, and
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}_{a,b}}(n)}{n^s} = \frac{1}{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)}H(s),\quad \Bigl(\sigma>\max\Bigl\{\frac{3ab}{2(a+b)},a-\frac{19}{40}\Bigr\}\Bigr)\]
for Theorem 2, where in both cases $(\mathcal{R}, \mathcal{N})$ is a good prime system and $H(s)$ an absolutely convergent product.
The proof of Theorem 1 is longer, mainly because of the presence of the pole at $s=b$ in (8.1) that deserves a special treatment. It is why we begin with the proof of Theorem 2.
\newline
\noindent
{\em Proof of Theorem 2.}\, Let $a$ and $b$ be two real numbers satisfying (1.14) and let $(\mathcal{R},\mathcal{N})$ be a good g-prime system. Thanks to Zhang, we know such a system exists. Let $Q:=Q_{a,b} = \mathcal{R}^{1/a}\cup \mathcal{R}^{1/b}$, choose $\varepsilon$ such that $0<\varepsilon<\frac{a}{4}$, and define
\[ h = \max\Bigl\{ 1-\frac{a}{2}+2\varepsilon, \frac{21}{40}\Bigr\}.\tag{8.2}\]
By using Lemma 6.2, (1.14), (8.2) and Lemma 6.1, we have, for $x$ large enough
\begin{align*}
Q(x+x^h)-Q(x) & = R^{1/a}(x+x^h)-R^{1/a}(x) + R^{1/b}(x+x^h)-R^{1/b}(x)\\
& = \frac{x^{a+h-1}}{\log x} +\frac{x^{b+h-1}}{\log x} + O(x^{\max\{ a/2 + \varepsilon, a+2h-2\} })\\
& \sim \frac{x^{a+h-1}}{\log x} \le \frac{x^h}{12\log x} \le \pi(x+x^h)-\pi (x).
\end{align*}
Moreover, as $\mathcal{R}$ is good, we also have $Q(x)\ll x^a$ and
\[ \lim_{x\to\infty} (\pi(x)-Q(x))=\infty.\]
Put $\sigma(a) = \max\{\frac{a}{2}, a-\frac{19}{40}\}$. By Lemma 6.3, there exists a set $\mathcal{P}:=\mathcal{P}_{a,b}$ of ordinary primes, and a bijection $p:\mathcal{Q}\to\mathcal{P}$ such that
\[ H^*(s):= \frac{\zeta_{\mathcal{Q}}(s)}{\zeta_{\mathcal{P}}(s)} = \frac{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{P}}(s)}\]
is an absolutely convergent product for $\sigma>\sigma(a)+2\varepsilon$.
As $\varepsilon>0$ can be chosen as small as we please, it follows that
\[ \frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)} = \frac{H(s)}{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)},\quad (\sigma>\sigma(a))\tag{8.3}\]
where $H(s)$ is again an absolutely convergent product for $\sigma>\sigma(a)$.
We denote by $\mathcal{M}_{a,b}$ the multiset of integers associated to the multiset of primes $\mathcal{Q}=\mathcal{Q}_{a,b}$. With the notation of section 7, we have for any $c\in (\frac{1}{2},1)$,
\[ \mathcal{Q}_{1,c}=\mathcal{Q}_c \quad\mbox{ and }\quad \mathcal{M}_{1,c} = \mathcal{M}_c.\]
Let $c=b/a$. For $\sigma>a$, we have the following formula where the generalized Euler products and the generalized Dirichlet series are normally convergent for $\sigma\ge a+\varepsilon$ for any fixed $\varepsilon>0$.
\[ \frac{1}{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)} = \frac{1}{\zeta_{\mathcal{Q}_{a,b}}(s)} = \sum_{\tilde{m}\in\mathcal{M}_{a,b}} \frac{\mu_{\mathcal{Q}_{a,b}}(\tilde{m})}{\tilde{m}^s} = \sum_{m\in\mathcal{M}_c} \frac{\mu_{\mathcal{Q}_c}(m)}{m^{s/a}}.\tag{8.4}\]
Let $A(x) = \sum_{m\le x} \mu_{\mathcal{Q}_c}(m)$ (where $m\in\mathcal{M}_c$). For $\sigma> \frac{3c}{2(1+c)}$, we have by Abel summation
\[ \sum_{\tiny \begin{array}{c} m \le x \\ m\in \mathcal{M}_c\end{array} } \frac{\mu_{\mathcal{Q}_c}(m)}{m^\sigma} = \frac{A(x)}{x^\sigma} + \sigma\int_1^x \frac{A(t)}{t^{\sigma+1}}\, dt = \sigma\int_1^\infty \frac{A(t)}{t^{\sigma+1}}\, dt +o(1)\]
by Lemma 7.3(i). It follows that
\[ \sigma_c\Bigl(\sum_{m\in\mathcal{M}_c} \frac{\mu_{\mathcal{Q}_c}(m)}{m^{s/a}}\Bigr)\le \frac{3ab}{2(a+b)}.\tag{8.5}\]
We have, normally for $\sigma\ge a+\varepsilon$ for all fixed $\varepsilon>0$
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}}(n)}{n^s} = \frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)}.\tag{8.6}\]
Combining (8.6), (8.3), (8.4) and (8.5) gives
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}}(n)}{n^s} =\frac{H(s)}{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)} , \qquad (\sigma>\max\{ \sigma(a), \frac{3ab}{2(a+b)}\})\tag{8.7}\]
and this proves that the abscissa of absolute convergence of the above series is $a$. By (1.14), we have $\max\{ \sigma(a), \frac{3ab}{2(a+b)}\} = \max\{ \frac{3ab}{2(a+b)}, a-\frac{19}{40}\}<b<a$ and the result follows.
\newline\phantom{a}
$\Box$\newline
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As $\varepsilon>0$ can be chosen as small as we please, it follows that
\[ \frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)} = \frac{H(s)}{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)},\quad (\sigma>\sigma(a))\tag{8.3}\]
where $H(s)$ is again an absolutely convergent product for $\sigma>\sigma(a)$.
We denote by $\mathcal{M}_{a,b}$ the multiset of integers associated to the multiset of primes $\mathcal{Q}=\mathcal{Q}_{a,b}$. With the notation of section 7, we have for any $c\in (\frac{1}{2},1)$,
\[ \mathcal{Q}_{1,c}=\mathcal{Q}_c \quad\mbox{ and }\quad \mathcal{M}_{1,c} = \mathcal{M}_c.\]
Let $c=b/a$. For $\sigma>a$, we have the following formula where the generalized Euler products and the generalized Dirichlet series are normally convergent for $\sigma\ge a+\varepsilon$ for any fixed $\varepsilon>0$.
\[ \frac{1}{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)} = \frac{1}{\zeta_{\mathcal{Q}_{a,b}}(s)} = \sum_{\tilde{m}\in\mathcal{M}_{a,b}} \frac{\mu_{\mathcal{Q}_{a,b}}(\tilde{m})}{\tilde{m}^s} = \sum_{m\in\mathcal{M}_c} \frac{\mu_{\mathcal{Q}_c}(m)}{m^{s/a}}.\tag{8.4}\]
Let $A(x) = \sum_{m\le x} \mu_{\mathcal{Q}_c}(m)$ (where $m\in\mathcal{M}_c$). For $\sigma> \frac{3c}{2(1+c)}$, we have by Abel summation
\[ \sum_{\tiny \begin{array}{c} m \le x \\ m\in \mathcal{M}_c\end{array} } \frac{\mu_{\mathcal{Q}_c}(m)}{m^\sigma} = \frac{A(x)}{x^\sigma} + \sigma\int_1^x \frac{A(t)}{t^{\sigma+1}}\, dt = \sigma\int_1^\infty \frac{A(t)}{t^{\sigma+1}}\, dt +o(1)\]
by Lemma 7.3(i). It follows that
\[ \sigma_c\Bigl(\sum_{m\in\mathcal{M}_c} \frac{\mu_{\mathcal{Q}_c}(m)}{m^{s/a}}\Bigr)\le \frac{3ab}{2(a+b)}.\tag{8.5}\]
We have, normally for $\sigma\ge a+\varepsilon$ for all fixed $\varepsilon>0$
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}}(n)}{n^s} = \frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)}.\tag{8.6}\]
Combining (8.6), (8.3), (8.4) and (8.5) gives
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}}(n)}{n^s} =\frac{H(s)}{\zeta_{\mathcal{R}}(s/a)\zeta_{\mathcal{R}}(s/b)} , \qquad (\sigma>\max\{ \sigma(a), \frac{3ab}{2(a+b)}\})\tag{8.7}\]
and this proves that the abscissa of absolute convergence of the above series is $a$. By (1.14), we have $\max\{ \sigma(a), \frac{3ab}{2(a+b)}\} = \max\{ \frac{3ab}{2(a+b)}, a-\frac{19}{40}\}<b<a$ and the result follows.
\newline\phantom{a}
$\Box$\newline
\noindent
{\em Proof of Theorem 1.}\, Let $a$ and $b$ be two real numbers satisfying (1.12) and let $(\mathcal{R},\mathcal{N})$ be a good g-prime system. Thanks to Zhang, we know such a system exists. Choose $\varepsilon$ such that $0<\varepsilon<\frac{a}{4}$, and define
\[ h = 1-\frac{a}{2}+2\varepsilon.\tag{8.8}\]
By using Lemma 6.2, (1.12) and (8.8), we have, for $x$ large enough
\begin{align*}
R^{1/b}(x+x^h)-R^{1/b}(x) & = \frac{x^{b+h-1}}{\log x} + O(x^{\max\{ b/2 + \varepsilon, b+2h-2\} })\\
& \le \frac{x^{a+h-1}}{\log x} - O(x^{\max\{ a/2 + \varepsilon, a+2h-2\} }) \le R^{1/a}(x+x^h)-R^{1/a}(x).
\end{align*}
Moreover, as $(\mathcal{R},\mathcal{N})$ is good, we also have $R^{1/b}(x)\ll x^b$ and
\[ \lim_{x\to\infty} (R^{1/a}(x)-R^{1/b}(x))=\infty.\]
By applying Lemma 6.3, there exists an injection $I:\mathcal{R}^{1/b}\to \mathcal{R}^{1/a}$ such that
\[ \frac{\zeta_{\mathcal{R}^{1/b}}(s)}{\zeta_{Im(I)}(s)} = \frac{\zeta_{\mathcal{R}}(s/b)}{\zeta_{Im(I)}(s)}\tag{8.9}\]
is an absolutely convergent product for $\sigma>\max\{b-\frac{a}{2}+2\varepsilon, 0\}$.
Let $\mathcal{Q} = \mathcal{R}^{1/a} \setminus {\rm Im}(I)$ and define a new $h$ by
\[ h = \max\Bigl\{ 1-\frac{a}{2}+2\varepsilon, \frac{21}{40}\Bigr\}.\tag{8.10}\]
By using Lemma 6.2, (1.12), (8.10) and finally Lemma 6.1, we have, for $x$ large enough
\begin{align*}
Q(x+x^h)-Q(x) & \le R^{1/a}(x+x^h)-R^{1/a}(x) = \frac{x^{a+h-1}}{\log x} + O(x^{\max\{ a/2 + \varepsilon, a+2h-2\} })\\
& \le \frac{x^h}{12\log x} \le \pi(x+x^h)-\pi (x).
\end{align*}
Moreover, as $\mathcal{R}$ is good, we also have $Q(x)\ll x^a$ and
\[ \lim_{x\to\infty} (\pi(x)-Q(x))=\infty.\]
Recall the notation $\sigma(a) = \max\{ \frac{a}{2}, a-\frac{19}{40}\}$. By Lemma 6.3, there exists a set $\mathcal{P}=\mathcal{P}_{a,b}$ of ordinary primes and a bijection
\[ p:\mathcal{Q}\to \mathcal{P}\]
such that the function
\[ \frac{\zeta_{\mathcal{Q}}(s)}{\zeta_{\mathcal{P}}(s)} = \frac{\zeta_{\mathcal{R}}(s/a)}{\zeta_{{\rm Im}(I)}(s)\zeta_{\mathcal{P}}(s)} \]
is an absolutely convergent product for $\sigma>\sigma(a)+2\varepsilon$. We have $\sigma(a)\ge a/2 > b-a/2$ by (1.12). By (8.9), it follows that
\[ \frac{1}{\zeta_{\mathcal{P}}(s)} = \frac{\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{R}}(s/a)}H^*(s), \quad (\sigma>\sigma(a)+2\varepsilon)\]
where $H^*(s)$ is an absolutely convergent product for $\sigma>\sigma(a)+2\varepsilon$. As $\varepsilon>0$ can be chosen as small as we please, it follows that
\[ \frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)} = \frac{H(s)\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{R}}(s/a)},\quad (\sigma>\sigma(a))\tag{8.11}\]
where
\[ H(s) = \frac{H^*(s)\zeta_{\mathcal{R}}(2s/a)}{H^*(2s)\zeta_{\mathcal{R}}(2s/b)} \]
is again an absolutely convergent product for $\sigma>\sigma(a)$.
Let $c = b/a$. We have the following formula where both sides converge normally for $\sigma\ge a+\varepsilon$ for any fixed $\varepsilon>0$.
\[ \frac{\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{R}}(s/a)} = \sum_{k,l\in\mathcal{N}} \frac{\mu_{\mathcal{R}}(k)}{(kl^{1/c})^{s/a}}.\tag{8.12}\]
Let
\[ A(x) = \sum_{\tiny \begin{array}{c} k\ell^{1/c} \le x \\ k,\ell\in \mathcal{N}\end{array} } \mu_{\mathcal{R}}(k).\]
By Abel summation, for $\sigma>c$,
\[ \sum_{\tiny \begin{array}{c} m \le x \\ m=kl^{1/c}\in \mathcal{M}_c\end{array} } \frac{\mu_{\mathcal{R}}(k)}{m^\sigma} = \frac{A(x)}{x^\sigma} + \sigma\int_1^x \frac{A(t)}{t^{\sigma+1}}\, dt = \sigma\int_1^\infty \frac{A(t)}{t^{\sigma+1}}\, dt +o(1)\]
by Lemma 7.3(ii). It follows that the abscissa of convergence of the series in (8.12) is at most $b$. As $b$ is a pole, the abscissa is indeed $b$. We have, normally for $\sigma\ge a+\varepsilon$ for all fixed $\varepsilon>0$
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}}(n)}{n^s} = \frac{\zeta_{\mathcal{P}}(2s)}{\zeta_{\mathcal{P}}(s)}.\tag{8.14}\]
Combining (8.14), (8.11), (8.12) and (8.13) we get
\[ \sum_{n=1}^\infty \frac{\lambda_{\mathcal{P}}(n)}{n^s} =\frac{\zeta_{\mathcal{R}}(s/b)}{\zeta_{\mathcal{R}}(s/a)}H(s) , \qquad (\sigma>\max\{ \sigma(a), b\})\tag{8.15}\]
and this proves that the abscissa of absolute convergence of the above series is $a$.
But by (1.12), we have $b>\sigma(a)$. Thus (8.15) is actually true for $\sigma>b$. It follows that $s=a$ is the only zero in $\mathbb{C}_b$ and this zero is simple. As $b$ is a pole it follows that the abscissa of convergence is $b$.
Moreover, for $t\in\mathbb{R}$, we have
\[ \lim_{\sigma\to b+} \frac{\zeta_{\mathcal{R}}(\frac{\sigma+it}{b})}{\zeta_{\mathcal{R}}(\frac{\sigma+it}{a})}H(\sigma+it) =
\frac{\zeta_{\mathcal{R}}(1+\frac{it}{b})}{\zeta_{\mathcal{R}}(\frac{b+it}{a})}H(b+it)\ne 0.\]
By Abel's Theorem, it follows that if the series in (8.15) converges at $b+it$, the sum cannot be 0. Thus we have $Z_{\mathcal{P}} = \{a\}$, which concludes the proof of Theorem 1.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf {\large 9. Open questions related to GRH-RH}}\newline
Let us recall that one of the classical statements equivalent to the Generalized Riemann Hypothesis (GRH) is the following: {\em for every Dirichlet character $\chi$, the meromorphic function $L_\chi(s)$ does not vanish in $\mathbb{C}_{\frac{1}{2}}$. }
The Dirichlet series defining $\zeta(s)$ and more generally $L_\chi(s)$ with $\chi$ a principal Dirichlet character are not convergent in the critical strip. As only the zeros of Dirichlet series themselves (and not of their meromorphic continuation) are studied here, it leads us to introduce GRH$\setminus$RH: {\em for every non-principal Dirichlet character $\chi$, the Dirichlet series $L_\chi(s)$ does not vanish in $\mathbb{C}_{\frac{1}{2}}$.}\newline
Let us recall the sets $V$ and $W$ of section 1(f). Theorem 3 says $V\supset [0,\frac{1}{2}]$ and, under RH, $W\supset [0,\frac{1}{2}]$. We wonder if these inclusions are equalities. Write
\[ V=\Bigl[0,\frac{1}{2}\Bigr] \tag{$B_c$}\]
and
\[ W=\Bigl[0,\frac{1}{2}\Bigr] \tag{$B_a$}\]
We shall see that either of these implies GRH$\setminus$RH.\newline
Let us also recall the two statements mentioned in \cite{KS1} which imply GRH$\setminus$RH.
\[ \mbox{\em For every completely multiplicative $f$, we have } \sum_{n\le x} f(n) =\Omega\Bigl(\frac{1}{\sqrt{x}}\Bigr).\tag{$A$}\]
Now, let $\sigma_n(f)$ and $\sigma_p(f)$ denote the abscissa of convergence of $\sum_{n=1}^\infty \frac{f(n)}{n^s}$ and
$\sum_p \frac{f(p)}{p^s}$ respectively. Write
\[ \mbox{\em For every completely multiplicative $f$, we have $\sigma_p(f)\le \sigma_n(f)+\frac{1}{2}$}.\tag{$C$}\]
These five statements are related in the following way:
\[ \left. \begin{array}{cl} A \implies & B_c \implies\\ & B_a \implies \\ & C \implies \end{array} \right\} \mbox{ GRH$\setminus$RH}\]
{\em Proof.}\, $A\implies B_c$\newline
Let us suppose $B_c$ is false. Then there exists a completely multiplicative function $f(n)$ and a zero $\beta+i\gamma$ of $\sum_{n=1}^\infty \frac{f(n)}{n^s}$ (with abscissa of convergence $\sigma_c$) such that $\beta-\sigma_c>\frac{1}{2}$.
By writing $f_1(n) = f(n)n^{-\sigma_c-i\gamma}$ we have that
\[ \sum_{n=1}^\infty \frac{f_1(n)}{n^s}\]
has abscissa of convergence zero, and the series vanishes at $s=\beta>\frac{1}{2}$. Defining $A(x) = \sum_{n\le x}f_1(n)$, we have $A(x)\ll x^\varepsilon$ for all $\varepsilon>0$ and by Abel summation that
\[ \sum_{n>x} \frac{f_1(n)}{n^\beta}\ll \frac{1}{x^{\beta-\varepsilon}}.\]
As $\sum_{n=1}^\infty \frac{f_1(n)}{n^\beta}=0$, we also have
\[ \sum_{n\le x} \frac{f_1(n)}{n^\beta}\ll \frac{1}{x^{\beta-\varepsilon}}.\]
Thus $g(n):=f_1(n)n^{-\beta}$ is completely multiplicative but does not satisfy $A$.
\newline
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But by (1.12), we have $b>\sigma(a)$. Thus (8.15) is actually true for $\sigma>b$. It follows that $s=a$ is the only zero in $\mathbb{C}_b$ and this zero is simple. As $b$ is a pole it follows that the abscissa of convergence is $b$.
Moreover, for $t\in\mathbb{R}$, we have
\[ \lim_{\sigma\to b+} \frac{\zeta_{\mathcal{R}}(\frac{\sigma+it}{b})}{\zeta_{\mathcal{R}}(\frac{\sigma+it}{a})}H(\sigma+it) =
\frac{\zeta_{\mathcal{R}}(1+\frac{it}{b})}{\zeta_{\mathcal{R}}(\frac{b+it}{a})}H(b+it)\ne 0.\]
By Abel's Theorem, it follows that if the series in (8.15) converges at $b+it$, the sum cannot be 0. Thus we have $Z_{\mathcal{P}} = \{a\}$, which concludes the proof of Theorem 1.
\newline\phantom{a}
$\Box$\newline
\noindent
{\bf {\large 9. Open questions related to GRH-RH}}\newline
Let us recall that one of the classical statements equivalent to the Generalized Riemann Hypothesis (GRH) is the following: {\em for every Dirichlet character $\chi$, the meromorphic function $L_\chi(s)$ does not vanish in $\mathbb{C}_{\frac{1}{2}}$. }
The Dirichlet series defining $\zeta(s)$ and more generally $L_\chi(s)$ with $\chi$ a principal Dirichlet character are not convergent in the critical strip. As only the zeros of Dirichlet series themselves (and not of their meromorphic continuation) are studied here, it leads us to introduce GRH$\setminus$RH: {\em for every non-principal Dirichlet character $\chi$, the Dirichlet series $L_\chi(s)$ does not vanish in $\mathbb{C}_{\frac{1}{2}}$.}\newline
Let us recall the sets $V$ and $W$ of section 1(f). Theorem 3 says $V\supset [0,\frac{1}{2}]$ and, under RH, $W\supset [0,\frac{1}{2}]$. We wonder if these inclusions are equalities. Write
\[ V=\Bigl[0,\frac{1}{2}\Bigr] \tag{$B_c$}\]
and
\[ W=\Bigl[0,\frac{1}{2}\Bigr] \tag{$B_a$}\]
We shall see that either of these implies GRH$\setminus$RH.\newline
Let us also recall the two statements mentioned in \cite{KS1} which imply GRH$\setminus$RH.
\[ \mbox{\em For every completely multiplicative $f$, we have } \sum_{n\le x} f(n) =\Omega\Bigl(\frac{1}{\sqrt{x}}\Bigr).\tag{$A$}\]
Now, let $\sigma_n(f)$ and $\sigma_p(f)$ denote the abscissa of convergence of $\sum_{n=1}^\infty \frac{f(n)}{n^s}$ and
$\sum_p \frac{f(p)}{p^s}$ respectively. Write
\[ \mbox{\em For every completely multiplicative $f$, we have $\sigma_p(f)\le \sigma_n(f)+\frac{1}{2}$}.\tag{$C$}\]
These five statements are related in the following way:
\[ \left. \begin{array}{cl} A \implies & B_c \implies\\ & B_a \implies \\ & C \implies \end{array} \right\} \mbox{ GRH$\setminus$RH}\]
{\em Proof.}\, $A\implies B_c$\newline
Let us suppose $B_c$ is false. Then there exists a completely multiplicative function $f(n)$ and a zero $\beta+i\gamma$ of $\sum_{n=1}^\infty \frac{f(n)}{n^s}$ (with abscissa of convergence $\sigma_c$) such that $\beta-\sigma_c>\frac{1}{2}$.
By writing $f_1(n) = f(n)n^{-\sigma_c-i\gamma}$ we have that
\[ \sum_{n=1}^\infty \frac{f_1(n)}{n^s}\]
has abscissa of convergence zero, and the series vanishes at $s=\beta>\frac{1}{2}$. Defining $A(x) = \sum_{n\le x}f_1(n)$, we have $A(x)\ll x^\varepsilon$ for all $\varepsilon>0$ and by Abel summation that
\[ \sum_{n>x} \frac{f_1(n)}{n^\beta}\ll \frac{1}{x^{\beta-\varepsilon}}.\]
As $\sum_{n=1}^\infty \frac{f_1(n)}{n^\beta}=0$, we also have
\[ \sum_{n\le x} \frac{f_1(n)}{n^\beta}\ll \frac{1}{x^{\beta-\varepsilon}}.\]
Thus $g(n):=f_1(n)n^{-\beta}$ is completely multiplicative but does not satisfy $A$.
\newline
\noindent
$B_c\implies$ GRH$\setminus$RH\newline
Suppose GRH$\setminus$RH is false. Then there exists a non-principal character $\chi$ and a zero $\beta+i\gamma$ of $L_\chi(s)$ with $\beta>\frac{1}{2}$. As $L_\chi(s)$ has $\sigma_c=0$, it follows that $\beta\in V$ and $B_c$ is false. \newline
\noindent
$B_a\implies$ GRH$\setminus$RH\newline
The proof is similar to the above. Suppose GRH$\setminus$RH is false. Then there exists a non-principal character $\chi$ and a zero $\beta+i\gamma$ of $L_\chi(s)$ with $\beta<\frac{1}{2}$. As $L_\chi(s)$ has $\sigma_a=1$, it follows that $1-\beta\in W$ and, since $1-\beta>\frac{1}{2}$, $B_a$ is false. \newline
\noindent
$C\implies$ GRH$\setminus$RH\newline
Suppose $C$ is true. Let $\chi$ be a non-principal character. Note that $\sigma_n(\chi)=0$. By $C$, we have
\[ \sum_{p\le x} \chi(p) \ll x^{\frac{1}{2}+\varepsilon}\quad\mbox{ for all $\varepsilon>0$}.\]
But this is an equivalent form of RH for $L_\chi(s)$; i.e. GRH$\setminus$RH follows.
\newline\phantom{a}
$\Box$\newline
\pagebreak
\noindent
{\bf Question}\, Among the four statements $A,B_c,B_a$ and $C$, which are true and which are false?
\noindent
{\bf Acknowledgements}\newline
We had stimulating discussions on the mathematics in and around this paper with Kristian Seip. We thank him for them.\newline
{\small
}
\end{document}
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\begin{document}
\title{BDD-Based Algorithm for SCC Decomposition\texorpdfstring{\\}{ }of Edge-Coloured Graphs}
\author[N.~Bene\v{s}]{Nikola Bene\v{s}}
\address{Masaryk University, Brno, Czech Republic}
\email{\{\texttt{xbenes3},\texttt{brim},\texttt{xpastva},\texttt{safranek}\}\texttt{@fi.muni.cz}}
\author[L.~Brim]{Lubo\v{s} Brim}
\author[S.~Pastva]{Samuel Pastva}
\author[D.~Šafránek]{David \v{S}afr{\'a}nek}
\begin{abstract}
Edge-coloured directed graphs provide an essential structure for modelling and analysis of complex systems arising in many scientific disciplines (e.g. feature-oriented systems, gene regulatory networks, etc.). One of the fundamental problems for edge-coloured graphs is the detection of strongly connected components, or SCCs.
The size of edge-coloured graphs appearing in practice can be enormous both in the number of vertices and colours. The large number of vertices prevents us from analysing such graphs using explicit SCC detection algorithms, such as Tarjan's, which motivates the use of a symbolic approach. However, the large number of colours also renders existing symbolic SCC detection algorithms impractical.
This paper proposes a novel algorithm that symbolically computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs $O(p\cdot n\cdot\log n)$ symbolic steps, where $p$ is the number of colours and $n$ is the number of vertices.
We evaluate the algorithm using an experimental implementation based on binary decision diagrams (BDDs). Specifically, we use our implementation to explore the SCCs of a large collection of coloured graphs (up to $2^{48}$) obtained from Boolean networks -- a~modelling framework commonly appearing in systems biology.
\end{abstract}
\maketitle
\section*{Introduction}\label{S:one}
In many scientific disciplines, the processing of massive data
sets represents one of the most important computational tasks.
A variety of these data sets can be modelled in terms of very large multi-graphs, augmented by a specific collection of application-dependent edge attributes. These attributes are often abstractly referred to as colours, and the resulting formalism is called an \emph{edge-coloured graph}~\cite{BANGJENSEN1997,bookgraphs79}. Geographic information systems, telecommunications traffic, or internet networks are prime examples of data that are best represented as such edge-coloured graphs.
For instance, in social networks, coloured edges can be used to link together groups of nodes related by some specific criteria (Sports, Health, Technology, Religion, etc.). In software engineering, one often speaks about feature-oriented systems~\cite{classen2010model}. In this case, colours represent possible combinations of features, altering the system's behaviour.
Our interest in processing huge edge-coloured graphs is primarily motivated by applications taken from systems biology~\cite{tcbb,GiacobbeGGHPP17} and genetics~\cite{DORNINGER94} where we have to deal not only with giant graphs as measured by the number of vertices and edges but also with large sets of colours. In this case, the graph colours represent valuations of numerous parameters that influence the dynamics of a~biological system~\cite{tcbb,BattPCGMJ10,Bernot05}.
Fundamental graph algorithms such as breadth-first search, spanning tree construction, shortest paths, decomposition into strongly connected components (SCCs), etc., are building blocks of many practical applications. For the edge-coloured graphs, the primary research focus so far has been on some of the ``classical'' coloured graph problems, like the determination of the chromatic index, finding sub-graphs with a specified colour property (the coloured version of the k-linked problem), alternating edge-coloured cycles and paths, rainbow cliques, monochromatic cliques and cycles, etc.~\cite{Das,Akbari,Alon,BANGJENSEN1997,Thomason,Kano}.
To the best of our knowledge, we are not aware of any work on SCC decomposition specifically for edge-coloured graphs, even though this problem has many important applications. For example, in biological systems, strongly connected components represent the so called attractors of the system. In this case, a specific focus is given to terminal (or bottom) SCCs, but non-terminal (transient) SCCs can also be detrimental to the system's long-term behaviour~\cite{long-lived-transients}. Overall, SCCs play an essential role in determining the system's biological properties, since they may correspond, for example, to the specific phenotypes expressed by a~cell~\cite{choo2016efficient}.
The valuation of parameters (e.g. the presence of certain genes or external stimuli) in such systems is then represented as edge colours in the state-transition graph. The knowledge of SCCs and how their structure depends on parameters is vital for understanding various biological phenomena~\cite{deritei2016principles,Li16}.
Other applications where investigation of attractors is crucial include predictions of the global climate change~\cite{Steffen18} or predictions of spreading of infectious diseases such as COVID-19~\cite{Matouk20}.
There is a serious computational problem related to the processing of massive edge-coloured graphs (or even the non-coloured ones) that significantly affects the tractability of SCC decomposition. The graphs often cannot be handled using standard (explicit) representations, since they are too large to be kept in the main memory. Various approaches have been considered to deal with such giant graphs: distributed-memory computation, symbolic data structures for graph representation, or storing the graphs in external memory. We review these approaches in more detail in the related work section.
In~\cite{cmsb2017,ICFEM19} we have initially attacked the SCC decomposition problem for massive edge-coloured graphs by developing a parallel, semi-symbolic algorithm for detection of bottom SCCs. The algorithm uses symbolic structures to represent sets of parameters, while the graph itself is represented explicitly. However, the results have shown that the parallel semi-symbolic algorithm is often not sufficient to tackle graphs representing real-world problems practically. These findings have motivated us to propose a new, entirely symbolic approach.
In this paper, we consider \emph{edge-coloured multi-digraphs}, i.e., multi-digraphs such that each directed edge has a colour and no two parallel (i.e., joining the same pair of vertices) edges have the same colour. Here, we refer to such graphs simply as \emph{coloured graphs}. For coloured graphs, we can define several notions of strongly connected components involving colours. We consider the simplest case, where the SCCs are \emph{monochromatic}, that is all their edges have the same colour~\cite{Kiraly14}. This choice is motivated by the application in systems biology, as mentioned above.
\paragraph{Contribution}
We propose a novel fully symbolic algorithm for detecting \emph{all}
monochromatic strongly connected components in a coloured graph. This algorithm is in practice significantly faster than what is achievable by na\"ively executing a symbolic SCC decomposition
algorithm for each colour separately. This is because in many applications,
the edges are largely shared among individual
colours~\cite{tcbb} and our algorithm is capable of exploiting this fact.
The algorithm conceptually follows the \emph{lock-step} reachability
approach by Bloem et al.~\cite{bloem2000} for purely monochromatic digraphs. The key
new ingredients behind our algorithm are a~careful orchestration of
the forward and backward reachability for different colours, and
a~colour-aware selection of the pivot set.
\subsection*{Structure of the paper} In Section~1, we recall the notions of strongly connected components and edge-coloured digraphs, and we state the coloured SCC decomposition problem. In Section~2, we first briefly introduce the forward-backward decomposition algorithm and the lock-step algorithm for monochromatic graphs. After that, we present the coloured SCC decomposition algorithm together with the proof of correctness and complexity analysis.
In Section~\ref{section:boolean-networks}, we introduce Boolean networks, discuss their symbolic encoding, and show how they can be translated into coloured graphs suitable for SCC-decomposition. Subsequently, Section~\ref{section:implementation} discusses several practical improvements to the main algorithm (saturation, trimming, and parallelism) which help it scale to larger models, and thus be more practically viable. Finally, Section~4 evaluates the main algorithm (including the improved variants) using a collection of large, real-world Boolean networks. A conclusion is provided in the last section.
This article is an extended version of an article that appeared in the Proceedings of TACAS 2021~\cite{tacas21}. We extend the information provided in the TACAS Proceedings with more in-depth technical details of the algorithm and related proof, including explanation of its key steps. Moreover, we extend the implementation and evaluation sections to give the reader more information on how the performance of the algorithm can be improved, and how the algorithm performs using a variety of real-world case studies.
\subsection*{Related Work}
The detection of SCCs in (monochromatic) digraphs is a well-known problem
computable in linear time.
Best serial (explicit) algorithms are
Kosaraju-Sharir~\cite{SHARIR1981} and Tarjan~\cite{Tarjan}, which are both inherently based on depth-first search. However, these algorithms do not scale for large graphs, e.g., those encountered in model-checking, when using explicit graph representation.
Therefore, alternative approaches to such SCC decomposition have been proposed (e.g. I/O efficient, parallel, or symbolic algorithms).
The algorithm of Jiang~\cite{JIANG1993} gives an I/O-efficient alternative
based on a~combination of depth-first and breadth-first search.
Efficient parallel, distributed-memory algorithms avoid the inherently sequential DFS step~\cite{REIF}
in several different ways. The Forward-Backward algorithm~\cite{FWBW} employs
a~divide-and-conquer approach relying on picking a pivot state and
splitting the graph in three independent (SCC-closed) parts. The
approach of Orzan~\cite{Orzan} uses a different distribution scheme
called a colouring transformation, employing a set of prioritised colours to
split the graph into many parts at once. The OWCTY-Backward-Forward (OBF) approach is
proposed in~\cite{Barnat09}. It recursively splits the graph in a number of independent sub-graphs called OBF slices and applies to each slice the One-Way-Catch-Them-Young (OWCTY) technique. In~\cite{Slota14}, the authors utilise
variants of the Forward-Backward and Orzan's algorithms for optimal
execution on shared-memory multi-core platforms. Finally, Bloemen et
al.~\cite{Bloemen16} present an on-the-fly parallel algorithm utilising a swarm of DFS searches, showing promising speed-up for large graphs containing large SCCs. On another end, GPU-accelerated approaches to computing
SCCs have been addressed for example in~\cite{BarnatBBC11,HRO13,LI2014,Dragan14}.
Computing SCCs of (monochromatic) digraphs symbolically is another way to handle giant graphs and has been thoroughly explored
in literature. As in the case of efficient parallelisation, depth-first
search is not feasible in the symbolic framework~\cite{GentiliniPP08}. In~consequence, many DFS-based algorithms cannot be
easily revised to work with symbolically represented graphs. An algorithm based on forward and backward reachability performing $\mathcal{O}(n^2)$ symbolic steps was presented by Xie and Beerel in~\cite{xie2000}. Bloem et al.~present an improved $\mathcal{O}(n \cdot \log n)$ algorithm in~\cite{bloem2000}. Finally, an $\mathcal{O}(n)$ algorithm was presented by Gentilini et al.~in~\cite{gentilini2003,GentiliniPP08}. This bound has been proven to be tight in~\cite{chatterjee2018}. In~\cite{chatterjee2018}, the authors argue that the algorithm from~\cite{gentilini2003} is optimal even when considering more fine-grained complexity criteria, like the diameter of the graph and the diameters of the individual components. Ciardo et al.~\cite{Ciardo11} use the idea of saturation~\cite{Ciardo06} to speed up state exploration within the Xie-Beerel algorithm, and show a saturation-based technique for computing the transitive closure of the graph's edge relation.
Besides these generic algorithms, there have also been symbolic SCC decomposition methods to deal with large graphs generated specifically by Boolean networks~\cite{MizeraIEEE,YUAN2019}. However, these primarily target detection of bottom SCCs. Methods in this area are also often incomplete, for example focusing on detection of single-state or small bottom SCCs~\cite{zhang-small-attractors}. As such, they generally perform better than an exhaustive symbolic SCC detection in their respective application domains, but are inherently limited in scope.
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\section{Problem Definition}\label{S:two}
As we have already stated in the introductory section, the SCC decomposition problem for edge-coloured graphs has remained mostly unexplored until now. We thus start this paper by introducing and formalising the notion of \emph{coloured SCC decomposition} itself and state some of its basic properties.
Before giving exact definitions, it might be instructive
to discuss the substance of the coloured SCC decomposition intuitively. There are
several ways of capturing the notion of a~``coloured connected component''.
One of them is that of a colour-connectivity first introduced by Saad~\cite{Saad92}. It is based on
alternating paths in which successive edges differ in colour. However, there is no
unique, universally acceptable notion of a coloured component.
In the biological applications we have in mind (i.e.~Boolean networks), we want to identify a coloured component as a~coloured collection of SCCs---a~collection where for every colour there is
a set of all relevant monochromatic SCCs. Such a setting leads us to represent SCCs
in the form of a relation. To that end, we first introduce such a relation for
monochromatic graphs (Section~\ref{sec:monographs}) and afterwards extend it
to edge-coloured graphs (Section~\ref{sec:colourgraphs}). The relation-based
approach gives us also the advantage of allowing a feasible symbolic encoding
of the problem.
\subsection{Graphs and Strongly Connected Components}
\label{sec:monographs}
Let us first recall the standard definitions of a~directed graph and its
strongly connected components:
\begin{defi}\label{def:graph}
A \emph{directed graph} is a tuple $G = (V, E)$ where $V$ is a set of graph \emph{vertices} and $E \subseteq V \times V$ is a set of graph \emph{edges}.
\end{defi}
We are going to use the word \emph{graph} to mean \emph{directed graph} in
the following.
We write $u \to v$ when $(u, v) \in E$ and $u \to^* v$ when $(u, v) \in E^*$,
the reflexive and transitive closure of $E$. We say that $v$ is
\emph{reachable} from $u$ if $u \to^* v$.
The reachability relation allows us to decompose a~graph into strongly
connected components, defined as follows:
\begin{defi}
In a~graph $G = (V, E)$, a \emph{strongly connected component} (SCC)
is a~maximal set $W \subseteq V$ such that for all $u, v \in W$, $u
\to^* v$ and $v \to^* u$. For a fixed $v \in V$, we write $SCC(G, v)$
to denote the SCC of $G$ that contains $v$.
\end{defi}
If the graph $G$ is clear from the context, we can simply write $SCC(v)$.
A set of vertices $S \subseteq V$ is said to be \emph{SCC-closed} if every SCC
$W$ is either fully contained inside $S$ ($W \subseteq S$), or in its
complement ($W \subseteq V \setminus S$). Notice that given a vertex $v$, the
set of all vertices reachable from $v$, as well as the set of all vertices
that can reach $v$, are both SCC-closed.
A pivotal problem in computer science is to find the SCC decomposition of~$G$.
As mentioned above, we represent the decomposition in the form of an
\emph{equivalence relation} $\ensuremath{R_\mathit{scc}}$ such that the individual SCCs are exactly
the equivalence classes of $\ensuremath{R_\mathit{scc}}$. The relation-based formulation of the SCC
decomposition problem is the following:
\begin{problem2}[SCC decomposition]
Given a graph $G = (V, E)$, find the~\emph{SCC decomposition relation}
$\ensuremath{R_\mathit{scc}} \subseteq V \times V$ such that $(u,v) \in \ensuremath{R_\mathit{scc}}$ if and only if $SCC(u) = SCC(v)$.
\end{problem2}
Note that $SCC(u)$ can be obtained by fixing the first attribute of $\ensuremath{R_\mathit{scc}}$, i.e.
$SCC(u) = \{ v \mid (u, v) \in \ensuremath{R_\mathit{scc}} \}$.
We refer to such operation as \emph{section} and denote it in the following way: $SCC(u) = \ensuremath{R_\mathit{scc}}(u, \_)$ (the concept is properly formalised later as part of Fig.~\ref{tab:operations}).
Here, $u$ is the specific value of an attribute at which the section is taken,
and $\_$ is used in place of the attributes that remain unchanged. Such
notation naturally extends to arbitrary relations.
\subsection{Coloured SCC Decomposition Problem}\label{sec:colourgraphs}
We now lift the formal framework to the coloured setting. An edge-coloured graph can be
seen as a succinct representation of several different graphs, all sharing the same
set of vertices.
To emphasise the difference from the standard graphs (i.e. Definition~\ref{def:graph}), we sometimes call the standard graphs
\emph{monochromatic}.
\begin{defi}
An \emph{edge-coloured directed multi-graph} (coloured graph for short) is a
tuple $\ensuremath{\mathfrak G} = (V, C, E)$ where $V$ is a set of vertices, $C$ is a set of
colours and $E \subseteq V \times C \times V$ is a~coloured edge relation.
\end{defi}
We also write $u \xrightarrow{c} v$ whenever $(u, c, v) \in E$ and
use $\creach{c}$ to denote the reflexive and transitive closure
of $\xrightarrow{c}$. We say that $v$ is $c$-reachable from $u$ if
$u \creach{c} v$, i.e.~there is a~path from $u$ to $v$ using
only $c$-coloured edges.
By fixing a~colour $c \in C$ and keeping only the $c$-coloured edges (with the
colour attribute removed), we obtain a~monochromatic graph
$\ensuremath{\mathfrak G}(c) = (V, \{(u, v) \mid (u, c, v) \in E\})$. We call this graph the
\emph{monochromatisation of $\ensuremath{\mathfrak G}$ with respect to $c$}.
Intuitively, one can view the elements of $C$ as a type of graph
parametrisation where the edge structure of the graph changes based on the
specific $c \in C$.
The SCC decomposition relation $\ensuremath{R_\mathit{scc}}$ is extended to the coloured SCC
decomposition relation $\ensuremath{\mathfrak R_\mathit{scc}}$ by relating every colour $c\in C$ with all
SCCs of the monochromatisation $\ensuremath{\mathfrak G}(c)$. In consequence, the SCC decomposition
problem is then lifted to the coloured SCC decomposition problem as follows:
\begin{problem2}[Coloured SCC decomposition]
Given a coloured graph $\ensuremath{\mathfrak G} = (V, C, E)$, find the \emph{coloured SCC
decomposition relation} $\ensuremath{\mathfrak R_\mathit{scc}} \subseteq V \times C \times V$ satisfying
$(u,c,v) \in \ensuremath{\mathfrak R_\mathit{scc}}$ if and only if $(u,v) \in \ensuremath{R_\mathit{scc}}$ of $\ensuremath{\mathfrak G}(c)$.
\end{problem2}
From this definition, we can immediately observe the following properties
about the relationship of $\ensuremath{\mathfrak R_\mathit{scc}}$ with the terms which we have defined before:
\begin{itemize}
\item $\ensuremath{R_\mathit{scc}}$ of a monochromatisation $\ensuremath{\mathfrak G}(c)$ is exactly the section
$\ensuremath{\mathfrak R_\mathit{scc}}(\_, c, \_)$;
\item $SCC(\ensuremath{\mathfrak G}(c), v)$ is exactly the section $\ensuremath{\mathfrak R_\mathit{scc}}(v, c, \_)$, or equivalently, $\ensuremath{\mathfrak R_\mathit{scc}}(\_, c, v)$ (since $\ensuremath{\mathfrak R_\mathit{scc}}$ and $\ensuremath{R_\mathit{scc}}$ are symmetric with regards to $V$).
\end{itemize}
From this, it should be immediately apparent that $\ensuremath{\mathfrak R_\mathit{scc}}$ contains all
components of the underlying monochromatisations.
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\section{Algorithm}
Conceptually, our algorithm follows the \emph{lock-step} reachability approach by Bloem~\cite{bloem2000} for monochromatic graphs. The lock-step algorithm itself is based on the basic forward-backward decomposition algorithm~\cite{xie2000}.
In this section, we first briefly introduce these two algorithms to explain better the key ideas behind our approach and, in particular, to explain the main difficulties encountered in employing the concepts of these algorithms to edge-coloured graphs.
Although the algorithms were originally presented as producing a~set of SCCs,
we reformulate them slightly using the equivalent relation-based approach as
explained in the previous section.
After that, we present the coloured SCC decomposition algorithm.
However, before we dive into the algorithmics, let us first briefly discuss
the computation model we are using.
\subsection{Symbolic Computation Model}\label{ssec:symb}
As a complexity measure of our algorithm, we consider the number of
symbolic steps, or more specifically, symbolic set and relation operations that the
algorithm performs. As is customary, we assume that sets of vertices ($V$) and colours ($C$) can be represented symbolically (for example, using reduced ordered binary decision diagrams~\cite{bryant86}) as well as any relations over these sets. In particular, we often talk about \emph{coloured vertex sets}, by which we mean the subsets of $V \times C$.
\begin{figure}
\caption{Summary of symbolic operations that appear in the presented algorithms. The derived operations can be implemented using the standard and relational operations. However, typically they also have a slightly more efficient direct implementations.}
\label{tab:operations}
\end{figure}
Aside from normal set operations (union, intersection, difference, product and element selection), we also require some basic relational operations, all of which we outline in Figure~\ref{tab:operations}. These extra operations tend to appear in other applications as well (such as symbolic model checking~\cite{BurchCMDH92}), and are thus typically already available in mature symbolic computation packages.
Finally, there are several derived operators that are partially
specific to our application to coloured graphs. However, these can be
constructed using standard set and relation operations.
The intuitive meaning of the derived operators is as follows:
$\textsc{Colours}$ returns all the colours that appear in the given coloured
vertex set. $\textsc{Pre}$ and $\textsc{Post}$ compute the pre- and post-image
of a (monochromatic or coloured) set of vertices, i.e.~the set of successors
or predecessors of all the vertices in the given set, respectively.
Finally, $\textsc{Join}$ takes a~coloured vertex set $A$ and computes
the set $\{(u, c, v) \mid (u, c) \in A, (v, c) \in A\}$.
\subsection{Forward-Backward Algorithm}
To symbolically compute the SCCs of a~graph $G = (V, E)$, Xie and
Beerel~\cite{xie2000} observed that for any vertex $v \in V$,
the intersection $W = F \cap B$
of the forward reachable vertices $F = \{ v' \in V \mid v \to^* v' \}$ and the
backward reachable vertices $B = \{ v' \in V \mid v' \to^* v \}$ is exactly
the strongly connected component of $G$ which contains $v$.
The algorithm thus picks an arbitrary \emph{pivot} $v \in V$, and divides the
vertices of the graph into four disjoint sets: $W$, $F \setminus W$, $B
\setminus W$ and $V \setminus (F \cup B)$. This is illustrated graphically in Figure~\ref{fig:algorithms}~(left). The set $W$ is then immediately
reported as an SCC of the graph, and added into the component relation: $\ensuremath{R_\mathit{scc}}
\gets \ensuremath{R_\mathit{scc}} \cup (W\times W)$. It is easy to see that every other SCC is
fully contained within one of the three remaining sets (they are SCC-closed), and the algorithm thus
recursively repeats this process independently in each set.
The correctness of the algorithm follows from the initial observation and the
fact that every vertex eventually appears in $W$ (either as a~pivot or as
a~result of $F \cap B$). In the worst case, the algorithm performs $O(|V|^2)$
symbolic steps, since every vertex is picked as a~pivot at most once and the
computation of $F$ and $B$ requires at most $O(|V|)$
\textsc{Pre}/\textsc{Post} operations.
\begin{figure}
\caption{Illustration of the difference between the forward-backward algorithm (left) and the lock-step algorithm (right). On the left, we fully compute both backward ($B$) and forward ($F$) reachable sets from the pivot $v$, identifying $W$ as $F \cap B$. On the right, without loss of generality, assume $F$ is fully computed first. It thus becomes converged ($Con$) and the computation of $B$ ($Non$) is stopped before it is fully explored. }
\label{fig:algorithms}
\end{figure}
\subsection{Lock-step Algorithm}
To improve the efficiency of the forward-backward algorithm, the lock-step
approach~\cite{bloem2000} uses another important observation:
To compute $W$, it is not
necessary to fully compute both $F$ and $B$; only the smaller (in terms of diameter) of the two sets
needs to be entirely known. With this observation, the computation of $F$ and
$B$ can be modified in the following way:
Instead of computing $F$ and $B$ one after the other, the computation is
\emph{interleaved} in a step-by-step manner (dovetailing). When one of the
sets is fully computed, the computation of the second set is stopped.
Let us call the computed set \emph{converged} and denote it by $\ensuremath{\mathit{Con}}$, and the
unfinished set \emph{non-converged} and denote it by $\ensuremath{\mathit{Non}}$. This situation is illustrated in Figure~\ref{fig:algorithms}~(right).
However, even when $\ensuremath{\mathit{Con}}$ is fully known, we still need to finish the
computation of states in $\ensuremath{\mathit{Non}}$ that are inside $\ensuremath{\mathit{Con}}$ to discover the
whole component $W$. This is necessary if there are vertices $w$ in $W$ whose
forward distance from $v$ (i.e.~the length of the path $v \to^* w$) is short
while their backward distance (the length of the path $w \to^* v$) is long,
or vice versa. Such vertices are thus only discovered in one of the two
reachability procedures and still need to be discovered by the other one
to identify the whole component.
However,
an important observation is that only the vertices already inside $\ensuremath{\mathit{Con}}$ need
to be considered in this phase.
After this, the SCC can be identified and reported just as in the
forward-backward algorithm. Finally, the recursion now continues in sets $\ensuremath{\mathit{Con}}
\setminus W$ and $V \setminus \ensuremath{\mathit{Con}}$. This is due to $\ensuremath{\mathit{Non}}$ being not fully
computed; we cannot guarantee that no SCC overlaps outside of $\ensuremath{\mathit{Non}}$ ($\ensuremath{\mathit{Non}}$ is not necessarily SCC-closed).
The algorithm is still correct because every vertex is eventually either
picked as a pivot or discovered in some $W$. Furthermore, due to the way
$\ensuremath{\mathit{Con}}$ and $\ensuremath{\mathit{Non}}$ are computed guarantees that $W$ is still a whole SCC\@.
In terms of complexity, the algorithm performs $O(|V| \cdot \log |V|)$
symbolic steps in the worst case. To see why this is true, we may observe that
every vertex appears in $W$ exactly once, and that the smaller of the two
sets $\ensuremath{\mathit{Con}} \setminus W$ and $V \setminus \ensuremath{\mathit{Con}}$, let us call it $S$,
is always smaller than $\frac{|V|}{2}$. The authors then argue that
the price of every iteration can be attributed (up to a multiplicative
constant) to the vertices in $S \cup W$ and that every vertex appears in $S$
at most $O(\log |V|)$-times.
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\subsection{Coloured Lock-step Algorithm}
When developing an algorithm for coloured graphs, one needs to deal with multiple
challenges which do not appear for monochromatic graphs and require
careful consideration. In the following, we refer to the pseudocode in
Algorithm~\ref{algo:symbolic}.
\begin{algorithm}
\SetKwProg{Fn}{Function}{}{}
\Fn{\upshape\textsc{ColouredSCC}$(\ensuremath{\mathfrak G} = (V, C, E))$}{
$\ensuremath{\mathfrak R_\mathit{scc}} \subseteq (V \times C \times V) \gets \emptyset$\;
$\textsc{Decomposition}(\ensuremath{\mathfrak G}, \ensuremath{\mathfrak R_\mathit{scc}}, V \times C)$\;
\Return $\ensuremath{\mathfrak R_\mathit{scc}}$\;
}
\BlankLine
\Fn{\upshape\textsc{Decomposition}$(\ensuremath{\mathfrak G} = (V, C, E), \ensuremath{\mathfrak R_\mathit{scc}} \subseteq (V \times C \times V), \mathcal{V} \subseteq (V \times C))$}{
\lIf{$\mathcal{V} = \emptyset$}{\Return}
$\ensuremath{\mathcal F}, \ensuremath{\mathcal B}, \ensuremath{\mathcal F}f, \ensuremath{\mathcal B}f \subseteq (V \times C) \gets \textsc{Pivots}(\mathcal{V})$\;
$\ensuremath{\mathcal F}u, \ensuremath{\mathcal B}u \subseteq (V \times C) \gets \emptyset$\;
$\ensuremath{F_\mathit{lock}}, \ensuremath{B_\mathit{lock}} \subseteq C \gets \emptyset$\;
\While{$\ensuremath{F_\mathit{lock}} \cup \ensuremath{B_\mathit{lock}} \subset \textsc{Colours}(\mathcal{V})$}{\label{alg:ls-start}
$\ensuremath{\mathcal F}f \gets (\textsc{Post}(\ensuremath{\mathfrak G}, \ensuremath{\mathcal F}f) \cap \mathcal V) \setminus \ensuremath{\mathcal F}$\;\label{alg:lockstep-post}
$\ensuremath{\mathcal B}f \gets (\textsc{Pre}(\ensuremath{\mathfrak G}, \ensuremath{\mathcal B}f) \cap \mathcal V) \setminus \ensuremath{\mathcal B}$\;\label{alg:lockstep-pre}
$\ensuremath{F_\mathit{lock}} \gets \ensuremath{F_\mathit{lock}} \cup (\textsc{Colours}(\mathcal V) \setminus \textsc{Colours}(\ensuremath{\mathcal F}f) \setminus \ensuremath{B_\mathit{lock}})$\;
$\ensuremath{B_\mathit{lock}} \gets \ensuremath{B_\mathit{lock}} \cup (\textsc{Colours}(\mathcal V) \setminus \textsc{Colours}(\ensuremath{\mathcal B}f) \setminus \ensuremath{F_\mathit{lock}})$\;\label{alg:lock}
$\ensuremath{\mathcal F}u \gets \ensuremath{\mathcal F}u \cup (\ensuremath{\mathcal F}f \cap (V \times \ensuremath{B_\mathit{lock}}))$\;
$\ensuremath{\mathcal B}u \gets \ensuremath{\mathcal B}u \cup (\ensuremath{\mathcal B}f \cap (V \times \ensuremath{F_\mathit{lock}}))$\;
$\ensuremath{\mathcal F}f \gets \ensuremath{\mathcal F}f \setminus (V \times \ensuremath{B_\mathit{lock}})$\;
$\ensuremath{\mathcal B}f \gets \ensuremath{\mathcal B}f \setminus (V \times \ensuremath{F_\mathit{lock}})$\;
$\ensuremath{\mathcal F} \gets \ensuremath{\mathcal F} \cup \ensuremath{\mathcal F}f$\;
$\ensuremath{\mathcal B} \gets \ensuremath{\mathcal B} \cup \ensuremath{\mathcal B}f$\;
}\label{alg:ls-end}
$\ensuremath{\mathcal{C}on} \subseteq V \times C \gets (\ensuremath{\mathcal F} \cap (V \times \ensuremath{F_\mathit{lock}})) \cup (\ensuremath{\mathcal B} \cap (V \times \ensuremath{B_\mathit{lock}}))$\;\label{alg:con}
$\ensuremath{\mathcal F}f \gets \ensuremath{\mathcal F}u \cap \ensuremath{\mathcal{C}on}$\;\label{alg:con-apply}
$\ensuremath{\mathcal B}f \gets \ensuremath{\mathcal B}u \cap \ensuremath{\mathcal{C}on}$\;\label{alg:con-apply-2}
\While{$\ensuremath{\mathcal F}f \ne \emptyset \lor \ensuremath{\mathcal B}f \ne \emptyset$}{\label{alg:rest-start}
$\ensuremath{\mathcal F}f \gets (\textsc{Post}(\ensuremath{\mathfrak G}, \ensuremath{\mathcal F}f) \cap \ensuremath{\mathcal{C}on}) \setminus \ensuremath{\mathcal F}$\;
$\ensuremath{\mathcal B}f \gets (\textsc{Pre}(\ensuremath{\mathfrak G}, \ensuremath{\mathcal B}f) \cap \ensuremath{\mathcal{C}on}) \setminus \ensuremath{\mathcal B}$\;
$\ensuremath{\mathcal F} \gets \ensuremath{\mathcal F} \cup \ensuremath{\mathcal F}f$\;
$\ensuremath{\mathcal B} \gets \ensuremath{\mathcal B} \cup \ensuremath{\mathcal B}f$\;
}\label{alg:rest-end}
$\ensuremath{\mathcal W} \subseteq V \times C \gets \ensuremath{\mathcal F} \cap \ensuremath{\mathcal B}$\;\label{alg:scc}
$\ensuremath{\mathfrak R_\mathit{scc}} \gets \ensuremath{\mathfrak R_\mathit{scc}} \cup \textsc{Join}(\ensuremath{\mathcal W})$\;
$\textsc{Decomposition}(\ensuremath{\mathfrak G}, \ensuremath{\mathfrak R_\mathit{scc}}, \mathcal{V} \setminus \ensuremath{\mathcal{C}on})$\;
$\textsc{Decomposition}(\ensuremath{\mathfrak G}, \ensuremath{\mathfrak R_\mathit{scc}}, \ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W})$\;
}
\BlankLine
\Fn{\upshape\textsc{Pivots}$(\mathcal{V})$}{
$\mathcal{P} \subseteq (V \times C) \gets \emptyset$; $\mathcal{V}' \subseteq (V \times C) \gets \mathcal{V}$\;
\While{$\mathcal{V}' \ne \emptyset$}{
$(v, c) \gets \textsc{Pick}(\mathcal{V}')$\;
$\mathcal{P} \gets \mathcal{P} \cup ( \{ v \} \times \sigma_1(v, \mathcal{V'})) $\;
$\mathcal{V'} \gets \mathcal{V'} \setminus (V \times \textsc{Colours}(\mathcal{P}))$\;
}
\Return $\mathcal P$\;
}
\caption{Symbolic Coloured SCC Decomposition}\label{algo:symbolic}
\end{algorithm}
An important observation is that the structure of components in the graph can change arbitrarily with
respect to the graph colours. In consequence, our algorithm cannot simply operate with sets of
graph vertices as the normal algorithm would. To that end, we use the
notion of coloured vertex sets as introduced in
Section~\ref{ssec:symb} where the symbolic operations we perform on
these sets have been described.
\paragraph{Pivot selection} Initially, the algorithm starts with all vertices and colours, i.e.~the full
set $V \times C$. However, as the components are discovered, the intermediate results $\mathcal{V}$ may contain different vertices appearing only for certain subsets of $C$. As a result, we often cannot pick a~single pivot vertex that
would be valid for all considered colours. Instead, we aim to pick a~\emph{pivot set} $P \subseteq V \times C$ such that for every colour
that still appears in $\mathcal V$,
the set contains \emph{exactly} one vertex. Alternatively, one can also
view the pivot set as a (partial) function from $C$ to $V$. This is done
in the \textsc{Pivots} function. In the following discussion of the algorithm,
we write $c$-coloured pivot to mean the vertex $u$ such that
$(u, c)$ is found in the coloured set returned by \textsc{Pivots}
in the current iteration (for all colours still present in $\mathcal{V}$).
Please note that the presented \textsc{Pivots} routine is rather naive, as it has to explicitly iterate all the pivot vertices, whose number can be substantial in the worst case. However, as presented, it should be easy to implement for basically any type of coloured graphs, regardless of the underlying representation. In the implementation section, we show \textsc{Pivots} can be re-implemented in the domain of BDDs such that it is guaranteed to always require only $\mathcal{O}(\log|V|)$ symbolic operations.
\paragraph{Coloured lock-step (phase one)} The lock-step reachability procedure also cannot operate as in a standard
graph. First of all, there can be colours where the forward reachability
converges first, as well as colours where this happens for backward
reachability. The algorithm thus has to account for both options
simultaneously. Second, for each colour, the reachability can converge in a
different number of steps. To deal with this problem, we introduce the
$\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$ variables. These store the mutually disjoint sets of
colours for which the forward and backward reachability procedures have already
converged. The lock-step procedure then terminates when $\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$
contain all the colours that appear in $\mathcal V$.
Throughout the algorithm, we keep track of several coloured-set variables.
The first two, $\ensuremath{\mathcal F}$ and $\ensuremath{\mathcal B}$, represent the forward and backward reachable
sets, respectively.
This means that for every colour $c$ present in $\mathcal V$, if $u$ is the
$c$-coloured pivot, every $(v, c) \in \ensuremath{\mathcal F}$ satisfies $u \creach{c} v$
and every $(v, c) \in \ensuremath{\mathcal B}$ satisfies $v \creach{c} u$.
Furthermore, if $c \in \ensuremath{F_\mathit{lock}}$ then $\ensuremath{\mathcal F}$ contains exactly all such pairs;
similarly for $\ensuremath{B_\mathit{lock}}$ and $\ensuremath{\mathcal B}$.
We say that a coloured vertex pair $(v, c)$ has been \emph{forward expanded}
or \emph{backward expanded} in the current iteration of the algorithm, if
there has been a call to the \textsc{Post} or \textsc{Pre} symbolic operation
with $(v, c)$ being an element of the coloured set argument.
To track which reachable coloured vertices are to be expanded later,
also called the \emph{frontiers} of the reachability sets,
we have the four variables $\ensuremath{\mathcal F}f$, $\ensuremath{\mathcal F}u$, $\ensuremath{\mathcal B}f$, $\ensuremath{\mathcal B}u$.
The frontier of $\ensuremath{\mathcal F}$ is the union $\ensuremath{\mathcal F}f \cup \ensuremath{\mathcal F}u$. The sets $\ensuremath{\mathcal F}f$ and $\ensuremath{\mathcal F}u$
are disjoint: $\ensuremath{\mathcal F}f$ involves those colours for which the
lock-step reachability procedure has not finished yet, i.e.~the colours that
are neither in $\ensuremath{F_\mathit{lock}}$ nor in $\ensuremath{B_\mathit{lock}}$,
while $\ensuremath{\mathcal F}u$ represents
the part of the frontier whose exploration is currently paused due to the fact
that its colours are in $\ensuremath{B_\mathit{lock}}$.
Note that there may be no pair $(v, c)$ of the forward frontier with
$c \in \ensuremath{F_\mathit{lock}}$ as that means that the exploration of the $c$-coloured
forward-reachable set is complete.
A symmetric role is played by the sets $\ensuremath{\mathcal B}f$ and $\ensuremath{\mathcal B}u$.
In the first while loop (lines~\ref{alg:ls-start}--\ref{alg:ls-end}), we
compute the reachability sets in the lock-step manner. Once a~reachability set
is completed for some colours (i.e.,~there are no vertices to expand with those
colours), we add the colours to the corresponding $\ensuremath{F_\mathit{lock}}$ or $\ensuremath{B_\mathit{lock}}$
variable. Note that we ensure that $\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$ remain disjoint even
if the two reachability procedures converged at the same time for certain
colours---see line~\ref{alg:lock}.
We use $\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$ to split the newly computed frontier sets into
the parts that are to be expanded in the next iteration ($\ensuremath{\mathcal F}f$, $\ensuremath{\mathcal B}f$)
and the parts currently left unexpanded ($\ensuremath{\mathcal F}u$, $\ensuremath{\mathcal B}u$).
Note that during the computation of $\textsc{Post}$ and $\textsc{Pre}$ on lines~\ref{alg:lockstep-post} and~\ref{alg:lockstep-pre}, we intersect the resulting set with $\mathcal{V}$. This step is not necessary for correctness, but as the algorithm divides $V \times C$ into smaller sets in each recursive call to \textsc{Decomposition}, it can happen that the set of states \emph{reachable} from $\mathcal{V}$ is substantially larger than $\mathcal{V}$ itself. In such cases, this intersection effectively restricts the computation of $\textsc{Post}$ and $\textsc{Pre}$ to the sub-graph of $\ensuremath{\mathfrak G}$ induced by $\mathcal{V}$.
\paragraph{Component identification (phase two)} After the first while loop terminates, we compute the set $\ensuremath{\mathcal{C}on}$ that is an analogue
for the converged set of the original lock-step algorithm
(line~\ref{alg:con}).
As already suggested above and unlike the original
algorithm, this set cannot be just $\ensuremath{\mathcal F}$ or $\ensuremath{\mathcal B}$, but is instead a~mixture of
both, depending on the converged colours. To~compute this set, we use the
$\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$ variables.
Once $\ensuremath{\mathcal{C}on}$ is computed, $\ensuremath{\mathcal F}f$ and $\ensuremath{\mathcal B}f$ are restarted using the converged portion of $\ensuremath{\mathcal F}u$ and $\ensuremath{\mathcal B}u$ (lines~\ref{alg:con-apply} and~\ref{alg:con-apply-2}). The second while loop (lines~\ref{alg:rest-start}--\ref{alg:rest-end}) can then
complete the unfinished forward and backward reachability set, now restricted to
the inside of the converged set. The intersection of $\ensuremath{\mathcal F}$ and $\ensuremath{\mathcal B}$ then
forms a~coloured set $\ensuremath{\mathcal W}$ with the property that for all
$c \in \textsc{Colours}(\mathcal V)$, $\ensuremath{\mathcal W}(\_, c)$ is a~strongly
connected component of $\ensuremath{\mathfrak G}(c)$. We create the corresponding relation
using the \textsc{Join} operation, add this relation to the resulting
$\ensuremath{\mathfrak R_\mathit{scc}}$, and recursively call the whole procedure with
$\mathcal V \setminus \ensuremath{\mathcal{C}on}$ and $\ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W}$ as the
base sets.
\paragraph{Comments on the coloured approach} Let us note that there is possibly another approach to processing coloured graphs. Instead of trying
to work with all colours still appearing in the coloured vertex set at once,
we could fork a~new recursive procedure whenever the colour set
splits due to the differences in the graph structure. For example, instead
of picking multiple coloured vertices as pivots, one could pick a single
vertex with a valid subset of colours and then address the remaining colours
in a separate recursive call. Similarly, instead of a single recursive $\textsc{Decomposition}$ call with $\ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W}$, we could consider two calls, one with the $\mathcal{F}$ portion of $\ensuremath{\mathcal{C}on}$ and the other with the $\mathcal{B}$ portion of $\ensuremath{\mathcal{C}on}$ (note that these are colour-disjoint since each colour can converge only in one of the two sets).
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We say that a coloured vertex pair $(v, c)$ has been \emph{forward expanded}
or \emph{backward expanded} in the current iteration of the algorithm, if
there has been a call to the \textsc{Post} or \textsc{Pre} symbolic operation
with $(v, c)$ being an element of the coloured set argument.
To track which reachable coloured vertices are to be expanded later,
also called the \emph{frontiers} of the reachability sets,
we have the four variables $\ensuremath{\mathcal F}f$, $\ensuremath{\mathcal F}u$, $\ensuremath{\mathcal B}f$, $\ensuremath{\mathcal B}u$.
The frontier of $\ensuremath{\mathcal F}$ is the union $\ensuremath{\mathcal F}f \cup \ensuremath{\mathcal F}u$. The sets $\ensuremath{\mathcal F}f$ and $\ensuremath{\mathcal F}u$
are disjoint: $\ensuremath{\mathcal F}f$ involves those colours for which the
lock-step reachability procedure has not finished yet, i.e.~the colours that
are neither in $\ensuremath{F_\mathit{lock}}$ nor in $\ensuremath{B_\mathit{lock}}$,
while $\ensuremath{\mathcal F}u$ represents
the part of the frontier whose exploration is currently paused due to the fact
that its colours are in $\ensuremath{B_\mathit{lock}}$.
Note that there may be no pair $(v, c)$ of the forward frontier with
$c \in \ensuremath{F_\mathit{lock}}$ as that means that the exploration of the $c$-coloured
forward-reachable set is complete.
A symmetric role is played by the sets $\ensuremath{\mathcal B}f$ and $\ensuremath{\mathcal B}u$.
In the first while loop (lines~\ref{alg:ls-start}--\ref{alg:ls-end}), we
compute the reachability sets in the lock-step manner. Once a~reachability set
is completed for some colours (i.e.,~there are no vertices to expand with those
colours), we add the colours to the corresponding $\ensuremath{F_\mathit{lock}}$ or $\ensuremath{B_\mathit{lock}}$
variable. Note that we ensure that $\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$ remain disjoint even
if the two reachability procedures converged at the same time for certain
colours---see line~\ref{alg:lock}.
We use $\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$ to split the newly computed frontier sets into
the parts that are to be expanded in the next iteration ($\ensuremath{\mathcal F}f$, $\ensuremath{\mathcal B}f$)
and the parts currently left unexpanded ($\ensuremath{\mathcal F}u$, $\ensuremath{\mathcal B}u$).
Note that during the computation of $\textsc{Post}$ and $\textsc{Pre}$ on lines~\ref{alg:lockstep-post} and~\ref{alg:lockstep-pre}, we intersect the resulting set with $\mathcal{V}$. This step is not necessary for correctness, but as the algorithm divides $V \times C$ into smaller sets in each recursive call to \textsc{Decomposition}, it can happen that the set of states \emph{reachable} from $\mathcal{V}$ is substantially larger than $\mathcal{V}$ itself. In such cases, this intersection effectively restricts the computation of $\textsc{Post}$ and $\textsc{Pre}$ to the sub-graph of $\ensuremath{\mathfrak G}$ induced by $\mathcal{V}$.
\paragraph{Component identification (phase two)} After the first while loop terminates, we compute the set $\ensuremath{\mathcal{C}on}$ that is an analogue
for the converged set of the original lock-step algorithm
(line~\ref{alg:con}).
As already suggested above and unlike the original
algorithm, this set cannot be just $\ensuremath{\mathcal F}$ or $\ensuremath{\mathcal B}$, but is instead a~mixture of
both, depending on the converged colours. To~compute this set, we use the
$\ensuremath{F_\mathit{lock}}$ and $\ensuremath{B_\mathit{lock}}$ variables.
Once $\ensuremath{\mathcal{C}on}$ is computed, $\ensuremath{\mathcal F}f$ and $\ensuremath{\mathcal B}f$ are restarted using the converged portion of $\ensuremath{\mathcal F}u$ and $\ensuremath{\mathcal B}u$ (lines~\ref{alg:con-apply} and~\ref{alg:con-apply-2}). The second while loop (lines~\ref{alg:rest-start}--\ref{alg:rest-end}) can then
complete the unfinished forward and backward reachability set, now restricted to
the inside of the converged set. The intersection of $\ensuremath{\mathcal F}$ and $\ensuremath{\mathcal B}$ then
forms a~coloured set $\ensuremath{\mathcal W}$ with the property that for all
$c \in \textsc{Colours}(\mathcal V)$, $\ensuremath{\mathcal W}(\_, c)$ is a~strongly
connected component of $\ensuremath{\mathfrak G}(c)$. We create the corresponding relation
using the \textsc{Join} operation, add this relation to the resulting
$\ensuremath{\mathfrak R_\mathit{scc}}$, and recursively call the whole procedure with
$\mathcal V \setminus \ensuremath{\mathcal{C}on}$ and $\ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W}$ as the
base sets.
\paragraph{Comments on the coloured approach} Let us note that there is possibly another approach to processing coloured graphs. Instead of trying
to work with all colours still appearing in the coloured vertex set at once,
we could fork a~new recursive procedure whenever the colour set
splits due to the differences in the graph structure. For example, instead
of picking multiple coloured vertices as pivots, one could pick a single
vertex with a valid subset of colours and then address the remaining colours
in a separate recursive call. Similarly, instead of a single recursive $\textsc{Decomposition}$ call with $\ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W}$, we could consider two calls, one with the $\mathcal{F}$ portion of $\ensuremath{\mathcal{C}on}$ and the other with the $\mathcal{B}$ portion of $\ensuremath{\mathcal{C}on}$ (note that these are colour-disjoint since each colour can converge only in one of the two sets).
While such an approach could be to some extent
beneficial in a massively parallel environment where each recursive call can
be executed independently on a~new CPU, the amount of forking in large systems
will soon become unreasonable. More importantly, it defeats the purpose of
the symbolic representation, which aims to minimise the number of symbolic
operations.
\begin{figure}
\caption{An illustration of the algorithm execution.
There are two colours: ${\color{red}
\label{fig:example}
\end{figure}
\paragraph{Example} The execution of one iteration of the algorithm is illustrated in
Figure~\ref{fig:example}. Here, we have an edge-coloured graph with six vertices
and two colours (red and blue).
The top-most picture represents the initial situation after we have chosen
the pivots; in this case, $\{(b, \mathit{blue}), (b, \mathit{red})\}$.
The next four rows illustrate the first phase (the first while loop) of the
algorithm. After the second iteration of the loop, the blue colour becomes
locked in $\ensuremath{F_\mathit{lock}}$, and thus $(f, \mathit{blue})$ is not expanded in the
backward reachability procedure. This is illustrated by its dashed outline.
After the third iteration of the loop,
the red colour becomes locked in $\ensuremath{B_\mathit{lock}}$ and thus the first phase ends.
In the second phase, both reachability procedures continue from the paused
coloured vertices (dashed outlines); the result is seen in the fifth row.
The intersection of the two reachable sets (i.e.~the coloured set $\mathcal W$)
is then illustrated in the bottom-most picture.
The algorithm would now continue with the coloured sets
$\mathcal{V} \setminus \ensuremath{\mathcal{C}on} = \{ (a, \mathit{blue}), (d, \mathit{blue}) \}$
and $\ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W} = \{ (c, \mathit{red}) \}$.
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\subsection{Correctness and Complexity of the Coloured Lock-step Algorithm}
\begin{thm}\label{thm:correctness}
Let $\ensuremath{\mathfrak G} = (V, C, E)$ be a~coloured graph. The coloured lock-step
algorithm terminates and computes the coloured SCC decomposition relation~$\ensuremath{\mathfrak R_\mathit{scc}}$.
\end{thm}
\begin{proof}
We first show that the set $\ensuremath{\mathcal W}$ computed in line~\ref{alg:scc}
indeed contains one SCC for every colour $c \in \textsc{Colours}(\mathcal V)$
and that the recursive calls of \textsc{Decomposition} preserve the property
that $\mathcal V$ is SCC-closed with respect to all colours.
Let us assume that $\mathcal V$ is SCC-closed and let us take an arbitrary
$c \in \textsc{Colours}(\mathcal V)$. The function \textsc{Pivots} chooses
a~set that contains exactly one pair whose colour is $c$, let us call this
pair $(v, c)$. Let us further assume that $c$ is assigned into $\ensuremath{F_\mathit{lock}}$
first (the case with $\ensuremath{B_\mathit{lock}}$ is completely symmetric).
Let us now choose an arbitrary vertex $w$ such that $v$ and $w$ are in the
same SCC of $\ensuremath{\mathfrak G}(c)$, i.e.~$v \to^* w$ and $w \to^* v$.
As the first while loop
finishes, $\ensuremath{\mathcal F}$ contains all the pairs of the form $(u, c) \in \mathcal V$
where $u$ is reachable from $v$ in $\ensuremath{\mathfrak G}(c)$. Thus, it also contains $(w, c)$
due to the fact that $\mathcal V$ is SCC-closed.
Now, either $(w, c) \in \ensuremath{\mathcal B}$, or there exists a vertex $x$ such that
$w \to^* x$, $x \to^* v$ in $\ensuremath{\mathfrak G}(c)$ and $x \in \ensuremath{\mathcal B}u$.
This means that $(w, c)$ is added to $\ensuremath{\mathcal B}$ in the second while loop.
In both cases, both $(v, c)$ and $(w, c)$ are then added to
$\ensuremath{\mathcal W}$. As the vertex choices were arbitrary, this proves that
the SCC of $v$ in $\ensuremath{\mathfrak G}(c)$ is contained in $\ensuremath{\mathcal W}$.
Furthermore, if $(y, c) \in \ensuremath{\mathcal W}$ for an arbitrary $y$, then
$v \to^* y$ and $y \to^* v$ in $\ensuremath{\mathfrak G}(c)$, which means that $y$ is in
$SCC(\ensuremath{\mathfrak G}(c), v)$. This proves that $\ensuremath{\mathcal W}$ contains exactly one
SCC for every colour $c \in \textsc{Colours}(\mathcal V)$.
We now argue that $\ensuremath{\mathcal{C}on}$ is SCC-closed with respect to all colours.
This immediately implies that both $\mathcal V \setminus \ensuremath{\mathcal{C}on}$ and
$\ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W}$ are SCC-closed.
Let us assume that there is a~colour $c \in \textsc{Colours}(\mathcal V)$
and two vertices $v$, $w$ in the same SCC of $\ensuremath{\mathfrak G}(c)$ such that
$(v, c) \in \ensuremath{\mathcal{C}on}$, but $(w, c) \not\in \ensuremath{\mathcal{C}on}$.
Let us assume that $c \in \ensuremath{F_\mathit{lock}}$ (as above, the case of $\ensuremath{B_\mathit{lock}}$ is
completely symmetrical).
Then $(v, c) \in \ensuremath{\mathcal F}$ after the first while loop finishes. This also means
that $(w, c) \in \ensuremath{\mathcal F}$ as the forward reachability procedure is completed for
$c$ and thus $(w, c) \in \ensuremath{\mathcal{C}on}$, a~contradiction.
What remains is to show that the algorithm terminates and that every
SCC is eventually found. Termination is trivially proved by the fact
that size of the set $\mathcal V$ always decreases in recursive calls:
both $\ensuremath{\mathcal W}$ and $\ensuremath{\mathcal{C}on}$ are non-empty because they contain the
initial pivot set as a~subset.
Clearly, a representant of every SCC of every monochromatisation $\ensuremath{\mathfrak G}(c)$ is
eventually chosen as a~pivot. Together with the above reasoning, this implies
that the algorithm is correct.
\end{proof}
\begin{thm}\label{thm:complexity}
Let $|V|$ be the number of vertices in the coloured graph and let
$|C|$ be the number of colours. The coloured lock-step algorithm
performs at most $\mathcal O(|C|\cdot |V|\cdot \log |V|)$ symbolic steps.
\end{thm}
\begin{proof}
Let us first note that all the derived operations defined in
Figure~\ref{tab:operations} use only a~constant number of the basic symbolic
operations. As we are considering asymptotic complexity here, we can view all
the operations in Figure~\ref{tab:operations} as elementary symbolic steps.
We first make the observation that each vertex may be chosen as a~part of the
pivot set at most $|C|$ times. Clearly, once a~vertex is included in the pivot
set with a~set of colours $C'$, then, $\{v\} \times C'$ is a subset of first $\ensuremath{\mathcal{C}on}$, and later $\ensuremath{\mathcal W}$
(due to the monotonicity of the construction of $\mathcal F$ and $\mathcal B$). Therefore, the elements of $\{v\} \times C'$ do not appear in subsequent recursive
calls. Since a single vertex-colour pair cannot be returned by \textsc{Pivots} twice, it means that the total cumulative complexity of all the calls to the \textsc{Pivots} routine is bounded by $O(|C|\cdot|V|)$. We can therefore exclude them from the rest of the complexity analysis.
We now consider the complexity of a~single call to \textsc{Decomposition}
without the subsequent recursive calls. Let us now select one of the
colours for which the lock-step reachability procedure
(lines~\ref{alg:ls-start}--\ref{alg:ls-end}) finished last, i.e.,~one of the
colours that have been added to $\ensuremath{F_\mathit{lock}}$ or $\ensuremath{B_\mathit{lock}}$ in the final
iteration of the loop. Let us call this colour $c$.
Recall that $\sigma_2(c, \mathcal X)$ is the set of vertices with colour
$c$ in a~coloured set $\mathcal X$.
Let us denote by $W$ the monochromatic SCC discovered for $c$, i.e. $W := \sigma_2(c, \ensuremath{\mathcal W})$, and let
$S$ be the smaller of $\sigma_2(c, \mathcal V \setminus \ensuremath{\mathcal{C}on})$
and $\sigma_2(c, \ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W})$.
Clearly $S$ contains at most $|V|/2$ vertices.
Let $k = |S \cup W|$.
We now argue that the number of symbolic steps in a~given call
(without the recursive calls) is bounded by $\mathcal O(k)$. This is because in a lock-step algorithm, the call to \textsc{Decomposition} must explore the discovered SCC itself (i.e. $W$), and the smaller of the forward or backward reachable sets from this SCC (i.e. $S$) -- intuitively, its complexity should be thus bounded by the size of these two sets.
Assume w.l.o.g.~that $c \in \ensuremath{F_\mathit{lock}}$ (a completely symmetric argument solves
the case $c \in \ensuremath{B_\mathit{lock}}$).
Then after the first while loop finishes, we have $\sigma_2(c, \ensuremath{\mathcal{C}on}) = \sigma_2(c, \mathcal F)$.
If $S$ is $\sigma_2(c, \ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W})$
then $k$ is the size of $\sigma_2(c, \mathcal F)$ (and thus also $\sigma_2(c, \ensuremath{\mathcal{C}on})$), since $\sigma_2(c, \mathcal{F})$ consists of $\sigma_2(c, \ensuremath{\mathcal{C}on} \setminus \ensuremath{\mathcal W})$ (the set $S$) and $\sigma_2(c, \ensuremath{\mathcal W})$ (the discovered SCC).
Each iteration of the first while loop puts at least one vertex with
colour $c$ into $\mathcal F$; otherwise $c$ would not have finished in the last iteration. This means that the loop runs for at most
$k$ iterations. This also means that the size of $\sigma_2(x, \mathcal X)$ for
all colours $x$ and $\mathcal X \in \{ \mathcal F, \mathcal B \}$ is also
bounded by $k$ after the first while loop finishes, which in turn means that the second while loop cannot make more
than $O(k)$ steps.
If $S$ is $\sigma_2(c, \mathcal V \setminus \ensuremath{\mathcal{C}on})$ instead,
let us define $B := \sigma_2(c, \mathcal B)$ right after the first
while loop has finished. We know that $B \subseteq S \cup W$:
if a vertex $v$ was in $B \setminus S$, then it would have to be in $\ensuremath{\mathcal{C}on}$ (i.e. $(v, c) \in \ensuremath{\mathcal{C}on}$). Due to our initial assumption of $c \in \ensuremath{F_\mathit{lock}}$ (w.l.o.g), we then also have $(v,c) \in \mathcal
F$ which dictates $v \in W$. Consequently, we see that any vertex $v \in B$ must be either in $S$ or in $W$, arriving at $B \subseteq S \cup W$.
Again, each iteration of the first while loop puts at least one vertex
with colour $c$ into $\mathcal B$; otherwise $c$ would have been
in $\ensuremath{B_\mathit{lock}}$ before it appeared in $\ensuremath{F_\mathit{lock}}$. Similarly to the previous
case, this means that both while loops run for at most $O(k)$ steps.
The rest of the argument uses amortised reasoning, in a~way similar to the
proof in~\cite{bloem2000}. Note that each vertex is going to be an element of
the set $W$ as described above at most $|C|$ times (once for each colour).
Furthermore, each vertex is going to be an element of the set $S$ as described
above at most $|C|\cdot\log|V|$ times: for each colour, the vertex can be an
element of the smaller of the two sets at most $\log|V|$ times. As the cost of
each single call can be charged to the vertices in $S \cup W$ as explained
above, it is sufficient to charge each vertex the total cost of
$|C| +|C|\cdot\log|V|$. Together, this means that the total number of symbolic
steps is bounded by $O(|C|\cdot|V|\cdot\log|V|)$.
\end{proof}
Note that the upper bound established by Theorem~\ref{thm:complexity} is
no better than the one we would get if we split the coloured graph into
its monochromatic constituents and processed each
separately using the original lock-step algorithm~\cite{bloem2000}.
We remark, however, that the practical complexity of the coloured approach can be
much smaller. Indeed, the complexity analysis in the previous proof focused
on a~single colour, omitting the fact that SCCs for many other colours are
found at the same time. In cases where the edges are largely shared among
the colours, which is true in many applications, the coloured algorithm has the
potential to significantly outperform the parameter-scan approach. The
situation is similar to that of the coloured model checking; see the
observations made in~\cite{tcbb}.
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\section{Symbolic Computation with Boolean Networks}
\label{section:boolean-networks}
The algorithm as presented in the previous section is completely agnostic to the properties of the underlying system, as long one provides an implementation of all the necessary symbolic operations. However, to empirically test its performance, we need to pick such an implementation, which typically entails analysis of a specific class of systems.
In this paper, we consider Boolean networks~\cite{KAUFFMAN,Bernot05,SCHWAB,THOMAS}, specifically asynchronous Boolean networks, which represent a popular discrete modelling framework in systems biology~\cite{Brim2013, Grieb_2015}. Due to incomplete biological knowledge, the dynamics of a Boolean network can by often only partially known. This uncertainty can be then captured using coloured directed graphs. In this section, we introduce Boolean networks and show how they can be translated into coloured graphs suitable for SCC-decomposition.
Asynchronous Boolean networks are especially challenging for symbolic analysis. It is a well-known fact, that using symbolic structures (e.g BDDs) to explore very large state spaces gives good results for synchronous systems, but shows its limits when trying to tackle asynchronicity (see e.g.~\cite{DBLP:conf/forte/CouvreurT05}).
\subsection{Boolean networks with inputs} A Boolean network (BN), as the name suggests, consists of $n$ Boolean \emph{variables} $s_1, \ldots, s_n$ which together describe the state of the network. The dynamics of the network can also depend on additional $m$ Boolean \emph{inputs} $c_1, \ldots, c_m$ (sometimes also called \emph{constants}, or \emph{logical parameters}), whose value is assumed to be fixed, but generally unknown. The valuations of these inputs correspond to the colours of our Kripke structure.
Each network variable $s_i$ is equipped with a Boolean update function $b_i: \{0,1\}^n \times \{0,1\}^m \to \{0,1\}$ that updates the variable based on the state of the network, and the values of its inputs. We assume that the variables are updated \emph{asynchronously}, meaning that during every state transition, exactly one variable is updated.
Such a network with inputs defines a coloured graph where $V = \{0,1\}^n$, $C = \{0,1\}^m$, and for every $c \in C$, we have that $u \xrightarrow{c} v$ if and only if $u \not= v$ and $v = u[u_i \mapsto b_i(u, c)]$ for some $i \in [1,n]$. That is, $v$ is equal to $u$ where the $i$-th variable is updated with the output of function $b_i$. Because all variables and inputs are Boolean, this structure has a~fairly straightforward symbolic representation in terms of binary decision diagrams, as we later demonstrate.
Note that in practice, we often work within a~subset of biologically relevant colours, denoted as~$Valid$ (i.e. not every possible valuation of $c_1, \ldots, c_m$ may be biologically admissible). In the algorithms, this is implicitly reflected such that the set of all possible colours $C$ corresponds to the set $Valid$ instead of the set of \emph{all} possible valuation (i.e. $\{0,1\}^m$) if demanded by the application at hand.
\subsection{Partially specified Boolean networks} A Boolean network with inputs allows us to easily encode a wide range of biochemical systems in a~machine friendly format. However, for systems with a high degree of uncertainty, it often fails to capture this uncertainty in a~way understandable to a human reader.
To mitigate this issue, we consider \emph{partially specified} Boolean networks that allow us to explicitly mark parts of the update functions as unknown. Specifically, let us assume that $f_1^{(a_1)}$, $f_2^{(a_2)}$, $\ldots$ are symbols standing in for some uninterpreted (fixed but arbitrary) Boolean functions (here, $a_i$ denotes their arity). A partially specified Boolean network then consists of $n$ Boolean variables and $p$ uninterpreted Boolean functions. In such a~network, every update function $b'_i$ is specified as a Boolean expression that can use the function symbols $f_1, \ldots, f_p$.
This type of formalism is often easier to comprehend, as the uncertainty in dynamics is tied to the update functions instead of inputs (if desired, input can be still expressed using uninterpreted functions of arity zero). It is not immediately clear how such a~network should be represented symbolically though.
One option is to translate a partially specified network into a~BN with inputs. Any uninterpreted function $f_i^{(a)}$ can be encoded in terms of $2^a$ Boolean inputs $c_1^{i}, \ldots, c_{2^a}^i$ if we consider that input $r_j^i$ denotes the output of $f_i^{(a)}$ in the $j$-th row of its truth table. Formally, this translation can be achieved using a~repeated application of the following expansion rule:
\begin{align*}
f(\alpha_1, \ldots, \alpha_a) \equiv (\alpha_1 \Rightarrow f'_1(\alpha_2, \ldots, \alpha_a)) \land (\neg\alpha_1 \Rightarrow f'_2(\alpha_2, \ldots, \alpha_a))
\end{align*}
Here, $f'_1$ and $f'_2$ are fresh uninterpreted functions of arity $a-1$, and $\alpha_i$ are arbitrary Boolean expressions. Using this rule, we can always convert a partially specified network to a Boolean network with inputs. The number of inputs will be exponential with respect to the arity of the employed uninterpreted functions though (since each application of the rule replaces one uninterpreted function with two, and the depth of the recursive expansion is the arity $a$).
For example, consider the following partially specified Boolean network:
\begin{align*}
b'_1 &:= x_1 \land f_1^{(1)}(x_2)\\
b'_2 &:= \neg x_1 \lor f_2^{(2)}(x_1, x_3)\\
b'_3 &:= (f_3^{(0)} \Leftrightarrow x_3) \land f_2^{(2)}(\neg x_1, x_2)
\end{align*}
It uses three uninterpreted Boolean functions $f_1^{(1)}$, $f_2^{(2)}$, and $f_3^{(0)}$. After performing the aforementioned expansion, and simplifying the resulting expressions slightly for readability, we obtain the following network with logical inputs:
\begin{align*}
b_1(x, c) =&~x_1 \land (x_2 \Rightarrow c^{1}_{[1]}) \land (\neg x_2 \Rightarrow c^{1}_{[0]})\\
b_2(x, c) =&~\neg x_2 \lor (((x_1 \land x_3) \Rightarrow c^{2}_{[1,1]}) \land ((x_1 \land \neg x_3) \Rightarrow c^{2}_{[1,0]})\\&\hspace{15.5pt}\land~((\neg x_1 \land x_3) \Rightarrow c^{2}_{[0,1]}) \land ((\neg x_1 \land \neg x_3) \Rightarrow c^{2}_{[0,0]})) \\
b_3(x, c) =&~(c^{3} \Leftrightarrow x_3) \land ((\neg x_1 \land x_2) \Rightarrow c^{2}_{[1,1]}) \land ((\neg x_1 \land \neg x_2) \Rightarrow c^{2}_{[1,0]})\\&\hspace{57pt}\land((x_1 \land x_2) \Rightarrow c^{2}_{[0,1]}) \land ((x_1 \land \neg x_2) \Rightarrow c^{2}_{[0,0]})
\end{align*}
Here, each $c^i_j$ corresponds to one truth table row of $f_i$, such that $j$ describes the input vector corresponding to said row (i.e. $c^{1}_{[0,1]}$ represents the value of $f_1(0, 1)$).
\subsection{Symbolic Representation of BNs}
As a symbolic representation, a natural choice are Reduced Ordered Binary Decision Diagrams (ROBDD, or simply BDD)~\cite{bryant86}, which can concisely encode Boolean functions or relations of Boolean vectors. Specifically, out implementation leverages the internal tools and libraries provided by the tool AEON~\cite{aeon}.
Since a Boolean network consists of $n$ Boolean variables and $m$ Boolean inputs, any subset of $V$, $C$, or a relation $X \subseteq V \times C$ (a coloured set of vertices) can be seen as a Boolean formula over the network variables and inputs. That is, each network variable and logical input corresponds to one decision variable of the BDD. Here, a pair $(s,c)$ belongs to such a~relation iff it represents a satisfying assignment of this formula $X$. For relations of higher arity, fresh decision variables are created for each component of the relation. Standard set operations as described in Fig.~\ref{tab:operations} then correspond to logical operations on such formulae ($\land \equiv \cap$, $\lor \equiv \cup$, etc.).
Relation operations are similarly implementable using BDD primitives. In particular, existential quantification of a single decision variable (e.g. $\exists s_i . X$ or $\exists c_j . X$) is a native operation on BDDs. Consequently, existential quantification on relations (as well as \textsc{Colours}) is simply a quantification over all decision variables encoding the specific relation component (i.e. all network variables for $V$, or all logical inputs for $C$). Finally, \textsc{Swap} only influences the way in which a BDD is interpreted -- the actual structure of the BDD is unaffected.
To encode the network dynamics, notice that every update function $b_i$ can be directly represented as a separate BDD. From such BDDs, we can build one large BDD describing the whole coloured transition relation, which is traditionally used for the computation of \textsc{Pre} and \textsc{Post}. But the symbolic representation of such relation is often prohibitively complex for asynchronous systems. Instead, we compute \textsc{Pre} and \textsc{Post} using partial results for individual variables, which uses more symbolic operations but is less likely to cause a~blow-up in the size of the BDD:
\begin{align*}
\textsc{VarPost}(\ensuremath{\mathfrak G}, i, \mathcal{X}) & = (\mathcal{X} \land (b_i \centernot\Leftrightarrow s_i))[s_i \mapsto \neg s_i]\\
\textsc{VarPre}(\ensuremath{\mathfrak G}, i, \mathcal{X}) & = \mathcal{X}[s_i \mapsto \neg s_i] \land (b_i \centernot\Leftrightarrow s_i)\\
\textsc{Post}(\ensuremath{\mathfrak G}, \mathcal{X}) & = \bigvee_{i \in [1,n]} \textsc{VarPost}(i, \mathcal{X})\\
\textsc{Pre}(\ensuremath{\mathfrak G}, \mathcal{X}) & = \bigvee_{i \in [1,n]} \textsc{VarPre}(i, \mathcal{X})
\end{align*}
Here, $[s_i \mapsto \neg s_i]$ is the standard substitution operation, which we use to flip the value of variable $s_i$ in the resulting formula if it does not agree with the output of $b_i$. Note that this operation can be also implemented structurally directly on the BDD by exchanging the children of decision nodes conditioning on $s_i$. Also note that sub-formulae that do not depend on $X$ can be pre-computed once for the whole run of the algorithm, and the version of $\textsc{Pre}$ and $\textsc{Post}$ for monochromatic graphs can be implemented in exactly the same way.
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\section{Implementation}
\label{section:implementation}
Finally, let us discuss a number of technical improvements which our algorithm employs in practice, and whose impact we consider in the evaluation section.
\subsection{Pivot Selection}
In Algorithm~\ref{algo:symbolic}, we gave a naive implementation of the $\textsc{Pivots}(\mathcal{X})$ function. Here, we show how to implement it for BDDs in a much more concise way. Note that our approach uses the notation we established earlier for Boolean networks, but is generally applicable to any set or relation of bit-vectors represented using BDDs.
First, notice that for a single network variable, we can define a similar operation, which we call $\textsc{Pick}(i, \mathcal{X})$:
\begin{equation*}
\textsc{Pick}(i, \mathcal{X}) = \mathcal{X} \setminus (\mathcal{X} \land \neg s_i)[s_i \gets \neg s_i]
\end{equation*}
Here, we first restrict $\mathcal{X}$ only to the valuations which have $s_i = \mathit{false}$, and then invert the value of $s_i$ (resulting in $s_i$ being always $\mathit{true}$ in the set). Once we subtract these valuations from $\mathcal{X}$, the resulting set then contains a valuation with $s_i = \mathit{true}$ only if it does \emph{not} contain the same valuation with $s_i = \mathit{false}$. Intuitively, for any valuation of the remaining BDD decision variables (i.e. $s_j$ and $c_j$ in our case) that is in $\mathcal{X}$, we just picked a single unique value of $s_i$ (while preferring the value $s_i = \mathit{false}$).
However, observe that we cannot simply apply $\textsc{Pick}$ to every network variable alone to obtain the result of \textsc{Pivots}. Intuitively, the problem lies in the fact that \textsc{Pick} selects a witness for each variable in isolation, while $\textsc{Pivots}$ considers all network variables as interconnected. We resolve this problem using a different equation, one which eliminates the picked variable in the recursive invocation:
\begin{align*}
\textsc{Pivots}(\mathcal{X}) &= \textsc{F}(\mathcal{X}, s_1, \ldots, s_n)\\
\textsc{F}(\mathcal{X}, s_1) &= \textsc{Pick}(1, \mathcal{X})\\
\textsc{F}(\mathcal{X}, s_1, \ldots, s_k) &= \textsc{Pick}(k, \mathcal{X}) \cap \textsc{F}(\exists s_k. \mathcal{X}, s_1, \ldots, s_{k-1})
\end{align*}
In this equation, the final case $\textsc{F}(\mathcal{X}, s_1)$ is clearly correct, since it simply defers to $\textsc{Pick}(i, \mathcal{X})$. However, to understand why the recursive case $\textsc{F}(\mathcal{X}, s_1, \ldots, s_k)$ is correct, observe the following: Assume that the set $\mathcal{Y} = \textsc{F}(\exists s_k. \mathcal{X}, s_1, \ldots, s_{k-1})$ is computed correctly. That is, for any valuation of the remaining variables, $\mathcal{Y}$ contains a single unique \emph{incomplete witness} valuation of variables $s_1, \ldots, s_{k-1}$. Now, since the BDD representing $\exists s_k . \mathcal{X}$ does not depend on $s_k$, each such unique \emph{witness} must be included in $\mathcal{Y}$ twice: once with $s_k = \mathit{true}$ and once with $s_k = \mathit{false}$. In other words, a single \emph{witness} valuation of $s_1, \ldots, s_{k-1}$ must be tied to two different valuations of the remaining variables, and these valuations are differentiated only by the variable $s_k$.
Now, one of these two valuations is necessarily included in the set $\textsc{Pick}(k, \mathcal{X})$. The other is either missing from $\mathcal{X}$ altogether, or is eliminated by $\textsc{Pick}(k, \mathcal{X})$. As such, computing $\textsc{Pick}(k, \mathcal{X}) \cap \mathcal{Y}$ extends the witness from $s_1, \ldots, s_{k-1}$ to $s_1, \ldots, s_{k}$ by eliminating one of the two aforementioned occurrences of the \emph{incomplete witness}.
Observe that, as opposed to the original naive implementation of \textsc{Pivots}, this implementation only requires $\mathcal{O}(n)$ (i.e. $\mathcal{O}(\log |V|)$) symbolic operations in any case.
\subsection{Saturation}
In~\cite{Ciardo06}, and later in greater detail within~\cite{Ciardo11}, Ciardo et al. show that when the system is asynchronous, it may be much easier to compute reachable sets (and consequently SCCs) by applying only one transition (e.g. denoted $t_1$) at a time. Once applying $t_1$ cannot add new states to the reachable set, another transition (e.g denoted $t_2$) can be considered, respecting the order in which the affected variables appear in the symbolic data structure (Ciardo et al. employ multivalued decision diagrams, but the principle also applies to BDDs). If the application of other transitions causes that we can again add new states using $t_1$, the process starts anew and $t_1$ is ``saturated'' again.
In the comparison presented in~\cite{Ciardo11}, only the Xie-Beerel $\mathcal{O}(|V|^2)$ algorithm is used with saturation enabled, while the lock-step algorithm is used as given in~\cite{bloem2000}. However, we argue that saturation can be also beneficial in the lock-step algorithm.
\paragraph{Asymptotic complexity} Unfortunately, combining lock-step with saturation disrupts the $\mathcal{O}(|V| \cdot \log|V|)$ asymptotic complexity of the algorithm. To see why this is the case, observe that classical symbolic reachability (i.e. a fixed-point algorithm iterating the $\textsc{Post}$ procedure) requires $\mathcal{O}(|V|)$ steps to explore a graph. Meanwhile, a reachability procedure employing saturation needs $\mathcal{O}(|V||T|)$ operations, where $|T|$ is the number of distinct transitions.
This is caused by the fact that saturation needs to check up to $|T|$ transitions to discover a vertex. For example, consider an asynchronous graph employing transitions $t_1, \ldots, t_n$ such that $t_1$ and $t_n$ are alternated on a path of length $\mathcal{O}(|V|)$. Between considering $t_1$ and $t_n$, saturation will attempt each of the $|T|$ transitions, which are useless on this path, but still consume a symbolic operation.
Consequently, this $|T|$ factor trickles down into the complexity of both the Xie-Beerel and lock-step algorithms if saturation is used, as both ultimately rely on some form of reachability to discover the graph vertices. The complexity of the coloured algorithms is then similarly affected.
\paragraph{Saturation and lock-step} The main idea of how saturation is applied in a coloured lock-step algorithm (for Boolean networks) is shown in Algorithm~\ref{algo:saturation}. The algorithm presents a helper function which performs \emph{one reachability step}, similar to what is performed by the $\textsc{Post}$ function. However, in this algorithm, only one transition is fired for each colour (we assume the iteration follows the order of variables as they appear in the symbolic representation, which benefits saturation). Additionally, a set $R$ of colours that could not perform a step is computed. A similar procedure can be considered for backwards reachability, simply replacing $\textsc{VarPost}$ with $\textsc{VarPre}$.
Note that there is a slight discrepancy between Algorithm~\ref{algo:saturation} and the intuitive description of saturation that we gave earlier. In particular, we see that during a \textsc{NextStep} operation, a transition for each variable is triggered at most once, as opposed to the original description, where a transitions are fired repeatedly. This is caused by the simple nature of Boolean networks: In a BN, a single transition always modifies a single Boolean variable. Consequently, no new states can be discovered by firing a single transition multiple times in sequence. For other asynchronous systems, $\textsc{VarPost}$ may need to be modify to apply the corresponding transition repeatedly.
Additionally, note that we use the set $R$ to ensure that $\textsc{VarPost}$ (i.e. firing of a single transition) is executed only for colours for which we have not found a successor yet using some of the previously considered transitions. This is necessary to ensure that in each invocation of \textsc{NextStep}, each colour present in $\mathcal{F}$ is either advanced by one step (using exactly one transition), or is reported as converged within the returned set $R$.
Using this process, we can replace the $\textsc{Pre}/\textsc{Post}$ procedures in the main lock-step algorithm (lines 11 and 12 of Algorithm~\ref{algo:symbolic}). The $R$ sets computed here are then used to update $F_{lock}$ and $B_{lock}$ (lines 13 and 14), as they exactly represent the converged colours that do not need further computation. A similar modification is necessary for the second while loop (lines 25-29), but here the sets of remaining colours $R$ are not needed.
\begin{algorithm}
\SetKwProg{Fn}{Function}{}{}
\Fn{\textsc{NextStep}$(\mathfrak{G}, \mathcal{F})$}{
$R \gets \textsc{Colours}(\mathcal{F})$\;
\For{$\var{A} \in \ensuremath{\mathit{Var}}$}{
$\mathcal{S} \gets \textsc{VarPost}(\mathfrak{G}, \var{A}, (\mathcal{F} \cap V) \times R)$\;
$R \gets R \setminus \textsc{Colours}(\mathcal{S})$\;
$\mathcal{F} \gets \mathcal{F} \cup \mathcal{S}$\;
\If{$R = \emptyset$}{\textbf{break}\;}
}
\Return $\mathcal (\mathcal{F}, R)$\;
}
\caption{Main idea of the lock-step-saturation approach. The algorithm extends $\mathcal{F}$ with one additional reachability step, and returns a set of colours locked in this iteration ($R$).}\label{algo:saturation}
\end{algorithm}
\subsection{Trimming and Parallelism}
Most graphs typically contain a large number of trivial SCCs that introduce unnecessary overhead to the main algorithm. To avoid this overhead, we additionally perform a trimming step before each invocation of \textsc{Decomposition}. Trimming consists of repeatedly removing all vertices which have no outgoing or no incoming edges and is employed by most symbolic SCC algorithms on standard directed graphs as well.
The coloured analogue of trimming is straightforward, as it can be achieved using \textsc{Pre} and \textsc{Post} operations just as in the non-coloured case. For a coloured set of vertices $\mathcal{V}$, operation $\textsc{Post}(\ensuremath{\mathfrak G}, \textsc{Pre}(\ensuremath{\mathfrak G}, \mathcal{V}) \cap \mathcal{V}) \cap \mathcal{V}$ returns only the vertices which have at least one predecessor in $\mathcal{V}$. The successor variant simply exchanges the \textsc{Post} and \textsc{Pre} operations.
As such, applying this operation to each $\mathcal{V}$ until a fixed-point is reached before \textsc{Decomposition} is invoked eliminates the undesired trivial SCCs. Since the total number of steps performed collectively by all such fixed-point computations is bounded by $|C||V|$ (the total number of removable vertex-colour pairs), this does not impact the overall asymptotic complexity of the algorithm.
In some cases, we have observed that the symbolic representation is able to handle the SCC computation but explodes during trimming. The algorithm then times-out during trimming, even though useful information about SCCs could be obtained if the trimming was skipped or postponed. To avoid this issue, we enforce an extra condition that a trimming procedure is terminated prematurely if the computed BDDs are more than twice the size (in terms of BDD decision nodes) of the initial set.
Additionally, the lock-step algorithm can be rather trivially parallelised. The recursive \textsc{Decomposition} calls operate on independent coloured vertex sets and can be therefore deferred to separate threads. Since the body of the \textsc{Decomposition} method is rather complex, this can be done easily with a queue guarded by a mutex which is shared between all threads (i.e. the synchronisation overhead is negligible due to the long running time of \textsc{Decomposition}). Finally, a simple termination detection procedure is needed to ensure that idle threads do not terminate prematurely while decomposition is still running.
Note that most BDD packages are not internally thread-safe, as they share decision node memory across different BDD objects. In our experiments, this aspect is handled by cloning the set $\mathcal{V}$ corresponding to each recursive invocation, plus the symbolic representation of the BN necessary to compute $\textsc{Post}$ and $\textsc{Pre}$. As such, the memory used to represent BDDs manipulated by each thread is completely independent from other threads.
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\section{Experimental Evaluation}
To test the algorithm, we compiled a benchmark set of Boolean networks from the CellCollective~\cite{helikar2012cell} and GINsim~\cite{chaouiya2012} model databases. Since the models in these databases contain fully specified networks, uninterpreted functions were introduced into existing models by pseudo-randomly erasing parts of the existing update functions.
While this process is to some extent artificial, we believe it to be a good approximation of the model development process, where at some point, the structure of the network is already established, but its dynamics are still not fully determined. Using this process, we obtained a collection of networks ranging between $2^{20}$ and $2^{50}$ in the size of the coloured graph (i.e. $|V \times C|$). Note that for each graph, we consider only a subset of possible input valuations that is biologically relevant with respect to the established network structure. For example, the first model (i.e.~\cite{sanchez2017modeling}) admits $2^{48}$ input valuations, but only $2^{19}$ are biologically relevant due to constraints on function monotonicity.
A complete overview of the employed models is given in Table~\ref{tab:models}. For each model, we give the number of discovered non-trivial components as an interval, because each colour can correspond to a different number of components. We employ a 24h timeout for all experiments.
\begin{table}
\caption{The considered benchmark models. Here, $n$ is the number of BN variables, $m$ is the number of logical inputs (after expansion of uninterpreted functions), $|C|$ is the number of all biologically relevant colours (input valuations), and $|V \times C|$ is the size of the whole biologically relevant coloured state space. Finally, \#SCC gives the number of detected non-trivial SCCs. Note that this number varies depending on input valuation, and is thus given as a range.}
\label{tab:models}
\setlength\tabcolsep{8 pt}
\renewcommand{1.5}{1.5}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
\textbf{Model name} & $n$ & $m$ & $|C|$ & $|V \times C|$ & \#SCC \\\hline
{\small Asymmetric Cell Division~\cite{sanchez2017modeling}} & $5$ & $48$ & $\sim2^{19}$ & $\sim2^{24}$ & 1-13 \\ \hline
{\small Reduced TCR Signalisation~\cite{klamt2006methodology}} & $10$ & $46$ & $\sim2^{14}$ & $\sim2^{24}$ & 36-115 \\ \hline
{\small Budding Yeast (Orlando)~\cite{orlando2008global}} & $9$ & $54$ & $\sim2^{16}$ & $\sim2^{27}$ & 1-16 \\ \hline
{\small Budding Yeast (Irons)~\cite{irons2009logical}} & $18$ & $44$ & $\sim2^{17}$ & $\sim2^{35}$ & 2-5568 \\ \hline
{\small Tumor Cell Migration~\cite{cohen2015mathematical}} & $20$ & $44$ & $\sim2^{15}$ & $\sim2^{35}$ & 436-379308 \\ \hline
{\small T-cell Differentiation~\cite{mendoza2006method}} & $23$ & $40$ & $\sim2^{15}$ & $\sim2^{38}$ & 41728-43264 \\ \hline
{\small WG Signalling Pathway~\cite{mbodj2013logical}} & $26$ & $38$ & $\sim2^{22}$ & $\sim2^{48}$ & 0 \\ \hline
{\small Full TCR Signalisation~\cite{klamt2006methodology}} & $30$ & $48$ & $\sim2^{17}$ & $\sim2^{47}$ & 48-1087 \\ \hline
\end{tabular}
\end{table}
\begin{table}
\caption{Overview of runtime for different version of the SCC detection algorithm. The times (\texttt{hours:minutes:seconds}) refer to the total runtime of the SCC decomposition procedure for the basic lock-step, lock-step with saturation, and lock-step with saturation and parallelism, with \texttt{DNF} representing a time-out after 24-hours. }\label{tab:results}
\centering
\setlength\tabcolsep{8 pt}
\renewcommand{1.5}{1.5}
\begin{tabular} { | c | c | c | c | }
\hline
\textbf{Model Name} & \textbf{Parallel} & \textbf{Satur.} & \textbf{Lock-step} \\ \hline
{\small Asymmetric Cell Division~\cite{sanchez2017modeling}} & \texttt{00:05} & \texttt{00:10} & \texttt{00:15} \\ \hline
{\small Reduced TCR Signalisation~\cite{klamt2006methodology}} & \texttt{00:04} & \texttt{00:45} & \texttt{01:12} \\ \hline
{\small Budding Yeast (Orlando)~\cite{orlando2008global}} & \texttt{06:29} & \texttt{06:50} & \texttt{11:21} \\ \hline
{\small Budding Yeast (Irons)~\cite{irons2009logical}} & \texttt{15:14} & \texttt{2:53:16} & \texttt{3:28:44} \\ \hline
{\small Tumor Cell Migration~\cite{cohen2015mathematical}} & \texttt{40:10} & \texttt{18:34:16} & \texttt{DNF} \\ \hline
{\small T-cell Differentiation~\cite{mendoza2006method}} & \texttt{16:10:41} & \texttt{DNF} & \texttt{DNF} \\ \hline
{\small WG Signalling Pathway~\cite{mbodj2013logical}} & \texttt{1:18:38} & \texttt{1:23:37} & \texttt{1:42:12} \\ \hline
{\small Full TCR Signalisation~\cite{klamt2006methodology}} & \texttt{4:49:04} & \texttt{DNF} & \texttt{DNF} \\ \hline
\end{tabular}
\end{table}
The experiments were performed on a 32-core AMD Threadripper workstation with 64GB of RAM memory. All tested models are available in our source code repository.\footnotemark[3] \footnotetext[3]{\url{https://github.com/sybila/biodivine-lib-param-bn/tree/lmcs}} Note that the smaller models ($<2^{30}$) should be easy to process even on a less powerful machine; however, the larger models can require substantial amount of memory.
For each model, we have tested the lock-step algorithm as presented in the main part of this paper (\emph{Lock-step} in Table~\ref{tab:results}), an enhanced version with saturation enabled (\emph{Satur.} in Table~\ref{tab:results}), and a parallel implementation which also includes saturation (\emph{Parallel} in Table~\ref{tab:results}). In all algorithms, we employ the trimming optimisation.
From the results, we can see that parallelisation improves the performance of the algorithm significantly: in case of models with a large number of SCCs, we see an up-to 30x speed-up, comparing \emph{Parallel} and \emph{Satur.} in Table~\ref{tab:results}. On the other hand, when the number of SCCs is small (such as~\cite{orlando2008global}), the speed-up is understandably minimal, since the number of independent recursive calls is also small.
As expected, the total number of SCCs has a significant impact on the performance of the algorithm (e.g.~\cite{irons2009logical} and~\cite{cohen2015mathematical}) overall, since the number of calls to \textsc{Decomposition} increases. Furthermore, we see that our ``coloured saturation'' indeed provides a performance benefit. However, this improvement is mostly incremental.
After further analysis, we discovered that the whole algorithm is often limited by the performance of the trimming procedure, rather than reachability procedures though. In~particular, the use of saturation has significantly reduced the size of symbolic representation during computation of reachability, however the symbolic representation still performs rather poorly (at least for Boolean networks) during trimming. This limits the performance of the whole method, since all the considered graphs contain a large portion of trivial SCCs. Furthermore, in many cases the number of iterations needed to completely trim a set of states is substantial. This leads us to believe there is still space for improvement in terms of SCC detection in large Boolean networks, even without parameters.
Finally, we examined the benefit of processing all colours simultaneously versus a naive parameter scan approach, where each monochromatic case is handled separately. To do so, we considered various pseudo-random monochromatisations of the studied models and processed these using our algorithm. Here, we observe that for the four models with at least $20$ variables, no computation for any of the monochromatic models finished in under one second (with T-cell differentiation typically requiring more than one minute due to the relatively large number of components).
Consequently, we can extrapolate that computing the full coloured SCC decomposition using such naive parameter scan would require more than 10 ours for each model (and $10+$ days in the case of T-cell differentiation). This approach could be to some extent beneficial in a massively parallel environment (hundreds or thousands of CPUs), but the coloured approach clearly scales better in setups where resources are more limited.
\section{Conclusions}
This paper presents a fully symbolic algorithm for detecting all monochromatic strongly connected components in edge-coloured graphs. The work has been motivated by systems sciences, namely systems biology, where the need for efficient automated analysis of components in large graphs with a large sets of coloured edges is emerging. The algorithm combines several ideas inspired by existing state-of-the-art algorithms for SCC decomposition in a~non-trivial way. We believe this is the first fully symbolic algorithm aiming to solve the problem efficiently.
The experimental evaluation has shown that the algorithm can handle large, real-world systems that would be otherwise too large to fit into the memory of a conventional workstation ($>2^{32}$), and that the performance of the algorithm can be further improved using saturation and parallelisation. Finally, the algorithm has a strong potential to be significantly faster
than iterating a standard algorithm for SCC decomposition executed on all monochromatic sub-graphs one-by-one.
\end{document}
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\begin{document}
\begin{abstract}
The \emph{Wiener index} of a finite graph $G$ is the sum over all pairs $(p,q)$ of vertices of $G$ of the distance between $p$ and $q$. When $P$ is a finite poset, we define its \emph{Wiener index} as the Wiener index of the graph of its Hasse diagram. In this paper, we find exact expressions for the Wiener indices of the distributive lattices of order ideals in minuscule posets.
For infinite families of such posets, we also provide results on the asymptotic distribution of the distance between two random order ideals.
\end{abstract}
\maketitle
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\section{Introduction}
\subsection{Background: the Wiener index of the noncrossing partition lattice}
Let $\mathrm{NC}(n)$ be the lattice of noncrossing partitions
of $n$.
In the paper~\cite{goulden2020asymptotics},
motivated by problems about meanders and meandric systems, Goulden, Nica, and Puder raised the following question: what is the average distance between two (uniform)
random partitions in $\mathrm{NC}(n)$?
The question was answered for large $n$ by Th\'evenin
and the second author in~\cite{feray2022components},
where it was proved that this average distance behaves as $\kappa\, n$ for some constant $\kappa$.
It is natural to ask similar questions for other families of posets, looking either for an exact nice formula or for an asymptotic answer.
When the number of elements is known (which is the case for $\mathrm{NC}(n)$),
one can equivalently ask for the sum of distances between all pairs of elements.
In general, let $G=(V,E)$ be a finite connected graph, and for $p,q\in V$, write $d(p,q)$ for the distance in $G$ from $p$ to $q$. The defn{Wiener index} of $G$ is defined to be
\begin{equation}\label{eq:dist}
d(G):=\sum_{(p,q) \in V \times V} d(p,q).
\end{equation}
This definition has its origin as the \emph{Wiener index} predicting the boiling point of certain organic compounds~\cite{wiener1947structural,wikiwiener,rouvray2002rich}, and it has also been called the \emph{distance} of the graph $G$~\cite{entringer1976distance}. When $P$ is a poset, we define $d(P):=d(G(P))$ for the Wiener index of the Hasse diagram $G(P)$ of $P$ (that is, the vertices of $G(P)$ are the elements of $P$, and there is an edge in $G(P)$ between $p$ and $q$ when there is a cover relation in $P$ between $p$ and $q$).
In the case of the noncrossing partition lattice, the results in~\cite{feray2022components} imply that
\[d(\mathrm{NC}(n)) \sim |\mathrm{NC}(n)|^2 \kappa \, n \sim \frac{\kappa\, 8^n}{\pi n^2},\]
but no exact enumeration appears possible.
(There does not even seem
to be a simple formula for $\kappa$;
see the discussion in~\cite{feray2022components}
and the related open problem in~\cite{ober2022}.)
\subsection{Wiener indices of other lattices}
Computer experiments suggest that there are few nontrivial families of graphs $\{G_n\}_{n\geq 1}$ of combinatorial objects for which it is possible to find \emph{exact} formulas for the Wiener index.
First, there are families with elementary exact solutions, for which $|G_n|$ is a relatively small polynomial in $n$ (for example: path graphs, grid graphs, etc.),
or in which each graph $G_n$ has a transitive underlying symmetry group (for example: the weak order on a finite Coxeter group, a boolean lattice or hypercube, etc.). A short list of examples is given in~\cite{weissteinwiener}.
There is, however, one class of posets in algebraic combinatorics that demonstrates consistently exceptional enumerative behavior: the minuscule lattices~\cite{proctor1984bruhat}. For example, both the number of elements and the number of maximal chains in a minuscule lattice have simple (uniformly stated and proven) product formulas, and the minuscule lattices are well understood from the perspective of dynamical algebraic combinatorics (promotion, rowmotion, etc.)~\cite{striker2012promotion,hopkins2020order}.
It turns out that the Wiener indices of minuscule lattices also admit simple exact formulas,
and one of the goal of the present paper is to establish such formulas.
For completeness, we first recall the definition of minuscule lattices
(note that we will only use here their classification, and not the algebraic definition). Let $\mathfrak{g}$ be a complex simple Lie group with Weyl group $W$. Fix a set $Phi^+$ of positive roots of $\mathfrak g$, and let $\Lambda^+$ be the set of dominant weights. The finite-dimensional irreducible complex representations $V_\lambda$ of $\mathfrak{g}$ are indexed by dominant weights $\lambda \in \Lambda^+$; $\lambda$ is called defn{minuscule} if the $W$-orbit of $\lambda$ is the set of \emph{all} weights in $V_\lambda$. The minuscule weights are exactly those fundamental weights whose corresponding simple roots appear exactly once in the simple root expansion of the highest root. For more information, we refer the reader to~\cite{stanley1980weyl,proctor1984bruhat}.
For $\lambda$ minuscule, define a poset on the weights in $V_\lambda$ by introducing a cover relation $\mu \lessdot \nu$ whenever $\mu + \alpha = \nu$ for some simple root $\alpha \in Phi^+$. This poset on $V_\lambda$ is a distributive lattice, which we call a defn{minuscule lattice}~\cite{proctor1984bruhat}.
There are three infinite families of minuscule lattices---the order ideals in:
\begin{itemize}
\item a rectangle (type $A$; in type $B$, minuscule lattices are chains and thus particular cases of rectangles),
\item a shifted staircase (types $C$ and $D$), and
\item a ``double tailed diamond'' (type $D$)
\end{itemize}
---as well as two exceptional minuscule lattices (of types $E_6$ and $E_7$).
In this paper, we show that the Wiener indices of the infinite families of minuscule lattices admit simple product formulas, although we regrettably have been unable to find a unifying expression for these formulas. We also provide information about the asymptotic distribution
of the distance between random elements in these posets (in the rectangle and shifted staircase cases),
and we give a method for computing the higher moments exactly.
\subsection{Rectangles}
Write $J(P)$ for the lattice of order ideals in a finite poset $P$, ordered by inclusion. By Birhoff's representation theorem, any distributive lattice is of this form. In~\Cref{sec:rect_bijections,sec:proof_rect_thm,sec:proof_rect_cors}, we consider the Wiener index of $P_{m,k}=J([m]\times[k])$, the lattice of order ideals in an $m \times k$ rectangle.
In this case (and, more generally, for
any distributive lattice), it is easy to see that $d(p,q)=|p \vartriangle q|$, where $p \vartriangle q = (p \backslash q) \cup (q \backslash p)$ is the symmetric difference of the order ideals $p$ and $q$.
It will be convenient to draw the elements of $P_{m,k}$ as lattice paths from $(0,0)$ to $(m+k,m-k)$ using steps of the form $U=(1,1)$ and $D=(1,-1)$.
Writing $p_i,q_i$ for the heights of $p$ and $q$ after the ends of their $i$th steps, the number of squares in column $i$ between the lattice paths $p$ and $q$ is given as $\left|\frac{q_i-p_i}{2}\right|$, so that
\begin{equation}\label{eq:distance}
d(p,q) = \left|\frac{q_1-p_1}{2}\right|+\left|\frac{q_2-p_2}{2}\right|+\cdots+\left|\frac{q_{m+k}-p_{m+k}}{2}\right|.
\end{equation}
\begin{example}
The graph $G(P_{2,2})$ is drawn in \Cref{fig:2x2}. Its Wiener index is
\begin{align*}
56 = \frac{4}{18}\binom{10}{5}=&(0{+}1{+}2{+}2{+}3{+}4){+}(1{+}0{+}1{+}1{+}2{+}3){+}(2{+}1{+}0{+}2{+}1{+}2){+}\\&{+}(2{+}1{+}2{+}0{+}1{+}2){+}(3{+}2{+}1{+}1{+}0{+}1){+}(4{+}3{+}2{+}2{+}1{+}0).
\end{align*}
\end{example}
\begin{figure}
\caption{The Hasse diagram $G(P_{2,2}
\label{fig:2x2}
\end{figure}
We have three results that completely describe the Wiener index of the lattice of order ideals of a rectangle.
\begin{theorem}\label{thm:gf}
The generating function for the Wiener index of all posets $P_{m,k}=J([m]\times[k])$ is given by
\begin{equation}\label{eq:total_geo}
\sum_{m=0}^\infty \sum_{k=0}^\infty d(P_{m,k})x^m y^k = \frac{2xy}{(x^2 - 2xy + y^2 - 2x - 2y + 1)^2}.
\end{equation}
\end{theorem}
This theorem is obtained via classical first return decomposition for lattice paths.
The fact that this generating series is rational comes as a surprise,
since several intermediate computation steps involve algebraic non-rational functions.
Extracting the coefficient of $x^m y^k$ from~\Cref{eq:total_geo}, we obtain a formula for $d(P_{m,k})$.
\begin{corollary}\label{cor:coeff}
The Wiener index of $P_{m,k}$ is
\[d(P_{m,k})=\frac{mk}{4m+4k+2}\binom{2m+2k+2}{2k+1}.\]
\end{corollary}
For fixed $\alpha$, we obtain the asymptotic expected value of $d(p,q)$ in a $(\alpha n) \times n$ rectangle as $n \to \infty$.
To keep notation simple, we assume throughout the paper that $\alpha n$ is an integer
(otherwise, it suffices to replace $\alpha n$
by its integer value).
\begin{corollary}\label{cor:asymptotic}
We have \[\frac{d(P_{\alpha n,n})}{|P_{\alpha n,n}|^2} \sim \frac{\sqrt{\pi \alpha (1+\alpha)}}{4} n^{3/2} \text{ as } n \to \infty.\]
\end{corollary}
In~\Cref{sec:asymptotics}, we also describe in this regime the asymptotic distribution of the distance $D_{\alpha,n}$ between
two independent uniform random elements of $P_{\alpha n,n}$.
\begin{proposition}
\label{prop:cv_law_distance}
The random variable $n^{-3/2} D_{\alpha, n}$ converges in distribution
and in moments to $\sqrt{2 \alpha (1+\alpha)} \cdot \int_0^1 |B_0(t)| dt$,
where $B_0(t)$ is a Brownian bridge on $[0,1]$.
\end{proposition}
Informally, a Brownian bridge on $[0,1]$ is a Brownian motion conditioned to have value $0$ at time $1$.
Alternatively,
if $B$ is a Brownian motion, then $B_0(t):=B(t)-tB(1)$ is a Brownian bridge.
Brownian bridges have been extensively studied in the probabilistic litterature.
In particular, much is known on the random variable $\int_0^1 |B_0(t)| dt$;
see \cite[Section 20]{janson2007area} for a survey of results including numerous references.
In particular, a table of the first few moments can be found in \cite[Table 2]{janson2007area}.
We copy here the first three:
\[ \mathbb E\left[\int_0^1 |B_0(t)| dt\right] = \frac14 \sqrt{\frac{\pi}2},
\quad \mathbb E\left[\left(\int_0^1 |B_0(t)| dt\right)^2\right] = \frac7{60},
\quad \mathbb E\left[\left(\int_0^1 |B_0(t)| dt\right)^3\right] = \frac{21}{512} \sqrt{\frac{\pi}2}.\]
Together with \cref{prop:cv_law_distance}, this implies
\begin{align*}
\frac1{|P_{\alpha n,n}|^2} \sum_{p,q \in P_{\alpha n,n}} d(p,q) =\mathbb E[D_{\alpha,n}] &\sim \frac{\sqrt{\pi \alpha (1+\alpha)}}{4} n^{3/2},\\
\frac1{|P_{\alpha n,n}|^2} \sum_{p,q \in P_{\alpha n,n}} d(p,q)^2 =\mathbb E[D_{\alpha,n}^2]&\sim \frac{7}{30} \alpha (1+\alpha) n^{3},\\
\frac1{|P_{\alpha n,n}|^2} \sum_{p,q \in P_{\alpha n,n}} d(p,q)^3 =\mathbb E[D_{\alpha,n}^3]&\sim \frac{21}{256} \sqrt{\pi} \alpha^{3/2} (1+\alpha)^{3/2} n^{9/2}.
\end{align*}
Note that the first estimate is nothing but \cref{cor:asymptotic}. This gives a second derivation
of this asymptotic result, which does not go through the exact expression.
Exact expressions for such higher moments can also be obtained through combinatorial means,
see \cref{sec:higher_moments} for a derivation of the second moment.
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\subsection{Shifted staircases}
In~\Cref{sec:bij_stair,sec:proof_stair}, we consider the Wiener index of $Q_n$, the distributive lattice of order ideals in the $n$th defn{shifted staircase} poset. Explicitly, $Q_n$ is the set of order ideals in the poset $\{(i,j):1\leq i\leq j\leq n\}$ under componentwise ordering. The following results determine $d(Q_n)$ exactly.
The elements of $Q_n$ can be represented
as lattice paths starting at $(0,0)$,
ending somewhere on the line $x=n$, and using steps
of the form $U=(1,1)$ and $D=(1,-1)$.
In particular, $|Q_n|=2^n$.
Similarly as for rectangles,
if $p$ and $q$ are elements in $Q_n$,
writing $p_i,q_i$ for the heights of $p$ and $q$ after the ends of their $i$th steps, we have
\begin{equation}
\label{eq:distance_shifted}
d(p,q) = \frac12 \sum_{i=1}^n \left|\frac{q_i-p_i}{2}\right|.
\end{equation}
\begin{example}
The graph $G(Q_{3})$ is plotted in \Cref{fig:ss3}. Its Wiener index is
\begin{align*}
140 = \frac{6\cdot 7}{3}\binom{5}{3}=&24+18+14+14+14+14+18+24.
\end{align*}
\end{example}
\begin{figure}
\caption{The Hasse diagram $G(Q_{3}
\label{fig:ss3}
\end{figure}
\begin{theorem}\label{thm:gf_SS}
The generating function for the Wiener index of all lattices $Q_n$ is given by
\begin{equation}\label{eq:total_geo_SS}
\sum_{n=0}^\infty d(Q_n)x^n = \frac{8x\left(1+\sqrt{1-4x}-x(3+\sqrt{1-4x})\right)}{(1-4x)(1-4x+\sqrt{1-4x})^3}.
\end{equation}
\end{theorem}
\begin{corollary}
\label{cor:Wiener_JSn}
The Wiener index of $Q_n$ is \[d(Q_n)=\frac{2n(2n+1)}{3}\binom{2n-1}{n}.\]
Consequently, as $n$ tends to $+\infty$, we have
$d(Q_n) \sim \frac{2}{3\sqrt \pi} 4^n n^{3/2}$.
\end{corollary}
In~\Cref{sec:asymptotics}, we turn to the asymptotic distribution of the distance $E_n$ between two random order ideals of
$Q_n$.
\begin{proposition}
\label{prop:cv_law_distance_shifted}
The random variable $n^{-3/2} E_n$ converges in distribution
and in moments to $\frac{1}{\sqrt{2}} \cdot \int_0^1 |B(t)| dt$,
where $B(t)$ is a Brownian motion on $[0,1]$.
\end{proposition}
Again, much is known about the random variable
$\int_0^1 |B(t)| dt$, and a comprehensive literature review appears in~\cite[Section 21]{janson2007area}. In particular, the first few moments are given in~\cite[Table 3]{janson2007area}:
\[ \mathbb E\left[\int_0^1 |B(t)| dt\right] = \frac23 \sqrt{\frac2{\pi}},
\quad \mathbb E\left[\left(\int_0^1 |B(t)| dt\right)^2\right] = \frac3{8},
\quad \mathbb E\left[\left(\int_0^1 |B(t)| dt\right)^3\right] = \frac{263}{630} \sqrt{\frac2{\pi}}.\]
Together with \cref{prop:cv_law_distance_shifted}, this implies
\begin{align*}
\frac1{|Q_n|^2} \sum_{p,q \in Q_n} d(p,q) =\mathbb E[E_n]&\sim \frac{2}{3 \sqrt \pi} n^{3/2},\\
\frac1{|Q_n|^2} \sum_{p,q \in Q_n} d(p,q)^2 =\mathbb E[E_n^2] &\sim \frac{3}{16} n^{3},\\
\frac1{|Q_n|^2} \sum_{p,q \in Q_n} d(p,q)^3 =\mathbb E[E_n^3]&\sim \frac{263}{1260 \sqrt{\pi}}
n^{9/2}.
\end{align*}
Again, recalling that $|Q_n|=2^n$,
this allows us
to recover the asymptotic
behaviour of $d(Q_n)$ given in \cref{cor:Wiener_JSn}
without going through its exact expression.
\subsection{The remaining minuscule lattices}
The Wiener indices of the remaining minuscule lattices are simple calculations.
Let $R_n$ be the $n$th ``double tailed diamond''---that is, the distributive lattice of order ideals in the minuscule poset of type $D_n$ corresponding to the first fundamental weight.
\begin{theorem}\label{thm:other_mins}
We have
\[d(R_n)=\frac{2}{3} (n+3) \left(4 n^2+9 n+8\right).\] The minuscule lattices of types $E_6$ and $E_7$ have Wiener indices $3584$ and $24048$, respectively.
\end{theorem}
The proof in the case of $R_n$ is elementary and left to the reader.
The cases of $E_6$ and $E_7$ are treated by computer.
The expressions for Wiener indices given in \Cref{cor:coeff}, \Cref{cor:Wiener_JSn}, and \Cref{thm:other_mins} suggest that there may be a uniform formula for $d(P)$ for $P$ a minuscule lattice---but we regrettably have been unable to find such an expression.
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\section{Lattice path bijections}\label{sec:rect_bijections}
We continue to use the steps $U=(1,1)$ and $D=(1,-1)$, but we will also make use of two versions (or colors) of the step $(1,0)$, denoted $O_1$ and $O_2$. For $(p,q) \in P_{k,n-k} \times P_{k,n-k}$, define a lattice path $A(p,q)$ using the following dictionary between the $i$th pair of steps in $(p, q)$ and the $i$th step in $A(p,q)$:
\begin{equation}\label{eq:bija}
\begin{array}{|c|c|c|c|} \hline
(p,q) & A(p,q) & \left|\frac{q_{i+1}-p_{i+1}}{2}\right|-\left|\frac{q_i-p_i}{2}\right| & r_{i+1}-r_i\\ \hline
(D,U) & U & +1 & +1\\
(U,D) & D & -1 & -1\\
(U,U) & O_1 & 0 & 0\\
(D,D) & O_2 & 0 & 0 \\\hline
\end{array}.
\end{equation}
Two examples of this bijection are illustrated in~\Cref{fig:bija}. Given a lattice path $r$ of length $n$ with steps from the set $\{U,D,O_1,O_2\}$, write $r_i$ for the height (i.e.\, the $y$-coordinate) of $r$ at the end of its $i$th step. The unsigned area between $r$ and the $x$-axis is $d(r)=|r_0|+|r_1|+|r_2|+\cdots+|r_{n-1}|+\frac{1}{2}|r_n|$. Let us also write $\overlined(r)=|r_0|+|r_1|+|r_2|+\cdots+|r_{n}|=d(r)+\frac{1}{2}|r_n|$. Then, comparing with~\Cref{eq:distance}, it is clear that $d(p,q) = d(A(p,q))$: certainly $\frac{q_0-p_0}{2}=0=r_0$, so suppose that $\frac{q_i-p_i}{2} = r_i$; then the difference in height at the $(i+1)$st step in $A(p,q)$ matches the difference in height at the $(i+1)$st steps of $p$ and $q$, as shown in the rightmost two columns of~\eqref{eq:bija}.
\begin{figure}
\caption{Illustration of the bijection $A$ from the table in~\eqref{eq:bija}
\label{fig:bija}
\end{figure}
Write the set of all ordered pairs of paths in $P_{k.n-k}$ as \[P^\times_{k,n-k}:=P_{k,n-k} \times P_{k,n-k},\] and denote the restriction of $P^\times_{k,n-k}$ to those pairs $(p,q)$ with $p\leq q$ as \[P^\leq_{k,n-k} := \{(p,q) \in P^\times_{k,n-k} : p \leq q\}.\] We begin by converting the total area between pairs of paths in $P^\times_{k,n-k}$ and $P^\leq_{k,n-k}$ into the (unsigned) area under a single Motzkin path.
\begin{definition}
Write $\mathcal{M}b$ for the set of defn{bilateral Motzkin paths}---that is, lattice paths from $(0,0)$ to $(n,0)$ for some $n \in \mathbb{Z}_{\geq 0}$ that use step set $\{U,D,O_1,O_2\}$. We write $\mathcal{M}b_{n,k}$ for the set of bilateral Motzkin paths that end at $(n,0)$ and use exactly $k$ steps of the form $U$ or $O_1$. A defn{bicolored Motzkin path} is a bilateral Motzkin path that stays weakly above the $x$-axis. Write $\mathcal{M}$ (resp. $\mathcal{M}_{n,k}$) for the set of bicolored Motzkin paths in $\mathcal{M}b$ (resp. $\mathcal{M}b_{n,k}$).
\end{definition}
\begin{proposition}\label{prop:bija}
The map $A\colonP^\times_{n,n-k}\to\mathcal{M}b_{n,k}$ is a bijection satisfying $d(p,q) = d(A(p,q))$, and it restricts to a bijection from $P^\leq_{n,n-k}$ to $\mathcal{M}_{n,k}$.
\end{proposition}
\begin{proof}
The dictionary in~\eqref{eq:bija} shows that $A$ is a bijection. When $p \leq q$, $A(p,q)$ never goes below the $x$-axis, so $A$ maps $P^\leq_{n,n-k}$ onto $\mathcal{M}_{n,k}$. The claim about $d$ was proven immediately after~\eqref{eq:bija}.
\end{proof}
\section{Proof of~\Cref{thm:gf}}\label{sec:proof_rect_thm}
In this section, we use recurrences on Motzkin paths and their associated generating functions to deduce~\Cref{thm:gf}.
\subsection{Bicolored Motzkin paths}
Define the generating function for bicolored Motzkin paths to be
\[\mathcal{M}(x,u) = \sum_{n \geq 0 } \sum_{k \geq 0} |\mathcal{M}_{n,k}| x^{n} u^k,\]
so that the coefficient of $x^n u^k$ counts bicolored Motzkin paths of total length $n$ with exactly $k$ steps of the form $U$ and $O_1$.
\begin{proposition} \label{prop:m_decomp}
The generating function $\mathcal{M}(x,u)$ satisfies the functional equation
\[\mathcal{M}(x,u)=1+x(1+u)\mathcal{M}(x,u)+ux^2 \mathcal{M}(x,u)^2,\]
with the explicit solution
\[\mathcal{M}(x,u) = \frac{1-(u + 1)x - \sqrt{(u^2 - 2u + 1)x^2 - 2(u + 1)x + 1}}{2ux^2}.\]
\end{proposition}
\begin{proof}
The functional equation comes from decomposing a lattice path $r\in \mathcal{M}$ by first return to the $x$-axis: $r$ is empty; or $r$ starts with an $O_1$ step; or $r$ starts with an $O_2$ step; or $r$ starts with a $U$ step. This is illustrated in~\Cref{fig:m_decomp}. The explicit solution is easily obtained by solving this quadratic equation in $\mathcal{M}(x,u)$.
\end{proof}
\begin{figure}
\caption{The decompositions of lattice paths in $\mathcal{M}
\label{fig:m_decomp}
\end{figure}
Define the generating function for the total area of paths in $\mathcal{M}$ by \[\mathcal{M}m(x,u) := \sum_{n \geq 0 } \sum_{k \geq 0} x^n u^k \sum_{p \in \mathcal{M}_{n,k}} d(p).\]
\begin{proposition} \label{prop:mm_decomp}
The generating function $\mathcal{M}m(x,u)$ satisfies the functional equation
\[\mathcal{M}m(x,u)=x(1+u)\mathcal{M}m(x,u)+ux^2(2\mathcal{M}(x,u)\mathcal{M}m(x,u)+\mathcal{M}(x,u) \frac{d}{dx}(x\mathcal{M}(x,u)),\]
with the explicit solution
\[\mathcal{M}m(x,u) = \frac{\left(u^2+1\right) x^2-(u+1) x+((u +1)x-1) \left(\sqrt{(u-1)^2 x^2-2 (u+1) x+1}-1\right)}{2 u x^2 \left((u-1)^2 x^2-2 (u+1) x+1\right)}.\]
\end{proposition}
\begin{proof}
As in~\Cref{prop:m_decomp}, the functional equation comes from decomposing a lattice path $r$ in $\mathcal{M}m$ by first return to the $x$-axis: either $r$ is empty (in which case it contributes zero area); or $r$ starts with an $O_1$ step; or $r$ starts with an $O_2$ step; or $r$ starts with a $U$ step. This is illustrated in~\Cref{fig:m_decomp}. The explicit solution is easily obtained by using the explicit form of $\mathcal{M}(x,u)$ from~\Cref{prop:m_decomp} and solving the linear equation in $\mathcal{M}m(x,u)$.
\end{proof}
\subsection{Bilateral Motzkin paths}
Define the generating function for bilateral Motzkin paths to be
\[\mathcal{M}b(x,u) = \sum_{n \geq 0 } \sum_{k \geq 0} |\mathcal{M}b_{n,k}| x^{n} u^k,\]
so that the coefficient of $x^n u^k$ counts bilateral Motzkin paths of total length $n$ with exactly $k$ steps of the form $U$ or $O_1$.
\begin{proposition}\label{prop:mb_decomp}
The generating function $\mathcal{M}b(x,u)$ satisfies the functional equation
\[\mathcal{M}b(x,u)=1+x(1+u)\mathcal{M}b(x,u)+2u x^2 \mathcal{M}(x,u) \mathcal{M}b(x,u),\]
with the explicit solution
\[\mathcal{M}b(x,u) = (u^2x^2 - 2ux^2 - 2ux + x^2 - 2x + 1)^{-1/2}.\]
\end{proposition}
\begin{proof}
As in~\Cref{prop:m_decomp}, the functional equation comes from decomposing a lattice path $r$ in $\mathcal{M}b$ by first return to the $x$-axis: $r$ is empty; or $r$ starts with an $O_1$ step; or $r$ starts with an $O_2$ step; or $r$ starts with a $U$ or $D$ step. This is illustrated in~\Cref{fig:m_decomp}. The explicit solution is easily obtained by using the explicit form of $\mathcal{M}(x,u)$ from~\Cref{prop:m_decomp} and solving the linear equation in $\mathcal{M}b(x,u)$.
\end{proof}
\begin{remark}
Because $\mathcal{M}b_{n,k}$ encodes $P^\times_{k,n-k}$ by Proposition~\ref{prop:bija}, we have
\begin{equation}
\label{eq:expansion_bilateral}
\mathcal{M}b(x,u)=\sum_{n,k\geq 0}\binom{n}{k}^2x^nu^k.
\end{equation}
\end{remark}
Define the generating function for the total area of paths in $\mathcal{M}b$ to be
\[\mathcal{M}bm(x,u) := \sum_{n \geq 0 } \sum_{k \geq 0} x^n u^k \sum_{p \in \mathcal{M}b_{n,k}} d(p).\]
\begin{proposition}\label{prop:mbm_decomp}
The generating function $\mathcal{M}bm(x,u)$ satisfies the functional equation
\[\mathcal{M}bm(x,u)=x(1+u)\mathcal{M}bm(x,u)+2ux^2(\mathcal{M}(x,u)\mathcal{M}bm(x,u)+\mathcal{M}m(x,u)\mathcal{M}b(x,u)+\mathcal{M}b(x,u)\frac{d}{dx}(x\mathcal{M}(x,u)),\]
with the explicit solution
\begin{equation}\label{eq:total_geo_sub}
\mathcal{M}bm(x,u) = \frac{2 u x^2}{\left((u-1)^2 x^2-2 (u+1) x+1\right)^2}.
\end{equation}
\end{proposition}
\begin{proof}
As in~\Cref{prop:mm_decomp}, the functional equation comes from decomposing a lattice path $r$ in $\mathcal{M}bm$ by first return to the $x$-axis: if $r$ is empty, then it counts for zero area; otherwise, $r$ starts with an $O_1$ step; or $r$ starts with an $O_2$ step; or $r$ starts with a $U$ or $D$ step. This is illustrated in~\Cref{fig:m_decomp}. The explicit solution is easily obtained by using the explicit forms of $\mathcal{M}(x,u)$, $\mathcal{M}b(x,u)$, and $\mathcal{M}m(x,u)$ from~\Cref{prop:m_decomp,prop:mb_decomp,prop:mm_decomp} and solving the linear equation in $\mathcal{M}bm(x,u)$.
\end{proof}
Substituting $u=y/x$ into~\Cref{eq:total_geo_sub}, we obtain~\Cref{eq:total_geo} and thus complete the proof of~\Cref{thm:gf} for the generating function for the Wiener indices of the lattices $P_{m,k}=J([m]\times[k])$.
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\section{Proofs of~\Cref{cor:coeff,cor:asymptotic}}\label{sec:proof_rect_cors}
We note that
\[x^2 - 2xy + y^2 - 2x - 2y + 1 = (q - t - 1) (q - t + 1) (q + t - 1) (q + t + 1),\]
where $q^2=x$ and $t^2=y$. Then, by performing a partial fraction decomposition with a computer algebra system, we get
\begin{align} \label{eq:qt}
\nonumber \frac{2xy}{(x^2 - 2xy + y^2 - 2x - 2y + 1)^2} &=
\frac{1}{32} \left(
\frac{1}{(-1-t+q)^2}
+\frac{1}{(1-t+q)^2}
+\frac{1}{(-1+t+q)^2}
+\frac{1}{(1+t+q)^2}\right) \\
\nonumber &+\frac{1}{32t(t+1)}\left(
\frac{1+t+t^2}{(1+t-q)}
+\frac{1+t+t^2}{(1+t+q)}
\right)\\
&+\frac{1}{32t(t-1)}\left(
\frac{1-t+t^2}{(1-t+q)}
+\frac{-1+t-t^2}{(-1+t+q)}
\right).
\end{align}
Taking the $n$th coefficient in $q$ from the right-hand side of~\Cref{eq:qt} gives
\begin{align}\label{eq:qt2}
\nonumber \frac{n+1}{32}&\Big((1+t)^{-2-n}+(-1+t)^{-2-n}+(1-t)^{-2-n}+(-1-t)^{-2-n}\Big)\\
\nonumber +
\frac{1}{32t}& \Big(\left(t^2+t+1\right) (t+1)^{-n-2}+\left(t^2+t+1\right) (-t-1)^{-n-2}\Big)\\
+
\frac{1}{32t}& \Big(\left(-t^2+t-1\right) (1-t)^{-n-2}+\left(-t^2+t-1\right) (t-1)^{-n-2}\Big).
\end{align}
Taking the $j$th coefficient in $t$ from~\eqref{eq:qt2} and simplifying gives
\begin{align}\label{eq:qt3}
\frac{\left((-1)^n+1\right) \left((-1)^j+1\right)}{32} \left(
\frac{(n+j+1)!}{n!j!}-
\frac{(n+j)! ((n+1)^2+j^2+j (n+2))}{(n+1)! (j+1)!}\right).
\end{align}
Restricting~\eqref{eq:qt3} to $n=2m$ and $j=2k$ even, we obtain the desired expression in~\Cref{cor:coeff} for the coefficient of $q^{n} t^{m}$, giving the coefficient for $x^{m} y^{k}$:
\begin{align*}
&\frac{1}{8} \left(\frac{(n+j+1)!}{n!j!}-
\frac{ (n+j)! ((n+1)^2+j^2+j(n+2))}{(n+1)! (j+1)!}\right)\\
&=\frac{(n+j)!}{8 n!j!}\left(n+j+1-\frac{j^2+(n+1)^2+j (n+2)}{(n+1)(j+1)}\right)\\
&=\frac{nj}{8(n+j+1)}\binom{n+j+2}{j+1}\\
&=\frac{mk}{4m+4k+2}\binom{2m+2k+2}{2k+1}.
\end{align*}
Given the exact expression for $d(P_{m,k})$,~\Cref{cor:asymptotic} is routine using Stirling's asymptotic equivalent for factorials.
\section{Shifted staircases}\label{sec:bij_stair}
As for rectangles, we can view elements of $Q_n$ as lattice paths of length $n$ that start at $(0,0)$ and use steps of the form $U=(1,1)$ and $D=(1,-1)$. The main difference is that paths representing different order ideals can have different endpoints. Let \[Q_n^\times:=Q_n \times Q_n \quad\text{and}\quad Q_n^{\leq}:=\{(p,q)\in Q_n^\times:p\leq q\}.\]
\begin{definition}
Define a defn{bilateral Motzkin prefix} to be a lattice path that starts at $(0,0)$ and uses the steps of the form $U,D,O_1,O_2$. Let $\mathcal V$ denote the set of bilateral Motzkin prefixes, and let $\mathcal V_n$ be the set of bilateral Motzkin prefixes that use exactly $n$ steps. A defn{bicolored Motzkin prefix} is a bilateral Motzkin prefix that stays weakly above the $x$-axis. Write $\mathcal N$ for the set of bicolored Motzkin prefixes, and let $\mathcal N_n=\mathcal N\cap\mathcal V_n$.
\end{definition}
Throughout this section, we write $d(p,q)$ for the length of a geodesic between bilateral Motzkin prefixes $p$ and $q$ in the Hasse diagram of $Q_n$.
By applying the same rules as in the table in \eqref{eq:bija}, we can transform a pair $(p,q)\in Q_n\times Q_n$ into a path $A(p,q)$ that uses steps $U,D,O_1,O_2$. The path $A(p,q)$ is similar to a bilateral Motzkin path, except it does not necessarily end on the $x$-axis.
\begin{proposition}\label{prop:bija_SS}
The map $A\colonQ_n^\times\to\mathcal V_n$ is a bijection satisfying $d(p,q) = \overlined(A(p,q))$, and it restricts to a bijection from $Q_n^\leq$ to $\mathcal N_n$.
\end{proposition}
\begin{proof}
The proof is essentially the same as that of \Cref{prop:bija}.
\end{proof}
\section{Proof of \Cref{thm:gf_SS} and \Cref{cor:Wiener_JSn}}\label{sec:proof_stair}
\subsection{Bicolored Motzkin prefixes}
Let \[\mathcal N(x):=\sum_{n\geq 0}|\mathcal N_n|x^n\] be the generating function for bicolored Motzkin prefixes.
\begin{proposition}\label{prop:N}
The generating function $\mathcal N(x)$ satisfies the functional equation \[\mathcal N(x)=1+3x\mathcal N(x)+x^2\mathcal M(x,1)\mathcal N(x).\] Thus, \[\mathcal N(x)=\frac{2}{1-4x+\sqrt{1-4x}}.\]
\end{proposition}
\begin{proof}
The expression $1+x\mathcal N(x)$ counts (possibly empty) bicolored Motzkin prefixes that only touch the $x$-axis at $(0,0)$, while the expression $2x\mathcal N(x)+x^2\mathcal M(x,1)\mathcal N(x)$ counts bicolored Motzkin prefixes that touch the $x$-axis at some point other than $(0,0)$. This is illustrated on the first line of~\Cref{fig:n_decomp}. It is routine to derive the explicit solution from the functional equation and \Cref{prop:m_decomp}.
\end{proof}
\begin{figure}
\caption{The decompositions of lattice paths in $\mathcal N$, $\boldsymbol{\mathcal{N}
\label{fig:n_decomp}
\end{figure}
Define the generating function \[\boldsymbol{\mathcal{N}}(x):=\sum_{n\geq 0}x^n\sum_{p\in\mathcal N_n}\overlined(p).\]
\begin{proposition}\label{prop:Nn}
The generating function $\boldsymbol{\mathcal{N}}(x)$ satisfies the functional equation \begin{align*}
\boldsymbol{\mathcal{N}}(x)&=2x\boldsymbol{\mathcal{N}}(x)+x^2\mathcal{M}m(x,1)\mathcal N(x)+x^2\mathcal M(x,1)\mathcal N(x)+x^2\mathcal N(x)\frac{\partial}{\partial x}(x\mathcal M(x,1)) \\
&+x\boldsymbol{\mathcal{N}}(x)+x\frac{\partial}{\partial x}(x\mathcal N(x)).
\end{align*}
Thus, \[\boldsymbol{\mathcal{N}}(x)=\frac{4x\left(1+\sqrt{1-4x}-x(1-\sqrt{1-4x})\right)}{\sqrt{1-4x}(1-4x+\sqrt{1-4x})^3}.\]
\end{proposition}
\begin{proof}
Following the same ideas used in the proof of \Cref{prop:mm_decomp}, we find that \[2x\boldsymbol{\mathcal{N}}(x)+x^2\mathcal{M}m(x,1)\mathcal N(x)+x^2\mathcal M(x,1)\mathcal N(x)+x^2\mathcal N(x)\frac{\partial}{\partial x}(x\mathcal M(x,1))\] counts bicolored Motzkin prefixes that touch the $x$-axis at some point other than $(0,0)$, with each path $p$ weighted by $\overline d(p)$. The generating function for bicolored Motzkin prefixes that only touch the $x$-axis at $(0,0)$ (with each path $p$ weighted by $\overline d(p)$) is \[x\boldsymbol{\mathcal{N}}(x)+x\frac{\partial}{\partial x}(x\mathcal N(x)).\] This is illustrated on the second line of~\Cref{fig:n_decomp}.
This yields the functional equation, from which the explicit solution is straightforward to obtain via \Cref{prop:m_decomp,prop:mm_decomp,prop:N}.
\end{proof}
Let \[\mathcal V(x):=\sum_{n\geq 0}|\mathcal V_n|x^n.\]
\begin{proposition}\label{prop:V}
The generating function $\mathcal V(x)$ satisfies the functional equation \[\mathcal V(x)=1+2x\mathcal N(x)+2x\mathcal V(x)+2x^2\mathcal M(x,1)\mathcal V(x).\] Thus, \[\mathcal V(x)=\frac{1+\sqrt{1-4x}}{\sqrt{1-4x}(1-4x+\sqrt{1-4x})}.\]
\end{proposition}
\begin{proof}
The expression $1+2x\mathcal N(x)$ counts (possibly empty) bilateral Motzkin prefixes that only touch the $x$-axis at $(0,0)$, while the expression $2x\mathcal V(x)+2x^2\mathcal M(x,1)\mathcal V(x)$ counts bilateral Motzkin prefixes that touch the $x$-axis at some point other than $(0,0)$. This is illustrated on the third line of~\Cref{fig:n_decomp}. The explicit solution can be derived from the functional equation using \Cref{prop:m_decomp,prop:N}.
\end{proof}
Finally, consider the generating function \[\boldsymbol{\mathcal{V}}(x):=\sum_{n\geq 0}x^n\sum_{p\in\mathcal V_n}\overlined(p).\]
\begin{proposition}\label{prop:Vv}
The generating function $\boldsymbol{\mathcal{V}}(x)$ satisfies the functional equation \begin{align*}
\boldsymbol{\mathcal{V}}(x)&=2x\boldsymbol{\mathcal{V}}(x)+2x^2\mathcal{M}m(x,1)\mathcal V(x)+2x^2\mathcal M(x,1)\boldsymbol{\mathcal{V}}(x)+2x^2\mathcal V(x)\frac{\partial}{\partial x}(x\mathcal M(x,1)) \\
&+2x\boldsymbol{\mathcal{N}}(x)+2x\frac{\partial}{\partial x}(x\mathcal N(x)).
\end{align*}
Thus, \begin{equation}\label{eq:Vv}\boldsymbol{\mathcal{V}}(x)=\frac{8x\left(1+\sqrt{1-4x}-x(3+\sqrt{1-4x})\right)}{(1-4x)(1-4x+\sqrt{1-4x})^3}.\end{equation}
\end{proposition}
\begin{proof}
Following the same ideas used in the proof of \Cref{prop:Nn}, we find that \[2x\boldsymbol{\mathcal{V}}(x)+2x^2\mathcal{M}m(x,1)\mathcal V(x)+2x^2\mathcal M(x,1)\boldsymbol{\mathcal{V}}(x)+2x^2\mathcal V(x)\frac{\partial}{\partial x}(x\mathcal M(x,1))\] counts bilateral Motzkin prefixes that touch the $x$-axis at some point other than $(0,0)$, with each path $p$ weighted by $\overline d(p)$. Furthermore, the generating function for bilateral Motzkin prefixes that only touch the $x$-axis at $(0,0)$ (with each path $p$ weighted by $\overline d(p)$) is \[2x\boldsymbol{\mathcal{N}}(x)+2x\frac{\partial}{\partial x}(x\mathcal N(x)).\] This is illustrated on the fourth line of~\Cref{fig:n_decomp}.
This yields the functional equation, from which one can derive the explicit solution using \Cref{prop:m_decomp,prop:mm_decomp,,prop:N,,prop:V}.
\end{proof}
We conclude \Cref{cor:Wiener_JSn} by expanding the explicit generating function for $\boldsymbol{\mathcal{V}}(x)$ given in \Cref{eq:Vv} as \[\frac{8x\left(1+\sqrt{1-4x}-x(3+\sqrt{1-4x})\right)}{(1-4x)(1-4x+\sqrt{1-4x})^3} = \sum_{n\geq 0} a_n x^n.\]
Using a computer algebra system, the coefficients $a_n$ satisfy the difference equation
\[(2 n+3)2^{-2n-1} a_n - (4 n +5)2^{-2n-3} a_{n + 1} + (2 + 2 n)2^{-2n-5} a_{n + 2} = 0\]
with initial conditions $a_0=0$ and $a_1=2$. It is easily checked that $\frac{2 n (2 n + 1)}{3}\binom{2 n - 1}{n}$ satisfies this equation and initial conditions.
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\section{Asymptotic distributions}\label{sec:asymptotics}
In this section, we prove \cref{prop:cv_law_distance,prop:cv_law_distance_shifted}, which describe the asymptotic distribution
of the distance between 2 random points (also called $2$-point distance) in $P_{\alpha n,n}$ and $Q_n$, respectively.
We start with the case of shifted staircases, which is easier.
\subsection{2-point distance in $Q_n$}
Recall that the elements in $Q_n$
are exactly the lattice paths starting at (0,0),
ending somewhere on the line $x=n$, and using steps
of the form $U=(1,1)$ and $D=(1,-1)$.
Let $p^n$ and $q^n$ be independent uniform random elements in $Q_n$.
Seeing $p^n$ and $q^n$ as lattice paths, we write $p^n_i$
and $q^n_i$
for their heights after $i$ steps.
Clearly, for all $n \ge 1$ and $i \le n$,
one has $p^n_i=X_1+dots+X_i$ and
$q^n_i=Y_1+dots+Y_i$,
where $(X_j)_{j \ge 1}$ and $(Y_j)_{j \ge 1}$ are
independent sequences of i.i.d.~Rademacher random variables of parameter $1/2$.
Using \Cref{eq:distance_shifted}, we write
\[d(p^n,q^n)= \frac12 \sum_{i=1}^n |p^n_i-q^n_i|
= \frac{n}2 \int_0^1 \lvert\, p^n_{\lceil nt \rceil}-q^n_{\lceil nt \rceil}\rvert dt.\]
By Donsker's theorem, the processes
\[\left(\tfrac1{\sqrt n} p^n_{\lceil nt \rceil}\right)_{t \le 1} \text{ and }\left(\tfrac1{\sqrt n} q^n_{\lceil nt \rceil}\right)_{t \le 1} \]
converge in distribution to independent Brownian motions $(B_t)_{t \le 1}$ and $(B'_t)_{t \le 1}$
in Skorokhod space $D[0,1]$
(see \cite[Chapter 3]{billingsley_convergence} for background on Skorokhod space). Since integration is a continuous functional on $D[0,1]$,
we have
\[n^{-3/2} d(p^n,q^n)= \frac12
\int_0^1 \left\lvert n^{-1/2} p^n_{\lceil nt \rceil}-n^{-1/2} q^n_{\lceil nt \rceil}\right\rvert dt \stackrel{d}{\longrightarrow} \frac12
\int_0^1 |B_t-B'_t| dt,\]
where $\stackrel{d}{\longrightarrow}$ means convergence in distribution.
But $B_t-B'_t \stackrel{d}= \sqrt 2 \, B_t$,
proving that $n^{-3/2} d(p^n,q^n)$
converges in distribution to $\frac1{\sqrt 2} \int_0^1 |B_t| dt$, as claimed in \cref{prop:cv_law_distance_shifted}.
It remains to prove moment convergence. By \cite[Corollary of Theorem 25.12]{billingsley_probability}, it suffices to show that for each $s>1$, the sequence of $s$th moments of $n^{-3/2} d(p^n,q^n)$ is bounded as $n$ tends to $+\infty$.
We have
\[n^{-3/2} d(p^n,q^n) \le n^{-1/2} \max_{i \le n} |p^n_i|
+ n^{-1/2} \max_{i \le n} |q^n_i|.\]
Both terms in the upper bound are identically distributed, so we only consider the first one.
By Doob's maximal inequality,
we have
\[ \mathbb{E}\left[\left(\max_{i \le n} |p^n_i|\right)^s\right] \le \left(\frac{s}{s-1}\right)^s
\mathbb{E}\Big[|p^n_n|^s\Big].\]
Since $p^n_n$ is a sum of $n$ i.i.d.~{\em centered}
random variables, we have the following classical bound on its moments
(see, e.g., \cite{petrov1989moments}):
\[ \mathbb{E}\Big[|p^n_n|^s\Big]
\le C(s)\, n^{s/2}\, \mathbb{E}\big[ |X_1|^s \big],\]
where $C(s)$ is a constant depending only on $s$.
In particular the $s$th moment of $n^{-1/2} p^n_n$ is bounded (as $n$ tends to $+\infty$).
Consequently, the $s$th moment of
$n^{-1/2} \max_{i \le n} p^n_i$
is bounded, and that of $n^{-3/2} d(p^n,q^n)$
is as well. This proves that the convergence
of $n^{-3/2} d(p^n,q^n)$ to $\frac1{\sqrt 2} \int_0^1 |B_t| dt$ holds also in moments, concluding the proof of \cref{prop:cv_law_distance_shifted}. \qed
\subsection{2-point distance in $P_{\alpha n,n}$}
We now turn to the case of rectangles.
Let $p^n$ and $q^n$ be independent uniform random elements in $P_{\alpha n,n}$, seen as lattice paths
from $(0,0)$ to ${((\alpha+1)n, (\alpha-1)n)}$.
These paths $p^n$ and $q^n$ can be constructed as partial sums of sequences of i.i.d.~random variables {\em under some conditioning}.
To this end, let $(X_j)_{j \ge 1}$ and $(Y_j)_{j \ge 1}$ be
independent sequences of i.i.d.~Rademacher random variables of parameter $\alpha/(\alpha+1)$.
We also let $(\tilde X^n_j)_{j \ge 1}$ have the distribution
of $(X_j)_{j \ge 1}$ conditioned to the event $\sum_{j \le (\alpha+1)n} X_j=(\alpha-1)n$.
Then one has the equality in distribution
\[\big(p^n_i\big)_{i\le (\alpha+1)n} \stackrel{d}=
\left( \sum_{j \le i} \tilde X^n_j \right)_{i\le (\alpha+1)n}.\]
Recall that we are interested in the quantity
\begin{equation}\label{eq:distance_shifted_integral}
n^{-3/2} D_{\alpha,n} = n^{-3/2} d(p^n,q^n) =
n\frac1{2n^{3/2}} \sum_{i=1}^n |p^n_i-q^n_i|
= \frac{\alpha+1}{2 \sqrt n} \int_0^1 |p^n_{\lceil (\alpha+1)nt \rceil}-q^n_{\lceil (\alpha+1)nt \rceil}| dt.
\end{equation}
A version of Donsker's theorem for conditioned partial sums has been proved by Liggett \cite{liggett1968invariance} (see in particular the corollary of Theorem 4 there).
In our case, the centered process
\[\left(\tfrac1{\sigma \, \sqrt {(1+\alpha)n}} \big(p^n_{\lceil (\alpha+1)nt \rceil} - \lceil nt \rceil (\alpha-1) \big)\right)_{0 \le t \le 1} \]
converges in distribution to $B_0(t)$
in Skorokhod space $D[0,1]$,
where $\sigma^2=\mathrm{Var}(X_1)$ and $B_0(t)$ is a Brownian bridge.
A similar convergence result holds for $q^n$
with an independent Brownian bridge $B'_0(t)$.
Using the continuity of taking integrals on $D[0,1]$, the quantity in \eqref{eq:distance_shifted_integral}
converges in distribution to
\[ \frac12 (\alpha+1) \, \sigma\, \sqrt{\alpha+1} \int_0^1 |B_0(t) -B'_0(t)| dt.\]
An easy computation gives $\sigma=2\sqrt{\alpha}/(\alpha+1)$, while $B_0(t) -B'_0(t)\stackrel{d}=\sqrt{2} B_0(t)$ in distribution.
Consequently, $n^{-3/2} D_{\alpha,n}$ converges in distribution
to $\sqrt{2\alpha(\alpha+1)} \int_0^1 |B_0(t)| dt$, as claimed in \cref{prop:cv_law_distance}.
It remains to prove moment convergence.
As above, we shall prove that for any $s>1$, the random variable $n^{-3/2} D_{\alpha,n}$
has a bounded $s$th moment as $n$ tends to $+\infty$.
Using the convexity of the map $t \mapsto |t|^s$, we obtain
\begin{equation}
\label{eq:bounding_Ens}
n^{-s} D_{\alpha,n}^s = 2^{-s} \left(\frac1{n} \sum_{i \le (\alpha+1)n} |p^n_i-q^n_i|\right)^s
\le \frac{2^{-s}}n \sum_{i \le (\alpha+1)n}
|p^n_i-q^n_i|^s \le \frac{1}{n} \sum_{i \le (\alpha+1)n}
\frac{|\bar p^n_i|^s+|\bar q^n_i|^s}{2},
\end{equation}
where $\bar p^n_i=p^n_i -i \frac{\alpha-1}{\alpha+1}$
is the centered version of $p^n_i$ (and idem for $q$). Writing $\bar X_i=X_i - \frac{\alpha-1}{\alpha+1}$, we have
\[\mathbb{E}\big[ |\bar p^n_i|^s \bar]
= \mathbb{E}\left[ \bigg| \sum_{j \le i} \bar X_j \bigg|^s \Bigg| \sum_{j \le (\alpha+1)n} \bar X_j=0 \right]
= \sum_{k} |k|^s\, \mathbb P\left[ \sum_{j \le i} \bar X_j =k \Bigg| \sum_{j \le (\alpha+1)n} \bar X_j=0 \right],
\]
where the sum runs over possible values $k$ for $\sum_{j \le i} \bar X_j$.
Using the independence of the $\bar X_j$, we have
\begin{align*}
\mathbb P\left[ \sum_{j \le i} \bar X_j =k \Bigg| \sum_{j \le (\alpha+1)n} \bar X_j=0 \right] &=\frac{\mathbb P\left[ \sum_{j \le i} \bar X_j =k \ \wedge \ \sum_{j \le (\alpha+1)n} \bar X_j=0\right]}
{\mathbb P \left[\sum_{j \le (\alpha+1)n} \bar X_j=0 \right]} \\
& = \mathbb P\left[ \textstyle \sum_{j \le i} \bar X_j =k\right] \cdot
\frac{\mathbb P\left[ \sum_{i< j \le (\alpha+1)n} \bar X_j=-k\right]}{\mathbb P \left[\sum_{j \le (\alpha+1)n} \bar X_j=0 \right]}.
\end{align*}
Take $i \le (\alpha+1)n/2$.
The probabilities in the fraction can be evaluated asymptotically---uniformly in $k$---through the local limit theorem (see, e.g., \cite[Theorem 3.5.2]{durrett2019probability}), which yields
\begin{align*}
\mathbb P \left[\sum_{j \le (\alpha+1)n} \bar X_j=0 \right] &\sim \frac{2}{\sigma \sqrt{2 \pi (\alpha+1) n}};\\
\mathbb P\left[ \sum_{i< j \le (\alpha+1)n} \bar X_j=-k\right] &=
\frac{2 e^{-k^2/2((\alpha+1)n-i) \sigma^2}}{\sigma \sqrt{2 \pi ((\alpha+1) n-i)}} + o(n^{-1/2}) \le \frac{2+o(1)}{\sigma \sqrt{\pi (\alpha+1) n}}.
\end{align*}
In particular, the quotient is bounded by $2$ for $n$ large enough, uniformly in $k$.
Bringing everything together, we obtain that for $n$ large enough and $i \le (\alpha+1)n/2$,
\[\mathbb{E}\big[ |\bar p^n_i|^s \bar]
\le \sum_{k} |k|^s \cdot 2 \mathbb P\left[ \textstyle \sum_{j \le i} \bar X_j =k\right]
= 2 \mathbb{E}\left[ \bigg| \sum_{j \le i} \bar X_j \bigg|^s
\right].\]
Since the $\bar X_j$ are i.i.d.~{\em centered} random variables with finite moments, we have (see, e.g., \cite{petrov1989moments})
\[ \mathbb{E}\left[ \bigg| \sum_{j \le i} \bar X_j \bigg|^s
\right] \le C(s) i^{s/2} \mathbb{E}\big[ |X_1|^s \big],\]
where $C(s)$ is a constant depending only on $s$
(and $\alpha$ in the sequel),
which may change from line to line.
Therefore, for $n$ large enough and $i \le (\alpha+1)n/2$, we
have
\[\mathbb{E}\big[ |\bar p^n_i|^s \bar] \le C(s)\, n^{s/2}. \]
By symmetry, this holds also for $i \ge (\alpha+1)n/2$ (we have $p^n_i \stackrel{d}= p^n_{(\alpha+1)n-i}$ for all $i \le (\alpha+1)n$).
Going back to \eqref{eq:bounding_Ens}, we get
\[n^{-s} \mathbb{E} \big[ D_{\alpha,n}^s \big]
\le (\alpha+1)\, C(s)\, n^{s/2}.\]
Thus $n^{-3/2} D_{\alpha,n}$ has bounded moments,
and the convergence to
$\sqrt{2\alpha(\alpha+1)} \int_0^1 |B_0(t)| dt$
holds also in moments.
\cref{prop:cv_law_distance_shifted} is proved. \qed
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\section{Higher moments}
\label{sec:higher_moments}
Given a finite graph $G=(V,E)$ and a positive integer $r$, let $d^r(G)$ denote the moment $d^r(G)=\sum_{(p,q)\in V\times V}d(p,q)^r$.
The convergence of the distance between two random elements in distribution and in moments established in the previous section
yield some asymptotic estimates for $d^r(P_{\alpha n,n})$ and $d^r(Q_n)$.
In this section, we give an exact expression of $d^2(P_{k,n-k})$. The same method can, in principle, be used to compute the moments $d^r(P_{k,n-k})$ one by one. Similarly, one could use a similar method, drawing from the ideas in \Cref{sec:bij_stair,sec:proof_stair}, to compute the moments $d^r(Q_n)$. For the sake of brevity, we merely state the explicit formula for $d^2(Q_n)$ and omit the computation.
\begin{proposition} We have
\[d^2(P_{m,k}) = \frac{1}{30} \frac{m+k+1}{m+k} \binom{m+k}{m-1}\binom{m+k}{k-1} \left( 7mk^2 + 7m^2k + 3m^2 + 10mk+ 3k^2 + 3m + 3k + 4 \right)\] and \[d^2(Q_{n}) = 2^{2n-4} n \left(20 + 15 (n - 2) + 3 (n - 2)^2\right).\]
\end{proposition}
\begin{proof}
As mentioned above, we will only prove the first formula. Given a bilateral Motzkin path $p$, let $\mathrm{len}(p)$ denote the length of $p$, and let $\mathcal U(p)$ be the number of steps in $p$ of the form $U$ or $O_1$. Recall that \[\mathcal{W}(x,u)=\sum_{p\in\mathcal{W}}x^{\mathrm{len}(p)}u^{\mathcal U(p)},\quad \mathcal{W}w(x,u)=\sum_{p\in \mathcal{W}}x^{\mathrm{len}(p)}u^{U(p)}d(p),\] \[\mathcal{M}(x,u)=\sum_{p\in\mathcal{M}}x^{\mathrm{len}(p)}u^{\mathcal U(p)}, \quad\mathcal{M}m(x,u)=\sum_{p\in \mathcal{M}}x^{\mathrm{len}(p)}u^{U(p)}d(p).\] Let \[\mathbb{W}(x,u)=\sum_{p\in \mathcal{W}}x^{\mathrm{len}(p)}u^{U(p)}d(p)^2\quad\text{and}\quad\mathcal{M}MM(x,u)=\sum_{p\in \mathcal{M}}x^{\mathrm{len}(p)}u^{U(p)}d(p)^2.\] It follows from \Cref{prop:bija} that $\mathbb{W}(x,u)=\sum_{n\geq 0}\sum_{k\geq 0}d^2(P_{k,n-k})x^nu^k$.
The contribution to $\mathcal{M}MM(x,u)$ coming from paths that start with $O_1$ or $O_2$ is $x(u+1)\mathcal{M}MM(x,u)$. The other paths that contribute to $\mathcal{M}MM(x,u)$ begin with $U$ and have the form $UpDq$ for some $p,q\in\mathcal{M}$. We find that \begin{equation}\label{eq:Sigma}
\mathcal{M}MM(x,u)=x(u+1)\mathcal{M}MM(x,u)+ux^2\Sigma,
\end{equation}
where \[\Sigma=\sum_{p,q\in\mathcal{M}}x^{\mathrm{len}(p)+\mathrm{len}(q)}u^{\mathcal U(p)+\mathcal U(q)}(d(p)+d(q)+\mathrm{len}(p)+1)^2.\] We can write
\begin{align*}
(d(p)+d(q)+\mathrm{len}(p)+1)^2&=(d(p)^2+d(q)^2)+(\mathrm{len}(p)+1)^2+2d(p)(\mathrm{len}(p)+1) \\
&+2d(q)(\mathrm{len}(p)+1)+2d(p)d(q)
\end{align*}
to find that
\begin{align}\label{eq:Sigma2}
\Sigma&=2\mathcal{M}MM(x,u)\mathcal{M}(x,u)+\mathcal{M}(x,u)\frac{\partial}{\partial x}\left(x\frac{\partial}{\partial x}(x\mathcal{M}(x,u))\right)+2\mathcal{M}(x,u)\frac{\partial}{\partial x}(x\mathcal{M}m(x,u)) \nonumber\\
&+2\mathcal{M}m(x,u)\frac{\partial}{\partial x}(x\mathcal{M}(x,u))+2\mathcal{M}m(x,u)^2.
\end{align}
A similar argument yields the functional equation
\begin{equation}\label{eq:Sigma'}
\mathbb{W}(x,u)=x(u+1)\mathbb{W}(x,u)+2ux^2\Sigma',
\end{equation}
where
\begin{align}\label{eq:Sigma'2}
\Sigma'&=\sum_{\substack{p\in\mathcal{M} \\ q\in\mathcal{W}}}x^{\mathrm{len}(p)+\mathrm{len}(q)}u^{\mathcal U(p)+\mathcal U(q)}(d(p)+d(q)+\mathrm{len}(p)+1)^2 \nonumber\\
&=\mathcal{M}MM(x,u)\mathcal{W}(x,u)+\mathbb{W}(x,u)\mathcal{M}(x,u)+\mathcal{W}(x,u)\frac{\partial}{\partial x}\left(x\frac{\partial}{\partial x}(x\mathcal{M}(x,u))\right) \nonumber \\
&+2\mathcal{W}(x,u)\frac{\partial}{\partial x}(x\mathcal{M}m(x,u))+2\mathcal{W}w(x,u)\frac{\partial}{\partial x}(x\mathcal{M}(x,u))+2\mathcal{M}m(x,u)\mathcal{W}w(x,u).
\end{align}
We already computed explicit formulas for $\mathcal{M}(x,u)$, $\mathcal{M}m(x,u)$, $\mathcal{W}(x,u)$, and $\mathcal{W}w(x,u)$ in \Cref{prop:m_decomp,prop:mb_decomp,prop:mm_decomp,prop:mbm_decomp}. Combining those formulas with \Cref{eq:Sigma,eq:Sigma2,eq:Sigma',eq:Sigma'2}, we can derive the explicit formula
\[\mathbb{W}(x,u)=\frac{2 u x^2 \left((u-1)^2 (u+1) x^3-((u-8) u+1) x^2-(u+1) x+1\right)}{\left((u x+x-1)^2-4 u x^2\right)^{7/2}}.\] Setting $u=y/x$ and extracting coefficients yields the desired explicit formula for $d^2(P_{m,k})$.
\end{proof}
\begin{comment}
Generating function for nested lattice paths is
\[\mathcal{M}MM(x,u)=u x^2 \left(2 \mathcal{M}m(x,u) \frac{\partial (x M(x,u))}{\partial x}+2 M(x,u) \frac{\partial (x \mathcal{M}m(x,u))}{\partial x}+2 \mathcal{M}(x,u) \mathcal{M}MM(x,u)+\mathcal{M}(x,u) \frac{\partial }{\partial x}\left(x \frac{\partial (x \mathcal{M}(x,u))}{\partial x}\right)+2 \mathcal{M}m(x,u)^2\right)+(u+1) x \mathcal{M}MM(x,u).\]
Solving gives
\[\mathcal{M}MM(x,u)=\frac{1}{2}\left(\frac{20 u x^2}{\left((u x+x-1)^2-4 u x^2\right)^{5/2}}+\frac{2 u x}{\left((u x+x-1)^2-4 u x^2\right)^{3/2}}+\frac{2 x}{\left((u x+x-1)^2-4 u x^2\right)^{3/2}}+\frac{\sqrt{(u x+x-1)^2-4 u x^2}}{u (x-1) x^2}+\frac{1}{u x^2}+\frac{4}{(u x+x-1)^2-4 u x^2}+\frac{8 (u x+x-1)}{\left((u x+x-1)^2-4 u x^2\right)^2}+\frac{6}{\left((u x+x-1)^2-4 u x^2\right)^{3/2}}-\frac{u}{(x-1) \sqrt{(u x+x-1)^2-4 u x^2}}+\frac{3}{(x-1) \sqrt{(u x+x-1)^2-4 u x^2}}+\frac{1}{(x-1) x \sqrt{(u x+x-1)^2-4 u x^2}}\right.\]
Now the generating function for pairs of lattice paths is
\[\mathbb{W}(x,u)=2 u x^2 \left(2 \left(\mathcal{M}bm(x,u) \frac{\partial (x \mathcal{M}(x,u))}{\partial x}+\mathcal{M}b(x,u) \frac{\partial (x \mathcal{M}m(x,u))}{\partial x}+\mathcal{M}m(x,u) \mathcal{M}bm(x,u)\right)+\mathcal{M}b(x,u) \frac{\partial }{\partial x}\left(x \frac{\partial (x \mathcal{M}(x,u))}{\partial x}\right)+\mathcal{M}(x,u) \mathbb{W}(x,u)+\mathcal{M}MM(x,u) \mathcal{M}b(x,u)\right)+(u+1) x \mathbb{W}(x,u)\]
\[\mathbb{W}(x,u)=\frac{2 u x^2 \left((u-1)^2 (u+1) x^3-((u-8) u+1) x^2-(u+1) x+1\right)}{\left((u x+x-1)^2-4 u x^2\right)^{7/2}}.\]
And then extract coefficients...
\end{comment}
\section{Open problems}
Comparing the results of Proposition~\ref{prop:mb_decomp} and~\ref{prop:mbm_decomp}, we get the intriguing equation
\begin{equation}
\label{eq:interesting_identity}
\mathcal{M}bm=2ux^2\;\mathcal{M}b^4.
\end{equation}
A direct proof of this would be interesting in itself and could lead to a bijective proof of~\Cref{cor:coeff} via the explicit formula \eqref{eq:expansion_bilateral}. Recall that $\mathcal{M}bm$ counts pairs of paths where a cell in the symmetric difference is marked. The connected component where this cell occurs corresponds to a part where the two paths only meet at their beginning and end (this forms a \emph{parallelogram polyomino}), and the generating function $\mathcal{M}b^2$ naturally enumerates the rest of the paths. It follows that a bijective proof of \eqref{eq:interesting_identity} reduces to a bijective proof that $\mathcal{M}b^2$ enumerates the total area of parallelogram polyominoes. The specialization $u=1$ is known \cite{dellungo2004bijection}.
As minuscule lattices arise as the weak order on certain maximal parabolic quotients of finite Coxeter groups, it would be interesting to extend our results to other parabolic quotients. Minuscule lattices also appear as certain crystal graphs; one could also ask about the Wiener indices of more general crystals.
\renewcommand*{\bibliofont}{\normalsize}
\end{document}
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\begin{document}
\title{Small Entangled Quantum Worlds with a Simple Structure}
\author{E. D. Vol}
\email{[email protected]}
\affiliation{B. Verkin Institute for Low Temperature Physics and Engineering of the
National Academy of Sciences of Ukraine, 47, Nauky Ave., Kharkov 61103,
Ukraine.}
\begin{abstract}
We introduce the notion of a small quantum world (SQW) , which in
our opinion, is very helpful in the situations when an
experimenter's tools for preparing quantum states and (or) for
measuring the observables of a quantum system under study are
restricted due to some kind of reasons. In this case it is advisable
to use as original an appropriate subspace of complete Hilbert space
of states and respectively to utilize observables acting only in
this subspace. If this subspace possesses some additional
symmetries, the structure of a pure states set, as a rule, is
simpler by far than in general case. Moreover in such SQWs some
specific irreducible entangled states may appear. The similar states
could be very helpful in various tasks connected with quantum
informational applications. In the present paper main ideas outlined
above are developed in detail in the simple and instructive case of
a two-qubit system in which the accessible space of states possesses
the additional symmetry structure of permutation group of three
elements.
\end{abstract}
\maketitle
It is well known that there are two fundamental concepts in quantum theory,
namely states and observables, and respectively an experimenter has to deal
with two main procedures that realize these concepts, namely preparation of
the required state and measuring the observable of interest in the end of
the experiment. Note that among all states of quantum system its pure
states, which contain the maximal possible information about the system,
play the most important role. One of the main postulates of quantum theory
claims that there is exact mapping between pure states of a quantum system
under study and vectors of appropriate Hilbert space. For example, in the
case of the most simple quantum system, that is a qubit, the relevant
Hilbert space, representing its states, is two-dimensional and there is a
geometrically descriptive image of the states of such system by means of the
Bloch sphere. As is well known, an arbitrary state of the qubit (mixed or
pure) with a density matrix ${\hat{\rho}}$ can be represented in the form ${
\hat{\rho}}=\frac{1+\vec{P}\vec{\sigma}}{2}$, where $\hat{\sigma}_{i}$ $
(i=1,2,3)$ are the Pauli matrices and $\vec{P}$ is appropriate Bloch vector.
The pure states of a qubit for which the condition ${\hat{\rho}}^{2}={\hat{
\rho}}$ is satisfied are placed on the surface of the Bloch sphere with $
\vec{P}=1$ while all the rest (mixed) states are settled inside this sphere.
Unfortunately, in more complex situations, in particular for composite
quantum systems the problem of description of the set of pure states by
geometrically distinct way is unsolved till now. Furthermore, when one is
operating with the states of composite systems, the important problem of
determination their entanglement ( the quantity that just specifies the
nonlocal informational resource and distinguishes quantum communicational
systems from classical ones), can be solved exactly also only for two-qubit
systems. In spite of enormous number of papers devoted to this problem (see
for example the comprehensive review of the topic \cite{1}), this problem
remains open up to now. In this connection we propose as a first step to
consider more simple problem, namely, to study some special classes of
quantum systems in which both the set of accessible quantum states and the
set of observables that are available for measurement are restricted by some
additional conditions.Evidently we will talking about specific subspaces of
the total Hilbert space and appropriate algebras of observables acting in
these subspaces. From physical point of view it means that, due to various
reasons, capabilities of an experimenter for the quantum states preparation
and for performing arbitrary measurements are restricted. Nevertheless,
since in this approach the space of accessible states of the system remains
linear and closed, all postulates of quantum theory continue to be valid.
Henceforth we will define as a small quantum world (SQW) certain subspace of
states within the complete Hilbert space with the relevant algebra of
observables acting in this subspace. The main goal of the present paper is
to demonstrate on particular examples that the structure of states in such
SQWs could be simpler by far than the structure of states in the enveloping
large quantum world. At first let us briefly remind one of the known
examples of SQW, namely, so called X-states in two-qubit quantum systems
\cite{2}. In this case one assumes that density matrix of any accessible
state of the quantum system under study can be represented as ${\hat{\rho}}=
\begin{pmatrix}
\rho _{11} & 0 & 0 & \rho _{14} \\
0 & \rho _{22} & \rho _{23} & 0 \\
0 & \rho _{32} & \rho _{33} & 0 \\
\rho _{41} & 0 & 0 & \rho _{44}
\end{pmatrix}
$. The relevant basis of the algebra of observables acting on these states
consists of 8 operators (generators of the algebra). Let us write them
explicitly:1) $\hat{1}-$the unit operator ,2) another diagonal operator $
\hat{E}=
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
$, and in addition six operators $\hat{\lambda}_{i}$ and ${\hat{\tau}}_{i}$
\ ($i=1,2,3$) that are defined by analogy with the known Pauli
matrices,namely :
\begin{eqnarray*}
&&\hat{\lambda}_{1}=
\begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0
\end{pmatrix}
,\hat{\lambda}_{2}=
\begin{pmatrix}
0 & 0 & 0 & -i \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
i & 0 & 0 & 0
\end{pmatrix}
, \\
&&\hat{\lambda}_{3}=
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1
\end{pmatrix}
\end{eqnarray*}
and
\begin{eqnarray*}
&&{\hat{\tau}}_{1}=
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
,{\hat{\tau}}_{2}=
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & -i & 0 \\
0 & i & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
, \\
&&{\hat{\tau}}_{3}=
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
.
\end{eqnarray*}
It is easy to point out the complete system of algebraic relations
connecting the generators of this algebra, namely:
\begin{eqnarray}
&&{\hat{\lambda}_{i}}{\hat{\lambda}_{j}}=\frac{1+\hat{E}}{2}\delta
_{ij}+i\varepsilon _{ijk}{\hat{\lambda}_{k}}, \notag \label{1} \\
&&{\hat{\tau}_{i}}{\hat{\tau}_{j}}=\frac{1-\hat{E}}{2}\delta
_{ij}+i\varepsilon _{ijk}{\hat{\tau}_{k}}, \notag \\
&&{\hat{\lambda}_{i}}{\hat{\tau}_{j}}={\hat{\tau}_{j}}{\hat{\lambda}_{i}}=0,
\notag \\
&&\hat{E}{\hat{\lambda}_{i}}={\hat{\lambda}_{i}}\hat{E}={\hat{\lambda}_{i}}
\text{ and } \notag \\
&&\hat{E}{\hat{\tau}_{i}}={\hat{\tau}_{i}}\hat{E}=-{\hat{\tau}_{i}}
\end{eqnarray}
( where $\varepsilon _{ijk}$ is completely antisymmetric tensor).Any X-state
(that is its density matrix) can be represented as $\hat{\rho}=\frac{1+e\hat{
E}+P_{i}{\hat{\lambda}_{i}}+S_{i}{\hat{\tau}_{i}}}{4}.$(where $e,P_{i},S_{i}$
are appropriate numerical coefficients). It is easy to see that 4
eigenvalues of such matrix can be calculated explicitly and are equal to: $
\rho _{1,2}=\frac{1+e\pm \left\vert P\right\vert }{4}$ and $\rho _{3,4}=
\frac{1-e\pm \left\vert S\right\vert }{4}.$ Evidently, that one can specify
two disjoint classes of pure X-states in such SQW: 1) with $e=1$ , $
\left\vert P\right\vert =2,$ $\left\vert S\right\vert =0$ and 2) with $e=-1$
$\left\vert S\right\vert =2,$ $\left\vert P\right\vert =0$. Thus the
structure of pure states in this SQW turns out to be simpler by far than for
two-qubit states in general case. We will continue to study the properties
of X- states elsewhere, while the present paper will be devoted to another
interesting case, namely the SQWs that can be constructed on the basis of
the permutation group of four elements- $S_{4}$. It is well-known that due
to the principle of indistinguishability the permutation group plays the
fundamental and diverse role in quantum theory, but in this paper it will be
used only for the description of quantum states and their properties. With
reference to group $S_{4}$ it should be noted that among its 30 subgroups
there are 4 subgroups that are isomorphic to group $S_{3}$ (that is
permutation group of 3 elements).Exactly this subgroup can serve as
demonstrative and instructive example of the small (and entangled as we see
further) quantum world. Let us consider now the concrete realization of this
subgroup which leaves as invariant the fourth element (state) of the group
and construct the model of SQW based on this realization. In the standard
basis of two-qubit states the relevant algebra of observables acting in this
SQW consists of six generators, from which three (aside from unit operator)
are Hermitian and the rest two are unitary. Let us write down them in
explicit form. The three Hermitian operators are:
\begin{eqnarray}
{\hat{H}_{1}} &=&
\begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
,{\hat{H}_{2}}=
\begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
, \notag \label{2} \\
{\hat{H}_{3}} &=&
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\end{eqnarray}
and two unitary ones are
\begin{equation}
\hat{A}=
\begin{pmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\text{ , }{\hat{B}}=
\begin{pmatrix}
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
. \label{3}
\end{equation}
Note that unitary operators $\hat{A}$ and ${\hat{B}}$ are conjugate and
reciprocal to each other, that is $\hat{A}={\hat{B}^{+}}$ and $\hat{A}\hat{B}
={\hat{B}}\hat{A}=\hat{1}$. Let us point out the complete system of
relations for hermitian generators of the algebra: ${\hat{H}_{i}}^{2}=\hat{1}
$ ($i=1,2,3$), ${\hat{H}_{1}}{\hat{H}_{2}}={\hat{H}_{2}}{\hat{H}_{3}}={\hat{H
}_{3}}{\hat{H}_{1}}=\hat{A}$ and $\hat{H_{1}}{\hat{H}_{3}}={\hat{H}_{2}}{
\hat{H}_{1}}={\hat{H}_{3}}\hat{H_{2}}={\hat{B}}$. In addition let us give
also the algebraic relations connecting operators ${\hat{H}_{i}}$ with
unitary operators $\hat{A}$ \ which have the next form: ${\hat{H}_{1}}\hat{A}
={\hat{H}_{2}},$ ${\hat{H}_{2}}\hat{A}={\hat{H}_{3}},$ $\hat{H_{3}}\hat{A}={
\hat{H}_{1}}$ and $\hat{A}{\hat{H}_{1}}=\hat{H_{3}},\hat{A}{\hat{H}_{2}}={
\hat{H}_{1}},\hat{A}{\hat{H}_{3}}={\hat{H}_{2}}$ .The similar equations
including the matrix ${\hat{B}}$ can be obtained from these relations by
conjugation. There are also two relations, connecting operators $\widehat{A}$
and $\widehat{B\text{ }}$ : $\hat{A}^{2}={\hat{B}}$,and ${\hat{B}}^{2}=\hat{A
}$. The density matrix of any state belonging to this SQW may be represented
as $\hat{\rho}=\frac{k}{2}\hat{1}+l{\hat{H}_{1}}+m{\hat{H}_{2}}+n\hat{H_{3}}
+p\left( \hat{A}+{\hat{B}}\right) $, where $k,l,m,n,p$ are real numbers
satisfying the normalization condition: $k+l+m+n+p=\frac{1}{2}$. However, it
is easy to see that generators of the algebra ${\hat{H}_{i}}$, $\hat{A}$ and
${\hat{B}}$ are connected by the additional relation : $\hat{A}+{\hat{B}}=
\hat{C}-1$, where $\hat{C}\equiv {\hat{H}_{1}}+{\hat{H}_{2}}+{\hat{H}_{3}}$.
Thus the general representation for the density matrix in given SQW may be
written finally as :
\begin{equation}
\hat{\rho}=\frac{a}{2}+b{\hat{H}_{1}}+c{\hat{H}_{2}}+d{\hat{H}_{3}}
\label{4}
\end{equation}
with normalization condition: $a+b+c+d=\frac{1}{2}$.
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Note that unitary operators $\hat{A}$ and ${\hat{B}}$ are conjugate and
reciprocal to each other, that is $\hat{A}={\hat{B}^{+}}$ and $\hat{A}\hat{B}
={\hat{B}}\hat{A}=\hat{1}$. Let us point out the complete system of
relations for hermitian generators of the algebra: ${\hat{H}_{i}}^{2}=\hat{1}
$ ($i=1,2,3$), ${\hat{H}_{1}}{\hat{H}_{2}}={\hat{H}_{2}}{\hat{H}_{3}}={\hat{H
}_{3}}{\hat{H}_{1}}=\hat{A}$ and $\hat{H_{1}}{\hat{H}_{3}}={\hat{H}_{2}}{
\hat{H}_{1}}={\hat{H}_{3}}\hat{H_{2}}={\hat{B}}$. In addition let us give
also the algebraic relations connecting operators ${\hat{H}_{i}}$ with
unitary operators $\hat{A}$ \ which have the next form: ${\hat{H}_{1}}\hat{A}
={\hat{H}_{2}},$ ${\hat{H}_{2}}\hat{A}={\hat{H}_{3}},$ $\hat{H_{3}}\hat{A}={
\hat{H}_{1}}$ and $\hat{A}{\hat{H}_{1}}=\hat{H_{3}},\hat{A}{\hat{H}_{2}}={
\hat{H}_{1}},\hat{A}{\hat{H}_{3}}={\hat{H}_{2}}$ .The similar equations
including the matrix ${\hat{B}}$ can be obtained from these relations by
conjugation. There are also two relations, connecting operators $\widehat{A}$
and $\widehat{B\text{ }}$ : $\hat{A}^{2}={\hat{B}}$,and ${\hat{B}}^{2}=\hat{A
}$. The density matrix of any state belonging to this SQW may be represented
as $\hat{\rho}=\frac{k}{2}\hat{1}+l{\hat{H}_{1}}+m{\hat{H}_{2}}+n\hat{H_{3}}
+p\left( \hat{A}+{\hat{B}}\right) $, where $k,l,m,n,p$ are real numbers
satisfying the normalization condition: $k+l+m+n+p=\frac{1}{2}$. However, it
is easy to see that generators of the algebra ${\hat{H}_{i}}$, $\hat{A}$ and
${\hat{B}}$ are connected by the additional relation : $\hat{A}+{\hat{B}}=
\hat{C}-1$, where $\hat{C}\equiv {\hat{H}_{1}}+{\hat{H}_{2}}+{\hat{H}_{3}}$.
Thus the general representation for the density matrix in given SQW may be
written finally as :
\begin{equation}
\hat{\rho}=\frac{a}{2}+b{\hat{H}_{1}}+c{\hat{H}_{2}}+d{\hat{H}_{3}}
\label{4}
\end{equation}
with normalization condition: $a+b+c+d=\frac{1}{2}$.
It is worth noting that operator $\hat{C}$ commutes with all
generators of the algebra and hence is the Casimir operator of the
group $S_{3}$. Let us turn now to the question of clarifying the
structure of pure states in this SQW. Starting from the
representation \eqref{4} and using the defining condition for pure
states: ${\hat{\rho}}^{2}={\hat{\rho}}$, one, after elementary
calculations, can obtain the following restrictions on the
coefficients of the decomposition \eqref{4}, namely$:$
\begin{eqnarray}
&&\frac{a^{2}}{2}=\frac{a^{2}}{4}+b^{2}+c^{2}+c^{2}-(bc+bd+cd); \notag
\label{5} \\
&&b=ab+(bc+bd+cd); \notag \\
&&c=ac+(bc+bd+cd); \notag \\
&&d=ad+(bc+bd+cd).
\end{eqnarray}
It is clear that the only possibility to satisfy all relations \eqref{5} is
to put the coefficient $a$ equal to unit and impose on the coefficients $
b,c,d$ the next restriction: $b^{2}+c^{2}+d^{2}=\frac{1}{4}$ (together with
normalization condition $b+c+d=-\frac{1}{2}$). Thus, in the parameter space
of coefficients $b,c,d$ the set of pure states is the intersection of the
sphere centered in the origin, whose radius is equal to $\frac{1}{2}$, and
the plane satisfying the equation $b+c+d=-\frac{1}{2}$. Evidently it is a
circle. Further, it is well-known that in Hilbert space pure states form the
boundary of a convex set of all quantum states \cite{3}. Hence the result
obtained means that all mixed states of the system in this SQW must be
settled within the circle specified above. Thus the structure of quantum
states in the SQW under study is quite simple and obvious. In addition one
can point out \ the parametrization of pure states in this SQW by writing
the coefficients of \eqref{4} with $a=1$ in the next convenient form:
\begin{eqnarray}
b &=&-\frac{t\left( 1+t\right) }{2\left( 1+t+t^{2}\right) };c=-\frac{\left(
1+t\right) }{2\left( 1+t+t^{2}\right) }; \notag \label{6} \\
d &=&\frac{t}{2\left( 1+t+t^{2}\right) }
\end{eqnarray}
(with the only real parameter $t).$ It is easy to verify directly that two
relations $b+c+d=-\frac{1}{2}$ and $b^{2}+c^{2}+d^{2}=\frac{1}{4}$
characterizing pure states in this SQW are fulfilled automatically. Using
the parametrization \eqref{6} one can write down the normalized vector $
\left\vert \Psi \right\rangle $ corresponding to the density matrix of a
pure state, that is, if ${\hat{\rho}}=\left\vert \Psi \right\rangle
\left\langle \Psi \right\vert $, then the appropriate vector $\left\vert
\Psi \right\rangle $ can be represented as: $\left\vert \Psi \left( t\right)
\right\rangle =\frac{1}{\sqrt{2\left( 1+t+t^{2}\right) }}
\begin{pmatrix}
1+t \\
t \\
1 \\
0
\end{pmatrix}
.$
Since all information contained in quantum state of the system can be
extracted only by making appropriate measurements , it is useful to give a
simple experimental criterion of the state purity. To this end we assume
that the unknown quantum state has the above-mentioned form:
\begin{multline}
{\hat{\rho}}=\frac{1}{2}+b{\hat{H}_{1}}+c{\hat{H}_{2}}+d\hat{H_{3}}=
\label{7} \\
=
\begin{pmatrix}
\frac{1}{2}+d & b & c & 0 \\
b & \frac{1}{2}+c & d & 0 \\
c & d & \frac{1}{2}+b & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
.
\end{multline}
First of all let us find the eigenvalues of the density matrix \eqref{7}.
One can verify easily that ${\hat{\rho}}$ has two zero eigenvalues: the
first with eigenvector $\left\vert 0\right\rangle _{1}=
\begin{pmatrix}
0 \\
0 \\
0 \\
1
\end{pmatrix}
$ and the second with eigenvector $\left\vert 0\right\rangle _{2}=\frac{1}{
\sqrt{3}}
\begin{pmatrix}
1 \\
1 \\
1 \\
0
\end{pmatrix}
$. Besides them there are two nonzero eigenvalues $\mu _{1}$and $\mu _{2}$
that satisfy to the quadratic equation: $\mu ^{2}-\mu +3\left(
bc+bd+cd\right) =0$ with the solutions:
\begin{equation}
\mu _{1,2}=\frac{1\pm \sqrt[2]{1-12\left( bc+bd+cd\right) }}{2} \label{8}
\end{equation}
The expression \eqref{8} implies that coefficients $b,c,d$ aside from
normalization condition must satisfy to the inequality $0\leq \left(
bc+bd+cd\right) \leq \frac{1}{12}.$
Let us define now as usual the mean value of an arbitrary observable $\hat{A}
$ as $\left\langle \hat{A}\right\rangle =\mathrm{Tr}\left( \hat{ \rho } \hat{
A}\right)$. Using the representation \eqref{7} one can write down the mean
values of the following three selected observables:
1) $\left\langle H_{1}\right\rangle -1\equiv A_{1}=c+d+4b,$
2) $\left\langle H_{2}\right\rangle -1\equiv A_{2}=b+d+4c,$ and
3) $\left\langle H_{3}\right\rangle -1\equiv A_{3}=b+c+4d.$
Taking into account the condition $b+c+d=-\frac{1}{2}$ one can find that $\
\ \ \left\langle A_{1}+A_{2}+A_{3}\right\rangle =-3$, that is for all states
of SQW (with $a=1$) the mean value of the Casimir operator $\left\langle
C\right\rangle \equiv \left\langle H_{1}+H_{2}+H_{3}\right\rangle =0$. On
the other hand if one considers another useful quantity, namely $R\equiv
A_{1}^{2}+A_{2}^{2}+A_{3}^{2}=\left( c+d+4b\right) ^{2}+\left( b+d+4c\right)
^{2}+\left( b+c+4d\right) ^{2}$, then she(he) obtains that $R=\left( -\frac{1
}{2}+3b\right) ^{2}+\left( -\frac{1}{2}+3c\right) ^{2}+\left( -\frac{1}{2}
+3d\right) ^{2}=9\left( \frac{1}{4}+b^{2}+c^{2}+d^{2}\right) =\frac{9}{2}$.
Thus, for all pure states in SQW the following criterion of purity should be
true:
\begin{equation}
\left( \left\langle H_{1}\right\rangle -1\right) ^{2}+\left( \left\langle
H_{2}\right\rangle -1\right) ^{2}+\left( \left\langle H_{3}\right\rangle
-1\right) ^{2}=\frac{9}{2} \label{9}
\end{equation}
Let us go to the next question which we are interested in: how the
explicit expression for the entanglement of the states (both pure
and mixed) in this SQW looks? For the sake of simplicity as before
we are limited ourselves to studying the states for which the
coefficient $a$ in \eqref{4} is equal to unit and the appropriate
density matrix can be represented as
\begin{equation}
{\hat{\rho}}=\frac{1}{2}+b{\hat{H}_{1}}+c{\hat{H}_{2}}+d{\hat{H}_{2}}
\label{10}
\end{equation}
with a normalization condition: $b+c+d=-\frac{1}{2}$. As was explained
above, among these states there are certain pure states (for which $
b^{2}+c^{2}+d^{2}=\frac{1}{4}$) while all the rest are mixed. Let us
determine the entanglement of the state \eqref{10}. To this end in the case
of two-qubit arbitrary mixed state the well-known recipe was proposed by
Wootters in \cite{4}. This recipe reads as follows. First of all one must
construct the auxiliary matrix $\tilde{\rho}=\left( \sigma _{y}\otimes
\sigma _{y}\right) \rho ^{\ast }\left( \sigma _{y}\otimes \sigma _{y}\right)
$, where $\sigma _{y}$ $=
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}
$ is the Pauli matrix and ${\hat{\rho}^{\ast }}$ is the matrix
conjugated to given matrix ${\hat{\rho}} $ \eqref{10}. At the next
step one needs
to introduce another auxiliary matrix $\hat{\Omega}$ $=\hat{\rho}{\hat{\rho}}
^{\ast }$ (that is non-Hermitian and positive) and after that to find its
four eigenvalues $\Omega _{i}$ ($i=1,2,3,4$). If one arranges them in
decreasing order $\Omega _{1}\geq \Omega _{2}\geq \Omega _{3}\geq \Omega
_{4}\geq 0$, then according to the paper \cite{4} the entanglement of
formation $E(\widehat{\rho })$ for the state ${\hat{\rho}}$ $\ $ may be
calculated as follows:
\begin{multline}
E\left( {\hat{\rho}}\right) =-\frac{\left( 1+\sqrt{1-C^{2}}\right) }{2}\log
_{2}\frac{1+\sqrt{1-C^{2}}}{2}- \label{11} \\
-\frac{\left( 1-\sqrt{1-C^{2}}\right) }{2}\log _{2}\frac{\left( 1-\sqrt{
1-C^{2}}\right) }{2},
\end{multline}
where the concurrence
\begin{equation*}
C=C\left( {\hat{\rho}}\right) =\max \left\{ 0,\sqrt{\Omega _{1}}-\sqrt{
\Omega _{2}}-\sqrt{\Omega _{3}}-\sqrt{\Omega _{4}}\right\} .
\end{equation*}
Note that since $E\left( {\hat{\rho}}\right) $ is a monotonic and increasing
function of concurrence $C\left( \widehat{\rho }\right) $, we can restrict
ourselves to determination of the value of $C\left( \widehat{\rho }\right) $
that evidently ranges from zero to unit and defines the degree of
entanglement as well as $E\left( {\hat{\rho}}\right) $. Omitting elementary
calculations we give the final expressions for the auxiliary matrix $\hat{
\Omega}$ and its four eigenvalues, namely:
\begin{equation}
\hat{\Omega}=
\begin{pmatrix}
0 & b\left( \frac{1}{2}+b\right) +cd & bd+c\left( \frac{1}{2}+c\right) & -2bc
\\
0 & \left( \frac{1}{2}+b\right) \left( \frac{1}{2}+c\right) +d^{2} & 2\left(
\frac{1}{2}+c\right) d & 0 \\
0 & 2\left( \frac{1}{2}+b\right) d & & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
\label{12}
\end{equation}
The eigenvalues of matrix $\hat{\Omega}$ (in decreasing order) are equal to:
$\Omega _{1}=\left[ d+\sqrt{\left( \frac{1}{2}+b\right) \left( \frac{1}{2}
+c\right) }\right] ^{2},\Omega _{2}=\left[ d-\sqrt{\left( \frac{1}{2}
+b\right) \left( \frac{1}{2}+c\right) }\right] ^{2},\Omega _{3}=\Omega
_{4}=0.$
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Using the above-mentioned recipe for determination of entanglement one can
find the required result:
\begin{equation}
C\left( {\hat{\rho}}\right) =2\sqrt{\left( \frac{1}{2}+b\right) \left( \frac{
1}{2}+c\right) }. \label{13}
\end{equation}
It is worth noting that if the state \eqref{6} is pure, that is ${\hat{\rho}}
=\left\vert \Psi \right\rangle \left\langle \Psi \right\vert $, where $
\left\vert \Psi \right\rangle =\frac{1}{2\sqrt{1+t+t^{2}}}
\begin{pmatrix}
1+t \\
t \\
1 \\
0
\end{pmatrix}
$, then the expression \eqref{13} takes the form: $C\left\{ \Psi \right\} =
\frac{\left\vert t\right\vert }{1+t+t^{2}}$ which coincides with the
standard definition of concurrence of a pure state$\left\vert \Psi
\right\rangle $. We see that unlike of general case the entanglement
of states belonging to the SQW considered may be explicitly
expressed in terms of the coefficients of the given density matrix
only.Now we turn to the study another interesting problem, namely,
to clarify how much entanglement may be extracted from the given
state (pure or mixed) belonging to the SQW by means of various
measurements carried out on a system being in the state \eqref{10}.
Clearly, having in hands the simple expression for the amount of
entanglement contained in any quantum state of SQW \eqref{13}, this
problem may be easily solved. We consider here only the cases when
the measured observables are the basic observables that is
generators of the algebra $H_{1},H_{2},H_{3}$
Note that in view of relation ${\hat{H}_{i}}^{2}=1\left( \text{for all }
i=1,2,3\right) $ one can write a simple equation connecting two density
matrices,namely,density matrix ${\hat{\rho}_{0}}$ before the measurement and
the density matrix ${\hat{\rho}_{\infty }}$ after the measurement of the
observable ${\hat{H}_{i}}$. This equation reads as:
\begin{equation}
{\hat{\rho}_{\infty }}=\frac{{\hat{\rho}_{0}}+{\hat{H}_{i}}{\hat{\rho}_{0}}{
\hat{H}_{i}}}{2}. \label{14}
\end{equation}
Using Eq.\eqref{14} and the algebra of operators described above one can
consider separately three cases: the case I when the observable ${\hat{H}_{1}
}$ \ is measured and the initial state is ${\hat{\rho}_{0}}=\frac{1}{2}+b{
\hat{H}_{1}}+c{\hat{H}_{2}}+d{\hat{H}_{3}}$, the case II when the observable
${\hat{H}_{2}}$ is measured and the case III when observable ${\hat{H}_{3}}$
is measured. In the case I after the measurement the state of the system is:
${\hat{\rho}_{\infty }}^{I}=\frac{1}{2}+b{\hat{H}_{1}}+\frac{(c+d)}{2}\left(
{\hat{H}_{2}}+{\hat{H}_{3}}\right) $. Similarly in the case II the final
state is ${\hat{\rho}_{\infty }}^{II}=\frac{1}{2}+\frac{(b+d)}{2}\left( \hat{
H_{1}}+{\hat{H}_{3}}\right) +c{\hat{H}_{2}}$ and in the case III the final
stateof the system after measurement is ${\hat{\rho}_{\infty }}^{III}=\frac{1
}{2}+\frac{\left( b+c\right) }{2}\left( {\hat{H}_{1}}+{\hat{H}_{2}}\right) +d
{\hat{H}_{3}}$. Now we are interested in the maximum amount of entanglement
that can be extracted from initial state by these different measurements.
Let us calculate this maximum. Without loss of generality we may assume that
the initial state of the system is pure and can be represented by
parametrization \eqref{6}. Then the gain of entanglement caused by
measurement of ${\hat{H}_{1}}$ may be written as:
\begin{multline}
\Delta C_{I}=C\left\{ {\hat{\rho}_{\infty }}^{I}\right\} -C\left\{ {\hat{\rho
}_{0}}\right\} = \label{15} \\
=2\sqrt{\left( b+\frac{1}{2}\right) \left( \frac{c+d+1}{2}\right) }-2\sqrt{
\left( b+\frac{1}{2}\right) \left( c+\frac{1}{2}\right) }= \\
=\frac{1}{1+t+t^{2}}\left[ \sqrt{\frac{1+2t+2t^{2}}{2}}-\left\vert
t\right\vert \right] .
\end{multline}
Note that in the derivation of relation \eqref{15} we used the
parametrization \eqref{6} for coefficients $b,c,d$ of the initial pure
state. It is easy to see that maximum \eqref{15} is equal to $\frac{1}{\sqrt{
2}}$ and is reached when $t=0$. The required initial state in this case is $
\left\vert \Psi _{0}\right\rangle _{I}=\frac{1}{\sqrt{2}}
\begin{pmatrix}
1 \\
0 \\
1 \\
0
\end{pmatrix}
$. In the same way one can find that in the case II the gain of entanglement
can be represented as:
\begin{equation}
\Delta C_{II}=\frac{\left\vert t\right\vert }{1+t+t^{2}}\left[ \sqrt{\frac{
2+2t+t^{2}}{2}}-1\right] . \label{16}
\end{equation}
The maximum of \eqref{16} is reached when $t=\pm \infty $ and is equal to $
\frac{1}{\sqrt{2}}$ as well. The required initial state in this case is $
\left\vert \Psi _{0}\right\rangle _{II}=\frac{1}{\sqrt{2}}
\begin{pmatrix}
1 \\
\pm 1 \\
0 \\
0
\end{pmatrix}
$. However, in the case III, when the observable ${\hat{H}_{3}}$ is
measured, the result is somewhat distinct, namely, he gain of entanglement
caused by this measurement can be written as:
\begin{equation}
\Delta C_{III}=C\left\{ {\hat{\rho}_{\infty }}^{III}\right\} -C\left\{ {\hat{
\rho}_{0}}\right\} =\frac{1+t^{2}-2\left\vert t\right\vert }{2\left(
1+t+t^{2}\right) }. \label{17}
\end{equation}
It is easy to see that the maximum of \eqref{17} is reached when $t=0$ and
is equal to $\frac{1}{2}$. It is curious that, although optimal initial
states for cases I and III coincide, nevertheless, the extracted amount of
entanglement is larger in the first case. Now let us consider another
important and interesting feature of certain states belonging to the SQWs
that makes them potentially very helpful in various quantum informational
applications. We have in mind the existence of irreducible entangled states
both in SQW under examination and in many others SQWs as well. Really let us
consider the selected mixed state $IE$ (that is irreducible and entangled)
with density matrix ${\hat{\rho}}_{IE}=\frac{1}{2}-\frac{\hat{C}}{6}\equiv
\frac{1}{2}-\frac{\hat{H_{1}}+{\hat{H}_{2}}\text{+}{\hat{H}_{3}}}{6}$. It is
easy to see that this state belongs to the SQW because two necessary
conditions: $b+c+d=-\frac{1}{2}$ and $bc+bd+cd=\frac{1}{12}\leq \frac{1}{12}$
are satisfied. On the other hand, it is clear that this state is irreducible
(that is cannot be changed in time by any dynamical way or by means of
measurements) because the Casimir operator $\hat{C}={\hat{H}_{1}}+{\hat{H}
_{2}}+{\hat{H}_{3}}$ commutes with all generators of the algebra.
In addition note that this selected state is entangled with concurrence $
C_{IE}=\frac{2}{3}$. Thus we come to the conclusion that similar states ( in
the case when realization of the SQW would be possible) may be used as
long-lived keepers of entanglement stored in the system. The mentioned
feature of irreducible entangled states makes them indispensable for various
quantum informational applications. It should also be noted that although
the IE state is dynamically stable it can be achieved easily from other
states by means of appropriate measurements. For example if one takes the
initial state of the system in the form: ${\hat{\rho}_{0}}=\frac{1}{2}-\frac{
{\hat{H}_{1}}}{6}+c{\hat{H}_{2}}+d{\hat{H}_{3}}$ and then performs the
measurement of the observable $\hat{H_{1}}$ in this state then (if the
conditions: 1) $c+d=-\frac{1}{3}$ and 2) $cd\leq \frac{5}{36}$ hold true)
she(he) gets exactly the required IE state after the measurement.
Let us sum up the main results obtained in the present paper. We introduce
the notion of small quantum world (SQW) which allows one to study special
classes of composite quantum systems whose pure states and nonlocal
properties turn out to be simpler by far comparing with large quantum worlds
containing them as a small part. We are demonstrating that some features of
selected states belonging to these SQWs would be very helpful for performing
various quantum information tasks.
\end{document}
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\begin{document}
\title{Media Theory: Representations and Examples}
\author{Sergei~Ovchinnikov \\
Mathematics Department\\
San Francisco State University\\
San Francisco, CA 94132\\
[email protected]}
\date{\today}
\maketitle
\begin{abstract}
In this paper we develop a representational approach to media theory. We construct representations of media by well graded families of sets and partial cubes and establish the uniqueness of these representations. Two particular examples of media are also described in detail.
\noindent
\emph{Keywords:} Medium; Well graded family of sets; Partial cube
\end{abstract}
\section{Introduction}
A medium is a semigroup of transformations on a possibly infinite set of states, constrained by four axioms which are recalled in Section~2 of this paper. This concept was originally introduced by Jean--Claude Falmagne in his 1997 paper~\cite{jF97} as a model for the evolution of preferences of individuals (in a voting context, for example). As such it was applied to the analysis of opinion polls~\cite{mRjFbG99} (for closely related papers, see~\cite{jF96,jFjD97,jFmRbG97}). As shown by Falmagne and Ovchinnikov~\cite{jFsO02} and Doignon and Falmagne~\cite{jDjF97}, the concept of a medium provides an algebraic formulation for a variety of geometrical and combinatoric objects. The main theoretical developments so far can be found in~\cite{jF97,jFsO02,sOaD00} (see also Eppstein and Falmagne's paper~\cite{dE04} in this volume).
The purpose of this paper is to further develop our understanding of media. We focus in particular on representations of media by means of well graded families of sets and graphs. Our approach utilizes natural distance and betweenness structures of media, graphs, and families of sets. The main results of the paper show that, in some precise sense, any medium can be uniquely represented by a well graded family of sets or a partial cube. Two examples of infinite media are explored in detail in Sections~8 and~9.
\section{Preliminaries}
In this section we recall some definitions and theorems from~\cite{jF97}.
Let ${\cal S}$ be a set of \emph{states}. A \emph{token} (of information) is a function $\tau:{\cal S}\rightarrow{\cal S}$. We shall use the abbreviations $S\tau=\tau(S),$ and
$S\tau_1\cdots\tau_n=\tau_n[\ldots[\tau_1(S)]]$ for the function composition. We denote by $\tau_0$ the identity function on ${\cal S}$ and suppose that $\tau_0$ is not a token. Let ${\cal T}$ be a set of tokens on ${\cal S}$. The pair $({\cal S},{\cal T})$ is called a \emph{token system}. Two distinct states $S,T\in{\cal S}$ are \emph{adjacent} if $S\tau =T$ for some token $\tau\in{\cal T}$. To avoid trivialities, we assume that $|{\cal S}|>1$.
A token $\tau'$ is a \emph{reverse} of a token $\tau$ if for all distinct
$S,V\in{\cal S}$
$$
S\tau=V\quad\Leftrightarrow\quad V\tau'=S.
$$
A finite composition $\boldsymbol m=\tau_1\cdots\tau_n$ of not necessarily distinct tokens $\tau_1,\ldots,\tau_n$ such that $S\boldsymbol m=V$ is called a \emph{message producing} $V$ \emph{from} $S$. We write $\ell(\boldsymbol m)=n$ to denote the \emph{length} of $\boldsymbol m$. The \emph{content} of a message
$\boldsymbol m=\tau_1\ldots\tau_n$ is the set ${\cal C}(\boldsymbol m)$ of its distinct tokens. Thus, $|{\cal C}(\boldsymbol m)|\leq\ell(\boldsymbol m)$. A message $\boldsymbol m$ is \emph{effective} (resp. \emph{ineffective}) for a state $S$ if $S\boldsymbol m\neq S$ (resp. $S\boldsymbol m=S$). A message $\boldsymbol m=\tau_1\ldots\tau_n$ is \emph{stepwise effective} for $S$ if
$$
S\tau_1\ldots\tau_k\neq S\tau_0\ldots\tau_{k-1},\qquad 1\leq k\leq n.
$$
A message is called \emph{consistent} if it does not contain both a token and its reverse, and \emph{inconsistent} otherwise. A message which is both consistent and stepwise effective for some state $S$ is said to be \emph{straight} for $S$. A message $\boldsymbol m=\tau_1\ldots\tau_n$ is \emph{vacuous} if the set of indices $\{1,\ldots,n\}$ can be partitioned into pairs $\{i,j\},$ such that one of $\tau_i,\tau_j$ is a reverse of the other. Two messages $\boldsymbol m$ and $\boldsymbol n$ are \emph{jointly consistent} if $\boldsymbol{mn}$ (or, equivalently, $\boldsymbol{nm}$) is consistent.
The next definition introduces the main concept of media theory.
\begin{definition} \label{D:medium}
A token system is called a \emph{medium} if the following axioms are satisfied.
\begin{enumerate}
\item[]
\begin{enumerate}
\item[{\rm [M1]}] Every token $\tau$ has a unique reverse, which we denote by $\tilde{\tau}$.
\item[{\rm [M2]}] For any two distinct states $S,V,$ there is a consistent message transforming $S$ into $V$.
\item[{\rm [M3]}] A message which is stepwise effective for some state is ineffective for that state if and only if it is vacuous.
\item[{\rm [M4]}] Two straight messages producing the same state are jointly consistent.
\end{enumerate}
\end{enumerate}
\end{definition}
It is easy to verify that \lbrack M2\rbrack\ is equivalent to the following
axiom (cf.~\cite{jF97}, Theorem 1.7).
\begin{enumerate}
\item[]
\begin{enumerate}
\item[{\rm [M2*]}] For any two distinct states $S,V,$ there is a
straight message transforming $S$ into $V$.
\end{enumerate}
\end{enumerate}
We shall use this form of axiom [M2] in the paper.
Various properties of media have been established in~\cite{jF97}. First, we recall the concept of `content'.
\begin{definition}
Let $({\cal S},{\cal T})$ be a medium. For any state $S$, the \emph{content} of $S$ is the set $\widehat{S}$ of all tokens each of which is contained in at least one straight message producing $S$. The family
$\widehat{{\cal S}}=\{\widehat{S}\,|\,S\in{\cal S}\}$ is called the \emph{content family} of ${\cal S}$.
\end{definition}
The following theorems present results of theorems 1.14, 1.16, and 1.17 in~\cite{jF97}. For reader's convinience, we prove these results below.
\begin{theorem} \label{Theorem1.17-1}
For any token $\tau$ and any state $S$, we have either $\tau\in\widehat{S}$ or $\tilde{\tau}\in\widehat{S}$. Consequently, $|\widehat{S}|=|\widehat{V}|$ for any two states $S$ and $V$. {\rm (}$|A|$ stands for the cardinality of the set $A$.{\rm )}
\end{theorem}
\begin{proof}
Since $\tau$ is a token, there are two states $V$ and $W$ such that $W=V\tau$. By Axiom [M2*], there are straight messages $\boldsymbol m$ and $\boldsymbol n$ such that $S=V\boldsymbol m$ and $S=W\boldsymbol n$. By Axiom [M3], the message $\tau\boldsymbol n\widetilde{\boldsymbol m}$ is vacuous. Therefore, $\tilde{\tau}\in{\cal C}(\boldsymbol n)$ or $\tilde{\tau}\in{\cal C}(\widetilde{\boldsymbol m})$. It follows that $\tilde{\tau}\in\widehat{S}$ or $\tau\in\widehat{S}$. By Axiom [M4], we cannot have both $\tilde{\tau}\in\widehat{S}$ and $\tau\in\widehat{S}$.
\end{proof}
\begin{theorem} \label{Theorem1.16}
If $S$ and $V$ are two distinct states, with $S\boldsymbol m=V$ for some straight message $\boldsymbol m$, then \mbox{$\widehat{V}\setminus\widehat{S}={\cal C}(\boldsymbol m)$}.
\end{theorem}
\begin{proof}
If $\tau\in{\cal C}(\boldsymbol m)$, then $\tilde{\tau}\in{\cal C}(\widetilde{\boldsymbol m})$. Thus, $\tau\in\widehat{V}$ and $\tilde{\tau}\in\widehat{S}$. By Theorem~\ref{Theorem1.17-1}, the latter inclusion implies $\tau\notin\widehat{S}$. It follows that ${\cal C}(\boldsymbol m)\subseteq\widehat{V}\setminus\widehat{S}$.
If $\tau\in\widehat{V}\setminus\widehat{S}$, then there is a state $W$ and a straight message $\boldsymbol n$ such that $V=W\boldsymbol n$ and $\tau\in{\cal C}(\boldsymbol n)$. By Axiom [M2*], there is a straight message $\boldsymbol p$ such that $W=S\boldsymbol p$. By Axiom [M3], the message $\boldsymbol p\boldsymbol n\widetilde{\boldsymbol m}$ is vacuous, so $\tilde{\tau}\in{\cal C}(\boldsymbol p\boldsymbol n\widetilde{\boldsymbol m})$, that is, $\tau\in{\cal C}(\boldsymbol m\widetilde{\boldsymbol n}\widetilde{\boldsymbol p})$. But $\tau\notin{\cal C}(\widetilde{\boldsymbol p})$, since $S=W\widetilde{\boldsymbol p}$ and $\tau\notin\widehat{S}$, and $\tau\notin{\cal C}(\widetilde{\boldsymbol n})$, since $\tau\in{\cal C}(\boldsymbol n)$. Hence, $\tau\in{\cal C}(\boldsymbol m)$, that is, $\widehat{V}\setminus\widehat{S}\subseteq{\cal C}(\boldsymbol m)$.
\end{proof}
\begin{theorem} \label{Theorem1.17-2}
For any token $\tau$ and any state $S$, we have either $\tau\in\widehat{S}$ or $\tilde{\tau}\in\widehat{S}$. Moreover,
$$
S=V\quad\Leftrightarrow\quad\widehat{S}=\widehat{V}.
$$
\end{theorem}
\begin{proof}
Let $\widehat{S}=\widehat{V}$ and let $\boldsymbol m$ be a straigt message producing $V$ from $S$. By Theorem~\ref{Theorem1.16},
$$
\varnothing=\widehat{V}\setminus\widehat{S}={\cal C}(\boldsymbol m).
$$
Thus, $S=V$.
\end{proof}
\begin{theorem} \label{Theorem1.14}
Let $\boldsymbol m$ and $\boldsymbol n$ be two distinct straight messages transforming some state $S$. Then $S\boldsymbol m=S\boldsymbol n$ if and only if ${\cal C}(\boldsymbol m)={\cal C}(\boldsymbol n)$.
\end{theorem}
\begin{proof}
Suppose that $V=S\boldsymbol m=S\boldsymbol n$. By Theorem~\ref{Theorem1.16},
$$
{\cal C}(\boldsymbol m)=\widehat{V}\setminus\widehat{S}={\cal C}(\boldsymbol n).
$$
Suppose that ${\cal C}(\boldsymbol m)={\cal C}(\boldsymbol n)$ and let $V=S\boldsymbol m$ and $W=S\boldsymbol n$. By Theorem~\ref{Theorem1.16},
$$
\widehat{V}\Delta\widehat{S}={\cal C}(\boldsymbol m)\cup{\cal C}(\widetilde{\boldsymbol m})={\cal C}(\boldsymbol n)\cup{\cal C}(\widetilde{\boldsymbol n})=\widehat{W}\Delta\widehat{S},
$$
which implies $\widehat{V}=\widehat{W}$. By Theorem~\ref{Theorem1.17-2}, $V=W$.
\end{proof}
One particular class of media plays an important role in our constructions (cf.~\cite{jFsO02}).
\begin{definition} \label{D:complete-media}
A medium $({\cal S},{\cal T})$ is called \emph{complete} if for any state $S\in{\cal S}$ and token $\tau\in{\cal T}$, either $\tau$ or $\tilde{\tau}$ is effective on $S$.
\end{definition}
An important example of a complete medium is found below in Theorem~\ref{CompleteMedium}.
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\section{Isomorphisms, embeddings, and token subsystems}
The purpose of combinatorial media theory is to find and examine those properties of media that do not depend on a particular structure of individual states and tokens. For this purpose we introduce the concepts of embedding and isomorphism for token systems.
\begin{definition} \label{D:embedding}
Let $({\cal S},{\cal T})$ and $({\cal S}',{\cal T}')$ be two token systems. A pair $(\alpha,\beta)$ of one--to--one functions $\alpha:{\cal S}\rightarrow{\cal S}'$ and $\beta:{\cal T}\rightarrow{\cal T}'$ such that
$$
S\tau=T \quad \Leftrightarrow \quad \alpha\left(S\right)\beta\left(\tau\right)=\alpha\left(T\right)
$$
for all $S,T\in{\cal S}$, $\tau\in{\cal T}$ is called an \emph{embedding} of the token system $({\cal S},{\cal T})$ into the token system $({\cal S}',{\cal T}')$.
Token systems $({\cal S},{\cal T})$ and $({\cal S}',{\cal T}')$ are \emph{isomorphic} if there is an embedding $(\alpha,\beta)$ from $({\cal S},{\cal T})$ into $({\cal S}',{\cal T}')$ such that both $\alpha$ and $\beta$ are bijections.
\end{definition}
Clearly, if one of two isomorphic token systems is a medium, then the other one is also a medium.
A general remark is in order. If a token system $({\cal S},{\cal T})$ is a medium and $S\tau_1=S\tau_2\not=S$ for some state $S$, then, by axiom [M3], $\tau_1=\tau_2$. In particular, if $(\alpha,\beta)$ is an embedding of a medium into a medium, then $\beta(\tilde{\tau})=\widetilde{\beta(\tau)}$. We extend $\beta$ to the semigroup of messages by defining $\beta(\tau_1\cdots\tau_k)=\beta(\tau_1)\cdots\beta(\tau_k)$. Clearly, the image $\beta(\boldsymbol m)$ of a straight message $\boldsymbol m$ is a straight message.
Let $({\cal S},{\cal T})$ be a token system and ${\cal Q}$ be a subset of ${\cal S}$ consisting of more than two elements. The restriction of a token $\tau\in{\cal T}$ to ${\cal Q}$ is not necessarily a token on ${\cal Q}$. In order to construct a medium with the set of states ${\cal Q}$, we introduce the following concept.
\begin{definition} \label{D:restriction}
Let $({\cal S},{\cal T})$ be a token system, ${\cal Q}$ be a nonempty subset of ${\cal S}$, and
$\tau\in{\cal T}$. We define a \emph{reduction} of $\tau$ to ${\cal Q}$ by
$$
S\tau_{{\cal Q}} = \begin{cases}
S\tau & \text{if $S\tau\in{\cal Q}$,} \\
S & \text{if $S\tau\notin{\cal Q}$,}
\end{cases}
$$
for $S\in{\cal Q}$. A token system $({\cal Q},{\cal T}_{{\cal Q}})$ where
${\cal T}_{{\cal Q}}=\{\tau_{{\cal Q}}\}_{\tau\in{\cal T}} \setminus\{\tau_0\}$ is the set of all distinct reductions of tokens in ${\cal T}$ to ${\cal Q}$ different from the identity function $\tau_0$ on ${\cal Q}$, is said to be the \emph{reduction} of $({\cal S},{\cal T})$ to ${\cal Q}$.
We call $({\cal Q},{\cal T}_{{\cal Q}})$ a \emph{token subsystem} of $({\cal S},{\cal T})$. If both $({\cal S},{\cal T})$ and $({\cal Q},{\cal T}_{{\cal Q}})$ are media, we call
$({\cal Q},{\cal T}_{{\cal Q}})$ a \emph{submedium} of $({\cal S},{\cal T})$.
\end{definition}
A reduction of a medium is not necessarily a submedium of a given medium. Consider, for instance, the following medium:
{
}
\noindent
The set of tokens of the reduction of this medium to ${\cal Q}=\{P,R\}$ is empty. Thus this reduction is not a medium (Axiom [M2] is not satisfied).
The image $(\alpha({\cal S}),\beta({\cal T}))$ of a token system $({\cal S},{\cal T})$ under embedding $(\alpha,\beta):({\cal S},{\cal T})\rightarrow({\cal S}',{\cal T}')$ is not, generally speaking, the reduction of $({\cal S}',{\cal T}')$ to $\alpha({\cal S})$. Indeed, let ${\cal S}'={\cal S}$, and let ${\cal T}$ be a proper nonempty subset of ${\cal T}'$. Then the image of $({\cal S},{\cal T})$ under the identity embedding is not the reduction of $({\cal S},{\cal T}')$ to ${\cal S}$ (which is $({\cal S},{\cal T}')$ itself).
On the other hand, this is true in the case of media as the following proposition demonstrates.
\begin{proposition}
Let $(\alpha,\beta):({\cal S},{\cal T})\rightarrow({\cal S}',{\cal T}')$ be an embedding of a medium $({\cal S},{\cal T})$ into a medium $({\cal S}',{\cal T}')$. Then the reduction $(\alpha({\cal S}),{\cal T}'_{\alpha({\cal S})})$ is isomorphic to $({\cal S},{\cal T})$.
\end{proposition}
\begin{proof}
For $\tau\in{\cal T}$, we define $\beta'(\tau)=\beta(\tau)_{\alpha({\cal S})}$, the reduction of $\beta(\tau)$ to $\alpha({\cal S})$. Let $S\tau=T$ for $S\not=T$ in ${\cal S}$. Then $\alpha(S)\beta(\tau)=\alpha(T)$ for $\alpha(S)\not=\alpha(T)$ in $\alpha({\cal S})$. Hence, $\beta'$ maps ${\cal T}$ to ${\cal T}'_{\alpha({\cal S})}$.
Let us show that $(\alpha,\beta')$ is an isomorphism from $({\cal S},{\cal T})$ onto $(\alpha({\cal S}),{\cal T}'_{\alpha({\cal S})})$.
\begin{itemize}
\item[(i)] $\beta'$ is onto. Suppose $\tau'_{\alpha({\cal S})}\not=\tau_0$ for some $\tau'\in{\cal T}'$. Then there are $P\not= Q$ in ${\cal S}$ such that $\alpha(P)\tau'_{\alpha({\cal S})}=\alpha(P)\tau'=\alpha(Q)$. Let $Q=P\boldsymbol m$ where $\boldsymbol m$ is a straight message. We have
$$
\alpha(Q)=\alpha(P\boldsymbol m)=\alpha(P)\beta(\boldsymbol m)=\alpha(P)\tau',
$$
implying, by Theorem~\ref{Theorem1.16}, $\beta(\boldsymbol m)=\tau'$, since $\beta(\boldsymbol m)$ is a straight message. Hence, $\boldsymbol m=\tau$ for some $\tau\in{\cal T}$. Thus $\beta(\tau)=\tau'$, which implies
$$
\beta'(\tau)=\beta(\tau)_{\alpha({\cal S})}=\tau'_{\alpha({\cal S})}.
$$
\item[(ii)] $\beta'$ is one--to--one. Suppose $\beta'(\tau_1)=\beta'(\tau_2)$. Since $\beta'(\tau_1)$ and $\beta'(\tau_2)$ are tokens on $\alpha({\cal S})$ and $({\cal S}',{\cal T}')$ is a medium, we have $\beta(\tau_1)=\beta(\tau_2)$. Hence, $\tau_1=\tau_2$.
\item[(iii)]Finally,
$$
S\tau=T \quad \Leftrightarrow \quad \alpha\left(S\right)\beta'\left(\tau\right)=\alpha\left(T\right),
$$
since
$$
S\tau=T \quad \Leftrightarrow \quad \alpha\left(S\right)\beta\left(\tau\right)=\alpha\left(T\right).
$$
\end{itemize}
\end{proof}
We shall see later (Theorem~\ref{FiniteRepresentationTheorem}) that any medium is isomorphic to a submedium of a complete medium.
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\section{Families of sets representable as media}
In this section, the objects of our study are token systems which are defined by means of families of subsets of a given set $X$.
Let $\mathfrak P(X)=2^X$ be the family of all subsets of $X$ and let ${\cal G}=\cup\{\gamma_x,\tilde{\gamma}_x\}$ be the family of functions from $\mathfrak P(X)$ into $\mathfrak P(X)$ defined by
\begin{align*}
&\gamma_x:S\mapsto S\gamma_x = S\cup\{x\}, \\
&\tilde{\gamma}_x:S\mapsto S\tilde{\gamma}_x = S\setminus\{x\}
\end{align*}
for all $x\in X$.
It is clear that $(\mathfrak P(X),{\cal G})$ is a token system and that, for a given $x\in X$, the token $\tilde{\gamma}_x$ is the unique reverse of the token $\gamma_x$.
Let ${\cal F}$ be a nonempty family of subsets of $X$. In what follows, $({\cal F},{\cal G}_{\cal F})$ stands for the reduction of the token system $(\mathfrak P(X),{\cal G})$ to ${\cal F}\subseteq\mathfrak P(X)$.
In order to characterize token systems $({\cal F},{\cal G}_{\cal F})$ which are media, we introduce some geometric concepts in $\mathfrak P(X)$ (cf. \cite{kB73,vKsO75,sO83}).
\begin{definition} \label{D:line segment}
Given $P,Q\in \mathfrak P(X)$, the \emph{interval} $[P,Q]$ is defined by
\begin{equation*}
[P,Q]=\{R\in 2^X:P\cap Q\subseteq R\subseteq P\cup Q\}.
\end{equation*}
If $R\in[P,Q]$, we say that $R$ \emph{lies between $P$ and $Q$}.
A sequence $P=P_0,P_1,\ldots,P_n=Q$ of distinct elements of $\mathfrak P(X)$ is a \emph{line segment} between $P$ and $Q$ if
\begin{enumerate}
\item[\emph{L1.}] $P_i\in[P_k,P_m]$ for $k\leq i\leq m,$ and
\item[\emph{L2.}] $R\in[P_i,P_{i+1}]$ implies $R=P_i$ or $R=P_{i+1}$ for all $0\leq i\leq n-1$.
\end{enumerate}
The \emph{distance}
between $P$ and $Q$ is defined by
\begin{equation*}
d(P,Q) = \begin{cases}
|P\Delta Q|, &\text{if $P\Delta Q$ is a finite set,} \\
\infty, &\text{otherwise,}
\end{cases}
\end{equation*}
where $\Delta$ stands for the symmetric difference operation.
A binary relation $\sim$ on $\mathfrak P(X)$ is defined by
\begin{equation*}
P\sim Q\quad\Leftrightarrow\quad d(P,Q)<\infty.
\end{equation*}
The relation $\sim$ is an equivalence relation on $\mathfrak P(X)$. We denote $[S]$ the equivalence class of $\sim$ containing $S\in\mathfrak P(X)$. We also denote $\mathfrak PF(X)=[\varnothing]$, the family of all finite subsets of the set $X$.
\end{definition}
\begin{theorem} \label{DistanceTheorem}
Given $S\in \mathfrak P(X)$,
\begin{itemize}
\item[\emph{(i)}] The distance function $d$ defines a metric on $[S]$.
\item[\emph{(ii)}] $R$ lies between $P$ and $Q$ in $[S]$, that is, $R\in[P,Q],$ if and only if
\begin{equation*}
d(P,R)+d(R,Q)=d(P,Q).
\end{equation*}
\item[\emph{(iii)}] A sequence $P=P_0,P_1,\ldots,P_n=Q$ is a line segment between $P$ and $Q$ in $[S]$ if and only if $d(P,Q)=n$ and $d(P_i,P_{i+1})=1,\;0\leq i\leq n-1$.
\end{itemize}
\end{theorem}
\begin{proof}
(i) It is clear that $d(P,Q)\geq 0$ and $d(P,Q)=0$ if and only if $P=Q$, and that $d(P,Q)=d(Q,P)$ for all $P,Q\in\langle S\rangle$.
It remains to verify the triangle inequality. Let $S_1,S_2,S_3$ be three sets in $[S]$. Since these sets belong to the same equivalence class of the relation $\sim$, the following six sets
$$
V_i = (S_j\cap S_k)\setminus S_i,\;U_i = S_i\setminus(S_j\cup S_k)\quad\text{for $\{i,j,k\}=\{1,2,3\}$,}
$$
are finite. It is not difficult to verify that
$$
S_i\Delta S_j=U_i\cup V_j\cup U_j\cup V_i,
$$
with disjoint sets in the right hand side of the equality. We have
\begin{align} \label{so1 triangle id}
&\phantom{=(}|S_i\Delta S_j|+|S_j\Delta S_k|\notag\\
&=(|U_i|+|V_j|+|U_j|+|V_i|)+(|U_j|+|V_k|+|U_k|+|V_j|)\\
&=|S_i\Delta S_k|+2(|U_j|+|V_j|), \notag
\end{align}
which implies the triangle inequality.
(ii) By~(\ref{so1 triangle id}),
$$
|S_i\Delta S_j|+|S_j\Delta S_k|=|S_i\Delta S_k|
$$
if and only if $U_j=\varnothing$ and $V_j=\varnothing$, or, equivalently, if and only if
$$
S_i\cap S_k\subseteq S_j\subseteq S_i\cup S_j.
$$
(iii) (Necessity.) By Condition L2 of Definition~\ref{D:line segment},
$$
d(P_i,P_{i+1})=|P_i\Delta P_{i+1|}|=1.
$$
Indeed, if there were $x\in P_i\setminus P_{i+1}$ and $y\in P_{i+1}\setminus P_i$, then the set $R=(P_i\setminus\{x\})\cup\{y\}$ would lie strictly between $P_i$ and $P_{i+1}$, a contradiction. By Condition L1 of Definition~\ref{D:line segment} and part (ii) of the theorem, we have
$$
d(P,Q)=1+d(P_1,Q)=\cdots=\underbrace{1+1+\cdots+1}_{n}=n.
$$
(Sufficiency.) Let $P=P_0,P_1,\ldots,P_n=Q$ be a sequence of sets such that $d(P_i,P_{i+1})=1$ and $d(P,Q)=n$. Condition L2 of Definition~\ref{D:line segment} is clearly satisfied.
By the triangle inequality, we have
$$
d(P,P_i)\leq i,\;\;d(P_i,P_j)\leq j-i,\;\;d(P_j,Q)\leq n-j
$$
for $i<j$. Let us add these inequalities and use the triangle inequality again. We obtain
$$
n=d(P,Q)\leq d(P,P_i)+d(P_i,P_j)+d(P_j,Q)\leq n.
$$
It follows that $d(P_i,P_j)=j-i$ for all $i<j$. In particular,
$$
d(P_k,P_i)+d(P_i,P_m)=d(P_k,P_m)\quad\text{for $k\leq i\leq m$.}
$$
By part (ii) of the theorem, $P_i\in[P_k,P_m]$ for $k\leq i\leq m$, which proves Condition L1 of Definition~\ref{D:line segment}.
\end{proof}
The concept of a line segment seems to be similar to the concept of a straight message. The following lemma validates this intuition.
\begin{lemma} \label{StraightMessage}
Let $({\cal F},{\cal G}_{\cal F})$ be a token system and let $P$ and $Q$ be two distinct sets in ${\cal F}$. A message $\boldsymbol m=\tau_1\tau_2\cdots\tau_n$ producing $Q$ from $P$ is
straight if and only if
$$
P_0=P,\;P_1=P_0\tau_1,\;\ldots,\;P_n=P_{n-1}\tau_n=Q
$$
is a line segment between $P$ and $Q$.
\end{lemma}
\begin{proof}
(Necessity.) We use induction on $n=\ell(\boldsymbol m)$. Let $\boldsymbol m=\tau_1$. Since $\tau_1$ is either $\gamma_x$ or $\tilde{\gamma}_x$ for some $x\in X$ and effective, either $Q=P\cup\{x\}$ or $Q=P\setminus\{x\}$ and $Q\not= P$. Therefore, $d(P,Q)=1$, that is $\{P,Q\}$ is a line segment.
Now, let us assume that the statement holds for all straight messages $\boldsymbol n$ with $\ell(\boldsymbol n)=n-1$ and let $\boldsymbol m=\tau_1\tau_2\cdots\tau_n$ be a straight message producing $Q$ from $P$. Clearly, $\boldsymbol m_1=\tau_2\cdots\tau_n$ is a straight message producing $Q$ from $P_1=P\tau_1$ and $\ell(\boldsymbol m_1)=n-1$.
Suppose that $\tau_1=\gamma_x$ for some $x\in X$. Since $\boldsymbol m$ is stepwise effective, $x\notin P$. Since $\boldsymbol m$ is consistent, $x\in Q$. Therefore $P_1=P\tau_1=P\cup\{x\}\in [P,Q]$. Suppose that $\tau_1=\tilde{\gamma}_x$ for some $x\in X$. Since $\boldsymbol m$ is stepwise effective, $x\in P$. Since it is consistent, $x\notin Q$. Again, $P_1=P\setminus\{x\}\in[P,Q]$. In either case, $d(P,P_1)=1$. By Theorem~\ref{DistanceTheorem}(ii) and the induction hypothesis, $d(P,Q)=d(P,P_1)+d(P_1,Q)=n$. By the induction hypothesis and Theorem~\ref{DistanceTheorem}(iii), the sequence
\begin{equation*}
P_0=P,P_1=P_0\tau_1,\ldots,P_n=P_{n-1}\tau_n=Q
\end{equation*}
is a line segment between $P$ and $Q$.
(Sufficiency.) Let $P_0=P,P_1=P_0\tau_1,\ldots,P_n=P_{n-1}\tau_n=Q$ be a line segment between $P$ and $Q$ for some message $\boldsymbol m=\tau_1\cdots\tau_n$. Clearly, $\boldsymbol m$ is stepwise effective. To prove consistency, we use induction on $n$. The
statement is trivial for $n=1$. Suppose it holds for all messages of length less than $n$ and let $\boldsymbol m$ be a message of length $n$. Suppose $\boldsymbol m$ is inconsistent. By the induction hypothesis, this can occur only if either $\tau_1=\gamma_x$, $\tau_n=\tilde{\gamma}_x$ or $\tau_1=\tilde{\gamma}_x$, $\tau_n=\gamma_x$ for some $x\in X$. In the former case, $x\in P_1,\;x\notin P,\;x\notin Q$. In the latter case, $x\notin P_1,\;x\in P,\;x\in Q$. In both cases, $P_1\notin[P,Q]$, a
contradiction.
\end{proof}
The following theorem is an immediate consequence of Lemma~\ref{StraightMessage}.
\begin{theorem} \label{Fin[S]}
If $({\cal F},{\cal G}_{\cal F})$ is a medium, then ${\cal F}\subseteq [S]$ for some $S\in\mathfrak P(X)$.
\end{theorem}
Clearly, the converse of this theorem is not true. To characterize those token systems $({\cal F},{\cal G}_{\cal F})$ which are media, we use the concept of a well graded family of sets \cite{jDjF97}. (See also \cite{vKsO75,sO80} and \cite{sO83} where the same concept was introduced as a ``completeness condition''.)
\begin{definition}
A family ${\cal F}\subseteq\mathfrak P(X)$ is \emph{well graded} if for any two distinct sets $P$ and $Q$ in ${\cal F}$, there is a sequence of sets $P=R_0,R_1,\ldots,R_n=Q$ such that $d(R_{i-1},R_i)=1$ for $i=1,\ldots,n$ and $d(P,Q)=n$.
\end{definition}
In other words, ${\cal F}$ is a well graded family if for any two distinct elements $P,Q\in{\cal F}$ there is a line segment between $P$ and $Q$ in ${\cal F}$.
\begin{theorem} \label{T:wg gamma}
Let ${\cal F}$ be a well graded family of subsets of some set $X$. Then $x\in X$ defines tokens $\gamma_x,\tilde{\gamma}_x\in{\cal G}_{\cal F}$ if and only if $x\in\cup{\cal F}\setminus\cap{\cal F}$.
\end{theorem}
\begin{proof}
It is clear that elements of $X$ that are not in $\cup{\cal F}\setminus\cap{\cal F}$ do not define tokens in ${\cal G}_{\cal F}$.
Suppose that $x\in\cup{\cal F}\setminus\cap{\cal F}$. Then there are sets $P$ and $Q$ in ${\cal F}$ such that $x\in Q\setminus P$. Let $R_0,R_1,\ldots,R_n$ be a line segment in ${\cal F}$ between $P$ and $Q$. Then there is $i$ such that $R_{i+1}=R_i\cup\{x\}$ and $x\notin R_i$. Therefore, $R_{i+1}=R_i\gamma_x$ and $R_i=R_{i+1}\tilde{\gamma}_x$, that is, $\gamma_x,\tilde{\gamma}_x\in{\cal G}_{\cal F}$.
\end{proof}
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The following theorem is an immediate consequence of Lemma~\ref{StraightMessage}.
\begin{theorem} \label{Fin[S]}
If $({\cal F},{\cal G}_{\cal F})$ is a medium, then ${\cal F}\subseteq [S]$ for some $S\in\mathfrak P(X)$.
\end{theorem}
Clearly, the converse of this theorem is not true. To characterize those token systems $({\cal F},{\cal G}_{\cal F})$ which are media, we use the concept of a well graded family of sets \cite{jDjF97}. (See also \cite{vKsO75,sO80} and \cite{sO83} where the same concept was introduced as a ``completeness condition''.)
\begin{definition}
A family ${\cal F}\subseteq\mathfrak P(X)$ is \emph{well graded} if for any two distinct sets $P$ and $Q$ in ${\cal F}$, there is a sequence of sets $P=R_0,R_1,\ldots,R_n=Q$ such that $d(R_{i-1},R_i)=1$ for $i=1,\ldots,n$ and $d(P,Q)=n$.
\end{definition}
In other words, ${\cal F}$ is a well graded family if for any two distinct elements $P,Q\in{\cal F}$ there is a line segment between $P$ and $Q$ in ${\cal F}$.
\begin{theorem} \label{T:wg gamma}
Let ${\cal F}$ be a well graded family of subsets of some set $X$. Then $x\in X$ defines tokens $\gamma_x,\tilde{\gamma}_x\in{\cal G}_{\cal F}$ if and only if $x\in\cup{\cal F}\setminus\cap{\cal F}$.
\end{theorem}
\begin{proof}
It is clear that elements of $X$ that are not in $\cup{\cal F}\setminus\cap{\cal F}$ do not define tokens in ${\cal G}_{\cal F}$.
Suppose that $x\in\cup{\cal F}\setminus\cap{\cal F}$. Then there are sets $P$ and $Q$ in ${\cal F}$ such that $x\in Q\setminus P$. Let $R_0,R_1,\ldots,R_n$ be a line segment in ${\cal F}$ between $P$ and $Q$. Then there is $i$ such that $R_{i+1}=R_i\cup\{x\}$ and $x\notin R_i$. Therefore, $R_{i+1}=R_i\gamma_x$ and $R_i=R_{i+1}\tilde{\gamma}_x$, that is, $\gamma_x,\tilde{\gamma}_x\in{\cal G}_{\cal F}$.
\end{proof}
In the rest of the paper we assume that a well graded family ${\cal F}$ of subsets of $X$ defining a token system $({\cal F},{\cal G}_{\cal F})$ satisfies the following conditions:
\begin{equation} \label{E:wg gamma}
\cap{\cal F}=\varnothing\quad\text{and}\quad\cup{\cal F}=X.
\end{equation}
We have the following theorem (cf.~Theorem~4.2 in~\cite{sOaD00}).
\begin{theorem} \label{WellGradedMedia}
A token system $({\cal F},{\cal G}_{\cal F})$ is a medium if and only if ${\cal F}$ is a well graded family of subsets of $X$.
\end{theorem}
\begin{proof}
(Necessity.) Suppose $({\cal F},{\cal G}_{\cal F})$ is a medium. By axiom [M2*], for given $P,Q\in{\cal F}$, there is a straight message producing $Q$ from $P$. By Lemma~\ref{StraightMessage}, there is a line segment in ${\cal F}$ between $P$ and $Q$. Hence ${\cal F}$ is well graded.
(Sufficiency.) Let ${\cal F}$ be a well graded family of subsets of $X$. We need to show that the four axioms defining a medium are satisfied for $({\cal F},{\cal G}_{\cal F})$.
[M1]. Clearly, $\gamma_x$ and $\tilde{\gamma}_x$ are unique mutual reverses of each other.
[M2*]. Follows immediately from Lemma~\ref{StraightMessage}.
[M3]. (Necessity.) Let $\boldsymbol m$ be a message which is stepwise effective for $P\in{\cal F}$ and ineffective for this state, that is, $P\boldsymbol m=P$.
Let $\tau$ be a token in $\boldsymbol m$ such that $\tilde\tau\notin\boldsymbol m$. If $\tau=\gamma_x$ for some $x\in X$, then $x\notin P$ and $x\in P\boldsymbol m$, since $\boldsymbol m$ is stepwise effective for $P$ and $\tilde\gamma_x=\tilde\tau\notin\boldsymbol m$. We have a contradiction, since $P\boldsymbol m=P$. In a similar way, we obtain a contradiction assuming that $\tau=\tilde\gamma_x$. Thus, for each token $\tau$ in $\boldsymbol m$, there is an appearance of the reverse token $\tilde\tau$ in $\boldsymbol m$.
Because $\boldsymbol m$ is stepwise effective, the appearances of tokens $\tau$ and $\tilde{\tau}$ in $\boldsymbol m$ must alternate. Suppose that the sequence of appearances of $\tau$ and $\tilde{\tau}$ begins and ends with $\tau=\gamma_x$ (the argument is similar if $\tau=\tilde{\gamma}_x$). Since the message $\boldsymbol m$ is stepwise effective for $P$ and ineffective for this state, we must have $x\notin P$ and $x\in P\boldsymbol m=P$, a contradiction. It follows that $\boldsymbol m$ is vacuous.
(Sufficiency.) Let $\boldsymbol m$ be a vacuous message which is stepwise effective for some state $P$. Since $\boldsymbol m$ is vacuous, the number of appearances of $\gamma_x$ in $\boldsymbol m$ is equal to the number of appearances of $\tilde\gamma_x$ for any $x\in X$. Because $\boldsymbol m$ is stepwise effective, the appearances of tokens $\gamma_x$ and $\tilde{\gamma}_x$ in $\boldsymbol m$ must alternate. It follows that $x\in P$ if and only if $x\in P\boldsymbol m$, that is $P\boldsymbol m=P$. Thus the message $\boldsymbol m$ is ineffective for $P$.
[M4]. Suppose two straight messages $\boldsymbol m$ and $\boldsymbol n$ produce $R$ from $P$ and $Q$, respectively, that is, $R=P\boldsymbol m$ and $R=Q\boldsymbol n$. Let us assume that $\boldsymbol m$ and $\boldsymbol n$ are not jointly consistent, that is, that $\boldsymbol{mn}$ is inconsistent. Then there are two mutually reverse tokens $\tau$ and $\tilde{\tau}$ in $\boldsymbol{mn}$. Since $\boldsymbol m$ and $\boldsymbol n$ are straight messages, we may assume, without loss of generality, that $\tau=\gamma_x$ is in $\boldsymbol m$ and $\tilde{\tau}=\tilde{\gamma}_x$ is in $\boldsymbol n$ for some $x\in X$. Since $\boldsymbol m$ is straight, $x\in R$. Since $\boldsymbol n$ is straight, $x\notin R$, a contradiction.
\end{proof}
Clearly, for a given $S\in\mathfrak P(X)$, $[S]$ is a well graded family of subsets of $X$. Hence, $([S],{\cal G}_{[S]})$ is a medium. It is easy to see that any such medium is a complete medium (see Definition~\ref{D:complete-media}). The converse is also true as the following theorem asserts.
\begin{theorem} \label{CompleteMedium}
A medium in the form $({\cal F},{\cal G}_{\cal F})$ is complete if and only if ${\cal F}=[S]$ for some $S\in\mathfrak P(X)$.
\end{theorem}
\begin{proof}
We need to prove necessity only. Suppose that $({\cal F},{\cal G}_{\cal F})$ is a complete medium. By Theorem~\ref{Fin[S]}, ${\cal F}\subseteq[S]$ for some $S\in\mathfrak P(X)$. For a given $P\in[S]$, let $m=d(P,S)$. We prove that $P\in{\cal F}$ by induction on $m$.
Let $m=1$. Then either $P=S\cup\{x\}$ or $P=S\setminus\{x\}$ for some $x\in X$ and $P\not= S$. In the former case, $x\notin S$ implying that $\tilde{\gamma}_x$ is not effective on $S$. By completeness, $S\gamma_x= P$. Thus $P\in{\cal F}$. Similarly, if $P=S\setminus\{x\}$, then $x\in S$ and $S\tilde{\gamma}_x= P$.
Suppose that $Q\in{\cal F}$ for all $Q\in[S]$ such that $d(S,Q)=m$ and let $P$ be an element in $[S]$ such that $d(S,P)=m+1$. Then there exists $R\in{\cal F}$ such that $d(S,R)=m$ and $d(R,P)=1$. Since $[R]=[S]$, it follows from the argument in the previous paragraph that $P\in{\cal F}$.
\end{proof}
The following theorem shows that all media in the form $([S],{\cal G}_{[S]})$ are isomorphic.
\begin{theorem} \label{IsomorphismTheorem}
For any $S',S\in\mathfrak P(X)$, the media $([S'],{\cal G}_{[S']})$ and $([S],{\cal G}_{[S]})$ are isomorphic.
\end{theorem}
\begin{proof}
It suffices to consider the case when $S'=\varnothing$.
We define $\alpha:\mathfrak PF(X)\rightarrow[S]$ and $\beta:{\cal G}_{\mathfrak PF(X)}\rightarrow{\cal G}_{[S]}$ by
\begin{align*}
&\alpha(P)=P\Delta S, \\
&\beta(\tau)= \begin{cases}
\tilde{\tau} &\text{if $\tau=\gamma_x$ or $\tau=\tilde{\gamma}_x$ for $x\in S$,} \\
\tau &\text{if $\tau=\gamma_x$ or $\tau=\tilde{\gamma}_x$ for $x\notin S$.}
\end{cases}
\end{align*}
Clearly, $\alpha$ and $\beta$ are bijections. To prove that $P\tau=Q$ implies $\alpha(P)\beta(\tau)=\alpha(Q)$, let us consider the following cases:
\begin{enumerate}
\item $\tau=\gamma_x,\;x\in S$.
\begin{enumerate}
\item $x\in P$. Then $Q=P$ and
$$\alpha(P)\beta(\tau) = (P\Delta S)\setminus\{x\} = P\Delta S = Q\Delta S.$$
\item $x\notin P$. Then $Q=P\cup\{x\}$ and
$$\alpha(P)\beta(\tau) = (P\Delta S)\setminus\{x\} = P\Delta(S\setminus\{x\}) = Q\Delta S.$$
\end{enumerate}
\item $\tau=\gamma_x,\;x\notin S$.
\begin{enumerate}
\item $x\in P$. Then $Q=P$ and
$$\alpha(P)\beta(\tau) = (P\Delta S)\cup\{x\} = P\Delta S = Q\Delta S.$$
\item $x\notin P$. Then $Q=P\cup\{x\}$ and
$$\alpha(P)\beta(\tau) = (P\Delta S)\cup\{x\} = Q\Delta S.$$
\end{enumerate}
\end{enumerate}
A similar argument proves the theorem in the case when $\tau=\tilde{\gamma}_x$. It is also easy to verify the converse implication: $\alpha(P)\beta(\tau)=\alpha(Q)\;\Rightarrow\; P\tau=Q$.
\end{proof}
We summarize the results of this section as follows:
1.~A token system $({\cal F},{\cal G}_{\cal F})$ is a medium if and only if ${\cal F}$ is a well graded family of subsets of $X$.
2.~A complete medium in the form $({\cal F},{\cal G}_{\cal F})$ is $([S],{\cal G}_{[S]})$ for some $S\in\mathfrak P(X)$ and all such media are isomorphic.
3.~Any medium in the form $({\cal F},{\cal G}_{\cal F})$ is a submedium of a complete medium and isomorphic to a submedium of the complete medium $(\mathfrak PF(X),{\cal G}_{\mathfrak PF(X)})$ of all finite subsets of $X$.
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\section{The representation theorem}
In this section we show that any medium is isomorphic to a medium in the form $({\cal F},{\cal G}_{\cal F})$ where ${\cal F}$ is a well graded family of finite subsets of some set $X$. In our construction we employ the concept of `orientation'~\cite{jFsO02}.
\begin{definition} \label{D:orientation}
An \emph{orientation} of a medium
$({\cal S},{\cal T})$ is a partition of its set of tokens into two classes ${\cal T}^{+}$ and ${\cal T}^{-}$, respectively called \emph{positive} and \emph{negative}, such that for any $\tau\in{\cal T}$, we have
$$
\tau\in{\cal T}^{+}\quad\Leftrightarrow\quad\tilde{\tau}\in{\cal T}^{-}
$$
A medium $\left({\cal S},{\cal T}\right)$ equipped with an orientation $\{{\cal T}^{+},{\cal T}^{-}\}$ is said to be \emph{oriented} by $\{{\cal T}^{+},{\cal T}^{-}\}$, and tokens from ${\cal T}^{+}$
\emph{(}resp. ${\cal T}^{-}$\emph{)} are called \emph{positive} \emph{(}resp. \emph{negative}\emph{)}. The \emph{positive content} \emph{(}resp.~\emph{negative content}\emph{)} of a state $S$ is the set $\widehat{S}^{+}=\widehat{S}\cap{\cal T}^{+}$ \emph{(}resp. $\widehat{S}^{-}=\widehat{S}\cap{\cal T}^{-}$\emph{)} of its positive \emph{(}resp. negative\emph{)} tokens.
\end{definition}
Let $({\cal S},{\cal T})$ be a medium equipped with an orientation $\{{\cal T}^{+},{\cal T}^{-}\}$. In what follows, we show that $({\cal S},{\cal T})$ is isomorphic to the medium $({\cal F},{\cal G}_{\cal F})$ of the well graded family ${\cal F}=\{\widehat{S}^+\}_{S\in{\cal S}}$ of all positive contents.
\begin{lemma} \label{S+=T+}
\emph{(cf.~\cite{sOaD00})} For any two states $S,T\in{\cal S}$,
\begin{equation*}
\widehat{S}^+=\widehat{T}^+\quad\Leftrightarrow\quad S=T.
\end{equation*}
\end{lemma}
\begin{proof}
By Theorem~\ref{Theorem1.17-2}, it suffices to prove that $\widehat{S}^+=\widehat{T}^+$ implies $\widehat{S}=\widehat{T}$. Let $\tau\in\widehat{S}^-$. Then, by Theorem~\ref{Theorem1.17-1}, $\tilde{\tau}\notin\widehat{S}^+$. Hence, $\tilde{\tau}\notin\widehat{T}^+$ which implies $\tau\in\widehat{T}^-$. Therefore $\widehat{S}^-\subseteq\widehat{T}^-$. By symmetry, $\widehat{S}^-=\widehat{T}^-$. Hence, $\widehat{S}=\widehat{T}$.
\end{proof}
We define
\begin{equation} \label{alpha}
\alpha : S\mapsto \widehat{S}^+,
\end{equation}
for $S\in{\cal S}$. It follows from Lemma~\ref{S+=T+} that $\alpha$ is a bijection.
Suppose that $\tau\in\cap\,{\cal F}=\cap_{S\in{\cal S}}\widehat{S}^+$. There are $S,T\in{\cal S}$ such that $T=S\tau$. Then, by Theorem~\ref{Theorem1.16}, $\widehat{T}\setminus\widehat{S}=\{\tau\}$, that is, $\tau\notin\widehat{S}\supseteq\widehat{S}^+$. Hence, $\cap\,{\cal F}=\varnothing$.
Let $\tau\in{\cal T}^+$. There are $S,T\in{\cal S}$ such that $T=S\tau$. Then $\tau\in\widehat{T}^+$. Hence, $\cup\,{\cal F}={\cal T}^+$.
We define a mapping $\beta:{\cal T}\rightarrow{\cal G}_{\cal F}$ by
\begin{equation} \label{beta}
\beta(\tau)=\begin{cases}
\gamma_\tau &\text{if $\tau\in{\cal T}^+$,} \\
\tilde{\gamma}_{\tilde{\tau}} &\text{if $\tau\in{\cal T}^-$,}
\end{cases}
\end{equation}
and show that the pair $(\alpha,\beta)$, where $\alpha$ and $\beta$ are mappings defined, respectively, by (\ref{alpha}) and (\ref{beta}), is an isomorphism between the token systems $({\cal S},{\cal T})$ and $({\cal F},{\cal G}_{\cal F})$.
\begin{theorem} \label{RepresentationTheorem}
Let $({\cal S},{\cal T})$ be an oriented medium. For all $S,T\in{\cal S}$ and $\tau\in{\cal T}$,
\begin{equation*}
T=S\tau\quad\Leftrightarrow\quad\alpha(T)=\alpha(S)\beta(\tau),
\end{equation*}
that is, the token systems $({\cal S},{\cal T})$ and $({\cal F},{\cal G}_{\cal F})$ are isomorphic.
\end{theorem}
\begin{proof}
(i) Suppose that $\tau\in{\cal T}^+$. We need to prove that
\begin{equation*}
T=S\tau\quad\Leftrightarrow\quad\widehat{T}^+=\widehat{S}^+\gamma_\tau
\end{equation*}
for all $S,T\in{\cal S}$ and $\tau\in{\cal T}$.
Let us consider the following cases:
\begin{enumerate}
\item $\tau\in\widehat{S}^+$. Suppose $S\tau=T\not=S$. Then, by Theorem~\ref{Theorem1.16}, $\tau\in\widehat{T}\setminus\widehat{S}$, a contradiction. Hence, $T=S$. Clearly,
$\widehat{S}^+=\widehat{S}^+\cup\{\tau\}=\widehat{S}^+\gamma_\tau$.
\item $\tau\notin\widehat{S}^+,\,\widehat{S}^+\cup\{\tau\}\notin{\cal F}$. Suppose that $S\tau=T\not=S$. By Theorem~\ref{Theorem1.16}, $\widehat{T}\setminus\widehat{S}=\{\tau\}$ and $\widehat{S}\setminus\widehat{T}=\{\tilde{\tau}\}$. Since $\tau\in{\cal T}^+$, we have $\widehat{S}^+\cup\{\tau\}=\widehat{T}^+\in{\cal F}$, a contradiction. Hence, in this case, $S=S\tau$ and $\widehat{S}^+=\widehat{S}^+\gamma_\tau$.
\item $\tau\notin\widehat{S}^+,\,\widehat{S}^+\cup\{\tau\}\in{\cal F}$. Then there exists $T\in{\cal S}$ such that $\widehat{T}^+=\widehat{S}^+\cup\{\tau\}$. Thus $\tau\in\widehat{T}\setminus\widehat{S}$. Suppose that there is $\tau'\not=\tau$ which is also in $\widehat{T}\setminus\widehat{S}$. Then $\tau'$ is a negative token. Since $\tau'\notin\widehat{S}$, we have $\tilde{\tau}'\in\widehat{S}^+\subset\widehat{T}^+\subseteq\widehat{T}$. Hence, $\tau'\notin\widehat{T}$, a contradiction. It follows that $\widehat{T}\setminus\widehat{S}=\{\tau\}$. By Theorem~\ref{Theorem1.16}, $T=S\tau$. By the argument in item 2, $\widehat{T}^+=\widehat{S}^+\cup\{\tau\}=\widehat{S}^+\gamma_\tau$.
\end{enumerate}
(ii) Suppose that $\tau\in{\cal T}^-$. Then $\tilde{\tau}\in{\cal T}^+$. We need to prove that
$$
T=S\tau\quad \Leftrightarrow\quad\widehat{T}^+=\widehat{S}^+\tilde{\gamma}_{\tilde{\tau}}
$$
for all $S,T\in{\cal S}$ and $\tau\in{\cal T}$.
Let us consider the following cases:
\begin{enumerate}
\item $\tilde{\tau}\notin\widehat{S}^+$. Suppose that $S\tau=T\not=S$. Then $S=T\tilde{\tau}$ and, by Theorem~\ref{Theorem1.16}, $\tilde{\tau}\in\widehat{S}\setminus\widehat{T}$, a contradiction since $\tilde{\tau}$ is a positive token. Hence, $S=S\tau$. On the other hand, $\widehat{S}^+=\widehat{S}^+\setminus\{\tilde{\tau}\}=\widehat{S}^+\tilde{\gamma}_{\tilde{\tau}}$.
\item $\tilde{\tau}\in\widehat{S}^+,\,\widehat{S}^+\setminus\{\tilde{\tau}\}\notin{\cal F}$. Suppose again that $S\tau=T\not=S$. Then $S=T\tilde{\tau}$ and, by Theorem~\ref{Theorem1.16}, $\{\tilde{\tau}\}=\widehat{S}\setminus\widehat{T}$ and $\{\tau\}=\widehat{T}\setminus\widehat{S}$. Since $\tilde{\tau}$ is a positive token, we have $\widehat{S}^+\setminus\{\tilde{\tau}\}=\widehat{T}^+$, a contradiction. Hence, in this case, $S=S\tau$ and $\widehat{S}^+=\widehat{S}^+\tilde{\gamma}_{\tilde{\tau}}$.
\item $\tilde{\tau}\in\widehat{S}^+,\,\widehat{S}^+\setminus\{\tilde{\tau}\}\in{\cal F}$. There is $T\in{\cal S}$ such that $\widehat{T}^+=\widehat{S}^+\setminus\{\tilde{\tau}\}$. We have $\tau\notin\widehat{S}$, since $\tilde{\tau}\in\widehat{S}^+$, and $\tau\in\widehat{T}$, since $\tilde{\tau}\notin\widehat{T}^+$ and $\tilde{\tau}$ is a positive token. Hence, $\tau\in\widehat{T}\setminus\widehat{S}$. Suppose that there is $\tau'\not=\tau$ which is also in $\widehat{T}\setminus\widehat{S}$. Then $\tau'$ is a negative token. Since $\tau'\notin\widehat{S}$, we have $\tilde{\tau}'\in\widehat{S}^+$. Since $\tau'\not=\tau$, we have $\tilde{\tau}'\in\widehat{T}^+$. Hence, $\tau'\notin\widehat{T}$, a contradiction. It follows that $\widehat{T}\setminus\widehat{S}=\{\tau\}$. By Theorem~\ref{Theorem1.16}, $T=S\tau$. Clearly, $\widehat{T}^+=\widehat{S}^+\setminus\{\tau\}=\widehat{S}^+\tilde{\gamma}_{\tilde{\tau}}$.
\end{enumerate}
\end{proof}
Since $({\cal S},{\cal T})$ is a medium, the token system $({\cal F},{\cal G}_{\cal F})$ is also a medium. By Theorem~\ref{WellGradedMedia}, we have the following result.
\begin{corollary}
${\cal F}=\{\widehat{S}^+\}_{S\in{\cal S}}$ is a well graded family of subsets of ${\cal T}^+$.
\end{corollary}
Theorem~\ref{RepresentationTheorem} states that an oriented medium is isomorphic to the medium of its family of positive contents. By `forgetting' the orientation, one can say that any medium $({\cal S},{\cal T})$ is isomorphic to some medium $({\cal F},{\cal G}_{\cal F})$ of a well graded family ${\cal F}$ of sets. The following theorem is a stronger version of this result (cf.~Theorem~\ref{IsomorphismTheorem}).
\begin{theorem} \label{FiniteRepresentationTheorem}
Any medium $({\cal S},{\cal T})$ is isomorphic to a medium of a well graded family of finite subsets of some set $X$.
\end{theorem}
\begin{proof}
Let $S_0$ be a fixed state in ${\cal S}$. By Theorem~\ref{Theorem1.17-1}, the state $S_0$ defines an orientation $\{{\cal T}^{+},{\cal T}^{-}\}$ with ${\cal T}^-=\widehat{S}_0$ and ${\cal T}^{+}={\cal T}\setminus{\cal T}^-$. Let $\boldsymbol m$ be a straight message producing a state $S$ from the state $S_0$. By Theorem~\ref{Theorem1.16}, we have ${\cal C}(\boldsymbol m)=\widehat{S}\setminus\widehat{S}_0=\widehat{S}^+$. Thus, $\widehat{S}^+$ is a finite set. The statement of the theorem follows from Theorem~\ref{RepresentationTheorem}.
\end{proof}
\begin{remark}
{\rm An infinite oriented medium may have infinite positive contents of all its states. Consider, for instance, the medium $({\cal F},{\cal G}_{\cal F})$, where
$$
{\cal F}=\{S_n=]-\infty,n]:n\in\mathbb{Z}\}.
$$
Then each $\widehat{S}_n^+=\{\gamma_k\}_{k<n}$ is an infinite set. Nevertheless, by Theorem~\ref{FiniteRepresentationTheorem}, the medium $({\cal F},{\cal G}_{\cal F})$ is isomorphic to a medium of a well graded family of finite sets.
}
\end{remark}
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\section{Media and graphs}
In this section we study connections between media and graph theories.
\begin{definition}
Let $({\cal S},{\cal T})$ be a medium. We say that a graph $G=(V,E)$ \emph{represents} $({\cal S},{\cal T})$ if there is a bijection $\alpha:{\cal S}\rightarrow V$ such that an unordered pair of vertices $PQ$ is an edge of the graph if and only if $P\not=Q$ (no loops) and there is $\tau\in{\cal T}$ such that $\alpha^{-1}(P)\tau=\alpha^{-1}(Q)$.
A graph $G$ representing a medium $({\cal S},{\cal T})$ is the \emph{graph of the medium} $({\cal S},{\cal T})$ if the set of states ${\cal S}$ is the set of vertices of $G$, the mapping $\alpha$ is the identity, and the edges of $G$ are defined as above.
\end{definition}
Clearly, any graph which is isomorphic to a graph representing a token system $({\cal S},{\cal T})$, also represents $({\cal S},{\cal T})$, and isomorphic media are represented by isomorphic graphs.
The main goal of this section is to show that two media which are represented by isomorphic graphs are isomorphic (Theorem~\ref{Medium=Graph}).
First, we prove three lemmas about media and their graph representations.
\begin{lemma} \label{Lemma6.1}
Let $(\mathcal{S,T})$ be a medium, and suppose that: $\tau\in{\cal T}$ is a token; $S,T,P$ and $Q$ are states in ${\cal S}$ such that $S\tau=T$ and $P\tau=Q$. Let $\boldsymbol m$ and $\boldsymbol n$ be two straight messages producing $P$ from $S$ and $Q$ from $T$, respectively. Then $\boldsymbol m$ and $\boldsymbol n$ have equal contents and lengths, that is ${\cal C}(\boldsymbol m)={\cal C}(\boldsymbol n)$ and $\ell(\boldsymbol m)=\ell(\boldsymbol n)$, and $\boldsymbol m\tau$ and $\tau\boldsymbol n$ are straight messages for $S$.
\end{lemma}
\begin{proof}
The message $\boldsymbol m\tau\tilde{\boldsymbol n}\tilde{\tau}$ is stepwise effective for $S$ and ineffective for that state. By Axiom [M3], this message is vacuous. Hence, ${\cal C}(\boldsymbol m)={\cal C}(\boldsymbol n)$ and $\ell(\boldsymbol m)=\ell(\boldsymbol n)$.
Each of two straight messages $\boldsymbol m$ and $\tilde{\tau}$ produces $P$. By Axiom [M4], they are jointly consistent, that is, $\tau\notin{\cal C}(\boldsymbol m)$. Hence, $\boldsymbol m\tau$ is a straight message. Similarly, $\tau\boldsymbol n$ is a straight message.
\end{proof}
\begin{lemma} \label{Lemma6.2}
Let $S,T,P,Q$ be four distinct states such that $S\tau_1=T$, $P\tau_2=Q$ and $\boldsymbol m$ and $\boldsymbol n$ be two straight messages producing $P$ from $S$ and $Q$ from $T$, respectively. If the messages $\tau_1\boldsymbol n,\boldsymbol m\tau_2$, and $\tilde{\tau}_1\boldsymbol m$ are straight, then $\tau_1=\tau_2$.
\end{lemma}
\begin{proof}
Suppose that $\tau_1\not=\tau_2$. By Theorem~\ref{Theorem1.14}, ${\cal C}(\tau_1\boldsymbol n)={\cal C}(\boldsymbol m\tau_2)$. Hence, $\tau_1\in{\cal C}(\boldsymbol m)$, a contradiction, since we assumed that $\tilde{\tau}_1\boldsymbol m$ is a straight message.
\end{proof}
\begin{lemma} \label{Lemma6.3}
Let $({\cal S},{\cal T})$ be a medium, $G=(V,E)$ be a graph representing this medium, and $\alpha$ be the bijection ${\cal S}\rightarrow V$ defining the graph $G$. If $\boldsymbol m=\tau_1\cdots\tau_m$ is a straight message transforming a state $S$ into a state $T$, then the sequence of vertices $(\alpha(S_i))_{0\leq i\leq m}$, where $S_i=S\tau_0\tau_1\cdots\tau_i$, forms a shortest path joining $\alpha(S)$ and $\alpha(T)$ in $G$. Conversely, if a sequence $(\alpha(S_i))_{0\leq i\leq m}$ is a shortest path connecting $\alpha(S_0)=\alpha(S)$ and $\alpha(S_m)=\alpha(T)$, then $S\boldsymbol m= T$ for some straight message $\boldsymbol m$ of length $m$.
\end{lemma}
\begin{proof}
(Necessity.) Let $\alpha(P_0)=\alpha(S),\alpha(P_1),\ldots,\alpha(P_n)=\alpha(T)$ be a path in $G$ joining $\alpha(S)$ and $\alpha(T)$. There is a stepwise effective message $\boldsymbol n=\rho_1\cdots\rho_n$ such that $P_i=T\rho_1\cdots\rho_{n-i}$ for $0\leq i<n$. The message $\boldsymbol{mn}$ is stepwise effective for $S$ and ineffective for this state. By Axiom [M3], this message is vacuous. Since $\boldsymbol m$ is a straight message for $S$, we have $\ell(\boldsymbol m)\leq\ell(\boldsymbol n)$. It follows that $(\alpha(S_i))_{0\leq i\leq m}$ forms a shortest path joining $\alpha(S)$ and $\alpha(T)$ in $G$.
(Sufficiency.) Let $\alpha(S_0)=\alpha(S),\alpha(S_1),\ldots,\alpha(S_m)=\alpha(T)$ be a shortest path connecting vertices $\alpha(S)$ and $\alpha(T)$ in $G$. Then $S_i\tau_{i+1}=S_{i+1}$ for some tokens $\tau_i$, $1\leq i\leq m$. The message $\boldsymbol m=\tau_1\cdots\tau_m$ transforms the state $S$ into the state $T$. By the argument in the necessity part of the proof, $\boldsymbol m$ is a straight message for $S$.
\end{proof}
\begin{definition}\label{so1 cube and partial cube}
{\rm (cf.~\cite{dD73,wI00})} Let $X$ be a set. The graph ${\cal H}(X)$ is defined as follows: the set of vertices is the set $\mathfrak PF(X)$ of all finite subsets of $X$; two vertices $P$ and $Q$ are adjacent if the symmetric difference $P\Delta Q$ is a singleton. We say that ${\cal H}(X)$ is a \emph{cube on} $X$. Isometric subgraphs of the cube ${\cal H}(X)$, as well as graphs that are isometrically embeddable in ${\cal H}(X)$, are called \emph{partial cubes}.
\end{definition}
The proof of the following proposition is straightforward and omitted.
\begin{proposition} \label{cube=wg-family}
An induced subgraph $G=(V,E)$ of the cube ${\cal H}(X)$ is a partial cube if and only if $V$ is a well graded family of finite subsets of $X$. Then a shortest path in $G$ is a line segment in ${\cal H}(X)$ and the graph distance function $d$ on both ${\cal H}(X)$ and $G$ is given by
$$
d(P,Q)=|P\Delta Q|.
$$
\end{proposition}
The following theorem characterizes media in terms of their graphs.
\begin{theorem} \label{MediumGraph}
A graph $G$ represents a medium $({\cal S},{\cal T})$ if and only if $G$ is a partial cube.
\end{theorem}
We give two proofs of this important theorem. The first proof uses the representation theorem.
\begin{proof}
(Necessity.) By Theorem~\ref{FiniteRepresentationTheorem}, we may assume that the given medium is $({\cal F},{\cal G}_{\cal F})$ where ${\cal F}$ is a well graded family of finite subsets of some set $X$. By Proposition~\ref{cube=wg-family}, the graph of this medium is a partial cube.
(Sufficiency.) For a partial cube $G=(V,E)$ there is an isometric embedding $\alpha$ of $G$ into a cube $\mathcal{H}(X)$ for some set $X$. Vertices of $\alpha(G)$ form a well graded family of subsets of $X$. Then the medium $(\alpha(V),{\cal G}_{\alpha(V)})$ has $G$ as its graph.
\end{proof}
The second proof utilizes the Djokovi\'{c}--Winkler relation $\Theta$ on the set of edges of a graph. The definition and properties of this relation are found in the book~\cite{wI00}.
\begin{proof}
(Necessity.) We may assume that $G$ is the graph of $({\cal S},{\cal T})$. Let
$$
S,S_1,\ldots,S_n=S
$$
be a cycle of length $n$ in $G$. There is a stepwise effective message $\boldsymbol m$ such that $S\boldsymbol m=S$. By Axiom [M3], $\boldsymbol m$ is vacuous. Therefore, $\ell(\boldsymbol m)=n$ is an even number. Hence, $G$ is a bipartite graph.
For each edge $ST$ of $G$ there is a unique unordered pair of tokens $\{\tau,\tilde{\tau}\}$ such that $S\tau=T$ and $T\tilde{\tau}=S$. We denote $\sim$ the equivalence relation on the set of edges of $G$ defined by this correspondence. Let $ST\sim PQ$ and the notation is chosen such that $S\tau=T$ and $P\tau=Q$. Then, by lemmas~\ref{Lemma6.1} and~\ref{Lemma6.3},
\begin{equation} \label{Eq6.1}
d=d(S,P)=d(T,Q)=d(S,Q)-1=d(T,P)-1
\end{equation}
By Lemma~2.3 in~\cite{wI00}, $ST\Theta PQ$. On the other hand, if $ST\Theta PQ$ holds, then, by the same lemma, equation~(\ref{Eq6.1}) holds. By lemmas~\ref{Lemma6.2} and~\ref{Lemma6.3}, there is a token $\tau$ such that $S\tau=T$ and $P\tau=Q$, that is, $ST\sim PQ$. Thus $\sim\,=\Theta$, that is, $\Theta$ is an equivalence relation. By Theorem~2.10 in~\cite{wI00}, $G$ is a partial cube.
(Sufficiency.) We have already shown in the first proof that a partial cube is the graph of a medium.
\end{proof}
\begin{theorem} \label{Medium=Graph}
Two media are isomorphic if and only if the graphs representing these media are isomorphic.
\end{theorem}
\begin{proof}
(Necessity.) Clearly, graphs representing isomorphic media are isomorphic.
(Sufficiency.) Let $({\cal S},{\cal T})$ and $({\cal S}',{\cal T}')$ be two media and $G=(V,E)$ and $G'=(V',E')$ be two isomorphic graphs representing these media. Since $G$ and $G'$ are isomorphic, $G$ represents $({\cal S}',{\cal T}')$. Thus we need to show that two media represented by the same graph are isomorphic.
Since $G$ is a partial cube, it also represents a medium $({\cal F},{\cal G}_{\cal F})$ of a well graded family of subsets of some set $X$. We denote $\mu:{\cal S}\rightarrow V$ and $\nu:V\rightarrow{\cal F}$ the two bijections that define the graph representation $G$ of $({\cal S},{\cal T})$ and $({\cal F},{\cal G}_{\cal F})$, respectively. Then $\alpha=\nu\circ\mu$ is a bijection ${\cal S}\rightarrow{\cal F}$ such that
$$
S\tau = T\quad\Leftrightarrow\quad |\alpha(S)\Delta\alpha(T)|=1,
$$
for all $S\not= T$ in ${\cal S}$ and $\tau\in{\cal T}$.
Clearly, it suffices to prove that the media $({\cal S},{\cal T})$ and $({\cal F},{\cal G}_{\cal F})$ are isomorphic.
Let $\tau$ be a token in ${\cal T}$ and $S$ and $T$ be two distinct states in ${\cal S}$ such that $S\tau= T$. Then either $\alpha(T)=\alpha(S)\cup\{x\}$ for some $x\notin \alpha(S)$ or $\alpha(T)=\alpha(S)\setminus\{x\}$ for some $x\in \alpha(S)$. We define $\beta:{\cal T}\rightarrow{\cal G}_{\cal F}$ by
$$
\beta(\tau) = \begin{cases}
\gamma_x, &\text{if $\alpha(T)=\alpha(S)\cup\{x\}$ for some $x\notin\alpha(S)$,} \\
\tilde{\gamma}_x, &\text{if $\alpha(T)=\alpha(S)\setminus\{x\}$ for some $x\in\alpha(S)$.}
\end{cases}
$$
Let us show that $\beta$ does not depend on the choice of $S$ and $T$. We consider only the case when $\beta(\tau)=\tau_x$. The other case is treated similarly.
Let $P,Q$ be another pair of distinct states in ${\cal S}$ such that $P\tau= Q$, and let $P=S\boldsymbol m$ and $Q=T\boldsymbol n$ for some straight messages $\boldsymbol m$ and $\boldsymbol n$. By Lemma~\ref{Lemma6.1}, $\ell(\boldsymbol m)=\ell(\boldsymbol n)$. Then, by Lemma~\ref{Lemma6.3}, $d(\alpha(S),\alpha(P))=d(\alpha(T),\alpha(Q))$, and, by Lemma~\ref{Lemma6.1},
\begin{align*}
d(\alpha(S),\alpha(Q))&=d(\alpha(S),\alpha(T))+d(\alpha(T),\alpha(Q)),\\
d(\alpha(T),\alpha(P))&=d(\alpha(T),\alpha(S))+d(\alpha(S),\alpha(P)).
\end{align*}
By Theorem~\ref{DistanceTheorem},
\begin{gather*}
\alpha(S)\cap\alpha(Q)\;\subseteq\:\alpha(T)=\alpha(S)\cup\{x\}\;\subseteq\;\alpha(S)\cup\alpha(Q),\\
\alpha(T)\cap\alpha(P)=[\alpha(S)\cup\{x\}]\cap\alpha(P)\;\subseteq\;\alpha(S)\;\subseteq\;\alpha(S)\cup\alpha(P).
\end{gather*}
Since $x\notin\alpha(S)$, it follows that $x\in\alpha(Q)$ and $x\notin\alpha(P)$. Then
\begin{equation*}
\alpha(Q)=\alpha(P)\cup\{x\},
\end{equation*}
since $d(\alpha(P),\alpha(Q))=1$. Hence, the mapping $\beta:{\cal S}\rightarrow{\cal G}_{\cal F}$ is well defined.
Clearly, $\beta$ is a bijection satisfying the condition
$$
S\tau= T\quad\Leftrightarrow\quad\alpha(S)\beta(\tau)=\alpha(T).
$$
Therefore $(\alpha,\beta)$ is an isomorphism from $({\cal S},{\cal T})$ onto $({\cal F},{\cal G}_{\cal F})$.
\end{proof}
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\begin{theorem} \label{Medium=Graph}
Two media are isomorphic if and only if the graphs representing these media are isomorphic.
\end{theorem}
\begin{proof}
(Necessity.) Clearly, graphs representing isomorphic media are isomorphic.
(Sufficiency.) Let $({\cal S},{\cal T})$ and $({\cal S}',{\cal T}')$ be two media and $G=(V,E)$ and $G'=(V',E')$ be two isomorphic graphs representing these media. Since $G$ and $G'$ are isomorphic, $G$ represents $({\cal S}',{\cal T}')$. Thus we need to show that two media represented by the same graph are isomorphic.
Since $G$ is a partial cube, it also represents a medium $({\cal F},{\cal G}_{\cal F})$ of a well graded family of subsets of some set $X$. We denote $\mu:{\cal S}\rightarrow V$ and $\nu:V\rightarrow{\cal F}$ the two bijections that define the graph representation $G$ of $({\cal S},{\cal T})$ and $({\cal F},{\cal G}_{\cal F})$, respectively. Then $\alpha=\nu\circ\mu$ is a bijection ${\cal S}\rightarrow{\cal F}$ such that
$$
S\tau = T\quad\Leftrightarrow\quad |\alpha(S)\Delta\alpha(T)|=1,
$$
for all $S\not= T$ in ${\cal S}$ and $\tau\in{\cal T}$.
Clearly, it suffices to prove that the media $({\cal S},{\cal T})$ and $({\cal F},{\cal G}_{\cal F})$ are isomorphic.
Let $\tau$ be a token in ${\cal T}$ and $S$ and $T$ be two distinct states in ${\cal S}$ such that $S\tau= T$. Then either $\alpha(T)=\alpha(S)\cup\{x\}$ for some $x\notin \alpha(S)$ or $\alpha(T)=\alpha(S)\setminus\{x\}$ for some $x\in \alpha(S)$. We define $\beta:{\cal T}\rightarrow{\cal G}_{\cal F}$ by
$$
\beta(\tau) = \begin{cases}
\gamma_x, &\text{if $\alpha(T)=\alpha(S)\cup\{x\}$ for some $x\notin\alpha(S)$,} \\
\tilde{\gamma}_x, &\text{if $\alpha(T)=\alpha(S)\setminus\{x\}$ for some $x\in\alpha(S)$.}
\end{cases}
$$
Let us show that $\beta$ does not depend on the choice of $S$ and $T$. We consider only the case when $\beta(\tau)=\tau_x$. The other case is treated similarly.
Let $P,Q$ be another pair of distinct states in ${\cal S}$ such that $P\tau= Q$, and let $P=S\boldsymbol m$ and $Q=T\boldsymbol n$ for some straight messages $\boldsymbol m$ and $\boldsymbol n$. By Lemma~\ref{Lemma6.1}, $\ell(\boldsymbol m)=\ell(\boldsymbol n)$. Then, by Lemma~\ref{Lemma6.3}, $d(\alpha(S),\alpha(P))=d(\alpha(T),\alpha(Q))$, and, by Lemma~\ref{Lemma6.1},
\begin{align*}
d(\alpha(S),\alpha(Q))&=d(\alpha(S),\alpha(T))+d(\alpha(T),\alpha(Q)),\\
d(\alpha(T),\alpha(P))&=d(\alpha(T),\alpha(S))+d(\alpha(S),\alpha(P)).
\end{align*}
By Theorem~\ref{DistanceTheorem},
\begin{gather*}
\alpha(S)\cap\alpha(Q)\;\subseteq\:\alpha(T)=\alpha(S)\cup\{x\}\;\subseteq\;\alpha(S)\cup\alpha(Q),\\
\alpha(T)\cap\alpha(P)=[\alpha(S)\cup\{x\}]\cap\alpha(P)\;\subseteq\;\alpha(S)\;\subseteq\;\alpha(S)\cup\alpha(P).
\end{gather*}
Since $x\notin\alpha(S)$, it follows that $x\in\alpha(Q)$ and $x\notin\alpha(P)$. Then
\begin{equation*}
\alpha(Q)=\alpha(P)\cup\{x\},
\end{equation*}
since $d(\alpha(P),\alpha(Q))=1$. Hence, the mapping $\beta:{\cal S}\rightarrow{\cal G}_{\cal F}$ is well defined.
Clearly, $\beta$ is a bijection satisfying the condition
$$
S\tau= T\quad\Leftrightarrow\quad\alpha(S)\beta(\tau)=\alpha(T).
$$
Therefore $(\alpha,\beta)$ is an isomorphism from $({\cal S},{\cal T})$ onto $({\cal F},{\cal G}_{\cal F})$.
\end{proof}
We conclude this section with an example illustrating Theorem~\ref{Medium=Graph}.
\begin{example}
{\rm If $(\mathcal{S,T})$ and $(\mathcal{S',T'})$ are two finite isomorphic media, then $|{\cal S}|=|\mathcal{S'}|$ and $|{\cal T}|=|\mathcal{T'}|$. The converse, generally speaking, is not true. Consider, for instance, two media, $({\cal F},{\cal G}_{\cal F})$ and $({\cal F}',{\cal G}_{{\cal F}'})$, of well graded subsets of $X=\{a,b,c\}$ with
\begin{equation*}
{\cal F}=\{a,b,c,ab,ac,bc\}\quad\text{and}\quad{\cal F}'=\{a,c,ab,ac,bc,abc\}.
\end{equation*}
Their graphs, $G$ and $G'$, are shown in Figure~\ref{G and G'}.
{\begin{figure}
\caption{Graphs $G$ and $G'$.}
\label{G and G'}
\end{figure}
}
Clearly, these graphs are not isomorphic. Thus the media $({\cal F},{\cal G}_{\cal F})$ and $({\cal F}',{\cal G}_{{\cal F}'})$ are not isomorphic.
}
\end{example}
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\section{Uniqueness of media representations}
Theorem~\ref{FiniteRepresentationTheorem} asserts that any medium $({\cal S},{\cal T})$ is isomorphic to the medium $({\cal F},{\cal G}_{\cal F})$ of a well graded family ${\cal F}$ of finite subsets of some set $X$. In this section we show that this representation is unique in some precise sense.
Let $({\cal F}_1,{\cal G}_{{\cal F}_1})$ and $({\cal F}_2,{\cal G}_{{\cal F}_2})$ be two isomorphic representations of $({\cal S},{\cal T})$ with well graded families ${\cal F}_1$ and ${\cal F}_2$ of subsets of $X_1$ and $X_2$, respectively. By Theorem~\ref{T:wg gamma},
$$
|{\cal T}|=|{\cal G}_{{\cal F}_i}|=2|\cup{\cal F}_i\setminus\cap{\cal F}_i|\quad\text{for $i=1,2$}.
$$
Thus, without loss of generality, we may assume that ${\cal F}_1$ and ${\cal F}_2$ are well graded families of finite subsets of the same set $X$ and that they satisfy conditions~(\ref{E:wg gamma}). The graphs of the media $({\cal F}_1,{\cal G}_{{\cal F}_1})$ and $({\cal F}_2,{\cal G}_{{\cal F}_2})$ are isomorphic partial subcubes of the cube ${\cal H}(X)$. On the other hand, by theorems~\ref{Medium=Graph} and~\ref{MediumGraph}, isometric partial cubes represent isomorphic media.
We formulate the uniqueness problem geometrically as follows:
\begin{quote}
Show that any isometry between two partial subcubes of ${\cal H}(X)$ can be extended to an isometry of the cube ${\cal H}(X)$.
\end{quote}
In other words, we want to show that partial subcubes of ${\cal H}(X)$ are unique up to isometries of ${\cal H}(X)$ onto itself.
\begin{remark}
{\rm Note, that ${\cal H}(X)$ is not a fully homogeneous space~(as defined, for instance, in \cite{dB01}), that is, an isometry between two arbitrary subsets of ${\cal H}(X)$, generally speaking, cannot be extended to an isometry of the cube ${\cal H}(X)$. On the other hand, ${\cal H}(X)$ is a homogeneous metric space.
}
\end{remark}
A general remark is in order. Let $Y$ be a homogeneous metric space, $A$ and $B$ be two metric subspaces of $Y$, and $\alpha$ be an isometry from $A$ onto $B$. Let $c$ be a fixed point in $Y$. For a given $a\in A$, let $b=\alpha(a)\in B$. Since $Y$ is homogeneous, there are isometries $\beta$ and $\gamma$ of $Y$ such that $\beta(a)=c$ and $\gamma(b)=c$. Then $\lambda=\gamma\alpha\beta^{-1}$ is an isometry from $\beta(A)$ onto $\gamma(B)$ such that $\lambda(c)=c$. Clearly, $\alpha$ is extendable to an isometry of $Y$ if and only if $\lambda$ is extendable. Therefore, in the case of the space ${\cal H}(X)$, we may consider only well graded families of subsets containing the empty set $\varnothing$ and isometries between these families fixing this point.
In what follows, we assume that $\varnothing\in{\cal F}$ and $\cup{\cal F}=X$.
\begin{definition}
We define
$$
r_{\cal F}(x)=\min\{|A|: x\in A, A\in{\cal F}\}
$$
and, for $k\geq 1$,
$$
X_k^{\cal F}=\{x\in X: r_{\cal F}(x)=k\}.
$$
\end{definition}
We have $X_i^{\cal F}\cap X_j^{\cal F}=\varnothing$ for $i\not=j$, and $\cup_k X_k^{\cal F}=X$. Note that some of the sets $X_k^{\cal F}$ could be empty for $k>1$, although $X_1^{\cal F}$ is not empty, since, by the wellgradedness property, ${\cal F}$ contains at least one singleton (we assumed that $\varnothing\in{\cal F}$).
\begin{example}
{\rm Let $X=\{a,b,c\}$ and
$$
{\cal F}=\{\varnothing,\{a\},\{b\},\{a,b\},\{a,b,c\}\}.
$$
Clearly, ${\cal F}$ is well graded. We have $r_{\cal F}(a)=r_{\cal F}(b)=1,\;r_{\cal F}(c)=3$, and
$$
X_1^{\cal F}=\{a,b\},\;X_2^{\cal F}=\varnothing,\;X_3^{\cal F}=\{c\}.
$$
}
\end{example}
\begin{lemma}
For $A\in{\cal F}$ and $x\in A$, we have
\begin{equation} \label{OneLess}
r_{\cal F}(x)=|A|\quad\Rightarrow\quad A\setminus\{x\}\in{\cal F}.
\end{equation}
\end{lemma}
\begin{proof}
Let $k=|A|$. Since ${\cal F}$ is well graded, there is a nested sequence $\{A_i\}_{0\leq i\leq k}$ of distinct sets in ${\cal F}$ with $A_0=\varnothing$ and $A_k=A$. Since $r_{\cal F}(x)=k$, we have $x\notin A_i$ for $i<k$. Hence, $A\setminus \{x\}=A_{k-1}\in{\cal F}$.
\end{proof}
Let us recall (Theorem~\ref{DistanceTheorem}(ii)) that
\begin{equation} \label{so1 betweenness}
B\cap C\subseteq A\subseteq B\cup C\;\Leftrightarrow\; d(B,A)+d(A,C)=d(B,C)
\end{equation}
for all $A,B,C\in\mathfrak P(X)$.
It follows that isometries between two well graded families of sets ${\cal F}_1$ and ${\cal F}_2$ preserve the betweenness relation, that is,
\begin{equation} \label{so1 inclusion}
B\cap C\subseteq A\subseteq B\cup C\;\Leftrightarrow\;\alpha(B)\cap \alpha(C)\subseteq \alpha(A)\subseteq \alpha(B)\cup \alpha(C)
\end{equation}
for $A,B,C\in{\cal F}_1$ and an isometry $\alpha:{\cal F}_1\rightarrow{\cal F}_2$.
In the sequel, ${\cal F}_1$ and ${\cal F}_2$ are two well graded families of finite subsets of $X$ and $\alpha:{\cal F}_1\rightarrow{\cal F}_2$ is an isometry such that $\alpha(\varnothing)=\varnothing$.
\begin{definition}
We define a binary relation $\pi$ between on $X$ by means of the following construction. By~(\ref{OneLess}), for a given $x\in X$ there is $A\in{\cal F}_1$ such that $x\in A$, $r_{{\cal F}_1}(x)=|A|$, and $A\setminus\{x\}\in{\cal F}_1$.
Since $\varnothing\subseteq A\setminus\{x\}\subset A$, we have, by~(\ref{so1 inclusion}), $\alpha(A\setminus\{x\})\subset\alpha(A)$. Since $d\,(A\setminus\{x\},A)=1$, there is $y\in X,\;y\notin\alpha(A)$ such that $\alpha(A)=\alpha(A\setminus\{x\})\cup\{y\}$. In this case we say that $xy\in\pi$.
\end{definition}
\begin{lemma}
If $x\in X_k^{{\cal F}_1}$ and $xy\in\pi$, then $y\in X_k^{{\cal F}_2}$.
\end{lemma}
\begin{proof}
Let $A\in{\cal F}_1$ be a set of cardinality $k$ defining $r_{{\cal F}_1}(x)=k$.
Since
$$
|A|=d\,(\varnothing,A)=d\,(\varnothing,\alpha(A))=|\alpha(A)|\quad\text{and}\quad y\in\alpha(A),
$$
we have $r_{{\cal F}_2}(y)\leq k$. Suppose that $m=r_{{\cal F}_2}(y)<k$. Then, by~(\ref{OneLess}), there is $B\in{\cal F}_2$ such that $y\in B$, $|B|=m$, and $B\setminus\{y\}\in{\cal F}_2$. Clearly,
$$
\alpha(A\setminus\{x\})\cap B\subseteq\alpha(A)\subseteq\alpha(A\setminus\{x\})\cup B.
$$
By~(\ref{so1 inclusion}), we have
$$
(A\setminus\{x\})\cap\alpha^{-1}(B)\subseteq A\subseteq (A\setminus\{x\})\cup\alpha^{-1}(B).
$$
Thus, $x\in\alpha^{-1}(B)$, a contradiction, since $r_{{\cal F}_1}(x)=k$ and $|\alpha^{-1}(B)|=m<k$. It follows that $r_{{\cal F}_2}(y)=k$, that is, $y\in X_k^{{\cal F}_2}$.
\end{proof}
We proved that, for every $k\geq 1$, the restriction of $\pi$ to $X_k^{{\cal F}_1}$ is a relation $\pi_k$ between $X_k^{{\cal F}_1}$ and $X_k^{{\cal F}_2}$.
\begin{lemma}
The relation $\pi_k$ is a bijection for every $k\geq 1$.
\end{lemma}
\begin{proof}
Suppose that there are $z\not=y$ such that $xy\in\pi_k$ and $xz\in\pi_k$. Then, by~(\ref{OneLess}), there are two distinct sets $A,B\in{\cal F}_1$ such that
$$
k=r_{{\cal F}_1}(x)=|A|=|B|,\;\;A\setminus\{x\}\in{\cal F}_1,\;B\setminus\{x\}\in{\cal F}_1,
$$
and
$$
\alpha(A)=\alpha(A\setminus\{x\})+\{y\},\;\;\alpha(B)=\alpha(B\setminus\{x\})+\{z\}.
$$
We have
\begin{align*}
d\,(\alpha(A),\alpha(B))&=d\,(A,B)=d\,(A\setminus\{x\},B\setminus\{x\})\\
&=d\,(\alpha(A)\setminus\{y\},\alpha(B)\setminus\{z\}).
\end{align*}
Thus $y,z\in \alpha(A)\cap\alpha(B)$, that is, in particular, that $z\in \alpha(A)\setminus\{y\}$, a contradiction, because $r_{{\cal F}_2}(z)=k$ and $|\alpha(A)\setminus\{y\}|=k-1$.
By applying the above argument to $\alpha^{-1}$, we prove that $\pi_k$ is a bijection.
\end{proof}
It follows from the previous lemma that $\pi$ is a permutation on $X$.
\begin{lemma}
$\alpha(A)=\pi(A)$ for any $A\in{\cal F}_1$.
\end{lemma}
\begin{proof}
We prove this statement by induction on $k=|A|$. The case $k=1$ is trivial, since $\alpha(\{x\})=\{\pi_1(x)\}$ for $\{x\}\in\mathcal{F}_1$.
Suppose that $\alpha(A)=\pi(A)$ for all $A\in{\cal F}_1$ such that $|A|<k$. Let $A$ be a set in ${\cal F}_1$ of cardinality $k$. By the wellgradedness property, there is a nested sequence $\{A_i\}_{0\leq i\leq k}$ of distinct sets in ${\cal F}_1$ with $A_0=\varnothing$ and $A_k=A$. Thus, $A=A_{k-1}\cup\{x\}$ for some $x\notin A_{k-1}$. Clearly, $m=r_{{\cal F}_1}(x)\leq k$.
If $m=k$, then $\alpha(A)=\alpha(A_{k-1})\cup\{\pi(x)\}=\pi(A)$, by the definition of $\pi$ and the induction hypothesis.
Suppose now that $m<k$. There is a set $B\in{\cal F}_1$ containing $x$ such that $|B|=m$. By the wellgradedness property, there is a nested sequence $\{B_i\}_{0\leq i\leq m}$ of distinct sets in ${\cal F}_1$ with $B_0=\varnothing$ and $B_m=B$. We have $x\notin B_i$ for $i<m$, since $m=r_{{\cal F}_1}(x)$. Therefore, $B=B_{m-1}\cup\{x\}$. Clearly,
$$
B_{m-1}\cap A\subseteq B\subseteq B_{m-1}\cup A.
$$
By~(\ref{so1 inclusion}), we have
$$
\alpha(B)\subseteq \alpha(B_{m-1})\cup \alpha(A).
$$
Thus, by the induction hypothesis,
$$
\pi(B_{m-1})\cup\{\pi(x)\}=\pi(B)\subseteq \pi(B_{m-1})\cup \alpha(A).
$$
Hence, $\pi(x)\in\alpha(A)$. Since $\alpha(A)=\pi(A_{k-1})\cup\{y\}$ for $y\notin\pi(A_{k-1})$ and $x\notin A_{k-1}$, we have $y=\pi(x)$, that is, $\alpha(A)=\pi(A)$.
\end{proof}
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We proved that, for every $k\geq 1$, the restriction of $\pi$ to $X_k^{{\cal F}_1}$ is a relation $\pi_k$ between $X_k^{{\cal F}_1}$ and $X_k^{{\cal F}_2}$.
\begin{lemma}
The relation $\pi_k$ is a bijection for every $k\geq 1$.
\end{lemma}
\begin{proof}
Suppose that there are $z\not=y$ such that $xy\in\pi_k$ and $xz\in\pi_k$. Then, by~(\ref{OneLess}), there are two distinct sets $A,B\in{\cal F}_1$ such that
$$
k=r_{{\cal F}_1}(x)=|A|=|B|,\;\;A\setminus\{x\}\in{\cal F}_1,\;B\setminus\{x\}\in{\cal F}_1,
$$
and
$$
\alpha(A)=\alpha(A\setminus\{x\})+\{y\},\;\;\alpha(B)=\alpha(B\setminus\{x\})+\{z\}.
$$
We have
\begin{align*}
d\,(\alpha(A),\alpha(B))&=d\,(A,B)=d\,(A\setminus\{x\},B\setminus\{x\})\\
&=d\,(\alpha(A)\setminus\{y\},\alpha(B)\setminus\{z\}).
\end{align*}
Thus $y,z\in \alpha(A)\cap\alpha(B)$, that is, in particular, that $z\in \alpha(A)\setminus\{y\}$, a contradiction, because $r_{{\cal F}_2}(z)=k$ and $|\alpha(A)\setminus\{y\}|=k-1$.
By applying the above argument to $\alpha^{-1}$, we prove that $\pi_k$ is a bijection.
\end{proof}
It follows from the previous lemma that $\pi$ is a permutation on $X$.
\begin{lemma}
$\alpha(A)=\pi(A)$ for any $A\in{\cal F}_1$.
\end{lemma}
\begin{proof}
We prove this statement by induction on $k=|A|$. The case $k=1$ is trivial, since $\alpha(\{x\})=\{\pi_1(x)\}$ for $\{x\}\in\mathcal{F}_1$.
Suppose that $\alpha(A)=\pi(A)$ for all $A\in{\cal F}_1$ such that $|A|<k$. Let $A$ be a set in ${\cal F}_1$ of cardinality $k$. By the wellgradedness property, there is a nested sequence $\{A_i\}_{0\leq i\leq k}$ of distinct sets in ${\cal F}_1$ with $A_0=\varnothing$ and $A_k=A$. Thus, $A=A_{k-1}\cup\{x\}$ for some $x\notin A_{k-1}$. Clearly, $m=r_{{\cal F}_1}(x)\leq k$.
If $m=k$, then $\alpha(A)=\alpha(A_{k-1})\cup\{\pi(x)\}=\pi(A)$, by the definition of $\pi$ and the induction hypothesis.
Suppose now that $m<k$. There is a set $B\in{\cal F}_1$ containing $x$ such that $|B|=m$. By the wellgradedness property, there is a nested sequence $\{B_i\}_{0\leq i\leq m}$ of distinct sets in ${\cal F}_1$ with $B_0=\varnothing$ and $B_m=B$. We have $x\notin B_i$ for $i<m$, since $m=r_{{\cal F}_1}(x)$. Therefore, $B=B_{m-1}\cup\{x\}$. Clearly,
$$
B_{m-1}\cap A\subseteq B\subseteq B_{m-1}\cup A.
$$
By~(\ref{so1 inclusion}), we have
$$
\alpha(B)\subseteq \alpha(B_{m-1})\cup \alpha(A).
$$
Thus, by the induction hypothesis,
$$
\pi(B_{m-1})\cup\{\pi(x)\}=\pi(B)\subseteq \pi(B_{m-1})\cup \alpha(A).
$$
Hence, $\pi(x)\in\alpha(A)$. Since $\alpha(A)=\pi(A_{k-1})\cup\{y\}$ for $y\notin\pi(A_{k-1})$ and $x\notin A_{k-1}$, we have $y=\pi(x)$, that is, $\alpha(A)=\pi(A)$.
\end{proof}
In summary, we have the following theorem.
\begin{theorem} \label{WG homogeneity}
Any isometry between two partial subcubes of ${\cal H}(X)$ can be extended to an isometry of the cube ${\cal H}(X)$.
\end{theorem}
\begin{remark}
{\rm In the case of a finite set $X$ the previous theorem is a consequence of Theorem~19.1.2 in \cite{mD97}.}
\end{remark}
In the line of our arguments which led to the proof of Theorem~\ref{WG homogeneity} we used two kinds of isometries of ${\cal H}(X)$: isometries that map elements of ${\cal H}(X)$ to the empty set, and isometries defined by permutations on $X$. It is not difficult to show that these isometries generate the isometry group of ${\cal H}(X)$.
\begin{theorem} \label{so1 isometry group of cube}
The isometry group of ${\cal H}(X)$ is generated by permutations on the set $X$ and functions
$$
\alpha_A:S\mapsto S\Delta A,\quad S\in{\cal H}(X).
$$
\end{theorem}
\begin{proof}
Clearly, $\alpha_A$ is an isometry of ${\cal H}(X)$ and $\alpha_A(A)=\varnothing$. A permutation $\pi$ on $X$ defines an isometry $\hat{\pi}:{\cal H}(X)\rightarrow{\cal H}(X)$ by
$$
\hat{\pi}(S)=\{\pi(x): x\in S\}.
$$
Let $\alpha:{\cal H}(X)\rightarrow{\cal H}(X)$ be an isometry of ${\cal H}(X)$ and let $A=\alpha^{-1}(\varnothing)$. Then the isometry $\alpha_A\circ\alpha^{-1}$ fixes $\varnothing\in{\cal H}(X)$ and therefore defines a permutation $\pi:X\rightarrow X$ (singletons are on the distance $1$ from $\varnothing$). Let $\beta=\hat{\pi}^{-1}\circ\alpha_A\circ\alpha^{-1}$. Since $\alpha_A\circ\alpha^{-1}$ fixes $\varnothing$, we have $\alpha_A\circ\alpha^{-1}(\{x\})=\{\pi(x)\}$ for any $x\in X$. Hence, $\beta(\{x\})=\{x\}$ for all $x\in X$. For $S\in{\cal H}(X)$, we have
$$
|\beta(S)|=d(\beta(S),\varnothing)=d(S,\varnothing)=|S|,
$$
since $\beta(\varnothing)=\varnothing$. For any $x\in \beta(S)$, we have
$$
d(\{x\},S)=d(\{x\},\beta(S))=|\beta(S)|-1=|S|-1,
$$
which is possible only if $x\in S$. Thus $\beta(S)\subseteq S$. The same argument shows that $S\subseteq\beta(S)$. Thus $\beta$ is the identity mapping. It follows that $\alpha=\hat{\pi}^{-1}\circ\alpha_A$.
\end{proof}
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\section{Linear Media}
The representation theorem (Theorem~\ref{FiniteRepresentationTheorem}) is a powerful tool for constructing media. We illustrate an application of this theorem by constructing a medium of linear orderings on a given finite or infinite countable set $Z$.
Let $Z=\{a_1,a_2,\ldots\}$ be a fixed (finite or infinite) enumeration of elements of $Z$. This enumeration defines a particular irreflexive linear ordering on $Z$ that we will denote by $L_0$.
\begin{definition}
A binary relation $R$ on $Z$ is said to be \emph{locally finite} if there is $n\in\mathbb{N}$ such that the restriction of $R$ to $\{a_{n+1},a_{n+2},\ldots\}$ coincides with the restriction of $L_0$ to the same set.
\end{definition}
Let ${\cal L}{\cal O}$ be the set of all locally finite irreflexive linear orders on the set $Z$. Note that if $Z$ is a finite set, then ${\cal L}{\cal O}$ is the set of all linear orderings on $Z$.
As usual, for a given $L\in{\cal L}{\cal O}$, we say that $x$ \emph{covers} $y$ in $L$ if $yx\in L$ and there is no $z\in Z$ such that $yz\in L,zx\in L$. Here and below $xy$ stands for an ordered pair of elements $x,y\in Z$. In what follows all binary relations are assumed to be locally finite for a given enumeration of $Z$.
\begin{lemma} \label{LO1}
Let $L$ be a linear order on $Z$. Then $L'=(L\setminus yx)\cup xy$ is a linear order if and only if $x$ covers $y$ in $L$.
\end{lemma}
\begin{proof}
Suppose $L'$ is a linear order and there is $z$ such that $yz\in L$ and $zx\in L$. Then $yz\in L'$ and $zx\in L'$ implying $yx\in L'$, a contradiction.
Suppose $x$ covers $y$ in $L$. Let $uv\in L'$ and $vw\in L'$. We need to show that $uw\in L'$. There are three possible cases.
\begin{enumerate}
\item $uv=xy$ and $vw\in L,vw\not=yx$. Then $yw=vw\in L$ which implies $uw=xw\in L$, since $x$ covers $y$ in $L$. We have $uw\in L'$, since $uw=xw\not=yx$.
\item $vw=xy$ and $uv\in L,uv\not=yx$. Then $ux=uv\in L$ which implies $uw=uy\in L$, since $x$ covers $y$ in $L$. We have $uw\in L'$, since $uw=uy\not=yx$.
\item $uv\in L,uv\not=yx$ and $vw\in L,vw\not=yx$. Then $uw\in L$ and $uw\not= yx$, since $x$ covers $y$ in $L$. Therefore, $uw\in L'$.
\end{enumerate}
\end{proof}
We shall also need the following fact.
\begin{lemma} \label{LO2}
Let $P,Q$ and $R$ be complete asymmetric binary relations on $Z$. Then
\begin{equation*}
P\cap R = Q\cap R\quad\Leftrightarrow\quad P=Q.
\end{equation*}
\end{lemma}
\begin{proof}
Suppose that $P\cap R=Q\cap R$ and let $xy\in P$. If $xy\in R$, then $xy\in Q$. Otherwise, $yx\in R$. Since $yx\notin P$, we have $yx\notin Q$ implying $xy\in Q$. Thus $P\subseteq Q$. By symmetry, $P=Q$.
\end{proof}
For $L\in{\cal L}{\cal O}$, we define
\begin{equation*}
\alpha : L \mapsto L\cap L_0
\end{equation*}
By Lemma~\ref{LO2}, $\alpha$ is a one--to--one mapping from ${\cal L}{\cal O}$ onto the set $\alpha({\cal L}{\cal O})$ of partial orders.
Note that for any two locally finite binary relations $R$ and $Q$ on $Z$, the symmetric difference $R\Delta Q$ is a finite set. Thus the distance $d(R,Q)=|R\Delta Q|$ is a finite number. We use this fact in the proof of the following theorem.
\begin{theorem} \label{WGfamilyLO}
The family $\alpha({\cal L}{\cal O})$ is a well graded family of subsets of $L_0$.
\end{theorem}
\begin{proof}
Let $P,P'$ be two distinct partial orders in $\alpha({\cal L}{\cal O})$ and $L,L'$ be corresponding linear orders. It is easy to see that there is a pair $xy\in L$ such that $y$ covers $x$ and $xy\notin L'$. By Lemma~\ref{LO1}, $L''$ defined by
$$
L''=(L\setminus xy)\cup yx
$$
is a linear order. Then
$$
P''=L''\cap L_0 = [(L\cap L_0)\setminus(L_0\cap xy)]\cup(L_0\cap yx)=\begin{cases}
P\setminus xy, & \text{if $xy\in L_0$,} \\
P\cup yx, & \text{if $xy\notin L_0$,}
\end{cases}
$$
where $xy\in P$ if $xy\in L_0$ and $yx\notin P$ if $xy\notin L_0$. Hence, $P''\not= P$ and $d(P,P'')=1$. Clearly,
\begin{equation*}
L\cap L'\subseteq L''\subseteq L\cup L'.
\end{equation*}
Therefore
\begin{equation*}
P\cap P'=L\cap L'\cap L_0\subseteq P''=L''\cap L_0\subseteq (L\cup L')\cap L_0=P\cup P',
\end{equation*}
that is, $P''$ lies between $P$ and $P'$. Thus,
\begin{equation*}
d(P,P')=d(P,P'')+d(P'',P')=1+d(P'',P')
\end{equation*}
and the result follows by induction.
\end{proof}
Since ${\cal F}=\alpha({\cal L}{\cal O})$ is a well graded family of subsets of $X=L_0$, it is the set of states of the medium $({\cal F},{\cal G}_{\cal F})$ with tokens defined by
\begin{equation*}
P\rho_{xy} = \begin{cases}
P\cup xy, & \text{if $P\cup xy\in\alpha({\cal L}{\cal O})$,} \\
P, & \text{otherwise,}
\end{cases}
\end{equation*}
and
\begin{equation*}
P\tilde{\rho}_{xy} = \begin{cases}
P\setminus xy, & \text{if $P\setminus xy\in\alpha({\cal L}{\cal O})$,} \\
P, & \text{otherwise,}
\end{cases}
\end{equation*}
for $xy\in L_0$ and $P=L\cap L_0\in\alpha({\cal L}{\cal O})$.
We have
\begin{align*}
(L\cap L_0)\rho_{xy} & = \begin{cases}
(L\cap L_0)\cup xy, & \text{if $(L\cap L_0)\cup xy\in\alpha({\cal L}{\cal O})$,} \\
L\cap L_0, & \text{otherwise,}
\end{cases} \\
& = \begin{cases}
(L\cup xy)\cap L_0, & \text{if $(L\cup xy)\cap L_0=L'\cap L_0$,} \\
L\cap L_0, & \text{otherwise,}
\end{cases}
\end{align*}
where $L'$ is some linear order. Since $yx\notin L_0$, we have
\begin{equation*}
(L\cup xy)\cap L_0=[(L\setminus yx)\cup xy]\cap L_0 =L'\cap L_0.
\end{equation*}
By Lemma~\ref{LO2}, $(L\setminus yx)\cup xy$ is a linear order. We define
\begin{equation*}
L\tau_{xy}=\begin{cases}
(L\setminus yx)\cup xy & \text{if $x$ covers $y$ in $L$,} \\
L & \text{otherwise.}
\end{cases}
\end{equation*}
Then, for $xy\in L_0$,
\begin{equation*}
(L\cap L_0)\rho_{xy}=L\tau_{xy}\cap L_0.
\end{equation*}
A similar argument shows that, for $xy\in L_0$,
\begin{equation*}
(L\cap L_0)\tilde{\rho}_{xy}=L\tau_{yx}\cap L_0=L\tilde{\tau}_{xy}\cap L_0.
\end{equation*}
We obtained the set of tokens ${\cal T}=\{\tau_{xy}\}_{xy\in L_0}$ by `pulling back' tokens from the set ${\cal G}_{\cal F}$. The medium $({\cal L}{\cal O},{\cal T})$ is isomorphic to the medium $({\cal F},{\cal G}_{\cal F})$. In the case of a finite set $Z$, it is the \emph{linear medium} introduced in \cite{jF97}.
Simple examples show that $\alpha({\cal L}{\cal O})$ is a proper subset of the set of all partial orders contained in $L_0$. In the case of a finite set $Z$ this subset is characterized in the following theorem.
\begin{theorem}
Let $L$ be a linear order on a finite set $Z$ and $P\subseteq L$ be a partial order. Then $P=L\cap L'$, where $L'$ is a linear order, if and only if $P'=L\setminus P$ is a partial order.
\end{theorem}
\begin{proof}
(1) Suppose $P=L\cap L'$. It suffices to prove that $P'=L\setminus P$ is transitive. Let $(x,y),\,(y,z)\in P'$. Then $(x,z)\in L$. Suppose $(x,z)\notin P'$. Then $(x,z)\in P$, implying $(x,z)\in L'$. Since $(x,y),\,(y,z)\in P'$, we have $(x,y)\notin L'$ and $(y,z)\notin L'$, implying $(y,x),\,(z,y)\in L'$, implying $(z,x)\in L'$, a contradiction.
(2) Suppose now that $P$ and $P'=L\setminus P$ are partial orders. We define $L'=P\cup {P'}^{-1}$ and prove that thus defined $L'$ is a linear order.
Clearly, relations $P,P^{-1},P',{P'}^{-1}$ form a partition of $(Z\times Z)\setminus\Delta$. It follows that $L'$ is a complete and antisymmetric binary relation.
To prove transitivity, suppose $(x,y),\,(y,z)\in L'$. It suffices to consider only two cases:
(i) $(x,y)\in P,\;(y,z)\in {P'}^{-1}$. Suppose $(x,z)\notin L'$. Then $(z,x)\in L'$. Suppose $(z,x)\in P$. Since $(x,y)\in P$, we have $(z,y)\in P$, a contradiction, since $(z,y)\in P'$. Suppose $(z,x)\in {P'}^{-1}$. Then $(x,z)\in P'$ and $(z,y)\in P'$ imply $(x,y)\in P'$, a contradiction. Hence, $(x,z)\in L'$.
(ii) $(y,z)\in P,\;(x,y)\in {P'}^{-1}$. Suppose $(x,z)\notin L'$. Then, again, $(z,x)\in L'$. Suppose $(z,x)\in P$. Since $(y,z)\in P$, we have $(y,x)\in P$, a contradiction, since $(y,x)\in P'$. Suppose $(z,x)\in {P'}^{-1}$. Then $(x,z)\in P'$ and $(y,x)\in P'$ imply $(y,z)\in P'$, a contradiction. Hence, $(x,z)\in L'$.
Clearly $P=L\cap L'$.
\end{proof}
We conclude this section with a geometric illustration of Theorem~\ref{WGfamilyLO}.
{\begin{figure}
\caption{The diagram of ${\cal L}
\label{diagram LO}
\end{figure}
}
\begin{example}
{\rm Let $Z=\{1,2,3\}$. We represent linear orders on $Z$ by $3$--tuples. There are $3!=6$ different linear orders on $Z$:
\begin{equation*}
L_0=123,\quad L_1=213,\quad L_2=231,\quad L_3=321,\quad L_4=312,\quad L_5=132.
\end{equation*}
These relations are represented by the vertices of the diagram in Figure~\ref{diagram LO}.
One can compare this diagram with the diagram shown in Figure~5 in~\cite{jF97}.
The elements of ${\cal F}=\alpha({\cal L}{\cal O})$ are subsets of $X=L_0$:
\begin{gather*}
L_0=\{12,13,23\},\quad L_1\cap L_0=\{13,23\},\quad L_2\cap L_0=\{23\}, \\
L_3\cap L_0=\varnothing,\quad L_4\cap L_0=\{12\},\quad L_5\cap L_0=\{12,13\}.
\end{gather*}
These sets are represented as vertices of the cube on the set $L_0=\{12,13,23\}$ as shown in Figure~\ref{(LO,T)}.
}
\end{example}
{\begin{figure}
\caption{Partial cube representing $({\cal L}
\label{(LO,T)}
\end{figure}
}
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\section{Hyperplane arrangements}
In this section we consider an example of a medium suggested by Jean--Paul Doignon (see Example~2 in~\cite{jFsO02}).
Let $\mathcal{A}$ be a locally finite arrangements of affine hyperplanes in $\mathbb{R}^r$, that is a family of hyperplanes such that any open ball in $\mathbb{R}^r$ intersects only finite number of hyperplanes in $\mathcal{A}$~\cite[Ch.~V,\;\S 1]{nB02}. Clearly, there are only countably many hyperplanes in $\mathcal{A}$, so we can enumerate them, $\mathcal{A}=\{H_1,H_2,\ldots\}$. Every hyperplane is given by an affine linear function $\ell_i(\boldsymbol{x})=\sum_{j=1}^r a_{ij}x_j+b_i$, that is, $H_i=\{\boldsymbol{x}\in\mathbb{R}^r : \ell_i(\boldsymbol{x})=0\}$.
In what follows, we construct a token system $({\cal S},{\cal T})$ associated with an arrangement $\mathcal{A}$ and show that this system is a medium.
We define the set ${\cal S}$ of states to be the set of connected components of $\mathbb{R}^r\setminus\cup\,\mathcal{A}$. These components are called \emph{regions}~\cite{aB99} or \emph{chambers}~\cite{nB02} of $\mathcal{A}$. Each state $P\in{\cal S}$ is an interior of an $r$--dimensional polyhedron in $\mathbb{R}^r$.
To every hyperplane in $\mathcal{A}$ corresponds an ordered pair $(H,H')$ of open half spaces $H$ and $H'$ separated by this hyperplane. This ordered pair generates a transformation $\tau_{H,H'}$ of the states. Applying $\tau_{H,H'}$ to some state $P$ results in some other state $P'$ if $P\subseteq H,\;P'\subseteq H'$ and regions $P$ and $P'$ share a facet which is included in the hyperplane separating $H$ and $H'$; otherwise, the application of $\tau_{H,H'}$ to $P$ does not change $P$. We define the set ${\cal T}$ of tokens to be the set of all $\tau_{H,H'}$. Clearly, $\tau_{H,H'}$ and $\tau_{H',H}$ are reverses of each other.
\begin{theorem} \label{HyperplaneMedia}
$({\cal S},{\cal T})$ is a medium.
\end{theorem}
\begin{proof}
In order to prove that $({\cal S},{\cal T})$ is a medium, we show that it is isomorphic to a medium of a well graded family of sets. Let $J=\{1,2,\ldots\}$. We denote $H_i^+=\{\boldsymbol{x}\in\mathbb{R}^r : \ell_i(\boldsymbol{x})>0\}$ and $H_i^-=\{\boldsymbol{x}\in\mathbb{R}^r : \ell_i(\boldsymbol{x})<0\}$, open half spaces separated by $H_i$. Each region $P$ is an intersection of open half spaces corresponding to hyperplanes in $\mathcal{A}$. We define $J_P = \{j\in J : P\subseteq H_j^+\}$. Clearly, $P\mapsto J_P$ defines a bijection from ${\cal S}$ to ${\cal S}'=\{J_P:P\in{\cal S}\}$. It is also easy to see that $\cap\,{\cal S}'=\varnothing$ and $\cup\,{\cal S}'=J$.
Given $k\in J$, we define transformations $\tau_k$ and $\tilde{\tau}_k$ of ${\cal S}'$ as follows:
$$
J_P\tau_k = \begin{cases}
J_P\cup\{k\} &\text{if $H_k$ defines a facet of $P$,} \\
J_P &\text{otherwise},
\end{cases}
$$
and
$$
J_P\tilde{\tau}_k = \begin{cases}
J_P\setminus\{k\} &\text{if $H_k$ defines a facet of $P$}, \\
J_P &\text{otherwise}.
\end{cases}
$$
Let $P$ be a region of $\mathcal{A}$ and let $H_k$ be a hyperplane in $\mathcal{A}$ defining a facet of $P$. There is a unique region $P'$ sharing this facet with $P$. Moreover, $H_k$ is the only hyperplane separating $P$ and $P'$. It follows that $J_P\tau_k=J_{P'}$ if $k\notin J_P$ and $J_P\tilde{\tau}_k=J_{P'}$ if $k\in J_P$. Thus transformations $\tau_k$ and $\tilde{\tau}_k$ are well defined.
We denote ${\cal T}'$ the set of all transformations $\tau_i,\;\tilde{\tau}_i,\;i=1,\ldots,n$. Clearly, the correspondences $\tau_{H_i^+,H_i^-}\mapsto\tilde{\tau}_i$ and $\tau_{H_i^-,H_i^+}\mapsto\tau_i$ define a bijection from ${\cal T}$ to ${\cal T}'$. This bijection together with the bijection from ${\cal S}$ to ${\cal S}'$ given by $P\mapsto J_P$ define an isomorphism of two token systems, $({\cal S},{\cal T})$ and $({\cal S}',{\cal T}')$.
It remains to show that ${\cal S}'$ is a well graded family of subsets of $J$.
Clearly, $k\in J_P\Delta J_Q$ if and only if $H_k$ separates $P$ and $Q$. Since $\mathcal{A}$ is locally finite, there is a finite number of hyperplanes in $\mathcal{A}$ that separate two regions. Thus $J_P\Delta J_Q$ is a finite set for any two regions $P$ and $Q$. Let $d$ be the usual Hamming distance on ${\cal S}'$, i.e, $d(J_P,J_Q)=|J_P\Delta J_Q|$. Thus $d(J_P,J_Q)$ is equal to the number of hyperplanes in $\mathcal{A}$ separating $P$ and $Q$.
Let $\boldsymbol{p}\in P$ and $\boldsymbol{q}\in Q$ be points in two distinct regions $P$ and $Q$. The interval $[\boldsymbol{p},\boldsymbol{q}]$ has a single intersection point with any hyperplane separating $P$ and $Q$. Moreover, a simple topological argument shows that we can always choose $\boldsymbol{p}$ and $\boldsymbol{q}$ in such a way that different hyperplanes separating $P$ and $Q$ intersect $[\boldsymbol{p},\boldsymbol{q}]$ in different points. Let us number these points in the direction from $\boldsymbol{p}$ to $\boldsymbol{q}$ as follows
\begin{equation*}
\boldsymbol{r}_0 = \boldsymbol{p}, \boldsymbol{r}_1,\ldots,\boldsymbol{r}_{k+1}=\boldsymbol{q}.
\end{equation*}
Each open interval $(\boldsymbol{r}_i,\boldsymbol{r}_{i+1})$ is an intersection of $[\boldsymbol{p},\boldsymbol{q}]$ with some region which we denote $R_i$ (in particular, $R_0=P$ and $R_k=Q$). Moreover, by means of this construction, points $\boldsymbol{r}_i$ and $\boldsymbol{r}_{i+1}$ belong to facets of $R_i$. We conclude that regions $R_i$ and $R_{i+1}$ are adjacent, that is, share a facet, for all $i=0,\ldots,k-1$. Clearly, $d(J_{P_i},J_{P_{i+1}})=1$ for all $i=0,\ldots,k-1$ and $d(J_P,J_Q)=k$. Thus, ${\cal S}'$ is a well graded family of subsets of $J$.
\end{proof}
The \emph{region graph} $G$~\cite{aB99} of the arrangement $\mathcal{A}$ has ${\cal S}$ as the set of vertices; edges of $G$ are pairs of adjacent regions in ${\cal S}$. It follows from Theorem~\ref{HyperplaneMedia} that $G$ is a partial cube.
In the case of a finite arrangement $\mathcal{A}$, the graph $G$ is the \emph{tope graph} of the oriented matroid associated with the arrangement $\mathcal{A}$. It follows from Proposition~4.2.3 in~\cite{aB99} that $G$ is an isometric subgraph of the $n$--cube, where $n$ is the number of hyperplanes in $\mathcal{A}$. Thus our Theorem~\ref{HyperplaneMedia} is an infinite dimensional analog of this result.
To give geometric examples of infinite partial cubes, let us consider locally finite line arrangements $\mathcal{A}$ in the plane $\mathbb{R}^2$. The closures of the regions of a given $\mathcal{A}$ form a tiling~\cite{bG87,mS95} of the plane. The region graph of this tiling is the $1$--skeleton of the dual tiling.
\begin{example}
{\rm Let us consider a line arrangement $\mathcal{A}$ shown in Figure~\ref{hex mosaic} by dotted lines.
The regions of this line arrangement are equilateral triangles that form $(3^6)$ mosaic (an edge--to--edge planar tiling by regular polygons; for notations and terminology see, for instance,~\cite{mD02,bG87}). The $1$--skeleton of the orthogonally dual~\cite{mS95} mosaic $(6^3)$ is the region graph of $\mathcal{A}$. This graph is also known as the hexagonal lattice in the plane.
By Theorem~\ref{HyperplaneMedia}, the hexagonal lattice is an infinite partial cube. This lattice is isometrically embeddable into the graph of the cubical lattice $\mathbb{Z}^3$~\cite{mD02}.
}
\end{example}
{\begin{figure}
\caption{Hexagonal lattice ($(6^3)$ mosaic).}
\label{hex mosaic}
\end{figure}
}
\begin{example}
{\rm
Another example of an infinite partial cube is shown in Figure~\ref{truncated mosaic}. There, the region graph is the $1$--skeleton of $(4.8^2)$ mosaic also known~\cite{mD02} as the truncated net $(4^4)$. Like in the previous case, this mosaic is orthogonally dual to the tiling defined by the line arrangement shown in Figure~\ref{truncated mosaic} and the region graph can be isometrically embedded into $\mathbb{Z}^4$~\cite{mD02}.
}
\end{example}
{\begin{figure}
\caption{$(4.8)$ mosaic.}
\label{truncated mosaic}
\end{figure}
}
\begin{example}
{\rm
A more sophisticated example of an infinite partial cube was suggested by a referee. This is one of the Penrose rhombic tilings (see, for instance,~\cite{dB81,mS95}) a fragment of which is shown in Figure~\ref{Penrose}~\cite[Ch.~9]{sW99}. The construction suggested by de~Bruijn~\cite{dB81} demonstrates that the graph of this tiling is the region graph of a particular line arrangement known as a pentagrid. This graph is isometrically embeddable in $\mathbb{Z}^5$~\cite{dB81,mD02}.
}
\end{example}
{\begin{figure}
\caption{A Penrose rhombic tiling.}
\label{Penrose}
\end{figure}
}
\section*{Acknowledgments}
The author is grateful to Jean--Claude Falmagne for his careful reading of the original manuscript and many helpful suggestions, and to Jean--Paul Doignon for his comments on the results presented in Section~7. I also thank the referees for their constructive criticism.
\end{document}
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\begin{document}
\title[Boundedness of the Gaussian Riesz potentials ] {Boundedness of the Gaussian Riesz potentials on Gaussian variable Lebesgue spaces}
\author{Eduard Navas}
\address{Departamento de Matem\'aticas, Universidad Nacional Experimental Francisco de Miranda, Punto Fijo, Venezuela.}
\email{[email protected]}
\author{Ebner Pineda}
\address{Departamento de Matem\'{a}tica, Facultad de Ciencias Naturales y Matem\'aticas, ESPOL Guayaquil 09-01-5863, Ecuador.}
\email{[email protected]}
\author{Wilfredo~O.~Urbina}
\address{Department of Mathematics, Actuarial Sciences and Economics, Roosevelt University, Chicago, IL,
60605, USA.}
\email{[email protected]}
\subjclass[2010]{Primary 42B25, 42B35 ; Secondary 46E30, 47G10 }
\keywords{Gaussian harmonic analysis, variable Lebesgue spaces, Ornstein-Uhlenbeck semigroup, Riesz potentials.}
\begin{abstract}
In this paper we prove the boundedness of the Gaussian Riesz potentials $I_{\beta}$, for $\beta\geq 1$ on $L^{p(\cdot)}(\gamma_d)$, the Gaussian variable Lebesgue spaces under a certain additional condition of regularity on $p(\cdot)$ following \cite{DalSco}. Additionally, this result trivially gives us an alternative proof of the boundedness of Gaussian Riesz potentials $I_\beta$ on Gaussian Lebesgue spaces $L^p(\gamma_d)$.
\end{abstract}
\maketitle
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\section{Introduction and Preliminaries}
In the classical case, the Riesz potential of order $\beta>0$ is defined as negative fractional powers of the negative Laplacian $-\Delta = -\displaystyle\sum_{i=1}^d \frac{\partial^2}{\partial x_i^2}$,
\begin{equation}
(-\Delta)^{-\beta/2},
\end{equation}
which means, using Fourier transform, that
\begin{equation}
( (-\Delta)^{-\beta/2} f)\hat{}\;(\xi) = (2\pi |\xi| )^{-\beta} \hat{f}(\xi).
\end{equation}
for more details; see \cite{duo}, \cite{grafak}, \cite{st1}.\\
Analogously, the {\em Gaussian fractional integrals} or {\em Gaussian Riesz potentials} can be also defined as negative fractional powers of the {\em Ornstein-Uhlenbeck operator }
\begin{equation}
(-L)= - \frac12 \Delta + \langle x, \nabla_x \rangle=- \sum_{i=1}^d \Big[\frac{1}{2} \frac{\partial^2}{\partial x_i^2} + x_i \frac{\partial }{\partial x_i}\Big].
\end{equation}
However, since the Ornstein-Uhlenbeck operator has eigenvalue $0,$ the negative powers are not defined on all of $L^2(\gamma_d),$ and therefore we need to be more careful with the definition. Let us consider
$$\Pi_{0}f=f-\displaystyle\int_{\mathbb{R}^{d}}f(y)\gamma_{d}(dy),$$
for $f\in L^{2}(\gamma_{d})$, the orthogonal projection on the orthogonal complement of the eigenspace corresponding to the eigenvalue $0$.
\begin{defi}
The Gaussian Fractional Integral or Gaussian Riesz potential of order $\beta>0$, $I_\beta$, is defined spectrally as,
\begin{equation}\label{i1}
I_\beta=(-L)^{-\beta/2}\Pi_{0},
\end{equation}
which means that for any multi-index $\nu, \; |\nu|>0$ its action on the Hermite polynomial $\vec{H}_\nu$ is given by
\begin{equation}\label{RieszPotAct}
I_\beta \vec{H}_\nu(x)=\frac 1{\left|
\nu \right|^{\beta/2}}\vec{H}_\nu(x),
\end{equation}
and for $\nu=0=(0,...,0), \, I_{\beta}(\vec{H}_{0})=0.$
\end{defi}
By linearity, using the fact that the Hermite polynomials are an algebraic basis of $\mathcal{P}(\mathbb{R}^d),$ $I_\beta$ can be defined for any polynomial function $f(x) = \sum_{\nu} \widehat{f}_{\gamma_{d}}(\nu) \vec{H}_\nu(x),$ where $\widehat{f}_{\gamma_{d}}(\nu) = \frac{1}{\|\vec{H}_\nu\|_2} \int_{\mathbb{R}^d} f(t) \vec{H}_\nu (t) dt,$ as
\begin{equation}\label{RieszPotMult}
I_\beta f(x) = \sum_{\nu} \frac {\widehat{f}_{\gamma_{d}}(\nu)}{\left|\nu \right|^{\beta/2}}\vec{H}_\nu(x) = \sum_{k\geq 1} \frac 1{k^{\beta/2}} {\bf J}_k f(x),
\end{equation}
and similarly for $f \in L^2(\gamma_d),$ as the Hermite polynomials are an orthogonal basis of $L^2(\gamma_d).$\\
It can be proved that the Gaussian Riesz potential $I_\beta,$ $\beta>0,$ has the following integral representations, for $f$ a polynomial function or $f \in C^2_b(\mathbb{R}^d),$
\begin{equation}\label{RieszOUIntRep}
I_\beta f(x) =\frac 1{\Gamma(\beta/2)}\int_0^{\infty}
t^{\beta/2-1} T_t (I-{\bf J}_0)f(x) \,dt,
\end{equation}
with respect to the Ornstein-Uhlenbeck semigroup $\{T_t\}$, and
\begin{equation}\label{RieszPHIntRep}
I_\beta f(x) = \frac 1{\Gamma(\beta)}\int_0^{\infty}
t^{\beta-1}P_t (I-{\bf J}_0)f(x) \,dt,
\end{equation}
with respect to the Poisson-Hermite semigroup, $\{P_t\}$. \\
Therefore, from (\mathbb{R}f{RieszOUIntRep}) we have an explicit integral representation of $I_\beta$ as
\begin{equation}\label{RieszExplntRep}
I_{\beta}f(x)=\int\limits_{\mathbb{R}^{d}}N_{\beta/2}(x,y)f(y)dy\
\end{equation}
where the kernel $N_{\beta/2}$ is defined as
\begin{equation}\label{kernelRiesz}
N_{\beta/2}(x,y)=\frac{1}{\pi^{\frac{d}{2}}\Gamma(\frac{\beta}{2})}\int_{0}^{+\infty}
t^{\frac{\beta}{2}-1}\left(\frac{e^{-\frac{|y-e^{-t}x|^{2}}{1-e^{-2t}}}}{(1-e^{-2t})^{\frac{d}{2}}}-e^{-|y|^{2}}\right)dt
\end{equation}
For more details and background we refer to \cite{urbina2019}.\\
From (\mathbb{R}f{RieszPotMult}) it is clear that the Gaussian Riesz potentials $I_\beta$ are the simplest Meyer's multipliers (see for instance Theorem 6.2 of \cite{urbina2019}), since in this case
\begin{equation}\label{RieszPotMult2}
m(k) = \frac{1}{k^\beta} = h( \frac{1}{k^\beta}),
\end{equation}
with $h(x) =x,$ the identity function and therefore their $L^p(\gamma_d)$-boundedness follows immediately.\\
In this paper we prove that Gaussian Riesz potentials $I_\beta$, for $\beta\geq 1$ are also bounded in $L^{p(\cdot)}(\gamma_d)$, the Gaussian variable Lebesgue spaces for certain exponent functions $p(\cdot)$ that will be determine later. For completeness, we will briefly review the notion of variable Lebesgue spaces.\\
Given $\mu$ a Borel measure, any $\mu_{-}$measurable function $p(\cdot):\mathbb{R}^{d}\rightarrow [1,\infty]$ is an $exponent$ $function$; the set of all the exponent functions will be denoted by $\mathcal{P}(\mathbb{R}^{d},\mu)$. For $E\subset\mathbb{R}^{d}$ we set $$p_{-}(E)=\text{ess}\inf_{x\in E}p(x) \;\text{and}\; p_{+}(E)=\text{ess}\sup_{x\in E}p(x).$$
$\Omega_{\infty}=\lbrace x\in \Omega:p(x)=\infty\rbrace$.\\
We use the abbreviations $p_{+}=p_{+}(\mathbb{R}^{d})$ and $p_{-}=p_{-}(\mathbb{R}^{d})$.
\begin{defi}\label{deflogholder}
Let $E\subset \mathbb{R}^{d}$. We say that $\alpha(\cdot):E\rightarrow\mathbb{R}$ is locally log-H\"{o}lder continuous, and denote this by $\alpha(\cdot)\in LH_{0}(E)$, if there exists a constant $C_{1}>0$ such that
\begin{eqnarray*}
|\alpha(x)-\alpha(y)|&\leq&\frac{C_{1}}{log(e+\frac{1}{|x-y|})}
\end{eqnarray*}
for all $x,y\in E$. We say that $\alpha(\cdot)$ is log-H\"{o}lder continuous at infinity with base point at $x_{0}\in \mathbb{R}^{d}$, and denote this by $\alpha(\cdot)\in LH_{\infty}(E)$, if there exist constants $\alpha_{\infty}\in\mathbb{R}$ and $C_{2}>0$ such that
\begin{eqnarray*}
|\alpha(x)-\alpha_{\infty}|&\leq&\frac{C_{2}}{log(e+|x-x_{0}|)}
\end{eqnarray*}
for all $x\in E$. We say that $\alpha(\cdot)$ is log-H\"{o}lder continuous, and denote this by $\alpha(\cdot)\in LH(E)$ if both conditions are satisfied.
The maximum, $\max\{C_{1},C_{2}\}$ is called the log-H\"{o}lder constant of $\alpha(\cdot)$.
\end{defi}
\begin{defi}\label{defPdlog}
We denote $p(\cdot)\in\mathcal{P}_{d}^{log}(\mathbb{R}^{d})$, if $\frac{1}{p(\cdot)}$ is log-H\"{o}lder continuous and denote by $C_{log}(p)$ or $C_{log}$ the log-H\"{o}lder constant of $\frac{1}{p(\cdot)}$.
\end{defi}
\begin{defi}
For a $\mu_{-}$measurable function $f:\mathbb{R}^{d}\rightarrow \overline{\mathbb{R}}$, we define the modular \begin{equation}
\rho_{p(\cdot),\mu}(f)=\displaystyle\int_{\mathbb{R}^{d}\setminus\Omega_{\infty}}|f(x)|^{p(x)}\mu(dx)+\|f\|_{L^{\infty}(\Omega_{\infty},\mu)},
\end{equation}
\end{defi}
\begin{defi} The variable exponent Lebesgue space on $\mathbb{R}^{d}$, $L^{p(\cdot)}(\mathbb{R}^{d},\mu)$ consists on those $\mu\_$measurable functions $f$ for which there exists $\lambda>0$ such that $\rho_{p(\cdot),\mu}\left(\frac{f}{\lambda}\right)<\infty,$ i.e.
\begin{equation*}
L^{p(\cdot)}(\mathbb{R}^{d},\mu) =\left\{f:\mathbb{R}^{d}\to \overline{\mathbb{R}}: f \; \text{is}\; \mu_{-} \text{measurable and} \; \rho_{p(\cdot),\mu}\left(\frac{f}{\lambda}\right)<\infty, \; \text{for some} \;\lambda>0\right\}
\end{equation*}
and the norm
\begin{equation}
\|f\|_{L^{p(\cdot)}(\mathbb{R}^{d},\mu)}=\inf\left\{\lambda>0:\rho_{p(\cdot),\mu}(f/\lambda)\leq 1\right\}.
\end{equation}
\end{defi}
It is well known that, if $p(\cdot) \in L H\left(\mathbb{R}^{d}\right)$ with $1<p_{-} \leq p^{+}<\infty$ the classical Hardy-Littlewood maximal function $\mathcal{M}$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(\Bbb R^{d}),$ see \cite{dcruz1}. However, it is known that even though these are the sharpest possible point-wise conditions, they are not necessary. In \cite{LibroDenHarjHas} a necessary and sufficient condition is given for the $L^{p(\cdot)}$-boundedness of $\mathcal{M},$ but it is not an easy to work condition. The class $L H(\mathbb{R}^{d})$ is also sufficient for the boundedness on $L^{p(\cdot)}$-spaces of classical singular integrals of Calder\'on-Zygmund type, see \cite[Theorem 5.39]{dcruz}.\\
If $\mathcal{B}$ is a family of balls (or cubes) in $\mathbb{R}^{d}$, we say that $\mathcal {B}$ is $N$-finite if it has bounded overlappings for $N$, i.e., $\displaystyle\sum_{B\in\mathcal{B}}\chi_{B}(x)\leq N$ for all $x\in\mathbb {R}^{d}$; in other words, there is at most $N$ balls (resp. cubes) that intersect at the same time.\\
The following definition was introduced for the first time by Berezhno\v{\i} in \cite{Berez}, defined for a family of disjoint balls or cubes. In the context of variable spaces, it has been considered in \cite{LibroDenHarjHas}, allowing the family to have bounded overlappings.\\
\begin{defi}
Given an exponent $p(\cdot)\in\mathcal{P}(\mathbb{R}^{d})$, we will say that $p(\cdot)\in\mathcal{G}$, if for every family of balls (or cubes) $\mathcal{B}$ which is $N$-finite,
\begin{eqnarray*}
\sum_{B\in\mathcal{B}}||f\chi_{B}||_{p(\cdot)}||g\chi_{B}||_{p'(\cdot)} &\lesssim& ||f||_{p(\cdot)}||g||_{p'(\cdot)}
\end{eqnarray*}
for all functions $f\in L^{p(\cdot)}(\mathbb{R}^{d})$ and $g\in L^{p'(\cdot)}(\mathbb{R}^{d})$. The constant only depends on N.
\end{defi}
\begin{teo}[Teorema 7.3.22 of \cite{LibroDenHarjHas}]\label{implication1}
If $p(\cdot)\in LH(\mathbb{R}^{d})$, then $p(\cdot)\in\mathcal{G}$
\end{teo}
We will consider only variable Lebesgue spaces with respect to the Gaussian measure $\gamma_d,$ $L^{p(\cdot)}(\mathbb{R}^{d},\gamma_d).$ The next condition was introduced by E. Dalmasso and R. Scotto in \cite{DalSco}.
\begin{defi}\label{defipgamma}
Let $p(\cdot)\in\mathcal{P}(\mathbb{R}^{d},\gamma_{d})$, we say that $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$ if there exist constants $C_{\gamma_{d}}>0$ and $p_{\infty}\geq1$ such that
\begin{equation}
|p(x)-p_{\infty}|\leq\frac{C_{\gamma_{d}}}{|x|^{2}},
\end{equation}
for $x\in\mathbb{R}^{d}\setminus\{(0,0,\ldots,0)\}.$
\end{defi}
\begin{obs}\label{obs4.1}
If $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$, then $p(\cdot)\in LH_{\infty}(\mathbb{R}^{d})$
\end{obs}
\begin{lemma}[Lemma 2.5 of \cite{DalSco}]\label{lemaequiPgamma}
If $1<p_{-}\leq p_{+}<\infty,$ the following statements are equivalent:
\begin{itemize}
\item [(i)] $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$
\item [(ii)] There exists $p_{\infty}>1$ such that
\begin{eqnarray}
C_{1}^{-1}\leq e^{-|x|^{2}(p(x)/p_{\infty}-1)}\leq C_{1} &\;\;\hbox{and}\;\;& C_{2}^{-1}\leq e^{-|x|^{2}(p'(x)/p'_{\infty}-1)}\leq C_{2},
\end{eqnarray}
for all $x\in\mathbb{R}^{d}$, where $C_{1}=e^{C_{\gamma_{d}}/p_{\infty}}$ and $C_{2}=e^{C_{\gamma_{d}}p'_{-}/p_{\infty}}$.
\end{itemize}
\end{lemma}
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\begin{defi} The variable exponent Lebesgue space on $\mathbb{R}^{d}$, $L^{p(\cdot)}(\mathbb{R}^{d},\mu)$ consists on those $\mu\_$measurable functions $f$ for which there exists $\lambda>0$ such that $\rho_{p(\cdot),\mu}\left(\frac{f}{\lambda}\right)<\infty,$ i.e.
\begin{equation*}
L^{p(\cdot)}(\mathbb{R}^{d},\mu) =\left\{f:\mathbb{R}^{d}\to \overline{\mathbb{R}}: f \; \text{is}\; \mu_{-} \text{measurable and} \; \rho_{p(\cdot),\mu}\left(\frac{f}{\lambda}\right)<\infty, \; \text{for some} \;\lambda>0\right\}
\end{equation*}
and the norm
\begin{equation}
\|f\|_{L^{p(\cdot)}(\mathbb{R}^{d},\mu)}=\inf\left\{\lambda>0:\rho_{p(\cdot),\mu}(f/\lambda)\leq 1\right\}.
\end{equation}
\end{defi}
It is well known that, if $p(\cdot) \in L H\left(\mathbb{R}^{d}\right)$ with $1<p_{-} \leq p^{+}<\infty$ the classical Hardy-Littlewood maximal function $\mathcal{M}$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(\Bbb R^{d}),$ see \cite{dcruz1}. However, it is known that even though these are the sharpest possible point-wise conditions, they are not necessary. In \cite{LibroDenHarjHas} a necessary and sufficient condition is given for the $L^{p(\cdot)}$-boundedness of $\mathcal{M},$ but it is not an easy to work condition. The class $L H(\mathbb{R}^{d})$ is also sufficient for the boundedness on $L^{p(\cdot)}$-spaces of classical singular integrals of Calder\'on-Zygmund type, see \cite[Theorem 5.39]{dcruz}.\\
If $\mathcal{B}$ is a family of balls (or cubes) in $\mathbb{R}^{d}$, we say that $\mathcal {B}$ is $N$-finite if it has bounded overlappings for $N$, i.e., $\displaystyle\sum_{B\in\mathcal{B}}\chi_{B}(x)\leq N$ for all $x\in\mathbb {R}^{d}$; in other words, there is at most $N$ balls (resp. cubes) that intersect at the same time.\\
The following definition was introduced for the first time by Berezhno\v{\i} in \cite{Berez}, defined for a family of disjoint balls or cubes. In the context of variable spaces, it has been considered in \cite{LibroDenHarjHas}, allowing the family to have bounded overlappings.\\
\begin{defi}
Given an exponent $p(\cdot)\in\mathcal{P}(\mathbb{R}^{d})$, we will say that $p(\cdot)\in\mathcal{G}$, if for every family of balls (or cubes) $\mathcal{B}$ which is $N$-finite,
\begin{eqnarray*}
\sum_{B\in\mathcal{B}}||f\chi_{B}||_{p(\cdot)}||g\chi_{B}||_{p'(\cdot)} &\lesssim& ||f||_{p(\cdot)}||g||_{p'(\cdot)}
\end{eqnarray*}
for all functions $f\in L^{p(\cdot)}(\mathbb{R}^{d})$ and $g\in L^{p'(\cdot)}(\mathbb{R}^{d})$. The constant only depends on N.
\end{defi}
\begin{teo}[Teorema 7.3.22 of \cite{LibroDenHarjHas}]\label{implication1}
If $p(\cdot)\in LH(\mathbb{R}^{d})$, then $p(\cdot)\in\mathcal{G}$
\end{teo}
We will consider only variable Lebesgue spaces with respect to the Gaussian measure $\gamma_d,$ $L^{p(\cdot)}(\mathbb{R}^{d},\gamma_d).$ The next condition was introduced by E. Dalmasso and R. Scotto in \cite{DalSco}.
\begin{defi}\label{defipgamma}
Let $p(\cdot)\in\mathcal{P}(\mathbb{R}^{d},\gamma_{d})$, we say that $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$ if there exist constants $C_{\gamma_{d}}>0$ and $p_{\infty}\geq1$ such that
\begin{equation}
|p(x)-p_{\infty}|\leq\frac{C_{\gamma_{d}}}{|x|^{2}},
\end{equation}
for $x\in\mathbb{R}^{d}\setminus\{(0,0,\ldots,0)\}.$
\end{defi}
\begin{obs}\label{obs4.1}
If $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$, then $p(\cdot)\in LH_{\infty}(\mathbb{R}^{d})$
\end{obs}
\begin{lemma}[Lemma 2.5 of \cite{DalSco}]\label{lemaequiPgamma}
If $1<p_{-}\leq p_{+}<\infty,$ the following statements are equivalent:
\begin{itemize}
\item [(i)] $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$
\item [(ii)] There exists $p_{\infty}>1$ such that
\begin{eqnarray}
C_{1}^{-1}\leq e^{-|x|^{2}(p(x)/p_{\infty}-1)}\leq C_{1} &\;\;\hbox{and}\;\;& C_{2}^{-1}\leq e^{-|x|^{2}(p'(x)/p'_{\infty}-1)}\leq C_{2},
\end{eqnarray}
for all $x\in\mathbb{R}^{d}$, where $C_{1}=e^{C_{\gamma_{d}}/p_{\infty}}$ and $C_{2}=e^{C_{\gamma_{d}}p'_{-}/p_{\infty}}$.
\end{itemize}
\end{lemma}
Definition \mathbb{R}f{defipgamma} with Observation \mathbb{R}f{obs4.1} and Lemma \mathbb{R}f{lemaequiPgamma} end up strengthening the regularity conditions on the exponent functions $p(\cdot)$ to obtain the boundedness of the Ornstein-Uhlenbeck semigroup $\{T_{t}\}$, see \cite{MorPiUrb}. As a consequence of Theorem \mathbb{R}f{implication1}, we have
\begin{corollary}\label{solapamientoacotadoG}
If $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$, then $p(\cdot)\in\mathcal{G}.$
\end{corollary}
As we have already mentioned, the main result in this paper is the proof that the Gaussian Riesz Potentials $I_\beta$, for $\beta\geq 1$, are bounded on Gaussian variable Lebesgue spaces under the same condition of regularity on \(p(\cdot)\) considered by Dalmasso and Scotto \cite{DalSco}.
\begin{teo}\label{boundLpvarRieszPot}
Let $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ with $1<p_{-}\leq p_{+}<\infty$. Then for $\beta\geq 1$ there exists a constant $C>0,$ depending only on $p$, $\beta$ and the dimension $d$ such that
\begin{equation}
\| I_\beta f\|_{p(\cdot),\gamma_{d}} \leq C \| f\|_{p(\cdot),\gamma_{d}},
\end{equation}
for any $f \in L^{p(\cdot)}(\gamma_d).$ \\
\end{teo}
Trivially, Theorem \mathbb{R}f{boundLpvarRieszPot} give us an alternative proof of the boundedness of the Gaussian Riesz Potentials $I_\beta$, for $\beta\geq 1$ on Gaussian Lebesgue spaces $L^p(\gamma_d)$, by simply taking the exponent function constant, but the constant $C$ depends on $\beta$ and the dimension, which is weaker than the estimate obtained using Meyer's multiplier theorem mentioned above.
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\section{Proof of the main result.}
In order to prove our main result, Theorem \mathbb{R}f{boundLpvarRieszPot} we need some technical results.
\begin{lemma}\label{lema3.26CU}
Let $\rho(\cdot):\mathbb{R}^{d}\rightarrow[0,\infty)$ be such that $\rho(\cdot)\in LH_{\infty}(\mathbb{R}^{d})$, $0<\rho_{\infty}<\infty$, and let $R(x)=(e+|x|)^{-N}$, $N>d/\rho_{-}$. Then there exists a constant $C$ depending on $d$, $N$ and the $LH_{\infty}$ constant of $\rho(\cdot)$ such that given any set $E$ and
any function $F$ with $0\leq F(y)\leq 1$, for all $y\in E$,
\begin{eqnarray}
\int_{E}F^{\rho(y)}(y)dy &\leq& C\int_{E}F^{\rho_{\infty}}(y)dy + \int_{E}R^{\rho_{-}}(y)dy,\label{3.26.1} \\
\int_{E}F^{\rho_{\infty}}(y)dy &\leq& C\int_{E}F^{\rho(y)}(y)dy + \int_{E}R^{\rho^{-}}(y)dy.\label{3.26.2}
\end{eqnarray}
\end{lemma}
For the proof see Lemma 3.26 of \cite{dcruz}.
\begin{lemma}\label{lemgamma}
If $\alpha>0$, there exists a constant $C>0$ such that
\begin{equation}\label{desigualdadgamma}
\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\alpha-1}du =
C\Gamma(\alpha)<\infty
\end{equation}
\end{lemma}
\begin{proof}
Taking the change of variables $t=-log(\sqrt{1-u})$ then $u=1-e^{-2t}$\\
and $du=2e^{-2t}dt$. For $\alpha> 0$ we get
\begin{equation*}
\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\alpha-1}du =
2\int_{0}^{+\infty}t^{\alpha-1}e^{-2t}dt=C\Gamma(\alpha)<\infty
\end{equation*}
\end{proof}
\begin{lemma}\label{leminteg}
For $\beta>0$\\
\begin{enumerate}
\item[ i)]
\begin{equation}
\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{1}{\sqrt{1-u}}du\,<\infty.
\end{equation}
\item[ ii)]
\begin{equation}\label{gammaint}
\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u}du\,<\infty.
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}\quad\\
\begin{enumerate}
\item[ i)] Using H\"older's inequality with $p=\frac{3}{2}$, $q=3$ and Lemma \mathbb{R}f{lemgamma}, with $\alpha=\frac{3\beta}{2}+1$, we have that
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{du}{\sqrt{1-u}}&\leq
&\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{1}{\sqrt{1-u}}du\\
&\leq
&\left(\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{3\beta}{2}}du\right)^{1/3}
\left(\int_{0}^{1}\frac{du}{(1-u)^{3/4}}\right)^{2/3}\\
&=
&\left(\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{(\frac{3\beta}{2}+1)-1}du\right)^{1/3}
\left(\int_{0}^{1}\frac{du}{(1-u)^{3/4}}\right)^{2/3}\,<\infty
\end{eqnarray*}
\item[ii)] Let us rewrite the integral as
\begin{equation*}
\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u}du=\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)}{u}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}du,
\end{equation*}
since $\displaystyle\lim_{u\to
0}\frac{\left(-log(\sqrt{1-u})\right)}{u}=\frac{1}{2}$ and
$\displaystyle\frac{\left(-log(\sqrt{1-u})\right)}{u}$ is bounded in $(0,1/2],$ then we have we have by (\mathbb{R}f{desigualdadgamma})
\begin{eqnarray*}
\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u}du&=& \int_{0}^{1/2}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}\frac{\left(-log(\sqrt{1-u})\right)}{u}du\\
&\leq&
C\int_{0}^{1/2}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}du\,<
\infty.
\end{eqnarray*}
\end{enumerate}
\end{proof}
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We are now ready to prove the main result, Theorem \mathbb{R}f{boundLpvarRieszPot}.
\begin{proof}
As usual, we split the operator $I_\beta$ in its local part and its global part
\begin{equation*}
I_{\beta}f(x)=I_{\beta,L}f(x)+I_{\beta,G}f(x),
\end{equation*}
where
\begin{equation*}
I_{\beta,L}f(x)=I_{\beta}(f\chi_{B_{h}(\cdot)} )(x)
\end{equation*}
is the local part,
\begin{equation*}
I_{\beta,G}f(x)=I_{\beta}(f\chi_{B^{c}_{h}(\cdot)} )(x)
\end{equation*}
is the global part, and for $x\in \Bbb R^{d}$ by taking $m(x)=1\wedge \frac{1}{|x|}$,\\
$B_{h}(x):=\lbrace y\in \Bbb R^{d}:|x-y|<dm(x)\rbrace$ is an $hiperbolic$ $ball$ ($admissible$ $ball$).\\
Let us take
$\omega(s)=\displaystyle\frac{e^{-\frac{|y-e^{-s}x|^{2}}{1-e^{-2s}}}}{(1-e^{-2s})^{\frac{d}{2}}}$,
then, from (\mathbb{R}f{kernelRiesz})
\begin{eqnarray*}
N_{\beta/2}(x,y)&=&\frac{1}{\pi^{\frac{d}{2}}\Gamma(\frac{\beta}{2})}\int_{0}^{+\infty}
t^{\frac{\beta}{2}-1}\left(\omega(t)-\omega(+\infty)\right)dt\\
&=&\frac{1}{\pi^{\frac{d}{2}}\Gamma(\frac{\beta}{2})}\int_{0}^{+\infty}
t^{\frac{\beta}{2}-1}\left(-\int_{t}^{+\infty}\frac{\partial
\omega(s)}{\partial s}ds\right)dt.
\end{eqnarray*}
Thus, using Hardy's inequality, see \cite{st1}
$$
\left|N_{\beta/2}(x,y)\right|\leq \frac{1}{\pi^{\frac{d}{2}}\Gamma(\frac{\beta}{2})}\int_{0}^{+\infty}
t^{\frac{\beta}{2}-1}\int_{t}^{+\infty}\left|\frac{\partial
\omega(s)}{\partial s}\right|ds\;dt \leq
\frac{1}{\pi^{\frac{d}{2}}\Gamma(\frac{\beta}{2})}\frac{2}{\beta}\int_{0}^{+\infty}
s^{\frac{\beta}{2}}\left|\frac{\partial
\omega(s)}{\partial s}\right|ds.
$$
Now,
\begin{eqnarray*}
\frac{\partial \omega(s)}{\partial
s}&=&\frac{(1-e^{-2s})^{\frac{d}{2}}e^{-\frac{|y-e^{-s}x|^{2}}{1-e^{-2s}}}}{(1-e^{-2s})^{d}}
\left(\frac{-2(1-e^{-2s})(y-e^{-s}x)\cdot(e^{-s}x)+|y-e^{-s}x|^{2}e^{-2s}}{(1-e^{-2s})^{2}}\right)\\
&&\hspace{4.5cm}-\frac{1}{(1-e^{-2s})^{d}}\left(\frac{d}{2}e^{\frac{|y-e^{-s}x|^{2}}{1-e^{-2s}}}(1-e^{-2s})^{\frac{d}{2}-1}2e^{-2s}\right)\\
&=&\omega(s)\left(\frac{-2(1-e^{-2s})(y-e^{-s}x)\cdot(e^{-s}x)+|y-e^{-s}x|^{2}e^{-2s}}{(1-e^{-2s})^{2}}-\frac{de^{-2s}}{(1-e^{-2s})}\right).
\end{eqnarray*}
Then, taking $u=1-e^{-2s},\; du=2e^{-2s}ds,$ i.e., $e^{-s} = \sqrt{1-u},$ we have
\begin{eqnarray*}
\left|N_{\beta/2}(x,y)\right|&\leq
&\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}}}
\\
&&\hspace{0.2cm}\times\left(\frac{2u|y-\sqrt{1-u}x|\sqrt{1-u}|x|+|y-\sqrt{1-u}x|^{2}(1-u)}{u^{2}}+\frac{d(1-u)}{u}\right)\frac{du}{2(1-u)}\\
&=&\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}}}
\left(\frac{|y-\sqrt{1-u}x||x|}{u\sqrt{1-u}}+\frac{|y-\sqrt{1-u}x|^{2}}{2u^{2}}+\frac{d}{2u}\right)du\\
&=& \int_{0}^{1}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u^{\frac{d}{2}+1}}e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}
\frac{|y-\sqrt{1-u}x||x|}{\sqrt{1-u}}du \\
&& \hspace{1cm}+\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}+1}}\frac{|y-\sqrt{1-u}x|^{2}}{2u}du\\
&& \hspace{1.5cm}+\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}}}\frac{d}{2u} du\\
&&=I+II+III,
\end{eqnarray*}
where
\begin{equation}
I = \int_{0}^{1}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u^{\frac{d}{2}+1}}e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}
\frac{|y-\sqrt{1-u}x||x|}{\sqrt{1-u}}du,
\end{equation}
\begin{equation}
II= \int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}+1}}\frac{|y-\sqrt{1-u}x|^{2}}{2u}du,
\end{equation}
and
\begin{equation}
III = \int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}}}\frac{d}{2u} du.
\end{equation}
\begin{itemize}
\item Let us study the local part first. We need to bound each of the terms $I$, $II$ and $III$ in this part.
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\begin{itemize}
\item Let us study the local part first. We need to bound each of the terms $I$, $II$ and $III$ in this part.
For $I$, since we are in the local part
$|x-y| \leq \frac{d}{|x|}$, then we have $|x-y||x| \leq d $ therefore,
\begin{eqnarray}\label{ineqlocpart}
\nonumber |y-\sqrt{1-u}\,x|^{2} & \geq& (|y-x|-|x|(1-\sqrt{1-u}))^{2} \\
& \geq& |y-x|^{2}-2|x||y-x| \frac{u}{1+\sqrt{1-u}} \geq|y-x|^{2}-2 d \; u.
\end{eqnarray}
On the other hand, it is well known that, there exist $C>0$ such that\\
for any $x>0$, $\alpha \geq 0$ and $c>0$,
\begin{equation}\label{expineq}
x^\alpha e^{-cx^2} \leq C \hspace{0.5 cm}\cdot
\end{equation}
Thus, using (\mathbb{R}f{ineqlocpart}) and (\mathbb{R}f{expineq}) twice, with $\alpha =1,\; c= \frac12$ and $\alpha=d-1, c=\frac12$, we get
\begin{eqnarray*}
I&=
&\int_{0}^{1}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u^{\frac{d}{2}}}e^{-\frac{|y-\sqrt{1-u}x|^{2}}{2u}}
\left(\left|\frac{y-\sqrt{1-u}x}{\sqrt{u}}
\right|e^{-\frac{|y-\sqrt{1-u}x|^{2}}{2u}}\right)\frac{|x|}{\sqrt{u}\sqrt{1-u}}du\\
&\leq
&C|x|\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}
\frac{du}{\sqrt{u}\sqrt{1-u}}
\end{eqnarray*}
\begin{eqnarray*}
&=&
C|x|\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d-1}{2}}}
\frac{du}{u\sqrt{1-u}}\\
&\leq
&\frac{C|x|}{|x-y|^{d-1}}\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}
\frac{du}{u\sqrt{1-u}}.\\
\end{eqnarray*}
By Lemma \mathbb{R}f{leminteg} we get
\begin{eqnarray*}
&&\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}
\frac{du}{u\sqrt{1-u}}\\
&& \hspace{1cm} = \int_{0}^{1/2}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}
\frac{du}{u\sqrt{1-u}}\; +
\;\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}
\frac{du}{u\sqrt{1-u}}\\
&& \hspace{1cm} \leq C\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}}{u}
du\; +
\;C\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}
\frac{du}{\sqrt{1-u}}<\infty,
\end{eqnarray*}
since $\displaystyle\frac{1}{u}$ in bounded on $[1/2,1]$ and
$\displaystyle\frac{1}{\sqrt{1-u}}$ is bounded on $[0,1/2]$. Thus, we have
\begin{equation*}
I\leq \frac{C|x|}{|x-y|^{d-1}}.
\end{equation*}
For $II$, we use again (\mathbb{R}f{ineqlocpart}) and (\mathbb{R}f{expineq}) with $\alpha =2$ and $c=\frac12$ we have
\begin{eqnarray*}
II &=&
\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}+1}}\frac{|y-\sqrt{1-u}x|^{2}}{2u}du\\
&\leq &
C\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{2u}}}{u^{\frac{d}{2}+1}}du\\
&\leq
& C\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}+1}}du\\
&=&C \int_{0}^{1/2}\frac{\left(-log(\sqrt{1-u})\right)}{u}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}du\;+
\;\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}+1}}du.\\
\end{eqnarray*}
Since $\displaystyle\lim_{u\to
0}\frac{\left(-log(\sqrt{1-u})\right)}{u}=\frac{1}{2}$, this function is bounded on $[0,1/2]$ and
$\displaystyle\frac{1}{u}$ is bounded on $[1/2,1]$, thus we get
\begin{eqnarray*}
II&\leq
&C\int_{0}^{1/2}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}du\;+
\;C\int_{1/2}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}du\\
&\leq
&C\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}du\;+
\;C\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d}{2}}}du\\
&=&C\mathcal{K}_{2}(x-y)\;+\;CG_{2}(x-y),
\end{eqnarray*}
where
$$ \mathcal{K}_2(x) := \int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}-1}\frac{e^{-\frac{|x|^{2}}{2u}}}{u^{\frac{d}{2}}}du,$$
and
$$
G_{2}(x):=\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|x|^{2}}{2u}}}{u^{\frac{d-1}{2}}}\frac{du}{\sqrt{u}}.
$$
Again by (\mathbb{R}f{expineq})
\begin{eqnarray*}
G_{2}(x-y)&=&\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-x|^{2}}{2u}}}{u^{\frac{d-1}{2}}}\frac{du}{\sqrt{u}}\\
&\leq& \frac{C}{|x-y|^{d-1}}\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{du}{\sqrt{u}}.
\end{eqnarray*}
Now, by H\"older's inequality with $p=\frac{3}{2}$ y $q=3$ we have
\begin{equation*}
\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{du}{\sqrt{u}}\leq
\left(\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{3\beta}{2}}du\right)^{1/3}
\left(\int_{0}^{1}\frac{du}{u^{3/4}}\right)^{2/3}.
\end{equation*}
Thus, by Lemma (\mathbb{R}f{lemgamma}) we obtain
\begin{equation*}
\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{3\beta}{2}}du=\int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{(\frac{3\beta}{2}+1)-1}du\;<\;\infty
\end{equation*}
and trivially
$\displaystyle{\int_{0}^{1}\frac{du}{u^{3/4}}}< \infty.$\\
Finally for $III$, by analogous arguments as in $II$, we get
$$
III = \int_{0}^{1}\left(-log(\sqrt{1-u})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{|y-\sqrt{1-u}x|^{2}}{u}}}{u^{\frac{d}{2}}}\frac{d}{2u} du\leq C \mathcal{K}_{2}(x-y)\;+\;CG_{2}(x-y).
$$
Therefore,
\begin{eqnarray*}
\left|N_{\beta/2}(x,y)\right|&\leq & I+II+III\\
&\leq & \frac{C|x|}{|x-y|^{d-1}}\,+\,C\mathcal{K}_{2}(x-y)\,+\,\frac{C}{|x-y|^{d-1}}\\
&\leq & C\frac{|x|+1}{|x-y|^{d-1}}\,+\,C\mathcal{K}_{2}(x-y)= C\mathcal{K}_{3}(x,y)\,+\,C\mathcal{K}_{2}(x-y),
\end{eqnarray*}
where
$$ \mathcal{K}_{3}(x,y) := \frac{|x|+1}{|x-y|^{d-1}}.$$
Thus, using the above estimates, we conclude that the local part
$I_{\beta,L}$ can be bounded as
\begin{eqnarray*}
|I_{\beta,L} f(x)| &=& |I_{\beta} (f\chi_{B_h(\cdot)})(x)| = \Big| \int_{B_h(x)} N_{\beta/2} (x,y) f(y) \;dy \Big|\\
&\lesssim& \int_{B_h(x)} {\mathcal K}_3(x,y) |f(y)| \;dy + \int_{B_h(x)} {\mathcal K}_2(x-y) | f(y) |\;dy \\
&=& IV + V.\\
\end{eqnarray*}
Now, to bound IV and V, we need to take a countable family of admissible balls $ \mathcal{F}$ that satisfies the condition of Lemma 4.3 of \cite{urbina2019}. In particular, \(\mathcal{F}\) verifies
\begin{enumerate}
\item[i)] For each $B \in\mathcal{F}$ let \(\tilde{B}=2 B,\) then, the family of those balls $\tilde{\mathcal{F}}= \{B(0,1),\{\tilde{B}\}_{B \in \mathcal{F}}\}$ is a covering of \(\mathbb{R}^{d}\);
\item[ii)] \(\mathcal{F}\) has a bounded overlaps property;
\item[iii)] Every ball \(B \in \mathcal{F}\) is contained in an admissible ball, and therefore for
any pair \(x, y \in B, e^{-|x|^{2}} \sim e^{-|y|^{2}}\) with constants independent of \(B\)
\item[iv)] There exists a uniform positive constant \(C_{d}\) such that, if \(x \in B \in \mathcal{F}\) then \(B_h(x) \subset C_{d} B:=\hat{B} .\) Moreover, the collection \(\hat{\mathcal{F}}=\{\hat{B}\}_{B \in \mathcal{F}}\) also satisfies the properties ii) and iii).\\
\end{enumerate}
Given $B \in \mathcal{F},$ if $x \in B$ then $B_h(x) \subset
\hat{B}$, we get,
\begin{eqnarray*}
IV &=& (1+|x|) \sum_{k=0}^\infty \int_{2^{-(k+1)}C_d m(x) < |x-y| < 2^{-k}C_d m(x)} \frac{|f(y)| \chi_ {\hat{B}}}{|x-y|^{d-1}} dy\\
&\leq& C_d 2^d \mathcal{M}(f\chi_{\hat{B}})(x) (1+|x|) m(x)
\sum_{k=0}^\infty 2^{-(k+1)} \leq C
\mathcal{M}(f\chi_{\hat{B}})(x)\chi_{B}(x),
\end{eqnarray*}
where $\mathcal{M}(g)$ is the classical Hardy-Littlewood maximal
function of the function $g.$\\
On the other hand, let us consider the function
$\varphi(y) =\frac{1}{\pi^{d/2}} e^{-\frac{1}{2}|y|^2},$
then $\displaystyle\int_{\mathbb{R}^d} \varphi(y) dy =1$. It is well known that
$\varphi$ is a non-increasing radial function, and given $t>0,\,$
we rescale this function as $\varphi_{\sqrt{t}} (y) = t^{-d/2}
\varphi(y/\sqrt{t}),$ and, since $0\leq \varphi \in
L^1(\mathbb{R}^d),$ $\left\{\varphi_{\sqrt{t}}\right \}_{t>0}\;$ is the
classical (Gauss-Weiertrass) approximation of the identity in $\mathbb{R}^d.\,$ Then,
since\\
$$\displaystyle\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}
dt <\infty,$$
we get
\begin{eqnarray*}
V &=& \int_{B_h(x)} \mathcal{K}_2(x-y) | f(y) |\;dy =
\int_{B_h(x)} \Big(\int_0^1 \varphi_{\sqrt{t}}(x-y)
\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}
dt\Big) |f(y)| dy\\
&\leq& \int_{B_h(x)} \Big(\sup_{t>0} \varphi_{\sqrt{t}}(x-y)
\Big)
\Big(\int_0^1\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}
dt\Big) |f(y)| dy\\
&\leq& C \int_{B_h(x)} \Big(\sup_{t>0} \varphi_{\sqrt{t}}(x-y)
\Big) |f(y)| dy.
\end{eqnarray*}
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Therefore,
\begin{eqnarray*}
\left|N_{\beta/2}(x,y)\right|&\leq & I+II+III\\
&\leq & \frac{C|x|}{|x-y|^{d-1}}\,+\,C\mathcal{K}_{2}(x-y)\,+\,\frac{C}{|x-y|^{d-1}}\\
&\leq & C\frac{|x|+1}{|x-y|^{d-1}}\,+\,C\mathcal{K}_{2}(x-y)= C\mathcal{K}_{3}(x,y)\,+\,C\mathcal{K}_{2}(x-y),
\end{eqnarray*}
where
$$ \mathcal{K}_{3}(x,y) := \frac{|x|+1}{|x-y|^{d-1}}.$$
Thus, using the above estimates, we conclude that the local part
$I_{\beta,L}$ can be bounded as
\begin{eqnarray*}
|I_{\beta,L} f(x)| &=& |I_{\beta} (f\chi_{B_h(\cdot)})(x)| = \Big| \int_{B_h(x)} N_{\beta/2} (x,y) f(y) \;dy \Big|\\
&\lesssim& \int_{B_h(x)} {\mathcal K}_3(x,y) |f(y)| \;dy + \int_{B_h(x)} {\mathcal K}_2(x-y) | f(y) |\;dy \\
&=& IV + V.\\
\end{eqnarray*}
Now, to bound IV and V, we need to take a countable family of admissible balls $ \mathcal{F}$ that satisfies the condition of Lemma 4.3 of \cite{urbina2019}. In particular, \(\mathcal{F}\) verifies
\begin{enumerate}
\item[i)] For each $B \in\mathcal{F}$ let \(\tilde{B}=2 B,\) then, the family of those balls $\tilde{\mathcal{F}}= \{B(0,1),\{\tilde{B}\}_{B \in \mathcal{F}}\}$ is a covering of \(\mathbb{R}^{d}\);
\item[ii)] \(\mathcal{F}\) has a bounded overlaps property;
\item[iii)] Every ball \(B \in \mathcal{F}\) is contained in an admissible ball, and therefore for
any pair \(x, y \in B, e^{-|x|^{2}} \sim e^{-|y|^{2}}\) with constants independent of \(B\)
\item[iv)] There exists a uniform positive constant \(C_{d}\) such that, if \(x \in B \in \mathcal{F}\) then \(B_h(x) \subset C_{d} B:=\hat{B} .\) Moreover, the collection \(\hat{\mathcal{F}}=\{\hat{B}\}_{B \in \mathcal{F}}\) also satisfies the properties ii) and iii).\\
\end{enumerate}
Given $B \in \mathcal{F},$ if $x \in B$ then $B_h(x) \subset
\hat{B}$, we get,
\begin{eqnarray*}
IV &=& (1+|x|) \sum_{k=0}^\infty \int_{2^{-(k+1)}C_d m(x) < |x-y| < 2^{-k}C_d m(x)} \frac{|f(y)| \chi_ {\hat{B}}}{|x-y|^{d-1}} dy\\
&\leq& C_d 2^d \mathcal{M}(f\chi_{\hat{B}})(x) (1+|x|) m(x)
\sum_{k=0}^\infty 2^{-(k+1)} \leq C
\mathcal{M}(f\chi_{\hat{B}})(x)\chi_{B}(x),
\end{eqnarray*}
where $\mathcal{M}(g)$ is the classical Hardy-Littlewood maximal
function of the function $g.$\\
On the other hand, let us consider the function
$\varphi(y) =\frac{1}{\pi^{d/2}} e^{-\frac{1}{2}|y|^2},$
then $\displaystyle\int_{\mathbb{R}^d} \varphi(y) dy =1$. It is well known that
$\varphi$ is a non-increasing radial function, and given $t>0,\,$
we rescale this function as $\varphi_{\sqrt{t}} (y) = t^{-d/2}
\varphi(y/\sqrt{t}),$ and, since $0\leq \varphi \in
L^1(\mathbb{R}^d),$ $\left\{\varphi_{\sqrt{t}}\right \}_{t>0}\;$ is the
classical (Gauss-Weiertrass) approximation of the identity in $\mathbb{R}^d.\,$ Then,
since\\
$$\displaystyle\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}
dt <\infty,$$
we get
\begin{eqnarray*}
V &=& \int_{B_h(x)} \mathcal{K}_2(x-y) | f(y) |\;dy =
\int_{B_h(x)} \Big(\int_0^1 \varphi_{\sqrt{t}}(x-y)
\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}
dt\Big) |f(y)| dy\\
&\leq& \int_{B_h(x)} \Big(\sup_{t>0} \varphi_{\sqrt{t}}(x-y)
\Big)
\Big(\int_0^1\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}
dt\Big) |f(y)| dy\\
&\leq& C \int_{B_h(x)} \Big(\sup_{t>0} \varphi_{\sqrt{t}}(x-y)
\Big) |f(y)| dy.
\end{eqnarray*}
Again, using the family $\mathcal{F}$, if $x \in B$ then $B_h(x)
\subset \hat{B}$. By a similar argument as before and as in as result in Stein's book \cite[Chapter II \S4, Theorem 4]{st1},
\begin{eqnarray*}
V &=& \int_{B_h(x)} \mathcal{K}_2(x-y) | f(y) |\;dy \leq C
\int_{\mathbb{R}^d} \left(\sup_{t>0} \varphi_{\sqrt{t}}(x-y) \right)
|f(y)| \chi_{\hat{B}}(y) dy\\
&\lesssim& \sum_{B \in \mathcal{F}} \left| \left( \sup_{t>0}\varphi_{\sqrt{t}} * |f \chi_{\hat{B}}|\right)(x) \right| \chi_{B}(x)\leq \sum_{B \in \mathcal{F}} \mathcal{M}(f \chi_{\hat{B}})(x)
\chi_{B}(x),
\end{eqnarray*}
which yields,
$|I_{\beta,L}f(x)|\leq\displaystyle\sum_{B \in \mathcal{F}} \mathcal{M}(f \chi_{\hat{B}})(x)
\chi_{B}(x).$\\
Then, for \(f \in L^{p(\cdot)}\left(\mathbb{R}^{d}, \gamma_d\right)\) we will use the characterization of the norm by duality, we get
\begin{equation}\label{ineq2}
\left\|I_{\beta,L}(f)\right\|_{p(\cdot), \gamma_{d}} \leq 2 \sup _{\|g\|_{p^{\prime}(\cdot), \gamma_{d}} \leq 1} \int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_d(dx).
\end{equation}
Using the estimates above, we get
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_d(dx) &\lesssim&
\sum_{B \in \mathcal{F}} \int_{B} \mathcal{M}\left(f \chi_{\hat{B}(\cdot)}\right)(x)|g(x)| e^{-|x|^{2}} d x \\
&& \approx\sum_{B \in \mathcal{F}} e^{-\left|c_{B}\right|^{2}} \int_{B} \mathcal{M}\left(f \chi_{\hat{B}(\cdot)}\right)(x)|g(x)| d x,
\end{eqnarray*}
where \(c_{B}\) is the center of \(B\) and \(\hat{B}\) and we have used property iii) above, i.e. that over each ball of the family \(\mathcal{F},\) the values of \(\gamma_d\) are all equivalent.\\ Applying H\"older's inequality for \(p(\cdot)\) and \(p^{\prime}(\cdot)\) with respect of the Lebesgue measure and the boundedness of \(\mathcal{M}\) on $L^{p(\cdot)}\left(\mathbb{R}^{d}\right),$ we get
\begin{eqnarray}\label{ineq3}
\nonumber \int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_{d}(d x) &\lesssim & \sum_{B \in \mathcal{F}} e^{-\left|c_{B}\right|^{2}}\left\|\mathcal{M}\left(f \chi_{\hat{B}(\cdot)}\right) \chi_{B}\right\|_{p(\cdot)}\left\|g \chi_{B}\right\|_{p^{\prime}(\cdot)} \\
\nonumber &\lesssim & \sum_{B \in \mathcal{F}} \mathrm{e}^{-\left|c_{B}\right|^{2}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot)}\left\|g \chi_{B}\right\|_{p^{\prime}(\cdot)} \\
&=& \sum_{B \in \mathcal{F}} \mathrm{e}^{-\left|c_{B}\right|^{2} / p_{\infty}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot)} e^{-\left|c_{B}\right|^{2} / p_{\infty}^{\prime}}\left\|g \chi_{\hat{B}}\right\|_{p^{\prime}(\cdot)}.
\end{eqnarray}
Since \(p \in P_{\gamma_{d}}^{\infty}\left(\mathbb{R}^{d}\right)\) and \(p_{-}>1$ then $p'\in P_{\gamma_{d}}^{\infty}\left(\mathbb{R}^{d}\right) .\) Thus, from Lemma \(1.4,\) for
every \(x \in \mathbb{R}^{d}\)
\begin{equation}\label{equiv}
e^{-|x|^{2}\left(p(x) / p_{\infty}-1\right)} \leq C_{1} \text { and } e^{-|x|^{2}\left(p^{\prime}(x) / p_{\infty}^{\prime}-1\right)} \leq C_{2}.
\end{equation}
Moreover, since the values of the Gaussian measure \(\gamma_{d}\) are all equivalent on any ball $\hat{B} \in \tilde{\mathcal{F}}$, we have
\begin{eqnarray*}
\int_{\hat{B}}\left(\frac{|f(y)|}{e^{\left|c_{B}\right|^{2} / p_{\infty}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}}\right)^{p(y)} d y &\lesssim & \int_{\hat{B}}\left(\frac{|f(y)|}{\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}}\right)^{p(y)} e^{-|y|^{2}\left(p(y) / p_{\infty}-1\right)} \gamma_{d}(dy) \\
&\lesssim &\int_{\hat{B}}\left(\frac{|f(y)|}{\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}}\right)^{p(y)} \gamma_{d}(dy) \lesssim 1,
\end{eqnarray*}
which yields
$$
e^{-\left|c_{B}\right|^{2} / p_{\infty}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot)} \lesssim \left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}.
$$
Similarly, by the second inequality of (\mathbb{R}f{equiv}) we also get
$$
e^{-\left|c_{B}\right|^{2} / p_{\infty}^{\prime}}\left\|g \chi_{\hat{B}}\right\|_{p^{\prime}(\cdot)} \lesssim \left\|g \chi_{\hat{B}}\right\|_{p^{\prime}(\cdot), \gamma_{d}}.
$$
Replacing both estimates in (\mathbb{R}f{ineq3}) we obtain
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_{d}(d x) & \lesssim &\sum_{B \in \mathcal{F}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}\left\|g \chi_{\hat{B}}\right\|_{p^{\prime}(\cdot), \gamma_{d}} \\
&=& \sum_{B \in \mathcal{F}}\left\|f \chi_{\hat{B}} e^{-|\cdot|^{2} / p(\cdot) |}\right\|_{p(\cdot)}\left\|g \chi_{\hat{B}} e^{-|\cdot|^{2} / p^{\prime}(\cdot)}\right\|_{p^{\prime}(\cdot)}.
\end{eqnarray*}
Since the family of balls \(\hat{\mathcal{F}}\) has bounded overlaps, from Corollary 1.1 applied to \(f e^{-|\cdot|^{2} / p(\cdot)} \in L^{p(\cdot)}(\mathbb{R}^{d})\) and \(g e^{-|\cdot|^{2} / p^{\prime}(\cdot)} \in L^{p^{\prime}(\cdot)}(\mathbb{R}^{d}),\) it follows that
$$
\int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_{d}(d x) \leqslant\|f\|_{p(\cdot), \gamma_{d}}\|g\|_{p^{\prime}(\cdot), \gamma_{d}}.
$$
Taking supremum over all functions \(g\) with \(\|g\|_{p^{\prime}(\cdot), \gamma_{d}} \leq 1,\) from (\mathbb{R}f{ineq2}) we get finally
$$
\left\|I_{\beta,L}(f)\right\|_{p(\cdot), \gamma_{d}}\leq C\|f\|_{p(\cdot), \gamma_{d}}.\\
$$
Therefore, the local part $I_{\beta,L}$ is bounded in $L^{p(\cdot)}(\gamma_d)$.\\
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Moreover, since the values of the Gaussian measure \(\gamma_{d}\) are all equivalent on any ball $\hat{B} \in \tilde{\mathcal{F}}$, we have
\begin{eqnarray*}
\int_{\hat{B}}\left(\frac{|f(y)|}{e^{\left|c_{B}\right|^{2} / p_{\infty}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}}\right)^{p(y)} d y &\lesssim & \int_{\hat{B}}\left(\frac{|f(y)|}{\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}}\right)^{p(y)} e^{-|y|^{2}\left(p(y) / p_{\infty}-1\right)} \gamma_{d}(dy) \\
&\lesssim &\int_{\hat{B}}\left(\frac{|f(y)|}{\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}}\right)^{p(y)} \gamma_{d}(dy) \lesssim 1,
\end{eqnarray*}
which yields
$$
e^{-\left|c_{B}\right|^{2} / p_{\infty}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot)} \lesssim \left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}.
$$
Similarly, by the second inequality of (\mathbb{R}f{equiv}) we also get
$$
e^{-\left|c_{B}\right|^{2} / p_{\infty}^{\prime}}\left\|g \chi_{\hat{B}}\right\|_{p^{\prime}(\cdot)} \lesssim \left\|g \chi_{\hat{B}}\right\|_{p^{\prime}(\cdot), \gamma_{d}}.
$$
Replacing both estimates in (\mathbb{R}f{ineq3}) we obtain
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_{d}(d x) & \lesssim &\sum_{B \in \mathcal{F}}\left\|f \chi_{\hat{B}}\right\|_{p(\cdot), \gamma_{d}}\left\|g \chi_{\hat{B}}\right\|_{p^{\prime}(\cdot), \gamma_{d}} \\
&=& \sum_{B \in \mathcal{F}}\left\|f \chi_{\hat{B}} e^{-|\cdot|^{2} / p(\cdot) |}\right\|_{p(\cdot)}\left\|g \chi_{\hat{B}} e^{-|\cdot|^{2} / p^{\prime}(\cdot)}\right\|_{p^{\prime}(\cdot)}.
\end{eqnarray*}
Since the family of balls \(\hat{\mathcal{F}}\) has bounded overlaps, from Corollary 1.1 applied to \(f e^{-|\cdot|^{2} / p(\cdot)} \in L^{p(\cdot)}(\mathbb{R}^{d})\) and \(g e^{-|\cdot|^{2} / p^{\prime}(\cdot)} \in L^{p^{\prime}(\cdot)}(\mathbb{R}^{d}),\) it follows that
$$
\int_{\mathbb{R}^{d}}\left|I_{\beta,L}f(x)\right||g(x)| \gamma_{d}(d x) \leqslant\|f\|_{p(\cdot), \gamma_{d}}\|g\|_{p^{\prime}(\cdot), \gamma_{d}}.
$$
Taking supremum over all functions \(g\) with \(\|g\|_{p^{\prime}(\cdot), \gamma_{d}} \leq 1,\) from (\mathbb{R}f{ineq2}) we get finally
$$
\left\|I_{\beta,L}(f)\right\|_{p(\cdot), \gamma_{d}}\leq C\|f\|_{p(\cdot), \gamma_{d}}.\\
$$
Therefore, the local part $I_{\beta,L}$ is bounded in $L^{p(\cdot)}(\gamma_d)$.\\
\item Now, let us study the global part. Again, since
$$
\left|N_{\beta/2}(x,y)\right|\leq
I+II+III,
$$
we need to estimate each term in this part. As usual for the global part, the arguments are completely different but are based on the following technical result, obtained S. P\'erez \cite[Lemma 3.1]{pe}, see also \cite[\S4.5]{urbina2019}.To simplify the notation, in what follows we denote
$$
a=a(x, y):=|x|^{2}+|y|^{2}, b=b(x, y):=2\langle x, y\rangle \text {, }
$$
$$
u(t)=u(t ; x, y):=\frac{|y-\sqrt{1-t x}|^{2}}{t}=\frac{a}{t}-\frac{\sqrt{1-t}}{t} b-|x|^{2},
$$
$$t_{0}=2 \frac{\sqrt{a^{2}-b^{2}}}{a+\sqrt{a^{2}-b^{2}}},$$
then
$$
u\left(t_{0}\right)=\frac{\sqrt{a^{2}-b^{2}}}{2}+\frac{a}{2}-|x|^{2}=\frac{|y|^{2}-|x|^{2}}{2}+\frac{\sqrt{a^{2}-b^{2}}}{2}
$$
and
$$
t_{0} \sim \frac{\sqrt{a^{2}-b^{2}}}{a} \sim \frac{\sqrt{a-b}}{\sqrt{a+b}}=\frac{|x-y|}{|x+y|}.
$$
It is well known that $t_{0}<1$, the minimum of $u(t)$ is attained at
$t_{0}$ and
$$\frac{1}{t_{0}^{d/2}}\lesssim |x+y|^{d}.$$
For details and other properties of these terms, see \cite{pe}, \cite{TesSon} or \cite{urbina2019}.\\
Let us fix $x\in \Bbb R^{d}$ and consider $E_{x}=\lbrace y\in \Bbb R^{d}:b>0\rbrace$.\\
\begin{itemize}
\item \underline{Case $b\leq 0$}:\\
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First, for $0<
{$\Box $}silon <1$, using inequality (\mathbb{R}f{expineq}) we have
\begin{eqnarray*}
I&=&
\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-u(t)}}{t^{\frac{d}{2}}}\frac{\left|y-\sqrt{1-t}x\right||x|}{t\sqrt{1-t}}dt\\
&=&
|x|\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-
{$\Box $}silon
u(t)-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}\frac{\left|y-\sqrt{1-t}x\right|}{\sqrt{t}\sqrt{1-t}}dt\\
&\leq &
C_{
{$\Box $}silon}|x|\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}\frac{dt}{\sqrt{1-t}}.
\end{eqnarray*}
Since
$\displaystyle\frac{\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}}{\sqrt{1-t}}$
is continuous on $[0,1/2]$ and therefore bounded, we get
\begin{equation*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}\frac{dt}{\sqrt{1-t}}
\leq C_{\beta}
\int_{0}^{1/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}dt,
\end{equation*}
and since $0<t<1/2$, $\frac{1}{\sqrt{t}}<\frac{1}{t}$ and then, by (\mathbb{R}f{desint}), we get
\begin{eqnarray*}
\int_{0}^{1/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}dt&\leq
&e^{(1-
{$\Box $}silon)|x|^{2}}\int_{0}^{1/2}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d+1}{2}}}dt\\
&\leq
&e^{(1-
{$\Box $}silon)|x|^{2}}\int_{0}^{1/2}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt\\
&\leq
&e^{(1-
{$\Box $}silon)|x|^{2}}C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)a}= C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)|y|^{2}}.
\end{eqnarray*}
Analogously,
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}\frac{dt}{\sqrt{1-t}}
&\leq
&e^{(1-
{$\Box $}silon)|x|^{2}}\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d+1}{2}}}\frac{dt}{\sqrt{1-t}}\\
&\leq &C_{d}e^{(1-
{$\Box $}silon)|x|^{2}}\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{\sqrt{1-t}}dt,
\end{eqnarray*}
since $\displaystyle\frac{1}{t^{\frac{d+1}{2}}}$ is bounded on
$[1/2,1]$, also as $1/2<t<1$ we have $-\frac{a}{t}<-a$ and then by Lemma \mathbb{R}f{leminteg}
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{\sqrt{1-t}}dt&\leq
&e^{-(1-
{$\Box $}silon)a}\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{dt}{\sqrt{1-t}}\\
&=&C_{\beta}e^{-(1-
{$\Box $}silon)a}.
\end{eqnarray*}
Thus,
\begin{equation*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d+1}{2}}}\frac{dt}{\sqrt{1-t}}
\leq C_{d,\beta}
e^{(1-
{$\Box $}silon)|x|^{2}}e^{-(1-
{$\Box $}silon)a}=Ce^{-(1-
{$\Box $}silon)|y|^{2}}.
\end{equation*}
Therefore
$$I \leq C_
{$\Box $}silon |x| e^{-(1-
{$\Box $}silon)|y|^{2}}.$$
Now, using again inequality (\mathbb{R}f{expineq}), we get
\begin{eqnarray*}
II&=&
\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-u(t)}}{t^{\frac{d}{2}}}\frac{\left|y-\sqrt{1-t}x\right|^{2}}{2t^{2}}dt\\
&=&
\frac{1}{2}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-
{$\Box $}silon
u(t)-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}+1}}\frac{\left|y-\sqrt{1-t}x\right|^{2}}{t}dt\\
&\leq
&C_{
{$\Box $}silon}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}+1}}dt\\
&\leq
&C_{
{$\Box $}silon}e^{(1-
{$\Box $}silon)|x|^{2}}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt.\\
\end{eqnarray*}
Now,
\begin{equation*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt
\leq
C\int_{0}^{1/2}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt
\leq
C\int_{0}^{1}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt.
\end{equation*}
Therefore, by taking the change of variables $s=a(\frac{1}{t}-1)$, we get
\begin{eqnarray}\label{desint}
\nonumber\int_{0}^{1}\frac{e^{-(1-
{$\Box $}silon)\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt&=
&\int_{0}^{+\infty}e^{-(1-
{$\Box $}silon)(s+a)}\left(\frac{s+a}{a}\right)^{\frac{d}{2}+1}\frac{a}{\left(s+a\right)^{2}}ds\\
\nonumber&=
&\frac{e^{-(1-
{$\Box $}silon)a}}{a^{\frac{d}{2}}}\int_{0}^{+\infty}e^{-(1-
{$\Box $}silon)s}\left(s+a\right)^{\frac{d}{2}-1}ds\\
\nonumber &\leq
&\frac{Ce^{-(1-
{$\Box $}silon)a}}{a^{\frac{d}{2}}}\int_{0}^{+\infty}e^{-(1-
{$\Box $}silon)s}\left(s^{\frac{d}{2}-1}+a^{\frac{d}{2}-1}\right)ds\\
\nonumber &\leq &\frac{C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)a}}{a^{\frac{d}{2}}}\left(\Gamma\left(\frac{d}{2}\right)\;+\;a^{\frac{d}{2}-1}\right)\\
\nonumber &=&C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)a}\left(\frac{\Gamma\left(\frac{d}{2}\right)}{a^{\frac{d}{2}}}\;+\;\frac{1}{a}\right)\\
&\leq &C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)a},
\end{eqnarray}
since $a\geq\frac{d}{2}$. Analogously, by Lemma \mathbb{R}f{lemgamma}
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)}}{t^{\frac{d}{2}+1}}dt
&\leq &C\int_{ 1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}e^{-(1-
{$\Box $}silon)\frac{a}{t}}dt\\
&\leq &C\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}e^{-(1-
{$\Box $}silon)a}dt\\
&\leq&Ce^{-(1-
{$\Box $}silon)a}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}dt \\
&=& Ce^{-(1-
{$\Box $}silon)a}.
\end{eqnarray*}
Then,
\begin{equation*}
II\leq
C_{
{$\Box $}silon}e^{(1-
{$\Box $}silon)|x|^{2}}e^{-(1-
{$\Box $}silon)a}=C_{
{$\Box $}silon}e^{(1-
{$\Box $}silon)|x|^{2}}e^{-(1-
{$\Box $}silon)\left(|y|^{2}+|x|^{2}\right)}=C_{
{$\Box $}silon}e^{-(1-
{$\Box $}silon)|y|^{2}}.
\end{equation*}
Finally,
\begin{eqnarray*}
III&=&\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-u(t)}}{t^{\frac{d}{2}}}\frac{d}{2}\frac{dt}{t}\leq \frac{d}{2}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}e^{-\frac{a}{t}+|x|^{2}}\frac{dt}{t^{\frac{d}{2}+1}}\\
&= &Ce^{|x|^{2}}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt.\\
\end{eqnarray*}
Now,
\begin{equation*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt\leq
C\int_{0}^{1/2}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt\leq
C\int_{0}^{1}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt
\end{equation*}
since $\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}$ is
bounded in $[0,1/2],$ by taking
$$s=a(\frac{1}{t}-1), \;ds=-\frac{a}{t^{2}}dt,$$
we get
\begin{eqnarray*}
\int_{0}^{1}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt&=&
\int_{0}^{+\infty}e^{-(s+a)}\left(\frac{s+a}{a}\right)^{\frac{d}{2}+1}\left(\frac{a}{s+a}\right)^{2}\frac{ds}{a}\\
&=&\frac{e^{-|y|^{2}-|x|^{2}}}{a^{\frac{d}{2}}}\int_{0}^{+\infty}e^{-s}\left(s+a\right)^{\frac{d}{2}-1}ds.\\
\end{eqnarray*}
On the other hand,
\begin{equation*}
\left(s+a\right)^{\frac{d}{2}-1}=\frac{\left(s+a\right)^{\frac{d}{2}}}{s+a}\leq
C\frac{\left(s^{\frac{d}{2}}+a^{\frac{d}{2}}\right)}{s+a}\leq
C\left(\frac{s^{\frac{d}{2}}}{s}+\frac{a^{\frac{d}{2}}}{a}\right)
=C\left(s^{\frac{d}{2}-1}+a^{\frac{d}{2}-1}\right).
\end{equation*}
Therefore,
\begin{eqnarray*}
\int_{0}^{1}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt&\leq &
C\frac{e^{-|y|^{2}-|x|^{2}}}{a^{\frac{d}{2}}}\left(\int_{0}^{+\infty}e^{-s}s^{\frac{d}{2}-1}ds\;+
\;\int_{0}^{+\infty}e^{-s}a^{\frac{d}{2}-1}ds\right)\\
&=&C\frac{e^{-|y|^{2}-|x|^{2}}}{a^{\frac{d}{2}}}\left(\Gamma\left(\frac{d}{2}\right)\;+\;a^{\frac{d}{2}-1}\right)=Ce^{-|y|^{2}-|x|^{2}}\left(\frac{\Gamma\left(\frac{d}{2}\right)}{a^{\frac{d}{2}}}\;+\;\frac{1}{a}\right).
\end{eqnarray*}
Thus,
\begin{equation*}
\int_{0}^{1}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt\leq
Ce^{-|y|^{2}-|x|^{2}},\hspace{0.7cm}\mbox{as }a\geq\frac{d}{2}.
\end{equation*}
For $\frac{1}{2}<t<1,\hspace{0.3cm}-a>-\frac{a}{t}>-2a$. Hence, by Lemma \mathbb{R}f{lemgamma}
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-\frac{a}{t}}}{t^{\frac{d}{2}+1}}dt&\leq
&C\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}e^{-a}dt\\
&\leq
&Ce^{-a}\int_{0}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}dt\\
&=&Ce^{-|y|^{2}-|x|^{2}}.
\end{eqnarray*}
Then,
$$
III\leq Ce^{|x|^{2}}e^{-|y|^{2}-|x|^{2}}= Ce^{-|y|^{2}}\leq Ce^{-(1-
{$\Box $}silon)|y|^{2}}.
$$
In other words,
\begin{equation*}
I\leq C_{
{$\Box $}silon}|x|e^{-(1-
{$\Box $}silon)|y|^{2}}
\end{equation*}
and
\begin{equation*}
II, \; III\leq Ce^{-|y|^{2}}\leq Ce^{-(1-
{$\Box $}silon)|y|^{2}}.
\end{equation*}
Thus, for $b\leq 0$
\begin{equation*}
\left|N_{\beta/2}(x,y)\right|\leq
C_{
{$\Box $}silon}(|x|+1)e^{-(1-
{$\Box $}silon)|y|^{2}}.
\end{equation*}
Next, we take $0<
{$\Box $}silon<1/p'_{-}$ and $\tilde{
{$\Box $}silon}=1/p'_{-} -
{$\Box $}silon = 1-
{$\Box $}silon-1/p_{-}$. Then, $\tilde{
{$\Box $}silon}>0$ and $1-
{$\Box $}silon=\tilde{
{$\Box $}silon}+1/p_{-}$\,.Therefore, for $f\in L^{p(\cdot)}(\mathbb{R}^{d},\gamma_{d})$ with $\|f\|_{p,\gamma_{d}}=1$, using H\"older's inequality
\begin{eqnarray*}
\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E^{c}_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx) &\lesssim
&\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{\mathbb{R}^{d}}(|x|+1)e^{-(1-
{$\Box $}silon)|y|^{2}}|f(y)|dy\right)^{p(x)}\gamma_{d}(dx)
\\ &=
&\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{\mathbb{R}^{d}}e^{-(\tilde{
{$\Box $}silon}+1/p_{-})|y|^{2}}|f(y)|dy\right)^{p(x)}(|x|+1)^{p(x)}\gamma_{d}(dx)
\\ &\leq
&\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{\mathbb{R}^{d}}|f(y)|^{p_{-}}e^{-|y|^{2}}dy\right)^{p(x)/p_{-}}\left(\int\limits_{\mathbb{R}^{d}}e^{-\tilde{
{$\Box $}silon}|y|^{2}p'_{-}}dy\right)^{p(x)/p'_{-}}\\
&&\hspace{5.5cm} \times (|x|+1)^{p_{+}}\gamma_{d}(dx)
\\ &\lesssim
&\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{\mathbb{R}^{d}}|f(y)|^{p_{-}}e^{-|y|^{2}}dy\right)^{\frac{p(x)}{p_{-}}}(|x|+1)^{p_{+}}\gamma_{d}(dx),
\end{eqnarray*}
and, since $\rho_{p(\cdot),\gamma_{d}}(f)\leq 1$,
\begin{eqnarray*}
\int\limits_{\mathbb{R}^{d}}|f(y)|^{p_{-}}e^{-|y|^{2}}dy&\lesssim&\int\limits_{|f|\geq 1}|f(y)|^{p_{-}}\gamma_{d}(dy)+\int\limits_{|f|<1}|f(y)|^{p_{-}}\gamma_{d}(dy)\\
&\leq&\int\limits_{|f|\geq 1}|f(y)|^{p(y)}\gamma_{d}(dy)+1\leq\rho_{p(\cdot),\gamma_{d}}(f)+1\leq 2.
\end{eqnarray*}
Thus,
$$\int\limits_{\mathbb{R}^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E^{c}_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx) \lesssim
2^{\frac{p_{+}}{p_{-}}}\int\limits_{\mathbb{R}^{d}}(|x|+1)^{p_{+}}\gamma_{d}(dx)=C_{d,p}.$$
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Hence,
$$\|I_{\beta}(f\chi_{B_{h}^{c}(\cdot)\cap E^{c}_{(\cdot)}})\|_{p(\cdot),\gamma_{d}}\leq C_{d,p}.$$
\item \underline{Case $b>0$}:\\
In this case, $I$ is a very problematic term so we will discuss it at the end.\\
Again, by inequality (\mathbb{R}f{expineq})
\begin{eqnarray*}
II&=&\frac{1}{2}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-u(t)}}{t^{d/2+1}}\frac{\left|y-\sqrt{1-t}x\right|^{2}}{t}\,dt\\
&=&\frac{1}{2}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2+1}} e^{-
{$\Box $}silon u(t)}\frac{\left|y-\sqrt{1-t}x\right|^{2}}{t}\,dt\\
&\leq &C_{
{$\Box $}silon}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2+1}}\,dt.
\end{eqnarray*}
By using inequality (4.44) of \cite{urbina2019}, see also \cite{pe}, we get
\begin{equation}\label{phib0}
\frac{e^{-u(t)}}{t^{d/2}}\leq 2^{d}\frac{e^{-u(t_{0})}}{t_{0}^{d/2}}.
\end{equation}
Thus,
\begin{eqnarray}\label{phib1}
\nonumber \frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2}}&=&\left(\frac{e^{-u(t)}}{t^{d/2}}\right)^{1-
{$\Box $}silon} \frac{1}{t^{
{$\Box $}silon
d/2}}\\
&\lesssim
&\left(\frac{e^{-u(t_{0})}}{t_{0}^{d/2}}\right)^{1-
{$\Box $}silon} \frac{1}{t^{
{$\Box $}silon
d/2}}=\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}\frac{1}{t^{
{$\Box $}silon
d/2}}.
\end{eqnarray}
Now, splitting the above integral into two the integrals on $[0,1/2]$ and $[1/2,1]$; we have for the first integral using (\mathbb{R}f{phib1}),
\begin{eqnarray*}
\int^{1/2}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2+1}}\,dt&\lesssim
&\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}\int^{1/2}_{0}\frac{\left(-log(\sqrt{1-t})\right)^{\beta/2}}{t^{
{$\Box $}silon
d/2+1}}\,dt.
\end{eqnarray*}
Set $r=\min\{\frac{\beta}{4},\frac{1}{2}\},\,$ then
$0<r<1$ and by taking $
{$\Box $}silon>0$ such that $\frac{
{$\Box $}silon d}{2}<
\frac{\beta}{2}-r\,\,$ we get
\begin{equation*}
\lim_{t\to 0^{+}}\frac{\left(-log(\sqrt{1-t})\right)^{\beta/2}}{t^{
{$\Box $}silon
d/2+r}} = \lim_{t\to 0^{+}}\left[\frac{\left(-log(\sqrt{1-t})\right)}{t}\right]^{\beta/2}t^{\beta/2 -(
{$\Box $}silon d/2+r)}= 0,
\end{equation*}
thus, $\displaystyle\frac{\left(-log(\sqrt{1-t})\right)^{\beta/2}}{t^{
{$\Box $}silon
d/2+r}}$ is bounded on $(0,1/2]$,
and hence
\begin{equation*}
\int^{1/2}_{0}\frac{\left(-log(\sqrt{1-t})\right)^{\beta/2}}{t^{
{$\Box $}silon
d/2+1}}\,dt=\int^{1/2}_{0}\frac{\left(-log(\sqrt{1-t})\right)^{\beta/2}}{t^{
{$\Box $}silon
d/2+r}}\frac{dt}{t^{1-r}}\leq
C_{
{$\Box $}silon,\beta}\int^{1/2}_{0}\frac{dt}{t^{1-r}}=C_{
{$\Box $}silon,\beta}.
\end{equation*}
Then,
\begin{equation}\label{bceromedio}
\int^{1/2}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2+1}}\,dt\;\leq\;C_{
{$\Box $}silon,\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}.
\end{equation}
For the integral on $[1/2,1]$ we have, using again (\mathbb{R}f{phib1}) and Lemma \mathbb{R}f{lemgamma}
\begin{eqnarray*}
\int^{1}_{1/2}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2+1}}\,dt&=&
\int^{1}_{1/2}\left(-log(\sqrt{1-t})\right)^{\beta/2}\left(\frac{e^{-u(t)}}{t^{d/2}}\right)^{(1-
{$\Box $}silon)}\,\frac{dt}{t^{
{$\Box $}silon
d/2+1}}\\
&\leq
&C_{
{$\Box $}silon}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}\int^{1}_{1/2}\left(-log(\sqrt{1-t})\right)^{\beta/2}\,dt\\
&\leq
&C_{
{$\Box $}silon}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\,dt\\
&=&C_{
{$\Box $}silon,\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}.
\end{eqnarray*}
Therefore, since $t_{0}<1$ and
$\frac{d}{2}(1-
{$\Box $}silon)<\frac{d}{2},$ we get
$$
II\leq C_{
{$\Box $}silon,\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d(1-
{$\Box $}silon)/2}}
\leq C_{
{$\Box $}silon}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d/2}}.
$$
Now, using (\mathbb{R}f{phib0}) and (\mathbb{R}f{gammaint}) we get
\begin{eqnarray*}
III=\frac{1}{4}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-u(t)}}{t^{d/2}}\frac{dt}{t}&\lesssim&
\frac{e^{-u(t_{0})}}{t_{0}^{d/2}}\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{dt}{t}\\
&=&C_{\beta}\frac{e^{-u(t_{0})}}{t_{0}^{d/2}}\leq
C_{\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{d/2}}.
\end{eqnarray*}
Now, to estimate $I$, we use again inequality (\mathbb{R}f{expineq})
\begin{eqnarray*}
I&=&\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-u(t)}}{t^{d/2}}\frac{\left|y-\sqrt{1-t}x\right|}{\sqrt{t}}\frac{|x|}{\sqrt{1-t}}\frac{dt}{\sqrt{t}}\\
&\leq
&C_{
{$\Box $}silon}|x|\int^{1}_{0}\left(-log(\sqrt{1-t})\right)^{\beta/2}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{d/2}\sqrt{t}}\frac{dt}{\sqrt{1-t}}.
\end{eqnarray*}
Again, splitting the above integral into two the integrals on $[0,1/2]$ and $[1/2,1]$. For the second integral, using (\mathbb{R}f{phib0}), that $t \geq 1/2$ and Lemma \mathbb{R}f{leminteg}, we get
\begin{eqnarray*}
\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}}\sqrt{t}}\frac{dt}{\sqrt{1-t}}&\lesssim& e^{-(1-
{$\Box $}silon)u(t_{0})}\int_{1/2}^{1}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{dt}{\sqrt{1-t}}\\
&=&C_{\beta}e^{-(1-
{$\Box $}silon)u(t_{0})},
\end{eqnarray*}
Next, for the integral on $[0,1/2]$ we need to consider two cases:\\
\begin{itemize}
\item Case $\beta>0$ and $d=1$: by Lemma \mathbb{R}f{lemgamma} and the fact that $\frac{-log(\sqrt{1-t})}{t}$ is bounded on $(0, 1/2],$
\begin{eqnarray*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{1}{2}}\sqrt{t}}\frac{dt}{\sqrt{1-t}}
&=&\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t}\frac{dt}{\sqrt{1-t}}\\
&=&\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}\frac{\left(-log(\sqrt{1-t})\right)}{t}e^{-(1-
{$\Box $}silon)u(t)}\frac{dt}{\sqrt{1-t}}\\
&\leq&C_{\beta}\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}e^{-(1-
{$\Box $}silon)u(t)}\frac{dt}{\sqrt{1-t}}\\
&\leq&C_{\beta}e^{-(1-
{$\Box $}silon)u(t_{0})}\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}dt\\
&=&C_{\beta}e^{-(1-
{$\Box $}silon)u(t_{0})}.
\end{eqnarray*}
\item Case $\beta\geq 1$ and $d\geq 2$: by taking $
{$\Box $}silon <
\frac{2}{d}$
\begin{eqnarray*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}}\sqrt{t}}\frac{dt}{\sqrt{1-t}}
&\leq& C\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}}}\frac{dt}{\sqrt{1-t}}\\
&= &C\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}}{t^{\frac{(d-2)}{2}}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t\sqrt{1-t}}dt\\
&= &C\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}}{t^{\frac{d}{2}\frac{(d-2)}{d}}}e^{-(\frac{d-2}{d})u(t)}\frac{e^{(
{$\Box $}silon-\frac{2}{d})u(t)}}{t\sqrt{1-t}}dt\\
&\leq &C\frac{e^{-(\frac{d-2}{d})u(t_{0})}}{t_{0}^{\frac{d}{2}\frac{(d-2)}{d}}}\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}\frac{e^{(
{$\Box $}silon-\frac{2}{d})u(t)}}{t\sqrt{1-t}}dt.
\end{eqnarray*}
Since
$\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}$ is continuous
on $[0,1/2]$ for $\beta\geq 1$ and proceeding in analogous way as in Lemma 4.36 of \cite{urbina2019}, we get
\begin{eqnarray*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}}\sqrt{t}}\frac{dt}{\sqrt{1-t}} &\leq& C_{\beta}\frac{e^{-(\frac{d-2}{d})u(t_{0})}}{t_{0}^{\frac{d}{2}\frac{(d-2)}{d}}}\int_{0}^{1}\frac{e^{(
{$\Box $}silon-\frac{2}{d})u(t)}}{t\sqrt{1-t}}dt\\
&\leq &C_{\beta}\frac{e^{-(\frac{d-2}{d})u(t_{0})}}{t_{0}^{\frac{d}{2}-1}}e^{(
{$\Box $}silon-\frac{2}{d})u(t_{0})}\\
&= &C_{\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{\frac{d}{2}-1}}= C_{\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{\frac{d}{2}}}t_{0}.
\end{eqnarray*}
Thus,
$$I\leq C_{
{$\Box $}silon,\beta}\displaystyle|x|\left(1+\frac{1}{t_{0}^{\frac{d}{2}}}t_{0}\right)e^{-(1-
{$\Box $}silon)u(t_{0})}.$$
\end{itemize}
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Next, for the integral on $[0,1/2]$ we need to consider two cases:\\
\begin{itemize}
\item Case $\beta>0$ and $d=1$: by Lemma \mathbb{R}f{lemgamma} and the fact that $\frac{-log(\sqrt{1-t})}{t}$ is bounded on $(0, 1/2],$
\begin{eqnarray*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{1}{2}}\sqrt{t}}\frac{dt}{\sqrt{1-t}}
&=&\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t}\frac{dt}{\sqrt{1-t}}\\
&=&\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}\frac{\left(-log(\sqrt{1-t})\right)}{t}e^{-(1-
{$\Box $}silon)u(t)}\frac{dt}{\sqrt{1-t}}\\
&\leq&C_{\beta}\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}e^{-(1-
{$\Box $}silon)u(t)}\frac{dt}{\sqrt{1-t}}\\
&\leq&C_{\beta}e^{-(1-
{$\Box $}silon)u(t_{0})}\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}-1}dt\\
&=&C_{\beta}e^{-(1-
{$\Box $}silon)u(t_{0})}.
\end{eqnarray*}
\item Case $\beta\geq 1$ and $d\geq 2$: by taking $
{$\Box $}silon <
\frac{2}{d}$
\begin{eqnarray*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}}\sqrt{t}}\frac{dt}{\sqrt{1-t}}
&\leq& C\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}}}\frac{dt}{\sqrt{1-t}}\\
&= &C\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}}{t^{\frac{(d-2)}{2}}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t\sqrt{1-t}}dt\\
&= &C\int_{0}^{1/2}\frac{\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}}{t^{\frac{d}{2}\frac{(d-2)}{d}}}e^{-(\frac{d-2}{d})u(t)}\frac{e^{(
{$\Box $}silon-\frac{2}{d})u(t)}}{t\sqrt{1-t}}dt\\
&\leq &C\frac{e^{-(\frac{d-2}{d})u(t_{0})}}{t_{0}^{\frac{d}{2}\frac{(d-2)}{d}}}\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}\frac{e^{(
{$\Box $}silon-\frac{2}{d})u(t)}}{t\sqrt{1-t}}dt.
\end{eqnarray*}
Since
$\left(-log(\sqrt{1-t})\right)^{\frac{(\beta-1)}{2}}$ is continuous
on $[0,1/2]$ for $\beta\geq 1$ and proceeding in analogous way as in Lemma 4.36 of \cite{urbina2019}, we get
\begin{eqnarray*}
\int_{0}^{1/2}\left(-log(\sqrt{1-t})\right)^{\frac{\beta}{2}}\frac{e^{-(1-
{$\Box $}silon)u(t)}}{t^{\frac{d}{2}}\sqrt{t}}\frac{dt}{\sqrt{1-t}} &\leq& C_{\beta}\frac{e^{-(\frac{d-2}{d})u(t_{0})}}{t_{0}^{\frac{d}{2}\frac{(d-2)}{d}}}\int_{0}^{1}\frac{e^{(
{$\Box $}silon-\frac{2}{d})u(t)}}{t\sqrt{1-t}}dt\\
&\leq &C_{\beta}\frac{e^{-(\frac{d-2}{d})u(t_{0})}}{t_{0}^{\frac{d}{2}-1}}e^{(
{$\Box $}silon-\frac{2}{d})u(t_{0})}\\
&= &C_{\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{\frac{d}{2}-1}}= C_{\beta}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{\frac{d}{2}}}t_{0}.
\end{eqnarray*}
Thus,
$$I\leq C_{
{$\Box $}silon,\beta}\displaystyle|x|\left(1+\frac{1}{t_{0}^{\frac{d}{2}}}t_{0}\right)e^{-(1-
{$\Box $}silon)u(t_{0})}.$$
\end{itemize}
Finally, since $\displaystyle\frac{1}{t_{0}^{d/2}}\lesssim |x+y|^{d}$, $t_{0}<1$, and $|x|\leq |x+y|$ as $b>0$, we have
\begin{itemize}
\item For $|x|<1$,
\begin{eqnarray*}
I&\leq& C_{
{$\Box $}silon,\beta}\displaystyle|x|\left(1+\frac{1}{t_{0}^{\frac{d}{2}}}t_{0}\right)e^{-(1-
{$\Box $}silon)u(t_{0})}\leq C_{
{$\Box $}silon,\beta}\frac{1}{t_{0}^{\frac{d}{2}}}e^{-(1-
{$\Box $}silon)u(t_{0})}\\
&\leq& C_{
{$\Box $}silon,\beta}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}.
\end{eqnarray*}
\item For $|x|\geq 1$, since $b>0$, $|x|\leq |x+y|$
\begin{eqnarray*}
I&\leq & C_{
{$\Box $}silon,\beta}\displaystyle|x|\left(1+\frac{1}{t_{0}^{\frac{d}{2}}}t_{0}\right)e^{-(1-
{$\Box $}silon)u(t_{0})}\\
&=& C_{
{$\Box $}silon,\beta}|x|e^{-(1-
{$\Box $}silon)u(t_{0})}\,+\, C_{
{$\Box $}silon,\beta}|x|t_{0}\frac{e^{-(1-
{$\Box $}silon)u(t_{0})}}{t_{0}^{\frac{d}{2}}}\\
&\leq &
C_{
{$\Box $}silon,\beta}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}\,+\, C_{
{$\Box $}silon,\beta}t_{0}|x||x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}.
\end{eqnarray*}
\end{itemize}
Given that $t_{0}\leq C\frac{|x-y|}{|x+y|}$ and the fact that $|x|\leq |x+y| $ we get, for $|x-y|<1$,
$$
|x|t_{0}\leq C\frac{|x||x-y|}{|x+y|}\leq C.
$$
Thus, for $|x-y|<1$,
$$
I\leq
C_{
{$\Box $}silon,\beta}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})};
$$
and for $|x-y|\geq 1$,
$$
I\leq
C_{
{$\Box $}silon,\beta}|x+y|^{d+1}e^{-(1-
{$\Box $}silon)u(t_{0})}.
$$
Hence, we conclude that\\
\begin{itemize}
\item $\left|N_{\beta/2}(x,y)\right|\leq C_{
{$\Box $}silon,\beta}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})},$ \; for either $|x|\leq 1,$ or for $|x|\geq 1$ with $|x-y|<1$.\\
\item $\left|N_{\beta/2}(x,y)\right|\leq C_{
{$\Box $}silon,\beta}|x+y|^{d+1}e^{-(1-
{$\Box $}silon)u(t_{0})},$ \; for \; $|x|\geq 1$ with $|x-y|\geq 1$.\\
\end{itemize}
Now, as $b>0$, for $f\in L^{p(\cdot)}(\mathbb{R}^{d},\gamma_{d})$ with $\|f\|_{p(\cdot),\gamma_{d}}=1$, we have that
\begin{eqnarray*}
&& \int_{\mathbb{R}^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx)= \int_{|x|<1}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx)\\
&& \hspace{.5cm}+ \int_{|x|\geq 1}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|<1}\left|N_{\beta/2}(x,y)\right||f(y)|dy+\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|\geq 1}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx)
\end{eqnarray*}
\begin{eqnarray*}
&\leq& \int_{|x|<1}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx)\\
&+& C\int_{|x|\geq 1}\left(\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|<1}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}+\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|\geq 1}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\right)\gamma_{d}(dx)\\
&\leq&C_{
{$\Box $}silon,\beta}\int_{\Bbb R^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}|f(y)|dy\right)^{p(x)}\gamma_{d}(dx)\\
&& \hspace{0.75cm}+ C_{
{$\Box $}silon,\beta}\int_{\Bbb R^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|\geq 1}|x+y|^{d+1}e^{-(1-
{$\Box $}silon)u(t_{0})}|f(y)|dy\right)^{p(x)}\gamma_{d}(dx)\\
&=&C_{
{$\Box $}silon,\beta}\int_{\Bbb R^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}e^{|y|^{2}/p(y)-|x|^{2}/p(x)}|f(y)|e^{-|y|^{2}/p(y)}dy\right)^{p(x)}dx\\
&& \hspace{0.75cm}+ C_{
{$\Box $}silon,\beta}\int_{\Bbb R^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x},|x-y|\geq 1}|x+y|^{d+1}e^{-(1-
{$\Box $}silon)u(t_{0})}e^{|y|^{2}/p(y)-|x|^{2}/p(x)}|f(y)|e^{-|y|^{2}/p(y)}dy\right)^{p(x)}dx.
\end{eqnarray*}
Since $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$, we obtain that $e^{|y|^{2}/p(y)-|x|^{2}/p(x)}\approx e^{(|y|^{2}-|x|^{2})/p_{\infty}},$ and by the Cauchy-Schwartz inequality we have,
$\left| |y|^{2}-|x|^{2}\right|\leq|x+y| |x-y|,$
for all $x,y\in\mathbb{R}^{d}$. Therefore,
\begin{eqnarray*}
&&\int_{B^{c}_h(x)\cap E_{x}}|x+y|^{d}e^{-(1-
{$\Box $}silon)u(t_{0})}e^{|y|^{2}/p(y)-|x|^{2}/p(x)}|f(y)|e^{-|y|^{2}/p(y)}dy \hspace{3.5cm}\\
&& \hspace{6.5cm} \lesssim \int_{B^{c}_h(x)\cap E_{x}}P(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy,
\end{eqnarray*}
and \begin{eqnarray*}
&&\int_{B^{c}_h(x)\cap E_{x},|x-y|\geq 1}|x+y|^{d+1}e^{-(1-
{$\Box $}silon)u(t_{0})}e^{|y|^{2}/p(y)-|x|^{2}/p(x)}|f(y)|e^{-|y|^{2}/p(y)}dy \hspace{3.5cm}\\
&& \hspace{5.5cm} \lesssim \int_{B^{c}_h(x)\cap E_{x}}Q(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy,
\end{eqnarray*}
where
$$
P(x,y)=|x+y|^{d}e^{-\alpha_{\infty}|x+y||x-y|}, \; Q(x,y)=|x+y|^{d+1}e^{-\alpha_{\infty}|x+y|}$$
and
$$ \alpha_{\infty}=\left(\frac{1-
{$\Box $}silon}{2}-\left|\frac{1}{p_{\infty}}-\frac{1-
{$\Box $}silon}{2}\right|\right).$$
It is easy to see that $\alpha_{\infty}>0$ if $
{$\Box $}silon<1/p'_{\infty}$.
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Therefore, in order to make sense of all the estimates above we need to take
$$0<
{$\Box $}silon<\displaystyle\min\left\lbrace \frac{1}{d},\frac{1}{p'_{\infty}},\frac{2}{d}\left(\frac{\beta}{2}-r\right)\right\rbrace.$$
Observe that $P(x,y)$ is the same kernel considered in the proof of Theorem 3.5, page 416 of \cite{DalSco}, so we can conclude that
$$\displaystyle\int_{\Bbb R^{d}}\left(\int_{B^{c}_h(x)\cap E_{x}}P(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy\right)^{p(x)}dx\leq C.$$
On the other hand, it can be proved that $Q(x,y)$ is integrable on each variable and the value of each integral is independent of $x$ and $y$.
Now, we use an analogous argument as in \cite{DalSco} for $Q(x,y)$. Taking,
$$J=\displaystyle\int_{\Bbb R^{d}}Q(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy,$$
and using H\"older's inequality, we obtain
$$ J\lesssim\|Q(x,\cdot)\|_{p'(\cdot)}\|fe^{-|\cdot|^{2}/p(\cdot)}\|_{p(\cdot)}\leq \|Q(x,\cdot)\|_{p'(\cdot)}.$$
and,
\begin{eqnarray*}
\int_{\Bbb R^{d}}Q(x,y)^{p'(y)}dy&=&\int_{\Bbb R^{d}}|x+y|^{(d+1)p'(y)}e^{-\alpha_{\infty}|x+y|p'(y)}dy\\
&\leq&\int_{|x+y|<1}|x+y|^{d+1}e^{-\alpha_{\infty}|x+y|}dy +\int_{|x+y|\geq 1}|x+y|^{(d+1)p'_{+}}e^{-\alpha_{\infty}|x+y|}dy\\
&\leq&\int_{\Bbb R^{d}}\left(|z|^{d+1}+|z|^{(d+1)p'_{+}}\right)e^{-\alpha_{\infty}|z|}dz\\
&=&C_{p,d}.
\end{eqnarray*}
Thus, $J\lesssim\|Q(x,\cdot)\|_{p^{'}(\cdot)}\leq C_{p,d}$, and therefore
$$\frac{1}{C_{p,d}}\int_{\Bbb R^{d}}Q(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy\leq 1.$$
We set $g(y)=|f(y)|e^{-|y|^{2}/p(y)}=g_{1}(y)+g_{2}(y)$, where $g_{1}=g\chi_{\{g\geq 1\}}$ and $g_{2}=g\chi_{\{g<1\}}$, then
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}J^{p(x)}dx&=& \int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)|f(y)|e^{-|y|^{2}/p(y)}dy\right)^{p(x)}dx \\
&\lesssim& \int_{\mathbb{R}^{d}}\left(\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g(y)dy\right)^{p(x)}dx\\
&\lesssim& \int_{\mathbb{R}^{d}}\left(\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g_{1}(y)dy\right)^{p(x)}dx+\int_{\mathbb{R}^{d}}\left(\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g_{2}(y)dy\right)^{p(x)}dx.
\end{eqnarray*}
By H\"{o}lder's inequality and Fubini's theorem
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left(\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g_{1}(y)dy\right)^{p(x)}dx&\lesssim&\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g_{1}(y)dy\right)^{p_{-}}dx\\
&=&\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q^{\frac{1}{p'_{-}}}(x,y)Q^{\frac{1}{p_{-}}}(x,y)g_{1}(y)dy\right)^{p_{-}}dx\\
&\leq&\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)dy\right)^{p_{-}/p'_{-}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g^{p_{-}}_{1}(y)dy\right)dx\\
&=&C_{p}\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g^{p_{-}}_{1}(y)dy\right)dx\\
&=&C_{p}\int_{\mathbb{R}^{d}}g^{p_{-}}_{1}(y)\left(\int_{\mathbb{R}^{d}}Q(x,y)dx\right)dy=C_{p}\int_{\mathbb{R}^{d}}g^{p_{-}}_{1}(y)dy.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left(\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g_{1}(y)dy\right)^{p(x)}dx&\lesssim&\int_{\mathbb{R}^{d}}g^{p(y)}_{1}(y)dy\leq\int_{\mathbb{R}^{d}}|f(y)|^{p(y)}e^{-|y|^{2}}dy\\
&\lesssim& \rho_{p(\cdot),\gamma_{d}}(f)\leq 1.
\end{eqnarray*}
On the other hand, applying the inequality (\mathbb{R}f{3.26.1}) in Lemma \mathbb{R}f{lema3.26CU}, since\\
$G(x):=\displaystyle\frac{1}{C_{p,d}}\int_{\mathbb{R}^{d}}Q(x,y)g_{2}(y)dy\leq 1$, we obtain that
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}\frac{1}{C_{p,d}}Q(x,y)g_{2}(y)dy\right)^{p(x)}dx&=&\int_{\mathbb{R}^{d}}(G(x))^{p(x)}dx \\
&\leq& \int_{\mathbb{R}^{d}}(G(x))^{p_{\infty}}dx + \int_{\mathbb{R}^{d}}\frac{dx}{(e+|x|)^{dp_{-}}}\\
&\lesssim&\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g_{2}(y)dy\right)^{p_{\infty}}+C_{d,p}.
\end{eqnarray*}
Finally, to estimate the integral
$$\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g_{2}(y)dy\right)^{p_{\infty}}dx,$$
we proceed in an analogous way, applying H\"{o}lder's inequality to the exponent $p_{\infty}$, Fubbini's theorem and inequality (\mathbb{R}f{3.26.2}) in Lemma \mathbb{R}f{lema3.26CU}, we get
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}Q(x,y)g_{2}(y)dy\right)^{p_{\infty}}dx &\lesssim&\int_{\mathbb{R}^{d}}g_{2}^{p_{\infty}}(y)dy\\
&\leq&\int_{\mathbb{R}^{d}}g_{2}^{p(y)}(y)dy+\int_{\mathbb{R}^{d}}\frac{dy}{(e+|y|)^{dp_{-}}}\\
&\leq&\int_{\mathbb{R}^{d}}|f(y)|^{p(y)}e^{-|y|^{2}}dy+C_{d,p}\lesssim 1+C_{d,p}.
\end{eqnarray*}
Thus,
\begin{eqnarray*}
\int_{\mathbb{R}^{d}}\left(\int\limits_{B_{h}^{c}(x)\cap
E_{x}}\left|N_{\beta/2}(x,y)\right||f(y)|dy\right)^{p(x)}\gamma_{d}(dx)&\leq& C_{d,p}.
\end{eqnarray*}
With this, we obtain that $\|I_{\beta}(f\chi_{B_{h}^{c}(\cdot)\cap E_{(\cdot)}})\|_{p(\cdot),\gamma_{d}}\leq C_{d,p}$.\\
We conclude that
$$
\left\|I_{\beta,G}(f)\right\|_{p(\cdot), \gamma_{d}}=\left\|I_{\beta}\left(f \chi_{B^{c}_h(\cdot)}\right)\right\|_{p(\cdot), \gamma_{d}} \leq C,
$$\\
and by homogeneity of the norm,\\
$
\left\|I_{\beta,G}(f)\right\|_{p(\cdot), \gamma_{d}} \leq C\|f\|_{p(\cdot),\gamma_{d}}$, for all function $f\in L^{p(\cdot)}(\mathbb{R}^{d},\gamma_{d})$.\\
Therefore, the global part $I_{\beta,G}$ is bounded in $L^{p(\cdot)}(\gamma_d)$ and the proof is complete. \\
\end{itemize}
\end{itemize}
\end{proof}
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\end{document}
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\begin{document}
\title{
\leftline{Effective de la Vall\'e Poussin style bounds}
on the first Chebyshev function\\
}
\author{
\Large
Matt Visser\!\orcidMatt\!
}
\affiliation{School of Mathematics and Statistics, Victoria University of Wellington, \\
\null\qquad PO Box 600, Wellington 6140, New Zealand.}
\mathrm{e}mailAdd{[email protected]}
{\mathrm{d}}ef\vartheta{\vartheta}
{\mathrm{d}}ef{\mathcal{O}}{{\mathcal{O}}}
\abstract{
In 1898 Charles Jean de la Vall\'e Poussin, as part of his famed proof of the prime number theorem, developed an ineffective bound on the first Chebyshev function of the form:\\
\[
|\vartheta(x)-x| = {\mathcal{O}}\left(x \mathrm{e}xp(-K \sqrt{\ln x})\right).
\]
This bound holds for $x$ sufficiently large, $x\geq x_0$, and $K$ some unspecified positive constant.
To the best of my knowledge this bound has never been made effective --- I have never yet seen this bound made fully explicit, with precise values being given for $x_0$ and $K$.
Herein, using a number of effective results established over the past 50 years, I shall develop two very simple explicit fully effective bounds of this type:
\[
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over4} \sqrt{\ln x}\right);
\qquad (x\geq 2).
\]
\[
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over3} \sqrt{\ln x}\right);
\qquad (x\geq 3).
\]
Many other fully explicit bounds along these lines can easily be developed. \\
For instance one can trade off stringency against range of validity:
\[
|\vartheta(x)-x| < \;
{1\over 2} \; {x} \;\mathrm{e}xp\left( - {1\over4} \sqrt{\ln x}\right);
\qquad (x\geq 29),
\]
\[
|\vartheta(x)-x| < \;
{1\over 2} \; {x} \;\mathrm{e}xp\left( - {1\over3} \sqrt{\ln x}\right);
\qquad (x\geq 41).
\]
With hindsight, some of these effective bounds could have been established almost 50 years ago.
\noindent
{\sc Date:} 2 November 2022; \LaTeX-ed \today
\noindent{\sc Keywords}: Chebyshev $\vartheta$ function; effective bounds.
}
\maketitle
{\mathrm{d}}ef{\mathrm{tr}}{{\mathrm{tr}}}
{\mathrm{d}}ef{\mathrm{diag}}{{\mathrm{diag}}}
{\mathrm{d}}ef{\mathrm{cof}}{{\mathrm{cof}}}
{\mathrm{d}}ef{\mathrm{pdet}}{{\mathrm{pdet}}}
{\mathrm{d}}ef{\mathrm{d}}{{\mathrm{d}}}
\parindent0pt
\parskip7pt
{\mathrm{d}}ef{\scriptscriptstyle{\mathrm{Kerr}}}{{\scriptscriptstyle{\mathrm{Kerr}}}}
{\mathrm{d}}ef\mathrm{e}os{{\scriptscriptstyle{\mathrm{eos}}}}
\section{Introduction}
In 1898 Charles Jean de la Vall\'e Poussin developed an ineffective bound on the first Chebyshev function of the form~\cite{Poussin}:
\begin{equation}
\label{E:Poussin}
|\vartheta(x)-x| = {\mathcal{O}}\left(x \mathrm{e}xp(-K \sqrt{\ln x})\right).
\mathrm{e}nd{equation}
This bound holds for $x$ sufficiently large, $x\geq x_0$, and $K$ some unspecified positive constant.
Subsequent work over the last 50 years has developed a large number of related but distinct fully effective bounds of the form~\cite{Schoenfeld,Trudgian,Johnston-Yang,Fiori-et-al}:
\begin{equation}
|\vartheta(x)-x| < a \;x \;(\ln x)^b \; \mathrm{e}xp\left(-c\; \sqrt{\ln x}\right);
\qquad (x \geq x_0).
\mathrm{e}nd{equation}
\begin{itemize}
\item
For some widely applicable effective bounds of this type see Table I.\\
(A straightforward elementary numerical computation is required to determine the numerical coefficients in the Schoenfeld~\cite{Schoenfeld} and Trudgian~\cite{Trudgian} bounds.)
\item
For some asymptotically more stringent effective bounds of this type, but valid on significantly more restricted regions, see Table~II (based on reference~\cite{Johnston-Yang}), and the extensive tabulations in reference~\cite{Broadbent-et-al}.
\mathrm{e}nd{itemize}
\mathrm{e}nlargethispage{30pt}
What I have not yet seen is any attempt to take the effective bounds of Tables I and II and use them to
make the original de la Vall\'e Poussin bound fully effective.
Here are two particularly clean fully effective versions of the de la Vall\'e Poussin bound:
\begin{equation}
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over4} \sqrt{\ln x}\right);
\qquad (x\geq 2).
\mathrm{e}nd{equation}
\begin{equation}
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over3} \sqrt{\ln x}\right);
\qquad (x\geq 3).
\mathrm{e}nd{equation}
I shall explain how to derive these bounds below.
\begin{table}[!h]
\caption{Some widely applicable effective bounds.}
\begin{center}
\begin{tabular}{||c|c|c|c||c||}
\hline
\hline
$a$ & $b$ & $c$ & $x_0$ & Source \\
\hline
\hline
0.2196138920& 1/4 & 0.3219796502 & 101 & Schoenfeld~\cite{Schoenfeld}\\
\hline
\hline
0.2428127763 & 1/4 &0.3935970880 & 149 & Trudgian~\cite{Trudgian}\\
\hline
\hline
9.220226 & 3/2 & 0.8476836 & 2 & Fiori--Kadiri--Swidinsky~\cite{Fiori-et-al}\\
\hline
\hline
9.40 & 1.515 & 0.8274 & 2 & Johnston--Yang~\cite{Johnston-Yang}\\
\hline
\hline
\mathrm{e}nd{tabular}
\mathrm{e}nd{center}
\mathrm{e}nd{table}
\begin{table}[!htb]
\caption{Asymptotically stringent effective bounds valid on restricted regions~\cite{Johnston-Yang}.}
\begin{center}
\begin{tabular}{||c|c|c|c||}
\hline
\hline
$a$ & $b$ & $c$ & $x_0$ \\
\hline
\hline
8.87 & 1.514 & 0.8288 & $\mathrm{e}xp(3000)$ \\
8.16 & 1.512 & 0.8309 & $\mathrm{e}xp(4000)$ \\
7.66 & 1.511 & 0.8324 & $\mathrm{e}xp(5000)$ \\
7.23 & 1.510 & 0.8335 & $\mathrm{e}xp(6000)$ \\
7.00 & 1.510 & 0.8345 & $\mathrm{e}xp(7000)$ \\
6.79 & 1.509 & 0.8353 & $\mathrm{e}xp(8000)$ \\
6.59 & 1.509 & 0.8359 & $\mathrm{e}xp(9000)$ \\
6.73 & 1.509 & 0.8359 & $\mathrm{e}xp(10000)$ \\
\hline\hline
23.14 & 1.503 & 0.8659 & $\mathrm{e}xp(10^5)$ \\
38.58 & 1.502 & 1.0318 & $\mathrm{e}xp(10^6)$ \\
42.91 & 1.501 & 1.0706 & $\mathrm{e}xp(10^7)$ \\
44.42 & 1.501 & 1.0839 & $\mathrm{e}xp(10^8)$ \\
44.98 & 1.501 & 1.0886 & $\mathrm{e}xp(10^9)$ \\
45.18 & 1.501 & 1.0903 & $\mathrm{e}xp(10^{10})$ \\
\hline
\hline
\mathrm{e}nd{tabular}
\mathrm{e}nd{center}
\mathrm{e}nd{table}
\section{Strategy}
Note that for any $b>0$, $c>0$, and any $\tilde c\in(0,c)$, elementary calculus implies:
\begin{eqnarray}
(\ln x)^b \; \mathrm{e}xp\left(-c\; \sqrt{\ln x}\right)
&=& \left\{ (\ln x)^b \mathrm{e}xp\left(-[c-\tilde c] \; \sqrt{\ln x}\right) \right\} \mathrm{e}xp\left(-\tilde c\; \sqrt{\ln x}\right)
\nonumber\\
& \leq & \left\{ \left( 2b\over c-\tilde c \right)^{2b} \mathrm{e}xp(-2 b) \right\} \mathrm{e}xp\left(-\tilde c\; \sqrt{\ln x}\right).
\mathrm{e}nd{eqnarray}
The key observation here is that the quantity in braces is explicitly bounded,
and achieves a global maximum at
$x_{peak} = \mathrm{e}xp\left( \left[2b/(c-\tilde c)\right]^2\right)$.
Consequently we have the following lemma.
\paragraph{Lemma:}
\mathrm{e}mph{Any effective bound of the form
\begin{equation}
|\vartheta(x)-x| < a \;x \;(\ln x)^b \; \mathrm{e}xp\left(-c\; \sqrt{\ln x}\right);
\qquad (x \geq x_0).
\mathrm{e}nd{equation}
implies the existence of another effective bound of the de la Vall\'e Poussin form
\begin{equation}
|\vartheta(x)-x| < \tilde a \; x\; \mathrm{e}xp\left(-\tilde c\; \sqrt{\ln x}\right);
\qquad (x \geq x_*; \;\; x_* \leq x_0).
\mathrm{e}nd{equation}
Here $\tilde c$ is an arbitrary number in the interval $\tilde c \in (0,c)$ and
\begin{equation}
\tilde a = a \left( 2 b\over c-\tilde c \right)^{2b} \mathrm{e}xp(-2 b).
\mathrm{e}nd{equation}
Note that this new bound of the de la Vall\'e Poussin form certainly holds for $x>x_0$, but if $x_0$ is sufficiently small one might be able to widen the range of applicability to some new $x \geq x_*$ with $x_* \leq x_0$ by explicit computation.
}
We now apply this lemma to the various bounds explicated above.
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\section{Strategy}
Note that for any $b>0$, $c>0$, and any $\tilde c\in(0,c)$, elementary calculus implies:
\begin{eqnarray}
(\ln x)^b \; \mathrm{e}xp\left(-c\; \sqrt{\ln x}\right)
&=& \left\{ (\ln x)^b \mathrm{e}xp\left(-[c-\tilde c] \; \sqrt{\ln x}\right) \right\} \mathrm{e}xp\left(-\tilde c\; \sqrt{\ln x}\right)
\nonumber\\
& \leq & \left\{ \left( 2b\over c-\tilde c \right)^{2b} \mathrm{e}xp(-2 b) \right\} \mathrm{e}xp\left(-\tilde c\; \sqrt{\ln x}\right).
\mathrm{e}nd{eqnarray}
The key observation here is that the quantity in braces is explicitly bounded,
and achieves a global maximum at
$x_{peak} = \mathrm{e}xp\left( \left[2b/(c-\tilde c)\right]^2\right)$.
Consequently we have the following lemma.
\paragraph{Lemma:}
\mathrm{e}mph{Any effective bound of the form
\begin{equation}
|\vartheta(x)-x| < a \;x \;(\ln x)^b \; \mathrm{e}xp\left(-c\; \sqrt{\ln x}\right);
\qquad (x \geq x_0).
\mathrm{e}nd{equation}
implies the existence of another effective bound of the de la Vall\'e Poussin form
\begin{equation}
|\vartheta(x)-x| < \tilde a \; x\; \mathrm{e}xp\left(-\tilde c\; \sqrt{\ln x}\right);
\qquad (x \geq x_*; \;\; x_* \leq x_0).
\mathrm{e}nd{equation}
Here $\tilde c$ is an arbitrary number in the interval $\tilde c \in (0,c)$ and
\begin{equation}
\tilde a = a \left( 2 b\over c-\tilde c \right)^{2b} \mathrm{e}xp(-2 b).
\mathrm{e}nd{equation}
Note that this new bound of the de la Vall\'e Poussin form certainly holds for $x>x_0$, but if $x_0$ is sufficiently small one might be able to widen the range of applicability to some new $x \geq x_*$ with $x_* \leq x_0$ by explicit computation.
}
We now apply this lemma to the various bounds explicated above.
\section{Some effective bounds}
\mathrm{e}nlargethispage{20pt}
First let us consider some widely applicable derived bounds of the de la Vall\'e Poussin form, as presented in Table~III.
Note that the selection of a specific value of $\tilde c$ is a \mathrm{e}mph{choice}, and the computation of $\tilde a$ is then immediate --- there is an infinite number of other
effective
bounds of de la Vall\'e Poussin form that we could develop.
Determining $x_*$ then requires computationally checking low values of $x$.
\begin{table}[!htb]
\caption{Some widely applicable derived bounds of the de la Vall\'e Poussin form.}
\begin{center}
\begin{tabular}{||c|c|c||c||}
\hline
\hline
$\tilde a$ & $\tilde c$ & $x_*$ & Based on \\
\hline
\hline
0.3510691792& 1/4 & 59 & Schoenfeld~\cite{Schoenfeld}\\
\hline
\hline
0.2748124978 & 1/4 & 101 & Trudgian~\cite{Trudgian}\\
\hline
0.4242102935 & 1/3 & 59 & Trudgian~\cite{Trudgian}\\
\hline
\hline
295 & 1/2 & 2 & Fiori--Kadiri--Swidinsky~\cite{Fiori-et-al}\\
\hline
\hline
385 & 1/2 & 2 & Johnston--Yang~\cite{Johnston-Yang}\\
\hline
\hline
\mathrm{e}nd{tabular}
\mathrm{e}nd{center}
\mathrm{e}nd{table}
By now relaxing the prefactor $\tilde a$, one can increase the range of validity of the bound, (ie, decrease $x_*$). In this way, after some computation, one finds
\begin{equation}
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over4} \sqrt{\ln x}\right);
\qquad (x\geq 2).
\mathrm{e}nd{equation}
Note that this could in principle have been deduced as early as 1976, some 46 years ago,
from the work of Schoenfeld~\cite{Schoenfeld}.
Similarly
\begin{equation}
|\vartheta(x)-x| < \;
{x} \;\mathrm{e}xp\left( - {1\over3} \sqrt{\ln x}\right);
\qquad (x\geq 3).
\mathrm{e}nd{equation}
Note that this particular bound could in principle have been deduced as early as 2016,
some 6 years ago, from the work of Trudgian~\cite{Trudgian}.
For the other two widely applicable bounds, some preliminary experimental investigations \mathrm{e}mph{suggest} that it might be possible to reduce the numerical prefactors (295, 385) significantly --- but doing so would require rather different techniques from the elementary observations made above.
As always one can trade of stringency against range of validity. In this regard let me mention two specific examples
\begin{equation}
|\vartheta(x)-x| < \;
{1\over 2} \; {x} \;\mathrm{e}xp\left( - {1\over4} \sqrt{\ln x}\right);
\qquad (x\geq 29),
\mathrm{e}nd{equation}
and
\begin{equation}
|\vartheta(x)-x| < \;
{1\over 2} \; {x} \;\mathrm{e}xp\left( - {1\over3} \sqrt{\ln x}\right);
\qquad (x\geq 41).
\mathrm{e}nd{equation}
In contrast, for some asymptotically stringent bounds, based on the Johnston--Yang results presented in reference~\cite{Johnston-Yang}, consider Table~IV. Note that we can again make the exponential factor smaller, (by increasing $\tilde c$), at the cost of making the numerical prefactor $\tilde a$ larger.
For instance one can deduce the ineffective bound
\begin{equation}
|\vartheta(x)-x| = \;
{\mathcal{O}} \left( {x} \;\mathrm{e}xp\left( - \sqrt{\ln x}\right)\right),
\mathrm{e}nd{equation}
which can be made effective as (for instance):
\begin{equation}
|\vartheta(x)-x| = \;
83063 \;\; {x} \;\mathrm{e}xp\left( - \sqrt{\ln x}\right); \qquad \left(x> \mathrm{e}xp(10^{10})\right).
\mathrm{e}nd{equation}
Many variations on this theme can be developed.
\begin{table}[!htb]
\caption{Asymptotically stringent bounds of the de la Vall\'e Poussin form.}
\begin{center}
\begin{tabular}{||c|c|c||}
\hline
\hline
$\tilde a$ & $\tilde c$ & $x_*$ \\
\hline
\hline
357 & 1/2 & $\mathrm{e}xp(3000)$\\
320 & 1/2 & $\mathrm{e}xp(4000)$ \\
295 & 1/2 & $\mathrm{e}xp(5000)$ \\
274 & 1/2 & $\mathrm{e}xp(6000)$ \\
263 & 1/2 & $\mathrm{e}xp(7000)$ \\
252 & 1/2 & $\mathrm{e}xp(8000)$ \\
244 & 1/2 & $\mathrm{e}xp(9000)$ \\
249 & 1/2 & $\mathrm{e}xp(10000)$ \\
\hline
\hline
644 & 1/2 & $\mathrm{e}xp(10^5)$ \\
348 & 1/2 & $\mathrm{e}xp(10^6)$ \\
312 & 1/2 & $\mathrm{e}xp(10^7)$ \\
301 & 1/2 & $\mathrm{e}xp(10^8)$ \\
298 & 1/2 & $\mathrm{e}xp(10^9)$ \\
297 & 1/2 & $\mathrm{e}xp(10^{10})$ \\
\hline
\hline
1642333 & 1 & $\mathrm{e}xp(10^6)$ \\
165152 & 1 & $\mathrm{e}xp(10^7)$ \\
101831 & 1 & $\mathrm{e}xp(10^8)$ \\
87551 & 1 & $\mathrm{e}xp(10^9)$ \\
83063 & 1 & $\mathrm{e}xp(10^{10})$ \\
\hline
\hline
\mathrm{e}nd{tabular}
\mathrm{e}nd{center}
\label{default}
\mathrm{e}nd{table}
\section{Conclusions}\label{S:discussion}
With some hindsight, deriving effective bounds of the de la Vall\'e Poussin form is, (given various effective results obtained over the last 50 years~\cite{Schoenfeld,Trudgian,Johnston-Yang,Fiori-et-al,Broadbent-et-al}), seen to be almost trivial.
Certainly, (given the effective results reported in~\cite{Schoenfeld,Trudgian,Johnston-Yang,Fiori-et-al,Broadbent-et-al}), nothing deeper than elementary calculus and some slightly tedious numerical checking was required. On the other hand, conceptually it is very pleasant to see simple explicit and effective bounds of the de la Vall\'e Poussin form dropping out so nicely.
\begin{thebibliography}{99}
\newcommand{\arXiv}[1]{arXiv:~{\href{https://arxiv.org/abs/#1}{\color{blue}#1}}}
\bibitem{Poussin}
Charles Jean de la Vall\'e Poussin,\\
``Recherches analytiques sur la th\'eorie des nombres premiers'',\\ Ann. Soc. Scient.
Bruxelles, deuxi\'eme partie {\bf20}, (1896), pp. 183--256
\bibitem{Schoenfeld}
Lowell Schoenfeld,\\
``Sharper bounds
for the Chebyshev functions $\vartheta(x)$ and $\psi(x)$. II'', \\
Mathematics of Computation,
{\bf 30 \#134} (April 1976) 337--360.\\
{\mathrm{d}}oi{10.1090/S0025-5718-1976-0457374-X}
\bibitem{Trudgian}
Tim Trudgian,
``Updating the error term in the prime number theorem'',\\
The Ramanujan Journal {\bf 39} (2016) 225--236,
{\mathrm{d}}oi{10.1007/S11139-014-9656-6}.
\arXiv{1401.2689} [math.NT]
\bibitem{Johnston-Yang}
Daniel R. Johnston, Andrew Yang,\\
``Some explicit estimates for the error term in the prime number theorem'',\\
\arXiv{2204.01980} [math.NT]
\bibitem{Fiori-et-al}
Andrew Fiori, Habiba Kadiri, Joshua Swindisky,\\
``Sharper bounds for the error term in the Prime Number Theorem'',\\
\arXiv{2206.12557} [math.NT]
\bibitem{Broadbent-et-al}
S. Broadbent, H. Kadiri, A. Lumley, N. Ng, and K. Wilk. \\
``Sharper bounds for the Chebyshev function $\vartheta(x)$''. \\
Math. Comp. 90.331 (2021), pp. 2281--2315.
[\arXiv{2002.11068} [math.NT]]
\hrule\hrule\hrule
\mathrm{e}nd{thebibliography}
\mathrm{e}nd{document}
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\begin{document}
\title{The generalised Oberwolfach problem}
\begin{abstract}
We prove that any quasirandom dense large graph
in which all degrees are equal and even can be
decomposed into any given collection of two-factors
($2$-regular spanning subgraphs).
A special case of this result gives a new solution to
the Oberwolfach problem.
\end{abstract}
\section{Introduction}
At meals in the Oberwolfach Mathematical Institute,
the participants are seated at circular tables.
At an Oberwolfach meeting in 1967, Ringel (see \cite{LR})
asked whether there must exist a sequence of seating plans
so that every pair of participants
sit next to each other exactly once.
We assume, of course, that there are an odd number of participants,
as each participant sits next to two others in each meal.
The tables may have various sizes,
which we assume are the same at each meal.
\nib{Oberwolfach Problem (Ringel).}
Let $F$ be any two-factor (i.e.\ $2$-regular graph)
on $n$ vertices, where $n$ is odd.
Can the complete graph $K_n$ be
decomposed into copies of $F$?
We obtain a new solution of this problem for large $n$,
with a theorem that is more general in three respects:
(a) we can decompose any dense quasirandom graph
that is regular of even degree
(not just $K_n$ for $n$ odd),
(b) we can decompose into any prescribed
collection of two-factors
(not just copies of some fixed two-factor $F$),
(c) our theorem applies to directed graphs (digraphs).
We start by stating our result for undirected graphs.
We require the following quasirandomness definition.
We say that a graph $G$ on $n$ vertices is
$(\varepsilon,t)$-typical if every set $S$ of at most $t$ vertices
has $((1 \pm \varepsilon)d(G))^{|S|} n$ common neighbours, where
$d(G) = e(G) \tbinom{n}{2}^{-1}$ is the density of $G$.
\begin{theo}\leftarrowbel{mainundir}
For all $\alpha>0$ there exist $t,\varepsilon,n_0$ such that
any $(\varepsilon,t)$-typical graph on $n \ge n_0$ vertices
that is $2r$-regular for some integer $r > \alpha n$
can be decomposed into any family of $r$ two-factors.
\end{theo}
Theorem \ref{mainundir} implies some variant forms of the
Oberwolfach problem that have appeared in the literature,
such as the Hamilton--Waterloo Problem (two types of two-factors),
or that if $n$ is even then $K_n$ can be decomposed into a
perfect matching and any specified collection
of $n/2-1$ two-factors. More generally,
with parameters as in Theorem \ref{mainundir},
it is easy to deduce that
any $(\varepsilon,t)$-typical graph on $n \ge n_0$ vertices
that is $(2r+1)$-regular for some integer $r > \alpha n$
can be decomposed into a perfect matching
and any family of $r$ two-factors.
We will deduce Theorem~\ref{mainundir}
from the directed version below.
First we extend our definitions to digraphs.
We say that a digraph $G$ on $n$ vertices is
$(\varepsilon,t)$-typical if for every set $S=S^- \cup S^+$
of at most $t$ vertices
there are $((1 \pm \varepsilon)d(G))^{|S|} n$ vertices which
are both common inneighbours of $S^-$
and outneighbours of $S^+$, where
$d(G) = e(G) \tbinom{n}{2}^{-1}$ is the density of $G$.
We say that $G$ is $r$-regular if
$d^+_G(v)=d^-_G(v)=r$ for all $v \in V(G)$.
A \emph{one-factor} is a $1$-regular digraph;
equivalently, it is a union of
vertex-disjoint oriented cycles.
\begin{theo}\leftarrowbel{main}
For all $\alpha>0$ there exist $t,\varepsilon,n_0$ such that
any $(\varepsilon,t)$-typical digraph on $n \ge n_0$ vertices
that is $r$-regular for some integer $r > \alpha n$
can be decomposed into any family of $r$ one-factors.
\end{theo}
Theorem~\ref{mainundir} follows from Theorem~\ref{main}
and the observation that for any typical graph
that is regular of even degree
there exists an orientation
which is a regular typical digraph.
To see this, one can orient edges independently at random
and make a few modifications to obtain the required orientation.
(See Lemma~\ref{typ:split} below for a similar argument.)
While we were preparing this paper,
the Oberwolfach problem (for large $n$) was solved by
Glock, Joos, Kim, K\"uhn and Osthus~\cite{GJKKO}.
They also obtained a more general result that covers
the other undirected applications just mentioned,
but our result is more general than theirs
in the three respects mentioned above:
(a) we can decompose any dense typical regular graph
(whereas their result only applies to almost complete graphs),
(b) we can decompose into any collection of two-factors
(whereas they can allow for a collection of two-factors
provided that some fixed $F$ occurs $\Omega(n)$ times),
(c) our result also applies to digraphs
(whereas theirs is for undirected graphs).
There is a large literature on the Oberwolfach Problem,
of which we mention just a few highlights
(a more detailed history is given in \cite{GJKKO}).
The problem was solved for infinitely many $n$
by Bryant and Scharaschkin \cite{BS},
in the case when $F$ consists of two cycles by Traetta \cite{T},
and for cycles of equal length by
Alspach, Schellenberg, Stinson and Wagner \cite{ASSW}.
A related conjecture of Alspach that $K_n$ can be
decomposed into any collection of cycles each of length $\le n$
and total size $\tbinom{n}{2}$ was solved
by Bryant, Horsley and Pettersson \cite{BHP}.
There are several recent general results on approximate
decompositions that imply an approximate solution
to the generalised Oberwolfach Problem,
i.e.\ that any given collection of two-factors
can be embedded in a quasirandom graph
provided that a small fraction of the edges
can be left uncovered: we refer to the papers of
Allen, B\"ottcher, Hladk\'y and Piguet \cite{ABHP},
Ferber, Lee and Mousset \cite{FLM}
and Kim, K\"uhn, Osthus and Tyomkyn \cite{KKOT}.
\nib{Notation.}
Given a graph $G = (V,E)$,
when the underlying vertex set $V$ is clear,
we will also write $G$ for the set of edges.
So $|G|$ is the number of edges of $G$. Usually $|V|=n$.
The \emph{edge density} $d(G)$ of $G$ is $|G|/\tbinom{n}{2}$.
We write $N_G(x)$ for the neighbourhood of a vertex $x$ in $G$.
The degree of $x$ in $G$ is $d_G(x)=|N_G(x)|$.
For $A \subseteqseteq V(G)$, we write
$N_G(A) := \bigcap_{x \in A}N_G(x)$;
note that this is the common neighbourhood
of all vertices in $A$, not the neighbourhood of $A$.
In a directed graph $J$ with $x \in V(J)$,
we write $N_J^+(x)$ for the set of out-neighbours of $x$ in $G$
and $N_G^-(x)$ for the set of in-neighbours.
We let $d^\pm_G(A) := |N^\pm_G(A)|$.
We define common out/in-neighbourhoods
$N_J^\pm(A) = \bigcap_{x \in A} N_J^\pm (A).$
We say $G$ is \emph{$(\varepsilon,t)$-typical}
if $d_G(S) = ((1 \pm \varepsilon)d(G))^{|S|} n$
for all $S \subseteqseteq V(G)$ with $|S| \le t$.
We say that an event $E$ holds with high probability (whp)
if $\mb{P}(E) > 1 - \mbox{ex}p(-n^c)$ for some $c>0$ and $n>n_0(c)$.
We note that by a union bound for any fixed collection $\mc{E}$
of such events with $|\mc{E}|$ of polynomial growth
whp all $E \in \mc{E}$ hold simultaneously.
We omit floor and ceiling signs for clarity of exposition.
We write $a \ll b$ to mean $\forall\ b>0 \
\mbox{ex}ists\ a_0>0 \ \forall\ 0<a<a_0$.
We write $a \pm b$ for an unspecified number in $[a-b,a+b]$.
Throughout the vertex set $V$ will come with a cyclic order,
which we usually identify with the natural cyclic order
on $[n]=\{1,\dots,n\}$. For any $x \in V$ we write $x^+$
for the successor of $x$, so if $x \in [n]$ then
$x^+$ is $x+1$ if $x \ne n$ or $1$ if $x=n$.
We define the predecessor $x^-$ similarly. Given $x,y$ in $[n]$
we write $d(x,y)$ for their cyclic distance,
i.e.\ $d(x,y) = \min \{ |x-y|, n-|x-y| \}$.
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\section{Overview of the proof} \leftarrowbel{sec:over}
We will illustrate the ideas of our proof by starting
with a special case and becoming gradually more general.
Suppose first that we wish to decompose a typical dense (undirected)
$2r$-regular graph $G$ on $n$ vertices into $r$ triangle-factors
(i.e.\ two-factors in which each cycle is a triangle
-- we require $3 \mid n$ for this question to make sense).
The existence of such a decomposition
(also known as a resolvable triangle-decomposition of $G$)
follows from a recent result of the first author \cite{K2}
generalising the existence of designs (see \cite{Kexist})
to many other `design-like' problems. The proof in \cite{K2} goes
via the following auxiliary decomposition problem,
which also plays an important role in this paper.
Let $J$ be an auxiliary graph with $V(J)$ partitioned
as $V \cup W$, where $V=V(G)$ and $|W|=r$.
Let $J[V]=G$, $J[V,W]=V \times W$ and $J[W]=\emptyset$.
Note that a decomposition of $G$ into triangle-factors
is equivalent to a decomposition of $J$ into copies of $K_4$
each having $3$ vertices in $V$ and $1$ vertex in $W$.
Indeed, given such a decomposition of $J$, for each $w \in W$
we define a triangle-factor of $G$ by removing $w$ from all
copies of $K_4$ containing $w$ in the decomposition;
clearly every edge of $G$ appears in exactly one
of these triangle-factors. Conversely, any decomposition of $G$
into triangle-factors can be converted into a suitable
$K_4$-decomposition of $J$ by adding each $w \in W$ to one of
the triangle-factors (according to an arbitrary matching).
The auxiliary construction described above is quite flexible,
so a similar argument covers many other cases of our problem.
For example, decomposing $G$ into $C_\ell$-factors
(two-factors in which each cycle has length $\ell$)
is equivalent to decomposing $J$ into `wheels' $W_\ell$
with `rim' in $V$ and `hub' in $W$. (We obtain $W_\ell$ from
$C_\ell$, which is called the rim, by adding a new vertex,
called the hub, joined to every other vertex,
by edges that we call spokes.)
Such a decomposition exists by \cite{K2}.
We can encode our generalised Oberwolfach Problem
in full generality by introducing colours on the edges.
For each possible cycle length $\ell$ we introduce a colour,
which we also call $\ell$. For each $w \in W$, we denote its
corresponding factor by $F_w$, and suppose that it has $n^w_\ell$
cycles of length $\ell$ (where $\sum_\ell \ell n^w_\ell = n$).
We colour $J$ so that each $w \in W$ is incident to exactly
$n^w_\ell$ edges of colour $\ell$, and all other edges are uncoloured.
We colour each $W_\ell$ so that exactly one spoke has colour $\ell$
and all other edges are uncoloured. Then a decomposition of $G$ into
$\{F_w: w \in W\}$ is equivalent to a decomposition of $J$
into wheels with this colouring with rim in $V$ and hub in $W$.
Note that this equivalence does not depend on which edges of $J$
we colour, but to apply \cite{K2} we will require the colouring
to be suitably quasirandom. Another important constraint
in applying \cite{K2} is that the number of colours
and the size of the wheels should be bounded by an absolute constant.
Thus our generalised Oberwolfach Problem can only be solved by
direct reduction to \cite{K2} in the case that all factors
have all cycle lengths bounded by some absolute constant.
This now brings us to the crucial issue for this paper:
how can we encode two-factors with cycles of arbitrary length
by an auxiliary construction to which \cite{K2} applies?
Before describing this, we pass to an auxiliary problem
of decomposing a subgraph $G'$ of $G$ into graphs
$(G_w: w \in W)$, where each $G_w$ is a vertex-disjoint
union of paths with prescribed endpoints, lengths and vertex set.
More precisely, for each $w \in W$ we are given specified
lengths $(\ell^w_i: i \in I_w)$, vertex-pairs
$((x^w_i,y^w_i): i \in I_w)$, a forbidden set $Z_w$,
and we want each $G_w$ to be a union of vertex-disjoint
$x^w_i y^w_i$-paths of length $\ell^w_i$ for each $i \in I_w$
with $V(G_w)=V(G) \setminus Z_w$. We will arrive at this problem
having embedded some subgraphs $F'_w \subseteq F_w$ of each $w \in W$,
so the prescribed endpoints will be endpoints of
paths in $F'_w$ that need to be connected up to form cycles,
and $Z_w$ will consist of all vertices of degree $2$ in $F'_w$.
We assume that all lengths $\ell^w_i$ are divisible by $8$
(which is easy to ensure for long cycles).
We will translate the above path factor problem
into an equivalent problem of decomposing a certain
auxiliary two-coloured directed graph $J$,
with $V(J) = V \cup W$ as in the previous construction.
We call the two colours `$0$' (which means `uncoloured')
and `$K$' (which means `special'). Again, $J[W]=\emptyset$.
For now we defer discussion of $J[V,W]$
and describe the arcs of $J[V]$,
which are in bijection with the edges of $G$.
For colour $0$ this bijection simply corresponds
to a choice of orientation for edges, but for
colour $K$ we employ the following `twisting' construction.
We fix throughout a cyclic order of $V$,
and require that each arc $\overlinea{xy}$ of colour $K$ in $J$
comes from an edge $xy^+$ of $G$, where $y^+$ denotes
the successor of $y$ in the cyclic order.
Consider any directed $8$-cycle $C$ in $J$ with vertex sequence
$x_1 \dots x_8$, such that all arcs have colour $0$
except that $\overlinea{x_7 x_8}$ has colour $K$.
The edges in $G$ corresponding to $C$ form a path
with vertex sequence $x_8 x_1 \dots x_7 x_8^+$.
Now suppose we have a family of such cycles
$\mc{C} = (C^i: i \in I)$ where each $C^i$
has vertex sequence $x^i_1 \dots x^i_8$.
Call $\mc{C}$ compatible if
(i) its cycles are mutually vertex-disjoint, and
(ii) if any $(x^i_8)^+$ is used by a cycle in
$\mc{C}$ then it is some $x^j_8$.
Suppose $\mc{C}$ is compatible and let
$([x_j,y_j]: j \in J)$ denote the family of maximal
cyclic intervals contained in $\{x^i_8: i \in I\}$.
Then the edges of $G$ corresponding to the cycles of $\mc{C}$
form a family of vertex-disjoint paths $(P_j: j \in J)$, where
each $P_j$ is an $x_j y_j^+$-path whose vertex sequence is the
concatenation of vertex sequences of the $8$-paths as described
above for each cycle of $\mc{C}$ using a vertex of $[x_j,y_j]$.
The above construction allows us to pass from the path factor problem
to finding certain edge-disjoint compatible cycle families in $J$.
In order for our path factor problem to obey the constraints of
this encoding we require the prescribed vertex-pairs for each $w$
to define disjoint cyclic intervals $([x^w_i,(y^w_i)^-]: i \in I_w)$
of lengths $\ell^w_i/8$ (and also that no successor $y^w_i$
is contained in any of the other intervals for $w$).
We are thus introducing extra constraints into the path factor
problem that may affect up to $n/8$ vertices for each $w$,
but the flexibility on the remaining
vertices will be sufficient.
Now we can complete the description of the auxiliary graph $J$
and the decomposition problem that encodes the compatible cycle
family problem. We define $J[V]$ as above,
and $J[V,W]$ so that all arcs are directed
towards $W$, each in-neighbourhood $N^-_J(w)$ is obtained
from $V(G) \setminus Z_w$ by deleting the interval successors
$\{ y^w_i: i \in I_w \}$, all arcs $\overlinea{xw}$
with $x$ in an interval $[x^w_i,(y^w_i)^-]$ are coloured $K$,
and all other arcs of $J[V,W]$ are coloured $0$.
Finally, the compatible cycle family problem is equivalent
to decomposing $J$ into coloured directed wheels $\ova{W}^K_{\! 8}$,
obtained from $W_8$ by directing the rim cyclically,
directing all spokes towards the hub $w$, giving colour $K$
to one rim edge $\overlinea{xy}$ and one spoke $\overlinea{yw}$,
and colouring the other edges by $0$.
The deduction from \cite{K2} of the existence of wheel
decompositions is given in section \ref{sec:wheel}.
We now describe the strategy for the proof of Theorem \ref{main}.
The goal is to embed some parts of our two-factors
so that the remaining problem is of one of two special types
that has an encoding suitable for applying \cite{K2},
either a path factor problem
encoded as $\ova{W}^K_{\! 8}$-decomposition
or a $C_\ell$-factor problem
encoded as $\overlinea{W}_{\!\ell}$-decomposition
(we take the coloured wheel $W_\ell$ discussed above
for $C_\ell$-factors and introduce directions
as in $\ova{W}^K_{\! 8}$, which are not necessary
but convenient for giving a unified analysis).
We call a factor `long' if it has at least $n/2$ vertices
in cycles of length at least $K$ (as well as denoting the
special colour, $K$ is also used as a large constant length
threshold, above which we treat cycles using the special
twisting encoding as above). We call the other factors `short'.
We start by reducing to the case that all factors are long
or all factors are short. To do so, suppose first that
there are $\Omega(n)$ long factors and $\Omega(n)$ short factors.
Then we can randomly partition $G$ into typical graphs
$G^L$ and $G^S$, each of which is regular of the correct degree
(twice the number of long factors for $G^L$
and twice the number of short factors for $G^S$).
If there are $o(n)$ factors of either type then these can
be embedded one-by-one (by the blow-up lemma \cite{KSS}),
and then the remaining problem still satisfies the conditions
of Theorem \ref{main} (with slightly weaker typicality).
The short factor problem can be further reduced to the case
that there is some length $\ell^*$ such that each factor has
$\Omega(n)$ cycles of length $\ell^*$. Indeed, we can divide the
factors into a constant number of groups according to some
choice of cycle length that appears $\Omega(n)$ times in each factor
of the group. Any group of $o(n)$ factors can be embedded greedily,
so after taking a suitable random partition, it suffices
to show that the remaining groups can each be embedded
in a graph that is typical and regular of the correct degree.
Thus we can assume that we are in one of the following cases.
Case $K$: all factors are long,
our goal is to reduce to $\ova{W}^K_{\! 8}$-decomposition.
Case $\ell^*$: all factors have $\Omega(n)$ cycles of length $\ell^*$,
our goal is to reduce to $\ova{W}_{\! \ell^*}$-decomposition.
In any case, the reduction is achieved by applying an approximate decomposition result in a suitable random subgraph,
in which we embed a subgraph of each of our factors.
At this step, in Case $\ell^*$ we embed
all cycles of length $\ne \ell^*$,
and in Case $K$ we embed all short cycles
and some parts of the long cycles as needed
to reduce to a suitable path factor problem.
This approximate decomposition result is superficially
similar to the maximum degree $2$ case of
the blow-up lemma for approximate decompositions
due to Kim, K\"uhn, Osthus and Tyomkyn \cite{KKOT}.
However, it does not suffice to use their result,
as we require a decomposition that is compatible
with the conditions of our final decomposition problem
(into $\ova{W}^K_{\! 8}$ or $\ova{W}_{\! \ell^*}$),
so the sets of vertices of the partial factors
embedded in this step must be suitably quasirandom
and avoid the intervals needed for Case $K$.
Furthermore, we obtain the required approximate
decomposition by similar arguments to those
for the exact decomposition,
which does not add much extra work.
The technical heart of the paper is a randomised algorithm
(presented in section \ref{sec:alg}), which gives
a unified treatment of the cases described above.
It simultaneously (a) partitions almost all of $G$
into two graphs $G_1$ and $G_2$, and
(b) sets up auxiliary digraphs $J_1$ and $J_2$ such that
(i) an approximate wheel decomposition of $J_2$
gives an approximate decomposition of $G_2$
into the partial factors described above, and
(ii) the graph $G'_1$ of edges that are unused by
the approximate decomposition has an auxiliary digraph
that is a sufficiently small perturbation of $J_1$ that
it can still be used for the exact decomposition step.
The analysis of the algorithm falls naturally into two parts:
the choice of intervals (section \ref{sec:int}),
then regularity properties of an auxiliary
hypergraph defined by wheels (section \ref{sec:reg}).
The results of this analysis are applied
to show the existence of the various partial factor
decompositions discussed above:
the approximate step is in section \ref{sec:approx}
and the exact step in section \ref{sec:exact}.
Section \ref{sec:pf} combines
all the ingredients prepared in the previous sections
to produce the proof of our main theorem.
The final section contains some concluding remarks.
\begin{figure}
\caption{An overview of the proof.}
\end{figure}
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\section{Wheel decompositions} \leftarrowbel{sec:wheel}
In this section we describe the results we need on wheel
decompositions and how they follow from \cite{K2}.
We start by recalling the coloured wheels
described in section \ref{sec:over}.
For any $c \ge 3$, the uncoloured $c$-wheel consists
of a directed $c$-cycle (called the rim),
another vertex (called the hub),
and an arc from each rim vertex to the hub.
We obtain the coloured $c$-wheel $\ova{W}_{\! c}$
by giving all arcs colour $0$ except
that one of the spokes has colour $c$.
We obtain the special $c$-wheel $\ova{W}^K_{\! 8}c$
by giving all arcs colour $0$ except
that one rim edge $\overlinea{xy}$
and one spoke $\overlinea{yw}$ have colour $K$.
As discussed in section \ref{sec:over},
we will only use $\ova{W}^K_{\! 8}c$ with $c=8$, but here
we will consider the general configuration
so that the decomposition problems are quite similar.
We start by stating the result for $\ova{W}_{\! c}$.
\begin{center}
\includegraphics{figwheel}
\end{center}
\begin{theo} \leftarrowbel{decompL}
Let $n^{-1} \ll \delta \ll \omega \ll c^{-1}$ and $h=2^{50c^3}$.
Let $J = J^0 \cup J^c$ be a digraph with arcs coloured
$0$ or $c$, with $V(J)$ partitioned as $(V,W)$
where $\omega n \le |V|, |W| \le n$.
Then $J$ has a $\ova{W}_{\! c}$-decomposition such that
every hub lies in $W$ if the following hold:
{\em Divisibility:}
all arcs in $J[V]$ have colour $0$,
all arcs in $J[V,W]$ point towards $W$,
$d_J^-(v,V)=d_J^+(v,V)=d_J^+(v,W)$ for all $v \in V$,
and $d^-_J(w) = cd^-_{J^c}(w)$ for all $w \in W$.
{\em Regularity:}
each copy of $\ova{W}_{\! c}$ in $J$ has a weight
in $[\omega n^{1-c}, \omega^{-1} n^{1-c}]$ such that
for any arc $\overlinea{e}$ there is total weight $1 \pm \delta$
on wheels containing $\overlinea{e}$.
{\em Extendability:}
for all disjoint $A,B \subseteq V$ and $C \subseteq W$
each of size $\le h$ we have
$|N^+_{J^0}(A) \cap N^+_{J^c}(B) \cap W| \ge \omega n$
and $|N^+_{J^0}(A) \cap N^-_{J^0}(B)
\cap N^-_{J^{c'}}(C)| \ge \omega n$
for both $c' \in \{0,c\}$.
\end{theo}
Before stating our result on $\ova{W}^K_{\! 8}$-decompositions,
we recall that $V$ has a cyclic order, which we can
identify with the natural cyclic order on $[n]$,
and define the following separation properties.
\begin{defn} \leftarrowbel{def:sep}
For $1 \le x<y \le n$ the cyclic distance is
$d(x,y) = \min\{y-x,n+x-y\}$.
We say that $S \subseteq [n]$ is $d$-separated
if $d(a,a') \ge d$ for all distinct $a,a'$ in $S$.
For disjoint $S,S' \subseteq [n]$ we say $(S,S')$
is $d$-separated if $d(a,a') \ge d$
for all $a \in S$, $a' \in S'$.
\end{defn}
Now we state our result on $\ova{W}^K_{\! 8}$-decompositions.
We note that it only concerns digraphs $J$ such that
$d(x,y) \ge d$ for all $\overlinea{xy} \in J[V]$,
as this is implied by the regularity assumption.
Our proof of Theorem \ref{main} will require us
to only consider such $J$, so that we
can satisfy the extendability assumption.
\begin{theo} \leftarrowbel{decompK}
Let $n^{-1} \ll \delta \ll \omega \ll c^{-1}$.
Let $h=2^{50c^3}$ and $d \ll n$.
Let $J = J^0 \cup J^K$ be a digraph with arcs coloured
$0$ or $K$, with $V(J)$ partitioned as $(V,W)$
where $\omega n \le |V|, |W| \le n$, such that
all arcs in $J[V,W]$ point towards $W$ and $J[W]=\emptyset$.
Then $J$ has a $\ova{W}^K_{\! 8}c$-decomposition such that
every hub lies in $W$ if the following hold:
{\em Divisibility:}
$d^-_J(w) = cd^-_{J^K}(w)$ for all $w \in W$,
and for all $v \in V$ we have
$d_J^-(v,V)=d_J^+(v,V)=d_J^+(v,W)$
and $d^-_{J^K}(v,V)=d^+_{J^K}(v,W)$.
{\em Regularity:}
each $3d$-separated copy of $\ova{W}^K_{\! 8}c$ in $J$ has a weight
in $[\omega n^{1-c}, \omega^{-1} n^{1-c}]$ such that
for any arc $\overlinea{e}$ there is total weight $1 \pm \delta$
on wheels containing $\overlinea{e}$.
{\em Extendability:}
for all disjoint $A,B \subseteq V$ and $L \subseteq W$
each of size $\le h$,
for any $a, b, \ell \in \{0,K\}$ we have
$|N^+_{J^a}(A) \cap N^-_{J^b}(B)
\cap N^-_{J^\ell}(L)| \ge \omega n$,
and furthermore, if $(A,B)$ is $3d$-separated then
$|N^+_{J^0}(A) \cap N^+_{J^K}(B) \cap W| \ge \omega n$.
\end{theo}
For the remainder of this section we will explain
how Theorem \ref{decompK} follows from \cite{K2}
(we omit the similar and simpler details
for Theorem \ref{decompL}). We follow the exposition
in \cite{K3}, which deduces from \cite{K2} a general
result on coloured directed designs that we will apply here.
\subseteqsection{The functional encoding}
We encode any digraph $J$ by a set of functions $\mf{J}$,
where for each arc $\overlinea{ab} \in J$ we include in $\mf{J}$
the function $(1 \mapsto a, 2 \mapsto b)$, i.e.\ the
function $f:[2] \to V(J)$ with $f(1)=a$ and $f(2)=b$.
We will identify $\mf{J}$ with its characteristic vector,
i.e.\ $\mf{J}_f = 1_{f \in \mf{J}}$; if we want to emphasise
the vector interpretation we write $\underline{\mf{J}}$.
If $J$ has coloured arcs, and $\ell$ is a colour,
we write $J^\ell$ for the digraph in colour $\ell$,
which is encoded by $\mf{J}^\ell$.
We will consider decompositions by a coloured digraph $H$
defined as follows. We start with $\ova{W}^K_{\! 8}c$ on the vertex set
$[c+1]$, where we label the rim cycle by $[c]$ cyclically
(so $c+1$ is the hub) so that, writing $c_-=c-1$ and $c_+=c+1$,
$\overlinea{c_- c}$ and $\overlinea{c c_+}$ have colour $K$
and all other arcs have colour $0$.
We let $\mc{P}$ be the partition $([c],\{c_+\})$ of $[c+1]$.
We introduce new colours $0'$ and $K'$, and change the colours
of $\overlinea{c c_+}$ to $K'$ and of the other spokes to $0'$.
We do this so that $H$ is `$(\mc{P},\text{id})$-canonical'
in the sense of \cite[Definition 7.1]{K3};
specialised to our setting, the relevant properties
are that $H$ is an oriented graph
(with no multiple edges or $2$-cycles)
and that for each colour all of its arcs have one fixed
pattern with respect to $\mc{P}$
(specifically, for colours $0$ and $K$
all arcs are contained in $[c]$,
and for colours $0'$ and $K'$
all arcs are directed from $[c]$ to $\{c_+\}$).
Now we translate the $H$-decomposition problem
for a digraph $J$ into its functional encoding.
We will have a partition $\mc{Q}=(V,W)$ of $V(J)$,
and wish to decompose $J$ by copies $\phi(H)$ of $H$
such that $\phi([c]) \subseteq V$ and $\phi(c_+) \in W$
(i.e.\ wheels with hub in $W$),
and $\phi([c])$ is $3d$-separated
(in which case we will say that the graph $\phi(H)$ is $3d$-separated).
We think of the
functional encoding $\mf{J}$ as living inside a
`labelled complex' $\Phi$ of all possible partial
embeddings of $H$: we define $\Phi = (\Phi_B: B \subseteq [c+1])$,
where each $\Phi_B$ consists of all injections
$\phi:B \to V(J)$ such that $\phi(B \cap [c]) \subseteq V$,
$\phi(B \cap \{c_+\}) \subseteq W$ and $Im(\phi)$ is $3d$-separated.
The set of functional encodings of possible embeddings
of $H$ (if present in $\mf{J}$) is then
\[ H(\Phi) := \{ \phi \mf{H} : \phi \in \Phi_{[c+1]} \},
\quad \text{where }
\phi \mf{H} := \{ \phi \circ \theta: \theta \in \mf{H} \}.\]
The $H$-decomposition problem for $J$ is equivalent
to finding $\mc{H} \subseteq H(\Phi)$ with
$\sum \{ \underline{\mf{H}'}: \mf{H}' \in \mc{H} \} = \underline{\mf{J}}$,
or equivalently $\bigcup \mc{H} = \mf{J}$
(where if $\mf{J}$ has multiple edges
we consider a multiset union).
We call such $\mc{H}$ an $H$-decomposition in $\Phi$.
\subseteqsection{Regularity}
Now we will describe the hypotheses of the theorem
that will give us an $H$-decomposition in $\Phi$.
We start with regularity, which is simply the
functional encoding of the regularity assumption
in Theorem \ref{decompK}. Specifically, we say
$J$ is $(H,\delta,\omega)$-regular in $\Phi$ if there are
weights $y_\phi \in [\omega n^{1-c}, \omega^{-1} n^{1-c}]$
for each $\phi \in \Phi_{[c+1]}$
with $\phi \mf{H} \subseteq \mf{J}$ such that
$\sum_\phi y_\phi \underline{\phi \mf{H}} = (1 \pm \delta)\underline{\mf{J}}$.
\subseteqsection{Extendability}
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{\em Extendability:}
for all disjoint $A,B \subseteq V$ and $L \subseteq W$
each of size $\le h$,
for any $a, b, \ell \in \{0,K\}$ we have
$|N^+_{J^a}(A) \cap N^-_{J^b}(B)
\cap N^-_{J^\ell}(L)| \ge \omega n$,
and furthermore, if $(A,B)$ is $3d$-separated then
$|N^+_{J^0}(A) \cap N^+_{J^K}(B) \cap W| \ge \omega n$.
\end{theo}
For the remainder of this section we will explain
how Theorem \ref{decompK} follows from \cite{K2}
(we omit the similar and simpler details
for Theorem \ref{decompL}). We follow the exposition
in \cite{K3}, which deduces from \cite{K2} a general
result on coloured directed designs that we will apply here.
\subseteqsection{The functional encoding}
We encode any digraph $J$ by a set of functions $\mf{J}$,
where for each arc $\overlinea{ab} \in J$ we include in $\mf{J}$
the function $(1 \mapsto a, 2 \mapsto b)$, i.e.\ the
function $f:[2] \to V(J)$ with $f(1)=a$ and $f(2)=b$.
We will identify $\mf{J}$ with its characteristic vector,
i.e.\ $\mf{J}_f = 1_{f \in \mf{J}}$; if we want to emphasise
the vector interpretation we write $\underline{\mf{J}}$.
If $J$ has coloured arcs, and $\ell$ is a colour,
we write $J^\ell$ for the digraph in colour $\ell$,
which is encoded by $\mf{J}^\ell$.
We will consider decompositions by a coloured digraph $H$
defined as follows. We start with $\ova{W}^K_{\! 8}c$ on the vertex set
$[c+1]$, where we label the rim cycle by $[c]$ cyclically
(so $c+1$ is the hub) so that, writing $c_-=c-1$ and $c_+=c+1$,
$\overlinea{c_- c}$ and $\overlinea{c c_+}$ have colour $K$
and all other arcs have colour $0$.
We let $\mc{P}$ be the partition $([c],\{c_+\})$ of $[c+1]$.
We introduce new colours $0'$ and $K'$, and change the colours
of $\overlinea{c c_+}$ to $K'$ and of the other spokes to $0'$.
We do this so that $H$ is `$(\mc{P},\text{id})$-canonical'
in the sense of \cite[Definition 7.1]{K3};
specialised to our setting, the relevant properties
are that $H$ is an oriented graph
(with no multiple edges or $2$-cycles)
and that for each colour all of its arcs have one fixed
pattern with respect to $\mc{P}$
(specifically, for colours $0$ and $K$
all arcs are contained in $[c]$,
and for colours $0'$ and $K'$
all arcs are directed from $[c]$ to $\{c_+\}$).
Now we translate the $H$-decomposition problem
for a digraph $J$ into its functional encoding.
We will have a partition $\mc{Q}=(V,W)$ of $V(J)$,
and wish to decompose $J$ by copies $\phi(H)$ of $H$
such that $\phi([c]) \subseteq V$ and $\phi(c_+) \in W$
(i.e.\ wheels with hub in $W$),
and $\phi([c])$ is $3d$-separated
(in which case we will say that the graph $\phi(H)$ is $3d$-separated).
We think of the
functional encoding $\mf{J}$ as living inside a
`labelled complex' $\Phi$ of all possible partial
embeddings of $H$: we define $\Phi = (\Phi_B: B \subseteq [c+1])$,
where each $\Phi_B$ consists of all injections
$\phi:B \to V(J)$ such that $\phi(B \cap [c]) \subseteq V$,
$\phi(B \cap \{c_+\}) \subseteq W$ and $Im(\phi)$ is $3d$-separated.
The set of functional encodings of possible embeddings
of $H$ (if present in $\mf{J}$) is then
\[ H(\Phi) := \{ \phi \mf{H} : \phi \in \Phi_{[c+1]} \},
\quad \text{where }
\phi \mf{H} := \{ \phi \circ \theta: \theta \in \mf{H} \}.\]
The $H$-decomposition problem for $J$ is equivalent
to finding $\mc{H} \subseteq H(\Phi)$ with
$\sum \{ \underline{\mf{H}'}: \mf{H}' \in \mc{H} \} = \underline{\mf{J}}$,
or equivalently $\bigcup \mc{H} = \mf{J}$
(where if $\mf{J}$ has multiple edges
we consider a multiset union).
We call such $\mc{H}$ an $H$-decomposition in $\Phi$.
\subseteqsection{Regularity}
Now we will describe the hypotheses of the theorem
that will give us an $H$-decomposition in $\Phi$.
We start with regularity, which is simply the
functional encoding of the regularity assumption
in Theorem \ref{decompK}. Specifically, we say
$J$ is $(H,\delta,\omega)$-regular in $\Phi$ if there are
weights $y_\phi \in [\omega n^{1-c}, \omega^{-1} n^{1-c}]$
for each $\phi \in \Phi_{[c+1]}$
with $\phi \mf{H} \subseteq \mf{J}$ such that
$\sum_\phi y_\phi \underline{\phi \mf{H}} = (1 \pm \delta)\underline{\mf{J}}$.
\subseteqsection{Extendability}
Next we consider extendability, which we discuss
in a simplified setting that suffices for our purposes,
following \cite[Definition 7.3]{K3}.
The idea is that for any vertex $x$ of $H$
there should be many ways to extend certain sets
of partial embeddings of $H-x$ to embeddings of $H$.
Specifically, we say $(\Phi,J)$ is $(\omega,h,H)$-vertex-extendable
if for any $x \in [c+1]$ and disjoint $A_i \subseteq V \cup W$
for $i \in [c+1] \setminus \{x\}$ each of size $\le h$ such that
$(i \mapsto v_i: i \in [c+1] \setminus \{x\}) \in \Phi$
whenever each $v_i \in A_i$, there are at least $\omega n$
vertices $v$ such that
\begin{enumerate}
\item $(i \mapsto v_i: i \in [c+1]) \in \Phi$
whenever $v_x=v$ and $v_i \in A_i$ for each $i \ne x$, and
\item each $\mf{J}^\ell$ with $\ell \in \{0,K,0',K'\}$
contains all $(1 \mapsto v_1, 2 \mapsto v_2)$
where for some $\theta \in \mf{H}^\ell$ we have
($v_1=v \ \& \ v_2 \in A_{\theta(2)}$) or
($v_2=v \ \& \ v_1 \in A_{\theta(1)}$).
\end{enumerate}
Note that by definition of $\Phi$ this only concerns
maps $\phi$ such that $Im(\phi)$ is $3d$-separated.
To interpret (ii) we consider $4$ cases
according to the position of $x$ in the wheel.
\begin{description}
\item[$x=c+1$.]
For any pairwise $3d$-separated $A_i \subseteq V$, $i \in [c]$ of sizes
$\le h$ there are at least $\omega n$ vertices $v$
such that $\overlinea{v_c v} \in J^{K'}$ for all $v_c \in A_c$
and $\overlinea{v_i v} \in J^{0'}$ for all $v_i \in A_i$, $i \ne c$.
Equivalently, for any disjoint $A,B \subseteq V$
with $|A| \le h$ and $|B| \le (c-1)h$
such that $(A,B)$ is $3d$-separated we have
$|N^+_{J^{K'}}(A) \cap N^+_{J^{0'}}(B)| \ge \omega n$.
\item[$x=c$.]
For any pairwise $3d$-separated $A_i \subseteq V$, $i \in [c-1]$
and $A_{c+1} \subseteq W$ of sizes $\le h$
there are at least $\omega n$ vertices $v$ such that
$\overlinea{v v_{c+1}} \in J^{K'}$ for all $v_{c+1} \in A_{c+1}$,
$\overlinea{v_{c-1} v} \in J^K$ for all $v_{c-1} \in A_{c-1}$,
and $\overlinea{vv_1} \in J^0$ for all $v_1 \in A_1$.
Equivalently, for any disjoint $A,B \subseteq V$ and $C \subseteq W$
of sizes $\le h$ such that $(A,B)$ is $3d$-separated we have
$|N^+_{J^K}(A) \cap N^-_{J^0}(B) \cap N^-_{J^{K'}}(C)| \ge \omega n$.
\item[$x=c-1$.]
For any pairwise $3d$-separated
$A_i \subseteq V$, $i \in [c] \setminus \{c-1\}$
and $A_{c+1} \subseteq W$ of sizes $\le h$
there are at least $\omega n$ vertices $v$ such that
$\overlinea{v v_{c+1}} \in J^{0'}$ for all $v_{c+1} \in A_{c+1}$,
$\overlinea{v v_c} \in J^K$ for all $v_c \in A_c$,
and $\overlinea{v_{c-2} v} \in J^0$ for all $v_{c-2} \in A_{c-2}$.
Equivalently, for any disjoint $A,B \subseteq V$ and $C \subseteq W$
of sizes $\le h$ such that $(A,B)$ is $3d$-separated we have
$|N^-_{J^K}(A) \cap N^+_{J^0}(B) \cap N^-_{J^{0'}}(C)| \ge \omega n$.
\item[$x \in \brak{c-2}$.]
For any pairwise $3d$-separated
$A_i \subseteq V$, $i \in [c] \setminus \{x\}$
and $A_{c+1} \subseteq W$ of sizes $\le h$
there are at least $\omega n$ vertices $v$ such that
$\overlinea{v v_{c+1}} \in J^{0'}$ for all $v_{c+1} \in A_{c+1}$,
$\overlinea{v v_{x+1}} \in J^0$ for all $v_{x+1} \in A_{x+1}$,
and $\overlinea{v_{x-1} v} \in J^0$ for all $v_{x-1} \in A_{x-1}$,
where $A_0 := A_c$.
Equivalently, for any disjoint $A,B \subseteq V$ and $C \subseteq W$
of sizes $\le h$ such that $(A,B)$ is $3d$-separated we have
$|N^-_{J^0}(A) \cap N^+_{J^0}(B) \cap N^-_{J^{0'}}(C)| \ge \omega n$.
\end{description}
All of these conditions follow from the extendability
assumption in Theorem \ref{decompK}
(after renaming colours $0$ and $K$
in $J[V,W]$ as $0'$ and $K'$, and replacing $h$ with $(c-1)h$).
\subseteqsection{Divisibility}
It remains to consider divisibility;
we follow \cite[Definition 7.2]{K3}.
For integers $s \le t$ we write $I^s_t$
for the set of injections from $[s]$ to $[t]$.
We identify $V \cup W$ with $[n']$ for some $n'$.
For $0 \le i \le 2$, $\psi \in I^i_{n'}$, $\theta \in I^i_{c+1}$,
we define index vectors in $\mb{N}^2$ describing types
with respect to the partitions $\mc{P}$ or $\mc{Q}$: we write
$i_{\mc{P}}(\theta) = (|Im(\theta) \cap [c]|,|Im(\theta) \cap \{c_+\}|)$
and $i_{\mc{Q}}(\psi) = (|Im(\psi) \cap V|,|Im(\psi) \cap W|)$.
For example, for $\theta = (1 \mapsto c_-, 2 \mapsto c) \in \mf{H}$
we have $i_{\mc{P}}(\theta) = (2,0)$. We define degree vectors
$\mf{H}(\theta)^*$ and $\mf{J}(\psi)^*$ in $\mb{N}^{C \times I^i_2}$ by
\[ \mf{H}(\theta)^*_{\ell\pi}=|\mf{H}^\ell(\theta\pi^{-1})|
\ \ \text{ and } \ \
\mf{J}(\psi)^*_{\ell\pi}=|\mf{J}^\ell(\psi\pi^{-1})|, \]
where e.g.\ $\mf{H}^\ell(\theta\pi^{-1})$ denotes the set of
$\theta' \in \mf{H}^\ell$ having $\theta\pi^{-1}$ as a restriction.
Letting $\sgen{\cdot}$ denote the integer span of a set of vectors,
we say $J$ is $H$-divisible in $\Phi$ if
\[ \mf{J}(\psi)^* \in \sgen{\mf{H}(\theta)^*:
i_{\mc{P}}(\theta) = i_{\mc{Q}}(\psi) }
\ \ \text{ for all } \psi \in \Phi. \]
We refer to the divisibility conditions for index vectors
$(i_1,i_2)$ with $i_1+i_2=j$ as $j$-divisibility conditions,
where we assume $0 \le j \le 2$, as otherwise they are vacuous.
We describe these conditions concretely as follows.
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\subseteqsection{Divisibility}
It remains to consider divisibility;
we follow \cite[Definition 7.2]{K3}.
For integers $s \le t$ we write $I^s_t$
for the set of injections from $[s]$ to $[t]$.
We identify $V \cup W$ with $[n']$ for some $n'$.
For $0 \le i \le 2$, $\psi \in I^i_{n'}$, $\theta \in I^i_{c+1}$,
we define index vectors in $\mb{N}^2$ describing types
with respect to the partitions $\mc{P}$ or $\mc{Q}$: we write
$i_{\mc{P}}(\theta) = (|Im(\theta) \cap [c]|,|Im(\theta) \cap \{c_+\}|)$
and $i_{\mc{Q}}(\psi) = (|Im(\psi) \cap V|,|Im(\psi) \cap W|)$.
For example, for $\theta = (1 \mapsto c_-, 2 \mapsto c) \in \mf{H}$
we have $i_{\mc{P}}(\theta) = (2,0)$. We define degree vectors
$\mf{H}(\theta)^*$ and $\mf{J}(\psi)^*$ in $\mb{N}^{C \times I^i_2}$ by
\[ \mf{H}(\theta)^*_{\ell\pi}=|\mf{H}^\ell(\theta\pi^{-1})|
\ \ \text{ and } \ \
\mf{J}(\psi)^*_{\ell\pi}=|\mf{J}^\ell(\psi\pi^{-1})|, \]
where e.g.\ $\mf{H}^\ell(\theta\pi^{-1})$ denotes the set of
$\theta' \in \mf{H}^\ell$ having $\theta\pi^{-1}$ as a restriction.
Letting $\sgen{\cdot}$ denote the integer span of a set of vectors,
we say $J$ is $H$-divisible in $\Phi$ if
\[ \mf{J}(\psi)^* \in \sgen{\mf{H}(\theta)^*:
i_{\mc{P}}(\theta) = i_{\mc{Q}}(\psi) }
\ \ \text{ for all } \psi \in \Phi. \]
We refer to the divisibility conditions for index vectors
$(i_1,i_2)$ with $i_1+i_2=j$ as $j$-divisibility conditions,
where we assume $0 \le j \le 2$, as otherwise they are vacuous.
We describe these conditions concretely as follows.
{\bf $2$-divisibility.}
These conditions simply say that the arcs of $J$
have the same types with respect to $\mc{Q}$
as those of $H$ do with respect to $\mc{P}$,
i.e.\ all arcs of $J[V]$ have colour $0$ or $K$,
all arcs of $J[V,W]$ have colour $0'$ or $K'$, and $J[W]=\emptyset$.
To see this, consider any degree vector $\mf{H}(\theta)^*$
with $\theta \in I^2_{c+1}$. We write
$\text{id} = (1 \mapsto 1, 2 \mapsto 2)$ and
$(12) = (1 \mapsto 2, 2 \mapsto 1)$.
For any $\ell \in C$, $\pi \in \{\text{id},(12)\}$
we have $\mf{H}(\theta)^*_{\ell \pi}$ equal to $1$
if $(\ell,\pi)$ is the pair such that
$\theta \circ \pi^{-1} \in \mf{H}^\ell$
(there is at most one such pair)
or equal to $0$ otherwise.
For example, if $\theta = (1 \mapsto c, 2 \mapsto c_-)$
then $\mf{H}(\theta)^*_{\ell \pi}$ is $1$
if $(\ell,\pi)=(K,(12))$, otherwise $0$.
Thus $\mf{H}\sgen{(i_1,i_2)} :=
\sgen{\mf{H}(\theta)^*: i_{\mc{P}}(\theta) = (i_1,i_2)}$
consists of all integer vectors supported in coordinates
with colours in $\{0,K\}$ if $(i_1,i_2)=(2,0)$
or $\{0',K'\}$ if $(i_1,i_2)=(1,1)$, whereas
$\mf{H}\sgen{(0,2)}$ only contains the all-$0$ vector.
Therefore, the $2$-divisibility conditions say that
$\mf{J}(\psi)^*$ can be non-zero only at coordinates
with colours in $\{0,K\}$ if $i_{\mc{Q}}(\psi)=(2,0)$
or $\{0',K'\}$ if $i_{\mc{Q}}(\psi)=(1,1)$,
and $\mf{J}(\psi)^*=0$ if $i_{\mc{Q}}(\psi)=(0,2)$,
i.e.\ $J$ has the same arc types with respect to $\mc{Q}$
as $H$ with respect to $\mc{P}$.
{\bf $0$-divisibility.}
Writing $\emptyset$ for the function with empty domain,
all $\mf{H}(\emptyset)^*_{\ell \emptyset} = |\mf{H}^\ell|=|H^\ell|$,
and similarly for $J$, so the $0$-divisibility condition
is that for some integer $m$ all $|J^\ell|=m|H^\ell|$.
For our specific $H$, this is equivalent to
$|J[V]|=|J[V,W]|=c|J^c[V]|=c|J^c[V,W]|$.
{\bf $1$-divisibility.}
Given $\theta=(1 \mapsto a) \in I^1_{c+1}$
and $\ell \in C =\{0,K,0',K'\}$,
the two coordinates of $\mf{H}(\theta)^*$
corresponding to colour $\ell$
are the in/outdegrees of $a$ in $H^\ell$:
we have $\mf{H}(\theta)^*_{\ell \text{id}}
= |\mf{H}(1 \mapsto a)| = d^+_{H^\ell}(a)$
and $\mf{H}(\theta)^*_{\ell (12)}
= |\mf{H}(2 \mapsto a)| = d^-_{H^\ell}(a)$.
Similarly, for $\psi=(1 \mapsto v) \in I^1_{n'}$
the coordinates of $\mf{J}(\psi)^*$ corresponding
to colour $\ell$ are $d^\pm_{J^\ell}(v)$. We compute:
\begin{tabular}{LCCCCCCCC}
\mf{H}(1 \mapsto a)^* & d^+_{H^0}(a) & d^-_{H^0}(a)
& d^+_{H^K}(a) & d^-_{H^K}(a) & d^+_{H^{0'}}(a)
& d^-_{H^{0'}}(a) & d^+_{H^{K'}}(a) & d^-_{H^{K'}}(a) \\
a = c_+ & 0 & 0 & 0 & 0 & 0 & c-1 & 0 & 1 \\
a = c & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
a = c_- & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\
a \in [c-2] & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0
\end{tabular}
\begin{align*}
\text{ so } \ \ \sgen{\mf{H}(1 \mapsto c_+)^*}
& = \{ \bm{v} \in \mathbb{Z}^8 : v_1=v_2=v_3=v_4=v_5=v_7=0,
v_6=(c-1)v_8 \}, \text{ and } \\
\sgen{\mf{H}(1 \mapsto a)^*: a \in [c]}
& = \{ \bm{v} \in \mathbb{Z}^8 : v_2=v_5, v_4=v_7,
v_1+v_3=v_2+v_4, v_6=v_8=0 \}.
\end{align*}
For $w \in W$ the $1$-divisibility condition
is $\mf{J}(1 \mapsto w)^* \in \sgen{\mf{H}(1 \mapsto c_+)^*}$,
i.e.\ $d^-_{J^{0'}}(w) = (c-1)d^-_{J^{K'}}(w)$,
or equivalently $d^-_J(w) = cd^-_{J^{K'}}(w)$.
For $v \in V$ the $1$-divisibility condition
is $\mf{J}(1 \mapsto v)^* \in
\sgen{\mf{H}(1 \mapsto a)^*: a \in [c]}$,
which is equivalent to
$d^-_{J^K}(v) = d^+_{J^{K'}}(v)$
and $d^+_J(v,V) = d^-_J(v,V) = d^+_J(v,W)$.
All of these divisibility conditions follow from the
divisibility assumption in Theorem \ref{decompK}
(after renaming colours $0$ and $K$
in $J[V,W]$ as $0'$ and $K'$).
By the above discussion, Theorem \ref{decompK}
follows from the following special case
of \cite[Theorem 7.4]{K3}.
\begin{theo}
Let $n^{-1} \ll \delta \ll \omega \ll c^{-1}$.
Let $h=2^{50c^3}$ and $d \ll n$.
Let $J$ be a digraph with $V(J)$ partitioned as $(V,W)$
where $\omega n \le |V|, |W| \le n$, such that $J[W]=\emptyset$,
all arcs in $J[V,W]$ point towards $W$,
all arcs in $J[V]$ are coloured $0$ or $K$
and all arcs in $J[V,W]$ are coloured $0'$ or $K'$.
Let $\Phi = (\Phi_B: B \subseteq [c+1])$,
where $\Phi_B$ consists of all injections
$\phi:B \to V(J)$ such that $\phi(B \cap [c]) \subseteq V$,
$\phi(B \cap \{c_+\}) \subseteq W$ and $Im(\phi)$ is $3d$-separated.
Suppose $J$ is $H$-divisible in $\Phi$
and $(H,\delta,\omega)$-regular in $\Phi$ and
$(\Phi,J)$ is $(\omega,h,H)$-vertex-extendable.
Then $J$ has an $H$-decomposition in $\Phi$.
\end{theo}
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\section{The algorithm} \leftarrowbel{sec:alg}
Suppose we are in the setting of Theorem \ref{main}:
we are given a $(\varepsilon,t)$-typical $\alpha n$-regular digraph $G$
on $n$ vertices, where $n^{-1} \ll \varepsilon \ll t^{-1} \ll \alpha$,
and we need to decompose $G$ into some given
family $\mc{F}$ of $\alpha n$ oriented one-factors on $n$ vertices.
In this section we present an algorithm that partitions
almost all of $G$ into two digraphs $G_1$ and $G_2$,
and each factor $F_w$ into subfactors $F^1_w$ and $F^2_w$,
and also sets up auxiliary digraphs $J_1$ and $J_2$, such that
(i) an approximate wheel decomposition of $J_2$
gives an approximate decomposition of $G_2$
into partial factors that are roughly $\{F^2_w\}$,
(ii) given the approximate decomposition of $G_2$,
we can set up (via a small additional greedy embedding)
the remaining problem to be finding an exact decomposition
of a small perturbation $G'_1$ of $G_1$ into partial factors
that are roughly $\{F^1_w\}$, corresponding to a wheel
decomposition of a small perturbation $J'_1$ of $J_1$.
For most of the section we will describe and motivate
the algorithm; we then conclude with the formal statement.
We fix additional parameters with hierarchy
\begin{equation} \leftarrowbel{hierarchy}
n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll d^{-1}
\ll \eta \ll s^{-1} \ll L^{-1} \ll \alpha.
\end{equation}
For convenient reference later, we also make some comments
here regarding the roles of these additional parameters:
$\eta$ will be used to bound the number of vertices embedded greedily,
we consider a cycle `long' if it has length at least $K$,
and the cyclic intervals used to define the special colour $K$
will have sizes $d_i = d/(2s)^{i-1}$ with $i \in [2s+1]$.
By the reductions in section \ref{sec:red}, we will be able
to assume that we are in one of the following cases:
Case $K$: each $F \in \mc{F}$ has at least $n/2$ vertices
in cycles of length at least $K$,
Case $\ell^*$ with $\ell^* \in [3,L]$: each $F \in \mc{F}$
has $\ge L^{-3} n$ cycles of length $\ell^*$.
We write $\mc{F}=(F_w: w \in W)$, so $|W| = \alpha n$.
We partition each $F_w$ as $F^1_w \cup F^2_w$ as follows.
In Case $\ell^*$ we let $F^1_w$ consist of exactly
$L^{-3} n$ cycles of length $\ell^*$
(and then $F^2_w = F_w \setminus F^1_w$).
In Case $K$ we choose $F^1_w$
with $|F^1_w| - n/2 \in [0,2K]$
to consist of some cycles of length at least $K$
and at most one path of length at least $K$.
To see that this is possible,
consider any induced subgraph $F'_w$ of $F_w$
with $|F'_w|=n/2+K$ obtained by greedily adding cycles
of length at least $K$ until the size is at least $n/2 + K$,
and then deleting a (possibly empty) path from one cycle.
Let $P_1$ and $P_2$ denote the two paths
of the (possibly) split cycle, where $P_1 \in F'_w$.
If $|P_1|, |P_2| \ge K$ we let $F^1_w=F'_w$.
If $|P_1| < K$ we let $F^1_w=F'_w \setminus P_1$.
If $|P_2| < K$ we let $F^1_w=F'_w \cup P_2$.
In all cases, $F^1_w$ is as required.
The algorithm is randomised, so we start by defining
probability parameters. The graphs $G_1$ and $G_2$
are binomial random subdigraphs of $G$ of sizes
that are slightly less than one would expect
(we leave space for a greedy embedding that will occur
between the approximate decomposition step
and the exact decomposition step).
For each $w \in W$ we let
$p^g_w = (1-\eta) n^{-1}|F^g_w| + n^{-.2}$
(so $1-\eta \le p^1_w+p^2_w \le 1-L^{-3}\eta$).
We let $p_g = |W|^{-1}\sum_{w \in W} p^g_w$
(so $1-\eta \le p_1+p_2 \le 1-L^{-3}\eta$).
For each arc $e$ of $G$ independently we will let
$\mb{P}(e \in G_g) = p_g$ for $g \in [2]$.
We introduce further probabilities corresponding
to the cycle distributions of each $F^g_w$.
For $c<K$ we write $q^g_{w,c} n$ for the number
of cycles of length $c$ in $F^g_w$
and let $p^g_{w,c} = (1-\eta) q^g_{w,c}$.
We define $p^g_{w,K}$ so that $F^g_w$
has about $8p^g_{w,K}n$ vertices
not contained in cycles of length $<K$
(for technical reasons, we also ensure that
each $p^g_{w,K} \ge n^{-.2}$, which explains
the term $n^{-.2}$ in the definition of $p^g_w$).
Averaging over $W$ gives the corresponding probabilities
that describe the uses of arcs in each $G_g$:
we let $p^g_c = |W|^{-1} \sum_{w \in W} p^g_{w,c}$
so that for each $c<K$, the number of edges in $G_g$
allocated to cycles of length $c$ will be roughly
$\sum_{w \in W} cp^g_{w,c} n = |W| cp^g_c n
= \alpha cp^g_c n^2 = cp^g_c |G| + O(n)$,
and similarly, roughly $8p^g_K |G| + O(n)$ arcs
in $G_g$ will be allocated to long cycles.
The remainder of the algorithm is concerned
with the auxiliary digraphs $J_g$.
For any colour $c$, we let $J^c_g$
denote the arcs of colour $c$ in $J_g$.
We also write $J^*_g = \cup_{c \ne K} J^c_g$.
First we consider arcs within $J_g[V]$.
Throughout the paper, we fix a cyclic order on $V$,
which we choose uniformly at random.
For $v \in V$, let $v^+$ denote the successor of $v$
and $v^-$ denote the predecessor of $V$.
Arcs of the special colour $K$ should correspond
to $1/8$ of the factor arcs that are not in short cycles,
so should form a graph of density about $p^g_K$.
For each arc $\overlinea{xy} \in G_g$ not of the form
$\overlinea{zz}^+$ (to avoid loops, we don't mind double edges)
independently we assign $\overlinea{xy}$ to
colour $K$ with probability $p^g_K/p_g$
or colour $0$ with probability $p^g_*/p_g$
(where $p^g_K + p^g_*$ is slightly less than $p_g$).
If $\overlinea{xy}$ has colour $K$ we add
$\overlinea{xy}^-$ to $J^K_g$.
Now we consider $J_g[V,W]$.
These arcs are all directed from $V$ to $W$.
For each $w \in W$ and cycle length $c<K$,
there should be about $cp^g_{w,c} n$
vertices available for the $c$-cycles in $F^g_w$.
The colouring of $\ova{W}_{\! c}$ requires $1/c$-fraction
of these to be joined to $w$ in colour $c$, so we
should have $N^-_{J^c_g}(w) \approx p^g_{w,c} n$.
Similarly, there should be about $8p^g_{w,K} n$
vertices available for vertices of $F^g_w$ not in
short cycles, and the colouring of $\ova{W}^K_{\! 8}$ requires
$1/8$ of these to be joined to $w$ in colour $c$,
so we should have $N^-_{J^K_g}(w) \approx p^g_{w,K} n$.
These arcs are chosen randomly, not independently,
but according to a random collection of intervals,
of sizes $d_i = d/(2s)^{i-1}$ with $i \in [2s+1]$,
where $d$ is small enough that the resulting graph
is roughly typical, but large enough to give a good
upper bound on the number of vertices in long cycles
that become unused when they are chopped up into paths,
and so need to be embedded greedily.
These intervals must be chosen quite carefully,
because of the following somewhat subtle constraint.
Recall that in Case $K$ we will reduce to a path factor
problem in some subdigraph $H$ of $G$. This can only have
a solution if each vertex $x$ has degree
$d_H^\pm(x) = d_2(x) - d_{\pm}(x)$, where $d_2(x)$
is the number of path factors that will use $x$
and $d_-(x)$ (respectively $d_+(x)$)
is the number of these
in which $x$ is the start (respectively end).
The path factors will be obtained from
a set of arc-disjoint $\ova{W}^K_{\! 8}$'s, where for each $w \in W$,
its colour $K$ neighbourhood is given by a set of intervals
$([x^w_i,(y^w_i)^-]: i \in I_w)$, so its $\ova{W}^K_{\! 8}$'s will define
paths from $x^w_i$ to $y^w_i$. Thus in the auxiliary
digraph $J$, the degree of $x$ into $W$ must be
$d^+_J(x,W) = d_2(x) - d'_1(x)$, where $d'_1(x)$
is the number of path factors in which
$x$ is some successor $(y^w_i)^+$.
To relate these two formulae,
we note that a wheel decomposition of $J$ requires
$d^+_J(x,W)=d^+_J(x,V)=d^-_J(x,V)$
and $d^+_{J^K}(x,W)=d^-_{J^K}(x,V)$,
and that in the twisting construction,
$d^-_{J^K}(x^-,V)$ arcs of $H$ at $x$
are not counted by $d^-_J(x,V)$,
whereas $d^-_{J^K}(x,V)$ arcs of $H$
not at $x$ are counted by $d^-_J(x,V)$.
Writing $\Delta(x) = d^-_{J^K}(x^-,V) - d^-_{J^K}(x,V)
= d^+_{J^K}(x^-,W) - d^+_{J^K}(x,W)$,
we deduce $d_H^+(x)=d^+_J(x,V)$ and
$d_H^-(x) = d^-_J(x,V) + \Delta(x)$,
so we need $\Delta(x) = d'_1(x) - d_+(x)$
and $d_1'(x)=d_-(x)$.
So $\Delta(x)=d_-(x)-d_+(x)$.
We will ensure that both sides are always $0$
(taking $H$ equal to the digraph $G'_1$ in which we need
to solve the path factor problem), i.e.\
\begin{enumerate}
\item every vertex is used equally often
as a startpoint or as a successor of an interval, and
\item all vertices appear in some interval
for the same number of factors.
\end{enumerate}
To achieve this, we identify $V$ with $[n]$ under the
natural cyclic order, and select our intervals from
canonical sets $\mc{I}^i_j$, $i \in [2s+1]$, $j \in [d_i]$,
where each $\mc{I}^i_j$ is a partition of $[n]$
into $n/d_i \pm 1$ intervals of length at most $d_i$,
we have $\mc{I}^i_j \cap \mc{I}^i_{j'} = \emptyset$ for $j \ne j'$,
and for each $i$, every $v \in [n]$ occurs exactly once as
a startpoint of some interval in $\mc{I}^i = \cup_j \mc{I}^i_j$,
and also exactly once as a successor of some interval in $\mc{I}^i$.
The two conditions discussed in the previous paragraph will then
be satisfied if there are numbers $t_i$, $i \in [2s+1]$ such that
every interval in $\mc{I}^i$ is used by exactly $t_i$ factors.
Each $w$ will select intervals from some $\mc{I}^{i(w)}_{j(w)}$,
and these intervals must be non-consecutive,
so that the paths do not join up into longer paths.
This explains why we use several different interval sizes:
if we only used one size $d$ then a pair of vertices in $V$
at cyclic distance $d$ could never be both used for the same factor,
and so we would be unable to satisfy the conditions of the
wheel decomposition results in section \ref{sec:wheel}.
Now we describe how factors choose intervals.
For each $w \in W$, we start by independently choosing
$i=i(w) \in [2s+1]$ and $j=j(w) \in [d_i]$ uniformly at random.
Given $i$ and $j$, we activate each interval in $\mc{I}^i_j$
independently with probability $1/2$, and select any interval $I$
such that $I$ is activated, and its two neighbouring
intervals $I^\pm$ are not activated.
We thus obtain a random set of non-consecutive intervals where
each interval appears with probability $1/8$ (not independently).
We form random sets of intervals $\mc{X}^g_w$ where each
interval selected for $w$ is included in $\mc{X}^g_w$
independently with probability $8p^g_{w,K}$ (and is included
in at most one of $\mc{X}^1_w$ or $\mc{X}^2_w$).
Thus, given $w \in W_i := \{w': i(w')=i\}$, any interval
$I \in \mc{I}^i$ appears in $\mc{X}^g_w$ with probability
$p^g_{w,K}/d_i$. The events $\{I \in \mc{X}^g_w\}$ for $w \in W_i$
are independent, so whp about $\sum_{w \in W_i} p^g_{w,K}/d_i$
factors use $I$.
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To achieve this, we identify $V$ with $[n]$ under the
natural cyclic order, and select our intervals from
canonical sets $\mc{I}^i_j$, $i \in [2s+1]$, $j \in [d_i]$,
where each $\mc{I}^i_j$ is a partition of $[n]$
into $n/d_i \pm 1$ intervals of length at most $d_i$,
we have $\mc{I}^i_j \cap \mc{I}^i_{j'} = \emptyset$ for $j \ne j'$,
and for each $i$, every $v \in [n]$ occurs exactly once as
a startpoint of some interval in $\mc{I}^i = \cup_j \mc{I}^i_j$,
and also exactly once as a successor of some interval in $\mc{I}^i$.
The two conditions discussed in the previous paragraph will then
be satisfied if there are numbers $t_i$, $i \in [2s+1]$ such that
every interval in $\mc{I}^i$ is used by exactly $t_i$ factors.
Each $w$ will select intervals from some $\mc{I}^{i(w)}_{j(w)}$,
and these intervals must be non-consecutive,
so that the paths do not join up into longer paths.
This explains why we use several different interval sizes:
if we only used one size $d$ then a pair of vertices in $V$
at cyclic distance $d$ could never be both used for the same factor,
and so we would be unable to satisfy the conditions of the
wheel decomposition results in section \ref{sec:wheel}.
Now we describe how factors choose intervals.
For each $w \in W$, we start by independently choosing
$i=i(w) \in [2s+1]$ and $j=j(w) \in [d_i]$ uniformly at random.
Given $i$ and $j$, we activate each interval in $\mc{I}^i_j$
independently with probability $1/2$, and select any interval $I$
such that $I$ is activated, and its two neighbouring
intervals $I^\pm$ are not activated.
We thus obtain a random set of non-consecutive intervals where
each interval appears with probability $1/8$ (not independently).
We form random sets of intervals $\mc{X}^g_w$ where each
interval selected for $w$ is included in $\mc{X}^g_w$
independently with probability $8p^g_{w,K}$ (and is included
in at most one of $\mc{X}^1_w$ or $\mc{X}^2_w$).
Thus, given $w \in W_i := \{w': i(w')=i\}$, any interval
$I \in \mc{I}^i$ appears in $\mc{X}^g_w$ with probability
$p^g_{w,K}/d_i$. The events $\{I \in \mc{X}^g_w\}$ for $w \in W_i$
are independent, so whp about $\sum_{w \in W_i} p^g_{w,K}/d_i$
factors use $I$.
Our final sets of intervals $\mc{Y}^g_w$ are obtained from
$\mc{X}^g_w$ by removing a small number of intervals so that
every interval in $\mc{I}^i$ is used exactly $t^g_i$ times,
where $t^g_i$ is about $\sum_{w \in W_i} p^g_{w,K}/d_i$.
(We only need this property when $g=1$, but for uniformity
of the presentation we do the same thing for $g=2$.)
These intervals determine $J^K_g[V,W]$: we let
$N^-_{J^K_g}(w) = Y^g_w := \bigcup \mc{Y}^g_w$,
i.e.~the subset of $V$ which is
the union of the intervals in $\mc{Y}^g_w$.
As each $x$ is the startpoint of exactly one interval
in $\mc{I}^i$ it occurs as the startpoint of an interval
for exactly $t_g := \sum_i t^g_i$ factors; the same
statement holds for successors of intervals.
As each $x \in V$ appears in exactly one interval
in each $\mc{I}^i_j$ we deduce $d^+_{J^K_g}(x,W)
= \sum_{i=1}^{2s+1} \sum_{j=1}^{d_i} t^g_i
\approx \sum_{w \in W} p^g_{w,K} = |W| p^g_K$.
The other arcs of $J$ incident to $w$ will come from
$\overline{Y}_w := V \setminus \big( Y^1_w \cup Y^2_w
\cup (Y^1_w)^+ \cup (Y^2_w)^+ \big)$,
where $(Y^g_w)^+$ is the set of successors of intervals
in $\mc{Y}^g_w$ (these vertices are endpoints of paths so
should be avoided by the short cycles, and also by the $7/8$
of the paths not specified by the intervals).
We define $\overline{J}[V,W]$ by $N^-_{\overline{J}}(w)=\overline{Y}_w$.
For any $x \in V$ we will have
$\mb{P}(x \in Y_w^g) \approx
\mb{P}(x \in X_w^g) = p_{w,K}^g$
and $\mb{P}(x \in Y_w^g \mid w \in W_i) \approx
\mb{P}(x \in X_w^g \mid w \in W_i) = p_{w,K}^g/d_i$,
so $|\overline{Y}_w| \approx \overline{p}_w n$,
where $\overline{p}_w = 1 - \tfrac{d_i+1}{d_i} (p_{w,K}^1+p_{w,K}^2)$.
In $J^*_g = J_g \setminus J^K_g$ we require about $p^g_{w,*} n$
such arcs, where $p^g_{w,*} := p^g_w - p^g_{w,K}$,
and of these, for each cycle length $c<K$
we require about $p^g_{w,c} n$ arcs of colour $c$.
For each $x \in \overline{Y}_w$ independently we include
the arc $xw$ in at most one of the $J^*_g$
with probability $p^g_{w,*}/\overline{p}_w$,
which is a valid probability as
$p^1_{w,*} + p^2_{w,*}
= 1 - L^{-3}\eta - p_{w,K}^1 - p_{w,K}^2 < \overline{p}_w$.
Then we give each $xw \in J^*_g[V,W]$
colour $c$ with probability $p^g_{w,c}/p^g_{w,*}$.
In particular, $xw$ in $J^*_g$ is coloured $0$
with probability $p^g_{w,0}/p^g_{w,*}$, where
$p^g_{w,0} := p^g_{w,*} - \sum_{c=3}^{K-1} p^g_{w,c}$.
\subseteqsection{Formal statement of the algorithm}
The input to the algorithm consists of
an $\alpha n$-regular digraph $G$ on $V$,
a family $(F_w: w \in W)$ of $\alpha n$ oriented one-factors,
each partitioned as $F_w = F^1_w \cup F^2_w$,
and parameters satisfying
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1}
\ll d^{-1} \ll \eta \ll s^{-1} \ll L^{-1} \ll \alpha$.
We identify $V$ with $[n]$ according
to a uniformly random bijection and
adopt the natural cyclic order on $[n]$:
each $x \in [n]$ has successor $x^+=x+1$ (where $n+1$ means
$1$) and predecessor $x^-=x-1$ (where $0$ means $n$).
Let $d_i=d/(2s)^{i-1}$ for $i \in [2s+1]$.
We write $n=r_id_i+s_i$ with $r_i \in \mb{N}$
and $0 \le s_i < d_i$, and let
\[ P^i_j = \left\{ \begin{array}{ll}
\{ kd_i+j: 0 \le k \le r_i \}
& \text{if } j \in [s_i], \\
\{ kd_i+j: 0 \le k \le r_i-1 \}
& \text{if } j \in [d_i] \setminus [s_i].
\end{array} \right. \]
For each $i \in [s+1]$ and $j \in [d_i]$
we define a partition of $[n]$ into a family
of cyclic intervals $\mc{I}^i_j$ defined
as all $[a,b^-]$ where $a \in P^i_j$ and $b$ is
the next element of $P^i_j$ in the cyclic order.
(So $|\mc{I}^i_j|=n/d_i \pm 1$,
each $I \in \mc{I}^i_j$ has $|I| \le d_i$, and
$\mc{I}^i_j \cap \mc{I}^i_{j'} = \emptyset$ for $j \ne j'$.)
We let $\mc{I}^i = \cup_{j \in [d_i]} \mc{I}^i_j$.
(So for every $v \in [n]$,
exactly one $[a,b^-] \in \mc{I}^i$ has $a=v$,
and exactly one $[a,b^-] \in \mc{I}^i$ has $b=v$.)
Each $w \in W$ will be assigned $i(w) \in [2s+1]$.
For $c<K$ write $q^g_{w,c} n$ for the number
of cycles of length $c$ in $F^g_w$. Let
\begin{gather*}
p^g_w = (1-\eta) n^{-1}|F^g_w| + n^{-.2}, \qquad
p^g_{w,c} = (1-\eta) q^g_{w,c}
\ \text{ for } 3 \le c < K, \qquad
p^g_{w,K} = \tfrac{1}{8} \left( p^g_w
- \Sigma_{c=3}^{K-1} cp^g_{w,c} \right), \\
p^g_{w,*} = p^g_w - p^g_{w,K}, \qquad
p^g_{w,0} = p^g_{w,*} - \Sigma_{c=3}^{K-1} p^g_{w,c}, \qquad
p_{w,K} = p^1_{w,K} + p^2_{w,K}, \\
\overline{p}_w = 1 - \tfrac{d_{i(w)}+1}{d_{i(w)}} p_{w,K}, \qquad
p_g = |W|^{-1} \Sigma_{w \in W} p^g_w, \qquad
p^g_c = |W|^{-1} \Sigma_{w \in W} p^g_{w,c}
\ \text{ for } c \in [0,K] \cup \{*\}.
\end{gather*}
We complete the algorithm by applying the
following subroutines INTERVALS and DIGRAPH.
\begin{center}
INTERVALS \end{center}
\begin{enumerate}
\item For each $w \in W$ independently choose
$i(w) \in [2s+1]$ and $j(w) \in [d_{i(w)}]$
uniformly at random. Let $W_i = \{ w: i(w)=i \}$.
\item For each $w \in W$, let $\mc{A}_w$ include each interval
of $\mc{I}^{i(w)}_{j(w)}$ independently with probability $1/2$. \\
Let $\mc{S}_w$ consist of all $I \in \mc{A}_w$ such that
both neighbouring intervals $I^\pm$ of $I$ are not in $\mc{A}_w$.
\item Let $\mc{X}^g_w$, $g \in [2]$ be disjoint
with $\mb{P}(I \in \mc{X}^g_w)=8p^g_{w,K}$
independently for each $I \in \mc{S}_w$.
\item Let $t^g_i = \min \{ |\mc{X}^g(I)|: I \in \mc{I}^i \}$,
where $\mc{X}^g(I) := \{w \in W_i: I \in \mc{X}^g_w\}$,
and obtain $\mc{Y}^g_w \subseteq \mc{X}^g_w$ by deleting
each $I \in \mc{I}^i$, $i \in [2s+1]$ from
$|\mc{X}^g(I)|-t^g_i$ sets $\mc{X}^g_w$ with $w \in \mc{X}^g(I)$,
independently uniformly at random.
Write $\mc{Y}^g(I) := \{w \in W_i: I \in \mc{Y}^g_w\}$
(so $|\mc{Y}^g(I)|=t^g_i$ for $I \in \mc{I}^i$).
\end{enumerate}
\begin{center}
DIGRAPH \end{center}
\begin{enumerate}
\item Let $G_1$ and $G_2$ be arc-disjoint with
$\mb{P}(\overlinea{e} \in G_g) = p_g$ independently for each arc $\overlinea{e}$ of $G$.
\item For each $g \in [2]$ and $\overlinea{xy} \in G_g$ independently,
if $\overlinea{xy}$ is $\overlinea{zz}^-$ or $\overlinea{zz}^+$ for some $z$
add $\overlinea{xy}$ to $J^0_g$, otherwise choose exactly one of
$\mb{P}(\overlinea{xy} \in J^0_g) = p^g_*/p_g$ or
$\mb{P}(\overlinea{xy}^- \in J^K_g) = p^g_K/p_g$.
\item For each $w \in W$,
add $\overlinea{xw}$ to $J^K_g$
for each $x \in Y^g_w := \bigcup \mc{Y}^g_w$,
and add $\overlinea{xw}$ to $\overline{J}$
for each $x \in \overline{Y}_w := V \setminus ( Y^1_w \cup Y^2_w
\cup (Y^1_w)^+ \cup (Y^2_w)^+ )$.
\item For each arc $\overlinea{xw}$ of $\overline{J}[V,W]$ independently,
add $\overlinea{xw}$ to $J^*_g[V,W]$ with probability $p^g_{w,*}/\overline{p}_w$,
and give it exactly one colour $c \ne K$ (including $0$)
with probability $p^g_{w,c}/p^g_{w,*}$.
\end{enumerate}
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We complete the algorithm by applying the
following subroutines INTERVALS and DIGRAPH.
\begin{center}
INTERVALS \end{center}
\begin{enumerate}
\item For each $w \in W$ independently choose
$i(w) \in [2s+1]$ and $j(w) \in [d_{i(w)}]$
uniformly at random. Let $W_i = \{ w: i(w)=i \}$.
\item For each $w \in W$, let $\mc{A}_w$ include each interval
of $\mc{I}^{i(w)}_{j(w)}$ independently with probability $1/2$. \\
Let $\mc{S}_w$ consist of all $I \in \mc{A}_w$ such that
both neighbouring intervals $I^\pm$ of $I$ are not in $\mc{A}_w$.
\item Let $\mc{X}^g_w$, $g \in [2]$ be disjoint
with $\mb{P}(I \in \mc{X}^g_w)=8p^g_{w,K}$
independently for each $I \in \mc{S}_w$.
\item Let $t^g_i = \min \{ |\mc{X}^g(I)|: I \in \mc{I}^i \}$,
where $\mc{X}^g(I) := \{w \in W_i: I \in \mc{X}^g_w\}$,
and obtain $\mc{Y}^g_w \subseteq \mc{X}^g_w$ by deleting
each $I \in \mc{I}^i$, $i \in [2s+1]$ from
$|\mc{X}^g(I)|-t^g_i$ sets $\mc{X}^g_w$ with $w \in \mc{X}^g(I)$,
independently uniformly at random.
Write $\mc{Y}^g(I) := \{w \in W_i: I \in \mc{Y}^g_w\}$
(so $|\mc{Y}^g(I)|=t^g_i$ for $I \in \mc{I}^i$).
\end{enumerate}
\begin{center}
DIGRAPH \end{center}
\begin{enumerate}
\item Let $G_1$ and $G_2$ be arc-disjoint with
$\mb{P}(\overlinea{e} \in G_g) = p_g$ independently for each arc $\overlinea{e}$ of $G$.
\item For each $g \in [2]$ and $\overlinea{xy} \in G_g$ independently,
if $\overlinea{xy}$ is $\overlinea{zz}^-$ or $\overlinea{zz}^+$ for some $z$
add $\overlinea{xy}$ to $J^0_g$, otherwise choose exactly one of
$\mb{P}(\overlinea{xy} \in J^0_g) = p^g_*/p_g$ or
$\mb{P}(\overlinea{xy}^- \in J^K_g) = p^g_K/p_g$.
\item For each $w \in W$,
add $\overlinea{xw}$ to $J^K_g$
for each $x \in Y^g_w := \bigcup \mc{Y}^g_w$,
and add $\overlinea{xw}$ to $\overline{J}$
for each $x \in \overline{Y}_w := V \setminus ( Y^1_w \cup Y^2_w
\cup (Y^1_w)^+ \cup (Y^2_w)^+ )$.
\item For each arc $\overlinea{xw}$ of $\overline{J}[V,W]$ independently,
add $\overlinea{xw}$ to $J^*_g[V,W]$ with probability $p^g_{w,*}/\overline{p}_w$,
and give it exactly one colour $c \ne K$ (including $0$)
with probability $p^g_{w,c}/p^g_{w,*}$.
\end{enumerate}
We conclude this section by recording some estimates
on the algorithm parameters used throughout the paper.
\begin{gather*}
\text{In Case } K, \text{ all }
|F^g_w| = n/2 \pm 2K, \quad p^1_w, p^2_w > .49,
\quad p^1_{w,K} = p^1_w/8 > 1/17, \\
p^1_{w,c}=0 \text{ for } c \in [3,K-1],
\quad p^1_{w,*} = p^1_{w,0} = 7p^1_w/8 > 1/3
\quad \text{ and } \
p^2_{w,*} \ge p^2_{w,0} \ge 2p^2_w/3 > 1/4. \\
\text{In Case } \ell^*, \text{ all }
|F^1_w| = \ell^* L^{-3} n,
\quad |F^2_w| = n - \ell^* L^{-3} n,
\quad p^1_w > (1-\eta)\ell^*L^{-3} > 2L^{-3}, \\
p^2_w > 1 - 2L^{-2} > .9,
\quad p^1_{w,\ell^*} = p^1_w/\ell^* > .9L^{-3},
\quad p^1_{w,K}=n^{-.2}/8,
\quad p^1_{w,c}=0 \text{ for } c \in [3,K-1] \setminus \{\ell^*\}, \\
p^1_{w,*}>2L^{-3},
\quad p^1_{w,0} \ge 2p^1_{w,*}/3 > L^{-3}
\quad \text{ and } \
p^2_{w,*} \ge p^2_{w,0} \ge 2p^2_w/3 > .6.\\
\text{In both cases, } p^2_{w,K} \geq n^{-.2}/8.
\end{gather*}
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\section{Analysis I: intervals} \leftarrowbel{sec:int}
In this section we analyse the families of intervals chosen
by the INTERVALS subroutine in section \ref{sec:alg};
our goal is to establish various regularity and extendability
properties of $J^K_g[V,W]$ and $\overline{J}_g[V,W]$
(which are defined in step (iii) of DIGRAPH
but are completely determined by INTERVALS).
We also deduce some corresponding properties that follow
from these under the random choices in DIGRAPH.
Before starting the analysis, we state some standard results
on concentration of probability that will be used throughout
the remainder of the paper. We use the following classical
inequality of Bernstein (see e.g.\ \cite[(2.10)]{BLM})
on sums of bounded independent random variables.
(In the special case of a sum of independent indicator
variables we will simply refer to the `Chernoff bound'.)
\begin{lemma} \leftarrowbel{bernstein}
Let $X = \sum_{i=1}^n X_i$ be a sum of
independent random variables with each $|X_i|<b$.
Let $v = \sum_{i=1}^n \mb{E}(X_i^2)$.
Then $\mb{P}(|X-\mb{E}X|>t)
< 2e^{-t^2/2(v+bt/3)}$.
\end{lemma}
We also use McDiarmid's bounded differences inequality,
which follows from Azuma's martingale inequality
(see \cite[Theorem 6.2]{BLM}).
\begin{defn} \leftarrowbel{def:vary}
Suppose $f:S \to \mb{R}$ where $S = \prod_{i=1}^n S_i$
and $b = (b_1,\dots,b_n) \in \mb{R}^n$.
We say that $f$ is \emph{$b$-Lipschitz} if for any
$s,s' \in S$ that differ only in the $i$th coordinate
we have $|f(s)-f(s')| \le b_i$.
We also say that $f$ is \emph{$v$-varying}
where $v=\sum_{i=1}^n b_i^2/4$.
\end{defn}
\begin{lemma} \leftarrowbel{azuma}
Suppose $Z = (Z_1,\dots,Z_n)$ is a sequence
of independent random variables,
and $X=f(Z)$, where $f$ is $v$-varying.
Then $\mb{P}(|X-\mb{E}X|>t) \le 2e^{-t^2/2v}$.
\end{lemma}
The next lemma records various regularity and extendability
properties of $J^K_g[V,W]$ and $\overline{J}_g[V,W]$.
We recall that each $N^-_{J^K_g}(w) = Y^g_w$
and $N^-_{\overline{J}_g}(w) = \overline{Y}_w$, and also
our notation for common neighbourhoods,
e.g.\ $N^-_{J^K_g}(R) = \bigcap_{w \in R} N^-_{J^K_g}(w)$
in statement (iv). Statements (iv) and (v)
will be applied to $n^{O(1)}$ choices
of set $U$ or function $h$, so their conclusions apply
whp simultaneously to all these choices
(recalling our convention that `whp' refers to events
with exponentially small failure probability).
For $x \in V$ we write
$t^-_g(x)$ or $t^+_g(x)$ for the number of $w$
such that $x$ is the startpoint or successor
of an interval in $\mc{Y}^g_w$.
We also use the separation property
from Definition \ref{def:sep}.
\begin{lemma} \leftarrowbel{lem:int}
Let $g \in [2]$, $U \subseteq V$ and $h:W \to \mb{R}$
with each $|h(w)|<n^{.01}$. Then whp:
\begin{enumerate}
\item
$|\mc{Y}^g(I)| = t^g_i
= \tfrac{|W| p^g_K}{(2s+1)d_i} \pm n^{.51}$
for all $I \in \mc{I}^i$, $i \in [2s+1]$.
\item $d^+_{J^K_g}(x,W) = |W| p^g_K \pm n^{.52}$
and $t^\pm_g(x) = t_g := \sum_i t^g_i$
for each $x \in V$.
\item $d^-_{J^K_g}(w) = |Y^g_w| = p^g_{w,K} n \pm n^{3/4}$ and
$d^-_{\overline{J}}(w) = |\overline{Y}_w| = \overline{p}_w n \pm n^{3/4}$
for all $w \in W$.
\item For any disjoint $R,R' \subseteq W$ of sizes $\le s$ we have
\[ \bsize{U \cap N^-_{J^K_g}(R) \cap N^-_{\overline{J}}(R')}
= |U| \prod_{w \in R} p^g_{w,K}
\prod_{w \in R'} \overline{p}_w \pm 3sn^{3/4}. \]
\item Consider $H :=
\sum \bracc{ h(w): w \in N^+_{J^K_g}(S) \cap N^+_{\overline{J}}(S') }$
for disjoint $S,S' \subseteq V$ of sizes $\le s$.
\begin{align*}
& \text{If } S \cup S' \text{ is } 3d\text{-separated then }
H = \sum_{w \in W} (p^g_{w,K})^{|S|} \overline{p}_w^{|S'|} h(w)
\pm 5sn^{3/4}. \\
& \text{If } (S,S') \text{ is } 3d\text{-separated then }
H \ge 2^{-2s} \sum_{w \in W} (p^g_{w,K})^{|S|} h(w).
\end{align*}
\end{enumerate}
\end{lemma}
Write $X^g_w = \bigcup \mc{X}^g_w$
and $\overline{X}_w = V \setminus (X^1_w \cup X^2_w
\cup (X^1_w)^+ \cup (X^2_w)^+ )$.
In the proof we repeatedly use the observation that
if $S \cup S' \subseteq V$ is $3d$-separated and $w \in W$,
given $i(w)$ and $j(w)$,
the events $\{ \{x \in X^g_w\}: x \in S \}
\cup \{ \{x \in \overline{X}_w\}: x \in S' \}$ are independent,
as they are determined by disjoint sets
of random decisions in INTERVALS.
The weaker assumption that $(S,S')$ is $3d$-separated
only implies independence of $\{S \subseteq X^g_w\}$
and $\{S' \subseteq \overline{X}_w\}$. We also note that for any $S,S'$
the events $\{S \subseteq X^g_w \} \cap \{S' \subseteq \overline{X}_w\}$
are independent over $w \in W$.
\begin{proof}
For (i), consider any $I \in \mc{I}^i_j$
with $i \in [2s+1]$, $j \in [d_i]$.
For each $w \in W_i$ independently
we have $\mb{P}(j(w)=j)=1/d_i$,
$\mb{P}(I \in \mc{S}_w \mid j(w)=j)=1/8$,
$\mb{P}(I \in \mc{X}^g_w \mid I \in \mc{S}_w)=8p^g_{w,K}$,
so $\mb{P}(I \in \mc{X}^g_w) = p^g_{w,K}/d_i$.
As $\mb{P}(w \in W_i) = 1/(2s+1)$ for each $w \in W$
and $\sum_{w \in W} p^g_{w,K} = |W|p^g_K$,
by a Chernoff bound, whp
$|\mc{X}^g(I)| = \tfrac{|W| p^g_K}{(2s+1)d_i} \pm n^{.51}$.
This estimate holds for all such $I$, and so for
$t^g_i = \min \{ |\mc{X}^g(I)|: I \in \mc{I}^i \}$;
thus (i) holds.
For (ii), note that each $x \in V$
appears in exactly one interval in each $\mc{I}^i_j$,
so
$$
d^+_{J^K_g}(x,W) = \sum_{i=1}^{2s+1} \sum_{j=1}^{d_i}
\big( \tfrac{|W| p^g_K}{(2s+1)d_i} \pm n^{.51} \big)
= |W| p^g_K \pm n^{.52}.
$$
Next we recall that
INTERVALS chooses uniformly at random
$\mc{Y}^g(I) \subseteq \mc{X}^g(I)$ of size $t^g_i$.
The statements on $t^\pm_g(x)$ hold as
for each $i$ there is exactly one $[a,b] \in \mc{I}^i$ with $a=x$
and exactly one $[a,b] \in \mc{I}^i$ with $b^+=x$.
For future reference, we note that each
$|\mc{X}^g(I) \setminus \mc{Y}^g(I)| < 2n^{.51}$.
For (iii), consider any $w \in W$.
We start INTERVALS by choosing $i=i(w) \in [2s+1]$
and $j=j(w) \in [d_i]$ uniformly at random.
Given these choices, any $I \in \mc{I}^i_j$ appears in $\mc{S}_w$
if $I \in \mc{A}_w$ and $I^\pm \notin \mc{A}_w$;
this occurs with probability $1/8$, so
$\mb{E}|\mc{S}_w|=|\mc{I}^i_j|/8 = n/8d_i \pm 1$.
As $|\mc{S}_w|$ is a $3$-Lipschitz function of
the events $\{I \in \mc{A}_w\}$, $I \in \mc{I}^i_j$,
by Lemma \ref{azuma} whp $|\mc{S}_w| = n/8d_i \pm n^{.51}$.
Each $I \in \mc{S}_w$ is included in $\mc{X}^g_w$
independently with probability $8p^g_{w,K}$,
so by a Chernoff bound whp
$|\mc{X}^g_w| = p^g_{w,K}n/d_i \pm 2n^{.51}$.
For each $I \in \mc{X}^g_w$ independently we have
$I \in \mc{Y}^g_w$ with probability
$t^g_i / |\mc{X}^g(I)| = 1 \pm n^{-.27}$,
as $p^g_K \ge n^{-.2}$. Thus $d_i \mb{E}|\mc{Y}^g_w|
= p^g_{w,K}n \pm n^{.73}$, so by a Chernoff bound whp
$d^-_{J^K_g}(w) = |Y^g_w| = d_i|\mc{Y}^g_w| \pm d_i
= p^g_{w,K}n \pm 2n^{.73}$. We deduce
$d^-_{\overline{J}}(w) = n - \tfrac{d_i+1}{d_i} (|Y^1_w|+|Y^2_w|)
= \overline{p}_w n \pm n^{3/4}$, so (ii) holds.
We note that each $|Y^g_w| = |X^g_w| \pm n^{3/4}$
and $|\overline{Y}_w|=|\overline{X}_w| \pm n^{3/4}$.
For (iv), we first estimate the number $N$ of $u \in U$
such that $u \in X^g_w$ for all $w \in R$ and
$u \in \overline{X}_w$ for all $w \in R'$.
The actual quantity we need to estimate is obtained
by replacing `X' with `Y', and so differs in size
by at most $2sn^{3/4}$.
For each $u \in U$, we have independently
$\mb{P}(u \in X^g_w) = p^g_{w,K}$ for all $w \in R$
and $\mb{P}(u \in \overline{X}_w) = \overline{p}_w$ for all $w \in R'$,
so $\mb{E}N = |U| \prod_{w \in R} p^g_{w,K}
\prod_{w \in R'} \overline{p}_w$.
Indeed, given choices of $i=i(w)$ and $j=j(w)$,
letting $I$ be the unique interval in $\mc{I}_j^i$
whose successor is $u$, we have
$\mb{P}(u \in \overline{X}_w) = 1-
\sum_{g = 1}^{2}(\mb{P}(u \in X_w^g) + \mb{P}(I \in \mc{X}_w^g))
= \overline{p}_w$.
Now (iv) follows from Lemma \ref{azuma},
as $N$ is a $3d$-Lipschitz function of $\le 2n$
independent random decisions in INTERVALS.
For (v), we will estimate
$H' = \sum \{ h(w) : S \subseteq X^g_w, S' \subseteq \overline{X}_w \}$.
The actual quantity $H$ we need to estimate
is obtained from $H'$ by replacing `X' with `Y'.
We have $|H-H'| < 4sn^{3/4}$, as for each $i,j$ there are
$\le 2s$ intervals $I \in \mc{I}^i_j$
with $I \cap (S \cup S') \ne \emptyset$
each with $<2n^{.51}$ choices of
$w \in \mc{X}^g(I) \setminus \mc{Y}^g(I)$
each with $|h(w)| < n^{.01}$.
If $S \cup S'$ is $3d$-separated then
independently for all $w \in W$ we have
$\mb{P}(x \in X^g_w) = p^g_{w,K}$ for all $x \in S$
and $\mb{P}(x \in \overline{X}_w) = \overline{p}_w$ for all $x \in S'$;
the required estimates on $H'$ and so $H$
follow whp from Lemma \ref{bernstein}.
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For (iii), consider any $w \in W$.
We start INTERVALS by choosing $i=i(w) \in [2s+1]$
and $j=j(w) \in [d_i]$ uniformly at random.
Given these choices, any $I \in \mc{I}^i_j$ appears in $\mc{S}_w$
if $I \in \mc{A}_w$ and $I^\pm \notin \mc{A}_w$;
this occurs with probability $1/8$, so
$\mb{E}|\mc{S}_w|=|\mc{I}^i_j|/8 = n/8d_i \pm 1$.
As $|\mc{S}_w|$ is a $3$-Lipschitz function of
the events $\{I \in \mc{A}_w\}$, $I \in \mc{I}^i_j$,
by Lemma \ref{azuma} whp $|\mc{S}_w| = n/8d_i \pm n^{.51}$.
Each $I \in \mc{S}_w$ is included in $\mc{X}^g_w$
independently with probability $8p^g_{w,K}$,
so by a Chernoff bound whp
$|\mc{X}^g_w| = p^g_{w,K}n/d_i \pm 2n^{.51}$.
For each $I \in \mc{X}^g_w$ independently we have
$I \in \mc{Y}^g_w$ with probability
$t^g_i / |\mc{X}^g(I)| = 1 \pm n^{-.27}$,
as $p^g_K \ge n^{-.2}$. Thus $d_i \mb{E}|\mc{Y}^g_w|
= p^g_{w,K}n \pm n^{.73}$, so by a Chernoff bound whp
$d^-_{J^K_g}(w) = |Y^g_w| = d_i|\mc{Y}^g_w| \pm d_i
= p^g_{w,K}n \pm 2n^{.73}$. We deduce
$d^-_{\overline{J}}(w) = n - \tfrac{d_i+1}{d_i} (|Y^1_w|+|Y^2_w|)
= \overline{p}_w n \pm n^{3/4}$, so (ii) holds.
We note that each $|Y^g_w| = |X^g_w| \pm n^{3/4}$
and $|\overline{Y}_w|=|\overline{X}_w| \pm n^{3/4}$.
For (iv), we first estimate the number $N$ of $u \in U$
such that $u \in X^g_w$ for all $w \in R$ and
$u \in \overline{X}_w$ for all $w \in R'$.
The actual quantity we need to estimate is obtained
by replacing `X' with `Y', and so differs in size
by at most $2sn^{3/4}$.
For each $u \in U$, we have independently
$\mb{P}(u \in X^g_w) = p^g_{w,K}$ for all $w \in R$
and $\mb{P}(u \in \overline{X}_w) = \overline{p}_w$ for all $w \in R'$,
so $\mb{E}N = |U| \prod_{w \in R} p^g_{w,K}
\prod_{w \in R'} \overline{p}_w$.
Indeed, given choices of $i=i(w)$ and $j=j(w)$,
letting $I$ be the unique interval in $\mc{I}_j^i$
whose successor is $u$, we have
$\mb{P}(u \in \overline{X}_w) = 1-
\sum_{g = 1}^{2}(\mb{P}(u \in X_w^g) + \mb{P}(I \in \mc{X}_w^g))
= \overline{p}_w$.
Now (iv) follows from Lemma \ref{azuma},
as $N$ is a $3d$-Lipschitz function of $\le 2n$
independent random decisions in INTERVALS.
For (v), we will estimate
$H' = \sum \{ h(w) : S \subseteq X^g_w, S' \subseteq \overline{X}_w \}$.
The actual quantity $H$ we need to estimate
is obtained from $H'$ by replacing `X' with `Y'.
We have $|H-H'| < 4sn^{3/4}$, as for each $i,j$ there are
$\le 2s$ intervals $I \in \mc{I}^i_j$
with $I \cap (S \cup S') \ne \emptyset$
each with $<2n^{.51}$ choices of
$w \in \mc{X}^g(I) \setminus \mc{Y}^g(I)$
each with $|h(w)| < n^{.01}$.
If $S \cup S'$ is $3d$-separated then
independently for all $w \in W$ we have
$\mb{P}(x \in X^g_w) = p^g_{w,K}$ for all $x \in S$
and $\mb{P}(x \in \overline{X}_w) = \overline{p}_w$ for all $x \in S'$;
the required estimates on $H'$ and so $H$
follow whp from Lemma \ref{bernstein}.
Finally, we consider (v) when $(S,S')$ is $3d$-separated.
We fix $w \in W$, condition on $i(w)=i$ and $j(w)=j$,
and recall $\mb{P}(S \subseteq X^g_w, S' \subseteq \overline{X}_w)
= \mb{P}(S \subseteq X^g_w) \mb{P}(S' \subseteq \overline{X}_w)$.
We have the bound
$\mb{P}(S' \subseteq \overline{X}_w) \ge 2^{-s}$
from the event $I \notin \mc{A}_w$
for all $I \in \mc{I}^i_j$ with $I \cap S' \ne \emptyset$.
We claim that
$\mb{P}(S \subseteq X^g_w) > (5s)^{-1} (p^g_{w,K})^{|S|}$,
which by Lemma \ref{bernstein} suffices to complete the proof.
To prove the claim, we first note that if for some $\mc{I}^i_j$
no two vertices of $S$ lie in consecutive intervals then
$\mb{P}(S \subseteq X^g_w \mid i(w)=i, j(w)=j)
\ge (p^g_{w,K})^{|S|}$: indeed, the events
$\{I \in \mc{X}^g_w\}$ for $I \in \mc{I}^i_j$
with $I \cap S \ne \emptyset$ are positively correlated.
For $i \in [2s+1]$ let $J^i_s$ be the set of $j \in [d_i]$
for which some pair $x,x'$ of $S$ lie in consecutive intervals
of $\mc{I}^i_j$: we say $j$ is $i$-bad for $x,x'$.
We note that if $j$ is $i$-bad for some pair in $S$
then it is $i$-bad for some consecutive pair $x,x'$ in $S$
(i.e.\ $\{x,x'\} \cap S = \emptyset$).
It suffices to show that some $|J^i_s| < d_i/2$.
For this, we note that as $|S| \le s$ we can fix $i \in [2s+1]$
so that the cyclic distance between any pair of vertices
in $S$ is either $< d_{i+1}$ or $\ge d_{i-1}$.
There are no $i$-bad $j$ for any pair $x,x'$
with $d(x,x') \ge d_{i-1} = 2sd_i$.
Also, if $d(x,x')<d_{i+1}$ then $j$ is $i$-bad for $x,x'$
only if $\mc{I}^i_j$ contains an interval with an endpoint
in the cyclic interval $[x,x']$, so there are at most
$d_{i+1}$ such $j$. We deduce $|J^i_s| < sd_{i+1} = d_i/2$,
which completes the proof of the claim, and so of the lemma.
\end{proof}
| 2,141 | 61,149 |
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0.4961.10
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The next lemma contains similar statements to those in
the previous one concerning the colours and directions
introduced in DIGRAPH.
In (iii) we define $J^{K'}_g$ by $J^{K'}_g[V,W]=J^K_g[V,W]$ and
$\overlinea{uv} \in J^{K'}_g[V] \Leftrightarrow \overlinea{uv}^- \in J^K_g[V]$,
thus removing the twist: if for some arc $\overlinea{uv}$ of $G_g$
we add $\overlinea{uv}^-$ to $J^K_g$
then we add $\overlinea{uv}$ to $J^{K'}_g$.
\begin{lemma} \leftarrowbel{deg}
Let $g \in [2]$.
Write $q^g_0=p^g_*$, $q^g_{K'}=p^g_K$
and $q^g_c=0$ otherwise. Then whp:
\begin{enumerate}
\item For every $v \in V$
and $c \in [3,K] \cup \{0\}$ we have
$d^\pm_{J_g}(v,V) = p_g (1 \pm \varepsilon)\alpha n \pm n^{.6}$,
$d^\pm_{J^c_g}(v,V) = p^g_c (1 \pm \varepsilon) \alpha n \pm n^{.6}$,
$d^+_{J^c_g}(v,W) = p^g_c \alpha n \pm 2n^{3/4}$.
\item For every $w \in W$
and $c \in [3,K] \cup \{0\}$ we have
$d^-_{J^c_g}(w,V) = p^g_{w,c} n \pm 2n^{3/4}$.
\item For any mutually disjoint sets $R_c \subseteq W$
and $S^+_c, S^-_c \subseteq V$
for $c \in [3,K-1] \cup \{0,K'\}$
with $\sum_c |R_c| \le s$
and $\sum_c |S^\pm_c| \le s$ we have
\begin{align*} &\Big| \bigcap_c
\big( N^-_{J^c_g}(R_c) \cap
N^+_{J^c_g}(S^+_c) \cap
N^-_{J^c_g}(S^-_c) \big) \Big|\\
&= |N_G^+(\cup_c S_c^+) \cap N_G^-(\cup_c S_c^-)|
\prod_c \Big( (q^g_c)^{|S_c^+|+|S_c^-|}
\prod_{w \in R_c} p^g_{w,c} \Big)
\pm 4sn^{3/4}.
\end{align*}
\item Consider $H' :=
\big| W \cap N^+_{J^K_g}(S) \cap
\bigcap_c N^+_{J^c_g}(S_c) \big|$
for disjoint $S,S' \subseteq V$ of sizes $\le s$ with
$S'$ partitioned as $(S_c: c \in [3,K-1] \cup \{0\})$.
\begin{align*}
& \text{If } S \cup S' \text{ is } 3d\text{-separated then }
H' = \sum_{w \in W} (p^g_{w,K})^{|S|} \prod_c (p^g_{w,c})^{|S_c|}
\pm 6sn^{3/4}. \\
& \text{If } (S,S') \text{ is } 3d\text{-separated then }
H' + n^{.6} \ge 2^{-2s} \sum_{w \in W} (p^g_{w,K})^{|S|}
\prod_c (p^g_{w,c})^{|S_c|}.
\end{align*}
\end{enumerate}
\end{lemma}
\begin{proof}
All quantities considered are $1$-Lipschitz functions
of the random choices in DIGRAPH, so by Lemma \ref{azuma}
it suffices to estimate the expectations.
For (i), we recall that $G$ has vertex in- and outdegrees
$(1 \pm \varepsilon) \alpha n$, and for each $\overlinea{xy}$ in $G$
we have $\mb{P}(\overlinea{xy} \in J_g) = p_g$,
so $\mb{E}d^+_{J_g}(v,V) = p_g (1 \pm \varepsilon)\alpha n$.
The other expectations are similar, with slightly
modified calculations due to the twisting in colour $K$
and avoiding loops; for example,
$\mb{E}d^-_{J^K_g}(v,V) = p^g_K (d^-_G(v^+) \pm 1)
= p^g_K (1 \pm \varepsilon)\alpha n \pm 1$.
For (ii), we recall $d^-_{\overline{J}}(w) = \overline{p}_w n \pm n^{3/4}$
from Lemma \ref{lem:int}.iii, so for $c \ne K$ we have
$\mb{E}d^-_{J^K_c}(w) = p^g_{w,c}\overline{p}_w^{-1} d^-_{\overline{J}}(w)
= p^g_{w,c} n \pm n^{3/4}$. (The estimate for $c=K$
was already given in Lemma \ref{lem:int}.iii.)
For (iii), we first apply Lemma \ref{lem:int}.iv
with $U = N_G^+(\cup_c S_c^+) \cap N_G^-(\cup_c S_c^-)$,
$R = R_K$
and $R' = \cup_{c \ne K} R_c$ to obtain
\begin{align*} &\phantom{=}\bsize{N_G^+(\cup_c S_c^+) \cap N_G^-(\cup_c S_c^-) \cap N^-_{J^K_g}(R_K)
\cap N^-_{\overline{J}}(\cup_{c \ne K} R_c)}\\
&= |N_G^+(\cup_c S_c^+) \cap N_G^-(\cup_c S_c^-)| \prod_{w \in R_K} p^g_{w,K}
\prod_{w \in \cup_{c \ne K} R_c}
\overline{p}_w \pm 3sn^{3/4}. \end{align*}
For each vertex $v$ counted here independently we have
$\mb{P}(\overlinea{vw} \in J^c_g \mid \overlinea{vw} \in \overline{J})
= p^g_{w,c}/\overline{p}_w$ for all $w \in R_c$,
$\mb{P}(\overlinea{vx} \in J^c_g \mid \overlinea{vx} \in G)
= q^g_{c}$ for all $x \in S_c^-$ and
$\mb{P}(\overlinea{xv} \in J^c_g \mid \overlinea{xv} \in G)
= q^g_{c}$ for all $x \in S_c^+$,
so whp the stated bound for (iii) holds.
For (iv) we first consider
$H := |N^+_{J^K_g}(S) \cap N^+_{\overline{J}}(S')|$.
By Lemma \ref{lem:int}.v with $h(w)=1$,
if $S \cup S'$ is $3d$-separated then
$H = \sum_{w \in W} (p^g_{w,K})^{|S|} \overline{p}_w^{|S'|}
\pm 5sn^{3/4}$, and if $(S,S')$ is $3d$-separated then
$H \ge 2^{-2s} \sum_{w \in W} (p^g_{w,K})^{|S|}$.
For each vertex $w$ counted here independently we have
$\mb{P}(\overlinea{vw} \in J^c_g \mid \overlinea{vw} \in \overline{J})
= p^g_{w,c}/\overline{p}_w$ for all $v \in S_c$,
so whp the stated bound for (iv) holds.
\end{proof}
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0.4961.11
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\section{Analysis II: wheel regularity} \leftarrowbel{sec:reg}
In this section we show how to assign weights to wheels in
each $J_g$ so that for any arc $\overlinea{e}$ there is total weight
about $1$ on wheels containing $\overlinea{e}$, and furthermore
all weights on wheels with $c+1$ vertices are of order $n^{1-c}$.
This regularity property is an assumption in the wheel
decomposition results of section \ref{sec:wheel},
and is also sufficient in its own right for approximate
decompositions by a result of Kahn \cite{KaLP}.
The estimate for the total weight of wheels on an arc will hold
even if we add any new arc to $J_g$, which is useful
as we will need to consider small perturbations of $J_1$
due to arcs of $G$ not allocated to $G_1$ or $G_2$ or not
covered in the approximate decomposition of $G_2$.
We start by considering wheels $\ova{W}_{\! c}$ with $c<K$. Let
\[W^g_{w,c} = n^c p^g_{w,c} (p^g_{w,0})^{c-1} (\alpha p^g_*)^c.\]
The motivation for this formula is that it is about the
expected number of $\ova{W}_{\! c}$'s in $J_g$ using $w$.
For any arc $\overlinea{e}$ let $W^g_c(\overlinea{e})$
be the set of copies of $\ova{W}_{\! c}$ in $J_g$ with hub in $W$ using $\overlinea{e}$.
Let \[ \hat{W}^g_c(\overlinea{e}) = \sum \{
p^g_{w,c} n (W^g_{w,c})^{-1} :
\mc{W} \in W^g_c(\overlinea{e}), w \in V(\mc{W}) \}.\]
(If $p^g_{w,c}=0$ there are no such $\mc{W}$,
so $(W^g_{w,c})^{-1}$ is always defined when used.)
In the following lemma we calculate the total weights on arcs
due to copies of $\ova{W}_{\! c}$, although we note that we do not have
a good estimate for $\overlinea{xy} \in J^0_g[V]$ if $d(x,y)<3d$.
In $J_2$ we can ignore such arcs, as we only need an
approximate decomposition, whereas in $J_1$ we will cover
these by wheels greedily before finding the exact decomposition
-- this forms part of the perturbation referred to above.
\begin{lemma} \leftarrowbel{degWc}
Let $c' \in \{0,c\}$, $N_c=1$ and $N_0 = c-1$.
Then whp:
\begin{enumerate}
\item If $p_{w,c'}^g \neq 0$ and we add $\overlinea{xw}$ to $J^{c'}_g[V,W]$
then $\hat{W}^g_c(\overlinea{xw})
= (1 \pm 4\varepsilon) N_{c'} p^g_{w,c}/p^g_{w,c'} \pm n^{-.2}$.
\item If $d(x,y)\ge 3d$ and we add $\overlinea{xy}$ to $J^0_g[V]$
then $\hat{W}^g_c(\overlinea{xy})
= (1 \pm 4\varepsilon) cp^g_c/p^g_* \pm n^{-.2}$.
\end{enumerate}
\end{lemma}
\begin{proof}
As a preliminary step for counting copies of $\ova{W}_{\! c}$
we count $c$-prewheels, which we define to consist of a wheel
with oriented rim cycle in $G$ and all spokes in $\overline{J}$.
For any arc $\overlinea{e}$ we let $P_c(\overlinea{e})$ be the set of $c$-prewheels
using $\overlinea{e}$; we will estimate $|P_c(\overlinea{e})|$ using the analysis
of INTERVALS in Lemma \ref{lem:int}.
For (i), we estimate $|P_c(\overlinea{xw})|$ as follows.
We let $x=x_c$ and choose the other rim vertices
$x_1,\dots,x_{c-1}$ sequentially in cyclic order.
At $c-2$ steps we choose
$x_{i+1} \in N_G^+(x_i) \cap N^-_{\overline{J}}(w)$:
each has $\alpha n \overline{p}_w \pm 3sn^{3/4}$
options by Lemma \ref{lem:int}.iv
with $U=N_G^+(x_i)$, $R=\emptyset$, $R'=\{w\}$,
using $|N_G^+(x_i)|=\alpha n$ ($G$ is $\alpha n$-regular).
At the last step we choose
$x_{c-1} \in N_G^+(x_{c-2}) \cap N_G^-(x_c) \cap
N^-_{\overline{J}}(w)$, so similarly there are
$|N_G^+(x_{c-2}) \cap N_G^-(x_c)| \overline{p}_w \pm 3sn^{3/4}$
options, where $|N_G^+(x_{c-2}) \cap N_G^-(x_c)|
= ((1 \pm \varepsilon)\alpha)^2 n$ by typicality of $G$.
Thus $|P_c(\overlinea{xw})|
= (1 \pm 3\varepsilon) \alpha^c (\overline{p}_w n)^{c-1}$.
Now consider the case $c'=c$,
i.e.\ $\overlinea{xw}$ is added to $J^c[V,W]$.
For any $c$-prewheel containing $\overlinea{xw}$,
independently we include the cycle arcs in $J^0_g$
with probability $p^g_*$
and give each $\overlinea{x_i w}$ with $i \ne c$ colour $0$
with probability $p^g_{w,0}/\overline{p}_w$, so
$\mb{E}|W^g_c(\overlinea{xw})|
= (1 \pm 3\varepsilon) (\alpha p^g_*)^c (p^g_{w,0} n)^{c-1}
= (1 \pm 3\varepsilon) W^g_{w,c}/p^g_{w,c}n$.
Of these random decisions,
$\le 2n$ concern an arc containing one of $x,w$,
which affect $|W^g_c(\overlinea{xw})|$ by $O(n^{c-2})$,
and the others have effect $O(n^{c-3})$.
Thus $|W^g_c(\overlinea{xw})|$ is $O(n^{2c-3})$-varying,
so by Lemma \ref{azuma} whp $|W^g_c(\overlinea{xw})|
= (1 \pm 4\varepsilon) W^g_{w,c}/p^g_{w,c}n$,
i.e.\ $\hat{W}^g_c(\overlinea{xw}) = 1 \pm 4\varepsilon$.
When $c'=0$ we argue similarly.
Now $x$ can be any $x_i$ with $i \ne c$,
for which we have $c-1$ choices.
The probability factors are the same as in the previous
calculation, except that for $\overlinea{x_c w}$ we replace
$p^g_{w,0}/\overline{p}_w$ by $p^g_{w,c}/\overline{p}_w$.
Again, the stated estimate holds whp
by Lemma \ref{azuma}, so (i) holds.
For (ii), we write $\hat{W}^g_c(\overlinea{xy})
= \sum_{w \in W} \hat{W}^g_c(xyw)$,
where $\hat{W}^g_c(xyw)$ is the sum of $(W^g_{w,c})^{-1}$
over the set $W^g_c(xyw)$ of copies of $\ova{W}_{\! c}$ in $J_g$
using $\overlinea{xy}$, $\overlinea{xw}$ and $\overlinea{yw}$.
Fix $w \in N^+_{\overline{J}}(x) \cap N^+_{\overline{J}}(y)$ and
consider the number $|P_c(xyw)|$ of $c$-prewheels
using $\{\overlinea{xy},\overlinea{xw},\overlinea{yw}\}$. Choosing rim
vertices sequentially as in (i), now there are
$c-3$ steps with $\alpha n \overline{p}_w \pm 3sn^{3/4}$ options
and again $((1 \pm \varepsilon)\alpha)^2 \overline{p}_w n \pm 3sn^{3/4}$
options at the last step, so $|P_c(xyw)|
= (1 \pm 3\varepsilon) \alpha^{c-1} (\overline{p}_w n)^{c-2}$.
Now we consider which of these $c$-prewheels
extend to wheels in $W^g_c(xyw)$:
there are $c$ choices for the position of $\overlinea{xy}$ on the rim,
then some probabilities determined by independent random decisions:
the $c-1$ rim edges are each correct with probability $p^g_*$,
the spoke of colour $c$ with probability $p^g_{w,c}/\overline{p}_w$,
and the other $c-1$ spokes each with probability $p^g_{w,0}/\overline{p}_w$.
Therefore \[ \mb{E} \hat{W}^g_c(xyw)
= (1 \pm 3\varepsilon) c (\alpha p^g_*)^{c-1} p^g_{w,c} (p^g_{w,0})^{c-1}
\overline{p}_w^{-2} n^{c-2} p^g_{w,c} n (W^g_{w,c})^{-1}
= (1 \pm 3\varepsilon) c (\alpha p^g_*)^{-1} p^g_{w,c} n (\overline{p}_w n)^{-2}. \]
By Lemma \ref{azuma} whp $\hat{W}^g_c(\overlinea{xy})
= (1 \pm 3.1\varepsilon) c (\alpha p^g_* n)^{-1} H$,
with $H = \sum \{ p^g_{w,c} \overline{p}_w^{-2} :
w \in N^+_{\overline{J}}(x) \cap N^+_{\overline{J}}(y) \}$.
We estimate $H$ by Lemma \ref{lem:int}.v with $S=\emptyset$, $S'=\{x,y\}$
and $h(w) = p^g_{w,c}\overline{p}_w^{-2}$ (each $7/8 \le \overline{p}_w \le 1$).
As $S \cup S'$ is $3d$-separated,
whp $H = |W|p^g_c \pm 5sn^{3/4}$, giving
$\hat{W}^g_c(\overlinea{xy}) = (1 \pm 4\varepsilon) cp^g_c/p^g_* \pm n^{-.2}$.
\end{proof}
| 2,833 | 61,149 |
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0.4961.12
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Now we apply a similar analysis for $\ova{W}^K_{\! 8}$. Let
\[ W^g_{w,K} = n^8 \alpha p^g_K p^g_{w,K}
(\alpha p^g_* p^g_{w,0})^7 .\]
For any arc $\overlinea{e}$ let $W^g_K(\overlinea{e})$
be the set of copies of $\ova{W}^K_{\! 8}$ in $J_g$ using $\overlinea{e}$.
We define $\hat{W}^g_K(\overlinea{e})$ by
setting $c=K$ in $\hat{W}^g_c(\overlinea{e})$.
Now we calculate the total weights on arcs
due to copies of $\ova{W}^K_{\! 8}$. Note that we cannot give
a good estimate for $\overlinea{xy} \in J^K_g[V]$ if $d(x,y)<3d$.
We can ignore such arcs in $J_2$ (as mentioned above),
but in $J_1$ we will replace such arcs
by arcs of colour $0$ (modified by twisting)
-- this also forms part of the perturbation.
\begin{lemma} \leftarrowbel{degWK}
Let $c' \in \{0,K\}$, $N_K=1$,
$N_0 = 7$, $q^g_K = p^g_K$, $q^g_0=p^g_*$. Then whp:
\begin{enumerate}
\item If we add $\overlinea{xw}$ to
$J^{c'}_g[V,W]$ then $\hat{W}^g_K(\overlinea{xw})
= (1 \pm 4\varepsilon) N_{c'} p^g_{w,K}/p^g_{w,c'}$.
\item Suppose we add $\overlinea{xy}$ to $J^{c'}_g[V]$.
If $d(x,y)\ge 3d$ then $\hat{W}^g_c(\overlinea{xy})
= (1 \pm 4\varepsilon) N_{c'} q^g_K/q^g_{c'}$.\\
If $c'=0$ then
$\hat{W}^g_K(\overlinea{xy}) > 2^{-2s-1} p^g_K/p^g_*$.
\end{enumerate}
\end{lemma}
\begin{proof}
For (i), we start by counting $(K,g)$-prewheels, which
we define to consist of a hub $w \in W$ and an oriented $8$-path
in $G$ between $z$ and $z^+$ for some $z$ such that
$\overlinea{zw} \in J^K_g$ and $\overlinea{z'w} \in \overline{J}$
for all internal vertices $z'$ of the path.
For any arc $\overlinea{e}$ we let $P^g_K(\overlinea{e})$ be the set of
$(K,g)$-prewheels using $\overlinea{e}$.
To estimate $|P^g_K(\overlinea{xw})|$, suppose first that $c'=K$.
We require $z=x$. We choose the vertices of the path one by one.
At $6$ steps there are $\alpha n \overline{p}_w \pm 3sn^{3/4}$ options,
and at the last step
$((1 \pm \varepsilon)\alpha)^2 \overline{p}_w n \pm 3sn^{3/4}$
options of a common outneighbour of some vertex and $z^+$, so
$|P^g_K(\overlinea{xw})| = (1 \pm 3\varepsilon) \alpha^8 (\overline{p}_w n)^7$.
On the other hand, if $c'=0$ then there are $7$ choices
for the position of $x$ as an internal vertex,
dividing the path into two segments.
We construct one segment by choosing its vertices one by one,
and then do the same for the other segment, starting with one of
length $\le 4$ so that $\{z,z^+\}$ is not the last choice.
At the step where we choose $\{z,z^+\}$, there is some vertex $v$
on the path for which we need the arc $\overlinea{vz}$ or $\overlinea{vz}^+$.
We also require $z \in N^-_{J^K_g}(w)$.
The number of options is $\alpha n p^g_{w,K} \pm 3sn^{3/4}$
by Lemma \ref{lem:int}.iv, with
$R=\{w\}$, $R=\emptyset$ and $U=N_G^+(v)$
or $U=N_G^+(v)^- = \{z: \overlinea{vz}^+ \in G\}$.
There are also $5$ steps with
$\alpha n \overline{p}_w \pm 3sn^{3/4}$ options,
and at the last step
$((1 \pm \varepsilon)\alpha)^2 \overline{p}_w n \pm 3sn^{3/4}$ options,
so $|P^g_K(\overlinea{xw})| = (1 \pm 3\varepsilon)
7 \alpha^8 p^g_{w,K} (\overline{p}_w)^6 n^7$
(as $p^g_{w,K} \ge n^{-.2}/8$).
To estimate $|W^g_K(\overlinea{xw})|$, we first consider $c'=K$.
For any $(K,g)$-prewheel containing $\overlinea{xw}$,
independently we include the last path arc (to $z^+$)
in $J^K_g$ with probability $p^g_K$,
the other $7$ path arcs in $J^0_g$
with probability $p^g_*$,
and give $\overleftarrow{wz}'$ for each internal vertex $z'$ colour $0$
with probability $p^g_{w,0}/\overline{p}_w$, so
$$
\mb{E}|W^g_K(\overlinea{xw})|
= (1 \pm 3\varepsilon) \alpha p^g_K (\alpha p^g_* p^g_{w,0} n)^7
= (1 \pm 3\varepsilon) W^g_{w,K}/p^g_{w,K}n.
$$
As $|W^g_K(\overlinea{xw})|$ is $O(n^{13})$-varying,
by Lemma \ref{azuma} whp $|W^g_K(\overlinea{xw})|
= (1 \pm 3.1\varepsilon) W^g_{w,K}/p^g_{w,K}n \pm n^{6.51}$,
so $\hat{W}^g_K(\overlinea{xw}) = 1 \pm 4\varepsilon$
(using $p^g_K>n^{-.2}$).
For $c'=0$ we have a similar calculation.
Indeed, the path arcs are again correct with
probability $(p^g_*)^7 p^g_K$,
and the arcs $\overleftarrow{wz}'$ (now excluding $z'=x$)
are correct with probability $(p^g_{w,0}/\overline{p}_w)^6$,
so
$$
\mb{E}|W^g_K(\overlinea{xw})|
= (1 \pm 3\varepsilon) 7 \alpha p^g_K p^g_{w,K}
(p^g_{w,0})^6 (\alpha p^g_* n)^7
= (1 \pm 3\varepsilon) 7 W^g_{w,K}/p^g_{w,0}n.
$$
By Lemma \ref{azuma} whp
$|W^g_K(\overlinea{xw})| = (1 \pm 4\varepsilon) 7W^g_{w,K}/p^g_{w,0}n
\pm n^{6.51}$, so $\hat{W}^g_K(\overlinea{xw})
= (1 \pm 4\varepsilon) 7p^g_{w,K}/p^g_{w,0}$.
For (ii), we write $\hat{W}^g_K(\overlinea{xy})
= \sum_{w \in W} |\hat{W}^g_K(xyw)|$,
where $\hat{W}^g_K(xyw)$ is the sum of $(W^g_{w,K})^{-1}$
over the set $W^g_K(xyw)$ of copies of $\ova{W}^K_{\! 8}$ in $J_g$
using $\overlinea{xy}$, $\overlinea{xw}$ and $\overlinea{yw}$. For each $w$
we consider the set $P^g_K(xyw)$ of $(K,g)$-prewheels
using $\{\overlinea{xy},\overlinea{xw},\overlinea{yw}\}$
Suppose first that $\overlinea{xy}$ has colour $c'=K$.
We assume $d(x,y) \ge 3d$ (or there is nothing to prove).
We must have $y=z$ and in our prewheels the oriented $8$-paths
from $z$ to $z^+$ must end with the arc $\overlinea{xz}^+$,
corresponding to $\overlinea{xy} \in J^K$ under twisting.
We need $w \in N^+_{\overline{J}}(x) \cap N^+_{J^K_g}(y)$
so that $\overlinea{yw}$ has colour $K$ and $\overlinea{xw}$ can
receive colour $0$. Choosing rim vertices sequentially,
now $\{z,z'\}$ is already fixed, there are
$5$ steps with $\alpha n \overline{p}_w \pm 3sn^{3/4}$ options,
and at the last step
$((1 \pm \varepsilon)\alpha)^2 \overline{p}_w n \pm 3sn^{3/4}$ options,
so $|P^g_K(\overlinea{xw})| = (1 \pm 3\varepsilon)
\alpha^7 (\overline{p}_w)^6 n^6$.
Now consider which of these prewheels extend to wheels in $W^g_K(xyw)$,
according to the following independent random decisions:
the other $7$ arcs of the oriented $8$-path excluding $\overlinea{xy}$
are each correct with probability $p^g_*$,
we already have $\overlinea{yw} \in J^K_g$,
and for each of the $7$ internal vertices $z'$ we have
$\overlinea{z'w}$ correct with probability $p^g_{w,0}/\overline{p}_w$.
Therefore \[ \mb{E} \hat{W}^g_K(xyw)
= (1 \pm 3\varepsilon) (\alpha p^g_*)^7 (p^g_{w,0})^7
\overline{p}_w^{-1} n^6 p^g_{w,K} n (W^g_{w,K})^{-1}
= (1 \pm 3\varepsilon) (\alpha p^g_K \overline{p}_w n )^{-1}. \]
By Lemma \ref{azuma} whp $\hat{W}^g_K(\overlinea{xy})
= (1 \pm 3.1\varepsilon) (\alpha p^g_K n)^{-1} H \pm n^{-.2}$,
with $H = \sum \{ \overline{p}_w^{-1} :
w \in N^+_{\overline{J}}(x) \cap N^+_{J^K_g}(y) \}$.
We estimate $H$ by Lemma \ref{lem:int}.v
with $S=\{y\}$ and $S'=\{x\}$. As $d(x,y) \ge 3d$,
whp $H = |W| p^g_K \pm 5sn^{3/4}$, giving
$\hat{W}^g_K(\overlinea{xy}) = 1 \pm 4\varepsilon$.
Now suppose that $\overlinea{xy}$ has colour $c'=0$.
For the hub $w$ we require $\overlinea{yw} \in J^0$
and $\overlinea{xw}$ in $J^K$ or $J^0$.
We first consider the contribution
from $\overlinea{xw} \in J^K$, when the
first vertex of the oriented $8$-path must be $z=x$.
The estimate of $|P^g_K(\overlinea{xw})|$
is the same as when $c'=K$, and the probability factors
are the same except that the factor for the last path
edge (to $z^+$) is now $p^g_K$ instead of $p^g_*$.
If $d(x,y) \ge 3d$ then the same calculation with
Lemma \ref{azuma} and Lemma \ref{lem:int}.v
shows that the contribution to $\hat{W}^g_K(\overlinea{xy})$
from $w \in N^+_{\overline{J}}(x) \cap N^+_{J^K_g}(y) \}$
is $(1 \pm 4\varepsilon) (p^g_* n)^{-1}$.
Now we consider the contribution from $\overlinea{xw} \in J^0$.
There are $6$ positions for $\overlinea{xy}$ on the path
avoiding $\{z,z'\}$. The estimate of $|P^g_K(\overlinea{xw})|$
is the same as before except that one factor of $\overline{p}_w$
is replaced by $p^g_{w,K}$ (at the choice of $\{z,z'\}$).
The probability factors are the same as in the previous
calculation for $\overlinea{xw} \in J^K$, so
$\mb{E} \hat{W}^g_K(xyw) = (1 \pm 3\varepsilon) p^g_{w,K}
(\alpha p^g_* \overline{p}_w^2 n )^{-1}$.
By Lemma \ref{azuma} whp the contribution
to $\hat{W}^g_K(\overlinea{xy})$ from such $w$ is
$(1 \pm 3.1\varepsilon) 6 (\alpha p^g_* n)^{-1} H$,
with $H = \sum \{ h(w) :
w \in N^+_{\overline{J}}(x) \cap N^+_{\overline{J}}(y) \}$,
$h(w) = p^g_{w,K} (\overline{p}_w)^{-2}$.
We estimate $H$ by Lemma \ref{lem:int}.v with $S=\emptyset$, $S'=\{x,y\}$.
As $(S,S')$ is $3d$-separated (vacuously) whp
$H \ge 2^{-2s} \sum_{w \in W} h(w) = 2^{-2s} |W|p^g_K$,
so $\hat{W}^g_K(\overlinea{xy}) > 2^{-2s-1} p^g_K/p^g_*$.
Now suppose $d(x,y) \ge 3d$. Then $S \cup S'$
is $3d$-separated, so whp $H = |W|p^g_K \pm 5sn^{3/4}$.
The contribution here to $\hat{W}^g_K(\overlinea{xy})$ is
$(1 \pm 4\varepsilon) 6 p^g_K/p^g_* $, so altogether
$\hat{W}^g_K(\overlinea{xy}) = (1 \pm 4\varepsilon) 7 p^g_K/p^g_*$.
\end{proof}
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0.4961.13
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We combine the above estimates to deduce the main lemma
of this section, establishing wheel regularity.
Let \[ \hat{W}^g(\overlinea{e}) = \sum \{
\hat{W}^g_c(\overlinea{e}): c \in [3,K] \}.\]
\begin{lemma} \leftarrowbel{reg}
Suppose we add $\overlinea{e}$ to $J$ in any colour,
such that if $\overlinea{e} \in J[V]$
then $\overlinea{e}=\overlinea{xy}$ with $d(x,y) \ge 3d$,
and if $\overlinea{e}$ has a vertex in $W$ then it is an endvertex.
Then $\hat{W}^g(\overlinea{e}) = 1 \pm 5\varepsilon$.
\end{lemma}
\begin{proof}
By Lemmas \ref{degWc} and \ref{degWK} we can
analyse the various cases as follows.
\begin{itemize}
\item
If $\overlinea{e} \in J^c_g[V,W]$ with $c \ne 0$
then $\hat{W}^g(\overlinea{e}) = \hat{W}^g_c(\overlinea{e}) = 1 \pm 5\varepsilon$.
\item
If $\overlinea{xy} \in J^K_g[V]$ with $d(x,y) \ge 3d$ then
$\hat{W}^g(\overlinea{e}) = \hat{W}^g_K(\overlinea{e}) = 1 \pm 5\varepsilon$.
\item
If $\overlinea{e} \in J^0_g[V,W]$ then
$$
\hat{W}^g(\overlinea{e}) =
(1 \pm 4\varepsilon) 7 p^g_{w,K}/p^g_{w,0} + \textstyle\sum_{c=3}^{K-1}
\big( (1 \pm 4\varepsilon) (c-1) p^g_{w,c}/p^g_{w,0} \pm n^{-.2} \big)
= 1 \pm 5\varepsilon,
$$
as
$p^g_{w,0} = 7 p^g_{w,K} + \sum_{c=3}^{K-1} (c-1) p^g_{w,c}$.
\item
If $\overlinea{xy} \in J^0_g[V]$ with $d(x,y) \ge 3d$
then
$$
\hat{W}^g(\overlinea{e}) = (1 \pm 4\varepsilon) 7 p^g_K/p^g_*
+ \textstyle\sum_{c=3}^{K-1} \big(
(1 \pm 4\varepsilon) cp^g_c/p^g_* \pm n^{-.2} \big)
= 1 \pm 5\varepsilon,
$$
as $p^g_* = p_g - p^g_K
= 7p^g_K + \sum_{c=3}^{K-1} cp^g_c$.
\end{itemize}
\end{proof}
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0.4961.14
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\section{Approximate decomposition} \leftarrowbel{sec:approx}
Here we describe the approximate decomposition of $G_2$.
Recall that at the start of section \ref{sec:alg}
we partitioned each factor $F_w$ into subfactors
$F^1_w$ and $F^2_w$, that each $F^g_w$
has $q^g_{w,c} n$ cycles of length $c \in [3,K-1]$,
and $p^g_{w,c} = (1-\eta)q^g_{w,c}$.
We will embed almost all of each $F^2_w$ in $G_2$.
We say $F'_w \subseteq F^2_w$ is valid if it does not have
any independent arcs (i.e.\ arcs $\overlinea{xy}$ such that
both $x$ and $y$ have total degree $1$ in $F'_w$)
and if $F^2_w$ contains a path then $F'_w$
contains the arcs incident to each of its ends.
\begin{lemma} \leftarrowbel{lem:approx}
There are arc-disjoint digraphs $G^2_w \subseteq G_2$ for $w \in W$,
where each $G^2_w$ is a copy of some valid $F'_w \subseteq F^2_w$
with $V(G^2_w) \subseteq N^-_{J_2}(w)$, such that
\begin{enumerate}
\item $G_2^- = G_2 \setminus \bigcup_{w \in W} G^2_w$
has maximum degree at most $5d^{-1/3}n$,
\item the digraph $J_2^-$
obtained from $J_2[V,W]$ by deleting all $\overlinea{xw}$
with $x \in V(G^2_w)$ has maximum degree at most $5d^{-1/3}n$, and
\item any $x \in V$ has degree $1$ in $F'_w$
for at most $n/\sqrt{d}$ choices of $w$.
\end{enumerate}
\end{lemma}
\begin{proof}
Say that an arc $\overlinea{vw}$ with $v \in V$ and $w \in W$ is \emph{bad}
there is some $c \in [3,K-1]$ such that
$\overlinea{vw} \in J^c$ and $p^2_{w,c} < n^{-.1}$,
or $\overlinea{vw} \in J^K$ and $p^2_{w,K} < d^{-1/3}$.
The expected bad degree of $v \in V$
is at most $(Kn^{-.1}+d^{-1/3})n$
so by Chernoff bounds we can assume that
every $v \in V$ has bad degree at most $2d^{-1/3}n$.
Let $J'_2$ be obtained from $J_2$ by deleting
all bad arcs and
all $\overlinea{xy} \in J^K_2[V]$ with $d(x,y)<3d$.
We consider the auxiliary hypergraph $\mc{H}$
whose vertices are all arcs of $J'_2$ and
whose edges correspond to all copies of the
coloured wheels $\ova{W}^K_{\! 8}$ or $\ova{W}_{\! c}$ with $c \in [3,K-1]$.
We recall that
$W^g_{w,c} = n^c p^g_{w,c} (p^g_{w,0})^{c-1} (\alpha p^g_*)^c$
and $W^g_{w,K} = n^8 \alpha p^g_K p^g_{w,K}
(\alpha p^g_* p^g_{w,0})^7 $.
We assign weights
$(1-5\varepsilon)p^g_{w,c} n / (W^g_{w,c})^{-1}$
to each copy of any $\ova{W}_{\! c}$ (and to $\ova{W}^K_{\! 8}$ for $c=K$).
By Lemma \ref{reg}, the total weight
of wheels in $J_2$ on any arc $\overlinea{e}$ satisfies
$1-10\varepsilon < \hat{W}^g(\overlinea{e}) < 1$.
Thus the total weight of wheels in $J_2'$
on any arc $\overlinea{e}$ satisfies
$1-d^{-1/4} < \hat{W}^g(\overlinea{e}) < 1$,
as we deleted at most $2d^{-1/3}n^7$ (say) copies of $\ova{W}^K_{\! 8}$
on $\overlinea{e}$ using a deleted arc.
Note also that for any two arcs the total weight of wheels
containing both is at most $n^{-.7}$ (as $p^g_K \ge n^{-1/4}$).
Thus $\mc{H}$ satisfies the hypotheses of a result of Kahn \cite{KaLP}
on almost perfect matchings in weighted hypergraphs
that are approximately vertex regular and have small codegrees.
A special case of this result (slightly modified) implies that
for any collection $\mc{F}$ of at most $n^{100}$ (say) subsets
of $V(\mc{H})=J$ each of size at least $\sqrt{n}$ (say) we can find
a matching $M$ in $\mc{H}$ such that
$|F \setminus \bigcup M| < d^{-1/5} |F|$ for all $F \in \mc{F}$.
(This is immediate from \cite{KaLP} if $\mc{F}$ has constant size,
and a slight modification using better concentration inequalities
implies the stated version. Alternatively, one can reduce to the
problem to an unweighted version via a suitable random selection
of edges and then apply a result of Alon and Yuster \cite{AY}.)
This is also implied by a recent result of Ehard,
Glock and Joos~\cite{EGJ}.
We choose such a matching $M$ for the family $\mc{F}$
where for each $v \in V \cup W$ we include sets
$F_v = \{ \overlinea{e} \in J_2[V,W]: v \in \overlinea{e} \}$,
$F^K_v = \{ \overlinea{e} \in J^K_2[V,W]: v \in \overlinea{e} \}$,
and $F'_v = \{ \overlinea{e} \in J_2[V]: v \in \overlinea{e} \}$
(the last just for $v \in V$). This $\mc{F}$ is valid as
all $|F|>\sqrt{n}$ by Lemma \ref{deg}.
By construction
for all $c \in [3,K-1]$ every copy of $\ova{W}_{\! c}$ in $M$
with hub $w$ has $p^2_{w,c} \ge n^{-.1}$ and
every copy of $\ova{W}^K_{\! 8}$ in $M$
with hub $w$ has $p^2_{w,K} \ge nd^{-1/3}$.
For each $w$ we define $G^2_w$ to be the subgraph of
$G$ corresponding to the wheels in $M$ containing $w$,
where we take account of the twisting in colour $K$.
Thus $G^2_w$ contains the rim $c$-cycle
of any $c$-wheel in $M$ containing $w$,
and for any copy of $\ova{W}^K_{\! 8}$ in $M$ containing
$\overlinea{xw} \in J^K[V,W]$ we obtain
an oriented path of length $8$ from $x$ to $x^+$.
The maximum degree bounds in (i) and (ii) clearly hold.
Recalling that $N^-_{J_2}(w)$ is disjoint from
the set of interval successors $(Y^2_w)^+$,
we see that these cycles and paths are vertex-disjoint,
except that some paths may connect up to form longer paths,
which can be described as follows. Let $\mc{Y}'_w$ be
the set of maximal cyclic intervals $I$ such that
for every $x \in I$ there is a copy of $\ova{W}^K_{\! 8}$ in $M$
containing $\overlinea{xw} \in J^K[V,W]$. Then for each
$[a,b] \in \mc{Y}'_w$ we have a component of $G^2_w$
that is a path of length $8d(a,b)$ from $a$ to $b^+$.
All these paths have length at most $8d$, as each such $I$
is contained within an interval of $\mc{Y}^2_w$.
Furthermore, if $x \in V$ is an endpoint of some path
in $G^2_w$ then either $x$ is a startpoint or successor of
some interval in $\mc{Y}^2_w$, for which there are
at most $2t_2$ choices of $w$ by Lemma \ref{lem:int},
or $x^+ w \in F^K_{x^+} \setminus \bigcup M$,
or $x^- w \in F^K_{x^-} \setminus \bigcup M$,
giving at most $2n/K$ more choices of $w$,
for a total of at most $n/\sqrt{d}$ (say).
It remains to show that each $G^2_w$ is isomorphic
to some valid $F'_w \subseteq F_w$. First we show
for any $c \in [3,K-1]$ that whp each $G^2_w$
has at most $q^2_{w,c} n$ cycles of length $c$.
The number of $c$-cycles
is in $G^2_w$ is at most $|N^-_{J^c_2}(w)|$, which
by Chernoff bounds is whp $< p^2_{w,c} n + n^{.6}
= (1-\eta) q^2_{w,c} n + n^{.6} < q^2_{w,c} n$,
recalling that $p^2_{w,c} \ge n^{-.1}$.
Next we bound the total length $L_w$ of paths in $G^2_w$.
By Lemma \ref{lem:int} we have
$L_w \le 8|Y^2_w| < 8p^2_{w,K} n + 8n^{3/4}$.
Writing $L'_w$ for the total length of long
(length $\ge K$) cycles and paths in $F^2_w$,
we recall that $8p^2_{w,K} n
= p^2_w n - \sum_{c=3}^{K-1} cp^2_{w,c}n
= (1-\eta)(L'_w+n^{.8})$.
So since $p^2_{w,K} \ge d^{-1/3}n$, we have
$L'_w > 8d^{-1/3}n$ and $L_w < (1-\eta/2)L'_w$.
We embed the paths of $G^2_w$ into the long cycles and
paths in $F^2_w$ according to a greedy algorithm,
where in each step that we embed some path $P$ of $G^2_w$
we delete a path of length $|P|+4$ from $F^2_w$,
which we allocate to a copy of $P$ surrounded
on both sides by paths of length $2$ that we will not
include in $F'_2$ (so that $F'_2$ will be valid).
We choose such a path (if it exists) within a remaining
cycle or path of $G^2_w$, using an endpoint if it is
a path (so that we preserve the number of components).
Recalling that there are at most $n/\sqrt{d}$ endpoints
of paths in $G^2_w$, we thus allocate a total of at most
$2n/\sqrt{d}$ edges to the surrounding paths of length $2$.
Suppose for a contradiction that the algorithm gets stuck,
trying to embed some path $P$ in some remainder $R$.
Then all components of $R$ have size $\le |P|+5 \le 8d+5$.
All components of $G^2_w$ have size $\ge K$,
so $|R| \le (8d+5)|L'_w|/K$. However, we also have
$|R| \ge |L'_w|-|L_w|-2n/\sqrt{d}
\ge \eta |L'_w|/2 - 2n/\sqrt{d}$, which is a contradiction,
as $K^{-1} \ll d^{-1} \ll \eta$ and $L'_w > 8d^{-1/3}n$.
Thus the algorithm succeeds in constructing a valid
copy $F'_w$ of $G^2_w$ in $F^2_w$.
\end{proof}
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0.4961.15
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\section{Exact decomposition} \leftarrowbel{sec:exact}
This section contains the two exact decomposition results
that will conclude the proof in both Case $K$ and Case $\ell^*$.
We start by giving a common setting for both cases.
We say that $G'_1$ is a $\gamma$-perturbation of $G_1$ if
$|N_{G_1}^\pm(x) \bigtriangleup N_{G'_1}^\pm(x)| < \gamma n$ for any $x \in V$.
We say that $J'_1$ is a $\gamma$-perturbation of $J_1$
if $J'_1$ is obtained from $J_1$ by adding, deleting
or recolouring at most $\gamma n$ arcs at each vertex.
We will only consider perturbations which are
compatible in the sense that arcs added between $V$ and $W$
will point towards $W$,
and existing colours will be used.
\begin{set} \leftarrowbel{set}
Let $G'_1$ be an $\eta^{.9}$-perturbation of $G_1$.
Suppose for each $w \in W$ that $Z_w \subseteq V$
with $|Z_w \bigtriangleup (V \setminus N^-_{J^1}(w))| < 5\eta n$.
For $x \in V$ we write $Z(x)=\{w \in W: x \in Z_w\}$.
\end{set}
We start with the exact result for Case $\ell^*$, where we
recall that $F^1_w$ consists of exactly $L^{-3} n$ cycles of length
$\ell^*$, so $p^1_w = (1-\eta)\ell^* L^{-3} + n^{-.2}$,
$p^1_{w,\ell^*} = (1-\eta) L^{-3}$, $p^1_{w,K} = n^{-.2}/8$
and $p^1_{w,c}=0$ for $c \in [3,K-1]$.
\begin{lemma} \leftarrowbel{exactL}
Suppose in Setting \ref{set} and Case $\ell^*$ that
$d_{G'_1}^\pm(x)=|W|-|Z(x)|$ for all $x \in V$
and $\ell^*$ divides $n-|Z_w|$ for all $w \in W$.
Then $G'_1$ can be partitioned into graphs $(G^1_w: w \in W)$,
where each $G^1_w$ is an oriented $C_{\ell^*}$-factor
with $V(G^1_w) = V \setminus Z_w$.
\end{lemma}
\begin{proof}
We will show that there is a perturbation $J'_1$ of $J_1$
such that $J'_1[V] = G'_1$,
each $N^-_{J'_1}(w) = V \setminus Z_w$, and Theorem \ref{decompL}
applies to give a $\ova{W}_{\! \ell^*}$-decomposition of $J'_1$.
This will suffice, by taking each $G^1_w$ to consist of
the rim $\ell^*$-cycles of the copies of $\ova{W}_{\! \ell^*}$ containing $w$.
We construct $J'_1$ by starting with $J'_1=J_1$ and applying
a series of modifications as follows. First we delete all arcs
of $J'_1[V]$ corresponding to arcs of $G_1 \setminus G'_1$ and add
arcs of colour $0$
corresponding to arcs of $G'_1 \setminus G_1$.
Similarly, we delete all arcs $\overlinea{vw} \in J'_1[V,W]$
with $v \in N^-_{J_1}(w) \cap Z_w$ and add arcs $\overlinea{vw}$
of colour $0$ for each $v \in (V \setminus Z_w) \setminus N^-_{J_1}(w)$.
We also recolour any $\overlinea{vw} \in J'_1[V,W]$ of colour $K$
to have colour $0$ and replace any $\overlinea{xy}$ of colour $K$
in $J'_1[V]$ by $\overlinea{xy}^+$ of colour $0$.
As each $p^1_{w,K}=n^{-.2}/8$ in this case,
whp this affects at most $n^{.8}$ arcs at any vertex.
Now $J'_1[V]=G'_1$,
each $N^-_{J'_1}(w) = V \setminus Z_w$ and
$J'_1$ is a $\eta^{.8}$-perturbation of $J_1$.
We note for each $x \in V$ that $d_{J'_1}^\pm(x,V)
= d_{G'_1}^\pm(x) = |W|-|Z(x)| = d^+_{J'_1}(x,W)$,
so the divisibility conditions for $x \in V$ are satisfied.
Finally, to satisfy the divisibility conditions
for all $w \in W$ we recolour so that
$d^-_{(J'_1)^{\ell^*}}(w) = d^-_{J'_1}(w)/\ell^*$,
which is an integer, as $\ell^*$ divides
$d^-_{J'_1}(w) = n-|Z_w|$.
By Lemma \ref{deg} each $d^-_{J_1}(w) = p^1_w n \pm 2n^{3/4}$
and $d^-_{J_1^{\ell^*}}(w) = p^1_{w,\ell^*} n \pm 2n^{3/4}$,
where $p^1_w = \ell^* p^1_{w,\ell^*} + n^{-.2}$ in this case.
As $J'_1$ is an $\eta^{.8}$-perturbation of $J_1$,
we only need to recolour at most $2\eta^{.8} n$ arcs
at any vertex, so our final digraph $J'_1$
is a $3\eta^{.8}$-perturbation of $J_1$.
Next we consider the regularity condition of Theorem \ref{decompK}.
To each copy of $\ova{W}_{\! \ell^*}$ in $J'_1$ with hub $w$
we assign weight $p^1_{w,\ell^*} n/ W^g_{w,\ell^*} =
p^1_{w,0} n (\alpha p^1_{w,0} p^1_* n)^{-\ell^*}$,
which lies in $[n^{1-\ell^*}, L^L n^{1-\ell^*}]$.
We claim that for any arc $\overlinea{e}$ of $P'$ there is total weight
$1 \pm \eta^{.6}$ on wheels containing $\overlinea{e}$.
To see this, we compare the weight to $\hat{W}^1_{\ell^*}(\overlinea{e})$
as defined in section \ref{sec:reg},
which is $1 \pm 4\varepsilon$ by Lemma \ref{degWc}
(as $p^1_{w,0}=(\ell^*-1) p^1_{w,\ell^*}$
and $p^1_*=(\ell^*-1) p^1_{\ell^*}$).
The actual weight on $\overlinea{e}$ differs from this estimate only due
to wheels containing $\overlinea{e}$ that have another arc in $J'_1 \bigtriangleup J_1$.
There are at most $40\eta^{.7} n^{\ell^*-1}$ such wheels,
each affecting the weight by at most $L^L n^{\ell^*-1}$, so the claim holds.
Thus regularity holds with $\delta=\eta^{.6}$ and $\omega=L^{-L}$.
It remains to show that $J'_1$ satisfies the
extendability condition of Theorem \ref{decompL}.
Consider any disjoint $A,B \subseteq V$ and $C \subseteq W$
each of size $\le h$, where $h = 2^{50 (\ell^*)^3}$.
By Lemma \ref{deg}.iii,
for $c \in \{0,\ell^*\}$ we have
$$
|N^+_{J^0_1}(A) \cap N^-_{J^0_1}(B) \cap N^-_{J^c_1}(C)|
= |N_G^+(A) \cap N_G^-(B)| (p_*^1)^{|A|} (p_*^1)^{|B|}
\prod_{w \in C} p^1_{w,c} \pm 4sn^{3/4}
> (L^{-5} \alpha )^{2h} n,
$$
by typicality of $G$.
Also, by Lemma \ref{deg}.iv
(with $S=\emptyset$ and $S'=A \cup B$) we have
$|N^+_{J^0_1}(A) \cap N^+_{J^{\ell^*}_1}(B) \cap W|
\ge 2^{-2s} L^{-7h} |W|$, say.
The perturbation from $J_1$ to $J'_1$ affects these estimates
by at most $6h\eta^{.7}n < \eta^{.6}n$, so $J'_1$ satisfies
extendability with $\omega=L^{-L}$ as above.
Now Theorem \ref{decompL} applies to give
a $\ova{W}_{\! \ell^*}$-decomposition of $J'_1$,
which completes the proof.
\end{proof}
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Our second exact decomposition result concerns the path factors
with prescribed ends required for Case $K$.
We recall that each $F^1_w$ consists of cycles of length $\ge K$
and at most one path of of length $\ge K$
with $|F^1_w| - n/2 \in [0,2K]$,
and that $(Y^1_w)^-$ and $(Y^1_w)^+$ are the sets of
startpoints and successors of intervals in $\mc{Y}^1_w$.
We also recall from Lemma \ref{lem:int} that for each $x \in V$,
letting $t^\pm_1(x) = |\{w: x \in (Y^1_w)^\pm\}|$,
we have $t^+_1(x)=t^-_1(x)=t_1$.
After embedding $F^2_w$, and a greedy embedding connecting
the paths to $(Y^1_w)^-$ and $(Y^1_w)^+$, we will need
path factors $G^1_w$ as follows.
\begin{lemma} \leftarrowbel{exactK}
Suppose in Setting \ref{set} and Case $K$ that
$Z_w$ is disjoint from $Y^1_w \cup (Y^1_w)^+$
and $8|Y^1_w| = n-|Z_w|-|(Y^1_w)^+|$ for all $w \in W$,
and $d_{G'_1}^\pm(x)=|W|-t_1-|Z(x)|$ for all $x \in V$.
Then $G'_1$ can be partitioned into graphs $(G^1_w: w \in W)$,
such that each $G^1_w$ is a vertex-disjoint union of oriented paths
with $V(G^1_w) = V\setminus Z_w$,
where for each $[a,b] \in \mc{Y}^1_w$ there
is an $ab^+$-path of length $8d(a,b)$.
\end{lemma}
\begin{proof}
We will show that there is
a perturbation $P$ of $J_1$ such that
each $N^-_P(w) = V \setminus Z_w$ and $P[V]$
corresponds to $G'_1$ under twisting, and
a set $E$ of arc-disjoint copies of $\ova{W}^K_{\! 8}$ in $P$,
such that Theorem \ref{decompK} applies to give
a $\ova{W}^K_{\! 8}$-decomposition of $P' := P \setminus \bigcup E$.
This will suffice, by taking each $G^1_w$ to consist of the
union of the oriented $8$-paths that correspond under twisting to
the rim $8$-cycles of the copies of $\ova{W}^K_{\! 8}$ containing $w$.
We construct $P$ by starting with $P=J_1$ and applying
a series of modifications as follows. First we delete all arcs
of $P[V]$ corresponding to arcs of $G_1 \setminus G'_1$ and add
arcs of colour $0$
corresponding to arcs of $G'_1 \setminus G_1$.
Similarly, we delete all arcs $\overlinea{vw} \in P[V,W]$
with $v \in N^-_{J_1}(w) \cap Z_w$ and add arcs $\overlinea{vw}$
of colour $0$ for each
$v \in V \setminus (Z_w \cup (Y^1_w)^+ \cup N^-_{J_1}(w))$.
We also replace any $\overlinea{xy}$ of colour $K$ with $d(x,y)<3d$
by an arc $\overlinea{xy}^+$ of colour $0$;
this affects at most $6d$ arcs at each vertex.
Now $P[V]$ corresponds to $G'_1$ under twisting,
each $N^-_P(w) = V \setminus (Z_w \cup (Y^1_w)^+)$ and
$P$ is a $2\eta^{.9}$-perturbation of $J_1$.
We note that $P$ now satisfies the divisibility condition
$d^-_P(w) = 8|Y^1_w| = 8d^-_{P^K}(w)$, and for each $v \in V$
that $d^+_P(v,W) = |W|-t_1-|Z(x)| = d_P(v,V)/2$,
so $|P[V,W]| = |P[V]|$. We continue to modify $P$ to obtain
$|P^0[V,W]|=|P^0[V]|$ and $|P^K[V,W]|=|P^K[V]|$.
To do so, we will recolour arcs of $P[V]$ according to a
greedy algorithm, where if $|P^0[V]|>|P^0[V,W]|$ we replace
some $\overlinea{xy} \in P^0[V]$ by $\overlinea{xy}^- \in P^K[V]$,
or if $|P^0[V]|<|P^0[V,W]|$ we replace
some $\overlinea{xy} \in P^K[V]$ by $\overlinea{xy}^+ \in P^0[V]$.
This preserves $P[V]$ corresponding to $G'_1$ under twisting
and $|P[V]| = |P[V,W]|$, so if we ensure $|P^0[V,W]|=|P^0[V]|$,
we will also have $|P^K[V,W]|=|P^K[V]|$.
During the greedy algorithm, we choose the arc to recolour
arbitrarily, subject to avoiding the set $S$ of vertices
at which we have recoloured more than $\eta^{.8} n/2$ arcs.
The total number of recoloured arcs is at most
$||P[V,W]|-|P[V]|| \le ||J_1[V,W]|-|J_1[V]|| + 2\eta^{.9}n^2
< 3\eta^{.9}n^2$ (by Lemma \ref{deg}), so $|S|<12\eta^{.1}n$.
Thus the algorithm can be completed,
giving $P$ that is an $\eta^{.8}$-perturbation of $J_1$
with $|P^0[V,W]|=|P^0[V]|$ and $|P^K[V,W]|=|P^K[V]|$.
We will continue modifying $P[V]$ until it satisfies the
remaining degree divisibility conditions for each $v \in V$,
i.e.\ $d^+_P(v,V) = d^-_P(v,V) = d^+_P(v,W)$
and $d^-_{P^K}(v,V) = d^+_{P^K}(v,W)$.
To do so, we will reduce to $0$
the imbalance $\Delta' = \sum_{v \in V} \Delta'(v)$
with each $\Delta'(v) = |d^+_{P^K}(v,V)-d^+_{P^K}(v,W)|
+ |d^-_{P^K}(v,V)-d^+_{P^K}(v,W)|
$. We do not attempt
to control any $d^\pm_{P^0}(v,V)$, but nevertheless
the divisibility conditions will be satisfied when $\Delta'=0$.
To see this, note that if $\Delta'=0$ then clearly all
$d^+_{P^K}(v,V)=d^-_{P^K}(v,V)=d^+_{P^K}(v,W)$,
so it remains to
show that $d_P^-(v,V)=d_P^+(v,V)=d_P^+(v,W)$.
Here we recall the discussion in section \ref{sec:alg}
relating the choice of intervals to degree divisibility,
where (setting $H=G'_1$ and $J=P$) we noted that
$d_{G'_1}^+(v) = d_P^+(v,V)$ and
$d_{G_1'}^-(v) = d_P^-(v,V) + \Delta(v)$, with
$\Delta(v) = d^-_{P^K}(v^-,V) - d^-_{P^K}(v,V)
= d^+_{P^K}(v^-,W) - d^+_{P^K}(v,W)$.
By our choice of intervals
all $d^+_{P^K}(v,W)$ are equal to $t_1$,
so $\Delta(v)=0$ and $d_P^\pm(v,V) = d_{G'_1}^\pm(v)
= |W|-t_1-|Z(x)| = d^+_P(v,W)$, as required.
We have two types of reduction according to the two types
of term in the definition of $\Delta'(v)$:
\begin{enumerate}
\item If $\sum_v |d^-_{P^K}(v,V)-d^+_{P^K}(v,W)| > 0$
then we can choose $x,y$ in $V$ with
$d^-_{P^K}(x,V) > d^+_{P^K}(x,W)$ and
$d^-_{P^K}(y,V) < d^+_{P^K}(y,W)$.
We will find $z \in V$ such that
$\overlinea{zx} \in P^K$, $\overlinea{zy}^+ \in P^0$
and replace these arcs by
$\overlinea{zx}^+ \in P^0$, $\overlinea{zy} \in P^K$.
\item If $\sum_v |d^+_{P^K}(v,V)-d^+_{P^K}(v,W)| > 0$
then we can choose $x,y$ in $V$ with
$d^+_{P^K}(x,V) > d^+_{P^K}(x,W)$ and
$d^+_{P^K}(y,V) < d^+_{P^K}(y,W)$.
We will find $z \in V$ such that
$\overlinea{xz} \in P^K$, $\overlinea{yz}^+ \in P^0$
and replace these arcs by
$\overlinea{yz} \in P^K$, $\overlinea{xz}^+ \in P^0$.
\end{enumerate}
\begin{center}
\includegraphics{figZ}
\end{center}
Each of these operations preserves $P[V]$
corresponding to $G'_1$ under twisting and
reduces $\Delta'$.
To reduce $\Delta'$ to $0$ we apply a greedy algorithm
where in each step we apply one of the above operations.
We do not allow $z$ with $d(x,z)<3d+2$ or $d(y,z)<3d+2$
(to avoid creating close arcs in colour $K$)
or $z$ in the set $S'$ of vertices
that have played the role of $z$ at $\eta^{.7}n/2$
previous steps. The total number of steps is at most
$2\eta^{.8}n^2$, so $|S'| < 4\eta^{.1} n$.
To estimate the number of choices for $z$ at each step,
we apply Lemma \ref{deg}.iii to
$|N^-_{J^{K'}_1}(x^+) \cap N^-_{J^0_1}(y^+)|$
for operation (i),
$|N^+_{J^{K'}_1}(x) \cap N^+_{J^0_1}(y)|$ to find $z^+$ for (ii).
By typicality of $G$ this gives
at least $\alpha^2 n/9$ choices,
of which at most $5\eta^{.1} n$ are forbidden
by lying in $S$ or too close to $x$ or $y$,
or due to requiring an arc of $J_1 \setminus P$,
so some choice always exists.
Thus the algorithm can be completed, giving $P$
that is an $\eta^{.7}$-perturbation of $J_1$,
satisfies the divisibility conditions, and has
$P[V]$ corresponding to $G'_1$ under twisting.
Next we construct $E$ as a set of arc-disjoint
copies of $\ova{W}^K_{\! 8}$ that cover all $\overlinea{xy} \in P[V]$
with $d(x,y)<3d$. Note that all such $\overlinea{xy}$ have colour $0$.
We apply a greedy algorithm, where in each step that we consider
some $\overlinea{xy}$ we choose a copy of $\ova{W}^K_{\! 8}$ that is arc-disjoint
from all previous choices and does not use any vertex in the
set $S$ of vertices that have been used $.1d^2$ times.
Then $|S|.1d^2 < 27dn$, so this forbids
at most $270n^7/d$ choices of $\ova{W}^K_{\! 8}$.
By Lemma \ref{degWK} we have
$\hat{W}^1_K(\overlinea{xy}) > 2^{-2s-1} p^1_K/p^1_* > 2^{-3s}$,
so the number of choices is at least
$2^{-3s} \min_{w \in W} W^2_{w,K}/p^2_{w,K}n
> 2^{-4s} n^7$, say. Thus there is always some choice
that is not forbidden, so the algorithm can be completed.
We note that $\bigcup E$ has maximum degree
at most $d^2$ by definition of $S$, so $P' := P \setminus \bigcup E$
is a $2\eta^{.7}$-perturbation of $J_1$.
Furthermore, $P'$ satisfies the divisibility conditions,
as $P$ does and so does each $\ova{W}^K_{\! 8}$ in $E$.
Next we consider the regularity condition of Theorem \ref{decompK}.
To each $3d$-separated copy of $\ova{W}^K_{\! 8}$ in $P'$ with hub $w$
we assign weight $p^1_{w,K} n/ W^1_{w,K} =
(\alpha p^1_K (\alpha p^1_* p^1_{w,0} n)^7 )^{-1}$,
which lies in $[n^{-7}, L n^{-7}]$.
We claim that for any arc $\overlinea{e}$ of $P'$ there is total weight
$1 \pm \eta^{.6}$ on wheels containing $\overlinea{e}$.
To see this, we compare the weight to $\hat{W}^1_K(\overlinea{e})$
as defined in section \ref{sec:reg},
which is $1 \pm 4\varepsilon$ by Lemma \ref{degWK}
(as $\overlinea{e}$ is $3d$-separated,
$p^1_{w,0}=7p^1_{w,K}$ and $p^1_*=7p^1_K$).
The actual weight on $\overlinea{e}$ differs from this estimate only due
to wheels containing $\overlinea{e}$ that have another arc in $P' \bigtriangleup J_1$.
There are at most $40\eta^{.7} n^7$ such wheels,
each affecting the weight by at most $Ln^{-7}$, so the claim holds.
Thus regularity holds with $\delta=\eta^{.6}$ and $\omega=L^{-1}$.
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\end{enumerate}
\begin{center}
\includegraphics{figZ}
\end{center}
Each of these operations preserves $P[V]$
corresponding to $G'_1$ under twisting and
reduces $\Delta'$.
To reduce $\Delta'$ to $0$ we apply a greedy algorithm
where in each step we apply one of the above operations.
We do not allow $z$ with $d(x,z)<3d+2$ or $d(y,z)<3d+2$
(to avoid creating close arcs in colour $K$)
or $z$ in the set $S'$ of vertices
that have played the role of $z$ at $\eta^{.7}n/2$
previous steps. The total number of steps is at most
$2\eta^{.8}n^2$, so $|S'| < 4\eta^{.1} n$.
To estimate the number of choices for $z$ at each step,
we apply Lemma \ref{deg}.iii to
$|N^-_{J^{K'}_1}(x^+) \cap N^-_{J^0_1}(y^+)|$
for operation (i),
$|N^+_{J^{K'}_1}(x) \cap N^+_{J^0_1}(y)|$ to find $z^+$ for (ii).
By typicality of $G$ this gives
at least $\alpha^2 n/9$ choices,
of which at most $5\eta^{.1} n$ are forbidden
by lying in $S$ or too close to $x$ or $y$,
or due to requiring an arc of $J_1 \setminus P$,
so some choice always exists.
Thus the algorithm can be completed, giving $P$
that is an $\eta^{.7}$-perturbation of $J_1$,
satisfies the divisibility conditions, and has
$P[V]$ corresponding to $G'_1$ under twisting.
Next we construct $E$ as a set of arc-disjoint
copies of $\ova{W}^K_{\! 8}$ that cover all $\overlinea{xy} \in P[V]$
with $d(x,y)<3d$. Note that all such $\overlinea{xy}$ have colour $0$.
We apply a greedy algorithm, where in each step that we consider
some $\overlinea{xy}$ we choose a copy of $\ova{W}^K_{\! 8}$ that is arc-disjoint
from all previous choices and does not use any vertex in the
set $S$ of vertices that have been used $.1d^2$ times.
Then $|S|.1d^2 < 27dn$, so this forbids
at most $270n^7/d$ choices of $\ova{W}^K_{\! 8}$.
By Lemma \ref{degWK} we have
$\hat{W}^1_K(\overlinea{xy}) > 2^{-2s-1} p^1_K/p^1_* > 2^{-3s}$,
so the number of choices is at least
$2^{-3s} \min_{w \in W} W^2_{w,K}/p^2_{w,K}n
> 2^{-4s} n^7$, say. Thus there is always some choice
that is not forbidden, so the algorithm can be completed.
We note that $\bigcup E$ has maximum degree
at most $d^2$ by definition of $S$, so $P' := P \setminus \bigcup E$
is a $2\eta^{.7}$-perturbation of $J_1$.
Furthermore, $P'$ satisfies the divisibility conditions,
as $P$ does and so does each $\ova{W}^K_{\! 8}$ in $E$.
Next we consider the regularity condition of Theorem \ref{decompK}.
To each $3d$-separated copy of $\ova{W}^K_{\! 8}$ in $P'$ with hub $w$
we assign weight $p^1_{w,K} n/ W^1_{w,K} =
(\alpha p^1_K (\alpha p^1_* p^1_{w,0} n)^7 )^{-1}$,
which lies in $[n^{-7}, L n^{-7}]$.
We claim that for any arc $\overlinea{e}$ of $P'$ there is total weight
$1 \pm \eta^{.6}$ on wheels containing $\overlinea{e}$.
To see this, we compare the weight to $\hat{W}^1_K(\overlinea{e})$
as defined in section \ref{sec:reg},
which is $1 \pm 4\varepsilon$ by Lemma \ref{degWK}
(as $\overlinea{e}$ is $3d$-separated,
$p^1_{w,0}=7p^1_{w,K}$ and $p^1_*=7p^1_K$).
The actual weight on $\overlinea{e}$ differs from this estimate only due
to wheels containing $\overlinea{e}$ that have another arc in $P' \bigtriangleup J_1$.
There are at most $40\eta^{.7} n^7$ such wheels,
each affecting the weight by at most $Ln^{-7}$, so the claim holds.
Thus regularity holds with $\delta=\eta^{.6}$ and $\omega=L^{-1}$.
It remains to show that $P'$ satisfies the
extendability condition of Theorem \ref{decompK}.
Consider any disjoint $A,B \subseteq V$ and $L \subseteq W$
each of size $\le h$ and $a, b, \ell \in \{0,K\}$.
By Lemma \ref{deg}.iii we have
$|N^+_{J_1^a}(A) \cap N^-_{J_1^b}(B)
\cap N^-_{J_1^\ell}(L)|
= |N_G^+(A) \cap N_G^-(B)| (p_1^a)^{|A|} (p_1^b)^{|B|}
\prod_{w \in L} p^1_{w,\ell} \pm 4sn^{3/4}
> (10^{-3} \alpha )^{2h} n$, say.
Also, if $(A,B)$ is $3d$-separated then
by Lemma \ref{deg}.iv we have
$|N^+_{J_1^0}(A) \cap N^+_{J_1^K}(B) \cap W|
\ge 2^{-2s+10h} |W|$, say.
The perturbation from $J_1$ to $P'$ affects these estimates
by at most $6h\eta^{.7}n < \eta^{.6}n$, so $P'$ satisfies
extendability with $\omega=L^{-1}$ as above.
Now Theorem \ref{decompK} applies to give
a $\ova{W}^K_{\! 8}$-decomposition of $P'$,
which completes the proof.
\end{proof}
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\section{The proof} \leftarrowbel{sec:pf}
This section contains the proof of our main theorem.
We give the reduction to cases in the first subsection
and then the proof for both cases in the second subsection.
\subseteqsection{Reduction to cases} \leftarrowbel{sec:red}
In this subsection we formalise the reduction to cases
discussed in section \ref{sec:over}. For Theorem \ref{main},
we are given an $(\varepsilon,t)$-typical $\alpha n$-regular digraph $G$
on $n$ vertices, where $n^{-1} \ll \varepsilon \ll t^{-1} \ll \alpha$,
and we need to decompose $G$ into some given
family $\mc{F}$ of $\alpha n$ oriented one-factors on $n$ vertices.
We prove Theorem \ref{main} assuming that it holds in the
following cases with $t^{-1} \ll K^{-1} \ll \alpha$:
Case $K$: each $F \in \mc{F}$ has at least $n/2$ vertices
in cycles of length at least $K$,
Case $\ell$ for all $\ell \in [3,K-1]$:
each $F \in \mc{F}$ has $\ge K^{-3} n$
cycles of length $\ell$.
We will divide into subproblems via
the following partitioning lemma.
\begin{lemma} \leftarrowbel{typ:split}
Let $n^{-1} \ll \varepsilon \ll t^{-1} \ll \alpha_0$.
Suppose $G$ is an $(\varepsilon,t)$-typical $\alpha n$-regular digraph
on $n$ vertices and $\alpha = \sum_{i \in I} \alpha_i$
with each $\alpha_i > \alpha_0$. Then $G$ can be decomposed
into digraphs $(G_i: i \in I)$ on $V(G)$ such that
each $G_i$ is $(2\varepsilon,t)$-typical and $\alpha_i n$-regular.
\end{lemma}
\begin{proof}
We start by considering a random partition of $G$
into graphs $(G'_i: i \in I)$ where for each arc $\overlinea{e}$
independently we have $\mb{P}(\overlinea{e} \in G'_i)=\alpha_i/\alpha$.
We claim that whp each $G'_i$ is $(1.1\varepsilon,t)$-typical.
Indeed, this holds by Chernoff bounds,
as $\mb{E}d(G'_i) = \alpha_i d(G)/\alpha$ for each $i$,
so whp $d(G'_i) = \alpha_i \pm n^{-.4}$ (say),
and for any set $S=S_- \cup S_+$
of at most $t$ vertices, by typicality of $G$
we have $\mb{E} |N_{G'_i}^-(S_-) \cap N_{G'_i}^+(S_+)|
= (\alpha_i/\alpha)^{|S|} |N_G^-(S_-) \cap N_G^+(S_+)|
= ((1 \pm \varepsilon)d(G) \alpha_i/\alpha)^{|S|} n$,
so whp $|N_{G'_i}^-(S_-) \cap N_{G'_i}^+(S_+)|
= ((1 \pm 1.1\varepsilon) d(G'_i))^{|S|} n$,
Now we modify the partition to obtain $(G_i: i \in I)$,
by a greedy algorithm starting from all $G_i=G'_i$.
First we ensure that all $|G_i| = \alpha_i n^2$.
At any step, if this does not hold then some
$|G_i| > \alpha_i n^2$ and $|G_j| < \alpha_j n^2$.
We move an arc from $G_i$ to $G_j$, arbitrarily
subject to not moving more than $n^{.7}$ arcs
at any vertex. We move at most $n^{1.6}$ arcs,
so at most $2n^{.9}$ vertices become forbidden
during this algorithm. Hence the algorithm can
be completed to ensure that all $|G_i| = \alpha_i n^2$.
Each $|N_{G'_i}^-(S_-) \cap N_{G'_i}^+(S_+)|$
changes by at most $tn^{.7}$,
so each $G_i$ is now $(1.2\varepsilon,t)$-typical.
Let $\widetilde{G_i}$ be the undirected graph of $G_i$
(which could have parallel edges).
We will continue to modify the partition until each $\widetilde{G_i}$
is $2\alpha_i n$-regular, maintaining all $|G_i|=\alpha_i n^2$.
At each step we reduce the imbalance
$\sum_{i,x} |d_{\widetilde{G_i}}(x)-2\alpha_i n|$.
If some $\widetilde{G_i}$ is not $2\alpha_i n$-regular we have
some $d_{\widetilde{G_i}}(x) > 2\alpha_i n$ and $d_{\widetilde{G_i}}(y) < 2\alpha_i n$.
Considering the total degree of $x$,
there is some $j$ with $d_{\widetilde{G_j}}(x) < 2\alpha_j n$.
We will choose some $z$ with $xz \in \widetilde{G_i}$ and $yz \in \widetilde{G_j}$,
then move $xz$ to $\widetilde{G_j}$ and $yz$ to $\widetilde{G_i}$,
thus reducing the imbalance by at least $2$.
We will not choose $z$ in the set $L$ of vertices that
have played the role of $z$ at $n^{.8}$ previous steps.
We had all $d_{\widetilde{G_i}}(x) = 2(\alpha_i n \pm n^{.7})$
after the first algorithm, so this algorithm will have
at most $2n^{1.7}$ steps, giving $|L| < n^{.9}$.
By typicality, there are at least $3\alpha_i \alpha_j n$
choices of $z$, of which at most $2n^{.9}$ are forbidden
by $L$ or requiring an edge that has been moved,
so the algorithm to make each $\widetilde{G_i}$ be $2\alpha_i n$-regular
can be completed. Each $|N^-_{G_i}(S_-) \cap N^+_{G_i}(S_+)|$
changes by at most
$tn^{.8}$, so each $G_i$ is now $(1.1\varepsilon,t)$-typical.
We will continue to modify the partition until each $G_i$
is $\alpha_i n$-regular, maintaining all $d_{\widetilde{G_i}}(x)=2\alpha_i n$.
At each step we reduce the imbalance
$\sum_{i,x} |d_{G_i}^+(x)-\alpha_i n|$
(if it is $0$ then since total degrees $d_{\widetilde{G_i}}(x)$
are correct, $G_i$ is regular).
If it is not $0$ we have
some $d_{G_i}^+(x) > \alpha_i n$ and $d_{G_i}^+(y) < \alpha_i n$.
Again there is some $j$ with $d_{G_j}^+(x) < \alpha_j n$ and we
choose some $z$ with $\overlinea{xz} \in G_i$ and $\overlinea{yz} \in G_j$,
then move $\overlinea{xz}$ to $G_j$ and $\overlinea{yz}$ to $G_i$,
avoiding vertices $z$ which have played this role
at $n^{.9}$ previous steps.
By typicality we can find such $z$ at every step and
complete the algorithm.
Each $|N_{G_i}^-(S_-) \cap N_{G_i}^+(S_+)|$
changes by at most
$tn^{.9}$, so each $G_i$ is now $(2\varepsilon,t)$-typical.
\end{proof}
Factors of a type that is too rare
will be embedded greedily via the following lemma.
\begin{lemma} \leftarrowbel{typ:greedy}
Let $n^{-1} \ll \varepsilon \ll t^{-1} \ll \alpha$.
Suppose $G$ is an $(\varepsilon,t)$-typical $\alpha n$-regular digraph
on $n$ vertices and $\mc{F}$ is a family of at most $\varepsilon n$ oriented
one-factors. Then we can remove from $G$ a copy of each $F \in \mc{F}$
to leave a $(\sqrt{\varepsilon},t)$-typical $(\alpha n-|\mc{F}|)$-regular graph.
\end{lemma}
\begin{proof}
We embed the one-factors one by one. At each step,
the remaining graph $G'$ is obtained from $G$ by deleting
a graph that is regular of degree at most $2\varepsilon n$,
so is $(\sqrt{\varepsilon},t)$-typical. It is a standard argument
(which we omit) using the blow-up lemma
of Koml\'os, S\'ark\"ozy and Szemer\'edi \cite{KSS}
to show that any one-factor can be embedded in $G'$,
so the process can be completed.
\end{proof}
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Now we prove Theorem \ref{main}
assuming that it holds in the above cases.
We introduce new parameters $\alpha_1, \alpha_2, M_1', M_1, M_2, M_3$
with $\varepsilon \ll t^{-1} \ll M_3^{-1} \ll \alpha_2 \ll M_2^{-1}
\ll \alpha_1 \ll (M_1')^{-1} \ll M_1^{-1} \ll \alpha$.
For $\ell \in [3,M_2]$ let $\mc{F}_\ell$ consist of
all factors $F \in \mc{F}$ such that
$F$ has $\ge M_2^{-3} n$ cycles of length $\ell$
but $< M_2^{-3} n$ cycles of each smaller length.
Let $\mc{F}_2$ consist of all remaining factors in $\mc{F}$.
Note that each $F \in \mc{F}_2$ has fewer than $n/M_2$
vertices in cycles of length less than $M_2$,
so at least $(M_2-1)n/M_2$ in cycles of length at least $M_2$.
Let $B$ be the set of $\ell \in [3,M_2]$
such that $|\mc{F}_\ell| < \alpha_2 n$.
Then for $\ell \in I' := [3,M_2] \setminus B$ we have
$\beta_\ell := n^{-1} |\mc{F}_\ell| \ge \alpha_2$. Also,
writing $\mc{F}_B = \bigcup_{\ell \in B} \mc{F}_\ell$,
we have $\beta_B := n^{-1} |\mc{F}_B| < M_2\alpha_2 < \sqrt{\alpha_2}$.
Let $\mc{F}_1$ be the set of $F$ in $\mc{F}$ with
at least $n/2$ vertices in cycles of length $>M_1$.
We first consider the case
$\eta := n^{-1}|\mc{F}_1| \ge \alpha/2$.
Let $B^1 = B \cap [3,M_1]$,
$\mc{F}_{B^1} = \bigcup_{\ell \in B^1} \mc{F}_\ell$,
and $\beta_{B^1} := n^{-1} |\mc{F}_{B^1}| < \beta_B < \sqrt{\alpha_2}$.
We apply Lemma \ref{typ:split} with
$I = (I' \cap [3,M_1]) \cup \{1\}$, letting
$\alpha_\ell = \beta_\ell$ for all $\ell \in I' \cap [3,M_1]$
and $\alpha_1 = \eta + \beta_{B^1}$, thus decomposing $G$ into
$(2\varepsilon,t)$-typical $\alpha_i n$-regular digraphs $G_i$ on $V(G)$.
For each $\ell \in I' \cap [3,M_1]$ we decompose $G_\ell$ into
$\mc{F}_\ell$ by Case $\ell$ of Theorem \ref{main},
where in place of the parameters
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll \alpha$ we use
$n^{-1} \ll 2\varepsilon \ll t^{-1} \ll M_3^{-1} \ll \alpha_2$.
For $G_1$, we first embed $\mc{F}_{B^1}$ via
Lemma \ref{typ:greedy}, leaving an $\eta n$-regular
digraph $G'_1$ that is $(\varepsilon',t)$-typical
with $\alpha_2 \ll \varepsilon' \ll t{}^{-1} \ll M_2^{-1}$.
We then conclude the proof of this case by decomposing $G'_1$
into $\mc{F}_1$ by Case $K$ of Theorem \ref{main},
where in place of the parameters
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll \alpha$ we use
$n^{-1} \ll \varepsilon' \ll t{}^{-1} \ll M_1^{-1} \ll \eta$.
It remains to consider the case $\eta < \alpha/2$.
Here there are at least $\alpha n/2$ factors $F \in \mc{F}$
with at least $n/2$ vertices in cycles of length $\le M_1$,
so we can fix $\ell^* \in [M_1] \cap I'$
with $\beta_{\ell^*} > \alpha/2M_1$.
We consider two subcases according to $\beta_2 := n^{-1}|\mc{F}_2|$.
Suppose first that $\beta_2 < \alpha_1 n$.
We apply Lemma \ref{typ:split} with $I = I'$, letting
$\alpha_\ell = \beta_\ell$ for all $\ell \in I \setminus \{\ell^*\}$
and $\alpha_{\ell^*} = \beta_{\ell^*} + \beta_{B^1} + \beta_2$.
For each $\ell \in I \setminus \{\ell^*\}$ we decompose $G_\ell$
into $\mc{F}_\ell$ by Case $\ell$ of Theorem \ref{main},
where (as before) in place of the parameters
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll \alpha$ we use
$n^{-1} \ll 2\varepsilon \ll t^{-1} \ll M_3^{-1} \ll \alpha_2$.
For $G_{\ell^*}$ we first embed $\mc{F}_B \cup \mc{F}_2$
by Lemma \ref{typ:greedy}, leaving a $\beta_{\ell^*} n$-regular
digraph $G'_{\ell^*}$ that is $(\varepsilon',t)$-typical
with $\alpha_1 \ll \varepsilon' \ll t{}^{-1} \ll M_1^{-1}$.
We then complete the decomposition by decomposing $G'_{\ell^*}$
into $\mc{F}_{\ell^*}$ by Case $\ell^*$ of Theorem \ref{main},
where in place of the parameters
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll \alpha$ we use
$n^{-1} \ll \varepsilon' \ll t{}^{-1} \ll (M_1')^{-1} \ll \beta_{\ell^*}$.
It remains to consider the subcase $\beta_2 \ge \alpha_1 n$.
We apply Lemma \ref{typ:split} with $I = I' \cup \{2\}$, letting
$\alpha_\ell = \beta_\ell$ for all $\ell \in I \setminus \{\ell^*\}$
and $\alpha_{\ell^*} = \beta_{\ell^*} + \beta_{B^1}$.
The same argument as in the first subcase applies to
decompose $G_\ell$ into $\mc{F}_\ell$
for all $\ell \in I' \setminus \{\ell^*\}$,
and also to embed $\mc{F}_B$ in $G_{\ell^*}$
by Lemma \ref{typ:greedy} and decompose
the leave $G'_{\ell^*}$ into $\mc{F}_{\ell^*}$.
We complete the proof of this case,
and so of the entire reduction, by decomposing
$G_2$ into $\mc{F}_2$ by Case $K$ of Theorem \ref{main},
where in place of the parameters
$n^{-1} \ll \varepsilon \ll t^{-1} \ll K^{-1} \ll \alpha$ we use
$n^{-1} \ll 2\varepsilon \ll t^{-1} \ll M_2^{-1} \ll \beta_2$.
\subseteqsection{Proof of Theorem \ref{main}}
We are now ready to prove our main theorem.
We are given an $(\varepsilon,t)$-typical $\alpha n$-regular digraph $G$
on $n$ vertices, where $n^{-1} \ll \varepsilon \ll t^{-1} \ll \alpha$,
and we need to decompose $G$ into some given
family $\mc{F}$ of $\alpha n$ oriented one-factors on $n$ vertices.
By the reductions in section \ref{sec:red},
we can assume that we are in one of the
following cases with $t^{-1} \ll M^{-1} \ll \alpha$:
Case $K$: each $F \in \mc{F}$ has at least $n/2$ vertices
in cycles of length at least $M$,
Case $\ell^*$ with $\ell^* \in [3,M-1]$: each $F \in \mc{F}$
has $\ge M^{-3} n$ cycles of length $\ell^*$.
Here the parameters of section \ref{sec:red} are renamed:
$\ell$ is now $\ell^*$ so that
`$\ell$' is free to denote generic cycle lengths;
$K$ is now $M$, as we want $K$ to take different
values in each case: we introduce $M'$ with
$t^{-1} \ll M'{}^{-1} \ll M^{-1}$ and define
\[ K = \left\{ \begin{array}{ll}
M & \text{ in Case } K, \\
M' & \text{ in Case } \ell^*.
\end{array} \right. \]
We define a parameter $L$ by $L=M$ in Case $\ell^*$
(so $\ell^* \ll L \ll K$), or as a new parameter
with $K^{-1} \ll L^{-1} \ll \alpha$ in Case $K$.
We use these parameters to apply the algorithm
of section \ref{sec:alg} as in~(\ref{hierarchy}), so we can apply the
conclusions of the lemmas in
sections \ref{sec:int} to \ref{sec:exact}.
We recall that each factor $F_w$
is partitioned as $F^1_w \cup F^2_w$,
where $F^1_w$ either consists of exactly
$L^{-3} n$ cycles of length $\ell^*$ in Case $\ell^*$,
or in Case $K$ we have $|F^1_w| - n/2 \in [0,2K]$
and $F^1_w$ consists of cycles of length $\ge K$
and at most one path of length $\ge K$
(and then $F^2_w = F_w \setminus F^1_w$).
By Lemma \ref{lem:approx},
there are arc-disjoint digraphs $G^2_w \subseteq G_2$ for $w \in W$,
where each $G^2_w$ is a copy of some valid $F'_w \subseteq F^2_w$
with $V(G^2_w) \subseteq N^-_{J_2}(w)$, such that
\begin{enumerate}
\item $G_2^- = G_2 \setminus \bigcup_{w \in W} G^2_w$
has maximum degree at most $5d^{-1/3}n$,
\item the digraph $J_2^-$
obtained from $J_2[V,W]$ by deleting all $\overlinea{xw}$
with $x \in V(G^2_w)$ has maximum degree at most $5d^{-1/3}n$,
\item any $x \in V$ has degree $1$ in $F'_w$
for at most $n/\sqrt{d}$ choices of $w$.
\end{enumerate}
(Recall that `valid' means that $F'_w$
does not have any independent arcs,
and if $F^2_w$ contains a path then $F'_w$
contains the arcs incident to each of its ends.)
Note that (ii) implies for each $w \in W$ that
$|F'_w| \ge |N^-_{J_2}(w)| - 5d^{-1/3}n
> p^2_w n - 6d^{-1/3}n$ (by Lemma \ref{deg}),
so as $p^2_w n = (1-\eta)|F^2_w| + n^{.8}$
we have $|F^2_w \setminus F'_w| < \eta n$.
Next we will embed oriented graphs
$R_w = (F^2_w \setminus F'_w) \cup L_w$ for $w \in W$,
where $L_w \subseteq F^1_w$ is defined as follows.
In Case $\ell^*$ we let each $L_w$ consist of
$2\eta L^{-3} n$ cycles of length $\ell^*$.
In Case $K$ we partition each $F^1_w$ as $\mc{P}_w \cup L_w$,
where $\mc{P}_w$ is a valid vertex-disjoint union of paths,
such that for each $[a,b] \in \mc{Y}^1_w$ we have
an oriented path $P^{ab}_w$ in $\mc{P}_w$ of length $8d(a,b)$
(which we will embed as an $ab^+$-path).
To see that such a partition exists, we apply the same
argument as at the end of the proof of Lemma \ref{lem:approx}.
We consider a greedy algorithm, where at each step that we consider
some path $P^{ab}_w$ we delete a path of length $8d(a,b)+4$
from $F^1_w$, which we allocate as $P^{ab}_w$ surrounded on both
sides of paths of length $2$ that we add to $L_w$.
As $|\mc{Y}^1_w| < n/2d_{2s+1} = (2s)^{2s}n/2d$ we thus
allocate $< (2s)^{2s}n/d$ edges to $L_w$.
Suppose for contradiction that the algorithm gets stuck,
trying to embed some path $P$ in some remainder $Q_w$.
Then all components of $Q_w$ have size $\le 8d+5$.
All components of $F^1_w$ have size $\ge K$,
so $|Q_w| \le (8d+5)|F^1_w|/K < 5dn/K$.
However, we also have
$|Q_w| \ge |F^1_w|-|Y^1_w|-|L_w| \ge \eta n/3$,
as $|F^1_w| \ge n/2$ and
$|Y^1_w| = (1-\eta)n/2 \pm 2n^{3/4}$ by Lemma \ref{lem:int}.
This is a contradiction, so the algorithm finds a partition
$F^1_w = \mc{P}_w \cup L_w$ with $\mc{P}_w$ valid.
We note that each $|R_w| < 2\eta n$.
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Here the parameters of section \ref{sec:red} are renamed:
$\ell$ is now $\ell^*$ so that
`$\ell$' is free to denote generic cycle lengths;
$K$ is now $M$, as we want $K$ to take different
values in each case: we introduce $M'$ with
$t^{-1} \ll M'{}^{-1} \ll M^{-1}$ and define
\[ K = \left\{ \begin{array}{ll}
M & \text{ in Case } K, \\
M' & \text{ in Case } \ell^*.
\end{array} \right. \]
We define a parameter $L$ by $L=M$ in Case $\ell^*$
(so $\ell^* \ll L \ll K$), or as a new parameter
with $K^{-1} \ll L^{-1} \ll \alpha$ in Case $K$.
We use these parameters to apply the algorithm
of section \ref{sec:alg} as in~(\ref{hierarchy}), so we can apply the
conclusions of the lemmas in
sections \ref{sec:int} to \ref{sec:exact}.
We recall that each factor $F_w$
is partitioned as $F^1_w \cup F^2_w$,
where $F^1_w$ either consists of exactly
$L^{-3} n$ cycles of length $\ell^*$ in Case $\ell^*$,
or in Case $K$ we have $|F^1_w| - n/2 \in [0,2K]$
and $F^1_w$ consists of cycles of length $\ge K$
and at most one path of length $\ge K$
(and then $F^2_w = F_w \setminus F^1_w$).
By Lemma \ref{lem:approx},
there are arc-disjoint digraphs $G^2_w \subseteq G_2$ for $w \in W$,
where each $G^2_w$ is a copy of some valid $F'_w \subseteq F^2_w$
with $V(G^2_w) \subseteq N^-_{J_2}(w)$, such that
\begin{enumerate}
\item $G_2^- = G_2 \setminus \bigcup_{w \in W} G^2_w$
has maximum degree at most $5d^{-1/3}n$,
\item the digraph $J_2^-$
obtained from $J_2[V,W]$ by deleting all $\overlinea{xw}$
with $x \in V(G^2_w)$ has maximum degree at most $5d^{-1/3}n$,
\item any $x \in V$ has degree $1$ in $F'_w$
for at most $n/\sqrt{d}$ choices of $w$.
\end{enumerate}
(Recall that `valid' means that $F'_w$
does not have any independent arcs,
and if $F^2_w$ contains a path then $F'_w$
contains the arcs incident to each of its ends.)
Note that (ii) implies for each $w \in W$ that
$|F'_w| \ge |N^-_{J_2}(w)| - 5d^{-1/3}n
> p^2_w n - 6d^{-1/3}n$ (by Lemma \ref{deg}),
so as $p^2_w n = (1-\eta)|F^2_w| + n^{.8}$
we have $|F^2_w \setminus F'_w| < \eta n$.
Next we will embed oriented graphs
$R_w = (F^2_w \setminus F'_w) \cup L_w$ for $w \in W$,
where $L_w \subseteq F^1_w$ is defined as follows.
In Case $\ell^*$ we let each $L_w$ consist of
$2\eta L^{-3} n$ cycles of length $\ell^*$.
In Case $K$ we partition each $F^1_w$ as $\mc{P}_w \cup L_w$,
where $\mc{P}_w$ is a valid vertex-disjoint union of paths,
such that for each $[a,b] \in \mc{Y}^1_w$ we have
an oriented path $P^{ab}_w$ in $\mc{P}_w$ of length $8d(a,b)$
(which we will embed as an $ab^+$-path).
To see that such a partition exists, we apply the same
argument as at the end of the proof of Lemma \ref{lem:approx}.
We consider a greedy algorithm, where at each step that we consider
some path $P^{ab}_w$ we delete a path of length $8d(a,b)+4$
from $F^1_w$, which we allocate as $P^{ab}_w$ surrounded on both
sides of paths of length $2$ that we add to $L_w$.
As $|\mc{Y}^1_w| < n/2d_{2s+1} = (2s)^{2s}n/2d$ we thus
allocate $< (2s)^{2s}n/d$ edges to $L_w$.
Suppose for contradiction that the algorithm gets stuck,
trying to embed some path $P$ in some remainder $Q_w$.
Then all components of $Q_w$ have size $\le 8d+5$.
All components of $F^1_w$ have size $\ge K$,
so $|Q_w| \le (8d+5)|F^1_w|/K < 5dn/K$.
However, we also have
$|Q_w| \ge |F^1_w|-|Y^1_w|-|L_w| \ge \eta n/3$,
as $|F^1_w| \ge n/2$ and
$|Y^1_w| = (1-\eta)n/2 \pm 2n^{3/4}$ by Lemma \ref{lem:int}.
This is a contradiction, so the algorithm finds a partition
$F^1_w = \mc{P}_w \cup L_w$ with $\mc{P}_w$ valid.
We note that each $|R_w| < 2\eta n$.
Now we apply a greedy algorithm to construct arc-disjoint
embeddings $(\phi_w(R_w): w \in W)$ in $G_1$.
At each step we choose some $\phi_w(x) \in N^-_{J^1}(w)$
(which is disjoint from $G^2_w \subseteq N^-_{J_2}(w)$).
We require $\phi_w(x)$ to be an outneighbour of some
previously embedded $\phi_w(x_1)$ or both an outneighbour of
$\phi_w(x_1)$ and an inneighbour of $\phi_w(x_2)$
for some previously embedded images;
the latter occurs when we finish a cycle or a path
(the image under $\phi_w$ of the ends of the paths in $R_w$
have already been prescribed:
they are either images of endpoints of paths in $F'_w$
or startpoints / successors of intervals in $\mc{Y}^1_w$).
We also require $\phi_w(x)$ to be distinct from all
previously embedded $\phi_w(x_1)$ and not to lie
in the set $S$ of vertices that are already in the image
of $\phi_{w'}$ for at least $\eta^{.9} n/2$ choices of $w'$.
As $\eta^{.9} n |S|/2 \le \sum_{w \in W} |R_w| < 2\eta n^2$
we have $|S| < 4\eta^{.1} n$.
To see that it is possible to choose $\phi_w(x)$,
first note for any $v,v'$ in $V$ and $w \in W$ that
$|N_{G_1}^+(v) \cap N_{G_1}^-(v') \cap N^-_{J^1}(w)| > \alpha^2 n/3$,
by Lemma \ref{deg}.iii and typicality of $G$.
At most $|R_w|+|S| < 5\eta^{.1} n$ choices of $\phi_w(x)$
are forbidden due to using $S$ or some previously embedded
$\phi_w(x_1)$. Also, by definition of $S$, we have used
at most $\eta^{.9} n$ arcs at each of $v$ and $v'$ for
other embeddings $\phi_{w'}$, so this forbids
at most $2\eta^{.9} n$ choices of $\phi_w(x)$.
Thus the algorithm never gets stuck, so we can construct
$(\phi_w(R_w): w \in W)$ as required.
Let $G'_1 = G \setminus \bigcup_{w \in W} (G^2_w \cup R_w)$.
For each $w \in W$ let $Z_w$ be the set
of vertices of in- and outdegree $1$ in $G^2_w \cup R_w$.
We claim that $G'_1$ and $Z_w$ satisfy Setting \ref{set}.
To see this, first note that by definition of $S$ above
each $|N_{G_1}^\pm(x) \setminus N_{G'_1}^\pm(x)| < \eta^{.9} n/2$.
As $d_{G_2^-}^\pm(x) < 5d^{-1/3}n$ by (i) above and
(by Lemma~\ref{deg}) $d_G^\pm(x)-d_{G_1}^\pm(x)-d_{G_2}^\pm(x)
< (1-p_1-p_2)d_G^\pm(x) + n^{.6} < 2\eta n$
we have $|N_{G_1}^\pm(x) \bigtriangleup N_{G'_1}^\pm(x)| < \eta^{.9} n$,
so $G'_1$ is an $\eta^{.9}$-perturbation of $G_1$.
Also, as $|N^-_{J_2}(w) \setminus F'_w| \le 5d^{-1/3}n$,
$|R_w| < 2\eta n$ and $|V \setminus N^-_J(w)| < 2\eta n$
(the last by Lemma \ref{deg}) we have
$|Z_w \bigtriangleup (V \setminus N^-_{J^1}(w))| < 5\eta n$, as claimed.
In Case $\ell^*$, every vertex
has equal in- and outdegrees $0$ or $1$ in $G^2_w \cup R_w$
(it is a vertex-disjoint union of cycles) so
$d_{G'_1}^\pm(x)=|W|-|Z(x)|$ for all $x \in V$
and $\ell^*$ divides $n-|Z_w|$ for all $w \in W$.
Thus Lemma \ref{exactL} applies to partition $G'_1$
into graphs $(G^1_w: w \in W)$, where each $G^1_w$
is a $C_{\ell^*}$-factor with $V(G^1_w) = V \setminus Z_w$,
thus completing the proof of this case.
In Case $K$, a vertex $x$ has indegree (respectively outdegree)
$1$ in $G^2_w \cup R_w$
exactly when $x \in (Y^1_w)^-$
(respectively $(Y^1_w)^+$),
for which there are each $t_1$ choices of $w$,
so $d_{G'_1}^\pm(x)=|W|-t_1-|Z(x)|$ for all $x \in V$.
By construction, $Z_w$ is disjoint from $(Y^1_w)^- \cup (Y^1_w)^+$,
and the total length of paths required in the remaining path
factor problem satisfies
$8|Y^1_w| = n-|Z_w|-|(Y^1_w)^+|$ for all $w \in W$.
Thus Lemma \ref{exactK} applies to partition $G'_1$
into graphs $(G^1_w: w \in W)$,
such that each $G^1_w$ is a vertex-disjoint union of oriented paths
with $V(G^1_w) = V\setminus Z_w$,
where for each $[a,b] \in \mc{Y}^1_w$ there
is an $ab^+$-path of length $8d(a,b)$.
This completes the proof of this case,
and so of Theorem \ref{main}.
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\section{Concluding remarks} \leftarrowbel{sec:con}
As mentioned in the introduction,
our solution to the generalised Oberwolfach Problem
is more general than the result of \cite{GJKKO}
in three respects: it applies to any typical graph
(theirs is for almost complete graphs)
and to any collection of two-factors
(they need some fixed $F$ to occur $\Omega(n)$ times),
and it applies also to directed graphs.
Although there are some common elements
in both of our approaches (using \cite{K2}
for the exact step and some form of twisting),
the more general nature of our result
reflects a greater flexibility in our approach
that has further applications.
One such application is our recent proof \cite{KSringel}
that every quasirandom graph with
$n$ vertices and $rn$ edges can be
decomposed into $n$ copies of any fixed
tree with $r$ edges.
The case of the complete graph solves
Ringel's tree-packing conjecture~\cite{ringel}
(solved independently via different
methods by Montgomery, Pokrovskiy
and Sudakov~\cite{MPS3}).
A natural open problem raised in \cite{GJKKO}
is whether the generalised Oberwolfach problem
can be further generalised to decompositions of $K_n$
into any family of regular graphs of bounded degree
(where the total of the degrees is $n-1$).
\end{document}
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\begin{document}
\begin{frontmatter}
\title{A supplement on feathered gyrogroups \tnoteref{t1}}
\tnotetext[t1]{This research was supported by the National Natural Science Foundation of China (Nos. 12071199, 11661057), the Natural Science Foundation of Jiangxi Province, China (No. 20192ACBL20045).}
\author[M. Bao]{Meng Bao}
\ead{[email protected]}
\address[M. Bao]{College of Mathematics, Sichuan University, Chengdu 610064, China}
\author[X. Ling]{Xuewei Ling}
\ead{[email protected]}
\address[X. Ling]{Institute of Mathematics, Nanjing Normal University, Nanjing 210046, China}
\author[X. Xu]{Xiaoquan Xu\corref{mycorrespondingauthor}}
\cortext[mycorrespondingauthor]{Corresponding author.}
\ead{[email protected]}
\address[X. Xu]{Fujian Key Laboratory of Granular Computing and Applications, Minnan Normal University, Zhangzhou 363000, China}
\begin{abstract}
A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. It is shown that each compact subset of a topological gyrogroup with an $\omega^{\omega}$-base is metrizable, which deduces that if $G$ is a topological gyrogroup with an $\omega^{\omega}$-base and is a $k$-space, then it is sequential. Moreover, for a feathered strongly topological gyrogroup $G$, based on the characterization of feathered strongly topological gyrogroups, we show that if $G$ has countable $cs^{*}$-character, then it is metrizable; and it is also shown that $G$ has a compact resolution swallowing the compact sets if and only if $G$ contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is a Polish space.
\end{abstract}
\begin{keyword}
Topological gyrogroups, metrizable, $\omega^{\omega}$-base, feathered
\MSC 22A22; 54A20; 20N05; 18A32.
\end{keyword}
\end{frontmatter}
\section{Introduction}
In the field of Topological Algebra, topological groups are standard researching objects and have been studied for many years, see \cite{AA}. Moreover, the combination of topology and non-associative algebra has attracted the attention of many scholars. For example, in \cite{CZ}, Cai, Lin and He introduced and investigated the concept of paratopological left Bol loops and proved some results of paratopological groups can be extended to paratopological left Bol loops. Banakh and Repov\v s \cite{Banakh} studied many generalized metric properties in rectifiable spaces and topological lops and showed that a rectifiable space $X$ is metrizable if and only if it is sequential, has
countable $cs^{*}$-character, and contains no closed copy of the Fr\'echet-Urysohn fan $S_{\omega}$. In \cite{Shen2020}, Shen introduced paratopological left-loops and showed that every weakly first-countable paratopological left-loop is first-countable. As we all known, a gyrogroup as an important type of non-associative algebra has many applications in Geometry and Physics, in particular, in the study of the $c$-ball of relativistically admissible velocities with the Einstein velocity addition, see \cite{UA1988,UA2002,UA2005,UA}. Therefore, topological gyrogroups are very important topological spaces which were posed by Atiponrat \cite{AW}. Clearly, every topological group is a topological gyrogroup and each topological gyrogroup is a rectifiable space. The readers may consult \cite{AW1,AW2020,BL,BL2,BL3,BX2022,BZX,BZX2,WAS2020,ZBX} for more details about topological gyrogroups.
In this paper, we show that each compact subset of a topological gyrogroup with an $\omega^{\omega}$-base is metrizable, which deduces that if $G$ is a topological gyrogroup with an $\omega^{\omega}$-base and is a $k$-space, then it is sequential. Moreover, we mainly research some weakly first-countable properties in feathered strongly topological gyrogroups. Indeed, for further study on M\"{o}bius gyrogroups, Bao and Lin posed the concept of strongly topological gyrogroups and showed that a strongly topological gyrogroup $G$ is feathered if and only if it contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is metrizable. Based on the characterization of feathered strongly topological gyrogroups, they proved that each feathered strongly topological gyrogroup is paracompact. Also based on the characterization, we give some equivalent relationships of metrizability for strongly topological gyrogroups, such as every feathered strongly topological gyrogroup with countable $cs^{*}$-character is metrizable, and so on. It is also shown that for a feathered strongly topological gyrogroup $G$, $G$ has a compact resolution swallowing the compact sets if and only if $G$ contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is a Polish space. Some problems about topological gyrogroups with countable $cs^{*}$-character are posed.
\section{Preliminary}
Throughout this paper, all topological spaces are assumed to be Hausdorff, unless otherwise is explicitly stated. Let $\mathbb{N}$ be the set of all positive integers and $\omega$ the first infinite ordinal. The readers may consult \cite{AA, E, linbook1, UA} for notation and terminology not explicitly given here. Next we recall some definitions and facts.
\begin{definition}\cite{AW}
Let $G$ be a nonempty set, and let $\oplus: G\times G\rightarrow G$ be a binary operation on $G$. Then the pair $(G, \oplus)$ is called a {\it groupoid}. A function $f$ from a groupoid $(G_{1}, \oplus_{1})$ to a groupoid $(G_{2}, \oplus_{2})$ is called a {\it groupoid homomorphism} if $f(x\oplus_{1}y)=f(x)\oplus_{2} f(y)$ for any elements $x, y\in G_{1}$. Furthermore, a bijective groupoid homomorphism from a groupoid $(G, \oplus)$ to itself will be called a {\it groupoid automorphism}. We write $\mbox{Aut}(G, \oplus)$ for the set of all automorphisms of a groupoid $(G, \oplus)$.
\end{definition}
\begin{definition}\cite{UA}
Let $(G, \oplus)$ be a groupoid. The system $(G,\oplus)$ is called a {\it gyrogroup}, if its binary operation satisfies the following conditions:
$(G1)$ There exists a unique identity element $0\in G$ such that $0\oplus a=a=a\oplus0$ for all $a\in G$.
$(G2)$ For each $x\in G$, there exists a unique inverse element $\ominus x\in G$ such that $\ominus x \oplus x=0=x\oplus (\ominus x)$.
$(G3)$ For all $x, y\in G$, there exists $\mbox{gyr}[x, y]\in \mbox{Aut}(G, \oplus)$ with the property that $x\oplus (y\oplus z)=(x\oplus y)\oplus \mbox{gyr}[x, y](z)$ for all $z\in G$.
$(G4)$ For any $x, y\in G$, $\mbox{gyr}[x\oplus y, y]=\mbox{gyr}[x, y]$.
\end{definition}
Notice that a group is a gyrogroup $(G,\oplus)$ such that $\mbox{gyr}[x,y]$ is the identity function for all $x, y\in G$. The definition of a subgyrogroup is given as follows.
\begin{definition}\cite{ST}
Let $(G,\oplus)$ be a gyrogroup. A nonempty subset $H$ of $G$ is called a {\it subgyrogroup}, denoted
by $H\leq G$, if $H$ forms a gyrogroup under the operation inherited from $G$ and the restriction of $\mbox{gyr}[a,b]$ to $H$ is an automorphism of $H$ for all $a,b\in H$.
Furthermore, a subgyrogroup $H$ of $G$ is said to be an {\it $L$-subgyrogroup}, denoted
by $H\leq_{L} G$, if $\mbox{gyr}[a, h](H)=H$ for all $a\in G$ and $h\in H$.
\end{definition}
\begin{definition}\cite{AW}
A triple $(G, \tau, \oplus)$ is called a {\it topological gyrogroup} if the following statements hold:
(1) $(G, \tau)$ is a topological space.
(2) $(G, \oplus)$ is a gyrogroup.
(3) The binary operation $\oplus: G\times G\rightarrow G$ is jointly continuous while $G\times G$ is endowed with the product topology, and the operation of taking the inverse $\ominus (\cdot): G\rightarrow G$, i.e. $x\rightarrow \ominus x$, is also continuous.
\end{definition}
Obviously, every topological group is a topological gyrogroup. However, every topological gyrogroup whose gyrations are not identically equal to the identity is not a topological group.
\begin{example}\cite[Example 3]{AW}
The Einstein gyrogroup with the standard topology is a topological gyrogroup but not a topological group.
\end{example}
Then we recall some weakly first-countable concepts which are important in the following researches.
\begin{definition}\cite{BT,GK,LPT}
A point $x$ of a topological space $X$ is said to have a {\it neighborhood $\omega^{\omega}$-base} or a {\it local $\mathfrak{G}$-base} if there exists a base of neighborhoods at $x$ of the form $\{U_{\alpha}(x):\alpha \in \mathbb{N}^{\mathbb{N}}\}$ such that $U_{\beta}(x)\subset U_{\alpha}(x)$ for all elements $\alpha \leq \beta$ in $\mathbb{N}^{\mathbb{N}}$, where $\mathbb{N}^{\mathbb{N}}$ consisting of all functions from $\mathbb{N}$ to $\mathbb{N}$ is endowed with the natural partial order, ie., $f\leq g$ if and only if $f(n)\leq g(n)$ for all $n\in \mathbb{N}$. The space $X$ is said to have an {\it $\omega^{\omega}$-base} or a {\it $\mathfrak{G}$-base} if it has a neighborhood $\omega^{\omega}$-base or a local $\mathfrak{G}$-base at every point $x\in X$.
\end{definition}
\begin{definition}
Let $X$ be a topological space.
$(1)$\, $X$ is called a {\it sequential space} \cite{FS} if for each non-closed subset $A\subseteq X$, there are a point $x\in X\setminus A$ and a sequence in $A$ converging to $x$ in $X$.
$(2)$\, $X$ is called a {\it Fr\'{e}chet-Urysohn space} \cite{FS} if for any subset $A\subseteq X$ and $x\in \overline{A}$, there is a sequence in $A$ converging to $x$ in $X$.
$(3)$\, $X$ is called an {\it $\alpha_{7}$-space} \cite{BZ}, if for every point $x\in X$ and each sheaf $\{S_{n}:n\in\omega\}$ with the vertex $x$, there exists a sequence converging to some point $y\in X$ which meets infinitely many sequences $S_{n}$.
\end{definition}
A family $\mathcal{N}$ of subsets of a topological space $X$ is called a {\it $cs^{*}$-network at a point $x\in X$} \cite{GMZ} if for each sequence $(x_{n})_{n\in \mathbb{N}}$ in $X$ convergent to $x$ and for each neighborhood $O_{x}$ of $x$ there is a set $N\in \mathcal{N}$ such that $x\in N\subset O_{x}$ and the set $\{n\in \mathbb{N}:x_{n}\in N\}$ is infinite.
Then we give the concept of $cs^{*}$-character of a topological gyrogroup.
\begin{definition}\cite[Theorem 3.7]{BZX2}
Let $G$ be a topological gyrogroup, the $cs^{*}$-character of $G$ is the least cardinality of $cs^{*}$-network at the identity element $0$ of $G$.
\end{definition}
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\section{Weakly first-countable properties of topological gyrogroups}
In this section, it is shown that each compact subset of a topological gyrogroup with an $\omega^{\omega}$-base is metrizable, which deduces that if $G$ is a topological gyrogroup with an $\omega^{\omega}$-base and is a $k$-space, then it is sequential. Moreover, for a feathered strongly topological gyrogroup $G$, based on the characterization of feathered strongly topological gyrogroups, that is, a strongly topological gyrogroup is feathered if and only if it contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is metrizable, we give some equivalent relationships of metrizability for strongly topological gyrogroups, such as every feathered strongly topological gyrogroup with countable $cs^{*}$-character is metrizable.
\begin{definition}\cite[Definition 3.1]{GKL}
A family $\{K_{\alpha}:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ of compact sets of a topological space $X$ is called a {\it compact resolution} if $X=\bigcup \{K_{\alpha}:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ and $K_{\alpha}\subseteq K_{\beta}$ for all $\alpha \leq \beta$. In additionally, every compact set in $X$ is contained in some $K_{\alpha}$, we say that $\{K_{\alpha}:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ {\it swallows the compact sets} of $X$.
\end{definition}
It is well-known that each Polish space $X$ has a compact resolution swallowing the compact sets of $X$. Moreover, it was proved in \cite[Theorem 3.3]{CRJP} that if $X$ is a metrizable topological space, then $X$ is a Polish space if and only if $X$ has a compact resolution swallowing the compact sets of $X$. Then, it follows from \cite[Proposition 3.3]{TVV} that each hemicompact topological space has a compact resolution swallowing the compact sets and the property of having a compact resolution swallowing the compact sets is closed-hereditary and is closed under countable products.
\begin{theorem}\label{3compact}
Let $G$ be a topological gyrogroup which has an $\omega^{\omega}$-base, $K$ an arbitrary compact subset of $G$. Then $K$ is metrizable.
\end{theorem}
\begin{proof}
By the hypothesis, let $\{U_{\alpha}:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ be an open $\omega^{\omega}$-base in $G$. Without loss of generality, we assume that all sets $U_{\alpha}$ are symmetric. By \cite[Theorem 1]{CBOJ}, a compact space $K$ is metrizable if and only if $(K\times K)\setminus \Delta$ has a compact resolution swallowing its compact sets, where $\Delta =\{(x,x):x\in K\}$. Therefore, it suffices to show that the set $W=(K\times K)\setminus \Delta$ has a compact resolution which swallows its compact sets.
For each $\alpha\in \mathbb{N}^{\mathbb{N}}$, set $W_{\alpha}=\{(x,y)\in W,x\oplus (\ominus y)\not\in U_{\alpha}\}$. Then $W_{\alpha}$ is closed in $K\times K$, and hence it is compact for each $\alpha\in \mathbb{N}^{\mathbb{N}}$.
For each compact subset $C$ of $W$, $q(C)=\{x\oplus (\ominus y):(x,y)\in C\}$ is compact and does not contain the identity element $0$ of $G$. Since $\{U_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$ is a local base at $0$, for some $\alpha\in \mathbb{N}^{\mathbb{N}}$, we obtain $U_{\alpha}\cap q(C)=\emptyset$. Then $C\subseteq W_{\alpha}$. Thus, $\{W_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$ is a compact resolution swallowing the compact sets in $W$. We conclude that $K$ is metrizable.
\end{proof}
In \cite[Theorem 1.1]{Banakh}, Banakh showed that each non-metrizable sequential rectifiable space $X$ of countable $cs^{*}$-character contains a clopen rectifiable submetrizable $k_{\omega}$-subspace. Indeed, during the process of proof, it is not difficult to see that if $X$ is a non-metrizable sequential topological gyrogroup which has countable $cs^{*}$-character, then it contains an open and closed subgyrogroup which is a submetrizable $k_{\omega}$-space. For a topological space $X$, Chasco, Mart\'{i}n and Tarieladze in \cite[Lemma 1.5]{CMT} showed that if $X$ is sequential, then it is a $k$-space and if $X$ is a Hausdorff $k$-space and its compact subsets are sequential (in particular first countable or metrizable), then $X$ is sequential. Furthermore, it was proved in \cite[Theorem 3.8]{BZX2} that if a topological gyrogroup $G$ has an $\omega^{\omega}$-base, then it has countable $cs^{*}$-character. Therefore, by Theorem \ref{3compact} and these results, we obtain:
\begin{corollary}\label{k-sequential}
If a topological gyrogroup $G$ has an $\omega^{\omega}$-base, then the following conditions are equivalent.
\begin{enumerate}
\item $G$ is a $k$-space;
\item $G$ is sequential;
\item $G$ is metrizable or contains an open submetrizable $k_{\omega}$-subgyrogroup.
\end{enumerate}
\end{corollary}
In \cite[Theorem 3.5]{ZBX}, the authors showed that if $G$ is a sequential topological gyrogroup with an $\omega^{\omega}$-base, then $G$ has the strong Pytkeev property. Therefore, Corollary \ref{k-sequential} poses the following result directly.
\begin{theorem}\label{k-Pytkeev}
Let $G$ be a topological gyrogroup with an $\omega^{\omega}$-base. If $G$ is a $k$-space, then $G$ has the strong Pytkeev property.
\end{theorem}
Since for a Baire topological gyrogroup $G$, $G$ is metrizable if and only if it has the strong Pytkeev property, see \cite[Theorem 3.10]{ZBX}, Theorem \ref{k-Pytkeev} provides the following corollary.
\begin{corollary}
Let $G$ be a Baire topological gyrogroup. Then $G$ is metrizable if and only if $G$ is a $k$-space and has an $\omega^{\omega}$-base.
\end{corollary}
Theorem \ref{k-Pytkeev} also provides the other type of proof about the following result.
\begin{corollary}\cite[Corollary 3.6]{BZX2}
A topological gyrogroup $G$ is metrizable if and only if $G$ has an $\omega^{\omega}$-base and $G$ is also Fr\'echet-Urysohn.
\end{corollary}
\begin{proof}
The necessity is trivial, it suffices to prove the sufficiency.
Suppose that $G$ is a Fr\'echet-Urysohn topological gyrogroup and has an $\omega^{\omega}$-base. By Theorem \ref{k-Pytkeev}, $G$ has the strong Pytkeev property, hence has countable $cs^{*}$-character. It follows from \cite[Corollary 3.6]{BX2022} that every Fr\'echet-Urysohn topological gyrogroups with countable $cs^{*}$-character is metrizable.
\end{proof}
\begin{remark} A topological gyrogroup $G$ with an $\omega^{\omega}$-base has countable $cs^{*}$-character, see \cite[Theorem 3.8]{BZX2}, hence it is natural to consider the following question. If a topological gyrogroup $G$ is of countable $cs^{*}$-character and it is a $k$-space, then is $G$ sequential? Indeed, Shen in \cite[Example 4.5]{Shen2014} showed that there is a non-metrizable $snf$-countable topological group $X$ which is a $k$-space. Clearly, $X$ is not sequential. Furthermore, this example gives a negative answer to the question whether the $k$-property and sequentiality are equivalent for topological groups with countable $cs^{*}$-character posed in \cite{GKL} under Corollary 3.13.
\end{remark}
Then, we introduce the concept of strongly topological gyrogroups, which was first posed by Bao and Lin in \cite{BL}, and we investigate some weakly first-countable properties in feathered strongly topological gyrogroups.
\begin{definition}{\rm (\cite{BL})}\label{d11}
Let $G$ be a topological gyrogroup. We say that $G$ is a {\it strongly topological gyrogroup} if there exists a neighborhood base $\mathscr U$ of $0$ such that, for every $U\in \mathscr U$, $\mbox{gyr}[x, y](U)=U$ for any $x, y\in G$. For convenience, we say that $G$ is a strongly topological gyrogroup with neighborhood base $\mathscr U$ of $0$.
\end{definition}
For each $U\in \mathscr U$, we can set $V=U\cup (\ominus U)$. Then, $$\mbox{gyr}[x,y](V)=\mbox{gyr}[x, y](U\cup (\ominus U))=\mbox{gyr}[x, y](U)\cup (\ominus \mbox{gyr}[x, y](U))=U\cup (\ominus U)=V,$$ for all $x, y\in G$. Obviously, the family $\{U\cup(\ominus U): U\in \mathscr U\}$ is also a neighborhood base of $0$. Therefore, we may assume that $U$ is symmetric for each $U\in\mathscr U$ in Definition~\ref{d11}. Moreover, in the classical M\"{o}bius, Einstein, or Proper Velocity gyrogroups, we know that gyrations are indeed special rotations, however for an arbitrary gyrogroup, gyrations belong to the automorphism group of $G$ and need not be necessarily rotations.
In \cite{BL}, the authors proved that there is a strongly topological gyrogroup which is not a topological group.
\begin{example}\cite[Example 3.1]{BL}
The M\"{o}bius gyrogroup with the standard topology is a strongly topological gyrogroup but not a topological group.
\end{example}
A topological gyrogroup $G$ is {\it feathered} if it contains a non-empty compact set $K$ of countable character in $G$. It was proved in \cite[3.1 E(b) and 3.3 H(a)]{E} that every locally compact topological gyrogroup is feathered. Moreover, by \cite[Theorem 3.14]{BL}, we know that a strongly topological gyrogroup $G$ is feathered if and only if it contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is metrizable.
\begin{theorem}\label{feathered}
Let $G$ be a feathered strongly topological gyrogroup. Then $G$ has an $\omega^{\omega}$-base if and only if $G$ is metrizable.
\end{theorem}
\begin{proof}
Suppose that $G$ is a strongly topological gyrogroup and has an $\omega^{\omega}$-base. Then $G$ contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is metrizable. It follows from Theorem \ref{3compact} that the subgyrogroup $H$ is metrizable. Since each compact subset of a Hausdorff space is closed, it is clear that $H$ is a closed $L$-subgyrogroup of $G$. Then, by \cite[Corollary 4.3]{BZX}, if $G$ is a topological gyrogroup and $H$ is a closed $L$-subgyrogroup of $G$ and if the spaces $H$ and $G/H$ are metrizable, then the space $G$ is also metrizable. Therefore, we obtain that $G$ is a metrizable space.
\end{proof}
\begin{corollary}
Let $G$ be a locally compact strongly topological gyrogroup. Then $G$ has an $\omega^{\omega}$-base if and only if $G$ is metrizable.
\end{corollary}
Since every topological gyrogroup with an $\omega^{\omega}$-base has countable $cs^{*}$-character, it is natural to pose the following question.
\begin{question}\label{question-cs}
Let $G$ be a feathered strongly topological gyrogroup with countable $cs^{*}$-character. Is $G$ metrizable?
\end{question}
Then, we give a affirmative answer to Question \ref{question-cs}. We note that Uspenski\v\i \cite{UVV, UVV89} proved that compact rectifiable spaces are dyadic. Since every topological gyrogroup is a rectifiable space, it is trivial that each compact topological gyrogroup is dyadic. Moreover, Banakh and Zdomskyy in \cite[Proposition 7]{BZ} claimed that a dyadic compactum is metrizable if and only if it has countable $cs^{*}$-character.
\begin{proposition}
Every compact topological gyrogroup with countable $cs^{*}$-character is metrizable.
\end{proposition}
\begin{theorem}\label{csf-feath}
Every feathered strongly topological gyrogroup with countable $cs^{*}$-character is metrizable.
\end{theorem}
\begin{proof}
Since $G$ is a feathered strongly topological gyrogroup, there exists a compact $L$-subgyrogroup $H$ of $G$ such that the quotient space $G/H$ is metrizable. By the hypothesis, $G$ has countable $cs^{*}$-character, then $H$ also has countable $cs^{*}$-character, which deduces that the compact subgyrogroup $H$ with countable $cs^{*}$-character is metrizable, and it follows from \cite[Corollary 4.3]{BZX} that $G$ is metrizable.
\end{proof}
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\begin{corollary}
Let $G$ be a locally compact strongly topological gyrogroup. Then $G$ has an $\omega^{\omega}$-base if and only if $G$ is metrizable.
\end{corollary}
Since every topological gyrogroup with an $\omega^{\omega}$-base has countable $cs^{*}$-character, it is natural to pose the following question.
\begin{question}\label{question-cs}
Let $G$ be a feathered strongly topological gyrogroup with countable $cs^{*}$-character. Is $G$ metrizable?
\end{question}
Then, we give a affirmative answer to Question \ref{question-cs}. We note that Uspenski\v\i \cite{UVV, UVV89} proved that compact rectifiable spaces are dyadic. Since every topological gyrogroup is a rectifiable space, it is trivial that each compact topological gyrogroup is dyadic. Moreover, Banakh and Zdomskyy in \cite[Proposition 7]{BZ} claimed that a dyadic compactum is metrizable if and only if it has countable $cs^{*}$-character.
\begin{proposition}
Every compact topological gyrogroup with countable $cs^{*}$-character is metrizable.
\end{proposition}
\begin{theorem}\label{csf-feath}
Every feathered strongly topological gyrogroup with countable $cs^{*}$-character is metrizable.
\end{theorem}
\begin{proof}
Since $G$ is a feathered strongly topological gyrogroup, there exists a compact $L$-subgyrogroup $H$ of $G$ such that the quotient space $G/H$ is metrizable. By the hypothesis, $G$ has countable $cs^{*}$-character, then $H$ also has countable $cs^{*}$-character, which deduces that the compact subgyrogroup $H$ with countable $cs^{*}$-character is metrizable, and it follows from \cite[Corollary 4.3]{BZX} that $G$ is metrizable.
\end{proof}
\begin{corollary}
Every locally compact strongly topological gyrogroup with countable $cs^{*}$-character is metrizable.
\end{corollary}
\begin{theorem}
Let $G$ be a strongly topological gyrogroup. Then the following conditions are equivalent:
\begin{enumerate}
\item $G$ is metrizable;
\item $G$ is Fr\'echet-Urysohn and has countable $cs^{*}$-character;
\item $G$ is Fr\'echet-Urysohn and has an $\omega^{\omega}$-base;
\item $G$ is feathered and has countable $cs^{*}$-character;
\item $G$ is feathered and has an $\omega^{\omega}$-base.
\end{enumerate}
\end{theorem}
In the research of rectifiable spaces, Banakh and Repov\v s \cite[Lemma 5.1]{Banakh} showed that suppose that $G$ is a topological lop and $F\subseteq G$ is a subset containing the unit $e$ of $G$, then put $F_{1}=F$ and $F_{n+1}=F_{n}^{-1}F_{n}$ for $n\in \mathbb{N}$. If $F$ is a sequential space containing no closed topological copy of the Fr\'echet-Urysohn fan $S_{\omega}$ and each space $F_{n}$, $n\in \mathbb{N}$, has a countable $cs^{*}$-network at $e$, then $F$ has a countable $sb$-network at $e$; if $F$ is sequential and each space $F_{n},n\in \mathbb{N}$, has countable $sb$-network at $e$, then $F$ is first-countable at $e$. Then, Shen \cite[Proposition 2.6 and Theorem 2.7]{Shen2020} showed that every paratopological left-loop with $sb$-network is $sof$-countable and a sequential, regular paratopological left-loop $G$ with countable $cs^{*}$-network is first-countable if and only if $G$ contains no closed copy of $S_{\omega}$. These results in both of two articles can obtain the following results immediately.
\begin{proposition}\label{csf-snf}
If $G$ is a sequential topological gyrogroup with countable $cs^{*}$-network containing no closed copy of $S_{\omega}$, then $G$ has countable $sb$-network.
\end{proposition}
\begin{proposition}\label{snf-sof}
Every topological gyrogroup with countable $sb$-network is $sof$-countable.
\end{proposition}
\begin{theorem}
A strongly topological gyrogroup $G$ is metrizable if and only if $G$ is a $k$-space of countable pseudocharacter with countable $sb$-network.
\end{theorem}
\begin{proof}
The necessity is trivial, it suffices to claim the sufficiency.
Let a strongly topological gyrogroup $G$ be a $k$-space of countable pseudocharacter with countable $sb$-network. It follows from \cite[Theorem 4.3]{BL1} that every strongly topological gyrogroup with countable pseudocharacter is submetrizable. We obtain that every compact subset of $G$ is metrizable. Since $G$ is a $k$-space, it is easy to see that $G$ is sequential. Indeed, a space $X$ is first-countable if and only if $X$ is sequential and $sof$-countable, which deduces that $G$ is first-countable by Proposition \ref{snf-sof}, hence $G$ is metrizable.
\end{proof}
Recall that a continuous mapping $q:G\rightarrow H$ is called {\it compact-covering} if for every compact subset $K$ of $H$ there exists a compact subset $C$ of $G$ such that $q(C)=K$. Indeed, it was claimed in \cite[Theorem 3.8]{BL} that if $G$ is a topological gyrogroup and $H$ is a compact L-subgyrogroup of $G$, then the quotient mapping $\pi$ of $G$ onto the quotient space $G/H$ is perfect. Furthermore, if $f:X\rightarrow Y$ is a perfect mapping, then for every compact subspace $Z\subseteq Y$, the inverse image $f^{-1}(Z)$ is compact by \cite[Theorem 3.7.2]{E}. Therefore, if $H$ is a compact $L$-subgyrogroup of a topological gyrogroup $G$, then the quotient mapping $\pi$ of $G$ onto the quotient space $G/H$ is a compact covering mapping.
\begin{theorem}\label{Polish}
If $G$ is a feathered strongly topological gyrogroup, the followings are equivalent.
\begin{enumerate}
\item $G$ has a compact resolution swallowing the compact sets of $G$;
\item $G$ has a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is a Polish space.
\end{enumerate}
If (1) holds, then $G$ is $\check{C}$ech-complete.
\end{theorem}
\begin{proof}
(1)$\Rightarrow$(2). Let $\pi$ be the natural homomorphism from $G$ to its quotient topology on $G/H$. Since $G$ is a feathered strongly topological gyrogroup, it follows from \cite[Lemma 3.14]{BL} that $G$ contains a compact $L$-subgyrogroup $H$ such that $G/H$ is metrizable. Let $G$ have a compact resolution swallowing the compact sets of $G$, say $\mathcal{K}=\{K_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$. Then put $\mathcal{K}'=\{\pi (K_{\alpha}):\alpha\in \mathbb{N}^{\mathbb{N}}\}$. Then $\mathcal{K}'$ swallows the compact subsets of $G/H$. Indeed, if $K'$ is compact in $G/H$, then $\pi^{-1}(K')$ is compact in $G$. So there exists $\alpha\in \mathbb{N}^{\mathbb{N}}$ such that $\pi^{-1}(K')\subseteq K_{\alpha}$ and hence $K'\subseteq \pi (K_{\alpha})$. We know that $G/H$ is a Polish space by \cite[Theorem 3.3]{CRJP}.
(2)$\Rightarrow$(1). Since the space $G/H$ is a Polish space, it follows from \cite[Theorem 3.3]{CRJP} that $G/H$ has a compact resolution swallowing the compact sets of $G/H$, say $\mathcal{K}'=\{K'_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$. For every $\alpha\in \mathbb{N}^{\mathbb{N}}$, put $K_{\alpha}=\pi^{-1}(K'_{\alpha})$. Then $K_{\alpha}$ is a compact subset of $G$. Hence, $\mathcal{K}=\{K_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$ is a compact resolution. Let $C$ be a compact subset of $G$. Then there exists $\alpha\in \mathbb{N}^{\mathbb{N}}$ such that $\pi (C)\subseteq K'_{\alpha}$. Therefore, $C\subseteq K_{\alpha}$, and hence $\mathcal{K}$ swallows the compact sets of $G$. We conclude that $G$ has a compact resolution swallowing the compact sets of $G$.
By \cite[Theorem 3.17]{BL}, a strongly topological gyrogroup $G$ is $\check{C}$ech-complete if and only if it contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is metrizable by a complete metric. Since each Polish space is a complete metric space, we know that $G$ is $\check{C}$ech-complete.
\end{proof}
\begin{proposition}
Let $G$ be a topological gyrogroup and have a compact resolution swallowing the compact sets of $G$. If $q:G\rightarrow H$ is a quotient compact-covering map, then $H$ also has a compact resolution swallowing the compact sets of $H$.
\end{proposition}
\begin{proof}
Indeed, the result is trivial. If $\{K_{\alpha}:\alpha\in \mathbb{N}^{\mathbb{N}}\}$ is a compact resolution swallowing the compact sets of $G$, it is clear that $\{q(K_{\alpha}):\alpha\in \mathbb{N}^{\mathbb{N}}\}$ is a compact resolution swallowing the compact sets of $H$.
\end{proof}
\begin{proposition}
Let $G$ be a topological gyrogroup and a $k$-space. If $G$ has an $\omega^{\omega}$-base and also has a compact resolution swallowing the compact sets of $G$, then $G$ is either a Polish space or contains a submetrizable $k_{\omega}$-subgyrogroup.
\end{proposition}
\begin{proof}
Since topological gyrogroup $G$ is a $k$-space and has an $\omega^{\omega}$-base, we have that $G$ is metrizable or contains an open submetrizable $k_{\omega}$-subgyrogroup. Moreover, by \cite[Theorem 3.3]{CRJP}, in a metrizable space $X$, $X$ is a Polish space if and only if $X$ has a compact resolution swallowing the compact sets of $X$. Therefore, if $G$ is metrizable, we know that $G$ is a Polish space.
\end{proof}
In Theorems \ref{feathered}, \ref{csf-feath} and \ref{Polish}, it is clear that the characterization of feathered strongly topological gyrogroup playes an important role in the proof. However, we do not know whether the characterization of feathered holds in topological gyrogroups. If it holds in topological gyrogroups, all of Theorems \ref{feathered}, \ref{csf-feath} and \ref{Polish} can be extended to topological gyrogroups immediately.
\begin{question}
If $G$ is a feathered topological gyrogroup, is there a compact $L$-subgyrogroup of $G$ such that the quotient space $G/H$ metrizable?
\end{question}
A space $X$ is called {\it hemicompact} if $X=\bigcup_{n\in \mathbb{N}}X_{n}$, where $X_{n}$ is compact for any $n\in \mathbb{N}$ and for any compact $K\subseteq X$, there is $n\in \mathbb{N}$ such that $K\subseteq X_{n}$.
\begin{corollary}
Let $G$ be a locally compact strongly topological gyrogroup. Then $G$ has a compact resolution swallowing the compact sets of $G$ if and only if $G$ is hemicompact space.
\end{corollary}
\begin{proof}
Since each hemicompact topological space has a compact resolution swallowing the compact sets, it suffices to claim the necessity.
Suppose that $G$ has a compact resolution swallowing the compact sets of $G$. Since each locally compact topological gyrogroup is feathered, by Theorem \ref{Polish}, $G$ contains a compact $L$-subgyrogroup $H$ such that the locally compact space $G/H$ is second countable. Therefore, $G/H$ is hemicompact, and $G$ is also hemicompact.
\end{proof}
\begin{proposition}
Every Fr\'echet-Urysohn hemicompact topological gyrogroup is locally compact.
\end{proposition}
\begin{proof}
Let $G=\bigcup_{n\in \mathbb{N}}K_{n}$, where $\{K_{n}\}_{n}$ is an increasing sequence of compact subsets of $K$ containing the identity element $0$ such that every compact set in $G$ is contained in some $K_{n}$. Then we can find $n\in \mathbb{N}$ such that $K_{n}$ is a neighborhood of $0$. Suppose on the contrary that there is no $n$ such that $K_{n}$ is a neighborhood of $0$ for each $n\in \mathbb{N}$. Then for each $n\in \mathbb{N}$ and each neighborhood $U$ of $0$, there exists $x_{U,n}\in U\setminus K_{n}$. For each $n\in \mathbb{N}$, set $B_{n}=\{x_{U,n}:U\mbox{ is an open neighborhood of 0}\}$. Then $0\in \overline{B_{n}}$. Since $G$ is Fr\'echet-Urysohn, for each $n\in \mathbb{N}$, we can find an open neighborhood sequence $\{U_{n}(k)\}_{k}$ of $0$ such that $x_{U_{n}(k),n}\rightarrow 0$ at $k\rightarrow \infty$. Since every Fr\'echet-Urysohn topological gyrogroup is a strong $\alpha_{4}$-space by \cite[Lemma 3.3]{BZX2}, there exists strictly increasing sequences $(k_{p})_{p}$ and $(n_{p})_{p}$ such that $x_{U_{n_{p}}(k_{p}),n_{p}}\rightarrow 0$ at $p\rightarrow \infty$. Since the set $B=\{x_{U_{n_{p}}(k_{p}),n_{p}}:p\in \mathbb{N}\}\cup \{0\}$ is compact in $G$, we can find $m\in \mathbb{N}$ such that $B\subseteq K_{m}$, which is a contradiction . Therefore, $G$ is locally compact.
\end{proof}
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\section{metrizability of strongly topological gyrogroups}
In \cite{AW}, Atiponrat posed a question that is it true that the first-countability axiom implies that $G$ is metrizable for a topological gyrogroup $G$? Then Cai, Lin and He showed that every topological gyrogroup is a rectifiable space, which deduces that every first-countable (strongly) topological gyrogroup is metrizable. Indeed, Alexandra S. Gul'ko \cite[Theorem 3.2]{Alexan} proved that every first-countable $T_{0}$ rectifiable space is metrizable by the tool of strong development. Here, we give a direct construction to show that every first-countable strongly topological gyrogroup is metrizable.
\begin{lemma}\label{3.3.10}\cite[Lemma 3.12]{BL}
Let $G$ be a strongly topological gyrogroup with the symmetric neighborhood base $\mathscr{U}$ at $0$, and let $\{U_{n}: n\in\mathbb{N}\}$ and $\{V(m/2^{n}): n, m\in\mathbb{N}\}$ be two sequences of open neighborhoods satisfying the following conditions (1)-(5):
(1) $U_{n}\in\mathscr{U}$ for each $n\in \mathbb{N}$.
(2) $U_{n+1}\oplus U_{n+1}\subseteq U_{n}$, for each $n\in \mathbb{N}$.
(3) $V(1)=U_{0}$;
(4) For any $n\geq 1$, put $$V(1/2^{n})=U_{n}, V(2m/2^{n})=V(m/2^{n-1})$$ for $m=1,...,2^{n-1}$, and $$V((2m+1)/2^{n})=U_{n}\oplus V(m/2^{n-1})=V(1/2^{n})\oplus V(m/2^{n-1})$$ for each $m=1,...,2^{n-1}-1$;
(5) $V(m/2^{n})=G$ when $m>2^{n}$;
Then there exists a prenorm $N$ on $G$ that satisfies the following conditions:
(a) for any fixed $x, y\in G$, we have $N(\mbox{gyr}[x,y](z))=N(z)$ for any $z\in G$;
(b) for any $n\in \mathbb{N}$, $$\{x\in G: N(x)<1/2^{n}\}\subseteq U_{n}\subseteq\{x\in G: N(x)\leq 2/2^{n}\}.$$
\end{lemma}
\begin{theorem}\label{new-metric}
Every first-countable strongly topological gyrogroup is metrizable.
\end{theorem}
\begin{proof}
Let $G$ be a strongly topological gyrogroup with a symmetric neighborhood base $\mathscr U$. Since $G$ is first-countable, put $\{W_{n}:n\in \mathbb{N}\}$ a countable base at the identity element $0$. By induction, we obtain a sequence $\{U_{n}:n\in \mathbb{N}\}\subseteq \mathscr{U}$ such that $U_{n}\subseteq W_{n}$ and $U_{n+1}\oplus U_{n+1}\subseteq U_{n}$, for each $n\in \mathbb{N}$. It is easy to see that $\{U_{n}:n\in \mathbb{N}\}$ is also a base of $G$ at $0$. By Lemma \ref{3.3.10}, there exists a continuous prenorm $N$ on $G$ which satisfies $$N(\mbox{gyr}[x, y](z))=N(z)$$ for any $x, y, z\in G$ and $$\{x\in G: N(x)<1/2^{n}\}\subseteq U_{n}\subseteq \{x\in G: N(x)\leq 2/2^{n}\},$$ for each integer $n\geq 0$. Put $B_{N}(\varepsilon)=\{x\in G:N(x)<\varepsilon\}$, where $\varepsilon$ is a positive number. It is easy to see that $B_{N}(1/2^{n})$ also forms a base of $G$ at $0$.
Now, for arbitrary $x$ and $y$ in $G$, put $\varrho _{N}(x, y)=N(\ominus x\oplus y)$. Let us show that $\varrho _{N}$ is a metric on $G$.
(1) Clearly, $\varrho _{N}(x, y)=N(\ominus x\oplus y)\geq 0$, for every $x, y\in G$. At the same time, $\varrho _{N}(x, x)=N(0)=0$, for each $x\in G$. Assume that $$\varrho _{N}(x, y)=N(\ominus x\oplus y)=0.$$ Then, for each $n\in\mathbb{N}$, $$\ominus x\oplus y\in \{x\in G: N(x)<1/2^{n}\}\subseteq U_{n}.$$ Since $\{0\}=\bigcap _{n\in \mathbb{N}}U_{n}$, it follows that $\ominus x\oplus y=0$, that is, $x=y$.
(2) For every $x, y\in G$, $\varrho _{N}(y, x)=N(\ominus y\oplus x)=N(gyr[\ominus y,x](\ominus x\oplus y))=N(\ominus x\oplus y)=\varrho _{N}(x, y)$.
(3) For every $x, y, z\in G$, it follows from \cite[Theorem 2.11]{UA2005} that
\begin{eqnarray}
\varrho _{N}(x, y)&=&N(\ominus x\oplus y)\nonumber\\
&=&N((\ominus x\oplus z)\oplus \mbox{gyr}[\ominus x, z](\ominus z\oplus y))\nonumber\\
&\leq&N(\ominus x\oplus z)+N(\mbox{gyr}[\ominus x, z](\ominus z\oplus y))\nonumber\\
&=&N(\ominus x\oplus z)+N(\ominus z\oplus y)\nonumber\\
&=&\varrho _{N}(x, z)+\varrho _{N}(z, y)\nonumber
\end{eqnarray}
Thus, $\varrho _{N}$ is a metric on $G$.
Since $B_{N}(1/2^{n})$ forms a base of $G$ at $0$ and $G$ is homogeneous, for each $x\in G$, $B_{N}(1/2^{n})\oplus x$ constitutes a base of $G$ at $x$. Therefore, it is easy to see that the topology generated by metric $\varrho _{N}$ is coincide with the original topology of $G$. Hence, $G$ is metrizable.
\end{proof}
However, we do not know whether each topological gyrogroup has the similar result like Lemma \ref{3.3.10}, therefore, we can not give the direct construction that every first-countable topological gyrogroup is metrizable. Moreover, we pose the following questions.
\begin{question}
Let $G$ be a metrizable (strongly) topological gyrogroup and $H$ a closed $L$-subgyrogroup of $G$. Is the quotient space $G/H$ metrizable?
\end{question}
\begin{remark}
It was posed a question in \cite{BL} that if $G$ is a (strongly) topological gyrogroup with a countable pseudocharacter, is $G$ submetrizable? then Bao and Lin gave an affirmative answer to this question in \cite[Theorem 4.3]{BL1} by constructing a metric $\varrho _{N}(x, y)=N(\ominus x\oplus y)+N(\ominus y\oplus x)$ when $G$ is a strongly topological gyrogroup. Notice that the proof of Theorem \ref{new-metric} can be applied to show that every strongly topological gyrogroup with a countable pseudocharacter is submetrizable, that is, the metric in \cite[Theorem 4.3]{BL1} can be replaced by $\varrho _{N}(x, y)=N(\ominus x\oplus y)$.
\end{remark}
A subgyrogroup $H$ of a topological gyrogroup $G$ is called {\it inner (outer) neutral} if for every open neighborhood $U$ of $0$ in $G$, there exists an open neighborhood $V$ of $0$ such that $H\oplus V\subseteq U\oplus H$ ($V\oplus H\subseteq H\oplus U$).
\begin{question}
Let $G$ be a (strongly) topological gyrogroup and $H$ a closed inner neutral $L$-subgyrogroup of $G$. If the quotient space $G/H$ is first-countable, is it metrizable?
\end{question}
\begin{question}
Let $G$ be a feathered (strongly) topological gyrogroup and $H$ a closed $L$-subgyrogroup of $G$. If the quotient space $G/H$ is first-countable, is it metrizable?
\end{question}
\end{document}
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