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\section{Preliminaries and notation}\label{prels}
\subsection{}
Let $\mathcal{C}$ be a fusion category over $k$. A (\emph{right}) ${\mathcal C}$-\emph{module category} is a f\/inite semisimple $k$-linear abelian category $\mathcal{M}$ equipped with a bifunctor $\bar\otimes\colon \mathcal{M}\times \mathcal{C} \rightarrow\mathcal{M}$ and natural isomorphisms
\begin{gather*}\mu_{M, X,Y}\colon \ M \bar\otimes (X\otimes Y)\rightarrow (M\bar\otimes X)\bar\otimes Y,\qquad r_M\colon \ M \bar\otimes \textbf{1}\rightarrow M, \end{gather*}
$X, Y \in {\mathcal C}$, $M \in \mathcal M$, satisfying the following conditions:
\begin{gather} \label{modcat1} \mu_{M\bar\otimes X, Y, Z} \mu_{M, X, Y \otimes Z} ( \id_M \bar\otimes a_{X, Y, Z} ) =
(\mu_{M, X, Y} \otimes \id_Z ) \mu_{M, X\otimes Y, Z},\\
\label{modcat2} (r_M \otimes \id_Y ) \mu_{M, \textbf{1}, Y} = \id_M \bar\otimes l_Y,
\end{gather}
for all $M \in {\mathcal M}$, $X, Y \in {\mathcal C}$, where $a\colon \otimes \circ (\otimes \times \id_{\mathcal C}) \to \otimes \circ (\id_{\mathcal C} \times \otimes)$ and $l\colon \textbf{1} \otimes ? \to \id_{\mathcal C}$, denote the associativity and left unit constraints in ${\mathcal C}$, respectively.
Let $A$ be an algebra in ${\mathcal C}$. Then the category $_A{\mathcal C}$ of left $A$-modules in ${\mathcal C}$ is a right ${\mathcal C}$-module category with action $\bar\otimes\colon _A{\mathcal C} \times {\mathcal C} \to {\mathcal C}_A$, given by $M\bar\otimes X = M\otimes X$ endowed with the left $A$-module structure $(m_M \otimes \id_X) a_{A, M, X}^{-1}\colon A\otimes (M\otimes X) \to M\otimes X$, where $m_M\colon A\otimes M \to M$ is the $A$-module structure in $M$. The associativity constraint of $_A{\mathcal C}$ is given by $a^{-1}_{M, X, Y}\colon M\bar\otimes (X \otimes Y) \to (M\bar\otimes X) \bar\otimes Y$, for all $M \in {} _A{\mathcal C}$, $X, Y \in {\mathcal C}$.
A ${\mathcal C}$-module functor ${\mathcal M} \to {\mathcal M}'$ between right ${\mathcal C}$-module categories $({\mathcal M}, \bar\otimes)$ and $({\mathcal M}', \bar\otimes')$ is a pair $(F, \zeta)$, where $F\colon {\mathcal M}\to {\mathcal M}'$ is a functor and $\zeta_{M, X}\colon F(M \bar\otimes X) \to F(M) \bar\otimes' X$ is a natural isomorphism satisfying
\begin{gather}\label{uno-z} (\zeta_{M, X} \otimes \id_Y ) \zeta_{M \bar\otimes X, Y} F(\mu_{M, X, Y}) = {\mu'}_{F(M), X, Y} \zeta_{M, X \otimes Y},\\
\label{dos-z} {r'}_{F(M)} \zeta_{M, \textbf{1}} = F(r_M),
\end{gather}
for all $M \in {\mathcal M}$, $X, Y \in {\mathcal C}$.
Let ${\mathcal M}$ and ${\mathcal M}'$ be ${\mathcal C}$-module categories. An \emph{equivalence} of ${\mathcal C}$-module categories ${\mathcal M} \to {\mathcal M}'$ is a ${\mathcal C}$-module functor $(F, \zeta)\colon {\mathcal M}\to {\mathcal M}'$ such that $F$ is an equivalence of categories. If such an equivalence exists, ${\mathcal M}$ and ${\mathcal M}'$ are called \emph{equivalent ${\mathcal C}$-module categories}. A~${\mathcal C}$-module category is called \emph{indecomposable} if it is not equivalent to a direct sum of two nontrivial $\mathcal{C}$-submodule categories.
Let ${\mathcal M}, {\mathcal M}'$ be indecomposable ${\mathcal C}$-module categories. Then $\operatorname{Fun}_{\mathcal C}({\mathcal M}, {\mathcal M})$ is a fusion category with tensor product given by composition of functors and the category $\operatorname{Fun}_{\mathcal C}({\mathcal M}, {\mathcal M}')$ is an indecomposable module category over $\operatorname{Fun}_{\mathcal C}({\mathcal M}, {\mathcal M})$ in a natural way. If $A$ and $B$ are indecomposable algebras in ${\mathcal C}$ such that ${\mathcal M} \cong _A\!{\mathcal C}$ and ${\mathcal M}' \cong {}_B{\mathcal C}$, then $\operatorname{Fun}_{\mathcal C}({\mathcal M}, {\mathcal M})^{\rm op}$ is equivalent to the fusion category $_A{\mathcal C}_A$ of $(A, A)$-bimodules in ${\mathcal C}$ and there is an equivalence of $_A{\mathcal C}_A$-module categories $_B{\mathcal C}_A \cong \operatorname{Fun}_{\mathcal C}({\mathcal M}, {\mathcal M}')$, where $_B{\mathcal C}_A$ is the category of $(B, A)$-bimodules in ${\mathcal C}$.
\subsection{}\label{action-mc} Let ${\mathcal M}$ be a ${\mathcal C}$-module category. Every tensor autoequivalence $\rho\colon {\mathcal C} \to {\mathcal C}$ induces a ${\mathcal C}$-module category structure ${\mathcal M}^\rho$ on ${\mathcal M}$ in the form $M \bar\otimes^\rho X = M\bar\otimes \rho(X)$, with associativity constraint
\begin{gather*}\mu^\rho_{M, X, Y} = \mu_{M, \rho(X) \otimes \rho(Y)} \big(\id_M \bar\otimes {\rho^2_{X, Y}}^{-1}\big)\colon \ M \bar\otimes \rho(X\otimes Y) \to (M\bar\otimes \rho(X)) \bar\otimes \rho(Y),\end{gather*} for all $M \in {\mathcal M}$, $X, Y \in {\mathcal C}$, where $\rho^2_{X, Y}\colon \rho(X) \otimes \rho(Y) \to \rho(X \otimes Y)$ is the monoidal structure of~$\rho$. See \cite[Section~3.2]{nikshych}.
Suppose that $A$ is an algebra in ${\mathcal C}$. Then $\rho(A)$ is an algebra in ${\mathcal C}$ with multiplication
\begin{gather*}m_{\rho(A)} = \rho(m_A) \rho^2_{A, A}\colon \ \rho(A) \otimes \rho(A) \to \rho(A).\end{gather*}
The functor $\rho$ induces an equivalence of ${\mathcal C}$-module categories $_{\rho(A)}{\mathcal C} \to (_A{\mathcal C})^\rho$ with intertwining isomorphisms
\begin{gather*} {\rho^2_{M, X}}^{-1}\colon \ \rho(M \bar\otimes X) \to \rho(M) \bar\otimes^\rho X.\end{gather*}
\subsection{}\label{cgomega} Let $G$ be a f\/inite group.
Let $X$ be a $G$-module. Given an $n$-cochain $f \in C^n(G, X)$ (where $C^0(G, M) = M$), the coboundary of $f$ is the $(n+1)$-cochain $df = d^nf \in C^{n+1}(G, X)$ def\/ined by
\begin{gather*}d^nf(g_1, \dots, g_{n+1}) = g_1.f(g_2, \dots, g_{n+1}) + \sum_{i = 1}^{n}f(g_1, \dots, g_ig_{i+1}, \dots, g_n)\\
\hphantom{d^nf(g_1, \dots, g_{n+1}) =}{} + (-1)^{n+1} f(g_1, \dots, g_n),\end{gather*}
for all $g_1, \dots, g_{n+1} \in G$. The kernel of $d^n$ is denoted $Z^n(G, M)$; an element of $Z^n(G, M)$ is an $n$-cocycle. We have $d^nd^{n-1} = 0$, for all $n \geq 1$. The $n$th cohomology group of $G$ with coef\/f\/icients in $M$ is $H^n(G, M) = Z^n(G, M)/d^{n-1}(C^{n-1}(G, M))$. We shall write $f \equiv f'$ when the cochains $f, f' \in C^n(G, k^\times)$ dif\/fer by a~coboundary.
We shall assume that every cochain $f$ is \emph{normalized}, that is, $f(g_1, \dots, g_n) = 1$, whenever one of the arguments $g_1, \dots, g_n$ is the identity. If $H$ is a subgroup of $G$ and $f \in C^n(H, k^\times)$, we shall indicate by~$f^g$ the $n$-cochain in $^{g^{-1}\!}H$ given by $f^g(h_1, \dots, h_n) = f({}^g h_1, \dots, {}^g h_n)$, $h_1, \dots, h_n \in H$.
Let $\omega \colon G \times G \times G \to k^\times$ be a 3-cocycle on $G$. Let ${\mathcal C}(G, \omega)$ denote the fusion category of f\/inite-dimensional $G$-graded vector spaces with associativity constraint def\/ined, for all $U, V, W\in {\mathcal C}(G, \omega)$, as
\begin{gather*}a_{X, Y, Z} ((u \otimes v)\otimes w) = \omega^{-1}(g_1, g_2, g_3) u \otimes (v\otimes w),\end{gather*}
for all homogeneous vectors $u \in U_{g_1}$, $v \in V_{g_2}$, $w \in W_{g_3}$, $g_1, g_2, g_3 \in G$. Any pointed fusion category is equivalent to a~category of the form ${\mathcal C}(G, \omega)$.
A fusion category ${\mathcal C}$ is called \emph{group-theoretical} if it is categorically Morita equivalent to a~pointed fusion category. Equivalently, ${\mathcal C}$ is group-theoretical if and only if there exist a f\/inite group $G$ and a $3$-cocycle $\omega\colon G \times G \times G \to k^{\times}$ such that~${\mathcal C}$ is equivalent to the fusion category ${\mathcal C}(G, \omega, H, \psi) = _{A(H, \psi)\!}{\mathcal C}(G, \omega)_{A(H, \psi)}$, where $H$ is a subgroup of $G$ such that the class of $\omega\vert_{H\times H \times H}$ is trivial and $\psi\colon H \times H \to k^{\times}$ is a 2-cochain on~$H$ satisfying condition~\eqref{cond-alfa}.
Let ${\mathcal C}(G, \omega, H, \psi) \cong {\mathcal C}(G, \omega)^*_{{\mathcal M}_0(H, \psi)}$ be a group-theoretical fusion category. Then there is a bijective correspondence between equivalence classes of indecomposable ${\mathcal C}(G, \omega, H, \psi)$-module categories and equivalence classes of indecomposable ${\mathcal C}(G, \omega)$-module categories. This correspondence attaches to every indecomposable ${\mathcal C}(G, \omega)$-module category ${\mathcal M}$ the ${\mathcal C}(G, \omega, H, \psi)$-module category
\begin{gather*}{\mathcal M}(H, \psi) = {\mathbb F}un_{{\mathcal C}(G, \omega)}({\mathcal M}_0(H, \psi), {\mathcal M}). \end{gather*}
\section[Indecomposable module categories over ${\mathcal C}(G, \omega)$]{Indecomposable module categories over $\boldsymbol{{\mathcal C}(G, \omega)}$}\label{ptd-mc}
Throughout this section $G$ is a f\/inite group and $\omega \colon G \times G \times G \to k^\times$ is a 3-cocycle on $G$.
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\subsection{}\label{cgomega} Let $G$ be a f\/inite group.
Let $X$ be a $G$-module. Given an $n$-cochain $f \in C^n(G, X)$ (where $C^0(G, M) = M$), the coboundary of $f$ is the $(n+1)$-cochain $df = d^nf \in C^{n+1}(G, X)$ def\/ined by
\begin{gather*}d^nf(g_1, \dots, g_{n+1}) = g_1.f(g_2, \dots, g_{n+1}) + \sum_{i = 1}^{n}f(g_1, \dots, g_ig_{i+1}, \dots, g_n)\\
\hphantom{d^nf(g_1, \dots, g_{n+1}) =}{} + (-1)^{n+1} f(g_1, \dots, g_n),\end{gather*}
for all $g_1, \dots, g_{n+1} \in G$. The kernel of $d^n$ is denoted $Z^n(G, M)$; an element of $Z^n(G, M)$ is an $n$-cocycle. We have $d^nd^{n-1} = 0$, for all $n \geq 1$. The $n$th cohomology group of $G$ with coef\/f\/icients in $M$ is $H^n(G, M) = Z^n(G, M)/d^{n-1}(C^{n-1}(G, M))$. We shall write $f \equiv f'$ when the cochains $f, f' \in C^n(G, k^\times)$ dif\/fer by a~coboundary.
We shall assume that every cochain $f$ is \emph{normalized}, that is, $f(g_1, \dots, g_n) = 1$, whenever one of the arguments $g_1, \dots, g_n$ is the identity. If $H$ is a subgroup of $G$ and $f \in C^n(H, k^\times)$, we shall indicate by~$f^g$ the $n$-cochain in $^{g^{-1}\!}H$ given by $f^g(h_1, \dots, h_n) = f({}^g h_1, \dots, {}^g h_n)$, $h_1, \dots, h_n \in H$.
Let $\omega \colon G \times G \times G \to k^\times$ be a 3-cocycle on $G$. Let ${\mathcal C}(G, \omega)$ denote the fusion category of f\/inite-dimensional $G$-graded vector spaces with associativity constraint def\/ined, for all $U, V, W\in {\mathcal C}(G, \omega)$, as
\begin{gather*}a_{X, Y, Z} ((u \otimes v)\otimes w) = \omega^{-1}(g_1, g_2, g_3) u \otimes (v\otimes w),\end{gather*}
for all homogeneous vectors $u \in U_{g_1}$, $v \in V_{g_2}$, $w \in W_{g_3}$, $g_1, g_2, g_3 \in G$. Any pointed fusion category is equivalent to a~category of the form ${\mathcal C}(G, \omega)$.
A fusion category ${\mathcal C}$ is called \emph{group-theoretical} if it is categorically Morita equivalent to a~pointed fusion category. Equivalently, ${\mathcal C}$ is group-theoretical if and only if there exist a f\/inite group $G$ and a $3$-cocycle $\omega\colon G \times G \times G \to k^{\times}$ such that~${\mathcal C}$ is equivalent to the fusion category ${\mathcal C}(G, \omega, H, \psi) = _{A(H, \psi)\!}{\mathcal C}(G, \omega)_{A(H, \psi)}$, where $H$ is a subgroup of $G$ such that the class of $\omega\vert_{H\times H \times H}$ is trivial and $\psi\colon H \times H \to k^{\times}$ is a 2-cochain on~$H$ satisfying condition~\eqref{cond-alfa}.
Let ${\mathcal C}(G, \omega, H, \psi) \cong {\mathcal C}(G, \omega)^*_{{\mathcal M}_0(H, \psi)}$ be a group-theoretical fusion category. Then there is a bijective correspondence between equivalence classes of indecomposable ${\mathcal C}(G, \omega, H, \psi)$-module categories and equivalence classes of indecomposable ${\mathcal C}(G, \omega)$-module categories. This correspondence attaches to every indecomposable ${\mathcal C}(G, \omega)$-module category ${\mathcal M}$ the ${\mathcal C}(G, \omega, H, \psi)$-module category
\begin{gather*}{\mathcal M}(H, \psi) = {\mathbb F}un_{{\mathcal C}(G, \omega)}({\mathcal M}_0(H, \psi), {\mathcal M}). \end{gather*}
\section[Indecomposable module categories over ${\mathcal C}(G, \omega)$]{Indecomposable module categories over $\boldsymbol{{\mathcal C}(G, \omega)}$}\label{ptd-mc}
Throughout this section $G$ is a f\/inite group and $\omega \colon G \times G \times G \to k^\times$ is a 3-cocycle on $G$.
\subsection{}\label{adj-action} Let $g \in G$. Consider the 2-cochain $\Omega_g\colon G \times G \to k^{\times}$ given by
\begin{gather*}\Omega_g(g_1, g_2) = \frac{\omega({}^gg_1, {}^gg_2, g) \omega(g, g_1, g_2)}{\omega({}^gg_1, g, g_2)}.
\end{gather*}
For all $g \in G$ we have the relation
\begin{gather}\label{o-oy}d\Omega_g = \frac{\omega}{\omega^g}.
\end{gather}
Let ${\mathcal C} = {\mathcal C}(G, \omega)$ and let $g \in G$. For every object $V$ of ${\mathcal C}$ let ${}^g V$ be the object of ${\mathcal C}$ such that $^gV = V$ as a vector space with $G$-grading def\/ined as $({}^gV)_x = V_{^g x}$, $x \in G$. For every $g \in G$, we have a functor $\ad_g \colon {\mathcal C} \to {\mathcal C}$, given by $\ad_g(V) = {}^gV$ and $\ad_g(f) = f$, for every object $V$ and morphism $f$ of ${\mathcal C}$.
Relation~\eqref{o-oy} implies that $\ad_g$ is a tensor functor with monoidal structure def\/ined by
\begin{gather*}\big(\ad_g^2\big)_{U, V}\colon \ {}^gU \otimes {}^gV \to {}^g (U \otimes V), \qquad \big(\ad_g^2\big)_{U, V} (u \otimes v) = \Omega_g(h, h')^{-1} u\otimes v,\end{gather*}
for all $h, h' \in G$, and for all homogeneous vectors $u \in U_h$, $v \in V_{h'}$.
For every $g, g_1, g_2 \in G$, let $\gamma(g_1, g_2)\colon G \to k^\times$ be the map def\/ined in the form
\begin{gather*}
\gamma(g_1, g_2)(g) = \frac{\omega(g_1, g_2, g) \omega({}^{g_1g_2}g, g_1, g_2)}{\omega(g_1,{}^{g_2}g, g_2)}.
\end{gather*}
The following relation holds, for all $g_1, g_2 \in G$:
\begin{gather}\label{rel-omega}
\Omega_{g_1g_2} = \Omega_{g_1}^{g_2} \Omega_{g_2} d \gamma(g_1, g_2) .
\end{gather}
In this way, $\ad\colon \underline G \to \underline{\text{Aut}}_\otimes{\mathcal C}$, $\ad(g) = \big(\ad_g, \ad_g^2\big)$, gives rise to an action by tensor autoequivalences of $G$ on ${\mathcal C}$ where, for every $g, x \in G$, $V \in {\mathcal C}(G, \omega)$, the monoidal isomorphisms ${\ad^2}_V\colon {}^g({}^{g'}V) \to {}^{gg'}V$ are given by
\begin{gather*}\ad^2_V(v) = \gamma(g, g') (x) v,\end{gather*} for all homogeneous vectors $v \in V_x$, $h \in G$. The equivariantization ${\mathcal C}^G$ with respect to this action is equivalent to the category of f\/inite-dimensional representations of the twisted quantum double $D^\omega G$ (see \cite[Lemma~6.3]{naidu}).
For each $g \in G$, and for each ${\mathcal C}$-module category ${\mathcal M}$, let ${\mathcal M}^g$ denote the module category induced by the functor $\ad_g$ as in Section~\ref{action-mc}. Recall that the action of ${\mathcal C}$ on ${\mathcal M}^g$ is def\/ined by $M\bar\otimes^g V = M\bar\otimes ({}^gV)$, for all objects~$V$ of~${\mathcal C}$.
\begin{Lemma}\label{adj-triv} Let $g \in G$ and let ${\mathcal M}$ be a ${\mathcal C}$-module category. Then ${\mathcal M}^g \cong {\mathcal M}$ as ${\mathcal C}$-module categories.
\end{Lemma}
\begin{proof} For each $g \in G$, let $\{g\}$ denote the object of ${\mathcal C}$ such that $\{g\} = k$ with degree $g$. In what follows, by abuse of notation, we identify $\{g\} \otimes \{h\}$ and $\{gh\}$, $g, h \in G$, by means of the canonical isomorphisms of vector spaces.
Let $R_g\colon {\mathcal M}^g \to {\mathcal M}$ be the functor def\/ined by the right action of $\{g\}$: $R_g(M) = M \bar\otimes \{g\}$. Consider the natural isomorphism $\zeta\colon R_g \circ \bar\otimes^g \to \bar\otimes \circ (R_g \times \id_{\mathcal C})$, def\/ined as
\begin{gather*}\zeta_{M, V} = \mu_{M, \{g\}, V} \mu^{-1}_{M, {}^g V, \{g\}}\colon \ R_g(M\bar\otimes^g V) \to R_g(M) \bar\otimes V,
\end{gather*} for all objects $M$ of ${\mathcal M}$ and $V$ of ${\mathcal C}$, where $\mu$ is the associativity constraint of~${\mathcal M}$.
The functor $R_g$ is an equivalence of categories with quasi-inverse given by the functor $R_{g^{-1}}\colon$ ${\mathcal M} \to {\mathcal M}^g$.
A direct calculation, using the coherence conditions~\eqref{modcat1} and~\eqref{modcat2} for the module category~${\mathcal M}$, shows that $\zeta$ satisf\/ies conditions~\eqref{uno-z} and~\eqref{dos-z}. Hence $(R_g, \zeta)$ is a ${\mathcal C}$-module functor. Therefore ${\mathcal M}^g \cong {\mathcal M}$ as ${\mathcal C}$-module categories, as claimed. \end{proof}
\begin{Lemma}\label{gdea} Let $H$ be a subgroup of $G$ and let $\psi$ be a $2$-cochain on $H$ satisfying~\eqref{cond-alfa}. Let $A(H, \psi)$ denote the corresponding indecomposable algebra in~${\mathcal C}$. Then, for all $g \in G$, ${}^gA(H, \psi) \cong A({}^gH, \psi^{g^{-1}} \Omega_{g^{-1}})$ as algebras in~${\mathcal C}$.
\end{Lemma}
\begin{proof} By def\/inition, ${}^gA(H, \psi) = A\big({}^gH, \psi^{g^{-1}} \big(\Omega_{g}^{g^{-1}}\big)^{-1}\big)$. It follows from formula~\eqref{rel-omega} that $\big(\Omega_{g}^{g^{-1}}\big)^{-1}$ and $\Omega_{g^{-1}}$ dif\/fer by a coboundary. This implies the lemma. \end{proof}
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\subsection{} Let $H$, $L$ be subgroups of $G$ and let $\psi \in C^2(H, k^\times)$, $\xi \in C^2(L, k^\times)$, be 2-cochains such that $\omega\vert_{H \times H \times H} = d\psi$ and $\omega\vert_{L \times L \times L} = d\xi$.
Let $B$ be an object of the category $_{A(H, \psi)}{\mathcal C}_{A(L, \xi)}$ of $(A(H, \psi), A(L, \xi))$-bimodules in ${\mathcal C}$. For each $z\in G$, let $\pi_l(h)\colon B_z \to B_{hz}$ and $\pi_r(s)\colon B_z \to B_{zs}$, denote the linear maps induced by the actions of $h \in H$ and $s \in L$, respectively. Then the following relations hold, for all $h, h' \in H$, $s, s' \in L$:
\begin{gather}\label{uno}\pi_l(h)\pi_l(h') = \omega(h, h', z) \psi(h, h') \pi_l(hh'), \\
\label{dos}\pi_r(s')\pi_r(s) = \omega(z, s, s')^{-1} \xi(s, s') \pi_r(ss'),\\
\label{tres}\pi_l(h)\pi_r(s) = \omega(h, z, s) \pi_r(s)\pi_l(h).\end{gather}
\begin{Lemma}\label{alfa-g} Let $g \in G$ and let $B_g$ denote the homogeneous component of degree $g$ of $B$. Then the map $\pi\colon H \cap {}^gL \to \text{\rm GL}(B_g)$, defined as $\pi(x) = \pi_r\big({}^{g^{-1}}x\big)^{-1}\pi_l(x)$ is a projective representation of $H \cap {}^gL$ with cocycle~$\alpha_g$ given, for all $x, y \in H \cap {}^gL$, as follows:
\begin{gather*}
\alpha_g(x, y) = \psi(x, y) \xi^{-1}\big({}^{g^{-1}}x, {}^{g^{-1}}y\big) \frac{\omega(x, y, g) \omega\big(x, yg, {}^{g^{-1}}\big(y^{-1}\big)\big)}
{\omega\big(xyg, {}^{g^{-1}}\big(y^{-1}\big),{} ^{g^{-1}}\big(x^{-1}\big)\big)} du_g(x, y) \\
\hphantom{\alpha_g(x, y) =}{} \times \frac{\omega\big({}^{g^{-1} } y, {}^{g^{-1}}\big(y^{-1}\big), {}^{g^{-1}}\big(x^{-1}\big)\big)} {\omega\big({}^{g^{-1}}x, {} ^{g^{-1}}y, {} ^{g^{-1}}\big(y^{-1}x^{-1}\big)\big)},
\end{gather*}
where the $1$-cochain $u_g$ is defined as $u_g(x) = \omega\big(xg, {} ^{g^{-1}} x, {}^{g^{-1}} \big(x^{-1}\big)\big)$. \end{Lemma}
\begin{proof} It follows from~\eqref{dos} that $\pi_r(s)^{-1} = \omega\big(z, s, s^{-1}\big) \xi\big(s, s^{-1}\big)^{-1} \pi_r\big(s^{-1}\big)$, for all $z\in G$, $s \in L$. In addition, for all $h, h' \in L$, we have the following relation:
\begin{gather*}\xi \big({h'}^{-1}, h^{-1}\big) \xi (h, h') = df(h, h')\frac{\omega\big(h', {h'}^{-1}, h^{-1}\big)}
{\omega\big(h, h', {h'}^{-1}h^{-1}\big)},\end{gather*}
where $f$ is the 1-cochain given by $f(h) = \xi\big(h, h^{-1}\big)$. A straightforward computation, using this relation and conditions~\eqref{uno},~\eqref{dos} and~\eqref{tres}, shows that $\pi(x) \pi(y) = \alpha_g(x, y) \pi(xy)$, for all $x, y \in H \cap {}^gL$. This proves the lemma. \end{proof}
\begin{Remark}\label{rmk-alfag} Lemma \ref{alfa-g} is a~version of \cite[Proposition~3.2]{ostrik}, where it is shown that $B$ is a~simple object of~$_{A(H, \psi)}{\mathcal C}_{A(L, \xi)}$ if and only if~$B$ is supported on a single double coset $HgL$ and the projective representation~$\pi$ in the component~$B_g$ is irreducible.
\end{Remark}
For all $g \in G$, $\psi^g \Omega_g$ is a 2-cochain in $^{g^{-1}\!}\!H$ such that $\omega\vert_{{}^{g^{-1}}H \times {}^{g^{-1}}H \times {}^{g^{-1}} H} = d(\psi^g \Omega_g)$. Then the product $\xi^{-1} \psi^g \Omega_g$ def\/ines a 2-cocycle of ${}^{g^{-1}} H \cap L$.
\begin{Lemma}\label{rel-cociclos} The class of the $2$-cocycle $\big(\xi^{-1} \psi^g \Omega_g\big)^{g^{-1}}$ in $H^2(H\cap {}^gL, k^\times)$ coincides with the class of the $2$-cocycle $\alpha_g$ in Lemma~{\rm \ref{alfa-g}}.
\end{Lemma}
\begin{proof}
A direct calculation shows that for all $x, y \in G$,
\begin{gather*}\frac{\omega\big(y, y^{-1}, x^{-1}\big)}
{\omega\big(x, y, y^{-1}x^{-1}\big)}
\frac{\omega\big({}^gx, {}^gy, g\big) \omega\big({}^gx, {}^gyg, {} y^{-1}\big)}
{\omega\big({}^gx ^gyg, {}y^{-1}, x^{-1}\big)}
= \Omega_g(x, y) d\theta_g (x, y),\end{gather*}
where the 1-cochain $\theta_g$ is def\/ined as $\theta_g(x) = \omega\big(g, x, x^{-1}\big)^{-1}$. This implies that $\alpha_g^g \equiv \xi^{-1} \psi^g \Omega_g$, as was to be proved.
\end{proof}
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\subsection{}\label{demo} In this subsection we give a proof of the main result of this paper.
\begin{proof}[Proof of Theorem \ref{main}] Let $H, L$ be subgroups of $G$ and let $\psi \in C^2(H, k^{\times})$ and $\xi \in C^2(L, k^{\times})$ be 2-cochains satisfying condition~\eqref{cond-alfa}. Let $A(H, \psi)$, $A(L, \xi)$ be the associated algebras in ${\mathcal C}$ and let ${\mathcal M}_0(H, \psi)$, ${\mathcal M}_0(L, \xi)$ be the corresponding ${\mathcal C}$-module categories.
Let ${\mathcal M} = {\mathcal M}_0(L, \xi)$. For every $g \in G$, let ${\mathcal M}^g$ denote the module category induced by the autoequivalence $\ad_g\colon {\mathcal C} \to {\mathcal C}$.
The ${\mathcal C}$-module category ${\mathcal M}^g$ is equivalent to $_{^gA(L, \xi)}{\mathcal C}$. Hence, by Lemma~\ref{gdea}, ${\mathcal M}^g \cong {\mathcal M}_0\big({}^gL, \xi^{g^{-1}} \Omega_{g^{-1}}\big)$.
Suppose that there exists an element $g \in G$ such that $H = {}^gL$ and the class of the cocycle $\xi^{-1}\psi^g\Omega_g$ is trivial on $L$. Relation~\eqref{rel-omega} implies that $\Omega_g^{g^{-1}} = \Omega_{g^{-1}}^{-1}$, and thus the class of $\psi^{-1}\xi^{g^{-1}}\Omega_{g^{-1}}$ is trivial on~$H$. Then $\psi = \xi^{g^{-1}}\Omega_{g^{-1}} df$, for some 1-cochain $f \in C^1(H, k^{\times})$. Therefore $^gA(L, \xi) = A\big(H, \xi^{g^{-1}} \Omega_{g^{-1}}\big) \cong A(H, \psi)$ as algebras in ${\mathcal C}$. Thus we obtain equivalences of ${\mathcal C}$-module categories
\begin{gather*}{\mathcal M}_0(L, \xi) \cong {\mathcal M}_0(L, \xi)^g \cong {} _{^gA(L, \xi)}{\mathcal C} \cong {\mathcal M}_0(H, \psi),\end{gather*}
where the f\/irst equivalence is deduced from Lemma~\ref{adj-triv}.
Conversely, suppose that $F\colon {\mathcal M}_0(L, \xi) \to {\mathcal M}_0(H, \psi)$ is an equivalence of ${\mathcal C}$-module categories. Recall that there is an equivalence
\begin{gather*}\label{equiv-fun-1} {\mathbb F}un_{\mathcal C}\left({\mathcal M}_0(L, \xi), {\mathcal M}_0(H, \psi)\right) \cong {} _{A(H, \psi)}{\mathcal C}_{A(L, \xi)}.\end{gather*}
Under this equivalence, the functor $F$ corresponds to an object $B$ of $_{A(H, \psi)}{\mathcal C}_{A(L, \xi)}$ such that there exists an object $B'$ of $_{A(L, \xi)}{\mathcal C}_{A(H, \psi)}$ satisfying
\begin{gather}\label{dim-b}B \otimes_{A(L, \xi)}B' \cong A(H, \psi),\end{gather} as $A(H, \psi)$-bimodules in ${\mathcal C}$, and
\begin{gather}\label{bbprime}B' \otimes_{A(H, \psi)}B \cong A(L, \xi), \end{gather} as $A(L, \xi)$-bimodules in ${\mathcal C}$.
Let ${\mathbb F}Pdim_{A(H, \psi)}M$ denote the Frobenius--Perron dimension of an object $M$ of $_{A(H, \psi)}{\mathcal C}_{A(H, \psi)}$. Then we have
\begin{gather*}\dim M = \dim A(H, \psi) {\mathbb F}Pdim_{A(H, \psi)}M = |H| {\mathbb F}Pdim_{A(H, \psi)}M.\end{gather*}
Taking Frobenius--Perron dimensions in both sides of~\eqref{dim-b} and using this relation we obtain that $\dim \left(B \otimes_{A(L, \xi)}B'\right) = |H|$.
On the other hand, $\dim (B \otimes_{A(H, \psi)}B' ) = \frac{\dim B \dim B'}{\dim{A(L, \xi)}} = \frac{\dim B \dim B'}{|L|}$. Thus
\begin{gather}\label{dimbb'} \dim B \dim B' = |H| |L|. \end{gather}
Since $A(H, \psi)$ is an indecomposable algebra in ${\mathcal C}$, then it is a simple object of $_{A(H, \psi)}{\mathcal C}_{A(H, \psi)}$. Then~\eqref{bbprime} implies that $B$ is a simple object of $_{A(H, \psi)}{\mathcal C}_{A(L, \xi)}$ and $B'$ is a simple object of $_{A(L, \xi)}{\mathcal C}_{A(H, \psi)}$.
In view of \cite[Proposition~3.2]{ostrik}, the support of $B$ is a two sided $(H, L)$-double coset, that is, $B = \bigoplus_{(h,h') \in H \times L} B_{hgh'}$, where $g \in G$ is a representative of the double coset that supports~$B$. Moreover, the homogeneous component $B_g$ is an irreducible $\alpha_g$-projective representation of the group $^gL \cap H$, where the 2-cocycle $\alpha_g$ satisf\/ies~$\alpha_g \equiv \big(\xi^{-1}\psi^g \Omega_g\big)^{g^{-1}}$; see Remark~\ref{rmk-alfag} and Lemmas~\ref{alfa-g} and~\ref{rel-cociclos}.
Notice that the actions of $h \in H$ and $h'\in L$ induce isomorphisms of vector spaces $B_{g} \cong B_{hg}$ and $B_{g} \cong B_{gh'}$. Hence
\begin{gather}\label{dimb}\dim B = |HgL| \dim B_g = \frac{|H| |L|}{|H \cap {} ^gL|} \dim B_g = [H: H \cap {}^gL] |L| \dim B_g.\end{gather}
In particular, $\dim B \geq |L|$. Reversing the roles of $H$ and $L$, the same argument implies that $\dim B' \geq |H|$. Combined with relations~\eqref{dimbb'} and~\eqref{dimb} this implies
\begin{gather*}|H| |L| = \dim B \dim B' \geq |H| [H : H \cap {}^gL] |L| \dim B_g.\end{gather*}
Hence $[H\colon H \cap {}^gL] \dim B_g = 1$, and therefore $[H\colon H \cap {}^gL] = 1$ and $\dim B_g = 1$. The f\/irst condition means that $H \subseteq {}^gL$, while the second condition implies that the class of $\alpha_g$ is trivial in $H^2(H \cap {}^gL, k^\times)$. Since the rank of ${\mathcal M}_0(H, \psi)$ equals the index $[G:H]$ and the rank of ${\mathcal M}_0(H, \xi)$ equals the index $[G:L]$, then $|H| = |L|$. Thus we get that $H = {}^gL$ and that the class of the 2-cocycle~\eqref{cond-equiv} is trivial in $H^2(L, k^\times)$. This f\/inishes the proof of the theorem.
\end{proof}
\begin{Example}\label{kp}
Let $B_8$ be the 8-dimensional Kac Paljutkin Hopf algebra. The Hopf algebra $B_8$ f\/its into an exact sequence
\begin{gather*}
k \longrightarrow k^C \longrightarrow B_8 \longrightarrow kL \longrightarrow k,
\end{gather*}
where $C = \mathbb Z_2$ and $L = \mathbb Z_2 \times \mathbb Z_2$. See~\cite{masuoka-6-8}. This exact sequence gives rise to mutual actions by permutations
\begin{gather*}C \overset{\vartriangleleft}\longleftarrow C \times L
\overset{\vartriangleright}\longrightarrow L,\end{gather*}
and compatible cocycles $\tau\colon L \times L \to \big(k^C\big)^\times$, $\sigma\colon C \times C \to (k^L)^\times$, such that $B_8$ is isomorphic to the bicrossed product $kC {}^{\tau}\#_\sigma kL$. The data $\lhd$, $\rhd$, $\sigma$ and $\tau$ are explicitly determined in \cite[Proposition~3.11]{ma-contemp} as follows. Let $C = \langle x\colon x^2 = 1 \rangle$, $L = \langle z, t\colon z^2 = t^2 = ztz^{-1}t^{-1} = 1\rangle$. Then $\lhd\colon C\times L \to C$ is the trivial action of $L$ on $C$, $\rhd\colon C \times L \to L$ is the action def\/ined by $x \rhd z = z$ and $x\rhd t = zt$,
\begin{gather*}\tau_{x^n}\big(z^it^j, z^{i'}t^{j'}\big) = (-1)^{nji'},\end{gather*} for all $0\leq n, i, i', j, j' \leq 1$,
and
\begin{gather*}\sigma_{z^it^j}\big(x^n, x^{n'}\big) = (\sqrt{-1})^{j\big(\frac{n+n'-\langle n+n'\rangle}{2}\big)},\end{gather*}
for all $0 \leq i, j, n, n' \leq 1$, where $\langle n+n'\rangle$ denotes the remainder of $n+n'$ in the division by~$2$. Here we use the notation $\tau(a, a')(y) = : \tau_y(a, a')$ and, similarly, $\sigma(y, y')(a) = : \sigma_a(y, y')$, $a, a' \in L$, $y, y' \in C$.
In view of \cite[Theorem~3.3.5]{schauenburg} (see \cite[Proposition~4.3]{gttic}), the fusion category of f\/inite-dimen\-sio\-nal representations of $B_8^{\rm op} \cong B_8$ is equivalent to the category ${\mathcal C}(G, \omega, L, 1)$, where $G = L \rtimes C$ is the semidirect product with respect to the action~$\rhd$, and $\omega$ is the 3-cocycle arising from the pair $(\tau, \sigma)$ under one of the maps of the so-called \emph{Kac exact sequence} associated to the matched pair.
In this example $G$ is isomorphic to the dihedral group $D_8$ of order 8. The 3-cocycle $\omega$ is determined by the formula
\begin{gather}\label{kac}\omega\big(x^{n}z^{i}t^{j}, x^{n'}z^{i'}t^{j'}, x^{n''}z^{i''}t^{j''}\big) =
\tau_{x^n}\big(z^{i'}t^{j'}, x^{n'} \rhd z^{i''}t^{j''}\big) \sigma_{z^{i''}t^{j''}}\big(x^n, x^{n'}\big),\end{gather}
for all $0 \leq i, j, i', j', i'', j'', n, n', n'' \leq 1$.
Notice that $\omega\vert_{L \times L\times L} = 1$. Hence, for every 2-cocycle $\xi$ on $L$, the pair $(L, \xi)$ gives rise to an indecomposable ${\mathcal C}$-module category ${\mathcal M}(L, \xi)$.
Formula~\eqref{kac} implies that $\Omega_x\vert_{L \times L}$ is given by
\begin{gather*}\Omega_x\big(z^{i}t^{j}, z^{i'}t^{j'}\big) = (-1)^{ji'}, \qquad 0\leq i, i', j, j' \leq 1.\end{gather*}
Then $\Omega_x$ is a 2-cocycle representing the unique nontrivial cohomology class in $H^2(L, k^\times)$. By Theorem~\ref{main}, for any 2-cocycle~$\xi$ on $L$, ${\mathcal M}_0(L, 1)$ and ${\mathcal M}_0(L, \xi)$ are equivalent as ${\mathcal C}(G, \omega)$-module categories, and therefore so are the corresponding ${\mathcal C}$-module categories ${\mathcal M}(L, 1)$ and ${\mathcal M}(L, \xi)$. This implies that indecomposable ${\mathcal C}$-module categories are in this example parameterized by conjugacy classes of subgroups of $D_8$ on which $\omega$ has trivial restriction, as claimed in \cite[Section~6.4]{MN}.
\end{Example}
\LastPageEnding
\end{document}
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\begin{document}
\title{SHIP: A Scalable High-performance IPv6 Lookup Algorithm that Exploits Prefix Characteristics}
\author{Thibaut Stimpfling, Normand Belanger,
J.M. Pierre~Langlois~\IEEEmembership{~Member~IEEE}
and~Yvon~Savaria~\IEEEmembership{~Fellow~IEEE}\thanks{Thibaut Stimpfling, Normand Bélanger, J.M. Pierre Langlois, and Yvon Savaria are with École Polytechnique de Montréal (e-mail: \{thibaut.stimpfling, normand.belanger, pierre.langlois, yvon.savaria\}@polymtl.ca).}}
\maketitle
\begin{abstract}
Due to the emergence of new network applications, current IP lookup engines must support high-bandwidth, low lookup latency and the ongoing growth of IPv6 networks.
However, existing solutions are not designed to address jointly those three requirements.
This paper introduces SHIP, an IPv6 lookup algorithm that exploits prefix characteristics to build a two-level data structure designed to meet future application requirements. Using both prefix length distribution and prefix density, SHIP first clusters prefixes into groups sharing similar characteristics, then it builds a hybrid trie-tree for each prefix group. The compact and scalable data structure built can be stored in on-chip low-latency memories, and allows the traversal process to be parallelized and pipelined at each level in order to support high packet bandwidth.
Evaluated on real and synthetic prefix tables holding up to 580 k IPv6 prefixes, SHIP has a logarithmic scaling factor in terms of the number of memory accesses, and a linear memory consumption scaling. Using the largest synthetic prefix table, simulations show that compared to other well-known approaches, SHIP uses at least 44\% less memory per prefix, while reducing the memory latency by 61\%.
\end{abstract}
\begin{IEEEkeywords}
Algorithm, Routing, IPv6 Lookup, Networking.
\end{IEEEkeywords}
\IEEEpeerreviewmaketitle
\section{Introduction}
\IEEEPARstart{G}{lobal} IP traffic carried by networks is continuously growing,
as around a zettabyte total traffic is expected for the whole of 2016, and it is envisioned to increase threefold between 2015 and 2019~\cite{cisco_forecast}. To handle this increasing Internet traffic, network link working groups have ratified the 100-gigabit Ethernet standard (IEEE P802.3ba), and are studying the 400-gigabit Ethernet standard (IEEE P802.3bs). As a result, network nodes have to process packets at those line rates which puts pressure on IP address lookup engines used in the routing process. Indeed, less than $6$ ns is available to determine the IP address lookup result for an IPv6 packet~\cite{scalable_ipv6_vk}.
The IP lookup task consists of identifying the next hop information (NHI) to which a packet should be forwarded. The lookup process starts by extracting the destination IP field from the packet header, and then matching it against a list of entries stored in a lookup table, called the forwarding information base (FIB). Each entry in the lookup table represents a network defined by its prefix address. While a lookup key may match multiple entries in the FIB, only the longest prefix and its NHI are returned for result as IP lookup is based on the Longest Prefix Match (LPM)~\cite{high_performance_routers_switches}.
IP lookup algorithms and architectures that have been tailored for IPv4 technology are not performing well with IPv6~\cite{scalable_ipv6_vk, mem_eff}, due to the fourfold increase in the number of bits in IPv6 addresses over IPv4. Thus, dedicated IPv6 lookup methods are needed to support upcoming IPv6 traffic.
IP lookup engines must be optimized for high bandwidth, low latency, and scalability for two reasons. First, due to the convergence of wired and mobile networks, many future applications that require a high bandwidth and a low latency, such as virtual reality, remote object manipulation, eHealth, autonomous driving, and the Internet of Things, will be carried on both wired and mobile networks~\cite{5g_whitepaper}. Second, the number of IPv6 networks is expected to grow, and so is the size of the IPv6 routing tables, as IPv6 technology is still being deployed in production networks~\cite{potaroo,ris_raw_data}. However, current solutions presented in the literature are not jointly addressing these three performance requirements.
In this paper, we introduce SHIP: a Scalable and High Performance IPv6 lookup algorithm designed to meet current and future performance requirements. SHIP is built around the analysis of prefix characteristics. Two main contributions are presented: 1) two-level prefix grouping, that clusters prefixes in groups sharing common properties, based on the prefix length distribution and the prefix density, 2) a hybrid trie-tree tailored to handle prefix distribution variations.
SHIP builds a compact and scalable data structure that is suitable for on-chip low-latency memories, and allows the traversal process to be parallelized and pipelined at each level in order to support high packet bandwidth. SHIP stores 580 k prefixes and the associated NHI using less than $5.9$ MB of memory, with a linear memory consumption scaling. SHIP achieves logarithmic latency scaling and requires in the worst case 10 memory accesses per lookup. For both metrics, SHIP outperforms known methods by over 44\% for the memory footprint, and by over 61\% for the memory latency.
The remainder of this paper is organized as follows. Section~\ref{sec:related_work} introduces common approaches used for IP lookup and Section~\ref{sec:overview} gives an overview of SHIP. Then, two-level prefix grouping is presented in Section~\ref{sec:two_level_prefix_grouping}, while the proposed hybrid trie-tree is covered in Section~\ref{sec:trie_tree_hybrid_data_structure}. Section~\ref{sec:method} introduces the method and metrics used for performance evaluation and Section~\ref{sec:results} presents the simulation results. Section~\ref{sec:disscusion} shows that SHIP fulfills the properties for hardware implementability, and compares SHIP performance with other methods. Lastly, we conclude the work by summarizing our main results in Section~\ref{sec:conclusion}.
\section{Related Work}\label{sec:related_work}
Many data structures have been proposed for the LPM operation applied to IP addresses. We can classify them in four main types: hash tables, Bloom filters, tries and trees. Those data structures are encoding prefixes that are loosely structured. First, not only prefix length distribution is highly nonuniform, but it also varies with the prefix table used. Second, for any given prefix length, prefix density ranges from sparse to very dense. Thus, each of the four main data structures type comes with a different tradeoff between time and storage complexity.
Interest for IP lookup with hash table is twofold. First, a hash function aims at distributing uniformly a large number of entries over a number of bins, independently of the structure of the data stored. Second, a hash table provides $O(1)$ lookup time and $O(N)$ space complexity. However, a pure hash based LPM solution can require up to one hash table per IP prefix length. An alternative to reduce the number of hash tables is to use prefix expansion~\cite{flashlook}, but it increases memory consumption. Two main types of hash functions can be selected to build a hash table: perfect or non-perfect hash functions. A Hash table built with a perfect hash functions offers a fixed time complexity that is independent from the prefixes used as no collision is generated. Nevertheless, a perfect hash function cannot handle dynamic prefix tables, making it unattractive for a pure hash based LPM solution. On the other hand, a non-perfect hash function leads to collisions and cannot provide a fixed time complexity. Extra-matching sequences are required with collisions that drastically decrease performance~\cite{flashtrie_conf,distributed_bloom_filters}. In addition, not only the number of collisions is determined after the creation of the hash table but it also depends on the prefix distribution characteristics. In order to reduce the number of collisions independently of the characteristics of the prefix table used, a method has been proposed that exploits multiple hash tables~\cite{flashlook,flashtrie_conf}. This method divides the prefix table into groups of prefixes, and selects a hash function such that it minimizes the number of collisions within each prefix group~\cite{flashlook,flashtrie_conf}. Still, the hash function selection for each prefix group requires to probe all the hash functions, making it unattractive for dynamic prefix tables. Finally, no scaling evaluation has been completed in recent publications~\cite{flashlook, flashtrie_journal} making it unclear whether the proposed hash-based data structures can address forthcoming challenges.
Low-memory footprint hashing schemes known as Bloom filters have also been covered in the literature~\cite{distributed_bloom_filters,bloom_filters}. Bloom filters are used to select a subgroup of prefixes that may match the input IP address. However, Bloom filters suffer from two drawbacks. First, by design, this data structure generates false positives independent of the configuration parameters used. Thus, a Bloom filter can improve the average lookup time, but it can also lead to poor performance in the worst case, as many sub-groups need to be matched. Second, the selection of a hash function that minimizes the number of false positives is highly dependent of the prefix distribution characteristics used. Hence, its complexity is similar to that of of a hash function that minimizes the number of collisions in a regular hash table.
Tree solutions based on binary search trees (BST) or generalized B-trees have also been explored in~\cite{mem_eff,scalable_ipv6_vk}. Such data structures are tailored to store loosely structured data such as prefixes, as their time complexity is independent from the prefix distribution characteristics. Indeed, BST and 2-3 Trees have a time complexity of respectively $log_2(N)$ and $log_3(N)$, with $N$ being the number of entries~\cite{scalable_ipv6_vk}. Nevertheless, such data structure provides a solution at the cost of a large memory consumption. Indeed, each node stores a full-size prefix, leading to memory waste. Hence, their memory footprint makes them unsuitable for the very large prefix tables that are anticipated in future networks.
At the other end of the tree spectrum, decision-trees (D-Trees) have been proposed in~\cite{hicuts,scalable_packet_classification} for the field of packet classification. D-Trees were found to offer a good tradeoff between memory footprint and the number of memory accesses. However, no work has been conducted yet on using this data structure for IPv6 lookup.
The trie data structure, also known as radix tree, has regained interest with tree bitmap~\cite{bitmap_tree}. Indeed, a $k$-bit trie requires $k/W$ memory accesses, but has very poor memory efficiency when built with unevenly distributed prefixes. A tree bitmap improves the memory efficiency over a multi-bit trie, independently of the prefix distribution characteristics, by using a bitmap to encode each level of a multi-bit trie. However, tree bitmaps cannot be used with large strides, as the node size grows exponentially with the stride size, leading to multiple wide memory accesses to read a single node. An improved tree bitmap, the PC-trie, is proposed for the FlashTrie architecture~\cite{flashtrie_journal}. A PC-Trie reduces the size of bitmap nodes using a multi-level leaf pushing method. This data structure is used jointly with a pre-processing hashing stage to reduce the total number of memory accesses. Nevertheless, the main shortcoming of the Flashtrie architecture lies in its pre-processing hashing module. First, similar to other hashing solutions, its performance highly depends on the distribution characteristics of the prefixes used. Second, the hashing module does not scale well with the number of prefixes used.
At the other end of the spectrum of algorithmic solutions, TCAMs have been proposed as a pure hardware solution, achieving $O(1)$ lookup time by matching the input key simultaneously against all prefixes, independently of their distribution characteristics. However, these solutions use a very large amount of hardware resources, leading to large power consumption and high cost, and making them unattractive for routers holding a large number of prefixes~\cite{a_tcam_based_distributed_parallel,flashtrie_journal}.
Recently, information-theoretic and compressed data structures have been applied to IP lookup, yielding very compact data structures, that can handle a very large number of prefixes~\cite{compressing_ip_forwarding_tables}. Even though this work is limited to IPv4 addresses, it is an important shift in terms of concepts. However, the hardware implementation of the architecture achieves 7 million lookups per second. In order to support a 100-Gbps bandwidth, this would require many lookup engines, leading to a memory consumption that is similar or higher than previous trie or tree algorithms~\cite{flashlook,flashtrie_conf,flashtrie_journal,scalable_ipv6_vk}.
In summary, the existing data structures are not exploiting the full potential of the prefix distribution characteristics.
In addition, none of the existing data structures were shown to optimize jointly the time complexity, the storage complexity, and the scalability.
\section{SHIP Overview}\label{sec:overview}
SHIP consists of two procedures: the first one is used to build a two-level data structure, and the second one is used to traverse the two-level data structure. The procedure to build the data structure is called two-level prefix grouping, while the traversal procedure is called the lookup algorithm.
Two-level prefix grouping clusters prefixes upon their characteristics to build an efficient two-level data structure, presented in Fig~\ref{fig:global_lookup_seq}. At the first level, SHIP leverages the low density of the IPv6 prefix MSBs to divide prefixes into M address block bins (ABBs). A pointer to each ABB is stored in an N-entry hash table. At the second level, SHIP uses the uneven prefix length distribution to sort prefixes held in each ABB into K Prefix Length Sorted (PLS) groups. For each non-empty K$\cdot$M PLS groups, SHIP further exploits the prefix length distribution and the prefix density variation to encode prefixes into a hybrid trie-tree (HTT).
The lookup algorithm, which identifies the NHI associated to the longest prefix matched, is presented in Fig~\ref{fig:global_lookup_seq}. First, the MSBs of the destination IP address are hashed to select an ABB pointer stored in an $N$ entry hash table. The selected ABB pointer in this figure is held in the $n$-th entry of the hash table, represented with a dashed rectangle. This pointer identifies bin $m$, represented with a dashed rectangle. Second, the HTTs associated to each PLS group of the $m$-th bin, are traversed in parallel, using portions of the destination IP address. Each HTT can output a NHI, if a match occurs with its associated portion of the destination IP address. Thus, up to $K$ HHT results can occur, and a priority resolution module is used to select the NHI associated to the longest prefix.
\begin{figure}
\caption{SHIP two-level data structure organization and its lookup process with $M$ address block bins and $K$ prefix length sorting groups.}
\label{fig:global_lookup_seq}
\end{figure}
In the following section, we present the two-level prefix grouping procedure.
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\section{SHIP Overview}\label{sec:overview}
SHIP consists of two procedures: the first one is used to build a two-level data structure, and the second one is used to traverse the two-level data structure. The procedure to build the data structure is called two-level prefix grouping, while the traversal procedure is called the lookup algorithm.
Two-level prefix grouping clusters prefixes upon their characteristics to build an efficient two-level data structure, presented in Fig~\ref{fig:global_lookup_seq}. At the first level, SHIP leverages the low density of the IPv6 prefix MSBs to divide prefixes into M address block bins (ABBs). A pointer to each ABB is stored in an N-entry hash table. At the second level, SHIP uses the uneven prefix length distribution to sort prefixes held in each ABB into K Prefix Length Sorted (PLS) groups. For each non-empty K$\cdot$M PLS groups, SHIP further exploits the prefix length distribution and the prefix density variation to encode prefixes into a hybrid trie-tree (HTT).
The lookup algorithm, which identifies the NHI associated to the longest prefix matched, is presented in Fig~\ref{fig:global_lookup_seq}. First, the MSBs of the destination IP address are hashed to select an ABB pointer stored in an $N$ entry hash table. The selected ABB pointer in this figure is held in the $n$-th entry of the hash table, represented with a dashed rectangle. This pointer identifies bin $m$, represented with a dashed rectangle. Second, the HTTs associated to each PLS group of the $m$-th bin, are traversed in parallel, using portions of the destination IP address. Each HTT can output a NHI, if a match occurs with its associated portion of the destination IP address. Thus, up to $K$ HHT results can occur, and a priority resolution module is used to select the NHI associated to the longest prefix.
\begin{figure}
\caption{SHIP two-level data structure organization and its lookup process with $M$ address block bins and $K$ prefix length sorting groups.}
\label{fig:global_lookup_seq}
\end{figure}
In the following section, we present the two-level prefix grouping procedure.
\section{Two-level Prefix Grouping}\label{sec:two_level_prefix_grouping}
This section introduces two-level prefix grouping, that clusters and sort prefixes into groups, and then builds the two-level data structure. First, prefixes are binned and the first level of the two-level data structure is built with the address block binning method. Second, inside each bin, prefixes are sorted into groups, and then the HTTs are built with the prefix length sorting method.
\subsection{Address block binning}
The proposed address block binning method exploits both the structure of IP addresses and the low density of the IPv6 prefix MSBs to cluster the prefixes into bins, and then build the hash table used at the first level of SHIP data structure.
IPv6 addresses are structured into IP address blocks, managed by the Internet Assigned Numbers Authority (IANA) that assigns blocks of IPv6 addresses ranging in size from $/16$ to $/23$, that are then further divided into smaller address blocks. However, the prefix density on the first 23 bits is low, and the prefix distribution is sparse~\cite{global_unicast_address_assig}. Therefore, the address block binning method bins prefixes based on their first 23 bits. Before prefixes are clustered, all prefixes with a prefix length that is less than $/23$ are converted into $/23$. The pseudo-code used for this binning method is presented in Algorithm~\ref{alg:build_address_block_binning}. For each prefix held in the prefix table, this method checks whether a bin already exists for the first $23$ bits. If none exists, a new bin is created, and the prefix is added to the new bin. Otherwise, the prefix is simply added to the existing bin.
This method only keeps track of the $M$ created bins. The prefixes held in each bin are further grouped by the prefix length sorting method that is presented next.
Address block binning utilizes a perfect hash function~\cite{perfect_hashing_function} to store a pointer to each valid bin. Let $N$ be the size of the hash table used. The first 23 MSBs of the IP address represents the key of the perfect hash table. A perfect hashing function is chosen for three reasons. First, the number of valid keys on the first 23 bits is relatively small compared to the number of bits used, making hashing attractive. Second, a perfect hashing function is favoured when the data hashed is static, which is the case here, for the first 23 bits only, because it represents blocks of addresses allocated to regional internet registries that are unlikely to be updated on a short time scale. Finally, no resolution module is required because no collisions are generated.
\begin{algorithm}
\caption{Building the Address Block binning data structure}
\label{alg:build_address_block_binning}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Prefix table
\Ensure Address Block binning data structure
\For{each prefix held in the prefix table}
\State {Extract its 23 MSBs}
\If {no bin already exists for the extracted bits}
\parState{Create a bin that is associated to the value of the extracted bits}
\State{Add the current prefix to the bin}
\Else
\parState {Select the bin that is associated to the value of the extracted bits}
\State{Add the current prefix to the selected bin}
\EndIf
\EndFor
\State {Build a hash table that stores a pointer to each address block bin, with the 23 MSBs of each created bin as a key. Empty entries are holding invalid pointers.}
\State \textbf{return} {the hash table}
\end{algorithmic}
\end{algorithm}
The lookup procedure is presented in Algorithm~\ref{alg:lookup_address_block_binning}. It uses the 23 MSBs of the destination IP address as a key for the hash table and returns a pointer to an address block bin. If no valid pointer exists in the hash table for the key, then a null pointer is returned.
\begin{algorithm}
\caption{Lookup in the Address Block binning data structure}
\label{alg:lookup_address_block_binning}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require{Address Block binning hash table, destination IP Address}
\Ensure{Pointer to an address block bin}
\State{Extract the 23 MSBs of the destination IP address}
\State {Hash the extracted 23 MSBs}
\If{the hash table entry pointed by the hashed key holds a valid pointer}
\State{ \textbf{return} {pointer to the address block bin}}
\Else
\State{ \textbf{return} {null pointer}}
\EndIf
\end{algorithmic}
\end{algorithm}
The perfect hash table created with the address block binning method is an efficient data structure to perform a lookup on the first 23 MSBs of the IP address.
However, within the ABBs the prefix length distribution can be highly uneven, which degrades the performance of the hybrid trie-trees at the second level. Therefore, the prefix length sorting method, described next, is proposed to address that problem.
\subsection{Prefix length sorting}
Prefix length sorting (PLS) aims at reducing the impact of the uneven prefix length distribution on the number of overlaps between prefixes held in each address block bin. By reducing the number of prefix overlaps, the performance of the HTTs is improved, as it will be shown later. The PLS method sorts the prefixes held in each address block bin by their length, into $K$ PLS groups that cover disjoints prefix length ranges. Each range consists of contiguous prefix lengths that are associated to a large number of prefixes with respect to the prefix table size. For each PLS group, a hybrid trie-tree is built.
The number of PLS groups, $K$, is chosen to maximize the HTT's performance. As will be shown experimentally in section~\ref{sec:results}, beyond a threshold value, increasing the value of $K$ does not further improve performance. The prefix length range selection is based on the prefix length distribution and it is guided by two principles. First, to minimize prefix overlap, when a prefix length covers a large percentage of the total number of prefixes, this prefix length must be used as an upper bound of the considered group. Second, prefix lengths included in a group are selected such that group sizes are as balanced as possible.
To illustrate those two principles, an analysis of prefix length distribution using a real prefix table is presented in Fig.~\ref{fig:prefix_distribution}. The prefix table extracted from~\cite{ris_raw_data} holds approximately 25 k prefixes. The first $23$ prefix lengths are omitted in Fig.~\ref{fig:prefix_distribution}, as the address block binning method already bins prefixes on their $23$ MSBs. It can be observed in Fig.~\ref{fig:prefix_distribution} that the prefix lengths with the largest cardinality are $/32$ and $/48$ for this example. Applying the two principles of prefix length sorting to this example, the first group covers prefix lengths from $/24$ to $/32$, and the second group covers the second peak, from $/33$ to $/48$. Finally, all remaining prefix lengths, from $/49$ to $/64$ are left in the third prefix length sorting group.
\begin{figure}
\caption{The uneven prefix length distribution of real a prefix table used by the PLS method to create 3 PLS groups.}
\label{fig:prefix_distribution}
\end{figure}
For each of the $K$ PLS group created, an HTT is built. Thus, the lookup step associated to the prefix length sorting method consists of traversing the $K$ HTTs held in the selected address block bin.
To summarize, the created PLS groups cover disjoint prefix length ranges by construction. Therefore, the PLS method directly reduces prefix overlaps in each address block bin that increases the performance of HTT. However, within each PLS group, the prefix density variation remains uneven. Hence, a hybrid-trie tree is proposed that exploits the local prefix characteristics to build an efficient data structure.
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\section{Hybrid Trie-Tree data structure}\label{sec:trie_tree_hybrid_data_structure}
The hybrid trie-tree proposed in this work is designed to leverage the prefix density variation. This hybrid data structure uses a density-adaptive trie, and a reduced D-Tree leaf when the number of prefixes covered by a density-adaptive trie node is below a fixed threshold value. A description of the two data structures is first presented, then the procedure to build the hybrid trie-tree is formulated, and finally the lookup procedure is introduced.
\subsection{Density-Adaptive Trie}
The proposed density-adaptive trie is a data structure that is built upon the prefix density distribution. A density-adaptive trie combines a trie data structure with the Selective Node Merge (SNM) method.
While a trie or multi-bit trie creates equi-sized regions whose size is independent of the prefix density distribution, the proposed SNM method adapts the size of the equi-sized regions to the prefix density. Low-density equi-sized regions created with a trie data structure are merged into variable region sizes by the SNM method. Two equi-sized regions are merged if the total number of prefixes after merging is equal to the largest number of prefixes held by the two equi-sized regions, or if it is less than a fixed threshold value. The SNM method merges equi-sized regions both from the low indices to the highest ones and from the high indices to the lowest ones. For both directions, the SNM method first selects the lowest and highest index equi-sized regions, respectively. Second, it evaluates if each selected region can be merged with its next contiguous equi-sized region. The two steps are repeated until the selected region can no longer be merged with its next contiguous equi-sized region. Else, the two previous steps are repeated from the last equi-sized region that was left un-merged. The SNM method has two constraints with respect to the number of merged regions. First, each merged region covers a number of equi-sized regions that is restricted to powers of two, as the space covered by the merged region is described with the prefix notation. Second, the total number of merged regions is bounded by the size of a node header field of the adaptive trie.
By merging equi-sized regions together, the SNM method reduces the number of regions that are stored in the data structure. As a result, the SNM method improves the memory efficiency of the data structure.
The benefit of the SNM method on the memory efficiency is presented in Fig.~\ref{fig:with_selective_node_merge} for the first level of a multi-bit trie. As a reference, the first level of a multi-bit trie without the SNM method is also presented in Fig.~\ref{fig:wo_selective_node_merge}. In both figures, IP addresses are defined on $3$ bits for the following prefix set $P_{1} = 110/3$, $P_{2} = 111/3 $ and $P_{3} = 0/0$. In both figures, the region is initially partitioned into four equi-sized regions, each corresponding to a different bit combination, called $0$ to $3$. In Fig.~\ref{fig:with_selective_node_merge}, the SNM method merges the two leftmost equi-sized regions $0$ and $1$, separated by a dashed line, as they fulfill the constraints of SNM. In Fig.~\ref{fig:with_selective_node_merge}, not only the SNM method reduces the number of nodes held in memory by 25\% compared to the multi-bit trie presented in Fig.~\ref{fig:wo_selective_node_merge} but also prefix $P_{3}$ is replicated twice, that is a 33\% reduction of the prefix replication factor. As a result, the SNM method increases the memory efficiency of the multi-bit trie data structures.
\begin{figure}
\caption{Impact of Selective Node Merge on the replication factor for the first level of a trie data structure.}
\label{fig:with_selective_node_merge}
\label{fig:wo_selective_node_merge}
\label{fig:selective_node_merge_example}
\end{figure}
The regions that are traversed by the SNM method (merged or not) are stored in a SNM field of the adaptive-trie node header. The SNM field is divided into a $LtoH$ and $HtoL$ array. The $LtoH$ and $HtoL$ arrays hold the indices of the regions traversed respectively from low to high index values, and high to low index values. For each region traversed by the SNM method, merged or equi-sized, one index is stored either in the $LtoH$ or the $HtoL$ array. Indeed, as a merged region holds two or more multiple contiguous equi-sized regions, a merged region can be described with the indices of the first and the last equi-sized region it holds. In addition, the SNM method traverses the equi-sized regions contiguously. Therefore, the index of the last equi-sized region held in a merged region can be determined implicitly using the index of the next region traversed by the SNM method. The index value of a non-merged region is sufficient to fully describe it.
\subsection{Reduced D-Tree leaf}
A reduced D-Tree leaf is built when the number of prefixes held in a region of the density-adaptive trie is below a fixed threshold value \textit{b}. The proposed leaf is based on a D-tree leaf~\cite{hicuts,scalable_packet_classification} that is extended with the Leaf Size Reduction technique (LSR).
A D-Tree leaf is a bucket that stores the prefixes and their associated NHI held in a given region. A D-Tree leaf has a memory complexity and time complexity of $O(n)$ for $n$ prefixes stored. A D-Tree leaf is used for the regions at the bottom of the density-adaptive trie because of its higher memory efficiency in those regions. Indeed, we observed that most of the bottom level regions of an density-adaptive trie hold highly unevenly distributed prefixes. Moreover, a D-Tree leaf has better memory efficiency with highly unevenly distributed prefixes over a density-adaptive trie. Whereas a density-adaptive trie can create prefix replication, which reduces the memory efficiency, no prefix replication is created with a D-Tree leaf. However, the D-Tree leaf comes at the cost of higher time complexity compared to a density-adaptive trie.
As a consequence, the LSR technique is introduced to reduce the time complexity of a D-Tree leaf by reducing the amount of information stored in a D-Tree leaf. In fact, a D-Tree leaf stores entirely each prefix even the bits that have already been matched by the density-adaptive trie before reaching the leaf. On the other hand, the LSR technique stores in the reduced D-Tree leaf only the prefix bits that are left unmatched. To specify the number of bits that are left unmatched, a new LSR leaf header field is added, coded on $6$ bits. The LSR technique reduces the amount of information that is stored in each reduced D-Tree leaf. As a result, not only does the reduced D-Tree leaf requires fewer memory accesses but it also has a better memory efficiency over a D-Tree leaf.
\subsection{HTT build procedure}
The hybrid trie-tree build procedure is presented in Algorithm~\ref{alg:hybrid_data_structure_build_proc}, starting with the root region that holds all the prefixes (line $1$). If the number of prefixes stored in the root region is below a fixed threshold \textit{b}, then a reduced D-Tree leaf is built (line $2 - 3$). Else, the algorithm iteratively partitions this region into equi-sized regions (lines $4 - 10$). The SNM method is then applied on the equi-sized regions (line $11$). Next, for each region, if the number of prefixes is below the threshold value (line $12$), a reduced D-Tree leaf is built (line $13$), else a density-adaptive trie node is built (line $14 - 16$) and the region is again partitioned.
\begin{algorithm}
\caption{Hybrid Trie-Tree build procedure}
\label{alg:hybrid_data_structure_build_proc}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Prefix Table, stack Q
\Ensure Hybrid Trie-Tree
\State{Create a root region covering all prefixes \;}
\If{the number of prefixes held in that region is below the threshold value \textit{b}}
\State Create a reduced D-Tree leaf for those prefixes \;
\Else
\State Push the root region onto Q \;
\EndIf
\While{Q is not empty}
\parState{Remove the top node in Q and use it as the reference region}
\parState{Compute the number of partitions in the reference region}
\parState{Partition the reference region according to the previous step}
\parState{Apply the SNM method on the partitioned reference regions}
\For{each partitioned reference region}
\If{it holds a number of prefixes that is below the threshold value}
\State{Create a reduced D-Tree leaf for those prefixes}
\Else
\parState{Build an adaptive-density trie node for those prefixes}
\State Push this region onto Q
\EndIf
\EndFor
\EndWhile
\State \Return the Hybrid Trie-Tree \;
\end{algorithmic}
\end{algorithm}
The number of partitions in a region (line $9$ of Algorithm~\ref{alg:hybrid_data_structure_build_proc}) is computed by a greedy heuristic proposed in~\cite{hicuts}. The heuristic uses the prefix distribution to adapt the number of partitions, as expressed in Algorithm~\ref{alg:bi_heuristic}. An objective function, the \textit{Space Measurement (Sm)} is evaluated at each iteration (lines $4$ and $5$) and compared to a threshold value, the \textit{Space Measurement Factor (Smpf)} evaluated in the first step (line $1$). The number of partitions increases by a factor of two at each iteration (line $3$), until the value of the objective function \textit{Sm} (line $4$) becomes greater than the threshold value (line $5$). The objective function estimates the memory usage efficiency with the prefix replication factor by summing the number of prefixes held in each $j$ equi-sized region created $\sum_{j=0}^{N_{p}} Num_{Prefixes}(equi-sized region_{j} )$ (line $4$). The prefix replication factor is impacted by the prefix distribution. If prefixes are evenly distributed, the replication factor remains very low until the equi-sized regions become smaller than the average prefix size. Then, the prefix replication factor increases exponentially. Thus, to avoid over-partitioning a region if the replication factor remains low for many iterations, the number of partitions $N_{p}$ and the result of the previous iterations $Sm(N_{p-1})$ are used as a penalty term that is added to the objective function (line $4$). On the other hand, if prefixes are unevenly distributed, the prefix replication factor increases linearly until the largest prefixes in the region partitioned become slightly smaller compared to an equi-sized region. Passed this point, an exponential growth of the replication factor is observed. The heuristic creates fine-grained partition size in a dense region, and coarse-grained partition size in a sparse region.
\begin{algorithm}
\caption{Heuristic used to compute the number of partitions in a region}
\label{alg:bi_heuristic}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Region to be cut
\Ensure Number of partitions ($N_{p}$)
\label{alg:bi_obj_greedy}
\State $N_{p} = 1; Smpf = Num_{Prefixes} \cdot 8; Sm(N_{p}) = 0 ;$
\Do
\State $N_{p} = N_{p} \cdot 2; $
\parState{$ Sm(N_{p})= \sum_{j=0}^{N_{p}} Num_{Prefixes}(equi-sized region_{j}) + N_{p} + Sm(N_{p-1}) ; $}
\doWhile $Sm(N_{p}) \leq Smpf $ \\
\Return $N_{p}$
\end{algorithmic}
\end{algorithm}
The number of partitions in a region is a power of two. Thus, the base-2 logarithm of the number of partitions represents the number of bits from the IP address used to select the equi-sized region covering this IP address.
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\subsection{HTT build procedure}
The hybrid trie-tree build procedure is presented in Algorithm~\ref{alg:hybrid_data_structure_build_proc}, starting with the root region that holds all the prefixes (line $1$). If the number of prefixes stored in the root region is below a fixed threshold \textit{b}, then a reduced D-Tree leaf is built (line $2 - 3$). Else, the algorithm iteratively partitions this region into equi-sized regions (lines $4 - 10$). The SNM method is then applied on the equi-sized regions (line $11$). Next, for each region, if the number of prefixes is below the threshold value (line $12$), a reduced D-Tree leaf is built (line $13$), else a density-adaptive trie node is built (line $14 - 16$) and the region is again partitioned.
\begin{algorithm}
\caption{Hybrid Trie-Tree build procedure}
\label{alg:hybrid_data_structure_build_proc}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Prefix Table, stack Q
\Ensure Hybrid Trie-Tree
\State{Create a root region covering all prefixes \;}
\If{the number of prefixes held in that region is below the threshold value \textit{b}}
\State Create a reduced D-Tree leaf for those prefixes \;
\Else
\State Push the root region onto Q \;
\EndIf
\While{Q is not empty}
\parState{Remove the top node in Q and use it as the reference region}
\parState{Compute the number of partitions in the reference region}
\parState{Partition the reference region according to the previous step}
\parState{Apply the SNM method on the partitioned reference regions}
\For{each partitioned reference region}
\If{it holds a number of prefixes that is below the threshold value}
\State{Create a reduced D-Tree leaf for those prefixes}
\Else
\parState{Build an adaptive-density trie node for those prefixes}
\State Push this region onto Q
\EndIf
\EndFor
\EndWhile
\State \Return the Hybrid Trie-Tree \;
\end{algorithmic}
\end{algorithm}
The number of partitions in a region (line $9$ of Algorithm~\ref{alg:hybrid_data_structure_build_proc}) is computed by a greedy heuristic proposed in~\cite{hicuts}. The heuristic uses the prefix distribution to adapt the number of partitions, as expressed in Algorithm~\ref{alg:bi_heuristic}. An objective function, the \textit{Space Measurement (Sm)} is evaluated at each iteration (lines $4$ and $5$) and compared to a threshold value, the \textit{Space Measurement Factor (Smpf)} evaluated in the first step (line $1$). The number of partitions increases by a factor of two at each iteration (line $3$), until the value of the objective function \textit{Sm} (line $4$) becomes greater than the threshold value (line $5$). The objective function estimates the memory usage efficiency with the prefix replication factor by summing the number of prefixes held in each $j$ equi-sized region created $\sum_{j=0}^{N_{p}} Num_{Prefixes}(equi-sized region_{j} )$ (line $4$). The prefix replication factor is impacted by the prefix distribution. If prefixes are evenly distributed, the replication factor remains very low until the equi-sized regions become smaller than the average prefix size. Then, the prefix replication factor increases exponentially. Thus, to avoid over-partitioning a region if the replication factor remains low for many iterations, the number of partitions $N_{p}$ and the result of the previous iterations $Sm(N_{p-1})$ are used as a penalty term that is added to the objective function (line $4$). On the other hand, if prefixes are unevenly distributed, the prefix replication factor increases linearly until the largest prefixes in the region partitioned become slightly smaller compared to an equi-sized region. Passed this point, an exponential growth of the replication factor is observed. The heuristic creates fine-grained partition size in a dense region, and coarse-grained partition size in a sparse region.
\begin{algorithm}
\caption{Heuristic used to compute the number of partitions in a region}
\label{alg:bi_heuristic}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Region to be cut
\Ensure Number of partitions ($N_{p}$)
\label{alg:bi_obj_greedy}
\State $N_{p} = 1; Smpf = Num_{Prefixes} \cdot 8; Sm(N_{p}) = 0 ;$
\Do
\State $N_{p} = N_{p} \cdot 2; $
\parState{$ Sm(N_{p})= \sum_{j=0}^{N_{p}} Num_{Prefixes}(equi-sized region_{j}) + N_{p} + Sm(N_{p-1}) ; $}
\doWhile $Sm(N_{p}) \leq Smpf $ \\
\Return $N_{p}$
\end{algorithmic}
\end{algorithm}
The number of partitions in a region is a power of two. Thus, the base-2 logarithm of the number of partitions represents the number of bits from the IP address used to select the equi-sized region covering this IP address.
\subsection{HTT lookup procedure}\label{sec:lookup_fixed}
The hybrid trie-tree lookup algorithm starts with a traversal of the density-adaptive trie until a reduced D-Tree leaf is reached. Next, the reduced D-Tree leaf is traversed to identify the matching prefix and its NHI.
The traversal of the density-adaptive trie consists in computing the memory address of the child node that matches the destination IP address, calculated with Algorithm~\ref{alg:children_node_address}. This algorithm uses as input parameters the memory base address, the destination-IP bit-sequence, the $LtoH$ and the $HtoL$ arrays that are extracted from the node header. The SNM method can merge multiple equi-sized nodes into a single node in memory, and thus the destination-IP bit-sequences cannot be used directly as an index to the child node. Therefore, Algorithm~\ref{alg:children_node_address} computes for each destination-IP bit-sequence the number of equi-sized nodes that are skipped in memory based on the characteristics of the merged regions described in the $LtoH$ and the $HtoL$ arrays. The value of the destination-IP bit-sequence can point to a region that is either included 1) in a merged region described in the $LtoH$ array (line $1$), or 2) in a merged region described in the $HtoL$ array (line $4$), or 3) in a equi-sized region that has not been traversed by the SNM method (line $7$).
The following notation is introduced: $L$ represents the size of the $HtoL$ and $LtoH$ arrays, $LtoH [i]$ and $HtoL [i]$ are respectively the $i-th$ entry of the $LtoH$ and the $HtoL$ arrays. In the first case, each entry of the $LtoH$ array is traversed to find the closest $LtoH [i]$ that is less than or equal to the destination-IP bit-sequence (line $1$). The index of the matched child node is equal to $index_{LtoH}$ (line $2$), where $index_{LtoH}$ is the index of the $LtoH$ array that fulfills this condition. In the second case, each entry of the $HtoL$ array is similarly traversed to find the closest $HtoL [i]$ that is greater than or equal to the destination-IP bit-sequence (line $4$). The $index_{HtoL}$ in the $LtoH$ array that fulfills this condition is combined with the characteristics of the $LtoH$ and $HtoL$ arrays to compute the index of the selected child node (line $5$). In the third case, the algorithm evaluates only the number of equi-sized nodes that are skipped in memory based on the characteristics of the $LtoH$ array and the destination IP address bit sequence of the matched child node (line $7$). Finally, the index that is added to the base address points in memory to the matched child node.
\begin{algorithm}
\caption{Memory address of the matched child node using SNM method}
\label{alg:children_node_address}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Children base address, destination IP address bit sequence, $LtoH$ and $HtoL$ arrays
\Ensure Child node address
\Comment{Index included in a region using SNM method}
\If{destination IP address bit sequence $\leq LtoH [L-1]$}
\Comment{In LtoH array}
\State{Index = $Index_{LtoH}$}
\Else{
\If{destination IP address bit sequence $\geq HtoL[0] $}
\Comment{In HtoL array}
\parState{Index = $index_{HtoL} + HtoL [0] - LtoH [L-1]+ L-1$}
\Else{
\Comment{destination IP address bit sequence included in an equi-sized region that has not been traversed by the SNM method}
\parState{Index = destination IP address bit sequence $- LtoH [L-1]+ L-1$}
\EndIf}
}
\EndIf
\\
\Return{Child~node~address = base~address + Index}
\end{algorithmic}
\end{algorithm}
Algorithm~\ref{alg:children_node_address} is illustrated with Figures~\ref{fig:example_SNM} in which $L = 3$ and the destination IP address bit sequence is arbitrarily set to $10$. Based on Fig.~\ref{fig:example_SNM}, the destination IP address bit sequence $10$ matches the equi-sized region with the index $10$ before the SNM method is applied. However, after the SNM method is applied, the destination IP address bit sequence matches a merged node with the index $9$. Based on the SNM header, the destination IP address bit sequence $10$ is greater than both $LtoH [L-1]= 3$ and $HtoL [0]= 9$. Thus, we must identify the number of equi-sized nodes that are skipped in memory with the $LtoH$ and $HtoL$ arrays. Because $LtoH [L-1] = 3$, two equi-sized nodes have been merged. As one node is skipped in memory, any child index greater than $3$ is stored at offset $index - 1$ in memory. Moreover, the destination IP address bit sequence is greater than $HtoL [0]= 9$. However, $HtoL [1] = 10$, meaning that indices $9$ and $10$ are not merged, and no entry is skipped in memory for the first two regions held in the $HtoL$ array. As a consequence, only one node is skipped in memory, and thus the child node index is $10 - 1 = 9$.
\begin{figure}
\caption{SNM method applied to a region that holds 11 nodes after merging, and its associated SNM field}
\label{fig:example_SNM}
\end{figure}
\begin{algorithm}
\caption{Lookup in the Reduced D-tree leaf}
\label{alg:lookup_reduced_leaf}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Reduced D-Tree leaf, destination IP Address
\Ensure LPM and its NHI
\State {Parse the leaf header}
\State {Read the prefixes held in the leaf}
\For {each prefix held in the leaf}
\parState{Match the destination IP address against the selected prefix}
\If{Positive Match}
\State{Record the prefix length of the matched prefix}
\EndIf
\EndFor
\State{Identify the longest prefix match amongst all positive matches}\\
\Return {the longest prefix match and its NHI }
\end{algorithmic}
\end{algorithm}
The density-adaptive trie is traversed until a reduced D-Tree leaf is reached. The lookup procedure of a reduced D-Tree leaf is presented in Algorithm~\ref{alg:lookup_reduced_leaf}. The leaf header is first parsed, and then prefixes are read (lines $1$ to $2$). Next, all prefixes are matched against the destination IP address, and their prefix length is recorded if matches are positive (lines $3$ to $6$). When all the prefixes are matched, only the longest prefix match is returned with its NHI (lines $7-8$).
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\section{Performance Measurement Methodology}\label{sec:method}
This section describes the methodology used to evaluate SHIP performance using both real and synthetic prefix tables. Eleven real prefix tables were extracted using the RIS remote route collectors~\cite{ris_raw_data}, and each one holds approximately $25$ k prefixes. Each scenario, noted $rrc$ followed by a two-digit number, characterizes the location in the network of the remote route collector used. For prefix tables holding up to $580$ k entries, synthetic prefixes were generated with a method that uses IPv4 prefixes to generate IPv6 prefixes, in a one-to-one-mapping~\cite{non_random_generation}. The IPv4 prefixes used were also extracted from~\cite{ris_raw_data}. Using the IPv6 prefix table holding $580$ k prefixes, four smaller prefix tables were created, with a similar prefix length distribution, holding respectively $290$ k, $116$ k, $58$ k and $29$ k prefixes.
The performance of SHIP was evaluated using two metrics: the number of memory accesses to traverse its data structure and its memory consumption. For the two metrics, the performance is reported separately for the hash table used by the address block binning method, and the HTTs built by the prefix length sorting method. SHIP performance is characterized using $1$ to $6$ groups for two-level prefix grouping, and as a reference the performance of a single HTT without grouping is also presented.The number of groups is limited to six, as we have observed with simulations that increasing it further does not improve the performance.
For the evaluation of the number of memory accesses, it is assumed that the selected hybrid trie-trees within an address block bin are traversed in parallel, using dedicated traversal engines. Therefore, the reported number of memory accesses is the largest number of memory accesses of all the hybrid trie-trees amongst all address block bins. It is also assumed that the memory bus width is equal to a node size, in order to transfer one node per memory clock cycle.
The memory consumption is evaluated as the sum of all nodes held in the hybrid trie-tree for the prefix length sorting method, and of the size of the perfect hash table used for the address block binning method. In order to evaluate the data structure overhead, this metric is given in bytes per byte of prefix. This metric is evaluated as the size of the data structure divided by the size of the prefixes held in the prefix table.
The format and size of a non-terminal node and a leaf header used in a hybrid trie-tree are detailed respectively in Table~\ref{tab:node_format} and in Table~\ref{tab:SHIP_leaf_header}. The node type field, coded with $1$ bit, specifies whether the node is a leaf or a non-terminal node. The following fields are used only for non-terminal nodes. Up to $10$ bits can be matched at each node, corresponding to a node header field coded with $4$ bits. The fourth field is used for SNM, to store the index value of the traversed regions. Each index is restricted to $10$ bits, while the $HtoL$ and $LtoH$ arrays each store up to $5$ indices. The third field, coded in $16$ bits, stores the base address of the first child node associated with its parent's node.
\begin{table}[htb]
\renewcommand{1.3}{1.3}
\caption{ Non-terminal node header field sizes in bits}
\label{tab:node_format}
\centering
\begin{tabular}{|c|c|c|}
\hline
\bf Header Field & \bf Size \\
\hline
Node type & $1$\\
\hline
Number of cuts & $4$ \\
\hline
Pointer to child node & $16$ \\
\hline
Size of selective node merge array & $5 \cdot 10 + 5 \cdot 10$ \\
\hline
\end{tabular}
\end{table}
The leaf node format is presented in Table~\ref{tab:SHIP_leaf_header}. A leaf can be split over multiple nodes to store all its prefixes. Therefore, two bits are used in the leaf header to specify whether the current leaf node is a terminal leaf or not. The next field gives the number of prefixes stored in the leaf. It is coded with $4$ bits because in this work, the largest number of prefixes held in a leaf is set to $12$ for each hybrid trie-tree. The LSR field stores the number of bits that need to be matched, using 6 bits. If a leaf is split over multiple nodes, a pointer coded with $16$ bits points at the remaining nodes that are part of the leaf. Inside a leaf, prefixes are stored alongside their prefix length and with their NHI. The prefix length is coded with the number of bits specified by the LSR field while the NHI is coded with $8$ bits.
\begin{table}[htb]
\renewcommand{1.3}{1.3}
\caption{Leaf header field sizes in bits}
\label{tab:SHIP_leaf_header}
\centering
\begin{tabular}{|c|c|c|}
\hline
\bf Header Field & Size \\
\hline
Node type & $2$ \\
\hline
Number of prefixes stored & $4$ \\
\hline
LSR field & $6$ \\
\hline
Pointer to remaining leaf entries & $16$ \\
\hline
Prefix and NHI & Value specified in the LSR field + $8$ \\
\hline
\end{tabular}
\end{table}
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\section{Results}\label{sec:results}
SHIP performance is first evaluated using real prefixes, and then with synthetic prefixes, for both the number of memory accesses and the memory consumption.
\subsection{Real Prefixes}
The performance analysis is first made for the perfect hash table used by the address block binning method. In Table~\ref{tab:result_bin23_real},
the memory consumption and the number of memory accesses for the hash table are shown. The ABB method uses between $19$ kB and $24$ kB, that is between $0.7$ and $0.9$ bytes per prefix byte for the real prefix tables evaluated. The memory consumption is similar across all the scenarios tested as prefixes share most of the $23$ MSBs. On the other hand, the number of memory accesses is by construction independent of the number of prefixes used, and constant to $2$.
\begin{table}[htbp]
\renewcommand{1.3}{1.3}
\caption{Memory consumption of the address block binning method for real prefix tables}
\label{tab:result_bin23_real}
\centering
\begin{tabular}{|l|c|c|}
\hline
\bf Scenario & \bf Hashing Table size (kB) & \bf Memory Accesses \\
\hline
$rrc00$ & 20 & 2 \\
\hline
$rrc01$ & 19 & 2 \\
\hline
$rrc04$ & 24 & 2 \\
\hline
$rrc05$ & 19 & 2 \\
\hline
$rrc06$ & 19 & 2 \\
\hline
$rrc07$ & 20 & 2 \\
\hline
$rrc10$ & 21 & 2 \\
\hline
$rrc11$ & 20 & 2 \\
\hline
$rrc12$ & 20 & 2 \\
\hline
$rrc13$ & 22 & 2 \\
\hline
$rrc14$ & 21 & 2 \\
\hline
\end{tabular}
\end{table}
In Figures~\ref{mem_cons_real_prefix_table} and~\ref{mem_acc_real_prefix_table}, the performance of the HTTs is evaluated respectively on the memory consumption and the number of memory accesses. In both figures, $1$ to $6$ groups are used for two-level prefix grouping. As a reference, the performance of the HTT without grouping is also presented.
In Fig.~\ref{mem_cons_real_prefix_table}, the memory consumption of the HTTs ranges from $1.36$ to $1.60$ bytes per prefix byte for all scenarios, while it ranges between $1.22$ up to $3.15$ bytes per byte of prefix for a single HTT. Thus, using two-level prefix grouping, the overhead of the HTTs ranges from $0.36$ to $0.6$ byte per byte of prefix. However, a single HTT leads to an overhead of $0.85$ on average, and up to $3.15$ bytes per byte of prefix for scenario $rrc13$. Thus, two-level grouping reduces the memory consumption and smooths its variability, but it also reduces the hybrid trie-tree overhead.
Fig.~\ref{mem_cons_real_prefix_table} shows that increasing the number $K$ of groups up to three reduces the memory consumption. However, using more groups does not improve the memory consumption, and even worsens it. Indeed, it was observed experimentally that when increasing the value of $K$ most groups hold very few prefixes, leading to a hybrid trie-tree holding a single leaf with part of the allocated node memory left unused. Thus, using too many groups increases memory consumption.
\begin{figure}
\caption{Real prefix tables: impact of the number of groups on the memory consumption (a) and the number of memory accesses (b) of the HTTs.}
\label{mem_cons_real_prefix_table}
\label{mem_acc_real_prefix_table}
\label{fig:result_memory_consumption}
\end{figure}
It can be observed in Fig.~\ref{mem_acc_real_prefix_table} that the number of memory accesses to traverse the HTTs ranges from $6$ to $9$ with two-level prefix grouping, whereas it varies between $9$ and $18$ with a single HTT. So, two-level prefix grouping smooths the number of memory accesses variability, but it also reduces on average the number of memory accesses approximatively by a factor 2.
However, increasing the number $K$ of groups used by two-level prefix grouping, from $1$ to $6$, yields little gain on the number of memory accesses, as seen in Fig.~\ref{mem_acc_real_prefix_table}. Indeed, for most scenarios, one memory access is saved, and up to two memory accesses are saved in two scenarios, by increasing the number $K$ of groups from $1$ to $6$. Indeed, for each scenario, the performance is limited by a prefix length that cannot be divided in smaller sets by increasing the number of groups. Still, using two or more groups, in the worst case, $8$ memory accesses are required for all scenarios. The performance is similar across all scenarios evaluated, as few variations exist between the prefix groups created using two-level grouping for those scenarios.
\subsection{Synthetic Prefixes}
The complexity of the perfect hash table used for the address block binning method is presented in Table~\ref{tab:result_bin23} with synthetic prefix tables. It requires on average $2.7$ bytes per prefix byte for the $5$ scenarios tested, holding from $29$ k up to $580$ k prefixes. The perfect hash table used shows linear memory consumption scaling. For the number of memory accesses, its value is independent of the prefix table, and is equal to $2$.
\begin{table}[htbp]
\renewcommand{1.3}{1.3}
\caption{Cost of binning on the first 23 bits for synthetic prefix tables}
\label{tab:result_bin23}
\centering
\begin{tabular}{|l|c|c|}
\hline
\bf Prefix Table Size & \bf Hashing Table size (kB) & \bf Memory Accesses \\
\hline
580 k & 1282 & 2 \\
\hline
290 k & 642 & 2 \\
\hline
110 k & 322 & 2 \\
\hline
50 k & 162 & 2 \\
\hline
29 k & 82 & 2 \\
\hline
\end{tabular}
\end{table}
The performance of the HTTs with synthetic prefixes is evaluated for the number of memory accesses, the memory consumption, and the memory consumption scaling, respectively in Fig.~\ref{mem_acc_synth_prefix_table},~\ref{mem_cons_synth_prefix_table}, and~\ref{fig:mem_cons_synth_prefix_table_scaling}. For each of the three figures, $1$ to $6$ groups are used for two-level prefix grouping. The performance of the HTT without grouping is also presented in the three figures, and is used as a reference.
Two behaviors can be observed for the memory consumption in Fig.~\ref{mem_cons_synth_prefix_table}.
First, for prefix tables with $290$ k prefixes and more, it can be seen that two-level prefix grouping used with $2$ groups slightly decreases the memory consumption over a single HTT. Using this method with two groups, the HTTs consumes between $1.18$ and $1.09$ byte per byte of prefix, whereas the memory consumption for a single HTT lies between $1.18$ and $1.20$ byte per byte of prefix. However, increasing the number of groups to more than two does not improve memory efficiency, as it was observed that most prefix length sorting groups hold very few prefixes, leading to hybrid trie-tree holding a single leaf, with part of the allocated node memory that is left unused. Even though the memory consumption reduction brought by two-level prefix grouping over a single HTT is small for large synthetic prefix tables, it will be shown in this paper that the memory consumption remains lower when compared to other solutions. Moreover, it will be demonstrated that two-level prefix grouping reduces the number of memory accesses to traverse the HTT with the worst case performance over a single HTT, for all synthetic prefix table sizes. Second, for smaller prefix tables with up to $116$ k prefixes, a lower memory consumption is achieved using only a single HTT for two reasons. First, the synthetic prefixes used have fewer overlaps and are more distributed than real prefixes for small to medium size prefix tables, making two-level prefix grouping less advantageous in terms of memory consumption. Indeed, a larger number $M$ of address block bins has been observed compared to real prefix tables with respect to the number of prefixes held the prefix tables, for small and medium prefix tables. Thus, on average, each bin holds fewer prefixes compared to real prefix tables. As a consequence, we observe that the average and maximum number of prefixes held in each PLS group is smaller for prefix tables holding up to $116$ k prefixes. It then leads to hybrid trie-trees where the allocated leaf memory is less utilized, achieving lower memory efficiency and lower memory consumption.
\begin{figure}
\caption{Synthetic prefix tables: impact of the number of groups on the number of memory accesses (a), the memory consumption (b) and scaling (c) of the HTTs.}
\label{mem_acc_synth_prefix_table}
\label{mem_cons_synth_prefix_table}
\label{fig:mem_cons_synth_prefix_table_scaling}
\label{fig:result_memory_accesses}
\end{figure}
In order to observe the memory consumption scaling of the HTTs, Fig.~\ref{fig:mem_cons_synth_prefix_table_scaling} shows the total size of the HTTs using synthetic prefix tables, two-level prefix grouping, and a number $K$ of groups that ranges from $1$ to $6$. The memory consumption of the HTTs with and without two-level prefix grouping grows exponentially for prefix tables larger than $116$ k. However, because the abscissa uses a logarithmic scale, the memory consumption scaling of the proposed HTT is linear with and without two-level prefix grouping. In addition, the memory consumption of the HTTs is $4,753$ kB for the largest scenario with $580$ k prefixes, two-level prefix grouping, and $K = 2$.
Next, we analyze in Fig.~\ref{mem_acc_synth_prefix_table} the number of memory accesses required to traverse the HTT leading to the worst-case performance, for synthetic prefix tables, with two-level prefix grouping using $1$ to $6$ groups. It can be observed that two-level prefix grouping reduces the number of memory accesses over a single HTT, for all the number of groups and all prefix table sizes. The impact of two-level prefix grouping is more pronounced when using two groups or more, as the number of memory accesses is reduced by 40\% over a single HTT. Using more than $3$ groups does not further reduce the number of memory accesses as the group leading to the worst-case scenario cannot be reduced in size by increasing the number of groups. Finally, it can be observed in Fig.~\ref{mem_acc_synth_prefix_table} that the increase in the number of memory accesses for a search is at most logarithmic with the number of prefixes, since each curve is approximately linear and the x-axis is logarithmic.
The performance analysis presented for synthetic prefixes has shown that two-level prefix grouping improves the performance over a single HTT for the two metrics evaluated. Although the performance improvement of the memory consumption is limited to large prefix tables, using few groups, the number of memory accesses is reduced for all prefix table sizes and for all numbers of groups. In addition, it has been observed experimentally that the HTTs used with two-level prefix grouping have a linear memory consumption scaling, and a logarithmic scaling for the number of memory accesses. The hash table used in the address block binning method has shown to offer a linear memory consumption scaling and a fixed number of memory accesses. Thus, SHIP has a linear memory consumption scaling, and a logarithm scaling for the number of memory accesses.
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\section{Discussion}\label{sec:disscusion}
This section first demonstrates that SHIP is optimized for a fast hardware implementation. Then, the performance of SHIP is compared with previously reported results.
\subsection{SHIP hardware implementability}
We demonstrate that SHIP is optimized for a fast hardware implementation, as it complies with the following two properties; 1) pipeline-able and parallelizable processing to maximize the number of packets forwarded per second, 2) use of a data structure that can fit within on-chip memory to minimize the total memory latency.
A data structure traversal can be pipelined if it can be decomposed into a fixed number of stages, and for each stage both the information read from memory and the processing are fixed. First, the HTT traversal can be decomposed into a pipeline, where each pipeline stage is associated to a HTT level. Indeed, the next node to be traversed in the HTT depends only on the current node selected and the value of the packet header. Second, for each pipeline stage of the HTT both the information read from memory and the processing are fixed. Indeed, the information is stored in memory using a fixed node size for both the adaptive-density trie and the reduced D-Tree leaf. In addition, the processing of a node is constant for each data structure and depends only on its type, as presented in Section~\ref{sec:lookup_fixed}. As a result, the HTT traversal is pipeline-able. Moreover, the HTTs within the $K$ PLS groups are independent, thus their traversal is by nature parallelizable. As a consequence, by combining a parallel traversal of the HTTs with a pipelined traversal of each HTT, property $1$ is fulfilled. The hash table data structure used for the address block binning technique has been implemented in hardware in previous work~\cite{mem_eff}, and thus it already complies with property $1$.
For the second property, SHIP uses $5.9$ MB of memory for $580$ k prefixes, with two-level prefix grouping, and $K = 2$ . Therefore, SHIP data structure can fit within on-chip memory of the current generation of FPGAs and ASICs~\cite{altera,xilinx,pisa}. Hence, SHIP fulfills property $2$. As both the hash table used by two-level prefix grouping, and the hybrid trie-tree comply with properties $1$ and $2$ required for a fast hardware implementation, SHIP is optimized for a fast hardware implementation.
\subsection{Comparison with previously reported results}
\begin{table*}[htbp]
\renewcommand{1.3}{1.3}
\caption{Comparison Results}
\label{tab:result_comparison}
\centering
\begin{tabular}{|l|c|c|c|c|}
\hline
\bf Method & \bf Memory Consumption & \bf Latency (ns) & \multicolumn{2}{|c|}{\bf Complexity} \\
& \bf (in bytes per prefix) & & Memory Consumption & Memory latency \\
\hline
Tree-based \cite{scalable_ipv6_vk} & $19.0$ & 90 & $O(N)$ & $O(log_2 (N) ) \leq Latency \leq 2 \cdot O(log_3 (N) )$\\
\hline
CLIPS \cite{mem_eff} & $27.6$ & N/A & N/A & N/A \\
\hline
FlashTrie \cite{flashtrie_journal} & $124.2$ & 80 & N/A & N/A \\
\hline
FlashLook \cite{flashlook} & $1010.0$ & 90 & N/A & N/A \\
\hline
\bf SHIP & $10.64$ & 31 & $O(N)$ & $O(log(N))$ \\
\hline
\end{tabular}
\end{table*}
Table~\ref{tab:result_comparison} compares the performance of SHIP and previous work in terms of memory consumption and worst case memory latency. If available, the time and space complexity are also shown. In order to use a common metric between all reported results, the memory consumption is expressed in bytes per prefix, obtained by dividing the size of the data structure by the number of prefixes used. The memory latency is based on the worst-case number of memory accesses to traverse a data structure.
For the following comparison, it is assumed that on-chip SRAM memory running at 322 MHz~\cite{scalable_ipv6_vk} is used, and off-chip DDR3-1600 memory running at 200 MHz is used.
Using both synthetic and real benchmarks, SHIP requires in the worst case $10$ memory accesses, and consumes $5.9$ MB of memory for the largest prefix table, with $2$ groups for two-level prefix grouping. Hence, the memory latency to complete a lookup with on-chip memory is equal to $10 \cdot 3.1$ = 31 ns.
FlashTrie has a high memory consumption, as reported in Table~\ref{tab:result_comparison}. The results presented were reevaluated using the node size equation presented in~\cite{flashtrie_conf} due to incoherence with equations shown in~\cite{flashtrie_journal}. This algorithm leads to a memory consumption per prefix that is around $11 \times$ higher than the SHIP method, as multiple copies of the data structure have to be spread over DDR3 DRAM banks. In terms of latency, in the worst case, two on-chip memory accesses are required, followed by three DDR3 memory bursts. However, DRAM memory access comes at a high cost in terms of latency for the FlashTrie method. First, independently of the algorithm, a delay is incurred to send the address off-chip to be read by the DDR3 memory controller. Second, the latency to complete a burst access for a given bank, added to the maximum number of bank-activate commands that can be issued in a given period of time, limits the memory latency to $80$ ns and reduces the maximum lookup frequency to $84$ MHz. Thus, FlashTrie memory latency is $2.5 \times$ higher than SHIP.
The FlashLook~\cite{flashlook} architecture uses multiple copies of data structures in order to sustain a bandwidth of 100 Gbps, leading to a very large memory consumption compared to SHIP. Moreover, the memory consumption of this architecture is highly sensitive to the prefix distribution used. For the memory latency, in the worst case, when a collision is detected, two on-chip memory accesses are required, followed by three memory bursts pipelined in a single off-chip DRAM, leading to a total latency of~80~ns. The observed latency of SHIP is 61\% smaller. Finally, no scaling study is presented, making it difficult to appreciate the performance of FlashLook for future applications.
The method proposed in~\cite{scalable_ipv6_vk} uses a tree-based solution that requires $19$ bytes per prefix, which is $78\%$ larger than the proposed SHIP algorithm. Regarding the memory accesses, in the worst case, using a prefix table holding $580$ k prefixes, $22$ memory accesses are required, which is more than twice the number of memory accesses required by SHIP. In terms of latency, their implementation leads to a total latency of $90$ ns for a prefix table holding $580$ k prefixes, that is $2.9 \times$ higher than the proposed SHIP solution. Nevertheless, similar to SHIP, this solution has a logarithmic scaling factor in terms of memory accesses, and scales linearly in terms of memory consumption.
Finally, Tong et al.~\cite{mem_eff} present the CLIPS architecture~\cite{CLIPS} extended to IPv6. Their method uses $27.6$ bytes per prefix, which is about $2.5 \times$ larger than SHIP. The data structure is stored in both on-chip and off-chip memory, but the number of memory accesses per module is not presented by the authors, making it impossible to give an estimate of the memory latency. Finally, the scalability of this architecture has not been discussed by the authors.
These results show that SHIP reduces the memory consumption over other solutions and decreases the total memory latency to perform a lookup. It also offers a logarithmic scaling factor for the number of memory accesses, and it has a linear memory consumption scaling.
\section{Conclusion}\label{sec:conclusion}
In this paper, SHIP, a scalable and high performance IPv6 lookup algorithm, has been proposed to address current and future application performance requirements. SHIP exploits prefix characteristics to create a shallow and compact data structure. First, two-level prefix grouping leverages the prefix length distribution and prefix density to cluster prefixes into groups that share common characteristics. Then, for each prefix group, a hybrid trie-tree is built. The proposed hybrid trie-tree is tailored to handle local prefix density variations using a density-adaptive trie and a reduced D-Tree leaf structure.
Evaluated with real and synthetic prefix tables holding up to 580 k IPv6 prefixes, SHIP builds a compact data structure that can fit within current on-chip memory, with very low memory lookup latency. Even for the largest prefix table, the memory consumption per prefix is $10.64$ bytes, with a maximum number of $10$ on-chip memory accesses. Moreover, SHIP provides a logarithmic scaling factor in terms of the number of memory accesses and a linear memory consumption scaling. Compared to other approaches, SHIP uses at least 44\% less memory per prefix, while reducing the memory latency by 61\%.
\section*{Acknowledgments}
The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC), Prompt,
and Ericsson Canada for financial support to this research.
\end{document}
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\begin{document}
\title[Induced Representation]{Induced representations of Hilbert $C^*$-modules }
\author{Gh. Abbaspour tabadkan and S. Farhangi}
\address{Department of pure mathematics, school of mathematics and computer science, Damghan university, Damghan, Iran.}
\email{[email protected], [email protected].}
\subjclass[2010]{46L08, 46L05}
\keywords{Hilbert modules, Morita equivalent and Induced representations. }
\date{\today}
\begin{abstract}
In this paper, we define the notion of induced representations of a Hilbert $C^{*}$-module and we show that Morita equivalence of two Hilbert modules (in the sense of Moslehian and Joita \cite{JOI}), implies the equivalence of categories of non-degenerate representations of two Hilbert modules.
\end{abstract}
\maketitle
\section{Introduction}
The concept of Morita equivalence was first made by Morita \cite{MOR} in a purely algebraic content. Two unital rings are called Morita equivalent if their categories of left modules are equivalent.\\
This concept has been applied to many different categories in mathematics. And investigate the relationship between an "object", and its "representation theory".\\
In the category of $C^{*}$-algebras, Rieffel \cite{RIE1,RIE2} defined the notions of induced representations and (strong) Morita equivalence. The notion of induced representations of $C^*$-algebras, now called Rieffel induction, is to constructing functors between the categories of non-degenerate representations of two $C^*$-algebras. Bursztyn and Waldmann \cite{BUR}, generalized this notion to $*$-algebras and in 2005 Joita \cite{JOT1} introduced this notion for locally $C^*$-algebras.\\ Two $C^*$-algebras $A$ and $B$ are called Morita equivalent if there exist an $A-B$-imprimitivity. This notion is weaker than isomorphism. There are many valuable papers which study properties of $C^*$-algebras are invariant under the Morita equivalence. (see for examples \cite{ BEE, RIE1, Z})\\
The notion of Morita equivalence in the category of Hilbert $C^*$-modules is defined by Skeide \cite{MSK1} and Joita, Moslehian \cite{JOI}, in two different form. In \cite{JOI}, two Hilbert $A$-module $V$ and Hilbert $B$-module $W$ are called Morita equivalent if the $C^*$-algebras $K(V)$ and $K(W)$ are Morita equivalent as $C^*$-algebras. This notion is weaker than the notion of Morita equivalence defined by Skeide, where he called $V$ and $W$ Morita equivalent when the $C^*$-algebras $K(V)$ and $K(W)$ are isomorphic as $C^*$-algebras.\\
In this paper we show that the weaker notion of Morita equivalence is enough to Hilbert modules have same categories of non-degenerate representations.\\
In section 2, we fix our terminologies and discuss preliminaries about representations of Hilbert modules and Morita equivalence of $C^*$-algebras.\\ In section 3, we introduce the notion of induced representations for Hilbert modules
and we show that Morita equivalence Hilbert module in the sense of Joita and Moslehian \cite{JOI}, have same categories of non-degenerate representations. \\
----------------------------------------------------------------
\section{Preliminary}
A (right) Hilbert $C^*$-module $V$ over a $C^*$-algebra $A$ (or a Hilbert $A$-module ) is by definition
a linear space that is a right $A$-module, together with an $A$-valued inner
product $\langle . , . \rangle$ on $V \times V$ that is $A$-linear in the second and conjugate linear in the first variable, such that $V$ is a Banach space with the norm define by $\Vert x\Vert_{A}:= \Vert \langle x , x \rangle_{A} \Vert^{\frac{1}{2}}.$ A Hilbert $A$-module $V$ is a full Hilbert $A$-module if the ideal
\begin{center}
I = $span\lbrace\langle x , y \rangle_{A} ; x,y \in{X} \rbrace $
\end{center}
is dense in $A$. The notion of left Hilbert $A$-module is defined in similar way.\\
We denote the $C^*$-algebras of adjointable and compact operators on Hilbert $C^*$-module $V$ by $L(V)$ and $K(V)$, respectively. See \cite{LAN} for more details on Hilbert modules.\\
Now we have a quick review on the notion of Rieffel induction. Let $X$ be a right Hilbert $B$-module and let $\pi:B\rightarrow B(H)$ be a representation. Then $X\otimes_{alg}H$ is a Hilbert space with inner product
\begin{center}
$\langle x \otimes h,y \otimes k \rangle$:=$\langle \pi(\langle y,x\rangle_{B})h,k \rangle$
\end{center}
for $x,y \in X$ and $h,k \in H$ [ \cite{RAE}, Proposition 2.64 ].\\
If $A$ acts as adjointable operators on a Hilbert $B$-module $X$, and $\pi$ is a non-degenerate representation of $B$ on $H$. Then $Ind\pi$ defined by $$Ind\pi(a)(x\otimes_{B}h):=(ax)\otimes_{B}h$$ is a representation of $A$ on $X\otimes_{B}H$. If $X$ is non-degenerate as an $A$-module, then $Ind\pi$ is a non-degenerate representation of $A$ [ \cite{RAE}, Proposition 2.66 ].\\ This is a functor from the non-degenerate representations of $B$ to the non-degenerate representations of $A$. Now if we want to get back from representations of $A$ to representations of $B$, we need also an $A$-valued inner product on $X$. This lead us to the following definition.
\begin{defn}
An $A-B$\emph{-imprimitivity bimodule} is an $A-B$-bimodule such that:\\
$(a)$ $X$ is a full left Hilbert $A$-module, and is a full right Hilbert $B$-module;\\
$(b)$ for all $ x,y \in X$, $a \in A$, $b \in B$
\begin{center}
$\langle ax , y \rangle_{B}$=$ \langle x , a^{*}y \rangle_{B} $ and $_{A}\langle xb , y \rangle$=$ _{A}\langle x , yb^{*} \rangle $
\end{center}
$(c)$ for all $ x,y,z \in X$
\begin{center}
$_{A}\langle x , y \rangle z $=$ x \langle y,z \rangle_{B} $.
\end{center}
\end{defn}
If $X$ is an $A-B$-imprimitivity bimodule, let $\widetilde{X}$ be the conjugate vector space, so that there is by definition an additive bijection $b:X\rightarrow \widetilde{X}$ such that $b(\lambda x):= \overline{\lambda}b(x)$. Then $\widetilde{X}$ is a $B-A$-imprimitivity bimodule with\\
$\begin{array}{cc}
bb(x):=b(xb^{*}) & b(x)a:=b(a^{*}x) \\
_{B}\langle b(x),b(y)\rangle:=\langle x,y\rangle_{B} & \langle b(x),b(y)\rangle _{A}:=_{A} \langle x,y\rangle
\end{array}$\\
for $x,y \in X$, $a\in A$ and $b \in B$. $\widetilde{X}$ called the \emph{dual module} of $X$.
\begin{exa}
A Hilbert space $H$ is a $K(H)-\textbf{C}$-impirimitivity bimodule with $_{K(H)}\langle h,k \rangle $:=$h \otimes \overline{k}$, where $h \otimes \overline{k}$ denote the rank one operator $g \mapsto \langle g,k\rangle h$.
\end{exa}
\begin{exa}
Every $C^*$-algebra $A$ is an $A-A$-imprimitivity bimodule for the bimodule structure given by the multiplication in $A$, with the inner product $ _{A}\langle a , b \rangle$=$ab^{*}$ and $ \langle a , b \rangle_{A}$=$a^{*}b$ .
\end{exa}
Two $C^*$-algebras $A$ and $B$ are Morita equivalent if there is an $A-B$-imprimitivity bimodule $X$; we shall say that $X$ implements the Morita equivalence of $A$ and $B$ .\\
Morita equivalence is weaker than isomorphism. If $\varphi$ is an isomorphism of $A$ onto $B$, we can construct an imprimitivity bimodule $_{A}X_{B}$ with underlying space $B$ by
\begin{center}
$xb:=xb$, $ax:= \varphi(a)x$, $\langle x,y \rangle_{B}:=x^{*}y$ and $_{A}\langle x,y \rangle:=\varphi^{-1}(xy^{*})$.
\end{center}
Morita equivalence is an equivalence relation on $C^*$-algebras. If $A$ and $B$ are Morita equivalent then the functor mentioned above which comes from tensoring by $X$ has an inverse functor. In fact its inverse is functor comes from tensoring by $\widetilde{X}$, the dual module of $X$ [ \cite{RAE}, Proposition 3.29 ]. So $A$ and $B$ have the same categories of non-degenerate representations.\\
In this paper we will prove that two full Hilbert modules on Morita equivalent $C^*$-algebras have the same categories of non-degenerate representations. But let us first say some facts about representations of Hilbert modules.\\
Let $V$ and $W$ be Hilbert $C^{*}$-modules over $C^{*}$-algebras $A$ and $B$, respectively, and $ \varphi:A\longrightarrow B$ a morphism of $C^*$-algebras.\\
A map $\Phi :V \longrightarrow W$ is said to be a $\varphi$-morphism of Hilbert $C^*$-modules if\\ $ \langle \Phi(x), \Phi (y) \rangle=\varphi(\langle x,y \rangle)$ is satisfied for all $x,y \in V$.\\
A $\varphi$-morphism $ \Phi :V \longrightarrow B(H,K)$, where $ \varphi :A \longrightarrow B(H) $ is a representation of $A$ is called a representation of $V$. We will say that a representation $ \Phi :V \longrightarrow B(H,K)$ is a faithful representation of $V$ if $\Phi$ is injective.\\
Throughout the paper, when we say that $\Phi$ is a representation of $V$, we will assume that an associated representation of $A$ is denoted by the same small case letter $ \varphi$.\\
Let $ \Phi :V \longrightarrow B(H,K)$ be a representation of a Hilbert $A$-module $V$. $\Phi$ is said to be\emph{ non-degenerate} if $\overline{\Phi (V)H}$=$K$ and $\overline{\Phi (V)^{*}K}$=$H$. (Or equivalently, if $\xi_{1} \in H$,$\xi_{2} \in K$ are such that $\Phi (V)\xi_{1}=0$ and $\Phi(V)^{*}\xi_{2}=0$, then $\xi_{1}=0$ and $\xi_{2}=0$ ). If $\Phi$ is non-degenerate, then $\varphi$ is non-degenerate [ \cite{ARA}, Lemma 3.4 ].\\
Let $ \Phi :V \longrightarrow B(H,K)$ be a representation of a Hilbert $A$-module $V$ and $ K_{1} \prec H $ , $ K_{2} \prec K $ be closed subspaces.
A pair of subspaces $(K_{1},K_{2})$ is said to be $\Phi$\emph{-invariant} if
\begin{center}
$\Phi (V)K_{1}\subseteq K_{2}$ and $\Phi (V)^{*}K_{2}\subseteq K_{1}$.
\end{center}
$ \Phi $ is said to be \emph{irreducible} of $(0,0)$ and $(H,K)$ are the only $\Phi$-invariant pairs.\\
Two representations $\Phi_{i}: V \rightarrow B(H_{i};K_{i})$ of V, $i =1,2$ are said to be (unitarily) equivalent, if there are unitary operators $U_{1}:H_{1}\rightarrow H_{2}$ and $U_{2}:K_{1}\rightarrow K_{2}$; such that $U_{2}\Phi_{1}(v)=\Phi_{2}(v)U_{1}$
for all $v \in V$. For more details on representations of Hilbert modules see \cite{ARA}.\\
Finally we need the interior tensor product of Hilbert modules, we mention it here briefly. For more details one can refer to the Lance book \cite{LAN}.
Suppose that $V$ and $W$ are Hilbert $A$-module and Hilbert $B$-module, respectively, and $\rho :A \longrightarrow L(W)$ is a *-homomorphism, we can regard $W$ as a left $A$-module, the action being given by $(a,y) \longmapsto \rho(a)y$ for all $a \in A$, $y \in W$.\\
We can form the algebraic tensor product of $V$ and $W$ over $A$, $V \otimes_{alg} W$, which is a right $B$-module. The action of $B$ being given by $(x \otimes y)b$ := $x \otimes yb$ for $b \in B$.\\
In fact it is the quotient space of the vector space tensor product $V \otimes_{alg} W$ by the subspace generated by elements of the form
\begin{center}
$xa \otimes y-x \otimes \rho(a)y$, $(x \in V , y \in W , a \in A)$.
\end{center}
$V\otimes_{alg}W$ is an inner product $B$-module under the inner product
\begin{center}
$\langle x_{1} \otimes y_{1},x_{2} \otimes y_{2} \rangle$=$\langle y_{1},\rho( \langle x_{1},x_{2} \rangle)y_{2}\rangle$
\end{center}
for $x_{1},x_{2} \in V$, $y_{1},y_{2} \in W$.\\
And $V\otimes_{A}W$, which is called the interior tensor product of $V$ and $W$, obtained by completing $V\otimes_{alg}W$ with respect to this inner product.\\
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\section{Induced Representation}
In this section we discussed about Morita equivalence of Hilbert $C^{*}$-modules and speak about the notion of induced representation of a Hilbert $C^{*}$-module and then we prove the imprimitivity theorem for induced representations of Hilbert $C^{*}$-modules.
\begin{prop}\label{prop:ind}
Let $V$ and $W$ be two full Hilbert $C^{*}$-modules over $C^{*}$-algebras $A$ and $B$, respectively. Let $X$ be a $B$-module and $A$ acts as adjointable operators on Hilbert $C^{*}$-module $X$, and $\Phi : W \rightarrow B(H,K)$ is a non-degenerate representation. Then the formula,
\begin{center}
$ Ind_{X}\Phi(v)(x\otimes g) := v\otimes x \otimes h $
\end{center}
extends to give a representation of $V$ as bounded operator of Hilbert space $X\otimes_{B} H$ to Hilbert space $V\otimes_{A} X\otimes_{B} H$. If $X$ is non-degenerate as an $A$-module, then $Ind_{X}\Phi$ is a non-degenerate representation.
\end{prop}
\begin{proof}
Since $A$ acts as adjointable operators on the Hilbert $B$-module $X$, so we may construct interior tensor product $V\otimes_{A} X$, which is a $B$-module. Then $V\otimes_{A} X\otimes_{B} H$ and $X\otimes_{B} H $ are Hilbert spaces.\\
Let $\Phi :W\rightarrow B(H,K)$ be a non-degenerate representation, so there is a representation $\varphi :B\rightarrow B(H)$ such that $\langle \Phi(x),\Phi(y) \rangle$=$\varphi (\langle x,y \rangle_{B}) $ for all $x,y \in W$. $\varphi$ is non-degenerate so by Rieffel induction we get a non-degenerate representation,
\begin{center}
$Ind_{X}\varphi:A\rightarrow B(X\otimes_{B} H)$.
\end{center}
Now we want to construct a representation, $Ind_{X}\Phi$ of $V$.\\
The mapping $(x,h)\mapsto v\otimes x \otimes h$ is bilinear, thus there is a linear transformation $\eta_{v}: X\otimes_{alg}H \rightarrow V\otimes_{A} X\otimes_{B} H$ such that $\eta _{v}(x\otimes h)$=$v\otimes x \otimes h$.\\
To see that $\eta_{v}$ is bounded, as in the $C^*$-algebraic case, we may suppose that $\varphi$ is cyclic, with cyclic vector $h$. Then for any $x_{i} \in X , b_{i} \in B$ we have
\begin{align*}
\|\eta_{v}(\sum_{i=1}^{n}x_{i} \otimes \varphi(b_{i})h) \|^{2} &= \sum_{i} \sum_{j}\langle v\otimes x_{i} \otimes \varphi(b_{i})h,v\otimes x_{j} \otimes \varphi(b_{j})h \rangle \\
& = \sum_{i} \sum_{j}\langle x_{i} \otimes \varphi(b_{i})h,Ind_{X}\varphi (_{A}\langle v,v \rangle) x_{j} \otimes \varphi(b_{j})h \rangle \\
& = \sum_{i} \sum_{j}\langle x_{i} \otimes \varphi(b_{i})h,_{A}\langle v,v \rangle x_{j} \otimes \varphi(b_{j})h \rangle \\
& = \sum_{i} \sum_{j}\langle \varphi(b_{i})h,\varphi (\langle _{A}\langle v,v \rangle x_{i}, x_{j}\rangle_{B}) \varphi(b_{j})h \rangle \\
& = \sum_{i} \sum_{j}\langle h,\varphi(b_{i}^{*})\varphi (\langle _{A}\langle v,v \rangle x_{i}, x_{j}\rangle_{B})\varphi(b_{j})h \rangle \\
& = \sum_{i} \sum_{j}\langle h,\varphi (\langle _{A}\langle v,v \rangle x_{i}b_{i}, x_{j}b_{j}\rangle_{B})h \rangle \\ & = \sum_{i} \sum_{j}\langle h,\varphi(\langle _{A}\langle v,v \rangle^{\frac{1}{2}} x_{i}b_{i},_{A}\langle v,v \rangle^{\frac{1}{2}} x_{j}b_{j}\rangle_{B})h \rangle \\
& = \langle h,\varphi(\langle _{A}\langle v,v \rangle^\frac{1}{2}\sum_{i} x_{i}b_{i},_{A}\langle v,v \rangle^\frac{1}{2} \sum_{j} x_{j}b_{j}\rangle_{B})h \rangle \\
& \leq \|_{A}\langle v,v \rangle^\frac{1}{2}\|^{2}\langle h,\varphi(\langle \sum_{i} x_{i}b_{i}, \sum_{j}x_{j}b_{j}\rangle_{B})h \rangle \\
& = \|v\|_{A}^{2}\langle \sum_{i} x_{i}b_{i}\otimes h,\sum_{j} x_{j}b_{j}\otimes h \rangle \\
& = \|v\|_{A}^{2}\| \sum_{i} x_{i} \otimes \varphi(b_{i})h\|^{2} .
\end{align*}
So $\eta_{v}$ is bounded and $\|\eta_{v}\|^{2}\leq \|v\|_{A}^{2}$.\\
Hence $\eta_{v}$ extends to an operator $Ind_{X}\Phi(v)$ on $X\otimes_{B} H$ and we have\\
\begin{align*}
\langle x \otimes h,Ind_{X}\Phi^{*}(v) Ind_{X}\Phi(v^{'}) x^{'}\otimes h^{'} \rangle & = \langle Ind_{X}\Phi(v)(x\otimes h),Ind_{X}\Phi(v^{'})(x^{'}\otimes h^{'}) \rangle \\
& = \langle v\otimes x\otimes h,v^{'}\otimes x^{'}\otimes h^{'} \rangle \\
& = \langle x\otimes h, Ind_{X}\varphi(_{A}\langle v,v^{'}\rangle)x^{'}\otimes h^{'} \rangle .
\end{align*}
Thus $$\langle Ind_{X}\Phi(v), Ind_{X}\Phi(v^{'}) \rangle=Ind_{X}\Phi^{*}(v) Ind_{X}\Phi(v^{'})=Ind_{X}\varphi(_{A}\langle v,v^{'}\rangle).$$So $Ind_{X}\Phi:V \rightarrow B(X\otimes_{B} H,V\otimes_{A} X\otimes_{B} H)$ is an $Ind_{X}\varphi$-morphism and hence a representation of $V$.\\
Now we show that $Ind_{X}\Phi$ is non-degenerate. For this we must to show that\\ $\overline{Ind_{X}\Phi(V)X\otimes_{B} H}$=$V\otimes_{A} X\otimes_{B} H$ and $\overline{Ind_{X}\Phi(V)^{*}(V\otimes X\otimes h)}$=$X\otimes H$.\\
By definition of $Ind_{X}\Phi$ it is easy to see that $\overline{Ind_{X}\Phi(V)X\otimes_{B} H}$=$V\otimes_{A} X\otimes_{B} H$.\\
By hypotheses, $A$ acts as adjointable operators on Hilbert $C^{*}$-module $X$ and this action is non-degenerate, that is, $\overline{AX}$=$X$ and $V$ is full, $\overline{\langle V, V\rangle}$=$A$, so $\overline{\langle V,V\rangle X}$=$X$.\\
For all $x\otimes h\in X\otimes_{alg}H$ we have:
$$\|x\otimes h\|^{2}=|\langle h,(\varphi(\langle x,x\rangle_{B}))h\rangle |\leq \|\langle x,x\rangle_{B}\|\|h\|^{2}=\|x\|_{B}^{2}\|h\|^{2},$$
so if $\sum _{A}\langle v_{i},v^{'}_{i}\rangle x_{i}$ approximates $x$, then $\sum _{A}\langle v_{i},v^{'}_{i}\rangle x_{i}\otimes h$ approximates $x\otimes h$.\\
But, $\sum _{A}\langle v_{i},v^{'}_{i}\rangle x_{i}\otimes h$=$\sum Ind_{X}\varphi(_{A}\langle v_{i},v^{'}_{i}\rangle)x_{i}\otimes h$=$\sum Ind_{X}\Phi(v_{i})^{*}Ind_{X}\Phi(v^{'}_{i})(x_{i}\otimes h)$.\\
So every elementary tensor $x\otimes h$ in $X\otimes_{alg}H$ can be approximated by a sum of the form $\sum Ind_{X}\Phi(v_{i})^{*}Ind_{X}\Phi(v^{'}_{i})(x_{i})\otimes h$=$\sum Ind_{X}\Phi(v_{i})^{*}(v^{'}_{i}\otimes x_{i}\otimes h)$.\\
Thus $Ind_{X}\Phi$ is a non-degenerate representation.
\end{proof}
\begin{defn}
We call the representation $ Ind_{X}\Phi$ constructed above, the Rieffel-induced representation from $W$ to $V$ via $X$.
\end{defn}
\begin{prop}
Let $\Phi_{1} :W\rightarrow B(H_{1},K_{1})$ and $\Phi_{2} :W\rightarrow B(H_{2},K_{2})$ be two non-degenerate
representations. If $\Phi_{1}$ and $\Phi_{2}$ are unitarily equivalent, then $Ind_{X}\Phi_{1}$ and $Ind_{X}\Phi_{2}$ are unitarily equivalent.
\end{prop}
\begin{proof} Suppose $U_{1}:H_{1}\rightarrow H_{2}$ and $U_{2}:K_{1}\rightarrow K_{2}$ be unitary operators
such that $U_{2}\Phi_{1}(w)=\Phi_{2}(w)U_{1}$. Then $id_{X}\otimes U_{1}:X\otimes_{alg} H_{1}\rightarrow X\otimes_{alg} H_{2}$ given by $x\otimes h\mapsto x\otimes U_{1}(h)$ and $id_{V}\otimes id_{X}\otimes U_{2}:V\otimes_{A} X\otimes_{alg} H_{1}\rightarrow V\otimes_{A} X\otimes_{alg} H_{2}$ given by $v\otimes x\otimes h\mapsto v\otimes x\otimes U_{1}(h)$ may be extended to unitary operators $V_{1}$ from $X\otimes_{B} H_{1} $ onto $X\otimes_{B} H_{2} $ and $V_{2}$ from $V\otimes_{A}X\otimes_{B} H_{1}$ onto $V\otimes_{A}X\otimes_{B} H_{1}$ and moreover, $V_{2}Ind_{X}\Phi_{1}(v)=Ind_{X}\Phi_{2}(v)V_{1}$. So $Ind_{X}\Phi_{1}$ and $Ind_{X}\Phi_{2}$ are unitarily equivalent.
\end{proof}
\begin{cor}
Suppose $\Phi:W \rightarrow B(H,K)$ and $\oplus_{s}\Phi_{s}:W \rightarrow B(\oplus_{s}H_{s},K)$ are unitary equivalent, then $Ind_{X}\Phi:V\rightarrow B(X\otimes_{B}H,V\otimes_{A}X\otimes_{B}H)$ is unitary equivalent to $\oplus_{s}Ind_{X}\Phi_{s}:V\rightarrow B(X\otimes_{B}\oplus_{s}H_{s},V\otimes_{A}X\otimes_{B}\oplus_{s}H_{s})$.\\
\end{cor}
\begin{defn}[ \cite{JOI}, Definition 2.1 ]
Two Hilbert $C^*$-modules $V$ and $W$ , respectively, over $C^*$-algebras
$A$ and $B$ are called Morita equivalent, if the $C^*$-algebras $K(V)$ and $K(W)$ are Morita equivalent as $C^*$-algebras.
\end{defn}
It is well known that for Hilbert $C^*$-module $V$, $K(V)$ is Morita equivalent to $\overline{\langle V, V\rangle}$, so if $V$ and $W$ are full, then they are Morita equivalent if and only if their underlying $C^*$-algebras are Morita equivalent [ \cite{JOI}, Proposition 2.8 ].\\
The following theorem show that there is a bijection between non-degenerate representations of two Morita equivalent full Hilbert $C^*$-modules. The fullness property is not crucial, if necessary we can replace underlying $C^*$-algebra by a suitable ones, thus two Morita equivalent Hilbert modules in the above sense have same categories of non-degenerate representations.
\begin{thm} \label{thm:cor}
Suppose that $X$ is an $A-B$-imprimitivity bimodule, and $\Phi$ and $\Psi$ are non-degenerate representation of $W$ and $V$, respectively. Then $Ind_{\widetilde{X}}(Ind_{X}\Phi)$ is naturally unitary equivalent to $\Phi$, and $Ind_{X}(Ind_{\widetilde{X}}\Psi)$ is naturally unitary equivalent to $\Psi$.
\end{thm}
\begin{proof} If $\Phi:W\rightarrow B(H,K)$ is a non-degenerate representation by proposition \ref{prop:ind}, $Ind_{X}\Phi:V\rightarrow B(X\otimes_{B}H,V\otimes_{A}X\otimes_{B}H)$ is a non-degenerate representation of $V$.\\ Again usage of proposition \ref{prop:ind} to $Ind_{X}\Phi$ instead of $\Phi$, give us the following non-degenerate representation of $W$, $$Ind_{\widetilde{X}}(Ind_{X}\Phi):W\rightarrow B(\widetilde{X}\otimes_{A}X\otimes_{B}H,W\otimes_{B}\widetilde{X}\otimes_{A}X\otimes_{B}H).$$ Now we want to show that $\Phi$ is unitary equivalent to $Ind_{\widetilde{X}}(Ind_{X}\Phi)$. By the proof of Theorem 3.29 \cite{RAE}; $U_{1}:\widetilde{X}\otimes_{A}X\otimes_{B}H\rightarrow H$ defined by $b(x)\otimes y\otimes h\mapsto \varphi(\langle x,y \rangle_{B})h$ is a unitary operator.\\ We define $$U_{2}:W\otimes_{B}\widetilde{X}\otimes_{A}X\otimes_{B}H\rightarrow K$$ given by $$w\otimes b(x)\otimes y\otimes h\mapsto\Phi(w)\varphi(\langle x,y \rangle_{B})h.$$ $U_{2}$ is a unitary operator, and we have,
\begin{align*}
U_{2}Ind_{\widetilde{X}}(Ind_{X}\Phi(w))(b(x)\otimes y\otimes h)&=U_{2}(w\otimes \varphi(\langle x,y\rangle_{B})h)\\&=\Phi(w)\varphi(\langle x,y \rangle_{B})h \\&= \Phi(w)U_{1}(b(x)\otimes y\otimes h).
\end{align*}
So $$U_{2}Ind_{\widetilde{X}}(Ind_{X}\Phi(w))=\Phi(w)U_{1}.$$ Hence $\Phi$ and $Ind_{\widetilde{X}}(Ind_{X}\Phi)$ are unitary equivalent.\\
For the equivalence of $\Psi$ and $Ind_{X}(Ind_{\widetilde{X}}\Psi)$, apply the first part to $_{B}\widetilde{X}_{A}$ instead of $_{A}X_{B}$.
\end{proof}
The Financial Support of the Research Council of Damghan University
with the Grant Number 92093 is Acknowledged.
\end{document}
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\begin{document}
\begin{abstract}
In 2008, Haglund, Morse and Zabrocki \cite{classComp} formulated a Compositional form of the Shuffle Conjecture of Haglund {\em et al.} \cite{Shuffle}. In very recent work, Gorsky and Negut by combining their discoveries \cite{GorskyNegut}, \cite{NegutFlags} and \cite{NegutShuffle}, with the work of Schiffmann-Vasserot \cite{SchiffVassMac} and \cite{SchiffVassK} on the symmetric function side and the work of Hikita \cite{Hikita} and Gorsky-Mazin \cite{GorskyMazin} on the combinatorial side, were led to formulate an infinite family of conjectures that extend the original Shuffle Conjecture of \cite{Shuffle}. In fact, they formulated one conjecture for each pair $(m,n)$ of coprime integers. This work of Gorsky-Negut leads naturally to the question as to where the Compositional Shuffle Conjecture of Haglund-Morse-Zabrocki fits into these recent developments. Our discovery here is that there is a compositional extension of the Gorsky-Negut Shuffle Conjecture for each pair $(km,kn)$, with $(m,n)$ co-prime and $k > 1$.
\end{abstract}
\title{Compositional \lowercase{$(km,kn)$}
\operatorname{comp}arskip=0pt
{ \setcounter{tocdepth}{1}\operatorname{comp}arskip=0pt\footnotesize Teofcontents}
\operatorname{comp}arskip=8pt
\operatorname{comp}arindent=20pt
\section*{Introduction}
The subject of the present investigation has its origin, circa 1990, in a effort to obtain a representation theoretical setting for the Macdonald $q,t$-Kotska coefficients. This effort culminated in Haimain's proof, circa 2000, of the $n!$ conjecture (see \cite{natBigraded})
by means of the Algebraic Geometry of the Hilbert Scheme.
In the 1990's, a concerted effort by many researchers led to a variety of conjectures tying the theory of Macdonald Polynomials to
the representation theory of Diagonal Harmonics and the combinatorics
of parking functions. More recently, this subject has been literally flooded
with connections with other areas of mathematics such as: the Elliptic Hall Algebra of Shiffmann-Vasserot, the Algebraic Geometry of Springer Fibers of Hikita, the Double Affine Hecke Algebras of Cherednik, the HOMFLY polynomials, and the truly fascinating Shuffle Algebra of symmetric functions. This has brought to the fore a variety of symmetric function operators with close connection to the extended notion of \define{rational parking functions}. The present work results from an ongoing effort to express and deal with these new developments in a
language that is more accessible to the algebraic
combinatorial audience.
This area of investigation involves many aspects of symmetric function theory, including a central role played by Macdonald polynomials, as well as some of their closely related symmetric function operators.
One of the alluring characteristic features of these operators is that they appear to control in a rather surprising manner combinatorial properties of
rational parking functions. A close investigation of these connections led us to a variety of new discoveries and conjectures in this area which in turn
should open up a variety of open problems in
Algebraic Combinatorics as well as in the above
mentioned areas.
\section{The previous shuffle conjectures}
We begin by reviewing the statement of the Shuffle Conjecture of Haglund {\em et al.} (see \cite{Shuffle}). In Figure \ref{fig:Park2} we have an example of two convenient ways to represent a parking function: a two-line array and a tableau.
\begin{figure}
\caption{Two representations of a parking function}
\label{fig:Park2}
\end{figure}
The tableau on the right is constructed by first choosing a \hbox{Dyck path.} Recall that this is a path in the $n\times n$ lattice square that goes from $(0,0)$ to $(n,n)$ by \define{north} and \define{east} steps, always remaining weakly above the main diagonal (the shaded cells). The lattice cells adjacent and to the east of north steps are filled with {\define{cars} $ 1,2,\dots,n $} in a column-increasing manner. The numbers on the top of the two-line array are the cars as we read them by rows, from bottom to top. The numbers on the bottom of the two line array are the \define{area} numbers, which are obtained by successively counting the number of lattice cells between a \define{north} step and the main diagonal.
All the necessary statistics of a parking function $\operatorname{comp}ark$ can be immediately obtained from the corresponding two line array
$$
\operatorname{comp}ark:= \Big[\begin{matrix}
v_1 & v_2 & \cdots & v_n\\[-4pt]
u_1 & u_2 & \cdots & u_n\\
\end{matrix}\Big].$$
To begin we let
\begin{eqnarray*}
\operatorname{area}(\operatorname{comp}ark)&:=&\sum_{i=1}^n u_i,
\qquad {\rm and}\\
\operatorname{dinv}(\operatorname{comp}ark)&:=&
\sum_{1 \leq i<j \leq n } \raise 2pt\hbox{\large$\chi$}\big(\,( u_i=u_j\ \&\ v_i<v_j)\quad {\rm or}\quad
(u_i=u_j+1\ \&\ v_i>v_j)\,\big),
\end{eqnarray*}
where $\raise 2pt\hbox{\large$\chi$}(-)$ denotes the function that takes value $1$ if its argument is true, and $0$ otherwise.
Next we define $\sigma(\operatorname{comp}ark)$ to be the permutation obtained by successive right to left readings of the components of the vector $(v_1 , v_2 , \dots , v_n)$ according to decreasing values of $u_1 , u_2 , \dots, u_n$. Alternatively, $\sigma(\operatorname{comp}ark)$ is also obtained by reading the cars, in the tableau, from right to left by diagonals and from the highest diagonal to the lowest. Finally, we denote by $\operatorname{ides}(\operatorname{comp}ark)$ the descent set of the inverse of $\sigma(\operatorname{comp}ark)$.
This given, in \cite{Shuffle} Haglund {\em et al.} stated the following.
\begin{conj}[HHLRU-2005] \label{conjHHLRU}
For all $n \geq 1$,
\begin{equation} \label{HHLRU}
\nabla e_n(\mathbf{x}) \,=\, \sum_{\operatorname{comp}ark \in\mathrm{Park}_n } t^{\operatorname{area}(\operatorname{comp}ark) }q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)}[\mathbf{x}]
\end{equation}
\end{conj}
Here, $\mathrm{Park}_n$ stands for the set of all parking functions in the $n\times n$ lattice square. Moreover, for a subset $S \subseteq \{1,2,\cdots, ,n-1\}$, we denote by $F_S[\mathbf{x}]$ the corresponding Gessel fundamental quasi-symmetric function homogeneous of degree $n$ (see \cite{Gessel}). Finally, $\nabla$ is the symmetric function\footnote{See Section~\ref{secsymm} for a quick review of the usual tools for calculations with symmetric functions, and operators on them.} operator introduced in \cite{SciFi}, with eigenfunctions the modified Macdonald polynomial basis $\{\widetilde{H}_\mu[\mathbf{x};q,t] \}_{\mu}$, indexed by partitions $\mu$.
Let us now recall that a result of Gessel implies
that
a homogeneous symmetric function $f[\mathbf{x}]$ of degree $n$ has an expansion of the form
\begin{displaymath}
f[\mathbf{x}] \,=\, \sum_{\sigma \in S_n} c_\sigma F_{\operatorname{ides}(\sigma)}[\mathbf{x}],
\qquad\qquad ( \, \operatorname{ides}(\sigma) = \operatorname{des}(\sigma^{-1}) \, )
\end{displaymath}
if and only, if for all partitions $\mu =(\mu_1,\mu_2,\dots \mu_k)$ of $n$, we have
\begin{displaymath}
\left\langle f \,,\, h_\mu \right\rangle
\,=\, \sum_{\sigma \in S_n} c_\sigma \, \raise 2pt\hbox{\large$\chi$} \big(\sigma \in E_1\shuffle E_2 \shuffle \cdots \shuffle E_k \big),
\end{displaymath}
where $\langle- \,,\,- \rangle$ denotes the Hall scalar product of symmetric functions, $E_1, E_2, \dots, E_k$ are successive segments of the word $123\cdots n$ of respective lengths $\mu_1,\mu_2,\dots ,\mu_k$,
and the symbol $E_1\shuffle E_2 \shuffle \cdots \shuffle E_k$ denotes the collection of permutations obtained by shuffling in all possible ways the words $E_1,E_2,\dots ,E_k$.
Thus \operatorname{comp}ref{HHLRU} may be restated as
\begin{equation} \label{I.2}
\left\langle \nabla e_n \,,\, h_\mu \right\rangle
\, = \sum_{\operatorname{comp}ark \in\mathrm{Park}_n} t^{\operatorname{area} (\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} \raise 2pt\hbox{\large$\chi$} \big( \sigma(\operatorname{comp}ark) \in E_1\shuffle E_2 \shuffle \cdots \shuffle E_k \big)
\end{equation}
for all $ \mu \vdash n$, which is the original form of the Shuffle Conjecture. Recall that it is customary to write $\mu\vdash n$, when $\mu$ is a partition of $n$.
For about 5 years from its formulation, the Shuffle Conjecture appeared untouchable for lack of any recursion satisfied by both sides of the equality. However, in the fall of 2008, Haglund, Morse, and Zabrocki \cite{classComp} made a discovery that is nothing short of spectacular. They discovered that two slight deformations $\BC_a$ and $\BB_a$ of the well-known Hall-Littlewood operators, combined with $\nabla$, yield considerably finer versions of the Shuffle conjecture. For any $a\in\mathbb{N}$, they define the operators $\BC_a$ and $\BB_a$, acting on symmetric polynomials $f[\mathbf{x}]$, as follows.
\begin{eqnarray}
\BC_a f[\mathbf{x}] &=& (-q)^{1-a} f\!\left[\mathbf{x}-{(q-1)}/{(qz)}\right]\, \sum_{m \geq 0} z^m h_m[\mathbf{x}]\, \Big|_{z^a}, \qquad {\rm and} \label{defC}\\
\BB_b f[\mathbf{x}] &=& f\!\left[ \mathbf{x} + \epsilon\,(1-q)/z\right]\, \sum_{m \geq 0} z^m e_m[\mathbf{x}]\, \Big|_{z^b},
\end{eqnarray}
where $(-)\big|_{z^a}$ means that we take the coefficient of $z^a$ in the series considered. We use here ``plethystic'' notation which is described in more details in Section~\ref{secsymm}.
Haglund, Morse, and Zabrocki also introduce a new statistic on paths (or parking functions), the \define{return composition}
$$\operatorname{comp}(\operatorname{comp}ark)=(a_1,a_2,\dots, a_\ell),$$
whose parts are the sizes of the intervals between successive diagonal hits of the Dyck path of $\operatorname{comp}ark$, reading from left to right. As usual we write $\alpha\models n$, when $\alpha$ is a composition of $n$, {\em i.e.} $n=a_1+\ldots+a_\ell$ with the $a_i$ positive integers, and set $\BC_\alpha$ for the product $\BC_{a_1} \BC_{a_2} \cdots \BC_{a_\ell}$, with a similar convention for $\BB_\alpha$.
This given, their discoveries led them to state the following two conjectures\footnote{To make it clear that we are applying an operator to a constant function such as ``$\mathbf{1}$'', we add a dot between this operator and its argument.}.
\begin{conj} [HMZ-2008] \label{conjHMZC}
For any composition $\alpha$ of $n$,
$$
\nabla \BC_\alpha \cdot\mathbf{1} \,=\sum_{\operatorname{comp}(\operatorname{comp}ark)=\alpha} t^{\operatorname{area}(\operatorname{comp}ark)}q^{\operatorname{dinv}(\operatorname{comp}ark)}F_{\operatorname{ides}(\operatorname{comp}ark)}[\mathbf{x}],
$$
where the sum is over parking functions in the $n\times n$ lattice square with composition \hbox{equal to $\alpha$.}
\end{conj}
\begin{conj} [HMZ-2008]\label{conjHMZB}
For any composition $\alpha$ of $n$,
$$
\nabla \BB_\alpha \cdot\mathbf{1} \,=\sum_{\operatorname{comp}(\operatorname{comp}ark)\operatorname{comp}receq \alpha}t^{\operatorname{area}(\operatorname{comp}ark)}q^{\operatorname{dinv}_\alpha(\operatorname{comp}ark)}F_{\operatorname{ides}(\operatorname{comp}ark)}[\mathbf{x}],
$$
where ``$\operatorname{dinv}_\alpha$'' is a suitably $\alpha$-modified $\operatorname{dinv}$ statistic, and the sum is over parking functions in the $n\times n$ lattice square with composition finer than $\alpha$.
\end{conj}
We first discuss Conjecture \ref{conjHMZC}, referred to as the Compositional Shuffle Conjecture, and we will later come back to Conjecture 1.3. Yet another use of Gessel's Theorem shows that Conjecture \ref{conjHMZC} is equivalent to the family of identities
\begin{equation} \label{I.6}
\left\langle \nabla \BC_\alpha \cdot\mathbf{1}, h_\mu\right\rangle =
\sum_{\operatorname{comp}(\operatorname{comp}ark)=\alpha}
t^{\operatorname{area}(\operatorname{comp}ark)}q^{\operatorname{dinv}(\operatorname{comp}ark)} \raise 2pt\hbox{\large$\chi$} \big( \sigma(\operatorname{comp}ark) \in E_1\shuffle E_2 \shuffle \cdots \shuffle E_k \big),
\end{equation}
where, as before, the parts of $\mu$ correspond to the cardinalities of the $E_i$.
The fact that Conjecture \ref{conjHMZC} refines the Shuffle Conjecture is due to the identity
\begin{equation} \label{I.7}
\sum_{\alpha \models n} \BC_\alpha \cdot\mathbf{1} \,=\, e_n,
\end{equation}
hence summing \operatorname{comp}ref{I.6} over all compositions $\alpha\models n$ we obtain \operatorname{comp}ref{I.2}.
Our main contribution here is to show that a suitable extension of the Gorsky-Negut Conjectures (NG Conjectures) to the non-coprime case leads to the formulation of an infinite variety of new Compositional Shuffle conjectures, widely extending both the NG and the HMZ Conjectures. To state them we need to briefly review the Gorsky-Negut Conjectures in a manner that most closely resembles the classical Shuffle conjecture.
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\section{The coprime case} \label{sec:coprime}
Our main actors on the symmetric function side are the operators $\D_k$ and $\D_k^*$, introduced in \cite{qtCatPos}, whose action on a symmetric function $f[\mathbf{x}]$ are defined by setting respectively
\begin{eqnarray}
\D_k f[\mathbf{x}] &:=& f\!\left[\mathbf{x}+{M}/{z}\right] \sum_{i \geq 0}(-z)^i e_i[\mathbf{x}] \Big|_{z^k},\qquad {\rm and} \label{defD}\\
\D_k^*f[\mathbf{x}]&:=& f\!\left[\mathbf{x}-{\widetilde{M}}/{z}\right] \sum_{i \geq 0}z^i h_i[\mathbf{x}] \Big|_{z^k}.\label{defDetoile}
\end{eqnarray}
with $M:=(1-t)(1-q)$ and $\widetilde{M}:=(1-1/t)(1-1/q)$.
The focus of the present work is the algebra of symmetric function operators generated by the family $\{ \D_k \}_{k \geq 0}$. Its connection to the algebraic geometrical developments is that this algebra is a concrete realization of a portion of the Elliptic Hall Algebra studied Schiffmann and Vasserot in \cite{elliptic2}, \cite{SchiffVassMac}, and \cite{SchiffVassK}. Our conjectures are expressed in terms of a family of operators $\Qop_{a,b}$ indexed by pairs of positive integers $a,b$. Here, and in the following, we use the notation $\Qop_{km,kn}$, with $(m,n)$ a coprime pair of non-negative integers and $k$ an arbitrary positive integer. In other words, $k$ is the greatest common divisor of $a$ and $b$, and $(a,b)=(km,kn)$.
\begin{wrapfigure}[8]{R}{2.8cm} \centering
\vskip-5pt
\operatorname{des}sin{height=1.5 in}{dia35.pdf}
\end{wrapfigure}
Restricted to the coprime case, the definition of the operators $\Qop_{m,n}$ is first illustrated in a special case. For instance, to obtain $\Qop_{3,5}$ we start by drawing the $3\times 5$ lattice with its diagonal (the line $(0,0) \to (3,5)$) as depictedf in the adjacent figure. Then we look for the lattice point $(a,b)$ that is closest to and below the diagonal. In this case $(a,b)=(2,3)$. This yields the decomposition $(3,5)=(2,3)+(1,2)$, and unfolding the recursivity we get
\begin{eqnarray}
\Qop_{3,5} &=& \frac{1}{M} \left[ \Qop_{1,2} , \Qop_{2,3} \right]\nonumber\\
&=& \frac{1}{M} \left(
\Qop_{1,2} \Qop_{2,3} - \Qop_{2,3} \Qop_{1,2} \right).\displaywidth=\parshapelength\numexpr\prevgraf+2\relax \label{1.2}
\end{eqnarray}
\begin{wrapfigure}[5]{R}{2.8cm} \centering
\vskip-15pt
\operatorname{des}sin{height=1 in}{dia23.pdf}
\end{wrapfigure}
We must next work precisely in the same way with the $2\times 3$ rectangle, as indicated in the adjacent figure. We obtain the decomposition $(2,3)=(1,1)+(1,2)$ and recursively set
\begin{equation} \displaywidth=\parshapelength\numexpr\prevgraf+2\relax \label{1.3}
\Qop_{2,3} = \frac{1}{M} \left[ \Qop_{1,2} , \Qop_{1,1} \right]
= \frac{1}{M} \left(
\Qop_{1,2} \Qop_{1,1} - \Qop_{1,1} \Qop_{1,2} \right).
\end{equation}
Now, in this case, we are done, since it turns out that we can set
\begin{equation} \label{1.4}
\Qop_{1,k} = \D_k.
\end{equation}
In particular by combining \operatorname{comp}ref{1.2}, \operatorname{comp}ref{1.3} and \operatorname{comp}ref{1.4} we obtain
\begin{equation} \label{1.5}
\Qop_{3,5}
= \frac{1}{M^2}
\left( \D_2 \D_2 \D_1 - 2 \D_2 \D_1 \D_2 + \D_1 \D_2 \D_2 \right).
\end{equation}
To give a precise general definition of the $\Qop$ operators we use the following elementary number theoretical characterization of the closest lattice point $(a,b)$ below the line $(0,0) \to (m,n)$. We observe that by construction $(a,b)$ is coprime. See \cite{newPleth} for a proof.
\begin{prop} \label{prop3.3}
For any pair of coprime integers $m,n> 1$ there is a unique pair $a,b$ satisfying the following three conditions
\begin{equation} \label{3.18}
(1) \quad 1\leq a\leq m-1, \qquad\qquad
(2) \quad 1\leq b\leq n-1, \qquad\qquad
(3) \quad mb+1=na
\end{equation}
In particular, setting $(c,d):=(m,n)-(a,b)$ we will write, for $m,n>1$,
\begin{equation} \label{3.19}
\spl(m,n):= (a,b)+(c,d).
\end{equation}
Otherwise, we set
\begin{equation} \label{3.20}
a) \quad \spl(1,n):=(1,n-1)+(0,1),
\qquad
b) \quad \spl(m,1):=(1,0)+(m-1,1).
\end{equation}
\end{prop}
All pairs considered being coprime, we are now in a position to give the definition of the operators $\Qop_{m,n}$ (restricted for the moment to the coprime case) that is most suitable in the present writing.
\begin{defn} \label{defnQ}
For any coprime pair $(m,n)$, we set
\begin{equation} \label{eqdefnQ}
\Qop_{m,n} :=
\begin{cases}
\frac{1}{M}[\Qop_{c,d},\Qop_{a,b}] & \hbox{if }m>1 \hbox{ and } \spl(m,n)=(a,b)+(c,d), \\[6pt]
\D_n & \hbox{if }m=1.
\end{cases}
\end{equation}
\end{defn}
The combinatorial side of the upcoming conjecture is constructed in \cite{Hikita} by Hikita as the Frobenius characteristic of a bi-graded $S_n$ module whose precise definition is not needed in this development. For our purposes it is sufficient to directly define the \define{Hikita polynomial}, which we denote by $H_{m,n}[\mathbf{x};q,t]$, using a process that closely follows our present rendition of the right hand side of \operatorname{comp}ref{HHLRU}. That is, we set
\begin{equation} \label{defHikita}
H_{m,n}[\mathbf{x};q,t] :=
\sum_{\operatorname{comp}ark \in\mathrm{Park}_{m,n} } t^{\operatorname{area}(\operatorname{comp}ark) } q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)}[\mathbf{x}],
\end{equation}
with suitable definitions for all the ingredients occurring in this formula. We will start
with the collection of $(m,n)$-\define{parking functions} which we have denoted $\mathrm{Park}_{m,n}$. Again, a simple example will suffice.
\begin{figure}
\caption{First combinatorial ingredients for the Hikita polynomial.}
\label{fig:1to3}
\end{figure}
Figures \ref{fig:1to3} and \ref{fig:4to5} contain all the information needed to construct the polynomial $H_{7,9}[\mathbf{x};q,t]$. The first object in Figure~\ref{fig:1to3} is a $5 \times 7$ lattice rectangle with its main diagonal $(0,0) \to (5,7)$. In a darker color we have the lattice cells cut by the main diagonal, which we will call the \define{lattice diagonal}. Because of the coprimality of $(m,n)$, the main diagonal, and any line parallel to it, can touch at most a single lattice point inside the $m \times n$ lattice. Thus the main diagonal (except for its end points) remains interior to the lattice cells that it touches. Since the path joining the centers of the touched cells has $n-1$ north steps and $m-1$ east steps, it follows that the lattice diagonal has $m+n-1$ cells. This gives that the number of cells above (or below) the lattice diagonal is $(m-1)(n-1)/2$.
A path in the $m \times n$ lattice that proceeds by north and east steps from $(0,0)$ to $(m,n)$, always remaining weakly above the lattice diagonal, is said to be an $(m,n)$-Dyck path. For example, the second object in Figure \ref{fig:1to3} is a $(5,7)$-Dyck path. The number of cells between a path $\operatorname{comp}ath$ and the lattice diagonal is denoted $\operatorname{area}(\operatorname{comp}ath)$. In the third object of Figure \ref{fig:1to3}, we have an $(11,10)$-Dyck path. Notice that the collection of cells above the path may be viewed as an english Ferrers diagram. We also show there the \define{leg} and the \define{arm} of one of its cells (see Section~\ref{secsymm} for more details). Denoting by $\lambda(\operatorname{comp}ath)$ the Ferrers diagram above the path $\operatorname{comp}ath$, we define
\begin{equation} \label{1.8}
\operatorname{dinv}(\operatorname{comp}ath) := \sum_{c \in \lambda(\operatorname{comp}ath)} \raise 2pt\hbox{\large$\chi$}\!\left( \frac{\operatorname{arm}(c)}{\operatorname{leg}(c)+1} < \frac{m}{n} < \frac{\operatorname{arm}(c)+1}{\operatorname{leg}(c)}\right).
\end{equation}
As in the classical case an $(m,n)$-parking function is the tableau obtained by labeling the cells east of and adjacent to the north steps of an $(m,n)$-Dyck path with cars $1,2,\dots,n$ in a column-increasing manner. We denote by $\mathrm{Park}_{m,n}$ the set of $(m,n)$-parking functions. When $(m,n)$ is a pair of coprime integers, it is easy to show that there are $m^{n-1}$ such parking functions. For more on the coprime case, see \cite{armstrong}. We will discuss further aspects of the more general case in \cite{newPleth}, a paper in preparation.
\begin{figure}
\caption{Last combinatorial ingredients for the Hikita polynomial.}
\label{fig:4to5}
\end{figure}
The first object in Figure \ref{fig:4to5} gives a $(7,9)$-parking function and the second object gives a $7 \times 9$ table of \define{ranks}. In the general case, this table is obtained by placing in the \define{north-west} corner of the $m\times n$ lattice a number of one's choice. Here we have used $47=(m-1)(n-1)-1$, but the choice is immaterial. We then fill the cells by subtracting $n$ for each east step and adding $m$ for each north step. Denoting by $\operatorname{rank}(i)$ the \define{rank} of the cell that contains car $i$, we define the \define{temporary dinv} of an $(m,n)$-Parking function $\operatorname{comp}ark$ to be the statistic
\begin{equation} \label{1.9}
\operatorname{tdinv}(\operatorname{comp}ark) := \sum_{1\leq i<j \leq n} \raise 2pt\hbox{\large$\chi$}\!\left( \operatorname{rank}(i)< \operatorname{rank}(j) < \operatorname{rank}(i)+m \right).
\end{equation}
Next let us set for any $(m,n)$-path $\operatorname{comp}ath$
\begin{equation} \label{1.10}
\operatorname{maxtdinv}(\operatorname{comp}ath) := \max \{ \operatorname{tdinv}(\operatorname{comp}ark) : \operatorname{comp}ath(\operatorname{comp}ark)=\operatorname{comp}ath \},
\end{equation}
where the symbol $\operatorname{comp}ath(\operatorname{comp}ark)=\operatorname{comp}ath$ simply means that $\operatorname{comp}ark$ is obtained by labeling the path $\operatorname{comp}ath$. It will also be convenient to refer to $\operatorname{comp}ath(\operatorname{comp}ark)$ as the \define{support} of $\operatorname{comp}ark$. This given, we can now set
\begin{equation} \label{1.11}
\operatorname{dinv}(\operatorname{comp}ark) := \operatorname{dinv}(\operatorname{comp}ath(\operatorname{comp}ark)) + \operatorname{tdinv}(\operatorname{comp}ark) - \operatorname{maxtdinv}(\operatorname{comp}ath(\operatorname{comp}ark)).
\end{equation}
This is a reformulation of Hikita's definition of the \define{dinv} of an $(m,n)$-parking function first introduced by Gorsky and Mazin in \cite{GorskyMazin}.
To complete the construction of the Hikita polynomials we need the notion of the \define{word} of a parking function, which we denote by $\sigma(\operatorname{comp}ark)$. This is the permutation obtained by reading the cars of $\operatorname{comp}ark$ by decreasing ranks. Geometrically, $\sigma(\operatorname{comp}ark)$ can be obtained simply by having a line parallel to the main diagonal sweep the cars from left to right, reading a car the moment the moving line passes through the south end of its adjacent north step. For instance, for the parking function in Figure \ref{fig:4to5} we have $\sigma(\operatorname{comp}ark)= 784615923.$
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A path in the $m \times n$ lattice that proceeds by north and east steps from $(0,0)$ to $(m,n)$, always remaining weakly above the lattice diagonal, is said to be an $(m,n)$-Dyck path. For example, the second object in Figure \ref{fig:1to3} is a $(5,7)$-Dyck path. The number of cells between a path $\operatorname{comp}ath$ and the lattice diagonal is denoted $\operatorname{area}(\operatorname{comp}ath)$. In the third object of Figure \ref{fig:1to3}, we have an $(11,10)$-Dyck path. Notice that the collection of cells above the path may be viewed as an english Ferrers diagram. We also show there the \define{leg} and the \define{arm} of one of its cells (see Section~\ref{secsymm} for more details). Denoting by $\lambda(\operatorname{comp}ath)$ the Ferrers diagram above the path $\operatorname{comp}ath$, we define
\begin{equation} \label{1.8}
\operatorname{dinv}(\operatorname{comp}ath) := \sum_{c \in \lambda(\operatorname{comp}ath)} \raise 2pt\hbox{\large$\chi$}\!\left( \frac{\operatorname{arm}(c)}{\operatorname{leg}(c)+1} < \frac{m}{n} < \frac{\operatorname{arm}(c)+1}{\operatorname{leg}(c)}\right).
\end{equation}
As in the classical case an $(m,n)$-parking function is the tableau obtained by labeling the cells east of and adjacent to the north steps of an $(m,n)$-Dyck path with cars $1,2,\dots,n$ in a column-increasing manner. We denote by $\mathrm{Park}_{m,n}$ the set of $(m,n)$-parking functions. When $(m,n)$ is a pair of coprime integers, it is easy to show that there are $m^{n-1}$ such parking functions. For more on the coprime case, see \cite{armstrong}. We will discuss further aspects of the more general case in \cite{newPleth}, a paper in preparation.
\begin{figure}
\caption{Last combinatorial ingredients for the Hikita polynomial.}
\label{fig:4to5}
\end{figure}
The first object in Figure \ref{fig:4to5} gives a $(7,9)$-parking function and the second object gives a $7 \times 9$ table of \define{ranks}. In the general case, this table is obtained by placing in the \define{north-west} corner of the $m\times n$ lattice a number of one's choice. Here we have used $47=(m-1)(n-1)-1$, but the choice is immaterial. We then fill the cells by subtracting $n$ for each east step and adding $m$ for each north step. Denoting by $\operatorname{rank}(i)$ the \define{rank} of the cell that contains car $i$, we define the \define{temporary dinv} of an $(m,n)$-Parking function $\operatorname{comp}ark$ to be the statistic
\begin{equation} \label{1.9}
\operatorname{tdinv}(\operatorname{comp}ark) := \sum_{1\leq i<j \leq n} \raise 2pt\hbox{\large$\chi$}\!\left( \operatorname{rank}(i)< \operatorname{rank}(j) < \operatorname{rank}(i)+m \right).
\end{equation}
Next let us set for any $(m,n)$-path $\operatorname{comp}ath$
\begin{equation} \label{1.10}
\operatorname{maxtdinv}(\operatorname{comp}ath) := \max \{ \operatorname{tdinv}(\operatorname{comp}ark) : \operatorname{comp}ath(\operatorname{comp}ark)=\operatorname{comp}ath \},
\end{equation}
where the symbol $\operatorname{comp}ath(\operatorname{comp}ark)=\operatorname{comp}ath$ simply means that $\operatorname{comp}ark$ is obtained by labeling the path $\operatorname{comp}ath$. It will also be convenient to refer to $\operatorname{comp}ath(\operatorname{comp}ark)$ as the \define{support} of $\operatorname{comp}ark$. This given, we can now set
\begin{equation} \label{1.11}
\operatorname{dinv}(\operatorname{comp}ark) := \operatorname{dinv}(\operatorname{comp}ath(\operatorname{comp}ark)) + \operatorname{tdinv}(\operatorname{comp}ark) - \operatorname{maxtdinv}(\operatorname{comp}ath(\operatorname{comp}ark)).
\end{equation}
This is a reformulation of Hikita's definition of the \define{dinv} of an $(m,n)$-parking function first introduced by Gorsky and Mazin in \cite{GorskyMazin}.
To complete the construction of the Hikita polynomials we need the notion of the \define{word} of a parking function, which we denote by $\sigma(\operatorname{comp}ark)$. This is the permutation obtained by reading the cars of $\operatorname{comp}ark$ by decreasing ranks. Geometrically, $\sigma(\operatorname{comp}ark)$ can be obtained simply by having a line parallel to the main diagonal sweep the cars from left to right, reading a car the moment the moving line passes through the south end of its adjacent north step. For instance, for the parking function in Figure \ref{fig:4to5} we have $\sigma(\operatorname{comp}ark)= 784615923.$
Letting $\operatorname{ides}(\operatorname{comp}ark)$ denote the \define{descent set of the inverse} of the permutation $\sigma(\operatorname{comp}ark)$ and setting, as in the classical case, $\operatorname{area}(\operatorname{comp}ark) = \operatorname{area}(\operatorname{comp}ath(\operatorname{comp}ark))$, we finally have all of the ingredients necessary for \operatorname{comp}ref{defHikita} to be a complete definition of the Hikita polynomial. The Gorsky-Negut $(m,n)$-Shuffle Conjecture may now be stated as follows.
\begin{conj}[GN-2013] \label{conjNG}
For all coprime pairs of positive integers $(m,n)$, we have
\begin{equation} \label{1.12}
\Qop_{m,n} \cdot\mathbf{1}n = H_{m,n}[\mathbf{x};q,t].
\end{equation}
\end{conj}
Of course we can use the word ``Shuffle'' again since another use of Gessel's theorem allows us to rewrite \operatorname{comp}ref{1.12} in the equivalent form
$$
\left\langle \Qop_{m,n}\cdot\mathbf{1}n , h_\alpha \right\rangle = \hskip -10pt
\sum_{\operatorname{comp}ark \in\mathrm{Park}_{m,n}} \hskip -10pt
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} \raise 2pt\hbox{\large$\chi$} \left( \sigma(\operatorname{comp}ark) \in E_1\shuffle\cdots \shuffle E_k \right),\quad\hbox{for all}\quad \alpha\models n,
$$
where $a_i=|E_i|$ when $\alpha=(a_1,\ldots,a_k)$, and writing $h_\alpha$ for the product $h_{a_1}\cdots h_{a_k}$.
We must point out that it can be shown that \operatorname{comp}ref{1.12} reduces to \operatorname{comp}ref{HHLRU} when $m=n+1$. In fact, it easily follows from the definition in \operatorname{comp}ref{defnQ} that $\Qop_{n+1,n} = \nabla \D_n \nabla^{-1}.$ This, together with the fact that $\nabla^{-1}\cdot\mathbf{1} = 1$ and the definition in \operatorname{comp}ref{defD}, yields $\Qop_{n+1,n} \cdot\mathbf{1}n = \nabla e_n.$ The equality of the right hand sides of \operatorname{comp}ref{1.12} and \operatorname{comp}ref{HHLRU} for $m=n+1$ is obtained by a combinatorial argument which is not too difficult.
\operatorname{comp}agebreak
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\section{Our Compositional \texorpdfstring{$(km,kn)$}--Shuffle Conjectures}
The present developments result from theoretical and computer explorations of what takes place in the non-coprime case. Notice first that there is no difficulty in extending the definition of parking functions to the
$km \times kn$ lattice square, including the $\operatorname{area}$ statistic. Problems arise in extending the definition of the $\operatorname{dinv}$ and $\sigma$ statistics. Previous experience strongly suggested to use the symmetric function side as a guide to the construction of these two statistics. We will soon see that we may remove the coprimality condition in the definition of the $\Qop$ operators, thus allowing us to consider operators $\Qop_{km,kn}$ which, for $k>1$, may be simply obtained by bracketing two $\Qop$ operators indexed by coprime pairs.
However one quickly discovers, by a simple parking function count, that these $\Qop_{km,kn}$ operators do not provide the desired symmetric function side.
Our search for the natural extension
of the symmetric function side led us to focus on the following general construction of symmetric function operators indexed by non-coprime pairs $(km,kn)$.
This construction is based
on a simple commutator identity satisfied by the operators
$D_k$ and $D_k^*$ which shows that the $Q_{0,k}$ operator
is none other than multiplication by a rescaled plethystic version of the ordinary symmetric function $h_k$.
This implies that
the family
$\{\operatorname{comp}rod_i \Qop_{0,\lambda_i} \}_\lambda$
is a basis for the space of symmetric functions (viewed as multiplication operators).
Our definition also uses a commutativity property
(proved in [4]) between $\Qop$ operators indexed by collinear vectors, {\em i.e.} $\Qop_{km,kn}$ and $\Qop_{jm,jn}$ commute for all $k$, $j$, $m$ and $n$ (see Theorem~\ref{thmQind}).
For our purpose, it is convenient to denote by $\cdot\mathbf{1}derline{f}$ the operator of \define{multiplication by} $f$ for any symmetric function $f$.
We can now give our general construction.
\begin{alg} \label{algF}
Given any symmetric function $f$ that is homogeneous of degree $k$, and any coprime pair $(m,n)$, we proceed as follows
\begin{enumerate}\itemsep=-4pt
\item[]{\bf Step 1:} calculate the expansion
\begin{equation}
f = \sum_{\lambda \vdash k} c_\lambda(q,t)\, \operatorname{comp}rod_{i=1}^{\ell(\lambda)} \Qop_{0,\lambda_i},
\end{equation}
\item[]{\bf Step 2:} using the coefficients $c_\lambda(q,t)$, set
\begin{equation} \label{defnF}
\Bf_{km,kn} := \sum_{\lambda \vdash k} c_\lambda(q,t) \operatorname{comp}rod_{i=1}^{\ell(\lambda)}\Qop_{m \lambda_i, n\lambda_i}.
\end{equation}
\end{enumerate}
\end{alg}
Theoretical considerations reveal, and extensive computer experimentations confirm, that the operators that we should use to extend the \define{rational parking function} theory to all pairs $(km,kn)$, are none other than the operators $\Be_{km,kn}$ obtained by taking $f=e_k$ in Algorithm~\ref{algF}. This led us to look for the construction of natural extensions of the definitions of $\operatorname{dinv}(\operatorname{comp}ark)$ and $\sigma(\operatorname{comp}ark)$, that would ensure the validity of the following sequence of increasingly refined conjectures. The coarsest one of which is as follows.
\begin{conj}\label{conjE} For all coprime pair of positive integers $(m,n)$, and any $k\in\mathbb{N}$, we have
\begin{equation} \label{2.3}
\Be_{km,kn}\cdot {(-\mathbf{1})^{k(n+1)}}= \sum_{\operatorname{comp}ark \in\mathrm{Park}_{km,kn}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)},
\end{equation}
\end{conj}
\begin{wrapfigure}[12]{r}{4cm} \centering
\operatorname{des}sin{height=2.3 in}{multimn.pdf}
\end{wrapfigure}
To understand our first refinement, we focus on a special case. In the figure displayed on the right, we have depicted a $12 \times 20$ lattice. The pair in this case has a $\gcd$ of $4$. Thus here $(m,n)=(3,5)$ and $k=4$. Note that in the general case, $(km,kn)$-Dyck paths can hit the diagonal in $k-1$ places within the $km\times kn$ lattice square. In this case, in $3$ places. We have depicted here a Dyck path which hits the diagonal in the first and third places.
At this point the classical decomposition (discovered in \cite{qtCatPos})
\begin{equation}\label{edecompE}\displaywidth=\parshapelength\numexpr\prevgraf+2\relax
e_k= E_{1,k}+E_{2,k}+\cdots +E_{k,k},
\end{equation}
combined with extensive computer experimentations, suggested that we have the following refinement of Conjecture~\ref{conjE}.
\begin{conj}\label{conjEr} For all coprime pair of positive integers $(m,n)$, all $k\in\mathbb{N}$, and if $1\leq r\leq k$, we have
\begin{equation} \label{2.4}
\BE_{km,kn}^{r}\cdot (-\mathbf{1})^{k(n+1)} = \sum_{\operatorname{comp}ark \in\mathrm{Park}_{km,kn}^{r}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)},
\end{equation}
where $\BE_{km,kn}^{r}$ is the operator obtained by setting $f=E_{k,r}$ in Algorithm~\ref{algF}.
Here $\mathrm{Park}_{km,kn}^{r}$ denotes the set of parking functions, in the $km \times kn$ lattice, whose Dyck path hit the diagonal in $r$ places {\rm (}including $(0,0)${\rm )}.
\end{conj}
Clearly, \operatorname{comp}ref{edecompE} implies that Conjecture~\ref{conjE} follows from Conjecture~\ref{conjEr}. For example, the parking functions supported by the path in the above figure would be picked up by the operator $\BE_{4\times 3,4\times 5 }^{3}$.
Our ultimate refinement is suggested by the decomposition (proved in \cite{classComp})
\begin{equation} \label{2.5}
E_{k,r} = \sum_{\alpha \models k}
C_{\alpha_1}C_{\alpha_2}\cdots C_{\alpha_r}
\cdot\mathbf{1}.
\end{equation}
What emerges is the following most general conjecture that clearly subsumes our two previous conjectures, as well as Conjectures~\ref{conjHHLRU}, \ref{conjHMZC}, and \ref{conjNG}.
\begin{conj}[Compositional $(km,kn)$-Shuffle Conjecture] \label{conjBGLX}
For all compositions $\alpha=(a_1,a_2, \dots ,a_r)\models k$ we have
\begin{equation} \label{2.6}
\BC_{km,kn}^{(\alpha)}\cdot(-\mathbf{1})^{k(n+1)}
= \sum_{\operatorname{comp}ark \in\mathrm{Park}_{km,kn}^{(\alpha)}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)},
\end{equation}
where $\BC_{km,kn}^{(\alpha)}$ is the operator obtained by setting $f=\BC_\alpha\cdot\mathbf{1}$ in Algorithm~\ref{algF} and $\mathrm{Park}_{km,kn}^{(\alpha)}$ denotes the collection of parking functions in the $km\times kn$ lattice whose Dyck path hits the diagonal according to the composition $\alpha$.
\end{conj}
For example, the parking functions supported by the path in the above figure would be picked up by the operator $\BC_{4\times 3 ,4\times 5}^{(1,2,1)}$.
We will later see that an analogous conjecture may be stated for the operator $\BB_{km,kn}^{(\alpha)}$ obtained by taking $f=\BB_\alpha\cdot\mathbf{1}$ in our general Algorithm~\ref{algF}, with $\alpha$ any composition of $k$.
We will make extensive use in the sequel of a collection of results stated and perhaps even proved in the works of Schiffmann and Vasserot.
Unfortunately most of this material is written in a language that is nearly inaccessible to most practitioners of Algebraic Combinatorics. We were fortunate that the two young researchers E. Gorsky and A. Negut, in a period of several months, made us aware of some of the contents of the latter publications as well as the results in their papers (\cite{GorskyNegut}, \cite{NegutFlags} and \cite{NegutShuffle}) in a language we could understand. The present developments are based on these results. Nevertheless, for sake of completeness we have put together in \cite{newPleth} a purely Algebraic Combinatorial treatment of all the background needed here with proofs that use only the Macdonald polynomial ``tool kit'' derived in the 90's in \cite{SciFi}, \cite{IdPosCon}, \cite{plethMac} and\cite{explicit}, with some additional identities discovered in \cite{HLOpsPF}.
The remainder of this paper is divided into three further sections. In the next section we review some notation and recall some identities from Symmetric Function Theory, and our Macdonald polynomial tool kit.
This done, we state some basic identities that will be instrumental in extending the definition of the $\Qop$ operators to the non-coprime case.
In the following section we describe how the modular group $\mathrm{SL}_2(\mathbb{Z})$ acts on the operators $\Qop_{m,n}$ and use this action to justify our definition of the operators $\Qop_{km,kn}$. Elementary proofs that justify the uses we make of this action are given in \cite{newPleth}. Here we also show how these operators can be efficiently programmed on the computer. This done, we give a precise construction of the operators $\BC_{km,kn}^{(\alpha)}$ and $\BB_{km,kn}^{(\alpha)}$, and workout some examples. We also give a compelling argument which shows the inevitability of Conjecture \ref{conjBGLX}.
In the last section we complete our definitions for all the combinatorial ingredients occurring in the right hand sides of \operatorname{comp}ref{2.3}, \operatorname{comp}ref{2.4} and \operatorname{comp}ref{2.6}. Finally, we derive some consequences of our conjectures and discuss some possible further extensions.
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\section{Symmetric function basics, and necessary operators}\label{secsymm}
In dealing with symmetric function identities, especially those arising in the theory of Macdonald Polynomials, it is convenient and often indispensable to use plethystic notation. This device has a straightforward definition which can be implemented almost verbatim in any computer algebra software. We simply set for any expression $E = E(t_1,t_2 ,\dots )$ and any symmetric function $f$
\begin{equation} \label{3.1}
f[E] := \Qop_f(p_1,p_2, \dots )
\Big|_{p_k \to E( t_1^k,t_2^k,\dots )},
\end{equation}
where $(-)\big|_{p_k \to E( t_1^k,t_2^k,\dots )}$ means that we replace each $p_k$ by $E( t_1^k,t_2^k,\dots )$, for $k\geq 1$.
Here $\Qop_f$ stands for the polynomial yielding the expansion of $f$ in terms of the power basis. We say that we have a \define{plethystic substitution} of $E$ in $f$.
The above definition of plethystic substitutions implicitly requires that
$p_k[-E]= -p_k[E]$, and we say that this is the \define{plethystic} minus sign rule. This notwithstanding, we also need to carry out ``ordinary'' changes of signs. To distinguish the later from the plethystic minus sign, we obtain the \define{ordinary} sign change by multiplying our expressions by a new variable ``$\epsilon$'' which, outside of the plethystic bracket, is replaced by $-1$. Thus we have
\begin{displaymath}
p_k[\epsilon E]= \epsilon^k p_k[ E]= (-1)^k p_k[E].
\end{displaymath}
In particular we see that, with this notation, for any expression $E$ and any symmetric function $f$ we have
\begin{equation} \label{3.2}
(\omega f)[E]= f[-\epsilon E],
\end{equation}
where, as customary, ``$\omega$'' denotes the involution that interchanges the elementary and homogeneous symmetric function bases.
Many symmetric function identities can be considerably simplified by means of the $\Omega$-notation, allied with plethystic calculations. For any expression $E = E(t_1,t_2,\cdots )$ set
\begin{displaymath}
\Omega[E] := \exp\! \left(
\sum_{k \geq 1}{p_k[E]\over k}
\right) = \exp \! \left(
\sum_{k \geq 1} \frac{E(t_1^k,t_2^k,\cdots )}{k}
\right).
\end{displaymath}
In particular, for $\mathbf{x}=x_1+x_2+\cdots$, we see that
\begin{equation} \label{3.3}
\Omega[z\mathbf{x}]= \sum_{m \geq 0} z^m h_m[\mathbf{x}]
\end{equation}
and for $M=(1-t)(1-q)$ we have
\begin{equation} \label{3.4}
\Omega[-uM] = \frac{(1-u)(1-qtu)}{(1-tu)(1-qu)}.
\end{equation}
\begin{wrapfigure}[7]{r}{4.5cm} \centering
\vskip-10pt
\operatorname{des}sin{height=1.2 in}{ninesix.pdf}
\begin{picture}(0,0)(-3,0)\setlength{\cdot\mathbf{1}itlength}{3mm}
\operatorname{comp}ut(-2,7.8){$\scriptstyle\operatorname{arm}$}
\operatorname{comp}ut(-7,9){$\scriptstyle\operatorname{coarm}$}
\operatorname{comp}ut(-4.8,6.3){$\scriptstyle\operatorname{leg}$}
\operatorname{comp}ut(-3,10){$\scriptstyle\operatorname{coleg}$}
\end{picture}
\end{wrapfigure}
Drawing the cells of the Ferrers diagram of a partition $\mu$ as in \cite{Macdonald}, For a cell $c$ in $\mu$, (in symbols $c\in\mu$), we have parameters $\operatorname{leg}(c)$, and $\operatorname{arm}(c)$,
which respectively give the number of cells of $\mu$ strictly south of $c$, and strictly east of $c$.
Likewise we have parameters $\operatorname{coleg}(c)$, and $\operatorname{coarm}(c)$, which respectively give the number of cells of $\mu$ strictly north of $c$, and strictly west of $c$.
This is illustrated in the adjacent figure for the partition that sits above a path.
Denoting by $\mu'$ the conjugate of $\mu$, the basic ingredients we need to keep in mind here are
$$\begin{array}{lll}\displaystyle
\displaystyle n(\mu):= \sum_{k=1}^{\ell(\mu)} (k-1) \mu_k, \qquad
& \displaystyle T_\mu:= t^{n(\mu)}q^{n(\mu')},
\qquad M:=(1-t)(1-q),\\[8pt]
\displaystyle B_\mu(q,t):= \sum_{c \in \mu} t^{\operatorname{coleg}(c)} q^{\operatorname{coarm}(c)} ,
&\displaystyle \displaystyle\Pi_\mu(q,t):=\operatorname{comp}rod_{{c\in\mu\atop c\not=(0,0)}} (1-t^{\operatorname{coleg}(c)} q^{\operatorname{coarm}}),
\end{array}$$
and
$$w_\mu(q,t) := \operatorname{comp}rod_{c \in \mu} (q^{\operatorname{arm}(c)} - t^{\operatorname{leg}(c)+1})(t^{\operatorname{leg}(c)} - q^{\operatorname{arm}(c)+1})$$
Let us recall that the Hall scalar product is defined by setting
\begin{displaymath}
\left\langle p_\lambda, p_\mu \right\rangle\ := \
z_\mu \, \chi(\lambda=\mu),
\end{displaymath}
where $z_\mu$ gives the order of the stabilizer of a permutation with cycle structure $\mu$.
The Macdonald polynomials we work with here are the unique (\cite{natBigraded}) symmetric function basis $\{\widetilde{H}_\mu[\mathbf{x};q,t]\}_\mu$ which is upper-triangularly related (in dominance order) to the modified Schur basis $\{s_\lambda[\frac{\mathbf{x}}{t-1}] \}_\lambda$ and satisfies the orthogonality condition
\begin{equation} \label{3.5}
\left\langle \widetilde{H}_\lambda, \widetilde{H}_\mu \right\rangle_* =\ \raise 2pt\hbox{\large$\chi$}(\lambda=\mu)\, w_\mu(q,t),
\end{equation}
where $\left\langle-,- \right\rangle_*$ denotes a deformation of the Hall scalar product defined by setting
\begin{equation} \label{3.6}
\left\langle p_\lambda, p_\mu \right\rangle_*
:= (-1)^{|\mu|-\ell(\mu)} \operatorname{comp}rod_i (1-t^{\mu_i})(1-q^{\mu_i})
\, z_\mu \, \raise 2pt\hbox{\large$\chi$}(\lambda =\mu).
\end{equation}
We will use here the operator $\nabla$, introduced in \cite{SciFi}, obtained by setting
\begin{equation} \label{3.7}
\nabla \widetilde{H}_\mu[\mathbf{x};q,t]= T_\mu\, \widetilde{H}_\mu[\mathbf{x};q,t].
\end{equation}
We also set, for any symmetric function $f[\mathbf{x}]$,
\begin{equation} \label{3.8}
\Delta_f \widetilde{H}_\mu[\mathbf{x};q,t]= f[B_\mu]\, \widetilde{H}_\mu[\mathbf{x};q,t].
\end{equation}
These families of operators were intensively studied in the $90's$ (see \cite{IdPosCon} and \cite{explicit}) where they gave rise to a variety of conjectures, some of which are still open to this date. In particular it is shown in \cite{explicit} that the operators $\D_k$, $\D_k^*$, $\nabla$ and the modified Macdonald polynomials $\widetilde{H}_\mu[\mathbf{x};q,t]$ are related by the following identities.
\begin{equation} \label{formulaoper}
\begin{array}{clcl}
{\rm (i)} & \D_0 \widetilde{H}_\mu = -D_\mu(q,t)\, \widetilde{H}_\mu,
\qquad\qquad &
{\rm (i)}^* & \D_0^* \widetilde{H}_\mu = -D_\mu(1/q,1/t) \widetilde{H}_\mu,
\\[5pt]
{\rm (ii)} & \D_k \underline{e}_1 - \underline{e}_1 \D_k = M \D_{k+1} ,&
{\rm (ii)}^* & \D_k^* \underline{e}_1 - \underline{e}_1 \D_k^* = -\widetilde{M} \D_{k+1}^*,
\\[5pt]
{\rm (iii)} & \nabla \underline{e}_1 \nabla^{-1} = -\D_1, &
{\rm (iii )}^* & \nabla \D_1^* \nabla^{-1} = \underline{e}_1,
\\[5pt]
{\rm (iv)} & \nabla^{-1} \underline{e}_1^\operatorname{comp}erp \nabla = {\textstyle \frac{1}{M}} \D_{-1}, &
{\rm (iv )}^* & \nabla^{-1} \D_{-1}^* \nabla = -{\widetilde M}\, \underline{e}_1^\operatorname{comp}erp,
\end{array}
\end{equation}
with $\underline{e}_1^\operatorname{comp}erp$ denoting the Hall scalar product adjoint of multiplication by $e_1$, and
\begin{equation} \label{formDmu}
D_\mu(q,t)= MB_\mu(q,t)-1.
\end{equation}
We should mention that recursive applications of \operatorname{comp}ref{formulaoper} ${\rm (ii)}$ and ${\rm (ii)}^*$ give
\begin{eqnarray}
\D_{k} &=&
\frac{1}{M^k} \sum_{i=0}^k {k \choose r}(-1)^r \underline{e}_1^r \D_0 \underline{e}_1^{k-r},\qquad{\rm and}\\
\D_{k}^* &=&
\frac{1}{\widetilde{M}^k} \sum_{i=0}^k {k \choose r}(-1)^{k-r} \underline{e}_1^r \D_0^* \underline{e}_1^{k-r}.
\end{eqnarray}
For future use, it is convenient to set
\begin{eqnarray}
\Phi_k &:=& \nabla \D_k \nabla^{-1}
\qquad {\rm and}\label{3.12a} \\
\Psi_k &:=& -(qt)^{1-k} \nabla \D_k^* \nabla^{-1}.\label{3.12b}
\end{eqnarray}
The following identities are then immediate consequences of identities \operatorname{comp}ref{formulaoper}. See \cite{newPleth} for details.
\begin{prop} \label{propphipsi}
The operators $\Phi_k$ and $\Psi_k$ are uniquely determined by the recursions
\begin{equation} \label{3.13}
{\rm a)} \quad \Phi_{k+1} = \frac{1}{M}[ \D_1,\Phi_{k} ]
\qquad {\rm and} \qquad
{\rm b)} \quad \Psi_{k+1} = \frac{1}{M}[\Psi_{k},\D_1]
\end{equation}
with initial conditions
\begin{equation}
{\rm a)}\quad \Phi_1 = \frac{1}{M} [\D_1,\D_0]
\qquad {\rm and} \qquad
{\rm b)}\quad \Psi_1=-e_1.
\end{equation}
\end{prop}
Next, we must include the following fundamental identity, proved in \cite{newPleth}.
\begin{prop} \label{propDD}
For $a,b \in \mathbb{Z}$ with $n=a+b>0$ and any symmetric function $f[\mathbf{x}]$, we have
\begin{equation} \label{3.15}
\frac{1}{M} ( \D_a \D_b^* - \D_b^* \D_a) f[\mathbf{x}] = \frac{(qt)^b}{qt-1} h_{n} \!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] f[\mathbf{x}].
\end{equation}
\end{prop}
As a corollary we obtain the following.
\begin{prop} \label{propcrochetphipsi}
The operators $\Phi_k$ and $\Psi_k$, defined in \operatorname{comp}ref{3.12a} and \operatorname{comp}ref{3.12b}, satisfy the following identity when $a,b$ are any positive integers with sum equal to $n$.
\begin{equation} \label{3.16}
\frac{1}{M} [\Psi_b, \Phi_a] = \frac{qt}{qt-1} \nabla \underline{h}_n \!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] \nabla^{-1}.
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
The identity in \operatorname{comp}ref{3.15} essentially says that under the given hypotheses the operator $\frac{1}{M}(\D_b^* \D_a - \D_a \D_b^*)$ acts as multiplication by the symmetric function $\frac{(qt)^b}{qt-1} h_{n} \!\left[(1-qt)\mathbf{x}/(qt)\right]$. Thus, with our notational conventions, \operatorname{comp}ref{3.15} may be rewritten as
\begin{displaymath}
- \frac{(qt)^{1-b}}{M} \left( \D_b^* \D_a - \D_a \D_b^* \right) = \frac{qt}{qt-1}\ \underline{h}_n\left[\frac{1-qt}{qt}\,\mathbf{x}\right].
\end{displaymath}
Conjugating both sides by $\nabla$, and using
\operatorname{comp}ref{3.12a} and \operatorname{comp}ref{3.12b}, gives \operatorname{comp}ref{3.16}.
\end{proof}
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In the sequel, we will need to keep in mind the following identity which expresses the action of a sequence of $\D_k$ operators on a symmetric function $f[\mathbf{x}]$.
\begin{prop} \label{prop3.2} For all composition $\alpha=(a_1,a_2,\ldots,a_m)$ we have
\begin{equation}
\D_{a_m} \cdots \D_{a_2} \D_{a_1} f[\mathbf{x}]
= f\!\left[\mathbf{x}+{\textstyle{\sum_{i=1}^m {M}/{z_i}}}\right] \,
\frac{\Omega[-\mathbf{z}X] }{\mathbf{z}^\alpha} \,
\operatorname{comp}rod_{1\leq i<j \leq m} \Omega \left[-M \textstyle{z_i/ z_j} \right] \Big|_{\mathbf{z}^0},
\end{equation}
where, for $\mathbf{z}=z_1+\ldots +z_m$, we write $\mathbf{z}^\alpha=z_1^{a_1}\cdots z_m^{a_m}$, and in particular $\mathbf{z}^0=z_1^0z_2^0\cdots z_m^0$.
\end{prop}
\begin{proof}[\bf Proof]
It suffices to see what happens when we use \operatorname{comp}ref{defD} twice.
\begin{eqnarray*}
\D_{a_2} \D_{a_1}f[\mathbf{x}]
&=& \D_{a_2}f\!\left[\mathbf{x}+{\tfrac{M}{z_1}}\right] \Omega[-z_1\mathbf{x}] \Big|_{z_1^{a_1}}\\
&=& f\!\left[\mathbf{x}+{\textstyle \frac{M}{z_1}}+{\tfrac{M}{z_2}}\right]\,\Omega[-z_1(\mathbf{x}+{\tfrac{M}{z_2}})]\Omega[-z_2\mathbf{x}]\Big|_{z_1^{a_1}z_2^{a_2}} \\
&=& f\!\left[\mathbf{x}+{\tfrac{M}{z_1}}+{\tfrac{M}{z_2}}\right]\,\Omega[-z_1\mathbf{x}]\,\Omega[-z_2\mathbf{x}]\, \Omega[-Mz_1/ z_2]
\Big|_{z_1^{a_1}z_2^{a_2}}\\
&=& f\!\left[\mathbf{x}+{\tfrac{M}{z_1}}+{\tfrac{M}{z_2}}\right]\,\frac{\Omega[-(z_1+z_2)\mathbf{x}]}{z_1^{a_1}z_2^{a_2}}\, \Omega[-Mz_1/ z_2]
\Big|_{z_1^0z_2^0},
\end{eqnarray*}
and the pattern of the general result clearly emerges.
\end{proof}
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\section{The \texorpdfstring{$\mathrm{SL}_2[\mathbb{Z}]$}--action and the \texorpdfstring{$ \Qop$}- operators indexed by pairs \texorpdfstring{$(km,kn)$}.}
To extend the definition of the $\Qop$ operators to any non-coprime pairs of indices we need to make use of the action of $\mathrm{SL}_2[ \mathbb{Z} ]$ on the operators $\Qop_{m,n}$. In \cite{newPleth}, $\mathrm{SL}_2[\mathbb{Z}]$ is shown to act on the algebra generated by the $\D_k$ operators by setting, for its generators
\begin{equation} \label{4.1}
N:=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}
\qquad {\rm and} \qquad
S:=\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix},
\end{equation}
\begin{equation} \label{4.2}
N( \D_{k_1} \D_{k_2} \cdots \D_{k_r} )
= \nabla ( \D_{k_1} \D_{k_2} \cdots \D_{k_r} ) \nabla ^{-1},
\end{equation}
and
\begin{equation} \label{4.3}
S( \D_{k_1} \D_{k_2} \cdots \D_{k_r})
= \D_{k_1+1} \D_{k_2+1} \cdots \D_{k_r+1}.
\end{equation}
It is easily seen that \operatorname{comp}ref{4.2} is a well-defined action since any polynomial in the $\D_k$ that acts by zero on symmetric functions has an image under $N$ which also acts by zero. In \cite{newPleth}, the same property is shown to hold true for the action of $S$ as defined by \operatorname{comp}ref{4.3}.
Since $\D_k = \Qop_{1,k}$, and thus $S \Qop_{1,k}= \Qop_{1,k+1}$, it recursively follows that
\begin{equation} \label{4.4}
S\Qop_{m,n}= \Qop_{m,n+m}.
\end{equation}
On the other hand, it turns out that the property
\begin{equation} \label{4.5}
N\Qop_{m,n}=\Qop_{m+n,n},
\end{equation}
is a consequence of the following general result proved in \cite{newPleth}
\begin{prop} \label{prop4.1}
For any coprime pair $m,n$ we have
\begin{equation} \label{4.6}
\Qop_{m+n,n} = \nabla \Qop_{m,n} \nabla^{-1}.
\end{equation}
It then follows from \operatorname{comp}ref{4.4} and \operatorname{comp}ref{4.5} that for any $\Big[\begin{matrix} a & c \\[-2pt] b & d \end{matrix}\Big] \in \mathrm{SL}_2[\mathbb{Z}]$, we have
\begin{equation} \label{4.7}
\begin{bmatrix} a & c \\ b & d \end{bmatrix} \Qop_{m,n} = \Qop_{am+cn,bn+dn}.
\end{equation}
\end{prop}
The following identity has a variety of consequences
in the present development.
\begin{prop} \label{prop4.2}
For any $k \geq 1$ we have $\Qop_{k+1,k}= \Phi_k$ and $\Qop_{k-1,k} = \Psi_k$. In particular, for all pairs $a,b$, of positive integers with sum equal to $n$, it follows that
\begin{equation} \label{4.8}
\frac{1}{M}[\Qop_{b+1,b}, \Qop_{a-1,a}] =
\frac{qt}{qt-1} \nabla \underline{h}_n \!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] \nabla^{-1}.
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
In view of \operatorname{comp}ref{3.12a} and the second case of \operatorname{comp}ref{eqdefnQ}, the first equality is a special instance of \operatorname{comp}ref{4.6}. To prove the second equality, by Proposition \ref{propphipsi}), we only need to show that the operators $\Qop_{k-1,k}$ satisfy the same recursions and base cases as the $\Psi_k$ operators. To begin, note that since $\spl(k,k+1)=(1,1)+(k-1,k)$ it follows that
\begin{equation}\label{4.10}
\Qop_{k,k+1} = \frac{1}{M}\left[\Qop_{k-1,k},\Qop_{1,1} \right]
= \frac{1}{M}\left[\Qop_{k-1,k}, \D_1 \right],
\end{equation}
which is (\ref{3.13}b) for $\Qop_{k,k+1}$. However the base case is trivial since by definition $\Qop_{0,1}= -\underline{e}_1$.
The identity in \operatorname{comp}ref{4.10} is another way of stating \operatorname{comp}ref{3.16}.
\end{proof}
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\begin{figure}
\caption{The four splits of (12,8).}
\label{fig:quadrop}
\end{figure}
Our first application is best illustrated by an example. In Figure \ref{fig:quadrop} we have depicted $k$ versions of the $km\times kn$ rectangle for the case $k=4$ and $(m,n)=(3,2)$. These illustrate that there are $4$ ways to split the vector $(0,0)\to (4\times 3,4\times 2)$ by choosing a closest lattice point below the diagonal. Namely,
\begin{displaymath}
(12,8) = (2,1)+(10,7) = (5,3)+(7,5) = (8,5)+(4,3) = (11,7)+(1,1).
\end{displaymath}
Now, it turns out that the corresponding four bracketings
\begin{displaymath}
[\Qop_{10,7},\Qop_{2,1}], \qquad
[\Qop_{7,5},\Qop_{5,3}], \qquad
[\Qop_{4,3},\Qop_{8,5}], \qquad {\rm and} \qquad
[\Qop_{1,1},\Qop_{11,7}],
\end{displaymath}
give the same symmetric function operator. This is one of the consequences of the identity in \operatorname{comp}ref{4.8}. In fact, the reader should not have any difficulty checking that these four bracketings are the images of the bracketings in \operatorname{comp}ref{4.8} for $n=4$ by $\big[\begin{matrix}\scriptstyle 2 & \scriptstyle1\\[-5pt] \scriptstyle1 &\scriptstyle 1 \end{matrix}\big]$. Therefore they must also give the same symmetric function operator since our action of $\mathrm{SL}_2[\mathbb{Z}]$ preserves all the identities satisfied by the $\D_k$ operators.
In the general case if $\spl(m,n)=(a,b)+(c,d)$, the $k$ ways are given by
\begin{displaymath}
\left( (u-1)m+a,(u-1)n+b \right) + \left( (k-u)m+c,(k-u)n+d \right),
\end{displaymath}
with $u$ going from $1$ to $k$.
The definition of the $\Qop$ operators in the non-coprime case, as well as some of their remarkable properties, appear in the following result proved in \cite{newPleth}.
\begin{thm} \label{thmQind}
If $\spl(m,n)=(a,b)+(c,d)$ then we may set, for $k> 1$ and any $1\leq u\leq k$,
\begin{displaymath}
\Qop_{km,kn}= \frac{1}{M}
\left[\Qop_{ (k-u)m+c,(k-u)n+d},
\Qop_{(u-1)m+a,(u-1)n+b} \right].
\end{displaymath}
Moreover, letting
$\Gamma := \Big[\begin{matrix} a & c \\[-2pt] b & d\end{matrix}\Big]$
we also have\footnote{Notice $\Gamma \in \mathrm{SL}_2[\mathbb{Z}]$ since (3) of \operatorname{comp}ref{3.13} gives $ad-bc=1$.}
\begin{displaymath}
{\rm a)} \quad \Qop_{k,k}= \frac{qt}{qt-1}
\nabla \underline{h}_k \!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] \nabla^{-1},
\qquad {\rm and} \qquad
{\rm b)} \quad \Qop_{km,kn}= \Gamma \Qop_{k,k}.
\end{displaymath}
In particular, it follows that for any fixed $(m,n)$ the operators $\left\{ \Qop_{km,kn} \right\}_{k\geq1}$ form a commuting family.
\end{thm}
An immediate consequence of Theorem \ref{thmQind} is a very efficient recursive algorithm for computing the action of the operators $\Qop_{km,kn}$ on a symmetric function $f$.
Let us recall that the \define{Lie derivative} of an operator $X$ by an operator $Y$, which we will denote $\delta_Y X$, is simply defined by setting $\delta_Y X = [X,Y]:=XY-YX$. It follows, for instance, that
\begin{displaymath}
\delta_Y^2X= [[X,Y],Y], \qquad \delta_Y^3X= [[[X,Y],Y],Y], \qquad \ldots
\end{displaymath}
Now, our definition gives
\begin{eqnarray*}
\Qop_{2,1} &=& \frac{1}{M} [\Qop_{1,1},\Qop_{1,0}] =\frac{1}{M}[\D_1,\D_0], \qquad{\rm and}\\
\Qop_{3,1} &=& \frac{1}{M} [\Qop_{2,1},\Qop_{1,0}] = \frac{1}{M^2}[ [\D_1,\D_0], \D_0],
\end{eqnarray*}
and by induction we obtain
$$\Qop_{k,1} = \frac{1}{M^{k-1}}\, \delta_{\D_0}^{k-1} \D_1.$$
Thus the action of the matrix $S$ gives
$\Qop_{k,k+1} = \tfrac{1}{M^{k-1}} \delta_{\D_1}^{k-1} \D_2$.
In conclusion we may write
\begin{displaymath}
\Qop_{k,k} = \frac{1}{M}[\Qop_{k-1,k},\Qop_{1,0}]
= \frac{1}{M}[\Qop_{k-1,k},\D_0]
= \frac{1}{M^{k-1}}\left[ \delta_{\D_1}^{k-2} \D_2, \D_0 \right].
\end{displaymath}
This leads to the following recursive general construction of the operator $\Qop_{u,v}$.
\begin{alg} \label{alg1}
Given a pair $(u,v)$ of positive integers:
\begin{enumerate}\itemsep3pt
\item[] {\bf If} $u=1$ {\bf then} $\Qop_{u,v}:= \D_v$
\item[] \qquad {\bf else if} $u<v$ {\bf then} $\Qop_{u,v}:= S \Qop_{u,v-u}$
\item[] \qquad {\bf else if} $u>v$, {\bf then} $ \Qop_{u,v}:= N\Qop_{u-v,v}$
\item[] \qquad {\bf else} $\Qop_{u,v}:= \frac{1}{M^{u-1}} \left[ \delta_{\D_1}^{u-2} \D_2, \D_0 \right]$.
\end{enumerate}
\end{alg}
\noindent
This assumes that $S$ acts on a polynomial in the $\D$ operators by the replacement $\D_k \mapsto \D_{k+1}$, and $N$ acts by the replacement
$$\D_k\ \mapsto\ (-1)^{k}\frac{1}{M^{k}}\delta_{\D_1}^{k} \D_0.$$
We are now finally in a position to validate our construction (see Algorithm~\ref{algF}) of the operators $\BF_{km,kn}$. To this end, for any partition $\lambda =(\lambda_1,\lambda_2, \dots, \lambda_\ell)$, of length $\ell(\lambda)=\ell$, it is convenient to set
\begin{equation} \label{4.14}
h_\lambda[\mathbf{x};q,t] = \left(\frac{qt}{qt-1}\right)^\ell
\operatorname{comp}rod_{i=1}^\ell h_{\lambda_i}\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right].
\end{equation}
Notice that the collection $\left\{ h_\lambda[\mathbf{x};q,t] \right\}_\lambda$ is a symmetric function basis. Thus we may carry out step one of Algorithm~\ref{algF}.
It may be good to illustrate this in a special case. For instance, when $f=e_3$ we proceed as follows. Note first that for any two expressions $A,B$ we have
\begin{displaymath}
h_3[AB]= \sum_{\lambda \vdash 3} h_\lambda[A]\, m_\lambda[B].
\end{displaymath}
Letting $A=\mathbf{x}(1-qt)/(qt)$ and $B=qt/(qt-1)$ gives
\begin{displaymath}
(-1)^3 e_3[\mathbf{x}] = h_3[-\mathbf{x}] = \sum_{\lambda \vdash 3} h_\lambda[\mathbf{x};q,t]\, m_\lambda\! \left[ \frac{qt}{qt-1} \right] \left( \textstyle\frac{qt-1}{qt} \right)^{\ell(\lambda)}.
\end{displaymath}
Thus
\begin{equation} \label{4.16}
\Be_{3m,3n} = -\sum_{\lambda \vdash 3}
m_\lambda\! \left[ \frac{qt}{qt-1} \right]
\left(\frac{qt-1}{qt} \right)^{\ell(\lambda)}
\operatorname{comp}rod_{i=1}^{\ell(\lambda)} \Qop_{m \lambda_i, n \lambda_i}.
\end{equation}
Carrying this out gives
\begin{displaymath}
\Be_{3m,3n}=
-\frac{1}{[3]_{qt}[2]_{qt}} \Qop_{m,n}^3
-\frac{qt(2+qt)}{[3]_{qt}[2]_{qt}} \Qop_{m,n}\Qop_{2m,2n}
-\frac{(qt)^2}{[3]_{qt}} \Qop_{3m,3n},
\end{displaymath}
where for convenience we have set $[a]_{qt} = ({1-(qt)^a})/({1-qt})$.
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To illustrate, by direct computer assisted calculation, we get
\begin{eqnarray*}
\Be_{3,6}\cdot(-\mathbf{1}) &=& s_{{33}}[\mathbf{x}]+ ( {q}^{2}+qt+{t}^{2}+q+t ) s_{{321}}[\mathbf{x}]\\
&&\quad + ( {q}^{3}+{q}^{2}t+q{t}^{2}+{t}^{3}+qt ) s_{{3111}}[\mathbf{x}]\\
&&\quad + ( {q}^
{3}+{q}^{2}t+q{t}^{2}+{t}^{3}+qt ) s_{{222}}[\mathbf{x}]\\
&&\quad + ( {q}^{4}+{
q}^{3}t+{q}^{2}{t}^{2}+q{t}^{3}+{t}^{4}\\
&&\qquad\qquad+{q}^{3}+2\,{q}^{2}t+2\,q{t}^{2
}+{t}^{3}+{q}^{2}+qt+{t}^{2} ) s_{{2211}}[\mathbf{x}]\\
&&\quad + ( {q}^{5}+{q}
^{4}t+{q}^{3}{t}^{2}+{q}^{2}{t}^{3}+q{t}^{4}+{t}^{5}\\
&&\qquad\qquad+{q}^{4}+2\,{q}^{3
}t+2\,{q}^{2}{t}^{2}+2\,q{t}^{3}+{t}^{4}+{q}^{2}t+q{t}^{2} ) s_{
{21111}}[\mathbf{x}]\\
&&\quad + ( {q}^{6}+{q}^{5}t+{q}^{4}{t}^{2}+{q}^{3}{t}^{3}+{q
}^{2}{t}^{4}+q{t}^{5}+{t}^{6}\\
&&\qquad\qquad+{q}^{4}t+{q}^{3}{t}^{2}+{q}^{2}{t}^{3}+q
{t}^{4}+{q}^{2}{t}^{2} ) s_{{111111}}[\mathbf{x}]
\end{eqnarray*}
Conjecturally, the symmetric polynomial $\Be_{km,kn}\cdot(-\mathbf{1})^{k(n+1)}$ should be the Frobenius characteristic of a bi-graded $S_{kn}$ module.
In particular the two expressions
\begin{displaymath}
\left\langle \Be_{3,6}\cdot(-\mathbf{1}), e_1^6 \right\rangle
\qquad {\rm and} \qquad
\left\langle \Be_{3,6}\cdot(-\mathbf{1}), s_{1^6} \right\rangle
\end{displaymath}
should respectively give the Hilbert series of the corresponding $S_6$ module and the Hilbert series of its alternants.
\begin{figure}
\caption{Dyck paths in the $3 \times 6$ lattice.}
\label{fig:ntwelves}
\end{figure}
Our conjecture states that we should have (as in the case of the module of Diagonal Harmonics)
\begin{displaymath}
\left\langle \Be_{3,6}\cdot(-\mathbf{1}), e_1^6 \right\rangle
= \sum_{\operatorname{comp}ark\in\mathrm{Park}_{3,6}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)},
\qquad {\rm and} \qquad
\left\langle \Be_{3,6}\cdot(-\mathbf{1}), s_{1^6} \right\rangle
= \sum_{\operatorname{comp}ath \in \mathcal{D}_{3,6}} t^{\operatorname{area}(\operatorname{comp}ath)}q^{\operatorname{dinv}(\operatorname{comp}ath)},
\end{displaymath}
where the first sum is over all parking functions and the second is over all Dyck paths in the $3 \times 6$ lattice rectangle.
The first turns out to be the polynomial
\begin{eqnarray*}
&&{q}^{6}+{q}^{5}t+{q}^{4}{t}^{2}+{q}^{3}{t}^{3}+{q}^{2}{t}^{4}+q{t}^{5}
+{t}^{6}\\
&&\qquad +5\,{q}^{5}+6\,{q}^{4}t+6\,{q}^{3}{t}^{2}+6\,{q}^{2}{t}^{3}+6
\,q{t}^{4}+5\,{t}^{5}\\
&&\qquad +14\,{q}^{4}+19\,{q}^{3}t+20\,{q}^{2}{t}^{2}+19\,
q{t}^{3}+14\,{t}^{4}\\
&&\qquad +24\,{q}^{3}+38\,{q}^{2}t+38\,q{t}^{2}+24\,{t}^{3}
+\\
&&\qquad 25\,{q}^{2}+40\,qt+25\,{t}^{2}+16\,q+16\,t+5.
\end{eqnarray*}
Setting first $q=1$, we get
$${t}^{6}+6\,{t}^{5}+21\,{t}^{4}+50\,{t}^{3}+90\,{t}^{2}+120\,t+90,$$
which evaluates to $378$ at $t=1$; the second polynomial
\begin{equation} \label{4.17}
{q}^{6}+{q}^{5}t+{q}^{4}{t}^{2}+{q}^{3}{t}^{3}+{q
}^{2}{t}^{4}+q{t}^{5}+{t}^{6} +{q}^{4}t+{q}^{3}{t}^{2}+{q}^{2}{t}^{3}+q
{t}^{4}+{q}^{2}{t}^{2} ,
\end{equation}
evaluates to $12$, at $q=t=1$.
All this is beautifully confirmed on the combinatorial side. Indeed, there are 12 Dyck paths in the $3\times 6$ lattice, as presented in Figure \ref{fig:ntwelves}. A simple calculation verifies that the total number of column-increasing labelings of the north steps of these Dyck paths (as recorded in Figure~\ref{fig:ntwelves} below each path) by a permutation of $\{1,2,\dots,6\}$, is indeed $378$. One may carefully check that this coincidences still holds true when one takes into account the statistics area and dinv.
In the next section we give the construction of the parking function statistics that must be used to obtain the polynomial $\Be_{3,6}\cdot(-\mathbf{1})$ by purely combinatorial methods. Figure~\ref{fig:ntwelves} shows the result of a procedure that places a square in each lattice cell, above the path, that contributes a unit to the dinv of that path. Taking into account that the area is the number of lattice squares between the path and the lattice diagonal, the reader should have no difficulty seeing that the polynomial in \operatorname{comp}ref{4.17} is indeed the $q,t$-enumerator of the above Dyck paths by $\operatorname{dinv}$ and $\operatorname{area}$.
\begin{rmk} In retrospect, our construction of the operators $\BC_{km,kn}^{(\alpha)}$ has a certain degree of inevitability. In fact, since the multiplication operator
$$\frac{qt}{qt-1}\, \underline{h}_k\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right]$$
corresponds to the operator $\Qop_{0,k}$, and since we must have
\begin{displaymath}
\Qop_{k,k} = \nabla \Qop_{0,k} \nabla ^{-1} = \frac{qt}{qt-1} \nabla \underline{h}_k\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] \nabla ^{-1}
\end{displaymath}
by Proposition \ref{prop4.1},
then it becomes natural to set
\begin{displaymath}
\Bf_{k,k} := \nabla \cdot\mathbf{1}derline{f} \nabla ^{-1}.
\end{displaymath}
for any symmetric function $f$ homogeneous of degree $k$.
Therefore using the matrix $\Gamma$ of Theorem \ref{thmQind}, we obtain
\begin{equation} \label{4.18}
\Bf_{km,kn}= \Gamma \Bf_{k,k}
\end{equation}
In particular, it follows (choosing $f=e_k$) that
\begin{displaymath}
\Be_{k,k}\cdot\mathbf{1} = \nabla e_k\nabla^{-1} \cdot\mathbf{1}= \nabla e_k.
\end{displaymath}
The expansion
$e_k = \sum_{\alpha\models k} C_\alpha \cdot\mathbf{1}$
yields the decomposition $\Be_{k,k} = \sum_{\alpha\models k} \BC_{k ,k }^{(\alpha)}$,
and \operatorname{comp}ref{4.18} then yields
$$
\Be_{km,kn}= \sum_{\alpha\models k}\BC_{km ,kn }^{(\alpha)}.
$$
\qed
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\end{rmk}
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\section{The combinatorial side, further extensions and conjectures.}
Our construction of the parking function statistics in the non-coprime case closely follows what we did in section 1 with appropriate modifications necessary to
resolve conflicts that did not arise in the coprime case. For clarity we will present our definitions as a collection of algorithms which can be directly implemented on a computer.
The symmetric polynomials arising from the right hand sides of our conjectures may also be viewed as Frobenius characteristics of certain bi-graded $S_n$ modules. Indeed, they are shown to be by \cite{Hikita} in the coprime case. Later in this section we will present some conjectures to this effect.
As for Diagonal Harmonics (see \cite{Shuffle}), all these Frobenius Characteristics are sums of LLT polynomials. More precisely, a $(km,kn)$-Dyck path $\operatorname{comp}ath$ may be represented by a vector
\begin{equation} \label{5.1}
\mathbf{u}=(u_1,u_2,\dots,u_{kn}) \qquad
\quad {\rm with}\quad u_0=0,
\qquad{\rm and}\qquad u_{i-1} \leq u_i \leq (i-1)\frac{m}{n},
\end{equation}
for all $2\leq i\leq kn$.
This given, we set
\begin{equation} \label{5.2}
\mathrm{LLT}(m,n,k;\mathbf{u}) := \sum_{\operatorname{comp}ark \in\mathrm{Park}(\mathbf{u})}
t^{\operatorname{area}(\mathbf{u})} q^{\operatorname{dinv}(\mathbf{u}) + \operatorname{tdinv}(\operatorname{comp}ark) - \operatorname{maxtdinv}(\mathbf{u})} s_{\operatorname{pides}(\operatorname{comp}ark)}[\mathbf{x}],
\end{equation}
where $\mathrm{Park}(\mathbf{u})$ denotes the collection of parking functions supported by the path corresponding to $\mathbf{u}$. We also use here the Egge-Loehr-Warrington (see \cite{expansions}) result and substitute the Gessel fundamental by the Schur function indexed by $\operatorname{pides}(\operatorname{comp}ark)$ (the descent composition of the inverse of the permutation $\sigma(\operatorname{comp}ark)$).
Here $\sigma(\operatorname{comp}ark)$ and the other statistics used in \operatorname{comp}ref{5.2} are constructed according to the following algorithm.
\begin{alg} \label{algL}
\begin{enumerate}
\item Construct the collection $\mathrm{Park}(\mathbf{u})$ of vectors $\mathbf{v}=( v_1, v_2, \dots, v_{kn} )$ which are the permutations of $1, 2, \dots, kn$ that satisfy the conditions
\begin{displaymath}
u_{i-1}=u_i \quad\implies\quad v_{i-1}<v_{i}.
\end{displaymath}
\item Compute the $\operatorname{area}$ of the path, that is the number of full cells between the path and the
main diagonal of the $km\times kn$ rectangle, by the formula
\begin{displaymath}
\operatorname{area}(\mathbf{u})= (kmkn - km - kn + k)/2 - \sum_{i=1}^{kn} u_i .
\end{displaymath}
\item Denoting by $\lambda(\mathbf{u})$ the partition above the path, set
\begin{displaymath}
\operatorname{dinv}(\mathbf{u}) = \sum_{c \in \lambda(\mathbf{u})} \raise 2pt\hbox{\large$\chi$}\!\left( \frac{\operatorname{arm}(c)}{\operatorname{leg}(c)+1} \le \frac{m}{n} < \frac{\operatorname{arm}(c)+1}{\operatorname{leg}(c)}\right).
\end{displaymath}
\item Define the $\operatorname{rank}$ of the $i^{th}$ north step by $km(i-1) - kn u_i + u_i/(km+1)$ and accordingly use this number as the $\operatorname{rank}$ of car $v_i$, which we will denote as $\operatorname{rank}(v_i)$. This given, we set, for $\operatorname{comp}ark = \Big[\begin{matrix} \mathbf{u} \\[-3pt] \mathbf{v} \end{matrix}\Big]$
\begin{displaymath}
\operatorname{tdinv}(\operatorname{comp}ark) = \sum_{1\leq r<s \leq kn}
\raise 2pt\hbox{\large$\chi$}\left( \operatorname{rank}(r) < \operatorname{rank}(s) < \operatorname{rank}(r)+km \right).
\end{displaymath}
\item Define $\sigma(\operatorname{comp}ark)$ as the permutation of $1, 2, \dots, kn$ obtained by reading the cars by decreasing ranks. Let $\operatorname{pides}(\operatorname{comp}ark)$ be the composition whose first $kn-1$ partial sums give the descent set of the inverse of $\sigma(\operatorname{comp}ark)$.
\item Finally $\operatorname{maxtdinv}(\mathbf{u})$ may be computed as $\max \{\operatorname{tdinv}(\operatorname{comp}ark) : \operatorname{comp}ark \in\mathrm{Park}(\mathbf{u}) \}$, or more efficiently as the $\operatorname{tdinv}$ of the $\operatorname{comp}ark$ whose word $\sigma(\operatorname{comp}ark)$ is the reverse permutation $(kn)\cdots 3 2 1$.
\end{enumerate}
\end{alg}
This completes our definition of the polynomial $\mathrm{LLT}(m,n,k;\mathbf{u})$, which may be shown to expand as a linear combination of Schur functions with coefficients in $\mathbb{N}[q,t]$. It may be also be shown that, at $q=1$, the polynomial $\mathrm{LLT}(m,n,k;\mathbf{u})$ specializes to $t^{\operatorname{area}(\mathbf{u})}e_{\lambda(\mathbf{u})}[\mathbf{x}]$, with $\lambda(\mathbf{u})$ the partition giving the multiplicities of the components of $\mathbf{u}_{km,kn}-\mathbf{u}$, with $\mathbf{u}_{km,kn}$ the vector encoding the unique $0$-area $(km,kn)$-Dyck path.
With the above definition at hand, Conjecture \ref{conjBGLX} can be restated as
\begin{equation} \label{5.3}
\BC^{(\alpha)}_{km,kn}\cdot (-\mathbf{1})^{k(n+1)}
=\sum_{\mathbf{u} \in \mathcal{U}(\alpha)}\mathrm{LLT}(m,n,k;\mathbf{u})
\end{equation}
where $\mathcal{U}(\alpha)$ denotes the collection of all $\mathbf{u}$ vectors satisfying \operatorname{comp}ref{5.1} whose corresponding Dyck path hits the diagonal of the $km\times kn$ rectangle according to the composition $\alpha\models k$. Alternatively we can simply require that $\alpha$ be the composition of $k$ corresponding to the subset
\begin{displaymath}
\left\{ 1\leq i\leq k-1 : u_{ni+1}=i m \right\}.
\end{displaymath}
Note that, although the operators $\BB_b$ and $ \sum_{\beta \models b}
\BC_{\beta}$ are different, in view of definition \operatorname{comp}ref{defC} they take the same value on constant symmetric functions
\begin{equation} \label{5.4}
\BB_b \cdot\mathbf{1} = e_b[\mathbf{x}] = \sum_{\beta \models b}
\BC_{\beta} \cdot\mathbf{1}.
\end{equation}
This circumstance, combined with the commutativity relation (proved in \cite{classComp})
\begin{equation} \label{5.5}
\BB_b \BC_\gamma= q^{\ell(\gamma)} \BC_\gamma \BB_b,
\end{equation}
for all compositions $\gamma$,
enables us to derive a $(km, kn)$ version of the
Haglund-Morse-Zabrocki conjecture \ref{conjHMZB}. To see how this comes about we briefly reproduce an argument first given in \cite{classComp}.
Exploiting \operatorname{comp}ref{5.4} and \operatorname{comp}ref{5.5}, we calculate that
\begin{align*}
\BB_a \BB_b \BB_c \cdot\mathbf{1} &= \BB_a \BB_b \sum_{\gamma \models c} \BC_{\gamma} \cdot\mathbf{1} \\
&= \BB_a \sum_{\gamma \models c}q^{\ell(\gamma)} \BC_{\gamma} \BB_b \cdot\mathbf{1} \\
&= \BB_a \sum_{\gamma\models c} q^{\ell(\gamma)} \BC_{\gamma}\sum_{\beta \models b} \BC_{\beta} \cdot\mathbf{1} \\
&= \sum_{\gamma\models c} q^{2 \ell(\gamma)} \BC_{\gamma}\sum_{\beta\models b} q^{\ell(\beta)} \BC_{\beta}\sum_{\alpha \models a} \BC_{\alpha} \cdot\mathbf{1}.
\end{align*}
From this example one can easily derive that the following identity holds true in full generality.
\begin{prop} \label{5.1}
For any $\beta=(\beta_1,\beta_2,\dots ,\beta_k)$, we have
\begin{equation} \label{5.6}
\BB_{\beta} \cdot\mathbf{1}
= \sum_{\alpha \operatorname{comp}receq \beta} q^{c(\alpha,\beta)}
\BC_{\alpha} \cdot\mathbf{1}
\end{equation}
where $\alpha \operatorname{comp}receq \beta$ here means that $\alpha$ is a refinement of the reverse of $\beta$. That is $\alpha = \alpha^{(k)} \cdots \alpha^{(2)}\alpha^{(1)}$ with $\alpha^{(i)} \models \beta_{i}$, and in that case
\begin{displaymath}
c(\alpha,\beta) = \sum_{i=1}^k (i-1)\, \ell(\alpha^{(i)}).
\end{displaymath}
\end{prop}
Using \operatorname{comp}ref{5.6} we can easily derive the following.
\begin{prop} Assuming that Conjecture~\ref{conjBGLX} holds, then
for all compositions
$$\beta=(\beta_1,\beta_2,\dots ,\beta_{\ell)}\models k$$ we have
\begin{displaymath}
\BB_{km,kn}^{(\beta)}\cdot (-\mathbf{1})^{k(n+1)}
= \sum_{\alpha \operatorname{comp}receq \beta} q^{c(\alpha,\beta)}
\sum_{\operatorname{comp}ark \in\mathrm{Park}_{km,kn}^{(\alpha)}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)}.
\end{displaymath}
Here $\BB_{km,kn}^{(\beta)}$ is the operator obtained by setting $f=\BB_{\beta} \cdot\mathbf{1}$ in Algorithm~\ref{algF} and, as before, $\mathrm{Park}_{km,kn}^{(\alpha)}$ denotes the collection of parking functions in the $km\times kn$ lattice whose Dyck path hits the diagonal according to the composition $\alpha$.
\end{prop}
A natural question that arises next is what parking function interpretation may be given to the
polynomials $\Qop_{km,kn} \cdot(-\mathbf{1})^{k(n+1)}$. Our attempts to answer this question lead to a variety of interesting identities. We start with a known identity and two new ones.
\begin{thm} \label{thmQsquare} For all $n$, we have
\begin{eqnarray}
\Qop_{n,n+1} \cdot\mathbf{1}n &=& \nabla e_n,\qquad {\rm and}\label{5.7a}\\
\Qop_{n,n} \cdot\mathbf{1}n &=& (-qt)^{1-n}\nabla \Delta_{e_1} h_n=\Delta_{e_{n-1}} e_n.\label{5.7b}
\end{eqnarray}
\end{thm}
\begin{proof}[\bf Proof]
Since $\Qop_{n+1,n} = \nabla \Qop_{1,n}\nabla ^{-1}$ and $\Qop_{1,n} = \D_n$, then
\begin{eqnarray*}
\Qop_{n,n+1} \cdot\mathbf{1}n &=& \nabla \D_n \nabla ^{-1} \cdot\mathbf{1}n\\
&=& \nabla \D_n \cdot\mathbf{1}n.
\end{eqnarray*}
Recalling that $\nabla^{-1} \, \cdot\mathbf{1}n =(-1)^{n}$ and $\D_n \cdot\mathbf{1} = (-1)^n e_n$, this gives \operatorname{comp}ref{5.7a}. The proof of \operatorname{comp}ref{5.7b} is a bit more laborious. We will obtain it below, by combining a few auxiliary identities.
\end{proof}
\begin{prop} \label{prop5.2}
For any monomial $m$ and $\lambda \vdash n$
\begin{equation} \label{5.8}
s_\lambda[1-m] =
\begin{cases}
(-m)^k (1-m) & \hbox{\rm if}\ \lambda = (n-k,1^k)\ \hbox{\rm for some }\ 0\leq k\leq n-1,\\[4pt]
0 & \hbox{\rm otherwise.}
\end{cases}
\end{equation}
\end{prop}
\noindent
This is an easy consequence of the addition formula for Schur functions.
\begin{prop}\label{prop5.3} For all $n$,
\begin{equation} \label{5.9}
\frac{qt}{qt-1} h_n\! \left[ \frac{1-qt}{qt}\,\mathbf{x}\right]
= -(qt)^{1-n} \sum_{k=0}^{n-1} (-qt)^k s_{n-k,1^k}[\mathbf{x}].
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
The Cauchy formula gives
\begin{displaymath}
\frac{qt}{qt-1} h_n\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right]
= \frac{(qt)^{1-n}}{qt-1} \sum_{\lambda \vdash n} s_\lambda[\mathbf{x}] s_\lambda[1-qt]
\end{displaymath}
and \operatorname{comp}ref{5.8} with $m=qt$ proves \operatorname{comp}ref{5.9}.
\end{proof}
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\begin{prop} \label{prop5.2}
For any monomial $m$ and $\lambda \vdash n$
\begin{equation} \label{5.8}
s_\lambda[1-m] =
\begin{cases}
(-m)^k (1-m) & \hbox{\rm if}\ \lambda = (n-k,1^k)\ \hbox{\rm for some }\ 0\leq k\leq n-1,\\[4pt]
0 & \hbox{\rm otherwise.}
\end{cases}
\end{equation}
\end{prop}
\noindent
This is an easy consequence of the addition formula for Schur functions.
\begin{prop}\label{prop5.3} For all $n$,
\begin{equation} \label{5.9}
\frac{qt}{qt-1} h_n\! \left[ \frac{1-qt}{qt}\,\mathbf{x}\right]
= -(qt)^{1-n} \sum_{k=0}^{n-1} (-qt)^k s_{n-k,1^k}[\mathbf{x}].
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
The Cauchy formula gives
\begin{displaymath}
\frac{qt}{qt-1} h_n\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right]
= \frac{(qt)^{1-n}}{qt-1} \sum_{\lambda \vdash n} s_\lambda[\mathbf{x}] s_\lambda[1-qt]
\end{displaymath}
and \operatorname{comp}ref{5.8} with $m=qt$ proves \operatorname{comp}ref{5.9}.
\end{proof}
\noindent
Observe that when we set $qt=1$, the right hand side of \operatorname{comp}ref{5.9} specializes to $-p_n$.
\begin{prop} \label{prop5.4} For all $n$,
\begin{equation} \label{5.10}
\Delta_{e_1} h_n[\mathbf{x}] = \sum_{k=0}^{n-1} (-qt)^k\,s_{n-k,1^k}[\mathbf{x}].
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
By \operatorname{comp}ref{3.8} and \operatorname{comp}ref{formulaoper} $(i)$ we have $\Delta_{e_1} = (I-\D_0)/M$. Thus, by \operatorname{comp}ref{defD}
\begin{eqnarray*}
M \Delta_{e_1} h_n[\mathbf{x}]
&=& h_n[\mathbf{x}] - h_n[\mathbf{x}+{M}/{z}]\, \Omega[-z\mathbf{x}]\Big|_{z^0}\\
&=& - \sum_{k=1}^{n} (-1)^k h_{n-k}[\mathbf{x}] \, e_k[\mathbf{x}]\, h_k[M] \\
&=& - \sum_{k=1}^{n-1} (-1)^k s_{n-k,1^k}[\mathbf{x}]\, h_k[M] + \sum_{k=0}^{n-1} (-1)^{k} s_{n-k,1^k}[\mathbf{x}]\, h_{k+1}[M] \\
&=& \sum_{k=1}^{n-1} (-1)^k s_{n-k,1^k}[\mathbf{x}]\, (h_{k+1}[M]-h_{k}[M]) + s_n[\mathbf{x}] M.
\end{eqnarray*}
This proves \operatorname{comp}ref{5.10} since the Cauchy formula and \operatorname{comp}ref{5.8} give
\begin{eqnarray*}
h_n[M] &=& \sum_{k=0}^{n-1}(-t)^k(1-t)(-q)^k(1-q)\\
&=& M\sum_{k=0}^{n-1}(qt)^k.
\end{eqnarray*}
\end{proof}
Now, it is shown in \cite{explicit} that $e_n[\mathbf{x}]$ and $h_n[\mathbf{x}]$ have the expansions
\begin{align}
e_n[\mathbf{x}] &= \sum_{\mu \vdash n}\frac{M\, B_\mu(q,t) \Pi_\mu(q,t)}{w_\mu}\,\widetilde{H}_\mu[\mathbf{x};q,t], \qquad {\rm and} \label{5.11a}\\
h_n[\mathbf{x}] &= (-qt)^{n-1} \sum_{\mu\vdash n}\frac{M\,B_\mu(1/q,1/t)\, \Pi_\mu(q,t)}{w_\mu}\,\widetilde{H}_\mu[\mathbf{x};q,t]. \label{5.11b}
\end{align}
We are then ready to proceed with the rest of our proof.
\begin{proof}[\bf Proof][Proof of Theorem \ref{thmQsquare} continued]
The combination of \operatorname{comp}ref{5.9}, \operatorname{comp}ref{5.10} and
\operatorname{comp}ref{5.11a} gives
\begin{eqnarray}
\frac{qt}{qt-1} h_n\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right]
&=& -(qt)^{1-n} \Delta_{e_1} h_n\label{5.12}\\
&=& (-1)^{n} \sum_{\mu\vdash n} \frac{ M B_\mu(q,t) B_\mu(1/q,1/t)\, \Pi_\mu(q,t)}{w_\mu}\,\widetilde{H}_\mu[\mathbf{x};q,t] \label{5.13}
\end{eqnarray}
Now from Theorem \ref{thmQind} we derive that
\begin{eqnarray*}
\Qop_{n,n}\cdot\mathbf{1}n &=& \frac{qt}{qt-1} \nabla \underline{h}_n \!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] \nabla^{-1} \cdot\mathbf{1}n\\
&=& (-1)^n \frac{qt}{qt-1} \nabla h_n\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right].
\end{eqnarray*}
Thus \operatorname{comp}ref{5.12} gives
\begin{displaymath}
\Qop_{n,n} \cdot\mathbf{1}n = (-qt)^{1-n} \nabla \Delta_{e_1}h_n.
\end{displaymath}
This proves the first equality in \operatorname{comp}ref{5.7b}. The second equality in \operatorname{comp}ref{5.13} gives
\begin{equation} \label{5.13}
\Qop_{n,n} \cdot\mathbf{1}n = \sum_{\mu\vdash n}\frac{T_\mu\, M B_\mu(q,t) B_\mu(1/q,1/t)\, \Pi_\mu(q,t)}{w_\mu}\,\widetilde{H}_\mu[\mathbf{x};q,t].
\end{equation}
But it is not difficult to see that we may write (for any $\mu\vdash n$)
\begin{displaymath}
T_\mu \,B_\mu(1/q,1/t)= e_{n-1}\left[B_\mu(q,t)\right]
\end{displaymath}
and \operatorname{comp}ref{5.13} becomes (using \operatorname{comp}ref{5.11a})
\begin{eqnarray*}
\Qop_{n,n} \cdot\mathbf{1}n &=& \Delta_{e_{n-1}} \sum_{\mu\vdash n} \frac{ M B_\mu(q,t)\, \Pi_\mu(q,t)}{w_\mu}\,\widetilde{H}_\mu[\mathbf{x};q,t]\\
&=& \Delta_{e_{n-1}} e_n.
\end{eqnarray*}
\end{proof}
To obtain a combinatorial version of \operatorname{comp}ref{5.7b} we need some auxiliary facts.
\begin{prop} For all positive integers $a$ and $b$, we have
\begin{equation} \label{5.15}
\BC_a \BB_b \cdot\mathbf{1} = \BC_a e_b[\mathbf{x}] = (-1/q)^{a-1} s_{a,1^{b}}[\mathbf{x}] - (-1/q)^a s_{1+a,1^{b-1}}[\mathbf{x}],
\end{equation}
and
\begin{equation} \label{5.16}
\sum_{\beta\models n-a} \BC_a \BC_\beta \cdot\mathbf{1}
= (-1/q)^{a-1} s_{a,1^{n-a}}[\mathbf{x}] - (-1/q)^a s_{1+a,1^{n-a-1}}[\mathbf{x}].
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
The equality in \operatorname{comp}ref{5.4} gives the first equality in \operatorname{comp}ref{5.15} and the equivalence of \operatorname{comp}ref{5.15} to \operatorname{comp}ref{5.16}, for $b=n-a$. Using \operatorname{comp}ref{defC} and \operatorname{comp}ref{5.8} we derive that
\begin{eqnarray*}
\BC_a e_b[\mathbf{x}]
&=& (-1/q)^{a-1} \sum_{r=0}^b e_{b-r}[\mathbf{x}] (-1)^r\, h_r \left[ (1-1/q)/z \right]\, \Omega[z\mathbf{x}] \Big|_{z^a} \\
&=& (-1/{q})^{a-1} \left(e_b[\mathbf{x}]\, h_a[\mathbf{x}] + ( 1- 1/q) \sum_{r=1}^b (-1)^r e_{b-r}[\mathbf{x}]\, h_{r+a}[\mathbf{x}] \right) \\
&=& (-1/q)^{a-1} \Big(s_{a,1^{b}}[\mathbf{x}]+s_{a+1,1^{b-1}}[\mathbf{x}] + \\
&& \qquad\qquad + (1-1/q) \left( \textstyle \sum_{r=1}^b (-1)^r s_{r+a,1^{b-r}}[\mathbf{x}] -
\sum_{r=2}^{b} (-1)^r s_{r+a ,1^{b-r}}[\mathbf{x}] \right) \Big) \\
&=& (-1/q)^{a-1} \left(s_{a,1^{b}}[\mathbf{x}]+s_{a+1,1^{b-1}}[\mathbf{x}] - (1-1/q)\, s_{1+a,1^{b-1}}[\mathbf{x}] \right) \\
&=& (-1/q)^{a-1} s_{a,1^{b}}[\mathbf{x}]-(-1/q)^a s_{1+a,1^{b-1}}[\mathbf{x}].
\end{eqnarray*}
\end{proof}
The following conjectured identity is well known and is also stated in \cite{Shuffle}. We will derive it from Conjecture \ref{conjHMZC} for sake of completeness.
\begin{thm} \label{thm5.3}
Upon the validity of the Compositional Shuffle conjecture we have
\begin{equation} \label{5.17}
\nabla (-q)^{1-a} s_{a,1^{n-a}} =
\sum_{\operatorname{comp}ark \in\mathrm{Park}_{n, \geq a}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))},
\end{equation}
where the symbol $\mathrm{Park}_{n, \geq a}$ signifies that the sum is to be extended over the parking functions in the $n\times n$ lattice whose Dyck path's first return to the main diagonal occurs in a row $y \geq a$.
\end{thm}
\begin{proof}[\bf Proof]
An application of $\nabla$ to both sides of \operatorname{comp}ref{5.16} yields
\begin{equation} \label{5.18}
\sum_{\beta\models n-a} \nabla \BC_a \BC _\beta\cdot\mathbf{1}
= ( \tfrac{1}{q})^{a-1} \nabla (- 1)^{a-1} s_{a,1^{n-a}} - ( \tfrac{1}{q})^{a} \nabla (-1)^{a} s_{1+a,1^{n-a-1}}.
\end{equation}
Furthermore, \operatorname{comp}ref{5.17} is an immediate consequence of the fact that the Compositional Shuffle Conjecture implies
\begin{equation} \label{5.19}
\sum_{\beta\models n-a; } \nabla \, \BC_a \BC _\beta\cdot\mathbf{1}
= \sum_{\operatorname{comp}ark\in\mathrm{Park}_{n,= a}}
t^{\operatorname{area}(\operatorname{comp}ark))} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))},
\end{equation}
where the symbol $\mathrm{Park}_{n,= a}$ signifies that the sum is to be extended over the parking functions whose Dyck path's first return to the diagonal occurs
exactly at row $a$.
\end{proof}
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To obtain a combinatorial version of \operatorname{comp}ref{5.7b} we need some auxiliary facts.
\begin{prop} For all positive integers $a$ and $b$, we have
\begin{equation} \label{5.15}
\BC_a \BB_b \cdot\mathbf{1} = \BC_a e_b[\mathbf{x}] = (-1/q)^{a-1} s_{a,1^{b}}[\mathbf{x}] - (-1/q)^a s_{1+a,1^{b-1}}[\mathbf{x}],
\end{equation}
and
\begin{equation} \label{5.16}
\sum_{\beta\models n-a} \BC_a \BC_\beta \cdot\mathbf{1}
= (-1/q)^{a-1} s_{a,1^{n-a}}[\mathbf{x}] - (-1/q)^a s_{1+a,1^{n-a-1}}[\mathbf{x}].
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
The equality in \operatorname{comp}ref{5.4} gives the first equality in \operatorname{comp}ref{5.15} and the equivalence of \operatorname{comp}ref{5.15} to \operatorname{comp}ref{5.16}, for $b=n-a$. Using \operatorname{comp}ref{defC} and \operatorname{comp}ref{5.8} we derive that
\begin{eqnarray*}
\BC_a e_b[\mathbf{x}]
&=& (-1/q)^{a-1} \sum_{r=0}^b e_{b-r}[\mathbf{x}] (-1)^r\, h_r \left[ (1-1/q)/z \right]\, \Omega[z\mathbf{x}] \Big|_{z^a} \\
&=& (-1/{q})^{a-1} \left(e_b[\mathbf{x}]\, h_a[\mathbf{x}] + ( 1- 1/q) \sum_{r=1}^b (-1)^r e_{b-r}[\mathbf{x}]\, h_{r+a}[\mathbf{x}] \right) \\
&=& (-1/q)^{a-1} \Big(s_{a,1^{b}}[\mathbf{x}]+s_{a+1,1^{b-1}}[\mathbf{x}] + \\
&& \qquad\qquad + (1-1/q) \left( \textstyle \sum_{r=1}^b (-1)^r s_{r+a,1^{b-r}}[\mathbf{x}] -
\sum_{r=2}^{b} (-1)^r s_{r+a ,1^{b-r}}[\mathbf{x}] \right) \Big) \\
&=& (-1/q)^{a-1} \left(s_{a,1^{b}}[\mathbf{x}]+s_{a+1,1^{b-1}}[\mathbf{x}] - (1-1/q)\, s_{1+a,1^{b-1}}[\mathbf{x}] \right) \\
&=& (-1/q)^{a-1} s_{a,1^{b}}[\mathbf{x}]-(-1/q)^a s_{1+a,1^{b-1}}[\mathbf{x}].
\end{eqnarray*}
\end{proof}
The following conjectured identity is well known and is also stated in \cite{Shuffle}. We will derive it from Conjecture \ref{conjHMZC} for sake of completeness.
\begin{thm} \label{thm5.3}
Upon the validity of the Compositional Shuffle conjecture we have
\begin{equation} \label{5.17}
\nabla (-q)^{1-a} s_{a,1^{n-a}} =
\sum_{\operatorname{comp}ark \in\mathrm{Park}_{n, \geq a}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))},
\end{equation}
where the symbol $\mathrm{Park}_{n, \geq a}$ signifies that the sum is to be extended over the parking functions in the $n\times n$ lattice whose Dyck path's first return to the main diagonal occurs in a row $y \geq a$.
\end{thm}
\begin{proof}[\bf Proof]
An application of $\nabla$ to both sides of \operatorname{comp}ref{5.16} yields
\begin{equation} \label{5.18}
\sum_{\beta\models n-a} \nabla \BC_a \BC _\beta\cdot\mathbf{1}
= ( \tfrac{1}{q})^{a-1} \nabla (- 1)^{a-1} s_{a,1^{n-a}} - ( \tfrac{1}{q})^{a} \nabla (-1)^{a} s_{1+a,1^{n-a-1}}.
\end{equation}
Furthermore, \operatorname{comp}ref{5.17} is an immediate consequence of the fact that the Compositional Shuffle Conjecture implies
\begin{equation} \label{5.19}
\sum_{\beta\models n-a; } \nabla \, \BC_a \BC _\beta\cdot\mathbf{1}
= \sum_{\operatorname{comp}ark\in\mathrm{Park}_{n,= a}}
t^{\operatorname{area}(\operatorname{comp}ark))} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))},
\end{equation}
where the symbol $\mathrm{Park}_{n,= a}$ signifies that the sum is to be extended over the parking functions whose Dyck path's first return to the diagonal occurs
exactly at row $a$.
\end{proof}
All these findings lead us to the following surprising identity.
\begin{thm} \label{thm5.4}
Let $\operatorname{ret}(\operatorname{comp}ark)$ denote the first row where the supporting Dyck path of $\operatorname{comp}ark$ hits the diagonal. We have, for all $n\geq 1$, that
\begin{equation} \label{5.20}
\Qop_{n,n} \cdot\mathbf{1}n = \sum_{\operatorname{comp}ark\in\mathrm{Park}_{n}} [\operatorname{ret}(\operatorname{comp}ark)]_t\, t^{\operatorname{area}(\operatorname{comp}ark))-\operatorname{ret}(\operatorname{comp}ark)+1} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))}.
\end{equation}
\end{thm}
\begin{proof}[\bf Proof]
Combining \operatorname{comp}ref{5.7a} with \operatorname{comp}ref{5.10} gives
\begin{equation} \label{5.21}
\Qop_{n,n} \cdot\mathbf{1}n = (-qt)^{1-n} \sum_{k=0}^{n-1} \nabla (-qt)^k s_{n-k,1^k}.
\end{equation}
Now this may be rewritten as
\begin{equation} \label{5.22}
\Qop_{n,n} \cdot\mathbf{1}n
= (-qt)^{1-n} \sum_{a=1}^{n} \nabla (-qt)^{n-a} s_{a,1^{n-a}}
= \sum_{a=1}^{n} \nabla (-qt)^{1-a} s_{a,1^{n-a}},
\end{equation}
and \operatorname{comp}ref{5.17} gives
\begin{eqnarray}
\Qop_{n,n} \cdot\mathbf{1}n
&=& \sum_{a=1}^{n} t^{1-a} \sum_{\operatorname{comp}ark \in\mathrm{Park}_{n}} t^{\operatorname{area}(\operatorname{comp}ark))} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))}\, \raise 2pt\hbox{\large$\chi$}\left(\operatorname{ret}(\operatorname{comp}ark)\geq a \right) \label{5.23}\\
&=& \sum_{\operatorname{comp}ark\in\mathrm{Park}_{n}}
t^{\operatorname{area}(\operatorname{comp}ark))} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))} \sum_{a=1}^n t^{1-a}\, \raise 2pt\hbox{\large$\chi$} \left( a\leq \operatorname{ret}(\operatorname{comp}ark) \right).
\end{eqnarray}
This proves \operatorname{comp}ref{5.20}.
\end{proof}
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All these findings lead us to the following surprising identity.
\begin{thm} \label{thm5.4}
Let $\operatorname{ret}(\operatorname{comp}ark)$ denote the first row where the supporting Dyck path of $\operatorname{comp}ark$ hits the diagonal. We have, for all $n\geq 1$, that
\begin{equation} \label{5.20}
\Qop_{n,n} \cdot\mathbf{1}n = \sum_{\operatorname{comp}ark\in\mathrm{Park}_{n}} [\operatorname{ret}(\operatorname{comp}ark)]_t\, t^{\operatorname{area}(\operatorname{comp}ark))-\operatorname{ret}(\operatorname{comp}ark)+1} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))}.
\end{equation}
\end{thm}
\begin{proof}[\bf Proof]
Combining \operatorname{comp}ref{5.7a} with \operatorname{comp}ref{5.10} gives
\begin{equation} \label{5.21}
\Qop_{n,n} \cdot\mathbf{1}n = (-qt)^{1-n} \sum_{k=0}^{n-1} \nabla (-qt)^k s_{n-k,1^k}.
\end{equation}
Now this may be rewritten as
\begin{equation} \label{5.22}
\Qop_{n,n} \cdot\mathbf{1}n
= (-qt)^{1-n} \sum_{a=1}^{n} \nabla (-qt)^{n-a} s_{a,1^{n-a}}
= \sum_{a=1}^{n} \nabla (-qt)^{1-a} s_{a,1^{n-a}},
\end{equation}
and \operatorname{comp}ref{5.17} gives
\begin{eqnarray}
\Qop_{n,n} \cdot\mathbf{1}n
&=& \sum_{a=1}^{n} t^{1-a} \sum_{\operatorname{comp}ark \in\mathrm{Park}_{n}} t^{\operatorname{area}(\operatorname{comp}ark))} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))}\, \raise 2pt\hbox{\large$\chi$}\left(\operatorname{ret}(\operatorname{comp}ark)\geq a \right) \label{5.23}\\
&=& \sum_{\operatorname{comp}ark\in\mathrm{Park}_{n}}
t^{\operatorname{area}(\operatorname{comp}ark))} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))} \sum_{a=1}^n t^{1-a}\, \raise 2pt\hbox{\large$\chi$} \left( a\leq \operatorname{ret}(\operatorname{comp}ark) \right).
\end{eqnarray}
This proves \operatorname{comp}ref{5.20}.
\end{proof}
\begin{rmk} \label{rmk5.3}
Extensive computer experiments have revealed that the following difference is Schur positive
\begin{equation} \label{5.24}
\Be_{km+1,kn} \cdot (-\mathbf{1}) - t^{d(km,kn)} \Be_{km,kn} \cdot\mathbf{1},
\end{equation}
where $d(km,kn)$ is the number of integral points between the diagonals for $(km+1,kn)$ and $(km,kn)$. Assuming that $m\leq n$ for simplicity sake, this implies that the following difference is also Schur positive
\begin{equation} \label{5.25}
\Be_{kn,kn} \cdot\mathbf{1} - t^{a(km,kn)} \Be_{km,kn} \cdot\mathbf{1},
\end{equation}
with $a(km,kn)$ equal to the area between the diagonal $(km,kn)$ and the diagonal $(kn,kn)$. This suggests that there is a nice interpretation of $t^{a(km,kn)} \Be_{km,kn} \cdot\mathbf{1}$ as a new sub-module of the space of diagonal harmonic polynomials. We believe that we have a good candidate for this submodule, at least in the coprime case.
\end{rmk}
We terminate with some surprising observations concerning the so-called \define{Rational $(q,t)$-Catalan}. In the present notation, this remarkable generalization of the $q,t$-Catalan polynomial (see \cite{qtCatPos}) may be defined by setting, for a coprime pair $(m,n)$
\begin{equation} \label{5.26}
C_{m,n}(q,t):= \left\langle \Qop_{m,n}\cdot\mathbf{1}n, e_n\right\rangle_*.
\end{equation}
It is shown in \cite{NegutShuffle}, by methods which are still beyond our reach, that this polynomial may also be obtained by the following identity.
\begin{thm} [A. Negut] \label{thm5.8}
\begin{equation} \label{5.27}
C_{m,n}(q,t)= \operatorname{comp}rod_{i=1}^n \frac{1}{(1-z_i)z_i^{a_i(m,n)}}
\operatorname{comp}rod_{i=1}^{n-1} \frac{1}{(1-qtz_i/z_{i+1})}
\operatorname{comp}rod_{1\leq i<j\leq n}\Omega[-M z_i/z_j]
\Big|_{z_1^0z_2^0\cdots z_n^0},
\end{equation}
with
\begin{equation} \label{5.28}
a_i(m,n):=\left\lfloor i\frac{m}{n}\right\rfloor- \left\lfloor (i-1)\frac{m}{n}\right\rfloor.
\end{equation}
\end{thm}
By a parallel but distinct path Negut has obtained the same polynomial as a weighted sum of standard tableaux. However his version of this result turns out to be difficult to program on the computer. Fortunately, in \cite{constantTesler}, in a different but in a closely related context a similar sum over standard tableaux has been obtained. It turns out that basically the same method used in \cite{constantTesler} can be used in the present context to derive a standard tableaux expansion for $C_{m,n}(q,t)$ directly from \operatorname{comp}ref{5.27}.
Let us write
\begin{eqnarray*}
\mathbf{z}_{m,n}&:=&\operatorname{comp}rod_{i=1}^n z_i^{a_{n+1-i}(m,n)}, \qquad {\rm and\ then}\\
\mathcal{N}_{m,n} [\mathbf{z};q,t]&:=& \frac{\Omega[\mathbf{z}]}{\mathbf{z}_{m,n}}
\, \operatorname{comp}rod_{i=2}^{n} \frac{1}{(1-qt z_i/z_{i-1})}
\operatorname{comp}rod_{1\leq i <j \leq n} \Omega[-uMz_j/z_i],
\end{eqnarray*}
where $\mathbf{z}$ stands for the set of variables $z_1,z_2,\ldots,z_n$. Equivalently, $\mathbf{z}=z_1+z_2+\ldots+z_n$ in the plethystic setup.
The resulting rendition of the Negut's result can then be stated as follows.
\begin{prop} \label{thm5.9}
Let $T_n$ be the set of all standard tableaux with labels $1,2,\dots,n$.
For a given $T \in T_n$, we set $w_T(k) = q^{j-1} t^{i-1}$ if the label $k$ of $T$ is in the $i$-th row $j$-th column.
We also denote by $S_T$ the substitution set
\begin{equation} \label{5.29}
\{z_k = w_T^{-1}(k): 1\leq k\leq n \}.
\end{equation}
We have
\begin{equation} \label{5.30}
C_{m,n}(q,t) = \sum_{T\in T_n} \mathcal{N}_{m,n} [\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^n (1-z_i w_T(i)) \Big|_{S_T},
\end{equation}
where the sum ranges over all standard tableaux of size $n$, and the $S_T$ substitution should be carried out in the iterative manner. That is, we successively do the substitution for $z_1$ followed by cancellation, and then do the substitution
for $z_2$ followed by cancellation, and so on.
\end{prop}
\begin{proof}[\bf Proof]
For convenience, let us write $f(u)$ for $\Omega[-Mu]$ and $b_i$ for $a_{n+1-i}(m,n)$ this gives
$$
\mathcal{N}_{m,n} [\mathbf{z};q,t] =
\operatorname{comp}rod_{i=1}^n{1\over (1- z_i)z_i^{b_i}}\operatorname{comp}rod_{i=2}^n{1\over (1-qtz_i/z_{i-1})}\operatorname{comp}rod_{1\le i<j\le n}f(z_i/z_j).
$$
We will show by induction that
\begin{displaymath}
\mathcal{N}_{m,n}[\mathbf{z};q,t] \Big|_{z_1^0\cdots z_{d}^0} = \sum_{T} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^d (1-z_i w_T(i)) \Big|_{S_T},
\end{displaymath}
where $T$ ranges over all standard tableaux of size $d$. Then the proposition is just the $d=n$ case. The $d=1$ case is straightforward. Thus, assume the equality holds for $d-1$.
Now for any term corresponding to a tableau $T$ of size $d-1$, the factors containing $z_d$ are
\begin{eqnarray*}
&& \frac{1}{(1-z_d) z_d^{b_d}}\frac{1}{(1-qt\, {z_d/z_{d-1}})(1- qt \, {z_{d+1}/z_d})^{\chi(d<n)}}
\operatorname{comp}rod_{1\leq i <d } f({z_d/z_i})\operatorname{comp}rod_{d <j \leq n} f({z_j}/{z_d}) \Big|_{S_T} \\
&&=\ \frac{1}{(1-z_d) z_d^{b_d}}
\frac{1}{\left(1-qt\, z_d\,w_T(d-1)\right)\left(1- qt \,{z_{d+1}}/{z_d} \right)^{\chi(d<n)} }
\operatorname{comp}rod_{1\leq i <d } f(z_d\,w_T(i))\operatorname{comp}rod_{d <j \leq n} f({z_j}/{z_d})\\
&&=\ \frac{1}
{z_d^{b_d} }\, \Omega\!\Big[z_d(1+qt\,w_T(d-1) -M\sum_{i=1}^{d-1} w_T(i)) \Big]
\operatorname{comp}rod_{d <j \leq n} f({z_j}/{z_d})\times\frac{1}{\big(1- qt \, {z_{d+1} \over{z_d}} \big)^{\chi(d<n)} }.
\end{eqnarray*}
By a simple calculation, first carried out in \cite{plethMac} we may write
\begin{displaymath}
- M B_{\operatorname{sh}(T)} = \bigg(\sum_{(i,j) \in \OC{\operatorname{sh}(T)}} t^{i-1} q^{j-1} - \sum_{(i,j) \in \IC{\operatorname{sh}(T)}} t^i q^j \bigg) - 1
\end{displaymath}
where for a partition $\lambda$ we respectively denote by ``$\OC{\lambda}$'' and ``$\IC{\lambda}$'' the \define{outer} and \define{inner} corners of the Ferrers diagram of $\lambda$, as depicted in Figure~\ref{fig:corners} using the french convention.
\begin{figure}
\caption{Inner and outer corners of a partition.}
\label{fig:corners}
\end{figure}
This given, the rational function object of the constant term becomes the following proper rational function in $z_d$ (provided $b_d \geq -1$):
$$
\frac{1}{z_d^{b_d}}
\frac{ \operatorname{comp}rod_{(i,j) \in \IC{\operatorname{sh}(T)}} (1- t^i q^j z_{d}) }
{ (1-{ z_d}\, qt\, w_T(d-1))
\operatorname{comp}rod_{(i,j)\in \OC{\operatorname{sh}(T)}} (1- t^{i-1} q^{j-1} z_{d})}
\operatorname{comp}rod_{d <j \leq n}f({z_j}/{z_d})
\times
\frac{1}{\left(1- qt\, {z_{d+1} \over{z_d}} \right)^{\chi(d<n)}}.
$$
Since $d-1$ must appear in $T$ in an inner corner of $\operatorname{sh}(T)$, the factor $1-{ z_d}\, qt\, w_T(d-1)$ in the denominator cancels with a factor in the numerator.
Therefore, the only factors, in the denominator that contribute to the constant term\footnote{By the partial fraction algorithm in \cite{fastAlg}.} are those of the form $(1-q^{j-1}t^{i-1} z_{d})$, for $(i,j)$ an outer corner of $T$. For each such $(i,j)$, construct $T'$ by adding $d$ to $T$ at the cell $(i,j)$.
Thus, by the partial fraction algorithm, we obtain
\begin{eqnarray*}
\mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^{d-1} (1-z_i w_T(i)) \Big|_{S_T} \Big|_{z_d^0}
&=&\sum_{T'} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^d (1-z_i w_{T'}(i)) \Big|_{S_T} \Big|_{z_d={1\over w_{T'}(d)} }\\
&=&\sum_{T'} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^d (1-z_i w_{T'}(i)) \Big|_{S_{T'}},
\end{eqnarray*}
where the sum ranges over all $T'$ obtained from $T$ by adding $d$ at one of its outer corners.
Applying the above formula to all $T$ of size $d-1$, and using the induction hypothesis, we obtain:
\begin{eqnarray*}
\mathcal{N}_{m,n}[\mathbf{z};q,t] \Big|_{z_1^0\cdots z_{d}^0}
&=& \sum_{T} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^{d-1} (1-z_i w_T(i)) \Big|_{S_T} \Big|_{z_d^0} \\
&=& \sum_T \sum_{T'} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^d (1-z_i w_{T'}(i)) \Big|_{S_{T'}} \\
&=& \sum_{T'} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^d (1-z_i w_{T'}(i)) \Big|_{S_{T'}},
\end{eqnarray*}
where the final sum ranges over all $T'$ of size $d$.
\end{proof}
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\begin{rmk} \label{rmk5.4}
We can see that the above argument only needs $b_1\geq 0$, and $b_i\geq -1$ for $i=2,3,\dots,n$. Thus the equality of the right hand sides of \operatorname{comp}ref{5.27} and \operatorname{comp}ref{5.30} holds true also if the sequence $\{a_i(m,n)\}_{i=1}^n$ is replaced by any of these sequences. In fact computer data reveals that the constant term in \operatorname{comp}ref{5.27} yields a polynomial with positive integral coefficients for a variety of choices of $(b_1,b_2,\dots,b_n)$ replacing the sequence $\{a_{i}(m,n)\}_{i=1}^n$. Trying to investigate the nature of these sequences and the possible combinatorial interpretations of the resulting polynomial led to the following construction.
\end{rmk}
Given a path $\operatorname{comp}ath$ in the $m\times n$ lattice rectangle, we define the monomial of $\operatorname{comp}ath$ by setting
\begin{equation} \label{5.31}
\mathbf{z}_\operatorname{comp}ath:=\operatorname{comp}rod_{j=1}^n z_j^{e_j},
\end{equation}
where $e_j=e_j(\operatorname{comp}ath)$ gives the number of east steps taken by $\operatorname{comp}ath$ at height $j$. Note that, by the nature of \operatorname{comp}ref{5.31}, we are tacitly assuming that the path takes no east steps at height $0$. Note also that if $\operatorname{comp}ath$ remains above the diagonal $(0,0)\to (m,n)$ then for each east step $(i-1,j)\to(i,j)$ we must have $i/j \leq m/n$. In particular for the path $\operatorname{comp}ath_0$ that remains closest to the diagonal $(0,0)\to (m,n)$, the last east step at height $j$ must be given by $i=\lfloor j{m}/{n}\rfloor$. Thus
\begin{displaymath}
\mathbf{z}_{\operatorname{comp}ath_0}=\mathbf{z}_{m,n}=\operatorname{comp}rod_{j=1}^n
z_j^{ \lfloor j{m}/{n}\rfloor - \lfloor (j-1){m}/{n}\rfloor }.
\end{displaymath}
We can easily see that the series
\begin{displaymath}
\Omega[\mathbf{z}]= \operatorname{comp}rod_{j=1}^n\frac{1}{1- z_j}
\end{displaymath}
may be viewed as the generating function of all monomials of paths with north and east steps that end at height $n$ and start with a north step. We will refer to the later as the \define{NE-paths}. Our aim is to obtain a formula for the $q$-enumeration of the NE-paths in the $m\times n$ lattice rectangle that remain weakly above a given NE-path $\operatorname{comp}ath$.
Notice that if
\begin{equation} \label{5.32}
\mathbf{z}_\operatorname{comp}ath = z_{r_1}z_{r_2}\cdots z_{r_m},
\qquad {\rm and} \qquad
\mathbf{z}_\delta=z_{s_1}z_{s_2}\cdots z_{s_m}.
\end{equation}
Then $\delta$ remains weakly above $\operatorname{comp}ath$ if and only if
\begin{displaymath}
s_i \geq r_i \qquad\qquad \hbox{for }1\leq i\leq m.
\end{displaymath}
When this happens let us write $\delta \geq \operatorname{comp}ath$. This given, let us set
\begin{displaymath}
C_\operatorname{comp}ath(t):=\sum_{\delta \geq \operatorname{comp}ath} t^{\operatorname{area}(\delta / \operatorname{comp}ath)},
\end{displaymath}
where for $\operatorname{comp}ath$ and $\delta$ as in \operatorname{comp}ref{5.32}, we let $\operatorname{area}(\delta/\operatorname{comp}ath)$ denote the number of lattice cells between $\delta$ and $\operatorname{comp}ath$. In particular, for $\delta$ as in \operatorname{comp}ref{5.32}, we have
\begin{displaymath}
\operatorname{area}(\delta/\operatorname{comp}ath)= \sum_{i=1}^m (s_i-r_i).
\end{displaymath}
Now we have the following fact
\begin{prop} \label{prop5.5} For all path $\gamma$, we have
\begin{equation} \label{5.33}
\frac{\Omega[\mathbf{z}]}{\mathbf{z}_\operatorname{comp}ath}\,
\operatorname{comp}rod_{i=1}^{n-1} \frac{1}{1-tz_i/z_{i+1}}
\Big|_{z_1^0z_2^0\cdots z_n^0}
=\sum_{\delta \geq \operatorname{comp}ath} t^{\operatorname{area}(\delta/\operatorname{comp}ath)}.
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
Notice that each Laurent monomial produced by expansion of the product
\begin{equation} \label{5.34}
\operatorname{comp}rod_{i=1}^{n-1} \frac{1}{1-tz_i/z_{i+1}}=
\operatorname{comp}rod_{i=1}^{n-1} \left(1+t \frac{z_i}{z_{i+1}}+\left(t\frac{z_i}{z_{i+1}} \right)^2+\cdots
\right)
\end{equation}
may be written in the form
\begin{eqnarray} \label{5.35}
\operatorname{comp}rod_{i=1}^{n-1}\left(t \frac{z_i}{z_{i+1}}\right)^{c_i}
&=& \frac{z_1^{c_1}
\operatorname{comp}rod_{i=2}^{n-1} z_i^{(c_i-c_{i-1})^+} }
{z_{n}^{c_{n-1}} \operatorname{comp}rod_{i=2}^{n-1} z_i^{(c_i-c_{i-1})^-} }\ \operatorname{comp}rod_{i=1}^{n-1} t^{c_i} \nonumber\\[4pt]
&=& \frac{z_{a_1}z_{a_2}\cdots z_{a_\ell}}
{z_{b_1}z_{b_2}\cdots z_{b_\ell}}\ t^{\sum_{i=1}^{n-1}c_i}, \label{5.35}
\end{eqnarray}
with
\begin{eqnarray*}
\ell &=&c_1+\sum_{i=2}^{n-1}(c_i-c_{i-1})^+ = \sum_{i=2}^{n-1}(c_i-c_{i-1})^-+c_{n-1},\\
a_r &=& \min \left\{ j : c_1+\sum_{i=2}^j(c_i-c_{i-1})^+ \geq r \right\},
\qquad {\rm and} \\
b_r &=& \min \left\{ j : \sum_{i=2}^j(c_i-c_{i-1})^- \geq r\right\}.
\end{eqnarray*}
Since $j=a_r$ forces $c_j>0$, we see that the equality
\begin{displaymath}
c_1+\sum_{i=2}^j(c_i-c_{i-1})^+=c_j+\sum_{i=2}^j(c_i-c_{i-1})^-
\end{displaymath}
yields that in \operatorname{comp}ref{5.35} we must have
\begin{equation} \label{5.36}
a_r< b_r , \qquad \hbox{for}\qquad 1\leq r\leq \ell.
\end{equation}
Now for the ratio in \operatorname{comp}ref{5.35} to contribute to the constant term in \operatorname{comp}ref{5.33}, it is necessary and sufficient that the reciprocal of this ratio should come out of the expansion
\begin{equation} \label{5.37}
\frac{\Omega[\mathbf{z}]}{\mathbf{z}_\operatorname{comp}ath}
= \frac{1}{z_{r_1}z_{r_2}\cdots z_{r_m}}
\sum_{d_i\geq 0} z_1^{d_1}z_2^{d_2}\cdots z_n^{d_n}.
\end{equation}
That is for some $d_1,d_2,\dots,d_n$ we must have
\begin{equation} \label{5.38}
{ z_{r_1}z_{r_2}\cdots z_{r_m}\over z_1^{d_1}z_2^{d_2}\cdots z_n^{d_n}}
= {z_{a_1}z_{a_2}\cdots z_{a_\ell}\over z_{b_1}z_{b_2}\cdots z_{b_\ell}}.
\end{equation}
Notice that $z_{a_1}z_{a_2}\cdots z_{a_\ell}$ and $z_{b_1}z_{b_2}\cdots z_{b_\ell}$ have no factor in common, since from the second expression in \operatorname{comp}ref{5.35} we derive that each variable $z_i$ can appear only in one of these two monomials. Thus $z_{a_1}z_{a_2}\cdots z_{a_\ell}$ divides $z_{r_1}z_{r_2}\cdots z_{r_m}$ and $z_{b_1}z_{b_2}\cdots z_{b_\ell}$ divides $z_1^{d_1}z_2^{d_2}\cdots z_n^{d_n}$ and in particular $\ell \leq m$. But this, together with the inequalities in \operatorname{comp}ref{5.36} shows that we must have
\begin{displaymath}
z_1^{d_1}z_2^{d_2}\cdots z_n^{d_n}
= z_{s_1}z_{s_2} \cdots z_{s_m},
\qquad \hbox{with}\quad s_i\geq r_i \quad\hbox{for}\quad 1\leq i \leq m.
\end{displaymath}
In other words $z_1^{d_1}z_2^{d_2}\cdots z_n^{d_n}$ must be the monomial of a NE-path $\delta\geq \operatorname{comp}ath$. Moreover from the identity in \operatorname{comp}ref{5.38} we derive that
\begin{displaymath}
\operatorname{area}(\delta/\operatorname{comp}ath) = (b_1-a_1)+(b_2-a_2)+\cdots +(b_\ell - a_\ell)
= - \sum_{i=1}^\ell a_i + \sum_{i=1}^\ell b_i.
\end{displaymath}
Thus from the middle expression in 5.35 it follows that
\begin{eqnarray*}
\operatorname{area}(\delta/\operatorname{comp}ath)
&=& -c_1-\sum_{i=2}^{n-1}i(c_i-c_{i-1})^+
+ \sum_{i=2}^{n-1}i(c_i-c_{i-1})^-+nc_{n-1} \\
&=& -c_1-\Big(\sum_{i=2}^{n-1}i(c_i-c_{i-1}) \Big)+c_{n-1}\\
&=& -c_1- \sum_{i=2}^{n-1}i c_i + \sum_{i=1}^{n-2}(i+1) c_i+nc_{n-1}\\
&=&\sum_{i=1}^{n-1}c_i,
\end{eqnarray*}
which is precisely the power of $t$ contributed by the Laurent monomial in \operatorname{comp}ref{5.35}.
Finally, suppose that $\delta$ is a NE-path weakly above $\operatorname{comp}ath$ as in \operatorname{comp}ref{5.32}. This given, let us weight each lattice cell with southeast corner $(a,b)$ with the Laurent monomial $t z_b/z_{b+1}$. Then it is easily seen that for each fixed column $1\leq i\leq m$, the product of the weights of the lattice cells lying between $\delta$ and $\operatorname{comp}ath$ is precisely $t^{s_i-r_i}x_{r_i}/x_{s_i}$. Thus
\begin{displaymath}
\operatorname{comp}rod_{i=1}^m t^{s_i-r_i} \frac{x_{r_i}}{x_{s_i}}
= t^{\operatorname{area}(\delta/\operatorname{comp}ath)} \frac{\mathbf{z}_\operatorname{comp}ath}{\mathbf{z}_\delta}.
\end{displaymath}
Since the left hand side of this identity is in the form given in \operatorname{comp}ref{5.35}, we clearly see that every summand of $C_\operatorname{comp}ath(t)$ will come out of the constant term.
\end{proof}
\begin{rmk} \label{rmk5.5}
It is easy to see that, for $q=1$, the constant term in \operatorname{comp}ref{5.27}, reduces to the one in \operatorname{comp}ref{5.33} with $\operatorname{comp}ath=\operatorname{comp}ath_0$ (the closest path to the diagonal $(0,0)\to (m,n)$). This is simply due to the identity
\begin{displaymath}
\Omega[-uM]\Big|_{q=1}= {(1-u)(1-qtu)\over (1-tu)(1-qu)}\Big|_{q=1}= 1.
\end{displaymath}
Moreover, since the coprimality of the pair $(m,n)$ had no role in the proof of Proposition \ref{prop5.5}, we were led to the formulation of the following conjecture, widely supported by our computer data.
\end{rmk}
\begin{conj} \label{conjV}
For any pair of positive integers $(u,v)$ and any NE-path $\operatorname{comp}ath$ in the $u\times v$ lattice that remains weakly above the lattice diagonal $(0,0)\to (u,v)$ we have
\begin{equation} \label{5.39}
\frac{\Omega[\mathbf{z}]}{\mathbf{z}_\operatorname{comp}ath}\,
\operatorname{comp}rod_{i=1}^{n-1} \frac{1}{(1-qtz_i/z_{i+1})}
\operatorname{comp}rod_{1\leq i<j\leq n} \Omega[-M z_i/z_j]
\Big|_{z_1^0z_2^0\cdots z_n^0}
= \sum_{\delta \geq \operatorname{comp}ath} t^{\operatorname{area}((\delta/\operatorname{comp}ath)} q^{\operatorname{dinv}(\delta)},
\end{equation}
where $\operatorname{dinv}(\delta)$ is computed as in step (3) of Algorithm \ref{algL} for $(km,kn)=(u,v)$.
\end{conj}
Remarkably, the equality in \operatorname{comp}ref{5.39} is still unproven even for general coprime pairs $(m,n)$, except of course for the cases $m=n+1$ proved in \cite{qtCatPos}. What is really intriguing is to explain how the inclusion of the expression
\begin{displaymath}
\operatorname{comp}rod_{1\leq i<j\leq n}\Omega[-M z_i/z_j]
\end{displaymath}
accounts for the insertion of the factor
\begin{displaymath}
q^{\operatorname{dinv}(\operatorname{comp}ath)-\operatorname{area}(\delta/\operatorname{comp}ath)}
\end{displaymath}
in the right hand side of \operatorname{comp}ref{5.33}). A combinatorial explanation of this phenomenon would lead to an avalanche of consequences in this area, in addition to proving Conjecture \ref{conjV}.
\end{document}
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\begin{document}
\title{Three Approaches to the
Quantitative Definition of Information in an Individual Pure
Quantum State}
\author{Paul Vitanyi\thanks{Partially supported by the EU fifth framework
project QAIP, IST--1999--11234, the NoE QUIPROCONE IST--1999--29064,
the ESF QiT Programmme, and ESPRIT BRA IV NeuroCOLT II Working Group
EP 27150.
Part of this work was done
during the author's 1998 stay at Tokyo Institute of Technology,
Tokyo, Japan, as Gaikoku-Jin Kenkyuin at INCOCSAT. A preliminary version
was archived as quant-ph/9907035. Address:
CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands. Email:
{\tt [email protected]}}\\
CWI and University of Amsterdam}
\date{}
\maketitle
\thispagestyle{empty}
\begin{abstract}
In analogy of classical Kolmogorov complexity
we develop a theory of the algorithmic information in bits contained
in any one of continuously many pure quantum states:
quantum Kolmogorov complexity.
Classical Kolmogorov complexity
coincides with the new quantum Kolmogorov complexity restricted to
the classical domain. Quantum Kolmogorov complexity is upper
bounded and can be effectively approximated from above.
With high probability a quantum object is incompressible.
There are two alternative approaches possible:
to define the complexity as the length of the shortest
qubit program that effectively describes the object, and to
use classical descriptions with
computable real parameters.
\end{abstract}
\section{Introduction}
While Kolmogorov complexity is the accepted absolute measure
of information content in a {\em classical} individual finite object,
a similar absolute notion is needed for the information
content of a pure quantum state.
\footnote{
For definitions and theory of Kolmogorov complexity consult \cite{LiVi97},
and for quantum theory consult \cite{Pe95}.
}
Quantum theory assumes that every complex vector, except the null vector,
represents a realizable pure quantum state.\footnote{That is,
every complex vector that can be normalized to unit length.}
This leaves open the question
of how to design the equipment that prepares such a pure
state. While there are continuously many pure states in
a finite-dimensional complex vector space---corresponding to all vectors of
unit length---we can finitely
describe only a countable subset. Imposing effectiveness on
such descriptions leads to constructive procedures.
The most general such procedures satisfying universally agreed-upon
logical principles of effectiveness are quantum Turing machines, \cite{BV97}.
To define quantum Kolmogorov complexity
by way of quantum Turing machines leaves
essentially two options:
\begin{enumerate}
\item
We want to describe every quantum superposition exactly; or
\item
we want to take
into account the number of bits/qubits in the specification
as well the accuracy of the quantum state produced.
\end{enumerate}
We have to deal with three problems:
\begin{itemize}
\item
There are continuously many quantum Turing machines;
\item
There are continously many pure quantum states;
\item
There are continuously many qubit descriptions.
\end{itemize}
There are uncountably many quantum Turing machines
only if we allow arbitrary real rotations in the definition of
machines. Then, a quantum Turing machine can only be universal
in the sense that it can approximate the computation of an
arbitrary machine, \cite{BV97}. In descriptions using universal
quantum Turing machines
we would have to account for the closeness of approximation,
the number of steps required to get this precision, and the like.
In contrast, if we fix the rotation
of all contemplated machines to a single primitive rotation $\theta$
with $\cos \theta = 3/5$ and $\sin \theta = 4/5$ then there
are only countably many Turing machines and the universal machine
simulates the others exactly \cite{ADH97}.
Every quantum
Turing machine computation using arbitrary real rotations
can be approximated to any precision by machines with fixed
rotation $\theta$ but in general cannot be simulated
exactly---just like in the case of the simulation of
arbitrary quantum Turing machines by a universal
quantum Turing machine. Since exact simulation is impossible
by a fixed universal quantum Turing machine anyhow, but arbitrarily
close approximations are possible by Turing machines using
a fixed rotation like $\theta$, we are motivated to fix
$Q_1 , Q_2 , \ldots$ as a standard enumeration of
quantum Turing machines using only rotation $\theta$.
Our next question is whether we want programs
(descriptions) to be in classical bits
or in qubits?
The intuitive notion of computability requires
the programs to be classical. Namely, to prepare a quantum state
requires a physical apparatus that ``computes'' this quantum state
from classical specifications.
Since such specifications have effective descriptions,
every quantum state that can be prepared can
be described effectively in descriptions consisting of classical bits.
Descriptions consisting of arbitrary pure quantum states
allows noncomputable (or hard to compute)
information to be hidden in the bits of
the amplitudes.
In Definition~\ref{def.pqscomp} we call a pure quantum state {\em directly
computable} if there is a (classical) program such that
the universal quantum Turing machine computes that state from
the program and then halts in an appropriate fashion.
In a computational setting we naturally
require that directly computable pure quantum states can be
prepared.
By repeating the preparation we can obtain
arbitrarily many copies of the pure quantum state.
\footnote{See the discussion in \cite{Pe95}, pp. 49--51.
If descriptions are not effective then we are not going to use them in our
algorithms except possibly on inputs from an ``unprepared''
origin. Every quantum state used in a quantum computation
arises from some classically preparation or is possibly
captured from some unknown origin. If the latter, then we can consume
it as conditional side-information or an oracle.}
Restricting ourselves to an effective enumeration of
quantum Turing machines and classical descriptions
to describe by approximation continuously many pure quantum states is
reminiscent of the construction of continuously many real numbers
from Cauchy sequences of rational numbers, the rationals being
effectively enumerable.
The second approach
considers the shortest effective qubit description
of a pure quantum state. This can also be properly
formulated in terms of the conditional version
of the first approach. An advantage of this version is that the
upper bound on the complexity of a pure
quantum state is immediately given by the number of qubits involved in the
literal description of that pure quantum state.
The status
of incompressibility and degree of uncomputability is as yet
unknown and potentially a source of problems with this approach.
The third approach is to give programs for the $2^{n+1}$ real numbers
involved in the precise description of the $n$-qubit state. Then
the question reduces to the problem of describing lists of
real numbers.
In the classical situation there are also several variants
of Kolmogorov complexity that are very meaningful in their
respective settings: plain Kolmogorov complexity, prefix complexity,
monotone complexity, uniform complexity,
negative logarithm of universal measure, and so on \cite{LiVi97}.
It is therefore not surprising that in the more complicated situation
of quantum information several different choices of complexity
can be meaningful and unavoidable in different settings.
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\section{Classical Descriptions}
The complex
quantity $\bracket{x}{z}$ is the inner product of vectors $\ket{x}$
and $\ket{z}$.
Since pure quantum states $\ket{x}, \ket{z}$ have unit length,
$|\bracket{x}{z}| = | \cos \theta |$ where $\theta$ is the
angle between vectors $\ket{x}$ and $\ket{z}$
and $|\bracket{x}{z}|^2$ is the probability of outcome
$\ket{x}$ being
measured from state $\ket{z}$, \cite{Pe95}. The idea is as follows.
A {\em von Neumann measurement} is a decomposition of the Hilbert space
into subspaces that are mutually orthogonal, for example an
orthonormal basis is an observable. Physicists like to
specify observables as Hermitian matrices, where the
understanding is that the eigenspaces of the matrices
(which will always be orthogonal) are the actual subspaces.
When a measurement is performed, the state is projected
into one of the subspaces (with probability equal to the square of
the projection). So the subspaces correspond to the possible {\em outcomes}
of a measurement. In the above case we project $\ket{z}$ on outcome
$\ket{x}$ using projection $\shar{x}{x}$ resulting in
$\bracket{x}{z} \ket{x}$.
Our model of computation is a quantum
Turing machine with classical binary program $p$ on the input tape
and a quantum auxiliary input
on a special conditional input facility. We think of this auxiliary input
as being given as a pure quantum state $\ket{y}$
(in which case it can be used only once),
as a mixture density matrix $\rho$, or
(perhaps partially) as a classical program from which
it can be computed. In the last case, the classical program can of course
be used indefinitely often.\footnote{We can even allow that the conditional
information $y$ is infinite or noncomputable, or an oracle.
But we will not need this in the present paper.
}
It is therefore not only important {\em what} information is
given conditionally, but also {\em how} it is described---like
this is the sometimes the case in the classical version
of Kolmogorov complexity for other reasons that would additionally
hold in the quantum case.
We impose the condition that the set of {\em halting
programs} ${\cal P}_y = \{p: T(p | y) < \infty \}$ is {\em prefix-free}:
no program in ${\cal P}_y$ is a proper prefix of another program in
${\cal P}_y$. Put differently, the Turing machine scans all of a
halting program $p$ but never scans the bit following the last
bit of $p$: it is {\em self-delimiting}.
\footnote{One can also use a model were the input $p$ is delimited
by distinguished markers. Then the Turing machine always knows where
the input ends. In the self-delimiting case the endmarker must be
implicit in the halting program $p$ itself. This encoding of the
endmarker carries an inherent penalty in the form of increased length:
typically a prefix code of an $n$-length binary string has length
about $n+ \log n + 2 \log \log n$ bits, \cite{LiVi97}.}
\footnote{
There are two possible interpretations for the computation relation
$Q(p, y) = \ket{x}$. In the narrow interpretation
we require that $Q$ with $p$ on the input tape
and $y$ on the conditional tape halts with $\ket{x}$
on the output tape. In the wide interpretation we can
define pure quantum states by requiring
that for every precision $\delta > 0$ the computation
of $Q$ with $p$ on the input tape
and $y$ on the conditional tape
and $\delta$ on a tape where the precision is to be supplied
halts with $\ket{x'}$
on the output tape and $|\bracket{x}{x'}|^2 \geq 1-\delta$.
Such a notion of ``computable''
or ``recursive'' pure quantum states is similar to Turing's
notion of ``computable numbers.''
In the remainder of this section we use the narrow interpretation.
}
\begin{definition}
\rm
The {\em (self-delimiting) complexity} of $\ket{x}$
with respect to quantum Turing machine $Q$
with $y$ as conditional input given for free is
\[
K_Q (\ket{x} | y ) :=
\min_{p} \{ l(p) + \lceil - \log(|\bracket{z}{x}|^2) \rceil :
Q(p, y) = \ket{z} \}
\]
where $l(p)$ is the number of bits in the specification $p$,
$y$ is an input quantum state and
$\ket{z}$ is
the quantum state produced by the computation $Q(p, y)$,
and $\ket{x}$ is the target state that one is
trying to describe.
\end{definition}
\begin{theorem}\label{theo.inv}
There is a universal machine
\footnote{
We use ``$U$'' to denote a universal (quantum) Turing machine
rather than a unitary matrix.}
$U$ such that for all machines $Q$
there is a constant $c_Q$ (the length of the
description of the index of $Q$ in the enumeration)
such that for all quantum states $\ket{x}$ we have
$K_U (\ket{x} |y) \leq K_Q (\ket{x}|y) + c_Q$.
\end{theorem}
\begin{proof}
There is a universal quantum Turing machine $U$ in the standard enumeration
$Q_1 , Q_2, \ldots$ such that for every quantum Turing machine
$Q$ in the enumeration there is a self-delimiting program $i_Q$
(the index of $Q$) and $U(i_Q p , y) = Q(p,y)$ for all $p,y$.
Setting $c_Q = l(i_Q)$ proves the theorem.
\end{proof}
We fix once and for all a reference universal quantum Turing machine $U$
and define the quantum Kolmogorov complexity as
\begin{eqnarray*}
&& K (\ket{x} | y) := K_U (\ket{x}|y), \\
&& K (\ket{x}) := K_U (\ket{x} | \epsilon ),
\end{eqnarray*}
where $\epsilon$ denotes the absence of any conditional
information.
The definition is continuous:
If two quantum states are very close then their quantum Kolmogorov
complexities are very close. Furthermore, since we can approximate
every (pure quantum) state $\ket{x}$ to arbitrary closeness, \cite{BV97},
in particular, for every constant $\epsilon > 0$
we can compute a (pure quantum) state $\ket{z}$
such that
$|\bracket{z}{x}|^2 > 1-\epsilon$.
\footnote{We can view this as the probability of the possibly
noncomputable outcome $\ket{x}$
when executing projection $\shar{x}{x}$ on $\ket{z}$
and measuring outcome $\ket{x}$.}
For this definition to be
useful it should satisfy:
\begin{itemize}
\item
The complexity of a pure state that can be directly computed should be the
length of the shortest program that computes that state. (If the
complexity is less then this may lead to discontinuities when we restrict
quantum Kolmogorov complexity to the domain of classical objects.)
\item
The quantum Kolmogorov complexity of a classical object should
equal the classical Kolmogorov complexity of that object (up to
a constant additive term).
\item
The quantum Kolmogorov complexity of a quantum object should
have an upper bound. (This is necessary for the complexity
to be approximable from above, even if the quantum object is
available in as many copies as we require.)
\item
Most objects should be ``incompressible'' in terms of quantum
Kolmogorov complexity.
\item
In a probabilistic ensemble the expected quantum Kolmogorov
complexity should be about equal (or have another meaningful
relation) to the von Neumann entropy.
\footnote{In the classical case the average self-delimiting
Kolmogorov complexity
equals the Shannon entropy up to an additive constant depending
on the complexity of the distribution concerned.}
\end{itemize}
For a quantum system
$\ket{z}$
the quantity $P(x):= |\bracket{z}{x}|^2$ is the probability that
the system passes a test for $\ket{x}$, and vice versa.
The term $\lceil - \log(|\bracket{z}{x}|^2) \rceil$ can be viewed
as the
code word length to redescribe $\ket{x}$ given $\ket{z}$
and an orthonormal basis with $\ket{x}$ as one of the basis vectors
using
the well-known Shannon-Fano prefix code.
This works as follows: For every state $\ket{z}$ in
$N :=2^n$-dimensional Hilbert space
with basis vectors ${\cal B} = \{ \ket{e_0}, \ldots , \ket{e_{N-1}}\}$ we have
$\sum_{i=0}^{N-1} |\bracket{e_i }{z}|^2 =1$. If the basis has
$\ket{x}$ as one of the basis vectors, then we can
consider $\ket{z}$ as a random variable that assumes value $\ket{x}$
with probability $|\bracket{x}{z}|^2$. The Shannon-Fano code word
for $\ket{x}$ in the probabilistic ensemble
${\cal B}, (|\bracket{e_i}{z}|^2)_i$ is
based on the probability $|\bracket{x}{z}|^2$ of $\ket{x}$
given $\ket{z}$ and has length
$\lceil - \log(|\bracket{x}{z}|^2) \rceil$. Considering a canonical
method of constructing an orthonormal basis
${\cal B} = \ket{e_0}, \ldots, \ket{e_{N-1}}$
from a given basis
vector, we can choose ${\cal B}$ such that
$K({\cal B}) = \min_i \{ K(\ket{e_i}) \} +O(1)$.
The Shannon-Fano code is appropriate for our purpose since it is optimal
in that it achieves the least expected code word
length---the expectation taken over the probability of the
source words---up to 1 bit by Shannon's Noiseless Coding Theorem.
\subsection{Consistency with Classical Complexity}
Our proposal would not be useful if it were the case that for
a directly computable object the complexity is less than the
shortest program to compute that object. This would imply
that the code corresponding to the
probabilistic component in the description is possibly shorter than
the difference in program lengths for programs for an approximation
of the object and the
object itself. This would penalize definite description compared
to probabilistic description and in case of classical objects
would make quantum Kolmogorov complexity less than classical
Kolmogorov complexity.
\begin{theorem}\label{theo.equiv}
Let $U$ be the reference universal quantum Turing machine
and let $\ket{x}$ be a basis vector in a directly computable orthonormal
basis ${\cal B}$ given $y$: there is
a program $p$ such that $U(p, y)= \ket{x}$.
Then $K(\ket{x} | y)= \min_p \{l(p): U(p, y)= \ket{x} \}$
up to $K({\cal B}|y) +O(1)$.
\end{theorem}
\begin{proof}
Let $\ket{z}$ be such that
\[
K (\ket{x} | y ) =
\min_{q} \{ l(q) + \lceil - \log(|\bracket{z}{x}|^2) \rceil :
U(q, y) = \ket{z} \} .
\]
Denote the program $q$ that minimizes the righthand side
by $q_{\min}$
and the program $p$ that minimizes the expression in the statement
of the theorem by $p_{\min}$.
By running $U$ on all binary strings (candidate programs)
simultaneously dovetailed-fashion
\footnote{A {\em dovetailed} computation is a method related
to Cantor's diagonalization to run all programs alternatingly
in such a way that every program eventually makes progress. On
an list of programs $p_1, p_2, \ldots$ one divides
the overall computation into stages $k:=1,2, \ldots$.
In stage $k$ of the overall computation one
executes the $i$th computation step of every program $p_{k-i+1}$
for $i:=1, \ldots , k$.}
one can enumerate all objects that are directly computable given $y$
in order of their halting programs. Assume that $U$ is also
given a $K({\cal B}|y)$ length program $b$ to compute
${\cal B}$---that is, enumerate the basis
vectors in ${\cal B}$.
This way $q_{\min}$ computes
$\ket{z}$, the program $b$ computes ${\cal B}$.
Now since the vectors of ${\cal B}$ are mutually orthogonal
\[ \sum_{\ket{e} \in {\cal B}} | \bracket{z}{e}|^2 = 1 .
\]
Since $\ket{x}$ is one of the basis vectors
we have $- \log |\bracket{z}{x}|^2$ is the length of
a prefix code (the Shannon-Fano code) to compute $\ket{x}$ from $\ket{z}$
and ${\cal B}$.
Denoting this code by $r$ we have that the concatenation $q_{\min} b r$
is a program to compute $\ket{x}$: parse it into
$q_{\min}, b,$ and $r$ using the self-delimiting
property of $q_{\min}$ and $b$. Use
$q_{\min}$ to compute $\ket{z}$ and use $b$ to compute ${\cal B}$,
determine the
probabilities $|\bracket{z}{e}|^2$ for all basis vectors
$\ket{e}$ in ${\cal B}$. Determine the Shannon-Fano code words
for all the basis vectors from these probabilities.
Since $r$ is the code word for $\ket{x}$ we can now
decode $\ket{x}$. Therefore,
\[ l(q_{\min} ) + \lceil - \log(|\bracket{z}{x}|^2) \rceil
\geq l( p_{\min}) - K({\cal B}|y) - O(1) \]
which was what we had to prove.
\end{proof}
\begin{corollary}\label{cor.clasquant}
\rm
On classical objects (that is, the natural numbers
or finite binary strings that are all directly computable) the
quantum Kolmogorov complexity coincides up
to a fixed additional constant with the self-delimiting
Kolmogorov complexity since $K({\cal B}|n) = O(1)$ for the standard
classical basis ${\cal B}= \{0,1\}^n$.
\footnote{
This proof does not show that it coincide up to an additive constant term
with the original plain complexity defined by Kolmogorov, \cite{LiVi97},
based on Turing machines where the input is delited by distinguished markers.
The same proof for the plain Kolmogorov complexity shows
that it coincides up to a logarithmic additive term.
}
(We assume that the information about the dimensionality
of the Hilbert space is given conditionally.)
\end{corollary}
\begin{remark}
\rm
Fixed additional constants are no problem since
the complexity also varies by fixed additional constants due to the choice of
reference universal Turing machine.
\end{remark}
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0.80.2
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\subsection{Consistency with Classical Complexity}
Our proposal would not be useful if it were the case that for
a directly computable object the complexity is less than the
shortest program to compute that object. This would imply
that the code corresponding to the
probabilistic component in the description is possibly shorter than
the difference in program lengths for programs for an approximation
of the object and the
object itself. This would penalize definite description compared
to probabilistic description and in case of classical objects
would make quantum Kolmogorov complexity less than classical
Kolmogorov complexity.
\begin{theorem}\label{theo.equiv}
Let $U$ be the reference universal quantum Turing machine
and let $\ket{x}$ be a basis vector in a directly computable orthonormal
basis ${\cal B}$ given $y$: there is
a program $p$ such that $U(p, y)= \ket{x}$.
Then $K(\ket{x} | y)= \min_p \{l(p): U(p, y)= \ket{x} \}$
up to $K({\cal B}|y) +O(1)$.
\end{theorem}
\begin{proof}
Let $\ket{z}$ be such that
\[
K (\ket{x} | y ) =
\min_{q} \{ l(q) + \lceil - \log(|\bracket{z}{x}|^2) \rceil :
U(q, y) = \ket{z} \} .
\]
Denote the program $q$ that minimizes the righthand side
by $q_{\min}$
and the program $p$ that minimizes the expression in the statement
of the theorem by $p_{\min}$.
By running $U$ on all binary strings (candidate programs)
simultaneously dovetailed-fashion
\footnote{A {\em dovetailed} computation is a method related
to Cantor's diagonalization to run all programs alternatingly
in such a way that every program eventually makes progress. On
an list of programs $p_1, p_2, \ldots$ one divides
the overall computation into stages $k:=1,2, \ldots$.
In stage $k$ of the overall computation one
executes the $i$th computation step of every program $p_{k-i+1}$
for $i:=1, \ldots , k$.}
one can enumerate all objects that are directly computable given $y$
in order of their halting programs. Assume that $U$ is also
given a $K({\cal B}|y)$ length program $b$ to compute
${\cal B}$---that is, enumerate the basis
vectors in ${\cal B}$.
This way $q_{\min}$ computes
$\ket{z}$, the program $b$ computes ${\cal B}$.
Now since the vectors of ${\cal B}$ are mutually orthogonal
\[ \sum_{\ket{e} \in {\cal B}} | \bracket{z}{e}|^2 = 1 .
\]
Since $\ket{x}$ is one of the basis vectors
we have $- \log |\bracket{z}{x}|^2$ is the length of
a prefix code (the Shannon-Fano code) to compute $\ket{x}$ from $\ket{z}$
and ${\cal B}$.
Denoting this code by $r$ we have that the concatenation $q_{\min} b r$
is a program to compute $\ket{x}$: parse it into
$q_{\min}, b,$ and $r$ using the self-delimiting
property of $q_{\min}$ and $b$. Use
$q_{\min}$ to compute $\ket{z}$ and use $b$ to compute ${\cal B}$,
determine the
probabilities $|\bracket{z}{e}|^2$ for all basis vectors
$\ket{e}$ in ${\cal B}$. Determine the Shannon-Fano code words
for all the basis vectors from these probabilities.
Since $r$ is the code word for $\ket{x}$ we can now
decode $\ket{x}$. Therefore,
\[ l(q_{\min} ) + \lceil - \log(|\bracket{z}{x}|^2) \rceil
\geq l( p_{\min}) - K({\cal B}|y) - O(1) \]
which was what we had to prove.
\end{proof}
\begin{corollary}\label{cor.clasquant}
\rm
On classical objects (that is, the natural numbers
or finite binary strings that are all directly computable) the
quantum Kolmogorov complexity coincides up
to a fixed additional constant with the self-delimiting
Kolmogorov complexity since $K({\cal B}|n) = O(1)$ for the standard
classical basis ${\cal B}= \{0,1\}^n$.
\footnote{
This proof does not show that it coincide up to an additive constant term
with the original plain complexity defined by Kolmogorov, \cite{LiVi97},
based on Turing machines where the input is delited by distinguished markers.
The same proof for the plain Kolmogorov complexity shows
that it coincides up to a logarithmic additive term.
}
(We assume that the information about the dimensionality
of the Hilbert space is given conditionally.)
\end{corollary}
\begin{remark}
\rm
Fixed additional constants are no problem since
the complexity also varies by fixed additional constants due to the choice of
reference universal Turing machine.
\end{remark}
\subsection{Upper Bound on Complexity}
A priori, in the worst case $K(\ket{x} |n )$
is possibly $\infty$. We show that the worst-case has a $2n$ upper bound.
\begin{lemma}
For all $n$-qubit quantum states $\ket{x}$ we
have $K(\ket{x} |n)\leq 2n+O(1)$.
\end{lemma}
\begin{proof}
For every state $\ket{x}$ in $N :=2^n$-dimensional Hilbert space
with basis vectors $\ket{e_0}, \ldots , \ket{e_{N-1}}$ we have
$\sum_{i=0}^{N-1} |\bracket{e_i }{x}|^2 =1$. Hence there is an $i$
such that $|\bracket{e_i }{x}|^2 \geq 1/N$.
Let $p$ be a $K(i|n)+O(1)$-bit program to construct a
basis state $\ket{e_i}$ given $n$.
Then $l(p) \leq n + O(1)$.
Then $K ( \ket{x} |n ) \leq l(p) - \log (1/N) \leq 2n + O(1)$.
\end{proof}
\begin{comment}
\begin{remark}
\rm
One may think that $K(\ket{x} ) \leq 3n/2 + O(1)$.
Namely, for every diagonal $d \in \{ d_0, \ldots , d_{2^n - 1} \}$
for some basis vector $e$ we have $|\bracket{e}{d}|^2 \geq 1/2^{n/2}$.
Moreover, for every vector $x$ there is a $d$ such that
$| \bracket{e}{x}|^2 \geq | \bracket{e}{d}|^2$.
But the argument seems wrong because there is equal probability
for all basis vectors that a diagonal is observed?
\end{remark}
\end{comment}
\subsection{Computability}
In the classical case Kolmogorov complexity is not computable
but can be approximated from above by a computable process.
The non-cloning property prevents us from copying an unknown pure
quantum state given to us. Therefore, an approximation from
above that requires checking every output state against the
target state destroys the latter. To overcome the fragility of
the pure quantum target state one has to postulate that it
is available as an outcome in a measurement.
\begin{theorem}
Let $\ket{x}$ be the pure quantum state we want to describe.
{\rm (i)} The quantum Kolmogorov complexity $K(\ket{x})$ is not computable.
{\rm (ii)}
If we can repeatedly execute the projection $\shar{x}{x}$
and perform a measurement with outcome $\ket{x}$, then
the quantum Kolmogorov complexity $K(\ket{x})$
can be approximated
from above by a computable process with arbitrarily
small probability of error $\alpha$ of giving a too small value.
\end{theorem}
\begin{proof}
The uncomputability follows a fortiori from the classical case.
The semicomputability follows because we have established an upper
bound on the quantum Kolmogorov complexity, and we can simply
enumerate all halting classical programs up to that length by running their
computations dovetailed fashion. The idea is as follows:
Let the target state be $\ket{x}$
of $n$ qubits. Then, $K(\ket{x}|n) \leq 2n + O(1)$. (The
unconditional case $K(\ket{x})$ is similar
with $2n$ replaced by $2(n + \log n)$.)
We want to identify a program $x^*$ such that $p:=x^*$ minimizes
$l(p) - \log |\bracket{x}{U(p,n)}|^2$ among all candidate programs.
To identify it in the limit,
for some fixed $k$ satisfying (\ref{eq.alpha}) below
for given $n, \alpha , \epsilon$,
repeat the computation of every halting program
$p$ with $l(p) \leq 2n+O(1)$ at least $k$ times and perform the assumed
projection and measurement. For every halting program $p$ in the dovetailing
process we estimate the probability
$q :=|\bracket{x}{U(p,n)}|^2$ from the fraction $m/k$:
the fraction of $m$ positive outcomes out of $k$ measurements.
The probability that the estimate $m/k$ is off from the real
value $q$ by more than
an $\epsilon q$ is given by Chernoff's bound:
for
$0 \leq \epsilon \leq 1$,
\begin{equation}
\label{chernoff}
P ( |m- qk | > \epsilon qk )
\leq 2e^{ - \epsilon^2 qk /3}.
\end{equation}
This means that the probability that the deviation $|m/k - q|$
exceeds $\epsilon q$ vanishes exponentially with growing $k$.
Every candidate program $p$ satisfies
(\ref{chernoff}) with its own $q$ or $1-q$. There are $O(2^{2n})$
candidate programs $p$ and hence also $O(2^{2n})$ outcomes $U(p,n)$
with halting computations.
We use this estimate to upper bound the probability of error $\alpha$.
For given $k$, the probability
that {\em some} halting candidate program $p$ satisfies
$ |m- qk | > \epsilon qk$
is at most $\alpha$ with
\[ \alpha \leq \sum_{U(p,n) < \infty } 2e^{ - \epsilon^2 q k /3} .\]
The probability that {\em no} halting program does so is
at least $1- \alpha$. That is, with probability
at least $1-\alpha$ we have
\[ (1- \epsilon)q \leq \frac{m}{k} \leq (1+ \epsilon)q \]
for every halting program $p$.
It is convenient to restrict attention to the case that all $q$'s are large.
Without loss of generality,
if $q < \frac{1}{2}$ then consider $1-q$ instead of $q$.
Then,
\begin{equation}\label{eq.alpha}
\log \alpha \leq 2n- (\epsilon^2 k \log e )/ 6 +O(1).
\end{equation}
The approximation algorithm is as follows:
{\bf Step 0:} Set the required degree of approximation $\epsilon < 1/2$
and the number of trials $k$ to achieve the required probability of error $\alpha$.
{\bf Step 1:} Dovetail the running of all candidate programs until the
next halting program is enumerated.
Repeat the computation of the new halting program $k$ times
{\bf Step 2:} If there is more than one program $p$ that achieves the
current minimum then choose the program with the smaller length
(and hence least number of successfull observations).
If $p$ is the selected program with $m$ successes out of $k$ trials
then set the current approximation of $K(\ket{x})$ to
\[l(p) - \log \frac{m}{(1+\epsilon)k} .\]
This exceeds the proper value
of the approximation based on the real $q$ instead
of $m/k$ by at most 1 bit for all $\epsilon < 1$.
{\bf Step 3:} Goto {\bf Step 1}.
\end{proof}
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\subsection{Computability}
In the classical case Kolmogorov complexity is not computable
but can be approximated from above by a computable process.
The non-cloning property prevents us from copying an unknown pure
quantum state given to us. Therefore, an approximation from
above that requires checking every output state against the
target state destroys the latter. To overcome the fragility of
the pure quantum target state one has to postulate that it
is available as an outcome in a measurement.
\begin{theorem}
Let $\ket{x}$ be the pure quantum state we want to describe.
{\rm (i)} The quantum Kolmogorov complexity $K(\ket{x})$ is not computable.
{\rm (ii)}
If we can repeatedly execute the projection $\shar{x}{x}$
and perform a measurement with outcome $\ket{x}$, then
the quantum Kolmogorov complexity $K(\ket{x})$
can be approximated
from above by a computable process with arbitrarily
small probability of error $\alpha$ of giving a too small value.
\end{theorem}
\begin{proof}
The uncomputability follows a fortiori from the classical case.
The semicomputability follows because we have established an upper
bound on the quantum Kolmogorov complexity, and we can simply
enumerate all halting classical programs up to that length by running their
computations dovetailed fashion. The idea is as follows:
Let the target state be $\ket{x}$
of $n$ qubits. Then, $K(\ket{x}|n) \leq 2n + O(1)$. (The
unconditional case $K(\ket{x})$ is similar
with $2n$ replaced by $2(n + \log n)$.)
We want to identify a program $x^*$ such that $p:=x^*$ minimizes
$l(p) - \log |\bracket{x}{U(p,n)}|^2$ among all candidate programs.
To identify it in the limit,
for some fixed $k$ satisfying (\ref{eq.alpha}) below
for given $n, \alpha , \epsilon$,
repeat the computation of every halting program
$p$ with $l(p) \leq 2n+O(1)$ at least $k$ times and perform the assumed
projection and measurement. For every halting program $p$ in the dovetailing
process we estimate the probability
$q :=|\bracket{x}{U(p,n)}|^2$ from the fraction $m/k$:
the fraction of $m$ positive outcomes out of $k$ measurements.
The probability that the estimate $m/k$ is off from the real
value $q$ by more than
an $\epsilon q$ is given by Chernoff's bound:
for
$0 \leq \epsilon \leq 1$,
\begin{equation}
\label{chernoff}
P ( |m- qk | > \epsilon qk )
\leq 2e^{ - \epsilon^2 qk /3}.
\end{equation}
This means that the probability that the deviation $|m/k - q|$
exceeds $\epsilon q$ vanishes exponentially with growing $k$.
Every candidate program $p$ satisfies
(\ref{chernoff}) with its own $q$ or $1-q$. There are $O(2^{2n})$
candidate programs $p$ and hence also $O(2^{2n})$ outcomes $U(p,n)$
with halting computations.
We use this estimate to upper bound the probability of error $\alpha$.
For given $k$, the probability
that {\em some} halting candidate program $p$ satisfies
$ |m- qk | > \epsilon qk$
is at most $\alpha$ with
\[ \alpha \leq \sum_{U(p,n) < \infty } 2e^{ - \epsilon^2 q k /3} .\]
The probability that {\em no} halting program does so is
at least $1- \alpha$. That is, with probability
at least $1-\alpha$ we have
\[ (1- \epsilon)q \leq \frac{m}{k} \leq (1+ \epsilon)q \]
for every halting program $p$.
It is convenient to restrict attention to the case that all $q$'s are large.
Without loss of generality,
if $q < \frac{1}{2}$ then consider $1-q$ instead of $q$.
Then,
\begin{equation}\label{eq.alpha}
\log \alpha \leq 2n- (\epsilon^2 k \log e )/ 6 +O(1).
\end{equation}
The approximation algorithm is as follows:
{\bf Step 0:} Set the required degree of approximation $\epsilon < 1/2$
and the number of trials $k$ to achieve the required probability of error $\alpha$.
{\bf Step 1:} Dovetail the running of all candidate programs until the
next halting program is enumerated.
Repeat the computation of the new halting program $k$ times
{\bf Step 2:} If there is more than one program $p$ that achieves the
current minimum then choose the program with the smaller length
(and hence least number of successfull observations).
If $p$ is the selected program with $m$ successes out of $k$ trials
then set the current approximation of $K(\ket{x})$ to
\[l(p) - \log \frac{m}{(1+\epsilon)k} .\]
This exceeds the proper value
of the approximation based on the real $q$ instead
of $m/k$ by at most 1 bit for all $\epsilon < 1$.
{\bf Step 3:} Goto {\bf Step 1}.
\end{proof}
\subsection{Incompressibility}
\begin{definition}\label{def.pqscomp}
\rm
A pure quantum state $\ket{x}$ is {\em computable} if
$K(\ket{x}) < \infty$. Hence all finite-dimensional pure
quantum states are computable.
We call a pure quantum state {\em directly
computable} if there is a program $p$ such that
$U(p)= \ket{x}$.
\end{definition}
The standard orthonormal basis---consisting of all $n$-bit strings---of
the $2^n$-dimensional
Hilbert space ${\cal H}_N$ has
at least $2^n (1-2^{-c})$ basis vectors $\ket{e_i}$
that satisfy $K(\ket{e_i} |n) \geq n-c$. This is the standard counting argument
in \cite{LiVi97}. But what about nonclassical orthonormal bases?
\begin{lemma}\label{lem.lowb}
There is a (possibly nonclassical) orthonormal basis of the $2^n$-dimensional
Hilbert space ${\cal H}_N$ such
that at least $2^n (1-2^{-c})$ basis vectors $\ket{e_i}$
satisfy $K(\ket{e_i} |n) \geq n-c$.
\end{lemma}
\begin{proof}
Every orthonormal basis of ${\cal H}_N$
has $2^n$ basis vectors and there are at most
$m \leq \sum_{i=0}^{n-c-1} 2^i = 2^{n-c}-1$ programs of length less than
$n-c$. Hence there are at most $m$ programs
available to approximate the basis vectors.
We construct an orthonormal basis satisfying the lemma:
The set of directly computed pure quantum states
$\ket{x_0}, \ldots , \ket{x_{m-1}}$
span an $m'$-dimensional subspace ${\cal A}$ with $m' \leq m$
in the $2^n$-dimensional Hilbert space ${\cal H}_N$ such
that ${\cal H}_N = {\cal A} \oplus {\cal A}^{\perp}$.
Here ${\cal A}^{\perp}$ is a $(2^n - m')$-dimensional
subspace of ${\cal H}_N$ such that every vector in it is
perpendicular to every vector in ${\cal A}$.
We can write every element $\ket{x} \in {\cal H}_N$ as
\[
\sum_{i=0}^{m'-1} \alpha_i \ket{a_i}+ \sum_{i=0}^{2^n-m'-1} \beta_i \ket{b_i}
\]
where the $\ket{a_i}$'s form an orthonormal basis
of ${\cal A}$ and the $\ket{b_i}$'s form an
orthonormal basis of $ {\cal A}^{\perp}$ so that
the $\ket{a_i}$'s and $\ket{b_i}$'s form an orthonormal basis $K$
for ${\cal H}_N$. For every directly computable state
$\ket{x_j} \in {\cal A}$ and basis vector $\ket{b_i} \in A^{\perp}$ we have
$|\bracket{x_j}{b_i} |^2 = 0$
implying that
$K(\ket{x_j}|n) - \log | \bracket{x_j}{b_i} |^2 = \infty$
and therefore $K(\ket{b_i}|n) > n-c$
($0 \leq j < m, 0 \leq i < 2^n - m'$).
This proves the lemma.
\end{proof}
We generalize this lemma to arbitrary bases:
\begin{theorem}\label{theo.lowb}
Every orthonormal basis $\ket{e_0}, \dots ,
\ket{e_{2^n-1}}$ of the $2^n$-dimensional
Hilbert space ${\cal H}_N$ has
at least $2^n (1-2^{-c})$ basis vectors $\ket{e_i}$
that satisfy $K(\ket{e_i}|n) \geq n-c$.
\end{theorem}
\begin{proof}
Use the notation of the proof of Lemma~\ref{lem.lowb}.
Assume to the contrary that there are $>2^{n-c}$
basis vectors $\ket{e_i}$ with $K(\ket{e_i}|n) < n-c$.
Then at least two of them, say $\ket{e_0}$
and $\ket{e_1}$ and some
pure quantum state $\ket{x}$ directly computed from a $<(n-c)$-length program
satisfy
\begin{equation}\label{eq.ex}
K(\ket{e_i}|n) = K(\ket{x}|n) + \lceil - \log |\bracket{e_i}{x}|^2 \rceil .
\end{equation}
($i=0,1$). This means that $K(\ket{x}|n)<n-c-1$ since not both
$\ket{e_0}$ and $\ket{e_1}$ can be equal to $\ket{x}$.
Hence for every directly computed pure quantum state of complexity
$n-c-1$ there is at most one basis state of the same complexity
(in fact only if that basis state is identical with the directly
computed state.)
Now eliminate all directly computed pure quantum states $\ket{x}$ of
complexity $n-c-1$ together with the basis states $\ket{e}$
that stand in relation Equation~\ref{eq.ex}. We are now
left with $> 2^{n-c-1}$ basis states that stand in relation
of Equation~\ref{eq.ex} with the remaining at most
$2^{n-c-1}-1$ remaining directly computable pure quantum states
of complexity $\leq n-c-2$.
Repeating the same argument we end up with $>1$ basis vector
that stand in relation of Equation~\ref{eq.ex} with 0
directly computable pure quantum states of complexity $\leq 0$
which is impossible.
\end{proof}
\begin{corollary}
The uniform probability
$\Pr\{\ket{x}: K(\ket{x}|n) \geq n-c \} \geq 1-1/2^c$.
\end{corollary}
\begin{example}
\rm
We elucidate the role of the $- \log | \bracket{x}{z} |^2$
term.
Let $x$ be a random classical string with $K(x) \geq l(x)$
and let $y$ be a string obtained from $x$
by complementing one bit. It is known (Exercise 2.2.8 in \cite{LiVi97})
that for every such $x$ of length $n$ there is such a $y$ with complexity
$K(y|n) = n - \log n +O(1)$. Now let $\ket{z}$ be a pure quantum state which has
classical bits except the difference bit between $x$ and $y$ that has
equal probabilities of being observed as ``1'' and as ``0.''
We can prepare $\ket{z}$ by giving $y$ and the position of the
difference bit (in $\log n$ bits)
and therefore $K(\ket{z}|n) \leq n + O(1)$. Since from
$\ket{z}$ we have probability $\frac{1}{2}$ of obtaining $x$
by observing the particular bit in superposition
and $K(x|n) \geq n$ it follows $K(\ket{z} |n) \geq n + O(1)$ and
therefore $K(\ket{z} |n) = n + O(1)$.
\begin{comment}
Since $\ket{z}$ is
a directly computable state this is consistent with Corollary~\ref{cor.clasquant}.
\end{comment}
From $\ket{z}$ we have probability $\frac{1}{2}$ of obtaining $y$
by observing the particular bit in superposition which (correctly) yields that
$K(y|n) \leq n +O(1)$.
\end{example}
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\subsection{Conditional Complexity}
We have used the conditional complexity $K(\ket{x}|y)$
to mean the minimum sum of the length of a classical program to compute
$\ket{z}$ plus the negative logarithm of the probability of outcome
$\ket{x}$ when executing projection $\shar{x}{x}$ on $\ket{z}$
and measuring, given the pure quantum
state $y$ as input on a separate input tape.
In the quantum situation the notion of inputs consisting
of pure quantum states is subject to very special rules.
Firstly, if we are given an unknown pure quantum state $\ket{y}$ as
input it can be used only once, that is, it is irrevocably consumed
and lost in the computation. It cannot be copied or cloned without
destroying the original \cite{Pe95}. This phenomenon is
subject to the so-called {\em no-cloning theorem} and means that there is
a profound difference between giving a directly computable pure quantum
state as a classical program or giving it literally. Given as a
classical program we can prepare and use arbitrarily many copies of it.
Given as an (unknown) pure quantum state in superposition it can be
used as start of a computation only once---unless of course we
deal with an identity computation in which the input state is simply
transported to the output state. This latter computation nonetheless
destroys the input state.
If an unknown state $\ket{y}$ is given as input (in the conditional for example)
then the no-cloning theorem of quantum computing says it can be used
only {\em once}. Thus, for a non-classical pure quantum state $\ket{x}$
we have
\[ K(\ket{x},\ket{x} | \ket{x}) \leq K(\ket{x})+O(1) \]
rather than
$K(x,x|x)=O(1)$ as in the case for classical objects $x$. This
holds even if $\ket{x}$ is directly computable but is
given in the conditional in the form of an unknown pure quantum state. However,
if $\ket{x}$ is directly computable and
the conditional is a classical program to compute
this directly computable state, then
that program can be used over and over again.
In the previous example,
if the conditional $\ket{x}$ is directly computable, for example
by a classical program $p$, then we have
both $K(\ket{x}|p) = O(1)$ and
$ K(\ket{x},\ket{x} | p) = O(1)$.
In particular, for a classical program $p$ that
computes a directly computable state $\ket{x}$ we have
\[ K(\ket{x},\ket{x} | p) = O(1) .\]
It is important here to notice that a classical program for
computing a directly computable quantum state carries {\em more information}
than the directly computable quantum state itself---much like a
shortest program for a classical object carries more information than the
object itself. In the latter case it consists in partial information
about the halting problem. In the quantum case of a directly
computable pure state we have the additional
information that the state is directly computable {\em and}
in case of a shortest classical program additional information
about the halting problem.
\subsection{Sub-Additivity}
Quantum Kolmogorov complexity of directly computable pure
quantum states in simple orthonormal bases is {\em sub-additive}:
\begin{lemma}\label{lem.additive}
For directly computable $\ket{x}, \ket{y}$ both of which
belong to (possibly different) orthonormal bases of
Kolmogorov complexity $O(1)$ we have
\[ K(\ket{x}, \ket{y} ) \leq K(\ket{x}|\ket{y}) + K(\ket{y}) \]
up to an additive constant term.
\end{lemma}
\begin{proof}
By Theorem~\ref{theo.equiv} we there is a program $p_y$ to compute $\ket{y}$
with $l(p)= K(\ket{y})$ and a program
$p_{y \rightarrow x}$ to compute $\ket{x}$ from $\ket{y}$
with $l(p_{y \rightarrow x}) = K(\ket{x}|\ket{y})$ up
to additional constants. Use $p_y$ to
construct two copies of $\ket{y}$ and $p_{y \rightarrow x}$ to construct
$\ket{x}$ from one of the copies of $\ket{y}$.
The separation between
these concatenated binary programs is taken care of
by the self-delimiting property
of the subprograms. The additional constant term
takes care of the couple of $O(1)$-bit
programs that are required.
\end{proof}
\begin{comment}
\begin{definition}
\rm
Define the information $I(\ket{x}:\ket{y})$ in pure quantum state $\ket{x}$
about pure quantum state $\ket{y}$ by
$I(\ket{x}: \ket{y}):= K(\ket{x})-K(\ket{x}| \ket{y})$.
\end{definition}
The proof is identical to that of the same relations in the classical
case since we are dealing with directly computable states. The last displayed
equation is known as ``symmetry of information'' since
it states that the information in a classical program for $\ket{x}$
about a classical program for $\ket{y}$ is
the same as the information in a classical program
for $\ket{y}$ about a classical program
for $\ket{x}$ up to the
additional logarithmic term.
\[ I(\ket{x}:\ket{y}) = I(\ket{y}:\ket{x}) .\]
\end{comment}
\begin{remark}
\rm
In the classical case we have equality in the theorem (up
to an additive logarithmic term).
The proof of the remaining inequality, as given in the classical case,
doesn't hold directly for the quantum case. It would require
a decision procedure that establishes equality between
two pure quantum states without error. While the sub-additivity
property holds in case of directly computable states,
is easy to see that for the general case of pure states
the subadditivity property fails
due to the ``non-cloning'' property.
For example for pure states $\ket{x}$ that are not ``clonable'' we
have:
\[ K(\ket{x}, \ket{x} ) > K(\ket{x}| \ket{x}) + K(\ket{x}) =
K(\ket{x}) + O(1) .\]
\end{remark}
We additionally note:
\begin{lemma}
For all directly computable pure states $\ket{x}$ and $\ket{y}$ we have
$K(\ket{x}, \ket{y}) \leq K(\ket{y}) - \log | \bracket{x}{y}|^2$
up to an additive logarithmic term.
\end{lemma}
\begin{proof}
$ K(\ket{x}|\ket{y}) \leq - \log | \bracket{x}{y}|^2$ by the proof
of Theorem~\ref{theo.equiv}.
Then, the lemma follows by Lemma~\ref{lem.additive}.
\end{proof}
\begin{comment}
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\section{Application to QFA}
A one-way {\em quantum finite automaton} (QFA)
is the quantum version of the classical finite automaton.
We follow te definition in \cite{ANTSV99}.
A QFA has a finite set of {\em basis states} $Q$ consisting
of three disjoint subsets $Q_1$ (accepting states), $Q_0$ (rejecting
states) and $Q_n$ (non-halting states). The set $Q_h= Q_0 \bigcup Q_1$
is the set of halting states. The QFA has a finite
nonempty {\em input alphabet} $\Sigma$. The input is placed
between special input markers
$\#, \$ \notin \Sigma$ indicating the left end and the right end of the
input and the working alphabet is $\Gamma = \Sigma \bigcup \{\#,\$$.
For every $a \in \Gamma$ there is a unitary transformation $U_a$
on the space ${\cal C}^{|Q|}$ where ${\cal C}$ denotes the complex reals.
The QFA starts in a distinguished
basis state $q_0 \in Q$. A {\em state} of the QFA is a superposition
of basis states, in particular the QFA starts in state $\ket{q_0}$.
The computation of the QFA on input $\#a_1 \ldots a_n \$$
applies $U_{\#} U_{a_1} \ldots U_{a_n} U_{\$}$ in sequence to
$\ket{q_0}$ until a halting state occurs. In particular,
a transformation according to $a \in \Gamma$ does the following:
\begin{enumerate}
\item
If $\ket{\phi}$ is the current state then $\phi' := U_a (\phi)$.
\item
We measure $\phi'$ according to the observables
$E_i = \mbox{ span}\{\ket{q}: q \in Q_i \}$,
$i \in \{0,1,n\}$. The probability of observing $E_i$ equals
the squared norm of the projection of $\ket{\phi '}$ on $E_i$.
On measurement the superposition of the automaton collapses to the
projection of one of the spaces observed and is renormalized.
If we observe $E_1$ or $E_0$ then the input is accepted or rejected,
otherwise the computation continues.
\end{enumerate}
A QFA {\em accepts} or {\em rejects} a language $L \subseteq \Sigma^*$
with probability $p > \frac{1}{2}$ if it accepts every word in $L$
with probability at least $p$ and rejects every word not in $L$
with probability at least $p$.
A priori it is by no means obvious that QFA cannot accept nonregular
languages like $L_1 = \{0^k1^k: k > 0 \}$.
Using incompressibility this is very simple to show and in fact
we can give a characterization of QFA languages.
\begin{lemma}
The language $L_1$ is not accepted by a QFA.
\end{lemma}
\begin{proof}
Suppose a QFA accepts $L_1$. Then for every input word $0^k1^k$
we can do the following. First run the QFA until we have processed
all of $0^k$. Then the QFA is in a superposition say $\ket{\psi}$.
If we now continue running the QFA feeding it only $1$'s
then the first accepting state it meets must be after precisely $k$
such $1$'s with probability more than $\frac{1}{2}$.
The description of the QFA together with the description of $\ket{\psi}$
is a description of $KQ$-complexity $O(1)$. But we can choose $k$
such that $KQ(k) \geq \log k$: contradiction for $k$ large enough.
\end{proof}
\end{comment}
\begin{comment}
\appendix
\section{Potentially Useful Ideas}
\begin{remark}
\rm
The following arguments may be useful later:
Use the notation of the proof of Lemma~\ref{lem.lowb}.
Let $P$ be the linear operator that projects a state $\ket{x}$
on the subspace $A$: If $\ket{x} = \ket{x_A} + \ket{x_{A^{\perp}}}$
then $\ket{Px} = \ket{x_A}$. The {\em trace} of $P$ is defined by
$\trace{P} = \sum_{\ket{e} \in K} \bracket{e}{Pe}$ where
$K$ is an orthonormal basis of the Hilbert space ${\cal H}_N$.
It is known that the trace is invariant under change of $K$.
The orthonormal basis $K$ constructed in the proof of Lemma~\ref{lem.lowb}
demonstrates
\[ \trace{P} = \sum_{i=0}^{m'-1} \bracket{a_i}{a_i}=m' \leq 2^{n-c}-1 \] since
$\ket{Pa_i}=\ket{a_i}$ if $\ket{a_i}$ is an element of the orthonormal basis
for $A$ and $ \sum_{i=0}^{2^n -m'-1} \bracket{b_i}{Pb_i}=0$ since all the
$\ket{Pb_i}$'s are 0. Then, for every basis $K'= (\ket{e_0}, \ldots , \ket{e_{2^n-1}})$
we have
\[ \sum_{\ket{e} \in K'} \bracket{e}{Pe} \leq 2^{n-c}-1 , \]
where the $\bracket{e}{Pe}= | \bracket{e}{Pe}|^2 = \cos \theta$ where $\theta$ is the
angle between $\ket{e}$ and $\ket{Pe}$.
By the concavity of the logarithm function
\[ - \sum_{\ket{e} \in K'} \log | \bracket{e}{Pe}|^2 > 2^n c . \]
So if the directly computable vector $\ket{x^e}$
is involved in the shortest program for $\ket{e}$
\[ \sum_{\ket{e} \in K'} ( C(\ket{e}) \geq
\sum_{\ket{e} \in K'} ( C( \ket{x^e}) + c) ,\]
and
\begin{equation}\label{eq.ub1}
\sum_{\ket{e} \in K'} C(\ket{x^e}) \leq 2^n (n-c).
\end{equation}
Let $\alpha_i := |A_i|$ where $A_i := \{\ket{e}: \ket{x^e} = \ket{x_i} \}$.
That is, $\alpha_i$ is the number of basis vectors
that use $\ket{x_i}$ as directly computable vector involved in the
shortest program. Without loss of generality we can assume there is only
one such directly computable vector for every basis vector.
From Equation~\ref{eq.ub1} we have
\[ \sum_{i=0}^{2^{n-c}-1} \frac{ \alpha_i}{2^n} C(\ket{x_i} ) \leq n-c .\]
Note that $p_i = \alpha_i /2^n$ is the probability that a uniformly chosen
basis vector $e$ belongs to $A_i$. It is known that the $p_i$-expectation
of the complexity equals (up to an additive constant) the $p_i$-expectation
of $\log p_i$ (the entropy) \cite{LiVi97}:
\[ \sum_{i=0}^{2^{n-c}-1} p_i C(\ket{x_i} ) =
- ( \sum_{i=0}^{2^{n-c}-1} p_i \log p_i ) +O(1) \leq n-c+O(1) .\]
Consequently, the $p_i$-expectation of $\alpha_i$ is at least $2^{c-O(1)}$.
This means that if we uniformly at random pick a basis vector then
the expectation is that it belongs to an $A_i$ of size at least $2^{c-O(1)}$.
\end{remark}
\begin{theorem}
With uniform probability of at least $1-1/2^c$ an $n$-qubit quantum
state $\ket{x}$ satisfies $C(\ket{x}) \geq n-c$.
\end{theorem}
\begin{proof}
There are $2^n$ basis vectors and there are at most
$m \leq \sum_{i=0}^{n-c-1} 2^i = 2^{n-c}-1$ programs of length less than
$n-c$. Hence there are at most $m$ programs
available to approximate the basis vectors.
The set of directly computed pure quantum states is
$\ket{x_0}, \ldots , \ket{x_{m-1}}$
spanning a $m'$-dimensional subspace ${\cal A}$ with $m' \leq m$
in the $2^n$-dimensional Hilbert space ${\cal H}_N$ such
that ${\cal H}_N = {\cal A} \oplus {\cal A}^{\perp}$.
Here ${\cal A}^{\perp}$ is a $\geq (2^n - m')$-dimensional
subspace of ${\cal H}_N$ such that every vector in it is
perpendicular to every vector in ${\cal A}$. Consider the
directly computable states as observables.
We can write every element $\ket{x} \in {\cal H}_N$ as
\[
\sum_{i=0}^{m'-1} \alpha_i \ket{a_i}+ \sum_{i=0}^{2^n-m'-1} \beta_i \ket{b_i}
\]
where the $\ket{a_i}$'s form an orthonormal basis
of ${\cal A}$ and the $\ket{b_i}$'s form an
orthonormal basis of $ {\cal A}^{\perp}$.
Note that the $\ket{a_i}$'s and $\ket{b_i}$'s form an orthonormal basis
for ${\cal H}_N$. The uniform probability
that the state $\ket{x}$ will not be observed in ${\cal A}$ is
(by symmetry):
\[
\sum_{i=0}^{2^n-m'-1} | \beta_i \overline{\beta_i}|^2
= \frac{2^n-m'}{2^n} > 1 - 2^{-c}.
\]
Hence the uniform expectation of the probability that state $\ket{x}$
is observed in ${\cal A}$ at all, and hence the uniform expectation
of the maximal probability that it is observed as any vector in
${\cal A}$ is $< 2^{-c}$.
***********
Let $p_{i} = | \bracket{x}{x_i}|^2$ be the probability
of observing basis vector $\ket{x_i}$ when we are in state $\ket{x}$.
The set of $\ket{x}$ with complexity exceeding $n-c$ is
\[
\{\ket{x}: C (\ket{x_i}) - \log | \bracket{x}{x_i}|^2 \geq n-c
\mbox{ for all $i$ } (0 \leq i \leq m-1) \}.
\]
Since $C ( \ket{x_i} ) = C(i)+O(1)$,
if $\theta_i$ is the angle between the vectors
$\ket{x}$ and $\ket{x_i}$ then $\cos^2 \theta_i \leq 1/2^{n-c-C(i)-O(1)}$
such that
\[
-2^{-\frac{n-c-C(i)}{2}}
\leq \cos \theta_i
\leq 2^{-\frac{n-c-C(i)}{2}}
\]
for all $0 \leq i < 2^{n-c}-1$.
\end{proof}
\begin{remark}
\rm
What about $C(x,y)=C(x)+C(y|x)$ up to log term?
If the answer to previous remark is positive than
this equality holds certainly for directly computed
sattes and also wrt symmetry of information.
\end{remark}
\begin{remark}
\rm
What about the expected quantum KC? Is it equal
to the entropy of the probability distribution?
\end{remark}
\end{comment}
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\section{Qubit Descriptions}
One way to avoid two-part descriptions as we used above
is to allow qubit programs as input. This leads to the
following definitions, results, and problems.
\begin{definition}
\rm
The {\em qubit complexity}
of $\ket{x}$ with respect to quantum Turing machine $Q$
with $y$ as conditional input given for free is
\[
KQ_Q (\ket{x} | y ) :=
\min_{p} \{ l(\ket{p}) :
Q(\ket{p}, y) = \ket{x} \}
\]
where $l(\ket{p})$ is the number of qubits in
the qubit specification $\ket{p}$,
$\ket{p}$ is an input quantum state,
$y$ is given conditionally, and
$\ket{x}$ is
the quantum state produced by the computation $Q(\ket{p}, y)$:
the target state that one
describes.
\end{definition}
Note that here too
there are two possible interpretations for the computation relation
$Q(\ket{p}, y) = \ket{x}$. In the narrow interpretation
we require that $Q$ with $\ket{p}$ on the input tape
and $y$ on the conditional tape halts with $\ket{x}$
on the output tape. In the wide interpretation we require
that for every precision $\delta > 0$ the computation
of $Q$ with $\ket{p}$ on the input tape
and $y$ on the conditional tape
and $\delta$ on a tape where the precision is to be supplied
halts with $\ket{x'}$
on the output tape and $|\bracket{x}{x'}|^2 \geq 1-\delta$.
Additionally one can require that the approximation finishes
in a certain time, say, polynomial in
$l(\ket{x})$ and $1/\delta$.
In the remainder of this section we can allow either interpretation
(note that the ``narrow'' complexity will always be at least as
large as the ``wide''
complexity).
Fix an enumeration of quantum Turing machines like in Theorem~\ref{theo.inv},
this time with Turing machines that use qubit programs.
Just like before it is now straightforward to derive an Invariance Theorem:
\begin{theorem}
There is a universal machine $U$ such that for all machines $Q$
there is a constant $c$ (the length of a self-delimiting
encoding of the index of $Q$ in the enumeration)
such that for all quantum states $\ket{x}$ we have
$KQ_U (\ket{x} |y) \leq KQ_Q (\ket{x}|y) + c$.
\end{theorem}
We fix once and for all a reference universal quantum Turing machine $U$
and express the {\em qubit quantum Kolmogorov complexity} as
\begin{eqnarray*}
&& KQ (\ket{x} | y) := KQ_U (\ket{x}|y), \\
&& KQ (\ket{x}) := KQ_U (\ket{x} | \epsilon ),
\end{eqnarray*}
where $\epsilon$ indicates the absence of conditional
information (the conditional tape contains the ``quantum state''
with 0 qubits). We now have immediately:
\begin{lemma}
$KQ ( \ket{x} ) \leq l(\ket{x})+O(1)$.
\end{lemma}
\begin{proof}
Give the reference universal machine $\ket{1^n 0} \otimes \ket{x}$
as input where $n$ is the index of the identity quantum Turing machine
that transports the attached pure quantum state $\ket{x}$ to
the output.
\end{proof}
It is possible to define unconditional $KQ$-complexity
in terms of conditional $K$-complexity as follows:
Even for pure quantum states that are not directly computable from
effective descriptions we have
$K( \ket{x} | \ket{x}) = O(1)$. This naturaly gives:
\begin{lemma}
The qubit quantum Kolmogorov complexity of
$\ket{x}$ satisfies
\[ KQ ( \ket{x} ) = \min_{p}
\{ l( \ket{p}): K(\ket{x} | \ket{p} ) \} + O(1),\]
where $l(\ket{p})$ denotes the number of qubits in $\ket{p}$.
\end{lemma}
\begin{proof}
Transfer the conditional $\ket{p}$ to the input using an $O(1)$-bit
program.
\end{proof}
We can generalize this definition
to obtain conditional $KQ$-complexity.
\subsection{Potential Problems of Qubit Complexity}
While it is clear that (just as with the previous aproach)
the qubit complexity is not computable, it is unknown to the author
whether one can approximate the qubit complexity from above by
a computable process in any meaningful sense.
In particular, the dovetailing approach
we used in the first approach now doesn't seem applicable due
to the non-countability of the potentential qubit program candidates.
While it is clear that the qubit complexity of a pure quantum
state is at least 1, why would it need to be more than
one qubit since the probability amplitude can be any complex number?
In case the target pure quantum state is a classical binary string,
as observed by Harry Buhrman,
Holevo's theorem \cite{Pe95} tells us that on average one cannot transmit
more than $n$ bits of classical information by $n$-qubit messages
(without using entangled qubits on the side).
This suggests that for every $n$ there exist classical binary
strings of length $n$ that have qubit complexity at least $n$.
This of course leaves open the case of the non-classical pure quantum
states---a set of measure one---and
of how to prove incompressibility
of the overwhelming majority of states. These matters have since been
investigated by A. Berthiaume, S. Laplante, and W. van Dam
(paper in preparation).
\section{Real Descriptions}
A final version of quantum Kolmogorov complexity uses
computable real parameters to describe the pure quantum state
with complex probability amplitudes.
This requires two reals per complex probability amplitude, that is,
for $n$ qubits one requires $2^{n+1}$ real numbers in the worst case.
Since every computable real number may require a separate program,
a computable $n$ qubit state may require $2^{n+1}$ finite programs.
While this approach does not allow the development of a clean
theory in the sense of the previous approaches, it can be directly
developed in terms of algorithmic thermodynamics---an extension
of Kolmogorov complexity to randomness of infinite sequences
(such as binary expansions of real numbers)
in terms of coarse-graining and sequential Martin-L\"off tests, completely
analogous to Peter G\'acs theory \cite{Ga94,LiVi97}.
\end{document}
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\begin{document}
\vspace*{0.5in}
{\Large
\centerline {\bf Quantum Wavelet Transforms: Fast Algorithms}
\centerline {\bf and Complete Circuits\footnote{Presented at 1st NASA Int. Conf. on Quantum Computing and Communication, Palm Spring, CA, Feb. 17-21, 1998.}}
}
\centerline {\bf Amir Fijany and Colin P. Williams}
\centerline {\it Jet Propulsion Laboratory, California Institute of Technology}
\centerline {4800 Oak Grove Drive, Pasadena, CA 91109}
\centerline {Email: [email protected] \, and \, [email protected]}
\centerline {\bf Abstract}
The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the
multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies $D^{(4)}$ transforms. Our approach is to factor the classical operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.
\noindent {\it Key Words: Quantum Computing, Quantum Algorithms, Quantum
circuits, Wavelet Transforms}
\section{Introduction}
The field of quantum computing has undergone an explosion of activity over the past few years. Several important quantum algorithms are now known. Moreover, prototypical quantum computers have already been built using nuclear magnetic resonance [1, 2] and nonlinear optics technologies [3]. Such devices are far from being general-purpose computers.
Nevertheless, they constitute significant milestones along the road to practical quantum computing.
A quantum computer is a physical device whose natural evolution over time can be interpreted as the execution of a useful computation. The basic element of a quantum computer is the quantum bit or "qubit", implemented physically as the state of some convenient 2-state quantum system such as the spin of an electron. Whereas a classical bit must be either a 0 or a 1
at any instant, a qubit is allowed to be an arbitrary superposition of a 0 and a 1 simultaneously. To make a quantum memory register we simply consider the simultaneous state of (possibly entangled) tuples of qubits.
The state of a quantum memory register, or any other isolated quantum system, evolves in time according to some unitary operator. Hence, if the evolved state of a quantum memory register is interpreted as having implemented some computation, that computation must be describable as a unitary operator. If the quantum memory register consists of $n$ qubits, this operator will be represented, mathematically, as some $2^n \times 2^n$ dimensional unitary matrix.
Several quantum algorithms are now known, the most famous examples being Deutsch and Jozsa's algorithm for deciding whether a function is even or balanced [4], Shor's algorithm for factoring a composite integer [5] and Grover's algorithm for finding an item in an unstructured database [6]. However, the field is growing rapidly and new quantum algorithms are
being discovered every year. Some recent examples include Brassard, Hoyer, and Tapp's quantum algorithm for counting the number of solutions to a problem [7], Cerf, Grover and Williams quantum algorithm for solving NP-complete problems by nesting one quantum search within another [8] and van Dam, Hoyer, and Tapp's algorithm for distributed quantum computing [9].
The fact that quantum algorithms are describable in terms of unitary transformations is both good news and bad for quantum computing. The good news is that knowing that a quantum computer must perform a unitary transformation allows theorems to be proved about the tasks that quantum computers can and cannot do. For example, Zalka has proved that Grover's algorithm is optimal [10]. Aharonov, Kitaev, and Nisan have proved that a quantum algorithm that involves intermediate measurements is no more powerful than one that postpones all measurements until the end of the unitary evolution stage [11]. Both these proofs rely upon quantum algorithms being unitary transformations. On the other hand, the bad news is that many computations that we would like to perform are not originally described in terms of unitary operators. For example, a desired computation might be nonlinear, irreversible or both nonlinear and irreversible. As a unitary transformation must be linear and reversible we might need to be quite creative in encoding a desired computation on a quantum computer. Irreversibility can be handled by incorporating extra "ancilla" qubits that permit us to remember the input corresponding to each output. But nonlinear transformations are still problematic.
Fortunately, there is an important class of computations, the unitary transforms, such as the Fourier transform, Walsh-Hadamard transform and assorted wavelet transforms, that are describable, naturally, in terms of unitary operators. Of these, the Fourier and Walsh-Hadamard transforms have been the ones studied most extensively by the quantum computing community. In fact, the quantum Fourier transform (QFT) is now recognized as being pivotal in many known quantum algorithms [12]. The quantum Walsh-Hadamard transform is a critical component of both Shor's algorithm [5] and Grover's algorithm [6]. However, the wavelet transforms are every bit as useful as the Fourier transform, at least in the context of classical computing. For example, wavelet transforms are particularly suited to exposing the multi-scale structure of a signal. They are likely to be useful for quantum image processing and quantum data compression. It is natural therefore to consider how to achieve a quantum wavelet transform.
Starting with the unitary operator for the wavelet transform, the next step in the process of finding a quantum circuit that implements it, is to factor the wavelet operator into the direct sum, direct product and dot product of smaller unitary operators. These operators correspond to 1-qubit and 2-qubit quantum gates. For such a circuit to be physically realizable, the number of gates within it must be bounded above by a polynomial in the number of qubits, $n$. Finding such a factorization can be extremely challenging. For example, although there are known algebraic techniques for factoring an arbitrary $2^n \times 2^n$ operator, e.g. [13], they are guaranteed to produce $O(2^n)$, i.e., exponentially many, terms in the factorization. Hence, although such a factorization is mathematically valid, it is physically unrealizable because, when treated as a quantum circuit design, would require too many quantum gates. Indeed, Knill has {\it proved} that an arbitrary unitary matrix will require exponentially many quantum gates if we restrict ourselves to using only gates that correspond to all 1-qubit rotations and XOR [14]. It is therefore clear that the key enabling factor for achieving an efficient quantum implementation, i.e., with a polynomial time and space complexity, is to exploit the specific structure of the given unitary operator.
Perhaps the most striking example of the potential for achieving compact and efficient quantum circuits is the case of the Walsh-Hadamard transform. In quantum computing, this transform arises whenever a quantum register is loaded with all integers in the range 0 to $2^n-1$. Classically, application of the Walsh-Hadamard transform on a vector of length $2^n$
involves a complexity of $O(2^n)$. Yet, by exploiting the factorization of the Walsh-Hadamard operator in terms of the Kroenecker product, it can implemented with a complexity of $O(1)$ by $n$ identical 1-qubit quantum gates. Likewise, the classical FFT algorithm has been found to be implementable in a polynomial space and time complexity, quantum circuit
[15] (see also Sec. 2.3). However, exploitation of the operator structure arising in the wavelet transforms (and perhaps other unitary transforms) is more challenging.
A key technique, in classical computing, for exposing and exploiting specific structure of a given unitary transform is the use of permutation matrices. In fact, there is an extensive literature in classical computing on the use of permutation matrices for factorizing unitary transforms into simpler forms that enable efficient implementations to be
devised (see, for example, [16] and [17]). However, the underlying assumption in using the permutation matrices in classical computation is that they can be implemented easily and inexpensively. Indeed, they are considered so trivial that the cost of their implementation is often not included in the complexity analysis. This is because any permutation matrix
can be described by its effect on the ordering of the elements of a vector. Hence, it can simply be implemented by re-ordering the elements of the vector involving only data movement and without performing any arithmetic operations. As is shown in this paper, the permutation matrices also play a pivotal role in the factorization of the unitary operators that arise in the wavelet transforms. However, unlike the classical computing, the cost of implementation of the permutation matrices cannot be neglected in quantum computing. Indeed, the main issue in deriving feasible and efficient quantum circuits for the quantum wavelet transforms considered in this paper, is the design of efficient quantum circuits for certain permutation matrices. Note that, any permutation matrix acting on $n$ qubits can mathematically be represented by a $2^n \times 2^n$ unitary operator. Hence, it is possible to factor any permutation matrix by using general techniques such as [13] but this would lead to an exponential time and space complexity. However, the permutation matrices, due to their specific structure (i.e., sparsity pattern), represents a very special subclass of unitary matrices. Therefore, the key to achieve an efficient quantum implementation of permutation matrices is the exploitation of this specific structure.
In this paper, we first develop efficient quantum circuits for a set of permutation matrices arising in the development of the quantum wavelet transforms (and the quantum Fourier transform). We propose three techniques for an efficient quantum implementation of permutation matrices, depending on the permutation matrix considered. In the first technique, we show that a certain class of permutation matrices, designated as {\it qubit permutation matrices}, can directly be described by their effect on the ordering of qubits. This quantum description is very similar to classical description of the permutation matrices. We show that the {\it Perfect Shuffle} permutation matrix, designated as $\Pi_{2^n}$, and the {\it Bit Reversal} permutation matrix, designated as $P_{2^n}$, which arise in the quantum wavelet and Fourier transforms (as well as in many other classical computations) belong to this class. We present a new gate, designated as the {\it qubit swap gate} or $\Pi_4$, which can be used to directly derive efficient quantum circuits for implementation of the qubit permutation matrices. Interestingly, such circuits for quantum implementation of $\Pi_{2^n}$ and $P_{2^n}$ lead to new factorizations of these two permutation matrices which were not previously know in classical computation. A second technique is based on a {\it quantum arithmetic description} of permutation matrices. In particular, we consider the {\it downshift} permutation matrix, designated as $Q_{2^n}$, which plays a major role in derivation of quantum wavelet transforms and also frequently arises in many classical computations [16]. We show that a quantum description of $Q_{2^n}$ can be given as a {\it quantum arithmetic operator}. This description then allows the quantum implementation of $Q_{2^n}$ by using the quantum arithmetic circuits proposed in [18].
A third technique is based on developing totally new factorizations of the permutation matrices. This technique is the most case dependent, challenging, and even counterintuitive (from a classical computing point of view). For this technique, we again consider the permutation matrix $Q_{2^n}$ and we show that it can be factored in terms of FFT which then allows its implementation by using the circuits for QFT. More interestingly, however, we derive a recursive factorization of $Q_{2^n}$ which was not previously known in classical computation. This new factorization enables a direct and efficient implementation of $Q_{2^n}$. Our analysis of though a limited set of permutation matrices reveals some of the surprises of quantum computing in contrast to classical computing. That is, certain operations that are hard to implement in classical computing are much easier to implement on quantum computing and vice versa. As a specific example, while the classical implementation of $\Pi_{2^n}$ and $P_{2^n}$ are much harder (in terms of the data movement pattern) than $Q_{2^n}$, their quantum implementation is much easier and more straightforward than $Q_{2^n}$.
Given a wavelet kernel, its application is usually performed according to the packet or pyramid algorithms. Efficient quantum implementation of theses two algorithms requires efficient circuits for operators of the form $I_{2^{n-i}} \otimes \Pi_{2^i}$ and $\Pi_{2^i} \oplus I_{2^n - 2^i}$, for some $i$, where $\otimes$ and $\oplus$ designate, respectively, the kronecker product and the direct sum operator. We show that these operators can be efficiently implemented by using our proposed circuits for implementation of $\Pi_{2^i}$. We then consider two representative wavelet kernels, the Haar [17] and Daubechies $D^{(4)}$ [19] wavelets which have previously been considered by Hoyer [20]. For the Haar wavelet, we show that Hoyer's proposed solution is incomplete since it does not lead to a gate-level circuit and, consequently, it does not allow the analysis of time and space complexity. We propose a scheme for design of a complete gate-level circuit for the Haar wavelet and analyze its time and space complexity. For the Daubechies $D^{(4)}$ wavelet, we develop three new factorizations which lead to three gate-level circuits for its implementation. Interestingly, one of this factorization allows efficient implementation of Daubechies $D^{(4)}$ wavelets by using the circuit for QFT.
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\section{Efficient Quantum Circuits for two Fundamental Qubits Permutation Matrices: Perfect Shuffle and Bit-Reversal}
In this section, we develop quantum circuits for two fundamental permutation matrices, the perfect shuffle, $\Pi_{2^n}$, and the bit reversal, $P_{2^n}$, permutation matrices, which arise in quantum wavelet and Fourier transforms as well as many classical computations involving unitary transforms for signal and image processing [16]. For quantum computing, these two permutation matrices can directly be described in terms of their effect on ordering of qubits. This enables the design of efficient circuits for their implementation. Interestingly, these circuits lead to the discovery of new factorizations for these two permutation matrices.
\subsection{Perfect Shuffle Permutation Matrices}
A classical description of $\Pi_{2^n}$ can be given by describing its effect on a given vector. If $Z$ is a $2^n$-dimensional vector, then the vector $Y = \Pi_{2^n}Z$ is obtained by splitting $Z$ in half and then shuffling the top and bottom halves of the deck. Alternatively, a description of the matrix $\Pi_{2^n}$, in terms of its elements $\Pi_{ij}$, for $i$
and $j = 0, 1, \cdots, 2^n-1$, can be given as
\begin{equation}
\Pi_{ij} = \left\{ \begin{array}{ll}
1 & \mbox{ if $j = i/2$ and $i$ is even, or if $j = (i - 1)/2 +2^{n-1}$ and
$i$ is odd} \\
0 & \mbox{ otherwise}
\end{array}
\right.
\end{equation}
As first noted by Hoyer [20], a quantum description of $\Pi_{2^n}$ can be given by
\begin{equation}
\Pi_{2^n}: \, \vert a_{n-1} \, a_{n-2} \, \cdots \, a_1 \, a_0 \rangle \,
\longmapsto \vert a_0 \, a_{n-1} \, a_{n-2} \, \cdots a_1 \rangle
\end{equation}
That is, for quantum computation, $\Pi_{2^n}$ is the operator which performs the left qubit-shift operation on $n$ qubits. Note that, $\Pi_{2^n}^t$ ($t$ indicates the transpose) performs the right qubit-shift operation, i.e.,
\begin{equation}
\Pi_{2^n}^t: \, \vert a_{n-1} \, a_{n-2} \, \cdots a_1 \, a_0 \rangle \,
\longmapsto \vert a_{n-2} \, \cdots a_1 \, a_0 \, a_{n-1} \rangle
\end{equation}
\subsection{Bit-Reversal Permutation Matrices}
A classical description of $P_{2^n}$ can be given by describing its effect on a given vector. If $Z$ is a $2^n$-dimensional vector and $Y = P_{2^n}Z$, then $Y_i = Z_j$, for $i = 0, 1, \cdots, 2^n-1$, wherein $j$ is obtained by reversing the bits in the binary representation of index $i$. Therefore, a description of the matrix $P_{2^n}$, in terms of its elements $P_{ij}$, for $i$ and $j = 0, 1, \cdots, 2^n-1$, is given as
\begin{equation}
P_{ij} = \left\{ \begin{array}{cc}
1 & \mbox{if $j$ is bit reversal of $i$} \\
0 & \mbox{otherwise}
\end{array}
\right.
\end{equation}
A factorization of $P_{2^n}$ in terms of $\Pi_{2^i}$ is given as [16]
\begin{equation}
P_{2^n} = \Pi_{2^n}(I_2 \otimes \Pi_{2^{n-1}}) \cdots (I_{2^i} \otimes \Pi_{2^{n-i}}) \, \cdots (I_{2^{n-3}} \otimes \Pi_8)(I_{2^{n-2}} \otimes \Pi_4)
\end{equation}
A quantum description of $P_{2^n}$ is given as
\begin{equation}
P_{2^n}: \, \vert a_{n-1} \, a_{n-2}, \cdots a_1 \, a_0 \rangle \,
\longmapsto \vert a_0 \, a_1 \, \cdots a_{n-2} \, a_{n-1} \rangle
\end{equation}
That is, $P_{2^n}$ is the operator which reverses the order of $n$ qubits. This quantum description can be seen from the factorization of $P_{2^n}$, given by (5), and quantum description of permutation matrices $\Pi_{2^i}$. It is interesting to note that for classical computation the term "bit-reversal" refers to reversing the bits in the binary representation of index of the elements of a vector while, for quantum computation, the matrix $P_{2^n}$ literally performs a reversal of the order of qubits.
Note that, $P_{2^n}$ is symmetric, i.e., $P_{2^n} = P_{2^n}^t$ [16]. This can be also easily proved based on the quantum description of $P_{2^n}$ since if the qubits are reversed twice then the original ordering of the qubits is restored. This implies that, $P_{2^n}P_{2^n} = I_{2^n}$ and since $P_{2^n}$ is orthogonal, i.e., $P_{2^n}P_{2^n}^t = I_{2^n}$, it then follows that $P_{2^n} = P_{2^n}^t$.
\subsection{Quantum FFT and Bit-Reversal Permutation Matrix}
Here, we review the quantum FFT algorithm since it not only arises in derivation of the quantum wavelet transforms (see Sec. 4.3) but also it represents a case in which the roles of permutation matrices $\Pi_{2^n}$ and $P_{2^n}$ seems to have been overlooked in quantum computing literature.
The classical Cooley-Tukey FFT factorization for a $2^n$-dimensional vector is given by [16]
\begin{equation}
F_{2^n} = A_n A_{n-1} \cdots A_1 P_{2^n} =
{\underline F}_{2^n} P_{2^n}
\end{equation}
where $A_i = I_{2^{n-i}} \otimes B_{2^i}$, $B_{2^i} = \frac {1} {\sqrt {2}}
\left( \begin{array}{cc}
I_{2^{i-1}} & \Omega_{2^{i-1}} \\
I_{2^{i-1}} & - \Omega_{2^{i-1}}
\end{array} \right)
$ and $\Omega_{2^{i-1}} = \mbox {Diag} \{1, \, \omega_{2^i}, \, \omega_{2^i}^2, \, \ldots , \omega_{2^i}^{2^{i-1} -1} \}$ with
$\omega_{2^i} = e^{-{2 \iota \pi} \over {2^i}}$ and $\iota = \sqrt {-1}$. We have that $F_2 = W =
\frac {1} {\sqrt {2}}
\left( \begin{array}{cc}
1 & 1 \\
1 & - 1
\end{array} \right)
$. The operator
\begin{equation}
{\underline F}_{2^n} = A_n A_{n-1} \cdots A_1
\end{equation}
represents the computational kernel of Cooley-Tukey FFT while $P_{2^n}$ represents the permutation which needs to be performed on the elements of the input vector before feeding that vector into the computational kernel. Note that, the presence of $P_{2^n}$ in (7) is due to the accumulation of its factors, i.e., the terms $(I_{2^i} \otimes \Pi_{2^{n-i}})$, as given by (5).
The Gentleman-Sande FFT factorization is obtained by exploiting the symmetry of $F_{2^n}$ and transposing the Cooley-Tukey factorization [16] leading to
\begin{equation}
F_{2^n} = P_{2^n} A_1^t \cdots A_{n-1}^t A_n^t =
P_{2^n} {\underline F}_{2^n}^t
\end{equation}
where
\begin{equation}
{\underline F}_{2^n}^t = A_1^t \cdots A_{n-1}^t A_n^t
\end{equation}
represents the computational kernel of the Gentleman-Sande FFT while $P_{2^n}$ represents the permutation which needs to be performed to obtain the elements of the output vector in the correct order.
In [15] a quantum circuit for the implementation of ${\underline F}_{2^n}$, given by (8), is presented by developing a factorization of the operators $B_{2^i}$ as
\begin{equation}
B_{2^i} =
\frac {1} {\sqrt {2}}
\left( \begin{array}{cc}
I_{2^{i-1}} & \Omega_{2^{i-1}} \\
I_{2^{i-1}} & -\Omega_{2^{i-1}}
\end{array} \right) =
\frac {1} {\sqrt {2}}
\left( \begin{array}{cc}
I_{2^{i-1}} & I_{2^{i-1}} \\
I_{2^{i-1}} & - I_{2^{i-1}}
\end{array} \right)
\left( \begin{array}{cc}
I_{2^{i-1}} & 0 \\
0 & \Omega_{2^{i-1}}
\end{array} \right)
\end{equation}
Let $C_{2^i} =
\left( \begin{array}{cc}
I_{2^{i-1}} & 0 \\
0 & \Omega_{2^{i-1}}
\end{array} \right)
$. It then follows that
\begin{equation}
B_{2^i} = (W \otimes I_{2^{i-1}}) C_{2^i}
\end{equation}
\begin{equation}
A_i = I_{2^{n-i}} \otimes B_{2^i} =
(I_{2^{n-i}} \otimes W \otimes I_{2^{i-1}})(I_{2^{n-i}} \otimes C_{2^i})
\end{equation}
In [15] a factorization of the operators $C_{2^i}$ is developed as
\begin{equation}
C_{2^i} = \theta_{n-1, n-i}\theta_{n-2, n-i} \cdots \theta_{n-i+1, n-i}
\end{equation}
where $\theta_{jk}$ is a two-bit gate acting on $j$th and $k$th qubits.
Using (13)-(14) a circuit for implementation of (8) is developed in [15] and presented in Fig. 1. However, there is an error in the corresponding figure in [15] since it implies that, with a correct ordering of the input qubits, the output qubits are obtained in a reverse order. Note that, as can be seen from (7), the operator ${\underline F}_{2^n}$ performs the FFT operation and provides the output qubits in a correct order if the input qubits are presented in a reverse order.
The quantum circuit for Gentleman-Sande FFT can be obtained from the circuit of Fig. 1 by first reversing the order of gates that build the operator block $A_i$ (and thus building operators $A_i^t$) and then reversing the order of the blocks representing operators $A_i$. By using the Gentleman-Sande circuit, with the input qubits in the correct order the output qubits are obtained in reverse order.
For an efficient and correct implementation of the quantum FFT, one needs to take into account the ordering of the input and output qubits, particularly if the FFT is used as a block box in a quantum computation. If the FFT is used as a stand-alone block or as the last stage in the computation (and hence its output is sampled directly), then it is more efficient to use the Gentleman-Sande FFT since the ordering of the output qubits does not cause any problem. If the FFT is used as the first stage of the computation, then it is more efficient to use the Cooley-Tukey factorization by preparing the input qubits in a reverse order. Note that, as in classical computation, each or a combination of the Cooley-Tukey or Gentleman-Sande FFT factorization can be chosen in a given quantum computation to avoid explicit implementation of $P_{2^n}$ (or, any other mechanism) for reversing the order of qubits and hence achieve a greater efficiency. As an example, in Sec. 4.3 we will show that the use of the Cooley-Tukey rather than the Gentleman-Sande factorization leads to a greater efficiency in quantum implementation by eliminating the need for an explicit implementation of $P_{2^n}$ (or, any other mechanism) for reversing the order of qubits.
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\subsection{A Basic Quantum Gate for Efficient Implementation of Qubits Permutation Matrices}
If a permutation matrix can be described by its effect on the ordering of the qubits then it might be possible to devise circuits for its implementation directly. We call the class of such permutation matrices as "Qubit Permutation Matrices". A set of efficient and practically realizable circuits for implementation of Qubit Permutation Matrices can be built by using a new quantum gate, called
{\it the qubit swap gate}, $\Pi_4$, where \begin{equation}
\Pi_4 =
\left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array} \right)
\end{equation}
For quantum computation, $\Pi_4$ is the "qubit swap operator", i.e.,
\begin{equation}
\Pi_4: \, \vert a_1 \, a_0 \rangle \, \longmapsto \, \vert a_0 \, a_1 \rangle
\end{equation}
The $\Pi_4$ gate, shown in Fig. 2.a, can be implemented with three XOR (or Controlled-NOT) gates as shown in Fig. 2.b. The $\Pi_4$ gate offers two major advantages for practical implementation:
\begin{itemize}
\item It performs a local operation, i.e., swapping the two neighboring qubits. This locality can be advantageous in
practical realizations of quantum circuits, and
\item Given the fact that $\Pi_4$ can be implemented using three XOR (or, Controlled-NOT) gates, it is possible to implement conditional operators involving $\Pi_4$, for example, operators of the form $\Pi_4 \oplus I_{2^n - 4}$, by using Controlled$^k$-NOT gates [21].
\end{itemize}
A circuit for implementation of $\Pi_{2^n}$ by using $\Pi_4$ gates is shown in Fig. 3. This circuit is based on an intuitively simple idea of successive swapping of the neighboring qubits, and implements $\Pi_{2^n}$ with a complexity of $O(n)$ by using an $O(n)$ number of $\Pi_4$ gates. It is interesting to note that, this circuit leads to a new (to our knowledge) factorization of $\Pi_{2^n}$ in terms of $\Pi_4$ as
\begin{equation}
\Pi_{2^n} = (I_{2^{n-2}} \otimes \Pi_4)(I_{2^{n-3}} \otimes \Pi_4 \otimes I_2)
\cdots (I_{2^{n-i}} \otimes \Pi_4 \otimes I_{2^{i-2}}) \, \cdots \,
(I_2 \otimes \Pi_4 \otimes I_{2^{n-3}})(\Pi_4 \otimes I_{2^{n-2}})
\end{equation}
This new factorization of $\Pi_{2^n}$ is less efficient than other schemes (see, for example, [16]) for a {\it classical implementation} of $\Pi_{2^n}$. Interestingly, it is derived here as a result of our search for an efficient {\it quantum implementation} of $\Pi_{2^n}$, and in this sense it is only efficient for a quantum implementation. Note also, that a new (to our knowledge) recursive factorization of $\Pi_{2^i}$ directly results from Fig. (3) as
\begin{equation}
\Pi_{2^i} = (I_{2^{i-2}} \otimes \Pi_4)(\Pi_{2^{i-1}} \otimes I_2)
\end{equation}
A circuit for implementation of $P_{2^n}$ by using $\Pi_4$ gates is shown in Fig. 4. Again, this circuit is based on an intuitively simple idea, that is, successive and parallel swapping of the neighboring qubits, and implements $P_{2^n}$ with a complexity of $O(n)$ by using $O(n^2)$ $\Pi_4$ gates. This circuit leads to a new (to our knowledge) factorization of $P_{2^n}$ in terms of $\Pi_4$ as
\begin{equation}
P_{2^n} = ((\underbrace{\Pi_4 \otimes \Pi_4 \cdots \otimes \Pi_4}_{\frac{n}{2}})
(I_2 \otimes \underbrace{\Pi_4 \otimes \cdots \otimes \Pi_4}_{\frac{n}{2} -1} \otimes I_2))^{\frac{n}{2}}
\end{equation}
for $n$ even, and
\begin{equation}
P_{2^n} = ((I_2 \otimes \underbrace{\Pi_4 \otimes \cdots \otimes \Pi_4}_{\frac{n-1}{2}})
(\underbrace{\Pi_4 \otimes \cdots \, \Pi_4}_{\frac{n-1}{2}} \otimes I_2))^{\frac{n-1}{2}}
(I_2 \otimes \underbrace{\Pi_4 \otimes \cdots \otimes \Pi_4}_{\frac{n-1}{2}})
\end{equation}
for $n$ odd.
It should be emphasized that this new factorization of $P_{2^n}$ is less efficient than other schemes, e.g., the use of (5) for a {\it classical implementation} (see also [16] for further discussion). However, this factorization is more efficient for a {\it quantum implementation} of $P_{2^n}$. In fact, a quantum implementation of $P_{2^n}$ by using (5) and (17) will result in a complexity of $O(n^2)$ by using $O(n^2)$ $\Pi_4$ gates.
As will be shown, the development of {\it complete} and efficient circuits for implementation of wavelet transforms requires a mechanism for implementation of conditional operators of the forms $\Pi_{2^i} \oplus I_{2^n - 2^i}$ and $P_{2^i} \oplus I_{2^n - 2^i}$, for some $i$. The key enabling factor for a successful implementation of such conditional operators is the use of factorizations similar to (17) and (19)-(20) or, alternatively, circuits similar to those in Figures 3 and 4, along with the conditional operators involving $\Pi_4$ gates.
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\section{Quantum Wavelet Algorithms}
\subsection{Wavelet Pyramidal and Packet Algorithms}
Given a wavelet kernel, its corresponding wavelet transform is usually performed according to a packet algorithm (PAA) or a pyramid algorithm (PYA). The first step in devising quantum counterparts of these algorithms is the development of suitable factorizations. Consider the Daubechies fourth-order wavelet kernel of dimension $2^i$, denoted as $D^{(4)}_{2^i}$. First level factorizations of PAA and PYA for a $2^n$-dimensional vector are given as
\begin{equation}
PAA = (I_{2^{n-2}} \otimes D^{(4)}_4)(I_{2^{n-3}} \otimes \Pi_8) \cdots
(I_{2^{n-i}} \otimes D^{(4)}_{2^i})(I_{2^{n-i-1}} \otimes \Pi_{2^{i+1}}) \cdots (I_2 \otimes D^{(4)}_{2^{n-1}}) \Pi_{2^n}D^{(4)}_{2^n}
\end{equation}
\begin{equation}
PYA = (D^{(4)}_4 \oplus I_{2^n-4})(\Pi_8 \oplus I_{2^n-8}) \cdots
(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})(\Pi_{2^{i+1}} \oplus I_{2^n - 2^{i+1}}) \cdots
\Pi_{2^n}D^{(4)}_{2^n}
\end{equation}
These factorizations allow a first level analysis of the feasibility and efficiency of quantum implementations of the packet and pyramid algorithms. To see this, suppose we have a practically realizable and efficient, i.e., $O(i)$, quantum algorithm for implementation of $D^{(4)}_{2^i}$. For the packet algorithm, the operators $(I_{2^{n-i}} \otimes D^{(4)}_{2^i})$ can be directly and efficiently implemented by using the algorithm for $D^{(4)}_{2^i}$. Also, using the factorization of $\Pi_{2^i}$, given by (17), the operators $(I_{2^{n-i}} \otimes \Pi_{2^i})$ can be implemented efficiently in $O(i)$.
For the pyramid algorithm, the existence of an algorithm for $D^{(4)}_{2^i}$ does not automatically imply an efficient algorithm for implementation of the conditional operators $(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$. An example of such a case is discussed in Sec. 4.4. Thus, careful analysis is needed to establish both the feasibility and efficiency of implementation of the conditional operators $(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$ by using the algorithm for $D^{(4)}_{2^i}$. Note, however, that the conditional operators $(\Pi_{2^i} \oplus I_{2^n - 2^i})$ can be efficiently implemented in $O(i)$ by using the factorization in (17) and the conditional $\Pi_4$ gates.
The above analysis can be extended to any wavelet kernel (WK) and summarized as follows:
\begin{itemize}
\item Packet algorithm: A physically realizable and efficient algorithm for the WK along with the use of (17) leads to a physically realizable and efficient implementation of the packet algorithm.
\item Pyramid algorithm: A physically realizable and efficient algorithm for the WK does not automatically lead to an
implementation of the conditional operators involving WK (and hence the pyramid algorithm) but the conditional operators $(\Pi_{2^i} \oplus I_{2^n - 2^i})$ can be efficiently implemented by using the factorization in (17) and the conditional $\Pi_4$ gates.
\end{itemize}
\subsection{Haar Wavelet Factorization and Implementation}
The Haar transform can be defined from the Haar functions [17]. Hoyer [20] used a recursive definition of Haar matrices based on the {\it generalized Kronecker product} (see also [17] for similar definitions) and developed a factorization of $H_{2^n}$ as
\begin{eqnarray}
H_{2^n} = & (I_{2^{n-1}} \otimes W) \cdots (I_{2^{n-i}} \otimes W \oplus
I_{2^n - 2^{n-i+1}}) \, \cdots \, (W \oplus I_{2^n - 2}) \times \nonumber \\
& (\Pi_4 \oplus I_{2^n - 4}) \, \cdots \, (\Pi_{2^i} \oplus I_{2^n - 2^i}) \, \cdots \, (\Pi_{2^{n-1}} \oplus I_{2^{n-1}}) \Pi_{2^n}
\end{eqnarray}
Hoyer's circuit for implementation of (23) is shown in Fig 5. However, this represents an {\it incomplete} solution for
quantum implementation and subsequent complexity analysis of the Haar transform. To see this, let
\begin{equation}
H^{(1)}_{2^n} = (I_{2^{n-1}} \otimes W) \cdots (I_{2^{n-i}} \otimes W \oplus
I_{2^n - 2^{n-i+1}}) \, \cdots \, (W \oplus I_{2^n - 2})
\end{equation}
\begin{equation}
H^{(2)}_{2^n} = (\Pi_4 \oplus I_{2^n - 4}) \, \cdots \, (\Pi_{2^i} \oplus I_{2^n - 2^i}) \, \cdots \, (\Pi_{2^{n-1}} \oplus I_{2^{n-1}}) \Pi_{2^n}
\end{equation}
Clearly, the operator $H^{(1)}_{2^n}$ can be implemented in $O(n)$ by using $O(n)$ conditional $W$ gates. But the feasibility of practical implementation of the operator $H^{(2)}_{2^n}$ and its complexity (and consequently those of the factorization in (23)) cannot be assessed unless a mechanism for implementation of the terms $(\Pi_{2^i} \oplus I_{2^n - 2^i})$ is devised.
However, by using the factorizations and circuits similar to (17) and Figure 3, it can be easily shown that the operators $(\Pi_{2^i} \oplus I_{2^n - 2^i})$ can be implemented in $O(i)$ by using $O(i)$ conditional $\Pi_4$ gates (or, Controlled$^k$-NOT gates). This leads to the implementation of $H^{(2)}_{2^n}$ and consequently $H_{2^n}$ in $O(n^2)$ by using $O(n^2)$ gates. This represents not only the first practically feasible quantum circuit for implementation of $H_{2^n}$ but also the first complete analysis of complexity of its time and space (gates) quantum implementation. Note that, both operators $(I_{2^{n-i}} \otimes H_{2^i})$ and $(H_{2^i} \oplus I_{2^n - 2^i})$ can be directly and efficiently implemented by using the above algorithm and circuit for implementation of $H_{2^i}$. This implies both the feasibility and efficiency of the quantum implementation of the packet and pyramid algorithms by using our factorization for Haar wavelet kernel.
\subsection{Daubechies $D^{(4)}$ Wavelet and Hoyer's Factorization}
The Daubechies fourth-order wavelet kernel of dimension $2^n$ is given in a matrix form as [22]
\begin{equation}
D^{(4)}_{2^n} =
\left( \begin{array}{ccccccccccc}
c_0 & c_1 & c_2 & c_3 \\
c_3 & -c_2 & c_1 & -c_0 \\
& & c_0 & c_1 & c_2 & c_3 \\
& & c_3 & -c_2 & c_1 & -c_0 \\
\vdots & \vdots & & & & & \ddots \\
& & & & & & & c_0 & c_1 & c_2 & c_3 \\
& & & & & & & c_3 & -c_2 & c_1 & -c_0 \\
c_2 & c_3 & & & & & & & & c_0 & c_1 \\
c_1 & -c_0 & & & & & & & & c_3 & -c_2
\end{array} \right)
\end{equation}
where $c_0 = \frac {(1 + \sqrt {3})} {4 \sqrt {2}}$, $c_1 = \frac {(3 + \sqrt {3})} {4 \sqrt {2}}$, $c_2 = \frac {(3 - \sqrt {3})} {4 \sqrt {2}}$, and $c_3 = \frac {(1 - \sqrt {3})} {4 \sqrt {2}}$. For classical computation and given its sparse structure, the application of $D^{(4)}_{2^n}$ can be performed with an optimal cost of $O(2^n)$. However, the matrix $D^{(4)}_{2^n}$, as given by (26), is not suitable for a quantum implementation. To achieve a feasible and efficient quantum implementation, a suitable factorization of $D^{(4)}_{2^n}$ needs to be developed. Hoyer [20] proposed a factorization of $D^{(4)}_{2^n}$ as
\begin{equation}
D^{(4)}_{2^n} = (I_{2^{n-1}} \otimes C_1) S_{2^n}(I_{2^{n-1}} \otimes C_0)
\end{equation}
where
\begin{equation}
C_0 =
2 \left( \begin{array}{cc}
c_4 & -c_2 \\
-c_2 & c_4
\end{array} \right)
\mbox{ and }
C_1 = \frac{1}{2}
\left( \begin{array}{cc}
\frac{c_0}{c_4} & 1 \\
1 & \frac{c_1}{c_2}
\end{array} \right)
\end{equation}
and $S_{2^n}$ is a permutation matrix with a classical description given by
\begin{equation}
S_{ij} = \left\{ \begin{array}{cc}
1 & \mbox{ if $i = j$ and $i$ is even, or if $i+2 = j$ \, (mod $2^n$)} \\
0 & \mbox{ otherwise}
\end{array} \right.
\end{equation}
Hoyer's block-level circuit for implementation of (27) is shown in Figure 6. Clearly, the main issue for a practical quantum implementation and subsequent complexity analysis of (27) is the quantum implementation of matrix $S_{2^n}$. To this end,
Hoyer discovered a quantum arithmetic description of $S_{2^n}$ as
\begin{equation}
S_{2^n}: \, \vert a_{n-1} \, a_{n-2} \, \cdots a_1 \, a_0 \rangle \,
\longmapsto \vert b_{n-1} \, b_{n-2} \, \cdots b_1 \, b_0 \rangle
\end{equation}
where
\begin{equation}
b_i = \left\{ \begin{array}{cc}
a_i - 2 \mbox{ \, (mod $n$)}, & \mbox{if $i$ is odd} \\
a_i & \mbox{otherwise}
\end{array} \right.
\end{equation}
As suggested by Hoyer, this description of $S_{2^n}$ then allows its quantum implementation by using quantum arithmetic circuits of [18] with a complexity of $O(n)$. This algorithm can be directly extended for implementation of the operators
$(I_{2^{n-i}} \otimes D^{(4)}_{2^i})$ and hence the packet algorithm. However, the feasibility and efficiency of an implementation of the operators $(I_{2^{n-i}} \oplus D^{(4)}_{2^i})$ and thus the pyramid algorithm needs further analysis.
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\section{Fast Quantum Algorithms and Circuits for Implementation of Daubechies $D^{(4)}$ Wavelet}
In this section, we develop a new factorization of the Daubechies $D^{(4)}$ wavelet. This factorization leads to three new and efficient circuits, including one using the circuit for QFT, for implementation of Daubechies $D^{(4)}$ wavelet.
\subsection{A New Factorization of Daubechies $D^{(4)}$ Wavelet}
We develop a new factorization of the Daubechies $D^{(4)}$ wavelet transform by showing that the permutation matrix $S_{2^n}$ can be written as a product of two permutation matrices as
\begin{equation}
S_{2^n} = Q_{2^n}R_{2^n}
\end{equation}
where $Q_{2^n}$ is the {\it downshift permutation matrix} [16] given by
\begin{equation}
Q_{2^n} = \left( \begin{array}{ccccccc}
0 & 1 \\
0 & 0 & 1 \\
0 & 0 & 0 & 1 \\
\vdots & \vdots & \vdots & & \ddots \\
0 & 0 & \cdots & 0 & 0 & 1 \cr
1 & 0 & \cdots & 0 & 0 & 0
\end{array} \right)
\end{equation}
and $R_{2^n}$ is a permutation matrix given by
\begin{equation}
R_{2^n} = \left( \begin{array}{cccccccc}
0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
& \ddots & \ddots & \ddots & \ddots \\
& & & & & & 0 & 1 \cr
& & & & & & 1 & 0
\end{array} \right)
\end{equation}
The matrix $R_{2^n}$ can be written as
\begin{equation}
R_{2^n} = I_{2^{n-1}} \otimes N
\end{equation}
where $
N =
\left( \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \right)
$.
Substituting (35) and (32) into (27), a new factorization of $D^{(4)}_{2^n}$ is derived as
\begin{equation}
D^{(4)}_{2^n} = (I_{2^{n-1}} \otimes C_1) Q_{2^n}(I_{2^{n-1}} \otimes N)
(I_{2^{n-1}} \otimes C_0) = (I_{2^{n-1}} \otimes C_1) Q_{2^n}
(I_{2^{n-1}} \otimes C_0^\prime)
\end{equation}
where
\begin{equation}
C_0^\prime = N.C_0 = 2
\left( \begin{array}{cc}
-c_2 & c_4 \\
c_4 & -c_2
\end{array} \right)
\end{equation}
Fig. 7 shows a block-level implementation of (36). Clearly, the main issue for a practical quantum gate-level implementation and subsequent complexity analysis of (36) is the quantum implementation of matrix $Q_{2^n}$. In the following, we present three circuits for quantum implementation of matrix $Q_{2^n}$.
\subsection{Quantum Arithmetic Implementation of Permutation Matrix $Q_{2^n}$}
A first circuit for implementation of matrix $Q_{2^n}$ is developed based on its description as a {\it quantum arithmetic operator}. We have discovered such a quantum arithmetic description of $Q_{2^n}$ as
\begin{equation}
Q_{2^n}: \, \vert a_{n-1} \, a_{n-2} \, \cdots a_1 \, a_0 \rangle \,
\longmapsto \vert b_{n-1} \, b_{n-2} \, \cdots b_1 \, b_0 \rangle
\end{equation}
where
\begin{equation}
b_i = a_i - 1 \mbox { \, (mod $n$)}
\end{equation}
This description of $Q_{2^n}$ allows its quantum implementation by using quantum arithmetic circuit of [18] with a complexity of $O(n)$. Note, however, that the arithmetic description of $Q_{2^n}$ is simpler than that of $S_{2^n}$ since it does not involve conditional quantum arithmetic operations (i.e., the same operation is applied to all qubits). This algorithm for quantum implementation of $Q_{2^n}$ and hence $D^{(4)}_{2^n}$ can be directly extended for implementation of the operators $(I_{2^{n-i}} \otimes D^{(4)}_{2^i})$ and hence the packet algorithm. However, the feasibility and efficiency of an implementation of the operators $(I_{2^{n-i}} \oplus D^{(4)}_{2^i})$ and thus the pyramid algorithm needs further analysis.
\subsection{Quantum FFT Factorization of Permutation Matrix $Q_{2^n}$}
A direct and efficient factorization and subsequent circuit for implementation of $Q_{2^n}$ (and hence Daubechies $D^{(4)}$ wavelet) can be derived by using the FFT algorithm. This factorization is based on the observation that $Q_{2^n}$ can be described in terms of FFT as [16]
\begin{equation}
Q_{2^n} = F_{2^n} T_{2^n} F^*_{2^n}
\end{equation}
where $T_{2^n}$ is a diagonal matrix given as
$T_{2^n} = \mbox {Diag} \{1, \, \omega_{2^n}, \, \omega_{2^n}^2, \, \ldots ,
\omega_{2^n}^{2^n -1} \}$ with
$\omega_{2^n} = e^{{-2 \iota \pi} \over {2^n}}$ (* indicates conjugate transpose). As will be seen, it is more efficient to use the Cooley-Tukey factorization, given by (7), and write (40) as
\begin{equation}
Q_{2^n} = {\underline F}_{2^n} P_{2^n} T_{2^n}P_{2^n}{\underline F}^*_{2^n}
\end{equation}
It can be shown that the matrix $T_{2^n}$ has a factorization as
\begin{equation}
T_{2^n} = (G(\omega_{2^n}^{2^{n-1}}) \otimes I_{2^{n-1}}) \cdots
(I_{2^{i-1}} \otimes G(\omega_{2^n}^{2^{n-i}}) \otimes I_{2^{n-i}}) \cdots
(I_{2^{n-1}} \otimes G(\omega_{2^n}))
\end{equation}
where $G(\omega_{2^n}^k) = \mbox {Diag} \{1, \, \omega_{2^n}^k \} =
\left( \begin{array}{cc}
1 & 0 \\
0 & \omega_{2^n}^k
\end{array} \right)
$. This factorization leads to an efficient implementation of $T_{2^n}$ by using $n$ single qubit $G(\omega_{2^n}^k)$ gates as shown in Fig. 8. Together with the circuit for implementation of $P_{2^n}$ (Fig. 4) and the circuit for implementation of FFT (Fig. 1), they represent a complete gate-level implementation of $D^{(4)}_{2^n}$.
However, a more efficient circuit can be derived by avoiding the explicit implementation of $P_{2^n}$ by showing that the operator
\begin{equation}
P_{2^n}T_{2^n}P_{2^n} = P_{2^n}(G(\omega_{2^n}^{2^{n-1}}) \otimes I_{2^{n-1}})\cdots
(I_{2^{i-1}} \otimes G(\omega_{2^n}^{2^{n-i}}) \otimes I_{2^{n-i}}) \cdots
(I_{2^{n-1}} \otimes G(\omega_{2^n}))P_{2^n}
\end{equation}
can be efficiently implemented by simply reversing the order of gates in Fig. 8. This is established by the following lemma:
\noindent {\bf Lemma 1.}
\begin{equation}
P_{2^n}(G(\omega_{2^n}^{2^{n-1}}) \otimes I_{2^{n-1}}) =
(I_{2^{n-1}} \otimes G(\omega_{2^n}^{2^{n-1}}))P_{2^n}
\end{equation}
\begin{equation}
P_{2^n} (I_{2^{n-j}} \otimes G(\omega_{2^n}^{2^{j-1}}) \otimes I_{2^{j-1}}) =
(I_{2^{j-1}} \otimes G(\omega_{2^n}^{2^{j-1}}) \otimes I_{2^{n-j}})P_{2^n}
\end{equation}
\begin{equation}
P_{2^n}(I_{2^{n-1}} \otimes G(\omega_{2^n})) = (G(\omega_{2^n}) \otimes
I_{2^{n-1}})P_{2^n}
\end{equation}
\noindent {\bf\it Proof.} This lemma can be easily proved based on the physical interpretation of operations in (44)-(46). The left-hand side of (44) implies first an operation, i.e., application of $G(\omega_{2^n}^{2^{n-1}})$, on the last qubit and then application of $P_{2^n}$ on all the qubits, i.e., reversing the order of qubits. However, this is equivalent to first reversing the order of qubits, i.e., applying $P_{2^n}$, and then applying $G(\omega_{2^n}^{2^{n-1}})$, on the first qubit which is the operation described by the right-hand side of (44). Similarly, the left-hand side of (45) implies first application of $G(\omega_{2^n}^{2^{i-1}})$ on the $(n-i)$th qubit and then reversing the order of qubits. This is equivalent to first reversing the order of qubits and then applying $G(\omega_{2^n}^{2^{i-1}})$ on the $i$th qubit which is the operations described by the right hand side of (45). In a same fashion, the left hand side of (46) implies first application of $G(\omega_{2^n})$ on the first qubit and then reversing the order of qubits which is equivalent to first reversing the order of qubits and then applying $G(\omega_{2^n}^{2^{n-1}})$ on the last qubit, that is, the operations in right-hand side of (46).
Applying (44)-(46) to (43) from left to right and noting that, due to the symmetry of $P_{2^n}$, we have
$P_{2^n}P_{2^n} = I_{2^n}$, it then follows that
\begin{equation}
P_{2^n}T_{2^n}P_{2^n} = (I_{2^{n-1}} \otimes G(\omega_{2^n}^{2^{n-1}})) \cdots
(I_{2^{n-i}} \otimes G(\omega_{2^n}^{2^{n-i}}) \otimes I_{2^{i-1}}) \cdots
(G(\omega_{2^n}) \otimes I_{2^{n-1}})
\end{equation}
The circuit for implementation of (47) is shown in Fig.9 which, as can be seen, has been obtained by reversing the order of gates in Fig. 8. Note that, the use of (47), which is a direct consequence of using the Cooley-Tukey factorization, enables the implementation of (40) without explicit implementation of $ P_{2^n}$.
Using (40) and (47), the complexity of the implementation of $Q_{2^n}$ and thus $D^{(4)}_{2^n}$ is the same as of the quantum FFT, that is, $O(n^2)$ for an exact implementation and $O(nm)$ for an approximation of order $m$ [15]. Note
that, by using (47), (40), and (36) both operators $(I_{2^{n-i}} \otimes D^{(4)}_{2^i})$ and
$(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$ can be directly implemented. This implies both the feasibility and efficiency
of the quantum implementation of the packet and pyramid algorithms by using this algorithm for quantum implementation of $D^{(4)}_{2^n}$.
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\subsection{A Direct Recursive Factorization of Permutation Matrix $Q_{2^n}$}
A new direct and recursive factorization of $Q_{2^n}$ can be derived based on a similarity transformation of $Q_{2^n}$ by using $\Pi_{2^n}$ as
\begin{equation}
\Pi^t_{2^n}Q_{2^n}\Pi_{2^n} =
\left( \begin{array}{cc}
0 & I_{2^{n-1}} \\
Q_{2^{n-1}} & 0
\end{array} \right)
\end{equation}
which can be written as
\begin{equation}
\Pi^t_{2^n}Q_{2^n}\Pi_{2^n} =
\left( \begin{array}{cc}
0 & I_{2^{n-1}} \\
I_{2^{n-1}} & 0
\end{array} \right)
\left( \begin{array}{cc}
Q_{2^{n-1}} & 0 \\
0 & I_{2^{n-1}}
\end{array} \right) =
(N \otimes I_{2^{n-1}})(Q_{2^{n-1}} \oplus I_{2^{n-1}})
\end{equation}
from which $Q_{2^n}$ can be calculated as
\begin{equation}
Q_{2^n} = \Pi_{2^n}(N \otimes I_{2^{n-1}})(Q_{2^{n-1}} \oplus
I_{2^{n-1}})\Pi^t_{2^n}
\end{equation}
Replacing a similar factorization of $Q_{2^{n-1}}$ into (50), we get
\begin{equation}
Q_{2^n} = \Pi_{2^n}(N \otimes I_{2^{n-1}})
(\Pi_{2^{n-1}}(N \otimes I_{2^{n-2}})(Q_{2^{n-2}} \oplus I_{2^{n-2}}) \Pi^t_{2^{n-1}} \oplus I_{2^{n-1}})\Pi^t_{2^n}
\end{equation}
By using the identity
\begin{equation}
\Pi_{2^{n-1}} A \Pi^t_{2^{n-1}} \oplus I_{2^{n-1}} =
(I_2 \otimes \Pi_{2^{n-1}})(A \oplus I_{2^{n-1}})(I_2 \otimes \Pi^t_{2^{n-1}})
\end{equation}
for any matrix $A \varepsilon \Re^{2^{n-1} \times 2^{n-1}}$, (51) can be then written as
\begin{equation}
Q_{2^n} = \Pi_{2^n}(N \otimes I_{2^{n-1}})(I_2 \otimes \Pi_{2^{n-1}})
((N \otimes I_{2^{n-2}})(Q_{2^{n-2}} \oplus I_{2^{n-2}})\oplus I_{2^{n-1}})
(I_2 \otimes \Pi^t_{2^{n-1}})\Pi^t_{2^n}
\end{equation}
Using the identity
\begin{eqnarray}
(N \otimes I_{2^{n-2}})(Q_{2^{n-2}} \oplus I_{2^{n-2}})\oplus I_{2^{n-1}} & = &
(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}})(Q_{2^{n-2}} \oplus
I_{2^{n-2}} \oplus I_{2^{n-1}}) \nonumber \\
& = & (N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}})(Q_{2^{n-2}} \oplus I_{3.2^{n-2}})
\end{eqnarray}
(53) is now written as
\begin{equation}
Q_{2^n} = \Pi_{2^n}(N \otimes I_{2^{n-1}})(I_2 \otimes \Pi_{2^{n-1}})
(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}})(Q_{2^{n-2}} \oplus I_{2^n - 2^{n-2}})
(I_2 \otimes \Pi^t_{2^{n-1}})\Pi^t_{2^n}
\end{equation}
Repeating the same procedures for all $Q_{2^i}$, for $i = n-3$ to 1, and noting that $Q_2 = N$, it then follows
\begin{eqnarray}
Q_{2^n} & = \Pi_{2^n}(N \otimes I_{2^{n-1}})(I_2 \otimes \Pi_{2^{n-1}})
(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}})(I_4 \otimes \Pi_{2^{n-2}})
(N \otimes I_{2^{n-3}} \oplus I_{2^n - 2^{n-2}}) \cdots \nonumber \\
& (I_{2^{n-2}} \otimes \Pi_4)(N \otimes I_2 \oplus I_{2^n -4})
(N \oplus I_{2^n -2})(I_{2^{n-2}} \otimes \Pi^t_4) \cdots
(I_2 \otimes \Pi^t_{2^{n-1}})\Pi^t_{2^n}
\end{eqnarray}
The above expression of $Q_{2^n}$ can be further simplified by exploiting the fact that (see Appendix for the proof) every operator of the form $(I_{2^i} \otimes \Pi_{2^{n-i}})$, for $i = n-2$ to $1$, commutes with all operators of the form $(N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}})$, for $j = i$ to $1$. Using this commutative property, (56) can be now written as
\begin{eqnarray}
Q_{2^n} & = \Pi_{2^n}(I_2 \otimes \Pi_{2^{n-1}})(I_4 \otimes
\Pi_{2^{n-2}}) \cdots (I_{2^{n-2}} \otimes \Pi_{4}) (N \otimes
I_{2^{n-1}}) (N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}}) \cdots \nonumber \\
& \qquad (N \otimes I_2 \oplus I_{2^n -4}) (N \oplus I_{2^n
-2})(I_{2^{n-2}} \otimes \Pi^t_4) \cdots (I_2 \otimes
\Pi^t_{2^{n-1}})\Pi^t_{2^n}
\end{eqnarray}
Using the factorization of $P_{2^n}$ given in (5), we then have
\begin{equation}
Q_{2^n} = P_{2^n}(N \otimes I_{2^{n-1}})(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}}) \cdots (N \otimes I_2 \oplus I_{2^n -4})(N \oplus I_{2^n -2})P_{2^n}
\end{equation}
Substituting (58) into (36), a factorization of $D^{(4)}_{2^n}$ is then obtained as
\begin{equation}
D^{(4)}_{2^n} = (I_{2^{n-1}} \otimes C_1) P_{2^n}(N \otimes I_{2^{n-1}})(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}}) \cdots (N \otimes I_2 \oplus I_{2^n -4})(N \oplus I_{2^n -2})P_{2^n} (I_{2^{n-1}} \otimes C_0^\prime)
\end{equation}
Using Lemma 1, it then follows that
\begin{equation}
D^{(4)}_{2^n} = P_{2^n}(C_1 \otimes I_{2^{n-1}})(N \otimes I_{2^{n-1}})(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}}) \cdots (N \otimes I_2 \oplus I_{2^n -4})(N \oplus I_{2^n -2})(C_0^\prime \otimes I_{2^{n-1}})P_{2^n}
\end{equation}
A circuit for implementation of $D^{(4)}_{2^n}$, based on (60), is shown in Fig. 10. Together with the circuit for implementation of $P_{2^n}$, shown in Fig. 4, they represent a complete gate-level circuit for implementation of $D^{(4)}_{2^n}$ with an optimal complexity of $O(n)$.
Using (60) and (19)-(20), the operators $(I_{2^{n-i}} \otimes D^{(4)}_{2^i})$ can be directly and efficiently implemented with a complexity of $O(i)$. This implies both the feasibility and efficiency of the implementation of the packet algorithm by using this algorithm for $D^{(4)}_{2^n}$ wavelet kernel. However, this algorithm is less efficient for implementation of the operators $(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$ and hence the pyramid algorithm. To see this, note that, the implementation of the operators $(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$, by using (60), requires the implementation of the conditional operators $(P_{2^i} \oplus I_{2^n - 2^i})$. However, these conditional operators cannot be directly implemented by using (19) and (20). An alternative solution is to use the factorization of $P_{2^i}$ in (5) and the conditional operators $(\Pi_{2^i} \oplus I_{2^n - 2^i})$. However, this leads to a complexity of $O(i^2)$ for implementation of operators $(P_{2^i} \oplus I_{2^n - 2^i})$ and hence the operators $(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$. Therefore, while (60) is optimal for implementation of $D^{(4)}_{2^i}$ and the packet algorithm, it is not efficient for implementation of the pyramid algorithm.
It should be emphasized that this recursive factorization of $Q_{2^n}$, originated by the similarity transformation in (48) and given by (56) and (58), was not previously known in classical computing. Note that, the permutation matrices $\Pi_{2^n}$ and, particularly, $P_{2^n}$ are much harder (in terms of data movement pattern) for a classical implementation than $Q_{2^n}$. In this sense, such a factorization of $Q_{2^n}$ is rather counterintuitive from a classical computing point of view since it involves the use of permutation matrices $\Pi_{2^n}$ and $P_{2^n}$ and thus it is highly inefficient for a classical implementation.
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\section{Discussion and Conclusion}
In this paper, we developed fast algorithms and efficient circuits for quantum wavelet transforms. Assuming an efficient quantum circuit for a given wavelet kernel and starting with a high level description of the packet and pyramid algorithms, we analyzed the feasibility and efficiency of the implementation of the packet and pyramid algorithms by using the given wavelet kernel. We also developed efficient and complete gate-level circuits for two representative wavelet kernels, the Haar and Daubechies $D^{(4)}$ kernels. We gave the first complete time and space complexity analysis of the quantum Haar wavelet transform. We also described three complete circuits for Daubechies $D^{(4)}$ wavelet kernel. In particular, we showed that Daubechies $D^{(4)}$ kernel can be implemented by using the circuit for QFT. Given the problem of decoherence, exploitation of parallelism in quantum computation is a key issue in practical implementation of a given computation. To this end, we are currently analyzing the algorithms of this paper in terms of their parallel efficiency and developing more efficient parallel quantum wavelet algorithms.
As shown in this paper, permutation matrices play a pivotal role in the development of quantum wavelet transforms. In fact, not only they arise explicitly in the packet and pyramid algorithms but also they play a key role in factorization of wavelet kernels. For classical computing, the implementation of permutation matrices is trivial. However, for quantum computing, it represents a challenging task and demands new, unconventional, and even counterintuitive (from a classical computing view point) techniques. For example, note that most of the factorizations developed in paper for permutation matrices $\Pi_{2^n}$, $P_{2^n}$, and $Q_{2^n}$ were not previously known in classical computing and, in fact, they are not at all efficient for a classical implementation. Also, implementation of the permutation matrices reveals some of the surprises of quantum computing in contrast to classical computing. In the sense that, certain operations that are hard to implement in classical computing are easier to implement in quantum computing and vice versa. As a concrete example, note that while the classical implementation of permutation matrices $\Pi_{2^n}$ and (particularly) $P_{2^n}$ is much harder (in terms of data movement pattern) than the permutation matrix $Q_{2^n}$, their quantum implementation is much easier and more straightforward than $Q_{2^n}$.
In this paper, we focussed on the set of permutation matrices arising in the development of quantum wavelet transforms and analyzed three techniques for their quantum implementation. However, it is clear that the permutation matrices will also play a major role in deriving compact and efficient factorizations, i.e., with polynomial time and space complexity, for other unitary operators by exposing and exploiting their specific structure. Therefore, we believe strongly that a more systematic study of permutation matrices is needed in order to develop further insight into efficient techniques for their implementation in quantum circuits. Such a study might eventually lead to the discovery of new and more efficient approaches for the implementation of unitary transformations and therefore quantum computation.
\noindent {\bf Acknowledgement}
The research described in this paper was performed at the Jet Propulsion Laboratory (JPL), California Institute of Technology, under contract with National Aeronautics and Space Administration (NASA). This work was supported by the NASA/JPL Center for Integrated Space Microsystems (CISM), NASA/JPL Advanced Concepts Office, and NASA/JPL Autonomy and Information Technology Management Program.
\noindent {\bf Appendix: Commutation of the Operators $I_{2^i} \otimes \Pi_{2^{n-i}}$ with $N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}}$}
We first prove that every operator of the form $I_{2^i} \otimes \Pi_{2^{n-i}}$, for $i = n-2$ to $1$, commutes with all the operators of the form $N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}}$, for $j = i$ to $2$, by simply showing that
\begin{equation}
(I_{2^i} \otimes \Pi_{2^{n-i}})(N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^(n-j+1}) = (N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}})(I_{2^i} \otimes \Pi_{2^{n-i}})
\end{equation}
The matrix $I_{2^i} \otimes \Pi_{2^{n-i}}$ is a block diagonal matrix and therefore can be written as
\begin{equation}
I_{2^i} \otimes \Pi_{2^{n-i}} = I_2 \otimes \Pi_{2^{n-j}} \oplus I_{2^j - 2} \otimes \Pi_{2^{n-j}}
\end{equation}
It can be then shown that
\begin{equation}
(I_2 \otimes \Pi_{2^{n-j}} \oplus I_{2^j - 2} \otimes \Pi_{2^{n-j}})(N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}}) = N \otimes \Pi_{2^{n-j}} \oplus
I_{2^j - 2} \otimes \Pi_{2^{n-j}}
\end{equation}
and
\begin{equation}
(N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}})(I_2 \otimes \Pi_{2^{n-j}} \oplus I_{2^j - 2} \otimes \Pi_{2^{n-j}}) = N \otimes \Pi_{2^{n-j}} \oplus I_{2^j - 2} \otimes \Pi_{2^{n-j}}
\end{equation}
It now remains to show that every operator of the form $I_{2^i} \otimes \Pi_{2^{n-i}}$ commutes with the operator $N \otimes I_{2^{n-1}}$. This is simply proved by first using the fact that
\begin{equation}
I_{2^i} \otimes \Pi_{2^{n-i}} = I_2 \otimes (I_{2^{i-1}} \otimes \Pi_{2^{n-i}})
\end{equation}
and then showing that
\begin{equation}
(I_2 \otimes (I_{2^{i-1}} \otimes \Pi_{2^{n-i}}))( N \otimes I_{2^{n-1}}) =
(N \otimes I_{2^{n-1}})(I_2 \otimes (I_{2^{i-1}} \otimes \Pi_{2^{n-i}})) =
N \otimes I_{2^{i-1}} \otimes \Pi_{2^{n-i}}
\end{equation}
\input{epsf}
\begin{figure}
\caption{A circuit for implementation of quantum Fourier transform, QFT (from [15]).}
\label{fig:one}
\end{figure}
\begin{figure}
\caption{The $\Pi_4$ gate (a) and its implementation by using three XOR (Controlled-NOT) gates (b).}
\label{fig:two}
\end{figure}
\begin{figure}
\caption{A circuit for implementation of Perfect Shuffle permutation matrix, $\Pi_{2^n}
\label{fig:three}
\end{figure}
\begin{figure}
\caption{Circuits for implementation of Bit Reversal permutation matrix, $P_{2^n}
\label{fig:four}
\end{figure}
\begin{figure}
\caption{A block-level circuit for Haar wavelet (from [20]).}
\label{fig:five}
\end{figure}
\begin{figure}
\caption{A block-level circuit for implementation of Hoyer's factorization of $D^{(4)}
\label{fig:six}
\end{figure}
\begin{figure}
\caption{A block-level circuit for implementation of new factorization of $D^{(4)}
\label{fig:seven}
\end{figure}
\begin{figure}
\caption{A circuit for implementation of operator $T_{2^n}
\label{fig:eight}
\end{figure}
\begin{figure}
\caption{A circuit for implementation of operator $P_{2^n}
\label{fig:nine}
\end{figure}
\begin{figure}
\caption{A circuit for implementation of $D^{(4)}
\label{fig:ten}
\end{figure}
\end{document}
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\begin{document}
\title{A host-parasite model for a two-type cell population}
\begin{abstract}
We consider a host-parasite model for a population of cells that can be of two types, $\sfA$ or $\sfB$, and exhibits unilateral reproduction: while a $\sfB$-cell always splits into two cells of the same type, the two daughter cells of an $\sfA$-cell can be of any type. The random mechanism that describes how parasites within a cell multiply and are then shared into the daughter cells is allowed to depend on the hosting mother cell as well as its daughter cells. Focusing on the subpopulation of $\sfA$-cells and its parasites, our model differs from the single-type model recently studied by \textsc{Bansaye} \cite{Bansaye:08} in that the sharing mechanism may be biased towards one of the two types. Our main results are concerned with the nonextinctive case and provide information on the behavior, as $n\to\infty$, of the number $\sfA$-parasites in generation $n$ and the relative proportion of $\sfA$- and $\sfB$-cells in this generation which host a given number of parasites. As in \cite{Bansaye:08}, proofs will
make use of a so-called random cell line which, when conditioned to be of type $\sfA$, behaves like a branching process in random environment.
\end{abstract}
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\section{Introduction}\label{Section.Introduction}
The reciprocal adaptive genetic change of two antagonists (e.g.\ different species or genes)
through reciprocal selective pressures is known as host-parasite coevolution. It may be observed even in real-time under both, field and laboratory conditions, if reciprocal adaptations take place rapidly and generation times are short. For more information see e.g.\ \cite{Laine:09, Woolhouse:02}.
The present work studies a host-parasite branching model with two types of cells (the hosts), here called $\sfA$ and $\sfB$, and proliferating parasites colonizing the cells. Adopting a genealogical perspective, we are interested in the evolution of certain characteristics over generations and under the following assumptions on the reproductive behavior of cells and parasites. All cells behave independently and split into two daughter cells after one unit of time. The types of the daughter cells of a type-$\sfA$ cell are chosen in accordance with a random mechanism which is the same for all mother cells of this type whereas both daughter cells of a type-$\sfB$ cell are again of type $\sfB$. Parasites within a cell multiply in an iid manner to produce a random number of offspring the distribution of which may depend on the type of this cell as well as on those of its daughter cells. The same holds true for the random mechanism by which the offspring is shared into these daughter cells.
The described model grew out of a discussion with biologists in an attempt to provide a first very simple setup that allows to study coevolutionary adaptations, here due to the presence of two different cell types. It may also be viewed as a simple multi-type extension of a model studied by \textsc{Bansaye} \cite{Bansaye:08} which in turn forms a discrete-time version of a model introduced by \textsc{Kimmel} \cite{Kimmel:97}. Bansaye himself extended his results in \cite{Bansaye:09} by allowing immigration and random environments, the latter meaning that each cell chooses the reproduction law for the parasites it hosts in an iid manner.
Let us further mention related recent work by \textsc{Guyon} \cite{Guyon:07} who studied another discrete-time model with asymmetric sharing and obtained limit theorems under ergodic hypotheses which, however, exclude an extinction-explosion principle for the parasites which is valid in our model.
We continue with the introduction of some necessary notation which is similar to the one in \cite{Bansaye:08}. Making the usual assumption of starting from one ancestor cell, denoted as $\varnothing$, we put $\G_{0}:=\{\varnothing\}$, $\G_{n}:=\{0,1\}^{n}$ for $n\ge 1$, and let
\begin{equation*}
\T := \bigcup_{n\in\mathbb N_0} \G_n\quad\text{with}\quad\G_n := \{0,1\}^n
\end{equation*}
be the binary Ulam-Harris tree rooted at $\varnothing$ which provides the label set of all cells in the considered population. Plainly, $\G_{n}$ contains the labels of all cells of generation $n$. For any cell $v\in\T$, let $T_v\in\{\sfA,\sfB\}$ denote its type and $Z_v$ the number of parasites it contains. \emph{Unless stated otherwise, the ancestor cell is assumed to be of type $\sfA$ and to contain one parasite, i.e.}
\begin{equation}\label{SA1}\tag{SA1}
T_{\varnothing}=\sfA\quad\text{and}\quad Z_{\varnothing}=1.
\end{equation}
Then, for $\sft\in\{\sfA,\sfB\}$ and $n\ge 0$, define
\begin{equation*}
\G_n(\sft) := \{v\in\G_n : T_v=\sft\}\quad\text{and}\quad\G^*_n(\sft) := \{v\in\G_n (\sft) : Z_v>0\}
\end{equation*}
as the sets of type-$\sft$ cells and type-$\sft$ contaminated cells in generation $n$, respectively. The set of all contaminated cells in generation $n$ are denoted $\G^*_n$,
thus $\G^*_n = \G^*_n(\sfA)\cup\G^*_n(\sfB)$.
As common, we write $v_{1}...v_{n}$ for $v=(v_1,...,v_n)\in\G_n$, $uv$ for the concatenation of $u,v\in\T$, i.e.
\begin{equation*}
uv=u_1...,u_m v_1...v_n\ \ \text{if}\ u=u_1...u_m\ \text{and}\ v=v_1...v_n,
\end{equation*}
and $v|k$ for the ancestor of $v=v_{1}...v_{n}$ in generation $k\le n$, thus $v|k=v_1,...,v_k$. Finally, if $v|k=u$ for some $k$ and $u\ne v$, we write $u<v$.
The process $(T_v)_{v\in\T}$ is a Markov process indexed by the tree $\T$ as defined in \cite{BenPeres:94}. It has transition probabilities
\begin{align*}
&\mathbb{P}(T_{v0}=\sfx,T_{v1}=\sfy|T_v = \sfA) = p_{\sfx\sfy},
\quad (\sfx,\sfy)\in\{(\sfA,\sfA),(\sfA,\sfB),(\sfB,\sfB)\},\\[1ex]
&\mathbb{P}(T_{v0}=\sfB,T_{v1}=\sfB|T_v = \sfB) = 1,
\end{align*}
and we denote by
\begin{equation*}
p_0 := p_{\sfAA} + p_{\sfAB}=1-p_{\sfBB}\quad\text{and}\quad p_1 := p_{\sfAA}
\end{equation*}
the probabilities that the first and the second daughter cell are of type $\sfA$, respectively.
In order to rule out total segregation of type-$\sfA$ and type-$\sfB$ cells, which would just lead back to the model studied in \cite{Bansaye:08}, it will be assumed throughout that
\begin{equation}\label{SA2}\tag{SA2}
p_{\sfAA}<1.
\end{equation}
The sequence $(\#\G_n(\sfA))_{n\ge 0}$ obviously forming a Galton-Watson branching process with one ancestor (as $T_{\varnothing}=\sfA$) and mean
\begin{equation*}
\nu := p_{0}+p_{1}=2 p_{\sfAA}+p_{\sfAB}=1+(p_{\sfAA}-p_{\sfBB})<2,
\end{equation*}
it is a standard fact that (see e.g.\ \cite{Athreya+Ney:72})
\begin{equation*}
\#\G_n(\sfA)\rightarrow 0\text{ a.s.}\quad\text{iff}\quad p_{\sfAA}\leq p_{\sfBB}\quad\text{and}\quad p_{\sfAB}<1.
\end{equation*}
To describe the multiplication of parasites, let $Z_{v}$ denote the number of parasites in cell $v$ and, for $\sft\in\{\sfA,\sfB\}$, $\sfs\in\{\sfAA,\sfAB,\sfBB\}$, let
$$ \left(X^{(0)}_{k,v}(\sft, \sfs),X^{(1)}_{k,v}(\sft, \sfs)\right)_{k\in\mathbb N, v\in\T},\quad\sft\in\{\sfA,\sfB\},\ \sfs\in\{\sfAA,\sfAB,\sfBB\}$$
be independent families of iid $\mathbb N^2_0$-valued random vectors with respective generic copies $(X^{(0)}(\sft, \sfs),X^{(1)}(\sft, \sfs))$. If $v$ is of type $\sft$ and their daughter cells are of type $\sfx$ and $\sfy$, then $X^{(i)}_{k,v}(\sft, \sfx\!\sfy)$ gives the offspring number of the $k^{\rm th}$ parasite in cell $v$ that is shared into the daughter cell $vi$ of $v$. Since type-$\sfB$ cells can only produce daughter cells of the same type, we will write $(X^{(0)}_{k,v}(\sfB),X^{(1)}_{k,v}(\sfB))$ as shorthand for $(X^{(0)}_{k,v}(\sfB, \sfBB),X^{(1)}_{k,v}(\sfB, \sfBB))$. To avoid trivialities, it is always assumed hereafter that
\begin{equation}\label{SA3}\tag{SA3}
\mathbb{P}\left(X^{(0)}(\sfA, \sfAA) \leq 1,\ X^{(1)}(\sfA, \sfAA)\leq1\right)<1
\end{equation}
and
\begin{equation}\label{SA4}\tag{SA4}
\mathbb{P}\left(X^{(0)}(\sfB)\leq1,\ X^{(1)}(\sfB)\leq1\right)<1.
\end{equation}
Next, observe that
\begin{equation*}
(Z_{v0}, Z_{v1}) = \sum_{\sft\in\{\sfA,\sfB\}}\1_{\{T_{v}=\sft\}}\sum_{\sfs\in\{\sfAA,\sfAB,\sfBB\}}\1_{\{(T_{v0},T_{v1})=\sfs\}}\sum_{k=1}^{Z_v}(X^{(0)}_{k,v}(\sft, \sfs),X^{(1)}_{k,v}(\sft, \sfs)).
\end{equation*}
We put $\mu_{i,\sft}(\sfs) := \mathbb{E} X^{(i)}(\sft, \sfs)$ for $i\in\{0,1\}$ and $\sft,\sfs$ as before, write $\mu_{i,\sfB}$ as shorthand for $\mu_{i,\sfB}(\sfBB)$ and assume throughout that $\mu_{i,\sft}(\sfs)$ are finite and
\begin{equation}\label{SA5}\tag{SA5}
\mu_{0,\sfA}(\sfAA),\ \mu_{1,\sfA}(\sfAA),\ \mathbb{E}\left(\#\G^*_1(\sfB)\right)>0,\ \mu_{0,\sfB},\ \mu_{1,\sfB}\ >\ 0.
\end{equation}
The total number of parasites in cells of type $\sft\in\{\sfA, \sfB\}$ at generation $n$ is denoted by
\begin{equation*}
\mathcal{Z}_n(\sft) := \sum_{v\in\G_n(\sft)} Z_v,
\end{equation*}
and we put $\mathcal{Z}_n := \mathcal{Z}_n(\sfA)+\mathcal{Z}_n(\sfB)$, plainly the total number of all parasites at generation $n$. Both, $(\mathcal{Z}_n)_{n\ge 0}$ and $(\mathcal{Z}_n(\sfA))_{n\ge 0}$, are transient Markov chains with absorbing state $0$ and satisfy the extinction-explosion principle (see Section I.5 in \cite{Athreya+Ney:72} for a standard argument), i.e.
\begin{equation*}
\mathbb{P}(\mathcal{Z}_n\rightarrow 0)+\mathbb{P}(\mathcal{Z}_n\rightarrow\infty)=1\quad\text{and}\quad \mathbb{P}(\mathcal{Z}_n(\sfA)\rightarrow 0)+\mathbb{P}(\mathcal{Z}_n(\sfA)\rightarrow\infty)=1.
\end{equation*}
The extinction events are defined as
\begin{equation*}
\mathbb{E}xt := \{\mathcal{Z}_n\rightarrow 0\}\quad\text{and}\quad\mathbb{E}xt(\sft):=\{\mathcal{Z}_n(\sft)\rightarrow 0\},\quad\sft\in\{\sfA,\sfB\},
\end{equation*}
their complements by $\Surv$ and $\Surv(\sft)$, respectively.
As in \cite{Bansaye:08}, we are interested in the statistical properties of an infinite \emph{random cell line}, picked however from those lines consisting of $\sfA$-cells only.
This leads to a so-called \emph{random $\sfA$-cell line}. Since $\sfB$-cells produce only daughter cells of the same type, the properties of a random $\sfB$-cell line may be deduced from the afore-mentioned work and are therefore not studied hereafter.
For the definition of a random $\sfA$-cell line, a little more care than in \cite{Bansaye:08} is needed because cells occur in two types and parasitic reproduction may depend on the types of the host and both its daughter cells. On the other hand, we will show in Section \ref{Section.Preliminaries} that a random $\sfA$-cell line still behaves like a branching process in iid random environment (BPRE) which has been a fundamental observation in \cite{Bansaye:08} for a random cell line in the single-type situation.
Let $U=(U_n)_{n\in\mathbb N}$ be an iid sequence of symmetric Bernoulli variables independent of the parasitic evolution and put $V_{n}:=U_{1}...U_{n}$. Then
$$ \varnothing=:V_{0}\to V_{1}\to V_{2}\to...\to V_{n}\to... $$
provides us with a random cell line in the binary Ulam-Haris tree, and we denote by
\begin{equation*}
T_{[n]} = T_{V_n}\quad\text{and}\quad Z_{[n]} = Z_{V_n}\quad n\ge 0,
\end{equation*}
the cell types and the number of parasites along that random cell line. A random $\sfA$-cell line up to generation $n$ is obtained when $T_{[n]}=\sfA$, for then $T_{[k]}=\sfA$ for any
$k=0,...,n-1$ as well. As will be shown in Prop.\ \ref{prop:random cell line=BPRE}, the conditional law of $(Z_{[0]},...,Z_{[n]})$ given $T_{[n]}=\sfA$, i.e., given an $\sfA$-cell line up to generation $n$ is picked at random, equals the law of a certain BPRE $(Z_{k}(\sfA))_{k\ge 0}$ up to generation $n$, for each $n\in\N$. It should be clear that this cannot be generally true for the unconditional law of $(Z_{[0]},...,Z_{[n]})$, due to the multi-type structure of the cell population.
Aiming at a study of host-parasite coevolution in the framework of a multitype host population, our model may be viewed as the simplest possible alternative. There are only two types of host cells and reproduction is unilateral in the sense that cells of type $\sfA$ may give birth to both, $\sfA$- and $\sfB$-cells, but those of type $\sfB$ will never produce cells of the opposite type.
The basic idea behind this restriction is that of irreversible mutations that generate new types of cells but never lead back to already existing ones. Observe that our setup could readily be generalized without changing much the mathematical structure by allowing the occurrence of further irreversible mutations from cells of type $\sfB$ to cells of type $\mathsf{C}$, and so on.
The rest of this paper is organized as follows. We focus on the case of non-extinction of
contaminated $\sfA$-cells, that is $\mathbb{P}(\mathbb{E}xt(\sfA))<1$. Basic results on $\mathcal{Z}_n(\sfA)$, $Z_{[n]}$, $\#\G^*_n(\sfA)$ and $\#\G^*_n$ including the afore-mentioned one will be shown in Section \ref{Section.Preliminaries} and be partly instrumental for the proofs of our results on the asymptotic behavior of the relative proportion of contaminated cells with $k$ parasites within the population of all contaminated cells. These results are stated in Section \ref{Section.Results} and proved in Section \ref{Section.Proofs}. A glossary of the most important notation used throughout may be found at the end of this article.
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\section{Basic Results}\label{Section.Preliminaries}
We begin with a number of basic properties of and results about the quantities $\G^*_n(\sfA)$, $\G^*_n$, $\mathcal{Z}_n(\sfA)$ and $Z_{[n]}$.
\subsection{The random $\sfA$-cell line and its associated sequence $(Z_{[n]})_{n\ge 0}$}
In \cite{Bansaye:08}, a random cell line was obtained by simply picking a random path in the infinite binary Ulam-Harris tree representing the cell population. Due to the multi-type structure here, we must proceed in a different manner when restricting to a specific cell type, here type $\sfA$. In order to study the properties of a ''typical'' $\sfA$-cell in generation $n$ for large $n$, i.e., an $\sfA$-cell picked at random from this generation, a convenient (but not the only) way is to first pick at random a cell line up to generation $n$ from the full height $n$ binary tree as in \cite{Bansaye:08} and then to condition upon the event that the cell picked at generation $n$ is of type $\sfA$. This naturally leads to a random $\sfA$-cell line up to generation $n$, for $\sfA$-cells can only stem from cells of the same type. Then looking at the conditional distribution of the associated parasitic random vector $(Z_{[0]},...,Z_{[n]})$ leads to a BPRE not depending on $n$ and thus to an analogous situation as in
\cite{Bansaye:08}. The precise result is stated next.
\begin{Proposition}\label{prop:random cell line=BPRE}
Let $(Z_{n}(\sfA))_{n\ge 0}$ be a BPRE with one ancestor and iid environmental sequence
$(\Lambda_{n})_{n\ge 1}$ taking values in $\{\mathcal L(X^{(0)}(\sfA, \sfAA)), \mathcal L(X^{(1)}(\sfA, \sfAA)), \mathcal L(X^{(0)}(\sfA, \sfAB))\}$ such that
\begin{equation*}
\mathbb{P}\left(\Lambda_1=\mathcal L(X^{(0)}(\sfA, \sfAB))\right)=\frac{p_{\sfAB}}{\nu}\quad\text{and}\quad\mathbb{P}\left(\Lambda_1=\mathcal L(X^{(i)}(\sfA, \sfAA))\right)=\frac{p_{\sfAA}}{\nu},
\end{equation*}
for $i\in\{0,1\}$.
Then the conditional law of $(Z_{[0]},...,Z_{[n]})$ given $T_{[n]}=\sfA$ equals the law of $(Z_{0}(\sfA),...,Z_{n}(\sfA))$, for each $n\ge 0$.
\end{Proposition}
\begin{proof}
We use induction over $n$ and begin by noting that nothing has to be shown if $n=0$.
For $n\ge 1$ and $(z_0,...,z_n)\in\N^{n+1}_0$, we introduce the notation
\begin{equation*}
C_{z_0,...,z_n}:=\{(Z_{[0]},...,Z_{[n]})=(z_0,...,z_n)\}\quad\text{and}\quad C_{z_0,...,z_n}^{\sfA}:=C_{z_0,...,z_n}\cap\{T_{[n]}=\sfA\}
\end{equation*}
and note that
\begin{equation*}
\mathbb{P}\left(T_{[n]}=\sfA\right)=2^{-n}\, \mathbb{E}\left(\sum_{v\in\G_n}\1_{\{T_v=\sfA\}}\right)=\left(\frac{\nu}{2}\right)^n,
\end{equation*}
for each $n\in\N$, in particluar
$$ \mathbb{P}(T_{[n]}=\sfA|T_{[n-1]}=\sfA)=\frac{\mathbb{P}(T_{[n]}=\sfA)}{\mathbb{P}(T_{[n-1]}=\sfA)}=\frac{\nu}{2}. $$
Assuming the assertion holds for $n-1$ (inductive hypothesis), thus
$$ \mathbb{P}(C_{z_{0},...,z_{n-1}}|T_{[n-1]}=\sfA)=\mathbb{P}\left(Z_{0}(\sfA)=z_0,...,Z_{n-1}(\sfA)=z_{n-1}\right) $$
for any $(z_{0},...,z_{n-1})\in\N_{0}^{n}$, we infer with the help of the Markov property that
\begin{align*}
\mathbb{P}&\left((Z_{[0]},...,Z_{[n]})=(z_0,...,z_n)| T_{[n]}=\sfA\right)\\[1ex]
&=~\frac{\mathbb{P}(C^{\sfA}_{z_0,...,z_n})}{\mathbb{P}(T_{[n]}=\sfA)}\\[1ex]
&=~\mathbb{P}\left(C_{z_0,...,z_{n-1}}|T_{[n-1]}=\sfA\right)\,
\mathbb{P}(Z_{[n]}=z_n,T_{[n]}=\sfA|C^{\sfA}_{z_0,...,z_{n-1}})\,\frac{\mathbb{P}(T_{[n-1]}=
\sfA)}{\mathbb{P}(T_{[n]}=\sfA)}\\[1ex]
&=~\mathbb{P}\left(Z_{0}(\sfA)=z_0,...,Z_{n-1}(\sfA)=z_{n-1}\right)\,
\frac{\mathbb{P}(Z_{[1]}=z_n, T_{[1]}=\sfA | Z_{[0]}=z_{n-1},
T_{[0]}=\sfA)}{\mathbb{P}(T_{[n]}=\sfA|T_{[n-1]}=\sfA)}\\
&=~\mathbb{P}\left(Z_{0}(\sfA)=z_0,...,Z_{n-1}(\sfA)=z_{n-1}\right)\\
&\hspace{4ex}\times\frac{2}{\nu}\left(\frac{p_{\sfA\sfB}}{2}\left(\mathbb{P}^{X^{(0)}
(\sfA, \sfAB)}\right)^{*z_{n-1}}(\{z_n\})+\sum_{i\in\{0,1\}}\frac{p_{\sfA\sfA}}
{2}\left(\mathbb{P}^{X^{(i)}(\sfA, \sfAA)}\right)^{*z_{n-1}}(\{z_n\})\right)\\[1ex]
&=~\mathbb{P}\left(Z_{0}(\sfA)=z_0,...,Z_{n-1}(\sfA)=z_{n-1}\right)\,\mathbb{P}\left(Z_
{[n]}(\sfA)=z_{n}|Z_{[n-1]}(\sfA)=z_{n-1}\right)\\[1ex]
&=~\mathbb{P}\left(Z_{0}(\sfA)=z_0,...,Z_{n}(\sfA)=z_{n}\right).
\end{align*}
This proves the assertion.
\end{proof}
The connection between the distribution of $Z_{n}(\sfA)$ and the expected number of $\sfA$-cells in generation $n$ with a specified number of parasites is stated in the next result.
\begin{Proposition}\label{Prop:dist of Zn(A)}
For all $n\in\N$ and $k\in\N_{0}$,
\begin{equation}\label{Eq.BPRE.EG}
\mathbb{P}\left(Z_n(\sfA)=k\right)=\nu^{-n}\,\mathbb{E}\left(\#\{v\in\G_n(\sfA):Z_v=k\}\right),
\end{equation}
in particular
\begin{equation}\label{Eq.BPRE.EG,k>0}
\mathbb{P}\left(Z_n(\sfA)>0\right)=\nu^{-n}\,\mathbb{E}\#\G_{n}^{*}(\sfA).
\end{equation}
\end{Proposition}
\begin{proof}
For all $n,k\in\N$, we find that
\begin{align*}
\mathbb{E}\left(\#\{v\in\G_n(\sfA):Z_v=k\}\right)
~&=~\sum_{v\in\G_n}\mathbb{P}(Z_v=k, T_v=\sfA)\\[1ex]
&=~2^n\mathbb{P}\left(Z_{[n]}=k, T_{[n]}=\sfA\right)\\[1ex]
&=~2^n\mathbb{P}(T_{[n]}=\sfA)\mathbb{P}\left(Z_{[n]}=k| T_{[n]}=\sfA\right)\\[1ex]
&=~\nu^n\mathbb{P}\left(Z_{[n]}=k| T_{[n]}=\sfA\right)\\[1ex]
&=~\nu^n\mathbb{P}\left(Z_n(\sfA)=k\right),
\end{align*}
and this proves the result.
\end{proof}
For $n\in\N$ and $s\in [0,1]$, let
\begin{equation*}
f_n(s|\Lambda) := \mathbb{E}(s^{Z_n(\sfA)}|\Lambda)\quad\text{and}\quad f_n(s) := \mathbb{E} s^{Z_n(\sfA)}=\mathbb{E} f_{n}(s|\Lambda)
\end{equation*}
denote the quenched and annealed generating function of $Z_n(\sfA)$, respectively, where $\Lambda:=(\Lambda_{n})_{n\ge 1}$.
Then the theory of BPRE (see \cite{Athreya:71.1, Athreya:71.2, GeKeVa:03, Smith+Wilkinson:69} for more details) provides us with the following facts: For each $n\in\N$,
\begin{align*}
f_n(s|\Lambda)=g_{\Lambda_{1}}\circ...\circ g_{\Lambda_{n}}(s),
\quad g_{\lambda}(s):=\mathbb{E}(s^{Z_{1}(\sfA)}|\Lambda_{1}=\lambda)=\sum_{n\ge 0}\lambda_{n}s^{n}
\end{align*}
for any distribution $\lambda=(\lambda_{n})_{n\ge 0}$ on $\N_{0}$. Moreover, the $g_{\Lambda_{n}}$ are iid with
\begin{align*}
\mathbb{E} g_{\Lambda_{1}}'(1)&=\mathbb{E} Z_{1}(\sfA)\\
&=\frac{p_{\sfAA}}{\nu}\Big(\mu_{0,\sfA}(\sfAA)+\mu_{1,\sfA}(\sfAA)\Big)+\frac{p_{\sfAB}}{\nu}\mu_{0,\sfA}(\sfAB)=\frac{\gamma}{\nu},
\end{align*}
where
\begin{equation*}
\gamma := \mathbb{E}\mathcal{Z}_1(\sfA) = p_{\sfAA}\left(\mu_{0,\sfA}(\sfAA)+\mu_{1,\sfA}(\sfAA)\right)+p_{\sfAB}\mu_{0,\sfA}(\sfAB)
\end{equation*}
denotes the expected total number of parasites in cells of type $\sfA$ in the first generation (recall from \eqref{SA1} that $Z_{\varnothing}=Z_{\varnothing}(\sfA)=1$). As a consequence,
\begin{align*}
\mathbb{E}(Z_{[n]}|T_{[n]}=\sfA)&=\mathbb{E} Z_n(\sfA)
=f_n'(1)=\prod_{k=1}^{n}\mathbb{E} g_{\Lambda_{k}}'(1)=\left(\frac{\gamma}{\nu}\right)^n
\end{align*}
for each $n\in\N$. It is also well-known that $(Z_{n}(A))_{n\ge 0}$ dies out a.s., which in terms of $(Z_{[n]})_{n\ge 0}$ means that $\lim_{n\to\infty}\mathbb{P}(Z_{[n]}=0|T_{[n]}=\sfA)=1$, iff
\begin{align}\label{Eq.BPRE.Aussterben}
\mathbb{E}\log g_{\Lambda_{1}}'(1)=\frac{p_{\sfAA}}{\nu}\Big(\log\mu_{0,\sfA}(\sfAA)+\log
\mu_{1,\sfA}(\sfAA)\Big)+\frac{p_{\sfAB}}{\nu}\log\mu_{0,\sfA}(\sfAB)\le 0.
\end{align}
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\subsection{Properties of $\#\G^*_n(\sfA)$ and $\#\G^*_n$:}
We proceed to the statement of a number of results on the asymptotic behavior of $\G^*_n(\sfA)$ and $\G^*_n$ conditioned upon $\Surv(\sfA)$ and $\Surv$, respectively. It turns out that, if the number of parasites tends to infinity, then so does the number of contaminated cells.
\begin{Theorem}\label{Satz.ExplosionInfZellen} ${}$
\begin{enumerate}[(a)]
\item\label{Item.ExplosionInfZellenB} If $\mathbb{P}(\Surv(\sfA))>0$ and $p_{\sfAA}>0$, then $\mathbb{P}(\#\G^*_n(\sfA)\to\infty|\Surv(\sfA))=1$.
\item\label{Item.ExplosionInfZellenAlle} If $\mathbb{P}(\Surv)>0$, then $\mathbb{P}(\#\G^*_n\to\infty|\Surv)=1.$
\end{enumerate}
\end{Theorem}
\begin{proof}
The proof of assertion \eqref{Item.ExplosionInfZellenB} is the same as for Theorem 4.1 in \cite{Bansaye:08} and thus omitted.
\eqref{Item.ExplosionInfZellenAlle} We first note that, given $\Surv$, a contaminated $\sfB$-cell is eventually created with probability one and then spawns a single-type cell process (as $\mathbb{E}\mathcal{Z}_1(\sfB)>0$ by \eqref{SA5}). Hence the assertion follows again from Theorem 4.1 in \cite{Bansaye:08} if $\mu_{\sfB}:=\mu_{0,\sfB}+\mu_{1,\sfB}>1$.
Left with the case $\mu_{\sfB}\le 1$, it follows that
$$ \mathbb{P}(\Surv(\sfA)|\Surv)=1,$$
for otherwise, given $\Surv$, only $\sfB$-parasites would eventually be left with positive probability which however would die out almost surely. Next, $p_{\sfAA}>0$ leads back to
\eqref{Item.ExplosionInfZellenB} so that it remains to consider the situation when $p_{\sfAA}=0$. In this case there is a single line of $\sfA$-cells, namely $\varnothing\to 0\to 00\to ...$, and $(\mathcal{Z}_n(\sfA))_{n\ge 0}$ is an ordinary Galton-Watson branching process
tending $\mathbb{P}(\cdot|\Surv(\sfA))$-a.s.\ to infinity. For $n,k\in\N$, let
\begin{equation*}
\mathcal{Z}_k(n,\sfB) := \sum_{v\in\G_{n+k+1}(\sfB):v|n+1=0^{n}1}Z_v
\end{equation*}
denote the number of $\sfB$-parasites at generation $k$ sitting in cells of the subpopulation stemming from the cell $0^{n}1$, where $0^n:=0...0$ ($n$-times). Using $p_{\sfAB}=1$ and \eqref{SA5}, notably $\mu_{1,\sfA}(\sfAB)>0,\mu_{0,\sfB}>0$ and $\mu_{1,\sfB}>0$, it
is readily seen that
\begin{equation*}
\mathbb{P}\left(\lim_{n\to\infty}\mathcal{Z}_0(n-k,\sfB)=\infty|\Surv(\sfA)\right)=1
\end{equation*}
and thus
\begin{equation*}
\mathbb{P}\left(\lim_{n\to\infty}\mathcal{Z}_K(n-k,\sfB)=0|\Surv(\sfA)\right)=0
\end{equation*}
for all $K\in\N$ and $k\leq K$. Consequently,
\begin{align*}
&\mathbb{P}\left( \liminf_{n\to\infty}\#\G^*_n\leq K|\Surv(\sfA)\right)\\
&\hspace{1cm}\leq\ \mathbb{P}\left(\lim_{n\to\infty}\max_{0\le k\le K}
\mathcal{Z}_k(n-k,\sfB)=0|\Surv(\sfA)\right)\\
&\hspace{1cm}\leq\ \sum_{k=0}^{K}
\mathbb{P}\left(\lim_{n\to\infty}\mathcal{Z}_K(n-k,\sfB)=0|\Surv(\sfA)\right)\\
&\hspace{1cm}=~0
\end{align*}
for all $K\in\N$
\end{proof}
The next result provides us with the geometric rate at which the number of contaminated cells tends to infinity.
\begin{Theorem}\label{Satz.ErholungB} $(\nu^{-n}\#\G^*_n(\sfA))_{n\ge 0}$ is a non-negative supermartingale and therefore a.s.\ convergent to a random variable $L(\sfA)$ as $n\to\infty$. Furthermore,
\begin{enumerate}[(a)]
\item\label{Item.ErholungB.FSExtinction} $L(\sfA)=0$ a.s.\ iff $\mathbb{E}\log g_{\Lambda_
{1}}'(1)\leq 0$ or $\nu\leq1$
\item\label{Item.ErholungB.Extinction} $\mathbb{P}(L(\sfA)=0)<1$ implies $\{L(\sfA)=0\}=
\mathbb{E}xt(\sfA)$ a.s.
\end{enumerate}
\end{Theorem}
\begin{proof}
That $(\nu^{-n}\#\G^*_n(\sfA))_{n\ge 0}$ forms a supermartingale follows by an easy calculation and therefore a.s. convergence to an integrable random variable $L(\sfA)$ is ensured. This supermartingale is even uniformaly integrable in the case $\nu>1$, which follows because the obvious majorant $(\nu^{-n}\#\G_n(\sfA))_{n\ge 0}$ is a normalized Galton-Watson branching process having a reproduction law with finite variance and is thus $L^2$-bounded (see Section I.6 in \cite{Athreya+Ney:72}). Consequently, $(\nu^{-n}\#\G^*_n(\sfA))_{n\geq0}$ is uniformaly integrable and
\begin{align}\label{eq:EL(A)}
\mathbb{E} L(\sfA) = \lim_{n\to\infty}\mathbb{E}\frac{\#\G_{n}^{*}(\sfA)}{\nu^n}
= \lim_{n\to\infty}\mathbb{P}(Z_n(\sfA)>0),
\end{align}
the last equality following from \eqref{Eq.BPRE.EG,k>0} in Proposition \ref{Prop:dist of Zn(A)}.
As for \eqref{Item.ErholungB.FSExtinction}, $L(\sfA)=0$ a.s.\ occurs iff either $\nu\le 1$, in which case $\#\G_n^*(\sfA)\le\#\G_{n}(\sfA)=0$ eventually, or $\nu>1$ and $\mathbb{E}\log g_{\Lambda_{1}}'(1)\leq 0$, in which case almost certain extinction of $(Z_{n}(\sfA))_{n\ge 0}$ in combination with \eqref{eq:EL(A)} yields the conclusion.
\eqref{Item.ErholungB.Extinction} Defining $\tau_n = \inf\{m\in\N:\#\G^*_m(\sfA)\geq n\}$,
we find that
\begin{align*}
\mathbb{P}(L(\sfA)=0) &\leq~ \mathbb{P}(L(\sfA)=0|\tau_n<\infty)+\mathbb{P}(\tau_n=\infty)\\
&\leq\ \mathbb{P}\left(\bigcap_{k=1}^{\#\G^*_{\tau_n}(\sfA)}\{\#\G^*_{m,k}(\sfA)/\nu^m
\to0\}\bigg|\tau_n<\infty\right)+\mathbb{P}(\tau_n=\infty)\\
&\leq\ \mathbb{P}(L(\sfA)=0)^n+\mathbb{P}(\tau_n=\infty)
\end{align*}
for all $n\ge 1$, where the $\#\G^*_{m,k}(\sfA)$, $k\ge 1$, are independent copies of $\#\G^*_m(\sfA)$. Since $\mathbb{P}(L(\sfA)=0)<1$, Theorem \ref{Satz.ExplosionInfZellen} implies
\begin{equation*}
\mathbb{P}(L(\sfA)=0)\leq \lim_{n\to\infty}\mathbb{P}(\tau_n=\infty)=\mathbb{P}\left(\sup_{n\ge 1}
\#\G^*_{n}(\sfA)<\infty\right)=\mathbb{P}(\mathbb{E}xt(\sfA))
\end{equation*}
which in combination with $\mathbb{E}xt(\sfA)\subset\{L(\sfA)=0\}$ a.s. proves the assertion.
\end{proof}
Since $\nu<2$ and $(\nu^{-n}\#\G_{n}(\sfA))_{n\ge 0}$ is a nonnegative, a.s.\ convergent martingale, we see that $2^{-n}\#\G^{*}_n(\sfA)\le 2^{-n}\#\G_n(\sfA)\to 0$ a.s. and therefore
\begin{equation*}
\frac{\#\G^*_n}{2^n}\ \simeq\ \frac{\#\G^*_n(\sfB)}{2^n},\quad\text{as }n\to\infty.
\end{equation*}
that is, the asymptotic proportion of all contaminated cells is the same as the asymptotic proportion of contaminated $\sfB$-cells. Note also that
\begin{equation}\label{eq:T[n]to zero}
\mathbb{P}(T_{[n]}=\sfA)=\mathbb{E}\left(\frac{\#\G_n(\sfA)}{2^n}\right)\to 0,\quad\text{as }n\to\infty.
\end{equation}
Further information is provided by the next result.
\begin{Theorem}\label{Satz.Erholung}
There exists a r.v. $L\in[0,1]$ such that $\#\G^*_n/2^n\to L$ a.s. Furthermore,
\begin{enumerate}[(a)]
\item\label{Item.Erholung.AlleFSExtinction} $L=0$ a.s. iff
$\mu_{0,\sfB}\mu_{1,\sfB}\leq1$.
\item\label{Item.Erholung.AlleExtinction} If $\mathbb{P}(L=0)<1$, then $\{L=0\}=\mathbb{E}xt$
a.s.
\end{enumerate}
\end{Theorem}
\begin{proof}
The existence of $L$ follows because $2^{-n}\#\G^*_n$ is obviously decreasing.
As for \eqref{Item.Erholung.AlleFSExtinction}, suppose first that $\mu_{0,\sfB}\mu_{1,\sfB}\leq1$ and note that this is equivalent to almost sure extinction of a random $\sfB$-cell line, i.e.
$$ \lim_{n\to\infty}\mathbb{P}(Z_{[n]}>0|Z_{\varnothing}=k,T_{[0]}=\sfB)=0 $$
for any $k\in\N$. This follows because, starting from a $\sfB$-cell, we are in the one-type model studied in \cite{Bansaye:08}. There it is stated that $(Z_{[n]})_{n\geq0}$ forms a BPRE which dies out a.s.\ iff $\mu_{0,\sfB}\mu_{1,\sfB}\leq1$ (see \cite[Prop.\ 2.1]{Bansaye:08}). Fix any $\varepsilon>0$ and choose $m\in\N$ so large that $\mathbb{P}(T_{[m]}=\sfA)\leq\varepsilon$, which is possible by \eqref{eq:T[n]to zero}. Then, by the monotone convergence theorem, we find that for sufficiently large $K\in\N$
\begin{align*}
\mathbb{E} L ~&=~ \lim_{n\to\infty}\mathbb{P}(Z_{[n+m]}>0)\\
&\leq~ \lim_{n\to\infty}\mathbb{P}(Z_{[n+m]}>0, T_{[m]}=\sfB) + \varepsilon\\
&\leq~ \lim_{n\to\infty}\sum_{k\ge 0}\mathbb{P}(Z_{[n+m]}>0, Z_{[m]}=k,
T_{[m]}=\sfB) + \varepsilon\\
&\leq~ \lim_{n\to\infty}\sum_{k=0}^{K}\mathbb{P}(Z_{[n]}>0|Z_{[0]}=k,
T_{[0]}=\sfB) + 2\varepsilon\\
&\leq~ 2\varepsilon
\end{align*}
and thus $\mathbb{E} L=0$. For the converse, note that
\begin{align*}
0 ~&=~ \mathbb{E} L\\
&=~ \lim_{n\to\infty}\mathbb{P}(Z_{[n+1]}>0)\\
&\geq~ \lim_{n\to\infty}\mathbb{P}(Z_{[1]}>0, T_{[1]}=\sfB)\mathbb{P}(Z_{[n]}>0|T_{[0]}=\sfB)
\end{align*}
implies $0=\lim_{n\to\infty}\mathbb{P}(Z_{[n]}>0|T_{[0]}=\sfB)$ and thus $\mu_{0,\sfB}\mu_{1,\sfB}\leq1$ as well.
The proof of \eqref{Item.Erholung.AlleExtinction} follows along similar lines as Theorem \ref{Satz.ErholungB}\eqref{Item.ErholungB.Extinction} and is therefore omitted.
\end{proof}
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\subsection{Properties of $\mathcal{Z}_n(\sfA)$}
We continue with some results on $\mathcal{Z}_n(\sfA)$, the number of $\sfA$-parasites at generation $n$, and point out first that $(\gamma^{-n}\mathcal{Z}_n(\sfA))_{n\ge 0}$ constitutes a nonnegative, mean one martingale which is a.s.\ convergent to a finite random variable $W$. In particular, $\mathbb{E}\mathcal{Z}_n(\sfA)=\gamma^n$ for all $n\in\N_{0}$. If $\mathbb{E}\mathcal{Z}_1(\sfA)^2<\infty$, $\gamma>1$ and
$$ \hat\gamma\ :=\ \nu\,\mathbb{E} g^{\prime}_{\Lambda_1}(1)^2=p_{\sfAA}\left(\mu^2_{0,\sfA}(\sfAA)+\mu^2_{1,\sfA}(\sfAA)\right)+p_{\sfAB}\mu^2_{0,\sfA}(\sfAB)\ \leq\ \gamma, $$
then the martingale is further $L^{2}$-bounded as may be assessed by a straightforward but tedious computation. The main difference between a standard Galton-Watson process and the $\sfA$-parasite process $(\mathcal{Z}_n(\sfA))_{n\geq0}$ is the dependence of the offspring numbers of parasites living in the same cell, which (by some elementary calculations) leads to an additional term in the recursive formula for the variance, viz.
\begin{equation*}
\mathbb{V}ar\left(\mathcal{Z}_{n+1}(\sfA)\right)~=~\gamma^2\,\mathbb{V}ar\left(\mathcal{Z}_{n}(\sfA)\right)+\gamma^n\,\mathbb{V}ar(\mathcal{Z}_1(\sfA))+c_{1} \nu^n f_{n}''(1)
\end{equation*}
for all $n\geq0$ and some finite positive constant $c_{1}$. Here it should be recalled that $f_n(s)=\mathbb{E} s^{Z_n(\sfA)}$. Consequently, by calculating the second derivative of $f_{n}$ and using $\hat\gamma\le\gamma$, we obtain
\begin{equation*}
f^{\prime\prime}_n(1)\ =\ \mathbb{E} g^{\prime\prime}_{\Lambda_1}(1) \sum_{i=1}^n\left(\frac{\hat\gamma}{\nu}\right)^{n-i}\left(\frac{\gamma}{\nu}\right)^{i-1}\ \leq\ c_{2} n \left(\frac{\gamma}{\nu}\right)^n
\end{equation*}
for some finite positive constant $c_2$. A combination of this inequality with the above recursion for the variance of $\mathcal{Z}_{n}(\sfA)$ finally provides us with
\begin{equation*}
\mathbb{V}ar\left(\gamma^{-n}\mathcal{Z}_{n}(\sfA)\right)~\leq~1+\gamma^{-2}\sum_{k=0}^{\infty}\gamma^{-k}\big(\mathbb{V}ar(\mathcal{Z}_1(\sfA))+c_{1} c_{2} k\big)\ <\ \infty
\end{equation*}
for all $n\ge 0$ and thus the $L^2$-boundness of $(\gamma^{-n}\mathcal{Z}_{n}(\sfA))_{n\geq0}$.
Recalling that $(\mathcal{Z}_n(\sfA))_{n\ge 0}$ and $(\mathcal{Z}_n)_{n\ge 0}$ satisfy the extinction-explosion principle, the next theorem gives conditions for almost sure extinction, that is, for
$\mathbb{P}(\mathbb{E}xt(\sfA))=1$ and $\mathbb{P}(\mathbb{E}xt)=1$.
\begin{Theorem}\label{Satz.Aussterben}
\begin{enumerate}[(a)]
\item\label{Extinction_p=0} If $p_{\sfAA}=0$, then
\begin{equation*}
\mathbb{P}(\mathbb{E}xt(\sfA))=1\quad\text{iff}\quad \mu_{0,\sfA}(\sfAB)\le 1\quad\text{or}\quad\nu<1.
\end{equation*}
\item\label{Extinction_p>0} If $p_{\sfAA}>0$, then the following statements are
equivalent:
\begin{enumerate}[(1)]\setlength{\itemsep}{1ex}
\item\label{Item.Extinction.1} $\mathbb{P}(\mathbb{E}xt(\sfA))=1$
\item\label{Item.Extinction.EG} $\mathbb{E}\#\G^*_n(\sfA)\leq1$ for
all $n\in\N\quad$
\item\label{Item.Extinction.gamma} $\nu\leq1$, or
\begin{equation*}
\nu>1,\quad\mathbb{E}\log g_{\Lambda_{1}}'(1)<0\quad\text{and}\quad\inf_{0\leq\theta
\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\leq \frac{1}{\nu}.
\end{equation*}
\end{enumerate}
\item\label{Extinction_all} $\mathbb{P}(\mathbb{E}xt)=1\quad\text{iff}\quad \mathbb{P}(\mathbb{E}xt
(\sfA))=1\ \text{and}\ \mu_{0,\sfB}+\mu_{1,\sfB}\leq1$
\end{enumerate}
\end{Theorem}
\begin{Remark}
Let us point out the following useful facts before proceeding to the proof of the theorem. We first note that, if $\mathbb{E}\log g_{\Lambda_{1}}'(1)<0$ and $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)\leq 0$, then the convexity of
$\theta\mapsto\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}$ implies that
\begin{equation*}
\mathbb{E} g_{\Lambda_{1}}'(1)=\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}.
\end{equation*}
If $\mathbb{E} Z_{1}(\sfA)^{2}<\infty$, Geiger et al. \cite[Theorems 1.1--1.3]{GeKeVa:03} showed that
\begin{equation}\label{eq:Geiger survival estimate}
\mathbb{P}(Z_{n}(\sfA)>0)\simeq cn^{-\kappa}\left(\inf_{0\leq\theta\leq 1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\right)^{n}\quad\text{as }n\to\infty
\end{equation}
for some $c\in (0,\infty)$, where
\begin{itemize}
\item[] $\kappa=0$ if $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)<0$,
(strongly subcritical case)
\item[] $\kappa=1/2$ if $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)=0$,
(intermediately subcritical case)
\item[] $\kappa=3/2$ if $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)>0$,
(weakly subcritical case)
\end{itemize}
A combination of \eqref{Eq.BPRE.EG,k>0} and \eqref{eq:Geiger survival estimate} provides us with the asymptotic relation
\begin{equation}\label{eq:Gn and Geiger}
\mathbb{E}\#\G^*_{n}(\sfA)\simeq cn^{-\kappa}\nu^{n}\left(\inf_{0\leq\theta\leq 1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\right)^{n}\quad\text{as }n\to\infty,
\end{equation}
in particular (with $\mathbb{E} Z_{1}(\sfA)^{2}<\infty$ still being in force)
\begin{equation}\label{eq:Gn and Geiger2}
\inf_{0\leq\theta\leq 1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\le\frac{1}{\nu}\quad\text{if}\quad\sup_{n\ge 1}\mathbb{E}\#\G^*_{n}(\sfA)<\infty.
\end{equation}
\end{Remark}
\begin{proof}
\eqref{Extinction_p=0} If $p_{\sfAA}=0$ and $\nu=p_{\sfAB}=1$, each generation possesses exactly one $\sfA$-cell and $(\mathcal{Z}_n(\sfA))_{n\ge 0}$ thus forms a Galton-Watson branching process with offspring mean $\mu_{0,\sfA}(\sfAB)$ and positive offspring variance (by \eqref{SA3}). Hence a.s.\ extinction occurs iff $\mu_{0,\sfA}(\sfAB)\le 1$ as claimed. If $\nu<1$, type $\sfA$ cells die out a.s. and so do type $\sfA$ parasites.
``\eqref{Item.Extinction.1}$\Rightarrow$\eqref{Item.Extinction.EG}''
(by contraposition) We fix $m\in\N$ such that $\mathbb{E}\left(\#\G^*_m(\sfA)\right)>1$ and consider a supercritical Galton-Watson branching process $(S_n)_{n\ge 0}$ with $S_0=1$ and offspring distribution
\begin{equation*}
\mathbb{P}(S_1=k) = \mathbb{P}(\#\G^*_m(\sfA)=k),\quad k\in\N_{0}.
\end{equation*}
Obviously,
\begin{equation*}
\mathbb{P}(S_n>k)\leq \mathbb{P}(\#\G^*_{nm}(\sfA)>k)
\end{equation*}
for all $k,n\in\N_0$, hence
\begin{equation*}
\lim_{n\to\infty}\mathbb{P}(\#\G^*_{nm}(\sfA)>0)\geq\lim_{n\to\infty}\mathbb{P}(S_n>0)>0,
\end{equation*}
i.e.\ $\sfA$-parasites survive with positive probability.
``\eqref{Item.Extinction.EG}$\Rightarrow$\eqref{Item.Extinction.1}''
If $\mathbb{E}\#\G^*_n(\sfA)\leq1$ for all $n\in\N$, then Fatou's lemma implies
\begin{equation*}
1\geq\liminf_{n\to\infty} \mathbb{E}\#\G^*_n(\sfA)\geq \mathbb{E}\left(\liminf_{n\to\infty}\#\G^*_n
(\sfA)\right)
\end{equation*}
giving $\mathbb{P}(\mathbb{E}xt(\sfA))=1$ by an appeal to Theorem \ref{Satz.ExplosionInfZellen}.
``\eqref{Item.Extinction.gamma}$\Rightarrow$\eqref{Item.Extinction.1},\eqref{Item.Extinction.EG}''
If $\nu\leq1$ then $\mathbb{E}\#\G_{n}^{*}(\sfA)\le\mathbb{E}\#\G_n(\sfA)=\nu^{n}\le 1$ for all $n\in\N$. So let us consider the situation when
$$\nu>1,\quad\mathbb{E}\log g_{\Lambda_{1}}'(1)<0\quad\text{and}\quad\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\leq\frac{1}{\nu} $$
is valid. By \eqref{Eq.BPRE.EG,k>0},
\begin{equation*}
\mathbb{E}\#\G^*_n(\sfA) = \nu^n \mathbb{P}(Z_n(\sfA)>0)
\end{equation*}
for all $n\in\N$. We must distinguish three cases:
\textsc{Case A}. $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)\leq 0$.
By what has been pointed out in the above remark, we then infer
\begin{equation*}
\frac{\gamma}{\nu}=\mathbb{E} g_{\Lambda_{1}}'(1)=\inf_{0\leq\theta\leq1}\mathbb{E}
g_{\Lambda_{1}}'(1)^{\theta}\le\frac{1}{\nu}
\end{equation*}
and thus $\gamma\le 1$, which in turn entails
\begin{equation*}
\mathbb{E}\#\G^*_n(\sfA)\leq \mathbb{E}\mathcal{Z}_n(\sfA) = \gamma^n \leq 1
\end{equation*}
for all $n\in\N$ as required.
\textsc{Case B}. $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)>0$ and $\mathbb{E} Z_{1}(\sfA)^{2}<\infty$. Then, by \eqref{eq:Geiger survival estimate},
\begin{equation*}
\mathbb{P}(Z_n(\sfA)>0)\ \simeq\ c n^{-3/2} \left(\inf_{0\leq\theta\leq 1}
\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\right)^n\quad\text{as } n\to\infty
\end{equation*}
holds true for a suitable constant $c\in (0,\infty)$ and therefore
\begin{equation*}
0 = \lim_{n\to\infty} \nu^n \mathbb{P}(Z_n(\sfA)>0) = \liminf_{n\to\infty} \mathbb{E}\#\G^*_n(\sfA)
\geq \mathbb{E}\left(\liminf_{n\to\infty}\#\G^*_n(\sfA)\right),
\end{equation*}
implying $\mathbb{P}(\mathbb{E}xt(\sfA))=1$ by Theorem \ref{Satz.ExplosionInfZellen}.
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\begin{proof}
\eqref{Extinction_p=0} If $p_{\sfAA}=0$ and $\nu=p_{\sfAB}=1$, each generation possesses exactly one $\sfA$-cell and $(\mathcal{Z}_n(\sfA))_{n\ge 0}$ thus forms a Galton-Watson branching process with offspring mean $\mu_{0,\sfA}(\sfAB)$ and positive offspring variance (by \eqref{SA3}). Hence a.s.\ extinction occurs iff $\mu_{0,\sfA}(\sfAB)\le 1$ as claimed. If $\nu<1$, type $\sfA$ cells die out a.s. and so do type $\sfA$ parasites.
``\eqref{Item.Extinction.1}$\Rightarrow$\eqref{Item.Extinction.EG}''
(by contraposition) We fix $m\in\N$ such that $\mathbb{E}\left(\#\G^*_m(\sfA)\right)>1$ and consider a supercritical Galton-Watson branching process $(S_n)_{n\ge 0}$ with $S_0=1$ and offspring distribution
\begin{equation*}
\mathbb{P}(S_1=k) = \mathbb{P}(\#\G^*_m(\sfA)=k),\quad k\in\N_{0}.
\end{equation*}
Obviously,
\begin{equation*}
\mathbb{P}(S_n>k)\leq \mathbb{P}(\#\G^*_{nm}(\sfA)>k)
\end{equation*}
for all $k,n\in\N_0$, hence
\begin{equation*}
\lim_{n\to\infty}\mathbb{P}(\#\G^*_{nm}(\sfA)>0)\geq\lim_{n\to\infty}\mathbb{P}(S_n>0)>0,
\end{equation*}
i.e.\ $\sfA$-parasites survive with positive probability.
``\eqref{Item.Extinction.EG}$\Rightarrow$\eqref{Item.Extinction.1}''
If $\mathbb{E}\#\G^*_n(\sfA)\leq1$ for all $n\in\N$, then Fatou's lemma implies
\begin{equation*}
1\geq\liminf_{n\to\infty} \mathbb{E}\#\G^*_n(\sfA)\geq \mathbb{E}\left(\liminf_{n\to\infty}\#\G^*_n
(\sfA)\right)
\end{equation*}
giving $\mathbb{P}(\mathbb{E}xt(\sfA))=1$ by an appeal to Theorem \ref{Satz.ExplosionInfZellen}.
``\eqref{Item.Extinction.gamma}$\Rightarrow$\eqref{Item.Extinction.1},\eqref{Item.Extinction.EG}''
If $\nu\leq1$ then $\mathbb{E}\#\G_{n}^{*}(\sfA)\le\mathbb{E}\#\G_n(\sfA)=\nu^{n}\le 1$ for all $n\in\N$. So let us consider the situation when
$$\nu>1,\quad\mathbb{E}\log g_{\Lambda_{1}}'(1)<0\quad\text{and}\quad\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\leq\frac{1}{\nu} $$
is valid. By \eqref{Eq.BPRE.EG,k>0},
\begin{equation*}
\mathbb{E}\#\G^*_n(\sfA) = \nu^n \mathbb{P}(Z_n(\sfA)>0)
\end{equation*}
for all $n\in\N$. We must distinguish three cases:
\textsc{Case A}. $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)\leq 0$.
By what has been pointed out in the above remark, we then infer
\begin{equation*}
\frac{\gamma}{\nu}=\mathbb{E} g_{\Lambda_{1}}'(1)=\inf_{0\leq\theta\leq1}\mathbb{E}
g_{\Lambda_{1}}'(1)^{\theta}\le\frac{1}{\nu}
\end{equation*}
and thus $\gamma\le 1$, which in turn entails
\begin{equation*}
\mathbb{E}\#\G^*_n(\sfA)\leq \mathbb{E}\mathcal{Z}_n(\sfA) = \gamma^n \leq 1
\end{equation*}
for all $n\in\N$ as required.
\textsc{Case B}. $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)>0$ and $\mathbb{E} Z_{1}(\sfA)^{2}<\infty$. Then, by \eqref{eq:Geiger survival estimate},
\begin{equation*}
\mathbb{P}(Z_n(\sfA)>0)\ \simeq\ c n^{-3/2} \left(\inf_{0\leq\theta\leq 1}
\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\right)^n\quad\text{as } n\to\infty
\end{equation*}
holds true for a suitable constant $c\in (0,\infty)$ and therefore
\begin{equation*}
0 = \lim_{n\to\infty} \nu^n \mathbb{P}(Z_n(\sfA)>0) = \liminf_{n\to\infty} \mathbb{E}\#\G^*_n(\sfA)
\geq \mathbb{E}\left(\liminf_{n\to\infty}\#\G^*_n(\sfA)\right),
\end{equation*}
implying $\mathbb{P}(\mathbb{E}xt(\sfA))=1$ by Theorem \ref{Satz.ExplosionInfZellen}.
\textsc{Case C}. $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)>0$ and $\mathbb{E} Z_{1}(\sfA)^{2}=\infty$. Using contraposition, suppose that $\sup_{n\in\N}\mathbb{E}\#\G^*_n(\sfA)>1$.
Fix any vector $\alpha=(\alpha^{(u)}_{\sfs})_{u\in\{0,1\},\sfs\in\{\sfAA,\sfAB,\sfBB\}}$
of distributions on $\N_{0}$ satisfying
$$ \alpha_{\sfs,x}^{(u)}\ \le\ \mathbb{P}\left(X^{(u)}_{1,v}(\sfA,\sfs)=x\right)\quad\text{for }x\geq 1 $$
and $u,\sfs$ as stated, hence
$$ \alpha_{\sfs,0}^{(u)}\ \ge\ \mathbb{P}\left(X^{(u)}_{1,v}(\sfA,\sfs)=0\right)\quad\text{and}\quad\sum_{x\ge n}\alpha_{\sfs,x}^{(u)}\ \le\ \mathbb{P}\left(X^{(u)}_{1,v}(\sfA,\sfs)\ge n\right) $$
for each $n\ge 0$. Possibly after enlarging the underlying probability space, we can then construct a cell division process $(Z_{\alpha,v}, T_v)_{v\in\T}$ coupled with and of the same kind as $(Z_{v}, T_v)_{v\in\T}$ such that
\begin{align*}
&X^{(u)}_{\alpha, k,v}(\sfA,\sfs)\ \leq\ X^{(u)}_{k,v}(\sfA,\sfs)\quad\text{a.s.}\\
\text{and}\quad
&\mathbb{P}\left(X^{(u)}_{\alpha,k,v}(\sfA,\sfs) = x \right)\ =\ \alpha_{\sfs,x}^{(u)}
\end{align*}
for each $u\in\{0,1\}$, $\sfs\in\{\sfAA,\sfAB,\sfBB\}$, $v\in\T$, $k\geq1$ and $x\ge 1$.
To have $(Z_{\alpha,v}, T_v)_{v\in\T}$ completely defined, put also
$$ (X^{(0)}_{\alpha,k,v}(\sfB),X^{(1)}_{\alpha,k,v}(\sfB)):=(X^{(0)}_{k,v}(\sfB),X^{(1)}_{k,v}(\sfB)) $$
for all $v\in\T$ and $k\ge 1$. Then $Z_{\alpha,v}\leq Z_v$ a.s.\ and thus
\begin{equation}\label{Eq.GestutzterProzess.G}
\mathbb{E} g_{\alpha,\Lambda_{1}}'(1)^{\theta}\ \leq\ \mathbb{E} g_{\Lambda_{1}}'(1)^{\theta},\quad \theta
\in[0,1],
\end{equation}
where $Z_{\alpha,k}(\sfA)$ and $g_{\alpha,\Lambda_{1}}$ have the obvious meaning. Since the choice of $\alpha$ has no affect on the cell splitting process, we have $\nu_{\alpha}=\nu>1$, while \eqref{Eq.GestutzterProzess.G} ensures
\begin{equation}\label{eq:truncation gf}
\mathbb{E}\log g_{\alpha,\Lambda_{1}}'(1)\le\mathbb{E}\log g_{\Lambda_{1}}'(1)<0.
\end{equation}
For $N\in\N$ let $\alpha(N)=(\alpha_{\sfs}^{(u)}(N))_{u\in\{0,1\},\sfs\in\{\sfAA,\sfAB,\sfBB\}}$ be the vector specified by
\begin{equation*}
\alpha_{\sfs,x}^{(u)}(N)\ :=\
\begin{cases}
\mathbb{P}\left(X^{(u)}_{k,v}(\sfA,\sfs) = x \right), &\text{if $1\leq x\leq N$}\\
0, &\text{if $x>N$}.
\end{cases}
\end{equation*}
Then $\mathbb{E} Z_{\alpha(N),1}(\sfA)^2<\infty$ and we can fix $N\in\N$ such that $\sup_{n\in\N}\mathbb{E}\#\G^*_{\alpha(N),n}(\sfA)>1$, because $\#\G^*_{\alpha(N),n}(\sfA)\uparrow\#\G^*_{n}(\sfA)$ as $N\to\infty$. Then, by what has already been proved under Case B in combination with \eqref{Eq.GestutzterProzess.G},\eqref{eq:truncation gf} and $\nu_{\alpha(N)}>1$, we infer
\begin{equation*}
\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\geq\inf_{0\leq\theta\leq1}\mathbb{E} g_{\alpha(N),\Lambda_{1}}'(1)^{\theta}>\frac{1}{\nu}.
\end{equation*}
and thus violation of \eqref{Item.Extinction.EG}.
``\eqref{Item.Extinction.EG}$\Rightarrow$\eqref{Item.Extinction.gamma}''
Suppose $\mathbb{E}\#\G_{n}^{*}(\sfA)\le 1$ for all $n\in\N$ and further $\nu>1$ which, by
\eqref{Eq.BPRE.EG,k>0}, entails $\lim_{n\to\infty}\mathbb{P}(Z_{n}(\sfA)>0)=0$ and thus $\mathbb{E}\log g_{\Lambda_{1}}'(1)\le 0$. We must show that $\mathbb{E}\log g_{\Lambda_{1}}'(1)<0$ and $\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\le\nu^{-1}$. But given $\mathbb{E}\log g_{\Lambda_{1}}'(1)<0$, the second condition follows from \eqref{eq:Gn and Geiger2} if $\mathbb{E} Z_{1}(\sfA)^{2}<\infty$, and by a suitable ``$\alpha$-coupling'' as described under Case C above if $\mathbb{E} Z_{1}(\sfA)^{2}=\infty$. Hence it remains to rule out that $\mathbb{E}\log g_{\Lambda_{1}}'(1)=0$. Assuming the latter, we find with the help of Jensen's inequality that
\begin{equation*}
\inf_{0\leq\theta\leq 1}\log \mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\geq \inf_{0\leq\theta\leq1}\theta \,\mathbb{E}\log g_{\Lambda_{1}}'(1) = 0
\end{equation*}
or, equivalently,
\begin{equation*}
\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\geq 1>\frac{1}{\nu}
\end{equation*}
(which implies $\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}=1$).
Use once more a suitable ``$\alpha$-coupling'' and fix $\alpha$ in such a way that
\begin{equation*}
1=\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta} >\inf_{0\leq\theta\leq1}\mathbb{E} g_{\alpha,\Lambda_{1}}'(1)^{\theta}>\frac{1}{\nu}
\end{equation*}
which implies subcriticality of the associated BPRE $(Z_{\alpha,n}(\sfA))_{n\geq0}$.
By another appeal to \eqref{eq:Gn and Geiger2}, we thus arrive at the contradiction
\begin{equation*}
\sup_{n\in\N}\mathbb{E}\#\G^*_{n}(\sfA)\ge\sup_{n\in\N}\mathbb{E}\#\G^*_{\alpha,n}(\sfA)=\infty.
\end{equation*}
This completes the proof of \eqref{Extinction_p>0}.
\eqref{Extinction_all}
Since $\mathbb{E}xt\subseteq \mathbb{E}xt(\sfA)$, we see that $\mathbb{P}(\mathbb{E}xt)=1$ holds iff $\mathbb{P}(\mathbb{E}xt(\sfA))=1$
and the population of $\sfB$-parasites dies out a.s.\ as well. But the latter form a Galton-Watson branching process with offspring mean $\mu_{0,\sfB}+\mu_{1,\sfB}$ once all $\sfA$-parasites have disappeared and hence die out as well iff $\mu_{0,\sfB}+\mu_{1,\sfB}\leq1$.
\end{proof}
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\begin{Theorem}\label{Satz.W}
Assuming $\mathbb{P}(\Surv(\sfA))>0$ and thus particularly $\gamma>1$, the following assertions hold true:
\begin{enumerate}[(a)]
\item\label{Item.W>0} If $\mathbb{E}\mathcal{Z}_1(\sfA)^2<\infty$ and $\hat{\gamma}\leq\gamma$, then $\mathbb{P}(W>0)>0$ and $\mathbb{E} W=1$.
\item\label{Item.W=0} If $\mathbb{P}(W=0)<1$, then $\mathbb{E}xt(\sfA)=\{W=0\}$ a.s.
\end{enumerate}
\end{Theorem}
\begin{proof}
\eqref{Item.W>0}
As pointed out at the beginning of this subsection, $(\mathcal{Z}_n(\sfA)/\gamma^n)_n$ is a $L^2$-bounded martingale and thus uniformly integrable. It therefore converges in $L^1$ to its limit $W$ satisfying $\mathbb{E} W=1$ as well as $\mathbb{P}(W>0)>0$.
\eqref{Item.W=0} follows in the same manner as Theorem \ref{Satz.ErholungB}\eqref{Item.ErholungB.Extinction}.
\end{proof}
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\section{Relative proportions of contaminated cells}\label{Section.Results}
We now turn to a statement of our main results that are concerned with the long-run behavior of relative proportions of contaminated cells containing a given number of parasites, viz.
\begin{equation*}
F_k(n):=\frac{\#\{v\in\G^*_n|Z_v=k\}}{\#\G^*_n}
\end{equation*}
for $k\in\N$ and $n\to\infty$, and of the corresponding quantities when restricting to contaminated cells of a given type $\sft$, viz.
\begin{equation*}
F_k(n,\sft):=\frac{\#\{v\in\G^*_n(\sft)|Z_v=k\}}{\#\G^*_n(\sft)}
\end{equation*}
for $\sft\in\{\sfA,\sfB\}$. Note that
\begin{equation*}
F_k(n) = F_k(n,\sfA)\,\frac{\#\G^*_n(\sfA)}{\#\G^*_n}+F_k(n,\sfB)\,\frac{\#\G^*_n (\sfB)}{\#\G^*_n}.
\end{equation*}
Given survival of type-$\sfA$ parasites, i.e.\ conditioned upon the event $\Surv(\sfA)$, our results, devoted to regimes where at least one of $\sfA$- or $\sfB$-parasites multiply at a high rate, describe the limit behavior of $F_k(n,\sfA)$, $\#\G^*_n(\sfA)/\#\G^*_n$ and $F_k(n,\sfB)$, which depends on that of $\mathcal{Z}_n(\sfA)$ and the BPRE $Z_n(\sfA)$ in a crucial way.
For convenience, we define
\begin{equation*}
\mathbb{P}_{z,\sft} := \mathbb{P}(\cdot|Z_{\varnothing}=z, T_{\varnothing}=\sft),\quad z\in\N,\ \sft\in\{\sfA,\sfB\},
\end{equation*}
and use $\mathbb{E}_{z,\sft}$ for expectation under $\mathbb{P}_{z,\sft}$. Recalling that $\mathbb{P}$ stands for $\mathbb{P}_{1,\sfA}$,
we put $\mathbb{P}^{*}:=\mathbb{P}(\cdot|\Surv(\sfA))$ and, furthermore,
\begin{equation*}
\mathbb{P}^*_{z,\sft}:=\mathbb{P}_{z,\sft}(\cdot|\Surv(\sfA))\quad \text{and}\quad \mathbb{P}^n_{z,\sft}=\mathbb{P}_{z,\sft}(\cdot|\mathcal{Z}_n(\sfA)>0)
\end{equation*}
for $z\in\N$ and $\sft\in\{\sfA,\sfB\}$. Convergence in probability with respect to $\mathbb{P}^{*}$ is shortly expressed as $\mathop{\Prob^{*}\text{\rm -lim}\,}$.
Theorem \ref{Satz.Proportion.A1} deals with the situation when $\sfB$-parasites multiply at a high rate, viz.\
$$ \mu_{0,\sfB}\mu_{1,\sfB}>1, $$
In essence, it asserts that among all contaminated cells in generation $n$ those of type $\sfB$ prevail as $n\to\infty$. This may be surprising at first glance because multiplication of $\sfA$-parasites may also be high (or even higher), namely if
\begin{equation}\tag{SupC}\label{Eq.Supercritical}
\mu_{0,\sfA}(\sfAA)^{p_{\sfAA}}\mu_{1,\sfA}(\sfAA)^{p_{\sfAA}}\mu_{0,\sfA}(\sfAB)^{p_{\sfAB}}>1,
\end{equation}
i.e., if the BPRE $(Z_n(\sfA))_{n\ge 0}$ is supercritical. On the other hand, it should be recalled that the subpopulation of $\sfA$-cells grows at rate $\nu<2$ only, whereas the growth rate of $\sfB$-cells is 2. Hence, prevalence of $\sfB$-cells in the subpopulation of all contaminated cells is observed whenever $\#\G^*_n(\sfB)/\#\G_{n}(\sfB)$, the relative proportion of contaminated cells within the $n^{th}$ generation of all $\sfB$-cells, is asymptotically positive as $n\to\infty$.
\begin{Theorem}\label{Satz.Proportion.A1}
Assuming $\mu_{0,\sfB}\mu_{1,\sfB}>1$, the following assertions hold true:
\begin{enumerate}[(a)]
\item\label{Item.Satz.A1.G}
\begin{equation*}
\frac{\#\G^*_n(\sfA)}{\#\G^*_n}\rightarrow0\quad \mathbb{P}^*\text{-a.s.}
\end{equation*}
\item\label{Item.Satz.A1.FA} Conditioned upon survival of $\sfA$-cells, $F_k(n,
\sfB)$ converges to $0$ in probability for any $k\in\N$, i.e.
\begin{equation*}
\mathop{\Prob^{*}\text{\rm -lim}\,}_{n\to\infty} F_k(n,\sfB) = 0.
\end{equation*}
\end{enumerate}
\end{Theorem}
Properties attributed to a high multiplication rate of $\sfA$-parasites are given in Theorem \ref{Satz.Proportion.B1}. First of all, contaminated $\sfB$-cells still prevail in the long-run because, roughly speaking, highly infected $\sfA$-cells eventually produce highly infected $\sfB$-cells whose offspring $m$ generations later for any fixed $m$ are all contaminated (thus $2^{m}$ in number). Furthermore, the $F_{k}(n,\sfA)$ behave as described in \cite{Bansaye:08} for the single-type case: as $n\to\infty$, the number of parasites in any contaminated $\sfA$-cell in generation $n$ tends to infinity and $F_k(n,\sfA)$ to $0$ in probability. Finally, if we additionally assume that type-$\sfB$ parasites multiply faster than type-$\sfA$ parasites, i.e.
$$\mu_{\sfB}:=\mu_{0,\sfB}+\mu_{1,\sfB}>\gamma,$$
then type-$\sfB$ parasites become predominant and $F_k(n,\sfB)$ behaves again in Bansaye's single-type model \cite{Bansaye:08}.
\begin{Theorem}\label{Satz.Proportion.B1}
Assuming \eqref{Eq.Supercritical}, the following assertions hold true:
\begin{enumerate}[(a)]
\item\label{Item.Satz.B1.FB} Conditioned upon survival of $\sfA$-cells, $F_k(n,
\sfA)$ converges to $0$ in probability for any $k\in\N$, i.e.
\begin{equation*}
\mathop{\Prob^{*}\text{\rm -lim}\,}_{n\to\infty}F_k(n,\sfA)=0.
\end{equation*}
\item\label{Item.Satz.B1.G} \begin{equation*}
\mathop{\Prob^{*}\text{\rm -lim}\,}_{n\rightarrow\infty}\frac{\#\G^*_n(\sfA)}{\#\G^*_n}=0.
\end{equation*}
\item If $\mathbb{E}_{1,\sfB}\mathcal{Z}_1^2<\infty$, $\mu_{\sfB}>\gamma$ and $ \mu_{0,\sfB}\log\mu_{0,\sfB}+\mu_{1,\sfB}\log\mu_{1,\sfB}<0$,
then
\begin{equation*}
\mathop{\Prob^{*}\text{\rm -lim}\,}_{n\rightarrow\infty}F_k(n,\sfB)=\mathbb{P}(\mathcal{Y}(\sfB)=k)
\end{equation*}
for all $k\in\N$, where $\mathbb{P}(\mathcal{Y}(\sfB)=k)=\lim_{n\rightarrow\infty}
\mathbb{P}_{1,\sfB}(Z_{[n]}=k|Z_{[n]}>0)$.
\end{enumerate}
\end{Theorem}
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\section{Proofs}\label{Section.Proofs}
\begin{Beweis}[Theorem \ref{Satz.Proportion.A1}]
\eqref{Item.Satz.A1.G}
By Theorem \ref{Satz.Erholung}, $2^{-n}\#\G^*_n\to L$ $\mathbb{P}^{*}$-a.s. and $\mathbb{P}^{*}(L>0)=1$, while Theorem \ref{Satz.ErholungB} shows that $\nu^{-n}\#\G^*_n(\sfA)\to L(\sfA)$ $\mathbb{P}$-a.s.\ for an a.s. finite random variable $L(\sfA)$. Consequently,
\begin{equation*}
\frac{\#\G^*_n(\sfA)}{\#\G^*_n} ~=~ \left(\frac{\nu}{2}\right)^{n}\left(\frac{2^n}
{\#\G^*_n}\right)\left(\frac
{\#\G^*_n(\sfA)}{\nu^n}\right) ~\simeq~ \frac{1}{L}\left(\frac{\nu}{2}\right)^{n}
\frac{\#\G^*_n(\sfA)}{\nu^n}\to 0\quad \mathbb{P}^{*}\text{-a.s.}
\end{equation*}
as $n\to\infty$, for $\nu<2$.
\eqref{Item.Satz.A1.FA}
Fix arbitrary $\varepsilon, \delta>0$ and $K\in\N$ and define
\begin{equation*}
D_n := \left \{ \sum_{k=1}^{K}F_k(n,\sfB)>\delta \right \}\cap \Surv(\sfA).
\end{equation*}
By another appeal to Theorem \ref{Satz.Erholung}, $\#\G^*_n(\sfB)\geq 2^nL$ $\mathbb{P}^*$-a.s.\ for all $n\in\N$ and $L$ as above. It follows that
\begin{align*}
\#\{v\in\G_n(\sfB) : 0< Z_v\leq K\} ~\geq~ \delta\,\#\G^*_n(\sfB)\1_{D_n}
~\geq~ \delta\,2^n\,L\1_{D_n},
\end{align*}
and by taking the expectation, we obtain for $m\leq n$
\begin{align*}
\delta\, &\mathbb{E}\left( L\1_{D_n}\right)
~\leq~ \frac{1}{2^{n}}\,\mathbb{E}\left(\sum_{v\in\G_n}\1_{\{0<Z_v\leq K, T_v=\sfB
\}}\right)\\
&~\leq~ \frac{1}{2^{n}}\,\mathbb{E}\left(\sum_{v\in\G_n}\1_{\{0<Z_v\leq K, T_{v|m}
=\sfB\}}+\,\#\big\{v\in\G_n:T_{v|m}=\sfA, T_v=\sfB\big\}\right)\\
&~\leq~ \frac{1}{2^{n}}\,\sum_{v\in\G_n}\mathbb{P}\left(0<Z_v\leq K, T_{v|m}=
\sfB\right)+\frac{1}{2^{m}}\,\mathbb{E}\#\G_{m}(\sfA)\\
&~\leq~ \frac{1}{2^{n}}\,\sum_{z\ge 1}\sum_{v\in\G_n}\mathbb{P}\left(0<Z_v\leq
K, Z_{v|m}=z, T_{v|m}=\sfB\right)+\left(\frac{\nu}{2}\right)^m\\
&~\leq~ \sum_{z=1}^{\infty}\left(\sum_{u\in\G_m}\frac{\mathbb{P}(Z_u=z,
T_u=\sfB)}{2^{m}}\right)\Bigg(\sum_{u\in\G_{n-m}}\frac{\mathbb{P}_{z,\sfB}
(0<Z_v\leq K)}{2^{n-m}}\Bigg)+\left(\frac{\nu}{2}\right)^m\\
&~\leq~ \sum_{z=1}^{\infty}\mathbb{P}(Z_{[m]}=z, T_{[m]}=\sfB)\,\mathbb{P}_{z,\sfB}
\left(0<Z_{[n-m]}\leq K\right)+\left(\frac{\nu}{2}\right)^m.
\end{align*}
Since $\nu<2$ we can fix $m\in\N$ such that $(\nu/2)^{m}\le\varepsilon$.
Also fix $z_0\in\N$ such that
\begin{equation*}
\mathbb{P}(Z_{[m]}>z_0)\leq\varepsilon.
\end{equation*}
Then
\begin{align*}
\delta\,\mathbb{E}\left( L\1_{D_n}\right) ~&\leq~ \sum_{z\ge 1}\mathbb{P}(Z_{[m]}=z, T_
{[m]}=\sfB)\,\mathbb{P}_{z,\sfB}\left(0<Z_{[n-m]}\leq K\right)+\left(\frac{\nu}
{2}\right)^m\\
&\leq~\sum_{z=1}^{z_0}\mathbb{P}_{z,\sfB}\left(0<Z_{[n-m]}\leq K\right)
+2\varepsilon.
\end{align*}
But the last sum converges to zero as $n\to\infty$ because, under $\mathbb{P}_{z,\sfB}$,
$(Z_{[n]})_{n\ge 0}$ is a single-type BPRE (see \cite{Bansaye:08}) and thus satisfies the extinction-explosion principle. So we have shown that $\mathbb{E} L\1_{D_n}\to 0$
implying $\mathbb{P}(D_{n})\to 0$ because $L>0$ on $\Surv$. This completes the proof of the theorem.
\end{Beweis}
Turning to the proof of Theorem \ref{Satz.Proportion.B1}, we first note that part (a) can be directly inferred from Theorem 5.1 in \cite{Bansaye:08} after some minor modifications owing to the fact that $\sfA$-cells do not form a binary tree here but rather a Galton-Watson subtree of it. Thus left with the proof of parts (b) and (c), we first give an auxiliary lemma after the following notation:
For $v\in\G_n$ and $k\in\N$, let
\begin{equation*}
\G^*_k(\sft,v):=\{u\in\mathbb G^*_{n+k}(\sft):v<u\}
\end{equation*}
denote the set of all infected $\sft$-cells in generation $n+k$ stemming from $v$.
Let further be
\begin{equation*}
\mathbb G^*_n(\sfA,\sfB) := \{u\in\G^*_{n+1}(\sfB):T_{u|n}=\sfA\}
\end{equation*}
which is the set of all infected $\sfB$-cells in generation $n+1$ with mother cells of type $\sfA$.
\begin{Lemma}\label{Lemma.Proportion.B1}
If \eqref{Eq.Supercritical} holds true, then
\begin{equation*}
\mathop{\Prob^{*}\text{\rm -lim}\,}_{n\to\infty}\frac{\#\mathbb G^*_n(\sfA,\sfB)}{\#\mathbb G^*_n(\sfA)} ~=~ \beta ~>~0,
\end{equation*}
where $\beta := \lim_{z\to\infty}\mathbb{E}_{z,\sfA}\#\G^{*}_{1}(\sfB)$.
\end{Lemma}
\begin{proof}
Since $z\mapsto\mathbb{E}_{z,\sfA}\#\G^{*}_{1}(\sfB)$ is increasing and $\mathbb{E}_{1,\sfA}\#\G^{*}_{1}(\sfB)>0$ by our standing assumption \eqref{SA5}, we see that $\beta$ must be positive. Next observe that, for each $n\in\N$,
\begin{equation*}
\#\G^*_n(\sfA,\sfB) = \sum_{v\in\G^*_{n-1}(\sfA)}\#\G^*_1(\sfB,v),
\end{equation*}
where the $\#\G^*_1(\sfB,v)$ are conditionally independent given $\mathcal{Z}_n(\sfA)>0$. Since $\#\G^*_n(\sfA)\to\infty$ $\mathbb{P}^*$-a.s. (Theorem \ref{Satz.ExplosionInfZellen}) and $\mathbb{P}^n=\mathbb{P}(\cdot|\mathcal{Z}_n(\sfA)>0)\xrightarrow{w} \mathbb{P}^*$, it is not difficult to infer with the help of the SLLN that
\begin{equation*}
\frac{\#\mathbb G^*_n(\sfA,\sfB)}{\#\mathbb G^*_n(\sfA)} ~\overset{\mathbb{P}^*}\simeq~
\frac{1}{\#\G^*_n(\sfA)}\sum_{v\in\G^*_n(\sfA)}\mathbb{E}_{Z_v,\sfA}\#\G^*_1(\sfB),\quad n
\to\infty.
\end{equation*}
where $a_{n}\stackrel{\mathbb{P}^{*}}{\simeq}b_{n}$ means that $\mathbb{P}^{*}(a_{n}/b_{n}\to 1)=1$. Now use $\mathbb{E}_{z,\sfA}\#\G^*_1(\sfB)\uparrow\beta$ to infer the existence of a $z_0\in\N$ such that for all $z\geq z_0$
\begin{equation*}
\mathbb{E}_{z,\sfA}\#\G^*_1(\sfB) \geq \beta(1-\varepsilon).
\end{equation*}
After these observations we finally obtain by an appeal to Theorem \ref{Satz.Proportion.B1}(a) that
\begin{align*}
\beta~&\geq~\frac{1}{\#\G^*_n(\sfA)}\sum_{v\in\G^*_n(\sfA)}\mathbb{E}_{Z_v,\sfA}\#\G^*_1(\sfB)\\
&\geq~\sum_{z\geq z_0}\frac{F_z(n,\sfA)}{\#\{v\in\G^*_n(\sfA)|Z_v\geq z_0\}}\sum_{v\in\{u\in\G^*_n(\sfA)|Z_u\geq z_0\}}\mathbb{E}_{Z_v,\sfA}\#\G^*_1(\sfB)\\
&\geq~\beta(1-\varepsilon)\sum_{z\geq z_0}F_z(n,\sfA)\\
&\to~ \beta(1-\varepsilon),\quad n\to\infty.
\end{align*}
This completes the proof of the lemma.
\end{proof}
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\begin{Beweis}[Theorem \ref{Satz.Proportion.B1}(b) and (c)]
Let $\varepsilon>0$ and $N\in\N$. Then
\begin{align*}
\#\G^*_n(\sfB) ~&=~ \sum_{k=0}^{n-1}\sum_{v\in\G^*_k(\sfA,\sfB)}\#\mathbb
G^*_{n-k-1}(\sfB,v)\\
&\geq~\sum_{k=0}^{n-1}\sum_{v\in\{u\in\mathbb G^*_k(\sfA,\sfB) | Z_u\geq z
\}}\#\mathbb G^*_{n-k-1}(\sfB,v)\\
&\geq~\sum_{v\in\{u\in\mathbb G^*_{n-1-m}(\sfA,\sfB) | Z_u\geq z\}}\#\mathbb
G^*_{m}(\sfB,v)
\end{align*}
a.s. for all $n>m\ge 1$ and $z\in\N$, and thus
\begin{equation}\label{Eq.GA.nur ein m}
\begin{split}
\mathbb{P}^*&\left(\frac{\#\mathbb G^*_n(\sfA)}{\#\mathbb G^*_n}>\frac{1}{N
+1}\right)~=~\mathbb{P}^*\left(N\,\#\mathbb G^*_n(\sfA) >
\#\mathbb G^*_n(\sfB)\right)\\
&\leq~\mathbb{P}^*\left(N\,\#\mathbb G^*_n(\sfA)>\sum_{v\in\{u\in\mathbb G^*
_{n-1-m}(\sfA,\sfB) | Z_u\geq z\}}\#\mathbb G^*_{m}(\sfB,v)\right).
\end{split}
\end{equation}
Fix $m$ so large that
\begin{equation*}
2^{m}(1-\varepsilon)>\frac{4N}{\beta}.
\end{equation*}
Then, since
\begin{equation*}
\lim_{z\rightarrow\infty}\mathbb{P}_{z,\sfB}(\#\mathbb G^*_{m}=2^{m})=1,
\end{equation*}
there exists $z_0\in\N$ such that
\begin{equation*}
\mathbb{P}_{z,\sfB}(\#\mathbb G^*_{m}=2^{m})~\geq~1-\varepsilon
\end{equation*}
and therefore
\begin{equation}\label{Eq.Th.B1.1}
\mathbb{E}_{z,\sfB}\#\mathbb G^*_{m}~\geq~(1-\varepsilon)2^{m} ~>~\frac{4N}{\beta}
\end{equation}
for all $z\geq z_0$. Moreover, $\sum_{k\geq z_0}F_k(n,\sfA)\xrightarrow{\mathbb{P}^*} 1$
by part (a), whence
\begin{equation*}
\frac{\#\{v\in\G^*_n(\sfA,\sfB):Z_v\geq z_0\}}{\#\G^*_n(\sfA,\sfB)}\xrightarrow
{\mathbb{P}^*}1.
\end{equation*}
This together with Lemma \ref{Lemma.Proportion.B1} yields
\begin{equation*}
\frac{\#\{v\in\G^*_n(\sfA,\sfB):Z_v\geq z_0\}}{\#\G^*_n(\sfA)}\xrightarrow{\mathbb{P}^*}\beta
\end{equation*}
and thereupon
\begin{equation}\label{Eq.Th.B1.2}
\mathbb{P}^*\left(\frac{\#\{v\in\G^*_n(\sfA,\sfB):Z_v\geq z_0\}}{\#\G^*_n(\sfA)}\geq
\frac{\beta}{2}\right)\geq1-\varepsilon
\end{equation}
for all $n\geq n_0$ and some $n_{0}\in\N$. By combining \eqref{Eq.GA.nur ein m} and \eqref{Eq.Th.B1.2}, we now infer for all $n\geq n_0+m$
\begin{align*}
&\mathbb{P}^*\left(\frac{\#\mathbb G^*_n(\sfA)}{\#\mathbb G^*_n}>\frac{1}{N
+1}\right)\\
&\leq~\mathbb{P}^*\left(N\,\#\mathbb G^*_n(\sfA) > \sum_{v\in\{u\in\mathbb G^*_{n-1-
m}(\sfA,\sfB) : Z_u\geq z\}}\#\mathbb G^*_{m}(\sfB,v)\right)\\
&\leq~\mathbb{P}^*\left(\frac{2N}{\beta} > \frac{\sum_{v\in\{u\in\mathbb
G^*_{n-1-m}(\sfA,\sfB) : Z_u\geq z\}}\#\mathbb G^*_{m}(\sfB,v)}
{\displaystyle\#\{u\in\mathbb G^*_{n-1-m}(\sfA,\sfB):Z_u\geq z\}}\right)+\varepsilon\\
&\leq~\mathbb{P}^{\,n-m}\left(\frac{2N}{\beta} > \frac{\sum_{v\in\{u\in
\mathbb G^*_{n-1-m}(\sfA,\sfB):Z_u\geq z\}}\#\mathbb G^*_{m}(\sfB,v)}
{\displaystyle\#\{u\in\mathbb G^*_{n-1-m}(\sfA,\sfB):Z_u\geq z\}}\right)\frac
{\mathbb{P}(\mathcal{Z}_{n-m}(\sfA)>0)}{\mathbb{P}(\Surv(\sfA))}+\varepsilon\\
&\leq~\mathbb{P}^{\,n-m}\left(\frac{2N}{\beta} > \frac{\sum_{i=1}^{\#\{u
\in\mathbb G^*_{n-1-m}(\sfA,\sfB):Z_u\geq z\}}\mathcal G_{i,m}(z_0)}
{\displaystyle\#\{u\in\mathbb G^*_{n-1-m}(\sfA,\sfB):Z_u\geq z\}}\right)\frac
{\mathbb{P}(\mathcal{Z}_{n-m}(\sfA)>0)}{\mathbb{P}(\Surv(\sfA))}+\varepsilon\\
\end{align*}
where the $\mathcal G_{i,m}(z_0)$ are iid with the same law as $\#\{v\in\G^*_{m}(\sfB):Z_{\varnothing}=z_0, T_{\varnothing}=\sfB\}$. The LLN provides us with $n_1\geq n_0+m$ such that for all $n\geq n_1$
\begin{equation*}
\mathbb{P}^{n-m}\left(\frac{\sum_{i=1}^{\#\{u\in\mathbb G^*_{n-1-m}(\sfA,\sfB):
Z_u\geq z\}}\mathcal G_{i,m}(z_0)}{\#\{u\in\mathbb G^*_{n-1-m}(\sfA,
\sfB) :Z_u\geq z\}}\geq \mathbb{E}\mathcal G_{i,m}(z_0)/2\right)\geq 1-\varepsilon.
\end{equation*}
By combining this with \eqref{Eq.Th.B1.1}, we can further estimate in the above inequality
\begin{align*}
\mathbb{P}^*&\left(\frac{\#\mathbb G^*_n(\sfA)}{\#\mathbb G^*_n}>\frac{1}{N
+1}\right)\\
&\leq~\left(\mathbb{P}^{n-m}\left(\frac{2N}{\beta} > \mathbb{E}\mathcal G_{i,m}(z_0)/2 > \frac{2N}{\beta}\right) +\varepsilon\right)
\frac{\mathbb{P}(\mathcal{Z}_{n-m}(\sfA)>0)}{\mathbb{P}(\Surv(\sfA))}+\varepsilon\\
&=~\left(\frac{\mathbb{P}(\mathcal{Z}_{n-m}(\sfA)>0)}{\mathbb{P}(\Surv(\sfA))}+1\right)\varepsilon~\xrightarrow{n\to\infty}~2\varepsilon.
\end{align*}
This completes the proof of part (b).
As for (c), we will show that all conditions needed by Bansaye \cite{Bansaye:08} to prove his Theorem 5.2 are fulfilled. Our assertions then follow along the same arguments as provided there.
\textsc{Step 1:} $(\mu_{\sfB}^{-n}\mathcal{Z}_n(\sfB))_{n\ge 0}$ is a submartingale and converges a.s.\ to a finite limit $W(\sfB)$. The submartingale property follows from
\begin{align*}
&\mathbb{E}(\mathcal{Z}_{n+1}(\sfB)|\mathcal{Z}_n(\sfB))
~=~\mathbb{E}\left(\sum_{v\in\G^*_n}\big(Z_{v0}\1_{\{T_{v0}=\sfB\}}+Z_{v1}\1_{\{T_{v1}=\sfB\}}\big)\Bigg|\mathcal{Z}_n(\sfB)\right)\\
&=~\mathcal{Z}_n(\sfB)\mathbb{E}\left(X^{(0)}(\sfB)+X^{(1)}(\sfB)\right)+\mathbb{E}\left(\sum_{v\in\G^*_n(\sfA)}\hspace{-.2cm}\big(Z_{v0}\1_{\{T_{v0}=\sfB\}}+Z_{v1}\1_{\{T_{v1}=\sfB\}}\big)\Bigg|\mathcal{Z}_n(\sfB)\right)\\
&\geq~\mathcal{Z}_n(\sfB)\mu_{\sfB}
\end{align*}
for any $n\in\N$, while the a.s.\ convergence is a consequence of
\begin{equation*}
\sup_{n\in\N}\mathbb{E}\left(\frac{\mathcal{Z}_{n}(\sfB)}{\mu_{\sfB}^{n}}\right)~<~\infty
\end{equation*}
which, using our assumption $\gamma<\mu_{\sfB}$, follows from
\begin{align*}
\mathbb{E}\left(\frac{\mathcal{Z}_{n+1}(\sfB)}{\mu_{\sfB}^{n+1}}\right) ~&=~ \mathbb{E}\left(\frac{\mathcal{Z}_{n}(\sfB)}{\mu_{\sfB}^{n}}\right)+\mathbb{E}\left(\frac{1}{\mu_{\sfB}^{n+1}}\sum_{v\in\G^*_n(\sfA)}Z_{v0}\1_{\{T_{v0}=\sfB\}}+Z_{v1}\1_{\{T_{v1}=\sfB\}}\right)\\
&=~\mathbb{E}\left(\frac{\mathcal{Z}_{n}(\sfB)}{\mu_{\sfB}^{n}}\right)+\frac{1}{\mu_{\sfB}^{n+1}}\mathbb{E}\left(\mathcal{Z}_n(\sfA)\underbrace{\mathbb{E}(Z_{0}\1_{\{T_{0}=\sfB\}}+Z_{1}\1_{\{T_{1}=\sfB\}})}_{=:\mu_{\sfAB}}\right)\\
&=~\mathbb{E}\left(\frac{\mathcal{Z}_{n}(\sfB)}{\mu_{\sfB}^{n}}\right)+\frac{\mu_{\sfAB}}{\mu_{\sfB}}\left(\frac{\gamma}{\mu_{\sfB}}\right)^n\\
=...&=~\frac{\mu_{\sfAB}}{\mu_{\sfB}}\sum_{k=0}^n\left(\frac{\gamma}{\mu_{\sfB}}\right)^k\
\ \leq~\frac{\mu_{\sfAB}}{\mu_{\sfB}}\sum_{k=0}^{\infty}\left(\frac{\gamma}{\mu_{\sfB}}\right)^k~<~\infty
\end{align*}
for any $n\in\N$.
\textsc{Step 2:} $\{W(\sfB)=0\}=\mathbb{E}xt$ a.s.\\
The inclusion $\supseteq$ being trivial, we must only show that $\mathbb{P}(W(\sfB)>0)\geq\mathbb{P}(\Surv)$. For $i\ge 1$, let $(\mathcal{Z}_{i,n}(\sfB))_{n\ge 0}$ be iid copies of $(\mathcal{Z}_{n}(\sfB))_{n\ge 0}$ under $\mathbb{P}_{1,\sfB}$. Each $(\mathcal{Z}_{i,n}(\sfB))_{n\ge 0}$ forms a Galton-Watson process which dies out iff $\mu_{\sfB}^{-n}\mathcal{Z}_{i,n}(\sfB)\to 0$ (see \cite{Bansaye:08}). Then for all $m,N\in\N$, we obtain
\begin{align*}
\mathbb{P}(W(\sfB)>0) ~&=~\mathbb{P}\left(\lim_{n\to\infty}\frac{\mathcal{Z}_{m+n}(\sfB)}{\mu_{\sfB}^{m+n}}>0\right)\\
&\geq~\mathbb{P}\left(\lim_{n\to\infty}\frac{1}{\mu_{\sfB}^m}\sum_{i=1}^{\mathcal{Z}_{m}(\sfB)}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}>0\right)\\
&\geq~\mathbb{P}\left(\lim_{n\to\infty}\frac{1}{\mu_{\sfB}^{m}}\sum_{i=1}^{\mathcal{Z}_{m}(\sfB)}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}>0, \mathcal{Z}_m(\sfB)\geq N\right)\\
&\geq~\mathbb{P}\left(\lim_{n\to\infty}\sum_{i=1}^{N}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}>0, \mathcal{Z}_{m}(\sfB)\geq N\right)\\
&\geq~\mathbb{P}\left(\mathcal{Z}_m(\sfB)\geq N\right)-\mathbb{P}_{1,\sfB}\left(\lim_{n\to\infty}\sum_{i=1}^{N}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}=0\right)\\
&=~\mathbb{P}\left(\mathcal{Z}_{m}(\sfB)\geq N\right)-\mathbb{P}_{1,\sfB}\left(\lim_{n\to\infty}\frac{\mathcal{Z}_{n}(\sfB)}{\mu_{\sfB}^{n}}=0\right)^N\\
&=~\mathbb{P}\left(\mathcal{Z}_{m}(\sfB)\geq N|\Surv\right)\,\mathbb{P}(\Surv)-\mathbb{P}_{1,\sfB}\left(\mathbb{E}xt\right)^N.
\end{align*}
and then, upon letting $m$ and $N$ tend to infinity,
\begin{equation*}
\mathbb{P}(W(\sfB)>0)~\geq~\mathbb{P}(\Surv)
\end{equation*}
because $\mathbb{P}_{1,\sfB}\left(\mathbb{E}xt\right)<1$ and by Theorem \ref{Satz.ExplosionInfZellen}.
\textsc{Step 3:} $\sup_{n\ge 0}\mathbb{E}\xi_{n}<\infty$, where $\xi_{n}:=\left(\mu_{\sfB}/2\right)^{-n} Z_{[n]}$.
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\textsc{Step 2:} $\{W(\sfB)=0\}=\mathbb{E}xt$ a.s.\\
The inclusion $\supseteq$ being trivial, we must only show that $\mathbb{P}(W(\sfB)>0)\geq\mathbb{P}(\Surv)$. For $i\ge 1$, let $(\mathcal{Z}_{i,n}(\sfB))_{n\ge 0}$ be iid copies of $(\mathcal{Z}_{n}(\sfB))_{n\ge 0}$ under $\mathbb{P}_{1,\sfB}$. Each $(\mathcal{Z}_{i,n}(\sfB))_{n\ge 0}$ forms a Galton-Watson process which dies out iff $\mu_{\sfB}^{-n}\mathcal{Z}_{i,n}(\sfB)\to 0$ (see \cite{Bansaye:08}). Then for all $m,N\in\N$, we obtain
\begin{align*}
\mathbb{P}(W(\sfB)>0) ~&=~\mathbb{P}\left(\lim_{n\to\infty}\frac{\mathcal{Z}_{m+n}(\sfB)}{\mu_{\sfB}^{m+n}}>0\right)\\
&\geq~\mathbb{P}\left(\lim_{n\to\infty}\frac{1}{\mu_{\sfB}^m}\sum_{i=1}^{\mathcal{Z}_{m}(\sfB)}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}>0\right)\\
&\geq~\mathbb{P}\left(\lim_{n\to\infty}\frac{1}{\mu_{\sfB}^{m}}\sum_{i=1}^{\mathcal{Z}_{m}(\sfB)}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}>0, \mathcal{Z}_m(\sfB)\geq N\right)\\
&\geq~\mathbb{P}\left(\lim_{n\to\infty}\sum_{i=1}^{N}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}>0, \mathcal{Z}_{m}(\sfB)\geq N\right)\\
&\geq~\mathbb{P}\left(\mathcal{Z}_m(\sfB)\geq N\right)-\mathbb{P}_{1,\sfB}\left(\lim_{n\to\infty}\sum_{i=1}^{N}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}=0\right)\\
&=~\mathbb{P}\left(\mathcal{Z}_{m}(\sfB)\geq N\right)-\mathbb{P}_{1,\sfB}\left(\lim_{n\to\infty}\frac{\mathcal{Z}_{n}(\sfB)}{\mu_{\sfB}^{n}}=0\right)^N\\
&=~\mathbb{P}\left(\mathcal{Z}_{m}(\sfB)\geq N|\Surv\right)\,\mathbb{P}(\Surv)-\mathbb{P}_{1,\sfB}\left(\mathbb{E}xt\right)^N.
\end{align*}
and then, upon letting $m$ and $N$ tend to infinity,
\begin{equation*}
\mathbb{P}(W(\sfB)>0)~\geq~\mathbb{P}(\Surv)
\end{equation*}
because $\mathbb{P}_{1,\sfB}\left(\mathbb{E}xt\right)<1$ and by Theorem \ref{Satz.ExplosionInfZellen}.
\textsc{Step 3:} $\sup_{n\ge 0}\mathbb{E}\xi_{n}<\infty$, where $\xi_{n}:=\left(\mu_{\sfB}/2\right)^{-n} Z_{[n]}$.
\noindent
First, we note that $(Z_{[n]})_{n\ge 0}$, when starting with a $\sfB$-cell hosting one parasite (under $\mathbb{P}_{1,\sfB}$), is a BPRE with mean $\mu_{\sfB}/2$ (see \cite{Bansaye:08}). Second, we have
\begin{equation*}
\mathbb{E} Z_{[n]}\1_{\{T_{[n]}=\sfA\}} ~=~ \mathbb{P}(T_{[n]}=\sfA)\,\mathbb{E} Z_{n}(\sfA) ~=~ \left(\frac{\gamma}{2}\right)^n
\end{equation*}
and thus
\begin{align*}
\mathbb{E} Z_{[n]} ~&=~ \mathbb{E} Z_{[n]}\1_{\{T_{[n]}=\sfA\}}+\sum_{m=0}^{n-1}\mathbb{E} Z_{[n]}\1_{\{T_{[m]}=\sfA,T_{[m+1]}=\sfB\}}\\
&=~\left(\frac{\gamma}{2}\right)^n+\sum_{m=0}^{n-1}\mathbb{E} Z_{[m]}\1_{\{T_{[m]}=\sfA\}}\,\mathbb{E}_{1,\sfA}Z_{[1]}\1_{\{T_{[1]}=\sfB\}}\,\mathbb{E}_{1,\sfB}Z_{[n-m-1]}\\
&=~\left(\frac{\gamma}{2}\right)^n+\eta\sum_{m=0}^{n-1}\left(\frac{\gamma}{2}\right)^m\left(\frac{\mu_{\sfB}}{2}\right)^{n-m-1}
\end{align*}
for all $n\in\N$, where $\eta:=\mathbb{E}_{1,\sfA}Z_{[1]}\1_{\{T_{[1]}=\sfB\}}$. This implies
\begin{equation}\label{Eq.BPRE.l1beschr}
\sup_{n\in\N}\mathbb{E}\xi_n ~=~ \left(\frac{\gamma}{\mu_{\sfB}}\right)^n+\frac{2\eta}{\mu_{\sfB}}\sum_{m=0}^{n-1}\left(\frac{\gamma}{\mu_{\sfB}}\right)^m~\leq~c\sum_{m=0}^{\infty}\left(\frac{\gamma}{\mu_{\sfB}}\right)^m~<\infty
\end{equation}
for some $c<\infty$.
\textsc{Step 4:} $\lim_{K\to\infty}\sup_{n\ge 0}\mathbb{E}\xi_n\1_{\{Z_{[n]}\geq K\}}=0$.
\noindent
By our assumptions, $(Z_{[n]})_{n\ge 0}$, when starting in a $\sfB$-cell with one parasite, is a strongly subcritical BPRE with mean $\mu_{\sfB}/2$ (see \cite{Bansaye:08}). Hence, by \cite[Corollary 2.3]{Afanasyev+etal:05},
\begin{equation}\label{Eq.B1c.gi}
\lim_{K\to\infty}\sup_{n\ge 0}\mathbb{E}_{1,\sfB}\xi_n\1_{\{Z_{[n]}> K\}} ~=~0,
\end{equation}
which together with \eqref{Eq.BPRE.l1beschr} implies for $n,m\in\N$
\begin{align*}
\lim_{K\to\infty}\sup_{n\ge 0}\mathbb{E}\, &\xi_{n+m}\1_{\{Z_{[n+m]}>K\}}\\
~&\leq~ \lim_{K\to\infty}\sup_{n\ge 0}\mathbb{E}\,\xi_{n+m}\1_{\{Z_{[n+m]}>K\}}\1_{\{T_{[m]}=\sfB\}}+\sup_{n\ge 0}\mathbb{E}\,\xi_{n+m}\1_{\{T_{[m]}=\sfA\}}\\
&\leq~ \lim_{K\to\infty}\sup_{n\ge 0}\mathbb{E}_{1,\sfB}\xi_n\1_{\{Z_{[n]}>K\}}\,\mathbb{E} \xi_m+\mathbb{E} \xi_m\1_{\{T_{[m]}=\sfA\}}\,\sup_{n\in\N}\mathbb{E}\,\xi_n\\
&\leq~ \left(\frac{\gamma}{\mu_{\sfB}}\right)^m\sup_{n\in\N}\mathbb{E}\,\xi_n
\end{align*}
and the last expression can be made arbitrarily small by choosing $m$ sufficiently large, for $\gamma<\mu_{\sfB}$. This proves Step 4.
\textsc{Final step:}
Having verified all conditions needed for the proof of Theorem 5.2 in \cite{Bansaye:08}, one can essentially follow his arguments to prove Theorem \ref{Satz.Proportion.B1}(c). We refrain from supplying all details here and restrict ourselves to an outline of the main ideas. First use
what has been shown as \textsc{Step 1 - 4} to prove an analogue of \cite[Lemma 6.5]{Bansaye:08}, i.e. \textit{(control of filled-in cells)}
\begin{equation}\label{control.filled.in.cells}
\lim_{K\to\infty}\sup_{n,q\geq0}\mathbb{P}^*\left(\frac{\#\{v\in\G^*_{n+q}(\sfB):Z_{v|n}>K\}}{\#\G^*_{n+q}(\sfB)}\geq\eta\right)=0
\end{equation}
for all $\eta>0$, and of \cite[Prop.\ 6.4]{Bansaye:08}, i.e. \textit{(separation of descendants of parasites)}
\begin{equation}\label{sep.of.descendants}
\lim_{q\to\infty}\sup_{n\geq0}\mathbb{P}^*\left(\frac{\#\{v\in\G^*_{n+q}(\sfB):Z_{v|n}\leq K, N_n(v)\geq2\}}{\#\G^*_{n+q}(\sfB)}\geq\eta\right)=0
\end{equation}
for all $\eta>0, K\in\N$, where $N_n(v)$ denotes the number of parasites in cell $v|n$ with at least one descendant in cell $v$. In particular, \eqref{control.filled.in.cells} (with $q=0$) combined with $\#\G^*_n(\sfB)\to\infty$ $\mathbb{P}^*$-a.s. implies the existence of a $K_0\geq0$ such that for all $N\in\N$
\begin{equation}\label{concentration.of.parasites}
\lim_{n\to\infty}\inf_{K\geq K_0}\mathbb{P}^*\left(\sum_{v\in\G^*_n(\sfB)}Z_v\1_{\{Z_v\leq K\}}\geq N\right) = 1.
\end{equation}
Using \eqref{control.filled.in.cells} and \eqref{sep.of.descendants}, we infer that, for all $\eta, \varepsilon>0$, there exist $K_1\geq K_0$ and $q_0\in\N$ such that for all $n\in\N$
\begin{equation*}
\mathbb{P}^*\Bigg(\Bigg|F_k(n+q_0,\sfB)-\underbrace{\frac{\#\{v\in\G^*_{n+q_0}(\sfB) | Z_v=k, Z_{v|n}\leq K_1, N_n(v)=1\}}{\#\{v\in\G^*_{n+q_0}(\sfB) | Z_{v|n}\leq K_1, N_n(v)=1\}}}_{=: J_{n}}\Bigg|\geq\eta\Bigg)~\leq~\varepsilon.
\end{equation*}
Since $\#\G^*_n(\sfA)/\#\G^*_n(\sfB)\xrightarrow{\mathbb{P}^*}0$, we further get
\begin{align*}
J_{n} ~&\overset{\mathbb{P}^*}{{\underset{n\to\infty}{\simeq}}}~ \frac{\#\{v\in\G^*_{n+q_0}(\sfB) | Z_v=k, Z_{v|n}\leq K_1, T_{v|n}=\sfB, N_n(v)=1\}}{\#\{v\in\G^*_{n+q_0}(\sfB) | Z_{v|n}\leq K_1, T_{v|n}=\sfB, N_n(v)=1\}}
\end{align*}
as $n\to\infty$, which puts us in the same situation as in the proof of \cite[Thm.\ 5.2]{Bansaye:08}. Now, by using \eqref{concentration.of.parasites} and the LLN, we can identify the limit of $J_n$ which is in fact the same as in Step 1 of the proof of \cite[Thm.\ 5.2]{Bansaye:08}. A reproduction of the subsequent arguments from there finally establishes the result.
\end{Beweis}
\section*{Acknowledgment}
We are indebted to Joachim Kurtz (Institute of Evolutionary Biology, University of M\"unster)
for sharing with us his biological expertise of host-parasite coevolution and many fruitful discussions that helped us to develop the model studied in this paper.
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\section{Glossary}
\begin{tabular}[c]{p{0.25\textwidth} p{0.75\textwidth}}
$\T$ & cell tree\\
$\G_n$ & set of cells in generation $n$\\
$\G_n(\sft)$ & set of cells of type $\sft$ in gegeration $n$\\
$\G^*_n$ & set of contaminated cells in generation $n$\\
$\G^*_n(\sft)$ & set of contaminated cells of type $\sft$ in generation $n$\\
$T_v$ & type of cell $v$\\
$p_{\sfs}$ & probability that the daughter cell of an $\sfA$-cell is of type $\sfs$\\
$p_0$ & probability that the $1^{st}$ daughter cell of an $\sfA$-cell
is of type $\sfA$\\
$p_1$ & probability that the $2^{nd}$ daughter cell of an $\sfA$-cell
is of type $\sfA$\\
$\nu$ & mean number of type-$\sfA$ daughter cells of an $\sfA$-cell\\
$(X^{(0)}(\sfA,\sfs),X^{(1)}(\sfA,\sfs))$ & offspring numbers of an $\sfA$-parasite with
daughter cells of type $\sfs\in\{\sfAA,\sfAB,\sfBB\}$\\
$(X^{(0)}(\sfB),X^{(1)}(\sfB))$ & offspring numbers of a $\sfB$-parasite\\
$Z_v$ & number of parasites in cell $v$\\
$\mu_{i,\sft}(\sfs)$ & mean number of offspring of a $\sft$-parasite which goes in
daughter cell $i\in\{0,1\}$ if daughter cells are of type
$\sfs\in\{\sfAA,\sfAB,\sfBB\}$\\
$\mu_{i,\sfB}$ & mean offspring number of $\sfB$-parasites which go in
daughter cell $i\in\{0,1\}$\\
$\mu_{\sfB}$ & reproduction mean of a parasite in a $\sfB$-cell\\
$\mathcal{Z}_n$ & number of parasites in generation $n$\\
$\Z_n(\sft)$ & number of parasites in $\sft$-cells in generation $n$\\
$\mathbb{E}xt/\Surv$ & event of extinction/survival of parasites\\
$\mathbb{E}xt(\sft)/\Surv(\sft)$ & event of extinction/survival of $\sft$-parasites\\
$Z_{[n]}$ & number of parasites in a random cell in generation $n$\\
$Z_n(\sfA)$ & number of parasites of a random $\sfA$-cell in generation $n$\\
$f_{n}(s|\Lambda),\ f_{n}(s)$ &quenched and annealed generating function of $Z_{n}(\sfA)$, respectively\\
$g_{\Lambda_n}(s)$ & generating function giving the $n$-th reproduction law of the
process of a random $\sfA$ cell line\\
$\gamma$ & mean number of offspring of an $\sfA$-parasite which go in an $\sfA$-cell\\
$\hat\gamma$ & $:=\nu\,\mathbb{E} g^{\prime}_{\Lambda_1}(1)^2
=p_{\sfAA}\left(\mu^2_{0,\sfA}(\sfAA)+\mu^2_{1,\sfA}(\sfAA)\right)
+p_{\sfAB}\,\mu^2_{0,\sfA}(\sfAB)$\\
$\mathbb{P}_{z,\sft}$ & probability measure under which the process starts with one $\sft$-cell
containing $z$ parasites\\
$\mathbb{P}^*_{z,\sft}$ & the same as before but conditioned upon $\Surv(\sfA)$\\
$\mathbb{P}^n_{z,\sft}$ & the same as before but conditioned upon survival of
$\sfA$-parasites in generation $n$
\end{tabular}
\input{host-parasite.bbl}
\end{document}
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\begin{document}
\mkcoverpage
\setcounter{page}{1}
\title{Heuristics for Packing Semifluids}
\author{Jo{\~a}o Pedro Pedroso}
\date{June 2015}
\maketitle
\begin{abstract}
Physical properties of materials are seldom studied in the context of packing problems. In this work we study the behavior of semifluids: materials with particular characteristics, that share properties both with solids and with fluids. We describe the importance of some specific semifluids in an industrial context, and propose methods for tackling the problem of packing them, taking into account several practical requirements and physical constraints. Although the focus of this paper is on the computation of practical solutions, it also uncovers interesting mathematical properties of this problem, which differentiate it from other packing problems.
\ \\
\noindent \textbf{Keywords:}
Packing; Semifluid; Heuristics; Tree search.
\end{abstract}
\section{Introduction}
\label{sec:intro}
Semifluids are materials having characteristics of both fluids and solids. In the context of this paper, we will consider materials that cannot flow in one direction, though they are fluid in the other directions. As an example, consider tubes, which correspond to the industrial origin of this problem. Placed in a container, they can flow in the directions perpendicular to their length, but \emph{not} in the direction of their length (see Figure~\ref{fig:pipes1}). Assuming the tubes will be positioned perpendicularly to the Cartesian axes, depending on the direction of their placement they will flow either in the $x$ or in the $y$ dimension. Pipes, having positive radii, are imperfect semifluids, as they will not fully occupy the space available in the $z$ dimension; however, they approximate a perfect fluid as the radii becomes smaller. We will consider that the material is a perfect semifluid, and hence the volume occupied is constant and divisible.
This paper describes several possibilities for packing semifluids in a container, and presents heuristics for the variant which closer corresponds to an industrial application.
\section{Problem description}
\label{sec:problem}
Even though packing problems may be generalized into a single problem, they are usually divided in two categories: minimizing the number of bins, and maximizing the load to pack in a bin (see, \emph{e.g.}, \cite{Baldi20121205}). Given an index set $\mathcal{S}$ of semifluid items, with each item $i$ characterized by a fixed length $\ell_i$ and a volume $v_i$, and dimensions $D, W, H$ of containers, these two variants for the problem of packing a semifluid are:
\begin{enumerate}
\item bin packing variant: find the minimum number of containers to accommodate all the items;
\item knapsack variant: given, additionally, a value $w_i$ for the available volume $v_i$ of each item $i$, find the packing of maximum value that can be inserted in a container.
\end{enumerate}
In this paper we will focus on the knapsack variant.
\begin{figure}
\caption{A container accommodating a semifluid: tubes (left); coordinate system used (right).}
\label{fig:pipes1}
\end{figure}
\subsection{Semifluid packing problems}
There are several possibilities for packing a semifluid orthogonally in a container, as shown in Figure~\ref{fig:pipes2}. Both the length $\ell$ (corresponding to the length of the tubes) and the volume $v$ occupied by the semifluid are constant; in this figure, this means that $a \times b \times \ell = c \times d \times \ell = v$. Assuming that, except the container itself, there are no walls, a semifluid will take on all the available horizontal space in the direction where it freely flows. In the case presented, $a$ would take the depth $D$ of the container, and $c$ would take its width $W$, and hence the corresponding heights are $b = \frac{v}{D \ell}$ and $d = \frac{v}{W \ell}$.
\begin{figure}
\caption{Two possibilities for accommodating a semifluid in a container.}
\label{fig:pipes2}
\end{figure}
After a semifluid is placed, others may be put on top of it, but they must not protrude (as detailed next). Hence, one may think of the space above a semifluid as a ``container'', which can be filled up with the same rules as the original container; in this sense, this is a recursive problem.
Depending on the application, it may be allowed or not that, when packing a semifluid, it overflows others previously packed, as illustrated in Figure~\ref{fig:pipes3}. In general, allowing overflow makes packing solutions more difficult to implement in practice, and brings the problem more difficult to tackle; overflow will not be considered here.
\begin{figure}
\caption{Packing a semifluid without overflowing another previously packed (left), and overflowing it (right).}
\label{fig:pipes3}
\end{figure}
We will focus on packing semifluids by positioning the fixed dimension parallel to the $x$ axis, as shown in Figure~\ref{fig:pipes1}. This is the relevant variant when the container must be loaded from a lateral door at $x=D$: if the semifluids were rotated and be placed along the $y$ axis, they would flow out of the door.
An important, practical packing rule restricts what can be placed on top of what. Indeed, for cargo stability and for facilitating loading, it is usually acceptable that shorter tubes are placed on top of longer tubes, but not the inverse; more precisely, there must be no holders protruding with respect to holders below them.
In semifluid packing, any fraction of an item's available volume may be packed; this is major difference with respect to other packing problems.
We call the problem of maximizing the value of semifluids packed in the container in these conditions the \emph{basic semifluid packing problem}.
\subsection{Background}
\label{sec:background}
Three-dimensional packing has recently been studied under several different perspectives; a recent survey can be found in~\cite{Crainic2012}. The problem of allocating a given set of three-dimensional rectangular items to the minimum number of identical finite bins without overlapping has been addressed with tabu search in~\cite{Lodi2002410}: items are packed in several layers, the floor of the container being the first. A heuristic method for the situation where there is no requirement for packed boxes to form flat layers, keeping track of empty space seen from different perspectives and using a look-ahead scheme for positioning, is presented in~\cite{Lim2003471}.
However, the nature of the basic semifluid packing problem is rather different of these three-dimensional packing problems.
As will be seen later, there is more similarity between our problem and two-dimensional cutting. The most closely related problem is the orthogonal two-dimensional knapsack problem with guillotine patterns. Methods for tackling this problem are often based on a discretization of possible positions for the rectangles in the Cartesian plane (see, \emph{e.g.}, \cite{Puchinger2007,Elsa2010,dolatabadi2012}). A different approach is proposed in~\cite{fekete2007}, providing an exact algorithm for higher-dimensional orthogonal packing; the algorithm is based on bounding procedures which make use of dual feasible functions, within a tree search procedure. With respect to these problems, semifluid packing has the property that it is not required to pack all the available volume of each item; in rectangle packing, this would correspond to being able to cut some of the rectangles at the time of packing. Another difference between semifluid packing and previously studied problems concerns the requirement of no protuberance of items above others; this requirement is naturally respected in two-staged guillotine cuts, but usually is not enforced in general guillotine patterns.
To the best of our knowledge, basic semifluid packing or equivalent problems have not been studied before.
\subsection{Mathematical model}
We are not aware of previous attempts to formulate the semifluid packing problem as a mathematical optimization model, but there are some related problems. Integer programming models for two-dimensional two-stage bin packing problem have been proposed
in~\cite{Lodi2004} and extended by~\cite{Puchinger2007} to the three-stage problem. In both cases, decision variables are related to the assignment of the items to bins, stripes or stacks. Models for the related cutting stock problem, providing better linear relaxation bounds, are presented in~\cite{Elsa2010}, where a set of small rectangular items of given sizes is to be cut from a set of larger rectangular plates, in such a way that the total number of used plates is minimized. Despite some similarities, none of these models is adequate for our problem, mainly for two reasons: in semifluid packing the number of stages is in general much larger, and items may be partially assigned to a position (\emph{i.e.}, a position may hold a fraction of the available volume of an item).
The formulation proposed next is not compact, as it requires an exponential number of variables; however, it hopefully conveys the characteristics of the problem. For the sake of clarity, we start with a simplified model, and later describe how it could be extended to the general case; the simplification consists of assuming that only one stack of each item is allowed on each layer. A layer, in this context, is either the floor of the container or the space above a previously packed item. Figures \ref{fig:example} and~\ref{fig:instances} may be of help for visualizing the model.
The first set of binary variables indicates which items are packed in the first layer: $y_i=1$ if item $i$ is packed directly on the container, $y_i=0$ otherwise. To each variable $y_i$ there is a corresponding continuous variable $0 \leq x_i \leq 1$ which represents the fraction of item $i$ being packed at this place. Before introducing more variables, let us specify a constraint related to the length $D$ of the container, which limits the length of items packed in this layer:
\begin{alignat*}{27}
& \sum_i \ell_i y_i \leq D.
\end{alignat*}
Variable $x_i$ may be positive only if $y_i=1$:
\begin{alignat*}{27}
& x_i \leq y_i, && \quad \forall i.
\end{alignat*}
The height of the first layer is limited by the height of the container:
\begin{alignat*}{27}
& h_i x_i \leq H, && \quad \forall i,
\end{alignat*}
where $h_i = v_i/W$ is the total height that item $i$ would take on a container of width~$W$ (this will be later be replaced by a stronger constraint).
We now introduce variables concerning the placement of items $j$ on the second layer, \emph{i.e.}, directly above some previously packed item~$i$. Variables $y_{ij}$ are the indicators for this, and the corresponding $x_{ij}$ represent the fraction of $j$ packed at this place. The solution must, therefore, observe:
\begin{alignat*}{27}
& y_{ij} \leq y_{i}, && \quad \forall i,j,\\
& x_{ij} \leq y_{ij}, && \quad \forall i,j,\\
& \sum_{j} \ell_{ij} y_{ij} \leq \ell_{i} y_{i}, && \quad \forall i,
\end{alignat*}
where the last constraint limits the length of items placed directly above item~$i$. For each pair $i,j$ the height of the corresponding stack is limited to the height of the container:
\begin{alignat*}{27}
& h_i x_i + h_j x_{ij} \leq H, && \quad \forall i,j.
\end{alignat*}
The fraction of $i$ used in the two first layers is limited by one (this and the previous constraints will later be extended):
\begin{alignat*}{27}
& 0 \leq x_{i} + \sum_{j} x_{ji} \leq 1.
\end{alignat*}
We now have all the components to complete the model, by extending the number of layers. Notice that, as layers cannot protrude and items with identical length should be placed by decreasing value, there may be at most $N$ layers, where $N$ is the number of items. We may assume that the items are reversely ordered by length, \emph{i.e.}, $\ell_1 \geq \ell_2 \geq \ldots \geq \ell_N$; this allows us to define variables with indices $i,i'$ only for $i'>i$. Notice also that the number of indices indicates the level at which the item corresponding to a variable is being packed:
variables for layer $1\leq K\leq N$ will have $K$ indices $i,j,\ldots,m,n$, with $i < j < \ldots < m < n$.
The entire model is presented in Figure~\ref{fig:model}.
\begin{figure}
\caption{Mathematical optimization model.}
\label{eq:obj}
\label{eq:totalx}
\label{eq:D}
\label{eq:Dn}
\label{eq:x1}
\label{eq:xn}
\label{eq:y1}
\label{eq:yn}
\label{eq:H}
\label{eq:beginvar}
\label{eq:endvar}
\label{fig:model}
\end{figure}
Equations~(\ref{eq:totalx}) determines the total quantity of item $i$ packed, which allows determining the total value packed in~(\ref{eq:obj}).
Constraints (\ref{eq:D}) to~(\ref{eq:Dn}) guarantee that the total length of what is packed on top of the container, or of a packed item, does exceed the respective lengths.
Constraints (\ref{eq:x1}) to~(\ref{eq:xn}) allow a positive quantity of an item to packed only if the corresponding indicator variable is equal to~1.
Constraints (\ref{eq:y1}) to~(\ref{eq:yn}) allow packing only on top of previously packed items.
Inequalities (\ref{eq:H}) determines the height of all stacks, and limits it to the height of the container.
Finally, (\ref{eq:beginvar}) to~(\ref{eq:endvar}) define the domain for each of the variables.
This model is rather clumsy, but it is not yet complete: it does not take into account the possibility of packing several stacks of each item on a given layer. For make the model complete one would have to create, for each layer and each compatible item, a number of variables equal to the number of times that item would fit in the layer, if it was packed alone. It is obvious that direct usage of this model is implausible, except for a rather small number of items; realistic usage would require a column generation approach.
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\section{Heuristic and complete search}
\label{sec:heur}
For solving the basic semifluid packing problem, we firstly propose a heuristic method --- which will later be improved --- for dividing a container into smaller parallelepipeds, which we call \emph{holders}. Each holder has a fixed depth, determined by the length of the semifluid it will accommodate. Due to the possibility for the semifluid to flow downwards, along the $z$ dimension, and also along the $y$ dimension, a semifluid will fully use the width of the physical container. The height of a filled holder is determined either by the volume of its semifluid or by the height of the physical container; in the latter case, the semifluid left over will possibly be packed in a different holder.
In this situation, one may think of the packing process as a division of, say, the container's wall at $y=0$, into rectangles. Each rectangle corresponds to the volume of a particular item when projected into the $y=0$ plane. For example, consider the placement of a semifluid as in Figure~\ref{fig:pipes1}; a projection of the volume occupied is represented as a rectangle, alike~1 in the left diagram of Figure~\ref{fig:example}. Upon placing this item, the container is divided into three partitions: one where item~1 is held (which is \emph{closed}, in the sense that it may not be used for other items), and the \emph{open} holders above (A) and besides the item~(B). Upon placing three more items in this example, the open holders are A, B, C, D in the right diagram of Figure~\ref{fig:example}.
\begin{figure}
\caption{Section of a container through the $y=0$ plane: open holders (shaded) after placing one item (left), and after placing four items (right).}
\label{fig:example}
\end{figure}
\subsection{Simple packing}
\label{packing}
A heuristic method for packing semifluids in these conditions can hence be though of as the process of choosing an item to pack, and an open holder for putting it (if some is available). For a semifluid of length $\ell$, candidate holders $j$ must have depth $D_j \geq \ell$. If the volume of a semifluid does not completely fit in the selected holder, the full height of the holder will be used (as for item 4 in the right diagram of Figure~\ref{fig:example}), and the remaining fluid is left to (possibly) pack later.
Given the characteristics of this problem, one might think of adapting known heuristics for bin packing and knapsack problems, as has been done for the two-dimensional knapsack problem (see, \emph{e.g.},~\cite{coffman1980,dolatabadi2012}); however, the geometric constraint forbidding longer lengths on top of shorter leads to possibly unexpected performance, as we will see shortly. Several alternative heuristic rules have been tried:
\begin{enumerate}
\item Best fit (BF): select the item/holder pair $(i,j)$ which leads to the minimum difference $D_j - \ell_i$, \emph{i.e.}, which leads to minimum currently unused space along~$x$;
\item Longest item first, first fit (LFF): select the longest item that can be packed in some open holder (\emph{i.e.}, item $i$ with largest $\ell_i$ for which there exists a holder $j$ such that $D_j - \ell_i \geq 0$), and insert it in the last open holder where it fits;
\item Longest item first, best fit (LBF): as LFF, but select the \emph{smallest} open holder in which the item fits;
\item Worthiest item first, first fit (WFF): as LFF, but select most valuable items (per unit volume) first;
\item Worthiest item first, best fit (WBF): as LBF, but select most valuable items first.
\end{enumerate}
\begin{algorithm}[htbp]
\begin{footnotesize}
\DontPrintSemicolon
\SetKwFunction{algo}{algo}\SetKwFunction{pack}{pack}
\SetKwFunction{algo}{algo}\SetKwFunction{h}{h}
\KwData{instance:
\begin{itemize}
\item set $\mathcal{S}$ of items to pack\;
\item item's length $\ell_i$, volume $v_i$, and value $w_i$, $\forall i \in \mathcal{S}$\;
\item physical container's width $W$, height $H$, and depth $D$;
\end{itemize}
}
\KwResult{
\begin{itemize}
\item set of holders $\mathcal{H}$ and their dimensions and position inside the container;
\item for each item $i$, the set $x_i$ of holders where it is packed.
\end{itemize}
}
\SetKwProg{myproc}{procedure}{}{}
\SetKw{Break}{break}
\myproc{\pack{$D, W, H, \mathcal{S}, \ell, v, w$}}{
$x_i \leftarrow \{\}, \quad \forall i \in \mathcal{S}$ \tcp*{initialize holders packing item $i$ as empty sets}
$\mathcal{H} \leftarrow \{$holder with dimensions $D \times W \times H\}$ \tcp*{open main holder}
\While{some item in $\mathcal{S}$ fits in an holder in $\mathcal{H}$}{
$(i,j) \leftarrow \h(\mathcal{S},\mathcal{H},\ell,v,w)$ \tcp*{heuristic choice of item $i$ and holder $j$} \label{alg:rule}
let $D_j, W_j, H_j$ be the current dimensions of holder $j$ \;
$z \leftarrow v_i / (\ell_i W_j)$ \;
\If(\tcp*[f]{all volume of $i$ fits}){$z \leq H_j$}{
$v_i \leftarrow 0$\;
$\mathcal{S} \leftarrow \mathcal{S} \setminus \{i\}$ \;
$(D_j,W_j,H_j) \leftarrow (\ell_i,W_j,z)$ \tcp*{adjust $j$'s dimensions}
}
\Else{
$v_i \leftarrow (v_i - \ell_i W_j H_j)$ \tcp*{update volume of $i$ remaining unpacked}
$(D_j,W_j,H_j) \leftarrow (\ell_i,W_j,H_j)$ \tcp*{adjust $j$'s dimensions}
}
$x_i \leftarrow x_i \cup \{j\}$ \tcp*{add $j$ to set of holders packing $i$}
$\mathcal{H} \leftarrow \mathcal{H} \setminus \{j\}$ \tcp*{remove $j$ from open holders}
\If{$D_j > \ell_i$}{
$\mathcal{H} \leftarrow \mathcal{H} \cup \{$holder with dimensions $(D_j-\ell_i) \times W_j \times H_j\}$ \tcp*{open holder besides $j$}
}
\If{$H_j > z$}{
$\mathcal{H} \leftarrow \mathcal{H} \cup \{$holder with dimensions $\ell_i \times W_j \times (H_j-z)\}$ \tcp*{open holder on top of $j$}
}
}
\Return{$x$}\;
}
\end{footnotesize}
\caption{Simple heuristic method for packing semifluids.}
\label{alg:pack}
\end{algorithm}
These rules are used in the heuristic method detailed in Algorithm~\ref{alg:pack}; we are abusing of notation, by allowing items and holders to be represented also by indices in their respective sets. The algorithm returns a map associating each item to the set of holders that contain it (which is empty for items that are not packed). The heuristic rule to be used is specified in line~\ref{alg:rule}, and holders are created accordingly in the subsequent lines. The algorithm iterates as long as there is an open holder where some unpacked item fits.
The full description of the computational setup is deferred to Section~\ref{sec:results}; for the time being, we just present in Table~\ref{tab:pack} a comparison of the solutions obtained with these simple rules on a set of 3000 test instances. We have counted the number of times that heuristic construction with a rule is \emph{strictly} better than with another, for all the combinations.
\begin{table}[h!tbp]
\centering
\caption{Comparison of simple rules for a data set of 3000 instances. Left table: $n_{ij}$, the number of times rule $i$ was strictly better (\emph{i.e.}, found a better solution) than rule $j$. Right table: $n_{ij} - n_{ji}$; positive values mean that rule on line $i$ is better for more instances than the rule in column $j$.}
\label{tab:pack}
\begin{tabular}{l|*{5}{r@{~~~~}}}
& BF & LFF & LBF & WFF & WBF \\\hline
BF & 0 & 225 & 51 & 2526 & 2397 \\
LFF & 338 & 0 & 101 & 2525 & 2396 \\
LBF & 338 & 237 & 0 & 2529 & 2404 \\
WFF & 339 & 342 & 336 & 0 & 91 \\
WBF & 398 & 401 & 391 & 1737 & 0 \\
\end{tabular}
~~~~~
\begin{tabular}{l|*{5}{r@{~~~~}}}
& BF & LFF & LBF & WFF & WBF \\\hline
BF & 0 & -113 & -287 & 2187 & 1999 \\
LFF & 113 & 0 & -136 & 2183 & 1995 \\
LBF & 287 & 136 & 0 & 2193 & 2013 \\
WFF & -2187 & -2183 & -2193 & 0 & -1646 \\
WBF & -1999 & -1995 & -2013 & 1646 & 0 \\
\end{tabular}
\end{table}
The results obtained are rather surprising: rules based on the value of the items, very effective for the knapsack problem, are clearly outclassed by rules based on the length of the semifluid. The simple rule of selecting the longest semifluid, independently of its value, and placing it in the open holder that leads to less used space along the $x$ axis (LBF) has generated the best results. This is the heuristic rule selected for comparison with more elaborate methods.
\subsection{Local ascent}
\label{sec:local}
The previous packing algorithm can be easily extended to encompass local ascent, as proposed in Algorithm~\ref{alg:ascent}. The idea is very simple: after finding a packing with the previous heuristics, attempt another construction forbidding items packed in the current solution, one at a time. As soon as an improving solution is found, it is adopted as incumbent (\emph{first-improve}). This process stops when all the neighbors of the current solution have been attempted, and they all lead to inferior solutions.
\begin{algorithm}[htbp]
\begin{footnotesize}
\DontPrintSemicolon
\SetKwFunction{algo}{algo}\SetKwFunction{ascent}{ascent}
\SetKwFunction{algo}{algo}\SetKwFunction{pack}{pack}
\SetKwProg{myproc}{procedure}{}{}
\SetKw{True}{true}
\SetKw{False}{false}
\SetKw{Break}{break}
\myproc{\ascent{$D, W, H, \mathcal{S}, \ell, v, w$}}{
$x \leftarrow \pack(D, W, H, \mathcal{S}, \ell, v, w)$ \;
let $\mathcal{I}$ be the set of items packed in $x$ \;
$\mathcal{T} \leftarrow \{\}$\;
\Repeat{not improved}{
improved = \False\;
\For{$i \in \mathcal{I} \setminus \mathcal{T}$}{
$\mathcal{T} \leftarrow \mathcal{T} \cup \{ i\}$ \;
$x' \leftarrow \pack(D, W, H, \mathcal{S}\setminus\{i\}, \ell, v, w)$ \;
\If{value of $x'$ is greater than value of $x$}{
$x \leftarrow x'$ \;
let $\mathcal{I}$ be the set of items packed in $x$ \;
improved = \True \;
\Break \;
}
}
}
\Return{$x$}\;
}
\caption{Local ascent for packing semifluids.}
\label{alg:ascent}
\end{footnotesize}
\end{algorithm}
This method is simple, and obviously finds a solution which is at least as good as that of Algorithm~\ref{alg:pack}. As local ascent is still very fast, it is suitable for demanding situations (\emph{e.g.}, interactive processes).
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\subsection{Local ascent}
\label{sec:local}
The previous packing algorithm can be easily extended to encompass local ascent, as proposed in Algorithm~\ref{alg:ascent}. The idea is very simple: after finding a packing with the previous heuristics, attempt another construction forbidding items packed in the current solution, one at a time. As soon as an improving solution is found, it is adopted as incumbent (\emph{first-improve}). This process stops when all the neighbors of the current solution have been attempted, and they all lead to inferior solutions.
\begin{algorithm}[htbp]
\begin{footnotesize}
\DontPrintSemicolon
\SetKwFunction{algo}{algo}\SetKwFunction{ascent}{ascent}
\SetKwFunction{algo}{algo}\SetKwFunction{pack}{pack}
\SetKwProg{myproc}{procedure}{}{}
\SetKw{True}{true}
\SetKw{False}{false}
\SetKw{Break}{break}
\myproc{\ascent{$D, W, H, \mathcal{S}, \ell, v, w$}}{
$x \leftarrow \pack(D, W, H, \mathcal{S}, \ell, v, w)$ \;
let $\mathcal{I}$ be the set of items packed in $x$ \;
$\mathcal{T} \leftarrow \{\}$\;
\Repeat{not improved}{
improved = \False\;
\For{$i \in \mathcal{I} \setminus \mathcal{T}$}{
$\mathcal{T} \leftarrow \mathcal{T} \cup \{ i\}$ \;
$x' \leftarrow \pack(D, W, H, \mathcal{S}\setminus\{i\}, \ell, v, w)$ \;
\If{value of $x'$ is greater than value of $x$}{
$x \leftarrow x'$ \;
let $\mathcal{I}$ be the set of items packed in $x$ \;
improved = \True \;
\Break \;
}
}
}
\Return{$x$}\;
}
\caption{Local ascent for packing semifluids.}
\label{alg:ascent}
\end{footnotesize}
\end{algorithm}
This method is simple, and obviously finds a solution which is at least as good as that of Algorithm~\ref{alg:pack}. As local ascent is still very fast, it is suitable for demanding situations (\emph{e.g.}, interactive processes).
\subsection{Complete search}
\label{sec:tree}
There are two reasons why the previous methods may be unsatisfactory. The first reason concerns some rare, small instances for which a better solution can easily be found by inspection; the second reason concerns proving that the solution found is optimal. We next propose some variants for doing complete search, based on tree search.
Let us start with a caveat. In the packing process we are considering, division of the semifluid occurs only when it does not fit vertically, and the amount left is possibly packed in another holder. However, it may be optimal to fill only a part of the available amount of a semifluid. This case is illustrated in Figure~\ref{fig:subopt}; if item 2 is more valuable than 1, it would be optimal to fill all the volume of item 2 over a part of 1, and leave the remaining 1 unpacked, as shown in the rightmost diagram. However, visited solutions in a complete tree search are only the leftmost and the one in the center; hence, an \emph{``optimum''} for tree search many not be truly optimal for the original problem.
\begin{figure}
\caption{An instance for which complete search does not find the optimum (shown in the rightmost diagram). A vertical section of the container is represented with a bold line, and the item left over is shown beside it.}
\label{fig:subopt}
\end{figure}
Complete search is an extension of Algorithm~\ref{alg:pack} where, instead of considering only packing the item chosen by the heuristic rule in line~\ref{alg:rule}, we consider all the possibilities of placing available items in open containers; each of these possibilities leads to a new node in the search tree. Notice that the branching factor is very large, and hence straightforward complete search is prohibitive even for small instances. Next, we present three relevant tree search alternatives for dealing with this difficulty; a visual insight of the differences between them is provided in Figure~\ref{fig:queue}.
\begin{figure}
\caption{Queueing methods: branch-and-bound (left), where nodes in the queue are sorted by their upper bound; breadth-first search with diving (center), where no information about about the nodes entering the queue is used (at each expansion, one node generated is the diving node); and limited discrepancy search (right), where nodes are sorted by discrepancy (at each expansion, nodes are generated in this order).}
\label{fig:queue}
\end{figure}
\subsubsection{Branch-and-bound}
\label{sec:bb}
Branch-and-bound (BB) is the standard method for searching a tree in optimization (see, \emph{e.g.},~\cite{lawler66} for an early survey). For a maximization problem, the comparison of an upper bound of the objective that can be reached from a given node, to a known lower bound of the objective, is used to eliminate from consideration parts of the search tree. The best solution visited so far is commonly used as the lower bound. In the case of the basic semifluid packing problem, an upper bound can be obtained by sorting the items by decreasing unit value, and filling the space still available in the container by this order, assuming no shape constraints (this is similar to the linear relaxation bound for the knapsack problem; see~\cite{martello1990}). For a given partial solution, holders that cannot be filled due to having no unpacked items that fit inside them are withdrawn from the list of open holders; their volume is subtracted from the space available when computing the corresponding upper bound.
Another important factor for having a reasonably effective branch-and-bound concerns avoiding symmetric, or otherwise equivalent solutions. This is done with the following rules:
\begin{itemize}
\item items placed at the same horizontal level must have increasing indices in the set $\mathcal{S}$ of semifluids to pack;
\item items placed on top of given item $i$ having the same length as $i$ cannot have a larger unit value than~$i$.
\end{itemize}
The main steps of the branch-and-bound algorithm are outlined in Algorithm~\ref{alg:bb} (see also Appendix~\ref{sec:data}). The algorithm is based on the iteration over elements in a queue ($Q$) until it becomes empty. Nodes whose upper bound is inferior to the objective value of the best known solution are discarded (line~\ref{alg:pruning}). Branching is carried out in lines \ref{alg:branchS}--\ref{alg:branchE}. As all the possible assignments of yet unpacked items to open holders must be considered, the main limitation of the algorithm concerns the large number of nodes added in these lines.
The algorithm has two parameters, limiting CPU time and the size of the queue. The latter is used when restricting the number of open nodes is required for keeping memory usage acceptable; in such cases, we provide the possibility of removing a part of the queue (\emph{chopping}, lines \ref{alg:chopS}--\ref{alg:chopE}). When this occurs, as well as when the time limit is reached, the solution returned may be not optimal. In the experiment reported in the Section~\ref{sec:results}, the maximum number of nodes is set to infinity, making CPU time the only factor limiting the search.
\newcommand{P}{P}
\newcommand{S}{S}
\newcommand{\textit{UB}}{\textit{UB}}
\newcommand{\textit{LB}}{\textit{LB}}
\newcommand{\textit{OPT}}{\textit{OPT}}
\begin{algorithm}[htbp!]
\SetKw{True}{true}
\SetKw{False}{false}
\SetKw{Continue}{continue}
\DontPrintSemicolon
create a queue $Q$ with one node (the root relaxation) \tcp*{Initialization}
set upper bound $\textit{UB} \leftarrow \infty$, lower bound $\textit{LB} \leftarrow -\infty$, optimality flag $\textit{OPT} \leftarrow \True$ \;
\Repeat{$Q = \{\}$ or time limit has been reached}{
select and remove from $Q$ node $k$ with largest $\textit{UB}$ \label{alg:pruning} \tcp*{Subproblem selection}
\If(\tcp*[f]{Pruning and fathoming}){$\textit{UB}^k \leq \textit{LB}$ or no items fit in open holders}{
\Continue
}
\ForEach(\tcp*[f]{Partitioning}){feasible assignment of unpacked items to open holders \label{alg:branchS}}{
add new node $n$ to $Q$\;
\If{$\textit{LB}^n > \textit{LB}$}{
update $\textit{LB} \leftarrow \textit{LB}^n$\; \label{alg:branchE}
}
}
\While(\tcp*[f]{Chopping}){size of $Q$ is larger than the allowed limit \label{alg:chopS}}{
remove from $Q$ node with smallest $\textit{UB}$\;
$\textit{OPT} \leftarrow \False$ \; \label{alg:chopE}
}
\If(\tcp*[f]{Termination}){time limit has been reached} {
$\textit{OPT} \leftarrow \False$ \;
}
}
\Return{solution that yielded $\textit{LB}$, with optimality flag $\textit{OPT}$}
\caption{Main steps of the branch-and-bound algorithm.}\label{alg:bb}
\end{algorithm}
\subsubsection{Breadth-first search with diving}
\label{sec:bfs}
As the branching factor is very large, standard branch-and-bound may not be allowed the time and space to produce a good solution, even for relatively small instances. Indeed, as will be seen in the next session, in a limited time the solution of branch-and-bound is often worse than that of the simple heuristics. For overcoming this issue, several alternatives have been proposed in the literature; these are usually based on \emph{diving} (see, \emph{e.g.}, \cite{achterberg2008,pochet2006}).
We firstly propose what we call \emph{breadth-first with diving (BFD)}, which consists of the following:
\begin{enumerate}
\item Keep two search queues: the main queue $Q$ and the \emph{diving queue} $R$;
\item If $R$ is not empty, at the current iteration explore the last element added to this queue (\emph{i.e.}, explore $R$ in a last in, first out manner);
\item If $R$ is empty, at the current iteration explore the first element added to $Q$ (\emph{i.e.}, explore $Q$ in a first in, first out manner);
\item When creating children of the current node, append the one that corresponds to the heuristic rule (LBF) to the queue $R$, and the remaining children (generated by decreasing item length) to $Q$.
\end{enumerate}
Hence, $R$ is searched in a last in, first out fashion, corresponding to the order of the LBF heuristic rule (longest item first, best fit container); therefore, the first leaf visited is the LBF solution. The exploration of $Q$ in a breadth-first (first in, first out) fashion introduces diversity in the search, which balances well with the intensive search of the dive; this is important for time-limited executions, where parts of the tree are left unexplored. Furthermore, quickly finding solutions of good quality allows pruning more nodes in the search tree. Notice that as long as the item list is initially sorted by length, we can generate new nodes to add to $Q$ without further sorting (however, sorting available items by value is required for computing the upper bound of a new node).
Diving does not interact well with the symmetry breaking rules: if the diving item was forbidden for avoiding symmetry, the first dive would be interrupted, and the corresponding heuristic solution would not be reached. In order to assure that we reach that solution, rules for avoiding symmetry are not enforced during diving.
In our implementation of BFS we are using the bounds described in Section~\ref{sec:bb}, which in most cases allow pruning significant parts of the search tree.
\subsubsection{Limited discrepancy search}
\label{sec:lds}
Another alternative to standard branch-and-bound is \emph{limited discrepancy search (LDS)}, where the tree is searched by increasing order of the \emph{number of violations} of the heuristic rule, as proposed in \cite{harvey1995}. This method has been attempted with the LBF heuristic rule, but the computation of discrepancy in this case requires sorting the moves available, using considerable computational time. A better alternative is to base the search in the longest item first, first fit rule (LFF); this allows a very quick expansion of nodes at each iteration, and exploring much larger parts of the tree in a limited time.
As in the standard version of LDS, this method uses a parameter specifying the discrepancy level above which search is abandoned. This usually allows adjusting the part of the tree that is explored to the resources available, as an alternative to simply interrupting the execution after a certain time has elapsed. We acknowledge that better solutions are often found with such an adjustment, and that memory usage will make the search impractical for long-running executions without limiting discrepancy; however, for an easier comparison with the other methods, we have set the discrepancy limit to infinity. Due to this choice, whenever LDS ends before reaching the limit CPU time, its solution is optimal.
In our implementation of LDS we are using the bounds described in Section~\ref{sec:bb}, which in most cases allow pruning significant parts of the search tree.
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\section{Computational results}
\label{sec:results}
In order to assess the performance of the methods proposed, we have created a set of instances based on the characteristics of the real-world application. Practical instances we are aware of are small, as is the number of different semifluid lengths (tubes are usually cut in standard lengths). Instances with more than 20 items go beyond the application's requirement, but are useful for testing the behavior of the different algorithms. Instances are classified into two main families:
\begin{itemize}
\item Easy instances: generated in such a way that in the optimum there are no items left unpacked; for these instances, an optimal solution completely occupying the container is known.
\item Hard instances: no optimum is known in advance; the volume of available items corresponds either to 100\% of the container (as for easy instances, though now it is unlikely that all items can be packed), or to 150\% of it.
\end{itemize}
The number of semifluids considered are 5, 10, 20, 50, and~100. Some instances have just a few distinct item lengths, other have more diverse lengths.
For each combination of these characteristics, 100 different instances have been generated, totaling 3000 instances. A visualization of instances from the easy and hard subsets, with corresponding optimal and heuristic solutions, is provided in Figure~\ref{fig:instances} (details on the instance generator are available in Appendix~\ref{sec:data}).
\begin{figure}
\caption{An optimal solution (left), and a heuristic solution (right) for instances with ten items: an easy instance (top) and a hard instance (bottom).}
\label{fig:instances}
\end{figure}
Our programs use exact arithmetic for all operations (hence, values in the instance files are written as fractions). All the executions were limited to 60 seconds of CPU time, and both the maximum number of nodes and the discrepancy limit were set to infinity.
We start recalling the comparison among simple heuristics (Table~\ref{tab:pack}). Having selected LBF, we now compare it to more elaborate methods in Table~\ref{tab:all}. As expected, local ascent is always at least as good as LBF, being strictly superior for a massive share of instances. As the CPU time limitation is rather severe, local ascent is also often better than tree search methods. The best results overall have been obtained by limited discrepancy search.
\begin{table}[h!tbp]
\centering
\caption{Comparison of simple rule (LBF), local ascent (LA), and tree search --- standard branch-and-bound version (BB), breadth-first search with diving (BFD), and limited discrepancy search (LDS) --- for a data set of 3000 instances. Left table: $n_{ij}$, the number of times method $i$ was strictly better (\emph{i.e.}, found a better solution) than method $j$. Right table: $n_{ij} - n_{ji}$; positive values mean the method on line $i$ is better for more instances than the method in column $j$.}
\label{tab:all}
\begin{tabular}{l|*{7}{r@{~~~}}}
& LBF & LA & BB & BFS & LDS \\\hline
LBF & 0 & 0 & 1627 & 306 & 42 \\
LA & 2041 & 0 & 1744 & 849 & 187 \\
BB & 909 & 633 & 0 & 108 & 101 \\
BFS & 1520 & 1092 & 1895 & 0 & 329 \\
LDS & 2007 & 1350 & 1915 & 1192 & 0 \\
\end{tabular}
~~~~~
\begin{tabular}{l|*{7}{r@{~~~}}}
& LBF & LA & BB & BFS & LDS \\\hline
LBF & 0 & -2041 & 718 & -1214 & -1965 \\
LA & 2041 & 0 & 1111 & -243 & -1163 \\
BB & -718 & -1111 & 0 & -1787 & -1814 \\
BFS & 1214 & 243 & 1787 & 0 & -863 \\
LDS & 1965 & 1163 & 1814 & 863 & 0 \\
\end{tabular}
\end{table}
Figure~\ref{fig:apercu} graphically summarizes the results obtained. On each sub-figure, results for instances of type easy and hard are separated into two rows. Each bar (or curve, in the bottom sub-figure) represents percentages or averages considering all the instances of each size. Methods considered are, as before, the simple heuristic rule (LBF), local ascent based on this rule, branch-and-bound, breadth-first search with diving, and limited discrepancy search; to each method corresponds a column in the top three sub-figures, and a line in the bottom sub-figure. The abscissa for the three top sub-figures is the instance size, and for the bottom sub-figure is the CPU time used. For each instance we have identified the best solution found by all the methods (which in some cases is optimal); on the top sub-figure, the ordinate is the percentage of instances for which each method finds such solution. The next set of plots shows the percentage of instances for which the search tree was completely explored (for the relevant methods). Follows a plot of the average CPU time used in the solution process, for all the instances of each size/type; the time for each run was limited to 60 seconds, but in many cases was smaller. Finally, the sub-figure in the bottom shows the evolution of the average value of the objective for the best solution found by each method, in terms of the CPU time used; here we can observe how the gap between the different methods progresses.
As can be seen in Figure~\ref{fig:apercu}, ``standard'' branch-and-bound (taking the node with the highest upper bound at each iteration) quickly becomes very limited, when the instance size increases; this is due to the very high branching factor. Crossing information on that figure with that of Table~\ref{tab:nnodes}, we see that when instance size increases, a very large number of nodes are open, but, due to the time limitation, only a small part of them can be explored. This can be observed for all except smallest, easy instances. For finding good solutions in a limited time, methods fully exploiting the heuristics (LA, BFS and LDS) have a much better performance. Note that, for larger values of the CPU limit, the number of open nodes may have to be limited for avoiding memory overflow.
\begin{figure}
\caption{Overall aper{\c c}
\label{fig:apercu}
\end{figure}
We have seen in Table~\ref{tab:all} that the method that is able to find strictly better solutions than the others for more instances is limited discrepancy search. This is corroborated by the evolution over time of the average solution, for all instances of a given size, presented at the bottom of Figure~\ref{fig:apercu}. The general tendency is to have LDS finding good solutions more quickly than the other tree search methods; however, near the CPU limit imposed, LDS is closely followed by BFD (\emph{e.g.}, for hard instances of size 50). In terms of the ability to complete the search, and hence to prove optimality, BFD and LDS are roughly equivalent; these methods appear to be considerably better than BB for easy instances, though slightly inferior for hard instances.
The main factor for LDS to be able to explore much more nodes than BFD is the ability to easily keep nodes organized by increasing discrepancy; for technical details, please consult the implementation code (see Appendix~\ref{sec:data}).
Another interesting observation concerns the performance of local ascent. For small instances, LA quickly finds the best solution (often proven optimal by tree search); however, LA is outclassed by tree search methods for mid-sized instances, to regain a relative good performance for large instances, as can be seen in the top graphic of Figure~\ref{fig:apercu}. This is because local ascent is very fast, and hence the time constraint is not limiting it in our experiment, even for large instances.
\begin{table}
\centering
\caption{Average number of nodes explored, remaining in the queue at the end of the search, and created, for each of the tree search methods.}
\label{tab:nnodes}
\sisetup{
round-integer-to-decimal,
table-format = 5.0,
round-mode = places,
round-precision = 0,
}
\begin{small}
\begin{tabular}{lr|*{3}S|*{3}S|*{3}S}
\multicolumn{2}{c|}{Instance} & \multicolumn{3}{c|}{Nodes explored} & \multicolumn{3}{c|}{Nodes in queue} & \multicolumn{3}{c}{Nodes created} \\
Type & Size & {BB} & {BFS} & {LDS} & {BB} & {BFS} & {LDS} & {BB} & {BFS} & {LDS}\\ \hline
easy & 5 & 10.38 & 5 & 25.96 & 0 & 0 & 0 & 27.84 & 16.545 & 16.695 \\
easy & 10 & 3193.46 & 11.97 & 103.195 & 4379.28 & 0 & 0 & 13790 & 82.42 & 78.09 \\
easy & 20 & 8113.42 & 165.94 & 1895.53 & 17430.8 & 998.705 & 3699.11 & 37510.8 & 3348.65 & 5502.44 \\
easy & 50 & 1973.79 & 139.17 & 6052.78 & 17811.1 & 6167.86 & 109864 & 23525.5 & 8231.42 & 115697 \\
easy & 100 & 574.07 & 54.485 & 4487.73 & 13399.5 & 7131.38 & 208455 & 14544.6 & 7811.54 & 212709 \\
hard & 5 & 1895.21 & 4691.01 & 9091.66 & 1279.15 & 2830.75 & 4108.38 & 4563.49 & 9964.85 & 12813.2 \\
hard & 10 & 5586.23 & 7454.29 & 28896.2 & 7979.26 & 12363.2 & 23270.6 & 20654& 32700.7 & 51551.2 \\
hard & 20 & 4951 & 3429.5 & 34053.7 & 25272.2 & 22851 & 73126.3 & 40219.9 & 40261.3 & 106639 \\
hard & 50 & 890.347 & 446.882 & 10487.9 & 26709.5 & 16993.2 & 268792 & 27603.2 & 18228.7 & 278513 \\
hard & 100 & 290.43 & 83.8375 & 4649.9 & 18534.9 & 7564.85 & 256590 & 18824.5 & 7675.5 & 260851 \\
\end{tabular}
\end{small}
\end{table}
\section{Conclusions}
\label{sec:conclusions}
Semifluids are materials having both fluid and solid characteristics. In this paper, we studied the problem of packing a particular type of semifluid which cannot flow in one direction, though it is fluid in the other directions; this is the case when tubes of a small diameter are packed in parallel. In this context, a packing item --- a semifluid --- is a set of identical tubes. Different items have different length and/or value, and any fraction of an item may be used, with the objective of obtaining the maximum value packed.
Given the assumption of continuity, \emph{i.e.}, that one may arbitrarily divide a given volume, the problem of packing a set of tubes of different lengths in a container is surprisingly difficult. This paper presents heuristics and complete search for the variant which closer corresponds to the industrial application: all the semifluids must be packed in the same direction, and a semifluid placed on top of another must not protrude. In this paper, we have considered divisions of the volume of an item only when it reaches the ceiling of the container.
Several methods, from simple heuristics to complete search, are proposed. The choice among them depends on the application. Simple heuristics are very quick, but often fail to find good solutions. Local ascent based on simple heuristics often finds very good solutions, and is likely to be the best method for large instances and limited CPU time. Among tree search methods, limited discrepancy search is often superior to the others, finding solutions of very good quality and frequently proving them optimal.
Semifluid packing under assumptions not considered in this paper is an interesting subject for future research; in particular, exploring different packing directions and the possibility of overflow. The complexity class of this problem is unknown; determining it is an interesting research topic. Another interesting research direction concerns developing compact mathematical models for optimization, taking full consideration of the possibility of packing fractions of each item.
The proposed heuristics could be extended and refined, in particular for taking into account the possibility of diversifying the point of division of an item into several packing places, further than the top of the container. Yet another unexplored possibility for improvement concerns using the objective value of a heuristic solution as a lower bound, at the root node.
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\section{Conclusions}
\label{sec:conclusions}
Semifluids are materials having both fluid and solid characteristics. In this paper, we studied the problem of packing a particular type of semifluid which cannot flow in one direction, though it is fluid in the other directions; this is the case when tubes of a small diameter are packed in parallel. In this context, a packing item --- a semifluid --- is a set of identical tubes. Different items have different length and/or value, and any fraction of an item may be used, with the objective of obtaining the maximum value packed.
Given the assumption of continuity, \emph{i.e.}, that one may arbitrarily divide a given volume, the problem of packing a set of tubes of different lengths in a container is surprisingly difficult. This paper presents heuristics and complete search for the variant which closer corresponds to the industrial application: all the semifluids must be packed in the same direction, and a semifluid placed on top of another must not protrude. In this paper, we have considered divisions of the volume of an item only when it reaches the ceiling of the container.
Several methods, from simple heuristics to complete search, are proposed. The choice among them depends on the application. Simple heuristics are very quick, but often fail to find good solutions. Local ascent based on simple heuristics often finds very good solutions, and is likely to be the best method for large instances and limited CPU time. Among tree search methods, limited discrepancy search is often superior to the others, finding solutions of very good quality and frequently proving them optimal.
Semifluid packing under assumptions not considered in this paper is an interesting subject for future research; in particular, exploring different packing directions and the possibility of overflow. The complexity class of this problem is unknown; determining it is an interesting research topic. Another interesting research direction concerns developing compact mathematical models for optimization, taking full consideration of the possibility of packing fractions of each item.
The proposed heuristics could be extended and refined, in particular for taking into account the possibility of diversifying the point of division of an item into several packing places, further than the top of the container. Yet another unexplored possibility for improvement concerns using the objective value of a heuristic solution as a lower bound, at the root node.
\section{Supplementary programs and data}
\label{sec:data}
Supplementary programs and data associated with this article can be found online at \url{http://www.dcc.fc.up.pt/~jpp/code/semifluid}. That page contains an implementation of all the algorithms described in this paper and the program used for generating the instances, as well as the generated data.
In the real-world application of this problem the number of different tube lengths in catalog is small. To a smaller extent, this is also true for other numeric values in the required data. We simulate this by limiting the number of digits in the random numbers generated: 3 digits in general, 2 or 3 digits for tube lengths. All the values are normalized, so that the container dimensions are $1 \times 1 \times 1$. As we use exact arithmetic for all operations, values generated and stored in the files are fractions. The combinations of parameters used for instance generation are summarized in Table~\ref{tab:data}.
\begin{table}[h!tbp]
\centering
\caption{Characteristics of benchmark instances used: for each set of parameters 100 independent random instances have been generated, totaling 3000 instances.}
\begin{tabular}{lccccc}
Type & Number of items & Digits in $\ell_i$ & Volume of items (\% of $D \times W \times H $) & Total \\ \hline
easy & 5, 10, 20, 50, 100 & 2, 3 & 100\% & 1000 instances\\
hard & 5, 10, 20, 50, 100 & 2, 3 & 100\%, 150\% & 2000 instances\\
\end{tabular}
\label{tab:data}
\end{table}
For hard instances no optimum is known in advance. These instances have been generated by simply drawing random numbers for lengths and volumes with the required number of digits, and afterwards updating the volumes so that the total volume will be the desired factor of the container's volume (in our data set, 1 or 3/2).
Easy instances have the space of the container completely filled. This is done by successive divisions of the container, as shown in Algorithm~\ref{alg:geneasy}. To each holder generated this way there will correspond a different item. Using this procedure, the total volume of items will always equal the volume of the container.
Instances closer to the real-world application that motivated this paper are hard instances with 10 to 20 items, two digits in $\ell_i$ and items occupying 100\% to 150\% of the container's volume.
\begin{algorithm}[htbp!]
\DontPrintSemicolon
create a holder $h$ with the size of the container\;
$\mathcal{H} \leftarrow [h]$ \;
\Repeat{number of holders is equal to the number of desired items}{
randomly select a holder $h$ from $\mathcal{H}$\;
randomly select $r$ with uniform distribution in $[0,1]$\;
\If(\tcp*[f]{With 50\% probability}){$r < 1/2$}{
\If{$h$ has no other holders on top}{
divide $h$ vertically\;
replace $h$ by the two newly created holders
}
}
\Else{
divide $h$ horizontally\;
replace $h$ by the two newly created holders
}
}
\caption{Main steps for generating an easy instance.}\label{alg:geneasy}
\end{algorithm}
\end{document}
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\begin{document}
\title[Infinite dimensional oscillatory integrals]{Infinite dimensional oscillatory integrals with polynomial phase and applications to high order heat-type equations}
\author{S. Mazzucchi}
\address{ Dipartimento di Matematica, Universit\`a di Trento, 38123 Povo, Italia}
\maketitle
\begin{abstract}
The definition of infinite dimensional Fresnel integrals is generalized to the case of polynomial phase functions of any degree and applied to the construction of a functional integral representation of the solution to a general class of high order heat-type equations.\\
\noindent {\it Key words:} Infinite dimensional integration, partial differential equations, representations of solutions.
\noindent {\it AMS classification }: 35C15, 35G05, 28C20, 47D06.
\end{abstract}
\vskip 1\baselineskip
\section{Introduction}
Functional integration is a powerful tool for the study of dynamical systems \cite{Sim}. The main example is the celebrated Feynman-Kac formula \eqref{Fey-Kac}, which provides a probabilistic representation of the solution to the heat equation
\begin{equation} \label{heat}
\left\{\begin{aligned}
\frac{\partial}{\partial t}u(t,x)&=\frac{1}{2}\Delta u(t,x) -V(x)u(t,x),\qquad t\in {\mathbb{R}}^+, x\in{\mathbb{R}}^d, \\
u (0,x)&=u _0(x),
\end{aligned}\right. \end{equation}
in terms of the expectation with respect to the distribution of the Wiener process $W$ starting at $x$ (see, e.g. \cite{KarSh}),
\begin{equation}\label{Fey-Kac}
u (t,x)={\mathbb{E}}^x[ e^{-\int_0^tV(W(s))ds}u_0(W(t))].\end{equation}
Formula \eqref{Fey-Kac} can be established under rather mild requirements on the potential $V$ and the initial datum $u_0$ (see, e.g. , \cite{Sim}) and provides an important instrument in the study of heat equation and its solutions.
More generally, an extensively developed theory relates stochastic processes with the solution to parabolic equations associated to second-order elliptic operators \cite{Dyn}.
However, that theory cannot be applied to more general PDEs such as, for instance, the Schr\"odinger equation
\begin{equation} \label{schroedinger}\left\{ \begin{aligned}
i\frac{\partial}{\partial t}u(t,x)& =-\frac{1}{2}\Delta u(t,x) +V(x)u(t,x),\qquad t\in {\mathbb{R}}^+, x\in{\mathbb{R}}^d\\
u (0,x)&=u _0(x)
\end{aligned}\right. \end{equation}
describing the time evolution of the state of a nonrelativistic quantum particle,
or also heat-type equations associated to high-order differential operators, such as for instance
\begin{equation}\label{eqLapl2}
\frac{\partial}{\partial t} u(t)=- \Delta^2u(t)-V(x)u(t,x).
\end{equation}
Indeed, a Markov process $\{X(s)\, :\, 0\leq s\leq t\}$ playing the same role for Eq. \eqref{schroedinger} or Eq. \eqref{eqLapl2} as the Brownian motion for the heat equation doesn't exist. Hence there is no ``generalized Feynman Kac formula"
\begin{eqnarray}\label{Fey-KacN}
u (t,x)&=&{\mathbb{E}}^x[ e^{-\int_0^tV(X(s))ds}u_0( X(t))]\\
&=&\int _{{\mathbb{R}}^[0,t]}e^{-\int_0^tV(\omega(s))ds}u_0( \omega (t))dP(\omega),\nonumber
\end{eqnarray} representing the solution of Eq. \eqref{schroedinger} or Eq. \eqref{eqLapl2} in terms of a (Lebesgue type) integral with respect to a probability measure $P$ on ${\mathbb{R}}^{[0,t]}$ associated to the process $X(s)$.\\
Contrarily to the heat equation case, for both Eq. \eqref{schroedinger} and Eq. \eqref{eqLapl2} the fundamental solution $G_t(x,y)$ is not real and positive, even in the simplest case $V\equiv 0$. In particular the Green function $G_t(x,y)$ of the Schr\"odinger equation is complex, while for the high-order heat-type equation \eqref{eqLapl2} $G_t(x,y)$ is real and attains both positive and negative values \cite{Hochberg1978}. Therefore it cannot be interpreted as the density of a transition probability measure.
As a troublesome consequence,
the complex (resp. signed) finitely-additive measure $\mu$ on $\Omega={\mathbb{R}}^{[0,t]}$ defined on the algebra of ``cylinder sets"
$I_k\subset\Omega$ (where $\Omega\equiv {\mathbb{R}}^{[0,+\infty)}$) of the form $$I_k:=\{\omega\in \Omega: \omega (t_j)\in [a_j,b_j], j=1,\dots k\},\quad 0<t_1<t_2<\dots t_k,$$ by
\begin{equation}\label{CylMeas}
\mu(I_k)=\int_{a_1}^{b_1}...\int_{a_k}^{b_k}\prod_{j=0}^{k-1}G_{t_{j+1}-t_j}(x_{j+1},x_j)dx_1...dx_{k},
\end{equation} doesn't extend to a corresponding $\sigma$-additive measure on the generated $\sigma$-algebra. As a matter of fact, if this measure existed, it would have infinite total variation.\\
This problem was addressed in 1960 by Cameron \cite{Cam} for the Schr\"odinger equation and by Krylov \cite{Kry} for Eq. \eqref{eqLapl2}. These results may be viewed as particular cases of a general theorem later established by E. Thomas \cite{Thomas}, extending Kolmogorov existence theorem to limits of projective systems of signed or complex measures, instead of probability ones.\\
In fact, these no-go results forbid a functional integral representation of the solution of Eq. \eqref{schroedinger} or Eq. \eqref{eqLapl2} in terms of a Lebesgue-type integral with respect to a $\sigma$-additive complex or signed measure with finite total variation. Consequently, the integral appearing in the generalized Feynman-Kac formula \eqref{Fey-KacN} has to be thought in a weaker sense. One possibility is the definition of the ``integral" in terms of a linear continuous functional on a suitable Banach algebra of ``integrable functions", in the spirit of Riez-Markov theorem, that provides a one-to-one correspondence between complex bounded measures (on suitable topological spaces $X$) and linear continuous functionals on $C_\infty(X)$ (the continuous functions on $X$ vanishing at $\infty$).\\
Referring to Schr\"odinger equation, this issue has been extensively studied, producing a number of different mathematical definitions of Feynman path integrals (see \cite{Ma} for an account). We mention in particular for future reference the {\em Parseval approach}, introduced by It\^o \cite{Ito1,Ito2} in the 60s and developed in the 70s by S. Albeverio and R. Hoegh-Krohn \cite{AlHK, AlHKMa}, and by D. Elworthy and A. Truman \cite{ELT}.
Dealing with the parabolic equation \eqref{eqLapl2} associated to the bilaplacian, various formulations have been proposed. One of the first was introduced by Krylov \cite{Kry} and extended by Hochberg \cite{Hochberg1978}. Defining a suitable stochastic pseudo-process whose transition probability function is not positive definite, the authors realized formula \eqref{Fey-KacN} in terms of the expectation with respect to a signed measure on ${\mathbb{R}}^{[0,t]}$ with infinite total variation. That is the reason way the integral in \eqref{Fey-KacN} is not defined in Lebesgue sense, but is meant as the limit of finite dimensional cylindrical approximations \cite{BeHocOr}.
It is worthwhile mentioning the work by D. Levin and T. Lyons relying on the ``rough paths" theory. Indeed, in \cite{LevLyo} the authors conjecture that the signed measure (with infinite total variation) associated to the Krylov-Hochberg pseudo-process could became finite if defined on a certain quotient space on the path space (two path paths are equivalent if they differ for reparametrization). \\
A different approach was proposed by Funaki \cite{Funaki1979} and continued by Burdzy \cite{Burdzy1993}. It is based
on the construction of a complex-valued stochastic process with dependent increments, obtained by composing two independent Brownian motions. In \cite{Funaki1979}, formula \eqref{Fey-KacN} with $V=0$ is realized as an integral with respect to a well defined positive probability measure on a complex space for a suitable class of analytic initial data $u_0$ at least. These results have been further developed in \cite{Funaki1979,HocOr96,OrZha} and are related to Bochner's subordination theory \cite{Boch}.
Complex-valued processes, related to PDEs of the form \eqref{eqLapl2}, were also proposed by other authors exploiting various techniques \cite{Burdzy1995,ManRy93,BurMan96,Sainty1992}. A new construction for the solution of a general class of high order heat-type equations has been recently proposed, where formula \eqref{Fey-KacN} has been realized as limit of expectations with respect to a sequence of suitable random walks in the complex plane \cite{BoMa14}.\\
We also mention a completely different approach proposed by R. L\'eandre \cite{Lea}, which shares some analogies with the mathematical construction of Feynman path integrals with the white-noise-calculus approach \cite{HKPS}.\\
It is worthwhile remarking that most of the results appearing in the literature are restricted to the cases where either $V=0$ or $V$ is linear. The construction of a generalized Feynman-Kac type formula is still lacking for the solution of high-order heat-type equations similar to \eqref{eqLapl2} with a more general $V$. \par
This work aims to construct a Feynman-Kac formula for the solution of a general class of high-order heat-type equations of the form
\begin{equation}\label{PDE-N}
\frac{\partial}{\partial t}u(t,x)=(-i)^p \alpha \frac{\partial^p}{\partial x^p}u(t,x)+V(x)u(t,x), \quad t\in [0, +\infty), \; x\in {\mathbb{R}},
\end{equation}
where $p\in{\mathbb{N}}$, $p>2$, $\alpha\in{\mathbb{C}}$ is a complex constant and $V:{\mathbb{R}}\to{\mathbb{C}}$ a continuous bounded function Fourier transform of a complex Borel measure on ${\mathbb{R}}$. \\
Adopting the {\em Fresnel integral } formulation of the mathematical definition of Feynman path integrals \cite{AlHKMa, AlHK}, we introduce infinite dimensional Fresnel integrals with polynomial phase, generalizing the existing results valid for quadratic phase functions.
If the phase function is an homogeneous polynomial of order $p$, we show in particular how this new kind of functional integral
is related to the fundamental solution of Eq. \eqref{PDE-N} with $V\equiv 0$. This relation will be eventually exploited in the proof of a functional integral representation of the solution of Eq. \eqref{PDE-N}, for a suitable class of potentials $V$ and initial data $u_0$, giving rise to a new type of generalized Feynman-Kac formula.
In section 2, a detailed study of the fundamental solution of Eq. \eqref{PDE-N} takes place in the case $V=0$. In section 3, we introduce the definition of infinite dimensional Fresnel integral with polynomial phase function showing that a particular example is related to the PDE \eqref{PDE-N} with $V\equiv 0$. In section 4, we build up a representation of the solution of \eqref{PDE-N} with $V\neq 0$ in terms of an infinite dimensional Fresnel integral.
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\section{The fundamental solution of high-order heat-type equations}
Let us consider the $p$-order heat-type equation:
\begin{equation} \label{PDE-p}\left\{ \begin{aligned}
\frac{\partial}{\partial t}u(t,x)&=(-i)^p \alpha \frac{\partial^p}{\partial x^p}u(t,x)\\
u(0,x) &= u_0(x),\qquad x\in{\mathbb{R}}, t\in [0,+\infty)
\end{aligned}\right. \end{equation}
where $p\in{\mathbb{N}}$, $p\geq 2$, and $\alpha\in{\mathbb{C}}$ is a complex constant. In the following we shall assume that $|e^{\alpha tx^p}|\leq 1$ for all $x\in{\mathbb{R}}$ and $ t\in [0,+\infty)$. In particular, if $p$ is even this condition is fulfilled if ${\mathbb{R}}ea(\alpha)\leq 0$, while if $p$ is odd then $\alpha$ will be taken purely imaginary.\\
In the case where $p=2$ and $\alpha \in {\mathbb{R}}$, $\alpha<0$, we obtain the heat equation, while for $p=2$ and $\alpha =i$ Eq. \eqref{PDE-p} is the Schr\"oedinger equation. Since both cases are extensively studied, in the following we shall mainly focus ourselves on the case where $p\geq 3$.
Let $G^p_t(x,y)$ be the fundamental solution of Eq.\eqref{PDE-p}. Given an initial datum $u_0$ belonging to the space $S({\mathbb{R}})$ of Schwartz test functions, the solution of the Cauchy problem \eqref{PDE-p} is given by:
\begin{equation}\label{Green}
u(t,x)=\int_{\mathbb{R}} G^p_t(x,y)u_0(y)dy.
\end{equation}
In particular the following equality holds: $$G^p_t(x-y)=g^p_t(x-y),$$ where $g^p_t\in S'({\mathbb{R}})$ is the Schwartz distribution defined by the Fourier transform
\begin{equation}\label{Green2}
g^p_t(x):=\frac{1}{2\pi}\int e^{ikx}e^{\alpha tk^p}dk,\qquad x\in{\mathbb{R}}.
\end{equation}
The following lemmas state some regularity properties of the distribution $g^p_t$ that will be used in the next section.
\begin{lemma}\label{lemma1}
The tempered distribution \eqref{Green2} is a $C^\infty$ function.
\end{lemma}
\begin{proof}
A priori $g^p_t$ is an element of $S'({\mathbb{R}})$, the Schwartz space of distribution, but we shall prove that $g^p_t$ is a $C^\infty$ function defined by an absolutely convergent Lebesgue integral. This can be easily proved in the case where $p$ is even and ${\mathbb{R}}ea(\alpha) <0$, since the function $k\mapsto e^{\alpha tk^p}$ is an element of $L^1({\mathbb{R}})$.\\
In the case where ${\mathbb{R}}ea(\alpha)= 0$, i.e. $\alpha=ic$ with $c\in{\mathbb{R}}$, the function $k\mapsto e^{\alpha tk^p}$ is not summable.
Let us denote by $\psi\in S'({\mathbb{R}})$ the tempered distribution defined by this map and by $\chi_{[-R,R]}$ the characteristic function of the interval $[-R,R]\subset {\mathbb{R}}$. By the convergence of $\chi_{[-R,R]}\psi$ to $ \psi $ in $S'({\mathbb{R}})$ as $R\to +\infty$ and the continuity of the Fourier transform as a map from $S'({\mathbb{R}})$ to $S'({\mathbb{R}})$ we have that $$g^p_t=\hat \psi=\lim _{R\to+\infty }\widehat{\chi_{[-R,R]}\psi}.$$
On the other hand, by a change in the integration path in the complex $k$-plane, in the case where $p$ is even and $c>0$ we have:
\begin{eqnarray}
g^p_t(x)&=&\lim_{R\to\infty}\frac{1}{2\pi}\int_{-R}^R e^{ikx}e^{ict k^p}dk=\lim_{R\to\infty}\frac{1}{2\pi}\int_{0}^R (e^{ikx}+e^{-ikx})e^{ict k^p}dk\nonumber\\
&=&\lim_{R\to\infty}\frac{e^{i\pi/2p}}{2\pi}\int_{0}^R (e^{ie^{i\pi/2p}kx}+e^{-ie^{i\pi/2p}kx})e^{-c tk^p}dk\nonumber\\
&=&\frac{e^{i\pi/2p}}{2\pi} \int_{\mathbb{R}} e^{ie^{i\pi/2p}kx}e^{-ct k^p}dk,\label{G1}
\end{eqnarray}
while in the case where $p$ is even and $c<0$:
\begin{equation}
g^p_t(x)=\frac{e^{-i\pi/2p}}{2\pi} \int_{\mathbb{R}} e^{ie^{-i\pi/2p}kx}e^{-c tk^p}dk.\label{G2}
\end{equation}
In the case $p$ is odd, a different integration contour in the complex $k-$plane yields the following representation:
\begin{eqnarray}
g^p_t(x)&=&\lim_{R\to\infty}\frac{1}{2\pi}\int_{-R}^R e^{ikx}e^{ict k^p}dk\nonumber\\
&=&\frac{1}{2\pi}\int_{{\mathbb{R}}+i\eta} e^{ixz}e^{ic tz^p}dz\label{G3}
\end{eqnarray}
where $\eta>0$ if $c>0$ while $\eta<0$ if $c<0$. The integrand in the second line of \eqref{G3} is absolutely convergent since $ |e^{ict ({\mathbb{R}}ea(z)+i\eta)^p}|\sim e^{-ct\eta ({\mathbb{R}}ea(z))^{p-1}}$ as $|{\mathbb{R}}ea(z)|\to\infty$. \\
Eventually representations \eqref{G1}, \eqref{G2} and \eqref{G3} show that $g^p_t$ is a $C^\infty $ function of the variable $x$.
\end{proof}
\begin{Remark}
The proof of lemma \eqref{lemma1} shows that $g^p_t:{\mathbb{R}}\to {\mathbb{C}}$ can be extended to an entire analytic function of $z\in{\mathbb{C}}$.
The analyticity of $g^p_t$ follows by the application of Fubini's and Morera's theorems.
\end{Remark}
\begin{Remark}
A formula similar to \eqref{G1} has also been proved in \cite{AlMa2005} and applied to the study of some asymptotic properties of finite dimensional Fresnel integral with polynomial phase function.
\end{Remark}
The following lemma relies on the study of the detailed asymptotic behaviour of $g^p_t(x)$ for $x\to \infty$.
\begin{lemma}\label{lemma1-asy}
The function $g^p_t$ is bounded. In particular if $p$ is even and ${\mathbb{R}}ea(\alpha)<0$ then $g^p_t\in L^1({\mathbb{R}})$.
\end{lemma}
\begin{proof}
By lemma \ref{lemma1} the function $g^p_t$ is continuous, hence the proof of its boundedness can be based only on the study of its asymptotic behavior for $x\to \infty$. This task is accomplished by means of the stationary phase method \cite{Mur,Hor}.\\
For $x\to +\infty$, a change of variables in \eqref{Green2} gives:
\begin{equation}\label{int1}
g^p_t(x)=\frac{x^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{x^{p/p-1}(i\xi+\alpha t\xi^p)}d\xi=\frac{x^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{x^{p/p-1}\phi(\xi)}d\xi,
\end{equation}
$\phi:{\mathbb{R}}\to {\mathbb{C}}$ being the complex phase function
$$\phi(\xi)=i\xi+\alpha t\xi^p, \qquad \xi\in{\mathbb{R}}.$$
If either ${\mathbb{R}}ea (\alpha) \neq 0$ or
$p$ is odd and $\alpha =ic$, with $c\in {\mathbb{R}}^+$, then the phase function $\phi$ has no stationary points on the real line, i.e. there are no real solutions of the equation $\phi'(\xi)=0$. In this cases an integration by parts argument yelds:
$$\int e^{x^{\frac{p}{p-1}}\phi(\xi)}d\xi=\int \frac{1}{x^{\frac{p}{p-1}}\phi'(\xi)} \frac{d}{d\xi} e^{x^{\frac{p}{p-1}}\phi(\xi)}d\xi=\frac{1}{x^{\frac{p}{p-1}}}\int e^{x^{\frac{p}{p-1}}\phi(\xi)}\frac{\phi''(\xi)}{(\phi'(\xi))^2}d\xi.$$
By iterating this procedure we obtain that for all $N\in {\mathbb{N}}$:
$$g^p_t(x)\stackrel{x\to +\infty}{<<}(x^{\frac{p}{p-1}})^{-N}.$$
In the case where $\alpha =ic$ with $c\in {\mathbb{R}}$, Eq. \eqref{int1} can be written as
\begin{equation}\nonumber
g^p_t(x)=\frac{x^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{ix^{p/p-1}(\xi+c t\xi^p)}d\xi, \qquad x>0.
\end{equation}
If $p$ is even, an application of the stationary phase method \cite{Mur,Hor} gives:
\begin{eqnarray}
g^p_t(x)&=&\frac{x^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{ix^{p/p-1}(\xi+ct\xi^p)}d\xi \nonumber\\
&\stackrel{x\to +\infty}{\sim}& e^{sign (c)i\frac{\pi}{4}} \frac{x^{\frac{2-p}{2(p-1)}}}{\sqrt{2\pi}} e^{-ix^{p/p-1}\frac{p-1}{p}\left(\frac{1}{pct}\right)^{1/p-1}} \sqrt{ \frac{ (pct)^{ \frac{p-2}{p-1}} }{|c|tp(p-1) } } .\nonumber
\end{eqnarray}
In the case where $p$ is odd and $c<0$, the same technique yields:
\begin{eqnarray}
g^p_t(x)&=&\frac{x^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{ix^{p/p-1}(\xi+ct\xi^p)}d\xi \nonumber\\
&\stackrel{x\to +\infty}{\sim} & e^{-i\frac{\pi}{4}}(p-1)^{-1/2}(p|c|t)^{-\frac{1}{2(p-1)}} \frac{x^{\frac{2-p}{2(p-1)}}}{\sqrt{2\pi}} e^{ix^{p/p-1}\frac{p-1}{p}\left(-\frac{1}{pct}\right)^{\frac{1}{p-1}}}.\nonumber
\end{eqnarray}
The case where $x\to -\infty$ can be studied in the same way. In particular, if $p$ is an even integer the behaviour of $g^p_t$ for $x\to -\infty$ coincides with the one for $x\to +\infty$.\\
For $p$ odd and
$x<0$, a change of variable argument gives:
$$g^p_t(x)=\frac{(-x)^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{i(-x)^{p/p-1}(-\xi+ct\xi^p)}d\xi. $$
If $c<0$ then the phase function $\phi(\xi)=-\xi+ct\xi^p$ has no real stationary points, hence
$$g^p_t(x)\stackrel{x\to -\infty}{<<}x^{-N}, \qquad \forall N\in{\mathbb{N}}.$$ In the case where $c>0$ and $x\to -\infty$ the stationary phase method yields
$$g^p_t(x)\stackrel{x\to -\infty}{\sim} e^{i\frac{\pi}{4}}(p-1)^{-1/2}(pct)^{-\frac{1}{2(p-1)}} \frac{(-x)^{\frac{2-p}{2(p-1)}}}{\sqrt{2\pi}}e^{i(-x)^{p/p-1}\frac{1-p}{p}(pct)^{-\frac{1}{p-1}}}.$$
Eventually these results give the boundedness of the function $g^p_t$. Furthermore, if $p$ is even and ${\mathbb{R}}ea(\alpha)<0$ then $g^p_t$ is even summable.
\end{proof}
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\section{Infinite dimensional Fresnel integrals with polynomial phase}\label{sez2}
Classical oscillatory integrals on ${\mathbb{R}}^n$ are objects of this form
\begin{equation}\label{int-osc}\int _{{\mathbb{R}}^n}f(x)e^{i\Phi(x)}dx,
\end{equation}
where $\Phi$ and $ f$ are complex Borel functions. The interesting case where the {\em phase function} $\Phi$ is real valued has been extensively studied in connection with the theory of Fourier integral operator \cite{Hor}. If the function $f$ is not summable the integral \eqref{int-osc} is not defined in Lebesgue sense. In \cite{Hor}, H\"ormander proposes and exploits an alternative definition which can handle the case where $f\notin L^1({\mathbb{R}}^n)$. We present here a formulation of H\"ormander's definition of oscillatory integral, which was applied to the mathematical construction of Feynman path integrals in \cite{ELT,AlBr}.
\begin{definition}\label{def-int-osc}Let $f:{\mathbb{R}}^n\to {\mathbb{C}}$ and $\Phi:{\mathbb{R}}^n\to {\mathbb{R}}$ be Borel functions. Assuming that:
\begin{enumerate}
\item for any Schwartz test function $\phi\in S({\mathbb{R}}^n)$ such that $\phi(0)=1$ the function $g_\epsilon (x):=\phi(\epsilon x)f(x)e^{i\Phi(x)}$ is summable,
\item the limit $\lim_{\epsilon \to 0}\int g_\epsilon (x)dx $ exists and is independent of $\phi$.
\end{enumerate}
Then the {\em oscillatory integral } $ \int^o _{{\mathbb{R}}^n}f(x)e^{i\Phi(x)}dx$ is defined as:
$$\int^o _{{\mathbb{R}}^n}f(x)e^{i\Phi(x)}dx:=\lim_{\epsilon \to 0}\int_{{\mathbb{R}}^n} \phi(\epsilon x)f(x)e^{i\Phi(x)}dx$$
\end{definition}
In the case where $f\in L^1({\mathbb{R}}^n)$ the oscillatory integral reduces to a Lebesgue integral, i.e. $\int^o _{{\mathbb{R}}^n}f(x)e^{i\Phi(x)}dx =\int _{{\mathbb{R}}^n}f(x)e^{i\Phi(x)}dx$. \\
Definition \ref{def-int-osc} gives sense to classical Fresnel integrals such as
$\int_{{\mathbb{R}}^n}f(x)e^{\frac{i}{2}\|x\|^2} dx$ which are extensively applied in the theory of wave diffraction. In particular, for $f=1$ definition \eqref{int-osc} yields the equality $\int_{{\mathbb{R}}^n}e^{\frac{i}{2}\|x\|^2} dx=(2\pi i)^{n/2}$.\\
In \cite{AlHKMa} oscillatory integration is generalized to the case where ${\mathbb{R}}^n$ is replaced by a real separable Hilbert space $({\mathcal{H}}, \langle \;,\;\rangle)$ and the definition of {\em infinite dimensional Fresnel integral} is introduced. The construction relies upon a generalization of the Parseval equality
\begin{equation}\label{Parseval1}
\int_{{\mathbb{R}}^n}\frac{e^{\frac{i}{2}\|x\|^2}}{(2\pi i)^{n/2}}f(x)dx=\int_{{\mathbb{R}}^n}e^{-\frac{i}{2}\|x\|^2}\hat f(x)dx,
\end{equation}
(valid for Schwartz test functions functions $f\in S({\mathbb{R}}^n)$, where $\hat f(x)=\int_{{\mathbb{R}}^n}e^{ixy}f(y)dy$). In fact (see \cite{ELT}) equality \eqref{Parseval1} can be generalized to the case the function $f:{\mathbb{R}}^n\to {\mathbb{C}}$ is the Fourier transform of a complex bounded Borel measure $\mu_f$ on ${\mathbb{R}}^n$, giving the following Parseval equality for the oscillatory integral
\begin{equation}\label{Parseval2}
\int^o_{{\mathbb{R}}^n}\frac{e^{\frac{i}{2}\|x\|^2}}{(2\pi i)^{n/2}}f(x)dx=\int_{{\mathbb{R}}^n}e^{-\frac{i}{2}\|x\|^2}d\mu_f(x),
\end{equation}
with $f(x)=\int_{{\mathbb{R}}^n}e^{ixy}d\mu_f(y)$.
Formula \eqref{Parseval2} is crucial for the extension of oscillatory integration theory to an infinite dimensional setting.\\
Let us introduce the Banach space ${\mathcal{M}}({\mathcal{H}})$ of complex Borel measures on ${\mathcal{H}}$ with finite total variation, endowed with the total variation norm $\|\mu\|_{{\mathcal{M}}({\mathcal{H}})}$. ${\mathcal{M}}({\mathcal{H}})$ is a commutative Banach algebra under convolution, the unit being the Dirac point measure at $0$.\\ Let ${\mathcal{F}}({\mathcal{H}})$ be the space of complex functions $f:{\mathcal{H}}\to {\mathbb{C}}$ the form:
\begin{equation}\label{Fo}f(x)=\int_{\mathcal{H}} e^{i\langle x,y\rangle}d\mu(y)\equiv \hat \mu (x),\qquad x\in {\mathcal{H}}\end{equation}
for some $\mu \in {\mathcal{M}}({\mathcal{H}})$. The map ${\mathcal{F}}:{\mathcal{M}}({\mathcal{H}})\to {\mathcal{F}}({\mathcal{H}})$ sending a complex measure $\mu \in{\mathcal{M}}({\mathcal{H}})$ to its Fourier transform $\hat \mu$ defined by Eq. \eqref{Fo} is linear and one to one.
By endowing the space ${\mathcal{F}}({\mathcal{H}})$ with the norm $\|f\|_{\mathcal{F}}:=\|{\mathcal{F}}^{-1}(f)\|_{{\mathcal{M}}({\mathcal{H}})}$, ${\mathcal{F}}({\mathcal{H}})$ becomes a commutative Banach algebra of continuous functions and the map ${\mathcal{F}}:{\mathcal{M}}({\mathcal{H}})\to {\mathcal{F}}({\mathcal{H}})$ is an isometry.\\
In \cite{AlHKMa,AlHK,ELT} the Parseval equality \eqref{Parseval2} is generalized to the case where $f\in {\mathcal{F}}({\mathcal{H}})$. The {\em infinite dimensional Fresnel integral} of a function $f\in {\mathcal{F}}({\mathcal{H}})$ is denoted by $\widetilde{ \int} e^{\frac{i}{2}\|x\|^2}f(x)dx$ and defined as
\begin{equation}\label{Parseval2}
\widetilde{\int} e^{\frac{i}{2}\|x\|^2}f(x)dx:=\int_{{\mathcal{H}}}e^{-\frac{i}{2}\|x\|^2}d\mu(x),
\end{equation}
where $f(x)=\int_{\mathcal{H}} e^{i\langle x,y\rangle}d\mu(y)$ and
the right hand side of \eqref{Parseval2} is a well defined (absolutely convergent) Lebesgue integral.\\
Infinite dimensional Fresnel integrals have been successfully applied to the representation of the solution of Schr\"odinger equation \eqref{schroedinger} (see i.e. \cite{AlHKMa,Ma} and references therein). Let us denote with ${\mathcal{H}}_t$ the real Hilbert space of absolutely continuous paths $\gamma:[0,t]\to{\mathbb{R}}^d$, such that $\int_0^t\dot\gamma(s)^2ds<\infty$ and $\gamma(t)=0$. The inner product in ${\mathcal{H}}_t$ is defined as $\langle\gamma, \eta\rangle=\int_0^t\dot \gamma(s)\dot \eta(s)ds$. By assuming that the initial datum $u_0$ and the potential $V$ in Eq. \eqref{schroedinger} belong to ${\mathcal{F}}({\mathbb{R}}^d)$, it is possible to prove that the function on ${\mathcal{H}}_t$:
$$\gamma\mapsto u_0(\gamma(0)+x)e^{-i\int_0^tV(\gamma(s)+x)ds},\qquad \gamma\in H_t,\; x\in {\mathbb{R}}^d,$$
belongs to ${\mathcal{F}}({\mathcal{H}}_t)$. Further the infinite dimensional Fresnel integral
$$\widetilde{\int} e^{\frac{i}{2}\|\gamma\|^2}e^{-i\int_0^tV(\gamma(s)+x)ds}u_0(\gamma(0)+x)d\gamma$$
provides a functional integral representation of the solution to the Schr\"odinger equation \eqref{schroedinger}.\\
A partial generalization of the definition of infinite dimensional Fresnel integrals and of formula \eqref{Parseval2} was developed in \cite{AlMa2}, where the quadratic phase function $\Phi(x)=\frac{i}{2}\|x\|^2$ was replaced with a fourth order polynomial. This new functional integral allows the mathematical definition of the Feynman path integrals for the Schr\"odinger equation with a quartic-oscillator potential \cite{AlMa2,AlMa3,Ma2008}.\par
In the following we are going to generalize the definition in \eqref{Parseval2} to polynomial phase functions of any order and apply these {\em generalized Fresnel integrals} to the construction of a Feynman-Kac formula for the solution of high-order heat-type equations \eqref{PDE-N}.
Let us consider a real separable Banach space $({\mathcal{B}}, \|\,\|)$. Let ${\mathcal{M}}({\mathcal{B}})$ be the space of complex bounded variation measures on ${\mathcal{B}}$, endowed with the total variation norm. As remarked above, ${\mathcal{M}}({\mathcal{B}})$ is a Banach algebra under convolution. Let ${\mathcal{B}}^*$ be the topological dual of ${\mathcal{B}}$ and ${\mathcal{F}}({\mathcal{B}})$ the Banach algebra of complex-valued functions $f:{\mathcal{B}}^*\to {\mathbb{C}}$ of the form
\begin{equation}\label{Fo-Ba}f(x)=\int_{{\mathcal{B}}}e^{i\langle x,y\rangle}d\mu(y)\equiv \hat\mu(x), \qquad x\in {\mathcal{B}}^*,\, \mu\in {\mathcal{M}}({\mathcal{B}}), \end{equation}
where $\langle \, ,\,\rangle$ denotes the dual pairing between ${\mathcal{B}}$ and ${\mathcal{B}}^*$. The space ${\mathcal{F}}({\mathcal{B}})$ endowed with the norm $\|\hat \mu\|_{{\mathcal{F}}}:=\|\mu\|_{{\mathcal{M}}({\mathcal{B}})}$ and the pointwise multiplication is a Banach algebra of functions.\\
In the following we are going to define a class of linear continuous functionals on ${\mathcal{F}}({\mathcal{B}})$, by generalizing the construction of infinite dimensional Fresnel integrals defined by Eq. \eqref{Parseval2}.
\begin{definition}
Let $\Phi:{\mathcal{B}}\to {\mathbb{C}}$ be a continuous map such that ${\mathbb{R}}ea (\Phi_p(x))\leq 0$ for all $x\in{\mathcal{B}}$.
The infinite dimensional Fresnel integral on ${\mathcal{B}}^*$ with phase function $\Phi$ is the functional $I_{\Phi}:{\mathcal{F}}({\mathcal{B}})\to {\mathbb{C}}$, given by
\begin{equation}\label{Parseval-p}I_{\Phi}(f):=\int_{\mathcal{B}} e^{\Phi(x)}d\mu(x), \qquad f\in{\mathcal{F}}({\mathcal{B}}), f=\hat\mu.\end{equation}
\end{definition}
By construction, the functional $I_{\Phi}$ is linear and continuous, indeed:
$$|I_{\Phi_p}(f)|\leq \int_{\mathcal{B}} |e^{\Phi_p}|d|\mu|(x)\leq \|\mu\|=\|f\|_{{\mathcal{F}} }$$
Further $I_{\Phi}$ is normalized, i.e., $I_{\Phi_p}(1)=1$.
We summarize these properties in the following proposition.
\begin{proposition}
The space ${\mathcal{F}}({\mathcal{B}})$ of Fresnel integrable functions is a Banach function algebra in the norm $\|\,\|_{\mathcal{F}}$. The infinite dimensional Fresnel integral with phase function $\Phi$ is a continuous bounded linear functional $I_{\Phi}:{\mathcal{F}}({\mathcal{B}})\to {\mathbb{C}}$ such that $|I_{\Phi_p}(f)|\leq \|f\|_{{\mathcal{F}} }$ and $I_{\Phi_p}(1)=1$.
\end{proposition}
We can now present an interesting example of infinite dimensional Fresnel integral with polynomial phase function.\par
Fixed a $p\in {\mathbb{N}}$, with $p\geq 2$, let us consider the Banach space
${\mathcal{B}}_p$ of absolutely continuous maps $\gamma:[0,t]\to{\mathbb{R}}$, with $\gamma(t)=0$ and a weak derivative $\dot \gamma $ belonging to $L^p([0,t])$, endowed with the norm:
$$\|\gamma\|_{{\mathcal{B}}_p}=\left(\int_0^t|\dot \gamma(s)|^pds\right)^{1/p}. $$
The application $T:{\mathcal{B}}_p\to L^p([0,t])$ mapping an element $\gamma\in {\mathcal{B}}_p$ to its weak derivative $\dot \gamma \in L^p([0,t])$ is an isomorphism and its inverse $T^{-1}:L^p([0,t])\to{\mathcal{B}}_p$ is given by:
\begin{equation}\label{T-1}
T^{-1}(v)(s)=-\int_s^t v(u)du\qquad v\in L^p([0,t]) .
\end{equation}
Analogously the dual space ${\mathcal{B}}_p^*$ is isomorphic to $ L^q([0,t])=(L^{p}([0,t]))^*$, with $\frac{1}{p}+\frac{1}{q}=1$, and the pairing $\langle \eta, \gamma\rangle$ between $\eta\in{\mathcal{B}}_p^*$ and $\gamma\in {\mathcal{B}}_p$ can be written in the following form:
$$\langle \eta, \gamma\rangle=\int_0^t\dot \eta(s)\dot\gamma(s)ds\qquad \dot\eta\in L_{q}([0,t]), \gamma\in{\mathcal{B}}_p.$$
Further ${\mathcal{B}}_p^*$ is isomorphic to ${\mathcal{B}}_q$.\\
Let us consider the space ${\mathcal{F}}({\mathcal{B}}_q) $ of functions $f:{\mathcal{B}}_q\to{\mathbb{C}}$ of the form
$$f(\eta)=\int_{{\mathcal{B}}_p}e^{i\int_0^t\dot \eta(s)\dot \gamma(s)ds}d\mu_f(\gamma),\, \quad \eta \in {\mathcal{B}}_q, \mu_f\in {\mathcal{M}}({\mathcal{B}}_p).$$
Let $\Phi_p:{\mathcal{B}}_p\to{\mathbb{C}}$ be the phase function defined as
$$\Phi_p(\gamma):=(-1)^p\alpha\int_0^t\dot\gamma(s)^pds,$$
where $\alpha\in{\mathbb{C}}$ is a complex constant such that
\begin{itemize}
\item ${\mathbb{R}}ea(\alpha)\leq 0$ if $p$ is even,
\item ${\mathbb{R}}ea(\alpha)= 0$ if $p$ is odd.
\end{itemize}
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Let us consider a real separable Banach space $({\mathcal{B}}, \|\,\|)$. Let ${\mathcal{M}}({\mathcal{B}})$ be the space of complex bounded variation measures on ${\mathcal{B}}$, endowed with the total variation norm. As remarked above, ${\mathcal{M}}({\mathcal{B}})$ is a Banach algebra under convolution. Let ${\mathcal{B}}^*$ be the topological dual of ${\mathcal{B}}$ and ${\mathcal{F}}({\mathcal{B}})$ the Banach algebra of complex-valued functions $f:{\mathcal{B}}^*\to {\mathbb{C}}$ of the form
\begin{equation}\label{Fo-Ba}f(x)=\int_{{\mathcal{B}}}e^{i\langle x,y\rangle}d\mu(y)\equiv \hat\mu(x), \qquad x\in {\mathcal{B}}^*,\, \mu\in {\mathcal{M}}({\mathcal{B}}), \end{equation}
where $\langle \, ,\,\rangle$ denotes the dual pairing between ${\mathcal{B}}$ and ${\mathcal{B}}^*$. The space ${\mathcal{F}}({\mathcal{B}})$ endowed with the norm $\|\hat \mu\|_{{\mathcal{F}}}:=\|\mu\|_{{\mathcal{M}}({\mathcal{B}})}$ and the pointwise multiplication is a Banach algebra of functions.\\
In the following we are going to define a class of linear continuous functionals on ${\mathcal{F}}({\mathcal{B}})$, by generalizing the construction of infinite dimensional Fresnel integrals defined by Eq. \eqref{Parseval2}.
\begin{definition}
Let $\Phi:{\mathcal{B}}\to {\mathbb{C}}$ be a continuous map such that ${\mathbb{R}}ea (\Phi_p(x))\leq 0$ for all $x\in{\mathcal{B}}$.
The infinite dimensional Fresnel integral on ${\mathcal{B}}^*$ with phase function $\Phi$ is the functional $I_{\Phi}:{\mathcal{F}}({\mathcal{B}})\to {\mathbb{C}}$, given by
\begin{equation}\label{Parseval-p}I_{\Phi}(f):=\int_{\mathcal{B}} e^{\Phi(x)}d\mu(x), \qquad f\in{\mathcal{F}}({\mathcal{B}}), f=\hat\mu.\end{equation}
\end{definition}
By construction, the functional $I_{\Phi}$ is linear and continuous, indeed:
$$|I_{\Phi_p}(f)|\leq \int_{\mathcal{B}} |e^{\Phi_p}|d|\mu|(x)\leq \|\mu\|=\|f\|_{{\mathcal{F}} }$$
Further $I_{\Phi}$ is normalized, i.e., $I_{\Phi_p}(1)=1$.
We summarize these properties in the following proposition.
\begin{proposition}
The space ${\mathcal{F}}({\mathcal{B}})$ of Fresnel integrable functions is a Banach function algebra in the norm $\|\,\|_{\mathcal{F}}$. The infinite dimensional Fresnel integral with phase function $\Phi$ is a continuous bounded linear functional $I_{\Phi}:{\mathcal{F}}({\mathcal{B}})\to {\mathbb{C}}$ such that $|I_{\Phi_p}(f)|\leq \|f\|_{{\mathcal{F}} }$ and $I_{\Phi_p}(1)=1$.
\end{proposition}
We can now present an interesting example of infinite dimensional Fresnel integral with polynomial phase function.\par
Fixed a $p\in {\mathbb{N}}$, with $p\geq 2$, let us consider the Banach space
${\mathcal{B}}_p$ of absolutely continuous maps $\gamma:[0,t]\to{\mathbb{R}}$, with $\gamma(t)=0$ and a weak derivative $\dot \gamma $ belonging to $L^p([0,t])$, endowed with the norm:
$$\|\gamma\|_{{\mathcal{B}}_p}=\left(\int_0^t|\dot \gamma(s)|^pds\right)^{1/p}. $$
The application $T:{\mathcal{B}}_p\to L^p([0,t])$ mapping an element $\gamma\in {\mathcal{B}}_p$ to its weak derivative $\dot \gamma \in L^p([0,t])$ is an isomorphism and its inverse $T^{-1}:L^p([0,t])\to{\mathcal{B}}_p$ is given by:
\begin{equation}\label{T-1}
T^{-1}(v)(s)=-\int_s^t v(u)du\qquad v\in L^p([0,t]) .
\end{equation}
Analogously the dual space ${\mathcal{B}}_p^*$ is isomorphic to $ L^q([0,t])=(L^{p}([0,t]))^*$, with $\frac{1}{p}+\frac{1}{q}=1$, and the pairing $\langle \eta, \gamma\rangle$ between $\eta\in{\mathcal{B}}_p^*$ and $\gamma\in {\mathcal{B}}_p$ can be written in the following form:
$$\langle \eta, \gamma\rangle=\int_0^t\dot \eta(s)\dot\gamma(s)ds\qquad \dot\eta\in L_{q}([0,t]), \gamma\in{\mathcal{B}}_p.$$
Further ${\mathcal{B}}_p^*$ is isomorphic to ${\mathcal{B}}_q$.\\
Let us consider the space ${\mathcal{F}}({\mathcal{B}}_q) $ of functions $f:{\mathcal{B}}_q\to{\mathbb{C}}$ of the form
$$f(\eta)=\int_{{\mathcal{B}}_p}e^{i\int_0^t\dot \eta(s)\dot \gamma(s)ds}d\mu_f(\gamma),\, \quad \eta \in {\mathcal{B}}_q, \mu_f\in {\mathcal{M}}({\mathcal{B}}_p).$$
Let $\Phi_p:{\mathcal{B}}_p\to{\mathbb{C}}$ be the phase function defined as
$$\Phi_p(\gamma):=(-1)^p\alpha\int_0^t\dot\gamma(s)^pds,$$
where $\alpha\in{\mathbb{C}}$ is a complex constant such that
\begin{itemize}
\item ${\mathbb{R}}ea(\alpha)\leq 0$ if $p$ is even,
\item ${\mathbb{R}}ea(\alpha)= 0$ if $p$ is odd.
\end{itemize}
The infinite dimensional Fresnel integral on $B_q$ with phase function $\Phi_p$ is the functional $I _{\Phi_p} : {\mathcal{F}}({\mathcal{B}}_q)\to{\mathbb{C}}$ given by
\begin{equation}\label{funzI-p}
I _{\Phi_p}(f)=\int_{{\mathcal{B}}_p}e^{(-1)^p\alpha \int_0^t\dot \gamma(s)^pds}d\mu_f(\gamma), \quad f\in {\mathcal{F}}({\mathcal{B}}_q),\,f=\hat\mu_f.
\end{equation}
The following lemma states an interesting connection between the functional \eqref{funzI-p} and the high-order PDE \eqref{PDE-p}.
\begin{lemma}\label{lemmacyl}
Let $f: {\mathcal{B}}_q\to{\mathbb{C}}$ be a cylinder function of the
following form:
$$f(\eta)=F(\eta(t_1), \eta(t_2), ...,\eta(t_n)),\qquad \eta \in {\mathcal{B}}_q,$$ with $0\leq t_1<t_2<...<t_n< t$ and $F:{\mathbb{R}}^n\to {\mathbb{C}}$, $F\in {\mathcal{F}}({\mathbb{R}}^n)$:
$$F(x_1,x_2, ..., x_n)=\int_{{\mathbb{R}}^n}e^{i\sum_{k=1}^ny_kx_k}d\nu_F(y_1,...,y_n), \qquad \nu_F\in{\mathcal{M}}({\mathbb{R}}^n).$$
Then $f\in{\mathcal{F}}({\mathcal{B}}_p)$ and its infinite dimensional Fresnel integral with phase function $\Phi_p$ is given by
\begin{equation}\label{IntIphi-p}
I _{\Phi_p}(f)=\int^o_{{\mathbb{R}}^n}F(x_1,x_2, ...,x_n)\Pi_{k=1}^nG^p_{t_{k+1}-t_{k}}(x_{k+1},x_{k})dx_1...dx_n,
\end{equation}
where $x_{n+1}\equiv 0$, $t_{n+1}\equiv t$ , $G^p_s$ is the fundamental solution \eqref{Green} of the high order heat-type equation \eqref{PDE-p} and the integral on the right hand side of \eqref{IntIphi-p} is an oscillatory integral in the sense of definition \ref{def-int-osc}.
\end{lemma}
\begin{Remark}
In the case $p$ is even and ${\mathbb{R}}ea(\alpha) <0$ the integral \eqref{IntIphi-p} is an absolutely convergent Lebesgue integral because of the boundedness of the function $F\in {\mathcal{F}}({\mathbb{R}}^n)$ and the summability of the function $g^p_t$ stated in lemma \ref{lemma1-asy}.
\end{Remark}
\begin{proof}[Proof of lemma \ref{lemmacyl}]
The proof that $f\in{\mathcal{F}}({\mathcal{B}}_p)$ follows froms the explicit form of the function $f$
$$f(\eta)=F(\eta(t_1), \eta(t_2), ...,\eta(t_n))=\int_{{\mathbb{R}}^n}e^{i\sum_{k=1}^ny_k\eta(t_k)}d\nu_F(y_1,...,y_n)),\quad \eta\in {\mathcal{B}}_q.$$
and the identity
$$e^{iy\eta(s)}=\int _{{\mathcal{B}}_p}e^{i\langle \eta,\gamma \rangle}\delta_{yv_{s}}(\gamma),$$ where $v_s\in{\mathcal{B}}_p$ is the vector of ${\mathcal{B}}_p$ defined by $$\langle \eta, v_s\rangle=\eta(s), \qquad \forall \eta \in {\mathcal{B}}_q,$$ which can be explicitly written as $$v_s(\tau)=\chi_{[0,s]}(t-s)+\chi_{(s,t]}(t-\tau)s.$$
By the definition of the functional $I_{\Phi_p}$ we have
\begin{eqnarray}
I _{\Phi_p}(f)
&=&\int_{{\mathbb{R}}^n} e^{(-1)^p\alpha \int_0^t \left(\sum_{k=1}^ny_k\dot v_{t_k}(\tau)\right)^p d\tau}d\nu_F(y_1,...,y_n)\nonumber\\
&=&\int_{{\mathbb{R}}^n} e^{\alpha \int_0^t \left(\sum_{k=1}^ny_k\chi_{(t_k,t]}(\tau)\right)^p d\tau}d\nu_F(y_1,...,y_n)\nonumber\\
&=&\int_{{\mathbb{R}}^n} e^{\alpha \int_0^t \left(\sum_{k=1}^n\chi_{(t_k,t_{k+1}]}(\tau)\sum_{j=1}^ky_j\right)^p d\tau}d\nu_F(y_1,...,y_n)\nonumber\\
&=&\int_{{\mathbb{R}}^n} e^{\alpha\sum_{k=1}^n ( \sum_{j=1}^k y_j)^p(t_{k+1}-t_{k})}d\nu_F(y_1,...,y_n)\label{eq16}
\end{eqnarray}
On the other hand the last line of Eq. \eqref{eq16} coincides with the oscillatory integral
\begin{equation}\int^o_{{\mathbb{R}}^n}F(x_1,x_2, ...,x_n)\Pi_{k=1}^nG^p_{t_{k+1}-t_{k}}(x_{k+1},x_{k})dx_1...dx_n.
\end{equation}
Indeed, taken an arbitrary test function $\phi\in S({\mathbb{R}}^n)$ such that $\phi(0)=1$, the the function $F_\epsilon:{\mathbb{R}}^n\to{\mathbb{C}}$
$$F_\epsilon (x_1,x_2, ...,x_n)\equiv F(x_1,x_2, ...,x_n)\phi(\epsilon x_1,\epsilon x_2, ...,\epsilon x_n))\Pi_{k=1}^nG^p_{t_{k+1}-t_{k}}(x_{k+1},x_{k})$$
is summable because of the boundedness of $F\in{\mathcal{F}}({\mathbb{R}}^n)$ and the decaying properties at infinity stated in lemma \ref{lemma1-asy}. Further a change of variable argument and Fubini theorem yield:
$$\int_{{\mathbb{R}}^n}F_\epsilon(x)dx
=\int_{{\mathbb{R}}^n} \left( \int_{{\mathbb{R}}^n}e^{\alpha\sum_{k=1}^n(t_{k+1}-t_k)(\sum_{j=1}^ky_j+\epsilon \xi_j)^p}\hat \phi (\xi
)d\xi\right) d\nu_F(y),$$
where $\phi(x)=\int_{{\mathbb{R}}^n}e^{ix \xi }\hat \phi (\xi)d\xi$. By dominated convergence theorem and the condition $\phi(0)=\int_{{\mathbb{R}}^n}\hat \phi (\xi)d\xi=1$, we eventually obtain
$$\lim_{\epsilon\to 0}\int_{{\mathbb{R}}^n}F_\epsilon(x)dx =\int_{{\mathbb{R}}^n} e^{\alpha\sum_{k=1}^n(t_{k+1}-t_k)(\sum_{j=1}^ky_j)^p}d\nu_F(y),$$
\end{proof}
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\begin{corollary}\label{cor1}
Let $u_0\in {\mathcal{F}}({\mathbb{R}})$. Then the cylinder function $f_0:{\mathcal{B}}_q\to{\mathbb{C}}$ defined by
$$f_0(\eta):=u_0(x+\eta(0)), \qquad x\in {\mathbb{R}}, \eta\in {\mathcal{B}}_q,$$
belongs to ${\mathcal{F}}({\mathcal{B}}_q)$ and its infinite dimensional Fresnel integral with phase function $\Phi_p$ provides a representation for the solution of the Cauchy problem \eqref{PDE-p}, in the sense that the function
$u(t,x):=I_{\Phi_p}(f_0)$ has the form \begin{equation}\label{sol-free}u(t,x)=\int^o _{\mathbb{R}} G_t(x,y)u_0(y)dy.\end{equation}
In the case $p$ is even and ${\mathbb{R}}ea(\alpha)<0$ then the integral \eqref{sol-free} is absolutely convergent, while in the general case it is meant in the oscillatory sense of definition \ref{def-int-osc}.
\end{corollary}
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\section{A generalized Feynman-Kac formula}
In the present section, we consider a Cauchy problem of the form
\begin{equation} \label{PDE-p-V}\left\{ \begin{aligned}
\frac{\partial}{\partial t}u(t,x)&=(-i)^p \alpha \frac{\partial^p}{\partial x^p}u(t,x)+V(x)u(t,x)\\
u(0,x) &= u_0(x),\qquad x\in{\mathbb{R}}, t\in [0,+\infty)
\end{aligned}\right. \end{equation}
where $p\in{\mathbb{N}}$, $p\geq 2$, and $\alpha\in{\mathbb{C}}$ is a complex constant such that $|e^{\alpha tx^p}|\leq 1$ forall $x\in{\mathbb{R}}, t\in [0,+\infty)$, while $V:{\mathbb{R}}\to{\mathbb{C}}$ is a bounded continuous function. Under these assumption the Cauchy problem \eqref{PDE-p-V} is well posed in $L^2({\mathbb{R}})$. Indeed the operator ${\mathcal{D}}_p:D({\mathcal{D}}_p)\subset L^2({\mathbb{R}})\to L^2({\mathbb{R}})$ defined by
\begin{eqnarray*}
D({\mathcal{D}}_p)&:=& H^p=\{u\in L^2({\mathbb{R}}), k\mapsto k^p\hat u(k)\in L^2 ({\mathbb{R}}) \},\\
\widehat{{\mathcal{D}}_pu}(k)&:=&k^p\hat u(k), \, u\in D({\mathcal{D}}_p),
\end{eqnarray*}
($\hat u$ denoting the Fourier transform of $u$) is self-adjoint. For $\alpha\in{\mathbb{C}}$, with $|e^{\alpha tx^p}|\leq 1$ forall $x\in{\mathbb{R}}, t\in [0,+\infty)$, one has that the operator $A:=\alpha D_p$ generates a strongly continuous semigroup $(e^{tA})_{t\geq 0}$ on $L^2(R)$. By denoting with $B:L^2({\mathbb{R}})\to L^2({\mathbb{R}})$ the bounded multiplication operator defined by
$$Bu(x)=V(x)u(x), \qquad u\in L^2({\mathbb{R}}),$$
one has that the operator sum $A+B:D(A)\subset L^2({\mathbb{R}})\to L^2({\mathbb{R}})$ generates a strongly continuous semigroup $(T(t))_{t\geq 0}$ on $L^2({\mathbb{R}})$. Moreover, given a $u\in L^2({\mathbb{R}})$, the vector $T(t)u$
can be computed by means of the convergent (in the $L^2({\mathbb{R}})$-norm) Dyson series (see \cite{HiPhi}, Th. 13.4.1):
\begin{equation}\label{Dyson}
T(t)u=\sum_{n=0}^\infty S_n(t)u,
\end{equation}
where $S_0(t)u=e^{tA}u$ and $S_n(t)u=\int_0^t e^{(t-s)A}VS_{n-1}(s)uds$.
By passing to a subsequence, the series above converges also a.e. in $x\in{\mathbb{R}}$ giving
\begin{multline}\label{Dyson2}
T(t)u (x)=\\
=\sum_{n=0}^\infty \;\; \idotsint\limits _{0\leq s_1 \leq \dots \leq s_n \leq t }\int_{{\mathbb{R}}^{n+1}}V(x_1)\dots V(x_n) G_{t-s_n}(x,x_n)
G_{s_n-s_{n-1}}(x_n,x_{n-1})\\ \dots G_{s_1}(x_1,x_0)
u_0(x_0)dx_0\dots dx_n\,ds_1\dots ds_n ,\qquad a.e.\; x\in {\mathbb{R}}.
\end{multline}
Under suitable assumptions on the initial datum $u_0$ and the potential $V$, we are going to construct a representation of the solution of equation \eqref{PDE-p-V} in $L^2({\mathbb{R}})$ in terms of an infinite dimensional oscillatory integral with polynomial phase.
\begin{teorema}
Let $u_0\in {\mathcal{F}}({\mathbb{R}})\cap L^2({\mathbb{R}})$ and $V\in {\mathcal{F}}({\mathbb{R}})$, with $u_0(x)=\int_{\mathbb{R}} e^{ixy}d\mu_0(y)$ and $V(x)=\int_{\mathbb{R}} e^{ixy}d\nu(y)$, $\mu_0,\nu\in{\mathcal{M}}({\mathbb{R}})$. Then the functional $f_{t,x}:{\mathcal{B}}_q\to{\mathbb{C}}$ defined by
\begin{equation}\label{f-tx}
f_{t,x}(\eta):=u_0(x+\eta(0))e^{\int_0^tV(x+\eta(s))ds}, \qquad x\in {\mathbb{R}}, \eta\in {\mathcal{B}}_q,
\end{equation}
belongs to ${\mathcal{F}}({\mathcal{B}}_q)$ and its infinite dimensional Fresnel integral with phase function $\Phi_p$ provides a representation for the solution of the Cauchy problem \eqref{PDE-p-V}.
\end{teorema}
\begin{Remark}
By Plancherel's theorem the assumption that $u_0\in {\mathcal{F}}({\mathbb{R}})\cap L^2({\mathbb{R}})$ is equivalent to the fact that $u_0$ is the Fourier transform of a function $\hat u_0\in L^1({\mathbb{R}})\cap L^2({\mathbb{R}})$.
\end{Remark}
\begin{proof}
Let $\mu_V\in {\mathcal{M}}({\mathcal{B}}_p)$ be the measure defined by
$$\int_{{\mathcal{B}}_p}f(\gamma)d\mu_V(\gamma)=\int_0^t\int_{\mathbb{R}} e^{ixy}f(y\,v_s)d\nu(y)ds, \qquad f\in C_b({\mathcal{B}}_p),$$
where $v_s\in {\mathcal{B}}_p$ is the function $v_s(\tau)=\chi_{[0,s]}(\tau)(t-s)+\chi_{(s,t]}(t-\tau)s$. One can easily verify that $\|\mu_V\|_{{\mathcal{M}}({\mathcal{B}}_p)}\leq t\|\nu\|_{{\mathcal{M}}({\mathbb{R}})}$ and the map $\eta \in {\mathcal{B}}_q\mapsto \int_0^tV(x+\eta(s))ds$ is the Fourier transform of $\mu_V$. Analogously the map $\eta \in {\mathcal{B}}_q\mapsto \exp(\int_0^tV(x+\eta(s))ds) $ is the Fourier transform of the measure $\nu_V\in {\mathcal{M}}({\mathcal{B}}_p)$ given by $\nu_V=\sum _{n=0}^\infty\frac{1}{n!}\mu_V^{*n}$, where $\mu_V^{*n}$ denotes the $n$-fold convolution of $\mu_V$ with itself.
The series is convergent in the ${\mathcal{M}}({\mathcal{B}}_p)$-norm and one has $\|\nu_V\|_{{\mathcal{M}}({\mathcal{B}}_p)}\leq e^{t\|\nu\|_{{\mathcal{M}}({\mathbb{R}})}}$. Further, by lemma \ref{lemmacyl} the cylinder function $\eta\mapsto u_0(x+\eta(0))$, $\eta \in {\mathcal{B}}_q$, is an element of ${\mathcal{F}}({\mathcal{B}}_q)$. More precisely, it is the Fourier transform of the measure $\nu_{u_0}$ defined by
$$\int_{{\mathcal{B}}_p}f(\gamma)d\nu_{u_0}(\gamma)=\int _{\mathbb{R}} e^{ixy}f(y\,v_0)d\mu_0(y), \qquad f\in C_b({\mathcal{B}}_p).$$
We can then conclude that the map
$f_{t,x}:{\mathcal{B}}_q\to{\mathbb{C}}$ defined by \eqref{f-tx} belongs to ${\mathcal{F}}({\mathcal{B}}_q)$ and its infinite dimensional Fresnel integral $ I_{\Phi_p}(f_{t,x})$ with phase function $\Phi_p$ is given by
\begin{multline}\nonumber
\sum_{n=0}^\infty \frac{1}{n!}\int_{{\mathcal{B}}_p}e^{(-1)^p\alpha \int_0^y \dot \gamma(s)^pds}d\nu_{u_0}*\mu_V*\cdots *\mu_V=\\
\sum_{n=0}^\infty \frac{1}{n!}\int_0^t...\int_0^t I_{\Phi_p}\big(u_0(x+\eta(0))V(x+\eta(s_1))\dots V(x+\eta(s_n))\big)ds_1 \cdots ds_n
\end{multline}
By the symmetry of the integrand the latter is equal to
$$\sum_{n=0}^\infty\; \idotsint\limits _{0\leq s_1 \leq \dots \leq s_n \leq t } I_{\Phi_p}\big(u_0(x+\eta(0))V(x+\eta(s_1))\dots V(x+\eta(s_n))\big)ds_1 \cdots ds_n$$
By lemma \ref{lemmacyl} we eventually obtain
\begin{multline}\nonumber \sum_{n=0}^\infty\; \idotsint\limits _{0\leq s_1 \leq \dots \leq s_n \leq t }\int_{{\mathbb{R}}^{n+1}} u_0(x+x_0)V(x+x_1)\dots V(x+x_n)G_{s_1}(x_1,x_0)\\G_{s_2-s_1}(x_2,x_1)\dots G_{t-s_n}(0,x_{n})dx_0 dx_1\cdots dx_n ds_1 \cdots ds_n,
\end{multline}
that coincides with the Dyson series \eqref{Dyson2} for the solution of the high-order PDE \eqref{PDE-p-V}, as one can easily verify by means of a change of variables argument.
\end{proof}
\section*{Acknowledgments}
Many interesting discussions with Prof. S. Albeverio, S. Bonaccorsi, G. Da Prato and L.Tubaro are gratefully acknowledged, as well as the financial support of CIRM-Fondazione Bruno Kessler to the project { \em Functional integration and applications to quantum dynamical systems}.
\end{document}
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\begin{document}
\begin{flushleft}
JOURNAL OF COMBINATORIAL THEORY, Series B \textbf{42}, 146-155 (1987)
\end{flushleft}
$$$$$$$$
\begin{center}
\textbf{\large Circuit Preserving Edge Maps II}
\end{center}
\begin{center}
\textbf{Jon Henry Sanders }
\end{center}
$$$$
\begin{center}
JHS Consulting jon\[email protected]\\
\end{center}
\noindent\par In Chapter 1 of this article we prove the following. Let $f:G\rightarrow G^\prime$ be a \textit{circuit surjection,} i.e., a mapping of the edge set of $G$ onto the edge set of $G^\prime$ which maps circuits of $G$ onto circuits of $G^\prime,$ where $G,G^\prime$ are graphs without loops or multiple edges and $G^\prime$ has no isolated vertices. We show that if $G$ is assumed finite and 3-connected, then $f$ is induced by a vertex isomorphism. If $G$ is assumed 3-connected but not necessarily finite and $G^\prime$ is assumed to not be a circuit, then $f$ is induced by a vertex isomorphism. Examples of circuit surjections $f:G\rightarrow G^\prime$ where $G^\prime$ is a circuit and $G$ is an infinite graph of arbitrarily large connectivity are given. In general if we assume $G$ two-connected and $G^\prime$ not a circuit then any circuit surjection $f:G \rightarrow G^\prime$ may be written as the composite of three maps, $f(G)=q(h(k(G))),$ where $k$ is a $1-1$ onto edge map which preserves circuits in both directions (the``2-isomorphism'' of Whitney(\textit{Amer. J. Math.} 55(1993), 245-254 ) when $G$ is finite), $h$ is an onto edge \textit{circuit injection} (a 1-1 circuit surjection). Let $f: G\rightarrow M$ be a 1-1 onto mapping of the edges of $G$ onto the cells of $M$ which takes circuits of $G$ onto circuits of $M$ where $G$ is a graph with no isolated vertices, $M$ a matroid. If there exists a circuit $C$ of $M$ which is not the image of a circuit in $G$, we call $f$ \textit{nontrivial}, otherwise \textit{trivial}. In Chapter 2 we show the following. Let $G$ be a graph of even order. Then the statement `` no nontrivial map $f: g\rightarrow M$ exists, where $M$ is a binary matroid,'' is equivalent to ``$G$ is Hamiltonian.'' If $G$ is a graph of odd order, then the statement ``no nontrivial map $f:G\rightarrow M$ exists, where $M$ is a binary matroid'' is equivalent to ``$G$ is almost Hamiltonian'', where we define a graph $G$ of order $n$ to be \textit{almost Hamiltonian} if every subset of vertices of order $n-1$ is contained in some circuit of $G$.
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\textbf{INTRODUCTION AND PRELIMINARY DEFINITIONS}
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\noindent\par The results obtained in this paper grew from an attempt to generalize the main theorem of [1]. There it was shown that any \textit{circuit injection} (a 1-1 onto edge map $f$ such that if $C$ is a circuit then $f(C)$ is a circuit from a 3-connected (not necessarily finite)) graph $G$ onto a graph $G^\prime$ is induced by a vertex isomorphism, where $G^\prime$ is assumed to not have any isolated vertices. In the present article we examine the situation when the 1-1 condition is dropped (Chapter 1). An interesting result then is that the theorem remains true for finite (3-connected ) graphs $G$ but not for infinite $G$.
\par In Chapter 2 we retain the 1-1 condition but allow the image of $f$ to be first an arbitrary matroid and second a binary matroid.
\par Throughout this paper we will assume that graphs are undirected without loops or multiple edges and not necessarily finite unless otherwise stated. We will denote the set of edges of a graph $G$ by $E(G)$ and the set of vertices of $G$ by $V(G)$. We will also use the notation $G=(V,E)$ to indicate $V=V(G), E=E(G)$ when $G$ is a graph. The graph $G: A$ will be the graph with edge set $A$ and vertex set $V(G)$. The abuse of language of referring to a set of edges $S$ as a graph (usually a subgraph of a given graph) will be tolerated where it is understood that the set of vertices of such a graph is simply the set of all vertices adjacent to any edge of $S$.
\par A subgraph $P$ of a graph $G$ is a \textit{suspended chain} of $G$ if $|V|\geq 3, |V|$ finite and there exists two distinct vertices $v_1,v_2\in V$, the endpoints of $P$ such that $\deg_Pv_1=1,~~ \deg_P v_2=1$, and $\deg_Pv=\deg_Gv=2$ for $v\in V, v\neq v_1,v_2$, where $V=V(P).$ We shall also refer to the set of edges of $P$ as a suspended chain. The notation $\mathscr{C}(v)$ will be used to indicated the set of edges adjacent to the vertex $v$ in a given graph.
\par A \textit{circuit surjection f} of $G$ onto $G^\prime,$ denoted by $f:G\rightarrow G^\prime$, is an onto map of the edge set of $G$ onto the edge set of $G^\prime$ such that if $C$ is a circuit of $G$ then $f(C)$ is a circuit of $G^\prime$. We also understand the terminology $f:G\rightarrow G^\prime$ is a circuit surjrction to preclude the possibility of $G^\prime$ having isolated vertices.
\chapter{\small 1. CIRCUIT SURJECTIONS ONTO GRAPHS}
\numberwithin{theorem}{chapter}
\begin{lemma}
Let $f:G\rightarrow G^\prime$ be circuit surjection where $G$ is 2-connected and $G^\prime$ is not a circuit. Let $e$ be an edge of $G^\prime$. Then if $C$ is circuit of $G$ such that $C$ contains at least one element of $f^{-1}(e)$ then $C$ contains every element of $f^{-1}(e).$
\end{lemma}
\noindent\par\textit{Proof.}~~~~ First, we note that $G^\prime$ is 2-connected since if $e_1,e_2$ are two distinct edges of $G^\prime$ then $f(C)$ is a circuit which contains $e_1$ and $e_2$ where $C$ is any circuit of $G$ which contains $h_1,h_2$ such that $h_1\in f^{-1}(e_1),h_2\in f^{-1}(e_2).$ Let $v_1,v_2$ be the vertices adjacent to $e$. Let $P(v_1,v^\prime)$ be a path in $G^\prime$ of minimal length such that $v^\prime$ is a vertex of degree greater than 2. Define $S=\mathscr{C}(v^\prime)-\{h\}$ if $v^\prime\neq v_1,~~ S=\mathscr{C}(v^\prime)-\{e\}$ if $v^\prime=v_1,$ where $h$ is the edge in $P(v_1,v^\prime)$ adjacent to $v^\prime.$\\
FACT 1.~~ Any circuit of $G^\prime$ which contains $e$ must contain one and only one element of $S$.
\par Let $a_\alpha,\alpha\in I$ be the elements of $S$ and let $A=f^{-1}(e),A_\alpha=f^{-1}(a_\alpha),\alpha\in I.$ Then Fact 1 implies\\
FACT 2.~~ If $C\cap A\neq\O$ for $C$ a circuit of $G$ then $C\cap A_\alpha\neq\O$ is true for one and only $\alpha\in I.$
\par Let $C_0$ be a circuit which contains an edge of $A$. We will show that the assumption $C_0\not\supset A$ leads to a contradiction of Fact2. Denote by $B$ the unique set $A_{\alpha 0}, \alpha_0\in I$ such that $C_0\cap A_{\alpha 0}\neq\O.$ Let $D=A_{\alpha 1}, \alpha_1\neq \alpha_0$ (since $|I|=|S|\geq 2$, this is possible) and let $d\in D$. Since $G$ is 2-connected and $d\notin C_0$ there is a path $P_3(q_0,q_1),d\in P_3(q_0,q_1)$ where $q_0,q_1$ are distinct vertices of $C_0$ and $P_3(q_0,q_1)$ is edge disjoint from $C_0.$ Denote by $P_1(q_0,q_1)$ and $P_2(q_0,q_1)$ the two paths such that $C_0=P_1(q_0,q_1)\cup P_2(q_0,q_1).$ Now $P_i\cap A\neq\O$ and $P_i\cap B\neq\O$ is not possible, $i=1$ or $2$, since then $P_3\cup P_i$ would be a circuit which violates Fact 2. Thus $P_i\cap A\neq\O (P_i\cap B=\O),$ and $P_j\cap B\neq\O (P_j\cap A=\O)$ where either $i=1, j=2,$ or $j=1, i=2,$ say, the former (Fig. 1).
\par Suppose now there exists an edge $k\in A,k\notin C_0$. Now $k\in P_3$ is impossible since if that were the case then $P_3\cup P_2$ would be a circuit which violates Fact 2. Thus $k$ is edge disjoint from $G^{\prime\prime}$, where $G^{\prime\prime}$ is the subgraph of $G$ consisting of $P_3\cup P_1\cup P_2.$ Since $G$ is 2-connected there exists a path $P_4(t_0,t_1)$ in $G$ such that $k\in P_4(t_0,t_1),t_0,t_1$ are distinct vertices of $G^{\prime\prime}$ and $P_4(t_0,t_1)$ is edge disjoint from $G^{\prime\prime}$. We now show that no matter where $t_0,t_1$ fall on $G^{\prime\prime}$ a contradiction to Fact 2 arises. For if $G^{\prime\prime}$ has a $t_0-t_1$ path $P_5$ disjoint from $B\cup D,$ then $P_4\cup P_5$ is a circuit intersecting $A$ and hence $P_4$ intersects some $A_\alpha$. Since $P_4$ can be extended to a circuit intersecting $B$ (resp. $D$) this contradicts Fact 2. If $G^{\prime\prime}$ has no such path $P_5$, then it has a $t_0-t_1$ path intersecting both $B$ and $D$ and that path union $P_4$ contradicts Fact 2.\\
\begin{theorem}
Let $f:G\rightarrow H$ be a circuit surjection, where $G$ is 2-connected and $H$ is not a circuit. Then $f$ is the composite of three maps $f(G)=g(h(k(G))),$ where $k$ is a 1-1 onto edge map which preserves circuits in both directions (a ``2-isomorphism'' of [8] when $G$ is finite), $h$ is an onto edge map obtained by replacing suspended chains by single edges (which preserves circuits in both directions) and $q$ is a circuit injection.
\end{theorem}
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\includegraphics[scale=1.0]{Images/fig1.jpg}
\noindent\par We note that the theorem implies that $f^{-1}(e)$ is a finite set for each edge of $H$ and thus $H$ must be infinite if $G$ is infinite.
\par Theorem 1.1 follows from the fact that (by Lemma 1.1) for any $e\in H$, any two edges of $f^{-1}(e)$ form a minimal cut set (cocycle) It is apparent that $f^{-1}(e)$ can thus be transformed into a suspended chain by a sequence of 2-switchings. This establishes Theorem 1.1 for finite $G$. Theorem 1.1 also holds for infinite $G$ by the same method used in Theorem 4.1 of [3] (where Whitney's 2-isomorphism theorem [8] is extended to the infinite case).
\begin{theorem}
Let $f:G\rightarrow G^\prime$ be a circuit surjection, where $G$ is finite and 3-connected. Then $f$ is induced by a vertex isomorphism.
\end{theorem}
\noindent\par\textit{Proof.}~~~~ We will show that $G^\prime$ cannot be a circuit. For assume $G^\prime$ is a $k$-circuit, $k\geq3.$ Write $G=(V,E)$ and $|V|=n$. Now $f^{-1}(G^\prime-\{e\})$ contains no circuit and thus $|f^{-1}(G^\prime-\{e_i\})|<n,i=1,\ldots,k,$ ~~ where $e_1,\ldots,e_k$ are the edges of $G^\prime.$ But each of $G,$ i.e., each element of $E$ occurs in exactly $k-1$ of the $k$ sets $f^{-1}(G^\prime-\{e_i\},i=1),\ldots,k,$ and $E=\displaystyle \bigcup_{i=1,\ldots,k} f^{-1}(G^\prime-\{e_i\}).$ Thus $(k-1)|E|<kn,$ or $|E|<(k/(k-1))n,$ and thus $|E|<\frac{3}{2}n.$ But $|E|\geq\frac{3}{2}n$ for any (finite) graph each vertex of which is of degree three or greater and thus for any 3-connected finite graph,$\Rightarrow\Leftarrow$. Thus $G^\prime$ cannot be a circuit. Theorem 1.1 thus implies that $f$ is 1-1 so the result follows from [1].
\begin{theorem}
Let $f:G\rightarrow G^\prime$ be a circuit surjection, where $G$ is 3-connected, not necessarily finite and $G^\prime$ is not a circuit. Then $f$ is induced by a vertex isomorphism.
\end{theorem}
\noindent\par\textit{Proof.}~~~~ Theorem 1.1 implies that $f$ must be a 1-1 map so the result follows from [1].\\
\textit{Construction}\\
\noindent\par An $n$-connected graph which has a circuit surjection onto a 3-circuit may be obtained from a sequence of disjoint 2-way infinite paths $P_1,P_2,\ldots,$ such that each vertex of $P_i$ is ``connected'' to $P_{i+1}$ by a tree as indicated in Fig. 2 for $n=4$. (The mapping which takes each edge labeled $i$ onto $e_i,i=1,2,3,$ defines the circuit surjection onto the 3-circuit with edges $e_1,e_2,$ and $e_3$)
\chapter{\small 2. CIRCUIT INJECTIONS ONTO MATROIDS}
\textit{Terminology and Notation}\\
\noindent\par A \textit{matroid} $M$ is an ordered pair of sets $\{S,\mathscr{C}\},$ where $S\neq\O,\mathscr{C}\subseteq 2^S$, which satisfies the following two axioms. Axiom I. $A,B\in\mathscr{C}, A\subseteq B$ implies $A=B$. Axiom II. $A,B\in\mathscr{C}, a\in A\cap B,~~ b\in(A\cup B)-(A\cap B)$ implies there
\includegraphics{Images/fig22.jpg}
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\begin{theorem}
Let $f:G\rightarrow H$ be a circuit surjection, where $G$ is 2-connected and $H$ is not a circuit. Then $f$ is the composite of three maps $f(G)=g(h(k(G))),$ where $k$ is a 1-1 onto edge map which preserves circuits in both directions (a ``2-isomorphism'' of [8] when $G$ is finite), $h$ is an onto edge map obtained by replacing suspended chains by single edges (which preserves circuits in both directions) and $q$ is a circuit injection.
\end{theorem}
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\includegraphics[scale=1.0]{Images/fig1.jpg}
\noindent\par We note that the theorem implies that $f^{-1}(e)$ is a finite set for each edge of $H$ and thus $H$ must be infinite if $G$ is infinite.
\par Theorem 1.1 follows from the fact that (by Lemma 1.1) for any $e\in H$, any two edges of $f^{-1}(e)$ form a minimal cut set (cocycle) It is apparent that $f^{-1}(e)$ can thus be transformed into a suspended chain by a sequence of 2-switchings. This establishes Theorem 1.1 for finite $G$. Theorem 1.1 also holds for infinite $G$ by the same method used in Theorem 4.1 of [3] (where Whitney's 2-isomorphism theorem [8] is extended to the infinite case).
\begin{theorem}
Let $f:G\rightarrow G^\prime$ be a circuit surjection, where $G$ is finite and 3-connected. Then $f$ is induced by a vertex isomorphism.
\end{theorem}
\noindent\par\textit{Proof.}~~~~ We will show that $G^\prime$ cannot be a circuit. For assume $G^\prime$ is a $k$-circuit, $k\geq3.$ Write $G=(V,E)$ and $|V|=n$. Now $f^{-1}(G^\prime-\{e\})$ contains no circuit and thus $|f^{-1}(G^\prime-\{e_i\})|<n,i=1,\ldots,k,$ ~~ where $e_1,\ldots,e_k$ are the edges of $G^\prime.$ But each of $G,$ i.e., each element of $E$ occurs in exactly $k-1$ of the $k$ sets $f^{-1}(G^\prime-\{e_i\},i=1),\ldots,k,$ and $E=\displaystyle \bigcup_{i=1,\ldots,k} f^{-1}(G^\prime-\{e_i\}).$ Thus $(k-1)|E|<kn,$ or $|E|<(k/(k-1))n,$ and thus $|E|<\frac{3}{2}n.$ But $|E|\geq\frac{3}{2}n$ for any (finite) graph each vertex of which is of degree three or greater and thus for any 3-connected finite graph,$\Rightarrow\Leftarrow$. Thus $G^\prime$ cannot be a circuit. Theorem 1.1 thus implies that $f$ is 1-1 so the result follows from [1].
\begin{theorem}
Let $f:G\rightarrow G^\prime$ be a circuit surjection, where $G$ is 3-connected, not necessarily finite and $G^\prime$ is not a circuit. Then $f$ is induced by a vertex isomorphism.
\end{theorem}
\noindent\par\textit{Proof.}~~~~ Theorem 1.1 implies that $f$ must be a 1-1 map so the result follows from [1].\\
\textit{Construction}\\
\noindent\par An $n$-connected graph which has a circuit surjection onto a 3-circuit may be obtained from a sequence of disjoint 2-way infinite paths $P_1,P_2,\ldots,$ such that each vertex of $P_i$ is ``connected'' to $P_{i+1}$ by a tree as indicated in Fig. 2 for $n=4$. (The mapping which takes each edge labeled $i$ onto $e_i,i=1,2,3,$ defines the circuit surjection onto the 3-circuit with edges $e_1,e_2,$ and $e_3$)
\chapter{\small 2. CIRCUIT INJECTIONS ONTO MATROIDS}
\textit{Terminology and Notation}\\
\noindent\par A \textit{matroid} $M$ is an ordered pair of sets $\{S,\mathscr{C}\},$ where $S\neq\O,\mathscr{C}\subseteq 2^S$, which satisfies the following two axioms. Axiom I. $A,B\in\mathscr{C}, A\subseteq B$ implies $A=B$. Axiom II. $A,B\in\mathscr{C}, a\in A\cap B,~~ b\in(A\cup B)-(A\cap B)$ implies there
\includegraphics{Images/fig22.jpg}
exists $D\in\mathscr{C}$ such that $D\subseteq A\cup B,a\notin D, b\in D.$ The elements of $S$ are called the cells of $M$, the elements of $\mathscr{C}$ are called the circuits of $M.$
\par The matroid associated with a graph $G_M$, is the matroid whose cells are the edges of $G$ and whose circuits are the circuits of $G$.
\par Let $M=\{S,\mathscr{C}\}, M^\prime=\{S^\prime,\mathscr{C}^\prime\}$ be matroids, and let $f:S\rightarrow S^\prime$ be a 1-1 onto map such that $f(A)\in\mathscr{C}^\prime$ whenever $A\in\mathscr{C}$. Such an $f$ is called a circuit injection of $M$ onto $M^\prime$ denoted by $f:M\rightarrow M^\prime.$ The circuit injection injection $f$ is called nontrivial if there exists $B\in\mathscr{C}^\prime$ such that $B\neq f(A)$ for all $A\in\mathscr{C}.$
\par We can assume without loss of generality that $S=S^\prime, f$ is the identity map and $\mathscr{C}\subseteq\mathscr{C}^\prime$ for a circuit injection $f$. Then $f$ is nontrivial if $\mathscr{C}$ is properly contained in $\mathscr{C}^\prime$.
\par We denote by $A\oplus B$ the $mod$ 2 $addition$ of set $A$ and $B$ which is defined to be the set $(A\cup B)-(a\cap B).$
\par A matroid $(S,\mathscr{C})$ is a binary matroid if for all $A,B\in\mathscr{C}, A\oplus B =\displaystyle \bigcup_{i=1}^k C_i$ for $C_i\in \mathscr{C}, i=1,\ldots,k,~~ C_i\cap C_j=\O, i\neq j,1\leq i, j\leq k$. Given a set $S$ and an arbitrary set $\mathscr{C}\subseteq 2^S$ we denote by $<\mathscr{C}>$ the collection of all sets $A$ such that there exists $k\geq1, C_1,\ldots, C_k\in\mathscr{C}$ and $A=C_1\oplus\cdots\oplus C_k.$
\par We denote by $<\mathscr{C}>_{\min}$ the minimal elements of $<\mathscr{C}>,$~~i.e., the elements $A\in<\mathscr{C}>$ such that $B\in<\mathscr{C}>, B\subseteq A\Rightarrow B=A$. A useful theorem of matroid theory [5, Sects. 1 and 5.3] is that $\{S,<\mathscr{C}>_{\min}\}$ is a binary matroid for arbitrary $\mathscr{C}\subseteq 2^S.$
\par We denote the rank of a matroid by $r(M)$. If $A\in\mathscr{C}$ exists such that $|A|=r(M)+1$ we call $A$ a $Hamiltonian$ $circuit$ of $M$, and we call $M$ $Hamiltonian$.\\
\textit{Condition for Trivial/Nontrivial Circuit Injections}\\
\par We would like to establish conditions on a graph $G$ such that all circuit injections $f:G_M\rightarrow N$ are trivial, where $N$ is first assumed to be an arbitrary matroid and second assumed to be a binary matroid. (We note that if $N$ is assumed to be a graphic matroid, i.e., $N=G^\prime_M$ for some graph $G^\prime$ then the theorem of [1] implies that $G$ 3-connected is a condition when ensures no nontrivial circuit injection exists).
\par Since the addition of an isolated vertex to a graph $G$ has no effect on $G_M$ we assume (without loss of generality) that $G$ has no isolated vertices throughout this section to simplify the statements of the theorems.\\
\textit{Remark.}~~~~ The fact that if $M$ is a Hamiltonian matroid (or in particular $G_M$, where $G$ is a Hamiltonian graph) then the only circuit injections $f: M\rightarrow M^\prime$ are trivial, where $M^\prime$ is an arbitrary matroid follows from the fact that $r(M^\prime)=r(M)$ in this case. The converse is also easily established as follows.
\begin{theorem}
If $G$ is a non-Hamiltonian matroid (or in particular the matroid associated with a graph without Hamiltonian circuits) there exists a nontrivial circuit injection $f:G\rightarrow M$, where $M$ is a (not in general binary) matroid.
\end{theorem}
\noindent\par\textit{Proof.}~~~~ Let the cells of $M$ be the edges of $G$; let the circuits of $M$ be $\mathscr{C}\cup\mathscr{L},$ where $\mathscr{C}$ is the set of circuits of $G$ and $\mathscr{L}$ is the set of all bases of $G$, and let $f$ be the identity map. Then $f$ is a nontrivial circuit injection (the matroid $M$ is the so-called truncation of $G$ see [7]).
\par \textit{Remark}.~~~~ Since matroids of arbitrarily large connectivity exist without Hamiltonian circuits (the duals of complete graphs are one example \footnote{We take the definition of connectivity for matroids from [4, 6]. A property of this definition is that the connectivity of a matroid equals the connectivity of its dual and also the connectivity of the matroid $G^n_M$ associated with the complete graph on n vertices $G^n$ approaches $\infty $ as $n \rightarrow \infty $. Thus the duals of the complete graphs have arbitrarily large connectivity.}) there is no general matroid analogue to the result of [1]. We note that $M$ is never a binary matroid in the construction of Theorem 2.1.
\par A more interesting result is obtained when we restrict $M$ to be an arbitrary matroid, $G$ a graphic matroid.\\
\par DEFINITION.~~~~Let the order of a graph $G$ be $n$. We say $G$ is almost Hamiltonian if every subset of $n-1$ vertices is contained in a circuit.
\begin{theorem}
Let the order of $G$ be even. Then ``no nontrivial circuit injection $f$ exists, $f:G_M\rightarrow B,$ where $B$ is binary'' is true iff $G$ is Hamiltonian. Let the order of $G$ be odd. Then ``no nontrivial circuit injection $f:G_M\rightarrow B$ exists where $B$ is binary'' is true iff $G$ is almost Hamiltonian.
\end{theorem}
We abbreviate ``no nontrivial circuit injection $f:G_M\rightarrow B$ exists, where $B$ is binary'' by saying ``$G$ has no nontrivial map.'' To prove the theorem we need the following
\begin{lemma}
$G$ has no nontrivial map implies ``if $v_1,\ldots,v_n$ are vertices of odd degree in $S$, for any subgraph $S$ of $G$, then there exists a circuit $C$ of $G$ such that $v_1,\ldots,v_n$ are vertices of C.''
\end{lemma}
\noindent\par\textit{Proof.}~~~~ Let $\mathscr{C}$ be the set of circuits of $G$, $S$ a subset of edges of $G$. Let $\mathscr{C}^\prime=<\mathscr{C}\cup\{S\}>_{\min}$. Then $f:\{E,\mathscr{C}\}\rightarrow\{E,\mathscr{C}^\prime\},$ where $f$ is the identity map, will be a circuit injection unless $\mathscr{C}\not\subseteq\mathscr{C}^\prime$, i.e., unless there exists $A\in<\mathscr{C}\cup\{S\}>_{\min},C\in\mathscr{C}$ and $A$ is properly contained in $C$, i.e., unless
\begin{equation}
S\oplus C_1\oplus\cdots\oplus C_k\subset C \qquad\qquad \mbox{ for }\qquad C_i\in\mathscr{C}, i=1,\ldots,k.
\end{equation}
Now if $S$ has a vertex $v$ of odd degree in $S$ then $\mathscr{C}\neq<\mathscr{C}\cup\{S\}>_{\min}$ so $f$ will be a nontrivial circuit injection unless $(2.1)$ holds. But $v$ of odd degree in $S$ implies $v$ will be of odd degree in $S\oplus C_1\oplus\cdots\oplus C_k$ and thus $v$ must be contained in $C$. (If vertex $q$ is of even degree in $S$ then all edges adjacent to it could cancel in $S\oplus C_1\oplus\cdots\oplus C_k$ and thus $q\notin C$ is possible).
\begin{corollary}
$G$ has no nontrivial map implies $G$ is 2-connected.
\end{corollary}
\noindent\par\textit{Proof.}~~~~ We show given $q_1\neq q_2,$ vertices of $G$, there exist $C\in\mathscr{C}$ with $q_1,q_2$ vertices of $C$. First assume there exists the edge $e=(q_1,q_2)$ in $G$. Then taking $S=\{e\}$ in the hypothesis of Lemma 2.3 yields $C$. Otherwise choose an edge a adjacent to $q_1$ and an edge $b$ adjacent to $q_2$ (since $G$ has no isolated vertices this is possible) and put $S=\{a,b\}$ to get $C$.
\par We prove the implications of Theorem 2.2 separately in the following two lemmas.
\begin{lemma}
$|G|=2N$ and $G$ has no nontrivial map $\Rightarrow G$ is Hamiltonian; $|G|=2N+1$ and $G$ has no nontrivial map $\Rightarrow G$ is almost Hamiltonian.
\end{lemma}
\noindent\par\textit{Proof.}~~~~ Let $C$ be a circuit of $G$ and let $G$ have no nontrivial map, $|G|$ odd or even.
\par FACT 1.~~ If $C$ is even and there exist two distinct vertices $v_1,v_2$ of $G$ not on $C$ then $C$ is not of maximal order.
\par \textit{Proof of Fact 1.}~~ Let $q_1,q_2$ be two distinct vertices of $C$. Then by Menger's Theorem (since $G$ is 2-connected ) there exists a pair of vertex disjoint paths $P(v_1,q_1), P(v_2,q_2)$ or $P(v_1,q_2), P(v_2,q_1)$. In either case there exists a pair of distinct vertices $v_1^\prime,v_2^\prime$ not on $C$ such that $(v_1^\prime, q_1),(v_2^\prime,q_2)$ are edges of $G$. If $q_1,q_2$ are separated by an odd (even) number of edges in $C$ there exists a subgraph of $G$ having $|C|+2$ odd vertices as in Fig. 3(A) (3(B)) and thus $C$ is not maximal by Lemma 2.1.
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\par DEFINITION.~~~~Let the order of a graph $G$ be $n$. We say $G$ is almost Hamiltonian if every subset of $n-1$ vertices is contained in a circuit.
\begin{theorem}
Let the order of $G$ be even. Then ``no nontrivial circuit injection $f$ exists, $f:G_M\rightarrow B,$ where $B$ is binary'' is true iff $G$ is Hamiltonian. Let the order of $G$ be odd. Then ``no nontrivial circuit injection $f:G_M\rightarrow B$ exists where $B$ is binary'' is true iff $G$ is almost Hamiltonian.
\end{theorem}
We abbreviate ``no nontrivial circuit injection $f:G_M\rightarrow B$ exists, where $B$ is binary'' by saying ``$G$ has no nontrivial map.'' To prove the theorem we need the following
\begin{lemma}
$G$ has no nontrivial map implies ``if $v_1,\ldots,v_n$ are vertices of odd degree in $S$, for any subgraph $S$ of $G$, then there exists a circuit $C$ of $G$ such that $v_1,\ldots,v_n$ are vertices of C.''
\end{lemma}
\noindent\par\textit{Proof.}~~~~ Let $\mathscr{C}$ be the set of circuits of $G$, $S$ a subset of edges of $G$. Let $\mathscr{C}^\prime=<\mathscr{C}\cup\{S\}>_{\min}$. Then $f:\{E,\mathscr{C}\}\rightarrow\{E,\mathscr{C}^\prime\},$ where $f$ is the identity map, will be a circuit injection unless $\mathscr{C}\not\subseteq\mathscr{C}^\prime$, i.e., unless there exists $A\in<\mathscr{C}\cup\{S\}>_{\min},C\in\mathscr{C}$ and $A$ is properly contained in $C$, i.e., unless
\begin{equation}
S\oplus C_1\oplus\cdots\oplus C_k\subset C \qquad\qquad \mbox{ for }\qquad C_i\in\mathscr{C}, i=1,\ldots,k.
\end{equation}
Now if $S$ has a vertex $v$ of odd degree in $S$ then $\mathscr{C}\neq<\mathscr{C}\cup\{S\}>_{\min}$ so $f$ will be a nontrivial circuit injection unless $(2.1)$ holds. But $v$ of odd degree in $S$ implies $v$ will be of odd degree in $S\oplus C_1\oplus\cdots\oplus C_k$ and thus $v$ must be contained in $C$. (If vertex $q$ is of even degree in $S$ then all edges adjacent to it could cancel in $S\oplus C_1\oplus\cdots\oplus C_k$ and thus $q\notin C$ is possible).
\begin{corollary}
$G$ has no nontrivial map implies $G$ is 2-connected.
\end{corollary}
\noindent\par\textit{Proof.}~~~~ We show given $q_1\neq q_2,$ vertices of $G$, there exist $C\in\mathscr{C}$ with $q_1,q_2$ vertices of $C$. First assume there exists the edge $e=(q_1,q_2)$ in $G$. Then taking $S=\{e\}$ in the hypothesis of Lemma 2.3 yields $C$. Otherwise choose an edge a adjacent to $q_1$ and an edge $b$ adjacent to $q_2$ (since $G$ has no isolated vertices this is possible) and put $S=\{a,b\}$ to get $C$.
\par We prove the implications of Theorem 2.2 separately in the following two lemmas.
\begin{lemma}
$|G|=2N$ and $G$ has no nontrivial map $\Rightarrow G$ is Hamiltonian; $|G|=2N+1$ and $G$ has no nontrivial map $\Rightarrow G$ is almost Hamiltonian.
\end{lemma}
\noindent\par\textit{Proof.}~~~~ Let $C$ be a circuit of $G$ and let $G$ have no nontrivial map, $|G|$ odd or even.
\par FACT 1.~~ If $C$ is even and there exist two distinct vertices $v_1,v_2$ of $G$ not on $C$ then $C$ is not of maximal order.
\par \textit{Proof of Fact 1.}~~ Let $q_1,q_2$ be two distinct vertices of $C$. Then by Menger's Theorem (since $G$ is 2-connected ) there exists a pair of vertex disjoint paths $P(v_1,q_1), P(v_2,q_2)$ or $P(v_1,q_2), P(v_2,q_1)$. In either case there exists a pair of distinct vertices $v_1^\prime,v_2^\prime$ not on $C$ such that $(v_1^\prime, q_1),(v_2^\prime,q_2)$ are edges of $G$. If $q_1,q_2$ are separated by an odd (even) number of edges in $C$ there exists a subgraph of $G$ having $|C|+2$ odd vertices as in Fig. 3(A) (3(B)) and thus $C$ is not maximal by Lemma 2.1.
\begin{center}
\includegraphics[width=1\textwidth]{Images/fig3.jpg}
\end{center}
$$$$$$$$
\begin{flushright}
\includegraphics{Images/fig4.jpg}
\end{flushright}
\par FACT 2.~~ If $|C|$ is odd and there exists a vertex $v_1\in G$ not on $C$ then $C$ is not maximal.
\par \textit{Proof of Fact 2.}~~ By the connectivity of $G$ we have $(v_1,q)$ is an edge for some vertex $q$ on $C$. We construct a subgraph having $|C|+1$ odd vertices as in Fig. 4 and apply Lemma 2.1.
\par If $|G|=2N$, Facts 1 and 2 imply that a circuit of maximal length is a Hamiltonian circuit. If $|G|=2N+1$, Facts 1 and 2 imply either $G$ is Hamiltonian (in which case it is also almost Hamiltonian) or a maximal circuit is of length $2N$. Let $C$ be a circuit of length $2N,v$ the vertex of $G$ not on $C, q$ a vertex on $C$ such that $(v,q)$ is an edge. We can find a subgraph of $G$ all vertices of which are of odd degree containing $v$ and all other vertices of $C$ other than an arbitrary vertex $v^\prime$ of $C$ as in Fig. 5. Thus $G$ is almost Hamiltonian by Lemma 2.1.
\begin{lemma}
Let $G$ be an almost Hamiltonian graph, $|G|=2N+1$. Then $G$ has no nontrivial map. Let $G$ be a Hamiltonian graph, $|G|=2N$. Then $G$ has no nontrivial map.
\end{lemma}
\noindent \par \textit{Proof.}~~~~ Case 1. $|G|=2N+1.$ Suppose otherwise, i.e., let $f:(E,\mathscr{C})\rightarrow(E,\mathscr{C}^\prime)$ be a nontrivial circuit injection, where $E$ are the edges of $G,\mathscr{C}$ are the circuits of G, and $\mathscr{C}^\prime$ properly contains $\mathscr{C}.$ Let $C$ be a circuit of $G$, \\
\includegraphics[width=1\textwidth]{Images/fig5.jpg}
$$$$
$|C|=2N, q$ a vertex of $G$ not on $C,e^\prime$ an edge of $G$ adjacent to $q$ and some vertex $v$ of $C$, and $e$ an edge of $C$ adjacent to $v$.
\par Then $P=(C-\{e\})\cup\{e^\prime\}$ is a Hamiltonian path of $G$ (i.e., a path which contains every vertex) and $P$ is a dependent set of $\{E,\mathscr{C}^\prime\}$(since otherwise $r(E,\mathscr{C})=r(E,\mathscr{C}^\prime)=2N$ and $f$ must be trivial). Let $T\in\mathscr{C}^\prime, T\notin\mathscr{C}, T\subseteq P.$ Now $T$ has at most $2N$ odd vertices, $v_1,\ldots,v_s$, since the sum of the degrees of all the vertices of $T$ is even and $T$ has at most $2N+1$ vertices. Let $C^\prime$ be a circuit of $G$ which contains $v_1,\ldots,v_s$. Let $T\subseteq T$ be the set of edges of $T$ not contained in $C^\prime.$ Then $T^\prime\subseteq P$ is the union of vertex disjoint paths $P_1,\ldots,P_k$ and the endpoints $b_i,e_i$ of $P_i$ are on $C^\prime$.Let $C_i^\prime$ be one of the two paths in $C^\prime$ with endpoints $b_i,e_i$ of $P_i$ are on $C^\prime$. Let $C_i^\prime$ be one of the two paths in $C^\prime$ with endpoints $b_i,e_i$ and define $k$ circuits of $G$ by $C_i=C_i^\prime\cup P_i, i=1,\ldots,k.$ Then $T\oplus C_i\oplus\cdots\oplus C_k\subseteq C^\prime$ contradicting the definition of $T$.\\
\par Case 2.~~ $|G|=2N$. If $G$ is Hamiltonian of arbitrary order then $G$ has no nontrivial map as noted in an earlier remark.
\par Lemmas 2.2 and 2.3 establish Theorem 2.2. The existence of almost Hamiltonian graphs of odd order which are not Hamiltonian is shown in [2]. Thus there are graphs which are not Hamiltonian for which no nontrivial map exists.
\par\textit{Remark}.~~ The duals of the matroids of complete graphs of order 5 or more provide a counter example to the assertion that an $n$ exists such that if a binary matroid $M$ has a connectivity $n$ no nontrivial map $f:M\rightarrow M^\prime$ exists, where $M^\prime$ is a binary matroid. For if $G_n$ is the complete graph of $n$ vertices let $M_n^\prime=<B_n\cup\{E_n\}>_{\min},$ where $E_n=E(G_n)$ and $B_n$ is the set of bonds of $G_n.$ Then $f:M_n\rightarrow M_n^\prime,$ where $M_n$ is the dual of $G_n$, and $f$ is the identity map, is a nontrivial map, since $a\oplus E_n\not\subset b$ for $a,b\in B_n$ when $n\geq 5$ and $a_1\oplus\cdots\oplus a_k$ where $a_i\in B_n,$ $1\leq i \leq k.$
\section*{\centering\small ACKNOWLEDGMENT}
The author would like to thank the referee for many helpful suggestions.
$$$$
\section*{\centering\small REFERENCES}
\begin{description}
\item 1. J.H. SANDERS AND D. SANDERS, Circuit preserving edge maps,\textit{J. Combin. Theory Ser}. B \textbf{22} (1977),91-96.
\item 2. C. THOMASSEN, Planner and infinite hypohamiltonian and hypotraceable graphs, \textit{Discrete math.} \textbf{14 } (1976),377-389.
\item 3. C. THOMASSEN, Duality of infinite graphs, \textit{J. Combin. Theory Ser.} B \textbf{33} (1982), 137-160.
\item 4. W.T. TUTTE, Menger's theorem for matroids, \textit{J. Res. Nat. Bur. Standards} B \textbf{69}(1964, 49-53).
\item 5. W.T. TUTTE, Lectures on matroids,\textit{J. Res. Nat. Bur. Standards} B \textbf{68}(1965),1-47.
\item 6. W.T. TUTTE, Connectivity in matroids, \textit{Canad. J. Math.} \textbf{18} (1966), 1301-1324
\item 7. D.J. A. WELSH, ``Matroid Theory,'' Academic Press, London/ New York, 1976.
\item 8. H. WHITNEY, 2- isomorphic graphs, \textit{Amer. J. Math.} \textbf{55}(1933), 245-254.
\end{description}
\end{document}
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\begin{document}
\title{The rationality of dynamical zeta functions and Woods Hole fixed point formula}
\begin{abstract}
For one variable rational function $\phi\in K(z)$ over a field $K$, we can define a discrete dynamical system by regarding $\phi$ as a self morphism of $\mathbb{P}^{1}_K$. Hatjispyros and Vivaldi defined a dynamical zeta function for this dynamical system using multipliers of periodic points, that is, an invariant which indicates the local behavior of dynamical systems. In this paper, we prove the rationality of dynamical zeta functions of this type for a large class of rational functions $\phi\in K(z)$. The proof here relies on Woods Hole fixed point formula and some basic facts on the trace of a linear map acting on cohomology of a coherent sheaf on $\mathbb{P}^{1}_{K}$.
\end{abstract}
\tableofcontents
\section{Introduction}
In this paper, we study the rationality of the dynamical zeta function introduced by Hatjispyros and Vivaldi in \cite{Hatjispyros-Vivaldi}. In this section, we introduce the dynamical zeta function $\dzeta{m}{\phi}$ without explaining details and state our main result.
Let $K$ be an algebraically closed field with characteristic 0. For a rational function $\phi\in K(z)$ and a non-negative integer $m\in\mathbb{Z}_{\ge 0}$, we define the \textit{dynamical zeta function} as
\[\dzeta{m}{\phi}=\exp\left(\sum_{n=1}^{\infty}\frac{t^n}{n}\sum_{x\in \per{n}{\phi}}\mult{\phi^n}{x}^m\right)\in K\llbracket t\rrbracket.\]
$\per{n}{\phi}=\{x\in \mathbb{P}^{1}_{K} : \phi^{n}(x)=x\}=\fixpt{\phi^n}$ is the set of periodic points of period $n$ and
\begin{align*}
\mult{\phi}{x}=
\begin{cases}
\phi'(x) & \text{if } x\neq \infty \\
\psi'(0) & \text{if } x=\infty
\end{cases},
\end{align*}
where $\psi(z)=1/\phi(1/z)$, is the \textit{multiplier} of the fixed point $x\in \fixpt{\phi}$.
For $m=0$, $\dzeta{0}{\phi}$ is Artin-Mazur zeta function defined by Artin and Mazur in \cite{Artin-Mazur}.
\[\dzeta{0}{\phi}=\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n}\# \per{n}{\phi}\right).\]
Hinkkanen showed that $\dzeta{0}{\phi}$ is a rational function for $K=\mathbb{C}$ in \cite[Theorem 1]{Hinkkanen}. More precisely, he showed that there exists $N\in\mathbb{Z}_{\ge 0}$, $p_1,\dots p_N,q_1,\dots, q_N, l_1,\dots, l_N \in \mathbb{Z}_{\ge 0}$ such that
\[\dzeta{0}{\phi}=\frac{1}{(1-t)(1-dt)}\prod_{i=1}^{N}(1-t^{p_i q_i})^{l_i}.\]
He used techniques from complex dynamics. Lee pointed out the Hinkkanen's result is still valid for any algebraically closed field $K$ with characteristic $0$ in \cite[Theorem 1.1.]{Lee}
For $m>0$, Hatjispyros and Vivaldi showed the rationality of $\dzeta{1}{\phi}$ for a polynomial of special type $\phi(z)=z^d+c$ in \cite[Lemma 3.1.]{Hatjispyros-Vivaldi} and conjectured that $\dzeta{m}{\phi}$ is a rational function for quadratic polynomials in the same paper. Predrag Cvitanovic, Kim Hansen, Juri Rolf, and G{\'{a}}bor Vattay showed the rationality of $\dzeta{m}{\phi}$ for quadratic polynomials in \cite[Section 4.1.]{Cvitanovic}. Eremenko and Levin showed that $\dzeta{1}{\phi}$ is a rational function for each polynomial in \cite[Lemma 1]{Eremenko-Levin}.
For $m>0$, the rationality of $\dzeta{m}{\phi}$ in previous results is restricted to the case that $\phi$ is a polynomial. Our main result is the rationality of the zeta function $\dzeta{m}{\phi}$ for a rational function $\phi\in K(z)$ which is not supposed to be a polynomial.
\begin{thm}
Let $K$ be an algebraiclly closed field with characteristic $0$ and let $\phi \in K(z)$ be a rational function of degree $\ge 2$. Assume that $\phi$ has no periodic point with multiplier 1. Then the dynamical zeta function $\dzeta{m}{\phi}$ is a rational function over $K$ for all $m\in \mathbb{Z}_{\ge 0}$ .
\end{thm}
This paper will be organized as follows. In Section \ref{preliminaries}, we recall some basic notions of the discrete dynamics on $\mathbb{P}^{1}$ and some results on dynamical zeta functions. We devote Section \ref{woodshole} to recall the Woods Hole fixed point formula, which is the key tool of our proof for the main theorem. In Section \ref{main}, we prove the main theorem after some observation. In Section \ref{examples}, We construct some examples of rational function $\phi\in \mathbb{C}(z)$ that we can apply our main theorem and we give the explicit formula of $\dzeta{1}{\phi}$ for such $\phi$.
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\section{Prelimiaries}\label{preliminaries}
In this section, we recall some basic notions on dynamical systems on $\mathbb{P}^{1}$ associated with a rational function and we see some examples and results of dynamical zeta functions. Notation in this section follows Silverman's textbook \cite{Silverman}.
\subsection{Periodic points and its multipliers}
Let $K$ be a field. For a rational function $\phi\in K(z)$, we can regard $\phi$ as an endomorphism $\phi:\mathbb{P}^{1}_K\to \mathbb{P}^{1}_K$. For any integer $n\ge 0$, $\phi^n:\mathbb{P}^{1}_K\to\mathbb{P}^{1}_K$ denotes the \textit{n-th iterate} of $\phi$,
\begin{align*}
\phi^n = \underbrace{\phi\circ\phi\circ \cdots \circ \phi }_{n}.
\end{align*}
\begin{defin}
Let $\phi \in K(z)$ be a rational function.
\begin{itemize}
\item[$(1)$]A point $x\in\mathbb{P}^{1}_K$ is said to be a \textit{fixed point} of $\phi$ if $\phi(x)=x$. We denote by $\fixpt{\phi}$ the set of fixed points of $\phi$.
\item[$(2)$]We say a point $x\in\mathbb{P}^{1}_K$ is a \textit{periodic point} of period $n$ if $\phi^{n}(x)=x$. We denote by $\per{n}{\phi}=\fixpt{\phi^n}$ the set of periodic points of period $n$.
\item[$(3)$]A periodic point $x$ is said to be \textit{of minimal period} $n$ if $x\in \per{n}{\phi}$ and $x\not\in \per{m}{\phi}$ for any $0<m<n$. We denote by $\mathrm{Per}_{d}^{**}(\phi)$ the set of periodic points of minimal period $n$.
\item[$(4)$]The \textit{forward orbit} of $x\in\mathbb{P}^{1}_K$ is defined by $\mathcal{O}_{\phi}(x)=\{\phi^{n}(x)\colon n\in\mathbb{Z}_{\ge 0}\}$.
\end{itemize}
\end{defin}
It is easy to show that we can decompose $\per{n}{\phi}$ into the disjoint union $\per{n}{\phi}=\bigcup_{d|n}\mathrm{Per}_{d}^{**}(\phi)$. Next, we define the multiplier of fixed points. Recall that the linear fractional transformation $\theta(z)=\dfrac{az+b}{cz+d}$ for $ \begin{pmatrix}a & b\\ c & d\end{pmatrix}\in\mathrm{GL}_{2}(K)$ defines an automorphism of $\mathbb{P}^{1}_K$.
\begin{defin}
For $\phi\in K(z)$ and $\theta\in \mathrm{PGL}_2(K)$ the \textit{linear conjugate} of $\phi$ by $\theta$ is the map $\theta\circ\phi\circ \theta^{-1}$.
\end{defin}
The linear conjugation of $\phi$ at $\theta$ yields the following commutative diagram.
\begin{center}
\begin{tikzcd}
\mathbb{P}^{1}_K \arrow[d,"\theta"']\arrow[r,"\phi"] & \mathbb{P}^{1}_K \arrow[d,"\theta"]& \\
\mathbb{P}^{1}_K \arrow[r,"\theta\circ\phi\circ \theta^{-1}"'] & \mathbb{P}^{1}_K.
\end{tikzcd}
\end{center}
Given $x\in \fixpt{\phi}\setminus\{\infty\}$, we define the \textit{multiplier} $\mult{\phi}{x}$ of $\phi$ at $x$ by $\mult{\phi}{x}=\phi'(x)$. The multiplier is invariant under the linear conjugation.
\begin{prop}
Let $\phi\in K(z)$ be a rational function and $\theta\in\mathrm{PGL}_2(K)$ be a fractional linear transformation. Then the followings hold.
\begin{itemize}
\item[$(1)$]$\fixpt{\theta\circ\phi\circ \theta^{-1}}=\theta(\fixpt{\phi})$.
\item[$(2)$]For any $x\in\fixpt{\phi}$ , $\mult{\theta\circ\phi\circ \theta^{-1}}{\theta(x)}=\mult{\phi}{x}$ if $\theta(x) \neq \infty$.
\end{itemize}
\end{prop}
\begin{proof}
The first statement is trivial. The second statement is an easy consequence of the chain rule.
\end{proof}
Using this property, we can extend the definition of the multiplier as below.
\begin{defin}
Let $\phi\in K(z)$ be a rational function. We define the \textit{multiplier} of $\phi$ at each $x\in\fixpt{\phi}$ as
\begin{align*}
\mult{\phi}{x}=
\begin{cases}
\phi'(x) & \text{if } x\neq \infty \\
\psi'(0) & \text{if } x=\infty
\end{cases},
\end{align*}
where $\psi(z)=1/\phi(1/z)$, which is the linear conjugation of $\phi$ at $\theta(z)=1/z$.
\end{defin}
\subsection{Dynamical zeta function}
\begin{defin}
Let $K$ be an algebraically closed field with characteristic 0. For a rational function $\phi\in K(z)$ and a non-negative integer $m\in\mathbb{Z}_{\ge 0}$, we define the \textit{dynamical zeta function} as
\[\dzeta{m}{\phi}=\exp\left(\sum_{n=1}^{\infty}\frac{t^n}{n}\sum_{x\in \per{n}{\phi}}\mult{\phi^n}{x}^m\right)\in K\llbracket t\rrbracket.\]
\end{defin}
\begin{rem}
Since the multiplier is invariant under the linear conjugation, $\dzeta{m}{\phi}$ is also invariant under the linear conjugation.
\end{rem}
In \cite[Section 2]{Hatjispyros-Vivaldi}, Hatjispyros and Vivaldi pointed out that the dynamical zeta function $\dzeta{m}{\phi}$ has the Eulerian product as follows. We denote by $\mathrm{Per}_{n}^{**}(\phi)/\sim$ the quotient set of $\mathrm{Per}_{n}^{**}(\phi)$ by the equivalent relation $x\sim y \Leftrightarrow \mathcal{O}_{\phi}(x)=\mathcal{O}_{\phi}(y)$.
\begin{thm}
\[\dzeta{m}{\phi}=\prod_{n=1}^{\infty}\prod_{x\in \mathrm{Per}_{n}^{**}(\phi)/\sim} (1-\mult{\phi^n}{x}^{m}t^{n})^{-1}\]
where the second product runs over a complete system of representatives of $\mathrm{Per}_{n}^{**}(\phi)/\sim$.
\end{thm}
\begin{proof}
Using the chain rule, we have
\begin{itemize}
\item $\mult{\phi^{md}}{x}=\mult{\phi^{d}}{x}^m$ for any $x\in \per{d}{\phi}$,
\item $\mult{\phi^n}{x}=\mult{\phi^n}{y}$ if $x\sim y$.
\end{itemize}
Since we can decompose $\per{n}{\phi}$ into the disjoint union $\per{n}{\phi}=\bigcup_{d|n}\mathrm{Per}_{d}^{**}(\phi)$, we obtain
\begin{align*}
\sum_{x\in \per{n}{\phi}}\mult{\phi^n}{x}^m &= \sum_{d|n}\sum_{x\in \mathrm{Per}_{d}^{**}(\phi)}\mult{\phi^n}{x}^m\\
&=\sum_{d|n}\sum_{x\in \mathrm{Per}_{d}^{**}(\phi)/\sim} d(\mult{\phi^{d}}{x}^m)^{n/d}.
\end{align*}
Therefore,
\begin{align*}
\dzeta{m}{\phi}&=\exp\left(\sum_{n=1}^{\infty}\sum_{d|n}\sum_{\mathrm{Per}_{d}^{**}(\phi)/\sim} \frac{(\mult{\phi^{d}}{x}^m)^{n/d}}{n/d}t^n\right)\\
&=\exp\left(\sum_{l=1}^{\infty}\sum_{d=1}^{\infty}\sum_{\mathrm{Per}_{d}^{**}(\phi)/\sim} \frac{(\mult{\phi^{d}}{x}^mt^d)^{l}}{l}\right)\\
&=\exp\left(-\sum_{d=1}^{\infty}\sum_{\mathrm{Per}_{d}^{**}(\phi)/\sim}\log (1-\mult{\phi^d}{x}^mt^d)\right)\\
&=\prod_{d=1}^{\infty}\prod_{\mathrm{Per}_{d}^{**}(\phi)/\sim}(1-\mult{\phi^d}{x}^mt^d)^{-1}.\\
\end{align*}
\end{proof}
\begin{ex} Let $\phi(z)=z^d$ for an integer $d\ge 2$. Since $\phi^n(z)=z^{d^{n}}$, we have $\per{n}{\phi}=\{0\}\cup \mu_{d^{n}-1}=\{0\}\cup \{\zeta\in K \colon \zeta^{d^{n}-1}=1\}$. Therefore,
\begin{align*}
\mult{\phi^{n}}{x}=
\begin{cases}
0 & (x=0) \\
d^n & (x\in \mu_{d^{n}-1})
\end{cases}
\end{align*}
and
\[\sum_{x\in\per{n}{\phi}} \mult{\phi^{n}}{x}^m=(d^n-1)d^{nm}=d^{n(m+1)}-d^{nm}.\]
So we obtain
\[\dzeta{m}{\phi}=\frac{1-d^{m} t}{1-d^{m+1} t}.\]
\end{ex}
\begin{ex}
Let $T_{d}$ be the \textit{d-th Chebyshev polynomial} satisfying $2\cos (d\theta)=T_d(2\cos \theta)$. It is well known that $T_{d}^{n}=T_{d^n}$, $\per{n}{T_d}=\{\zeta+\zeta^{-1}\colon \zeta \in \mu_{d-1}\}\cup\{\zeta+\zeta^{-1}\colon \zeta \in \mu_{d+1}\}\cup\{\infty\}$ and
\begin{align*}
\mult{T_d}{x}=
\begin{dcases}
d & (x=\zeta+\zeta^{-1} , \zeta\in \mu_{d-1}\setminus\{\pm 1\})\\
-d & (x=\zeta+\zeta^{-1} , \zeta\in \mu_{d+1}\setminus\{\pm 1\})\\
d^2 & (x= \pm 2)\\
0 & (x=\infty)
\end{dcases}.
\end{align*}
For details, see \cite[Section~6.2]{Silverman}. Therefore, we obtain
\begin{align*}
\dzeta{m}{T_d}=
\begin{dcases}
\frac{1-d^{m}t}{1-d^{2m}t} \frac{1}{1-d^{m+1}t} & (d:\text{even},m:\text{even})\\
\frac{1-d^{m}t}{1-d^{2m}t} & (d:\text{even},m:\text{odd})\\
\left(\frac{1-d^{m}t}{1-d^{2m}t}\right)^2 \frac{1}{1-d^{m+1}t} & (d:\text{odd},m:\text{even})\\
\frac{1-d^{m}t}{(1-d^{2m}t)^2} & (d:\text{odd},m:\text{odd})
\end{dcases}.
\end{align*}
This result is firstly obtained by Hatjispyros in \cite[Theorem 1]{Hatjispyros}.
\end{ex}
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\section{Woods Hole fixed point formula}\label{woodshole}
In this section, we review the Woods Hole fixed point formula which is the key tool of our proof. This formula is also called \textit{Atiyah-Bott fixed point formula} since Atiyah and Bott proved this formula in differential geometry in \cite[Theorem A]{AtBo} and \cite[Theorem A]{AB}. A purely algebraic proof can be found in SGA5 \cite[Expos{\'{e}} III, Corollaire 6.12]{SGA5} and in \cite[Theorem A.4.]{Taelman}. For details and other applications of this formula, see \cite{SGA5}, \cite{Taelman}, \cite{Beauville}, \cite{Kond}, and \cite{Ramirez}.
Let $X$ be a Noetherian scheme over an algebraically closed field $K$, $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $X$, and $\varphi: \mathcal{G}\to\mathcal{F}$ is a sheaf homomorphism. For $x\in X$, we define $\mathcal{F}(x)=\mathcal{F}_x\otimes_{\mathcal{O}_{X,x}}\mathcal{O}_{X,x}/\mathfrak{m}_x$. Then we obtain a natural $\mathcal{O}_{X,x}/\mathfrak{m}_x$-linear map $\varphi(x): \mathcal{G}(x)\to \mathcal{F}(x)$. If $f:X\to X$ is an endomorphism and $\mathcal{G}=f^{*}\mathcal{F}$, we have a map $\widetilde{\varphi}^{(p)}:H^{p}(X,\mathcal{F})\to H^{p}(X,\mathcal{F})$ satisfying
\begin{center}
\begin{tikzcd}
H^{p}(X,\mathcal{F}) \arrow[rd,"\widetilde{\varphi}^{(p)}"]\arrow[d]& \\
H^{p}(X,f^{*}\mathcal{F}) \arrow[r]& H^{p}(X,\mathcal{F})
\end{tikzcd}.
\end{center}
$H^{p}(X,\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$ is the pull-back on cohomology and $H^{p}(X,f^{*}\mathcal{F})\to H^{p}(X,\mathcal{F})$ is the homomorphism induced by $\varphi:f^{*}\mathcal{F}\to\mathcal{F}$. Note that the pull-back $H^{p}(X,\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$ is decomposed as $H^{p}(X,\mathcal{F})\to H^{p}(X,f_{*}f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$, where $H^{p}(X,\mathcal{F})\to H^{p}(X,f_{*}f^{*}\mathcal{F})$ is induced by the sheaf homomorphism $\mathcal{F}\to f_{*}f^{*}\mathcal{F}$. Moreover, note that if $f$ is affine, $H^{p}(X,f_{*}f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$ is an isomorphism \cite[III, Excercise 8.2.]{Hartshorne}.
\begin{thm}(Woods Hole Fixed Point Formula, \cite[Theorem A.4.]{Taelman})
Let $X$ be a smooth proper scheme over an algebraically closed field $K$ and let $f: X \to X$ be an endomorphism. Let $\mathcal{F}$ be a locally free $\mathcal{O}_{X}$ module of finite rank and let $\varphi: f^{*}\mathcal{F}\to \mathcal{F}$ be a homomorphism of $\mathcal{O}_{X}$ module. Assume that the graph $\Gamma_{f}\subset X\times X$ and the diagonal $\Delta\subset X\times X$ intersect transversally in $X\times X$. Then the following identity holds.
\[\sum_{p}(-1)^{p}\tr{\widetilde{\varphi}^{(p)}}{H^{p}(X,\mathcal{F})}=\sum_{x\in \fixpt{f}}\frac{\tr{\varphi(x)}{\mathcal{F}(x)}}{\det (1-df(x)\colon \Omega_{X/K}(x))} \]
where $df:f^{*}\Omega_X\to\Omega_X$ is the differential of $f$.
\end{thm}
For a morphism $f:X\to Y$, the differential $df:f^{*}\Omega_Y\to\Omega_X$ locally comes from
\[ \Omega_B\otimes_{B}A\to \Omega_A; db\otimes a\mapsto ad(\varphi(b)),\]
where $\spec{B}\subset X$ and $\spec{A} \subset Y$ are affine open subsets and $f$ corresponds to a ring homomorphism $\varphi: B\to A$.
The statement that $\Gamma_{f}\subset X\times X$ and the diagonal $\Delta\subset X\times X$ intersect transversally in $X\times X$ means that $\Gamma_{f}$ and $\Delta$ meet with intersection multiplicity $1$ at each point $x\in \Gamma_{f}\cap\Delta$.
\begin{rem}
It is known that the followings are equivalent.
\begin{itemize}
\item[$(1)$]The graph $\Gamma_{f}\subset X\times X$ and the diagonal $\Delta\subset X\times X$ intersect transversally in $X\times X$.
\item[$(2)$]$\det (1-df(x))\neq 0$ for all $x$ satisfying $f(x)=x$.
\end{itemize}
For details, see \cite[Proposition I.1]{Beauville}. Therefore, in the case that $X=\mathbb{P}^{1}_K$ and $\phi\in K(z)$, the transversality condition equals to that $\phi$ has no fixed point with multiplier $1$.
\end{rem}
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\section{Main theorem}\label{main}
In this section, we prove, by applying Woods Hole fixed point formula, the rationality of dynamical zeta functions $\dzeta{m}{\phi}$ for each \textit{completely transversal} rational function $\phi \in K(z)$.
\subsection{Notation}\label{preparation}
\begin{defin}
Let $K$ be an algebraiclly closed field with characteristic $0$ and let $\phi \in K(z)$ be a rational function. We say that $\phi$ is \textit{completely transversal} if $\mult{\phi^n}{x}\neq 1$ for any $n\in\mathbb{Z}_{\ge 0}$ and $x\in\per{n}{\phi}$.
\end{defin}
\begin{defin}
For $\phi\in K(z)$ with no fixed point with multiplier $1$ and $m\in\mathbb{Z}_{\ge 0}$, we define
\[T_{m}(\phi)=\sum_{x\in \fixpt{\phi}}\frac{\mult{\phi}{x}^{m}}{1-\mult{\phi}{x}}.\]
\end{defin}
\begin{rem}\label{decomposition}
Note that
\[ \sum_{x\in\fixpt{\phi}}\mult{\phi}{x}^{m}=T_{m}(\phi)-T_{m+1}(\phi)\]
for all rational functions $\phi \in K(z)$ with no fixed point with multiplier $1$. So we have
\[ \sum_{x\in\per{n}{\phi}}\mult{\phi^n}{x}^{m}=T_{m}(\phi^n)-T_{m+1}(\phi^n)\]
for all $n\in\mathbb{Z}_{>0}$ if $\phi$ is completely transversal.
\end{rem}
\begin{defin}
Let $K$ be an algebraically closed field with characteristic $0$ and let $\phi \in K(z)$ be a completely transversal rational function of degree $\ge 2$.
We define $\lzeta{m}{\phi}\in K\llbracket t\rrbracket$ by
\[\lzeta{m}{\phi}=\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} T_{m}(\phi^n)\right).\]
\end{defin}
\begin{rem}\label{quotient}
By Remark \ref{decomposition}, we have
\[\dzeta{m}{\phi}=\frac{\lzeta{m}{\phi}}{\lzeta{m+1}{\phi}}\]
if $\phi$ is completely transversal.
\end{rem}
We state the main theorem as below. A proof will given in Section \ref{conclution} after some observation.
\begin{thm}\label{maintheorem}
Let $K$ be an algebraiclly closed field with characteristic $0$ and let $\phi \in K(z)$ be a completely transversal rational function of degree $\ge 2$. Then the following hold.
\[\lzeta{m}{\phi}=\dfrac{\det (1-tD_m^1(\phi))}{\det (1-tD_{m}^0(\phi))}\in K(t),\]
where $D_m^p(\phi): H^{p}(\mathbb{P}^{1}_K,\Omega_{\mathbb{P}^{1}_K}^{\otimes m})\to H^{p}(\mathbb{P}^{1}_K,\Omega_{\mathbb{P}^{1}_K}^{\otimes m})$ is the map associated to the sheaf homomorphism $(d\phi)^{\otimes m}:{\phi}^{*}\Omega_{\mathbb{P}^{1}_K}^{\otimes m}\to \Omega_{\mathbb{P}^{1}_K}^{\otimes m}$.
Especially, $\dzeta{m}{\phi}\in K(t)$ for any $m\in\mathbb{Z}_{\ge0}$.
\end{thm}
\begin{rem}
For $m>0$, we have
\[\dzeta{m}{\phi}=\frac{\dyndet{m}}{\dyndet{m+1}}\]
since $\Omega_{\mathbb{P}^{1}_K} \cong \mathcal{O}_{\mathbb{P}^{1}_K}(-2)$, $\Omega_{\mathbb{P}^{1}_K}^{\otimes m} \cong \mathcal{O}_{\mathbb{P}^{1}_K}(-2m)$ and
\begin{align*}
\dim_K H^{i}(\mathbb{P}^{1}_K,\mathcal{O}_{\mathbb{P}^{1}_K}(-2m))=
\begin{cases}
2m-1 & (i=1)\\
0 & (\text{otherwise}).
\end{cases}
\end{align*}
\end{rem}
\subsection{Connecton between multipliers and sheaf cohomologies}
\begin{lem}\label{lem_localvsgrobal}
Let $\phi \in K(z)$ be a rational function with no fixed point with multiplier $1$. Then for any positive integer $m\in\mathbb{Z}_{>0}$, the following holds.
\[T_m(\phi)=\sum_{i=0}^{1}(-1)^{i}\tr{D_m^i(\phi)}{\cohomo{i}{m}},\]
where $D_m^i(\phi):\cohomo{i}{m}\to\cohomo{i}{m}$ is the $K$-linear map assosiated to the sheaf homomorphism $(d\phi)^{\otimes m}:\phi^{*}\Omega_{\mathbb{P}^{1}_K}^{\otimes m}\to \Omega_{\mathbb{P}^{1}_K}^{\otimes m}$.
\end{lem}
\begin{proof}
After changing a coordinate, we may assume that $\infty \not\in \fixpt{\phi}$. We apply Woods Hole fixed point formula for $X=\mathbb{P}^{1}_K$, $\mathcal{F}=\Omega_{X}^{\otimes m}$, $f=\phi$ and $\varphi=(d\phi)^{\otimes m}:\phi^{*}\Omega_{X}^{\otimes m}\to\Omega_{X}^{\otimes m}$ and obtain
\[\sum_{x\in\fixpt{\phi}}\frac{\tr{d\phi^{\otimes m}(x)}{\Omega_{\mathbb{P}^{1}_K}^{\otimes m}(x)}}{\det (1-d\phi(x)\colon \Omega_{\mathbb{P}^{1}_K}(x))}=\sum_{i=0}^{1}(-1)^{i}\tr{D_m^i(\phi)}{\cohomo{i}{m}}.\]
For $x\in\fixpt{\phi}$, we take a local parameter $t=z-x$ of $\mathbb{P}^{1}_K$ at $x$. Then we have \begin{itemize}
\item $\mathcal{O}_{\mathbb{P}^{1}_K,x}\cong K[z]_{(t)}$,
\item $\mathcal{O}_{\mathbb{P}^{1}_K,x}/\mathfrak{m}_{x}\cong K$ via the evalutation at $t=0$, and
\item $\Omega_{\mathbb{P}^{1}_K,x}\cong K[z]_{(t)}dt$.
\end{itemize}
Therefore, we have
\[(d\phi)_{x}(dt)=d(\phi^{*}t)=d\phi(t+x)=\phi'(t+x)dt\]
and $\Omega_{\mathbb{P}^{1}_K}(x)=K[z]_{(t)}dt\otimes_{K[z]_{(t)}}(K[z]_{(t)}/(t)K[z]_{(t)})\cong Kdt$ via $a(t)dt\otimes 1\mapsto a(0)dt$. So
\[d\phi(x)(dt)=\phi'(x)dt=\mult{\phi}{x}dt.\]
This yields that $d\phi(x)=\mult{\phi}{x} \mathrm{id}$. So we have
\begin{align*}
T_m(\phi)&=\sum_{x\in \fixpt{\phi}}\frac{\mult{\phi}{x}^{m}}{1-\mult{\phi}{x}}\\
&=\sum_{x\in\fixpt{\phi}}\frac{\tr{d\phi^{\otimes m}(x)}{\Omega_{\mathbb{P}^{1}_K}^{\otimes m}(x)}}{\det (1-d\phi(x)\colon \Omega_{\mathbb{P}^{1}_K}(x))}.
\end{align*}
Summing up, we conclude
\[T_m(\phi)=\sum_{i=0}^{1}(-1)^{i}\tr{D_m^i(\phi)}{\cohomo{i}{m}}.\]
\end{proof}
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\subsection{Lemmas on sheaf cohomologies}
We prepare some lemmas for cohomology of coherent sheaves on schemes. First we recall that there are an adjunction $f^{*}\dashv f_{*}$ for a morphism $f:X\to Y$ of schemes and there are two natural transformations $\varepsilon_{f}: f^{*}f_{*} \to 1$ called the \textit{counit} and $\eta_{f}: 1\to f_{*}f^{*}$ called the \textit{unit}. Although it is abusing notation, we use the same notation $\eta_{f}$ for corresponding sheaf homomorphism $\eta_{f}:\mathcal{F}\to f_{*}f^{*}\mathcal{F}$ for a coherent sheaf $\mathcal{F}$ on $Y$.
The next lemma shows that $\eta_{f}$ has the functoriality as following.
\begin{lem}\label{lem_adjunction}
Let $X$,$Y$ and $Z$ be Noetherian schemes and $f:X \to Y$ and $g:Y \to Z$ be morphisms . Let $\mathcal{F}$ be a quasi-coherent sheaf on $Z$. Then the following diagram commutes.
\begin{center}
\begin{tikzcd}
\mathcal{F} \arrow[rd,"\eta_{g\circ f}"]\arrow[d,"\eta_{g}"']& \\
g_{*}g^{*}\mathcal{F} \arrow[r,"g_{*}\eta_{f}"']& g_{*}f_{*}f^{*}g^{*}\mathcal{F}.
\end{tikzcd}
\end{center}
\end{lem}
\begin{proof}
Since the problem is local, we may assume that $X$,$Y$ and $Z$ are affine. We assume that $X=\spec{A}$, $Y=\spec{B}$, $Z=\spec{C}$ and $f:X \to Y$ and $g:Y \to Z$ come from $\varphi: B\to A$ and $\psi: C\to B$, respectively. We use the notation ${}_{C}N$ (\textit{resp}. ${}_{C}L$) for $B$-module $N$(\textit{resp}. $A$-module $L$) if we regard $N$ (\textit{resp}. $L$) as $C$-module via $\psi: C\to B$ (\textit{resp}. $\phi\circ\psi:C\to A$).
Then there exsists a $C$ module $M$ such that $\mathcal{F}=\tilde{M}$ and we have $g_{*}g^{*}\mathcal{F}= {}_{C}(M\otimes_{C}B)^{\sim}$, $g_{*}f_{*}f^{*}g^{*}\mathcal{F}={}_{C}(M\otimes_{C}A)^{\sim}={}_{C}((M\otimes_{C}B)\otimes_B A)^{\sim}$. So the desired commutative diagram comes from the following diagrams.
\begin{center}
\begin{tikzcd}
M \arrow[d] \arrow[r] & {}_{C}(M\otimes_{C}A) \arrow[d, no head, equal] & m \arrow[d, maps to] \arrow[r, maps to] & m\otimes 1 \arrow[d, maps to] \\
{}_{C}(M\otimes_{C}B) \arrow[r] & {}_{C}((M\otimes_CB)\otimes_BA), & m\otimes 1 \arrow[r, maps to] & (m\otimes 1)\otimes 1 .
\end{tikzcd}
\end{center}
\end{proof}
The next lemma shows that the differential $df$ has the functoriality as following.
\begin{lem}\label{lem_differential}
Let $X$,$Y$ and $Z$ be Noetherian schemes over $K$ and $f:X \to Y$ and $g:Y \to Z$ be morphisms over $K$. Then the following diagram commutes.
\begin{center}
\begin{tikzcd}
f^{*}g^{*}\Omega_{Z} \arrow[rd,"d(g\circ f)"]\arrow[d,"f^{*}dg"']& \\
f^{*}\Omega_{Y} \arrow[r,"df"']& \Omega_X.
\end{tikzcd}
\end{center}
\end{lem}
\begin{proof}
Since the problem is local, we may assume that $X$,$Y$ and $Z$ are affine. We assume that $X=\spec{A}$, $Y=\spec{B}$, $Z=\spec{C}$ and $f:X \to Y$ and $g:Y \to Z$ come from $\varphi: B\to A$ and $\psi: C\to B$, respectively. Then $df:f^{*}\Omega_{Y}\to\Omega_{X}$, $dg:g^{*}\Omega_{Z}\to\Omega_{Y}$ and $d(g\circ f):f^{*}g^{*}\Omega_{Z}\to\Omega_{X}$ correspond to $\Omega_B\otimes_{B}A\to \Omega_A; db\otimes a\mapsto ad(\varphi(b))$, $\Omega_C\otimes_{C}B\to \Omega_B; dc\otimes b\mapsto bd(\psi(c))$ and $\Omega_C\otimes_{C}A\to \Omega_A; dc\otimes a\mapsto ad(\varphi\psi(c))$, respectively. So the desired commutative diagram comes from the following diagrams.
\begin{center}
\begin{tikzcd}
\Omega_{C}\otimes_CA \arrow[d] \arrow[rd] & & dc\otimes 1 \arrow[d, maps to] \arrow[rd, maps to] & \\
\Omega_{B}\otimes_BA \arrow[r] & \Omega_A, & d(\psi(c))\otimes 1 \arrow[r, maps to] & d(\varphi\psi(c))\otimes 1.
\end{tikzcd}
\end{center}
\end{proof}
\begin{lem}\label{comparison}
Let $\mathcal{A}$ be an abelian category and $f:A\to B$ be a morphism in $\mathcal{A}$. Let $0\to A\to I^{\bullet}$ and $0\to B\to J^{\bullet}$ be complexes in $\mathcal{A}$. If each $J^{n}$ is injective, and if $0\to A\to I^{\bullet}$ is exact, then there exists a chain map $I^{\bullet}\to J^{\bullet}$ making the following diagram commute.
\begin{center}
\begin{tikzcd}
0 \arrow[r] & A \arrow[r] \arrow[d, "f"] & I^{0} \arrow[r] \arrow[d] & I^{1} \arrow[r] \arrow[d] & {\cdots} \\
0 \arrow[r] & B \arrow[r] & J^{0} \arrow[r] & J^{1} \arrow[r] & {\cdots}.
\end{tikzcd}
\end{center}
Moreover, any two such chain maps are homotopic.
\end{lem}
\begin{proof}
See \cite[Theorem 6.16]{Rotman}.
\end{proof}
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The next lemma shows that the differential $df$ has the functoriality as following.
\begin{lem}\label{lem_differential}
Let $X$,$Y$ and $Z$ be Noetherian schemes over $K$ and $f:X \to Y$ and $g:Y \to Z$ be morphisms over $K$. Then the following diagram commutes.
\begin{center}
\begin{tikzcd}
f^{*}g^{*}\Omega_{Z} \arrow[rd,"d(g\circ f)"]\arrow[d,"f^{*}dg"']& \\
f^{*}\Omega_{Y} \arrow[r,"df"']& \Omega_X.
\end{tikzcd}
\end{center}
\end{lem}
\begin{proof}
Since the problem is local, we may assume that $X$,$Y$ and $Z$ are affine. We assume that $X=\spec{A}$, $Y=\spec{B}$, $Z=\spec{C}$ and $f:X \to Y$ and $g:Y \to Z$ come from $\varphi: B\to A$ and $\psi: C\to B$, respectively. Then $df:f^{*}\Omega_{Y}\to\Omega_{X}$, $dg:g^{*}\Omega_{Z}\to\Omega_{Y}$ and $d(g\circ f):f^{*}g^{*}\Omega_{Z}\to\Omega_{X}$ correspond to $\Omega_B\otimes_{B}A\to \Omega_A; db\otimes a\mapsto ad(\varphi(b))$, $\Omega_C\otimes_{C}B\to \Omega_B; dc\otimes b\mapsto bd(\psi(c))$ and $\Omega_C\otimes_{C}A\to \Omega_A; dc\otimes a\mapsto ad(\varphi\psi(c))$, respectively. So the desired commutative diagram comes from the following diagrams.
\begin{center}
\begin{tikzcd}
\Omega_{C}\otimes_CA \arrow[d] \arrow[rd] & & dc\otimes 1 \arrow[d, maps to] \arrow[rd, maps to] & \\
\Omega_{B}\otimes_BA \arrow[r] & \Omega_A, & d(\psi(c))\otimes 1 \arrow[r, maps to] & d(\varphi\psi(c))\otimes 1.
\end{tikzcd}
\end{center}
\end{proof}
\begin{lem}\label{comparison}
Let $\mathcal{A}$ be an abelian category and $f:A\to B$ be a morphism in $\mathcal{A}$. Let $0\to A\to I^{\bullet}$ and $0\to B\to J^{\bullet}$ be complexes in $\mathcal{A}$. If each $J^{n}$ is injective, and if $0\to A\to I^{\bullet}$ is exact, then there exists a chain map $I^{\bullet}\to J^{\bullet}$ making the following diagram commute.
\begin{center}
\begin{tikzcd}
0 \arrow[r] & A \arrow[r] \arrow[d, "f"] & I^{0} \arrow[r] \arrow[d] & I^{1} \arrow[r] \arrow[d] & {\cdots} \\
0 \arrow[r] & B \arrow[r] & J^{0} \arrow[r] & J^{1} \arrow[r] & {\cdots}.
\end{tikzcd}
\end{center}
Moreover, any two such chain maps are homotopic.
\end{lem}
\begin{proof}
See \cite[Theorem 6.16]{Rotman}.
\end{proof}
We review how the map $H^{p}(Y,f_{*}f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$ is constructed in general. Let $0\to f^{*}\mathcal{F}\to J^{\bullet}$ and $0\to f_{*}f^{*}\mathcal{F}\to I^{\bullet}$ be injective resolutions of $f^{*}\mathcal{F}$ and $f_{*}f^{*}\mathcal{F}$, respectively. Applying the functor $f_{*}$, we obtain a complex $0\to f_{*}f^{*}\mathcal{F}\to f_{*}J^{\bullet}$. Note that the complex $f_{*}J^{\bullet}$ consists of injective sheaves. By Lemma \ref{comparison}, we obtain a chain map $I^{\bullet}\to f_{*}J^{\bullet}$ and this chain map gives the map $H^{p}(Y,f_{*}f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$.
The next lemma shows that the map $H^{p}(Y,f_{*}f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$ has the functoriality as following.
\begin{lem}\label{functoriality_of_pull-back}
\begin{itemize}
\item[$(1)$]Let $X$ and $Y$ be Noetherian schemes and $f:X\to Y$ be a morpshim. Let $\mathcal{F}$ and $\mathcal{G}$ be sheaves of $\mathcal{O}_{Y}$ module and $\varphi : \mathcal{F}\to \mathcal{G}$ be a morphism of $\mathcal{O}_{Y}$ module. Then the following diagram commutes.
\begin{center}
\begin{tikzcd}
{H^{p}(Y,f_{*}f^{*}\mathcal{F})} \arrow[d] \arrow[r] & {H^{p}(X,f^{*}\mathcal{F})} \arrow[d] \\
{H^{p}(Y,f_{*}f^{*}\mathcal{G})} \arrow[r] & {H^{p}(X,f^{*}\mathcal{G})},
\end{tikzcd}
\end{center}
where $H^{p}(Y,f_{*}f^{*}\mathcal{F})\to H^{p}(Y,f_{*}f^{*}\mathcal{G})$ and $H^{p}(X,f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{G})$ are induced by $f_{*}f^{*}\varphi : f_{*}f^{*}\mathcal{F}\to f_{*}f^{*}\mathcal{G}$ and $f^{*}\varphi: f^{*}\mathcal{F}\to f^{*}\mathcal{G}$, respectively.
\item[$(2)$]Let $X$,$Y$ and $Z$ be Noetherian schemes and let $f:X \to Y$ and $g:Y \to Z$ be morphisms. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{Z}$ module. Then the following diagram commutes.
\begin{center}
\begin{tikzcd}
{H^{p}(Z,g_{*}f_{*}f^{*}g^{*}\mathcal{F})} \arrow[d] \arrow[rd] & \\
{H^{p}(Y,f_{*}f^{*}g^{*}\mathcal{G})} \arrow[r] & {H^{p}(X,f^{*}g^{*}\mathcal{F})}.
\end{tikzcd}
\end{center}
\end{itemize}
\end{lem}
\begin{proof}
For the first claim, let $0\to f^{*}\mathcal{F}\to I^{\bullet}$ and $0\to f^{*}\mathcal{G}\to J^{\bullet}$ be injective resolutions of $f^{*}\mathcal{F}$ and $f^{*}\mathcal{G}$, respectively. Then we obtain complexes $0\to f_{*}f^{*}\mathcal{F}\to f_{*}I^{\bullet}$ and $0\to f_{*}f^{*}\mathcal{G}\to f_{*}I^{\bullet}$. By Lemma \ref{comparison}, we have a chain map $I^{\bullet}\to J^{\bullet}$ which makes the following diagram commute.
\begin{center}
\begin{tikzcd}
0 \arrow[r] & f^{*}\mathcal{F} \arrow[r] \arrow[d, "f^{*}\varphi"] & I^{0} \arrow[r] \arrow[d] & I^{1} \arrow[r] \arrow[d] & {\cdots} \\
0 \arrow[r] & f^{*}\mathcal{G} \arrow[r] & J^{0} \arrow[r] & J^{1} \arrow[r] & {\cdots}.
\end{tikzcd}
\end{center}
Applying the functor $f_{*}$, we obtain two complexes $0\to f_{*}f^{*}\mathcal{F}\to f_{*}I^{\bullet}$ and $0\to f_{*}f^{*}\mathcal{G}\to f_{*}J^{\bullet}$ and a chain map $f_{*}I^{\bullet}\to f_{*}J^{\bullet}$. Note that $f_{*}I^{\bullet}$ and $f_{*}J^{\bullet}$ consist of injective sheaves. Let $0\to f_{*}f^{*}\mathcal{F}\to I^{'\bullet}$ and $0\to f_{*}f^{*}\mathcal{G}\to J^{'\bullet}$ be injective resolutions of $f_{*}f^{*}\mathcal{F}$ and $f_{*}f^{*}\mathcal{G}$, respectively. By Lemma \ref{comparison}, we have chain maps $I^{'\bullet}\to J^{'\bullet}$, $I^{'\bullet}\to f_{*}I^{\bullet}$, and $J^{'\bullet}\to f_{*}J^{\bullet}$. Moreover, since the chain map $I^{'\bullet}\to f_{*}J^{\bullet}$ is unique up to homotopy, the following diagram commutes up to homotopy.
\begin{center}
\begin{tikzcd}
I^{'\bullet} \arrow[d] \arrow[r] & f_{*}I^{\bullet} \arrow[d] \\
J^{'\bullet} \arrow[r] & f_{*}J^{\bullet}.
\end{tikzcd}
\end{center}
Therefore, the following diagram commutes.
\begin{center}
\begin{tikzcd}
{H^{p}(Y,f_{*}f^{*}\mathcal{F})} \arrow[d] \arrow[r] & {H^{p}(X,f^{*}\mathcal{F})} \arrow[d] \\
{H^{p}(Y,f_{*}f^{*}\mathcal{G})} \arrow[r] & {H^{p}(X,f^{*}\mathcal{G})}.
\end{tikzcd}
\end{center}
Next we prove the second claim. Let $0\to f^{*}g^{*}\mathcal{F}\to K^{\bullet}$ be an injective resolution of $f^{*}g^{*}\mathcal{F}$. Applying the functor $f_{*}$, we have a complex $0\to f_{*}f^{*}g^{*}\mathcal{F}\to f_{*}K^{\bullet}$. Let $0\to f_{*}f^{*}g^{*}\mathcal{F}\to J^{\bullet}$ be an injective resolution of $f_{*}f^{*}g^{*}\mathcal{F}$. By Lemma \ref{comparison}, we have a chain map $J^{\bullet}\to f_{*}K^{\bullet}$ which makes the following diagram commute.
\begin{center}
\begin{tikzcd}
0 \arrow[r] & f_{*}f^{*}g^{*}\mathcal{F} \arrow[r] \arrow[d, equal] & J^{0} \arrow[r] \arrow[d] & J^{1} \arrow[r] \arrow[d] & {\cdots} \\
0 \arrow[r] & f_{*}f^{*}g^{*}\mathcal{F} \arrow[r] & f_{*}K^{0} \arrow[r] & f_{*}K^{1} \arrow[r] & {\cdots}.
\end{tikzcd}
\end{center}
For an injective resolution $0\to g_{*}f^{*}f^{*}g^{*}\mathcal{F}\to I^{\bullet}$ of $g_{*}f^{*}f^{*}g^{*}\mathcal{F}$, we obtain chain maps $I^{\bullet}\to g_{*}J^{\bullet}$ and $I^{\bullet}\to g_{*}f_{*}K^{\bullet}$. Since the chain map $I^{\bullet}\to g_{*}f_{*}K^{\bullet}$ is unique up to homotopy, the following diagram commutes up to homotopy.
\begin{center}
\begin{tikzcd}
I^{\bullet} \arrow[d] \arrow[rd] & \\
g_{*}J^{\bullet} \arrow[r] & g_{*}f_{*}K^{\bullet}.
\end{tikzcd}
\end{center}
Therefore, the following diagram commutes.
\begin{center}
\begin{tikzcd}
{H^{p}(Z,g_{*}f_{*}f^{*}g^{*}\mathcal{F})} \arrow[d] \arrow[rd] & \\
{H^{p}(Y,f_{*}f^{*}g^{*}\mathcal{G})} \arrow[r] & {H^{p}(X,f^{*}g^{*}\mathcal{F})}.
\end{tikzcd}
\end{center}
\end{proof}
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\begin{prop}\label{lem_cohomology}
Let $X$ be a Noetherian scheme over $K$ and let $f:X\to X$ and $g:X\to X$ be endomorphisms over $K$. Then for all $p,m \in \mathbb{Z}_{\ge 0}$ the following diagram commutes.
\begin{center}
\begin{tikzcd}
H^{p}(X,\Omega_{X}^{\otimes m}) \arrow[d, "D_m^p(g)"'] \arrow[rd, "D_m^p(g\circ f)"]& \\
H^{p}(X,\Omega_{X}^{\otimes m}) \arrow[r, "D_m^p(f)"'] & H^{p}(X,\Omega_{X}^{\otimes m}),
\end{tikzcd}
\end{center}
where $D_m^p(f): H^{p}(X,\Omega_{X}^{\otimes m})\to H^{p}(X,\Omega_{X}^{\otimes m})$ is the map associated to the sheaf homomorphism $(df)^{\otimes m}:f^{*}\Omega_{X}^{\otimes m}\to \Omega_{X}^{\otimes m}$.
\end{prop}
\begin{proof}
By combining Lemma \ref{lem_adjunction}, Lemma \ref{lem_differential}, and the naturality of unit $\eta_{f}: 1\to f_{*}f^{*}$, we have following three commutative diagrams.
\begin{center}
\begin{tikzcd}
\Omega_{X}^{\otimes m} \arrow[d, "\eta_{g}"'] \arrow[rd, "\eta_{g\circ f}"] & & & \\
g_{*}g^{*}\Omega_{X}^{\otimes m} \arrow[r, "g_{*}\eta_{f}"'] & g_{*}f_{*}f^{*}g^{*}\Omega_{X}^{\otimes m}, & & \\
g^{*}\Omega_{X}^{\otimes m} \arrow[d, "dg^{\otimes m}"'] \arrow[r, "\eta_{f}"] & f_{*}f^{*}g^{*}\Omega_{X}^{\otimes m} \arrow[d, "f_{*}f^{*}dg^{\otimes m}"] & f^{*}g^{*}\Omega_{X}^{\otimes m} \arrow[d, "f^{*}dg^{\otimes m}"'] \arrow[rd, "d{(g \circ f)^{\otimes m}}"] & \\
\Omega_{X}^{\otimes m} \arrow[r, "\eta_{f}"'] & f_{*}f^{*}\Omega_{X}^{\otimes m}, & f^{*}\Omega_{X}^{\otimes m} \arrow[r, "df^{\otimes m}"'] & \Omega_{X}^{\otimes m}.
\end{tikzcd}
\end{center}
Taking cohomology $H^{p}(X,-)$ and using Lemma \ref{functoriality_of_pull-back}, we obtain
\begin{center}
\begin{tikzcd}
{H^{p}(X,\Omega_{X}^{\otimes m})} \arrow[d] \arrow[rd] \arrow[dd, "D_{m}^{p}(g)"', bend right=67] \arrow[rrdd, "D_{m}^{p}(g\circ f)", bend left] & & \\
{H^{p}(X,g^{*}\Omega_{X}^{\otimes m})} \arrow[d] \arrow[r] & {H^{p}(X,g^{*}f^{*}\Omega_{X}^{\otimes m})} \arrow[d] \arrow[rd] & \\
{H^{p}(X,\Omega_{X}^{\otimes m})} \arrow[r] \arrow[rr, "D_{m}^{p}(f)"', bend right] & {H^{p}(X,f^{*}\Omega_{X}^{\otimes m})} \arrow[r] & {H^{p}(X,\Omega_{X}^{\otimes m})}.
\end{tikzcd}
\end{center}.
\end{proof}
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\subsection{Conclusion of the proof}\label{conclution}
Now we are ready to prove Theorem \ref{maintheorem}.
\begin{proof}
Using Lemma \ref{lem_localvsgrobal}, we obtain
\begin{align*}
T_m(\phi^n)&=\sum_{i=0}^{1}(-1)^i\tr{D_m^i(\phi^n)}{\cohomo{i}{m}}.
\end{align*}
Therefore,
\begin{align*}
\lzeta{m}{\phi}=\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} \sum_{i=0}^{1}(-1)^{i}\mathrm{tr}D_m^i(\phi^n)\right).
\end{align*}
Using Proposition \ref{lem_cohomology}, we have $D_m^i(\phi^n)=D_m^i(\phi)^n$. Note that $D_m^i(\phi)$ is a $K$-linear operator acting on the finite dimensional $K$-vector space $\cohomo{i}{m}$. Therefore, we obtain
\begin{align*}
&\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} \sum_{i=0}^{1}(-1)^{i}\mathrm{tr}D_m^i(\phi^n)\right)\\
=&\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} \sum_{i=0}^{1}(-1)^{i}\mathrm{tr}D_m^i(\phi)^n\right)\\
=&\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} \mathrm{tr}D_m^0(\phi)^n\right)\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} \mathrm{tr}D_m^1(\phi)^n\right)^{-1}\\
=&\frac{\det (1-tD_m^1(\phi))}{\det (1-tD_{m}^0(\phi))}\in K(t).
\end{align*}
Summing up, we conclude
\[\lzeta{m}{\phi}=\dfrac{\det (1-tD_m^1(\phi))}{\det (1-tD_{m}^0(\phi))} .\]
\end{proof}
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\section{Examples}\label{examples}
In this section, we construct some examples of completely transversal rational functions and calculate an explicit form of $\dzeta{m}{\phi}$ using Theorem \ref{maintheorem}. We consider the case $K=\mathbb{C}$ since we use the results on complex dynamics.
\begin{defin}
Let $x$ be a periodic point with minimal period $n$ for a rational function $\phi\in \mathbb{C}(z)$, and $\lambda=\mult{\phi^n}{x}$ be the multiplier of $\phi$ at $x$. Then $x$ is:
\begin{itemize}
\item[$(1)$]\textit{attracting} if $|\lambda|<1$;
\item[$(2)$]\textit{repelling} if $|\lambda|>1$;
\item[$(3)$]\textit{rationally indifferent} if $\lambda$ is a root of unity;
\item[$(4)$]\textit{irrationally indifferent} if $|\lambda|=1$, but $\lambda$ is not a root of unity.
\end{itemize}
\end{defin}
\begin{defin}
Let $\phi\in\mathbb{C}(z)$ be a rational function and $x\in\per{n}{\phi}$. For a point $c\in\mathbb{P}^{1}_{\mathbb{C}}$, we say that $c$ \textit{is attracted to the orbit of} $x$ if there exists $i\in\{0,1,\dots,n-1\}$ such that $\displaystyle \lim_{m\to\infty}\phi^{mn}(c)=\phi^{i}(x)$ with respect to the classical topology of $\mathbb{P}^{1}_{\mathbb{C}}$.
\end{defin}
\begin{thm}\label{att_rat_ind}
Let $\phi\in\mathbb{C}(z)$ be a rational function. If $x$ is a periodic point of $\phi$ and $x$ is attracting or rationally indifferent, then there exists a critical point $c$ of $\phi$ which is attracted to the orbit of $x$.
\end{thm}
\begin{proof}
See \cite[Theorem 9.3.1. and Theorem 9.3.2.]{Beardon}.
\end{proof}
\begin{rem}\label{crit}
If $\phi(z)=F(z)/G(z)$ in lowest terms, the \textit{degree} of $\phi$ is $\deg \phi=\mathrm{max}\{\deg F,\deg G\}$. By Riemann-Hurwitz formula, $\phi$ has at most $2d-2$ critical points. See \cite[Section 1.2]{Silverman}.
\end{rem}
Using Theorem \ref{att_rat_ind}, we can find a sufficient condition for the complete transversality.
\begin{cor}\label{sufficient}
Let $\phi\in\mathbb{C}(z)$ be a rational function of degree $d$. If $\phi$ has $2d-2$ attracting periodic points whose orbits are pairwise distinct, then $\phi$ is completely transversal.
\end{cor}
\begin{proof}
By the definition of the complete transversality, $\phi$ is completely transversal if $\phi$ has no rationally indifferent periodic points. By Theorem \ref{att_rat_ind} and \ref{crit}, $\phi$ has no rationally indifferent periodic points if $\phi$ has $2d-2$ attracting periodic points whose orbits are pairwise distinct.
\end{proof}
We can construct completely transversal rational functions using Corollary \ref{sufficient}.
\begin{ex}
Let $\lambda_0,\lambda_{\infty}\in \mathbb{C}$ be complex numbers such that $|\lambda_0|<1$ and $|\lambda_{\infty}|<1$. We define $\phi\in\mathbb{C}(z)$ by
\[\phi(z)=\frac{z^2+\lambda_0 z}{\lambda_{\infty} z+1}\in\mathbb{C}(z).\]
Then we have
\[\fixpt{\phi}=\left\{0,\infty,\alpha=\frac{1-\lambda_{0}}{1-\lambda_{\infty}}\right\}.\]
The multipliers of $\phi$ at $0$ and $\infty$ are $\mult{\phi}{0}=\lambda_0$ and $\mult{\phi}{\infty}=\lambda_{\infty}$, respectively. Since $|\lambda_0|<1$ and $|\lambda_{\infty}|<1$, $\phi$ has two attracting fixed points. On the other hand, $\phi$ has at most $2=2d-2$ critical points since $d=\deg \phi =2$. Therefore, $\phi$ is completely transversal.
\end{ex}
\begin{rem}
All nonconstant polynomials have a fixed point at $\infty$ and the multiplier at $\infty$ is $0$. Therefore, if $\psi\in\mathbb{C}(z)$ is conjugate to a polynomial, then $\psi$ has a fixed point with multiplier $0$. Since $\mult{\phi}{0}=\lambda_0$, $\mult{\phi}{\infty}=\lambda_{\infty}$, and $\mult{\phi}{\alpha}=\dfrac{2-\lambda_{0}-\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}}$, $\phi$ is not conjugate to any polynomial if $\lambda_{0}\lambda_{\infty}\neq 0$.
\end{rem}
Next, we calculate the dynamical zeta function $\dzeta{1}{\phi}$ using the formula
\[\dzeta{1}{\phi}=\frac{\dyndet{1}}{\dyndet{2}}.\]
\begin{ex}
We use {\v{C}}ech cohomology to compute the linear maps $D_{1}^{1}(\phi)$ and $D_{2}^{1}(\phi)$ explicitly.
We define $F_{0}(z),F_{\infty}(z)\in \mathbb{C}[z]$, and $G_0(w)\in\mathbb{C}[w]$ by $F_{0}(z)=z+\lambda_{0}$, $F_{\infty}(z)=\lambda_{\infty} z +1$, and $G_0(w)=\lambda_{0} w +1$, respectively. Note that $\phi(z)=zF_{0}/F_{\infty}$ and $G_0(1/z)=F_{0}(z)/z$. We take open coverings $\mathcal{U}=\{U_0, U_1\}$ and $\mathcal{V}=\{V_0, V_1\}$ of $\mathbb{P}^{1}$, where $U_0=\mathbb{P}^{1}\setminus\{\infty\}=\spe{\mathbb{C}[z]}$, $U_1=\mathbb{P}^{1}\setminus\{0\}=\spe{\mathbb{C}[w]}$, $V_0=\phi^{-1}(U_0)=\mathbb{P}^{1}\setminus\{\infty, -\lambda_{\infty}^{-1}\}=\spe{\mathbb{C}[z,F_{\infty}^{-1}]}$, and $V_1=\phi^{-1}(U_1)=\mathbb{P}^{1}\setminus\{0, -\lambda_{0}\}=\spe{\mathbb{C}[w,G_0^{-1}]}$. Then the differential induces the map $\check{H}^{1}(\mathcal{U},\Omega_{\mathbb{P}^{1}}^{\otimes m})\to \check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ and this map is identified with $D_{m}^{1}(\phi)$ via the isomrphisms $H^{1}(\mathbb{P}^{1},\Omega_{\mathbb{P}^{1}}^{\otimes m})\cong \check{H}^{1}(\mathcal{U},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ and $H^{1}(\mathbb{P}^{1},\Omega_{\mathbb{P}^{1}}^{\otimes m})\cong \check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}}^{\otimes m})$.
$\check{H}^{1}(\mathcal{U},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ is the first cohomology of the complex
\[\mathbb{C}[z](dz)^{\otimes m}\times \mathbb{C}[w](dw)^{\otimes m}\to \mathbb{C}[z^{\pm 1}]\dform{m}.\]
$\check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ is the first cohomology of the complex
\[\mathbb{C}[z,F_{\infty}^{-1}](dz)^{\otimes m}\times \mathbb{C}[w,G_{0}^{-1}](dw)^{\otimes m}\to \mathbb{C}[z^{\pm 1},F_{0}^
{-1},F_{\infty}^{-1}]\dform{m}.\]
Note that $\dfrac{z^m a(z)}{F_{\infty}^i}\left(\dfrac{dz}{z}\right)^{\otimes m}=0$ and $\dfrac{b(1/z)}{z^{m-j}F_{0}^j}\left(\dfrac{dz}{z}\right)^{\otimes m}=0$ in $\check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ for all $a(z)\in\mathbb{C}[z]$ and $b(w)\in\mathbb{C}[w]$ since they are the image of $\dfrac{a(z)}{F_{\infty}^i}(dz)^{\otimes m}$ and $\dfrac{b(w)}{G_0^j}(dw)^{\otimes m}$, respectively.
Both $\check{H}^{1}(\mathcal{U},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ and $\check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ have a $\mathbb{C}$-basis
\[\left\{ z^i \dform{m} \colon |i|<m\right \}.\]
The image of $z^i\left(\dfrac{dz}{z}\right)^{\otimes m}$ by $(d\phi)^{\otimes m}$ is
\begin{align*}
(d\phi)^{\otimes m}\left(z^{i}\dform{m}\right)&=\phi^{i}\frac{d\phi}{\phi}\\
&=z^i\frac{F_{0}^{i}}{F_{\infty}^{i}}\left(1+\frac{(1-\lambda_{0}\lambda_{\infty})z}{F_{0}F_{\infty}}\right)^{m}\dform{m}\\
&=\sum_{j=0}^{m}\binom{m}{j}(1-\lambda_{0}\lambda_{\infty})^j\frac{z^{i+j}}{F_{0}^{j-i} F_{\infty}^{i+j}}\dform{m}.
\end{align*}
We can compute $\dfrac{z^{i+j}}{F_{0}^{j-i} F_{\infty}^{i+j}}\left(\dfrac{dz}{z}\right)^{\otimes m}$ by
\[ \frac{\lambda_{0}}{F_{0}}=-\sum_{n=1}^{m-1}\left(\frac{-\lambda_{0}}{z}\right)^n-\frac{(-\lambda_{0})^m}{z^{m-1}F_{0}},\]
\[ \frac{1}{F_{\infty}}=\sum_{n=0}^{m-1}(-\lambda_{\infty} z)^n+\frac{(-\lambda_{\infty} z)^{m}}{F_{\infty}}, \,\text{and}\]
\[ \frac{(1-\lambda_{0}\lambda_{\infty})z}{F_{0}F_{\infty}}=\frac{1}{F_{\infty}}-\frac{\lambda_{0}}{F_{0}}.\]
For example, if $m=1$ then $\dfrac{1}{F_{\infty}}\dfrac{dz}{z}=\dfrac{dz}{z}$ and $\dfrac{\lambda_{0}}{F_{0}}\dfrac{dz}{z}=0$ in $\check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}})$. Therefore,
\begin{align*}
d\phi\left(\frac{dz}{z}\right)&=\sum_{j=0}^{1}\binom{1}{j}(1-\lambda_{0}\lambda_{\infty})^j\frac{z^{j}}{F_{0}^{j} F_{\infty}^{j}}\frac{dz}{z}\\
&=\frac{dz}{z}+\frac{(1-\lambda_{0}\lambda_{\infty})z}{F_{0}F_{\infty}}\frac{dz}{z}\\
&=\frac{dz}{z}+\left(\frac{1}{F_{\infty}}-\frac{\lambda_{0}}{F_{0}}\right)\frac{dz}{z}\\
&=2\frac{dz}{z}.
\end{align*}
The characteristic polynomial of $D_{1}^{1}(\phi)$ is $\det(1-tD_{1}^{1}(\phi))=1-2t$.
A similar but complicated computation shows that the representation matrix of $D_{2}^{1}(\phi)$ with respect to the basis $\{z^i(dz/z)^{\otimes 2} \colon |i|<2\}$ is
\begin{align*}
\begin{pmatrix}
-\dfrac{\lambda_{0}\lambda_{\infty}^{2}}{1-\lambda_{0}\lambda_{\infty}} & -2\lambda_{0}\dfrac{2-\lambda_{0}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}} & -\dfrac{\lambda_{0}^{3}}{1-\lambda_{0}\lambda_{\infty}} \\[8pt]
\dfrac{\lambda_{\infty}^{2}}{1-\lambda_{0}\lambda_{\infty}} & 2\dfrac{2-\lambda_{0}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}} & \dfrac{\lambda_{0}^{2}}{1-\lambda_{0}\lambda_{\infty}} \\
-\dfrac{\lambda_{0}^{2}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}} & -2\lambda_{\infty}\dfrac{2-\lambda_{0}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}} &
-\dfrac{\lambda_{0}^{2}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}} \\
\end{pmatrix}
=
\begin{pmatrix}
\lambda_{0} A & \lambda_{0} B & \lambda_{0} C\\
-A & -B & -C\\
\lambda_{\infty} A & \lambda_{\infty} B & \lambda_{\infty} C\\
\end{pmatrix},
\end{align*}
where
\[A=-\frac{\lambda_{\infty}^2}{1-\lambda_{0}\lambda_{\infty}},\, B=-2 \dfrac{2-\lambda_{0}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}}, \text{and}\, C=-\dfrac{\lambda_{0}^{2}}{1-\lambda_{0}\lambda_{\infty}}.\]
Therefore, the characteristic polynomial of $D_{2}^{1}(\phi)$ is
\begin{align*}
\det(1-tD_{2}^{1}(\phi))&=1+(B-\lambda_{0} A- \lambda_{\infty} C)t\\
&=1-\left(2+\lambda_{0}+\lambda_{\infty}+\frac{2-\lambda_{0}-\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}}\right)t\\
&=1-(2+\sigma_{1}(\phi))t,
\end{align*}
where $\sigma_{1}(\phi)=\mult{\phi}{0}+\mult{\phi}{\infty}+\mult{\phi}{\alpha}$ is the sum of the multipliers at fixed points of $\phi$.
Summing up, we obtain an explicit formula for $\dzeta{1}{\phi}$.
\begin{align*}
\dzeta{1}{\phi}&=\frac{\dyndet{1}}{\dyndet{2}}\\
&=\frac{1-2t}{1-(2+\sigma_1(\phi))t}.
\end{align*}
\end{ex}
\end{document}
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\begin{document}
\maketitle
\centerline{\scshape{Silvia Frassu$^{\sharp,*}$, Tongxing Li$^{\natural}$ \and Giuseppe Viglialoro$^{\sharp}$}}
{
\centerline{$^\sharp$Dipartimento di Matematica e Informatica}
\centerline{Universit\`{a} di Cagliari}
\centerline{Via Ospedale 72, 09124. Cagliari (Italy)}
}
{
\centerline{$^{\natural}$School of Control Science and Engineering}
\centerline{Shandong University}
\centerline{Jinan, Shandong, 250061 (P. R. China)}
}
\begin{abstract}
We enter the details of two recent articles concerning as many chemotaxis models, one nonlinear and the other linear, and both with produced chemoattractant and saturated chemorepellent. More precisely, we are referring respectively to the papers ``Boundedness in a nonlinear attraction-repulsion Keller--Segel system with production and consumption'', by S. Frassu, C. van der Mee and G. Viglialoro [\textit{J. Math.\ Anal.\ Appl.} {\bf 504}(2):125428, 2021] and ``Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent'', by S. Frassu and G. Viglialoro [\textit{Nonlinear Anal. }{\bf 213}:112505, 2021]. These works, when properly analyzed, leave open room for some improvement of their results. We generalize the outcomes of the mentioned articles, establish other statements and put all the claims together; in particular, we select the sharpest ones and schematize them. Moreover, we complement our research also when logistic sources are considered in the overall study.
\end{abstract}
\section{Preamble}
For details and discussions on the meaning of the forthcoming model, especially in the frame of chemotaxis phenomena and related variants, as well as for mathematical motivations and connected state of the art, we refer to \cite{FrassuCorViglialoro,frassuviglialoro}. These articles will be often cited throughout this work.
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\section{Preamble}
For details and discussions on the meaning of the forthcoming model, especially in the frame of chemotaxis phenomena and related variants, as well as for mathematical motivations and connected state of the art, we refer to \cite{FrassuCorViglialoro,frassuviglialoro}. These articles will be often cited throughout this work.
\section{Presentation of the Theorems}\label{IntroSection}
Let $\Omega \subset \mathbb R} \def\N{\mathbb N^n$, $n \geq 2$, be a bounded and smooth domain, $\chi,\xi,\delta>0$, $m_1,m_2,m_3\in\mathbb R} \def\N{\mathbb N$, $f(u), g(u)$ and $h(u)$ be reasonably regular functions generalizing the prototypes $f(u)=K u^\alpha$, $g(u)=\gamma u^l$, and $h(u)=k u - \mu u^{\beta}$ with $K,\gamma, \mu>0$, $k \in \mathbb R} \def\N{\mathbb N$ and suitable $\alpha, l, \beta>0$. Once nonnegative initial configurations $u_0$ and $v_0$ are fixed, we aim at deriving sufficient conditions involving the above data so to ensure that the following attraction-repulsion chemotaxis model
\begin{equation}\label{problem}
\begin{cases}
u_t= \nabla \cdot ((u+1)^{m_1-1}\nabla u - \chi u(u+1)^{m_2-1}\nabla v+\xi u(u+1)^{m_3-1}\nabla w) + h(u) & \text{ in } \Omega \times (0,T_{\rm{max}}),\\
v_t=\Delta v-f(u)v & \text{ in } \Omega \times (0,T_{\rm{max}}),\\
0= \Delta w - \delta w + g(u)& \text{ in } \Omega \times (0,T_{\rm{max}}),\\
u_{\nu}=v_{\nu}=w_{\nu}=0 & \text{ on } \partial \Omega \times (0,T_{\rm{max}}),\\
u(x,0)=u_0(x), \; v(x,0)=v_0(x) & x \in \bar\Omega,
\end{cases}
\end{equation}
admits classical solutions which are global and uniformly bounded in time. Specifically, we look for nonnegative functions $u=u(x,t), v=v(x,t), w=w(x,t)$ defined for $(x,t) \in \bar{\Omega}\times [0,T_{\rm{max}})$, and $T_{\rm{max}}=\infty$, with the properties that
\begin{equation}\label{ClassicalAndGlobability}
\begin{cases}
u,v\in C^0(\bar{\Omega}\times [0,\infty))\cap C^{2,1}(\bar{\Omega}\times (0,\infty)), w\in C^0(\bar{\Omega}\times [0,\infty))\cap C^{2,0}(\bar{\Omega}\times (0,\infty)),\\ (u,v,w) \in (L^\infty((0, \infty);L^{\infty}(\Omega)))^3,
\end{cases}
\end{equation}
and pointwisely satisfying all the relations in problem \eqref{problem}.
To this scope, let $f$, $g$ and $h$ be such that
\begin{equation}\label{f}
f,g \in C^1(\mathbb R} \def\N{\mathbb N) \quad \textrm{with} \quad 0\leq f(s)\leq Ks^{\alpha} \textrm{ and } \gamma s^l\leq g(s)\leq \gamma s(s+1)^{l-1},\; \textrm{for some}\; K,\,\gamma,\,\alpha>0,\, l\geq 1 \quad \textrm{and all } s \geq 0,
\end{equation}
and
\begin{equation}\label{h}
h \in C^1(\mathbb R} \def\N{\mathbb N) \quad \textrm{with} \quad h(0)\geq 0 \textrm{ and } h(s)\leq k s-\mu s^{\beta}, \quad \textrm{for some}\quad k \in \mathbb R} \def\N{\mathbb N,\,\mu>0,\, \beta>1\, \quad \textrm{and all } s \geq 0.
\end{equation}
Then we prove these two theorems.
\begin{theorem}\label{MainTheorem}
Let $\Omega$ be a smooth and bounded domain of $\mathbb{R}^n$, with $n\geq 2$, $\chi, \xi, \delta$ positive reals, $l \geq 1$ and $h \equiv 0$. Moreover, for $\alpha >0$ and $m_1, m_2, m_3 \in \mathbb R} \def\N{\mathbb N$, let $f$ and $g$ fulfill \eqref{f} for each of the following cases:
\begin{enumerate}[label=$A_{\roman*}$)]
\item \label{A1} $\alpha \in \left(0, \frac{1}{n}\right]$ and $m_1>\min\left\{2m_2-(m_3+l),\max\left\{2m_2-1,\frac{n-2}{n}\right\}, m_2 - \frac{1}{n}\right\}=:\mathcal{A}$,
\item \label{A2} $\alpha \in \left(\frac{1}{n},\frac{2}{n}\right)$ and $m_1>m_2 + \alpha - \frac{2}{n}=:\mathcal{B}$,
\item \label{A3} $\alpha \in \left[\frac{2}{n},1\right]$ and $m_1>m_2 + \frac{n\alpha-2}{n\alpha-1}=:\mathcal{C}$.
\end{enumerate}
Then for any initial data $(u_0,v_0)\in (W^{1,\infty}(\Omega))^2$, with $u_0, v_0\geq 0$ on $\bar{\Omega}$, there exists a unique triplet $(u,v,w)$ of nonnegative functions, uniformly bounded in time and classically solving problem \eqref{problem}.
\end{theorem}
\begin{theorem}\label{MainTheorem1}
Under the same hypotheses of Theorem \ref{MainTheorem} and $\beta>1$, let $h$ comply with \eqref{h}.
Moreover, for $\alpha >0$ and $m_1, m_2, m_3 \in \mathbb R} \def\N{\mathbb N$, let $f$ and $g$ fulfill \eqref{f} for each of the following cases:
\begin{enumerate}[label=$A_{\roman*}$)]
\setcounter{enumi}{3}
\item \label{A4} $\alpha \in \left(0, \frac{1}{n}\right]$ and $m_1>\min\left\{2m_2-(m_3+l), 2m_2-\beta\right\}=:\mathcal{D}$,
\item \label{A5} $\alpha \in \left(\frac{1}{n},1\right)$ and $m_1>\min\left\{2m_2+1-(m_3+l), 2m_2+1-\beta\right\}=:\mathcal{E}$.
\end{enumerate}
Then the same claim holds true.
\end{theorem}
When the logistic term $h$ does not take part in the model, problem \eqref{problem} has been already analyzed in \cite{FrassuCorViglialoro} for the nonlinear diffusion and sensitivities case, and in \cite{frassuviglialoro} for the linear scenario; nevertheless, in these papers only small values of $\alpha$ are considered. Precisely, for $\alpha$ belonging to $(0,\frac{1}{2}+\frac{1}{n}),$ boundedness is ensured:
\begin{itemize}
\item in \cite[Theorem 2.1]{FrassuCorViglialoro} for $m_1,m_2,m_3\in \mathbb R} \def\N{\mathbb N$ and $l\geq 1$, under the assumption
$$m_1>\min\left\{2m_2+1-(m_3+l),\max\left\{2m_2,\frac{n-2}{n}\right\}\right\}=:\mathcal{F};$$
\item in \cite[Theorem 2.1]{frassuviglialoro} for either $m_1=m_2=m_3=l=1$, under the assumption
$$\xi>\left(\frac{8}{n}
\frac{2^\frac{2}{n}\frac{2}{n}^{n+1}(\frac{2}{n}-1)(n^2+n)}{(\frac{2}{n}+1)^{\frac{2}{n}+1}}\right)^\frac{2}{n}\lVert v_0\rVert_{L^\infty(\Omega)}^\frac{4}{n}=:\mathcal{G},$$
or in \cite[Theorem 2.2]{frassuviglialoro} for $m_1=m_2=m_3=1$ and any $l>1$.
\end{itemize}
In light of Theorems \ref{MainTheorem} and \ref{MainTheorem1}, herein we develop an analysis dealing also with values of $\alpha$ larger than $\frac{1}{2}+\frac{1}{n}$. Additionally, for $\alpha$ belonging to some sub-intervals of $(0,\frac{1}{2}+\frac{1}{n})$ we improve \cite[Theorem 2.1]{FrassuCorViglialoro} and \cite[Theorems 2.1 and 2.2]{frassuviglialoro}. On the other hand, the introduction of $h$ allows us to obtain further generalizations and/or claims.
All this aspects are put together into Table \ref{Table_ResultUnified}. It, when possible, also indicates which of the assumptions taken from \cite[Theorem 2.1]{FrassuCorViglialoro}, \cite[Theorems 2.1 and 2.2]{frassuviglialoro}, and Theorems \ref{MainTheorem} and \ref{MainTheorem1} are the mildest leading to boundedness.
\setlength\extrarowheight{4.8pt}
\begin{table}[h!]
\makegapedcells
\centering
\begin{tabular}{ccccccccccc}
\hline
$m_2$ &$m_3$&$l$&\multicolumn{1}{c|}{$\alpha$}&\multicolumn{1}{c|}{$m_1$}&$\chi$&$\xi$& Reference & \quad \quad Implication\\
\hline
$1$ &$1$&$1$&\multicolumn{1}{c|}{$[\frac{2}{n},1)$}&\multicolumn{1}{c|}{$1$}&$\mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$ >\mathcal{G}$}& Remark \ref{Alpha}, generalizing \cite[Th. 2.1]{frassuviglialoro}\\
$1$ &$1$&$>1$&\multicolumn{1}{c|}{$ [\frac{2}{n},1)$}&\multicolumn{1}{c|}{$1$}&$ \mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}& Remark \ref{Alpha}, generalizing \cite[Th. 2.2]{frassuviglialoro}\\
$ \mathbb R} \def\N{\mathbb N$ &$\mathbb R} \def\N{\mathbb N$&$\geq 1$&\multicolumn{1}{c|}{$ (\frac{1}{n},1)$}&\multicolumn{1}{c|}{$>\mathcal{F}$}&$\mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}& Remark \ref{Alpha}, generalizing \cite[Th. 2.1]{FrassuCorViglialoro}\\
$ \mathbb R} \def\N{\mathbb N$ &$\mathbb R} \def\N{\mathbb N$&$\geq 1$&\multicolumn{1}{c|}{$ (0,\frac{1}{n}]$}&\multicolumn{1}{c|}{$>\mathcal{A}$}&$\mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem}& \quad \quad $\ast$ and $\ast\ast$\\
$\mathbb R} \def\N{\mathbb N$& $\mathbb R} \def\N{\mathbb N$&$\geq 1$&\multicolumn{1}{c|}{$(\frac{1}{n},\frac{2}{n})$}&\multicolumn{1}{c|}{$>\mathcal{B}$}&$\mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem}& \quad \quad $\ast$\\
$\mathbb R} \def\N{\mathbb N$& $\mathbb R} \def\N{\mathbb N$&$\geq 1$&\multicolumn{1}{c|}{$[\frac{2}{n},1]$}&\multicolumn{1}{c|}{$>\mathcal{C}$}&$\mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem}\\
\hline
$m_2$ &$m_3$&$l$&$\beta$&\multicolumn{1}{c|}{$\alpha$}&\multicolumn{1}{c|}{$m_1$}&$\chi$&$\xi$&$k$&$\mu$& Reference \\
\hline
$1$ &$1$&$1$&$>2$&\multicolumn{1}{c|}{$(\frac{1}{n},1)$}&\multicolumn{1}{c|}{$1$}&$\mathbb R} \def\N{\mathbb N^+$&$\mathbb R} \def\N{\mathbb N^+$&$ \mathbb R} \def\N{\mathbb N$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem1}\\
$ \mathbb R} \def\N{\mathbb N$ &$\mathbb R} \def\N{\mathbb N$&$\geq 1$&$>1$&\multicolumn{1}{c|}{$ (0,\frac{1}{n}]$}&\multicolumn{1}{c|}{$>\mathcal{D}$}&$\mathbb R} \def\N{\mathbb N^+$&$ \mathbb R} \def\N{\mathbb N^+$&$ \mathbb R} \def\N{\mathbb N$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem1}\\
$\mathbb R} \def\N{\mathbb N$& $\mathbb R} \def\N{\mathbb N$&$\geq 1$&$>1$&\multicolumn{1}{c|}{$(\frac{1}{n},1)$}&\multicolumn{1}{c|}{$>\mathcal{E}$}&$\mathbb R} \def\N{\mathbb N^+$&$ \mathbb R} \def\N{\mathbb N^+$&$ \mathbb R} \def\N{\mathbb N$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem1}\\
\hline
\end{tabular}
\caption{Schematization collecting the ranges of the parameters involved in model \eqref{problem} for which boundedness of its solutions is established for any fixed initial distribution $u_0$ and $v_0$. The symbol $\ast$ stands for ``improves \cite[Th. 2.1]{frassuviglialoro} and recovers \cite[Th. 2.2]{frassuviglialoro}'' and $\ast\ast$ for ``improves \cite[Th. 2.1]{FrassuCorViglialoro}''. ($\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}, \mathcal{E}, \mathcal{F}$ are defined above.)
}\label{Table_ResultUnified}
\end{table}
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0.87.2
|
\section{Local well posedness, boundedness criterion, main estimates and analysis of parameters}\label{SectionLocalInTime}
For $\Omega$, $\chi,\xi,\delta$, $m_1,m_2,m_3$ and $f, g, h$ as above, from now on with $u, v, w \geq 0$ we refer to functions of $(x,t) \in \bar{\Omega}\times [0,T_{\rm{max}})$, for some finite $T_{\rm{max}}$, classically solving problem \eqref{problem} when nonnegative initial data $(u_0,v_0)\in (W^{1,\infty}(\Omega))^2$ are provided. In particular, $u$ satisfies
\begin{equation}\label{massConservation}
\int_\Omega u(x, t)dx \leq m_0 \quad \textrm{for all }\, t \in (0,T_{\rm{max}}),
\end{equation}
whilst $v$ is such that
\begin{equation*}\label{MaxPrincV}
0 \leq v\leq \lVert v_0\rVert_{L^\infty(\Omega)}\quad \textrm{in}\quad \Omega \times (0,T_{\rm{max}}).
\end{equation*}
Further, globality and boundedness of $(u,v,w)$ (in the sense of \eqref{ClassicalAndGlobability}) are ensured whenever (boundedness criterion) the $u$-component belongs to $L^\infty((0,T_{\rm{max}});L^p(\Omega))$, with $p>1$ arbitrarily large, and uniformly with respect $t\in (0,T_{\rm{max}})$.
These basic statements can be proved by standard reasoning; in particular, when $h\equiv 0$ they verbatim follow from \cite[Lemmas 4.1 and 4.2]{FrassuCorViglialoro} and relation \eqref{massConservation} is the well-known mass conservation property. Conversely, in the presence of the logistic terms $h$ as in \eqref{h}, some straightforward adjustments have to be considered and the $L^1$-bound of $u$ is consequence of an integration of the first equation in \eqref{problem} and an application of the H\"{o}lder inequality: precisely for $k_+=\max\{k,0\}$
\[
\frac{d}{dt} \int_\Omega u = \int_\Omega h(u) =k \int_\Omega u - \mu \int_\Omega u^{\beta} \leq k_+ \int_\Omega u - \frac{\mu}{|\Omega|^{\beta-1}} \left(\int_\Omega u\right)^{\beta}
\quad \textrm{for all }\, t \in (0,T_{\rm{max}}),
\]
and we can conclude by invoking an ODI-comparison argument.
In our computations, beyond the above positions, some uniform bounds of $\|v(\cdot,t)\|_{W^{1,s}(\Omega)}$ are required. In this sense, the following lemma gets
the most out from $L^p$-$L^q$ (parabolic) maximal regularity; this is a cornerstone and for some small values of $\alpha$ the succeeding $W^{1,s}$-estimates are sharper than the $W^{1,2}$-estimates derived in \cite{FrassuCorViglialoro,frassuviglialoro}, and therein employed.
\begin{lemma}\label{LocalV}
There exists $c_0>0$ such that $v$ fulfills
\begin{equation}\label{Cg}
\int_\Omega |\nabla v(\cdot, t)|^s\leq c_0 \quad \textrm{on } \, (0,T_{\rm{max}})
\begin{cases}
\; \textrm{for all } s \in [1,\infty) & \textrm{if } \alpha \in \left(0, \frac{1}{n}\right],\\
\; \textrm{for all } s \in \left[1, \frac{n}{(n\alpha-1)}\right) & \textrm{if } \alpha \in \left(\frac{1}{n},1\right].
\end{cases}
\end{equation}
\begin{proof}
For each $\alpha \in (0,1]$, there is $\rho >\frac{1}{2}$ such that for all $s \in \left[\frac{1}{\alpha},\frac{n}{(n\alpha-1)_+}\right)$ we have
$\frac{1}{2}<\rho <1-\frac{n}{2}\big(\alpha-\frac{1}{s}\big)$. From $1-\rho-\frac{n}{2}\big(\alpha-\frac{1}{s}\big)>0$, the claim follows invoking properties related to the Neumann heat semigroup, exactly as done in the second part of \cite[Lemma 5.1]{FrassuCorViglialoro}.
\end{proof}
\end{lemma}
We will make use of this technical result.
\begin{lemma}\label{LemmaCoefficientAiAndExponents}
Let $n\in \N$, with $n\geq 2$, $m_1>\frac{n-2}{n}$, $m_2,m_3\in \mathbb R} \def\N{\mathbb N$ and $\alpha \in (0,1]$. Then there is $s \in [1,\infty)$, such that for proper
$p,q\in[1,\infty)$, $\theta$ and $\theta'$, $\mu$ and $\mu'$ conjugate exponents, we have that
\begin{align}
a_1&= \frac{\frac{m_1+p-1}{2}\left(1-\frac{1}{(p+2m_2-m_1-1)\theta}\right)}{\frac{m_1+p-1}{2}+\frac{1}{n}-\frac{1}{2}},
& a_2&=\frac{q\left(\frac{1}{s}-\frac{1}{2\theta'}\right)}{\frac{q}{s}+\frac{1}{n}-\frac{1}{2}}, \nonumber \\
a_3 &= \frac{\frac{m_1+p-1}{2}\left(1-\frac{1}{2\alpha\mu}\right)}{\frac{m_1+p-1}{2}+\frac{1}{n}-\frac{1}{2}},
& a_4 &= \frac{q\left(\frac{1}{s}-\frac{1}{2(q-1)\mu'}\right)}{\frac{q}{s}+\frac{1}{n}-\frac{1}{2}},\nonumber\\
\kappa_1 & =\frac{\frac{p}{2}\left(1- \frac{1}{p}\right)}{\frac{m_1+p-1}{2}+\frac{1}{n}-\frac{1}{2}} \nonumber, & \kappa_2 & = \frac{q - \frac{1}{2}}{q+\frac{1}{n}-\frac{1}{2}},
\end{align}
belong to the interval $(0,1)$. If, additionally,
\begin{equation}\label{Restrizionem1-m2-alphaPiccolo}
\alpha \in \left(0,\frac{1}{n}\right] \; \text{and}\; m_2-m_1<\frac{1}{n},
\end{equation}
\begin{equation}\label{Restrizionem1-m2-alphaGrande}
\alpha \in \left(\frac{1}{n},\frac{2}{n}\right) \; \text{and}\; m_2-m_1<\frac{2}{n}-\alpha,
\end{equation}
or
\begin{equation}\label{Restrizionem1-m2-alphaGrandeBis}
\alpha \in \left[\frac{2}{n},1\right] \; \text{and}\; m_2-m_1<\frac{2-n\alpha}{n\alpha-1},
\end{equation}
these futher relations hold true:
\begin{equation*}\label{MainInequalityExponents}
\beta_1 + \gamma_1 =\frac{p+2m_2-m_1-1}{m_1+p-1}a_1+\frac{1}{q}a_2\in (0,1) \;\textrm{ and }\; \beta_2 + \gamma_2= \frac{2 \alpha }{m_1+p-1}a_3+\frac{q-1}{q}a_4
\in (0,1).
\end{equation*}
\begin{proof}
For any $s\geq 1$, let $\theta'>\max\left\{\frac{n}{2},\frac{s}{2}\right\}$ and $\mu>\max\left\{\frac{1}{2\alpha},\frac{n}{2}\right\}$. Thereafter, for
\begin{equation}\label{Prt_q}
\begin{cases}
q > \max \left\{\frac{(n-2)}{n}\theta', \frac{s}{2\mu'}+1\right\} \\
p>\max \left\{2-\frac{2}{n}-m_1,\frac{1}{\theta}-2m_2+m_1+1, \frac{(2m_2-m_1-1)(n-2)\theta-nm_1+n}{n-(n-2)\theta},\frac{2\alpha \mu(n-2)}{n} -m_1+1\right\},
\end{cases}
\end{equation}
it can be seen that $a_i,k_2\in (0,1)$, for any $i=1,2,3,4.$ On the other hand, $k_1\in (0,1)$ also thanks to the assumption $m_1>\frac{n-2}{n}.$
As to the second part, we distinguish three cases: $\alpha \in \left(0,\frac{1}{n}\right]$, $\alpha \in \left(\frac{1}{n},\frac{2}{n}\right)$ and
$\alpha \in \left[\frac{2}{n},1\right].$ (We insert Figure \ref{FigureSpiegazioneLemma} to clarify the proof, by focusing on the relation involving the values of $\alpha$, $s$ and
$\theta'$ in terms of assumptions \eqref{Restrizionem1-m2-alphaPiccolo}, \eqref{Restrizionem1-m2-alphaGrande}, \eqref{Restrizionem1-m2-alphaGrandeBis}.)
\begin{itemize}
\item [$\circ$] $\alpha \in \left(0,\frac{1}{n}\right]$.
For $s>\frac{2\mu'}{2\mu'-1}$ arbitrarily large, consistently with \eqref{Prt_q}, we take $p=q=s$ and $\theta'=s\omega$, for some $\omega>\frac{1}{2}$. Computations provide
\begin{equation*}
0< \beta_1+\gamma_1=\frac{s+2m_2-m_1-1-\frac{1}{\theta}}{m_1+s-2+\frac{2}{n}}+\frac{2-\frac{1}{\omega}}{s+\frac{2s}{n}},
\end{equation*}
and
\begin{equation*}
0< \beta_2+\gamma_2=\frac{2\alpha-\frac{1}{\mu}}{m_1+s-2+\frac{2}{n}}+\frac{2s-2-\frac{s}{\mu'}}{s+\frac{2s}{n}}.
\end{equation*}
In light of the above positions, the largeness of $s$ infers $\theta$ arbitrarily close to $1$. Further, by choosing $\omega$ approaching $\frac{1}{2}$, continuity arguments imply that $\beta_1+\gamma_1<1$ whenever restriction \eqref{Restrizionem1-m2-alphaPiccolo} is satisfied, whereas $\beta_2+\gamma_2<1$ comes from $\mu>\frac{n}{2}.$
\item [$\circ$] $\alpha \in \left(\frac{1}{n},\frac{2}{n}\right).$ First let $s$ be arbitrarily close to $\frac{n}{n\alpha-1}$ and let $q=\frac{p}{2}$ fulfill \eqref{Prt_q}. Then, in these circumstances it holds that $\max\left\{\frac{s}{2},\frac{n}{2}\right\}=\frac{s}{2}$, so that restriction on $\theta'$ reads $\theta'>\frac{s}{2}$. Subsequently,
\begin{equation*}
0< \beta_1+\gamma_1=\frac{p+2m_2-m_1-1-\frac{1}{\theta}}{m_1+p-2+\frac{2}{n}}+\frac{2-\frac{s}{\theta'}}{p+\frac{2s}{n}-s},
\end{equation*}
and
\begin{equation*}
0< \beta_2+\gamma_2=\frac{2\alpha-\frac{1}{\mu}}{m_1+p-2+\frac{2}{n}}+\frac{p-2-\frac{s}{\mu'}}{p+\frac{2s}{n}-s}.
\end{equation*}
Since from $\theta'>\frac{s}{2}$ we have that $\theta'$ approaches $\frac{n}{2(n\alpha-1)}$, similar arguments used above imply that upon enlarging $p$ one can see that condition \eqref{Restrizionem1-m2-alphaGrande} leads to $\beta_1+\gamma_1<1$. On the other hand, in order to have $\beta_2+\gamma_2<1$ we have to invoke the above constrain on $\mu$, i.e., $\mu>\frac{1}{2\alpha}.$
\item [$\circ$] $\alpha \in \left[\frac{2}{n},1\right].$ We only have to consider in the previous case that now $\theta'>\frac{n}{2}$, so concluding thanks to
\eqref{Restrizionem1-m2-alphaGrandeBis}.
\end{itemize}
\end{proof}
\end{lemma}
\begin{figure}
\caption{The colored lines, functions of $\alpha$, represent the supremum of the difference $m_2-m_1$ for some space dimension $n$. Moreover, for the sub-intervals $(0,1/n]$, $(1/n,2/n)$ and $[2/n,1]$ of $\alpha$, the corresponding range of $s$ and choice of $\theta'$ are also indicated.}
\label{FigureSpiegazioneLemma}
\end{figure}
\begin{remark}\label{RemarkOnS}
In view of its importance in the computations, we have to point out that from the above lemma $s$ can be chosen arbitrarily large only when $\alpha \in \left(0,\frac{1}{n}\right]$.
In particular, as we will see, in such an interval the terms $\int_{\Omega} (u+1)^{p+2m_2-m_1-1} \vert \nabla v\lvert^2$ and $\int_{\Omega} (u+1)^{2\alpha} \vert \nabla v\lvert^{2(q-1)}$, appearing in our reasoning, can be treated in two alternative ways: either invoking the Young inequality or the Gagliardo--Nirenberg one.
\end{remark}
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0.87.3
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\section{A priori estimates and proof of the Theorems}\label{EstimatesAndProofSection}
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0.87.4
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\subsection{The non-logistic case}\label{NonLog}
Recalling the globality criterion mentioned in $\S$\ref{SectionLocalInTime}, let us define the functional $y(t):=\int_\Omega (u+1)^p + \int_\Omega |\nabla v|^{2q}$,
with $p,q>1$ properly large (and, when required, with $p=q$), and let us dedicate to derive the desired uniform bound of $\int_\Omega u^p$.
In the spirit of Remark \ref{RemarkOnS}, let us start by analyzing the evolution in time of the functional $y(t)$ by relying on the Young inequality.
\begin{lemma}\label{Estim_general_For_u^p_nablav^2qLemma}
Let $\alpha \in \left(0,\frac{1}{n}\right]$. If $m_1,m_2,m_3 \in \mathbb R} \def\N{\mathbb N$ comply with either $m_1>2m_2-(m_3+l)$ or $m_1>\max\left\{2m_2-1,\frac{n-2}{n}\right\}$,
then there exist $p, q>1$ such that $(u,v,w)$ satisfies for some $c_{16}, c_{17}, c_{18}>0$
\begin{equation}\label{MainInequality}
\frac{d}{dt} \left(\int_\Omega (u+1)^p + \int_\Omega |\nabla v|^{2q}\right) + c_{16} \int_\Omega |\nabla |\nabla v|^q|^2 + c_{17} \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2 \leq c_{18} \quad \text{on } (0,T_{\rm{max}}).
\end{equation}
\begin{proof}
Let $p=q>1$ sufficiently large; moreover, in view of Remark \ref{RemarkOnS}, from now on, when necessary we will tacitly enlarge these parameters.
In the first part of the proof we focus on the estimate of the term $\frac{d}{dt} \int_\Omega (u+1)^p$.
Standard testing procedures provide
\begin{equation*}
\begin{split}
\frac{d}{dt} \int_\Omega (u+1)^p =\int_\Omega p(u+1)^{p-1}u_t &= -p(p-1) \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2
+p(p-1)\chi \int_\Omega u(u+1)^{m_2+p-3} \nabla u \cdot \nabla v \\
&-p(p-1)\xi \int_\Omega u(u+1)^{m_3+p-3} \nabla u \cdot \nabla w \quad \text{on } (0,T_{\rm{max}}).
\end{split}
\end{equation*}
By reasoning as in \cite[Lemma 5.2]{FrassuCorViglialoro}, we obtain for $\epsilon_1, \epsilon_2, \tilde{\sigma}$ positive, and for all $t \in (0,T_{\rm{max}})$, some $c_1>0$ such that
\begin{equation}\label{Estim_1_For_u^p}
\begin{split}
\frac{d}{dt} \int_\Omega (u+1)^p & \leq -p(p-1) \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 +p(p-1)\chi \int_\Omega u(u+1)^{m_2+p-3} \nabla u \cdot \nabla v \\
&+ \left(\epsilon_1 + \tilde{\sigma} - \frac{p(p-1)\xi \gamma}{2^{p-1}(m_3+p-1)} \right) \int_\Omega (u+1)^{m_3+p+l-1}
+ \epsilon_2 \int_\Omega |\nabla (u+1)^\frac{m_1+p-1}{2}|^2 + c_1.
\end{split}
\end{equation}
Let us now discuss the cases $m_1>2m_2-(m_3+l)$ and $m_1>\max\left\{2m_2-1,\frac{n-2}{n}\right\}$, respectively.
A double application of the Young inequality in \eqref{Estim_1_For_u^p} and bound \eqref{Cg} give
\begin{equation}\label{Young}
\begin{split}
&p(p-1)\chi \int_\Omega u (u+1)^{m_2+p-3} \nabla u \cdot \nabla v \leq \epsilon_3 \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2
+ c_2 \int_\Omega (u+1)^{p+2m_2-m_1-1} |\nabla v|^2\\
& \leq \epsilon_3 \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 + \epsilon_4 \int_\Omega |\nabla v|^{s} + c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}}\\
& \leq \epsilon_3 \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 + c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}} + c_4 \quad \text{on }\, (0,T_{\rm{max}}),
\end{split}
\end{equation}
with $\epsilon_3, \epsilon_4 >0$ and some positive $c_2, c_3, c_4$.
From $m_1> 2m_2-(m_3+l)$, we have $\frac{(p+2m_2-m_1-1)s}{s-2} < (m_3+p+l-1)$, and for every $\epsilon_5>0$, Young's inequality yields some $c_5>0$ entailing
\begin{equation}\label{Young1}
c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}} \leq \epsilon_5 \int_\Omega (u+1)^{m_3+p+l-1} + c_5 \quad \text{for all }\, (0,T_{\rm{max}}).
\end{equation}
Now, we note that $m_1>2 m_2-1$ implies $\frac{(p+2m_2-m_1-1)s}{s-2} < p$, and the Young inequality allows us to rephrase \eqref{Young1} in an alternative way:
\begin{equation}\label{Young4}
c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}} \leq \epsilon_6 \int_\Omega (u+1)^p + c_6 \quad \text{on }\, (0,T_{\rm{max}}),
\end{equation}
with $\epsilon_6>0$ and positive $c_6$.
Further, an application of the Gagliardo--Nirenberg inequality and property \eqref{massConservation} yield
\begin{equation*}\label{Theta_2}
\theta=\frac{\frac{n(m_1+p-1)}{2}\left(1-\frac{1}{p}\right)}{1-\frac{n}{2}+\frac{n(m_1+p-1)}{2}}\in (0,1),
\end{equation*}
so giving for $c_7, c_8>0$
\begin{equation*}
\begin{split}
\int_{\Omega} (u+1)^p&= \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2p}{m_1+p-1}}(\Omega)}^{\frac{2p}{m_1+p-1}}\\
&\leq c_7 \|\nabla (u+1)^{\frac{m_1+p-1}{2}}\|_{L^2(\Omega)}^{\frac{2p}{m_1+p-1}\theta} \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{2p}{m_1+p-1}(1-\theta)} + c_7 \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{2p}{m_1+p-1}}\\
& \leq c_8 \Big(\int_\Omega |\nabla (u+1)^\frac{m_1+p-1}{2}|^2\Big)^{\kappa_1}+ c_8 \quad \text{ for all } t \in(0,T_{\rm{max}}).
\end{split}
\end{equation*}
Since $\kappa_1 \in (0,1)$ (see Lemma \ref{LemmaCoefficientAiAndExponents}), for any positive $\epsilon_7$ thanks to the Young inequality we arrive for some positive $c_9>0$ at
\begin{equation}\label{GN2}
\begin{split}
\epsilon_6 \int_\Omega (u+1)^p &\leq \epsilon_7 \int_\Omega |\nabla (u+1)^\frac{m_1+p-1}{2}|^2 + c_9 \quad \text{on } (0,T_{\rm{max}}).
\end{split}
\end{equation}
By plugging estimate \eqref{Young} into relation \eqref{Estim_1_For_u^p}, and by relying on bound \eqref{Young1} (or, alternatively to \eqref{Young1}, relations \eqref{Young4} and \eqref{GN2}), infer for appropriate $\tilde{\epsilon}_1, \tilde{\epsilon}_2>0$ and proper $c_{10}>0$
\begin{equation}\label{ClaimU}
\begin{split}
\frac{d}{dt} \int_\Omega (u+1)^p &\leq \left(-\frac{4p(p-1)}{(m_1+p-1)^2} + \tilde{\epsilon}_1 \right) \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\\
&+ \left(\tilde{\epsilon}_2 - \frac{p(p-1)\xi \gamma}{2^{p-1}(m_3+p-1)}\right) \int_\Omega (u+1)^{m_3+p+l-1} + c_{10} \quad \text{for all } t \in (0,T_{\rm{max}}),
\end{split}
\end{equation}
where we also exploited that
\begin{equation}\label{GradU}
\int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 = \frac{4}{(m_1+p-1)^2} \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2 \quad \text{on } (0,T_{\rm{max}}).
\end{equation}
Now, as to the term $\frac{d}{dt} \int_\Omega |\nabla v|^{2q}$ of the functional $y(t)$, reasoning similarly as in \cite[Lemma 5.3]{FrassuCorViglialoro}, we obtain for some
$c_{11}, c_{12}>0$
\begin{equation}\label{Estim_gradV}
\frac{d}{dt}\int_\Omega |\nabla v|^{2q}+ q \int_\Omega |\nabla v|^{2q-2} |D^2v|^2 \leq c_{11} \int_\Omega u^{2\alpha} |\nabla v|^{2q-2} +c_{12} \quad \textrm{on } \; (0,T_{\rm{max}}).
\end{equation}
Moreover, Young's inequalities and bound \eqref{Cg} give for every arbitrary $\epsilon_8, \epsilon_9>0$ and some $c_{13}, c_{14}, c_{15}>0$
\begin{equation}\label{Estimat_nablav^2q+2}
\begin{split}
&c_{11} \int_\Omega u^{2\alpha} |\nabla v|^{2q-2} \leq \epsilon_8 \int_\Omega (u+1)^{m_3+p+l-1} + c_{13} \int_\Omega |\nabla v|^{\frac{2(q-1)(m_3+p+l-1)}{m_3+p+l-1-2\alpha}}\\
& \leq \epsilon_8 \int_\Omega (u+1)^{m_3+p+l-1} + \epsilon_9 \int_\Omega |\nabla v|^s + c_{14}
\leq \epsilon_8 \int_\Omega (u+1)^{m_3+p+l-1} + c_{15} \quad \textrm{for all} \quad t \in (0,T_{\rm{max}}).
\end{split}
\end{equation}
Therefore, by inserting relation \eqref{Estimat_nablav^2q+2} into \eqref{Estim_gradV} and adding \eqref{ClaimU}, we have the claim for a proper choice of
$\tilde{\epsilon}_1, \tilde{\epsilon}_2, \epsilon_8$ and some positive $c_{16}, c_{17}, c_{18}$, also by taking into account the relation (see \cite[page 17]{FrassuCorViglialoro})
\begin{equation}\label{GradV}
\vert \nabla \lvert \nabla v\rvert^q\rvert^2=\frac{q^2}{4}\lvert \nabla v \rvert^{2q-4}\vert \nabla \lvert \nabla v\rvert^2\rvert^2=q^2\lvert \nabla v \rvert^{2q-4}\lvert D^2v \nabla v \rvert^2\leq q^2|\nabla v|^{2q-2} |D^2v|^2.
\end{equation}
\end{proof}
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\end{lemma}
Let us now turn our attention when, as mentioned before, the Gagliardo--Nirenberg inequality is employed. In this case, we can derive information not only for
$\alpha \in \left(0,\frac{1}{n}\right]$ but also for $\alpha \in \left(\frac{1}{n},1\right]$.
\begin{lemma}\label{Met_GN}
If $m_1,m_2\in \mathbb R} \def\N{\mathbb N$ and $\alpha>0$ are taken accordingly to \eqref{Restrizionem1-m2-alphaPiccolo}, \eqref{Restrizionem1-m2-alphaGrande},
\eqref{Restrizionem1-m2-alphaGrandeBis}, then there exist $p, q>1$ such that $(u,v,w)$ satisfies a similar inequality as in \eqref{MainInequality}.
\begin{proof}
For $s$, $p$ and $q$ taken accordingly to Lemma \ref{LemmaCoefficientAiAndExponents} (in particular, $p=q$ for $\alpha \in \left(0,\frac{1}{n}\right]$, and $q=\frac{p}{2}$
for $\alpha \in \left(\frac{1}{n},1\right]$), let $\theta, \theta', \mu, \mu'$, $a_1,a_2, a_3, a_4$ and $\beta_1, \beta_2, \gamma_1, \gamma_2$ be therein defined.
With a view to Lemma \ref{Estim_general_For_u^p_nablav^2qLemma}, by manipulating relation \eqref{Estim_1_For_u^p} and focusing on the first inequality in \eqref{Young} and on \eqref{Estim_gradV}, proper $\epsilon_1, \tilde{\sigma}$ lead to
\begin{equation}\label{Somma}
\begin{split}
&\frac{d}{dt} \left(\int_\Omega (u+1)^p + \int_\Omega |\nabla v|^{2q}\right) + q \int_\Omega |\nabla v|^{2q-2} |D^2v|^2
\leq \left(-\frac{4p(p-1)}{(m_1+p-1)^2} + \tilde{\epsilon}_1 \right) \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\\
&+ c_2 \int_\Omega (u+1)^{p+2m_2-m_1-1} |\nabla v|^2 + c_{11} \int_\Omega u^{2\alpha} |\nabla v|^{2q-2} +c_{19} \quad \textrm{on } \; (0,T_{\rm{max}}),
\end{split}
\end{equation}
for some $c_{19}>0$ (we also used relation \eqref{GradU}).
In this way, we can estimate the second and third integral on the right-hand side of \eqref{Somma} by applying the H\"{o}lder inequality so to have
\begin{equation} \label{H1}
\int_{\Omega} (u+1)^{p+2m_2-m_1-1} |\nabla v|^2 \leq \left(\int_{\Omega} (u+1)^{(p+2m_2-m_1-1)\theta}\right)^{\frac{1}{\theta}}
\left(\int_{\Omega} |\nabla v|^{2 \theta'}\right)^{\frac{1}{\theta'}} \quad \textrm{ on } (0, T_{\rm{max}}),
\end{equation}
and
\begin{equation} \label{H2}
\int_{\Omega} (u+1)^{2\alpha} |\nabla v|^{2q-2} \leq
\left(\int_{\Omega} (u+1)^{2\alpha\mu}\right)^{\frac{1}{\mu}} \left(\int_{\Omega} |\nabla v|^{2(q-1)\mu'}\right)^{\frac{1}{\mu'}}\quad \textrm{ on } (0,T_{\rm{max}}).
\end{equation}
By invoking the Gagliardo--Nirenberg inequality and bound \eqref{massConservation}, we obtain for some $c_{20}, c_{21}>0$
\begin{align}\label{a1}
&\left(\int_{\Omega} (u+1)^{(p+2m_2-m_1-1)\theta}\right)^{\frac{1}{\theta}}= \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2(p+2m_2-m_1-1)}{m_1+p-1}\theta}(\Omega)}^{\frac{2(p+2m_2-m_1-1)}{m_1+p-1}}\\ \nonumber
& \leq c_{20} \|\nabla(u+1)^{\frac{m_1+p-1}{2}}\|_{L^2(\Omega)}^{\frac{2(p+2m_2-m_1-1)}{m_1+p-1} a_1} \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{2(p+2m_2-m_1-1)}{m_1+p-1} (1-a_1)} + c_{20} \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{2(p+2m_2-m_1-1)}{m_1+p-1}} \\ \nonumber
&\leq c_{21} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_1}+ c_{21} \quad \textrm{ for all } \,t\in(0,T_{\rm{max}}),
\end{align}
and for some $c_{22}, c_{23}>0$
\begin{align}\label{a3}
&\left(\int_{\Omega} (u+1)^{2\alpha\mu}\right)^{\frac{1}{\mu}}= \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{4\alpha\mu}{m_1+p-1}}(\Omega)}^{\frac{4\alpha}{m_1+p-1}}\\ \nonumber
& \leq c_{22} \|\nabla(u+1)^{\frac{m_1+p-1}{2}}\|_{L^2(\Omega)}^{\frac{4\alpha}{m_1+p-1} a_3} \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{4\alpha}{m_1+p-1} (1-a_3)} + c_{22} \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{4\alpha}{m_1+p-1}}\\ \nonumber
&\leq c_{23} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_2}+ c_{23}
\quad \textrm{for all }\, t \in (0,T_{\rm{max}}).
\end{align}
In a similar way, we can again apply the Gagliardo--Nirenberg inequality and bound \eqref{Cg} and get for some $c_{24}, c_{25}>0$
\begin{align}\label{a2}
&\left(\int_{\Omega} |\nabla v|^{2 \theta'}\right)^{\frac{1}{\theta'}} =\| |\nabla v|^q \|_{L^{\frac{2 \theta'}{q}}(\Omega)}^{\frac{2}{q}}\leq c_{24} \|\nabla |\nabla v|^q\|_{L^2(\Omega)}^{\frac{2}{q}a_2} \| |\nabla v|^q\|_{L^{\frac{s}{q}}(\Omega)}^{\frac{2}{q}(1-a_2)} + c_{24} \| |\nabla v|^q\|_{L^{\frac{s}{q}}(\Omega)}^{\frac{2}{q}} \\ \nonumber
& \leq c_{25} \left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_1}+ c_{25}
\quad \textrm{ for all } t\in (0, T_{\rm{max}}),
\end{align}
and for some $c_{26}, c_{27}>0$
\begin{align}\label{a4}
&\left(\int_{\Omega} |\nabla v|^{2(q-1) \mu'}\right)^{\frac{1}{\mu'}} =\| |\nabla v|^q \|_{L^{\frac{2(q-1)}{q}\mu'}(\Omega)}^{\frac{2(q-1)}{q}}
\leq c_{26} \|\nabla |\nabla v|^q\|_{L^2(\Omega)}^{\frac{2(q-1)}{q} a_4} \| |\nabla v|^q\|_{L^{\frac{s}{q}}(\Omega)}^{\frac{2(q-1)}{q} (1-a_4)}
+ c_{26} \| |\nabla v|^q\|_{L^{\frac{s}{q}}(\Omega)}^{\frac{2(q-1)}{q}} \\ \nonumber
&\leq c_{27} \left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_2}+ c_{27} \quad \textrm{ for every } t\in (0,T_{\rm{max}}).
\end{align}
By plugging \eqref{H1} and \eqref{H2} into \eqref{Somma} and taking into account \eqref{a1}, \eqref{a3}, \eqref{a2}, \eqref{a4}, we deduce for a proper
$\tilde{\epsilon}_1$, once inequality \eqref{GradV} is considered, the following estimate for some $c_{28}, c_{29}, c_{30}, c_{31}, c_{32}>0$:
\begin{equation}\label{Somma1}
\begin{split}
&\frac{d}{dt} \left(\int_\Omega (u+1)^p + \int_\Omega |\nabla v|^{2q}\right) + c_{28} \int_\Omega |\nabla |\nabla v|^q|^2
+ c_{29} \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2 - c_{30}\\
&\leq c_{31} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_1}
\left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_1} + c_{31} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_1}\\
& + c_{31} \left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_1}
+ c_{32} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_2} \left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_2}\\
&+ c_{32} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_2} + c_{32} \left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_2} \quad \text{on }
(0,T_{\rm{max}}).
\end{split}
\end{equation}
Since by Lemma \ref{LemmaCoefficientAiAndExponents} we have that $\beta_1 + \gamma_1 <1$ and $\beta_2 + \gamma_2 <1$, and in particular $\beta_1, \gamma_1, \beta_2, \gamma_2 \in (0,1)$, we can treat the two integral products and the remaining four addenda of the right-hand side in a such way that eventually they are absorbed by the two integral terms involving the gradients in the left one. More exactly, to the products we apply
\[
a^{d_1}b^{d_2} \leq \epsilon(a+b)+c \quad \textrm{with } a,b\geq0, d_1,d_2 >0 \; \textrm{such that } d_1+d_2<1, \; \textrm{for all } \epsilon>0 \; \textrm{and some } c>0
\]
(achievable by means of applications of Young's inequality), and to the other terms the Young inequality. In this way, the resulting linear combination of $\int_{\Omega} |\nabla |\nabla v|^q|^2$ and $\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2$ can be turned into $\frac{c_{28}}{2} \int_{\Omega} |\nabla |\nabla v|^q|^2 + \frac{c_{29}}{2} \int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2$, which coming back to \eqref{Somma1} infers what claimed.
\end{proof}
\end{lemma}
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\subsection{The logistic case}\label{Log}
For the logistic case we retrace part of the computations above connected to the usage of the Young inequality only.
\begin{lemma}\label{Estim_general_For_u^p_nablav^2qLemmaLog}
If $m_1,m_2,m_3 \in \mathbb R} \def\N{\mathbb N$ comply with $m_1>2m_2-(m_3+l)$ or $m_1>2m_2-\beta$ whenever $\alpha \in \left(0, \frac{1}{n}\right]$, or
$m_1>2m_2 +1-(m_3+l)$ or $m_1>2m_2+1-\beta$ whenever $\alpha \in \left(\frac{1}{n},1\right)$, then there exist $p,q>1$ such that $(u,v,w)$ satisfies
a similar inequality as in \eqref{MainInequality}.
\begin{proof}
As in Lemma \ref{Estim_general_For_u^p_nablav^2qLemma}, in view of inequality \eqref{Young} and the properties of the logistic $h$ in \eqref{h}, relation \eqref{Estim_1_For_u^p} now becomes for some positive $c_{33}$
\begin{equation}\label{Estim_1_For_u^p1}
\begin{split}
\frac{d}{dt} \int_\Omega (u+1)^p & \leq (-p(p-1)+\epsilon_3) \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 +c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}}\\
&+ \left(\epsilon_1 + \tilde{\sigma} - \frac{p(p-1)\xi \gamma}{2^{p-1}(m_3+p-1)} \right) \int_\Omega (u+1)^{m_3+p+l-1}
+ \epsilon_2 \int_\Omega |\nabla (u+1)^\frac{m_1+p-1}{2}|^2\\
&+ pk_+ \int_\Omega (u+1)^p - p \mu \int_\Omega (u+1)^{p-1}u^{\beta} + c_{33} \quad \text{ for all } t \in (0,T_{\rm{max}}).
\end{split}
\end{equation}
Applying the inequality $(A+B)^p \leq 2^{p-1} (A^p+B^p)$ with $A,B \geq 0$ and $p>1$ to the last integral in \eqref{Estim_1_For_u^p1}, implies that
$-u^{\beta} \leq -\frac{1}{2^{\beta-1}} (u+1)^{\beta}+1$; therefore
\begin{equation}\label{beta}
- p \mu \int_\Omega (u+1)^{p-1} u^{\beta} \leq -\frac{p \mu}{2^{\beta-1}} \int_\Omega (u+1)^{p-1+\beta} + p \mu \int_\Omega (u+1)^{p-1} \quad \text{on } (0,T_{\rm{max}}).
\end{equation}
Henceforth, by taking into account the Young inequality, we have that for $t \in (0,T_{\rm{max}})$
\begin{equation} \label{k}
pk_+ \int_\Omega (u+1)^p \leq \delta_1 \int_\Omega (u+1)^{p-1+\beta} + c_{34} \quad \text{and} \quad
p \mu \int_\Omega (u+1)^{p-1} \leq \delta_2 \int_\Omega (u+1)^{p-1+\beta} + c_{35},
\end{equation}
with $\delta_1, \delta_2 >0$ and some $c_{34}, c_{35} >0$.
\quad \textbf{Case $1$}: $\alpha \in \left(0, \frac{1}{n}\right]$ and $m_1>2m_2-(m_3+l)$ or $m_1>2m_2-\beta$.
For $m_1> 2m_2-(m_3+l)$ we refer to Lemma \ref{Estim_general_For_u^p_nablav^2qLemma} and we take in mind inequality \eqref{Young1}.
Conversely, when $m_1>2 m_2-\beta$, we have that (recall $s$ may be arbitrary large) $\frac{(p+2m_2-m_1-1)s}{s-2} < p-1+\beta$, and by means of the Young inequality estimate \eqref{Young1} can alternatively read
\begin{equation}\label{young4}
c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}} \leq \delta_3 \int_\Omega (u+1)^{p-1+\beta} + c_{36} \quad \text{on }\, (0,T_{\rm{max}}),
\end{equation}
with $\delta_3>0$ and positive $c_{36}$.
By inserting estimates \eqref{beta} and \eqref{k} into relation \eqref{Estim_1_For_u^p1}, as well as taking into account \eqref{Young1}
(or, alternatively to \eqref{Young1}, bound \eqref{young4}), for suitable $\hat{\epsilon}, \tilde{\epsilon}_2, \tilde{\delta} >0$ and some $c_{37}>0$ we arrive at
\begin{equation*}\label{ClaimUlog}
\begin{split}
\frac{d}{dt} \int_\Omega (u+1)^p &\leq \left(-\frac{4p(p-1)}{(m_1+p-1)^2} + \hat{\epsilon}\right) \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2
+ \left(\tilde{\epsilon_2} - \frac{p(p-1)\xi \gamma}{2^{p-1}(m_3+p-1)} \right) \int_\Omega (u+1)^{m_3+p+l-1}\\
& + \left(\tilde{\delta} - \frac{p \mu}{2^{\beta-1}} \right) \int_\Omega (u+1)^{p-1+\beta} + c_{37} \quad \text{ for all } t \in (0,T_{\rm{max}}),
\end{split}
\end{equation*}
where we used again relation \eqref{GradU}. We conclude reasoning exactly as in the second part of the proof of Lemma \ref{Estim_general_For_u^p_nablav^2qLemma} and by choosing suitable
$\hat{\epsilon}, \tilde{\epsilon}_2, \tilde{\delta}, \epsilon_8$.
\textbf{Case $2$}: $\alpha \in \left(\frac{1}{n},1\right)$ and $m_1>2m_2 +1-(m_3+l)$ or $m_1>2m_2+1-\beta$.
Accordingly to Remark \ref{RemarkOnS}, since now $s$ cannot increase arbitrarily, relations \eqref{Young1} and \eqref{young4} have to be differently discussed.
In particular, for some $\bar{c}_1>0$ we can estimate relation \eqref{Young} as follows:
\begin{equation*}
\begin{split}
&p(p-1)\chi \int_\Omega u (u+1)^{m_2+p-3} \nabla u \cdot \nabla v \leq \epsilon_3 \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2
+ c_2 \int_\Omega (u+1)^{p+2m_2-m_1-1} |\nabla v|^2\\
& \leq \epsilon_3 \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 + \epsilon_4 \int_\Omega |\nabla v|^{2(p+1)} + \bar{c}_1 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)(p+1)}{p}}
\quad \text{on }\, (0,T_{\rm{max}}).
\end{split}
\end{equation*}
Now, if $m_1>2m_2+1-(m_3+l)$, then some $p$ sufficiently large infers to $\frac{(p+2m_2-m_1-1)(p+1)}{p}< p+m_3+l-1$, so that for any positive $\bar{\epsilon}_1$
and some $\bar{c}_2>0$ we have
\[
\bar{c}_1 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)(p+1)}{p}} \leq \bar{\epsilon}_1 \int_\Omega (u+1)^{p+m_3+l-1} + \bar{c}_2 \quad \text{on } (0,T_{\rm{max}}).
\]
Conversely, and in a similar way, for $m_1>2m_2+1-\beta$ we have for any positive $\bar{\epsilon}_2$ and some $\bar{c}_3>0$
\[
\bar{c}_1 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)(p+1)}{p}} \leq \bar{\epsilon}_2 \int_\Omega (u+1)^{p-1+\beta} + \bar{c}_3 \quad \text{on } (0,T_{\rm{max}}).
\]
The remaining part of the proof follows as the previous case, by taking into account \cite[Lemma 5.3-Lemma 5.4]{FrassuCorViglialoro} for the term dealing with
$ \int_\Omega |\nabla v|^{2(p+1)}$.
\end{proof}
\end{lemma}
As a by-product of what now obtained we are in a position to conclude.
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\subsection{Proof of Theorems \ref{MainTheorem} and \ref{MainTheorem1}}
\begin{proof}
Let $(u_0,v_0) \in (W^{1,\infty}(\Omega))^2$ with $u_0, v_0 \geq 0$ on $\bar{\Omega}$. For $f$ and $g$ as in \eqref{f} and, respectively, for $f$, $g$ as in \eqref{f} and
$h$ as in \eqref{h}, let $\alpha >0$ and let $m_1,m_2,m_3 \in \mathbb R} \def\N{\mathbb N$ comply with \ref{A1}, \ref{A2} and \ref{A3}, respectively, \ref{A4} and \ref{A5}. Then, we refer to Lemmas \ref{Estim_general_For_u^p_nablav^2qLemma} and \ref{Met_GN}, respectively, Lemma \ref{Estim_general_For_u^p_nablav^2qLemmaLog} and obtain for some
$C_1,C_2,C_3>0$
\begin{equation}\label{Estim_general_For_y_2}
y'(t) + C_1 \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2 + C_2 \int_\Omega |\nabla |\nabla v|^q|^2 \leq C_3
\quad \text{ on } (0, T_{\rm{max}}).
\end{equation}
Successively, the Gagliardo--Nirenberg inequality again makes that some positive constants $c_{38}, c_{39}$ imply from the one hand
\begin{equation*}\label{G_N2}
\int_\Omega (u+1)^p \leq c_{38} \Big(\int_\Omega |\nabla (u+1)^\frac{m_1+p-1}{2}|^2\Big)^{\kappa_1}+ c_{38} \quad \text{for all } t \in (0,T_{\rm{max}}),
\end{equation*}
(as already done in \eqref{GN2}), and from the other
\begin{equation}\label{Estim_Nabla nabla v^q}
\int_\Omega \lvert \nabla v\rvert^{2q}=\lvert \lvert \lvert \nabla v\rvert^q\lvert \lvert_{L^2(\Omega)}^2
\leq c_{39} \lvert \lvert\nabla \lvert \nabla v \rvert^q\rvert \lvert_{L^2(\Omega)}^{2\kappa_2} \lvert \lvert\lvert \nabla v \rvert^q\lvert \lvert_{L^\frac{1}{q}(\Omega)}^{2(1-\kappa_2)} +c_{39} \lvert \lvert \lvert \nabla v \rvert^q\lvert \lvert^2_{L^\frac{1}{q}(\Omega)}\quad \textrm{on } (0,T_{\rm{max}}),
\end{equation}
with $\kappa_2$ already defined in Lemma \ref{LemmaCoefficientAiAndExponents}.
Subsequently, the $L^s$-bound of $\nabla v$ in \eqref{Cg} infers some $c_{40}>0$ such that
\begin{equation*}\label{Estim_Nabla nabla v^p^2}
\int_\Omega \lvert \nabla v\rvert^{2q}\leq c_{40} \Big(\int_\Omega \lvert \nabla \lvert \nabla v \rvert^q\rvert^2\Big)^{\kappa_2}+c_{40} \quad \text{for all } t \in (0,T_{\rm{max}}).
\end{equation*}
In the same flavour of \cite[Lemma 5.4]{FrassuCorViglialoro}, by using the estimates involving $\int_\Omega (u+1)^p$ and $\int_\Omega \rvert\nabla v\lvert^{2q}$,
relation \eqref{Estim_general_For_y_2} provides positive constants $c_{41}$ and $c_{42}$, and $\kappa=\min\{\frac{1}{\kappa_1},\frac{1}{\kappa_2}\}$ such that
\begin{equation*}\label{MainInitialProblemWithM}
\begin{cases}
y'(t)\leq c_{41}-c_{42} y^{\kappa}(t)\quad \textrm{for all } t \in (0,T_{\rm{max}}),\\
y(0)=\int_\Omega (u_0+1)^p+ \int_\Omega |\nabla v_0|^{2q}.
\end{cases}
\end{equation*}
Finally, ODE comparison principles imply $u \in L^\infty((0,T_{\rm{max}});L^p(\Omega))$, and the conclusion is a consequence of the boundedness criterion in
$\S$\ref{SectionLocalInTime}.
\end{proof}
\begin{remark}[On the validity of the theorems in \cite{FrassuCorViglialoro} and \cite{frassuviglialoro} for $\alpha \geq \frac{1}{2}+\frac{1}{n}$]\label{Alpha}
In the proofs of \cite[Theorem 2.1]{FrassuCorViglialoro} and \cite[Theorems 2.1 and 2.2]{frassuviglialoro}, it is seen that the $L^2$ uniform estimate of $\nabla v$ is used to control some integral on $\partial \Omega$ (and this allows us to avoid to restrict our study to convex domains), as well as to deal with the term
$\int_\Omega \rvert \nabla v \lvert^{2p}$ with the Gagliardo--Nirenberg inequality; for instance we are referring to \cite[(28) and (39)]{frassuviglialoro}, respectively.
Such finiteness of $\int_\Omega \rvert \nabla v \lvert^2$ is related to the values of $\alpha$ in these articles:
$\alpha \in \left(0, \frac{1}{2}+\frac{1}{n}\right)$ (see \cite[Lemma 4.1]{frassuviglialoro}). Apparently only $\nabla v \in L^\infty((0,T_{\rm{max}});L^1(\Omega))$ suffices to address these issues. Indeed, as far as the topological property of $\Omega$ is concerned, we can invoke \cite[(3.10) of Proposition 8]{YokotaEtAlNonCONVEX} with $s=1$; on the other hand, for the question tied to the employment of the Gagliardo--Nirenberg inequality, we may operate as done in \eqref{Estim_Nabla nabla v^q}. As a consequence, in view of Lemma \ref{LocalV}, we have that $\nabla v \in L^\infty((0,T_{\rm{max}});L^1(\Omega))$, so that \cite[Theorem 2.1]{FrassuCorViglialoro} and \cite[Theorems 2.1 and 2.2]{frassuviglialoro} hold true for any $\alpha \in (0,1)$.
\end{remark}
\subsubsection*{Acknowledgments}
SF and GV are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and are partially supported by the research project \emph{Evolutive and Stationary Partial Differential Equations with a Focus on Biomathematics}, funded by Fondazione di Sardegna (2019). GV is partially supported by MIUR (Italian Ministry of Education, University and Research) Prin 2017 \emph{Nonlinear Differential Problems via Variational, Topological and Set-valued Methods} (Grant Number: 2017AYM8XW). TL is partially supported by NNSF of P. R. China (Grant No. 61503171), CPSF (Grant No. 2015M582091), and NSF of Shandong Province (Grant No. ZR2016JL021).
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\begin{document}
\mainmatter
\title{Efficiency in Multi-objective Games}
\titlerunning{Efficiency in Multi-objective Games}
\author{Anisse Ismaili$^{1,2}$}
\authorrunning{Anisse Ismaili}
\institute{$^1$ Universit\'{e} Pierre et Marie Curie,
Univ Paris 06, UMR 7606, LIP6, F-75005, Paris, France\\
$^2$ Paris Dauphine University, Place du Mal de Lattre de Tassigny,
75775 Paris Cedex 16, France\\
\email{[email protected]}}
\maketitle
\vspace*{-0.5cm}
\begin{abstract}
In a multi-objective game, each agent individually evaluates each overall action-profile on multiple objectives. I generalize the price of anarchy to multi-objective games and provide a polynomial-time algorithm to assess it$^1$.\\
This work asserts that policies on tobacco promote a higher economic efficiency.
\end{abstract}
\vspace*{-0.8cm}
\section{Introduction}
Economic agents, for each individual decision, make a trade off between multiple objectives, like for instance: time, resources, goods, financial income, sustainability, happiness and life. This motivated the introduction of a super-class of games: multi-objective (MO) games \cite{blackwell1956analog,shapley1959equilibrium}. Each agent evaluates each overall action profile by a \emph{vector}. His individual preference is a \emph{partial} rationality modelled by the Pareto-dominance. It induces Pareto-Nash-equilibria (PN) as the overall selfish outcomes.
Furthermore, concerning economic models, such vectorial evaluations are a humble backtrack from the intrinsic and subjective theories of value, towards a non-theory of value where the evaluations are maintained vectorial, in order to enable partial rationalities and to avoid losses of information in the model.
In this more realistic (behaviourally less assumptive) framework, in order to avoid critical losses of information on the several objectives in the model, thoroughly computing efficiency is a tremendous necessity \cite{madeley1999big,sloan2004price}.
The literature on MO games is disparate and will be presented where relevant.
After the preliminaries below, Section \ref{sec:mopoa} generalizes the \textit{coordination ratio} (CR, better known as ``price of anarchy'') to MO games. Section \ref{sec:application} applies it to the efficiency of tobacco economy. Section \ref{sec:computation} provides algorithms\footnote{Appendix \ref{app:smooth} shows that ``smoothness'' analysis \cite{roughgarden2009intrinsic} cannot be applied to MO games.} to assess the MO-CR.
Let $N=\{1,\ldots,n\}$ denote the \textit{set of agents}.
Let $A^i$ denote each agent $i$'s \textit{action-set} (discrete, finite).
Each agent $i$ decides an \textit{action} $a^i\in A^i$.
Given a subset of agents $M\subseteq N$, let $A^M$ denote $\times_{i\in M} A^i$ and let $A=A^N$ denote the \textit{set of overall action-profiles}.
Let ${\mathcal{O}}=\{1,,\ldots,d\}$ denote the \textit{set of all the objectives}, with $d$ fixed.
Let $v^i: A\rightarrow \mathbb{R}p$ denote an agent $i$'s \textit{individual MO evaluation function}, which maps each overall action-profile $a=(a^1,\ldots,a^n)\in A$ to an MO evaluation $v^i(a)\in\mathbb{R}p$. Hence, agent $i$'s evaluation for objective $k$ is $v^i_k(a)\in\mathbb{R}_+$. Given an overall action-profile $a\in A$, $a^M$ is the restriction of $a$ to $A^M$, and $a^{-i}$ to $A^{N\setminus\{i\}}$.
\begin{definition}\quad A Multi-objective Game (MOG) is a tuple $\left(N, \{A^i\}_{i\in N}, {\mathcal{O}}, \{v^i\}_{i\in N}\right)$.
\end{definition}
For instance, MO games encompass single-objective (discrete) optimization problems, MO optimization problems and non-cooperative games.
Assuming $\alpha=|A^i|\in\mathbb{N}$ for each agent, the representation of an MOG requires $n\alpha^n$ $d$-dimensional vectors.
Let us now supply the vectors with a preference relation. Assuming a \textit{maximization} setting, given $x,y\in\mathbb{R}p$, the following relations state respectively that $y$ (\ref{eq:wpp}) weakly-Pareto-dominates and (\ref{eq:pp}) Pareto-dominates $x$:
\begin{eqnarray}
y\succsim x &\hspace{0.5cm}\Leftrightarrow\hspace{0.5cm}& \forall k\in{\mathcal{O}},~~ y_k\geq x_k
\label{eq:wpp}\\
y\succ x &\Leftrightarrow & \forall k\in{\mathcal{O}},~~ y_k\geq x_k\text{~~and~~}\exists k\in{\mathcal{O}},~~ y_k> x_k
\label{eq:pp}
\end{eqnarray}
The Pareto-dominance is a \emph{partial} order, inducing a multiplicity of Pareto-efficient outcomes. Formally, the set of efficient vectors is defined as follows:
\begin{definition}[Pareto-efficiency]
For $Y\subseteq\mathbb{R}p$, the efficient vectors $\text{EFF}[Y]\subseteq Y$ are:
$$\text{EFF}[Y]=\{y^\ast\in Y~~|~~ \forall y\in Y, \mbox{~not~} (y\succ y^\ast)\}$$
\end{definition}
(Similarly, let $\text{WST}[Y]=\{y^{-}\in Y\mbox{~s.t.~} \forall y\in Y, \mbox{~not~} (y^{-}\succ y)\}$ denote the subset of worst vectors.)
Pareto-efficiency enables to define as efficient all the trade-offs that cannot be improved on one objective without being downgraded on another one, that is: the best compromises between objectives (see e.g. Figure \ref{fig:eff}).
At the individual scale, Pareto-efficiency defines a \textit{partial rationality}, enabling to model behaviours that single-objective (SO) games would not model consistently.
\begin{definition}[Pareto-Nash equilibrium \cite{shapley1959equilibrium}]\label{def:PE}
In an MOG, an action-profile $a\in A$ is a Pareto-Nash equilibrium (denoted by $a\in\text{PN}$), if and only if, for each agent $i\in N$:
$$v^i(a^i,a^{-i})\quad\in\quad\text{EFF}\left[\quad v^i(A^i,a^{-i})\quad\right]$$
where $v^i(A^i,a^{-i})$ denotes $\{v^i(b^i,a^{-i})\mid b^i\in A^i\}$.
\end{definition}
Pareto-Nash equilibria encompass most behaviourally possible action-profiles. For instance, whatever an agent's subjective linear positive weighted combination of the objectives, his decision is Pareto-efficient. One can distinguish behavioural objectives inducing $\text{PN}$ and also objectives on which to focus an efficiency study.
{\it Equilibrium existence.}
In many sound probabilistic settings \cite{daskalakis2011connectivity,dresher1970probability,rinott2000number}, Pareto efficiency is not demanding on the conditions of individual rationality, hence there are multiple Pareto-efficient responses. Consequently, pure PN are numerous in average: $|\text{PN}|\in\Theta(\alpha^{\frac{d-1}{d}n})$, justifying their existence in a probabilistic manner. Furthermore, in MO games with MO potentials \cite{monderer1996potential,patrone2007multicriteria,rosenthal1973class}, the existence is guaranteed.
\begin{example}[A didactic toy-example in Ocean Shores]
Five shops in Ocean Shores (the nodes) can decide upon two activities: renting bikes or buggies, selling clams or fruits, etc. Each agent evaluates his local action-profile depending on the actions of his inner-neighbours and according to two objectives: financial revenue and sustainability.\\
\hspace*{-31cm}\includegraphics[scale=0.8]{Ocean4.pdf}\\
For instance, we have $(b^1,b^2,a^3,b^4,b^5)\in\text{PN}$, since each of these individual actions, given the adversary local action profile (column), is Pareto-efficient among the two actions of the agent (row). Even if the relative values of the objectives cannot be certainly ascertained, all the subjectively efficient vectors are encompassed by the individual Pareto-efficiency. In this MO game, there are $13$ Pareto-Nash-equilibria, which utilitarian evaluations are depicted in Figure \ref{fig:eff} (Section \ref{sec:mopoa}).
\label{ex:Ocean}
\end{example}
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\section{The Multi-objective Coordination Ratio}
\label{sec:mopoa}
It is well known in game theory that an equilibrium can be overall inefficient with regard to the sum of the individual evaluations. This loss of efficiency is measured by the {\it coordination ratio}\footnote{As Smoothness \cite{roughgarden2009intrinsic} cannot be applied to MO games, I cannot use the term \textit{Price of Anarchy}.} (CR) \cite{aland2006exact,awerbuch2005price,christodoulou2005price,guo2005price,koutsoupias1999worst,roughgarden2009intrinsic,roughgarden2007introduction} $\min[u(PE)]/\max[u(A)]$. Regrettably, when focusing on one sole objective (e.g. making money or a higher GDP), there are losses of efficiency that are not measured (e.g. non-sustainability of productions or production of addictive carcinogens). This appeals for a more thorough analysis of the loss of efficiency at equilibrium and the definition of a {\it multi-objective} coordination ratio.
The utilitarian social welfare $u:A\rightarrow\mathbb{R}p$ is a vector-valued function measuring social welfare with respect to the $d$ objectives: $u(a)=\sum_{i\in N} v^i(a)$, excluding the purely behavioural objectives that cause irrationality \cite{sloan2004price}.
Given a function $f:A\rightarrow Z$, the \textit{image set} $f(E)$ of a subset $E\subseteq A$ is defined by $f(E)=\{f(a)|a\in E\}\subseteq Z$. Given $\rho,y,z\in\mathbb{R}p$, the vector $\rho\star y\in\mathbb{R}p$ is defined by $\forall k\in{\mathcal{O}}, (\rho\star y)_k=\rho_k y_k$ and the vector $y / z\in\mathbb{R}p$ is defined by $\forall k\in{\mathcal{O}}, (y / z)_k=y_k / z_k$.
For $x\in\mathbb{R}p$, $x\star Y$ denotes $\{x\star y\in\mathbb{R}p ~~|~~ y\in Y\}$.
Given $x\in\mathbb{R}p$, $\mathcal{C}(x)$ denotes $\{y\in\mathbb{R}p~|~x\succsim y\}$.
I also introduce\footnote{To enable ratios, one can do the minor assumption $\mathcal{F}\subseteq\mathbb{R}pp$.} the notations $\mathcal{E}$ and $\mathcal{F}$, illustrated in Figures \ref{fig:eff} and \ref{fig:mopoa}:
\noindent
\begin{minipage}{0.58\columnwidth}
\noindent
\begin{itemize}
\item \textcolor{darkblue}{$\mathcal{A}=u(A)$} the set of \textit{outcomes}. $\textcolor{darkblue}{(\bullet)}$
\item \textcolor{darkgreen}{$\mathcal{E}=u(\text{PN})$} the \textit{equilibria outcomes}. $\textcolor{darkgreen}{(\blacklozenge)}$
\item \textcolor{darkred}{$\mathcal{F}=\text{EFF}[u(A)]$} the \textit{efficient outcomes}. $\textcolor{darkred}{(\times)}$
\end{itemize}
For SO games, the worst-case efficiency of equilibria is measured by the CR $\min[u(PE)]/\max[u(A)]$. However, for MO games, there are many equilibria and optima, and a ratio of the (green) \textit{set} \textcolor{darkgreen}{$\mathcal{E}$} over the (red) \textit{set} \textcolor{darkred}{$\mathcal{F}$} is not defined yet and ought to maintain the information on each objective without introducing dictatorial choices.
\end{minipage}
~~~
\begin{minipage}{0.38\columnwidth}
\centering
\includegraphics[scale=0.36]{utilitarian_ocean4.png}
\captionof{figure}{The bi-objective utilitarian vectors of Ocean Shores}
\label{fig:eff}
\end{minipage}\\[0.5ex]
I introduce a multi-objective CR.
Firstly, the efficiency of one equilibrium $y\in\mathcal{E}$ is quantified without taking side with any efficient outcome, by defining with flexibility and no dictatorship, a \textit{disjunctive set} of guaranteed ratios of efficiency $R[y,\mathcal{F}]=\bigcup_{z\in\mathcal{F}}\mathcal{C}(y/z)$.
Secondly, in MOGs, in average, there are many Pareto-Nash-equilibria. An efficiency \textit{guarantee} $\rho\in\mathbb{R}p$, must hold \textit{for each} equilibrium-outcome, inducing the conjunctive definition of the set of guaranteed ratios $R[\mathcal{E},\mathcal{F}]=\bigcap_{y\in\mathcal{E}} R[y,\mathcal{F}]$.
Technically, $R[\mathcal{E},\mathcal{F}]$ only depends on $\text{WST}[\mathcal{E}]$ and $\mathcal{F}$.
Finally, if two bounds on the efficiency $\rho$ and $\rho'$ are such that $\rho\succ\rho'$, then $\rho'$ brings no more information, hence, MO-CR is defined by using $\text{EFF}$ on the guaranteed efficiency ratios $R[\text{WST}[\mathcal{E}],\mathcal{F}]$.
This MO-CR satisfies a set of key properties detailed in Appendix \ref{app:axioms}.
\begin{definition}[MO Coordination Ratio]\label{def:mopoa}
Given an MOG, a vector $\rho\in\mathbb{R}p$ bounds the MOG's inefficiency if and only if it holds that:\quad
$\forall y\in\mathcal{E},\quad \exists z\in\mathcal{F},\quad y/z\succsim \rho$.
Consequently, the set of guaranteed ratios is defined by:
$$R[\mathcal{E},\mathcal{F}]\quad=\quad\bigcap_{y\in\mathcal{E}}\bigcup_{z\in\mathcal{F}}\mathcal{C}(y/z)$$
and the MO-CR is defined by:\quad
$
\text{MO-CR}[\mathcal{E},\mathcal{F}]=\text{EFF}[R[\text{WST}[\mathcal{E}],\mathcal{F}]]
$
\end{definition}
\begin{example}[The Efficiency ratios of Example \ref{ex:Ocean}]\label{ex:mopoashores}
I depict the efficiency ratios of Ocean Shores (intersected with $[0,1]^d$) which depend of $\text{WST}[\mathcal{E}]=\{(30,53), (40,38)\}$ and $\mathcal{F}=\{(46,61),\ldots, (69,31)\}$.
The part below the red line corresponds to $R[(30,53),\mathcal{F}]$, the part below the blue line to $R[(40,38),\mathcal{F}]$ and the yellow part below both lines is the conjunction on both equilibria $R[\text{WST}[\mathcal{E}],\mathcal{F}]$.
The freedom degree of deciding what
\vspace*{0.1cm}
\noindent
\begin{minipage}{0.55\columnwidth}
the overall efficiency should be is left free (no dictatorship) which results in several ratios in the MO-CR.
Firstly, for each $\rho\in R[\mathcal{E},\mathcal{F}]$, we have $\rho_1\leq 65\%$. Hence, whatever the choices of overall efficiency, one cannot guarantee more than \textit{65\% of efficiency on objective 1}.
Secondly, there are some subjectivities for which the efficiency on objective 2 is already total (100\%, if not more) while situation on objective 1 is worse and only 50\% can be obtained.
Thirdly, from 50\% to 65\% of subjective efficiency on objective 1, the various subjectivities range the efficiency on objective 2 from 100\% to 75\%.
\end{minipage}
~~
\begin{minipage}{0.43\columnwidth}
\includegraphics[scale=0.44]{yellow.pdf}
\captionof{figure}{The MO-CR of Ocean Shores}
\end{minipage}
\end{example}
\vspace*{1cm}
\noindent
\begin{minipage}{0.46\columnwidth}
\centering
\includegraphics[scale=0.8]{MOCR.pdf}
\end{minipage}
~~~
\begin{minipage}{0.5\columnwidth}
Having $\rho$ in $\text{MO-CR}$ means that for each $y\in\mathcal{E}$, there is an efficient outcome $z^{(y)}\in\mathcal{F}$ such that $y$ dominates $\rho\star z^{(y)}$. In other words, if $\rho\in R[\mathcal{E},\mathcal{F}]$, then each equilibrium satisfies the ratio of efficiency $\rho$. This means that equilibria-outcomes are at least as good as $\rho\star\mathcal{F}$. That is: $\mathcal{E}\subseteq(\rho\star\mathcal{F})+\mathbb{R}p$. Moreover, since $\rho$ is tight, $\mathcal{E}$ sticks to $\rho\star\mathcal{F}$.
\end{minipage}
\captionof{figure}{$\rho\in\text{MO-CR}$ bounds below $\mathcal{E}$'s inefficiency:\quad $\mathcal{E}~~\subseteq~~(\rho\star\mathcal{F})+\mathbb{R}p$}
\label{fig:mopoa}~\\
\section{Application to Tobacco Economy}\label{sec:application}
Tobacco consumption is a striking example of economic inefficiency induced by bounded rationalities.
According to the World Health Organisation \cite{world2011report}, 17.000 humans die each day of smoking related diseases (one person per 5 seconds). Meanwhile, addictive satisfaction and the financial revenue of the tobacco industry fosters consumption and production.
According to the subjective theory of value \cite{walras1896elements}, some economists would say: ``Since consumers value the product, then the industry creates value.'' According to other health economists \cite{sloan2004price}, most consumers become addict before age 18, and as adults, would prefer a healthier life, but fail to opt-out.
\hspace*{0.5cm}The theory of MO games, based on a non-theory of value, just maintains vectorial evaluations and properly considers dollars, addiction and life expectancy as distinct objectives, with PN equilibria encompassing the relevant behaviours, even irrational.
We \textbf{modelled} the tobacco industry and its consumers \cite{globalissuestobacco,madeley1999big} by a succinct MOG, with the help of (..) the association\footnote{I am grateful to Cl\'{e}mence Cagnat-Lardeau for her help on modelling tobacco economy.} ``\textit{Alliance contre le tabac}''.
The set of agents is $N=\{\text{industry}, \nu \text{ consumers}\}$, where there are about $\nu=6.10^9$ prospective consumers.
Each consumer decides in $A^{\text{consumer}}=\{\text{not-smoking},\text{smoking}\}$ and cares about money, his addictive pleasure, and living. The industry only cares about money and decides in $A^{\text{industry}}=\{\text{not-active},\text{active},\text{advertise\&active}\}$. We have ${\mathcal{O}}=\{\text{money},\text{reward},\text{life-expectancy}\}$.
The tables below depict the evaluation vectors (over a life-time and ordered as in ${\mathcal{O}}$) of one prospective consumer and the evaluations of the industry with respect to the number $\theta\in\{0,\ldots,\nu\}$ of consumers who decide to smoke. The money budget (already an aggregation) is expressed in kilo-dollars\footnote{Note that most states set the prices of tobacco, hence prices do not follow supply/demand.}\footnote{These numbers differ from \cite{sloan2004price} which aggregates everything (e.g. life expectancy) into money.}; the addictive reward is on an ordinal scale $\{1,2,3,4\}$; life-expectancy is in years.
\begin{center}
\begin{tabular}{r|ccc}
$v^{\text{consumer}}$ & not-active & active & advertise\&active\\
\hline
not-smoking &
$(48,1,75)$ & $(48,1,75)$ & $(48,1,75)$\\
smoking &
$(48,1,75)$ & $(12,3,65)$ & $(0,4,55)$\\
\multicolumn{4}{c}{}\\
$v^{\text{industry}}(\theta)$ & not-active & active & advertise\&active \\
\hline
$(\nu-\theta)\times$ &
$(0,-,-)$ & $(0,-,-)$ & $(0,-,-)$ \\
$+$ \hspace*{0.8cm}$\theta~~\times$ &
$(0,-,-)$ & $(26,-,-)$ & $(36,-,-)$ \\
\end{tabular}
\end{center}
\textbf{Pareto-Nash equilibria.} If the industry is active, then for the consumer, deciding to smoke or not depends on how the consumer subjectively values/weighs money, addiction and life expectancy: both decisions are encompassed by Pareto-efficiency. For the industry, advertise\&active is a dominant strategy. Consequently, Pareto-Nash-equilibria are all the action-profiles in which the industry decides advertise\&active.
\textbf{Efficiency.}
Since addiction is irrational (detailed in Appendix \ref{app:tabac}), I focus on money and life-expectancy.
We have
$
\mathcal{E}
=
\{\theta(36,55)+(\nu-\theta)(48,75) \mid 0\leq\theta\leq \nu\}
$
and
$\mathcal{F}=\{\nu(48,75)\}$,
where $\nu$ is the world's population, and $\theta$ the number of smokers. Since $\text{WST}[\mathcal{E}]=\{(36,55)\}$, the MO-CR is the singleton $\{(75\%,73\%)\}$: in the worst case, we lose 12k\$ and 20 years of life-expectancy per-consumer. These Pareto-Nash-equilibria are the worst action-profiles for money and life-expectancy, a critical information that was not lost by this MOG and its MO-CR.
\textbf{Practical lessons.} Advertising tobacco fosters consumption. The association ``\textit{Alliance contre le tabac}'' passed a law for standardized neutral packets (April 3rd 2015), in order to annihilate all the benefits of branding, but only in France. The model indicates that:
\begin{center}
\emph{This law will promote a higher economic efficiency}.
\end{center}
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\section{Application to Tobacco Economy}\label{sec:application}
Tobacco consumption is a striking example of economic inefficiency induced by bounded rationalities.
According to the World Health Organisation \cite{world2011report}, 17.000 humans die each day of smoking related diseases (one person per 5 seconds). Meanwhile, addictive satisfaction and the financial revenue of the tobacco industry fosters consumption and production.
According to the subjective theory of value \cite{walras1896elements}, some economists would say: ``Since consumers value the product, then the industry creates value.'' According to other health economists \cite{sloan2004price}, most consumers become addict before age 18, and as adults, would prefer a healthier life, but fail to opt-out.
\hspace*{0.5cm}The theory of MO games, based on a non-theory of value, just maintains vectorial evaluations and properly considers dollars, addiction and life expectancy as distinct objectives, with PN equilibria encompassing the relevant behaviours, even irrational.
We \textbf{modelled} the tobacco industry and its consumers \cite{globalissuestobacco,madeley1999big} by a succinct MOG, with the help of (..) the association\footnote{I am grateful to Cl\'{e}mence Cagnat-Lardeau for her help on modelling tobacco economy.} ``\textit{Alliance contre le tabac}''.
The set of agents is $N=\{\text{industry}, \nu \text{ consumers}\}$, where there are about $\nu=6.10^9$ prospective consumers.
Each consumer decides in $A^{\text{consumer}}=\{\text{not-smoking},\text{smoking}\}$ and cares about money, his addictive pleasure, and living. The industry only cares about money and decides in $A^{\text{industry}}=\{\text{not-active},\text{active},\text{advertise\&active}\}$. We have ${\mathcal{O}}=\{\text{money},\text{reward},\text{life-expectancy}\}$.
The tables below depict the evaluation vectors (over a life-time and ordered as in ${\mathcal{O}}$) of one prospective consumer and the evaluations of the industry with respect to the number $\theta\in\{0,\ldots,\nu\}$ of consumers who decide to smoke. The money budget (already an aggregation) is expressed in kilo-dollars\footnote{Note that most states set the prices of tobacco, hence prices do not follow supply/demand.}\footnote{These numbers differ from \cite{sloan2004price} which aggregates everything (e.g. life expectancy) into money.}; the addictive reward is on an ordinal scale $\{1,2,3,4\}$; life-expectancy is in years.
\begin{center}
\begin{tabular}{r|ccc}
$v^{\text{consumer}}$ & not-active & active & advertise\&active\\
\hline
not-smoking &
$(48,1,75)$ & $(48,1,75)$ & $(48,1,75)$\\
smoking &
$(48,1,75)$ & $(12,3,65)$ & $(0,4,55)$\\
\multicolumn{4}{c}{}\\
$v^{\text{industry}}(\theta)$ & not-active & active & advertise\&active \\
\hline
$(\nu-\theta)\times$ &
$(0,-,-)$ & $(0,-,-)$ & $(0,-,-)$ \\
$+$ \hspace*{0.8cm}$\theta~~\times$ &
$(0,-,-)$ & $(26,-,-)$ & $(36,-,-)$ \\
\end{tabular}
\end{center}
\textbf{Pareto-Nash equilibria.} If the industry is active, then for the consumer, deciding to smoke or not depends on how the consumer subjectively values/weighs money, addiction and life expectancy: both decisions are encompassed by Pareto-efficiency. For the industry, advertise\&active is a dominant strategy. Consequently, Pareto-Nash-equilibria are all the action-profiles in which the industry decides advertise\&active.
\textbf{Efficiency.}
Since addiction is irrational (detailed in Appendix \ref{app:tabac}), I focus on money and life-expectancy.
We have
$
\mathcal{E}
=
\{\theta(36,55)+(\nu-\theta)(48,75) \mid 0\leq\theta\leq \nu\}
$
and
$\mathcal{F}=\{\nu(48,75)\}$,
where $\nu$ is the world's population, and $\theta$ the number of smokers. Since $\text{WST}[\mathcal{E}]=\{(36,55)\}$, the MO-CR is the singleton $\{(75\%,73\%)\}$: in the worst case, we lose 12k\$ and 20 years of life-expectancy per-consumer. These Pareto-Nash-equilibria are the worst action-profiles for money and life-expectancy, a critical information that was not lost by this MOG and its MO-CR.
\textbf{Practical lessons.} Advertising tobacco fosters consumption. The association ``\textit{Alliance contre le tabac}'' passed a law for standardized neutral packets (April 3rd 2015), in order to annihilate all the benefits of branding, but only in France. The model indicates that:
\begin{center}
\emph{This law will promote a higher economic efficiency}.
\end{center}
\section{Computation of the MO-CR}
\label{sec:computation}
In this section, I provide a polynomial-time algorithm for the computation of MO-CR which relies on a very general procedure based on two phases:
\begin{enumerate}
\item Given a MOG, compute the worst equilibria $\text{WST}[\mathcal{E}]$ and the efficient outcomes $\mathcal{F}$.
\item Given $\text{WST}[\mathcal{E}]$ and $\mathcal{F}$, compute $\text{MO-CR}=\text{EFF}[~R[~\text{WST}[\mathcal{E}]~,~\mathcal{F}~]~]$.
\end{enumerate}
Depending on the input (normal form or compact representation), it adapts as follows.
\subsection{Computation of the MO-CR for Multi-objective Normal Forms}
For a MOG given in MO normal form (which representation length is $L=n\alpha^n d$), Phase 1 (computing $\text{WST}[\mathcal{E}]$ and $\mathcal{F}$) is easy and takes time $O(L^2)$ (see Appendix \ref{app:proof}). For $d=2$, this lowers to $O(L\log_2(L))$. Let us denote the sizes of the outputs $q=|\text{WST}[\mathcal{E}]|$ and $m=|\mathcal{F}|$. For normal forms, it holds that $q,m=O(|\mathcal{A}|)=O(L)$.
\hspace*{0.5cm}For Phase 2, at first glance, the development of the intersection of unions $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\cap_{y\in\text{WST}[\mathcal{E}]}\cup_{z\in\mathcal{F}}\mathcal{C}(y/z)$ causes an exponential $m^q$. But fortunately, one can compute the $\text{MO-CR}$ in polynomial time. Below, $D^t$ is a set of vectors. Given two vectors $x,y\in\mathbb{R}p$,
let $x\wedge y$ denote the vector defined by $\forall k\in{\mathcal{O}},~(x\wedge y)_k=\min\{x_k,y_k\}$ and recall that $\forall k\in{\mathcal{O}},~(x/y)_k=x_k/y_k$. Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} is the development of $\cap_{y\in\text{WST}[\mathcal{E}]}\cup_{z\in\mathcal{F}}\mathcal{C}(y/z)$, on a set-algebra of cone-unions.
Appendix \ref{app:proof} shows that Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} takes time $O((qm)^{2d-1}d)$, or $O((qm)^{2}\log_2(qm))$ for $d=2$.
\vspace*{-0.4cm}
\restylealgo{ruled}
\begin{algorithm}[!h]
\KwIn{$\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^q\}$ and $\mathcal{F}=\{z^1,\ldots,z^m\}$}
\KwOut{$\text{MO-CR}=\text{EFF}[R[\text{WST}[\mathcal{E}],\mathcal{F}]]$}
\ \\ [-1.5ex]
{\bf create } $D^1\leftarrow \{y^1/z\in\mathbb{R}p~|~z\in\mathcal{F}\}$\\
\For{$t=2,\ldots,q$}{
$D^t\leftarrow\text{EFF}[\{\rho ~\wedge~ (y^t/z)~~|~~\rho\in D^{t-1},~~z\in\mathcal{F}\}]$
}
{\bf return } $D^q$
\caption{Computing MO-CR in polynomial-time $O((qm)^{2d-1}d)$}
\label{alg:givenEFoutputMOPoA:polynomial}
\end{algorithm}
\vspace*{-0.4cm}
Having specified Phase 1 and 2 for normal forms, Theorem 1 follows:
\begin{theorem}[Computation of MO-CR]\label{th:master}
Given a MO normal form, one can compute the MO-CR in polynomial time $O(L^{4d-2})$.\quad If $d=2$, it lowers to $O(L^4\log_2(L))$.
\end{theorem}
\subsection{Computation of the MO-CR for Multi-objective Compact Representations}
Compact representations of massively multi-agent games (e.g. MO graphical games, MO action-graph games) have a representation length $L$ that is polynomial with respect to the number of agents $n$ and the sizes of the action-sets $\alpha$. As $q=|\text{WST}[\mathcal{E}]|$ and $m=|\mathcal{F}|$ can be exponentials $\alpha^n$ of this representation length, compact representations are algorithmically more challenging, leaving open the computation of $\text{WST}[\mathcal{E}]$ and $\mathcal{F}$ in Phase 1, and complicating the use of Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} in Phase 2.
To overcome this, one can do MO approximations \cite{papadimitriou2000approximability}, by implementing an approximate Phase 1 which precision transfers to Phase 2 in polynomial time, as follows.
\begin{lemma}\label{lem:approx} Given $\varepsilon_1,\varepsilon_2>0$ and approximations $E$ of $\mathcal{E}$ and $F$ of $\mathcal{F}$ in the sense that:
\begin{eqnarray}
\forall y\in\mathcal{E}, \exists y'\in E,\quad y\succsim y' &
\quad \text{and}\quad &
\forall y'\in E, \exists y\in\mathcal{E},\quad (1+\varepsilon_1)y'\succsim y\label{eq:approx:E}\\
\forall z'\in F, \exists z\in\mathcal{F},\quad z'\succsim z &
\quad \text{and}\quad &
\forall z\in\mathcal{F}, \exists z'\in F,\quad (1+\varepsilon_2)z\succsim z'\label{eq:approx:F}
\end{eqnarray}
it holds that $R[E,F]\subseteq R[\mathcal{E},\mathcal{F}]$ and:
\begin{eqnarray}
\forall \rho\in R[\mathcal{E},\mathcal{F}], \exists \rho'\in R[E,F],\quad (1+\varepsilon_1)(1+\varepsilon_2)\rho'\succsim\rho\label{eq:approx:R}
\end{eqnarray}
\end{lemma}
Equations (\ref{eq:approx:E}) and (\ref{eq:approx:F}) state approximation bounds. Equations (\ref{eq:approx:E}) state that $(1+\varepsilon_1)^{-1}\mathcal{E}$ bounds below $E$ which bounds below $\mathcal{E}$. Equations (\ref{eq:approx:F}) state that $\mathcal{F}$ bounds below $F$ which bounds below $(1+\varepsilon_2)\mathcal{F}$. Crucially, whatever the sizes of $\mathcal{E}$ and $\mathcal{F}$, there exist such approximations $E$ and $F$ that are $O((1/\varepsilon_1)^{d-1})$ and $O((1/\varepsilon_2)^{d-1})$ sized \cite{papadimitriou2000approximability}, yielding the approximation scheme below.
\begin{theorem}[Approximation Scheme for MO-CR]\label{th:approx}
Given a compact MOG of representation length $L$, precisions $\varepsilon_1,\varepsilon_2>0$ and two algorithms to compute approximations $E$ of $\mathcal{E}$ and $F$ of $\mathcal{F}$ in the sense of Equations (\ref{eq:approx:E}) and (\ref{eq:approx:F}) that take time $\theta_{\mathcal{E}}(\varepsilon_1,L)$ and $\theta_{\mathcal{F}}(\varepsilon_2,L)$, one can approximate $R[\mathcal{E},\mathcal{F}]$ in the sense of Equation (\ref{eq:approx:R}) in time:
$$O\left(\theta_{\mathcal{E}}(\varepsilon_1,L) \quad+\quad \theta_{\mathcal{F}}(\varepsilon_2,L) \quad+\quad {(\varepsilon_1 \varepsilon_2)^{-(d-1)(2d-1)}}\right)$$
\end{theorem}
For MO graphical games, Phase 1 could be instantiated with approximate junction-tree algorithms on MO graphical models \cite{dubus2009multiobjective}. For MO symmetric action-graph games, in the same fashion, one could generalize existing algorithms \cite{jiang2007computing}. More generally, for $\text{WST}[\mathcal{E}]$ and $\mathcal{F}$, one can also use meta-heuristics with experimental guarantees.
\section{Prospects}
Multi-objective games can be used as a behaviourally more realistic framework to model a wide set of games occurring in business situations ranging from carpooling websites to combinatorial auctions.
Also, studying the efficiency of MO generalizations of routing or Cournot-competitions \cite{guo2005price} could provide realistic economic insights.
\appendix
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\section{Prospects}
Multi-objective games can be used as a behaviourally more realistic framework to model a wide set of games occurring in business situations ranging from carpooling websites to combinatorial auctions.
Also, studying the efficiency of MO generalizations of routing or Cournot-competitions \cite{guo2005price} could provide realistic economic insights.
\appendix
\section{Why Smoothness will not work on multi-objective games}\label{app:smooth}
Most single-objective price of anarchy analytic results rely on a smoothness-analysis \cite{roughgarden2009intrinsic}. A crucial step for ``smoothness'' is to sum the best response inequalities: For a single-objective game and an equilibrium $a\in\text{PN}$, from the best-response conditions $\forall i\in N, \forall b^i\in A^i, v^i(a)\geq v^i(b^i,a^{-i})$, one has:
$\forall b\in A, \sum_{i=1}^{n} v^i(a)\geq\sum_{i=1}^{n} v^i(b^i,a^{-i})$.
However, the Pareto-Nash-equilibrium conditions are rather:
$\forall i\in N, \forall b^i\in A^i, v^i(b^i,a^{-i}) \not\succ v^i(a)$.
As shown in the following counter-example, such $\not\succ$ relations cannot be summed:
$$
\left(\begin{array}{c} 2\\4 \end{array}\right)
\not\succ
\left(\begin{array}{c} 3\\1 \end{array}\right)
\text{ and }
\left(\begin{array}{c} 3\\1 \end{array}\right)
\not\succ
\left(\begin{array}{c} 1\\2 \end{array}\right)
~~~~~~~~\text{ but }~~~~~~~~
\left(\begin{array}{c} 2\\4 \end{array}\right)
+
\left(\begin{array}{c} 3\\1 \end{array}\right)
\succ
\left(\begin{array}{c} 3\\1 \end{array}\right)
+
\left(\begin{array}{c} 1\\2 \end{array}\right)
$$
Consequently, Smoothness-analysis does not encompass Pareto-Nash equilibria, regardless of the efficiency measurement chosen.
\section{Properties of the Multi-objective Coordination Ratio}\label{app:axioms}
The Multi-objective Coordination Ratio fulfils a list of key good properties for the thorough measurement of the multi-objective efficiency of MO games.
\subsection{Worst case guarantee on equilibria outcomes\\ and No dictatorship on efficient outcomes}
Each vectorial efficiency ratio that the MO-CR states, bounds below the efficiency for each equilibrium outcome, compared to an existing efficient outcome:
$$
\forall \rho\in\text{MO-CR}[\mathcal{E},\mathcal{F}],\quad
\forall y\in\mathcal{E},\quad
\exists z\in\mathcal{F},\quad
y/z\succsim\rho
$$
The process of measuring efficiency by MO-CR does not imply any choice in $\mathcal{F}$ that would impose a point-of-view telling what efficiency should be (e.g. no five-year plans).
\subsection{Multi-objective ratio-scale}
Given $\mathcal{E}$, $\mathcal{F}$ and $r\in\mathbb{R}pp$, it holds that:
\begin{eqnarray}
\text{MO-CR}[\mathcal{E},\mathcal{F}] &\quad\subseteq\quad & \mathbb{R}p\label{eq:ratio:6}\\
\text{MO-CR}[\{(0,\ldots,0)\},\mathcal{F}] & \quad=\quad & \{(0,\ldots,0)\}\label{eq:ratio:7}\\
\text{MO-CR}[r\star\mathcal{E},\mathcal{F}] & \quad=\quad & r\star\text{MO-CR}[\mathcal{E},\mathcal{F}]\label{eq:ratio:8}\\
\text{MO-CR}[\mathcal{E},r\star\mathcal{F}] & \quad=\quad & \text{MO-CR}[\mathcal{E},\mathcal{F}]/r\label{eq:ratio:9}\\
\mathcal{E}\subseteq\mathcal{F} & \quad\Leftrightarrow\quad & (1,\ldots,1)\in \text{MO-CR}[\mathcal{E},\mathcal{F}]\label{eq:ratio:10}
\end{eqnarray}
Equation (\ref{eq:ratio:6}) states that the MO-CR is expressed in the multi-objective space. It is worth noting that while $\text{MO-CR}[\mathcal{E},\mathcal{F}]\subseteq[0,1]^d$ is a more classical choice, MO-CR also allows for measurements of over-efficiencies. (E.g. if $\mathcal{F}$ is a family-car and $\mathcal{E}$ is a Lamborghini, then there is over-efficiency on the speed objective.)
\hspace*{0.5cm} Equations (\ref{eq:ratio:7}), (\ref{eq:ratio:8}) and (\ref{eq:ratio:9}) state that MO-CR is sensitive on each objective to multiplications of the outcomes. For instance, if $\mathcal{E}$ is three times better on objective $k$, then so is MO-CR. If there are twice better opportunities of efficiency in $\mathcal{F}$ on objective $k'$, then MO-CR is one half on objective $k'$. In other words, the efficiency of each objective independently reflects into the MO-CR in a ratio-scale.
\hspace*{0.5cm} If all equilibria outcomes are efficient (i.e. $\mathcal{E}\subseteq\mathcal{F}$), then this must imply that according to the MO-CR, the MO game is fully efficient, that is: $(1,\ldots,1)\in \text{MO-CR}[\mathcal{E},\mathcal{F}]$. The MO-CR seems to be the only multi-objective ratio-scale measurement that fulfils Equation (\ref{eq:ratio:10}) while being a worst case guarantee on equilibria outcomes with no dictatorship on what efficiency should be.
\hspace*{0.5cm} It is also worth noting that MO-CR is MO-monotonic with respect to $\mathcal{E}$ and $\mathcal{F}$. For $X,Y\subseteq\mathbb{R}p$, let $X\unrhd Y$ denote $Y\subseteq\mathcal{C}(X)$ where $\mathcal{C}(X)=\cup_{x\in X}\mathcal{C}(x)$ (i.e. $X$ dominates $Y$). Then it holds that:
\begin{eqnarray}
\mathcal{E}\unrhd\mathcal{E}' &\quad\Rightarrow\quad & \text{MO-CR}[\mathcal{E},\mathcal{F}]\unrhd\text{MO-CR}[\mathcal{E}',\mathcal{F}]\\
\mathcal{F}\unrhd\mathcal{F}' &\quad\Rightarrow\quad & \text{MO-CR}[\mathcal{E},\mathcal{F}']\unrhd\text{MO-CR}[\mathcal{E},\mathcal{F}]
\end{eqnarray}
\section{Bounded Rationality in Tobacco Consumption}\label{app:tabac}
According to the \textit{intrinsic theory of value} \cite{adam1776inquiry}, the value of a cigarette objectively amounts to the quantities of raw materials used for its production, or is the combination of the labour times put into it \cite{marx1867kapital}. However, each economic agent needs to keep the freedom to evaluate and act how he pleases, in order to keep his good will and some economic efficiency, as observed in the end of the Soviet Union.
According to the \textit{subjective theory of value} \cite{adam1776inquiry}, the value of a cigarette amounts to the price an agent is willing to pay for it. Since the consumers value the product, then the industry creates value \cite{walras1896elements}.
However, this disregards what the disastrous consequence is on life expectancy, belittles 7.500.000 deaths-per-year and emphasizes the bounded rationality of behaviours.
While for some health economists, consuming a cigarette is a rational choice, as one values pleasure more than life expectancy, for others, consumers are stuck into addiction before becoming adults. The truth is likely between these two extreme points of view \cite{sloan2004price}: Economic agents discount the future at a rate of 6\% per-year, hence a day of life in 40 years is valued 10 times less than now, leading to overweighting the actual smoking pleasure and to irrational behaviours with respect to preferences over a full lifetime. Agents behave according to objectives (e.g. addictive satisfaction) that they would avoid if they had the full experience of their lifetime (e.g. a lung cancer with probability $1/2$) and a sufficient will (e.g. quit smoking). Time discounting also explains other non-sustainable behaviours like over-fishing catastrophes.
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\section{Proofs}\label{app:proof}
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\subsection{Phase 1 for Normal Forms, Correctness of Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} and Theorem \ref{th:master}}
\label{sec:mopoa:poly}
Phase 1 is easy, if the MOG is given in normal form. The MOG is made of the MO evaluations of each agent on each action-profile, that is: $O(n\alpha^n)$ vectors. Hence, the computation of $u(A)$ requires
for each $a\in A$, the addition of $n$ vectors. Therefore, the computation of $u(A)$ takes time $O(\alpha^n n d)$ (linear in the size of the input) and yields $O(\alpha^n)$ vectors. The computation of $\mathcal{F}=\text{EFF}[u(A)]$ given $u(A)$ takes time $O(|u(A)|^2 d)=O(\alpha^{2n} d)$. To conclude, the computation of $\mathcal{F}$ takes time $O(n \alpha^n d + \alpha^{2n} d)$, which is polynomial (quadratic) in the size of the input. If $d=2$, this can be significantly lowered to $O(n \alpha^n d + \alpha^{n}\log_2(\alpha^n)d)=O(n\alpha^n \log_2(\alpha) )$.
\hspace*{0.7cm}The computation of $\text{WST}[\mathcal{E}]$ can be achieved by first computing $\text{PN}$. For this purpose, for each agent $i\in N$ and each adversary action profile $a^{-i}\in A^{-i}$, one has to compute which individual actions give a Pareto-efficient evaluation in $v^i(A^i,a^{-i})$, in order to mark which action-profiles can be a $\text{PN}$ from $i$'s point of view. Overall, computing $\text{PN}$ takes time $O(n \alpha^{n-1} \alpha^2 d)$. Using back $u(A)$, computing $\mathcal{E}=u(\text{PN})$ is straightforward. Again, the computation of $\text{WST}[\mathcal{E}]$ given $\mathcal{E}$ takes time $O(|\mathcal{E}|^2 d)=O(\alpha^{2n} d)$. To sum up, the computation of $\text{WST}[\mathcal{E}]$ takes time $O(n \alpha^{n+1} d + \alpha^{2n} d)$. If $d=2$, this lowers to $O(n\alpha^n \log_2(\alpha))$.\\
\hspace*{0.7cm}In order to compute $\text{MO-CR}=\text{EFF}[R[\text{WST}[\mathcal{E}],\mathcal{F}]]$,
let us study the structure of $\bigcap_{y\in\text{WST}[\mathcal{E}]}\bigcup_{z\in\mathcal{F}}\mathcal{C}(y/z)$, by restricting a set-algebra to the following objects:
\begin{definition}[Cone-Union]
\label{def:coneunion}
For a set of vectors $X\subseteq\mathbb{R}p$, the Cone-Union $\mathcal{C}(X)$ is:
$$
\mathcal{C}(X)
~~=~~\bigcup_{x\in X}\mathcal{C}(x)
~~~~=\{y\in\mathbb{R}p ~~|~~ \exists x\in X, x\succsim y\}
$$
Let $\mathcal{C}$ denote the set of all cone-unions of $\mathbb{R}p$.
\end{definition}
To define an algebra on $\mathcal{C}$, one can supply $\mathcal{C}$ with $\cup$ and $\cap$.
\begin{lemma}[On the Set-Algebra $(\mathcal{C},\cup,\cap)$]\label{prop:algebra}~\\
Given two descriptions of cone-unions $X^1,X^2\subseteq\mathbb{R}p$, we have:
$$\mathcal{C}(X^1)\cup\mathcal{C}(X^2)=\mathcal{C}( X^1\cup X^2 )$$
Given two descriptions of cones $x^1,x^2\in\mathbb{R}p$, we have:
$$\mathcal{C}(x^1)\cap\mathcal{C}(x^2)=\mathcal{C}(x^1\wedge x^2)$$
where $x^1\wedge x^2\in\mathbb{R}p$ is: $\forall k\in{\mathcal{O}}, (x^1\wedge x^2)_k=\min\{x^1_k,x^2_k\}$.\\
Given two descriptions of cone-unions $X^1,X^2\subseteq\mathbb{R}p$, we have:
\begin{eqnarray*}
\mathcal{C}(X^1)\cap\mathcal{C}(X^2)
&=&\left(\cup_{x^1\in X^1}\mathcal{C}(x^1)\right)\cap\left(\cup_{x^2\in X^2}\mathcal{C}(x^2)\right)\\
&=&\bigcup_{(x^1,x^2)\in X^1\times X^2}\mathcal{C}(x^1)\cap\mathcal{C}(x^2)\\
&=&\bigcup_{(x^1,x^2)\in X^1\times X^2}\mathcal{C}(x^1\wedge x^2)\\
&=&\mathcal{C}( X^1\wedge X^2 )
\end{eqnarray*}
where $X^1\wedge X^2=\{x^1\wedge x^2~|~x^1\in X^1,~x^2\in X^2\}\subseteq\mathbb{R}p$.\\
Therefore, $(\mathcal{C},\cup,\cap)$ is stable, and then is a set-algebra.
\end{lemma}
\begin{proof}
The three properties derive from set calculus.
\end{proof}
The main consequence of Lemma \ref{prop:algebra} is that $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\cap_{y\in\text{WST}[\mathcal{E}]}\cup_{z\in\mathcal{F}}\mathcal{C}(y/z)$ is a cone-union. Moreover, one can do the development for $\cap_{y\in\text{WST}[\mathcal{E}]}\cup_{z\in\mathcal{F}}\mathcal{C}(y/z)$ within the cone-unions, using distributions and developments.
\begin{remark}\label{rk:app:cone}
For a finite set $X\subseteq\mathbb{R}p$, we have: $\mathcal{C}(X)=\mathcal{C}(\text{EFF}[X])$.
\end{remark}
\begin{proof}
Firstly, we prove $\mathcal{C}(X)\subseteq\mathcal{C}(\text{EFF}[X])$.
If $y\in\mathcal{C}(X)$, then there exists $x\in X$ such that $x\succsim y$. There are two cases, $x\in\text{EFF}[X]$ and $x\not\in\text{EFF}[X]$. If $x\in\text{EFF}[X]$, then $y\in\mathcal{C}(\text{EFF}[X])$, by definition of a cone-union.
Otherwise, if $x\not\in\text{EFF}[X]$, then there exists $z\in X$ such that $z\succ x$. And since $X$ is finite, we can find such a $z$ in $\text{EFF}[X]$, by iteratively taking $z'\succ z$ until $z\in\text{EFF}[X]$, which will happen because $X$ is finite and $\succ$ is transitive and irreflexive. Hence, there exists $z\in\text{EFF}[X]$ such that $z\succ x\succsim y$ and then $z\succsim y$. Consequently, $y\in\mathcal{C}(\text{EFF}[X])$, by definition of a cone-union.
Conversely, $Y\subseteq X\Rightarrow \mathcal{C}(Y)\subseteq\mathcal{C}(X)$ proves $\mathcal{C}(\text{EFF}[X])\subseteq\mathcal{C}(X)$.
\end{proof}
As a consequence of Remark \ref{rk:app:cone}, for $x\in\mathbb{R}p$, a simple cone $\mathcal{C}(x)$ is fully described by its summit $x$. The main consequence of this remark is that $\mathcal{C}(X)$ can be fully described and represented by $\text{EFF}[X]$.
For instance, since $R[\text{WST}[\mathcal{E}],\mathcal{F}]$ is a cone-union (thanks to Lemma \ref{prop:algebra}), and since $\text{MO-CR}=\text{EFF}[R[\text{WST}[\mathcal{E}],\mathcal{F}]]$ (by definition of the MO-CR), then $R[\text{WST}[\mathcal{E}],\mathcal{F}]$ is fully represented (as a cone-union) by the MO-CR, which means that $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\mathcal{C}(\text{MO-CR})$.\\
Recall that $q=|\text{WST}[\mathcal{E}]|$ and $m=|\mathcal{F}|$. In this subsection, we also denote $\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^q\}$ and $\mathcal{F}=\{z^1,\ldots,z^m\}$.
Let $\mathcal{A}_q^m$ denote the set of functions $\pi$ from $\{1,\ldots,q\}$ to $\{1,\ldots,m\}$. (We have: $|\mathcal{A}_q^m|=m^q$.)
\begin{corollary}[The cone-union of MO-CR]~\\
Given $\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^q\}$ and $\mathcal{F}=\{z^1,\ldots,z^m\}$, we have:
$$R[\text{WST}[\mathcal{E}],\mathcal{F}]=\bigcup_{\pi\in\mathcal{A}_q^m}\bigcap_{t=1}^{q} \mathcal{C}(y^t / z^{\pi(t)})$$
and therefore:
$$\text{MO-CR}=\text{EFF}\left[\left\{\bigwedge\nolimits_{t=1}^{q}y^t / z^{\pi(t)}~~|~~\pi\in\mathcal{A}_q^m\right\}\right]$$
\end{corollary}
\begin{proof}
For the first statement, just think to a development. We write down $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\cap_{y\in\text{WST}[\mathcal{E}]}\cup_{z\in\mathcal{F}}\mathcal{C}(y/z)$ into the layers just below. There is one layer per $y^t$ in $\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^t,\ldots,y^q\}$:
$$
\begin{array}{ccccccccccl}
& ( & \mathcal{C}(\frac{y^1}{z^1}) & \cup & \mathcal{C}(\frac{y^1}{z^2}) & \cup & \ldots & \cup & \mathcal{C}(\frac{y^1}{z^m})& )&\text{layer 1}\\
\bigcap & ( & \mathcal{C}(\frac{y^2}{z^1}) & \cup & \mathcal{C}(\frac{y^2}{z^2}) & \cup & \ldots & \cup & \mathcal{C}(\frac{y^2}{z^m})& )&\text{layer 2}\\
&&&&&\vdots\\
\bigcap & ( & \mathcal{C}(\frac{y^q}{z^1}) & \cup & \mathcal{C}(\frac{y^q}{z^2}) & \cup & \ldots & \cup & \mathcal{C}(\frac{y^q}{z^m})& )&\text{layer q}
\end{array}
$$
Imagine the simple cones as vertices and imagine edges going from each vertex of layer $t$ to each vertex of the next layer $(t+1)$. The development into a union outputs as many intersection-terms as paths from the first layer to the last one.
Let the function $\pi:\{1,\ldots,q\}\rightarrow\{1,\ldots,m\}$ denote a path from layer $1$ to layer $q$, where $\pi(t)$ is the vertex chosen in layer $t$.
Consequently, in the result of the development into an union, each term is an intersection $\bigcap_{t=1}^{q} \mathcal{C}(y^t / z^{\pi(t)})$.
The second statement results from the first statement, Lemma \ref{prop:algebra} and Remark \ref{rk:app:cone}.
\begin{eqnarray*}
R[\text{WST}[\mathcal{E}],\mathcal{F}]
&=&\bigcup_{\pi\in\mathcal{A}_q^m}\bigcap_{t=1}^{q} \mathcal{C}(y^t / z^{\pi(t)})\\
&=&\bigcup_{\pi\in\mathcal{A}_q^m} \mathcal{C}\left(\bigwedge_{t=1}^{q} y^t / z^{\pi(t)}\right)\\
&=&\mathcal{C}\left(\left\{\bigwedge\limits_{t=1}^{q}y^t / z^{\pi(t)}~~|~~\pi\in\mathcal{A}_q^m\right\}\right)
\end{eqnarray*}
That $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\mathcal{C}(\text{MO-CR})$ concludes the proof.
\end{proof}
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As a consequence of Remark \ref{rk:app:cone}, for $x\in\mathbb{R}p$, a simple cone $\mathcal{C}(x)$ is fully described by its summit $x$. The main consequence of this remark is that $\mathcal{C}(X)$ can be fully described and represented by $\text{EFF}[X]$.
For instance, since $R[\text{WST}[\mathcal{E}],\mathcal{F}]$ is a cone-union (thanks to Lemma \ref{prop:algebra}), and since $\text{MO-CR}=\text{EFF}[R[\text{WST}[\mathcal{E}],\mathcal{F}]]$ (by definition of the MO-CR), then $R[\text{WST}[\mathcal{E}],\mathcal{F}]$ is fully represented (as a cone-union) by the MO-CR, which means that $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\mathcal{C}(\text{MO-CR})$.\\
Recall that $q=|\text{WST}[\mathcal{E}]|$ and $m=|\mathcal{F}|$. In this subsection, we also denote $\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^q\}$ and $\mathcal{F}=\{z^1,\ldots,z^m\}$.
Let $\mathcal{A}_q^m$ denote the set of functions $\pi$ from $\{1,\ldots,q\}$ to $\{1,\ldots,m\}$. (We have: $|\mathcal{A}_q^m|=m^q$.)
\begin{corollary}[The cone-union of MO-CR]~\\
Given $\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^q\}$ and $\mathcal{F}=\{z^1,\ldots,z^m\}$, we have:
$$R[\text{WST}[\mathcal{E}],\mathcal{F}]=\bigcup_{\pi\in\mathcal{A}_q^m}\bigcap_{t=1}^{q} \mathcal{C}(y^t / z^{\pi(t)})$$
and therefore:
$$\text{MO-CR}=\text{EFF}\left[\left\{\bigwedge\nolimits_{t=1}^{q}y^t / z^{\pi(t)}~~|~~\pi\in\mathcal{A}_q^m\right\}\right]$$
\end{corollary}
\begin{proof}
For the first statement, just think to a development. We write down $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\cap_{y\in\text{WST}[\mathcal{E}]}\cup_{z\in\mathcal{F}}\mathcal{C}(y/z)$ into the layers just below. There is one layer per $y^t$ in $\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^t,\ldots,y^q\}$:
$$
\begin{array}{ccccccccccl}
& ( & \mathcal{C}(\frac{y^1}{z^1}) & \cup & \mathcal{C}(\frac{y^1}{z^2}) & \cup & \ldots & \cup & \mathcal{C}(\frac{y^1}{z^m})& )&\text{layer 1}\\
\bigcap & ( & \mathcal{C}(\frac{y^2}{z^1}) & \cup & \mathcal{C}(\frac{y^2}{z^2}) & \cup & \ldots & \cup & \mathcal{C}(\frac{y^2}{z^m})& )&\text{layer 2}\\
&&&&&\vdots\\
\bigcap & ( & \mathcal{C}(\frac{y^q}{z^1}) & \cup & \mathcal{C}(\frac{y^q}{z^2}) & \cup & \ldots & \cup & \mathcal{C}(\frac{y^q}{z^m})& )&\text{layer q}
\end{array}
$$
Imagine the simple cones as vertices and imagine edges going from each vertex of layer $t$ to each vertex of the next layer $(t+1)$. The development into a union outputs as many intersection-terms as paths from the first layer to the last one.
Let the function $\pi:\{1,\ldots,q\}\rightarrow\{1,\ldots,m\}$ denote a path from layer $1$ to layer $q$, where $\pi(t)$ is the vertex chosen in layer $t$.
Consequently, in the result of the development into an union, each term is an intersection $\bigcap_{t=1}^{q} \mathcal{C}(y^t / z^{\pi(t)})$.
The second statement results from the first statement, Lemma \ref{prop:algebra} and Remark \ref{rk:app:cone}.
\begin{eqnarray*}
R[\text{WST}[\mathcal{E}],\mathcal{F}]
&=&\bigcup_{\pi\in\mathcal{A}_q^m}\bigcap_{t=1}^{q} \mathcal{C}(y^t / z^{\pi(t)})\\
&=&\bigcup_{\pi\in\mathcal{A}_q^m} \mathcal{C}\left(\bigwedge_{t=1}^{q} y^t / z^{\pi(t)}\right)\\
&=&\mathcal{C}\left(\left\{\bigwedge\limits_{t=1}^{q}y^t / z^{\pi(t)}~~|~~\pi\in\mathcal{A}_q^m\right\}\right)
\end{eqnarray*}
That $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\mathcal{C}(\text{MO-CR})$ concludes the proof.
\end{proof}
Ultimately, this proves the \textbf{correctness} of Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} for the computation of MO-CR, given $\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^q\}$ and $\mathcal{F}=\{z^1,\ldots,z^m\}$. It consists in the iterative development of the intersection $R(\mathcal{E},\mathcal{F})$, which can be seen as dynamic programming on the paths of the layer graph.
For $k\in\{1,\ldots,q\}$, we denote $D^t$ the description of the cone-union corresponding to the intersection:
$$\mathcal{C}(D^t)=\cap_{l=1}^{t} \cup_{z\in\mathcal{F}} \mathcal{C}(y^l/z)$$
Recursively, for $t>1$, $\mathcal{C}(D^t)=\mathcal{C}(D^{t-1})~\cap~(\cup_{z\in\mathcal{F}}~\mathcal{C}(y^{t}/z))$.
From Lemma \ref{prop:algebra} and Remark \ref{rk:app:cone}, in order to develop, we
then have to iterate the following:
$$
D^t=\text{EFF}[\{\rho ~\wedge~ (y^t/z)~~|~~\rho\in D^{t-1},~~z\in\mathcal{F}\}]
$$
We now proceed with the \textbf{time complexity} of Algorithm \ref{alg:givenEFoutputMOPoA:polynomial}. At first glance, since there are $m^q$ paths in the layer graph, then there are $O(m^q)$ elements in MO-CR. Fortunately, they are much less, because we have:
\begin{theorem}[MO-CR is polynomially-sized]~\\
\label{th:mopoa:poly}
Given a MOG and denoting $d=|{\mathcal{O}}|$, $q=|\text{WST}[\mathcal{E}]|$ and $m=|\mathcal{F}|$, we have:
$$|\text{MO-CR}|\leq (qm)^{d-1}$$
\end{theorem}
\begin{proof}
Given $\rho\in\text{MO-CR}$, for some $\pi\in\mathcal{A}_q^m$, we have $\rho=\bigwedge\nolimits_{t=1}^{q}y^t / z^{\pi(t)}$, and then $\forall k\in{\mathcal{O}}, \rho_k=\min_{t=1\ldots q}\{y^t_k / z^{\pi(t)}_k\}$. Therefore, $\rho_k$ is exactly realized by the $k$th component of at least one cone summit $y^t / z^{\pi(t)}$ in the layer graph (that is a vertex in the layer-graph above). Consequently, there are at most as many possible values for the $k$th component of $\rho$, as the number of vertices in the layer graph, that is $qm$. This holds for the $d$ components of $\rho$; hence there are at most $(qm)^d$ vectors in MO-CR. More precisely, by Lemma \ref{lem:eff} (below), since MO-CR is an efficient set, then there are at most $(qm)^{d-1}$ vectors in MO-CR.
\end{proof}
\begin{lemma}\label{lem:eff}
Let $Y\subseteq\mathbb{R}p$ be a set of vectors, with at most $M$ values on each component:
$$|~\text{EFF}[Y]~|\leq M^{d-1}$$
\end{lemma}
\begin{proof} At most $M^{d-1}$ valuations are realized on the $d-1$ first components. If you fix the $d-1$ first components, there is at most one Pareto-efficient vector which maximizes the last component.
\end{proof}
In Algorithm \ref{alg:givenEFoutputMOPoA:polynomial}, there are $\Theta(q)$ steps. At each step $t$, from Theorem \ref{th:mopoa:poly}, we know that $|D^{t-1}|\leq (qm)^{d-1}$. Hence, $|\{\rho ~\wedge~ (y^t/z)~~|~~\rho\in D^{t-1},~~z\in\mathcal{F}\}|\leq q^{d-1} m^d$, and the computation of the efficient set $D^t$ requires time $O((q^{d-1} m^d)^2 d)$.
However, by using an insertion process, since there are at most $B=|D^{t}|\leq (qm)^{d-1}$ Pareto-efficient vectors at each insertion, then we only need $O(q^{d-1} m^d \times (qm)^{d-1})$ Pareto-comparisons. If $d=2$, time lowers to $O(q^{d-1} m^d\log_2(q m) d^2)=O(q m^2\log_2(q m))$.
Ultimately, Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} takes $q$ steps and then time $O(q (q^{d-1} m^d)(qm)^{d-1} d)=O((qm)^{2d-1}d)$. If $d=2$, this lowers to $O((q m)^2\log_2(q m))$.
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\subsection{Approximations: Proof of Lemma \ref{lem:approx} and Theorem \ref{th:approx}}
\begin{proof}
(1) First, let us show $R[E,F]\subseteq R[\text{WST}[\mathcal{E}],\mathcal{F}]$. Let $\rho'$ be a ratio of $R[E,F]$ and let us show that:
$$
\forall y\in \text{WST}[\mathcal{E}],~~
\exists z\in \mathcal{F},~~
\text{ s.t.: } y\succsim\rho'\star z
$$
Take $y\in\text{WST}[\mathcal{E}]$. From Equation (\ref{eq:approx:E}) (first condition), there is a $y'\in E$ such that $y\succsim y'$.
From Definition \ref{def:mopoa}, there is a $z'$ such that $y'\succsim \rho'\star z'$. From Equation (\ref{eq:approx:F}) on $z'$ (first condition), there exists $z\in\mathcal{F}$ such that $z'\succsim z$. Recap: $y\succsim y'\succsim \rho'\star z'\succsim \rho'\star z$.
(2) Then, let $\rho$ be a ratio of $R[\text{WST}[\mathcal{E}],\mathcal{F}]$,
and let us show that $\rho'=(1+\varepsilon_1)^{-1}(1+\varepsilon_2)^{-1}\rho$~~ is in $R[E,F]$, that is:
$$
\forall y'\in E,~~
\exists z'\in F,~~
(1+\varepsilon_1) y'\succsim (1+\varepsilon_2)^{-1} \rho\star z'
$$
Take an element $y'$ of $E$.
From Equation (\ref{eq:approx:E}) (second condition), there is $y\in\text{WST}[\mathcal{E}]$ such that $(1+\varepsilon_1)y'\succsim y$.
From Definition \ref{def:mopoa}, there is $z\in\mathcal{F}$ such that $y\succsim\rho\star z$.
From Equation (\ref{eq:approx:E}) on $z$ (second condition), there exists $z'\in F$ s.t. $z\succsim (1+\varepsilon_2)^{-1} z'$.
Recap: $(1+\varepsilon_1)y'\succsim y\succsim \rho\star z\succsim (1+\varepsilon_2)^{-1} \rho\star z'$.
\end{proof}
\begin{proof}[Theorem \ref{th:approx}]
For the first claim, since Algorithm \ref{alg:givenEFoutputMOPoA:polynomial}, given $E$ and $F$, outputs the MO-PoA corresponding to $R[E,F]$, by Theorem \ref{th:approx}, Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} outputs an $((1+\varepsilon_1)(1+\varepsilon_2))$-covering of $R(\text{WST}[\mathcal{E}],\mathcal{F})$.
For the second claim, from Lemma \ref{lem:approx}, applying Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} on $E$ and $F$ outputs an $((1+\varepsilon_1)(1+\varepsilon_2))$-covering of $R(\text{WST}[\mathcal{E}],\mathcal{F})$.
Moreover, since we have $|E|=O((1/\varepsilon_1)^{d-1})$ and $|F|=O((1/\varepsilon_2)^{d-1})$, Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} takes time $O\left(d/(\varepsilon_1\varepsilon_2)^{(d-1)(2d-1)}\right)$.
\end{proof}
\end{document}
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\begin{document}
\maketitle
\begin{abstract} Finsler metrics are direct generalizations of Riemannian metrics such that the quadratic Riemannian indicatrices in the tangent spaces of a manifold are replaced by more general convex bodies as unit spheres. A linear connection on the base manifold is called compatible with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called generalized Berwald manifolds \cite{Wag1}. Compatible linear connections are the solutions of the so-called compatibility equations containing the components of the torsion tensor as unknown quantities. Although there are some theoretical results for the solvability of the compatibility equations (monochromatic Finsler metrics \cite{BM}, extremal compatible linear connections and algorithmic solutions \cite{V14}), it is very hard to solve in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique. Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor. It is proved \cite{V10} (see also \cite{V11}) that such a compatible linear connection must be uniquely determined.
The original proof is based on averaging in the sense that the 1-form in the decomposition of the torsion tensor can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. In what follows we present a new proof for the unicity result by using linear algebra and some basic facts about convex bodies. We present an explicit formula for the solution without integration. The method has a new contribution to the problem as well: necessary conditions of the solvability are formulated in terms of intrinsic equations without unknown quantities. They are sufficient if and only if the solution depends only on the position.
\end{abstract}
\section{Compatibility equations in Finsler geometry}
Let $M$ be a smooth connected manifold with a local coordinate system $u^1, \ldots, u^n$. The induced local coordinate system on the tangent manifold $TM$ consists of the functions $x^1, \ldots, x^n$ and $y^1, \ldots, y^n$ given by $x^i(v):=u^i\circ \pi (v)=u^i(p)=:p^i$, where $\pi \colon TM \to M$ is the canonical projection and $y^i(v)=v(u^i)$, $i=1, \ldots, n$. Throughout the paper, we will use the shorthand notations
\begin{equation}
\partial_i := \dfrac{\partial}{\partial x^i} \hspace{1cm} \text{and} \hspace{1cm} \dot{\partial}_i:= \dfrac{\partial}{\partial y^i}.
\end{equation}
A Finsler metric \cite{BSC} on a manifold is a smoothly varying family of Minkowski norms in the tangent spaces. It is a direct generalization of Riemannian metrics, with inner products (quadratic indicatrices) in the tangent spaces replaced by Minkowski norms (smooth strictly convex bodies).
\begin{defi} A \textbf{Finsler metric} is a non-negative continuous function $F\colon TM\to \mathbb{R}$ satisfying the following conditions: $\displaystyle{F}$ is smooth on the complement of the zero section (\emph{regularity}), $\displaystyle{F(tv)=tF(v)}$ for all $\displaystyle{t> 0}$ (\emph{positive homogeneity}), $F(v)= 0$ if and only if $v={\bf 0}$ (\emph{definiteness}) and the Hessian $\displaystyle{g_{ij}=\dot{\partial}_{i}\dot{\partial}_{j}E}$ of the energy function $E=F^2/2$ is positive definite at all non-zero elements $\displaystyle{v\in T_pM}$ (\emph{strong convexity}). The pair $(M,F)$ is called a \textbf{Finsler manifold}.
\end{defi}
On a Riemannian manifold we obviously have compatible linear connections in the sense that the induced parallel transports preserve the Riemannian length of tangent vectors (metric linear connections). Following the classical Christoffel process it is clear that such a linear connection is uniquely determined by the torsion tensor. In contrast to the Riemannian case, non-Riemannian Finsler manifolds admitting compatible linear connections form a special class of spaces in Finsler geometry. They are called generalized Berwald manifolds \cite{Wag1}. It is known that some Finsler manifolds do not admit any compatible linear connections because of topological constraints, some have infinitely many compatible linear connections and it can also happen that the compatible linear connection is uniquely determined \cite{VOM}, see also \cite{RandersGBM}.
\begin{defi} A linear connection is \textbf{compatible} with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called \textbf{generalized Berwald manifolds}.
\end{defi}
In terms of local coordinates, equations
\begin{equation}
\label{ceq_christoffel}
X_i^h F := \partial_i F-y^j \left( {\Gamma}^k_{ij}\circ \pi \right) \dot{\partial}_k F=0 \hspace{1cm} (i=1,\dots,n)
\end{equation}
form necessary and sufficient conditions for a linear connection $\nabla$ to be compatible with the Finsler metric $F$. Equations (\ref{ceq_christoffel}) are called \textbf{compatibility equations} or CEQ for short. The fundamental result of generalized Berwald manifold theory states that a compatible linear connection $\nabla$ is always Riemann metrizable \cite{V5}, i.e. $\nabla$ must be a metric linear connection with respect to a Riemannian metric $\gamma$. Such a Riemannian metric can be given by integration of $g_{ij}$ on the indicatrix hypersurfaces \cite{V5}, see also \cite{Cram} and \cite{Mat1}. It is the so-called \textbf{averaged Riemannian metric}. Therefore CEQ can be reformulated by replacing the Christoffel symbols $\Gamma_{ij}^k$ by the torsion tensor components \cite{V14}, see also \cite{RandersGBM}. Using the horizontal vector fields
\[ X_i^{h*}:=\partial_i -y^j \left( {\Gamma}^{k*}_{ij}\circ \pi \right) \dot{\partial}_k \hspace{1cm} (i=1,\dots,n), \]
where ${\Gamma}^{k*}_{ij}$ are the Christoffel symbols of the L\'{e}vi-Civita connection of the averaged Riemannian metric $\gamma$, CEQ takes the form
\begin{equation}
y^j \left(T^{l}_{jk}\gamma^{kr}\gamma_{il}+T^{l}_{ik}\gamma^{kr}\gamma_{jl}-T_{ij}^r\right) \dot{\partial}_r F=-2X_i^{h^*}F\hspace{1cm} (i=1,\dots,n).
\end{equation}
The unknown quantities $T_{ab}^c$ are the torsion tensor components of the compatible linear connection.
In what follows we are going to use normal coordinates with respect to the Riemannian metric $\gamma$ around a given point $p\in M$. The coordinate vector fields $\partial/\partial u^1, \dots, \partial/\partial u^n$ form an orthonormal basis in $T_p M$, i.e. $\gamma_{ij}(p)=\delta_{ij}$ and ${\Gamma}^{k*}_{ij}(p)=0$. Therefore $X_i^{h*}(v)=\partial_i(v)$ for any $v\in T_pM$ and CEQ takes the form
\begin{equation}
\label{CEQ-tors2}
\sum_{a<b,c} \sigma_{ab;i}^c T_{ab}^{c} = -2\partial_i F \hspace{1cm} (i=1, \dots, n),
\end{equation}
where the coefficients are
\begin{equation}
\label{CEQ-coeff2}
\sigma_{ab;i}^c := \delta_i^a f_{cb} + \delta_i^b f_{ac} + \delta_i^c f_{ab}, \hspace{1cm} f_{ij} := y^i \dot{\partial}_j F - y^j \dot{\partial}_i F.
\end{equation}
If none of the indices $a,b,c$ are equal to $i$ then $\sigma_{ab;i}^c=0$. Otherwise the table shows the possible cases, where indices are separated according to their values (equal indices are put into the same cell and different cells contain different values).
\begin{center}
{\tabulinesep=1pt \begin{tabu} {|c||c|c|c||l|}
\hline
& \multicolumn{3}{c||}{\textrm{indices}} & \textrm{the coefficients} \\
\hline
\hline
1. & $i=a$ & $b$ & $c$ & $\sigma_{ib;i}^{c}=f_{cb}$ \\
\hline
2. & $i=a$ & $b=c$ & & $\sigma_{ib;i}^{b}=0$ \\
\hline
3. & $i=b$ & $a$ & $c$ & $\sigma_{ai;i}^{c}=f_{ac}$ \\
\hline
4. & $i=b$ & $a=c$ & & $\sigma_{ai;i}^{a}=0$ \\
\hline
5. & $i=c$ & $a$ & $b$ & $\sigma_{ab;i}^{i}=f_{ab}$ \\
\hline
6. & $i=a=c$ & $b$ & & $\sigma_{ib;i}^{i}=2f_{ib}$ \\
\hline
7. & $i=b=c$ & $a$ & & $\sigma_{ai;i}^{i}=2f_{ai}$ \\
\hline
\end{tabu} }
\end{center}
Therefore the $i$-th compatibility equation at the point $p$ is
\begin{equation} \label{CEQ-general}
\sideset{}{'}\sum_{a} 2f_{ia} T_{ia}^i + \sideset{}{'}\sum_{a<b} f_{ab} \left( T_{ib}^a + T_{ab}^i + T_{ai}^b \right) = -2\, \partial_i F,
\end{equation}
where the primed summation means summing for $a \neq i$ in the first one, and in the second one, $a$ and $b$ where $i\neq a, b$.
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\section{The geometry of the tangent spaces}
The following table shows a panoramic view about the geometric structures of the tangent space $T_pM$ due to the simultaneously existing Finsler and Riemannian metrics.
\begin{center}
\begin{tabularx}{\textwidth}{|Y|Y|Y|}
\hline
\textbf{Finsler structure} & & \textbf{Riemannian structure} \\
\hline
Minkowski norm & metric on $T_p M$ & Euclidean norm and inner product \\
\hline
Finslerian spheres $F_p(\lambda)$ & level sets of $\lambda\in \mathbb{R}_+$ & Euclidean spheres $R_p(\lambda)$ \\
\hline
$\mathcal{F}_v$ & tangent hyperplanes of level sets at $v\in T_pM$ & $\mathcal{R}_v$ \\
\hline
$\mathcal{LF}_v:= \mathcal{F}_v - v$ & linear tangent hyperplanes at $v\in T_pM$ & $\mathcal{LR}_v := \mathcal{R}_v - v$ \\
\hline
$G := \mathrm{grad} \, F = [ \dot{\partial}_1 F, \dots, \dot{\partial}_n F]$ & normal vector fields (w.r.t. $\gamma$) & $C := [y^1, \dots, y^n]$ \\
\hline
\end{tabularx}
\end{center}
Both gradient vector fields $G$ and $C$ are nonzero everywhere on $T_p^{\circ} M := T_p M \backslash \{ \textbf{0} \}$. For every element $v \in T_p^{\circ} M$ the tangent hyperplanes of the Finslerian and the Riemannian (Euclidean) spheres passing through $v$ can be related as follows.
\begin{itemize}
\item If $\mathcal{F}_v=\mathcal{R}_v$, i.e. $G_v \parallel C_v$, we call the point $v$ a \textbf{vertical contact point} of the metrics.
\item If $\mathcal{F}_v \neq \mathcal{R}_v$, i.e. $G_v$ and $C_v$ are linearly independent, then the intersection $\mathcal{LF}_v \cap \mathcal{LR}_v$ is the orthogonal complement of $\mathrm{span}(C_v, G_v)$, and thus $\mathcal{F}_v \cap \mathcal{R}_v$ is an affine subspace of dimension $n-2$.
\end{itemize}
Let us define the vector field
\begin{equation} \label{H-def}
H_v := \left[ \partial_1 F(v), \dots, \partial_n F(v) \right].
\end{equation}
An element $v \in T_p^{\circ} M$ is a \textbf{horizontal contact point} of the metrics if $H_v$ is the zero vector. It can be easily seen that if $T_p M$ is a vertical contact tangent space, i. e. all of its non-zero elements are vertical contact, then the Finsler metric is a scalar multiple of $\gamma$ at the point $p\in M$. The quadratic indicatrix of a generalized Berwald metric at a single point means quadratic indicatrices at all points of the (connected) base manifold because the tangent spaces are related by linear isometries due to the parallel transports with respect to the compatible linear connection with the Finsler metric. Therefore such a generalized Berwald manifold reduces to a Riemannian manifold. At a horizontal contact point $v\in T_pM$, equations of CEQ are homogeneous. If $T_p M$ is a horizontal contact tangent space, i. e. all of its non-zero elements are horizontal contact, then $T \equiv 0$ is a solution of CEQ at $p\in M$.
\subsection{A useful family of vector fields} Let us define the vector fields
\begin{equation} \label{rowvector}
f_i(v) := \left[ f_{i1}(v), f_{i2}(v), \dots, f_{in}(v) \right]^T \hspace{1cm} (i=1,\dots, n)
\end{equation}
on $T_p^{\circ} M$ to help in proving some elementary properties and solving CEQ.
\begin{lemm} \label{rowvectorspan} For any $v \in T_p^{\circ} M$, we have $\mathrm{span}(f_1(v), \dots, f_n(v)) \subseteq \mathrm{span}(G_v, C_v)$.
\end{lemm}
\begin{proof} Observe that $f_i$ can be written as
\begin{equation} \label{rowvector-lincomb}
f_i = \begin{bmatrix} f_{i1} \\ f_{i2} \\ \vdots \\ f_{in}\end{bmatrix} =
\begin{bmatrix} y^i \dot{\partial}_1 F - y^1 \dot{\partial}_i F \\ y^i \dot{\partial}_2 F - y^2 \dot{\partial}_i F \\ \vdots \\ y^i \dot{\partial}_n F - y^n \dot{\partial}_i F \end{bmatrix} =
y^i \begin{bmatrix} \dot{\partial}_1 F \\ \dot{\partial}_2 F \\ \vdots \\ \dot{\partial}_n F \end{bmatrix} - \dot{\partial}_i F \begin{bmatrix}
y^1 \\ y^2 \\ \vdots \\ y^n \end{bmatrix},
\end{equation}
i.e. $f_i = y^i \cdot G - \dot{\partial}_i F \cdot C$.
\end{proof}
\begin{lemm} \label{rowvector-vertcont} At a vertical contact point $v \in T_p^{\circ} M$, $f_i(v)=0$ and CEQ takes the form $0=\partial_i F(v)$ $(i=1, \ldots, n)$.
\end{lemm}
\begin{proof} If $v$ is vertically contact, then $G_v = \lambda C_v$ for some nonzero $\lambda \in \mathbb{R}$. Substituting into \eqref{rowvector-lincomb}, we get
\[ f_i(v) = v^i \cdot \lambda C_v - \lambda v^i \cdot C_v = 0 \hspace{1cm} (i=1, \dots, n), \]
i.e. the coordinates of $f_i$ are all zero at $v$ and the reformulation \eqref{CEQ-general} of CEQ implies the statement.
\end{proof}
\begin{cor} In order for CEQ to have a solution, all vertically contact points must be horizontal contact.
\end{cor}
\begin{lemm} \label{rowvector-notvertcont} At a not vertically contact $v \in T_p^{\circ} M$ there are indices such that $f_{ij}(v)\neq 0$ and the vectors $f_i$ and $f_j$ are linearly independent over some neighborhood $U$ of $v$ in $T_pM$.
\end{lemm}
\begin{proof} Suppose that all the $f_{ij}(v)$, and consequently, all the vectors $f_i(v)$ are zero. Since $v$ is not vertically contact, $G_v$ and $C_v$ are linearly independent. In particular, neither of them is the zero vector, so one of their coordinates is nonzero, meaning that for some index $i$, \eqref{rowvector-lincomb} gives the zero vector as a linear combination of the independent vectors $C_v$ and $G_v$ with nonzero coefficients. This is a contradiction, so there must be an $f_{ij}(v)$, and thus two vectors $f_i(v)$ and $f_j(v)$ different from zero. Since the matrix
\[ \begin{bmatrix} f_i \\ f_j \end{bmatrix} =
\begin{bmatrix} f_{i1} & \dots & 0 & \dots & f_{ij} & \dots & f_{in} \\ f_{j1} & \dots & f_{ji} & \dots & 0 & \dots & f_{jn}\\ \end{bmatrix} \]
has rank 2 at $v$ (choose the $i$-th and $j$-th columns), they are linearly independent at $v$ and the same is true at the points of some adequately small neighborhood of $v$ in $T_pM$ by a continuity argument.
\end{proof}
\begin{cor} \label{rowvector-ultimate} At a point $v \in T_p^{\circ} M$ the vectors defined by \eqref{rowvector} span the subspace
\[ \mathrm{span}(f_1(v), \dots, f_n(v)) = \left\{\begin{array}{cl}
\{ \bm{0} \} & \text{if} \ v \ \text{is vertically contact}, \\
\mathrm{span}(G_v, C_v) & \text{if} \ v \ \text{is not vertically contact}.
\end{array}\right. \]
\end{cor}
\begin{proof} For any $v \in T_p^{\circ} M$, we have $\mathrm{span}(f_1(v), \dots, f_n(v)) \subseteq \mathrm{span}(G_v, C_v)$ by Lemma \ref{rowvectorspan}. If $v$ is vertically contact, all the vectors $f_i(v)$ are zero according to Lemma \ref{rowvector-vertcont}. If not, there are 2 independent vectors among them according to Lemma \ref{rowvector-notvertcont}, thus generating the whole $\mathrm{span}(G_v, C_v)$.
\end{proof}
\begin{lemm} \label{rowvector-basis} At a not vertically contact $v \in T_p^{\circ} M$, let us choose indices $i \neq j$ such that $f_{ij}(v)\neq 0$. Then $(f_i, f_j)$ is a basis of $\mathrm{span}(G, C)$ over some neighborhood $U$ of $v$ in $T_pM$ and
\begin{equation}\label{rowvector-lincomb2}
f_k = \dfrac{f_{kj}}{f_{ij}} \cdot f_i + \dfrac{f_{ik}}{f_{ij}} \cdot f_j \hspace{1cm} (k=1, \dots, n)
\end{equation}
at any point of $U$.
\end{lemm}
\begin{proof} By Lemma \ref{rowvector-notvertcont}, we know that $(f_i, f_j)$ is a basis of $\mathrm{span}(G, C)$ at the points of $U$. Let us choose an index $k \in \{ 1, \dots, n\}$ and write $f_k = \lambda_1 f_i + \lambda_2 f_j$. By \eqref{rowvector-lincomb}, we can write that
\[f_k= y^k \cdot G - \dot{\partial}_k F \cdot C = \lambda_1 \left( y^i \cdot G - \dot{\partial}_i F \cdot C \right) + \lambda_2 \left( y^j \cdot G - \dot{\partial}_j F \cdot C \right). \]
By comparing the coefficients in the basis $(G, C)$,
\[ \begin{bmatrix} y^k \\ \dot{\partial}_k F \end{bmatrix} = \begin{bmatrix} y^i & y^j \\ \dot{\partial}_i F & \dot{\partial}_j F \end{bmatrix} \cdot \begin{bmatrix} \lambda_1 \\ \lambda_2
\end{bmatrix} \]
and, consequently,
\[ \begin{bmatrix} \lambda_1 \\ \lambda_2
\end{bmatrix} = \dfrac{1}{f_{ij}} \begin{bmatrix} \dot{\partial}_j F & -y^j \\ -\dot{\partial}_i F & y^i \end{bmatrix} \cdot \begin{bmatrix} y^k \\ \dot{\partial}_k F \end{bmatrix} = \dfrac{1}{f_{ij}} \begin{bmatrix} y^k \dot{\partial}_j F - y^j \dot{\partial}_k F \\ y^i \dot{\partial}_k F - y^k \dot{\partial}_i F \end{bmatrix}. \qedhere \]
\end{proof}
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\section{The semi-symmetric compatible linear connection and its unicity}
Although there are some theoretical results for the solvability of the compatibility equations (monochromatic Finsler metrics \cite{BM}, extremal compatible linear connections and algorithmic solutions \cite{V14}), it is very hard to solve in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique. Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor.
\begin{defi} A linear connection is called \textbf{semi-symmetric} if its torsion tensor can be written as
\begin{equation}\label{semi-symm-tors}
T(X,Y) = \rho(Y) X - \rho(X)Y
\end{equation}
for some differential 1-form $\rho$ on the base manifold.
\end{defi}
It is proved \cite{V10} that a semi-symmetric compatible linear connection must be uniquely determined.
\begin{thm}\label{original} \emph{\cite{V10}} A non-Riemannian Finsler manifold admits at most one semi-symmetric compatible linear connection.
\end{thm}
The original proof is based on averaging in the sense that the 1-form $\rho$ can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. In what follows we present a new proof for the unicity result by using linear algebra and some basic facts about convex bodies. We present an explicit formula for the solution without integration. The method has a new contribution to the problem as well: necessary conditions of the solvability are formulated in terms of intrinsic equations without unknown quantities. They are sufficient if and only if the solution depends only on the position.
\subsection{The proof of Theorem \ref{original}} Since
$$T(\partial/\partial u^i, \partial/\partial u^j) = \rho(\partial/\partial u^j) \partial/\partial u^i - \rho(\partial/\partial u^i) \partial/\partial u^j=$$
$$= \rho_j \partial/\partial u^i - \rho_i \partial/\partial u^j = \left( \delta_i^k \rho_j - \delta_j^k \rho_i \right) \partial/\partial u^k,$$
the torsion components are
\begin{equation} \label{semi-symm-torscomp}
T_{ij}^k = \delta_i^k \rho_j - \delta_j^k \rho_i.
\end{equation}
In particular, all torsion components with 3 different indices are zero, and
\[ T_{ij}^i = \delta_i^i \rho_j - \delta_j^i \rho_i =\rho_j \hspace{1cm }(j \neq i). \]
Substituting the torsion components into the general form \eqref{CEQ-general} of CEQ at a point $p$, it takes the (matrix) form
\begin{equation} \label{CEQ}
{\tabulinesep=2pt
\begin{tabu} to .5\textwidth {|ccccc|c|}
\hline
\rho_1 & \rho_2 & \rho_3 & \cdots & \rho_n & \text{RHS} \\
\hline
0 & f_{12} & f_{13} & \cdots & f_{1n} & -\partial_1 F \\
f_{21} & 0 & f_{23} & \cdots & f_{2n} & -\partial_2 F \\
f_{31} & f_{32} & 0 & \cdots & f_{3n} & -\partial_3 F \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
f_{n1} & f_{n2} & f_{n3} & \cdots & 0 & -\partial_n F \\
\hline
\end{tabu}}
\end{equation}
The problem is to solve \eqref{CEQ} for $\rho_1, \dots, \rho_n$, considered as the coordinates of a vector $\rho \in T_p M$, as $v$ ranges over $T_p^{\circ} M$. To prove Theorem \ref{original} it is enough to consider the homogeneous version H-CEQ with vanishing right hand side of \eqref{CEQ}. We are going to verify that the only solution of H-CEQ is $\rho_1=\dots=\rho_n=0$. Since the rows of the matrix on the left hand side are exactly the vectors $f_1, \dots, f_n$ defined in \eqref{rowvector}, solving H-CEQ at a fixed element $v$ means finding the orthogonal complement of $\mathrm{span}(f_1(v), \dots, f_n(v))$. By Corollary \ref{rowvector-ultimate},
\begin{itemize}
\item it is $T_p M$ for any vertically contact element $v$,
\item it is the orthogonal complement of $\mathrm{span}(G_v, C_v)$ for any not vertically contact element $v$, i.e. the intersection $\mathcal{LF}_v \cap \mathcal{LR}_v$ of the linear tangent hyperplanes of the Finslerian and Riemannian spheres.
\end{itemize}
The solution of H-CEQ at the point $p$ is the intersection of all the solution spaces as the element $v$ ranges over $T_p^{\circ} M$. Note that the homogeneity of the coefficients imply that it is enough to consider the intersection of all the solution spaces as the element $v$ ranges over the Finslerian (or the Riemannian) unit sphere.
\begin{itemize}
\item If all the elements of $T_p M$ are vertically contact and the Finsler manifold admits a compatible (semi-symmetric) linear connection $\nabla$, then it is a Riemannian manifold because the linear isometries via the parallel transports with respect to $\nabla$ extend the quadratic Finslerian (esp. Riemannian) indicatrix at the point $p$ to the entire (connected) manifold.
\item If there is a not vertically contact element $v$, then, by a continuity argument, we can consider a neighborhood $U\subseteq T_pM$ containing not vertically contact elements. For the solution vector $\rho$ we have
\[ \rho \in \bigcap_{v \in U} \left( \mathcal{LF}_v \cap \mathcal{LR}_v \right) \subseteq \big(\bigcap_{v \in U} \mathcal{LF}_v \big) \cap \big(\bigcap_{v \in U} \mathcal{LR}_v \big). \]
It is clear that the right hand side contains only the zero vector because the normal vectors at the points of any open set on the boundary of a Euclidean sphere (or any smooth strictly convex body) span the entire space. Therefore $\rho=0$ is the only solution of H-CEQ at $p$ and, consequently, CEQ admits at most one solution for the components of the torsion tensor of a semi-symmetric linear connection point by point. \qedhere
\end{itemize}
\subsection{Intrinsic equations and $v$-solvability of CEQ} In this section we investigate \eqref{CEQ} evaluated at non-zero tangent vectors in $T_pM$:
\begin{equation} \label{v-CEQ}
{\tabulinesep=2pt
\begin{tabu} to .5\textwidth {|ccccc|c|}
\hline
\rho_1 & \rho_2 & \rho_3 & \cdots & \rho_n & \text{RHS} \\
\hline
0 & f_{12}(v) & f_{13}(v) & \cdots & f_{1n}(v) & -\partial_1 F(v) \\
f_{21}(v) & 0 & f_{23}(v) & \cdots & f_{2n}(v) & -\partial_2 F(v) \\
f_{31} (v)& f_{32}(v) & 0 & \cdots & f_{3n}(v) & -\partial_3 F(v) \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
f_{n1} (v)& f_{n2}(v) & f_{n3} (v)& \cdots & 0 & -\partial_n F(v) \\
\hline
\end{tabu}}
\end{equation}
\begin{defi} The system of the compatibility equations is called \textbf{$v$-solvable} for $\rho$ at the point $p\in M$ if \eqref{v-CEQ} is solvable for any non-zero element $v\in T_pM$.
\end{defi}
\begin{rem}
The system of the compatibility equations is $v$-solvable if and only if all vertical contact vectors are horizontal contact. It is an obvious necessary condition because the coefficient matrix of CEQ is zero at a vertically contact point and the system must be homogeneous with vanishing horizontal derivatives of the Finsler metric with respect to the compatible linear connection. The sufficiency is based on the idea of the extremal compatible linear connection \cite{V14}. The extremal solution typically depends on the reference element $v\in T_pM$ but does not take a decomposable form in general. Therefore $v$-solvability for $\rho$ needs additional conditions. Using Corollary \ref{rowvector-ultimate} and basic linear algebra we can formulate the following characterizations of $v$-solvability in case of semi-symmetric compatible linear connections.
\end{rem}
\begin{lemm} \label{v-solvability1} The system of the compatibility equations is $v$-solvable for $\rho$ at the point $p\in M$ if and only if the following are satisfied:
\begin{itemize}
\item all vertically contact elements are also horizontal contact in $T_pM$ and
\item the rank of the augmented matrix of system \eqref{v-CEQ} is 2 at all not vertically contact elements $v\in T_pM$, i.e. $H_v \in \mathrm{span}(f_1(v), \dots, f_n(v))$, where the vector $H_v$ is defined by formula \eqref{H-def}.
\end{itemize}
\end{lemm}
\begin{prop} \label{v-solvability2} The system of the compatibility equations is $v$-solvable for $\rho$ at the point $p\in M$ if and only if the following are satisfied:
\begin{itemize}
\item all vertically contact elements are also horizontal contact in $T_pM$ and
\item for any triplets of distinct indices $i,j,k$ we have
\begin{equation} \label{solv-symm}
f_{ij}(v) \, \partial_k F (v)+ f_{jk}(v) \, \partial_i F (v)+ f_{ki}(v) \, \partial_j F(v) = 0
\end{equation}
provided that $f_{ij}(v)\neq 0$.
\end{itemize}
\end{prop}
\begin{proof} If $v$ is vertically contact, \eqref{solv-symm} stands trivially. Otherwise we are going to show that it is equivalent to the augmented matrix having rank 2. Suppose that $v$ is not vertically contact and choose indices $i\neq j$ such that $f_{ij}(v)\neq 0$ and $(f_i(v), f_j(v))$ is a basis of $\mathrm{span}(G_v, C_v)$. By Lemma \ref{rowvector-basis}, if $k\neq i, j$ then
\[ f_k = \dfrac{f_{kj}}{f_{ij}} \cdot f_i + \dfrac{f_{ik}}{f_{ij}} \cdot f_j \hspace{1cm} (k=1, \dots, n). \]
In other words we can eliminate the $k$-th row for any $k\neq i, j$. The elimination must also yield zeroes on the right-hand side of \eqref{v-CEQ} to have a solution, i.e. for $k \neq i, j$ we must have
\[ \begin{array}{c}
-\partial_k F - \dfrac{f_{kj}}{f_{ij}} \cdot (-\partial_i F) - \dfrac{f_{ik}}{f_{ij}} \cdot (-\partial_j F) = 0 \\[12pt]
f_{ij} \, \partial_k F - f_{kj} \, \partial_i F - f_{ik} \, \partial_j F = 0.
\end{array} \]
Equation \eqref{solv-symm} follows by interchanging the indices in $f_{kj}$ and $f_{ik}$. Since the coefficient matrix is of maximal rank $2$, the augmented matrix is of maximal rank $2$ as the extension of the coefficient matrix with $-\partial_i F$ and $-\partial_j F$ in the corresponding rows.
\end{proof}
\begin{rem} Equations \eqref{solv-symm} do not contain unknown quantities. They are intrinsic conditions of the solvability. Taking $f_{ij}(v)\neq 0$ for some fixed indices $i$ and $j$ at a not vertically contact element, they provide $n-2$ equations to be automatically satisfied because $k=1, \ldots, n$, but $k\neq i, j$. The missing equations are
\begin{equation} \label{CEQ-elim}
\left.\begin{array}{rcl}
\dotprod{f_i,\rho} & = & -\partial_i F \\[4pt]
\dotprod{f_j,\rho} & = & -\partial_j F \\
\end{array}\right\},
\end{equation}
where $\dotprod{f_i,\rho}$ and $ \dotprod{f_j,\rho}$ stand for the inner product at $p\in M$ coming from the Riemannian metric $\gamma$. They provide the only possible solution $\rho$ in an explicit form.
\end{rem}
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\subsection{The only possible solution of CEQ at the point $p\in M$} \label{lastsect} Recall that if the tangent space at $p\in M$ contains only vertically contact non-zero elements (vertically contact tangent space) and the Finsler manifold admits a compatible (semi-symmetric) linear connection $\nabla$, then it is a Riemannian manifold because the linear isometries via the parallel transports with respect to $\nabla$ extend the quadratic Finslerian (esp. Riemannian) indicatrix at the point $p$ to the entire (connected) manifold. Therefore we present the solution of CEQ in the generic case of non-Riemannian Finsler manifolds. Let us choose a not vertically contact element $v \in T_p^{\circ} M$ and indices $i\neq j$ with $f_{ij}(v)\neq 0$, i.e. $f_i, f_j$ are linearly independent over some neighborhood $U$ of $v$ in $T_pM$. Using \eqref{rowvector-lincomb}, the eliminated form \eqref{CEQ-elim} of CEQ gives that
\[ {\arraycolsep=1pt \left.\begin{array}{ccccl}
y^i \dotprod{G,\rho} & - & \dot{\partial}_i F \dotprod{C,\rho} & = & -\partial_i F \\[5pt]
y^j \dotprod{G,\rho} & - & \dot{\partial}_j F \dotprod{C,\rho} & = & -\partial_j F \\
\end{array}\right\} } \Longleftrightarrow
\begin{bmatrix} y^i & -\dot{\partial}_i F \\[5pt] y^j & -\dot{\partial}_j F \end{bmatrix} \cdot \begin{bmatrix} \dotprod{G,\rho} \\[5pt] \dotprod{C,\rho} \end{bmatrix} = \begin{bmatrix}
-\partial_i F \\[5pt] -\partial_j F
\end{bmatrix}, \]
and, consequently,
\[ \begin{bmatrix} \dotprod{G,\rho} \\[5pt] \dotprod{C,\rho} \end{bmatrix} = \dfrac{1}{f_{ji}}\begin{bmatrix}
-\dot{\partial}_j F &\dot{\partial}_i F \\[5pt] -y^j & y^i
\end{bmatrix} \cdot \begin{bmatrix}
-\partial_i F \\[5pt] -\partial_j F
\end{bmatrix}. \]
We are going to concentrate on the second row
\begin{equation} \label{eq-c}
\dotprod{C,\rho} = \dfrac{1}{f_{ji}} \left( y^j \partial_i F - y^i \partial_j F \right) =: \dfrac{f_{ji}^h}{f_{ji}}
\end{equation}
at the points of the open neighborhood $U$ around $v$. Let us choose a value $\varepsilon>0$ such that all the elements
\[ \begin{array}{rcccl}
w_1 & := & v-\varepsilon \cdot \partial/\partial u^1(p) & = & [v^1-\varepsilon, v^2, v^3, \dots, v^n] \\
w_2 & := & v-\varepsilon \cdot \partial/\partial u^2(p) &= & [v^1, v^2-\varepsilon, v^3, \dots, v^n] \\
& \vdots &&& \\
w_n & := & v-\varepsilon \cdot \partial/\partial u^n(p) & = & [v^1, v^2, v^3, \dots, v^n-\varepsilon]
\end{array} \]
are contained in $U$. Then \eqref{eq-c} implies the system
\begin{equation} \label{ceq-indep} \begin{bmatrix}
v^1-\varepsilon & v^2 & v^3 & \cdots & v^n \\[4pt]
v^1 & v^2-\varepsilon & v^3 & \dots & v^n \\[4pt]
\vdots & \vdots & \vdots & \ddots & \vdots \\[4pt]
v^1 & v^2 & v^3 & \dots & v^n-\varepsilon
\end{bmatrix} \cdot \begin{bmatrix} \rho_1 \\[4pt] \rho_2 \\[4pt] \vdots \\[4pt] \rho_n \end{bmatrix} = \begin{bmatrix}
f^h_{ji}/f_{ji}(w_1) \\[4pt] f^h_{ji}/f_{ji}(w_2) \\[4pt] \vdots \\[4pt] f^h_{ji}/f_{ji}(w_n) \end{bmatrix}
\end{equation}
of linear equations. Using the notation
\[ V:= \begin{bmatrix}
v^1 & v^2 & \cdots & v^n \\
\vdots & \vdots & \ddots & \vdots \\
v^1 & v^2 & \cdots & v^n
\end{bmatrix}, \]
we have to investigate the regularity of the matrix $V-\varepsilon I$, where $I$ denotes the identity matrix of the same type as $V$.
\begin{lemm} The matrix $V-\varepsilon I$ is regular if and only if $\varepsilon \notin \{ 0, \widetilde{v}:=v^1+\dots+v^n \}$.
\end{lemm}
\begin{proof} The determinant $\mathrm{det}(V-\varepsilon I)$ is the characteristic polynomial of $V$ with $\varepsilon$ as the variable. It is zero if and only if $\varepsilon$ is an eigenvalue of $V$. Consider the transpose of $V$ as the matrix of a linear transformation $\varphi$ (the eigenvalues are the same as those of $V$). Since the image of $\varphi$ is the line generated by $v$, it follows that $\varphi$ has a kernel of dimension $n-1$ and $v$ is an eigenvector corresponding to the eigenvalue $\widetilde{v}:=v^1+\dots+v^n$ because of
\[ \varphi(v)= \begin{bmatrix}
v^1 & \cdots & v^1 \\ \vdots & \ddots & \vdots \\ v^n & \cdots & v^n \end{bmatrix} \cdot \begin{bmatrix} v^1 \\ \vdots \\ v^n\end{bmatrix} = \begin{bmatrix}
v^1 (v^1+\dots+v^n) \\ \vdots \\v^n(v^1+\dots+v^n)
\end{bmatrix} = \widetilde{v} \cdot v. \qedhere\]
\end{proof}
\begin{lemm} Choosing $\varepsilon \notin \{ 0, \widetilde{v}:=v^1+\dots+v^n\}$, we have
\[ \left[ V-\varepsilon I \right]^{-1} = \dfrac{1}{(\widetilde{v}-\varepsilon)\varepsilon} \left[ V-(\widetilde{v}-\varepsilon) I \right]. \]
\end{lemm}
\begin{proof} We shall prove the formula
\[ \left[V-\varepsilon I\right] \cdot \left[ V-(\widetilde{v}-\varepsilon)I\right] = (\widetilde{v}-\varepsilon)\varepsilon I.\]
Rearranging the left-hand side,
\[ V \cdot V - (\widetilde{v}-\varepsilon)V-\varepsilon V + \varepsilon (\widetilde{v}-\varepsilon)I=(\widetilde{v}-\varepsilon)\varepsilon I \]
because of $V\cdot V=\widetilde{v} \cdot V$.
\end{proof}
Returning to \eqref{ceq-indep},
\[\begin{bmatrix} \rho_1 \\[4pt] \rho_2 \\[4pt] \vdots \\[4pt] \rho_n \end{bmatrix} = \dfrac{1}{(\widetilde{v}-\varepsilon)\varepsilon} \begin{bmatrix}
v^1+\varepsilon-\widetilde{v} & v^2 & \cdots & v^n \\[4pt]
v^1 & v^2+\varepsilon-\widetilde{v} & \dots & v^n \\[4pt]
\vdots & \vdots & \ddots & \vdots \\[4pt]
v^1 & v^2 & \dots & v^n+\varepsilon-\widetilde{v}
\end{bmatrix} \cdot \begin{bmatrix}
f^h_{ji}/f_{ji}(w_1) \\[4pt] f^h_{ji}/f_{ji}(w_2) \\[4pt] \vdots \\[4pt] f^h_{ji}/f_{ji}(w_n) \end{bmatrix}. \]
\begin{thm} If a non-Riemannian Finsler manifold admits a semi-symmetric compatible linear connection, then the only possible values of the components $\rho_k$ in formula \eqref{semi-symm-torscomp} for its torsion at the point $p\in M$ are
\begin{equation}
\rho_k = \dfrac{1}{\varepsilon} \left( \dfrac{1}{\widetilde{v}-\varepsilon} \sum_{l=1}^n v^l \dfrac{f^h_{ji}}{f_{ji}}(w_l) - \dfrac{f^h_{ji}}{f_{ji}}(w_k) \right),
\end{equation}
where,
\begin{itemize}
\item $v=[v^1, \dots, v^n]$ is a not vertically contact vector in $T^{\circ}_p M$,
\item $\varepsilon \notin \{ 0, \widetilde{v}:=v^1+\dots+v^n\}$ is a radius of a closed ball around $v$ in $T_pM$ all of whose elements are also not vertically contact,
\end{itemize}
\begin{itemize}[itemsep=6pt]
\item $w_i=[v^1, \dots, v^{i-1}, v^i-\varepsilon, v^{i+1}, \dots, v^n]$, $i=1, \ldots, n$,
\item $f^h_{ji}= y^j \partial_i F - y^i \partial_j F = y^j \dfrac{\partial F}{\partial x^i} - y^i \dfrac{\partial F}{\partial x^j}$,
\item $f_{ji}= y^j \dot{\partial}_i F - y^i \dot{\partial}_j F = y^j \dfrac{\partial F}{\partial y^i} - y^i \dfrac{\partial F}{\partial y^j}$ and
\item the coordinates on the base manifold form a normal coordinate system with respect to the averaged Riemannian metric $\gamma$ around the point $p\in M$.
\end{itemize}
\end{thm}
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\section{Acknowledgments}
Márk Oláh is supported by the UNKP-21-3 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund of Hungary.
\end{document}
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\begin{document}
\maketitle
\newif\ifarx
\arxfalse
\begin{abstract}
In the field of reinforcement learning (RL), agents are often tasked with solving a variety of problems differing only in their reward functions.
In order to quickly obtain solutions to unseen problems with new reward functions, a popular approach involves functional composition of previously solved tasks.
However, previous work using such functional composition has primarily focused on specific instances of composition functions whose limiting assumptions allow for exact zero-shot composition.
Our work unifies these examples and provides a more general framework for compositionality in both standard and entropy-regularized RL.
We find that, for a broad class of functions, the optimal solution for the composite task of interest can be related to the known primitive task solutions. Specifically, we present double-sided inequalities relating the optimal composite value function to the value functions for the primitive tasks. We also show that the regret of using a zero-shot policy can be bounded for this class of functions.
The derived bounds can be used to develop clipping approaches for reducing uncertainty during training, allowing agents to quickly adapt to new tasks.
\end{abstract}
\mathcal{S}ection{Introduction}\label{sec:intro}
Reinforcement learning has seen great success recently, but still suffers from poor sample complexity and task generalization.
Generalizing and transferring domain knowledge to similar tasks remains a major challenge in the field.
To combat this, different methods of transfer learning have been proposed; such as the option framework \citep{sutton_4room, barreto2019option}, successor features \citep{dayan1993improving, barreto_sf, hunt_diverg, nemecek}, and functional composition \citep{Todorov, Haarnoja2018, peng_MCP, boolean_stoch, vanNiekerk}. In this work, we focus on the latter method of ``compositionality'' for transfer learning.
Research in compositionality has focused on the development of approaches to combine previously learned optimal behaviors to obtain solutions to new tasks. In the process, many instances of functional composition in the literature have required limiting assumptions on the dynamics and allowable class of reward functions (goal-based rewards in \citep{boolean}) in order to derive exact results. Furthermore, previous work has focused on isolated examples of particular functions in either standard or entropy-regularized RL and a framework for studying a general class of composition functions without limiting assumptions is currently lacking. One of the main contributions of our work is to provide a unifying general framework to study compositionality in reinforcement learning.
In our approach, we focus on ``primitive'' tasks which differ only in their associated reward functions.
More specifically, we consider those downstream tasks whose reward functions can be written as a global function of the known source tasks' reward functions.
To maintain generality, we do not assume that transition dynamics are deterministic. We also do not assume that reward functions are limited to the goal-based setting, in which there are a limited number of absorbing ``goal'' states \citep{Todorov, vanNiekerk} defining the primitive task. Given the generality of this setting, we cannot expect to obtain exact solutions for compositions as in prior work. Instead, we provide a class of functions which can be used to obtain approximate solutions and bounds on the corresponding downstream tasks.
Given the solutions to a set of primitive tasks, we show that it is possible to leverage such information to obtain approximate solutions for a large class of compositely-defined tasks.
To do so, we relate the solution of the downstream composite task to the solved primitive (source) tasks. Specifically, we derive relations on the optimal value function of interest. From such a relation, a ``zero-shot'' (i.e. not requiring further training) policy can be extracted for use in the composite domain of interest.
We then show that the suboptimality (regret) of this zero-shot policy is upper bounded.
Our results support the idea that RL agents can focus on obtaining domain knowledge for simpler tasks, and later use this knowledge to effectively solve more difficult tasks. The primary contributions of the present work are as follows:
\textbf{Main contributions}
\begin{itemize}
\item Establishing a general framework for analyzing reward transformations and compositions for the case of stochastic dynamics, globally varying reward structures, and continuing tasks.
\item Derivation of bounds on the respective optimal value functions for transformed and composite tasks.
\item Demonstration of zero-shot approximate solutions and value-based clipping of new tasks based on the known optimal solutions for primitive tasks.
\end{itemize}
\mathcal{S}ection{Background}
In this work, we analyze the case of finite, discrete state and action spaces, with the Markov Decision Process (MDP) model \citep{suttonBook}. Let $\Delta(X)$ represent the set of probability distributions over $X$. Then the MDP is represented as a tuple $\mathcal{T}=\langle \mathcal{S},\mathcal{A},p, \mu, r,\gamma \rangle$ where $\mathcal{S}$ is the set of available states; $\mathcal{A}$ is the set of possible actions; $p: \mathcal{S} \times \mathcal{A} \to \Delta(\mathcal{S})$ is a transition function describing the system dynamics; $\mu \in \Delta(\mathcal{S})$ is the initial state distribution; $r: \mathcal{S} \times \mathcal{A} \to \mathbb{R}$ is a (bounded) reward function which associates a reward (or cost) with each state-action pair; and $\gamma \in (0,1)$ is a discount factor which discounts future rewards and guarantees the convergence of total reward for infinitely long trajectories ($T \to \infty$).
In ``standard'' (un-regularized) RL, the agent maximizes an objective function which is the expected future reward:
\begin{equation}\label{eq:std_rl_obj}
J(\pi) = \underset{\tau \mathcal{S}im{} p, \pi}{\mathbb{E}}
\left[ \mathcal{S}um_{t=0}^{\infty} \gamma^{t} r(s_t,a_t) \right].
\end{equation}
This objective has since been generalized for the setting of entropy-regularized RL \citep{ZiebartThesis, LevineTutorial}, which augments the standard RL objective in Eq. \eqref{eq:std_rl_obj} by appending an entropic regularization term for the policy:
\begin{equation}
J(\pi)=
\underset{\tau \mathcal{S}im{} p, \pi}{\mathbb{E}}
\left[ \mathcal{S}um_{t=0}^{\infty} \gamma^{t} \left( r(s_t,a_t) - f\left(r(s,a)\right)ac{1}{\beta}
\log\left(f\left(r(s,a)\right)ac{\pi(a_t|s_t)}{\pi_0(a_t|s_t)} \right) \right) \right]
\label{eq:ent_rl_obj}
\end{equation}
where $\pi_0: \mathcal{S} \to \Delta(\mathcal{A})$ is the fixed prior policy. The inverse temperature parameter, $\beta \in (0, \infty)$, regulates the contribution of entropic costs relative to the accumulated rewards.
The additional entropic control cost discourages the agent from choosing policies that deviate too much from the prior policy. Importantly, entropy-regularized MDPs lead to stochastic optimal policies that are provably robust to perturbations of rewards and dynamics \citep{eysenbach}; making them a more suitable choice for real-world problems.
By ``solution to the RL problem'', we hereon refer to the corresponding \textit{optimal} action-value function $Q(s,a)$ from which an \textit{optimal} control policy can be derived: $\pi(s) \in \text{argmax}_a Q(s,a)$ for standard RL; and $\pi(a|s) \propto \exp(\beta Q(s,a))$ for entropy-regularized RL. Note that these definitions are consistent with the limit $\beta \to \infty$ in which the standard RL objective is recovered from Eq.~\eqref{eq:ent_rl_obj}. For both standard and entropy-regularized RL, the optimal $Q$-function can be obtained by iterating a recursive Bellman equation. For standard RL, the Bellman optimality equation is given by \citep{suttonBook}:
\begin{equation}
Q(s,a) = r(s,a) + \gamma \mathbb{E}_{s' \mathcal{S}im{} p(\cdot|s,a)} \max_{a'} \left( Q(s',a') \right)
\end{equation}
The entropy term in the objective function for entropy-regularized RL modifies the previous optimality equation to \citep{ZiebartThesis, Haarnoja_SAC}:
\begin{equation}
Q(s,a) = r(s,a) + f\left(r(s,a)\right)ac{\gamma}{\beta} \E_{s'} \log \mathbb{E}_{a' \mathcal{S}im{} \pi_0(\cdot|s')} e^{ \beta Q(s',a') }
\end{equation}
One of the primary goals of research in compositionality and transfer learning is deriving results for the optimal $Q$ function for new tasks based on the known optimal $Q$ function(s) for primitive tasks. There exist many forms of composition and transfer learning in RL, as discussed by \cite{taylor_survey}. In this paper, we focus on the case of concurrent skill composition by a single agent as opposed to an options-based approach \citep{sutton_4room}, or other hierarchical compositions \citep{hierarchy, saxe_hier}. We elaborate on this point with the definitions below.
To formalize our problem setup, we adopt the relevant definitions provided by \citep{ourAAAI}:
\begin{definition}
A \textbf{primitive RL task} is specified by an MDP $\mathcal{T} = \langle \mathcal{S},\mathcal{A},p,r, \gamma \rangle$ for which the optimal $Q$ function is known.
\end{definition}
In this work, we focus on primitive tasks with general reward functions, i.e. including both goal-based (sparse rewards on absorbing sets such as in the linearly solvable MDP framework of \citep{Todorov, boolean, vanNiekerk}) \textit{and} arbitrary reward landscapes \citep{Haarnoja2018}.
\begin{definition}
The \textbf{transformation of an RL task} is defined by its (bounded and continuous) transformation function: $f: \mathbb{R} \to \mathbb{R}$ and a primitive task $\mathcal{T}$. The transformed task shares the same states, actions, dynamics, and discount factor as $\mathcal{T}$ but has a transformed reward function $\widetilde{r}(s,a) = f(r(s,a))$.
\end{definition}
\begin{definition}
The \textbf{composition of $M$ RL tasks} is defined by a (bounded and continuous) function $F: \mathbb{R}^M \to \mathbb{R}$ and a set of primitive tasks $\{\mathcal{T}^{(k)}\}$. The \textbf{composite} RL task is defined by a new reward function $\widetilde{r}(s,a)~=~F(\{r^{(k)}(s,a)\})$; and shares the same states, actions, dynamics, and discount factor as all the primitive RL tasks.
\end{definition}
Finally, we define the Transfer Library, the set of functions which obey the hypotheses of our subsequent results (see Sections \ref{sec:std_lemmas} and \ref{sec:entropy-regularized_lemmas}). This definition serves to facilitate the general discussion of results obtained.
\begin{definition}
Given a set of primitive tasks $\{\mathcal{T}^{(k)}\}$, the \textbf{Transfer Library}, denoted by $\mathcal{T}L$, is the set of all transformation (or composition, when $M > 1$) functions $f$ which admit double-sided bounds (see Sections \ref{sec:std_lemmas} and \ref{sec:entropy-regularized_lemmas})
on the composite task's optimal $Q$ function ($\widetilde{Q}$).
\end{definition}
Specifically, $\mathcal{T}L = \{f\ | f\ \textrm{satisfies Lemma 4.1 or 4.3}\}$ for standard RL and $\mathcal{T}L = \{f\ | f\ \textrm{satisfies Lemma 5.1 or 5.3}\}$ for entropy-regularized RL.
We have empirically found (cf. Fig.~\ref{fig:tasks} and Supplementary Material) that by using the derived bounds for the optimal value function, the agent can learn the optimal policy more efficiently for tasks in the Transfer Library.
\mathcal{S}ection{Previous Work}\label{sec:prev_work}
There is much previous work concerning compositionality and transfer learning in reinforcement learning. In this section we will give a brief overview by highlighting the work most relevant for the current discussion.
In this work, we focus on value-based composition; rather than policy-based composition \cite{peng_MCP}, features-based composition \citep{barreto_sf}, or hierarchical (e.g. options-based) composition \citep{alver, sutton_4room, barreto2019option}.
Value based methods of composition use the optimal value functions of lower-level or simpler ``primitive'' tasks to derive an approximation (or in some cases exact solution) for the composite task of interest.
In the optimal control framework, \citep{Todorov}
has shown that optimal value functions can be composed exactly for linearly-solvable MDPs with a $\textrm{LogSumExp}$ or ``soft OR'' composition over primitive tasks; assuming that tasks share the same absorbing set (boundary states). With a similar assumption of the shared absorbing set, \citep{boolean} show that exact optimal value functions for Boolean compositions may be recovered from primitive task solutions; thereby allowing an exponential improvement in knowledge acquisition.
In more recent work, in the context of MaxEnt RL, \cite{Haarnoja2018} have shown that linear convex-weighted compositions in stochastic environments result in a bound on optimal value functions, and the policy extracted from this zero-shot bound is indeed useful for solving the composite task. The same premise of convex-weighted reward structures was studied by \cite{hunt_diverg} where the difference between the bound of \citep{Haarnoja2018} and the optimal value function can itself be learned, effectively tightening the bound until convergence. This notion of a corrective function was subsequently generalized by \cite{ourAAAI} to allow for arbitrary functions of composition in entropy-regularized RL.
Other authors have considered the question of linear task decomposition, for instance \citep{barreto_sf} where a convex weight vector over learned \textit{features} can be calculated to solve the transfer problem over linearly-decomposable reward functions in standard RL. More recent developments on this line of research include \citep{hong2022bilinear} where a more general ``bilinear value decomposition'', conditioned on various goals, is learned. In \citep{kimconstrained}, the authors consider the successor features (SFs) framework of \citep{barreto_sf}, and propose lower and upper bounds on the optimal value function of interest. They show that by replacing standard generalized policy improvement (GPI) with a constrained version which respects their bounds, they are able to transfer knowledge more successfully to future tasks in the successor features framework.
With our reduced assumptions (any constant dynamics, constant discount factor, any rewards) it is not generally possible to solve the transformed or composed tasks based only on primitive knowledge. Nevertheless, we are able to derive bounds on the optimal $Q$-functions in both standard and entropy-regularized RL, from which we can immediately derive policies which fare well in the transformed and composed problem settings. Additionally, we are able to prove that the derived policies have a bounded regret, in a similar form as \cite{Haarnoja2018}'s Theorem 1; but in a more general setting.
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0.90.1
|
To formalize our problem setup, we adopt the relevant definitions provided by \citep{ourAAAI}:
\begin{definition}
A \textbf{primitive RL task} is specified by an MDP $\mathcal{T} = \langle \mathcal{S},\mathcal{A},p,r, \gamma \rangle$ for which the optimal $Q$ function is known.
\end{definition}
In this work, we focus on primitive tasks with general reward functions, i.e. including both goal-based (sparse rewards on absorbing sets such as in the linearly solvable MDP framework of \citep{Todorov, boolean, vanNiekerk}) \textit{and} arbitrary reward landscapes \citep{Haarnoja2018}.
\begin{definition}
The \textbf{transformation of an RL task} is defined by its (bounded and continuous) transformation function: $f: \mathbb{R} \to \mathbb{R}$ and a primitive task $\mathcal{T}$. The transformed task shares the same states, actions, dynamics, and discount factor as $\mathcal{T}$ but has a transformed reward function $\widetilde{r}(s,a) = f(r(s,a))$.
\end{definition}
\begin{definition}
The \textbf{composition of $M$ RL tasks} is defined by a (bounded and continuous) function $F: \mathbb{R}^M \to \mathbb{R}$ and a set of primitive tasks $\{\mathcal{T}^{(k)}\}$. The \textbf{composite} RL task is defined by a new reward function $\widetilde{r}(s,a)~=~F(\{r^{(k)}(s,a)\})$; and shares the same states, actions, dynamics, and discount factor as all the primitive RL tasks.
\end{definition}
Finally, we define the Transfer Library, the set of functions which obey the hypotheses of our subsequent results (see Sections \ref{sec:std_lemmas} and \ref{sec:entropy-regularized_lemmas}). This definition serves to facilitate the general discussion of results obtained.
\begin{definition}
Given a set of primitive tasks $\{\mathcal{T}^{(k)}\}$, the \textbf{Transfer Library}, denoted by $\mathcal{T}L$, is the set of all transformation (or composition, when $M > 1$) functions $f$ which admit double-sided bounds (see Sections \ref{sec:std_lemmas} and \ref{sec:entropy-regularized_lemmas})
on the composite task's optimal $Q$ function ($\widetilde{Q}$).
\end{definition}
Specifically, $\mathcal{T}L = \{f\ | f\ \textrm{satisfies Lemma 4.1 or 4.3}\}$ for standard RL and $\mathcal{T}L = \{f\ | f\ \textrm{satisfies Lemma 5.1 or 5.3}\}$ for entropy-regularized RL.
We have empirically found (cf. Fig.~\ref{fig:tasks} and Supplementary Material) that by using the derived bounds for the optimal value function, the agent can learn the optimal policy more efficiently for tasks in the Transfer Library.
\mathcal{S}ection{Previous Work}\label{sec:prev_work}
There is much previous work concerning compositionality and transfer learning in reinforcement learning. In this section we will give a brief overview by highlighting the work most relevant for the current discussion.
In this work, we focus on value-based composition; rather than policy-based composition \cite{peng_MCP}, features-based composition \citep{barreto_sf}, or hierarchical (e.g. options-based) composition \citep{alver, sutton_4room, barreto2019option}.
Value based methods of composition use the optimal value functions of lower-level or simpler ``primitive'' tasks to derive an approximation (or in some cases exact solution) for the composite task of interest.
In the optimal control framework, \citep{Todorov}
has shown that optimal value functions can be composed exactly for linearly-solvable MDPs with a $\textrm{LogSumExp}$ or ``soft OR'' composition over primitive tasks; assuming that tasks share the same absorbing set (boundary states). With a similar assumption of the shared absorbing set, \citep{boolean} show that exact optimal value functions for Boolean compositions may be recovered from primitive task solutions; thereby allowing an exponential improvement in knowledge acquisition.
In more recent work, in the context of MaxEnt RL, \cite{Haarnoja2018} have shown that linear convex-weighted compositions in stochastic environments result in a bound on optimal value functions, and the policy extracted from this zero-shot bound is indeed useful for solving the composite task. The same premise of convex-weighted reward structures was studied by \cite{hunt_diverg} where the difference between the bound of \citep{Haarnoja2018} and the optimal value function can itself be learned, effectively tightening the bound until convergence. This notion of a corrective function was subsequently generalized by \cite{ourAAAI} to allow for arbitrary functions of composition in entropy-regularized RL.
Other authors have considered the question of linear task decomposition, for instance \citep{barreto_sf} where a convex weight vector over learned \textit{features} can be calculated to solve the transfer problem over linearly-decomposable reward functions in standard RL. More recent developments on this line of research include \citep{hong2022bilinear} where a more general ``bilinear value decomposition'', conditioned on various goals, is learned. In \citep{kimconstrained}, the authors consider the successor features (SFs) framework of \citep{barreto_sf}, and propose lower and upper bounds on the optimal value function of interest. They show that by replacing standard generalized policy improvement (GPI) with a constrained version which respects their bounds, they are able to transfer knowledge more successfully to future tasks in the successor features framework.
With our reduced assumptions (any constant dynamics, constant discount factor, any rewards) it is not generally possible to solve the transformed or composed tasks based only on primitive knowledge. Nevertheless, we are able to derive bounds on the optimal $Q$-functions in both standard and entropy-regularized RL, from which we can immediately derive policies which fare well in the transformed and composed problem settings. Additionally, we are able to prove that the derived policies have a bounded regret, in a similar form as \cite{Haarnoja2018}'s Theorem 1; but in a more general setting.
\mathcal{S}ection{Standard RL}\label{sec:std_lemmas}
\mathcal{S}ubsection{Transformation of Primitive Task}
In this section, we consider \textbf{transformations of a primitive task} in the ``standard'' (un-regularized) RL setting. We assume a solved primitive task is given with reward function $r(s,a)$. Transforming this underlying reward function gives rise to a new reward function, $f(r(s,a))$ which specifies a new RL task to solve. All other variables defining the MDP ($ \mathcal{S}, \mathcal{A}, p, \mu, \gamma $) are assumed to be fixed. In this new setting, we consider how to use the solution to the primitive task (that is, with rewards $r$) to inform the solution of the new, transfer task (that is, with rewards $f(r)$). The set of all applicable functions $f$ for which we can derive bounds, forms the aforementioned \textit{Transfer Library} with respect to the primitive task, for standard RL.
For a general class of transformations of reward functions (as defined below), we show that the optimal value function for the transformed task is bounded by an analogous functional transformation of the optimal value function for the primitive task. (The proofs for all theoretical results are provided in the Supplementary Material.)
We use the following definitions in the subsequent (standard RL) results: Let $X$ be the codomain for the $Q$ function of the primitive task ($Q: \mathcal{S} \times \mathcal{A} \to X \mathcal{S}ubseteq \mathbb{R}$). Let $V_f$ denote the state-value function derived from the transformation function $f(Q)$: $V_f(s) = \max_a f(Q(s,a))$.
\begin{lemma}[Convex Conditions]\label{thm:convex_cond_std}
Given a primitive task with discount factor $\gamma$ and a bounded, continuous transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is convex on its domain $X$\footnote{This condition is not required for deterministic dynamics.\label{dynamics condition}};
\item $f$ is sublinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \leq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \leq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \max_{a} \mathcal{Q}(s,a) \right) \leq \max_{a}~f\left( \mathcal{Q}(s,a) \right)$ for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to X.$\footnote{Although this condition is automatically satisfied, it allows for a smoother connection to the analogous hypotheses in Lemmas \ref{thm:concave_cond_std}, \ref{thm:forward_cond_entropy-regularized}, \ref{thm:reverse_cond_entropy-regularized} and compositional results in the Supplementary Material.}
\end{enumerate}
then the optimal action-value function for the transformed rewards, $\widetilde{Q}$, is now related to the optimal action-value function with respect to the original rewards by:
\begin{equation}\label{eqn:convex_std}
f(Q(s,a)) \leq \widetilde{Q}(s,a) \leq f(Q(s,a)) + C(s,a)
\end{equation}
where $C(s,a)$ is the optimal value function for a task with reward
\begin{equation}\label{eq:std_convex_C_def}
r_C(s,a) = f(r(s,a)) + \gamma \E_{s' \mathcal{S}im{} p } V_f(s') - f(Q(s,a)).
\end{equation}
that is, $C$ satisfies the following recursive equation:
\begin{equation}
C(s,a) = r_C(s,a) + \gamma \E_{s' \mathcal{S}im{} p} \max_{a'} C(s',a').
\end{equation}
\end{lemma}
With this result, we have a double-sided bound on the values of the optimal $Q$-function for the composite task.
In particular, the lower bound ($f(Q)$) provides a zero-shot approximation for the optimal $Q$-function. It is thus of interest to analyze how well a policy $\pi_f$ extracted from such an estimate ($f(Q)$) might perform.
To this end, we provide the following result which bounds the suboptimality of $\pi_f$ as compared to the optimal policy.
\begin{lemma}
Consider the value of the policy $\pi_f(s) = \max_{a} f(Q(s,a))$ on the transformed task of interest, denoted by $\widetilde{Q}^{\pi_f}(s,a)$.
The sub-optimality of $\pi_f$ is then upper bounded by:
\begin{equation}
\widetilde{Q}(s,a) - \widetilde{Q}^{\pi_f}(s,a) \leq D(s,a)
\end{equation}
where $D$ is the value of the policy $\pi_f$ in a task with reward
\begin{align*}
r_D(s,a) = \gamma \E_{s'\mathcal{S}im{} p}\E_{a' \mathcal{S}im{} \pi_f} &\biggr[ \max_{b} \big\{ f(Q(s',b)) + C(s',b) \big\} \\ &- f(Q(s',a')) \biggr]
\end{align*}
that is, $D$ satisfies the following recursive equation:
\begin{equation}
D(s,a) = r_D(s,a) + \gamma \E_{s'\mathcal{S}im{}p} \E_{a' \mathcal{S}im{} \pi_f} D(s',a').
\end{equation}
\end{lemma}
Interestingly, the previous result shows that for functions $f$ admitting a tight double-sided bound (that is, a relatively small value of $C$), the associated zero-shot policy $\pi_f$ can be expected to perform near-optimally in the composite domain.
Another class of functions for which general bounds can be derived arises when $f$ satisfies the following ``reverse'' conditions.
\begin{lemma}[Concave Conditions]\label{thm:concave_cond_std}
Given a primitive task with discount factor $\gamma$ and a bounded, continuous transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is concave on its domain $X$\textsuperscript{\ref{dynamics condition}};
\item $f$ is superlinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \geq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \geq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \max_{a} \mathcal{Q}(s,a) \right) \geq \max_{a}~f\left( \mathcal{Q}(s,a) \right)$ for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to X.$
\end{enumerate}
then the optimal action-value functions are now related in the following way:
\begin{equation}\label{eqn:concave_std}
f(Q(s,a)) - \hat{C}(s,a) \leq \widetilde{Q}(s,a) \leq f(Q(s,a))
\end{equation}
where $\hat{C}$ is the optimal value function for a task with reward
\begin{equation}
\hat{r}_C(s,a) = f(Q(s,a)) - f(r(s,a)) - \gamma \E_{s' \mathcal{S}im{} p} V_f(s')
\end{equation}
\end{lemma}
One obvious way to satisfy the final condition in the preceding lemma is to consider functions $f(x)$ which are monotonically increasing.
Note that the definitions of $C$ and $\hat{C}$ guarantee them to be positive, as is required for the bounds to be meaningful (this statement is shown explicitly in the Supplementary Material).
Furthermore, by again considering the derived policy $\pi_f(a|s)$, we next provide a similar result for concave conditions, noting the difference in definitions between $D$ and $\hat{D}$.
\begin{lemma}
Consider the value of the policy $\pi_f(s) = \max_{a} f(Q(s,a))$ on the transformed task of interest, denoted by $\widetilde{Q}^{\pi_f}(s,a)$.
The sub-optimality of $\pi_f$ is then upper bounded by:
\begin{equation}
\widetilde{Q}(s,a) - \widetilde{Q}^{\pi_f}(s,a) \leq \hat{D}(s,a)
\end{equation}
where $\hat{D}$ is the value of the policy $\pi_f$ in a task with reward
\begin{equation*}
\hat{r}_D = \gamma \underset{s'\mathcal{S}im{} p\ }{\mathbb{E}} \underset{a' \mathcal{S}im{} \pi_f}{\mathbb{E}} \biggr[ V_f(s') - f(Q(s',a')) + \hat{C}(s',a') \biggr]
\end{equation*}
\end{lemma}
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