arXiv subjects

Mohammad Ali Ahmadpoor

Publications and source records attributed to Mohammad Ali Ahmadpoor.

6 recordsLinked to original sources

arXiv2024

Uniqueness of optimal plans for multi-marginal mass transport problems via a reduction argument

For a family of probability spaces $\{(X_k,\mathcal{B}_{X_k},μ_k)\}_{k=1}^N$ and a cost function $c: X_1\times\cdots\times X_N\to \mathbb{R}$ we consider the Monge-Kantorovich problem \begin{align*}\tag{MK}\label{MONKANT} \inf_{λ\inΠ(μ_1,\ldots,μ_N)}\int_{\prod_{k=1}^N X_k}c\,dλ. \end{align*} Then for each ordered subset $\mathcal{P}=\{i_1,\ldots,i_p\}\subsetneq\{1,...,N\}$ with $p\geq 2$ we create a new cost function $c_\mathcal{P}$ corresponding to the original cost function $c$ defined on $\prod_{k=1}^p X_{i_k}$. This new cost function $c_\mathcal{P}$ enjoys many of the features of the original cost $c$ while it has the property that any optimal plan $λ$ of \eqref{MONKANT} restricted to $\prod_{k=1}^p X_{i_k}$ is also an optimal plan to the problem \begin{align*}\tag{RMK}\label{REDMONKANT} \inf_{τ\inΠ(μ_{i_1},\ldotsμ_{i_p})}\int_{\prod_{k=1}^p X_{i_k}}c_{\mathcal{P}}\,dτ. \end{align*} Our main contribution in this paper is to show that, for appropriate choices of index set $\mathcal{P}$, one can recover the optimal plans of \eqref{MONKANT} from \eqref{REDMONKANT}. In particular, we study situations in which the problem \eqref{MONKANT} admits a unique solution depending on the uniqueness of the solution for the lower marginal problems of the form \eqref{REDMONKANT}. This allows us to prove many uniqueness results for multi-marginal problems when the unique optimal plan is not necessarily induced by a map. To this end, we extensively benefit from disintegration theorems and the $c$-extremality notions. Moreover, by employing this argument, besides recovering many standard results on the subject including the pioneering work of Gangbo-Świ\c ech, several new applications will be demonstrated to evince the applicability of this argument.

math.OC↗

arXiv2024

Stratified Monge-Kantorovich optimal transport problems

In this paper, we investigate Monge-Kantorovich problems for which the absolute continuity of marginals is relaxed. For $X,Y\subseteq\mathbb{R}^{n+1}$ let $(X,\mathcal{B}_X,μ)$ and $(Y,\mathcal{B}_Y,ν)$ be two Borel probability spaces, $c:X\times Y\to\mathbb{R}$ be a cost function, and consider the problem \begin{align*}\tag{MKP}\label{MKPEQ} \inf\left\{\int_{X\times Y} c(x,y)\,dλ :\ λ\inΠ(μ,ν) \right\}. \end{align*} Inspired by the seminal paper \cite{GANGBOMCCANN2} with applications in shape recognition problem, we first consider \eqref{MKPEQ} for the cost $c(x,y)=h(x-y)$ with $h$ strictly convex defined on the multi-layers target space \begin{align*} X=\overline{X}\times\{\overline{x}\},\quad\text{and}\quad Y=\bigcup_{k=1}^K \left(\overline{Y}_{k}\times \{\overline{y}_k\}\right), \end{align*} where $\overline{X}, \overline{Y}_{k}\subseteq \mathbb{R}^{n}$ for $k\in \{1,\ldots,K\},$ $\overline{x}\in \mathbb{R}$, and $\{\overline{y}_1,..., \overline{y}_K\}\subseteq \mathbb{R}$. Here, we assume that $μ|_\overline{X}\ll\mathcal{L}^n$ (the Lebesgue measure on $\mathbb{R}^n$), but $μ$ is singular w.r.t. $\mathcal{L}^{n+1}$. When $K=1$, this translates to the standard \eqref{MKPEQ} for which the unique solution is concentrated on a map. We show that for $K\geq 2,$ the solution is still unique but it concentrates on the graph of several maps. Next, we study \eqref{MKPEQ} for a closed subset $X\subseteq \mathbb{R}^{n+1}$ and its $n$-dimensional submanifold $X_0$ with the first marginal of the form \begin{align*} \int_X f(x)\,dμ(x)=\int_X f(x)α(x)\,d\mathcal{L}^{n+1}(x)+\int_{X_0} f(x_0)\,d S(x_0),\ \ \forall f\in C_b(X). \end{align*} Here, $S$ is a measure on $X_0$ such that $S\ll \mathcal{L}^{n}$ on each coordinate chart of $X_0$. This can be seen as a two-layers problem as the measure $μ$ charges both $n$- and $n+1$-dimensional subsets.

math.OC↗

arXiv2023

Weighted homomorphisms on the $p$-analog of the Fourier-Stieltjes algebra induced by piecewise affine maps

In this paper, for $p\in(1,\infty)$ we study $p$-complete boundedness of weighted homomorphisms on the $p$-analog of the Fourier-Stieltjes algebras, $B_p(G)$, based on the $p$-operator space structure defined by the authors. Here, for a locally compact group $G$, the space $B_p(G)$ stands for Runde's definition of the $p$-analog of the Fourier-Stieltjes algebra and the implemented $p$-operator space structure is come from the duality between $B_p(G)$ and the algebra of universal $p$-pseudofunctions, $UPF_p(G)$. It is established that the homomorphism $Φ_α:B_p(G)\to B_p(H)$, defined by $Φ(u)=u\circα$ on $Y$ and zero otherwise, is $p$-completely contractive when the continuous and proper map $α:Y\subseteq H\to G$ is affine, and it is $p$-completely bounded whenever $α$ is piecewise affine map. Moreover, we assume that $Y$ belongs to the coset ring generated by open and amenable subgroups of $H$. To obtain the result, by utilizing the properties of $QSL_p$-spaces and representations on them, the relation between $B_p(G/N)$ and a closed subalgebra of $B_p(G)$ is shown, where $N$ is a closed normal subgroup of $G$. Additionally, $p$-complete boundedness of several well-known maps on such algebras are obtained.

math.FA↗

arXiv2020

On the functorial properties of the $p$-analog of the Fourier-Stieltjes algebras

In this paper, some known results will be generalized. Firstly, the idempotent theorem on the Fourier-Stieltjes algebra will be promoted and linked to the $p$-analog of such an algebra. Next, the $p$-analog of the $π$-Fourier space introduced by Arsac will be given, and by taking advantage of the theory of ultra filters, the connection between the dual space of the algebra of $p$-pseudofunctions and the $p$-analog of the $π$-Fourier space, will be fully investigated. As the main result, one of the significant and applicable functorial properties of the $p$-analog of the Fourier-Stieltjes algebras will be achieved.

math.FA↗

arXiv2020

A note on the $p$-operator space structure of the $p$-analog of the Fourier-Stieltjes algebra

In this paper one of the possible $p$-operator space structures of the $p$-analog of the Fourier-Stieltjes algebra will be introduced, and to some extend will be studied. This special sort of operator structure will be given from the predual of this Fourier type algebra, that is the algebra of universal $p$-pseudofunctions. Furthermore, some applicable and expected results will be proven. Current paper can be considered as a new gate into the collection of problems around the $p$-analog of the Fourier-Stieltjes algebra, in the $p$-operator space structure point of view.

math.FA↗

arXiv2020

On the functorial properties of the p-analog of the Fourier-Stieltjes algebras and their homomorphisms

In this paper, we follow two main goals. In the first attempt, we give some functorial properties of the $p$-analog of the Fourier-Stieltjes algebras in which we generalize some previously existed definitions and theorems in Arsac and Cowling's works, to utilize them to prove $p$-complete boundedness of some well-known maps on these algebras. In the second part, as an application of these generalizations, we prove $p$-completely boundedness of homomorphisms which are induced by continuous and proper piecewise affine maps that is a generalization of Ilie's work on Figà-Talamanca-Herz algebras.

math.FA↗