arXiv subjects
Publications and source records attributed to Gershon Wolansky.
Optimal mass transport is described by an approximation of transport cost via semi-discrete costs. The notions of optimal partition and optimal strong partition are given as well. We also suggest an algorithm for computation of Optimal Transport for general cost functions induced by an action, an asymptotic error estimate and several numerical examples of optimal partitions.
In this paper we generalize the classical theorem of Thue about the optimal circular disc packing in the plane. We are given a family of circular discs, not necessarily of equal radii, with the property that the inflation of every disc by a factor of $2$ around its center does not contain any center of another disc in the family (notice that this implies that the family of discs is a packing). We show that in this case the density of the given packing is at most $\fracπ{2\sqrt{3}}$, which is the density of the optimal unit disc packing. This result is used to obtain a discrete approximation to the Entropy functional in two dimensional domain.
Let $X$ a probability measure space and $ψ_1....ψ_N$ measurable, real valued functions on $X$. Consider all possible partitions of $X$ into $N$ disjoint subdomains $X_i$ on which $\int_{X_i}ψ_i$ are prescribed. We address the question of characterizing the set $(m_1,,,m_N) \in R^N$ for which there exists a partition $X_1, ...X_N$ of $X$ satisfying $\int_{X_i}ψ_i= m_i$ and discuss some optimization problems on this set of partitions. The relation of this problem to semi-discrete version of optimal mass transportation is discussed as well.
Let $X$ a probability measure space and $ψ_1....ψ_N$ measurable, real valued functions on $X$. Consider all possible partitions of $X$ into $N$ disjoint subdomains $X_i$ on which $\int_{X_i}ψ_i$ are prescribed. We address the question of characterizing the set $(m_1,,,m_N) \in \R^N$ for which there exists a partition $X_1, ... X_N$ of $X$ satisfying $\int_{X_i}ψ_i= m_i$ and discuss some optimization problems on this set of partitions. The relation of this problem to semi-discrete version of optimal mass transportation is discussed as well.
The optimal mass transportation was introduced by Monge some 200 years ago and is, today, the source of large number of results in analysis, geometry and convexity. Here I investigate a new, surprising link between optimal transformations obtained by different Lagrangian actions on Riemannian manifolds. As a special case, for any pair of non-negative measures $λ^+,λ^-$ of equal mass $$ W_1(λ^-, λ^+)= \lim_{\eps\to 0} \eps^{-1}\inf_μ W_p(μ+\epsλ^-, μ+\epsλ^+)$$ where $W_p$, $p\geq 1$ is the Wasserstein distance and the infimum is over the set of probability measures in the ambient space.
It is shown that optimal network plans can be obtained, naturally, as a limit of easier problems of point allocations. These problems are obtained by minimizing the mass transportation on the set of atomic measures of prescribed number of atoms. Each of these problems can be solved by minimizing a function of a finite, prescribed number of variables.
A new Combinatorial Ricci curvature and Laplacian operators for grayscale images are introduced and tested on 2D synthetic, natural and medical images. Analogue formulae for voxels are also obtained. These notions are based upon more general concepts developed by R. Forman. Further applications, in particular a fitting Ricci flow, are discussed.
We prove a concentration result of a Bloch eigenstate in a periodic channel under a constant gauge. In the semi-classical limit $h--> 0$ these eigenstates concentrate near a maximizer of the scalar potential of the associated Schrodinger operator, provided the constant gauge converges to a critical value from above. This is in contrast with the ground states which concentrate for any gauge in this limit near a minimizer of the scalar potential.
A diffusion's induced transport is defined for a linear model of a Fokker-Plank equation under periodic boundary conditions in one-dimensional geometry. The flow is generated by a diffusion and a periodic deriving force induced by a velocity potential. An inverse problem is suggested for evaluating the deriving force in terms of the response function associated with the flow. It is also shown that the inverse problem can be partially solved under some simplifying assumption.
Given a probability measure $μ$ on the $n-$torus $T^n$ and a rotation vector $k\in R^n$, we ask wether there exists a minimizer to the integral $\int_{T^n} |\gradϕ+k|^2 dμ$. This problem leads, naturally, to a class of elliptic PDE and to an optimal transportation (Monge-Kantorovich) class of problems on the torus. It is also related to higher dimensional Aubry-Mather theory, dealing with invariant sets of periodic Lagrangians, and is known as the "Weak-KAM theory".
his paper proposes a sensible definition of a deformation metric between 2-dimensional surfaces obtained from each other by an area preserving (incompressible) mapping, and an algorithm for obtaining this metric, as well as the optimal deformation.
We study a simple system of two hyperbolic semi-linear equations, inspired by the Einstein equations. The system, which was introduced in gr-qc/0612136, is a model for singularity formation inside black holes. We show for a particular case of the equations that the system demonstrates a finite time blowup. The singularity that is formed is a null singularity. Then we show that in this particular case the singularity has features that are analogous to known features of models of black-hole interiors - which describe the inner-horizon instability. Our simple system may provide insight into the formation of null singularities inside spinning or charged black holes.
The continuous limit of large systems of particles of finite size on the line is described. The particles are assumed to move freely and stick under collision, to form compound particles whose mass and size is the sum of the masses and sizes of the particles before collision, and whose velocity is determined by conservation of linear momentum.
We study the discreteness of the spectrum of Schrodinger operators which are defined on N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their eigenvalues using operators on a one-dimensional tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N-dimensional tree. We show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero. Moreover, the projections to the one-dimensional tree of eigenfunctions of the N-dimensional Laplace operator converge to the corresponding eigenfunctions of the one-dimensional limit operator.
The optimal (Monge-Kantorovich) transportation problem is discussed from several points of view. The Lagrangian formulation extends the action of the {\em Lagrangian} $L(v,x,t)$ from the set of orbits in $\R^n$ to a set of measure-valued orbits. The {\em Eulerian}, dual formulation leads an optimization problem on the set of sub-solutions of the corresponding Hamilton-Jacobi equation. Finally, the Monge problem and its Kantorovich relaxation are obtained by reducing the optimization problem to the set of measure preserving mappings and two point distribution measures subjected to an appropriately defined cost function. In this paper we concentrate on mechanical Lagrangians $L=|v|^2/2+P(x,t)$ leading, in general, to a non-homogeneous cost function. The main results yield existence of a unique {\em flow} of homomorphisms which transport the optimal measure valued orbit of the extended Lagrangian, as well as the existence of an optimal solution to the dual Euler problem and its relation to the Monge- and Kantorovich formulations.
We consider the Schrödinger-Poisson system in the attractive (plasma physics) Coulomb case. Given a steady state from a certain class we prove its nonlinear stability, using an appropriately defined energy-Casimir functional as Lyapunov function. To obtain such steady states we start with a given Casimir functional and construct a new functional which is in some sense dual to the corresponding energy-Casimir functional. This dual functional has a unique maximizer which is a steady state of the Schrödinger-Poisson system and lies in the stability class. The steady states are parametrized by the equation of state, giving the occupation probabilities of the quantum states as a strictly decreasing function of their energy levels.