arXiv subjects

Gershon Wolansky

Publications and source records attributed to Gershon Wolansky.

At least 19 recordsLinked to original sources

arXiv2026

Weighted least action principle for Maxwell equations

The Fermat principle of least time determines the path of a single ray of light given its initial and final positions and the medium electromagnetic properties. However, a single ray is not a measurable physical object. We derive here an upgraded variational principle for the geometric optics limit of Maxwell equations in an arbitrary medium, based on measuring the intensity of the wave on two planes. The principle provides the complete Fresnel rays bundle connecting associated points at the first and second planes. One of the applications of the present theory is to use the reciprocity between Fresnel rays and phase normals to determine the phase of an electromagnetic wave from two intensity measurements.

math-ph↗

arXiv2026

Self perimeter of convex sets

This paper introduces a natural definition for the volume of the unit ball in $n$-dimensional normed spaces $\mathbb{R}^n$. This definition preserves the Euclidean relation $P(B)/V(B)=n$ between the perimiter and the volume of the unit ball $B$ in $R^n$. We show that this volume definition is invariant under origin-preserving affine transformations and polar duality. For $n=2$, we derive an explicit integral formula for the self-perimeter of the unit ball, extend it to non-centrally symmetric sets;. The construction is extended to $\mathbb{R}^n$ via a recursive integration over the boundary, utilizing $(n-1)$-dimensional volumes of planar intersections. Finally, we pose and discuss an Alexandrov-type problem for the associated surface measure, providing perturbative solutions in the 2D case. In particular we prove that, generically, any perturbation of the surface measure of the Euclidean 2-D disk yields a 4-fold symmetric convex set in the leading order.

math.MG↗

arXiv2026

Trimmed branching random walk and a free obstacle problem

Consider $N$ particles performing random walks on the $ε$-grid $(εZ)^d$, $ε>0$ with branching and density-dependent selection: When one of the particles branches, a particle is removed from the most populated site. The walks are assumed to be asymptotic, as $ε\to0$, to diffusion processes of the form \[ dX_i(t)=b(X_i(t))dt+\sqrt{2}dW_i(t), \] for $b$ a given vector field. Denoting $L^*=Δ-\nabla\cdot(b\,\cdot)$, the hydrodynamic limit, as $N\to\infty$ followed by $ε\to0$, is characterized in terms of a parabolic free obstacle problem \[ \partial_t u=L^*u+u-β\] where $β$ is a measure on $R^d\times[0,\infty)$ supported on $\{(x,t):u(x,t)=|u(\cdot,t)|_\infty\}$. Here, the unknowns are $u$, the mass density, and $β$, the removal measure, for which $t\mapstoβ(R^d\times[0,t])$ is prescribed. This is analogous to the well-understood relation between particle systems with spatial selection and free boundary problems, but the techniques require quite different ideas. The key ingredients of the proof include PDE uniqueness for continuous densities and a uniform-in-$ε$ estimate on modulus of continuity of prelimit densities. The work gives rise to open problems such as ``flat top'' versus ``sharp top'' solutions, which are discussed based on concrete examples.

math.PR↗

arXiv2024

Dual formulation for constraint solutions of the multi-state Choquard equation

The Choquard equation is a partial differential equation that has gained significant interest and attention in recent decades. It is a nonlinear equation that combines elements of both the Laplace and Schrödinger operators, and it arises frequently in the study of numerous physical phenomena, from condensed matter physics to nonlinear optics. In particular, the steady states of the Choquard equation were thoroughly investigated using a variational functional acting on the wave functions. In this article, we introduce a dual formulation for the variational functional in terms of the potential indiced by the wave function, and use it to explore the existence of steady states of a multi-state version the Choquard equation in critical and sub-critical cases.

math-ph↗

arXiv2024

Invariance principle and McKean-Vlasov limit for randomized load balancing in heavy traffic

We consider a load balancing model where a Poisson stream of jobs arrive at a system of many servers whose service time distribution possesses a finite second moment. A small fraction of arrivals pass through the so called power-of-choice algorithm, which assigns a job to the shortest among $\ell$, $\ell\ge 2$, randomly chosen queues, and the remaining jobs are assigned to queues chosen uniformly at random. The system is analyzed at critical load in an asymptotic regime where both the number of servers and the usual heavy traffic parameter associated with individual queue lengths grow to infinity. The first main result is a hydrodynamic limit, where the empirical measure of the diffusively normalized queue lengths is shown to converge to a path in measure space whose density is given by the unique solution of a parabolic PDE with nonlocal coefficients. Further, two forms of an invariance principle are proved, corresponding to two different assumptions on the initial distribution, where individual normalized queue lengths converge weakly to solutions of SDE. In one of these results, the limit is given by a McKean-Vlasov SDE, and propagation of chaos holds. The McKean-Vlasov limit is closely related to limit results for Brownian particles on $\mathbb{R}_+$ interacting through their rank (with a specific interaction). However, an entirely different set of tools is required, as the collection of $n$ prelimit particles does not obey a Markovian evolution on $\mathbb{R}_+^n$.

math.PR↗

arXiv2024

Functional Dimensionality of Koopman Eigenfunction Space

This work presents the general form solution of Koopman Partial Differential Equation and shows that its functional dimensionality is finite. The dimensionality is as the dimensionality of the dynamics. Thus, the representation of nonlinear dynamics as a linear one with a finite set of Koopman eigenfunctions without error is possible. This formulation justifies the flowbox statement and provides a simple numerical method to find such representation.

math.AP↗

arXiv2023

The linearized Poisson-Nernst-Planck system as heat flow on the interval under non-local boundary conditions

The linearized of the Poisson-Nernst-Planck (PNP) equation under closed ends around a neutral state is studied. It is reduced to a damped heat equation under non-local boundary conditions, which leads to a stochastic interpretation of the linearized equation as a Brownian particle which jump and is reflected, at Poisson distributed time, to one of the end points of the channel, with a probability which is proportional to its distance from this end point. An explicit expansion of the heat kernel reveals the eigenvalues and eigenstates of both the PNP equation and its adjoint. For this, we take advantage of the representation of the resulvent operator and recover the heat kernel by applying the inverse Laplace transform.

math-ph↗

arXiv2020

Semi-discrete optimal transport

In the current book I suggest an off-road path to the subject of optimal transport. I tried to avoid prior knowledge of analysis, PDE theory and functional analysis, as much as possible. Thus I concentrate on discrete and semi-discrete cases, and always assume compactness for the underlying spaces. However, some fundamental knowledge of measure theory and convexity is unavoidable. In order to make it as self-contained as possible I included an appendix with some basic definitions and results. I believe that any graduate student in mathematics, as well as advanced undergraduate students, can read and understand this book. Some chapters (in particular in Parts II\&III ) can also be interesting for experts. Starting with the the most fundamental, fully discrete problem I attempted to place optimal transport as a particular case of the celebrated stable marriage problem. From there we proceed to the partition problem, which can be formulated as a transport from a continuous space to a discrete one. Applications to information theory and game theory (cooperative and non-cooperative) are introduced as well. Finally, the general case of transport between two compact measure spaces is introduced as a coupling between two semi-discrete transports.

math.OC↗

arXiv2020

On semi-discrete sub-partitions of vector-valued measures

We introduce a concept of optimal transport for vector-valued measures and its dual formulation. In this note we concentrate on the semi-discrete case and show some fundamental differences between the scalar and vector cases. A manifestation of this difference is the possibility of non-existence of optimal solution for the dual problem for feasible primer problems.

math.OC↗

arXiv2020

On the critical mass Patlak-Keller-Segel system for multi-species populations: global existence and infinite time aggregation

We study the global in time existence and long time asymptotics of solutions to the parabolic-elliptic Patlak-Keller-Segel system for the multi-species populations in the whole Euclidean space $\mathbb{R}^2.$ We prove that at the borderline case of critical mass there exists a global {\it free energy solution} subject to initial data with finite entropy and second moment. Moreover, we show that as time $t$ approaches to infinity, all the components of the solutions concentrate in the form of a Dirac measure at a single point. Our approach utilizes the gradient flow structure in Wasserstein space in the spirit of De Giorgi's minimizing movement or the JKO-schemes. Due to the critical mass, the minimization problem in JKO-schemes may not admit a solution in general. We find a necessary and sufficient criterion for which any minimizing sequence remains uniformly bounded in an appropriate topology to ensure the existence of a minimizer.

math.AP↗

arXiv2019

Is the mailing Gilbert-Steiner problem convex?

A convexification of the mailing version of the finite Gilbert problem for optimal networks is introduced. It is ia convex functional on the set of probability measures subject to the Wasserstein $p-$ metric. The minimizer of this convex functional is a measure supported in a graph. If this graph is a tree (i.e contains no cycles) then this tree is also a minimum of the corresponding mailing Gilbert problem. A numerical algorithm for the implementation of the convexified Gilbert-mailing problem is also suggested, based on entropic regularization.

math.OC↗

arXiv2019

On Patlak-Keller-Segel system for several populations: a gradient flow approach

We study the global in time existence of solutions to the parabolic-elliptic Patlak-Keller-Segel system of multi-species populations. We prove that if the initial mass satisfies an appropriate notion of sub-criticality, then the system has a solution defined for all time. We explore the gradient flow structure in the Wasserstein space to study the question of existence. Moreover, we show that the obtained solution satisfies energy dissipation inequality.

math.AP↗

arXiv2018

Happy family of stable marriages

Some aspects of the problem of stable marriage are discussed. There are two distinguished marriage plans: the fully transferable case, where money can be transferred between the participants, and the fully non transferable case where each participant has its own rigid preference list regarding the other gender. We continue to discuss intermediate partial transferable cases. Partial transferable plans can be approached as either special cases of cooperative games using the notion of a core, or as a generalization of the cyclical monotonicity property of the fully transferable case (fake promises). We shall introduced these two approaches, and prove the existence of stable marriage for the fully transferable and non-transferable plans.

econ.EM↗

arXiv2018

Self-similar solutions of decaying Keller-Segel systems for several populations

It is known that solutions of the parabolic elliptic Keller-Segel equations in the two dimensional plane decay, as time goes to infinity, provided the initial data admits sub-critical mass and finite second moments, while such solution concentrate, as $t\rightarrow\infty$, in the critical mass. In the sub-critical case this decay can be resolved by a steady, self-similar solution, while no such self similar solution is known to exist for the concentration in the critical case. This paper is motivated by the Keller-Segel system of several interacting populations, under the existence of an additional drift for each component which decays in time at the rate $O(1/\sqrt{t})$. We show that self-similar solutions always exists in the sub-critical case, while the existence of such self-similar solution in the critical case depends on the gap between the decaying drifts for each of the components. For this, we study the conditions for existence/non existence of solutions for the corresponding Liouville's systems, which, in turn, is related to the existence/non existence of minimizers to a corresponding Free Energy functional.

math.AP↗

arXiv2017

Contact angles of liquid drops subjected to a rough boundary

The contact angle of a liquid drop on a rigid surface is determined by the classical theory of Young-Laplace. For chemically homogeneous surfaces, this angle is a constant. We study the minimal-energy configurations of liquid drops on rough surfaces. Here the actual angle is still constant for homogeneous surfaces, but the apparent angle can fluctuate widely. A limit theorem is introduced for minimal energy configuration, where the rigid surface converges to a smooth one, but the roughness parameter is kept constant. It turns out that the limit of minimal energy configurations correspond to liquid drop on a smooth surface with an appropriately defined effective chemical interaction energy. It turns out that the effective chemical interaction depends linearly on the roughness in a certain range of parameters, corresponding to full wetting. Outside this range the most stable configuration corresponds to a partial wetting and the effective interaction energy depends on the geometry in an essential way. This result partially justifies and extends Wenzel and Cassie's laws and can be used to deduce the actual inclination angle in the most stable state, where the apparent one is known by measurement. This, in turn, may be applied to deduce the roughness parameter if the interfacial energy is known, or visa versa.

math-ph↗

arXiv2016

From optimal transportation to optimal teleportation

The object of this paper is to study estimates of $ε^{-q}W_p(μ+εν, μ)$ for small $ε>0$. Here $W_p$ is the Wasserstein metric on positive measures, $p>1$, $μ$ is a probability measure and $ν$ a signed, neutral measure ($\int dν=0$). In [W1] we proved uniform (in $ε$) estimates for $q=1$ provided $\int ϕdν$ can be controlled in terms of the $\int|\nablaϕ|^{p/(p-1)}dμ$, for any smooth function $ϕ$. In this paper we extend the results to the case where such a control fails. This is the case where if, e.g. $μ$ has a disconnected support, or if the dimension of $μ$ , $d$ (to be defined) is larger or equal $p/(p-1)$. In the later case we get such an estimate provided $1/p+1/d\not=1$ for $q=\min(1, 1/p+1/d)$. If $1/p+1/d=1$ we get a log-Lipschitz estimate. As an application we obtain Hölder estimates in $W_p$ for curves of probability measures which are absolutely continuous in the total variation norm . In case the support of $μ$ is disconnected (corresponding to $d=\infty$) we obtain sharp estimates for $q=1/p$ ("optimal teleportation"): $$ \lim_{ε\rightarrow 0}ε^{-1/p}W_p(μ, μ+εν) = \|ν\|_μ$$ where $\|ν\|_μ$ is expressed in terms of optimal transport on a metric graph, determined only by the relative distances between the connected components of the support of $μ$, and the weights of the measure $ν$ in each connected component of this support.

math.PR↗

arXiv2016

Chemotactic systems in the presence of conflicts: a new functional inequality

The evolution of a chemotactic system involving a population of cells attracted to self-produced chemicals is described by the Keller-Segel system. In spacial dimension 2, this system demonstrates a balance between the spreading effect of diffusion and the concentration due to self-attraction. As a result, there exists a critical "mass" (i.e. total cell's population) above which the solution of this system collapses in a finite time, while below this critical mass there is global existence in time. The existence of this critical mass is related to a functional inequality known as the Moser-Trudinger inequality. An extension of the Keller-Segel model to several cells populations was considered before in the literature. Here we review some of these results and, in particular, consider the case of conflict between two populations, that is, when population one attracts population two, while, at the same time, population two repels population one. This assumption leads to a new functional inequality which generalizes the Moser-Trudinger inequality.

math.AP↗

arXiv2016

On optimal partitions, individual values and cooperative games: Will a wiser agent always produce a higer value?

We consider an optimal partition of resources (e.g. consumers) between several agents (e.g. experts), given utility functions ("wisdoms") for the agents and their capacities. This problem is a variant of optimal transport (Monge-Kantorovich) between two measure spaces where one of the measures is discrete (capacities) and the costs of transport are the wisdoms of the agents. We concentrate on the individual value for each agent under optimal partition and show that, counter-intuitively, this value may decrease if the agent's wisdom is increased. Sufficient and necessary conditions for increment of the individual values will be given, independently of the other agents. The sharpness of these conditions is also discussed. Motivated by the above we define a cooperative game based on optimal partition and investigate conditions for the existence of a core for this game, guaranteeing the stability of the grand coalition.

math.OC↗