arXiv subjects

Dmitry Kaliuzhnyi-Verbovetskyi

Publications and source records attributed to Dmitry Kaliuzhnyi-Verbovetskyi.

5 recordsLinked to original sources

arXiv2020

Integrability of Free Noncommutative Functions

Noncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity properties of usual analytic functions) and admit a good difference-differential calculus. Noncommutative functions appear naturally in a large variety of settings: noncommutative algebra, systems and control, spectral theory, and free probability. Starting with pioneering work of J.L. Taylor, the theory was further developed by D.-V. Voiculescu, and established itself in recent years as a new and extremely active research area. The goal of the present paper is to establish a noncommutative analog of the Frobenius integrability theorem: we give necessary and sufficient conditions for higher order free noncommutative functions to have an antiderivative.

math.FA↗

arXiv2019

Sparse Monte Carlo method for nonlocal diffusion problems

A class of evolution equations with nonlocal diffusion is considered in this work. These are integro-differential equations arising as models of propagation phenomena in continuum media with nonlocal interactions including neural tissue, porous media flow, peridynamics, models with fractional diffusion, as well as continuum limits of interacting dynamical systems. The principal challenge of numerical integration of nonlocal systems stems from the lack of spatial regularity of the data and solutions intrinsic to nonlocal models. To overcome this problem we propose a semidiscrete numerical scheme based on the combination of sparse Monte Carlo and discontinuous Galerkin methods. An important feature of our method is sparsity. Sparse sampling of points in the Monte Carlo approximation of the nonlocal term allows to use fewer discretization points without compromising the accuracy. We prove convergence of the numerical method and estimate the rate of convergence. There are two principal ingredients in the error of the numerical method related to the use of Monte Calro and Galerkin approximations respectively. We analyze both errors. Two representative examples of discontinuous kernels are presented. The first example features a kernel with a singularity, while the kernel in the second example experiences jump discontinuity. We show how the information about the singularity in the former case and the geometry of the discontinuity set in the latter translate into the rate of convergence of the numerical procedure. In addition, we illustrate the rate of convergence estimate with a numerical example of an initial value problem, for which an explicit analytic solution is available. Numerical results are consistent with analytical estimates.

math.NA↗

arXiv2017

The mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limit

The Kuramoto model (KM) of coupled phase oscillators on graphs provides the most influential framework for studying collective dynamics and synchronization. It exhibits a rich repertoire of dynamical regimes. Since the work of Strogatz and Mirollo, the mean field equation derived in the limit as the number of oscillators in the KM goes to infinity, has been the key to understanding a number of interesting effects, including the onset of synchronization and chimera states. In this work, we study the mathematical basis of the mean field equation as an approximation of the discrete KM. Specifically, we extend the Neunzert's method of rigorous justification of the mean field equation to cover interacting dynamical systems on graphs. We then apply it to the KM on convergent graph sequences with non-Lipschitz limit. This family of graphs includes many graphs that are of interest in applications, e.g., nearest-neighbor and small-world graphs.

math.DS↗

arXiv2016

The semilinear heat equation on sparse random graphs

Using the theory of $L^p$-graphons (Borgs et al, 2014), we derive and rigorously justify the continuum limit for systems of differential equations on sparse random graphs. Specifically, we show that the solutions of the initial value problems for the discrete models can be approximated by those of an appropriate nonlocal diffusion equation. Our results apply to a range of spatially extended dynamical models of different physical, biological, social, and economic networks. Importantly, our assumptions cover network topologies featured in many important real-world networks. In particular, we derive the continuum limit for coupled dynamical systems on power law graphs. The latter is the main motivation for this work.

math.DS↗

arXiv2013

Scattering systems with several evolutions and formal reproducing kernel Hilbert spaces

A Schur-class function in $d$ variables is defined to be an analytic contractive-operator valued function on the unit polydisk. Such a function is said to be in the Schur--Agler class if it is contractive when evaluated on any commutative $d$-tuple of strict contractions on a Hilbert space. It is known that the Schur--Agler class is a strictly proper subclass of the Schur class if the number of variables $d$ is more than two. The Schur--Agler class is also characterized as those functions arising as the transfer function of a certain type (Givone--Roesser) of conservative multidimensional linear system. Previous work of the authors identified the Schur--Agler class as those Schur-class functions which arise as the scattering matrix for a certain type of (not necessarily minimal) Lax--Phillips multievolution scattering system having some additional geometric structure. The present paper links this additional geometric scattering structure directly with a known reproducing-kernel characterization of the Schur--Agler class. We use extensively the technique of formal reproducing kernel Hilbert spaces that was previously introduced by the authors and that allows us to manipulate formal power series in several commuting variables and their inverses (e.g., Fourier series of elements of $L^2$ on a torus) in the same way as one manipulates analytic functions in the usual setting of reproducing kernel Hilbert spaces.

math.FA↗