arXiv subjects

V. Gol'dshtein

Publications and source records attributed to V. Gol'dshtein.

18 recordsLinked to original sources

arXiv2024

On the de Rham homomorphism for $L_π$-cohomologies

We study the procedure of regularization in the context of the Lipschitz version of de Rham calculus on metric simplicial complexes with bounded geometry. It provides us with the machinery to handle the de Rham homomorphism for $L_π$-cohomologies. In this respect, we obtain the condition resolving the question of triviality of the kernel for de Rham homomorphism. In particular, we specify the non-trivial cohomology classes explicitly for a sequence of parameters $π= \langle p_0,\,p_1,\dots,\,p_n\rangle$ missing non-increasing monotonicity.

math.DG↗

arXiv2018

On Conformal Spectral Gap Estimates of the Dirichlet-Laplacian

We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains $Ω\subset\mathbb R^2$. With the help of these estimates we obtain asymptotically sharp inequalities of ratios of eigenvalues in the frameworks of the Payne-Pólya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains in terms of conformal (hyperbolic) geometry.

math.AP↗

arXiv2018

Composition Operators on Sobolev Spaces and Neumann Eigenvalues

In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. The lower estimates of the first non-trivial Neumann eigenvalues of the $p$-Laplace operator in cusp domains $Ω\subset\mathbb R^n$, $n\geq 2$, are given.

math.AP↗

arXiv2017

On the First Eigenvalue of the Degenerate $p$-Laplace Operator in Non-Convex Domains

In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate $p$-Laplace operator, $p>2$, in a large class of non-convex domains. This study is based on applications of the geometric theory of composition operators on Sobolev spaces that permits us to estimates constants of Poincaré-Sobolev inequalities and as an application to derive lower estimates of the first non-trivial eigenvalues for the Alhfors domains (i.e. to quasidiscs). This class of domains includes some snowflakes type domains with fractal boundaries.

math.AP↗

arXiv2017

Sobolev Extension Operators and Neumann Eigenvalues

In this paper we apply estimates of the norms of Sobolev extension operators to the spectral estimates of of the first nontrivial Neumann eigenvalue of the Laplace operator in non-convex extension domains. As a consequence we obtain a connection between resonant frequencies of free membranes and the smallest-circle problem (initially proposed by J.~J.~Sylvester in 1857).

math.AP↗

arXiv2017

Space Quasiconformal Mappings and Neumann Eigenvalues in Fractal Domains

We study the variation of the Neumann eigenvalues of the $p$-Laplace operator under quasiconformal perturbations of space domains. This study allows to obtain lower estimates of the Neumann eigenvalues in fractal type domains. The suggested approach is based on the geometric theory of composition operators in connections with the quasiconformal mapping theory.

math.AP↗

arXiv2017

On the First Eigenvalues of Free Vibrating Membrane in Conformal Regular Domains

In 1961 G.Polya published a paper about the eigenvalues of vibrating membrane. The "free vibrating membrane"' corresponds to the Neumann-Laplace operator in bounded plane domains. In this paper we obtain estimates for the first eigenvalue of this operator in a large class of domains that we call as conformal regular domains, that includes convex domains, John domains etc... On the base of our estimates we conjecture that the eigenvalues of the Neumann-Laplace operator depend on the hyperbolic metrics of plane domains. We propose a new method for the estimates that is based on weighted Poincaré-Sobolev inequalities obtained by the authors recently.

math.AP↗

arXiv2017

Spectral Properties of the Neumann-Laplace operator in Quasiconformal Regular Domains

In this paper we study spectral properties of the Neumann-Laplace operator in planar quasiconformal regular domains $Ω\subset\mathbb R^2$. This study is based on the quasiconformal theory of composition operators on Sobolev spaces. Using the composition operators theory we obtain estimates of constants in Poincaré-Sobolev inequalities and as a consequence lower estimates of the first non-trivial eigenvalue of the Neumann-Laplace operator in planar quasiconformal regular domains.

math.AP↗

arXiv2016

Fast-slow vector fields of reaction-diffusion systems

A geometrically invariant concept of fast-slow vector fields perturbed by transport terms (describing molecular diffusion processes) is proposed in this paper. It is an extension of our concept of singularly perturbed vector fields to reaction-diffusion systems. Fast-slow vector fields can be represented locally as "singularly perturbed systems of PDE". The paper focuses on possible ways of original models decomposition to fast and slow subsystems. We demonstrate that transport terms can be neglected (under reasonable physical assumptions) for fast subsystem. A method of analysis of slow subsystem as an evolution of singularly perturbed profiles along slow invariant manifolds was discussed in our previous work \cite{BCMGM2016}. This paper is motivated by an algorithm of reaction-diffusion manifolds (REDIM) \cite{BM2007,BM2009,MB2011}. It can be considered as its theoretical justification extending it from a practical algorithm to a robust computational procedure under some reasonable physical assumptions. A practical application of the proposed algorithm for numerical treatment of reaction-diffusion systems is demonstrated.

math.NA↗

arXiv2016

Singularly Perturbed Profiles

In the current paper the so-called REaction-DIffusion Manifold (REDIM) method of model reduction is discussed within the framework of standard singular perturbation theory. According to the REDIM a reduced model for the system describing a reacting flow (accounting for chemical reaction, advection and molecular diffusion) is represented by a low-dimensional manifold, which is embedded in the system state space and approximates the evolution of the system solution profiles in space and in time. This pure geometric construction is reviewed by using Singular Perturbed System (SPS) theory as the only possibility to formalize, to justify and to verify the suggested methodology. The REDIM is studied as a correction by the diffusion of the slow invariant manifold defined for a pure homogeneous system. A main result of the study is an estimation of this correction to the slow invariant manifold. A benchmark model of Mechaelis-Menten is extended to the system with the standard diffusion described by the Laplacian and used as an illustration and for validation of analytic results.

math.NA↗

arXiv2016

Conformal Spectral Stability Estimates for the Neumann Laplacian

We study the eigenvalue problem for the Neumann-Laplace operator in conformal regular planar domains $Ω\subset\mathbb{C}$. Conformal regular domains support the Poincaré inequality and this allows us to estimate the variation of the eigenvalues of the Neumann Laplacian upon domain perturbation via energy type integrals. Boundaries of such domains can have any Hausdorff dimension between one and two.

math.AP↗

arXiv2015

About Bifurcational Parametric Simplification

A concept of "critical" simplification was proposed by Yablonsky and Lazman in 1996 for the oxidation of carbon monoxide over a platinum catalyst using a Langmuir-Hinshelwood mechanism. The main observation was a simplification of the mechanism at ignition and extinction points. The critical simplification is an example of a much more general phenomenon that we call \emph{a bifurcational parametric simplification}. Ignition and extinction points are points of equilibrium multiplicity bifurcations, i.e., they are points of a corresponding bifurcation set for parameters. Any bifurcation produces a dependence between system parameters. This is a mathematical explanation and/or justification of the "parametric simplification". It leads us to a conjecture that "maximal bifurcational parametric simplification" corresponds to the "maximal bifurcation complexity." This conjecture can have practical applications for experimental study, because at points of "maximal bifurcation complexity" the number of independent system parameters is minimal and all other parameters can be evaluated analytically or numerically. We illustrate this method by the case of the simplest possible bifurcation, that is a multiplicity bifurcation of equilibrium and we apply this analysis to the Langmuir mechanism. Our analytical study is based on a coordinate-free version of the method of invariant manifolds (proposed recently in 2006 ). As a result we obtain a more accurate description of the "critical (parametric) simplifications." With the help of the "bifurcational parametric simplification" kinetic mechanisms and reaction rate parameters may be readily identified from a monoparametric experiment (reaction rate vs. reaction parameter).

physics.chem-ph↗

arXiv2013

Universal conformal weights on Sobolev spaces

The Riemann Mapping Theorem states existence of a conformal homeomorphism $φ$ of a simply connected plane domain $Ω\subset\mathbb C$ with non-empty boundary onto the unit disc $\mathbb D\subset \mathbb C$. In the first part of the paper we study embeddings of Sobolev spaces $\overset{\circ}{W_{p}^{1}}(Ω)$ into weighted Lebesgue spaces $L_{q}(Ω,h)$ with an {}"universal" weight that is Jacobian of $φ$ i.e. $h(z):=J(z,φ)=| φ'(z)|^2$. Weighted Lebesgue spaces with such weights depend only on a conformal structure of $Ω$. By this reason we call the weights $h(z)$ conformal weights. In the second part of the paper we prove compactness of embeddings of Sobolev spaces $\overset{\circ}{W_{2}^{1}}(Ω)$ into $L_{q}(Ω,h)$ for any $1\leq q<\infty$. With the help of Brennan's conjecture we extend these results to Sobolev spaces $\overset{\circ}{W_{p}^{1}}(Ω)$. In this case $q$ is not arbitrary and depends on $p$ and the summability exponent for Brennan's conjecture. Applications to elliptic boundary value problems are demonstrated in the last part of the paper.

math.FA↗

arXiv2009

Sobolev Homeomorphisms and Composition Operators

We study invertibility of bounded composition operators of Sobolev spaces. The problem is closely connected with the theory of mappings of finite distortion. If a homeomorphism $φ$ of Euclidean domains $D$ and $D'$ generates by the composition rule $φ^{\ast}f=f\circφ$ a bounded composition operator of Sobolev spaces $φ^{\ast}: L^1_{\infty}(D')\to L^1_p(D)$, $p>n-1$, has finite distortion and Luzin $N$-property then its inverse $φ^{-1}$ generates the bounded composition operator from $L^1_{p'}(D)$, $p'=p/(p-n+1)$, into $L^1_{1}(D')$.

math.CV↗

arXiv2008

Sobolev homeomorphisms and Poincare inequality

We study global regularity properties of Sobolev homeomorphisms on $n$-dimensional Riemannian manifolds under the assumption of $p$-integrability of its first weak derivatives in degree $p\geq n-1$. We prove that inverse homeomorphisms have integrable first weak derivatives. For the case $p>n$ we obtain necessary conditions for existence of Sobolev homeomorphisms between manifolds. These necessary conditions based on Poincaré type inequality: $$ \inf_{c\in \mathbb R} \|u-c\mid L_{\infty}(M)\|\leq K \|u\mid L^1_{\infty}(M)\|. $$ As a corollary we obtain the following geometrical necessary condition: {\em If there exists a Sobolev homeomorphisms $ϕ: M \to M'$, $ϕ\in W^1_p(M, M')$, $p>n$, $J(x,ϕ)\ne 0$ a. e. in $M$, of compact smooth Riemannian manifold $M$ onto Riemannian manifold $M'$ then the manifold $M'$ has finite geodesic diameter.}}

math.FA↗

arXiv2007

Weighted Sobolev spaces and embedding theorems

In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A_p-condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion.) The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic Hölder singularities demonstrate sharpness of our machinery comparatively with known results.

math.FA↗

arXiv2006

An indicator for community structure

An indicator for presence of community structure in networks is suggested. It allows one to check whether such structures can exist, in principle, in any particular network, without a need to apply computationally cost algorithms. In this way we exclude a large class of networks that do not possess any community structure.

physics.soc-ph↗

arXiv2004

Vulnerability and Hierarchy of Complex Networks

We suggest an approach to study hierarchy, especially hidden one, of complex networks based on the analysis of their vulnerability. Two quantities are proposed as a measure of network hierarchy. The first one is the system vulnerability V. We show that being quite suitable for regular networks this characteristic does not allow one to estimate the hierarchy of large random networks. The second quantity is a relative variance h of the system vulnerability that allows us to characterize a "natural" hierarchy level of random networks. We find that hierarchical properties of random networks depend crucially on a ratio between the number of nodes and the number of edges. We note that any graph with a transitive isometry group action (i.e. an absolutely symmetric graph) is not hierarchical. Breaking such a symmetry leads to appearance of hierarchy.

cond-mat.dis-nn↗