Supersymmetry, Witten Complex and Asymptotics for Directional Lyapunov Exponents in Zd

. We study the long distance asymptotic for random walks in random potentials. We use the analytical formulation. We relate the problem to Witten Laplacians via using supersymmetry. We obtain the asymptotic for the directional Lyapunov exponents.     

The inviscid limit for the complex Ginzburg-Landau equation

We study the inviscid limit of the complex Ginzburg-Landau equation. We observe that the solutions for the complex Ginzburg-Landau equation converge to the corresponding solutions for the nonlinear Schrödinger equation. We give its convergence rate. We estimate the integral forms of solutions for two equations.     

Stable exact solutions in cosmological models with two scalar fields

We consider the stability of isotropic solutions for two-field models in the Bianchi I metric. We prove that the sufficient conditions for Lyapunov stability in the Friedmann-Robertson-Walker metric ensure the stability under anisotropic perturbations in the Bianchi I metric and also under perturbations of the energy density for cold dark matter. We find sufficient conditions for the Lyapunov stability of isotropic fixed points for the system of Einstein equations. We use the superpotential method to construct stable kink-type solutions and obtain sufficient conditions on the superpotential for the Lyapunov stability of the corresponding exact solutions. We analyze the stability of isotropic kink-type solutions for models related to string field theory.     

Solvability of the stationary heat convection problem for a high-viscosity fluid

We study the stationary heat convection problem in the Boussinesq approximation. We derive a priori estimates for its solution. We prove existence and uniqueness theorems for a weak solution of the problem and analyze the smoothness of a weak solution for raised smoothness of the problem data. We consider the two- and three-dimensional cases.     

Nonequilibrium Statistical Operator Method and Generalized Kinetic Equations

We consider some principal problems of nonequilibrium statistical thermodynamics in the framework of the Zubarev nonequilibrium statistical operator approach. We present a brief comparative analysis of some approaches to describing irreversible processes based on the concept of nonequilibrium Gibbs ensembles and their applicability to describing nonequilibrium processes. We discuss the derivation of generalized kinetic equations for a system in a heat bath. We obtain and analyze a damped Schrödinger-type equation for a dynamical system in a heat bath. We study the dynamical behavior of a particle in a medium taking the dissipation effects into account. We consider the scattering problem for neutrons in a nonequilibrium medium and derive a generalized Van Hove formula. We show that the nonequilibrium statistical operator method is an effective, convenient tool for describing irreversible processes in condensed matter.     

Weakly analytic sets for function spaces

We define and study weakly analytic sets for function spaces. This concept generalizes the concept of weak analyticity for function algebras. We also define weak analyticity for the space of affine functions using the ideas of convexity. We show that it coincides with the same concept regarding it as a function space. We give several examples to illustrate the concepts.     

The distribution of the maximum likelihood-ratio statistic used for testing a normal sample for three outliers

We study the exact distribution of the likelihood-ratio statistic used for testing a normal sample for three upper (lower) outliers. We obtain recursive correlations for the integral distribution function of this statistic. We apply the obtained correlations for calculating critical values of the likelihood-ratio statistic which appear to be close to critical values of this statistic simulated by the Monte Carlo method. We give an example of the joint use of the likelihood-ratio statistic for testing a sample for more than one outlier.     

Uniform convexity and the splitting problem for selections

We continue to investigate cases when the Repovš–Semenov splitting problem for selections has an affirmative solution for continuous set-valued mappings. We consider the situation in infinite-dimensional uniformly convex Banach spaces. We use the notion of Polyak of uniform convexity and modulus of uniform convexity for arbitrary convex sets (not necessary balls). We study general geometric properties of uniformly convex sets. We also obtain an affirmative solution of the splitting problem for selections of certain set-valued mappings with uniformly convex images.     

Quasi-averages in the solution of the classification problem for equilibriums of condensed media with a spontaneously broken symmetry

We develop a statistical approach for solving the classification problem for equilibriums of degenerate condensed media. We introduce generators of unbroken and spatial symmetries of an equilibrium and use them to derive the classification equations for the order parameter. We elucidate the mechanism of the appearance of additional thermodynamic parameters characterizing both homogeneous and inhomogeneous equilibriums. We solve the classification problem for equilibriums of various liquid crystals analytically.     

On uniqueness sets for multiple Walsh series

We study uniqueness sets for multiple Walsh series under ρ-regular (or bounded) convergence in rectangles. We prove that a countable set is a uniqueness set for such a series under this convergence. We construct a class of perfect uniqueness sets for multiple Walsh series under this convergence. We show that the notion of index of a perfect set does not solve the problem of whether this set belongs to the class of uniqueness sets. We note that the results of this paper remain valid for several rearranged multiple Walsh series.     

Stochastic center of systems of stochastic differential equations on the plane

We study a stochastic analogy of the famous center problem of Dulac for quadratic differential equations in the plane. We introduce the concept of center for systems of stochastic differential equations of It\^o''s type on the plane, called stochastic center. We derive a criterion for the existence of such a center. We apply it to obtain necessary and sufficient conditions for quadratic stochastic differential equations in dimension 2.     

On null and alternative distribution of statistic of two-side discordancy test for an extreme outlier

We find the joint distribution of Grubbs statistics for a normal sample. Those statistics are standardized maximum and standardized minimum. We note some properties of the joint distribution function. We apply the joint distribution function and find the exact distribution of the test statistic which uses in two-sided discordancy test for an extreme outlier. We obtain recursive relationships for the distribution function of the statistic, which uses in two-sided discordancy test. We obtain the region of critical values of the statistic, where the significance level of criteria equals the double significance level of the Grubbs criteria. We apply the joint distribution of Grubbs statistics and find the power function for the criteria in the case of a normal sample with a single outlier.     

Inverse problems for quasi-variational inequalities

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.     

Economical Iterative Algorithms for Solving Stationary Problems of Mathematical Physics

We construct and investigate additive iterative methods of complete approximation for solving stationary problems of mathematical physics. We prove the convergence of the proposed methods and obtain error estimates without the requirement of commutativity of the decomposition operators. We provide the results of a computational experiment for a three-dimensional boundary-value problem. We consider possible generalizations of algorithms for equations with mixed derivatives and Navier–Stokes equation systems.     

A distance-based,misclassification rate adjusted classifier for multiclass,high-dimensional data

In this paper, we consider a scale adjusted-type distance-based classifier for high-dimensional data. We first give such a classifier that can ensure high accuracy in misclassification rates for two-class classification. We show that the classifier is not only consistent but also asymptotically normal for high-dimensional data. We provide sample size determination so that misclassification rates are no more than a prespecified value. We propose a classification procedure called the misclassification rate adjusted classifier. We further develop the classifier to multiclass classification. We show that the classifier can still enjoy asymptotic properties and ensure high accuracy in misclassification rates for multiclass classification. Finally, we demonstrate the proposed classifier in actual data analyses by using a microarray data set.     

Blow-up solutions of cubic differential systems

We investigate the existence problem for blow-up solutions of cubic differential systems. We find sets of initial values of the blow-up solutions. We also discuss a method of finding upper estimates for the blow-up time of these solutions. Our approach can be applied to systems of partial differential equations. We apply this approach to the Cauchy-Dirichlet problem for systems of semilinear heat equations with cubic nonlinearities.     

Bounds and Characterizations of Authentication/Secrecy Schemes

We consider authentication/secrecy schemes from the information theoretic approach. We extend results on unconditionally secure authentication schemes and then consider unconditionally secure authentication schemes that offer perfect L-fold secrecy. We consider both ordered and unordered secrecy. We establish entropy bounds on the encoding rules for authentication schemes with these types of secrecy. We provide some combinatorial characterizations and constructions for authentication schemes having perfect L-fold secrecy that meet these bounds.     

On distribution of Grubbs’ statistics in case of normal sample with outlier

We investigate one-sided Grubbs’ statistics for a normal sample. Those statistics are standardized maximum and standardized minimum, i.e., studentized extreme deviation statistics. We consider the case of the sample when there is one abnormal observation (outlier), unknown to what number according. The outlier differs from other observations in values of population mean and dispersion. We obtain recursive relationships for the marginal distribution function of one-sided Grubbs’ statistics. We find asymptotic formulas for marginal distribution functions. We obtain recursive relationships for the joint distribution function of one-sided Grubbs’ statistics and investigate its properties.     

A two-phase flow with a viscous and an inviscid fluid

We study the free boundary between a viscous and an inviscid fluid satisfying the Navier-Stokes and Euler equations respectively. Surface tension is incorporated. We read the equations as a free boundary problem for one viscous fluid with a nonlocal boundary force. We decompose the pressure distribution in the inviscid fluid into two contributions. A positivity result for the first, and a compactness property for the second contribution are dervied. We prove a short time existence theorem for the two-phase problem     

Periodic non-autonomous second-order dynamical systems

We study the existence of periodic solutions for a second-order non-autonomous dynamical system. We give three sets of hypotheses which guarantee the existence of non-constant solutions. We were able to weaken the hypotheses considerably from those used previously for such systems. We employ a saddle point theorem using linking methods.