Convergence of a Stochastic Method for the Modeling of Polymeric Fluids

We present a convergence analysis of a stochastic method for numerical modeling of complex fluids using Brownian configuration fields (BCF) for shear flows. The analysis takes into account the special structure of the stochastic partial differential equations for shear flows. We establish the optimal rate of convergence. We also analyze the nature of the error by providing its leading order asymptotics.     

Distribution of Deficit at Ruin for a PDMP Insurance Risk Model

In this paper we consider the risk process described by a piecewise deterministic Markov processes(PDMP). We mainly discuss the distribution of the deficit at ruin for the risk process. We derive the integrodifferential equation satisfied by this distribution. We obtain the explicit expressions for it for certain choices of the claim amount distribution.     

New Estimates for the Rate of Convergence of the Method of Subspace Corrections

We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections. We provide upper bounds and in a special case, a lower bound for preconditioners defined via the method of successive subspace corrections.     

Development of the Berezin Number Inequalities

We present new bounds for the Berezin number inequalities which improve on the existing bounds. We also obtain bounds for the Berezin norm of operators as well as the sum of two operators.     

The method of <Emphasis Type="Italic">A</Emphasis>-harmonic approximation and boundary regularity for nonlinear elliptic systems under the natural growth condition

We consider the questions of boundary regularity for weak solutions of second-order nonlinear elliptic systems under the natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. The proof yields directly the optimal regularity for the solution in this neighborhood. This result is new for the situation under the natural growth conditions.     

The mixed problem for degenerate elliptic equations of second order

The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the existence of solutions of the above problem for elliptic equations by the above estimates and the method of parameter extension. We use the complex method, namely first discuss the corresponding problem for degenerate elliptic complex equations of first order, afterwards discuss the above problem for degenerate elliptic equations of second order.     

Stable Solution of Nonlinear Age-structured Forest Evolution System

This paper studies the dynamical behavior of a class of total area dependent nonlinear age-structured forest evolution model. We give the problem of equal value for the forest system, and discuss the stable solution of system. We obtained the necessary and sufficient conditions for there exists the stable solution.     

Orbital-Free Density Functional Theory for Molecular Structure Calculations

We give here an overview of the orbital-free density functional theory that is used for modeling atoms and molecules.We review typical approximations to the kinetic energy,exchange-correlation corrections to the kinetic and Hartree energies, and constructions of the pseudopotentials.We discuss numerical discretizations for the orbital-free methods and include several numerical results for illustrations.     

Sharp Error Estimate for Maximum Likelihood Estimator of Nonstationary Diffusion Processes

We consider the maximum likelihood estimator of the unknown parameter in a class of nonstationary diffusion processes. We give further a precise estimate for the error of the estimator.     

A blow-up criterion for compressible viscous heat-conductive flows

We study an initial boundary value problem for the Navier-Stokes equations of compressible viscous heat-conductive fluids in a 2-D periodic domain or the unit square domain. We establish a blow-up criterion for the local strong solutions in terms of the gradient of the velocity only, which coincides with the famous Beale-Kato-Majda criterion for ideal incompressible flows.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Invariants of Characteristics of Some Systems of Partial Differential Equations

We introduce the notion of an invariant of characteristics for a system of first-order partial differential equations. We prove that the existence of invariants is connected with passiveness of some systems. We describe a few methods for construction of new invariants from those already known. We give a scheme for application of the invariants to reduction and integration of systems of partial differential equations. As an application we consider the equation of gas dynamics.     

Contingent derivatives and regularization for noncoercive inverse problems

Abstract

We study the inverse problem of parameter identification in noncoercive variational problems that commonly appear in applied models. We examine the differentiability of the set-valued parameter-to-solution map using the first-order and the second-order contingent derivatives. We explore the inverse problem using the output least-squares and the modified output least-squares objectives. By regularizing the noncoercive variational problem, we obtain a single-valued regularized parameter-to-solution map and investigate its smoothness and boundedness. We also consider optimization problems using the output least-squares and the modified output least-squares objectives for the regularized variational problem. We give a complete convergence analysis showing that for the output least-squares and the modified output least-squares, the regularized minimization problems approximate the original optimization problems suitably. We also provide the first-order and the second-order adjoint method for the computation of the first-order and the second-order derivatives of the output least-squares objective. We provide discrete formulas for the gradient and the Hessian calculation and present numerical results.