Asymptotic error estimates of a linearized projection-difference method for a differential equation with a monotone operator

We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A(t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A(t). The method is applied to a model parabolic problem.     

Galerkin method for a third-order differential-operator equation

We study the Galerkin method for a third-order differential-operator equation with self-adjoint leading operator A and subordinate linear operator K(t) in a separable Hilbert space. We prove a theorem on the existence and uniqueness of a strong solution of the original problem. We derive estimates for the accuracy of the approximate solutions constructed by the Galerkin method. An application of the suggested method to the solution of a model problem is described.     

Smooth diffeomorphisms of the plane with stable periodic points in a neighborhood of a homoclinic point

We consider self-diffeomorphisms of the plane of the class C ^r (1 ?? r < ??) with a fixed hyperbolic point and a nontransversal point homoclinic to it. We present a method for constructing a set of diffeomorphisms for which the neighborhood of a homoclinic point contains countably many stable periodic points with characteristic exponents bounded away from zero.     

How to change a boundary condition in a problem to make the problem have a prescribed spectrum

We consider a boundary value problem for an ordinary differential equation of order n with a spectral parameter in n boundary conditions. We suggest a method for changing one of the boundary conditions so as to make the problem have a prescribed spectrum.     

Asymptotic expansions for the distribution of the crossing number of a strip by a Markov-modulated random walk

We obtain complete asymptotic expansions for the distribution of the crossing number of a strip in n steps by sample paths of a random walk defined on a finite Markov chain. We assume that the Cramér condition holds for the distribution of jumps and the width of the strip grows with n. The method consists in finding factorization representations of the moment generating functions of the distributions under study, isolating the main terms in the asymptotics of the representations, and inverting those main terms by the modified saddle-point method.     

The use of first integrals in the problem of optimal control of a linear system with a quadratic functional

We propose a method for constructing an optimal control of a linear system in a variational problem with fixed time for the control process, fixed endpoints of the phase trajectory, and a quadratic functional. The method is based on the use of first integrals of the equations of unperturbed motion. We obtain sufficient conditions for complete controllability of the linear nonstationary system.     

The cyclicity of a cubic system with nonradical Bautin ideal

We present a method for investigating the cyclicity of an elementary focus or center of a polynomial system of differential equations by means of complexification of the system and application of algorithms of computational algebra, showing an approach to treating the case that the Bautin ideal B of focus quantities is not a radical ideal (more precisely, when the ideal BK is not radical, where BK is the ideal generated by the shortest initial string of focus quantities that, like the Bautin ideal, determines the center variety). We illustrate the method with a family of cubic systems.     

Numerical simulations for a seawater intrusion problem in a free aquifer

We simulate a sharp interface model issuing from a seawater intrusion problem in a free aquifer. We model the evolution of the sea front and of the upper free surface of the aquifer. We use a P1 finite element method for the space discretization combined with a semi-implicit in time scheme.     

Group classification of a family of second-order differential equations

We find the group of equivalence transformations for equations of the form y^″=A(x)y^′+F(y), where A and F are arbitrary functions. We then give a complete group classification of this family of equations using a direct method of analysis, together with the equivalence transformations.     

Blowup of solutions of a Korteweg-de Vries-type equation

We investigate the nonlinear third-order differential equation (u_xx ? u)_t + u _xxx + uu_x = 0 describing the processes in semiconductors with a strong spatial dispersion. We study the problem of the existence of global solutions and obtain sufficient conditions for the absence of global solutions for some initial boundary value problems corresponding to this equation. We consider examples of solution blowup for initial boundary value and Cauchy problems. We use the Mitidieri-Pokhozhaev nonlinear capacity method.     

A new inversion-free method for a rational matrix equation

Motivated by the classical Newton-Schulz method for finding the inverse of a nonsingular matrix, we develop a new inversion-free method for obtaining the minimal Hermitian positive definite solution of the matrix rational equation X+A^∗X^-1A=I, where I is the identity matrix and A is a given nonsingular matrix. We present convergence results and discuss stability properties when the method starts from the available matrix AA^∗. We also present numerical results to compare our proposal with some previously developed inversion-free techniques for solving the same rational matrix equation.     

Computing p-values in conditional independence models for a contingency table

We present a Markov chain Monte Carlo (MCMC) method for generating Markov chains using Markov bases for conditional independence models for a four-way contingency table. We then describe a Markov basis characterized by Markov properties associated with a given conditional independence model and show how to use the Markov basis to generate random tables of a Markov chain. The estimates of exact p-values can be obtained from random tables generated by the MCMC method. Numerical experiments examine the performance of the proposed MCMC method in comparison with the χ ² approximation using large sparse contingency tables.     

Homogenization of a Ginzburg–Landau model for a nematic liquid crystal with inclusions

We consider a nonlinear homogenization problem for a Ginzburg–Landau functional with a (positive or negative) surface energy term describing a nematic liquid crystal with inclusions. Assuming that inclusions are separated by distances of the same order ɛ as their size, we find a limiting functional as ɛ approaches zero. We generalize the variational method of mesocharacteristics to show that a corresponding homogenized problem for arbitrary, periodic or non-periodic geometries is described by an anisotropic Ginzburg–Landau functional. We obtain computational formulas for material characteristics of an effective medium. As a byproduct of our analysis, we show that the limiting functional is a Γ-limit of a sequence of Ginzburg–Landau functionals. Furthermore, we prove that a cross-term corresponding to interactions between the bulk and the surface energy terms does not appear at the leading order in the homogenized limit.     

An application of Lemke's method to a class of Markov decision problems

This paper presents an application of Lemke's method to a class of Markov decision problems, appearing in the optimal stopping problems, and other well-known optimization problems. We consider a special case of the Markov decision problems with finitely many states, where the agent can choose one of the alternatives; getting a fixed reward immediately or paying the penalty for one term. We show that the problem can be reduced to a linear complementarity problem that can be solved by Lemke's method with the number of iterations less than the number of states. The reduced linear complementarity problem does not necessarily satisfy the copositive-plus condition. Nevertheless we show that the Lemke's method succeeds in solving the problem by proving that the problem satisfies a necessary and sufficient condition for the extended Lemke's method to compute a solution in the piecewise linear complementarity problem.     

An iterative method for a system of linear complementarity problems with perturbations and interval data

In this paper, we introduce a total step method for solving a system of linear complementarity problems with perturbations and interval data. It is applied to two interval matrices [A] and [B] and two interval vectors [b] and [c]. We prove that the sequence generated by the total step method converges to ([x^∗],[y^∗]) which includes the solution set for the system of linear complementarity problems defined by any fixed A∈[A],B∈[B],b∈[b] and c∈[c]. We also consider a modification of the method and show that, if we start with two interval vectors containing the limits, then the iterates contain the limits. We close our paper with two examples which illustrate our theoretical results.     

Estimates for the lowest eigenvalue of a star graph

We derive new estimates for the lowest eigenvalue of the Schrödinger operator associated with a star graph in R². We achieve this by a variational method and a procedure for identifying test functions which are sympathetic to the geometry of the star graph.     

Stability of stationary solutions of a semilinear parabolic partial differential equation

We consider a parabolic partial differential equation u_t = u_xx + f(u) on a compact interval of spatial variable x with Dirichlet boundary conditions. The stability of stationary solutions of this system is studied by the use of Liapunov's second method. We obtain necessary and sufficient conditions for the stability, asymptotic stability, neutral stability, instability, and conditional stability. These conditions are closely connected with the conditions for the existence of the stationary solutions.     

Isoperimetric properties of Euclidean boundary moments of a simply connected domain

We consider integral functionals of a simply connected domain which depend on the distance to the domain boundary. We prove an isoperimetric inequality generalizing theorems derived by the Schwarz symmetrization method. For L ^p -norms of the distance function we prove an analog of the Payne inequality for the torsional rigidity of the domain. In compare with the Payne inequality we find new extremal domains different from a disk.     

On a problem of nonoverlapping domains

We solve the problem of finding the range E of some functional on the class of pairs of functions univalent in the system of the disk and the interior of the disk for the arbitrary parameters characterizing the functional. We prove that E is connected and bounded. Using the method of internal variations and the parametric method, we find the equation of the boundary of E. The obtained results extend Lebedev’s study [1].