On the subset sum problem over finite fields

The subset sum problem over finite fields is a well-known NP-complete problem. It arises naturally from decoding generalized Reed–Solomon codes. In this paper, we study the number of solutions of the subset sum problem from a mathematical point of view. In several interesting cases, we obtain explicit or asymptotic formulas for the solution number. As a consequence, we obtain some results on the decoding problem of Reed–Solomon codes.     

Asymptotic Integration of Parabolic Problems with Large High-Frequency Summands

We develop the averaging method theory for parabolic problems with rapidly oscillating summands some of which are large, i.e., proportional to the square root of the frequency of oscillations. In this case the corresponding averaged problems do not coincide in general with those obtained by the traditional averaging, i.e., by formally averaging the summands of the initial problem (since the principal term of the asymptotic expansion of a solution to the latter problem is not in general a solution to the so-obtained problem). In this article we consider the question of time periodic solutions to the first boundary value problem for a semilinear parabolic equation of an arbitrary order 2k whose nonlinear terms, including the large, depend on the derivatives of the unknown up to the order k-1. We construct the averaged problem and the formal asymptotic expansion of a solution. When the large summands depend on the unknown rather than its derivatives we justify the averaging method and the complete asymptotic expansion of a solution.Original Russian Text Copyright © 2005 Levenshtam V. B.The author was supported by the Russian Foundation for Basic Research (Grant 01-01-00678) and the Program “ Universities of Russia” (UR.04.01.029).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 805–821, July–August, 2005.     

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 1. Linearization on an antisymmetric solution

In this paper our objective is to provide physically reasonable solutions for the stationary Navier–Stokes equations in a two-dimensional domain with two outlets to infinity, a semi-strip Π₋ and a half-plane K. The same problem in an aperture domain, i.e. in a domain with two half-plane outlets to infinity, has been studied but only under symmetry restrictions on the data. Here, we assume that the main asymptotic term of the solution takes an antisymmetric form in K and apply the technique of weighted spaces with detached asymptotics, i.e. we use spaces where the functions have prescribed asymptotic forms in the outlets.After first showing that the corresponding Stokes problem admits a unique solution if and only if certain compatibility conditions are satisfied, we write the Navier–Stokes equations as a perturbation of the Stokes problem and the crucial compatibility condition as an algebraic equation by which the flux becomes determined. Assuming that the coefficient of the main (antisymmetric) asymptotic term of the solution in K does not vanish and that the data are sufficiently small, we use a contraction principle to solve the Navier–Stokes system coupled with the algebraic equation.Finally, we discuss the ill-posedness of the Navier–Stokes problem with prescribed flux.     

Global classical solution of quasi-stationary Stefan free boundary problem

We consider an one-phase quasi-stationary Stefan problem (Hele–Shaw problem) in multidimensional case. Under some reasonable conditions we prove that the problem has a classical solution globally in time. The method can be used in two-phase problem as well. We also discuss asymptotic behavior of solution as t→+∞. The method developed here can be extended to a general class of free boundary problems.     

Approximation of Continuous Functions by de la Vallée-Poussin Operators

For the upper bounds of the deviations of a function defined on the entire real line from the corresponding values of the de la Vallée-Poussin operators, we find asymptotic equalities that give a solution of the well-known Kolmogorov–Nikol'skii problem.     

Properties of solutions of a nonlinear system of equations

The objective of this paper is to study a highly general model system of nonlinear equations that describe propagation of nerve impulses in a cell. We prove time-local existence of a classical solution, derive sufficient conditions for the existence of a time-global solution, and consider the question of eventual smoothing of discontinuous initial values. The large-time asymptotic expansion of the solutions of the Cauchy problem is constructed. The coefficients of the principal term of the asymptotic expression for the solution of the Cauchy problem are computed in terms of the Fourier transform of the initial values.The study has been partially supported by the Russian Foundation of Basic Research (93-011-131).Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 3, pp. 64–93, 1993.     

Perturbation of Solutions with Moving Singularities

We study the Cauchy problem for a nonlinear second-order differential equation with a small parameter in the case where the exact solution has a power singularity depending on a small parameter. We propose an asymptotic method similar to the Krylov–Bogoliubov method for localizing the singularity up to the accuracy of any order and construct an asymptotic expansion of the solution in the domain of regular behavior.     

Asymptotic behavior of the solution of the cauchy problem for perturbed Klein-Fock-Gordon equation

This paper deals with the Cauchy problem for the perturbed Klein-Fock-Gordon equation. The principal term of an asymptotic expansion of the solution is constructed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 115–121, 1987.     

Linear continuous operators for the Stieltjes moment problem in Gelfand–Shilov spaces

We obtain linear continuous operators providing a solution to the Stieltjes moment problem in the framework of Gelfand–Shilov spaces of rapidly decreasing smooth functions. The construction rests on an interpolation procedure due to R. Estrada for general rapidly decreasing smooth functions, and adapted by S.-Y. Chung, D. Kim and Y. Yeom to the case of Gelfand–Shilov spaces. It requires a linear continuous version of the so-called Borel–Ritt–Gevrey theorem in asymptotic theory.     

Asymptotics of a solution of a discontinuous singularly perturbed Cauchy problem

We construct the asymptotic expansion of a solution of the Cauchy problem for a singularly perturbed system of differential equations whose right-hand side is discontinuous on a certain surface. We consider the case where the surface of discontinuity is crossed and estimate the remainder of the constructed asymptotic expansion.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1502–1508, November, 1994.The present work was supported by the Ukrainian State Committee on Science and Technology.     

Short-wavelength asymptotic solution of the problem of a moving point source in a time-dependent inhomogeneous medium

A uniform short-wavelength asymptotic solution is obtained for the problem of the field of a point source moving with subsonic velocity in a time-dependent inhomogeneous medium; the solution is valid both in a small neighborhood of the source and at a large distance from it.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 42–51, 1987.     

A Christoffel–Darboux formula for multiple orthogonal polynomials

Bleher and Kuijlaars recently showed that the eigenvalue correlations from matrix ensembles with external source can be expressed by means of a kernel built out of special multiple orthogonal polynomials. We derive a Christoffel–Darboux formula for this kernel for general multiple orthogonal polynomials. In addition, we show that the formula can be written in terms of the solution of the Riemann–Hilbert problem for multiple orthogonal polynomials, which will be useful for asymptotic analysis.     

Construction of an asymptotic expansion for the solution of a linear system of equations of neutral type with a small delay in the absence of a boundary layer

In this paper we simplify the algorithm for constructing the asymptotic expansion for the solution of a linear system of neutral type at a large distance from the origin. After using the Laplace transformation to determine the asymptotic expansion near the initial point, we succeed in reducing the problem of determining the initial conditions to the computation of the residues of certain functions for which we have recurrence formulas.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 109–113, July, 1969.     

Periodic solutions of monotone systems with delay

A class of nonlinear delay differential systems with a monotone nonlinearity is studied. Questions of solvability of the basic initial value problem, justification of the Galërkin method of finding a periodic solution, and of asymptotic behavior of the solution in the case of small or large parameter are considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 659–668, May, 1990.     

Asymptotic solution of a problem with one and two cusp points

The perturbation method and the Langer transform method are applied to obtain a second approximate solution of the problem with one and two cusp points. As an example, Langer type asymptotic formulas are derived for Bessel functions.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 42–50, 1989.     

Asymptotic properties of a solution of a boundary-value problem

The asymptotic behavior of the solution of a boundary-value problem for the equation u_txx+ u_x =f when the time tends to infinity is investigated. It is proved that the time mean of the solution tends to a stationary solution everywhere except in a boundary region at the left end of the interval.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 273–284, September, 1970.     

Problem of an elastic half-plane with a crack emerging orthogonally at the boundary

The problem of the half-plane, in which a finite crack emerges orthogonally at the boundary, is studied. On the edges of the crack a self-balancing load is applied. A detailed investigation is carried out for an integral equation with respect to the unknown derivative of the displacement jump, to which the problem can be reduced. The exact solution of the integral equation is constructed with the aid of the Mellin transform and the Riemann boundary value problem for the halfplane. The asymptotic behavior of the solution at both ends of the crack is elucidated. First the asymptotic behavior of the solution at the point of emergence of the crack is obtained and the dependence of this asymptotic behavior on the type of the load is established. For a special form of the load one obtains a simple expression of the stress intensity coefficient. In the case of a general load, the asymptotic behavior is used for the construction of an effective approximate solution on the basis of the method of orthogonal polynomials. As a result, the problem reduces to an infinite algebraic system, solvable by the reduction method.Translated from Dinamicheskie Sistemy, No. 4, pp. 45–51, 1985.     

Verification of asymptotic solutions for one-dimensional nonlinear parabolic equations

The present article proves a result that is new for partial differential equations. According to this result, the solution of the Cauchy problem for a nonlinear parabolic equation with variable, slowly changing coefficients will turn into (asymptotically approach) a special asymptotic solution, either a solution or a kink, for high values of t.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 10–16, September, 1992.     

Asymptotics of small spacings for subsonic nonstationary streamlined flow of a thin airfoil close to a solid boundary

In an asymptotic approximation of small spacings an analytic solution to the problem on harmonic oscillations of a thin airfoil which is moving with a subsonic velocity near a solid plane boundary is given. Results of a computation of the lifting force are given.Translated from Dinamicheskie Sistemy, No. 7, pp. 48–53, 1988.     

Approximation of Classes of Analytic Functions by de la Vallée-Poussin Sums

For upper bounds of the deviations of de la Vallée-Poussin sums taken over classes of functions that admit analytic extensions to a fixed strip of the complex plane, we obtain asymptotic equalities. In certain cases, these equalities give a solution of the corresponding Kolmogorov–Nikol'skii problem.