Showing papers in "Journal of Mathematical Sciences in 1995"

Linear recurring sequences over rings and modules

Abstract: Here we present some fundamental concept and results of the theory of linear recurring sequences over rings and modules and their applications. Of course, the authors give in more detail those results that are close to their mathematical interests. In particular, an attempt has been made to construct a general algebraic theory of k-LRS over modules, paying explicit attention to periodic k-sequences, to properties of linear recurrences over finite rings and especially over Galois rings, and also to methods of constructing codes baed on such recurrences.

Nonlinear scattering: the states which are close to a soliton

Abstract: We assume that the nonlinear Schroedinger equation with sufficiently general nonlinearity admits solutions of the soliton type. The Cauchy problem with initial data close to a soliton is considered. We also assume that the linearization of the equation in the vicinity of the soliton possesses only a real spectrum. The main result claims that the asymptotic behavior of the solution as t→+∞ is given by the sum of a soliton with deformed parameters and a dispersive tail, i.e., a solution of the linear Schroedinger equation. The case of the minimal spectrum has been considered in the previous paper. Bibliography: 1 title.

Nonlocal problems for the equations of motion of Kelvin-Voight fluids

Abstract: For the equations of the motion of Kelvin-Voight fluids (0.1) and for the semilinear abstract differential Eqs. (0.11)–(0.17) arising in the theory of Sobolev equations (to which Eqs. (0.1) also belong), we study the four following nonlocal problems: 1) the solvability of the initial boundary problem for Eqs. (0.1) and the Cauchy problem for Eqs. (0.11)–(0.17) on the semiaxis 0

Turning Points of Linear Systems and Double Asymptotics of the Painlevé Transcendents

Abstract: Recently, on the basis of the isomonodromy deformation method (IDM) [1], a considerable success has been achieved in the description of asymptotic properties of the Painleve transcendents especially in obtaining so-called connection formulae [2, 3, 4, 5], Moreover, there is no problem in using the same IDM technique for obtaining asymptotic formulae of Boutroux type in the complex domain, if the asymptotic behavior of a solution is given on a possible special ray (the analogs of Stokes rays) [6, 7]. It is also possible to obtain the justification of IDM asymptotic formulae by intrinsic means of the IDM. For different approaches to this problem, we refer to [8, 9, 10].

Weakly nonlinear solutions of equationP 1 2

Abstract: Using the isomonodromic deformation method, we study the equation P 1 2 , $$\frac{1}{{10}}y^{(4)} + y''y + \frac{1}{2}(y')^2 + y^3 = x$$ , which is the first higher equation in the hierarchy of the first Painleve equation. We construct the asymptotics of its weakly nonlinear solutions for x→∞ along the Stokes rays, and those of the real regular solutions for x→±∞. Bibliography: 11 titles.

Matrix models of two-dimensional quantum gravity and isomonodromic solutions of “discrete Painlevé” equations

Abstract: A method of studing of double scaling limits for the two-dimensional string models of quantum gravity is stated. It is actually shown that the study of such limits reduces to the isomonodromic deformation method for integrable discrete equations. A relationship is indicated between the “universality” and isomonodromy properties of the model. It is shown that the partition function of the model is the τ-function associated with the fourth Painleve equation (ℙ4), and also the Volterra chain. We consider in detail the properties of the Backlund transformations for ℙ4. Bibliography: 19 titles.

A generalized Green's formula for elliptic problems in domains with edges

Abstract: The usual Green's formula connected with the operator of a boundary-value problem fails when both of the solutions u and v that occur in it have singularities that are too strong at a conic point or at an edge on the boundary of the domain. We deduce a generalized Green's formula that acquires an additional bilinear form in u and v and is determined by the coefficients in the expansion of solutions near singularities of the boundary. We obtain improved asymptotic representations of solutions in a neighborhood of an edge of positive dimension, which together with the generalized Green's formula makes it possible, for example, to describe the infinite-dimensional kernel of the operator of an elliptic problem in a domain with edge. Bibliography: 14 titles.

Symmetric solutions for the first and second Painlevé equations

Abstract: For the equations P1 and P2, particular solutions of special form are given, for which the Cauchy problem is effectively solved, i.e., for the parameters describing their behavior near zero and infinity the connection formulas are obtained. Bibliography: 6 titles.

Asymptotic expansions of eigenvalues and eigenfunctions for elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube

Abstract: We consider spectral boundary value problems of Steklov, Neumann, and Dirichlet types for second-order elliptic operators with e-periodic coefficients in a perforated cube; the coefficients of the differential equations are assumed to satisfy some symmetry conditions. Complete asymptotic expansions with respect to the small parameter e are constructed for eigenvalues and eigenfunctions of the said problems. Bibliography: 24 titles.

Lp-estimates of the gradients of solutions of initial/boundary-value problems for quasilinear parabolic systems

Abstract: We prove that the gradients of solutions of parabolic systems of equations (linear and quasi-linear systems with quadratic nonlinearity with respect to the gradient) satisfying first of second boundary conditions are in Lp, p>2. It is assumed that the data of the problems belong to anisotropic spaces. The proof is based on the application of results on reverse Holder inequalities obtained previously by the author. Bibliography: 6 titles.

Positive definite probability densities and probability distributions

Abstract: In the first short survey article [H.-J. Rossberg,Wiss. Z. Karl Marx Univ.,37, 366–374 (1988)] the author indicated that positive definite probability densities deserve attention and that the investigation of this class leads to serious new mathematical problems. Later [R. Riedel and H.-J. Rossberg,Metrika, (submitted for publication)] it turned out that it is worthwhile to consider an even larger class of distribution functions, namely those having characteristic function f≥0; they were first briefly considered in [P. Levy,C. R. Acad. Sci.,265, 249–252 (1967)]. We survey the basic ideas and the present state of the developing new theory; moreover, we formulate open problems. Some new results are proved. For the sake of brevity we do not enter into a discussion of continuation problems and limit theorems; for these subjects see [H.-J. Rossberg,Appendix to Gnedenko B. W., Einfuhrung in die Wahrscheinlichkeitstheorie, 9th ed., Akademie-Verlag, Berlin (1991)], [H.-J. Rossberg,Math. Nachr.,141, 227–232 (1989)], and [H.-J. Rossberg,Lect. Notes Math.,1412, 296–308 (1989)].

Monodromy map and classical r -matrices

Abstract: A regularization of the Poisson brackets of the monodromy matrices based on the method of zero range potentials is proposed. Classification of the regularized brackets is reduced to the classical Yang-Baxter identity for the square of the initial Lie algebra. Bibliography: 11 titles.

A fast direct method of solving Hermitian fourth-order finite-element schemes for the Poisson equation

Abstract: The method of separation of variables (the Fourier method) is widely applied in solving difference boundary-value problems The popularity of this method is due primarily to the fact that in combination with the fast discrete Fourier transform it makes it possible to construct an algorithm with an asymptotic number of arithmetic operations close to optimal The author has used the method of separation of variables to construct algorithms for solving the grid schemes of the finite-element method with second-order precision for the Poisson equation in a rectangular domain The method used previously can be applied to Lagrangian and Hermitian finite element schemes fo arbitrary order of precision In the present paper we give genrealizations of this method using the example of Hermitian fourth-order finite-element schemes

Boundary estimates for the first-order derivatives of a solution to a nondivergent parabolic equation with composite right-hand side and coefficients of lower-order derivatives

Abstract: A linear parabolic equation with special singularities is studied. A priori boundary estimates are established for the maximum of the modulus of the gradient of a solution and for the Holder constants as well. These estimates depend linearly on the functions appearing on the right-hand side of the equation. Bibliography: 7 titles.

Convergence of weighted sums of independent random variables

Abstract: Let {Xk} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s Fn(x) of sums $$S_n = \Sigma _{k = 1}^{m_k } a_{nk}^{1/a} X_k - A_n $$ is studied, where 0 0, 1≤k≤mn, and, as n→∞, bothmax 1≤k≤mna nk→0 and $$\Sigma _{k = 1}^{m_n } a_k \to 1$$ . It is shown that such convergence, with suitably chosen An's and necessarily stable limit laws, holds for all such arrays {αnk} provided it holds for the special case αnk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence of the moments of the sequence {Fn(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums $$S_n = \left( {1/b_n } \right)\Sigma _{k = 1}^n a_k X_k - A_n $$ , where an≠0, bn>0, andmax 1≤k≤n ak/bn→0. The strong law is said to hold if there are constants An for which Sn→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which bn/|an| 0. The main result is as follows. If the strong law holds,EN (|X1|)<∞. If $$\int_{ - \infty }^\infty {\left| x \right|^p } \int_{\left| x \right|}^\infty {N\left( y \right)} /y^{p + 1} dydF< \infty $$ for some 0

Finite-gap attractors and transition processes of the shock-wave type in integrable systems

Abstract: An attractor interpretation of finite-gap solutions in integrable KdV-type systems is suggested. Certain shock-wave problems in these systems are discussed. Bibliography: 25 titles.

Probabilistic methods for obtaining asymptotic formulas for generalized Stirling numbers

Abstract: In the present paper probabilistic methods are used to study the asymptotic behavior of Stirling numbers.

Construction of Cauchy integrals for one class of nonanalytic functions of complex variables

Abstract: For some values of characteristics of p-analytic and (p, q)-analytic functions, the values of kernels of Cauchy integrals are found in an explicit form involving elementary and special functions. The application of the Cauchy integral for deriving the Muskhelishvili equation in the Stokes problem on a slow axially symmetric flow of a viscous liquid is demonstrated. Bibliography: 9 titles.

An approximate method of determining the stress intensity factors for two-dimensional multiply connected bodies with rectilinear cracks

Abstract: We propose a new method of determining the stress intensity factors based on the representation of rectilinear cracks as elliptic holes with a zero axis and using the classical complex potentials. We calculate the stress intensity factors in terms of the limiting values of the potentials as the points tend to the ends of a crack. The efficiency of the method is shown using examples.

Noncommutative differential geometry related to the Young-Baxter equation

Abstract: An analogue of the differential calculus associated with a unitary solution of the quantum Young-Baxter equation is constructed. An example of a ring sheaf is considered in which local solutions of the Young-Baxter quantum equation are defined but there is no global section. Bibliography: 13 titles.

Influence of viscosity on the structure of shock waves in the MKdV model

Abstract: A “non-self-adjoint” integrable MKdV model with boundary conditions of “step-like” type is considered. The time-asymptotic behavior of the solution for the Cauchy problem is obtained. A similar problem for the dissipative perturbation of this problem is discussed. Bibliography: 10 titles.