On Recognition by Spectrum of Finite Simple Linear Groups over Fields of Characteristic 2

A finite group G is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group H having the same spectrum as G is isomorphic to G. We prove that the simple linear groups L _n (2^k) are recognizable by spectrum for n = 2^m ≥ 32.Original Russian Text Copyright © 2005 Vasil’ev A. V. and Grechkoseeva M. A.The authors were supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program “ Development of the Scientific Potential of Higher School” of the Ministry for Education of the Russian Federation (Grant 8294), the Program “Universities of Russia” (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 749–758, July–August, 2005.     

Asymptotic behavior of solutions to the Dirichlet problem for parabolic equations in domains with singularities

We study solutions of the Dirichlet problem for a second-order parabolic equation with variable coefficients in domains with nonsmooth lateral surface. The asymptotic expansion of the solution in powers of the parabolic distance is obtained in a neighborhood of a singular point of the boundary. The exponents in this expansion are poles of the resolvent of an operator pencil associated with the model problem obtained by freezing the coefficients at the singular point. The main point of the paper is in proving that the resolvent is meromorphic and in estimating it. In the one-dimensional case, the poles of the resolvent satisfy a transcendental equation and can be expressed via parabolic cylinder functions.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 12–23, January, 1996.This work was partially supported by the Russian Foundation for Basic Research under grant No. 242 93-01-16035.     

Kantorovich theorem and justification of asymptotics of solutions of differential equations with large high-frequency summands

This paper considers systems of ordinary differential equations of the 1st order whose terms oscillate with a high frequency ω, among which there exist those proportional to , the large high-frequency summands. For the problem on periodic solutions, the author justifies the averaging method and establishes a posteriori bounds for the error of partial sums of the complete asymptotic expansion for the solution. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

Stability Estimates in Inverse Problems for Hyperbolic Equations

We discuss here a method for investigating inverse problems for hyperbolic equations based on combining an asymptotic expansion of solutions to the equations in a neighborhood of a characteristic surface, the connections between coefficients of the expansion and the unknown coefficients of the equation and the estimates of a solution to the Cauchy problem in terms of the data on a time-like surface. We give some applications of this method to a number of inverse problems. Stability results to the inverse problems are given. This work was partly supported by the Russian Foundation for Basic Research (Grant No. 05-01-00171). Received: December 2005     

The leading term of the spectral asymptotics for the Kohn-Laplace operator in a bounded domain

We prove the Weyl asymptotic formula for the number of eigenvalues of the Kohn-Laplace operator on a Heisenberg group and write out the leading term of asymptotics. The method of study is based on estimates of the Green function for the Dirichlet problem for the corresponding parabolic operator and makes use of the classical Hardy-Littlewood Tauberian theorem.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 493–505, October, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00443 and No. 96-01-00503.     

Szego-Type Limit Theorems for Generalized Discrete Convolution Operators

We study the asymptotic behavior of the averaged f-trace of a truncated generalized multidimensional discrete convolution operator as the truncation domain expands. By definition, the averaged f-trace of a finite-dimensional operator A is equal to , where n is the dimension of the space in which the operator A acts, the set of numbers γ_k, k = 1,..., n, is the complete collection of eigenvalues of the operator A, counting multiplicity; a generalized discrete convolution is an operator from the closure of the algebra generated by discrete convolution operators and by operators of multiplication by functions admitting a continuous continuation onto the sphere at infinity.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 265–277.Original Russian Text Copyright © 2005 by I. B. Simonenko.     

About the Admissible Predicates on Admissible Sets

We study the admissible predicates, i.e., the predicates having the property that their addition to the signature of an admissible set preserves the property “to be an admissible set.” We show that the family of these predicates is much wider than the family of Δ-predicates. We also construct a family of admissible predicates of cardinality 2^ω such that the addition of an arbitrary pair of predicates of this family to the signature of an admissible set violates the admissibility of the latter as well as other examples of families of admissible predicates.Original Russian Text Copyright © 2005 Morozov A. S.The author was supported by the International Russian-German Program (Grant RFRB-DFG 01-01-04003), the Russian Science Support Foundation, and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant 2112.2003.1).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 841–850, July–August, 2005.     

On asymptotics of solutions of parabolic equations with nonlocal high-order terms

In the paper, we study the Cauchy problem for second-order differential-difference parabolic equations containing translation operators acting to the high-order derivatives with respect to spatial variables. We construct the integral representation of the solution and investigate its long-term behavior. We prove theorems on asymptotic closeness of the constructed solution and the Cauchy problem solutions for classical parabolic equations; in particular, conditions of the stabilization of the solution are obtained. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 143–183, 2005.     

Effective Solution of the Carleman Problem for Some Groups of Divergent Type

We consider a Fuchsian group of the first kind containing only hyperbolic transformations. We obtain an effective solution (explicit in terms of transformations of the group) to the Carleman boundary value problem on a fundamental polygon where the inverse shift is induced by the generating transformations of the group.Original Russian Text Copyright © 2005 Aksent’eva E. P. and Garif’yanov F. N.The authors were supported by the Russian Foundation for Basic Research (Grant 02-01-00914).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 723–733, July–August, 2005.     

On Blowup of a Solution to a Sobolev-Type Equation with a Nonlocal Source

We consider the initial-boundary value problem for a dissipative nonlinear wave equation of the Sobolev type with a nonlocal cubic source. For this problem we obtain a sufficient condition for blowup of a strong generalized solution. Moreover, we obtain a two-sided estimate for the blowup time of the solution.Original Russian Text Copyright © 2005 Korpusov M. O. and Sveshnikov A. G.The authors were supported by the Russian Foundation for Basic Research (Grants 02-01-00253; 02-01-06038) and a grant of the Project Junior Scientists of Russia.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 567–578, May–June, 2005.     

The Degree Spectra of Definable Relations on Boolean Algebras

We study some questions concerning the structure of the spectra of the sets of atoms and atomless elements in a computable Boolean algebra. We prove that if the spectrum of the set of atoms contains a 1-low degree then it contains a computable degree. We show also that in a computable Boolean algebra of characteristic (1, 1, 0) whose set of atoms is computable the spectrum of the atomless ideal consists of all Π ₀ ² degrees.Original Russian Text Copyright © 2005 Semukhin P. M.The author was supported by the Russian Foundation for Basic Research (Grant 02-01-00593), the Leading Scientific Schools of the Russian Federation (Grant NSh-2112.2003.1), and the Program “Universities of Russia” (Grant UR.04.01.013).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 928–941, July–August, 2005.     

Linear parabolic problem: High-frequency asymptotics in the critical case

A second-order linear parabolic problem with high-frequency terms is considered. The elliptic operator of the corresponding limiting (averaged) problem is assumed to be degenerate. A complete formal asymptotic expansion of a time-periodic solution of the perturbed problem is constructed.     

Asymptotics of Bounded-at-Infinity Solutions of the Principal Resonance Equation

We consider a system of six nonlinear differential equations obtained by averaging fast forced oscillations. The main result consist in the construction of the asymptotics at infinity for the general solution with bounded amplitudes. We show that the structure of asymptotic series depends on the parameters so that the coefficients of the series vary in jumps on the resonance set.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 85–97.Original Russian Text Copyright © 2005 by L. A. Kalyakin, Yu. Yu. Bagderina.     

A Generalization of the Euler Gamma Function

We define a generalized Euler gamma function Λ^β(z), where the product is taken over powers of integers rather than integers themselves. Studying the associated spectral functions and in particular the zeta function, we obtain the main properties of Λ^β(z) and its asymptotic expansion for large values of the argument.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 87–91, 2005Original Russian Text Copyright © by M. Spreafico     

Necessary Conditions for the Localization of the Spectrum of the Sturm-Liouville Problem on a Curve

We consider the Sturm-Liouville operator on a convex smooth curve lying in the complex plane and connecting the points 0 and 1. We prove that if the eigenvalues λ_k with large numbers are localized near a single ray, then this ray is the positive real semiaxis. Moreover, if the eigenvalues λ_k are numbered with algebraic multiplicities taken into account, then λ_k ∼ π · k as k → +∞.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 72–84.Original Russian Text Copyright © 2005 by Kh. K. Ishkin.     

k-Best Solutions under Distance Constraints in Valuated -Matroids

In discrete maximization problems one usually wants to find an optimal solution. However, in several topics like “alignments,” “automatic speech recognition,” and “computer chess” people are interested to find thekbest solutions for somek ≥ 2. We demand that theksolutions obey certain distance constraints to avoid that thekalternatives are too similar. Several results for valuated -matroids are presented, some of them concerning time complexity of algorithms.     

Asymptotic expansions for partial solutions of the sixth Painlevé equation

A formalism for an averaging method for the Painlevé equations, in particular, the sixth equation, is developed. The problem is to describe the asymptotic behavior of the sixth Painlevé transcendental in the case where the module of the independent variable tends to infinity. The corresponding expansions contain an elliptic function (ansatz) in the principal term. The parameters of this function depend on the variable because of the modulation equation. The elliptic ansatz and the modulation equation for the sixth Painlevé equation are obtained in their explicit form. A partial solution of the modulation equation leading to a previously unknown asymptotic expansion for the partial solution of the sixth Painlevé equation is obtained.     

The asymptotic expansion of the solution to the Cauchy problem for a parabolic equation in a perforated space

We consider the Cauchy problem for a second-order parabolic equation with rapidly oscillating coefficients of the forma(ɛ ⁻ x,ɛ ^−2k t) in a perforated space ℝ _ɛ ⁿ . We construct a complete asymptotic expansion for the solution of the said problem and obtain an estimate for the remainder term in this expansion. Bibliography: 8 titles. To my dear Teacher, Olga Arsenievna Oleinik This research was supported in part by Grant MIE000 from the International Science Foundation. Translated from Trudy Seminara imeni I. G. Petrovskogo. No. 19, pp. 000-000. 0000.     

On the structure of solutions of a class of hyperbolic systems with several spatial variables in the far field

An asymptotic expansion of the solution to the Cauchy problem for a class of hyperbolic weakly nonlinear systems with many spatial variables is constructed. A parabolic quasilinear equation describing the behavior of the solution at asymptotically large values of the independent variables is obtained. The pseudo-diffusion processes that depend on the relationship between the number of equations and the number of spatial variables are analyzed. The structure of the subspace in which there are pseudo-diffusion evolution processes of the solution in the far field is described.     

Homotopy Theories of Algebras over Operads

Homotopy theories of algebras over operads, including operads over “little n-cubes,” are defined. Spectral sequences are constructed and the corresponding homotopy groups are calculated.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 278–285.Original Russian Text Copyright © 2005 by V. A. Smirnov.