Some Spectral Properties of One Sturm-Liouville Type Problem with Discontinuous Weight

We consider a discontinuous weight Sturm-Liouville equation together with eigenparameter dependent boundary conditions and two supplementary transmission conditions at the point of discontinuity. We extend and generalize some approaches and results of the classic regular Sturm-Liouville problems to the similar problems with discontinuities. In particular, we introduce a special Hilbert space formulation in such a way that the problem under consideration can be interpreted as an eigenvalue problem for a suitable selfadjoint operator, construct the Green’s function and resolvent operator, and derive asymptotic formulas for eigenvalues and normalized eigenfunctions.Original Russian Text Copyright © 2005 Mukhtarov O. Sh. and Kadakal M.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 860–875, July–August, 2005.     

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.     

<Emphasis Type="Italic">n</Emphasis>-Lie Structures That Are Generated by Wronskians

We study the (k + 1)-Lie structures, k-left commutative and homotopy (k + 1)-Lie structures with multiplication generated by Wronskians and prove that the nontrivial structures of n-Lie algebras appear only in the case of small characteristic.Original Russian Text Copyright © 2005 Dzhumadil’daev A. S.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 759–773, July–August, 2005.     

On R-Universal Functions

We prove that the ρ-resemblance type of an R-universal function (R ≠ = ∅, ℕ) consists of one isomorphism type, but of more than one ρ-recursive isomorphism type.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 259–264.Original Russian Text Copyright © 2005 by E. A. Polyakov.     

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.     

Convergence Conditions for Interpolation Fractions at the Nodes Distinct from the Singular Points of a Function

We consider an interpolation process for the class of functions with finitely many singular points by means of the rational functions whose poles coincide with the singular points of the function under interpolation. The interpolation nodes constitute a triangular matrix and are distinct from the singular points of the function. We find a necessary and sufficient condition for uniform convergence of sequences of interpolation fractions to the function under interpolation on every compact set disjoint from the singular points of the function and other conditions for convergence.Original Russian Text Copyright © 2005 Lipchinskii A. G.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 822–833, July–August, 2005.     

Decidability of the Admissibility Problem for Inference Rules in Some <Emphasis Type="Italic">S</Emphasis>5<Subscript>t</Subscript>-Logics

We examine some many-modal logics extending S5_t, t ∈ N, for decidability w.r.t. admissibility of inference rules, and for the logics in question, we prove an algorithmic criterion determining whether the inference rules in them are admissible.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 438–458, July–August, 2005.     

On Fractional Moments of Dirichlet <Emphasis Type="Italic">L</Emphasis>-Functions

We obtain upper and lower bounds of the same order for fractional moments of Dirichlet L-functions.Supported by a grant from the Lithuanian Science and Studies Fundation.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 208–228, April–June, 2005.     

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.     

Path calculus in the modal logic S4

We describe the tableau and inverse calculi for the propositional modal logic S4. The formulas are treated as sets of paths. We obtain the upper bound for the number of applications of rules in the deduction tree.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 117–126, January–March, 2005.Translated by R. Lapinskas     

The Riemannian Structure of John and Uniform Domains in ℝ<Superscript>n</Superscript>

Using the theory of pre-ends, we study the boundary and metric properties of John and uniform domains in the Euclidean n-space. We obtain some results on the metric Riemannian structure of these classes of domains. We prove that the family of John domains is closed under the class of homeomorphisms quasi-isometric in the intrinsic Riemannian metric and the family of uniform domains is closed under the class of bi-Lipschitz mappings.Original Russian Text Copyright © 2005 Karmazin A. P.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 786–804, July–August, 2005.     

Korn’s Inequalities for Junctions of Elastic Bodies with Thin Plates

We prove the anisotropic Korn inequalities for an elastic junction of a massive body with thin plates clamped along parts of the lateral surfaces. The distribution of the weight factors in the norms under consideration depends essentially on the disposition of the plates, the way they are clamped to the body, and their relative rigidity (as compared with the body).Original Russian Text Copyright © 2005 Nazarov S. A.The author was supported by the Russian Foundation for Basic Research (Grant 03-01-00835).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 876–889, July–August, 2005.     

Dyadic Mappings and Dyadic Superparacompact Topological Groups

We introduce the notion of a dyadic superparacompactum which generalizes the classical notion of a dyadic bicompactum and give an analog of V. I. Kuz’minov and L. N. Ivanovskii’s Theorem on dyadicity of the space of a bicompact topological group for superparacompact groups. Moreover, we generalize L. S. Pontryagin’s Theorem on existence of an open bicompact subgroup in each neighborhood of unity of a locally bicompact totally disconnected topological group.Original Russian Text Copyright © 2005 Musaev D. K.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 851–859, July–August, 2005.     

Special Monodromy Groups and the Riemann-Hilbert Problem for the Riemann Equation

In this paper, we solve the Riemann-Hilbert problem for the Riemann equation and for the hypergeometric equation. We describe all possible representations of the monodromy of the Riemann equation. We show that if the monodromy of the Riemann equation belongs to SL(2, ℂ), then it can be realized not only by the Riemann equation, but also by the more special Riemann-Sturm-Liouville equation. For the hypergeometric equation, we construct a criterion for its monodromy group to belong to SL(2, ℤ).__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 753–767.Original Russian Text Copyright ©2005 by V. A. Poberezhnyi.     

Factorization Representations in the Boundary Crossing Problems for Random Walks on a Markov Chain

Let τ be some stopping time for a random walk S _n defined on transitions of a finite Markov chain and let τ(t) be the first passage time across the level t which occurs after τ. We prove a theorem that establishes a connection between the dual Laplace-Stieltjes transforms of the joint distributions of (τ, S _τ) and (τ(t), S _τ(t)). This result applies to the study of the number of crossings of a strip by sample paths of a random walk.Original Russian Text Copyright © 2005 Lotov V. I. and Orlova N. G.The authors were partially supported by the Russian Foundation for Basic Research (Grant 05-01-00810) and the Grant Council of the President of the Russian Federation (Grant NSh-2139.2003.1).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 833–840, July–August, 2005.     

The Logarithmic Asymptotic Expansions for the Norms of Evaluation Functionals

Let μ be a compactly supported finite Borel measure in ℂ, and let Π_n be the space of holomorphic polynomials of degree at most n furnished with the norm of L ²(μ). We study the logarithmic asymptotic expansions of the norms of the evaluation functionals that relate to polynomials p ∈ Π_n their values at a point z ∈ ℂ. The main results demonstrate how the asymptotic behavior depends on regularity of the complement of the support of μ and the Stahl-Totik regularity of the measure. In particular, we study the cases of pointwise and μ-a.e. convergence as n → ∞.Original Russian Text Copyright © 2005 Dovgoshei A. A., Abdullaev F., and Kucukaslan M.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 774–785, July–August, 2005.     

On the Self-Similar Jordan Arcs Admitting Structure Parametrization

We study the attractors γ of a finite system of contraction similarities S _j (j = 1,..., m) in ℝ^d which are Jordan arcs. We prove that if a system possesses a structure parametrization (ℐ,ϕ) and ℱ(ℐ) is the associated family of ℐ then we have one of the following cases:1. The identity mapping Id does not belong to the closure of ℱ(ℐ). Then (if properly rearranged) is a Jordan zipper.2. The identity mapping Id is a limit point of ℱ(ℐ). Then the arc γ is a straight line segment.3. The identity mapping Id is an isolated point of .We construct an example of a self-similar Jordan curve which implements the third case.Original Russian Text Copyright © 2005 Aseev V. V. and Tetenov A. V.The authors were supported by the Program “Universities of Russia” (Grant UR.04.01.456).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 733–748, July–August, 2005.