On the Lipschitz Constant for a Nonisometric Bi-Lipschitz Transformation of a Cantor Set

A bi-Lipschitz continuous mapping of a space X is a bijection such that , where . We write if f is a Lipschitz (bi-Lipschitz) mapping of X into itself and denote by the set of all bi-Lipschitz mappings of X that are not isometry. Thus, if and blip . For X we consider a standard Cantor set K on the real line (with standard metric). The main result of this paper is formulated as follows: where Bibliography: 2 titles.     

On the Uniqueness of the Recovery of Parameters of the Maxwell System from Dynamical Boundary Data

The paper deals with the problem of recovering the parameters (functions) and of the Maxwell dynamical system

Invariant and quasi-invariant transformations of measures that correspond to stable processes with independent increments

Let ξ(t), t ∈ [0, 1], be a strictly stable process with an index of stability α ∈ (0, 2). By we denote the law of ξ in the Skorokhod space [0,1]. For arbitrary strictly stable process ξ, we construct a semigroup of transformations of [0, 1] for which the measure is quasi-invariant. For strictly stable processes with positive and negative jumps, we construct a group of transformations of [0, 1] for which the measure is quasi-invariant. In the symmetric case, this group is a group of invariant transformations. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 135–149.     

Sharp Constant in a Jackson-Type Inequality for Approximation by Positive Linear Operators

In what follows, C is the space of -periodic continuous real-valued functions with uniform norm, is the first continuity modulus of a function with step h, H _n is the set of trigonometric polynomials of order at most n, is the set of linear positive operators (i.e., of operators such that for every ), is the space of square-integrable functions on ,

Regular Unitarily Invariant Spaces on the Complex Sphere

Let K be a compact space, let X be a closed subspace of C(K), and let be a positive measure on K. The triple is said to be regular if, for any positive function and for any , there exists a function such that on K and . The case where K is the unit sphere in and the subspace X is invariant with respect to the unitary group is investigated. Sufficient spectral conditions and a necessary condition for the regularity of a triple are obtained. Connections with compactness of certain Hankel operators and applications to interpolation problems are presented. Bibliography: 16 titles.     

On One Extremal Problem for a Seminorm on the Space l<Subscript>1</Subscript> with Weight

Let be a nondecreasing sequence of positive numbers and let l _1,α be the space of real sequences for which . We associate every sequence ξ from l _1,α with a sequence , where ϕ(·) is a permutation of the natural series such that , j ∈ ℕ. If p is a bounded seminorm on l _1,α and , then

Large deviations of the steady-state distribution of reflected processes with applications to queueing systems

We consider a Skorohod map which takes paths in to paths which stay in the positive orthant . We let be the domain of definition of . A convex and lower semi-continuous function and a set are given. We are concerned with the calculation of the infimum of the value for t ⩾ 0 and absolutely continuous subject to the conditions and . We show that such minimization problems characterize large deviation asymptotics of tail probabilities of the steady-state distribution of certain reflected processes. We approximate the infimum by a sequence of finite-dimensional minimization problems. This approximation allows to formulate an algorithm for the calculation of the infimum and to derive analytical bounds for its value. Several applications are discussed including large deviations of generalized processor sharing and large deviations of heavily loaded queueing networks. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Spectrum of the C*-Algebra of Pseudodifferential Boundary Value Problems on a Manifold with Smooth Edges

The C ^*-algebra generated by the operators of pseudodifferential boundary value problems on a manifold with smooth closed disjoint edges and boundary is studied. The operators act in the space L ₂( ) L ₂( ). The goal of this paper is to describe all (up to an equivalence) irreducible representations of the algebra Bibliography: 12 titles.     

An asymptotic analysis method for the linearly shell theory

In this paper, we consider a linearly elastic shell, i.e. a three-dimensional linearly elastic body with a small thickness denoted by 2ε, which is clamped along its part of the lateral boundary and subjected to the regular loads. In the linear case, one can use the two-dimensional models of Ciarlet or Koiter to calculate the displacement for the shell. Some error estimates between the approximate solution of these models and the three-dimensional displacement vector field of a flexural or membrane shell have been obtained. Here we give a new model for a linear and nonlinear shell, prove that there exists a unique solution U of the two-dimensional variational problem and construct a three-dimensional approximate solutions UKT(x,ξ) in terms of U: We also provide the error estimates between our model and the three-dimensional displacement vector field :‖u-UKT‖1,Ω≤C∈r,r=3/2, an elliptic membrane, r = 1/2, a general membrane, where C is a constant dependent only upon the data‖u‖3,Ω,‖UKT‖3,Ω,θ.     

On the representation of differentiations by Hamiltonians,operating from an algebra of local observable spin systems

Let and be algebras of local and quasilocal observable spin systems corresponding to the group Zr, be a differentiation invariant with respect to displacements. The question of representation of D in the form of formal Hamiltonian formed by the displacements of an elementx ε is considered. It is shown that such a representation exists if the condition holds, where means an element obtained from the elements [T_kX,a] by some r-multiple process of summation. Translated from Matematicheskii Zametki, Vol. 21, No. 1, pp. 93–98, January, 1977.     

Existence of Positive Solutions for a Class of Higher Order Neutral Functional Differential Equations

The higher order neutral functional differential equation

Quasi-Invariant Measures and Irreducible Representations of the Inductive Limit of Special Linear Groups

Unitary representations of the group are constructed. The construction uses G-quasi-invariant measures on some G-spaces that are subspaces of the space of two-way infinite real matrices. We give a criterion for the irreducibility of these representations.     

Sharp Estimates for Deviations of Linear Approximation Methods for Periodic Functions by Linear Combinations of Moduli of Continuity of Different Order

Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that is odd, . Then

On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II

This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)])

Categories with homotopy determine a closed model structure

In a category with homotopy (Definition 1.1), one can define a natural concept of (co)fibrations and weak equivalences (Sec. 2) such that some properties of a closed model category hold. If is a complete and cocomplete category (with respect to finite limits) with simplicial homotopy (Definition 1.3), one achieves a full closed model structure in (Theorem 4.6). For each category with homotopy , there exists a category with simplicial homotopy (see Sec. 5) (the simplicial envelope of ). If is already a complete category with simplicial homotopy, then is Quillen equivalent to (Theorem 5.9). __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 41, Topology and Its Applications, 2006.     

Relatively free products of Magnus groups

The class of Magnus -groups is closed under the operation of the -product. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 199–213, 2004.     

Universal weight systems and the Kontsevich integral for the unknot

The value of the -weight system for the Kontsevich integral of the unknot is found. This value is a normalizing factor used in calculating knot invariants. A weight system is a function defined on the space of chordal diagrams and taking values in the center of the universal enveloping algebra . __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Problems on Extremal Decomposition of the Riemann Sphere

We apply a variant of the method of the extremal metric to some problems concerning extremal decompositions and related problems. Let be a system of distinct points on and let be the family of all systems of nonoverlapping simply connected domains on such that . Let

A new hyperbolic equation possessing a zero-curvature representation

Using a direct procedure of calculation of zero-curvature representations (ZCR), we find a previously unknown hyperbolic equation which possesses an -valued ZCR. This ZCR admits no parameters and is not reducible to a proper subalgebra of . __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.