Coercive Lp-estimates for solutions to the initial-boundary-value problem for the generalized stokes equations in a half-space

The initial-boundary-value problem in a half-space with the boundary condition is considered for a class of systems with constant coefficients. In particular, this class contains the nonstationary Stokes equations. To solve the problem, some coercive estimates are obtained in the L_p-norms. For the presented results are new even for the Stokes equations. Bibliography: 6 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 243–270.     

Solutions of the stationary Navier-Stokes system of equations with an infinite Dirichlet integral

In unbounded domains of the three-dimensional Euclidean space, having several exits _i at infinity of a sufficiently general form, one finds the solutions of the stationary Navier-Stokes system, equal to zero on the boundary of the domain , having arbitrary flow rates d_i through each exit _i, i=1,..., , and having an unbounded Dirichlet integral . One gives sufficient conditions for the existence of a solution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 251–263, 1982.     

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.     

Rate-of-convergence bound for difference schemes for the system of equilibrium equations of a rigidly clamped nonhomogeneous anisotropic elastic solid

Exact difference scheme operators are used to construct a difference scheme for a system of equilibrium equations of a rigidly clamped nonhomogeneous anisotropic elastic solid. The rate-of-convergence bound .Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 14–19, 1987.     

Extremal Decompositions of Riemann Surfaces and Quasiconformal Mappings of Special Form. II

Let be a finite Riemann surface. For a quadratic differential on associated with a certain problem on extremal decomposition of into n domains, a parametric family of quasiconformal mappings , is defined. These mappings map the domains of the extremal decomposition of onto the domains of the extremal decomposition of ._K This allows one to study the functional dependence of the problem on the parameters. Bibliography: 6 titles.     

On the two-dimensional initial boundary value problem for the Navier-Stokes equations with discontinuous boundary data

We consider the initial boundary value problem for the Navier-Stokes equations with boundary conditions . We assume that may have jump discontinuities at finitely many points ξ1;. . .,ξm of the boundary ϖΩ of a bounded domain Ω ⊂ ℝ². We prove that this problem has a unique generalized solution in a finite time interval or for small initial and boundary data. The solution is found in a class of vector fields with infinite energy integral. The case of a moving boundary is also considered. Bibliography: 11 titles. Dedicated to O. A. Ladyzhenskaya on the occasion of her 70^th birthday. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 159–178, 1992. Translated by E. V. Frolova.     

Some estimates for the oscillation of the deformation gradient

As a measure of deformation we can take the difference D - R, where D is the deformation gradient of the mapping and R is the deformation gradient of the mapping , which represents some proper rigid motion. In this article, the norm is estimated by means of the scalar measure e( ) of nonlinear strain. First, the estimates are given for a deformation W ^1,p() satisfying the condition . Then we deduce the estimate in the case that (x) is a bi-Lipschitzian deformation and .     

Sequence Spaces l p,q in Probabilistic Characterizations of Weak Type Operators

We study operators (not necessarily linear) defined on a quasi-Bahach space X and taking values in the space of real-valued Lebesgue-measurable functions. Factorization theorems for linear and superlinear operators with values in the space are proved with the help of the Lorentz sequence spaces . Sequences of functions belonging to fixed bounded sets in the spaces are characterized for and . The possibility of distinguishing weak type operators (bounded in the space ) from operators factorizable through is obtained in terms of sequences of independent random variables. A criterion under which an operator is symmetrically bounded in order in , is established. Some refinements of the above-mentioned results are obtained for translation shift-invariant sets and operators. Bibliography: 30 titles.     

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 ,

The Spectrum of Lévy Constants for Quadratic Irrationalities and Class Numbers of Real Quadratic Fields

Let h(d) be the class number of the field and let be the Lévy constant. A connection between these constants is studied. It is proved that if d is large, then the value h(d) increases, roughly speaking, at the rate as grows. A similar result is obtained in the case where the value is close to , i.e., to the least possible value. In addition, it is shown that the interval contains no values of for prime p such that p 3 (mod 4). As a corollary, a new criterion for the equality h(d)=1 is obtained. Bibliography: 14 titles.     

Common invariant subspaces for collections of operators

Let be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that is finitely quasinilpotent at a vectorx ₀X whenever for any finite subset of the joint spectral radius of atx ₀ is equal 0. If such collection contains a non-zero compact operator, then and its commutant have a common non-trivial invariant, subspace. If in addition, is a collection of positive operators on a Banach lattice, then has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then has a common non-trivial invariant closed ideal.This work was supported by the Research Ministry of Slovenia.     

Quasisimilar Weak Contractions Have Isomorphic Lattices of Invariant Subspaces

A contraction T acting on a Hilbert space H is called a weak contraction if the spectrum of T does not cover the unit disk and the operator I-T ^* T is of trace class. Operators T₁:H₁ H₁ and T₂:H₂ H₂ are called quasisimilar if there exist operators >X:H₁ H₂ and Y:H₂ H₁ such that T₂X=XT₁, YT₂=T₁Y, and X and Y have zero kernels and dense ranges. It is proved that if two weak contractions T₁ and T₂ acting on separable spaces H₁ and H₂ are quasisimilar, then there exists an operator X:H₁ H₂ such that XT₁=T₂X and the mapping , where E=clos XE for E Lat T₁, is a lattice isomorphism. An example is given of two quasisimilar weak contractions such that for any isomorphism , its inverse is not equal to for a (bounded) operator Y. Bibliography: 4 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

Invariants of classCk for finite coxeter groups and their representations in terms of anisotropic spaces

This article is devoted to the study of representations of C^k { } functions f invariant with respect to finite Coxeter groups W in the form f=F op, where p is a base in the algebra of W-invariant polynomials. We examine the lowering of smoothness of F as compared with f and conclude that this lowering has an anisotropic nature and that, more precisely, at each point Po it is described by a vector { }. We examine the cases W=A_n, B_n, D_n, { }; in each case the greatest component { } of { } is equal to the Coxeter number of the stabilizer W_yo of the point yo, where po=p(yo). Bibliography: 22 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 46–70. Translated by S. Yu. Pilyugin.     

Direct Theorem about Approximation on a Family of Two Segments

Introduce the notation: , is the union of two segments [-1,1] and [-1 + ,1+ ], is a noninteger number, is the Hölder class with exponent on The following result announced by the authors in [J. Math. Sci. 117 (2003), No. 3] is proved. There exist numbers a ₁ ( ) , b ₁ ( ) 0 depending only on such that for any there exists a polynomial , such that . Bibliography: 11 titles.     

Geometry of Two- and Three-Dimensional Minkowski Spaces

A class of centrally-symmetric convex 12-topes (12-hedrons) in is described such that for an arbitrary prescribed norm on each polyhedron in the class can be inscribed in (circumscribed about) the -ball via an affine transformation, and this can be done with large degree of freedom. Bibliography: 5 titles.     

Zeros of H ∞-Functions on Hyperplanes in \mathbb{B} n

Let ⁿ be the unit ball in ℂⁿ, n ≥ 2. Let T_α = {z ∈ ⁿ : (z, a) = |a|²} for a ∈ ⁿ and denote for a discrete set A in ⁿ. We find a sharp necessary condition for a set A to be a part of the zero-set for a function in H^∞( ⁿ). Bibliography 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 272–278.     

Gray codes from antimatroids

We show three main results concerning Hamiltonicity of graphs derived from antimatroids. These results provide Gray codes for the feasible sets and basic words of antimatroids.For antimatroid (E, ), letJ( ) denote the graph whose vertices are the sets of , where two vertices are adjacent if the corresponding sets differ by one element. DefineJ( ;k) to be the subgraph ofJ( )² induced by the sets in with exactlyk elements. Both graphsJ( ) andJ( ;k) are connected, and the former is bipartite.We show that there is a Hamiltonian cycle inJ( )×K ₂. As a consequence, the ideals of any poset % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFpepuaaa!414C!\[\mathcal{P}\] may be listed in such a way that successive ideals differ by at most two elements. We also show thatJ( ;k) has a Hamilton path if (E, ) is the poset antimatroid of a series-parallel poset.Similarly, we show thatG( )×K ₂ is Hamiltonian, whereG( ) is the basic word graph of a language antimatroid (E, ). This result was known previously for poset antimatroids.Research supported in part by NSERC.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A3379.     

A Property of Purely Hyperbolic Fuchsian Groups of the Second Kind

Let G be a purely hyperbolic, finitely generated, nonelementary Fuchsian group of the second kind. If is the limit set of the group G, then the function belongs to the class BMO. This follows from the fact that is porous, proved in the paper. Bibliography: 4 titles.