The product of finitely based varieties of lattice-ordered groups

The question as to whether a product of two finitely based varieties of lattice-ordered groups is finitely based is considered. It is proved that varieties and are finitely based; here is a variety of lattice-ordered groups defined by identities [x ⁿ,y ⁿ] =e and [[x,y] z, [x ₁,y ₁] z ₁] =e; is a variety of lattice-ordered nilpotent groups of class s, defined by an identity [x ₁,x ₂,...,x _(s+1)] =e; V is an arbitrary finitely based variety of lattice-ordered groups. Translated fromAlgebra i Logika, Vol. 33, No. 3, pp. 255–263, May–June, 1994.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1524.     

Primitive Inner Illuminating Systems for Convex Bodies

A set X of boundary points of a (possibly unbounded) convex body KE ^d illuminating K from within is called primitive if no proper subset of X still illuminates K from within. We prove that for such a primitive set X of an unbounded, convex set KE ^d (distinct from a cone) one has X=2 if d=2, X6 if d=3, and that there is no upper bound for X if d4.     

Critical determinant of the region ∣x∣p+∣y∣p<1 (supplement to the paper “on the application of the electronic computer to the proof of a conjecture of Minkowski from the geometry of numbers,” Zap. Nauchn. Sem. Lomi,71, 163–180, 1977)

Supplement to the paper cited in the title. One presents the results of the calculations on an electronic computer, by the method developed in that paper, proving Minkowski's conjecture regarding the critical determinant of the region x^p+y^p<1 for all real numbers p>2.035 and for 1.01p1.99.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, pp. 29–32, 1979.     

k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

Application of a computer to the proof of a conjecture of Minkowski in the geometry of numbers

A computer-assisted proof is given of Minkowski's conjecture on the critical determinant of the region x^p+y^p<1 in the cases 1.03p 1.9745, p2.40, p2.577.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 163–180, 1977.     

On the optimal mapping of distributions

We consider the problem of mappingXY, whereX andY have given distributions, so as to minimize the expected value of X–Y². This is equivalent to finding the joint distribution of the random variable (X, Y), with specified marginal distributions forX andY, such that the expected value of X–Y² is minimized. We give a sufficient condition for the minimizing joint distribution and supply numerical results for two special cases.     

A note on rational approximation rate for Müntz system {x λn } with λ n ↘ 0

[Zho2] {x ⁿ } , _n 0 n .

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Supported in part by an NSERC Postdoctoral Fellowship and a CRF grant of University of Alberta.     

Subgroups of a finite group whose algebra of invariants is a complete intersection

Let G be a finite subgroup of GL(V), where V is a finite-dimensional vector space over the field K and char KG. We show that if the algebra of invariants K(V)^G of the symmetric algebra of V is a complete intersection then K(V)^H is also a complete intersection for all subgroups H of G such that H={ Gv (v)=v for all v V^H}.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 63–67, 1982.     

A representation of symmetric functions in Carleman-Gevrey spaces

For a symmetric function t(x)(x^d) one investigates the representation, where _j(x) is the elementary symmetric polynomial of degree j. Let be the closure of the domain in ^d, let be a numerical sequence such that (n) does not decrease, let be the Carleman-Gevrey space, i.e. the collection of functions (n+1)/(n) such that for any bounded subdomain there exists a constant tC() with which one has the inequality _x t(x)H⁺¹!() (x*#x03A9;'). Let S be the image of ^d under the mapping x(₁(x), ..., _d(x)). One proves the following theorem: For any tk(^d) there exists such that, if and only if (n)(nd)ⁿ⁺¹, where is some positive number, independent of n.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 116–126, 1986.     

Eigenvalue distribution of some fractal semi-elliptic differential operators: Combinatorial approach

We obtain the sharp order of growth of the eigenvalue distribution function for the operator in the anisotropic Sobolev space , generated by the quadratic form _Q u² d, whereQ² is the unit square and is a probability self-affine fractal measure onQ. The geometry of Supp should be in a certain way consistent with the parameterst ₁ ,t ₂ .     

Existence of solutions for the Navier-Stokes equations,having an infinite dissipation of energy,in a class of domains with noncompact boundaries

In the domain ³ with two exits at infinity, ₁ = {x:x<g ₁(x ₃),x ₃ > 2} and ₂ = {x:0<x ₃<g ₃(x), x>2}, one investigates the stationary system of Navier-Stokes equations under the boundary condition. One proves existence and uniqueness theorems for the solution of this problem with an infinite Dirichlet integral, under a given flow of the velocity vector across the cross sections of the exits at infinity. As an example one considers the case wheng _i(t) 1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 180–202, 1981.     

A Lower Bound for the Height of a Rational Function at <Emphasis Type="Italic">S</Emphasis>-unit Points

Let a,b be given, multiplicatively independent positive integers and let >0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(n) for g.c.d.(aⁿ–1, bⁿ–1); shortly afterwards we generalized this to the estimate g.c.d.(u–1,v–1)v) for multiplicatively independent S-units u,vZ. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g.c.d. to pairs of rational functions other than {u–1,v–1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G_m².     

Vector Equilibrium Problems Under Asymptotic Analysis

Given a closed convex set K in Rⁿ; a vector function F:K×K R^m; a closed convex (not necessarily pointed) cone P(x) in ^m with non-empty interior, PP(x) Ø, various existence results to the problemfind xK such that F(x,y)- int P(x) y K under P(x)-convexity/lower semicontinuity of F(x,) and pseudomonotonicity on F, are established. Moreover, under a stronger pseudomonotonicity assumption on F (which reduces to the previous one in case m=1), some characterizations of the non-emptiness of the solution set are given. Also, several alternative necessary and/or sufficient conditions for the solution set to be non-empty and compact are presented. However, the solution set fails to be convex in general. A sufficient condition to the solution set to be a singleton is also stated. The classical case P(x)=^m ₊ is specially discussed by assuming semi-strict quasiconvexity. The results are then applied to vector variational inequalities and minimization problems. Our approach is based upon the computing of certain cones containing particular recession directions of K and F.     

A convolution metric in the space of measures and ε-isometries in Lp

For >0, 2,4,6,... on the set of those Borel measures on , such that x^pd(x)<, one introduces a metric, characterizing the nearness of the convolutions x^p* and x^p*. From the convergence of a sequence of probability measures in this metric there follows its convergence in the Kantorovich-Rubinshtein metric. From here one derives theorems on the approximation of -isometries: if H is a finite-dimensional subspace in L_p, then there exists a continuous function _H(), such that for any linear -isometric operator THL_p there exists a linear isometry UH L_p, such that T–U<_H().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 151–156, 1987.     

Produktsätze für Verfahren zur Limitierung von Doppelfolgen

{p _mn } - ₀₀>0, (1, 1) (1.1) (1.2). {s _mn } J _p - ( bJ _p -lims _mn =), (1.3) 0<x,y<1 p _s (, )/p(x, y) x, y 1-. {r _mn } - , (1.5) 0<, <1. N _rp - , (1.6). , bJ _p -lims _mn = bJ _q -lim(N _rps) _mn =. J _p - . , .     

Cauchy problem for a semilinear wave equation. II

The existence and the uniqueness of a weak solution of the Cauchy problem, for a second-order semilinear pseudodifferential hyperbolic equation on a smooth Riemannian manifold (of dimension n3) without boundary, is proved. In particular, for the equation ü–u+u^p–1=0 one has unique solvability for the Cauchy problem when 1(n+2)/(n–2).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 38–85, 1990.     

Combinatorics and flexure

Combinatorial identities, trigonometric formulas, together with complex variable techniques are used to derive exact and closed expressions for the six flexure functions of certain isotropic cylinders under flexure. The cross sections are bounded either by the closed curvesr=a cosⁿ (/n) (–<) or the closed curvesr=asin(/n)ⁿ(–<), wheren isa positive integer (n>1).

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Résumé Des identiteés combinatoires et des formules trigonométriques avec des techniques de variables complexes sont utilisées pour dériver des expressions exactes et simples pour les six fonctions de flexion de quelques cylindres isotropiques. Les sections sont limitées par les courbes ferméesr=a cosⁿ /n(–) et les courbesr=asin/nⁿ(–) où est un entier positif (n>1).

     

On the classical logarithmic barrier function method for a class of smooth convex programming problems

In this paper, we describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming. It is assumed that the objective and constraint functions fulfill the so-called relative Lipschitz condition, with Lipschitz constantM>0.In our method, we do line searches along the Newton direction with respect to the strictly convex logarithmic barrier function if we are far away from the central trajectory. If we are sufficiently close to this path, with respect to a certain metric, we reduce the barrier parameter. We prove that the number of iterations required by the algorithm to converge to an -optimal solution isO((1+M ²) log) orO((1+M ²)nlog), depending on the updating scheme for the lower bound.on leave from Eötvös University, Budapest, Hungary.     

Uniqueness of Nelson's Diffusions II: Infinite Dimensional Setting and Applications

Under mild condition on the modulus = of the time independent wave function , we prove that the generalized Schrödinger operator = + 2 (, ·)/ (or the generator of Nelson's diffusion) defined on a good space of test-functions on a general Polish space, generates a unique semigroup of class (C _o) in L ¹. This result reinforces the known results on the essential Markovian self-adjointness in different contexts and extends our previous works in the finite dimensional Euclidean space setting. In particular it can be applied to the ground or excited state diffusion associated with an usual Schr\"odinger operator , and to stochastic quantization of several Euclidean quantum fields.     

A Minimal Triangulation of the Hopf Map and its Application

We give a minimal triangulation : S ₁₂ ³ S ₄ ² of the Hopf map h:S ³S ² and use it to obtain a new construction of the 9-vertex complex projective plane.