Extremal Distances between Sections of Convex Bodies

Let K, D be centrally symmetric convex bodies in Let k < n and let d_k(K, D) be the smallest Banach–Mazur distance between k-dimensional sections of K and D. Define

A Limit Result for U-Statistics of Binary Variables

Define , where is a symmetric U-type statistic, H _k() is the Hermite polynomial of degree k, and {X, X _n, n1} are independent identically distributed binary random variables with Pr(X{–1, 1}})=1. We show that according as EX=0 or EX0, respectively.     

Quasilinearization and Rate of Convergence for Higher-Order Nonlinear Periodic Boundary-Value Problems

We study the convergence of a sequence of approximate solutions for thefollowing higher-order nonlinear periodic boundary–value problem:

On the asymptotic joint distribution of an unbounded number of sample extremes

Summary Convergence of the sample maximum to a nondegenerate random variable, as the sample sizen, implies the convergence in distribution of thek largest sample extremes to ak-dimensional random vectorM _k , for all fixedk. If we letk=k(n),k/n0, then a question arises in a natural way: how fast cank grow so that asymptotic probability statements are unaffected when sample extremes are replaced byM _k . We answer this question for two classes of events-the class of all Lebesgue sets inR ^k and the class of events of the form .     

Hypersurfaces of restricted type in Minkowski space

A submanifold M _n ^r of Minkowski space is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of to the tangent space of M _n ^r at every point of M _n ^r . In this paper we completely classify hypersurfaces of restricted type in . More precisely, we prove that a hypersurface of is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S ^k × , S _k ¹ × , H ^k × , S _n ¹ , H ⁿ , with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium.     

The Maximum Size Of Graphs With A Unique<Emphasis Type="Italic">k</Emphasis>- Factor

For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved for even n and , respectively. Recently, Johann confirmed the following conjectures of Hendry: for and kn even and for n = 2kq, where q is a positive integer. In this paper we prove for and kn even, and we determine m(n, 3).     

Convergence Analysis of Some Methods for Minimizing a Nonsmooth Convex Function

In this paper, we analyze a class of methods for minimizing a proper lower semicontinuous extended-valued convex function . Instead of the original objective function f, we employ a convex approximation f _{k + 1} at the kth iteration. Some global convergence rate estimates are obtained. We illustrate our approach by proposing (i) a new family of proximal point algorithms which possesses the global convergence rate estimate even it the iteration points are calculated approximately, where are the proximal parameters, and (ii) a variant proximal bundle method. Applications to stochastic programs are discussed.     

On the Boundedness of a Recurrence Sequence in a Banach Space

We investigate the problem of the boundedness of the following recurrence sequence in a Banach space B: where |y _n} and | _n } are sequences bounded in B, and A _k, k 1, are linear bounded operators. We prove that if, for any > 0, the condition is satisfied, then the sequence |x _n} is bounded for arbitrary bounded sequences |y _n} and | _n } if and only if the operator has the continuous inverse for every z C, |z| 1.     

Sur les plus grands facteurs premiers d'un entier

Let and, for each integern such that (n)k, denote byP _k (n) itsk ^th largest prime factor. Further, given a set of primesQ of positive density <1 satisfying a certain regularity condition, defineP(n, Q), as the largest prime divisor ofn belonging toQ, assuming thatP(n,Q)=+ if no such prime factor exists. We provide estimates of , fork2, and of . We also study the median value of the functionP(n,Q) and that of the functionP _k (n) for eachk1.     

An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Let be the affine Lie algebra associated to the simple finite-dimensional Lie algebra . We consider the tensor product of the loop -module associated to the irreducible finite-dimensional -module V() and the irreducible highest weight -module L _k,. Then L _k, can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,) be the corresponding -bimodule. We prove that if the -module is zero, then the -module is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.     

Hamiltonicity in balancedk-partite graphs

One of the earliest results about hamiltonian graphs was given by Dirac. He showed that if a graphG has orderp and minimum degree at least thenG is hamiltonian. Moon and Moser showed that a balanced bipartite graph (the two partite sets have the same order)G has orderp and minimum degree more than thenG is hamiltonian. In this paper, their idea is generalized tok-partite graphs and the following result is obtained: LetG be a balancedk-partite graph with orderp = kn. If the minimum degree

On <Emphasis Type="Italic">k</Emphasis>-th Power of Upper Bound Graphs

We consider k-th power of upper bound graphs. According to the characterization of upper bound graphs, we obtain a characterization of k-th power of upper bound graphs. That is, for a connected upper bound graph G, G^k is an upper bound graph if and only if for any pair of A^k -simplicial vertices s₁, s₂ such that , there exists a G^k -simplicial vertex s satisfying the conditions: and . Furthermore we also get some properties on squares of upper bound graphs.AMS Subject Classification: 05C62.     

A Note on Extremes of Concomitants of Order Statistics

Let be a sequence of identically distributed variables. We study the asymptotic distribution of , where Y _[r:n] denotes the concomitant of the rth order statistic X _r:n , corresponding to , and is held fixed while . Conditions are given for the and to have the same asymptotic behavior as that we would apply if were i.i.d. The result is illustrated with a simple linear regression model , where is a stationary sequence with extremal index .     

A new limit theorem for Gaussian stochastic processes possessing the Baxter property

Let(t) be a Gaussian stochastic process with zero mean and correlation functionR(t, s), and let . We obtain conditions for the weak convergence of the sequence of stochastic processes ,t [0, 1] and we find the form of the limiting distribution. As a corollary, in the stationary case we find the limiting distribution asn for .Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 90–97, 1987.     

A Necessary and Sufficient Condition for the Existence of Some Ternary [n, k, d] Codes Meeting the Greismer Bound

Let k and d be any integers such that k 4 and . Then there exist two integers and in {0,1,2} such that . The purpose of this paper is to prove that (1) in the case k 5 and (,) = (0,1), there exists a ternary code meeting the Griesmer bound if and only if and (2) in the case k 4 and (,) = (0,2) or (1,1), there is no ternary code meeting the Griesmer bound for any integers k and d and (3) in the case k 5 and , there is no projective ternary code for any integers k and such that 1k-3, where and for any integer i 0. In the special case k=6, it follows from (1) that there is no ternary linear code with parameters [233,6,154] , [234,6,155] or [237,6,157] which are new results.     

Elementary symmetric polynomials of increasing order

Summary The asymptotic behaviour of elementary symmetric polynomials S _n ^(k) of order k, based on n independent and identically distributed random variables X ₁,..., X _n,is investigated for the case that both k and n get large. If , then the distribution function of a suitably normalised S _n ^(k) is shown to converge to a standard normal limit. The speed of this convergence to normality is of order , provided and certain natural moment assumptions are imposed. This order bound is sharp, and cannot be inferred from one of the existing Berry-Esseen bounds for U-statistics. If k at the rate n ^1/2 then a non-normal weak limit appears, provided the X _i's are positive and S _n ^(k) is standardised appropriately. On the other hand, if k at a rate faster than n ^1/2 then it is shown that for positive X _j'sthere exists no linear norming which causes S _n ^(k) to converge weakly to a nondegenerate weak limit.     

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 the Relation between Curvature,Diameter, and Volume of a Complete Riemannian Manifold

In this note, we prove that if N is a compact totally geodesic submanifold of a complete Riemannian manifold M, g whose sectional curvature K satisfies the relation K ≥ k > 0, then for any point m ∈ M. In the case where dim M = 2, the Gaussian curvature K satisfies the relation K ≥ k ≥ 0, and γ is of length l, we get Vol (M, g) ≤ if k ≠ 0 and Vol (M, g ≤ 2ldiam (M) if k = 0.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1576–1583, November, 2004.     

Analysis of general quadrature methods for integral equations of the second kind

Summary This paper is concerned with a class of approximation methods for integral equations of the form , wherea andb are finite,f andy are continuous and the kernelk may be weakly singular. The methods are characterized by approximate equations of the form ; such methods include the Nyström method and a variety of product-integration methods. A general convergence theory is developed for methods of this type. In suitable cases it has the feature that its application to a specific method depends only on a knowledge of convergence properties of the underlying quadrature rule. The theory is used to deduce convergence results, some of them new, for a number of specific methods.Work supported by the U.S. Department of Energy