On the general solution of a nonsymmetric partial difference functional equation analogous to the wave equation

We give the general solution of the nonsymmetric partial difference functional equationf(x + t,y) + f(x – t,y) – 2f(x,y)/t ² =f(x,y + s) + f(x,y – s) – 2f(x,y)/s ² (N) analogous to the well-known wave equation ( ²/x ² – ²/y ²)f(x,y) = 0 with the aid of generalized polynomials when no regularity assumptions are imposed onf. The result is as follows. Theorem.Let R be the set of all real numbers. A function f: R × R R satisfies the functional equation (N)for all x, y R, s, t R\{0}, and s t if and only if there exist

Contracting endomorphisms and Gorenstein modules

A finite module M over a noetherian local ring R is said to be Gorenstein if Extⁱ(k, M) = 0 for all i ≠ dim R. An endomorphism φ: R → R of rings is called contracting if for some i ≥ 1. Letting ^φR denote the R-module R with action induced by φ, we prove: A finite R-module M is Gorenstein if and only if Hom_R(^φR, M) ≅ M and Extⁱ_R(^φR, M) = 0 for 1 ≤ i ≤ depth R. Received: 7 December 2007     

Can you cover your shadows?

A subsetX of thed-dimensional Euclidean space ^d can cover its shadows inR ^d , if every orthogonal projection ofX onto a (d–1)-dimensional linear subspace of ^d is contained in some congruent copy ofX. Whereas every two-dimensional convex discC R ^d has this property, no (d–1)-polytope does, provided thatd>-4.     

On some additive mappings in rings with involution

Summary LetR be a *-ring. We study an additive mappingD: R R satisfyingD(x ²) =D(x)x ^* +xD(x) for allx R.It is shown that, in caseR contains the unit element, the element 1/2, and an invertible skew-hermitian element which lies in the center ofR, then there exists ana R such thatD(x) = ax ^* – xa for allx R. IfR is a noncommutative prime real algebra, thenD is linear. In our main result we prove that a noncommutative prime ring of characteristic different from 2 is normal (i.e.xx ^* =x ^* x for allx R) if and only if there exists a nonzero additive mappingD: R R satisfyingD(x ²) =D(x)x ^* +xD(x) and [D(x), x] = 0 for allx R. This result is in the spirit of the well-known theorem of E. Posner, which states that the existence of a nonzero derivationD on a prime ringR, such that [D(x), x] lies in the center ofR for allx R, forcesR to be commutative.     

Small ball probabilities for a Wiener process under weighted sup-norms,with an application to the supremum of bessel local times

LetR be the radial part of ad-dimensional Wiener process, starting from 0. In this paper, small ball probabilities are evaluated for sup_0<11(t ^–p R(t)) and sup _t 0(e ^–1 R(t)), withp[0, 1/2]. Chung's law of the iterated logarithm is established for the supremum of the local times of a two-dimensional Bessel process.     

Riesz Exponential Families on Symmetric Cones

Let E be a simple Euclidean Jordan algebra of rank r and let be its symmetric cone. Given a Jordan frame on E, the generalized power _s (– ^–1) defined on – is the Laplace transform of some positive measure R _s on E if and only if s is in a given subset of R ^r . The aim of this paper is to study the natural exponential families (NEFs) F(R _s ) associated to the measures R _s . We give a condition on s so that R _s generates a NEF, we calculate the variance function of F(R _s ) and we show that a NEF F on E is invariant by the triangular group if and only if there exists s in such that either F=F(R _s ) or F is the image of F(R _s ) under the map x–x.     

Distribution of Cardinalities of Row Spaces of Boolean Matrices of Order n

Let R(A) denote the row space of a Boolean matrix A of order n. We show that if n 7, then the cardinality |R(A)| (2^n–1 - 2^n–5, 2^n–1 - 2^n–6) U (2^n–1 - 2^n–6, 2^n–1). This result confirms a conjecture in [1].AMS Subject Classification (1991): 05B20 06E05 15A36Support partially by the Postdoctoral Science Foundation of China.Dedicated to Professor Chao Ko on the occasion of his 90th birthday     

On Non-Linear Rayleigh quotients

The stability with respect top of the non-linear eigenvalue problem div(|u| ^p–2u)+|u| ^p–2 u=0 is studied.     

Biharmonic equations with asymptotically linear nonlinearities

This article considers the equation △2u = f(x, u)with boundary conditions either u|(a)Ω = (a)u/(a)n|(a)Ω = 0 or u|(a)Ω = △u|(a)Ω = 0, where f(x,t) is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in RN, N ＞ 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f(x, t).     

On simplexes inscribed in a hypersurface

The sufficient conditions are obtained for the existence, on a hyper surface M Rⁿ, of k points whose convex hull forms a (k–1)-dimensional simplex, homothetic to a given simplex Rⁿ. In particular, it is shown that if M is a smooth hypersurface, homeomorphic to a sphere, such points will exist for any simplex Rⁿ. The proofs are based on simple topological considerations. There are six references.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 81–89, January, 1969.     

Real <Emphasis Type="Italic">AW</Emphasis><Superscript>*</Superscript>-Algebras of type I

Let R be a real AW ^*-algebra, and suppose that its complexification M = R + iR is also a (complex) AW ^*-algebra. We prove that R is of type I if and only if so is M.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 79–81, 2004Original Russian Text Copyright © by Sh. A. Ayupov     

Steiner systems andn −1-isomorphisms

In this work we introduce the concept of n ^–1-isomorphism between Steiner systems (this coincides with the concept of isomorphism whenever n=1).Precisely two Steiner systems S₁ and S₂ are said to be n^–1-isomorphic if there exist n partial systems S _i ⁽¹⁾ ,...,S _i ⁽ⁿ⁾ contained in Si, i.{1,2},such that S ₁ ^(k) and S ₂ ^(k) are isomorphic for each k{1,..., n}.The n^–1-isomorphisms are also used to study nets replacements, see Ostrom [8], and to study the transformation methods of designs and other incidence structures introduced in [9] and generalized in [1] and [10].Work done under the auspicies of G.N.S.A.G.A. supported by 40% grants of M.U.R.S.T.     

Some positive cotes numbers for the chebyshev weight function

We show that, for the Chebyshev weight function (1–x ²)^–1/2, the Cotes numbers for the quadrature rule with nodes at the zeros of the ultraspherical polynomialP _n ^/() are nonnegative if and only if –1/2<1.     

Beste Quadraturformeln für Integrale mit einer Gewichtsfunktion

Let be a fixed matrix with elements that are 0 or 1 and letX be a fixed set ofm+1 different knots. The problem is to find necessary and sufficient conditions for (E, X) to guarantee the existence of a quadrature formula with a remainder term of type for any choice of a weight functionw(t) and satisfyingR(f)=0 forf a polynomial of degree at mostn–1. The result generalizes the corresponding result ofI. J. Schoenberg for the special case of quasi-Lagrange-matricesE. —in case of the existence ofR it is possible to calculate the best quadrature formulaR ^* in the sense ofSard by integrating splines of degree 2n–1. But ifE contains onlyn ones it is sufficient to integrate polynomials of degreen–1.     

Noncommutative invariants of bialgebras

Suppose that H is a bialgebra over a field C and R = CV is the tensor algebra of the C-space V endowed with the structure of an H-module algebra, so that V is a submodule of the H-module R, R^H is the algebra of H-invariants, and W, the support of the algebra R^H, is the smallest subspace of the C-space V such that . The main result of the paper is the theorem stating that if the algebra of H-invariants R^H is finitely generated, then the support of R^H is a finite-dimensional submodule of the H-module V, whose elements are H-semi-invariants of the same weight.Translated fromAlgebra i Logika, Vol. 33, No. 6, pp. 654–680, November–December, 1994.Supported by the Soros Foundation (grant RPS000) and the Russian Foundation for Fundamental Research (grant No. 93-01-16171).     

Fourier series of functions whose Hankel transform is supported on [0, 1]

LetJ denote the Bessel function of order . For >–1, the system x^–/2–1/2J_+2n+1(x^1/2, n=0, 1, 2,..., is orthogonal onL ²((0, ),x dx). In this paper we study the mean convergence of Fourier series with respect to this system for functions whose Hankel transform is supported on [0, 1].Communicated by Mourad Ismail.     

An existence result for variational problems and applications to some nonlinear hyperbolic and elliptic equations involving critical Sobolev exponents

Summary We prove the existence of nontrivial solutions for nonlinear equations of the type Lu=g(x, u) + ¦u¦^¯p–2u, ¯ p > 2, where L is a continuous self- adjoint linear operator in a Hilbert space H and ug(x, u) is a lower order perturbation of ¦u¦^¯p–1. We assume that ¯p is the critical exponent in the sense that the embedding H L^p, If is compact for 1p<¯p and is continuous (not necessarily compact) for p=¯p. From this result we deduce, for example, that u_tt -u- u=¦u¦^2/Nu, u L²(S^N×S¹) has at least one pair (–u, u) of solutions nonconstant with respect to t, provided that is sufficiently close to some eigenvalue of _tt–.Work supported by M.P.I. Italy (fondi 40%, 60%) and by G.N.A.F.A. of C.N.R.     

Equivariant framability of involutions on homotopy spheres

Let G be the group Z₂. Denote byR ^n,k theR ^n+k with non trivial G-action on the first n coordinates. Let ^n,k be the trivial bundle with fibreR ^n,k. We say that a G-manifold M is (n,k)-framable if t(M)= =^n,k in KO_G(M) with t(M) the tangent bundle of M. We show that if G acts on a homotopy sphere ^n+k such that the fixed point set is a k-dimensional homotopy sphere then is (n,k)-framable.     

Local times for the norm of a multidimensional Brown motion

Let w(t) be a Brownian motion inR ⁿ , letf be an arbitrary norm of the spaceR ⁿ , and let Z(f)=f(w(t)). For the random process Z(t) one establishes the existence of a local time (x, ), square integrable with respect to the probability measure P.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 181–184, 1989.     

Operator Machines on Directed Graphs

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator ${R\,:\,X\longrightarrow\, X}We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X? X{R\,:\,X\longrightarrow\, X} such that the set

A = {x ? X : ||Rn x||? ￥}A = \{x \in X\,:\,{\left|\left|{R^n x}\right|\right|}\rightarrow \infty\}