Universal correlations in finite posets

Posets A, BX×X, with X finite, are said to be universally correlated (AB) if, for all posets R over X, (i.e., all posets RY×Y with XY), we have P(RA) P(RB)P(RAB) P(R). Here P(RA), for instance, is the probability that a randomly chosen bijection from Y to the totally ordered set with |Y| elements is a linear extension of RA. We show that AB iff, for all posets R over X, P(RA) P(RB)P(RAB) P(R(AB)).Winkler proved a theorem giving a necessary and sufficient condition for AB. We suggest an alteration to his proof, and give another condition equivalent to AB.Daykin defined the pair (A, B) to be universally negatively correlated (A B) if, for all posets R over X, P(RA) P(RB)P(RAB) P(R(AB)). He suggested a condition for AB. We give a counterexample to that conjecture, and establish the correct condition. We write AB if, for all posets R over X, P(RA) P(RB)P(RAB) P(R). We give a necessary and sufficient condition for AB.We also give constructive techniques for listing all pairs (A, B) satisfying each of the relations AB, AB, and AB.     

Einstein Metrics with Nonpositive Sectional Curvature on Extensions of Lie Algebras of Heisenberg Type

The purpose of this article is to study some simply connected Lie groups with left invariant Einstein metric, negative Einstein constant and nonpositive sectional curvature. These Lie groups are classified if their associated metric Lie algebra s is of Iwasawa type and s = An₁n₂...n_r, where all n_iare Lie algebras of Heisenberg type with [[n_i,n_j] = {0} for ij. The most important ideas of the article are based on a construction method for Einstein spaces introduced by Wolter in 1991. By this method some new examples of Einstein spaces with nonpositive curvature are constructed. In another part of the article it is shown that Damek-Ricci spaces have negative sectional curvature if and only if they are symmetric spaces.     

Round-off error analysis of iterations for large linear systems

Summary We deal with the rounding error analysis of successive approximation iterations for the solution of large linear systemsA _x =b. We prove that Jacobi, Richardson, Gauss-Seidel and SOR iterations arenumerically stable wheneverA=A ^*>0 andA has PropertyA. This means that the computed resultx _k approximates the exact solution with relative error of order A·A ^–1 where is the relative computer precision. However with the exception of Gauss-Seidel iteration the residual vector Ax _k –b is of order A² A ^–1 and hence the remaining three iterations arenot well-behaved.This work was partly done during the author's visit at Carnegie-Mellon University and it was supported in part by the Office of Naval Research under Contract N00014-76-C-0370; NR 044-422 and by the National Science Foundation under Grant MCS75-222-55     

Some Landau's type inequalities for infinitesimal generators

Summary Lett T(t) be a strongly continuous contraction semigroup on a complex Banach space and letA be its infinitesimal generator. We prove that, forx D(A ³), the following inequalities hold true: Ax³ 243/8 x²A ³ x, A ² x 24 xA ³ x². Ift T(t) is a contraction group (resp. cosine function) we get the analogous but better inequalities with constants 9/8 and 3 (resp. 81/40 and 72/25) instead of 243/8 and 24. We consider also uniformly bounded semigroups, groups and cosine functions.     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

On extremal perturbations of semi-Fredholm operators

Given a bounded linear operatorA in an infinite dimensional Banach space and a compact subset of a connected component of its semi-Fredholm domain, we construct a finite rank operatorF such that –A+F is bounded below (or surjective) for each ,F ²=0 and rankF=max min{dimN(–A), codimR(–A)}, if ind(–A)0 (or ind(–A)0, respectively) for each .     

Semimodular lattices with isomorphic graphs

Two discrete modular lattice and have isomorphic graphs if and only if is of the form A × and is of the form A × for some lattices A and and . We prove that for discrete semimodular lattices and this latter condition holds if and only if and have isomorphic graphs and the isomorphism preserves the order on all cover-preserving sublattices of which are isomorphic to the seven-element, semimodular, nonmodular lattice (see Figure 1). This answers in the affirmative a question posed by J. Jakubik.     

Ergodic properties of Poisson processes with almost periodic intensity

Summary We discuss in this paper a non-homogeneous Poisson process A driven by an almost periodic intensity function. We give the stationary version A ^* and the Palm version A ⁰ corresponding to A ^*. Let (T _i ,i) be the inter-point distance sequence in A and (T _i ⁰ ,i) in A ⁰. We prove that forj, the sequence (T _i+j,i) converges in distribution to (T _i ⁰ ,i). If the intensity function is periodic then the convergence is in variation.     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(), s())} > 0 in linear programming at both = 0 and = . We establish the existence of a partition (B ,N ) of the index set { 1, ,n } such thatx _i() ands _j () as fori B , andj N , andx _N (),s _B () converge to weighted analytic centers of certain polytopes. For allk 1, we show that thekth order derivativesx ^(k) () ands ^(k) () converge when 0 and . Consequently, the derivatives of each order are bounded in the interval (0, ). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k 2) converge to zero when .     

The central limit theorem for Chébli-Trimèche hypergroups

Let (S _nn>-1) be a random walk on a hypergroup ( ₊ , *), i.e., a Markov chain with transition kernelN(x, A) = _x * (A), where is a fixed probability measure on ₊ such that the second moment exists. Then depending on the growth of the hypergroup two situations can occur: when ( ₊ , *) is of exponential growth then it is shown thatS _n is asymptotically normal. In the case of polynomial growth {more precisely, if the densityA of the Haar measure of ( ₊ , *) satisfies lim[A()/A()]=}, the normalized variablesS _n/[n Var()/(+1)]^1/2 converge to a Rayleigh distribution with parameter .     

Representations of distributive lattice-ordered semigroups with binary relations

LetR(, , ¦) denote the class of all algebras isomorphic to ones whose elements are binary relations and whose operations are union, intersection, and relation composition (or relative product) of relations. We prove thatR(, , ¦) is not a variety and is not finitely axiomatizable. LetDLOS denote the class of all structures (A, , , ) where (A, , ) is a distributive lattice, (A, ) is a semigroup and is additive w.r.t. . We prove thatDLOS is the variety generated byR(, , ¦), and moreover, if (A, , , ) DLOS then it is representable whenever we disregard one of its operations.Presented by Boris M. Schein.Research supported by Hungarian National Foundation for Scientific Research grant No. 1810.     

A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

Canonical Semigroups of States and Cocycles for the Group of Automorphisms of a Homogeneous Tree

Let G=A ut(T) be the group of automorphisms of a homogeneous tree and let d(v,gv) denote the natural tree distance. Fix a base vertex e in T. The function (g)=exp(–d(e,ge)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator d/d|₌₀=log() is conditionally positive definite but not positive definite. Hence, log() corresponds to a nontrivial cocycle (g): GH in some representation space H . In contrast with the case of PGL(2,), the representation is not irreducible.Let _o (g) be the derivative of the spherical function corresponding to the complementary series of A ut(T). We show that –d(e,ge) and _o (g) come from cohomologous cocycles. Moreover, _o is associated to one of the two (irreducible) special representations of A ut(T).     

Discretely normed Abelian groups

A discrete norm on an Abelian groupA is a non-negative function · A which satisfies the triangle inequality, is homogenous with respect to scaling ofA by and is bounded away from 0 onA/{0}.A countable Abelian group is discretely normed if and only if the group is free.     

On ideals of the group algebra of a free group of degree of freedom two over the field of complex numbers

In this paper, we prove the existence of an element of the group algebra A=F of a free groupF with two generatorsx andy over the field of complex numbersC such that, for any complexa andb for which ¦a¦=¦b¦=1, we haveA _a,b ()A=0, where _a,b ( is an automorphism ofA that mapsx,y intoax, by, respectively. Thus, we give a negative answer to question 12.46 of P. A. Linnel from Kourovka Notebook.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No.4, pp. 571–572, April, 1995.     

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

The continuous solution of a functional equation of Abel

Summary The functional equation(x) + (y) = (xf(y) + yf(x)) (1) for the unknown functionsf, and mapping reals into reals appears in the title of N. H. Abel's paper [1] from 1827 and its differentiable solutions are given there. In 1900 D. Hilbert pointed to (1), and to other functional equations considered by Abel, in the second part of his fifth problem. He asked if these equations could be solved without, for instance, assumption of differentiability of given and unknown functions. Hilbert's question was recalled by J. Aczél in 1987, during the 25th International Symposium on Functional Equations in Hamburg-Rissen. In particular Aczél asked for all continuous solutions of (1). An answer to his question is contained in our paper. We determine all continuous functionsf: I ,: A _f (I × I) and: I that satisfy (1). HereI denotes a real interval containing 0 andA _f (x,y) := xf(y) + yf(x), x, y I. The list contains not only the differentiable solutions, implicitly described by Abel, but also some nondifferentiable ones.Applying some results of C. T. Ng and A. Járai we are able to obtain even a more general result. For instance, the assertion (i.e. the list of solutions) remains unchanged if we replace continuity of and by local boundedness of orf(0)I from above or below. Strengthening a bit the assumptions onf we can preserve a large part of the assertion requiring only the measurability of either orf(0)I.     

Some results about the spectrum of commutative Banach algebras under the weak topology and applications

LetA be a commutative Banach algebra with a nonempty spectrum A. By weak we denote the relative weak topology induced on A by (A ^*,A ^**). In this note we study some properties of the topological space (A, weak) and present some applications of the results obtained and tools used to amenability, weakly compact homomorphisms, weakly compact subsets of the spectrum of the uniform algebras and to a characterization of the synthesizable ideals of the algebraA.