On Gauss quadrature formulas with equal weights

Summary It is well known that the Chebyshev weight function (1–x ²)^–1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln. In this paper we describe all weight functions for which thenm point Gauss quadrature formula has equal weights for alln, wherem is fixed.     

An honest stiff tree-like algebra

We construct in ZFC a compactificationb of integers such that the Boolean algebra Clopen(b) is tree-like and every one-to-one continuous mapping as well as every continuous onto mapping fromb tob is a homeomorphism, and every homeomorphism fromb onto itself is identity at all except finitely many points from .Presented by J. Mycielski.The research was supported by the grant GAUK 350.The author owes sincere thanks to the referee, whose comments bridged many gaps in the first version of the present paper.     

Almost sure behaviour ofF-valued random fields

Summary LetX _n, n ^d be a field of independent random variables taking values in a semi-normed measurable vector spaceF. For a broad class of fields _n, ^d of positive numbers, the almost sure behaviour of _knX_k/_n, n ^d is studied. The main result allows us to deduce some new and well-known theorems for fields of independentF random variables from related results for fields of independent real random variables.Supported in part by the Youth Science Foundation of China, No. 19001018Supported by the National Natural Science Foundation of China     

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.     

Positivity Preserving Forms have the Fatou Property

We prove that if (,D) is a positivity preserving form on L ² (E;m), and if (u _n)_n is a sequence in D() converging m-almost everywhere to u L ² (E;m), then (u,u) lim inf_n (u _n ,u _n ).     

Path Regularity and Regularity in Mean of 1st Order Processes

We show that a 1st order stochastic process which is C ⁿ in mean, for some n{} on an interval T has a version whose paths lie in C ^n–1(T).     

Compact groups having almost discrete orbit hypergroups

LetG be a compact group of automorphism acting continuously on a compact groupH. Then the orbit spaceH ^G is a compact hypergroup. We characterize, all solvable groupsH and compact automorphism groupsG for whichH ^G is almost discrete, i.e.,H ^G is homeomorphic to the one-point-compactification of . It turns out that thenH is isomorphic either to the infinite direct product (p) of the cyclic groups (p) or to _p ⁿ ( _p the group of allp-adic numbers) for some primep and some . The almost discrete orbit hypergroupsH ^G are determined explicitly for some examples.     

Convergence of diagonally stationary sequences in convex optimization

LetX be a real normed linear space,f, f ⁿ, n , be extended real-valued proper closed convex functions onX. A sequence {x _n} inX is called diagonally stationary for {f ⁿ} if for alln there existsx* _n f ⁿ (x _n) such that x* _{n *} 0. Such sequences arise in approximation methods for the problem of minimizingf. Some general convergence results based upon variational convergence theory and appropriate equi-well-posedness are presented.     

Über beschränkte Mengen in induktiven Limiten topologischer Vektorräume

In topological linear spaces convex hulls of bounded sets are, in general, not bounded. The question arises whether there is at least for every bounded set B a sequence {|} of strictly positive numbers such that the set { _l ⁿ _v B|n} is bounded. When this obtains, bounded sets share several of the properties known in locally convex spaces. The main result of this note is an example of a countable inductive limit of complete metrizable topological linear spaces which is neither regular nor sequentially complete and also fails to have the above bounded summability property.     

Aufeinanderfolgende Elemente in multiplikativen Zahlenmengen

LetM be a multiplicative set with 1M andmnM if and only ifmM,nM for (m,n)=1. It is shown by elementary means that there exists the asymptotic density of the setM(M–1) for every multiplicative setM. The density is positive if and only ifM possesses a positive density and 2M for some . This result is slightly generalized to sums over multiplicative functionsf with |f|1.     

Percolation in half-spaces: equality of critical densities and continuity of the percolation probability

Renormalization arguments are developed and applied to independent nearest-neighbor percolation on various subsets of ^d ,d2, yielding:

On Sums of Projections

In the paper, for all n, we describe the set _n of all real numbers admitting a collection of projections P ₁,...,P _n on a Hilbert space H such that _k=1 ⁿ P _k= I (I is the identity operator on H) and study the problem to find all collections of this kind for a given _n .     

Nielsen Methods and Groups Acting on Hyperbolic Spaces

We show that for any n & in there exists a constant C(n) such that any n-generated group G which acts by isometries on a -hyperbolic space (with >0) is either free or has a nontrivial element with translation length at most C(n).     

Homothetic floating bodies

We show that a convex bodyK in ⁿ is homothetic to an ellipsoid if there is a sequence { _k }k converging to 0 so thatK is homothetic to its floating bodiesK _k.Supported by NSF grant DMS-9108003.     

Quasi-Differential Posets and Cover Functions of Distributive Lattices II: A Problem in Stanley’s <Emphasis Type="Italic">Enumerative Combinatorics</Emphasis>

A distributive lattice L with 0 is finitary if every interval is finite. A function f:₀₀ is a cover function for L if every element with n lower covers has f(n) upper covers. All non-decreasing cover functions have been characterized by the author ([2]), settling a 1975 conjecture of Richard P. Stanley. In this paper, all finitary distributive lattices with cover functions are characterized. A problem in Stanleys Enumerative Combinatorics is thus solved. 2000 Mathematics Subject Classification. 06A07, 06B05, 06D99, 11B39     

Cauchy Problems in Weighted Lebesgue Spaces

Global solvability and asymptotics of semilinear parabolic Cauchy problems in ⁿ are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over ⁿ, n . In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.     

Invariance principles for random walks on hypergroups on ℝ2 and ℕ

Let (X _n:n) be i.i.d. with finite variance and values in a hypergroupK:=₊ or and _j=1 ⁿ X _j be the randomized sum of these random variables. It is shown that the processes converge in distribution to a Gaussian process in the caseK=₊, that the processes converge towards a Bessel process on ₊ in the case of polynomial growth of the hypergroupK=₊ or , and that in the case of exponential growth converges towards a Brownian motion asn.     

OnC *-algebras having linear,polynomial and subexponential growth

Summary Answering a question of Voiculescu [16, Problem 5.9], we show thatC ^*-algebras having filtrations (A _n)_n satisfying the condition lim sup_n ln dimA _n/n=0 (in particular having subexponential growth), are nuclear.For the case of linear growth we obtain the following particular result: letX be a finite dimensional self-adjoint generating system of aC ^*-algebraA such that dim (span (X ⁿ +1))1+dim (span (X ⁿ )), then there exist a finite dimensionalC ^*-algebraC having only irreducible representations of dimension 1 and aC ^*-algebraB, which is generated by a single self-adjoint element, such thatACB.Some other results are given on linear growth and we show that there exist singly generatedC ^*-algebras such that the growth of the filtration (span (X ⁿ ))_n is polynomial, whereX={x,x ^*, 1} is a generating system, and such that in every neighbourhood ofx there exists an invertibley such thatY={1,y,y ^*} is a generating system whose associated filtration (span (Y ⁿ ))_n doesn't satisfy the previous condition of Voiculescu, and in particular does not have subexponential growth.Oblatum 19-X-1991Work partially supported by DFGAllocataire M.R.T., Université Aix-Marseille II (France)     

Durchschnitte von Pjateckij-Shapiro-Folgen

The distribution of integers and prime numbers in sequences of the formF _c1F _c2 is investigated. HereF _c={[_n ^c]:n } withc>1.