Explicit matchings in the middle levels of the Boolean lattice

New classes of explicit matchings for the bipartite graph (k) consisting of the middle two levels of the Boolean lattice on 2k+1 elements are constructed and counted. This research is part of an ongoing effort to show that (k) is Hamiltonian.Supported by Office of Naval Research contract N00014-85K-0494.Supported by National Science Foundation grant DMS-8041281.     

Self-Converse Directed BIBDs with Block Size Four

A directed balanced incomplete block design (or D B(k,;v)) (X,) is called self-converse if there is an isomorphic mapping f from (X,) to (X,^–1), where ^–1={B ^–1:B} and B ^–1=(x _k ,x _{k –1},,x ₂,x ₁) for B=(x ₁,x ₂,,x _{k –1},x _k ). In this paper, we give the existence spectrum for self-converse D B(4,1;v). AMS Classification:05BResearch supported in part by NSFC Grant 10071002 and SRFDP under No. 20010004001     

The caratheodory number for thek-core

Thek-core of the setS ⁿ is the intersection of the convex hull of all setsA S with ¦SA¦<-k. The Caratheodory number of thek-core is the smallest integerf (d,k) with the property thatx core _{kS, S} ⁿ implies the existence of a subsetT S such thatx core_kT and ¦T¦f (d, k). In this paper various properties off(d, k) are established.Research of this author was partially supported by Hungarian National Science Foundation grant no. 1812.     

The Ramsey property for collections of sequences not containing all arithmetic progressions

A family of sequences has the Ramsey property if for every positive integerk, there exists a least positive integerf (k) such that for every 2-coloring of {1,2, ...,f (k)} there is a monochromatick-term member of . For fixed integersm > 1 and 0 q < m, let _q(m) be the collection of those increasing sequences of positive integers {x ₁,..., x_k} such thatx _i+1 – x_i q(modm) for 1 i k – 1. Fort a fixed positive integer, denote byA _t the collection of those arithmetic progressions having constant differencet. Landman and Long showed that for allm 2 and 1 q < m, _q(m) does not have the Ramsey property, while _q(m) A _m does. We extend these results to various finite unions of _q(m) 's andA _t 's. We show that for allm 2, _q=1 ^m–1 _q(m) does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form _q(m) ( _{t T} A _t) to have the Ramsey property. We determine when collections of the form _a(m₁) _b(m₂) have the Ramsey property. We extend this to the study of arbitrary finite unions of _q(m)'s. In all cases considered for which has the Ramsey property, upper bounds are given forf .     

On the typicality of linear Hamiltonian systems with exponential separateness

Linear Hamiltonian systems of ordinary differential equations,are considered, where f ^t is a continuous action of the group R on a complete metric space , A is an element of the space S _H ,consisting of the continuous bounded mappings of into the set of all pseudosymmetric matrices and endowed with the uniform convergence metric. By ₁ (A, x) _2n (A, x) we denote the Lyapunov exponents of such systems. The typicality (in the Baire sense) in the space S _H × is proved for those pairs (A, x) for which one has the alternative: either _k (A, x) = _k+1 (A, x) or the linear subspace of the solutions of the corresponding system with exponents less than _k (A, x) is exponentially separated from any of its algebraic complements in the space of all the solutions of the system. From here, in particular, there follows the typicality of the formulated alternative for linear Hamiltonian systems with continuous quasiperiodic coefficients (with the same frequency module) and also for linear Hamiltonian systems with continuous almost periodic coefficients. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 137–181, 1992.     

Bijectivity of -endomorphisms of ℬ(H) and the unilateral shift

LetH be a separable infinite-dimensional complex Hilbert space. We prove that if : (H)(H) is a^*-preserving ring homomorphism whose range contains a rank-one operator and an operator with dense range, then is an isometric linear or conjugate-linear algebra automorphism of (H). In particular, if the unilateral shift is contained in the range of a^*-endomorphism of (H), then is bijective.Research partially supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA 1652 and K&H Bank Ltd., Universitas Foundation.     

On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.     

Maximal Classes of Topological Spaces and Domains Determined by Function Spaces

In this paper the question of what classes A of T ₀-spaces should be paired with classes of domains in order that all function spaces [AB] for AA and B are -compact domains is considered. It is shown that core compact spaces are paired with bounded complete domains and a class of topological spaces called RW-spaces (with finitely many components) is paired with the class of -compact pointed L-domains (L-domains).     

Homotopy Classification of Categorical Torsors

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G,) over a small category . We show a natural bijection between the set of equivalence classes of such torsors and [B({}),B(G,)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with _i (Y)=0 for all i1,2.     

A Lévy Type Martingale Convergence Theorem for Random Sets with Unbounded Values

Given a nondecreasing sequence ( _n ) of sub--fields and a real or vector valued random variable f, the Lévy Martingale convergence Theorem (LMCT) asserts that E(f/ _n ) converges to E(f/) almost surely and in L ¹, where stands for the -field generated by the _n . In the present paper, we study the validity of the multivalued analog this theorem for a random set F whose values are members of (X), the space of nonempty closed sets of a Banach space X, when (X) is endowed either with the Painlevé–Kuratowski convergence or its infinite dimensional extensions. We deduce epi-convergence results for integrands via the epigraphical multifunctions. As it is known, these results are useful for approximating optimization problems. The method relies on countability supportness hypotheses which are shown to hold when the values of the random set E(F/ _n ) do not contain any line. On the other hand, since the values of F are not assumed to be bounded, conditions involving barrier and asymptotic cones are shown to be necessary. Moreover, we discuss the relations with other multivalued martingale convergence theorems and provide examples showing the role of the hypotheses. Even in the finite dimensional setting, our results are new or subsume already existing ones.