Convex Hulls in Singular Spaces of Negative Curvature

The paper gives a simple example of a complete CAT(–1)-space containing a set S with the following property: the boundary at infinity CH(S)of the convex hull of S differs from S by an isolated point. In contrast to this it is shown that if S is a union of finitely many convex subsets of a complete CAT(–1)-space X, then CH(S) = S. Moreover, this identity holds without restrictions on S if CH is replaced by some notion of almost convex hull.     

Strongly almost disjoint sets and weakly uniform bases

A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of for singular is proved. CECA is used to show that certain ``almost point-' families can be refined to point- families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of ``every first countable -space with a weakly uniform base has a point-countable base.'

     

The set of idempotents in the weakly almost periodic compactification of the integers is not closed

This paper answers negatively the question of whether the sets of idempotents in the weakly almost periodic compactifications of and are closed.

     

Convergence of Jamison-Type Weighted Sums of Pairwise Negatively Quadrant Dependent Random Variables

Abstract Under very general weight function,we discuss the convergence of Jamison-type weighted sums ofpairwise negatively quadrant dependent(NQD)r.v.'s.The results on.i.i.d.setting of [3] and [1] are extendedand generalized.As corollaries,we obtain some results of [11].     

ON EXPONENTIAL STABILITY OF NON-AUTONOMOUS STOCHASTIC SEMILINEAR EVOLUTION EQUATIONS

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Density estimation for a class of stationary nonlinear processes

Let {X _t ;t∈ℤ be a strictly stationary nonlinear process of the formX _t =ε _t +∑ _r=1 ^∞ W _rt , whereW _rt can be written as a functiong _r (ε _t−1,...ε _t-r-q ), {ε _t ;t∈ℤ is a sequence of independent and identically distributed (i.i.d.) random variables withE|ε₁| ^g < ∞ for some γ>0 andq≥0 is fixed integer. Under certain mild regularity conditions ofg _r and {ε _t } we then show thatX ₁ has a density functionf and that the standard kernel type estimator baded on a realization {X ₁,...,X _n } from {X _t } is, asymptotically, normal and converges a.s. tof(x) asn→∞. The research of this author was partially carried out while he was a research scholar, on a sabbatical leave, at the Department of Statistics and Probability, Michigan State University.     

Generalized almost periodic and ergodic solutions of linear differential equations on the half-line in Banach spaces

The Bohl-Bohr-Amerio-Kadets theorem states that the indefinite integral y=Pφ of an almost periodic (ap) is again ap if y is bounded and the Banach space X does not contain a subspace isomorphic to c₀. This is here generalized in several directions: Instead of it holds also for φ defined only on a half-line , instead of ap functions abstract classes with suitable properties are admissible, can be weakened to φ in some “mean” class , then ; here contains all f∈L¹_loc with in for all h>0 (usually strictly); furthermore, instead of boundedness of y mean boundedness, y in some , or in , ergodic functions, suffices. The Loomis-Doss result on the almost periodicity of a bounded Ψ for which all differences Ψ(t+h)−Ψ(t) are ap for h>0 is extended analogously, also to higher order differences. Studying “difference spaces” in this connection, we obtain decompositions of the form: Any bounded measurable function is the sum of a bounded ergodic function and the indefinite integral of a bounded ergodic function. The Bohr-Neugebauer result on the almost periodicity of bounded solutions y of linear differential equations P(D)y=φ of degree m with ap φ is extended similarly for ; then provided, for example, y is in some with U=L^∞ or is totally ergodic and, for the half-line, Reλ?0 for all eigenvalues P(λ)=0. Analogous results hold for systems of linear differential equations. Special case: φ bounded and Pφ ergodic implies Pφ bounded. If all Reλ>0, there exists a unique solution y growing not too fast; this y is in if , for quite general .     

Generalized Tensor Structures of Kawaguchi Spaces

We prove that, in a generalized Kawaguchi space, there exist intrinsic almost product and almost complex structures associated to a metric of this space. We derive conditions for the integrability of these structures and find compatible affine connections.     

Weierstrass functions with random phases

Consider the function

where 1$">, , and is a non-constant 1-periodic Lipschitz function. The phases are chosen independently with respect to the uniform probability measure on . We prove that with probability one, we can choose a sequence of scales such that for every interval of length , the oscillation of satisfies . Moreover, the inequality is almost surely true at every scale. When is a transcendental number, these results can be improved: the minoration is true for every choice of the phases and at every scale.

     

On regular branching processes with infinite mean

A martingale, previously used to prove the classical almost sure convergence of the normed supercritical Galton-Watson branching process with finite mean without using probability generating functions, is here used to study similar behaviour for certain processes with infinite mean.