Necessary and sufficient conditions for Hölder continuity of Gaussian processes

The continuity of Gaussian processes is an extensively studied topic and it culminates in Talagrand’s notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note we study the Hölder continuity of Gaussian processes. It turns out that necessary and sufficient conditions can be stated in a simple form that is a variant of the celebrated Kolmogorov–Čentsov condition.     

The research of thirteen students at Howard University

Results from, and problems related to, the dissertations of my Ph.D. students are discussed.     

On openness and surjectivity of lifted frame homomorphisms

Given a completely regular frame L, let, as usual, βL, λL and υL denote, respectively, the Stone-?ech compactification, the universal Lindelöfication and the Hewitt realcompactification of L. Let γ denote any of the functors β, λ or υ. It is well known that any frame homomorphism h:L→M has a unique “lift” to a frame homomorphism hγ:γL→γM such that σM⋅hγ=h⋅σL, where the σ-maps are effected by join. We find a condition on h such that if h satisfies it, then h is open iff its lift hγ is open. Furthermore, the same condition ensures that hγ is nearly open iff h is nearly open. This latter result is, in fact, a special case of a more general phenomenon. In the last part of the paper we investigate when hυ is surjective. The instances when hβ or hλ is surjective are known. It turns out that the surjectivity of the lifted map hυ:υL→υM captures Blair's notion of υ-embedding in the sense that a subspace S of a Tychonoff space X is υ-embedded iff the lifted map υ(Oi):υ(OX)→υ(OS) is surjective, where i:S→X is the subspace embedding.     

Spaces N ∪ R and their dimensions

About spaces N∪$\text{R}$ (see [2, Exercise 5I]), the following are proved: (1) dim $\text{N∪}\text{R}\text{= dim β(N∪}\text{R}\text{)?N∪}\text{R}\text{,(2)if|β(N∪}\text{R}\text{)?N∪}\text{R}\text{|<2}{}^{\mathrm{?}}{}^{\mathrm{o}}$, then no real-valued continuous fu ction on N∪$\text{R}$ is onto (and hence, dim $\text{N∪}\text{R}\text{=0}$), (3) any compact metric space without isolated points is homeomorphic to some $\text{β(N∪}\text{R}\text{)?N∪}\text{R}$ and (4)there are spaces X,X₁ and X₂ of the form N∪$\text{R}$ such that X=X₁∪X₂,X₂andX₂ are zero sets of X, and dim X=n, dimX₁=dimX₂=0, where n=1,2,… or ∞.     

集合论在拟仿紧性中的应用

1977年,刘应明引进拟仿紧性并证明了下述集论拓扑学的结果：在假设$2^{\omega_1}>2^{\omega}$下每一个可分正规的拟仿紧空间是仿紧空间.我们进一步证明假设$2^{\omega_1}>2^{\omega}$等价于每一个可分正规的拟仿紧空间是仿紧空间,获得了拟仿紧性的一个独立性结果.