弱^＊局部一致凸空间的一些性质

讨论了弱^*局部一致凸空间的一些等价定义和性质,以及乘积空间的弱^*凸部一致凸的传递性。     

Ultrafilter Completeness in {{\varepsilon}} -approach Nearness Spaces

This paper presents a new approach to the proof of the Niemytzki–Tychonoff theorem for symmetric topological spaces. The proof uses the concept of completeness in ${\varepsilon}$ -approach nearness spaces which was introduced by Peters and Tiwari (Appl Math Lett 25:1544–1547, 2012), and of clusters that are a generalization of Cauchy sequences.     

Global Existence of Solutions to the Coupled Einstein and Maxwell-Higgs System in the Spherical Symmetry

We prove the global unique existence of classical solutions to the Einstein equations coupled with Maxwell-Higgs system for small initial data under the spherical symmetry. We also obtain the decay estimates of the solutions, and find that the corresponding space-time is time-like and null geodesically complete toward the future. For the proof we reduce the system to a single first order integrodifferential equation, and use the contraction mapping theorem in the appropriate function spaces. We also obtain the completeness of space-time along the future directed time-like lines exterior to a region which resembles the even horizon of the Reissner-Nordström black hole.     

Translation representations for automorphic solutions of the wave equation in non-euclidean spaces. III

In Part III we show that the translation representations defined in Part I, and shown to be complete in Part II, are incoming and outgoing in the sense of propagation of signals along rays. We give a new proof of the absolute continuity of the spectrum below -¼n², and point out its implications for local energy decay. We give a new proof of completeness in dimensions 2 and 3 when energy is positive. Finally, we define and prove completeness of the translation representations when the metric is perturbed on a compact set.     

Approximation of the Müntz-Szász type in weight spaces L p and zeroes of functions of Bergman classes in a half-plane

We study the completeness of the system of exponents exp(?λ _n t), Re λ _n > 0, in spaces L ^p with the power weight on the semiaxis ?₊. We prove a sufficient condition for the completeness; one can treat it as a modification of the well-known Szász condition. With p = 2 it is unimprovable (in a sense). The proof is based on the results (which are also obtained in this paper) on the distribution of zeroes of functions of the Bergman classes in a half-plane.     

Countable ultraproducts without CH

An important application of ultrafilters is in the ultraproduct construction in model theory. In this paper we study ultraproducts of countable structures, whose universe we assume is ω, using ultrafilters on a countable index set, which we also assume to be ω. Many of the properties of the ultraproduct are in fact inherent properties of the ultrafilter. For example, if we take a sequence of countable linear orders without maximal element, then their ultraproduct will have no maximal element, and we can ask what its cofinality is. This cardinal depends only on the ultrafilter; it does not depend on what linear orders comprise the factors.     

On Strict Extensions of Nearness Spaces

Strict extensions of nearness spaces are constructed as spaces of round Cauchy filters. Morita's simple extension is identified as the strict extension generated by Morita-generated filters. Carlson's B-completeness is compared with Herrlich completeness and completeness of Morita T-uniformities. The three completeness concepts are shown to be equivalent in regular nearness spaces.     

Translation representation for automorphic solutions of the wave equation in non-euclidean spaces,IV

In Part I of this series of papers we have defined the incoming and outgoing translation representations for automorphic solutions of the hyperbolic wave equations; in Part II we have proved the completeness of these representations when the fundamental polyhedron F has a finite number of sides with a finite or infinite volume, but is not compact. In Part IV we present a proof of completeness which is simpler than our original proof contained in Section 7 of Part II for the case when F has cusps of less than maximal rank; and we supply a proof for the case, not covered in Section 7, when the parabolic subgroup associated with such cusps contains twists.     

Three‐query PCPs with perfect completeness over non‐Boolean domains

We study non‐Boolean PCPs that have perfect completeness and query three positions in the proof. For the case when the proof consists of values from a domain of size d for some integer constant d ≥ 2, we construct a nonadaptive PCP with perfect completeness and soundness d^?1 + d^?2 + ?, for any constant ? > 0, and an adaptive PCP with perfect completeness and soundness d^?1 + ?, for any constant ? > 0. The latter PCP can be converted into a nonadaptive PCP with perfect completeness and soundness d^?1 + ?, for any constant ? > 0, where four positions are read from the proof. These results match the best known constructions for the case d = 2 and our proofs also show that the particular predicates we use in our PCPs are nonapproximable beyond the random assignment threshold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Universal algebra in a closed category

The development of finitary universal algebra is carried out in a suitable closed category called a π-category. The π-categories are characterized by their completeness and cocompleteness and some product-colimit commutativities. We establish the existence of left adjoints to algebraic functors, completeness and cocompleteness of algebraic categories, a structure-semantics adjunction, a characterization theory for algebraic categories and the existence of the theory generated by a presentation. The conditions on the closed category are sufficiently weak to be satisfied by any (complete and cocomplete) cartesian closed category, semi-additive category, commutatively algebraic category and also the categories of semi-normed spaces, normed spaces and Banach spaces.     

四元数Hilbert空间中Riesz基的刻画*

四元数Hilbert空间在应用物理科学特别是量子物理中占有重要地位.本文讨论四元数Hilbert空间的框架理论, 在四元数Hilbert空间中引入了Riesz基的概念, 在此基础上刻画了Riesz基,给出了它们的一些等价条件; 特别地, 得到了四元数Hilbert空间中的一个序列是Riesz基的充要条件是它是一个具有双正交序列的完备Bessel序列,且它的双正交序列也是一个完备Bessel序列; 并进一步证明了双正交序列中一个序列的完备性可以从特征刻画中去除.文中举例说明了双正交性、完备性和Bessel性质之间的关系.     

Bishop's Lemma

Bishop's Lemma is a centrepiece in the development of constructive analysis. We show that

• 1. its proof requires some form of the axiom of choice; and that
• 2. the completeness requirement in Bishop's Lemma can be weakened and that there is a vast class of non‐complete spaces that Bishop's Lemma applies to.

     

Proofs and Countermodels in Non-Classical Logics

Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame class is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait–Schütte–Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope.     

From concentration to logarithmic Sobolev and Poincaré inequalities

We give a new proof of the fact that Gaussian concentration implies the logarithmic Sobolev inequality when the curvature is bounded from below, and also that exponential concentration implies Poincaré inequality under null curvature condition. Our proof holds on non-smooth structures, such as length spaces, and provides a universal control of the constants. We also give a new proof of the equivalence between dimension free Gaussian concentration and Talagrand's transport inequality.     

A RESULT OF SUZUKI TYPE IN PARTIAL G-METRIC SPACES

Recently, Suzuki [T. Suzuki characterizes metric completeness, Proc. A generalized Banach contractlon principle that Amer. Math. Soc. 136 （2008）, 1861-1869] proved a fixed point theorem that is a generalization of the Banach contraction principle and char- acterizes the metric completeness. Paesano and Vetro [D. Paesano and P. Vetro, Suzuki＇s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topology Appl., 159 （2012）, 911-920] proved an analogous fixed point result for a self-mapping on a partial metric space that characterizes the partial metric O- completeness. In this article, we introduce the notion of partial G-metric spaces and prove a result of Suzuki type in the setting of partial G-metric spaces. We deduce also a result of common fixed point.     

Abstract capacitary estimates and the completeness and separability of certain classes of non-locally convex topological vector spaces

We are concerned with establishing completeness and separability criteria for large classes of topological vector spaces which are typically non-locally convex, including Lebesgue-like spaces, Lorentz spaces, Orlicz spaces, mixed-normed spaces, tent spaces, and discrete Triebel–Lizorkin and Besov spaces. For vector spaces of measurable functions we also derive pointwise convergence results. Our approach relies on abstract capacitary estimates and works in certain cases of interest even in the absence of a background measure space and/or of a vector space structure.     

Morse-Palais lemma for nonsmooth functionals on normed spaces

Using elementary differential calculus we get a version of the Morse-Palais lemma. Since we do not use powerful tools in functional analysis such as the implicit theorem or flows and deformations in Banach spaces, our result does not require the -smoothness of functions nor the completeness of spaces. Therefore it is stronger than the classical one but its proof is very simple.

     

Completeness in generalized ultrametric spaces

We analyze the relationship between four notions of completeness for Γ-ultrametric spaces. The notions we consider include Cauchy completeness, strong Cauchy completeness, spherical completeness and injectivity. In the process we show that the category of Γ-ultrametric spaces is equivalent to the category of flabby separated presheaves on Γ ^op .     

Weakly Cauchy Filters and Quasi-Uniform Completeness

It is well-known that the notion of a Smyth complete quasi-uniform space provides an appropriate notion of completeness to study many interesting quasi-metric spaces which appear in theoretical computer science. We observe that several of these spaces actually possess a stronger form of completeness based on the use of weakly Cauchy filters in the sense of H. H. Corson and we develop a theory of completion and completeness for this kind of filters. In parallel, we also study a more general notion of completeness based on the use of certain stable filters. Thus our results extend and generalize important theorems of Á. Császár, J. R. Isbell and N. R. Howes on uniform completeness.