Hilbert spaces expanded with a unitary operator

We study Hilbert spaces expanded with a unitary operator with a countable spectrum. We show that the theory of such a structure is ω ‐stable and admits quantifier elimination. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reduced products and sheaves of metric structures

We introduce reduced products in continuous logic, and provide ways to establish satisfaction in the reduced product based on the satisfaction in its fibers, and conversely, on the lines that Horn and Chang did for reduced products in classical logic. We also consider a definition of a sheaf of metric structures, endow its stalks with a metric prestructure, and determine what filters can be used so that a collection of reduced products of arbitrary structures forms a sheaf on a given index space.     

Static search games played over graphs and general metric spaces

We define a general game which forms a basis for modelling situations of static search and concealment over regions with spatial structure. The game involves two players, the searching player and the concealing player, and is played over a metric space. Each player simultaneously chooses to deploy at a point in the space; the searching player receiving a payoff of 1 if his opponent lies within a predetermined radius r of his position, the concealing player receiving a payoff of 1 otherwise. The concepts of dominance and equivalence of strategies are examined in the context of this game, before focusing on the more specific case of the game played over a graph. Methods are presented to simplify the analysis of such games, both by means of the iterated elimination of dominated strategies and through consideration of automorphisms of the graph. Lower and upper bounds on the value of the game are presented and optimal mixed strategies are calculated for games played over a particular family of graphs.     

A Ramsey theorem for metric spaces

We show the Ramsey property of the metric spaces $(V,d)$ where $1\le d(x,y)\le 3$ ($x\ne y$).     

Topologies on homeomorphism spaces of certain metric spaces

A new proof for the equivalence of several topologies on homeomorphism groups over certain metric spaces X is given, which is based on the metric of X.     

Improved bounds in the metric cotype inequality for Banach spaces

It is shown that if (X,‖⋅X_‖) is a Banach space with Rademacher cotype q then for every integer n there exists an even integer such that for every we have

(1)     

Characterizations of Sobolev inequalities on metric spaces

We discuss Maz'ya type isocapacitary characterizations of Sobolev inequalities on metric measure spaces.     

Generalized metric spaces with algebraic structures

We discuss generalized metrizable properties on paratopological groups and topological groups. It is proved in this paper that a first-countable paratopological group which is a β-space is developable; and we construct a Hausdorff, separable, non-metrizable paratopological group which is developable. We consider paratopological (topological) groups determined by a point-countable first-countable subspaces and give partial answers to Arhangel'skii's conjecture; Nogura-Shakhmatov-Tanaka's question (Nogura et al., 1993 [23]). We also give a negative answer to a question in Cao et al. (in press) [10]. Finally, remainders of topological groups and paratopological groups are discussed and Arhangel'skii's Theorem (Arhangel'skii, 2007 [3]) is improved.     

Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D^∞(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D^∞(X) is compared with the space LIP^∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D^∞(X) with the Newtonian-Sobolev space N1,∞(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D^∞(X)=N1,∞(X).     

Localization and tensorization properties of the curvature-dimension condition for metric measure spaces

This paper is devoted to the analysis of metric measure spaces satisfying locally the curvature-dimension condition CD(K,N) introduced by the second author and also studied by Lott & Villani. We prove that the local version of CD(K,N) is equivalent to a global condition CD^∗(K,N), slightly weaker than the (usual, global) curvature-dimension condition. This so-called reduced curvature-dimension condition CD^∗(K,N) has the local-to-global property. We also prove the tensorization property for CD^∗(K,N). As an application we conclude that the fundamental group π₁(M,x₀) of a metric measure space (M,d,m) is finite whenever it satisfies locally the curvature-dimension condition CD(K,N) with positive K and finite N.     

Convergence results for obstacle problems on metric spaces

Let X be a complete metric space equipped with a doubling Borel measure supporting a p-Poincaré inequality. We obtain various convergence results for the single and double obstacle problems on open subsets of X. In particular, we consider single and double obstacle problems with fixed obstacles and boundary data on an increasing sequence of open sets.     

A note on unbounded metric temporal logic over dense time domains

We investigate the consequences of removing the infinitary axiom and rules from a previously defined proof system for a fragment of propositional metric temporal logic over dense time (see [1]). (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hybrid fixed points of multivalued operators in metric spaces with applications

This paper deals with the existence of hybrid fixed points involving two multivalued operators in a complete metric space. Our claim is also illustrated with applications to a class of integral inclusions for proving the existence results.     

Boole代数上的度量结构及其在命题逻辑中的应用

设B是一个Boole代数, Ω是从B到Boole代数{0,1}的全体同态之集,μ是Ω上的概率测度.本文基于μ在B中引入了元素的尺寸概念以及元素对之间的相似度概念,并由此在B上建立了度量结构.作为应用,本文改进了新近提出的命题逻辑中的近似推理理论.     

Fixed point theorems of upper semicontinuous multivalued mappings with applications in hyperconvex metric spaces

In the present paper, we establish two fixed point theorems for upper semicontinuous multivalued mappings in hyperconvex metric spaces and apply these to study coincidence point problems and minimax problems.