On the upper semicontinuity of the Wu metric

We discuss continuity and upper semicontinuity of the Wu pseudometric.

     

Decomposition of finite pseudometric spaces

Here we define decomposable pseudometrics. A pseudometric is decomposable if it can be represented as the sum of two pseudometrics that are obtained in a way other than the multiplication all distances by a positive factor. We consider spaces consisting ofn points. We prove that there exist a finite number of indecomposable pseudometrics (that is, a basis) such that any pseudometric is a linear combination of basic pseudometrics with nonnegative coefficients. Forn ≤ 7, the basic pseudometrics are listed. A decomposability test is derived for finite pseudometric spaces. We also establish some other conditions of decomposability and indecomposability. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 225–234, February, 1998.     

Ptolemaic spaces

We show that equilateral pseudolinear quadruples are in a certain sense extremally non-Ptolemaic metric spaces. We find four-point pseudometric spaces that are maximally Ptolemaic.     

On the connectedness of isomorphism classes

This paper is concerned with some topological properties of a notion of distance between normed linear spaces which is due to Banach and Mazur [1], We are interested in a pseudometric space associated naturally with this notion of distance, and our main result is that this pseudometric space is always pathwise connected.The results in this paper were taken from the author's doctoral dissertation written at the University of Maryland under the direction of Robert Whitley. The research was partially supported by the N.S.F.     

Kobayashi–Royden pseudometric versus Lempert function

We give an example showing that the Kobayashi–Royden pseudometric for a pseudoconvex domain is, in general, not the derivative of the Lempert function.     

Pre-Schwarz导数意义下区域之间的距离

Lehto曾用Schwarz导数定义了边界多于一点的两个单连通区域的Mbius等价类之间的"距离",并猜测它是一个距离.但最近Bozin和Markovic否定了这一猜想.一个自然的问题就是:在Pre-Schwarz导数意义相应情况如何?用Pre-Schwarz导数给出了边界多于一点的两个单连通区域的仿射等价类之间的"距离",并证明了这样定义的"距离"是一个伪距离,即使将其限制在由具有解析边界的单连通区域的仿射等价类空间上也是如此.     

A unified approach to fuzzy pattern recognition

Goldfarb has proposed a unified approach to pattern recognition. In Goldfarb's approach the data from a pseudometric space are isometrically embedded in a pseudo-Euclidean space. The data representation is built for every particular data set. The resulting data space will be data dependent. The aim of this paper is to extend Goldfarb's approach for fuzzy clustering. A fuzzy clustering procedure for a pseudometric data set is given. A generalisation of this algorithm to obtain the cluster substructure of a fuzzy class is also proposed.     

Hofer’s metric on the space of Lagrangian submanifolds and wrapped Floer homology

In this paper, we study the relationship between wrapped Floer homology and displaceability of a Lagrangian submanifold which we call vanishing theorem of wrapped Floer homology. We also use this theorem to study Hofer’s pseudometric on the space of Lagrangian submanifolds. We prove an inequality, the Lagrangian version of the inequality of Gromov width and displacement energy, which is called energy-capacity inequality.     

Common Fixed Point Theory for Multivalued Contractive Maps of Reich Type in Uniform Spaces

This article presents common fixed point theory for multivalued maps defined on complete gauge spaces. Our analysis is elementary and relies only on properties of the generalized Hausdorff pseudometric.     

Pseudoconvex regions of finite D’Angelo type in four dimensional almost complex manifolds

Let D be a J-pseudoconvex region in a smooth almost complex manifold (M, J) of real dimension four. We construct a local peak J-plurisubharmonic function at every point p ∈ bD of finite D’Angelo type. As applications we give local estimates of the Kobayashi pseudometric, implying the local Kobayashi hyperbolicity of D at p. In case the point p is of D’Angelo type less than or equal to four, or the approach is nontangential, we provide sharp estimates of the Kobayashi pseudometric.     

Discontinuity of the Lempert Function and the Kobayashi-Royden Metric of the Spectral Ball

Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given. This work was initialized during the stay of the first and second named authors at the Jagiellonian University, Kraków in October, 2006, supported by the EGIDE program. They wish to thank both institutions. The third author was supported by the KBN research grant No. 1 PO3A 005 28.     

Fefferman’s mapping theorem on almost complex manifolds in complex dimension two

We give a necessary and sufficient condition for the smooth extension of a diffeomorphism between smooth strictly pseudoconvex domains in four real dimensional almost complex manifolds (see Theorem 1.1). The proof is mainly based on a reflection principle for pseudoholomorphic discs, on precise estimates of the Kobayashi-Royden infinitesimal pseudometric and on the scaling method in almost complex manifolds.Mathematics Subject Classification (2000): 32H02,53C15     

Multidimensional Persistence and Noise

In this paper, we study multidimensional persistence modules (Carlsson and Zomorodian in Discrete Comput Geom 42(1):71–93, 2009; Lesnick in Found Comput Math 15(3):613–650, 2015) via what we call tame functors and noise systems. A noise system leads to a pseudometric topology on the category of tame functors. We show how this pseudometric can be used to identify persistent features of compact multidimensional persistence modules. To count such features, we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For one-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.     

Nonexpansive algebras

A nonexpansive algebra is a pseudometric algebra in which the operations are all nonexpansive. We study such algebras, particularly in the case of algebras in permutable and n-permutable varieties, leading to new characterizations of such varieties. Free nonexpansive algebras are also investigated. This paper is dedicated to Walter Taylor. Received September 27, 2005; accepted in final form February 4, 2006. The author would like to thank the referee for the many helpful suggestions.     

Sigma-compactness of metric boolean algebras and uniform convergence of frequencies to probabilities

The topological properties of a pseudometric space defined by a measure are investigated. Criteria of compactness and σ-compactness of this space are proved. A new sufficient condition for the uniform convergence (over an event class) of frequencies to probabilities is proved as a corollary.     

Conjugate families of mappings in pointwise metric fuzzy lattices

In this paper some relations among the axioms of pointwise metric for fuzzy lattices introduced by Shi are studied. By means of the conjugate families of mappings, some representation results are presented concerning metric, pseudometric and pseudo-quasi-metric fuzzy lattices. These results reconcile the metric lattice theory generated via concept of neighborhoodwith the one generated via concept of remote-neighborhood.     

Fundamental polytopes of metric trees via parallel connections of matroids

We tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik (2010).In this paper we consider a hyperplane arrangement associated to every split pseudometric and, for tree-like metrics, we study the combinatorics of its underlying matroid.

• •We give explicit formulas for the face numbers of fundamental polytopes and Lipschitz polytopes of all tree-like metrics.
• •We characterize the metric trees for which the fundamental polytope is simplicial.

     

Smooth kinematic-type manifolds

We propose a general approach for describing different causality-type relations on smooth manifolds. The causality structure can be defined either axiomatically (by a cone in the tangent space) or by a pseudometric with the signature (+−...−) or (+−...−0...0). In the latter case, the manifold acquires the structure of a fibered space with “absolute simultaneity” fibers. The smooth structure (atlas) of the manifold is directly related to its causal structure. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 2, pp. 264–281, May, 1999.     

On the Metrizability of Fuzzy Neighborhood Spaces

A fuzzy neighborhood space is said to be pseudometrizable if, and only if, the associated fuzzy topology is induced by a probabilistic pseudometric under. By the aid of the concept of quasi-inverse functions it will be shown that a fuzzy neighborhood space is pseudometrizable if, and only if, every associated level-topology is pseudometrizable and a family of generating pseudometrics is monotone decreasing.