A fixed point theorem for a family of mappings in a fuzzy metric space

In this paper we give a common fixed point theorem for a family of mappings of a G-complete fuzzy metric space (X, M, *) into itself. From this result we deduce a common fixed point theorem for a family of mappings of a complete metric space (X, d) into itself. Supported by University of Palermo.     

Equivariant Embeddings of the n-Dimensional Space into a Hilbert Space

The present paper is devoted to the study of equivariant embeddings of the n-dimensional space into a Hilbert space. We consider a representation of a group of similarities. The existence of a cocycle for this representation implies the existence of an isometric embedding of a metric group into the Hilbert space. Then we describe all cocycles of a representation of the additive group of real numbers and construct an embedding of the n-dimensional space with metric d(x,y)=|x-y| into the Hilbert space. Bibliography: 5 titles.     

On completion of fuzzy metric spaces

Completions of fuzzy metric spaces (in the sense of George and Veeramani) are discussed. A complete fuzzy metric space Y is said to be a˜fuzzy metric completion of a˜given fuzzy metric space X if X is isometric to a˜dense subspace of Y. We present an example of a˜fuzzy metric space that does not admit any fuzzy metric completion. However, we prove that every standard fuzzy metric space has an (up to isometry) unique fuzzy metric completion. We also show that for each fuzzy metric space there is an (up to uniform isomorphism) unique complete fuzzy metric space that contains a˜dense subspace uniformly isomorphic to it.     

Note on Weakly n -Dimensional Spaces

 Weakly n-dimensional spaces were first distinguished by Karl Menger. In this note we shall discuss three topics concerning this class of spaces: universal spaces, products, and the sum theorem. We prove that there is a universal space for the class of all weakly n-dimensional spaces, present a simpler proof of Tomaszewski’s result about the dimension of a product of weakly n-dimensional spaces, and show that there is an n-dimensional space which admits a pairwise disjoint countable closed cover by weakly n-dimensional subspaces but is not weakly n-dimensional itself.     

Embeddings of Gromov hyperbolic spaces

It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant.?Another embedding theorem states that any -hyperbolic metric space embeds isometrically into a complete geodesic -hyperbolic space.?The relation of a Gromov hyperbolic space to its boundary is further investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasi-isometries. Submitted: October 1998, Revised version: January 1999.     

Advances in metric embedding theory

Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The mathematical theory of metric embedding is well studied in both pure and applied analysis and has more recently been a source of interest for computer scientists as well. Most of this work is focused on the development of bi-Lipschitz mappings between metric spaces. In this paper we present new concepts in metric embeddings as well as new embedding methods for metric spaces. We focus on finite metric spaces, however some of the concepts and methods are applicable in other settings as well.One of the main cornerstones in finite metric embedding theory is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with distortion. Bourgain?s result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is natural to ask: can an embedding do much better in terms of the average distortion? Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs.In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain?s theorem. In fact, our embedding possesses a much stronger property. We define the ?q-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible ?q-distortion for all 1?q?∞simultaneously.The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension of the host space. The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result improved the bound on the dimension which can be derived from Bourgain?s embedding. Our embedding methods achieve better dimension, and in fact, shed new light on another fundamental question in metric embedding, which is: whether the embedding dimension of a metric space is related to its intrinsic dimension? I.e., whether the dimension in which it can be embedded in some real normed space is related to the intrinsic dimension which is reflected by the inherent geometry of the space, measured by the space?s doubling dimension. The existence of such an embedding was conjectured by Assouad,⁴and was later posed as an open problem in several papers. Our embeddings give the first positive result of this type showing any finite metric space obtains a low distortion (and constant average distortion) embedding in Euclidean space in dimension proportional to its doubling dimension.Underlying our results is a novel embedding method. Probabilistic metric decomposition techniques have played a central role in the field of finite metric embedding in recent years. Here we introduce a novel notion of probabilistic metric decompositions which comes particularly natural in the context of embedding. Our new methodology provides a unified approach to all known results on embedding of arbitrary finite metric spaces. Moreover, as described above, with some additional ideas they allow to get far stronger results.The results presented in this paper⁵have been the basis for further developments both within the field of metric embedding and in other areas such as graph theory, distributed computing and algorithms. We present a comprehensive study of the notions and concepts introduced here and provide additional extensions, related results and some examples of algorithmic applications.     

An asymptotic minimax theorem of order n

The asymptotic minimax theorem of LeCam and Hájek is refined by inclusion of terms of order n^−1/2. This renders more precise informations about the local properties of superefficient estimator-sequences.     

An (α ,β)-Optimal solution concept in Fuzzy Optimization problems

We propose an (α,β)-optimal solution concept of fuzzy optimization problem based on the possibility and necessity measures. It is well known that the set of all fuzzy numbers can be embedded into a Banach space isometrically and isomorphically. Inspired by this embedding theorem, we can transform the fuzzy optimization problem into a biobjective programming problem by applying the embedding function to the original fuzzy optimization problem. Then the (α,β)-optimal solutions of fuzzy optimization problem can be obtained by solving its corresponding biobjective programming problem. We also consider the fuzzy optimization problem with fuzzy coefficients (i.e., the coefficients are assumed as fuzzy numbers). Under a setting of core value of fuzzy numbers, we provide the Karush–Kuhn–Tucker optimality conditions and show that the optimal solution of its corresponding crisp optimization problem (the usual optimization problem) is also a (1,1)-optimal solution of the original fuzzy optimization problem.     

Ranking fuzzy numbers with maximizing set and minimizing set

Up to now, these are five methods of ranking n fuzzy numbers in order, but these methods contain some confusions and occasionally conflict with intuition. This paper introduces the concept of maximizing set and minimizing set to decide the ordering value of each fuzzy number and uses these values to determine the order of the n fuzzy numbers. In addition, we give a method for calculating the ordering value of each fuzzy number with triangular, trapezoidal, and two-sided drum-like shaped membership functions.     

n维双曲空间和n维球面空间中的正弦定理及应用

本文研究了n维双曲空间和n维球面空间中单形的正弦定理和相关几何不等式. 应用距离几何的理论和方法, 给出了n维双曲空间和n维球面空间中一种新形式的正弦定理, 利用建立的正弦定理获得了Hadamard 型和Veljan-Korchmaros型不等式. 另外, 建立了涉及两个n维双曲单形和n维球面单形的"度量加"的一些几何不等式.     

非绝对模糊积分,绝对可积性与积分的绝对值不等式

关于模糊数值函数的绝对可积性及对应的积分不等式,无论在何种绝对值意义下是值得讨论的. 本文借助于模糊数空间到具体的 Banach 空间上的嵌入定理和模糊非绝对积分的刻划定理, 讨论了模糊数值函数的绝对可积性及对应的积分不等式,得到了若干个充分必要条件, 并举出了一些反例.     

关于square-full数上的函数q_k~((e))(n)的均值估计

本文研究了指数k-free数的特征函数q_k~((e))(n)(k≥3)在square-full数集中的均值估计问题.利用黎曼Zeta函数的性质以及留数定理,获得了该均值的渐近公式,推广了q_k~((e))(n)在整数集中的均值估计相关结果.     

Fréchet Embeddings of Negative Type Metrics

We show that every n-point metric of negative type (in particular, every n-point subset of L ₁) admits a Fréchet embedding into Euclidean space with distortion , a result which is tight up to the O(log log n) factor, even for Euclidean metrics. This strengthens our recent work on the Euclidean distortion of metrics of negative into Euclidean space. S. Arora supported by David and Lucile Packard Fellowship and NSF grant CCR-0205594. J.R. Lee supported by NSF grant CCR-0121555, NSF 0514993, NSF 0528414 and an NSF Graduate Research Fellowship.     

Controlled fuzzy metric spaces and some related fixed point results

In this paper, we introduce a new extension in the subject of fuzzy metric, called controlled fuzzy metric space. This notion is a generalization of fuzzy b‐metric space. Also, we prove a Banach‐type fixed point theorem and a new fixed point theorem for some self‐mappings satisfying fuzzy ψ ‐contraction condition that is more general than existing theorems. Furthermore, we establish some examples about our main results.     

随机模糊集及其落影大数定理

本文给出随机模糊集的定义,并证明了随机模糊集的落影大数定理,为模糊集值统计提供了理论依据。     

On some results of analysis for fuzzy metric spaces

A necessary and sufficient condition for a fuzzy metric space to be complete is given. We prove that a subspace of a separable fuzzy metric space is separable and every separable fuzzy metric space is second countable. Uniform limit theorem is generalized to fuzzy metric spaces.     

Similarities in Fuzzy Regression Models

The solutions of a fuzzy regression model are obtained by converting the problem into a linear programming problem. For each level h, h[0, 1), there exists a solution. In this paper, we study the set of all the solutions to the fuzzy regression model that comes from a set of data as a metric space with an appropriate metric on it. We define a similarity ratio that allows us to compare the spaces of solutions of a fuzzy regression model that come from different sets of data. We also give an application using data sets concerning the GNP–money relationship.     

Lower bound for Ln/2 curvature norm and its application

We control the number of critical points of a height function arising from the Nash isometric embedding of a compact Riemanniann-manifoldM. The L^n/2 curvature norm ∥R∥ and a similar scalar ∥R∥ are introduced and their integralR(M) andR(M) overM. We prove thatR(M) is bounded below by a constant depending only onn and the Betti numbers ofM. Thus a new sphere theorem is proved by eliminating allith Betti numbers fori = 1, .…n −1. The emphasis is that our sphere theorem imposes no restriction on the range of curvature. Research partially supported by Grant-in-Aid for General Scientific Research, grant no. 07454018.     

Symmetry analysis of the charged squashed Kaluza–Klein black hole metric

In this research article, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza–Klein black hole space‐time in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space‐time metric using Lie point symmetries is presented, and then the n‐dimensional optimal system of this space‐time metric, for n = 1,…,4, are computed. It is shown that there is no any n‐dimensional optimal system of Lie symmetry subalgebra associated to the system of geodesic for n≥5. Then the point symmetries of the one‐parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian that means Noether symmetries are found, and then the conservation laws associated to the system of geodesic equations are calculated via Noether's theorem. Copyright © 2015 John Wiley & Sons, Ltd.