(a) Show that the 3-pointed star metric (S, d) does not embed isometrically (i.e. with distortion 1) into `2. The star metric has four points S = {c, v1, v2, v3} (a center and 3 leaves) with d(c, vi) = 1 for i = 1, 2, 3 and d(vi, vj) = 2 for i 6= j. (b) Show that (S, d) does embed isometrically into `1. (c) Let T = (V, E) be a graph-theoretic tree, and let dT be the (unweighted) shortest path m...

We introduced a general class of contraction maps on a metric space, called Acontractions (that includes the contractions originally studied by R. Kannan, M. S. Khan at el, R. Bianchini, and S. Reich), and extended some common fixed point theorems on M. S. Khan’s contractions to general self maps of a metric space satisfying certain A-contraction type condition; in this paper, we prove exact an...

Let S be a surface of finite type; that is, a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we classify (globally) geodesic rays in the moduli space M(S) of Riemann surfaces, endowed with the Teichmüller metric, and we determine precisely how pairs of rays asymptote. We then use these results to relate two important but disparate topics in the stud...

several practical problems of hypotheses testing can be under a general linear model analysis of variance which would be examined. in analysis of variance, when the response random variable y , has linear relationship with several random variables x, another important model as analysis of covariance can be used. in this paper, assuming that y is fuzzy and using dlr metric, a method for testing ...

For a class of fuzzy metric spaces (in the sense of George and Veeramani) with an H-type t-norm,  we present a method to construct a metric on a  fuzzy metric space. The induced metric space shares many important properties with the given fuzzy metric space.  Specifically, they generate the same topology, and have the same completeness. Our results can give the constructive proofs to some probl...

In this paper, we propose a new definition of intuitionistic fuzzyquasi-metric and pseudo-metric spaces based on intuitionistic fuzzy points. Weprove some properties of intuitionistic fuzzy quasi- metric and pseudo-metricspaces, and show that every intuitionistic fuzzy pseudo-metric space is intuitionisticfuzzy regular and intuitionistic fuzzy completely normal and henceintuitionistic fuzzy nor...

We define a family of pseudo metrics on B and study elementary properties of the associated metric spaces. As an application we prove that, for any a > 0 and for any countable-to-one function f from (S, dE) to [0, a], the set NMnf = {x ∈ S 2 | ∃y ∈ S such that f(x)− f(y) > ndE(x, y)} is uncountable for all n ∈ N, where dE is the standard Euclidean metric on S = { (x, y, z) ∈ R | x + y + z = 1 }

Relations of convergence of real sequences and convergence of metric spaces are investigated. An abstract intermediate value theorem for two closed sets in the range is presented. At the end, it is proven that an arc connecting the west minimal point and the east maximal point in a simple closed curve must be identical to the upper arc or lower arc of the closed curve. provide the notation and ...

we prove that every r-quadratic metric of scalar flag curvature with a dimension greater than twois of constant flag curvature. then we show that generalized douglas-weyl metrics contain r-quadraticmetrics as a special case, but the class of r-quadratic metric is not closed under projective transformations