scholarly journals A De Bruijn–Erdős Theorem for Chordal Graphs

10.37236/3527 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Laurent Beaudou ◽  
Adrian Bondy ◽  
Xiaomin Chen ◽  
Ehsan Chiniforooshan ◽  
Maria Chudnovsky ◽  
...  
Keyword(s):  
Metric Spaces ◽  
Chordal Graphs ◽  
Combinatorial Theorem ◽  
De Bruijn ◽  
Special Case

A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chávtal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces induced by connected chordal graphs.

scholarly journals New fixed point theorems for orthogonal $F_m$-contractions in incomplete $m$-metric spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 405-412
Author(s):  
M. Mehmood ◽  
H. Isik ◽  
F. Uddin ◽  
A. Shoaib
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Special Case

In this paper, we introduce the concept of orthogonal $m$-metric spaces and by using $F_m$-contraction in orthogonal $m$-metric spaces, we give the concept of orthogonal $F_m$-contraction (briefly, $\bot_{F_m}$-contraction) and investigate fixed point results for such mappings. Many existing results in the literature appear to be special case of results proved in this paper. An example to support our main results is also mentioned.

A Theorem on Nets

1966 ◽  
Vol 9 (3) ◽  
pp. 343-346
Author(s):  
M. Shimrat
Keyword(s):  
Metric Spaces ◽  
Countable Number ◽  
Simple Manner ◽  
Compact Spaces ◽  
Special Case

It is well-known that Tychonoff's theorem on the product of compact spaces may be proved, for the special case of a countable number of metric spaces X1, X2…, Xn,…, in the following simple manner.

scholarly journals TESTING EMBEDDABILITY BETWEEN METRIC SPACES

2009 ◽  
Vol 20 (02) ◽  
pp. 313-329
Author(s):  
CHING-LUEH CHANG ◽  
YUH-DAUH LYUU ◽  
YEN-WU TI
Keyword(s):  
Metric Space ◽  
Upper Bound ◽  
Metric Spaces ◽  
Graph Isomorphism ◽  
Upper And Lower Bounds ◽  
Wrong Answer ◽  
Probability Of Error ◽  
Finite Metric Space ◽  
Finite Metric Spaces ◽  
Special Case

Let L ≥ 1, ε > 0 be real numbers, (M, d) be a finite metric space and (N, ρ) be a metric space. A query to a metric space consists of a pair of points and asks for the distance between these points. We study the number of queries to metric spaces (M, d) and (N, ρ) needed to decide whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into N, ρ). When (M, d) is ∊-far from being L-bilipschitz embeddable into (N, ρ), we allow an o(1) probability of error (i.e., returning the wrong answer "L-bilipschitz embeddable"). However, no error is allowed when (M, d) is L-bilipschitz embeddable into (N, ρ). That is, algorithms with only one-sided errors are studied in this paper. When |M| ≤ |N| are both finite, we give an upper bound of [Formula: see text] on the number of queries for determining with one-sided error whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into (N, ρ). For the special case of finite |M| = |N|, the above upper bound evaluates to [Formula: see text]. We also prove a lower bound of Ω(|N|3/2) for the special case when |M| = |N| are finite and L = 1, which coincides with testing isometry between finite metric spaces. For finite |M| = |N|, the upper and lower bounds thus match up to a multiplicative factor of at most [Formula: see text], which depends only sublogarithmically in |N|. We also investigate the case when (N, ρ) is not necessarily finite. Our results are based on techniques developed in an earlier work on testing graph isomorphism.

scholarly journals Best proximity point theorems for G-proximal weak contractions in complete metric spaces endowed with graphs

2018 ◽  
Vol 34 (1) ◽  
pp. 65-75
Author(s):  
CHALONGCHAI KLANARONG ◽  
◽  
SUTHEP SUANTAI ◽  
Keyword(s):  
Directed Graph ◽  
Metric Space ◽  
Metric Spaces ◽  
Best Proximity Point ◽  
Complete Metric Space ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Complete Metric Spaces ◽  
Partially Ordered ◽  
Special Case

In this paper, the existence of best proximity point theorems for two new types of nonlinear non-self mappings in a complete metric space endowed with a directed graph are established. Our main results extend and generalize many known results in the literatures. As a special case of the main results, the best proximity point theorems on partially ordered sets are obtained.

scholarly journals A de Bruijn-Erdős theorem for 1–2 metric spaces

2014 ◽  
Vol 64 (1) ◽  
pp. 45-51 ◽  
Author(s):  
Vašek Chvátal
Keyword(s):  
Metric Spaces ◽  
De Bruijn

The Infinite Server Problem

2021 ◽  
Vol 17 (3) ◽  
pp. 1-23
Author(s):  
Christian Coester ◽  
Elias Koutsoupias ◽  
Philip Lazos
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Metric Spaces ◽  
Bounded Interval ◽  
Weighted Trees ◽  
Infinite Server ◽  
Tight Connection ◽  
Server Problem ◽  
Double Coverage ◽  
Special Case

We study a variant of the k -server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the resource augmentation version of the k -server problem, also known as the (h,k) -server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k) -server problem has bounded competitive ratio for some k = O ( h ). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which holds even for the line and some simple weighted stars. It implies the same lower bound for the (h,k) -server problem on the line, even when k/h → ∞, improving on the previous known bounds of 2 for the line and 2.4 for general metrics. For weighted trees and layered graphs, we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case.

scholarly journals Asymptotics of the Average Height of $2$–Watermelons with a Wall

10.37236/982 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Markus Fulmek
Keyword(s):  
Average Height ◽  
Exact Enumeration ◽  
Standard Methods ◽  
Dyck Paths ◽  
Classical Work ◽  
De Bruijn ◽  
Enumeration Formula ◽  
Special Case

We generalize the classical work of de Bruijn, Knuth and Rice (giving the asymptotics of the average height of Dyck paths of length $n$) to the case of $p$–watermelons with a wall (i.e., to a certain family of $p$ nonintersecting Dyck paths; simple Dyck paths being the special case $p=1$.) An exact enumeration formula for the average height is easily obtained by standard methods and well–known results. However, straightforwardly computing the asymptotics turns out to be quite complicated. Therefore, we work out the details only for the simple case $p=2$.

scholarly journals Some Remarks on Transformations in Metric Spaces

1965 ◽  
Vol 8 (5) ◽  
pp. 659-666 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Fixed Point ◽  
Partial Differential Equations ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Metric Spaces ◽  
Present Note ◽  
Existence Theorems ◽  
Special Case

Recently A. Haimovici [1] has proved a general fixed point theorem of transformations in metric spaces from which he obtained existence theorems for certain types of ordinary and partial differential equations. However, both the result and the proof are given for a rather special case. One of the purposes of this present note is to put his result on a more concrete basis and give a stronger characterization of the kind of transformations used in [l]. (Theorem 3).

On disjointness of dynamical systems

1979 ◽  
Vol 85 (3) ◽  
pp. 477-491 ◽  
Author(s):  
J. Auslander ◽  
Y. N. Dowker
Keyword(s):  
Dynamical System ◽  
Metric Spaces ◽  
Probability Measures ◽  
Compact Metric Spaces ◽  
Probability Spaces ◽  
Dynamical Properties ◽  
Borel Probability ◽  
The Given ◽  
Special Case ◽  
Measure Preserving Transformations

By a dynamical system we mean one of several related objects: measure preserving transformations on probability spaces (processes), self homeomorphisms of compact metric spaces (compact systems), or a combination of these, namely compact systems provided with invariant Borel probability measures. It is the latter, which we call compact processes, which will be of most interest in this paper. In particular, we will study the dynamical properties of the product of two processes with respect to compatible measures – those measures which project to the given measures on the component spaces. This leads to the notion of disjointness of two processes – the only compatible measure is the product measure. As an application we obtain a theorem, a special case of which gives rise to a class of transformations which preserve normal sequences. Finally, we study a topological analog (topological disjointness) and briefly consider the relation between the two notions of disjointness.

scholarly journals A de Bruijn - Erdos theorem and metric spaces

2011 ◽  
Vol Vol. 13 no. 1 (Combinatorics) ◽  
Author(s):  
Ehsan Chiniforooshan ◽  
Vasek Chvatal
Keyword(s):  
Metric Spaces ◽  
International Audience ◽  
De Bruijn

Combinatorics International audience De Bruijn and Erdos proved that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this theorem in the framework of metric spaces. We provide partial results in this direction.

Sign in / Sign up

Export Citation Format

Share Document