scholarly journals Invariant Bipartite Random Graphs on Rd

2014 ◽  
Vol 51 (03) ◽  
pp. 769-779
Author(s):  
Fabio Lopes
Keyword(s):  
Point Processes ◽  
Random Number ◽  
Probability Distributions ◽  
Sufficient Conditions ◽  
Stable Marriage ◽  
Simple Point ◽  
Stable Marriage Problem ◽  
Blue Point ◽  
Translation Invariant ◽  
Positive Integers

Suppose that red and blue points occur inRdaccording to two simple point processes with finite intensities λRand λB, respectively. Furthermore, let ν and μ be two probability distributions on the strictly positive integers with means ν̅ and μ̅, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law ν (μ). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law ν or μ depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if λRν̅ = λBμ̅, including the case when ν̅ = μ̅ = ∞ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on ν and μ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and Häggström.

scholarly journals Invariant Bipartite Random Graphs on Rd

2014 ◽  
Vol 51 (3) ◽  
pp. 769-779
Author(s):  
Fabio Lopes
Keyword(s):  
Point Processes ◽  
Random Number ◽  
Probability Distributions ◽  
Sufficient Conditions ◽  
Stable Marriage ◽  
Simple Point ◽  
Stable Marriage Problem ◽  
Blue Point ◽  
Translation Invariant ◽  
Positive Integers

Suppose that red and blue points occur in Rd according to two simple point processes with finite intensities λR and λB, respectively. Furthermore, let ν and μ be two probability distributions on the strictly positive integers with means ν̅ and μ̅, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law ν (μ). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law ν or μ depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if λRν̅ = λBμ̅, including the case when ν̅ = μ̅ = ∞ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on ν and μ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and Häggström.

Compensator conditions for stochastic ordering of point processes

1991 ◽  
Vol 28 (04) ◽  
pp. 751-761 ◽  
Author(s):  
A. Kwieciński ◽  
R. Szekli
Keyword(s):  
Probability Space ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Stochastic Ordering ◽  
Half Line ◽  
Simple Point ◽  
Martingale Approach ◽  
Random Elements ◽  
Markov Point Processes ◽  
Positive Half

Sufficient conditions are given under which two simple point processes on the positive half-line can be stochastically compared as random elements of D(0,∞) or R∞ + Using a martingale approach to point processes, the conditions are proposed via a compensator function family. Appropriate versions of the processes being compared are constructed on the same probability space. The results are illustrated by replacement policies and semi-Markov point processes.

Weight-Equitable Subdivision of Red and Blue Points in the Plane

2018 ◽  
Vol 28 (01) ◽  
pp. 39-56 ◽  
Author(s):  
Jude Buot ◽  
Mikio Kano
Keyword(s):  
General Position ◽  
Sufficient Conditions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Total Weight ◽  
Blue Point ◽  
Disjoint Sets ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Let [Formula: see text] and [Formula: see text] be two disjoint sets of red points and blue points, respectively, in the plane in general position. Assign a weight [Formula: see text] to each red point and a weight [Formula: see text] to each blue point, where [Formula: see text] and [Formula: see text] are positive integers. Define the weight of a region in the plane as the sum of the weights of red and blue points in it. We give necessary and sufficient conditions for the existence of a line that bisects the weight of the plane whenever the total weight [Formula: see text] is [Formula: see text], for some integer [Formula: see text]. Moreover, we look closely into the special case where [Formula: see text] and [Formula: see text] since this case is important to generate a weight-equitable subdivision of the plane. Among other results, we show that for any configuration of [Formula: see text] with total weight [Formula: see text], for some integer [Formula: see text] and odd integer [Formula: see text], the plane can be subdivided into [Formula: see text] convex regions of weight [Formula: see text] if and only if [Formula: see text]. Using the proofs of the main result, we also give a polynomial time algorithm in finding a weight-equitable subdivision in the plane.

Compensator conditions for stochastic ordering of point processes

1991 ◽  
Vol 28 (4) ◽  
pp. 751-761 ◽  
Author(s):  
A. Kwieciński ◽  
R. Szekli
Keyword(s):  
Probability Space ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Stochastic Ordering ◽  
Half Line ◽  
Simple Point ◽  
Martingale Approach ◽  
Random Elements ◽  
Markov Point Processes ◽  
Positive Half

Sufficient conditions are given under which two simple point processes on the positive half-line can be stochastically compared as random elements of D(0,∞) or R∞+ Using a martingale approach to point processes, the conditions are proposed via a compensator function family. Appropriate versions of the processes being compared are constructed on the same probability space. The results are illustrated by replacement policies and semi-Markov point processes.

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

A model for interaction of a Poisson and a renewal process and its relation with queuing theory

1972 ◽  
Vol 9 (2) ◽  
pp. 451-456 ◽  
Author(s):  
Lennart Råde
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Queuing Theory ◽  
Random Number ◽  
Sufficient Conditions ◽  
Stieltjes Transform ◽  
Time Intervals ◽  
Response Process ◽  
Necessary And Sufficient ◽  
Special Case

This paper discusses the response process when a Poisson process interacts with a renewal process in such a way that one or more points of the Poisson process eliminate a random number of consecutive points of the renewal process. A queuing situation is devised such that the c.d.f. of the length of the busy period is the same as the c.d.f. of the length of time intervals of the renewal response process. The Laplace-Stieltjes transform is obtained and from this the expectation of the time intervals of the response process is derived. For a special case necessary and sufficient conditions for the response process to be a Poisson process are found.

scholarly journals Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

Sign in / Sign up

Export Citation Format

Share Document