scholarly journals On the One Dimensional Poisson Random Geometric Graph

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
L. Decreusefond ◽  
E. Ferraz
Keyword(s):  
Poisson Process ◽  
Laplace Transforms ◽  
Geometric Graph ◽  
Connected Components ◽  
Random Geometric Graph ◽  
One Dimensional ◽  
Bounded Interval ◽  
The One

Given a Poisson process on a bounded interval, its random geometric graph is the graph whose vertices are the points of the Poisson process, and edges exist between two points if and only if their distance is less than a fixed given threshold. We compute explicitly the distribution of the number of connected components of this graph. The proof relies on inverting some Laplace transforms.

scholarly journals Topological properties of the one dimensional exponential random geometric graph

2008 ◽  
Vol 32 (2) ◽  
pp. 181-204 ◽  
Author(s):  
Bhupendra Gupta ◽  
Srikanth K. Iyer ◽  
D. Manjunath
Keyword(s):  
Geometric Graph ◽  
Topological Properties ◽  
Random Geometric Graph ◽  
One Dimensional ◽  
The One

On inference in a one-dimensional mosaic and an M/G/∞ queue

1985 ◽  
Vol 17 (01) ◽  
pp. 210-229
Author(s):  
Peter Hall
Keyword(s):  
Poisson Process ◽  
Segment Length ◽  
One Dimensional ◽  
Estimation Procedures ◽  
The One

Suppose segments are distributed at random along a line, their locations being determined by a Poisson process. In the case where segment length is fixed, we compare efficiencies of several different estimates of Poisson intensity. The case of random segment length is also considered, and there we study estimation procedures based on empiric properties. The one-dimensional mosaic may be viewed as an M/G/∞ queue.

scholarly journals Fractional negative binomial and Pólya processes

2018 ◽  
Vol 38 (1) ◽  
pp. 77-101
Author(s):  
Palaniappan Vellai Samy ◽  
Aditya Maheshwari
Keyword(s):  
Poisson Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Infinitely Divisible ◽  
One Dimensional ◽  
Fractional Poisson Process ◽  
Negative Binomial Process ◽  
The One ◽  
Definition Of ◽  
Binomial Process

In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

scholarly journals Revisiting the 1D and 2D Laplace Transforms

2020 ◽  
Author(s):  
Manuel Duarte Ortigueira ◽  
José Tenreiro Machado
Keyword(s):  
Transfer Function ◽  
Laplace Transform ◽  
Linear Systems ◽  
Laplace Transforms ◽  
Dimensional Case ◽  
Initial Condition ◽  
Two Dimensional ◽  
One Dimensional ◽  
The One

This paper reviews the unilateral and bilateral, one- and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one- and two-dimensional cases with the bilateral Laplace transform. General two-dimensional linear systems are introduced and the corresponding transfer function defined.

On the superposition of m-dimensional point processes

1968 ◽  
Vol 5 (1) ◽  
pp. 169-176 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Natural Extension ◽  
Renewal Processes ◽  
Independent Vector ◽  
One Dimensional ◽  
Independent Components ◽  
Markov Renewal Processes ◽  
The One ◽  
Vector Valued

Consider n independent vector valued point processes. Superposition is defined component by component as a natural extension of the definition for the one-dimensional case. Under proper conditions as n → ∞, it is shown that the superposed process is a many-dimensional Poisson process with independent components. The results are applied to the superposition of Markov renewal processes.

scholarly journals Connectivity of random k-nearest-neighbour graphs

2005 ◽  
Vol 37 (01) ◽  
pp. 1-24 ◽  
Author(s):  
Paul Balister ◽  
Béla Bollobás ◽  
Amites Sarkar ◽  
Mark Walters
Keyword(s):  
Poisson Process ◽  
Geometric Graph ◽  
Upper Bounds ◽  
Related Question ◽  
Nearest Neighbour ◽  
Lower And Upper Bounds ◽  
Random Geometric Graph ◽  
Nearest Neighbours ◽  
Closed Disc

Let 𝓅 be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞.

scholarly journals Global Components of Positive Bounded Variation Solutions of a One-Dimensional Indefinite Quasilinear Neumann Problem

2019 ◽  
Vol 19 (3) ◽  
pp. 437-473 ◽  
Author(s):  
Julian López-Gómez ◽  
Pierpaolo Omari
Keyword(s):  
Neumann Problem ◽  
Positive Solutions ◽  
Bounded Variation ◽  
Connected Components ◽  
One Dimensional ◽  
Linearized Problem ◽  
Bounded Variation Functions ◽  
The One ◽  
First Time ◽  
Prime 2

Abstract This paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem \begin{dcases}-\Bigg{(}\frac{u^{\prime}}{\sqrt{1+{u^{\prime}}^{2}}}\Bigg{)}^{% \prime}=\lambda a(x)f(u)\quad\text{in }(0,1),\\ u^{\prime}(0)=0,\quad u^{\prime}(1)=0,\end{dcases} where {\lambda\in\mathbb{R}} is a parameter, {a\in L^{\infty}(0,1)} changes sign, and {f\in C^{1}(\mathbb{R})} is positive in {(0,+\infty)} . The attention is focused on the case {f(0)=0} and {f^{\prime}(0)=1} , where we can prove, likely for the first time in the literature, a bifurcation result for this problem in the space of bounded variation functions. Namely, the existence of global connected components of the set of the positive solutions, emanating from the line of the trivial solutions at the two principal eigenvalues of the linearized problem around 0, is established. The solutions in these components are regular, as long as they are small, while they may develop jump singularities at the nodes of the weight function a, as they become larger, thus showing the possible coexistence along the same component of regular and singular solutions.

scholarly journals A critical constant for the k nearest-neighbour model

2009 ◽  
Vol 41 (01) ◽  
pp. 1-12 ◽  
Author(s):  
Paul Balister ◽  
Béla Bollobás ◽  
Amites Sarkar ◽  
Mark Walters
Keyword(s):  
Poisson Process ◽  
Geometric Graph ◽  
Nearest Neighbour ◽  
Random Geometric Graph ◽  
Fixed Integer ◽  
Nearest Neighbours ◽  
Critical Constant

Let 𝒫 be a Poisson process of intensity 1 in a square S n of area n. For a fixed integer k, join every point of 𝒫 to its k nearest neighbours, creating an undirected random geometric graph G n,k . We prove that there exists a critical constant c crit such that, for c < c crit, G n,⌊c log n⌋ is disconnected with probability tending to 1 as n → ∞ and, for c > c crit, G n,⌊c log n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).

On inference in a one-dimensional mosaic and an M/G/∞ queue

1985 ◽  
Vol 17 (1) ◽  
pp. 210-229 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Poisson Process ◽  
Segment Length ◽  
One Dimensional ◽  
Estimation Procedures ◽  
The One

Suppose segments are distributed at random along a line, their locations being determined by a Poisson process. In the case where segment length is fixed, we compare efficiencies of several different estimates of Poisson intensity. The case of random segment length is also considered, and there we study estimation procedures based on empiric properties. The one-dimensional mosaic may be viewed as an M/G/∞ queue.

scholarly journals A critical constant for the k nearest-neighbour model

2009 ◽  
Vol 41 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Paul Balister ◽  
Béla Bollobás ◽  
Amites Sarkar ◽  
Mark Walters
Keyword(s):  
Poisson Process ◽  
Geometric Graph ◽  
Nearest Neighbour ◽  
Random Geometric Graph ◽  
Fixed Integer ◽  
Nearest Neighbours ◽  
Critical Constant

Let 𝒫 be a Poisson process of intensity 1 in a square Sn of area n. For a fixed integer k, join every point of 𝒫 to its k nearest neighbours, creating an undirected random geometric graph Gn,k. We prove that there exists a critical constant ccrit such that, for c < ccrit, Gn,⌊c log n⌋ is disconnected with probability tending to 1 as n → ∞ and, for c > ccrit, Gn,⌊c log n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).

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