scholarly journals Refined convergence for the Boolean model

2009 ◽  
Vol 41 (04) ◽  
pp. 940-957 ◽  
Author(s):  
Pierre Calka ◽  
Julien Michel ◽  
Katy Paroux
Keyword(s):  
Probability Space ◽  
Order Term ◽  
Convergence Result ◽  
Boolean Model ◽  
Random Sets ◽  
Two Dimensional ◽  
Convergence In Distribution ◽  
Line Process ◽  
The Difference ◽  
Poisson Line Process

In Michel and Paroux (2003) the authors proposed a new proof of a well-known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process (see, e.g. Hall (1985)). In this paper we investigate the second-order term in this convergence when the two-dimensional Boolean model and the Poisson line process are coupled on the same probability space. We consider the particular case where the grains are discs with random radii. A precise coupling between the Boolean model and the Poisson line process is first established. A result of directional convergence in distribution for the difference of the two sets involved is then derived. Eventually, we show the convergence of the process, measuring the difference between the two random sets, once rescaled, as a function of the direction.

scholarly journals Refined convergence for the Boolean model

2009 ◽  
Vol 41 (4) ◽  
pp. 940-957 ◽  
Author(s):  
Pierre Calka ◽  
Julien Michel ◽  
Katy Paroux
Keyword(s):  
Probability Space ◽  
Order Term ◽  
Convergence Result ◽  
Boolean Model ◽  
Random Sets ◽  
Two Dimensional ◽  
Convergence In Distribution ◽  
Line Process ◽  
The Difference ◽  
Poisson Line Process

In Michel and Paroux (2003) the authors proposed a new proof of a well-known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process (see, e.g. Hall (1985)). In this paper we investigate the second-order term in this convergence when the two-dimensional Boolean model and the Poisson line process are coupled on the same probability space. We consider the particular case where the grains are discs with random radii. A precise coupling between the Boolean model and the Poisson line process is first established. A result of directional convergence in distribution for the difference of the two sets involved is then derived. Eventually, we show the convergence of the process, measuring the difference between the two random sets, once rescaled, as a function of the direction.

Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson-Voronoi tessellation and a Poisson line process

2003 ◽  
Vol 35 (3) ◽  
pp. 551-562 ◽  
Author(s):  
Pierre Calka
Keyword(s):  
Fundamental Domain ◽  
Joint Distribution ◽  
Voronoi Tessellation ◽  
Two Dimensional ◽  
Integral Expression ◽  
Geometric Characteristics ◽  
Typical Cell ◽  
Line Process ◽  
Poisson Line Process

In this paper, we give an explicit integral expression for the joint distribution of the number and the respective positions of the sides of the typical cell 𝒞 of a two-dimensional Poisson-Voronoi tessellation. We deduce from it precise formulae for the distributions of the principal geometric characteristics of 𝒞 (area, perimeter, area of the fundamental domain). We also adapt the method to the Crofton cell and the empirical (or typical) cell of a Poisson line process.

The distributions of the smallest disks containing the Poisson-Voronoi typical cell and the Crofton cell in the plane

2002 ◽  
Vol 34 (04) ◽  
pp. 702-717 ◽  
Author(s):  
Pierre Calka
Keyword(s):  
Joint Distribution ◽  
Voronoi Tessellation ◽  
Two Dimensional ◽  
Conditional Probabilities ◽  
Polygonal Cell ◽  
Typical Particle ◽  
Covering Properties ◽  
Typical Cell ◽  
Line Process ◽  
Poisson Line Process

Among the disks centered at a typical particle of the two-dimensional Poisson-Voronoi tessellation, letRmbe the radius of the largest included within the polygonal cell associated with that particle andRMbe the radius of the smallest containing that polygonal cell. In this article, we obtain the joint distribution ofRmandRM. This result is derived from the covering properties of the circle due to Stevens, Siegel and Holst. The same method works for studying the Crofton cell associated with the Poisson line process in the plane. The computation of the conditional probabilities P{RM≥r+s|Rm=r} reveals the circular property of the Poisson-Voronoi typical cells (as well as the Crofton cells) having a ‘large’ in-disk.

The distributions of the smallest disks containing the Poisson-Voronoi typical cell and the Crofton cell in the plane

2002 ◽  
Vol 34 (4) ◽  
pp. 702-717 ◽  
Author(s):  
Pierre Calka
Keyword(s):  
Joint Distribution ◽  
Voronoi Tessellation ◽  
Two Dimensional ◽  
Conditional Probabilities ◽  
Polygonal Cell ◽  
Typical Particle ◽  
Covering Properties ◽  
Typical Cell ◽  
Line Process ◽  
Poisson Line Process

Among the disks centered at a typical particle of the two-dimensional Poisson-Voronoi tessellation, let Rm be the radius of the largest included within the polygonal cell associated with that particle and RM be the radius of the smallest containing that polygonal cell. In this article, we obtain the joint distribution of Rm and RM. This result is derived from the covering properties of the circle due to Stevens, Siegel and Holst. The same method works for studying the Crofton cell associated with the Poisson line process in the plane. The computation of the conditional probabilities P{RM ≥ r + s | Rm = r} reveals the circular property of the Poisson-Voronoi typical cells (as well as the Crofton cells) having a ‘large’ in-disk.

Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson-Voronoi tessellation and a Poisson line process

2003 ◽  
Vol 35 (03) ◽  
pp. 551-562 ◽  
Author(s):  
Pierre Calka
Keyword(s):  
Fundamental Domain ◽  
Joint Distribution ◽  
Voronoi Tessellation ◽  
Two Dimensional ◽  
Integral Expression ◽  
Geometric Characteristics ◽  
Typical Cell ◽  
Line Process ◽  
Poisson Line Process

In this paper, we give an explicit integral expression for the joint distribution of the number and the respective positions of the sides of the typical cell 𝒞 of a two-dimensional Poisson-Voronoi tessellation. We deduce from it precise formulae for the distributions of the principal geometric characteristics of 𝒞 (area, perimeter, area of the fundamental domain). We also adapt the method to the Crofton cell and the empirical (or typical) cell of a Poisson line process.

Extended two-dimensional belief function based on divergence measurement

2020 ◽  
pp. 1-8
Author(s):  
Jianping Fan ◽  
Jing Wang ◽  
Meiqin Wu
Keyword(s):  
Belief Function ◽  
Distribution Functions ◽  
Divergence Measure ◽  
Uncertain Information ◽  
Belief Functions ◽  
Two Dimensional ◽  
Probability Distribution Functions ◽  
Basic Probability ◽  
The Difference ◽  
Supporting Degree

The two-dimensional belief function (TDBF = (mA, mB)) uses a pair of ordered basic probability distribution functions to describe and process uncertain information. Among them, mB includes support degree, non-support degree and reliability unmeasured degree of mA. So it is more abundant and reasonable than the traditional discount coefficient and expresses the evaluation value of experts. However, only considering that the expert’s assessment is single and one-sided, we also need to consider the influence between the belief function itself. The difference in belief function can measure the difference between two belief functions, based on which the supporting degree, non-supporting degree and unmeasured degree of reliability of the evidence are calculated. Based on the divergence measure of belief function, this paper proposes an extended two-dimensional belief function, which can solve some evidence conflict problems and is more objective and better solve a class of problems that TDBF cannot handle. Finally, numerical examples illustrate its effectiveness and rationality.

Theory of ultrasonic attenuation in one and two dimensional metals

1976 ◽  
Vol 54 (14) ◽  
pp. 1454-1460 ◽  
Author(s):  
T. Tiedje ◽  
R. R. Haering
Keyword(s):  
Ultrasonic Attenuation ◽  
Three Dimensional ◽  
Electronic Systems ◽  
Two Dimensional ◽  
Temperature Dependent ◽  
One Dimensional ◽  
The Difference

The theory of ultrasonic attenuation in metals is extended so that it applies to quasi one and two dimensional electronic systems. It is shown that the attenuation in such systems differs significantly from the well-known results for three dimensional systems. The difference is particularly marked for one dimensional systems, for which the attenuation is shown to be strongly temperature dependent.

NUMERICAL SOLUTION OF TWO-DIMENSIONAL PROBLEMS OF THERMOVISCOELASTICITY

2021 ◽  
Vol 335 (1) ◽  
pp. 60-64
Author(s):  
M. Bukenov ◽  
Ye. Mukhametov
Keyword(s):  
A Priori ◽  
Elastic Collision ◽  
Qualitative Behavior ◽  
Stress Profile ◽  
Two Dimensional ◽  
A Priori Information ◽  
Deformable Solids ◽  
Methods Of Calculation ◽  
The Difference ◽  
Priori Information

This paper considers the numerical implementation of two-dimensional thermoviscoelastic waves. The elastic collision of an aluminum cylinder with a two-layer plate of aluminum and iron is considered. In work [1] the difference schemes and algorithm of their realization are given. The most complete reviews of the main methods of calculation of transients in deformable solids can be found in [2, 3, 4], which also indicates the need and importance of generalized studies on the comparative evaluation of different methods and identification of the areas of their most rational application. In the analysis and physical interpretation of numerical results in this work it is also useful to use a priori information about the qualitative behavior of the solution and all kinds of information about the physics of the phenomena under study. Here is the stage of evolution of contact resistance of collision – plate, stress profile.

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