A distributed clustering process

1991 ◽  
Vol 28 (04) ◽  
pp. 737-750 ◽  
Author(s):  
E. G. Coffman ◽  
P.-J. Courtois ◽  
E. N. Gilbert ◽  
Ph. Piret
Keyword(s):  
Probability Distributions ◽  
Distributed Clustering ◽  
Random Real ◽  
Real Numbers ◽  
Flow Process ◽  
Integer Points ◽  
The Laplace Transform ◽  
The Given ◽  
General Graphs ◽  
Total Resource

The points of a graph G will form clusters as a result of a flow process. Initially, points i of G own resources xi which are i.i.d. random real numbers. Afterwards, resources flow between points, but always from a point to a neighbor that has accumulated a larger total resource. Thus points with small resource tend to lose it and points with large resource tend to gain. Eventually the flow stops with only two kinds of points, nulls with no resource left and absorbers with such large resource that no neighbor can take it. The final resource at an absorber is a sum of certain initial resources xi , and the corresponding points i form one cluster. Analytical results are obtainable when G is the chain of integer points on the line. Probability distributions are derived for the distance between consecutive absorbers and the size of a cluster. Indeed these distributions do not involve the given distribution for the xi. The Laplace transform of the distribution of final resources at absorbers is derived but the distribution itself is obtained by a simulation. For general graphs G only partial results are obtained.

A distributed clustering process

1991 ◽  
Vol 28 (4) ◽  
pp. 737-750 ◽  
Author(s):  
E. G. Coffman ◽  
P.-J. Courtois ◽  
E. N. Gilbert ◽  
Ph. Piret
Keyword(s):  
Probability Distributions ◽  
Distributed Clustering ◽  
Random Real ◽  
Real Numbers ◽  
Flow Process ◽  
Integer Points ◽  
The Laplace Transform ◽  
The Given ◽  
General Graphs ◽  
Total Resource

The points of a graph G will form clusters as a result of a flow process. Initially, points i of G own resources xi which are i.i.d. random real numbers. Afterwards, resources flow between points, but always from a point to a neighbor that has accumulated a larger total resource. Thus points with small resource tend to lose it and points with large resource tend to gain. Eventually the flow stops with only two kinds of points, nulls with no resource left and absorbers with such large resource that no neighbor can take it. The final resource at an absorber is a sum of certain initial resources xi, and the corresponding points i form one cluster. Analytical results are obtainable when G is the chain of integer points on the line. Probability distributions are derived for the distance between consecutive absorbers and the size of a cluster. Indeed these distributions do not involve the given distribution for the xi. The Laplace transform of the distribution of final resources at absorbers is derived but the distribution itself is obtained by a simulation. For general graphs G only partial results are obtained.

scholarly journals On a distributed clustering process of Coffman, Courtois, Gilbert and Piret

1998 ◽  
Vol 35 (4) ◽  
pp. 919-924 ◽  
Author(s):  
J. van den Berg ◽  
A. Ermakov
Keyword(s):  
Monte Carlo Simulation ◽  
Continuous Distribution ◽  
Square Lattice ◽  
Main Question ◽  
Distributed Clustering ◽  
Flow Process ◽  
Integer Points ◽  
Overwhelming Evidence ◽  
Higher Dimensional ◽  
Finite Problem

Coffman, Courtois, Gilbert and Piret (1991) have introduced a flow process in graphs, where each vertex gets an initial random resource, and where at each time vertices with large resources tend to attract resources from neighbours. The initial resources are assumed to be i.i.d., with a continuous distribution.We are mainly interested in the following question: does, with probability 1, the resource of each vertex change only finitely many times?Coffmanet al.concentrate mainly on the case where the graph corresponds with the integer points on the line, in which case it is easily seen that the answer to the above question is positive. For higher-dimensional lattices they make general remarks which suggest that the answer to the above question is still positive. However, no proof seems to be known.We restrict to the case of the square lattice, and, by a percolation approach, we reduce the question above to the question whether a certain quantity, which can be obtained from afinitecomputation, is sufficiently small. This computation is, however, still too long to be executed in an acceptable time. We therefore apply Monte Carlo simulation for this finite problem, which gives overwhelming evidence that, for the square lattice, the answer to the main question is positive.

scholarly journals On a distributed clustering process of Coffman, Courtois, Gilbert and Piret

1998 ◽  
Vol 35 (04) ◽  
pp. 919-924
Author(s):  
J. van den Berg ◽  
A. Ermakov
Keyword(s):  
Monte Carlo Simulation ◽  
Continuous Distribution ◽  
Square Lattice ◽  
Main Question ◽  
Distributed Clustering ◽  
Flow Process ◽  
Integer Points ◽  
Overwhelming Evidence ◽  
Higher Dimensional ◽  
Finite Problem

Coffman, Courtois, Gilbert and Piret (1991) have introduced a flow process in graphs, where each vertex gets an initial random resource, and where at each time vertices with large resources tend to attract resources from neighbours. The initial resources are assumed to be i.i.d., with a continuous distribution. We are mainly interested in the following question: does, with probability 1, the resource of each vertex change only finitely many times? Coffman et al. concentrate mainly on the case where the graph corresponds with the integer points on the line, in which case it is easily seen that the answer to the above question is positive. For higher-dimensional lattices they make general remarks which suggest that the answer to the above question is still positive. However, no proof seems to be known. We restrict to the case of the square lattice, and, by a percolation approach, we reduce the question above to the question whether a certain quantity, which can be obtained from a finite computation, is sufficiently small. This computation is, however, still too long to be executed in an acceptable time. We therefore apply Monte Carlo simulation for this finite problem, which gives overwhelming evidence that, for the square lattice, the answer to the main question is positive.

scholarly journals On Certain Approximation Problem Connected with the Sums of Subseries

2013 ◽  
Vol 55 (1) ◽  
pp. 37-45
Author(s):  
Roman Wituła ◽  
Konrad Kaczmarek ◽  
Edyta Hetmaniok ◽  
Damian Słota
Keyword(s):  
Approximation Problem ◽  
Real Numbers ◽  
The Real ◽  
The Given

Abstract In this paper a problem of approximating the real numbers by using the series of real numbers is considered. It is proven that if the given family of sequences of real numbers satisfies some conditions of set-theoretical nature, like being closed under initial subsequences and (additionally) possessing properties of adding and removing elements, then it automatically possesses some approximating properties, like, for example, reaching supremum of the set of sums of subseries.

Innovative Genetic Algorithmic Approach to Select Potential Patches Enclosing Real and Complex Zeros of Nonlinear Equation

2017 ◽  
Vol 6 (2) ◽  
pp. 18-37 ◽  
Author(s):  
Vijaya Lakshmi V. Nadimpalli ◽  
Rajeev Wankar ◽  
Raghavendra Rao Chillarige
Keyword(s):  
Genetic Algorithm ◽  
Nonlinear Equation ◽  
Complex Plane ◽  
Search Space ◽  
Regions Of Interest ◽  
Real Numbers ◽  
Algorithmic Approach ◽  
Space Reduction ◽  
Complex Zeros ◽  
The Given

In this article, an innovative Genetic Algorithm is proposed to find potential patches enclosing roots of real valued function f:R→R. As roots of f can be real as well as complex, the function is reframed on to complex plane by writing it as f(z). Thus, the problem now is transformed to finding potential patches (rectangles in C) enclosing z such that f(z)=0, which is resolved into two components as real and imaginary parts. The proposed GA generates two random populations of real numbers for the real and imaginary parts in the given regions of interest and no other initial guesses are needed. This is the prominent advantage of the method in contrast to various other methods. Additionally, the proposed ‘Refinement technique' aids in the exhaustive coverage of potential patches enclosing roots and reinforces the selected potential rectangles to be narrow, resulting in significant search space reduction. The method works efficiently even when the roots are closely packed. A set of benchmark functions are presented and the results show the effectiveness and robustness of the new method.

Exact Solutions for Unsteady Magnetohydrodynamic Oscillatory Flow of a Maxwell Fluid in a Porous Medium

2013 ◽  
Vol 68 (10-11) ◽  
pp. 635-645 ◽  
Author(s):  
Ilyas Khan ◽  
Farhad Ali ◽  
Sharidan Shafie ◽  
Keyword(s):  
Steady State ◽  
Exact Solutions ◽  
Mhd Flow ◽  
Maxwell Fluid ◽  
Transform Method ◽  
Transient Solutions ◽  
The Laplace Transform ◽  
Porous Half Space ◽  
The Given ◽  
Steady State Solutions

In this paper, exact solutions of velocity and stresses are obtained for the magnetohydrodynamic (MHD) flow of a Maxwell fluid in a porous half space by the Laplace transform method. The flows are caused by the cosine and sine oscillations of a plate. The derived steady and transient solutions satisfy the involved differential equations and the given conditions. Graphs for steady-state and transient velocities are plotted and discussed. It is found that for a large value of the time t, the transient solutions disappear, and the motion is described by the corresponding steady-state solutions.

Spaces on which every Continuous Map into a given Space is Constant

1986 ◽  
Vol 38 (6) ◽  
pp. 1281-1298 ◽  
Author(s):  
S. Iliadis ◽  
V. Tzannes
Keyword(s):  
Topological Spaces ◽  
Real Numbers ◽  
Continuous Map ◽  
Countable Space ◽  
Continuous Maps ◽  
Usual Topology ◽  
The Given ◽  
Connected Spaces

This paper is concerned with topological spaces whose continuous maps into a given space R are constant, as well as with spaces having this property locally. We call these spaces R-monolithic and locally R-monolithic, respectively.Spaces with such properties have been considered in [1], [5]-[7], [10], [11], [22], [28], [31], where with the exception of [10], the given space R is the set of real-numbers with the usual topology. Obviously, for a countable space, connectedness is equivalent to the property that every continuous real-valued map is constant. Countable connected (locally connected) spaces have been constructed in papers [2]-[4], [8], [9], [11]-[21], [23]-[26], [30].

Differences, Derivatives, and Decreasing Rearrangements

1967 ◽  
Vol 19 ◽  
pp. 1153-1178 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Finite Sequence ◽  
Real Numbers ◽  
Decreasing Rearrangement ◽  
Image Position ◽  
Decreasing Rearrangements ◽  
The Given

The decreasing rearrangement of a finite sequence a1, a2, … , an of real numbers is a second sequence aπ(1), aπ(2), … , aπ(n), where π(l), π(2), … , π(n) is a permutation of 1, 2, … , n and(1, p. 260). The kth term of the rearranged sequence will be denoted by . Thus the terms of the rearranged sequence correspond to and are equal to those of the given sequence ak, but are arranged in descending (non-increasing) order.

scholarly journals A generalisation of Dirichlet's multiple integral

1956 ◽  
Vol 40 ◽  
pp. 16-18
Author(s):  
S. K. Lakshmana Rao
Keyword(s):  
Laplace Transform ◽  
Mellin Transform ◽  
Multiple Integral ◽  
Real Numbers ◽  
General Integral ◽  
The Laplace Transform ◽  
Image Position

The well-known multiple integralwhere Rn is the region defined by x1 ≥ 0, x2 ≥ 0, …., xn ≥ 0, x1 + x2 + …. + xn ≤ 1, and where a0, a1, …, an are positive constants, can be evaluated either in the classical way using the Dirichlet transformation or by the use of the Laplace transform. I. J. Good has considered a more general integral and has proved the following result by induction:—If f1(t), f2(t), …, fn(t) are Lebesgue measurable for 0 ≤ t ≤ 1, m1, m2, …., mn, mn+1 (= 0) are real numbers, Mr = m1 + m2 + … + mr, x1, x2, …, xn are non-negative variables and Xr = x1 + x2 + … + xr, thenIt does not seem to be possible to establish this relation by employing the Laplace transform, but we show below that it can be obtained using the Mellin transform.

Sign in / Sign up

Export Citation Format

Share Document