Coefficients of ergodicity: structure and applications

1979 ◽  
Vol 11 (3) ◽  
pp. 576-590 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Stochastic Matrix ◽  
Numerical Examples ◽  
Localization Property ◽  
Spectrum Localization

The concept of ‘coefficient of ergodicity’, τ(P), for a finite stochastic matrix P, is developed from a standpoint more general and less standard than hitherto, albeit synthesized from ideas in existing literature. Several versions of such a coefficient are studied theoretically and by numerical examples, and usefulness in applications compared from viewpoints which include the degree to which extension to more general matrices is possible. Attention is given to the less familiar spectrum localization property: where λ is any non-unit eigenvalue of P. The essential purpose is exposition and unification, with the aid of simple informal proofs.

Coefficients of ergodicity: structure and applications

1979 ◽  
Vol 11 (03) ◽  
pp. 576-590 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Stochastic Matrix ◽  
Numerical Examples ◽  
Localization Property ◽  
Spectrum Localization

The concept of ‘coefficient of ergodicity’, τ(P), for a finite stochastic matrixP, is developed from a standpoint more general and less standard than hitherto, albeit synthesized from ideas in existing literature. Several versions of such a coefficient are studied theoretically and by numerical examples, and usefulness in applications compared from viewpoints which include the degree to which extension to more general matrices is possible. Attention is given to the less familiar spectrum localization property:where λ is any non-unit eigenvalue ofP.The essential purpose is exposition and unification, with the aid of simple informal proofs.

Coefficients of ergodicity with respect to vector norms

1983 ◽  
Vol 20 (2) ◽  
pp. 277-287 ◽  
Author(s):  
Choon-Peng Tan
Keyword(s):  
Transition Matrix ◽  
Ergodic Markov Chain ◽  
Numerical Examples ◽  
Localization Property ◽  
New Class ◽  
Geometric Convergence ◽  
Functional Forms ◽  
Ergodicity Coefficients ◽  
Spectrum Localization ◽  
Vector Norms

The stationary distribution may be used to estimate the rate of geometric convergence to ergodicity for a finite homogeneous ergodic Markov chain. This is done by invoking the spectrum localization property of a new class of ergodicity coefficients defined with respect to column vector norms for the transition matrix P. Explicit functional forms in terms of the entries of P are obtained for these coefficients with respect to the l∞ and l1, norms, and comparison in performance with various known coefficients is made with the aid of numerical examples.

Coefficients of ergodicity with respect to vector norms

1983 ◽  
Vol 20 (02) ◽  
pp. 277-287 ◽  
Author(s):  
Choon-Peng Tan
Keyword(s):  
Transition Matrix ◽  
Ergodic Markov Chain ◽  
Numerical Examples ◽  
Localization Property ◽  
New Class ◽  
Geometric Convergence ◽  
Functional Forms ◽  
Ergodicity Coefficients ◽  
Spectrum Localization ◽  
Vector Norms

The stationary distribution may be used to estimate the rate of geometric convergence to ergodicity for a finite homogeneous ergodic Markov chain. This is done by invoking the spectrum localization property of a new class of ergodicity coefficients defined with respect to column vector norms for the transition matrix P. Explicit functional forms in terms of the entries of P are obtained for these coefficients with respect to the l∞ and l 1, norms, and comparison in performance with various known coefficients is made with the aid of numerical examples.

Conservative solute transport with periodic velocity and sinusoidal source concentration in semi-infinite porous media: An analytical solution

2014 ◽  
Vol 6 (1) ◽  
pp. 1024-1031
Author(s):  
R R Yadav ◽  
Gulrana Gulrana ◽  
Dilip Kumar Jaiswal
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Laplace Transformation ◽  
Seepage Velocity ◽  
Transformation Technique ◽  
Numerical Examples ◽  
One Dimensional ◽  
Porous Domain ◽  
Conservative Solute Transport ◽  
Source Concentration

The present paper has been focused mainly towards understanding of the various parameters affecting the transport of conservative solutes in horizontally semi-infinite porous media. A model is presented for simulating one-dimensional transport of solute considering the porous medium to be homogeneous, isotropic and adsorbing nature under the influence of periodic seepage velocity. Initially the porous domain is not solute free. The solute is initially introduced from a sinusoidal point source. The transport equation is solved analytically by using Laplace Transformation Technique. Alternate as an illustration; solutions for the present problem are illustrated by numerical examples and graphs.

Stability of Implicit Difference Equations Generated by Parabolic Functional Differential Problems

2007 ◽  
Vol 7 (1) ◽  
pp. 68-82
Author(s):  
K. Kropielnicka
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Initial Boundary ◽  
Numerical Examples ◽  
Convergence Results ◽  
Nonlinear Parabolic ◽  
Implicit Difference Methods ◽  
Neumann Type ◽  
Difference Methods ◽  
Functional Differential

AbstractA general class of implicit difference methods for nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type is constructed. Convergence results are proved by means of consistency and stability arguments. It is assumed that given functions satisfy nonlinear estimates of Perron type with respect to functional variables. Differential equations with deviated variables and differential integral problems can be obtained from a general model by specializing given operators. The results are illustrated by numerical examples.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

Assessing maximal allowable levels for simultaneous skin and eyes exposure to laser irradiation of various wavelengths

2019 ◽  
pp. 35-38
Author(s):  
Boris N. Rakhmanov ◽  
Vladimir I. Kezik ◽  
Vladimir T. Kibovsky ◽  
Valentin M. Ponomarev
Keyword(s):  
Total Energy ◽  
Laser Irradiation ◽  
Customs Union ◽  
Economic Community ◽  
Laser Safety ◽  
Numerical Examples ◽  
Allowable Level ◽  
Correct Formula

Introduction.Evidences prove falseness of formula determining maximal allowable level of total energy of laser irradiation in case when eyes or skin are simultaneously exposed to several irradiation sources with various wavelengths. The formula was mentioned in actual «Sanitary rules and regulations for lasers construction and exploitation» Nо 5804–91 and in SanPiN 2.2.4.3359–16, that in a part of VIII section «Laser irradiation atworkplace» are latest acting regulation document on laser safety. SanPiN 2.2.4.13–2–2006 of Belarus Republic and regulation document Nо 299 of Customs Union Commission of Eurasia Economic Community on 28/05/2010 appeared to contain other, more correct formula determining the same maximal allowable level.Objectivewas to improve regulation basis in laser safety by correcting mistakes made previously in regulation documents.Deducing formulae.The article presents thorough and consistent deducing a formula to determine total energy of laser irradiation in case when eyes or skin are simultaneously and jointly exposed to several irradiation sources with various wavelengths. The efforts resulted in the formula that agreed with formulae presented in the regulation document on laser safety of Belarus Republic and in the regulation document Nо 299 of Customs Union Commission of Eurasia Economic Community on 28/05/2010.Discussion.Correctness of the obtained formula is supported by numerical examples and by comparison with other formulae used in regulation documents on hygienic regulation of other acting factors.Conclusion.Results of the work are summarized, and emphasis is made on its value for solving problems of improving regulation basis for laser safety.

Encapsulation and the C++ class

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Program Design ◽  
Internal Representation ◽  
Floating Point ◽  
Complex Class ◽  
Numerical Examples ◽  
Important Notion ◽  
Standard Library

Built-in datatypes and C++ classes are introduced in this chapter, and discussed in relation to the important notion of encapsulation, which refers to the separation between the internal representation of the datatype and the operations to which it responds. Encapsulation later becomes an important consideration in the design of custom C++ classes that programmers develop themselves. It is illustrated with built-in floating-point datatypes float and double and with the complex class from the C++ standard library. While a sophisticated programmer is aware of the internal representation of data and its resulting limitations, encapsulation allows one to consider these as details and frees one to think at a higher level of program design. Some simple numerical examples are discussed in the text and in the exercises.

scholarly journals A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws

2018 ◽  
Vol 40 (1) ◽  
pp. 405-421 ◽  
Author(s):  
N Chatterjee ◽  
U S Fjordholm
Keyword(s):  
Initial Data ◽  
Conservation Laws ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Numerical Examples ◽  
Continuity Equations ◽  
Multiple Dimensions ◽  
Nonlocal Conservation Laws ◽  
Volume Method ◽  
Singular Data

Abstract We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

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