scholarly journals Multivariate semi-markov matrices

1972 ◽  
Vol 13 (1) ◽  
pp. 107-113 ◽  
Author(s):  
Marcel F. Neuts ◽  
Peter Purdue
Keyword(s):  
Probability Theory ◽  
Probability Distribution ◽  
Convex Function ◽  
Maximal Eigenvalue ◽  
Partial Derivatives ◽  
The Matrix ◽  
Stieltjes Transforms ◽  
Quantities Of Interest

Finite matrices with entries pij Fij (x1,…, xk), where {pij} is stochastic and Fij(.) is a k-variate probability distribution are discussed. It is shown that the matrix of k-variate Laplace-Stieltjes transforms of the Pij Fij(x1, …, xk) has a Perron-Frobenius eigenvalue which is a convex function in k variables in a suitably defined region. The values of the partial derivatives near the origin of this maximal eigenvalue are exhibited. They are quantities of interest in a variety of applications in Probability theory.

Some exact results for dams with Markovian inputs

1976 ◽  
Vol 13 (02) ◽  
pp. 329-337
Author(s):  
Pyke Tin ◽  
R. M. Phatarfod
Keyword(s):  
Probability Distribution ◽  
Stationary Distribution ◽  
Special Form ◽  
Geometric Distribution ◽  
Transition Probabilities ◽  
Exact Results ◽  
Input Process ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Finite Dam

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj ) where p ij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.

A Novel Heuristic PageRank Algorithm in Web Search

2011 ◽  
Vol 216 ◽  
pp. 747-751
Author(s):  
Yan Li He
Keyword(s):  
Probability Distribution ◽  
Web Search ◽  
The Internet ◽  
Power Method ◽  
Web Page ◽  
Principal Eigenvector ◽  
Pagerank Algorithm ◽  
Empirical Distributions ◽  
The Matrix ◽  
The Web

With the booming development of the Internet, web search engines have become the most important Internet tools for retrieving information. PageRank computes the principal eigenvector of the matrix describing the hyperlinks in the web using the famous power method. Based on empirical distributions of Web page degrees, we derived analytically the probability distribution for the PageRank metric. We found out that it follows the familiar inverse polynomial law reported for Web page degrees.

Some exact results for dams with Markovian inputs

1976 ◽  
Vol 13 (2) ◽  
pp. 329-337 ◽  
Author(s):  
Pyke Tin ◽  
R. M. Phatarfod
Keyword(s):  
Probability Distribution ◽  
Stationary Distribution ◽  
Special Form ◽  
Geometric Distribution ◽  
Transition Probabilities ◽  
Exact Results ◽  
Input Process ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Finite Dam

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj) where pij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.

scholarly journals A linear complementarity problem involving a subgradient

1988 ◽  
Vol 37 (3) ◽  
pp. 345-351 ◽  
Author(s):  
J. Parida ◽  
A. Sen ◽  
A. Kumar
Keyword(s):  
Convex Function ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Nonlinear Complementarity Problem ◽  
Linear Complementarity ◽  
Existence Of A Solution ◽  
Nonlinear Complementarity ◽  
The Matrix ◽  
Square Matrix

A linear complementarity problem, involving a given square matrix and vector, is generalised by including an element of the subdifferential of a convex function. The existence of a solution to this nonlinear complementarity problem is shown, under various conditions on the matrix. An application to convex nonlinear nondifferentiable programs is presented.

scholarly journals Hermite-Hadamard, Jensen, and Fractional Integral Inequalities for Generalized P -Convex Stochastic Processes

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Fangfang Ma ◽  
Waqas Nazeer ◽  
Mamoona Ghafoor
Keyword(s):  
Probability Theory ◽  
Convex Function ◽  
Stochastic Processes ◽  
Fractional Integral ◽  
Probabilistic Models ◽  
Random Variable ◽  
Expected Value ◽  
Integral Inequalities ◽  
Stochastic Convexity ◽  
Convex Stochastic Processes

The stochastic process is one of the important branches of probability theory which deals with probabilistic models that evolve over time. It starts with probability postulates and includes a captivating arrangement of conclusions from those postulates. In probability theory, a convex function applied on the expected value of a random variable is always bounded above by the expected value of the convex function of that random variable. The purpose of this note is to introduce the class of generalized p -convex stochastic processes. Some well-known results of generalized p -convex functions such as Hermite-Hadamard, Jensen, and fractional integral inequalities are extended for generalized p -stochastic convexity.

The methodological apparatus for planning and regulating trends in high-tech production costs under risk

2020 ◽  
Vol 16 (3) ◽  
pp. 400-414
Author(s):  
A.V. Leonov ◽  
A.Yu. Pronin
Keyword(s):  
Mathematical Modeling ◽  
Probability Theory ◽  
Economic Evaluation ◽  
Probability Distribution ◽  
Comprehensive Approach ◽  
Production Costs ◽  
Financial Resources ◽  
High Tech ◽  
Distribution Laws ◽  
The Given

Subject. The article discusses the methodological apparatus for assessing how many financial resources are needed, and optimize their gradual consumption in high-tech production projects. Results of the projects can be forecasted only as a probable estimate, since there is great uncertainty due to a multitude of random factors. Objectives. The study aims to form the methodological apparatus to assess the amount of financial resources needed under risk and substantiate the reasonable strategy for consuming them in high-tech production projects within a given period of time. The apparatus is to allow for quick adjustments of the financing plan and estimation of project expenditures. Methods. We applied the comprehensive approach to planning and regulating trends in high-tech production costs, methods of economic and mathematical modeling and the probability theory. Results. We reviewed methods used to assess how many financial resources are needed through demand probability distribution laws. Based on them, we devised the interval technique for regulating cost trends so as to substantiate the reasonable strategy for the performance of projects with the desired probability within the given period of time. What distinguishes the interval technique is that it provides the overall vision of the period, during which the project will be performed, and the probability of the advanced prediction of a shortage or excess of financial resources. Conclusions and Relevance. The methodological apparatus proposed herein will facilitate the technological and economic evaluation of various options of high-tech production and choose those ones which ensure the best use of financial resources, quickly regulate the economic dynamism throughout the high-tech production phases in line with a variety of factors, which randomly emerge at certain phases.

Estimation in SEM: A Concrete Example

2007 ◽  
Vol 32 (1) ◽  
pp. 110-120 ◽  
Author(s):  
John M. Ferron ◽  
Melinda R. Hess
Keyword(s):  
Structural Equation ◽  
Likelihood Estimation ◽  
Equation Model ◽  
Equation Modeling ◽  
Unknown Parameters ◽  
Technical Literature ◽  
Partial Derivatives ◽  
First Order ◽  
The Matrix ◽  
Definition Of

A concrete example is used to illustrate maximum likelihood estimation of a structural equation model with two unknown parameters. The fitting function is found for the example, as are the vector of first-order partial derivatives, the matrix of second-order partial derivatives, and the estimates obtained from each iteration of the Newton-Raphson algorithm. The goal is to provide a concrete illustration to help those learning structural equation modeling bridge the gap between the verbal descriptions of estimation procedures and the mathematical definition of these procedures provided in the technical literature.

scholarly journals Second Order Perturbations of Elliptic Elements with Respect to the Initial Ones

1999 ◽  
Vol 172 ◽  
pp. 453-454
Author(s):  
F.J. Marco Castillo ◽  
M.J. Martínez Usó ◽  
J.A. López Ortí
Keyword(s):  
Quadratic Form ◽  
Second Order ◽  
Initial Instant ◽  
Orbital Elements ◽  
Partial Derivatives ◽  
Orbital Parameters ◽  
The Matrix ◽  
The Individual ◽  
Derivatives Of

AbstractThe following paper is devoted to the theoretical exposition of the obtention of second order perturbations of elliptic elements and is a follow-up of previous papers (Marco et al., 1996; Marco et al., 1997) where the hypothesis was made that the matrix of the partial derivatives of the orbital elements with respect to the initial ones is the identity matrix at the initial instant only. So, we must compute them through the integration of Lagrange planetary equations and their partial derivatives.Such developments have been applied to the individual corrections of orbits together with the correction of the reference system through the minimization of a quadratic form obtained from the linearized residual. In this state two new targets emerged: 1.To be sure that the most suitable quadratic form was to be considered.2.To provide a wider vision of the behavior of the different orbital parameters in time.Both aims may be accomplished through the consideration of the second order partial derivatives of the elliptic orbital elements with respect to the initial ones.

THEORY OF RANDOM ADVECTION IN TWO DIMENSIONS

1996 ◽  
Vol 10 (18n19) ◽  
pp. 2273-2309 ◽  
Author(s):  
M. CHERTKOV ◽  
G. FALKOVICH ◽  
I. KOLOKOLOV ◽  
V. LEBEDEV
Keyword(s):  
Velocity Field ◽  
Probability Distribution ◽  
Probability Distributions ◽  
Classical Problem ◽  
Passive Scalar ◽  
Two Dimensions ◽  
Change Of Variables ◽  
The Matrix ◽  
Two Dimensional Flow ◽  
Non Gaussian

The steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described at scales less than the correlation length of the flow and larger than the diffusion scale. The probability distribution of the scalar is expressed via the probability distribution of the line stretching rate. The description of the line stretching can be reduced to the classical problem of studying the product of many matrices with a unit determinant. We found a change of variables which allows one to map the matrix problem into a scalar one and to prove thus a central limit theorem for the statistics of the stretching rate. The proof is valid for any finite correlation time of the velocity field. Whatever be the statistics of the velocity field, the statistics of the passive scalar in the inertial interval of scales is shown to approach Gaussianity as one increases the Peclet number Pe (the ratio of the pumping scale to the diffusion one). The first n < ln (Pe) simultaneous correlation functions are expressed via the flux of the squared scalar and only one unknown factor depending on the velocity field: the mean stretching rate. That factor can be calculated analytically for the limiting cases. The non-Gaussian tails of the probability distributions at finite Pe are found to be exponential.

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