In an earlier paper (4) it was shown how to define for any matrix a unique generalization of the inverse of a non-singular matrix. The purpose of the present note is to give a further application which has relevance to the statistical problem of finding ‘best’ approximate solutions of inconsistent systems of equations by the method of least squares. Some suggestions for computing this generalized inverse are also given.
Pure mathematics, Applied Mathematics, General Mathematics, Mehler–Heine formula, Legendre's equation, Legendre function, Neumann series, Associated Legendre polynomials, symbols.namesake, symbols, Abelian von Neumann algebra, Legendre's constant, Legendre polynomials, Mathematics
Abstract
§1. Introductory. The formulawhere w is zero or a positive integer and | ζ | > 1, was given by F. E. Neumann “Crelle's Journal, XXXVII (1848), p. 24”. In § 2 of this paper some related formulae are given; the extension to the case when n is not integral is dealt with in § 3; while in § 4 the corresponding formulae for the Associated Legendre Functions when the sum of the degree and the order is a positive integer are established.
symbols.namesake, Partial differential equation, Integro-differential equation, General Mathematics, symbols, Applied mathematics, Nyström method, Daniell integral, Electric-field integral equation, Summation equation, Integral equation, Volterra integral equation, Mathematics
Abstract
In a recent paper Cooke [1] obtained a solution of the integral equationby using the identityand the technique, first used by Copson, of interchanging the orders of integration and hence reducing the problem to that of the successive solution of two Abel integral equations. It is also shown in [1] that the above identity can also be used to solve the dual series equationsThe kernel in equation (1) is a particular member of a general class of kernels which the author [6] has shown to be such that the resulting integral equation is directly soluble by using Copson's technique. The particular example of equation (1) is given in [6] and the identity of equation (2) was used by the author [7] to obtain the solution of equation (3).
Statistics and Probability, Discrete mathematics, Queueing theory, General Mathematics, Applied Mathematics, Real-time computing, Markov jump process, Motion (geometry), Poisson process, Management Science and Operations Research, Computer Science Applications, symbols.namesake, Distribution (mathematics), symbols, Jump, Invariant (mathematics), Variety (universal algebra), State distribution, Mathematics
Abstract
Let {Xt}t>0 be a Markov jump process and {Tn}n=0∞ a subset of the jump epochs of the process {Xt}t>0. The main results of this paper are computable formulae for the transient and invariant distributions of {XTn}n=0∞ and {XTn̄}n=0∞. We also characterize when the invariant distributions of {Xt}t>0 and {XTn}n=0∞ are the same and show that in this case {Tn}n=0∞ is a Poisson process. The results are applied to a variety of discrete-slate queueing networks to obtain their state distribution as seen by customers in arrival streams, departure streams, and traffic streams on the arcs. The main conclusion drawn is that for many traffic streams, the invariant distribution seen by customers in that stream coincides with the invariant distribution of {Xt}t>0 provided the customer in motion is excluded.
Statistics and Probability, symbols.namesake, General Mathematics, symbols, Applied mathematics, Poisson process, Statistics, Probability and Uncertainty, Characterization (materials science), Mathematics
Abstract
In a recent paper Rényi (1964) has obtained the following characterization of the Poisson process.
Published
1971
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.