This research consists of two parts. The first part examines the posterior probability integrals for a family of linear models which arises from the work of Hart, Koen and Lombard (2003). Applying Laplace's method to these integrals is not entirely straightforward. One of the requirements is to analyze the asymptotic behavior of the information matrices as the sample size tends to infinity. This requires a number of analytic tricks, including viewing our covariance matrices as tending to differential operators. The use of differential operators and their Green's functions can provide a convenient and systematic method to asymptotically invert the covariance matrices. Once we have found the asymptotic behavior of the information matrices, we will see that in most cases BIC provides a reasonable approximation to the log of the posterior probability and Laplace's method gives more terms in the expansion and hence provides a slightly better approximation. In other cases, a number of pathologies will arise. We will see that in one case, BIC does not provide an asymptotically consistent estimate of the posterior probability; however, the more general Laplace's method will provide such an estimate. In another case, we will see that a naive application of Laplace's method will give a misleading answer and Laplace's method must be adapted to give the correct answer. The second part uses numerical methods to compute the "exact" posterior probabilities and compare them to the approximations arising from BIC and Laplace's method.

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No description available.

The purpose of this study is to find analytical solutions to Laplacian field problems rela-tive to arbitrary configurations of spheres based on novel translational addition theorems derived specifically for scalar Laplacian functions. These theorems are used to express in analytic form the fields due to individual spheres in system of coordinates attached to other spheres, thus allowing for the exact boundary conditions to be imposed. In the literature, translational addition theorems are available for scalar cylindrical and spherical wave functions. Such theorems are not directly available for the general solu-tion of the Laplace equation. This thesis presents the derivation of the required translational addition theorems for the general solution of Laplace equation in spherical coordinates and then the application of these theorems to find analytical solutions to some electrostatic and magnetostatic field problems relative to arbitrarily located spheres. Computation results for electric and magnetic spheres have been generated and numerical results are compared with the re-sults obtained by other methods available in the literature for two sphere systems. Such numerical data, of known accuracy, are also useful for validating various approximate numerical methods.

The purpose of this study is to find analytical solutions to Laplacian field problems rela-tive to arbitrary configurations of spheres based on novel translational addition theorems derived specifically for scalar Laplacian functions. These theorems are used to express in analytic form the fields due to individual spheres in system of coordinates attached to other spheres, thus allowing for the exact boundary conditions to be imposed. In the literature, translational addition theorems are available for scalar cylindrical and spherical wave functions. Such theorems are not directly available for the general solu-tion of the Laplace equation. This thesis presents the derivation of the required translational addition theorems for the general solution of Laplace equation in spherical coordinates and then the application of these theorems to find analytical solutions to some electrostatic and magnetostatic field problems relative to arbitrarily located spheres. Computation results for electric and magnetic spheres have been generated and numerical results are compared with the re-sults obtained by other methods available in the literature for two sphere systems. Such numerical data, of known accuracy, are also useful for validating various approximate numerical methods.

Thesis (Ph. D.)--Oregon State University, 1965. / Typescript. Includes bibliographical references (leaves 57-59). Also available on the World Wide Web.

Thesis (M.Sc.) -- University of Adelaide, School of Applied Mathematics, 2003. / "October 13, 2003" Bibliography: leaves 102-105.

Thesis (M.S.)--Oklahoma State University. / Bibliography: leaves 80-81.

University, Diss., 2005--Bonn.

The present paper is a study of a class of polynomial set defined by means of a generating function of the form G(2xt- t²) for the choice G(u) = 0 F1(--;α;u). The paper contains some interesting results in the form of recurrence relations, generating functions, finite series of product of polynomials, hypergeometric form, relationship with Shively 's pseudo Laguerre and other polynomials, integral representation, fractional integral and Laplace transform of the polynomial.