Abstract:

The theories that have been employed to derive the macroscopic differential equations that describe solute transport through porous media are reviewed critically. These foundational theories may be grouped into three classes: (1) those based in fluid mechanics, (2) those based in kinematic approaches employing the mathematics of the theory of Markov processes, and (3) those based in a formal analogy between statistical thermodynamics and hydrodynamic dispersion. It is shown that the theories of class 1 have had to employ highly artificial models of a porous medium in order to produce a well-defined velocity field in the pore space that can be analysed rigorously or have had to assume that well-defined solutions of the equations of fluid mechanics exist in the pore space of a natural porous medium and then adopt an ad hoc definition of the solute difusivity tensor. The theories of class 2 do not require the validity of fluid mechanics but they suffer from the absence of a firm dynamical basis, at the molecular level, for the stochastic properties they attribute to the velocity of a solute molecule, or they ignore dynamics altogether and make kinematic assumptions directly on the position process of a solute molecule. The theories of class 3 have been purely formal in nature, with an unclear physical content, or have been no different in content from empirically based theories that make use of the analogy between heat and matter flow at the macroscopic level. It is concluded that none of the existing foundational theories has yet achieved the objectives of: (1) deriving, in a physically meaningful and mathematically rigorous fashion, the macroscopic differential equations of solute transport theory, and (2) elucidating the structure of the empirical coefficients appearing in these equations. © 1979.

Abstract:

For all p > 2, k > p, a size-and-reflection-shape space S R Σp, 0^k of k-ads in general position in R^p, invariant under translation, rotation and reflection, is shown to be a smooth manifold and is equivariantly embedded in a space of symmetric matrices, allowing a nonparametric statistical analysis based on extrinsic means. Equivariant embeddings are also given for the reflection-shape-manifold R Σp, 0^k, a space of orbits of scaled k-ads in general position under the group of isometries of R^p, providing a methodology for statistical analysis of three-dimensional images and a resolution of the mathematical problems inherent in the use of the Kendall shape spaces in p-dimensions, p > 2. The Veronese embedding of the planar Kendall shape manifold Σ2^k is extended to an equivariant embedding of the size-and-shape manifold S Σ2^k, which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory. © 2009 Elsevier Inc. All rights reserved.

Abstract:

In this paper we prove the existence, uniqueness and stability of the invariant distribution of a random dynamical system in which the admissible family of laws of motion consists of monotone maps from a closed subset of a finite dimensional Euclidean space into itself. © 2008 Springer-Verlag.