Mathematics

It is well known that the classic Galerkin finite-element method is unstable when applied to hyperbolic conservation laws, such as the Euler equations for compressible flow. Adding a diffusion term to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use. An elegant solution devised by Eitan Tadmor for spectral methods is to add diffusion only to the high frequency modes of the solution, which stabilizes the method without the sacrifice of accuracy. We incorporate this idea into the finite-element framework by using hierarchical functions as a multi-frequency basis. The result is a new finite element method for solving hyperbolic conservation laws. For this method, we are able to prove convergence for a one-dimensional scalar conservation Law; Numerical results are presented for one- and two-dimensional hyperbolic conservation laws.

The Schrodinger partial differential equation is reduced to an equivalent problem in the calculus of variations and this problem is then directly solved by the use of trial wave functions;A proper anti-symmetric wave function, containing the interactions of the electrons, is set up. The energy integral and normalizing the integral are found in terms of parameters which enter in the wave function. These parameters are varied until the value of the energy W is a minimum;A method for finding the best values of the parameters by means of simple trial wave functions and direct differentiation is described;Evaluation of an elliptic integral is given in terms of Legendre Polynomials and methods for integrating several troublesome exponential integrals are given;The value of the energy is found within .2 percent of the experimental value. This value could be improved by use of a more complicated wave function and additional parameters. (Abstract shortened by UMI.)

The intricately interwoven basins of attraction stemming from Newton's Method applied to a simple complex polynomial are a common sight in fractal, dynamical systems, and numerical analysis literature. In this work, the author investigates how this workhorse of root-finding algorithms works for complex polynomials, in addition to a variety of other settings, from the simple, one-dimensional real function with a simple root, to the infinite-dimension Banach space. The rapid, quadratic convergence of Newton's method to a simple root is well known, but this performance is not guaranteed for all roots and for all starting points. Damping is one modification to the Newton algorithm that can be used to overcome difficulties in global convergence. We explore computationally how this damping affects the fractal geometry of the Newton basins of attraction for a simple function.

We consider the kinetics of irreversible film growth in solid-on-solid models with various restrictions on the adsorption (or growth) sites. We show how the master equations for the probabilities of subconfigurations of filled sites can be analyzed exactly to obtain coverages and spatial correlations for the first several layers. These provide an efficient framework for analysis of the early-stage growth kinetics, and indicate rapid attainment of asymptotic behavior. We illustrate the (1+1)- and (2+1)-dimensional cases for the simplest restricted solid-on-solid condition, and various modifications.

A poset (P,≤) with maximum 1 is called compact (C) iff sup W exists and is not 1 for any nonempty well ordered subset W of (P - 1). P is classically compact (CC) iff whenever sup S = 1 for some S ⊆ P, then sup F = 1 for some finite F ⊆ S. The product ∏[superscript] P[subscript] i (ordered coordinatewise) is C (resp., CC) iff the subset of all tuples with finitely many nonunit entries is C (CC). Among other results we show ∏[superscript] P[subscript] i is C(CC) iff each P[subscript] i is C(CC);A poset P is called a T[subscript] i-Topology poset iff P is isomorphic to the open-set lattice of some T[subscript] i-topology. We state necessary and sufficient conditions that a finite poset P must satisfy to be a T[subscript] O, T[subscript] F, T[subscript] Y, T[subscript] DD, T[subscript] FF, or T[subscript]1 topology poset. We show that an infinite poset P is a T[subscript] D-topology poset iff (i) P is a complete distributive lattice (ii) with join-infinite distributivity and (iii) P has a representational witnessed collection [phi] of completely prime filters. [phi] is said to be witnessed iff for any F ϵ [phi] there is an x ϵ F where for any G ϵ [phi] it is the case that x ϵ G iff G is not a proper superset of F;A brief discussion of the set theoretical axioms M and SM is made in order to introduce a method of construction of unramified symmetric models of ZFA + (-AC).

Using a construction due to A. Young, H. Boerner gives a prescription for writing down the matrices for the natural (integral) irreducible representation of the symmetric group S(,n) H. Boerner, Representations of Groups with Special Consideration for the Needs of Modern Physics, North Holland Publishing Co., Amsterdam, 1963, p. 119. Writing down the matrices using this prescription is rather tedious, and becomes computationally impossible for large n (n (GREATERTHEQ) 10) because of the need to calculate chains;This dissertation greatly simplifies the computation of the matrices for the natural (integral) irreducible representation of the symmetric group S(,n) by eliminating the need to calculate chains. The calculation of the chains is replaced by the simple act of setting up and inverting an upper triangular matrix, A(I). In many cases a set of tableaux which is not necessarily standard can be chosen so that A(I) is the identity matrix. Such a set of tableaux is called a complete set of orthogonal tableaux, and yields an equivalent representation of S(,n) in which all entries of the matrices are +1, -1, or 0.