Physical Sciences and Mathematics Commons^™

Small And Large Scale Limits Of Multifractal Stochastic Processes With Applications, Jennifer Laurie Sinclair

Doctoral Dissertations

Various classes of multifractal processes, that is processes that display different properties at different scales, are studied. Most of the processes examined in this work exhibit stable trends at small scales and Gaussian trends at large scales, although the opposite can also occur. Many natural phenomena exhibit a fractal structure depending on some scaling factor, such as space or time. Thus, these types of processes have many useful modeling applications, including Biology and Economics. First, generalized tempered stable processes are defined and studied, following the original work on tempered stable processes by Jan Rosinski [16]. Generalized tempered stable processes encompass …

Two-Step Variations For Processes Driven By Fractional Brownian Motion With Application In Testing For Jumps From The High Frequency Data, Shiying Si

Doctoral Dissertations

In this dissertation we introduce the realized two-step variation of stochastic processes and develop its asymptotic theory for processes based on fractional Brownian motion and on more general Gaussian processes with stationary increments. The realized two-step variation is analogous to the realized 1, 1-order bipower variation introduced by Barndorff-Nielsen and Shephard [8] but mathematically is simpler to deal with. The powerful techniques of Wiener/Itˆo/Malliavin calculus for establishing limit laws play a key rule in our proofs. We include some stochastic simulations as an illustration of our theory. As a result of our study, we provide test statistics for testing for …

Optimal Control Applied To Population And Disease Models, Rachael Lynn Miller Neilan

Doctoral Dissertations

This dissertation considers the use of optimal control theory in population models for the purpose of characterizing strategies of control which minimize an invasive or infected population with the least cost. Three different models and optimal control problems are presented. Each model describes population dynamics via a system of differential equations and includes the effects of one or more control methods.

The first model is a system of two ordinary differential equations describing dynamics between a native population and an invasive population. Population growth terms are functions of the control, constructed so that the value of the control may affect …

Countable Groups As Fundamental Groups Of Compacta In Four-Dimensional Euclidean Space, Ziga Virk

Doctoral Dissertations

This dissertation addresses the question of realization of countable groups as funda- mental groups of continuum. In first chapter we discuss classical realizations in the category of CW complexes. We introduce Eilenberg-Maclane spaces and their topological properties. The second chapter provides recent developments on realization question such as those of Shelah, Keesling, ... The third chapter proves the realization theorem for countable groups. The re- sulting space is compact path connected, connected subspace of four dimensional Euclidean space.

Some Congruence Modulo 2 Statements Of Primitive Conway Vassiliev Invariants., James M. Dawson

Masters Theses

Polynomial knot invariants can often be used to define Vassiliev invariants on singu- lar knots. Here Vassiliev invariants form the Conway, Jones, HOMFLY, and Kauffman polynomials are explored. Also, some explanation is given about how symbols of the Jones and Conway polynomial can evaluated on suitable chord diagrams. These in- variants are further used to find expressions that are congruent modulo 2 to some low degree invariants derived from the Primitive Conway polynomial.