Physical Sciences and Mathematics Commons^™

Three Population Models Applied To Competition, Disease And Invasion, Erika Asano

Doctoral Dissertations

In this work, we present three diffrent types of population models. The first two models are examined in the context of optimal control problems. The third involves the construction of an invasion model using a significant amount of data.

The first model describes the interaction of three populations, motivated by a combat scenario. One of the three populations can switch the mode of alliance with the other two populations between cooperation and competition. The other two populations always compete with each other. In this system of parabolic partial differential equations, the control is the function which measures the strength of …

Two Biological Applications Of Optimal Control To Hybrid Differential Equations And Elliptic Partial Differential Equations, Wandi Ding

Doctoral Dissertations

In this dissertation, we investigate optimal control of hybrid differential equations and elliptic partial differential equations with two biological applications. We prove the existence of an optimal control for which the objective functional is maximized. The goal is to characterize the optimal control in terms of the solution of the opti- mality system. The optimality system consists of the state equations coupled with the adjoint equations. To obtain the optimality system we differentiate the objective functional with respect to the control. This process is applied to studying two prob- lems: one is a type of hybrid system involving ordinary differential …

Cohomological Dimension With Respect To Nonabelian Groups, Atish Jyoti Mitra

Doctoral Dissertations

This dissertation addresses three aspects of cohomological dimension of metric spaces with respect to nonabelian groups.

In the first part we examine when the Eilenberg-Maclane space (n = 1) of the abelianization of a solvable group being an absolute extensor of a metric space implies the Eilenberg-Maclane space of the group itself is an absolute extensor. We also give an elementary approach to this problem in the case of nilpotent groups and 2-dimensional metric spaces.

The next part of the dissertation is devoted to generalizations of the Cencelj- Dranishnikov theorems relating extension properties of nilpotent CW complexes to its homology …

Adaptive Discontinuous Galerkin Finite Element Methods For Second And Fourth Order Elliptic Partial Differential Equations, Michael A. Saum

Doctoral Dissertations

A unified mathematical and computational framework for implementation of an adaptive discontinuous Galerkin (DG) finite element method (FEM) is developed using the symmetric interior penalty formulation to obtain numerical approximations to solutions of second and fourth order elliptic partial differential equations. The DG-FEM formulation implemented allows for h-adaptivity and has the capability to work with linear, quadratic, cubic, and quartic polynomials on triangular elements in two dimensions. Two different formulations of DG are implemented based on how fluxes are represented on interior edges and comparisons are made. Explicit representations of two a posteriori error estimators, a residual based type and …

Packings Of Conformal Preimages Of Circles, Matthew Edward Cathey

Doctoral Dissertations

This dissertation provides existence and uniqueness results for packings of conformal preimages of circles in the unit disk. Examples are given showing how these results can be applied in more general situations, such as finite- and infinite-to-one covers of the punctured plane.

Optimal Control Of Partial Di®Erential Equations And Variational Inequalities, Volodymyr Hrynkiv

Doctoral Dissertations

This dissertation deals with optimal control of mathematical models described by partial differential equations and variational inequalities. It consists of two parts. In the first part, optimal control of a two dimensional steady state thermistor problem is considered. The thermistor problem is described by a system of two nonlinear elliptic partial differential equations coupled with some boundary conditions. The boundary conditions show how the thermistor is connected to its surroundings. Based on physical considerations, an objective functional to be minimized is introduced and the convective boundary coefficient is taken to be a control. Existence and uniqueness of the optimal control …

A Numerical Method For Obtaining An Optimal Temperature Distribution In A Three-Dimensional Triple-Layered Skin Structure Embedded With Multi-Level Blood Vessels, Xingui Tang

Doctoral Dissertations

The research related to hyperthermia has stimulated a lot of interest in recent years because of its application in cancer treatment. When heating the tumor tissue, the crucial problem is keeping the temperature of the surrounding normal tissue below a certain threshold in order to avoid the damage to the normal tissue. Hence, it is important to obtain the temperature field of the entire region during the treatment. The objective of this dissertation is to develop a numerical method for obtaining an optimal temperature distribution in a 3D triple-layered skin structure embedded with multi-level blood vessels where the surface of …