Digital Commons Network^™

Covering Maps In The Uniform Category, Brendon L. Labuz

Doctoral Dissertations

First, generalized uniform covering maps are classified in terms of subgroups of the uniform fundamental group. Hausdorff, locally uniform joinable, and chain connected covering spaces of a uniformly locally joinable chain connected space are classifieded in terms of closed subgroups of its uniform fundamental group. If the space is also semilocally simply uniform joinable, uniform covering spaces are classified in terms of all subgroups of its uniform fundamental group. Next it is shown that the inverse limit of a strong Mittag-Leer inverse system of Hausdorff uniform covering spaces is a generalized uniform covering space. The question of the converse is …

Numerical Methods For Fully Nonlinear Second Order Partial Differential Equations, Michael Joseph Neilan

Doctoral Dissertations

This dissertation concerns the numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). The numerical methods and analysis are based on a new concept of weak solutions called moment solutions, which unlike viscosity solutions, are defined by a constructive method called the vanishing moment method. The main idea of the vanishing moment method is to approximate fully nonlinear second order PDEs by a family of fourth order quasi-linear PDEs. Because the method is constructive, we can develop a wealth of convergent numerical discretization methods to approximate fully nonlinear second order PDEs. We first study the numerical …

Nonlinear Dissipative Wave Equations With Space-Time Dependent Potentials, Maisa Khader

Doctoral Dissertations

We study the long time behavior of solutions of the wave equations with absorption abs (u(t, x))[superscript p]⁻¹u(t, x) and variable damping a(t, x)u[subscript t](t, x), where p belongs to (1, n + 2/n - 2) and a(t, x) ~ a₀(1 + abs(x))⁻[superscript alpha](1 + t)⁻[superscript beta] for large abs x and t, a₀ > 0, for alpha belongs to (-infinity, 1), beta belongs to (-1, 1). We established decay estimates for the energy, L² and L[superscript p]⁺¹ norm of the solutions. 1. For alpha belongs to [0, 1), beta belongs to (-1, 1) and alpha + beta belongs to (0, …

Invariant Frechet Algebras On Bounded Symmetric Domains, Oleg Eroshkin

Doctoral Dissertations

Let D be a bounded domain in the complex vector space Cn . We say that D is symmetric iff, given any two points p, q ∈ D, there is a biholomorphism &phis;, which interchanges p and q. These domains were classified abstractly by Elie Cartan in his general study of symmetric spaces, and were canonically realized in Cn by Harish-Chandra. They include polydisks and Siegel domains.

Let D be a bounded symmetric domain in Cn , and G be the largest connected group of biholomorphic automorphisms of D. The algebra C( D) of all continuous (not necessarily bounded) complex-valued …

Wavelet Regression With Long Memory Infinite Moving Average Errors, Juan Liu

Doctoral Dissertations

For more than a decade there has been great interest in wavelets and wavelet-based methods. Among the most successful applications of wavelets is nonparametric statistical estimation, following the pioneering work of Donoho and Johnstone (1994, 1995) and Donoho et al. (1995). In this thesis, we consider the wavelet-based estimators of the mean regression function with long memory infinite moving average errors, and investigate the rates of convergence of estimators based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence rates within a logarithmic term over a large class of non-smooth functions that involve …

Kadison -Singer Algebras With Applications To Von Neumann Algebras, Mohan Ravichandran

Doctoral Dissertations

I develop the theory of Kadison-Singer algebras, introduced recently by Ge and Yuan. I prove basic structure theorems, construct several new examples and explore connections to other areas of operator algebras. In chapter 1, I survey those aspects of the theory of non-selfadjoint algebras that are relevant to this work. In chapter 2, I define Kadison-Singer algebras and give different proofs of results of Ge-Yuan, which will be further extended in the last chapter. In chapter 3, I analyse in detail a class of elementary Kadison-Singer algebras that contain Hinfinity and describe their lattices of projections. In chapter 4, I …

Mathematics Of Double-Walled Carbon Nanotube Model: Asymptotic Spectral And Stability Analysis, Miriam Rojas-Arenaza

Doctoral Dissertations

This dissertation is devoted to analytical study of a contemporary model of a double-walled carbon nano-tube. Carbon nano-tubes have been considered outstanding candidates to innovate and to promote emerging technologies, due to their remarkable chemical, mechanical, and physical properties. For these technologies, there is a need to develop mathematical models that capture the nature of the responses of these structures under a variety of physical conditions. Developing these models is challenging because the behavior lies on the borderline between classical and quantum systems. The main goal of the present dissertation is to prove mathematically rigorous results concerning the vibrational behavior …

The Process Of Making Meaning: The Interplay Between Teachers' Knowledge Of Mathematical Proofs And Their Classroom Practices, Megan Paddack

Doctoral Dissertations

The purpose of this study was to investigate and describe how middle school mathematics teachers make meaning of proofs and the process of proving in the context of their classroom practices. A framework of making meaning, created by the researcher, guided the data collection and analysis phases of the study. This framework describes the five central aspects of the process of making meaning: knowledge, beliefs, utilization of knowledge, interconnections of practice and knowledge, and making sense of past knowledge and current experiences. The utilization of a qualitative research methodology that combined ethnographic fieldwork and discourse analysis allowed the researcher to …

Kadison -Singer Algebras, Wei Yuan

Doctoral Dissertations

In this dissertation, we defined a new class of non selfadjoint operator algebras---Kadison-Singer algebras or KS-algebras for simplicity. These algebras combine triangularity, reflexivity and von Neumann algebra property into one consideration. Generally speaking, KS-algebras are reflexive, maximal triangular with respect to its "diagonal subalgebra". Many selfadjoint features are preserved in them and concepts can be borrowed directly from the theory of von Neumann algebras. In fact, a more direct connection of KS-algebras and von Neumann algebras is through the lattice of invariant projections of a KS-algebra. The lattice is reflexive and "minimally generating" in the sense that it generates the …