Now showing items 68-87 of 96

• #### The Positive Semidefinite Rank of Matrices and Polytopes ﻿

The positive semidefinite (psd) rank of a nonnegative <italic>p</italic> × <italic>q</italic> matrix <italic>M</italic> is defined to be the smallest integer <italic>k</italic> such that there exist <italic>k</italic> × ...
• #### Problems in Algebraic Vision ﻿

This thesis studies several fundamental mathematical problems that arise from computer vision using techniques in algebraic geometry and optimization. Chapters 2 and 3 consider the fundamental question of the existence of ...
• #### Problems in computational algebra and integer programming ﻿

(2007)
This thesis is a compendium of several projects that span the gap between commutative algebra and geometric combinatorics: one in tropical geometry, one in computational algebra, and two in discrete geometry and integer ...
• #### Qualitative stability properties of matrices ﻿

(1985)
A matrix A is sign stable if some matrix with the same sign pattern of positive, negative, and zero entries has all eigenvalues with negative real parts. It is potentially stable if it is not sign unstable. The system A x ...
• #### The Radiative Transfer Equation in Photoacoustic Imaging ﻿

(2013-07-25)
Photoacoustic tomography is a rapidly developing medical imaging technique that combines optical and ultrasound imaging to exploit the high contrast and high resolution of the respective individual modalities. Mathematically, ...
• #### Random recursion ﻿

We study the limiting behavior of three stochastic processes. Two are interacting particle systems, the frog model and coalescing random walk. We work out transience and recurrence properties on various graphs. The last ...
• #### The regularity of Loewner curves ﻿

The Loewner differential equation, a classical tool that has attracted recent attention due to Schramm-Loewner evolution (SLE), provides a unique way of encoding a simple 2-dimensional curve into a continuous 1-dimensional ...
• #### Results on singularities of pairs ﻿

Singularities of algebraic varieties have been studied extensively, and recently also the properties of singularities of pairs have been investigated. This thesis presents several results on singularities of different kinds ...
• #### Self-shrinking Solutions to Mean Curvature Flow ﻿

We construct new examples of self-shrinking solutions to mean curvature flow. We first construct an immersed and non-embedded sphere self-shrinker. This result verifies numerical evidence dating back to the 1980's and shows ...
• #### Selmer groups for elliptic curves with isogenies of prime degree ﻿

(2003)
The Mordell-Weil theorem states that the points of an elliptic curve defined over a number field form a finitely generated, abelian group. The rank of this group, generally referred to as the rank of the elliptic curve, ...
• #### Sheaves on support varieties and varieties of elementary subalgebras ﻿

We present several results about two closely related types of objects: the projectivized scheme $\PG$ of one parameter subgroups of an infinitesimal group scheme $G$ and the variety $\bE(\fg)$ of maximal elementary subalgebras ...
• #### Shimura Degrees for Elliptic Curves over Number Fields ﻿

A crowning achievement of Number theory in the 20th century is a theorem of Wiles which states that for an elliptic curve E over <bold>Q</bold> of conductor N, there is a non-constant map from the modular curve of level N ...
• #### Smoothness of Loewner Slits ﻿

(2013-02-25)
In this dissertation, we show that the chordal Loewner differential equation with C^{beta} driving function generates a C^{beta + 1/2} slit for 1/2 < beta <= 2, except when beta = 3/2 the slit is only proved to be weakly C^{1,1}.
• #### Some cohomology of finite general linear groups ﻿

We prove that the degree r(2p − 3) cohomology of any (untwisted) finite group of Lie type over F_(p^r), with coefficients in characteristic p, is nonzero as long as its Coxeter number is at most p. We do this by providing ...
• #### Some Inverse Problems in Analysis and Geometry ﻿

The aim of a typical inverse problem is to recover the interior properties of a medium by making measurements only on the boundary. These types of problems are motivated by geophysics, medical imaging and quantum mechanics ...
• #### Some Linear and Nonlinear Geometric Inverse Problems ﻿

Inverse problems is an area at the interface of several disciplines and has become a prominent research topic due to its potential applications. A wide range of these problems can be formulated under various geometric ...
• #### Spectral Theory of Z^d Substitutions ﻿

In this paper, we generalize and develop results of Queffelec allowing us to characterize the spectrum of an aperiodic substitution in Z^d by describing the Fourier coefficients of mutually singular measures of pure type ...

(1996)
• #### Strichartz estimates for wave equations with coefficients of Sobolev regularity ﻿

(2005)
Wave packet techniques provide an effective method for proving Strichartz estimates on solutions to wave equations whose coefficients are not smooth. In this work, such methods are used to show Strichartz inequalities for ...