Browsing Mathematics by Title
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LamWilliams Markov chains on symmetric groups
(20140224)This paper reviews the current state of the LamWilliams conjectures on a multivariate Markov chain on the symmetric group S_n. We start with Lam's work on random core partitions which led to a remarkable Markov chain on ... 
Local cohomology at generic singularities of Schubert varieties in cominuscule flag varieties
(20120913)We study the local cohomology of a Schubert variety in a cominuscule flag variety at a generic singularity. Let $G$ be a reductive complex algebraic group with a Borel subgroup $B$, let $W$ be the Weyl group of $G$ with ... 
Local Set Approximation: Infinitesimal to Local Theorems for Sets in Euclidean Space and Applications
In this thesis we develop the theory of Local Set Approximation (LSA), a framework which arises naturally from the study of sets with singularities. That is, we describe the local structure of a set A in Euclidean space ... 
Markov chain mixting time, card shuffling and spin systems dynamics
(20130725)The mixing time of a Markov chain describes how fast the Markov chain converges to its stationary distribution. In this thesis, we survey some of the knowledge and main tools available in this field by looking at examples. ... 
Markov partitions for hyperbolic toral automorphisms
(1992)The study of the dynamical properties of hyperbolic toral automorphisms is simplified when the automorphisms are represented as shifts of finite type. The conventional method used to represent such an automorphism symbolically ... 
Mathematical Aspects of Gerrymandering
(20131114)Every 10 years the United States performs a census, and this census determines how many members of congress will represent each state. Then begins an unfortunate battle, as cartographers manipulate the boundaries for ... 
Matrix free methods for large scale optimization
Sequential quadratic optimization (SQP) methods are widely used to solve largescale nonlinear optimization problems. We build two matrixfree methods for approximately solving exact penalty subproblems that arise when ... 
Noninterior pathfollowing methods for complementarity problems
(1998)Because of its excellent numerical performance, noninterior path following methods (also called smoothing methods) have become an important class of methods for solving complementarity problems. However, no rate of ... 
The nonexistence of certain free prop extensions and capitulation in a family of dihedral extensions of Q
(1996)$\doubz\sbsp{p}{d}$extensions are a natural way to extend Iwasawa's theory of $\doubz\sb{p}$extensions. A further extension would be to look at free prop extensions which though nonabelian, can be studied by looking at ... 
Nonholonomic EulerPoincaré equations and stability in Chaplygin's sphere
(2000)A method of reducing several classes of nonholonomic mechanical systems that are defined on semidirect products of Lie groups is developed. The method reduces the Lagranged'Alembert principle to obtain a reduced constrained ... 
On numerics and inverse problems
In this thesis, two projects in inverse problems are described. The first concerns a simple mathematical model of synthetic aperture radar with undirected beam, modeled as a 2D circular Radon transform with centers restricted ... 
On Particle Interaction Models
(20140224)This dissertation deals with three problems in Stochastic Analysis which broadly involve interactions, either between particles (Chapters 1 and 2), or between particles and the boundary of a C2 domain (Chapter 3). In Chapter ... 
On singularities of generic projection hypersurfaces
(2006)The present work studies singularities of hypersurfaces arising from generic projections of smooth projective varieties, in the context of Du Bois and semi log canonical singularities. It is demonstrated that Du Bois ... 
On special Lagrangian equations
(20140224)In this paper we study the special Lagrangian equation and related equations. Special Lagrangian equation originates in the special Lagrangian geometry by HarveyLawson [HL1]. In subcritical phases, we construct singular ... 
On the mod 2 general linear group homology of totally real number rings
(1997)We study the mod 2 homology of the general linear group of rings of integers in totally real number fields. In particular, for certain such rings R, we construct a space JKR and show that the mod 2 homology of JKR is a ... 
The OrnsteinUhlenbeck Process In Neural DecisionMaking: Mathematical Foundations And Simulations Suggesting The Adaptiveness Of Robustly Integrating Stochastic Neural Evidence
(20130225)This master's thesis reviews the concepts behind a stochastic process known as the OrnsteinUhlenbeck Process, and then uses that process as a way to investigate neural decision making. In particular, MATLAB simulations ... 
Path algebras and monomial algebras of finite GKdimension as noncommutative homogeneous coordinate rings
This thesis sets out to understand the categories QGr A where A is either a monomial algebra or a path algebra of finite GelfandKirillov dimension. The principle questions are: \begin{enumerate} \item What is the structure ... 
Permutation diagrams in symmetric function theory and Schubert calculus
A fundamental invariant of a permutation is its inversion set, or diagram. Natural machinery in the representation theory of symmetric groups produces a symmetric function from any finite subset of <bold>N</bold><super>2</super>, ... 
Polynomials in Multiview Geometry
(20130417)We study multiview geometry and some of its applications through the use of polynomials. A threedimensional world point gives rise to n ≥ 2 twodimensional projections in n given cameras. The object of focus in this ... 
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> × ...