Browsing Mathematics by Title
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Boundary Harnack Principle for StableLike Processes
We establish the boundary Harnack principle for certain classes of symmetric stablelike processes in $\mathbf{R}^d$ on arbitrary open sets as well as censored stablelike processes on $\mathcal{C}^{1,1}$domains. Using ... 
Brownian Motion on Spaces with Varying Dimension
In this thesis we introduce and study Brownian motion with or without drift on state spaces with varying dimension. Starting with a concrete such state space that is the plane with an infinite pole on it, we construct a ... 
The C*algebra of a finite T_0 topological space
We are concerned with the following motivating question: how can one extend the classical GelfandNaimark theorem to the simplest nonHausdorff topological spaces? Our model space is a finite $T_0$ topological space, or ... 
Classification of connected Hopf algebras up to primecube dimension
We classify all connected Hopf algebras up to p^3 dimension over an algebraically closed field of characteristic p>0 under the mild restriction such that in dimension p^3, we only work over odd primes p when the primitive ... 
Combinatorial Laguerre Series
(20140224)We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials. We describe how such a series can be computed by ... 
Competing Brownian Particles
Consider a finite system of N Brownian particles on the real line. Rank them from bottom to top: the (currently) lowest particle has rank 1, the second lowest has rank 2, etc., up to the top particle, which has rank N. The ... 
Computational aspects of modular parametrizations of elliptic curves
\abstract{ We investigate computational problems related to modular parametrizations of elliptic curves defined over $\mathbb{Q}$. We develop algorithms to compute the Mazur SwinnertonDyer critical subgroup of elliptic ... 
Conformal welding of uniform random trees
A conformally balanced tree is an embedding of a given planar map into the plane with constraints on the harmonic measure of its edges such that the resulting set is unique up to scale and rotation. Bishop (2013) showed ... 
Connections Between Lanczos Iteration and Orthogonal Polynomials
(20100110)In this thesis we examine the connections between orthogonal polynomials and the Lanczos algorithm for tridiagonalizing a Hermitian matrix. The Lanczos algorithm provides an easy way to calculate and to estimate the ... 
Convergence and approximation for primaldual methods in largescale optimization
(1990)Largescale problems in convex optimization often can be reformulated in primaldual (minimax) representations having special decomposition properties. Approximation of the resulting highdimensional problems by restriction ... 
Convex Optimization over Probability Measures
The thesis studies convex optimization over the Banach space of regular Borel measures on a compact set. The focus is on problems where the variables are constrained to be probability measures. Applications include ... 
Convexity, convergence and feedback in optimal control
(2000)The results of this thesis are oriented towards the study of convex problems of optimal control in the extended piecewise linearquadratic format. Such format greatly extends the classical linearquadratic regulator problem ... 
Deformation invariance of rational pairs
Rational pairs, recently introduced by Kollár and Kovács, generalize rational singularities to pairs (X,D). Here X is a normal variety and D is a reduced divisor on X. Integral to the definition of a rational pair is the ... 
Deformations of Categories of Coherent Sheaves and FourierMukai Transforms
(20130725)In modern algebraic geometry, an algebraic variety is often studied by way of its category of coherent sheaves or derived category. Recent work by Toda has shown that infinitesimal deformations of the category of coherent ... 
Dual Equivalence Graphs and their Applications
In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's ... 
Eigenvalue fluctuations for random regular graphs
One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are <italic>universal</italic>. We probe the edges of universality by studying ... 
Eigenvalue Fluctuations of Random Matrices beyond the Gaussian Universality Class
(20131114)The goal of this thesis is to develop one of the threads of what is known in random matrix theory as universality, which essentially is that a large class of matrices generalizing the Gaussian matrices (certain Wigner ... 
An electrodynamic inverse problem in chiral media
(1998)We consider the inverse problem of determining the electromagnetic material parameters of a body from information obtainable only at the boundary of the body; such information comes in the form of a boundary map which we ... 
Elliptic Inverse Problems
Inverse problems arise in various areas of science and engineering including medical imaging, computer vision, geophysics, solid mechanics, astronomy, and so forth. A wide range of these problems involve elliptic operators. ... 
Essential spanning forests and electric networks in groups
(1997)Let $\Gamma$ be a Cayley graph of a finitely generated group G. Subgraphs which contain all vertices of $\Gamma ,$ have no cycles, and no finite connected components are called essential spanning forests. The set ${\cal ...