Browsing Mathematics by Title
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A Geometric Perspective on FirstPassage Competition
(20130417)We study the macroscopic geometry of firstpassage competition on the integer lattice <bold>Z</bold><super>d</super>, with a particular interest in describing the behavior when one species initially occupies the exterior ... 
Graded group schemes
We define graded group schemes and graded group varieties and develop their theory. We give a generalization of the result that connected graded bialgebras are graded Hopf algebra. Our result is given for a broader class ... 
Grothendieck Duality on Diagrams of Schemes
The Du Bois complex and Du Bois singularities, which extend results of Hodge theory to singular complex varieties, can be defined in terms of a cubical hyperresolution. In this dissertation I further develop the language ... 
The Grothendieck Groups of Module Categories over Coherent Algebras
Let <italic>k</italic> be a field and <italic>B</italic> either a finitely generated free <italic>k</italic>algebra, or a regular <italic>k</italic>algebra of global dimension two with at least three generators, generated ... 
Heat Kernel Estimates for Markov Processes Associated with TimeDependent Dirichlet Forms
In this paper, timeinhomogeneous stablelike processes are investigated. We establish the relation between the transition operators and timedependent parabolic equations, as well as upper heat kernel estimates. 
Hopf algebras of finite GelfandKirillov dimension
(20131114)We study Hopf algebras of finite GelfandKirillov dimension. By analyzing the free pointed Hopf algebra F(t), we show the existence of certain Hopf subalgebras of pointed Hopf domains H whose GKdimension are finite and ... 
Hybrid inverse problems
(20131114)Inverse problems arise in different disciplines including exploration geophysics, medical imaging and nondestructive evaluation. In some settings, a single modality displays either high contrast or high resolution but not ... 
Hybrid Inverse Problems Arising from AcoustoElectric Coupling
(20131114)We study hybrid imaging techniques that aim to overcome the illposedness of the Electric Impedance Tomography using the acoustoelectric coupling. For the isotropic case, we consider the problem of reconstructing the ... 
Interacting particle systems with partial annihilation through membranes
This thesis studies the <italic>hydrodynamic limit</italic> and the <italic>fluctuation limit</italic> for a class of interacting particle systems in domains. These systems are introduced to model the transport of positive ... 
Inverse BoundaryValue Problems on an Infinite Slab
In this work, we study the stability aspect of two inverse boundaryvalue problems (IBVPs) on an infinite slab with partial data. The uniqueness aspects of these IBVPs were considered and studied by Li and Uhlmann for the ... 
Inverse Problems for Scalar Elliptic Equations and Systems
In this thesis, we discuss inverse boundary value problems for scalar equations and for systems. First we introduce the famous Calder\'on problem and its recent developments. We focus on deriving the stability estimate of ... 
An Inverse Source Problem in Radiative Transfer
(20130225)We consider the inverse source problem for the radiative transfer equation, under various assumptions on the scattering and absorption parameters, as well as on the accessible data. In such a setup, we measure the outgoing ... 
Inverse transport with angularly averaged measurements
(2008)The inverse problem in radiative transfer is considered. The measurement setup involves controlling incoming radiation at the boundary of a convex domain in Rn and measuring the outgoing radiation in an attempt to ... 
Isolated Curves for Hyperelliptic Curve Cryptography
(20130225)We introduce the notion of isolated genus two curves. There is no known efficient algorithm to explicitly construct isogenies between two genus two curves with large conductor gap. Thus there is no known way of transporting ... 
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 ...