Browsing Mathematics by Title
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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 ... 
Estimating Norms of Matrix Functions using Numerical Ranges
(20131114)We study Crouzeix's conjecture: for any polynomial p and any square matrix A, the spectral norm of the matrix p(A) is at most double of the supremum norm of the polynomial p on the numerical range of the matrix A. 
FiniteDifference Methods for SecondOrder Wave Equations with Reduced Dispersion Errors
Finite Difference (FD) schemes have been used widely in computing approximations for partial differential equations for wave propagation, as they are simple, flexible and robust. However, even for stable and accurate ... 
FiniteDifference Methods for the Wave Equation with Reduced Dispersion Errors
(20130417)A new methodology was proposed in Finkelstein and Kastner (2007,2008) to derive finitedifference (FD) schemes in the joint timespace domain to reduce dispersion error. The key idea is that the true dispersion relation ... 
Four Problems in Probability and Optimization
(20140224)This thesis studies bootstrap percolation, a problem in probability, as well as several topics in the application of sums of squares to combinatorial optimization. In the chapter on percolation, we bound the critical ... 
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 ... 
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 ... 
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 ...