Browsing School of Mathematics Theses and Dissertations by Issue Date
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Numerical Estimates for Arm Exponents and the Acceptance Profile in TwoDimensional Invasion Percolation
(Georgia Institute of Technology, 20200505)The main object of this thesis is to numerically estimate some conjectured arm exponents when there exist a number of open paths and closed dual paths that extend to the boundary of different sizes of boxes centering at ... 
Finding and certifying numerical roots of systems of equations
(Georgia Institute of Technology, 20200501)Numerical algebraic geometry studies methods to approach problems in algebraic geometry numerically. Especially, finding roots of systems of equations using theory in algebraic geometry involves symbolic algorithm which ... 
Small torsion generating sets for mapping class groups
(Georgia Institute of Technology, 20200427)A surface of genus g has many symmetries. These form the surface’s mapping class group Mod(S_g), which is finitely generated. The most commonly used generating sets for Mod(S_g) are comprised of infinite order elements ... 
The MaxwellPauli Equations
(Georgia Institute of Technology, 20200319)We study the quantum mechanical manybody problem of N ≥ 1 nonrelativistic electrons with spin interacting with their selfgenerated classical electromagnetic field and K ≥ 0 static nuclei. We model the dynamics of the ... 
Randomness as a tool for modeling and uncovering structure
(Georgia Institute of Technology, 20200312)This thesis contains four main research directions, united by the themes of using randomness to (i) construct structure and (ii) uncover structure. Randomness has long been used for these tasks. Random models are defined ... 
6connected graphs are twothree linked
(Georgia Institute of Technology, 20191111)Let $G$ be a graph and $a_0, a_1, a_2, b_1,$ and $b_2$ be distinct vertices of $G$. Motivated by their work on Four Color Theorem, Hadwiger's conjecture for $K_6$, and J\o rgensen's conjecture, Robertson and Seymour asked ... 
The proxy point method for rankstructured matrices
(Georgia Institute of Technology, 20191106)Rankstructured matrix representations, e.g., $\mathcal{H}^2$ and HSS, are commonly used to reduce computation and storage cost for dense matrices defined by interactions between many bodies. The main bottleneck for their ... 
TOPICS ON THE LENGTH OF THE LONGEST COMMON SUBSEQUENCES, WITH BLOCKS, IN BINARY RANDOM WORDS
(Georgia Institute of Technology, 20190827)The study of LIn, the length of the longest increasing subsequences, and of LCIn, the length of the longest common and increasing subsequences in random words is classical in computer science and bioinformatics, and has ... 
Quantum torus methods for Kauffman bracket skein modules
(Georgia Institute of Technology, 20190822)We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to ... 
Lattice points, oriented matroids, and zonotopes
(Georgia Institute of Technology, 20190726)The first half of this dissertation concerns the following problem: Given a lattice in R^d which refines the integer lattice Z^d, what can be said about the distribution of the lattice points inside of the halfopen unit ... 
The polaron hydrogenic atom in a strong magnetic field
(Georgia Institute of Technology, 20190719)It is shown that: (1) The groundstate electron density of a polaron bound in a Coulomb potential and exposed to a homogeneous magnetic field of strength B–with its transverse electron coordinates integrated out and when ... 
On the independent spanning tree conjectures and related problems
(Georgia Institute of Technology, 20190717)We say that trees with common root are (edge)independent if, for any vertex in their intersection, the paths to the root induced by each tree are internally (edge)disjoint. The relationship between graph (edge)connectivity ... 
On a classical solution to the master equation of a first order mean field game
(Georgia Institute of Technology, 20190711)For a first order (deterministic) meanfield game with nonlocal couplings, a classical solution is constructed for the associated, socalled master equation, a partial differential equation in infinite dimensional space ... 
Topics in dynamical systems
(Georgia Institute of Technology, 20190614)The thesis consists of two parts. the first one is dealing with isosspectral transformations and the second one with the phenomenon of local immunodeficiency. Isospectral transformations (IT) of matrices and networks allow ... 
The applications of discrete optimal transport in path planning and data clustering
(Georgia Institute of Technology, 20190515)Optimal transport introduces the concept of Wasserstein distance, which has been widely used in various applications in computational mathematics, machine learning as well as many areas in engineering. Meanwhile, control ... 
Percolation theory: The complement of the infinite cluster & the acceptance profile of the invasion percolation
(Georgia Institute of Technology, 20190507)In independent bond percolation with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite cluster? GrimmettHolroydKozma ... 
Textclassification methods and the mathematical theory of Principal Components
(Georgia Institute of Technology, 20190422)This thesis studies three topics. First of all, in text classification, one may use Principal Components Analysis (PCA) as a dimension reduction technique, or with few topics even as unsupervised classification method. We ... 
Coloring graphs with no k5subdivision: disjoint paths in graphs
(Georgia Institute of Technology, 20190327)The Four Color Theorem states that every planar graph is 4colorable. Hajos conjectured that for any positive integer k, every graph containing no K_{k+1}subdivision is kcolorable. However, Catlin disproved Hajos conjecture ... 
Comparison of sequences generated by a hidden Markov model
(Georgia Institute of Technology, 20190326)The length $LC_n$ of the longest common subsequences of two strings $X = (X_1, \ldots, X_n)$ and $Y = (Y_1, \ldots, Y_n)$ is way to measure the similarity between $X$ and $Y$. We study the asymptotic behavior of $LC_n$ ... 
Topics on the longest common subsequences: Simulations, computations, and variance
(Georgia Institute of Technology, 20181107)The study of the longest common subsequences (LCSs) of two random words/strings is classical in computer science and bioinformatics. A problem of particular probabilistic interest is to determine the limiting behavior of ...