School of Mathematics Theses and Dissertations
Original work in partial fulfillment of the requirements for a graduate degree from the School of Mathematics.
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Approximate Schauder Frames for Banach Sequence Spaces
(Georgia Institute of Technology, 20210429)The main topics of this thesis concern two types of approximate Schauder frames for the Banach sequence space l_1^n. The first main topic pertains to finiteunit norm tight frames (FUNTFs) for the finitedimensional real ... 
Mathematical and Datadriven Pattern Representation with Applications in Image Processing, Computer Graphics, and Infinite Dimensional Dynamical Data Mining
(Georgia Institute of Technology, 20210430)Patterns represent the spatial or temporal regularities intrinsic to various phenomena in nature, society, art, and science. From rigid ones with welldefined generative rules to flexible ones implied by unstructured data, ... 
ON SCALABLE AND FAST LANGEVINDYNAMICSBASED SAMPLING ALGORITHMS
(Georgia Institute of Technology, 20210428)Langevin dynamicsbased sampling algorithms are arguably among the most widelyused Markov Chain Monte Carlo (MCMC) methods. Two main directions of the modern study of MCMC methods are (i) How to scale MCMC methods to big ... 
On the stationary and uniformlyrotating solutions of active scalar equations
(Georgia Institute of Technology, 20210413)In this thesis, we study qualitative and quantitative properties of stationary/uniformly rotating solutions of the 2D incompressible Euler equation and the generalized Surface Quasi Geostrophic (SQG) equations. The main ... 
Numerical Estimation of Several Topological Quantities of the First Passage Percolation Model
(Georgia Institute of Technology, 20210413)In this thesis, our main goal is to use numerical simulations to study some quantities related to the growing set B(t). Motivated by prior works, we mainly study quantities including the boundary size, the hole size, and ... 
Interaction energies, lattices, and designs
(Georgia Institute of Technology, 20200521)This thesis has four chapters. The ﬁrst three concern the location of mass on spheres or projective space, to minimize energies. For the Columb potential on the unit sphere, this is a classical problem, related to arranging ... 
RayleighTaylor instability with heat transfer
(Georgia Institute of Technology, 20200513)In this thesis, the RayleighTaylor instability effect with heat transfer in the setting of the NavierStokes equations, given threedimensional and incompressible fluids, is investigated. Under suitable initial and boundary ... 
Legendrian large cables and nonuniformly thick knots
We define the notion of a knot type having Legendrian large cables and show that having this property implies that the knot type is not uniformly thick. Moreover, there are solid tori in this knot type that do not thicken ... 
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 ... 
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 ... 
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 ... 
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 ... 
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 ... 
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 ... 
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 ... 
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 ... 
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 ...