Browsing School of Mathematics Theses and Dissertations by Issue Date
Now showing items 1-20 of 424
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On difference graph covers and the local dimension of the Boolean lattice
(Georgia Institute of Technology, 2022-05-03)N/A -
Duality Between Deep Learning And Algorithm Design
(Georgia Institute of Technology, 2022-05-02)This thesis introduces 'Duality Between Deep Learning And Algorithm Design'. Deep learning is a data-driven method, whereas conventional algorithm design is a knowledge-driven method. Based on their connections and ... -
Dual representations of polynomial modules with applications to partial differential equations
(Georgia Institute of Technology, 2022-05-02)In 1939, Wolfgang Gröbner proposed using differential operators to represent ideals in a polynomial ring. Using Macaulay inverse systems, he showed a one-to-one correspondence between primary ideals whose variety is a ... -
Numerical computation and analysis related to optimal transport theory
(Georgia Institute of Technology, 2022-04-28)In this thesis we apply the optimal transport (OT) theory to various disciplines of applied and computational mathematics such as scientific computing, numerical analysis, and dynamical systems. The research consists of ... -
Heat kernel and geometry of metric measure spaces with Ricci curvature lower bounds
(Georgia Institute of Technology, 2022-04-28)The thesis is a study of geometric properties of non-collapsed metric measure spaces with Ricci curvature lower bounds. We establish some characterizations of non-collapsed spaces and as a consequence, solve the De ... -
A Self-limiting Hawkes Process: Interpretation, Estimation, and Use in Modeling
(Georgia Institute of Technology, 2022-04-28)Many real life processes that we would like to model have a self-exciting property, i.e. the occurrence of one event causes a temporary spike in the probability of other events occurring nearby in space and time. Examples ... -
The Extremal Function for K10 Minors
(Georgia Institute of Technology, 2021-12-15)We prove that every graph on n >= 8 vertices and at least 8n-35 edges either has a K10 minor or is isomorphic to some graph included in a few families of exceptional graphs. -
Decay of Entropy and Information for multidimensional Kac models
(Georgia Institute of Technology, 2021-12-14)We study the approach to equilibrium in relative entropy of systems of gas particles modeled via the Kac master equation in arbitrary dimensions. First, we study the Kac system coupled to a thermostat, and secondly connected ... -
Domains of analyticity and Gevrey estimates of tori in weakly dissipative systems
(Georgia Institute of Technology, 2021-12-07)We consider the problem of following quasi-periodic tori in perturbations of Hamiltonian systems which involve friction and external forcing. In the first part, we study a family of dissipative standard maps of the cylinder ... -
Classification fo tight contact structures on the Weeks manifold
(Georgia Institute of Technology, 2021-08-18)One of important problems in 3-dimensional contact geometry is to figure out which 3-manifolds admit tight contact structures and classify tight contact structures on the manifolds which admit tight contact structures. It ... -
A combinatorial approach to biological structures and networks in predictive medicine
(Georgia Institute of Technology, 2021-08-09)This work concerns the study of combinatorial models for biological structures and networks as motivated by questions in predictive medicine. Through multiple examples, the power of combinatorial models to simplify problems ... -
Fast Algorithm for Invariant Circle and their Stable Manifolds: Rigorous Results and Efficient Implementations
(Georgia Institute of Technology, 2021-08-04)In this thesis, we present, analyze, and implement a quadratically convergent algorithm to compute the invariant circle and the foliation by stable manifolds for 2-dimensional maps. The 2-dimensional maps we are considering ... -
Persistence of Invariant Objects under Delay Perturbations
(Georgia Institute of Technology, 2021-07-30)In this dissertation, we investigate functional differential equations which come from adding delay-related perturbations to ODEs or evolutionary PDEs. Despite the singular nature of the perturbations, when the perturbations ... -
Multiscale Problems in Mechanics: Spin Dynamics, Structure-Preserving Integration, and Data-Driven Methods
(Georgia Institute of Technology, 2021-07-28)This thesis focuses on analyzing the physics and designing multiscale methods for nonlinear dynamics in mechanical systems, such as those in astronomy. The planetary systems (e.g., the Solar System) are of great interest ... -
Algorithmic approaches to problems in probabilistic combinatorics
(Georgia Institute of Technology, 2021-07-27)The probabilistic method is one of the most powerful tools in combinatorics; it has been used to show the existence of many hard-to-construct objects with exciting properties. It also attracts broad interests in designing ... -
Polyhedral and tropical geometry in nonlinear algebra
(Georgia Institute of Technology, 2021-07-26)This dissertation consists of four chapters on various topics in nonlinear algebra. Particularly, it focuses on solving algebraic problems and polynomial systems through the use of combinatorial tools. Chapter one gives a ... -
Applications of monodromy in solving polynomial systems
(Georgia Institute of Technology, 2021-07-14)Polynomial systems of equations that occur in applications frequently have a special structure. Part of that structure can be captured by an associated Galois/monodromy group. This makes numerical homotopy continuation ... -
Physical Billiards and Open Dynamical Systems
(Georgia Institute of Technology, 2021-07-14)This thesis consists of four works in dynamical systems with a focus on billiards. In the first part, we consider open dynamical systems, where there exists at least a hole of positive measure in the phase space which some ... -
Mathematical and Data-driven Pattern Representation with Applications in Image Processing, Computer Graphics, and Infinite Dimensional Dynamical Data Mining
(Georgia Institute of Technology, 2021-04-30)Patterns represent the spatial or temporal regularities intrinsic to various phenomena in nature, society, art, and science. From rigid ones with well-defined generative rules to flexible ones implied by unstructured data, ... -
Approximate Schauder Frames for Banach Sequence Spaces
(Georgia Institute of Technology, 2021-04-29)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 finite-unit norm tight frames (FUNTFs) for the finite-dimensional real ...