Browsing School of Mathematics Theses and Dissertations by Issue Date
Now showing items 1-20 of 440
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Trellis Decoding And Applications For Quantum Error Correction
(Georgia Institute of Technology, 2022-08-08)Compact, graphical representations of error-correcting codes called trellises are a crucial tool in classical coding theory, establishing both theoretical properties and performance metrics for practical use. The idea was ... -
NON-SEPARATING PATHS IN GRAPHS
(Georgia Institute of Technology, 2022-08-01)When developing a theory for 3-connected graphs, Tutte showed that for any 3-connected graph G and any three vertices a, b, c of G, G-c has an a-b path P such that G-P is connected. We call paths non-separating if their ... -
Application of The Circle Method in Five Number Theory Problems
(Georgia Institute of Technology, 2022-08-01)This thesis consists of three applications of the circle method in number theory problems. In the first chapter, we study a question of Graham. Are there infinitely many integers $n$ for which the central binomial coefficient ... -
Algebraic and semi-algebraic invariants on quadrics
(Georgia Institute of Technology, 2022-07-30)This dissertation consists of two topics concerning algebraic and semi-algebraic invariants on quadrics. The ranks of the minimal graded free resolution of square-free quadratic monomial ideals can be investigated ... -
Shortest closed curve to inspect a sphere
(Georgia Institute of Technology, 2022-07-30)We show that in Euclidean 3-space any closed curve γ which lies outside the unit sphere and contains the sphere within its convex hull has length ≥ 4π. Equality holds only when γ is composed of 4 semicircles of length π, ... -
Matching problems in hypergraphs
(Georgia Institute of Technology, 2022-07-30)Kühn, Osthus, and Treglown and, independently, Khan proved that if H is a 3-uniform hypergraph on n vertices, where n is a multiple of 3 and large, and the minimum vertex degree of H is greater than {(n-1) choose 2} - {2n/3 ... -
Contact geometric theory of Anosov flows in dimension three and related topics
(Georgia Institute of Technology, 2022-07-20)This thesis consists of the author's work on the contact and symplectic geometric theory of Anosov flows in low dimensions, as well as the related topics from Riemannian geometry. This includes the study of the interplay ... -
Factorization theorems and canonical representations for generating functions of special sums
(Georgia Institute of Technology, 2022-07-19)This manuscript explores many convolution (restricted summation) type sequences via certain types of matrix based factorizations that can be used to express their generating functions. The last primary (non-appendix) section ... -
On embeddings of 3-manifolds in symplectic 4-manifolds
(Georgia Institute of Technology, 2022-07-13)We proposed a conjecture that every 3-manifolds smoothly embedded in some closed symplectic 4-manifolds. This work shows that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed ... -
Learning Dynamics from Data Using Optimal Transport Techniques and Applications
(Georgia Institute of Technology, 2022-07-08)Optimal Transport has been studied widely in recent years, the concept of Wasserstein distance brings a lot of applications in computational mathematics, machine learning, engineering, even finance areas. Meanwhile, people ... -
Erdos-Posa theorems for undirected group-labelled graphs
(Georgia Institute of Technology, 2022-06-14)Erdős and Pósa proved in 1965 that cycles satisfy an approximate packing-covering duality. Finding analogous approximate dualities for other families of graphs has since become a highly active area of research due in part ... -
New Numerical And Computational Methods Leveraging Dynamical Systems Theory For Multi-Body Astrodynamics
(Georgia Institute of Technology, 2022-06-01)Many proposed interplanetary space missions, including Europa Lander and Dragonfly, involve trajectory design in environments where multiple large bodies exert gravitational influence on the spacecraft, such as the Jovian ... -
A Novel Delay Differential Equation Model of the Germinal Center Reaction and an Algorithm for Minimum Length Surveillance Paths
(Georgia Institute of Technology, 2022-05-19)The humoral adaptive immune system in vertebrates includes a process called the germinal center reaction in which B-cells rapidly increase their binding affinity to an antigen that is part of a pathogen. A fraction of ... -
Invariance principle of random matrix
(Georgia Institute of Technology, 2022-05-16)Random matrix has been found useful in many real-world applications. The celebrated Johnson-Lindenstrauss lemma states that certain geometric structure of deterministic vectors is preserved when projecting high dimensional ... -
On difference graph covers and the local dimension of the Boolean lattice
(Georgia Institute of Technology, 2022-05-03)N/A -
CAPILLARY GRAVITY WATER WAVE LINEARIZED AT MONOTONE SHEAR FLOWS: EIGENVALUE AND INVISCID DAMPING
(Georgia Institute of Technology, 2022-05-03)This work is concerned with the two dimensional capillary gravity water waves of finite depth $x_2 \in (-h, 0)$ linearized at a uniformly monotonic shear flow $U(x_2)$. We focus on the eigenvalue distribution and linear ... -
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 ... -
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 ... -
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 ...