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School of Mathematics

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College of Sciences

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https://finding-aids.library.gatech.edu/agents/corporate_entities/245

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Publication Search Results

Now showing 1 - 10 of 107

6-connected graphs are two-three linked

(Georgia Institute of Technology, 2019-11-11) Xie, Shijie

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 when does $G$ contain disjoint connected subgraphs $G_1, G_2$, such that $\{a_0, a_1, a_2\}\subseteq V(G_1)$ and $\{b_1, b_2\}\subseteq V(G_2)$. We prove that if $G$ is 6-connected then such $G_1,G_2$ exist. Joint work with Robin Thomas and Xingxing Yu.
The proxy point method for rank-structured matrices

(Georgia Institute of Technology, 2019-11-06) Xing, Xin

Rank-structured 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 application is the expensive computation required to represent a matrix in a rank-structured matrix format which involves compressing specific matrix blocks into low-rank form. This dissertation is mainly about the study and application of a hybrid analytic-algebraic compression method, called \textit{the proxy point method}. This work uncovers the full strength of this presently underutilized method that could potentially resolve the above bottleneck for all rank-structured matrix techniques. As a result, this work could extend the applicability and improve the performance of rank-structured matrix techniques and thus facilitate dense matrix computations in a wider range of scientific computing problems, such as particle simulations, numerical solution of integral equations, and Gaussian processes. Application of the proxy point method in practice is presently very limited. Only two special instances of the method have been used heuristically to compress interaction blocks defined by specific kernel functions over points. We address several critical problems of the proxy point method which limit its applicability. A general form of the method is then proposed, paving the way for its wider application in the construction of different rank-structured matrix representations with kernel functions that are more general than those usually used. In addition to kernel-defined interactions between points, we further extend the applicability of the proxy point method to compress the interactions between charge distributions in quantum chemistry calculations. Specifically, we propose a variant of the proxy point method to efficiently construct an $\mathcal{H}^2$ matrix representation of the four-dimensional electron repulsion integral tensor. The linear-scaling matrix-vector multiplication algorithm for the constructed $\mathcal{H}^2$ matrix is then used for fast Coulomb matrix construction which is an important step in many quantum chemical methods. Two additional contributions related to $\mathcal{H}^2$ and HSS matrices are also presented. First, we explain the exact equivalence between $\mathcal{H}^2$ matrices and the fast multipole method (FMM). This equivalence has not been rigorously studied in the literature. Numerical comparisons between FMM and $\mathcal{H}^2$ matrices based on the proxy point method are also provided, showing the relative advantages and disadvantages of the two methods. Second, we consider the application of HSS approximations as preconditioners for symmetric positive definite (SPD) matrices. Preserving positive definiteness is essential for rank-structured matrix approximations to be used efficiently in various algorithms and applications, e.g., the preconditioned conjugate gradient method. We propose two methods for constructing HSS approximations to an SPD matrix that preserve positive definiteness.
Topics on the length of the longest common subsequences, with blocks, in binary random words

(Georgia Institute of Technology, 2019-08-27) Zhang, Yuze

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 been well explored over the last few decades. This dissertation studies a generalization of LCIn for two binary random words, namely, it analyzes the asymptotic behavior of LCbBn, the length of the longest common subsequences containing a fixed number, b, of blocks. We first prove that after proper centerings and scalings, LCbBn, for two sequences of i.i.d. Bernoulli random variables with possibly two different parameters, converges in law towards limits we identify. This dissertation also includes an alternative approach to the one-sequence LbBn problem, and Monte-Carlo simulations on the asymptotics of LCbBn and on the growth order of the limiting functional, as well as several extensions of the LCbBn problem to the Markov context and some connection with percolation theory.
Quantum torus methods for Kauffman bracket skein modules

(Georgia Institute of Technology, 2019-08-22) Paprocki, Jonathan Michael

We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3-manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to quantum Teichmuller space, or filtering the algebra and obtaining an associated graded algebra that is a monomial subalgebra of a quantum torus. We utilize the former method to generalize the Chebyshev homomorphism of Bonahon and Wong between skein algebras of surfaces to a Chebyshev-Frobenius homomorphism between skein modules of marked 3-manifolds, in the course of which we define a surgery theory, and whose image we show is either transparent or (skew)-transparent. The latter method is used to show that skein algebras of surfaces are maximal orders, which implies a refined unicity theorem, shows that SL2C-character varieties are normal, and suggests a conjecture on how this result may be utilized for topological quantum compiling.
Lattice points, oriented matroids, and zonotopes

(Georgia Institute of Technology, 2019-07-26) Celaya, Marcel Luis

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 half-open unit cube [0,1)^d? This question is of interest in discrete geometry, especially integral polytopes and Ehrhart theory. We observe a combinatorial description of the linear span of these points, and give a formula for the dimension of this span. The proofs of these results use methods from classical multiplicative number theory. In the second half of the dissertation, we investigate oriented matroids from the point of view of tropical geometry. Given an oriented matroid, we describe, in detail, a polyhedral complex which plays the role of the Bergman complex for ordinary matroids. We show how this complex can be used to give a new proof of the celebrated Bohne-Dress theorem on tilings of zonotopes by zonotopes with an approach which relies on a novel interpretation of the chirotope of an oriented matroid.
The polaron hydrogenic atom in a strong magnetic field

(Georgia Institute of Technology, 2019-07-19) Ghanta, Rohan

It is shown that: (1) The ground-state 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 scaled appropriately with the magnetic field strength–converges pointwise and in a weak sense as B → ∞ to the square of a hyperbolic secant function; (2) The ground state of a polaron bound in a symmetric Mexican hat- type potential, scaled appropriately with the electron-phonon coupling parameter, is unique and therefore rotation-invariant, but the minimizers of the corresponding Pekar problem are nonradial; in the strong-coupling limit under the assumption that these minimizers are unique up to rotation the ground-state electron density–when scaled appropriately with the electron-phonon coupling strength–converges in a weak sense to a rotational average of their densities.
On the independent spanning tree conjectures and related problems

(Georgia Institute of Technology, 2019-07-17) Hoyer, Alexander

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 and the existence of (edge-)independent spanning trees is explored. The (Edge-)Independent Spanning Tree Conjecture states that every k-(edge-)connected graph has k-(edge-)independent spanning trees with arbitrary root. We prove the case k = 4 of the Edge-Independent Spanning Tree Conjecture using a graph decomposition similar to an ear decomposition, and give polynomial-time algorithms to construct the decomposition and the trees. We provide alternate geometric proofs for the cases k = 3 of both the Independent Spanning Tree Conjecture and Edge-Independent Spanning Tree Conjecture by embedding the vertices or edges in a 2-simplex, and conjecture higher-dimension generalizations. We provide a partial result towards a generalization of the Independent Spanning Tree Conjecture, in which local connectivity between the root and a vertex set S implies the existence of trees whose independence properties hold only in S. Finally, we prove and generalize a theorem of Györi and Lovász on partitioning a k-connected graph, and give polynomial-time algorithms for the cases k = 2, 3, 4 using the graph decompositions used to prove the corresponding cases of the Independent Spanning Tree Conjecture.
On a classical solution to the master equation of a first order mean field game

(Georgia Institute of Technology, 2019-07-11) Mayorga Tatarin, Sergio

For a first order (deterministic) mean-field game with nonlocal couplings, a classical solution is constructed for the associated, so-called master equation, a partial differential equation in infinite- dimensional space with a nonlocal term, assuming the time horizon is sufficiently small and the coefficients are smooth enough, without convexity conditions on the individual Hamiltonian. The couplings, albeit smooth, are not assumed to derive from a potential, which makes the result currently the most general one for short time horizon. The approach to obtain the master equation is inspired by that of Gangbo and Święch [GŚ15] for the problem in which the Hamiltonian is quadratic and the couplings derive from a potential, but we use a non-variational method and require further results of the calculus on the Wasserstein space that has been advanced recently by Gangbo et. al. [GC17; Gan18].
Topics in dynamical systems

(Georgia Institute of Technology, 2019-06-14) Shu, Longmei

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 for compression of either object while keeping all the information about their eigenvalues and eigenvectors. Chapter 1 analyzes what happens to generalized eigenvectors under isospectral transformations and to what extent the initial network can be reconstructed from its compressed image under IT. We also generalize and essentially simplify the proof that eigenvectors are invariant under isospectral transformations and generalize and clarify the notion of spectral equivalence of networks. In the recently developed theory of isospectral transformations of networks isospectral compressions are performed with respect to some chosen characteristics (attributes) of the network's nodes (edges). Each isospectral compression (when a certain characteristic is fixed) defines a dynamical system on the space of all networks. Chapter 2 shows that any orbit of this dynamical system which starts at any finite network (as the initial point of this orbit) converges to an attractor. This attractor is a smaller network where the chosen characteristic has the same value for all nodes (or edges). We demonstrate that isospectral compressions of one and the same network defined by different characteristics of nodes (or edges) may converge to the same as well as to different attractors. It is also shown that a collection of networks may be spectrally equivalent with respect to some network characteristic but nonequivalent with respect to another. These results suggest a new constructive approach which allows to analyze and compare the topologies of different networks. Some basic aspects of the recently discovered phenomenon of local immunodeficiency generated by antigenic cooperation in cross-immunoreactivity (CR) networks are investigated in chapter 3. We prove that stable with respect to perturbations local immunodeficiency (LI) already occurs in very small networks and under general conditions on their parameters. Therefore our results are applicable not only to Hepatitis C where CR networks are known to be large, but also to other diseases with CR. A major necessary feature of such networks is the non-homogeneity of their topology. It is also shown that one can construct larger CR networks with stable LI by using small networks with stable LI as their building blocks. Our results imply that stable LI occurs in networks with quite general topologies. In particular, the scale-free property of a CR network, assumed previously, is not required.
The applications of discrete optimal transport in path planning and data clustering

(Georgia Institute of Technology, 2019-05-15) Zhai, Haoyan

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 theory and path planning is an active branch in mathematics and robotics, focusing on algorithms that calculate feasible or optimal paths for robotic systems. In this thesis, we use the properties of the gradient flows induced by Wasserstein-2 metric to design algorithms to handle different types of path planning and control problems. Also, we define the Wasserstein K-means problems on graphs and propose an efficient algorithm to solve it. First of all, we provide an algorithm to handle the path planning problem in unknown environments. We develop a deterministic approach with finite-step convergence guarantee. Also, there is a theoretical relation between this algorithm and the Fokker-Planck equations, which bounds the searching region of the algorithm. We use numerical examples to show the efficiency of the algorithm as well as to support the theoretical results. Then, we generalize the algorithm to solve the general control problem in the unknown environments and similar convergence results can be proven. Besides, there is an evidence that the algorithm is guided by the evolution of Fokker-Planck equation, and we use experiments to demonstrate our theorems. We move on to study the optimal path planning in flow field. In this case, the objective function, the traveling time or kinetic energy, is to be minimized with a given flow field. Following the idea of method of evolving junctions, we first transform the original infinite dimensional optimal control into a finite dimensional global optimization problem by introducing junctions located only on the discontinuity positions of the dynamics. To handle the global optimization, intermittent diffusion is used here to guarantee the completeness of the method. At last, we define the discrete Wasserstein-(1,p) distance that depends on the graph structure. With this distance function, we further propose the Wasserstein K-means problem on a general graph and provide an algorithm in the framework of Lloyd method. The key part of the algorithm is the calculation of discrete Wasserstein-(1,p) distance and the gradient flow induced by Wasserstein-2 metric to solve an optimization with objective function being a linear combination of Wasserstein-(1.p) distance. Examples and simulation results are provided in the thesis.