Organizational Unit:
College of Sciences
College of Sciences
Permanent Link
Research Organization Registry ID
Description
Previous Names
Parent Organization
Parent Organization
Includes Organization(s)
Organizational Unit
Organizational Unit
Organizational Unit
Organizational Unit
Organizational Unit
ArchiveSpace Name Record
Publication Search Results
Now showing
1 - 10 of 52
-
ItemTrellis Decoding And Applications For Quantum Error Correction(Georgia Institute of Technology, 2022-08-08) Sabo, EricCompact, 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 extended to quantum error-correcting codes by Ollivier and Tillich in 2005. Here, we use their foundation to establish a practical decoder able to compute the maximum-likely error for any stabilizer code over a finite field of prime dimension. We define a canonical form for the stabilizer group and use it to classify the internal structure of the graph. Similarities and differences between the classical and quantum theories are discussed throughout. Numerical results are presented which match or outperform current state-of-the-art decoding techniques. New construction techniques for large trellises are developed and practical implementations discussed. We then define a dual trellis and use algebraic graph theory to solve the maximum-likely coset problem for any stabilizer code over a finite field of prime dimension at minimum added cost. Classical trellis theory makes occasional theoretical use of a graph product called the trellis product. We establish the relationship between the trellis product and the standard graph products and use it to provide a closed form expression for the resulting graph, allowing it to be used in practice. We explore its properties and classify all idempotents. The special structure of the trellis allows us to present a factorization procedure for the product, which is much simpler than that of the standard products. Finally, we turn to an algorithmic study of the trellis and explore what coding-theoretic information can be extracted assuming no other information about the code is available. In the process, we present a state-of-the-art algorithm for computing the minimum distance for any stabilizer code over a finite field of prime dimension. We also define a new weight enumerator for stabilizer codes over F_2 incorporating the phases of each stabilizer and provide a trellis-based algorithm to compute it.
-
ItemApplication of The Circle Method in Five Number Theory Problems(Georgia Institute of Technology, 2022-08-01) Mousavi, Seyyed HamedThis 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 $\binom{2n}{n}$ is relatively prime to $105 = 3\cdot 5 \cdot 7$? By Kummer's Theorem, this is the same to ask if there are infinitely many integers $n$, so that $n$ added to itself base $3$, $5$, or $7$, has no carries. A probabilistic heuristic of Pommerance predicts that there should be infinitely many such integers $n$. We establish a result of statistical nature supporting Pommerance's heuristic. The proof consists of the Fourier analysis method, as well as geometrically bypassing an old conjecture about the primes. In the second chapter, we discover an unexpected cancellation on the sums involving the exponential functions. Applying this theorem on the first terms of the Ramanujan-Hardy-Rademacher expansion for the partition function gives us a natural proof of a ``weak" pentagonal number theorem. We find several similar upper bounds for many different partition functions. Additionally, we prove another set of ``weak" pentagonal number theorems for the primes, which allows us to count the number of primes in certain intervals with small error. Finally, we show an approximate solution to the Prouhet-Tarry-Escott problem using a similar technique. The core of the proofs is an involved circle method argument. The third chapter of this thesis is about endpoint scale independent $\ell^p-$improving inequality for averages over the prime numbers. The primes are almost full-dimensional, hence one expects improving estimates for all $p >1$. Those are known, and relatively easy to establish. The endpoint estimates are far more involved however, engaging for instance Siegel zeros, in the unconditional case, and the Generalized Riemann Hypothesis (GRH) in the general case. Assuming GRH, we prove the sharpest possible bound up to a constant. Unconditionally, we prove the same inequality up to a logarithmic factor. The proof is based on a circle method argument, and utilizing smooth numbers to gain additional control of Ramanujan sums.
-
ItemNON-SEPARATING PATHS IN GRAPHS(Georgia Institute of Technology, 2022-08-01) Qian, YingjieWhen 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 removal results in a graph satisfying a certain connectivity constraint. There is a series of work on non-separating paths in graphs and their applications. For any graph G and distinct vertices a,b,c,d in V(G), we give a structural characterization for G not containing a path A from a to b and avoiding c and d such that removing A from G results in a 2-connected graph. Using this structure theorem, we construct a 7-connected such graph. We will also discuss potential applications to other problems, including the 3-linkage conjecture made by Thomassen in 1980. This is based on joint work with Shijie Xie and Xingxing Yu.
-
ItemMatching problems in hypergraphs(Georgia Institute of Technology, 2022-07-30) Yuan, XiaofanKühn, 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 choose 2}, then H contains a perfect matching. We show that for sufficiently large n divisible by 3, if F_1, ..., F_{n/3} are 3-uniform hypergraphs with a common vertex set and the minimum vertex degree in each F_i is greater than {(n-1) choose 2} - {2n/3 choose 2} for i = 1, ..., n/3, then the family {F_1, ..., F_{n/3}} admits a rainbow matching, i.e., a matching consisting of one edge from each F_i. This is done by converting the rainbow matching problem to a perfect matching problem in a special class of uniform hypergraphs. We also prove that, for any integers k, l with k >= 3 and k/2 < l <= k-1, there exists a positive real μ such that, for all sufficiently large integers m, n satisfying n/k - μn <= m <= n/k - 1 - (1 - l/k){ceil of (k - l)/(2l - k)}, if H is a k-uniform hypergraph on n vertices and the minimum l-degree of H is greater than {(n-l) choose (k-l)} - {(n-l-m) choose (k-l)}, then H has a matching of size m+1. This improves upon an earlier result of Hàn, Person, and Schacht for the range k/2 < l <= k-1. In many cases, our result gives tight bound on the minimum l-degree of H for near perfect matchings. For example, when l >= 2k/3, n ≡ r (mod k), 0 <= r < k, and r + l >= k, we can take m to be the minimum integer at least n/k - 2.
-
ItemAlgebraic and semi-algebraic invariants on quadrics(Georgia Institute of Technology, 2022-07-30) Jung, JaewooThis 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 combinatorially. We study the bounds on the algebraic invariant, Castelnuovo-Mumford regularity, of the quadratic ideals in terms of properties on the corresponding simple graphs. Our main theorem is the graph decomposition theorem that provides a bound on the regularity of a quadratic monomial ideal. By combining the main theorem with results in structural graph theory, we proved, improved, and generalized many of the known bounds on the regularity of square-free quadratic monomial ideals. The Hankel index of a real variety is a semi-algebraic invariant that quantifies the (structural) difference between nonnegative quadrics and sums of squares on the variety. This project is motivated by an intriguing (lower) bound of the Hankel index of a variety by an algebraic invariant, the Green-Lazarsfeld index, of the variety. We study the Hankel index of the image of the projection of rational normal curves away from a point. As a result, we found a new rank of the center of the projection which detects the Hankel index of the rational curves. It turns out that the rational curves are the first class of examples that the lower bound of the Hankel index by the Green-Lazarsfeld index is strict.
-
ItemShortest closed curve to inspect a sphere(Georgia Institute of Technology, 2022-07-30) Wenk, James F.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, arranged in the shape of a baseball seam, as conjectured by V. A. Zalgaller in 1996.
-
ItemContact geometric theory of Anosov flows in dimension three and related topics(Georgia Institute of Technology, 2022-07-20) Hozoori, SurenaThis 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 between various geometric, topological and dynamical features of such flows. After reviewing some basic elements from the theory of contact and symplectic structures in low dimensions, we discuss a characterization of Anosov flows on three dimensional manifolds, purely in terms of those geometric structure. This is based on the previous observations of Mitsumatsu and Eliashberg-Thurston in the mid 90s, and in the context of a larger class of dynamics, namely projectively Anosov flows. Our improvement of those observations, which have been left unexplored to a great extent in our view, facilitates employing new geometric tools to the study of questions about (projectively) Anosov flows and vice versa. We then discuss another characterization of Anosov three flows, in terms of the associated underlying Reeb dynamics. Beside the contact topological consequences of this result, it sheds light on contact geometric interpretation of the existence of an invariant volume form for these flows, a condition which is well known to have deep consequences in the dynamics of the flow from the viewpoint of the long term behavior of the flow (transitivity) and measure theory (ergodicity). The implications of these results on various related theories, namely, Liouville geometry, the theory of contact hyperbolas and bi-contact surgery, are discussed as well. As contact Anosov flows are an important and well studied special case of volume preserving Anosov flows, we also make new observation regarding these flows, utilizing the associated Conley-Zehnder indices of their periodic orbits, a classical tool from the field of contact dynamics. We finally discuss some Riemannian geometric motivations in the study of contact Anosov flows in dimension three. In particular, this bridges our study to the curvature properties of Riemannian structures, which are compatible with a given contact manifold. Our study of the curvature in this context goes beyond the study of Anosov dynamics, although has implications on the topic. In particular, we investigate a natural curvature realization for compatible Riemannian structures, namely Ricci-Reeb realization problem. The majority of the results in this manuscript, with the exception of some parts of Chapter 5, can be found in the author's previous papers.
-
ItemFactorization theorems and canonical representations for generating functions of special sums(Georgia Institute of Technology, 2022-07-19) Schmidt, Maxie DionThis 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 of the thesis explores the topic of how to best rigorously define a so-termed "canonically best" matrix based factorization for a given class of convolution sum sequences. The notion of a canonical factorization for the generating function of such sequences needs to match the qualitative properties we find in the factorization theorems for Lambert series generating functions (LGFs). The expected qualitatively most expressive expansion we find in the LGF case results naturally from algebraic constructions of the underlying LGF series type. We propose a precise quantitative requirement to generalize this notion in terms of optimal cross-correlation statistics for certain sequences that define the matrix based factorizations of the generating function expansions we study. We finally pose a few conjectures on the types of matrix factorizations we expect to find when we are able to attain the maximal (respectively minimal) correlation statistic for a given sum type.
-
ItemOn embeddings of 3-manifolds in symplectic 4-manifolds(Georgia Institute of Technology, 2022-07-13) Mukherjee, AnubhavWe 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 symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exists 3-manifolds which cannot smoothly embed in a way that appropriately respect orientations in any symplectic manifold with weakly convex boundary.
-
ItemLearning Dynamics from Data Using Optimal Transport Techniques and Applications(Georgia Institute of Technology, 2022-07-08) Ma, ShaojunOptimal 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 are gradually realizing that as the amount of data as well as the needs of utilizing data increase vastly, data-driven models have great potentials in real-world applications. In this thesis, we apply the theories of OT and design data-driven algorithms to form and compute various OT problems. We also build a framework to learn inverse OT problem. Furthermore, we develop OT and deep learning based models to solve stochastic differential equations, optimal control, mean field games related problems, all in data-driven settings. In Chapter 2, we provide necessary mathematical concepts and results that form the basis of this thesis. It contains brief surveys of optimal transport, stochastic differential equations, Fokker-Planck equations, deep learning, optimal controls and mean field games. Chapter 3 to Chapter 5 present several scalable algorithms to handle optimal transport problems within different settings. Specifically, Chapter 3 shows a new saddle scheme and learning strategy for computing the Wasserstein geodesic, as well as the Wasserstein distance and OT map between two probability distributions in high dimensions. We parametrize the map and Lagrange multipliers as neural networks. We demonstrate the performance of our algorithms through a series of experiments with both synthetic and realistic data. Chapter 4 presents a scalable algorithm for computing the Monge map between two probability distributions since computing the Monge maps remains challenging, in spite of the rapid developments of the numerical methods for optimal transport problems. Similarly, we formulate the problem as a mini-max problem and solve it via deep learning. The performance of our algorithms is demonstrated through a series of experiments with both synthetic and realistic data. In Chapter 5 we study OT problem in an inverse view, which we also call Inverse OT (IOT) problem. IOT also refers to the problem of learning the cost function for OT from observed transport plan or its samples. We derive an unconstrained convex optimization formulation of the inverse OT problem. We provide a comprehensive characterization of the properties of inverse OT, including uniqueness of solutions. We also develop two numerical algorithms, one is a fast matrix scaling method based on the Sinkhorn-Knopp algorithm for discrete OT, and the other one is a learning based algorithm that parameterizes the cost function as a deep neural network for continuous OT. Our numerical results demonstrate promising efficiency and accuracy advantages of the proposed algorithms over existing state-of-the-art methods. In Chapter 6 we propose a novel method using the weak form of Fokker Planck Equation (FPE) --- a partial differential equation --- to describe the density evolution of data in a sampled form, which is then combined with Wasserstein generative adversarial network (WGAN) in the training process. In such a sample-based framework we are able to learn the nonlinear dynamics from aggregate data without explicitly solving FPE. We demonstrate our approach in the context of a series of synthetic and real-world data sets. Chapter 7 introduces the application of OT and neural networks in optimal density control. Particularly, we parametrize the control strategy via neural networks, and provide an algorithm to learn the strategy that can drive samples following one distribution to new locations following target distribution. We demonstrate our method in both synthetic and realistic experiments, where we also consider perturbation fields. Finally Chapter 8 presents applications of mean field game in generative modeling and finance area. With more details, we build a GAN framework upon mean field game to generate desired distribution starting with white noise, we also investigate its connection to OT. Moreover, we apply mean field game theories to study the equilibrium trading price in stock markets, we demonstrate the theoretical result by conducting experiments on real trading data.