Mathematics Commons^™

Near-Optimal Learning Of Tree-Structured Distributions By Chow-Liu, Arnab Bhattacharyya, Sutanu Gayen, Eric Price, N. V. Vinodchandran

Department of Mathematics: Faculty Publications

We provide finite sample guarantees for the classical Chow-Liu algorithm (IEEE Trans. Inform. Theory, 1968) to learn a tree-structured graphical model of a distribution. For a distribution P on Σⁿ and a tree T on n nodes, we say T is an ε-approximate tree for P if there is a T-structured distribution Q such that D(P || Q) is at most ε more than the best possible tree-structured distribution for P. We show that if P itself is tree-structured, then the Chow-Liu algorithm with the plug-in estimator for mutual information with eO (|Σ| …

Algorithms Related To Triangle Groups, Bao The Pham

LSU Doctoral Dissertations

Given a finite index subgroup of $\PSL_2(\Z)$, one can talk about the different properties of this subgroup. These properties have been studied extensively in an attempt to classify these subgroups. Tim Hsu created an algorithm to determine whether a subgroup is a congruence subgroup by using permutations \cite{hsu}. Lang, Lim, and Tan also created an algorithm to determine if a subgroup is a congruence subgroup by using Farey Symbols \cite{llt}. Sebbar classified torsion-free congruence subgroups of genus 0 \cite{sebbar}. Pauli and Cummins computed and tabulated all congruence subgroups of genus less than 24 \cite{ps}. However, there are still some problems …

Towards The Homotopy Type Of The Morse Complex, Connor Donovan

Mathematics Summer Fellows

Mathematicians have long been interested in studying the properties of simplicial complexes. In 1998, Robin Forman developed gradient vector fields as a tool to study these complexes. Having gradient vector fields to study these simplicial complexes, in 2005, Chari and Joswig discovered the Morse complex, a complex consisting of all gradient vector fields on a fixed complex. Although the Morse complex has been studied since 2005, there is little information regarding its homotopy type for different simplicial complexes. Pursuing our curiosity of the topic, we extend a result by Ayala et. al., stating that the pure Morse complex of a …

Positive Solutions For The Fractional Schrödinger Equations With Logarithmic And Critical Non-Linearities, Haining Fan, Zhaosheng Feng, Xingjie Yan

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we study a class of fractional Schrödinger equations involving logarithmic and critical non-linearities on an unbounded domain, and show that such an equation with positive or sign-changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik–Schnirelmann category, we investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets related to the weight potential.

An Analysis Of Capillary Flow In Finite Length Interior Corners, Samuel Shaw Mohler

Dissertations and Theses

We analyze the mathematical robustness of slow massively parallel interior corner flows in low gravity environments. An interior corner provides a preferential orientation in low gravity environments. This is a luxury usually only found on earth. It also provides a passive pumping mechanism due to geometry of a conduit. The driving force for this flow is a pressure difference due to local surface curvature gradients. An alternative reasoning is that due to the geometrical constraints the interior corner surface energy is unbounded below. This results in the liquid wicking into corners indefinitely. Interior corner flow's main quantity of interest is …

Odd Annular Bar-Natan Category And Gl(1|1), Casey Necheles

Dissertations - ALL

We introduce two monoidal supercategories: the odd dotted Temperley-Lieb category TLo,•(δ), which is a generalization of the odd Temperley-Lieb category studied by Brun-dan and Ellis in [5], and the odd annular Bar-Natan category BNo(A), which generalizes the odd Bar-Natan category studied by Putyra in [16]. We then show there is an equivalence of categories between them if δ=0. We use this equivalence to better understand the action of the Lie superalgebra gl(1|1) on the odd Khovanov homology of a knot in a thickened annulus found by Grigsby and Wehrli in [7].

Regularity Of The D-Bar-Neumann Operator, The Bergman Projection And The Canonical Solution Operator Of D-Bar-Equation, Muzhi Jin

Dissertations - ALL

We begin this thesis by a brief introduction to the $\bar{\partial}$-problem in several complex variables and the classical $L^2$ Theorem of $\bar{\partial}$. We then introduce the $\bar{\partial}$-Neumann operator on pseudoconvex domains and give a description of the relation between the $\bar{\partial}$-Neumann operator, the canonical solution operator, the Bergman projection and the Hankel operator. Meanwhile, the study of these operators is deeply related to the regularity of the $\bar{\partial}$-problem. In this thesis, we focus on the regularity, compactness and boundedness of these operators.

In chapter 2, we study the weakly pseudoconvex points on the boundary of a class of Hartogs domains. …

On Communication For Distributed Babai Point Computation, Maiara F. Bollauf, Vinay A. Vaishampayan, Sueli I.R. Costa

Publications and Research

We present a communication-efficient distributed protocol for computing the Babai point, an approximate nearest point for a random vector X∈Rn in a given lattice. We show that the protocol is optimal in the sense that it minimizes the sum rate when the components of X are mutually independent. We then investigate the error probability, i.e. the probability that the Babai point does not coincide with the nearest lattice point, motivated by the fact that for some cases, a distributed algorithm for finding the Babai point is sufficient for finding the nearest lattice point itself. Two different probability models for X …

Multiple Baseline Interrupted Time Series: Describing Changes In New Mexico Medicaid Behavioral Health Home Patients’ Care, Jessica Reno

Mathematics & Statistics ETDs

In 2016, the CareLink New Mexico behavioral health homes program began enrolling Medicaid recipients with the goal of increasing care coordination, improving access to services, and decreasing long-term costs of care for adults with serious mental illness (SMI) and children with severe emotional disturbance (SED). To evaluate these aims, a retrospective interrupted time series study using Medicaid claims data was designed. First, a comparable subset of non-enrolled individuals was selected from the pool of Medicaid recipients with SMI or SED using propensity score matching. Then, segmented regression was applied to three outcomes: total Medicaid charges, number of outpatient behavioral health …

Optimal Transport Driven Bayesian Inversion With Application To Signal Processing, Elijah F. Perez

Mathematics & Statistics ETDs

This paper will outline a Debiased Sinkhorn Divergence driven Bayesian inversion framework. Conventionally, a Gaussian Driven Bayesian framework is used when performing Bayesian inversion. A major issue with this Gaussian framework is that the Gaussian likelihood, driven by the L2 norm, is not affected by phase shift in a given signal. This issue has been addressed in [1] using a Wasserstein framework. However, the Wasserstein framework still has an issue because it assumes statistical independence when multidimensional signals are analyzed. This assumption of statistical independence cannot always be made when analyzing signals where multiple detectors are recording one event, say …

Graph-Theoretic Partitioning Of Rnas And Classification Of Pseudoknots-Ii, Louis Petingi

Publications and Research

Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In previous works, a linear-time algorithm was introduced to partition dual graphs into maximally connected components called blocks and determine whether each block contains a pseudoknot or not. As pseudoknots can not be contained into two different blocks, this characterization allow us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Moreover we have extended the partitioning algorithm by classifying a pseudoknot as either recursive or non-recursive in order to continue with our research …

The Unit Generalized Log Burr Xii Distribution: Properties And Application, Fiaz Ahmad Bhatti, Azeem Ali, Gholamhossein Hamedani, Mustafa Ç. Korkmaz, Munir Ahmad

Mathematical and Statistical Science Faculty Research and Publications

In this paper, a three-parameter bounded unit distribution with a flexible hazard rate called the unit generalized log Burr XII (UGLBXII) distribution is derived. To show the importance of the proposed distribution, we establish some of its mathematical properties such as random number generator, ordinary moments, generalized TL moments, conditional moments, reliability and uncertainty measures. We characterize the UGLBXII distribution via innovative techniques. We also present the bivariate‐ and multivariate‐type distributions via Morgenstern (Mor) family and via Clayton family. Six estimation methods such as the maximum likelihood, maximum product spacings, least squares, weighted least squares, Cramer-von Mises and Anderson-Darling methods …

Terrorism And Counter-Terrorism: A Game Theoretic Approach., Aditya Bhan Dr.

Doctoral Theses

Microeconomic concepts are often applied to analyze the phenomenon of terrorism, and gametheoretic frameworks are commonly applied throughout the existing literature. This is unsurprising, since real-world scenarios involving strategic interactions are best formulated using game theory because such formulations not only provide deep insights into the web of incentives and disincentives within which each player operates and consequent predictive paradigms, but also enable the creation of policy architectures based on anticipation of players‟ actions and/or alteration of players‟ incentives/disincentives to encourage/discourage certain behavioural outcomes. This thesis also applies game-theoretic frameworks to analyze some issues within the domain of terrorism and …

Holomorphic Functions, Relativistic Sum, Blaschke Products And Superoscillations, Daniel Alpay, Fabrizio Colombo, Stefano Pinton, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The notion of superoscillation is a particular case of that one of supershift. In the recent years, superoscillating functions, that appear for example in weak values in quantum mechanics, have become an interesting and independent field of research in complex analysis and in the theory of infinite order differential operators. The aim of this paper is to study some infinite order differential operators acting on entire functions which naturally arise in the study of superoscillating functions. Such operators are of particular interest because they are associated …

Injective And Projective Semimodules Over Involutive Semirings, Peter Jipsen, Sara Vanucci

Mathematics, Physics, and Computer Science Faculty Articles and Research

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective leads to a necessary and sufficient condition for the interval [d,1] to be a subalgebra of an involutive residuated lattice, where d is the dualizing element. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for …

Discrete Dynamics Of Dynamic Neural Fields, Eddy Kwessi

Mathematics Faculty Research

Large and small cortexes of the brain are known to contain vast amounts of neurons that interact with one another. They thus form a continuum of active neural networks whose dynamics are yet to be fully understood. One way to model these activities is to use dynamic neural fields which are mathematical models that approximately describe the behavior of these congregations of neurons. These models have been used in neuroinformatics, neuroscience, robotics, and network analysis to understand not only brain functions or brain diseases, but also learning and brain plasticity. In their theoretical forms, they are given as ordinary or …

Irreducibility And Galois Groups Of Random Polynomials, Hanson Hao, Eli Navarro, Henri Stern

Rose-Hulman Undergraduate Mathematics Journal

In 2015, I. Rivin introduced an effective method to bound the number of irreducible integral polynomials with fixed degree d and height at most N. In this paper, we give a brief summary of this result and discuss the precision of Rivin's arguments for special classes of polynomials. We also give elementary proofs of classic results on Galois groups of cubic trinomials.

Lebesgue Measure Preserving Thompson Monoid And Its Properties Of Decomposition And Generators, William Li

Rose-Hulman Undergraduate Mathematics Journal

This paper defines the Lebesgue measure preserving Thompson monoid, denoted by G, which is modeled on the Thompson group F except that the elements of G preserve the Lebesgue measure and can be non-invertible. The paper shows that any element of the monoid G is the composition of a finite number of basic elements of the monoid G and the generators of the Thompson group F. However, unlike the Thompson group F, the monoid G is not finitely generated. The paper then defines equivalence classes of the monoid G, use them to construct a monoid H …

Repeat Length Of Patterns On Weaving Products, Zhuochen Liu

Rose-Hulman Undergraduate Mathematics Journal

On weaving products such as fabrics and silk, people use interlacing strands to create artistic patterns. Repeated patterns form aesthetically pleasing products. This research is a mathematical modeling of weaving products in the real world by using cellular automata. The research is conducted by observing the evolution of the model to better understand products in the real world. Specifically, this research focuses on the repeat length of a weaving pattern given the rule of generating it and the configuration of the starting row. Previous studies have shown the range of the repeat length in specific situations. This paper will generalize …

Disjointness Of Linear Fractional Actions On Serre Trees, Henry W. Talbott

Rose-Hulman Undergraduate Mathematics Journal

Serre showed that, for a discrete valuation field, the group of linear fractional transformations acts on an infinite regular tree with vertex degree determined by the residue degree of the field. Since the p-adics and the polynomials over the finite field of order p act on isomorphic trees, we may ask whether pairs of actions from these two groups are ever conjugate as tree automorphisms. We analyze permutations induced on finite vertex sets, and show a permutation classification result for actions by these linear fractional transformation groups. We prove that actions by specific subgroups of these groups are conjugate only …