Mathematics Commons^™

Some Np-Complete Edge Packing And Partitioning Problems In Planar Graphs, Jed Yang

Communications on Number Theory and Combinatorial Theory

Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not share edges. Bernáth and Király proved that this decision problem is NP-complete and asked if the same result holds when restricting to planar graphs. Similarly, they showed that the packing problem with a spanning tree and a path between two distinguished vertices is NP-complete. They also established the NP-completeness of the partitioning problem of determining whether the edge set of a graph can be …

Adjusting Environmental Performance Index Scores Toaccount For Gross National Income, Rory Black

Research and Creativity Symposium

I examined the Environmental Performance Index (EPI) report assembled by Yale University to address the need for environmental action. In this report, Yale University ranks countries across the globe on their environmental performance based on various indicators. I was interested in determining if all countries were ranked on a fair scale. Yale University acknowledges that each country varies in their economic resources measured by GDP but did not adjust for that in their scoring. I utilized factor analysis and cluster analysis to first analyze patterns and groupings that appeared the countries. I then standardized the original rankings with respect to …

Modeling And Simulation Of Microscopic Fibers In A Viscous Fluid, William Mitchell

Mathematics Colloquium Series

In biology, the movements of tiny structures often rely on the mechanical properties of long, thin tubes. For example, bacteria swim by rotating their flagella, and in cell division (mitosis) the two copies of the DNA must be pulled apart by microtubules. To understand these processes it is very tempting to take advantage of the large aspect ratio of the thin structures, for example by modeling them as one-dimensional curves rather than as more complicated objects with volume and surface area. This kind of shortcut saves a lot of work! I will describe one standard and widely used tool known …

Additional Fay Identities Of The Extended Toda Hierarchy, Yu Wan

Rose-Hulman Undergraduate Mathematics Journal

The focus of this paper is the extended Toda Lattice hierarchy, an infinite system of partial differential equations arising from the Toda lattice equation. We begin by giving the definition of the extended Toda hierarchy and its explicit bilinear equation, following Takasaki’s construction. We then derive a series of new Fay identities. Finally, we discover a general formula for one type of Fay identity.

Anticipating Stochastic Integrals And Related Linear Stochastic Differential Equations, Sudip Sinha

LSU Doctoral Dissertations

Itô’s stochastic calculus revolutionized the field of stochastic analysis and has found numerous applications in a wide variety of disciplines. Itô’s theory, even though quite general, cannot handle anticipating stochastic processes as integrands. There have been considerable efforts within the mathematical community to extend Itô’s calculus to account for anticipation. The Ayed–Kuo integral — introduced in 2008 — is one of the most recent developments. It is arguably the most accessible among the theories extending Itô’s calculus — relying solely on probabilistic methods. In this dissertation, we look at the recent advances in this area, highlighting our contributions. First, we …

Machine Learning In Support Of Student Success, Rachel Rucker

Undergraduate Research Conference

Our goal is to predict whether a student will finish the semester on academic probation by mid-term using university data.

Survival Analysis, Darrell Charles

2022 Student Academic Showcase

Survival analysis is a branch of statistics used by engineers to estimate the reliability of some mechanical system before it ultimately fails. In particular, survival analysis is used to estimate the lifespan of consumer goods. Previous efforts have studied the estimated lifespan of civilian vehicles based on a set of common attributes. In this preliminary report, we focus on the American market over 1990-2020. For our study, the variables of interest are the manufacturer, body type, number of owners, mileage, and age of the vehicle.

Relative Energy Comparison For Various Water Clusters Using Mp2, Df-Mp2, And Ccsd(T):Mp2 Methods, Qihang Wang

Honors Theses

The study of water clusters is an important area of research in many disciplines, such as biology, physical chemistry, and environmental studies. However, due to the difficulty in studying larger water clusters, such as clathrate hydrates, it is beneficial to obtain accurate descriptions of smaller water clusters to use as models for larger systems via computational methods. By starting with small water clusters, such as (H2O)₆, and moving into larger systems it is possible to build up data on various water structures that can determine the energetics of the various geometries within a certain number of water molecules. …

Fock And Hardy Spaces: Clifford Appell Case, Daniel Alpay, Kamal Diki, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we study a specific system of Clifford–Appell polynomials and, in particular, their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows us to obtain various function spaces by specifying a suitable sequence of real numbers. We focus on the Fock and Hardy cases in this setting, and we study the action of the Fueter mapping and its range.

Eigenfunction Restriction Estimates For Curves With Nonvanishing Geodesic Curvatures In Compact Riemannian Surfaces With Nonpositive Sectional Curvatures, Chamsol Park

Mathematics & Statistics ETDs

For 2 ≤ p < 4, we study the Lp norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact 2-dimensional Riemannian manifolds. Burq, G\´erard, and Tzvetkov [12], and Hu [21] found eigenfunction restriction estimates for a curve with nonvanishing geodesic curvatures. We will explain how the proof of the known estimates helps us to consider the case where the given smooth compact Riemannian manifold has nonpositive sectional curvatures. For p = 4, we will also obtain a logarithmic analogous estimate, by using arguments in Xi and Zhang [37], Sogge [33], and Bourgain [10]. At the end of this dissertation, we will talk about a future work, which is a follow up study for higher dimensional analogues of the above curve cases.

A New Perspective On A Polynomial Time Knot Polynomial, Robert John Quarles

LSU Doctoral Dissertations

In this work we consider the Z1(K) polynomial time knot polynomial defined and
described by Dror Bar-Natan and Roland van der Veen in their 2018 paper ”A polynomial time knot polynomial”. We first look at some of the basic properties of Z1(K), and develop an invariant of diagrams Ψm(D) related to this polynomial. We use this invariant as a model to prove how Z1(K) acts under the connected sum operation. We then discuss the effect of mirroring the knot on Z1(K), and described a geometric interpretation of some of the building blocks of the invariant. We then use these to …

General Stochastic Calculus And Applications, Pujan Shrestha

LSU Doctoral Dissertations

In 1942, K. Itô published his pioneering paper on stochastic integration with respect to Brownian motion. This work led to the framework for Itô calculus. Note that, Itô calculus is limited in working with knowledge from the future. There have been many generalizations of the stochastic integral in being able to do so. In 2008, W. Ayed and H.-H. Kuo introduced a new stochastic integral by splitting the integrand into the adaptive part and the counterpart called instantly independent. In this doctoral work, we conduct deeper research into the Ayed–Kuo stochastic integral and corresponding anticipating stochastic calculus.

We provide a …

Modeling The Extended Low And Sudden High Periods Of Activity In Infectious Diseases, Dana Strong

Student Research Symposium

This research explores how the aggregation of individuals during disease spread may, in conjunction with noise, explain how some diseases’ trajectories spend some time at low numbers after which there is a sudden outbreak.

Modeling The Spread Of Curly Top Disease In Tomato Crops, Rachel Frantz

Student Research Symposium

Curly Top disease (CT), caused by a family of curtoviruses, infects a wide variety of agricultural crops. Historically, CT has caused extensive damage in tomato crops resulting in substantial economic loss for the tomato industry. Control methods for CT are scarce, and methods for predicting and assessing the scope of CT outbreaks are limited. In this paper, we formulate two theoretical models, a deterministic model and a stochastic model, for the spread of CT in a heterogeneous environment consisting of beets (preferred host) and tomatoes. The models are composed of two susceptible classes and two infected classes (infectious beets and …

Manifold Alignment With Inter-Domain Diffusion, Andres Duque

Student Research Symposium

The integration of multimodal data presents a challenge in cases when the study of a given phenomena by different instruments or conditions generates distinct but related domains. In this paper, we propose Diffusion Transport Alignment (DTA) a semi-supervised manifold alignment method that exploits prior correspondence knowledge between distinct data views. DTA finds a bijection between two domains, which by assumption, share a similar geometrical structure coming from the same underlying data generating process. We empirically demonstrate the effectiveness of our method to integrate multimodal data, as well as how it can improve the performance of machine learning tasks, otherwise less …

The Outdoor Fashion Industry Is Not Sustainable, Joshua Hillam

Student Research Symposium

Outdoor fashion companies are responsible for a large amount of waste and pollution worldwide. I believe they are ethically bound to make changes in their manufacturing process to decrease their footprint because their products are used in outdoor environments that are impacted negatively by waste and pollution. The fashion industry as a whole is responsible for 10% of the carbon emissions worldwide. Research was conducted by first understanding what being sustainable means from the EPA, from individuals who work in the fashion industry, and from companies that advocate for more sustainable clothing. I then investigated what materials are the most …

Analyzing Suicidal Text Using Natural Language Processing, Cassandra Barton

Student Research Symposium

Using Natural Language Processing (NLP), we are able to analyze text from suicidal individuals. This can be done using a variety of methods. I analyzed a dataset of a girl named Victoria that died by suicide. I used a machine learning method to train a different dataset and tested it on her diary entries to classify her text into two categories: suicidal vs non-suicidal. I used topic modeling to find out unique topics in each subset. I also found a pattern in her diary entries. NLP allows us to help individuals that are suicidal and their family members and close …

Finding Higher Order Interactions Using Local Corex, Thomas Kerby

Student Research Symposium

In applications such as financial markets, social networks, and gene expression data, the variables often interact in complex ways. Yet accurately characterizing pairwise variable interactions can be a difficult task, let alone efficiently characterizing complex higher-order interactions, which is an unsolved problem. This difficulty is exacerbated when variable interactions change across the data. For example, gene interactions in single-cell RNA-sequencing (scRNA-seq) data will typically differ from one cell type to another. To solve these problems, we propose a new method called Local Correlation Explanation (CorEx). Local CorEx captures higher-order variable interactions at a local scale by first clustering data points …

State Estimation—The Role Of Reduced Models, Albert Cohen, Wolfgang Dahmen, Ron Devore

Faculty Publications

The exploration of complex physical or technological processes usually requires exploiting available information from different sources: (i) physical laws often represented as a family of parameter dependent partial differential equations and (ii) data provided by measurement devices or sensors. The amount of sensors is typically limited and data acquisition may be expensive and in some cases even harmful. This article reviews some recent developments for this “small-data” scenario where inversion is strongly aggravated by the typically large parametric dimensionality. The proposed concepts may be viewed as exploring alternatives to Bayesian inversion in favor of more deterministic accuracy quantification related …

Intra-Hour Solar Forecasting Using Cloud Dynamics Features Extracted From Ground-Based Infrared Sky Images, Guillermo Terrén-Serrano

Electrical and Computer Engineering ETDs

Due to the increasing use of photovoltaic systems, power grids are vulnerable to the projection of shadows from moving clouds. An intra-hour solar forecast provides power grids with the capability of automatically controlling the dispatch of energy, reducing the additional cost for a guaranteed, reliable supply of energy (i.e., energy storage). This dissertation introduces a novel sky imager consisting of a long-wave radiometric infrared camera and a visible light camera with a fisheye lens. The imager is mounted on a solar tracker to maintain the Sun in the center of the images throughout the day, reducing the scattering effect produced …

Characterizations Of Certain Classes Of Graphs And Matroids, Jagdeep Singh

LSU Doctoral Dissertations

``If a theorem about graphs can be expressed in terms of edges and cycles only, it probably exemplifies a more general theorem about matroids." Most of my work draws inspiration from this assertion, made by Tutte in 1979.

In 2004, Ehrenfeucht, Harju and Rozenberg proved that all graphs can be constructed from complete graphs via a sequence of the operations of complementation, switching edges and non-edges at a vertex, and local complementation. In Chapter 2, we consider the binary matroid analogue of each of these graph operations. We prove that the analogue of the result of Ehrenfeucht et. al. does …

Ensemble Kernel Density Estimation, Ethan Ancell

Student Research Symposium

The problem of estimating a probability density function from data has many applications in machine learning and data science. Nonparametric estimators are useful in this context as they require relatively few assumptions on the densities. Unfortunately, standard nonparametric methods such as kernel density estimationtend to converge slowly to the true value in high dimensions as a function of the data sample size. Recent work has shown that optimally weighted ensembles of nonparametric estimators can be used to achieve a fast convergence rate when estimating information theoretic functionals such as information divergence. We explore the extension of this theory to density …

Development Of Particle Image Velocimetry Learning Tool: Learnpiv.Org, Kevin Roberts

Student Research Symposium

Particle Image Velocimetry (PIV) is an experimental measurement technique used for quantitative and visual analysis of the velocity distribution in a flow field. Researchers conduct PIV by using lasers to illuminate particles in a flow field of interest, digital cameras to record images of these illuminated particles, and PIV software to analyze the recorded images, producing a velocity vector field. Applying proper techniques to gather useful images and identifying PIV processing algorithm parameters to the collected images is a challenging task for novice PIV users who are unaware of the inner workings of PIV algorithms. Recognizing this problem, we created …

Economic Losses From Covid-19 Cases In The Philippines: A Dynamic Model Of Health And Economic Policy Trade-Offs, Elvira P. De Lara-Tuprio, Ma. Regina Justina E. Estuar, Joselito T. Sescon, Cymon Kayle Lubangco, Rolly Czar Joseph T. Castillo, Timothy Robin Y. Teng, Lenard Paulo V. Tamayo, Jay Michael R. Macalalag, Gerome M. Vedeja

Mathematics Faculty Publications

The COVID-19 pandemic forced governments globally to impose lockdown measures and mobility restrictions to curb the transmission of the virus. As economies slowly reopen, governments face a trade-off between implementing economic recovery and health policy measures to control the spread of the virus and to ensure it will not overwhelm the health system. We developed a mathematical model that measures the economic losses due to the spread of the disease and due to different lockdown policies. This is done by extending the subnational SEIR model to include two differential equations that capture economic losses due to COVID-19 infection and due …

Neumann Problems For P-Harmonic Functions, And Induced Nonlocal Operators In Metric Measure Spaces, Luca Capogna, Josh Kline, Riikka Korte, Nageswari Shanmugalingam, Marie Snipes

Mathematics Sciences: Faculty Publications

Following ideas of Caffarelli and Silvestre in [20], and using recent progress in hyperbolic fillings, we define fractional p-Laplacians (−∆p) θ with 0 < θ < 1 on any compact, doubling metric measure space (Z, d, ν), and prove existence, regularity and stability for the non-homogenous non-local equation (−∆p) θu = f. These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for p-Laplacians ∆p, 1 < p < ∞, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Our work also includes as special cases much of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that (Z, d, ν) supports a Poincaré inequality.

Unavoidable Structures In Large And Infinite Graphs, Sarah Allred

LSU Doctoral Dissertations

In this work, we present results on the unavoidable structures in large connected and large 2-connected graphs. For the relation of induced subgraphs, Ramsey proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either K_r or K_r. It is well known that, for every positive integer r, every sufficiently large connected graph contains an induced subgraph isomorphic to one of K_r, K_1,r, and P_r. We prove an analogous result for 2-connected graphs. Similarly, for infinite graphs, every infinite connected graph contains an induced subgraph …

Hyperbolic Quotients Of Projection Complexes, Matt Clay, Johanna Mangahas

Mathematical Sciences Faculty Publications and Presentations

This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex. We show that under certain conditions the quotient complex is δ-hyperbolic. Additionally, under certain circumstances, we show that if the original …

Why Rectified Linear Unit Is Efficient In Machine Learning: One More Explanation, Barnabas Bede, Vladik Kreinovich, Uyen Pham

Departmental Technical Reports (CS)

In many applications, in particular, in econometric application, deep learning techniques are very effective. In this paper, we provide a new explanation for why rectified linear units -- the main units of deep learning -- are so effective. This explanation is similar to the usual explanation of why Gaussian (normal) distributions are ubiquitous -- namely, it is based on an appropriate limit theorem.

Semantic Completeness Of Intuitionistic Predicate Logic In A Fully Constructive Meta-Theory, Ian Ray

Masters Theses & Specialist Projects

A constructive proof of the semantic completeness of intuitionistic predicate logic is explored using set-generated complete Heyting Algebra. We work in a constructive set theory that avoids impredicative axioms; for this reason the result is not only intuitionistic but fully constructive. We provide background that makes the thesis accessible to the uninitiated.

Hypothesis Testing Of Student's Mathematical Performance In Relation To The Covid-19 Pandemic, Elizabeth Foster

Mathematics Senior Capstone Papers

The COVID-19 Pandemic affected everyone, especially schools and education due to the rapid switch from traditional face-to-face classes to online and hybrid classes. The goal of this study is to see how the pandemic affected students’ grades during distance learning when compared to before. Using hypothesis testing, we will analyze data from 13 sections of MATH 243 classes to determine if the averages went down during the pandemic, specifically in hybrid class settings. Based on the raw data of the test grades in relation to the COVID-19 Pandemic, a significant difference can be found in test scores before and during …