Mathematics Commons^™

Factors That Relate To Academic Excellence, Matthew Lemoine

Mathematics Senior Capstone Papers

The American College Testing (ACT) provides university admissions offices a look into a student’s academic ability so that the university can determine admittance and scholarships. The premise of this study is to see whether this is the most effective way to determine academic ability, using the Logistic Regression Model. The data was collected from October 21, 2021 to December 31, 2021 at Louisiana Tech University. Using the Logistic Regression Model, predictions of academic excellence are obtained based on the parameters outlined in this paper. After the parameters for the Logistic Regression Model are calculated, conclusions can be drawn that the …

A Study Of Nash Equilibrium In Discrete Cases, Hallye Leleux

Mathematics Senior Capstone Papers

In game theory, Nash Equilibrium refers to a set of strategies in a non-cooperative game such that no player can benefit from changing their strategy. This paper examines the payoff matrices of the notable discrete games Battle of the Sexes, Matching Pennies, and The Prisoner’s Dilemma. We then create a custom discrete two player case and determine the pure and mixed strategy Nash Equilibria. After calculating the probabilities for each mixed strategy, we then determine the resulting payoff for each player and show how deviating from that strategy results in a lower payoff.

Statistical Analysis Of Home-Field Advantage In European Soccer, Francisco Simoncini

Mathematics Senior Capstone Papers

Home-field advantage in the sports world is commonly known as a huge factor in determining the final score of any sports game. By using a one-tailed, two-sample t-test, I am analyzing how playing at home generally affects a European soccer team’s performance. I am looking at how these performances are affected across three domestic leagues in Europe. I predict that when teams play at home, there will be a significant difference in their win ratio and goals scored.

Implementing A Key Exchange Protocol Using Isogenies On Elliptic Curves In Python, Thomas Schwartzenburg

Mathematics Senior Capstone Papers

The study of Elliptic Curves continues to reveal more secure and efficient methods to encrypt information. For example, in De Feo, Jao, and Plˆut’s paper Towards Quantum- Resistant Cryptosystems From Supersingular Elliptic Curve Isogenies, they provide an abstract key exchange protocol using isogonies on supersingular elliptic curves. The main idea of this research is to present an accurate manifestation of the abstract protocol constructed by De Feo et al., which will be realized using Python. This protocol is founded in the Diffie-Hellman algorithm using isogenies on Elliptic Curves as its primary operation. Background on Elliptic Curve Cryptography will also be …

Predicting Box Office Revenues Using Differential Equation Models, Brenden David

Mathematics Senior Capstone Papers

The growing popularity of the internet has increased exponentially in the past few decades. This sudden increase in popularity allows for users to gain access to more information than was thought possible in the past, including access to box-office movie revenues of previously released movies. Since we now have access to thousands of movie’s box office revenues, we can analyze some sort of trend within specific genres of movies. With these trends we develop a differential equations model that will predict the box office revenues of a movie that is in its early stages of the box office. This model …

Flood Risk Prediction In Southern Louisiana Using Support Vector Machines, Claire Dorsett

Mathematics Senior Capstone Papers

Flooding in Southern Louisiana is a growing concern as violent weather becomes more frequent. According to the Environmental Protection Agency (EPA), Louisiana soils have become drier and annual rainfall trends have increased, which may lead to more severe flooding in the coming years. In light of this change in weather trends, flood prediction has become increasingly important for the public’s safety. The focus of this paper is to apply the Support Vector Machine (SVM), a machine learning technique, to classify flood risks based on water gage height, wind speed and direction, and time of the year. In this paper, the …

Optimal Approximants And Orthogonal Polynomials In Several Variables, Meredith Sargent, Alan A. Sola

Mathematical Sciences Faculty Publications and Presentations

We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

Game-Theoretic Approach Explains -- On The Qualitative Level -- The Antigenic Map Of Covid-19 Variants, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

To effectively defend the population against future variants of Covid-19, it is important to be able to predict how it will evolve. For this purpose, it is necessary to understand the logic behind its evolution so far. At first glance, this evolution looks random and thus, difficult to predict. However, we show that already a simple game-theoretic model can actually explain -- on the qualitative level -- how this virus mutated so far.

A New Application Of The Central Limit Theorem, Kenneth Winters

Selected Honors Theses

This paper discusses the Central Limit Theorem (CLT) and its applications. The paper gives an introduction to what the CLT is and how it can be applied to real life. Additionally, the paper gives a conceptual understanding of the theorem through various examples and visuals. The paper discusses the applications of the CLT in fields such as computer science, psychology, and political science. The author then suggests a new mathematical theorem as an application of the CLT and provides a proof of the theorem. The new theorem relates to expected value and probabilities of random variables and provides a link …

Why Constraint Interval Arithmetic Works Well: A Theorem Explains Empirical Success, Barnabas Bede, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Often, we are interested in a quantity that is difficult or impossible to measure directly, e.g., tomorrow's temperature. To estimate this quantity, we measure auxiliary easier-to-measure quantities that are related to the desired ones by a known dependence, and use the known relation to estimate the desired quantity. Measurements are never absolutely accurate, there is always a measurement error, i.e., a non-zero difference between the measurement result and the actual (unknown) value of the corresponding quantity. In many practical situations, the only information that we have about each measurement error is the bound on its absolute value. In such situations, …

When Is Deep Learning Better And When Is Shallow Learning Better: Qualitative Analysis, Salvador Robles Herrera, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, deep neural networks work better than the traditional "shallow" ones, however, in some cases, the shallow neural networks lead to better results. At present, deciding which type of neural networks will work better is mostly done by trial and error. It is therefore desirable to come up with some criterion of when deep learning is better and when shallow is better. In this paper, we argue that this depends on whether the corresponding situation has natural symmetries: if it does, we expect deep learning to work better, otherwise we expect shallow learning to be more effective. …

Vertex Identification In Graphs, Yuya Kono

Dissertations

In recent decades, there has been increased interest in studying ways touniquely identify the vertices of a given graph. Two of best-known vertex identification methods are referred to as the metric dimension introduced by Slater and by Harary, and Melter in the1970s, and the partition dimension introduced by Chartrand in 2000. The first method deals with locating an ordered set of vertices in a connected graph G and uniquely identifying each vertex of G by means of the distance between this vertex and the vertices in the ordered set. The second method deals with partitioning the vertex set of a …

Irregular Orbital Domination In Graphs, Peter E. Broe

Dissertations

In recent decades, domination in graphs has become a popular area of study due in large degree to its applications to modern society and the mathematical beauty of the topic. While this area evidently began with the work of Claude Berge in 1958 and of Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of a survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then a large number of variations of domination have surfaced and provided numerous applications to different areas of science and real-life problems. Among these variations are …

A New Metaphor: How Artificial Intelligence Links Legal Reasoning And Mathematical Thinking, Melissa E. Love Koenig, Colleen Mandell

Marquette Law Review

Artificial intelligence’s (AI’s) impact on the legal community expands exponentially each year. As AI advances, lawyers have more powerful tools to enhance their ability to research and analyze the law, as well as to draft contracts and other legal documents. Lawyers are already using tools powered by AI and are learning to shift their methodologies to take advantage of these enhancements. To continue to grow into their shifting role, lawyers should understand the relationship between AI, mathematics, and legal reasoning.

Some Properties And Applications Of Spaces Of Modular Forms With Eta-Multiplier, Cuyler Daniel Warnock

Theses and Dissertations

This dissertation considers two topics. In the first part of the dissertation, we prove the existence of fourteen congruences for the $p$-core partition function of the form given by Garvan in \cite{G1}. Different from the congruences given by Garvan, each of the congruences we give yield infinitely many congruences of the form $$a_p(\ell\cdot p^{t+1} \cdot n + p^t \cdot k - \delta_p) \equiv 0 \pmod \ell.$$ For example, if $t \geq 0$ and $\sfrac{m}{n}$ is the Jacobi symbol, then we prove $$a_7(7^t \cdot n - 2) \equiv 0 \pmod 5, \text{ \ \ if $\bfrac{n}{5} = 1$ and $\bfrac{n}{7} = …

Gödel's Incompleteness Theorems, Derick Swarey

Senior Honors Theses

The Incompleteness Theorems of Kurt Godel are very famous both within and outside of mathematics. They focus on independence and consistency within mathematics and hence a more thorough understanding of these is beneficial to their study. The proofs of the theorems involve many ideas which may be unfamiliar to many, including those of formal systems, Godel numbering, and recursive functions and relations. The arguments themselves mirror the Liar’s Paradox in that Godel constructs a statement asserting its own unprovability and then shows that such a statement and its negation must both be independent of the system, otherwise the system is …

Gl(1|1) Graph Connections, Andrea Bourque

Honors Capstones

No abstract provided.

Opers On The Projective Line, Wronskian Relations, And The Bethe Ansatz, Ty J. Brinson

Honors Capstones

No abstract provided.

Finite Subdivision Rules For Matings Of Quadratic Thurston Maps With Few Postcritical Points, Jeremiah Zonio

Undergraduate Theses

A finite subdivision rule is set of instructions for repeatedly subdividing a partitioning of a given space. This turns out to be incredibly useful when attempting to describe a process known as polynomial mating. Polynomial mating is a way of gluing together two spaces which two polynomials may act upon such that the glued borders of each space respects the dynamics described by each polynomial. For matings of Misiurewicz polynomials, the spaces we are gluing together are 1-dimensional and are thus all border. This poses a conceptual difficulty which this paper attempts to resolve by using finite subdivison rules to …

The Trace Of T2 Takes No Repeated Values, Liubomir Chiriac, Andrei Jorza

Mathematics and Statistics Faculty Publications and Presentations

We prove that the trace of the Hecke operator T₂" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">T2 acting on the vector space of cusp forms of level one takes no repeated values, except for 0, which only occurs when the space is trivial.

A Super Fast Algorithm For Estimating Sample Entropy, Weifeng Liu, Ying Jiang, Yuesheng Xu

Mathematics & Statistics Faculty Publications

: Sample entropy, an approximation of the Kolmogorov entropy, was proposed to characterize complexity of a time series, which is essentially defined as − log(B/A), where B denotes the number of matched template pairs with length m and A denotes the number of matched template pairs with m + 1, for a predetermined positive integer m. It has been widely used to analyze physiological signals. As computing sample entropy is time consuming, the box-assisted, bucket-assisted, x-sort, assisted sliding box, and kd-tree-based algorithms were proposed to accelerate its computation. These algorithms require O(N²) or …

Remotely Close: An Investigation Of The Student Experience In First-Year Mathematics Courses During The Covid-19 Pandemic, Sawyer Smith

Honors Program: Senior Projects (Public)

The realm of education was shaken by the onset of the COVID-19 pandemic in 2020. It had drastic effects on the way that courses were delivered to students, and the way that students were getting their education at the collegiate level. At the University of Nebraska – Lincoln, the pandemic dramatically changed the way that first-year mathematics courses looked for students. By Spring 2021, students had the opportunity to take their first-year math courses either in-person or virtually. This project sought to identify differences between the two methods of course delivery during the Spring 2021 semester, regarding interaction with peers …

Quadratic Neural Network Architecture As Evaluated Relative To Conventional Neural Network Architecture, Reid Taylor

Senior Theses

Current work in the field of deep learning and neural networks revolves around several variations of the same mathematical model for associative learning. These variations, while significant and exceptionally applicable in the real world, fail to push the limits of modern computational prowess. This research does just that: by leveraging high order tensors in place of 2nd order tensors, quadratic neural networks can be developed and can allow for substantially more complex machine learning models which allow for self-interactions of collected and analyzed data. This research shows the theorization and development of mathematical model necessary for such an idea to …

Motivic Classes Of Degeneracy Loci And Pointed Brill-Noether Varieties, D. Anderson, Linda Chen, N. Tarasca

Mathematics & Statistics Faculty Works

Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern–Schwartz–MacPherson classes, K-theory classes, and Cappell–Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch χ_y-genus of classical and one-pointed Brill–Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.

Amphichiral Knots With Large 4-Genus, Allison N. Miller

Mathematics & Statistics Faculty Works

For each we give g > 0 infinitely many knots that are strongly negative amphichiral, hence rationally slice and representing 2-torsion in the smooth concordance group, yet which do not bound any locally flatly embedded surface in the 4-ball with genus less than or equal to g. Our examples also allow us to answer a question about the four-dimensional clasp number of knots.

Advances And Applications Of Optimal Polynomial Approximants, Raymond Centner

USF Tampa Graduate Theses and Dissertations

The history of optimal polynomial approximants (OPAs) dates back to the engineering literature of the 1970s. Here, these polynomials were studied in the context of the Hardy space H^2(X), where X denotes the open unit disk D or the bidisk D^2. Under certain conditions, it was thought that these polynomials had all of their zeros outside the closure of X. Hence, it was suggested that these polynomials could be used to design a stable digital filter. In recent mathematics literature, OPAs have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the …

Parallel Resource Defined Fitness Sharing: A Study On Parallel Optimizations For Niching Algorithms, Blayne A. Rogers

Masters Theses

The exploitation of niches by genetic algorithms (GAs) is a computationally expensive, but effective, methodology for solving complex open problems and real-world applications. Niching, differentiated on the modality of sharing, casts problems in terms of the specific resources available. These concepts arise from the broader natural algorithms that encapsulate the ideas and theories used in artificial intelligence. In remediating the computational costs, a study on exploiting niche-defined parallel structures is performed in the contest of the resourcedefined fitness sharing (RFS) algorithm.

Sharing is a natural algorithm paradigm that emulates the use of resources within an environment or population. Defining these …

Varieties Of Nonassociative Rings Of Bol-Moufang Type, Ronald E. White

All NMU Master's Theses

In this paper we investigate Bol-Moufang identities in a more general and very natural setting, \textit{nonassociative rings}.

We first introduce and define common algebras. We then explore the varieties of nonassociative rings of Bol-Moufang type. We explore two separate cases, the first where we consider binary rings, rings in which we make no assumption of it's structure. The second case we explore are rings in which, $2x=0$ implies $x=0$.

Quandles That Are Knot Quandles, Jason Haskell

All NMU Master's Theses

There are many papers that introduce the relationship between knots and quandles which are written tersely and focus mainly on applications or implications. Here, we will take time to explain in depth how to derive quandles from oriented knots. Starting with an rigorous introduction to what a knot is and what a quandle is, we will also define the Fundamental Quandle of a knot and the relationship between colorings of a knot and the homomorphisms from an arbitrary quandle to a Fundamental Quandle. Then using this foundation, we will examine two sets of knots that produce quandles that contain subquandles …

Kissing The Archimedeans, Anthony Webb

All NMU Master's Theses

In this paper the three dimensional kissing problem will be related to the Platonic and Archimedean solids. On each polyhedra presented their vertices will have spheres expanding such that the center of each of these outer spheres are the vertices of the polyhedron, and these outer spheres will continue to expand until they become tangent to each other. The ratio will be found between the radius of each outer sphere, and the radius of an inner sphere such that each inner sphere's center is the circumcenter of the polyhedron, and the inner sphere is tangent to each outer sphere. Every …