Mathematics Commons^™

Hodge Classes In The Cohomology Of Local Systems, Xiaojiang Cheng

Arts & Sciences Graduate Student Theses and Dissertations

Let $S$ be an arithmetic quotient of a Hermitian symmetric domain and $X/S$ be a family of varieties over $S$. One interesting problem is to find the Hodge classes of $X$, and if possible, to prove the Hodge conjecture for $X$. Using techniques from automorphic forms, we studied the Hodge conjecture for certain families of varieties over arithmetic quotients of balls and the Siegel domain of degree two. As a byproduct, we derived formulas for Hodge numbers in terms of automorphic forms.

Determination Of Output Composition In Reaction-Advection-Diffusion Systems And Improving Language Model Performance With Re-Tuning, Eric Joseph Pasewark

Arts & Sciences Graduate Student Theses and Dissertations

There are 2 main subjects studied in this thesis. The first is on modeling chemical reactions. We formulate the problem of determining how much product is formed from a reactor and model this problem using metric graphs. We develop an efficient method to explicitly solve the problem on metric graphs. We work through examples by hand and with code to solve the problem. The second subject is a novel method to improve language model performance on compositional tasks. Our method teaches a language model to break a given problem into different subproblems, create prompts for these, and then solve the …

Using Multi-Scale Spatial Models Of Dendritic Ecosystems To Infer Abundance Of A Stream Salmonid, Xinyi Lu, Yoichiro Kanno, George P. Valentine, Jacob M. Rash, Mevin B. Hooten

Mathematics and Statistics Faculty Publications

1. Understanding patterns of species abundance is essential for planning landscape-level conservation. The complex hierarchies of dendritic ecosystems result in different levels of heterogeneity at distinct geographic scales. Species responses to dynamic environmental drivers may also vary spatially depending on their interactions with landscape features. Monitoring abundance by explicitly quantifying their spatial and temporal variation is important for strategic management.

2. We analysed brook trout (Salvelinus fontinalis) count data collected from 173 sites in western North Carolina between 1989 and 2015. We developed a Bayesian hierarchical model that used single- and multi-pass electro-fishing data and characterized their respective …

Studying Hilbert’S 10th Problem Via Explicit Elliptic Curves, Debanjana Kundu, Antonio Lei, Florian Sprung

School of Mathematical & Statistical Sciences Faculty Publications

N. García-Fritz and H. Pasten showed that Hilbert’s 10th problem is unsolvable in the ring of integers of number fields of the form Q(p–√3,−q−−−√) for positive proportions of primes p and q. We improve their proportions and extend their results to the case of number fields of the form Q(p–√3,Dq−−−√) , where D belongs to an explicit family of positive square-free integers. We achieve this by using multiple elliptic curves, and replace their Iwasawa theory arguments by a more direct method.

Ua12/2/1 College Heights Herald: Ways Of The Future, Wku Student Affairs

WKU Administration Documents

Magazine edition of the College Heights Herald for the period April 8 - May 5, 2024.

• Anderson, Alexandria. Letter from the Editor – Science, Technology & Business
• Anderson, Alexandria. Women in STEM Make Advances
• Novoselia, Mariia. Turnover on Fountain Square: Local Businesses Share Concerns & Success Stories
• SOKY Rentals Offers New Apartments
• Walk2Campus: Where Home Meets Community
• Bush, Camden. Can a Baseball Team Reduce Their Goals to a Math Problem?
• Hawkins, Kaylee. Student Business Owners Share Experiences – Gatara Townsend, Andrew Garrett, Brittiny Sadler
• Ivory, Larkin. Bikewalk BG, WKU CHHS Offer Chances for Healthy Lifestyle – Health & Human Services

Calculations From On The Existence Of Periodic Traveling-Wave Solutions To Certain Systems Of Nonlinear, Dispersive Wave Equations, Jacob Daniels

Mathematics and Statistics Student Research and Class Projects

In the field of nonlinear waves, particular interest is given to periodic traveling-wave solutions of nonlinear, dispersive wave equations. This thesis aims to determine the existence of periodic traveling-wave solutions for several systems of water wave equations. These systems are the Schr¨odinger KdV-KdV, Schr¨odinger BBM-BBM, Schr¨odinger KdV-BBM, and Schr¨odinger BBM-KdV systems, and the abcd-system. In particular, it is shown that periodic traveling-wave solutions exist and are explicitly given in terms of cnoidal, the Jacobi elliptic function. Certain solitary-wave solutions are also established as a limiting case of the periodic traveling-wave solutions, that is, as the elliptic modulus approaches one.

An Alternate Proof For The Top-Heavy Conjecture On Partition Lattices Using Shellability, Brian Macdonald, Josh Hallam

Honors Thesis

A partially ordered set, or poset, is governed by an ordering that may or may not relate any pair of objects in the set. Both the bonds of a graph and the partitions of a set are partially ordered, and their poset structure can be depicted visually in a Hasse diagram. The partitions of {1, 2, ..., n} form a particularly important poset known as the partition lattice Πn. It is isomorphic to the bond lattice of the complete graph Kn, making it a special case of the family of bond lattices. Dowling and Wilson’s 1975 Top-Heavy Conjecture states that …

Boolean Group Structure In Class Groups Of Positive Definite Quadratic Forms Of Primitive Discriminant, Christopher Albert Hudert Jr.

Departmental Honors & Graduate Capstone Projects

It is possible to completely describe the representation of any integer by binary quadratic forms of a given discriminant when the discriminant’s class group is a Boolean group (also known as an elementary abelian 2-group). For other discriminants, we can partially describe the representation using the structure of the class group. The goal of the present project is to find whether any class group with 32 elements and a primitive positive definite discriminant is a Boolean group. We find that no such class group is Boolean.

A Mceliece Cryptosystem, Using Permutation Error-Correcting Codes, Fiona Smith

CSB and SJU Distinguished Thesis

Using existing methods of cryptography, we can encrypt messages through the internet. However, these methods are vulnerable to attacks done by a quantum computer, which are a rising threat to security. In this thesis I discuss a possible method of encryption, secure against quantum attacks, using permutation groups and coding theory.

Representations Of Gender In Math-Related Films, Jacob Gathje

CSB and SJU Distinguished Thesis

This project analyzes how four popular math-related films - Hidden Figures, Mean Girls, Good Will Hunting, and A Beautiful Mind - either follow, resist, or reconfigure gender stereotypes in mathematics. It includes close readings of specific scenes in each of the films, along with broader analysis of the effects of how women and men are represented differently. It concludes forward-looking focus, providing suggestions for how future math-related movies can depict a more realistic and inclusive version of the field of mathematics. Ideally, this will help improve one part of the larger issue of gender disparities in math.

Study Of Nanofluid Flow In A Stationary Cone–Disk System With Temperature-Dependent Viscosity And Thermal Conductivity, Anagha Susan John, Mahanthesh Basavarajappa, Igor V. Shevchuk

School of Mathematical & Statistical Sciences Faculty Publications

The substantial temperature gradient experienced by systems operating at relatively high temperatures significantly impacts the transport characteristics of fluids. Hence, considering temperature-dependent fluid properties is critical for obtaining realistic prediction of fluid behavior and optimizing system performance. The current study focuses on the flow of nanofluids in a stationary cone–disk system (SCDS), taking into account temperature-dependent thermal conductivity and viscosity. The influence of Brownian motion, thermophoresis, and Rosseland radiative flux on the heat transport features are also examined. The Reynolds model for viscosity and Chiam's model for thermal conductivity are employed. The Navier–Stokes equation, the energy equation, the incompressibility condition, …

Approval Gap Of Weighted K-Majority Tournaments, Jeremy Coste, Breeann Flesch, Joshua D. Laison, Erin Mcnicholas, Dane Miyata

Theory & Applications of Graphs

A $k$-majority tournament $T$ on a finite set of vertices $V$ is defined by a set of $2k-1$ linear orders on $V$, with an edge $u \to v$ in $T$ if $u>v$ in a majority of the linear orders. We think of the linear orders as voter preferences and the vertices of $T$ as candidates, with an edge $u \to v$ in $T$ if a majority of voters prefer candidate $u$ to candidate $v$. In this paper we introduce weighted $k$-majority tournaments, with each edge $u \to v$ weighted by the number of voters preferring $u$.

We define the …

Representations Of Solutions Of Time-Fractional Multi-Order Systems Of Differential-Operator Equations, Sabir Umarov

Mathematics Faculty Publications

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order …

Topics In The Study Of The Pragmatic Functions Of Phonetic Reduction In Dialog, Nigel G. Ward, Carlos A. Ortega

Departmental Technical Reports (CS)

Reduced articulatory precision is common in speech, but for dialog its acoustic properties and pragmatic functions have been little studied. We here try to remedy this gap. This technical report contains content that was omitted from the journal article (Ward et. al, submitted). Specifically, we here report 1) lessons learned about annotating for perceived reduction, 2) the finding that, unlike in read speech, the correlates of reduction in dialog include high pitch, wide pitch range, and intensity, and 3) a baseline model for predicting reduction in dialog, using simple acoustic/prosodic features, that achieves correlations with human perceptions of 0.24 for …

How To Make A Neural Network Learn From A Small Number Of Examples -- And Learn Fast: An Idea, Chitta Baral, Vladik Kreinovich

Departmental Technical Reports (CS)

Current deep learning techniques have led to spectacular results, but they still have limitations. One of them is that, in contrast to humans who can learn from a few examples and learn fast, modern deep learning techniques require a large amount of data to learn, and they take a long time to train. In this paper, we show that neural networks do have a potential to learn from a small number of examples -- and learn fast. We speculate that the corresponding idea may already be implicitly implemented in Large Language Models -- which may partially explain their (somewhat mysterious) …

Counting Hamming-Graceful Labelings Of Paths, Ashka Dalal

Mathematical Sciences Technical Reports (MSTR)

Let Γ be a graph of m edges and n vertices. A Hamming-graceful labeling of Γ labels vertices with binary strings of length m and the edge labels are induced by the Hamming distance between vertex labels. It is known that all paths have Hamming-graceful labelings, thus the question arises, how many possible labelings exist for a path of a given size. We develop an algebraic way to generate labelings, conjecture a method for counting, prove this for small examples, and verify larger examples using a Python program.

Statistical Modeling Of Right-Censored Spatial Data Using Gaussian Random Fields, Fathima Z. Sainul Abdeen, Akim Adekpedjou, Sophie Dabo Niang

Mathematics and Statistics Faculty Research & Creative Works

Consider a Fixed Number of Clustered Areas Identified by their Geographical Coordinates that Are Monitored for the Occurrences of an Event Such as a Pandemic, Epidemic, or Migration. Data Collected on Units at All Areas Include Covariates and Environmental Factors. We Apply a Probit Transformation to the Time to Event and Embed an Isotropic Spatial Correlation Function into Our Models for Better Modeling as Compared to Existing Methodologies that Use Frailty or Copula. Composite Likelihood Technique is Employed for the Construction of a Multivariate Gaussian Random Field that Preserves the Spatial Correlation Function. the Data Are Analyzed using Counting Process …

Key Benefits Of Small Group Instruction For Diverse Learners, Lydia Mcevoy

Master's Theses

Utilizing a mixed method approach this research study investigated the effects of small group instruction on the learning of diverse learners. Informed by a preliminary literature review that supports the use of small-group instruction, the researcher conducted a small-scale action research project to focus on three diverse learners in a 1st-grade classroom over four weeks. One of the findings of this project shows that small group instruction helps promote social and emotional skills as students feel more comfortable interacting with peers in a small group rather than in a whole group. Another finding indicates that students feel more encouraged by …

On Distortion Of Surface Groups In Right-Angled Artin Groups, Lucas Bridges

Mathematical Sciences Undergraduate Honors Theses

Surfaces have long been a topic of interest for scholars inside and outside of mathe- matics. In a topological sense, surfaces are spaces which appear flat on a local scale. Surfaces in this sense have a restricted set of properties, including the behavior of loops around a surface, codified in the fundamental group.

All but 3 surface groups have been shown to embed into a class of groups called right-angled Artin groups. The method through which these embeddings are created places large restrictions on all homomorphisms from surface groups to right-angled Artin groups.

One such restriction on these homomorphisms is …

Fundamental 2-Quandles Of The M(1/2, 1/3, 1/3; K) Montesino Knots, Alex Abrams

Mathematics Theses

Every oriented knot has an associated fundamental quandle. Except in two simple cases, these quandles have infinite order. For every integer n>1, there is a quotient of the fundamental quandle called the n-quandle of the knot. In some cases, these n-quandles may be finite. In the case of the M(1/2,1/3,1/3; k) Montesino knot family, the 2-quandle is finite. In this paper, we investigate the structure of the 2-quandle and prove its size.