Mathematics Commons^™

Distance-Dependent Connectivity In The Brain Facilitates High Dynamical And Structural Complexity, Victor J. Barranca

Mathematics & Statistics Faculty Works

Recent experiments have revealed that the inter-regional connectivity of the cerebral cortex exhibits strengths spanning over several orders of magnitude and decaying with distance. We demonstrate this to be a fundamental organizing feature that fosters high complexity in both connectivity structure and network dynamics, achieving an advantageous balance between integration and differentiation of information. This is verified through analysis of a multi-scale neuronal network model with nonlinear integrate-and-fire dynamics, incorporating inter-regional connection strengths decaying exponentially with spatial separation at the macroscale as well as small-world local connectivity at the microscale. Through numerical simulation and optimization over the model parameterspace, we …

Twas-Ctl: A Robust And Efficient Method For Multi-Tissue Transcriptome-Wide Association Studies Using Cross-Tissue Learners, Md Mutasim Billah, Hairong Wei, Fengzhu Sun, Kui Zhang

Michigan Tech Publications

The advent of transcriptome-wide association studies (TWAS) has expanded the classical genome-wide association study (GWAS) framework by integrating gene expression with genetic variation to identify trait-associated variants. While multi-tissue TWAS approaches improve statistical power over single-tissue models, existing methods often lose information during result aggregation and require intensive computation. Here, we present TWAS-CTL (Cross-Tissue Learner), a novel framework that leverages heterogeneous gene expression across tissues by adaptively reweighting and optimizing multiple single-tissue learners. Simulations demonstrate that TWAS-CTL achieves higher statistical power than the leading method, UTMOST, while maintaining proper type I error control and reducing computational time by over half. …

Perron Theorem Application To A Second-Order Differential Equation With Small Advances, Jason Hendrix

Theses and Dissertations

We analyze a delayed second order differential equation with small delay parameters, proving spectral stability via the characteristic quasi-polynomial and establishing uniform bounds on derivatives of the Green’s function to ensure sign preservation under perturbation, providing a foundation for monotone iteration methods. These results aim to advance the functional-analytic framework for traveling wave solutions in delayed reaction-diffusion systems.

Lipscomb University Parking Garage Analysis, Morgan Hayes, Reagan Maxwell, Daniel Diaz Tortolero, Dominick Dingus, Michael Ent

Student Scholar Symposium

Many students struggle to easily find parking during the school day at Lipscomb University, and this problem will only be emphasized with the potential removal of the nearby Stokes parking lot (containing 255 spaces). To combat this need and help provide additional room for university growth, we propose the addition of a parking garage west of the Fields engineering building, between Belmont Boulevard and Grandview Drive. This garage would be separated from the current garage behind Fields and be slightly larger, containing about 500 spaces over five levels. We evaluate the current number of parking spaces needed at Lipscomb based …

Mathematical Developments In American Spacecraft Technology, Tylah Grace Lynn, Lauren Reece Terry

Student Scholar Symposium

No abstract provided.

Analysis Of Solutions To Research Methods: Navigating The Infinite Content Of Mathematics, Jesus Navarro

Student Scholar Symposium

In the infinite realm of mathematical research, understanding and communicating complex concepts require structured and logical approaches. However, a variety of research methods are employed in mathematical studies; these may differ in approach, historical context, and other important considerations which can contribute to confusion and a lack of effective communication. The genesis of this research began with personal confusion during my research on "Counting the Uncountable: Life in a World Without Numbers", setting the stage for the central challenge: How do mathematicians explain their research? To answer this issue, we analyzed a variety of mathematical research articles to highlight their …

Exploration Of Euclidean Distances In Five-Dimensional Color Space As Proof-Of-Concept For Identification Of Promising Alternatives To Platinum-Based Cancer Drug Candidates, Kiomi Tanakura, Elena G. Harvey

Student Scholar Symposium

Cisplatin, cis-Pt(NH₃)₂Cl₂, is one of the most successful anticancer drugs of all time. However, it has some negative aspects, including toxicity to healthy cells, side effects, and cost. Our goal in this project is to design metal-based anticancer drugs that would be less expensive, less toxic, and more efficient for cancer patients than cisplatin. To begin this research we calculated Euclidean distances - as used by computerized recognition systems - to compare the properties of colors within seasonal palettes that are proven and visible to the eye. These Euclidean distances were then used …

Abstract By Shambhav Shriraam, Shambhav Shriraam

STEM for Success Showcase

No abstract provided.

Discussion Of The Collatz Conjecture, Morgan Hayes

Student Scholar Symposium

The Collatz conjecture was introduced in 1937, but, despite the efforts of many mathematicians, it has yet to be proven. It appears to have very little connection to other areas of mathematics, although some aspects of it allude to questions similar to those asked in studies of prime numbers and the Riemann Hypothesis. The nature of the Collatz sequence allows it to be useful in cryptography and semi-random number generation. Some, including Craig Alan Feinstein, believe a formal proof of the conjecture is not possible. Others believe mathematics has not been developed enough to resolve the issue. To enter into …

Face Value: A Computational Approach To Subjective Impressions Of Faces, Kevin Kpankou

Undergraduate Research Symposium

Various computational models of first impressions have been developed to uncover the mechanisms driving these judgments. However, the implicit notion of a singular ``human'' often overlooks meaningful individual differences in beliefs, attitudes, and associations, as well as culturally grounded group-level constructs. In this paper, we extend Cultural Consensus Theory (CCT) to estimate culturally shared beliefs about faces by incorporating latent constructs structured around interpretable facial features extracted via computer vision algorithms. We apply our model to a large-scale dataset of people’s first impressions of faces. Our approach reveals a robust mapping between facial features and culturally constructed impressions, allowing us …

Predicting The Attitudes Towards Mathematical Modeling Of Primary Math Teachers: The Role Of Self-Efficacy And Motivation, Brenden Rawls

Doctoral Dissertations and Projects

The purpose of this non-experimental quantitative predictive correlational study was to investigate the extent to which efficacy for pedagogy in mathematics, efficacy for teaching mathematics content, effectance motivation in mathematics, years of experience, and access to statistics, data science, or mathematical modeling courses predict attitudes toward constructivist practices in mathematical modeling of Virginia primary school mathematics teachers. The goal of the research was to understand what affective characteristics support effective adoption Virginia's mathematics curriculum change. Higher levels of mathematics teacher self-efficacy have been associated with constructivist instructional practices and higher levels of student achievement. The sample included kindergarten through fifth-grade …

The Mechanics And Consequences Of The Löwenheim-Skolem And Compactness Theorems, Adam Murphy

Honors Program Theses and Projects

The compactness theorem for first-order logic says that for a set of first-order sen-tences to be satisfiable, it’s enough for every finite subset to be satisfiable. We’ll see how the compactness theorem arises from two methods of model construction: first by ultraproducts, and second by Henkin’s method of constructing models from constant symbols. We’ll explore several representative applications of compactness, such as how it can be used to demonstrate that a property is not expressible, establishing limitations on first-order expressibility. The Löwenheim-Skolem theorem says that if an infinite set Γ of first-order sentences has an infinite model, then for all …

Stability Analysis Of Thermohaline Convection With A Time-Varying Shear Flow Using The Lyapunov Method, Kalin Kochnev

Honors Scholar Theses

This work applies the Lyapunov method to identify instabilities and compute the growth rate of a linear time-varying system. The linear system studied describes cold fresh water on top of hot salty water with a periodically time-varying background shear flow. A time-dependent weighting matrix is employed to construct a Lyapunov function candidate. The resulting linear matrix inequalities are discretized in time using the forward Euler method. As the number of temporal discretization points increases, the growth rate predicted by the Lyapunov method or Floquet theory, used for comparison, will converge to the same value obtained from numerical simulations. Furthermore, the …

There Is No Obstruction To A Euclidean Proof For The Fourth Postulate, José Gil-Férez, Piotr Błaszczyk, M. Andrew Moshier, Alberto Naibo, Jean-Michel Salanskis

Mathematics, Physics, and Computer Science Faculty Articles and Research

In Gil-Férez et al. (2025), we presented a proof of Postulate 4 using purely Euclidean techniques, against which Blåsjö (2025) raised several objections. In this note, we offer linguistic, textual, historical, and mathematical evidence that demonstrate that all these objections are baseless.

A Comprehensive Study Of Clique Graphs And Clique Regular Graphs, Connor Phillips

Senior Honors Projects, 2020-current

If Γ is a graph for which every edge is in exactly one clique of order ω, then one can form a new graph with vertex set equal to these cliques.  This is a generalization of the line graph of Γ.  We discover many general results and classifications related to these clique graphs that will be useful to researchers studying graphs with this property. In particular, we find bounds on the spectrum of Γ (with exact results when Γ is k-regular) and some complete classifications when Γ is strongly regular. We apply our results to derive novel information about the …

Kakeya Sets Over The Heisenberg Group, Gabriel Jacob Gress

Mathematics & Statistics ETDs

The Hausdorff dimension of Kakeya sets is an interesting problem where the two-dimensional case can be proven directly, but even obtaining bounds on the higher dimension analogues can require highly technical machinery. The difficulty of the general case has inspired analysts to look at Kakeya sets from non-Euclidean viewpoints.

In this paper, we explore a construction of the Kakeya set in the first Heisenberg group H^1. By utilizing the sub-Riemannian manifold of H^1 we can apply tools in geometric measure theory which at this time cannot be applied in R^3. We restate and provide a detailed proof for a sharp …

On Sharpest Tail Bounds For Functions Of Tail Bounded Random Variables, Stephen Harrison

Mathematics & Statistics ETDs

Consider n real/complex, independent/dependent random variables with respective tail bounds and g a measurable function of the r.v.’s. Consider f the “sharpest” tail bound of g (sharpest in the sense, if f were any less, then for some X1, ..., Xn satisfying the conditions, g(X1, ..., Xn) would not satisfy the tail f). Significant research has been done to approximate f often with high accuracy. These results are often of the form, for g in this family, and tail bounds of Xk in this family, f is bounded by some f′ with high accuracy. However, the question “what would it …

Constructing Genotype And Phenotype Network Helps Reveal Disease Heritability And Phenome-Wide Association Studies, Xuewei Cao, Lirong Zhu, Xiaoyu Liang, Shuanglin Zhang, Qiuying Sha

Michigan Tech Publications

Background

Analyses of a bipartite Genotype and Phenotype Network (GPN), linking the genetic variants and phenotypes based on statistical associations, provide an integrative approach to elucidate the complexities of genetic relationships across diseases and identify pleiotropic loci. In this study, we assess contributions to constructing a well-defined GPN with a clear representation of genetic associations by comparing the network properties with a random network, including connectivity, centrality, and community structure. Then, we extend our discussion to include two applications of bipartite GPN in disease heritability enrichment analysis and phenome-wide association studies (PheWAS).

Results

We construct network topology annotations of genetic …

Free Probability Theory Over The Scaled Hyperbolic Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a notion of free probability over the scaled hyperbolic numbers. Scaled hypercomplex numbers {D_t} _t∈R are constructed as sub-structures of scaled hypercomplex numbers {H_t} _t∈R under the scales (or, the moments) of the set R of real numbers.We show that if t < 0, then the classical free probability theory covers our free probability on {D_t} _{t< 0}; if t > 0, then our free probability on {D_t} _t>0 is represented by the free probability over the classical hyperbolic numbers D = D₁; and if t = 0, then the free probability on D₀ is actually over the …

From Textbook To Classroom: Gaps In Mathematical Representation And Translation In Nepali Education, Deepak Basyal

Mathematics and Statistics

Translating real-world problem contexts into mathematical symbols is a key skill in curricula worldwide. This study examines Nepali students’ abilities to translate words into symbols and how Nepali textbooks from earlier grades support the development of these skills. To investigate this phenomenon, a sequential explanatory study was conducted with a test for 10th graders and an analysis of Grades 6—9 textbooks. Results show that students struggle to convert real-world contexts into mathematical symbols, while textbooks provide limited opportunities for practice. In particular, students have difficulty translating symbols back into real-world situations, suggesting a link between textbook opportunities and achievement. These …