Digital Commons Network^™

Dataless Neural Networks For Boolean Satisfiability And Network Optimization, Andrew Evan Gautier

Graduate Theses, Dissertations, and Problem Reports (ETD)

Combinatorial optimization problems (COPs) require searching over a finite solution space subject to constraints, with the goal of satisfying an objective function. They arise in operations research, scheduling, resource allocation, circuit design, and many other fields. Many problems in combinatorial optimization (including satisfiability and network design) are NP-hard. Traditionally, researchers have built approximate solvers that return near- optimal solutions efficiently by developing increasingly sophisticated heuristics and meta- heuristics. Deep learning has provided new opportunities for improving combinatorial solvers by leveraging neural guidance to prune the search space. Traditional neural networks have distinct drawbacks in this context: separate training and …

Adaptive Multi-Grade Deep Learning For Highly Oscillatory Fredholm Integral Equations Of The Second Kind, Jie Jiang, Yuesheng Xu

Mathematics & Statistics Faculty Publications

This paper studies the use of Multi-Grade Deep Learning (MGDL) for solving highly oscillatory Fredholm integral equations of the second kind. We provide rigorous error analyses of continuous and discrete MGDL models, showing that the discrete model retains the convergence and stability of its continuous counterpart under sufficiently small quadrature error. We identify the DNN training error as the primary source of approximation error, motivating a novel adaptive MGDL algorithm that selects the network grade based on training performance. Numerical experiments with highly oscillatory (including wavenumber 500) and singular solutions confirm the accuracy, effectiveness and robustness of the proposed approach.

A New Functional Setting For Term Structure Modeling Using The Health-Jarrow-Morton Framework, Michael Pokojovy, Ebenezer Nkum, Thomas M. Fullerton Jr.

Mathematics & Statistics Faculty Publications

The well-known Heath–Jarrow–Morton (HJM) framework provides a universal and efficacious instrument for modeling the stochastic evolution of an entire yield curve by explaining the interest rate dynamics in continuous time under no-arbitrage conditions. Existing implementations involve exponentially weighted function spaces as theoretical settings for the former stochastic evolution. While the choice of weight can have a drastic effect on model calibration and subsequent forecasting, it cannot be estimated from market data and does not allow for any objective interpretation. The proposed approach does not have this shortcoming as it adopts a suitably designed unweighted function space. The HJM equation is …

The Second Minimum Size Of A Finite Subspace Partition, Esmeralda Năstase, Papa Sissokho

Faculty Publications – Mathematics

Let V = V (d, q) denote the vector space of dimension d over Fq . A subspace partition P of V , also known as a vector space partition, is a collection of nonempty subspaces of V such that each nonzero vector of V is in exactly one subspace of P. Motivated by applications of minimum blocking sets and maximal partial t-spreads, Beutelspacher (Geom Dedic 9:425– 449, 1980) determined in a lemma the minimum possible size δ(d) over all (nontrivial) subspace partitions of V . In Heden et al. (Des Codes Cryptogr 64:265–274, 2012) and N˘astase …

Second And Fifth Graders’ Changes In Strategies For Integer Addition, Mahtob Aqazade, Laura Bofferding

Faculty Publications – Mathematics

Research on integer addition often attributes students’ performance to their grade level while ignoring the role of prior whole number knowledge in influencing their learning difficulties. To explore whether these difficulties are related to their grade level or persistence in using rules based on their whole-number experiences, we first paired three second-grade students with three fifth-grade students who had similar integer addition strategies on their pretest. Then, we compared their strategy changes and performance after small-group sessions and whole-class activities around integer addition (i.e., instructional intervention). Although all pairs gained from the pretest to posttest, one fifth grader still relied …

Connected Fair Detachments Of Hypergraphs I, Amin Bahmanian

Faculty Publications – Mathematics

Let G be a hypergraph whose edges are colored. An (α, n)-detachment of G is a hypergraph obtained by splitting a vertex α into n vertices, say α1, . . . , αn , and sharing the incident edges among the subvertices. A detachment is fair if the degree of vertices and multiplicity of edges are shared as evenly as possible among the subvertices within the whole hypergraph as well as within each color class. In this paper we solve an open problem from the 1970s by finding necessary and sufficient conditions under which a k-edge-colored hypergraph …

Exponentially Many Correspondence Colourings Of Planar And Locally Planar Graphs, Luke Postle, Evelyne Smith-Roberge

Faculty Publications – Mathematics

We show that there exists a constant c>0 such that if G is a planar graph with a 5-correspondence assignment (L,M), then G has at least 2c⋅v(G) distinct (L,M)-colourings. This confirms a conjecture of Langhede and Thomassen. More broadly, we introduce a general method showing how hyperbolicity theorems for certain families of critical graphs can be used to derive lower bounds on the number of colourings of the associated class of planar graphs. Hence our main result follows from this method plus a technical theorem (that we proved in a previous paper) involving the hyperbolicity …

(R2151) Error Estimates Of Barycentric Lagrange Interpolation, Alvira Yawar, Swarnima Bahadur

Applications and Applied Mathematics: An International Journal (AAM)

Barycentric interpolation, which comes from Lagrange interpolation, is a useful method in numerical analysis. In this research paper, we explain how the barycentric interpolation formula is derived and discuss its features. We compare its stability and performance with the traditional Lagrange formula. First, we show how to get the barycentric formula from the Lagrange polynomial and present it as a rational function. We also provide an estimate of the error. Then, we use numerical examples to show that the barycentric formula is more stable and works better, especially when the degree of interpolation is high. Our results show that the …

Optimal Control Of Stochastic Systems: A Numerical Study Of The Stochastic Linear Quadratic Regulator Framework, Gülşen Orucova Büyüköz, Yaprak Güldoğan Dericioğlu, Tuğçem Partal

Mathematical Modelling and Numerical Simulation with Applications

Optimal control of stochastic linear systems is fundamental in control theory, with applications in robotics, finance, and engineering. The Stochastic Linear Quadratic Regulator (SLQR) derives optimal feedback laws via the Riccati equation but requires numerical discretization of the resulting stochastic dynamics. Despite extensive studies on numerical methods for stochastic differential equations, their performance within the SLQR framework remains insufficiently explored. This study compares two predictor–corrector schemes of different orders: the Order 1.0 Predictor-Corrector (PC) method and the Order 2.0 Weak PC method. A one-dimensional linear quadratic problem with a closed-form solution enables precise error evaluation against the analytical trajectory. Convergence …

Exploring The Metric Dimension Of The Graph Representing A Four-Ring Polycyclic Aromatic Hydrocarbon: Interstellar 1-Cyanopyrene, Andrei Matthew V. Robino, Sebastian Alex C. Traviña, Karl Benedict P. Narag, Mark Anthony B. Casao, Helix Raven C. Llanes, Azriel C. Alejo, Jonel C. Celamor

Sinaya: A Philippine Journal for Senior High School Teachers and Students

A graph's metric dimension is a parameter that denotes the minimum possible number of vertices in a subset that gives each vertex in the graph a unique representation. Despite extensive research on the metric dimension of polycyclic aromatic hydrocarbons (PAHs), there is currently no focused analysis on the metric basis and metric dimension of 1-cyanopyrene. In chemistry, graphs can be used to depict molecular structures. In this study, graph theory, specifically the concept of metric dimensions, is applied to the molecular structure of 1-cyanopyrene, which was detected in space for the first time in 2024. 1-cyanopyrene is a molecule made …

Locally Integral Involutive Po-Semigroups, José Gil-Férez, Peter Jipsen, Melissa Sugimoto

Mathematics, Physics, and Computer Science Faculty Articles and Research

We show that every locally integral involutive partially ordered semigroup (ipo-semigroup) A=(A,≤,⋅,∼,−), and in particular every locally integral involutive semiring, decomposes in a unique way into a family {Ap:p∈A+} of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are unital semirings. Moreover, we show that there is a family of monoid homomorphisms Φ={φpq:Ap→Aq:p≤q}, indexed over the positive cone (A+,≤), so that the structure of A can be recovered as a glueing ∫ΦAp of its integral components along Φ. Reciprocally, we give necessary and sufficient conditions so that the Płonka sum of any family …

Integration Of Islamic Values In Mathematics Learning, Ai Mulya, Surya Faradiba

Jurnal Pendidikan Sains

This study aims to identify the integration of Islamic values in mathematics learning and its contribution to student character development. The method used is a systematic literature study by reviewing relevant accredited scientific articles. The results of the study indicate that the integration of Islamic values is carried out through the preparation of Islamic-context-based mathematics problems, linking concepts to verses of the Qur'an and hadith, the application of ethnomathematics with Islamic cultural nuances, and the introduction of Islamic economic concepts. The dominant values that emerge include honesty, responsibility, discipline, justice, and piety. These findings indicate that the integration of Islamic …

An Exploratory Approach For Teaching Systems Of Linear Odes With Real Repeated Eigenvalues, Biyong Luo, Li Zhang

CODEE Journal

In this paper, we present a project that aims to pique students' interest and facilitate their engagement in learning. Students will be encouraged to take an active role in their learning of systems of linear ODEs with real repeated Eigenvalues. Through discussion and completion of a set of guided questions given in the project, students will gain a deeper understanding of the topic. Also, students will use the technology tool Bluffton App to draw solution trajectories in a phase plane, and then learn necessary skills for explaining the movements of the trajectories and their long term behaviors. This project may …

Integrable Discrete Massive Thirring Model, Junchao Chen, Bao-Feng Feng

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we are concerned with integrable semi- and fully discrete analogues of the massive Thirring model in light core coordinates. By using the Hirota’s bilinear approach and the Kadomtsev-Petviashvili (KP) hierarchy reduction method, we propose both the semi- and fully discrete massive Thirring models and construct their multi-bright soliton solutions.

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 …

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.

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 …

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 …