Digital Commons Network^™

Completing Multi-Latin Rectangles Via Factors With Prescribed Degrees In Bipartite Graphs, Amin Bahmanian

Faculty Publications – Mathematics

Let Q be an n x n array whose top left r x s sub‐array L is filled with a set of k different symbols such that each cell of L contains λ symbols. In this note, we find conditions under which each empty cell of Q can be filled with λ symbols in such a way that the total number of occurrences of each symbol is prescribed and that each symbol
occurs at most λ times in each row and column of Q. To prove this result, we establish a new criterion for a bipartite graph to have …

Developments On Quantum Phase Space Dynamics And Foundations, Gabriel Nowaskie

Mahurin Honors College Capstone Experience/Thesis Projects

This work explores the development and unification of quantum theory within phase space, with a particular focus on new mathematical structures and solution techniques that extend the standard quantum formalism. Building upon the Quantum Phase Space Representation (QPSR) introduced by Torres-Vega and Frederick, we present several advancements that address longstanding gaps in the formulation and solvability of quantum systems in this framework. We introduce the Half-Transform Ansatz, a novel method for solving the Time-Independent Schrodinger Equation by recasting it into a hypergeometric form, enabling the application of the Nikiforov-Uvarov method. This approach is demonstrated through the analysis of quarkonium systems …

Algebra Structures For The Koszul Homology Of Minimal Intersections, Kathryn A. Grebel

Mathematics Dissertations - Archive

The Koszul homology of a local ring is a powerful tool in commutative algebra as it provides information on the structure and properties of the ring. In this research, we explore the relationship between quotients of regular local rings and their Koszul homology algebra. One such relationship is detailed by the Tate-Assmus theorem, which asserts, in part, that a ring is a complete intersection if and only if the Koszul homology is generated by its degree 1 homology elements. An objective of this research is to examine and identify the properties of a minimal intersection and its Koszul homology algebra. …

Two Network Flow Problems: Volume Inequalities For Flow Polytopes Of Full Directed Acyclic Graphs; Optimal Additions To The Low-Stress Bike Network In Lexington, Kentucky, James F. Mcelroy

Theses and Dissertations--Mathematics

This dissertation addresses two distinct problems related by their foundation in network flows. The first problem concerns volumes of flow polytopes of directed acyclic graphs with out-degree sequence (3,2,...,2,0). It is proved that there is an interchange operation on the edge set of these graphs that induces a partial order on the graphs isomorphic to a Boolean algebra, and that moving up through this partial order decreases (weakly) the volumes of the corresponding flow polytopes. This result is reinterpreted in the context of linear extensions for posets that are bipartite non-crossing trees.

The second problem develops a discrete optimization model …

Novel Generative And Language Model Architectures With Applications, Edison Mucllari

Theses and Dissertations--Mathematics

This dissertation investigates novel architectures to address fundamental challenges in machine learning, particularly focusing on transformer models, recurrent neural networks, GAN and continual learning and their applications in natural language processing and computer vision. We propose the Neumann-Cayley Gated Recurrent Unit (NC-GRU), which leverages a Neumann series-based Scaled Cayley transformation to maintain orthogonal weight matrices, effectively mitigating exploding gradients problems while improving long-term memory retention across prediction tasks. We demonstrate the practical applications of NC-GRU by implementing our proposed architecture into an autoencoder to derive neural molecular fingerprints. Building upon these advancements, we turn our attention to the transformer architecture, …

Blow-Ups And Projectivized Toric Vector Bundles, Sara Church

Theses and Dissertations--Mathematics

This dissertation is set in the intersection of toric geometry, tropical geometry, and the theory of vector bundles. We focus on the geometry of projectivized toric vector bundles and their connections to Mori dream spaces, matroid theory, and tropical geometry. We generalize previous results on the quotient construction of Gonzalez, Hering, Payne, and Suss (GHPS) by introducing a new approach to describing the geometry of these bundles via associated blow-ups. Additionally, we examine tautological bundles arising from representable matroids and establish connections between their geometry and the wonderful compactification. Finally, we consider the case where Klyachko filtrations have maximal steps …

The Computational Algebra Of Conformal Blocks, Casey B. Hill

Theses and Dissertations--Mathematics

Conformal blocks are objects in quantum field theory that arise from conformal trans- formations, which are symmetries that preserve angles but not length. This aspect of conformal field theory has various interactions with algebraic geometry.

In this dissertation, we explore the underlying algebra and geometry of spaces and algebras of conformal blocks over SLn. We then use this information along with techniques from combinatorial commutative algebra, algebraic geometry, and representation theory to find a presentation of the algebra of SL4-conformal blocks. With this presentation, we then use computational methods to learn about some of the geometric properties of this algebra.

Schaper Numbers, Palindrome Partitions, And Symmetric Functions, With Applications To Characters Of The Symmetric Group, Karlee J. Westrem

Dissertations, Master's Theses and Master's Reports

Finding the decomposition numbers for the symmetric group is a difficult problem and has led to many different research directions. For a given Specht module, the Schaper sum formula can be refined with knowledge of the Schaper number for the particular partition to give more precise information about the decomposition numbers. In Chapter 2, we give a combinatorial formula for the Schaper number when the partition has at most two columns. At the end, we provide a conjecture for the Schaper number for the partition of shaper (3^n).

Chapter 3 covers the joint work with my advisor, David Hemmer, and …

The Combinatorics Of Integer Partitions Enumerated By Some Exotic Weights, Hunter Waldron

Dissertations, Master's Theses and Master's Reports

Identities of integer partitions generally state that two dissimilar appearing families of partitions are in fact equinumerous when both are restricted to any fixed size. Euler's theorem is a classic example of such an identity, which equates the number of partitions with odd parts to the number of partitions with distinct parts. Lately, analogs of known partition identities involving weights other than size have begun to attract research interest. This dissertation is an investigation of two such weights. In Chapter 2, we study Schmidt weights, which count only parts with indices belonging to some given subset of the positive integers. …

Measuring The Similarity Between Trees Of Different Order, Camilo Morales

HMC Senior Theses

Graphs encode relationships between data. However, due to their versatility, it is often difficult to generalize the notion of similarity between two graphs using a distance function. Since graphs can represent various data sets, specific distance metrics need to be tailored for questions we are interested in answering or the data set we are working with. This project is motivated by ongoing investigations into the evolution of female gender representation in mathematics. By building off previous work that has taken data from the Mathematics Genealogy Project and modeled this evolution of representation via a tree, we would like to develop …

Enhancement Of Mechanical, Structural, And Electrical Properties In Advanced Composites And Vat Photopolymerized 3d Printing Nanocomposites, Poom Narongdej

CGU Theses & Dissertations

Advanced composites have gained significant attention across various industries, including aerospace, automotive, clean energy, and healthcare, owing to their exceptional mechanical properties and versatility. Fiber-reinforced polymer (FRP) composites, particularly those reinforced with carbon fibers, are extensively used as structural materials in spacecraft, aircraft, high-performance vehicles, and wind turbines due to their high strength-to-weight ratios, stiffness, durability, and tailorable mechanical characteristics. In healthcare, the advent of additive manufacturing (3D printing) has expanded the utility of advanced composites, enabling precise customization of components to meet patient-specific needs while offering design flexibility and ease of fabrication. Despite these advantages, several challenges hinder the …

Mathematics And Determinism: Chaos, Quantum Mechanics, And The Limits Of Predictive Structure, Jackson T. Salumbides

CMC Senior Theses

This thesis examines the relationship between mathematics and determinism by analyzing how chaos theory, quantum mechanics, and formal mathematical limits challenge traditional conceptions of predictability and causal structure. Chaos theory shows that deterministic systems can exhibit practical unpredictability due to sensitivity to initial conditions. Quantum mechanics introduces probabilistic outcomes that complicate deterministic interpretation, though alternative frameworks such as Bohmian mechanics and superdeterminism attempt to restore determinism at conceptual cost. Additionally, results from mathematical logic, including Gödel’s incompleteness theorems and Turing’s undecidability, demonstrate intrinsic limitations on what can be deduced or computed, even in fully deterministic systems. By synthesizing these areas, …

The Effect Of Electrode Geometry On Excited Species Production In Atmospheric Pressure Air-Hydrogen Streamer Discharge, Shirshak Kumar Dhali, Stuart Reyes

Electrical & Computer Engineering Faculty Publications

When a gas is overvolted at or near atmospheric pressure, it results in a streamer discharge formation. Electrode geometries exert significant impact on the electrical breakdown of gases by altering the spatial profile of the electric field. In many applications the efficient generation of radicals is critical and is determined by the characteristics of the streamer discharge. We examine the effect of electrode geometry on the streamer characteristics and the production of radicals. This is performed for three different electrode geometries: plane–plane, pin–plane, and pin–pin. A two-dimensional rotationally symmetric fluid model is used for the streamer discharge simulation in the …

On Linear Invariants Of Hypergraphs, Clara Chaplin

Honors Theses

We introduce linear invariants of hypergraphs as a way to study hypergraphs by their tensor representations. Our primary research goal is to determine what information linear invariants capture about the hypergraphs they arise from. We first investigate the centroid, which is shown to determine the connected components of a hypergraph. Next, we study the derivations of a hypergraph, and use this linear invariant to define a quotient operator $Q_\mathrm{Der}$ on the collection of all hypergraphs. This operator is shown to be a closure operator in that $Q_\mathrm{Der}(Q_\mathrm{Der}(\mathcal{H}))=Q_\mathrm{Der}(\mathcal{H})$ for any hypergraph $\mathcal{H}$. We apply the operator $Q_\mathrm{Der}$ to synthetically generated hypergraphs, …

Developing Mathematical Maturity By Solving A Given Quadratic Equation, Armando A. Amador, Nieves Angulo, Juan B. Lacay

Publications and Research

This article explores the instructional challenges faced by mathematics educators when addressing students’ limited background knowledge and underdeveloped mathematical maturity, two critical barriers to success in college-level algebra. Using the quadratic equation 2x2+x−3=0, which is reducible over the field of rational numbers ℚ ⊂ ℝ, we analyze how diverse solution methods—factoring, completing the square, and the quadratic formula—can support learning. An additional strategy, “Slide and Divide,” was introduced to promote procedural fluency and conceptual flexibility. To gain insight into students’ evolving mathematical maturity, an online survey was conducted to students at various levels of mathematics. The survey captured their perspectives …

Partial Group Divisible 3-Designs, Apiwat Peereeyaphat

Chulalongkorn University Theses and Dissertations (Chula ETD)

We introduce a generalization of group divisible 3-designs, 3-GDDs, with two groups and two indices. A partial group divisible 3-design, 3-PGDD(gı + g2, k; A, (21, 112), is an ordered pair (GrU G2, B) where G, and G2 are disjoint finite sets called groups) of size g1 and 92, respectively; and B is a collection of k-subsets (called blocks) of G, UG, such that every 3-subset of G; occurs in exactly 1 blocks in B, and every i elements of G and j elements of G2 occur together in exactly Mig blocks in B for i. i € (1,2} and …

Uncertain Labeling Graphs And Uncertain Graph Classes (With Survey For Various Uncertain Sets), Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Graph theory, a branch of mathematics, studies the relationships between entities using vertices and edges. Uncertain Graph Theory has emerged within this field to model the uncertainties present in real-world networks. Graph labeling involves assigning labels, typically integers, to the vertices or edges of a graph according to specific rules or constraints. This paper introduces the concept of the Turiyam Neutrosophic Labeling Graph, which extends the traditional graph framework by incorporating four membership values—truth, indeterminacy, falsity, and a liberal state—at each vertex and edge. This approach enables a more nuanced representation of complex relationships. Additionally, we discuss the Single-Valued Pentapartitioned …

Symbolic Hyperplithogenic Set, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Concepts such as Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets have been widely investigated for tackling uncertainty, with numerous applications explored across various domains. As extensions of the Plithogenic Set, the HyperPlithogenic Set and the SuperHyperPlithogenic Set are also recognized. A Symbolic Plithogenic Set (SPS) is a structured set defined by symbolic components 𝑃𝑖 and coefficients 𝑎𝑖 , enabling flexible algebraic operations under a specified prevalence order. In this paper, we examine concepts including the Symbolic HyperPlithogenic Set and the Symbolic 𝑛-SuperhyperPlithogenic Set.

Some Types Of Hyperneutrosophic Set (4): Cubic, Trapozoidal, Q-Rung Orthopair, Overset, Underset, And Offset, Florentin Smarandache, Takaaki Fujita

Branch Mathematics and Statistics Faculty and Staff Publications

This paper builds upon the foundational work presented in [38–40]. The Neutrosophic Set provides a comprehensive mathematical framework for managing uncertainty, defined by three membership functions: truth, indeterminacy, and falsity. Recent advancements have introduced extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Set, which are specifically designed to address increasingly complex and multidimensional problems. The formal definitions of these sets are available in [30]. In this paper, we extend the Neutrosophic Cubic Set, Trapezoidal Neutrosophic Set, q-Rung Orthopair Neutrosophic Set, Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset using the frameworks of the Hyperneutrosophic Set and the SuperHyperneutrosophic Set. Furthermore, …

Some Types Of Hyperneutrosophic Set (3): Dynamic, Quadripartitioned, Pentapartitioned, Heptapartitioned, M-Polar, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper builds upon the foundation established in [50, 51]. The Neutrosophic Set provides a robust mathematical framework for handling uncertainty, defined by three membership functions: truth, indeterminacy, and falsity. Recent developments have introduced extensions such as the Hyperneutrosophic Set and SuperHyperneutrosophic Set to tackle increasingly complex and multidimensional problems. In this study, we explore further extensions, including the Dynamic Neutrosophic Set, Quadripartitioned Neutrosophic Set, Pentapartitioned Neutrosophic Set, Heptapartitioned Neutrosophic Set, and m-Polar Neutrosophic Set, to address advanced challenges and applications.

Plithogenic Duplets And Plithogenic Triplets, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

A Neutrosophic Set is a mathematical framework that represents degrees of truth, indeterminacy, and falsehood to address uncertainty in membership values [41, 42]. In contrast, a Plithogenic Set extends this concept by incorporating attributes, their possible values, and the corresponding degrees of appurtenance and contradiction [50]. Among the related concepts of Neutrosophic Sets, Neutrosophic Duplets and Neutrosophic Triplets are well-known. This paper defines Plithogenic Duplets and Plithogenic Triplets as extensions of these concepts using the Plithogenic Set framework and briefly examines their relationship with existing concepts.

Some Types Of Hyperneutrosophic Set (2): Complex, Single-Valued Triangular, Fermatean, And Linguistic Sets, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper is a continuation of the work presented in [35]. The Neutrosophic Set provides a mathematical framework for managing uncertainty, characterized by three membership functions: truth, indeterminacy, and falsity. Recent advancements have introduced extensions such as the Hyperneutrosophic Set and SuperHyperneutrosophic Set to address more complex and multidimensional challenges. In this study, we extend the Complex Neutrosophic Set, Single-Valued Triangular Neutrosophic Set, Fermatean Neutrosophic Set, and Linguistic Neutrosophic Set within the frameworks of Hyperneutrosophic Sets and SuperHyperneutrosophic Sets. Furthermore, we investigate their mathematical structures and analyze their connections with other set-theoretic concepts.

Soft Directed N-Superhypergraphs With Some Real-World Applications, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper introduces the Directed Soft Super Hyper Graph, a unified framework for modeling complex, multi-layered directed networks. It combines directionality, recursive hyperstructure, and soft-set parameterization to address the integration of Soft Super HyperGraphs and Directed SuperHyperGraphs, which remains largely unexplored. The paper provides formal definitions, core operations, and real-world examples, such as urban infrastructure and transportation networks, to demonstrate the framework's effectiveness in managing deep hierarchies and uncertain relationships simultaneously.

Beyond Dialectics, Paradoxes, And Binary Logic, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Philosophy, long defined by its pursuit of truth, has historically been a battleground for dichotomies: truth vs. falsehood, materialism vs. idealism, reason vs. emotion. These oppositions often provide a framework for understanding philosophical discourse, but they fail to capture the full nuances of reality. To challenge these binary oppositions, I introduced the neutrosophic perspective in philosophy, rooted in Mathematics, and Many-Valued Logics.1 By emphasizing the interrelation of affirmation, negation, and neutrality, neutrosophy allows for the reconciliation of seemingly irreconcilable viewpoints, providing a new lens through which to reinterpret age-old philosophical questions.

A New Simulation Framework For Analyzing Neutrosophic Data In Experimental Design, Muhammad Aslam, Nasrullah Khan, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

A recent simulation-based classical analysis has been developed for interval data. However, a review of the literature indicates that these existing simulations have notable limitations and fail to conform to the neutrosophic statistical framework. In this paper, we propose a novel simulation process designed to analyze neutrosophic data within an appropriate and rigorous neutrosophic framework. We demonstrate that the proposed simulation is more comprehensive and aligns closely with the principles of neutrosophic theory. The results will be obtained through simulation and compared with those of existing methods, with the expectation that the proposed approach provides substantial improvements and is better …

A Plausible Formal Correspondence Between Tetrahedral Condensates/Tsc And Pt-Symmetric Crystals Model Of Cmns (Aka. Low-Energy Nuclear Reactions), Victor Christianto, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Akito Takahashi's Tetrahedral Symmetric Condensate (TSC) model, detailed in several of his earlier works1 proposes a mechanism for condensed matter nuclear science (CMNS) aka. low-energy nuclear reactions (LENR) within palladium lattices. The model centres on the formation of a tetrahedral cluster of deuterons, enhancing the probability of nuclear fusion. Here, we explore the possibility of extending this framework by considering the TSC within a more general crystalline solid with tetrahedral symmetry, and by approximating the screening potential experienced by the deuterons using PT-symmetric potentials.

Moore Graphs, Trevor Saxton

Williams Honors College, Honors Research Projects

A Moore graph is a simple regular graph, with n vertices, degree d, and diameter k, that satisfies the Moore bound: n = 1 + d (d − 1)k − 1 d − 2 . There are graphs for which the bound is met and in which existence and uniqueness are known. For k = 2 it is known that the Moore bound is achieved for d = 2, 3, 7, with the case of d = 57 conjectured to exist. For k = 3 the bound is achieved for only d = 3 [5]. Due to the construction of …

Investigation Of Dynamic Adsorption And Desorption Of Polymer Nanogel In Porous Media Through Microfluidics, Junchen Liu, Fuqiao Bai, Abdulaziz A. Almakimi, Mingzhen Wei, Xiaoming He, Ibnelwaleed A. Hussein, Baojun Bai

Mathematics and Statistics Faculty Research & Creative Works

Understanding the transport and retention of elastic nanogel and microgel particles in porous media has been a significant research subject for decades, essential to the application of enhanced oil recovery (EOR). However, a lack of dynamic adsorption and desorption studies, in which the kinetics in porous media are seldom investigated, hinders the design and application of polymer nanogel in underground porous media. In this work, we visualized and quantified the transport and dynamic adsorption of polymer nanogel in 3D glass micromodels that were manufactured by packing glass beads in capillaries. Calibrating the linearity of fluorescence intensity to concentration, we calculated …

Analyzing Car Theft Trends In Central Texas: A Comparative Study Of Waco, College Station, And Killeen, Daniel Njogu

Williams Honors College, Honors Research Projects

This study examines motor vehicle theft (MVT) trends from 2019 to 2023 in three Central Texas cities—Waco, College Station, and Killeen—using temporal analysis, geospatial hotspot mapping, and make/model data. In Killeen, thefts generally rose over the period, with notable peaks in October and on Mondays. College Station saw an overall decline in thefts but experienced a seasonal spike each March, and Waco’s thefts increased until around 2021 before beginning to fall. Local festivals—such as the Spirit of Texas in College Station and the Heart O’ Texas Fair in Waco—appear to coincide with these seasonal upticks. Hyundais and Kias were most …

0th Order Solutions Of The Wavefunctions For The Quantum Elliptical Box And Microstrip Antenna, Nishtha Tikalal

Honors Undergraduate Theses

For a quantum particle confined to a two-dimensional elliptical box or electromagnetic wave in a microstrip antenna, geometrical and boundary condition interplay result in a spectrum of spatial patterns. Due to the asymmetrical nature of the ellipse, we are faced with continuous symmetry reductions, leaving both degenerate and nondegenerate solutions. Here, we present a complete derivation of an analytical solution and visualizations of the fundamental wavefunctions for both Dirichlet and Neumann boundary conditions respectively corresponding to the quantum elliptical box and the elliptical microstrip antenna.

We demonstrate that the eigenmodes, governed by eccentricity, directly correspond to the modal field distributions …