Mathematics Commons^™

Balanced Multi-Party Tournament Designs, Parsa Nematollahe

Honors College Theses

This paper introduces Multi-Party Tournament (MPT) designs that generalize established combinatorial structures, including Whist, Pitch, and Generalized Whist tournament designs. This work will formally define MPTs, establish the fundamental properties of resolvability, fullness, and balance, and formulate a mathematical and algorithmic foundation for multi-party tournament scheduling. The primary contributions of this research are the presentation of necessary and sufficient existence conditions for MPTs across various properties and parameters, the identification of connections between MPTs and other fields of mathematics such as combinatorial design theory, graph theory, and probability theory, and the investigation of MPT construction algorithms, including tree-search, finite-field constructions, …

Geovig And Purevig: Geometry-Aware Architectures For Efficient Computer Vision, Omar Ismail

Theses and Dissertations (Comprehensive)

Deploying deep learning models for medical image analysis on mobile devices requires a balance between inference latency, memory footprint, and delineating anatomical boundaries with high accuracy. While Convolutional Neural Networks (CNNs) and mobile Vision Transformers (ViTs) offer efficiency, they often struggle to model the irregular, non-local geometric structures inherent in biological tissues without incurring prohibitive computational costs. In this thesis, we introduce GeoViG (Geometric Vision Graph), an architecture that bridges the gap between efficient grid-based processing and explicit Geometric Deep Learning. GeoViG introduces a novel transition from high-resolution pixel grids to low-resolution dynamic graphs via a SpreadEdgePool operator, a geometry-aware …

Near Real-Time Adaptive Isotropic And Anisotropic Image-To-Mesh Conversion For Cerebral Aneurysm Simulations, Kevin Garner, Chander Sadasivan, Nikos Chrisochoides

Computer Science Faculty Publications

This paper presents two performance optimization techniques for a mesh adaptation method that is designed to help streamline the discretization of complex vascular geometries within the numerical modeling process. This method is integrated into a pipeline with an image-to-mesh conversion tool to generate adaptive anisotropic meshes from segmented medical images. The pipeline is shown to satisfy quality, fidelity, smoothness, and robustness requirements while providing near real-time performance for medical image-to-mesh conversion. Tested with two brain aneurysm cases and utilizing up to 96 CPU cores within a single, multicore node on Purdue University’s Anvil supercomputer, the parallel adaptive anisotropic meshing method …

All Games Have Equilibria, Arthur Paul Pedersen, M. Ali Khan, Maxwell B. Stinchcombe

Publications and Research

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same …

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 Parallel-In-Time Direct Inverse Method For Nonlinear Differential Equations, Nail K. Yamaleev, Subhash Paudel

Mathematics & Statistics Faculty Publications

We propose a new method for parallelization of the first-order backward difference discretization (BDF1) of the first-order time derivative in nonlinear partial differential equations, such as conservation law equations. The time derivative term is discretized by using the method of lines based on the implicit BDF1 scheme, while the inviscid and viscous terms are approximated by conventional 2nd-order central discretizations of the 1st- and 2nd-order derivatives in each spatial direction. The global system of nonlinear discrete equations in the space-time domain is solved by the Newton method for all time levels simultaneously. For the BDF1 discretization, this all-at-once system at …

Logistic-T Multinomial Mixture Model For Clustering For Microbiome Data, Wenshu Dai, Yuan Fang, Sanjeena Subedi

Mathematics & Statistics Faculty Publications

The logistic-normal multinomial distribution has been used for modelling microbiome data obtained from high-throughput sequencing technologies, which are compositional in nature. A logistic-normal multinomial distribution is a hierarchical multinomial distribution that assumes the latent variable which are the additive log-ratio (ALR) transformed proportions in a multinomial distribution follows a Gaussian distribution. Model-based clustering algorithms have also been developed for clustering microbiome data based on the logistic-normal models. However, the Gaussian assumption may violated when the ALR transformed variable exhibit heavy-tailed distributions or has outliers. Our study introduces a novel mixture of logistic-t multinomial models that effectively address these challenges. Utilizing …

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 …

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 …

Systematics And Systems Theory: Reconstructability Analysis Of The Tetrad, Martin Zwick

Complex Systems Faculty Publications and Presentations

This talk discusses the relationship between systems theory, specifically Reconstructability Analysis, and Systematics, a systems theory-like framework of number symbolism developed by John G. Bennett, which he presented in his four-volume magnum opus, The Dramatic Universe. The talk, given to a community of people interested in Bennett's ideas, focuses on Martin Zwick's paper "Ideas and Graphs: the Tetrad of Activity" archived at https://archives.pdx.edu/ds/psu/36249.

Universal Systems Simulation Via Constraint Hypergraphs With Applications To Digital Twins, John Morris

All Dissertations

The characterization of systems encompasses a variety of modeling frameworks designed to capture specific behaviors and components of various system domains. Whatever the framework, the core elements of a system representation are the information of the system and a description of how that information is related. The relations in deterministic systems are functions, which, when composed to form executable processes, can be used to simulate system data. A declarative modeling framework is one that encodes mechanisms for preparing these simulations within the model structure, allowing an external agent to form the execution processes required for a given context. To date, …

Content And Consequences: Impact Of Representation In Stem Higher Education Instructional Content On Marginalized Students, Nichole Ventura

Doctorate in Education

This qualitative study examined representation of historically marginalized students in STEM instructional content at the higher education level and its impact on their learning experiences. Despite growing diversity initiatives in STEM enrollment, curricular materials often fail to reflect the identities of underrepresented students. Using critical theory and interpretivist approaches, this research investigated how representation—or its absence—shapes students' sense of belonging, academic identity formation, and persistence. Through semi-structured interviews with undergraduate students from historically marginalized backgrounds, and purposeful sampling, this study captured the lived experiences of students engaging with STEM instructional materials. Interview protocols explored how students perceive their representation in …

A Leslie System For A Demographic Simulation: From An Actuarial Point Of View, David Kings

Electronic Theses and Dissertations

This thesis develops a discrete stochastic linear systems interpretation of age–stage demographic evolution grounded in Leslie operators and realized in a discrete-event simulation implemented with salabim. The central claim is that one annual cycle of the simulation constitutes a cone-preserving, stochastic affine transformation on a high- dimensional population state vector indexed by age, sex, marital status, household type, employment, and education, and that the composition of yearly operators yields a random matrix product whose top Lyapunov exponent is the stochastic counterpart of the Perron–Frobenius growth rate (Caswell, 2001; Tuljapurkar, 1997)[1, 2]. The actuarial bridge is constructed by mapping simulated survival …

Modeling Synaptic Dysfunction As Neural Contagion: A Graph-Based Sedr Framework For Simulating Signal Spread, Michelle Marfo, Dr. Padmanabhan Seshaiyer, Alonso Ogueda-Oliva

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Modeling The Cancer Cell Growth Predictions Based On Classical Mathematical Models With Physics-Informed Neural Network, Widodo Samyono

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Modeling The Probability Of N Clonal Rosettes In A Bromeliaceae Genetic Individual, Erin N. Bodine, Layla K. Lammers

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Tight Spherical Embeddings (Updated Version), Thomas E. Cecil, Patrick J. Ryan

Mathematics and Computer Science Department Faculty Scholarship

This is an updated version of the paper [14] which appeared in the proceedings of the 1979 Berlin Colloquium on Global Differential Geometry. This paper contains the original exposition together with some notes by the authors made in 2025 (as indicated in the text) that give references to descriptions of progress made in the field since the time of the original version of the paper. The main result of this paper is that every compact isoparametric hypersurface Mⁿ ⊂ Sⁿ⁺¹ ⊂ Rⁿ⁺² is tight, i.e., every non-degenerate linear height function ℓ_p, p ∈ …

Memoir On A General Property Of A Very Extensive Class Of Transcendental Functions, Niels Henrik Abel 1802--1829, John Little

Mathematics and Computer Science Department Faculty Scholarship

We present this new commentary and translation anticipating the 200th anniversary of the work, commonly known as Abel's ``Paris memoir.'' This is recognized today as one of Abel's most original and influential works. It is significant mostly because it marked the first appearance of a form of a result in the theory of algebraic curves and Riemann surfaces that has come to be known as ``Abel's theorem.'' However, Abel's original understanding of the meaning and context of his result was quite different from the typical modern formulation and the development of the modern understanding has been a long and tortuous …

Constructions Of Compact Dupin Hypersurfaces With Non-Constant Lie Curvatures, Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

A hypersurface M in the unit sphere Sⁿ ⊂ Rⁿ⁺¹ is Dupin if along each curvature surface of M, the corresponding principal curvature is constant. If the number g of distinct principal curvatures is constant on M, then M is called proper Dupin. In this expository paper, we give a detailed description of two important types of constructions of compact proper Dupin hypersurfaces in Sⁿ. One construction was published in 1989 by Pinkall and Thorbergsson [35], and the second was published in 1989 by Miyaoka and Ozawa [26]. Both types of examples have the …

Sugar: A Sequence Unfolding Based Transformer Model For Group Activity Recognition, Yash U. Gondkar

Graduate Masters Theses

Large Language Models have improved significantly in the past couple of years due to the adoption of transformers. However, transformers still find it challenging to process videos due to limited context size caused by their quadratic computing cost. Therefore, we studied a booming field in machine learning which powers applications like social scene analysis and video surveillance systems called Group Activity Recognition (GAR). We found that recent models were able to achieve more than 90% accuracy on popular datasets like the Volleyball dataset, however, it turned out that even they relied on transformers.

Therefore, in this work, we developed a …

A Backtracking Algorithm For Determining The Existence Of Regular Graphs Of Specified Girth And Excess, Stetson Ray Bosecker

Doctoral Dissertations

The study of cages focuses on finding (k, g)-graphs of minimal order. This dissertation generalizes the problem of finding cages to the determination of graphs with specified excess, thereby broadening the significance of the results. The (k, g, ε)-graph problem seeks to determine the existence or nonexistence of k-regular graphs with girth g and excess ε = n(G)−M(k, g) (where M(k, g) represents the Moore bound for cage graphs). Motivated by heuristic methods used to determine properties within the study of cages, we present a backtracking algorithm capable of constructing (k, g, ε)-graphs or determining their nonexistence. Chapter 2 provides …

Privacy-Preserving Structure Learning For Geospatial Data Using Information-Theoretic Dependency Measures, Ahmed Mudhish

Doctoral Dissertations

This dissertation proposes a privacy-preserving framework for structure learning in Bayesian networks (BNs) that addresses the challenges of distributed geospatial data face. Geospatial datasets often exhibit region-specific patterns such as sparsity and nonlinear dependencies. These patterns undermine the effectiveness of traditional machine learning models. Additionally, learned BN structures may reveal sensitive relationships in the generated graph by BNs. These relationships pose a significant privacy risk if reverse-engineered. To address these issues, three novel algorithms are introduced. First, the Selective Naïve Bayes with HSIC (SNB-HSIC) algorithm applies a kernel-based dependency measure to filter redundant and irrelevant features in sparse datasets, improving …

Traffic Prediction For Research And Education Networks: Anomaly-Aware Deep Learning And Benchmarking, Mohammad Arafath Uddin Shariff

School of Computing: Dissertations, Theses, and Student Research

Research and Education Networks (RENs) and High-Performance Computing (HPC) environments are critical infrastructures for modern scientific discovery, demanding sustained high-throughput and low-latency data transfers. Unlike commercial networks, RENs exhibit unique traffic characteristics, including predominant “elephant flows,” inherent burstiness, and complex temporal-spatial dynamics often decoupled from human-driven cycles. Traditional traffic forecasting methods, tailored for commercial Wide Area Networks (WANs), consistently fail to capture these distinct REN dynamics, leading to inefficient resource management and potential impediments to scientific progress.

This thesis addresses this critical gap by developing and validating a robust, scalable, and anomaly-aware traffic forecasting framework specifically tailored for REN/HPC networks. …

Rethinking Iterative Proportional Fitting: Scalable And Hybrid Approaches To Joint Distribution Fitting, William Ofosu Agyapong

Open Access Theses & Dissertations

The Iterative Proportional Fitting (IPF) algorithm is widely used in contingency table estimation, survey weighting, and synthetic population generation due to its simplicity and strong theoretical foundation for matching observed marginal distributions. However, in high-dimensional settings, IPF faces substantial computational and memory demands, as well as statistical instability caused by sparse contingency tables. Moreover, IPF is less useful in modern population synthesis tasks that require both scalability and realism because, despite its superiority in matching known marginal distributions, it cannot produce realistic out-of-sample data points. To address these limitations, we first propose a blockwise IPF framework, in which the feature …

Laser Scan Path Design For Controlled Microstructure In Additive Manufacturing With Integrated Reduced-Order Phase-Field Modeling And Deep Reinforcement Learning, Augustine Twumasi

Open Access Theses & Dissertations

Laser Powder Bed Fusion (L-PBF) is a well-established additive manufacturing technique for fabricating intricate metal components with exceptional precision. A significant challenge in L-PBF is the formation of complex microstructures that influence final material properties. We propose a physics-guided, machine learning-aided approach to optimize scan paths for desired microstructure outcomes, such as equiaxed grains. We employed a phase-field method (PFM) to model the evolution of the crystalline grain structure. To reduce computational costs, we trained a surrogate machine learning model, a 3D U-Net convolutional neural network, using single-track phase-field simulations with varying laser powers to predict crystalline grain orientations based …

Algebraic Multigrid Methods For Nonsymmetric And Indefinite Problems: Theory And Applications, Ahsan Ali

Mathematics & Statistics ETDs

Algebraic multigrid (AMG) is a well-established and highly efficient solver for symmetric positive definite (SPD) systems arising from elliptic and parabolic PDEs, while nonsymmetric systems from hyperbolic PDEs remain a significant challenge. This dissertation develops AMG methods and theory for nonsymmetric problems. First, we develop a novel approach combining mode constraints from energy-minimization AMG with local approximations of ideal restriction in $\ell$AIR, resulting in constrained $\ell$AIR (C$\ell$AIR), which demonstrates scalable convergence across advective and diffusive problems. Second, we extend optimal AMG theory by deriving spectral radius estimates for the two-grid error transfer operator using matrix-induced orthogonality, enabling convergence predictions for …

Modern Procedural Terrain Generation Techniques And Their Background, Hunter A. Barton

2025 Symposium

Procedural terrain generation has become a staple in many digital environments, enabling the automated creation of large-scale and realistic landscapes for applications such as video games and movies. This paper provides an in-depth look at smooth noise functions and their use for terrain generation, as well as an overview of some more modern methods of generation. A method utilizing machine learning stlye transfer was reproduced for this paper with some alterations to improve visualization and realism.

Exact Sampling Of The Six-Vertex Model Using Coupling From The Past, Malaeka Amir

DePaul Discoveries

This paper aims to explore the six-vertex model through simulations designed to investigate the behavior of configurations under specific domain wall boundary conditions. To generate random configurations, we employ the Markov Chain Monte Carlo method while addressing the challenge of mixing times by utilizing the Coupling from the Past (CFTP) algorithm. Implemented in Python, our approach leverages CFTP to ensure exact sampling, avoiding the uncertainty of convergence in traditional Monte Carlo methods. We explore the monotonicity property within this framework and prove that it is only maintained by the steps of this algorithm for very particular values of the parameters.

Exploring Communication In Multi-Agent Cooperative Reinforcement Learning, Matthew Kalarickal

Math and Computer Science Honors Theses

This work focuses on communication strategies within a cooperative multi-agent reinforcement learning system. The goal is to explore how communication can be used among agents to potentially improve performance. The research operates within the scope of “learning tasks with communication,” where the primary aim is to solve domain-specific tasks through information exchange using explicit communication protocols. Three distinct communication strategies were implemented and explored: Combinatorial Ghost, Feature Sharing Ghost, and Move Sharing Ghost. Fully Centralized Training and Execution and Centralized Training with Decentralized Execution training approaches were based on the different communication strategy used. Performance metrics were recorded for these …

Machine Learning: Neural Networking With Relu And Optimization, Aidan Redmond Brownell

Undergraduate Theses, Capstones, and Recitals

At its core, learning is an algorithmic process: it begins with input data, undergoes a series of transformations or computations, and yields an output intended to solve a specific task. This output is then compared against a target or desired result, and the internal mechanisms are updated based on how well the output aligns with expectations. While this feedback-driven process occurs almost effortlessly in humans, it is a far more structured, deliberate, and computationally intensive undertaking for machines.