Mathematics Commons^™

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 …

Mathematics And Political Ideology: A Historical Argument Against The Cultural Understanding Of Mathematics As An Apolitical Field, And A Journalistic Report On The Impact Of The Trump Administration On American Mathematics In 2026, Philo Judson

Pitzer Senior Theses

The interplay between political ideology and mathematics is a recurring theme throughout the history of the academic field. Mathematics has been used as a tool of empire, while mathematicians have been elevated by states as symbols of national genius for political prestige. Political ideologies have also shaped the field from within, both through the suppression of mathematical thought and the enforcement of mathematical authority. Since 2025, the Trump administration has launched large-scale political attacks on American institutions of higher education, which include lawsuits, discrimination investigations, and the removal of federal funding from certain institutions that don’t adhere to the administration’s …

Uncertainty Quantification, Propagation & Conjunction Assessment In Orbital Mechanics Using Generalized Polynomial Chaos Expansion & 2-Dimensional Conjunction Plane Analysis Techniques, Monalisa Karim

Mechanical and Aerospace Engineering Theses

Uncertainties, that are inherent to dynamic models, can be associated with state initial conditions, force modelling errors, navigation and actuation errors. In system modelling stochastic differential equations are used to represent dynamic phenomena with uncertainties, for which the solutions are probability density functions of quantities of interest characterizing the realization of the stochastic processes. In Polynomial Chaos Expansion (PCE) propagation, these solutions are represented as weighted sums of multivariate spectral polynomials that are functions of the input random variables. Generalized polynomial chaos expansion (gPC) is an extension to the original homogenous PCE which projects the random solution onto a basis …

Adaptive Control For A Robotic Bipedal Device Using A Hybrid Discrete-Continuous Reinforcement Learning Strategy, Karla Rincon-Martinez, Wen Yu, Isaac Chairez

Mathematics Faculty Publications

This research develops and implements a novel reinforcement learning (RL) architecture to address the trajectory-tracking problem in bipedal robotic systems under articulated-joint constraints. The proposed RL framework extends previously designed adaptive controllers characterized by state-dependent gain structures. The learning mechanism comprises two hierarchical adaptation layers: the first employs an adaptive dynamic programming (ADP) formulation to approximate the Bellman value function using a class of continuous-time dynamic neural networks. In contrast, the second uses an iterative optimization scheme based on the deep deterministic policy gradient (DDPG) algorithm. The resulting control strategy minimizes a robust performance index defined over the tracking trajectories …

A Note On Asymptotics Of Estimators For Axially Symmetric Processes On The Sphere, Haimeng Zhang, Chunfeng Huang, Xiaohuan Xue, A.L.A.R.R. Thanuja, Bukola O. Adaramola

Research, Publications & Creative Work

Axially symmetric processes, those stationary in longitude but nonstationary across latitude, provide a flexible and physically meaningful class of models for global environmental data. Despite their wide use, the asymptotic properties of classical method-of-moments (MOM) estimators for these processes remain largely unexamined. In this work, we investigate MOM estimators of covariances and cross-variograms for axially symmetric Gaussian processes observed on regular latitude-longitude grids. First, we show that MOM covariance estimators are asymptotically biased. We then examine MOM estimators of cross-variograms, and prove that they are unbiased. However, using the block circulant structure of the covariance matrix and its Fourier diagonalization, …

A Survey Of Ai Methods For Geometry Preparation And Mesh Generation In Engineering Simulation, Steven Owen, Nathan Brown, Nikos Chrisochoides, Rao Garimella, Xianfeng Gu, Franck Ledoux, Na Lei, Roshan Quadros, Navamita Ray, Nick Winovich, Yongjie Jessica Zhang

Computer Science Faculty Publications

Artificial intelligence is beginning to reduce the manual effort in the CAD-to-mesh pipeline. Written for meshing and geometry practitioners with limited AI background, this survey organizes recent work by workflow step. We cover part classification and segmentation, mesh quality prediction, and defeaturing. We review AI guidance for unstructured meshing, block-structured meshing in 2D and 3D, and volumetric parameterization, including reconstruction from implicit or sampled geometry. We also discuss parallel mesh generation and scripting automation via reinforcement learning and large language models. Across these topics, AI complements established geometry and meshing algorithms rather than replacing them. We conclude with practical lessons …

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 …

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, …

Math Anxiety, Math Self-Concept And Math Self-Efficacy: A Study Of The Jingle-Jangle Fallacies, Marsha Natasha Durrant-Walker

Dissertations

Problem

The overlap and lack of clear distinction among the constructs of math anxiety, math self-concept, and math self-efficacy presents issues for research and practice. The literature reveals that math anxiety is closely linked to math self-concept (Klee et al., 2022). Additionally, math self-concept and math self-efficacy often overlap and are not easily distinguishable (Kranzler & Pajares, 1997; Pajares & Miller, 1994; Pajares & Urdan, 1996). Each of these constructs has been shown to play a critical role in student math achievement (Timmerman et al., 2016). -- When constructs are not defined or measured distinctly, inconsistencies may emerge in research …

Analyzing Sleep Architecture And Brain State Transitions Via Hidden Markov Models On Fmri Data, Brileigh Jay Cates

Masters Theses

Sleep is associated with systematic changes in brain activity and functional connectivity observable in functional magnetic resonance imagining (fMRI) signals. Because subjects often fall asleep during resting-state experiments, the absence of vigilance monitoring can confound the interpretation of resting-state dynamics. Although electroencephalography (EEG) is the gold standard for sleep staging, simultaneous EEG-fMRI acquisition is not always feasible.

This study investigates whether sleep stages can be inferred directly from fMRI using a probabilistic latent-state framework. Hidden Markov Models (HMMs) are applied to blood-oxygen-level-dependent (BOLD) time series to identify latent brain states and their temporal transitions. Inferred states are aligned with EEG-derived …

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 …

An Energy-Stable Mixed Cg-Dg Scheme For A Full Shliomis Phase Field Model Of Two-Phase Ferrofluid Flows, Mahdi Gharehbaygloo

Doctoral Dissertations

"Ferrofluids are magnetic nanoparticle suspensions whose motion couples surface tension, flow field, magnetostatics, and magnetization dynamics. This dissertation develops, analyzes, and validates an energy-stable finite element method for a two-phase ferrofluid model that couples the Cahn-Hilliard equations with the full Shliomis model of single-phase ferrofluids, retaining its damping torque term, magnetic torque term, and magnetic stress term.

The spatial discretization is a mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) formulation. It uses continuous ��2 elements for the phase field, chemical potential, velocity, and magnetostatic potential, discontinuous ��2 elements for the magnetization, and discontinuous ��1 elements for the pressure. The …

Using Provided Guided Notes In Coordinated Introductory First-Year Mathematics Courses, Jennifer L. Huber

Mathematics Dissertations

The goal of this study is to investigate how standardized guided notes shape instructional practices and student engagement in coordinated introductory first-year college mathematics courses at a large public university. The researcher explored three multi-section introductory mathematics courses with overlapping learning objectives. Each course required students to purchase a student workbook as part of the instructional materials for the class. The instructors taught primarily from the workbook containing guided notes created by a former coordinator of the course. The researcher used a mixed-methods approach. Instructors and students participated in surveys, class observations and provided class meeting notes. Instructors shared additional …

Pursuer Evader Surveillance Game Control Theory And Motion Planning, Zijie Mu

Honors Theses

This thesis studies the pursuer evader surveillance game with a triangular obstacle in the short-term. In the game, the pursuer aims to maintain surveillance of the evader as long as possible while the evader aims to break surveillance in a finite time. We classify the player strategies into ideal ones and best admissible ones. The outcome of the game is determined by line of sight. We reduce the 4D game to a 3D game with an upward motion for a small time interval to terminate the game. When the evader starts outside the threshold, we show that there exists an …

Investigating The Connection Between Als Through The Mutation R522s In The Rna Binding Protein, Dennia Estrella-Vargas, Lydia Uptain

Mathematics

Amyotrophic lateral sclerosis (ALS) is a fatal disease that causes the deterioration of motor neurons , death is usually due to respiratory paralysis. The variant R522S was chosen because it is near a hot spot of pathogenic variants. It is an arginine-to-serine swap, this swap is present in pathogenic variants near the 522 position, such as R514S, R521S, R524S. Recent evidence suggests that arginine-deficiency can influence disease progression.

On Stripping And Antipodes In Motivic Steenrod Algebras, Joshua A. Peterson

Theses and Dissertations--Mathematics

We generalize the stripping process to the (mod 2) $\mathbb{C}$- and $\mathbb{R}$-motivic settings. Throughout, we include discussion on how the process changes and the difficulties moving to more general settings. We also introduce antipodes and consider what a potential $\mathbb{R}$-motivic analogue may look like. Finally, we elaborate on how the results may be used in future work to generalize a nilpotence result of Walker and Wood.

Extensions Between Modules Defined By Lattice Paths In The Preprojective Algebra, Chloe Napier

Theses and Dissertations--Mathematics

In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. The main goal is to find a combinatorial description for when certain cluster variables are compatible, or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc, and Schröer that answer this question in the case of …

A Character Theory For Loop Representations Of Symmetric Monoidal Bicategories, Jordan Sawdy

Theses and Dissertations--Mathematics

Character theory arises in many distinct fields of mathematics, but its many instantiations often share a few key features: they arise in contexts where one object is acted on or parametrized by another, and they are often computed via "trace-like" formulas. Focusing on these properties, we present a categorical formalism for constructing such characters. We first define a notion of "loop representation" for symmetric monoidal bicategories, then build a character for such representations via the canonical symmetric monoidal trace. We then show that this character defines a symmetric monoidal functor which satisfies commutativity properties with respect to both restriction- and …