Mathematics Commons^™

Guessing Games To Determine 1 Of Several Secret Numbers, Kyle Mckee, Julia Osmun, Dori Schlutt

Mathematics Undergraduate Research

A guessing game is a two-player game between a Responder and a Questioner. In a guessing game, the Responder is thinking of x ≥ 1 secret numbers in a set S of size n. The Questioner is tasked with asking questions of the form: “How many numbers are in Q ⊆ S?” until the Questioner knows at least 1 of the Responder’s secret numbers. We find a game-winning strategy for the Questioner with minimal possible questions for games with x = 2. We also find lower-bounds for x = 2 and x = 3.

Offline Guessing Games With Two Numbers, Justin Sciullo, Lisa Shen, Sarah Zaske

Mathematics Undergraduate Research

In an offline guessing game, there is a player called the Questioner and a player called the Responder. The Responder first picks two distinct numbers from the set {1, 2, 3, . . . , n}. The Questioner then creates a set of questions of the form “How many of your numbers are in the set q_i ⊆ {1, 2, 3, . . . , n}?” and sends them to the Responder who answers them. The Questioner wins if they can guess the Responder’s numbers no matter which numbers the Responder chose. The Responder wins otherwise. We …

Analyzing Patterns In Chicago Motor Vehicle Crashes Using Time-Series Techniques, Christina Trotta

Senior Honors Theses and Projects

This project explores time series forecasting of daily traffic crash rates in Chicago from 2018 to 2024, with a focus on understanding how past crash patterns and external conditions influence future risk. The primary research question asks: To what extent does yesterday’s crash rate help predict today’s? Using a combination of Holt-Winters exponential smoothing, Prophet forecasting, and SARIMAX models, we assess the role of autoregression, seasonality, and exogenous variables such as weather and roadway conditions. Daily crash data was cleaned, aggregated, and enriched with engineered features including holiday indicators, weather metrics from O’Hare and Midway airports, and binary flags for …

Sickle-Cell Genotyping Cost Analysis In Amua And R, Nicholas P. Haley

Senior Honors Theses and Projects

Sickle cell disease (SCD) is a prevalent genetic disorder in the United States, with a significant economic burden due to the high costs of care, especially from chronic red blood cell (RBC) transfusions. These transfusions carry risks such as alloimmunization, which can lead to complications like delayed hemolytic transfusion reactions (DHTRs), further increasing healthcare costs in addition to increasing suGering. This study aims to aid in the evaluation of cost-eGective strategies for SCD management using Markov models, currently within TreeAge Pro software, which is proprietary. A decision model was adapted from the Kacker et al. (2013) study, which analyzed the …

Fairness-Aware And Culturally Adaptive Machine Learning For Predicting Adolescent Substance Use, Stephanie Nworgu

Senior Honors Theses and Projects

Early initiation of substance use during adolescence poses significant risks to long-term health, educational attainment, and social outcomes, making early identification a critical public health priority. Machine learning models have increasingly been used to predict substance use risk; however, many such models do not explicitly examine whether predictive performance differs across demographic groups. This senior project examines the fairness of logistic regression models used to predict first-time alcohol use among adolescents. Using nationally representative survey data from the Youth Risk Behavior Surveillance System (YRBSS), pooled across the 2017, 2019, 2021, and 2023 survey cycles, this study develops logistic regression–based predictive …

Generalizations Of Finiteness Conditions And Extension Monads In Algebras With Infinitely Many Or Infinitary Operations, Danielle Christienne Bowerman

Doctoral Dissertations

In this work, we extend the results of finiteness conditions and extension monads found in Insall from finitely many finitary operations to infinitely many finitary operations, as well as touching on infinitary operations. We also examine varieties of algebras, including the notion of strong varieties introduced in Insall, and common constructions of extension monads in varieties of algebras. We see that for finite collections of algebras of the same signature, the extension monad operation on a variety of algebras commutes with the direct product operation, and all retractions from an enlargement or extension monad are trivial. We also see that …

Optimal Control And Structurally-Informed Gradient Optimization Of A Custom 4-Dof Rigid-Body, Brock Marcinczyk, Logan E. Beaver

Mechanical & Aerospace Engineering Faculty Publications

This work develops a control-centric framework for a custom 4-DOF rigid-body manipulator by coupling a reduced-order Pontryagin’s Maximum Principle (PMP) controller with a physics-informed Gradient Descent stage. The reduced PMP model provides a closed-form optimal control law for the joint accelerations, while the Gradient Descent module determines the corresponding time horizons by minimizing a cost functional built directly from the full Rigid-Body Dynamics. Structural-mechanics reaction analysis is used only to initialize feasible joint velocities—most critically the azimuthal component—ensuring that the optimizer begins in a physically admissible region. The resulting kinematic trajectories and dynamically consistent time horizons are then supplied to …

Mathematical Modeling Of Cancer Tumor Evolution, Belgacem Al-Azem

CGU Theses & Dissertations

Several mathematical models of cancer tumor evolution are presented. The aim of this endeavor is to write a thesis that fosters numerical and analytical understanding of certain mathematical models in the area of oncology as well as to offer potential predictive tools of therapeutic and clinical applicability. In this work, we give a panoramic view of the cancer phenotype, for to understand cancer, we need to "see" its biophysics. Indeed, the discovery of oncogenes led to the thinking that cancer tumor is perhaps a genetic disease. But the role of angiogenesis and other microenvironment-related discoveries (such as the role of …

Multipliers On Weighted Sequence Spaces, Gilbert Acheampong, Raymond Cheng

Mathematics & Statistics Faculty Publications

The space ℓp,α of complex sequences a = (a0, a1,a2,...) for which

[[formula omitted]]

is studied. Each such sequence can be identified with the analytic function with power series

[[formula omitted]]

In this setting, the point evaluation and the difference quotient mappings are shown to be bounded; the cases are identified in which ℓp,α is boundedly contained in ℓr,β. Conditions on the parameters are derived for the analytic functions of ℓp,α to have radial limits almost everywhere on the boundary, and for ℓp,α to be an algebra. Smoothness properties of the boundary function are investigated. Basic properties of multipliers on …

Golden Spirals Everywhere?, John Adam

Mathematics & Statistics Faculty Publications

The article explores different types of spirals, including Archimedean, hyperbolic, and logarithmic spirals, with a focus on the golden ratio and golden spirals. It discusses the misconception that golden rectangles and spirals can be found in various natural and man-made objects, emphasizing the importance of understanding the properties of logarithmic spirals. The text provides mathematical equations for logarithmic spirals and poses questions for readers to explore the concept further. The author, John Adam, invites readers to engage in Fermi Questions and submit ideas for consideration.

Model-Free Organization Of Patient Reported Outcomes Data: Geometrical Rep-Resentation Of The Modified Compartmen-Talization Method, Manasi Sheth, N. Rao Chaganty

Mathematics & Statistics Faculty Publications

There is a recent advancement in the field of mathematics and statistics to understand the geometry or connectedness of the data due to the massive amounts of data being generated. The data provided for analyses are usually very large and need to be organized and minimized in order to make it more useful and meaningful. In biostatistics or medical field, it is important for patients to have access to high-quality, safe and effective and/ or efficacious medical products. It is quite necessary to ascertain that the patients and their care-partners stay at the center of the regulatory decision-making process. In …

A Decision-Space Model Explains Context-Specific Decision Making, Dirk W. Beck, Cory N. Heaton, Luis D. Davila, Lara I. Rakocevic, Sabrina M. Drammis, Danil Tyulmankov, Atanu Giri, Shreeya Umashankar Beck, Qingyang Zhang, Michael Pokojovy, Kenichiro Negishi, Alexis A. Salcido, Neftali F. Reyes, Andrea Y. Macias, Serina A. Batson, Paulina Vara, Raquel J. Ibáñez Alcalá, Safa B. Hossain, Graham L. Waller, Laura E. O'Dell, Travis M. Moschak, Ki A. Goosens, Alexander Friedman

Mathematics & Statistics Faculty Publications

Optimal decision-making requires consideration of internal and external contexts. Biased decision-making is a transdiagnostic symptom of neuropsychiatric disorders. We created a computational model demonstrating how the striosome compartment of the striatum constructs a context-dependent mathematical space for decision-making computations, and how the matrix compartment uses this space to define action value. The model explains multiple experimental results and unifies other theories like reward prediction error, roles of the direct versus indirect pathways, and roles of the striosome versus matrix, under one framework. We also found, through new analyses, that striosome and matrix neurons increase their synchrony during difficult tasks, caused …

A Bridge Too Low: Solutions For Fermi Questions, May 2025, John Adam

Mathematics & Statistics Faculty Publications

The article discusses a low bridge in Keswick, England, near the River Greta, with an arch shaped like a semiellipse. It presents Fermi questions related to the bridge's dimensions, such as the maximum distance a person of a certain height can walk under it without hitting their head. The solutions involve mathematical calculations and approximations, including the use of formulas provided by the Indian mathematician Srinivasa Ramanujan.

Untangling The Classification Of Surfaces: An Accessibility-Centered Perspective On Topology, Charlotte N. Richards

Pitzer Senior Theses

Mathematics has clear benefits in education, from preparing students for future careers to teaching them how to problem-solve. While mathematics achievement has been falling in recent decades, students claim that the problem is not the mathematics itself, but the ‘boring’ classroom material that feels removed from real life. More advanced mathematics topics, such as topology, could offer a solution, as their applications lie in countless fields. However, topology has been restricted to upper-level mathematics, disregarding the potential benefits of making this material broadly reachable for a junior high-school audience. In this paper, we analyze five different proofs of the theorem …

The Algebra Behind Magic, Lois Carpenter

Undergraduate Research Awards

Card Tricks have long been a staple in the common magician’s repertoire, and while many tricks can be explained through sleight of hand alone, others rely on seemingly random shuffling methods that leave the magician with significant control over the deck. Applied card magic (frequently referred to as ‘cheating’) makes significant use of this ability. Thus, the utility of this topic is clear- anyone with basic mastery of perfect shuffles has complete control over the arrangement of cards in a deck, and with it a fundamental advantage against other players in any game of cards. While most perfect shuffles are …

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, …