Mathematics Commons^™

Gaps In Knowledge: Topological Insights Into The Structure Of Science, Gavin Engelstad

Mathematics, Statistics, and Computer Science Honors Projects

Understanding scientific development is essential to ascertaining the mechanisms leading us into the future. Building this understanding requires both methodological developments and empirical research. This thesis contributes in both aspects using a topological approach to examine scientific knowledge. The first section presents a new algorithm to find optimal cycle representatives for homological features in complex networks, a context for which we demonstrate existing algorithms can be inadequate. The second section applies a number of topological methods, including our cycle optimization algorithm, to data on individual scientific fields, demonstrating the value of topological approaches and highlighting new insights about how science …

Enhancing The Accuracy Of Image Classification For Degenerative Brain Diseases With Cnn Ensemble Models Using Mel-Spectrograms, Sang-Ha Sung, Michael Pokojovy, Do-Young Kang, Woo-Yong Bae, Yeon-Jae Hong, Sangjin Kim

Mathematics & Statistics Faculty Publications

Alzheimer’s disease (AD) and Parkinson’s disease (PD) are prevalent neurodegenerative disorders among the elderly, leading to cognitive decline and motor impairments. As the population ages, the prevalence of these neurodegenerative disorders is increasing, providing motivation for active research in this area. However, most studies are conducted using brain imaging, with relatively few studies utilizing voice data. Using voice data offers advantages in accessibility compared to brain imaging analysis. This study introduces a novel ensemble-based classification model that utilizes Mel spectrograms and Convolutional Neural Networks (CNNs) to distinguish between healthy individuals (NM), AD, and PD patients. A total of 700 voice …

Optimizing Decision-Making In A Cerebral Palsy Model Using Reinforcement Learning, Richard Ampah

Pitzer Senior Theses

This study presents an original interdisciplinary investigation into how reinforcement learning (RL) can model motor and cognitive defects and potentially improve motor and cognitive functions in individuals with cerebral palsy (CP), a non-progressive neurological disorder that impairs movement and adaptability. Integrating computational neuroscience and machine learning, the research applies policy gradient methods and Markov Decision Processes (MDPs) to simulate adaptive learning in agents with and without CP-related constraints.

The central aim is to compare the cumulative rewards of optimal policies, derived from value iteration, and human-like learning policies using the REINFORCE algorithm, both with and without the Bellman baseline. The …

Bytes, Banter, And The Bible: An Interdisciplinary Account Of Objective Meaning, Cameron Bonin

Senior Honors Theses

The claim that the Bible has objective meaning is contested in a postmodern world. This claim can be more persuasively defended when it is addressed by insights from multiple disciplines. In particular, the field of computer science is apt to illuminate the concept of meaning through its reflection on the nature of languages and its concern with the accurate transmission of information. By synthesizing insights from the field of computer science, such as that of Claude Shannon, with Nicholas Wolterstorff’s use of speech-act theory, the concept of meaning can be understood more clearly. Consequently, this synthesis assists in answering questions …

Estimating The Gender Wage Gap: A Comparative Analysis Of Different Estimators, Xinran Zhang

Mathematics, Statistics, and Computer Science Honors Projects

The gender wage gap between males and females has been well studied by labor economists. We take a multi-prong approach to evaluate three estimators —a regression-imputation estimator, a weighting estimator, and a doubly robust estimator—in estimating the gender wage gap. Using the Panel Study of Income Dynamics, we conduct an empirical study of the estimators’ performances. In a simulation study, we evaluate the properties of estimators and study whether bootstrapping is an appropriate measure of the uncertainty of each estimator. The findings show that while the estimators provide different results, the doubly robust estimator provides reliable and consistent results under …

Analyzing Political Sentiment On Micro-Blogging Data: A Lexicon And Machine Learning Approach To The 2024 U.S. Presidential Election, Ava Grey

CMC Senior Theses

This paper explores the trends in sentiment towards U.S. presidential candidates Kamala Harris and Donald Trump through micro-blogging social media text during the five months leading up to the election. Two datasets of varying sizes and origins were used to contextualize and validate analysis findings. The analyses include both a lexicon-based approach and a machine learning predictive method. Common sentiment analysis techniques like term frequency, term frequency inverse, various lexicons, and n-grams were utilized during the lexicon approach. During the modeling, a random forest was utilized in addition to the methods used during the lexicon approach. Results showed that overall …

Securing Secrets: Exploring The Aes Encryption And Key Security Capabilities Of Chatgpt, Kayla Taylor

Student Works

The development and increasing accessibility of generative artificial intelligence (AI) tools and large language models (LLMs) have allowed cryptographers to explore a variety of cryptanalysis problems in dynamic and interactive ways. Prompt engineering, the process by which input text is tested and refined to elicit a desired response from LLMs, is a nascent area of research that remains largely unexplored in many contexts, including cryptography. This study will explore the potential applications and limitations of prompt engineering in the context of Advanced Encryption Standard (AES) encryption and key security with OpenAI’s ChatGPT (GPT-4o) through two main objectives: First, given a …

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, …

Extremal Trees For Random Walks, Ben Bridenbaugh

Mathematics, Statistics, and Computer Science Honors Projects

A random walk is a sequence of adjacent vertices that are chosen uniformly at random from the neighbors of the previous vertex. An access time is the average length of time that a random walk takes to reach a target probability distribution from a starting probability distribution, given an optimal stopping rule. This paper deals with characterizing the trees of diameter d and on n vertices that extremize three different types of access times.

Machine Learning Methods For Intrusion Detection And Response In Network Security, Ayomide Oyemaja

College of Graduate Studies: Theses & Dissertations

Intrusion Detection Systems (IDS) play a crucial role in computer network security by identifying malicious activities and potential cyberattacks. This thesis combines machine learning and cybersecurity by applying Reinforcement Learning (RL) in intrusion detection and response using the NSL-KDD dataset.

We designed and implemented a Q-learning framework where an agent learns to classify network traffic over time by interacting with the environment and receiving rewards based on detection accuracy. We also look at the importance of feature selection and classification techniques and how effective they are in improving model performance, reducing the complexity of computation, and producing more desirable results. …

Action This Day: The Mathematics And Machinations That Bested The German Enigma, Jonah Weinbaum

Dartmouth College Master’s Theses

This thesis presents a comprehensive and chronological overview of cryptographic techniques designed to break Enigma, beginning in 1932 and culminating in the creation of the Turing-Welchman Bombe. We discuss the mathematical theory and electromechanical implements used to decode one of history's greatest ciphers.

Reexamining the Bombe through the lens of modern group theory, we critique Alan Turing's estimation of the number of "stops" that the Bombe produces for various plaintext-ciphertext pairing structures. To address its limitations, we introduce a new framework for estimating the number of stops by extending John Dixon's theorem concerning the probability that uniformly distributed elements of …

Studies On Convexity Of Dnf Formulae, Josue A. Ruiz

Electronic Theses & Dissertations (2024 - present)

In this dissertation, we investigate the problem of determining whether a Boolean formula given in disjunctive normal form (DNF) is convex. Although Boolean formulas have various applications, our research focuses on the practical application for rule-based access control policies, where policies are often expressed as a set of Boolean rules. Understanding the structural properties of such formulas is crucial for determining whether a policy can be efficiently represented within a specific access control model.

The main contribution of this research is the conception and analysis of convexity derived from the “gap problem.” In this context, convexity is characterized by the …

Isochronous And Period-Doubling Diagrams For Symplectic Maps Of The Plane, T. Zolkin, S. Nagaitsev, I. Morozov, S. Kladov, Y. -K. Kim

Physics Faculty Publications

Symplectic mappings of the plane serve as key models for exploring the fundamental nature of complex behavior in nonlinear systems. Central to this exploration is the effective visualization of stability regimes, which enables the interpretation of how systems evolve under varying conditions. While the area-preserving quadratic Hénon map has received significant theoretical attention, a comprehensive description of its mixed parameter-space dynamics remain lacking. This limitation arises from early attempts to reduce the full two-dimensional phase space to a one-dimensional projection, a simplification that resulted in the loss of important dynamical features. Consequently, there is a clear need for a more …

Ms-Yolo: Infrared Object Detection For Edge Deployment Via Mobilenetv4 And Slideloss, Jiali Zhang, Thomas S. White, Haoliang Zhang, Wenqing Hu, Donald C. Wunsch, Jian Liu

Mathematics and Statistics Faculty Research & Creative Works

Infrared imaging has emerged as a robust solution for urban object detection under low-light and adverse weather conditions, offering significant advantages over traditional visible-light cameras. However, challenges such as class imbalance, thermal noise, and computational constraints can significantly hinder model performance in practical settings. To address these issues, we evaluate multiple YOLO variants on the FLIR ADAS V2 dataset, ultimately selecting YOLOv8 as our baseline due to its balanced accuracy and efficiency. Building on this foundation, we present MS-YOLO (MobileNetv4 and SlideLoss based on YOLO), which replaces YOLOv8's CSPDarknet backbone with the more efficient MobileNetV4, reducing computational overhead by 1.5% …

A Bibliographic And Topic Modeling Analysis Of The P-Adic Theory Literature Using Latent Dirichlet Allocation, Humberto Llinás, Ismael Gutiérrez, Anselmo Torresblanca, Javier De La Hoz, Brian Llinás

Computer Science Faculty Publications

P-adic analysis, introduced by Kurt Hensel in the early 20th century, has developed into a fundamental area of mathematical research with broad applications in number theory, algebraic geometry, and mathematical physics. This study aims to examine the thematic evolution and scholarly impact of p-adic research through a comprehensive topic modeling and bibliometric analysis. Using classical bibliometric techniques (e.g., performance analysis, co-authorship, and co-citation networks) combined with Latent Dirichlet Allocation (LDA), we analyzed 7388 peer-reviewed documents published between 1965 and 2024. The computational workflow was conducted using R (version 4.4.1) and VOSviewer (version 1.6.20), which enabled the identification of 20 distinct …

The Impact Of Loss Function Topology On Gradient Descent, Robert B. Skudnig Jr.

Theses and Dissertations

Gradient descent is a popular optimization method that utilizes a model’s prediction error to iteratively improve its parameters for a given task. The functions that measure this error can be defined to align with the user’s goals and sometimes satisfy metric or norm properties. It is common for these functions to measure over Rn, but any differentiable space allows for gradient descent to occur. There has been some research investigating the influence of topological spaces on optimization methods, but it is a limited field of study. This thesis further explores this phenomenon by applying a transformation prediction model to multiple …

Empirical Analysis Of Political Districting Splitability Via Uniform Spanning Trees In Polynomial Time, Brooke C. Feinberg

Scripps Senior Theses

This work expands a recently proven conjecture that a polynomial fraction of all uniform spanning trees (USTs) are splittable into k balanced partitions on grid graphs to real-world political districting plans. We investigate whether similar structural properties hold for the planar dual graphs of U.S. counties (cnty) and tracts (t), using Wilson’s algorithm to generate uniform random spanning trees and Breadth- First Search (BFS) to check for splitability into balanced partitions. Our empirical findings suggest that real-world districting plans can be split into 2-balanced, connected partitions in a fraction of polynomial time. This result highlights the potential for scalable redistricting …

Calculation And Statistical Analysis Of Wins Above Replacement, Joshua Taylor

Departmental Honors & Graduate Capstone Projects

The Wins Above Replacement (WAR) statistic in Major League Baseball is a prominent metric used to estimate player value by quantifying all aspects of play in terms of wins added to a baseball team. We will use R to calculate WAR for all players from 1871 to 2012 and use data from those years to construct multivariate predictive models to attempt to estimate WAR for players from 2013 to 2024. We find strong correlations between predicted and actual WAR values for most models, with the exception of the polynomial predictive model for non-qualified pitchers.

Applications Of Neural Networks In Parkinson’S Disease Diagnosis, Saladin Minhaaj

Theses

Parkinson's disease (PD) is a complex and debilitating neurodegenerative disorder that affects millions of people worldwide. Early and accurate diagnosis is crucial for effective treatment and management of PD. This thesis explores the application of neural networks in PD diagnosis, leveraging their ability to learn patterns from large datasets and make accurate predictions.

Thesis provides an overview of PD, including its symptoms, diagnosis, and current challenges in diagnosis. We then delve into the fundamentals of neural networks, including supervised learning, mathematical interpretations, and parametric models. This research focuses on the development of neural network models that can accurately diagnose PD …

Why Linear Faults Have Fewer Earthquakes: A Geometric Explanation, Julio C. Urenda, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Earthquakes usually occur in the vicinity of fault lines. Until recently, geophysical analysis implied that the fault shape should not strongly affect the frequency of its earthquakes. However, recent statistical analysis has shown that faults whose shape is close to linear experience much fewer earthquakes than faults of more complex shape. Based on this empirical fact, researchers have adjusted the corresponding geophysical models, so the updated models do explain this newly discovered phenomenon. The experience of geophysics shows that the updated model will probably need to be updated again when new data appears. It is therefore desirable to come up …

Fair Economic Division: How To Modify Shapley Value To Take Into Account That Different People Have Different Productivity, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

Purpose: When several participants, working together, gained some amount of money, what is the fair way to distribute this amount between them? This is the problem that the future Nobelist Lloyd Shapley was working on when he proposed what is now called the Shapley value -- a division uniquely determined by natural fairness assumptions. However, this solutions is not universal: it assumes that all participants are equal -- in particular, that they have equal productivity. In practice, people have different productivity levels, and these productivity levels can differ a lot: e.g., some software engineers are several times more productive than …

How Shapley Value And Its Generalizations Can Help In The Analysis Of Complex Engineering Systems And What Next, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

For a complex engineering system -- such as a city's street network -- it is important to predict how its functionality is decreased when some of these components break down, and, if repairs are needed and repairs budget is limited, which subset of the set of components should be repaired first to maximize the resulting functionality. For systems with a large number of components, the number of possible subsets is astronomical, we cannot try to simulate all these subsets. So, the natural idea is to approximate the actual dependence of functionality on the subset by a simple expression -- linear …

What Is Optimal Granularity When Estimating Reliability Of A Complex Engineering Systems, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

For complex engineering systems, the usual way to estimate their reliability is to run simulations. If the resulting estimate does not satisfy the desired reliability level, we must replace some components with more reliable and again run simulations. This can take several iterations, so the required computation time often becomes unrealistically long. It is known that it is possible to speed up computations if components belong to a few types, and components of each type are identical. So, a natural idea to deal with the general case is to use the general granularity idea, i.e., to group components with similar …

Is Energy Local? Counterintuitive Non-Locality Of Energy In General Relativity Can Be Naturally Explained On The Newtonian Level, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

From the physics viewpoint, energy is the ability to perform work. To estimate how much work we can perform, physicists developed several formalisms. For example, for the fields, once we know the Lagrangian, we can find the energy density and, by integrating it, estimate the overall energy of the field. Usually, this adequately describe how much work this field can perform. However, there is an exception -- gravitational field in General Relativity. The known formalism to compute its energy density leads to 0 -- and by integrating this 0, we get a counterintuitive conclusion that the overall energy of the …

Logarithmic Number System Is Optimal For Ai Computations: Theoretical Explanation Of Empirical Success, Olga Kosheleva, Vladik Kreinovich, Christoph Lauter, Kristalys Ruiz-Rohena

Departmental Technical Reports (CS)

Everyone knows the success story of machine-learning AI. However, the current AI tools are not perfect. We know how to make them better: every time we increase the amount of computations by the order of magnitude, we get a drastic improvement in the performance of the resulting machine learning tools. Training modern AI system requires a tremendous amount of computations -- that already take a lot of time. So, to increase the number of computations, we need to make each computation step faster. One way to do that is to use low-precision arithmetic operations, e.g., with 1 byte per real …

Algotric: Symmetric And Asymmetric Encryption Algorithms For Cryptography – A Comparative Analysis In Ai Era, Naresh Kshetri, Mir Mehedi Rahman, Md Masud Rana, Omar Faruq Osama, James Hutson

Faculty Scholarship

The increasing integration of artificial intelligence (AI) within cybersecurity has necessitated stronger encryption methods to ensure data security. This paper presents a comparative analysis of symmetric (SE) and asymmetric encryption (AE) algorithms, focusing on their role in securing sensitive information in AI-driven environments. Through an in-depth study of various encryption algorithms such as AES, RSA, and others, this research evaluates the efficiency, complexity, and security of these algorithms within modern cybersecurity frameworks. Utilizing both qualitative and quantitative analysis, this research explores the historical evolution of encryption algorithms and their growing relevance in AI applications. The comparison of SE and AE …

Computational Representation, Analysis And Verification Of Requirements In Engineering Design And Systems Engineering, Chandan Kumar Sahu

All Dissertations

Systems are developed to satisfy a set of requirements derived from stakeholders’ needs, defining the problem space for which the system is created as a feasible solution. The system design process begins with eliciting these requirements and concludes with validating whether the created system meets them. Requirements engineering (RE) encompasses elicitation, representation, analysis, documentation, verification, and validation. However, challenges in RE, such as imprecision in natural language (NL), proprietary restrictions, and a lack of standardized quality metrics, hinder the creation of well-formed and comprehensive requirements. These challenges complicate formalization and analysis of requirements.

This dissertation addresses these challenges by proposing …

Decoding Neural Networks: An Information-Theoretic Guide To Interpretability, Error Analysis And Efficiency, Mackenzie J. Meni

Theses and Dissertations

This dissertation addresses critical challenges in neural network design by leveraging entropy-based techniques to improve model efficiency, interpretability, and bias reduction. Focusing on the unique demands of computer vision applications, particularly object detection and classification for real-time systems, this work introduces a series of innovative methods centered on information theory. At the core of these methods is the Probabilistic Explanations of Entropic Knowledge (PEEK) framework, a tool developed to analyze and visualize entropy distributions across feature maps. PEEK offers insights into information flow within neural networks, making it possible to pinpoint layers that contribute meaningfully to decision-making or identify those …

A Dynamical Systems Approach For Modeling Malware Propagating Through A Network And Potential Solutions Towards Mitigating Spread, James Johnson

Cybersecurity Undergraduate Research Showcase

Many people draw close parallels between malware propagating through a network and an epidemic spreading through a population. Epidemics are often modeled by a Susceptible-Infected-Recovered (SIR) model, in which a similar system of equations can model the spread of a virus through a computer network, and can be simplified when making assumptions about the network itself and its fixed number of nodes and edges. In this instance, malware propagating in a network also should reflect the network it is propagating through, in which the dynamical system will factor in the nodes of the network and their properties. The system itself …

Telu Activation Function For Fast And Stable Deep Learning, Alfredo Fernandez

USF Tampa Graduate Theses and Dissertations

We propose the Hyperbolic Tangent Exponential Linear Unit (TeLU), a neural network hidden activation function defined as $TeLU(x)=x \cdot tanh(e^x)$. TeLU’s design is grounded in the core principles of key activation functions, achieving strong convergence by closely approximating the identity function in its active region while effectively mitigating the vanishing gradient problem in its saturating region. Its simple formulation enhances computational efficiency, leading to improvements in scalability and convergence speed. Unlike many modern activation functions, TeLU seamlessly combines the simplicity and effectiveness of ReLU with the smoothness and analytic properties essential for learning stability in deep neural networks. TeLU’s ability …