Mathematics Commons^™

Some Results In Thermodynamic Formalism, C. Evans Hedges

Electronic Theses and Dissertations

This dissertation investigates several key questions at the intersection of dynamical systems, computability theory, and thermodynamic formalism. In the symbolic setting, we establish novel results regarding the statistical properties of equilibrium states, deriving bounds on probabilities of configurations and relating these bounds to the Gibbs property through the homoclinic relation. Additionally, we examine the computability of thermodynamic quantities such as pressure, ground state energy, and residual entropy. We show that topological pressure is computable from above for general subshifts and computable for strongly irreducible shifts, with similar results extending to ground state energy and residual entropy.

Extending beyond subshifts, we …

A Study On Fuzzy Time-Series And Its Applications To Stock Price Forecasting, Takeshi Stormer

University Honors Theses

Fuzzy mathematics looks to incorporate the vagueness that exists within the real world, specifically regarding imprecise classes, or non-numerical information expressed as "linguistic" variables. Since most traditional mathematical theories do not have the ability to be applied with the exactness that is otherwise seen in mathematics. As such, there had been many applications of fuzzy mathematics throughout many different fields of mathematics, including that of forecasting. By exploring the fundamentals of fuzzy mathematics, including fuzzy sets, operations of fuzzy sets, the surface level introduction to fuzzy logic, fuzzy relations, operations of fuzzy relations, and fuzzy time-series, this work looks to …

A Spatial Modeling Framework For Identifying Multidimensional Risk Factors Of Colorectal Cancer, Johnna K. Berryhill

Student Theses and Dissertations

Colorectal cancer (CRC) is one of the most common and preventable cancers, yet significant disparities in screening, incidence, and late-stage diagnosis persist across different geographic and socioeconomic contexts. This thesis presents a spatial modeling framework for identifying multidimensional risk factors associated with CRC outcomes across various geographic scales. Using county- and census-tract-level data from national and state sources, the study integrates demographic, socioeconomic, environmental, built environment, and behavioral predictors to analyze geographic variability in CRC risk factors. Advanced spatial modeling techniques, including Geographically Weighted Regression (GWR) and Geographically Weighted Random Forest (GW-RF), are employed to capture spatial heterogeneity in these …

Notes On The Invariance Of Tautness Under Lie Sphere Transformations, Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

An embedding ϕ : V → Sⁿ of a compact, connected manifold V into the unit sphere Sⁿ ⊂ Rⁿ⁺¹ is said to be taut, if every nondegenerate spherical distance function d_p, p ∈ Sⁿ, is a perfect Morse function on V , i.e., it has the minimum number of critical points on V required by the Morse inequalities. In these notes, we give an exposition of the proof of the invariance of tautness under Lie sphere transformations due to ´Alvarez Paiva. First we extend the definition of tautness of submanifolds of S …

Class Groups And Selmer Groups In Special Families, Debanjana Kundu, Abhishek _

School of Mathematical & Statistical Sciences Faculty Publications

We explore the relationship between (3-isogeny induced) Selmer group of an elliptic curve and the (3 part of) the ideal class group, over certain non-abelian number fields.

Temporal Modeling And Forecasting Of Blood Glucose Dynamics In Individuals With Diabetes Mellitus, Mj Ruff

University Honors Theses

People living with Diabetes Mellitus face significant health risks, including an increased likelihood of heart disease, stroke, and fluctuations in blood glucose levels. The unpredictable nature of glucose levels can lead to dangerous conditions such as ketoacidosis and hypoglycemia. This study employs advanced time series analysis tools to forecast the glucose levels for an individual diagnosed with Type 1 Diabetes Mellitus.

A Proposal To Explore Geometry With Geogebra: Graphical Exploration And Formal Demonstration Of The Collinearity Of The Barycenters Of A Polygon, Saulo Mosquera Lopez, Marlio Paredes, Walter Castro

School of Mathematical & Statistical Sciences Faculty Publications

This paper illustrates an example of a mathematical activity that teachers and students can replicate to create an experience that resembles professional mathematical activity. We extend the property “Consider a triangle ABC, any straight line and let A’, B’, C’ be the reflections of the points A, B, C on the straight line then the barycenters of the triangles ABC, A’BC, AB’C and ABC’ are collinear and the line of collinearity is perpendicular to the straight line” for any quadrilateral. It is proved that there are four additional triangles, for a total of eight, whose barycenters are collinear and that …

On Excursions Associated With A Certain Local Time Of Simple Symmetric Random Walks, With Applications, Takahiko Fujita, Naohiro Yoshida

Journal of Stochastic Analysis

In this note, some applications of excursions associated with a certain local time of simple symmetric random walks are presented. Specifically, the excursions are applied to calculate some probability distributions of interest regarding the random walks. Furthermore, a solution of the Skorokhod embedding problem for random walks is obtained through the excursions.

Forecasting Influenza Rates Using Machine Learning: A Study Of Chatgpt's Predictive Accuracy, Sara Saleh

University Honors Theses

This study evaluates ChatGPT's ability to forecast influenza rates, such as the number of flu cases, hospitalizations, and death during peak season periods using CDC data, and comparing forecasts against actual results to calculate statistical accuracy and consistency. Influenza forecasting is essential for public health planning, but traditional methods may not always provide timely or accurate predictions. In this research study, ChatGPT was utilized to predict the influenza rates for the following week based on the previous week's data obtained from the FluView surveillance system. The predicted rates were compared to the actual influenza rates to assess the model's overall …

Tilings In The 3 Dimensional Lattice With L-Tetrominoes, Ian N. Bridges

Rose-Hulman Undergraduate Mathematics Journal

We consider three dimensional L-tetrominoes. We show that there exists at least one way to tile every three dimensional rectangle whose side lengths are at least $3$ and area is congruent to $1 \pmod 4$ such that one square goes untiled. In addition, we show that every three dimensional rectangle is tileable provided one side has length at least $2$ and the other is a multiple of $4$.

The Other Side Of The Equation: De-Simplification, A Prerequisite For Calculus, Stephen L. Brown

ACMS Conference Proceedings 2005

No abstract provided.

Ceno: Non-Uniform, Segment And Parallel Zero-Knowledge Virtual Machine, Tianyi Liu, Zhenfei Zhang, Yuncong Zhang, Wenqing Hu, Ye Zhang

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we explore a novel Zero-knowledge Virtual Machine (zkVM) framework leveraging succinct, non-interactive zero-knowledge proofs for verifiable computation over any code. Our approach divides the proof of program execution into two stages. In the first stage, the process breaks down program execution into segments, identifying and grouping identical sections. These segments are then proved through data-parallel circuits that allow for varying amounts of duplication. In the subsequent stage, the verifier examines these segment proofs, reconstructing the program's control and data flow based on the segments' duplication number and the original program. The second stage can be further attested …

Wavelet-Based Multi-Step Methods For Systems Of Differential Equations, Rashad Assad Hijji

Theses

Wavelets have been widely used in many areas of engineering and mathematics, including the development of multistep algorithms to solve initial value problems (IVPs) in the context of the Galerkin method using Daubechies' wavelets. The main scope of our work is to build a comprehensive framework for solving Systems of Differential equations using the compactly supported wavelets proposed by I. Daubechies. Wavelets are mathematical functions that decompose data into distinct frequency components, and each element is analyzed with a resolution that matches its scale. Compact support of Daubechies wavelets is key in allowing them to be computationally efficient for high-dimensional …

The Degrees Of The Irreducible Representations Of The Symmetric Group With Respect To Primes, Vanessa F. Beeler

Master's Theses

In this thesis, we explore the relationship between the degrees of irreducible matrix representations of the symmetric group and prime numbers. We start by building up the required background necessary to construct our results. We dive into topics of discrete mathematics including matrix representations, characters of representations, integer partitions, conjugacy classes, class functions, and tableaux. Some other key ingredients in our work include using character tables to classify the irreducible rep- resentations of the symmetric group and the hook-length formula to easily compute the degrees of these representations. Once we have laid the groundwork for our in- vestigation, we begin …

Topiarism: The Kernel Embedding Of Distributions Applied To Modern Portfolio Theory, Stephen G. Cook

Master's Theses

The method of kernel embedding of distributions on a set $\Omega$ into a reproducing kernel Hilbert space is a method of studying the space of measures on a set $\Omega$ using Hilbert space geometry. Because positively weighted portfolios can be interpreted as nonnegative probability measures on the space of assets, we are able to apply this technique to portfolio theory. In this thesis, we discuss the theory of "topiarism", the study of positively weighted probability measures on compact sets under the kernel embedding of distributions. Given a specific payoff function $\psi$ on the set of assets $\Omega$, we optimize a …

Tailoring Evaluations Of Chronic Rhinosinusitis: Understanding Sleep And Its Effect On Memory Through Actigraphy, Donyea Moore, Rachel Nolte, Yitong Pepper Huang, Shreya Maharana, Pavan Nataraj, Bichun Ouyang, Mahboobeh Mahdavinia

Mathematics Sciences: Faculty Publications

Background/Objectives: Chronic rhinosinusitis (CRS) is a persistent inflammatory condition of the sinonasal mucosa lasting for at least three months. For patients, CRS-related sleep disturbances can significantly disrupt circadian rhythms, leading to further health complications such as cognitive impairment. Despite the well-documented sleep disturbances associated with CRS, there is limited research on objective assessment methods. Additionally, the severity of these issues can vary among patients. This study aims to assess sleep quality and timing in CRS patients and investigate their impact on cognition, providing guidance for personalized and tailored assessment and management of CRS. Methods: Our case–control study compares sleep patterns …

Radiation-Induced Cardiotoxicity In Hypertensive Salt-Sensitive Rats: A Feasibility Study, Dayeong An, Alison Kriegel, Suresh N. Kumar, Heather A. Himburg, Brian Fish, S. Klawikowski, Daniel B. Rowe, Marek Lenarczyk, John Baker, El Sayed H. Ibrahim

Mathematical and Statistical Science Faculty Research and Publications

Radiation therapy (RT) plays a vital role in managing thoracic cancers, though it can lead to adverse effects, including significant cardiotoxicity. Understanding the risk factors like hypertension in RT is important for patient prognosis and management. A Dahl salt-sensitive (SS) female rat model was used to study hypertension effect on RT-induced cardiotoxicity. Rats were fed a high-salt diet to induce hypertension and then divided into RT and sham groups. The RT group received 24 Gy of whole-heart irradiation. Cardiac function was evaluated using MRI and blood pressure measurements at baseline, 8 weeks and 12 weeks post-RT. Histological examination was performed …

Property Testing Ai: An Efficient Frontier, Paul Sopher Lintilhac

Dartmouth College Ph.D Dissertations

In this dissertation, we take a step towards addressing the major problem of a lack of standardized and rigorous approaches to testing and evaluation of AI systems. Taking inspiration from both the fields of Property Testing and Property Based Testing (for programs), we develop a novel taxonomy of partially overlapping classes of properties of AI systems, including simple properties, compound properties, higher order properties, data relation properties, and architecture-utility properties. We argue that this taxonomy categorizes a diverse set of AI traits -- including accuracy, fairness, robustness, monotonicity, point-wise and global privacy properties, sensitivity, and more -- according to the …

Stability Insights From Modeling Chronic Myelogenous Leukemia, Giovani Thai

Master's Theses

This thesis centers around a model for chronic myelogenous leukemia (CML) as it behaves under imatinib treatment, a common medication for CML patients, and the anti-leukemia immune response. The dynamics are represented with a system of nonlinear delay-differential equations first constructed by Kim et al. in 2008, capturing population changes of T-cells and various CML growth stages. We investigate stability in both the clinical and mathematical sense. Through numerical simulations, we computationally incorporate a supplementary treatment plan to determine its effectiveness in aiding immune response and medication in achieving remission and full elimination. The primary goal is to conduct a …

A Realizability Approach To Constructing Higher Types Via Classifiers, Benjamin Carrick Logsdon

Dartmouth College Ph.D Dissertations

We construct an interpretation of higher types into Peano arithmetic, showing in particular that every model of PA is a model of higher types. This is a reversal of Gödel’s Dialectica construction. We also define the classifier degrees, a degree structure which subsumes the Turing degrees, the enumeration degrees, and the many-one degrees. The classifier degrees boast a rich structure and many well-behaved operations.