Mathematics Commons^™

Editorial Insights: Reflections On The Volume And Charge To The Field, Heather Howell, Meghan Shaughnessy, Barbara Stengal, Carrie Wilkerson Lee, Liza Bondurant, Bima Sapkota, Gregory Benoit, Yvonne Lai

School of Mathematical & Statistical Sciences Faculty Publications

In this, the final chapter of Section 3, the authors document an interview that took place in June of 2024, as the editorial team wrapped up reviews of the chapters submitted to this volume. The interview was led by two of the advisory board members, Meghan Shaughnessy and Barbara Stengal, and included the editorial group, Heather Howell, Carrie Wilkerson Lee, Liza Bondurant, and Bima Sapkota, and other members of the advisory board, Greg Benoit and Yvonne Lai. The motivation to create this interview-based chapter was to capture some of the authors' own learning and reflection as they brought the volume …

Generalizations Of Some Q-Product Identities Of Ramanujan And Others, Timothy J. Huber, James Mclaughlin, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

By considering certain limiting cases of a WP-Bailey chain discovered by Andrews, and also limiting cases of certain classical summation formulae for basic hypergeometric series, we derive new expressions for certain Lambert series in terms of basic hypergeometric series. In some cases, the resulting series involve an arbitrary Bailey pair. This allows for the derivation of new basic hypergeometric expansions for some q-products and series that Ramanujan expressed in terms of Lambert series. Some of Ramanujan’s identities are extended to more general relations containing one or more free parameters.

Local Limit Theorems On Finitely Generated Abelian Groups, Yutong Yan

Honors Theses

In this thesis, we classify the pointwise behavior of finite-range random walks on finitely generated abelian groups in terms of local limit theorems. Random walks are central objects of research in probability theory, and the theory has found applications in statistics, physics, and even card shuffling. One significant topic in this line of study is random walks on finitely generated groups. Starting from the pioneering work of G. Pólya and H. Kesten, random walks on finitely generated groups have been studied extensively. However, many notable results on the subject (local limit theorems, for example) make assumptions about periodicity and irreducibility …

Theoretical And Applied Considerations In Dynamical Sampling, Brendan Miller

Graduate Research Theses & Dissertations

Dynamical sampling is an area of mathematics which studies the theory and applications of signals evolving through time, as well as the stable recovery of said signals from spatiotemporal measurements. This manuscript is devoted to problems which arise as theoretical considerations in signal recovery and as applications of using space-time measurements as opposed to static measurements.

For the former, we consider the problem of recovering an evolving signal via space-time measurements at irregular time intervals. To do so, we generalize the classic Kadec $1/4$ theorem using techniques of Banach $L^1(\R)$-modules. For the latter, we show how dynamical samples can be …

Methods For Finding High Quality And Optimal Solutions Of Binary Quadratic Optimization Problems, James Haas

Graduate Research Theses & Dissertations

BiqAlps is an extension of the BiqCrunch project that leverages its bounding procedures within a computer cluster environment. The primary goal of this work was to investigate whether distributing the branch-and-bound tree search across multiple CPUs could improve solve times. While some problem instances demonstrated speedups of up to fivefold, others showed no measurable improvement, revealing that parallelization benefits are problem-dependent.

Beyond parallelization, BiqAlps introduces enhanced heuristics for generating high-quality initial solutions and refining existing ones. Additionally, the framework explores the impact of propagating triangle inequality cuts to child nodes during search, with the aim of accelerating bounding processes and …

Cubic Blaschke Products, Alexandra Hill

Graduate Research Theses & Dissertations

It is well known that Blaschke products can be classified as elliptic, hyperbolic, or parabolic. Based on the classification of the Blaschke product, we know that the Julia set is either the entire unit circle or a Cantor subset of the unit circle. For unicritical Blaschke products of degree d, it has been shown that the parabolic functions are parameterized by an epicycloid with d-1 cusps, with the interior of the epicycloid giving rise to the elliptic parameters, and the relative complement of the epicycloid within the unit disk providing the hyperbolic parameters. In this dissertation, we study in more …

Green's Functions And Fixed Points Of Quasiregular Maps, Mark Broderius

Graduate Research Theses & Dissertations

This dissertation is on the dynamics of planar quasiregular maps of degree 2. These maps have three parameters. One of these parameters determines the distortion, one of them determines the orientation, and the last one is a shift parameter. Our first objective is to construct a new dynamical version of Green's functions for these maps. These functions provide information about the escaping sets of our quasiregular maps and are used to prove topological results about them. We show that the boundary of the escaping set is either connected or consists of infinitely many components. Additionally, we prove that if one …

Facial Reduction Of Semidefinite Relaxations Of Binary Quadratic Problems, Ian Sugrue

Graduate Research Theses & Dissertations

A standard technique for bounding a combinatorial optimization problem is to form a semidefinite relaxation of the problem by "lifting" the problem into a higher dimensional space of symmetric matrices. For some combinatorial problems such as maximum k-cluster, the resulting semidefinite relaxation is not strictly feasible. In the context of the maximum k-cluster problem, we present an equivalent, strictly feasible semidefinite relaxation by way of the dimensionality reduction technique known as facial reduction. We provide a recipe to form this semidefinite relaxation directly, bypassing the lifting and facial reduction steps entirely. We then present our work on generalizing this result, …

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 …

The Effect Of Electrode Geometry On Excited Species Production In Atmospheric Pressure Air-Hydrogen Streamer Discharge, Shirshak Kumar Dhali, Stuart Reyes

Electrical & Computer Engineering Faculty Publications

When a gas is overvolted at or near atmospheric pressure, it results in a streamer discharge formation. Electrode geometries exert significant impact on the electrical breakdown of gases by altering the spatial profile of the electric field. In many applications the efficient generation of radicals is critical and is determined by the characteristics of the streamer discharge. We examine the effect of electrode geometry on the streamer characteristics and the production of radicals. This is performed for three different electrode geometries: plane–plane, pin–plane, and pin–pin. A two-dimensional rotationally symmetric fluid model is used for the streamer discharge simulation in the …

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 …

A Question Of Transparency, John Adam

Mathematics & Statistics Faculty Publications

Question 1: Why is it easier to see through rain than fog?

Start thinking about this by imagining a fixed volume (V) of water being dispersed into, say, N identical droplets of diameter d. Surface area and volume considerations should lead to the answer in terms of V and d.

Question 2: (a) How "long" (in meters) might such a rain shower or fog bank be?

Hint: Suppose you are looking along a linear stack of S cubes with 1-m sides (through the rain or fog). Each cube contains N drops. If p is the visibility (i.e., the fraction of …

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 …

Multipliers Between ℓᴾ Spaces, Raymond Cheng

Mathematics & Statistics Faculty Publications

For 0 < p ⩽ ∞ and 0 < r ⩽ ∞, the space 𝔐_p,r of (coefficient) multipliers from ℓ^p and ℓ^r is completely characterized. This is elementary in most instances. The interesting case 0 < r < p < ∞ requires more effort, and it is shown that a sequence of complex numbers belongs to 𝔐_p,r if and only if the sequence of their absolute values has a non increasing rearrangement (h₀,h₁,h₂,...) satisfying

(∞

Σ (k +1)^(p-r)/p (h^r_k - h^r_k+1)^1/r) < ∞

k = 0

In that case, the expression on the left is the norm of the multiplier, and it is a compact operator. Further upper and lower bounds are given for the multiplier norm.

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.

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 …

Development Of Conceptual Scaffolding-Assisted Formative Assessment For Enhancing Understanding Of Temperature And Heat, Ana Lailatul Fitriyah, Sentot Kusairi, Markus Diantoro

Jurnal Pendidikan Sains

This research aims to develop formative assessment media assisted by conceptual scaffolding to improve students’ understanding of temperature and heat, test its validity and practicality, and gauge student responses. Utilizing the ADDIE model, the research follows five stages: Analysis, Design, Development, Implementation, and Evaluation. In the Analysis stage, a literature review was conducted to identify critical components, including formative assessment, conceptual scaffolding, concept mastery, and previous research. The Design stage involved identifying users, defining competencies and indicators, creating a storyboard, integrating assessment components, and determining mastery levels. During the Development stage, the assessment media were developed into a functional product. …

Composite Volume Learning Media Based On Augmented Reality To Enhance Students’ Visual-Spatial Intelligence, Ria Kurniawati, Susiswo Susiswo, Syamsul Hadi

Jurnal Pendidikan Sains

Augmented Reality (AR) technology, a three-dimensional visual tool that integrates virtual objects or environments into real-world displays in real time (Azuma et al., 2001), is particularly useful for grasping complex concepts such as the composite volume of solids, which often confuses students when they need to decompose solids into their individual components. While previous research has predominantly focused on the introduction of solids, spatial nets, and the volume of singular shapes, this study aims to develop composite volume learning media based on AR to enhance students’ visual-spatial intelligence in a valid, practical, and effective manner, aligning with their concrete operational …

A Novel Format For Math Circles: Circles With No Center, Frederick A. Peck, Ian P. Renga, Nick Grener, David R. Erickson, Matt B. Roscoe, Ke Wu

Journal of Math Circles

This paper shares a novel format for Math Circles, which we call “circles with no center.” Circles with no center do not have a traditional facilitator. The problem or activity is novel to everyone in the room. All members engage in mathematical inquiry, and everyone takes responsibility for running the gathering. The “no center” format has benefits for participants and organizers. For participants, the format offers more freedom and more responsibility. For organizers, the format is easier to organize, more fun, and involves less pressure. Our assessment of a 3-day retreat for teachers suggests participants may prefer a circle with …

Understanding Women, Infant And Children Feeding Decisions In Emergencies: The Case Of Covid-19 And The Formula Shortage, Davida Mensah, Edmund F. Agyemang, Constance Gewa

School of Mathematical & Statistical Sciences Faculty Publications

Background: The advantages of breastfeeding and its role in emergencies are undeniable. Studies show that many women in developed countries choose formula over breastmilk, even in emergencies. Nevertheless, no one knows if postpartum women under the Women, Infants and Children (WIC) program turned out to breastfeed or re-lactate their infants during these crises. This study aims to better understand WIC participants' perceptions about breastfeeding their babies in emergencies.

Methods: A chi-square test of independence was used to assess the association between breastfeeding choice and demographic characteristics and potential covariates of respondents. A post hoc analysis using Dunn's multiple comparisons test …

Effects Of Missing Data Imputation Methods On Univariate Blood Pressure Time Series Data Analysis And Forecasting With Arima And Lstm, Nicholas Niako, Jesus D. Melgarejo, Gladys E. Maestre, Kristina Vatcheva

School of Mathematical & Statistical Sciences Faculty Publications

Background

Missing observations within the univariate time series are common in real-life and cause analytical problems in the flow of the analysis. Imputation of missing values is an inevitable step in every incomplete univariate time series. Most of the existing studies focus on comparing the distributions of imputed data. There is a gap of knowledge on how different imputation methods for univariate time series affect the forecasting performance of time series models. We evaluated the prediction performance of autoregressive integrated moving average (ARIMA) and long short-term memory (LSTM) network models on imputed time series data using ten different imputation techniques. …

Apex Graphs And Cographs, Jagdeep Singh, Vaidy Sivaraman, Thomas Zaslavsky

Theory & Applications of Graphs

A class G of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by G^{apex} the class of graphs G that contain a vertex v such that G − v is in G. Borowiecki, Drgas-Burchardt, and Sidorowicz proved that if a hereditary class G has finitely many forbidden induced subgraphs, then so does G^{apex}. We provide an elementary proof of this result.

The hereditary class of cographs consists of all graphs G that can be generated from K_1 using complementation and disjoint union. A graph is an apex cograph if it contains a …

Parallel Circles: A Math Club As A University Math Circle, Sarah C. Cobb, Marcos Lopez

Journal of Math Circles

This paper will discuss a monthly social event for the University Math Club. Over the past several years, the event has been shaped by the philosophy of a Student Math Circle. We will discuss the event's organization and the ways it has benefited from the influence of the Math Circle community.

On The Power Series Method For Solitons And Peakons Of Nonlinear Pde, Erika Olivares, Vesselin Vatchev

School of Mathematical & Statistical Sciences Faculty Publications

In this paper we discuss the power series method to obtain traveling wave solutions of nonlinear PDE in one space variable. The method, including a recursive procedure for the coefficients and error estimate for the partial sums, is presented for the Camassa-Holm equation. Numerical examples of approximate single soliton and single peakon solutions are included.

Existence And Uniqueness Of Solutions Of Fourth Order Iterative Boundary Value Problems, Zach Whaley

Theses and Dissertations

In this dissertation, we study the fourth-order iterative differential equation \begin{displaymath} x^{(4)}(t) = \f(t, x(t), x^{[2]}(t), \dots, x^{[m]}(t)) \end{displaymath} where $x^{[2]}(t) = x(x(t))$ and $x^{[j]}(t) = x(x^{[j - 1]}(t))$ for $j > 2$. We consider the above equation with multiple sets of boundary conditions, and we state results on the existence and uniqueness of solutions for each set of boundary conditions. In Chapter 2, the boundary conditions are conjugate boundary conditions, \begin{align*} &x(-a) = -a, \ x'(-a) = b, \ x''(-a) = c, \ x(a) = a \\ &x(-a) = -a, \ x(a) = a, \ x'(a) = b, \ x''(a) …

Mathematical Modelling Of Disease Outbreak, Favour Christian, Matthew Molloy

SURE Journal: Science Undergraduate Research Experience Journal

Establishing a model framework for more research necessitates a thorough understanding of the causes, distribution, prevalence, and evolution of infectious illnesses. The main mathematical concept used in this modelling simulation is ordinary differential equations (ODEs). The purpose of this study was to investigate the significance of the many criteria linked to a zombie virus spread. The zombie framework provides an accessible and relatively simple representation of the nature of infectious disease spread, allowing for tractable assumptions and the development of more complex situations.

The models are designed around a zombie outbreak in which the zombie virus is spread through a …

Theory And Algorithms To Learn, Propagate, And Exploit Uncertainty For Stochastic Optimal Control Of Dynamical Systems, Vignesh Sivaramakrishnan

Electrical and Computer Engineering ETDs

Non-Gaussian uncertainty frequently arises in learning and control problems involving stochastic dynamical systems, particularly in autonomous vehicles, UAVs, satellites, and robotics. In this dissertation, we propose a new framework that leverages characteristic functions that provides a frequency-domain representation of random variables. The dissertation is structured into three key areas. First, we address model-based stochastic optimal control for linear systems with non-Gaussian noise, demonstrating that characteristic functions can be used to enforce chance constraints and control systems toward desired distributions. Second, we explore data-driven stochastic control, utilizing empirical characteristic functions to handle systems with unknown disturbances. In addition, we derive several …

Operations On Submodules With The Multiplicative And Quotient Properties, Jiekai Pang

Mathematics & Statistics ETDs

Inspired by the works of Petro, Epstein, Vassilev, and Morre, this thesis aims to study the generalized definitions of the semiprime operation, weakly prime operation, and standard closure operation on rings, that is, the multiplicative operation, weakly multiplicative operation, and standardly multiplicative operation on submodules. Then, we will use Matlis duality to induce the dual notions of these definitions on submodules of Matlis-dualizable Artinian modules. In order to understand the dual notion of the standardly multiplicative operation, that is, the standardly quotient operation, we will classify the operations on the injective hull of residue field of the ring K[[x,y]]/(xy) which …