Mathematics Commons^™

Toward Mapping Multiphase Multicomponent Mixtures With Neural Networks, Kristen L. Hallas, Melissa De Jesus, Christine J. Wu, Jianzhi Li, Jason Bernstein, Philip C. Myint

School of Mathematical & Statistical Sciences Faculty Publications

Equation of state (EOS) tables are commonly used in hydrodynamic simulations of high-pressure, high-temperature phenomena in fields like planetary science, astrophysics, and high-energy-density science. However, generating and storing EOS tables for multiphase, multicomponent mixtures over a wide range of pressures and temperatures is computationally infeasible due to their memory-intensive nature. To address this issue, we have developed a neural network-based machine learning model to predict new EOS tables for binary mixtures. In particular, a deep feedforward neural network trained on a set of ten EOS tables at particular mixture compositions is able to predict nine new (hold-out) EOS tables at …

A Finite Element Model For Thermomechanical Stress-Strain Fields In Transversely Isotropic Strain-Limiting Materials, Saugata Ghosh, Dambaru Bhatta, S. M. Mallikarjunaiah

School of Mathematical & Statistical Sciences Faculty Publications

This paper presents a comprehensive computational framework for investigating thermo-elastic fracture in transversely isotropic materials, where classical linear elasticity fails to predict physically realistic behavior near stress concentrations. We address the challenge of unphysical strain singularities at crack tips by employing a strain-limiting theory of elasticity. This theory is characterized by an algebraically nonlinear constitutive relationship between stress and strain, which intrinsically enforces a limit on the norm of the strain tensor. This approach allows the development of very large stresses, as expected near a crack tip, while ensuring that the corresponding strains remain physically bounded. A loosely coupled system …

Study Of A Nonlinear Delayed Parabolic Model For Prion Disease Dynamics With The Unfolded Protein Response, Gangadhara Boregowda, Laurent Pujo-Menjouet, Zhaosheng Feng, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Prion diseases are neurodegenerative disorders characterized by the dynamic spread of misfolded toxic proteins in the brain. In this process, the normal cellular prion protein (PrPC) produced by neurons misfolds into a toxic form known as scrapie prion protein (PrPSc). These misfolded proteins propagate through the brain by converting healthy prions into their toxic form. This biological mechanism can be modeled by a system of nonlinear parabolic partial differential equations, accompanied by a nonlinear delayed integral boundary condition. Our primary objective is to establish the existence of nonnegative classical solutions to this system. Furthermore, we derive a priori estimates for …

Quantum Mechanics, Non-Locality, And The Space Discreteness Hypothesis, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

The space discreteness hypothesis asserts that the nature of space at short distances is radically different from that at large distances. Based on the Bronstein inequality, here, we use a totally disconnected topological space X as a model for the physical space at short distances. However, we consider the time as a real variable. In this framework, the Dirac–von Neumann formalism can be used. This discreteness hypothesis implies that given two different points in space, there is no continuous curve (a world line) joining them. Consequently, this hypothesis is not compatible with the theory of relativity. We propose R×(R×X)3 as …

Bayesian Variable Selection In High-Dimensional Ordinal Quantile Regression Models, Mai Dao, Md. Sakhawat Hossain, Zhuanzhuan Ma

School of Mathematical & Statistical Sciences Faculty Publications

Quantile regression (QR) provides a flexible statistical framework for modeling the entire conditional distribution of the response variable, making it useful for analysis in various fields. Despite its advantages, existing methods for QR often encounter numerical challenges in high-dimensional settings, especially for those with ordinal responses. In this paper, we use a latent-response framework to construct a Bayesian hierarchical model to conduct parameter estimation and variable selection for ordinal QR. Using the asymmetric Laplace working likelihood and the horseshoe prior for the regression coefficients, we obtain the posterior samples to be screened by the sequential two-means clustering process to identify …

Cultural Traits May Replace Human Mobility Data In Forecasting Covid-19 Mortality: A Deep Learning Approach, Saif Abbas, Tamer Oraby, Michael G. Tyshenko, Samit Bhattacharyya

School of Mathematical & Statistical Sciences Faculty Publications

The COVID-19 pandemic highlighted the need for accurate epidemic forecasting to support public health decision-making. Most existing approaches depend heavily on human mobility data, while largely neglecting population behavior shaped by socio-cultural norms. In this study, we analyze daily COVID-19 mortality and Google mobility data from 72 countries during the first 130 d of the pandemic, a period characterized by high uncertainty and behavioral heterogeneity. In particular, we examine whether Hofstede’s country-level cultural dimensions can serve as latent behavioral forecasters of mortality in lieu of dynamic mobility indicators. Using 100 d for training and 30 d for forecasting, we employ …

A Bayesian-Optimized Ensemble Deep Learning Framework For Automated Detection And Classification Of Retinal Diseases In Ghana Using Oct Images, Gifty Duah, Eric Nyarko, Gideon Nana Amo, Theophilus Dwamena Frimpong, Anani Lotsi

School of Mathematical & Statistical Sciences Faculty Publications

Retinal diseases pose a significant global health challenge due to their potential to cause severe visual impairment and blindness. This study aimed to develop a robust deep learning ensemble framework for the automated detection and classification of retinal diseases from optical coherence tomography (OCT) images. This study used OCT images from WATBORG Eye Services in Ghana, including glaucoma, macular edema, posterior vitreous detachment (PVD), and healthy eyes. The data preprocessing steps included augmentation, resizing, and one-hot encoding. The dataset was divided into training (56%), validation (14%), and testing (30%) sets using stratified sampling. Six convolutional neural network (CNN) architectures, Visual …

Comparative Analysis Of Traditional And Deep Learning Time Series Architectures For Influenza A Infectious Disease Forecasting, Edmund Fosu Agyemang, Hansapani Rodrigo, Vincent Agbenyeavu

School of Mathematical & Statistical Sciences Faculty Publications

Influenza A remains a major cause of respiratory mortality worldwide, motivating accurate forecasting to support timely preparedness and resource allocation. This study presents a comparative evaluation of two traditional seasonal time series baselines, ARIMA and Holt–Winters exponential smoothing (ETS), and six deep learning (DL) architectures (Simple RNN, LSTM, GRU, BiLSTM, BiGRU, and a Transformer) for forecasting monthly Influenza A case counts in the United States. Data from January 2009 to December 2023 were analyzed, using January 2009 to December 2022 for training and January 2023 to December 2023 for out-of-sample testing. Models were tuned using a validation split and assessed …

Toward A Didactical Phenomenology For The Completeness Axiom, Sean Larsen, Tenchita Alzaga Elizondo, Kristen Vroom, Stephen Strand Ii

School of Mathematical & Statistical Sciences Faculty Publications

The study is part of an instructional design project focused on introductory real analysis. The goal of the project is to develop a theoretically grounded and empirically supported instructional approach that builds on students’ experiences in the calculus sequence to engage them in the reinvention of the rigorous foundations of the calculus. An essential aspect of this foundation is the completeness of the real numbers. Drawing on the didactical phenomenology heuristic from the theory of Realistic Mathematics Education (RME), we conducted an iterative instructional design study focused on the completeness axiom. The work proceeded in two phases. First, we conducted …

Bayesian Subgroup Learning Of Spatially Resolved Transcriptomics Data, Hou-Cheng Yang, Huimin Li, Guanyu Hu, Qiwei Li

School of Mathematical & Statistical Sciences Faculty Publications

Recent advancements in spatially resolved transcriptomics (SRT) technologies have enabled the comprehensive molecular and spatial characterization of single cells, providing valuable insights into the cellular organization of tissues. SRT techniques, such as single-molecule fluorescence in situ hybridization (FISH)-based methods (e.g., seqFISH, STARmap) and next-generation sequencing (NGS)-based methods (e.g., spatial transcriptomics, 10x Visium), allow for the measurement of gene expression across large populations of cells or tissue spots. These approaches generate high-dimensional data that integrate both molecular profiles and spatial context, which is crucial for understanding tissue structure and function in areas like development, neuroscience, and cancer biology. Identifying spatially variable …

Scattering From Analytic And Piecewise Analytic Inhomogeneities, Narek Hovsepyan, Michael S. Vogelius

School of Mathematical & Statistical Sciences Faculty Publications

We study scattering for the linear Helmholtz operator in two dimensions and develop a technique which can be used to ascertain scattering of a given incident wave from very regular inhomogeneities. This technique is then applied to a number of interesting examples.

Generating Live Heatmaps Of Edr Data Through A Spatiotemporal Weighting, Joel T. Williams, C. Sean Bohun, Alberto Fornaci, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

This paper introduces a novel method for generating live heatmaps of eddy dissipation rate (EDR) data through a spatiotemporal weighting designed to enhance turbulence visualization in aviation. As more flight data become available, approaches relying solely on in-flight EDR measurements have the potential to accurately nowcast and visualize turbulence with low computational cost. The proposed method significantly improves the turbulence visualization capabilities of common commercial aircraft. This is particularly valuable for pilot decision-making and trip planning, enhancing flight safety and operational efficiency. This approach also incorporates an innovative uncertainty threshold, which refrains from predicting when there are insufficient data, thereby …

Fostering Innovation At The Intersection Of Maker Education And Extended Reality (Xr), Jewoong Moon, Yong Ju Jung, Soo Hyeon Kim, Younggon Bae, Bertrand Schneider

School of Mathematical & Statistical Sciences Faculty Publications

No abstract provided.

Hyper-Bishops, Hyper-Rooks, And Hyper-Queens: Percentage Of Safe Squares On Higher Dimensional Chess Boards, Caroline Cashman, Joseph Cooper, Raul Marquez, Steven J. Miller, Jenna Shuffelton

School of Mathematical & Statistical Sciences Faculty Publications

Chess has inspired an abundance of mathematical problems, especially in combinatorics and probability. One such problem, initially studied by Miller, Sheng, and Turek, considers the proportion of safe spaces when randomly placing n rooks on an 𝑛×𝑛 chess board. They show that as n approaches infinity, the proportion of safe spaces converges to 1/𝑒2. We first generalize their results to bishops and queens. This problem is significantly more interesting and difficult; while a rook attacks the same number of spaces regardless of its position, this is not so for bishops and queens. We prove that the proportion of safe spaces …

Toward The Salmon Prize: A Computational Certificate For The M5, M6, And M9 Equation Families Of Σ4(P3 × P3 × P3), John Trevino

Theatre Faculty Publications

We present a computational certificate for three families of explicit polyno- mial equations whose zero locus contains the fourth secant variety σ4(P3×P3×P3) inside P63. The three families are: 192 degree-5 Strassen commutation equa- tions (M5); 160 degree-6 equations lifted from the Bates–Oeding generators of σ4(P2 × P2 × P3) via the Landsberg–Manivel–Friedland theorem (M6); and 64 degree-9 Ottaviani 9 × 9 determinant equations (M9). All 416 generators are explicit polynomials in the coordinate ring Z[Zijk | i, j, k ∈ {0, 1, 2, 3}]. We verify by exact integer arithmetic that every generator vanishes on rank-4 test tensors (six independent …

Data Driven Monitoring And Control Of Laser Powder Bed Fusion Process, Jose De Jesus Galarza

Theses and Dissertations

The Laser Powder Bed Fusion Process (LPBF) has been one of the main processes of additive manufacturing, enabling the manufacturing of complex geometries, customization, and lightweight parts. Modern LPBF processes have integrated monitoring systems that capture the light emissions per layer for quality assurance. However, standard defect detection algorithms have not yet achieved the high precision required due to the inherently variable nature of the signal, insufficient data for model training, and the confounding effects of the print.

The processes still have some challenges, such as characterizing the roughness from the build parameters alone, improving the pore detection using the …

I Am Still Learning, Too!: Pre-Service Teachers' Experiences Of Mathematics Anxiety In An Inquiry-Based Mathematics Education Course At Utrgv, Jeremiah James Adriano Dela Rosa

Theses and Dissertations

Mathematics anxiety is a prevalent issue in mathematics education that negatively impacts students’ learning, performance, and engagement in mathematics. Prior research suggests that mathematics anxiety is often shaped by students’ experiences and emotional responses within the classroom environment.

The purpose of this study is to explore how Pre-Service Teachers experience mathematics anxiety in an Inquiry-Based Mathematics Education (IBME) classroom. This study employed a qualitative research design supported by descriptive survey data collected through the Abbreviated Mathematics Anxiety Scale (AMAS), selected components of the Fennema-Sherman Mathematics Anxiety Scale (FSMAS), and semi-structured interviews. The survey instruments were used to provide descriptive background …

Fixed Perimeter Analogues Of Some Partition Results, Gabriel Gray, Emily Payne, Holly Swisher, Ren Watson

School of Mathematical & Statistical Sciences Faculty Publications

Euler's partition identity states that the number of partitions of n into odd parts is equal to the number of partitions of n into distinct parts. Strikingly, Straub proved in 2016 that this identity also holds when counting partitions of any size with largest hook length (perimeter) n. This has inspired further investigation of partition identities and inequalities in the fixed perimeter setting. Here, we explore fixed perimeter analogues of some well-known partition results inspired by Euler's partition identity.

Combinatorial Statistics Witnessing An Infinite Family Of Congruences For A Sum Of Partition Functions, Jena M. Gregory

Theses and Dissertations

In 2007, Kronholm established The Interval Theorem, infinite families of congruences in arithmetic progression, modulo any prime ��, for ��(��, ��), the function enumerating the partitions of �� into parts whose sizes come from the set {1, 2, … , ��}. In 2022, Eichhorn, Kronholm, and Larsen proved there are combinatorial statistics described in terms of the multiplicities of the part sizes that witness Kronholm’s Interval Theorem. Here, “witness" means given a congruence of the form ��(��, ��) ≡ 0 (mod ��), we can use these statistics to classify the set of partitions of �� into �� equally sized subsets …

Bounding The Average Kissing Number, Including New Bounds For Three-Dimensional Binary Sphere Packings, Mark William Bockhaus

Theses and Dissertations

The average kissing number is defined as the supremum over all sphere packings of the value: two times the number of tangencies divided by the number balls in the packing. In this paper, we present a survey of the literature about bounding the average kissing number, beginning with the first non-trivial results, through the most up-to-date bounds. We then turn our focus to binary sphere packings: those which contain spheres of two different radii. We improve upon known bounds for the average kissing number for binary sphere packings in three-dimensions and find exact bounds for many packings.

Bayesian Change-Point Detection In Stock And Cryptocurrency Markets Using Shrinkage Priors, Yosalin Sanchez

Theses and Dissertations

Financial markets often undergo abrupt structural changes driven by political, economic, and geopolitical events, leading to substantial volatility. Detecting such change-points is crucial for identifying structural breaks, improving risk management, and enhancing forecasting performance in financial time series. This study proposes a Bayesian change-point detection framework that incorporates both the t-shrinkage prior and the Horseshoe shrinkage prior. These priors enforce strong regularization on successive differences in mean parameters, enabling the identification of piecewise constant structures in time series data. Posterior inference is conducted using Markov Chain Monte Carlo (MCMC) methods, specifically a Gibbs sampling algorithm, which iteratively samples from the …

On The Structure Of The Homotopy Lie Algebra Of Local Rings, Dawson M. Strong

Theses and Dissertations

This thesis investigates the construction and homological properties of the homotopy Lie algebra π(R) of a commutative local ring (R,m,k). Drawing upon the theoretical framework of differential graded (DG) algebras, we first establish the theory of minimal free resolutions and other standard topics in homological algebra. The core of this work details the iterative construction of the acyclic closure R⟨Y⟩ of k over R, which is achieved by the systematic adjunction of exterior and divided power variables to eliminate cycles in homology. We demonstrate that this acyclic closure serves as a minimal free resolution and provides the means to define …

Theory And Simulations Of Delayed Stochastic And Deterministic Models Of Prion Diseases, Gangadhara Boregowda, Omar Sharif, Daniel Gutierrez Iii, Allegra Simmons, Laurent Pujo-Menjouet, Tamer Oraby, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Neurodegenerative diseases (NDs), such as Alzheimer’s, Parkinson’s, and prion diseases, are characterized by the dynamical spread of toxic proteins through the brain. In prion diseases, cellular prion protein (PrPC), produced by neurons, misfolds into a toxic form, known as scrapie prion protein (PrPSc). PrPSc induces neuronal stress which ultimately leads to cell death. In this paper, we develop mathematical models for the progression of prion diseases, incorporating a cellular defense mechanism that introduces a delay term affecting protein translation and a volatility term accounting for unaccounted biological factors influencing the system. We also extend the model to capture the spatial …

Emergent Storylines That Influence Positions And Mathematical Status In Collaborative Small-Group Proof Activity, Brittney M. Ellis, Tenchita Alzaga Elizondo

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we used positioning theory to examine storylines that emerged in students’ discourse as they collaborated on a proof construction task. We purposefully selected a case of group work from an inquiry-oriented introduction to proof course as prior analyses showed it was highly collaborative (Alzaga Elizondo, 2022), yet power dynamics seemed unbalanced. We hypothesized that positioning theory could provide a useful lens to interrogate such power dynamics. Through this analysis, we identified several implicit storylines that influenced the interaction related to the nature of proofs, the nature of mathematics, writing proofs, the role of an external authority, …

Dissection Of The Quintuple Product, With Applications, Tim Huber, James Mclaughlin, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

This work considers the m-dissection (for m≢0(mod3)">m≢0(mod3)) of the general quintuple productQ(z,q)=(z,q/z,q;q)∞(qz2,q/z2;q2)∞.">Q(z,q)=(z,q/z,q;q)∞(qz2,q/z2;q2)∞.Multiple novel applications arise from this m-dissection. For example, we derive the general partition identityDS(mn+(m2−1)/24)=(−1)(m+1)/6bm(n), for all n≥0,">DS(mn+(m2−1)/24)=(−1)(m+1)/6bm(n), for all n≥0,where m≡5(mod6)">m≡5(mod6) is a square-free positive integer relatively prime to 6; DS(n)">DS(n) is defined, for S the set of positive integers containing no multiples of m, to be the number of partitions of n into an even number of distinct parts from S minus the number of partitions of n into an odd number of distinct parts from S; and …

Soliton Solutions To The Coupled Sasa–Satsuma Equation Under Mixed Boundary Conditions, Changyan Shi, Xiyao Chen, Guangxiong Zhang, Chengfa Wu, Bao-Feng Feng

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we derive general bright–dark soliton solutions to the coupled Sasa–Satsuma (CSS) equation using the Kadomtsev–Petviashvili reduction method. Since the CSS equation is a special case of the four-component Hirota equation, our approach begins with the construction of two-bright-two-dark soliton solutions for the four-component Hirota equation. By imposing specific parameter constraints, these solutions are subsequently reduced to the bright–dark soliton solutions of the CSS equation. Finally, the dynamical behaviours of the one- and two-bright–dark soliton solutions are thoroughly analysed and illustrated.

Patterns Of Multimorbidity Among Low-Income Adults Who Smoke With Implications For Tailored Interventions: A Cluster Analysis Using A Mixture Of Bernoulli Model, Monique T. Cano, Michael R. Lindstrom, Oscar F. Rojas Perez, Ricardo F. Muñoz

School of Mathematical & Statistical Sciences Faculty Publications

Introduction: Smoking cigarettes remains a leading modifiable risk factor for preventable health conditions. In the United States, the health burden of smoking disproportionately impacts low-income individuals. Multimorbidity is common in this group, complicating treatment and worsening outcomes. Identifying multimorbidity clusters can support targeted, individualized interventions. This study aimed to identify multimorbidity clusters among individuals who smoke and experience economic hardship and provide clinical recommendations to enhance health outcomes.

Method: Individuals who smoke and experience economic hardship (N = 60) were recruited from the San Francisco Health Network (SFHN) and were assessed for physical and mental conditions. Cluster analysis was …

Measuring What Matters: Specifications Grading And Latin* Students’ Mathematics Identity, Luis Miguel Fernández, Mayra Ortiz Galarza, Cristina Villalobos, Martha Asare

School of Mathematical & Statistical Sciences Faculty Publications

This study examined Specifications Grading, an alternative grading system emphasizing clearly defined learning outcomes and revision, and mathematics identity among 846 Latin* Calculus I students at a Hispanic-Serving Institution. Mathematics identity, comprising competence/performance, recognition, and interest, was measured at the beginning and end of the semester. Repeated-measures analyses indicated stable competence/performance and recognition alongside declines in interest. Specifications Grading was associated with increased mathematics identity, and multilingual students experienced smaller declines than their peers overall.

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Este estudio examinó la calificación por especificaciones, un sistema de evaluación alternativo que enfatiza resultados de aprendizaje claramente definidos y oportunidades estructuradas de revisión, …

Vaccination Games Of Boundedly Rational Parents Toward New Childhood Immunization, Wei Yin, Martial L. Ndeffo-Mbah, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

Infectious diseases harm societies through disease-induced morbidity, mortality, loss of productivity, and inequality. Thus, controlling and preventing them is critical for public health and societal well-being. However, societies can hinder efforts to control the spread of diseases by failing to adhere to public health recommendations, such as through vaccine hesitancy. Various disease-transmission models have been utilized to help policymakers respond to (re)emerging outbreaks. The usefulness of such models in assessing the effectiveness of public health policies is significantly dependent on human behavior. This paper introduces a new model of parental behavior toward a new childhood immunization. The model incorporates societal …

Identical Vanishing Of Coefficients In The Series Expansion Of Eta Quotients, Modulo 4, 9 And 25, Tim Huber, James Mclaughlin, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

Let A(q)=:∑∞n=0anqn and B(q)=:∑∞n=0bnqn be two eta quotients. In some previous papers, the present authors considered the problem of when

an=0⟺bn=0.

In the present paper we consider the “mod m” version of this problem, i.e. for which eta quotients A(q) and B(q) and for which integers m>1 do we have (non-trivially) that

an≡0(modm)⟺bn≡0(modm)?

(We say “non-trivially” as there are trivial situations where an≡bn(modm) for all n≥0). The m for which we found non-trivial (in the sense just mentioned) results were m=p2, p=2,3 and 5. For m=4 and m=9, we found results which …