Mathematics Commons^™

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.

Koba-Nielsen Local Zeta Functions, Convex Subsets, And Generalized Selberg-Mehta-Macdonald And Dotsenko-Fateev-Like Integrals, Willem Veys, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

The Koba-Nielsen local zeta functions are integrals depending on several complex parameters, used to regularize the Koba-Nielsen string amplitudes. These integrals are convergent and admit meromorphic continuations in the complex parameters. In the original case, the integration is carried out on the n-dimensional Euclidean space. In this work, the integration is over a variety of (bounded or unbounded) convex subsets; the resulting integrals also admit meromorphic continuations in the complex parameters. We describe the meromorphic continuation's polar locus explicitly, using the technique of embedded resolution. This result can be reinterpreted as saying that the meromorphic continuations are weighted sums of …

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 …

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 …

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.

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.

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 …

Hybrid Vision Transformer–Cnn Framework For Alzheimer’S Disease Cell Type Classification: A Comparative Study With Vision–Language Models, Md Easin Hasan, Md Tahmid Hasan Fuad, Omar Sharif, Amy Wagler

School of Mathematical & Statistical Sciences Faculty Publications

Accurate identification of Alzheimer’s disease (AD)-related cellular characteristics from microscopy images is essential for understanding neurodegenerative mechanisms at the cellular level. While most computational approaches focus on macroscopic neuroimaging modalities, cell type classification from microscopy remains relatively underexplored. In this study, we propose a hybrid vision transformer–convolutional neural network (ViT–CNN) framework that integrates DeiT-Small and EfficientNet-B7 to classify three AD-related cell types—astrocytes, cortical neurons, and SH-SY5Y neuroblastoma cells—from phase-contrast microscopy images. We perform a comparative evaluation against conventional CNN architectures (DenseNet, ResNet, InceptionNet, and MobileNet) and prompt-based multimodal vision–language models (GPT-5, GPT-4o, and Gemini 2.5-Flash) using zero-shot, few-shot, and chain-of-thought …

Discrete Quantization On Spherical Geometries: Explicit Models, Computations, And Didactic Exposition, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

This article presents a comprehensive and analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric. Focusing on highly symmetric configurations on the unit sphere 𝕊2, we investigate three explicit models of discrete uniform distributions and derive closed-form expressions for their optimal quantizers and corresponding mean square quantization errors. (I) For N equally spaced points on the equator, we obtain exact error formulas for both divisible and non-divisible cases 𝑛 ∤𝑁, demonstrating that optimal Voronoi cells form contiguous arcs with midpoint representatives. (II) For two antipodally symmetric small circles at latitudes ±𝜙0, each with M …

Systematic Synthesis Of Crispr/Cas Applications For Enhancing Salt Tolerance In Crops: A Decade Of Progress And Challenges, Xindi Sun, Fusheng Wu, Zhuanzhuan Ma, Guohao Liang, Shumei Chen, Xutong Hu, Shugao Fan, Ying Zhao

School of Mathematical & Statistical Sciences Faculty Publications

Soil salinity is a major constraint on global crop productivity, driving the need for salt-tolerant varieties. While CRISPR-Cas genome editing offers targeted solutions for trait improvement, significant biological and technical bottlenecks limit its application in conferring salt stress resilience. This systematic summarizes findings from 83 peer-reviewed studies (2015–2024) employing CRISPR/Cas technologies to improve salt tolerance in five major crops (rice, wheat, maize, sorghum, barley). Our systematic review reveals that early single-gene edits achieved modest gains (30–50% Na⁺ exclusion) but often showed limited yield gains in field settings, potentially due to compensatory regulation and environmental variation. The literature suggests that multiplex …

Flips In Two-Dimensional Hypertriangulations, Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian

School of Mathematical & Statistical Sciences Faculty Publications

We study flips in hypertriangulations of planar points sets. Here a level-k hypertriangulation of n points in the plane is a subdivision induced by the projection of a k-hypersimplex, which is the convex hull of the barycenters of the (k−1)-dimensional faces of the standard (n−1)-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.

Constrained Quantization For Probability Distributions, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

In this work, we extend the classical framework of quantization for Borel probability measures defined on normed spaces ℝ𝑘 by introducing and analyzing the notions of the nth constrained quantization error, constrained quantization dimension, and constrained quantization coefficient. These concepts generalize the well-established nth quantization error, quantization dimension, and quantization coefficient, which are traditionally considered in the unconstrained setting and thereby broaden the scope of quantization theory. A key distinction between the unconstrained and constrained frameworks lies in the structural properties of optimal quantizers. In the unconstrained setting, if the support of P contains at least n elements, then the …

Demographic And Other Correlates Of Non-Prescription Drug Use Among College Students During The Covid-19 Pandemic, Subi Gandhi, Sidketa Fofana, Md Rafiul Islam, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

Background and objectives: Substance use among college students in the U.S. remains a pressing concern and may have intensified during the COVID-19 pandemic due to increased stress, uncertainty, and academic disruptions. This study investigates the relationship between non-prescription drug use and various demographic, mental health, and behavioral factors among college students during the pandemic's early stages.

Methods: Data were collected through online and in-person surveys in the summer semester of 2021. Behavioral health was assessed using validated instruments: the Patient Health Questionnaire-9 (PHQ-9) for depression and the Drug Abuse Screening Test-20 (DAST-20) for substance use. Demographic and behavioral variables were …

Recurrence Of Locally Perturbed Random Walks With Long Jumps, Tamer Oraby, András Telcs

School of Mathematical & Statistical Sciences Faculty Publications

We study the effect of local perturbations on the recurrence of random walks with long jumps. Such walks serve as discrete models for infinite-horizon Lorentz processes, in which a particle can take arbitrarily long steps in specific directions. Motivated by a question of Sinai in the finite-horizon case and its extension by Szász to the infinite-horizon setting, we give recurrence and transience criteria for long-jump walks on Z2 and certain classes of graphs, and we prove that local perturbations in a bounded region do not change the recurrence property. Our proofs combine the Markov chain approach with the electrical network …

Rogue Peakon, Posedness And Blow-Up Phenomenon For An Integrable Camassa–Holm Type Equation, Mingxuan Zhu, Zhenteng Zeng, Zaihong Jiang, Baoqiang Xia, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we study an integrable Camassa–Holm (CH) type equation with quadratic nonlinearity. The CH type equation is shown integrable through a Lax pair, and particularly the equation is found to possess a new kind of peaked soliton (peakon) solution – called rogue peakon, that is given in a rational form with some logarithmic function, but not a regular traveling wave. We also provide multi-rogue peakon solutions. Furthermore, we discuss the local well-posedness of the solution in the Besov space 𝐵𝑠𝑝,𝑟 with 1 ≤ p, r ≤ ∞, 𝑠 >max⁡{1+1/𝑝,3/2} or 𝐵3/22,1 , and then prove the ill-posedness …

Optimal Quantization Of Finite Uniform Data On The Sphere, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

This paper develops a systematic and geometric theory of optimal quantization on the unit sphere 𝕊2, focusing on finite uniform probability distributions supported on the spherical surface—rather than on lower-dimensional geodesic subsets such as circles or arcs. We first establish the existence of optimal sets of n-means and characterize them through centroidal spherical Voronoi tessellations. Three fundamental structural results are obtained. First, a cluster-purity theorem shows that when the support consists of well-separated components, each optimal Voronoi region remains confined to a single component. Second, a ring allocation (discrete water-filling) theorem provides an explicit rule describing how optimal representatives are …

Computational Insights Into Orthotropic Fracture: Crack-Tip Fields In Strain-Limiting Materials Under Non-Uniform Loads, Saugata Ghosh, Dambaru Bhatta, S. M. Mallikarjunaiah

School of Mathematical & Statistical Sciences Faculty Publications

A finite element framework is presented for analyzing crack-tip phenomena in transversely isotropic, strain-limiting elastic materials. Mechanical response is characterized by an algebraically nonlinear constitutive model, relating stress to linearized strain. Non-physical strain singularities at the crack apex are mitigated, ensuring bounded strain magnitudes. This methodology significantly advances boundary value problem (BVP) formulation, especially for first-order approximate theories. For a transversely isotropic elastic solid with a crack, the governing equilibrium equation, derived from linear momentum balance and the nonlinear constitutive model, is reduced to a second-order, vector-valued, quasi-linear elliptic BVP. This BVP is solved using a robust numerical scheme combining …

The Classical Limit In Geometric Quantization By Group Extension, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Physical systems as a rule are associated with a symmetry group. The group approach to geometric quantization makes use of this to introduce a quantization by means of group extension. This procedure is discussed and applied to a physical system whose group law has its origin with the Galilean group. The main intention is to investigate the classical limit and its relationship under this approach to geometric quantization. The classical quantization conditions are obtained based on a deeper foundation.

Strong Q-Analogues For Values Of The Dirichlet Beta Function, Ankush Goswami, Tim Huber

School of Mathematical & Statistical Sciences Faculty Publications

An infinite class of relations between modular forms is constructed that generalizes evaluations of the Dirichlet beta function at odd positive integers. The work is motivated by a base case appearing in Ramanujan’s Notebooks and a parallel construction for the Riemann zeta function. The identities are shown to be strong q-analogues by virtue of their reduction to the classical beta evaluations as 𝑞→1− and explicit evaluations at CM points for |𝑞|< 1. We also determine asymptotic formulas for the Fourier coefficients of the associated modular forms.

A Conceptual Framework For Cognitive Engagement And Human Agency In Ai-Mediated Learning, Sam Rhodes, Zuhal Yilmaz, Terrie Galanti, Younggon Bae, Victoria Delaney, Zareen Gul Aga, Stephen Hutt

School of Mathematical & Statistical Sciences Faculty Publications

The purpose of this chapter is to propose a conceptual framework that unpacks AI use in instructional practice across three interrelated dimensions: human agency, cognitive demand, and AI orchestration. We ground this work in historical perspectives on technology's mediating role in education. We then propose the conceptual framework and three characterizations of the use of AI in school instruction: cognitive load reducer, cognitive capacity builder, and cognitive companion. We then illustrate a worked example of each, including a curricular example and sample student prompts, and conclude with questions that educators might ask themselves as they consider if and how to …

Optimal Quantization On Spherical Surfaces: Continuous And Discrete Models—A Beginner-Friendly Expository Study, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets ofthe sphere, emphasizing both continuous and discrete settings. We first present a detailedgeometric and analytical foundation for intrinsic quantization on the unit sphere, includingdefinitions of great and small circles, spherical triangles, geodesic distance, Slerp interpolation,the Fréchet mean, spherical Voronoi regions, centroid conditions, and quantizationdimensions. Building upon this framework, we develop explicit continuous and discretequantization models on spherical curves, namely great circles, small circles, and greatcircular arcs—supported by rigorous derivations and pedagogical exposition. For uniformcontinuous distributions, we compute optimal …

Gain-Delay Decoupling In Human Stimulus Frequency Otoacoustic Emissions Reveals Constraints On Cochlear Nonlinear Amplification, Yoshita Sharma, Arturo Moleti, Renata Sisto, Teresa Botti, Hansapani Rodrigo, Sri Mishra

School of Mathematical & Statistical Sciences Faculty Publications

Purpose: The nonlinear cochlear amplifier, driven by outer hair cells, underlies the remarkable sensitivity and frequency selectivity of the mammalian auditory system. Stimulus frequency otoacoustic emissions (SFOAEs) provide a noninvasive window into these active cochlear processes, yet the relationship between emission gain and delay across stimulus levels remains incompletely understood. This study examined the level dependence of SFOAEs in normal-hearing human listeners to characterize cochlear nonlinear response properties. We tested how emission gain and delay vary with stimulus level and estimated the frequency of the apical–basal transition associated with the breakdown of the approximate local scaling symmetry.

Methods: SFOAEs were …