Mathematics Commons^™

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 …

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 …

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.

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 …

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.

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 …

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.

Integrable Discrete Massive Thirring Model, Junchao Chen, Bao-Feng Feng

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we are concerned with integrable semi- and fully discrete analogues of the massive Thirring model in light core coordinates. By using the Hirota’s bilinear approach and the Kadomtsev-Petviashvili (KP) hierarchy reduction method, we propose both the semi- and fully discrete massive Thirring models and construct their multi-bright soliton solutions.

Orbital Stability Of Solitary Wave Solutions Of The Generalized Benjamin Equation, Leijin Cao, Binhua Feng, Zhaosheng Feng, Yichun Mo

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we study the existence and stability of solitary wave solutions for the generalized Benjamin equation in both the L2-critical and L2-supercritical cases by applying the variational methods and the non-homogeneous Gagliardo–Nirenberg inequality. Our main results generalize and complement the existing results in the literature.

Estimation Of 3d Facial Dynamics With Nonlinear Filters For Position Tracking, Thoa Thieu, Roderick Melnik

School of Mathematical & Statistical Sciences Faculty Publications

This study presents a comparative evaluation of three nonlinear state estimation filters, the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and Particle Filter (PF), for the task of 3D facial landmark tracking. Using a publicly available dataset, we assess each filter's performance under both deterministic (noise-free) and stochastic (noisy) conditions. Metrics such as mean squared error (MSE), convergence rates of state and covariance estimates, and consistency over time are used to quantify tracking performance. Results show that the EKF consistently outperforms the UKF and PF, achieving faster convergence and lower estimation error, particularly in scenarios characterized by mild nonlinearity. …

Is Noise Exposure Associated With Impaired Extended High Frequency Hearing Despite A Normal Audiogram? A Systematic Review And Meta-Analysis, Sajana Aryal, Monica Trevino, Hansapani Rodrigo, Srikanta K. Mishra

School of Mathematical & Statistical Sciences Faculty Publications

Understanding the initial signature of noise-induced auditory damage remains a significant priority. Animal models suggest the cochlear base is particularly vulnerable to noise, raising the possibility that early-stage noise exposure could be linked to basal cochlear dysfunction, even when thresholds at 0.25-8 kHz are normal. To investigate this in humans, we conducted a meta-analysis following a systematic review, examining the association between noise exposure and hearing in frequencies from 9 to 20 kHz as a marker for basal cochlear dysfunction. Systematic review and meta-analysis followed PRISMA guidelines and the PICOS framework. Studies on noise exposure and hearing in the 9 …

Insecticide-Treated Net Use And Elimination Of Malaria In Sub-Saharan African Countries: Assessing The Global Technical Strategy Using An Evolutionary Game Approach, Laxmi, Tamer Oraby, Michael G. Tyshenko, Ina Danquah, Samit Bhattacharyya

School of Mathematical & Statistical Sciences Faculty Publications

Background: Malaria continues to be a major public health challenge in Sub-Saharan Africa (SSA), where the majority of the countries have not met the World Health Assembly's endorsed Global Technical Strategy (GTS) milestones in 2020 for malaria reduction. Insecticide-treated net (ITN) usage is a well-established and effective intervention, often outperforming other measures such as indoor residual spraying (IRS). However, multiple survey studies have reported improper use of ITNs across various SSA countries. This misuse likely poses an important barrier to the intervention's success, although it remains a largely untested hypothesis.

Methods: We developed a behaviour-incidence model and statistical analysis of …

Variants Of Conway Checkers And K-Nacci Jumping, Glenn Bruda, Joseph Cooper, Kareem Jaber, Raul Marquez, Steven J. Miller

School of Mathematical & Statistical Sciences Faculty Publications

Conway Checkers is a game played with a checker placed in each square of the lower half of an infinite checkerboard. Pieces move by jumping over an adjacent checker, removing the checker jumped over. Conway showed that it is not possible to reach row 5 in finitely many moves by weighting each cell in the board by powers of the golden ratio such that no move increases the total weight.

Other authors have considered the game played on many different boards, including generalizing the standard game to higher dimensions. We work on a board of arbitrary dimension, where we allow …

Connections On Manifolds: A Modern Approach To Classical Differential Geometry, Jed Allen Wilshire

Theses and Dissertations

The purpose of this thesis is to connect modern differential geometry concepts to the classical differential geometry study of surfaces, specifically spheres. This thesis will cover the topics of Riemannian manifolds, tangent spaces, vectors, vector fields, the connection of Levi-Civita, the Christoffel symbols, and Riemannian curvature. Each concept will be demonstrated on the well known unit sphere manifold.

A Computational Approach To Ramanujan-Type Congruences For Products Of Level 13, Aiden Michael Lalonde

Theses and Dissertations

In this thesis, we find products that are modular forms of level 13 whose Fourier coefficients satisfy certain congruences. These products are constructed in terms of integer powers of Klein forms and the Dedekind eta function. The products that are modular forms for Γ1 (13) have vectors of exponents that are elements of a bounded polytope. A parameterization for this polytope allows us to find a complete set of products that are modular forms for Γ1 (13). The corresponding graded ring of modular forms containing these products is constructed from a finite set of generators. These generators satisfy a finite …

Quantization For The Mixtures Of Overlap Probability Distributions, Asha Barua, Angelina Chavera, Ivan Djordjevic, Valerie Manzano, Sergio Soto Quintero, Mrinal Kanti Roychowdhury, Hilda Tejeda

School of Mathematical & Statistical Sciences Faculty Publications

Optimal quantization for mixed distributions has emerged as a compelling area of study. In this work, we have focused on a mixed distribution formed from two uniform distributions with partially overlapping supports. For this class of distributions, we have examined the structure of optimal sets of n-means and the corresponding nth quantization errors for all positive integers n. Initially, we explicitly determined the optimal sets and quantization errors for 1 < = n < = 6. Subsequently, we established several key lemmas and propositions and proposed an algorithm that facilitates the computation of optimal n-means and quantization errors for all n >= 5. Numerical results are also presented to illustrate the application of the algorithm in deriving these quantities. The findings of this study offer valuable insight and serve as a …

Sign Patterns Of Certain Infinite Products, Zeyu Huang, Timothy J. Huber, James Mclaughlin, Pengjun Wang, Yan Xu, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

The signs of Fourier coefficients of certain eta quotients are determined by dissecting expansions for theta functions and by applying a general dissection formula for certain classes of quintuple products. A characterization is given for the coefficient sign patterns for

(qi;qi)∞(qp;qp)∞

for integers i>1 and primes p>3. The sign analysis for this quotient addresses and extends a conjecture of Bringmann et al. for the coefficients of (q2;q2)∞(q5;q5)−1∞. The sign distribution for additional classes of eta quotients is considered. This addresses multiple conjectures posed by Bringmann et al.

Comparing Machine Learning, Deep Learning, And Reinforcement Learning Performance In Culex Pipiens Predictive Modeling, Wei Yin, Sanad H. Ragab, Michael G. Tyshenko, Teresa Patricia Feria-Arroyo, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

Several machine learning (ML) and deep learning (DL) methods have been used to predict the presence of species in classification problems. Another set of methods, called reinforcement learning (RL), has been used in training agents to perform various tasks, but not in predicting species distribution. Culex pipiens (Diptera: Culicidae), commonly known as the common house mosquito, is a globally distributed species prevalent in temperate and subtropical regions. They serve as a primary vector for West Nile Virus (WNV), a mosquito-borne pathogen that affects humans and other animals. The study objective is to compare the performance of logistic regression, random forest …

Pfaffian Solution For Dark–Dark Soliton To The Coupled Complex Modified Korteweg–De Vries Equation, Chenxi Li, Xiaochuan Liu, Bao-Feng Feng

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we study the coupled complex modified Korteweg–de Vries (ccmKdV) equation by combining the Hirota’s method and the Kadomtsev–Petviashvili (KP) reduction method. First, we show that the bilinear form of the ccmKdV equation under nonzero boundary condition is linked to the discrete BKP hierarchy through Miwa transformation. Based on this finding, we construct the dark–dark soliton solution in the pfaffian form. The dynamical behaviors for one- and two-soliton are analyzed and illustrated.

Ecophysiological Benefits Of Industrial Organic Wastes On Alfalfa Yield And Stress Mitigation In Saline Soils, Shugao Fan, Zhuanzhuan Ma, Jiawei Wu, Tianhui Yang, Jincheng Hao, Shumei Chen, Xutong Hu, Ying Zhao

School of Mathematical & Statistical Sciences Faculty Publications

Rising soil salinity and water scarcity demand sustainable solutions to maintain crop productivity. This study evaluated four agro-industrial organic amendments—fish emulsion, neem cake, coffee grounds, and seaweed extract on alfalfa (Medicago sativa) under combined salinity and drought stress, assessing their effects on soil respiration, plant physiology, and yield. Soil CO₂ efflux varied significantly among treatments: coffee grounds induced the highest microbial activity (6.2 μmol m⁻² s⁻¹), suggesting rapid decomposition, while neem cake exhibited stable emissions (3.8 μmol m⁻² s⁻¹), indicating slower organic matter turnover. Fish emulsion showed intermediate dynamics, peaking at 4.8 μmol m⁻² s⁻¹ by week 4, …

Paclobutrazol Enhances Tall Fescue Salt Tolerance Via Physiological And Root System Architecture Modulation, Shugao Fan, Xindi Sun, Guohao Liang, Zhuanzhuan Ma, Jincheng Hao, Jiawei Wu, Ying Zhao

School of Mathematical & Statistical Sciences Faculty Publications

Background: Salinity represents a major global constraint on crop productivity. Promoting the cultivation of tall fescue in saline environments offers not only nutritional advantages for livestock but also enhances its potential for ornamental use. In this mesocosm study, we examined the effects of paclobutrazol (PBZ) on tall fescue performance under salt stress, focusing on key physiological traits to evaluate salt tolerance.

Results: Under high salt stress, paclobutrazol application increased the total number of lateral roots by 85%, from 18.39 to 34.04, and widened their growth angle by 24%, from 24.10° to 29.88°, fundamentally enhancing topsoil exploration. This reconfigured root system …

Causal Predictive Modeling Of Survival Of Lung And Bronchus Cancer Patients Diagnosed During 2010–2011 In Texas, Zeinab Mohamed, Sidketa Fofana, Everardo Cobos, Manish K. Tripathi, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

Background: Lung and Bronchus cancer is the most fatal type of cancer in the United States. According to the American Cancer Society, there were more than 127,000 deaths from lung cancer in 2023. Lung cancer care cost 23.8 billion dollars in 2020. In Texas, only 22.8% of lung cancer patients survived 5 years or more past diagnosis based on 2012-2018 data.

Aim: This study evaluates the survival length of lung and bronchus cancer patients in Texas using advanced statistical and machine learning methods applied to an 11-year cohort study from Surveillance, Epidemiology, and End Results Program. It also quantifies the …

Existence, Uniqueness, Stability And Exact Expression Of Periodic Traveling Wave Solutions Of The Higher-Order Cubic-Quintic Nonlinear Schrödinger Equation, Nan Wang, Yuqian Zhou, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, the high-order cubic-quintic nonlinear Schrödinger equation is considered. By overcoming the difficulties caused by its non-integrability, we completely explore the existence, uniqueness and stability of the limit cycle of the equation. It enables us to accurately give the explicit parameter representation of its periodic wave solution in a special form for the first time. These results provide not only a new idea of obtaining periodic solutions in complex nonlinear systems, but also the important theoretical support for understanding related wave phenomena.

Follow Me Into Math: Near-Peer Influencers Take The Stage, Aaron T. Wilson, Sergey Grigorian, Mayra Ortiz Galarza, Xiaohui Wang

School of Mathematical & Statistical Sciences Faculty Publications

Young people like to be entertained, and to entertain others. What if students were to invest enough of their time and effort into learning topics from our mathematics courses to become experts in the eyes of their peers? In this study we positioned college students and high school students as mathematical performers on the stage in front of their nearpeers, for whom they became math influencers, developing and presenting MathShows. The paper shows readers how we positioned students on the stage, using a 7-step MathShow Design Process informed by a Neer-peer Mathematical Mentoring framework. Of import for educators, researchers and …

Rogue Wave Patterns Associated With Adler–Moser Polynomials In The Nonlocal Nonlinear Schrödinger Equation, Tingyou Wang, Zhenyun Qin, Gui Mu, Fuzhuang Zheng, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, novel rogue wave patterns in the nonlocal nonlinear Schrödinger equation (NLS) are investigated by means of asymptotic analysis, including heart-pentagon, oval-triangle, and fan-triangle. It is demonstrated that when multiple free parameters get considerably large, rogue wave patterns can approximately be predicted by the root structures of Adler–Moser polynomials. These polynomials, which extend the Yablonskii–Vorob’ev polynomial hierarchy, exhibit richer geometric shapes in their root distributions. The (x,t)-plane is partitioned into three regions and through a combination of asymptotic results in different regions, unreported rogue wave patterns can be probed. Predicted solutions are compared with true rogue waves in …