Mathematics Commons^™

Tau-Function Formulation For Bright, Dark Soliton And Breather Solutions To The Massive Thirring Model, Junchao Chen, Bao-Feng Feng

School of Mathematical & Statistical Sciences Faculty Publications

In the present paper, we are concerned with the link between the Kadomtsev–Petviashvili–Toda (KP–Toda) hierarchy and the massive Thirring (MT) model. First, we bilinearize the MT model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two-component KP–Toda hierarchy, we derive multibright solution to the MT model. Then, considering a set of bilinear equations of the single-component KP–Toda hierarchy, multidark soliton and multibreather solutions to the MT model are constructed by imposing constraints on the parameters in two types of tau function, respectively. The dynamics and properties of one- and two-soliton for bright, …

Probabilistic Solutions Of Fractional Differential And Partial Differential Equations And Their Monte Carlo Simulations, Tamer Oraby, Erwin Suazo, Harrinson Arrubla

School of Mathematical & Statistical Sciences Faculty Publications

The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical approach based on Monte Carlo integration to simulate solutions of fractional ordinary and partial differential equations. Thirdly, we show that this approach allows us to find the fundamental solutions for fractional partial differential equations (PDEs), in which the fractional derivative in time is in the Caputo sense and the fractional in space one is in the Riesz–Feller sense. Lastly, using Riccati equation, we …

Long-Time Asymptotics Of A Complex Cubic Camassa-Holm Equation, Hongyi Zhang, Yufeng Zhang, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

n this paper, we study the Cauchy problem of the following complex cubic Camassa-Holm (ccCH) equation mt=bux+12[m(|u|2−|ux|2)]x−12m(uu¯x−uxu¯),m=u−uxx,

where b is an arbitrary real constant. %By applying ∂¯-steepest descent method, l Long-time asymptotics of the equation is obtained through the ∂¯-steepest descent method. Firstly, based on the spectral analysis of the Lax pair and scattering matrix, the solution of the equation is able to be constructed %can be expressed by %the solution of via solving the corresponding Riemann-Hilbert problem (RHP). Then, we present %obtain different long time asymptotic expansions of the solution u(y,t) in different space-time solitonic regions of ξ=y/t. The …

Resurgence Of Habiro Elements, Samuel Crew, Ankush Goswami, Robert Osburn

School of Mathematical & Statistical Sciences Faculty Publications

We prove resurgence properties for the Borel transform of elements in the Habiro ring which satisfy a general type of strange identity. As an application, we provide evidence for (and against) conjectures in quantum topology due to Costin and Garoufalidis.

Classification On Boundary-Equilibria And Singular Continuums Of Continuous Piecewise Linear Systems, Hebai Chen, Zhaosheng Feng, Hao Yang, Linfeng Zhou

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we show that any switching hypersurface of n -dimensional continuous piecewise linear systems is an (n−1) -dimensional hyperplane. For two-dimensional continuous piecewise linear systems, we present local phase portraits and indices near the boundary equilibria (i.e. equilibria at the switching line) and singular continuum (i.e. continuum of nonisolated equilibria) between two parallel switching lines. The index of singular continuum is defined. Then we show that boundary-equilibria and singular continuums can appear with many parallel switching lines.

Quantization Dimension For Inhomogeneous Bi-Lipschitz Ifs, Amit Priyadarshi, Mrinal Kanti Roychowdhury, Manuj Verma

School of Mathematical & Statistical Sciences Faculty Publications

Let ν be a Borel probability measure on a d-dimensional Euclidean space Rd, d≥1, with a compact support, and let (p0,p1,p2,…,pN) be a probability vector with pj>0 for 1≤j≤N. Let {Sj:1≤j≤N} be a set of contractive mappings on Rd. Then, a Borel probability measure μ on Rd such that μ=∑Nj=1pjμ∘S−1j+p0ν is called an inhomogeneous measure, also known as a condensation measure on Rd. For a given r∈(0,+∞), the quantization dimension of order r, if it exists, denoted by Dr(μ), of a Borel probability measure μ on Rd represents the speed at which the nth quantization error of order r …

Quantization For A Set Of Discrete Distributions On The Set Of Natural Numbers, Juan Gomez, Haily Martinez, Mrinal Kanti Roychowdhury, Alexis Salazar, Daniel J. Vallez

School of Mathematical & Statistical Sciences Faculty Publications

The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. In this paper, first we state and prove a theorem, and then give a conjecture. We verify the conjecture by a few examples. Assuming that the conjecture is true, for a set of discrete distributions on the set of natural numbers we have calculated the optimal sets of n-means and the nth quantization errors for all positive integers n. In addition, the quantization dimension is also calculated.

Note On Illuminating Constant Width Bodies, Alexey Glazyrin

School of Mathematical & Statistical Sciences Faculty Publications

Recently, Arman, Bondarenko, and Prymak constructed a constant width body in R n whose illumination number is exponential in n. In this note, we improve their bound by generalizing the construction. In particular, we construct a constant width body in R n whose illumination number is at least (τ + o(1))n, where τ ≈ 1.047.

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

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, for a Borel probability measure P on a Euclidean space Rk, we extend the definitions of nth unconstrained quantization error, unconstrained quantization dimension, and unconstrained quantization coefficient, which traditionally in the literature known as nth quantization error, quantization dimension, and quantization coefficient, to the definitions of nth constrained quantization error, constrained quantization dimension, and constrained quantization coefficient. The work in this paper extends the theory of quantization and opens a new area of research. In unconstrained quantization, the elements in an optimal set are the conditional expectations in their own Voronoi regions, and it is not true …

Final Report: Conceptualizing Ethics, Authenticity, And Efficacy Of Simulations In Teacher Education, Carrie Wilkerson Lee, Liza Bondurant, Bima Sapkota, Heather Howell, Yvonne Lai

School of Mathematical & Statistical Sciences Faculty Publications

This working group was a continuation of working groups in 2019 and 2021 that initially aimed to focus on equity in simulations of practice in mathematics teacher education. We began by discussing our conceptualizations of simulations and equity. Next, we reflected on the lack of work that currently exists at the intersection of simulations and equity as well as our limited collective expertise in this space. We proposed the following areas of potential research: Access, Design, Affective Domains, Teaching Practices, Assessment, Critical Conversations. Attendees self-selected into focus groups and met to discuss their current work and how future work could …

Approximations Of Practice And Equity: Surveying The Spaces, Bima Sapkota, Carrie Lee, Liza Bondurant, Heather Howell

School of Mathematical & Statistical Sciences Faculty Publications

After three years of meeting as a working group at PME-NA, we will be editing a book on how several models of approximations of practice support or hinder the creation of equitable mathematics learning and thinking spaces. Participants and working group leaders will serve as critical friends as we use a fishbowl format and small groups to provide spaces for refinement and challenge. Participants will leave with a network of thought partners and a draft chapter. At the end of the working group, we will identify overarching themes about affordances and limitations of several models of digital and non-digital forms …

Borsuk–Ulam Type Theorems For G-Spaces With Applications To Tucker Type Lemmas, Oleg R. Musin, Alexey Yu. Volovikov

School of Mathematical & Statistical Sciences Faculty Publications

In this paper we consider several generalizations of the Borsuk–Ulam theorem for G-spaces and apply these results to Tucker type lemmas for G-simplicial complexes and PL-manifolds.

Equity In Focus : Investigating Gender Disparities In Glioblastoma Via Propensity Score Matching, Solomon Eshun

School of Mathematical & Statistical Sciences Faculty Publications

Gender disparities in health outcomes have garnered significant attention, prompting investigations into their underlying causes. Glioblastoma (GBM), a devastating and highly aggressive form of brain tumor, serves as a case for such inquiries. Despite the mounting evidence on gender disparities in GBM outcomes, investigations specific at the molecular level remain scarce and often limited by confounding biases in observational studies. In this study, I aimed to investigate the gender-related differences in GBM outcomes using propensity score matching (PSM) to control for potential confounding variables. The data used was accessed from the Cancer Genome Atlas (TCGA), encompassing factors such as gender, …

Brooks' Theorem For 2-Fold Coloring, Jacob A. White

School of Mathematical & Statistical Sciences Faculty Publications

The two-fold chromatic number of a graph is the minimum number of colors needed to ensure that there is a way to color the graph so that each vertex gets two distinct colors, and adjacent vertices have no colors in common. The Ore degree is the maximum sum of degrees of an edge in a graph. We prove that, for 2-connected graphs, the two-fold chromatic number is at most the Ore degree, unless G is a complete graph or an odd cycle.

Preferential Stiffness And The Crack-Tip Fields Of An Elastic Porous Solid Based On The Density-Dependent Moduli Model, Hyun C. Yoon, S. M. Mallikarjunaiah, Dambaru Bhatta

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we study the preferential stiffness and the crack-tip fields for an elastic porous solid of which material properties are dependent upon the density. Such a description is necessary to describe the failure that can be caused by damaged pores in many porous bodies such as ceramics, concrete and human bones. To that end, we revisit a new class of implicit constitutive relations under the assumption of small deformation. Although the constitutive relationship \textit{appears linear} in both the Cauchy stress and linearized strain, the governing equation bestowed from the balance of linear momentum results in a quasi-linear partial …

Predicting The Outcomes Of Internet-Based Cognitive Behavioral Therapy For Tinnitus: Applications Of Artificial Neural Network And Support Vector Machine, Hansapani Rodrigo, Eldré W. Beukes, Gerhard Andersson, Vinaya Manchaiah

School of Mathematical & Statistical Sciences Faculty Publications

Purpose:

Internet-based cognitive behavioral therapy (ICBT) has been found to be effective for tinnitus management, although there is limited understanding about who will benefit the most from ICBT. Traditional statistical models have largely failed to identify the nonlinear associations and hence find strong predictors of success with ICBT. This study aimed at examining the use of an artificial neural network (ANN) and support vector machine (SVM) to identify variables associated with treatment success in ICBT for tinnitus.

Method:

The study involved a secondary analysis of data from 228 individuals who had completed ICBT in previous intervention studies. A 13-point reduction …

Congruences For Consecutive Coefficients Of Gaussian Polynomials With Crank Statistics, Dennis Eichhorn, Lydia Engle, Brandt Kronholm

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we establish infinite families of congruences in consecutive arithmetic progressions modulo any odd prime ℓ for the function p ( n , m , N ) , which enumerates the partitions of n into at most m parts with no part larger than N . We also treat the function p ( n , m , ( a , b ] ) , which bounds the largest part above and below, and obtain similar infinite families of congruences.

For m ≤ 4 and ℓ = 3 , simple combinatorial statistics called "cranks" witness these congruences. We prove …

Local Well-Posedness Of The Cauchy Problem For A P -Adic Nagumo-Type Equation, L. F. Chacón-Cortés, C. A. Garcia-Bibiano, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

We introduce a new family of p -adic nonlinear evolution equations. We establish the local well-posedness of the Cauchy problem for these equations in Sobolev-type spaces. For a certain subfamily, we show that the blow-up phenomenon occurs and provide numerical simulations showing this phenomenon.

Non-Archimedean Quantum Mechanics Via Quantum Groups, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

We present a new non-Archimedean realization of the Fock representation of the q-oscillator algebras where the creation and annihilation operators act on complex-valued functions, which are defined on a non-Archimedean local field of arbitrary characteristic, for instance, the field of p-adic numbers. This new realization implies that many quantum models constructed using q-oscillator algebras are non-Archimedean models, in particular, p-adic quantum models. In this framework, we select a q-deformation of the Heisenberg uncertainty relation and construct the corresponding q-deformed Schrödinger equations. In this way we construct a p-adic quantum mechanics which is a …

Qualitative Analysis For A Two-Component Peakon System With Cubic Nonlinearity, Shaojie Yang, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

This paper is devoted to studying a two-component peakon system with cubic nonlinearity, which is a two-component extension of the cubic Camassa–Holm equation. We first discuss the local well-posedness for the Cauchy problem of the system. Then, in light of a fine structure of the system, we present the precise blow-up scenario for strong solutions to the system and derive a new blow-up result with respect to initial data. Finally, peakon solutions are discussed as well.

Developing Awareness Around Language Practices In The Elementary Bilingual Mathematics Classroom, Gladys Krause, Melissa Adams Corral, Luz A. Maldonado Rodríguez

Teaching and Learning Faculty Publications

This study contributes to efforts to characterize teaching that is responsive to children’s mathematical ideas and linguistic repertoire. Building on translanguaging, defined in this article as a pedagogical practice that facilitates students’ expression of their understanding using their own language practices, and on the literature surrounding children’s mathematical thinking, we present an example of a one-on-one interview and of the circulating portion of a mathematics class from a second grade classroom. We use these examples to foreground instructional practices, for researchers and practitioners, that highlight a shift from a simplified view of conveying mathematics as instruction in symbology and formal …

Fractional Sobolev Embedding With Radial Potential, Yu Su, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

We present some embedding results of fractional Ds,2(RN) space with radial potential. To overcome the difficulty of non-compactness, we obtain the Lions-type theorems by means of the refined Sobolev inequality with the Morrey norm. Then we apply the obtained results to study the existence of ground state solution for fractional equations with radial potential.

Bi-Dbar-Approach For A Coupled Shifted Nonlocal Dispersionless System, Junyi Zhu, Kaiwen Shao, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

We propose a Bi-Dbar approach and apply it to the extended coupled shifted nonlocal dispersionless system. We introduce the nonlocal reduction to solve the coupled shifted nonlocal dispersionless system. Since no enough constraint conditions can be found to curb the norming contants in the Dbar data, the “solutions” obtained by the Dbar dressing method, in general, do not admit the coupled shifted nonlocal dispersionless system. In the Bi-Dbar approach to the extended coupled shifted nonlocal dispersionless system, the norming constants are free. The constraint conditions on the norming constants are determined by the general nonlocal reduction, and the solutions of …

Functional Consequences Of Extended High Frequency Hearing Impairment: Evidence From The Speech, Spatial, And Qualities Of Hearing Scale, Udit Saxena, Srikanta K. Mishra, Hansapani Rodrigo, Moumita Choudhury

School of Mathematical & Statistical Sciences Faculty Publications

Hearing loss in the extended high frequencies, despite a normal audiogram, could affect speech-in-noise recognition. However, it is not known if extended high frequency (EHF) hearing loss is associated with everyday listening and communication deficits. The present study aimed to determine the functional effects of impaired EHF hearing among adults using the Speech, Spatial, and Qualities of Hearing Scale (SSQ). A secondary objective was to evaluate the relationship between objective (speech-in-noise recognition) and subjective (SSQ) measures of hearing function. Listeners with EHF hearing loss provided lower SSQ ratings compared with their EHF-normal counterparts. The lower ratings could not be attributed …

Alternative Siar Models For Infectious Diseases And Applications In The Study Of Non-Compliance, Marcelo Bongarti, Luke Diego Galvan, Lawford Hatcher, Michael R. Lindstrom, Christian Parkinson, Chuntian Wang, Andrea L. Bertozzi

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we use modified versions of the SIAR model for epidemics to propose two ways of understanding and quantifying the effect of non-compliance to non-pharmaceutical intervention measures on the spread of an infectious disease. The SIAR model distinguishes between symptomatic infected (I) and asymptomatic infected (A) populations. One modification, which is simpler, assumes a known proportion of the population does not comply with government mandates such as quarantining and social-distancing. In a more sophisticated approach, the modified model treats non-compliant behavior as a social contagion. We theoretically explore different scenarios such as the occurrence …

General Breather And Rogue Wave Solutions To The Complex Short Pulse Equation, Bao-Feng Feng, Ruyun Ma, Yujuan Zhang

School of Mathematical & Statistical Sciences Faculty Publications

In the present paper, we attempt to construct both the breather and rogue wave solutions to the focusing complex short pulse (fCSP) equation via the KP–Toda reduction method. Following a procedure of reducing the bilinear equations satisfied by tau functions of Kadomtsev–Petviashvili (KP)–Toda hierarchy to the ones of the fCSP equation with nonzero boundary condition, we first deduce the general breather solution of the fCSP equation starting from a specially arranged tau-function of the KP–Toda hierarchy, then we construct and prove the Nth order rogue wave solutions of the fCSP equation and express them in two different but equivalent forms …

Study Of Flow Of Buongiorno Nanofluid In A Conical Gap Between A Cone And A Disk, Mahanthesh Basavarajappa, Dambaru Bhatta

School of Mathematical & Statistical Sciences Faculty Publications

The cone–disk apparatus consists of a cone that touches the disk at its apex and is used in medical evices, viscosimeters, conical diffusers, etc. Theoretically, a three-dimensional flow of a nanofluid in a conical gap of a cone–disk apparatus is studied for four different physical configurations. Buongiorno nanofluid model, consisting of thermophoresis and Brownian diffusion mechanisms, is used to describe the convective heat transport of the nanofluid. The continuity equation, the Navier–Stokes momentum equation, the heat equation, and the conservation of nanoparticle volume fraction equation constitute the governing system for the flow of nanofluids. The Lie group approach is used …

Multi-Scale Hybridized Topic Modeling: A Pipeline For Analyzing Unstructured Text Datasets Via Topic Modeling, Keyi Cheng, Stefan Inzer, Adrian Leung, Xiaoxian Shen, Michael Perlmutter, Michael Lindstrom, Joyce Chew, Todd Presner, Deanna Needell

School of Mathematical & Statistical Sciences Faculty Publications

We propose a multi-scale hybridized topic modeling method to find hidden topics from transcribed interviews more accurately and efficiently than traditional topic modeling methods. Our multi-scale hybridized topic modeling method (MSHTM) approaches data at different scales and performs topic modeling in a hierarchical way utilizing first a classical method, Nonnegative Matrix Factorization, and then a transformer-based method, BERTopic. It harnesses the strengths of both NMF and BERTopic. Our method can help researchers and the public better extract and interpret the interview information. Additionally, it provides insights for new indexing systems based on the topic level. We then deploy our method …

Study Of Multilayer Flow Of A Bi-Viscous Bingham Fluid Sandwiched Between Hybrid Nanofluid In A Vertical Slab With Nonlinear Boussinesq Approximation, Mahanthesh Basavarajappa, Shruthy Myson, Kuppalapalle Vajravelu

School of Mathematical & Statistical Sciences Faculty Publications

Bi-Viscosity Bingham plastic fluids are used to understand the rheological characteristics of pigment-oil suspensions, polymeric gels, emulsions, heavy oil, etc. High-temperature applications in many industrial and engineering problems, linear density-temperature variation is inadequate to describe convective heat transport. Therefore, the characteristics of the nonlinear convective flow of a Bi-Viscosity Bingham Fluid (BVBF) through three layers in a vertical slab are studied. The two outer layers of the oil-based hybrid nanofluid and the intermediate layer of BVBF are considered. The thermal buoyancy force is governed by the nonlinear Boussinesq approximation. Continuity of heat flux, velocity, shear stress, and temperature are imposed …

The Spectral Picture And Joint Spectral Radius Of The Generalized Spherical Aluthge Transform, Chafiq Benhida, Raul E. Curto, Sang Hoon Lee, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

For an arbitrary commuting d--tuple $\bT$ of Hilbert space operators, we fully determine the spectral picture of the generalized spherical Aluthge transform $\dbT$ and we prove that the spectral radius of $\bT$ can be calculated from the norms of the iterates of $\dbT$. \ Let $\bm{T} \equiv (T_1,\cdots,T_d)$ be a commuting d--tuple of bounded operators acting on an infinite dimensional separable Hilbert space, let P:=T∗1T1+⋯+T∗dTd−−−−−−−−−−−−−−−√, and let ⎛⎝⎜⎜T1⋮Td⎞⎠⎟⎟=⎛⎝⎜⎜V1⋮Vd⎞⎠⎟⎟P be the canonical polar decomposition, with (V1,⋯,Vd) a (joint) partial isometry and ⋂i=1dkerTi=⋂i=1dkerVi=kerP. \medskip For 0≤t≤1, we define the generalized spherical Aluthge transform of $\bm{T}$ by \Delta_t(\bm{T}):=(P^t V_1P^{1-t}, \cdots, P^t V_dP^{1-t}). We …