Mathematics Commons^™

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 …

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 and Statistical Sciences Faculty Publications and Presentations

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.

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 …

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Presentations

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 …

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 …

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 …

Comparative Analysis Of All-Terrain Vehicles, Motorcycle And Automobile-Related Trauma In A Rural Border Community Of The Usa, Haissam S. Elzaim, Kristina Vatcheva, Annelyn Torres-Reveron, Gregery Pequeno, Monica Michelle Betancourt-Garcia

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Introduction: There is widespread use of all-terrain vehicles (ATVs) in the USA for both work-related and recreational activities. In this study, we aimed to determine the difference in injury severity, Glasgow Coma scales and length of stay between ATV-related injuries and injuries sustained from motorcycles (MOTOs) and automobiles (AUTOs).

Methods: We retrospectively analysed ATV, MOTO and AUTO injuries from a Level 2 Trauma Center between 01 January 2015 and 31 August 2020. Proportional odds regression analyses, as well as multivariable regression models, were used to analyse the data.

Results: There were significantly more male and paediatric patients that suffered ATV-related …

Optimal Quantization For Some Triadic Uniform Cantor Distributions With Exact Bounds, Mrinal Kanti Roychowdhury

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Let {Sj:1≤j≤3} be a set of three contractive similarity mappings such that Sj(x)=rx+j−12(1−r) for all x∈R, and 1≤j≤3, where 0

Let {Sj:1≤j≤3}">{Sj:1≤j≤3}{Sj:1≤j≤3} be a set of three contractive similarity mappings such that Sj(x)=rx+j−12(1−r)">Sj(x)=rx+j−12(1−r)Sj(x)=rx+j−12(1−r) for all x∈R">x∈Rx∈R, and 1≤j≤3">1≤j≤31≤j≤3, where 0P has support the Cantor set generated by the similarity mappings Sj">SjSj for 1≤j≤3">1≤j≤31≤j≤3. Let r0=0.1622776602">r0=0.1622776602r0=0.1622776602, and r1=0.2317626315">r1=0.2317626315r1=0.2317626315 (which are ten digit rational approximations of two real numbers). In this paper, for 00n-means and the nth quantization errors for the triadic uniform Cantor distribution P for all positive integers n≥2">n≥2n≥2. Previously, …

On The Size Of Maximal Binary Codes With 2, 3, And 4 Distances, Alexander Barg, Alexey Glazyrin, Wei-Jiun Kao, Ching-Yi Lai, Pin-Chieh Tseng, Wei-Hsuan Yu

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We address the maximum size of binary codes and binary constant weight codes with few distances. Previous works established a number of bounds for these quantities as well as the exact values for a range of small code lengths. As our main results, we determine the exact size of maximal binary codes with two distances for all lengths n≥6 as well as the exact size of maximal binary constant weight codes with 2,3, and 4 distances for several values of the weight and for all but small lengths.

Covering By Planks And Avoiding Zeros Of Polynomials, Alexey Glazyrin, Roman Karasev, Alexandr Polyanskii

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball, we show that for every real nonzero d-variate polynomial P of degree n⁠, there exists a point in the …

On Equivariant Flag F-Vectors For Balanced Relative Simplicial Complexes, Jacob A. White

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We study the equivariant flag f-vector and equivariant flag h-vector of a balanced relative simplicial complex with respect to a group action. When the complex satisfies Serre's condition (Sℓ), we show that the equivariant flag h-vector, the equivariant h-vector, and the equivariant f-vector satisfy several inequalities.
We apply these results to the study of P-partitions of double posets, and weak colorings of mixed graphs.

A Mesh-Free Method Using Piecewise Deep Neural Network For Elliptic Interface Problems, Cuiyu He, Xiaozhe Hu, Lin Mu

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the interface, we employ different neural networks for each sub-domain. By reformulating the interface problem as a least-squares problem, we discretize the objective function using mean squared error via sampling and solve the proposed deep least-squares method by standard training algorithms such as stochastic gradient descent. The discretized objective function utilizes only the point-wise information on the sampling points and thus no underlying mesh …

Definite Condition Of The Evolutionary (P)Over-Right-Arrow(X)-Laplacian Equation, Huashui Zhan, Zhaosheng Feng

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

For the nonlinear degenerate parabolic equations, how to find an appropriate boundary value condition to ensure the well-posedness of weak solution has been an interesting and challenging problem. In this paper, we develop the general characteristic function method to study the stability of weak solutions based on a partial boundary value condition.

Darboux Transformation And Solitonic Solution To The Coupled Complex Short Pulse Equation, Bao-Feng Feng, Liming Ling

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright soliton, dark-soliton, breather and rogue wave solutions to the CCSP equation. In case of vanishing boundary condition (VBC), we perform the inverse scattering analysis to understand the soliton solution better. Breather and rogue wave solutions are constructed in case of non-vanishing boundary condition (NVBC). Moreover, we conduct a modulational instability (MI) analysis based on the method of squared eigenfunctions, whose result confirms the condition for the existence of rogue …

Statistical Analysis Methods Applied To Early Outpatient Covid-19 Treatment Case Series Data, Eleftherios Gkioulekas, Peter A. Mccullough, Vladimir Zelenko

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

When confronted with a public health emergency, significant innovative treatment protocols can sometimes be discovered by medical doctors at the front lines based on repurposed medications. We propose a statistical framework for analyzing the case series of patients treated with such new protocols, that enables a comparison with our prior knowledge of expected outcomes, in the absence of treatment. The goal of the proposed methodology is not to provide a precise measurement of treatment efficacy, but to establish the existence of treatment efficacy, in order to facilitate the binary decision of whether the treatment protocol should be adopted on an …

Towards Fast Weak Adversarial Training To Solve High Dimensional Parabolic Partial Differential Equations Using Xnode-Wan, Paul Valsecchi Oliva, Yue Wu, Cuiyu He, Hao Ni

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Due to the curse of dimensionality, solving high dimensional parabolic partial differential equations (PDEs) has been a challenging problem for decades. Recently, a weak adversarial network (WAN) proposed in Zang et al. (2020)[17] offered a flexible and computationally efficient approach to tackle this problem defined on arbitrary domains by leveraging the weak solution. WAN reformulates the PDE problem as a generative adversarial network, where the weak solution (primal network) and the test function (adversarial network) are parameterized by the multi-layer deep neural networks (DNNs). However, it is not yet clear whether DNNs are the most effective model for the parabolic …

A Comparison Of Statistical Methods For Modeling Count Data With An Application To Hospital Length Of Stay, Gustavo Fernandez, Kristina Vatcheva

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Background

Hospital length of stay (LOS) is a key indicator of hospital care management efficiency, cost of care, and hospital planning. Hospital LOS is often used as a measure of a post-medical procedure outcome, as a guide to the benefit of a treatment of interest, or as an important risk factor for adverse events. Therefore, understanding hospital LOS variability is always an important healthcare focus. Hospital LOS data can be treated as count data, with discrete and non-negative values, typically right skewed, and often exhibiting excessive zeros. In this study, we compared the performance of the Poisson, negative binomial (NB), …

Substitution Tilings With Transcendental Inflation Factor, Dirk Frettlöh, Alexey Garber, Neil Mañibo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

For any λ>2, we construct a substitution on an infinite alphabet which gives rise to a substitution tiling with inflation factor λ. In particular, we obtain the first class of examples of substitutive systems with transcendental inflation factors. We also show that both the associated subshift and tiling dynamical systems are strictly ergodic, which is related to the quasicompactness of the underlying substitution operator.

Diederich-Fornæss Index And Global Regularity In The ∂-Neumann Problem: Domains With Comparable Levi Eigenvalues, Bingyuan Liu, Emil J. Straube

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Let Ω be a smooth bounded pseudoconvex domain in Cn. Let 1≤q0≤(n−1). We show that if q0--sums of eigenvalues of the Levi form are comparable, then if the Diederich--Fornæss index of Ω is 1, the ∂¯¯¯--Neumann operators Nq and the Bergman projections Pq−1 are regular in Sobolev norms for q0≤q≤n. In particular, for domains in C2, Diederich--Fornæss index 1 implies global regularity in the ∂¯¯¯--Neumann problem.

A Gpu Accelerated Genetic Algorithm For The Construction Of Hadamard Matrices, Andras Balogh, Raven Ruiz

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We use a genetic algorithm to construct Hadamard Matrices. The initial population of random matrices is generated to have a balanced number of +1 and −1 entries in each column except the first column with all +1. Several fitness functions are implemented in order to find the most effective one. The crossover process creates offspring matrix population by exchanging columns of the parent matrix population. The mutation process flips +1 and −1 entry pairs in random columns. The use of CuPylibrary in Python on graphics processing units enables us to handle populations of thousands of matrices and matrix operations in …

Quantization For A Nonuniform Triadic Cantor Distribution, Asha Barua

Theses and Dissertations

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. Let P be a Borel probability measure on R such that P := 1/4 P◦ S1−1 + 1\2 P◦ S2−1 + 1/4 P◦ S3−1, where S1, S2 and S3 are three contractive similarity mappings such that Sj(x) = 1/5x+2(j−1)/5, for all x ∈ R. For this probability measure, in this thesis, we determine the optimal sets of n-means and the nth quantization errors for …