Mathematics Commons^™

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.

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 …

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 & Statistical Sciences Faculty Publications

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 & Statistical Sciences Faculty Publications

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 0Phas 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, Roychowdhury …

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

School of Mathematical & Statistical Sciences Faculty Publications

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 & Statistical Sciences Faculty Publications

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 …

Eigen’S Paradox And The Quasispecies Model In A Non-Archimedean Framework, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

In this article we present a new p -adic generalization of the Eigen–Schuster model where the genomes (sequences) are represented by words written in the alphabet { 0 , 1 , … , p − 1 } , where p is a prime number, with a time variable length. The time evolution of the concentration of a given sequence is controlled by a p -adic evolution equation. The long term behavior of the concentration of a sequence depends on a fitness function f , a mutation measure Q , and an initial concentration distribution. The new model provides essentially two …

Norm And Numerical Radius Of Single Operators Through Tools And Techniques From Multivariable Operator Theory, Raul E. Curto, Sang Hoon Lee, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

We employ tools and techniques from multivariable operator theory to obtain new proofs and extensions of well known inequalities regarding the norm and the numerical radius of elementary operators defined on the C⁎" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; position: relative;">C⁎C⁎–algebra of all bounded operators on Hilbert space, or on the ⁎–ideal of Hilbert-Schmidt operators. In the process, we provide new insights on the study of Heinz-type inequalities related to the arithmetic-geometric mean inequality, and generalize results of several authors, including R. Bhatia, G. Corach, C. Davis, F. Kittaneh, and M.S. Moslehian. To estimate the norm, our approach …

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

School of Mathematical & Statistical Sciences Faculty Publications

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 …

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

School of Mathematical & Statistical Sciences Faculty Publications

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.

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

School of Mathematical & Statistical Sciences Faculty Publications

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 & Statistical Sciences Faculty Publications

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 & Statistical Sciences Faculty Publications

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), …

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

School of Mathematical & Statistical Sciences Faculty Publications

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 Probability Distribution Generated By An Infinite Iterated Function System, Lakshmi Roychowdhury, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Quantization for probability distributions concerns the best approximation of a d-dimensional probability distribution P by a discrete probability with a given number n of supporting points. In this paper, we have considered a probability measure generated by an infinite iterated function system associated with a probability vector on ℝ. For such a probability measure P, an induction formula to determine the optimal sets of n-means and the nth quantization error for every natural number n is given. In addition, using the induction formula we give some results and observations about the optimal sets of n-means for all n ≥ 2.

The Cartesian Analytical Solutions For The N-Dimensional Compressible Navier-Stokes Equations With Density-Dependent Viscosity, Engui Fan, Zhijun Qiao, Manwai Yuen

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the N-dimensional compressible Navier-Stokes equations with density-dependent viscosity, based on the matrix and curve integration theory. Two exact solutions are obtained by solving the reduced systems.

Navigating Mathematics Teacher Preparation During A Time Of Crisis, Zareen G. Rahman, Rani Satyam, Younggon Bae

School of Mathematical & Statistical Sciences Faculty Publications

In this paper we highlight the experience of a mathematics teacher educator (MTE) and their preservice teachers (PTs) in a middle school mathematics methods course during the 2020 shift to online instruction due to the COVID-19 pandemic. We believe it is valuable to report how the MTE reflected on their instructional decision-making in response to this massive transition to remote instruction. We also report that PTs needed support and guidance to employ new teaching practices they had learned in the methods course instead of reverting to familiar teaching methods.

Ultrametric Diffusion, Rugged Energy Landscapes And Transition Networks, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

In this article we introduce the ultrametric networks which are p-adic continuous analogues of the standard Markov state models constructed using master equations. A p-adic transition network (or an ultrametric network) is a model of a complex system consisting of a hierarchical energy landscape, a Markov process on the energy landscape, and a master equation. We focus on networks where the transition rates between two different basins are constant functions, and the jumping process inside of each basin is controlled by a p-adic radial function. We solve explicitly the Cauchy problem for the master equation attached to this type of …

Monotonicity Properties Of Functionals Under Ricci Flow On Manifolds Without And With Boundary, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, the idea of the Ricci flow is introduced and its significance and importance to related problems in mathematics had been discussed. Several functionals are defined and their behavior is studied under Ricci flow. A unique minimizer is shown to exist for one of the functionals. This functional evaluated at the minimizer is strictly increasing. The results for the first functional considered are extended to manifold with boundary. Finally, two physically motivated examples are presented.