Mathematics Commons^™

Bayesian Modelling Of Nonlinear Poisson Regression With Artificial Neural Networks, Hansapani Rodrigo, Chris P. Tsokos

School of Mathematical & Statistical Sciences Faculty Publications

Modelling and prediction of count and rate responses have substantial usage in many fields, including health, finance, social, etc. Conventionally, linear Poisson regression models have been widely used to model these responses. However, the linearity assumption of the systematic component of linear Poisson regression models restricts their capability of handling complex data patterns. In this regard, it is important to develop nonlinear Poisson regression models to capture the inherent variability within the count data. In this study, we introduce a probabilistically driven nonlinear Poisson regression model with Bayesian artificial neural networks (ANN) to model count and rate data. This new …

Cartan’S Approach To Second Order Ordinary Differential Equations, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

In his work on projective connections, Cartan discusses his theory of second order differential equations. It is the aim here to look at how a normal projective connection can be constructed and how it relates to the geometry of a single second order differential equation. The calculations are presented in some detail in order to highlight the use of gauge conditions

Hiv-Associated Neurocognitive Disorder (Hand) Biomarker Identification: Significance Analysis Of Microarrays And Two Persuasive Approaches With Random Forest, Hansapani Rodrigo, Bryan Martinez, Roberto De La Garza, Upal Roy

School of Mathematical & Statistical Sciences Faculty Publications

Background: HIV Associated Neurological Disorders (HAND) is relatively common among people with HIV-1 infection, even those taking combined antiretroviral treatment (cART). Genome-wide screening of transcription regulation in brain tissue helps in identifying substantial abnormalities present in patients’ gene transcripts and to discover possible biomarkers for HAND. This study explores the possibility of identifying differentially expressed (DE) genes, which can serve as potential biomarkers to detect HAND. In this study, we have investigated the gene expression levels of three subject groups with different impairment levels of HAND along with a control group in three distinct brain sectors: white matter, frontal cortex, …

Decomposition Of 2-Soliton Solutions For The Good Boussinesq Equations, Vesselin Vatchev

School of Mathematical & Statistical Sciences Faculty Publications

We consider decompositions of two-soliton solutions for the good Boussinesq equation obtained by the Hirota method and the Wronskian technique. The explicit forms of the components are used to study the dynamics of 2-soliton solutions. An interpretation in the context of eigenvalue problems arising from KdV type equations and transport equations is considered. Numerical examples are included.

Finite Lifespan Of Solutions Of The Semilinear Wave Equation In The Einstein–De Sitter Spacetime, Anahit Galstian, Karen Yagdjian

School of Mathematical & Statistical Sciences Faculty Publications

We examine the solutions of the semilinear wave equation, and, in particular, of the φq model of quantum field theory in the curved space-time. More exactly, for 1 < q < 4 we prove that the solution of the massless self-interacting scalar field equation in the Einstein-de Sitter universe has finite lifespan.

The Quantization Of The Standard Triadic Cantor Distribution, Mrinal Kanti Roychowdhury

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. For a given k ≥ 2, let {Sj : 1 ≤ j ≤ k} be a set of k contractive similarity mappings such that Sj(x) = 1 2k−1x + 2(j−1) 2k−1 for all x ∈ R, and let P = 1 k Pk j=1 P ◦ S−1 j . Then, P is a unique Borel probability measure on R such that P has support the Cantor set generated by the similarity mappings …

On The Voronoi Conjecture For Combinatorially Voronoi Parallelohedra In Dimension 5, Mathieu Dutour Sikiric, Alexey Garber, Alexander Magazinov

School of Mathematical & Statistical Sciences Faculty Publications

In a recent paper, Garber, Gavrilyuk, and Magazinov [Discrete Comput. Geom., 53 (2015), pp. 245--260] proposed a sufficient combinatorial condition for a parallelohedron to be affinely Voronoi. We show that this condition holds for all 5-dimensional Voronoi parallelohedra. Consequently, the Voronoi conjecture in $\mathbb{R}^5$ holds if and only if every 5-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron $P$ is combinatorially Voronoi, we mean that $P$ is combinatorially equivalent to a Dirichlet--Voronoi polytope for some lattice $\Lambda$, and this combinatorial equivalence is naturally translated into equivalence of the tiling by copies of $P$ with …

Investigation Of Constant Volume And Constant Flux Initial Conditions On Bidensity Particle-Laden Slurries On An Incline, Dominic Diaz, Jessica Bojorque, Joshua Crasto, Margaret Koulikov, Tameez Lati, Aviva Prins, Andrew Shapiro, Clover Ye, David Arnold, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Particle-laden slurries are pervasive in both natural and industrial settings, whenever particles are suspended or transported in a fluid. Previous literature has investigated the case of a single species of negatively buoyant particles suspended in a viscous fluid. On an incline, three distinct regimes emerge depending on the particle concentration and inclination angle: settled (where particles settle and there is a pure fluid front), well-mixed (where particle concentration is constant throughout), and ridged (where a particle-rich ridge leads the flow). Recently, the same three regimes were also found for constant volume two species bidensity slurries. We extend the literature on …

High-Order Rogue Waves Of A Long-Wave–Short-Wave Model Of Newell Type, Jungchao Chen, Liangyuan Chen, Bao-Feng Feng, Ken-Ichi Maruno

School of Mathematical & Statistical Sciences Faculty Publications

The long-wave–short-wave (LWSW) model of Newell type is an integrable model describing the interaction between the gravity wave (long wave) and the capillary wave (short wave) for the surface wave of deep water under certain resonance conditions. In the present paper, we are concerned with rogue-wave solutions to the LWSW model of Newell type. By combining the Hirota’s bilinear method and the KP hierarchy reduction, we construct its general rational solution expressed by the determinant. It is found that the fundamental rogue wave for the short wave can be classified into three different patterns: bright, intermediate, and dark states, whereas …

Experimental Demonstration Of Deterministic Chaos In A Waste Oil Biodiesel Semi-Industrial Furnace Combustion System, Shengyang Gao, Fashe Li, Qintai Xiao, Jianxin Xu, Huage Wang, Hua Wang

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, the nonlinear dynamic characteristics of the oxygen-enriched combustion of waste oil biodiesel in semi-industrial furnaces were tested by the power spectrum, phase space reconstruction, the largest Lyapunov exponents, and the 0-1 test method. To express the influences of the system parameters, experiments were carried out under different oxygen content conditions (21%, 25%, 28%, 31%, and 33%). Higher oxygen enrichment degrees contribute to finer combustion sufficiency, which produces flames with high luminance. Flame luminance and temperature can be represented by different gray scale values of flame images. The chaotic characteristics of gray scale time series under different oxygen …

The Aluthge Transform Of Unilateral Weighted Shifts And The Square Root Problem For Finitely Atomic Measures, Raul E. Curto, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

In this paper we consider the following Square Root Problem for measures: Given a positive probability Borel measure μ (supported on an interval [a,b]⊆R+), does there exist a positive Borel measure ν such that μ=ν*ν holds? (Here * denotes the multiplicative convolution, properly defined on R+.) This problem is intimately connected to the subnormality of the Aluthge transform of a unilateral weighted shift. We develop a criterion to test whether a measure μ admits a square root, and we provide a concrete solution for the case of a finitely atomic measure having at most five atoms. In addition, we sharpen …

Diabetes And Its Effect On Abdominal Aortic Aneurysm Growth Rate In Hispanic Patients, Monica Betancourt-Garcia, Kristina Vatcheva, Amrit Thakur, Prateek Gupta, Eugene Postevka, Ricardo Martinez, R. Armour Forse

School of Mathematical & Statistical Sciences Faculty Publications

Background

The growth rate of abdominal aortic aneurysms (AAA) can vary depending on age, baseline diameter, blood pressure, race, and history of smoking. Paradoxically, previous studies show evidence of a protective effect of diabetes on the rate of AAA expansion despite its well-established role in the morbidity and mortality of cardiovascular disease. This study aims to investigate the impact diabetes plays on AAA growth within a Hispanic population.

Methods

Data were collected from patients who were predominantly Mexican-American at a single hospital site. Baseline and follow-up measures for AAA diameter were obtained from serial imaging studies. Demographics, medical history, the …

Multi‐Breather And High‐Order Rogue Waves For The Nonlinear Schrödinger Equation On The Elliptic Function Background, Bao-Feng Feng, Liming Ling, Daisuke A. Takahashi

School of Mathematical & Statistical Sciences Faculty Publications

We construct the multi‐breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and nontrivial phase‐modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multibreather solutions with different velocities in the limit , where the solution in the neighborhood of each breather tends to the simple one‐breather solution. …

Joint Spectra Of Spherical Aluthge Transforms Of Commuting N-Tuples Of Hilbert Space Operators, Chafiq Benhida, Raul E. Curto, Sang Hoon Lee, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

Let be a commuting n-tuple of operators on a Hilbert space , and let be its canonical joint polar decomposition (i.e. ⁎ ⁎ , a joint partial isometry, and ). The spherical Aluthge transform of T is the (necessarily commuting) n-tuple . We prove that , where denotes the Taylor spectrum. We do this in two stages: away from the origin, we use tools and techniques from criss-cross commutativity; at the origin, we show that the left invertibility of T or implies the invertibility of P. As a consequence, we can readily extend our main result to other spectral systems …

Fast Equilibration Dynamics Of Viscous Particle-Laden Flow In An Inclined Channel, Jeffrey Wong, Michael R. Lindstrom, Andrea L. Bertozzi

School of Mathematical & Statistical Sciences Faculty Publications

A viscous suspension of negatively buoyant particles released into a wide, open channel on an incline will stratify in the normal direction as it flows. We model the early dynamics of this stratification under the effects of sedimentation and shear-induced migration. Prior work focuses on the behaviour after equilibration where the bulk suspension either separates into two distinct fronts (settled) or forms a single, particle-laden front (ridged), depending on whether the initial concentration of particles exceeds a critical threshold. From past experiments, it is also clear that this equilibration time scale grows considerably near the critical concentration. This paper models …

Shape Equations For Two-Dimensional Manifolds Through A Moving Frame Variational Approach, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

A variational approach is given which can be applied to functionals of a general form to determine a corresponding Euler–Lagrange or shape equation. It is the intention to formulate the theory in detail based on a moving frame approach. It is then applied to a functional of a general form which depends on both the mean and Gaussian curvatures as well as the area and volume elements of the manifold. Only the case of a two-dimensional closed manifold is considered. The first variation of the functional is calculated in terms of the variations of the basic variables of the manifold. …

Global Existence And Propagation Speed For A Generalized Camassa-Holm Model With Both Dissipation And Dispersion, Qiaoyi Hu, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we study a generalized Camassa–Holm (gCH) model with both dissipation and dispersion, which has (N+1)-order nonlinearities and includes the following three integrable equations: the Camassa–Holm, the Degasperis–Procesi, and the Novikov equations, as its reductions. We first present the local well-posedness and a precise blow-up scenario of the Cauchy problem for the gCH equation. Then, we provide several sufficient conditions that guarantee the global existence of the strong solutions to the gCH equation. Finally, we investigate the propagation speed for the gCH equation when the initial data are compactly supported.

Properties Of A Class Of Mean Dependent On A Parameter, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Some properties of a new class of binary symmetric mean Mp(a,b) which depends on two positive numbers a and b, as well as a positive parameter p, are investigated. The logarithmic mean and arithmetic mean are two members of this class. It is shown that, for all values of the parameter p, the set of p-dependent means Mp(a,b) is bounded above by the mean M∞(a,b).

Metric Geometry And The Determination Of The Bohmian Quantum Potential, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

A geometric approach to quantum mechanics which is formulated in terms of Finsler geometry is developed. It is shown that quantum mechanics can be formulated in terms of Finsler configuration space trajectories which obey Newton-like evolution but in the presence of an additional kind of potential. This additional quantum potential which was obtained first by Bohm has the consequence of contributing to the forces driving the system. This geometric picture accounts for many aspects of quantum dynamics and leads to a more natural interpretation. It is found for example that dynamics can be accounted for by incorporating quantum effects into …

Optimal Quantization For The Cantor Distribution Generated By Infinite Similutudes, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Let P be a Borel probability measure on ℝ generated by an infinite system of similarity mappings {Sj : j ∈ ℕ} such that P=Σ∞j=112jP∘S−1j, where for each j ∈ ℕ and x ∈ ℝ, Sj(x)=13jx+1−13j−1. Then, the support of P is the dyadic Cantor set C generated by the similarity mappings f1, f2 : ℝ → ℝ such that f1(x) = 1/3x and f2(x) = 1/3x+ 2/3 for all x ∈ ℝ. In this paper, using the infinite system of similarity mappings {Sj : j ∈ ℕ} associated with the probability vector (12,122,...), for all n ∈ ℕ, we …

Spherically Quasinormal Pairs Of Commuting Operators, Raul E. Curto, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

We first discuss the spherical Aluthge and spherical Duggal transforms for commuting pairs of operators on Hilbert space. Second, we study the fixed points of these transforms, which are the spherically quasinormal commuting pairs. In the case of commuting 2-variable weighted shifts, we prove that spherically quasinormal pairs are intimately related to spherically isometric pairs. We show that each spherically quasinormal 2-variable weighted shift is completely determined by a subnormal unilateral weighted shift (either the 0-th row or the 0-th column in the weight diagram). We then focus our attention on the case when this unilateral weighted shift is recursively …

“Diabesity” Among Latinos: A Culturally Relevant Mathematics Activity By A Mathematics Teacher Candidate, Olga Ramirez, Cherie Mccollough

School of Mathematical & Statistical Sciences Faculty Publications

This paper summarizes a culturally relevant mathematics project conducted by a mentored secondary-level Latino teacher candidate (TC) as a requirement for an undergraduate mathematics project course at a south Texas university. The culturally relevant math project involved four secondary Latino students and their parents. The topic was “diabesity”; that is, diabetes and obesity among Latinos southern region in the United States of America with an emphasis on body mass index, proper diet, and exercise. Remarks by the faculty mentor and a faculty consultant about the TC work and reflections regarding this project are also included and provide perspectives on how …

The Generalized Weierstrass System In Three-Dimensional Euclidean Space, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

In this chapter, some recent advances in the area of generalized Weierstrass representations will be given. This is an approach to the theory of surfaces in Euclidean three space. Weierstrass representations permit the explicit construction of surfaces in the designated space. The discussion proceeds in a novel and introductory manner. The inducing formulas for the coordinates of a surface are derived and important conservation laws are formulated. These lead to the inducing mechanism of a surface in terms of solutions to a system of two-dimensional Dirac equations. A set of fundamental forms as well as expressions for the mean and …

Attitudes Toward Using And Teaching Confidence Intervals: A Latent Profile Analysis On Elementary Statistics Instructors, Hyung Won Kim, Woo Jin Kim, Aaron T. Wilson, Ho Kyoung Ko

School of Mathematical & Statistical Sciences Faculty Publications

The use of confidence intervals (CIs) for making a statistical inference is gaining popularity in research communities. To evaluate college statistics instructors’ readiness to teach CIs, this study explores their attitudes toward teaching CIs in elementary statistics courses, and toward using CIs in inferential statistics. Data were collected with a survey that classifies instructors’ attitudes on the basis of three previously established pedagogical components: affective, cognitive, and behavioral. Based on the survey responses from 270 participants, we created three profiles (subgroups) via latent profile analysis, and identified each profile’s pattern of attitudes toward CIs and common characteristics of the instructors …

On The Regularity Radius Of Delone Sets In R3, Nikolay Dolbilin, Alexey Garber, Undine Leopold, Egon Schulte

School of Mathematical & Statistical Sciences Faculty Publications

We complete the proof of the upper bound ρ^3≤10R for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, summing up the results obtained earlier, and adding the missing cases, we show that if all 10R-clusters of a Delone set X with parameters (r,R) are equivalent, then X is a regular system.

Blocks In The Category Of Finite-Dimensional Representations Of Principal W-Algebra For Q(2), Elena Poletaeva

School of Mathematical & Statistical Sciences Faculty Publications

We describe the blocks in the category of finite-dimensional representations of the principal finite W-algebra for the Lie superalgebra Q(2).

The Near-Peer Mathematical Mentoring Cycle: Studying The Impact Of Outreach On High School Students’ Attitudes Toward Mathematics, Aaron T. Wilson, Sergey Grigorian

School of Mathematical & Statistical Sciences Faculty Publications

College students may be seen as near-peers to high school students and high school students are often able to see themselves in the college students who are but one step ahead. This nearness in maturity and educational level may place college students in a particularly powerful position when it comes to reaching out to high school students to promote higher education in math and science. In this study college students gave dynamic mathematics outreach presentations, MathShows, to minority and low-income high school students in a mid-sized public school district on the U.S. border with Mexico. The study investigated the impacts …

On A Class Of Parameter Dependent Series Generalizing Euler’S Constant, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Series which depend on a parameter and generalize the constant discovered by Euler are introduced and studied. Convergence results are established. An infinite series expansion is obtained from these generalized formulas which can be used to evaluate the generalized constant. Euler’s constant can be obtained as a special case. Some asymptotic results are formulated and limits of some closely related sequences are given.

Connections Between Pre-Service Teachers’ Mathematical Dispositions And Self-Efficacy For Teaching Mathematics, Jasmine M. Cruz, Aaron T. Wilson, Xiaohui Wang

School of Mathematical & Statistical Sciences Faculty Publications

Mathematical dispositions (MD) and self-efficacy for teaching mathematics (SEFTM) are important elements of teachers‟ beliefs that significantly influence their behaviors and educational practices in the classroom. This study looked at relationships between pre-service teachers‟ (PSTs) MD and their SEFTM in connection with other descriptors related to their progress through the teacher education program and prior mathematical experiences. Survey data were collected from 238 PSTs at a midsized university in the US. Results include the finding that PSTs‟ intended grade-level and subject approximately mirror their MD and SEFTM: those planning to be elementary school (generalist) teachers were slightly behind both middle …

On The Origin Of Crystallinity: A Lower Bound For The Regularity Radius Of Delone Sets, Igor A. Baburin, Mikhail M. Bouniaev, Nikolay Dolbilin, Nikolay Yu. Erokhovets, Alexey Garber, Sergey V. Krivovichev, Egon Schulte

School of Mathematical & Statistical Sciences Faculty Publications

The mathematical conditions for the origin of long-range order or crystallinity in ideal crystals are one of the very fundamental problems of modern crystallography. It is widely believed that the (global) regularity of crystals is a consequence of `local order', in particular the repetition of local fragments, but the exact mathematical theory of this phenomenon is poorly known. In particular, most mathematical models for quasicrystals, for example Penrose tiling, have repetitive local fragments, but are not (globally) regular. The universal abstract models of any atomic arrangements are Delone sets, which are uniformly distributed discrete point sets in Euclidean d space. …