Mathematics Commons^™

Optimal Control With Manf Treatment Of Photoreceptor Degeneration, Erika T. Camacho, Suzanne Lenhart, Luis A. Melara, Cristina Villalobos, Stephen Wirkus

School of Mathematical & Statistical Sciences Faculty Publications

People afflicted with diseases such as retinitis pigmentosa and age-related macular degeneration experience a decline in vision due to photoreceptor degeneration, which is currently unstoppable and irreversible. Currently there is no cure for diseases linked to photoreceptor degeneration. Recent experimental work showed that mesencephalic astrocyte-derived neurotrophic factor (MANF) can reduce neuron death and, in particular, photoreceptor death by reducing the number of cells that undergo apoptosis. In this work, we build on an existing system of ordinary differential equations that represent photoreceptor interactions and incorporate MANF treatment for three experimental mouse models having undergone varying degrees of photoreceptor degeneration. Using …

Optimal Quantization Via Dynamics, Joseph Rosenblatt, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Quantization for probability distributions refers broadly to estimating a given probability measure by a discrete probability measure supported by a finite number of points. We consider general geometric approaches to quantization using stationary processes arising in dynamical systems, followed by a discussion of the special cases of stationary processes: random processes and Diophantine processes. We are interested in how close stationary process can be to giving optimal n-means and nth optimal mean distortion errors. We also consider different ways of measuring the degree of approximation by quantization, and their advantages and disadvantages in these different contexts.

The Role Of The Asymmetric Ekman Dissipation Term On The Energetics Of The Two-Layer Quasi-Geostrophic Model, Eleftherios Gkioulekas

School of Mathematical & Statistical Sciences Faculty Publications

In the two-layer quasi-geostrophic model, the friction between the flow at the lower layer and the surface boundary layer, placed beneath the lower layer, is modeled by the Ekman term, which is a linear dissipation term with respect to the horizontal velocity at the lower layer. The Ekman term appears in the governing equations asymmetrically; it is placed at the lower layer, but does not appear at the upper layer. A variation, proposed by Phillips and Salmon, uses extrapolation to place the Ekman term between the lower layer and the surface boundary layer, or at the surface boundary layer. We …

Classification Of Radial Solutions To Equations Related To Caffarelli–Kohn–Nirenberg Inequalities, John Villavert

School of Mathematical & Statistical Sciences Faculty Publications

This article studies the qualitative and quantitative properties of radial solutions to an elliptic equation related to the Euler–Lagrange equations for certain sharp Caffarelli–Kohn–Nirenberg inequalities. Namely, we examine the equation

−div(|x|aDu)=|x|bup, u>0, in RN, where p>1, N≥2, N−2+a≥0 and b>−N. The main results establish the properties of radially symmetric solutions including existence, uniqueness, and classification results as well as results on the asymptotic and intersecting behaviour of such solutions.

Delaunay Surfaces Expressed In Terms Of A Cartan Moving Frame, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Delaunay surfaces are investigated by using a moving frame approach. These surfaces correspond to surfaces of revolution in the Euclidean three-space. A set of basic one-forms is defined. Moving frame equations can be formulated and studied. Related differential equation which depend on variables relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case in which the mean curvature is constant, these functions can be expressed in terms of Jacobi elliptic functions.

Quasinormality Of Powers Of Commuting Pairs Of Bounded Operators, Raul E. Curto, Sang H. Lee, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

We study jointly quasinormal and spherically quasinormal pairs of commuting operators on Hilbert space, as well as their powers. We first prove that, up to a constant multiple, the only jointly quasinormal 2-variable weighted shift is the Helton-Howe shift. Second, we show that a left invertible subnormal operator T whose square T2 is quasinormal must be quasinormal. Third, we generalize a characterization of quasinormality for subnormal operators in terms of their normal extensions to the case of commuting subnormal n-tuples. Fourth, we show that if a 2-variable weighted shift W(α,β) and its powers W(α,β)(2,1) and W(α,β)(1,2) are all spherically quasinormal, …

Surviving Mathematics, Nathalie M. Luna

School of Mathematical & Statistical Sciences Faculty Publications

In this essay written in an informal voice, the author shares the ups and downs of her experience in academia. She shares her motivation to study mathematics, her undergraduate experience in Puerto Rico, and her graduate experience in South Texas.

Topological Pressure And Fractal Dimensions Of Cookie-Cutter-Like Sets, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

The cookie-cutter-like set is defined as the limit set of a sequence of classical cookie-cutter mappings. For this cookie-cutter set it is shown that the topological pressure function exists, and that the fractal dimensions such as the Hausdorff dimension, the packing dimension and the box-counting dimension are all equal to the unique zero h of the pressure function. Moreover, it is shown that the h-dimensional Hausdorff measure and the h-dimensional packing measure are finite and positive.

Effect Of Hydraulic Resistivity On A Weakly Nonlinear Thermal Flow In A Porous Layer, Dambaru Bhatta, Daniel N. Riahi

School of Mathematical & Statistical Sciences Faculty Publications

Heat and mass transfer through porous media has been a topic of research interest because of its importance in various applications. The flow system in porous media is modelled by a set of partial differential equations. The momentum equation which is derived from Darcy’s law contains a resistivity parameter. We investigate the effect of hydraulic resistivity on a weakly nonlinear thermal flow in a horizontal porous layer. The present study is a realistic study of nonlinear convection flow with variable resistivity whose rate of variation is arbitrary in general. This is a first step for considering more general problems in …

On Cohen-Macaulay Hopf Monoids In Species, Jacob A. White

School of Mathematical & Statistical Sciences Faculty Publications

We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant arising from a linearized Hopf monoid, we show that under certain conditions it is the Hilbert polynomial of a relative simplicial complex. If the Hopf monoid is Cohen- Macaulay, we give necessary and sufficient conditions for the corresponding relative simplicial complex to be relatively Cohen-Macaulay, which implies that the polynomial has a nonnegative h-vector. We apply our results to the weak and strong chromatic polynomials of …

Screening Potential Citrus Rootstocks For Phytophthora Nicotianae Tolerance, Madhurababu Kunta, Sandy Chavez, Zenaida Viloria, Hilda S. Del Rio, Madhavi Devanaboina, George Yanev, Jong-Won Park, Eliezer S. Louzada

School of Mathematical & Statistical Sciences Faculty Publications

Seeds from four citrus rootstocks including sour orange, Bitters-C22 citrandarin, Sarawak pummelo 3 Rio Red grapefruit, and Sarawak pummelo 3Bower mandarin were exposed to high inoculum levels of Phytophthora nicotianae to screen for tolerance. Inoculation of pregerminated seeds (PGIS) and non-PGIS was carried out. The average P. nicotianae propagule counts from the soil samples where these seedlings were raised ranged from 424 to 1361 colony forming units/cm3. The proportion of live to dead plants was recorded at 11months postinoculation, which showed that Sarawak3Bower performed significantly better than other rootstocks. Evaluation of the rootstocks 18 months postinoculation resulted in only one …

Stability Of Anisotropic Parabolic Equations Without Boundary Conditions, Huashui Zhan, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

No abstract provided.

Forced (2 + 1)-Dimensional Discrete Three-Wave Equation, Junyi Zhu, Sishou Zhou, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

We generalize the ∂-dressing method to investigate a (2 + 1)-dimensional lattice, which can be regarded as a forced (2 + 1)-dimensional discrete three-wave equation. The soliton solutions to the (2 + 1)-dimensional lattice are given through constructing different symmetry conditions. The asymptotic analysis of one-soliton solution is discussed. For the soliton solution, the forces are zero.

Flows Of Co-Closed G_{2}-Structures, Sergey Grigorian

School of Mathematical & Statistical Sciences Faculty Publications

We survey recent progress in the study of G_{2}-structure Laplacian coflows, that is, heat flows of co-closed G_{2}-structures. We introduce the properties of the original Laplacian coflow of G_{2}-structures as well as the modified coflow, reviewing short-time existence and uniqueness results for the modified coflow and well as recent Shi-type estimates that apply to a more general class of G_{2}-structure flows.

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, …

Assessing Program Outcomes Of An M.Ed. Curriculum And Instruction Program: A Comparison Of Face-To-Face To Completely Online Deliverables, James A. Telese, Gregory Chamblee

Teaching and Learning Faculty Publications

Many mathematics education degree programs, especially at the graduate level, are now transitioning to an online format. There is a need to document how mathematics content and content pedagogy are assessed in an online environment. The objectives of this chapter are to document how a public higher education institution in Texas transitioned their master's degree program for mathematics teachers from a face-to-face program to an online program and how this transition impacted the assessment process related to the learning of content and pedagogical content knowledge.

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 …

Professional Development For Mathematics Teacher Education Faculty: Need And Design, Billy Jackson, Shandy Hauk, Jenq Jong Tsay, Alma Ramirez

School of Mathematical & Statistical Sciences Faculty Publications

The purpose of this report is to share a conceptual model useful in the design of professional learning about teaching for university mathematics faculty. The model is illustrated by examples from a particular design effort: the development of an online shortcourse for faculty new to teaching mathematics courses for prospective primary school teachers. How novice mathematics teacher educators grow as instructors is an emerging area of research and development in the United States. At the same time, it is well established that effective instructional design of any course, including a course for faculty, requires breadth first: understanding and anticipating the …

Constructing Associative Rings From Certain Hypergroups, Oscar Gonzalez

Theses and Dissertations

Tight hypergroups give rise to associative rings if the so-called general normality condition holds\cite{ARTICLE:1}. We consider four examples of tight hypergroups which do not satisfy the general normality condition and show that they still give rise to associative rings. Our examples are HM332(24),HM10353(32), HM10933(32) and HM10941(32)[1].

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. …