Mathematics Commons^™

Mathematical Modeling Of Nonlinear Blood Glucose-Insulin Dynamics With Beta Cells Effect, Gabriela Urbina, Daniel N. Riahi, Dambaru Bhatta

School of Mathematical & Statistical Sciences Faculty Publications

We consider mathematical modeling of blood glucose-insulin regulatory system with the additional effect of the secreted insulin by the pancreatic beta cells and in the presence of an external energy input to such system. Such modeling system is investigated to determine the time-dependent nonlinear dynamics that take place by the quantities, which represent the glucose and insulin concentrations in the blood, insulin action as well as in the absence or presence of secreted insulin due to the pancreatic beta cells. Using both analytical and numerical procedures, we determine such quantities versus time for both diabetes patients and normal human and …

An Effective Method To Obtain Contour Of Fisheye Images Based On Explicit Level Set Method, Xuegang Wu, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

Obtaining the effective contour from an image taken by fisheye lens is important for the following transactions. Many studies try to develop suitable methods to get accurate contours of fisheye images. Using the traditional level set method (CV model) is hard to meet the desire task that the final segmentation region is a circle. Therefore, the preprocessing of fisheye images and the improvement of traditional level set method are redesigned to get a final circular segmentation which may be suitable to other applications. In this paper, we use the local entropy method to make the value of pixels be even …

A Study On Quantization Dimension In Complete Metric Spaces, Mrinal Kanti Roychowdhury, S. Verma

School of Mathematical & Statistical Sciences Faculty Publications

The primary objective of the present paper is to develop the theory of quantization dimension of an invariant measure associated with an iterated function system consisting of finite number of contractive infinitesimal similitudes in a complete metric space. This generalizes the known results on quantization dimension of self-similar measures in the Euclidean space to a complete metric space. In the last part, continuity of quantization dimension is discussed.

Association Of Visceral Adipose Tissue And Subclinical Atherosclerosis In Us-Born Mexican Americans But Not First Generation Immigrants, Clarence Gill, Miryoung Lee, Kristina Vatcheva, Nahid Rianon, Beverly Smulevitz, David D. Mcpherson, Joseph B. Mccormick, Susan P. Fisher-Hoch, Susan T. Laing

School of Mathematical & Statistical Sciences Faculty Publications

BACKGROUND: Excess visceral adipose tissue (VAT) is a primary driver for the cardiometabolic complications of obesity; VATassociated cardiovascular disease risk varies by race, but most studies have been done on Non-Hispanics. This study aimed to evaluate the clinical and metabolic correlates of VAT, its association with subclinical atherosclerosis, and the factors affecting this association in Mexican Americans.

METHODS AND RESULTS: Participants (n=527) were drawn from the Cameron County Hispanic Cohort (CCHC), on whom a carotid ultrasound to assess carotid intima media thickness and a dual-energy X-ray absorptiometry scan to assess for VAT were obtained. Those in the highest quartiles of …

Periodic Triangulations Of Zn, Mathieu Dutour Sikiric, Alexey Garber

School of Mathematical & Statistical Sciences Faculty Publications

We consider in this work triangulations of Z n that are periodic along Z n . They generalize the triangulations obtained from Delaunay tessellations of lattices. In certain cases we impose additional restrictions on such triangulations such as regularity or invariance under central symmetry with respect to the origin; both properties hold for Delaunay tessellations of lattices. Full enumeration of such periodic triangulations is obtained for dimension at most 4 . In dimension 5 several new phenomena happen: there are centrally-symmetric triangulations that are not Delaunay, there are non-regular triangulations (it could happen in dimension 4 ) and a given …

Geometric And Measure-Theoretic Shrinking Targets In Dynamical Systems, Joseph Rosenblatt, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described.

Entropy In Quantum Mechanics And Applications To Nonequilibrium Thermodynamics, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Classical formulations of the entropy concept and its interpretation are introduced. This is to motivate the definition of the quantum von Neumann entropy. Some general properties of quantum entropy are developed, such as the quantum entropy which always increases. The current state of the area that includes thermodynamics and quantum mechanics is reviewed. This interaction shall be critical for the development of nonequilibrium thermodynamics. The Jarzynski inequality is developed in two separate but related ways. The nature of irreversibility and its role in physics are considered as well. Finally, a specific quantum spin model is defined and is studied in …

The Global Existence Of Small Self-Interacting Scalar Field Propagating In The Contracting Universe, Anahit Galstian, Karen Yagdjian

School of Mathematical & Statistical Sciences Faculty Publications

We present a condition on the self-interaction term that guaranties the existence of the global in time solution of the Cauchy problem for the semilinear Klein-Gordon equation in the Friedmann-Lamaˆitre-Robertson-Walker model of the contracting universe. For the Klein- Gordon equation with the Higgs potential we give a lower estimate for the lifespan of solution.

Analysis Of The Healthcare Mers-Cov Outbreak In King Abdulaziz Medical Center, Riyadh, Saudi Arabia, June–August 2015 Using A Seir Ward Transmission Model, Tamer Oraby, Michael G. Tyshenko, Hanan H. Balkhy, Yasar Tasnif, Adriana Quiroz-Gaspar, Zeinab Mohamed, Ayesha Araya, Susie Elsaadany, Eman Al-Mazroa, Mohammed A. Alhelail, Yaseen M. Arabi, Mustafa Al-Zoughool

School of Mathematical & Statistical Sciences Faculty Publications

Middle East respiratory syndrome coronavirus (MERS-CoV) is an emerging zoonotic coronavirus that has a tendency to cause significant healthcare outbreaks among patients with serious comorbidities. We analyzed hospital data from the MERS-CoV outbreak in King Abdulaziz Medical Center, Riyadh, Saudi Arabia, June–August 2015 using the susceptible-exposed-infectious-recovered (SEIR) ward transmission model. The SEIR compartmental model considers several areas within the hospital where transmission occurred. We use a system of ordinary differential equations that incorporates the following units: emergency department (ED), out-patient clinic, intensive care unit, and hospital wards, where each area has its own carrying capacity and distinguishes the transmission by …

A (2+1)-Dimensional Sine-Gordon And Sinh-Gordon Equations With Symmetries And Kink Wave Solutions, Gangwei Wang, Kaitong Yang, Haicheng Gu, Fei Guan, A. H. Kara

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, a (2+1)-dimensional sine-Gordon equation and a sinh-Gordon equation are derived from the well-known AKNS system. Based on the Hirota bilinear method and Lie symmetry analysis, kink wave solutions and travelingwave solutions of the (2+1)-dimensional sine-Gordon equation are constructed. The traveling wave solutions of the (2+1)-dimensional sinh-Gordon equation can also be provided in a similar manner. Meanwhile, conservation laws are derived.

Ramanujan’S Theorem And Highest Abundant Numbers, Oleg R. Musin

School of Mathematical & Statistical Sciences Faculty Publications

In 1915, Ramanujan proved asymptotic inequalities for the sum of divisors function, assuming the Riemann hypothesis (RH). We consider a strong version of Ramanujan’s theorem and define highest abundant numbers that are extreme with respect to the Ramanujan and Robin inequalities. Properties of these numbers are very different depending on whether the RH is true or false.

Integer Versus Fractional Order Seir Deterministic And Stochastic Models Of Measles, Md Rafiul Islam, Angela Peace, Daniel Medina, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations (FDEs) as approximations of some type of fractional nonlinear birth and death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While ordinary differential equations (ODEs) …

On Persistence Of Superoscillations For Schrödinger Equation With Time-Dependent Quadratic Hamiltonians, Elijah Lee Hight, Jose Palacio, Tamer Oraby, Erwin Suazo

School of Mathematical & Statistical Sciences Faculty Publications

In this work, we prove the persistence in time of superoscillations for the Schrödinger equation with time-dependent coefficients. In order to prove the persistence of superoscillations, we have conditioned the coefficients to satisfy a Riccati system, and we have expressed the solution as a convolution operator in terms of solutions of this Riccati system. Further, we have solved explicitly the Cauchy initial value problem with three different kinds of superoscillatory initial data. The operator is defined on a space of entire functions. Particular examples include Caldirola-Kanai and degenerate parametric harmonic oscillator Hamiltonians, and other examples could include Hamiltonians not self-adjoint. …

Introductory Chapter: Dynamical Symmetries And Quantum Chaos, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

No abstract provided.

Gram Determinant Solutions To Nonlocal Integrable Discrete Nonlinear Schrödinger Equations Via The Pair Reduction, Junchao Chen, Bao-Feng Feng, Yongyang Jin

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, Gram determinant solutions of local and nonlocal integrable discrete nonlinear Schrödinger (IDNLS) equations are studied via the pair reduction. A generalized IDNLS equation is firstly introduced which possesses the single Casorati determinant solution. Two kinds of Gram determinant solutions are presented from Casorati determinant ones due to the gauge freedom. The different pair constraint conditions for wave numbers are imposed and then solutions of local and nonlocal IDNLS equations are derived in terms of Gram determinant.

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 …