Mathematics Commons^™

Complete Integrability And Discretization Of Euler Top And Manakov Top, Austin Marstaller

Theses and Dissertations

The Euler top is a completely integrable system with physical system implications and the Manakov top is its four-dimensional extension. We are concerned about their complete integrability and the preservation of this property under a specific discretization known as the Hirota-Kimura Discretization. Surprisingly, it is not guaranteed that under any discretization the conserved quantities are preserved and therefore they must be discovered. In this work we construct the Poisson bracket and Lax pair for each system and provide the Lie algebra background needed to do such such constructions.

The Period Of The Coefficients Of The Gaussian Polynomial[N+33], Arturo J. Martinez

Theses and Dissertations

Definition 1. For any N, the central coefficient(s) of [N+33] is denoted by C0(N) and the coefficient that is x ''away" from the central coefficient(s) of [N+33] is denoted by Cx(N).

In [1] the following result is proved:

Theorem 2. The central coefficient(s) of the Gaussian polynomial [N+33] are described by the generating function

[Special characters omitted]

This generating function has period 4.

The main goal of this thesis is to generalize Theorem 0.2 by way of proving the following conjecture:

Conjecture 3. For any x the …

Embedding Parallelohedra Into Primitive Cubic Networks And Structural Automata Description, Mikhail M. Bouniaev, Sergey V. Krivovichev

School of Mathematical & Statistical Sciences Faculty Publications

The main goal of the paper is to contribute to the agenda of developing an algorithmic model for crystallization and measuring the complexity of crystals by constructing embeddings of 3D parallelohedra into a primitive cubic network (pcu net). It is proved that any parallelohedron P as well as tiling by P, except the rhombic dodecahedron, can be embedded into the 3D pcu net. It is proved that for the rhombic dodecahedron embedding into the 3D pcu net does not exist; however, embedding into the 4D pcu net exists. The question of how many ways the embedding of a parallelohedron can …

Drug Delivery In Catheterized Arterial Blood Flow With Atherosclerosis, Saulo Orizaga, Daniel N. Riahi, Jose R. Soto

School of Mathematical & Statistical Sciences Faculty Publications

We study the problem of drug delivery in a catheterized artery in the presence of atherosclerosis. The problem is modeled in the context of a two-phase flow system which consists of red blood cells and blood plasma. The coupled differential equations for fluid (plasma) and particles (red cells) are solved for the relevant quantities in the reasonable limits. The drug delivery problem is modeled with a partial differential equation that is developed in terms of the drug concentration, blood plasma velocity, hematocrit value and the diffusion coefficient of the drug/fluid. A conservative-implicit finite difference scheme is develop in order to …

Pipe Flow Of Newtonian And Non-Newtonian Fluids, Erick Sanchez

Theses and Dissertations

We consider an incompressible, viscous fluid in a cylindrical pipe. We obtain velocity profile for both Newtonian fluid and non-Newtonian fluids such as shear-thinning, shear- thickening and Bingham plastic fluids. The flow is governed by the equation of continuity (conservation of mass) and the momentum equation. After presenting the governing system in the cylindrical coordinate system and assuming that the flow is due to the pressure drop and wall shear stress, we derive the expressions for the velocity component in the axial direction for these cases. Some computational results of the velocity profiles for various cases are presented. We will …

Quantization For A Mixture Of Uniform Distributions Associated With Probability Vectors, Mrinal Kanti Roychowdhury, Wasiela Salinas

School of Mathematical & Statistical Sciences Faculty Publications

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixtures of probability distributions, also known as mixed distributions, are an exciting new area for optimal quantization. In this paper, we investigate the optimal quantization for three different mixed distributions generated by uniform distributions associated with probability vectors

A Single-Center Comparison Using Exoskeleton Rehabilitation For Cerebrovascular Accidents And Traumatic Brain Injury In A Cohort Of Hispanic Patients, Lisa R. Trevino, Kristina Vatcheva, Michael E. Auer, Angela Morales, Lama M. Abdurrahman, Sarajova Viswamitra, Annelyn Torres-Reveron

School of Mathematical & Statistical Sciences Faculty Publications

Background Traumatic brain injury (TBI) is one of the leading causes of disability in the United States. The EKSO GT Bionics® (EKSO®) is a robotic exoskeleton approved by the Federal Drug Administration (FDA) for rehabilitation following a cerebrovascular accident (CVA or stroke) and recently received approval for use in patients with TBI. The aim of the study was to examine if the use of exoskeleton rehabilitation in patients with TBI will produce beneficial outcomes.

Methods This retrospective chart-review reports the use of the (EKSO®) robotic device in the rehabilitation of patients with TBI compared to patients with CVA. We utilized …

Homotopy Groups And Quantitative Sperner-Type Lemma, Oleg R. Musin

School of Mathematical & Statistical Sciences Faculty Publications

We consider a generalization of Sperner's lemma for triangulations of m-discs whose vertices are colored in at most m colors. A coloring on the boundary (m-1)-sphere defines an element in the corresponding homotopy group of the sphere. Depending on this invariant, a lower bound is obtained for the number of fully colored simplexes. In particular, if the Hopf invariant is nonzero on the boundary of 4-disk, then there are at least 9 fully colored tetrahedra and if the Hopf invariant is d, then the lower bound is 3d + 3.

Concrete Polytopes May Not Tile The Space, Alexey Garber, Igor Pak

School of Mathematical & Statistical Sciences Faculty Publications

Brandolini et al. conjectured in (Preprint, 2019) that all concrete lattice polytopes can multitile the space. We disprove this conjecture in a strong form, by constructing an infinite family of counterexamples in ℝ3 .

Burnside Chromatic Polynomials Of Group-Invariant Graphs, Jacob A. White

School of Mathematical & Statistical Sciences Faculty Publications

We introduce the Burnside chromatic polynomial of a graph that is invariant under a group action. This is a generalization of the Q-chromatic function Zaslavsky introduced for gain graphs. Given a group G acting on a graph G and a G-set X, a proper X-coloring is a function with no monochromatic edge orbit. The set of proper colorings is a G-set which induces a polynomial function from the Burnside ring of G to itself. In this paper, we study many properties of the Burnside chromatic polynomial, answering some questions of Zaslavsky

Characterizing Quantum Physics Students’ Conceptual And Procedural Knowledge Of The Characteristic Equation, Kaitlyn Stephens Serbin, Brigitte Johana Sánchez Robayo, Julia Victoria Truman, Kevin Lee Watson, Megan Wawro

School of Mathematical & Statistical Sciences Faculty Publications

Research on student understanding of eigentheory in linear algebra has expanded recently, yet few studies address student understanding of the Characteristic Equation. In this study, we explore quantum physics students’ conceptual and procedural knowledge of deriving and using the Characteristic Equation. We developed the Conceptual and Procedural Knowledge framework for classifying the quality of students’ conceptual and procedural knowledge of both deriving and using the Characteristic Equation along a continuum. Most students exhibited deeper conceptual and procedural knowledge of using the Characteristic Equation than of deriving the Characteristic Equation. Furthermore, most students demonstrated deeper procedural knowledge than conceptual knowledge of …

Level 17 Ramanujan-Sato Series, Timothy Huber, Daniel Schultz, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

Two level 17 modular functions r=q2∏n=1∞(1−qn)(n17),s=q2∏n=1∞(1−q17n)3(1−qn)3 are used to construct a new class of Ramanujan–Sato series for 1/π. The expansions are induced by modular identities similar to those level of 5 and 13 appearing in Ramanujan’s Notebooks. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived.

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 …

Mathematical Modeling Of Nonlinear Dynamics Of Blood Hormones On The Regulatory System, Gabriela Urbina

Theses and Dissertations

We study a mathematical modeling of nonlinear dynamics of blood hormones, which includes glucose and insulin. On Chapter I, II, III and IV, we introduce this work, analyze an effect of the secreted insulin by the pancreatic beta cells and glucagon hormones and state concluding remarks, respectively. This model considers the time evolution of nonlinear dynamics of the equations for glucose, glucagon and insulin concentrations plus insulin and glucagon actions and the secreted insulin as a result of elevation of glucose in the blood plasma. Using both analytical and numerical procedures, we determine such quantities using different parameters for different …

Partial Differential Equations In Curved Spacetimes, Jorge A. Garcia

Theses and Dissertations

It is the ambition of this thesis to analyze in a concise and coherent manner the idiosyncratic nature of partial differential equations and their mathematical structure in distinct curved spacetimes. In our work special interest is taken in quantum fields dwelling within the de-Sitter geometry. In Chapters I, II, III, and IV, a meticulous study of general relativity is undertaken with one of its solutions derived, an introduction of quantum mechanics is posed, the relativistic quantum theory of fermions is defined, and a “merging” of the former chapters and results are considered, respectively. With what has been derived we seek …

Solitary And Periodic Wave Solutions For Several Short Wave Model Equations, Andrey V. Stukopin

Theses and Dissertations

We study the periodic and solitary wave solutions to several short wave model equations arising from a so-called $\beta$-family equation for $\beta=1,2,4$. These are integrable cases which possess Lax pair and multi-soliton solutions. By phase plane analysis, either the loop or cuspon type solutions are predicted. Then, by introducing a hodograph, or reciprocal, transformation, a coupled system is derived for each $\beta$. Applying a travelling wave setting, we are able to find the periodic solutions exactly expressed in terms of Jacobi Elliptic functions. In the limiting cases of modulus k=1, they all converge to the known solitary waves.

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.