Mathematics Commons^™

Arithmetic Properties Of Septic Partition Functions, Timothy Huber, Mayra Huerta, Nathaniel Mayes

School of Mathematical & Statistical Sciences Faculty Publications

Congruences and related identities are derived for a set of colored and weighted partition functions whose generating functions generate the graded algebra of integer weight modular forms of level seven. The work determines a general strategy for identifying and proving identities and associated congruences for modular forms on the principal congruence subgroup of level 7. Ramanujan's partition congruence modulo 7 serves as a prototype for the process used to prove new congruences for modular forms of level 7.

An Algorithm To Compute Cvts For Finitely Generated Cantor Distributions, Carl P. Dettmann, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions with respect to a given probability measure. CVT is a fundamental notion that has a wide spectrum of applications in computational science and engineering. In this paper, an algorithm is given to obtain the CVTs with n-generators to level m, for any positive integers m and n, of any Cantor set generated by a pair of self-similar mappings given by S1(x)=r1x and S2(x)=r2x+(1−r2) for x∈R, where r1,r2>0 and r1+r20 and p1+p2=1.

Analysis Of Intervention Effectiveness Using Early Outbreak Transmission Dynamics To Guide Future Pandemic Management And Decision-Making In Kuwait, Michael G. Tyshenko, Tamer Oraby, Joseph C. Longenecker, Harri Vainio, Janvier Gasana, Walid Q. Alali, Mohammad Alseaidan, Susie Elsaadany, Mustafa Al-Zoughool

School of Mathematical & Statistical Sciences Faculty Publications

Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) is a World Health Organization designated pandemic that can result in severe symptoms and death that disproportionately affects older patients or those with comorbidities. Kuwait reported its first imported cases of COVID-19 on February 24, 2020. Analysis of data from the first three months of community transmission of the COVID-19 outbreak in Kuwait can provide important guidance for decision-making when dealing with future SARS-CoV-2 epidemic wave management. The analysis of intervention scenarios can help to evaluate the possible impacts of various outbreak control measures going forward which aim to reduce the effective reproduction …

Internet-Based Cognitive–Behavioural Therapy For Tinnitus: Secondary Analysis To Examine Predictors Of Outcomes, Hansapani Rodrigo, Eldré W. Beukes, Gerhard Andersson, Vinaya Manchaiah

School of Mathematical & Statistical Sciences Faculty Publications

Objectives The current study examined predictors of outcomes of internet-based cognitive–behavioural therapy (ICBT) for individuals with tinnitus.

Design Secondary analysis of intervention studies.

Setting Internet-based guided tinnitus intervention provided in the UK.

Participants 228 individuals who underwent ICBT.

Interventions ICBT.

Primary and secondary outcome measures The key predictor variables included demographic, tinnitus, hearing-related and treatment-related variables as well as clinical factors (eg, anxiety, depression, insomnia), which can have an impact on the treatment outcome. A 13-point reduction in Tinnitus Functional Index (TFI) scores has been defined as a successful outcome.

Results Of the 228 subjects who were included in the …

Majorization And Minimal Energy On Spheres, Oleg R. Musin

School of Mathematical & Statistical Sciences Faculty Publications

In the present paper, we consider the majorization theorem (also known as Karamata's inequality) and the respective minima of the majorization (the so-called M-sets) for f-energy potentials of m-point configurations on the unit sphere. In particular, we show the optimality of regular simplexes, describe some M-sets of small cardinality, define and discuss spherical f-designs.

Tri-Hamiltonian Duality System Of Merola-Ragnisco-Tu Equation, Mengxia Zhang, Bao-Feng Feng, Jiawen Liu

School of Mathematical & Statistical Sciences Faculty Publications

The tri-Hamiltonian method is applied to the Merola-Ragnisco-Tu equation. This enables us to construct a new integrable system, whose continuum limit is related to the AKNS system. Moreover, the system is proved to have linear spectral problems (Lax pair), bi-Hamiltonian structure and Darboux-Bäcklund transformation. Through the Darboux-Bäcklund transformation, we construct some exact solutions for the system.

Quantization For Uniform Distributions On Hexagonal, Semicircular, And Elliptical Curves, Gabriela Pena, Hansapani Rodrigo, Mrinal Kanti Roychowdhury, Josef A. Sifuentes, Erwin Suazo

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon, and then investigated the optimal sets of n-means and the nth quantization errors for all positive integers n. We give an exact formula to determine them, if n is of the form n = 6k for some positive integer k. We further calculate the quantization dimension, the quantization coefficient, and show that the quantization dimension is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number. Then, we define a mixture of two uniform distributions on …

Hexagon Tilings Of The Plane That Are Not Edge-To-Edge, D. Frettlöh, Alexey Glazyrin, Z. Lángi

School of Mathematical & Statistical Sciences Faculty Publications

An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer k≥3, there exists a normal tiling of the Euclidean plane by convex hexagons of unit area with exactly k irregular vertices. Using the same approach we show that there are normal edge-to-edge tilings of the plane by hexagons of unit area and exactly k many n-gons (n>6) of unit area. A result of Akopyan yields an upper bound for k depending on the maximal diameter …

Combining Assumptions And Graphical Network Into Gene Expression Data Analysis, Demba Fofana, E. O. George, Dale Brown

School of Mathematical & Statistical Sciences Faculty Publications

Background

Analyzing gene expression data rigorously requires taking assumptions into consideration but also relies on using information about network relations that exist among genes. Combining these different elements cannot only improve statistical power, but also provide a better framework through which gene expression can be properly analyzed.

Material and methods

We propose a novel statistical model that combines assumptions and gene network information into the analysis. Assumptions are important since every test statistic is valid only when required assumptions hold. So, we propose hybrid p-values and show that, under the null hypothesis of primary interest, these p-values are …

Estimation Of The Healthcare Waste Generation During Covid-19 Pandemic In Bangladesh, Tamal Chowdhury, Hemal Chowdhury, Md Salman Rahman, Nazia Hossain, Ashfaq Ahmed, Sadiq M. Sait

School of Mathematical & Statistical Sciences Faculty Publications

COVID-19 pandemic-borne wastes imposed a severe threat to human lives as well as the total environment. Improper handling of these wastes increases the possibility of future transmission. Therefore, immediate actions are required from both local and international authorities to mitigate the amount of waste generation and ensure proper disposal of these wastes, especially for low-income and developing countries where solid waste management is challenging. In this study, an attempt is made to estimate healthcare waste generated during the COVID-19 pandemic in Bangladesh. This study includes infected, ICU, deceased, isolated and quarantined patients as the primary sources of medical waste. Results …

Characterizations Of Student, Instructor, And Textbook Discourse Related To Basis And Change Of Basis In Quantum Mechanics, Kaitlyn Stephens Serbin, Megan Wawro, Rebecah Storms

School of Mathematical & Statistical Sciences Faculty Publications

Communities develop social languages in which utterances take on culturally specific situated meanings. As physics students interact in their classroom, they can learn the broader physics community’s social language by co-constructing meanings with their instructors. We provide an exposition of a systematic and productive use of idiosyncratic, socially acquired language in two classroom communities that we consider to be subcultures of the broader community of physicists. We perform a discourse analysis on twelve quantum mechanics students, two instructors, and the course text related to statements about basis and change of basis within a spin-½ probability problem. We classify the utterances’ …

Schrödinger-Poisson Systems With Singular Potential And Critical Exponent, Senli Liu, Haibo Chen, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

In this article we study the Schrödinger-Poisson system−∆ u+ V (| x|) u+ λφu= f (u), x∈ R3,−∆ φ= u2, x∈ R3, where V is a singular potential with the parameter α and the nonlinearity f satisfies critical growth. By applying a generalized version of Lions-type theorem and the Nehari manifold theory, we establish the existence of the nonnegative ground state solution when λ= 0. By the perturbation method, we obtain a nontrivial solution to above system when λ= 0.

Exponential And Hypoexponential Distributions: Some Characterizations, George Yanev

School of Mathematical & Statistical Sciences Faculty Publications

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some n ≥ 2, X1, X2, . . . , Xn are independent copies of a random variable X with unknown distribution F and a specific linear combination of Xj ’s has hypoexponential distribution, then F is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently …

On Coupled Reaction Diffusion Equations And Their Applications, Juan J. Huerta

Theses and Dissertations

Reaction-diffusion equations are nonlinear partial differential equations that have been used extensively in mathematical modeling. An interesting case in this type of equation is the Fisher-Kolmogorov system, which has been used to study a low-grade glioma, a group of primary brain tumors. In the first part of this thesis, a stochastic version of the Fisher-Kolmogorov system will be studied, and exact and numerical solutions will be presented.

The second part of this thesis will show how the speed of information propagation affects disease spread and vaccination uptake through networks in epidemics. In this model, the information reaches different people at …

Bivariate Markov Chain Model Of Irritable Bowel Syndrome (Ibs) Subtypes And Abdominal Pain, Ricardo Reyna Jr.

Theses and Dissertations

Researchers use stochastic models like continuous-time Markov chains (CTMC) to model progression of morbidities of public health impact, like HIV and Hepatitis C. Most of the research in that area is done for a single disease. In this research, we use a bivariate continuous-time Markov chain (CTMC) to model progression of co-morbidities. In particular, we use a bivariate CTMC to model the joint progression of Irritable Bowel Syndrome (IBS) and abdominal pain. Symptoms of IBS are known to change throughout the duration of the disorder. Hence, patients are normally asked to make a journal of the stool type, symptoms, and …

Existence And Stability Of The Doubly Nonlinear Anisotropic Parabolic Equation, Huashui Zhan, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we are concerned with a doubly nonlinear anisotropic parabolic equation, in which the diffusion coefficient and the variable exponent depend on the time variable t. Under certain conditions, the existence of weak solution is proved by applying the parabolically regularized method. Based on a partial boundary value condition, the stability of weak solution is also investigated.

Functional Kernel Density Estimation: Point And Fourier Approaches To Time Series Anomaly Detection, Michael R. Lindstrom, Hyuntae Jung, Denis Larocque

School of Mathematical & Statistical Sciences Faculty Publications

We present an unsupervised method to detect anomalous time series among a collection of time series. To do so, we extend traditional Kernel Density Estimation for estimating probability distributions in Euclidean space to Hilbert spaces. The estimated probability densities we derive can be obtained formally through treating each series as a point in a Hilbert space, placing a kernel at those points, and summing the kernels (a “point approach”), or through using Kernel Density Estimation to approximate the distributions of Fourier mode coefficients to infer a probability density (a “Fourier approach”). We refer to these approaches as Functional Kernel Density …

Subnormality Of Powers Of Multivariable Weighted Shifts, Sang Hoon Lee, Woo Young Lee, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

Given a pair of commuting subnormal Hilbert space operators, the Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for the existence of a commuting pair of normal extensions of and ; in other words, is a subnormal pair. The LPCS is a longstanding open problem in the operator theory. In this paper, we consider the LPCS of a class of powers of -variable weighted shifts. Our main theorem states that if a “corner” of a 2-variable weighted shift is subnormal, then is subnormal if and only if a power is subnormal for some . As a …

Repeated Minimizers Of $P$-Frame Energies, Alexey Glazyrin, Josiah Park

School of Mathematical & Statistical Sciences Faculty Publications

For a collection of $N$ unit vectors ${X}=\{x_i\}_{i=1}^N$, define the $p$-frame energy of ${X}$ as the quantity $\sum_{i\neq j} |\langle x_i,x_j \rangle|^p$. In this paper, we connect the problem of minimizing this value to another optimization problem, thus giving new lower bounds for such energies. In particular, for $p<2$, we prove that this energy is at least $2(N-d) p^{-\frac p 2} (2-p)^{\frac {p-2} 2}$ which is sharp for $d\leq N\leq 2d$ and $p=1$. We also prove that for $1\leq m

Classical And Quantum Integrability: A Formulation That Admits Quantum Chaos, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

The concept of integrability of a quantum system is developed and studied. By formulating the concepts of quantum degree of freedom and quantum phase space, a realization of the dynamics is achieved. For a quantum system with a dynamical group G in one of its unitary irreducible representative carrier spaces, the quantum phase space is a finite topological space. It is isomorphic to a coset space G=R by means of the unitary exponential mapping, where R is the maximal stability subgroup of a fixed state in the carrier space. This approach has the distinct advantage of exhibiting consistency between classical …

Invited Article By M. Gidea Extreme Events And Emergency Scales, Veniamin Smirnov, Zhuanzhuan Ma, Dimitri Volchenkov

School of Mathematical & Statistical Sciences Faculty Publications

An event is extreme if its magnitude exceeds the threshold. A choice of a threshold is subject to uncertainty caused by a method, the size of available data, a hypothesis on statistics, etc. We assess the degree of uncertainty by the Shannon's entropy calculated on the probability that the threshold changes at any given time. If the amount of data is not sufficient, an observer is in the state of Lewis Carroll's Red Queen who said "When you say hill, I could show you hills, in comparison with which you'd call that a valley". If we have enough data, the …

Contact Graphs Of Ball Packings, Alexey Glazyrin

School of Mathematical & Statistical Sciences Faculty Publications

A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in R3 is not greater than 13.955. We also find new upper bounds for the average degree of contact graphs in R4 and R5.

Energy Spectrum Of Linear Internal Wave Field In The Vicinity Of Continental Slope, Ranis N. Ibragimov, Austin Biondi, Nathan Arndt, Maria Castillo, Guang Lin, Vesselin Vatchev, Sheng Zhang

School of Mathematical & Statistical Sciences Faculty Publications

The purpose of the research was to investigate two-dimensional modeling of efficiency of mixing, resulting from the reflection of a linear internal wave field (IWF) off a continental slope. Efficiency of deep ocean mixing was associated with the energy balance of the radiating IWF into an interior of the ocean in the vicinity of a sloping bottom topography. Since waves are generated not only at the fundamental frequency but also at all of its harmonics ωn = nω less than buoyancy frequency N and greater than Coriolis frequency f, our analysis includes, in general, an infinite number of …

On Triangular Paperfolding Patterns, Alexey Garber

School of Mathematical & Statistical Sciences Faculty Publications

We introduce patterns on a triangular grid generated by paperfolding operations. We show that in case these patterns are defined using a periodic sequence of foldings, they can also be generated using substitution rules and compute eigenvalues and eigenvectors of the corresponding matrices. We also prove that densities of all basic triangles are equal in these patterns.

Local Dimensions And Quantization Dimensions In Dynamical Systems, Mrinal Kanti Roychowdhury, Bilel Selmi

School of Mathematical & Statistical Sciences Faculty Publications

Let μ be a Borel probability measure generated by a hyperbolic recurrent iterated function system defined on a nonempty compact subset of Rk. We study the Hausdorff and the packing dimensions, and the quantization dimensions of μ with respect to the geometric mean error. The results establish the connections with various dimensions of the measure μ and generalize many known results about local dimensions and quantization dimensions of measures.

Learning To Leverage Obstacles, Resources, And Strategies In Math Classes With Multilingual Learners, Aaron T. Wilson, Hyung Won Kim, Mayra Ortiz Galarza, Josef A. Sifuentes

School of Mathematical & Statistical Sciences Faculty Publications

Mathematics teachers must be ready for diverse classrooms, where students who are multilingual learners (MLs) bring new dimensions to the teaching and learning. While MLs face obstacles to learning particular to their linguistic and cultural background, they also bring resources and strengths to bear that can be applied to teaching and learning. We have developed a Challenge-Based Instructional activity to help teachers leverage their experiences of teaching math to ML students, to better understand the obstacles and resources, and to select more effective pedagogical strategies particular to this context. This paper reports the benefits teachers gain in implementing this activity.

Multiple Positive Solutions To The Fractional Kirchhoff Problem With Critical Indefinite Nonlinearities, Jie Yang, Haibo Chen, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

This article concerns the existence and multiplicity of positive solutions to the fractional Kirchhoff equation with critical indefinite nonlinearities by applying the Nehari manifold approach and fibering maps.

Qualitative Features Of A Nonlinear, Nonlocal, Agent-Based Pde Model With Applications To Homelessness, Michael R. Lindstrom, Andrea L. Bertozzi

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we develop a continuum model for the movement of agents on a lattice, taking into account location desirability, local and far-range migration, and localized entry and exit rates. Specifically, our motivation is to qualitatively describe the homeless population in Los Angeles. The model takes the form of a fully nonlinear, nonlocal, non-degenerate parabolic partial differential equation. We derive the model and prove useful properties of smooth solutions, including uniqueness and L2 -stability under certain hypotheses. We also illustrate numerical solutions to the model and find that a simple model can be qualitatively similar in behavior to observed …

Solving Parametric Radical Equations With Depth 2 Rigorously Using The Restriction Set Method, Eleftherios Gkioulekas

School of Mathematical & Statistical Sciences Faculty Publications

We review the history and previous literature on radical equations and present the rigorous solution theory for radical equations of depth 2, continuing a previous study of radical equations of depth 1. Radical equations of depth 2 are equations where the unknown variable appears under at least one square root and where two steps are needed to eliminate all radicals appearing in the equation. We state and prove theorems for all three equation forms with depth 2 that give the solution set of all real-valued solutions. The theorems are shown via the restriction set method that uses inequality restrictions to …

Tilings By Hexagonal Prisms And Embeddings Into Primitive Cubic Networks, Mikhail M. Bouniaev, Nikolay Dolbilin, Mikhail I. Shtogrin

School of Mathematical & Statistical Sciences Faculty Publications

All possible combinatorial embeddings into primitive cubic networks of arbitrary tilings of 3D space by pairwise congruent and parallel regular hexagonal prisms are discussed and classified.