Mathematics Commons^™

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 …

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.

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 …

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.