Physical Sciences and Mathematics Commons^™

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 …

Contact Graphs Of Ball Packings, Alexey Glazyrin

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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.

Extreme Events And Emergency Scales, Veniamin Smirnov, Zhuanzhuan Ma, Dimitri Volchenkov

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 …

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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.

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 and Statistical Sciences Faculty Publications and Presentations

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 …

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 and Statistical Sciences Faculty Publications and Presentations

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 …

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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.

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 …

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 …

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 …

Coordinating Stem Core Courses For Student Success, Cristina Villalobos, Hyung Won Kim, Timothy J. Huber, Roger Knobel, Shaghayegh Setayesh, Lekshmi Sasidharan, Anahit Galstyan, Andras Balogh

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Research indicates multi-section coordination improves the academic performance of students in STEM education. This paper describes the process of coordination in Precalculus, Calculus 1, and Calculus 2 courses undertaken by a large department that grew from the merger of two institutions through a pilot program, and a project grant. Components introduced in the project courses are documented, including collaborative problem-solving sessions, student learning assistants, Q&A sessions, and additional technology resources. Preliminary data is provided on the impacts of the initiative on student success. The study findings provide a template for coordination, faculty buy-in, and increased student engagement at similar institutions …

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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.

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 and Statistical Sciences Faculty Publications and Presentations

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 …

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 and Statistical Sciences Faculty Publications and Presentations

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 …

Periodic Triangulations Of Zn, Mathieu Dutour Sikiric, Alexey Garber

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 …

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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.