Mathematics Commons^™

Be The Change: Re-Ignite Student’S Passion For Problem Solving & Mathematics, Joseph Bolz, Marti Shirley

Faculty Publications & Research

Outcomes!

• I will be able to facilitate an open ended and open middle problems in the classroom, complete with questioning techniques and student prompts.
• I will experience how to modify a textbook problem into a richer open task that encourages discussion and multiple approaches, with a focus on tailoring them to real world applications.
• I will leave with a list of questions, perfect for posing to students while engaged in the problem solving process, and also useful for any class period when students are lacking persistence

An Introduction To Shape Dynamics, Patrick R. Kerrigan

Physics

Shape Dynamics (SD) is a new fundamental framework of physics which seeks to remove any non-relational notions from its methodology. importantly it does away with a background space-time and replaces it with a conceptual framework meant to reflect direct observables and recognize how measurements are taken. It is a theory of pure relationalism, and is based on different first principles then General Relativity (GR). This paper investigates how SD assertions affect dynamics of the three body problem, then outlines the shape reduction framework in a general setting.

The Graphs That Have Antivoltages Using Groups Of Small Order, Vaidy Sivaraman, Dan Slilaty

Mathematics and Statistics Faculty Publications

Given a group Γ of order at most six, we characterize the graphs that have Γ-antivoltages and also determine the list of minor-minimal graphs that have no Γ-antivoltage. Our characterizations yield polynomial-time recognition algorithms for such graphs.

Biorthogonal-Wavelet-Based Method For Numerical Solution Of Volterra Integral Equations, Mutaz Mohammad

All Works

© 2019 by the authors. Framelets theory has been well studied in many applications in image processing, data recovery and computational analysis due to the key properties of framelets such as sparse representation and accuracy in coefficients recovery in the area of numerical and computational theory. This work is devoted to shedding some light on the benefits of using such framelets in the area of numerical computations of integral equations. We introduce a new numerical method for solving Volterra integral equations. It is based on pseudo-spline quasi-affine tight framelet systems generated via the oblique extension principles. The resulting system is …

On The Equality Case Of The Ramanujan Conjecture For Hilbert Modular Forms, Liubomir Chiriac

Mathematics and Statistics Faculty Publications and Presentations

The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations π on GL(2) asserts that |av(π)| ≤ 2. We prove that this inequality is strict if π is generated by a CM Hilbert modular form of parallel weight two and v is a finite place of degree one. Equivalently, the Satake parameters of πv are necessarily distinct. We also give examples where the equality case does occur for primes of degree two.

A Generalization Of Schroter's Formula To George Andrews, On His 80th Birthday, James Mclaughlin

Mathematics Faculty Publications

We prove a generalization of Schroter's formula to a product of an arbitrary number of Jacobi triple products. It is then shown that many of the well-known identities involving Jacobi triple products (for example the Quintuple Product Identity, the Septuple Product Identity, and Winquist's Identity) all then follow as special cases of this general identity. Various other general identities, for example certain expansions of (q; q)(infinity) and (q; q)(infinity)(k), k >= 3, as combinations of Jacobi triple products, are also proved.

A Multi-Step Approach To Modeling The 24-Hour Daily Profiles Of Electricity Load Using Daily Splines, Abdelmonaem Jornaz, V. A. Samaranayake

Mathematics and Statistics Faculty Research & Creative Works

Forecasting of real-time electricity load has been an important research topic over many years. Electricity load is driven by many factors, including economic conditions and weather. Furthermore, the demand for electricity varies with time, with different hours of the day and different days of the week having an effect on the load. This paper proposes a hybrid load-forecasting method that combines classical time series formulations with cubic splines to model electricity load. It is shown that this approach produces a model capable of making short-term forecasts with reasonable accuracy. In contrast to forecasting models that utilize a multitude of regressor …

"Big Ideas" In Secondary Mathematics Education Programs, Eryn Stehr, Hyunyi Jung, Jill Newton

Mathematical and Statistical Science Faculty Research and Publications

Although research and policy documents provide recommendations to inform secondary mathematics teacher preparation, no single study has addressed the “big ideas” of courses in multiple programs and how those big ideas may be interpreted through the lens of recent research and policy documents. To answer this need, we focused on big ideas and course objectives from three courses (i.e., Linear Algebra, Secondary Mathematics Methods, and Teaching in a Diversity Society) taught in four secondary mathematics programs. Major themes emerged related to mathematical content, pedagogy, and issues of equity. We describe findings related to big ideas, course objectives, and their connections …

An Explicit Finite Volume Numerical Scheme For 2d Elastic Wave Propagation, Mihhail Berezovski, Arkadi Berezovski

Publications

The construction of the two-dimensional finite volume numerical scheme based on the representation of computational cells as thermodynamic systems is presented explicitly. The main advantage of the scheme is an accurate implementation of conditions at interfaces and boundaries. It is demonstrated that boundary conditions influence the wave motion even in the simple case of a homogeneous waveguide.

Conceptual Models For Integer Addition And Subtraction, Nicole M. Wessman-Enzinger, Edward S. Mooney

Faculty Publications - College of Education

In this article, we report the findings of a study conducted with 6 Grade 8 students in the United States. The students posed stories for open number sentences involving addition and subtraction of integers. We analysed the stories posed by the students to build models that describe the conceptual structures behind these posed stories – the conceptual models for integer addition and subtraction. These four conceptual models for thinking about and using integer addition and subtraction include Bookkeeping, Counterbalance, Relativity, and Translation, and are generated from the students’ posed stories. We also provide profiles of conceptual model use for two …

Reduced Bias For Respondent Driven Sampling: Accounting For Non-Uniform Edge Sampling Probabilities In People Who Inject Drugs In Mauritius, Miles Q. Ott, Krista J. Gile, Matthew T. Harrison, Lisa G. Johnston, Joseph W. Hogan

Statistical and Data Sciences: Faculty Publications

People who inject drugs are an important population to study in order to reduce transmission of blood-borne illnesses including HIV and Hepatitis. In this paper we estimate the HIV and Hepatitis C prevalence among people who inject drugs, as well as the proportion of people who inject drugs who are female in Mauritius. Respondent driven sampling (RDS), a widely adopted link-tracing sampling design used to collect samples from hard-to-reach human populations, was used to collect this sample. The random walk approximation underlying many common RDS estimators assumes that each social relation (edge) in the underlying social network has an equal …

Experimental Demonstration Of Deterministic Chaos In A Waste Oil Biodiesel Semi-Industrial Furnace Combustion System, Shengyang Gao, Fashe Li, Qintai Xiao, Jianxin Xu, Huage Wang, Hua Wang

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, the nonlinear dynamic characteristics of the oxygen-enriched combustion of waste oil biodiesel in semi-industrial furnaces were tested by the power spectrum, phase space reconstruction, the largest Lyapunov exponents, and the 0-1 test method. To express the influences of the system parameters, experiments were carried out under different oxygen content conditions (21%, 25%, 28%, 31%, and 33%). Higher oxygen enrichment degrees contribute to finer combustion sufficiency, which produces flames with high luminance. Flame luminance and temperature can be represented by different gray scale values of flame images. The chaotic characteristics of gray scale time series under different oxygen …

Diabetes And Its Effect On Abdominal Aortic Aneurysm Growth Rate In Hispanic Patients, Monica Betancourt-Garcia, Kristina Vatcheva, Amrit Thakur, Prateek Gupta, Eugene Postevka, Ricardo Martinez, R. Armour Forse

School of Mathematical & Statistical Sciences Faculty Publications

Background

The growth rate of abdominal aortic aneurysms (AAA) can vary depending on age, baseline diameter, blood pressure, race, and history of smoking. Paradoxically, previous studies show evidence of a protective effect of diabetes on the rate of AAA expansion despite its well-established role in the morbidity and mortality of cardiovascular disease. This study aims to investigate the impact diabetes plays on AAA growth within a Hispanic population.

Methods

Data were collected from patients who were predominantly Mexican-American at a single hospital site. Baseline and follow-up measures for AAA diameter were obtained from serial imaging studies. Demographics, medical history, the …

The Fock Space As A De Branges–Rovnyak Space, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

We show that de Branges–Rovnyak spaces include as special cases a number of spaces, such as the Hardy space, the Fock space, the Hardy–Sobolev space and the Dirichlet space. We present a general framework in which all these spaces can be obtained by specializing a sequence that appears in the construction. We show how to exploit this approach to solve interpolation problems in the Fock space.

Student Perceptions Of Learning Introductory Mathematics In An Online Environment In Higher Education, Jamie Lynn Brooks

Doctoral Dissertations and Projects

The purpose of this transcendental phenomenological study was to describe the essence of student perception of learning introductory mathematics courses in an online environment at the college level. The central research question was, “What are the lived experiences of students who have completed introductory college mathematics courses in the online learning environment?” The phenomenon described was that of the beliefs and attitudes of the students who participated in introductory mathematics courses on the college level. The ideas explored were if students believe they learn effectively in this environment and how they believe they can best learn. Student beliefs and attitudes …

From Cymatics To Sound Therapy: Their Role In Spirituality And Consciousness Research, Victor Christianto, Kasan Susilo, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Sound is one of the types of waves that can be felt by the sense of hearing (ears). In physics, the definition of sound is something that is produced from objects that vibrate. Objects that produce sound are called sound sources. The sound source that vibrates will vibrate the molecules into the air around it. Sound is mechanical compression or longitudinal waves that propagate through the medium. This medium or intermediate agent can be liquid, solid, gas. So, sound waves can propagate for example in water, coal, or air. Most sounds are a combination of various vibratory signals composed of …

Centrosymmetric Stochastic Matrices, Lei Cao, Darian Mclaren, Sarah Plosker

Mathematics Faculty Articles

We consider the convex set Γ_m,n of m×n stochastic matrices and the convex set Γ^π_m,n ⊂Γ_m,n of m×n centrosymmetric stochastic matrices (stochastic matrices that are symmetric under rotation by 180 degrees). For Γ_m,n, we demonstrate a Birkhoff theorem for its extreme points and create a basis from certain (0,1)-matrices. For Γ^π_m,n, we characterize its extreme points and create bases, whose construction depends on the parity of m, using our basis construction for stochastic matrices. For each of Γ_m,n and Γ^π_m,n, we further characterize their extreme …

On The Linear Independence Of Finite Wavelet Systems Generated By Schwartz Functions Or Functions With Certain Behavior At Infinity, Abdelkrim Bourouihiya

Mathematics Faculty Articles

One of the motivations to state HRT conjecture on the linear independence of finite Gabor systems was the fact that there are linearly dependent Finite Wavelet Systems (FWS). Meanwhile, there are also many examples of linearly independent FWS, some of which are presented in this paper. We prove the linear independence of every three point FWS generated by a nonzero Schwartz function and with any number of points if the FWS is generated by a nonzero Schwartz function, for which the absolute value of the Fourier transform is decreasing at infinity. We also prove the linear independence of any FWS …

Improving Foresight Predictions In The 2002-2018 Nfl Regular-Seasons: A Classic Tale Of Quantity Vs. Quality, Eduardo C. Balreira, Brian K. Miceli

Mathematics Faculty Research

Utilizing a modied Bradley-Terry model, we develop a method of making foresight predictions of 2002-2018 NFL games by incorporating a home-eld parameter into previously established ranking models. Knowing only the home team and score of each contest, and taking into account previous predictions, we optimize this parameter considering one of two things: the quantity of correct picks to date or the quality of predictions to date as measured by a quadratic scoring
function. Our main results establish that optimization of quality-rather than quantity-when making a prediction has higher overall accuracy.

Across The Atlantic: Service-Learning In Spain And Morocco, Lauren Ward

Purdue Journal of Service-Learning and International Engagement

Purdue provides many activities in service-learning each year, and though they are varied experiences, many of the same lessons can be learned. I had the opportunity to participate in two service-learning study abroad trips while at Purdue- the first to Spain and Morocco, and the second to Haiti. While on these trips, I was involved in projects that seemed very different. In Morocco, my group taught high school students about the history of mathematics during the Islamic Golden Age and how mathematics is utilized in Purdue research. In Haiti, I worked with my teammates to teach water sanitation and storage …

Shifted Third Kind Chebyshev Operational Matrix To Solve Bvps Over Infinite Interval, Bushra E. Khashem

Emirates Journal for Engineering Research

The main purpose of this research is to solve boundary value problems (BVPs) with an infinite number of boundary conditions. By reducing the infinite interval to finite interval that is large and approximating the variable using finite difference method, the resulting boundary value problem is reduced to linear system of algebraic equations with unknown shifted third kind chebychev coefficients. The applications are demonstrated via test examples.

How Mathematics Can Help Winning The War Against Cancer?, Peng Feng

Mathematics Colloquium Series

In this talk, I will present a few mathematical models that aims to understand how our immune system interact with cancer cells. In particular, we focus on a model that studies the role or regulatory T cells. Recent advance in the field of regulatory T cell reveals that it plays a vital role during immunotherapy. For example, a higher ratio between regulatory T cells and effector T cells within tumor tissue is associated with worse prognoses in many cancers, including ovarian cancer (Leffers et al., 2009), lung cancer (Tao et al., 2012), glioblastoma (Sayour et al., 2015). On the other …

Compressive Sensing Inference Of Neuronal Network Connectivity In Balanced Neuronal Dynamics, Victor J. Barranca, D. Zhou

Mathematics & Statistics Faculty Works

Determining the structure of a network is of central importance to understanding its function in both neuroscience and applied mathematics. However, recovering the structural connectivity of neuronal networks remains a fundamental challenge both theoretically and experimentally. While neuronal networks operate in certain dynamical regimes, which may influence their connectivity reconstruction, there is widespread experimental evidence of a balanced neuronal operating state in which strong excitatory and inhibitory inputs are dynamically adjusted such that neuronal voltages primarily remain near resting potential. Utilizing the dynamics of model neurons in such a balanced regime in conjunction with the ubiquitous sparse connectivity structure of …

Forcibly-Biconnected Graphical Degree Sequences: Decision Algorithms And Enumerative Results, Kai Wang

Theory & Applications of Graphs

We present an algorithm to test whether a given graphical degree sequence is forcibly biconnected. The worst case time complexity of the algorithm is shown to be exponential but it is still much better than the previous basic algorithm for this problem. We show through experimental evaluations that the algorithm is efficient on average. We also adapt the classic algorithm of Ruskey et al. and that of Barnes and Savage to obtain some enumerative results about forcibly biconnected graphical degree sequences of given length n and forcibly biconnected graphical partitions of given even integer n. Based on these enumerative …

Laplacian Spectral Characterization Of Signed Sun Graphs, Fatemeh Motialah, Mohammad Hassan Shirdareh Haghighi

Theory & Applications of Graphs

A sun SG_n is a graph of order 2n consisting of a cycle C_n, n ≥ 3, to each vertex of it a pendant edge is attached. In this paper, we prove that unbalanced signed sun graphs are determined by their Laplacian spectra. Also we show that a balanced signed sun graph is determined by its Laplacian spectrum if and only if n is odd.

A Remark On How A Consciousness Model And Entanglement Can Lead Us To Quantum Communication, Victor Christianto, Robert N. Boyd, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In a recent paper, we describe how a model of quantum communication based on combining consciousness experiment and entanglement can serve as impetus to stop 5G-caused diseases. Therefore, in this paper we will discuss how entanglement can be explained in terms of quantum theory. This short review may be considered as an effort to bring QM into real problem solving, i.e. telecommunication.

Multi‐Breather And High‐Order Rogue Waves For The Nonlinear Schrödinger Equation On The Elliptic Function Background, Bao-Feng Feng, Liming Ling, Daisuke A. Takahashi

School of Mathematical & Statistical Sciences Faculty Publications

We construct the multi‐breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and nontrivial phase‐modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multibreather solutions with different velocities in the limit , where the solution in the neighborhood of each breather tends to the simple one‐breather solution. …

The Grass Grows Green In Virginia: A Grassroots Effort Leading To Comprehensive Change In Removing Mathematics Barriers For Students., Patricia Parker

Inquiry: The Journal of the Virginia Community Colleges

The Virginia Community College System (VCCS) embarked on a comprehensive mathematics pathways project in October 2015 with a move from design to implementation in spring 2017. The VCCS Mathematics Pathways Project (VMPP) aimed not only to develop strategies to improve retention and completion, but also to address foundational barriers to students’ success. This grassroots effort involved collaboration among all 23 community colleges, over 200 mathematics faculty, and staff from career and technical support departments. Collaboration extended to the K–12 and university sectors, professional organizations, publishers, and foundations. VMPP goals focused on creating structured mathematics pathway courses for all program levels, …

Fluids In Music: The Mathematics Of Pan’S Flutes, Bogdan Nita, Sajan Ramanathan

Department of Mathematics Faculty Scholarship and Creative Works

We discuss the mathematics behind the Pan’s flute. We analyze how the sound is created, the relationship between the notes that the pipes produce, their frequencies and the length of the pipes. We find an equation which models the curve that appears at the bottom of any Pan’s flute due to the different pipe lengths.

Why Does Ramanujan, The Man Who Knew Infinity, Matter?, Ken Ono

Dalrymple Lecture Series

Dr. Ken Ono is the Thomas Jefferson Professor of Mathematics at the University of Virginia, the Asa Griggs Candler Professor of Mathematics at Emory University, and the vice president of the American Mathematical Society.He is an associate producer of the film The Man Who Knew Infinity starring Dev Patel and Jeremy Irons about Srinivasa Ramanujan, a self-trained two-time college dropout who left behind three notebooks filled with equations that mathematicians are still trying to figure out today. Ramanujan claimed that his ideas came to him as visions from an Indian goddess. This lecture is about why Ramanujan matters. The answers …