Applied Mathematics Commons^™

Comic Books That Teach Mathematics, Bruce Kessler

Mathematics Faculty Publications

During the 2008--2009 academic year, the author embarked on an extremely non-standard curriculum path: developing comic books with embedded mathematics appropriate for 3rd through 6th grade students. With the help of an education professor to measure impact, an elementary-school principal, and talented undergraduate illustrators, this project came to fruition and the comics were implemented in elementary classrooms at Cumberland Trace Elementary in the Warren County School System in Bowling Green, Kentucky. This manuscript gives the history of this idea, the difficulties of developing the content of the comics and getting them illustrated, and the implementation plan in the school.

My Trig Book, Bruce Kessler

Mathematics Faculty Publications

This is the MATH 117 Trigonometry text developed by Dr. Bruce Kessler for the Gatton Academy of Math and Science at Western Kentucky University for the Academy sections of the course. The text has also been used in two online course offerings.

Thermocapillary Effects In Driven Dewetting And Self-Assembly Of Pulsed Laser-Irradiated Metallic Films, Agegnehu Atena, Mikhail Khenner

Mathematics Faculty Publications

In this paper the lubrication-type dynamical model is developed of a molten, pulsed laser-irradiated metallic film. The heat transfer problem that incorporates the absorbed heat from a single beam or interfering beams is solved analytically. Using this temperature field, we derive the 3D long-wave evolution PDE for the film height. To get insights into dynamics of dewetting, we study the 2D version of the evolution equation by means of a linear stability analysis and by numerical simulations. The stabilizing and destabilizing effects of various system parameters, such as the peak laser beam intensity, the film optical thickness, the Biot and …

My Trig Book, Bruce Kessler

Mathematics Faculty Publications

This is the MATH 117 Trigonometry text developed by Dr. Bruce Kessler for the Gatton Academy of Math and Science at Western Kentucky University for the Math and Science sections of the course. The text has also been used in two online course offerings.

Poisson Structures Of Equations Associated With Groups Of Diffeomorphisms, Rossen Ivanov

Conference papers

A class of equations describing the geodesic flow for a right-invariant metric on the group of diffeomorphisms of Rn is reviewed from the viewpoint of their Lie-Poisson structures. A subclass of these equations is analogous to the Euler equations in hydrodynamics (for n = 3), preserving the volume element of the domain of fluid flow. An example in n = 1 dimension is the Camassa-Holm equation, which is a geodesic flow equation on the group of diffeomorphisms, preserving the H1 metric.

Two Component Integrable Systems Modelling Shallow Water Waves, Rossen Ivanov

Conference papers

Our aim is to describe the derivation of shallow water model equations for the constant vorticity case and to demonstrate how these equations can be related to two integrable systems: a two component integrable generalization of the Camassa-Holm equation and the Kaup - Boussinesq system.

Stability Of Traveling Waves In Thin Liquid Films Driven By Gravity And Surfactant, Ellen Peterson, Michael Shearer, Thomas P. Witelski, Rachel Levy

All HMC Faculty Publications and Research

A thin layer of fluid flowing down a solid planar surface has a free surface height described by a nonlinear PDE derived via the lubrication approximation from the Navier Stokes equations. For thin films, surface tension plays an important role both in providing a significant driving force and in smoothing the free surface. Surfactant molecules on the free surface tend to reduce surface tension, setting up gradients that modify the shape of the free surface. In earlier work [12, 13J a traveling wave was found in which the free surface undergoes three sharp transitions, or internal layers, and the surfactant …

Confidence Wagering During Mathematics And Science Testing., Brady Jack, Chia-Ju Liu, Hoan-Lin Chiu, James Shymansky

Educator Preparation & Leadership Faculty Works

No abstract provided.

The Beauty Of Mathematics And The Mathematics Of Beauty: Continued Fractions And The Golden Ratio, Jessica Tush

Inquiry: The University of Arkansas Undergraduate Research Journal

This project begins with a look at the history of simple continued fractions and how we have arrived where we are today. We then move through a study of simple continued fractions, beginning first with rational numbers and moving to irrational numbers. Continuing further in the pursuit of joining mathematics and art, we define the specific continued fraction that gives rise to the Fibonacci sequence and the Golden Ratio~ (phi, pronounced 'Jai"). These two notions form a direct link to art and the properties that we hope to examine. I have taken an analytic approach to showing that the Golden …

Singular Points Of Real Quartic And Quintic Curves, David A. Weinberg, Nicholas J. Willis

Faculty Publications - Department of Mathematics

There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D. A. Gudkov. There are nine types of singular points for irreducible complex quartic curves and ten types of singular points for reducible complex quartic curves. There are 42 types of real singular points for irreducible real quintic curves and 49 types of real singular points for irreducible real quintic curves. The classification of real singular points for irreducible real quintic curves is originally due to Golubina and Tai. There are …

A Direct Solution Of The Robin Inverse Problem, Weifu Fang, Suxing Zeng

Mathematics and Statistics Faculty Publications

We present a direct, linear boundary integral equation method for the inverse problem of recovering the Robin coefficient from a single partial boundary measurement of the solution to the Laplace equation.

Rules And Insights: Connecting The Mathematical And Linguistic Abilities Of C.S. Lewis, Kim Regnier Jongerius

ACMS Journal 2008-2009

While most biographical works on C.S. Lewis give passing reference to Lewis' problems with elementary mathematics, few have made an attempt at diagnosing the difficulty or exploring its impact on his writing. A careful study of family correspondence, however, makes it clear that his learning difficulties were not with mathematics alone and suggests connections between attitudes toward and abilities in both mathematics and language. This paper will make these connections clear and will illustrate their ties to Lewis' effective mathematical references.

Computing Prime Harmonic Sums, Eric Bach, Dominic Klyve, Jonathan P. Sorenson

Scholarship and Professional Work - LAS

We discuss a method for computing Σ �≤� 1/�, using time about �2/3 and space about �1/3. It is based on the Meissel-Lehmer algorithm for computing the prime-counting function �(�), which was adapted and improved by Lagarias, Miller, and Odlyzko. We used this algorithm to determine the first point at which the prime harmonic sum first crosses.

Intermediate-Complexity Biological Modeling Framework For Nutrient Cycling In Lakes Based On Physical Structure, Michael Clay Rigley

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Mathematical models for the change in concentration of total dissolved nitrogen (TDN) in mountain lakes are developed based on the dynamics of coupled, well-mixed containers. Each includes a stratified lake structure without the complexity of a full fluid model. A lake is divided into a suite of compartments based on physical structure: warm upper layer (epilimnion), cold inflow and insertion layer (metalimnion), cold lower layer (hypolimnion), and a warm shallow shelf. With the compartments as the framework and literature values for uptake rates, death rates, half-saturation constants, and sinking rates, systems of equations are written for three models. The first …

A Note On The Positive Solutions Of An Inhomogeneous Elliptic Equation On Rn, Yinbin Deng, Yi Li, Fen Yang

Mathematics and Statistics Faculty Publications

This paper is contributed to the elliptic equation

(0.1) Δu+K(|x|)up+μf(|x|)=0,

where p>1, x∈Rn, n⩾3, and μ⩾0 is a constant. We study the structure of positive radial solutions of (0.1) and obtain the uniqueness of solution decaying faster than r−m at ∞ if μ is small enough under some assumptions on K and f, where m is the slow decay rate.

Periodic Traveling Waves In Sirs Endemic Models, Tong Li, Yi Li, Herbert W. Hethcote

Mathematics and Statistics Faculty Publications

Mathematical models are used to determine if infection wave fronts could occur by traveling geographically in a loop around a region or continent. These infection wave fronts arise by Hopf bifurcation for some spatial models for infectious disease transmission with distributed-contacts. Periodic traveling waves are shown to exist for the spatial analog of the SIRS endemic model, in which the temporary immunity is described by a delay, but they do not exist in a similar spatial SIRS endemic model without a delay. Specifically, we found that the ratio of the delay ω in the recovered class and the average infectious …

An H-Adaptive Finite-Element Technique For Constructing 3d Wind Fields, Darrell Pepper, Xiuling Wang

Mechanical Engineering Faculty Research

An h-adaptive, mass-consistent finite-element model (FEM) has been developed for constructing 3D wind fields over irregular terrain utilizing sparse meteorological tower data. The element size in the computational domain is dynamically controlled by an a posteriori error estimator based on the L2 norm. In the h-adaptive FEM algorithm, large element sizes are typically associated with smooth flow regions and small errors; small element sizes are attributed to fast-changing flow regions and large errors. The adaptive procedure employed in this model uses mesh refinement–unrefinement to satisfy error criteria. Results are presented for wind fields using sparse data obtained from two regions …

Two Component Integrable Systems Modelling Shallow Water Waves: The Constant Vorticity Case, Rossen Ivanov

Articles

In this contribution we describe the role of several two-component integrable systems in the classical problem of shallow water waves. The starting point in our derivation is the Euler equation for an incompressible fluid, the equation of mass conservation, the simplest bottom and surface conditions and the constant vorticity condition. The approximate model equations are generated by introduction of suitable scalings and by truncating asymptotic expansions of the quantities to appropriate order. The so obtained equations can be related to three different integrable systems: a two component generalization of the Camassa-Holm equation, the Zakharov-Ito system and the Kaup-Boussinesq system. The …

Characterization Of Compactly Supported Renable Splines With Integer Matrix, Tian-Xiao He, Yujing Guana

Scholarship

Let M be an integer matrix with absolute values of all its eigenvalues being greater than 1. We give a characterization of compactly supported M-refinable splines f and the conditions that the shifts of f form a Riesz basis.

A New Elasticity Element Made For Enforcing Weak Stress Symmetry, Bernardo Cockburn, Jay Gopalakrishnan, Johnny Guzmán

Mathematics and Statistics Faculty Publications and Presentations

We introduce a new mixed method for linear elasticity. The novelty is a simplicial element for the approximate stress. For every positive integer k, the row-wise divergence of the element space spans the set of polynomials of total degree k. The degrees of freedom are suited to achieve continuity of the normal stresses. What makes the element distinctive is that its dimension is the smallest required for enforcing a weak symmetry condition on the approximate stress. This is achieved using certain "bubble matrices", which are special divergence-free matrix-valued polynomials. We prove that the approximation error is of order k + …

Nonlinear Dynamics Of Infant Sitting Postural Control, Joan E. Deffeyes

Department of Psychology: Dissertations, Theses, and Student Research

Sitting is one of the first developmental milestones that an infant achieves. Thus measurements of sitting posture present an opportunity to assess sensorimotor development at a young age, in order to identify infants who might benefit from therapeutic intervention, and to monitor the efficacy of the intervention. Sitting postural sway data was collected using a force plate from infants with typical development, and from infants with delayed development, where the delay in development was due to cerebral palsy in most of the infants in the study. The center of pressure time series from the infant sitting was subjected to a …

Coarser Connected Topologies And Non-Normality Points, Lynne Yengulalp

Mathematics Faculty Publications

We investigate two topics, coarser connected topologies and non-normality points. The motivating question in the first topic is:

Question 0.0.1. When does a space have a coarser connected topology with a nice topological property? We will discuss some results when the property is Hausdorff and prove that if X is a non-compact metric space that has weight at least c, then it has a coarser connected metrizable topology. The second topic is concerned with the following question:

Question 0.0.2. When is a point y ∈ β X\X a non-normality point of β X\X? We will discuss the question in the …

A Two-Population Insurgency In Colombia: Quasi-Predator-Prey Models - A Trend Towards Simplicity, John A. Adam, John A. Sokolowski, Catherine M. Banks

Mathematics & Statistics Faculty Publications

A sequence of analytic mathematical models has been developed in the context of the "low-level insurgency" in Colombia, from 1993 to the present. They are based on generalizations of the two-population "predator-prey" model commonly applied in ecological modeling, and interestingly, the less sophisticated models yield more insight into the problem than the more complicated ones, but the formalism is available to adapt the model "upwards" in the event that more data becomes available, or as the situation increases in complexity. Specifically, so-called "forcing terms" were included initially in the coupled differential equations to represent the effects of government policies towards …

Polynomial Extension Operators. Part Ii, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

Mathematics and Statistics Faculty Publications and Presentations

Consider the tangential trace of a vector polynomial on the surface of a tetrahedron. We construct an extension operator that extends such a trace function into a polynomial on the tetrahedron. This operator can be continuously extended to the trace space of H(curl ). Furthermore, it satisfies a commutativity property with an extension operator we constructed in Part I of this series. Such extensions are a fundamental ingredient of high order finite element analysis.

The Derivation Of Hybridizable Discontinuous Galerkin Methods For Stokes Flow, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. We present four ways of hybridizing the methods, which differ by the choice of the globally coupled unknowns. Classical methods for the Stokes equations can be thought of as limiting cases of these new methods.

Unified Hybridization Of Discontinuous Galerkin, Mixed, And Continuous Galerkin Methods For Second Order Elliptic Problems, Bernardo Cockburn, Jay Gopalakrishnan, Raytcho Lazarov

Mathematics and Statistics Faculty Publications and Presentations

We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continu- ous Galerkin, non-conforming and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. Since the associated matrix is sparse, symmetric and positive definite, these methods can be efficiently implemented. Moreover, the …

Stability Of Linear Flocks On A Ring Road, J. J. P. Veerman, Carlos Martins Da Fonseca

Mathematics and Statistics Faculty Publications and Presentations

We discuss some stability problems when each agent of a linear flock on the line interacts with its two nearest neighbors (one on either side).

A Single Particle Impact Model For Motion In Avalanches, J. J. P. Veerman, Dacian Daescu, M. J. Romero-Vallés, P. J. Torres

Mathematics and Statistics Faculty Publications and Presentations

We describe the global behavior of the dynamics of a particle bouncing down an inclined staircase. For small inclinations all orbits eventually stop (independent of the initial condition). For large enough inclinations all orbits end up accelerating indefinitely (also independent of the initial conditions). There is an interval of inclinations of positive length between these two. In that interval the behavior of an orbit depends on its initial condition. In addition to stopping and accelerating orbits, there are also orbits with speeds bounded away from both zero and infinity. A second hallmark of the dynamics is that the orbits going …

Biwave Maps Into Manifolds, Yuan-Jen Chiang

Mathematics Articles

We generalize wave maps to biwave maps. We prove that the composition of a biwave map and a totally geodesic map is a biwave map. We give examples of biwave nonwave maps. We show that if f is a biwave map into a Riemannian manifold under certain circumstance, then f is a wave map. We verify that if f is a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then f is a wave map. We finally obtain a theorem involving an unstable biwave map.

Phase Transitions In Materials With Thermal Memory: The Case Of Unequal Conductivities, John Murrough Golden

Articles

A model for thermally induced phase transitions in materials with thermal memory was recently proposed, where the equations determining heatflow were assumed to be the same in both phases. In this work, the model is generalized to the case of phase dependent heatflow relations. The temperature (or coldness) gradient is decomposed into two parts, each zero on one phase and equal to the temperature (or coldness) gradient on the other. However, they vary smoothly over the transition zone. These are treated as separate independent quantities in the derivation of field equations from thermodynamics. Heat flux is given by an integral …