Mathematics Commons^™

Global Well-Posedness And Asymptotic Behavior Of A Class Of Initial-Boundary-Value Problems Of The Kdv Equation On A Finite Domain, Ivonne Rivas, Muhammad Usman, Bingyu Zhang

Mathematics Faculty Publications

In this paper, we study a class of initial boundary value problem (IBVP) of the Korteweg- de Vries equation posed on a ?nite interval with nonhomogeneous boundary conditions. The IBVP is known to be locally well-posed, but its global L2 a priori estimate is not available and therefore it is not clear whether its solutions exist globally or blow up in finite time. It is shown in this paper that the solutions exist globally as long as their initial value and the associated boundary data are small, and moreover, those solutions decay exponentially if their boundary data decay exponentially.

Making Algebra More Accessible: How Steep Can It Be For Teachers?, Diana Cheng, Polina Sabinin

Mathematics Faculty Publications

Teacher educators need to support middle grades teachers in developing mathematical knowledge for teaching algebraic concepts. In particular, teachers should become familiar with possible introductions and sequencing to the concept of slope, and common middle school students’ limited conceptions about measuring the steepness of an incline. Steepness can be expressed directly in terms of an angle or indirectly as a slope. Encouraging middle school students to find a measure of steepness using a ratio may help support students’ transition to multiplicative thinking. This mixed – methods study explores middle school students’ responses in solving a comparison problem involving the steepness …

Linear Systems Of Fractional Nabla Difference Equations, Ferhan M. Atici, Paul W. Eloe

Mathematics Faculty Publications

In this paper we shall consider a linear system of fractional nabla difference equations with constant coefficients. We shall construct the fundamental matrix for the homogeneous system and the causal Green’s function for the nonhomogeneous system. We employ transform methods and series methods, and we illustrate analogies with classical first-order differential or difference equations. We shall close the paper with an asymptotic result that follows from the analysis of a half-order nabla difference equation.

Research In Mathematics Educational Technology: Current Trends And Future Demands, Shannon O. Driskell, Robert N. Ronau, Christopher R. Rakes, Sarah B. Bush, Margaret L. Niess, David K. Pugalee

Mathematics Faculty Publications

This systematic review of mathematics educational technology literature identified 1356 manuscripts addressing the integration of educational technology into mathematics instruction. The manuscripts were analyzed using three frameworks (Research Design, Teacher Knowledge, and TPACK) and three supplementary lenses (Data Sources, Outcomes, and NCTM Principles) to produce a database to support future research syntheses and meta-analyses. Preliminary analyses of student and teacher outcomes (e.g., knowledge, cognition, affect, and performance) suggest that the effects of incorporating graphing calculator and dynamic geometry technologies have been abundantly studied; however, the usefulness of the results was often limited by missing information regarding measures of validity, reliability, …

Positive Solutions Of Nonlocal Boundary Value Problem For Higher Order Fractional Differential System, Mujeeb Ur Rehman, Rahmat Ali Khan, Paul W. Eloe

Mathematics Faculty Publications

In this paper, we study existence and multiplicity results for a coupled system of nonlinear nonlocal boundary value problems for higher order fractional differential equations of the type (see PDF) where (see PDF) is Caputo fractional derivative. We employ the Guo-Krasnosel’skii fixed point theorem to establish existence and multiplicity results for positive solutions. We derive explicit intervals for the parameters _ and μ for which the system possess the positive solutions or multiple positive solutions. Examples are included to show the applicability of the main results.

Semi-Direct Galois Covers Of The Affine Line, Linda Gruendken, Laura L. Hall-Seelig, Bo-Hae Im, Ekin Ozman, Rachel Pries, Katherine Stevenson

Mathematics Faculty Publications

Let k be an algebraically closed field of characteristic p > 0. Let G be a semi-direct product of the form (Z/`Z) b o Z/pZ where b is a positive integer and ` is a prime distinct from p. In this paper, we study Galois covers ψ : Z → P 1 k ramified only over ∞ with Galois group G. We find the minimal genus of a curve Z which admits a covering map of this form and we give an explicit formula for this genus in terms of ` and p. The minimal genus occurs when b equals the …

Tilings, Patterns And Technology, Ma. Louise Antonette N. De Las Peñas, Angela Fatima H. Guzon

Mathematics Faculty Publications

In this paper we discuss various situations where tilings and patterns, with the aid of technology, facilitate the teaching of mathematics and serve as tools in understanding and developing new mathematical ideas. We also illustrate how technology makes possible cultural connections in the study of mathematics using Islamic tilings and patterns.

Operation Comics: The Story Continues, Bruce Kessler, Janet Tassell, Tressa Tullis

Mathematics Faculty Publications

During the 2008-2009 academic year, the author K. wrote three issues of Operation Comics, a comic book with embedded mathematics content appropriate for 4th through 6th grade students. Several printed comics were placed in Cumberland Trace Elementary in the Warren County School System in Bowling Green, Kentucky, US. The author Ta. was enlisted to measure the impact of the comics on the attitudes and motivation of the students using the comics. A preliminary report was given by K. at the 2009 Bridges Banff Conference, and the written report appeared in the proceedings. Since then, data has been collected on the …

Regression Model Fitting With Quadratic Term Leads To Different Conclusion In Economic Analysis Of Washington State Smoking Ban, Marshal Ma, Scott Mcclintock

Mathematics Faculty Publications

No abstract provided.

Composition Operators Whose Symbols Have Orthogonal Powers, Valentin Matache

Mathematics Faculty Publications

Composition operators on the Hilbert Hardy space H2 whose symbols are analytic selfmaps of the open unit disk having orthogonal powers are considered. The spectra and essential spectra of such operators are described. In the general case of an arbitrary analytic selfmap of the open unit disk, it is proved that the composition operator induced by that map has essential spectral radius less than 1 if and only if the map under consideration is a non–inner map with a fixed point in the unit disk. The canonical decomposition of a non–unitary composition contraction is determined.

Boolean Modeling Of Biochemical Networks, Tomáš Helikar, Naomi Kochi, John Konvalina, Jim A. Rogers

Mathematics Faculty Publications

The use of modeling to observe and analyze the mechanisms of complex biochemical network function is becoming an important methodological tool in the systems biology era. Number of different approaches to model these networks have been utilized-- they range from analysis of static connection graphs to dynamical models based on kinetic interaction data. Dynamical models have a distinct appeal in that they make it possible to observe these networks in action, but they also pose a distinct challenge in that they require detailed information describing how the individual components of these networks interact in living cells. Because this level of …

Fully Nonlinear Boundary Value Problems With Impulse, Paul Eloe, Muhammad Usman

Mathematics Faculty Publications

An impulsive boundary value problem with nonlinear boundary conditions for a second order ordinary differential equation is studied. In particular, sufficient conditions are provided so that a compression- expansion cone theoretic fixed point theorem can be applied to imply the existence of positive solutions. The nonlinear forcing term is assumed to satisfy usual sublinear or superlinear growth as t → ∞ or t → 0 +. The nonlinear impulse terms and the nonlinear boundary terms are assumed to satisfy the analogous asymptotic behavior.

Prospective Teachers' Use Of Representations In Solving Statistical Tasks With Dynamic Statistical Software, Hollylynne Lee, Shannon O. Driskell, Suzanne R. Harper, Keith R. Leatham, Gladis Kersaint, Robin L. Angotti

Mathematics Faculty Publications

This study examined a random stratified sample (n=62) of prospective teachers' work across eight institutions on three tasks that utilized dynamic statistical software. Our work was guided by considering how teachers may utilize their statistical knowledge and technological statistical knowledge to engage in cycles of investigation. Although teachers did not tend to take full advantage of dynamic linking capabilities, they utilized a large variety of graphical representations and often added statistical measures or other augmentations to graphs as part of their analysis.

Non-Normality Points Of Β X\X, William Fleissner, Lynne Yengulalp

Mathematics Faculty Publications

We seek conditions implying that (β X\X) \ {y} is not normal. Our main theorem: Assume GCH and all uniform ultrafilters are regular. If X is a locally compact metrizable space without isolated points, then (β X\X) \ {y} is not normal for all y ∈ β X\X. In preparing to prove this theorem, we generalize the notions “uniform”, “regular”, and “good” from set ultrafilters to z-ultrafilters. We discuss non-normality points of the product of a discrete space and the real line. We topologically embed a nonstandard real line into the remainder of this product space.

Algorithms For Area Preserving Flows, Catherine Kublik, Selim Esedoglu, Jeffrey A. Fessler

Mathematics Faculty Publications

We propose efficient and accurate algorithms for computing certain area preserving geometric motions of curves in the plane, such as area preserving motion by curvature. These schemes are based on a new class of diffusion generated motion algorithms using signed distance functions. In particular, they alternate two very simple and fast operations, namely convolution with the Gaussian kernel and construction of the distance function, to generate the desired geometric flow in an unconditionally stable manner. We present applications of these area preserving flows to large scale simulations of coarsening.

Upper And Lower Solutions For Regime-Switching Diffusions With Applications In Financial Mathematics, Paul W. Eloe, R. H. Liu

Mathematics Faculty Publications

This paper develops a method of upper and lower solutions for a general system of second-order ordinary differential equations with two-point boundary conditions. Our motivation of study stems from a class of financial mathematics problems under regime-switching diffusion models. Two examples are double barrier option valuation and optimal selling rules in asset trading. We establish the existence of a unique C2 solution of the two-point boundary value problem. We construct monotone sequences of upper and lower solutions that are shown to converge to the unique solution of the boundary value problem. This construction provides a feasible numerical method to compute …

Optimal Intervals For Uniqueness Of Solutions For Nonlocal Boundary Value Problems, Paul W. Eloe, Johnny Henderson

Mathematics Faculty Publications

For the nth order differential equation (see PDF for equation) we obtain optimal bounds on the length of intervals on which solutions are unique for certain nonlocal three point boundary value problems. These bounds are obtained through an application of the Pontryagin Maximum Principle from the theory of optimal control.

The Unimodality Of Pure O-Sequences Of Type Three In Three Variables, Bernadette Boyle

Mathematics Faculty Publications

We will give a positive answer for the unimodality of the Hilbert functions in the smallest open case, that of Artinian level monomial algebras of type three in three variables.

Traveling Wave Solutions Of Burgers’ Equation For Power-Law Non- Newtonian Flows, Dongming Wei, Harry Borden

Mathematics Faculty Publications

In this work we present some analytic and semi-analytic traveling wave solutions of a generalized Burgers' equation for unidirectional flow of power-law non-Newtonian fluids. The solutions include the corresponding well-known traveling wave solution of the Burgers' equation for Newtonian flows. We also derive estimates of shock thickness for the power-law flows.

Coarser Connected Metrizable Topologies, Lynne Yengulalp

Mathematics Faculty Publications

We show that every metric space, X, with w(⩾) c has a coarser connected metrizable topology.

An Algorithm For Wavelet–Based Elemental Spectrum Analysis, Bruce Kessler

Mathematics Faculty Publications

At the previous Approximation Theory XII meeting, I discussed some preliminary work with the Applied Physics Institute at Western Kentucky University in using multiwavelets to provide an objective analysis of gamma-ray spectrum generated from fast neutron bombardment of objects, for the purpose of identifying the elemental composition of the object. The method discussed at the time worked moderately well with the limited amount of data provided, but subsequent use with data sets of different compounds and with different detectors brought to light serious flaws with its implementation.

This talk will illustrate those issues and will address how they have been …

When Is The Numerical Range Of A Nilpotent Matrix Circular?, Valentin Matache, Mihaela Teodora Matache

Mathematics Faculty Publications

The problem formulated in the title is investigated. The case of nilpotent matrices of size at most 4 allows a unitary treatment. The numerical range of a nilpotent matrix M of size at most 4 is circular if and only if the traces tr M^∗M² and tr M^∗M³ are null. The situation becomes more complicated as soon as the size is 5. The conditions under which a 5×5nilpotent matrix has circular numerical range are thoroughly discussed.

Improved Automated Monitoring And New Analysis Algorithm For Circadean Phototaxis Rhythms In Chlamydomonas, Christa Gaskill, Jennifer Forbes-Stovall, Bruce Kessler, Mike Young, Claire A. Rinehart, Sigrid Jacobshagen

Mathematics Faculty Publications

Automated monitoring of circadian rhythms is an efficient way of gaining insight into oscillation parameters like period and phase for the underlying pacemaker of the circadian clock. Measurement of the circadian rhythm of phototaxis (swimming towards light) exhibited by the green alga Chlamydomonas reinhardtii has been automated by directing a narrow and dim light beam through a culture at regular intervals and determining the decrease in light transmittance due to the accumulation of cells in the beam. In this study, the monitoring process was optimized by constructing a new computercontrolled measuring machine that limits the test beam to wavelengths reported …

Parallelization Of The Wolff Single-Cluster Algorithm, Jevgenijs Kaupužs, Jānis Rimšāns, Roderick V.N. Melnik

Mathematics Faculty Publications

A parallel [open multiprocessing (OpenMP)] implementation of the Wolff single-cluster algorithm has been developed and tested for the three-dimensional (3D) Ising model. The developed procedure is generalizable to other lattice spin models and its effectiveness depends on the specific application at hand. The applicability of the developed methodology is discussed in the context of the applications, where a sophisticated shuffling scheme is used to generate pseudorandom numbers of high quality, and an iterative method is applied to find the critical temperature of the 3D Ising model with a great accuracy. For the lattice with linear size L=1024, we have …

The Probabilistic Zeta Function, Bret Benesh

Mathematics Faculty Publications

This paper is a summary of results on the P_G(s) function, which is the reciprocal of the probabilistic zeta function for finite groups. This function gives the probability that s randomly chosen elements generate a group G, and information about the structure of the group G is embedded in it.

Positive Solutions For A System Of Singular Second Order Nonlocal Boundary Value Problems, Naseer Ahmad Asif, Paul W. Eloe, Rahmat Ali Khan

Mathematics Faculty Publications

Sufficient conditions for the existence of positive solutions for a coupled system of nonlinear nonlocal boundary value problems of the type (see PDF for details) are obtained. The nonlinearities (see PDF) are continuous and may be singular at t = 0, t = 1, x = 0, or y = 0. … An example is provided to illustrate the results.

Periodic Solutions Of Neutral Delay Integral Equations Of Advanced Type, Muhammad Islam, Nasrin Sultana, James Booth

Mathematics Faculty Publications

We study the existence of continuous periodic solutions of a neutral delay integral equation of advanced type. In the analysis we employ three fixed point theorems: Banach, Krasnosel'skii, and Krasnosel'skii-Schaefer. Krasnosel'skii-Schaefer fixed point theorem requires an a priori bound on all solutions. We employ a Liapunov type method to obtain such bound.

Conformal Classification Of (K, Μ)-Contact Manifolds, Ramesh Sharma, Luc Vrancken

Mathematics Faculty Publications

First we improve a result of Tanno that says "If a conformal vector field on a contact metric manifold M is a strictly infinitesimal contact transformation, then it is an infinitesimal automorphism of M" by waiving the "strictness" in the hypothesis. Next, we prove that a (k, μ)-contact manifold admitting a non-Killing conformal vector field is either Sasakian or has k = –n – 1, μ = 1 in dimension > 3; and Sasakian or flat in dimension 3. In particular, we show that (i) among all compact simply connected (k, μ)-contact manifolds of dimension > 3, only the unit sphere S2n+1 …

A New Nonlinear Classifier With A Penalized Signed Fuzzy Measure Using Effective Genetic Algorithm, Julia Hua Fang, Maria L. Rizzo, Honggang Wang, Kimberly Espy, Zhenyuan Wang

Mathematics Faculty Publications

This paper proposes a new nonlinear classifier based on a generalized Choquet integral with signed fuzzy measures to enhance the classification accuracy and power by capturing all possible interactions among two or more attributes. This generalized approach was developed to address unsolved Choquet-integral classification issues such as allowing for flexible location of projection lines in n-dimensional space, automatic search for the least misclassification rate based on Choquet distance, and penalty on misclassified points. A special genetic algorithm is designed to implement this classification optimization with fast convergence. Both the numerical experiment and empirical case studies show that this generalized …

On Hyperplanes And Semispaces In Max-Min Convex Geometry, Viorel Nitica, Sergeĭ Sergeev

Mathematics Faculty Publications

No abstract provided.