Mathematics Commons^™

B ̈Acklund Transformations For Darboux Integrable Equations, Ian M. Anderson

Mathematics and Statistics Faculty Presentations

Lie symmetry reduction is typically viewed as an integration method for differential systems of finite type, that is, systems of ordinary differential equations.

In this talk I shall present two new, recent applications of Lie symmetry reduction to the study of partial differential equations.

The first gives a remarkably simple method for constructing B ̈acklund transformations.

The second also gives a simple, very general method for constructing Darboux integrable equations.

The combination of these result in a new method for constructing B ̈acklund transformations for Darboux integrable equations.

The utility of this group theoretic approach will be illustrated by a …

On The Classification Of Warped Product Einstein Metrics, Chenxu He, Peter Petersen, William Wylie

Mathematics - All Scholarship

In this paper we take the perspective introduced by Case-Shu-Wei of studying warped product Einstein metrics through the equation for the Ricci curvature of the base space. They call this equation on the base the m-Quasi Einstein equation, but we will also call it the (lambda,n+m)-Einstein equation. In this paper we extend the work of Case-Shu-Wei and some earlier work of Kim-Kim to allow the base to have non-empty boundary. This is a natural case to consider since a manifold without boundary often occurs as a warped product over a manifold with boundary, and in this case we get some …

Evolution Of Ligand Specificity In Vertebrate Corticosteroid Receptors, Adam S. Arterbery, Daniel J. Fergus, Elizabeth A. Fogarty, John Mayberry, David L. Deitcher, W. L. Kraus, Andrew H. Bass

College of the Pacific Faculty Articles

Background. Corticosteroid receptors include mineralocorticoid (MR) and glucocorticoid (GR) receptors. Teleost fishes have a single MR and duplicate GRs that show variable sensitivities to mineralocorticoids and glucocorticoids. How these receptors compare functionally to tetrapod MR and GR, and the evolutionary significance of maintaining two GRs, remains unclear. Results. We used up to seven steroids (including aldosterone, cortisol and 11-deoxycorticosterone [DOC]) to compare the ligand specificity of the ligand binding domains of corticosteroid receptors between a mammal (Mus musculus) and the midshipman fish (Porichthys notatus), a teleost model for steroid regulation of neural and behavioral plasticity. Variation in mineralocorticoid sensitivity was …

First Radar Observations In The Vicinity Of The Plasmapause Of Pulsed Ionospheric Flows Generated By Bursty Bulk Flows, N. A. Frissell, J. B. H. Baker, J. M. Ruohoniemi, L. B. N. Clausen, Z. C. Kale, I. J. Rae, L. Kepko, K. Oksavik, R. A. Greenwald, M. L. West

Department of Mathematics Faculty Scholarship and Creative Works

[1] Recent expansion of the SuperDARN network to mid-latitudes and the addition of a new high-time resolution mode provides new opportunities to observe mid-latitude ultra-low frequency waves and other ionospheric sub-auroral features at high temporal resolution. On 22 February 2008, the Blackstone SuperDARN radar and THEMIS ground magnetometers simultaneously observed substorm Pi2 pulsations. Similarities in measurements from the Blackstone radar and a magnetometer at Remus suggest a common generating mechanism. Cross-phase analysis of magnetometer data places these measurements at the ionospheric projection of the plasmapause, while fine spatial and temporal details of the radar data show evidence of field line …

Which Chessboards Have A Closed Knight's Tour Within The Rectangular Prism?, Joseph Demaio, Mathew Bindia

Faculty Articles

A closed knight's tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. In 1991 Schwenk completely classified the m x n rectangular chessboards that admit a closed knight's tour. In honor of the upcoming twentieth anniversary of the publication of Schwenk's paper, this article extends his result by classifying the i x j x k rectangular prisms that admit a closed knight's tour.

Convex Polyhedra Realizing Given Face Areas, Joseph O'Rourke

Computer Science: Faculty Publications

Given n ≥ 4 positive real numbers, we prove in this note that they are the face areas of a convex polyhedron if and only if the largest number is not more than the sum of the others.

On The Erdos-Straus Conjecture, Eugen J. Ionascu

Faculty Bibliography

Paul Erdos conjectured that for every n ∈ N, n ≥ 2, there exist a, b, c natural numbers, not necessarily distinct, so that 4 n = 1 a + 1 b + 1 c (see [3]). In this paper we prove an extension of Mordell’s theorem and formulate a conjecture which is stronger than Erdos’ conjecture.

Regular Tetrahedra Whose Vertices Have Integer Coordinates, Eugen J. Ionascu

Faculty Bibliography

In this paper we introduce theoretical arguments for constructing a procedure that allows one to find the number of all regular tetrahedra that have coordinates in the set {0, 1, ..., n}. The terms of this sequence are twice the values of the sequence A103158 in the Online Encyclopedia of Integer Sequences [16]. These results lead to the consideration of an infinite graph having fractal nature which is tightly connected to the set of orthogonal 3-by-3 matrices with rational coefficients. The vertices of this graph are the primitive integer solutions of the Diophantine equation a 2 + b 2 + …

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 …

Extending A Mips Simulator For Fpga Support, Pete Sanderson

Mathematics Faculty Scholarship

This project enriches computer organization education by providing another avenue for execution of MIPS code. The initial and existing avenue is the MARS simulator for MIPS code, a well-established open source product for educational environments, in use at universities and colleges worldwide (http://www.cs.missouristate.edu/MARS). The new avenue is a special adaptation of MIPS circuitry on a Field Programmable Gate Array (FPGA) platform. FPGAs such as the Altera DE2 are very attractive in an educational environment because of their flexibility, re-use, and low cost. The FPGA MIPS circuit includes VGA graphical output for real-time display of the MIPS machine state, including registers, …

Balanced Pod For Model Reduction Of Linear Pde Systems: Convergence Theory, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

We consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems. The algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations. The algorithm is related to the proper orthogonal decomposition, and it was first proposed for systems of ordinary differential equations by Rowley (Int. J. Bifurc. Chaos Appl. Sci. Eng. 15(3), 997-1013, 2005). For the convergence analysis, we consider the algorithm in terms of the Hankel operator of the system, rather than the product of the system Gramians as originally proposed by Rowley. For …

Balanced Pod For Linear Pde Robust Control Computations, John R. Singler, Belinda A. Batten

Mathematics and Statistics Faculty Research & Creative Works

A mathematical model of a physical system is never perfect; therefore, robust control laws are necessary for guaranteed stabilization of the nominal model and also "nearby" systems, including hopefully the actual physical system. We consider the computation of a robust control law for large-scale nite dimensional linear systems and a class of linear distributed parameter systems. The controller is robust with respect to left coprime factor perturbations of the nominal system. We present an algorithm based on balanced proper orthogonal decomposition to compute the nonstandard features of this robust control law. Convergence theory is given, and numerical results are presented …

A Model Based Feedback Controller For Wing-Twist Via Piezoceramic Actuation, John R. Singler, Belinda A. Batten

Mathematics and Statistics Faculty Research & Creative Works

In this paper we present a model for a rubber plate with piezoceramic actuators to represent a bioinspired flexible wing. Using a Galerkin based finite element approximation to the system, we compute a linear quadratic based tracking control for piezoelectric actuators placed along both leading and trailing edges. Using these piezoceramic devices, we demonstrate the effectiveness of model based feedback control in achieving a desired wing tip position; this modified shape is analogous to aircraft roll moment generation via wing twist.

Convergent Snapshot Algorithms For Infinite-Dimensional Lyapunov Equations, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

We consider two algorithms to approximate the solution Z of a class of stable operator Lyapunov equations of the form AZ + ZA* + BB* = 0. The algorithms utilize time snapshots of solutions of certain linear infinite-dimensional differential equations to construct the approximations. Matrix approximations of the operators a and B are not required and the algorithms are applicable as long as the rank of B is relatively small. The first algorithm produces an optimal low-rank approximate solution using proper orthogonal decomposition. The second algorithm approximates the product of the solution with a few vectors and can be implemented …

Sub-Supersolution Method In Variational Inequalities With Multivalued Operators Given By Integrals, Vy Khoi Le

Mathematics and Statistics Faculty Research & Creative Works

No abstract provided.

Tangential Extremal Principles For Finite And Infinite Systems Of Sets, I: Basic Theory, Boris S. Mordukhovich, Hung M. Phan

Mathematics Research Reports

In this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, unified under the name of tangential extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. The first part of the paper concerns the basic theory of tangential extremal principles while the second part presents applications to problems of semi-infinite programming and multiobjective optimization.

Theory Of "Art" : Qualitative In Nature And Quaintintative In Practice, Richard Merritt

Theses, Dissertations and Capstones

The primary focus of this work is to discuss the mechanical action of the Marshall Differential Analyzer ("ART") in a mathematical context. Differential Analyzers are primarily used to calculate the solutions to Differential Equations. A full account of the uses of Marshall's Differential Analyzer is given. Furthermore, a pure mathematical justification of how the machine integrates (mechanical integration) is provided via an application of the Riemann Integral. Many simple example problems are detailed ultimately leading up to the calculation of nonlinear problems, more specifically, a nonlinear oscillator.

Solutions To Dynamic Equations On Varying Times Scales, Sher B. Chhetri

Theses, Dissertations and Capstones

Note: The mathematical symbols could not be represented. See the abstract in the thesis for complete text.

Infinitesimal Time Scale Calculus, Tom Cuchta

Theses, Dissertations and Capstones

Calculus has historically been fragmented into multiple distinct theories such as differential calculus, difference calculus, quantum calculus, and many others. These theories are all about the concept of what it means to "change", but in various contexts. Time scales calculus (introduced by Stefan Hilger in 1988) is a synthesis and extension of all the various calculi into a single theory. Calculus was originally approached with "infinitely small numbers" which fell out of use because the use of these numbers could not be justified. In 1960, Abraham Robinson introduced hyperreal numbers, a justification for their use, and therefore the original approach …

Predictors Of Student Outcomes In Developmental Math At A Public Community And Technical College, Linda Darlene Hunt

Theses, Dissertations and Capstones

With the wide range of abilities of community college students, proper course placement is crucial. Therefore, having better predictors of success can help improve placement of students for their achievement. This study analyzed student predictors, instructor predictors, and classroom predictors in relation to student final exam score and student final grade in Elementary Algebra and Intermediate Algebra classes. Student predictors included gender, ACT math score, SAT math score, community college enrollment, math pretest score, and ASC grade. Instructor predictors included gender, employment status, Mozart music use, and ALEKS software use. Classroom predictors included time of day, number of class meetings …

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, …

Global Stability Of Complex-Valued Neural Networks On Time Scales, Martin Bohner, V. Sree Hari Rao, Suman Sanyal

Mathematics and Statistics Faculty Research & Creative Works

In this paper, activation dynamics of complex-valued neural networks are studied on general time scales. Besides presenting conditions guaranteeing the existence of a unique equilibrium pattern, its global exponential stability is discussed. Some numerical examples for different time scales are given in order to highlight the results. © 2011 Foundation for Scientific Research and Technological Innovation.

On The Equivariant Cohomology Of Rotation Groups And Stiefel Manifolds, William Kronholm

Mathematics

In this paper, we compute the RO(Z/2)-graded equivariant cohomology of rotation groups and Stiefel manifolds with particular involutions.

Topological Symmetry Groups Of K4r+3, Dwayne Chambers '12, Erica Flapan, John D. O'Brien

Pomona Faculty Publications and Research

We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of a complete graph of the form K4r+3^in S³.

A Space-Filling, Nonregular Tetrahedron, Margaret Cagle, Joyce Frost, Christine Latulippe, Darryl H. Yong

All HMC Faculty Publications and Research

This activity is an investigation of a special nonregular tetrahedron that can be arranged to fill space without leaving any internal gaps in the same way that certain planar figures tessellate the plane. These tetrahedra can be connected together with hinges to make fun and interesting puzzles. More background information can be found in the paper "An Amazing, Space-Filling, Non-Regular Tetrahedron" by Joyce Frost and Peg Cagle, published by the IAS/Park City Mathematics Institute (available at mathforum.org/pcmi/hstp/resources/dodeca/).

Ken Kunen: Algebraist, Michael Kinyon

Mathematics Preprint Series

Ken Kunen is justifiably best known for his work in set theory and topology. What I would guess many of his friends and students in those areas do not know is that Ken also did important work in algebra, especially in quasigroup and loop theory. In fact, I think it is not an exaggeration to say that his work in loop theory, both alone and in collaboration, revolutionized the field. In this paper, I would like to describe some of his accomplishments to nonspecialists. My point of view is personal, of course, and so I will give the most attention …

Excess Porteous, Coherent Porteous, And The Hyperelliptic Locus In M3, Thomas S. Bleier

Mathematics - Dissertations

In the moduli space of curves of genus 3, the locus of hyperelliptic curves forms a divisor, that is a closed subscheme of codimension 1. J. Harris and I. Morrison compute an expression for the class of this divisor, in the Chow ring of the moduli space, using a map of vector bundles and by applying the Thom-Porteous formula to determine an expression for a certain degeneracy locus of this map. One would like to extend their idea in order to compute an expression for the divisor associated to the closure of the hyperelliptic locus, in the Chow ring of …

Mixed Problems And Layer Potentials For Harmonic And Biharmonic Functions, Moises Venouziou

Mathematics - Dissertations

The mixed problem is to find a harmonic or biharmonic function having prescribed Dirichlet data on one part of the boundary and prescribed Neumann data on the remainder. One must make a choice as to the required boundary regularity of solutions. When only weak regularity conditions are imposed, the harmonic mixed problem has been solved on smooth domains in the plane by Wendland, Stephan, and Hsiao. Significant advances were later made on Lipschitz domains by Ott and Brown. The strain of requiring a square-integrable gradient on the boundary, however, forces a strong geometric restriction on the domain. Well-known counterexamples by …

Mathematical Knowledge For Teaching Teachers: The Case Of Multiplication And Division Of Fractions, Dana E. Olanoff

Mathematics - Dissertations

This study attempts to answer the question, What is the mathematical knowledge required by teachers of elementary mathematics content courses in the area of multiplication and division of fractions? Beginning in the mid-1980s, when Shulman (1986) introduced the idea of pedagogical content knowledge, researchers have been looking at the knowledge needed to teach in a variety of different content areas. One area that has garnered much of the research is that of mathematics. Researchers have developed frameworks for what they call mathematical knowledge for teaching, but there has been little work done looking at the knowledge requirements for teachers of …