Physical Sciences and Mathematics Commons^™

Newton, Maclaurin, And The Authority Of Mathematics, Judith V. Grabiner

Pitzer Faculty Publications and Research

Sir Isaac Newton revolutionized physics and astronomy in his Principia. How did he do it? Would his method work on any area of inquiry, not only in science, but also about society and religion? We look at how some Newtonians, most notably Colin Maclaurin, combined sophisticated mathematical modeling and empirical data in what has come to be called the "Newtonian Style." We argue that this style was responsible not only for Maclaurin’s scientific success but for his ability to solve problems ranging from taxation to insurance to theology. We show how Maclaurin’s work strengthened the prestige of Newtonianism and …

Weighted Canonical Forms Of Nonlinear Single-Input Control Systems With Noncontrollable Linearization, Issa Amadou Tall, Witold Respondek

Miscellaneous (presentations, translations, interviews, etc)

We propose a weighted canonical form for single-input systems with noncontrollable first order approximation under the action of formal feedback transformations. This weighted canonical form is based on associating different weights to the linearly controllable and linearly noncontrollable parts of the system. We prove that two systems are formally feedback equivalent if and only if their weighted canonical forms coincide up to a diffeomorphism whose restriction to the linearly controllable part is identity.

Algorithms For Computing The Distributions Of Sums Of Discrete Random Variables, D. L. Evans, Lawrence Leemis

Arts & Sciences Articles

We present algorithms for computing the probability density function of the sum of two independent discrete random variables, along with an implementation of the algorithm in a computer algebra system. Some examples illustrate the utility of this algorithm.

Two High School Teachers’ Initial Use Of Geometers Sketchpad: Issues Of Implementation, Kathryn G. Shafer

Dissertations

This study documents two teachers' instructional practices as they learn and incorporate Geometer's Sketchpad (GSP) activities into their curriculum. The researcher (as a mentor) worked closely with two high school geometry teachers following a case study methodology. As a secondary goal, this study examined the use of a mentor as a professional development model to assist classroom teachers in the learning and implementation of Geometer'sSketchpad. Data were gathered through interviews, surveys, observations, classroom videotape, participant journal entries, as well as researcher field notes.

Results are reported in the areas of goals of instruction, method of instruction, role of the teacher, …

Generalised Surfaces In ℝ³, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

The correspondence between 2-parameter families of oriented lines in ℝ³ and surfaces in Tℙ¹ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences generated by global sections of Tℙ¹ are investigated, and a number of theorems are proven that generalise results for closed convex surfaces in ℝ³.

Primitive Ideals Of Semigroup Graded Rings, Hema Gopalakrishnan

Mathematics Faculty Publications

Prime ideals of strong semigroup graded rings have been characterized by Bell, Stalder and Teply for some important classes of semigroups. The prime ideals correspond to certain families of ideals of the component rings called prime families. In this paper, we define the notion of a primitive family and show that primitive ideals of such rings correspond to primitive families of ideals of the component rings.

A Multiple Linear Regression Analysis On Mathematics Placement At The University Of Tennessee, Knoxville, Steven K. Moore

Masters Theses

This paper looks at solving the least squares problem and then using that theory to solve a simple multiple regression analysis for placing students into mathematics courses at the University of Tennessee, Knoxville.

The first chapter is an introduction and a basic outline of the problem. Chapter 2 outlines the theory behind solving an actual least squares problem for both the general case and for the particular case of nonsingular matrices. Chapter 3 discusses assumptions that must be made when running the analysis.

In chapters 4 and 5 computers are discussed and software output is explained. Finally, Chapter 6 gives …

Teaching Fractions, Natalie Miles

Mahurin Honors College Capstone Experience/Thesis Projects

Because fractions are a vital part of mathematics instruction in schools and can be found in real life, it is important that all students understand and be able to utilize them. Teachers should choose strategies that will work best for their individual students, taking into account the various ways their students learn and the fraction knowledge students already possess. To effectively teach fractions, teachers must find ways to prompt student interest. Teachers can accomplish this through the use of technology. games. and manipulatives. Effective teachers must take into account the cultural diversity of their students. Teachers must be able to …

An Investigation Of Secondary Teachers’ Knowledge Of Rate Of Change In The Context Of Teaching A Standards-Based Curriculum, Jihwa Noh

Dissertations

This study investigated teachers' mathematical content knowledge and pedagogical content knowledge with respect to rate of change in the context of teaching a Standards-based high school mathematics curriculum that emphasizes rate of change as a central theme, the Core-Plus Mathematics Project (CPMP) materials. A framework was designed to provide a comprehensive guide for analyzing different aspects of rate of change knowledge incorporating existing frameworks relative to rate of change, NCTM recommendations described in Curriculum and Evaluation Standards for School Mathematics andPrinciples and Standards for School Mathematics (NCTM, 2000), and research related to pedagogical understanding of rate of change.

Data …

How (West) Hollywood Adds Up: A Queer Theoretical View Of Mathematics And Mathematicians In Film, Christopher D. Goff

College of the Pacific Faculty Presentations

No abstract provided.

Outerplanar Crossing Numbers, The Circular Arrangement Problem And Isoperimetric Functions, Eva Czabarka, Ondrej Sykora, Laszlo A. Szekely, Imrich Vrt'o

Faculty Publications

We extend the lower bound in [15] for the outerplanar crossing number (in other terminologies also called convex, circular and one-page book crossing number) to a more general setting. In this setting we can show a better lower bound for the outerplanar crossing number of hypercubes than the best lower bound for the planar crossing number. We exhibit further sequences of graphs, whose outerplanar crossing number exceeds by a factor of log n the planar crossing number of the graph. We study the circular arrangement problem, as a lower bound for the linear arrangement problem, in a general fashion. We …

Elemental Principles Of T-Topos, Goro Kato

Mathematics

In this paper, a sheaf-theoretic approach toward fundamental problems in quantum physics is made. For example, the particle-wave duality depends upon whether or not a presheaf is evaluated at a specified object. The t-topos theoretic interpretations of double-slit interference, uncertainty principle(s), and the EPR-type non-locality are given. As will be explained, there are more than one type of uncertainty principle: the absolute uncertainty principle coming from the direct limit object corresponding to the refinements of coverings, the uncertainty coming from a micromorphism of shortest observable states, and the uncertainty of the observation image. A sheaf theoretic approach …

Optimal Control Of Semilinear Evolution Inclusions Via Discrete Approximations, Boris S. Mordukhovich, Dong Wang

Mathematics Research Reports

This paper studies a Mayer type optimal control problem with general endpoint constraints for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces. First, we construct a sequence of discrete approximations to the original optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions …

A Conversation With R. Clifford Blair On The Occasion Of His Retirement, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

An interview was conducted on 23 November 2003 with R. Clifford Blair on the occasion on his retirement from the University of South Florida. This article is based on that interview. Biographical sketches and images of members of his academic genealogy are provided.

A Genetic Algorithm Hybrid For Constructing Optimal Response Surface Designs, David Drain, W. Matthew Carlyle, Douglas C. Montgomery, Connie Borror, Christine Anderson-Cook

Mathematics and Statistics Faculty Research & Creative Works

Hybrid heuristic optimization methods can discover efficient experiment designs in situations where traditional designs cannot be applied, exchange methods are ineffective, and simple heuristics like simulated annealing fail to find good solutions. One such heuristic hybrid is GASA (genetic algorithm-simulated annealing), developed to take advantage of the exploratory power of the genetic algorithm, while utilizing the local optimum exploitive properties of simulated annealing. the successful application of this method is demonstrated in a difficult design problem with multiple optimization criteria in an irregularly shaped design region. Copyright © 2004 John Wiley & Sons, Ltd.

The Shallow Water Equations In Lagrangian Coordinates, J. L. Mead

Mathematics Faculty Publications and Presentations

Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. However, using Lagrangian coordinates results in solving a highly nonlinear partial differential equation. The nonlinearity is mainly due to the Jacobian of the coordinate …

How Cellular Movement Determines The Collective Force Generated By The Dictyostelium Discoideum Slug, J. C. Dallon, H. G. Othmer

Faculty Publications

How the collective motion of cells in a biological tissue originates in the behavior of a collection of individuals, each of which responds to the chemical and mechanical signals it receives from neighbors, is still poorly understood. Here we study this question for a particular system, the slug stage of the cellular slime mold Dictyostelium discoideum. We investigate how cells in the interior of a migrating slug can effectively transmit stress to the substrate and thereby contribute to the overall motive force. Theoretical analysis suggests necessary conditions on the behavior of individual cells, and computational results shed light on experimental …

Unfolding Smooth Prismatoids, Nadia Benbernou, Patricia Cahn, Joseph O'Rourke

Computer Science: Faculty Publications

We define a notion for unfolding smooth, ruled surfaces, and prove that every smooth prismatoid (the convex hull of two smooth curves lying in parallel planes), has a nonoverlapping “volcano unfolding.” These unfoldings keep the base intact, unfold the sides outward, splayed around the base, and attach the top to the tip of some side rib. Our result answers a question for smooth prismatoids whose analog for polyhedral prismatoids remains unsolved.

Fibrations And Contact Structures, Hamidou Dathe, Philippe Rukimbira

Department of Mathematics and Statistics

We prove that a closed 3-dimensional manifold is a torus bundle over the circle if and only if it carries a closed nonsingular 1-form which is linearly deformable into contact forms.

A Spanning Tree Model For Khovanov Homology, Stephan Wehrli

Mathematics - All Scholarship

We use a spanning tree model to prove a result of E. S. Lee on the support of Khovanov homology of alternating knots.

Measurement Of The Generalized Forward Spin Polarizabilities Of The Neutron, M. Amarian, L. Auerbach, T. Averett, J. Berthot, P. Bertin, W. Bertozzi, T. Black, E. Brash, D. Brown, E. Burtin, J. Calarco, G. Cates, Z. Chai, J. P. Chen, Seonho Choi, E. Chudakov, E. Cisbani, C. W. De Jager, A. Deur, R. Disalvo, S. Dieterich, P. Djawotho, J. M. Finn, K. Fissum, H. Fonvieille, S. Frullani, H. Gao, J. Gao, F. Garibaldi, A. Gasparian, S. Gilad, R. Gilman, A. Glamazdin, C. Glashausser, E. Goldberg, J. Gomez, V. Gorbenko, J. O. Hansen, B. Hersman, R. Holmes, G. M. Huber, E. Hughes, B. Humensky, S. W. Korsch, K. Kramer, K. Kumar, G. Kumbartzki, M. Kuss, Enkeleida K. Lakuriqi, G. Laveissiere, J. Lerose, M. Liang, N. Liyanage, G. Lolos, S. Malov, J. Marroncle, K. Mccormick, R. Mckeown, Z. E. Meziani, R. Michaels, J. Mitchell, Z. Papandreou, T. Pavlin, G. G. Petratos, D. Pripstein, D. Prout, R. Ransome, Y. Roblin, D. Rowntree, M. Rvachev, F. Sabatie, A. Saha, K. Slifer, P. Souder, T. Saito, S. Strauch, R. Suleiman, K. Takahashi, S. Teijiro, L. Todor, H. Tsubota, H. Ueno, G. Urciuoli, R. Van Der Meer, P. Vernin, H. Voskanian, B. Wojtsekhowski, F. Xiong, W. Xu, J. C. Yang, B. Zhang, P. A. Zolnierczuk

Enkeleida K. Lakuriqi

The generalized forward spin polarizabilities γ0 and δLT f the neutron have been extracted for the first time in a Q2 range from 0.1 to 0.9 GeV2. Since γ0 is sensitive to nucleon resonances and δLT is insensitive to the Δ resonance, it is expected that the pair of forward spin polarizabilities should provide benchmark tests of the current understanding of the chiral dynamics of QCD. The new results on δLT how significant disagreement with chiral perturbation theory calculations, while the data for γ0 at low Q2 re in good agreement with a next-to-leading-order …

Voronoi Diagrams, Michael Mumm

The Mathematics Enthusiast

Suppose we have a finite number of distinct points in the plane. We refer to these points as sites. We wish to partition the plane into disjoint regions called cells, each of which contains exactly one site, so that all other points within a cell are closer to that cell's site than to any other site.

Tme Volume 1, Number 2

The Mathematics Enthusiast

No abstract provided.

Infinite Prandtl Number Limit Of Rayleigh-Bénard Convection, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

We rigorously justify the infinite Prandtl number model of convection as the limit of the Boussinesq approximation to the Rayleigh-Bénard convection as the Prandtl number approaches infinity. This is a singular limit problem involving an initial layer. © 2004 Wiley Periodicals, Inc.

Variational Stability And Marginal Functions Via Generalized Differentiation, Boris S. Mordukhovich, Nguyen Mau Nam

Mathematics Research Reports

Robust Lipschitzian properties of set-valued mappings and marginal functions play a crucial role in many aspects of variational analysis and its applications, especially for issues related to variational stability and optimizatiou. We develop an approach to variational stability based on generalized differentiation. The principal achievements of this paper include new results on coderivative calculus for set-valued mappings and singular subdifferentials of marginal functions in infinite dimensions with their extended applications to Lipschitzian stability. In this way we derive efficient conditions ensuring the preservation of Lipschitzian and related properties for set-valued mappings under various operations, with the exact bound/modulus estimates, as …

2004 (Fall), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2004 Fall Colloquium

Generating Sequences Of Clique-Symmetric Graphs Via Eulerian Digraphs, John P. Mcsorley, Thomas D. Porter

Articles and Preprints

Let {G_p1,G_p2, . . .} be an infinite sequence of graphs with G_pn having pn vertices. This sequence is called K_p-removable if G_p1 ≅ K_p, and G_pn − S ≅ G_p(n−1) for every n ≥ 2 and every vertex subset S of G_pn that induces a K_p. Each graph in such a sequence has a high degree of symmetry: every way of removing the vertices of any fixed number of disjoint K_p’s yields the same …

Author Information

The Mathematics Enthusiast

No abstract provided.

The Morley Trisector Theorem, Grant Swicegood

The Mathematics Enthusiast

This paper deals with an unannounced theorem by Frank Morley that he originally published amid a collection of other, more general, theorems. Having intrigued mathematicians for the past century, it is now simply referred to as Morley’s trisector theorem:

The three intersections of the angles of a triangle, lying near the three sides respectively, form an equilateral triangle.

Regular Polytopes, Jonathan Comes

The Mathematics Enthusiast

In the last proposition of the Elements Euclid proved that there are only five regular polyhedra, namely the tetrahedron, octahedron, icosahedron, cube, and dodecahedron. To show there can be no more than five he used the fact that in a polyhedra, the sum of the interior angles of the faces which meet at each vertex must be less than 360.