Mathematics Commons^™

Partitioning Regular Polygons Into Circular Pieces Ii: Nonconvex Partitions, Mirela Damian, Joseph O'Rourke

Computer Science: Faculty Publications

We explore optimal circular nonconvex partitions of regular k-gons. The circularity of a polygon is measured by its aspect ratio: the ratio of the radii of the smallest circumscribing circle to the largest inscribed disk. An optimal circular partition minimizes the maximum ratio over all pieces in the partition. We show that the equilateral triangle has an optimal 4-piece nonconvex partition, the square an optimal 13-piece nonconvex partition, and the pentagon has an optimal nonconvex partition with more than 20 thousand pieces. For hexagons and beyond, we provide a general algorithm that approaches optimality, but does not achieve it.

Hyperbolic Sets That Are Not Locally Maximal, Todd L. Fisher

Faculty Publications

This paper addresses the following topics relating to the structure of hyperbolic sets: First, hyperbolic sets that are not contained in locally maximal hyperbolic sets. Second, the existence of a Markov partition for a hyperbolic set. We construct new examples of hyperbolic sets which are not contained in locally maximal hyperbolic sets. The first example is robust under perturbations and can be constructed on any compact manifold of dimension greater than one. The second example is robust, topologically transitive, and constructed on a 4-dimensional manifold. The third example is volume preserving and constructed on R4. We show that every hyperbolic …

Wave Scattering From Infinite Cylindrical Obstacles Of Arbitrary Cross-Section, Matthew B. Weber

Theses and Dissertations

In this work the scattering of an incident plane wave propagating along a plane perpendicular to the xy-plane is studied. The wave is scattered from an infinitely long cylindrical object of arbitrary cross-section. Due to the arbitrary geometry of the obstacle, a finite differences numerical method is employed to approximate the solution of the scattering problems. The wave equation is expressed in terms of generalized curvilinear coordinates. Boundary conforming grids are generated using elliptic grid generators. Then, a explicit marching in time scheme is implemented over these grids. It is found that as time grows the numerical solution converges to …

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 …

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.

Strict Feedforward Form And Symmetries Of Nonlinear Control Systems, Witold Respondek, Issa Amadou Tall

Miscellaneous (presentations, translations, interviews, etc)

We establish a relation between strict feedforward form and symmetries of nonlinear control systems. We prove that a system is feedback equivalent to the strict feedforward form if and only if it gives rise to a sequence of systems, such that each element of the sequence, firstly, possesses an infinitesimal symmetry and, secondly, it is the factor system of the preceding one, i.e., is reduced from the preceding one by its symmetry. We also propose a strict feedforward normal form and prove that a smooth strict feedforward system can be smoothly brought to that form.

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.

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 …

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, …

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 …

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 …

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 …

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 …

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 …

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.

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.

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 …

Editorial, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

The Nature Of Proof In Today's Classroom. Book Review Of The Nature Of Proof, Erica Lane

The Mathematics Enthusiast

Harold Fawcett. The Nature of Proof. New York City: Bureau of Publications, Columbia University, 1938. Re-printed by the National Council of Teachers of Mathematics in 1995.

Throughout the world, children are taught new ideas and concepts in a variety of ways, and the advocates of the different approaches claim that their method is the most effective and profitable for the students. In addition to the different teaching styles, each individual learns the material in a distinct manner. Teachers, therefore, must be aware of the diverse techniques of both teaching and learning in an effort to optimize the learning experience …

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.

Mathematics Placement Test: Helping Students Succeed, Norma Rueda, Carole Sokolowski

Mathematics Faculty Publications

A study was conducted at Merrimack College in Massachusetts to compare the grades of students who took the recommended course as determined by their mathematics placement exam score and those who did not follow this recommendation. The goal was to decide whether the mathematics placement exam used at Merrimack College was effective in placing students in the appropriate mathematics class. During five years, first-year students who took a mathematics course in the fall semester were categorized into four groups: those who took the recommended course, those who took an easier course than recommended, those who took a course more difficult …

A (Not So) Complex Solution To A² + B² = Cⁿ, Arnold M. Adelberg, Arthur T. Benjamin, David I. Rudel '99

All HMC Faculty Publications and Research

No abstract provided in this article.

Random Walks On The Torus With Several Generators, Timothy Prescott '02, Francis E. Su

All HMC Faculty Publications and Research

Given n vectors {_i} ∈ [0, 1)^d, consider a random walk on the d-dimensional torus ^d = ℝ^d/ℤ^d generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*^k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*^k) ≥ C₁k^−n/2, where C₁ = C(n, d) is …

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.