Physical Sciences and Mathematics Commons^™

Level Sets Of Functions And Symmetry Sets Of Surface Sections, André Diatta, Peter Giblin, Brendan Guilfoyle, Wilhelm Klingenberg

Conference Papers

We prove that the level sets of a real C s function of two variables near a non-degenerate critical point are of class C [s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at an elliptic or hyperbolic point, and in particular at an umbilic point. We go on to use the results to study symmetry sets of the planar sections. We also analyse one of the cases coming from a degenerate critical point, corresponding to an elliptic cusp of Gauss on a surface, where the …

An Excel Application For Valuing European Options With Monte Carlo Analysis, Tom Arnold, Stephen C. Henry

Finance Faculty Publications

By developing the basic intuition of how Monte Carlo simulation works within an Excel spreadsheet framework, this paper allows the undergraduate student to use Monte Carlo simulation techniques to price European style options without additional sophisticated software. Further, the skills and intuition developed provide the basis for much more complex simulation techniques.

Response To Gutstein Generalized- A Philosophical Debate, Eric Gutstein

The Mathematics Enthusiast

It is a pleasure, challenge, and an honor to respond to the thoughtful and innovative debate started by Braver, Micklus, Bradley, van Spronsen, Allen, & Campbell on teaching mathematics for social justice. They take seriously the issues in, and raise many interesting views about, my article, Teaching and Learning Mathematics for Social Justice in an Urban Latino School (JRME, January, 2003). I would like to respond to (connected) two points in particular: the relationship of functional to critical literacies, and the relationship of “critical thinking” in mathematics to learning mathematics for social justice. However, I would first like to clarify …

Meet The Authors

The Mathematics Enthusiast

No abstract provided.

Cardano's Solution To The Cubic: A Mathematical Soap Opera, Kacee Ballou

The Mathematics Enthusiast

Through this research into the solution of the cubic, I hope to learn the thoughts of mathematicians during this period of history. This knowledge must be conveyed in a way that is historically and mathematically accurate, yet comprehensible and readily available to use in a class of eighth grade math students. It is my wish that as we begin to delve into the world of algebraic and graphical solutions to the problems in the eighth grade text, students will be able to make a connection with this information to that they are currently using to solve their problems. They should …

Commentary On Ballou's Paper: Galois - The Myths And The Man, Catherine Degrandpre

The Mathematics Enthusiast

mathematics curriculum. As seen in her paper, the solution of the cubic essentially involves the use of a clever substitution in the general cubic in order to reduce it into a cubic without the square term, which in turn is factorable as a quadratic provided one makes another substitution. A natural question to ask ourselves, which mathematicians in the colorful theory of equations undoubtedly did as well, is how far can we push this technique of cle ver substitutions to solve higher degree equations. These substitutions are called Tschirnhaus transformations and have a pattern of the form x = y …

Action: Tell Me More Albert!, Jerry T. Baty, Virginia A.H. Mcclendon

The Mathematics Enthusiast

The new school year opens with the instructor cheerfully announcing to the students on the first day, “Good morning, class! Welcome to the fun of a new language called algebra”. He continues, “Today we are going to have a conversation about your success in this classroom. First, is the area of mathematics called algebra, necessary for your future?"

None of the students responds.

Call For Papers

The Mathematics Enthusiast

No abstract provided.

A Preface To Gutstein Generalized: A Philosophical Debate, Johnny W. Lott

The Mathematics Enthusiast

The National Council of Teachers of Mathematics (NCTM) has taken a very firm stand on the issue of equity in mathematics education. Not only is equity incorporated in the Curriculum and Evaluation Standards for School Mathematics (1989) and Principles and Standards for School Mathematics (2000), equity is the first plank of NCTM’s legislative platform.

Taxi Cab Geometry: History And Applications!, Chip Reinhardt

The Mathematics Enthusiast

We will explore three real life situations proposed in Eugene F. Kraus e's book Taxicab Geometry. First a dispatcher for Ideal City Police Department receives a report of an accident at X = (-1,4). There are two police cars located in the area. Car C is at (2,1) and car D is at (-1,-1). Which car should be sent? Second there are three high schools in Ideal city. Roosevelt at (2,1), Franklin at ( -3,-3) and Jefferson at (-6,-1). Draw in school district boundaries so that each student in Ideal City attends the school closet to them. For the third …

Tme Volume 2, Number 1

The Mathematics Enthusiast

No abstract provided.

Actions Of Sl(N,Z) On Homology Spheres, Kamlesh Parwani

Kamlesh Parwani

Any continuous action of SL(n,Z), where n > 2, on a r-dimensional mod 2 homology sphere factors through a finite group action if r < n - 1. In particular, any continuous action of SL(n+2,Z) on the n-dimensional sphere factors through a finite group action.

Structure Preserving Algorithms For Computing The Symplectic Singular Value Decom Position, Archara Chaiyakarn

Dissertations

In this thesis we develop two types of structure preserving Jacobi algorithms for com puting the symplectic singular value decomposition of real symplectic matrices and complex symplectic matrices. Unlike general purpose algorithms, these algorithms produce symplectic structure in all factors of the singular value decomposition.

Our first algorithm uses the relation between the singular value decomposition and the polar decomposition to reduce the problem of finding the symplectic singular value decomposition to th a t of calculating the structured spectral decomposition of a doubly structured m atrix. A Jacobi-like m ethod is developed to compute this doubly structured spectral decomposition. …

Stratification And Domination In Graphs And Digraphs, Ralucca M. Gera

Dissertations

In this thesis we combine the idea of stratification with the one of domination in graphs and digraphs, respectively.

A graph is 2-stratified if its vertex set is partitioned into two classes, where the vertices in one class are colored red and those in the other class are colored blue. Let F be a 2-stratified graph rooted at some blue vertex v . An F -coloring of a graph G is a red-blue coloring of the vertices of G in which every blue vertexu belongs to a copy of F rooted at u . The F -domination number γ …

A Topological Representation Theorem For Oriented Matroids, Jürgen Bokowski, Simon King, Sussane Mock, Ileana Streinu

Computer Science: Faculty Publications

We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Schönflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved.

Global Optimality Conditions In Mathematical Programming And Optimal Control, Pariwat Pacheenburawaa

Dissertations

We derive new first-order necessary and sufficient optimality conditions characterizing global minimizers in mathematical programming and optimal controlproblems. These conditions are based on level sets of an objective functional and they do not assume special structure of a problem (convexity, linearity, etc.). For a mathematical programming problem of minimization of a smooth functional on some compact convex set with equality nonlinear constraints, we derive first-order optimality conditions in the form of a generalized Lagrange multiplier rule. This rule should hold for any point from the level set of the objective functional corresponding to a global minimizer. We demonstrate that these …

Statistical Analysis Of Longitudinal And Multivariate Discrete Data, Deepak Mav

Mathematics & Statistics Theses & Dissertations

Correlated multivariate Poisson and binary variables occur naturally in medical, biological and epidemiological longitudinal studies. Modeling and simulating such variables is difficult because the correlations are restricted by the marginal means via Fréchet bounds in a complicated way. In this dissertation we will first discuss partially specified models and methods for estimating the regression and correlation parameters. We derive the asymptotic distributions of these parameter estimates. Using simulations based on extensions of the algorithm due to Sim (1993, Journal of Statistical Computation and Simulation, 47, pp. 1–10), we study the performance of these estimates using infeasibility, coverage probabilities of the …

The Straggling Green's Function Method For Ion Transport, Steven Andrew Walker

Mathematics & Statistics Theses & Dissertations

For many years work has been conducted on developing a concise theory and method for HZE ion transport capable of being validated in the laboratory. Previous attempts have ignored dispersion and energy downshift associated with nuclear fragmentation and energy and range straggling. Here we present a Green's function approach to ion transport that incorporates these missing elements. This work forms the basis for a new version of GRNTRN, a Green's function transport code. Comparisons of GRNTRN predictions and laboratory results for an ⁵⁶Fe ion beam with average energy at the target of one GeV/amu or more are presented for …

Maximizing The Chances Of A Color Match, Ramin Naimi, Roberto Pelayo

Ramin Naimi

No abstract provided.

Keys To Successful Learning, Michael J. White, '09, Leah A. Nillas

Leah A. Nillas

A nationwide push for the use of technology has influenced classroom teaching in the United States. This study examined the use of technology in elementary classrooms. Specific aspects of technology such as PowerPoint and word processing were examined in relation to developing skills in writing. Data from 46 fifth and third grade students as well as self-evaluation interviews with teachers were used to understand the overall comprehensive value of technology in teaching how to write. I expect my results to support previous research findings which assert that using computer technology in writing motivates students to do better work and is …

Some Sober Conceptions Of Mathematical Truth, Marco Panza

MPP Published Research

It is not sufficient to supply an instance of Tarski’s schema, ⌈“p” is true if and only if p⌉ for a certain statement in order to get a definition of truth for this statement and thus fix a truth-condition for it. A definition of the truth of a statement x of a language L is a bi-conditional whose two members are two statements of a meta-language L’. Tarski’s schema simply suggests that a definition of truth for a certain segment x of a language L consists in a statement of the form: ⌈v(x) is true if and only if τ(x)⌉, …

The Norm Of A Composition Operator With Linear Symbol Acting On The Dirichlet Space, Christopher Hammond

Mathematics Faculty Publications

We obtain a representation for the norm of a composition operator on the Dirichlet space induced by a map of the form φ(z)=az+b. We compare this result to an upper bound for ‖Cφ‖ that is valid whenever φ is univalent. Our work relies heavily on an adjoint formula recently discovered by Gallardo-Gutiérrez and Montes-Rodríguez.

Relative Equilibria For The Generalized Rigid Body, Antonio Hernandez-Garduno, Jeffrey K. Lawson, Jerrold E. Marsden

Mathematics Faculty Research

This paper gives necessary and sufficient conditions for the (n-dimensional) generalized free rigid body to be in a state of relative equilibrium. The conditions generalize those for the case of the three-dimensional free rigid body, namely that the body is in relative equilibrium if and only if its angular velocity and angular momentum align, that is, if the body rotates about one of its principal axes. For the n-dimensional rigid body in the Manakov formulation, these conditions have a similar interpretation. We use this result to state and prove a generalized Saari’s Conjecture (usually stated for the …

Combinatorial Proofs Of Fermat's, Lucas's, And Wilson's Theorems, Peter G. Anderson, Arthur T. Benjamin, Jeremy A. Rouse

All HMC Faculty Publications and Research

No abstract provided in this article.

Schwinger Pair Creation Of Kaluza-Klein Particles: Pair Creation Without Tunneling, Tamar Friedmann, Herman Verlinde

Mathematics Sciences: Faculty Publications

We study Schwinger pair creation of charged Kaluza-Klein (KK) particles from a static KK electric field. We find that the gravitational backreaction of the electric field on the geometry—which is incorporated via the electric KK-Melvin solution—prevents the electrostatic potential from overcoming the rest mass of the KK particles, thus impeding the tunneling mechanism which is often thought of as responsible for the pair creation. However, we find that pair creation still occurs with a finite rate formally similar to the classic Schwinger result, but via an apparently different mechanism, involving a combination of the Unruh effect and vacuum polarization due …

Error Analysis Of Variable Degree Mixed Methods For Elliptic Problems Via Hybridization, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new …

The Terwilliger Algebra Of An Almost-Bipartite P- And Q-Polynomial Association Scheme, John S. Caughman Iv, Mark S. Maclean, Paul M. Terwilliger

Mathematics and Statistics Faculty Publications and Presentations

Let Y denote a D-class symmetric association scheme with D≥3, and suppose Y is almost-bipartite P- and Q-polynomial. Let x denote a vertex of Y and let T=T(x) denote the corresponding Terwilliger algebra. We prove that any irreducible T-module W is both thin and dual thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T-module is determined by two parameters, the dual endpoint and diameter of W. We find a recurrence which gives the multiplicities with which the …

Binomial Identities With Pascalgt, Tyler J. Evans

Tyler Evans

No abstract provided.

Cryptographic And Combinatorial Properties Of Boolean Functions And S-Boxes., Kishan Chand Gupta Dr.

Doctoral Theses

In this thesis we study combinatorial aspects of Boolean functions and S-boxes with impor- tant cryptographic properties and construct new functions possesing such properties. These have possible applications in the design of private key (symmetric key) cryptosystems.Symmetric key cryptosystems are broadly divided into two classes.1. Stream Ciphers,2. Block Ciphers.Some recent proposals of stream ciphers are SNOW [37], SCREAM [52], TURING (98], MUGI (117), HBB (102], RABBIT (9), HELIX (38] and some proposals of block ciphers are DES, AES, RC6 [97), MARS (12], SERPENT (6], TWOFISH (104].In stream cipher cryptography a pseudorandom sequence of bits of length cqual to the message …

Morphological Tower: A Tool For Multi-Scale Image Processing., Susanta Mukhopadhyay Dr.

Doctoral Theses

An image is a recorded replication of natural scene or objects using suitable sensor and recording media. The visual quality of the recorded image may be enhanced using various types of low-level processing namely noise smoothing, contrast enhancement. The images of the same scene recorded by several sensors reveal more inforination but in their own respective ways. This advantage of multi-sensor imaging system is real- ized through the fusion of the multimodal images. One more important higher-level processing is segmentation where the image is decomposed into a set of meaningful regions ( e.g. objects and background). An image, in general, …