
		Talks in the Department of Mathematical Sciences (Spring 2007) Dept. of Mathematical Sciences Seminar Joseph Brennan, Florida Central University Wednesday, April 25, 2:45-3:45pm in Room RI-222 Title: Differential and Diophantine Equations Abstract: Often it is thought that differential equations and Diophantine equations represent opposite extremes with regard to solutions of equations. This talk will focus on the surprisingly close connections that exist between these systems. Following the work of Almkvist, Beukers, Edwards, and Zagier, it will be shown that the solutions of differential equations can provide methods for the solution of Diophantine equations. The problem that the talk will use to highlight these relationships is the one of finding the integral solutions of x^2+y^4=z^5. CSAM Seminar in Mathematical Sciences Dr. David Helfand, Chair Astronomy Department, Columbia University Thursday, April 19, 4:00pm in Sokol Seminar Room (Science Hall) Title: Universal Timekeeper: Reconstructing history atom by atom Abstract: By utilizing the basic building blocks of matter as imperturbable little clocks, we are now able to reconstruct in quantitative detail a remarkable range of human and natural events. From a detailed history of human diet and the Earth's climate to the events surrounding the origin of the Solar System and the history of the Universe itself, the universal timekeepers provide us with a precise chronology from the beginning of time to the moment humans emerge to contemplate such questions. Dept. of Mathematical Sciences Seminar Derek Habermas, SUNY Potsdam Wednesday, March 28, 2:45-3:45pm in Room RI-222 Title: Triangular Factorization and Symmetric Spaces or How to disassemble a sphere with matrices Abstract: Often mathematicians will study certain objects by analyzing the types of functions that "act" on that object. Whether we are talking about continuous functions acting on the real line, or the group D3 acting on a molecule of NH3 (ammonia), or orthogonal matrices acting on a sphere, studying sets of functions can help us understand the objects they act on. Another way to understand complicated objects is to see how they can be "built" out of simpler objects, like a house is built out of nails, boards, and sheet rock. We will see how matrices "act" on geometric objects (also known as Lie groups acting on symmetric spaces), and how factoring matrices can show how these symmetric spaces are built out of their "building blocks." Dept. of Mathematical Sciences Seminar Michael Siegel, NJIT Wednesday, March 7, 2:45-3:45pm in Room RI-222 Title: Calculation of complex singular solutions to the 3D incompressible Euler equations. Abstract: The incompressible Euler equations are the fundamental equations of fluid dynamics and describe the flow of inviscid, incompressible liquids. The question of finite time singularity formation for the 3D Euler equations from smooth initial data has been an important open problem of mathematics and physics for over 50 years. We describe an approach for the construction of singular solutions to the 3D Euler equations for complex initial data. The solutions take the form of complex traveling waves with imaginary wave speed. We also discuss a semi-analytic approach to the problem of Euler singularities based on numerical computation of the complex traveling wave solutions, followed by perturbation construction of a real solution. The perturbation analysis depends on a small amplitude of the singularity in the traveling wave solution; techniques for producing such a small amplitude are described. Dept. of Mathematical Sciences Seminar Jim Kennis, Columbia University Wednesday, February 21, 10:15-11:15am in Room RI-232 Title: Probabilistic Misconceptions Across Age and Gender Abstract: You are likely to be familiar with the probabilistic misconception called the gambler's fallacy. The gambler's fallacy or "negative recency effect" is the notion about the fairness of the laws of chance. A person that uses the gambler's fallacy when making probabilistic judgments believes that any deviation from the mean will be cancelled by a corresponding deviation in the opposite direction. For example, if a fair coin is tossed five times and the outcomes are HHHHH, a person that uses the gambler's fallacy would believe that a tail is the most likely outcome on the next coin toss, even though the laws of chance give a 50/50 chance for the next toss to be a head or a tail. The use of the gambler's fallacy is one of several misconceptions that were examined in this investigation. This presentation will discuss the results from a study that explored urban high school students' (n = 426) abilities in the probabilistic areas of simple and compound events, representativeness heuristic, availability heuristic, effect of sample size, conjunction of events, and conditional probability. Moreover, data were analyzed to examine the evolution of these misconceptions across age and gender differences; some of the results are quite surprising. Dept. of Mathematical Sciences Seminar Iwan Elsta, Ohio State University Tuesday, February 20, 11:30-12:30pm in Room RI-232 Title: College students' understanding of rational exponents Abstract: The study examines the explanations and justifications novice and expert college students have of the concept of rational and negative exponents. A conjecture for transforming the teaching of exponents is proposed. The central claim of the conjecture is that the teaching and learning of exponents can be improved through the study of the concepts of rate of growth and factors of multiplication and a thorough study of roots and powers of factors of multiplication. A teaching experiment is conducted to test and refine the conjecture. It is investigated how the student's construction and development of the concept of rational and negative exponents can be modeled, and what the role is of the laws of exponents in the process of developing rational and negative exponents. The preliminary results suggest that students do not base their understanding of rational or negative exponents primarily on the patterns of the laws of exponents. Memorized rules and cues from the notation of exponents are more evident in their explanations. The transformation of the rate of growth into a factor of multiplication and the recognition of the central role of these factors seem to be a difficult part of the learning process. The procedure for calculating decimal exponents seems a promising way to bring the components of the construction process together. Tentative conclusions are that the definitions of exponents can be integrated into a coherent concept that preserves the initial repeated multiplication concept as an intuitive foundation for learning exponents. The study of percentages, roots and powers has to be intensified and connected to exponents to enable young students to learn with understanding. Dept. of Mathematical Sciences Seminar Jason Williford, Worcester Polytechnic Institute Friday, February 16, 11:30-12:30pm in Room RI-222 Title: On the independence number of the Erdvs-Rinyi graphs and related graphs Abstract: In this presentation we will consider the question of determining the independence number of Erdvs-Rinyi graphs, graphs which can be thought of graph theoretical representations of the finite classical projective planes. This problem is equivalent to finding the maximum size of a family of mutually non-orthogonal 1-dimensional subspaces in a vector space of dimension three over the finite field of order q where q is a prime power. Using eigenvalue and constructive techniques, asymptotically sharp bounds will be given. The independence numbers of the related Projective Norm graphs and Unitary polarity graphs will also be discussed, and for the latter, exact values will be given. This is joint work with Dhruv Mubayi of the University of Illinois at Chicago. Dept. of Mathematical Sciences Seminar Allison McCulloch, Rutgers University Thursday, February 15, 1:00-2:15pm in Room RI-222 Title: Teddy bear or tool: Students' perspectives on graphing calculator use Abstract: This talk presents findings from a mixed methods study on students' perspectives of the impact that graphing calculator use has on their independent mathematical experiences. Data includes a survey of Advanced Placement Calculus students (n = 111) past experiences and common practices concerning graphing calculator use and in-depth case studies of six representative students chosen from the survey participants. The case studies are constructed from data collected from three interviews: a general background interview, a task-based interview, and a stimulated response follow-up interview. The participants report that the graphing calculator impacts their mathematical experiences in manners that fall into four categories: (1) it provides a means for changing the cognitive demand of a task, (2) it allows for actions that impact one's mathematical security, (3) it provides a means for mathematical play, and (4) it allows for more control over time management. Though these four categories are seemingly unrelated, they all have an affective impact on the students' mathematical experiences. Dept. of Mathematical Sciences Seminar Rachael Welder, Montana State University Tuesday, February 13, 11:30-12:30pm in Room RI-232 Title: Preservice Elementary Teachers Mathematical Content Knowledge of Prerequisite Algebra Skills Abstract: Increasingly more district and state high school graduation requirements are including algebra, creating the need for all students, no longer just the college-bound, to be algebra proficient. Despite algebra's significance, the National Assessment of Educational Progress shows a deficiency in the algebra achievement of U.S. students. Research suggests that for students to succeed in Algebra I (or an equivalent first algebra course), it is vital they master prerequisite algebra concepts throughout their K-8 mathematics education: (1) numbers (and numerical operations), (2) ratios/proportions, (3) the order of operations, (4) equality, (5) patterning, (6) algebraic symbolism (including letter usage), (7) algebraic equations, (8) functions, and (9) graphing. Research illustrates that student achievement is effected by teachers' knowledge, requiring elementary and middle school (K-8) teachers to have satisfactory knowledge of prerequisite algebra concepts. The theoretical framework for the knowledge for teaching mathematics built for this study suggests that the mathematical content knowledge needed for teaching consists of specialized content knowledge in addition to common content knowledge. Specialized mathematical content knowledge extends beyond solving mathematical problems to encompass how and why mathematical procedures work and an awareness of structuring and representing mathematical content for learners. The effects of an undergraduate mathematics content course for elementary education students on preservice teachers' common and specialized content knowledge of prerequisite algebra concepts was investigated, using a pre-experimental one-group pretest-posttest design. A quantitative, 51-item, multiple-choice instrument, developed specifically to measure both types of content knowledge with respect to prerequisite algebra concepts, was conscientiously constructed from the Learning Mathematics for Teaching Project's Content Knowledge for Teaching Mathematics Measures question bank. This instrument was administered to all students enrolled in Mathematics for Elementary Teachers I (n = 48), at Montana State University, during the first and last weeks of the Fall 2006 semester. Matched pairs t-tests, comparing pretest and posttest scores within the single sample, show significant gains (p = .000) in both common and specialized content knowledge and in all tested aspects of prerequisite algebra knowledge (numbers and equations/functions). Results also suggest a significant correlation (r = .716, p = .000) between the common and specialized content knowledge of preservice teachers. Lastly, a one-parameter linear model was constructed to predict the number of participants to incorrectly answer each item on the instrument, based on item difficulty. Items missed by notably more or less students than predicted by the linear model were identified and analyzed. The one item students performed better than expected on addressed common content knowledge regarding a linear graph. The set of troublesome items addressed both common and specialized content knowledge of reading, writing, and representing functions in a variety of contexts and using ratios to write and solve proportions. Dept. of Mathematical Sciences Seminar Amy Myers, Saint Joseph's University Friday, February 9, 11:15-12:15 in Room RI-222 Title: Bad Squares on Board Games Abstract: Imagine a board game (such as Monopoly) in which the roll of two dice determines the number of squares we move forward on a given turn. A particularly "bad" square (TWO hotels on Boardwalk!) looms n squares ahead of our current square. What is the probability that we land on it? In this talk we consider a variation of this problem, and extend it to both "one-sided" random walks and to compositions of integers that "avoid" other such compositions. Dept. of Mathematical Sciences Seminar Mike Ferrara, University of Colorado at Denver Thursday, February 8, 11:30-12:30 in Room RI-232 Title: Two Variants of the Turan Problem Abstract: Let H be a graph. The extremal (or Turan) number of H, denoted ex(n,H) is the minimum number of edges needed to assure that every graph G with n vertices and at least ex(n,H) edges contains H as a subgraph. The problem of determining ex(n,H) is called the Turan problem, and is the classic example of a problem in extremal graph theory. We will begin with a discussion of this problem and of extremal problems in general, including an investigation of Turan's classic 1941 theorem. Next, we will examine two notions from extremal theory that are closely related to the Turan problem: potentially H-graphic sequences and H-saturated graphs of minimum size. We will give results from both areas, focusing on some common techniques used to solve these types of problems. Dept. of Mathematical Sciences Seminar Jonathan Cutler, University of Nebraska-Lincoln Wednesday, February 7, 11:30-12:30 in Room RI-232 Title: On the number of complete bipartite subgraphs of a graph Abstract: Entropy methods have recently been employed by Kahn to bound the number of independent sets in a regular bipartite graph. Galvin and Tetali noted that these methods extend to bounding the number of homomorphisms from a regular bipartite graph. Our research grows out of a conjecture of Galvin and Kahn, and examines the very special case of homomorphisms into a fully-looped path of length two. We, in fact, look at the problem in the complementary graph, where a homomorphism then corresponds to a complete bipartite subgraph. Thus, our question becomes an extremal one: which graph on a given number of vertices and edges has the most complete bipartite subgraphs? We answer this and show that some rather interesting extremal behavior exists. This is joint work with Jamie Radcliffe. Dept. of Mathematical Sciences Seminar Demetrios Papageorgiou, New Jersey Institute of Technology Wednesday, January 24, 2:45-3:45pm in Room RI-222 Title: Mathematical Problems in Interfacial Electrohydrodynamics Abstract: In many applications that involve fluids separated by an interface, one is interested in identifying ways to either completely remove any instability and wavy oscillations (as in coating flows, for example), or to enhance the instability and hence heat or mass transfer, for instance. One way to achieve this is by imposing electric fields to the fluid systems. The modeling, analytical and computational aspects of such moving boundary problems are very challenging. I will present some novel micro and nano-fluidic applications that have been suggested for micro-circuit manufacture and which are based on interfacial electrohydrodynamic phenomena. Some experiments in micro-fluidic devices performed in our lab at NJIT will also be shown. I will then discuss the fundamental problem of the evolution of a fluid-air interface wetting a substrate when a vertical electric field acts. The methodology will be that of asymptotic analysis leading to nonlinear evolution equations, their numerical computation and the derivation of rigorous analytical results such as global existence, uniqueness and estimation of the dimension of inertial manifolds.

Talks in the Department of Mathematical Sciences (Spring 2007)