SCHOOL OF SCIENCE

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Tuesday, September 8, 2009

An Uncountable Number of Solutions for a Boundary Value Problem from Fluid Mechanics

This talk will investigate a boundary value problem governing Marangoni convection over a flat surface.  The problem involves a third order nonlinear ordinary differential equation, and the existence and properties of the solutions to this equation will be investigated using techniques from calculus.  The nature of the solutions will be shown to depend on a parameter, k, measuring the temperature gradient in the fluid.  For some values of k there will be a unique solution, for other values there will be precisely two solutions.  There will also be a range of k where uncountably many solutions exist.

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Tuesday, September 22, 2009

Analysis of Stagnation Flow Toward a Stretching Cylinder

In this talk, we will investigate a nonlinear boundary value problem governing fluid flow toward a stretching permeable cylinder.  We will prove the existence of a solution for all values of the Reynolds number and for both suction and injection.  We will also present uniqueness results in the case of a monotonically decreasing solution.  Finally, we will discuss a priori bounds on the skin friction.

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Tuesday, October 6, 2009

A Class of Nonlocal, Nonlinear, Elliptic Integral-Differential Equations  Part 1

In 2005, existence and uniqueness results for a nonlinear partial differential equation that arises in physical models of thermodynamical equilibrium via Coulomb potential was established. In this talk we discuss a numerical method that was developed for a special case of this equation. We first consider the existence and uniqueness of the analytic problem using a fixed point argument and the contraction mapping theorem.  Next, we evaluate the solution of the numerical problem via a finite difference scheme. From there, the existence and convergence of the approximate solution will be addressed as well as a uniqueness argument, which requires some additional restrictions. Finally, we conclude the work with some numerical examples where an interval-halving technique was implemented. How essential ideas from Calculus and fundamental analysis are used in this work will be mentioned throughout the talk.

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Tuesday, October 13, 2009

A Class of Nonlocal, Nonlinear, Elliptic Integral-Differential Equations  Part 2

This talk will discuss how the analysis of the equation discussed in Part I lead to the development of a numerical method for a natural extension problem. We begin by establishing a priori estimates and the existence and uniqueness of the solution to the nonlinear auxiliary problem via the Schauder fixed point theorem. We then prove the existence of the solution to the nonlinear auxiliary problem by defining a continuous compact mapping and utilizing a priori estimates. From this analysis, we then prove the existence and uniqueness of the problem defined above via the Brouwer fixed point theorem. Next, we analyze a discretization of the above problem and show that a solution to the nonlinear difference problem exists and is unique, and that the numerical procedure converges with minimal error. We conclude with some examples of the numerical process. As in Part I, how essential ideas from Calculus and fundamental analysis are used in this work will be mentioned throughout the talk.

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Tuesday, October 20, 2009

The Distance Between Two Partitions

This talk is motivated by a problem involving air traffic management.  The United States is partitioned into different sectors, where flight controllers manage particular sectors.  In order to save cost, these sectors change with time since there are less flights to manage in the middle of the night versus mid day.  A distance function was needed to tell how close two partitions were to one another.  In the talk, we will develop the Hausdorff distance and show how it was extended to the case of partitions.

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Thursday, October 29, 2009

The Isomorphism Between Two Simple Finite Groups

This talk will be accessible to students who have some linear (matrix) Algebra.  I will start by defining two simple groups, GL (3,2) and PSL (2,7).  I will use simple combinatorics to count the order of the two finite groups which is 168.  It is proven by some high power mathematical theorem that any two simple groups of order 168 are isomorphic.  I will demonstrate how to efficiently get 3 generators in each group and hence, a natural homomorphism between the two groups which turns out to be an isomorphism.

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Tuesday, November 3, 2009

A Nagata-like Theorem for Cp(X,Z)

The study of Cp(X), the ring of real-valued continuous functions endowed with the topology of pointwise convergence, became popular in the late 1960's, however, some of the basic relations were arrived at twenty years before that.  In 1949 Nagata released a paper containing a result that can be translated into the following: Cp(X) is topologically isomorphic to Cp(Y) if and only if X is homeomorphic to Y.  In this talk we will show the special case of Cp(X,Z), the ring of integer-valued continuous functions endowed with pointwise topology, and what is needed in the proof of the parallel result.    

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Tuesday, November 10, 2009

Train-tracks and Automorphisms of Free Groups

Free groups are a fundamental object in group theory and behave as a kind of noncommutative discrete vector space.  This correspondence shows that the theory of the automorphisms (symmetries) of free groups is akin to linear algebra, but with a noncommutative twist.  From the 1930's through the 1970's, automorphisms of free groups were mostly studied using algebraic and combinatorial methods. It was in the 1980's that Stallings and others noticed that these methods could be placed in a topological and geometric framework which made the proofs and techniques of previous research more transparent.   Since then, most research on automorphisms of free groups heavily relies on this topological framework.
 
In this talk, I will describe a portion of this theory, namely train-track structures as discovered by Bestvina and Handel.  In light of the analogy with linear algebra, train-tracks should be viewed as a Jordan canonical form of an automorphism.

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Tuesday, November 17, 2009

An Application of Uniform Integrability

A function is (Lebesgue) integrable over the real line if and only if for every positive epsilon there exists a number C dependent only on epsilon so that the measure of the set of inputs for which the output is greater than C is less than epsilon.  A family, K, of integrable functions is uniformly integrable whenever for each epsilon, a "uniform" C will work for every element of K.  It turns out that the Walsh-Fourier series are completely characterized by the presence of a uniformly integrable sub-sequence of its sequence of partial sums.  We will review some facts about uniform integrability, explore the Walsh Functions in the Paley ordering, and demonstrate the characterization.

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Tuesday, December 1, 2009

Perturbation methods have proven to be an invaluable analytical tool in applied mathematics.  These methods provide a systematic way on how to construct an approximation of the solution to a problem that is otherwise intractable.  Examples of problems in which these methods have been employed are viscous fluid flow, celestial mechanics, and nonlinear oscillations to name a few.
 
In this talk, I will introduce the fundamental ideas used in constructing asymptotic approximations to solutions of algebraic equations as well as differential equations.  I will present elementary examples that are amenable to the regular perturbation theory and then discuss when this theory fails, requiring the use of singular perturbation theory.  In particular, I will present the method of matched asymptotic expansion, first developed by Prandtl in 1904, to solve a problem in fluid mechanics.

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Tuesday, December 8, 2009

Two Topologies on Lattices

In this talk I will discuss some basic notions from lattice theory and  introduce a particular class of complete lattices, namely, frames.  Frames were introduced in the 1930’s and were commonly known as complete Brouwerian lattices at that time.  Later, in the 1970’s the theory of frames evolved as a generalization of topologies.  As a result, several topological concepts have been generalized in terms of frames.  The main goal will be to discuss the set of minimal prime elements of a frame endowed with two well-known topologies, they being the Zariski (or hull-kernel) topology and the inverse topology.  We will observe the interplay between these two topologies and talk about some properties that they satisfy.

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Friday, January 22, 2010

NAEP: Do Pre-Service Teachers Measure Up?

In acknowledging the importance of translating among mathematical representations, Romberg, Fennema and Carpenter (1993) emphasize the need for research that accentuates students and teachers’ understandings of translations among multiple representations of functions.  In fact, it is suggested that competence of translations consists of being able to operate within two perspectives:  process and  object.  The process perspective entails viewing a line (or an equation) as a set of individual points that are related in a fixed way (ordered pairs).  Object perspective entails viewing a line (or an equation) as an object that can be manipulated as a whole (Moschkovich, Arcavi & Schoenfeld, 1993). Research regarding students’ knowledge and or illustrations of process and object perspectives (e.g., Schoenfeld, Smith & Arcavi, 1993 and Knuth, 2000) provide insight into how students view algebraic and  graphical representations and how such ways of thinking may promote or inhibit their attempts in  making translations among representations.  While this information is considered important, such information concerning teachers’ knowledge is limited.   Accordingly, in this presentation, we will discuss solution methods of pre-service  mathematics teachers as they attempt problems that call for translations.

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Tuesday, January 27, 2010

A Study of Teacher Knowledge as Secondary Mathematics Teachers Use a New Technology

During this research presentation, I will describe the background, theoretical framework, methodology, and preliminary findings of my dissertation research  project. Effective use of technology to teach mathematics requires a teacher who is knowledgeable about the technology as well as how to integrate it during mathematics instruction (Kaput, 1992; Laborde, 2001; Mitchell, Bailey, & Monroe, 2007; Ruthven & Hennessy, 2002).  I qualitatively investigate the following research questions: (1) What pedagogical content knowledge (PCK) do secondary teachers draw from when they begin to implement a new technology in their mathematics instruction? and (2) What orientations do teachers hold about teaching mathematics with a new technology?

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Tuesday, February 2, 2010

Some Insights Into the Sheffer B-Type 1 Orthogonal Polynomial Sequences

In this talk we first discuss the importance of characterizing orthogonal polynomial sequences (OPS); namely, extracting OPS from generating functions. We then discuss some of the key elements involved in the characterization process and supplement this discussion with classical examples. From there, we briefly explain that in 1939, I.M. Sheffer proved that every polynomial sequence belongs to one and only one type, and extensively developed properties of the B-Type 0 polynomial sequences determining which sets are also orthogonal. Sheffer subsequently generalized his classification method to the case of arbitrary B-Type k, k=1,2,3,…, by constructing a generalized generating function. Although extensive research has been done on characterizing polynomial sequences, no analysis has yet been published on higher-order Sheffer classes (k>0). The crux of this talk is to present an overview of a preliminary analysis of a special case of the B- Type 1 (k=1) class, which is an extension of the B-Type 0 class, in order to determine which sets, if any, are also orthogonal. We conclude with a conjecture based on this work and a brief discussion of extensions and future research.

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Monday, February 8, 2010

Student Perspectives on Collaboration in their High School Mathematics Class

Recommendations from high school mathematics reform documents and curricula often include the use of collaborative approaches to learning, where students work together to generate solutions to tasks that are posed to the group rather than individuals.   However, teachers often struggle to implement these recommendations, resulting in situations where students merely copy each others' answers, fail to make mathematical progress while working together, or continue to depend on the teacher as the sole provider of mathematical knowledge.  In this presentation, I will describe a qualitative research project which investigates students perspectives on working collaboratively in one mathematics class, revealing some of their beliefs about mathematics, their goals for working together, and some of their reasons for assuming various roles during collaborative activity.  I will also share some conjectures about how the teachers strategies for structuring collaboration may have influenced their perspectives.

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Tuesday, February 9, 2010

Mathematica 7 in Education and Research

Please contact Daniel Galiffa (djg34@psu.edu or Extension 6090) for more information.

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Wednesday, February 10, 2010

Investing in our Teachers: What Types of Professional Development Lead to the Highest Student Gains in Mathematics Achievement?

It has been well documented that students’ math achievement in the United States is dreadfully below established standards; only 40% of 4th graders were rated as at least proficient on the National Assessment of Educational Progress (NAEP) in 2009. This is further illustrated by results of the TIMSS, which places American students behind their international counterparts and future global economic competition. To rectify this, much attention has been given to improving teacher quality through professional development. However, resources such as time and money are still limited. As such, having some evidence that professional development works and, especially, some understanding of what the focus of the professional development should be is highly imperative. This study addresses two questions, which build upon each other, in an attempt to fulfill this need: First, is there an effect of professional development on elementary students’ math achievement; and, second, if there is an effect, what particular professional foci lead to the greatest difference in student achievement? These questions will be answered through the use of meta-analytic techniques.

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Updated December 3, 2009
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