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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Updated November 20, 2009
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