## Spring 2007

[1/29] [1/31] [2/1] [2/2] [2/6] [2/7] [2/8] [2/9] [2/27] [3/6] [3/21] [4/3] [4/24] [4/26] [4/30] [5/1] [5/2] [5/3]

**5/3: A Brief Introduction to Nonexpansive Mappings
**Brian Lins

Department of Mathematics

Rutgers University, New Brunswick

Nonexpansive mappings are an important tool in mathematical analysis. I will give a brief introduction to the theory of nonexpansive maps in normed vector spaces. In particular, I will discuss the fixed points and periodic points of these maps. Along the way, I will introduce several concepts from functional analysis. I will also mention some interesting unsolved problems and applications.

**5/2: Complex Dynamics Lacking Period 2 Orbits
**Rika Hagihara

Department of Mathematics

University of North Carolina at Chapel Hill

Most polynomials and rational maps can be easily solved for periodic points of all periods. In this talk we will study polynomials and rational maps that are missing period 2 points. We will introduce the parameter space of such maps, and investigate how the different kinds of dynamics are reflected in the parameter space. We will also compare the parameter space of the quadratic rational maps without period 2 points with the Mandelbrot set, the parameter space for all polynomials of degree 2, to see the similarities and difference.

**5/1: Honors thesis defense: Coefficient Shifting to Improve Glover's Linearization of the Heaviest k-Subgraph Problem
**Jared A. Lease ('07)

Dickinson College

Operations research (OR) is an interdisciplinary analytical approach to decision making. This thesis is concerned with a particular area of OR known as 0-1 quadratic programming. We model the heaviest k-subgraph problem as a 0-1 quadratic program and solve with commercial software. We develop a heuristic, based on the shifting of the quadratic objective coefficients, that attempts to improve the solution time. A detailed computational study shows that our heuristic reduces the solution time by 45-60% when compared to the standard modeling techniques.

**4/30: Honors thesis defense: The Exploration of a Dynamical System on a Circle
**Paul Winkler ('07)

Dickinson College

The itineraries of dynamical systems alone can tell us a great deal about the system itself. We will discuss the itineraries for the system known as the circle map and just what kind of information can be extracted from them. The uniqueness of these itineraries will also come into question as it is the motivating factor for studying them. The answer to some of these questions may be surprising.

**4/26: Honors thesis defense: Small World Structures in Evolved Neural Networks
**Scott McHugh ('07)

Dickinson College

Where do concepts from graph theory, sociology, biology, and computer science converge? What forces drive the small world phenomenon, wherein any two people on the planet may be connected through only 6 acquaintances? In order to answer these questions, a genetic algorithm was used to evolve abstract representations of the human brain (known as neural networks) over time in the hopes of developing a small world structure. In this talk I’ll be presenting my research into the role of small world networks in nature, as well as the motivations for uncovering the factors which lead to these networks. Further, I will discuss the experimental frameworks previously used to explore these questions, the approach taken here, and close with a presentation of my results thus far.

**4/24: Honors thesis defense: Dynamics of the Derivative of the Weierstrass Elliptic Function
**Jeff Goldsmith ('07)

Dickinson College

The Weierstrass elliptic function is a classic function from the history of mathematics, and has been a source of study since the late 1800’s. My research focused on the dynamics of the derivative of this function; that is, finding places where the derivative followed a repeating pattern, called a cycle, and where the derivative’s behavior was random or chaotic. I also looked for symmetries in the areas where the function showed these different kinds of behavior. Finally, I looked into the effect of the derivative’s parameter, called a lattice, on the behavior of the cycle: for some lattices, the derivative acts chaotically everywhere, and for other lattices, the derivative follows a pattern in many places.

**4/3: Shocks, Waves, Fans and the Method of Characteristics
**Dr. Linda Smolka

Department of Mathematics

Bucknell University

First order conservation laws can be used to model many complicated processes, such as, the adsorption of nutrients in the digestive tract, the flow of traffic or the flow of a thin film on a solid substrate. The unknown quantity in such a law depends on space and time, and the conservation law itself is a partial differential equation, that is, an equation containing partial derivatives in space and time of the unknown function. Using concepts from calculus and ordinary differential equations we show how to derive a traveling wave solution to a first order conservation law using the method of characteristics. We'll also discuss other possible solutions such as shocks and rarefaction fans and see an example of a conservation law for a thin film flow.

**3/21: Happy Birthday Pi--A Tribute to a Fascinating Number on its 300th Birthday (sort of)
**Alan Levine

Department of Mathematics

Franklin and Marshall College

It has been know for 4000 years that the ratio of the circumference to the diameter of a circle is a constant. Many attempts have been made to determine the "value" of that constant. We will look at a variety of those attempts, ranging from ancient Egypt to modern times. Along the way, we'll see many places in which pi makes an unexpected and fascinating appearance.

**3/6: Autonomic Computing Concepts and Directions
**Jim Whitmore

Executive IT Architect

Autonomic Computing - Self Managing Systems

As Information Technology business solutions increase in complexity, so does the supporting computing infrastructure. This talk will examine the challenges for deploying and managing computing infrastructure in business environments and describe how recent research and innovation in the area of autonomic computing proposes to address the situation.

**2/27: Careers in IT
**A panel of Alumni will discuss their careers in the IT field and provide insight into the industry now and where it is going.

Alumni panelists include:

Justin Parsley '04, a Computer Science and Political Science Major, currently a Systems Engineer at BNN Technologies

Cindy Harrison '96, a Political Science and Spanish Major, currently the Associate Director of IT at Cadwalader, Wickersham and Taft, LLP.

Kathy L. Clawson '98, a Computer Science Major, currently a Manager with Accenture, LTD

Amy Warnagiris '96, an English Major, currently with SAP as a Systems Administrator.

(Co-sponsored by the Career Center, Computer Science and Political Science Departments)

**2/9: Most Things Change, but One Remains the Same: Fixed Point Theorems and Shapes in Space
**Ryan Higginbottom

Department of Mathematics and Computer Science

Kalamazoo College

A function f has a fixed point at x=a if f(a)=a. The basic question we will ponder is this: Under what circumstances can you guarantee that a function will have a fixed point? We will begin by examining fixed points in the context of standard Calculus I functions. We will also discuss functions whose domain and range are slightly more complicated and how these complications affect the existence of a fixed point guarantee. In the course of this talk, we will come across the famous Brouwer Fixed Point Theorem and some of its applications. (If time permits, we may touch on other famous fixed-point theorems as well.) This talk should be very accessible to any student with a good memory of their introductory Calculus classes.

**2/8: Where in the World are All the Number Systems?
**Ulrica Wilson

Department of Mathematics

University of California, San Diego

In mathematics, much attention is given to organizing knowledge so that it is readily accessible to all those who wish to build upon its foundation. A primary goal for such organization is classifying the objects studied. In the 19th century topologists classified 2-dimensional manifolds and in recent years there has been a general consensus that the 3-dimensional manifolds have been classified. More recently algebraists have classified the finite simple groups. In this talk I will describe the problem of classifying some familiar and some not so familiar number systems.

**2/7: When Is Enough Enough?
**Sara Sprenkle

Department of Computer and Information Sciences

University of Delaware

Testing software is an arduous, time-consuming task: 40-60% of an application's development cycle is spent in testing. Software testers are torn between two conflicting goals: ensuring that the application meets high quality standards, i.e., is bug-free, and delivering the application quickly to demanding customers. To strike a balance between these conflicting demands, testers must decide--with confidence--when they have performed sufficient testing. In this presentation, we will discuss some of the issues in and approaches to deciding when testing is sufficient.

**2/6: Random Walks and the Gambler’s Ruin
**Jeff Forrester

Department of Mathematics and Computer Science

Dickinson College

The classical gambler’s ruin problem analyzes the chances of winning and the expected length of play in a (possibly unfair) game of chance between two individuals. This problem is a member of a class of probability models known as one-dimensional random walks. Such models have important applications in genetic algorithms, diffusion problems, population dynamics and financial systems. In this talk, the classical ruin problem is solved using difference equations with appropriate boundary conditions. The results are then extended to incorporate play against an infinitely rich adversary (casino) with some surprising results. Time permitting, random walks in 2 and 3 dimensions will be introduced and briefly discussed.

**2/2: Can Computers See?
**John MacCormick

Microsoft Research Silicon Valley

In the last few years it has become commonplace for computer users to do simple video and image processing on their own computers. But can a computer actually understand the images it processes? In short, can computers see? Most people would reply "obviously not" - and yet the field of computer vision research has made some remarkable strides in recent years, meaning that the answer to this question is no longer obvious. I will give a survey of some of the recent progress, including breakthroughs in object recognition, image segmentation, and visual tracking. We should also have time to do some actual computer science, coming to grips with one of the basic statistical tools of computer vision research: Bayes' rule. Finally, I will circle back to our original question and discuss whether or not computers really can see - now or in the future.

**2/1: Paradigms and Algorithms for Parallel Programming
**Mike Jochen

Department of Computer and Information Sciences

University of Delaware

There exist many problems which simply are not solvable within a reasonable amount of time on even the most powerful of single processor machines. Dividing these problems (either by data, or computation, or both) into smaller, more manageable sub-problems, can make the time to a solution more reasonable through computation in parallel on multiple processors. Through example and student participation, I will present an overview on designing an algorithm to solve a computation in parallel, focusing on the issues that often times have an adverse effect on run time performance.

**1/31: Projective Space
**Diane Davis

Department of Mathematics

Colorado State University

What happens if we allow infinity as just another point on the real number line? Suddenly weird things happen, like parallel lines intersecting and functions finally meeting their asymptotes. In projective space, mathematicians are able to include "points at infinity" in a coordinate system so that when we look from a different perspective, infinity becomes a point like any other. In this talk, we will introduce the projective line and projective plane. We will discuss how to make sense of them and why they help complete our intuition of what should happen if we could reach infinity.

**1/29: How many Twin Primes are there? The Search for Brun's Constant
**Dominic Klyve

Department of Mathematics

Dartmouth College

Determining the number of twin primes is one of the most easily-understood open questions in mathematics, and one of the hardest to answer. Nevertheless, there are many facts about twin primes that we can prove, and there are many interesting conjectures to explore. We shall examine several of these, and then focus on one fairly new method of determining how many twin primes there are. We shall show that the sum of the reciprocals of the twin primes converges, and then see how much we can say about the value of this sum, called Brun's Constant.