October 2, 2003
“The Best Way to Knock 'em Down”
Dr. Arthur Benjamin, Harvey Mudd College

“Knock 'em Down” is a game of dice that is so easy to learn that it is being played in classrooms around the world as a way to develop students' intuition about probability. However, as our analysis will show, lurking underneath this deceptively simple game are many surprising and highly unintuitive results.
Disclaimer: Professor Benjamin takes no responsibility for any scams or “get rich quick” schemes that students may learn by applying the ideas of this talk!

October 16, 2003
“The Spread of Branching Diffusion Under Stabilizing Drift.”
Dr. Olga Korosteleva, CSU Long Beach

A one-dimensional branching diffusion with a stabilizing drift is considered. We prove a theorem on the in-probability asymptotic behavior of the right frontier of the process over a time interval [0,t]. In her talk, the speaker assumes the knowledge of basic probability distributions and the definition of a Poisson process.

October 23, 2003
“How Graphics Illuminate Deep Ideas of Mathematics”
Dr. John de Pillis, UC Riverside

How Graphics Illuminate Deep Ideas of Mathematics
1. SPECIAL RELATIVITY: specifically, the following three paradoxes will be discussed:
(a) The twin paradox: Time travel into the future.
(b) The Train-in-the-Tunnel paradox: What happened to simultaneity?
(c) The pea-shooter paradox: why velocities do not add "correctly."
2. AXIOM of CHOICE (AC):
It has been said (e.g., in a review by Lester Dubins in the MAA Monthly in the 1950's) that Alfred Tarski thought the Axiom of Choice was rubbish. He therefore developed the Banach-Tarski paradox to prove that ridiculous results can arise using this "rubbish."
We present a weak form of B-T where a full proof is accessible.
3. MONTY HALL PROBLEM
There is a long history of the best strategy to use on "Let's Make a Deal." A contestant
must choose one of three doors, one of which hides a valuable prize. The odds of winning are 1 in 3. Now Monty Hall, the MC, opens an empty remaining door. (The odds
of winning are 1 in 2!?) Should the contestant switch to the remaining door? Does it matter? Hear Monty Hall's response to a mathematician who, in the 1970's, showed that if you always switch, your odds of winning grow to 2 out of 3!
4. WAVE PROPAGATION
How can a graphic show why the speed of a propagated wave in a rope does not
change its when you add more energy to the rope (increase the wave's frequency)?

October 30, 2003
“La Cupola di Santa Maria del Fiore”
Dr. Mario Martelli, Claremont McKenna College

Known as Brunelleschi's cupola, from the name of the Renaissance architect who designed it and supervised its realization from 1420 to 1436, it is regarded as one of the most beautiful and innovative constructions of all times. Brunelleschi's genius invented a method to build the octagonal cupola without centering. Afraid that a competitor would steal his idea, and replace him as the capomastro, Brunelleschi never talked to anyone or wrote anything about his method. He let the posterity struggle with understanding it. Nobody so far has been able to fully penetrate Brunelleschi's secret. There are however three aspects of the construction which may be illustrated today for the first time thanks to the joint efforts of Giuseppe Conti, a mathematician and former student of mine at the University of Firenze, Massimo Ricci, an architect from the same University and myself. I will describe the three ideas, illustrate their decisive role, and talk about the importance played by the work of the three mathematicians who assisted Brunelleschi.

November 13, 2003
“Transient Probability Functions of Birth-Death Processes”
Carrie Mortensen and Dr. Alan Krinik, Cal Poly Pomona

Transient probability functions of finite birth-death processes are determined using a new approach. This method uses randomization, the Cayley-Hamilton Theorem, linear recurrence relations, along with some useful numerical analysis results. Our approach generalizes to obtain transient probability functions of finite birth-death processes with catastrophes.
How many different paths are there that go from state 2 to state 3 in a total of nine steps if we move (stepwise) according to the transitions pictured in the following diagram?

A pretty, combinatoric solution of problems of this type is presented and discussed.

February 2, 2004
“A Model of Tumor and Lymphocyte Interactions”
Dr. Amy H. Lin, San Francisco State University

The interactions between a solid tumor and the immune system are described both prior to and after neovascularization by a predator-prey model, and predictions about tumor behavior in a host are made. Trajectory analysis of phase-plane portraits as well as standard perturbation analysis show that most system steady states are unstable but that stability is theoretically possible. Reasonable parameter value estimation enables meaningful analysis of system behavior, and Mathematica is used to simulate model
dynamics. The model accounts for many observed tumor behaviors, and regions of uncontrolled tumor growth, tumor extinction in finite time, and irreversible lymphocyte decline are found either analytically or numerically. A better understanding of tumor-immune dynamics is obtained, allowing for improved research on treatment specifically in the area of immunotherapy.

February 5, 2004
“Efficient and Accurate Methods for Computing Lyapunov Characteristic Exponents”
Dr. Hubertus von Bremen, USC

Lyapunov Characteristic Exponents (LCEs) play an important role in the study of nonlinear dynamical systems. They provide a way to characterize the asymptotic behavior of nonlinear dynamical systems by giving a measure of the exponential growth (or shrinkage) of perturbations about a nominal trajectory. LCEs are often used to establish the presence of chaotic motion. The increased interest in exploring the possibility of chaotic motions in all sorts of dynamical systems has spawned a concomitant interest in computational methods for determining LCEs, since most often, it is the occurrence of positive LCEs that signals the presence of chaotic motion. New approaches for the accurate and efficient computation of LCEs will be given for discrete and continuous dynamical systems. For discrete dynamical systems, an approach based on Householder reflector matrices will be presented. For continuous dynamical systems, a method based on the fact that an orthogonal matrix can be expressed as the matrix exponential of a skew symmetric matrix will be presented

February 12, 2004
“An Average Approach to Fluid Dynamics”
Mr. James P. Peirce, UC-Davis

The study of fluid motion has developed into a major field of mathematical research. A fundamental problem is the following: if you start out with a nice smooth vector field describing the flow of a fluid, it will often get complicated as turbulence develops. In three dimensions, nobody knows whether the solution exists for all time, or whether it develops singularities and becomes undefined. In fact, numerical evidence hints at the latter! So one would like to know whether solutions exist for all time and remain smooth - or at least find conditions under which this is the case. One approach is to model the large scale motion of the fluid while averaging the small, computationally unresolvable scales, of the classical equations. The heart of this talk will be the discussion of one such averaged model. A number of new results will be introduced and a broad description of the techniques used to prove the existence (sometimes global existence!) of solutions to these models will be given.

February 16, 2004
“The Inverse Scattering Problem in Anisotropic Media”
Ms. Borislava Gutarts, UCLA

Inverse Problems have found many applications in the last twenty years: from Medical Diagnostics to the Oil Exploration. In this talk two kinds of inverse problems will be defined: the Inverse Scattering Problem and the Inverse Boundary Value Problem. The interesting fact is that the 2-dimensional case is the hardest. I will state my uniqueness result for the Inverse Scattering Problem for Anisotropic Wave Equation in 2-D, at Fixed Energy. I will also outline some of the solved and open problems in the field of Inverse Scattering.

February 19, 2004
“Will We Elect Whom We Really Want?”
Dr. Donald Saari

Elections happen every day. But could it be that because of our voting procedures, the wrong person wins? This is an issue first studied by the French mathematicians in the 1780s. In this expository talk you will be shown that, in general, election outcomes need not reflect the views of the voters. Then, it is described how mathematics helps us find procedures that give legitimate outcomes. Be prepared to leave this lecture worrying about whether the correct person won in your last election.

April 8 , 2004
“Chen's Fundamental Inequalities in Submanifold Geometry”
Dr. Bogdan Suceava, CSU Fullerton

April 15 , 2004
“Proof and Problem Posing about Periodic Functions”
Dr. Greisy Winicki-Landman, Cal Poly Pomona

During the seminar I will present some episodes from a mathematics teacher's experience stressing her students' exposure to the value of asking questions and to the different roles played by proof in mathematics. It comes up as a way to struggle with a "distorted perspective of mathematical creativity as being always purely deductive" (De Villiers, 1997, p.15) and "the false impression sometimes created that mathematicians are only problems solvers who spend most of their time trying to solve already given problems" (ibid).

April 22 , 2004
“The Mathematics of Juggling”
Dr. Will Murray, CSU Long Beach

The art of juggling has been popular for thousands of years, but it wasonly recently that a natural mathematical notation arose to describejuggling patterns. Discovered simultaneously in three places (two inCalifornia!), site-swap notation has, surprisingly, led to the creation of many new juggling patterns. Using site-swap, Ron Graham and others proved a deep combinatorial theorem that the number of juggling patterns of period n with at most b balls is. After illustrating site-swap with juggling clubs and balls, I will give an elementary counting proof of this theorem that borrows ideas from the juggling. I will mention some possible extensions of the theorem that are still open.

April 29 , 2004
“Thomas Harriot's Treatise on Figurate Numbers, Finite Differences,
and Interpolation Formulas”
Dr. Janet Beery, University of Redlands

Thomas Harriot (1560-1621) may be best known as the navigator and scientist for Sir Walter Raleigh's 1585--1586 expedition to the Virginia Colony, but he also was the leading English mathematician of his day. Harriot made groundbreaking discoveries in a wide range of mathematical sciences, including algebra, geometry, navigation, astronomy, and optics. He published only one work during his lifetime, "A Briefe and True Report of the New Found Land of Virginia" (1588), but at his death left thousands of manuscript pages of mathematics, including at least two sets that appear to have been ready for press, a very complete theory of equations and a much shorter treatise entitled "De Numeris Triangularibus et inde De Progressionibus Arithmeticis." We shall examine the contents of this latter treatise, survey some of Harriot's other mathematical work, and review Harriot's very interesting life. Harriot's mathematical work is striking both in its content---he obtained many results generally credited to later mathematicians---and in its highly visual and symbolic presentation. Whether your mathematical interests are pure, applied, and/or educational, there is something in Harriot's work for you!

May 6, 2004
“Solitary waves in heterogeneous elastic materials”
Dr. Darryl Yong, Harvey Mudd College

Solitary waves are interesting nonlinear phenomena that have been found to play a role in particle physics, oceanography, optics and many other fields. After a bit of history of how the first solitary wave was observed, I will discuss how Randy LeVeque (University of Washington) and I have numerically observed solitary waves in heterogeneous nonlinear elastic materials. We also present some analytical evidence for these waves using multiple scale homogenization technique.

May 13, 2004
“Accessible Questions in Pebbling and Pegging”
Dr. Cindy Wyels, California Lutheran University

Suppose you have a graph with piles of pebbles on some of the vertices. You may move pebbles around by removing two pebbles from one vertex, then placing one pebble on an adjacent vertex (and throwing the second pebble away). How many pebbles do you need to start with to guarantee that you can move a pebble to any vertex? Does it matter how the pebbles were placed initially? Come explore graph pebbling and a related topic, graph pegging. You'll learn enough to investigate open research questions in this fun area of mathematics.

May 20, 2004
“Calculus and Planimeters”
Dr. Serban Raianu, CSU Dominguez Hills

According to the Merriam-Webster dictionary, a planimeter is ``an instrument for measuring the area of a plane figure by tracing its boundary line''. Even without knowing how a planimeter works, it is clear from the definition that the idea behind it is that one can compute the area of a figure just by ``walking'' on the boundary. For someone who has taken calculus, this immediately suggests Green's Theorem. The aim of this talk is to clarify why this principle works. We do this by using points of view from linear algebra to elementary plane geometry in order to obtain an intuitive justification for Green's Theorem. The talk is based on joint work with Paul Davis.

May 27, 2004
“The Mathematics of Peg Solitaire”
Dr. Amber Rosin, Cal Poly Pomona

We will consider the game peg solitaire, more commonly known as Hi-Q.  The instructions for the game claim that if you can end up with one peg in the center of the board, you are a genius, and if you end up with one peg anywhere else on the board, you are merely outstanding.  We will use the context of peg solitaire to introduce (or review as the case may be) the concept of a group.  We will then use a particular group, called the Klein four group, to show that the scoring for Hi-Q would be more accurate if it said:  If you end up with one peg in the center, you are a genius and if you end up with a peg anywhere else on the board, you are a moron.  We will aos consider different versions of the game on different boards.  Most importantly, there will be M&M's.

June 3, 2004
“How Well Can Random Walkers Rank College Football Teams?”
Dr. Mason Porter, Georgia Institute of Technology

We develop a one-parameter family of ranking systems for NCAA Division I-A football teams based on a collection of voters, each with a single vote, executing independent random walks on a network defined by the teams (vertices) and the games played (edges). The virtue of this class of ranking systems lies in the simplicity of its explanation. We discuss the statistical properties of the randomly walking voters and relate them to the community structure of the underlying network. We compare the results of these rankings for recent seasons with Bowl Championship Series standings. To better understand this ranking system, we also examine the asymptotic behavior of the aggregate of walkers. Finally, we consider possible generalizations to this ranking algorithm. This work was performed by Thomas Callaghan as an undergraduate research project under the supervision of Peter Mucha and me.