Computing Singular Values with Lanczos Routines by Tim Eddo, CSUSM Math.
The Lanczos method is an iterative process that tridiagonalizes an n-by-n symmetric matrix. At each iteration, the generated tridiagonal matrix contains successively better approximations to the eigenvalues of the symmetric matrix. I will discuss various Lanczos-based algorithms that can be used to estimate the large singular values of an m-by-n matrix A. Each iteration of the Lanczos-based algorithm produces successively better approximations to the large singular values of the rectangular matrix. I will discuss some complications of Lanczos-based algorithms due to finite precision, along with some remedies that allow us to retain the fast and reliable convergence properties of Lanczos-based algorithms.

Can you hear me now? Transmitting information mathematically by Gina Sanders, CSUSM Math.
In modern society, information is wealth, and it's transmission is becoming an essential process. How can information be transmitted reliably across a 'noisy' channel where essential data may be lost or changed? Coding Theory deals mathematically with the design of error-correcting codes for the reliable transmission of data across noisy channels. In this talk I will define what a code is in a purely mathematical sense; give examples of general codes; define error-correcting codes and show what it means for a code to be perfect.

So, why do we divide by n-1? Standard deviation vs. Sample standard deviation by Tina Shinsato, CSUSM Math.
In almost any statistics book one can find formulas for the standard deviation, σ as well as the sample standard deviation, s. Although they appear to be similar, there is a slight difference. The sample standard deviation is divided by n-1 instead of n, where n represents the sample size. In practice, if the sample is large it really should not make a difference whether we divide by n or n - 1. But in theory, formulas do not magically appear without good reason. After setting up a bit of background on sampling, point estimates, and testing the validity of point estimates, the answer to the question, so, why do we divide by n-1 will be answered.

The Padres Playoff hopes by Amber Puha, CSUSM Math.
As the 2004 Major League Baseball season was winding down, North County Times Senior Sports Columnist Steve Scholfield was beginning to feel the optimism of a strong Padres rebound slip away. The Padres, who finished the 2003 season with the worst record in the entire National League, were actually still in contention for the wildcard as the 2004 season was coming to a close. Even though the Padres had improved enough to be wildcard contenders, Scholfield knew that their chances of making the playoffs were becoming slimmer by the day. Scholfield wanted to quantify that statement for his article in the NCT and contacted me to see if she could help. I used basic combinatorial probability theory to determined that, at that point in the season, the Padres had a 1 in 2,000 chance of making the playoffs. In the talk, I will do the following: explain what it means to be in the wildcard race; discuss what questions I asked Scholfield; explain how I set up the probability model; show how I did the calculations.

Blackjack and house exposure by Erin Bossemeyer, CSUSM Math.
There are numerous books on the theory of blackjack. Most of the books are in favor of the player, for example, counting cards, and betting strategies. What does a casino do if they want to know the odds in their favor? Hire a Math major! This summer I worked for a local casino calculating probabilities of the house's exposure. The casino wanted to raise the maximum bet from $2,000 to $5,000, and wanted to know how much money they were risking in doing so. In the talk I will discuss the way I answered the casino's question.

From postage stamps to Chicken McNuggets: A fast solution to an old integer programming problem by Stan Wagon, Macalester College.
The Frobenius number is easily described in terms of Chicken McNuggets from McDo nald's. One can purchase packs of 6, 9, or 20 McNuggets. So one can buy 35 McNuggets, but one cannot buy, say 13, or 43 of them. However, every number beyond 43 is representable. So 43 is called the Frobenius number of 6, 9, and 20.
In this talk I will show how the Frobenius problem can be reinterpreted as a shortest path problem in a certain symmetric graph. The symmetry can be used to develop algorithms that are very efficient at finding the Frobenius number and solving the equation with a particular target. (Joint work with Dale Beihoffer and Albert Nijenhuis.)
Tournament games and constructions by Chuck Buchwald, CSUSM Math.
Consider a game played on a graph G by two persons who alternately select distinct vertices v₁, v₂, .... such that, for i > 1, v_i is adjacent to v_i-1. The last player able to select a vertex wins. Does the player who moves first (Player I) or the player who moves second (Player II) have a winning strategy? If so, what is it? Besides answering this question, we aim to show some results for a similar game played on tournaments and a generalized game played on sets. The goals are 1) to determine if there is a winning strategy for either player, 2) to determine for which tournaments there is a wining strategy for Player I, 3) to determine for which tournaments there is a wining strategy for Player II, and 4) to describe such strategies.
In 1978, Dr. Reid conjectured that every finite non-empty set S of non-negative integers is the score set for some tournament. In fact, he proved this when |S| = 1, 2, 3. Later in 1984, M. Hager showed the claim was true when |S| = 4, 5 and finally in 1987 Yao Tian-xing proved the conjecture for any positive integer |S| by an arithmetical argument. In other words, given a set S = {a₁, a₂, ..., a_n}, we know there exists a tournament in which every vertex has a score chosen from this set. Furthermore we know that every integer from this set is the score of some vertex in this tournament. Our goal is to obtain constructions for these tournaments (independent of LandauÕs Theorem) for at least |S| = 2, 3.

Apathic colorings of graphs by Gina Sanders, CSUSM Math.
Graph coloring is one of the most active research areas in graph theory. A proper coloring of a graph G is a coloring of the vertices so that no edge is monochromatic. A refinement of such a coloring is to additionally require that the graph has no 2-colored path on 4 vertices as a subgraph -- this refinement is called an apathic coloring of G or a star-coloring of G. My talk will explain why apathic colorings are interesting to study and the history of the area. I will present known bounds and discuss interesting questions that remain open, including the questions I will attempt to answer in my thesis.

Does the golf handicap create a fair game? by Tina Shinsato, CSUSM Math.
Over the past century golf has become an obsession. With the popularity of any sport, the opportunity to gamble will usually come into play. It is clear that a match between a casual golfer and an avid golfer would not be fair. Therefore a method was created to equalize the game: the golf handicap. This opened the door for tournaments between amateurs as well side bets between neighbors thus leading to the arguable question; does the golf handicap create a fair game? Using data from the Cal State San Marcos Golf team I will present my current findings through the use of statistical tests. Background on point estimation and interval estimation will pave the way for hypothesis testing. We will develop the students' t-distribution and show how that can be used to draw conclusions for our hypothesis. In addition, the questions that arose while I collected data will be considered as extensions in which I hope to predict the winning percentages between two people based on their previous record.

M-ideals in Banach Function Algebras by Heydar Zahedani, CSUSM Math.
Alfsen and Effros introduced M-ideals for a Banach space and showed that the M-ideals behave in many ways like the closed ideals of a C*-algebra. Smith and Ward investigated the M-ideals structure of a Banach algebra and proved that M-ideals in a unital Banach algebra A are subalgebras, that M-ideals are exactly the closed two-sided ideals when A is a C*-algebra and if A is commutative then every M-ideal in A is a closed ideal. Hirsberg showed that the M-ideals in a uniform algebra are closed ideals of all functions vanishing on a p-set. Curtis and Figa-Talamanca introduced the strong hull notion and showed that the strong hull of a Banach function algebra exhibits many of the properties of peak sets for uniform algebras. We investigate relations between M-ideals and strong hulls in a Banach fuction algebra and give a characterization of Banach function algebras in which the closed ideals are M-ideals.

A model of pre-state warfare: you show me yours and I'll show you mine. by Robert Rider, CSUSM Economics.
We present a model of pre-state warfare, which is based on a team production problem. Bands of warriors engage in contests over some prize. When we examine prehistoric war we observe that much of this early conflict was ritualized display and fighting. This is very similar to observations of animal contests. Our primeval practice of warfare had close connection to the behavior of our animal ancestors. Although most disputes were settled in these display and fight conflicts, fighting could escalate. Most casualties, though, were inflicted by ambush, which would result when escalation occurred. We model this early warfare by examining the Nash equilibria of this game. We then examine a refinement of the Nash equilibrium by employing the concept of evolutionary stable strategy equilibrium. The latter solution concept comes from evolutionary biology and seems to be appropriate given the close connection between animal contests and human contests. Two factors that are keys to our results are the value of the prize and the individual marginal cost of fighting. The paper employs concepts from game theory, conflict theory, behavioral ecology (biology) and history. The paper is the first in a series of works. The ultimate objective of our agenda is to understand the boundary between pre-state warfare and state warfare, "military horizon." A crucial difference between the two is the presence of organized violence in the latter. We want to understand why this significant change occurred.

Analytic center cutting plane method: theory, extensions, and applications by Mohammad Oskoorouchi, CSUSM Business.
Many large-scale optimization problems can be cast as nondifferentiable optimization (NDO) and solved more efficiently by NDO techniques. Amongst several techniques for solving NDO problems, cutting plane methods have many advantages when applied to large models. The general idea of cutting plane algorithms is discussed. In particular, we present a cutting plane technique that uses the analytic center (ACCPM) as query points. The extension of ACCPM to nonpolyhedral models such as semidefinite programming is discussed. We present our computational result when ACCPM is implemented on some combinatorial optimization problems such as the Max-Cut problem.

On convergence of formal CR mappings by Robert Juhlin, UCSD Math.
CR geometry is the study of real submanifolds in complex space and mappings between such manifolds by holomorphic mappings of the space surrounding the manifolds. Given two real submanifolds in complex space, one natural question to ask is whether we can find a nontrivial holomorphic map such that the image of the first manifold (locally) lies in the second. If the manifolds are real-analytic, the condition for the existence of such a mapping can be written as a power-series identity. To answer this existence question, it may be advantageous to initially drop the convergence condition on the power series of the holomorphic mapping, leaving us with a formal power series identity. The corresponding mapping is called a formal mapping.
In this talk I will discuss the problem of finding a formal mapping between two real-analytic manifolds and some other instances where formal mappings occur. I will also discuss under which circumstances formal mappings are necessarily convergent, and I will introduce some of the main techniques for proving convergence.

A hybrid algorithm for computing singular values by Tim Eddo, CSUSM Math.
Abstract not provided.

Does the golf handicap create a fair game? by Tina Shinsato, CSUSM Math.
Over the past century golf has become an obsession. With the popularity of any sport, the opportunity to gamble will usually come into play. It is clear that a match between a casual golfer and an avid golfer would not be fair. Therefore a method was created to equalize the game: the golf handicap system. This opened the door for tournaments between amateurs as well side bets between neighbors thus leading to the arguable question; Does the golf handicap create a fair game? Using data from a local golf country club, the question of fairness will be addressed in terms of the player against the course rating and the player randomly paired with another player. Background on point estimation and interval estimation will pave the way for hypothesis testing. Through the use of the t-test, 1-proportion Z-test and Wilcoxon matched pairs test, inferences about the hanidcap system will be made based on the statistical anaylsis.