abstracts05

Research Seminars are the most important portion of the SUMSRI program. Professional mathematicians and statisticians contribute problems and lead seminars. At least one of the seminar leaders is African American; the other two are Miami faculty from either the Department of Mathematics and Statistics. Before arriving in Oxford, the students are mailed a card listing each seminar area and prerequisites. They are asked to rank the topics and return the card. They are then assigned an area based on their choice. We hope to give each person his or her first choice, but this is not always possible. During the first four weeks of the program, each seminar director presents a series of lectures to the seminar participants in their area of expertise and assign research problems. These problems are challenging and at the same time easy enough for a very good undergraduate student to get partial results. Each student chooses a problem to work on and consults the appropriate professional. The seminar leaders are asked to meet with their students every day except Friday during the first four weeks and at least twice a week during the last three weeks. We strongly encourage students to work in groups. At the end of the program, the students give an oral presentation on their results and write a paper. These papers are included in an online Journal published by the Institute. (All mathematics between dollar signs is written in LaTeX mathematical typesetting language.)

Selected Papers in Number Theory, Abstract Algebra and Multivariate Statistics

Dr. Edray Goins, Kathleen Ansaldi,
Jennifer George, Kevin Mugo, Allison
Ford, and Lakeshia Legette (Graduate Assistant)

In Search of an 8: Rank Computations on a Family of Quartic Curves

The number theory group considered the family of elliptic curves $y^2=(1-x^2)(1-k^2x^2)$ for rational numbers $k/neq -1,0,1$. Every rational elliptic curve with torsion subgroup either $Z_2 \times Z_4$ or $Z_2 \times Z_8$ is birationally equivalent to this quartic curve for some k. We use this canonical form to search for such curves with large rank.
Our algorithm consists of the following steps. We compute a list of rational k by considering those associated to a given list of rational points (x,y). We then eliminate certain k by considering the associated 2-Selmer groups. Finally, we use Cremona’s mwrank to find the ranks. Using these steps, we found two elliptic curves with Mordell-Weil group $E(\mathbb Q) \backsimeq Z_2 \times Z_4 \times \mathbb Z^6$.

Line Graphs of Zero Divisor Graphs
Let $L(\Gamma(\mathbb Z_n))$ be the line graph of $\Gamma(\mathbb Z_n)$. The authors determine when $\overline{\Gamma(\mathbb Z_n)}$ and $L(\Gamma(\mathbb Z_n))$ are Eulerian. Moreover, studies are done on the diameter, girth, trees, planarity, center, eccentricity, clique, chromatic number, and the existence of Hamiltonian cycles for $L(\Gamma(\mathbb Z_n))$.

Dr. Reza Akhtar, Natalia Cordova, Clyde Gholston, Helen Hauser, Camil Aponte, Nathan Mims, Patrice Johnson, and Leigh Cobbs (Graduate Assistant)

The Structure of Zero-Divisor Graphs
Let $\Gamma(\mathbb Z_n)$ be the zero-divisor graph whose vertices are the nonzero zero-divisors of $\mathbb Z_n$, and such that two vertices u, v are adjacent if n divides uv. Here, the authors investigate the size of the maximum clique in $\Gamma(\mathbb Z_n)$. This leads to results concerning a conjecture posed by S. Hedetniemi, the core of $\Gamma(\mathbb Z_n)$, vertex colorings of $\Gamma(\mathbb Z_n)$ and $\overline{\Gamma(\mathbb Z_n)}$, and values of n for which $\overline{\Gamma(\mathbb Z_n)}$ is Hamiltonian. Additional work is done to determine the cases in which $\Gamma(\mathbb Z_n)$ is Eulerian.

Dr. Vasant Waikar, AdriAnne Demski, Joshua Svenson, Janelle Jones, Monique Owens, and Shenek Heyward (Graduate Assistant)

A Multivariate Statistical Analysis of Female Empowerment

As women of the world struggle for equality there is a need for ways of measuring progress. We explore the empowerment of women using multivariate statistical techniques such as factor analysis and discriminant analysis. We hope to classify countries into two populations, one where women are empowered and the other where women are not. We simplify this process by reducing the dimensionality of the data from 13 variables to a smaller collection of underlying factors.

A Multivariate Statistical Analysis of Substance Abuse in the United States

Where do the major drug problems occur in this country among the states? How are social and economic factors related to substance abuse in the states? We approach these questions with multivariate statistics. By using factor analysis, we distinguish the underlying factors of a collection of variables related to substance abuse. With discriminant analysis, we design a rule for classifying states as either having a major drug problem or minor drug problem.