Research Seminars. The most important portion of the program is the research seminars, where professional mathematicians and statisticians contribute problems. At least one of the seminar leaders will be an African American; the other two will be Miami faculty from either the Department of Mathematics and Statistics or the Department of Systems Analysis. 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. The paper will be included in an online journal published by the Institute.

Selected Abstracts from Student Papers

The Algebra seminar students looked into the Cayley Addition Tables for Z_n.

Joy Coleman, Margaret Hall, Duke Hutchings and Megan Ruhnke worked on this problem in the summer of 1999. In the June-July 1999 American Mathematical Monthly, Hunter Snevily stated, "Few mathematical objects could be considered more simple than the Cayley addition table of Zn but we show that even these simple objects have some interesting yet unproved properties." He then proposed the following conjecture:  Let n be any positive odd integer. Then, for any k * {1, …, n}, the k * k submatrix of the Cayley addition table of Zn contains a latin transversal.

Megan & Duke at banquet

The Linear Programming seminar students have looked at different aspects of the traveling salesman problem. The traditional traveling salesman problem involves minimizing the total distance traveled for a salesman who needs to visit a given number of cities (visiting each city only once) and then return home.

Kathleen gives final presentation of her work with Rehka

Kathleen Bellino and Rehka Narasimhan examined the case where a company is interested in building a warehouse or distribution center that will deliver products to a set of customers on a regular basis. Many factors are involved when choosing a building site for a warehouse. For example, the cost of making deliveries from the location site is important. Thus, for a fixed number of clients, the company might want to know the cost of supplying them with a product from several potential warehouse sites. If there are n customers, this requires the solution of a traveling salesman problem to n cities from various starting points. This motivates one to consider the following problem. Let (ai,bi) denote the location of city i in the xy plane, and let H = (x,y) denote a movable point in the plane, which we will refer to as the home point. For each position of H, solve the traveling salesman problem, which starts at H, visits the n cities, and returns to H. Now, color the points in the plane so that two points get the same color if their traveling salesman tours visit the cities in the same order. Clearly, only a finite number of colors will be required for any n. Kathleen and Rehka determined some interesting properties of the regions defined by these colors.

James Williams looked at the traveling salesman problem of 4 cities in three-dimensional space with a movable homeport. By assigning a unique color to each of the optimal tours, and coloring the homepoint with the color of the tour that is optimal for that home point, for each homepoint in three-dimensional space, some interesting determinations can be made about the nature of the colored regions.

The Statistics seminar students have looked at a number of topics.

In 1999, Rebekkah Dann, Lynn Holmes, Rachel Kahlenberg and Bethany Lyles used the randomized response method to look at how many students might cheat on tests using graphing calculators. Graphing calculators allow students to perform tedious mathematical calculations with great ease and considerably shorten the amount of time needed to work some difficult problems. However, it is possible to store information, such as formulas or definitions, in graphing calculators and use this information to cheat on exams. In order to address this issue, an unrelated-question randomized response experiment was conducted at Miami University in Oxford, Ohio. To compare the percentages of students that have cheated on exams using graphing calculators among different departments, samples were taken from among mathematics, chemistry, and physics students. The unrelated-question randomized response method applies to this situation because some people may feel uncomfortable responding truthfully to direct statements regarding sensitive issues, such as cheating on exams. Relative to standard randomized response, this method yields a smaller variance. The smaller variance given by the unrelated-question randomized response method allows a shorter confidence interval to be constructed.

Rebekkah & Lynn give their final presentations