Department of Mathematics

West Chester University

**Mathematics Information**

Office: Room 101

25 University Avenue

West Chester, PA 19383

Phone (610) 436-2440

Fax (610) 738-0578

Email: Department Chair

Each Thursday there will be a mathematics seminar (usually in UNA 127 from 3:15-4:15), while colloquium talks will normally be on a Wednesday (usually in UNA 158 from 3:15-4:15).

These seminars/colloquium talks may be by visiting speakers, WCU faculty, or WCU students, and are open to all interested students and faculty.

Send an e-mail to jmclaughl@wcupa.edu, if you would like to be on the e-mail list to receive advance notice of upcoming talks.

Previous Semesters: Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Summer 2006, Spring 2006,

**Thursday, January 22nd, 2009**

3:15 - 4:15 pm in UNA 125

**Professor Lin Tan** (West Chester University)

**WHAT IS THE OMEGA PROCESS AFTER ALL?, Part II**

We will show the essence of Macmahon's Omega Process in his Combinatorial Analysis in an entirely elementary fashion, inspired by results from Diophantine equations and Diophantine inequalities, for $\Omega_=$ and $\Omega_{\geq}$ respectively, by Gordan (an invariant theorist) and Dickson (a number theorist), among others. We will use Macmahon's list, Jimmy's triangle problem and some example from Andrews, Paule and Reise as our launching pad to illustrate the idea, in particular what the denominators and numerators in the corresponding generating functions account for.

**Thursday, January 29th, 2009**

3:15 - 4:15 pm in UNA 125

**Professor Lin Tan** (West Chester University)

**WHAT IS THE OMEGA PROCESS AFTER ALL?, Part II**

We will show the essence of Macmahon's Omega Process in his Combinatorial Analysis in an entirely elementary fashion, inspired by results from Diophantine equations and Diophantine inequalities, for $\Omega_=$ and $\Omega_{\geq}$ respectively, by Gordan (an invariant theorist) and Dickson (a number theorist), among others. We will use Macmahon's list, Jimmy's triangle problem and some example from Andrews, Paule and Reise as our launching pad to illustrate the idea, in particular what the denominators and numerators in the corresponding generating functions account for.

**Wednsday, February 4th, 2009**

3:20 - 4:10 pm in UNA 158

**Professor Viorel Nitica** (West Chester University)

**RIGIDITY OF ABELIAN COCYCLES **

We will define the cohomology of a dynamical systems and will summarize several typical results.

**Thursday, February 5th, 2009**

3:15 - 4:15 pm in UNA 125

**Professor Lin Tan** (West Chester University)

**A Module approach to the Omega Process**

I am going to finish MacMahon's examples this week, opting for the approach via module structure, the approach of my choice at the moment.

**Wednsday, February 11th, 2009**

3:20 - 4:10 pm in UNA 158

**XImena Catepillan** (Millersville University) and

**Professor Waclaw Szymanski** (West Chester University)

**INCA CULTURE AND MATHEMATICS**

Ancient Inca, one of the Pre-Columbian civilizations, developed an advanced culture. A part of that culture was mathematics based on decimal system. They used mathematics, in particular, to build their 14,000 mile road structure and monumental architecture. After an introduction to the Inca culture some algorithms believed to be used by Inca to do computations with the help of a ``yupana" - an ancient calculating device - will be presented, as well as classroom activities for the course ``Mathematics in Non-European Cultures" for non-mathematics and science majors offered at Millersville University. All, including the students, are welcome.

**Thursday, February 12th, 2009**

6:00 pm in UNA 162

**Alex Kouassi, Ph.D.** (Group Director of Statistics, Schering-Plough Research Institute)

Clinical trials are an essential component of drug development. They are instrumental in providing patients in need with safe and effective therapies. But for a clinical trial to weed out unsafe or ineffective drug compounds, the clinical team must

- Select the proper endpoint and adequate design in order to answer as precisely as possible the intended clinical question. Make use of appropriate randomization and blindness methods to ensure baseline comparability and control allocation as well as assessment biases. The target population must be carefully chosen to ensure a high degree of homogeneity, and the treatment schedule must be well planned in order to detect acute toxicities with short duration.
- Adopt sound and valid statistical analysis methods. Ensure that interim analysis, if any, is properly planned with clear stopping rules and a well defined data monitoring committee. In addition, ensure that the handling of missing data is well specified and all multiplicity issues are addressed.

In this presentation, we discuss some basic clinical trial designs and examine some special topics that often arise during the conduct and analysis of clinical trials. We close with a few words on the importance of the design of a clinical trial as well as the role of the statistician in the conduct of the trial.

**Thursday, February 19th, 2009**

3:15 - 4:15 pm in UNA 125

**Professor Lin Tan** (West Chester University)

**A q-Analogue of the Omega Operator**

I will discribe a q-analogue of MacMahon's generating functions in which when all the variables (other than q) are set to 1, you get the Hilbert-Serre series. (We'll start with symmetric functions as an appetizer.)

**Wednsday, February 25th, 2009**

3:20 - 4:10 pm in UNA 125

**Professor Shiv Gupta** (West Chester University)

**Interesting Rational Triangles**

**Thursday, February 26th, 2009**

3:15 - 4:15 pm in UNA 125

**Professor Lin Tan** (West Chester University)

**The Omega Process and Continued Fractions.**

I will be talking about the continued fraction method of getting a partial fraction decomposition of the generating function (among two other methods) for treating the Omega when the coefficients of the Diophantine inequalities are large.

If there is time, I will talk about Omega corresponding to the inhomogeneous Diophantine inequalities/equations after that.

**Tuesday, March 10th, 2009 **

3:15 - 4:15 pm in UNA 125

**Professor Scott Parsell** (Butler University, Indianapolis)

** Analytic Methods for Diophantine Problems**

Wiles and Taylor recently proved the long-standing conjecture that for every integer $k \geq 3$ the Fermat equation $x^k+y^k=z^k$ has no integral solutions other than the obvious ones with $xyz=0$. This provides an example of the general philosophy that diophantine equations in few variables should have few (if any) non-trivial solutions. Here the notion of ``few variables" is viewed in relation to the degree of the problem---the Fermat equation of course has many non-trivial solutions when $k=2$. When the number of variables is sufficiently large in terms of the degree, one expects a diophantine equation to have many non-trivial solutions unless there is some obvious reason why it should not. We will describe an analytic method, originally developed by Hardy and Littlewood, that verifies this expectation for a large class of interesting problems. For example, the method can be used to show that every positive integer is the sum of a bounded number of $k$th powers and that every sufficiently large odd integer is the sum of three primes. We plan to describe the essential features and standard applications of the method and then highlight some recent developments and open problems in this area.

**Friday, March 13th, 2009**

3:15 - 4:15 pm in UNA 125

**Professor Miron Bekker** (University of Missouri-Kansas City)

**Positive Definite Kernels and Functions and Some Applications**

Positive definite kernels and functions play important role in many problems of pure and applied mathematics. In our talk we define notions of a positive definite kernel and a positive definite function and consider some of their applications as well as extension problems. Special attention is paid to positive definite generalized Toeplitz kernels.

**Thursday, March 19th, 2009**

3:15 - 4:15 pm in UNA 125

**Professor Fangyun Yang** (University of California, Riverside)

**Index Theory and Its Applications**

Given a linear partial differential system, usually it is very hard or impossible to find the solutions. The critical insight of Atiyah and Singer is to ask a slightly different question: how many solutions are there? And The Atiyah Singer Index Theorem gives a nice answer: the number of solutions depends on the shape (topology) of the region we are working on. In this talk, I will first explain Atiyah Singer Index Theorem. Then I will talk about some of its generalizations on manifolds with singularities.

**Monday, March 23rd, 2009**

3:15 - 4:15 pm in UNA 125

**Professor Silvius Klein** (University of California, Irvine)

**Index Theory and Its Applications**

I will describe the discrete quasi-periodic Schroedinger operator and some of its expected spectral properties (predicted in solid state physics for disordered systems that this operator models). I will mention some of the known results for the case of real analytic potential functions. Then, I will discuss my own results and research projects for the case of more general potential functions. I will also describe a simpler model (that of a one-dimensional quasi-crystal) that could become an undergraduate research project.

**Wednsday, March 25th, 2009**

3:20 - 4:10 pm in UNA 162

**Jean Pedersen** (Santa Clara University)

We will begin by showing how a systematic folding procedure, used on a straight strip of paper can produce approximations to regular N-gons, for any N ≥ 3. Braided models constructed from the folded strips of paper will be displayed to show the geometric utility of this folding procedure.

We will then follow the paper-folding ideas in a natural way to discover some amazing facts about numbers. In particular, our discussion will lead to a very elementary algorithm for calculating, for a given odd positive integer b, the smallest positive integer k such that b will exactly divide either 2 k +1 or 2 k —1. And the algorithm determines whether the "+" or the "—" applies. Following this idea further we will be able to discover an even more surprising fact about numbers -- that you will only learn about if you attend the talk.

The models will be taken apart at the end of the talk demonstrating my "proof by destruction" that the tetrahedron can be braided with 2 strips, the cube with 3 strips, the octahedron with 4 strips, the icosahedron with 5 strips, and the dodecahedron with 6 strips.

Jean Pedersen is Professor of Mathematics and Computer Science at Santa Clara University. She is the author (or co-author) of over 200 articles and 7 books. She was the 1997 recipient of a Distinguished Teaching Award from the Northern California Section of the Mathematical Association of America.

**For further information e-mail mfisher@wcupa.edu or sgupta@wcupa.edu**

**Thursday, April 9th, 2009**

3:15 - 4:15 pm in UNA 125

**Professor Lin Tan** (West Chester University)

**THE OMEGA PROCESS - THE CONCLUSION**

Professor Tan concludes his series of talks on the Omega operator by describing a "Master Theorem" that encompasses most of the identities Mac Mahon derives from the Omega process.

**Thursday, April 16th, 2009**

3:15 - 4:15 pm in UNA 125

**Professor Peter Zimmer** (West Chester University)

**An Introduction to Dirichlet Characters and L-functions**

Modular forms and automophric forms are examples of quantities that have both an Euler product and a functional equation. The study of modular forms and automorphic forms incorporate algebra, complex analysis, number theory and representation theory. To see how such an object comes about, we will first consider Dirichlet characters and L-functions. Dirichlet characters are representations of Abelian groups and L-functions are similar to Riemann's zeta function. We will introduce Dirichlet characters and L-functions and derive their Euler product and functional equation.

**Wednsday, April 22nd, 2009**

3:20 - 4:10 pm in UNA 162

**Doron Zeilberger** (Rutgers University)

In a recent fascinating article in American Scientist, Theodore Hill described a cointossing game whose pay-off is the ratio of heads to the total number of throws. If right now you have 5 heads and 3 tails, should you stop, collecting 5/8, or should you keep playing, hoping to get a better score? No one knows. [Joint work with Luis Medina]

Doron Zeilberger is a Professor of Mathematics at Rutgers University. He received his PhD from the Weizmann Institute of Science (Israel) in 1976. Professor Zeilberger has made important contributions to the fields of hypergeometric summation and q-Series. The so-called Wilf-Zeilberger pair and Zeilberger's algorithm are indispensable tools for summing hypergeometric series, and these techniques are used extensively in modern computer algebra software. He was also the first to prove the elusive result in combinatorial theory known as the alternating sign matrix conjecture, as well as a number of related propositions. Professor Zeilberger was a co-recipient of the 1998 Steele Prize of the American Mathematical Society for his research on hyper- geometric summation and the recipient of the 2004 Euler Medal of the Institute of Combinatorics and its Applications.