Spring 1999
 Jan 11 Joel V. Brawley Clemson University A War with No Battles! Jan 19 Shuhong Gao Clemson University A New Algorithm for Factoring Bivariate Polynomials Jan 26 Shuhong Gao Clemson University Continued Feb  2 Arthur Sherk University of Toronto SPREADS IN PG(5,3) Feb  9 Douglas R. Shier Clemson University Spanning Trees: Let Me Count the Ways Feb 16 Robert E. Jamison Clemson University Singular Sum Covers of Finite Cyclic Groups Feb 26 E. Rodney Canfield University of Georgia Large Antichains in Posets of Partitions Mar  2 Neil Calkin Clemson University Factorizing the Generalized Vandermonde Matrix Mar  16 None Spring break Mar  23 Mark A. Fitch Clemson University Slope Critical Configurations Mar  30 Donald Mills Clemson University Associative Rational Functions in Two Variables Apr    6 Mark A. Fitch Clemson University Slope Critical Configurations Apr  13 Kirsten Mackenzie-Fleming Central Michigan University An infinite family of non-embeddable  2-(2\lambda+2,\lambda+1,\lambda) Designs

January 11, 1999

TITLE:                    A War with No Battles!
SPEAKER:            Professor  Joel V. Brawley
AFFILIATION:  Clemson University

ABSTRACT:

In the August 21, 1994 issue of Parade Magazine'', a writer to the Ask Marilyn'' column of Marilyn Vos Savant posed a probability question related to an old card game called "War" that many of us played as children.  It is not surprising that the question was not answered by Marilyn because it turns out to be a generalization of the classical "problème des recontres"
which asks for the probability that a random permutation of the numbers  $1, 2, ..., n$ places no element in its own position.  In this talk, I will give an analytical solution to the Marilyn problem, discuss some related computational results, and describe a "poor man's" variation of the game whose solution is quite different.

This talk represents joint work with Doug Shier and Neil Calkin and will be presented (hopefully) at a level accessible to Junior-Senior undergraduate mathematics majors. It will last approximately 25 minutes and will be a trial run of the 25 minute presentation I will give in San Antonio on January 15 as a winner of the Hamio'' award.

January 19, 1999

TITLE:                    A New Algorithm for Factoring Bivariate Polynomials
SPEAKER:            Professor Shuhong Gao
AFFILIATION:  Clemson University

ABSTRACT:

We present a new algorithm for factoring bivariate polynomials which was just discovered during the past Christmas.  Our method is based on a differential equation, which gives a system of linear equations. We characterize the solution space of the linear system and show how to factor polynomials from a basis of the solution space.

The problem of factoring polynomials has been studied in various forms for hundreds of years. It is not only basic in many areas of
Mathematics but also important in computer algebra systems (such as Maple and Mathematica). Dramatic progresses on the algorithmic aspect have been made in the past few decays, particularly for univariate polynomials. For bivariate polynomials, many algorithms have been proposed using Hensel lifting, Newton approximation, interpolation, LLL lattice basis reduction or function spaces, but their running times are lagging far behind that for univariate polynomials. Our method is simple but yet gives a theory parallel to that of univariate polynomials and is faster than all previously known algorithms.

Febuary 2, 1999

SPEAKER:           Professor  Arthur Sherk
AFFILIATION:  University of Toronto

ABSTRACT:

A spread in PG(5,3), the projective space of dimension 5 over GF(3), is  a set of 28 planes that partition the space. We present a classification of such spreads, using a geometrical realization of spread sets.  The classification leads to the discovery of some spreads which are believed to be new, and also gives some promise of a complete enumeration of spreads in PG(5,3).

February 9, 1999

TITLE:                   Spanning Trees: Let Me Count the Ways
SPEAKER:             Professor Douglas R. Shier
AFFILIATION:   Clemson University

ABSTRACT:

This expository talk discusses a very simple counting problem with an equally simple answer. What is surprising, and perhaps remarkable, is the number of different ways to solve this problem. As a result, we are guided on a mini-tour of combinatorics, from inclusion-exclusion to recursion to determinants.

The talk is intended to be accessible to all students, regardless of background. The objective is to illustrate the variety of different approaches that can be used to analyze a seemingly  innocent problem.

February 16, 1999

TITLE:                    Singular Sum Covers of Finite Cyclic Groups
SPEAKER:             Professor Robert E. Jamison
AFFILIATION:   Clemson University

ABSTRACT:

A subset S of an abelian group A is a sum cover of A iff every element  of A is a sum of two elements of S.  That is, A = S+S where
S+S := {x+y: x and y are in S}.  For example, {0,1,2,3,4,5,10,15,20,25} is a sum cover of the cyclic group Z30.  A holomorphy  of A is a map of the form  x --> f(x) + t where f is an automorphism of A and t is any element of A.  It is easy to check that these maps are bijective and carry sum covers to sum covers.  It is natural to say that two sum covers are equivalent iff there is a holomorphy that carries one into the other. The question motivating this talk is this:

(Q)   In a cyclic group Zn, is every sum cover equivalent to
one that contains 0 and 1?

Although a complete answer is not known, it is known that the answer is "yes" for all n less than 2,310 but "no" for n = 2,310 = 2.3.5.7.11. The smallest value for which the question remains open is n = 10,010 = 2.5.7.11.13.

It is possible to use the Chinese Remainder Theorem to convert the  question into an essentially combinatorial problem.  This reformulation will be presented and the results will be presented in this general context.

February 26, 1999

TITLE:                    Large Antichains in Posets of Partitions
SPEAKER:             Professor  E. Rodney Canfield
AFFILIATION:   University of Georgia

ABSTRACT:
A partition of the set [n] = {1, 2, ..., n} is a collection of pairwise disjoint subsets, called blocks, whose union is [n]. The partitions of an n-element set form a poset under the relation of refinement.  The subcollection of partitions having exactly k blocks constitutes an antichain, that is a collection no two of whose elements are related by refinement.  The size of this antichain is S(n,k), a Stirling number of the second kind.   In this talk we will review some of the research concerning the relative size of the largest antichain, (for fixed n)
compared to the largest Stirling number, max_k S(n,k).  We will also consider the analogous problem for partitions of an integer.  (A partition of an integer $n$ is a multi-set of positive integers whose sum is n; the partial order of interest is again refinement.)  Several open questions will be included.

March 2, 1999

TITLE:                   Factorizing the Generalized Vandermonde Matrix
SPEAKER:            Professor Neil Calkin
AFFILIATION:   Clemson University

ABSTRACT:

It is well known that the ratio of the determinants of a generalized Vandermonde matrix and the corresponding Vandermonde matrix gives the Schur symmetric functions.  We prove that the generalized matrix factors, one of the factors being the Vandermonde matrix, and thus give a new derivation of the determinantal form for the Schur functions.

The material in the talk will be completely elementary: we will prove all facts we need about symmetric functions and Vandermonde matrices from scratch.

March 23, 1999

TITLE:                  Slope Critical Configurations
SPEAKERs:          Mark A. Fitch
AFFILIATION: Clemson University

ABSTRACTs:

In 1970, P. R. Scott conjectured, and later P. Ungar proved, that $n$  non-collinear points in the plane determine at least $n-1$ directions. The goal is to obtain information on the structure of the critical sets where the minimum of $n-1$ is attained. Such a set consists of an odd number of points distributed in a regular way onto spokes' radiating out from  a centrex' point. If the number of spoke-pairs is even, the set must be centrally symmetric. One method of studying a critical configuration $X$ is based on identifying a canonical list of $n-1$ distinct slopes in $X$. Since $X$ is critical, these must account for all the slopes in $X$.  All other slopes must be equated in some way with slopes in the list. Solutions to the resulting set of equations yield locally critical'  configurations and some critical configurations.

March 30, 1999

TITLE:                    Associative Rational Functions in Two Variables
SPEAKER:              Donald Mills
AFFILIATION:    Clemson University

ABSTRACT:

Let k be a field and k(x,y) the field of rational functions in two variables over k. For any R in k(x,y), we say R is associative if R(R(x,y),z)=R(x,R(y,z)) in k(x,y,z). Associative polynomials are known in the literature; in this talk, we determine all associative rational functions over k. The rational functions we present, when regarded as power series in the variables x and y, are special
manifestations of one dimensional formal group laws. We describe the nature of those groups whose binary operations are given by rational functions, and we show how one can use MAPLE to generate such groups.

This is joint work with Joel Brawley and Shuhong Gao.

April 6, 1999

TITLE:                    Slope Critical Configurations
SPEAKER:            Mark A. Fitch
AFFILIATION:   Clemson University

ABSTRACT:

A collection of n points, not all collinear, in the affine plane is  slope critical iff the points generate exactly n-1 slopes.

At present there are four infinite families of slope critical configurations known, plus 102 sporadic examples.  Of these, only four fail to be centrally symmetric. The goal of the present talk is to use algebraic and computational techniques to examine the possible structure of non-centrally symmetric slope critical configurations.

Such a set consists of an odd number of points. These points are distributed in a more or less regular way onto spokes' radiating out from a centrex' point. For every point P1 in the configuration there exists a point P2 such that the centrex z lies on the segment from P1 to P2 and divides it in the ratio \rho, where \rho is a constant depending on the configuration but not on the point or  spoke chosen.  Note that \rho = 1 iff the configuration is centrally symmetric.

Three consecutive spokes are called a wedge.' Using the slope matching technique  developed by Jamison, wedges for non-centrally symmetric configurations having  a single point on each spoke can be characterized by rational functions in terms of the parameter \rho. These characterizations in turn can be used to generate conditions on `locally' slope critical configurations which include slope critical configurations. A special application of these techniques is a family of double n-gons which are shown to be locally slope critical but, with two exceptions, not slope critical.

April 13, 1999

TITLE:                   An infinite family of non-embeddable  2-(2\lambda+2,\lambda+1,\lambda) Designs
SPEAKER:            Professor Kirsten Mackenzie-Fleming
AFFILIATION:   Central Michigan University

ABSTRACT:

The parameters 2-(2\lambda+2,\lambda+1,\lambda) are those of a residual Hadamard 2-(4\lambda+3,2\lambda+1,\lambda) design.  J.H. van Lint,  H.C.A. van Tilborg and J. R. Wiekma have shown that all  2-(2\lambda+2,\lambda+1,\lambda) designs with \lambda<=4 are embeddable. The existence of non-embeddable Hadamard 2-designs has been determined for the cases \lambda = 5 (J.H. van Lint), \lambda = 6 (V.D. Tonchev), and \lambda = 7 (V.D. Tonchev). In this talk an infinite family of  non-embeddable 2-(2\lambda+2,\lambda+1,\lambda) designs,  \lambda = 3(2^m) - 1, m >= 1 is constructed.