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May 5 - 7, 2008
Wilfrid Laurier University
Room N1002, Faculty of Science Building
WWCA 2008 Schedule of Talks
Invited Talks
- Georgy P. Egorychev, Russia
Title: The method of coefficients: applications in ring theory and the Collatz problem
- George Andrews, Pennsylvania State University, USA
Title: Old and New Thoughts on the Rogers-Ramanujan Identities
Abstract:
The Rogers-Ramanujan identities first came to prominence in the
early part of the 20th century when Ramanujan conjectured them and later
found them fully proved in an old and very much neglected paper by L.J.
Rogers.
In this talk, we shall provide some account of what happened
subsequently and will conclude with a look at recent discoveries and
ongoing research.
- Ira Gessel, Brandeis University, USA
Title: The Method of Coefficients
Abstract:
In his 1977 book (Integral Representation and the Computation of Combinatorial Sums)
and in other publications, G. P. Egorychev developed a method for evaluating
combinatorial sums by representing them as coefficients of generating functions.
In this talk I will discuss some of the applications of this method and its
relation to other approaches.
- Leonid Gurvits, Los Alamos National Laboratory, USA
Title:
Van der Waerden/Schrijver-Valiant like Conjectures
and Stable (aka Hyperbolic) Homogeneous Polynomials: One Theorem for all.
Abstract (linked pdf file)
- Michiel Hazewinkel, CWI, Netherlands
Title: Niceness theorems
Abstract (linked pdf file)
- I-Chiau Huang, Taiwan
Title: Power Series, differentials and residues
Abstract:
Motivated by Egorychev's work on integral representations,
algebraic realization of the analytic procedure obtaining residues has
been considered. The realization enhances the method of generating
functions by algebraic notions of differentials. Such a viewpoint
applies to formal power series rings and others.
- Peter Paule, RISC, Johannes Kepler University Linz, Austria
Title: Combinatorial Multi-Sums: Algorithmic Approaches from Egorychev to WZ
Abstract:
The talk presents a survey of algorithmic
methods for proving combinatorial identities involving
multiple-sums.
- Marko Petkovsek, University of Ljubljana, Slovenia
Title: Subanalytic hypergeometric and P-recursive summation
Abstract:
Abstract: Given a difference operator L, hypergeometric and P-recursive
summation algorithms such as Gosper's algorithm or the Accurate Summation
algorithm of Abramov and van Hoeij return a summing operator R such that
Ry is an antidifference of those y which satisfy Ly = 0. However,
this is to be understood in the algebraic sense, and care should be taken
before Ry is used in the Fundamental Theorem of Discrete Calculus to
compute a definite sum. We show here that correct results will be obtained
when the values of y are taken from a meromorphic function.
This is joint work with Sergei A. Abramov.
- Herbert Wilf, University of Pennsylvania, USA
Title: The permanent importance of the permanent function
Abstract:
Dr. Egorychev gave a proof of the celebrated van der Waerden
conjecture on the permanent of a doubly stochastic matrix nearly
thirty years ago. We will trace the further developments in this
important subject since then, including the conjecture of Minc and its
proof and refinements and their applications.
- Doron Zeilberger, Rutgers University, USA
Title: Integral Representations from Euler to Egorychev
Abstract:
Integral Representations are very powerful, especially
when handled by such giants as Lehonard Euler and Georgy Egorychev.
When interfaced with computers and the so-called Wilf-Zeilberger
method they get yet-more powerful.
Contributed Talks
- Kathie Cameron, Wilfrid Laurier University, Canada
Title: Coflow, Covering Vertices by Directed Circuits,
and a Lower Bound on the Stability Number of a Graph
Abstract:
Let G be a digraph, and for each edge e of G, let d(e) be a non-negative integer.
The capacity, d(C), of a dicircuit C means the sum of the d(e)'s of the edges in C.
A version of the Coflow Theorem (1982) says:
Coflow Theorem. The maximum size of a subset S of vertices of digraph G such that
each dicircuit C of G contains at most d(C) members of S equals the minimum
of the sum of the capacities of any subset H of dicircuits of G plus the number
of vertices of G which are not in a dicircuit of H.
When we proved the Coflow Theorem, we hoped to prove the following
conjecture made by Gallai in 1963:
Gallai's Conjecture. For any digraph G such that each edge and each vertex is in a dicircuit,
the maximum number of vertices in G no two of which are joined by an edge is at least as
big as the minimum number of dicircuits which together cover all the vertices.
However, we were missing the following:
Lemma. For any digraph G such that each edge and each vertex is in a dicircuit,
G contains a set F of edges such that G-F is acyclic and every edge of G
is in some dicircuit which contains exactly one edge of F.
We recently learned that Knuth proved this lemma in 1974. The Coflow Theorem together
with Knuth's Lemma provides a proof of Gallai's Conjecture different from that recently
published by Bessy and Thomassé.
This is joint work with Jack Edmonds.
- Angele Hamel, Wilfrid Laurier University, Canada
Title: Restricted Cauchy Identities
Abstract:
There are many classical identities dating back to Littlewood involving
sums of Schur functions in a single set of variables. The well-known
Cauchy identity also gives an expression for the sum of the product of
Schur functions in x variables and y variables. More recent identities
concern the sums of Schur functions with a restricted number of parts
(e.g. work of Regev, Gessel, and others) and there are connections to the
theory of random matrices. Here we derive a new identity for the sum of
Schur functions with a restricted number of parts and also generalize this
to some restricted Cauchy identities.
This is joint work with R.C. King.
- Chinh Hoang, Wilfrid Laurier University, Canada
Title: A note on k-colourability of P_5-free graphs
Abstract:
A polynomial time algorithm that determines whether or not, for a
fixed k, a P_5-free graph can be k-colored is presented in
this paper. If such a coloring exists, the algorithm will produce
a valid k-coloring.
This is joint work with Marcin Kaminski, Vadim Lozin, J. Sawada, and X. Shu.
- Ilias Kotsireas, Wilfrid Laurier University, Canada
Title: Approaches to the Hadamard conjecture and a question on Permanents of Hadamard matrices
Abstract:
A Hadamard matrix H in an n by n matrix with -1,+1 elements such that
H H^t = n In, where t denotes transposition and In denotes the n by n
identity matrix. Hadamard matrices arise in Combinatorics,
Coding Theory, Cryptography, Statistics, Telecommunications and other areas.
A necessary condition for the existence of a Hadamard matrix is that
n = 1, 2, or a multiple of 4. The sufficiency of this statement, i.e. that
for every n a multiple of 4 there exists a Hadamard matrix of order n is
the celebrated Hadamard conjecture. The smallest order n for which a
Hadamard matrix is still unknown, is n = 668. A number of approaches to
tackle the Hadamard conjecture have been suggested in the last 120 years.
We will discuss some old and new approaches to the Hadamard conjecture.
These approaches are based on sequences with zero periodic/non periodic
autocorrelation functions, Computational Algebra and Coding Theory. A well-known
open problem asks whether the permanent of a Hadamard matrix can vanish for n >= 4.
The answer is negative for 4 <= n <= 28.
- Eugene Zima, Wilfrid Laurier University, Canada
Title: Divide and sum
Abstract:
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