Wilfrid Laurier University
Waterloo Workshop on Computer Algebra
May 5 - 7, 2008
Wilfrid Laurier University CARGO Contact
Invited Speakers
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
    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
    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
    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
    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: 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
    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
    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
    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
    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
    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
    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

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