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Communications libres
Org: Peter Hoffman (Waterloo)
[PDF]
- KATHIE CAMERON, Wilfrid Laurier University
Recent Progress in Colouring Perfect Graphs
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Berge defined a graph to be perfect if for every induced subgraph, the
minimum number of colours required in a vertex colouring equals the
maximum number of vertices in a clique. In 2002, Chudnovsky,
Robertson, Seymour and Thomas proved Berge's 40-year-old Strong
Perfect Graph Conjecture: a graph is perfect if and only if it
contains no odd holes or odd antiholes. A "proof from the book" of
this result might be a combinatorial polytime algorithm, which for any
graph, finds a clique and colouring the same size, or else finds an
odd hole or an odd antihole (or some other easily recognizable
combinatorial obstruction to being perfect). In view of precedents,
such an algorithm might be simpler than the Chudnovsky-Seymour and
Cornuejols-Liu-Vuskovich algorithms for recognizing perfect graphs
since it could end up giving a clique and a colouring of the same size
in a non-perfect graph. I will report on some recent progress on
special classes of graphs in joint work with Jack Edmonds, Elaine
Eschen, Chinh Hoang and R. Sritharan.
- CRISTIAN ENACHE, Département de mathématiques et de statistique,
Université Laval, Cité universitaire, Québec (Québec)
G1K 7P4
Spatial decay bounds and continous dependence on the data for
the solution of a semilinear heat equation in a long
cylindrical region
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In this work we study a semilinear heat equation in a long cylindrical
region for which the far end and the lateral surface are held at zero
temperature and a nonzero temperature is applied at the near end. In
other words, the specific domain we consider is a finite cylinder
W: = D×[0,L], where D is a bounded convex domain in the
(x1,x2)-plane with smooth boundary ¶D Î C2,e, the generators of the cylinder are parallel to the
x3-axis and its length is L. The specific problem we consider
is the following initial boundary value problem
|
(1.1) |
ì
ï
ï
í
ï
ï
î
|
| |
| |
x Î ¶WL È ¶Wlat, t Î (0,T), |
|
| |
| |
|
|
| (1) |
where ¶W0 : = D ×{0}, ¶WL : = D×{L}, ¶Wlat : = ¶D ×(0,L).
We also assume that h(x1,x2,t) is a prescribed non-negative
function with h(x1,x2,0)=0 and f is a non-negative function
satisfying the following conditions
|
(1.2) \undersets® 0 |
lim
| |
f(s)
s
|
exists, f¢ (s) £ p(s) and f¢¢ (s) £ q(s) for s ³ 0, |
| (2) |
where p(s) and q(s) are some non-decreasing function of s. We
are interested on the spatial decay bounds for the solution of the
semilinear heat equation (1.1) and on its continous dependence with
respect to the data at the near end of the cylinder. Since the
solution u(x,t) of the problem (1.1) can blow up at some
point in space time, our aim is to derive sufficient conditions on the
data which will guarantee that the solution remains bounded and
moreover, under such conditions, we will obtain some explicit spatial
decay bounds for the solution, its cross-sectional derivatives and its
temporal derivative. We will also prove that the solution depends
continously on the data h(x1,x2,t) at the near end of the cylinder.
- JOSHUA MACARTHUR, Dalhousie University, Halifax, Nova Scotia B3H 3J5
Computation and Application of the Fundamental Invariants of
Vector Spaces of Conformal Killing Vectors
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Let C(M) be the vector space of conformal Killing vectors
defined on a pseudo-Riemannian manifold M of constant curvature.
Consider the action of the isometry group I(M) on C(M).
If we employ the method of infinitesimal generators, the problem of
finding fundamental invariants and covariants reduces to solving a
system of first order linear homogeneous PDEs.
In theory, the method of characteristics may be used to find
solutions, however in practice it proves ineffective due to the sheer
size of the system. Alternatively, if the invariants or covariants
may be represented by polynomials, the problem reduces further to
solving a system of linear equations.
The successful application of this alternative to find invariants for
all such C(M) in dimensions 3, 4 and 5 will be
discussed. In addition, the cases where these invariants have been
used to distinguish between equivalence classes in certain
C(M) will be shown.
- MARNI MISHNA, The Fields Institute for Mathematical Research
Holonomic sequences and walks in the quarter plane
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In this talk we will consider the nature of the generating series for
different families of walks in the quarter plane. In particular, we
consider combinatorial criteria which ensure the holonomy of the
counting sequences, that is, when sequences satisfying linear
recurrences with polynomial coefficients. We will also ponder the
nature of series associated to different classes of formal languages.
Work in collaboration with Mireille Bousquet-Melou and Mike Zabrocki.
- MARCO POLLANEN, Trent University, Peterborough, ON K9J 7B8
Algebraic Curves for Non-Uniform Pseudo-Random Sequence
Generation
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Pseudo-random numbers are a critical part of modern computing,
especially for use in simulations and cryptography, and consequently
there are a myriad of algorithms for creating uniform pseudo-random
sequences. However, many simulations ultimately require non-uniform
random sequences. In this talk we introduce a new method to directly
generate, without transformation of rejection, some non-uniform
pseudo-random sequences. This method is a group-theoretic analogue of
linear congruential pseudo-random number generation. We provide
examples of such sequences, involving computations in Jacobian groups
of plane algebraic curves, that have both good theoretical and
statistical properties.
- CHRISTIAN ROETTGER, Iowa State University, Math Dept., 472 Carver Hall, Ames,
IA 50011, USA
Periodic points classify some families of Markov shifts
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Consider the space XG of doubly-indexed sequences over a finite
abelian group G satisfying
for all integers s,t. The left and downward shift induce an action
of Z2 on XG. Recently, we could prove a conjecture by
Ward stating that the periodic point data of XG determine the group
G up to isomorphism. Our approach is to view XG as the set of
sequences annihilated by T-(S+1) where S,T stand for the two shift
actions, and to study algebraic sets of sequences via their
annihilators in the polynomial ring Z[S,T]. We will sketch
a proof of this theorem and show that our method extends to many other
spaces defined by linear recurrences over groups. Key words are
Galois Rings, Teichmuller systems and Wieferich primes. The p-adic
representation of binomial and multinomial coefficients comes into
play.
- NAHID SULTANA, Kobe University, Department of Mathematics, 1-1, Rokkodai,
Nada-ku, Kobe 657-8501, Japan
Explicit conformal parametrization of Delaunay surfaces in
space forms
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"Delaunay surfaces" are translationally-periodic constant mean
curvature (CMC) surfaces of revolution. We compute explicit conformal
parametrizations of Delaunay surfaces in each of the three space forms
Euclidean 3-space R3, spherical 3-space S3 and
hyperbolic 3-space H3 by using the generalized Weierstrass
type representation for CMC surfaces established by J. Dorfmeister,
F. Pedit and H. Wu. This method is commonly called the DPW method,
and is a method based on integrable systems techniques. We show that
these parametrizations are in full agreement with those of the more
classical approach. The DPW method is certainly not the simplest way
to derive such parametrizations, but the DPW gives a means to
construct other CMC surfaces (such as trinoids and perturbed Delaunay
surfaces) that the classical methods have not given. The Delaunay
surfaces are an important base for constructing other CMC surfaces
using the DPW method, so explicitly understanding how the DPW method
makes Delaunay surfaces is valuable.
- GRANT WOODS, Dept. of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2
Regular rings of functions with Tychonoff domain
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Let X denote a Tychonoff space, C(X) denote its ring of
real-valued continuous functions, and bX denote X re-topologized
by using its zero-sets as a base for the open sets. Then C(bX) is a
(von Neumann) regular ring. Let G(X) denote the smallest regular
subring of C(bX) that contains C(X). Then X is called an
RG-space if G(X) = C(bX).
In this talk we discuss some recent results concerning RG-spaces. Here
is a non-exhaustive sampling:
(a) Countably compact RG-spaces, and "small" pseudocompact
RG-spaces, must be compact (and hence scattered and of finite
Cantor-Bendixon degree).
(b) There exist almost compact spaces of Cantor-Bendixon
degree 2 that are not compact.
(c) An RG-space must have a dense subspace of "very weak
P-points" (i.e., points not in the closure of any
countable discrete set), but there exists a countable space that is
not RG but consists entirely of very weak P-points.
This talk summarizes joint research with M. Hrusak and R. Raphael.
- JIN YUE, Dalhousie University, Halifax, Nova Scotia, B3H 3J5
A moving frames technique and the invariant theory of Killing
tensors
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In this talk I will discuss an application of the inductive version of
the moving frames method due to Irina Kogan to the invariant theory of
Killing tensors. The method is successfully employed to solve the
problem of the determination of isometry group invariants (covariants)
of Killing tensors of arbitrary valence defined in the Minkowski
plane.
This is joint work with Roman Smirnov.
- QIJI ZHU, Western Michigan University
Helly's Intersection Theorem on Manifolds of Nonpositive
Curvature
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We give a generalization of the classical Helly's theorem on
intersection of convex sets in RN for the case of manifolds of
nonpositive curvature. In particular, we show that if any N+1 sets
from a family of closed convex sets on N-dimensional
Cartan-Hadamard manifold contain a common point, and at least one of
set is compact then all sets from this family contain a common point.
Our proof use a variational argument. In RN this proof is rather
straightforward yet it seems new to us. The generalization to
manifolds of nonpositive curvature relies on tools for nonsmooth
analysis on smooth manifolds that we developed recently.
This is joint research with Yuri Ledyaev and Jay Treiman.
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