


Contributed Papers Session / Communications libres Org: William G. Brown (McGill)
 RYUICHI ASHINO, Mathematical Sciences, Osaka Kyoiku University, Kashiwara,
Osaka 5828582, Japan
A preprocessing design for the discrete multiwavelet
transform of irregularly sampled data

Unshifted and shifted multiscaling functions are used as mathematical
models for curve fitting of irregularly sampled data. A
preprocessing design for the discrete multiwavelet transform based on
this curve fitting method is proposed. This preprocessing procedure
combined with multiwavelet neural networks for dataadaptive curve
fitting is shown to perform well in the case of high resolution. In
the case of low resolution it is more accurate than numerical
integration and cheaper than matrix inversion.
This is joint work with Akira Morimoto and Rémi Vaillancourt.
 CHANTAL BUTEAU, Brock University, St. Catharines, Ontario L2S 3A1
Topological Spaces of Music Motives: An Application of
T_{0}Spaces

We present a topological model of motivic analysis of music that
mainly consists of identifying in a music composition the short
sequence of notes melodically generating, by its repetitions
and variations, the whole score. Our approach particularly solves the
problem of motif similarity for different cardinalities. Using the
music concepts of melodic contourmap on the set of all
possible motives, of gestaltorbit of a motif for a given
group action, and of motif similaritypseudometric for
motives with same cardinality, we introduce a topological structure on
the set of all motives. The crucial step in our approach is
the definition of motif neighborhoods that links motives with
different cardinalities. In the resulting T_{0}space, the
identification of the wanted germinal (generator) motif corresponds to
the determination of the motif with most `dense' neighborhood
given a similarity threshold. We illustrate our approach with Bach's
"Art of Fugue" for which we investigated the still debated problem
on the length of its main theme. Others' approaches based on the
socalled (music) Set Theory can be consistently redefined
and extended in our generic topological structure.
This model is part of the local theory of the socalled
Mathematical Music Theory developed by G. Mazzola in his book
The Topos of MusicGeometric Logic of Concepts, Theory, and
Performance, and which is based on local charts and global discrete
varieties of tones in specific parameter spaces.
 ADELA N. COMANICI, University of Houston, Department of Mathematics
Transition from Rotating Waves to Modulated Rotating Waves on
a Sphere

Spiral waves are spatiotemporal patterns that have been observed in
numerous physical situations, ranging from BelousovZhabotinsky
chemical reactions to cardiac tissue. The global geometry of the
heart is closer to a sphere than to a plane. Therefore, it would be
important to study spiral wave dynamics in a context where the
symmetries are that of a sphere instead of a plane. Also, rigidly
rotating spiral waves are rotating waves. In this talk we consider an
SO(3)equivariant reactiondiffusion system on a sphere:

¶u
¶t

(t,x) = DDu (t,x) + F 
æ è

u(t,x), l 
ö ø

on r S^{2}, 
 (1) 
u = (u_{1},u_{2},...,u_{N}) with N ³ 1, D is the diffusion
matrix and F = (F_{1},F_{2},...,F_{N}) is a sufficiently smooth
function.
For l = 0 we consider a relative equilibrium SO(3) u_{0} with
trivial isotropy S_{u0} = I_{3} consisting of rotating waves.
By Hopf bifurcation from rotating waves we generically get modulated
rotating waves. We show that there exists a decomposition of these
modulated rotating waves in two parts: a primary frequency vector
part, e^{X(l)t} and a
[(2p)/(w_{l})]periodic part, B(t,l). We
present a way of contructing X(l) and B(t,l) using the
BCH formula in SO(3) and the solution Z(t,l) to the initial
value problem on some interval [0,[(2p)/(nw_{l})]], with integer n > 0 independent of
l:


= 
é ë

I_{3} + 
1
2

Z+ 
æ è


1
Z^{2}

 


ö ø

Z^{2} 
ù û


®
X^{G} (t,l)

, 
  (2) 
 
  (3) 
This decomposition allows us to show that the quasiperiodic
meandering of these modulated rotating waves is possible and, in fact,
that there are three types of motions for the tips of these modulated
rotating waves. Some numerical results obtained with Maple will be
shown.
In the case of a resonant Hopf bifurcation for two parameters
l and m, there exists a branch of modulated rotating waves
with primary frequency vectors orthogonal to the frequency vector of
the rotating wave undergoing the Hopf bifurcation.
 BRUCE GILLIGAN, University of Regina
Kähler homogeneous manifolds

Suppose G is a (connected) complex Lie group and H is a closed
complex subgroup such that
(a) G/H does not have any nonconstant holomorphic functions
and
(b) H is not contained in any proper parabolic subgroup
of G.
In this setting D. N. Akhiezer asked whether every analytic
hypersurface in G that is Hinvariant is also G¢invariant,
where G¢ denotes the commutator of G. Positive answers are known
if G/H is compact or if G is solvable, semisimple, or a direct
product R×S, where R is the radical of G and S is a Levi
factor.
The problem reduces to Kähler homogeneous spaces of the form
G/G satisfying (a) and (b), where G is a discrete
subgroup of G, a situation that is of interest in its own right.
The goal is to show that G/G is a complex Lie group with no
nonconstant holomorphic functions. We call such a complex Lie group a
Cousin group, in honour of P. Cousin. In this talk we will
outline our recent results on this problem.
If the orbits of the radical R of G have no nonconstant
holomorphic functions themselves, then we showed S={e}. It follows
that G/G is a Cousin group. If the Rorbits do have
nonconstant holomorphic functions, then there is an induced fibration
G/G 
I/G
®

G/I = 
^
G

/ 
^
G

where 
^
G

: = G/I^{°} and 
^
G

: = I/I^{°}. 

We will indicate our methods used to handle the case where [^(G)] = [^(R)] ×[^(S)] is a direct product, and where [^(G)] is a
linear algebraic group whose radical is a vector group with a maximal
semisimple subgroup of [^(G)] acting on this radical via a linear
representation that has no nonzero invariant vectors. An example of
such a group [^(G)] is the affine group.
 FEDERICO INCITTI, KTH, Math Dept, Lindstedts väg 25, 10044 Stockholm
(Sweden)
Combinatorial invariance for rank 5 intervals in the
symmetric group

The wellknown "combinatorial invariance conjecture" states that,
given a Coxeter group W, ordered by Bruhat order, and given two
elements u,v Î W, with u < v, the KazhdanLusztig polynomial
associated with u,v supposedly depends only on the unlabelled
abstract poset [u,v].
This conjecture was known to be true for intervals of rank £ 4.
In this talk we consider the first open case, proving that it is true
for intervals of rank 5 in the symmetric group.
 JOSEPH KHOURY, University of Ottawa, 585 King Edward Avenue, Ottawa,
Ontario K1N 6N5
A note on elementary derivations

Let R be a UFD containing a field of characteristic 0, and
B_{m} = R[Y_{1},...,Y_{m}] be a polynomial ring over R. It was
conjectured in [1] that if D is an Relementary
monomial derivation of B_{3} such that kerD is a finitely
generated Ralgebra then the generators of kerD can be
chosen to be linear in the Y_{i}'s. In this paper, we prove that
this does not hold for B_{4}. We also investigate Relementary
derivations D of B_{m} satisfying one or the other of the
following conditions:
(i) D is standard.
(ii) kerD is generated over R by linear constants.
(iii) D is fixpointfree.
(iv) kerD is finitely generated as an Ralgebra.
(v) D is surjective.
(vi) The rank of D is strictly less than m.
References
 [1]

J. Khoury,
On some properties of locally nilpotent derivations in
dimension six.
J. Pure Appl. Algebra (1) 156(2001), 6979.
 RENATA OTÁHALOVÁ, Mathematical Institute of the Silesian University in Opava, Na Rybnícku 1, 746 01 Opava, Czech Republic
The simplest subspace of generators of matrix algebras

The simplest subspaces of generators of the complex 2 ×2 and 3 ×3  matrix
algebras are found. Simplicity is measured by the
number of parameters in the free resolution of the algebra. The result for 2 ×2  matrices
was presented at the competition SVOC in
Banská Bystrica last year and provided Clifford generators (Pauli matrices) as a solution.
The problem of the 3 ×3 matrix algebra M _{3} (C) is not a straightforward
extension of the case M_{2}(C), as there
are no Clifford generators. It is shown here that another wellknown subspace of generators, the
Lie algebra su(2) provides the solution.
 EMILE PELLETIER, University of Ottawa
Instrument DeSynthesis Using Wavelets

Our point of departure is the concept of `additive synthesis', which
is the traditional explanation for the individual of `timbre' or
`colour' of the sound of the various musical instruments.
When an instrument sounds a note, one hears the note as if by itself,
but this is not what is physically happening. What is in fact
occurring is a complex waveform featuring a collection of harmonic
frequencies, referred to as the spectrum.
A synthesizer attempts to imitate the sound of a particular instrument
by replicating the amplitudes of its harmonics.
We use the term `desynthesis' to refer to the inverse procedure,
computerized instrument identification.
We describe an experiment that we designed and executed with MATLAB to
explore the hypothesis that a computer will be able to recognize an
instrument by its characteristic timbre.
The idea of applying wavelets to analyze music comes naturally since
music consists of sound waves, and wavelets are wave shaped functions.
We propose a mathematical model that can take certain musical
instrument's attack and decay features into account that utilizes
Malvar wavelets: Super Malvar wavelets. Super wavelets are
superpositions of ordinary wavelets in some linear combination that
can be treated as a wavelet in itself.
This provides us with a way to both represent the harmonics that are
present in the notes and to deal with their individual attack and
decay envelopes.
Instrument recognition is a problem that has had some research done by
researchers in varying fields such as musicology, but also mathematics
and computer science. We present a method that uses Super Malvar
wavelets to identify the four musical instruments: the clarinet, the
oboe, the trumpet and the violin. By using Super Malvar wavelets in
this regard we hypothesize that we will have improved accuracy in the
identification process because we will be considering the instrument's
harmonic attack and the decay nonsynchrony characteristics.
Nonsynchrony of the harmonics is a term that refers to the physical
property that the higher numbered harmonics ( > 1) appear later in the
attack portion of a note and disappear sooner during the decay than
the fundamental or first harmonic.

