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Contributed Papers / Communications libres (Org: Walter Burgess and/et Abdellah Sebbar)
- MARTIN ARGERAMI, Department of Mathematics and Statistics, University of Regina,
Regina, Saskatchewan S4S 0A2
The Schur-Horn Theorem in II1 factors
[PDF] -
The Schur-Horn theorem expresses a relation, in real n-space, of two
apparently unrelated notions: majorization and convexity. In finite
dimensional von Neumann algebras, it can be expressed in terms of
maximal abelian subalgebras (masas) of factors:
Let M be a finite dimensional factor, A Ì M a masa in M, and
let P be the unique conditional expectation from M to A. Then,
for every a Î A,
P |
æ è
|
UM(a) |
ö ø
|
=co{vav*:v Î N(A)}, |
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where UM is the unitary group of M and N(A)
is the normalizer of A.
The author believes (and has evidence, although not a proof) that the
Schur-Horn property characterizes masas of finite dimensional factors
(that is, if a subalgebra of a finite dimensional factor satisfies the
Schur-Horn relation, then it is a masa).
II1 factors are a natural setting to try the Schur-Horn property,
because there always exist a normal conditional expectation onto any
masa, and also because the main notion used in the finite dimensional
proof is majorization, which can be reasonably extended to the setting
of II1 factor, thanks to the existence of the trace. One cannot
expect masas in II1 factors to be characterized by the Schur-Horn
property (it fails clearly for singular masas) but it may, for
instance, characterize regular masas.
In our attempts to prove this theorem, we have had to work with the
notion of majorization in II1 factors, and several interesting
characterizations appear for unitary orbits of states and
automorphisms.
- MURRAY BREMNER, Department of Mathematics, University of Saskatchewan,
Saskatoon, Saskatchewan S7N 5E6
Quantization of Lie and Jordan triple systems
[PDF] -
This talk will show how the decomposition of the group ring of the
symmetric group into a direct sum of full matrix subrings can be used
to give a complete classification of n-ary operations. Roughly
speaking, row equivalence of matrices corresponds to quasi-equivalence
of operations. In particular, the Lie and Jordan products represent
the two non-trivial quasi-equivalence classes of binary operations.
For ternary operations, there are infinitely many quasi-equivalence
classes, which divide into eight classes, and four infinite families of
classes each with a single parameter. The Lie triple product is
contained in one of the infinite classes, and the other operations in
the class can be regarded as quantizations of that product. Similar
remarks apply to the Jordan triple product. For special values of the
parameter, the operation satisfies an identity of degree 5. This
identifies some new ternary operations which define varieties of triple
systems, similar to Lie and Jordan triple systems, which seem to be an
interesting direction for further research.
- VAHID DABBAGHIAN, School of Mathematics and Statistics, Carleton University,
Ottawa, Ontario K1S 5B6
An efficient algorithm to construct representations of
finite groups
[PDF] -
Let G be a finite group. It is easy to compute the character of G
corresponding to a given complex representation, but much more
difficult to compute a representation affording a given character. In
part this is due to the fact that a class of equivalent representations
contains no natural canonical representation.
Although there is a large literature devoted to computing
representations, and methods are known for particular classes of
groups, no general method has been proposed which is practical for any
but very small groups. For example, the function
``IrreducibleRepresentationsDixon'' which is supplied in the latest
version 4.3 of the computer algebra system GAP, is very slow in
computing representations for even moderately sized groups and fails to
compute a representation in many cases.
We shall describe an algorithm to compute an irreducible matrix
representation R which affords a given character c of a given
group G. The algorithm uses properties of the structure of G which
can be computed efficiently by a program such as GAP, theoretical
results from representation theory, theorems from group theory
(including the classification of finite simple groups), and linear
algebra. The algorithm has been implemented in GAP and appears to
work well for a general group G when the character supplied has
degree up to about 30.
- HANS HEINIG, McMaster University, Hamilton, Ontario L8S 4K1
Exponential inequalities with weights
[PDF] -
Characterizations of weight pairs are obtained for which a class of
weighted exponential-logarithmic integral inequalities are satisfied.
Special cases yield weight characterizations for the geometric mean
operator as well as operators of Riemann-Liouville and Laplace type.
- TREVOR JONES, University of New Brunswick
Computing some nonstandard Betti numbers
[PDF] -
In previous work, we established an abstract generalized Gauss-Bonnet
theorem for surfaces. We discuss the computation of our nonstandard
Betti numbers in the theorem, at least for the special case of Riemann
surfaces of constant curvature. By the uniformization theorem we may
as well consider hyperbolic space modulo a discrete subgroup of
isometries and we discuss the case of hyperbolic space in detail.
- VALENTINA KIRITCHENKO, University of Toronto, Toronto, Ontario M5S 3G3
A Gauss-Bonnet theorem for constructible sheaves on reductive
groups
[PDF] -
We prove an analog of the Gauss-Bonnet formula for constructible
sheaves on reductive groups. This formula holds for all constructible
sheaves equivariant under the adjoint action and expresses the Euler
characteristic of a sheaf in terms of its characteristic cycle. As a
corollary from this formula we get that if a perverse sheaf on a
reductive group is equivariant under the adjoint action, then its Euler
characteristic is nonnegative.
- KUNQUAN LAN, Ryerson University
Multiple positive solutions of higher order boundary value
problems
[PDF] -
We consider the existence of one or several nonzero positive solutions
for a higher order nonlinear ordinary differential equation with
n-sets of separated boundary conditions. The boundary value problems
can be changed into a Hammerstein integral equation with a suitable
kernel. We shall show that the kernel has upper and lower bounds. This
enables us not only to exhibit a new property of positive solutions for
the boundary value problems but also to derive new results on these
boundary value problems from the well-known results on the existence of
one or several positive solutions of Hammerstein integral equations
with singularities obtained by the author recently. This avoids
utilizing the theory of fixed point index for compact maps defined on
cones directly.
- CLAUDE LEVESQUE, Département de mathématiques et de statistique, Université
Laval, Québec G1K 7P4
A fundamental system of units for some fields of degree 9
[PDF] -
We exhibit a fundamental system of units for some families of composita
of two pure cubic number fields of the form Q(w, q) where
w3 and q3 are certain positive integers.
- RAVIL MOUKHOMETOV, Ottawa
On problem of the reconstruction of metric on the Riemannian
manifold
[PDF] -
On a compact Riemannian manifold M with a boundary D we consider
the problem of the reconstruction of the Riemannian metric g if are
known the lengths of geodesics with endpoints on the boundary D of
M. First results, namely an uniqueness and stability for this
non-linear problem in a general formulation, have been got by the
auther in 1977. In present time the auther obtains some results when
geodesics are reflected from the part L of the boundary D of M
and also for the linearized problem that is the integral geometry
problem. The auther also gets in connection of this problem some
formula: the symplectic volume and from here the Riemannian volume of
M is expressed only by the lengths of geodesics with endpoints on the
boundary D without L. In this formula the metric g is unknown
and the manifold M is not known as the part L of the boundary D
is unknown. The considered problems refer to the Riemannian geometry,
also to the symplectic geometry (in proofs is used 1-contact form,
see: C. Godbillon, Geometrie Differentielle et Mecanique
Analytique). The near similar problems are also used in geophysics
(structure of Earth) and the obtained results may stimulate the
investigations of the new problems.
- AMRITANSHU PRASAD, CRM, CP 6128, Succursale centre-ville, Montreal, Quebec H3C
3J7
Almost unramified automorphic representations for slpit
groups over Fq(t)
[PDF] -
Let G be a split reductive group over a finite field Fq.
Let F=Fq(t) and A be the adeles of F. We
describe the local constituents at each valuation of F of all the
irreducible representations of G(A) that occur discretely in
L2(G(F)\G(A)) and have non-zero vectors
invariant under the compact open subgroup K of G(A) which
is a product Iwahori subgroups at two valuations of F and maximal
compact subgroups at all the other valuations.
This is done by showing, firstly, that the space generated by the
representations mentioned above is spanned by residues of Eisenstein
series associated to an unramified automorphic characters of a maximal
split torus of G. This imposes consistency conditions on the local
constituents at different valuations of a fixed automorphic
representation. An earlier result of the speaker describes the local
constituents of the aforementioned representations at the two places
where the local factor of K is Iwahori. This, together with the
consistency conditions uniquely determines the local constituents at
all places of the automorphic representation.
- RONALD SKLAR, St. John's University, Jamaica, New York 11439, USA
Mathematical techniques for circumventing query-set-size
control in a statistical database
[PDF] -
This talk will discuss the method of attacking a statistical database
by the use of a tracker. Different types of trackers will discussed,
and it will be shown that one of the commonly used techniques, namely
query-set-size control, for protecting a statistical database can
easily be subverted by the use of a tracker. The simple mathematics
employed makes this interesting and timely topic one that can be
successfully incorporated into a mathematics liberal arts course for
non-majors.
- VLADLEN TIMORIN, University of Toronto, Tornoto, Ontario M5S 3G3
Circles and Clifford algebras
[PDF] -
We study smooth maps from an open subset of Rm to an open
subset of Rn that take germs of straight lines to germs of
circles. ``Degenerate circles'', i.e. lines or points, are
counted as circles. We give a construction of such maps based on
representations of Clifford algebras. It provides a complete purely
geometric description of Clifford algebras representations in terms of
circles.
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