26. CJM 2001 (vol 53 pp. 758)
 Goulden, I. P.; Jackson, D. M.; Latour, F. G.

Inequivalent Transitive Factorizations into Transpositions
The question of counting minimal factorizations of permutations into
transpositions that act transitively on a set has been studied extensively
in the geometrical setting of ramified coverings of the sphere and in the
algebraic setting of symmetric functions.
It is natural, however, from a combinatorial point of view to ask how such
results are affected by counting up to equivalence of factorizations, where
two factorizations are equivalent if they differ only by the interchange of
adjacent factors that commute. We obtain an explicit and elegant result for
the number of such factorizations of permutations with precisely two
factors. The approach used is a combinatorial one that rests on two
constructions.
We believe that this approach, and the combinatorial primitives that have
been developed for the ``cut and join'' analysis, will also assist with the
general case.
Keywords:transitive, transposition, factorization, commutation, cutandjoin Categories:05C38, 15A15, 05A15, 15A18 

27. CJM 2001 (vol 53 pp. 470)
 Bauschke, Heinz H.; Güler, Osman; Lewis, Adrian S.; Sendov, Hristo S.

Hyperbolic Polynomials and Convex Analysis
A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to
a given vector $d$ if the univariate polynomial $t \mapsto p(xtd)$
has all real roots for all vectors $x$. Motivated by partial
differential equations, G{\aa}rding proved in 1951 that the largest
such root is a convex function of $x$, and showed various ways of
constructing new hyperbolic polynomials. We present a powerful new
such construction, and use it to generalize G{\aa}rding's result to
arbitrary symmetric functions of the roots. Many classical and recent
inequalities follow easily. We develop various convexanalytic tools
for such symmetric functions, of interest in interiorpoint methods
for optimization problems over related cones.
Keywords:convex analysis, eigenvalue, G{\aa}rding's inequality, hyperbolic barrier function, hyperbolic polynomial, hyperbolicity cone, interiorpoint method, semidefinite program, singular value, symmetric function Categories:90C25, 15A45, 52A41 

28. CJM 2000 (vol 52 pp. 141)
 Li, ChiKwong; Tam, TinYau

Numerical Ranges Arising from Simple Lie Algebras
A unified formulation is given to various generalizations of the
classical numerical range including the $c$numerical range,
congruence numerical range, $q$numerical range and von Neumann
range. Attention is given to those cases having connections with
classical simple real Lie algebras. Convexity and inclusion
relation involving those generalized numerical ranges are
investigated. The underlying geometry is emphasized.
Keywords:numerical range, convexity, inclusion relation Categories:15A60, 17B20 

29. CJM 2000 (vol 52 pp. 197)
 Radjavi, Heydar

Sublinearity and Other Spectral Conditions on a Semigroup
Subadditivity, sublinearity, submultiplicativity, and other
conditions are considered for spectra of pairs of operators on a
Hilbert space. Sublinearity, for example, is a weakening of the
wellknown property~$L$ and means $\sigma(A+\lambda B) \subseteq
\sigma(A) + \lambda \sigma(B)$ for all scalars $\lambda$. The
effect of these conditions is examined on commutativity,
reducibility, and triangularizability of multiplicative semigroups
of operators. A sample result is that sublinearity of spectra
implies simultaneous triangularizability for a semigroup of compact
operators.
Categories:47A15, 47D03, 15A30, 20A20, 47A10, 47B10 

30. CJM 1999 (vol 51 pp. 506)
31. CJM 1998 (vol 50 pp. 1323)
 Morales, Jorge

L'invariant de HasseWitt de la forme de Killing
Nous montrons que l'invariant de HasseWitt de la forme de Killing
d'une alg{\`e}bre de Lie semisimple $L$ s'exprime {\`a} l'aide de
l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de
$L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines
positives), et des invariants associ{\'e}s au groupe des
sym{\'e}tries du diagramme de Dynkin de $L$.
Categories:11E04, 11E72, 17B10, 17B20, 11E88, 15A66 

32. CJM 1998 (vol 50 pp. 929)
 Broer, Abraham

Decomposition varieties in semisimple Lie algebras
The notion of decompositon class in a semisimple Lie algebra is a
common generalization of nilpotent orbits and the set of
regular semisimple elements. We prove that the closure of a
decomposition class has many properties in common with nilpotent
varieties, \eg, its normalization has rational singularities.
The famous Grothendieck simultaneous resolution is related to the
decomposition class of regular semisimple elements. We study the
properties of the analogous commutative diagrams associated to
an arbitrary decomposition class.
Categories:14L30, 14M17, 15A30, 17B45 

33. CJM 1997 (vol 49 pp. 865)
 Goulden, I. P.; Jackson, D. M.

Maps in locally orientable surfaces and integrals over real symmetric surfaces
The genus series for maps is the generating series for the
number of rooted maps with a given number of vertices and
faces of each degree, and a given number of edges. It captures
topological information about surfaces, and appears in questions
arising in statistical mechanics, topology, group rings,
and certain aspects of free probability theory. An expression
has been given previously for the genus series for maps in
locally orientable surfaces in terms of zonal polynomials. The
purpose of this paper is to derive an integral representation
for the genus series. We then show how this can be used in
conjunction with integration techniques to determine the genus
series for monopoles in locally orientable surfaces. This
complements the analogous result for monopoles in orientable
surfaces previously obtained by Harer and Zagier. A conjecture,
subsequently proved by Okounkov, is given for the evaluation
of an expectation operator acting on the Jack symmetric function.
It specialises to known results for Schur functions and zonal
polynomials.
Categories:05C30, 05A15, 05E05, 15A52 

34. CJM 1997 (vol 49 pp. 840)
 Rodman, Leiba

NonHermitian solutions of algebraic Riccati equation
Nonhermitian solutions of algebraic matrix Riccati
equations (of the continuous and discrete types) are studied. Existence
is proved of nonhermitian solutions with given upper bounds of the
ranks of the skewhermitian parts, under the sign controllability
hypothesis.
Categories:15A99, 15A63, 93C60 
