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26. 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 well-known 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

27. CJM 2000 (vol 52 pp. 141)

Li, Chi-Kwong; Tam, Tin-Yau
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

28. CJM 1999 (vol 51 pp. 506)

Elduque, A.; Iltyakov, A. V.
On Polynomial Invariants of Exceptional Simple Algebraic Groups
We study polynomial invariants of systems of vectors with respect to exceptional simple algebraic groups in their minimal linear representations. For each type we prove that the algebra of invariants is integral over the subalgebra of trace polynomials for a suitable algebraic system (\cf\ \cite{Schw1}, \cite{Schw2}, \cite{Ilt}).

Categories:15A72, 17C20

29. CJM 1998 (vol 50 pp. 1323)

Morales, Jorge
L'invariant de Hasse-Witt de la forme de Killing
Nous montrons que l'invariant de Hasse-Witt de la forme de Killing d'une alg{\`e}bre de Lie semi-simple $L$ 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

30. 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

31. 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

32. CJM 1997 (vol 49 pp. 840)

Rodman, Leiba
Non-Hermitian solutions of algebraic Riccati equation
Non-hermitian solutions of algebraic matrix Riccati equations (of the continuous and discrete types) are studied. Existence is proved of non-hermitian solutions with given upper bounds of the ranks of the skew-hermitian parts, under the sign controllability hypothesis.

Categories:15A99, 15A63, 93C60
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