This paper contains results on the structure of generic local polar curves $P(\tau)$ of a reduced germ of a complex analytic plane curve $C$. We first prove a decomposition theorem of the branches of $P(\tau)$ into bunches. By construction, all the branches in the same bunch have the same contact with all branches of $C$. It follows that the Puiseux expansions of all the branches of a given bunch coincide with that of a branch of $C$ up to an order depending upon the bunch. An initial part of these expansions is therefore independant of $\tau$. In the second part, we study to what extent the contact of the branches of $C$ with the branches of $P(\tau)$ determines the topological type of $C$. We build from these contacts a matrix which is determined by the topological type of $C$ and determines it. In an appendix, we explain how to recover, from the results of the first part, the L\^e--Michel--Weber theorem on the behaviour of polar curves in an embedded resolution of singularities of $C$. Cet article pr\'esente des r\'esultats sur la structure des courbes polaires locales g\'en\'eriques $P(\tau)$ d'un germe r\'eduit de courbe analytique complexe plane $C$. Nous d\'emontrons d'abord un th\'eor\`eme de d\'ecomposition en paquets des branches de $P(\tau)$. Par construction, toutes les branches d'un m\^eme paquet ont le m\^eme contact avec chacune des branches de $C$. L'ensemble de ces paquets est index\'e par un graphe qui ne d\'epend que de la topologie de la courbe $C$ donn\'ee. En cons\'equence, le d\'eveloppement de Puiseux de toutes les branches d'un m\^eme paquet de $P(\tau)$ coincide avec celui d'une branche de $C$ jusqu'\`a un ordre d\'ependant du paquet. Une partie initiale de ce d\'eveloppement est donc ind\'ependante de $\tau$. Dans la deuxi\`eme partie de ce travail, nous \'etudions dans quelle mesure le contact avec les branches de $C$ des branches de $P(\tau)$ d\'etermine le type topologique de $C$. Nous construisons \`a partir de tous ces contacts une matrice qui ne d\'epend que du type topologique de $C$ et le d\'etermine. Dans un appendice, nous montrons comment retrouver, \`a partir des r\'esultats prouv\'es dans la premi\`ere partie, le th\'eor\`eme de L\^e--Michel--Weber sur le comportement des polaires dans une r\'esolution plong\'ee des singularit\'es de $C$. E-mail: [email protected] 1991 Mathematics Subject Classification: 14H20, 32S10.

Theorem 1 asserts that \emph{in a finitely generated prosoluble group, every subgroup of finite index is open}. This generalises an old result of Serre about pro-$p$ groups. It follows by a standard argument from Theorem 2: \emph{in a $d$-generator finite soluble group, every element of the derived group is equal to a product of $72d^2 +46d$ commutators}. This result about finite soluble groups is proved by induction on the order of the group, and is elementary though rather intricate. The essence of the proof lies in reducing the problem to one about the number of solutions of quadratic equations over a finite field. Corollaries include the following. \emph{Let $\Gamma$ be a finitely generated prosoluble group. Then each term of the lower central series of $\Gamma$ and each power subgroup $\Gamma ^n$ is closed}. 1991 Mathematics Subject Classification: 20E18, 20D10.

Suppose that $N$ is a normal $p$-subgroup of a finite group $G$ and let $G^0$ be the set of elements of $G$ whose $p$-part lies in $N$. We prove the existence of a canonical basis ${\rm IBr}(G, N)$ of the space of complex class functions of $G$ defined on $G^0$, such that the restriction $\chi^0$ of any irreducible complex character $\chi$ of $G$ is a linear combination $\sum_{\phi\in{\rm IBr}(G, N)} d_{\chi \phi} \phi$ of the elements of this basis, where the $d_{\chi \phi}$ are non-negative integers. Furthermore, if we write $\Phi_\phi=\sum_{\chi} d_{\chi \phi}\chi$, then the $\Phi_\phi$ form the K\"ulshammer--Robinson ${\Bbb Z}$-basis of the ${\Bbb Z}$-module generated by the characters afforded by the $N$-projective $RG$-modules, where $R$ is a certain complete discrete valuation ring. By using these `decomposition numbers', it is possible to define a linking in the set of the irreducible complex characters of $G$. 1991 Mathematics Subject Classification: 20C15, 20C20.

The authors consider the system of forced differential equations with variable delays $$x'(t) + \sum^N_{j=1}B_j(t)x(t-\tau_j(t)) = F(t)\eqno(*)$$ where $B_j(t)$ is a continuous $n\times n$ matrix on ${{\Bbb R}^+}$, $F\in C({{\Bbb R}^+, {\Bbb R}^n})$ and $\tau \in C({{\Bbb R}^+, {\Bbb R}^+})$. Using Razumikhin-type techniques and Liapunov's direct method, they establish conditions to ensure the ultimate boundedness and the global attractivity of solutions of $(*)$, and when $F(t) \equiv 0$, the asymptotic stability of the zero solution. Under those same conditions, they also show that $\int^{+\infty}_0\sum_{j=1}^{N}|B_j(t)|\,dt = +\infty$ is a necessary and sufficient condition for all of the above properties to hold. 1991 Mathematics Subject Classification: 34K15, 34C10.

The unitary highest weight modules for $G={\rm U}(1,q)$, which occur as irreducible subrepresentations of the oscillator representation on a Fock space ${\mathcal F}$, can each be realized on a space of polynomial-valued functions over the bounded realization ${\bf B}^q$ of $G/K$. This is achieved via an integral transform constructed by L.~Mantini. A decomposition of these representations into $K$-types is given, including an explicit description of how Mantini's transform behaves on $K$-types. An inverse is produced for the transform, thus giving unitary structures for the geometric realizations of the unitary highest-weight modules over $G/K$. 1991 Mathematics Subject Classification: 22E45, 22E70.

We develop a notion of Morita equivalence for general C$^{\ast}$-correspondences over C$^{\ast}$-algebras. We show that if two correspondences are Morita equivalent, then the tensor algebras built from them are strongly Morita equivalent in the sense developed by Blecher, Muhly and Paulsen. Also, the Toeplitz algebras are strongly Morita equivalent in the sense of Rieffel, as are the Cuntz--Pimsner algebras. Conversely, if the tensor algebras are strongly Morita equivalent, and if the correspondences are aperiodic in a fashion that generalizes the notion of aperiodicity for automorphisms of C$^{\ast}$-algebras, then the correspondences are Morita equivalent. This generalizes a venerated theorem of Arveson on algebraic conjugacy invariants for ergodic, measure-preserving transformations. The notion of aperiodicity, which also generalizes the concept of full Connes spectrum for automorphisms, is explored; its role in the ideal theory of tensor algebras and in the theory of their automorphisms is investigated. 1991 Mathematics Subject Classification: 46H10, 46H20, 46H99, 46M99, 47D15, 47D25.

A vector $x$ in a Banach space $\cal B$ is called hypercyclic for a bounded linear operator $T: \cal B \rightarrow \cal B $ if the orbit $\{T^n x : n \geq 1 \}$ is dense in $\cal B$. We prove that if $T$ is hereditarily hypercyclic and its essential spectrum meets the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for $T$. The converse is also true even if $T$ is not hereditarily hypercyclic. In this way, we improve and extend to Banach spaces a recent result for Hilbert spaces. We also investigate the corresponding problem for supercyclic operators. In this case we obtain a description of the norm closure of the class of all supercyclic operators that have an infinite-dimensional closed subspace of hypercyclic vectors. Moreover, for certain kinds of supercyclic operators we can characterize when they have an infinite-dimensional closed subspace of supercyclic vectors. 1991 Mathematics Subject Classification: 47B38, 47B99.

We classify the finite-dimensional quotient Hopf algebras of the deformed algebra of functions of the general linear group (over an algebraically closed field of zero characteristic). This gives an interesting class of Hopf algebras if the deformation parameter is a root of unity (of odd order). We investigate the properties of these Hopf algebras and construct a new counterexample to Kaplansky's tenth conjecture. E-mail: [email protected] 1991 Mathematics Subject Classification: 81R50, 16W30, 17B37.

The variation of the Hodge structure of a Calabi--Yau 3-fold induces a canonical K\"ahler metric on its Kuranishi moduli space, known as the Weil--Petersson metric. Similarly, {\it special} pseudo-K\"ahler manifolds correspond to certain (abstract) variations of Hodge structure which generalize the above example. We give the classification of homogeneous special pseudo-K\"ahler manifolds of semisimple groups with compact stabilizer. 1991 Mathematics Subject Classification: 53C30, 53C25.

A finite Borel measure $\mu$ on $\mathbb R^d$ is called R-O varying with index $F$ if there exist a $\operatorname{GL}(\mathbb R^d)$-valued function $f$ varying regularly with index $(-F)$, an increasing function $k : (0,\infty)\to (0,\infty)$ with $k(t)\to\infty$ and $k(t+1)/k(t)\to c\geq 1$ as $t\to\infty$, and a $\sigma$-finite measure $\phi$ on $\mathbb R^d\setminus\{0\}$ such that \[k(t)\cdot(f(k(t))\mu)\to\phi\quad \text{as $t\to\infty$.}\] R-O varying measures generalize regularly varying measures introduced by Meerschaert (see M.~M.~Meerschaert, `Regular variation in $\mathbb R^k$', {\it Proc.\ Amer.\ Math.\ Soc.} 102 (1988) 341--348) and have numerous applications in limit theorems for probability measures. For an R-O varying measure $\mu$ and $-\infty<\infty$ let \begin{equation*} \begin{split} V_a(t,\theta) &= \int_{|\langle x,\theta\rangle|>t} |\langle x,\theta\rangle|^a\,d\mu(x),\\ U_b(t,\theta) &= \int_{|\langle x,\theta\rangle|\leq t} |\langle x,\theta\rangle|^b\,d\mu(x) \end{split} \end{equation*} denote the tail- and truncated moment functions of $\mu$ in the direction $\|\theta\|=1$. The purpose of this paper is to show that R-O variation of a measure implies sharp bounds on the growth rate of the tail- and truncated moment functions depending on the real parts of the eigenvalues of the index $F$ along a compact set of directions. Furthermore, bounds on the ratio of these functions for certain values of $a$ and $b$ are obtained. 1991 Mathematics Subject Classification: 60B10, 28C15.