When we speak of the semigroup of a topological space X, we mean S(X) the semigroup of all continuous self maps of X. Let h be a homeomorphism from a topological space X onto a topological space Y. It is immediate that the mapping which sends f ∊ S(X) into h º f º h−1 is an isomorphism from the semigroup of X onto the semigroup of Y. More generally, let h be a continuous function from X into Y and k a continuous function from Y into X such that k º h is the identity map on X. One easily verifies that the mapping which sends f into h º f º k is a monomorphism from S(X) into S(Y). Now for “most” spaces X and Y, every isomorphism from S(X) onto S(Y) is induced by a homeomorphism from X onto Y. Indeed, a number of the early papers dealing with S(X) were devoted to establishing this fact.

In this paper we study the existence problem for topological actions of finite groups on euclidean spaces Rn+k which are free outside a fixed point set Rk (embedded as a vector subspace). We refer to such an action as a semi-free action on (Rn+k, Rn) and note that all our actions will be assumed orientation-preserving.

Suppose the finite group π acts semi-freely on (Rn+k, Rn), then it acts freely on (Rn+k – Rn) = Sn–l × Rk+1. Since this space is homotopy equivalent to Sn–l, π will have periodic integral cohomology and n will be a multiple of the period. In fact the orbit space

is a finitely-dominated Poincaré complex of formal dimension n – 1 with π1W = π and as considered by Swan [41].

Let S-L denote a singular Sturm-Liouville system on the half line with homogeneous boundary conditions, possessing a discrete negative and continuous positive spectrum. Let A be the S-L operator and Sα(f; x) the S-L eigenfunction expansion associated with the resolvent operator (z – A)–1, z complex. That is, Sα(f; x) denotes the resolvent summability means with weight function z(z – λ)–1 (or (1 + tλ)–1 where t = – 1/z).

We first study the problem of determining when

(1)

where is the Green's function associated with a certain perturbation of our system.

Let U denote the open unit disc of the complex plane, and φ a holomorphic function taking U into itself. In this paper we study the linear composition operator Cφ defined by Cφf = f º φ for f holomorphic on U. Our goal is to determine, in terms of geometric properties of φ, when Cφ is a compact operator on the Hardy and Bergman spaces of φ. For Bergman spaces we solve the problem completely in terms of the angular derivative of φ, and for a slightly restricted class of φ (which includes the univalent ones) we obtain the same solution for the Hardy spaces Hp (0 < p < ∞). We are able to use these results to provide interesting new examples and to give unified explanations of some previously discovered phenomena.

In the sequel, πn will denote the class of real polynomials of degree at most n and ‖f(x)‖∞ the L∞-norm of a function on [–l, +1].

In a series of recent papers, Saff and Varga studied the properties of the so-called incomplete polynomials; that is to say polynomials of the form

where sl and s2 are fixed integers and q ∊ πn.

In there, they define the constrained Tchebychev polynomial as being, up to a multiplicative constant, the solution of the following minimization problem

Let a and n be positive integers such that f(x) = xn + a is irreducible over the integers. A conjecture made by Bouniakowsky [4] in 1857 would imply that there exist infinitely many integers x such that f(x) is prime. An even stronger conjecture of Bateman and Horn [1, 2] would imply that

where π(x;f) is the number of integers m with 0 ≧ m ≧ x for which f(m) is prime, and

where w(p) is the number of solutions of the congruence

Except for the trivial case n = 1, neither of these conjectures has ever been resolved.

Soient Lp(X, , μ) les espaces de Banach usuels associés à un espace mesuré fini ou α-fini (X, , μ), p étant un nombre réel compris entre un et l'infini (1 < p < ∞). Notons Lp, l'espace Lp[0, 1]. Un opérateur T:Lp → Lp est dit être à puissances bornées sur Lp si

La convergence presque sure de la suite de fonctions

a été étudiée dans L2 pour T contraction [8], [1] et pour T inversible à puissances bornées dans , 1 < p ≧ 2 [9].

Let Γ be a differentiable curve in a real projective plane P2 met by every line of P2 at a finite number of points. The singular points of Γ are inflections, cusps (cusps of the first kind) and beaks (cusps of the second kind). Let n1(Γ), n2(Γ) and n3(Γ) be the number of these points in Γ respectively. Then Γ is non-singular if

otherwise, Γ is singular.

We wish to determine when T is singular and then find the minimum value of n(Γ). A history and an analysis of this problem were presented in [1] and [2]. It was shown that we may assume that Γ is a curve of even order (even degree if Γ is algebraic), met by every line in P2. Then if Γ does not contain any multiple points or if Γ contains only a certain type of multiple point, Γ is singular. Presently, we complete this investigation

Preface. The impetus for this study arose from the belief that the structure of a C*-algebra is illuminated by an understanding of the manner in which abelian subalgebras embed in it. Posed in its full generality, the question concerning abelian subalgebras would seem impossible to answer. A notion of diagonal subalgebra is, however, proposed which has the virtue that one can associate a “topological”; invariant to the pair consisting of the diagonal and the ambient algebra, from which these algebras may be retrieved.

In the setting of von Neumann algebras, the analogous question was addressed in the seminal work of Feldman and Moore [13]. Their definition of Cartan subalgebra permits the abstraction of a complete invariant consisting of a Borel equivalence relation together with a certain cohomology class on the relation from which the Cartan pair may be recovered. Our development parallels theirs in spirit; differences in substance derive from the topological flavor of C*-theory.

Le problème de la construction d'enveloppes d'holomorphie globales ou d'enveloppes polynomialement convexes n'a été que très peu abordé et peu d'exemples explicites sont connus. Il semble donc qu'il soit intéressant de se limiter d'abord au calcul d'enveloppes d'ensembles homogènes, par l'action d'un groupe agissant par transformations holomorphes. C'est le cas des sphères unité de Cn, des polytores par exemple. En s'inspirant des actions de groupes discutés par Ehrenpreis [2] ou par Hua [3], nous considérons ici l'action des groupes SU(2) et U(2) sur C3 identifié à l'ensemble des matrices symétriques complexes 2 × 2. L'action de SU(2) sur C3 est en fait l'action canonique de SU(2) sur une famille de quadriques dégénérant vers la quadrique singulière lorsque A tends vers 0. L'étude des orbites est équivalente à l'étude des équations de Cauchy Riemann sur SU(2), invariantes à gauche. Le spectre de l'algèbre des fonctions annulant une telle équation sur SU(2), qui sont invariantes par l'application antipodale est alors un domaine d'holomorphie de l'une de ces quadriques qui est d'ailleurs topologiquement non trivial.