In [4] we showed that one can tell whether a submeasure on a Boolean algebra has a control measure or is pathological by comparing the Fréchet-Nikodym topology it generates to the universal measure topology of Graves. We then wondered if a submeasure could be decomposed into a part with a control measure and a part which is pathological or zero. This led to the problem of finding a Lebesgue decomposition for a submeasure on an algebra of sets with respect to a Fréchet-Nikodym topology.

In [6] Drewnowski proved a Lebesgue decomposition theorem for exhaustive submeasures with respect to “additivities” and a similar theorem for exhaustive Fréchet-Nikodym topologies. He asked if an exhaustive Fréchet-Nikodym topology could be decomposed with respect to another Fréchet-Nikodym topology. In [12] Traynor showed that the answer is “yes”.

Let G be a group and φ an automorphism of G. We say that φ is a pseudo-identity (pi) if, for each x ∈ G, there exists a finitely generated (fg) subgroup K = Kx(φ) of G such that x ∈ K and φ|K is an automorphism of K. It has been shown [1, 3] that such special automorphisms of abelian or nilpotent groups play an important role in homotopy theory; and it was indicated in [2] that their purely algebraic properties might well repay study.

The following facts about pi's are elementary.

PROPOSITION 0.1. Let φ be an automorphism of G and let n be a non-zerointeger. Then φ is pi if and only if φn is pi.

PROPOSITION 0.2. Let φ be a pseudo-identity of G and a an automorphismof G. Then αφα-1 is a pseudo-identity.

On étudie, dans ce travail, l'hamiltonien − ½ Δ + V suivant deux points de vue distincts. Le premier, dans le paragraphe 1, consiste à faire une hypothèse de nature asymptotique sur le potentiel V : on suppose qu'en dehors d'une boule B, il est à croissance au plus polynômiale. On étudie la fonction de Green de H dans le complémentaire de la boule B précédente. La méthode consiste à utiliser le processus de générateur ½V stoppé sur le bord de B avec V jouant un rôle de création ou d'annihilation. On projette par une fonction d'exhaustion et on utilise un théorème de comparaison. C'est par un calcul élémentaire sur une équation du second ordre que l'on calcule les poids des espaces entre lesquels opère la fonction de Green du complémentaire de B définie par H.

In the early days of quantum mechanics, Weyl asked the following question. Let λ be a non-zero real number, ℋa separable Hilbert space. Given certain (unbounded) operators W1,…,Wn,W1+, …, Wn+ on ℋ satisfying

(on a dense subspace D of ℋ) with all other commutators vanishing. Given also a function where ζ ∈ Cn. Let W = (W1 …, Wn) W+ = (W1+ …, Wn+). How does one associate to f an operator f(W, W+)? (Actually, Weyl phrased the question in terms of p = Re ζ, q = Im ζ, P = Re W, Q = Im W+ which represent momentum and position. In this paper, however, we wish to exploit the unitary group on Cn and so prefer complex notation.)

The measure (x1x2x3)2adm(x) on the unit sphere in R3 is invariant under sign-changes and permutations of the coordinates; here dm denotes the rotation-invariant surface measure. The more general measure

corresponds to the measure

on the triangle

(where ). Appell ([1] Chap. VI) constructed a basis of polynomials of degree n in v1, v2 orthogonal to all polynomials of lower degree, and a biorthogonal set for the case γ = 0. Later Fackerell and Littler [6] found a biorthogonal set for Appell's polynomials for γ ≠ 0. Meanwhile Pronol [10] had constructed an orthogonal basis in terms of Jacobi polynomials.

Let M and N be connected Hermitian manifolds of dimensions m and n with Hermitian metrics hM and hN, respectively. Then the space ℓ(M, N) of continuous mappings between M and N endowed with the compact-open topology is second countable so that a metric can be furnished in ℓ(M, N) which induces the compact-open topology. A sequence {fn} in ℓ(M, N) converges to a n f in ℓ(M, N) in this topology if and only if fn converges to f uniformly on compact subsets of M. It is then an easy consequence of the Cauchy integral formula to show that the space ℋ(M, N) of holomorphic mappings f:M → N is a closed subspace of ℓ(M, N).

In this paper, generalizing the classical notions of normal functions, Bloch functions, regular sequences and P-point sequences of one complex variable to the mappings in ℋ(M, N), see also [25], we obtain various relations which exist between these notions.

In this paper we are concerned with the space BMOA of analytic functions of bounded mean oscillation for Riemann surfaces, and it is shown that for any analytic function on a Riemann surface the area of its range set bounds the square of its BMO norm, from which it is seen as an immediate corollary that the space BMOA includes the space AD of analytic functions with finite Dirichlet integrals.

Let R be an open Riemann surface which possesses a Green's function, i.e., R ∉ OG, and f b e an analytic function defined on R. The Dirichletintegral DR(f) = D(f) of f on R is defined by

1.1

and we denote by AD(R) the space of all functions f analytic on R for which D(f) < +∞.

Let R denote the set of real numbers. For f:R → R and h ∈ R, the difference function Δhf is defined by

The function H:R → R is called additive if it satisfies Cauchy's equation

Let ℱ be a class of real valued functions defined on R. ℱ is said to have the difference property if, for every function f:R → R satisfying Δhf ∈ ℱ for every h ∈ R, there exists an additive function H such that f — H ∈ ℱ

It was conjectured by P. Erdos that the class of continuous functions has the difference property. This conjecture was proved by N. G. de Bruijn in [1], where the difference property of several other classes was verified as well. (For other references, see [6].)