L’extension aux fonctions algébroïdes du théorème de Picard est faite par Rémoundos [7] et après quelques années, celle de la théorie de Nevanlinna aux fonctions algébroïdes dans |z| < + ∞ est donnée par Selberg [9], Ullrich [14] et Valiron [16]. Dans ce mémoire, on considère l’allure des fonctions algébroïdes d’après la méthode dernière.

Let D be the unit disk | z | < 1 and let f be a complex-valued function continuous in D. The cluster set C(f, eiθ) of f at eiθ is defined by

The aim of this paper is to investigate some applications of a certain universal problem. The universal problem deals with categories C which for every object A have some ‘‘structure” on the set C(A, A) of endomorphisms of A and calls for a universal solution, relative to this structure, that is associated with C(A, A) and centralizes the set of automorphisms of A. The commutative version of this universal problem asks for a universal solution, relative to the said structure, that abelianizes the canonical monoid structure of C(A, A).

In this note we shall denote by R a hyperbolic Riemann surface. Let HP′(R) be the totality of harmonic functions u on R such that every subharmonic function | u | has a harmonic majorant on R. The class HP′(R) forms a vector lattice under the lattice operations:

Let us consider a stochastically continuous, separable and measurable stationary Gaussian process X = {X(t), − ∞ < t < ∞} with mean zero and with the covariance function p(t) = EX(t + s)X(s). The conditions for continuity of paths have been studied by many authors from various viewpoints. For example, Dudley [3] studied from the viewpoint of ε-entropy and Kahane [5] showed the necessary and sufficient condition in some special case, using the rather neat method of Fourier series.

Let C and D be the unit circle and the open unit disk respectively. We shall use p(z,z′) to represent the non-Euclidean distance [3, p. 263] between the two points z and z′ in D, and X(w, w′) to represent the chordal distance between the two points w and w′ on the Riemann Sphere Ω.

Let k be a complete field of characteristic 0 whose topology is defined by a discrete valuation and let T be an algebraic torus of dimension d defined over k. As is well known, T has a splitting field K which is a finite Galois extension of k with Galois group . For a ring R, denote by TR the subgroup of R-rational points of T. Then TK and T0K, DK being a valuation ring of K, become -modules in the usual manner.

Let Γ be a Fuchsian group (of finite type) acting on the upper half plane. To each parabolic cusp Ki (i = 1, …, h), corresponds a Eisenstein serie

For a closed and bounded set E in the complex plane, let A(E) denote the collection of all functions continuous on E and analytic on E°, its interior; let R(E) denote the collection of all functions which are uniform limits on E of rational functions with poles outside E. Then let A denote the collection of all closed, bounded sets for which A(E) = R(E). The purpose of this paper is to formulate a condition on a set, which is essentially of a geometric nature, in order that the set belong to A. Then using approximation techniques, we shall construct a meromorphic function having a certain boundary behavior on a perfect set; this answers a question raised in [1].

Using the notation in [1], the Lumer-Phillips theorem (3. 1 of [2]) is refined to single parameter groups in real Banach space and real Hilbert space. The theory can be extended to complex spaces.

The notion of module together with many other concepts in abstract algebra we owe to Dedekind [2]. He recognized that the ring of integers OK of a number field was a free Z-module. When the extension K/F is Galois, it is known that K has an algebraic normal basis over F. A fractional ideal of K is a Galois module if and only if it is an ambiguous ideal. Hilbert [4, §§105-112] used the existence of a normal basis for certain rings of integers to develop the theory of root numbers — their decomposition already having been studied by Kummer.