Dans l’espace de Dirichlet il existe des mesures des condensateurs et inversement un certain espace de fonctions est léespace de Dirichlet séil y existe des mesures des condensateurs. Dans ce mémoire nous considérons le problème d’existence des mesures des condensateurs par rapport ä un noyau positif et continu au sens large, symétrique ou non.

Deny introduced in [4] the notion of functional spaces by generalizing Dirichlet spaces. In this paper, we shall give the following necessary and sufficient conditions for a functional space to be a real Dirichlet space.

Let be a regular functional space with respect to a locally compact Hausdorff space X and a positive measure ξ in X. The following four conditions are equivalent.

In der Differentialgleichung w″+a1(z)w‣w‣+a0(z)w=0 seien die Koeffizienten aj(z) ganze Funktionen. Dann ist jede Lösung w(z) eine ganze Funktion und genau dann von endlicher Ordnung, wenn die Koeffizienten Polynome sind. In diesem Fall gilt für die Ordnung

die Beziehung λ(w)= m/2, m ≧ 0 ganzzahlig.

Il y a quelques années, CHOQUET a démontré [5], en théorie classique du potentiel (ou pour certains noyaux) que l’ensemble E des points où un ensemble e est effilé est contenu dans un ouvert ω, tel que ω∩e soit de capacité arbitrairement petite. Cela contient les résultats-elefs, connus depuis longtemps que E∩e est polaire et que si e est ouvert, p. ex. borné, E ∩ ∂e est de mesure harmonique nulle en tout point de e. La partie nouvelle du résultat général est que les points de Ce où e est effilé peuvent être enfermés dans un ouvert ω tel que ω∩e soit de capacité arbitrairement petite et l’effilement n’intervient plus pour un ensemble qu’aux points du complémentaire, d’une façon équivalente à la notion de topologie fine.

We shall consider a class of Markov processes (n(t), x(t)) with the continuous time parameter t∈e[0, ∞), whose state space is {1, 2,..., N}×R1 . We shall assume that the processes are spacially homogeneous with respect to X∈R1 . In detail, our assumption is that the transition function

Fij(x,t) = P(n(t) = j, x(t)≦x|n(0) = i,x(0) = 0), t > 0, 1≦i, j,≦N, x∈R 1satisfies following conditions (1, 1)~(1,4).

Let W be a plane domain such that ∞∉W. Given a point a∉W and a boundary component C of W, consider the family consisting of all the functions f satisfying the following conditions: f is regular and univalent in W, f(a) = 0, f′(a) = l, and the image f(C) of C under f is the outer boundary component of the image domain f(W). Set

and

Recently a number of authors have studied the application of Grunsky’s coefficient inequalities to the study of the Bieberbach conjecture for the class of normalized regular univalent functions f(z) in the unit circle |z|< 1

Charzynski and Schiffer [2] applied this result to give an elementary proof of the inequality | a4 | Ȧ 4. One of the present authors [8] proved that if a2 is real non-negative then A natural first step in the study of the inequality for a coefficient is to prove local maximality for a2 near to 2.

Let be a transitive group of degree n, and let be the stabilizer of a symbol in ®. Then we owe to J.S. Frame the following remarkable relations between the lengths n i of the orbits of and the degrees fi of the absolutely irreducible components of the permutation matrix representation * of :

(A) If the irreducible constituents of * are all different, then the rational number

is an integer, where k is the number of the orbits of

Suppose Ω is a simply connected domain which is mapped conformally onto a disk. A much studied problem is the behavior of the mapping function at an accessible boundary point P of Ω, in particular the question, under what conditions the map is ‘ “conformai” at such a point (a) in the sense that angles are preserved as P is approached from Ω (“semi-conformality” at P) and (b) the dilatation at P is finite and positive. In his fundamental paper [8] in 1936, A. Ostrowski established a necessary and sufficient condition (depending on the geometry of the domain only) for the validity of the first property which subsumes all previous results and establishes a definitive solution of this problem.

In this paper we study certain injective resolutions for some Hopf algebras. An injective resolution for a coalgebra A with augmentation η is defined to be such an exact sequence that

where K is the basic field, Xn are injective A-comodules and δn are morphism of A-comodules.

In a recent note bearing the same title (SIAM Journ. of Numerical Analysis, Vol. 3, 1966), I showed that there existed Weierstrass zeta functions having a vanishing pseudo-period. Cf. line 8, p. 376 of Saks-Zygmund, Analytic Functions. Professor Peter Henrici kindly pointed out to me that the mere existence of a Weierstrass zeta function with a vanishing pseudo-period was readily concluded from a classical formula of the theory of elliptic functions (Hurwitz-Courant, Funktionentheorie, 5th ed., p. 210) which reduces the question to the study of the zeros of

The electric capacity of a conductor in the 3-dimensional euclidean space is defined as the ratio of a positive charge given to the conductor and the potential on its surface. The notion of capacity was defined mathematically first by N. Wiener [7] and developed by C. de la Vallée Poussin, O. Frostman and others. For the history we refer to Frostman’s thesis [2]. Recently studies were made on different definitions of capacity and related notions. We refer to M. Ohtsuka [4] and G. Choquet [1], for instance. In the present paper we shall investigate further some relations among various kinds of capacity and related notions. A part of the results was announced in a lecture of the author in 1962.

Soit E un espace vectoriel topologique localement convexe, E un ensemble fermé dans E et f une application complètement continue de E dans E. M. Nagumo a montré, après avoir défini le degré topologique d(b, E, I—f) de l’application I—f de E en un point b ∉ (I—f)(∂E), la validité des propriétés importantes dont jouit le degré topologique. Mais dans sa démonstration du théorème de produit, il reste encore une lacune à combler.

Un opérateur différentiel P(x,D) à coefficients C∞ est dit hypoelliptique si toute solution de l’équation

P(x,D)u=f

est indéfiniment derivable quand f∈C∞. II est bien connu qu’un opérateur formellement hypoelliptique est hypoelliptique (dont la démonstration a été donnée par plusieurs auteurs). D’autre part, Trèves [2] a introduit une condition suffisante sous laquelle un opérateur est hypoelliptique et qui englobe le cas formellement hypoelliptique (voir Définition 1).

This article is a continuation and completion of our study in the previous one [1] in which we gave the defining equations of abelian varieties projectively embedded by theta functions of level three. These defining equations are nothing else the canonical generators of theta relations for theta functions with ⅓—characteristics. In the present article the canonical generators of 3 theta relations will be given for the wider class of theta functions with rational characteristics.

In this paper we consider the following quasi-linear parabolic equations

Lu = ut — div A(x, t, u, ux)+B(x, t, u, ux= 0,

where A is a given vector function of the variables x,t,u,uX , and B is a given scalar function of the some variables. We assume that they are difined in the rectangle

Es bezeichne Rn(n≥2) den durch den Alexandroff-Punkt kompaktifizierten n-dimensionalen euklidischen Raum, [xl ,......,xn], ein Koordinatensystem von Rn, x = (x1 ,......,xn ) einen (endlichen) Punkt von Rn und

dessen Norm. Wir betrachten ein im Endlichen liegendes Gebiet G von Rn , definieren auf G einen linearen Operator

Consider the positive and twice continuously differentiable solutions u of the heat equation

in an open t-strip Ω = Rn×(0,T) for some T>0, where Rn is the n-dimensional Euclidean space.

In this note, we prove a theorem of Fatou type on u and, as its application, the uniqueness theorem for the Cauchy problem of ( 1 ).

Theory of real and time continuous martingales has been developed recently by P. Meyer [8, 9]. Let be a square integrable martingale on a probability space P. He showed that there exists an increasing process ‹X›t such that

This paper contains part of the author’s Ph.D. dissertation directed by Frederick Bagemihl at Wayne State University. The research was supported by a grant from the Michigan Institute of Science and Technology.

Alice Roth has made an extensive study of entire meromorphic functions with prescribed behavior along half rays emanating from the origin (6). The question arose whether analogous results could be found for functions meromorphic in the unit disk with the same behavior prescribed along an exhaustive class of spirals emanating from the origin. In this paper, I present a class of spirals which satisfactorily fills this role. However, I make no claim to the effect that only this class will suffice.