Für einen relativ-galoischen Zahlkörper K über k bezeichnen wir mit G(K/k) die zugehörige galoische Gruppe und mit (K: k) dem Erweiterungsgrade. Ferner seien bzw. K* die gröBten, unverzweigten Erweiterungskörper von K, die beziehungsweise die folgenden Eigenschaften besitzen: die galoische Gruppe abelsch; die galoische Gruppe ist im Zentrum von enthalten; K* ist das Kompositum von K und einem über k abelschen Erweiterungskörper.

We establish the minimal model theory for $\mathbb{Q}$-factorial log surfaces and log canonical surfaces in Fujiki’s class ${\mathcal{C}}$.

The defect relation gives the best-possible estimate, where f is a linearly non-degenerate holomorphic curve in Pn(C) and H1, …, Hq are hyperplanes in Pn(C) which are in general position. However, the case of moving hyperplanes has ever got only n(n + 1) instead of n + 1 (Stoll [4]) and it has not yet been known whether this bound is best-possible or not. In this paper we shall give some particular cases which have the bound n + 1.

The existence of condenser potentials was established in the framework of Dirichlet spaces by Beurling and Deny, cf. Deny [5] or Landkof [10], simply by choosing the potential of minimal energy within a certain convex set. This same idea works for non-symmetric Dirichlet spaces, cf. Bliedtner [3].

This is the second half of the paper entitled “On augmented Schottky spaces and automorphic forms”. In the first half, we introduced new coordinates to the Schottky space, and defined the augmented Schottky space by using the new coordinates. Furthermore we considered relations between the augmented Schottky space and Riemann surfaces with or without nodes.

Soit φ un polynome symétrique de q variables (q ≥ 1), homogène de degré p (p entier ≥ 2), à coefficients complexes. Soient (α1, α2, . . . , p) et (β1, β2, . . . , βp) deux familles de p nombres complexes, tous non nuls, et telles que toutes les différences soient également non nulles pour i ≠ j. Soient enfin (λ1, λ2, . . . , λq) et (μ1, μ2, . . . , μq) deux familles de q nombres complexes. On a alors:

THÉORÈME 1.

.

The exponential decay of the local energy for wave equations in exterior domains of the odd dimensional space has been proved in [1] ~ [6] etc. under the Dirichlet boundary condition and in [5], [7] under the Neumann condition and the other conditions. In this paper, we shall consider this problem for the following equation:

Let X = {X(A); A ∈ Q} be a Lévy’s Brownian motion with the basic time parameter space Q, where Q is taken to be the n-dimensional metric space Qn,k of constant curvature (2 ≤ n ≤ ∞, — ∞ < k: < ∞), i.e., Q is one of

(a) Euclidean space for k = 0, (b) sphere for k > 0 and

(c) real hyperbolic space for K < 0.

The increment X(A) — X(B) is, by definition, Gaussian in distribution and has mean 0 and variance d(A, B), the distance between A and B. The existence of such a Gaussian random field is well known ([3], [4], [16] and [23]).

Let M be a compact connected C∞ manifold with a smooth Riemannian foliation ℱ. That is, (M, ℱ) admits a bundle-like metric in the sense of [7]. In [4] it is shown that if all leaves of ℱ are closed without holonomy, then the space of leaves M/ℱ of the foliation is a manifold and the natural projection M → M/ℱ is a locally trivial fibre space. In the present work we show that for certain of the Riemannian homogeneous foliations, holonomy is the only obstruction to the foliation being a fibration.

Soit h(m) le nombre des classes de , m > 0. Dans [1], [3] et [5] des conditions suffisantes pour que h(m) > 1 sont établies, lorsque m = 4q2+ 1, m = q2 + 4 et m + q2 ± 2. Dans le présent travail on dégage, par l’algorithme de Chatelet, des cycles principaux canoniques d’idéaux réduits pour ces cas. On retrouve alors les résultats de [1], [3] et [5] et on démontre des critères de divisibilité de h(m) dont certains se trouvent dans [4].

Assume that G is a graph with edge ideal $I(G)$ and star packing number $\alpha _2(G)$ . We denote the sth symbolic power of $I(G)$ by $I(G)^{(s)}$ . It is shown that the inequality $ \operatorname {\mathrm {depth}} S/(I(G)^{(s)})\geq \alpha _2(G)-s+1$ is true for every chordal graph G and every integer $s\geq 1$ . Moreover, it is proved that for any graph G, we have $ \operatorname {\mathrm {depth}} S/(I(G)^{(2)})\geq \alpha _2(G)-1$ .

Let (φ1(u), …, φn(u)) be a system of holomorphic functions whose Wronskian does not vanish at origin, where holomorphic functions mean functions holomorphic around origin.

Let Rn+1 = Rn × R be the (n + 1)-dimensional Euclidean space (n ≥ 1). For X ∈ Rn+1, we write X = (x, t) with x ∈ Rn and t ∈ R. We consider parabolic operators of the following form:

(1)

La question posée par Noshiro dans [5] pour les ensembles d’accumulation relatifs à des fonctions pseudo-analytiques classiques a été résolue par (T.) Yosida [12] premièrement et ensuite, le présent auteur [8] a étendu son résultat (théorème 1 de [12]) au cas où un nombre fini ou dénombrable de surfaces de Riemann, qui sont des surfaces de recouvrement, sont transformées dans une autre surface de Riemann par une application continue à dérivées partielles continues sauf en un ensemble fermé de mesure linéaire nulle, ayant pour image un ensemble de mesure linéaire également nulle; le quotient de dilatation n’y est pas nécessairement borné mais subit certaine condition, et les ensembles d’accumulation y sont définis pour les éléments frontières de Kerékjartô-Stoïlow. Une généralisation d’un théorème étoilé de Gross obtenue dans [7] a été utilisée dans sa démonstration. L’auteur a continué son étude sur ce théorème étoilé et un de ses résultats établis dans [9] lui permettra d’améliorer les résultats de [8] dans le présent mémoire. L’idée fondamentale est la même que dans [8] mais nous la récrirons, en nous reportant à [8] le moins possible.

Let G be a locally compact group. Under a neighbourhood U we mean a symmetric (i.e. U = U−1) neighbourhood of the identity e, with the compact closure Ū. If there exists a neighbourhood U containing no subgroup other than the identity group, we say that G has no small subgroup.

Let E be an infinite dimensional real nuclear space and H be its completion by a continuous Hilbertian norm ║ ║ of E. Then we have the relation

where E* is the conjugate space of E. Consider a function C(ξ) on E defined by the formula

(1)

In the 1980s, Franks, Pugh and Shub raised the question “Given any subshift of finite type σA: ΣA → ΣA is there a non-singular Smale flow (or an NS flow for short) on S3 with the suspension of σA as a basic set?” (See [5] and [12]).

In 1952, the late Professor T. Nakayama succeeded in constructing the Galois theory for finite dimensional simple ring extensions [7]. And, we believe, the theory was essentially due to the following proposition: If a simple ring A is Galois and finite over a simple subring B then A is B′-A-completely reducible for any simple intermediate ring B′ of A/B [7, Lemmas 1.1 and 1.2]. Moreover, as was established in [5], Nakayama’s idea was still efficient in considering the infinite dimensional Galois theory of simple rings.

Let g(f), S(f), g*λ(f) be the Littlewood-Paley g function, Lusin area function, and Littlewood-Paley g*λ(f) function of f, respectively. In 1990 Chen Jiecheng and Wang Silei showed that if, for a BMO function f, one of the above functions is finite for a single point in ℝn, then it is finite a.e. on ℝn, and BMO boundedness holds. Recently, Sun Yongzhong extended this result to the case of Campanato spaces (i.e. Morrey spaces, BMO, and Lipschitz spaces). One of us improved his g*λ(f) result further, and treated parametrized Marcinkiewicz functions with Lipschitz kernel μρ(f), μρs(f) and μλ*,ρ(f). In this paper, we show that the same results hold also in the case of rough kernel satisfying Lp-Dini type condition.

We give a discussion on the properties of Δa(x) (− 1 < a < 0), which is a generalization of the error term Δ(x) in the Dirichlet divisor problem. In particular, we study its oscillatory nature and investigate the gaps between its sign-changes for −½ ≤ a < 0.