Let C and C′ be two simple closed curves in the complex z-plane which have no point in common and surround the origin. Denote by D the annulus bounded by C and C′. Consider a family {γ} of rectifiable curves γ in D and the family P of all non-negative lower semi-continuous functions ρ = ρ(z) in D. Put

Let S denote the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, G the Galois group of the quotient field extension, and f an element of Z2(G,U(S)) where U(S) denotes the multiplicative group of units of S. A crossed product Δ(f, S, G) whose radical is generated as a left ideal by the prime element II of S is an hereditary order according to the Corollary to Thm. 2. 2 of [2], and we call such a crossed product a II-principal hereditary order. In previous papers the author has studied II-principal hereditary orders Δ(f, S, G) for tamely and wildly ramified extensions S of R (see [10] and [11]). The purpose of this paper is to study II-principal hereditary orders Δ(f, S, G) with no restriction on the extension S of R.

In this article, we prove a finiteness result on the number of log minimal models for 3-folds in $\operatorname{char}p>5$. We then use this result to prove a version of Batyrev’s conjecture on the structure of nef cone of curves on 3-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_{X}$ of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on $X$.

In the classification problems of manifolds, the connected sums of sphere bundles over spheres appear frequently. In fact, the manifolds with certain tangential and homotopy properties come to such connected sums (cf. Tamura [15], [16], Ishimoto [6], [8], [9]). Motivated by those, in this paper and the subsequent paper, we classify connected sums of sphere bundles over spheres up to homotopy equivalence by extending the results of I. M. James and J. H. C. Whitehead [10], [11], which correspond to the case that the connected sums of the above happen to be single sums. We also use Wall [17] in the case when the fibres and the base spaces of bundles are same dimensional.

Un idéal entier I d’un corps quadratique réel k est dit primitif lorsque n ∈ N* et (n) divise I impliquent n = 1. Un idéal entier est primitif si et seulement si il vérifie les trois conditions suivantes:

(i) il n’est divisible par aucun idéal premier inerte,

(ii) il n’est divisible par le carré d’aucun idéal premier ramifié,

(iii) si il est divisible par un idéal premier totalement décomposé, alors il n’est pas divisible par son idéal premier conjugué.

In [1], Maass proved that the Dirichlet series associated with Siegel modular form satisfies a function equation. In this paper, we try to generalize the above fact to the indefinite case.

We construct distinctive surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we prove that a complete noncompact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary if the manifold M is not less curved than a noncompact model surface of revolution and if the total curvature of the model surface is finite and less than 2π. By the first result mentioned above, the second result covers a much wider class of manifolds than that of complete noncompact Riemannian manifolds whose sectional curvatures are bounded from below by a constant.

We consider a positively graded noetherian domain R = ⊕n∈NoRn for which R0 is essentially of finite type over a perfect field K of positive characteristic and we assume that the generic fibre of the natural morphism π: Y = Proj(R) → Y0 = Spec(R0) is geometrically connected, geometrically normal and of dimension > 1. Then we give bounds on the “ranks” of the n-th homogeneous part H2(R)n of the second local cohomology module of R with respect to R+:= ⊕m>0Rm for n < 0. If Y is in addition normal, we shall see that the R0-modules H2R+ (R)n are torsion-free for all n < 0 and in this case our bounds on the ranks furnish a vanishing result. From these results we get bounds on the first cohomology of ample invertible sheaves in positive characteristic.

Let R be a commutative Noetherian local ring. This paper deals with the problem asking whether R is Gorenstein if the nth syzygy module of the residue class field of R has a non-trivial direct summand of finite G-dimension for some n. It is proved that if n is at most two then it is true, and moreover, the structure of the ring R is determined essentially uniquely.

Let K be a finite Galois extension of the rational number field Q and G a Gal(K/Q)-stable finite subgroup of GLn(OK). We have shown that G is of A-type in several cases under some restrictions on K. In this paper, we show that it is true for n = 2 without any restrictions on K.

The universal character of the primitive logic LO in the sense that popular logics such as the lower classical predicate logic LK, the intuitionistic predicate logic LJ, Johansson’s minimal predicate logic LM, etc. can be faithfully interpreted in LO is very remarkable even from the view point of mechanical proof-checking. Since LO is very simple, deductions in LO could be mechanized in a simple form if a suitable formalism for LO is found out. Main purpose of this paper is to introduce a practical formalism for LO, practical in the sense that it is suitable at least for mechanical proof-checking business.

Dans cette note, nous étudions les anneaux d’intégrité primaires et complètement intégralement clos. Krull [1] a posé la question suivante: est-ce qu’un anneau d’intégrité primaire et complètemente intégralement clos est nécessairement un anneau de valuation?

Let X be a locally compact and σ-compact Abelian group and ξ be the Haar measure of X. A positive Radon measure N on X is called a convolution kernel when we regard it as a kernel of potentials of convolution type. M. Itô [4], [6] characterized the convolution kernel which satisfies the domination principle. The purpose of this paper is to characterize the relative domination principle for the convolution kernels.

We will study holomorphic mappings

from a connected complex manifold M of dimension m to a projective algebraic manifold N of dimension n. Assume first that N is of general type, i.e.

where KN→N is the canonical bundle of N. If KN is positive, then N is of general type.

At first, we define three relations ⊇, =, and ⊃ in connection with a pair of logics L and L* as follows:

• L ⊇ L*, if and only if every proposition provable in L* is also provable in L;

• L = L*, if and only if L ⊇ L* and L* ⊇ L;

• L ⊃ L*, if and only if L, ⊇ L* but not L* ⊇ L.

In dieser Note soil die Struktur des Quasinormalteilers vom Exponenten p der endlichen p-Gruppe untersucht werden. Ferner sollen einige Beispiele sowohl im regularen Falle als auch im nicht regulären Falle angeführt werden, die zeigen, daß das erst gesagte Ergebnis im allgemeinen Quasinormalteiler nicht immer zutrifft.

In this note we shall extend, for arbitrary n, Pesin’s [11] bidimensional definition for quasiconformal mappings and establish its equivalence with Gehring’s [7] and Väisälä’s [15] definitions.

In this paper, we give a characterization of the simultaneous unitarizability of any finite set of SL(2, ℂ)-valued functions on and determine all possible ways of the unitarization. Such matrix functions can be regarded as images of the generators for the fundamental group of a surface in an -family, and the results of this paper have applications in the construction of constant mean curvature surfaces in space.

We study the asymptotic behavior in Sobolev norm of the local time of the d-dimensional fractional Brownian motion with N-parameters when the space variable tends to zero, both for the fixed time case and when simultaneously time tends to infinity and space variable to zero.

In a previous paper [4], we studied a class of algebras over a commutative ring R which we called semisimple algebras. Here we shall study simple algebras.