G. Rabinowitsch [3] generalized the concept of the Euclidean algorithm and proved a theorem on a criterion in order that the class number of an imaginary quadratic number field is equal to one:

Theorem. It is necessary and sufficient for the class number of an imaginary quadratic number fieldD = 1 — 4m, m > 0, to be one that x2 — x + m is prime for any integer x such that 1 ≤ x ≤ m — 2.

We study some conjectures on the endomorphism algebras of the co-homology of Deligne-Lusztig varieties which are a refinement of those of [BMi].

Our object of study is a rational map  defined by homogeneous forms $g_{1},\ldots ,g_{n}$ , of the same degree $d$ , in the homogeneous coordinate ring $R=k[x_{1},\ldots ,x_{s}]$ of $\mathbb{P}_{k}^{s-1}$ . Our goal is to relate properties of $\unicode[STIX]{x1D6F9}$ , of the homogeneous coordinate ring $A=k[g_{1},\ldots ,g_{n}]$ of the variety parameterized by $\unicode[STIX]{x1D6F9}$ , and of the Rees algebra ${\mathcal{R}}(I)$ , the bihomogeneous coordinate ring of the graph of $\unicode[STIX]{x1D6F9}$ . For a regular map $\unicode[STIX]{x1D6F9}$ , for instance, we prove that ${\mathcal{R}}(I)$ satisfies Serre’s condition $R_{i}$ , for some $i>0$ , if and only if $A$ satisfies $R_{i-1}$ and $\unicode[STIX]{x1D6F9}$ is birational onto its image. Thus, in particular, $\unicode[STIX]{x1D6F9}$ is birational onto its image if and only if ${\mathcal{R}}(I)$ satisfies $R_{1}$ . Either condition has implications for the shape of the core, namely, $\text{core}(I)$ is the multiplier ideal of $I^{s}$ and $\text{core}(I)=(x_{1},\ldots ,x_{s})^{sd-s+1}.$ Conversely, for $s=2$ , either equality for the core implies birationality. In addition, by means of the generalized rows of the syzygy matrix of $g_{1},\ldots ,g_{n}$ , we give an explicit method to reduce the nonbirational case to the birational one when $s=2$ .

Dans ce travail nous étudierons le problème de classifier les espaces fibres principaux holomorphes de groupe GL(m, C) sur un tore complexe admettant des connexions holomorphes i.e. la classification des espaces fibres vectoriels holomorphes sur un tore complexe admettant des connexions holomorphes. Récemment M. Matsushima [6] a démontré qu’un espace fibré vectoriel holomorphe sur un tore complexe admettant une connexion holomorphe de fibre de dimension 2 possède nécessairement une connexion holomorphe intégrable c.à.d. une connexion holomorphe dont la forme de courbure est nulle. D’autre part d’après Atiyah [1] on sait qu’un espace fibré holomorphe de base M et de groupe structural G admettant une connexion holomorphe intégrable est uu espace fibré principal associé au revêtement universel de la base M par une représentation du groupe fondamental de M dans G et vice versa.

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.

Let H(m) denote the class number of the field where Q is the rational number field and ζm is a primitive m-th root of unity for a positive rational integer m.

The definition of semistable laws was originally given by Lévy in [11]. Two books, Kagan, Linnik and Rao [6] and Ramachandran and Lau [13], call a probability measure on R “semistable” when it is nondegenerate and its characteristic function (ch.f.) f(z) does not vanish on R and satisfies a functional equation of the form

for some real numbers b (0 < | b | <1) and c > 1.

In his book P. Lévy discussed certain permutation groups of natural numbers in connection with the theory of functional analysis. Among them the group , called the Lévy group after T. Hida, has been studied along with Hida’s theory of white noise analysis and has become very important keeping profound contact with the Lévy Laplacian which is an infinite dimensional analogue of the ordinary Laplacian.

Let X be a smooth rational surface. In this paper, we prove the rationality of the moduli space MX,L(2; c1; c2) of rank two L-stable vector bundles E on X with det (E) = c1 ∈ Pic(X) and c2(E) = c2 ≫ 0.

The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on n-dimensional smooth projective varieties X with an n-block collection B which generates the bounded derived category To this end, we use the theory of n-blocks and Beilinson type spectral sequence to define the notion of regularity of a coherent sheaf F on X with respect to the n-block collection B. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we compare our definition of regularity with previous ones. In particular, we show that in case of coherent sheaves on ℙn and for the n-block collection Castelnuovo-Mumford regularity and our new definition of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a multiprojective space ℙn1x…x ℙnr with respect to a suitable n1 +…+ nr-block collection and we compare it with the multigraded variant of the Castelnuovo-Mumford regularity given by Hoffman and Wang in [14].

En continuant l’étude d’un problème soulevé dans [3] et plus tard abordé par un de nous dans [1] et [2], nous avons publié deux “abstracts” (voir [6] et [7]) sur certains calculs propositionnels appellés Jn, 1 ≤ n ≤ 5, et leurs relations avec le postulat de la séparation. On développe maintenant les systèmes Jn, 1 ≤ n ≤ 5, et on esquisse la construction des calculs correspondants de prédicats de premier ordre. Enfin, on démontre la non trivialité des théories correspondantes des ensembles quand on utilise des postulats comme ceux de Zermelo-Fraenkel, mais avec le schéma de la séparation formulé sans les restrictions ad hoc pour éviter les paradoxes.

Let V be a projectively embeddable complete non-singular variety of dimension n > 1. Let f be a projective embedding of V, U a non-singular variety, W a non-singular variety and φ a morphism of W onto U such that φ-1(u0) = f(V) for some point u0 of U. Denote by ∑(V) the set of all those complete non-singular fibres φ-1(u), u ∈ U, as we consider all possible (f, U, W). Suppose that we call members of ∑(V) (algebraic) deformations of V and propose to study ∑(V) from the stand point of algebraic geometry, as a generalization of the case of curves. This has been taken up at least locally by Kodaira, Spencer, Kuranishi and others in the case of characteristic 0 from a little more general point of view of complex manifolds (cf. [9] and references given in [16]).

We compute a class of Sarkisov links from Fano 3-folds embedded in weighted Grassmannians using explicit methods for describing graded rings associated to a variation of geometric invariant theory (GIT) quotient.

In this note we show in § 1, as the main result, that any connected Lie subgroup of the general linear group GL(n, R) can be realized as the holonomy group of a linear connection, i.e. the homogeneous holonomy group of the associeted affine connection, defined on an affine space of dimension n (n ≧ 2).

Several kinds of generalizations of classical Abel Theorem in algebraic curves are known, for example see [12] and [13]. It seems to the author these are all regarded as local relations among rational differential forms. In this article we shall try to generalize Abel Theorem for integrals of rational forms in some specific cases where these can be described in terms of hyper-logarithms (for the definition see [3] and [4], Theorem 2). Trigonometric functions have been generalized to higher dimensional cases by L. Schläfli who has obtained a very important variational formula related to the volume of a spherical simplex [16].

We develop the algebraic theory of log abelian varieties. This is Part II of our series of papers on log abelian varieties, and is an algebraic counterpart of the previous Part I ([6]), where we developed the analytic theory of log abelian varieties.

In 1953 N. Itô defined the conjugate rank of a finite group as the number of distinct sizes, not equal to 1, of the conjugacy classes of the group [7].

Recently, R. Rentschler [4] proved that if k is a field of characteristic 0, any action of the additive group Ga on the affine plane is equivalent to an action of the form t(x, y) = (x, y + tf(x), where t∈Ga(k), (x, y)∈k2 and f∈k[X].

For every r-th order Weil functor TA, we introduce the underlying k-th order Weil functors We deduce that is an affine bundle for every manifold M. Generalizing the classical concept of contact element by C. Ehresmann, we define the bundle of contact elements of type A on M and we describe some affine properties of this bundle.

It is our object in this paper to show that the recent results of K. Ito and M. Nisio [4] on the oscillation function of Gaussian processes on [0,1] are valid for Gaussian processes with a general multiparameter “time” set T. Except in extending Theorem 4 of [4] where we assume T to be the d-dimensional cube, in all other cases we allow T to be a separable metric space. Despite the generality of the time set, the proofs are achieved essentially using the method of the above mentioned authors. However, in Theorem 1 below we find the use of Lemma 6 of [5] more convenient than the approach via orthogonal expansions and Kolmogorov’s zero-one law as is done in [4].