It is proved that, if is a proper holomorphic map with one-dimensional fibers and a covering map, a point t ∈ T has a neighbourhood U such that is holomorphically convex if and only if is holomorphically convex.

Let X be a 3-dimensional terminal singularity of index ≥ 2. We study projective birational morphisms ϕ: Y → X such that the exceptional divisor of ϕ consists of all prime divisors with discrepancies < 1 (resp. ≤ 1) over X.

It is known classically that abelian varieties of dimension one over the field of complex numbers may be expressed by non-singular Hesse’s canonical cubic plane curves, The purpose of the present paper is to generalize this idea to higher dimensional case.

Let be a Kac-Moody-Borcherds algebra on a field associated to a symetrizable matrix and with Cartan subalgebra . Let be an ad -invariant subalgebra such that the restriction to of the standard bilinear form is nondegenerate. We show that the root system Ψ of is a subsystem according to [Ba] of . Moreover, if a subsystem Ω satisfies some conditions (i.e. Ω is “réduit et presque-clos”) of Ψ, we construct inside of a Kac-Moody-Borcherds algebra with root system Ω.

Let k be a subfield of . We prove similar results in the case of an action of a finite group of k-semi-automorphisms. In particular, we obtain a generalization to the Kac-Moody case of a result by Borel and Tits.

Let be an almost-k-split form of a Kac-Moody algebra. We construct a Kac-Moody k-algebra with root system similar to the system of (save on some multiples of certain roots).

Let p denote an odd prime integer and let q = pf where f is a positive integer. Let ℋ denote the projective symplectic group PSp(4,q), the Dickson group G2(q), or the Steinberg “triality twisted” group over a field Fq of q elements. Then ℋ is simple and the Sylow 2-subgroups of ℋ have centers of order 2 so that involutions which centralize a Sylow 2-subgroup of ℋ form a single conjugacy class of ℋ.

Mr. Z. Kuramochi has recently obtained the following very interesting result concerning the classification of Riemann surfaces [1]:

If we delete from a Riemann surface belonging to the class an arbitrary compact set, the resulting surface remains in the class .

In the present paper we indicate a new method of proving this theorem, simultaneously for the cases B and D.

In this paper we attempt to set up a notion of homomorphism for continuous pseudogroups and show that the kernel exists (as a continuous pseudogroup) in the transitive case. This paper is really an extension of the paper by Kuranishi and Rodrigues [11] which essentially examines the question of the existence (as a continuous pseudogroup) of an image of a homomorphism. A certain amount of overlap in definitions and statements of results was unavoidable, especially in sections 2 and 3, but for many proofs and constructions the reader is referred to that paper. For the basic notions of the theory of continuous pseudogroups as used in section 4, see Kuranishi [9] and for the terminology of the Cartan-Kãhler theory used in section 5, see Kuranishi [6] Fuller expositions may be found in Cartan [I] Kàhler [4], Kumpera [5], Kuranishi [7], and Schouten and v.d. Kulk [13].

Let $A$ be an expansive dilation on $\mathbb{R}^{n}$ and $\unicode[STIX]{x1D711}:\mathbb{R}^{n}\times [0,\infty )\rightarrow [0,\infty )$ an anisotropic growth function. In this article, the authors introduce the anisotropic weak Musielak–Orlicz Hardy space $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ via the nontangential grand maximal function and then obtain its Littlewood–Paley characterizations in terms of the anisotropic Lusin-area function, $g$-function or $g_{\unicode[STIX]{x1D706}}^{\ast }$-function, respectively. All these characterizations for anisotropic weak Hardy spaces $\mathit{WH}_{A}^{p}(\mathbb{R}^{n})$ (namely, $\unicode[STIX]{x1D711}(x,t):=t^{p}$ for all $t\in [0,\infty )$ and $x\in \mathbb{R}^{n}$ with $p\in (0,1]$) are new. Moreover, the range of $\unicode[STIX]{x1D706}$ in the anisotropic $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ coincides with the best known range of the $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of classical Hardy space $H^{p}(\mathbb{R}^{n})$ or its weighted variants, where $p\in (0,1]$.

Soit X un groupe abélien localement compact, non-compact, séparé et dénombrable à l’infini. On désignera par ξ une mesure de Haar fixée sur X. Pour les noyaux de convolution réels sur X, le principe semi-complet du maximum est fondamental dans la théorie du potentiel. Soit N un noyau de convolution réel sur X vérifiant le principe semi-complet du maximum (désigné par Nє(PSM)). Pour déterminer l’allure de N à l’infini, la N-réduite ηN, δ de N à l’infini δ joue un rôle essentiel. Pour une exhaustion de compacts de est décroissante et est une mesure de Radon réelle sur X ou bien, pour une fonction finie et continue dans X à support compact quelconque, est la iV-réduite de N sur CKn. Dans ce cas, est indépendant du choix de Posons

Let S be the product space of finite interval [—r, r] and two points Θ = {θ±1}.

In the previous paper [6], we have considered flows on the measure space (Ω, ℬ, P), which was the projective limit of a certain subspace (Ωn, ℬn, Pn) of the measure space (Sn, ℬ(Sn), Pn), where Sn is the (n − 1) · sphere with radius and Pn is the uniform probability distribution over Sn.

The purpose of this paper is to discuss some ergodic properties of the stepping stone model proposed by Kimura, M. [4] and developed by Weiss, G.H. and Kimura, M. [12]. Our model to be discussed in this paper involves selection in addition to mutation and migration which are dealt with in [4], [12]. Because of the additional factor selection, the stochastic process describing our model becomes complicated and presents particularly interesting profound structure of the random phenomena in question.

A holomorphic n × n-matric automorphic factor with respect to a lattice L in Cg means a system of holomorphic n × n-matrices {μα(z) | α ∈ L} such that

Several years ago, K. Oikawa, my colleague, investigated the properties of Schottky coverings of closed Riemann surfaces, leaving an interesting problem as open [3]:

Does exist a Schottky covering between the basic surface and a given planar covering?

Some important formal systems are really developable from a finite number of axioms in the lower classical predicate logic LK or in the intuitionistic predicate logic LJ. Any system of this kind can be developed in LK (or in LJ) from the single conjunction of all the axioms of the system.

Let Q be the rational number field, K/Q be a finite Galois extension with the Galois group G, and let CK be the ideal class group of K in the wider sense. We consider CK as a G-module. Denote by I the augmentation ideal of the group ring of G over the ring of rational integers. Then CK/I(CK) is called the central ideal class group of K, which is the maximal factor group of CK on which G acts trivially. A. Fröhlich [3, 41 rationally determined the central ideal class group of a complete Abelian field over Q whose degree is some power of a prime. The proof is based on Theorems 3 and 4 of Fröhlich [2]. D. Garbanati [6] recently gave an algorithm which will produce the l-invariants of the central ideal class group of an Abelian extension over Q for each prime l dividing its order.

In a recent paper [HI2] the first named author and S. Ikeda have investigated Gorenstein properties under blowing up of height two equimultiple ideals from the arithmetical point of view. The problem is to what extent does the Gorenstein property of the Rees algebra R(a) = of an equimultiple ideal a of a local ring (A, m) describe the structure of A and a. One result is that if A is a generalized Cohen-Mac aulay ring (i.e. a ring of finite local cohomology) with dim (A) ⊇ 4 and if there is an height two equimultiple prime ideal p such that R(p) is Gorenstein, then A is Gorenstein and p is generated by a regular sequence (see [HI2], Theorem 2.6). That has led to the question whether this result is still valid for dim (∧) = 3. ∧ partial answer was given in [HI2], Proposition 2.10, where “equimultiplicity” was replaced by the stronger condition “p/p2 is flat over A/p”

In a recent paper [1], H. Ozeki has extended the author’s results [2] on local and infinitesimal holonomy groups for connections in linear fiber bundles (whose fiber is a vector space) to general fiber bundles whose structure group is a Lie group. Ozeki’s Lemma 7 (corresponding to the author’s Lemma 5.2) appears to be rather crucial in the development, while the proof is somewhat involved.—This note intends to present a more direct proof of the lemma, stating a case in which the local holonomy group H*(x)and the infinitesimal holonomy group H’(x)coincide:

We give a notion of ordinary Enriques surfaces and their canonical lifts in any positive characteristic, and we prove Torelli-type results for this class of Enriques surfaces.

Recently A. Morimoto [1] has proved the Takens conjecture in the tolerance stability by using the notion of pseudo-orbits and the stochastic stability. He also characterized group automorphisms of a torus to be stochastically stable and clarified the relations to other stabilities.

In this paper we shall give the condition for spherical or projective linear transformations to be stochastically stable.