Le but de la présente note est de clarifier la notion de variété riemanien-ne ou affine symétrique (et plus généralement, localement reductive) du point de vue de la théorie de connexions dans les espaces fibres. Nous donnerons quelques caractérisations des espaces localement symétriques, ou réductives (Th. 1. 2 et 3). On trouvera la définition de dérivation covariante de tenseurs de courbure et de torsion dans la terminologie de la théorie de connexions dans les espaces fibres. Cette définition peut être généralisée à la dérivation covariante d’un tenseur quelconque. Les résultats concernant la dérivation covariante seront publiés ailleurs. Enfin on retrouvera des résultats de Borel-Lichnerowicz. Même si l’on ne trouvera pas de résultats nouveaux dans cette note, on comprendra mieux les espaces symétriques par la théorie de connexion de Cartan.

Let G be a simple, connected, compact and simply-connected Lie group. If k is the field with characteristic zero, then the algebra of cohomology H*(G ; k) is the exterior algebra generated by the elements x1, …, xl of the odd dimension n1, …, nl; the integer l is the rank of G and is the dimension of G.

Discrete linear Weingarten surfaces in space forms are characterized as special discrete $\unicode[STIX]{x1D6FA}$-nets, a discrete analogue of Demoulin’s $\unicode[STIX]{x1D6FA}$-surfaces. It is shown that the Lie-geometric deformation of $\unicode[STIX]{x1D6FA}$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known Lawson correspondence in the constant mean curvature case.

In his paper [11] Shimura defined a correspondence between modular forms of half integral weight and modular forms of integral weight. To each pair (t, f(z)), consisting of a square-free integer t ≥ 1 and a cusp form of weight k/2 (k odd, ≥ 3), level N (divisible by 4) and character ϰ, he associated a certain function f(t)(z) (Ft(z) in Shimura’s notation).

We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations.

We continue to study l-adic iterated integrals introduced in the first part. We shall show that the l-adic iterated integrals satisfy essentially the same functional equations as the classical complex iterated integrals. Next we are studying l-adic analogs of classical polylogarithms.

The following theorem on infinite class field towers is well-known.

The present paper is devoted to the study of differential geometry of Kaehlerian homogeneous spaces. In section 1 we deal with the canonical decomposition of a simply connected complete Kaehlerian space and that of its largest connected group of automorphisms. We know that a simply connected complete Riemannian space V is the product of Riemannian spaces V0, V1, …, Vn, where V0 is a Euclidean space and V1, …,Vn are not locally flat and their homogeneous holonomy groups are irreducible [2]. Moreover, if V is homogeneous, so are all Vk [10]. We shall show that if V is Kaehlerian space with real analytic metric (resp. Kaehlerian homogeneous space), each factor Vk is also Kaehlerian (resp. Kaehlerian homogeneous) and that V is the product of V0, V1, …, Vn as Kaehlerian space. We call this decomposition the de Rham decomposition of the Kaehlerian space V. Although this result is supposedly known, there is no published proof as yet. Using this decomposition theorem we shall show that the largest connected group of automorphisms of a simply connected complete Kaehlerian space with real analytic metric is the direct product of those of the factors of the de Rham decomposition. In the Riemannian case this result has be been established in [3] by one of the authors of the present paper.

Let, for any n ≥ 0,

.

R. Apéry’s proof of the irrationality of ζ(2) and ζ(3) made use of these numbers (see [10]). As a result, many properties of the Apéry numbers were found (see [l]-[9]). In particular, Beukers and Stienstra showed the interesting congruence (see [11, Theorem 13.1]).

The term “simplest” field has been used to describe certain totally real, cyclic number fields of degrees 2, 3, 4, 5, 6, and 8. For each of these degrees, the fields are defined by a one-parameter family of polynomials with constant term ±1. The regulator of these “simplest” fields is small in an asymptotic sense: in consequence, the class number of these fields tends to be large.

We show that a continuous map of a metric space is pointwise recurrent and has the pseudo orbit tracing property if and only if the map is uniformly conjugate to an adding-machine-like map restricted to some invariant subset.

Let H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let , where p, q> 0, α = (α1,…,αn) with αj > -1, j =1,..., n, be the class of all measurable functions f defined on Dn such that

where Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on . We provide a characterization for a function f being in . Using the characterization we prove the following result: Let p> 1, then the Cesàro operator is bounded on the space .

In Part I we introduced the notion of 2. Frobenius extensions of a ring, as a generalization of Kasch’s [10] Frobenius extensions and hence of classical Frobenius algebras. We proved, in I, bilinear (or sesqui-linear, rather, to follow Bourbaki’s terminology) form and scalar product characterizations of Frobenius extensions in such extended sense, generalizing Kasch’s and classical case, and then studied homological dimensions in them, generalizing and refining the results in Eilenberg-Nakayama [4] and Hirata [6]. Dual bases were considered in case of quasi-free (2.) Frobenius extensions Also the case of a semi-primary or S-ring ground ring was studied.

Dans la théorie du potentiel par rapport à la noyau-fonction continue (au sens large), les théorèmes de Ninomiya et ceux de Kishi sont très originels (cf. [9] et [7]). Dans cet article, en généralisant la notion du noyau, on se propose de discuter quelques théorèmes analoques.

We shall introduce the main theorems of value distribution theory in the most general case of complex dimension one: analytic mappings of arbitrary Riemann surfaces into arbitrary Riemann surfaces. The case of mappings of arbitrary Riemann surfaces into closed Riemann surfaces was discussed in [41]. Earlier literature on analytic mappings is listed in the Bibliography.

R. Brauer not only laid the foundations of modular representation theory of finite groups, he also raised a number of questions and made conjectures (see [1], [2] for instance) which since then have attracted the interest of many people working in the field and continue to guide the research efforts to a good extent. One of these is known as the “Height zero conjecture”. It may be stated as follows:

Let R be a field and S a separable algebraic closure of R with galois group R. In [8] Harrison succeeded in describing R/′R in terms of R only. More precisely, he constructed a certain complex (R, Q/Z) and proved Homc, where Homc denotes continuous homomorphisms and H2 stands for the second cohomology group of the complex . In this paper, which is mainly expository in nature, we reexamine Harrison’s proof and show how [8] connects with Kummer theory and the theory of galois algebras [16]. We emphasize that most of the ideas on which this paper is based originate in [8].

We formulate a generalization of K. Takeuchi’s method to classify smooth Fano 3-folds and use it to give a list of numerical possibilities of ℚ-Fano 3-folds X with Pic X = ℤ(−2KX) and h0(−KX) ≥ 4 containing index 2 points P such that (X, P) ≃ ({xy + z2 + ua = 0}/ℤ2(1, 1, 1, 0), o) for some a ∈ ℕ. In particular we prove that then (–KX)3 ≤ 15 and h0(–KX) ≤ 10. Moreover we show that such an X is birational to a simpler Mori fiber space.

Let P and Q be the stochastic transition operators of two time-homogeneous, irreducible Markov chains with countable, discrete state spaces X and Y, respectively. On the Cartesian product Z = X x Y, define a transition operator of the form Ra = a·P + (1 — a) · Q, 0 < a < 1, where P is considered to act on the first variable and Q on the second. The principal purpose of this paper is to describe the minimal Martin boundary of Ra (consisting of the minimal positive eigenfunctions of Ra with respect to some eigenvalue t, also called t-harmonic functions) in terms of the minimal Martin boundaries of P and Q.

In this paper we shall study the holomorphic sectional curvature of the Bergman metric on a domain