A comprehensive study of the generalized Lambert series $\sum _{n=1}^{\infty }\frac{n^{N-2h}\text{exp}(-an^{N}x)}{1-\text{exp}(-n^{N}x)},0<a\leqslant 1,~x>0$, $N\in \mathbb{N}$ and $h\in \mathbb{Z}$, is undertaken. Several new transformations of this series are derived using a deep result on Raabe’s cosine transform that we obtain here. Three of these transformations lead to two-parameter generalizations of Ramanujan’s famous formula for $\unicode[STIX]{x1D701}(2m+1)$ for $m>0$, the transformation formula for the logarithm of the Dedekind eta function and Wigert’s formula for $\unicode[STIX]{x1D701}(1/N),N$ even. Numerous important special cases of our transformations are derived, for example, a result generalizing the modular relation between the Eisenstein series $E_{2}(z)$ and $E_{2}(-1/z)$. An identity relating $\unicode[STIX]{x1D701}(2N+1),\unicode[STIX]{x1D701}(4N+1),\ldots ,\unicode[STIX]{x1D701}(2Nm+1)$ is obtained for $N$ odd and $m\in \mathbb{N}$. In particular, this gives a beautiful relation between $\unicode[STIX]{x1D701}(3),\unicode[STIX]{x1D701}(5),\unicode[STIX]{x1D701}(7),\unicode[STIX]{x1D701}(9)$ and $\unicode[STIX]{x1D701}(11)$. New results involving infinite series of hyperbolic functions with $n^{2}$ in their arguments, which are analogous to those of Ramanujan and Klusch, are obtained.

This paper investigates the existence of positive solutions of nonresonant singular boundary value problem of second order differential equations. A necessary and sufficient condition for the existence of C[0, 1] positive solutions as well as C1[0, 1] positive solutions is given by means of the method of lower and upper solutions with the fixed point theorems.

1. When a permutation group G on a set Ω is given, a transitive extension G of G is defined to be a transitive permutation group on the set Γ which is a union of Ω and a new point ∞ such that the stabilizer of ∞ in G1 is isomorphic to G as a permutation group on Ω. The purpose of this paper is to prove that many known simple groups which can be represented as doubly transitive groups admit no transitive extension. Precise statement is found in Theorem 2. For example, the simple groups discovered by Ree [5] do not admit transitive extensions. Theorem 2 includes also a recent result of D. R. Hughes [3] which states that the unitary group U3(q) q>2 does not admit a transitive extension. As an application we prove a recent theorem of H. Nagao [4], which generalizes a theorem of Wielandt on the non-existence of 8-transitive permutation groups not containing the alternating groups under Schreier’s conjecture.

In the late sixties and early seventies the operation of weak normalization was formally introduced first in the case of analytic spaces and later in the abstract scheme setting (cf. [6] & [4]). The notion arose from a classification problem. An unfortunate phenomenon in this area occurs when one tries to parametrize algebraic objects associated with a space by an algebraic variety; the resulting variety is, in general, not uniquely determined and may, for example, depend on the choice of coordinates. Under certain conditions one does know that the normalization of the parameter variety is unique. The price one pays for passing to the normalization is that often this variety no longer parametrizes what it was intended to; one point on the original parameter variety may split into several in the normalization. This problem is avoided if one passes instead to the weak normalization of the original variety. One then obtains a variety homeomorphic to the original variety with universal mapping properties that guarantee uniqueness.

Ce travail est une suite du mémoire [8] paru dans ce journal. Dans [8], on a étudié la structure du groupe connexe d’isotropie d’un espace homogène de Stein G/H d’un groupe de Lie semi-simple complexe et connexe G et on a montré que le groupe H est égal au complexifié d’un sous-groupe compact maximal de H ([8], Théorème 3). Le but essentiel de ce travail est d’étendre ce résultat au cas d’un espace homogène de Stein d’un groupe de Lie complexe et connexe quelconque.

Let G be a finite group of odd order with an automorphism θ of order 2. (We use without further reference the fact, established by W. Feit and J. G. Thompson, that all groups of odd order are soluble.) Let Gθ denote the subgroup of G formed by the elements fixed under θ. It is an elementary result that if Gθ = 1 then G is abelian. But if we merely postulate that Gθ be cyclic, the structure of G may be considerably more complicated—indeed G may have arbitrarily large soluble length.

We proved that any hyperconvex manifold has a complete Bergman metric.

In the ring Ok of algebraic integers of a number field K the group Ik of ideals of Ok modulo the subgroup Pk of principal ideals is a finite abelian group of order hk, the class number of K. The determination of this number is an outstanding problem of algebraic number theory.

It is well known that a doubly transitive group has an irreducible character χ1 such that χ1(R) = α(R) − 1 for any element R of and a quadruply transitive group has irreducible characters χ3 and χ3 such that χ2(R) = where α(R) and β(R) are respectively the numbers of one cycles and two cycles contained in R. G. Frobenius was led to this fact in the connection with characters of the symmetric groups and he proved the following interesting theorem: if a permutation group of degree n is t-ply transitive, then any irreducible character of the symmetric group of degree n with dimension at most equal to is an irreducible character of .

We investigate period maps of polarized variations of Hodge structures of weight one or two. We treat the case when the period domains are bounded symmetric domains. We deal with a relationship between canonical extensions of some Hodge bundles and automorphic forms. As applications, we obtain a canonical bundle formula for certain algebraic fiber spaces, such as Abelian fibrations, K3 fibrations, and solve Iitaka’s famous conjecture Cn,m for some algebraic fiber spaces.

A linear Lie algebra is called toroidal if it is abelian and consists of semi-simple transformations. The maximum, t(L), of the dimensions of the toroidal subalgebras of the derivation algebra, Δ(L), is an invariant of L. This paper is mainly concerned with the relation between the magnitude of t(L) for nilpotent L and the structures of L and Δ(L).

D’abord nous définissons les ensembles de Cantor généralisés. Soient k1, k2, … des nombres entiers supérieurs à 1 et soient p1, p2, … des nombres finis quelconques également supérieurs à 1. On pose lq=l/(kqpq). Soit I un intervalle de longueur d > 0. On enlève de I (kq — l) intervalles de même longueur tels qu’il reste kq intervalles de même longueur lqd. On appelle cette opération la q-opération appliquée à I. On commence par appliquer l’1-opération à [0,1], on applique la 2-opération à chacun des intervalles qui restent, puis on applique la 3-opération à chacun des intervalles qui restent, et ainsi de suite. On appellera l’ensemble limite restant un ensemble de Cantor généralisé dans E1 et le notera F = F(kq, pq). Notre définition dans En = {(x1, …, xn)}, l’espace euclidien à n dimensions (n ≧ 2), est la suivante: Soit un ensemble de Cantor généralisé défini sur l’axe de xj; nous appellerons l’ensemble produit F = F1 × … × Fn un ensemble de Cantor généralisé dans En (n ≧ 2). II sera appelé symétrique si F1 = … =Fn.

The aim of this paper is to extend and unify several results concerning complete ideals in 2-dimensional regular local rings by using the theory of joint reductions and mixed multiplicities. The theory of complete ideals in a 2-dimensional regular local ring was developed by Zariski in his 1938 paper [Z]. This theory is presented in a simpler and general form in [ZS, Appendix 5] and [H2].

We consider the system F4 (a, b, c) of differential equations annihilating Appell's hypergeometric series F4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric series F4. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation of F4(a, b, c) and the twisted period relations for the fundamental systems of solutions of F4.

The aim of the present work is to determine the Galois group of the maximal abelian extension ΩA over an algebraic number field Ω of finite degree, which we fix once for all.

Let Z be a continuous character of the Galois group of ΩA/Ω. Then, by class field theory, the character Z is also regarded as a character of the idele group of Ω. We call such Z character of Ω. For our purpose, it suffices to determine the group Xl of the characters of Ω whose orders are powers of a prime number l.

Affine Kac-Moody algebras represent a well-trodden and well-understood littoral beyond which stretches the vast, chaotic, and poorly-understood ocean of indefinite Kac-Moody algebras. The simplest indefinite Kac-Moody algebras are the rank 2 Kac-Moody algebras (a) (a ≥ 3) with symmetric Cartan matrix , which form part of the class known as hyperbolic Kac-Moody algebras. In this paper, we probe deeply into the structure of those algebras (a), the e. coli of indefinite Kac-Moody algebras. Using Berman-Moody’s formula ([BM]), we derive a purely combinatorial closed form formula for the root multiplicities of the algebra (a), and illustrate some of the rich relationships that exist among root multiplicities, both within a single algebra and between different algebras in the class. We also give an explicit description of the root system of the algebra (a). As a by-product, we obtain a simple algorithm to find the integral points on certain hyperbolas.

Let K be a field and (K) the Brauer group of K. It consists of the similarity classes of finite central simple algebras over K. For any field extension F/K there is a natural mapping (K) → (F) which is obtained by assigning to each central simple algebra A/K the tensor product which is a central simple algebra over F. The kernel of this map is the relative Brauer group (F/K), consisting of those A ∈(K) which are split by F.

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.

The object of this paper is an investigation of existence problems and Dirichlet problems on an abstract Riemann surface in the sense of Weyl-Radó or on a covering surface over it, and of boundary correspondence in the conformal mapping of the surface.

By explicit studying on theta-multipliers (i.e. the multipliers which appear in theta-transformation formulas under general modular substitutions), we can naturally get the reciprocity law of Gauss sums or quadratic residue symbols. This remarkable fact, by Cauchy, Kronecker, Hecke and others, is very classical, but its theoretical meaning has not been sufficiently clear yet.