In this paper, Hinkkanen's problem (1984) is completely solved, i.e., it is shown that any meromorphic function $f$ is determined by its zeros and poles and the zeros of ${{f}^{\left( j \right)}}$ for $j=1,2,3,4$.

Let SO*(2n) be the group of quaternionic n × n matrices A satisfying A*JA = J, where J is a fixed skew-hermitian invertible matrix. An element R ∊ SO*(2n) is called a reflection if R2 = In and R — In has rank one. We assume that n ≧ 2, in which case S*(2n) is generated by reflections. The length of A ∊ SO*(2n) is the smallest integer k(≧0) such that A can be written as A = R1R2 … Rk where R1, …, Rk are reflections. In this paper, for each A ∊ SO*(2n), we compute its length l(A). Set r(A) = rank (A — In). Already in Section 3 we are able to show that the difference δ = l(A) – r(A) can take only three values 0, 1, or 2. The remainder of the paper deals with the problem of separating these three possibilities. The main results are stated in Section 4 and proved in Section 6. The intermediate Section 5 consists of a sequence of lemmas which are needed for the proof. Clearly l(A) depends only on the conjugacy class of A and the main results in Section 4 are stated in terms of conjugacy classes. For the description of conjugacy classes in SO*(2n) we refer the reader to [1]. The present paper relies heavily on our previous paper [5] where the analogous problem was solved for the groups U(p, q). It is worth remarking that only the various lemmas from that paper were used but not the main theorem.

A space X is locally planar if each point of X has a neighborhood which is embeddable in the plane. If X is a closed, locally planar subset of E3, we will say that X is locally tame if each point of X has a neighborhood in X which lies on a tame disk in E3; if every cell-like subset of X has such a neighborhood, we say that X is strongly locally tame.

This paper sets forth conditions sufficient that the generalized Lototsky method (F, dn) be regular and Mercerian. If the dn's are real and of constant sign, then the conditions are also necessary. Moreover, it follows that if f is a polynomial, then under the same conditions the method (f,dn) is equivalent to the Sonnenschein method generated by f . Various related results are also given.

Definition 2.1. Let f be a nonconstant function holomorphic on the closed unit disk and let be a complex sequence with f (l) + dn ≠ 0. Suppose

For every ( differentiable) bundle E over a manifold M, Jk(E) denotes the set of all k-jets of local (differentiable) sections of the bundle E. Jk(E) is a bundle over M such that if X is a section of E, then

is a (differentiable) section of Jk(E). If E is a vector bundle, Jk(E) is a vector bundle and we have the canonical exact sequence of vector bundles

where Sk(T*) is the symmetric Whitney tensor product of the cotangent vector bundle T* of M. and π is the canonical morphism which associates to each k-jet of section its jet of inferior order.

We explicitly find regulators of an infinite family $\{{{L}_{m}}\}$ of the simplest quartic function fields with a parameter $m$ in a polynomial ring ${{\mathbb{F}}_{q}}\left( t \right)$, where ${{\mathbb{F}}_{q}}$ is the finite field of order $q$ with odd characteristic. In fact, this infinite family of the simplest quartic function fields are subfields of maximal real subfields of cyclotomic function fields having the same conductors. We obtain a lower bound on the class numbers of the family $\{{{L}_{m}}\}$ and some result on the divisibility of the divisor class numbers of cyclotomic function fields that contain $\{{{L}_{m}}\}$ as their subfields. Furthermore, we find an explicit criterion for the characterization of splitting types of all the primes of the rational function field ${{\mathbb{F}}_{q}}\left( t \right)$ in $\{{{L}_{m}}\}$.

In achieving his celebrated results on the homology groups of fibre spaces, J. P. Serre used the exact couple of a fibring defined by J. Leray. One of his main tools was the so-called two-term condition on the E2 term of this exact couple, which, if satisfied, yielded exact sequences, such as those of Gysin and Wang (see (5), Chapter IX). H. Fédérer, in (3), defined an exact couple (X, F, v) on the mapping space M(X, Y) = {ƒ:X → Y|X, Y are spaces and ƒ is continuous} with the compact-open topology, where X is a locally finite CW complex and Y is arc-connected and n-simple for all n.

This study of r-separating sets was originally motivated by the maximum-flow-minimum-cut theorem of finite networks [1 ; 2]. In working toward a continuous version of the maximum-flow-minimum-cut theorem from the usual discrete version, one is led directly to the notion of r-connectedness of a set as defined below. This notion of r-connectedness also has a very simple intuitive interpretation. Intuitively speaking, a set of points in the plane is r-connected if a person starting at any one point of the set is able to reach any other point of the set by jumping from point to point within the set, but never jumping a distance exceeding r in any one jump.

Nous renvoyons le lecteur à [4] pour toutes les définitions non expliquées ci-dessous.

Un demi-groupe inverse S est un demi-groupe pour lequel pour tout élément a ∊ S il existe un seul inverse a–1, c'est-à-dire un élément a–l qui satisfait aux conditions aa–1a = a et a–1aa–1 = a–1. Une application λ : S → S, x → λx est appelée une translation à gauche si λ(xy) = (λx)y; une application ρ : S → S, x → xρ est dite translation à droite si (xy)ρ = x(yρ). Les translations à gauche (à droite) forment un demi-groupe Λ(S) [P(S)]. L'ensemble

forme un sous-demi-groupe de Λ(S) × P(S) qui est appelé l'enveloppe de translations de S. Si pour tout a ∊ S on définit λa : S → S, x → ax, et ρa : S → S, x → xa, on appelle

la partie interne de Ω(S).

In a recent paper Mostert and Shields (4) showed that if a space homeomorphic to the non-negative real numbers is a certain type of topological semigroup, then the semigroup must be that of the non-negative real numbers with the usual multiplication. Somewhat earlier Faucett (2) showed that if a compact connected ordered space is a suitably restricted topological semigroup, then it must be both topologically and algebraically the same as the unit interval of real numbers with its usual multiplication.

In studying certain binary relations on topological spaces there have become known (see, in particular, Wallace (5) and the author (3)) a number of properties analogous to those possessed by topological semigroups. Because of these analogous properties between relations and semigroups the author was motivated by the general nature of the Faucett and Mostert-Shields results (that is, that the multiplication assumed turned out to be the same as the usual multiplication) to feel that certain relations on a connected ordered space should turn out to be the same as the orders whose order topologies are the topology on the space.

Let M be a complex manifold of dimension n which admits a Kähler metric, and let D be a relatively compact domain on M whose boundary B is a C ∞ submanifold of M of real codimension one. The object of this paper is to use the potential theory associated with the Laplace-Beltrami operator on M to characterize the continuous functions on B which have holomorphic extensions to D.

The enumeration of the H-equivalence classes of multiplications on a space is a topic of current interest. In this paper we try to study the H-equivalence classes of multiplications on a CW complex X with finitely many non-vanishing homotopy groups, by using the Postnikov decomposition of X and multiplier arguments [1; 4], This paper presents a way to compute the set of H-equivalence classes of multiplications on X from the knowledge of certain quotient sets of H*(B Λ B, ∑) and some homotopy equivalences of B, where B represents the spaces in the Postnikov decomposition of X, and ∑ denotes abelian groups corresponding to the homotopy groups of X.

In recent years the harmonic analysis of n-fold (n > 2) metaplectic coverings of GL2 has played an increasingly important role in certain aspects of algebraic number theory. In large part this has been inspired by the pioneering work of Kubota (see [3] for example); as an application one could cite the solution by Heath-Brown and Patterson [3] to a question of Kummer's on the distribution of the arguments of cubic Gauss sums. In that paper, Eisenstein series on the 3-fold metaplectic cover of GL2(A) play a crucial role.

The object of this note is to point out that the theory of Eisenstein series can be made to work for a wide class of finite central coverings. Indeed, once the assumptions are made, the usual theory carries over readily, and one obtains a spectral decomposition of the appropriate L2-space of functions; this is done in Section 2 of this paper.

Dans une note récente (1), Szekeres a donné le théorème suivant:

Sif est une fonction réelle, définie, strictement croissante et ayant la deuxième dérivée continue sur (a, b) ( — ∞ ≤ a < b ≤ ≤) alors il existe une fonction réelle ϕ, définie et strictement croissante sur (a, b), et une fonction réelle ψ définie et strictement croissante sur (ϕ(a), ϕ(b)), telles que ϕ˝,(x) ≥ 0 pour a < x < b, ψ ˝(u) < 0 pour ϕ(a) < u < ϕ(b) et f(x) = ψ(ϕ(x)) pour a < x < b.

Dans la démonstration de ce théorème, Szekeres utilise une expression où la dérivée f‘(x) apparaît au dénominateur, ce qui exige qu'on ait f'(x) ≠ 0 pour a < x < b, condition qui ne figure pas dans l'énoncé du théorème. Il est naturel alors de poser le problème si le théorème ci-dessus reste en vigueur si cette condition n'est pas remplie. Il y a lieu aussi de se demander si les autres conditions, par exemple l'existence et la continuité def“(x) sont nécessaires pour qu'une fonction strictement croissante puisse être représentée comme une superposition de deux fonctions croissantes, l'une convexe, l'autre concave.

In this paper we note a simple connection between the (ƒ, dn) method of summability (defined by Smith (8)) and a composition of [F, dn] (defined by Jakimovski (4)) and Sonnenschein methods (9; 10). This connection is then used to supply some sufficient conditions for the regularity of (ƒ, dn) methods by using known regularity conditions for various [F, dn] and Sonnenschein methods. Finally, the connection is further exploited to obtain information about the Lebesgue constants for a certain class of [F, dn] methods by investigating related (ƒ, dn) methods.

2. Definitions and the regularity theorem. The (ƒ, dn) method of summability is defined by Smith (8) essentially as follows. Letƒ(z) be a nonconstant entire function satisfying ƒ (l) = 1 and let {dn} be a sequence of complex numbers satisfying dt ≠ – 1, di ≠ – ƒ(0) (i ≧ 1).

The Orlicz-Pettis type theorems are essentially statements about the equivalence of strong and weak subseries convergence of series in topological groups. The corresponding problem of the equivalence of subfamily summability for summable families was examined for instance in [2] and (16). We give here another uncountable generalization of the Orlicz-Pettis theorem in the sense that we assume less than subfamily summability and obtain straightforward generalizations of the classical case, which include or imply in a trivial manner—at least for polar topologies on groups—all known theorems “from weak to strong σ-additivity”.

A classical question in algebraic geometry is whether a given projection of a projective space induces an isomorphism on a given closed subvariety. To answer it, one investigates secant lines to the subvariety. There has been a lot of recent activity in this field ([12], [14],[18], [21], [23]): see [14] and [12] for references).

An obvious generalization of the secant lines is provided by the secant r-planes, which intersect a given closed subvariety in r + 1 linearly independent points. The closure of the set of these secant r-planes is the secant variety, and the aim of this paper is to determine its rational equivalence class in the case of curves. There is an extensive classical literature about this problem.

Let Ω be a cyclic extension of degree l over the field Σ. Correcting an error which for some time had been haunting the literature, Hasse (1, p. 272) noted that for l = 2, the field Ω may contain a unit ξ such that

.

Let be a Hilbert space and denote the collection of (bounded, linear) operators on by . Throughout this paper, the term ‘algebra’ will refer to a subalgebra of ; unless otherwise stated, it will not be assumed to contain I or to be closed in any topology.

An algebra is said to be transitive if it has no non-trivial invariant subspaces. The following lemma has revolutionized the study of transitive algebras. For a pr∞f and a general discussion of its implications, the reader is referred to [5].