The Picard group Γ = PSL2 (Z(i)) is the group of linear transformations

with a, b, c, d Gaussian integers.

Γ is of interest both as an abstract group and in automorphic function theory [10]. In [10] Waldinger constructed a subgroup H of finite index which is a generalized free product, while in [1] Fine showed that T is a semidirect product with the subgroup H, contained as a subgroup of finite index in the normal factor.

Let be the standard (n + 1) simplex with its standard triangulation. By the Generalized Poincare Conjecture, if n ≧ 5 and is a smooth homotopy w-sphere, then there exists a smooth triangulation , where K is a suitable subdivision of . On the other hand, in [3], N. Kuiper proves the following theorem.

We assume familiarity with the notation and contents of Conway [2] and Edge [3]. That the Mathieu group is a subgroup of the simple group PSU (6, 22) appears to have been first recognized by Conway and is consequent upon his identification of ·222 with PSU (6, 22). Although we know of no proof of this identification in the literature, several proofs exist in the folklore of the subject: for example, N. Patterson showed one of the authors a proof that depends on ·222 being a Fischer group, hence on consideration of order, isomorphic to PSU (6, 22). There is another proof which relies on McLaughlin's work on rank three groups.

N. Wiener [12] introduced the idea of the class Vp. A 2 π-periodic function ƒ is said to have bounded p-variation Vp(f)(1 ≦ p < ∞), or to belong to the class Vp, if

Let R be a commutative ring with an identity. An ideal A of R is called a multiplication ideal if for every ideal B ⊆ A there exists an ideal C such that B = AC. A ring R is called a multiplication ring if all its ideals are multiplication ideals. A ring R is called an almost multiplication ring if RM is a multiplication ring for every maximal ideal M of R. Multiplication rings and almost multiplication rings have been extensively studied—for example, see [4; 8; 9; 11; 12; 15; and 16].

In this paper we develop a theory of maximal and minimal operators and their duals associated with the system

In this paper we consider the problem of extending the Weierstrass division theorem to quasianalytic local rings of germs of functions of k real variables which properly contain the local ring of germs of analytic functions. After some background material (§ 2) and some technical preliminaries (§ 3), we show by examples that the so-called generic division theorem fails in such rings if k ≧ 1 and that the Weierstrass division theorem fails in such rings if k ≧ 2 (§ 4).

The notion of pure subgroups is due to Prufer [7]. It has proven extremely useful in establishing structural properties of abelian groups. In a recent paper [9], Waterhouse introduced the concept of a pure subfield of a purely inseparable extension. Let L be a purely inseparable modular extension of k, and let K be an intermediate field. K is called pure if K and k(Lpn) are linearly disjoint over k(Kpn) for all n. Waterhouse used this concept to establish the existence of basic subfields [9].

Let M denote the formally symmetric, second-order differential expression given by, for suitably differentiable complex-valued functions ƒ,

The coefficients p and q are real-valued, Lebesgue measurable on the halfclosed, half-open interval [a, b) of the real line, with - ∞ < a < b ≦ ∞, and satisfy the basic conditions:

Let A be a commutative real Banach algebra with unit, and MA its maximal ideal space. The existence of the Silov boundary SA for A was established in [5] by resorting to the complexification of A. We give here an intrinsic proof of this result which exhibits the close connection between the absolute values and the real parts of ‘functions’ in A (Theorem 1.3).

Consider the biaxially symmetric potential equation

where α, β > — 1/2. If 2α + 1 and 2 β + 1 are non-negative integers and if X corresponds to the hypercircle

then the biaxisymmetric Laplace equation in E2(α+ β+2),

and (1.1) are equivalent.

The graphs considered in this paper are finite and have no loops or multiple edges. If G is such a graph, we denote its vertex set by VG and its edge set by EG. If X and Y are disjoint subsets of VG, we define δ (X, Y) to be the set of edges of G that join a vertex in X to one in Y.

We prove that is quasi-primal, then every algebra in HSPhas a pure embedding into a product of finite algebras. For a general theory of varieties for which every can be purely embedded in an equationally compact algebra , and for all notions not explained here, the reader is referred to [38; 6; or 5]. This theorem was known for Boolean algebras simply as a corollary of the Stone representation theorem and the fact that in the variety of Boolean algebras, all embeddings are pure [2].

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 material in this paper is taken from the author's doctoral dissertation [2]. We will use the terminology and notation of [3]. Let us recall those terms which will be needed here.

We define a restricted linear space (RLS) as a finite set of p elements, called points, of which q subsets, called lines, are distinguished so that the following axioms hold:

(RLS-1) Any two distinct points u, v belong to exactly one common line uv.

Let R be a ring with unit element and G a finite group. We denote by RG the group ring of the group G over R and by U(RG) the group of units of this group ring.

The study of the nilpotency of U(RG) has been the subject of several papers.

Let Mn be a Riemannian manifold of dimension n ≧ 2 and class C3, (gtj) the symmetric matrix of the positive definite metric of Mn, and (gij) the inverse matrix of (gtj), and denote by and R = gijRij the operator of covariant differentiation with respect to gij, the Riemann tensor, the Ricci tensor and the scalar curvature of Mn respectively.

The isometries of the Hardy spaces Hv (0 < p < ∞, p ≠ 2) of the unit disc were determined by Forelli in 1964 [3]. For p = 1 the result had been found earlier by deLeeuw, Rudin and Wermer [2]. For several variables the state of affairs at present is this: For the polydisc the isometries of Hp onto itself have been characterized by Schneider [13]. For the unit ball the same result was proved in the case p > 2 by Forelli [4]. Finally in [12] Rudin removed the restriction p > 2 and also established some results about isometries of Hp of the ball and the polydisc into itself.

Soit Q(x) une fonction positive paire continûment différentiable pour — ∞ < x < ∞ telle que xQ’ (x) soit croissante pour x > 0 et que Q‘(x) → ∞ lorsque x → co. Désignons par Qn l'unique racine positive de l'équation xQ‘(x) = n. L'approximation polynomiale des fonctions continues relativement au poids e-Q(x) a été étudiée par de nombreux mathématiciens. Le résultat suivant est une conséquence d'un théorème de l'un des auteurs (voir Freud [1]): si F(x) est une fonction réelle continue pour —∞ < x < ∞ qui satisfait une condition de Lipschitz d'ordre a où 0 < a < 1, il existe une suite de polynômes ﹛Pn(x)﹜ où deg Pn ≦ n telle que

Section 1 of this paper is concerned with the effect on conjugate and σ-points of various perturbations of q(x) for differential equations of the form