Let K be an algebraic field of degree n over the rational field, and let b be the ground ideal (différente) of the field. Let

For 1 ≦ p < ∞ let Hp denote the familiar Hardy space of analytic functions on the open unit disc D and let ‖·‖ denote the Hp norm. Let Sp denote the space of analytic functions f on D such that f′ ∊ Hp. In this paper we will describe the linear isometries of Sp into itself when Sp is equipped with either of two norms. The first norm we consider is given by

(1)

and the second by

(2)

(It is well known [1, Theorem 3.11] that f′ ∊ Hp implies continuity for f on D, the closure of D. Thus (2) actually defines a norm on Sp.) In the former case, with the norm defined by (1), we will show that an isometry of Sp induces, in a sense to be made precise in Section 2, an isometry of Hp and that Forelli's characterization [2] of the isometries of Hp can thus be used to describe the isometries of Hp.

Let G be a group. Denote by ZG the group ring of G over the integers and by Δ = Δ(G) the augmentation ideal of ZG, that is, the kernel of the augmentation map ϵ : ZG → Z defined by . Then Δ is a free abelian group with a free basis . A filtration of Δ is a sequence

A body moves in a rarefied medium composed of point particles at rest. The particles make elastic reflections when colliding with the body surface and do not interact with each other. We consider a generalization of Newton’s minimal resistance problem: given two bounded convex bodies ${{C}_{1}}$ and ${{C}_{2}}$ such that ${{C}_{1}}\,\subset \,{{C}_{2}}\,\subset \,{{\mathbb{R}}^{3}}$ and $\partial {{C}_{1}}\,\cap \,\partial {{C}_{2}}\,=\,\varnothing $, minimize the resistance in the class of connected bodies $B$ such that ${{C}_{1}}\,\subset \,B\,\subset \,{{C}_{2}}$. We prove that the infimum of resistance is zero; that is, there exist “almost perfectly streamlined” bodies.

The aim of this paper is to extend to a suitable class of topological semigroups parts of well-defined theory of representations of topological groups. In attempting to obtain these results it was soon realized that no general theory was likely to be obtainable for all locally compact semigroups. The reason for this is the absence of any analogue of the group algebra Ll(G). So the theory in this paper is restricted to a certain family of topological semigroups. In this account we shall only give the details of those parts of proofs which depart from the standard proofs of analogous theorems for groups.

On a locally compact semigroup S the algebra of all μ ∊ M(S) for which the mapping and of S to M(S) (where denotes the point mass at x) are continuous when M(S) has the weak topology was first studied in the sequence of papers [1, 2, 3] by A. C. and J. W. Baker.

Exterior powers of the adjoint representation of a complex semisimple Lie algebra are decomposed into irreducible representations, to varying degrees of satisfaction.

Let L/K be a finite Galois extension in characteristic ≠ 2, and consider a non-split Galois theoretical embedding problem over L/K with cyclic kernel of order 2. In this paper, we prove that if the Galois group of L/K is the direct product of two subgroups, the obstruction to solving the embedding problem can be expressed as the product of the obstructions to related embedding problems over the corresponding subextensions of L/K and certain quaternion algebra factors in the Brauer group of K. In connection with this, the obstructions to realising non-abelian groups of order 8 and 16 as Galois groups over fields of characteristic ≠ 2 are calculated, and these obstructions are used to consider automatic realisations between groups of order 4, 8 and 16.

Given an algebra A, the elements of A induce linear operators on A by left and right multiplication. Various authors have studied Banach algebras A with the property that some or all of these multiplication maps are completely continuous operators on A ; see (1-5). In (3)1. Kaplansky defined an element u of a Banach algebra A to be completely continuous if the maps a ⟶ ua and a ⟶ au, a ∊ A, are completely continuous linear operators.

1. Introduction. Let X be a Banach space (B-space). A sequence {s(i)} in X is unconditionally summable if and only if every rearrangement of the series Σis(i) is convergent. The set of unconditionally summable sequences in X will be written as U(X). In this paper several classes of summable sequences in X will be compared with one another. Each class to be considered is identical with U(X) when X has finite dimension.

We call a topological space X minspectral if it is homeomorphic to the space of minimal prime ideals of a commutative ring A in the usual (hull-kernel or Zariski) topology (see [2, p. 111]). Note that if A has an identity, is a subspace of Spec A (as defined in [1, p. 124]). It is well known that a minspectral space is Hausdorff and has a clopen basis (and hence is completely regular). We give here a topological characterization of the minspectral spaces, and we show that all minspectral spaces can actually be obtained from rings with identity and that open (but not closed) subspaces of minspectral spaces are minspectral (Theorem 1, Proposition 5).

This paper is concerned with the numbers which are relatively prime to a given positive integer

where the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

1. Introduction. In this paper, infinite fields K will be discussed which satisfy the following condition :

CONDITION C: for any polynomial p(t, x) in x, coefficients in K(t), t transcendental with respect to K, which has no zeros in K (t), there is a t* in K for which p(t*, x) has no zeros in K.

By a labelled graph we shall mean a set of “nodes,“ distinguishable from one another and denoted by A1, A2, …, and a collection of “edges” viz., pairs of nodes. We say that an edge “joins” the pair of nodes which specifies it. We further stipulate that at most one edge joins any two nodes, and that no edge joins a node to itself.

By a “colouring” of a graph in k colours we shall mean a mapping of the nodes of the graph onto a set of k colours C1, C2, …, Ck such that no two nodes which are joined by an edge are mapped onto the same colour. A graph so coloured in exactly k colours will be called a k-coloured graph. Since it is usually possible to colour a graph in more than one way, there will, in general, be many k-coloured graphs corresponding to a given graph.

The genus series is the generating series for the number of maps (inequivalent two-cell embeddings of graphs), in locally orientable surfaces, closed and without boundary, with respect to vertex- and face-degrees, number of edges and genus. A hypermap is a face two-colourable map. An expression for the genus series for (rooted) hypermaps is derived in terms of zonal polynomials by using a double coset algebra in conjunction with an encoding of a map as a triple of matchings. The expression is analogous to the one obtained for orientable surfaces in terms of Schur functions.

We show that if is a holomorphic function in the Dirichlet space of the unit disk, then almost all of its randomizations are multipliers of that space. This parallels a known result for lacunary power series, which also has a version for smoothness classes: every lacunary Dirichlet series lies in the Lipschitz class Lip1/2 of functions obeying a Lipschitz condition with exponent 1/2. However, unlike the lacunary situation, no corresponding “almost sure” Lipschitz result is possible for random series: we exhibit a Dirichlet function with norandomization in Lip1/2. We complement this result with a “best possible” sufficient condition for randomizations to belong almost surely to Lip1/2. Versions of our results hold for weighted Dirichlet spaces, and much of our work is carried out in this more general setting.

A much-studied equation in recent years has been the second order nonlinear ordinary differential equation

where q and f are continuous on the real line and, in addition, f is monotone increasing with yf(y) > 0 for y ≠ 0. Although the original interest in (1) lay largely with the case that q﹛t) ≧ 0 for all large values of t, a number of papers have recently appeared in which this sign restriction is removed.

Dans (6), l'auteur a défini, pour chaque entier n ≥ 0, un groupe fini Gn d'ordre q3(q —1)(q3 + 1), où q = 32n+1. Les groupes G1, G2, . . . sont simples, tandis que est simple.

En réalisant G = Gn comme un groupe de permutations, opérant à droite, de l'ensemble de q3 + 1 classes à droite modulo le normalisateur N(P) d'un 3-sous-groupe de Sylow de G, on voit aisément que G satisfait aux conditions (0.1)-(0.4) suivantes :

• (0.1) G est un groupe de permutations doublement transitif d'un ensemble E de m + 1 lettres, où m ≥ 3;

• (0.2) si a et b sont deux lettres distinctes quelconques de E, le sous-groupe formé de la totalité des permutations dans G qui laissent invariantes a et b contient une, et une seule, permutation ≠ 1 qui laisse au moins 3 lettres invariantes;

• (0.3) toute involution dans G laisse au moins 3 lettres invariantes (une involution dans un group est par définition un élément d'ordre 2);

• (0.4) m est impair.

Noting that certain restrictions are placed on homeomorphisms of the pseudo-arc, since it is hereditarily indecomposable, in 1955 [4] R. H. Bing asked if the identity is the only stable homeomorphism of the pseudo-arc. In this paper we prove the following theorem.

THEOREM. Let U be an open subset of the pseudo-arc P. Let p and q be distinct points of P such that the subcontinuum M irreducible between p and q does not intersect cl(U). Then there exists a homeomorphism h : P → P with h (p) = q andh│U = 1U.

1. Definitions. A chain C is a collection of open sets C = {C(i))}i≧n such that C(i) ∩ C(j) ≠ ∅ if and only if |i – j| ≦ 1, cl|(C(i)) ∩ cl(C(j)) ≠ ∅ if and only if |i – j| ≦ 1, C(0) – C(l) ≠ ∅, and C(n) – C(n – 1) ≠ ∅. Each C(i) is called a link of C.

Examples of simple, separable, unital, purely infinite ${{C}^{*}}$-algebras are constructed, including:

1. (1) some that are not approximately divisible;

2. (2) those that arise as crossed products of any of a certain class of ${{C}^{*}}$-algebras by any of a certain class of non–unital endomorphisms;

3. (3) those that arise as reduced free products of pairs of ${{C}^{*}}$-algebras with respect to any from a certain class of states.

A lattice ∧ in euclidean n-space, En is a group of vectors under vector addition generated by n independent vectors, X1, X2 … , Xn, called a basis for the lattice. The absolute value of the n × n determinant the rows of which are the co-ordinates of a basis is called the determinant of the lattice and is denoted by d(∧). For any lattice ∧ there is a unique minimal positive number r such that, if spheres of radius r are placed with centres at all points of ∧, the entire space is covered. The density of this covering may be defined as (Jnrn)/(d(∧)) where Jn is the volume of the unit sphere in n-dimensional euclidean space. This density will be denoted by θn(∧). The density of the most efficient lattice covering of n-space by spheres, θn is the absolute minimum of θn(∧) considered as a function from the space of all lattices to the real numbers.