To each representation ρ on a transitive permutation group P of the group G = π(S – k) of an (ordered and oriented) link k = k1 ∪ k2 ∪ … ∪ kμ in the oriented 3-sphere S there is associated an oriented open 3-manifold M = Mρ(k), the covering space of S – k that belongs to ρ. The points 01, 02, … that lie over the base point o may be indexed in such a way that the elements g of G into which the paths from oi to oj project are represented by the permutations gρ of the form , and this property characterizes M. Of course M does not depend on the actual indices assigned to the points o1, o2, … but only on the equivalence class of ρ, where two representations ρ of G onto P and ρ′ of G onto P′ are equivalent when there is an inner automorphism θ of some symmetric group in which both P and P′ are contained which is such that ρ′ = θρ.

1. The Cesàro methods for ordinary [9, p. 17; 6, p. 96] and for absolute [9, p. 25] summation of infinite series can be generalized by the Riesz methods [7, p. 21; 12; 9, p. 52; 6, p. 86; 5, p. 2] and by “the generalized Cesàro methods“ introduced by Burkill [4] and Borwein and Russell [3]. (Also cf. [2]; for another generalization, see [8].) These generalizations raise the question as to their equivalence.

We shall consider series

(1)

with complex terms an. Throughout, we will assume that

(2)

and we call (1) Riesz summable to a sum s relative to the type λ = (λn) and to the order κ, or summable (R, λ, κ) to s briefly, if the Riesz means

(of the partial sums of (1)) tend to s as x → ∞.

1. We will use the terminology of part I [9], including the general assumptions of [9, § 1]. In that paper we had proved that |R, λ, κ| = |C, λ, κ| in case that κ is an integer. Now, we turn to non-integral orders κ.

As to ordinary summation, the following inclusion relations (in the customary sense; see [9, end of § 1]) for non-integral κ have been established so far. (Since we are comparing Riesz methods of the same type λ and order κ only, (R, λ, κ) is written (R), etc., for the moment.) (R) ⊆ (C) is a result by Borwein and Russell [2]. (C) ⊆ (R) was proved by Jurkat [3] in the case 0 < κ < 1, and, after Borwein [1], it holds in the case 1 < κ < 2 if

(1)

(2)

(i.e. decreases in the wide sense).

The approximately continuous integral which includes the Lebesgue integral has been considered by Burkill [1] and Ridder [3; 4]. I [2] also defined the AD-integral of this kind which is more general than the AP-integral of Burkill [1] and the Denjoy integral in the wide sense. But this integral is equivalent to the β-integral of Ridder.

Our aim in this paper is to characterize the AD-integral in the following way: The AD-integral is the least general approximately continuous integral (Definition 1) which includes the Lebesgue integral and fulfils the Cauchy and Harnack conditions (Definition 2).

The set operations under consideration are Borel operations and Souslin's operation (). With respect to a given family of sets and in a setting free of any topological structure there are defined three Borel families (Definitions 3.1) and the family of Souslin sets (Definition 4.1). Conditions on an initial family are determined under which iteration of the Borel operations with Souslin's operation () on the initial family and the families successively produced results in a non-decreasing sequence of families of analytic sets (Theorem 5.2.1 and Definition 3.5). A classification of families of analytic sets with respect to an initial family of sets is indicated in a manner analogous to the familiar classification of Borel sets (Definition 5.3).

1. Recently a number of functions have been shown to satisfy relations on polytopes similar to the classic Euler relation. Much of this work has been done by Shephard, and an excellent summary of results of this type may be found in [11]. For such functions, only continuity (with respect to the Hausdorff metric) is required to assure that it is a valuation, and the relationship between these two concepts was explored in [8]. It is our aim in this paper to extend the results obtained there to illustrate the relationship between valuations and the Euler relation on cell complexes.

To fix our notions, we will suppose that everything takes place in a given finite-dimensional Euclidean space X.

A polytope is the convex hull of a finite set of points and will be referred to as a d-polytope if it has dimension d. Polytopes have faces of all dimensions from 0 to d – 1 and each of these is in turn a polytope. A k-dimensional face will be termed simply a k-face.

Suppose that R is a commutative ring and G is a finite abelian group. In § 2 we review the definition of E(R, G) (T(R, G)), the group of all (commutative) Galois extensions S of R with Galois group G. We discuss the properties of these groups as functors of G and give an example which exhibits some of the pathological properties of the functor E(R, – ). In § 3 we display a homomorphism from E(R, G) to Pic (R(G)); we use this homomorphism to prove that if S is commutative, G has exponent m, and R(G) has Serre dimension 0 or 1, then a direct sum of m copies of S is isomorphic as a G-module to a direct sum of m copies of R(G). (This result is related to [5, Theorem 4.2], where it is shown that if S is a free R-module and G is any finite group with n elements, then Sn is isomorphic to R(G)n as G-modules.) We also give some examples of Galois extensions without normal bases.

Let R be a commutative ring with unity and let G be a group. The group ring RG is a free R-module having the elements of G as a basis, with multiplication induced by

The first theorem in this paper deals with idempotents in RG and improves a result of Connell. In the second section we consider the Jacobson radical of RG, and we prove a theorem about a class of algebras that includes RG when G is locally finite and R is an algebraically closed field of characteristic zero. The last theorem shows that if R is a field and G is a finite nilpotent group, then RG determines RP for every Sylow subgroup P of G, regardless of the characteristic of R.

An action is a continuous function α: T × X → X, where T is a semigroup, X is a Hausdorff space, and α(t1, α(t2, x)) = α(t1,t2x) for all t1, t2 ∈ T and x ∈ X . If, for an action α, Q(α) = {x ∈ X| α(T × {x}) = X} is non-empty, then α is called a point-transitive action. Our aim in this note is to classify the point-transitive actions of the unit interval with the usual, nil, or min multiplications.

The reader is referred to [5; 7; 9] for information concerning the general theory of semigroups. All semigroups which are considered here are compact and Abelian and all spaces are compact Hausdorff. Actions by semigroups have been studied in [1; 3; 8].

In [7] a Perron-like integral was denned in an arbitrary topological space and many of its basic properties were established. In this paper we shall show (the theorem in § 2) that in a suitable setting the integral from [7] includes a class of so-called singular integrals, i.e., generalized forms of the Cauchy principal value of an integral. Thus, the powerful machinery of Perron integration, e.g., the monotone and dominant convergence theorems, can be automatically applied to these singular integrals.

During the last few years, Branko Grünbaum, Micha Perles, and others have made extensive use of Gale transforms and Gale diagrams in investigating the properties of convex polytopes. Up to the present, this technique has been applied almost entirely in connection with combinatorial and enumeration problems. In this paper we begin by showing that Gale transforms are also useful in investigating properties of an essentially metrical nature, namely the symmetries of a convex polytope. Our main result here (Theorem (10)) is that, in a manner that will be made precise later, the symmetry group of a polytope can be represented faithfully by the symmetry group of a Gale transform of its vertices. If a d-polytope P ⊂ Ed has an axis of symmetry A (that is, A is a linear subspace of Ed such that the reflection in A is a symmetry of P), then it is called axi-symmetric. Using Gale transforms we are able to determine, in a simple manner, the possible numbers and dimensions of axes of symmetry of axi-symmetric polytopes.

Soit Y un espace localement convexe complet, la topologie étant définie par une famille de semi-normes {pα}. Soit A un opérateur fermé de domaine D(A) dense dans Y. Nous considérons dans cet article des fonctions u de la variable réelle t à valeurs dans Y. On dit que u(t) ∈ Cl[(a, b); Y] est une solution au sens classique de l'équation différentielle abstraite (ou opérationnelle)

(1)

si pour tout t ∈ (a, b), u(t) ∈ D(A), et satisfait la relation

élément de Y, f étant une fonction continue à valeurs dans Y.

Dans leur important mémoire [1], Agmon et Nirenberg ont étudié en détail les propriétés des solutions d'une telle équation dans le cas où Y est un espace de Banach.

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).

The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.

In this paper we consider Banach space automorphisms of m, the space of bounded sequences, which map c, the space of convergent sequences, into itself. In particular, we consider the problem of determining which maps from C0, the space of sequences converging to 0, to c can be extended to such automorphisms.

The origin of this note lies in an incorrect conjecture of mine. If the automorphism T: m → m is given by a matrix, that is, a sequence of elements of ll, and if T is conservative, that is, T(c) ⊂ c, then T(c) = c. That is, T restricted to c is an automorphism of c. We had hoped this would hold even if T were not a matrix. We can see, for example, that if the conservative automorphism T is bounded on the unit cube of m by 1 and ρ, where , then T(c) = c. However, in general it is possible for a conservative automorphism of m to map c properly into c.

Bernstein's classical integral representation theorem for completely monotonie functions can be proved most elegantly, on a commutative semigroup with identity, by the integral version of the Kreĭn-Milman theorem [2]. The key to this approach is the identification (as exponentials) of the extremal points of the normalized completely monotonie functions. Alternate proofs of this identification are given in § 1. The first (Corollary 1.3) is based on the Kreĭn-Milman theorem and the second (see remarks following Corollary 1.5) is derived from elementary analytic techniques. Other interesting facts about completely monotonie functions are mentioned in passing. For example, we observe that the normalized completely monotonie functions form a simplex (Corollary 1.4). In Corollary 1.6 we note that the product of completely monotonie functions corresponds to the convolution of their representing measures. Thus the normalized completely monotonie functions form an affine semigroup [3].

In this paper, we consider the extent to which a local Noether lattice (ℒ, M) is characterized by the sub-multiplicative lattice, denoted δℒ, of M-primary elements. (Here we use the notation (ℒ, M) to indicate that M is the maximal element of ℒ.) In particular, we call ℒM-complete if, given any decreasing sequence {Ai} of elements and any n ≧ 1, it follows that Ai ≦ A V Mn for large i, where A = ΛAi And we show that, given two Mi-complete local Noether lattices (ℒ1, M1) and (ℒ2, M2), with δℒ1 ≅ δℒ2, it follows that ℒ1 ≅ ℒ2. Further, we show that any local Noether lattice (ℒ, M) is a sublattice of a local Noether lattice (ℒ*, M) which is M-complete and such that δℒ = δℒ*.

Let C be a full subcategory of T, the category of based topological spaces and based maps, and let Cn be the corresponding category of n-tuples. Let S, T: Tn → T be covariant functors which respect homotopy classes and let u, v: S → T be natural transformations, u and v are homotopic inC, denoted u ≃ v(C), if uX ≃ vX: SX → TX (X ∈ Cn), that is to say, for every X ∈ C, uX and vX are homotopic (all homotopies are required to respect base points), u and v are naturally homotopic inC, denoted u ≃n v; (C), if there exist morphisms

such that, for every X ∈ C, utX is a homotopy from uX to vX and such that, for every t ∈ I, ut:S → T is a natural transformation.

Let {zk},0 < |zk| < 1, be a given sequence of points in the open unit disc D = {z: |z| < 1} and let E be its set of limit points on the unit circle T. In this note we consider the problem of finding conditions on the sequence {zk} which will ensure the existence of a function f analytic in D satisfying

(A)

and whose derivative f′ belongs to the Hardy class H1 or, alternatively, whose derivatives of all orders are bounded in D. We shall prove the following two theorems.

THEOREM 1. If

(1)

(2)

and

(3)

then there is a function f analytic in D which satisfies (A) and its derivative f′ belongs to H1.

In [3] Schafer has defined generalized standard rings as rings satisfying the identities

(1)

(2)

(3)

and observed that these identities imply (y,y, (x, z)) = 0 and if the characteristic is not three, (x, y, x2) = 0. Schafer determined the structure of simple, finite-dimensional generalized standard algebras of characteristic not two or three by showing that they must be either commutative, Jordan, or alternative.

Previously one of us [2] had studied accessible rings, which are defined by the identities (x,y,z) + (z,x,y) – (x,z,y) = 0 and ((w,x), y,z) = 0.

In this paper, we shall construct a vector space, called the (G, σ) space, which generalizes the tensor space, the Grassman space, and the symmetric space. Then we shall determine a necessary and sufficient condition that the (G, σ) product of the vectors x1, x2, …, xn is zero.

1. Let G be a permutation group on I = {1, 2, …, n} and F, an arbitrary field. Let σ be a linear character of G, i.e., σ is a homomorphism of G into the multiplicative group F* of F.

For each i ∈ I, let Vi be a finite-dimensional vector space over F. Consider the Cartesian product W = V1 × V2 × … × Vn.

1.1. Definition. W is called a G-set if and only if Vi = Vg(i) for all i ∊ I, and for all g ∊ G.