In the theory of vector valued functions there is a theorem which states that if a function from a compact interval I into a normed linear space X is of weak bounded variation, then it is of bounded variation. The proof uses in a straightforward way the Uniform Boundedness Principle (see [2, p. 60]). The present paper grew from the question of whether an analogous theorem holds for absolutely continuous functions. The answer is in the negative, and an example will be given (Theorem 7). But it will also be shown that if X is weakly sequentially complete (e.g. an Lp space, 1 ≦ p < ∞ ), then a weakly absolutely continuous point function from / into X is absolutely continuous. The method of proof involves the construction of a countably additive set function in the standard Lebesgue-Stieltjes fashion.

The paper is divided into three parts. In Section 1 extensions of finitely additive, absolutely continuous set functions are carried out in an abstract setting. Section 2 applies this to vector valued (point) functions on the real line.

Soit A un anneau intègre de corps des fractions K, on s'interesse ici aux polynômes de K[X] qui prennent sur A leurs valeurs dans A. Ils forment bien évidemment un sous anneau de K[X] qui contient A[X], nous le notons As.

Cet article reprend, dans un langage moderne, les idées de deux publications assez anciennes, dues à Polya [3] et Ostrowsky [4], en 1919. Les auteurs montraient que si A est principal, As est un A-module libre qui admet sur A une base formée de polynômes de degré croissant ; si A est plus généralement l'anneau des entiers d'un corps de nombres ils mettaient en évidence une suite d'idéaux fractionnaires de A. En exposant le cas de l'anneau A principal, pour un anneau de valisation discrète dans le premier paragraphe, puis en généralisant aux anneaux de Dedekind dans notre second paragraphe, nous pouvons énoncer de façon plus explicite, que As est un A-module projectif, que l'on peut décomposer canoniquement en somme d'idéaux fractionnaires de A.

A sequence {zn} in D = {z: |z| < 1} is a Blaschke sequence if and only if

If 0 appears m times in {zn} then

is the Blaschke product defined by {zn}. The set of all Blaschke products will be denoted by . If B ∊ it is well-known that B is regular in D, and |B(z, {zn})| < 1 when z ∊ D.

For a given pair of values p in (0, ∞) and q in [0, ∞) we denote by ℐ(p, g) the class of all Blaschke products B(z, {zn}) such that

as r → 1 — 0. In the case q ≦ max(p — 1,0) the classes of functions B and ℐ(p, q) are identical: this is a particular case of an elementary theorem for functions subharmonic in a disc, the analogous theorem for functions subharmonic in a half-plane appearing in [1],

Etant un polynôme Φ ∊ R[X1 … ,Xn] à n indéterminés, Φ est dit de poids homogène m si

avec m =

Si E est un fibre vectoriel complexe de rang n (i.e., n est la dimension sur C de la fibre Ex) sur une variété differentiate orientée compacte M de dimension m, à tout polynôme Φ à n indéterminés de poids homogène m, on associe un nombre de Chern:

où [M] est le cycle fondamental défini par l'orientation de M et Ci(E) la ième classe de Chern de E. Dans le cas où le poids homogène de Φ est différent de la dimension de M, on pose: Φ(E, M) = 0.

Hempel [6, Theorem 2] proved that if S is a tame 2-sphere in E3 and f is a map of E3 onto itself such that f|S is a homeomorphism and f(E3 - S) = E3- f(S), then f(S) is tame. Boyd [4] has shown that the converse is false; in fact, if S is any 2-sphere in E3, then there is a monotone map f of E3 onto itself such that f |S is a homeomorphism, f(E3 — S) = E3 — f(S), and f(S) is tame.

It is the purpose of this paper to prove that the corresponding converse for simple closed curves in E3 is also false. We show in Theorem 4 that if J is any simple closed curve in a closed orientable 3-manifold M3, then there is a monotone map f : M3 → S3 such that f |J is a homeomorphism, f(J) is tame and unknotted, and f(M3 - J) = S3 - f(J).

In Theorem 1 of § 2, we construct a cube-with-handles neighbourhood of a simple closed curve in an orientable 3-manifold.

In [7] Varadarajan denned the notion of a cyclic map f : A → X. The collection of all homotopy classes of such cyclic maps forms the Gottlieb subset G(A, X) of [A, X]. If A = S1 this reduces to the group G(X, X0) of Gottlieb [5]. We show that a cyclic map f maps ΩA into the centre of ΩX in the sense of Ganea [4]. If A and X are both suspensions, we then show that if f : A → X maps ΩA into the centre of ΩX, then f is cyclic. Thus for maps from suspensions to suspensions, Varadarajan's cyclic maps are just those maps considered by Ganea. We also define G (Σ4, ΣX) in terms of the generalized Whitehead product [1], This gives the computations for G(Sn+k, Sn) in terms of Whitehead products in π2n+k(Sn).

We work in the category of spaces with base points and having the homotopy type of countable CW-complexes. All maps and homotopies are with respect to base points. For simplicity, we shall frequently use the same symbol for a map and its homotopy class.

Let D be an integral domain with identity, and let R be a commutative ring. If n is a positive integer, R will be said to have property (n), (n)′, or (n)″ according as

property (n): For any x, y ∊ R, (x, y)n = (xn, yn).

property (n)′ : For any x ∊ R and any ideal A of R such that xn ∊ An, it follows that x ∊ A.

property (n)′ : For any ideals A, B of R, (A ∩ B)n = An ∩ Bn.

J. Ohm introduced property (n) in [7] in connection with the question: If n ≦ 2 and if D has property (n), must D be a Prüfer domain? (The integral domain D with identity is a Prüfer domain if each nonzero finitely generated ideal of D is invertible; equivalently, DP is a valuation ring for each proper prime ideal P of D.) Prior to Ohm's paper, it was known that if D has property (2) and if D is integrally closed, then D is Prüfer.

Let (M, g) be a C∞ Riemannian manifold and A be the field of symmetric endomorphisms corresponding to the Ricci tensor S; that is,

We consider a condition weaker than the requirement that A be parallel (▽ A = 0), namely, that the “second exterior covariant derivative” vanish ( ▽x▽YA — ▽Y ▽XA — ▽[X,Y]A = 0), which by the classical interchange formula reduces to the property

where R(X, Y) is the curvature transformation determined by the vector fields X and Y.

The property (P) is equivalent to

To see this we observe first that a skew symmetric and a symmetric endomorphism commute if and only if their product is skew symmetric.

It is easy to verify that any connected graph G with maximum degree s has chromatic number χ(G) ≦ 1 + s. In [1], R. L. Brooks proved that χ(G) ≦ s, unless s = 2 and G is an odd cycle or s > 2 and G is the complete graph Ks+1. This was the first significant theorem connecting the structure of a graph with its chromatic number. For s ≦ 4, Brooks' theorem says that every connected s-chromatic graph other than Ks contains a vertex of degree > s — 1. An equivalent formulation can be given in terms of s-critical graphs. A graph G is said to be s-critical if χ(G) = s, but every proper subgraph has chromatic number less than s. Each scritical graph has minimum degree ≦ s — 1. We can now restate Brooks' theorem: if an s-critical graph, s ≦ 4, is not Ks and has p vertices and q edges, then 2q ≦ (s — l)p + 1. Dirac [2] significantly generalized the theorem of Brooks by showing that 2q ≦ (s — 1)£ + s — 3 and that this result is best possible.

In a recent paper by Douglas Underwood [8] several definitions of “associated prime divisors” were discussed and shown to be unique. In this note we produce a fifth type, which is due to W. Krull, and is found in his classical paper [2] and further discussed by B. Banaschewski in [1]. Historically this characterization considerably predates the other four definitions.

Throughout this note, R denotes a commutative ring with unity, and all ideals and elements are assumed to be in such a ring. We shall let upper case letters, most frequently the beginning of the alphabet, denote ideals and lower case letters, elements of R. On the whole, our terminology will be that of [9]. We do, however, take exception with [9] in two instances, viz.

The β-construction assigns to each complex representation φ of the compact Lie group G a unique element β(φ) in (G). For the details of this construction the reader is referred to [1] or [5]. The purpose of the present paper is to determine some of the properties of the element β(φ) in terms of the invariants of the representation φ. More precisely, we consider the following question. Let G be a simple, simply-connected compact Lie group and let f : S3 →G be a Lie group homomorphism. Then (S3) ⋍ Z with generator x = β(φ1), φ1 the fundamental representation of S3 , so that if φ is a representation of G,f*(φ) = n(φ)x, where n(φ) is an integer depending on φ and f . The problem is to determine n(φ).

Since G is simple and simply-connected we may assume that ch2, the component of the Chern character in dimension 4 takes its values in H4(SG,Z)≅Z. Let u be a generator of H4(SG,Z) so that ch2(β (φ)) = m(φ)u, m(φ) an integer depending on φ.

We will say that a space X has property (*) if and only if each discrete subset of X is realcompact; i.e., the cardinality of each discrete subset of X is nonmeasurable. In [8], Shirota shows that a completely regular T1-space X is realcompact if and only if X has property (*) and X is complete with respect to some uniformity. In [7], Moran, using measure theoretic techniques, shows that any normal metacompact T1-space with property (*) is realcompact.

Let D be an ultrafilter on I, and let k be a cardinal. D is said to be k-descendingly incomplete (k-d.i.) if there exists a chain Xα : α < k of elements of D such that α < β → Xα ⊆ Xβ and Xα = ϕ. Such a chain will be called a k-chain for D. The notion of k-descending incompleteness is due to Chang [3].

In this paper we explore the relationship between the cardinality of the ultrapower kI/D and the existence of certain chains on D. Since we deal so much with questions of size, we do not ordinarily make a notational distinction between a set and its cardinality. Where such a distinction is useful, the cardinality of a set A will be denoted by |A|.

Throughout, we assume all rings are associative with identity and all modules are unitary. See [7] for undefined terms and [3] for all homological concepts.

Let R be a ring, E(R) the injective envelope of RR, and H =HomR(E(R),E(R)). Then we obtain a bimodule RE(R)H. Let Q = HomH(E(R), E(R)). Q is called the maximal left quotient ring of R. Q has the property that if p, q ∈ Q, p ≠ 0, then there exists r ∈ R such that rp ≠ 0, rq ∈ R, i.e., Q is a ring of left quotients of R.

A left ideal I of R is dense if for every x,y ∈ R,x ≠ 0, there exists r ∈ R such that rx ≠ 0, ry ∈ I. An alternate description of Q is Q = {x ∈ E(RR) : (R : x) is a dense left ideal of R{, where (R : x) = {r ∈ R : rx ∈ R}.

The left singular ideal of R is Zl(R) = {r ∈ R : lR(r) is an essential left ideal of R}, where lR(r) = {x ∈ R : xr = 0}. If Zl(R) = (0), then Q is a left self-injective von Neumann regular ring [7, § 4.5]. Most of the previous work on maximal left quotient rings has been done in this case.

Let G be a finite p-group, having a faithful character χ of degree f. The object of this paper is to bound the number, d(G), of generators in a minimal generating set for G in terms of χ and in particular in terms of f. This problem was raised by D. M. Goldschmidt, and solved by him in the case that G has nilpotence class 2.

This paper is a continuation of [1] where we began the study of intertwining analytic Toeplitz operators. Recall that X intertwines two operators A and B if XA = BX. Let H2 be the Hilbert space of analytic functions in the open unit disk D for which the functions fr(θ) = f(reiθ) are bounded in the L2 norm, and H∞ be the set of bounded functions in H2. For φ ∊ Hφ, Tφ (or Tφ(z)) is the analytic Toeplitz operator defined on H2 by the relation (Tφf)(z) = φ(z)f(z). For φ ∊ H∞, we shall denote {φ(z): |z| < 1} by Range (φ) or φ(D). Then where and σ(Tφ) = Closure(φ(D)) [1]. If φ ∊ H∞ maps D into D, then we define the composition operator Cφ on H2 by the relation (Cφf) (z) = f(φ(z)). J. Ryff has shown [11, Theorem 1] that Cφ, is a bounded linear operator on H2.

In [3] Lissner defined a class of rings called outer product rings, (OP-rings). These are commutative rings R with identity for which every exterior vector is decomposable, i.e., with .

If we look only at those vectors whose co-ordinates with respect to any basis of generate the unit ideal in R and consider those rings R for which all vectors of this type are decomposable, we obtain the class of rings which have been referred to as Her mite-rings (H-rings, see also Lissner [3]). This class of H-rings evidently contains the class of OP-rings.

Let X be a locally compact Hausdorff space, and let C*(X) denote the space of real-valued bounded continuous functions on X. An interesting and important property of the strict topology β on C*(X) was proved by Buck [2]: the dual space of (C*(X), β) has a natural representation as the space of bounded regular Borel measures on X.

Now suppose that X is completely regular (all topological spaces are assumed to be Hausdorff in this paper). Again it seems natural to seek locally convex topologies on the space C*(X) whose dual spaces are (via the integration pairing) significant classes of measures.

A group G is right-ordered if it can be totally ordered so that for any a, b, c in G, a < b implies that ac < bc. Right-ordered groups, considered as order preserving automorphisms of ordered sets, were studied by Cohn in [4]; but the first systematic study of the structure of these groups was made by Conrad in [5] where he gave several natural characterizations of right-ordered groups. We mention here that the class of right-ordered groups is precisely the subgroup closure of the class of lattice ordered groups (see [6], [7], [9] or [10]).

A super-reflexive Banach space is defined to be a Banach space B which has the property that no non-reflexive Banach space is finitely representable in B. Super-reflexivity is invariant under isomorphisms; a Banach space B is super-reflexive if and only if B* is super-reflexive. This concept has many equivalent formulations, some of which have been studied previously.