Multipliers of a difference set are of great importance in existence theorems, since they enable us to reject many configurations en bloc. (For a description of such theorems, see Mann (1).) The following theorem, which determines those cyclic group difference sets for which −1 is a multiplier, has been proved before by different methods (see, for example, Yamamoto(2) and Johnsen(3); and a more elementary matrix proof by Brualdi(4)) but the following ‘elementary’ proof may be of interest.

Suppose that E is a compact subset of a topological linear space ℒ. Then the convex kernel K, of E, is such that a point k belongs to K if every point of E can be seen, via E, from k. Valentine (l) has asked for conditions on E which ensure that the convex kernel K, of E, consists of exactly one point, and in this note we give such a condition. If A, B, C are three subsets of E, we use (A, B, C) to denote the set of those points of E, which can be seen, via E, from a triad of points a, b, c, with a ∈ A, b ∈ B, c ∈ C. We shall say that E has the property if, whenever A is a line segment and B, C are points of E which are not collinear with any point of A, the set (A, B, C) has linear dimension of at most one, and degenerates to a single point whenever A is a point.

In a recent paper (1) Effros defined a topology for the hyperspace of nonempty closed sets of a topological space X, and showed that, if X is polonais, the Borel structure of is standard. We shall show that this conclusion will also follow if X is σ-compact and metrizable.

Introduction. Let h*(X, A) be an extraordinary cohomology theory defined on the category of finite CW-pairs, that is, a cohomology theory satisfying all the axioms of Eilenberg and Steenrod (7) except the dimension axiom. This paper has three objects: to define and investigate the properties of the associated mod p cohomology theory h*(X, A; Zp) (see also (9)), to show that (with certain restrictions) if h*(X, A) is a multiplicative cohomology theory, so is h*(X, A. Zp), and lastly to generalize the notion of Bockstein coboundary homomorphisms and the apparatus of the Bockstein spectral sequence (see (4)) to an extraordinary cohomology theory.

1. A Banach space E over the complex field C is said to be a Banach inner-product (BIP) space if there exists a mapping 〈.,.〉 of E × E into C satisfying:

(i) 〈x, x〉 ≥ 0 (x ∈ E) with equality only if x = 0;

(ii) 〈x, y〉 = 〈y, x〉 (x, y ∈ E);

(iii) 〈x + λy, z〉 = 〈x, z〉 + λ〈y, z〉 (x, y, z ∈ E, λ ∈ C);

(iv) 〈x, x〉 ≤ k2 ‖x‖2 (x ∈ E),

where k is a fixed positive number. Thus 〈.,.〉 is an inner product on E, which induces a norm ‖·‖1 by the relation

Recently Isbell(4) and Ward (6) have given examples of pairs of uniformities inducing the same topology on the hyperspace of subsets of a uniform space. Smith (5) has shown that, if two uniformities induce the same topology on the hyperspace, they must be equal in proximity. The author, in (2), has introduced a relation, ‘height’, between uniformities which is in a sense dual to proximity. It is the purpose of this paper to show that, if two uniformities are equal in proximity and height, then they induce the same topology on the hyperspace. Although Ward's uniformities are not equal in height, an example of two distinct uniformities equal in proximity and height is given in (3). In fact, a slight modification of that example will give an example of a countably infinite number of distinct uniformities, all of which are equal in proximity and height.

Introduction. For applications in differential topology it is desirable to be able to find restrictions on the possible real and complex vector bundles over a given manifold. These restrictions usually state that the top component of some specific rational multinomial in the various characteristic classes is an integral multiple of the fundamental cocycle. For example, in proving non-embedding or non-immersion theorems (compare (1); (10)) one could test whether the normal bundle is consistent with these integrality conditions.

It is well known that if M is a vector bundle over a space Y, and r: X → Y is a map, then there is an ‘induced’ vector bundle r*M over X. It is also known that a vector bundle L over X determines a ‘direct image’ vector bundle r* L over Y provided that r is a finite covering map ((1), § 1).

This paper is devoted to results concerning the connexion between the ‘abstract’ preK spaces of (l) and the ‘concrete’ Köthe spaces of (3). These results, though in part independent, are not truly original, being in particular known to the authors of (2); but as far as I am aware they have not appeared in print in quite the form they take here.

We extend the notion of topological degree to a suitable class of continuous mappings of a locally convex Hausdorff topological vector space into itself which are not completely continuous. This extension and the resulting existence theorems for non-linear equations are further extended to product spaces. Our extension is developed with application to quantum field theory in mind; this application is made elsewhere.

Introduction. The object of this paper is to obtain an explicit formula for the zeta function of an arbitrary non-singular cubic surface over a finite field. Let k denote the finite field of q elements, and kn the field of qn elements which is the unique algebraic extension of k of degree n. Let be a non-singular variety defined over k, and for each n > 0 let be the number of points defined over kn which lie on . The zeta function of is given by

Dwork has shown in (3) that for any this is a rational function of q−s; and in particular it follows from the results he proves in (4) that if is a non-singular cubic surface then

and hence also

Here the numbers w o depend only on q and on .

Let X be a compact differentiable manifold of dimension 2m. A differentiable map from X to euclidean (2m + t)-space is an immersion if its Jacobian has rank 2m at each point of X; it is an embedding if it is also one–one. The existence of such an embedding or immersion implies that the characteristic classes of X satisfy certain integrality conditions; these can be used to obtain lower bounds for the integer t. In a similar way many other geometric properties of X can be deduced from a single integrality theorem involving characteristic classes of various vector bundles over X (see for instance (5)).

L-structure in a Banach space X was defined in (3) by L-projections, that is projections P satisfying

for all x ∈ X. The significance of L-structure is shown by the following facts: (1) All L-projections on X commute and together form a complete Boolean algebra. (2) X can be isometrically represented as a vector-valued L1 on a measure space constructed from the Boolean algebra of its L-projections (2). (3) L1-spaces in the ordinary sense are characterized among Banach spaces by properties equivalent to having so many L-projections that the representation in (2) is everywhere one-dimensional.

The nth symmetric product of a topological space, X, is defined to be the quotient of the Cartesian product Xn by the action of the symmetric group which permutes the factors. Even if X is a manifold, this product is, in general, not a manifold. The purpose of this note is to determine these products when X is the circle, S1, and to show that they are manifolds with boundary.

Introduction. Let Xi (i= 1, 2, 3,…) be a sequence of independent and identically distributed random variables with law ℒ(X) and write The Kolmogorov-Marcinkiewicz strong law of large numbers (Loève(6), p. 243) has the following statement:

If E|X|r < ∞, then with cr = 0 or EX according as r 1 or r ≥ 1.

In this paper I shall be considering the space of linear maps between two (perfect) Riesz spaces, and shall show how certain topological properties of these maps are related to the natural order structure of the space. The fundamental result is (e) of section 4, certain special cases of which have been treated in (4) and (5), using a less elliptic method of proof. Probably the most interesting new result in the present paper is section 10 in the special case of both L and M× being L1 spaces (so that |σ| (L, L×) and |σ| (M×, M) are the norm topologies ((2 b), section 7) and Λ(L; M×) is the space of norm-continuous linear maps).

There comes a time in the development of a cohomology theory when a discussion of cohomology operations becomes necessary. In the case of complex K-theory, the subject of the present paper, such operations have of course already been investigated by Adams (see (2)), so that any further discussion might appear superfluous. Powerful as Adams's results are, however, the situation still leaves something to be desired: it is not known just what other operations can be defined in K-theory, and it is an inconvenience from the standpoint of stable homotopy theory that Adams's operations are not themselves stable.

A K-space (E, τ) is a linear space E of sequences with a locally convex topology τ for which the inclusion map: (E, τ) → (ω, product topology) is continuous. In (2) topological properties of K-spaces were determined directly from properties of the space E and the topology τ. It is, however, very natural to consider duality properties of K-spaces and the purpose of this paper is to determine some of these properties.

Let R be the field of rational numbers, {x} = {x1, z2, …}, {y} = {y1, y2, …} be two countably infinite sets of variables and t an indeterminate. Let (λ) = (λ1, λ2, …, λm) be a partition of n. Then Littlewood (5) has shown that

can be expressed in the form

where Qλ(x, t) and Qλ(y, t) denote certain symmetric functions on the sets {x} and {y} respectively. In addition

where is the partition of n conjugate to (λ). In fact, Littlewood (5) showed that

where the summation is over all terms obtained by permutations of the variables xi (i = 1, 2, …) and

.

If L is a subcomplex of a simplicial complex K, we say that L is obtained from K by an elementary simplical collapse if K − L consists of a simplex σ of some dimension d together with one ‘free’ face of σ, i.e. a face τ of dimension d − 1 which is a face of no other simplex of K except σ. Such a collapse is said to take place through σ from τ. If L can be obtained from K by a finite sequence of elementary simplicial collapses we say that K simplicially collapses (s-collapses) onto L, denoted by K ↘ L

If K is regarded as being embedded in some Euclidean space we shall for convenience of notation fail to distinguish between K and its underlying polyhedron.