In his important paper (1), Lorentz defined the space f of almost convergent sequences, using the idea of Banach limits. If x ∈ l∞(R), the space of bounded real sequences, and

where the inf is taken over all sets n(1), n(2), …, n(r) of natural numbers, then a Banach limit L may be defined as a linear functional on l∞(R) which satisfies

Let α be a positive integer, and El, …, Eα Hadamard sets of positive integers. It is shown that E = E1 + … + Eα determines a Littlewood–Paley decomposition of Z.

Suppose that is a Hadamard set of positive integers such that nj+1/nj ≥ 2 for all j. Let α be a positive integer, and

We show that F(α) also determines a Littlewood-Paley decomposition of Z.

I continue to investigate Riesz spaces E with the property that every positive linear map from E to an Archimedean Riesz space is sequentially order-continuous. In order to give a criterion for the product of such spaces to be another, we are forced to investigate their internal structure, and to develop an ordinal hierarchy of such spaces.

Alfsen and Andersen(2) defined the centre of the complete order-unit space A(K) associated with a compact convex set K to be the set of functions in A(K) which multiply with A(K) pointwise on the extreme boundary of K, thereby generalizing the concept of centres of C*-algebras. It is therefore possible to extend this definition to include the space A (K; B) of continuous affine functions of K into a Banach algebra B. Such spaces arise in the theory of weak tensor products E ⊗λB of B with a Banach space E, which may be embedded in A(K; B) where K is the unit ball of E* in the weak* topology. Andersen and Atkinson(4) considered multipliers in A(K; B) and showed that if B is unital, then the multipliers are precisely those functions which are continuous in the facial topology on the extreme boundary. It is shown here that this result extends to non-unital Banach algebras with trivial left annihilator.

Of all sets lying within a given convex plane set S, which one gives the greatest ratio of area A to perimeter P? This was posed to one of us by E. Bombieri for the case when S is a square. He asserted that T. Bang used an easy upper bound on A/P in an elementary proof of the Prime Number Theorem. After solving this problem from scratch, we learned from M. Silver that Besicovitch (1) had solved a closely related problem, which we call Besicovitch's problem and which will be discussed shortly. Later, we found that the problem of maximizing A/P had occurred to Garvin (4, 5) and that he solved the problem when S is a triangle. Garvin inspired DeMar (2) to consider and solve Besicovitch's problem for the triangle and for polygons. DeMar found that Steiner (8, p. 166) had solved Besicovitch's problem for triangles, but Steiner was only interested in subsets which touched all sides of a polygon (8, p. 168). More recently, J. Wills has referred us to Herz and Kaapke (6) who use Besicovitch's results to solve our problem of maximizing A/P for polygons circumscribed about a circle.

In this paper we will show that certain elements of order p (p an odd prime) on the 2-line of the Adams-Novikov spectral sequence support non-trivial differentials and therefore do not detect elements in the stable homotopy groups of spheres. These elements are analogous to the so-called Arf invariant elements of order 2, hence the title. However, it is evident that the methods presented here do not extend to the prime 2.

The lack of a satisfactory notion of the dual of a compact semigroup S precludes a simple concrete characterization of the closed translation invariant subspaces of C(S). In this article we develop an abstract theory of such spaces (satisfying a weak version of the Banach approximation property) in terms of ‘injective coalgebras’. We characterize their dual spaces as those Banach algebras whose closed unit balls are compact semigroups under the weak star topology. Injective Hopf algebras are shown to be the spaces C(G), G a compact group.

In this paper we investigate the procedure of blowing up a non-normal variety V in its conductor ideal (denned in Section 1). If V is a hypersurface then corresponds to the subadjunction conditions of Italian algebraic geometry (see (4), section 15), and so we would expect the blow up of V in , denoted , to be closely connected with the normalization Ṽ.

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies

When A1 and A2 are Banach algebras, their algebraic tensor product A1 ⊗ A2 has a natural multiplication. In this paper we investigate when the condition that A1 and A2 are ℒp-spaces constrains this multiplication to extend to the injective tensor product A1A2, making it a Banach algebra.

The transfer has long been a fundamental tool in the study of group cohomology. In recent years, symmetric groups and a geometric version of the transfer have begun to play an important role in stable homotopy theory (2, 5). Thus, motivated by geometric considerations, we have been led to investigate the transfer homomorphism

in group homology, where n is the nth symmetric group, (n, p) is a p-Sylow sub-group and simple coefficients are taken in /p (the integers modulo a prime p). In this paper, we obtain an explicit characterization (Theorem 3·8) of this homomorphism. Roughly speaking, elements in H*(n) are expressible in terms of the wreath product k ∫ l → n (n = kl) and the ordinary product k × n−k→ n. We show that tr* preserves the form of these elements.

The Lorenz attractor is a strange attractor which has been proposed as an explicit model for turbulence ((4), compare (5)). First studied by E. N. Lorenz as a truncation of the Navier-Stokes equations (2), it has since attracted the attention of mathematicians because of its particularly interesting dynamical properties.

Let A be the complex Banach algebra of all bounded continuous complex-valued functions on the closed unit ball of a complex Banach space X, analytic on the open unit ball, with sup norm. For a class of spaces X which contains all infinite dimensional complex reflexive spaces we prove the existence of non-compact peak interpolation sets for A. We prove some related interpolation theorems for vector-valued functions and present some applications to the ranges of analytic maps between Banach spaces. We also show that in general peak interpolation sets for A do not exist.

The purpose of this paper is to give a proof of the following splitting theorem in stable homotopy theory. We assume all spaces are localized at a fixed prime p. Let k be the symmetric group on {1, …, k}, Q(.) = lim ΩnΣn(.), and QkS0, k ∈ , denote the components of QS0.

The search for a result such as the one presented in this note was motivated by an application in the theory of Markov branching processes. The limiting behaviour of a Markov branching process is determined mainly by properties of the set of its first moments, usually given as a semigroup of non-negative, linear-bounded operators on a Banach space. The principal case is that in which these operators are in some sense primitive. If the underlying space is finite-dimensional, the case of primitivity is described to complete satisfaction by Perron's theorem. For more general spaces we have the well known extension of Perron's result by Kreĭn and Rutman (8). Unfortunately, the use of this extension in the theory of general branching processes has so far not led to limit theorems as strong as the best results known in the finite-dimensional case. At least for this specific purpose the classical Kreĭn-Rutman theorem seems to be too crude. In fact, already the simplest branching diffusions on bounded domains (3) suggest a more refined, though necessarily less general extension of Perron's theorem.

A recently proved upper bound for the permanents of (0,1) matrices is used to improve the Fowler-Rushbrooke upper bound for the constant λd occurring in the d-dimensional dimer problem, d ≥ 3.

In (2) Adams gave a splitting of complex K-theory with coefficients in the ring R(d) of rationals a/b such that b contains no prime p with p ≡ 1 (mod d). The splitting comes from a complete set of projection operators on K(X; R(d)). One of the operators is then used to obtain a stable, multiplicative idempotent ε on complex cobordism with coefficients in the same ring R(d) and hence a splitting of the representing spectrum MUR(d). However, the idempotent is initially defined over the rational numbers and work is needed to show that it actually gives an operation on MUR(d). Since Novikov (6) has shown that multiplicative cobordism operations are distinguished by their values on the generator ω ∈ MU2CP∞, it is natural to seek an explicit formula for ε(ω) which wi11 show that ε gives an operation over the subring R(d).

Our problem and the motive for attacking it are the same as in (1). Using a different method, we obtain a result admitting disconnected domains and higher dimensions without symmetry assumptions. For a detailed introduction we refer to (1).

The effect of non-linearity on standing edge waves is studied on the basis of shallow water theory. Four problems are considered: the decay of free edge waves and the forcing of edge waves by an incident wave of double the frequency, a synchronous incident wave and by a side-wall wavemaker. Hysteresis effects are predicted for all types of forcing.

Let Yn ⇒ Y∞ be a sequence of random variables converging in distribution, or more generally a sequenceof random elements of a suitable metric space whose distributions are converging weakly. Let τn → ∞ be positive integer-valued random variables. If {τn} and {Yn} are independent, it is trivial that