For a continuous function f(x) on the reals or on the circle T (continuous and 2π periodic) we say that f(x) belongs to the generalized Lipschitz class, denoted by f ∈ Lip* α, if

where and Δhf(x) = f(x + ½h)−f(x−½h). For a convolution approximation process given by

where

we shall investigate equivalence relations between the asymptotic behaviour of (d/dx)rAn(f, x) and f ∈ Lip* α.

The purpose of this paper is to show that 3n annihilates the 3-primary component of the homotopy groups of the 2n + 1-dimensional sphere. In the terminology of (2) and (3), S2n+1 has exponent 3n at 3.

In fact, a stronger result is proved. Localize at 3 and let Ω2nS2n + 1〈2n + 1〉 denote the 2n-fold loop space of the 2n + 1-connected cover of S2n+1. Then Ω2nS2n + 1〈2n + 1〉 has a null homotopic 3n-th power map.

To a finite-dimensional real representation V of a finite group G there are associated its Stiefel–Whitney classes wk (V) (k = 1, 2, 3, …) in the cohomology groups Hk(G; ). ( is the field with two elements.) The total Stiefel-Whitney class

in the ring H*(G; is natural with respect to G in the obvious sense, and, in addition,

(a) exponential, i.e. w(V ⊕ W) = w(V).w(W),and

(b) stable, i.e. w(V) = 1 when F is a trivial representation.

If V is a symplectic space, each affine contact transformation is shown to define an automorphism of a certain algebra of Schwartz functions on V. This automorphism turns out to be a generalized inner automorphism and allows one to associate to the contact transformation a tempered distribution which can be found explicitly. It is shown by means of various examples that these distributions contain much information about the corresponding quantized system. For example the energy levels and stationary states of the harmonic oscillator can be deduced from the general formulae. Generalizations to other locally compact abelian groups V are described. Some connections with the theory of Fourier integral operators and with the characters of certain projective representations are also outlined.

Let X be a topological space with base-point. The algebraic K-theory AX of X is a space invented by Waldhausen in (15) for use in geometric topology. It can be defined in two ways, which I shall call geometric and ring-theoretic; Steinberger ((12)) has shown them to be equivalent.

The geometric method ((15), corollary to lemma 2·1) gives AX as the group-completion of the geometric realization of a permutative category. It follows from the machinery of May ((4), 4) or Segal (11) that AX is an infinite loop space in a well-defined way ((6)).

A complete normed real Jordan algebra A with 1 is said to be a JB-algebra if and only if, for all a, b ∈ A,

where o denotes the Jordan product on A. If A is a Banach dual space then A is known as a JBW-algebra. In this case, by Theorem 3·9 of (7),

where Aex is an exceptional Jordan algebra and Asp is Jordan isomorphic to a JW-algebra (i.e. a weakly closed Jordan algebra of self-adjoint operators on a complex Hilbert space). Theorem 3–9 of (7) gives a description of Aex and so in this paper attention will be restricted to JW-algebras. Such an algebra A is said to be of type I if there exists an idempotent e in A with central cover equal to 1 such that e is abelian i.e. such that eAe is commutative as a set of operators. (See § 7 of (11) for further details.) A type I algebra A is said to be of type In, where n is some cardinal number, if there exists an orthogonal family {eα: α ∈ Λ} of n abelian projections in A such that Σαeα = 1 and such that, for each α ≠ β, there exists a symmetry sαβ in A with eβ = sαβeαsαβ. (A symmetry is an element sαβ of A such that ). By theorems 15 and 16 of (11) any type I JW-algebra A can be uniquely decomposed into a direct sum of type In(α) JW-algebras for suitable cardinal numbers n(α). Thus in order to investigate the structure of type I JW-algebras it is sufficient to consider type In JW-algebras.

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).

This paper is a continuation of (6). We will use the definitions and notation of (6) freely. We continue our study of the restrictions imposed on the mod p cohomology of finite H-spaces by the requirement that is a U(M) algebra. As in (6) we will restrict our attention to the case where p is an odd prime. As we observed in (6), the restriction proved there, that the pth power map on is trivial, may not really require the hypothesis of . However, the restrictions proved in this paper are not valid without the assumption that . Indeed, they actually characterize the known mod odd finite H-spaces which have U(M) algebras.

Let A be the Steenrod algebra over the field F2. For any graded A-module M we construct here a small chain complex of A-modules EM. This complex computes the homology of the Steenrod algebra with coefficients in M, in the sense that:

The complexes F2 ⊗AEM are comparable in size but slightly larger than those that arise from the Λ-algebra(4). We are motivated to introduce them because they can be used to prove an embedding theorem for the homology of the Steenrod algebra. In fact, we construct a functor K on the category of graded. A-modules, and using the complexes EM, prove that there is a natural, one-one map:

Let i: M↬ℝn+1 be a self-transverse immersion of a compact closed smooth n-dimensional manifold in (n + 1)-dimensional Euclidean space. A point of ℝn+1 is an r-fold intersection point of the immersion if it is the image under i of (at least) r distinct points of the manifold. The self-transversality of i implies that the set of r-fold intersection points is the image of an immersion of a manifold of dimension n+1-r (the empty set if r > n + 1). In particular, the set of (n + l)-fold intersection points is finite of order, say, θ(i). In this paper we are concerned with the set of values of θ(i) for (self-transverse) immersions of all (compact closed smooth) manifolds of given dimension n.

Suppose that we are given a random sample of size n chosen according to the uniform distribution on the unit interval. Let Zn(x) = Zn(x, ω) be the length of the unique left-closed and right-open sample spacing that contains x. The purpose of this paper is to examine the almost sure, and exceptional, growth rates of the process {Zn}. The typical maximum growth rate and the growth rate of the maximum can be of quite different orders of magnitude as is shown by the following two results.

Theorem 2. With probability one we have

for almost all x.

Theorem 3. With probability one we have

The theory of unfoldings of singularities, or ‘elementary catastrophe theory’ (Thom(10), Poston and Stewart(9), Golubitsky and Guillemin(3), Gibson(2)) has been generalized by Golubitsky and Schaeffer(5,6), providing a powerful method for analysing imperfect bifurcation. One recent application by Golubitsky, Keyfitz, and Schaeffer(4) concerns the ‘explosion peninsula’ in chemical reactions such as that between hydrogen and oxygen. They show that the ‘thermal-chainbranching model’ of Gray and Yang(7,8) is capable of reproducing the necessary qualitative features, a result for which only heuristic arguments had previously been available.

For high-dimensional simple knots we give two theorems concerning unique factorization into irreducible knots, and provide examples to show that the hypotheses are necessary in each case.

A method is presented whereby all locally defined conserved currents of the Klein-Gordon field are found. The mathematical background to the method includes a generalization of the Poincaré lemma of the calculus of exterior differential forms. It is found that the only conserved currents are essentially a countably infinite set of functions, bilinear in the field, together with a single current in the case where the mass is zero. The usual energy-momentum tensor is included amongst these functions. The method does not depend on the use of any canonical formulation of the field theory.

A great deal is known about infinitely divisible distributions on [0, ∞). In this paper, a simple mapping which can be used to extend this information to more general convolution algebras is examined. Some examples are given where this approach has proved fruitful.

In a previous paper (4), the author showed that the martingale approach to point processes may be used to derive Poisson distributional limit theorems. Here this approach is extended to Cox convergence, general increasing processes are considered, and a regularity condition needed in (4) is removed (that of calculability). In Section 3 we give examples to show the need for the various hypotheses used in the main theorem of Section 2. This theorem is applied to various (dependent) thinnings and compound-ings of point processes in Section 4.

In this paper the major concern is in investigating various divisibility properties of the random variable W, the limit of the Heyde martingale in supercritical branching processes. Throughout the paper ø(s) will be used to denote the Laplace-Stieltjes transform of W.

Necessary and sufficient conditions for the local existence of a metric compatible with the affine connection are obtained in terms of the Riemann tensor and its first-order covariant derivatives in a generic affine manifold with torsion. In case these conditions are satisfied, the solutions of the metric are given in terms of integrals and are unique up to a constant scale factor. Some global conditions are also obtained and discussed.

Some models for spatial point processes are difficult to simulate directly and are most easily realized as the equilibrium distribution of certain spatial-temporal Markov processes. This paper examines the convergence of such processes, concentrating mainly on the ‘hard core’ case when the points represent the centres of non-overlapping discs. Coupling methods from the theory of Markov chains are used to establish sufficient conditions for the processes to converge to the required equilibrium, and to give a lower bound on the rate of convergence. One technique used is to couple processes when they become close in a suitable metric.

Let I be a countable set, with discrete topology, and let X = (Xn), Y = (Yn) be stationary stochastic processes taking values in I. To a probabilist, the natural topology on processes (strictly speaking, on distributions of processes) is weak convergence: