This note concerns a question of elementary calculus. Given a smooth composite function u = g o f [with values u(x) = g(f(x))], we write explicit formulae for its derivatives, of arbitrary order, in terms of derivatives of f and g. We consider (A) the general case,

in which E, F and G are Banach spaces, and U, V are open sets; (B) the finite-dimensional case E = ℝM and F = ℝN, where ℝM denotes real M-dimensional Euclidean space; and (C) the particular case of (B) (due to restricting g to part of an M-dimensional surface in ℝM + 1) in which N = M + 1 and u(x)= g(x, φ (x)), so that φ denotes a real-valued (scalar-valued) function of x = (x1, …, xM).

Let = (Ai;i ∈ I) be an indexed family of sets. The family is said to contain the family = (Bj;j ∈ J) if there is a one-to-one map f from J into I such that Bj = Af(i) for all j in J. The indexed family = (Ai;i ∈ I) is said to be a Δ (λ)-family if |I| = λ and Ai ∩ Aj = Ak ∩ Ai for all pairs i, j and k, l from I. The family is said to be a (λ, k)-family if |I| = λ and |Ai| = k for all Ai.

A subset, X, of R belongs to the family T(a) if |X| ≥ 2 and, for all x and y belonging to X, ax + (1 − a) y ∈ X, where a ∈ R.

I consider the problem of determining for which values of a > 1 all elements of T(a) are dense in R.

Let A = (aij) be a non-negative n × n matrix, that is, aij ≥ 0, i, j = 1, …, n; n > 1. We write A ≥ 0. Let r = r(A) be the spectral radius of A; assume r > 0 throughout to avoid trivial cases. Let be the mth derivative of r with respect to the element aij, all other elements of A being held constant.

In (1) it is remarked that theorem 5·10 is an analogue, for endomorphisms of a free group, of a result of A. Shenitzer ((3), theorem 2) about automorphisms. With this in mind, D. Solitar asked the author if there might also be an analogue for the corollary on page 276 of(3). There is such an analogue and, after a preliminary discussion of definitions and notation, we will describe it.

In (1) we showed that a subgroup closed Fitting formation is a primitive saturated formation, and in (2) we showed that a subgroup closed and metanilpotent Fitting class is a formation. Whether or not a subgroup closed Fitting class is always a formation is a question that has plagued us ever since. The purpose of this paper is to prove

In (2), Erdös and Straus pose the following two conjectures.

If G is a finite group of order n and X is a subset of G consisting of pairwise non-commuting elements, then |X| ≤ n/2 + 1.

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.

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.

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.

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 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.

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).

In (5) a weak convergence result for U-statistics was obtained as a special case of a reverse martingale theorem; in (7) Miller and Sen obtained another such result for U-statistics by a direct argument. As they stand these results are not very closely connected, since one is concerned with U-statistics Uk for k ≥ n, while the other deals with Uk for k ≤ n, but if one instead thinks of k as unrestricted and transforms the random functions Xn which enter into one of these results into new functions Yn by setting Yn(t) = tXn(t−1) one finds that the Yn are (aside from variations in interpolated values) just the functions with which the other result is concerned. As the limiting Wiener process W is well-known to have the property that tW(t−1) is another Wiener process it is not too surprising that both results should hold, and part of the purpose of this paper is to provide a general framework within which the relationship between these results will become clear. A second purpose is to illustrate the simplification that the martingale property brings to weak convergence studies; this is shown both in the U-statistic example and in a new proof of the convergence of the empirical process.

In a previous paper this author used elementary mathematics to describe some simple connexions between multi-valued Legendre transformations and those elementary catastrophes having one ‘behaviour’ variable, which belong to a family of polynomials called cuspoids (see Sewell (13)). The Legendre singularities permitted there are isolated and of dimension one, i.e. they exist where a Hessian matrix exceptionally has co-rank one (not more) and elsewhere has full rank (co-rank zero).

A new method for generating tensorial derivatives of tensor functions is proposed. The method is based on the use of tensors as absolute entities along with the advantages offered by their decomposition on an orthonormal base.

Some aspects are considered of the Einstein vacuum field equations for the diagonal form of the general static metric. A power series expansion is considered in which in the first non-trivial order one gets seven equations for four unknowns. This is reduced to a single Poisson equation in which the source is given in terms of a harmonic function and some arbitrary functions of two variables. A similar reduction is carried out in detail for the next non-trivial order. The approximation scheme can be extended to arbitrarily high orders in principle. The connexion of the diagonal form with the Weyl solutions and the merits of the diagonal and non-diagonal forms are briefly discussed. A general solution of one of the field equations is presented.

It is shown in this paper that the conservation laws of non-local continuum mechanics are equivalent to, and can be derived from, the conservation of energy and the invariance conditions under superposed rigid body motions. Also, the theory of non-local thermoelasticity is reconsidered in the light of recent developments in thermodynamics, taking invariance conditions fully into account.

The initial-value problem of waves generated by a moving oscillatory surface pressure against a vertical cliff is solved and a uniform asymptotic analysis of the unsteady state is given. The same problem with no cliff is solved by Kaplan, and by Debnath and Rosenblat but their solutions are not uniform.

Fluid is contained in a rotating annulus of rectangular cross-section which is subject to a radial temperature gradient along the base and is insulated around its upper surfaces. If E is an Ekman number, which is assumed small, a finite amplitude cellular motion of wavelength ∼ E⅓ develops near the inner sidewall of the annulus by an exchange process if the local Rayleigh number, R, exceeds its critical value Rc and the Prandtl number of the fluid, σ, is greater than a critical value, σc (Daniels and Stewartson(5)). In the present study we consider the remaining range of values 0 < σ < σc for which overstability is preferred, occurring at a value of the Rayleigh number, Rc0, less than Rc. Disturbances to the basic flow near the sidewall now oscillate in time with frequency ∼ E⅓ and are amplified if R exceeds Rc0, the vertical cellular velocity attaining values ∼ E in a region of extent ∼ E near the sidewall. On the time-scale ∼ E disturbances take the form of two wave-like components which travel in opposite directions with the characteristic group velocity of the system. The first is simply convected away from the sidewall into the stably stratified interior and decays, while the second travels towards the sidewall where it is reflected. The reflexion coefficient is determined by the dynamics of an E⅓ boundary layer where an oscillatory motion is generated by the incident wave.