The fact that for two or more real positive quantities there exist three well-known algebraic means, the Arithmetic, the Geometric, and the Harmonic, which stand in a fixed order of magnitude independent of the quantities operated on, suggests the question whether there may not be other algebraic means that stand in a definite order of magnitude with reference to those just named and to one another. The following paper supplies an affirmative answer to the question. The results given in the first section are, so far as I know, novel; some of those in the second section are well known, but I hope some freshness may be apparent in their treatment here.

Let represent any permutation of the first n natural numbers. If we take the first number, a1, and put it last, we get a new permutation; and if we perform this operation (n−1) times, we get the (n−1) permutations

A central trace on an order-unit Banach space A(K) is a centre-valued module homomorphism invariant under the group of symmetries of A(K).

The concept of central traces has been crucial in the theory of types for convex sets established in (4), (5). In von Neumann algebras, they are precisely the canonical centre-valued traces and their existence hinges on a fundamental theorem (Dixmier's approximation process) in von Neumann algebras. On the other hand, the existence of central traces in finite dimensional spaces is an easy consequence of Ryll-Nardzewski's fixed point theorem (5).

Consider the initial boundary value problem

In the context of the heat conduction problem, this models the case where the heat flux across the ends at the rod is a function of the temperature. If the heat exchange between the rod and its surroundings is purely by convection, then one commonly assumes that f is a linear function of the difference in temperatures between the ends of the rod and that of the surroundings, (Newton's law of cooling). For the case of purely radiative transfer of energy a fourth power law for the function f is usual, (Stefan's law).

A note on this subject was read to the Edinburgh Mathematical Society in June 1951. Subsequently I had the benefit of conversations and correspondence with J. G. Brennan and I. R. Porteous, and substantial contributions from them are incorporated here, more than are acknowledged in detail below.

The presence of a non-steady state of temperature in an elastic solid gives rise to an additional term in the generalised Hooke's Law connecting the stress and strain tensors and terms involving the time rate of change of the dilatation. This time-dependent dilatation may produce so-called thermoelastic stress waves. The present note is concerned with the effects produced by these additional terms in a simple situation, in which the elastic solid is regarded as a thin plate of infinite extent. The distribution of temperature in the plate is produced by a point heat source of Dirac type.

In the following discussion we shall assume that pn≧0, qn≧0 for all n and that qn + 1 > qn → ∞. The (J, pn, qn) method of summation is defined as follows.

The series with the partial sum sn, is called summable (J, pn, qn) to s, and we write if the series

and converge to the sum functions p*(x) and p(s)(x) respectively for 0<x<1 and if τ(x) = p(s)(x)/p*(x)→s as x→1–0.

The purpose of this work is to establish a priori C2, α estimates for mesh function solutions of nonlinear positive difference equations in fully nonlinear form on a uniform mesh, where the fully nonlinear finite-difference operator ℱh is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We deal here with the special case that the operator does not depend explicitly upon the independent variables. We do this by discretizing the approach of Evans for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger. The result in this special case forms the basis for a more general result in part II. We also derive the discrete interpolation inequalities needed to obtain estimates for the interior C2, α semi-norm in terms of the C0 norm.