Many papers have been written on transformations of sequences which can be written as

where is a function of a finite or infinite number of variables for fixed m. If the number of variables is finite it becomes large with m.

Almost all these papers are on linear transformations of the sr, e.g. Cesàro and Abel means, but there are obvious transformations of non-linear form which are regular, and it is a theorem on these which is considered here.

It is easy to see (cf. Theorem 1 below) that the centrality of all the nilpotent elements of a given associative ring implies the centrality of every idempotent element; and (Theorem 7) these two properties are in fact equivalent in any regular ring. We establish in this note various conditions, some necessary and some sufficient, for the centrality of nilpotent or idempotent elements in the wider class of π-regular rings (in Theorems 1, 2, 3 and 4 the rings in question are not even required to be π-regular).

1. The Gontcharoff interpolation series

where

has been studied in various special cases. For example, if an = a0 (all n), (1.0) reduces to the Taylor expansion of F(z). If an = (−1)n, J. M. Whittaker showed that the series (1.0) converges to F(z) provided F(z) is an integral function whose maximum modulus satisfies

the constant ¼π being the “best possible”. In the case |an| ≤ 1, I have shown that the series converges to F(z) provided F(z) is an integral function whose maximum modulus satisfies

and that while ·7259 is not the “best possible” constant here, it cannot be replaced by a number as great as ·7378.

1. V. Bernstein, N. Levinson, R. P. Boas, Jr., and others have investigated under what conditions on the sequence

for all functions f(z) regular and of suitably restricted growth in the halfplane x≥0. (For references see [1].)

In this note the restrictions on f(z) are that f(z) is regular, and not identically zero in continuous in and that for some positive B

In a recent paper [1] of Bell, an abstract inversion principle has been formulated for inverting a type of finite series by employing operators. Bell's result involves Baker's general principle of cross-classification [2], Dedekind-Möbius inversion, L.C.M. inversion and some known generalisations. The purpose of this note is to introduce operators of negative degree and to formulate an inversion principle which covers more cases than Bell's.

In the first paper of this series (L.Q.I)1 we have shown that the logarithmetic LQ of a finite quasigroup Q is a quasigroup with respect to addition and that it is a subdirect union of the logarithmetics of the elements of Q.

In this second part we shall discuss further the structure of LQ in its additive aspect, and obtain results concerning the order N of LQ. For plain quasigroups (§3) the structure of LQ(-\-) is studied in more detail and it is shown that N is a power of n, the order of Q.

In a previous paper [Spain, Proc. Roy. Soc. Edinburgh, Vol. LX (1940), 134], I have shown that the application of the cardinal function to the problem of interpolating the derivatives yields the result

This formula is valid for x > a (the constant of integration), and R(n) < 0. The analytical continuation for R(n) ≥ 0. is indicated in the paper just quoted. The first term is the familiar expression for a fractional derivative, but the second term is not Riemann's complementary function. Furthermore, this result is unsatisfactory because it is impossible to perform the repeated operation of a fractional derivative of a fractional derivative.

In the Proceedings of the Edinburgh Mathematical Society, 1948, there appear two papers by Lars Gårding and Turnbull respectively (Gårding [1], Turnbull [2]) which formulate the theory of Cayley and Capelli operators associated with symmetric matrices. Turnbull derives the modification, appropriate to symmetric matrices, of Capelli's Theorem, which states that (taking a third order operator for the sake of ease in writing)

where the symbol (xyz)ijk stands for the determinant

with a similar meaning for the determinantal differential operator, while the symbols are polarisations (Capelli [1]; cf. Turnbull [1], p. 116). Gårding's theorem deals with the effect of such modified Capelli operators on powers of the determinant of the symmetric matrix in question. The subject of this note is an alternative derivation of the modified Capelli theorem and of Gårding's theorem.

The 6Ψ6 summation theorem was first proved by Bailey1, who deduced it indirectly from a transformation of a well-poised 8Φ7 series into two 4Φ3 series. No direct proof of the theorem has been published, and, since it has interesting applications in the proofs of various identities which occur in combinatory analysis, for example the A series of Rogers2 and some elegant identities due to Ramanujan3, we give two new proofs of the theorem in this paper.

It is shown that every spherically symmetric metric can be transformed into the isotropic form. As illustration an example is given.

This note presents a proof of the following proposition:

Theorem. If Pythagorean orthogonality is homogeneous in a normed linear space T then T is an abstract Euclidean space.

The theorem was originally stated and proved by R. C. James ([1], Theorem 5. 2) who systematically discusses various characterisations of a Euclidean space in terms of concepts of orthogonality. I came across the result independently and the proof which I constructed is a simplified version of that of James. The hypothesis of the theorem may be stated in the form:

Since a normed linear space is known to be Euclidean if the parallelogram law:

is valid throughout the space (see [2]), it is evidently sufficient to show that (l) implies (2).

The study of non-associative algebras led to the investigation of identities connecting powers of elements of such algebras. Thus Etherington1 (1941, 1949, 1951) introduced the concept of the logarithmetic of an algebra, defining it roughly as “ the arithmetic of the indices of the general element”.

We consider a medium in which the equation satisfied by a disturbance ø(x,t) is capable of solutions of the form sin(vx—wt), with v and ω real but not necessarily of the same sign. If the phase velocity U = ω/v is not of constant magnitude the medium is said to be dispersive, and the group velocity V may conveniently be defined as dω/v, an expression easily re-written in other familiar forms. From this definition the two physical interpretations of V may easily be seen. In one interpretation we consider a superposition of two harmonic waves with slightly different v (and ω); the velocity of advance of the group form, being the velocity of advance of a point at which the two waves have a constant phase difference, is dω/v.

The present paper is a continuation of the work initiated in [l]-[5]. In [5] I gave an expansion of the form

for the second order C.F. associated with

where U8, V8, W8 satisfy a fourth-order recurrence relation, there being a similar expansion for third order C.F.'s. I shall now give simple expressions for U8, V8, W8 (or related forms) in terms of χ2s(Z1), χ2s (Z2), ω2s(Z1), ω2s(Z2), where

and show that there is a remarkable relation between the recurrence formula for the first order C.F. and that satisfied by U3, V3, W3. The generalised form of these results will be stated and proved.

1. Jacobi obtained his well-known formulae by a purely algebraic method, but it was not until H. J. S. Smith had obtained them similarly, but by the use of a more symmetrical notation, that they were put into the form by which they are known today.

An integer n may be partitioned into the set of integers (α1, α2, …, αl), where Whenever αi−1>αi the substitution of αi+1 for αi, where αi may be αi+1≡0, would give a partition of n+1 whilst the substitution of αi−1—1 for αi−1 would give a partition of n—1; partitions of n+1 and of n—1 so arising will be said to be associated with the given partition of n.

We define

where ap ≠ aq when p ≠ q. If N = Σλi, then the partition (λ1, λ2, …, λn) of N with λ1 ≥ λ2 ≥ … ≥ λn is denoted by (λ) and we set

All partitions will be in descending order and the usual notation for repeated parts will be used.

In his book on Hydrodynamics, Lamb obtained a solution for the potential flow of an incompressible fluid through a circular hole in a plane wall. More recently Sneddon (Fourier Transforms, New York, 1951) obtained Lamb's solution by an elegant application of Hankel transforms.

Since the streamlines in this solution are symmetric about the wall, it is not of particular physical interest. In this note, Sneddon's method is used to give a solution in which the fluid is infinite in extent on one side of the aperture but issues as a jet of finite diameter on the other side.

Our subject is a set of equations

where the uj(n)(j = 1, 2, …, k) are k “unknown” functions of the integer variable n, the zi(n) (i = 1, 2, … h) are h “known” functions of n, and the Aij(n) are hk “known” operators

which are polynomials in E, each of fixed order pij but with coefficients which may vary with n. E is the usual operator defined by

Our first task is to determine whether the equations (1) are self-consistent. Secondly, if they are self-consistent, we ask what follows from them for a given subset of the unknowns, e.g. for (uj+1, …, uk) in other words we wish to eliminate (u1, …, uj). In particular we wish to eliminate all the variables but one, say uk. We shall in fact find that either uk is arbitrary or else that it has only to satisfy a single linear recurrence relation : and the order of that relation is of interest to us. Thirdly, we ask that reduction to standard form is possible, with or without a transformation of the unknowns themselves.

Let σk(x) be the number of integers n≤x which are the product of just k prime factors, so that

and let πk(x) be the number of such n for which all the pi are different.