Our object in this note is to construct rings such that

More generally we prove the following embedding theorem (Theorem 5.1) from which rings satisfying (I) are easily constructed:

Any algebra over a field F which contains no zero-divisors or unit-element may be embedded in an algebra over F satisfying (I).

Some examples of the bending of a plane elastic plate by transverse forces applied at isolated points are first considered. The plate is infinite and is bounded internally by a circular edge along which it is clamped; simple expressions are found for the displacement.

The analogous hydrodynamical problem is that of the steady flow of viscous incompressible liquid past a fixed circular cylinder; the equation for the stream function is in the same form as the equation for the displacement of a plate due to a distributed force of amount Z per unit area. The inertia terms in the hydrodynamical problem correspond to Z. In slow motion there is no stream function with the correct form at infinity, because in this case the inertia terms are ignored so that in the plate problem Z = 0. The effect of a transverse force system can be most simply illustrated by supposing that the forces are concentrated at isolated points; in the corresponding stream functions a simple form of allowance for inertia is therefore made, and they display some of the features of steady flow past a cylinder.

Suppose throughout that l, an (n = 0, 1, …) are arbitrary complex numbers, that α is a fixed positive number and that x is a variable in the interval [0,µ]. Let

A point x in the real or complex space lpis an infinite sequence,(x1, x2, x3,…) of real or complex numbers such that is convergent. Here p ≥ 1 and we write

The unit sphere S consists of all points x ε lp for which ¶ x ¶ ≤ 1. The sphere of radius a≥ ≤ 0 and centre y is denoted by Sa(y) and consists of all points x ε lp such that ¶ x - y ¶ ≤ a. The sphere Sa(y) is contained in S if and only if ¶ y ¶≤1 - a, and the two spheres Sa(y) and Sa(z) do not overlap if and only if

¶ y- z ¶≥ 2a

The statement that a finite or infinite number of spheres Sa (y) of fixed radius a can be packed in S means that each sphere Sa (y) is contained in S and that no two such spheres overlap.

Unless the contrary is stated, all matrices are understood to be complex and of type n × n. The transposed conjugate of A is denoted by A*. The non-negative square roots of the characteristic roots of A*A are called the singular values of A; they will be denoted by st(A), i = 1, …, n, where s1(A)≥…≥ sn(A). The symbol [A]k denotes the k × k submatrix standing in the upper left-hand corner of A. We shall write Ei(z1, …, zn) for the j-th elementary symmetric function of z1..., zn, and E1(A) for the j-th elementary symmetric function of the characteristic roots of A. It is understood that, throughout, 1≥j≥k≥n.

Many authors have proved results deducing an asymptotic expansion of

for large from the behaviour of f(t), when f(t) is regular in an appropriate part of the complex t-plane. For example, if, for some k > 0 and some Am, αm

for all large such that R(t) > C, then, as ⃗ ∞ in a suitable sector in the z-plane, we have

where Z is an appropriate value of z1/z.

In this note we consider a problem which is suggested by a paper of W. Feller. A plane set A lies in a circle C of centre O and radius 1, and is such that the linear measure of the intersection of A with any straight line does not exceed 21, where 0 < l < 1. To find the upper bound of the plane measure of A. Both linear measure and plane measure are taken in the sense of Lebesgue, and they will be denoted by m and m2 respectively.

A method of solving Oseen's equations for the flow of viscous fluid past a cylinder was devised by Bairstow, Cave and Lang [1], who found a doublet solution of the equations and reproduced the flow by a distribution of these doublets over the surface of the cylinder. The same method is used here to deal with axisymmetric flow and an integral equation is given to determine the density of the distribution. The particular case of the flow past a sphere at low Reynolds numbers is solved by this method.

An expression for the Stokes's stream-function for a slow steady axisymmetrical motion of viscous fluid due to a hydrodynamical distribution in the presence of a sphere has been given by the author [1] in terms of the stream-function for the motion due to the same distribution in unbounded fluid, this latter motion being assumed irrotational. The author has since been able to obtain a more general expression for the stream-function by not assuming the motion in unbounded fluid to be irrotational, this result being given in his Ph.D. dissertation [2]. Hasimoto [3] has since given these results, the second in a different form from that of the author. Since the general expression derived by the author is considerably simpler than that found by Hasimoto, it is given in this note. A corresponding result for the motion in a region bounded by a plane is also given.

In § 2 a number of infinite series of E-functions are summed by expressing the E-functions as Barnes integrals and interchanging the order of summation and integration.

The Barnes integral employed is

where | amp z | < π and the integral is taken up the η;-axis, with loops, if necessary, to ensure that the origin lies to the left of the contour and the points α1, α2,… αp to the right of the contour. Zero and negative integral values of the α's and p's are excluded, and the α's must not differ by integral values. When p < q + 1 the contour is bent to the left at each end.

In the course of some recent work on Fourier series [5, 6] I had occasion to use a number of integral inequalities which were generalizations or limiting cases of known results. These inequalities may perhaps have other applications, and it seems worth while to collect them together in a separate note with one or two further results of a similar nature.

For any number k, used as an index (exponent), and such that K > 1, we write k' = k′(k–1), so that k and k′ are conjugate indices in the sense of Hölder's inequality.

The first discussion of the propagation of elastic waves in a thick plate was given by Lamb [1] for the two-dimensional problem of a harmonic wave travelling in a direction parallel to the medial plane of the plate. Lamb derived equations relating the thickness of the plate to the phase velocities of two types of wave, one symmetric with respect to the medial plane and the other antisymmetric. The symmetric modes of propagation introduced by Lamb have been studied by Holden [2] and the antisymmetric modes have been studied by Osborne and Hart [3]. More recently Pursey [4] has shown how the amplitude of the disturbance is related to a given distribution of stress, varying harmonically with time, applied to the free surfaces of the plate; two types of source are considered by Pursey, one producing a two-dimensional field of the Lamb type, and the other having circular symmetry about an axis normal to the surface of the plate.

The formula to be established is

where | amp z < π, z ≠ 0, R(α + β) > R(ς) > R(alpha;) > 0, R(ς - β) > 0.

In proving (1) use will be m ade of the two following formulae.

In a recent paper [1], the theory of non-linear semi-special permutations has been developed. A method for describing such permutations, on a given range [n], is to be found in §§2, 3. This method consists in choosing a proper divisor s of n, determining all the semispecial permutations with principal number s, and then making stake all its possible values. In this way the totality of non-linear semi-special permutations on [n] may be obtained.

Guided by an observation of Hausdorff ([4]; reproduced by Whittaker and Robinson [6, pp. 177–178]), I pointed out a long time ago [7] that his “Fourier“ treatment of certain products can be systematized so as to apply to an inclusive class of infinite convolutions. Recently I noticed [8] that an appropriate application of this method supplies the following curious result on gamma-quotients:

Corresponding to every index θ on the range 0 < θ < 1, there exists on the line - ∞ < t < ∞ a monotone function μ = μθ = μθ(t) in terms of which the identity

holds on the half-plane Re z > -1 and so, in particular, on the half-line z ≧0.

In § 2 a number of integrals in which the integrand contains a product of a hypergeometric function and an E-function will be evaluated. The following formulae will be employed in the proofs.

If ρ +σ = α + β + γ + 1, and if α, β or γ is zero or a negative integer,

this is Sallschütz's theorem [1].

If R(γ - ½α - ½β)< - ½,

This theorem was given by Wastson [2] for negative integral values of α and later by Whipple [3] for general values α.

R(γ) > 0,

This formula was given by Whipple [3]

l is a positive integer,

This note is concerned with an inequality for even order positive definite hermitian matrices together with an application to vector spaces.

The abbreviations p.d. and p.s-d. are used for positive definite and positive semi-definite respectively. An asterisk denotes the conjugate transpose of a matrix.

For a series Σan with partial sums An=a0 + a1 + … + an(n ≥ 0), supposed to be real in this note, we define, in a generally accepted notation ([2], pp. 7, 9, 94–98), the following transforms:

(C, α) sequence-transform,

(H, k)sequence-transform (k = 0, 1, 2, …):

(A: C, α)fnunction-transform,

In some recent work on uniformization [2], I found it necessary to consider a regular branched covering Riemann surface Ȓ of a given Riemann surface Rf, where Rf is an unlimited branched, but not necessarily regular, covering surface of a portion Rz of the extended complex z-plane Z(2-sphere). The branching of Ȓ over Rf had to be chosen so that Ȓ was regular over Rz, since the uniformization of the functions on Rf is then simpler; in particular, the Schwarzian derivative is then a single-valued function of z.