The formula to be proved is

In proving (1) the following formulae are required:

where p≥q + 1, (1).

In § 2 a product of two modified Bessel Functions of the Second Kind is expressed as an integral with a function of the same type as a factor of the integrand. In § 3 an integral involving a product of these functions, regarded as functions of their orders, is evaluated in terms of another function of this kind. These results were suggested by a study of Mellin's inversion formula.

The modern algebraic treatment of geometry in projective spaces focuses attention on the properties of homogeneous ideals in polynomial and power-series rings. This inevitably raises questions concerning how far ordinary ideal theory needs to be modified if only homogeneous ideals are to be regarded as significant. In practice, one can usually answer any particular question of this type without undue difficulty when it arises but, it seems to the author, the topic has enough intrinsic interest to merit a connected discussion by itself.

A point x in real Hilbert space is represented by an infinite sequence (x1, x2, x3, …) of real numbers such that

is convergent. The unit “sphere“ S consists of all points × for which ‖x‖ ≤ 1. The sphere of radius a and centre y is denoted by Sa(y) and consists of all points × for which ‖x−y‖ ≤ a.

The formula to be proved is

Recently R. S. Varma [11] gave the generalisation

for the Laplace transform

Since

(1) gives (2) when k = − m + ½.

We shall represent (1) by

and as usual, (2) will be denoted by

Groups that can be represented as the product of two proper subgroups have been studied extensively; one of the latest contributions is a paper by Wielandt (8), in which references to previous work can be found. In the case where the two proper subgroups have only the unit element in common, we adopt the term ‘general product’introduced by Neumann (1).

It was remarked by Liouville in 1844 that there is an obvious limit to the accuracy with which algebraic numbers can be approximated by rational numbers; if α is an algebraic number of degree n (at least 2) then

for all rational numbers h/q, where A is a positive number depending only on α.

Certain results ([7], [8], [10], [11]) suggest that there should be some principle of duality in homotopy theory. Among other things one is led to expect that cohomotopy groups will appear as dual to homotopy groups. But the fact that a cohomotopy group πn(X), unlike πn(X), is only defined if dim X ≤ 2n—2 is a serious obstacle to the formulation of such a principle. However, the set of S-maps (i.e.S-homotopy classes [11]) X → Y is a group for every pair of spaces X, Y. Therefore, this difficulty does not appear in S-theory [11].

An analogy is drawn between the slow steady rotation of a solid of revolution about its axis in a viscous fluid and the motion, with constant velocity, of the solid along its axis in a perfect fluid. The rotation of a sphere in viscous fluid is then discussed using a method due in this case to Bickley [1] and here further developed.

In this note it is shown that

(i) there is a plane convex set of minimal width λ which does not contain any convex set whose width is constant and greater than λ/(3—31/2),

(ii) every plane convex set of minimal width λ contains a convex set whose width is constant and is equal to λ/(3—31/2).

The main object of this paper is to show that an indefinite quadratic form in four or more variables with integral coefficients represents all integers except those excluded by congruence considerations. The subject of the representation of integers by quadratic forms is one on which little is known, and I believe this result to be new, although I can deduce it in an elementary way from the classical theory.

A well-known method of calculating an approximate value of an eigenvalue λ of a self-adjoint operator H employs an approximation, say w, to the corresponding eigenfunction ψ. Since

the exact value of λ is given by

In a recent paper [4], I introduced the notion of recursive formal Lie groups (of infinite dimension) over a field of characteristic p > 0, and studied a particular class of such groups, the groups of hyperexponential type; these can be characterized either as being (recursively) isomorphic to a special group of that class, the hyperexponential group, or by simple conditions on their Lie hyperalgebra. An interesting example of a group of that class is the additive Witt group W, whose “infinitesimal” structure can therefore be considered as known, at least “up to an isomorphism”. However, the intrinsic importance of the Witt group (which, as well known, is the “formalization”, so to speak, of the additive group of a p-adic field) leads one to think that it may be worth while to study in greater detail that group itself, instead of being content with the mere existence of an unspecified isomorphism with the hyperexponential group. This is what we intend to do here; it turns out that, although it seems hopeless to write down explicitly the group law of the Witt group, the multiplication table of its hyperalgebra is, on the contrary, as simple as one could hope, and is, in fact, identical to that of the hyperalgebra of the hyperexponential group (although the two groups are distinct). Moreover, this leads to a new and quite unexpected definition of the Witt group, which links it still closer to the hyperexponential group, and provides a well-determined isomorphism between the two groups.

Minkowski's fundamental theorem and the Minkowski-Hlawka theorem play basic complementary roles in the Geometry of Numbers. Blichfeldt showed essentially that Minkowski's fundamental theorem was a simple consequence of a more general theorem, in which the convex body was replaced by any measurable set and the lattice was replaced by a discrete set of points having a definite asymptotic density. Hlawka himself showed that the Minkowski-Hlawka theorem could be proved in a slightly modified form, when the star body was replaced by any measurable set, but he did not replace the lattice by a more general set of points.

A method of assessing the accuracy of an approximation to the steady two-dimensional flow of viscous incompressible liquid past a flat plate is developed. The approximate solution, with stream-function ψ1 is closely related to the boundary-layer solution, and is expressed in terms of the function used in that theory. The conditions at the surface of the plate and at infinity are exactly satisfied by ψ1 but the equation of (finite) motion is not exactly satisfied. However, ψ1 is an exact solution of a problem in fluid motion, if a body force, of appropriate magnitude depending on ψ1 is assumed to act on the fluid. The magnitude of this force provides a criterion of the accuracy of ψ1, and some use has been made [5] of this means of estimating accuracy. This criterion is, in effect, only qualitative, since it is not possible to make a numerical estimate of the effect on the motion of a body force which is non-conservative and acts over the whole field of flow.

The general features of the interaction of hydrodynamic and electromagnetic effects, when an electrically conducting fluid medium is in motion in the presence of a magnetic field, are now well established. A discussion of these effects in which the equations of the system are set out and an analogy is drawn between the behaviour of the magnetic field and that of vorticity in a viscous fluid, is given by Batchelor [1]. The development of the subject may be divided broadly into two sections. First, there are the problems of engineering interest, such as flow of conducting fluid in pipes, with associated problems of stability, and secondly there are the problems of astrophysical and geophysical interest.

Graham Higman (1) has investigated quasi-order [(2), p. 4] relations a ≤ b on a set S which, in his terminology, have the following finite basis property:

If A ⊂ S, then there is a finite set B ⊂ A such that, given any a α A, there is some b α B satisfying b ≤ a.