How many digraphs are isomorphic with their own converses? Our object is to derive a formula for the counting polynomial dp′(x) which has as the coefficient of xq, the number of “self-converse” digraphs with p points and q lines. Such a digraph D has the property that its converse digraph D′ (obtained from D by reversing the orientation of all lines) is isomorphic to D. The derivation uses the classical enumeration theorem of Pólya [9[ as applied to a restriction of the power group [6] wherein the permutations act only on 1–1 functions.

In this paper is derived Lagrange's expansion with remainder for a weak function whose argument satisfies an implicit relation. A necessary and sufficient condition is given for the associated infinite series expansion.

Let C be a compact convex set in En, not necessarily containing interior points. The Steiner point of K has been defined (see Shephard [6]) as

Although the theory of supersonic inviscid flow past thin wings and slender bodies has received an enormous amount of attention in recent years, comparatively little attention has been paid to the interaction between wings and finite bodies. The principal reason stems from the extensive numerical work that has to be done in order to form a detailed picture of the flow properties—even with the present generation of highspeed computers the student of this problem is faced with a formidable task. The main effort in this interaction problem has been put into the special case when the body is a circular cylinder symmetrically disposed towards the oncoming flow, and here Nielsen (1951) and Stewartson (1951, unpublished) have shown that a formal solution can be obtained in a simple way in terms of Laplace transforms. Interpretation of these formulae however was still a formidable task, but has been greatly facilitated by the publication of an extensive set of tables of the basic functions (Nielsen 1957). Using these tables a number of workers (Randall 1965, Chan and Sheppard 1965, Treadgold, unpublished, and others) have successfully computed pressure distributions on the wing and the body, except for the neighbourhoods of the lines separating the interaction regions from the remainder of the wing and body. The aim of this paper is to provide formulae for the pressure distributions in these neighbourhoods in order to enable the solutions already obtained to be completed.

A famous conjecture of Hardy and Littlewood [4] stated that all sufficiently large integers n could be represented in the form

where p is a rational prime and x, y are integers. G. K. Stanley [9] showed that this result held for “almost all” integers n if one assumed a hypothesis concerning the zeros of L-functions similar to, though weaker than, the extended Riemann hypothesis.

In a complete separable metric space the Souslin sets coincide with the analytic sets and the projection properties of the Souslin sets follow from those of the analytic sets. The projection properties of Souslin sets do not seem to have been discussed in more general circumstances when this equivalence to the analytic sets breaks down. The object of this note is to contribute the following result, that applies in these circumstances, and which will be used by Willmott in forthcoming work on the theory of uniformization.

The problem of a penny-shaped crack which is totally embedded in an isotropic material is treated by the theory of linear elasticity. It is shown that for a prescribed crack surface displacement due to compressive stresses on the surface, stress singularities of order higher than the usual inverse square root are possible. It is also demonstrated that for all physically admissible crack surface stresses the singularity can only be of the inverse square root order and that the shape of the crack tip must be elliptical.

It was conjectured by Mordell [6] that the Hasse principle holds for cubic surfaces in 3-dimensional projective space other than cones†: i.e., that such a surface defined over the rational field 0 has a rational point whenever it has points defined over every p-adic field Qp. This conjecture was verified for singular cubic surfaces by Skolem [11” and for surfaces

with

by Selmer [9]: but it was disproved for cubic surfaces in general by Swinnerton-Dyer [12] (see also Mordell [7]). It therefore becomes of interest to specify fairly wide classes of cubic surfaces for which the Hasse principle does hold. It was shown independently by F. Châtєlet and by Swinnerton-Dyer (both, apparently, unpublished) that this is the case when it contains a set of either 3 or 6 mutually skew lines which are rational as a whole (and trivially true when there is a rational pair of lines, since then there are always rational points). Selmer [9] conjectures on the basis of numerical evidence that the Hasse principle is also true for all surfaces of the type (1). It is the object of this note to disprove this by showing that the Hasse principle fails for

In 1934 Gelfond [2] and Schneider [6] proved, independently, that the logarithm of an algebraic number to an algebraic base, other than 0 or 1, is either rational or transcendental and thereby solved the famous seventh problem of Hilbert. Among the many subsequent developments (cf. [4, 7, 8]), Gelfond [3] obtained, by means of a refinement of the method of proof, a positive lower bound for the absolute value of β1 log α1+β2 log α2, where β1, β2 denote algebraic numbers, not both 0, and α1,α1, denote algebraic numbers not 0 or 1, with log α1/log α2 irrational. Of particular interest is the special case in which β1,β2 denote integers. In this case it is easy to obtain a trivial positive lower bound (cf. [1; Lemma 2]), and the existence of a non-trivial bound follows from the Thue–Siegel–Roth theorem (see [4; Ch. I]). But Gelfond's result improves substantially on the former, and, unlike the latter, it is derived by an effective method of proof. Gelfond [4; p. 177] remarked that an analogous theorem for linear forms in arbitrarily many logarithms of algebraic numbers would be of great value for the solution of some apparently very difficult problems of number theory. It is the object of this paper to establish such a result.

A study is made of the solutions of the differential equation f′″ + ff″ = 1 subject to the initial conditions f(0) = f′(0) = 0.

It is well known that every convex polytope in d-dimensional euclidean space Ed can be approximated arbitrarily closely, in the Hausdorff sense, by convex polytopes whose faces are simplexes (see [2, Section 4.5]). In this paper we prove some generalizations of this result, investigating the possibility of approximating a given d-polytope (d-dimensional convex polytope) by polytopes whose facets (faces of d − 1 dimensions) are all of some prescribed type.

1.1. Let A = (aμν) be a normal triangular matrix, i.e., one for which aμμ ≠ 0 (μ ≥ 0), aμν = 0 (ν > μ).

where (i) 0≤m<n, (ii) Rμ>0 (μ≥0), (iii) K is a constant, depending on the matrix A and the sequence {Rμ}, but independent of m, n and the finite sequence {sν}.

Let H be any closed bounded convex set in En, and -H be its reflection in the origin. Then the vector sum K = H+ (−H) has the origin as centre and is called the difference set of H. Clearly every closed bounded convex set K with centre at the origin is the difference set of ½K. Excluding this trivial case, we define such a set K to be reducible if it is the difference set of some H which is not homothetic to K.

We wish to prove a theorem concerning the average values for the functions Ln(2u)e−u, 0 ≤x≤u < ∞, n=0,1,2, …, where Ln is the n-th Laguerre polynomial. Such functions will be called Laguerre functions with domain truncated at x.

By a permutation mapping in a finite projective plane π is meant a one-to-one mapping σ: P→l of the points of π onto the lines of π with the property that corresponding elements are incident. The simplest aspects of such mappings are discussed in this note.

In this paper solutions are given for two problems on the motion of a dusty gas, using the formulation of Saffman [1]. The gas containing a uniform distribution of dust, occupies the semi-infinite space above a rigid plane boundary. The motion induced in the dusty gas is considered in the two cases when the plane moves parallel to itself (i) in simple harmonic motion, and (ii) impulsively from rest with uniform velocity. In case (i) the change in phase velocity and the decay of oscillatory waves are noted as functions of the mass concentration of dust f. In case (ii) the problem is solved by use of the Laplace transform, some velocity distributions are calculated for f=0·2, and it is shown that the shear layer thickness is decreased by the factor (1+f)−1/2 at large times.

In this paper, solutions of the ordinary non-linear differential equation

are considered. This equation arises in the theory of both axisymmetric and two-dimensional viscous jets falling under gravity. In (1.1), y represents the first approximation to the velocity along the axis of symmetry, and x is a measure of the distance along this axis. Accordingly one of the conditions that the solution must satisfy is, that it cannot have a singularity at a finite value of x. The other condition to be imposed is that y must vanish at x = 0. For a derivation of this equation the reader is referred to Brown [1] or Clarke [2].

The dependence of the skin friction on the parameter β for the reversed flow solutions found by Stewartson of the Falkner-Skan equation f′′′ + ff″ + β (1−f′2)=0 is determined in the limit as β→0−.