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is a circular cylinder symmetrically disposed towards the oncoming flow, and here Nielsen (1951) and Stewartson (1951, unpublished) have shown that a formal solution can be obtained in a simple way in terms of Laplace transforms.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 Using these tables a number of workers (Randall 1965, Chan and Sheppard 1965, Treadgold, unpublished, and others) have successfully computed pressure distributions on the wing and the body, except for the neighbourhoods of the lines separating the interaction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.

W. Feit [1], N. Itô [2] and M. Suzuki [3] have determined all doubly transitive groups with the property that only the identity fixes three symbols. It is of interest to the theory of projective planes to determine whether any of these groups contain a sharply doubly transitive subset (see Definition 1). It is found that if such a group G contains such a subset R then R is a normal subgroup of G, i.e. R is a doubly transitive normal subgroup of G in which only the identity fixes two symbols.

A topology on a set X is defined by specifying a family of its subsets which has the properties (i) arbitrary set intersections of members of belong to , (ii) finite set unions of members of belong to and (iii) the empty set □ and the set X each belong to . The members of are called the closed subsets of X. If X is any subset of X then denotes the closure of X, that is, the set intersection of all closed subsets which contain X, however when X = {x} contains one point only we will denote by . The pair (X, ) is called a topological space or, in what follows, a T-space. By a T-lattice we mean a complete distributive lattice of sets in which arbitrary g.l.b. means arbitrary set intersection, finite l.u.b. means finite set union and which contains the empty set □ It is well-known, for example Birkhoff [1], that if (X, ) is a T-space and the members of are partially ordered by set inclusion then is a T-lattice.

In a recent paper, Maher [2] proved that for any algebraic number field K of degree n and discriminant d there exists a constant C depending only on n and d such that for any ceiling λP of K there exists a basis α1 …, αn of the corresponding ideal αλ such that .

Let (X, B, m) be a measure space and let f(x) be a real-valued or complex-valued measurable function on X. A non-negative measurable function s(x) will be said to dominate f(x) provided |f(x)| ≦ s(x) for almost all x in X. The function s(x) will be said to dominate the sequence {f(x)}n∈N, N = {1, 2,…}, provided it dominates each fn(x) in the sequence. Unless otherwise specified, each integral will be over X with respect to m.

In this paper we obtain the sum of some infinite series involving hypergeometric functions of one or more variables.

It is known that to every proper homogeneous Lorentz transformation there corresponds a unique proper complex rotation in a three-dimensional complex linear vector space, the elements of which are here called “rotors”. Equivalently one has a one-one correspondence between rotors and self- dual bi-vectors in space-time (w-space). Rotor calculus fully exploits this correspondence, just as spinor calculus exploits the correspondence between real world vectors and hermitian spinors; and its formal starting point is the definition of certain covariant connecting quantities τAkl which transform as vectors under transformations in rotor space (r-space) and as tensors of valence 2 under transformations in w-space.

Let G be a finite group of odd order with an automorphism ω of order 2. The Feit-Thompson theorem implies that G is soluble and this is assumed throughout the paper. Let Gω denote the subgroup of G consisting of those elements fixed by ω. If F(G) denotes the Fitting subgroup of G then the upper Fitting series of G is defined by F1(G) = F(G) and Fr+1(G) = the inverse image in G of F(G/Fr(G)). G(r) denotes the rth derived group of G. The principal result of this paper may now be stated as follows: THEOREM 1. Let G be a group of odd order with an automorphism ω of order 2. Suppose that Gω is nilpotent, and that G(r)ω = 1. Then G(r) is nilpotent and G = F3(G).

In the development of the rotor calculus presented in a previous paper space-time was taken to be flat. This work is now extended to the case of curved space-times, which, in the first instance, is taken to be Riemannian. (The calculus bears at times a strong formal resemblance to the spinor analysis of Infeld and van der Waerden, but it is in fact developed quite independently of this.) Owing to the fact that all general relations had earlier already been written in a generally covariant form they may be taken over unchanged into the present context. In particular,

now serves as a defining relation for the connecting quantities τAkl. A linear rotor connection is introduced, and the covariant derivative of a rotor defined. The covariant constancy of the τAkl establishes the relation between the linear connections in w-space and in r-space. A rotor curvature tensor is considered alongside a number of other curvature objects. Next, conformal transformations are dealt with, of which duality rotations may be considered as a special case. This leads naturally to a gauge-covariant generalization of the whole calculus. A so-called rotor-derivative is defined, and some general relations involving such derivatives investigated. The relation of the rotor curvature tensor to the spin-curvature tensor is touched upon, after which the introduction of “geodesic frames” is considered. After this general theory some points concerning the Maxwell field are dealt with, which is followed by some work concerning basic quadratic and cubic invariants. Finally, a certain basic symmetric rotor is re-considered in the context of the classification of Weyl tensors, and the idea of a canonical representation suggested.

We say that a subgroup H is an n-th maximal subgroup of G if there exists a chain of subgroups G = G0 > G1 > … > Gn = H such that each Gi is a maximal subgroup of Gi-1, i = 1, 2, …, n. The purpose of this note is to classify all finite simple groups with the property that every third maximal subgroup is nilpotent.