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 It was shown independently by F. Châtєlet and by Swinnerton-Dyer (both, apparently, unpublished) that this is the case when it contains a set of either 3 or 6 mutually skew lines which are rational as a whole (and trivially true when there is a rational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

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−.

1. Certain geometric properties of the valuation theory were considered by O. Zariski in [7]. We have proved some related results in [1] and we consider further similar problems in this paper.

Let V be an irreducible algebraic primal situated in Sd, where d≥3. Throughout the ground field is the field K of complex numbers. For simplicity we assume that V lies in an affine space Ad with coordinates x1,…,xd. Let O be a point on V not at infinity and we take it to be the origin of Ad. Apply a monoidal transformation to V with O as the basis; We obtain thereby a (d−l)-fold V1 lying on a non-singular d-fold U1 situated in an affine space of dimension N1 Since V and V1 are birationally equivalent, we may identify their function fields and thus we denote their common function field by Σ.

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

Let A = {a1, …, an} and B={b1, …, bn} be two sets of n elements, and let R be a set of ordered pairs (ai, bi), or in other words a relation defined on A × B. By a map between A and B under R we mean a one-to-one correspondence between A and B such that if bi, corresponds to ai then (ai, bj) is one of the pairs in R.

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}.

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.

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.

The object of this paper is to give a simple proof of Menger's famous theorem [1] for undirected and for directed graphs. Proofs of this theorem have been given by D. König [2], G. Nöbeling [3], G. Hajós [4], T. Gallai [5], P. Erdös [6] and 0. Ore [7]. The present proof is shorter, and formulated to apply to directed and undirected graphs equally. The term, graph is to be understood to mean either a finite undirected graph or a finite directed graph throughout. (A: B) denotes either an undirected edge between the two vertices A and B or a directed edge from A to B according to whether undirected or directed graphs are considered. G1 and G2 being two non-empty disjoint graphs, G1: G2-edge denotes an undirected edge between a vertex of G1 and a vertex of G2 or a directed edge from a vertex of G1 to a vertex of G2 as the case may be, and G1: G2-path denotes a path with one end-vertex in G1 and one in G2 having no intermediate vertex in G1 + G2, undirected or directed from the end in G1 to the end in G2, as the case may be. (A set of isolated vertices is a graph.)

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 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 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.

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.

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

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].

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.

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.

Suppose, in n-dimensional Euclidean space, a sequence of disjoint closed spheres is packed into the open unit n-cube In in such a way as to ensure that the residual set has zero volume. Then, of course, is convergent and it can be shown, for n = 2, that is divergent. Let the exponent of convergence of the packing be the supremum of those real numbers t such that is divergent and let tn denote inf, where the infimum is taken over all packings which satisfy the above conditions. In recent work Z. A. Melzak [1] has been interested in finding estimates for in two-dimensional space. He has shown that t may take the value 2 and has produced some estimates for the Apollonius packing of disks. In this note we produce what is, perhaps, the most interesting estimate for tn by showing that tn is greater than n-1. Let sn denote the exact lower bound of the Besicovitch dimensions of every residual set which is formed by a packing of spheres in In.

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.