Let

be a quadratic form with integral coefficients, and suppose the equation

has a solution in integers x1…, xn, not all 0. It was proved by Cassels [2] that there is such a solution, which satisfies the estimate

where F = max|fij|. It was later observed by Birch and Davenport [1] that the result can be stated in a slightly more general form. Let

be a quadratic form which assumes only integral values at the points (x1 …, x2) of an n-dimensional lattice Λ of determinant Δ. Suppose there is some point of Λ, other than the origin, at which ø = 0. Then there is such a point for which also

where Φ = max |øij|.

In [1], A. H. Stone proved that for a cardinal number k ≥ 1 a set with a transitive relation can be partitioned into k cofinal subsets provided each element of the set has at least k successors. Using methods quite different from those of Stone, we show that for k ≥ ℵ0 the same condition on successors guarantees that a set on which there are defined not more than k transitive relations can be partitioned into k sets each of which is cofinal with respect to each of the relations. We also show that such a partition exists even if some of the relations are not transitive as long as the non-transitive relations have no more than k elements.

Let R be a commutative ring with identity, and let U be a unitary commutative R-algebra with identity. In [1] Gilmer defines the (l/n)th power (n a positive integer) of a valuation ideal R when R is a domain. Sections 2, 3 of the present note are devoted to the study of an extension of this notion to positive rational powers of an arbitrary R-submodule of U.

When a viscous incompressible fluid with uniform conditions at upstream infinity flows past a paraboloid of revolution, the boundary-layer equations admit of a similarity solution which leads to the ordinary differential equation (see Miller [1])

with boundary conditions

This paper uses systems of image sources to construct suitable generalised Green's functions for considering small amplitude short surface waves due to an oscillating immersed sphere. A sphere of radius a is half-immersed in a fluid under gravity and is making vertical oscillations of small constant amplitude and period 2π/σ about this position. It is required to find the fluid motion, and in particular the virtual mass and wavemaking coefficients. For sufficiently small amplitudes the motion depends non-trivially on only a single dimensionless parameter

where g is the gravitational acceleration. This was shown by Ursell in an unpublished U. S. Navy report in which the methods of an earlier paper (Ursell, 1949) were adapted from cylindrical to spherical symmetry and resolved the conflict between Havelock (1955) and Barakat (1962) in favour of the former. Ursell showed that the virtual mass coefficient is

i.e. infinitely increasing initially so that Barakat's results must be incorrect near N = 0. However, although existence is proved for all N, the same difficulty arises as with the heaving cylinder, namely computation is only practical for values of N up to about 1. In the cylindrical case, Ursell (1953) developed a method for finding the asymptotic solution for large N and here it will be adapted, surprisingly perhaps, to deal with the spherical case. Ursell (1954) then published a formal solution which gave the same virtual mass as the rigorous treatment and if this formal method is applied to the heaving sphere, the virtual mass coefficient obtained is

In a paper of the same title [3] Ch. Pommerenke and the author proved several results concerning the distances of Fekete points. In the present paper I will show that the same methods can be adapted to give an answer to a problem which we could not solve at the time.

Let E be a continuum and n ≥ 4 a given positive integer. A system of points z1, …, zn ∈ E that maximizes

is called a system of Fekete points. Such a system may not be unique.

If K is a set in n-dimensional Euclidean space En, n ≥ 2, with a non-empty interior, then a point p of the interior of K is called a pseudo centre of K provided each two-dimensional flat through p intersects K in a section centrally symmetric about some point, not necessarily coinciding with p. A pseudo centre p of K is called a false centre if K is not centrally symmetric about p. Rogers [5] showed that a convex body (compact convex set with interior points) with a pseudo centre necessarily has a true centre of symmetry. But, as each interior point of an ellipsoid is a pseudo centre, the true centre need not necessarily coincide with the pseudo centre. Rogers conjectured that, for n ≥ 3, a convex body K with a false centre is necessarily an ellipsoid. In this paper we prove this conjecture.

Let fn be a positive-definite n-ary quadratic form, with real coefficients. By the minimum of fn, denoted by min fn, is meant as usual the least value of fn(x1, …, xn) for integers xi not all 0. A minimum point of fn is a point x = (x1, …, xn), with integer coordinates, at which fn takes its minimum value. Let Δ (> 0) be the determinant of a set of n minimum points of fn; then in [1] it was proved that

where γn is the Hermite constant. Enough is known about γn to deduce from (1.1), as in [1], that

In a recent paper [1] I made the following

Definition. The function f: ℤ+ ∪ {0} → ℤ is a pseudo-polynomial if

for all integers n ≥ 0, k ≥ 1.

A tree is a connected graph that has no cycles. If x is any endnode of a tree, then the limb determined by x is the unique path that joins x with the nearest node other than x that does not have degree two in the tree; let l(x) denote the length of this path. (For definitions and results not given here see [2] or [3].) Different endnodes determine different limbs with one exception; when the tree is a path then both endnodes determine the same limb, namely, the tree itself. Our object here is to investigate the distribution of the length of limbs of trees Tn chosen at random from the set of nn-2 trees with n labelled nodes; in particular, it will follow from our results that the length of the longest limb in most trees Tn is approximately log n when n is large.

Let a 3-connected graph G be embedded in an orientable surface of genus g in such a way that the connected components of its complement in the surface are topological discs. We denote by vk(G) the number of k-valent vertices of G and by pk(G) the number of k-gonal faces of the map defined by the embedded graph. For the numbers vk(G) and pk(G), it follows from Euler's formula that

The following theorem has been proved by Bambah, Rogers and Zassenhaus [1],

Theorem A. Let K be a closed convex domain with a centre. Let

be points such that:

(i) the polygon A0 A1 … An is a Jordan polygon bounding a closed domain π of area a(π);

(ii) for each r, 0 ≤ r ≤ n, there is a point common to K + An-1 and K + Ar;

(iii) the points An+1, …. An+m are in the interior of π;

(iv) for each point X of π, there exists an Ar, 1 ≤ r ≤ n + m, such that X ∈ K + Ar and the line segment XAr is in π. Then

where t(K) is the area of the largest triangle contained in K.

A one parameter solution of th equation

was given by Ryley in 1825. Others were found by Richmond and myself. In the Landau memorial volume [1] recently published, there appears a joint paper, with the above title, of Davenport and Landau dating from 1935. They prove that if n is a positive integer, then positive rational solutions of (1) exist with denominators of order of magnitude O(n2). Their proof depended upon a two parameter solution of (1) due to Richmond and is very complicated.

Let x1, …, xn be linear forms in u1, …, un with real coefficients, and (for simplicity) determinant 1. Given a form (that is, a homogeneous polynomial) F(x1, …, xn), we can ask the following question: do there exist, for arbitrary real α1, …, αn, integers ul, …, un such that

where C is a suitable number independent of α1, …, αn and of the particular linear forms x1 …, xn? In two well-known cases this is true: namely when

and when

where 1 ≤ r ≤ n − 1.

In this note we shall be mainly concerned with convex bodies of constant width in En that are invariant under the group of congruences that leave invariant a regular simplex with its centre of gravity at the origin. We first show that there are many such convex bodies. This follows, by showing that any set S ot diameter 1 that is invariant under a group of congruences about the origin, is contained in a convex body of constant width 1 that is invariant under the group.

If L is a set of disjoint closed line segments in En, let E(L) denote the end set of L, i.e. the set of end points of members of L.

In [1, 2] V. L. Klee and M. Martin proved the following lemma:

IfLis a disjoint set of closed line segments in E2 such that E(L) is compact, then E(L) has zero 2-measure.

If ω, ω′ are two points in the extended complex plane then the distance between their stereographic projections on the Riemann sphere is

with an obvious interpretation if ω or ω′ is ∞. is called the distance between ω and ω′ in the spherical or in the chordal metric.

This note examines one particular feature of the motion produced by an infinitely long insulating circular cylinder moving parallel to its axis in a conducting fluid permeated by a uniform transverse magnetic field. The particular aspect examined is that of determining, for large values of the Hartmann number, the flow in the neighbourhood of those points on the cylinder where the applied field is tangential.

The problem of determining the fields in these regions is very similar to that of calculating the acoustic potential in the neighbourhood of glancing incidence for a plane wave normally incident on a sound soft circular cylinder. An elegant method of treating this latter problem has been given by Jones [1] and the purpose of this note is to indicate the way in which Jones' method has to be modified to treat the magnetohydrodynamic problem. The basis of the approach is to determine a solution which satisfies the boundary conditions in the tangential regions and this can be achieved by conventional integral representation methods.

This note is concerned with those linear viscoelastic materials which are known to be compatible with thermodynamics in a sense to be denned below. It is proved that equality of its instantaneous and equilibrium elastic moduli is a necessary and sufficient condition for such a material to be elastic.

Plummer (see [2; p. 69]) conjectured that the square of every block is hamiltonian, and this has just been proved by Fleischner [1]. It was shown by Karaganis [3] that the cube of every connected graph, and hence the cube of every tree, is hamiltonian. Our present object is to characterize those trees whose square is hamiltonian in three equivalent ways.

We follow the terminology and notation of the book [2]. In particular, the following concepts are used in stating our main result. A graph is hamiltonian if it has a cycle containing all its points. The graph with the same points as G, in which two points are adjacent if their distance in G is at most 2, is denoted by G2 and is called the square of G. The subdivision graph S(G) is formed (Figure 1) by inserting a point of degree two on each line of G.