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|.

Problem I was raised (oral communication) by Goffman some three years ago, and I found the example then. Problem II was raised at about the same time by Topsøe; Christensen has given an affirmative answer for spaces satisfying certain additional conditions.

It is easy to see that if the answer to problem I were affirmative then so would be that to problem II; therefore our counter-example for problem II implies the existence of one for problem I. It is also possible that another counter-example for problem I could be found by analysing the construction of Dieudonn6 in [1], which is also concerned (implicitly) with a failure of Vitali's theorem. Nevertheless our construction may be of independent interest.

An example will be given of a compact linear set K which is of zero or non-σ-finite measure for every translation-invariant Borel measure. This answers a question asked me by C. Dellacherie.

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.

In [5] Knowles proved the following.

Every compact Hausdorff space with no isolated points admits a non-atomic measure.

This note is concerned with the converse problem in a more general set up. Here we deal with certain properties of the family of completely regular spaces admitting no continuous measures. In §3 it is shown that this family contains spaces with no isolated points, thus theorem (1.1) does not generalize to completely regular spaces. In §4 a canonical decomposition of the compact members of the above family into discrete subspaces is obtained, and it is shown that these spaces are metrizable whenever they satisfy the first axiom of countability.

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.

The concepts of bounded paracompactness and bounded metacompactness were introduced and studied in [3]. In this paper we define bounded full normality and show that this concept is equivalent to bounded paracompactness.

It is known that countable paracompactness and countable metacompactness are equivalent properties in a normal space. We show that bounded countable metacompactness is equivalent to bounded countable paracompactness in a normal space and that bounded countable paracompactness is equivalent to bounded paracompactness in a hereditarily paracompact space.

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

The method of the large sieve has played a very important role in number theory. It turns out that estimates of exponential sums are of basic importance for large sieve inequalities. Let

be an exponential sum with complex coefficients c(n). It follows from Theorem 1 of Bombieri and Davenport [1] that

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

Let G and H be Hausdorff locally compact groups. By R(G, H) we denote the space of continuous homomorphisms of G into H equipped with the compact-open topology, namely that which is generated by subsets of the form

where K is any compact subset of G and U is any open subset of H. Further, let R0(G, H) be the subset of R(G, H) consisting of elements r satisfying the conditions that the quotient space H/r(G) is compact and r is proper, i.e., the action of G on H defined as the action by left translations of r(G) on H is proper. It has been shown by H. Abels [2] that if G and H are such that at least one of them has a compact defining subset then R0(G,H) is open in R(G,H). Moreover, for each r0 ∈ R0(G,H) there exists a neighbourhood M and a compact subset F of H such that r(G) F = H for all r ∈ M and for each compact subset K of H the union is a relatively compact subset of G. It is furthermore shown in [1] that if G contains no small subgroups and H is connected then the subset of R0(G, H) consisting of isomorphisms of G into H is open in R(G, H). These results, in the case in which H is a connected Lie Group and G is a discrete group, have been established by Weil in [5] and [6] appendix 1.

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.

The initial structure of wing-body interaction in supersonic flow was first investigated by Nielsen (1951), who analysed the steady supersonic inviscid flow past a proto-type wing-body combination consisting of an unswept thin wing at incidence and lying approximately in the axial plane of a non-lifting cylindrical body with a circular cross-section. In particular, he obtained the first two terms in the series expansion of the velocity potential at the start of the root chord where the wing and body meet. More recently, further theoretical studies which relate the problem to similar problems in diffraction theory have been carried out by Stewartson (1966), Jones (1967), Waechter (1969), and Clark (1970). In addition, Stewartson (1968) has extended Nielsen's formula for the following cases:

(a) the lifting wing with rounded leading edge,

(b) the flat plate wing at incidence with supersonic leading edge,

(c) the flat plate wing at incidence with subsonic leading edge in both symmetrical and antisymmetrical cases.

If G is a finite group with classifying space BG and complex representation ring R(G), one defines the topological filtration on R(G) by means of the natural map Ψ: R(G) → KU(BG).

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

The purpose of this paper is to demonstrate that a number of properties of independence spaces are of finite character, thus making it possible to easily generalise known theorems for finite spaces, or matroids, to independence spaces on infinite sets.

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.

Certain properties of ascending sequences of quotient rings of points of algebraic varieties were investigated by O. Zariski in [4] and some related problems were discussed in [3].

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.