A reduction method due to R. Rado [7] yields an elegant proof of the Halls' theorems on transversals of a family of sets (see for example [4]). We use here this method to give simple new proofs of some basic theorems on common transversals of a pair of families of sets.

Let τ(n; k, h) denote the number of divisors of n which are congruent to h (mod k), and τ(n; k) the number of divisors of n prime to k, so that

Let

Erdős [1] proved that, if ε and η are fixed arbitrary positive numbers, then for almost all integers n ≤ x, we have

provided

Hall and Sudbery [2] showed that it is sufficient that

and apart from the ε, this upper bound for k is best possible, for it is clear that k must not exceed the normal order of τ(n). For n ≤ x, Hardy and Ramanujan [3] showed that this normal order is (log x)log 2 = 2 log log x.

The idea of obtaining bounds on the order of approximation of algebraic functions by rational functions apparently goes back to Maillet [8]. It is very easy, using an argument analogous to that used by Liouville for algebraic numbers, to see that if w(z) is any algebraic function of degree n ≥ 2 over Q[z] then for all pairs of polynomials r(z) and s(z) with s(z) ≢ 0 we have

The main purpose of this paper is to evaluate the Whitehead group K1 (Zπ) of the quaternion group π of order 8. We show that the natural mapping of the units ± π of Zπ into K1(Zπ) induces an isomorphism between K1(Zπ) and ± V, where V, Klein's 4-group, is the commutator quotient group of π.

For any subsets E, F of a group G, written additively, we write

and

Let En+1, for some integer n ≥ 0, be the (n + 1)-dimensional Euclidean space, and denote by Sn the standard n–sphere in En+1, . It is convenient to introduce the (–1)-dimensional sphere , where denotes the empty set. By an i-dimensional subsphere T of Sn, i = 0 n, we understand the intersection of Sn with some (i+1)-dimensional subspace of En+1. The affine hull of T always contains, with this definition, the origin of En+1. is the unique (–1)-dimensional subsphere of Sn. By the spherical hull, sph X, of a set , we understand the intersection of all subspheres of Sn containing X. Further we set dim X: = dim sph X. The interior, the boundary and the complement of an arbitrary set , with respect to Sn, shall be denoted by int X, bd X and cpl X. Finally we define the relative interior rel int X to be the interior of with respect to the usual topology sphZ . For each (n–1)-dimensional subsphere of Sn defines two closed hemispheres of Sn, whose common boundary it is. The two hemispheres of the sphere Sº are denned to be the two one-pointed subsets of Sº. A subset is called a closed (spherical) polytope, if it is the intersection of finitely many closed hemispheres, and, if, in addition, it does not contain a subsphere of Sn. is called an i-dimensional, relatively open polytope, , or shortly an i-open polytope, if there exists a closed polytope such that dim P = i and Q = rel int P. is called a closed polyhedron, if it is a finite union of closed polytopes P1 …, Pr. The empty set is the only (–1)-dimensional closed polyhedron of Sn. We denote by the set of all closed polyhedra of Sn. is called an i-open polyhedron, for some , if there are finitely many i-open polytopes Q1 …, Qr in Sn such that , and dim . By we denote the set of all i-open polyhedra. Clearly for all , and each i-dimensional subsphere of Sn, , belongs to and to , For each i-dimensional subsphere T of Sn, set . A map is defined by , for all , and, for all .

By a plane tree or rooted plane tree is meant a realization of a tree or rooted tree by points and arcs in the plane. By an isomorphism between two plane trees or rooted plane trees is meant an isomorphism in the usual sense for such trees which preserves the clockwise cyclic order of the edges about each node. In [2] Harary, Prins, and Tutte demonstrate how Polya's Theorem may be used to obtain formal expressions for the enumerating functions for unrooted, rooted, and other species of plane trees. In the process they obtain the explicit formula

for the number of nonisomorphic planted plane trees with n ≥ 1 edges. (A planted tree is a rooted tree with root at a node of degree one.)

Introduction. Gale diagrams, and similar techniques, have been used in several recent investigations into the combinatorial properties of convex polytopes. Here we describe other applications, namely to problems concerning sections and projections of polytopes of a given combinatorial type. For example, in §4 we shall show that every n–dimensional convex polytope with at most n + 2 vertices possesses the universal shadow boundary (usb) property: If is a subcomplex of the boundary complex ℬ (P) of P, and set is homeomorphic to an (n – 2)-sphere, then there exists a polytope P', combinatorially isomorphic to P, such that the complex corresponding to is a shadow boundary of P'. We shall also show that this is the best possible result in that polytopes with n + 3 or more vertices do not, in general, have the usb property. A second application of the method, to be described in §5, is to the construction of non-realisable complexes.

Abstract. Let g(n) denote the largest number of positive integers not exceeding n such that the l.c.m. of any two of them does not exceed n. In this paper we obtain an improvement over a previously known upper estimate for g(n).

Introduction. Suppose that M i s a set in a cartesian product space X × Y. If X and Y are both R1 and M is a borel set, K. Kunugui [1] and W. J. Arsenin [2] have shown that those points which lie on sections {x} × Y which meet M in a nonempty Kσ set project onto a set in X which is the complement of an analytic set. Somewhat later, W. J. Arsenin and A. A. Lyapunov [3] reproduced the proof from [2], still apparently without being aware of Kunugui's results. Although, as we shall see later, Kunugui's proof seems to be incorrect, it does contain an ingenious lemma (Lemma 5 below). Actually [2] also contains this lemma in a disguised form that seems, to me at least, to be less satisfactory. One of the purposes of this article is to give a promised proof, see D. G. Larman [4] of Kunugui's theorem in slightly more general circumstances by combining ideas of Kunugui [1] with those of P. Novikoflf [5].

Abstract. Bose-Chaudhuri-Hocquenghem and Justesen codes are used to pack equa spheres in n–dimensional Euclidean space with density Δ satisfying

for all sufficiently large n of the form m 2m, where m is a power of 4. These appear to be the densest packings yet constructed in high dimensional space.

Integral geometry is the study of measures of sets of geometric figures. Commonly a measure of this sort is an integral of a density or differential form; the density is determined by the type of figure, but is independent of the particular set of such figures to which the measure is assigned. As one of the simplest examples, the area of a plane convex point set K is the integral over K of the density dx dy for points with Cartesian co-ordinates x, y. But when we assign a Hausdorff linear measure to the set of boundary points of K, we obtain a measure of quite another sort. This is representable as a Stieltjes integral of arc length density; here the density depends on the choice of K. The examples suggest examining measures for other sets of figures, where each such set is made up of all those figures from a certain class which support, in some sense, a convex body. Further, the examples lead us to expect that measures of this kind will appear as integrals of densities which may depend on the choices of . Here we treat a question of the type just described: to determine a measure for sets of q–flats which support a convex body.

Viscoelastic and viscoinelastic flows through straight ducts of different cross-sectional shapes have engaged the attention of a number of workers. These results may be found in all standard texts on the subject. The viscoelastic flows present a more complex analysis than the viscous flows, and for this reason a viscoelastic solution can rarely be obtained in an explicit and usable form. In this note, we obtain a simple formula relating the drag per unit length of the duct, the pressure gradient, the cross-sectional area, and the rate of flux across the section for a duct of arbitrary cross-section without explicitly solving for the physical components of velocity. This result may be of interest to workers experimenting with the non-Newtonian fluids. The recent work of Nigam and Arunachalam [1], which gives a similar result for the drag in viscous flows, can be deduced very easily from the present analysis.

In an earlier paper [10], I presented a base for the kernel, and in particular for the null space of I - λo K ((p.39). I remarked then that the base did not quite correspond to the arrangement given by Zaanen (cf. [13; p. 280], [14; pp. 339, 342]). The purpose of the present paper is to present an alternative base which does correspond to Zaanen's arrangement.

It is shown that the equations governing the motion of a set of circular vortex rings, separated by distances large compared with their dimensions, can be written in canonical form.

Introduction. Let F be a subset of a group G (written additively) and let κ be a positive integer. Following [1] we shall write

and

Introduction. This paper is concerned with an interpretation of some experiments on certain axisymmetric inertial oscillations of a rotating fluid in a thick spherical shell. These oscillations are the counterparts to the ones reported by Aldridge and Toomre† (1969) for a full sphere of rotating fluid. A description of the arrangements for both experiments is presented in A & T; some of the measured eigen-frequencies for spherical shells of rotating fluid given by Aldridge (1967) are presented here for comparison with values calculated from a variational principle.

We shall be concerned with two boundary value problems for the Falkner-Skan Equation

when –β is a small positive number. Our interest is in solutions of (1) which exhibit “reversed flow”; that is, solutions f such that f′(x) < 0 for small positive values of x. The boundary conditions which we wish to consider are

and