In a paper [1] published in 1956, David Gale introduced the idea of representing a convex polytope by a diagram (now called a Gale diagram). Later work by Gale, T. S. Motzkin, and more recently by M. A. Perles and B. Grünbaum,has shown the importance of this idea. Using it, a large number of results which were formerly inaccessible have been proved. For an account of Gale diagrams and their applications, the reader is referred to Grünbaum's recent book [2; §5.4 and §6.3] and we shall largely follow the notations used there.

Let Cn be the n-dimensional complex number space of the complex variables z1,…, zn and be the unit hypersphere. Further, let G be the group of all holomorphic automorphisms of K, then G is a n(n + 2)-dimensional real Lie group. In [3] the author has proved that for any P∈∂K (the boundary of K) there exists a decomposition of G in the form , where G0 is the group of all analytic rotations about the origin and is a 2n-dimensional real Lie group, whose underlying topological space is K, which acts transitively on K and P is a fixed point of all elements of .

Let V = V2n be the Segre product variety of two n–dimensional complex projective spaces. Then any Cremona transformation of Sn into (regarded as an irreducible algebraic system of ∞n ordered pairs of points) is represented on V by an irreducible n-dimensional subvariety H which satisfies (on V) the algebraic equivalence

where Si, j is a subvariety of V and m1, …, mn–1 are positive integers. We call m1, …, mn–1 the characters of T noting that, numerically,

Let E be a compact subset of R2, of Hausdorff dimension s > 0 and for each real number t let Ft be the linear set {x1 + tx2: (x1x2) ∈ E}. In this note we shall prove

THEOREM. If s ≤ 1 then Ft has dimension ≥ s, excepting numbers t in a set of dimension ≤ s. If s > 1 then Ft, has positive Lebesgue measure, excepting numbers t in a set of Lebesgue measure 0.

The Integral Equation. Consider a periodic wave moving with constant velocity c from right to left on the surface of an inviscid, incompressible fluid which is at rest at infinity. The motion is assumed to be irrotational and two-dimensional. The bottom is horizontal, and the depth of the undisturbed fluid is h.

As a consequence of the methods developed in the earlier papers of this series [1, 2, 3], an effective algorithm has recently been established for solving many Diophantine equations in two unknowns (see [4,5,6,7]). The algorithm leads to an explicit bound for the size of all the solutions, and, in principle therefore, it enables any specific equation of the type considered to be fully resolved by a finite amount of computation. On examining the various estimates occurring in the course of the exposition, however, it at once became apparent that the computation would involve a very large number of operations and would scarcely be practicable even with a modern machine. It was clear, on the other hand, that a modified version of the fundamental inequality involving the logarithms of algebraic numbers would much facilitate the computational work, and it was in the light of this observation that the researches discussed herein were begun. The object has been to obtain a theorem of an essentially practical nature which may be found useful in application to a wide variety of different problems. The result which we shall establish is neither the most precise nor the most general that can be obtained in this direction, but it would seem to be the most serviceable of its kind, and it would apparently make feasible many calculations which would otherwise have seemed quite out of the question.

Some diophantine equations in three variables with only finitely many solutions, 113–120. In formula (1) of Theorem 1 and in Corollary 1 read “min” fir “max”

Let T(n, k) denote the number of trees n with n labelled nodes of which exactly k have degree two. We shall derive a formula for T(n, k) and then determine the asymptotic behaviour of T(n,0); this will enable us to calculate the limiting distribution of Xn the number of nodes of degree two in a random tree n. Rényi [5] has treated the corresponding problem for nodes of degree one in random trees.

Let D be an integral domain with identity having quotient field K. A non-zero fractional ideal F of D is said to be divisorial if F is an intersection of principal fractional ideals of D[4; 2]. Equivalently, F is divisorial if there is a non-zero fractional ideal E of D such that

Divisorial ideals arose in the investigations of Van der Waerden, Artin, and Krull in the 1930's and were called v-ideals by Krull [9; 118]. The concept has played an important role in the development of multiplicative ideal theory.

In the nonoscillation theory of ordinary differential equations the asymptotic behaviour of the solutions has often been described by exhibiting asymptotic expansions for these solutions. A fundamental illustration of this technique may be found in Hille [6] wherein the linearly independent solutions of a second order homogeneous differential equation were described by single termed asymptotic expansions. For the dominant solution, this result was successively extended by Waltman in [8] for a second order equation and in [9] for an nth order equation. A further generalization of these results appears in [3] where a complete nth order nonhomogeneous nonlinear differential equation was considered; again, asymptotic representations were given to describe the behaviour of the solutions of the differential equation. Moore and Nehari [7], Wong [10], [11], and Hale and Onuchic [2], also use asymptotic representations in discussing the behaviour of the solutions of certain differential equations. All of the above results are essentially perturbation problems with the unperturbed linear differential equation having the form y(n) = h(t) for some n and h(t).

The higher dimensional concepts corresponding to trees are developed and studied. In order to enumerate these 2-dimensional structures called 2-trees, a dissimilarity characteristic theory is investigated. By an appropriate application of certain combinatorial techniques, generating functions are obtained for the number of 2-trees. These are specialized to count those 2-trees embeddable in the plane, thus providing a new approach to the old problem of determining the number of triangulations of a polygon.

Let Ω be a compact metric space of non-σ-finite Λh-measure, where Λh denotes the Hausdorff measure corresponding to some continuous increasing real function h(t), denned for t ≥ 0 and with h(t) > 0 and h(t) > 0 for t > 0. C. A. Rogers has drawn attention to two problems that have remained unsolved for some years, and which seem to present considerable difficulty:

I. Does Ω necessarily have subsets of finite positive Λh-measure?

II. Does Ω. necessarily contain a system ofdisjoint closed subsets each of non-σ-finite Λh-measure ?

The answers are affirmative when Ω is a subset of a Euclidean space (see [1], [2], [3], [4], [5]); recently D. G. Larman ([8], [9]) was able to extend each of these results to the case of the “finite-dimensional” compact metric spaces that he introduced in [7]. Both of his proofs depend on the detailed structure of such spaces; the aim of the present paper is to establish a general theorem, from which it follows in particular that Larman's second result is a consequence of his first, and that an affirmative answer to I would imply an affirmative answer to II. (Since it seems likely that I must be answered negatively in general, and II affirmatively, the present theorem may well be of only ephemeral significance; the nature of the proof reinforces this expectation.)

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

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

When the motion of a viscous fluid around a gas bubble is discussed, it is frequently assumed, especially for flows at low Reynolds numbers, that the bubble takes on a spherical shape in three dimensions or a circular cross-section in a two-dimensional flow. If this assumption is made, arid the gas within the bubble is assumed to have negligible density and viscosity, then the problem of finding the exterior flow is mathematically overdetermined and it is not obvious that a solution to the problem exists. Moreover, if such a solution does exist, then the over-determination of the system should, in general, give rise to relationships between the flow parameters, that is, certain conditions must be satisfied to ensure the existence of a solution. It is the purpose of this paper to derive these conditions in the case of a two-dimensional Stokes flow. The problem is generalised to the extent that part of the circular boundary is taken to be rigid, on which the no-slip condition is to be satisfied and part is to be a free streamline, on which stress conditions are to be satisfied. The conditions for the existence of a solution to this problem are derived and the solution is found in closed form. The method of solution is that of reducing the problem to one of a mixed boundary-value problem in analytic function theory. The classical solutions for the Stokes flow around a circular bubble and around a rigid circle are then easily derived as limiting cases.

The problem of estimating accurately the order of magnitude of the least primitive root g(p) to a large prime modulus p is as yet unsolved. The first non-trivial estimate was obtained by I. M. Vinogradov (see [5]) who in about 1919 showed that

The integral equation

occurs in the theory of the motion of an electrically conducting fluid, occupying the space between two electrodes at different potentials, in the presence of a strong magnetic fluid [Hunt and Malcolm 1]. In this theory a solution is required in which

and one of the principal properties of the solution to be determined is

Christoffel's problem, in its classical form, asks for the determination of necessary and sufficient conditions on a function φ, defined over the unit spherical surface Ω, in order that there exist a convex body K for which φ (u) is the sum of the principal radii of curvature at that boundary point of K where the outer unit normal is u. The figures Ω and K are in Euclidean n-dimensional space (n ≥ 3). It is assumed that φ is continuously differentiable and that K is of sufficient smoothness. A solution of Christoffe's problem was given in [6]. Yet that treatment is rather unsatisfactory in that the smoothness restrictions are set by the method rather than the problem, cf. [5; p. 60]. The present paper overcomes this defect. To do this it is first necessary to generalize the original problem so as to seek conditions on a measure M, defined over the Borel sets of Ω, in order that M be a first order area function for a convex body K. When K has sufficient smoothness, then φ is the Radon-Nikodym derivative of M with respect to surface area measure on Ω. It is this generalized Christoffel problem which is solved in what follows.

If θ is a real algebraic number of degree r ≥ 2, there is a computable number c = c(θ) > 0 such that

for all rational numbers p/q (q > 0). This follows directly from the definition of an algebraic number, as was shown by Liouville in 1843; and if r = 2 there is no more to be said. Axel Thue was the first to prove a stronger result when r ≥ 3; he showed that if

there are at most finitely many rational numbers p/q that satisfy