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.

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.

In recent years, functions of n variables whose partial derivatives are measures, have been found to retain the properties of functions of bounded variation of one variable to a remarkable degree [e.g., G1, G2, G3, K, Z, and especially the announcement F].

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.

Let G be any group and G′ its derived, then G/G′—the group G made abelian—will be denoted by Ga. Over any ring R, denote by E2(R) the group generated by the matrices as x ranges over R; the structure of E2(R)a has been described in a recent theorem [2; Th. 9.3] for certain rings R, the “quasi-free rings for GE2” (cf. §2 below). Now over a commutative Euclidean domain, E2(R) is just the special linear group SL2(R); this suggests applying the theorem to the ring of integers in a Euclidean number field. However, the only number fields whose rings were shown to be quasi-free for GE2 in [2] were the non-Euclidean imaginary quadratic fields. In fact that leaves the application of Th. 9.3 of [2] to the ring of Gaussian integers unjustified (I am indebted to J.-P. Serre for drawing this oversight to my attention). In order to justify this application one would have to show either (a) that the Gaussian integers are quasi-free for GE2., or (b) that Th. 9.3 of [2] holds under weaker hypotheses which are satisfied by the Gaussian integers. Our object in this note is to establish (b)–indeed our only course, since the Gaussian integers turn out to be not quasi-free.

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.

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,

The object of this paper is first to generalize the basic inequality of the large sieve method to exponential sums in many variables, and then to deduce results for algebraic number fields that are analogous to known results for the rational field.

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 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 .

It has been shown by Mirsky and Perfect [1] that the theorem of Rado [2], linking matroid theory and transversal theory, has important applications in combinatorial theory. In this note I use it to obtain necessary and sufficient conditions for two families of sets to have a common transversal containing a given set, and then I show how it may be used to obtain a variant of a well-known theorem that was obtained by Hoffman and Kuhn [3] using linear programming methods.

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.

Let (X, ≺) denote a non-empty (partially) ordered set, or more generally a non-empty set X with an arbitrary transitive relation ≺ on it. The relation ≺ will be fixed throughout what follows, so to simplify the notation we often write (X, ≺) as X. A successor of x ∈ X is an element y ∈ X such that x ≺ y; thus x may or may not be a successor of itself. As usual, a subset A ⊂ X is cofinal if each x ∈ X has a successor in A. A partition of X is a family of (pairwise) disjoint nonempty subsets of X whose union is X.

It was conjectured by Artin [1] that each non-zero integer a unequal to +1, −1 or a perfect square is a primitive root for infinitely many primes p. More precisely, denoting by Na(x) the number of primes p ≤ x for which a is a primitive root, he conjectured that

where c(a) is a positive constant. This conjecture has recently been proved by C. Hooley [2] under the assumption that the Riemann hypothesis holds for fields of the type .

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”