Let f be a positive-definite ternary quadratic form with integer coefficients; by c(f), the class-number of f, is meant the number of classes in the genus of f. The object of this paper is to find all the f with c(f) = 1; these f are the ones for which , where f′ is an arbitrary ternary form and ∼, denote equivalence and semi-equivalence respectively. Trivially, it suffices to find the primitive f with c(f) = 1.

In a recent paper [1], R. A. Rankin proved that

where τ(n) is Ramanujan's arithmetical function,

The system of equations governing the stress in a shear-strained prismatical body are examined and it is shown that, provided the stress-strain law adopts certain multi-parameter forms, the system may be reduced to the Cauchy-Riemann equations. Integration of the system is then immediate and the analysis of the stress concentration round certain notches thereby facilitated.

The notion of stability at infinity for an infinite finitely presented group with one end was introduced in [14], and for groups stable at infinity, the end invariant e was defined and studied for some (non-trivial) direct products. In this paper we study the corresponding problem for extensions.

In a recent article [2] D. G. Larman and C. A. Rogers proved the following two results in Descriptive Set Theory (where R = the space of real numbers): (1) There is no analytic set in the plane R2, which is universal for the countable closed subsets of R; (2) there is no Borel set in R2, which is universal for the countable Gδ subsets of R. Recall that, if b is a class of subsets of a space X, a set U ⊆ X × X is called universal for b if (a) for each x ∈ X, Ux = def {y : (x, y) ∈ U} ∈ b, and (b) for each A ∈ b there is an x such that A = Ux. (Larman and Rogers have also shown that in both cases coanalytic universal sets exist.)

Stewartson and Rickard [1] have shown that Rossby waves in thin spherical shells of an inviscid fluid contain singularities at the critical circles. These may be removed by introducing another wave containing square-root singularities in the velocity on the characteristics which by reflection touch the inner boundary at the critical circles. Stewartson and Walton [2] continued this wave all round the shell and showed it to be an inertial wave. Here we consider the effect of a weak kinematic viscosity v on these waves.

Hitherto, little use has been made of the well-known representation

of the Riemann zeta-function, ζ(s), within the critical strip l = {s:s = σ + it, 0 < σ < 1}. Certain variants of (1) have been used to deduce the functional equation of ζ(s), while a simple consequence of (1) itself is that ζ(s) does not vanish on the positive real axis.

In an article [2] in a volume of papers dedicated to the memory of Edmund Landau, Heilbronn investigates the following problem on continued fractions, posed by Dr. J. Gillis:

Let a and N be natural numbers such that 1 ≤ a < N and (a, N) = 1. Then there exist unique natural numbers ci such that

In [1], the author considered the distribution of rational integers which are norms of elements of a given algebraic number field K, and in particular obtained the result

where A(K), B(K) and C(K) are positive parameters depending only on K, as does the implied constant in the O-term. In fact 0 < B(K) < 1, and B(K) is a rational number related to the Galois groups Gal and Gal, where is the normal hull of .

The following theorem appears in Weyl's famous memoir [3] of 1916.

THEOREM A. Let λ1 ≤ λ2 ≤ … ≤ λn ≤ … be an increasing sequence of positive integers. Suppose that, of the numbers λ1, … λn, the first h1 are equal to each other, then the following h2 and so on, and finally that the last hm coincide. Let hj. If

the sequenceis uniformly distributed (mod 1)for almost all x, in the Lebesgue sense.

We begin by introducing some notations which will be used throughout the paper. Let denote a sequence of distinct positive integers. Let B denote a measurable subset of [0, 1) and {x} the fractional part of x. We write

so that f has period 1 on the real line; and we write

for the number of times {mkx} falls in B, minus the expected value of this number (m(B) ≤ Lebesgue measure of B).

By a zonotope we mean any set in Euclidean n-dimensional space Rn which can be written as a Minkowski (vector) sum of a finite number of line segments. A zonotope is a convex centrally-symmetric polytope, and all its faces are zonotopes. Familiar examples of three-dimensional zonotopes include the cube, rhombic dodecahedron, elongated dodecahedron (Figure 1) and truncated octahedron. Photographs of models of more complicated examples appear in [1, Plate II].

This paper deals with the problem (raised by J. Browkin) of how many ring generators are needed for the ring of integers of a given algebraic number field. I show that the number of generators needed for the integers of a field of degree n is less than (logn/log2) + 1, and that if 2 splits completely in the field the number of generators needed is in fact the largest integer less than (logn/log2) + 1. These results follow from a computable formula (that depends only on how the small primes factorize in the field) for the number of generators of the ring of integers of a given field. This formula has the single drawback that when it yields “one” two generators may be needed, and I show that there are fields of arbitrarily high degree for which this happens.

Two different approaches to a probability problem involving convex polytopes lead to a geometric proof of an integral geometric result about mixed surface areas. The proof can be modified to cover the corresponding results about mixed volumes.

The cohomology of the symmetric groups with coefficients in a field has been studied by several authors, see [6] and [7] for example, but hardly anything has been published with the integers as coefficients. Given the connection between the infinite symmetric group and the classifying space BG for stable spherical fibrations, the computation of is an interesting problem, and the purpose of this paper is to solve the first non-trivial case, n = 4. (The symmetric groups on 2 and 3 letters have cohomology of period four, which is generated by c1 and c2 of the permutation matrix representation, [8].)

Let K3 be a non-Galois cubic extension of the rationals and let K6 be its normal closure. Under K6 there is a unique quadratic field K2. For i = 2, 3, 6 we define Cli to be the 3-class group of Ki and ri; to be the rank of Cli. In an earlier paper we examined the structure of Cl3 when K2 is complex and K6/K2 is unramified. In this paper we remove these restrictions and obtain similar results.

A sequence {an} of integers is said to be primitive if whenever i ≠ j. For example, if n is any positive integer, the sequence

is primitive. This is an important example in the light of the elementary result (see [2; p. 244]) that if 0 < a1 < a2 < … < ar ≤ 2n is primitive then necessarily r ≤ n; i.e. at most half of the positive integers ≤ 2n can be members of the sequence. Besicovitch [2; p. 257] has obtained the surprising result that, given ε > 0, there exists an infinite primitive sequence {ai} such that

where A(n) denotes the number of ai ≤ n.

Recent years have witnessed a significant development of the theory of Prüfer domains; there are many known characterizations of such domains within the class of integral domains with identity or the class of integrally closed domains—for example, see [6; Exer. 12, p. 93] or [10; Chap. 4]. E. Bastida and R. Gilmer have recorded in [4] a number of open questions concerning Prüfer domains that are of the following form:

If D is an integral domain with identity with property E, is the integral closure of D a Prüfer domain?

Specifically, the questions listed by Bastida and Gilmer were first raised in [13], [11], [7], and [12].