Let f(x) be a trigonometric polynomial with N (≥2) non-zero coefficients of absolute value not less than 1. In this paper it is proved that the L1 norm of f exceeds a fixed positive multiple of (logN/log log N)½. This result improves a previous one due to H. Davenport and P. J. Cohen (the same bound with exponent ¼).

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.

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.

Definitions. We say that h is a Hausdorff measure function if it satisfies the following conditions:

(i) for all x > 0,

(ii) h(x) → 0 as x → 0,

(iii) h(x) is monotonic increasing.

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.

Let Zn denote the set of all ordered n-tuples of integers. Let us call any finite subset of Zn a body in Zn, and any finite set of bodies in Zn a family in Zn.

Consider the following problem:

Give a decision procedure which for any family ℱ in Zn decides the following.

In [3] it was proved that every convex Borel (= Baire) set in a finite dimensional real Banach space can be obtained, starting from the closed (or compact) convex sets, by the iteration of countable increasing unions and countable decreasing intersections.

In §2 of this note we define some concepts of the descriptive theory of convex sets in locally convex spaces. We prove several theorems, which are analogous to the standard theorems of the descriptive theory of sets in topological spaces.

If K is a set in n-dimensional Euclidean space En, n ≥ 2, with non-empty interior, then a point p of En is called a pseudo-centre of K provided tha each two dimensional flat through p intersects K in a section, which is either empt or centrally symmetric about some point, not necessarily coinciding with p. A pseudo centre p is called a false centre of K, if K is not centrally symmetric about p. A dee result of Aitchison, Petty and Rogers [1] asserts that, if K is a convex body in E and p is a false centre of K, with p in the interior, int K, of K, then K is an ellipsoid Recently J. Höbinger [2] extended this theorem to any smooth convex body K witl a false centre p anywhere in En. At a recent meeting in Oberwolfach, he asked if the condition of smoothness can be omitted and the purpose of this note is to prove sucl a result.

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].)

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].

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.

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].

1. W. Blaschke's kinematic formula in the integral geometry of Euclidean n-dimensional space gives a weighted measure to the set of positions in which a mobile figure K1 overlaps a fixed figure K0. In the simplest case, K0 and K1 are compact convex sets and all positions are equally weighted; we give this in more detail. Let Wq denote the q-th Quermassintegral of K1: Steiner's formula for the volume V of the vector sum K1 + λB of K1 and a ball of radius λ defines these set functions by the equation

see [4; p. 214]. Blaschke's formula [4; p. 243] gives

as the measure, to within a normalization, of overlapping positions of K1 relative to K0.

The Stefan problem is a particular free boundary problem for the heat equation which arises in the investigation of the melting of solids. In the case of one space dimension there are numerous results available concerning the existence, uniqueness, and stability of the solution [c.f. 6]. However the case of several space variables is considerably more difficult. This is due in large part to the fact that the geometry of the problem can become quite complicated, and smooth initial and boundary data do not necessarily lead to smooth solutions. In particular, under heating, a connected solid can melt into two (or more) disconnected solids, thus leading to a problem in which the free boundary varies in a discontinuous manner. These difficulties have motivated several researchers to look for “weak” solutions to the Stefan problem [c.f. 1, 3, 5]. Although this approach is quite general and leads to numerical schemes for solving the problem under consideration, there are several drawbacks to this method, among them being the fact that no information is obtained concerning the structure of the interphase boundary, nor is there much information on the regularity of these weak solutions.