In [3, 4] we showed how the use of a random-walk analogue can be made to yield non-trivial information about the behaviour of certain trigonometric sums in one variable. Our aim here is to show how our method can be adapted to yield similar results for a broad class of trigonometric sums in several variables. Let

be a polynomial in v independent variables with integral coefficients. We choose integers n ≥ 0, d ≥ 1 and p ≥ 2 with p prime, and assume that f(x) has total degree ≤ d + 1. We shall consider the problem of obtaining non-trivial upper bounds for the absolute value of sums of the type

where P = {1, 2, …, p} and f is non-constant.

A set A ⊂ ωω is called compactly if, for every compact K ⊂ ωω, A ∩ K is . Consider the proposition that every compactly set is . (AD implies that it is true, ZFC + CH implies that it is false.) We are concerned here with whether this is consistent with ZFC, particularly when n = 1. In the case of sets (that is, analytic sets), this consistency question is due to Fremlin (see [7], page 483, problem 18). Kunen and Miller [3] have proved the following two theorems.

Let N be a positive integer. We enquire whether there is a quadratic f(t)=at2 + bt + c, not identically square, which takes square values for N consecutive integers t. In [1] it was shown that this was possible, in what we shall call the symmetric case, if N = 8. A better treatment of this case appears here, together with a discussion of the nonsymmetric case, where we may take N = 7. There is no evidence that these are the best possible results.

Formal propositional logic describing the laws of constructive (intuitionistic) reasoning was first proposed in 1930 by Heyting. It is obtained from classical pro-positional calculus by deleting the Law of Excluded Middle, and it is usually referred to as Heyting's (intuitionistic) propositional calculus ([9], §§23, 19) (we write HPP in short). Formal logic involving predicates and quantifiers based on HPP is called Heyting's (intuitionistic) predicate calculus ([9], §§31, 19) (we write HPR in short).

An excellent introduction to the metric theory of diophantine approximation is provided by [19], where, in chapter 1·7, the reader may find a discussion of the first two problems considered in this paper. Our initial question concerns the number of solutions of the inequality

for almost all α(in the sense of Lebesgue measure on ℝ). Here ∥ ∥ denotes distance to a nearest integer, {βr}, {ar} are given sequences of reals and distinct integers respectively, and f is a function taking values in [0, ½] and with Σf(r) divergent (for convenience we write ℱ for the set of all such functions). It is reasonable to expect that, for almost all α and with some additional constraint on f, the number of solutions of (1) is asymptotically equal to

as k tends to infinity.

1. For an integer v > 1, we define P(v) to be the greatest prime factor of v and we write P(1) = 1. Let m ≥ 0 and k ≥ 2 be integers. Let d1, …, dt with t ≥ 2 be distinct integers in the interval [1, k]. For integers l ≥ 2, y > 0 and b > 0 with P(b) ≤ k, we consider the equation

Put

so that ½ < vt ≤ ¾. If α > 1 and kα < m ≤ kl, then equation (1) implies that

for 1 ≤ i ≤ t and hence

Let Γ be a subgroup of PSL2 (ℤ) of finite index. In this note we are concerned with the arithmetic nature of the Fourier coefficients of holomorphic Eisenstein series on Γ.

The purpose of this note is to extend Brumer's characterization of pro-p groups of cohomological dimension two ([1], corollary 5·3) to presentations more general than free ones. The result is then used to rid a proof in [7] of certain field theoretic ingredients. As a by-product we complete a result of Tsvetkov [6] about group extensions obtained by omitting relations.

One of the inequalities recently found by Cochran and Lee([1], theorem 1) is the following.

If γ and p are real numbers with p > 0, and f(t) is measurable and non-negative on (0, ∞), then

Cusp singularities were introduced and described in detail in Hirzebruch's fundamental paper [3] (se recall some of the basic results in § 1 below). They form a natural and well-behaved class, included in Laufer's ‘minimally elliptic’ singularities [5]. Those which occur on hypersurfaces appear also as hyperbolic singularities in Arnold's [1] classification of 1-modal singularities.

Bollobás and Erdös[1] have posed the problem:

If a is irrational, show that for 1 ≤ i < j ≤ p the number of integers t with 1 ≤ t ≤ p such that {(t–i)2a} < d and {(t–j)2a} < d, 0 < d < 1, is d2p + o(p) uniformly in i, j.

This paper aims to provide a complete resolution of the general norm form equation over function fields of positive characteristic. In a previous paper [4] we studied norm forms in the simpler case of zero characteristic; that study forms the starting point for the present investigations. Diophantine problems over function fields of positive characteristic were first investigated by Armitage in 1968 [1], who clamied to have established an analogue of the Thue–Siegel–Roth–Uchiyama theorem for such fields. This claim was refuted by Osgood in 1975 [6], who also derived a correct analogue of Thue's approximation theorem. in 1983 a different attack was made on Diophantine problems over function fields, the principal weapon being a bound [2] for the heights of the solutions of the unit equation

Let A be an associative ring with 1. For any natural number n, let GLnA denote the group of invertible n by n matrices over A, and let EnA be the subgroup generated by all elementary matrices ai, j, where aεA and 1 ≤ i ≡ j ≤ n. For any (two-sided) ideal B of A, let GLnB be the kernel of the canonical homomorphism GLnA→GLn(A/B) and Gn(A, B) the inverse image of the centre of GLn(A/B) (when n > 1, the centre consists of scalar matrices over the centre of the ring A/B). Let EnB denote the subgroup of GLnB generated by its elementary matrices, and let En(A, B) be the normal subgroup of EnA generated by EnB (when n > 2, the group GLn(A, B) is generated by matrices of the form ai, jbi, j(−a)i, j with aA, b in B, i ≡ j, see [7]). In particuler,

is the centre of GLnA

In [6] the first author introduced some combinatorial concepts involving Young diagrams corresponding to partitions with distinct parts and applied them to the projective representations of the symmetric group Sn. A conjecture concerning the p-block structure of the projective representations of Sn was formulated in terms of these concepts which corresponds to the well-known, but long proved, Nakayama ‘conjecture’ for the p-block structure of the linear representations of Sn. This conjecture has recently been proved by Humphreys [1].

An action of a group, G, on a surface, F, consists of a homomorphism

ø: G → Homeo (F).

We will restrict our discussion to finite groups acting on closed, connected, orientable surfaces, with ø(g) orientation-preserving for all g ε G. In addition we will consider only effective (ø is injective) free actions. Free means that ø(g) is fixed-point-free for all g ε G, g ≠ 1. This paper addresses the classification of such actions.

A manifold M is said to be aspherical if its universal covering space is contractible. Farrell and Hsiang have conjectured [3]:

Conjecture A. (Topological rigidity of aspherical manifolds.) Any homotopy equivalence f: N → M between closed aspherical manifolds is homotopic to a homeomorphism,

and its analogue in algebraic K-theory:

Conjecture B. The Whitehead groups Whj(π1M)(j ≥ 0) of the fundamental group of a closed aspherical manifold M vanish.

Diagrams have been used in group theory by numerous authors, and have led to significant results (see [4] and the references cited there). The idea of applying diagrams to semigroups seems to be more recent [3, 7, 8]. In the present paper we discuss semi group diagrams and use them to obtain results concerning the word problem for one-relator semigroups. The word problem for one-relator groups has been solved by Magnus [6], but the analogous question for semigroups remains open. We are not able to solve the problem in full generality, but have obtained some partial results.

We provide, for N ≥ 1, a solution to the word problem for the semigroup S having the presentation {a, b|a = abaNb}. For N = 2, this is example 7 in Howie and Pride [1]. Since solving the word problem for {a, b|a = abaNb} is equivalent to solving the word problem for {a, b|a = baNba}, the result here should be regarded as a special case of a more general result of Oganesyan [2], who solves the word problem for {a, b|a = bA}, A arbitrary.

Alexander [1] showed that an oriented link K in S3 can always be represented as a closed braid. Later Markov [5] described (without full details) how any two such representations of K are related. In her book [3], Birman gives an extensive description, with a detailed combinatorial proof of both these results.

In this paper we will be concerned with the problem of describing the Jacobson radical of the semigroup algebra K[S] of an arbitrary semigroup S over a field K in the case where this algebra satisfies a polynomial identity. Recently, Munn characterized the radical of commutative semigroup algebras [9]. The key to his result was to show that, in this situation, the radical must be a nilideal. We are going to extend the latter to the case of PI-semigroup algebras. Further, we characterize the radical by means of the properties of S or, more precisely, by some groups derived from S. For this purpose we will exploit earlier results leading towards a characterization of semigroup algebras satisfying polynomial identities [5], [15], which generalized the well known case of group algebras (cf. [13], chap. 5).