The purpose of this article is to give the proof of a theorem announced in [1]. We refer the reader to [1] for the terminology used here.

One of the substantial differences between real and complex analysis is the behaviour of pointwise sequential limits of functions. It is well known that, if f(z) is a bounded analytic function in D = {z∈ C: |z| < 1}, then there exists a sequence {pn(z): n = 1,2,…} of polynomials such that

(i) ‖Pn‖ ≤ ‖f‖ for all n = 1,2,3,…, and

(ii) for each z ∈ D, Pn(Z) → f(z) as n → ∞,

where we have used the notation

Our investigation starts from the question: is the even dimensional cohomology of the non-abelian group of order p3 and exponent p generated by Chern classes? From the computation of the complete cohomology ring in [8] one quickly sees that the essential problem is to express elements of the form cor (γk), γ ∈ H2 (k, ℤ), K a subgroup of index p, in terms of Chern classes. For a more general pair of groups (K ≤ G) it is known, see [7] that the best for which one can hope is a description of some multiple of cor (γk) in this way. Our first theorem shows that, under suitable hypotheses (satisfied in particular by the example of order p3) the numerical factor may be removed. Thus

In [2], Sawyer considers a closed, central, convex region K which is such that, however it is displaced in the plane, a point of the integral lattice is covered. He shows that the area A(K) of K satisfies . We prove here a result in the opposite direction.

Let Ω be an algebraic number field, and let NΩ ⊂ ℚ be the group of norms of fractional ideals of Ω. Then NΩ is a subgroup of the positive rationals; the latter is the direct sum of a denumerable infinity of infinite cyclic groups, and so it is free abelian; thus NΩ is free abelian, and, since it is not finitely generated, we must have qua abstract groups. The purpose of this paper is, in the first place, to find a “metrical ” way of distinguishing these isomorphic groups, and, to this end, we introduce the notion of Farey density, defined as follows; let X be a positive integer, and consider the Farey section ℱ(X) of order X, thus the set of all reduced positive fractions with denominator < X; then the quotient

measures the proportion of elements of ℱ(X) which are in NΩ, and, as X → ∞, it gives a measure of the “density ” of fractional ideal norms in the rational interval (0, 1).

The notion of nonatomicity of a measure on a Boolean σ-algebra is an important concept in measure theory. What could be an appropriate analogue of this notion for charges defined on Boolean algebras is one of the topics dealt with in this paper. Analogous to the decomposition of a measure on a Boolean σ-algebra into atomic and nonatomic parts, no decomposition of charges is available in the literature. We provide a simple proof of such a decomposition. Next, we study the conditions under which a Boolean algebra admits certain types of charges. These conditions lead us to give a characterisation of superatomic Boolean algebras. Babiker' [1] almost discrete spaces are connected with superatomic Boolean algebras and a generalisation of one of his theorems is obtained. A counterexample is also provided to disprove one of his theorems. Finally, denseness problems of certain types of charges are studied.

The letters a, b, n, m, t (perhaps with suffixes) always denote natural numbers. A, B, S denote finite sets of natural numbers. |A| stands for the cardinality of A.

For a given constant c > 1 we say that the set A has property α(c) if there are at most c|A| differences a − b ≥ 0 for a, b ∈ A.

In this paper is proved a formula for the number of simplicial neighbourly d-polytopes with d + 3 vertices, when d is odd.

Let pn denote the n-th prime number, and let

It follows from the prime number theorem that E ≤ 1. Erdős used a sieve argument to show that E < 1, and Rankin and Ricci gave the explicit estimates E ≤ 57/59 and E ≤ 15/16 respectively. A more powerful approach used by Hardy and Little-wood and by Rankin depended on hypotheses about the zeros of L-functions until Bombieri's Theorem was found. With its aid Bombieri and Davenport [1] proved that

A fuller history, with bibliography, is to be found in [1]

Among several interesting analogues of Chebyshev's problem about the largest prime factor of

there is the question of the largest prime factor of

where a is a given non-zero integer and the product is taken over positive primes p. The latter subject appears to have been first treated by Goldfeld [2] and Motohashi [6], who showed that, if Px be the greatest prime factor in question, then there exists a constant such that Px > xθ for all sufficiently large x. Their method, which involved the use of both Bombieri's theorem and the Brun-Titchmarsh theorem, had some affinity with the earlier treatments of Chebyshev's original problem.

A. J. W. Hilton [5] conjectured that if P, Q are collections of subsets of a finite set S, with |S| = n, and |P| > 2n−2, |Q| ≥ 2n−2, then for some A ∈ P, B ∈ Q we have A ⊆ B or B ⊆ A. We here show that this assertion, indeed a stronger one, can be deduced from a result of D. J. Kleitman. We then give another proof of a recent result also proved by Lovász and by Schönheim.

Let S(x) be a trigonometric polynomial,

where N > 0 and M are integers, the an are arbitrary complex numbers, and e(x) = e2πix. In its basic form, the large sieve of Linnik and Rényi is an inequality of the form

Let G be a finite connected graph with no loops or multiple edges. The point-connectivity K(G) of G is the minimum number of points whose removal results in a disconnected or trivial graph. Similarly, the line-connectivity λ(G) of G is the minimum number of lines whose removal results in a disconnected or trivial graph. For the complete graph Kp we have

This paper gives an algorithm for generating all the solutions in integers x0, x1…, xn of the inequality

where 1, α1, …,αn are numbers, linearly independent over the rationals, in a real algebraic number field of degree n + 1 ≥ 3 and c is any sufficiently large positive constant. It is well known [2, p. 79] that if c is small enough, then (1) has no integer solutions with x1…, xn not all zero.

In this note we prove that every convex Borel set in a finite dimensional real Banach space can be obtained, starting from the compact convex sets, by the iteration of countable increasing unions and countable decreasing intersections. This question was first raised by V. Klee [1, p. 451]. It was answered affirmatively by Klee for R2 in [2, pp. 109–111] and for R3 by D. G. Larman in [4]. C. A. Rogers has given an equivalent formulation of the question for Rn in [6].

The flow in the wake of a finite rotating disc was recently investigated by Leslie [1], who showed that the structure of the wake was very similar to that found for the two-dimensional flow past the trailing edge of a flat plate by Goldstein. Here, we investigate the inner region where the Goldstein wake joins onto the flow over the plate itself. Because there is no pressure jump across the rim at the edge of the Ekman layer, no third deck is necessary, so this inner region has width O (E½) and contains just two decks, the lower thickness O(E⅔) and the upper with thickness O(E½); E is the Ekman number for the flow. The resultant differential equation in the lower deck for the radial and axial velocities is uncoupled from that for the azimuthal velocity, and has exactly the same form as the equation found by Stewartson [4] for the lower deck in the flat plate problem cited above.

It has been known for many years that a laminar boundary layer adjoining a supersonic stream is in principle capable of spontaneously evolving from an unseparated to a separated form in which the boundary layer is detached from the wall and in between a slowly moving reversed flow is set up. The phenomenon arises because any thickening of the boundary layer tends to induce an adverse pressure gradient which in turn accelerates the thickening of the boundary layer and hence its evolution from the normal (Blasius) form.

Let a1 …, ar be elements of a commutative ring R with identity. When the R-length of

is finite and R is Noetherian the multiplicity e(a1 …, ar\R) can be defined in one of three ways. Starting either from the earliest definition by means of Hilbert functions (see Samuel [4]) or from the latest by induction on r(see Wright [5]) it can be shown that e(a1 …, ar\R) depends only on the ideal

and the number r of generators. The other definition due to Auslander and Buchsbaum [1] is as the Euler-Poincare characteristic of a Koszul complex. The main purpose of this note is to investigate the extent to which the separate homology modules and the Koszul complex itself are independent of the sequence a1 …, ar. First it is shown that when A = (a1, …, ar) and B = (b1 …, br) are related by a reversible R-linear transformation the two Koszul complexes are isomorphic. A special case of the final theorem states that if A and B are minimal generating sets of the same ideal and R is a local ring then again the Koszul complexes are isomorphic.