In a typical counter-example construction in geometric measure theory, starting from some initial set one obtains by successive reductions a decreasing sequence of sets Fn, whose intersection has some required property; it is desired that ∩ Fn shall have large Hausdorf F dimension. It has long been known that this can often be accomplished by making each Fn+1 sufficiently “dense” in Fn. Our first theorem expresses this intuitive idea in a precise form that we believe to be both new and potentially useful, if only for simplifying the exposition in such cases. Our second theorem uses just such a construction to solve the problem that originally stimulated this work: can a Borel set in ℝk have Hausdorff dimension k and yet for continuum-many directions in every angle have at most one point on each line in that direction? The set of such directions must have measure zero, since in fact in almost all directions there are lines that meet the Borel set (of dimension k) in a set of dimension 1: this can easily be deduced from Theorem 6.6 of Mattila [5], which generalized Marstrand's result [4] for the case k = 2.

A convex polytope is a zonotope, if, and only if, its support function satisfies Hlawka's inequality. It follows that a finite dimensional real space with piecewise linear norm is isometrically isomorphic to a subspace of an L1 space, if, and only if, it has the quadrilateral property

Steady plane periodic gravity waves on the surface of an ideal liquid flowing over a horizontal bottom are considered. The flow is rotational with a vorticity distribution ω(ψ) and has flux Q. Let R/g denote the total head and S the flow force of the wavetrain. The diagrams (Fig. 1) show combinations of R and S which are possible in the general case. (We normalise so that Q = 1 throughout. The axes are R/R* and S/S*, where the suffix * refers to the critical flow.) It is proved that no waves are possible below γ+ or to the right of γ here γ+ corresponds to unidirectional supercritical streams, and thus is the best possible barrier, while γ is a barrier to the right of the line of waves of greatest height. Bounds on wave properties are found in the process of establishing the above results. When ω ≡ 0 these bounds were conjectured by Benjamin and Lighthi U (1954) and established in Keady and Norbury (1975). The generalisation to flows with vorticity is accomplished under the condition that , where hc is the height of the crest of the wave.

Let f be a positive-definite quadratic form with integer coefficients, and denote by c(f) (≥ 1) the class-number of f, that is, the number of classes in the genus of f. I showed in [4] that c(f) ≥ 2 for every f in n ≥ 11 variables; the transformations of [3] were used to make the problem easier. I have since sought to find all the one-class n-ary genera with 3 ≤ n ≤ 10 (the case n = 1 is trivial, and n = 2 is very difficult).

Let ‖x‖ denote the distance of x from the nearest integer. We are going to use a method of W. M. Schmidt [4] to prove three theorems.

Consider two convex bodies K, K′ in Euclidean space En and paint subsets β, β′ on the boundaries of K and K′. Now assume that K′ undergoes random motion in such a way that it touches K.

This paper contains a derivation of Lagrange's expansion with remainder for a weak function of several independent variables each satisfying an implicit relation. We also provide necessary and sufficient conditions for the associated infinite series expansion.

The effects on a boundary layer of thickness O(LR−1/2) (where L is a typical streamwise lengthscale, and R is the Reynolds number) of a small unsteady hump at the wall is considered. The hump is of height O(LR−5/8) and length O(LR−3/8), and outside the boundary layer is potential flow. Three different regimes of unsteadiness parameter are considered, leading to a description of the flow over the complete spectrum for this size of excrescence.

We exhibit (§2) an example of §a compact Hausdorff space supporting a Radon probability measure μ and a continuous map ø : X → I, when I is the closed unit interval, for which the image measure ø(μ) is Lebesgue measur m with the properties:

(i) there exists an open set G ⊂ X for which ø(G) is not m-measurable;

(ii) μ is a non-atomic non-completion regular measure;

(iii) the measure algebras (X, μ) and (I, m) are isomorphic but for no choice c sets B ⊂ X, B′ ⊂ I of measure zero are and homeomorphic

(iv) there exists a selection p : I → X (i.e. p(t) ∊ ø−1(t) for all t ∊ I) which i Borel m-measurable, but there is no Lusin m-measurable selection.

The aim of this paper is to prove the following

Theorem A. Let K be an ordered, locally finite, simplicial complex, considered as a category, let L be a subcomplex, and let F : K → PL be a functor. Then

(i) the geometric realisation 〈F〉 of F has a natural PL structure in which 〈F|L〉 is a subpolyhedron, and, in particular,

(ii) 〈F〉 admits a triangulation by a locally finite simplicial complex in which 〈F|L〉 is triangulated as a subcomplex.

In this paper we prove two analogues of the following theorem:

If is a collection of distinct subsets of {1, …, m} such that , i ≠ j ⇒ Ai ∩ Aj ≠ ∅, Ai ∪ Aj ≠ {1, …, m} then .

§1. Let pi denote the i-th prime number, and let

The prime number theorem implies Er ≤ r. Bombieri and Davenport [1] showed that

improving earlier results of Erdős, Rankin and Ricci. Their basic result (corresponding to Lemma 1 below) counts pairs of primes differing by 2n with some weight t(n).

Let K/k be a normal extension of algebraic number fields whose Galois group G is a Frobenius group. Then K/k is said to be a Frobenius extension. Most of the structure of the unit group and of the ideal class group of K is determined by that; of the subfields fixed by the Frobenius kernel N and by a complement F. Here this is investigated when G is a maximal or metacyclic Frobenius group. In particular, the results apply firstly to the normal closure of where a ∊ k and p is a rational prime, and, secondly, when G is a dihedral group of order 2n for an odd integer n. A. Scholz, taking n = p = 3, was the first to consider this problem.

An arithmetic function f(n) is said to be additive, if it satisfies the relation f(ab) = f(a) + f(b), for every pair of coprime integers a and b; and stronglya dditive if, in addition, f(pm) = f(p) for every prime-power pm.

In this paper we continue our study of the values taken by Euler's ø-function begun in [1]–[3]. Let ør(n) be the iterated ø-function, that is ør(n) = ø{ør-1(n} where ø1 = ø1. Let

The question, whether a given element of a C*-algebra has an image of rank one in some faithful representation, was studied in [3]. Such elements were characterised there by the property of being “single” (as defined below). As was pointed out in [3], Section 5, this criterion fails for general Banach algebras and the purpose of this paper is to provide a stronger condition giving the required representation property for any semi-simple Banach algebra.

It is well-known that the σ -algebra of Borel subsets of a metric space coincides with the smallest family of sets which contains the open sets and is closed under countable intersections and countable disjoint unions «3, Th.3, p. 348». A deeper and less known result of Sierpiński is that for separable metric spaces the family of open sets may be replaced by the family of closed sets in the above result «16, p. 272–275» (and «17, p. 51» for the real line). This paper gives an in depth analysis of these and related generation processes. Several abstract formulations, generalizations and limiting examples are given.

An incident sound field is scattered by a semi-infinite rigid screen with periodically arranged slits or circular apertures and an approximate solution is sought when the slit (or aperture) width is small and the wavelength is large compared with the separation. An integral equation formulation is used to show that the scattering properties of the screens are equivalent to those of a homogeneous compliant plate. The effective compliance is estimated and is found to be essentially uniform over the plate, with corrections close to the edges.

Let Xt be a Lévy process in Rd, d dimensional euclidean space. That is Xt is a strong Markov process whose transition function, P(t, x, A), satisfies: