This paper is a sequel to [1], but can be read independently. However, we do quote one of the results of [1] in the proof of our theorem here.

As in [1], we define the continuantK(a1, a2, …, an) formed from the positive integers a1 …,an to be the denominator qn of the continued fraction

where pn/qn is in lowest terms.

The isometries of the space of convex bodies of Ed with respect to the symmetric-difference metric are precisely the mappings generated by measurepreserving affinities of Ed.

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.

An isomorphic factorisation of a digraph D is a partition of its arcs into mutually isomorphic subgraphs. If such a factorisation of D into exactly t parts exists, then t must divide the number of arcs in D. This is called the divisibility condition. It is shown conversely that the divisibility condition ensures the existence of an isomorphic factorisation into t parts in the case of any complete digraph. The sufficiency of the divisibility condition is also investigated for complete m-partite digraphs. It is shown to suffice when m = 2 and t is odd, but counterexamples are provided when m = 2 and t is even, and when m = 3 and either t = 2 or t is odd.

I have committed a couple of elementary errors which relate to the proof of Lemma 2.3 of the paper with the above title (Mathematika, 22 (1975), 60–70). These are easily corrected, and the remainder of the paper is not affected by them.

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.

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.

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 .

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.

The study of plasma instabilities has led to the question whether a certain third order linear differential equation involving a parameter p has solutions which vanish as x → ± ∞. Assuming existence, it is first easily shown that Rep must be positive and then, after a Fourier transform has changed the equation to one of second order, standard comparison equation techniques are used to obtain a contradiction, valid for large enough p.

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.