We consider a Markoff spectrum for the set of indefinite binary quadratic forms with real coefficients which represent zero non-trivially. As was done for the classical Markoff spectrum, we show that 1/3 is the largest accumulation point of the set and explicitly determine the countably infinite number of elements greater than 1/3. Unlike the situation for the classical Markoff spectrum, there is a countably infinite number of limit points greater than 1/3.

From time to time results about partitioning a given set into subsets have been established. (See for example [2], [3].) We consider here the reverse problem of forming the union of three sets in a certain best possible way. For simplicity we work in Euclidean n-space, En. Let mX denote the measure of the set X.

Positive definite quadratic forms are associated with pointlattices in the following way. If x ∊ Zm and H = MT for a real m × m matrix M, then xTHx is the square of the distance from the origin to the point Mx of MZm (equally, to W Mx ∊ W MZm, for orthogonal W).

It is known that if Λ is a self-dual lattice in ℝn, then

.

If equality holds the lattice is called extremal. In this paper we find all the extremal lattices: there are unique lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8, 12, 14, 15, 23, 24 and no others.

One of the important properties of the Dirichlet L-functions is that they may be expanded as Euler products. Unfortunately there is no corresponding expansion for the Kubota-Leopoldt p-adic L-functions since the infinite products involved do not converge. In this paper we give an explicit formula which shows how any finite number of Euler factors may be factored off in a natural manner. Of course, Euler factors may be factored from any function; but we show that, among p-adic analogues of Dirichlet series with periodic coefficients, the p-adic L-functions are precisely the ones which allow this to be done in a natural way which we call a weak Euler product. This might indicate that weak Euler products are the best that can be hoped for. Finally we make a few remarks concerning other analytic properties of the p-adic L-functions.

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 .

In a previous publication [1] we considered the structure of a fragmented ring and of its multiplicative semigroup. In the present publication we consider, in a purely semigroup-theoretic context, some ideas related to those in [1]. As shown by an example, the semigroups we consider need not be the multiplicative semigroup of a ring.

It is known that the class of all reachable states in boundary control of systems described by parabolic equations in one space dimension is independent of the time during which control is applied. This result is generalized here to systems governed by the heat equation in an arbitrary number of space variables.

We develop various facets of the theory of quadratic forms on a Hilbert space suggested by a criterion of Kato which characterizes closed forms in terms of lower semicontinuity.