Problems of classifying and enumerating types of plane patterns, tilings, and other repeating geometrical structures have interested mathematicians, crystallographers and others for many years. Recently we have formulated the general principles that seem to underlie many of the published treatments of these topics, and so have been able to put on a mathematical basis classification criteria often justified mainly on intuitive grounds. In other words, we can now decide whether or not two given patterns are of the same “type”, at one of a number of different possible levels of classification, without relying on some vaguely expressed distinction based on a “feeling” as to whether the objects in question should be regarded as of distinguishable kinds, or not.

Let h1(x1, …, xn), …, hs(x1, …, xn) be polynomials with integer coefficients. We give conditions on these polynomials which guarantee the existence, for all sufficiently large primes p, of small solutions to the system of congruences

Previous investigations of this problem include those of Mordell [10], Chalk and Williams [5], and Smith [14]. Smith's main result, which encompasses the other results, can be stated as follows.

The classification of conies in the real or complex projective plane under the action of the appropriate group is simple and well known. We consider here the more complicated question of classifying conies in the complex projective plane ℂℙ2 under the action of the real projective group PGL (3, ℝ).

Two basic approaches have been used to develop explicit formulae for the number of classes in a genus of binary quadratic lattices over an algebraic number field. Analytic machinery in the form of the Minkowski-Siegel Mass Formula or the Tamagawa number of an algebraic group was employed by Pfeuffer [13] and Shyr [17] to obtain such a formula for maximal positive definite lattices over totally real number fields. On the other hand, Peters [10] observed that a formula applicable to maximal lattices over any number field can be deduced by algebraic methods from the theory of quadratic field extensions. Using group-theoretic techniques set up by the present authors [3] along with the calculation of certain local unit indices, Korner [6] derived the corresponding formula for non-maximal lattices.

We calculate the minimum numbers of k-dimensional flats and cells of any Euclidean d-arrangement of n hyperplanes. The bounds are obtained by calculating lower bounds for the values of the doubly indexed Whitney numbers of a basepointed geometric lattice of rank r with n points. Additional geometric results concern the minimum number of cells of a Euclidean or projective arrangement met by a subspace in general position and the minimum number of non-Radon partitions of a Euclidean point set. We include remarks on the relationship between Euclidean arrangements and basepointed geometric lattices and on the minimum numbers of cells of arrangements with a bounded region.

In [10] the following generalization of the Erdös–Ko–Rado [6] theorem was proved:

If 1 ≤ h ≤ m/2 andare r sets of h-subsets of {1,…, m} such that

then

In Section 1 below I describe two measures of the complexity of a binary relation. J The theorem says that these two measures never disagree very much. Both measures of complexity arose in connection with Saharon Shelah's notion [5] of a stable firstt order theory; Shelah showed in effect that one measure is finite, if, and only if, the other is finite too. This follows trivially from the theorem below. I confess my main aim was not to get the extra information which the theorem provides, but to eliminate Shelah's use of uncountable cardinals, which seemed strangely heavy machinery for proving a purely finitary result. Section 2 below explains the modeltheoretic setting.

Since my article McMullen [1980] has appeared, Professors S. S. Ryškov and B. B. Venkov have drawn my attention to two previously published papers. B. A. Venkov [1954] proves my main Theorem 1 (and its corollary Theorem 2) by methods apparently very similar to mine (1 have not checked all the details), while A. D. Aleksandrov [1954] generalizes Venkov's result to tilings of spaces of constant curvature by polytopes (not necessarily convex) congruent to ones in some finite collection. I am happy to acknowledge their priority.

Let Mn be a smooth, compact and strictly convex, embedded hypersurface of Rn + 1 (n ≥ 1), an ovaloid for short. By “strictly convex” we mean that the Gauss-Kronecker curvature where ki are the principal curvatures with respect to the inner unit normal field, is everywhere positive. It is well knpwn [5, p. 41] that, for such a hypersurface, the spherical-image mapping is a diffeomorphism onto the unit hypersphere. Furthermore, Mn is the boundary of an open bounded convex body, which we shall call the interior of Mn.

Much recent attention has been given to geometric representation of elements of the stable homotopy groups of spheres, π*s A particular example concerns non-singular bilinear maps ℝm+1 × ℝn+1 → ℝm+n+1−p; on restriction and normalisation these become biskew maps Sm × Sn ℝ Sm+n-p;. Now the Hopf construction ℋ applied to any map f: Sm × Sn → Sm+n-p yields

In [7[ a functor Ext is defined in terms of C*-extensions. It is a covariant functor from the homotopy category of compact, metrizable spaces to abelian groups. Further details are given in [7, 8, 9, 11]. From [7, 14] Ext extends to a Steenrod homology theory, Ext*, which may be identified with the one associated with unitary K-theory. Since Lie groups are fundamental to K-theory (see [2, p. 24]) one might expect Ext(G) to be of interest when G is a Lie group.

Let (X, ℱ, μ) be a topological measure space with X a completely regular Hausdorff space and ℱ the σ-algebra of all μ-measurable sets, containing all the Baire sets of X. Consider the following two conditions on (X, ℱ, μ).

A metric space (X, ρ) is called precompact, if, for every ε > 0, there is a finite ε-cover (a covering by sets of diameter ≤ ε). The space (X, ρ) is separable if for every e there is a countable ε-cover. There should be some in-between condition. We say that (X, ρ) has fine covers, if, for every ε > 0, there exists a countable ε-cover (U1, U2, …), such that the diameter ∂(Ui) tends to zero as i → ∞. In fact, Goodey [1] has related this property to Hausdorff dimension. We show that a space with fine covers need not be σ-precompact and that on any complete metrizable non-σ-compact space X there is a metric ρ* such that (X, ρ*) has no fine cover.

I am grateful to Professor K. Prachar for pointing out to me that there is a mistake in the proof of Theorem 2 in my paper “On the distribution of primes in short intervals” [Mathematika, 23 (1976), 4–9]. The mistake is in the assertion on p. 6 that, if 1 ≤ μ/λ < 4, the result is trivial. The corrected version reads as follows.

§1. An initial and boundary value problem. In this article we study the solution of an initial and boundary value problem of dynamic linear thermoelasticity in which both inertia and the coupling between mechanical and thermal effects are retained.

In 1963 Mathematika published a note [2[ in which I “proved” that the equation f(x1, …, xm) = 1 could always be solved in algebraic integers f(x1, …, xm), whenever f(x1, …, xm) was a homogeneous polynomial of degree n ≥ 1, with algebraic integers of greatest common divisor 1 as coefficients. This “proof” was so good that it had to be corrected in [3]. Since neither I nor anyone else had any use for this result, these papers dropped into the decent obscurity reserved for dead ends in mathematical research. They presumably would have remained there had not Cantor [1] recently started looking at similar results. He discovered, to his surprise and mine, that the entire article [2] had been anticipated by Skolem in a 1934 monograph ]4] which, apparently, had also languished in obscurity. The only consolation I can draw from this is the observation that, if I was unaware of Skolem's article, he was unaware of Steinitz's work [5] of 1911, which he duplicated in Theorems 5 and 6 of [4]. The moral of this story is that any working mathematician would rather prove something himself than try to find it in any but the most accessible literature.

A function

analytic and univalent in U = {z: |z| < 1} is said to be starlike there, if f(U) is f starshaped with respect to the origin, that is, if w ε f(U) implies tw ε f(U) for 0 ≤t ≤ 1. We denote by S* the class of all such functions. The Koebe function; k(z) = z(l – z)-2, z ε U, maps U onto the complex plane minus a slit along the I negative real axis from - ¼ to ∞, and thus belongs to the class S*. Recently Leung [4] has shown that, if

then, for f ε S*,

for every p > 0.

Unless stated otherwise all quadratic forms have rational integer coefficients and all representations are integral representations. For positive binary quadratic forms of the same discriminant it is known that two such forms are equivalent provided they represent the same integers. See, for instance, [Ki2], and for a sharper extension [W2]. On the other hand, in the quaternary case these value-sets are far from characterizing the forms even within a genus. It is therefore natural to ask for positive ternary forms the corresponding question, whose answer appears to be unknown.

This note contains characterizations of those sigma-fields for which sigma-finiteness is a necessary condition in the Radon-Nikodym Theorem.

Our purpose is to consider those σ-fields for which σ-finiteness is a necessary condition in the Radon–Nikodym Theorem. We first prove a measure theoretic equivalence in the general case, and then use this to obtain an algebraic characterization in the case when the σ-field is the Borel field of a locally compact separable metric space. For undefined terminology we refer the reader to [1] for measure theoretic and [2] for algebraic properties.

By a measure, we mean a countably additive function from σ-field of sets or a Boolean σ-algebra into the non-negative extended real numbers. We will say that a measure μ on a σ-field of sets Σ is RN provided each μ-continuous finite measure on Σ has a Radon–Nikodym derivative in L1(μ).

New classes of pairs e, p are presented for which the Gauss sums corresponding to characters of order e over finite fields of characteristic p are pure, i.e., have a real power. Certain pure Gauss sums are explicitly evaluated.