Let K be a convex body (compact convex set with interior points) in d-dimensional euclidean space Ed, let D(K) denote its diameter, Δ(K) its minimal width, and

the number of lattice points (points of Ed with integer coordinates) in the interior of K. If G0(K) = 0, we call K lattice-point-free; in what follows, K will always be a lattice-point-free convex body.

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).

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.

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.

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.

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.

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.

We consider a body which occupies the open, bounded, regular region B, whose boundary is ∂B and whose closure is . We denote by da the element of surface area, by dυ the element of volume, and by n the outward unit normal. We suppose the behaviour of the body to be described by the equations of the quasi-static theory of homogeneous and isotropic thermoelasticity. These equations, which are obtained from the equations of the dynamical theory (see, for example, Carlson [1], Chadwick [2] or Boley and Weiner [3]) by omitting the inertial term pű from the right-hand side of the equation of motion (4), are:

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.

Existence and uniqueness of classical solutions are established for the dissipative quasigeostrophic equations of geophysical fluid dynamics, using a priori estimates and a Schauder fixed point theorem. The flow is periodic in both horizontal directions and is bounded above and below by rigid flat surfaces. The Reynolds analogy of unit turbulent Prandtl number is assumed. Existence is proved for an arbitrary finite time, if it is further assumed that the surface temperatures vanish. Without this additional assumption existence is guaranteed only for a certain finite time, which is inversely proportional to the norms of the sources and initial conditions.

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.

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.

We write e(x) for e2πix and let ‖x‖ denote the distance of x from the nearest integer. The notation A ≪ B will mean |A| ≤ C|B| where C is a positive constant depending at most on an arbitrary positive number ε, and on an integer k. The letter p always denotes a prime number. The main results of the present paper are as follows.

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.

In this paper the authors obtain a series transformation of concentric circles into concentric rectangles.

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 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.

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.

Consider a slab which is made from a homogeneous and isotropic thermoelastic material and which occupies the region 0 ≤ x ≤ a, where x, y, z are the usual rectangular cartesian coordinates. Suppose that the slab undergoes a motion in which the displacement vector is parallel to the x-axis and the displacement and the temperature are functions of the coordinate x and the time t ( ≥ 0) only. Suppose too that the faces of the slab are clamped, that the face x = 0 is maintained at a constant temperature, and that heat is supplied to unit area of the face x = a at a prescribed rate h(t).