A d-dimensional zonotope Z in Ed which is the vector sum of n line segments is linearly equivalent to the image of a regular n-cube under some orthogonal projection. The zonotope in En-d which is the image of the same cube under projection on to the orthogonal complementary subspace is said to be associated with Z. In this paper is proved a conjecture of G. C. Shephard, which asserts that, if Z tiles Ed by translation, with adjacent zonotopes meeting facet against facet, then tiles En-d in the same manner. A number of conditions, conjectured by Shephard and H. S. M. Coxeter to be equivalent to the tiling property, are also proved.

This paper arises from an attempt to generalise the following result of [1].

THEOREM (Armitage-Fröhlich). If K is a cyclic extension of Q of degree lr, where 2 has even order mod l, then

In this paper we continue our investigation of the topological filtration on the complex representation ring R(G) of a finite group, see [4] and [5]. To recall the basic definitions from (1): let

map a k-dimensional representation ζ to the (flat) vector bundle over the classifying space BG associated to the universal G-bundle by the G-structure on Ck. Then, if denotes the (2k − l)-skeleton of BG,

Let x, y, Q denote large real numbers, with x > y > Q. The object of this paper is to prove the following result

for arbitrary A > 0, with a larger value of Q than hitherto. The symbol denotes as usual the suppression of an absolute constant, π(N; q, a) denotes the number of prime numbers up to N which are congruent to a (mod q), and ϕ(q) denotes Euler's function.

A centrally symmetric d-polytope (d-dimensional convex polytope) P in Euclidean space Ed is called k-neighbourly provided every subset of k vertices of P, which does not contain two opposite vertices of P, is the set of vertices of a (k − 1)-simplex which is a face of P. Contrasting the situation of neighbourly polytopes without the symmetry assumption (see, e.g., Griinbaum [1; chap. 7]), it appears that the possible neighbourliness properties of centrally symmetric polytopes are rather restricted. For d ≥ 2 and n ≥ 1, let k(d, n) denote the greatest integer k, such that there exists a k-neighbourly, centrally symmetric d-polytope with 2(d + n) vertices. McMullen and Shephard [4] have shown that , and for n ≥ 3. They conjectured that

Let K be an algebraic number field. By a. full module in K [l,p.83] we mean a finitely-generated (necessarily free) subgroup M of the additive group of K whose rank is equal to the degree [K : ℚ] of K over the rational field ℚ. The intersection of M with ℤK, the ring of integers of K, is also a full module I, and we shall concern ourselves chiefly with the latter, in that we wish to count the number of rational integers in a given interval which can be expressed as the norms of elements of I. More precisely, we shall adapt the methods of [2] to prove the following

We shall prove the following

THEOREM 1. Let α1, …, αn be any positive algebraic numbers and let u1…, un, ν be positive integers, relatively prime in pairs, such that ν ≥ 2 and ui > v for at least one i (1 ≤ i ≤ n). Then for every ε > 0 there are only a finite number of positive integers v such that the inequality

is satisfied, where for real α we understand by ‖α‖ the distance of α from the nearest integer.

Let q be a power of an odd prime, [q] denote the Galois field GF(q) and write X(x) = xq − x. Let f(x) be a polynomial, having no linear factors, over [q], of positive degree, and write . Consider the continued fraction expansions

and

where the Ai(x) and aj,(x) are polynomials over [q] of degree ≥ 1 (if i ≥ 1, j ≥ 1). Plainly A0(x) = ao(x). Suppose that n = nf is the integer denned uniquely as the largest m such that

In this note we derive an implicit representation of the solution to the problem of plane, inviscid, irrotational flow from a symmetric nozzle of arbitrary wall shape. For the case in which the nozzle wall has a slope which is everywhere much less than unity, we are able to convert this implicit representation into an explicit one in an asymptotic sense (based upon the smallness of the wall slope). Particular attention is paid to the contraction ratio of the jet. This work is complementary to that of Lesser [2] and Larock [l].

Let K, K′ be two centrally symmetric convex bodies in En, with their centres at the origin o. Let Vr denote the r-dimensional volume function. A problem of H. Busemann and C. M. Petty [1], see also, H. Busemann [2] asks:—

“If, for each (n − 1)-dimensional subspace L of En,

does it follow that

If n = 2 or, if K is an ellipsoid, then Busemann [3] shows that it does follow. However we will show that, at least for n ≥ 12, the result does not hold for general centrally symmetric convex bodies K, even if K′ is an ellipsoid. We do not construct the counter example explicitly; instead we use a probabilistic argument to establish its existence.

Davies and Rogers [5] constructed a compact metric space Ω which is singular for a certain Hausdorff measure μh, in the sense that all subsets of Ω have μh-measure zero or infinity and μh(Ω) = ∞. (For a further study of this example see Boardman [3]). The interest lies in its extremely good descriptive character, which was lacking in the earlier examples given by Besicovitch [2] (a plane set singular for linear measure) and Choquet [4] (a plane set singular for any Hausdorff measure for which a segment has positive measure).

Motivated by questions of computational complexity, Rabin [7] introduced the notion of a complete proof of a system of inequalities. His work and the related paper of Spira [8] should interest geometers as well as computer scientists, for both papers involve convexity in an essential way. Spira's results concern the possibility of covering the intersection of a convex set C and a convex polyhedron Q with a finite collection P of polyhedra subject to certain conditions, while in Rabin's work the members of P may be more general than polyhedra. Both papers are interesting and treat important questions, but only Rabin's paper is correct in all respects. The present note contains counterexamples to some of Spira's results and establishes a correct version of one of them.

Let k be a knot of type K and with group πK. Let θ: nK → PSL(ℂ) = PSL (2, ℂ) be a parabolic representation (p-rep) as defined in [14]. We shall call the representation discrete when its image πKθ is a discrete subgroup of PSL(ℂ). It is known that PSL(ℂ) can be identified with the group of orientation preserving isometries of hyperbolic 3-space , and that each discrete subgroup of PSL(ℂ) acts discontinuously on . Hence each discrete p-rep θ has an associated orbit space . The present paper is a study of the general relations between the algebraic properties of a discrete image πKθ and the geometric properties of its orbit space.

Let G be any enumerable subset of the positive real numbers, with infinity as its only limit point. The purpose of this paper is to give a construction for a Lebesgue measurable set E ⊂ R+, with the following properties:

At high frequency, the leading terms in the virtual mass and wavemaking coefficients for a heaving, axisymmetric body depend only on the limit potential (Rhodes-Robinson, 1971). Here this result is applied to closed and open tori and solutions found in closed form. Some numerical values of the coefficients are tabulated.

Let Λ be a lattice in Cn such that the field of Abelian functions on the quotient space Cn/Λ is of transcendence degree n. This implies that is an algebraic extension of a field o of pure transcendence degree n. Thus there exists a vector A = (A1 …, An) of algebraically independent functions of the variable z = (z1, …, zn) and a function B = B(z), algebraic over

such that