Often stones on beaches pounded by waves wear into quite smooth, regular shapes, sometimes apparently ellipsoidal and even spherical [8]. This paper begins with an idealization of this wearing process for materials isotropic with respect to wear, then develops an equation governing the idealized process, and goes on to show that a stone which is initially convex and centrally symmetric tends to assume a spherical shape as a consequence of the governing equation. This conclusion is predicated on the assumption that the mathematical conditions describing the wearing process are those of a well-posed problem.

Asymmetry classes of convex bodies have been introduced and investigated by G. Ewald and G. C. Shephard [2], [3], [6]. These classes are defined as follows. Let denote the set of all convex bodies in n-dimensional Euclidean space ℝn. For K1, K2 ∊ write K1 ∼ K2 if there exist centrally symmetric convex bodies S1, S2 ∊ such that

where + denotes Minkowski addition. Then ∼ is an equivalence relation on and the corresponding classes are called asymmetry classes. The asymmetry class which contains K is denoted by [K].

1. A convex body K in n-dimensional Euclidean space En (nonvoid, compact, and convex subset of En) is uniquely determined, up to translations, by its mixed volumes with other convex bodies. Therefore one might expect that relations between two convex bodies K and L correspond to relations between the mixed volumes of K, L, and other bodies. In this paper we give conditions, formulated in terms of mixed volumes, which are necessary and sufficient for a certain property of decomposability, namely, the property that L is a summand of K, and we solve some related problems.

Abstract. In this paper is proved a formula for the number of neighbourly d-polytopes with d + 3 vertices, when d is odd.

Radon's theorem [8] asserts that, if X is a finite set of s points in Rd and s ≥ d + 2, then X admits a. Radon partition, that is, a partition {X1; X2} of X into disjoint subsets X1 and X2, such that

Blaschke [1] introduced the notion of maximal tetrahedra inscribed in two and three dimensional convex sets (maximal in the sense of volume). From this notion, he derived an inequality relating the volume of such maximal tetrahedra and the volume of the convex set, and used the inequality to characterize an ellipsoid and to obtain some results concerning isoperimetric inequalities.

Introduction. All the sets X considered in this paper are assumed to be compact convex sets in the Euclidean plane. We shall let K denote this class of sets. Problems concerning the division of such sets by three non-concurrent lines have been considered by Eggleston [1] on page 118 and by Grünbaum [2].

In the study of heat transfer to fluids flowing in pipes or channels, the inversion of a Fourier transform requires consideration of the zeros of a certain function. The principal contribution here is a proof that these zeros are all real and simple, properties assumed by previous authors.

Let be a = ℤ-order in A, a finite dimensional Q-algebra. K0() denotes the Grothendieck group of projective right -modules. locally isomorphic to } is a subgroup of K0() and is called the locally free classgroup of . (If = ℤΓ for some finite group Γ then as all ℤΓ projectives are locally free [12].)

Let K3 be a non-Galois cubic extension of the rationals and let K6 be its normal closure. Under K6 there is a unique quadratic field K2. For i = 2, 3, 6 we define C1i; to be the 3-class group of K2 and ri to be the rank of Cli. We suppose that K2 is complex and that K6/K2 is unramified. Our main result is

Let p be an odd prime, let q be a divisor of p – 1, and let α be a primitive q-th root modulo p. For each natural number r the metacyclic group Gr is defined by

Erdoső has asked whether there exists a positive constant c < 1 such that

where pn is the n-th prime. In 1943 Selberg «3» proved, on the assumption of the Riemann Hypothesis, that

uniformly for H > 0. By an elementary extension of Selberg's method we prove the following two results free of any unproved assumptions.

Let f(m;n) denote the largest integer so that, given any m integers a1 < … < am in [1, 2n], one can always choose f integers b1 < … < bf from [1, n], so that bi + bj = a1 (1 ≤ i ≤ j ≤ f; l ≤ l ≤ m) will never hold. Trivially f(m; n) ≥ n/ (m + 1). In this paper we shall attempt to improve upon this trivial bound by exploiting the possible irregularities of distribution of the sequence among certain congruence classes. One of our main results is

provided m ≥ log n. Related questions and results are also discussed.

For some integer modulus q let F be a subset of Z(q), the additive group of residues modulo q. As in «1» we shall write

and

In this paper I prove the following result:

Theorem. Let f(d) be a multiplicative function such that |f(d)| ≤ 1 and ∑{ l/p : p ε P} = ∞, where P denotes the set of primes p for which f{p) = −1, and let v(n) denote the number of distinct prime factors ofn. Then for almost all n,

where A is an arbitrary constant > 3/e.

In this paper it is shown that a simplicial action of Zp (p a prime) on an n-dimensional simplicial complex which is a Poincaré duality space of formal dimension n for Zp coefficients cannot have just one isolated fixed point.

§1. Introduction and Summary. Throughout X is a complete separable metric space. We write K1 for the family of non-empty compact subsets of X. K1 may be endowed with a metric (first introduced by Hausdorff) under which K1 is complete and separable. We shall make use of the subbase for this metrizable topology of K1 given by sets of the two forms

for U open in X (see Kuratowski [4] or E. Michael [9] for a discussion of topologies on the space of subsets of X). if we shall be concerned with sets in [0, 1] × X which are universal for ℋ. To define these let us make the convention that, for D ⊆ [0, 1] × X, we write

D is said to be universal for ℋ if

The stability against small disturbances which do not bend the vortex lines of a circular cylindrical vortex sheet of variable radius is examined. If the jump in tangential velocity across the sheet falls off with time faster than (time)–íthe disturbances are bounded (or grow less rapidly than linearly with time) so that the sheet is effectively stable. If the tangential velocity jump falls off with time slower than (time)–í the sheet is unstable.

The equations governing the evolution of the modulated amplitude of a pointcentred disturbance to a slightly supercritical flow are shown to have solutions which become infinite at a finite time and at a single point. The amplitude develops a sharp peak and the structure of this peak is found, for real and complex coefficients in the governing equations. Such a solution can only occur if the coefficients satisfy certain conditions

Using toroidal co-ordinates, an exact solution is derived for the velocity field induced in two immiscible semi-infinite viscous fluids, possessing a plane interface, by the slow rotation of a concave spherical lens, which is such that the circle of intersection of the composite spherical surfaces lies in the plane of the interface. The expression for the torque acting on the lens is derived, and this is shown to be reducible to an analytic closed form, when the lens degenerates into a spherical bowl and the fluids are identical.