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.

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 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

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

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.

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.

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

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

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

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.

§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

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.

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 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.

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].

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.