Introduction. This paper is concerned with an interpretation of some experiments on certain axisymmetric inertial oscillations of a rotating fluid in a thick spherical shell. These oscillations are the counterparts to the ones reported by Aldridge and Toomre† (1969) for a full sphere of rotating fluid. A description of the arrangements for both experiments is presented in A & T; some of the measured eigen-frequencies for spherical shells of rotating fluid given by Aldridge (1967) are presented here for comparison with values calculated from a variational principle.

In the present paper the researches initiated in the two earlier papers of this series are continued, and, by suitable generalizations of the techniques employed therein, solutions are obtained to some further well known problems from the theory of transcendental numbers. It will be proved, for example, that a non-vanishing linear form, with algebraic coefficients, in the logarithms of algebraic numbers, cannot be algebraic. This implies, in particular, that π + log α is transcendental for any algebraic number α ≠ 0, and also eα π+ß is transcendental for all algebraic numbers α, β with β ≠ 0.

The volume of the Lp-centroid body of a convex body K ⊂ ℝd is a convex function of a time-like parameter when each chord of K parallel to a fixed direction moves with constant speed. This fact is used to study extrema of some affine invariant functionals involving the volume of the Lp-centroid body and related to classical open problems like the slicing problem. Some variants of the Lp-Busemann-Petty centroid inequality are established. The reverse form of these inequalities is proved in the two-dimensional case.

Let Pn (n ≥ 0) be an n-polytope, that is, a convex polytope in n-dimensional euclidean space (Grünbaum [5], 3.1), and for 0 ≤ j ≤ n − 1 let be its j-faces. If Pn itself and Ø (the empty set) are also allowed to be faces of Pn, of dimensions n and − 1 respectively, then the set of faces of Pn forms a lattice partially ordered by inclusion ([5], 3.2). Two polytopes P1n and P2n are said to be combinatorially isomorphic, or of the same combinatorial type if their respective lattices of faces are isomorphic; that is, if there is a one–to–one correspondence between the set of faces of P1n and the set of faces of P2n which preserves the relation of inclusion ([5], 3.2). Similarly, any permutation of the set of faces of Pn which preserves inclusion will be called a (combinatorial) automorphism; it is clear that the set of automorphisms of Pn forms a group Γ(Pn), called the automorphism group of Pn.

Analytical and numerical properties are described for the free interaction and separation arising when the induced pressure and local displacement are equal, in reduced terms, for large Reynolds number flow. The interaction, known to apply to hypersonic flow, is shown to have possible relevance also to the origins of supercritical (Froude number > 1) hydraulic jumps in liquid layers flowing along horizontal walls. The main theoretical task is to obtain the ultimate behaviour far beyond the separation. An unusual structure is found to emerge there, involving a backward–moving wall layer with algebraically growing velocity at its outer edge, detached shear layer moving forward and, in between, reversed inertial flow uninfluenced directly by the adverse pressure gradient. As a result the pressure then increases like (distance)m, with m = 2(√(7)–2)/3 ( = 0.43050 …), and does not approach a plateau. Some more general properties of (Falkner–Skan) boundary layers with algebraic growth are also described.

About twenty years ago, in a note of the same title [2], I obtained the following result.

THEOREM 1. Let u and v be relatively prime integers satisfying u> v ≥ 2 and let ε be an arbitrarily small positive number. Suppose the inequality

is satisfied by an infinite sequence of positive integers n1 n2, … Then

Certain results ([7], [8], [10], [11]) suggest that there should be some principle of duality in homotopy theory. Among other things one is led to expect that cohomotopy groups will appear as dual to homotopy groups. But the fact that a cohomotopy group πn(X), unlike πn(X), is only defined if dim X ≤ 2n—2 is a serious obstacle to the formulation of such a principle. However, the set of S-maps (i.e.S-homotopy classes [11]) X → Y is a group for every pair of spaces X, Y. Therefore, this difficulty does not appear in S-theory [11].

Throughout this note the letters D and K denote a commutative integral domain with 1 and its field of fractions.

It was proved in a recent paper† that if au α1, …, αn denote non-zero algebraic numbers and if‡ logαn, …, log αn and and 2πi are linearly independent over the rationals then log α1, …, log αn are linearly independent over the field of all algebraic numbers. Further it was shown that if α1 …, αn are positive real algebraic numbers other than 1 and if β1, …, βn denote real algebraic numbers with 1, β1 …, βn linearly independent over the rationals then is transcendental.

We obtain lower bounds of the correct order of magnitude for the 2kth moment of the Riemann zeta function for all k≥1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the rational number k. Our new bounds are continuous in k, and thus extend also to the case when k is irrational. The method is a refinement of an approach of Rudnick and Soundararajan, and applies also to moments of L-functions in families.

If П is a k-dimensional vector subspace of Rn and E is a subset of Rn, let projп(E) denote the orthogonal projection of E onto П. Marstrand [8] and Kaufman [6] have developed results on the Hausdorff dimension and measure of projп(E) in terms of the dimension of E, leading to the very general theory of Mattila [11]. In particular, Mattila shows that if the Hausdorff dimension dim E of the Souslin set E is greater than k, then projп(E) has positive k-dimensional Lebesgue measure for almost all П ∈ Gn, k (in the sense of the usual normalized invariant measure on the Grassmann manifold Gn, k of k-dimensional subspaces of Rn).

The forms under discussion are integral positive definite quadratic forms in three variables. Such a form g is called regular if g represents every integer represented by the genus of g. This can be recast in elementary terms: g is regular if the solvability of g≡a (mod n) for every n implies the solvability of g = a.

Let K be a n-dimensional convex body. A lattice Λ will be called a covering lattice for K (or simply a covering lattice), if each point of space has at least one representation in the form k + g where k ε K and g ε Λ. The density ϑ(K, Λ) of the resultant covering of the whole of space by the bodies of the form K + g with g ε Λ is, quite naturally, defined to be the ratio V(K)/d(Λ), where V(K) is the volume of K and d(Λ) is the determinant of Λ (i.e. the volume of its fundamental parallepipeds). Clearly this density is at least 1. A number of authors, in particular H. Davenport [3], R. P. Bambah and K. F. Roth [2], G. L. Watson [8], C. A. Rogers [4], [5] and W. Schmidt [7], have either constructed or proved the existence of lattices for which the density ϑ(K, Λ) is reasonably small. But, when n is large, all the densities obtained are of the form cn where c is a constant greater than 1. The object of this note is to obtain some rather stronger results. In the general case we prove the existence of covering lattices Λ with

Where

as n→∞.

The distribution of squarefree binomial coefficients. For many years, Paul Erdős has asked intriguing questions concerning the prime divisors of binomial coefficients, and the powers to which they appear. It is evident that, if k is not too small, then must be highly composite in that it contains many prime factors and often to high powers. It is therefore of interest to enquire as to how infrequently is squarefree. One well-known conjecture, due to Erdős, is that is not squarefree once n > 4. Sarközy [Sz] proved this for sufficiently large n but here we return to and solve the original question.

The purpose of this paper is to take some first steps the investigation of the negative moments

where k>0 and 12, and the related discrete moments

where runs over the complex zeros of the zeta-function. We assume the Riemann hypothesis (RH) throughout; it then follows that Ik(, T) converges for every k > 0 when > but for no k = when =. We further note that Jk(T) is only defined for all T if all the zeros are simple and, in that case, Ik(, T) converges for all k<.

Let Q be a local ring and let q be an m-primary ideal of Q, where m is the maximal ideal of Q. With q we may associate a ring F(Q, q), termed the form ring of Q relative to the ideal q. If u1, …, um is a basis of q, and if B denotes the quotient ring Q/q, there is a homomorphism of the ring B[X1, …, Xm] of polynomials over B in indeterminates X1 …, Xm onto F(Q, q). The kernel of this homomorphism is a homogeneous ideal of B[X1 …, Xm]. Finally, if a is an ideal of Q there is a homomorphism of F(Q, q) onto F(Q/a, q+a/a). The kernel of this latter homomorphism will be termed the form ideal relative to q of a and denoted by ā.

Recent work on the mechanics of interacting continua has led to the formulation of linearized theories governing thermo-mechanical disturbances of small amplitude in mixtures of an elastic solid and a viscous fluid and of two elastic solids. These theories are well posed in the crude sense that the number of field and constitutive equations equals the number of field quantities to be determined, but the proper posing of initial and initial-boundary-value problems has not been studied. In this paper we prove, for both types of mixtures, the uniqueness of sufficiently smooth solutions of the field and constitutive equations subject to initial and boundary data which include conditions of direct physical significance. Sufficient conditions for uniqueness are given in the form of inequalities on the material constants appearing in the constitutive equations. These restrictions fall into two categories, one arising from the application of the entropy-production inequality, and therefore intrinsic to the theory, and the other representing constraints on the Helmholtz free energy of the mixture. Our results include as special cases uniqueness theorems for unsteady linearized compressible flow of a heat-conducting viscous fluid and for the linear theory of thermoelasticity.

In this note we give an answer to the following problem of Todorcevic: Find out the combinatorial essence behind the fact that the family ℋ of the ground-model infinite sets of integers in a Perfect-set forcing extension has the property that for any Borel f: [ℕ]ω → {0, 1} there exists an A ∈ ℋ such that f is constant on [A]ω (see [7], [13]). In other words, one needs to capture the combinatorial properties of the family ℋ of ground-model subsets of ℕ which assure that it diagonalizes all Borel partitions. It turns out that the notion which results from our analysis of this problem is a bit more optimal than the older notion of a “happy family” (or selective coideal) introduced by A.R.D. Mathias [16] long ago in order to extend the well-known theorems of Galvin–Prikry [6] and Silver [25] (see Theorems 3.1 and 4.1 below). We should remark that these Mathias-style extensions can indeed be as useful in the applications as the original partition theorems.

The stability of a two-dimensional vortex sheet against small disturbances in the plane of flow is examined. An integro-differential equation for the disturbances is derived and the possibility of solving it approximately is discussed. The approximation is equivalent to saying that short waves grow in a fashion determined by the local strength of the vortex sheet and it is shown that this need not be true throughout the evolution of the disturbance unless the growth rate is the same everywhere. It is possible for the disturbance on a distant part of the vortex sheet to control what happens locally, if the disturbance on the distant part is growing more rapidly. The approximate theory is applied to the tightly-wound spirals of aerodynamic interest and these are shown to be stable.

Let be a (finite) non-empty sequence of integers, and let K be a positive integer. The aim of the Brun-Selberg sieve is to obtain bounds for the “sifting function”

where z ≥ 2 is a real number and