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

where k>0 and12, and the related discrete moments

whereruns 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 zerosare simple and, in that case, Ik(, T) converges for all k<.

Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary, a new Brunn-Minkowski inequality is obtained for the volume of polar projection bodies.

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.

It has been known for many years that a laminar boundary layer adjoining a supersonic stream is in principle capable of spontaneously evolving from an unseparated to a separated form in which the boundary layer is detached from the wall and in between a slowly moving reversed flow is set up. The phenomenon arises because any thickening of the boundary layer tends to induce an adverse pressure gradient which in turn accelerates the thickening of the boundary layer and hence its evolution from the normal (Blasius) form.

The stability of a non-rotating column of liquid was discussed by Rayleigh [1] who showed that axisymmetric disturbances of wave-lengths greater than the circumference of the column decreased the surface area and the consequent release of surface energy enabled the disturbance to grow. Rayleigh [2] also considered the effect of viscosity on the same problem and found that for high viscosity the range of stability was unaltered. For disturbances which are the same in all planes perpendicular to the axis of the column, it is clear that any disturbance increases the length of the boundary of a plane section of the column and hence increases the surface energy. Thus the column is stable for these disturbances.

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.

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.

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.

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

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 main result concerning the stability of an inviscid liquid column, which is at rest, is due to Rayleigh [1], who showed that, taking account of capillarity at the surface, the column will be unstable to small axisymmetric disturbances whose wavelengths in the axial direction are greater than the circumference of the cross-section. The reason for such instabilities is simply that disturbances of these wavelengths decrease the surface area of the column and hence make available excess surface energy which goes into building up the disturbance. The effects of rotation on the stability of a column having a free surface do not appear to have been studied and this note establishes some simple results concerning the effect of plane two-dimensional disturbances on a rotating column. With regard to the stability of rotating fluids in general there is another result due to Rayleigh [2] stating that the fluid will be unstable if the numerical value of the circulation decreases with the radius at any point. But this result is again for axisymmetric disturbances, and hence does not necessarily bear any relationship to the results for plane disturbances which, in the case of an incompressible liquid as is assumed here, cannot be axisymmetric. A result more closely related to this work is that of Kelvin [3] who considered the stability of a column of uniform vorticity in a fluid otherwise free of vorticity. Such a column is stable to two-dimensional and to three-dimensional disturbances.

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.

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.

It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies Kt, is a convex function of t, thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santaló inequality for polytopes in ℝd that have at most d + 3 vertices.

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.

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.

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

In 1953 Erdős† proved in a characteristically ingenious manner that an irreducible‡; integral polynomial f(n) of degree r ≥ 3 represents (r – 1)-free integers (that is to say integers not divisible by an (r – 1)th power other than 1) infinitely often, provided that the obvious necessary condition be given that f(n) have no fixed (r – 1)th power divisors other than 1. His method did not, however, give a means for determining an asymptotic formula for N(x), the number of positive integers n not exceeding x for which f(n) is (r – 1)-free, nor did it even show that the positive integers n for which f(n) is (r – 1)-free had a positive lower density.

A little over a hundred years ago E. B. Christoffel in [6] asserted a proposition concerning the determination of a surface in Euclidean 3-space from a specification of its mean radius of curvature as a function of the outer normal direction. In that paper an assumption was made which limited the class of surfaces considered to be boundaries of convex bodies. The argument rested on the construction of functions describing the co-ordinates of surface points corresponding to outer normal directions as solutions of certain partial differential equations involving the mean radius of curvature. However, it was pointed out by A. D. Alexandrov [1], [2], that the conditions laid down by Christoffel on the preassigned mean radius of curvature function were not sufficient to ensure that that function actually be a mean radius of curvature function of a closed convex surface. Hence Christofrel's discussion is incomplete. Different and similarly incomplete treatments of Christoffelés problem were given by A. Hurwitz [9], D. Hilbert [8], T. Kubota [11], J. Favard [7], and W. Suss [13]. A succinct discussion of the question is in Busemann [5] but the footnote on p. 68, intended to rectify the discussion in Bonnesen and Fenchel [4], is also not correct. A sufficient, but not necessary condition was found by A. V. Pogorelov in [12].

In this paper we study various classes of centrally symmetric sets in d-dimensional Euclidean space Rd. As we will see, it is appropriate to focus our attention on those sets which have interior points.