We consider the spectral function, ρα(μ), for –∞<μ<∞ associated with the Sturm-Liouville equation

and the boundary condition

We suppose that q is a real-valued member of L1[0, ∞) and λ is a real parameter.

I discuss various necessary and sufficient conditions for a K-analytic space to be Souslin. In particular, I show that if the continuum hypothesis is true, then there is a non-Souslin K-analytic space in which every compact set is metrizable; while if Martin's Axiom is true and the continuum hypothesis is false, this is impossible.

Given ξ in [0, 1] let ξ = [0, a1, a2…] denote a simple continued fraction expansion of ξ Given the expansion ξ = [0, a1, a2, …] let

Thus with the conventions p1 = q0 = 1 and q−1 = p0 = 0 we have

§5. Some C(K) spaces that are not σ-fragmented. In [16] we remarked that the space (l∞, weak) is not σ-fragmented. Rosenthal [19] gives a number of conditions that ensure that a C(K) space contains an isomorphic copy of l∞. A Banach space is said to be injective if it is complemented in every Banach space containing it isomorphically. Rosenthal proves that every infinite dimensional injective Banach space contains a subspace isomorphic to l∞. It is known (see, for example, Day [2]) that C(K) is injective if K is extremally disconnected. Thus C(K) is not σ-fragmented when K is an infinite compact Hausdorff space that is extremally disconnected.

An origin-symmetric convex body K in ℝn is called an intersection body if its radial function ρK is the spherical Radon transform of a non-negative measure µ on the unit sphere Sn−1. When µ is a positive continuous function, K is called the intersection body of a star body. The notion of intersection body was introduced by Lutwak [L]. It played a key role in the solution of the Busemann-Petty problem, see [G1], [G2], [L], [Z1] and [Z2]. Koldobsky [K] showed that the cross-polytope is an intersection body. This indicates that the statement in [Z3] that no origin-symmetric convex polytope in ℝn (n > 3) is an intersection body is not correct. This paper will prove the weaker statement that no origin-symmetric convex polytope in ℝn (n > 3) is the intersection body of a star body.

In earlier treatments of the title problem it was found that it is impossible, in general, to obtain solutions of Stokes's equations of slow viscous flow in which the fluid velocity vanishes at infinity. It is shown here that the paradox can be resolved by the introduction of a resultant force on the cylinders. This enables the solution to be matched to an outer solution of the full Navier-Stokes equations.

Existence and uniqueness of classical solutions are established for the dissipative quasigeostrophic equations of geophysical fluid dynamics, using a priori estimates and a Schauder fixed point theorem. The flow is periodic in both horizontal directions and is bounded above and below by rigid flat surfaces. The Reynolds analogy of unit turbulent Prandtl number is assumed. Existence is proved for an arbitrary finite time, if it is further assumed that the surface temperatures vanish. Without this additional assumption existence is guaranteed only for a certain finite time, which is inversely proportional to the norms of the sources and initial conditions.

Let K be an arbitrary compact space and C(K) the space of continuous functions on K endowed with its natural supremum norm. We show that for any subset B of the unit sphere of C(K)* on which every function of C(K) attains its norm, a bounded subset A of C(K) is weakly compact if, and only if, it is compact for the topology tp(B) of pointwise convergence on B. It is also shown that this result can be extended to a large class of Banach spaces, which contains, for instance, all uniform algebras. Moreover we prove that the space (C(K), tp(B)) is an angelic space in the sense of D. H. Fremlin.

Given an integral lattice L and a hyperbolic decomposition of some quotient L/pL, there is a simple technique for obtaining other lattices of the same dimension and discriminant as L⊥ … ⊥L. When applied to the D4 and E8 root lattices, for example, this yields a new sphere packing in ℝ32, which is denser than those known up to now, and an extremal type II lattice in ℝ64.

Minkowski [1] first proved that the surface of a convex body in E3 can be approximated by a level surface of a convex analytic function. His proof is strikingly simple. His proof is also presented for En by Bonnesen-Fenchel [2, pp. 10–12[. We here prove that the same kind of result is achieved using level surfaces of convex non-negative polynomials. We give two types of approximation, one based on finite sums as Minkowski did, and the other using integration. Since these approximations may be used for other applications we also extend them and give special formulae when the surface is centrally symmetric.

For each index β in a set J let Aβ be a group. The free product

is by definition a group generated by the Aβ in which any two distinct reduced words represent distinct elements. Here a reduced word is either the expression 1 or an expression

with each “syllable” ai ≠ 1 and each βi ≠ βi+1.

Problems about partitions of cartesian products are common in mathematics. For example, in finite and infinite combinatorics, they keep emerging in Ramsay theory, where one seeks to show that, if a product is partitioned into finitely many parts, one part at least must contain a subset of a certain specified kind. In the transition from finite to infinite products, one usually imposes restrictions of a topological nature on the partition, in order to obtain theorems analogous to those which are valid in the finite case. (See, e.g., [G-P], [Si], [E].)

It is our purpose here to show that, using results already in the literature, it is easy to prove the following and similar theorems.

For every positive integer d, there exists an integer Ψ (d) such that if K is an algebraic number field of degree d over the field of rational numbers then every cubic form f(x1 x2, …, xn) over K, with n ≥ Ψ(d), has a non-trivial zero in K.

We use the Buchstab-Rosser sieve to derive an asymptotic formula for the distribution of those integers n for which the r numbers n + l1, n + l2,…, n + lr are all square-free. Our error estimate sharpens a similar result of Hall and is uniform in both r and maxl 1≤i≤r|li|.

Consider a homogeneous linear viscoelastic filament, one of whose ends is held fixed, while the other is subjected to a given traction τ(t) (t ≤ 0). I shall prove first that the total energy E(t), which is the sum of the kinetic and free energies, its derivative E′(t), and the rate of working W(t) satisfy

and

The purpose of this work is to investigate the relationship between Radon transforms and centrally symmetric convex bodies. Because of the injectivity properties of the Radon transform it is natural to consider transforms on the sphere separately from those on the higher order Grassmannians. Here we shall concentrate on the latter, whilst the former will be the subject of another article presently in preparation, Goodey and Weil [1991].

Marshall Hall has proved that every real number is representable as the sum of two continued fractions with partial quotients at most 4. This implies that for any real β1, β2 there exists a real α such that

for all integers x > 0 and y, where C is a positive constant. In this note I prove a generalization to r numbers β2, …, βr. The case r = 2 implies a result similar to Marshall Hall's but with a larger number (71) in place of 4.

In [3], D. H. Lehmer has analysed the incomplete Gaussian sum

where N and q are positive integers with N < q and e(x) is an abbreviation for e2πix. The crucial observation is that, for almost all values of N, Gq(N) is in the vicinity of the point ¼(1 + i)q1/2. This leads to sharp estimates of the shape Gq(N) = O(q½).

The problem of estimating accurately the order of magnitude of the least primitive root g(p) to a large prime modulus p is as yet unsolved. The first non-trivial estimate was obtained by I. M. Vinogradov (see [5]) who in about 1919 showed that

Let f = f(x, y, z) be a positive definite form of the type

where x, y, z are integral valued variables, and the coefficients a, …, t are integers whose highest common factor is 1. As the determinant of such a form may be fractional, I define

and

thus — C is the discriminant of the binary form f(x, y, 0), and the necessary and sufficient condition for f to be positive definite is that a > 0, C > 0, and d > 0.