A method of “inner and outer expansions” employed by Cox and Brenner (1967) is used to calculate the hydrodynamic force experienced by either of two identical small solid spheres, approaching each other, at low Reynolds number, with the same velocity along their line of centres, for the limiting case when the gap width between them tends to zero. Numerical results are compared with those furnished by the exact solution and an improved expression is then obtained for a similar problem arising from the motion of a small sphere towards an identical stationary sphere, in the limit of small gap widths.

Multiplications on spheres are studied in [11], [12] from the standpoint of homotopy theory. These multiplications are products with a unit element. The present paper deals with products in general. The investigation involves proving some results on the toric construction and the Whitehead product. These results also lead to theorems about the Stiefel manifold of unit tangent vectors to a sphere, originally proved by M. G. Barratt, which clear up some points in the homotopy theory of sphere bundles over spheres (see [15], [16]). They also enable us to prove that certain of the classical Lie groups are not homotopy-commutative.

The 2-ball property is shown to be transitive. Combining this with some results on the decomposability of convex bodies, we produce new examples of Banach spaces which contain proper semi-M-ideals. These semi-M-ideals are not hyperplanes, nor are they the direct sums of examples which are hyperplanes.

Let

and let p denote any prime. The p-content vp(f) of f is denned by

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of $t$-designs on such spaces are also considered (Theorem 2.3). In particular, it is shown that the optimal $t$-designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).

This note is a continuation of the articles [6] and [2]. In [1], trees with a given partition α = (a1; a2, …), where ai is the number of vertices (points) of valency (degree) i were enumerated. After the determination of the number of plane trees in [2], the number of planted plane trees with a given partition α was found explicitly in [6]. In the present note, the number of plane trees with a given partition is expressed as a function of the number of planted trees with a given partition. The method, which is not new, consists of an application of the enumeration techniques of Otter [3] and Pólya [4]; it was used in [1] and also by Riordan [5].

It is shown that in all dimensions n ≥ 11 there exists a lattice which is generated by its minimal vectors but in which no set of n minimal vectors forms a basis.

The following question of V. Stakhovskii was passed to us by N. Dolbilin [4]. Take the barycentric subdivision of a triangle to obtain six triangles, then take the barycentric subdivision of each of these six triangles and so on; is it true that the resulting collection of triangles is dense (up to similarities) in the space of all triangles? We shall show that it is, but that, nevertheless, the process leads almost surely to a flat triangle (that is, a triangle whose vertices are collinear).

For each algebraic integer α, let ℤα denote the ring of integers of the algebraic number field ℚ(α). There has been continuing interest in finding ring-theoretic conditions characterizing when ℤα coincides with its subring ℤ[α] (cf.[15,18,1,13,12]). One way to extend such work is to consider the intermediate ring ℤ[α]+, the seminormalization (in the sense of [17]) of ℤ[α] in ℤα. Indeed, if we let Iα denote the conductor (ℤ[α]: ℤα), then it is easy to see (cf. Proposition 3.1) that ⅂[α] = ℤα, if, and only if, ℤ[α]+ = ℤα and Iα is a radical ideal of Zα. The condition ℤ[α]+ = ℤα seems worthy of separate attention in view of recent results (cf. [3]) that seminormal rings generated by algebraic integers are “often” automatically of the form ℤα. We show in Proposition 3.3 that the condition ℤ[α]+ = ℤα is equivalent to several universal properties, including notably that the canonical closed surjection Spec (ℤα) → Spec (ℤ[α]) be universally open, be universally going-down, or be a universal homeomorphism.

Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj+1 is a square for 1≤i<j≤4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grinstead [4] gave alternative proofs. The construction of such sets originated with Diophantus who studied the problem when the ai are rational numbers. It is conjectured that there do not exist five positive integers whose pairwise products are all one less than the square of an integer. Recently Dujella [3] proved that there do not exist nine such integers. In this note we address the following related problem. Let V denote the set of pure powers, that is, the set of positive integers of the form xk with x and k positive integers and k>1. How large can a set of positive integers A be if aa′ + 1 is in V whenever a and a′ are distinct integers from A? We expect that there is an absolute bound for |A|, the cardinality of A. While we have not been able to establish this result, we have been able to prove that such sets cannot be very dense.

Let g(n) be a complex valued multiplicative function such that |g(n)| ≤ 1. In this paper we shall be concerned with the validity of the inequality

under the weak condition g(p)∈ for all primes p, where is a fixed subset of the closed unit disc Thus our point of view is similar to that of Halász [Hz 2] in that we seek a general inequality in terms of simple quantities, albeit g(p) may have a quite irregular distribution. We are not concerned here with the problem of asymptotic formulae for the sum on the left of (1) studied by (among others) Delange [D], Halász [Hz 1] and Wirsing [W].

The Stefan problem is a particular free boundary problem for the heat equation which arises in the investigation of the melting of solids. In the case of one space dimension there are numerous results available concerning the existence, uniqueness, and stability of the solution [c.f. 6]. However the case of several space variables is considerably more difficult. This is due in large part to the fact that the geometry of the problem can become quite complicated, and smooth initial and boundary data do not necessarily lead to smooth solutions. In particular, under heating, a connected solid can melt into two (or more) disconnected solids, thus leading to a problem in which the free boundary varies in a discontinuous manner. These difficulties have motivated several researchers to look for “weak” solutions to the Stefan problem [c.f. 1, 3, 5]. Although this approach is quite general and leads to numerical schemes for solving the problem under consideration, there are several drawbacks to this method, among them being the fact that no information is obtained concerning the structure of the interphase boundary, nor is there much information on the regularity of these weak solutions.

We shall deal throughout with regular matrix methods of summability defined by a stochastic matrix A = (ank). Thus

means

If C is a convex open set of a Hausdorff locally convex space, then for every compact set K ⊂ C, the closed convex hull of K is contained in C. It follows that the same is true if C is a countable intersection of convex open sets of E. However, as it is well known (and also follows from §7(a)) a convex of Gδ is not always a countable intersection of open convex sets. This paper is devoted to showing that the Gδ convex sets have some remarkable properties.

Central to the study of electromagnetic induction in turbulently moving conductors is the evolution of the ensemble averaged field, and so the subject of “mean field electrodynamics” has been born. It has provided a particularly fruitful approach to the dynamo problem, that is the study of the self-excitation of magnetic field by a moving fluid. It has been shown by Steenbeck, Krause and Rädler, in studies dating from 1966, that regeneration of field is efficient when the motions are sufficiently vigorous and lack mirror-symmetry. Using a commonly accepted approximation of the subject (the neglect of third order cumulants) which is valid when the correlation length and/or the correlation time of the turbulence are sufficiently small, and paying proper regard to a theorem due to Bochner, it is shown here that mirror-symmetric homogeneous turbulence cannot be regenerative.

Siegel defined an E-function to be a function which admits a power series expansion

with complex coefficients βn, belonging to some algebraic number field K (of finite degree over the rationals Q), satisfying the following conditions:

(i) Given ε > 0, the maximum of the absolute values of the conjugates of βn, satisfies

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

In this paper X(t) denotes Brownian motion on the line 0 ≤ t < ∞, E is a compact subset of (0, ∞) and F a compact subset of ( -∞, ∞). Then δ(E, F) is the supremum of the numbers c such that

In [4, 5] some lower and upper bounds for δ were found in terms of dim E and dim F, and it seemed possible that δ(E, F) could be determined entirely by these constants; this much is false, as the examples in our last paragraph will demonstrate. Here we shall show that δ(E, F) depends on a certain metric character η(E × F). However, η is not calculated relative to the Euclidean metric: the set F must be compressed to compensate for the oscillations of most paths X(t). Fortunately, η(E × F) can be calculated for a large enough class of sets E and F, by means of sequences of integers, to test any conjectures (and disprove most of them). In passing from η to δ we present a slight variation of Frostman's theory [3II].

We prove that, if (fn)n∈ω is a sequence of continuous functions on some recursively presentable Polish space, such that any pointwise cluster point of (fn)n∈ω is a Borel function, then there exists a -subsequence of (fn)n∈ω which is pointwise convergent. This is an effective version of a well known result of Bourgain, Fremlin and Talagrand.

If a scattered compact space K is such that its ω1-th derived set K(ω1) is empty then the Banach space ℒ(K) admits an equivalent locally uniformly convex norm.