A Banach space is c0-saturated if all of its closed infinite-dimensional subspaces contain an isomorph of c0. In this article, we study the stability of this property under the formation of direct sums and tensor products. Some of the results are: (1) a slightly more general version of the fact that c0-sums of c0-saturated spaces are c0-saturated; (2) C(K, E) is c0-saturated if both C(K) and E are; (3) the tensor product is c0-saturated, where JH is the James Hagler space

1. Introduction. Let X be a space and let E be a fibre space over E. A fibre-preserving map f: E → E determines, for each point x ∈ X, a map fx: Ex → Ex of the fibre over x. In a previous note (3) the situation was considered where fx is null-homotopic, for all x. In the present note we turn our attention to the situation where fx is homotopic to the identity on Ex, for all x ∈ X. If X admits a numerable categorical covering (as when X is an ANR) then such a fibre-preserving map f is a fibre homotopy equivalence, by the well-known theorem of Dold(1). Then the set Φ1(E) of fibre homotopy classes of such maps forms a normal subgroup of the group Φ*(E) of fibre homotopy classes of fibre homotopy equivalences. The purpose of this note is to prove

Theorem 1.1. Let X be a paracompact space of finite category. Let E be a fibre bundle over X of which the fibres are compact and path-connected ANR's. Then the Φ*(E)-growp Φ1(E) is Φ*(E)-nilpotent of class ≤ cat X.

In this paper we derive the Lorentz-Dirac equation of motion for a charged particle moving in an external electromagnetic field. We use Maxwell's electromagnetic field equations together with the assumptions (1) that all fields are retarded and (2) that the 4-force acting on the charged particle is a Lorentz 4-force. To define the self-field on the world-line of the charge we utilize a contour integral representation for the field due to A. W. Conway. This by-passes the need to define an ‘average field’. In an appendix the case of a scalar field is briefly discussed.

Approximating sequences have been extensively studied in many branches of mathematics, for example, in number theory (approximating real numbers by rationals) and in numerical analysis (approximations to functions by polynomials). In (1), A. Hyllengren introduced a type of approximating sequence ‘majorizing sequences’ which he used in solving a problem in complex analysis. In this note we study a very similar concept, which is general enough to be applicable to any separable metric space, and which turns out to have strong connexions with the theory of Hausdorff measures (as did Hyllengren's majorizing sequences).

Suppose throughout that {pn} is a sequence of non-negative numbers with p0 > 0, that

and that

Let {sn} be a sequence of real numbers.

Let 1 ≤ p < ∞ and let ω be a non-negative function defined on the unit circle T which satisfies the Ap condition of Muckenhoupt. The weighted Hardy space Hp(ω) consists of those functions f in the classical Hardy space H1 whose boundary values belong to Lp(ω). Recently McPhail (Studia Math. 96, 1990) has characterized those positive Borel measures μ on the unit disc Δ for which Hp(ω) is continuously contained in Lp(dμ). In this paper we study the question of finding necessary and sufficient conditions on a positive Borel measure μ on Δ for the differentiation operator D defined by Df = f′ to map Hp(ω) continuously into Lp(dμ). We prove that a necessary condition is that there exists a positive constant C such that

where for any interval I ⊂ T,

We prove that this condition is also sufficient in some cases, namely for 2 ≤ p < ∞ and ω(et0) = |θ|α, (|θ| ≤ π), – 1 < α < p – 1, but not in general. In the general case we prove the sufficiency of a condition which is slightly stronger than (A).

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:

Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that

(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;

(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

The rotation of a dielectric covered body, suspended between the poles of a Wimshurst machine, is investigated and found to be due to the accumulation of charge on the surface of the body, under the action of the brush discharge, and to the subsequent electrostatic repulsion of this charge. It is shown theoretically, in an appendix, that the rate of rotation is governed by the rate of decay of the surface charge, and is a maximum for a certain value of this rate of decay. The rotation of a chain hanging from an electrical machine into the dielectric layer of a Leyden jar (the inner conducting coating being removed) and similar electrostatic experiments are described, and shown to be due also to the superficial charging of a dielectric body, body, and depends upon the rate of decay of this surface charge. A similar explanation is shown to account for some other electrostatic phenomena.

A Hermitian form q on the dual space, *, of the Lie algebra, , of a simply connected complex Lie group, G, determines a sub-Laplacian, Δ, on G. Assuming Hörmander's condition for hypoellipticity, there is a smooth heat kernel measure, ρt, on G associated to etΔ/4. In a companion paper [6], we proved the existence of a unitary “Taylor” map from the space of holomorphic functions in L2(G, ρt) onto Jt0 (a subspace of) the dual of the universal enveloping algebra of . Here we give a very different proof of the surjectivity of the Taylor map under the assumption that G is nilpotent. This proof provides further insight into the structure of the Taylor map. In particular we show that the finite rank tensors are dense in Jt0 when the Lie algebra is graded and the Laplacian is adapted to the gradation. We also show how the Fourier–Wigner transform produces a natural family of holomorphic functions in L2(G, ρt), for appropriate t, when G is the complex Heisenberg group.

We work throughout in the smooth category. Homomorphisms of fundamental and homology groups are induced by inclusion. An n-knot, form n ≥ 1, is an embedded n-sphere K ⊂ Sn+2. A Seifert manifold for K is a compact, connected, orientable (n + 1)-manifold V ⊂ Sn+2 with boundary ∂V = K. By [9] Seifert manifolds always exist. As in [9] let Y denote Sn+2 split along V; Y is a compact manifold with ∂Y = V0 ∪ V1, where Vt ≈ V. We say that V is a minimal Seifert manifold for K if π1Vt → π1Y is a monomorphism for t = 0, 1. (Here and throughout basepoint considerations are suppressed.)

Wave-front singularities for the displacement functions, associated with the radiation of linear elastic waves from a point source embedded in a finitely strained two-dimensional elastic solid, are examined in detail. It is found that generally the singularities are of order d–½ where d measures distance away from the front. However, in certain exceptional cases singularities of order d–-n where n = ¼, ⅔, ¾, may be encountered.

The purpose of the present note is to prove the following two theorems:

Theorem 1. Let Q be an equicharacteristic local domain with maximal ideal m. Let a be any ideal of Q. Then the intersection of all integrally closed m-primary ideals of Q which contain a is the integral closure ā of a.

Theorem 2. If Q is as above, and if S denotes the set of valuations on the field of fractions F of Q which are associated with Q, then the intersection of the valuation rings belonging to valuations in S is the integral closure of Q.

Complex variable methods are applied to derive exact solutions in closed forms for the small deflexions of uniformly loaded thin isotropic plates bounded by regular curvilinear polygonal contours having n sides. The supported boundary is either clamped or has equal boundary cross-couples. For rectilinear edges the latter conditions agree with those for a simply supported boundary. The plates taken in the z-plane are conformally mapped on the unit circle in the ζ-plane by the general mapping function

In this paper, we construct Wakimoto modules for basic affine Lie superalgebras of type A(m−1, n−1)(1) and D(2, 1, a)(1). As an application, we compute the characters of irreducible highest weight modules at the critical level.

For numbers of increasing real functions f(x) with new integral inequalities. They generalize classical results. The proofs are short and simple being based on sequences.

1. The method generally employed in investigating the angular scattering of electrons in gases consists essentially of firing a directed beam of electrons into a “field-free” enclosure, containing the gas to be investigated at a pressure of a few hundredths or thousandths of a millimetre. The electrons in the beam will be scattered, on colliding with the gas molecules, through various angles. Those which are scattered between θ and θ + dθ are selected by means of a pair of slits which can be set at any desired angle to the electron beam. The electrons selected by these slits are then resolved by means of retarding potentials or electric or magnetic fields into those which have been elastically scattered and those which, on being scattered by the molecule, have excited or ionised it. By plotting the final current received by the Faraday cylinder against θ, we can obtain the angular scattering curve for the elastically or inelastically reflected electrons.

Clifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in previous papers by the authors. In this paper, we establish lower bounds for the Clifford indices for rank 3 bundles. As a consequence we show that, on smooth plane curves of degree at least 10, there exist non-generated bundles of rank 3 computing one of the Clifford indices.

On the asymptotic probability distribution for certain Markoff processes 46 (1950), 581.

By Walter Ledermann

Mr H. Schneider has drawn my attention to a mistake which occurs on p. 591 of the above paper.

The Stone-Weierstrass theorem gives very simple necessary and sufficient conditions for a subset A of the algebra of all real-valued continuous functions on the compact Hausdorff space X to generate a subalgebra dense in namely, this is so if and only if the functions of A strongly separate the points of X, in other words given any two distinct points of X there exists a function in A taking different values at these points, and given any point of X there exists a function in A non-zero there. In the case of the algebra of all complex-valued continuous functions on X, the same result holds provided that we consider the subalgebra generated by A together with Ā, the set of complex conjugates of the functions in A.

The problem of residual intersections is one of considerable importance in algebraic geometry. In its most general form the problem may be stated as follows. On a given variety V of d dimensions two varieties A and B, of respective dimensions k and k′, pass through a variety C whose dimension r is not less than r′ = k + k′ – d. It is required to determine the variety D of dimension r′ which forms the residual intersection of A and B. The classical paper on this subject is that of Severi*. He considers the case in which V is a linear space, and obtains a large variety of enumerative results connecting the characters of the residual intersection with those of the given loci.