A class of approximate stationary vacuum solutions are obtained by expanding the metric in powers of a certain parameter, and by solving explicitly the first few orders in terms of two harmonic functions ø and ψ. These solutions, to the order considered, reduce to the Weyl solutions when ψ = 0 and yield the Papapetrou solutions when ø = 0, and contain a subclass that is asymptotically flat and has non-zero mass. The physical interpretation and the possibility of proceeding further are briefly discussed.

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:

Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

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

A general theory of ‘strength’ for Hausdorff methods T of summability was given in the first part of this paper, to which we shall refer in the following as I. Those methods T which are regular (that is, at least as strong as ordinary convergence C), are defined by the linear transforms

of a given sequence (sk). Here ø(t) (that is, each of its real and imaginary parts) is of bounded variation (b.v.) in the closed interval 〈0, 1〉 [see I, (4)]. The additional condition for consistency (with C) is that ø(t) is continuous at t = 0, and that ø(1) − ø(0) = 1 [I, H. 3]. We write also T ˜ ø(t).

Let X be a complex normed space with dual space X′ and let T be a bounded linear operator on X. The numerical range of T is defined as

and the numerical radius is v(T) = sup {|ν: νε V(T)}. Most known results and problems concerning numerical range can be found in the notes by Bonsall and Duncan (5).

We investigate four properties related to an elliptic curve Et in Legendre form with parameter t: the curve Et has complex multiplication, E−t has complex multiplication, a point on Et with abscissa 2 is of finite order, and t is a root of unity. Combining all pairs of properties leads to six problems on unlikely intersections. Using a variety of techniques we solve these problems with varying degrees of effectivity (and for three of them we even present the list of all possible t).

We present a closed hyperbolic 3-manifold M with some surprising properties. The universal covering group of M is a normal torsion-free subgroup of minimal index in one of the nine Coxeter groups G, generated by the reflections in the faces of one of the nine Lannér-tetrahedra (bounded tetrahedra in hyperbolic 3-space all of whose dihedral angles are of the form π/n with n ∈ ℕ see [1] or [3]). The corresponding Coxeter group G splits as a semidirect product G = π1M⋉A, where A is a finite subgroup of G, and G is the only one of the nine Coxeter groups associated to the Lannér-tetrahedra which admits such a splitting (this follows using results in [4]). We derive a presentation of π1M and show that the first homology group H1(M) of M is isomorphic to ℚ11. This is in sharp contrast to other torsion-free (non-normal) subgroups of finite index in Coxeter groups constructed in [1] which all have finite first homology (though it is known that they are all virtually ℚ-representable (see [5], p. 434). It follows from our computations that the Heegaard genus of M is 11, and that there exists a Heegaard splitting of M of genus 11 invariant under the action of the group I+(M) ≌ S5 ⊕ ℚ2 of orientation-preserving isometries of M (we compute this group in [4]), so that the Heegaard genus of M is equal to the equivariant Heegaard genus of the action of I+(M) on M. Moreover M is maximally symmetric in the sense of [4, 6]: the order 120 of the subgroup of index 2 in I+(M) which preserves both handle-bodies of the Heegaard splitting is the maximal possible order of a group of orientation-preserving diffeomorphisms of a handle-body of genus 11. (This maximal order is 12(g—1) for a handle-body of genus g; see [7].) By taking the coverings Mq of M corresponding to the surjections π1M→H1(M) ≌ ℚ11→(ℚq)11 for q ∈ ℕ, we obtain explicitly an infinite series of maximally symmetric hyperbolic 3-manifolds.

Cusp singularities were introduced and described in detail in Hirzebruch's fundamental paper [3] (se recall some of the basic results in § 1 below). They form a natural and well-behaved class, included in Laufer's ‘minimally elliptic’ singularities [5]. Those which occur on hypersurfaces appear also as hyperbolic singularities in Arnold's [1] classification of 1-modal singularities.

It is possible to obtain approximate theoretical relations between the spectra of different atoms ionised to such an extent that they possess the same electron structure. In this paper these relations are used to extrapolate values of the terms of the spectra of various lithium-like and sodium-like atoms.

From the terms so estimated lines can be calculated, and in many cases correlated with lines observed by Millikan and Bowen in the hot spark spectrum of the corresponding element. Revised term values can then be calculated from the lines so identified.

The spectra of lithium-like atoms treated in this way include those from Be II up to O VI, and those of sodium-like atoms, the spectra P V and S VI.

It has recently been pointed out by Heitler(1) that the well-known discrepancy between the theoretical expression and the experimental results for the cross-section of scattering of charged mesons by nuclear particles can be removed by a proper consideration of the effect of radiation damping in the quantum theory. The radiation damping in quantum theory was first considered in complete detail for free electrons by Waller(2). A rigorous deduction of the integral equation set up by Waller was given by Heitler on the basis of a method developed by Góra. An alternative rigorous derivation of the integral equation has also been given by Wilson(3). Exact solutions of the integral equation for the simple scattering of mesons by nuclear particles have been found by Heitler in the non-relativistic approximation. An exact solution has not so far been given for physical problems in which the integral equations are complicated, but Wilson has given a general approximate formula for the scattering cross-section, which should be valid for all problems.

MacLane cohomology is an algebraic version of the topological Hochschild cohomology. Based on the computation of the third author (see Appendix) we obtain an interpretation of the third Mac Lane cohomology of rings using certain kind of crossed extensions of rings in the quadratic world. Actually we obtain two such interpretations corresponding to the two monoidal structures on the category of square groups.

We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ℝd is called hollow if it has a bounded complementary component. We show that for each d ⩾ 2 there exists a quasiregular map of transcendental type f: ℝd → ℝd with a quasi-Fatou component which is hollow.

Suppose that U is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if U is bounded, then U has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if U is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if J(f) has an isolated point, or if J(f) is not equal to the boundary of the fast escaping set. Finally, we deduce that if J(f) has a bounded component, then all components of J(f) are bounded.

1. In this note we give first our proof of a theorem (Theorem 1) which we stated in Note XIII. We then prove a new theorem (Theorem 2) which leads to another proof of the main theorem of Note XVII.

The first of these theorems requires some preliminary explanations. We are concerned with an integrable function f (θ) with the period 2π. We write

being the complex Fourier series of f (θ).

The idea of gauge-invariance in general relativity was first introduced by Weyl(1) who proposed that the field equations of gravitation should be invariant, not only under the general group of coordinate transformations, but also under the gauge-transformation

where is the symmetric metric tensor, is the symmetric affine connexion and λ(x8) is an arbitrary scalar function of the coordinates. In this way it was possible to introduce into the theory a four-vector Ak which in consequence of (1·1) transformed as

such that the six-vector remained an invariant quantity under the gauge-transformation. It was Weyl's hope that by widening the invariance properties gauge-transformation. It was Weyl's hope that by widening the invariance properties of general relativity in this way the vector Ak and its associated six-vector Fik could be interpreted as representing the electromagnetic field. However, no obvious or unique way of doing this was found. More recently (see Stephenson (2,3) and Higgs (4)) gaugeinvariant variational principles formed from Lagrangians quadratic in the Riemann—Christoffel curvature tensor and its contractions have been discussed by performing the variations with respect to the symetric and symetric independently (following the palatini method).

In 1858 Chebyshev, and some years later his students Korkin and Zolotarev, determined the polynomial which deviates least from zero among all polynomials of degree n with leading coefficient one with respect to the maximum- and the L1-norm, respectively; these are now called the Chebyshev polynomial of first and second kind.

The next natural step which is to find, at least asymptotically, the minimal polynomial with respect to a given weight function has not been settled until today. Indeed, Bernstein gave asymptotics for the minimum deviation of weighted minimal polynomials, Fekete and Walsh found nth root asymptotics and, recently, Lubinsky and Saff provided asymptotics outside [−1, 1]. But the main point of interest: the asymptotic representation of the weighted minimal polynomials on the interval of approximation [−1, 1] remained open. Here we settle this problem with respect to the maximum norm for weight functions whose second derivative is Lipα, α ∈ (0, 1), and with respect to the L1-norm under somewhat stronger differentiability conditions.

A space time is stationary if and only if it admits a congruence of time-like curves such that the distance between any two curves is time independent. The distance between two curves A and B, which are not necessarily infinitesimally close to each other, is defined as half the trip-time of a photon, which travels from A to B and back to A.

For a ε R, we define a subset V of Rn to be a-convex if x, y ε V implies

Clearly V is (1 — a)-convex iff it is a-convex: V is convex iff it is a-convex for all a ε [0, 1], and any set is 1-convex. We define the a-convex hull of V to be the intersection of all a-convex subsets of Rn containing V. Put D = 0,1} and D(a) the a-convex hull of D in R1. If V is a-convex and x, y ε V, then V must contain the points of the line through x, y given by {tx + (1 — t) y: t ε D(a)} and so in this paper we concentrate on the study of D(a).

In this paper, we consider the family of the p-groups of maximal class and study some invariants related to the conjugacy classes of these groups. This subject was first taken up in [7] and [6], whose notation and terminology we maintain and give after this introduction. In [7], the invariants σG and k(G) are found for the p-groups of maximal class satisfying |G| ≤ p7. Here, apart from extending those results to the case when G has order p8 or p9, we consider the problem in all its generality and propose to determine their values for as large a family of p-groups of maximal class as possible. In this direction, we get some general results in the first paragraph, from which we want to point out the following ones.

Some years ago Evelyn and Linfoot found an asymptotic formula, with estimation of remainder, for the number of representations of a large integer n as the sum of s r-free integers. For s ≥ 4 their formula was subsequently sharpened by Barham and Estermann, and for s = 3 recently by me.

We prove Du's positivity conjecture for the canonical basis of the q-Schur algebra, using elementary arguments and the positivity result for Lusztig's canonical basis for U+q(sln). We also describe a family of subalgebras of the q-Schur algebra, each of which is spanned by the canonical basis elements it contains.