Laws of large numbers and central limit theorems are proved for sums of general functions of m-spacings from general distributions. Explicit formulae are given for the norming constants. The results enable us to describe asymptotic properties of distributional tests under fixed alternatives. A generalization of Kimball's spacings test is considered in detail.

This paper proves that any knot is concordant to a prime knot; it thus solves Problem 13 of (3). In doing so it makes an exploration of a fairly general method of proving that a knot is a prime. Throughout, the word ‘knot’ means a knot of S1 in S3 (orientations being here irrelevant); occasionally reference will be made to the idea of a knotted arc spanning a 3-ball.

We consider the perturbation of the velocity of an inviscid incompressible fluid rotating about an axis with uniform angular velocity Ω, due to the slow uniform motion, after an impulsive start, of a sphere of radius a along the axis with velocity − V. This problem was first considered by Taylor (4), who obtained a family of solutions for the case of steady motion of a sphere along the axis of rotation of the fluid. All the solutions satisfied the boundary condition at the surface of the sphere and also the condition that the relative velocity should vanish at infinity. It was thought possible that the indeterminacy lay in the manner in which the motion was started. However, an experiment carried out in connexion with the problem showed that if V/aΩ was less than about 0·16 a column of liquid of the same diameter as the sphere was apparently pushed along in front of it. This agreed with Proudman's observation (3) that if the perturbation was small and steady it would be two-dimensional, while in Taylor's solutions the relative motion of the fluid was not small. Taylor (5) suggested that three possibilities presented themselves. Either the motion never becomes steady, or it becomes steady but not small, or it becomes steady and two-dimensional. Of these he preferred the latter, which seems to occur in practice.

Bethe's method is applied to determine statistically the rate of condensation of diatomic molecules which are adsorbed with dissociation to form an immobile film, the two atoms of one adsorbed molecule occupying two neighbouring sites. A lower limit to the number of gaps, or isolated unoccupied sites, in the final film is also obtained. The results calculated statistically agree with those obtained earlier by using models.

The relativistic Hamiltonian H for a particle in a field of potential energy V is given, in atomic units, by the formula

where any ρ commutes with any σ, any σ anticommutes with any other σ, ρ1 anticommutes with ρ3 and

In this paper we shall consider simultaneous approximations to algebraic numbers of the form for rational t1, …, tm, v. If t1, …, tm, are sufficiently close to 1 with respect to a common denominator we shall derive an effective inequality of the form max {|θ1 – p1 / q|, …, |θm – pm / q|} > cq–1–λ for all integers p1, …, pm, q with q > 0. The technique is used to derive the inequality

We use this to show that the only solution to the simultaneous equations

are the trivial ones.

0. Introduction. Let and be connected cohomology theories on the category of all CW spaces. In this paper we examine the relationship between natural homomorphisms (stable operations) of degree zero from to and natural homomorphisms from to in a number of cases of particular interest in topology – foremost the K-theories and the bundle theories classifying ‘surgery problems’. There are two parts to this. On the one hand we ask under what conditions a homomorphism from to extends to a stable operation and on the other hand when is a stable operation determined by its restriction to .

Let (R, M) be a commutative Noetherian local ring having an identity, and assume the residue field R/M is infinite. If I is an ideal in R, recall that an ideal J contained in I is called a reduction of I if JIn = In + 1 for some non-negative integer n. A reduction of J of I is called a minimal reduction of I if it does not properly contain another reduction of I. Reductions (and minimal reductions) were introduced and studied by Northcott and Rees[8]. If J is a reduction of I we define the reduction number of I with respect to J, denoted rJ(I), to be the smallest non-negative integer n such that JIn = In + 1 (note that rJ(I) = 0 if and only if J = I). The reduction number of I (sometimes referred to as the reduction exponent) is defined as r(I) = min{rj(I)|J ⊆ I is a minimal reduction of I}.

Let G denote a group of order a power of the prime p, and let G′ be the derived group of G. The lower central series of G will be written

For any subgroup H of G we denote by P(H) the subgroup of H generated by all elements xp as x runs through H, and by Φ(H) the Frattini subgroup of H. We write (H:Φ(H)) = pd(H); thus d(H) is the minimal number of generators of H.

The cross-sections for the elastic scattering of mesons by protons and electrons are calculated by analogy with Møller's theory of collisions between electrons, and the results are compared with those of Bhabha for the collisions with protons and electrons of Fermi-Dirac particles with the same mass as the meson. For collisions with electrons, the difference is inappreciable for energies of the meson less than 100 times its rest energy, but for collisions with protons the difference is important at much lower energies. The ratio of the corresponding cross-sections for scattering by a Coulomb field is found to tend to infinity as the square of the energy of the incident particle, giving a finite cross-section for the scattering of a meson of infinite energy by such a field. It is pointed out that, if an exact solution of this problem is possible for the meson case, the question of the validity of meson theory for high energies would be considerably clarified.

The theory of lattice deformations is presented in a new form, using the tensor calculus. The case of central forces is worked out in detail, and the results are applied to some simple hexagonal lattices. It is shown that the Bravais hexagonal lattice is unstable but the close-packed hexagonal lattice stable. The elastic constants of this lattice are calculated.

In the second part of this paper I pointed out how the presence of any singular point or points of the kind considered in the first part, on a rational surface, corresponds to a subgroup of the group of symmetry of the polytope which represents the properties of the system of base points in the plane representation of the surface. The subgroup is generated by reflexions, and may be the direct product of one or more factors (all the primes of symmetry in one factor being perpendicular to all those in any other factor). Each factor corresponds to a singular point on the surface, namely (in Coxeter's notation), a factor [ ] to a conic node C2, a factor [3n] to a binode Bn+2 (n ≥ 1), a factor [3n,1,1] to a unode Un+5, (n ≥ 1), and finally a factor [3n,2,1] to a unode The possible subgroups in the finite groups that arise have been enumerated by Coxeter; and we shall find that every subgroup generated by reflexions in the group of symmetry of the polytope in ε dimensions which represents ε base points corresponds to a possible configuration of the base points, in which just those rational curves of grade − 2 are actual which correspond to primes of symmetry belonging to the subgroup; without exception, for ε ≤ 6; with one exception—the subgroup [ ]7—for ε = 7; and with three exceptions—the subgroups [ ]7, [ ]8, and [31,1,1 × [ ]4—for ε = 8. Since moreover the system |k| of cubics passing simply through all the base points is in all these cases an actually existing system, for which all the rational curves of grade − 2 are fundamental, its projective model (or in the case ε = 8, in which |k| is only a pencil, the projective model of the system |2k|) provides a rational surface on which all the sets of curves corresponding to the subgroups in question actually appear as singular points.

We introduce a new approach to the local study of maximal surfaces in Lorentz–Minkowski space, based on a complex representation formula for this kind of surfaces. As an application we solve a certain Björling-type problem in Lorentz–Minkowski space and we obtain some results related to it. We also establish, springing from this complex representation, a way of introducing examples of maximal surfaces with interesting prescribed geometric properties. Further applications of the complex representation let us inspect some known results from a different perspective, and show how our approach can be used to classify certain families of maximal surfaces.

Given a set consisting of n1, objects of type 1, n2 of type 2, …, nk of type k, we denote by the number of possible derangements of the set i.e. permutations in which no object occupies a site originally occupied by an object of the same type. A formula is found for in terms of Laguerre polynomials, and some of its implications are considered.

A method, very suitable for use with an automatic computer, of solving the Hartree-Womersley approximation to the incompressible boundary-layer equation is developed. It is based on an iterative process and the Choleski method of solving a simultaneous set of linear algebraic equations. The programming of this method for an automatic computer is discussed. Tables of a solution of the boundary-layer equation in a region upstream of the separation point are given. In the upstream neighbourhood of separation this solution is compared with Goldstein's asymptotic solution and the agreement is good.

Introduction. Let Xi (i= 1, 2, 3,…) be a sequence of independent and identically distributed random variables with law ℒ(X) and write The Kolmogorov-Marcinkiewicz strong law of large numbers (Loève(6), p. 243) has the following statement:

If E|X|r < ∞, then with cr = 0 or EX according as r 1 or r ≥ 1.

1. Let V be an l-dimensional real vector space and let W be a finite subgroup of GL(V) generated by reflexions such that the space of W-invariant vectors in V is zero. Then W acts naturally on the symmetric algebra S of V preserving the natural grading . LetI be the ideal in S generated by the w-invariant elements in . The quotient algebra inherits the W-action and also a grading

is the image of Sk under S → S̅. It is well known that the W-module S̅ is isomorphic to the regular representation of W (see (3), ch. v, 5·2); in particular, S̅k = 0 for large k. (More precisely, S̅k = 0 for k > ν, where ν is the number of reflexions in W.)If ρ is an irreducible character of W, we denote by nk(ρ) the multiplicity of ρ in the W- module S̅k. The sequence n(ρ) = (n0(ρ), n1(ρ), n2(ρ), …) is an interesting invariant of the character ρ For example, in the study of unipotent classes in semisimple groups, one encounters the following question: what is the smallest k for which nk(ρ) 4= 0 (with ρ as above) and, then, what is nk(ρ) Also, the polynomial in q

can sometimes be interpreted as the dimension of an irreducible representation of a Chevalley group over the field with q elements. For these reasons it seems desirable to describe explicitly the sequence n(ρ) (or, equivalently, the polynomial Pρ(q)) for the various irreducible characters ρ of W. When W is a Weyl group of type Al, this is contained in the work of Steinberg (8); in the case where W is a Weyl group of type Bl or Dl this is done in (7),§2.

Let X be a closed, compact connected 2-manifold (a surface), which we will denote by O or N if we wish to stress that X is orientable or non-orientable. Let G(X) denote the group of all homeomorphisms X → X, D(X) the normal subgroup of homeomorphisms isotopic to the identity, and H(X) the factor group G(X)/D(X), i.e. the homeotopy group of X. The problem of determining generators for H(O) was considered by Lickorish in (7, 8), and the second of these papers specifies a finite set of generators of a particularly simple type. In (10) and (11) Lickorish considered the analogous problem for non-orientable surfaces, and, using Lickorish's partial results, Chilling-worth (4) determined a finite set of generators for H(N). While the generators obtained for H(O) and H(N) were strikingly similar, it was noteworthy that the techniques used in the two cases were different, and in particular that little use was made in the non-orientable case of the earlier results obtained on the orientable case. The purpose of this note is to show that the results of Lickorish and Chillingworth on non-orientable surfaces follow rather easily from the work in (7, 8) by an application of some ideas from the theory of covering spaces (2). Moreover, while Lickorish and Chillingworth sought only to find generators, we are able to show (Theorem 1) how in fact the entire structure of the group H(N) is determined by H(O), where O is an orientable double cover of N. Finally, we are able to determine defining relations for H(N) for the case where N is the connected sum of 3 projective planes (Theorem 3).

There are two natural analogues of the symmetric group on n symbols in the theory of semigroups, namely, the set of all mappings of a set of n symbols into itself, and the set of all partial transformations of such a set, with the obvious definitions of multiplication. We are concerned here with the latter system. This is an inverse semigroup, and accordingly we call it the ‘symmetric inverse semigroup’. It gives rise to a semisimple algebra over a field of characteristic zero or a prime greater than n, and its matrix representations over such a field are thus completely reducible.