If F(x0, x1, x2, x3) = 0 is the equation of a surface in space of three dimensions which has an ordinary isolated s-ple point O, then by means of the substitutions

where Φ0 = 0, Φ1 = 0, …, Φr = 0 are the equations of r + 1 linearly independent surfaces passing simply through O, F is transformed into a surface F′ in [r], on which to the point O of F there corresponds a simple curve γ. The points of γ arise from the points of F in the first neighbourhood of O, and in this simple case the genus of γ is ½ (s − 1) (s − 2). In the study of properties which are common to all members of an infinite family of birationally equivalent surfaces no distinction is made between O and γ, O being regarded as a curve which has become infinitesimal on the particular surface of the family in question.

In considering the geometric invariant theory of linear systems (pencils, nets, …) of geometrical objects (quadrics, cubic curves, binary forms,…) there are several ways to apply the basic theory. The object of this note is to point out the equivalence of different approaches and to apply this remark to a number of special cases.

In (5), Schmidt devised a procedure for completing a space-time M by adjoining a boundary ∂M called the b-boundary. Bosshard(1) and Johnson(4) have shown that certain Friedmann and Schwarzschild space-times have non-Hausdorff b-completions.

In (2), (3), the first author considered a modification of the b-completion which gives a Hausdorff completion in the Friedmann case. The modification used a parallelization of the space-time and can be given in terms of the linear connexion determined by the parallelization. We show that the completion obtained is the Cauchy completion in a particular Riemannian metric on the manifold and so is always Hausdorff.

1. In this paper I investigate the tangential properties of parametric surfaces, leading to the main result that the Lebesgue area is the Hausdorff two-dimensional measure of the set of points of the surface where an approximate tangential plane exists.

1. Few mod 2 finite simply-connected spaces are known to support an H-structure. These examples arise from products of Lie groups and seven spheres. It has been shown in (12) and (14) that the space of a mod 2 finite H-space of given rank can belong to only a finite number of different homotopy types. The situation at odd primes is very different; many examples of mod p finite H-spaces are known, such as p-localized odd-dimensional spheres. Therefore there are infinitely many homotopy types of prescribed rank. If one insists that the mod pH-space has a strictly associative multiplication, or equivalently has the homotopy type of a topological group, the picture is less clear. Examples have been constructed which do not arise from Lie groups and spheres but there are strong restrictions on just what can occur by these techniques, see (5). There are only finitely many known examples in each rank. We establish that there can in fact only be a finite number of possible homotopy types for the spaces.

It is shown that if T is a ternary ring of operators (TRO), X is a nondegenerate sub-TRO of T and there exists a contractive idempotent surjective map P: T → X then P has a unique, explicitly described extension to a conditional expectation between the associated linking algebras. A version of the result for W*-TROs is also presented and some applications mentioned.

Let be the 3-dimensional sphere and let be a smooth knot, i.e. K is the image of a regular diffeomorphism . We say that K is unknotted if there exists a diffeomorphism of onto the unit sphere of , that brings K to the equatorial circle otherwise we say that K is knotted.

In a recent paper of the same title (2), Conway considered the problem of dividing a square of side N, into squares of integral side. The case when we demand that all these squares are of unequal size has been considered in some detail by various authors ((l),(4),(6),(7)). Conway considered the case with the condition relaxed. Dudeney ((3), Problem 173) has considered the same problem for the special case N = 13; hence the title of this paper.

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.

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.

Throughout this paper q = 2h with h ≥ 4, and PG(r, q) stands for the r-dimensional projective space over the finite field GF(q) with q elements.

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.

Let C be a complex algebraic curve in the projective space ℙ2. The purpose of this paper is to calculate the fundamental group G of the complement of C in the case when C = X ∩ H1 ∩ … ∩ Hn−2, where

and Hi are generic hyperplanes (i = 1, … n − 2).

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.)

In this paper a quantum-mechanical variational principle is proposed, in which the functional varied is the precision, with which the time-dependent Schrödinger equation is satisfied by the wave functions of the given class. Another distinctive feature of our approach is that the independently varied functions are the time derivatives of the one-particle wave functions, of which the wave function of the system is constructed.

We explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak ω-categories, showing that the universal and canonical cofibrant replacement of the operad for strict ω-categories is precisely Leinster's operad for weak ω-categories.

Some topological identifications are introduced into the DS-spaces of Robinson and Trautman, and the symmetry and asymptotic symmetry groups of the resulting spaces are calculated. The latter have structures similar to the GBM group, each being the semi-direct product of a group of supertranslations by a group of classical type.

In this paper we shall consider contractible 2-dimensional cell complexes K with one 0-cell v, one l-cell a, and one 2-cell D, where the boundary of D is attached to a by a word in a and a−1. In order for such a complex K to be contractible, the attaching word in a and a−1 must have total degree ± 1. The simplest example occurs when the attaching word is a; in this case K is a 2-cell, and is of no interest since K is collapsible. If any other word of degree ± 1 is used, the resulting K is contractible but not collapsible.

The classical theory of electromagnetism with both electric and magnetic charges is discussed. Following Gamblin(l) we proceed from an action principle and investigate the resulting particle equations. A consistent theory in which electrically and magnetically charged particles interact amongst themselves and with each other, without any constraints, is shown to be possible, in contrast to the conclusions of Gamblin. The introduction of constraints is considered.

Let F be the finite field of q = pn elements and let F0 be its prime subfield; thus, card F0 = p. For polynomials f ∈ F[x] and non-principal additive characters η of F A. Weil (1) proved the estimate