In this paper, we prove that, for any prime p, a Bol loop of order 2p or of order p2 is necessarily a group, and we show that there exist exactly six non-associative Bol loops of order 8.

The nilpotence class of the free commutative Moufang loop on n generators (n > 3) is the maximum allowed by the Bruck-Slaby Theorem, namely n − 1. This is proved by setting up a presentation of an extension of the loop's multiplication group as a nilpotent group of class at most 2n − 2, and then using the Macdonald-Wamsley technique of nilpotent group theory to show that this class is exactly 2n − 2.

1. Introduction. This paper is concerned with proving identities in commutative Moufang loops. Many such identities were derived in chapter VIII of (1) in the course of demonstrating the local nilpotence of commutative Moufang loops. The results there are regarded as constituting the ‘first grammar of associators’: the reader is assumed to have a good knowledge of them. The current paper develops additional material required for the determination in (5) of the precise nilpotence class of the free commutative Moufang loop on any given finite number of generators. It is called a ‘grammar’ because it lists formal ways in which the language of associators works, and is merely meant to serve a reader of ‘literature’ in the language such as (5). However, it may be of interest for other purposes, such as answering Manin's question ((3), Vopros 10·3; (4), problem 10·2) on the 3-rank of the free commutative Moufang loop of exponent 3. There is also the problem raised below as to whether the Triple Argument Hypothesis is a consequence of the commutative Moufang loop laws. Finally, the Möbius function in Section 9 may tempt someone to look at lattice-theoretical aspects of associators.

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.

A torsion-free discrete subgroup G of PSL(2, C) acts as a group of isometries of hyperbolic 3-space H3. The resulting quotient manifold M has H3 as its universal covering space with G as the group of cover transformations. We shall give examples where M has finite hyperbolic volume and is a link complement in S3. In these examples, G is a subgroup of the Picard group and in most cases is given as an HNN extension or a free product with amalgamation of kleinian groups with fuchsian groups as amalgamated or conjugated subgroups.

A homomorph H is a normal Schunck class if and only if there exists a derived class χ such that H = χ*; moreover, in this case one has H′ = χ (for the definitions, see below). These results give to the derived classes a decisive significance on the study of the normal Schunck classes (see (5)). The aim of this paper is to study the homomorphs H such that H′ is a fixed derived class: we prove that these homomorphs compose a complete and distributive lattice for the inclusion relation (the maximum of this lattice being a normal Schunck class). We construct the greatest and the smallest formations whose derived class is given. We prove finally that, except in trivial cases, a normal Schunck class is not a formation.

0. Introduction. The concept of a nilpotent action on a nilpotent group was studied in (2), and was crucial to the definition of a nilpotent map (or nilpotent fibration) in (4). These ideas had, of course, already been treated systematically in (1)†. The concept found a new, and related, application to homotopy theory in (3), where the following situation was studied. Let X be a nilpotent space and let G be a group which ‘acts’on X. Here it is not strictly speaking necessary to specify the nature of the action, provided that there is an induced action of G on the homotopy and homology groups of X. Thus G might be a group of base-point preserving homeomorphisms of X, or a group of (based) homotopy classes of self-homotopy-equivalences of X. We then discussed in (3) questions related to the nilpotency of the induced actions of G on the homotopy and homology groups of X and also relativized the theory to a discussion of fibrations on which the group G acts; to obtain interesting results related to nilpotency, it turned out to be at least necessary to suppose the fibration F → E → B quasinilpotent (7), meaning that all spaces are connected and π1B acts nilpotently on the homology groups of F.

1. Introduction. Abstract length functions were first considered axiomatically in Lyndon (4). Length functions satisfying a less restrictive set of axioms were studied by Chiswell(1). Groups with a normal form structure (NFS-groups) were studied in Hurley (3) using a slightly different axiomatization to that used in this paper. Length functions naturally arise from normal form structures. In this paper we show that the length functions arising from normal form structures can be characterized axiomatic-ally by a set of axioms that consists of those of (1) together with one further axiom.

There are a number of theorems to the effect that spaces of the form ΩnσnX split stably into wedges of simpler spaces when X is connected (by which we mean pathwise connected). The proofs generally proceed by exploitation of combinatorially manageable approximations for ΩnσnX.

In the theory of the iteration of a rational function or transcendental entire function R(z) of the complex variable z we study the sequence of natural iterates, {Rn(z):n = 0, 1,…}, of R, where

The domain of definition of the iterates is , the extended complex plane (if R is rational), and (if R is entire transcendental) with the topology of the chordal metric and euclidean metric respectively. Fatou(5) and Julia(9) developed a global theory of the iteration of a rational function. In (6) Fatou extended the theory of (5) to transcendental entire functions. A central role is played in the theory by the F-set, F(R), of R, R rational or entire, which is defined to be the set of points at which the family of iterates do not form a normal family in the sense of Montel.

1. Introduction. Let G be a Fuchsian group which acts on the unit disc Δ. A fundamental region D for G acting in Δ is a subset of Δ such that D is open and connected and each point of Δ is G-equivalent to exactly one point in D or at least one point in D̅ (the closure of D in Δ). Throughout this paper we consider only fundamental regions which are (hyperbolically) convex. Beardon has shown (2) that it is possible for a convex fundamental region to have certain undesirable properties. It can happen that a convex region is not locally finite, i.e. there exist points of Δ where infinitely many G images of D accumulate. For a domain D we denote by F the set of such points.

1. introduction. A set P of disjoint paths in a graph G is a node-covering of G if every node of G is in a path in P. (For definitions not given here see (5) or (11).) The path (node-covering) number of G is the smallest possible number p(G) of paths in such a node-covering of G. The problem of determining the path number of a graph was considered in (1) and (4) for undirected graphs and in (2) for directed graphs. In particular, an algorithm for determining the path number of a tree was given in (1) and (4). If ℱ denotes some family of trees, let μ(n) = μℱ(n) denote the average value of p(T) over all trees T in ℱ with n nodes. Our object here is to show the existence and to determine the value of the path (node-covering) constant

for certain families ℱ of trees. The arguments are similar to those used in (7), (8), and (9), where some other parameters, associated with families of trees, were investigated.

The von Mises-Fisher density of parameter K is shown to be very similar to the spherical normal distribution of parameter υ = 1/K as k → ∞. In fact the two densities are closer by factor of K than one might expect.

The double cusp is the real, compact, unimodal singularity

see (2), (4). Functions in a universal unfolding of the double cusp can have nine non-degenerate critical points near the origin, but no more. Index considerations show that precisely four of the nine are saddles, and it has long been part of the folklore of singularity theory that one of the other five must be a maximum. Indeed, a standard form of the unfolded double cusp (1), (3) is a function having a pair of intersecting ellipses as one of its level curves; see Fig. 1(a). There are saddles at the four intersection points, a maximum inside the central quadrilateral, and a minimum inside each of the other four finite regions bounded by the ellipses. The rest of Fig. 1 suggests, however, that a deformation of this function (in which one of the saddles drops below the level of the other three) might turn the maximum into a fifth minimum. The following proposition shows that a function similar to the one in Fig. 1(d) can be realized in an unfolding of the double cusp.

1. Introduction. In recent years a good deal of progress has been made in obtaining solutions to practical problems concerned with reinforcement optimization of what, in practice, might be reinforced concrete or other fibre reinforced composites in bending (1). Almost all the solutions have been obtained using a linear expression relating bending moment to the amount of fibre required. In what follows a non-linear expression will be adopted; the expression is obtained by including the next term in a parameter series expansion.

The exterior problems of acoustics can be reduced to the solution of Fredholm integral equations of the second kind in many different ways. Some of these equations break down at certain values of the wavenumber but this difficulty can be reduced or even removed altogether by choosing a different (usually more complicated) integral equation. The present paper is concerned with a class of equations proposed recently by D. S. Jones. It is shown that the proof of one of his principal results can be much shortened.

We develop a theory of particle scattering in anisotropic media. That is, a medium in which the microstructure causes the mean free paths of the particles to become dependent on their direction of motion with respect to some fixed axis. The equation which results is similar to the normal, one-speed Boltzmann transport equation but has cross-sections which are functions of direction. This equation is solved for arbitrary cross-sectional dependence on direction in plane geometry.