This paper provides a framework for studying the general theory of the upper semi-lattice of degrees of unsolvability. A less obvious and more complicated extension of this framework has been initiated in our papers (17) and (18); these are concerned not with the general theory but with the Priority Method, for which a useful formal framework has long been sought. Our purpose here is partly to provide some background for these other papers, which should help explicate their less obvious aspects; but mainly we wish to illustrate the advantages of this framework from a purely expository point of view. There is an additional advantage in that a formal framework makes it possible to pose certain precise questions which were vague before, and even on occasion to answer them.

In this note we determine the greatest lower bound of the number of independent edges in a graph in terms of the number of vertices, the minimal degree and the maximal degree. More generally we examine the greatest lower bound of the number of independent edges in a K-connected (or λ-edge-connected) graph with given number of vertices and given minimal and maximal degrees. These results extend those of Erdös and Pósa (3) and Weinstein (7). Our proof makes heavy use of Berge's slight extension of Tutte's characterization of graphs with 1-factors.

The aim of this paper is to describe the possible structures for NEC (non-euclidean crystallographic) groups of the hyperbolic plane with non-compact quotient space. The case of compact quotient space was settled by Wilkie (6), and it has been shown by Hoare, Karrass and Solitar (3) that all subgroups of infinite index in a Wilkie group have a certain structure pattern. Their proof depends on showing that a subgroup of infinite index in a Wilkie group has a presentation satisfying certain restrictions on the occurrences of each generator. Our method here is to study the presentation of an arbitrary NEC group which is given by the classical construction of the Dirichlet region, and to prove that it satisfies the same restrictions as were found for the above subgroups of Wilkie groups.

The purpose of this paper is to study finite simple groups whose Sylow 2-subgroups are isomorphic to those of PSL(5, q) for some odd q. This work originally appeared in the author's Ph.D. Thesis, (14), and at that time, the structure of the centralizers of involutions was worked out – see below for details. Subsequently, in view of the fact that if q ≡ – 1 (mod 4), the Sylow 2-subgroup of PSL(5, q) has sectional 2-rank 4 (being, in fact, isomorphic to the wreath product of a semidihedral group by a cyclic group of order 2), the complete classification, at least for this case, was needed for Gorenstein and Harada's monumental work on sectional 2-rank at most 4, (8), and therefore Collins and Solomon completed the characterization of PSL(5, q) and PSU(5, q) (for all odd q) in a neat paper, (5). Combining our result with theirs, we are able to state the following Theorem.

In this paper we prove that the automorphism group A(N) of a free nilpotent group N of class 2 and finite rank n is complete, except when n is 1 or 3. Equivalently, the centre of A(N) is trivial and every automorphism of A(N) is inner, provided n ≠ 1 or 3. When n = 3, A(N) has an our automorphism of order 2, so A(A(N)) is a split extension of A(N) by . In this case, A(A(N)) is complete. These results provide some evidence supporting a conjecture of Gilbert Baumslag that the sequence

becomes periodic if N is a finitely generated nilpotent group.

Let k be a positive integer and F(x, k) denote the number of integers n < x which have a divisor in every residue class prime to k. Erdös (1) proved that for every fixed ε > 0, we have F(x, k) ˜ x when

and conjectured the following result, which we prove in this paper.

H. Läuchli (9) constructed, within a model of a weak form of set theory, an algebraic closure L of the field Q of rationals which had no real-closed subfield. Läuchli's construction is easily transferred to a model N of ZF (= Zermelo–Fraenkel set theory without the axiom of Choice), and it follows at once that neither of the two following statements is provable from ZF alone:

Every algebraic closure of Q has a real-closed subfield. (1)

There is, up to isomorphism, at most one algebraic closure of Q. (2)

It seems worthwhile to point out that a few of the theorems in (2) can, as indeed one would expect in a paper of such length and scope, be somewhat generalized or simplified. This note uses the notation of (2) and all references are to that paper.

In this paper we consider the classification problem for separable special simple J*-algebras (cf. (8)). We show, using a result of Ancochea, that if is the (finite-dimensional) Jordan algebra of all complex n × n matrices and ø a Jordan isomorphism of onto a special J*-algebra J then An can be given the structure of an H*-algebra such that ø is a *-preserving isomorphism of the J*-algebra onto J. This result enables us to construct explicitly a canonical basis for a finite-dimensional simple special J*-algebra isomorphic to a Jordan algebra of type I from which we also obtain canonical bases for special simple finite-dimensional J*-algebras isomorphic to Jordan algebras of type II and III.

We shall verify Miller's conjectured characterization of parts in measure algebras and obtain also a simpler proof of the main result of (2). The basic pattern of argument follows Miller while the chief novelty resides in the approximation method using Wiener–Ditkin sets which is introduced in section 2.

This paper concerns the existence and uniqueness of non-negative solutions of non-linear Volterra integral equations of the type

and

where the kernel function k(.,.) is non-negative and sufficiently smooth, and either 0 < p < 1 or – 1 < p < 1. We will consider also the corresponding Fredholm equations

and

Krickeberg (in (5) and (6)) showed that disintegration applied to invariant measures sometimes yields an integral representation which is useful in analysing the moment measures of point processes. His results, based on Bourbaki's disintegration theory, raised several questions. We refine the theory, using a more general disintegration theorem, and answer his questions by several examples. Finally we consider how far the enlarged theory is applicable in stochastic geometry.

An elaborated version of Zeeman's Catastrophe Machine (1), which physically illustrates the butterfly catastrophe, is described and analyzed.

The model is closely connected with a model by Lamb and Scully (10). Atoms described as two-level systems, initially in an incoherent superposition of the two levels, interact successively during a time T with an electromagnetic field of which only one mode is taken into consideration. In the limit as infinitely many atoms have interacted, it is shown that the field either approaches a thermal distribution or is excited to arbitrarily high Photon numbers according to whether or not the lower level of the atoms is initially more probable than the upper level. It is also shown that in any case the correlations between pure Photon number states converge to 0. If the atoms are initially in the upper level it is proved that the Photon number grows roughly as the square root of the number of atoms that have interacted. Throughout the discussion number-theoretical properties of T play a disturbing role. The last mentioned result in fact depends on a sharp (but arbitrary) value for T and is therefore disqualified for physical interpretation.

A previous application of the theory, to the nonlinear boundary-value problem for steady flows of a viscous fluid in a bounded domain, is first retraced in order to verify a general theorem concerning the indices of multiple solutions. Then, in §5, the bearing of the theorem on questions of the bifurcation of steady flows is discussed, and the conclusion is drawn that a transcritical form of bifurcation is virtually universal in practice. In §6 it is proved that a flow represented by a solution with index i = – 1 is necessarily unstable, and hence it appears that lack of uniqueness generally implies the existence of an unstable steady flow.