1. A group G is called characteristically simple if it has no proper non-trivial subgroups which are invariant under all automorphisms of G. It is known that if G is characteristically simple then each countable subgroup lies in a countable characteristically simple subgroup of G. A similar assertion holds for simple groups. These results were proved by Philip Hall in lectures in 1966, and further proofs appear in (4) and (6). For simple groups there is a well known and elementary result in the other direction: if every two-generator subgroup of a group G lies in a simple subgroup, then G is simple. These considerations prompt the question (first raised, I believe, by Philip Hall) whether a group G is necessarily characteristically simple if each countable subgroup lies in a characteristically simple subgroup.

1. Introduction. It is well known (see (5) and (6)) that if T is a 2-group containing an involution t such that |CT(t)| = 4, then T is dihedral, semidihedral or cyclic of order 4. Fomin(1) has studied 2-groups T containing an involution t such that |CT(t)| ≤ 8. He showed in particular that such a group has sectional 2-rank at most 4. In this paper we will prove the following:

Theorem A. Let T be a 2-group containing an involution t such that |CT(t)| ≤ 16. Then T has sectional 2-rank at most 6.

1. Introduction. In (2) we associated with a commutative ring R with identity the lattice ℒ(R) of rings S which satisfy (i) R ⊆ S ⊆ T, where T is the total quotient ring of R, and (ii) dS ⊆ R for some d ∈ R which is not a zero-divisor. The rings of ℒ(R) can be constructed by blowing up ideals of the set of ideals A of R which satisfy (i) 0: A = 0, and (ii) there exist integers m1n ≥ 1 and α ∈ An such that αAm = Am+n (see Propositions 2·3 and 2·5 of (2)).

1. Introduction. Let X be a space and let E be a fibre space over E. A fibre-preserving map f: E → E determines, for each point x ∈ X, a map fx: Ex → Ex of the fibre over x. In a previous note (3) the situation was considered where fx is null-homotopic, for all x. In the present note we turn our attention to the situation where fx is homotopic to the identity on Ex, for all x ∈ X. If X admits a numerable categorical covering (as when X is an ANR) then such a fibre-preserving map f is a fibre homotopy equivalence, by the well-known theorem of Dold(1). Then the set Φ1(E) of fibre homotopy classes of such maps forms a normal subgroup of the group Φ*(E) of fibre homotopy classes of fibre homotopy equivalences. The purpose of this note is to prove

Theorem 1.1. Let X be a paracompact space of finite category. Let E be a fibre bundle over X of which the fibres are compact and path-connected ANR's. Then the Φ*(E)-growp Φ1(E) is Φ*(E)-nilpotent of class ≤ cat X.

For any group G and x ∈ G, and any n ∈ ℕ (the natural numbers) let

i.e. the set of all nth roots of x in G.

In (4) we showed how to stably split certain spaces CX built up from a ‘coefficient system’ and a ‘Π-space’ X. Via an approximation theorem relating particular examples to loop spaces, there resulted stable splittings of Ωn σnX for all n ≥ 1 and all (path) connected based spaces X.

Let A denote the mod 2 Steenrod algebra. Let ℤ2[x, x−l] be the (graded) ring of finite Laurent series over ℤ2 in the variable x with dim (x) = 1. ℤ2[x, x−1] is a module over the Steenrod algebra A by

where are binomial coefficients modulo 2 and m > 0 is large compared with |k| and i. Let M be the A-submodule of ℤ2[x, x−1 ] generated by all powers xi with i ≠ −1. It is easy to see that ℤ2 [,x, x−1]/M ≅ σ−1ℤ2 (means ℤ2 on dimension − 1). Let ρ: ℤ2[x, x−1] → σ−1 ℤ2 be the projection map.

An H′-algebra which is not an operator algebra is constructed. This is used to prove that the class of operator algebras is not equal to the class of α-algebras for any tensor norm α.

1. Introduction. The main result of this paper is that the simple C*-algebra (5) has the Dixmier property. As a corollary of the proof we obtain that the relative commutant of the Choi algebra (4) in is trivial.

1. Introduction. Recently Kaplansky suggested the definition of a suitable Jordan analogue of B*-algebras, which we call J B*-algebras (see (10) and (11)). In this article, we give a characterization of those complex unital Banach Jordan algebras which are J B*-algebras in an equivalent norm. This is done by generalizing results of Bonsall ((3) and (4)) to give necessary and sufficient conditions on a real unital Banach Jordan algebra under which it is the self-adjoint part of a J B*-algebra in an equivalent norm. As a corollary we also obtain a characterization of the cones in a Banach Jordan algebra which are the set of positive elements of a J B*-algebra.

1. Introduction. The aim of this paper is to show that, in every complex Banach algebra with a one-sided or two-sided bounded approximate identity, there exists another bounded approximate identity of the same sort whose spectra lie close to the unit interval [0, 1].

1. Introduction. In 1966, Kuttner(4) constructed (D, α) summability methods for functions. In this paper, strong and absolute summabilities based on (D, α) summability methods are defined and some of their properties investigated.

0. Introduction. Asymptotic behaviour of distributions (Silva(8) and Benedetto (l)) plays a fundamental role in the analysis of singularities of generalized integral transforms. Such analysis in terms of Abelian theorems for the distributional Stieltjes transformation has been obtained by Misra (6) and Lavoine and Misra (3). To obtain his results Misra used some Abelian theorems (see Misra (6), theorems 3·1·1 and 4·1·1) for the Stieltjes transformation of functions which were essentially generalizations of the results given by Widder (9), pp. 183–185. The object of the present paper is to complete the work given by the authors in (3); Sections 2 and 3 of this paper are devoted to obtaining Abelian theorems concerning the distributional Stieltjes transformation in terms of the behaviour of this transform near the origin and at infinity.

1. Introduction. In this article we continue the work begun in (5). We will consider only finitely generated Fuchsian groups of the first kind. Let G be such a group acting on the unit disc Δ. A fundamental domain D for G is a connected open set with the property that any point of Δ is G-equivalent to exactly one point in D or at least one point in (the closure of D in Δ). A fundamental domain is said to be locally finite if there are no points in Δ where infinitely many G-images of D accumulate.

1. Introduction. Révész(8) has shown that if (fn) is a sequence of random variables, bounded in L2, there exists a subsequence (fnk) and a random variable f in L2 such that converges almost surely whenever . Komlós(5) has shown that if (fn) is a sequence of random variables, bounded in L1, then there is a subsequence (A*) with the property that the Cesàro averages of any subsequence converge almost surely. Subsequently Chatterji(2) showed that if (fn) is bounded in LP (where 0 < p ≤ 2) then there is a subsequence (gk) = (fnk) and f in Lp such that

almost surely for every sub-subsequence. All of these results are examples of subsequence principles: a sequence of random variables, satisfying an appropriate moment condition, has a subsequence which satisfies some property enjoyed by sequences of independent identically distributed random variables. Recently Aldous(1), using tightness arguments, has shown that for a general class of properties such a subsequence principle holds: in particular, the results listed above are all special cases of Aldous' principal result.

1. Simple linear clustering. One of the simplest clustering problems is the following: if n points are taken independently and uniformly at random on the interval I = [0, L], what is the probability Q = Q(n, a, L) that no two points are closer than a? The well-known answer

if (n − 1) a ≤ L, and Q = 0 otherwise, can be obtained in a variety of ways. Perhaps the simplest one is the original method of E.C.Molina(5). If x1,…,xn are the abscissas of the n points, then there are n! equiprobable orderings of the xi's consistent with the condition that no two points are closer than a. The probability of each one is

Let yi = xi − (i − 1) a for i = 1,…,n, then the above becomes

which is

We prove that the class of transition matrices for the finite state time-inhomogeneous birth and death processes coincides with the class of non-singular totally positive stochastic matrices. Thus we obtain a complete solution to the embedding problem for this class of Markov chains.

1. Introduction and summary. This paper offers yet another example of what probability theory can do for analysis. Using a Feynman-Kac formula derived in the theory of random evolutions (5), we find an expression (1) for the spectral radius r(A) of a finite square non-negative matrix A. This expression makes it very easy to study how r(A) behaves as a function of the diagonal elements of A.

1. Introduction. Hajnal (5) showed that under wide conditions a sequence of products H(1, q) = Mq…M1q = 1,2,…, of square non-negative matrices Mq approaches a sequence of positive matrices of rank 1. We call a product H(1,q) inhomogeneous if its factors M1,…, Mq are not necessarily all equal to one another. When the matrices Mq are members of an ‘ergodic set’, and x and y are positive vectors, the projective distance d(H(1,q)x, H(1,q)y) decays at least exponentially fast as q increases. An important condition on an ergodic set is that any product of g members from the set be a matrix in which all elements are (strictly) positive, where g is some fixed positive integer.

1. Introduction. The purpose of this paper is to study the global geometry of a space–time (M, g), which is related to the Lorentzian distance function induced on the manifold M by the Lorentzian structure. We will use the signature convention (−, +, …, +) for g and assume that (M, g) is time orientated. The first part of this paper deals with cut points and maximal geodesies, both of which were defined in (1) using the Lorentzian distance function in analogy to the standard concepts in Rie-mannian geometry. In (1), sections 2 and 3, some elementary properties of maximal geodesies were established. In particular, the principle that, for strongly causal space-times, a limit curve of a sequence of future-directed nonspacelike ‘almost maximal’ curves is a maximal geodesic was used to prove nonspacelike incompleteness ((1), theorem 6·3). Also null cut points were used to obtain results on null incompleteness ((1), section 5). In (2) we studied deeper properties of maximal geodesies and cut points using the technical tools developed in (1), sections 2 and 3. The first part of the present paper continues these investigations.