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Artstein has given a characterization of the distributions induced by selections of a random compact subset of a complete separable metric space. In this paper, Artstein's results are extended to spaces which may be neither metric nor separable.

Let us denote the group of based homotopy classes of seif-homotopy equivalences of a space X by E(X). We consider E0(X), the subgroup of E(X) consisting of elements which induce the identity map on homology. Dror and Zabrodsky have shown that E0(X) and the subgroup E#(X) consisting of elements inducing the identity on homotopy are both nilpotent groups for finite-dimensional nilpotent spaces, or finite-dimensional spaces respectively ([4], theorem D, theorem A). The theory of localization for nilpotent groups has been developed by several authors (see [8]). The aim of this paper is to prove the following theorem. The corresponding result for E#(X) is obtained in [9], theorem 0·1.

Let K be an oriented knot in the 3-sphere S3. An exterior of K is the closure of the complement of a regular neighbourhood of K, and is denoted by E(K). A Seifert surface for K is an oriented surface S( ⊂ S3) without closed components such that ∂S = K. We denote S ∩ E(K) by Ŝ, and we regard S as obtained from Ŝ by a radial extension. S is incompressible if Ŝ is incompressible in E(K). A tunnel for K is an embedded arc τ in S3 such that τ ∪ K = ∂τ. We denote τ ∪ E(K) by τ, and we regard τ as obtained from τ by a radial extension. Let τ1, τ2 be tunnels for K. We say that τ1 and τ2 are homeomorphic if there is a self-homeomorphism f of E(K) such that f(τ1) = τ2. The tunnels τ1 and τ2 are isotopic if τ1 is ambient isotopic to τ2 in E(K). Then the main result of this paper is as follows: Theorem. Let K be a knot in S3, and let τ1, τ2 be tunnels for K. Suppose that there are incompressible Seifert surfaces S1 S2 for K such that S1 ∪ S2 = K, and τi ⊂ Si (i = 1, 2). If τ1 and τ2 are isotopic, then there is an ambient isotopyhτ (0 ≤ t ≤ 1) of S3 such that ht(K) = K, and h1(τ1) = τ2.

This paper presents a uniqueness result for solutions to the general nonlinear coagulation-fragmentation equationwhereEquation (1·1) has many applications in the applied sciences (cf. [1, 3, 8, 13, 15]) and a brief physical interpretation can be found in Melzak [12] or the survey article by Drake[7]. c(x, t), for x ≥ 0, t ≥ 0, denotes the number of particles of size x at time t and the non-negative kernels K and F describe, respectively, the rates at which particles of size x coalesce with those of size y and particles of size (x + y) break-up into those of sizes x and y.

In this paper we improve the estimate for the lower bound of the Hausdorff dimension of certain sets of singular n-tuples in ℝn obtained by Baker in [1]. We begin with a brief discussion of the problem. More details can be found in [1]. For ease of comparison with Baker's results we will adopt a similar notation to that of [1].

Centralizers in fundamental groups of 3-manifolds are well understood because of their relationship with Seifert fibre spaces. Jaco and Shalen's book [4] provides detailed information about Seifert fibre spaces in 3-manifolds, and consequently about centralizers in their fundamental groups. It is the purpose of this note to record two group-theoretic properties, both easily deduced from results of Jaco and Shalen. Doubtless many other authors could have established the same results had they needed them. Our motivation for writing this paper is that these properties can be used as a basis for group-theoretic proofs of certain fundamental results in 3-manifold theory: Proposition 1 below can be used as a basis for a proof of the Torus Theorem (cf. [8]) and Proposition 2 for the Torus Decomposition Theorem (cf. [9]).

In p-primary stable homotopy theory, recent developments have shown the importance of the Morava K-theory spectra K(n) for positive integers n. A current major problem concerns the behaviour of the K(n)-cohomologies on the classifying spaces of finite groups and on related spaces. In this paper we show how to compute the Morava K-theory of extended power constructions Here Xp is the p-fold product of some space X and Cp is the cyclic group of order p. In particular, if we take X as the classifying space BG for some group G, then Dp(X) forms the classifying space for , the wreath product of G by Cp.

In [1] we used the alternating group A5 to produce a set of five permutations of order three acting on 12 letters; this set was normalized by A5 and generated the Mathieu group M12. We analogously used the linear group PSL2(7) to produce a set of seven involutions acting on 24 letters which was normalized by our PSL2(7) and generated M24. In this paper we describe the combinatorial and geometric interpretations of our generators, and aim to justify our use of the word ‘natural’. The generators for M12 are defined both as permutations of twelve 5-cycles in a conjugacy class of A5, and as permutations of the faces of a dodecahedron. The outer automorphism of M12 emerges as a bonus. The generators of M24 are defined analogously both as permutations of twenty-four 7-cycles in the action of PSL2(7) on seven letters, and as permutations of the 24 faces of a certain cubic graph drawn on a surface of genus three. In addition we mention the connection between the latter construction and the binary Golay code.

In 1922–23, Julia and Fatou proved that any 2 rational functions f and g of degree at least 2 such that f(g(z)) = g(f(z)), have the same Julia set. Baker then asked whether the result remains true for nonlinear entire functions. In this paper, we shall show that the answer to Baker's question is true for almost all nonlinear entire functions. The method we use is useful for solving functional equations. It actually allows us to find out all the entire functions g which permute with a given f which belongs to a very large class of entire functions.

In this paper, we will study the local cohomology modules HiI(S) of a polynomial ring S = k[x1, …, xn] with supports in a (radical) monomial ideal I. When S/I is a Cohen–Macaulay ring of dimension d (more generally, if Extn−d(S/I, [wfr ]S) is Cohen–Macaulay), we can ‘visualize’ a ℤn-graded minimal injective resolution of Hn−dI(S) using Stanley–Reisner's simplicial complex of I.

Let b ≥ 2 be an integer and let E / be a fixed elliptic curve. In this paper, we estimate the number of primes p ≤ x such that the number of points n E ( p ) on the reduction of E modulo p is a base b prime or pseudoprime. In particular, we improve previously known bounds which applied only to prime values of n E ( p ).

A Fox coloured link is a pair (L,ω), where L is a link in S3 and ω a simple and transitive representation of π1(S3∖L) onto the symmetric group Σ3 on three elements. Here, a representation is called simple if it sends the meridians to transpositions. By works of the first two authors, any Fox coloured link (L,ω) gives rise to a closed orientable 3-manifold M(L,ω) equipped with a 3-fold simple covering p:M(L,ω)→S3 branched over L, and any closed orientable 3-manifold is homeomorphic to an M(K,ω) for some Fox coloured knot (K,ω) [see H. M. Hilden, Bull. Amer. Math. Soc. 80 (1974), 1243–1244; J. M. Montesinos, Bull. Amer. Math. Soc. 80 (1974), 845–846;]. In [Adv. Geom. 3 (2003), no. 2, 191–225;], I. V. Izmestʹev and M. Joswig proved that a triangulation of S3 gives rise in a natural way to some graph G on S3 and a representation of π1(S3∖G) into the symmetric group Σm for some m≤4. They also proved that any pair (L,ω), where L is a link in S3 and ω a simple (not necessarily transitive) representation of π1(S3∖L) into the symmetric group Σ4, can be obtained from a triangulation of S3. The proof that Izmestʹev and Joswig gave of this result is non-constructive. In the paper under review, the authors give a constructive proof of the same result. In particular, given a pair (L,ω) consisting of a link L in S3 and a simple (not necessarily transitive) representation of π1(S3∖L) onto the symmetric group Σ4, they construct a triangulation of S3 that gives rise to (L,ω) in a natural way.

In this paper, we study the residual Julia sets of meromorphic functions. In fact, we prove that if a meromorphic function f belongs to the class S and its Julia set is locally connected, then the residual Julia set of f is empty if and only if its Fatou set F(f) has a completely invariant component or consists of only two components. We also show that if f is a meromorphic function which is not of the form fi + (z i fi) ik e g(z) , where k is a natural number, fi is a complex number and g is an entire function, then f has buried components provided that f has no completely invariant components and its Julia set J(f) is disconnected. Moreover, if F(f) has an inflnitely connected component, then the singleton buried components are dense in J(f). This generalizes a result of Baker and Dom¶‡nguez. Finally, we give some examples of meromorphic functions with buried points but without any buried components.

In this paper we consider the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. Using new exponential sums in tandem with a sieve method we are able to provide stronger “minor arc” estimates than previous authors, thereby improving the saving obtained in the exponent by a factor 8/7.

Multitype Moran sets are introduced in this paper. They appear naturally in the study of the structure of the quasi-crystal spectrum, and they generalize some known fractal structures such as self-similar sets, graph-direct sets and Moran sets. It is known that for any Moran set E with a bounded condition on contracting ratios, one has \[\dim_H E=s_*\le s^*=\dim_P E=\dimB E,\] where .

In this paper we present two versions in several variables of the following result:Theorem 1([2, 3]). Let f be a function in the disc algebra (more generally in H1). Then for every point z0 in the open unit disc, there is an interval I on the unit circle T such that f(z0) = 1/|I| ∫Ifdσ, where 0 < |I| ≤ 2π denotes the length of I and σ the Lebesgue measure on T.

In this paper we prove that for any odd prime p, a Moufang loop of order 2p2 is a group and there is exactly one non-associative right Bol loop of order 2p2.

The purpose of this paper is the proof of an almost everywhere version of the classical central limit theorem (CLT). As is well known, the latter states that for IID random variables Y1, Y2, … on a probability space (Ω, , P) with we have weak convergence of the distributions of to the standard normal distribution on ℝ. We recall that weak convergence of finite measures μn on a metric space S to a finite measure μ on S is defined to mean thatfor all bounded, continuous real functions on S. Equivalently, one may require the validity of (1·1) only for bounded, uniformly continuous real functions, or even for all bounded measurable real functions which are μ-a.e. continuous.

This paper treats the local study of singularities by means of their tangent cones, more specifically the study of graded rings associated to an ideal of a local ring. We recall some basic facts: let (R,) be a local ring,I, Jideals ofR, such thatJ⊆I; thenGR/J(I/J), the graded ring associated toI/J, is canonically isomorphic to the quotient ofGR(I) modulo a homogeneous ideal, which is calledJ*, and which is generated by the so-called ‘initial forms’ of the elements ofJ. Let us consider the following example: Letkbe a field,R=k[X, Y, Z](x, y, Z),I= (X, Y, Z)R, Jthe prime ideal generated byfl,f2wheref1=Y3−Z2,f2=YZ−X4. Thenand it is easily seen thatJ*properly contains the ideal generated by the initial formsf*1f*2off1,f2; namelyf*1= −Z2,f*2=YZand (Yf1+Zf2)* =Y4∉ (−Z2,YZ).

The transfer has long been a fundamental tool in the study of group cohomology. In recent years, symmetric groups and a geometric version of the transfer have begun to play an important role in stable homotopy theory (2, 5). Thus, motivated by geometric considerations, we have been led to investigate the transfer homomorphismin group homology, where n is the nth symmetric group, (n, p) is a p-Sylow sub-group and simple coefficients are taken in /p (the integers modulo a prime p). In this paper, we obtain an explicit characterization (Theorem 3·8) of this homomorphism. Roughly speaking, elements in H*(n) are expressible in terms of the wreath product k ∫ l → n (n = kl) and the ordinary product k × n−k→ n. We show that tr* preserves the form of these elements.

Let N > 1 and let AN be the polydisc algebra, i.e. the algebra of all continuous functions on the closed polydisc δ¯N ⊂ N, analytic on the open polydisc δN, with sup norm. Call a closed set F ⊂ δ¯N a peak interpolation set for AN if given any f ε C(F), f ≠ 0, there is an extension f ε AN of f such that ¦f˜(z)¦ < ‖ f ‖ (z ε δ¯N - F); call F a norm preserving interpolation set for AN if given any f ε C(F) there is an extension f˜ ε AN of f such that ‖f˜‖ = ‖f‖. The paper gives a complete description of norm preserving interpolation sets for AN in terms of peak interpolation sets for AM, M ≤ N.

Let S be a (jointly continuous) topological semigroup, C(S) the set of all bounded complex-valued continuous functions on S and M (S) the set of all bounded complex-valued Radon measures on S. Let (S) (or (S)) be the set of all µ ∈ M (S) such that x → │µ│ (x-1C) (or x → │µ│(Cx-1), respectively) is a continuous mapping of S into ℝ, for every compact set C ⊆ S, and . (Here │µ│ denotes the measure arising from the total variation of µ and the sets x-1C and Cx-1 are as defined in Section 2.) When S is locally compact the set Ma(S) was studied by A. C. and J. W. Baker in (1) and (2), by Sleijpen in (14), (15) and (16) and by us in (3). In this paper we show that some of the results of (1), (2), (14) and (15) remain valid for certain non-locally compact S and raise some new problems for such S.

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.

Throughout, the word knot means a subspace of the 3-sphere S3 homeomorphic with the 1-sphere S1. Any knot can be expressed as a connected sum of a finite number of prime knots in a unique way (13), we consider the trivial knot a non-prime knot. (For higher dimensional knots, factorization and uniqueness have been studied in (1).) However given a knot it is difficult to determine if it is prime or not. We prove that totally knotted knots, see definition in §2, are prime in theorem 1, give a class of examples in theorem 2 and investigate how the last result can be applied to the conjecture that the family Y of unknotting number one knots are prime. (See problem 2 in (5).) At the end, prime tangles as defined by W. B. R. Lickerish are used to prove that in a certain family of knots, related somewhat to Y, there is just one non-prime knot: the square knot. The paper should be interpreted as being in the piecewise linear category. Standard definitions of 3-manifolds and knot theory may be found in (6) and (11) respectively.

In this paper we will be concerned with the problem of describing the Jacobson radical of the semigroup algebraK[S] of an arbitrary semigroupSover a fieldKin the case where this algebra satisfies a polynomial identity. Recently, Munn characterized the radical of commutative semigroup algebras [9]. The key to his result was to show that, in this situation, the radical must be a nilideal. We are going to extend the latter to the case of PI-semigroup algebras. Further, we characterize the radical by means of the properties ofSor, more precisely, by some groups derived fromS. For this purpose we will exploit earlier results leading towards a characterization of semigroup algebras satisfying polynomial identities [5], [15], which generalized the well known case of group algebras (cf. [13], chap. 5).

The purpose of this paper is to characterize the Orlicz vector-valued function spaces containing a copy or a complemented copy of l1. Pisier proved in [13] that if a Banach space E contains no copy of l1, then the space Lp(S, Σ, μ, E) does not contain it either, for 1 < p < ∞. We extend this result to the case of Orlicz vector valued function spaces, by reducing the problem to the situation considered by Pisier. Next, we pass to study the problem of embedding l1 as a complemented subspace of LΦ(E). We obtain a complete characterization when E is a Banach lattice and only partial results in case of a general Banach space. We use here in a crucial way a result of E. Saab and P. Saab concerning the embedding of l1 as a complemented subspace of C(K, E), the Banach space of all the E-valued continuous functions on the compact Hausdorff space K (see [14]). Finally, we use these results to characterize several classes of Banach spaces for which LΦ(E) has some Banach space properties, namely the reciprocal Dunford-Pettis property and Pelczyński's V property.

For a given index set I, let us consider a family (Aν,: ν ∈ I) of subsets of a set E. In this note we deal with some aspects of the following question: to what extent is it possible to prescribe the cardinalities, or the order types in case E is ordered, of the sets Aν and of their pairwise intersections? In (1) the authors have shown that, given any regular cardinal a, there is a family of a+ sets of cardinal a whose pairwise intersections are arbitrarily prescribed to be either less than or equal to a. In Theorem 1 below we prove a stronger result which states that if a is regular, say a = ℵα, and if E is well-ordered and of order type , then one can find a+ subsets Aν, of E, each of type , whose pairwise intersections are arbitrarily prescribed to be either of type ωα or of a type less than ωα. By way of contrast, Theorem 2 below implies – this is its special case m = ℵω; n = ℵ2; p = ℵ0 – that, assuming the Generalized Continuum Hypothesis (GCH), there do not exist ℵω+1 sets Aν, each of cardinal at most ℵω such that ℵ2 of them have pairwise finite intersections, whereas all other pairs of sets Aν have a denumerable intersection. Theorem 3 gives another case in which some type of prescription of the sizes of the intersections cannot be satisfied. Finally, Theorem 4 asserts that in Theorem 3 the condition cfp ≠ cfm cannot be omitted. The paper concludes with some remarks on open questions.

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

In the course of their enumeration of all 4-dimensional space groups, Brown, Bülow, Neubüser, Wondratschek and Zassenhaus have introduced concepts appropriate to the study of n-dimensional crystallography for n ≥ 4 (see (2), (3)). One such concept is that of a crystal family. Families seem to be particularly useful as a framework within which to study higher dimensional crystallography, primarily because they determine a classification of all the standard crystallographic objects and overcome the traditional confusion over crystal systems (see (2); pp. 16–17, (9)). In this paper, techniques are developed for the determination of all rationally decomposable families in a given dimension from the indecomposable families of lower dimensions. These techniques place emphasis on three geometric invariants of families: the decomposition pattern; the canonical decomposition pattern; and the number of free parameters. This, it is felt, further reinforces their position as fundamental objects. The key result (Theorem 5.4) is:a family uniquely determines, and is uniquely determined by, the constituent families in the canonical decomposition.

In this paper, multigraphs will have no loops. For a multigraph G, the least number of colours needed to colour the edges of G in such a way that no two edges on the same vertex of G have the same colour, is called the edge-chromatic number, or the chromatic index, of G, and denoted χ′(G). It is clear that if Δ(G) denotes the maximum degree of G, then Δ(G) ≤ χ′(G). If Δ(G) = χ′(G), then G is Class 1, and otherwise G is Class 2.

The topology most often used on a totally ordered group (G, G x G (e.g., the lexicographic order) but the interval topology induced by such a total order is rarely used since the product topology has obvious advantages. Let ℝ(+) denote the real line with its usual order and Q(+) the subgroup of rational numbers. There is an order on Q x Q whose associated interval topology is the product topology, but no such order on ℝ x ℝ can be found. In this paper we characterize those pairs G, H of totally ordered groups such that there is a total order on G x H for which the interval topology is the product topology.

In this paper every space will be 2-local, for example BO will mean the 2-localization of the space usually denoted BO.

In this paper, we prove that if X is a strictly convex Lk-UR space, then X is an L-kR space. However, the converse need not be true. Also, for each k ≥ 2, there exists a Lk-UR space which is URED but is not L-(k – 1)R.

The centralizer ring of a permutation representation of a group appears in several contexts. In (19) and (20) Schur considered the situation where a permutation group G acting on a finite set Ω has a regular subgroup H. In this case Ω may be given the structure of H and the centralizer ring is isomorphic to a subring of the group ring of H. Schur used this in his investigations of B-groups. A group H is a B-group if whenever a permutation group G contains H as a regular subgroup then G is either imprimitive or doubly transitive. Surveys of the results known on B-groups are given in (28), ch. IV and (21), ch. 13. In (28), p. 75, remark F, it is noted that the existence of a regular subgroup is not necessary for many of the arguments. This paper may be regarded as an extension of this remark, but the approach here differs slightly from that suggested by Wielandt in that it appears to be more natural to work with transversals rather than cosets.

The original motivation for this paper was the question of primitivity for group rings of soluble groups of finite rank. At the end this is touched upon as an application of a theorem about prime ideals in such rings. If a group Γ acts on a set S we say an element is (Γ)-orbital if its orbit is finite and write ΔΓ(S) for the subset of such elements. The FC-radical of a group G, denoted by Δ(G), is just ΔG(G) where the action of G on itself is by conjugation.

LetGbe a finite group,Snbe the symmetric group onnsymbols andAnbe the corresponding alternating group. The conjugacy classes of the wreath productGSn(or monomial group as it is sometimes known) and the conjugacy classes ofGAnhave been described by Kerber (see [2] and [3]). The groupSnhas a double covernso that the faithful complex representations of this double cover may be regarded as protective representations ofSn. In Section 2, a particular double cover forGSnis constructed, the faithful complex representations of this group being the subject of a joint article with Peter Hoffman[1]. In the present paper, our task is to determine whether a conjugacy class ofGSncorresponds to one or to two conjugacy classes in the double cover ofGSn(and similarly forGAn). The main results, Theorems 1 and 2, are stated precisely in Section 2 and proved in Sections 3 and 4 respectively. The case whenG= 1 provides classical results of Schur ([5], Satz IV). WhenGis a cyclic group, Read [4] has determined the conjugacy classes, not just for our particular double cover, but for all possible double covers ofGSn.

Let V be the vector space of translations of a finite dimensional real affine space. The principal aim of this paper is to study (generally non-Euclidean) space groups whose point groups K are ‘linear’ Coxeter groups in the sense of Vinberg (4). This involves the investigation of lattices Λ in V left invariant by K and the calculation of cohomology groups H1(K, V/Λ) (3). The first problem is solved by generalizing classical concepts of ‘bases’ of root systems and their ‘weights’, while the second is carried out completely in the case when the Coxeter graph Γ of K contains only edges marked by 3. An important part in the calculation of H1(K, V/Λ) is then played by certain subgraphs of Γ which are complete multipartite graphs. The only subgraphs of this kind which correspond to finite Coxeter groups are of type Al× … × A1, A2, A3 or D4. This may help to explain why, in our earlier work on space groups with finite Coxeter point groups (3), (2), components of r belonging to these types played a rather mysterious exceptional role.

Smooth mappings defined on discriminant varieties of -versal unfoldings of isolated singularities arise in many interesting geometrical contexts, for example when classifying outlines of smooth surfaces in ℝ3 and their duals, or wave-front evolution [1, 2, 5]. In three previous papers we have classified various stable mappings on discriminants. When the isolated singularity is weighted homogeneous the discriminant is not a local smooth product, and this makes the classification of stable germs considerably easier than in general. Moreover, discriminants arising from weighted homogeneous singularities predominate in low dimensions, so such classifications are very useful for applications.

In (4) the classification of (complex, projective) cubic surfaces by the number and nature of their singularities is carried out. This gives a natural partition of the vector space of cubic surfaces (which we denote by H3(4, 1)). In this paper we investigate the differential geometric properties of this partition; we show that it provides a finite constructible stratification of H3(4,1) which, in the notation of (10), is Whitney (A) regular. In fact Whitney (B) regularity holds over each stratum other than E6, but this stratum of cubic cones has an exceptional (equianharmonic) orbit at which (B) regularity fails. It remains to be seen whether or not this is the only exceptional orbit.

In a 1975 paper [8], Heinig established the following three inequalities:where A = p/(p + s − λ) with p, s, λ real numbers satisfying p + s > λ,p > 0;where B = p/(2p + sp − λ −1) with p, s, λ real numbers satisfying 2p +sp > λ, + 1, p > 0;where is a sequence of nonnegative real numbers,and C = p[l + l/(p + s−λ)] with p, s, λ real numbers satisfying s > 0, p ≥ 1, and p +s > λ 0.

In their paper [5], Klein and Landau prove that given a symmetric ‘local semigroup’ of unbounded operators {T(t); t ≥ 0} on a Hilbert space, there exists a unique selfadjoint operator T such that T(t) is a restriction of e−tT, for each t ≥ 0. A similar representation theorem was proved earlier by Nussbaum [8]. The result of Klein and Landau was recently extended to the setting of reflexive Banach spaces by Kantorovitz ([4], theorem 2–3). More precisely, Kantorovitz presented necessary and sufficient conditions for a local semigroup of unbounded operators {T(t); t ≥ 0} to consist of restrictions of e−tT, t ≥ 0, for some unbounded spectral operator of scalar-type T with real spectrum (cf. [1] for the terminology).

In this paper, together with [7] and [8], we shall be concerned with estimating the number of solutions of the inequalityfor almost all α (in the sense of Lebesgue measure on Iℝ), where, and bothmandnare restricted to sets of number-theoretic interest. Our aim is to prove results analogous to the following theorem (an improvement given in [2] of an earlier result of Khintchine [10]) and its quantitative developments (for example, see [11, 12,6]):Let ψ(n) be a non-increasing positive function of a positive integer variable n. Then the inequality (1·1) has infinitely many, or only finitely many, solutions in integers to, n(n > 0) for almost all real α, according to whether the sumdiverges, or converges, respectively.

In this paper we classify the maximal subgroups of the smallest Fischer group Fi22, which is a simple group of order 64561 751 654 400 = 217. 39. 52. 7. 11. 13. Although this is not a complete classification into conjugacy classes, it is for most practical purposes almost as good, since the exceptional cases are very small groups.

An ordered topological space (E, τ, ≥) is a set E endowed with a topology τ and a partial order ≥. For such a space order separation axioms have been studied by Nachbin (6) and McCartan (3). In this paper we discuss the consequences for these axioms of the imposition, in turn, of four conditions on (E, τ, ≥), namely convexity (6), continuity, anticontinuity and bicontinuity (5).

The group Aut Γ of automorphisms of Rado's universal graph Γ (otherwise known as the ‘random’ graph: see [1]) and the corresponding groups Aut Γc for C a set of ‘colours’ with 2 ≤ |C| ≤ ℵ0, were studied in [4]. It was shown that Aut Γc is a simple group, and the possible cycle types of its members were classified. A natural extension of Aut Γc to a highly transitive permutation group on the same set is obtained by considering the ‘almost automorphisms’ of Γ. It is the purpose of the present paper to answer similar questions about the resulting group AAut Γc. Namely we shall classify its normal subgroups and the cycle types of its members. The main result on normal subgroups is summed up in Corollary 2·9, which says that the non-trivial normal subgroups of AAut Γc form a lattice isomorphic to the lattice of subgroups of the free Abelian group of rank n where n = |C| – 1, and for cycle types it will be shown that those occurring in AAut Γc are precisely the same as in Aut Γc except for those which are the product of finitely many cycles.

In the proof of the Freudenthal–Weil theorem in, for example (5), essential use is made of the fact that if G and H are compact analytic groups and ø: G → H is a continuous epimorphism then ø(Z(G)0) = Z(H)0 where the subscript 0 denotes the identity component of a topological group G and Z(G) its centre. Although this is sufficient for the proof of the Freudenthal–Weil theorem it raises the interesting question as to whether actually ø(Z(G)) = Z(H) (from which the above would follow) and, if so, in what generality this can be expected. The present paper deals with this question, in more general form, as well as certain of its structural consequences.

In 1967 Segal introduced the Dirichlet convolution (, h(n)), generalizing a method of Ingham developed in studies on the Prime Number Theorem. In this paper we establish necessary and sufficient conditions on the sequence h(n) in order that the convolution method (, h(n)) be conservative. Further conditions are established for the method to be absolutely conservative.

An array of random variables, indexed by a multidimensional parameter set, is said to be dissociated if the random variables are independent whenever their indexing sets are disjoint. The idea of dissociated random variables, which arises rather naturally in data analysis, was first studied by McGinley and Sibson(7). They proved a Strong Law of Large Numbers for dissociated random variables when their fourth moments are uniformly bounded. Silver man (8) extended the analysis of dissociated random variables by proving a Central Limit Theorem when the variables also satisfy certain symmetry relations. It is the aim of this paper to show that a Strong Law of Large Numbers (under more natural moment conditions), a Central Limit Theorem and in variance principle are consequences of the symmetry relations imposed by Silverman rather than the independence structure. To prove these results, reversed martingale techniques are employed and thus it is shown, in passing, how the well known Central Limit Theorem for U-statistics can be derived from the corresponding theorem for reversed martingales (as was conjectured by Loynes(6)).

The notion of packing measure, introduced in [12], [13] and [10], has been used by comparison with Hausdorif measure to study the regularity and rectifiability of sets in the plane [11]. Since a few technical mistakes can be found in [10], lemma 5·11 and [11], lemma 3·2, we wish in this paper to give the corresponding exact proofs, together with a natural development of this theory in higher dimensions.

This paper relates the concept of n-connection for graphs to Tutte's theory of n-connection for matroids (12). In particular, we show how Tutte's definition may be modified to give a matroid concept directly generalizing the graph-theoretic notion of n-connection.