Let G be a group. A subset D will be called a set of 3-transpositions if |x| = 2 for xεD and |xy| = 3 whenever x, yεD do not commute. We will call the set D closed if xDx−1 = D for each xεD. For each xεD, let

For each subset X of D, we denote by [X] the graph with vertex set X where two elements x, yεX are joined by an edge whenever they commute. We denote by (X) the complement graph; thus two elements x, yεX are joined by an edge of (X) whenever they do not commute.

Let D be a division ring with central subfield F, n a positive integer and G a subgroup of GL(n, D) such that the F-subalgebra F[G] generated by G is the full matrix algebra Dn×n. If G is soluble then Snider [9] proves that G is abelian by locally finite. He also shows that this locally finite image of G can be any locally finite group. Of course not every abelian by locally finite group is soluble. This suggests that Snider's conclusion should apply to some wider class of groups.

An ordinal is termed unsound if it has subsets An (nεω) such that un-countably many ordinals are realised as order types of sets of the form ∪ {An|nε a} where a ⊆ ω. It is shown that if ω1 is regular and then the least unsound ordinal is exactly but that if ω1 is regular and the least unsound ordinal, assuming one exists, is at least . Arguments due to Kechris and Woodin are presented showing that under the axiom of determinacy there is an unsound ordinal less than ω2. The relation between unsound ordinals and ideals on w is explored. The paper closes with a list of open problems.

Let (Dn) be the apollonian packing of a curvilinear triangle T, ρn the radius of Dn, E = T—U Dn the residual set, dim (E) its Hausdorff dimension. In this paper we give a new proof of the equality dim proved by Boyd [2]. Our technique is to construct a sequence of regular triangles covering E, and suitable measures μkcarried by E which allow us to apply a density theorem.

Let M (the mirror) be a plane oval (a smooth curve without inflexions), and let sεℝ2\M be the light source. Rays of light emanating from s are reflected by M, and the caustic by reflexion of M relative to s is the envelope of these reflected rays. In this article we suppose that M is generic (the precise assumption is stated later) and that s moves along a smooth curve in the plane; we are then able to describe how the local structure of the caustic changes. In order to state the result we recall a few facts from [3].

Let M be a plane oval (a smooth curve without inflexions). In this note we show that a generic such M (where the precise assumptions will be stated later) has to have at least one sextactic point, that is a point p where the unique conic touching M at p with at least 5-point contact actually has 6-point contact. This existence problem came into prominence whilst [2] was being written. It was hoped to use the existence of sextactic points to show that the Morse transition on a 1-parameter family of focoids with signature 0 or 2 could not occur. The problem proved to be remarkably stubborn, however. Indeed, the geometric interpretation of sextactic points as given in § 3 was totally unexpected.

For any locally compact topological group G let M(G) denote the topological semigroup of all probability (Borel) measures on G, furnished with the weak topology and with convolution as the multiplication. A Gauss semigroup on G is a homomorphism t→ μt of the strictly positive reals (under addition) into M(G) such that

(i) no μt is a point mesaure,

(ii) for each neighbourhood V of 1 in G we have

We define two notions of order for periodic orbits of circle maps, and describe some of the consequences of possession of a badly ordered periodic orbit.

A method of constructing projective representations of separable locally compact groups in reproducing kernel Hilbert spaces is presented, based on the generalized inducing process of Rieffel and Fell. Examples show that the method can be used to construct some well-known holomorphically induced representations. Some representations on cohomology spaces are also described.

For a classical knot K in the 3-shere, the unknotting number u(K) is defined to be the smallest number of crossing changes required to obtain the unknot, the minimum taken over all the regular projections. This dependence on projection makes the unknotting number a difficult knot invariant. While some algebriac methods exist to give a lower bound for u(K) the unknotting number for approximately one-sixth of the 84 knots with nine or fewer crossings remains unddetermined, see [9] or [7]. For an upper bound, one is usually forced to intellgently experiment knotting various projections of the knot under study. Usual practice is to work with a minimal crossingprojection; indeed, it has long been a ‘folk’ conjecture that the unknotting number is realized in such a projection ([5], p. 21). This note shows by example that this conjecture is false. This remarkable knot is rational, i.e. a 2-bridge knot, and hence alternating. Thus there is also the surprising result that the unknotting number of an alternating kniot is not necessarily realized in a minimal alternating projecting.

It was first proved by R. Lashof in [4], using the work of S. Cappell and J. Shaneson on four-dimensional surgeryu (see [1]), that there exist locally flat topological knots S3 ∪ S5 which are not smoothable. In [2] (compare also [6]) S. Cappell and J. Shaneson have constructed infinitely many non-smoothable locally fat topological knots as the fixed points of locally nice (= locally smoothable) Zp actions on S5, therefore giving non-trivial examples of locally smoothable but equivariantly non-smoothable actions of Zp on S5.

Let (Ω, ,μ) be a probability space, and let be a sub-sigma-algebra of . Let X be a uniformly convex Banach space. Let A =L∞(Ω, , μ X) denote the Banach space of (equivalence classes of) essentially bounded μ-Bochner integrable functions g: Ω.→ X, normed by the function ∥.∥∞ defined for g ∈ A by

(cf. [6] for a discussion of this space). Let B = L∞(Ω, , μ X), and let f ε A. A sufficient condition for g ε B to be a best L∞-approximation to f by elements of B is established herein.

Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. The (spatial) numerical range of an operator TεB(X) is defined as the set

If V(T) ⊂ ℝ, then T is called hermitian. More about numerical ranges may be found in [8] and [9].

In this note we introduce the concepts of Λ-Mackey sequence, Λ-Mackey convergence property, Λ-Schwartz family and associated Λ-Schwartz family and consider some applications of these to locally convex spaces. Hereby Λ denotes a Banach sequence space with the AK-property — the results of this paper generalize those in [4] where the case Λ = I1 is considered. We obtain a dual characterization of those locally convex spaces which satisfy the Λ-Mackey convergence property and characterize the dual Λ-Schwartz spaces in terms of the SM-property which is introduced in [10]. Finally, necessary and sufficient condition for a locally convex space to be ultra-bornological is proved.

Let μ be a finite complex Borel measure supported on the unit circle

.

In this paper, we are concerned with the characterization of the sets of functions satisfying the generalized mean value equation of the form

.

and for all ξ ∈ , | ξ | = R for some fixed R > 0.

Let Xnn ≥ 1, be i.i.d.r.v.'s and Sn = X1+…+Xn. Let be Sn minus the r terms of largest absoluete value. Maller proved that if coverages in distribution to N(0, 1) then so does (Sn/bn)−an, assuming that Xn have a continuous symmetric distribution. We show that his resul;t is true without these extra assumptions. Some related results are also given.

Laws of large numbers and central limit theorems are proved for sums of general functions of m-spacings from general distributions. Explicit formulae are given for the norming constants. The results enable us to describe asymptotic properties of distributional tests under fixed alternatives. A generalization of Kimball's spacings test is considered in detail.

A notion of ‘uniform ε-independence’ (u.ε.i.) is proposed for a sequence {Xn} successively indexed by random indices {τk}. The u.∊.i. property yields results other than those in the previous random indexing literature. Complementing the u.∊.i. property by suitable ‘approximation’ one recovers these previous results.

The uniformity of a certain shape density of a random ΠD tetrad relative to Lebesgue measure along part of the set corresponding to alignment is shown to be sufficient under certain technical conditions for the generating distribution to be the uniform distribution on a compact convex set. A counter-example is provided to show that this result fails for random IID triangles. The main theorem is used to characterize circular, elliptical and triangular uniform generators from the shape distribution of random IID tetrads.

Examples were exhibited in [4] of both reducible and irreducible symmetric operators (of deficiency index (1:1)) associated with − d2/dt2 in the Hilbert space L2(I) (I = [0,1). Such symmetric operators are determined by three linearly independent boundary conditions which define their domains as restrictions of the domain of the maximal operator associated with — d2/dt2.