A curve has increasing chords if AD ≥ BC for any four points A, B, C, D lying on the curve in that order. The length of such a curve that connects two points at distance 1 is at most 2π/Z in two dimensions, which is the optimal bound, and less than 30 in three dimensions.

Let d be a positive square-free number. Let Od be the ring of integers in Q(√ −d). The groups PSL(2, Od) are called collectively the Bianchi groups. The extended Bianchi group Bd is the maximal discrete extension of PSL(2, Od) in PSL(2, C). The group Bd acts by linear fractional transformations on the complex plane C. Let RBd be Bd with the generator θ, , adjoined. (RBd will be also called an extended Bianchi group (cf. [18])). A group is said to be reflective if it contains a Coxeter subgroup (i.e. a subgroup generated by reflections) of finite index. The groups RBd and their subgroups have been investigated in [3, 6, 13, 15, 16, 17, 18, 20, 21]. In 1892 Bianchi[3] proved that PGL(2, Od)⋊{θ} is reflective if d ≤ 19, d ╪ 14 or 17. Vinberg [18] proved that if the group RBd is reflective, then the orders of all elements of the ideal-class group of the field Q(√ −d) should divide 4. Shaiheev [16] proved that there are only finitely many reflective extended Bianchi groups and found all of them for d ≤ 30. Similar results are obtained for groups PGL(2, Od) ⋊ {θ} which, as Shvartsman [17] showed, are reflective only when d = 1, 2, 5, 6, 10, 13, 21, provided d = 1 or 2 (mod 4).

Our primary objective is to give necessary and sufficient conditions for a triangular subalgebra of a groupoid C-algebra to be semisimple, i.e. to have vanishing Jacobson radical. If, in addition, the subalgebra is the analytic subalgebra determined by a real-valued cocycle on the groupoid, then we can give an explicit description of the radical in terms of the cocycle. As a consequence of this analysis, we are able to determine when certain analytic crossed products are semisimple.

An ideal I in a ring R is primal if whenever J1, …, Jn is a finite set of ideals of R with J1 … Jn = {0} then Ji ⊆ I for at least one i ∈ {1, …, n}. If the ring is commutative then is is easily shown that each primal ideal contains a prime ideal. In this paper it is shown that in a separable semi-simple Banach algebra each primal ideal contains a prime ideal, and that the space of minimal prime ideals is compact and extremally disconnected in the hull-kernel topology. An example is given of an inseparable C-algebra with a primal ideal which does not contain a prime ideal.

The protective unitary representations of the mapping class groups of surfaces corresponding to the Jones–Witten topological quantum field theory for SU(2) are expressed as representations in algebras of skeins in the surface. The skein-theoretic construction of the representations uses neither Kirby's surgery theorem nor a presentation of the group. Using these representations and the Reidemeister–Singer classification of Heegaard splittings gives a proof of the existence of the moduli of the Witten invariants of 3-manifolds.

Let (R,m) be a local, noetherian, d-dimensional ring and let M be a finitely generated R-module. Since the local cohomology modules are artinian, is finitely generated for all i and j (see [4], Remark 1*middot;3 and 2·1). Grothendieck[2] made the following conjecture: If I is an ideal of a noetherian ring R, thenis finitely generated for all j.

Let L ⊂ S3 be a knot or a link. A tunnel system for L is a collection of disjoint arcs t1, … ts properly embedded in S3−N(L) so that is a handlebody. Alternatively one can view the t1, … ts ascocores of 2-handles which have to be attached to a handlebody in order to obtain S3−N(L). It follows immediately that this handlebody is of genus s+1. The tunnel number of L, denoted by t(L), is defined to be the minimal cardinality of all such collections.

All spaces are assumed to be locally path connected and semi-locally 1-connected. Let X be a connected topological group with identity e, and let be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for any point ẽ in with pẽ = e there is a unique structure of topological group on such that ẽ is the identity and is a morphism of groups. We say that the structure of topological group on X lifts to .

Let M be a closed manifold of dimension 3 and let Fol(M) be the space of codimension one C∞-foliations on M. A foliation ∈ Fol(M) is said to be Cr- structurally stable if there exists a neighbourhood V of in Fol(M) in the (Epstein) Cr-topology such that every foliation is topologically conjugate to , through a homeomorphism near to the identity. Some background on the problem of structural stability of foliations can be found in [8]. In this paper we shall be concerned with proper foliations, i.e. foliations all of whose leaves are proper.

Given a surface homeomorphism isotopic to the identity which is pseudo-Anosov relative to a finite set, we show that the sum of the Lefschetz numbers of periodic points of any period greater than one is non-negative. If this period is odd and greater than a number which depends only on the surface, the sum is zero. If we consider sequences of periods such that each element is twice that of its predecessor, then this sum is increasing beyond a certain point also depending on the surface. As a corollary, for each periodic orbit contained within the boundary of the surface there exists one of the same period contained in the interior.

Motivated by recent results on the space of compact operators between Banach spaces and by extensions of the Josefson–Nissenzweig theorem to Fréchet spaces, we investigate pairs of Fréchet spaces (E, F) such that every continuous linear map from E into F is Montel, i.e. it maps bounded subsets of E into relatively compact subsets of F. As a consequence of our results we characterize pairs of Köthe echelon spaces (E, F) such that the space of Montel operators from E into F is complemented in the space of all continuous linear maps from E into F.

This paper is a sequel to [6] and concerns the complex divisor function

which has had a number of applications in the analytic theory of numbers. Thus Hooley's Δ-function [8] defined by

satisfies the inequality

and Erdös' τ+-function, defined by

satisfies

This paper is a continuation of [4]; its aim is to extend the results of that paper to include abstract Sobolev spaces of higher order and even anisotropic spaces. Let Ω be a domain in ℝN, let X = X(Ω) and Y = Y(Ω) be Banach function spaces in the sense of Luxemburg (see [4] for details), and let W(X, Y) be the abstract Sobolev space consisting of all f ∈ X such that for each i ∈ {l, …, N} the distributional derivative belongs to Y; equipped with the norm

W(X, Y) is a Banach space. Given any weight function w on Ω, the triple [w, X, Y] is said to support the Poincaré inequality on Ω. if there is a positive constant K such that for all u ∈ W(X, Y)

the pair [X, Y] is said to support the Friedrichs inequality if there is a positive constant K such that for all u ∈ W0(X, Y) (the closure of in W(X, Y))

Several maximal inequalities of Kahane–Khintchine's type in certain Orlicz spaces are proved. The method relies upon Lévy's inequality and the technique established in [14] which is obtained by Haagerup–Young–Stechkin's best possible constants in the classical Khintchine inequalities. Moreover by using Donsker's invariance principle it is shown that the numerical constant in the inequality deduced by the method presented is nearly optimal: If is a Bernoulli sequence, and ‖ · ‖ψ denotes the Orlicz norm induced by the function then the following inequality is satisfied:

for all a1,…, an and all n ≥ 1, and the best possible numerical constant which can take the place of lies in the interval ]. Sharp estimates of this type are also deduced for some other maximal inequalities in Orlicz spaces discovered in this paper.