We find the adjoint of the Askey–Wilson divided difference operator with respect to the inner product on L2(–1, 1, (1– x2)½dx) defined as a Cauchy principal value and show that the Askey-Wilson polynomials are solutions of a q-Sturm–Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the Askey-Wilson operator.

Ankeny–Artin–Chowla obtained several congruences for the class number hk of a quadratic field K, some of which were also obtained by Kiselev. In particular, if the discriminant of K is a prime number p ≡ 1 (mod 4) and ε = t + u √p/2 is the fundamental unit of K, then

In this paper we conduct a systematic, computerized search for groups generated by small, but highly symmetric, sets of involutions. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. The techniques of symmetric generation developed elsewhere are described afresh, and the results are presented in a convenient tabular form.

Extra special p-groups are groups which are central extensions of ℤ/p by elementary abelian p-groups. The cohomology ring of these groups occupies an important place in equivariant cohomology theories and in representation theories. Quillen[6] decided the cohomology for p = 2 and Tezuka-Yagita [7] studied the varieties defined by its mod p cohomology for odd prime p. However, for applications we need more information about the behaviour of the Hochschild-Serre spectral sequence (see [3] p. 295, [1] p. 6).

Let Bn denote the Artin braid group on ‘n-strings[ and PBn its normal subgroup consisting of all the pure braids [Bi, Mo]. These groups have been considerably scrutinized by both topologists and algebraists [BL]. One question whose answer has so far eluded us is whether or not the Gassner representation G: PBn → Mn × n(λ), into the group of n-by-n matrices over , is faithful (see Section 1) [Bi; ·3] [Ga]. Recently the less discriminating Burau representation B: PBn → Mn × n(Z[t±1] ) was shown to have a non-trivial kernel for each n ≥ 6 [M, LP] but these techniques have not yet yielded an element of kernel(G). This paper is a partial step in that direction.

This paper is closely related to Bar-Natan's work, and fills in some of the gaps in [1]. Following his analogy of the extension of knot invariants to knots with double points to the notion of multivariate calculus on polynomials, we introduce a new notation which facilitates the formulation of a Leibniz type formula for the product of two Vassiliev invariants. This leads us to see how Bar-Natan's co-product of chord diagrams corresponds to multiplication of Vassiliev invariants. We also include a proof that the multiplication in is a consequence of Bar-Natan's 4T relation.

The last part of this paper consists of a proof that the space of weight systems is a sub-Hopf algebra of the space *, by means of the canonical projection.

The complete lattice generated by a totally atomic CSL ℒ and the projection lattice of a von Neumann algebra ℛ, commuting with ℒ, is reflexive. From this it follows that the strongly closed lattice generated by any CSL ℒ and the projection lattice of a properly infinite von Neumann algebra ℛ, commuting with ℒ, is reflexive.

Essential laminations in 3-manifolds, introduced in [GO], are a natural generalization of both incompressible surfaces and foliations without Reeb components. One of the main reasons why essential laminations are of particular interest is that many manifolds which contain no incompressible surfaces contain essential laminations. Indeed, there is some evidence that almost all irreducible manifolds with infinite fundamental group contain essential laminations [B1; D; GM; GO; H; N; R], the only known exceptions being Seifert manifolds [B2]. So essential laminations are potentially a quite powerful tool for studying non-Haken 3-manifolds.

Let X be an SQp-space, i.e. a quotient of a subspace of some Lp-space. Let B ⊂ B(X) be a subalgebra of all bounded operators on X and let I ⊂ B be a closed ideal. We show that the quotient algebra B/I is isometrically homomorphic to a subalgebra of B(Y) for some SQp-space Y. This generalizes a theorem of Bernard and Cole, corresponding to p = 2, which states that any quotient of an operator algebra is an operator algebra.

Let (fn) and (gn) be two sequences of random variables adapted to an increasing sequence of σ-algebras (ℱn) such that the conditional distributions of fn and gn given ℱn coincide. Suppose further that the sequence (gn) is conditionally independent. Then it is known that where the number C is a universal constant. The aim of this paper is to extend this result to certain classes of Orlicz and rearrangement invariant spaces. This paper includes fairly general techniques for obtaining rearrangement invariant inequalities from Orlicz norm inequalities.

In this paper, we will investigate discrete coactions on Hilbert C*-modules. In particular, we obtain a one-to-one correspondence between Hilbert C*-modules with discrete coactions and Hilbert C*-modules over the crossed products of the original C*-algebras which satisfies some nice properties (see 3·6 and 3·7). Then we will give some applications of this correspondence in the last three sections.

In Section 2, we give some results about discrete coactions on Hilbert C*-modules which mainly correspond to those about discrete coactions on C*-algebras (see [7]).

Let K be a knot in the 3-sphere S3, N(K) the regular neighbourhood of K and E(K) = cl(S3−N(K)) the exterior of K. The tunnel number t(K) is the minimum number of mutually disjoint arcs properly embedded in E(K) such that the complementary space of a regular neighbourhood of the arcs is a handlebody. We call the family of arcs satisfying this condition an unknotting tunnel system for K. In particular, we call it an unknotting tunnel if the system consists of a single arc.

For nilpotent spaces p-completion is well behaved and reasonably well understood. By p–completion we mean Bousfield–Kan completion with respect to the field Fp [BK]. For non-nilpotent spaces the completion process often has a chaotic effect, this is true even for small spaces. One knows, however, that the classifying space of a compact Lie group is Fp-good even though it is usually non-nilpotent.

We introduce an associative glueing operation ⊕q on the space of solutions of the Quantum Yang–Baxter Equations of Hecke type. The corresponding glueing operations for the associated quantum groups and quantum vector spaces are also found. The former involves 2×2 quantum matrices whose entries are themselves square or rectangular quantum matrices. The corresponding glueing operation for link-invariants is introduced and involves a state-sum model with Boltzmann weights determined by the link invariants to be glued. The standard su(n) solution, its associated quantum matrix group, quantum space and link-invariant arise at once by repeated glueing of the one-dimensional case.

Let Ω be a non-empty open set in ℝn with finite ‘volume’ (n-dimensional Lebesgue measure). Let be the Laplacian operator. Consider the eigenvalue problem (with Dirichlet boundary conditions):

where λ ∈ ℝ and u is a non-zero member of (the closure in the Sobolev space H1(Ω) of the set of smooth functions with compact support contained in Ω). It is well known that the values of λ∈ℝ for which (1·1) has a non-zero solution are positive and form a discrete set. Moreover, for each λ, the associated eigenspace is finite dimensional. Let the spectrum of (1·1) be denoted where 0 < λ1 ≤ λ2 ≤ … and where the multiplicity of each λ in the sequence is the dimension of the associated eigenspace. Let

We prove the existence of a piecewise analytic expanding map associated to certain Kleinian groups without parabolics acting in the 3-dimensional hyperbolic space. These groups have a fundamental domain ℛ with the property that the geodesic planes containing each face are part of the tesselation. We use this map together with the methods of thermodynamic formalism to give another proof that the Selberg zeta function for such groups has a meromorphic extension to ℂ.