Vector fields on complex quantum groups

Using previous results we construct theq-analogues of the left invariant vector fields of the quantum enveloping algebra corresponding to the complex Lie algebras of typeA _n–1 ,B _n ,C _n , andD _n . These quantum vector fields are functionals over the complex quantum groupA. In the special caseA ₁ it is shown that this Hopf algebra coincides withU _q sl(2, ).     

Complex quantum enveloping algebras as twisted tensor products

We introduce a *-structure on the quantum double and its dual in order to make contact with various approaches to the enveloping algebras of complex quantum groups. Furthermore, we introduce a canonical basis in the quantum double, its universalR-matrices and give its relation to subgroups in the dual Hopf algebra.     

Elastica hypoarealis

We examine the equilibria of a rigid loop in the plane, characterized by an energy functional quadratic in the curvature, subject to the constraints of fixed length and fixed enclosed area. Whereas the only non self-intersecting equilibrium corresponding to the fixed length constraint is the circle, the area constraint gives rise to distinct equilibria labeled by an integer. These configurations exhibit self-intersections and bifurcations as the area is reduced. In addition, not only can the Euler-Lagrange equation be integrated to provide a quadrature for the curvature but the embedding itself can be expressed as a local function of the curvature. Perturbations connecting equilibria are shown to satisfy a first order ODE which is readily solved. Analytical expressions for the energy as a function of the area are obtained in the limiting regimes. Received 18 October 2001 / Received in final form 31 May 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: capo@fis.cinvestav.mx RID="b" ID="b"e-mail: chryss@nuclecu.unam.mx RID="c" ID="c"e-mail: jemal@nuclecu.unam.mx     

Rheology of shear thickening suspensions and the effects of wall slip in torsional flow

The rheological characterisation of concentrated shear thickening materials suspensions is challenging, as complicated and occasionally discontinuous rheograms are produced. Wall slip is often apparent and when combined with a shear thickening fluid the usual means of calculating rim shear stress in torsional flow is inaccurate due to a more complex flow field. As the flow is no longer “controlled”, a rheological model must be assumed and the wall boundary conditions are redefined to allow for slip. A technique is described where, by examining the angular velocity response in very low torque experiments, it is possible to indirectly measure the wall slip velocity. The suspension is then tested at higher applied torques and different rheometer gaps. The results are integrated numerically to produce shear stress and shear rate values. This enables the measurement of true suspension bulk flow properties and wall slip velocity, with simple rheological models describing the observed complex rheograms.     

Homolytic Substitution Reaction at a Silicon Atom

Tris(trimethylsilyl)silane ( 2 ) is an efficient hydrosilylating agent of 1,6-dienes 1 operating via a free-radical chain mechanism. The intermediate alkyl radical 3 attacks the second olefinic bond of the diene 1 and forms cyclopentylalkyl radical 4 . Using a low concentration of the hydrogen donor 2 , a radical substitution reaction at the Si-center of cis- 4 occurs and yields the bicyclic silane 6 . The rate of this homolytic substitution at the Si-atom is 2.4 · 10⁵ s^-1 (80°).