It is shown that the compact matrix quantum groups SUq(2) are non-isomorphic to each other for q∈[−1, 1][setmn ]{0}, and that the compact matrix quantum groups SUq(n) are non-isomorphic to each other for q∈(0, 1]. Some invariants for compact quantum groups are also discussed.

The exact representation of symmetric polynomials on Banach spaces with symmetric basis and also on separable rearrangement-invariant function spaces over [0, 1] and [0, ∞) is given. As a consequence of this representation it is obtained that, among these spaces, [lscr]2n, L2n[0, 1], L2n[0, ∞) and L2n[0, ∞)∩L2m[0, ∞) where n, m are both integers are the only spaces that admit separating polynomials.

A weight system is defined from the (multivariable) Conway potential function. We also show that it can be calculated recursively by using five axioms.

It is proved that, for each pair (m, n) of non-negative integers, there is a Banach space [Xfr ] for which K0([Bscr]([Xfr ]))≅ℤm and K1([Bscr]([Xfr ]))≅ℤn. The K-groups of all closed ideals of operators contained in the ideal of strictly singular operators are computed, and some results about the existence of splittings of certain short exact sequences are derived.

We first establish a combinatorial result on deterministic real chains. This is then applied to prove a path transformation for chains with exchangeable increments. From this transformation we derive an identity on order statistics due to Port, together with some extensions. Then we give an interpretation of these results in continuous time. We extend some identities involving quantiles and occupation times for processes with exchangeable increments. In particular, this yields an extension of the uniform law for bridges obtained by Knight.

The equations of motion of an ideal incompressible liquid drop trapped between two parallel plates under the influence of surface tension and adhesion forces are studied. A main result of this paper is the proof that the equations of motion can be written in Hamiltonian form

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Here [Dscr] denotes a class of real-valued functions on the phase space [Nscr] of the system and the Hamiltonian H∈[Dscr] is the energy function of the system. This allows the derivation of an equation for the (dynamic) contact angle, in which the free fluid surface meets the plates. The behaviour of the dynamic contact angle is a point of great controversy in the capillarity literature and the derivation confirms one of the existing models. In the second part of the paper, which can be read independently, existence and stability questions for rigidly rotating drops are dealt with. The existence of solutions to the equations of motion that describe rotationally symmetric drops which rotate rigidly between the plates with constant angular velocity is proved. These solutions can be regarded as relative equilibria of a mechanical system with symmetry. Using ideas of the energy-momentum method of Lewis, Marsden and Simo, a stability criterion for this kind of motion is provided. To derive this criterion, the second derivative of the so-called augmented energy functional at the relative equilibrium in directions which are transversal to the group orbit of this equilibrium is studied. The stability criterion is applied to rigidly rotating drops of cylindrical shape. These represent solutions to the equations of motion in the case that no adhesion forces act along the plates. The result extends previous work of Vogel and Lewis.