The unstable tangent fibration of a Poincaré complex is defined so that it is consistent with the manifold case. It exists uniquely for each Poincaré complex X up to fibrewise homotopy equivalence and, furthermore, if a Poincaré embedding structure exists on the diagonal X→X×X, its normal fibration is the tangent fibration.

Recently Bost and Connes have studied an interesting C*-algebraic Hecke algebra arising in number theory. Here it is shown that this algebra can be realised as a semigroup crossed product, and be profitably studied using methods developed by the authors for analysing Toeplitz algebras. One main result is a characterisation of faithful representations of the Hecke algebra.

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.

In this paper we study quaternion-Kähler manifolds endowed with an isometric S1-action. We consider the corresponding moment map μ and prove that the only compact quaternion-Kähler manifold with positive scalar curvature which admits an isometric circle action free on μ−1(0) is the quaternionic projective space ℍPn.

Let f:M→N be a stable map between orientable 4-manifolds, where M is closed and N is stably parallelisable. It is shown that the signature of M vanishes if and only if there exists a stable map g:M→N homotopic to f which has only fold and cusp singularities. This together with results of Ando and Èliašberg shows that, in this situation, the Thom polynomials are the only obstructions to the elimination of the singularities except for the fold singularity. Also studied are some topological properties (including those of the discriminant set) of stable maps between 4-manifolds with only Ak-type singularities.

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.

Every compact, connected PL manifold Mn, with ∂Mn≠[emptyv ], collapses to a codimension-one subpolyhedron Qn−1, called a spine of Mn. The purpose of this paper is to prove that, if Qn−1 is appropriately chosen, one can reconstruct Mn from Qn−1, after taking the Cartesian product with an interval I=[0, 1].

Let F be a non-lattice distribution function which lies in the domain of attraction of a non-normal stable distribution. Exact uniform convergence rates are obtained for the convergence of the normalised partial sums of random variables with distribution F. Second order regular variation conditions are assumed related to the domain of attraction conditions.

In a previous article [11] we studied the central limit theorem for infinitesimal triangular arrays of probability measures on a Lie group. Since the work of Berg [2] and more recently of Bendikov [1] similar studies appeared to be urgent for the infinite-dimensional torus group and beyond that for the class of Lie projective groups which among others contains all compact groups. Consequently we started extending the existing theory within the enlarged framework of Lie projective groups. In the present contribution continuous convolution hemigroups (μ(s, t)) of probability measures on a Lie projective group G are investigated with respect to generation, representation and occurrence as limits of non-commutative infinitesimal triangular arrays.

The layout of our exposition is as follows. In Section 2 some facts on Lie projective groups G with Lie algebra [Lscr ](G), Lie system [Hfr ], projective basis (Xi)i∈I and corresponding projective weak coordinate system (xi)i∈I are collected to make the reader familiar with the setting. Section 3 contains the basic methodical result yielding the intended generalisations. It is shown in Proposition 3.3 that a hemigroup (μ(s, t)) in the set [Mfr ]1(G) of probability measures on G corresponds to a triplet (a, B, η) in the set ℙbv(ℝ+, G) of characteristics of bounded variation if and only if for each H∈[Hfr ] the projection (pH(μ(s, t))) in [Mfr ]1(G/H) corresponds to a certain triplet (aH, BH, ηH) in the set ℙbv(ℝ+, G/H). Here the correspondence between hemigroups and triplets is achieved via weak backward evolution equations with respect to (Xi)i∈I and (xi)i∈I (see Definition 3.2). Applying the reduction method based on this result, the convergence of infinitesimal arrays {μn[lscr ][ratio ](n,[lscr ])∈ℕ2} in [Mfr ]1(G) towards hemigroups (μ(s, t)) of continuous weak bounded variation is proved in Section 4. In fact, sufficient conditions involving the triplet (a, B, η) are given in order that the sequence (μn (s, t))n[ges ]1 of ‘scaled’ convolution products μn, kn(s)+1[midast ]…[midast ] μn, kn(t) converges weakly to (μ(s, t)). Moreover, necessary and sufficient conditions are given for the limiting hemigroup to be a diffusion hemigroup. Section 5 is devoted to the existence and uniqueness results leading to the one-to-one correspondence between the set ℙbv(ℝ+, G) and the set of hemigroups of continuous weak bounded variation on G. In Section 6 we illustrate the technique of Lie projectivity of the infinite-dimensional torus group and of the p-adic solenoidal group. Moreover, we sketch a little-known example of a hemigroup appearing in atomic physics and derive from it some thoughts towards a perturbation theory for hemigroups.

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

formula here

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.