Unitary representations of the braid group and corresponding link polynomials are constructed corresponding to each irreducible representation of a quantum double finite group algebra. Moreover the diagonal form of the braid generator is derived from which a general closed formula is obtained for link polynomials. As an example, link polynomials corresponding to certain induced representations of the symmetric group and its subgroups are determined explicitly.

In this paper we prove that the equivalence of anisotropic principal series of a free group Г related to different generator sets induces a Г-isomorphism between the related Cayley-graphs. As a consequence we obtain that a nontrivial change of generators for Г leads to inequivalent anisotropic principal series.

We consider the classifying set, denoted ℂ/˜ below, introduced by Mahler and we show that it can be endowed with non-discrete, Hausdorff topologies (even local distances) based on diophantine approximation properties of (complex) numbers. We then establish several results on the pointed topological space obtained, which could be dubbed Mahler's space (of first degree). We recover an analogue of Mahler's original classification by considering local distances to the special point (that is the class of all algebraic numbers). The main ingredients used here are diophantine properties in dimension zero over ℚ and non-standard analysis.

We determine explicitly the space of invariant Hermitian and Kähler metrics on the flag manifold. In particular, we show that a Killing metric is not Kähler. The Chern forms are also computed in terms of the Maurer–Cartan form, and this calculation is used to prove that the flag manifold is projective algebraic. An explicit projective embedding of the flag manifold is also given.

We introduce in this paper the space of bounded holomorphic functions on the open unit ball of a Banach space endowed with the strict topology. Some good properties of this topology are obtained. As applications, we prove some results on approximation by polynomials and a description of the continuous homomorphisms.

In this paper, we prove that there exist three finite sets Sj (j = 1, 2, 3) such that any two non-constant meromorphic functions f and g satisfying Ef(Sj) = Eg(Sj) for j = 1,2,3 must be identical. As a particular case of the above result, we obtain that there exist two finite sets Sj (j = 1, 2) such that any two non-constant entire functions f and g satisfying Ej(Sj) = Eg(Sj) for j = 1, 2 must be identical, which answers a question posed by Gross.

Some minimax inequalities are first proved both in the compact case and in the non-compact case using the concept of escaping sequences introduced by Border. Applications are given to deduce a generalisation of the Gale-Nikaido-Debreu Lemma due to Mehta and Tarafdar and to obtain a new generalisation of the Gale-Nikaido-Debreu Lemma from which the corresponding generalisation due to Grandmont is derived.

The Cahn-Hilliard system, a natural extension of the single Cahn-Hilliard equation in the case of multicomponent alloys, will be shown to generate a dissipative semigroup on the phase space ℋ = [H2(Ω)]m. Following Hale's ideas and based on the existence and form of the Lyapunov functional, our main result will be the existence of a global attractor on a subset of ℋ. New difficulties specific to the system case make our problem interesting.

In this paper we show that the method of calculating the Gaussian period polynomial which originated with Gauss can be replaced by a more general method based on formulas for Lagrange resolvants. The period polynomial of cyclic sextic fields of arbitrary conductor is determined by way of example.

We show that a Banach space has modulus of convexity of power type p if and only if best approximants to points from straight lines are uniformly strongly unique of order p. Assuming that the space is smooth, we derive a characterisation of the best simultaneous approximant to two elements, and use the characterisation to prove that p–type modulus of convexity implies order p strong unicity of the simultaneous approximant.

Every orthogonal array of strength s and of prime-power (or perhaps infinite) order q, has a well-defined collection of ranks r. Having rank r means that it can be constructed as a cone cut by qs hyperplanes in projective space of dimension r over a field of order q.

Two Lockett sections of Fitting classes of supersoluble groups are determined. One of these sections has only one member, a minimal supersoluble nonnilpotent Fitting class.

It is proved that under the following condition, the sum f of the double trigonometric series with coefficients cjk is integrable and the rectangular partial sums smn(f, x, y) converge to f in L1 norm:

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It is well known that non-zero nilpotent ideals in amenable Banach algebras must be infinite-dimensional. We show that under certain additional hypotheses such ideals cannot even have the approximation property.