A quandle is the algebraic distillation of the second and third Reidemeister moves, on unframed links. Quandles have been studied by many people including Kauffman[K], Fenn and Rourke[F-R] and Joyce [J]. Joyce defines the fundamental quandle, which is a classifying invariant of irreducible, unframed links. Fenn and Rourke, who study generalised quandles or racks in [F—R], show that the natural map from the fundamental quandle of a knot K to the knot group is never injective if K is a connected sum. We now prove that the natural map from the fundamental quandle to the fundamental group of the complement of a link in S3 is injective if and only if the link is prime.

We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in the notebooks [5]. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant γ.

The following problem was raised in the 1993 Miklós Schweitzer Mathematical Contest (see [4]). Let E be any connected set in the plane of diameter greater than 4, and let Z1, Z2, …, be any sequence of points on the plane. Then there is a point X ε E for which infinitely many of the products X¯Z¯1 · X¯Z¯n are greater than 1. Furthermore, the same is not necessarily true if the diameter of E is 4.

Let X be an algebraic (complete) variety over a fixed algebraically closed field k. To every Cartier divisor D on X, we can associate the graded k-algebra . As is known, for a semi-ample divisor D, R(X, D) is a finitely generated k-algebra (see [21] or [9]), while this property is no longer true for arbitrary nef and big divisors (see [21]).

Let p be an odd prime number, and let M2 be the extra-special p-group of order p5 and of exponent p2. For every p-group K, we denote by H*(K) the mod p cohomology of K. The purpose of this paper is to calculate the mod p cohomology groups of M2 and of Cp2* M2 (the central product of the cyclic group Cp2 of order p2 and M2).

We compute the subring of H*(CP∞ × CP∞; ) annihilated by the Steenrod algebra, , p being an odd prime. By calculating the subring's structure as a GL(2, )-space we may obtain information about the modular representations of that group.

A. M. Sinclair has proved that if is a semisimple Banach algebra then every continuous Jordan derivation from into is a derivation ([12, theorem 3·3]; ‘Jordan derivation’ is denned in Section 6 below). If is a Banach -bimodule one can consider Jordan derivations from into and ask whether Sinclair's theorem is still true. More recent work in this area appears in [1]. Simple examples show that it cannot hold for all modules and all semisimple algebras. However, for more restricted classes of algebras, including C*-algebras one does get a positive result and we develop two approaches. The first depends on symmetric amenability, a development of the theory of amenable Banach algebras which we present here for the first time in Sections 2, 3 and 4. A Banach algebra is symmetrically amenable if it has an approximate diagonal consisting of symmetric tensors. Most, but not all, amenable Banach algebras are symmetrically amenable and one can prove results for symmetric amenability similar to those in [8] for amenability. However, unlike amenability, symmetric amenability does not seem to have a concise homological characterisation. One of our results [Theorem 6·2] is that if is symmetrically amenable then every continuous Jordan derivation into an -bimodule is a derivation. Special techniques enable this result to be extended to other algebras, for example all C*-algebras. This approach to Jordan derivations appears in Section 6.

Let G be a locally compact group and F a left introverted subalgebra of C(G). For each of the algebras L1(G), M(G), F* and L∞(G)* we determine the finite-dimensional minimal left ideals of the algebra (if any); in some cases we also determine the finite-dimensional minimal two-sided ideals, and in certain cases show that all minimal ideals of the algebra are finite-dimensional.

The present work has its motivation in the papers [2] and [6] on distinguished Fréchet function spaces. Recall that a Fréchet space E is distinguished if it is the projective limit of a sequence of Banach spaces En such that the strong dual E′β is the inductive limit of the sequence of the duals E′n. Clearly, the property of being distinguished is inherited by complemented subspaces and in [6] Taskinen proved that the Fréchet function space C(R) ∩ L1(R) (intersection topology) is not distinguished, by showing that it contains a complemented subspace of Moscatelli type (see Section 1) that is not distinguished. Because of the criterion in [1], it is easy to decide when a Frechet space of Moscatelli type is distinguished. Using this, in [2], Bonet and Taskinen obtained that the spaces open in RN) are distinguished, by proving that they are isomorphic to complemented subspaces of distinguished Fréchet spaces of Moscatelli type.

Given a vector field X on the real plane, we study the influence of the curvature of the orbits of ẋ = X┴(x) in the stability of those of the system x˙ = X(x). We pay special attention to the case in which this curvature is negative in the whole plane. Under this assumption, we classify the possible critical points and give a criterion for a point to be globally asymptotically stable. In the general case, we also provide expressions for the first three derivatives of the Poincaré map associated to a periodic orbit in terms of geometrical quantities.

Convexity, monotonicity and smoothness properties of Köthe spaces of vector-valued functions are described.

We show that the dimension adim introduced by R. Kaufman (1987) coincides with the packing dimension Dim, but the dimension aDim introduced by Hu and Taylor [7] is different from the Hausdorff dimension. These results answer questions raised by Hu and Taylor.

The study of equivariant bifurcation problems via singularity theory (Golubitsky and Schaeffer[8], Golubitsky, Stewart and Schaeffer[9]) has been mainly concerned with models exhibiting spontaneous symmetry-breaking. The solutions of such bifurcation problems lose symmetry as the parameters vary, but the equations that they satisfy retain the same symmetry throughout.