The aim of this note is to present in the reflexive Banach space setting a natural and simple proof of the formula of the approximate subdifferential of a composite function.

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.

A non-pathological example is given of two topological spaces which have isomorphic homotopy groups, homology groups and cohomology ring and which cannot be distinguished from each other by the Whitehead product structure. A family of examples can be constructed likewise.

The abstract triangle groups Δ(2, 4, r) can be defined for any positive integer r by Δ(2, 4, r) = 〈x, y | x2 = y4 = (xy)r = 1〉. In this paper we show that for every r ≥ 6, all but finitely many of the alternating groups An can be obtained as quotients of Δ(2, 4, r).

A ring R is called right principally injective (right P-injective) if every R-linear map from a principal right ideal of R can be extended to R. If every ring homomorphic image of R is right P-injective, R is called completely right P-injective (right CP-injective). In this paper we characterise completely quasi-Frobenius rings in terms of CP-injectivity.

In this paper we investigate the situation where a group G has an abelian subgroup H with connected transversals. We show that if H is finite then G is solvable. We also investigate some special cases where the structure of H is very close to the structure of a cyclic group. Finally we apply our results to loop theory and we show that if the inner mapping group of a finite loop Q is abelian then Q is centrally nilpotent.

Nonsmooth calculus using the approximate subdifferential of Mordukhovich and loffe admits a sharper chain rule, hence sharper applications in optimisation, than does the generalised gradient of Clarke. We observe, however, that at a local minimum point of the composition of nonsmooth vector valued and real valued functions, the generalised gradient admits a special, relatively sharp chain rule, that yields sharper results than have been seen before in the context of the generalised gradient.

Let P be a class of topological groups such that every topological group is isomorphic to a topological subgroup of the direct product (with Tychonoff topology) of a subfamily of P. Then every Tychonoff space is homeomorphic to a subspace of a group from P.

It has recently been shown that a Banach space enjoys the weak fixed point property if it is ε0-inquadrate for some ε0 < 2 and has WORTH; that is, if then, ║xn — x║ — ║xn + x║ → 0, for all x. We establish the stronger conclusion of weak normal structure under the substantially weaker assumption that the space has WORTH and is ‘ε0-inquadrate in every direction’ for some ε0 < 2.

Some characterisations of generalised convex functions are established by means of Clarke's subdifferential and directional derivatives.

It has been conjectured by R. S. Ward that the self-dual Yang-Mills Equations (SDYMEs) form a “master system” in the sense that most known integrable ordinary and partial differential equations are obtainable as reductions. We systematically construct the group of symmetries of the SDYMEs on R4 with semisimple gauge group of finite dimension and show that this yields only the well known gauge and conformal symmetries.

In this note we characterise finitely generated self-projective R-modules M satisfying the property that every non-zero M-singular. R-module contains a non-zero M-injective submodule.

The Jacobson radical J(K[S]) of the semigroup ring K[S] of a cancellative semigroup S over a field K is studied. We show that, if J(K[S]) ≠ 0, then either S is a reversive semigroup or K[S] has many nilpotents and J(K[P]) ≠ 0 for a reversive subsemigroup P of S. This is used to prove that J(K[S]) = 0 for every unique product. semigroup S.