A formula is established for differentiating the composite F(ξ)○G(ξ) with respect to the parameter ξ where F and G are maps which assumetheir values in function spaces. This composite is denoted by . Use of this notation, together with the tangent functor T, enables the formula to be written in the variable-free form where π1 is a projection map and TF(ξ) = T(F(ξ)). Formal verificationof this formula is straightforward. It has application to many small divisor problems such as those which occur in celestial mechanics.

It is known that a strongly archimedean locally compact tight Riesz group without pseudozeros is essential Rm with the usual topology and tight order. We show that a locally compact tight Riesz group, (G, ≦), without pseudozeros, is algebraically and topologically isomorphic with Rm ⊕ D, where D is discrete. Rm ⊕ {0} is a clopen oideal; and we give necessary and sufficient conditions for G to be isomorphic with Rm ⊕ D in all respects. Further (G, ≦) contains an oideal isomorphic with Rm ⊕ ∑ℵZ and G is isomorphic with it if and only if (G, ≦, U) is interval-compact.

Let G be a group of order 2n and x, y ∈ G. We define the Commutator [x, y] as x−1y−1xy. Similarly, if X, Y are subsets of G, then [X, Y] denotes the sub-group genrated by all commutators of the form [x, y] where x ∈ X, y ∈ Y. Using this, we may define the lower central series of G inductively by The following results are well known.

If Do, D1, … are linear maps from an algebra A to an algebra B, both over the complexes, then {Do, D1, …} is a system of derivations if for all a, b in A and for all nonnegative integers k, we have Where C(k, i) is the binomial coefficient k!/i! (k—i)!. By (1.1) we see that Do must be a homomorphism and in case Do = I, where I is the identity map, D1 is a derivation and, for k ≧ 2, the Dk are higher derivations in the sense of Jacobson (1964), page 191. Gulick (1970), Theorem 4.2, proved that if A is a commutative regular semi-simple F-algebra with identity and {DO, D1, …} is a system of derivations from A to B = C(S(A)), the algebra of all continuous functions on the spectrum of A, where Dox = x, then the Dk are all continuous. Carpenter (1971), Theorem 5, shows that the regularity condition is unnecessary and Loy (1973) generalizes this a bit further. One of the many interesting features of systems of derivations is that they help determine analytic structure in Banach algebras (see for example, Miller (to appear)).

If G1,…,Gc are graphs without loops or multiple edges there is a smallest integer r(G1,…,Gc) such that if the edges of a complete graph Kn, with n ≧ r(G1,…,Gc), are painted arbitrarily with c colours the ith coloured subgraph contains Gi as a subgraph for at least one i. r(G1,…Gc) is called the Ramsey number of the graphs G1,…,Gc.

Let {Xt}t ∈ T be a family of real (R) random variable defined on a probability space (Ω, A P) and having the ranges in a subset S of R, that is, Xt (Ω) ⊂ S for all t. Let X be the mapping of Ω into the function space St such that for any ω ∈ Ω We shall write X = {Xt}t ∈ T and call X the random fundtion arising from {Xt}t ∈ T. It is well-known that any finite subfamily of {Xt}t∈T induces a “finite joint distribution” in ST, and according to Kolmogorov (1933) these finite joint distributions can be simultaneously extended to a probability measure Po on the Borel class Bo of subsets of ST. This extension is natural in the sense that for any B∈B0P0(B) turns out to be exactly P[X−1(B)]. The Kolmogorov extension Po has however a shortcoming in that its domain B0 is not broad enough to include many events of practical interest.

The theory of Yang-Mills fields is explicity formulated in terms of the theory of connections in principal and associated fiber bundles. Special attention is paid to the fiber bundles with Lorentz and Poincaré structures. Equations of the form ∇ψ = 0, where ∇ is a generalized convariant derivative, are shown to contain “mass” terms if the connection in the Poincaré fiber bundle is cononically associated to the connection in the Lorentz fiber bundle.

Theorem 6 of Wilde (1971) is in error. Følner's condition is necessary, but not sufficient, for amenability of a τ-semigroup. For clearly sufficiency would imply that every finite τ-semigroup is amenable, and the simple example on p. 503 of Wilde (1971) shows that this is not so. We refer the reader to Namioka (1964) for a lucid explanation of the fundamental ideas involved.