Some characterizations of nil radical and nil semisimple power series rings are given. The upper nil radical of a power series ring in an uncountable set of non-commutative indeterminates is completely described.

This paper shows that the idempotent generalized characters associated with a Raikov System generated by a K2 set in is contained in the closure of the characters D2 in ΔM(D2).

The classification of spaces of matrices of bounded rank is known to depend upon ‘primitive’ spaces, whose structure is considerably restricted. A characterisation of an infinite class of primitive spaces is given. The result is then applied to obtain a complete description of spaces whose matrices have rank at most 3.

Let D ⊂ Rn be a bounded domain and L: dom L ⊂ L2 (D) → L2 (D) be a self-adjoint operator of finite dimensional kernel. Let f: D × R → R be a function satisfying the Carathéodory condition. Assume that there are constants λ > 0 and δ ∈ [0, 1] such that and that .

Then with the aid of a generalized Krasnosel'skii's theorem it has been proved that under conditions exactly analogous to those of Landesman and Lazer there exists u ∈ L2(D) such that L(u)(x) = f(x, u(x)) for ∀x ∈ D. This result is then used to prove the existence of weak solutions of nonlinesr elliptic boundary value problems.

Other abstract results applicable to ordinary and partial differential equations have also been proved.

The functional least squares procedure of Chambers and Heathcote for estimating the slope parameter in a linear regression model is analysed. Strong uniform consistency for the family of these estimators is proved together with a necessary and sufficient condition for weak convergence in the space of continuous vector valued functions. These results are then used to develop the asymptotic normality of an adaptive version of the functional least squares estimator with minimum limiting variance.

Two index laws for fractional integrals and derivatives, which have been extensively studied by E. R. Love, are shown to be special cases of an index law for general powers of certain differential operators, by means of the theory developed in a previous paper. Discussion of the two index laws, which are rather different in appearance, can thus be unified.

For a non-constant entire or rational function f normalized by f(0) = 0, f′(0) = 1, f″(0) ≠ 0, which is not a Möbius tranformation, the existence of a sequence is established which has the properties . The result certainly implies f(0)= |f(0)|= 1, so these conditions cannot be omitted. The condition f″ (0)≠ 0 can be replaced by f(k)(0) ≠ 0 for some k ≥ 2.

Using the exponential map in multifunction context, the paper deduces a system of non-Hausdorff theorems which generalize all known Ascoli theorems for the space of continuous functions and the space of point-compact continuous multifunctions.

Algebras (A, ∧, ∨, ~, γ, 0, 1) of type (2,2,1,1,0,0) such that (A, ∧, ∨, ~, γ 0, 1) is a De Morgan algebra and γ is a lattice homomorphism from A into its center that satisfies one of the conditions (i) a ≤ γa or (ii) a ≤ ~ a ∧ γa are considered. The dual categories and the lattice of their subvarieties are determined, and applications to Lukasiewicz algebras are given.

The separable cyclic extension of rings is generalized to a separable noncyclic extension of rings: a crossed product with a factor set over a ring (not necessarily commutative). A representation of separable idempotents for a separable crossed product is obtained, and simplifications for some special factor sets are also given.

The elementary part of general topology is carried out in a system which is based on the arithmetically definable theory of the reals with definitions by definable induction (DDI), where a formal object is said to be definable if the quantifiers are restricted to the rationals, the names of the base members and the elements of the spaces.