In this paper, we present some new metrization theorems in terms of Heath g–functions.

We develop several iterative methods for computing generalized inverses using both first and second order optimization methods in C*-algebras. Known steepest descent iterative methods are generalized in C*-algebras. We introduce second order methods based on the minimization of the norms ‖Ax − b‖2 and ‖x‖2 by means of the known second order unconstrained minimization methods. We give several examples which illustrate our theory.

Let a1… ad be a basis of the Lie algebra g of a connected Lie group G and let M be a Lie subgroup of,G. If dx is a non-zero positive quasi-invariant regular Borel measure on the homogeneous space X = G/M and S: X × G → C is a continuous cocycle, then under a rather weak condition on dx and S there exists in a natural way a (weakly*) continuous representation U of G in Lp (X;dx) for all p ε [1,].

Let Ai be the infinitesimal generator with respect to U and the direction ai, for all i ∈ { 1… d}. We consider n–th order strongly elliptic operators H = ΣcαAα with complex coefficients cα. We show that the semigroup S generated by the closure of H has a reduced heat kernel K and we derive upper bounds for k and all its derivatives.

Let G be a compact abelian group and 1< p < ∞. It is known that the spectrum σ (Tψ) of a Fourier p–multiplier operator Tψ acting in Lp(G), may fail to coincide with its natural spectrum ψ(Г) if p ≠ 2; here Γ is the dual group to G and the bar denotes closure in C. Criteria are presented, based on geometric, topological and/or algebraic properties of the compact set σ(Tψ), which are sufficient to ensure that the equality σ(Tψ) = ψ(Г)holds.

A subgroup H of a finite group G is said to be c–supplemented in G if there exists a subgroup K of G such that G = HK and H∩K is contained in coreG (H). In this paper some results for finite p–nilpotent groups are given based on some subgroups of Pc–supplemented in G, where p is a prime factor of the order of G and P is a Sylow p–subgroup of G. We also give some applications of these results.

This paper presents two natural extensions of the topology of the space of scalar meromorphic functions M(Ω) described by Grosse-Erdmann in 1995 to spaces of vector-valued meromorphic functions M(ΩE). When E is locally complete and does not contain copies of ω we compare these topologies with the topology induced by the representation M (Ω, E) ≃ M(Ω)ε E recently obtained by Bonet, Maestre and the author.

We show that, for any positive integer k, there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes of elements of prime power order. This generalizes a result of E. Landau from 1903. The proof of our generalization makes use of the classification of finite simple groups.