For any positive integer q≧2, let Fq be a finite field with q elements, Fq ((z-1)) be the field of all formal Laurent series in an inderminate z, I denote the valuation ideal z-1Fq [[z-1]] in the ring of formal power series Fq ((z-1)) normalized by P(l) = 1. For any x ∈ I, let the series be the Engel expansin of Laurent series of x. Grabner and Knopfmacher have shown that the P-measure of the set A(α) = {x ∞ I: limn→∞ deg an(x)/n = ά} is l when α = q/(q -l), where deg an(x) is the degree of polynomial an(x). In this paper, we prove that for any α ≧ l, A(α) has Hausdorff dimension l. Among other thing we also show that for any integer m, the following set B(m) = {x ∈ l: deg an+1(x) - deg an(x) = m for any n ≧ l} has Hausdorff dimension 1.

The higher Lie characters of the symmetric group Sn arise from the Poincaré-Birkhoff-Witt basis of the free associative algebra. They are indexed by the partitions of n and sum up to the regular character of Sn. A combinatorial description of the multiplicities of their irreducible components is given. As a special case the Kraśkiewicz-Weyman result on the multiplicities of the classical Lie character is obtained.

Let A be a uniformly complete vector sublattice of an Archimedean semiprime f-algebra B and p ∈ {1, 2,…}. It is shown that the set ΠBp (A) = {f1 … fp: fk ∈ A, k = 1, …, p } is a uniformly complete vector sublattice of B. Moreover, if A is provided with an almost f-algebra multiplication * then there exists a positive operator Tp, from ΠBp(A) into A such that fi *…* fp = Tp(f1 …fp) for all f1…fp ∈ A.

As application, being given a uniformly complete almost f-algebra (A, *) and a natural number p ≧ 3, the set Π*p(A) = {f1 *… *fp: fk ∈ A, k = 1…p} is a uniformly complete semiprime f-algebra under the ordering and the multiplication inherited from A.

Generalizing earlier results of Katriňák, El-Assar and the present author we prove new structure theorems for l-algebras. We obtain necessary and sufficient conditions for the decomposition of an arbitrary bounded lattice into a direct product of (finitely) subdirectly irreducible lattices.

It is well known that c0(Z) is amenable and so its global dimension is zero. In this paper we will investigate the cyclic and Hochschild cohomology of Banach algebra c0 (Z, ω-1) and its unitisation with coefficients in its dual space, where ω is a weight on Z which satisfies inf {ω(i)} = 0.Moreover we show that the weak homological bi-dimension of c0 (Z, ω-1) is infinity.

We prove that in the character group of an abelian topological group, the topology associated (in a standard way) to the continuous convergence structure is the finest of all those which induce the topology of simple convergence on the corresponding equicontinuous subsets. If the starting group is furthermore metrizable (or even almost metrizable), we obtain that such a topology coincides with the compact-open topology. This result constitutes a generalization of the theorem of Banach-Dieudonné, which is well known in the theory of locally convex spaces.

We also characterize completeness, in the class of locally quasi-convex metrizable groups, by means of a property which we have called the quasi-convex compactness property, or briefly qcp (Section 3).

The connection between Clifford analysis and the Weyl functional calculus for a d-tuple of bounded selfadjoint operators is used to prove a geometric condition due to J. Bazer and D. H. Y. Yen for a point to be in the support of the Weyl functional calculus for a pair of hermitian matrices. Examples are exhibited in which the support has gaps.

An operator is said to be nice if its conjugate maps extreme points of the dual unit ball to extreme points. The classical Banach-Stone Theorem says that an isometry from a space of continuous functions on a compact Hausdorff space onto another such space is a weighted composition operator. One common proof of this result uses the fact that an isometry is a nice operator. We use extreme point methods and the notion of centralizer to characterize nice operators as operator weighted compositions on subspaces of spaces of continuous functions with values in a Banach space. Previous characterizations of isometries from a subspace M of C0( Q, X) into C0(K, Y) require Y to be strictly convex, but we are able to obtain some results without that assumption. Important use is made of a vector-valued version of the Choquet Boundary. We also characterize nice operators from one function module to another.