The existence of a maximizing vector for the numerical range of a compact operator

Let be a complex Lebesgue space with a unique duality map from to , the conjugate space of . Let be a compact operator on . This paper focuses on properties of and . We adapt an example due to Halmos to show that for , there is a compact operator on with the semi-open interval . So is not attained as a maximum. A corollary of the main result in this paper is that if , and , then is attained as a maximum.

     

Extensions of pairs of linear transformations between infinite-dimensional vector spaces

To a pair A, B:V→W of linear maps between complex vector spaces attach the pair (V, W) endowed with the operation (α, β)υ = (αA + βB)(υ), α,β ∈ C, υ ∈ V. A concept of rank, similar to the torsion-free rank of abelian groups, is definable for the systems (V, W). With appropriate morphisms, the systems from an abelian category and Ext¹ can be construed as a vector space valued functor. We find all the cases in which Ext¹ ((V, W), (X, Y)), with (X, Y), (V, W) indecomposable systems of rank 0 or 1, is finite-dimensional, and compute its dimension in these cases. This extends a former computation for finite-dimensional systems.     

The rank of a completion of a dedekind domain

Let D be a Dedekind domain with quotient field K. Let C_p be the completion of the localisationD_p , of D at a nonzero prime idealp, of D. Let r_p be the rank of C_p as a D-module, ier_p , is the dimension of the K-vector space K_p , = K? _DC_p . The following results on r_p are deduced from well-known theorems: if r_p is finite for at least one prime ideal p, then D is a discrete valuation ring; and D = C_p if _p = 1. If D is a discrete valuation ring, then r_p = dimExt(K, D) + 1. A module M is extensionless if every extension of M by M splits. The D-module r_C is an estensionless indecomposable module. If r_C is infinite for every nonzero prime ideal, it is shown that an estensionless D-module of finite rank is a direct sum or certain rank one modulcs.     

Halomonas sp. OKOH--a marine bacterium isolated from the bottom sediment of Algoa Bay--produces a polysaccharide bioflocculant: partial characterization and biochemical analysis of its properties

A bioflocculant-producing bacterium isolated from seawater was identified based on 16S rRNA gene nucleotide sequence to have 99% similarity to that of Halomonas sp. Au160H and the nucleotide sequence was deposited as Halomonas sp. OKOH (Genbank accession number is HQ875722). Influences of carbon source, nitrogen source, salt ions and pH on flocculating activity were investigated. The bioflocculant was optimally produced when glucose (87% flocculating activity) and urea (88% flocculating activity) were used as sources of carbon and nitrogen, respectively. Also, initial pH of 7.0 and Ca2? supported optimal production of the bioflocculant with flocculating activities of 87% respectively. Chemical analyses revealed the bioflocculant to be a polysaccharide.