Electrochemically controlled in-tube solid phase microextraction

We report a new in-tube solid phase microextraction approach named electrochemically controlled in-tube solid phase microextraction (EC in-tube SPME). This approach, which combined electrochemistry and in-tube SPME, led to decrease in total analysis time and increase in sensitivity. At first, pyrrole was elctropolymerized on the inner surface of a stainless steel tube. Then, the polypyrrole (PPy)-coated in-tube SPME was coupled on-line to liquid chromatography (HPLC) to achieve automated in-tube SPME–HPLC analysis. After the completion of EC-in-tube SPME–HPLC setup, the PPy-coated tube was used as working electrode for uptake of diclofenac as target analyte. Extraction ability of the tube in presence and in absence of applied electrical field was investigated. It was found that, under the same extraction conditions, the extraction efficiency could be greatly enhanced by using the constant potential. Important factors are also optimized. The detection limit (S/N = 3) and precision were 0.1 μg L⁻¹ and 4.4%, respectively.     

Burnside’s theorem in the setting of general fields

Archiv der Mathematik - We study the mean of the values of the zeta-function on a generalized arithmetic progression on the critical line.     

On Semitransitive Collections of Operators

A collection F of operators on a vector space V is said to be semitransitive if for every pair of nonzero vectors x and y in V there exists a member T of F such that either Tx = y or Ty = x (or both). We study semitransitive algebras and semigroups of operators. One of the main results is that if the underlying field is algebraically closed, then every semitransitive algebra of operators on a space of dimension n contains a nilpotent element of index n. Among other results on semitransitive semigroups, we show that if the rank of nonzero members of such a semigroup acting on an n-dimensional space is a constant k, then k divides n.     

An operator not satisfying Lomonosov's hypothesis

An example is presented of a Hilbert space operator such that no non-scalar operator that commutes with it commutes with a non-zero compact operator. This shows that Lomonosov's invariant subspace theorem does not apply to every operator.     

Invariant Subspaces for Semigroups of Algebraic Operators

T. Laffey showed (Linear and Multilinear Algebra6(1978), 269–305) that a semigroup of matrices is triangularizable if the ranks of all the commutators of elements of the semigroup are at most 1. Our main theorem is an extension of Hthis result to semigroups of algebraic operators on a Banach space. We also obtain a related theorem for a pair {A, B} of arbitrary bounded operators satisfying rank (AB−BA)=1 and several related conditions. In addition, it is shown that a semigroup of algebraically unipotent operators of bounded degree is triangularizable.     

Limitations on the size of semigroups of matrices

If a nonzero linear functional has finite, countable, or bounded range when restricted to an irreducible semigroup of complex matrices, it is shown that itself has the same property. Similar results are proven under the hypothesis that a nontrivial ideal of is finite, countable, or bounded.