Extensions of Banach Lie-Poisson spaces

The extension of Banach Lie-Poisson spaces is studied and linked to the extension of a special class of Banach Lie algebras. The case of W∗-algebras is given particular attention. Semidirect products and the extension of the restricted Banach Lie-Poisson space by the Banach Lie-Poisson space of compact operators are given as examples.     

Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I₂(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G₂, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in B_n-, D_n- and G₂-bracket algebras, and as an application, discover a Pieri-type formula in the B_n-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary B_n-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.     

Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern

The problem considered is that of determining the shape of aplane acoustically sound-soft obstacle from the knowledge ofthe far-field pattern for one time-harmonic incident field.An iterative procedure is proposed based on two boundary integralsrepresenting the incident field and the far-field pattern, respectively.Numerical examples are included which show that the proceduregives accurate numerical approximations in relatively few iterations.     

On reproducing kernels and quantization of states

Quantization of a mechanical system with the phase space a Kähler manifold is studied. It is shown that the calculation of the Feynman path integral for such a system is equivalent to finding the reproducing kernel function. The proposed approach is applied to a scalar massive conformal particle interacting with an external field which is described by deformation of a Hermitian line bundle structure.     

Determination of myo-inositol in a flow-injection system with co-immobilized enzyme reactors and amperometric detection

A selective and sensitive flow-injection system for the determination of myo-inositol (hexahydroxycyclohexane) is described. Inositol dehydrogenase, IDH, lactate dehydrogenase, LDH, and lactate oxidase, LOD, are co-immobilized on porous glass and used in a packed-bed enzyme reactor. myo-Inositol reacts to produce an equivalent amount of hydrogen peroxide, which oxidizes hexacyanoferrate(II) to hexacyanoferrate(III) in a second reactor containing immobilized peroxidase. The hexacyanoferrate(III) is then detected amperometrically at 0 mV vs. SCE in a flow-through detector. The system responds linearly to injected samples of myo-inositol (25 μl) in the concentration range 1–300 μM. The maximum throughput was 90 samples per hour. The IDH/LDH/LOD reactor was stable for at least 5 weeks.     

Intramolecular dimers: a new design strategy for fluorescence-quenched probes

Fluorogenic probes dual-labeled with reporter and quencher dyes use a change in fluorescence to monitor biochemical events (e.g., substrate binding or enzyme digestion). Such events change the reporter-quencher distance, which affects fluorescence. Recently, it is has been shown that static quenching through intramolecular dimers is an important mechanism that can sometimes be more efficient than F?rster resonance energy transfer (FRET).     

Studies of copper(II) sulphide ion-selective electrodes

Changes in liquid junction potentials in copper(II) solutions were measured when different reference electrodes were used. The slope and intercept of a calibration curve for a copper-selective electrode will depend on the selection of reference electrode. The condition of the electrode surface of an Orion copper-selective electrode was studied microscopically and the influence of redox potential on stability of the electrode against corrosion is discussed. Oxidizing solutions will produce pits at dislocations in the material and there will be a mixed electrode potential. The slope, stability, and speed of response are much lower when the surface contains pits. Diamond-polishing was shown to improve the electrode significantly.