Polarizations and differential calculus in affine spaces

Within the framework of mappings between affine spaces, the notion of nth polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as linearity, iterability, Leibniz and chain rules are shared - at the finite level - by the polarization operators. We give these results by means of explicit general formulae, which are valid at any order n, and are based on combinatorial identities. The infinitesimal limits of the nth polarizations of a function will yield its nth derivatives (without resorting to the usual recursive definition), and the afore-mentioned properties will be recovered directly in the limit. Polynomial functions will allow us to produce a coordinate free version of Taylor's formula.     

Über die magnetische Untersuchung der Zersetzungsprodukte des Kobalt(II)-hexacyano-ferrates

Zusammenfassung Kobalt(II)-hexacyanoferrat wird in der Luft zersetzt und die Zersetzungsprodukte werden analytisch, durch Röntgenaufnahmen und magnetische Messungen identifiziert.Die Rückstände, deren Zusammensetzung von der Zersetzungstemperatur abhängt, bestehen hauptsächlich aus Co, Co₃O₄, -Fe₂O₃, -Fe₂O₃, CoFe₂O₄ und CoO.     

Discontinuity of the Lempert Function and the Kobayashi-Royden Metric of the Spectral Ball

Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given. This work was initialized during the stay of the first and second named authors at the Jagiellonian University, Kraków in October, 2006, supported by the EGIDE program. They wish to thank both institutions. The third author was supported by the KBN research grant No. 1 PO3A 005 28.     

"Two-point" assembling of Zn(II) and Co(II) metalloporphyrins derivatized with a crown ether substituent in Langmuir and Langmuir-Blodgett films

The effect of "two-point" interactions of Zn(II) and Co(II) metalloporphyrins, bearing 15-crown-5 ether peripheral substituents, on their assembling in Langmuir and Langmuir-Blodgett (LB) films was investigated. That is, simultaneously, the central metal ion of the porphyrin was axially ligated by a nitrogen-containing ligand in the emerged part of the Langmuir film on one hand, and a suitably selected cation pertaining in the subphase solution was supramolecularly complexed by the crown ether moiety in the submerged part of the film on the other. The compression and polarity properties of the Langmuir films of the derivatized free-base 5,10,15-triphenyl-20-(benzo-15-crown-5)porphyrin, H2(TPMCP), and the corresponding cobalt(II) and zinc(II) metalloporphyrins, denoted as Co(TPMCP) and Zn(TPCMP), respectively, as well as inclusion complexes of the metalloporphyrins with selected cations were investigated. For the axial ligation of Zn(II) and Co(II), pyrazine (pyz) and 4,4'-bipyridnine (bpy) aromatic as well as piperazine (ppz) and 1,4-diazabicyclo[2.2.2]octane (DABCO) cyclic heteroaliphatic ligands were selected. The films were formed on the water subphase solution in the absence and presence of LiCl, NaCl, or NH4Cl. The Langmuir films were built of monolayer J-type aggregates of tilted porphyrin macrocycles. The porphyrins formed rather labile complexes with the cations in the subphase. Nevertheless, the XPS analysis revealed that these cations were LB transferred together with the porphyrins onto solid substrates. In the Co(TPMCP) Langmuir films formed on the water subphases, Co(II) was complexed by aromatic but not cyclic heteroaliphatic ligands, while, in these films formed on the NaCl subphase solutions, the metalloporphyrin was also complexed by DABCO. In Langmuir films spread on alkaline subphase solutions, both aromatic and heteroaliphatic ligands formed complexes with Co(TPMCP) of different stoichiometries. The X-ray reflectivity and GIXD measurements performed on selected LB films revealed some structure-building effects of the axial ligation.     

Langmuir-Blodgett films of a cationic zinc porphyrin-imidazole-functionalized fullerene dyad: formation and photoelectrochemical studies

Electron donor-acceptor dyad ensembles of a water-soluble cationic zinc porphyrin (viz., zinc tetrakis(N-methylpyridinium)porphyrin tetrachloride, Zn(TMPyP)) and a C60 derivative that bears an imidazole ligand (viz., 2-(phenylimidazolyl)fulleropyrrolidine, C60im) were assembled during the formation of Langmuir and then Langmuir-Blodgett (LB) films. Surface pressure versus surface area isotherms and surface pressure time profiles, as well as Brewster angle microscopic images documented that the Langmuir films formed were remarkably stable. Subsequently, these Langmuir films were transferred onto different solid substrates, by using the LB technique, for spectroscopic and photoelectrochemical characterization. The UV-vis spectroscopic investigations confirmed that the water-soluble Zn(TMPyP) was, indeed, transferred together with C60im in the LB films. Upon visible light illumination of these LB films, deposited on the ITO transparent conductive supports, a photocurrent generated in the C60im-Zn(TMPyP) system is ascribed to an efficient photoinduced electron transfer from the electron donor, porphyrin singlet excited-state to the electron acceptor, C60. Overall, internal photon-to-current efficiency, IPCE, of the photoanodic current generation (with ascorbate as a sacrificial electron donor) in the ITO/C60im-Zn(TMPyP)/ascorbate/Pt construct is over 5x larger than that of the photocathodic system (with methyl viologen, MV2+, as a sacrificial electron acceptor) in the ITO/Zn(TMPyP)-C60im/MV2+/Pt construct. Highly ordered film stacking favors vectorial electron transfer within the dyad, giving rise to the highest IPCE values of 2.5% determined for a photoanode that was composed of around 20 monolayer films.     

Reductive cleavage mechanism of methylcobalamin: elementary steps of Co-C bond breaking

Density functional theory has been applied to the investigation of the reductive cleavage mechanism of methylcobalamin (MeCbl). In the reductive cleavage of MeCbl, the Co-C bond is cleaved homolytically, and formation of the anion radical ([MeCbl]*-) reduces the dissociation energy by approximately 50%. Such dissociation energy lowering in [MeCbl]*- arises from the involvement of two electronic states: the initial state, which is formed upon electron addition, has dominant pi*corrin character, but when the Co-C bond is stretched the unpaired electron moves to the sigma*Co-C state, and the final cleavage involves the three-electron (sigma)2(sigma*)1 bond. The pi*corrin-sigma*Co-C states crossing does not take place at the equilibrium geometry of [MeCbl]*- but only when the Co-C bond is stretched to 2.3 A. In contrast to the neutral cofactor, the most energetically efficient cleavage of the Co-C bond is from the base-off form. The analysis of thermodynamic and kinetic data provides a rationale as to why Co-C cleavage in reduced form requires prior departure of the axial base. Finally, the possible connection of present work to B12 enzymatic catalysis and the involvement of anion-radical-like [MeCbl]*- species in relevant methyl transfer reactions is discussed.     

Quadrupole relaxation enhancement--application to molecular crystals

A general theory of field dependent spin-lattice relaxation for nuclei of the spin quantum number 1/2 (1H, 19F, 13C) caused by dipole-dipole interactions with neighboring quadrupolar nuclei (nuclei possessing a quadrupolar moment) is presented. The theory is valid for arbitrary motional conditions and should be treated as a quadrupolar counterpart of the paramagnetic relaxation enhancement theory. When the energy level splitting of the dipolar spin (I=1/2) matches one of the transition frequencies of the quadrupolar nuclei one can observe a local enhancement of the dipolar spin relaxation (referred to as "quadrupolar peaks"). To see such effects the dynamics modulating the spin interactions has to be relatively slow. This brings the system beyond the validity range of perturbation approaches and requires the stochastic Liouville equation to be applied. The presented theory describes the quadrupolar relaxation enhancement (QRE) for an arbitrary spin quantum number of the quadrupolar nuclei and includes the asymmetry of the quadrupolar coupling. It has been applied to interpret the shape of magnetization curves (amplitude of 1H magnetization versus magnetic field) for the molecular crystal [C3N2H5]6[Bi4Br18] ([C3N2H5]-imidazolium). The magnetization curves show several dips (local minima) attributed to 1H-14N quadrupolar relaxation enhancement effects. In addition, as a limiting case a perturbation approach to QRE has been presented and its validity conditions have been discussed.     

Dispersionsazofarbstoffe auf Aminobenzimidazolon-Basis

New dispersion azo dyes were prepared on the basis of 5-amino- and 5,6-diaminobenzimidazolone by diazotisation and coupling, esp. with -naphthol and naphthol AS. The new compounds are suitable for dying of polyamide, polyester, polyacrylonitrile and acetate fibers in pink, orange, brown and red-violet colours.     

Minimizing a Quadratic Cost Function of Waiting Times in Single-Machine Scheduling

This paper deals with scheduling n jobs on a single machine in order to minimize the weighted sum of squared waiting times of the jobs. We present a powerful decomposition mechanism, based on a precedence relation concept, that easily handles problems of the size n = 50 and 100 where the processing times and penalties are independently drawn from a uniform distribution. This mechanism is incorporated along with new branching rules in a branch-and-bound scheme that efficiently handles tough problems of the size 20 and 50.