Summierbarkeit der geometrischen Reihe auf vorgeschriebenen Mengen

In this paper we are concerned with the summability of the geometric series by matrix methods. We prove the following theorem: Suppose M_o:={z:|z|<1}, M₁, M₂, is a collection of countably many Lebesgue measureable, disjoint sets. For k=1,2, let f_k be a prescribed function, analytic on . Then there exists a triangular matrix , such that the V-transform {_n(z)} of the geometric series has the following properties: {_n(z)} converges compactly to on M_o; for k=1,2, there are sets B_k, such that has Lebesgue-measure zero and _n(z)f_k(z) for zB_k; if there is a set B^*, such that B^*M^* has Lebesgue-measure zero and {_n(z)} diverges for zB^*.     

Eine Erweiterung des Riemann-Lebesgue-Lemmas

It is proved that iff∈L ₁(?),f'∈L ₁(?) and ∫∣x∣ ⁱ ∣f(x)∣dx<∞ fori=1, ...,k?1 and ifA=(a _ij ) is a (k×k)-matrix with non-vanishing determinant, for $$\tilde f_A (\zeta ): = \smallint \exp (i\zeta _1 \sum\limits_{j = 1}^k {a_{1j} x^j } + ... + i\zeta _k \sum\limits_{j = 1}^k {a_{kj} x^j } )f(x)dx$$ the following relation holds: $$\tilde f_A (\zeta ) = O(\left\| \zeta \right\|)^{ - b_k } with b_k : = (\sum\limits_{j = 1}^k {j!)^{ - 1} } for k \in \mathbb{N}$$ .     

Tetranuclear [Cu-Ln]2 single molecule magnets: synthesis, structural and magnetic studies

A trianionic ligand H3L (2-hydroxy-N-(2-[[(2-hydroxyphenyl)methylene]amino]-2-methylpropyl)benzamide) with an inner N2O2 coordination site and an oxygen atom coming from an amide function not involved in this site yields monoanionic LCu- complexes that react with Ln(hfa)3 x 2H2O (hfa = hexafluoroacetylacetonato ligand) to give dinuclear Cu-Ln complexes that self-assemble into tetranuclear species, as demonstrated by the structural determination of the [LCuDy(hfa)2(dmf)2]2 complex. High-spin species are then isolated for two ferromagnetic interactions are active in the [Cu-Gd]2 entities, through the double phenoxo bridge (J = 3.2 cm(-1)) and through the single amide bridge (j = 0.54 cm(-1)). These interactions are still present in the [Cu-Tb]2 and [Cu-Dy]2 complexes which behave as single molecule magnets (SMMs), due to the introduction of anisotropic Ln ions in place of Gd ions.     

Redox-rich spin-spin-coupled semiquinoneruthenium dimers with intense near-IR absorption

Using the [RuCl(μ-tppz)ClRu](2+) [tppz = 2,3,5,6-tetrakis(2-pyridyl)pyrazine] platform for bridging two o-quinone/catecholate two-step redox systems (unsubstituted, Q(n), or 3,5- di-tert-butyl-substituted, DTBQ(n)), we have obtained the stable complexes [(Q(?-))Ru(II)Cl(μ-tppz)ClRu(II)(Q(?-))] (1) and the structurally characterized [(DTBQ(?-))Ru(II)Cl(μ-tppz)ClRu(II)(DTBQ(?-))] (2). The compounds exhibit mostly quinone-ligand-based redox activity within a narrow potential range, high-intensity near-IR absorptions (λ(max) ≈ 920 nm; ε > 50,000 M(-1) cm(-1)), and variable intra- and intermolecular spin-spin interactions. Density functional theory calculations, electron paramagnetic resonance (EPR), and spectroelectrochemical results (UV-vis-near-IR region) for three one-electron-reduction and two one-electron-oxidation processes were used to probe the electronic structures of the systems in the various accessible valence states. EPR spectroscopy of the singly charged doublet species showed semiquinone-type response for 1(+), 2(+), and 2(-), while 1 exhibits more metal based spin, a consequence of the easier reduction of Q as compared to DTBQ. Comparison with the analogous redox series involving a more basic N-phenyliminoquinone ligand reveals significant differences related to the shifted redox potentials, different space requirements, and different interactions between the metals and the quinone-type ligands. As a result, the tppz bridge is reduced here only after full reduction of the terminal quinone ligands to their catecholate states.     

Local and Collective Motions in Precise Polyolefins with Alkyl Branches: A Combination of 2H and 13C Solid‐State NMR Spectroscopy

Branching out : The mobility of linear polymers changes upon branching, which has a pronounced effect on processability and drawability. Regularly branched model polyolefins were studied by advanced solid‐state NMR spectroscopy, and twist defects around the branches in the crystalline regions are identified. For lower branch content, the twisting motions are decoupled; for higher content, collective motion is found (see picture).

     

The growth of nanoscale periodic and dot-like structures on the surface of stainless steel with femtosecond laser pulses in the dry and wet ambient environment

The present work deals with growth of nanoscale periodic and dot-like structures on the surface of stainless steel (SS) by the irradiation of femtosecond laser pulses. For this purpose Ti: Sapphire femtosecond laser pulses (wavelength of 800 nm, pulse length of 25 fs and pulse repetition rate of 1 kHz) were employed in a dry (air) and liquid confined (deionized water and ethanol) environments. The targets were exposed to 1000 succeeding pulses for various fluences ranging from 50 to 150 mJ?cm^?2. Nanoscale structures including ripples, and dots were observed by SEM analysis. The growth and dependence of structure-formation on the ambient environment and laser fluence in both central as well as peripheral ablated areas is systematically investigated. The development of nanostructures and nanoripples is correlated with structural analysis carried out by micro Raman spectroscopy.     

ESI-MS/MS library of 1,253 compounds for application in forensic and clinical toxicology

An electrospray ionization tandem mass spectrometry (ESI-MS/MS) library which contains over 5,600 spectra of 1,253 compounds relevant in clinical and forensic toxicology has been developed using a hybrid tandem mass spectrometer with a linear ion trap. Pure compound solutions—in some cases solutions made of tablets—were prepared and 1 to 2,000 ng of each compound were injected into the system using standard reversed-phase analytical columns with gradient elution. To obtain maximum mass spectral information enhanced product ion spectra were acquired with positive and/or negative ionization at low, medium, and high collision energies and additionally applying collision energy spread. In this mode, all product ions generated by the different collision energies are trapped in the linear ion trap prior to their detection. The applicability of the library for other types of hybrid tandem mass spectrometers with a linear ion trap of the same manufacturer as well as a standard triple-quadrupole tandem mass spectrometer has been investigated with a selection of compounds. The spectra of the developed library can be used to create methods for target analysis, either screening methods or quantitative procedures by generating transitions for multiple reaction monitoring. For those procedures, suitable transitions and convenient collision energies are selected from the library. It also has been utilized to identify compounds with a multi target screening approach for clinical and forensic toxicology with a standardized and automated system. The novel aspects compared to our former library produced with a standard triple-quadrupole mass spectrometer are the enlargement of the ESI-MS/MS library and the additional acquisition of spectra with collision energy spread.     

Metallorganische 5d6‐Übergangsmetallkomplexe von zwei 1‐Methyl‐(2‐alkylthiomethyl)‐1H‐benzimidazol‐Liganden: Strukturen und elektrochemische Oxidation

Organometallic 5d⁶ Transition Metal Complexes of 1‐Methyl‐(2‐alkylthiomethyl)‐1H‐benzimidazole Ligands: Structures and Electrochemical Oxidation The complexes [(mmb)Re(CO)₃Cl], [(mtb)Re(CO)₃Cl], [(mmb)OsCl(Cym)](PF₆) and [(Cym)OsCl(mtb)](PF₆) where Cym = p‐cymene, mmb = 1‐methyl‐(2‐methylthiomethyl)‐1H‐benzimidazole and mtb = 1‐methyl‐(2‐tert‐butylthiomethyl)‐1H‐benzimidazole were synthesized and, except for the latter, structurally characterized. In comparison with other late transition metal compounds of these N‐S chelate ligands the rhenium(I) systems exhibit a balanced coordination to both N and S donor atoms. Anodic one‐electron oxidation produces EPR‐silent rhenium(II) states whereas the osmium(III) species [(mmb)OsCl(Cym)]²⁺ could be identified via EPR and UV/VIS spectroelectrochemistry.