Lokale Auflösbarkeit von Differentialoperatoren erster Ordnung in einem kritischen Punkt

Ohne ZusammenfassungHerrn Elmar Thoma zum 60. Geburtstag gewidmet     

On the control of certain interacting populations

We study the time optimal control of the system $\text{x}\text{?}{}_1\text{= x}{}_1\text{?}{}_1\text{(x}{}_1\text{, x}{}_2\text{) + u}{}_1\text{(t) g}{}_1\text{(x}{}_1\text{),}\text{x}\text{?}{}_2\text{= x}{}_2\text{?}{}_2\text{(x}{}_1\text{, x}{}_2\text{) + u}{}_2\text{(t)g}{}_2\text{(x}{}_2\text{)}$, where x₁ is the size of the population of one species, x₂ is the population size of the second species, ?₁ and ?₂ are the fractional growth rates of the respective species, g₁ and g₂ are nowhere vanishing functions of class C¹(0, + ∞), and the control u(t) = (u₁(t), u₂(t)) takes on values in a closed rectangle. The functions ?₁ and ?₂ are chosen to represent prey-predator, competitive, and symbiotic interactions.We show, for the various interactions, that a time optimal control, if it exists, must be “bang-bang,” and give sufficient conditions for the controllability, and for the existence, of time optimal controls of the above system.     

Semiclassical theory of transport in antidot lattices

Motivated by a recent experiment by Weiss et al. [Phys. Rev. Lett. 70, 4118 (1993)], we present a detailed study of quantum transport in large antidot arrays whose classical dynamics is chaotic. We calculate the longitudinal and Hall conductivities semiclassically starting from the Kubo formula. The leading contribution reproduces the classical conductivity. In addition, we find oscillatory quantum corrections to the classical conductivity which are given in terms of the periodic orbits of the system. These periodic-orbit contributions provide a consistent explanation of the quantum oscillations in the magnetoconductivity observed by Weiss et al. We find that the phase of the oscillations with Fermi energy and magnetic field is given by the classical action of the periodic orbit. The amplitude is determined by the stability and the velocity correlations of the orbit. The amplitude also decreases exponentially with temperature on the scale of the inverse orbit traversal time/T . The Zeeman splitting leads to beating of the amplitude with magnetic field. We also present an analogous semiclassical derivation of Shubnikov-de Haas oscillations where the corresponding classical motion is integrable. We show that the quantum oscillations in antidot lattices and the Shubnikov-de Haas oscillations are closely related. Observation of both effects requires that the elastic and inelastic scattering lengths be larger than the lengths of the relevant periodic orbits. The amplitude of the quantum oscillations in antidot lattices is of a higher power in Planck's constant and hence smaller than that of Shubnikov-de Haas oscillations. In this sense, the quantum oscillations in the conductivity are a sensitive probe of chaos.This paper is dedicated to Prof. H. Wagner on the occasion of his 60th birthday     

Nucleoside und Nucleotide. Teil 11. Phosphorylierung von 1-(2′-Desoxy-β-D-ribofuranosyl)-2(1H)-pyridon und dessen Verhalten bei der Synthese von Dinucleotiden

Nucleosides and Nucleotides, Part 11. Phosphorylation of 1-(2′-Desoxy-β-D-ribofuranosyl)-2(1H)-pyridon and its Behaviour in the Synthesis of Dinucleotides The behaviour of the unnatural nucleoside 1-(2′-deoxy-β-D-ribofuranosyl)-2(1H)-pyridon (Π_d, 1 ) in the synthesis of dinucleotides with purine deoxynucleotides was studied. The optimized preparation of the protected dinucleoside phosphates (MeOTr) Π_d pG ( 5 ) and (MeOTr) Π_d pA ( 7 ) using the diester method of Khorana with DCC as condensing agent is described. The removal of the N-acyl- and p-methoxytrityl groups was effected by successive treatment with conc. ammonia solution and acetic acid/water 1:1 at 23° yielding the free dinucleoside phosphates Π_dpG_d ( 9 ) and Π_dpA_d ( 11 ). In a similar way, starting from (CNEt) pΠ_d( 15 ), the dinucleotides pΠ_dpG ( 16 ), pΠ_dpG_d ( 18 ), pΠ_dpA ( 17 ) and pΠ_dpA_d ( 19 ) were synthesized. The nucleotide 1-(5′-O-Phosphoryl-2′-deoxy-β-D-ribofuranosyl)-2(1H)-pyridon (pΠ_d, 3 ) was prepared in excellent yield by selective phosphorylation of Π_d ( 1 ) using phosphorylchloride in triethyl phosphate at ?40°. Deoxyadenosine was phosphorylated in the same way. The compounds were characterized by UV. spectroscopy, chromatography and enzymatic degradation.     

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Development of radiation-grafted FEP-g-polystyrene membranes: Some property–structure correlations

Property–structure correlation in proton exchange membranes, prepared by simultaneous radiation grafting of styrene into FEP films and their subsequent sulfonation, was evaluated. The distribution of ionic sites across the membrane matrix was determined by microprobe measurements. The properties of these membranes, such as ion exchange capacity, swelling and ionic resistivity as a function of the degree of grafting, were studied. The thermal stability of membranes was studied using thermogravimetric analysis and ion exchange capacity measurements. Membranes undergo considerable structural changes in terms of the increase in the ionic content, enhanced hydrophilicity and decrease in crystallinity with the increasing degree of grafting. A correlation between some physical properties and stuctural changes occurring during the membrane preparation was established.     

Evolution of Organized Lariat Ether Alcohol and Diol Hydrate Macrostructures in the Solid State

Lariat ether alcohols and diols derived from dibenzo-14-crown-4crystallize in the anhydrous form or as hydrogen-bondedmonohydrates or networks depending upon thepolyether ring substitutents and the composition of the aqueous organic crystallization solvent. Supramolecular structures are created when the lariat ether alcohol sym-(hydroxy)dibenzo-14-crown-4 (1) with ``sticky ends' of ether oxygens and hydroxyl protons, hydrogen bond to each other via bridging water molecules to form three-dimensional networks. For crystallization of 1 from 30% aqueous acetonitrile, this self-association is sufficiently strong to spontaneously generate a supramolecular structure with water channels.     

FLP-type nitrile activation and cyclic ether ring-opening by halo-borane nonagermanide-cluster Lewis acid–base pairs

Even though homoatomic nine-atom germanium clusters are known for two decades, their chemical properties are still rarely investigated. We now discovered that Zintl ion main group-element clusters possess a reactive lone pair of electrons, and we show a new pathway to bind ligands with functional groups to the [Ge₉] cluster core through Ge–C bond formation. We report on the reactivity of [Ge₉{Si(TMS)₃}₂]²⁻ (TMS = trimethylsilyl) towards a series of Lewis acidic bromo-boranes. The reaction of [Ge₉{Si(TMS)₃}₂]²⁻ and DAB^o-tol–Br (DAB = 1,3,2-diazaborolidine; o-tol = 2-methylphenyl) resulted, depending on the reaction protocol, either in the formation of [Ge₉{Si(TMS)₃}₂DAB^o-tol]⁻ (1a) with direct Ge–B interactions, or in [Ge₉{Si(TMS)₃}₂(CH₂)₄O–DAB^o-tol]⁻ (2a) featuring a ring-opened thf moiety. Ring opening reactions occur for all bulkier DAB^R–Br [R: o-xyl (2,6-dimethylphenyl), Mes (2,4,6-trimethylphenyl), Dipp (2,6-diisopropylphenyl)], DAB(ii)^Dipp–Br and acyclic (ⁱPr₂N)₂BBr without Ge–B bond formation as shown for the structural characterization of the ring-opened products of thf (3, 4) and trimethylene oxide (5). In contrast to thf, the activation of CH₃CN requires the simultaneous presence of Lewis-acid and Lewis-basic reactants allowing the formation of [Ge₉{Si(TMS)₃}₂CH₃C N–DAB^Mes]⁻ (6a). Within the presented compounds, 3 and 4 show an unusual substitution pattern of the three ligands at the [Ge₉] core in the solid state. The [Ge₉] cluster/borane systems correspond to intermolecular frustrated Lewis pairs (FLPs), in which the [Ge₉] cluster with several lone pairs represents the Lewis base, and the borane is the Lewis acid.

The reactivity of the lone pairs in polyhedral Zintl anions is shown by the reaction of the bis-silylated cluster [Ge₉{Si(TMS)₃}₂]²⁻ accomplishing cyclic-ether ring-opening or nitrile activation according to a FLP-like mechanism with bromo-boranes.     

Das unterschiedliche massenspektrometrische Fragmentierungsverhalten von Lysinmethylester und dessen N,N′-Diacetylderivat. 27. Mitteilung über das massenspektrometrische Verhalten von Stickstoffverbindungen

The Different Behaviour of Lysine Methyl Ester and its N,N′-Diacetyl Derivative under Electron Impact The base peak in the spectrum of lysine methyl ester is due to the fragment ion C₅H₁₀N (m/e 84), for which the cyclic structure g (Scheme 1) is deduced. During its formation from the [M-COOCH₃]-ion an equilibration of both nitrogen atoms takes place (ion c , Scheme 1). The cyclic nature of ion m/e 84 is in agreement with the intensity of the corresponding ions in the spectra of homologues of lysine methyl ester (Fig. 1). Although in comparison with lysine methyl ester ( 1 ) N,N′-diacetyl-lysine methyl ester ( 7 ) shows the same general fragmentation pathway with formation of the ions [M-COOCH₃] and [M-COOCH₃-H₂NCOCH₃] (m/e 126), the exact fragmentation mechanism proves to the different. Two mechanisms are discussed for the formation of the ion m/e 126 from 7 (Schemes 2 and 3). The results are based on the spectra of labelled derivatives.