Ein allgemeiner Vierscheitelsatz für ebene Jordankurven

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Über gewisse lineare Transformationen

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A higher-order large-scale regularity theory for random elliptic operators

We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields a in the context of stochastic homogenization. The large-scale regularity of a-harmonic functions is encoded by Liouville principles: The space of a-harmonic functions that grow at most like a polynomial of degree k has the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scale C^k,α-regularity theory, which in the present work is developed in the form of a corresponding C^k,α-“excess decay” estimate: For a given a-harmonic function u on a ball B_R, its energy distance on some ball B_r to the above space of a-harmonic functions that grow at most like a polynomial of degree k has the natural decay in the radius r above some minimal radius r₀.

Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realization a of the coefficient field, the couple (φ, σ) of scalar and vector potentials of the harmonic coordinates, where φ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct “kth-order correctors” and thereby the space of a-harmonic functions that grow at most like a polynomial of degree k, establish the above excess decay, and then the corresponding Liouville principle.     

Synthesis of melanervin from melaleuca quinquenervia,the first naturally occurring compound with a triphenylmethane structure

Melanervin from Melaleuca quinquenervia, the first naturally occurring compound with a triphenylmethane structure, has been synthesized and its structure thereby confirmed.     

Hydrogen-bonded networks in ethanol proton wires: IR spectra of (EtOH)qH+-Ln clusters (L = Ar/N2, q < or = 4, n < or = 5)

Isolated and microsolvated protonated ethanol clusters, (EtOH)qH+-Ln with L = Ar and N2, are characterized by infrared photodissociation (IRPD) spectroscopy in the 3 microm range and quantum chemical calculations. For comparison, also the spectrum of the protonated methanol dimer, (MeOH)2H+, is presented. The IRPD spectra carry the signature of H-bonded (EtOH)qH+ chain structures, in which the excess proton is either strongly localized on one or (nearly) equally shared between two EtOH molecules, corresponding to Eigen-type ion cores (EtOH2+ for q = 1, 3) or Zundel-type ion cores (EtOH-H+-HOEt for q = 2, 4), respectively. In contrast to neutral (EtOH)q clusters, no cyclic (EtOH)qH+ isomers are detected in the size range investigated (q < or = 4), indicative of the substantial impact of the excess proton on the properties of the H-bonded ethanol network. The acidity of the two terminal OH groups in the (EtOH)qH+ chains decreases with the length of the chain (q). Comparison between (ROH)qH+ with R = CH3 and C2H5 shows that the acidity of the terminal O-H groups increases with the length of the aliphatic rest (R). The most stable (EtOH)qH+-Ln clusters with n < or = 2 feature intermolecular H-bonds between the inert ligands and the two available terminal OH groups of the (EtOH)qH+ chain. Asymmetric microsolvation of (EtOH)qH+ with q = 2 and 4 promotes a switch from Zundel-type to Eigen-type cores, demonstrating that the fundamental structural motif of the (EtOH)qH+ proton wire sensitively depends on the environment. The strength of the H-bonds between L and (EtOH)qH+ is shown to provide a rather sensitive probe of the acidity of the terminal OH groups.     

Optical properties of anthracene single crystals in the excitonic region of the spectrum between 4 and 10.5 eV

The optical constants in the photon energy range between 4 and 10.5 eV for E ?b on (001) and E ?L and E ⊥ _L on the (010) plane are discussed. In particular the influence of macroscopic fields on the optical properties of anisotropic crystals is considered. For _E ? a on the (001) plane reflection spectra have been measured at various angles of incidence. These data and results obtained recently by Hymowitz and Clark for several artificially prepared crystal faces are discussed on the basis of the frequency dependend dielectric functions. New information on the directional dispersion of exciton bands is thus obtained.     

Über die Entstehung von Oligoanthrylen bei der milden Sulfonierung von Anthracen

The sulfonation of anthracene yields a variety of products and byproducts by different reaction paths depending on the reaction conditions. While trying to reduce the number of products formed by sulfonation under mild conditions, a new reaction was found: under certain sulfonation conditions anthracene is not sulfonated but reacts to form oligoanthrylenes. The highest average molecular weight of a fraction, insoluble in benzene, is about 3,000.