The mean square of Dirichlet series associated with automorphic forms

Letf be a non-holomorphic automorphic form of real weight and eigenvalue λ=1/4?ρ ², ?ρ≥0, which is defined with respect to a Fuchsian group of the first kind. Assume that ∞ is a cusp of this group and denote bya _∞,n,a _∞,n ,n ∈ ?, the Fourier coefficients off at ∞. Following Hecke and Maas we prove that under suitable assumptions the associated Dirichlet seriesL ⁺ (f, s) = ∑ _{n > 0} a _∞,n (n + μ_221E;)^?s andL ^? (f, s) = ∑ _{n < 0} a _∞,n |n + μ_221E;|^?s have meromorphic continuation in the entire complex plane and statisfy a certain functional equation (μ_∞ denotes the cusp parameter of the cusp ∞). We are interested in mean square estimates of these functions. Iff is not a cusp form we prove $$\int_0^T {|L^ \pm (f,\Re _\rho + it)|^2 dt = T(\log T)^a (B^ \pm + o(1)),}$$ wherea is either 1, 2 or 4, andB ^± is a constant. A similar result is true iff is a cusp form. In case of a congruence group the termo(1) can be replaced byO ((logT)^?1).     

L.P.-games with sufficiently many players

L.P.-games aren-person cooperative games arising from a linear program as studied by Owen and Billera/Raanan. Employing a version of nondegeneracy of games developed by Weidner and the author, we show that methods from the Geometry of Numbers are the suitable tool in order to obtain lower bounds for the number of players of each type that ensure that the core and the competitive equilibrium coincide.     

Electrophoretic separation of carrier-free radionuclides

The electrophoretic separation of radioactive rare earths on Cellogel in α-hydroxyisobutyric acid is described. Minute amounts of radioactive daughter nuclides are separated from the neutron-irradiated rare earth target materials, the ratio of the substances being 10⁵∶1 to 10⁷∶1. The suggested method is simple, rapid and suitable for analytical purposes.     

Studies on the Stereochemistry of 2-(Nitromethylidene)-Heterocycles

The ¹H-NMR spectra of 2-(nitromethylidene)pyrrolidine ( 7 ), 1-methyl-2-(nitromethylidene)imidazolidind ( 10 ) and 3-(nitromethylidene)tetrahydrothiazine ( 11 ) in CDCl₃ and (CD₃)₂SO indicate that these compounds have the intramolecularly H-bonded structures (Z)- 7 , (E)- 10 and (Z)- 11 while the N-methyl derivative 8 of 7 is (E)-configurated in both solvents. 1-Benzylamino-1-(methyltio)-2-nitroehtylene ( 13 ), an acylic model, has the H-bonded configuration (E)- 13 in CDCl₃ and in (CD₃)₂SO. 2-(Nitromethylidene)thiazolidine ( 3 ) has the (E)-configuration in CDCl₃ but exists in (CD₃)₂SO as a mixture of (Z)- and (E)-isomers with the former predominating. Both species are detected to varying proportions in a mixture of the two solvents. ¹⁵N-NMR spectroscopy of 3 ruled out unambiguously the nitronic acid structure 6 and the nitromethyleimine structure 5 . The N-methyl derivative 4 of 3 is (Z)-configurated in (CD₃)₂SO. Comparison of the olefinic proton shifts of (Z)- 3 and (Z)- 4 with those of analogues and also of 1,1-bis(methylti)-2-nitroethylene ( 12 ) shows decreased conjugation of the lone pair of electrons of the ring N-atom in (Z)- 3 and (Z)- 4 . This is also supported by ¹³C-NMR studies. Plausible explanations for the phenomenon are offered by postulating that the ring N-atoms are pyramidal in (Z)- 3 and (Z)- 4 and planar in other cases or, alternatively, that the conjugated nitroenamine system gets twisted due to steric interaction between the NO₂-group and the ring S-atom. Single-crystal X-ray studies of 3 and 8 show that the former exists in the (Z)-configuration and the latter in (E)-configuration; the ring N-atom in the former has slightly more pyramidal character than in the latter.     

Rheology of gelling polymers in the Zimm model

In order to study rheological properties of gelling systems in dilute solution, we investigate the viscosity and the normal stresses in the Zimm model [B. H. Zimm, J. Chem. Phys. 24, 269 (1956)]. for randomly cross-linked monomers. The distribution of cluster topologies and sizes is assumed to be given either by Erdos-Renyi random graphs or three-dimensional bond percolation. Within this model the critical behavior of the viscosity and of the first normal stress coefficient is determined by the power-law scaling of their averages over clusters of a given size n with n. We investigate these scaling relations numerically and conclude that the scaling exponents are independent of the hydrodynamic interaction strength. The numerically determined exponents agree well with experimental data for branched polymers. However, we show that this traditional model of polymer physics is not able to yield a critical divergence at the gel point of the viscosity for a polydisperse dilute solution of gelation clusters. A generally accepted scaling relation for the Zimm exponent of the viscosity is thereby disproved.     

Vtbrational spectra of the triclinic K3[ReO2(CN)4] single crystal

Infrared spectra of K₃[ReO₂(CN)₄] have been recorded using polarized radiation and oriented single crystals. The dichroic behaviour of the bands is analyzed and discussed on the basis of molecular geometry and the oriented molecular model. Experimentally determined dichroic ratios are used to obtain a full set of vibrational frequencies. By combining these data with those obtained from Raman spectra, the fundamental internal and lattice vibrations are assigned.     

Synthese und Struktur von Silber(II)-tetrafluoroaurat(III) Ag[AuF4]2

Synthesis and Structure of Silver(II) Tetrafluoroaurate(III) Ag[AuF₄]₂ Intensive green single crystals of Ag[AuF₄]₂ can be obtained by heating up micro crystalline Ag[AuF₄]₂ in autoclaves (p(F₂) ～ 200 bar, T ～ 400°C, t ～ 14 d). It crystallizes monoclinic, space group P2₁/n ? C; (No. 14) with a = 522.3(1), b = 1101.3(3), c = 550.5(2) pm, β = 94.98(3), Z = 2 and is isotypic with Pd[AuF₄]₂.