On a Formula for the L2 Wasserstein Metric between Measures on Euclidean and Hilbert Spaces

For a separable metric space (X, d) L^p Wasserstein metrics between probability measures μ and v on X are defined by where the infimum is taken over all probability measures η on X × X with marginal distributions μ and v, respectively. After mentioning some basic properties of these metrics as well as explicit formulae for X = R a formula for the L² Wasserstein metric with X = Rⁿ will be cited from [5], [9], and [21] and proved for any two probability measures of a family of elliptically contoured distributions. Finally this result will be generalized for Gaussian measures to the case of a separable Hilbert space.     

Properties of ITO films prepared by reactive magnetron sputtering

Indium tin oxide (ITO) thin films were deposited by mid frequency pulsed dual magnetron sputtering using a metallic alloy target with 10 wt.% tin in an atmosphere of argon and oxygen. The aim of the work was to study the interdependence of structural, electrical and optical properties of ITO films deposited in the reactive and transition target mode, respectively. The deposition rate in the transition mode exceeds the deposition rate in the reactive mode by a factor of six, a maximum value of 100 nm·m min⁻¹ could be achieved. This corresponds to a static deposition rate of 200 nm min⁻¹. The lowest electrical resistivity of 1.1·10⁻³ Ω cm was measured at samples deposited in the high oxygen flow range in the transition mode. The samples show a good transparency in the visible range corresponding to extinction coefficients being below 10⁻². X-ray diffraction was used to characterise crystalline structure as well as film stress. ITO films prepared in the transition mode show a slightly preferred orientation in (211) direction, whereas films deposited in the reactive mode are strongly (222) oriented. Compared to undoped In₂O₃ all samples have an enlarged lattice. The lattice strain perpendicular to the surface is about 0.8% and 2.0% for films grown in the transition and the reactive mode, respectively. Deposition in the transition mode introduces a biaxial film stress in the range of −300 MPa, while stress in reactive mode samples is −1500 MPa.     

Acyl- und Alkylidenphosphane. XXVIII. Synthese und Struktur von 1,3-Dibenzyl- und 1,3-Diethyl-2,4-bis(phenylimino)-1,3-diphosphetan

Acyl- and Alkylidenephosphines. XXVIII. Synthesis and Structure of 1,3-Dibenzyl- and 1,3-Diethyl-2,4-bis(phenylimino)-1,3-diphosphetane Catalyzed by small amounts of solid sodium hydroxide, the adducts 1a and 1b formed from benzyl- or ethylbis(trimethylsilyl)phosphine and phenylisocyanate, react at +20°C slowly to give hexamethldisiloxane and oligomeric [(phenylimino)methylidene]phosphines. In different solvents the benzyl compound was found to exist only as a mixture of [N,N′-(E)/(Z)]-isomeric 2,4-bis-(phenylimino)-1,3-diphosphetanes 2a with their alkyl groups at the phosphorus atoms in trans position, whereas in case of the ethyl derivative 2b a second pair of [N,N′-(E)/(Z)]-isomeric dimers with their substituents in cis position and two trimeric forms ( 3b and 4b ) could be detected in cyclopentane. [N,N′-(E)]-1r,3t-dibenzyl- ( 2a ) and [N,N′-(E)]-1r,3t-diethyl-2,4-bis(phenylimino)-1,3-diphosphetane 2b isolated from 1,2-dimethoxyethane or cyclopentane, crystallize in the monoclinic space group P2₁/c or P2₁/n, resp., with following dimensions of the unit cell determined at temperatures of measurement of +20 ± 3°C/?130 ± 3°C: a = 2145.4(1)/569.3(1); b = 568.1(2)/719.1(2); c = 1960.2(2)/2042.6(4) pm; β 99.43(1)°/95.03(2)°; Z = (2+2) and 2, resp. X-ray structure determinations (R_w = 0.034/0.041) show both molecules to be centrosymmetric. Characteristic rounded bond lengths (pm) and angles (°) are: endocyclic P? C 185/184; C? P? C 82/81; P? C? P 98/99; exocyclic P? C 186/184; C?N l27/127; C?N? C 121/11.