On the generalized Jacobi equation

The standard text-book Jacobi equation (equation of geodesic deviation) arises by linearizing the geodesic equation around some chosen geodesic, where the linearization is done with respect to the coordinates and the velocities. The generalized Jacobi equation, introduced by Hodgkinson in 1972 and further developed by Mashhoon and others, arises if the linearization is done only with respect to the coordinates, but not with respect to the velocities. The resulting equation has been studied by several authors in some detail for timelike geodesics in a Lorentzian manifold. Here we begin by briefly considering the generalized Jacobi equation on affine manifolds, without a metric; then we specify to lightlike geodesics in a Lorentzian manifold. We illustrate the latter case by considering particular lightlike geodesics (a) in Schwarzschild spacetime and (b) in a plane-wave spacetime.     

Preconditioning of One-Shot Pseudo-Timestepping Methods for Shape Optimization

One-shot pseudo-timestepping [1, 2] has proven to be a powerful approach for the solution of optimal design problems in engineering applications. It blends well in within pre-existing pseudo-timestepping simulation solvers. The goal of this approach is to transform the equations for the necessary optimality conditions of the optimal design problem into an ordinary differential equation (ODE) and then solve the pseudo-time embedded state, co-state and design equations all at once. Similar approaches have been considered in [4, 5]. An application of this method to a real-world example within the MEGADESIGN-Project has been presented in [3]. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Morphological changes in ABC triblock copolymers by chemical modification

Various SBT triblock copolymers with S being polystyrene, B being polybutadiene, and T being poly(tert-butyl methacrylate) and their saponified analogues, SBA with A being poly(methacrylic acid) are characterized in terms of their morphology. The chemical modification of the third block leads to a change of the overall morphology observed in solution cast films, which is interpreted as a consequence of the change of the incompatibility between the different components and the solvent.     

Polymerization of Cyclic Monomers, 10,

Methyl 2‐(bicyclo[3.1.0]hex‐1‐yl)acrylate ( 1 ) was synthesized from methyl 2‐(cyclopentene‐1‐yl)‐2‐hydroxyacetate by cyclopropanation, followed by oxidation to the corresponding bicyclic 2‐oxoacetate and Wittig olefination with methyltriphenylphosphonium bromide. Initiated with 2,2′‐azoisobutyronitrile at 65 °C in chlorobenzene, the radical homopolymerization of 1 occurred with opening of the cyclopropane ring leading to a polymer with a glass transition temperature of 90 °C. The reactivity of 1 in radical copolymerization was higher than that of methyl methacrylate.

     

The Stability of Metal N,N,N′,N′‐Tetrakis(2‐aminoethyl)ethane‐ 1,2‐diamine (=penten) Complexes and the X‐Ray Crystal Structure of [Tl(NO3)(penten)](NO3)2

Complex formation between N,N,N′,N′‐tetrakis(2‐aminoethyl)ethane‐1,2‐diamine (penten) and the metal ions Mn²⁺, Co²⁺, Cu²⁺, Zn²⁺, Cd²⁺, Hg²⁺, Ag⁺, Pb²⁺, and Tl³⁺ (in 1.00M NaNO₃ and 25°) was investigated by potentiometry and spectrophotometry. These are the first reported values of the stability constants for this ligand with Ag⁺, Pb²⁺, and Tl³⁺. The X‐ray crystal structure of [Tl(NO₃)(penten)](NO₃)₂ was determined. In this structure, Tl³⁺ shows a coordination number of seven made up of the six N‐donors and one O‐atom of NO.     

Wechselwirkungen in Kristallen, 176. Mitteilung,Redox‐Reaktionen von Hexahydropyren: Kristallstrukturen seiner Radikalanion‐Salze sowie von Trihydropyrenylium‐tetrachloroaluminat und Dichtefunktionaltheorie‐Berechnungen

Redox Reactions of Hexahydropyrene: Crystal Structures of Its Radical‐Anion Salts as well as of Trihydropyrenylium Tetrachloroaluminate and Density‐Functional‐Theory Calculations Hexahydropyrene 1 , a doubly propane‐1,3‐diyl‐bridged peri‐naphthalene derivative wtih 10 π‐electrons allows both oxidation to its cation as well as reduction to its radical‐anion salts, which could be crystallized and structurally characterized – a rather rare case for small unsaturated hydrocarbons. The unexpected formally threefold dehydrogenation by the oxidizing system AlCl₃/H₂CCl₂ (Bock's reagent) generated the hitherto unknown 1,2,3‐trihydropyrene cation in two polymorphic crystals, which contain 12 π‐electrons delocalized over three anellated six‐membered rings comprising 13 π‐centers. Structural comparison of the altogether four crystallized redox products [K⁺_solv][M^.−] 2a , [K⁺_solv][M^.−]_∞ 2b , [Na⁺_solv][M^.−] 2c , and [(M−3 H)⁺][AlCl⁻₄] 3 with the neutral hydrocarbon 1 reveals only small differences in bond lengths and angles, but establishes solvation contacts, π(η⁶)⋅⋅⋅K⁺ coordination in the polymer 2b , the flattening of one molecular half in the trihydropyren cation of 3 and ten H‐bonds CH⋅⋅⋅Cl to the AlCl₄⁻ counter anion of 3 . DFT/NBO Charge distributions, calculated based on the experimental structural parameters, show charge accumulation in the propanediyl bridges as well as in the peripheral naphthalene C−C bonds of the radical anions. The largest changes result expectedly for the formally triply dehydrogenated 1 , i.e. the trihydropyren cation of 3 , with two slightly positive and partly considerably less negative π‐centers.