Congruence,similarity, and symmetries of geometric objects

We consider the problem of computing geometric transformations (rotation, translation, reflexion) that map a point setA exactly or approximately into a point setB. We derive efficient algorithms for various cases (Euclidean or maximum metric, translation or rotation, or general congruence).Part of this research was supported by the DFG under Grants Me 620/6-1 and Al 253/1-1.     

Stationary Fokker-Planck equation applied to fission dynamics

By use of both analytical and numerical techniques, we study the relaxation of time-dependent solutions of the Fokker-Planck equation for an inverted oscillator to Kramers' stationary solution. This is done by integrating over all time the time-dependent solutions for given initial conditions at the saddle point to obtain stationary solutions, whose densities and higher velocity moments are compared as functions of the coordinate with the corresponding quantities calculated from Kramers' stationary solution. For large values of the coordinate an a symptotic expansion of the density is obtained, but for general values of the coordinate and for higher velocity moments the time integration must be done numerically. With increasing dissipation the relaxation to Kramers' stationary solution occurs at successively smaller values of the coordinate. By use of Kramers' stationary solution, we derive analytical expressions as functions of nuclear temperature and dissipation strength for several quantities of interest in fission dynamics, including the mean time from the saddle point to scission, the mean fission-fragment kinetic energy at the scission point and the contribution to the variance in the fission-fragment kinetic energy resulting from fluctuations in the fission degree of freedom. We apply these results to some examples that have been studied experimentally, including the mean saddle-to-scission time for the heavy-ion-induced fission of the compound nucleus ¹⁶⁸Yb and the mean fission-fragment kinetic energy at scission and the contribution to its variance for the α-particle-induced fission of the compound nucleus ²¹³At.     

η3-Allylnickel alkoxides and their use as homogeneous and heterogeneous catalysts for the dimerization of olefins

η³-Allylnickel alkoxides {η³-C₃H₅NiOR}₂ (R = Me, Et, i-Pr, Ph, SiPh₃) may be activated by gaseous boron trifluoride (BF₃) to give active catalysts for the dimerization of propene in homogeneous phase. In CH₂Cl₂ at ?20 °C catalytic turnover numbers of 5000 mol propene(mol Ni)^?1h^?1 were measured. The nature of the OR group influences both the catalytic activity and the oligomerization product distribution. The ratio of methylpentenes to dimethylbutenes in the dimer fraction may be controlled by the presence of additional phosphine ligands at the nickel atom. The nickel alkoxide precursor was heterogenized on alumina to give {Al₂O₃}–O–Ni–(η³-C₃H₅). Subsequent activation using gaseous BF₃ generates a powerful heterogeneous olefin dimerization catalyst which converts 50 × 10³ mol propene (mol Ni)^?1 at ?10° to ?5°C in a batchwise process and 143 × 10³ mol propene (mol Ni)^?1 continuously to give 75% dimers and 25% higher oligomers. The solvent-free treatment of oxide supports, e.g. alumina or silica, with gaseous BF₃ produces strong ‘solid acids’. The activated hydroxyl groups on the support surface serve as effective anchor sites for organometallic complexes to form heterogenous catalysts. By reaction of Ni(cod)₂ with {Al₂O₃}O(BF₃)H or {SiO₂}O(BF₃)H, η¹, η²-cyclo-octenylnickel–O fragments may be fixed to the surface. In the absence of halogenated solvents, the resulting catalysts, e.g. {SiO₂}O–(BF₃)–Ni–(η¹, η²-C₈H₁₃), dimerize propene continuously at +5°C at the rate of 800 × 10³ mol liquid propene (mol Ni)^?1.     

Conformal Geodesics on Vacuum Space-times

 A classification of discrete integrable systems on quad–graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three–dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature representation with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three–leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three–dimensional integrable systems and to the quantum context are also discussed. Received: 14 February 2002 / Accepted: 22 September 2002 Published online: 8 January 2003 Acknowledgements. This research was partly supported by DFG (Deutsche Forschungsgemeinschaft) in the frame of SFB 288 ``Differential Geometry and Quantum Physics'. V.A. was also supported by the RFBR grant 02-01-00144. He thanks TU Berlin for warm hospitality during the visit when part of this work has been fulfilled. Communicated by L. Takhtajan