Approximate momentum conservation for spatial semidiscretizations of semilinear wave equations

Summary. We prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions, analytic nonlinearity, and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize. Our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic. The method of proof is that of backward error analysis, whereby we construct a modified equation which is itself Lagrangian and translation invariant, and therefore also conserves momentum. This modified equation interpolates the semidiscrete system for all time, and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system. These properties directly imply approximate momentum conservation for the semidiscrete system. We also consider discretizations that are not variational as well as discretizations on non-uniform grids. Through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum.Mathematics Subject Classification (2000): 65M20, 58J70, 70H33     

Convergent synthesis of fully functionalized ring C allocolchicinoids. Benzannulation approach

[reaction: see text]. A novel convergent approach to fully functionalized ring C allocolchicinoids is developed which is based on the benzannulation reaction of Fischer carbene complexes with alkynes. The efficacy of this strategy was established with the conversion of bromide 1a (R1 = Me, R2 = H) to the biaryl phenol 3a (R = Me, R(L) = Pr, R(S) = H) via the carbene complex 2a. Bromide 1b (R1 = t-Bu, R2 = OMe) was then used for the analogous preparation of the diastereomeric allocolchicinoids 3b (R = Me, R(L) = Pr, R(S) = H).     

Picosecond X-ray diffraction probed transient structural changes in organic solids

In this Letter, we report on the experimental characterization of the geometry of short-lived electronically excited states in organic solids by time-resolved x-ray diffraction. Here, the structure factor of the organic crystal is measured as a function of time. Since this technique gives complete structural information, it is a very useful tool for learning more about atom motions on the excited-state energy surface-"beyond" the broad band typical of conventional spectroscopy. Although we used molecular crystals rather than free molecules, the compounds show detectable transient structural changes on the ps to ns time scale in our study.     

Photo-induced pyridine substitution in cis-[Ru(bpy)(2)(py)(2)]Cl(2): a snapshot by time-resolved X-ray solution scattering

Determination of transient structures in light-induced processes is a challenging goal for time-resolved techniques. Such techniques are becoming successful in detecting ultrafast structural changes in molecules and do not require the presence of probe-like groups. Here, we demonstrate that TR-WAXS (Time-Resolved Wide Angle X-ray Scattering) can be successfully employed to study the photochemistry of cis-[Ru(bpy)(2)(py)(2)]Cl(2), a mononuclear ruthenium complex of interest in the field of photoactivatable anticancer agents. TR-WAXS is able to detect the release of a pyridine ligand and the coordination of a solvent molecule on a faster timescale than 800 ns of laser excitation. The direct measurement of the photodissociation of pyridine is a major advance in the field of time-resolved techniques allowing detection, for the first time, of the release of a multiatomic ligand formed by low Z atoms. These data demonstrate that TR-WAXS is a powerful technique for studying rapid ligand substitution processes involving photoactive metal complexes of biological interest.     

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