Sudoku’s french ancestors

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.     

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Discrete duality finite volume schemes on general meshes, introduced by Hermeline and Domelevo and Omnès for the Laplace equation, are proposed for nonlinear diffusion problems in 2D with nonhomogeneous Dirichlet boundary condition. This approach allows the discretization of non linear fluxes in such a way that the discrete operator inherits the key properties of the continuous one. Furthermore, it is well adapted to very general meshes including the case of nonconformal locally refined meshes. We show that the approximate solution exists and is unique, which is not obvious since the scheme is nonlinear. We prove that, for general W^?1,p′(Ω) source term and W^1‐(1/p),p(?Ω) boundary data, the approximate solution and its discrete gradient converge strongly towards the exact solution and its gradient, respectively, in appropriate Lebesgue spaces. Finally, error estimates are given in the case where the solution is assumed to be in W^2,p(Ω). Numerical examples are given, including those on locally refined meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Dynamic absorption effects in KrF* amplifiers

The results of transient loss measurements performed in a self-sustained discharge KrF^* amplifier are reported. Analysis of these results gives a minimum value of 20 for the effective gain to loss ratiog ₀/_eff, indicating that efficient extraction of energy in subpicosecond KrF^* amplifiers in the 1 J range should be achievable.     

Broadband and band-selective IMPRESS-gHMBC: compensation of refocusing inefficiency with synchronized inversion sweep

Compensation of refocusing inefficiency in a gHMBC experiment by replacing the rectangular pi pulse with a pair of adiabatic pulses with synchronized inversion sweep (CRISIS) significantly improves the performance of the gHMBC experiment. The CRISIS-gHMBC experiment retains the pure absorptive shapes in F1 and hence results in better lineshape and higher resolution than the current versions of magnitude mode gHMBC spectra. When used as a broadband experiment, CRISIS-gHMBC, owing to better refocusing efficiency of the adiabatic pulse pairs, gives improved performance across the 13C spectral width. Moreover, it is shown that CRISIS-gHMBC is a robust and improved alternative and when used along with the IMPRESS (Improved Resolution using Symmetrically Shifted pulses) technique further increases the sensitivity and resolution without additional experimental time. The IMPRESS-CRISIS combination is demonstrated for broadband gHMBC and band-selective gHMBC experiments. The ICbs-gHMBC [IMPRESS-CRISIS-band-selective gHMBC] experiment is an attractive and better alternative to individual band-selective gHMBC.