Spherical basis

Abstract: SUMMARY The elastic wave equation in spherical coordinates is solved by a Chebyshev spectral method. In the algorithm presented the singularities in the governing equations are avoided by centring the physical domain around the equator. The highly accurate pseudospectral (PS) derivative operators reduce the required grid size compared to ¢nite diierence (FD) algorithms. The non-staggered grid scheme allows easy extension to general material anisotropy without additional interpolations being required as in staggered FD schemes. The boundary conditions previously derived for curvilinear coordinate systems can be applied directly to the velocity vector and stress tensor in the spherical basis. The algorithm is applied to the problem of a double-couple source located in a high-velocity region at the top of the mantle (slab). The synthetic seismograms show azimuth-dependent traveltime and waveform eiects which are likely to be observable in regions where subduction takes place. Such techniques are important in modelling the full-wave characteristics of the Earth’s 3-D structure and in providing accurate reference solutions for 3-D global models.

Abstract: Energy evaluation using fast Fourier transforms (FFTs) enables sampling billions of putative complex structures and hence revolutionized rigid protein-protein docking. However, in current methods, efficient acceleration is achieved only in either the translational or the rotational subspace. Developing an efficient and accurate docking method that expands FFT-based sampling to five rotational coordinates is an extensively studied but still unsolved problem. The algorithm presented here retains the accuracy of earlier methods but yields at least 10-fold speedup. The improvement is due to two innovations. First, the search space is treated as the product manifold [Formula: see text], where [Formula: see text] is the rotation group representing the space of the rotating ligand, and [Formula: see text] is the space spanned by the two Euler angles that define the orientation of the vector from the center of the fixed receptor toward the center of the ligand. This representation enables the use of efficient FFT methods developed for [Formula: see text] Second, we select the centers of highly populated clusters of docked structures, rather than the lowest energy conformations, as predictions of the complex, and hence there is no need for very high accuracy in energy evaluation. Therefore, it is sufficient to use a limited number of spherical basis functions in the Fourier space, which increases the efficiency of sampling while retaining the accuracy of docking results. A major advantage of the method is that, in contrast to classical approaches, increasing the number of correlation function terms is computationally inexpensive, which enables using complex energy functions for scoring.

Abstract: A practical introduction to coherence and coherence transfer pathways in NMR is provided using the density matrix and product-operator formalisms. The concept of coherence in NMR is first introduced using the density matrix formalism. Pictorial and quantitative representations of coherence are provided using product operators in the Cartesian basis. A simple transformation from the Cartesian basis to a spherical basis is used to follow coherence order during multiple-pulse experiments with coherence transfer pathways. The rules are introduced for designing phase cycles to eliminate unwanted coherences and pathways. An appendix contains a computer program written in Mathematica for performing product-operator calculations interactively.