Search Results

This chapter considers two ways of employing a spatial resolution that varies with position within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. It first provides an overview of nonuniform grids before discussing coordinate mapping as an alternative way of achieving spatial discretization. It then describes an approach for treating both the vertical and horizontal directions with simple finite-difference methods: defining a streamfunction, which automatically satisfies mass conservation, and solving for vorticity via the curl of the momentum conservation equation. It also explains the use of the Chebyshev–Fourier method to simulate the convection or gravity wave problem by employing spectral methods in both the horizontal and vertical directions. Finally, it looks at the basic ideas and some issues that need to be addressed with respect to parallel processing as well as choices that need to be made when designing a parallel code.

This chapter suggests a general working framework for experimental study of large-scale dynamics of EEG, including interpretation of data recorded at different spatial and temporal scales. It views source dynamics as a stochastic (random) process having statistical properties that change with changes in behavior and cognition. It avoids any attempts to choose the “best” methods of EEG or SSVEP data analysis because any such optimization requires prior knowledge of the underlying source dynamics. Fourier-based methods are emphasized to estimate power, phase, coherence, and closely related dynamic measures in distinct frequency bands. Additional methods include estimates of phase velocity across the scalp and frequency-wavenumber spectral analysis. The latter is expressed in terms of spherical harmonics, the natural spatial functions for dynamics on a spherical surface. This approach is used to pick out individual Schumann resonances as an example, in preparation for a similar application to EEG and SSVEP presented in Chapter 10.

This chapter describes how the three-dimensional electron density distribution can be reconstructed from measured one-dimensional projections (the directional Compton profiles) by the Fourier-Bessel reconstruction method, which is based upon an expansion of the momentum density and the reciprocal form factor in lattice harmonics. The propagation of random errors and the optimization of the experiment are discussed and illustrated with published results.

This chapter describes general approaches to the ab initio solution of crystal structures from X-ray or neutron powder diffraction data. The steps in the process, unit cell determination and indexing, intensity extraction, space group determination, structure solution, and structure refinement are described. Indexing methods such as zone indexing, exhaustive methods or recently developed whole pattern methods, and the use of a figure of merit (M₂₀) are presented. Intensity extraction is shown to be reasonably straightforward but for the problem of peak overlap that occurs in powder patterns. The phase problem makes structure solution more difficult: Fourier and Patterson methods, direct methods, or global optimization methods (simulated annealing, genetic algorithms) are brought to bear. The chapter concludes with a section on advanced refinement techniques, including the interpretation of displacement and site occupancy parameters, and the use of constraints. The discussion is illustrated by frequent reference to structure solution for the Ruddlesden-Popper compound Ca₃Ti₂O₇.

This chapter discusses the sensory-motor origins of involuntary head movements. These movements can be regarded as a continuous sequence of deviations and succeeding corrections effected by sensory-motor mechanisms. This is important in the maintenance of static equilibrium in the vertical posture. The concept of pendulum models of equilibrium dynamics is also discussed in this chapter. It is used to represent the whole-body equilibrium in the upright posture. This model belongs to the general class of second-order dynamic systems. Moreover, the spectral analysis of involuntary head movements is also employed in this chapter. Utilizing the Fourier spectral methods, the head movement is decomposed into a sum of sinusoidal components of different frequencies and amplitudes. Based on the discussion of results, it is concluded that quantitative analysis of movement behaviors would provide a more useful measure of understanding dynamic properties than Fourier methods or other methods.

This chapter begins by proving some important properties of (i) conductors in electrostatic equilibrium, and (ii) harmonic functions. These results underpin most of the remaining questions of Chapter 3. The coefficients of capacitance for an arbitrary arrangement of conductors are introduced at an early stage, and numerical calculations then follow in a number of subsequent questions. Some important techniques (both analytical and numerical) for finding solutions to Laplace’s equation are considered. These include: the Fourier method, the relaxation method, themethod of images and the method of conformal transformation. All of these are discussed in some detail, and with appropriate examples.