Generating multidimensional

Saltzman, J. A Variational Method for Generating Multidimensional Adaptive Grids Courant Mathematics and Computing Laboratory, New York University, 1982. 

More complicated structural situations are observed in di-halogen-rich adducts of chalcogen-donor molecules when further soft-soft halogen halogen interactions intervene, thus generating multidimensional networks (see Chapter 13.1). A clear example is represented by the structure of the compound (dithizone)2 7I2 46 Based on structural data the authors have described the repeating unit in this polymeric compound as a... 

In this section we explore the second possibility to generate multidimensional PES, i.e. a Taylor expansion in terms of normal mode coordinate with respect to the geometry of the stable structure. Including terms up to fourth order we have (using dimensionless coordinates)... 

A. Dravnieks, Odor quality semantically generated multidimensional profiles are stable, Science, 1982, 218, 799-801. 

The other general way of resolving powder patterns from different chemical sites is to generate multidimensional NMR spectra in which the desired powder patterns (or magic-angle spinning sideband patterns) are resolved in one dimension, separated according to (for instance) isotropic chemical shift in another dimension. These techniques are discussed below in the relevant section for each type of nuclear spin interaction. 

Generating Multidimensional Free Energy Surfaces via Adiabatic Free Energy Dynamics... 

In actual experiments, techniques of heavy isotope tags and rapid pulses of radio waves, highly resolved, can generate multidimensional spectra. Correspondingly, theoretical calculations can use a relativistic effect to obtain results for comparison as well as for confirming the model proposed. 

These multidimensional analyses do not necessarily predict overall generator performance or operating characteristics significantly more accurately than do the qua si-one-dimensional analyses, which are more economical to mn. Thus the latter are used for general channel design calculations, and the more sophisticated codes mainly to deal with more detailed aspects of channel operation. For example, current concentrations at electrode edges can be predicted by use of the more sophisticated codes. This allows appropriate electrode design for the condition. 

The combustion-flow interactions should be central in the computation of combustion-generated flow fields. This interaction is fundamentally multidimensional, and can only be computed by the most sophisticated numerical methods. A simpler approach is only possible if the concept of a gas explosion is drastically simplified. The consequence is that the fundamental mechanism of blast generation, the combustion-flow interaction, cannot be modeled with the simplified approach. In this case flame propagation must be formalized as a heat-addition zone that propagates at some prescribed speed. 

The notions of random variable and mathematical expectation also cany over to the multidimensional case. A function of n + m-real variables is called a random variable when it is used to generate a new time function Z(t) from the time functions X(t) and Y(t) by means of the equation... 

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