Multi-dimensional

Berezhkovskii A M and Zitserman V Yu 1993 Multi-dimensional Kramers theory of the reaction rate with highly... 

In NMR, the magnetization in the xy plane is detected, so it is the expectation value of the operator that is measured. This is just the unweighted sum of all the operators for the individual spins . It may be a fimction of several time variables (multi-dimensional experiments), including the tune during the acquisition. 

The single-surface assumption, known also as the Bom-Oppenlieimer approximation, implies that the nuclei are described by a single wavefiinction where a is a multi-dimensional vector describing the nuclear... 

The result of this approximation is that each mode is subject to an effective average potential created by all the expectation values of the other modes. Usually the modes are propagated self-consistently. The effective potentials governing die evolution of the mean-field modes will change in time as the system evolves. The advantage of this method is that a multi-dimensional problem is reduced to several one-dimensional problems. 

For concreteness, assume that we consider several particles described together by a multi-dimensional... 

K. Bolton, W. Hase, and G, Peslherbe, in Modem methods for multi-dimensional dynamics computations in chemistry, D. Thompson, ed.. World Scientific, Singapore, 1998, pp, 143-189,... 

Other methods for identifying multi-dimensional reaction paths arc based on stochastic dynamics. For example, a reaction path can be found by opti-... 

Many of the species involved in the endogenous metabolism can undergo a multitude of transformations, have many reaction channels open, and by the same token, can be produced in many reactions. In other words, biochemical pathways represent a multi-dimensional space that has to be explored with novel techniques to appreciate and elucidate the full scope of this dynamic reaction system. 

Analogous intei-polation procedures involving higher numbers of sampling points than the two ends used in the above example provide higher-order approximations for unknown functions over one-dimensiona elements. The method can also be extended to two- and three-dimensional elements. In general, an interpolated function over a multi-dimensional element Q is expressed as... 

As the number of elements in the mesh increases the sparse banded nature of the global set of equations becomes increasingly more apparent. However, as Equation (6,4) shows, unlike the one-dimensional examples given in Chapter 2, the bandwidth in the coefficient matrix in multi-dimensional problems is not constant and the main band may include zeros in its interior terms. It is of course desirable to minimize the bandwidth and, as far as possible, prevent the appearance of zeros inside the band. The order of node numbering during... 

For multi-dimensional potential energy surfaces a convenient measure of the gradient vector is the root-mean-square (RMS) gradient described by... 

ND Socci, IN Onuchic, PC Wolynes. Protein folding mechanisms and the multi-dimensional folding funnel. Protein Eng 32 136-158, 1998. 

In any design problem there is a multi-dimensional spaee of possible solutions. Some of these solutions may be aeeeptable, but not the best (loeal optima) and there may exist a single best solution (global optimum). 

The combination of non-ideal phase behaviour of solutions, the non-linearity of particle formation kinetics, the multi-dimensionality of crystals, their interactions and difficulties of modelling, instrumentation and measurement have conspired to make crystallizer control a formidable engineering challenge. Various aspects of achieving control of crystallizers have been reviewed by Rawlings etal. (1993) and Rohani (2001), respectively. 

Strehlow, R. A. 1970. Multi-dimensional detonation wave structure. Astronautica Acta 15 345-357. 

Van den Berg, A. C. 1990. BLAST—A code for numerical simulation of multi-dimensional blast effects. TNO Prins Maurits Laboratory report. 

Spalding, D. B. 1981. A general purpose computer program for multi-dimensional one- and two-phase flow. Mathematics and Computers in Simulation, IMACS, XXll. 267-276. 

E. Guichard, A. Kustermann and A. Mosandl, Chkal flavour compounds from apricots. Distr ibution of -) lactones enantiomers and stereodifferentiation of dihydroactinidiolide using multi-dimensional gas chromatogr aphy , 7. Chromatogr. 498 396-401 (1990). 

J. A. Apffel, T. V. Alfredson and R. E. Majors, Automated on-line multi-dimensional high performance liquid chromatographic techniques for the clean-up and analysis of water-soluble samples , J. Chromatogr. 206 43-57 (1981). 

Meanwhile orbitals cannot be observed either directly, indirectly since they have no physical reality contrary to the recent claims in Nature magazine and other journals to the effect that some d orbitals in copper oxide had been directly imaged (Scerri, 2000). Orbitals as used in ab initio calculations are mathematical figments that exist, if anything, in a multi-dimensional Hilbert space.19 Electron density is altogether different since it is a well-defined observable and exists in real three-dimensional space, a feature which some theorists point to as a virtue of density functional methods. 

The Gaussian Process.—A gaussian process was defined in the last section to be a process all of whose finite-order distributions are multi-dimensional gaussian distributions. This means that the multi-dimensional characteristic function of Px.fK must be of the form... 

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