Multidimensional coordinate system

The assumption we make is that similar samples will define points in 5-dimensional space that lie close to each other. The problem is then reduced to one of discriminating between geometric regions or "categories" in this multidimensional coordinate system. This is achieved by the introduction of a suitable decision vector ... 

From Mass Spectra to Multidimensional Coordinate Systems... 

In this section we give a practical, non-mathematical approach to the understanding of PCA and HC analysis. The most crucial point for the understanding of either technique is how mass spectra can be represented as multidimensional coordinate systems. 

Through a transformation that is equivalent to a rigid rotation of the multidimensional coordinate system, a new coordinate system is produced in which the new coordinate directions are linear combinations of the original directions. The rotation is made in such a way that the data (i.e. spectra) along each new coordinate direction is uncorrelated with data in the other directions. In this way, most of the variation in the data can be represented by just a few coordinates surprisingly only 2 to 4 coordinates are usually needed. 

In the last section we considered one-dimensional heat conduction and assumed heat conduction in other directions to be negligible. Most heat transfer problems encountered iu practice can be approximated as being onedimensional, and we mostly deal with such problems in tliis text. However, this is not always the case, and sometimes we need to consider heat transfer in other directions as well. In such cases heal conduction is said to be multidimensional, and in this section we develop the governing differential equation in such systems in rectangular, cylindrical, and spherical coordinate systems. 

For a complete treatment of a laser-driven molecule, one must solve the many-body, multidimensional time-dependent Schrodinger equation (TDSE). This represents a tremendous task and direct wavepacket simulations of nuclear and electronic motions under an intense laser pulse is presently restricted to a few bodies (at most three or four) and/or to a model of low dimensionality [27]. For a more general treatment, an approximate separation of variables between electrons (fast subsystem) and nuclei (slow subsystem) is customarily made, in the spirit of the BO approximation. To lay out the ideas underlying this approximation as adapted to field-driven molecular dynamics, we will consider from now on a molecule consisting of Nn nuclei (labeled a, p,...) and Ne electrons (labeled /, j,...), with position vectors Ro, and r respectively, defined in the center of mass (rotating) body-fixed coordinate system, in a classical field E(f) of the form Eof t) cos cot). The full semiclassical length gauge Hamiltonian is written, for a system of electrons and nuclei, as [4]... 

Although the ID DVR s are useful, die use of direct product DVR s ftn multidimensional problems is much mhighly advantageous. There are three reasons for this. First the Hamiltonian matrix in the muld-dimensional DVR is easy to ctmstruct Secmid, for a DVR in an ordumormal coordinate system, the Hamiltonian is sparse. Third, die "low... 

The reaction path (RP) is a projection of the valley path of the PES onto the coordinate space, i.e., it is a line in the multidimensional coordinate space connecting a particular minimum from the pool of minima representing the reactant system, with a minimum representing a metastable state or with a minimum from the product system along points of lowest potential energy. Hence, it describes the "minimum energy path" (MEP). As outlined above, "reaction coordinate" (RC) and "reaction path" (RP) are inherently synonymous to characterize the MEP. However, the terms RC and MEP have been introduced to illustrate microscopic [5,6] as well as macroscopic [6] energy sizes on the... 

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