Subset

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

In die potential section shown on the right hand side of figure A3,13,4 the subset of A energy states is... 

The general task is to trace the evolution of the third order polarization of the material created by each of the above 12 Raman field operators. For brevity, we choose to select only the subset of eight that is based on two colours only—a situation that is connnon to almost all of the Raman spectroscopies. Tliree-coloiir Raman studies are rather rare, but are most interesting, as demonstrated at both third and fifth order by the work in Wright s laboratory [21, 22, 23 and 24]- That work anticipates variations that include infrared resonances and the birth of doubly resonant vibrational spectroscopy (DOVE) and its two-dimensional Fourier transfomi representations analogous to 2D NMR [25]. 

The development of tunable, narrow-bandwidtli dye laser sources in tire early 1970s gave spectroscopists a new tool for selectively exciting small subsets of molecules witliin inhomogeneously broadened ensembles in tire solid state. The teclmique of fluorescence line-narrowing [1, 2 and 3] takes advantage of tire fact tliat relatively rigid chromophoric... 

As discussed in Section II.A, the adiabatic electronic wave functions and depend on the nuclear coordinates Rx only through the subset... 

As written, Eq. (52) depends on all the (infinite number of) adiabatic electi onic states. Fortunately, the inverse dependence of the coupling strength on energy separation means that it is possible to separate the complete set of states into manifolds that effeetively do not interact with one another. In particular, Baer has recendy shown [54] that Eq. (57), and hence Eq. (58) also holds in the subset of mutually coupled states. This finding has important consequences for the use of diabatic states explored below. 

While this derivation uses a complete set of adiabatic states, it has been shown [54] that this equation is also valid in a subset of mutually coupled states that do not interact with the other states. 

The long term behavior of any system (3) is described by so-called invariant measures a probability measure /r is invariant, iff fi f B)) = ft(B) for all measurable subsets B C F. The associated invariant sets are defined by the property that B = f B). Throughout the paper we will restrict our attention to so-called SBR-measures (cf [16]), which are robust with respect to stochastic perturbations. Such measures are the only ones of physical interest. In view of the above considerations about modelling in terms of probabilities, the following interpretation will be crucial given an invariant measure n and a measurable set B C F, the value /r(B) may be understood as the probability of finding the system within B. 

From a mathematical point of view, conformations are special subsets of phase space a) invariant sets of MD systems, which correspond to infinite durations of stay (or relaxation times) and contain all subsets associated with different conformations, b) almost invariant sets, which correspond to finite relaxation times and consist of conformational subsets. In order to characterize the dynamics of a system, these subsets are the interesting objects. As already mentioned above, invariant measures are fixed points of the Frobenius-Perron operator or, equivalently, eigenmodes of the Frobenius-Perron operator associated with eigenvalue exactly 1. In view of this property, almost invariant sets will be understood to be connected with eigenmodes associated with (real) eigenvalues close (but not equal) to 1 - an idea recently developed in [6]. 

Setting up the Frobenius-Perron operator with respect to this subset. 

Covering of Energy Cells Assume that the energy cells under consideration are compact sets and the stepsize r is fixed. We want to construct a collection B of boxes in phase space such that the union Q of these subsets is a covering of the energy cell we focus on. To this end, consider... 

As the number of conformations increases exponentially with the number of rotatable bonds, for most molecules it is not feasible to take all possible conformations into account. However, a balanced sampling of the conformational space should be ensured if only subsets arc being considered. In order to restrict the number of geometries output, while retaining a maximum of conformational diversity, ROTATE offers the possibility of classifying the remaining conformations, i.c., similar conformations can be combined into classes. The classification is based on the RMS deviation between the conformations, either in Cartesian (RMS y 7if [A]) or torsion space in [ ], The RMS threshold, which decides whether two... 

Perhaps the best idea is to compute as many descriptors as possible and then to select an optimal subset by applying sophisticated techniques, discussed below. 

Once the quality of the dataset is defined, the next task is to improve it. Again, one has to remove outliers, find out and remove redundant objects (as they deliver no additional information), and finally, select the optimal subset of descriptors. 

Let us consider a system S with n objects. Suppose we have a criterion which enables us to distribute the objects into different subsets of S. One condition is that no object can belong to any two different subsets. Once the distribution is complete, we may have m subsets containing ni objects, correspondingly, so that LiH = n and I = 1, 2, m. 

Whatever the criterion is, we may have the following two extreme situations. The first one occurs when all the objects fall into the same subset (such subsets are known in discrete algebra as classes of equivalence). The second is when each subset contains one, and only one, object. 

As we should remember now, we distribute the objects into subsets in accordance with some criterion, not having known even the number of subsets themselves. That is why the evaluation of data complexity is still a challenging problem. 

There was a time when one could use only a few molecular descriptors, which were simple topological indices. The 1990s brought myriads of new descriptors [11]. Now it is difficult even to have an idea of how many molecular desaiptors are at one s disposal. Therefore, the crucial problem is the choice of the optimal subset among those available. 

The idea behind this approach is simple. First, we compose the characteristic vector from all the descriptors we can compute. Then, we define the maximum length of the optimal subset, i.e., the input vector we shall actually use during modeling. As is mentioned in Section 9.7, there is always some threshold beyond which an inaease in the dimensionality of the input vector decreases the predictive power of the model. Note that the correlation coefficient will always be improved with an increase in the input vector dimensionality. 

It may be of interest to readers that all three methods mentioned above resulted in the same optimal subset of descriptors for the well-known Selwood dataset, which has become a de-facto standard in testing new approaches in this field [20]. 

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