Optimal correspondence

Only in a few cases are test samples measurable without any treatment. As a rule, test samples have to be transformed into a measurable form that optimally corresponds to the demands of the measuring technique. Therefore, sample preparation is a procedure that converts a test sample into a measuring sample. Whereas test samples represent the material in its original form, measuring samples embodies a form that is able to interact with the measuring system in an optimum way. In this sense, measuring samples can be solutions, extracts, pellets, and melt-down samples, but also definite surface layers and volumes in case of micro- and nanoprobe techniques. 

Allowing all excitations in the minimal basis valence space and performing the full optimization corresponds to a [8,8]-CASSCF wave function (Section 4.6). Similarly, the SCVB wave function in eq. (7.10) may be improved by adding ionic VB structures like and CH /H this corresponds to exciting an electron from one of the singly -occupied VB orbitals into another VB orbital, thereby making it doubly occupied. The ... 

Does the formal definition of ho as a bound wavefunction have an optimal correspondence wifh fhe description of the resonance state in terms of an initially localized wavepacket ... 

When solving an optimal control problem, it has to be kept in mind that several local optima may exist. Consider for example a problem with a single control function. The objective functional value may be locally optimal, i.e., optimal only in a vicinity of the obtained optimal control function. In another location within the space of all admissible control functions, the objective functional may again be locally optimal corresponding to some other optimal control function. This new optimal objective functional value may be better or worse than, or, even the same as the previous one. 

Choice of the optimal (corresponding to the minimum of additive function F x,d)) parameters of the tree branches by the DP method. The process of stepwise search for solutions on the basis of this method is described by the functional equations... 

The two subsets A(P) and B(Q) define a correspondence, and p = IA(P)I = IB(Q)I is called the correspondence length. Once the optimal correspondence is defined, it is easy to find the optimal rotation and translation using the rigid-body transformation algorithm described earlier. The concept of optimal correspondence, however, requires more explanation. It is clear that p = l defines a trivial solution to the protein superposition problem Any point of A can be aligned with any point of B, with a cRMS of 0. In practice, we are interested in finding the largest possible value for p under the condition that A(P) and B Q) remain similar. ... 

Scoring Functions for Protein Structure Superposition Because the concept of optimal correspondence is ambiguous, the protein structure superposition problem is not uniquely defined. Instead, finding the best superposition of two proteins corresponds to a family of optimization problems, which are specified by the weight given to the similarity (preferably a small deviation between the two subsets), and the correspondence length (preferably large). 

Given a rigid transformation, it should be possible to find an optimal correspondence in polynomial time. 

The STRUCTAL score ST is amenable to dynamic programming and therefore can be used to find an optimal correspondence in time and space requirements of order 0 n ), for any given rigid transformation r. The score of this optimal correspondence is denoted as STopt(r). It validates condition 1. The validity of condition 2 is derived from a lemma given by Kolodny and Linial, which states that for all s, a finite set G = G(s) of rigid transformation exists, such that for every choice of a rigid transformation r, a transformation tg in G(e) exists such that STopt r) — STopt rG) < s, and cardinal(G) = IGI is polynomial in n. 

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