Rigorous Analytic Results

The DKH transformation has been subjected to rigorous mathematical analysis. In particular, Siedentop and collaborators presented many analytical results [651,652]. The variational stability of the DKH2 Hamiltonian was analyzed [641] as well as the convergence of the DKH series [642]. 

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Despite the shortage of rigorous analytical results, information has been accumulating regarding various properties of self-avoiding walks, and it now seems possible to put forward a coherent pattern of general behavior. This is the aim of the present paper. No attempt will be made to provide complete references to the extensive literature, but a sufficient number of key papers will be cited to enable anyone interested to do so for himself,... 

Equation (1) is the most general and rigorous analytical result. However, it does not take account of the aggregation of similar defects and hence it is applicable only at not very large irradiation doses (up to a concentration of defects (l/2)no, where no is concentration at saturation). In fact, here the existence of clusters of similar defects is allowed, but it is actually assumed that these clusters are statistical fluctuations of the Poisson distribution of similar defects, which does not reflect a real pattern of cluster formation with a substantially non-Poisson spectrum of fluctuations. It is assumed implicitly in equation (1) that, after each event of creating a new pair of defects, the entire system of defects is stirred to attain the Poisson distribution. In the case of the absence of the defect correlation in genetic pairs we arrive at equation (2). 

This work discusses some rigorous analytical results for resonant states. It will be mainly concerned with full discrete resonant expansions, i.e., the second class of expansions mentioned above. We shall restrict the discussion to coherent (elastic) processes in one (ID) and three (3D) dimensions. It is worth mentioning, however, that there is work extending the formalism of resonant states to two dimensions (2D) [35,36] and also describing incoherent processes [35, 37]. 

Below we outline the quantitative results that have been gained by the theoreticians without recourse to rigorous analytical treatment of governing equations. The conditions under which a porous catalyst particle may exhibit multiple steady states have been conceived by Aris (2), Luss (5), and others (9) ... 

The ultimate objective of the method validation process is to produce the best analytical results possible. To obtain such results, all of the variables of the method should be considered, including sampling procedure, sample preparation steps, type of chromatographic sorbent, mobile phase, and detection. The extent of the validation rigor depends on the purpose of the method. The primary focus of this section will be the validation of chromatographic methods. 

Early in our work we frequently obtained erratic analytical results which were subsequently traced to cross-contamination between samples. In order to eliminate this source of error we instituted use of disposable pipettes and adopted a far more rigorous cleaning procedure for that glassware which is reused. All of our glassware is also silylated by the vapor-phase procedure recently reported by Fenimore, et. al. (5). 

Reliability of analytical results generated in a QC laboratory depends on the analytical method and procedure used. Unless the method and procedure are validated, the numerical results remain as an estimate and cannot be used for rigorous compliance judgment. The quality of the pharmaceutical product must be assured from the very beginning of the manufacturing activities, in-... 

All samples were analyzed within 6 hours after field work since degradation can occur and a rigorous time schedule was required to insure proper analytical results. Recoveries from 10 and 50 pg of field applied carbaryl standard solutions onto gauze pads were 101... 

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