Second-order point process definition

Now, as in the case of the energy, up to this point, we have worked with the nonsmooth expression for the electronic density. However, in order to incorporate the second-order effects associated with the charge transfer processes, one can make use of a smooth quadratic interpolation. That is, with the two definitions given in Equations 2.23 and 2.24, the electronic density change Ap(r) due to the electron transfer AN, when the external potential v(r) is kept fixed, may be approximated through a second-order Taylor series expansion of the electronic density as a function of the number of electrons,... 

Kolmogorov [84, 85] introduce the concept of structure functions describing processes with stationary—or homogeneous increments. By definition, the second order velocity structure function is the covariance of the difference in velocity between two points in space. A consequence of isotropy is that the structure function can be expressed in terms of a single scalar fimction. According to the similarity hypotheses of Kolmogorov, the scalar function can be expressed as = 5v ... 

The second difference relates to the definition of a cutoff time point for the evaluation of the difference factor and the Rescigno index. When cumulative data are available, evaluation of the difference factor or the Rescigno index usually requires a reference data set in order to define the cutoff time point for index evaluation (30). For the evaluation of fl and the , i.e., when the difference factor and the Rescigno index are evaluated from non-cumulative data, this difficulty does not exist, provided that the release process has been monitored up to the end (i.e., until dissolution of the drug is complete). At this point, it is worth mentioning that a similar conclusion cannot be drawn for the similarity factor (31) because application of this index to non-cumulative data is set apart by the careful scaling procedure required, in addition to the existence of a reference data set. The reason is that this index can continue to change even after dissolution of both products is complete. 

We shall presently describe the derivation of certain coherent excitations on the basis of definite physical models. Such developments show that random energy supplied to a certain system need not lead to heating but may result in the excitation of ordered (coherent) states. This need not imply that other models could not lead to similar excitations, i.e., that a multicausal situation exists— in other words that experimental verification of the excitation need not necessarily be considered as proof of a particular model. Clearly a situation then arises which requires close collaboration between theory and experiment. Thus, e.g., theory has predicted certain coherent excitations and experiment verifies their existance. This presents a development from the point of view of physics. From the point of view of biology, however, we may ask a question that is prohibited in physics what is the purpose of these excitations Evidence will be presented later in this chapter for the first (physical) stage, but the second, biological question has hardly been touched yet. Its solution will, of course, lead back again to physics, i.e., a certain process has certain consequences. 

This analysis can, for example, be applied to multistep radioactive decay reactions and to isomerization reactions. In such multistep processes, every step is by definition a first-order process. An example of multistep radioactive decay is the Actinium series (see Lederer et ah, 1968), in which Bi alpha-decays to ° T1, which beta-decays to ° Pb with respective half-lives of 2.14 and 4.77 min. Therefore, in this two-step consecutive process, k J ki =/9 = 2.14/4.77 = 0.449, very close to the Acme point. Similarly, in the Radium series, Pb beta-decays to which beta-decays to Po, which then alpha-decays very rapidly (with a half-life of only 0.16 ms) to ° Pb. This multistep decay can be closely approximated by two steps, the first with a half-life of 27 min, the second with a half-life... 

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