Regular state

Thus, the problem on the growth of a block copolymer chain in the course of the interphase radical copolymerization may be formulated in terms of a stochastic process with two regular states corresponding to two types of terminal units (i.e. active centers) of a macroradical. The fact of independent formation of its blocks means in terms of a stochastic process the independence of times ta of the uninterrupted residence in every a-th stay of any realization of this process. Stochastic processes possessing such a property have been scrutinized in the Renewal Theory [75]. On the basis of the main ideas of this theory, the set of kinetic equations describing the interphase copolymerization have been derived [74],... 

Based on the above qualitative discussion of the characteristics of the Jt-A curves, we now provide a useful kinetic model for the molecular aggregation. Let [S] and [D] be the concentration of PhDA2-8 molecules in the "regular" and "aggregated states, respectively. The "regular" state corresponds to the usual conformation of a surfactant molecule at the interface, i.e., the hydrophilic head faces the hydrophilic environment and the hydrophobic tail is expelled from the interface towards the hydrophobic environment. Then the aggregation can be described as... 

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A positive Aso denotes a regular state (Figure 14). In regular states the sublevel with smallest J or quantum number is lowest in energy. 

By computing the values of the generator function for 0 —> 0 (relations (4.123) and (4.124)), we can observe similarities (identities) between both relations. Indeed, we corroborate that these functions come from a process with identical behaviour and we have a correct asymptotic transformation of the original model. We can conclude that in the case when the transition matrix of probabilities has a regular state, the generator function of the polystochastic chain process when n —> goes from one generator function to a Markov chain related with the model that is, for the present discussion, characterized by relation (4.123)... 

FUTA determines covered employment. FUTA also imposes certain requirements on state programmes, but, as mentioned earlier, the states generally determine individual qualification requirements, disqualification provisions, eligibility, weekly benefit amounts, duration of benefits, and the state tax structure used to finance all of the regular state benefits and half of the extended benefits. 

The appearance of regular states embedded in a sea of chaotic states occurs for all quantum systems and may be attributed to near adiabaticity.66,69-73 This concept of adiabaticity, which is crucial for the understanding of many observations on realistic systems, is discussed subsequently. At this point we just note that in the stadium case the regular states are the quantum analogues of the (zero measure) classical periodic orbits. We will present, later in this chapter an heuristic argument for the existence of regular states for all quantum systems that have no continuous spectrum. 

In this section we have discussed the concept of chaotic versus regular states, from the spatial correlation function viewpoint. Below, we make use of the tools developed in this section to examine the role played by these qualitatively different states in reaction dynamics. 

Contrary to classical mechanics, quantum mechanics is never purely chaotic contamination by regular states exists for even the most chaotic systems.64,66... 

For intermediate systems, such as the HF dimer, mode specificity can be attributed to the differences in bonding of the two hydrogens.85 Likewise, in the NO dimer the two neighboring states with drastically different lifetimes correspond to two types of stretch vibrations. In the larger systems, however, such direct identification of the regular states is not always possible. Moreover, it is clear83,86 that the slowly decaying states are superimposed on other states with much faster decay, a behavior that can be anticipated theoretically. 

In this case the doorway component of the total wavefunction determines the dissociation rates. Based on our previous discussion, we anticipate that the decay rates will fluctuate considerably for the regular states, since the relative magnitude of the doorway channel component is expected to vary drastically from one state to another. In contrast, chaotic states should have a more homogeneous representation of the doorway component. As a result, chaotic states yield a diminished sensitivity of the decay rate to the exact eigenstate studied. Furthermore, individual T, are expected to be insensitive to the identity of the doorway channel. 

Figure 6.18. Transition from case (a) to case (b) for the rotational levels of a regular state.

For NO the FIi/2 fine structure component lies lower than the 113/2 this is called a regular state. Each level in figure 6.18 is actually doubly-degenerate, because of the yl-doubling, which is indeed resolved in the rotational spectrum of NO. The transition from case (a) to case (b) coupling, which is also called spin-decoupling,... 

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