Population equations

Depending on the method of pumping, the population of may be achieved by — Sq or S2 — Sq absorption processes, labelled 1 and 2 in Figure 9.18, or both. Following either process collisional relaxation to the lower vibrational levels of is rapid by process 3 or 4 for example the vibrational-rotational relaxation of process 3 takes of the order of 10 ps. Following relaxation the distribution among the levels of is that corresponding to thermal equilibrium, that is, there is a Boltzmann population (Equation 2.11). 

The standard deviation. s° for the sample corresponds to the true standard deviation O for the whole population in the same way that the mean x of the sample corresponds to the arithmetic average [L for the whole population. Equation (9-70) can be written more compactly as... 

Equation (3.14.2.26) is the novel population equation, which describes the cell population with inhibition or promotion.19,20... 

The positive version applies to p-type materials and the negative expression to n-type materials. Note that n()/nd increases as the number of defects falls, so that largest values of a are to be found in materials with the smallest defect populations. Equation (S5.1) can be written in an alternative form. Write ... 

Several estimates of turnover by zooplankton are available. Fluxes of 0.07-0.18 xg of P/L per day were estimated by Scavia et al. (7). Busch and Brooks (32) determined zooplankton excretion rates of 0.4-1.4 xg of P/L per day for summer epilimnetic populations. Equating of these turnover rates to a 6-month period and a 25-m photic zone results in areal fluxes in the range of 500-5000 mg of P/m2. 

Now we derive population equations by considering a large population of identical neurons, which does not mean that they have necessarily the same morphology but only the same dynamics, which is still a strong hypothesis however. By large we... 

The three terms In the denominators of (5) and (6) (and the numerator of (1)) come about, respectively, from the population (or depopulation) of a o or l h level following the absorption of one, two and three successive photons. The termination of the numerators of (5) and (6) at the third order arises from the fact that in reference 6 the authors considered only the first three vibrational levels of the electronic ground state to be significantly populated and truncated the solution with the v" = 2 level. Since three or fewer members of the antistokes progression are seen this assumption seems to suffice. When more vibrational levels are significantly populated, equations (5) and (6) must be enlarged to contain terms of higher order. 

The zeroth order moments of the volume averaged bubble population equations, i.e., the balances on the total bubble density in flowing and stationary foam, have the form of the usual transport equations and can be readily incorporated into a suitable reservoir simulator. 

In the corresponding single-population equation for v, we denote the positive equilibrium, which exists if and only if 5( -1->0, by v. For the full system (2.1), both (m,0) and (0, v) will be equilibria. 

With this transformation, the linear regression model, using the ordinary least-squares method of determination, is valid. However, to employ it, we need to know the population serial correlation coefficient, P. We estimate it by r. The population Equation 3.9 through Equation 3.11 will be changed to population estimates ... 

From the population equation (1.63) the overall rate constant k is obtained using equation (1.49) ... 

Consistent discussions of electron density distributions in cumulenes have to rely on semiempirical CNDO/S electron densities or ab initio STO-3G atomic populations (Equations 25-28). 

The population equation for an MSMPR crystallizer operated at steady state, with crystal growth rate independent of size (dG/dL) = 0, may be written (equation 9.11)... 

Generally, the distribution of droplet sizes in flow can be obtained as a solution of the generalized Smoluchowski (balance population) equation describing the competition between the droplet breakup and coalescence. Various approximate approaches to the solution of the equation with various expressions for breakup and coalescence frequencies have been used in the hteratnre (101,105-115). For rather long mixing in batch mixers, achievement of a steady state in the droplet size distribution is assumed. For mixing in extruders, development of the droplet... 

The population of crystals in a crystallizer can be summarized by the moments of the population (Equation 9.3). The zeroth and the third moments describe the number of particles and the mass of crystals in the crystallizer ... 

Le Bourlot et al (1987) have indeed reconsidered the rotational excitation of Cj by solving the equilibrium populations equations (eq. I) with inclusion of the intercombination transitions in the formalism with the recent molecular data reported in 4. They found two different classes of models describing satisfactorily the observations of the first rotational levels of H, and the various rotational levels of C, (figure 3), depending on the value of the interstellar radiation Held, a very critical parameter. 

A rearrangement of this term (Equations 9-12) demonstrates the similarities of the inhibitory term and the term relating the fractional association between substrate and enzyme population (Equation 3). 

While both populations are equivalent in principle, being related by a unitary transformation, one of them may be more clo.sely related to experiment than the other. For example, if there are dipole. selection rules forbidding the optical transition to or from a subset of the interacting electronic states, these selection rules are usually obeyed to a much larger extent in the diabatic basis than in the adiabatic ba.si.s. Then the diabatic electronic populations are monitored via the intensities of spontaneous and induced emission (the adiabatic populations may be more relevant if the optical transition takes place within the interacting manifold). More specifically, in the limit of ideally short pump and probe pulses the time-resolved pump-probe signal as a function of the delay time has been shown to be proportional to the diabatic population, equation (51). For the more realistic case of finite pulse durations the situation is more complex. In the present article we leave these problems aside and focus on the purely intramolecular aspects of the vibronic dynamics. The various aspects associated with their detection in real time have been surveyed in a recent review article. ... 

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