Corresponding times

The entire simulation mn consists of NSTEP long steps each step consists of nstep shorter sub-steps. DT and dt are the corresponding time steps, DT = nstep dt. 

From experimentally determined values of the integral of Equation 3-226, plot these at corresponding times as shown in Figure 3-18. 

Borides Metal solvent, Atomic ratio Max. and corresponding time Cooling rate ... 

The excitation energy and dynamic properties are evaluated from the time-averaged derivatives of the corresponding time-dependent energy functionals [11, 23-25]. However, a more straightforward way to define dynamic properties is through an expectation value of the corresponding properties over a state / ... 

The formulas derived in the time-independent framework can be easily transferred into the corresponding time-dependent solutions. The formulas in the time-independent linear potential model, for example, provide the formulas in the time-dependent quadratic potential model in which the two time-dependent diabatic quadratic potentials are coupled by a constant diabatic coupling [1, 13, 147]. The classically forbidden transitions in the time-independent framework correspond to the diabatically avoided crossing case in the time-dependent framework. One more thing to note is that the nonadiabatic tunneling (NT) type of transition does not show up and only the LZ type appears in the time-dependent problems, since time is unidirectional. 

The Dg-value is a process-describing unit expressed in terms of minutes at 121.1 °C (originally 250°F) or a corresponding time-temperature relationship to produce the same complete spore-killing effect. 

In the accumulation process explained in the previous section, data points collected during several scans and measured at corresponding time values, are added. One could also consider to accumulate the values of a number of data points in a small segment or window in the same scan. This is the principle of smoothing, which is explained in more detail below. 

An even more serious problem concerns the corresponding time scales on the most microscopic level, vibrations of bond lengths and bond angles have characteristic times of approx. rvib 10-13 s somewhat slower are the jumps over the barriers of the torsional potential (Fig. 1.3), which take place with a time constant of typically cj-1 10-11 s. On the semi-microscopic level, the time that a polymer coil needs to equilibrate its configuration is at least a factor of the order larger, where Np is the degree of polymerization, t = cj 1Np. This formula applies for the Rouse model [21,22], i. e., for non-... 

We also emphasize that the MD model does include the vibrational motions of bond, and torsional angles (in the minima of the respective potentials) but, somehow, these small scale fast motions are rapidly damped out in the melt, and do not affect the motion on the nanometer scale (and for corresponding times) significantly. 

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

Since the model is stable, all the roots of P(s), whether they be real or complex, have negative real parts and their corresponding time domain terms will decay away exponentially. Thus if we are only interested in the time domain response at sufficiently large times, we only need to consider the partial fraction expansion of the two purely sinusoidal terms associated with the input ... 

For a given system and closed-loop gain displayed in the root locus plot, we can generate its corresponding time response (step and impulse) and frequency response (Bode, Nyquist, and Nichols) plots. 

For the case of drops of the heavier liquid (2) dispersed in the lighter liquid (1), the corresponding time required for the maximum radial travel from the surface (iq) to the interface (r ) is... 

We want to approximate AM(s) by a Pade approximant Am s). The functional form of Am(s) should be such that the corresponding time-dependent function Am(t) is a series of N exponentials describing the relation of AM(t) to AM(oo) = 0. This implies that Am(s) is an [N — 1. /V]-Pade approximant that can be written in the form... 

The average rate of mass loss is calculated from the amount of mass lost and the corresponding time period. The calculations in Table I at 573 K represent the average mass loss of isothermal dehydrochlorination. Thus, the values in Table I (3.4 %/min for blue conduit, 2.9 %/min for grey conduit and 2.3 %/min for wire coating) represent a reasonable estimate of the mass loss rate of the PVC products in a fire, at a temperature not exceeding 563 K. 

In spectroscopy we may distinguish two types of process, adiabatic and vertical. Adiabatic excitation energies are by definition thermodynamic ones, and they are usually further defined to refer to at 0° K. In practice, at least for electronic spectroscopy, one is more likely to observe vertical processes, because of the Franck-Condon principle. The simplest principle for understandings solvation effects on vertical electronic transitions is the two-response-time model in which the solvent is assumed to have a fast response time associated with electronic polarization and a slow response time associated with translational, librational, and vibrational motions of the nuclei.92 One assumes that electronic excitation is slow compared with electronic response but fast compared with nuclear response. The latter assumption is quite reasonable, but the former is questionable since the time scale of electronic excitation is quite comparable to solvent electronic polarization (consider, e.g., the excitation of a 4.5 eV n — n carbonyl transition in a solvent whose frequency response is centered at 10 eV the corresponding time scales are 10 15 s and 2 x 10 15 s respectively). A theory that takes account of the similarity of these time scales would be very difficult, involving explicit electron correlation between the solute and the macroscopic solvent. One can, however, treat the limit where the solvent electronic response is fast compared to solute electronic transitions this is called the direct reaction field (DRF). 49,93 The accurate answer must lie somewhere between the SCRF and DRF limits 94 nevertheless one can obtain very useful results with a two-time-scale version of the more manageable SCRF limit, as illustrated by a very successful recent treatment... 

The equation says that a time interval measured in the rest system is always longer than the corresponding time interval observed in a system in which the particle is not at rest. This is an example of time dilation. 

In this section, the numerical solutions of the MINLP-model and of the MILP-model as presented in Sections 7.4 and 7.5 are compared with respect to their solution quality (measured by the objective values) and the required solution effort (measured by the computing time). In order to compare the MILP-solution with the MINLP-solution, the optimized values for the start times of polymerizations tn, the recipe assignments W, and the total holdups Mnr are inserted into the MINLP-model and the objective is calculated. To guarantee comparability of the results, the models were stated with identical initial conditions, namely t° = 0, = 2 Vk, pf = 0 Vs, and ra = 0.4 Vs (i.e., the variables defined at the beginning of the corresponding time axes are fixed to the indicated values). For the algorithmic solution procedure, all variables were initialized by 1 (i.e., the search for optimal values starts at values of 1 ), and none of the solvers was specifically customized. 

FIGURE 6.49 Time course obtained for evaluation of product formation in a protein A kinase and Kemptide system. Values of peak areas obtained for product are plotted against corresponding time points used during evaluation. 

The observed changes of rotation and the corresponding times should he substituted in the following equation for unimolecular reactions, and the constant K should he calculated. 

Model-independent techniques compare data pairs observed at corresponding time values, where time is only a class effect, as in a paired -test or in an ANOVA. A data-poor set of only two or three observations, originating from routine quality control of an immediate-release dosage form, cannot be treated other than model independent. 

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