Variation of constants method

This leads to the Dirac variation-of-constants method [10]. Although generall> successful, an unsatisfying feature of this method can be seen when we consider an adiabatically switched-on static perturbation. By the adiabatic theorem [11] the perturbed wave function as t — -)-< has the form... 

As in self-consistent wavefunction functional methods, the linear variation of constants method is commonly employed when solving the KS equations. That is, the <]> values are approximated by... 

The solution for this previous differential equation is more complex. With the condition that at time zero [PJ=0, the solution can be found by using the variation of constants method [70] ... 

This formula can be motivated by applying the variation of constants formula to ihtli = -t- (H(y) — H"). The method can be implemented... 

There are many variations of this method. To illustrate the procedure, a variation developed by Rosenbrock will be discussed. It is one of the best optimization methods known8,7 when there is no experimental error. The method is also very useful for determining constants in kinetic and thermodynamic equations that are highly nonlinear. An example of this type of application is given in reference 9. 

In a variation of this method, Tencer and Stein (1978), mixed the isotopic quasi-racemate to near, but not exactly, zero rotation so that at a certain time, tz, the observed optical rotation of the reaction mixture was zero. The equations for this type of kinetic experiment enable one to calculate the difference between the individual isotopic rate constants from tz and the ratio of rate constants (the KIE) from te and tz provided that the ratio of the initial rotations for the two isotopic substrates is known. Usually it is preferable to... 

Several modifications of the method are described in the literature (Artursson and Karlsson 1991 Hidalgo and Borchardt 1989 and many others). Modifications include cell culture medium, time of cultivation and frequence of medium change, variations of trypsinization methods and others. In an industrial environment cell cultivation methods are maintained over many years constant to reduce variability and ensure constant results in quality assessment protocols. Additionally to quality control parameters like TEER and permeability markers expression levels of major enzymes and transporters are checked. 

Bawn et al. (1950) and Malcolm et al. (1969) have used variations of this method. The apparatus consists of a U-tube manometer connected at the bend to a mercury reservoir. One arm of the manometer is attached to a vessel containing pure solvent and the other to a vessel containing the polymer and solvent. Both vessels are immersed in a constant temperature bath maintained at a temperature below ambient. Vapor pressures of the pure solvent are measured by evacuating the solution side of the system. 

A variation of these titrations is used to determine stoichiometry and formation constant(s) of complexes. In these methods, typically the metal concentration and total volume of solutions are kept constant, but the ligand-to-metal ion ratio is continuously varied. From a plot of absorbance versus mole fraction, the stoichiometry between the metal and ligand can be obtained. Numerous variations of this method are adopted to obtain stoichiometry and formation constants or binding constants in many biochemical determinations. 

What we can really infer from these data is that the fraction measured by the Menzel and Vacarro (4) variation includes those compounds that are middling diflBcult to oxidize or volatilize and that the distribution of this fraction is one of high values in the surface waters with low and constant values, within the large variability of the method, throughout the deep oceans. The extra 30% measured by the Sharp (27) variation of the method consists of those compounds degraded to carbon dioxide... 

THE TREATMENT OF A TIME-DEPENDENT PERTURBATION BY THE METHOD OF VARIATION OF CONSTANTS... 

There have been developed two essentially different wave-mechanical perturbation theories. The first of these, due to Schrodinger, provides an approximate method of calculating energy values and wave functions for the stationary states of a system under the influence of a constant (time-independent) perturbation. We have discussed this theory in Chapter VI. The second perturbation theory, which we shall-treat in the following paragraphs, deals with the time behavior of a system under the influence of a perturbation it permits us to discuss such questions as the probability of transition of the system from one unperturbed stationary state to another as the result of the perturbation. (In Section 40 we shall apply the theory to the problem of the emission and absorption of radiation.) The theory was developed by Dirac.1 It is often called the theory of the variation of constants the reason for this name will be evident from the following discussion. 

The same probabilities are given directly by our application of the method of variation of constants. The probability of transition to states of considerably different energy as the result of a small perturbation acting for a short time is very small, and we have neglected these transitions. Our calculation shows that the probability of finding the system in the state B depends on the value of t in the way given by Equation 41-12, varying harmonically between the limits 0 and 1. 

Let us consider a system composed of five harmonic oscillators, all with the same characteristic frequency v, which are coupled with one another by weak interactions. The set of product wave functions 4 (a) k(6) (c) (d) k(e) can be used to construct approximate wave functions for the system by the use of the method of variation of constants (Chap. XI). Here (a), ... 

Since this is a linear differential equation of the first order, it can be solved by the standard method of the variation of constants (see ref. 19... 

Direct synthesis is perhaps the single most widely used method of preparation. This method involves nucleating and growing the metal hydroxide layer by mixing an aqueous solution containing the salts of two metal ions, in the presence of the desired anion, and a base 50% sodium hydroxide is particularly useful for this purpose, since the common ion effect keeps it relatively free of carbonate (see later). It has been demonstrated that LDH materials form in preference to a mixture of the individual metal hydroxides (68) and that, in the case of aluminum as the trivalent cation, they generally do so through an aluminum hydroxide intermediate. Variations of this method include titration at constant or varied pH and buffered precipitation. 

A variation of this method consists of carrying ont the reaction withont the isolation of a component, i.e. using a large amount for each component so that all the amounts remain constant during the whole reaction. In this case, the reaction occurs within an extremely small fractional extent. 

That all four methods give a different result for the concentration of analyte underscores the importance of choosing a proper blank but does not tell us which of the methods is correct. In fact, the variation within each method for the reported concentration of analyte indicates that none of these four methods has adequately corrected for the blank. Since the three samples were drawn from the same source, they must have the same true concentration of analyte. Since all four methods predict concentrations of analyte that are dependent on the size of the sample, we can conclude that none of these blank corrections has accounted for an underlying constant source of determinate error. 

The concentric cylinder viscometer described in Sec. 2.3, as well as numerous other possible instruments, can also be used to measure solution viscosity. The apparatus shown in Fig. 9.6 and its variations are the most widely used for this purpose, however. One limitation of this method is the fact that the velocity gradient is not constant, but varies with r in this type of instrument, as noted in connection with Eq. (9.26). Since we are not considering shear-dependent viscosity in this chapter, we shall ignore this limitation. 

Early models used a value for that remained constant throughout the day. However, measurements show that the deposition velocity increases during the day as surface heating increases atmospheric turbulence and hence diffusion, and plant stomatal activity increases (50—52). More recent models take this variation of into account. In one approach, the first step is to estimate the upper limit for in terms of the transport processes alone. This value is then modified to account for surface interaction, because the earth s surface is not a perfect sink for all pollutants. This method has led to what is referred to as the resistance model (52,53) that represents as the analogue of an electrical conductance... 

Normalization is a preprocessing method often appHed to spectral data. It makes the lengths of all of the data vectors the same. Thus the sum of the squares of the elements of the data vectors is constant for all samples in the set. If is this sum for the unnormalized sample /, then to normalize the data vectors to the constant m, each element of the data vector would be multiphed by vnj.yj. A common example of this method is normalizing the area under a set of curves to unit area. AppHcation of this method effectively removes the variance in a data set because of arbitrary differences in magnitudes of a set of measurements when such variation is not meaningful and would obscure the significant variance. 

Two variations of the technique exists isocratic elution, when the mobile phase composition is kept constant, and gradient elution, when the mobile phase composition is varied during the separation. Isocratic elution is often the method of choice for analysis and in process apphcations when the retention characteristics of the solutes to be separated are similar and not dramaticallv sensitive to vei y small changes in operating conditions. Isocratic elution is also generally practical for systems where the equilibrium isotherm is linear or nearly hnear. In all cases, isocratic elution results in a dilution of the separated produces. 

Although the RSF contains matrix-dependent quantities, their variations are damped to some extent by virtue of taking ratios, and in practice the RSF is assumed constant for low concentrations of A (e. g. <1 atom%). It can be evaluated from measurements on a well-characterized set of standards containing A in known dilute concentrations. The accuracy of the method, however, is not as high as in laser-SNMS and XPS. 

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