Derivative approximation notation

Thus, in view of the inequality m2 mu we derive a physically obvious result the motion of a light particle 2 can approximately be regarded as vibrations about a relatively immobile particle 1. The squared frequency of these vibrations would be equal to ku/m2 thus differing (by virtue of the approximate equality (A 1.36)) from h2/m2 by the terms of the order m/M. Besides, the second formula in (A1.37) shows that the approximate notation of the sought-for frequency as knlm is preferable because it is valid accurate to terms of the order (m/M)2. 

Table A.l. rnfi (3.14) for multi-point first derivatives. The notation y[(n) means the approximation at point, i using n points nmnftered 1. .. n...

The mixed, v t — % notation here has historic causes.) The Schrodinger equation is obtained from the nuclear Lagrangean by functionally deriving the latter with respect to t /. To get the exact form of the Schrodinger equation, we must let N in Eq. (95) to be equal to the dimension of the electronic Hilbert space (viz., 00), but we shall soon come to study approximations in which N is finite and even small (e.g., 2 or 3). The appropriate nuclear Lagrangean density is for an arbitrary electronic states... 

We proceed to the estimation of the order of approximation for scheme (II) under the agreement that u = u(x,t) possesses a number of derivatives in X and t necessary in this connection for performing current and subsequent manipulations. Within the notations... 

Before comparing these predictions regarding the critical point with experimental results, we may profitably examine the binodial curve of the two-component phase diagram required by theory. The following useful approximate relationship between the composition V2 of the more dilute phase and the ratio y = V2/v2 of the compositions of the two phases may be derived (see Appendix A) by substituting Eq. (XII-26) on either side of the first of the equilibrium conditions (1), using the notation V2 for the volume fraction in the more dilute phase and V2 for that in the more concentrated phase, and similarly substituting Eq. (XII-32) for fX2 and y,2 in the second of these conditions ... 

Similar to irreversible reactions, biochemical interconversions with only one substrate and product are mathematically simple to evaluate however, the majority of enzymes correspond to bi- or multisubstrate reactions. In this case, the overall rate equations can be derived using similar techniques as described above. However, there is a large variety of ways to bind and dissociate multiple substrates and products from an enzyme, resulting in a combinatorial number of possible rate equations, additionally complicated by a rather diverse notation employed within the literature. We also note that the derivation of explicit overall rate equation for multisubstrate reactions by means of the steady-state approximation is a tedious procedure, involving lengthy (and sometimes unintelligible) expressions in terms of elementary rate constants. See Ref. [139] for a more detailed discussion. Nonetheless, as the functional form of typical rate equations will be of importance for the parameterization of metabolic networks in Section VIII, we briefly touch upon the most common mechanisms. 

Restricting ourselves to the rapid equilibrium approximation (as opposed to the steady-state approximation) and adopting the notation of Cleland [158 160], the most common enzyme-kinetic mechanisms are shown in Fig. 8. In multisubstrate reactions, the number of participating reactants in either direction is designated by the prefixes Uni, Bi, or Ter. As an example, consider the Random Bi Bi Mechanism, depicted in Fig. 8a. Following the derivation in Ref. [161], we assume that the overall reaction is described by vrbb = k+ [EAB — k EPQ. Using the conservation of total enzyme... 

The above approximations to a first derivative used only two points, which sets a limit on the approximation order. By using more points, higher-order approximations can be achieved. In the context of this book, forward and backward multi-point formulae are of special interest, as well as some asymmetric and centra] multi-point ones. To this end, a notation will be defined here. Figure 3.2 shows the same curve as Fig. 3.1 but now seven points are marked on it. The notation to be used is as follows. If a derivative is approximated using the n values yi. . yn, lying at the x-values. iq. ..xn (intervals h) and applied at the point (Xi,yi), then it will be denoted as y (n) (for a first derivative) and y"(n) (for a second derivative). 

In this way, the coefficients for any y((n) can be calculated. Table A.l in Appendix A shows them all, as whole numbers m/3j, where m is the multiplier mentioned above. For each n, the Table shows forward differences (at index 1), backward derivatives (at index n) and derivatives applying at points between the two ends. For n up to 6, all possible forms are included, as they will be needed later, while for n = 7, only the forward and backward formulae are shown, as only these are needed. In case the reader wonders why all this is of interest the forms y[(n) will be used to approximate the current in simulations (see the next section) the backward forms y n(n) will be used in the section on the BDF method in Chaps. 4 and 9, and the intermediate forms shown in the Table will be used for the Kimble White (high-order) start of the BDF method, also described in these chapters. The coefficients have a long history. Collatz [169] derived some of them in 1935 and presents more of them in [170]. Bickley tabulated a number of them in 1941 [88], The three-point current approximation, essentially y((3) in the present notation, was first used in electrochemistry by Randles [460] (preempted by two years by Eyres et al. [225] for heat flow simulations), then by Heinze et al. [301], and schemes of up to seven-point were provided in [133]. 

First we consider the current approximation presented in the above two sections. A question left untouched, for example, the equation for the current approximation (3.25) above, is just what terms were dropped when generating a particular form. The order of what was dropped is given in Sect. 3.3, but not extended to actual higher terms. This must be done now. Bieniasz [108] presents a table of these and we can write the first few of these. For this, it is convenient to use a more compact notation for the higher derivatives let... 

The derivatives in eq 188 are approximated by central differences. This leads to a discrete representation of the problem in the form of linear equations. For j = 1 and j = N we have slightly modified equations, since here the boundary conditions have to be considered. Because the concentration of three isomers has to be calculated for each of the N volume elements as a whole, we have a system of N x 3 linear equations. This can be expressed in matrix notation ... 

In this chapter we collect and present, without derivation, in explicit, Anal form the relevant phase-integral quantities and their partial derivatives with respect to E and Z expressed in terms of complete elliptic integrals for the first, third and fifth order of the phase-integral approximation. For the first- and third-order approximations some of the formulas were first derived by means of analytical calculations, and then all formulas were obtained by means of a computer program. In practical calculations it is most convenient to work with real quantities. For the phase-integral quantities associated with the r -equation we therefore give different formulas for the sub-barrier and the super-barrier cases. As in Chapter 6 we use instead of L2 , L2n, K2n the notations LAn+1 >, L( 2n+1 KAn+l). 

It has been demonstrated in an earlier paper that one can derive a mixed quantum-classical propagation scheme based on the TDSCF approximation [47]. The TDSCF scheme has been extensively used in many studies [48,27,49] and shall be resumed briefly in order to introduce the notations. 

The implicit field dependence is yet not apparent, since the parameters A in Eq. (100) are chosen freely. Let us therefore denote the set of parameters which are optimized in the presence of the field by A. The implicit dependence then becomes obvious, as we will then have a corresponding equation to that of Eq. (88) for approximate wave functions. In the following derivations, we will use the notation... 

However, we recollect from the derivation of the single phase -equation in sect 1.2.7 that the rigorous definition of this production term can be approximated by the generalized Boussinesq hypothesis, in the following manner (conveniently written in tensor notation) ... 

This is a common but confusing notation. In fact, it is concepmally more correct to classify these methods according to the terms excluded by the approximation. With the latter approach, a method that approximates the NDF with a constant value within each interval can be denoted a first-order method, since when approximating the NDF as a Taylor series the first term excluded is the first-order derivative term. 

This is an expression originally derived by John Ramshaw under certain approximations that one cannot expect to be exactly satisfied. However, as shown in Ref. 4, these approximations need not be assumed in the case of a cylindrically symmetric reference system to obtain (2.15). In contrast to (2.12), on the other hand, (2.15) will not hold if the reference system has more general symmetry. An Appendix to Ref. 5 gave a general reduction of (2.12) that yields an expression for e in terms of c(12) for all symmetries. Subsequently Ramshaw himself derived an equivalent general reduced form of (2.12) and (2.9) in somewhat different notation. 

For a miscible blend of two polymers, an expression giving I(q) was derived by deGennes by using a theoretical technique called random phase approximation. The discussion of this technique or the derivation of I q) unfortunately requires an excursion into a realm of polymer physics far beyond the scope of the present book, and we are here left with simply quoting the result. Interested readers are advised to consult other textbooks (deGennes1 and Doi4) for the derivation. The expression for I(q) derived by deGennes can be written, in our present notation, as... 

Fig. 2. Preparation of heterobifunctional PEG derivatives. The abbreviation PEG for this scheme is defined with structure 1, and differs from the way this abbreviation is used in the remainder of the paper. The notation PEG(X, Y) refers to the statistical mature of X-PEG-X, X-PEG-Y, and Y-PEG-Y, created in step 1, which was intentionally carried out aiming for approximately 50% conversion of OH group to Cl groups. All subsequent chemical steps were quantitative, and ion-exchange chromatography resulted in the isolation of pure heterobifunctional species 2, of defined structure (see box). The indicated chemistry was carried out for both PEG-2000, n = 45 (see Section 2.6) and PEG-4000, n = 90 (comparable to text Experimental w/v ratios were kept constant so that reaction concentrations were halved), with comparable yields. In other work, not reported herein, alternative approaches were used to generate statistical PEG mixtures that included

---