Series approximations

Approximate solutions to Eq. 11-12 have been obtained in two forms. The first, given by Lord Rayleigh [13], is that of a series approximation. The derivation is not repeated here, but for the case of a nearly spherical meniscus, that is, r h, expansion around a deviation function led to the equation... 

Figure 5.86. Tanks-in-series approximation of a tubular reactor.

For the parameters used to obtain the results in Fig. 3, X 0.6 so the mean free path is comparable to the cell length. If X -C 1, the correspondence between the analytical expression for D in Eq. (43) and the simulation results breaks down. Figure 4a plots the deviation of the simulated values of D from Do as a function of X. For small X values there is a strong discrepancy, which may be attributed to correlations that are not accounted for in Do, which assumes that collisions are uncorrelated in the time x. For very small mean free paths, there is a high probability that two or more particles will occupy the same collision volume at different time steps, an effect that is not accounted for in the geometric series approximation that leads to Do. The origins of such corrections have been studied [19-22]. 

Alternative methods, such as correcting the nonlinearity though the application of an appropriate physical theory as we described above, may do as well or even better than a Taylor series approximation, but a rigorous theory is not always available. Even in... 

The error in this linear approximation approaches zero proportionally to (Ax)2 as Ax approaches zero. Given initial values for the variables, all nonlinear functions in the problem are linearized and replaced by their linear Taylor series approximations at this initial point. The variables in the resulting LP are the Ax/s, representing changes from the base vaiues. In addition, upper and lower bounds (called step bounds) are imposed on these change variables because the linear approximation is reasonably accurate only in some neighborhood of the initial point. 

As discussed in Section (8.2), Equations (8.64) and (8.65) is a set of (n + m) nonlinear equations in the n unknowns x and tn unknown multipliers A.. Assume we have some initial guess at a solution (x,A). To solve Equations (8.64)-(8.65) by Newton s method, we replace each equation by its first-order Taylor series approximation about (x,A). The linearization of (8.64) with respect to x and A (the arguments are suppressed)... 

Given an initial guess x0 for x, Newton s method is used to solve Equation (8.84) for x by replacing the left-hand sidex>f (8.84) by its first-order Taylor series approximation at x0 ... 

The factor f reduces the oscillation amplitude symmetrically about R - R0, facilitating straightforward calculation of polymer refractive index from quantities measured directly from the waveform (3,). When r12 is not small, as in the plasma etching of thin polymer films, the first order power series approximation is inadequate. For example, for a plasma/poly(methyl-methacrylate)/silicon system, r12 = -0.196 and r23 = -0.442. The waveform for a uniformly etching film is no longer purely sinusoidal in time but contains other harmonic components. In addition, amplitude reduction through the f factor does not preserve the vertical median R0 making the film refractive index calculation non-trivial. 

This series approximation can be easily generated on a digital computer. 

The first approximation of Kasai and Oosawa (1969) has been improved by the efforts of two research groups. Zeeberg et al. (1980) employed an elegant approach with difference equations to the equilibrium exchange problem, and, through the use of a Taylor s series approximation as well as Stirling s approximation, obtained the following solution ... 

T. Lu, and D. Yevick, Comparative Evaluation of a Novel Series Approximation for Electromagnetic Fields at Dielectric Comers With Boundary Element Method Applications, Journa/ of Lightwave Technology 22, 1426-1432 (2004). 

Under various circumstances, it may be justified to replace either the exponential or T0 by their series approximations, thereby simplifying this result still further. ... 

Curve d -asc, but corrected to most stable level with Smax, giving energy of ground-state of / set below external reference. Latter, within compound series, approximately fixed at appropriate level w.r.t. p- and d-bands. 

Each property depends on only one (rather than the expected two) degrees of freedom, and each becomes pathological (divergent) in the limit of small T or P, respectively. For solids and liquids, aP and f3T are rather insensitive to P, T variations, so low-order Taylor series approximations may be adequate. For gases, however, it is generally necessary to differentiate an accurate equation of state to obtain a realistic (P, T) dependence of aP, fiT. 

The x - extends over all the solute species and is designated as xB, which we have previously used to designate the mole fraction of the single solute in a binary solution. At a concentration small enough to make the ideally dilute solution approximation, it is usually sufficient to use only the first term in the Taylor series approximation of ln(l — xB),4... 

In the small particle size regime, two equivalent formulations lead to the interpretation of the data in terms of ratios of moments of the particle size distribution or in terms of powers of the D32 average (equations 15 and 20). It is clear that in either case a sufficient number of terms in the series has to be included in order to account for the behavior of the extinction as function of a. The number of terms required cannot be decided a priori, rather the data itself has to dictate how many terms in the power series approximation the measurements can detect. 

This appendix reports the weights for the moments of the particle size distribution obtained from an eight order Taylor Series approximation to the scattering efficiency for the anomalous diffraction case... 

In descriptions of this problem, the names of Randles [460] and Sevclk [505] are prominent. They both worked on the problem and reported their work in 1948. Randles was in fact the first to do electrochemical simulation, as he solved this system by explicit finite differences (and using a three-point current approximation), referring to Emmons [218]. Sevclk attempted to solve the system analytically, using two different methods. The second of these was by Laplace transformation, which today is the standard method. He arrived at (9.116) and then applied a series approximation for the current. Galus writes [257] that there was an error in a constant. Other analytical solutions were described (see Galus and Bard and Faulkner for references), all in the form of series, which themselves require quite some computation to evaluate. 

Dirac equation, but an approximate perturbation treatment of the Schrodin-ger equation, due to Sommerfeld, gives the same result In the energy E = T + V, the kinetic energy Tnonre = p2/2me should be replaced by a power series approximation ... 

In some sitrrations, no theoretically motivated analytic form y = fix) is apparent, and low-order series approximations do not provide a good fit. Yon should then be cautious about using high-order empirical series fitting forms, since they may yield very poor... 

There exist (4, 5, 8, 9, 27) simple direct relations, between isotope effect, structure, and force field, which do not necessarily require a complete knowledge of all molecular parameters and avoid the solution of the secular equation. These relations are, however, approximations restricted to limited ranges of temperature. [Newer approximation methods, based on expansions in Jacobi polynomials, are applicable over wide ranges of temperatures (6, i6).] In the past, before the ready availability of fast digital computers, tests of the validity of these approximations were usually fairly limited in nature, but recent extensive tests on model calculations of kinetic isotope effects have been carried out 23, 28). In addition, extensive tests of power-series approximations (not considered in the present paper) have now been performed (6,16). 

In many practical situations we have to compute a function / (A) of an x TV matrix A. A popular way of computing a matrix function is through the truncated Taylor series approximation. The conditions under which a matrix function / (A) has a Taylor series representation are given by the following theorem (Golub and Van Loan, 1996). 

Laboratory tests have also shown that cesium and strontium in the effluent from the two plant-scale ion-exchange columns (Zeolon-900 and Amberlite-200) can be further removed by the use of an additional Amberlite-200 and Zeolon-900 column in series. Approximately 9000 column volumes (1.6 X 10 gal for 24 ft of resin) of effluent from the first ion-exchange unit can be processed before the radioactivity concentration guide (RCG) controlled area release limit for each radioisotope is reached and before either column would need to be regenerated. 

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