First-order expansion

Here, as the surface morphology changes with dissolution, the concentration distribution in the diffusion layer also changes. This influence is exhibited by the first-order expansion outside the double layer... 

Similarly, the first-order expansion of the p° and a of Eq. (5.1) is, respectively, responsible for IR absorption and Raman scattering. According to the parity, one can easily understand that selection mles for hyper-Raman scattering are rather similar to those for IR [17,18]. Moreover, some of the silent modes, which are IR- and Raman-inactive vibrational modes, can be allowed in hyper-Raman scattering because of the nonlinearity. Incidentally, hyper-Raman-active modes and Raman-active modes are mutually exclusive in centrosymmetric molecules. Similar to Raman spectroscopy, hyper-Raman spectroscopy is feasible by visible excitation. Therefore, hyper-Raman spectroscopy can, in principle, be used as an alternative for IR spectroscopy, especially in IR-opaque media such as an aqueous solution [103]. Moreover, its spatial resolution, caused by the diffraction limit, is expected to be much better than IR microscopy. 

We casually ignore the possibility of a more accurate second order expansion. That s because the higher order terms are nonlinear, and we need a linear approximation. Needless to say that with a first order expansion, it is acceptable only if h is sufficiently close to hs. 

An often-used method consists of expanding f in Eq. (8.1) as a Taylor series about a certain vector that is close to x(r). In particular, if a first-order expansion is carried out on the current estimate of the state vector, we obtain... 

Isotopic fractionation provides illustrative examples of first-order expansions of unknown functions. In general, the mass spectrometric measurement r/ of the ratio between two isotopes of mass m( and m, of the same element, differs from the natural value R/. Only a very small fraction of the original sample produces ions and different processes taking place in different parts of the mass spectrometer act differently on the sensitivity of each isotope. We assume that instrumental isotopic fractionation is mass-dependent. 

Some equations such as/(x)=0 cannot be explicitly solved for x. If multiple solutions are not expected in a narrow range, Newton s method is often simple to implement and has faster convergence than the natural method of interval splitting. The method is recursive and uses the first-order expansion off (x) in the vicinity of the fcth guess... 

The Euler method has little practical value, but forms the basis for most of the more elaborate methods. It consists in a first-order expansion of the derivative. The approximation at step t[Pg.129]

Two similar sets of derivatives can be written by substituting 67 and 68 to 66. In order to solve the system for cpg, P pie, and P , we select a first guess of these variables, which we group into a 3x1 vector Xq. Inserting this guess into each equation produces a 3x1 vector y, out of the three p values and a 3 x3 matrix Jo out of the nine derivatives. The updating Newton-Raphson step uses the first-order expansion of the pvalues about the current value Xo of x as ... 

When we now insert the expansion from equation (55) on both sides of equation (66), relations between the expansion coefficients in equation (55) ensue. For example, from equation (66) the first-order expansion coefficients in the expression for fi must satisfy... 

The above relation is very similar to the first-order expansion with respect to Rew of the polynomial fitting of numerically obtained data for heat transfer in a square channel with one porous and three solid walls (Hwang et al., 1990). 

Now consider how the stream-function field, which is defined in the two-dimensional z-r plane, relates to the mass flow through dA. Assume that two points in the z-r plane, 1 and 2, define the extremities of the differential line that when rotated in 0 becomes the area. Further assume that the dimensions of dA are sufficiently small that a first-order expansion of d V is valid,... 

As regards the double layer charging, the first-order expansion of the charge density—potential relationship is... 

The reverse flux can be obtained by using a first-order expansion of the concentration in the (3 plane, so that... 

The first-order expansion employed will be valid under all usual conditions. 

The first-order expansion of this equation gives the result... 

In the exciton-photon interaction, the translational molecular motions have negligible effects owing to the small amplitude of the translation compared to the optical wavelength. In contrast, the molecular rotations may cause an important variation of the transition dipole the librations may be strongly coupled to the incident photon via its coupling to the exciton. If DX(R) is the transition dipole of an a molecule in a unit cell, the first-order expansion in the libration coordinate 8 around the u axis will give... 

Likewise, we shall not pursue here in detail the fundamental work of Watson [24,25] who introduced the mass dependence or restructure, where for the first time an explicit compensation of part of the rovib contributions was attempted, based on the first-order expansion of the in terms of atomic masses. On the basis of Watson s work, Nakata and colleagues [26] have developed the complementary or -method where, by a judicious selection of complementary sets of isotopomers, the effects of also the second-order terms of the expansion can be almost completely removed, so that the restructure is expected to be very nearly equal to the r,-structure. In practice, the rm- and -methods are limited to very small molecules. 

The conventionally quoted result of the exact kinetic theory [5], [9] is that obtained from a first-order expansion in Sonine polynomials, namely. 

Results of Approximations for In b u). The optimum (y,8) and the corresponding RMSE for the case of L = 0 (Equation 44) are tabulated in Tables II and III, respectively. Similar tabulations for the case of L = 5 are given in Tables IV and V. In Tables II and IV, the upper and lower numbers for each range and order are the optimum values of y and 8, respectively. The first order expansion depends only on the ratio y/8 and not the individual values of y and 8. For convenience, therefore, the value unity has been entered for 8 for all the one-term expansions in... 

The NONMEM program implements two alternative estimation methods, the first-order conditional estimation and the Laplacian methods. The first-order conditional estimation (FOCE) method uses a first-order expansion about conditional estimates (empirical Bayes estimates) of interindividual random effects, rather than about zero. In this respect, it is like the conditional first-order method of Lindstrom and Bates.f Unlike the latter, which is iterative, a single objective function is minimized, achieving a similar effect as with iteration. The Laplacian method uses second-order expansions about the conditional estimates of the random effects. ... 

A first order expansion of the energy from one set of action variables to another Is then possible using the expression ... 

In the context of this section, a slightly dense gas is a gas properly described by the first-order expansion in the density, i.e., up to the linear term in (2.49). Before analyzing the content of the coefficient B(R R") in the expansion of g(R), let us demonstrate its origin by considering a system of exactly three particles. Putting AT=3 in (2.48), we get... 

Figure 2.3. The form of g R) for hard-sphere particles [a — 1), using the first-order expansion in the density (equation 2.68). The three curves correspond to p — 0.1 (lower), p — 0.4 (intermediate), and p — 0.9 (upper).

In practice, the most important set of thermodynamic variables is of course T, P, pA, employed in (6.34). However, relation (6.33) is also useful and has enjoyed considerable attention in osmotic experiments where pB is kept constant. This set of variables provides relations which bear a remarkable analogy to the virial expansion of various quantities of real gases. We demonstrate this point by extracting the first-order expansion of the osmotic pressure n in the solute density pA. This can be obtained by the use of the thermodynamic relation... 

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