Multiple expansion, methods

Expansion methods are often used for measuring gas densities. In these methods, a sample is expanded from a small volume to a larger volume (where the ratio of volumes is accurately known), holding the temperature constant and measuring the pressure ratio. Typically, multiple expansions are used (a successive expansion technique known as the Burnett method is popular), with the final state being at a pressure sufficiently low that the density is accurately known by other means (such as correction of the ideal-gas law by the second virial coefficient). The Burnett expansion method may achieve uncertainties in density as low as 0.01%. 

All the explanations given previously are rather intuitive and correspond to physics of the problem. However, the expression (27.17) has its mathematical basis. This is multiple-scale method or derivative-expansion method [9,26]. 

To find in (3-54), the Heaviside rule cannot be used for multiplication by s + 2), because 5 = -2 causes the second term on the right side to be unbounded, rather than 0 as desired. We therefore employ the Heaviside expansion method for the other two coefficients (tt2 and a3) that can be evaluated normally and then solve for by arbitrarily selecting some other value of 5. Multiplying (3-54) by (5 + if and letting 5 = -2 yields... 

When disperse phase of the coarse emulsion wets the membrane wall and suitable surfactants are dissolved in both liquid phases, the process results in a phase inversion namely a coarse OAV emulsion is inverted into a fine W/O emulsion (Figure 6.1c), and vice versa (Suzuki et al, 1999). The main advantage of this method is that a fine emulsion can be easily prepared from a low concentration coarse emulsion at high rates. For polytetrafluoroethylene (PTFE) membrane filters with a mean pore size of 1 im, the maximum dispersed phase volume fraction in phase-inverted emulsions was 0.9 and 0.84 for O/W and W/O emulsions, respectively (Suzuki et al., 1999). Flow-induced phase-inversion (FIPI) phenomenon was observed earlier by Akay (1998) who used a multiple expansion-contraction static mixer (MECSM) consisting of a series of short capillaries with flow dividers. Hino et al. (2000) and Kawashima et al. (1991) inverted a W/O/W emulsion made up of liquid paraffin. Span 80 (a hydrophobic surfactant), and Tween 20 (a hydrophilic surfactant) into a W/ O emulsion by extrusion through polycarbonate membranes with a mean pore sizes of 3 and 8 im. Inside the membrane pores, surfactant molecules are oriented with their hydrophobic groups toward the wall surface and with hydro-phihc groups toward the solubilized water molecules as a result of a lamellar structure formed inside the pores. The structure ruptured at the pore outlets. 

Gas dynamic approaches for growing microclusters in the gas phase, with the production of metal smokes, have been known for some time, but in general this method has been limited to relatively large particles of mean diameters greater than 2-5 nm. This technique has recently been extended by Andres and co-workers [59] with the development of a multiple expansion cluster source which has enabled the generation of copper clusters with mean sizes claimed to be controllable from 0.2 to 2.5 nm. 

The conceptually simplest approach to solve for the -matrix elements is to require the wavefimction to have the fonn of equation (B3.4.4). supplemented by a bound function which vanishes in the asymptote [32, 33, 34 and 35] This approach is analogous to the fiill configuration-mteraction (Cl) expansion in electronic structure calculations, except that now one is expanding the nuclear wavefimction. While successfiti for intennediate size problems, the resulting matrices are not very sparse because of the use of multiple coordinate systems, so that this type of method is prohibitively expensive for diatom-diatom reactions at high energies. 

Table 7.1 presents us with something of a dilemma. We would obviously desire to explore i much of the phase space as possible but this may be compromised by the need for a sma time step. One possible approach is to use a multiple time step method. The underlyir rationale is that certain interactions evolve more rapidly with rime than other interaction The twin-range method (Section 6.7.1) is a crude type of multiple time step approach, i that interactions involving atoms between the lower and upper cutoff distance remai constant and change only when the neighbour list is updated. However, this approac can lead to an accumulation of numerical errors in calculated properties. A more soph sticated approach is to approximate the forces due to these atoms using a Taylor seri< expansion [Streett et al. 1978] ... 

The SSW form an ideal expansion set as their shape is determined by the crystal structure. Hence only a few are required. This expansion can be formulated in both real and reciprocal space, which should make the method applicable to non periodic systems. When formulated in real space all the matrix multiplications and inversions become 0(N). This makes the method comparably fast for cells large than the localisation length of the SSW. In addition once the expansion is made, Poisson s equation can be solved exactly, and the integrals over the intersitital region can be calculated exactly. 

One element of database generation that is a key consideration is whether to expand the representative compounds to include alternative tautomers, protonated and deprotonated forms of the molecule, and also to enumerate stereochemistry fully if not specified in the input. Depending on the molecules in question and the options considered, these can lead to a 10-fold increase in the size of the database to be explored. However, such an expansion is necessary if methods are used that are sensitive to such chemical precision (e.g., docking). For 3D similarity searching, it is sometimes more efficient to consider various modifications to the query, leading to multiple searches against a smaller database. 

The next question is how to find the partial fractions in Eq. (2-25). One of the techniques is the so-called Heaviside expansion, a fairly straightforward algebraic method. We will illustrate three important cases with respect to the roots of the polynomial in the denominator (1) distinct real roots, (2) complex conjugate roots, and (3) multiple (or repeated) roots. In a given problem, we can have a combination of any of the above. Yes, we need to know how to do them all. 

Figure 18A shows the overlaid multiplicity-edited GHSQC and 60 Hz 1,1-ADEQUATE spectra of posaconazole (47). As will be noted from an inspection of the overlaid spectra, there is an overlap of the C46 and C47 resonances of the aliphatic side chain attached to the triazolone ring that can be seen more clearly in the expansion shown in Figure 18B. In contrast, when the data are subjected to GIC processing with power = 0.5, the overlap between the C46 and C47 resonances is clearly resolved (Figure 18C). In addition, the weak correlation between the C3 and C4 resonances of the tetrahydrofuryl moiety in the structure is also observed despite the fact that this correlation was not visible in the overlaid spectrum shown in A. This feature of the spectrum can be attributed to the sensitivity enhancement inherent to the covariance processing method.50...

The Newton-Raphson method for multiple equations also starts with truncated Taylor expansions. For the system... 

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... 

Spectroscopic applications usually require us to go beyond single-point electronic energy calculations or structure optimizations. Scans of the potential energy hypersurface or at least Taylor expansions around stationary points are needed to extract nuclear dynamics information. If spectral intensity information is required, dipole moment or polarizability hypersurfaces [202] have to be developed as well. If multiple relevant minima exist on the potential energy hyper surface, efficient methods to explore them are needed [203, 204],... 

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