Variable phase

These terms are analogous to those on p. 265 of [7], It will be noted that the symbol c has been reinstated as in Section VI.F, so as to facilitate the order of magnitude estimation in the nearly nonrelativistic limit. We now proceed based on Eq. (168) as it stands, since the transformation of Eq. (168) to modulus and phase variables and functional derivation gives rather involved expressions and will not be set out here. 

The term ff denotes the number of independent phase variables that should be specified in order to establish all of the intensive properties of each phase present. The phase variables refer to the intensive properties of the system such as temperature (T), pressure (P), composition of the mixture (e.g., mole fractions, x ), etc. As an example, consider the triple point of water at which all three phases—ice, liquid water, and water vapor—coexist in equilibrium. According to the phase rule,... 

As both phases occupy the full flow field concurrently, two sets of conservation equations correspond to these two phases and must be complemented by the set of interfacial jump conditions (discontinuities). A further topological law, relating the void fraction, a, to the phase variables, was needed to compensate for the loss of information due to model simplification (Boure, 1976). One assumption that is often used is the equality of the mean pressures of the two phases, ... 

Weiser, J.N., Shchepetov, M. and Chong, S.T. (1997) Decoration of lipo-polysaccharide with phosphorylcholine a phase-variable characteristic of Haemophilus influenzae. Infection and Immunity 65, 943-950. 

Phase and antigenic variation in bacteria. These may be mediated by slipped-strand mispairing of repeat sequences that occur in these phase variable genes or by gene duplication events. 

To compare Cl with ° we rewrite the latter in terms of the phase variables introduced in Eq. (163)... 

A firm understanding of the chromatographic system is expected of the analyst who developed an analytical procedure. However, when this procedure is transferred to another analyst, problems may occur. The problems may stem from differences in the chromatographic system, column variance, temperature fluctuations, mobile phase variability, and other factors. The standard means to ensure that the procedure transfers (technology transfer) successfully is through the use of system suitability... 

We begin our analysis by rewriting (6.2.7) in terms of the phase variables u, p as a system of two first-order equations ... 

We define a variable called Phase which represents the current offset into the waveform, which has both an integer and a fractional part. The integer part of the Phase variable is denoted integerPart (Phase). ... 

The meaning of the quantity 3 will be clarified in Section II.B. Since f0 could be chosen as one of the phase variables, the ensemble average of the integrand in Eq. (24) also vanishes ... 

The quantities h, (p0 and /, v(/o constitute (to within a constant multiplier) two pairs of the canonically conjugate arbitrary constants. Therefore we may choose them as the phase variables while averaging over T in (27). We note that these quantities refer to a local phase space corresponding to any chosen direction of the symmetry axis. Hence, integration performed in the overall phase space dTls corresponding to an isotropic fluid should additionally include averaging over all possible inclinations 0 of the symmetry axis C to the a.c. field vector E. Thus,... 

If strong collisions are rare (e.g., if y 1), another important factor—an anharmonicity of a molecular ensemble—plays an important role. This notion, introduced in GT, Section VII with respect to a parabolic potential well (or the well close to the latter), means dependence of the period on the phase variables h and l. 

In the case of a parabolic well the period is independent on the phase variables, the anharmonicity vanishes, and the bandwidth is nonzero only due to strong collisions. The more a potential profile differs from the parabolic one, the larger the anharmonicity and the wider the absorption band. The intensity of the absorption peak should then decrease since in accord with the Gordon rules (see, e.g., GT, Section III.G or see Section VIIA.4 in the present chapter) in an isotropic medium the integrated absorption does not depend on parameters of the model. 

Instead of h and /, we choose as phase variables the two components g and/of energy h given by... 

In view of Eq. (65) the phase-variable differential (which we shall apply in Section III.B) is... 

Taking into account Eq. (66), we express them through the phase variables g and /. Then we have... 

The function (208), found for motion in a plane with / = 0, is employed in this approximation for an ensemble of dipoles, characterized by both phase variables, h and l, distributed in the phase region (198). 

Unlike Section V.E, now we consider directly the spectral function of an isotropic spatial ensemble, for which an average over the phase variable h and l should be found. In this case L z) = K (z) + 2K (z). Omitting again the first term, we have in view40 of Eqs. (3.62) and (3.71) in GT ... 

We take coordinate ro and velocity v0 as the steady state phase variables both are determined at an infinite interval [—oo, oo]. The mean (vo) = 0 the mean (rj)) is found by using Eq. (323a) ... 

Retention and selectivity in HIC depend on temperature in addition to the nature of the mobile and stationary phases the strength of the hydrophobic interaction increases with increasing temperature.38 The primary stationary-phase variables are the ligand density, structure, and hydrophobicity. The more hydrophobic the stationary phase, the greater is the retention of the analytes. If the stationary phase is too hydrophobic, however, the analytes may become denatured. Some analytes may only be handled satisfactorily on hydrophilic stationary phases.12... 

The set of stochastic equations given by (3.37) is equivalent (in the linear case) to equations (3.11) with the memory functions defined in Section 3.3, but, in contrast to equations (3.11), set (3.37) is written as a set of Markov stochastic equations. This enables us to determine the variables that describe the collective motion of the set of macromolecules. In this particular approximation, the interaction between neighbouring macromolecules ensures that the phase variables of the elementary motion are co-ordinates, velocities, and some other vector variables - the extra forces. This set of phase variables describes the dynamics of the entire set of entangled macromolecules. Note that the Markovian representation of the equation of macromolecular dynamics cannot be made for any arbitrary case, but only for some simple approximations of the memory functions. We are considering the case with a single relaxation time, but generalisation for a case with a few relaxation times is possible. 

These three equations are merely algebraic combinations of Eq. 3.18. Inspection suggests that the activities of H20, Mg2+, Fe3, and H4Si04 are candidates for the dependent aqueous-phase variable in a predominance diagram. To preserve comparability with Fig. 3.5, (H,Si04) will be chosen, with the other three activities fixed as before. These choices reduce the general log K equations... 

A FORTRAN program is available from the authors. For each system, uj are assigned for species in phases expected at equilibrium. As uj become small for all species in a phase, variables are retained and new values computed using Powell s method. 

All of these motions yield the same value for the time average of any function p) of the phase variables.98... 

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