Mutual independence

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

For any proposed suppression system design, it is necessary to ascribe with confidence an effective worst-case suppressed maximum explosion overpressure Pred.max- Provided that the suppressed explosion overpressure is less than the process equipment pressure shoclc resistance and provided further that this projected suppression is achieved with a sufficient margin of safety, explosion protection security is assured. These two criteria are mutually independent, but both must be satisfied if a suppression system is to be deployed to provide industrial explosion protection. 

In common with all multidimensional separations, two-dimensional GC has a requirement that target analytes are subjected to two or more mutually independent separation steps and that the components remain separated until completion of the overall procedure. Essentially, the effluent from a primary column is reanalysed by a second column of differing stationary phase selectivity. Since often enhancing the peak capacity of the analytical system is the main goal of the coupling, it is the relationship between the peak capacities of the individual dimensions that is crucial. Giddings (2) outlined the concepts of peak capacity product and it is this function that results in such powerful two-dimensional GC separations. 

Extended- and folded-chain crystallization are mutually independent processes, and extended-chain crystallization can take place prior to folded-chain crystallization [6,19-25]. 

The fundamental assumption of Hansch analysis is that substituent values are additive. This implies that substituents are mutually independent, i.e. that the effect of a substitution group at one position in the parent molecule is independent of substitution groups at other positions. The assumption of additivity is violated, for example, when hydrogen bonding occurs between two adjacent substituents. 

The energy of interaction between the components is conveniently described by means of three mutually independent Flory Huggins parameters x X ° and xP° Indices refer to the respective component pairs. 

Standard theories of steric stabilisation assume that the free energy of the system can be expressed as a sum of two terms, volume restriction and osmotic, which are mutually independent. 

As we have supposed here that the ions were mutually independent, the total current is given by ... 

In addition to this, and in contrast with the homogeneous case discussed in Section 5.2.2, the diffusion of P and Q is therefore not perturbed by any homogeneous reaction. If, furthermore, the P/Q electron transfer at the electrode is fast and thus obeys Nernst s law, the diffusive contribution to the current in equations (5.11) and (5.12) is simply equal to the reversible diffusion-controlled Nernstian response, idif, discussed in Section 1.2. The mutual independence of the diffusive and catalytic contributions to the current, expressed as... 

In this particular case, the equations of motion for the X and Y directions are mutually independent and therefore can be integrated separately. Furthermore, because the frictional term is now a linear function of velocity, the sign will automatically adjust to take account of whether motion is downwards or upwards. 

In some sequences, such as the FFC variety of the Jeener-Broekaert sequence, the RF pulses applied during the preparatory sub-sequence need to be coordinated in phase with those applied during the detection subsequence. In such cases the preparatory and detection sub-sequence sections of the FFC sequence are no longer mutually independent. 

Because P and T are arbitrarily fixed and the conformations of the G-X loops of the various phases are mutually independent, no more than two phases may be stable in any part of the compositional space. 

According to equation 3.168, the energy properties of the four components are not mutually independent but, once the Gibbs free energy is fixed for three of them, the energy of the fourth is constrained by the reciprocal energy term... 

This calculation was performed under the assumption of mutual independence between the 2 variables. What about other sorts of dependencies Excluding the case where one variable is a direct function of the other (which should be modeled directly in the assessment), the following statistical situations are commonly encountered in analyses ... 

Contain mutually independent samples Let s review these items one at a time. 

Every physical relationship between n physical quantities can be reduced to a relationship between m = n — r mutually independent dimensionless groups, whereby r stands for the rank of the dimensional matrix, made up of the physical quantities in question and generally equal to the number of the basic quantities contained in them. ... 

The present modified form of Maxwell s equations in vacuo is based on two mutually independent hypotheses ... 

Whether starting from in vivo or in vitro data, and from animal or human data, extrapolation to clinical effects is currently performed by combinations of mutually independent factors (treated as independent variables), or even by simple monofactorial equations. Such an approach is clearly not erroneous, witness its many fair predictions, but it is a mere approximation. 

In other words, the molecules are dissociated. The dissociation of the gas is only partial, because the faster-moving molecules break down first. There is a kind of struggle for existence the slower-moving molecules are the fittest to survive. It is, however, still possible for the atoms of hydrogen and oxygen to unite each to each forming elementary molecules H2 and 02 but if a still higher temp, be applied, even these combinations become unstable, and the two-atom molecules dissociate into one-atom molecules, for the atoms themselves then become mutually independent free-rovers. 

---