Inequality expression

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

The first important step in this direction was taken by Harker and Kasper (1948), who derived relations between pairs or small groups of reflections in a centrosymmetric structure in the form of inequality expressions. The simplest of these says that if Uhkl is the unitary structure amplitude —the structure amplitude expressed as a fraction of what it would be if the waves from all atoms were exactly in phase with each other f—then... 

The determination of the value of fCj) allows checking the inequalities expressed by Equations (Al) and (AT). For the runs for which the previous inequalities do not hold. Equation (19) must be used for determining the only unknown parameter contained in it, that is, the Cl,o value. 

In fluid d mamics there is no specific use of the transport equation for entropy other than being a physical condition indicating whether a constitutive relation proposed has a sound physical basis or not (nevertheless, this may be a constraint of great importance in many situations). In this connexion we usually think of the second law of thermodynamics as providing an inequality, expressing the observation that irreversible phenomena lead to entropy production. 

The entropy equation can now be used to express the Clausius form of the second law of thermodynamics for open flow systems (e.g., [7] [145], p. 126). The inequality expresses that irreversible phenomena (diffusive momentum... 

In its simplest form the Schwarz inequality expressed an obvious relation between the products of magnitudes of two real vectors ci and C2 and their scalar product... 

This inequality expresses the notion that the error must be falling fast enough for the decreasing proportional term to overcome the effect of a still-increasing integral term. 

Substituting Zf together with W and A into the inequality expression results in the following system of inequality constraints ... 

This inequality expresses the fact that only the 4-fold square planar or the 2-fold apical Cul-0 coordination are favorable in the flame of the model [fig. 46b, (a) and (c)]. Other theoretical calculations have shown that the 3-fold coordination [fig. 46b, (b)] has higher energy (Burdett and Kulkami 1989) and decreases the hole count in the planes and Tc (Ceder et al. 1991b). We note that, nevertheless, this eoordination appears at the ends of chain fragments. Ignoring the intra-chain disorder is one of the drawbacks of this model. 

While the "more than" and "less than" aspects of the rate breakpoint labels in the columns of the tariff table may not match exactly the inequalities expressed in (4.14), one can see from Figure 4.5 that this will not really matter in the final analysis.) Having specified the breakpoint/, the freight charge for this shipment is given by... 

The important inequality expressed by (40) shows for nondegenerate fre-quentnes that no frequency can be decreased by the increase of any that is, the decrease of the mass of any atom. J he inequality follows from the fact that the kinetic energy can never be ncgatii C for any nonnegative values of the )x, since it is defined as... 

The results obtained as the relative orientation of the transmission axes of the polarizers was varied from 0° to 90°, were found to be in agreement with the quantum mechanical prediction as expressed in Eq. (26). Also, the results at (a, b) = 22.5° and (a, b) = 67.5° combined with those with both sets of polarizer plates removed gave i = 0.300 0.008, in clear violation of the Freedman form of the BCHSH inequality expressed in Eq. (25), and in agreement with the quantum mechanical prediction t QM = 0.301 0.007 obtained from Eq. (26). 

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