The function H Z

The function H Z). Equations (35) and (36) depend on the instantaneous state distribution Z only through the function 

The works of Boltzmann have revealed the widespread significance of this function.1 8 

The symbols dH(Z)/dt and AH(Z)/At. As it will be shown in Section 14b, the interchange of a certain time difference quotient of H(Z) with a differential quotient has caused a considerable amount of confusion in the discussion of the If-theorem. One can avoid such a confusion by making the following remarks  

The curve describing the time behavior of a certain F(Z) is always a step function.127 The time differential quotient therefore can assume only the following three values  

The time element At used in the kinetic theory of gases, and in particular in the ff-theorem, although very 

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At any point where the functions h(z) are zero, the Lagrange func tion equals the objective function. 

This relation requires the function h (z) to decrease rather rapidly with increasing z and finally to approach zero in the limit of infinite z, because the numerators in Eqs. (107) and (108) are rather insensitive to the variable (z/a ) for large z. Consequently, if we adopted Eq. (110) for a2. the quantity A2Mj[rf would become a decreasing function of z in the region of large z. In other words, A2M/[ >]] as a function of z would have to display a maximum at some value of z. On the other hand, according to Eq. (109), the variable (z/af) is bounded by the lower limit... 

If X is large and positive, and the function h(z) takes its maximum value at the lower limit of integration z = a in the integral... 

A trivial reversible CA consists of a collection of completely isolated systems each site contains only itself in its neighborhood. Since a7 = 1, the rule table and site value sets have the same cardinality. In particular, if the function 21,—> Z is invertible (i.e. if 4> serves merely to permute the elements of the set Zk) then the global CA system is itself reversible. More formally, writing < = tt H/ci where... 

This relation is only valid for a crystal with isotropic /-factor. The effect of crystal anisotropy will be treated in Sect. 4.6.2. The function h(6) describes the probability of finding an angle 6 between the direction of the z-axis and the y-ray propagation. In a powder sample, there is a random distribution of the principal axes system of the EFG, and with h 6) = 1, we expect the intensity ratio to be I2J li = I, that is, an asymmetric Mossbauer spectrum. In this case, it is not possible to determine the sign of the quadmpole coupling constant eQV. For a single crystal, where h ) = — 6o) 5 delta-function), the intensity ratio takes the form... 

The equality constraints composed of the mass and heat balances and the performance equations in each subsystem, thermodynamic properties of the flows, and specifications for design are represented by the functions h which are in the form of n equations with m+n variables. These equations are easily arranged in the order of precedence based on structural analysis. The number of independent variables (parameters), y, corresponds to the degrees of freedom in the system. When the value of the parameters is given, n equations are solved with respect to n variables, z. Thereupon, the inequality constraints, if any, are checked and the objective functions are calculated. Therefore, the problem is rewritten simply as follows ... 

To estimate the additional systematic uncertainty which originates from the unknown term of order a(Za)7 we studied a sensitivity of the fit (I) to introduction of some perturbation function h(z)... 

As with the generalized compressibility-factor correlation, the complexity ol the functions (H f/RZ. H Y/RZ. (S /R, and S Y/R preclude then general representation by simple equations. However, the correlation for Z basec on generalized virial coefficients and valid at low pressures can be extended U the residual properties. The equation relating Z to the functions and ia derived in Sec. 3.6 from Eqs. (3.46) and (3.47) ... 

To determine the 2n moments by a finite part of a spectrum, it is possible to approximate the inelastic cross section using (2.34). The function q(Z) changes on a constant ih h > 0). This constant and the moments may be determined with coincidence between the experimental curve and the constructed approximation. If the function (r) conserves sign up to distances that are comparable with the molecular diameter, the line of losses will appear only in the Stokes (anti-Stokes) neighborhood of the respective molecular term. In this case Ig e) is nonzero only on the negative (positive) half-axis, and the function q Z) = ih/ — Z (= ih/ /Z), h>0,is an appropriate choice. 

We consider a typical thin-film configuration like that sketched in Fig. 5-5. We have already noted that any curvature of the boundaries will play no role in the leading-order thin-film (or lubrication) approximations provided that the radius of curvature is large compared with the gap width, and there is therefore no loss of generality in assuming one of the two surfaces to be flat. For convenience, we denote the coordinate perpendicular to this surface as z. The other boundary is also a solid surface of fixed shape, with the thickness of the fluid layer between them defined by the function, h( s. t), which can be expressed as a ftmction of the two spatial coordinates that lie in the plane parallel to the lower, flat wall, as well as a function of time. Because the most appropriate coordinate system will depend on the particular problem, it is convenient to maintain the symbolic representation of the... 

The variable r extends through the range —1 to +1, which is the range traversed by the argument z — cos of the associated Legendre functions P ml of Section 19. With Hylleraas we expand the function H(i ) in terms of these functions, writing... 

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