Polynomial dependence

Figure 8.2 shows a good linear correlation between the -AH° values and the H- H distances found for dihydrogen-bonded complexes formed by [A1H4] , [BH4] , [GaH4] , and Cp Fe(dhpe)H with different proton donors [5,7]. In this case the bonding energy increases from 5 kcal/mol to 12 kcal/mol and the H- -H distance changes between 1.3 and 1.7 A. It is probable that the linearity of this relationship is connected with the relatively narrow diapasons mentioned above. In fact, larger diapasons for both parameters lead to the polynomial dependence [12] shown in Figure 8.3. This pattern includes the dihydrogen-bonded complexes...

A linear dependence of i/4 (yeak with in the case of discs and spheres (equations shown in the graphs) can be seen in this figure which coincides with those previously reported [6, 21]. For cylinders and bands, polynomial dependences are obtained showing a more complex behavior. Note that in the planar-limiting behavior (i.e., < G —> 0), the four geometries logically tend to the constant limit... 

Jacobi polynomials depends on the particular problem at hand. If one is concerned with physical insight into an isotope eflEect, the expansion should be carried out using the minimum number of terms. That is, a set of polynomials which make the expansion for the system converge as rapidly as possible should be chosen. On the other hand, if the expansion is being used to evaluate thermodynamic quantities numerically, the limitation on the number of terms carried in the expansion is not as critical. For efficiency of computation, one would still like to have reasonably rapid convergence, and as has been demonstrated in the previous sections, this can be achieved by use of a single Jacobi polynomial, such as the shifted Chebyshev of the first kind, for the entire range of i/-values. 

Several other details should be mentioned. Note, that the optimization is subject to variation of the coefficients of the linear combinations over the original set of projections of the 5 x hga functions onto the regular orbit cage. It is important that only linear combinations over the projections of the distinct a, b, c, d and e sets of polynomials listed in Table Al.l are applied. The transformation constraints that lead to these polynomials depend on this matching of the different order polynomials. This is a general observation and should be maintained with all the other sets of polynomials. 

Examples of dependences of cr on 77 for selected values of q are shown in Figure 3.8. Note that the coefficients of the polynomial depend only on the known moments mt, and not on the weights and abscissas. Thus, can be computed first, followed by the moments ml, using Eq. (3.95), from which the weights and abscissas are found using the Wheeler algorithm. In the limit where 5 = 0 (i.e. = 3 and q = 0), all of the moments mk are... 

In this Sect.4.9 we discuss Eqs. (4.156), (4.171) concerning chemical reactions in a regular linear fluids mixture (see end of Sect. 4.6), i.e. with linear transport phenomena. This model gives the (non-linear) dependence of chemical reaction rates on temperature and densities (i.e. on molar concentrations (4.288)) only (4.156), which is (at least approximately) assumed in classical chemical kinetics [132, 157]. Here, assuming additionally polynomial dependence of rates on concentrations, we deduce the basic law of chemical kinetics (homogeneous, i.e. in one fluid (gas, liquid) phase) called also the mass action law of chemical kinetics, by purely phenomenological means [56, 66, 79, 162, 163]. 

The specific heat of dry solid Cg is usually presented as a polynomial dependence of temperature. 

Since all finite domains can be expressed in the range [0,1] through a linear transformation, we will consider the Jacobi polynomials defined in this domain. This is critical because the orthogonality condition for the polynomials depends on the choice of the domain. The Jacobi polynomial is a solution to a class of second order differential equation defined by Eq. 3.145. 

The Legendre polynomials P ix) are tabulated for x = 0-1. In our case, the polynomials depend on angle d with x = cost and the integration should be made from K to 0. For even or odd m the polynomials are even or odd functions of cost), respectively ... 

In this case, one expects (cf equation 32) that (M2) has a polynomial dependence on cross-link density or shear modulus. 

Here, N is the number of qubit copies. To the scheme to work, this force must be comparable to the minimum force detectable by MRFM. For small polarization, the number of detectable qubits depends exponentially on p, just like in the liquid-state approach. However, for p 0.6 and above, there is a crossover from exponential to polynomial dependence ofnoap n l + p)/ l — p) (Figure 7.5). This is the main result of Ladd and co-workers proposal, for it means the system is scalable. Therefore, the usefulness of the scheme relies on the possibility to produce a large enough initial polarization, but there is no need of single spin detection and other difficulties present in the previous model. 

Calculated from the polynomial dependence of the radii as a function of the lanthanide number. 

Different functional dependences may be used for gradient-index AR layers. The simplest one is linear, but various polynomial dependences are met, for instance cubic or quintic. Sinusoidal dependence is also used. Of these, quintic dependence has been reported as the closest to optimum [164]. 

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