Polynominals

The polynomial approach is the logical expansion of the binomial approach. It is useful for the calculation of isotopic distributions of polyisotopic elements or for formulas composed of several non-monoisotopic elements. [2,14] In general, the isotopic distribution of a molecule can be described by a product of polynominals... 

C. Extensions of helices having sensitivities of ca. 2 mg mm-- - were read to 0.01 mm with a precision of 0.02 g sorbate per 100 g sorbent. However, because of experimental uncertainties described later, the precision of uptakes is seldom better than 0.1 g/100 g, and sometimes worse. The helices were calibrated at five-degree temperature intervals between 5 and 50°C, and at 100-mg mass intervals up to U00 mg calibration formulae comprising functions of both independent variables were found by a polynominal least-squares fit of all the data. Pressures were measured with large-bore mercury manometers. 

The enumeration of Kekule structures for rectangle-shaped benzenoids is treated. Combinatorial formulas for K (the Kekule structure count) are derived by several methods. The oblate rectangles, Rj(m, n), with fixed values of m are treated most extensively and used to exemplify different procedures based on the method of fragmentation (chopping, summation), a fully computerized method (fitting of polynominal coefficients), application of the John-Sachs theorem, and the transfer-matrix method. For Rj(m, n) with fixed values of n the relevant recurrence relations are accounted for, and general explicit combinatorial K formulas are reported. Finally a class of multiple coronoids, the perforated oblate rectangles, is considered in order to exemplify a perfectly explicit combinatorial K formula, an expression for arbitraty values of the parameters m and n. 

Local gas holdup is well described as a function of radial position with a third-order polynominal equation. 

The radial distribution of interfacial area for a two-phase system is shown in Figure 11. As discussed earlier, the gas holdup fraction is a strong function of radial position, but the Sauter mean bubble size is a less pronounced function of radial position. Consequently, the radial distribution of interfacial area has a shape similar to the gas holdup radial profile. Therefore, the inter facial area is well described with a third order polynominal equation. 

This partial averaging of the signals can be easily explained by reference to the dependence of the linewidths on the value of nij. For an S = 1/2 spin system, the peak-to-peak linewidths ABp.p of the first derivative signal can be expressed as a polynominal in mj. 

P2-13 Using POLYMATH, MatLab, Mathmatica, or some other software package, first fit the -ta versus X to a polynominal (i.e., = flo... 

In the theory of nonradiative transitions, the following methods were applied for the sums in Eqn. (50) the method of generating polynominals [154], operational calculus [155], and the matrix density method [13]. The latter seems to be the most justifiable in the case of deviations from Condon approximation, which are important in this problem. It should be emphasized that the problem is not confined to the account of some members of the fast subsystem matrix element expansion in the slow subsystem coordinates [156]. The F(K) function is too sharp for such expansion to be satisfactory, though the expansion of the exponent index (38) remains tolerable. 

Selected blends of styrene-acrylonitrile copolymer (30 to 55%), a styrene-butadiene copolymer grafted with styrene and acrylonitrile (45 to 70%), and a coal-tar pitch (0 to 25%), were prepared. Physical properties of the experimental blends were determined and statistical techniques were used to develop empirical equations relating these properties to blend composition. Scheff canonical polynominal models and response surfaces provided a thorough understanding of the mixture system. These models were used to determine the amount of coal-tar pitch that could be incorporated into ABS compounds that would still meet ASTM requirements for various pipe-material designations. ... 

The collocation points are chosen as the zeros of the global orthogonal polynomial (Jacobi polynominal) / ) " (m) for slab geometry, where u is defined by ... 

The fiEcoh. value of the -CH- and -CH3 groups from the -CH(CH2-CH3) side-chain of poly(1-butene), is according to Table 7.1 3430 + 9826 = 13256 J/mol. This value has to be multiplied with 0.896 to decrease the calculated Tg-value from -2l°C for polypropylene to -31°C for poly(1-butene), (n 1). The reduction factors calculated in this way are plotted as a function of the number of CH2 units in Figure 7.1. The third order polynominal curve fit has a correlation coefficient of 0.9844 and permits calculation of the following average reduction factors ... 

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