Computer-generated equations

These two systems are equivalent. Let us look at the equivalence of variables. The computer-generated equations from the bond graph use different sets of state variables. Substituting the physical variable definitions of the bond graph variables, position (q) and momentum p), in the bond graph system should produce the other state variable system. Consider then the first row of (11.19) equation (11.14) should be equivalent to (11.7) ... 

Newer, published CHARMM parameter sets override some of the combination rule generated parameters for some atom type pairs. These parameters are found in the file pointed to by the 6-12PairVDW entry for the parameter set, usually called npr.txt(dbf). The values of Ay and By for these are computed using equations (22) and (23) on page 178 by setting the 6-12PairVDWFormat entry to RStarEpsilon. 

A graphic or tabular data display can be generated for any z-statistic value given a population correlation coefficient, p. This is accomplished by using the Fisher s Z transformation (i.e., the Z-statistic) computation as (equation 60-20)... 

R.I. Jennrich and P.B. Right, Fitting systems of linear differential equations using computer generated exact derivatives, Technometrics,... 

Similar calculations, based on the same principles, were carried out for spherical particles. Since the Poisson-Boltzmann equation cannot be integrated analytically in spherical symmetry, a numerical integration was performed. The computer-generated numerical tables of reduced potential as a function of reduced distance of Loeb, Wiersema, and... 

Figures 2, 3, 4, 5, 6, 7, and 8 show computer-generated plots of the absolute errors of the approximation of In b u) by Equation 52 as functions of u. Plots are shown for ranges 27t, Stt,. . . , 87t, respectively. Each figure consists of four separate frames, one for every order. Each frame contains three error curves one obtained by the best polynomial given in Table IV which has L = 5, another by T (x) with L = 5, and the third by Tn x) with L = 0. The error curves for In b(n) evaluated by the least-squares-analysis are indistinguishable from the curves for the best polynomial with L = 5. The curves in Figures 2, 3, 4, 5, 6, 7, and 8 clearly illustrate the oscillatory behavior of the error obtained with the expansions they show error amplitudes, regions of maxima and minima.

Fig. 4.17. An example of a fully optimized computer-generated multivariate Horner equation. The repro-model obtained consists of a set of such FORTRAN functions. This function calculates the temperature 10 s later, given the temperature and mole fractions...

Figure 1 Computer-generated curves of the flux vs. increasing concentrations of substrate transported by either a simple diffusion process fit to the equation J = 5(S) or by a facilitated diffusion process fit to the equation J = 100(S)/(0.4 + S).

We present quantitative, computer-generated plots of the solutions to the particle-in-a-box models in two and three dimensions and use these examples to introduce contour plots and three-dimensional isosurfaces as tools for visual representation of wave functions. We show our students how to obtain physical insight into quantum behavior from these plots without relying on equations. In the succeeding chapters we expect them to use this skill repeatedly to interpret quantitative plots for more complex cases. 

To illustrate the kinds of phenomena that can arise. Figure 7.6.1 shows a computer-generated solution of the van der Pol equation in the (x,x) phase plane, for e = 0.1 and an initial condition close to the origin, The trajectory is a slowly winding spiral it takes many cycles for the amplitude to grow substantially. Eventually... 

These expressions are substituted in the summation equation to generate Equation 2.50. The value obtained for a is used to update the mixed K-values, and all the computations are repeated until the convergence is achieved. The numerical calculations at convergence are shown below ... 

The dynamic behavior of the intact system is characterized by the solution of Eqn. (30). In some cases this can be obtained as an explicit solution in terms of elementary mathematical functions (e.g., Voit and Savageau, 1984). However, in most instances there is no solution of this type, and one must rely on computer-generated solutions in which particular numbers for rate constants, kinetic orders, and initial values of the concentration variables must be specified. Even if some of these numbers are unknown for a particular system, one can explore the potential repertoire of dynamic behavior by systematically varying the values of the parameters and solving the resulting equations for the system (e.g., see Irvine and Savageau, I985a,b). 

Clymans, P. J., and Froment, G. F., Computer generation of reaction paths and rate equations in the thermal cracking of normal and branched paraffins, Comp. Chem. Png. 8(2), 137-142 (1984). 

Curve fitting techniques and the computer generation of degradation curves for single and mixed parameters have been described (15). The equations representing NSSK when initiation is first order have been derived for chain termination that is first order in impurities (16). 

The constants in Equation 3 originally were determined by Alder et al. (3) to fit their computer-generated data. Chen and Kreglewski (J) redetermined the constants based on data on argon. The latter set of constants is used in this chapter. Since their values have been reported in Ref. I, they will not be repeated here. 

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