Symbolic calculations

Figure 3.50 Extending kinetic explosion (squares) and thermal explosion limits by using a micro reactor with 300 pm channel diameter (filled symbols). Calculated values for (circles) and 7 3 = (triangles). Comparison with 1 m...

Figure 40.6. Selectivity to A -containing products (full symbols, calculated with respect to n-hexane converted) and to N2 (open symbols, calculated with respect to ammonia converted) as a function of temperature for catalysts with composition Sn/V/Nb/Sb x/0.2/1/3. Symbols as in Figure 40.5.

Figure 1. Apparent subcutaneous glycaemia (closed symbols) calculated using one point calibration compared to the actual plasma glycaemia (open symbols), measured by conventional method. Upper pannel shows the sensor response after 4 hours of implantation and the lower one after 28 hours of implantation. Conditions intramuscular administration of glucagon (2.6 ig) at time zero followed by insulin (100 mU) at time 30. ...

Using the symbolic calculation engine available in Matlab ,2 we obtained the following description of the intermediate dynamics of the reactor-condenser process ... 

Figure 1.33. Advancing and receding contact angles vs surface fraction of Mo, for a triple line moving along the two directions of the composite surface shown on Figure 1.32. Open symbols calculated data, full symbols experimental data from work reported in (Naidich et al. 1995).

As an example of a symbolic calculation. Fig. 5 displays a Mathcad solution of one 4X4 block of the secular determinant of a Hiickel molecular orbital calculation done in Exp. 41 for orf/ro-benzosemiquinone (compare with Table 41-2). Mathcad is a software package for numerical analysis but also makes use of a subset of the symbohc routines of Maple. The algebraic expansion of the determinant is generated and solved with two... 

Open symbols - Measured Closed symbols - calculated... 

Figure 57 Relationship between log and log C, for cP and PP. (O) cPj (A) cP ( ) CP5 (O) cPg (x) PP. Open symbols calculated by use of the reported experimental data [74] closed symbols this work.

In order to understand formal relationships the basic coupling of two angular momenta is expressed in terms of the Clebsch—Gordan coefficients. In setting up the algebra for computation it is more convenient to use the 3-j symbols. Calculations of the Wigner symbols and the Clebsch—Gordan coefficients are found in subroutine libraries. See, for example, Soper (1989). 

Figure 3 Mantle evolution defined by the initial snf of juvenile granites and their zircon separates. The snf compositions (closed symbols) have been recalculated using the most recently proposed Lu decay constant of Bizzarro et al. (2003). For comparison are shown the initial values (open symbols) calculated using the Blichert-Toft and Albarede (1997) decay constant. Note that the change of decay constant results in an increase in calculated initial SHf compositions of some Early Archean samples. The current database suggests rapid Lu/Hf fractionation in the early Earth and the possibility of episodic changes in mantle evolution similar to that seen in SNd evolution (sources Vervoort and Blichert-Toft, 1999 Amelin et al., 1999, 2000 ...

Figure 17.17 Evaluation of the separation area of the Troger s base enantiomers. Solid line Anal5Ttical Solution with the Henry coefficients derived from the analytical chromatograms. Dashed line Analytical Solution with the Best Fit Langmuir Isotherm Data. Symbols +, calculated Umits of the separation area based on the correct isotherm and the true porosity of each column o calculated limits of the separation area based on the average column porosity experimental conditions of the experiments performed. K. Mihlbachler, A. Seidel-Morgenstem, G. Guiochon, AlChE J., 50 (2004) 611 (Fig. 6). Reproduced by permission of the American Institute of Chemical Engineers. 1997 AlChE. All rights reserved.

This shows that = assigned the value 2 to x whereas = did not assign 2 to y. One can use Maple to do numerical and symbolic calculations. A few examples... 

Exact results of combinatorial optimization of PWCs structure are presented in the Table. The data were obtained by different methods. For small and middle-size clusters (N < 28), we used a complete enumeration method. For large clusters (N = 36, 60), the data were obtained using symbolic calculations and max-plus algebra. Characteristics of two lowest energy levels for the SWEB model are presented. For water fullerene N = 60) on the base of SWEB model, the data obtained only for ground state level without isomorphism analysis. Values in italics correspond to ground sates of the SWB model. For example, the... 

List Processing (LISP) is a programming language that uses an interpreter for symbolic calculations based on single-scalar values (atoms) and associative arrays (lists). 

When induction operators of high-order multipoles are taken into account intensity calculations tend to become very cumbersome [30,31]. We propose a relatively easy way of performing these calculations using the irreducible spherical tensor theory of multipole light scattering [e.g., Eqs. (6) and (7)] together with symbolic calculations of the Wigner coefficients by computer. 

It would be relatively easy to extend here our computer symbolic calculations to the hyperpolarizability part of the pair polarizability [see Eqs. (5) and (7)]. However, from all our numerical computations done for N2, C02, and CF4, it results that nonlinear part of the pair polarizability has a weak influence on the resulting spectrum (for details, see Refs. 8, 13, and 15-18). Bearing in mind these results in this review, we restrict our discussion to multipolar light scattering mechanisms. Formula (22) allows us to write the following simple symbolic program in Mathematica calculating the analytical form of the autocorrelation function (16) for a selected dipole-arbitrary order multipole induction operator ... 

FIG. 7 IT vs. L curves for finite disk-shaped substrates. Both tip and substrate reactions are diffusion-controlled. Filled symbols calculated using multidimensional integral equations open symbols are from ADI simulation. The h (as/a) values are as indicated. Dashed line is simulation for H = oo from Ref. 1. The lines through the symbols are drawn as a guide. (Reprinted with permission from Ref. 3. Copyright 1992 American Chemical Society.)... 

Figure 13.18. Comparison between model and measurements for clays dispersed in polymers. Viscosities observed as a function of shear rate by Krishnamoorti et al [46] for dispersions of silicate platelets (weight fractions of 0.06 and 0.13) in poly(dimethyl siloxane) at T=301K are indicated with symbols. Calculated results, assuming platelets to be monodisperse flexible cylinders with aspect ratio Af=(thickness/diameter)=0.01, are indicated as lines, (a) Relative viscosity=r)(dispersion)/r (polymer). (b) Dispersion viscosity, r)(dispersion).

In a more complicated situation, the following method is good form the derivatives with respect to kg, m, and s. Equate the resulting equations at kg = m = s = 1. This method is suitable for a symbolic calculator and subsequently a session with the program DERIVE(r) [29] is given that does the job for the above example. 

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