Software Mathematica

Finally, I wish to express my thanks and admiration to Wolfram Research for creating the Mathematica software. I have xsed Mathematica for simplifying many mathematical expressions and for most of the graphical illustrations. 

By calculating ms using Eq. 18, the color change that passes through the gray point can be obtained. QxQy, J, and ms values can be calculated, for instance, by the Mathematica software (Wolfram Research, Champaign, IL, USA) with a personal computer. 

Numerical calculations using MATHEMATICA software were made based on a theoretical model which assumes flow distribution in circular pipes under laminar conditions as described by the Bernoulli equation and applies an electrical circuit model based on Ohm s law [164],... 

By applying symbolic computation of Mathematica software, a mathematical mechanization of ADM based on the principle of parameterization, was carried out to reduce the solution for approximate expressions of model equations (A.l) and (A.2). The code of parameterized ADM is listed below. 

As demonstrated by Mikhailov and Cotta [9] the eigenvalues could be computed with specified working precision by using Mathematica software system [10], but the Mathematica rule given in [9] needs a small modification to by applicable for high-order eigenvalues. 

The solutions given here, are special cases from the general results for temperature distribution, average temperature and Nusselt numbers presented in the book [20]. Nevertheless all formulae have been derived again by using Mathematica software system [10]. 

The boundary condition at r=0 is commonly written as u [0] = 0. For this condition Mathematica software system is not able to find the velocity distribution. The correct condition at r=0 is the limit of Sr u [r] multiplied by the surface 2 tt r 1 when r - >0 to be zero ... 

The roots of (63) gives the desired eigenvalues. The FindRoot function of Mathematica software system calculates these roots starting from the values given by the asymptotic formula on p.ll3 of the book [20]. Fig. 5 shows the seconds per eigenvalue spend on 3 Gz computer to find 100 roots of a slightly modified eq.(63). The first 50 roots are computed much faster than the last 50 roots. 

Hgure 4 Dimensioniess current derived for a simpie reversible charge transfer process using the polylogarithm function in MATHEMATICA. Part A Calculation of the dimensionless current A at dimensionless potential = 15. Part B Calculation of the table of / values at values -10,0, and 10. Part C Calculated A - graph over the potential range of -10 to -i-15 dimensionless units. Note The command in line 2, part A, has to precede commands in subsequent parts. (Reprinted with permission from Mocak J and Bond AM (2004) Use of MATHEMATICA software for theoretical analysis of linear sweep voltammograms. Journal of Electroanalytical Chemistry XV. 191-202 Elsevier.)... 

For example, use Mathematica software to do this. Let us check the conservation of normalization (i.e., its... 

Mocak J, Bond A (2004) Use of MATHEMATICA software for theoretical analysis of linear sweep voltammograms. J Hectroanal Chem 561 191-202... 

Equations (4.8)-(4.10) have been solved in simple steady state shear flow using Mathematica software (Leonov and Chen 2010). The stress components are expressed as function of shear rate y with the values of constitutive parameters 00, a,p, r, r2,Xe, and t o. Here Oq and t]o represent relaxation time and zero shear viscosity respectively. The other parameters XgandXv represent the tumbling for elasticity and viscosity. Rest of the characteristic parameters a,p,ri,r2 represent anisotropic properties of liquid crystal polymers. Among the eight parameters, only relaxation time and zero shear viscosity are determined from experimental data. The other six parameters can be obtained from cinve fitting data using the Mathematica software. 

Modern mathematical software, such as Mathematica, allows us to compute symbolically the mean square deviation of this approximation from the exact acceleration, integrated over the feasible region, differentiate the resulting expression symbolically with respect to the parameters a and b, set the results to zero and solve the equations symbolically, and simplify the whole lot to find the following remarkably simple expressions... 

Mathematica is a technical software program from Wolfram Research (Champaign, Illinois), With this versatile tool it is possible to draw beautiful Klein bottles as discussed in Chapter 5. The following is a standard recipe for creating Klein bottle shapes using Mathematica. 

A fundamentally different type of simulation is offered by science-based learning environments. Such environments incorporate some general-purpose mathematical engine that either represents nature directly or that can be programmed to represent nature. Examples are Mathematica (7) and some similar programs (8-9) for general analytical modeling in the physical sciences and Interactive Physics for introductory classical mechanics (10). Mathematica and Interactive Physics can be applied to countless topics, as opposed to the narrow focus of Flash-based simulations. Even more importantly, Mathematica and Interactive Physics are open-ended in that the software may accommodate unscripted inquiries and follow-up questions. 

Note The tedium of completing such exercises, as well as following many derivations in this book, is reduced by the use of symbolic mathematical software. We recommend that students gain a working familiarity with at least one package such as Mathematica , MATLAB , Mathcad , or the public-domain package MAXIMA. 

In practice, the solution of polynomial equations is problematic if no simple roots are found by trial and error. In such circumstances the graphical method may be used or, in the cases of a quadratic or cubic equation, there exist algebraic formulae for determining the roots. Alternatively, computer algebra software (such as Maple or Mathematica, for example) can be used to solve such equations... 

Generally the rheometer manufacturer s software is adequate, although files may be exported to spreadsheet programs or calculation packages (e.g., Mathematica, Wolfram Research). 

Mathematica, version 5.1 software for technical computation Wolfram Research Champaign, IL 2004. 

The SBML homepage (http //sbml.org) provides an impressive list of software implementing the notation. The site offers downloads of software libraries that enable easy incorporation to several programming languages (C, C++, Java, Python) as well as mathematics packages (Matlab, Mathematica), and this accessibility has significantly contributed to its success. 

A Mathematica calculation of Franck-Condon factors that determine electronic transition intensities of I2 is presented in Chapter III, and program statements for this are illustrated for I2 in Fig. III-6. In this fignre, note the dramatic differences between the intensity patterns predicted for the harmonic oscillator and Morse cases and compare these patterns with those seen in your absorption spectra. If yon have access to this software, yon might examine the changes in the harmonic-oscillator and Morse-oscillator wavefnnctions for different v, v" choices. A calcnlation of the relative emission intensities from the v = 25, 40, or 43 level conld also be done for comparison with emission spectra obtained with a mercury lamp or with a krypton- or argon-ion laser, hi contrast to the smooth variation in the intensity factors seen in the absorption spectra, wide variations are observed in relative emission to v" odd and even valnes, and this can be contrasted with the calcnlated intensities. Note that, if accnrate relative comparisons are to be made with experimental intensities, the theoretical intensity factor from the Mathematica program for each transition of wavennmber valne v shonld be mnltiphed by v for absorption and for emission. ... 

Throughout this book, we have seen that when more than one species is involved in a process or when energy balances are required, several balance equations must be derived and solved simultaneously. For steady-state systems the equations are algebraic, but when the systems are transient, simultaneous differential equations must be solved. For the simplest systems, analytical solutions may be obtained by hand, but more commonly numerical solutions are required. Software packages that solve general systems of ordinary differential equations— such as Mathematica , Maple , Matlab , TK-Solver , Polymath , and EZ-Solve —are readily obtained for most computers. Other software packages have been designed specifically to simulate transient chemical processes. Some of these dynamic process simulators run in conjunction with the steady-state flowsheet simulators mentioned in Chapter 10 (e.g.. SPEEDUP, which runs with Aspen Plus, and a dynamic component of HYSYS ) and so have access to physical property databases and thermodynamic correlations. 

Kirkwood and Buff [15] obtained expressions for those quantities in compact matrix forms. For binary mixtures, Kirkwood and Buff provided the results listed in Appendix A. Starting from the matrix form and employing the algebraic software Mathematica [16], analytical expressions for the partial molar volumes, the isothermal compressibility and the derivatives of the chemical potentials for ternary mixtures were obtained by us. They are listed in Appendix B together with the expressions at infinite dilution for the partial molar volumes and isothermal compressibility. 

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