POLYMATH

For our purposes, the best of the various memoirs of Goldschmidt are a lecture by the British crystallographer and polymath John Desmond Bernal (Bernal 1949),... 

In this exereise, the fraetional eonversion for the 4-min isothermal bateh operation is 77.1%. Obviously, the required answer is obtained through trial-and-error. Equation 5-45 ean be ineorporated into a eomputer program or mathematieal paekage sueh as Mathead, Maple, Polymath, or Mathematiea. 

The three men whose work later in the nineteenth century was crucial in bringing clarity to this principle were two Germans, the physician Julius Robert Mayer and the great polymath Hermann von Helmholtz, and the British amateur scientist James Joule. In a lecture delivered by Helmholtz on February 7, 1854, in Konigsberg on The Interaction of Natural Forces, ... 

C.W. Kilmister, (ed) SCHRODINGER Centenary celebration of a polymath, 1987, University Press, Cambridge. 

Of the following list, the first six items are the principal ones used here. All but TableCurvc are currently available at reduced prices to students, but most of the required curvefitting also can be done with POLYMATH or AXUM. The last three items do analytical as well as numerical work. 

POLYMATH. AIChE Cache Corp, P O Box 7939, Austin TX 78713-7939. Polynomial and cubic spline curvefitting, multiple linear regression, simultaneous ODEs, simultaneous linear and nonlinear algebraic equations, matrix manipulations, integration and differentiation of tabular data by way of curve fit of the data. 

SEQS simultaneous equation solver. CET, PO Box 2029, Norman OK 73070. Twenty simultaneous equations can be solved. The program seems to be superior to the one in POLYMATH. 

Graphical representation and corresponding equations of tabular data are of value for interpolation, for revealing behavior patterns, and other purposes. The most complete software for this purpose is TableCurve. More limited but adequate for the present book are POLYMATH and MathCad. All spreadsheet software, for instance QuattroPro, can make plots of all kinds. In... 

These topics are explained in specialist books and briefly by CHAPRA CANALE. Polynomial and multilinear regression programs are in POLYMATH and AXUM, nonlinear in CONSTANTINIDES and AXUM. No periodic data are regressed in the present collection. 

Examples Polynomial regression is applied in problem Pi.03.02. Several examples of POLYMATH multilinear regression are in sections P3.06, P3.08 and P3.10. A non-linear regression is worked out in PI.02.07. 

These are solved by matrix manipulations. Programs are in POLYMATH, CONSTANTINIDES AND CHAPRA CANALE. When the number of equations is not large, a manual procedure can be used to eliminate one variable at a time by reduction of the leading coefficients to unity and appropriate additions and subtractions of the equations. 

Softwares for numerical integration of equations include the calculator HP-32SII, POLYMATH, CONSTANTINIDES AND CHAPRA CANALE. The last of these also can handle tabular data with variable spacing. POLYMATH fits a polynomial to the tabular data and then integrates. A comparison is made in problem PI.03.03 of the integration of an equation by the trapezoidal and Runge-Kutta rules. One hundred intervals with the trapezoidal rule takes little time and the result is usually accurate enough, so it is often convenient to standardize on this number. 

Both POLYMATH and CONSTANTINIDES use this method and also a fourth order Runge-Kutta method. Other methods are available in other software, but these two are adequate for the present book. 

POLYMATH multilinear regression could be used to find the constants w, p and q but the problem will be solved directly. 

Catalytic oxidation of carbon monoxide is studied in problem P3.07.06. The change in pressure is given as a function of the time in the first two columns of the table. Polynomials of several degrees are fitted to the data and the derivatives, -dP/dt, are found at selected points by POLYMATH. Values derived from degrees 3, 4 and 5 agree fairly well. 

POLYMATH ODE is used for the integration in preference to Constantinides ODE because it allows printout of algebraic as well as differential variables. In the tabulation, x = C, f = 0 and e = tj. 

Method (b). Numerical integration of the four equations, (3) to (6) is accomplished with ODE, either Constantinides or POLYMATH. Alternately, the material balances (11 and (2) and only the two differential equations (3) and (4) can be solved together. This procedure is better carried out with POLYMATH ODE. 

The concentration data are curve fitted by the equation shown with the graph. The derivative is obtained by differentiation and is tabulated in column 3. Linear regression by POLYMATH gives the relation In r = -3.8712 + 1.5612 In [HZJ or... 

The exponents a and b could be found independently by POLYMATH multilinear regression if data were available with unequal starting... 

The numerical values are found from the data with POLYMATH multilinear regression. Hence,... 

POLYMATH multilinear regression handles the rate equation in linearized form, In r = a - b/T + q ln(C/C0)... 

Use POLYMATH multilinear regression with the equation in the form,... 

The numbers are obtained by POLYMATH multilinear regression. The plot indicates only poor fit of the data. The negative constant in the denominator also casts doubt on the correlation. 

The POLYMATH graph show the fit to be good, but the constants do not seem to have physical significance. 

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