Digital Computed Curves

The method is quite effective, but is not widely used now because of the ubiquity of digital computers. Zuman and Patel - 36. show circuit designs for some kinetic schemes. Williams and Bruice made good use of the analog computer in their study of the reduction of pyruvate by 1,5-dihydroflavin. In this simulation eight rate constants were evaluated variations in these parameters of 5% yielded discemibly poorer curve fits. 

Before the advent of the digital computer high-order and nonlinear functions were impractical at best, and without a graphics-plotter much time is needed to draw a curve. Interpolation, particularly in the form X = /(y), is neither transparent nor straightforward if confidence limits are requested. 

Both the openloop and the closedloop frequency-response curves can be easily generated on a digital computer by using the complex variables and functions discussed in ( han. 12, Tlie freauencv-resnnnse curves for the closedloon... 

The experimental data from a pulse test are usually two continuous curves of x and m recorded as functions of time. A reasonable number of points are selected from these curves and fed into the digital computer. We will discuss later what a reasonable number is. 

Numerical analysis is important in digital-computer work from another viewpoint. Sometimes it is necessary to express complex functional relationships in a simpler form. Occasionally relationships may be given in a graphical or tabular form not directly suitable for processing on digital equipment. In these situations numerical methods for curve fitting and interpolation are techniques which will necessarily be employed. 

This system of equations was solved by a predictor-corrector method for several values of a, / , and y using a digital computer. It was not possible to examine values of ft above 50 (a = 0.001, y = 0) as the method broke down because of accumulated errors. Up to these values, although a step is formed in the extent of reaction vs. time curve, the rate of acceleration in the third phase of the reaction was much slower than observed in the experiments. 

An IBM 7090 digital computer was used to calculate all rate curves. Calculated curves for the finite bath studies were printed out in the form Ca/C0 vs. r. These curves were then compared with the experimental CJC0 vs. t curves, and D was calculated by a trial and error procedure. This approach was modified slightly in order to calculate the rate curves for the infinite bath studies. The value for A" taken for calculation purposes, in this case 1000 cc., was very much greater than the actual value, thus producing the effect of an essentially constant Ctt with time. In this case, the integral value... 

Monte Carlo Method The Monte Carlo method makes use of random numbers. A digital computer can be used to generate pseudorandom numbers in the range from 0 to 1. To describe the use of random numbers, let us consider the frequency distribution curve of a particular factor, e.g., sales volume. Each value of the sales volume has a certain probability of occurrence. The cumulative probability of that value (or less) being realized is a number in the range from 0 to 1. Thus, a random number in the same range can be used to select a random value of the sales volume. 

Bring, M. The Simulation and Analysis by Digital Computer of Biochemical Systems in Terms of Kinetic Models II. Curve-Fitting Procedures. J. Theoret. Biol. 17, 430 (1967). 

By proper treatment of the linear potential sweep data, the voltammetric i-E (or i-t) curves can be transformed into forms, closely resembling the steady-state voltammetric curves, which are frequently more convenient for further data processing. This transformation makes use of the convolution principle, (A.1.21), and has been facilitated by the availability of digital computers for the processing and acquisition of data. The solution of the diffusion equation for semi-infinite linear diffusion conditions and for species O initially present at a concentration Cq yields, for any electrochemical technique, the following expression (see equations 6.2.4 to 6.2.6) ... 

The MTE was solved using a finite difference technique on a digital computer. It was necessary to supply a constitutive relation for the gas-fluid pair that was being studied. For 02-blood, the dissociation curve of Severinghaus (37) was used in a tabulated form. Because of its high solubility, C02 was assumed to be physically dissolved in blood without considering its dissociation curve. 

The obvious advantage of the automated thermobalance system over existing instruments is the ability to determine the mass-loss curves of eight successive samples. Operation of the instrument is completely automatic, and once the cycle is begun the instrument does not require the attention of the operator until the eighth sample curve is completed. The instrument should find use for the routine TG examination of a large number of samples, each to be studied under identical thermal conditions. Because the system is completely automated, data reduction or control by a small digital computer could easily be accomplished (see Chapter 14.)... 

Both the openloop and the closedloop frequency response curves can be easily generated on a digital computer by using the complex variables and functions in FORTRAN discussed in Chapter 10 or by using MATLAB software. The frequency response curves for the closedloop servo transfer function can also be fairly easily found graphically by using a Nichols chart. This chart was developed many years ago, before computers were available, and was widely used because it greatly facilitated the conversion of openloop frequency response to closedloop frequency response. 

Although these equations are coupled and nonlinear, numerical solutions can be obtained on the digital computer. Figure 1 shows the results of such a numerical integration procedure applied to the set of Eqs. (44). We see that all phases of the batch growth curve, including a lag phase, are exhibited by... 

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