Numerical digitized curves

Notice that those distribution functions that satisfy Eq. (4-179) still constitute a convex set, so that optimization of the E,R curve is still straightforward by numerical methods. It is to be observed that the choice of an F(x) satisfying a constraint such as Eq. (4-179) defines an ensemble of codes the individual codes in the ensemble will not necessarily satisfy the constraint. This is unimportant practically since each digit of each code word is chosen independently over the ensemble thus it is most unlikely that the average power of a code will differ drastically from the average power of the ensemble. It is possible to combine the central limit theorem and the techniques used in the last two paragraphs of Section 4.7 to show that a code exists for which each code word satisfies... 

When the three coefficients oc12, oc13, and a23 are known, the coexistence curve can be found by simultaneous solution of Eqs. (119) and (120). A numerical iterative technique given by Hennico and Vermeulen (HI) was used by Balder for performing these calculations with a digital electronic computer. 

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. 

The culprit is digital overflow, a problem discussed in section 8.12, where we also describe a way to bypass it. Our remedy is based on the observation that the quantity described in (6.6-1) is always well within the capabilities of the spreadsheet (after all, the total area under the curve is 1, and the peaks are not all that narrow), while individual terms in its numerator and... 

There are numerous automatic titrators that employ potentiometric end-point detection. They usually can automatically record the first or second derivative of the titration curve and read out the end-point volume. The sample is placed in the titration vessel, and the titrant, drawn from a reservoir, is placed in a syringe-driven buret. The volume is digitally read from the displacement of the syringe plunger by the electronic driver. Titrators may also employ photometric detection of indicator color changes. An automatic titrator is shown in Figure 14.7. Automatic titrators make volumetric analyses rapid, reproducible, and convenient. While instrumental methods provide many advantages, classical volumetric analyses are still widely used and are very useful, especially for major constituents, for example, in the pharmaceutical industry. 

In addition, one needs the appropriate i-E characteristic (i.e., for a reversible, totally irreversible, or quasireversible reaction). The resulting nonlinear integral equation must be evaluated numerically. Alternatively, the problem can be addressed by digital simulation techniques. Figures 8.3.1 and 8.3.2 illuminate the effects of different relative contributions of double-layer charging on if (at constant /) and on the E-t curves of a nemstian reaction. The charging contribution is represented there by the dimensionless parameter, K, defined as... 

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... 

More exact theory for sphere-cap tips was developed by using digital simulations [29,30]. Selzer and Mandler solved numerically a time-dependent 2D axisymmetric diffusion problem for a sphere-cap electrode [29a]. They used the same approach to develop comprehensive time-dependent and steady-state SECM theory for a hemispherical tip approaching either a conductive or an insulating substrate [29b]. Daniele and coworkers [30] focused on diffusion-controlled steady-state responses and calculated approach curves for a broad range of H and various RG values. For RG= 10, the... 

In this expository article, the basic mathematical model of some simple electrochemical processes was discussed. The model is based on the concept of conservation of charge within the electrolyte. The boundary conditions, on the other hand, are problem-specific. The subject of electrode kinetics is central to the proper specification of the boundary conditions. In their most general form, the conditions are nonlinear, leading to a nonlinear boundary value problem. This is closely tied to the nonlinear polarization curves. The analytical solution of the mathematical model is formidable and for moderately simple two-dimensional regions is impossible to obtain. The only feasible approach is numerical simulation. The use of high-speed digital computers is an essential tool in solving such problems. 

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