Single equation for

Methods for finding roots of numerical nonlinear equations first involve making estimates and subsequently improving them by some systematic procedure, ideal problems for implementation on computers. Many scientific calculators also have built-in capability for finding roots of single equations, for instance the HP-15C or HP-32SII. 

As an alternative to the use of Table 5.1, it is convenient to be able to use a single equation for the calculation of n over the whole range of values of Ga or Re 0 of interest. Thus, Garside and Al-Dibouni(57) have proposed ... 

In this equation, we introduced the notation < F r) > = J d rF(r). Equation (15) is a single equation for two unknowns, Cd and Ck- Therefore they depend on a single arbitrary (positive) constant C such that... 

With this notation, a consecutive exclusion of s, (i = 1,2,3) from (4.3.22a-d) yields a single equation for j of the form ... 

If this result is used to substitute for a in (11.3) we obtain a single equation for the one independent concentration, b ... 

The fact that there are now two equations, viz. (8.12) and (8.9), implies that no longer is y(t) determined by /(0) alone, but by the initial vaue of a2 as well. One might hope that, after a short initial transient time, o2 adjusts itself by rapidly approaching an asymptotic value depending on the instantaneous y(t) alone, so that a renormalized equation for / holds after the initial transient. However, this is not the case the time scale on which a2 approaches (8.10) is determined by the coefficient a in (8.9) and is therefore comparable to the rate at which / itself varies, see (8.7). There is no separation of time scales and therefore no single equation for / by itself. 

XVI have shown that the effects of the individual substituents on the NH stretching frequency are additive and independent of the position and type of the other substituents and that the position of the absorption band may be represented by a single equation. For example, using the following equation and the constants given in Table XVI one would predict the NH stretching frequency of 3-ethoxycarbonyl-2-methyl-4,5-diphenylpyrrole to be 3455 cm-1 (the observed value is 3456 cm-1). 

Substituting this relationship into the solid mass balance, Eq. 2, one obtains a single equation for cA ... 

For large negative AG, the transfer becomes irreversible Wb < W. Setting Wb = 0 and W = W(r), we obtain under this condition a single equation for the survival probability of the initially excited state ... 

All the above data have been brought together by Whitaker [35] to develop a single equation for gases and liquids (lowing past spheres ... 

For creeping flow case (sRe l) we can neglect the convective terms and obtain the single equation for dimensionless liquid layer thickness h of elliptic type ... 

Starting from the set of equations (4.3) and (4.4), which are linearized about the saddle point, we follow Grote and Hynes and derive a single equation for the reactive mode by formally solving Eq. (4.4), then inserting the solution for Xj(t) into Eq. (4.3). The result is... 

This equation and (70) can then be combined into the following single equation for the curve ... 

Subtract the two equations in problem 6 to obtain a single equation for rj = Oi - O2. Obtain the boundary conditions for rj and arrive at an analytical solution for rj using the matrix exponential method. 

Exercise 6A.9. Show how to set up a single equation for the general reaction 2 = 0 assuming that the diffusion coefficients of the Aj all... 

Hence, eliminating /q by cross-differentiating and combining (4-84a) and (4-84b), we obtain a single equation for fi ... 

Because the stability analysis can now be focused exclusively on 2D disturbances, we can again use the stream-function transformation (12-312) to express the linear stability problem in terms of a single equation for the disturbance function [Pg.877]

The equation accounts for 89% of the variance in the potency, and the standard error is at the level of the reported experimental error. DiPaolo showed that a plot of potency versus reveals nonlinear dependence with a maximum point. However, a quadratic form is not able to unite the data for all three classes of compounds into a single equation. It is the combination of the reciprocal of x together with Xp that gives a single equation for the three chemical classes. Examination of the equation indicates that the maximum occurs at X = 4.5 and Xp = 0.863, values near to those for butyl propionate and propyl butyrate. 

By insertion of Af, obtained from equation (5.176) and obtained from using equation (5.169) and (5.171) into equation (5.175), a single equation for the total flux of the diffusive and viscous flow mechanisms is obtained ... 

In the simplest case, tba solid phase is pure 2 (no solubility of solvent in the solid), Eq- (1.6-11) does not apply, and in Eq. (1 -6-12) we may set/f equal to/i. the fogacity of pure solid 2. The (single) equation for gas-solid equilibrium is then... 

Directions a) To 5 c.c. of ferrous ammonium sulphate solution add potassium permanganate solution very slowly, a few drops at a time toward the end, until a slight excess is present, as indicated by the color of the solution. (1) What is the precipitate (2) Write hypothetical equations, similar to those written for potassium dichromate in Experiment 145, to show the composition of potassium permanganate as being potassium oxide and an acid anhydride, and to show the reduction of the anhydride as it occurs in alkaline or neutral solution. (3) Write equations for the remainder of the hypothetical steps in the oxidation of the ferrous salt and combine them into a single equation for the reaction. 

We obtain a single equation for S(r,l). Any variation of the temperature profile with time is determined from (6) with a very slowly varying The equation is of a convective-diffusive... 

The diagonal exchange term exactly cancels the spurious Coulomb selfinteraction term if both Coulomb and exchange summations are complete. This is the real reason why it is possible to generate a single equation for all the orbitals in the single-determinant wavefunction. ... 

Attempt to obtain a single equation for all the C . In this way we may be sure of the mutual orthogonality of all the orbitals which has been assumed throughout the derivation. 

Let us consider a single equation for the time being. If we denote with s the maximum acceptable error, the maximum integration step / new with the (p - l)-order method is... 

Most models focus on specific forms of corrosion and hence cannot be used in all environments with equal confidence. These models are usually fashioned and tweaked to either fit as accurately as possible to field data and results (which in themselves may not be entirely accurate for several reasons) or to fit results of laboratory tests, which may be similar in trend but frequently differ in value from observations made in the field. Neither model attempts to use a combination of numerical lab data and knowledge and heuristic field data and experience to make a prediction. Another approach to the corrosion prediction problem is to develop a model that does not result in a single equation for the whole domain of the parameter value but rather focuses on utilizing a set of equations to extract a prediction from data values that are as similar as possible to those of the problem. This reduces the domain of generalization of the equations since for each new problem, a new set of data values (which are similar to those of the problem) are used to extract a corrosion rate value. 

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