Laplace parameter

Laplace parameter, stoichiometric number, surface site solid (phase) slope instantaneous fractional yield of P with respect to A, equation 5.2-8 substrate split point (of streams) entropy, J K-1... 

The Laplace transform parameter in this chapter is denoted by s which is now common practice in the control field. Previous editions of this book and other volumes of this edition employ p as the Laplace parameter. 

Laplace parameter. Hence under conditions where

We can now replace the Laplace parameter p by j(o to obtain our final result... 

In the case of a dispersion model, for instance, transfer function G(p) is given as Eq. (6-59). Transfer function is defined as the ratio of the Laplace transform of the elution concentration curve and that of the input concentration curve, the latter of which is a constant in the case of impulse input. Laplace parameter, p, is a complex variable but if a response curve, C(r), is transformed by using Eq. (6-8) by assuming p as a real parameter, then the resultant C(p) gives a transfer function G(p) by dividing by the size of pulse, M, in a real plane. C(p) is then compared with the solution of basic equations obtained in a Laplace domain. 

Notice the use of A with lower indexes for the derivatives of the pgf values with respect to the logarithms of dummy Laplace parameters, as well as of X for their moments, a useful convention which will be encountered often in the rest of this chapter. 

Experimentally, the absorbance A(5) of a band is measured as a function of the angle of incidence B and thus of S. Two techniques can be used to determine a(z). A functional form can be assumed for a(z) and Eqs. 2 and 3 used to calculate the Laplace transform A(5) as a function of 8 [4]. Variable parameters in the assumed form of a(z) are adjusted to obtain the best fit of A(5) to the experimental data. Another approach is to directly compute the inverse Laplace transform of A(5) [3,5]. Programs to compute inverse Laplace transforms are available [6]. 

The temperature 0, corresponding to the transform 9, may now be found by reference to tables of the Laplace transform. It is first necessary, however, to evaluate the constants B] and S2 using the boundary conditions for the particular problem since these constants will in general involve the parameter p which was introduced in the transformation. 

It is interesting to note that independent, direct calculations of the PMC transients by Ramakrishna and Rangarajan (the time-dependent generation term considered in the transport equation and solved by Laplace transformation) have yielded an analogous inverse root dependence of the PMC transient lifetime on the electrode potential.37 This shows that our simple derivation from stationary equations is sufficiently reliable. It is interesting that these authors do not discuss a lifetime maximum for their formula, such as that observed near the onset of photocurrents (Fig. 22). Their complicated formula may still contain this information for certain parameter constellations, but it is applicable only for moderate flash intensities. 

Solution Define the Laplace transform of C t) with respect to the transform parameter as... 

Thus, the fraction unreacted is the Laplace transform with respect to the transform parameter k of the differential distribution function. 

It is worth noting here that the same estimate for no( ) was established before for ATM with optimal set of Chebyshev s parameters, but other formulas were used to specify r] in terms of and A - If R = —A, where A is the difference Laplace operator, and the Dirichlet problem is posed on a square grid in a unit square, then... 

With this tensor structure of the kernel, 2D Laplace inversion can be performed in two steps along each dimension separately [50]. Even though such procedure is applicable when the signal-to-noise ratio is good, the resulting spectrum, however, tends to be noisy [50]. Furthermore, it is not dear how the regularization parameters should be chosen. 

Secondly, although stable solutions covering the entire temporal range of interest are attainable, the spectra may not be well resolved that is, for a given dataset and noise, a limit exists on the smallest resolvable structure (or separation of structures) in the Laplace inversion spectrum [54]. Estimates can be made on this resolution parameter based on a singular-value decomposition analysis of K and the signal-to-noise ratio of the data [56], It is important to keep in mind the concept of the spectral resolution in order to interpret the LI results, such as DDIF, properly. 

We now derive the time-domain solutions of first and second order differential equations. It is not that we want to do the inverse transform, but comparing the time-domain solution with its Laplace transform helps our learning process. What we hope to establish is a better feel between pole positions and dynamic characteristics. We also want to see how different parameters affect the time-domain solution. The results are useful in control analysis and in measuring model parameters. At the end of the chapter, dead time, reduced order model, and the effect of zeros will be discussed. 

If Q (E) is differentiable in the ordinary sense the partition function of a generalized ensemble with m intensive parameters is the m-fold Laplace transform of the microcanonical partition function e, ... 

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