Modified Laplace transform

Application of the odd-number sampling modified Laplace transform (MET)... 

Thus, using the modified kernel, the set of transport equations given by (236) can be solved for any real system, and again the inverse Laplace transformation becomes an unnecessary procedure. This removes a considerable... 

When the deadtime in a process is an integer multiple of the sampling period, the function in the Laplace domain converts easily into z in the z domain, where dead time D = kT. When the dead time is not an integer multiple of the sampling period, we can use modified z transforms to handle the situation. 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

This equation can be solved by Laplace transform techniques and Mt expressed as modified spherical Bessel functions [28]. However, because the boundary conditions on M are radically symmetric, only the / = 0 (i.e. S-wave) component is of interest. 

For the very low density varieties of the cases shown in Figure 2 and, more particularly, Figure 4 (curve 7), for which initiation is slow compared to both termination (release from end of template) and polymerization, a simpler treatment, in which the interference of one ribosome with another is totally neglected, should suffice. In this case an equation of the form of Eq. (1), herein only applied to the problem of DNA synthesis, should be valid, but Eqs. (2) and (3) should be modified to account for repetitive initiation at site 1 and continuing release from site K, respectively Eqs. (4) and (6) will not apply. In the even more restricted (but perhaps biochemically relevant) case in which, in addition to neglecting ribosome interference, one may also neglect the back reaction (kb x 0), one may solve this system of equations (Eq. (1), plus Eqs. (2) and (3) modified as described) very easily by taking Laplace transforms.13 This is the only case with repetitive initiation for which we have been able to find solutions for the transient, as well as steady state, behavior. 

Although small, this is a principal disadvantage of the simplest integral theory. The near-contact density of the products nonlinear in c is lost in the lowest-order approximation to this parameter. However, the nonzero contribution to this region is provided by a modified encounter theory outlined in Section XII. The chief merit of MET is that the argument of the Laplace transformation of n r,t) in (3.311) is shifted from 1 /td to 1/xd + ck. As a result, in the limit xD = oo we have instead of (3.313) [133] ... 

The modified sensor response can be obtained by using Laplace transformation of equations (7.17)-(7.22) with kT substituted using equation (7.28) to yield (Appendix 2) ... 

Laplace Transforms. When Eqs. (2.16.2) and (2.16.3) are modified by using a real transform kernel exp(—kx) instead of exp(zfcx), then we have the Laplace transform ... 

Here we show how the modified Kubo formula (187) for p(co) leads to a relation between the (Laplace transformed) mean-square displacement and the z-dependent mobility (z denotes the Laplace variable). This out-of-equilibrium generalized Stokes-Einstein relation makes explicit use of the function (go) involved in the modified Kubo formula (187), a quantity which is not identical to the effective temperature 7,eff(co) however re T (co) can be deduced from this using the identity (189). Interestingly, this way of obtaining the effective temperature is completely general (i.e., it is not restricted to large times and small frequencies). It is therefore well adapted to the analysis of the experimental results [12]. 

In addition to the enhanced diffusivity effect, another issue needs to be taken into account when considering stationary-phase mass transfer in CEC with porous particles. The velocity difference between the pore and interstitial space may be small in CEC. Under such conditions the rate of mass transfer between the interstitial and pore space cannot be very important for the total separation efficiency, as the driving mechanism for peak broadening, i.e., the difference in mobile-phase velocity within and outside the particles, is absent. This effect on the plate height contribution II, s has been termed the equilibrium effect [35], How to account for this effect in the plate height equation is still open to debate. Using a modified mass balance equation and Laplace transformation, we first arrived at the following expression for Hc,s, which accounts for both the effective diffusivity and the equilibrium effect [18] ... 

We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model q — 24C - - 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two t5q>es of adsorption sites, Eq. 6.65a is modified to take into accoimt the kinetics of adsorption-desorption on these two site types. 

The last term in Eq. (A-3) is the Laplace transform of E(a) Aweq(a), where a plays the role of t in Eq. (A-4) and t plays the role of s in Eq. (A-4). Thus the problem is reduced to this experimental observations give us the Laplace transform of the desired distribution function [modified by multiplication by Aweq(A)], We want the distribution function. The required calculation is a numerical inverse Laplace transform. This is clearly feasible, and numerical techniques are discussed in the literature 56). It is not a simple matter to carry out, however, and the accuracy requirements on the data are likely to be stringent. No direct method of computing the distribution from frequency response is known, although the step response can be computed from the frequency response by standard techniques. In view of the foregoing discussion, it appears that in principle, at least, the distribution can be computed from the frequency response. 

We now construct the rate expressions in the nonstationary case. The Laplace transform of the modified rates Cd,r(f) yields [62]... 

To apply Laplace transforms to practical problems of integro-differential equations, we must develop a formal procedure for operating on such functions. We consider the ordinary first derivative, leading to the general case of nth order derivatives. As we shall see, this procedure will require initial conditions on the function and its successive derivatives. This means of course that only initial value problems can be treated by Laplace transforms, a serious shortcoming of the method. It is also implicit that the derivative functions considered are valid only for continuous functions. If the function sustains a perturbation, such as a step change, within the range of independent variable considered, then the transform of the derivatives must be modified to account for this, as we will show later. 

In terms of this modified Damkdhler number, the well-mixed dense phase behaves exactly as it should—namely, like a stirred tank. It may be shown that if f(t) is the residence time distribution of the bubble phase [in our plug flow model it is 5(t - 1)] the exponential e " in Equations 19 and 20 need only be replaced by the Laplace transform of f(t) and Tr playing the role of the transform variable. 

This method was later critically reviewed and modified by Barsoukov et al. [94]. They showed that the weakness of the direct Laplace transform lies in its large sensitivity to noise. Instead of direct numerical integration, Eq. (2.21), they proposed instead to fit first the time-domain data to a carrier function, which could then be directly transformed. The operator impedance of stable systems is... 

The volume integral specified in (3) is over a cylindrical region of unit height and infinite radial extent. Equation (6.34) may be reduced to the modified Bessel equation of order zero by the application of the Laplace transform on the age variable t. If as before 0 is the transform of Q, then the transform of (6.34) is... 

The actual signal which reaches the active circuitry of the preamplifier is modified from by the input circuitry and is given by (Laplace transform... 

The integration constant C is, strictly speaking, infinity for the two parameter model with no UV cut-off. However, since equation (32) involves only the ratio of inverse Laplace transforms, both numerator and denominator in equation (32) can be modified with a common multiplicative constant. That is, one can redefine... 

The Laplace invert transform h(t) of Eq. (18) gives the shape of the impulse response, in this case an exponraitially modified asymmetric chromatographic peak. The p moment of h(t) can be determined by ... 

The system equation along with boundary and initial conditions were solved analytically using a Laplace integral transform (1.) a computer program by Cleary (14) was modified for use in this study. 

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