Main Integration Methods

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

This integration method can be optimized for the ASC in two steps. The first is to construct the code so that vectorization over each set of equations occurs. Here the main problem is the decision process associated with the application of the "stiff" or "normal" formulas to each equation. If these formulas are implemented in the usual fashion with an IF test in the appropriate DO Loops the smooth flow of contiguous data from core through the CPU will be inhibited and scalar code will result. Optimization of this process can be accomplished by calculating both formulas and applying a multiplicative factor 0 or 1. The following example of Fortran code illustrates this technique. 

We emphasize that the results obtained by us, as well as those obtained by the other authors quoted in this chapter, are obtained by neglecting the fine structure corrections. This is not a serious disadvantage for us, since our main intention has been to compare the accuracy obtainable by the phase-integral method with the accuracy obtainable by other methods of computation. For the experimental data corresponding to the theoretical values presented in this chapter we refer to the publications mentioned in this chapter. 

However, the authors do not claim that these three main strategic lines in company of CETO functions constitute the unique way nor the best path to solve the molecular integral problem directed to find plausible substitutes of GTO functions. Other integration methods to deal with the present discussion can be used and analyzed, for instance Fourier, Laplace or Gauss transform methodology or any other possible choices and techniques available in the modem mathematical panoply. 

The main difficulty in MO calculations consists in evaluation of the multi-center integrals. The DV-integration method is one of the powerful methods to avoid such difficulty and has been successfully used in the MO method. When this technique is applied to the calculations of the transition matrix elements, it makes the evaluations of x-ray emission rates in complex molecules easier. 

The usual inverse gas chromatography, in which the stationary phase is the main object of investigation, is a classical elution method that neglects the mass transfer phenomena it does not take into account the sorption effect and it is also influenced by the carrier gas flow. In contrast to the integration method, the new methodology... 

Zsako [29] has suggested sub-classification of integral methods on the basis of the means of evaluation of the temperature integral in equation (5.4). The three main approaches are the use of (i) numerical values of/>(x) (ii) series approximations for p(x) and (iii) approximations to obtain an expression which can be integrated. 

At this point it should be kept in mind that S is an unknown function of x. The velocity function, Vx(x, y), is expressed in terms of the unknown function S(x). The main advantage of the integral method is that it is easier to obtain a solution for S x) than to solve the Navier Stokes equations for Vx x,y). A drawback is that we are using an approximate velocity field which to some extent reduces the accuracy of the result. 

The second method, the differential method, employs the rate equation in its differential, unintegrated, form. Values of dcjdt are obtained from a plot of c against by taking slopes, and these are directly compared with the rate equation. The main difficulty with this method is that slopes cannot always be obtained very accurately, but in spite of this drawback the method is on the whole the more reliable one and unlike the integration method it does not lead to any particular difficulties when there are complexities in the kinetic behavior. 

Thus, the stated above results allow one to make two main conclusions. Firstly, fractional differentiation and integration methods (the Eq. (68)) allow to calculate polymer chain molecular characteristics as precisely as other existing at present computational techniques. Secondly, accounting for the change of macromolecular coil structure, that is, its dynamics, at the external conditions variation, is needed for the correct calculation of the indicated characteristics. 

The so-called integrated method is a combination of fixed translation method and conversion angle-conversion method this method mainly used when there is major water damage in front of blind tunnel. 

The main advantage of the boundary integral methods, compared with other methods (e.g., FD, FE), is that for a number of multiphase flow problems, its implementation involves integration on the interfaces only. Thus, discretization is required only for the interfaces, which allows for higher accuracy and performance, especially in three-dimensional simulations. An important feature of the mathematical model used in the Bl method is that the velocity at a given time instant depends only on the position of the interfaces at that time instant. 

The main advantage of a boundary integral method is that it involves integration only on the interfaces. The main problem, however, regarding the numerical implementation of Eq. 20 is due to the singularities of the kernels G and T at X = Xq, which is described in detail elsewhere [3]. 

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