Point-simultaneous method

With all of the point-successive methods, only one array in the computer is needed to store both o k) and 6 k+1) because the element o k) is never again needed once o k+1) is computed. This contrasts with the point-simultaneous methods, where, for each n, the elements of 6 k) must be available corresponding to the full range over which [s] m is finite. 

We have also learned that the point-successive methods need not demand more computer memory than point-simultaneous methods, and that the seemingly inherent asymmetry of the point-successive methods can be overcome. Furthermore, the linear methods described in this section are... 

Recent work on a production basis involving 5-fim spectra of 12CD3F has verified line-center frequency stability under deconvolution. Large numbers of experimental records of 12CD3F, simultaneously recorded with CO in the sample, were calibrated before and after deconvolution (point-simultaneous methods, Jansson algorithm). No systematic differences were detected in comparisons of the before and after frequencies of nonblended absorption lines. That is, the variance was consistent with the optomechanical precision of the spectrometer and the mean deviation summed to approximately zero, validating the frequency calibration of the deconvolved data. 

Production of net-shape siUca (qv) components serves as an example of sol—gel processing methods. A siUca gel may be formed by network growth from an array of discrete coUoidal particles (method 1) or by formation of an intercoimected three-dimensional network by the simultaneous hydrolysis and polycondensation of a chemical precursor (methods 2 and 3). When the pore Hquid is removed as a gas phase from the intercoimected soHd gel network under supercritical conditions (critical-point drying, method 2), the soHd network does not coUapse and a low density aerogel is produced. Aerogels can have pore volumes as large as 98% and densities as low as 80 kg/m (12,19). 

An alternative means of calculating reaction pathways is employed in so-called global methods. These methods treat the entire path as a succession of points [71] which are found simultaneously. Methods of this type (for example the conjugate peak refinement algorithm [72], available in the TRAVEL module of CHARMM, which has the advantage of requiring only first derivatives of the energy) have been used to determine reaction paths in a number of proteins [4, 73]. 

The proper way is to use nonlinear regression with all four points simultaneously, by which method the values found are - 0.100 and k2 = 0.050. Alternatively, take arbitrary pairs of points and solve the simultaneous equations. The tabulation shows... 

The Molecular Surface (MS) first introduced by Richards (19) was chosen as the 3D space where the MLP will be calculated. MS specifically refers to a molecular envelope accessible by a solvent molecule. Unlike the solvent accessible surface (20), which is defined by the center of a spherical probe as it is rolled over a molecule, the MS (19), or Connolly surface (21) is traced by the inwardfacing surface of the spherical probe (Fig. 2). The MS consists of three types of faces, namely contact, saddle, and concave reentrant, where the spherical probe touches molecule atoms at one, two, or three points, simultaneously. Calculation of molecular properties on the MS and integration of a function over the MS require a numerical representation of the MS as a manifold S(Mk, nk, dsk), where Mk, nk, dsk are, respectively, the coordinates, the normal vector, and the area of a small element of the MS. Among the published computational methods for a triangulated MS (22,23), the method proposed by Connolly (21,24) was used because it provides a numerical presentation of the MS as a collection of dot coordinates and outward normal vectors. In order to build the 3D-logP descriptor independent from the calculation parameters of the MS, the precision of the MS area computation was first estimated as a function of the point density and the probe radius parameters. When varying... 

The three methods described in the preceding paragraphs each offer distinct advantages and disadvantages. The first and most obvious difference between the methods is the distinction between the sequential methods (sequential simplex and prisma method) and the simultaneous method (mixture design). With the sequential method some experiments are performed, these are evaluated, and on the basis of this evaluation new design points are selected, these are evaluated etc. With the simultaneous... 

Repeated application of this equation constitutes the point-simultaneous relaxation method, also known as the point Jacobi method or the method of simultaneous displacements. 

This is called point-simultaneous overrelaxation. If we set k = [s]nn, we have obtained the discrete formulation of Van Cittert s method. This connection between Van Cittert s method and the classic iterative methods of solving simultaneous equations was demonstrated in an earlier work (Jansson, 1968, 1970). 

The nonlinear iterative methods described here are based on the linear relaxation methods developed in Sections III. C.l and III.C.2 of Chapter 3. Initially, the correction term was set equal to zero in regions where o(k) was nonphysical. To illustrate this, we may rewrite the point-simultaneous equation [Chapter 3, Eq. (23)] with a relaxation parameter that depends on the estimate d(k) ... 

There is an important disadvantage stemming from variable separation and use of the FFT, however. One is limited to the use of the point simultaneous procedure in the determination of the coefficients. That is, all of the coefficients must be computed in an iteration before they can be resubstituted back in the iterative equations. With the point successive method, the improved coefficient determined by one equation is substituted in the succeeding equation to render additional improvement. With the test data treated in this work, the point simultaneous procedure converged much more slowly than the point successive method. 

FTIR is a simultaneous method in that signal processing is global. Each spectral point is calculated from all the given data... 

A subset of simultaneous methods which overcomes the difficulty of mapping complex response surfaces by an exhaustive series of experiments are the interpretive methods, in which retention surfaces are modeled using a minimum number of experimental data points. Retention surfaces thus obtained for the individual solutes are then used to calculate (via computer) the total response surface according to some predetermined criterion. The total response surface is then searched for the optimum. 

In the following two sections we will describe two kinds of interpretive methods. In section 5.5.1 we will discuss simultaneous methods, which involve a fixed experimental design. In the iterative procedures of section 5.5.2, an initial design that consists of a minimum number of experiments is used and the location of the next data point is calculated during the optimization process. 

There are two basic families of solution techniques for linear algebraic equations Direct- and iterative methods. A well known example of direct methods is Gaussian elimination. The simultaneous storage of all coefficients of the set of equations in core memory is required. Iterative methods are based on the repeated application of a relatively simple algorithm leading to eventual convergence after a number of repetitions (iterations). Well known examples are the Jacobi and Gauss-Seidel point-by-point iteration methods. 

EXAMPLE 2.9 BUBBLE POINT AND DEW POINT TEMPERATURES BY THE SIMULTANEOUS METHOD... 

Using the simultaneous method, calculate the bubble point and dew point temperatures at 35 kPa of a mixture of 60% mole benzene (1) and 40% mole toluene (2). Assume Raoult s law applies, and use the vapor pressure data in Example 2.8. 

The simultaneous method is applied in this problem with one modification the calculated variables include the vapor mole fractions of both components and the temperature, such that the total number of calculated variables is C + 1 = 3. This is balanced by including the vapor mole fraction summation equation along with the equilibrium relations. By Raoult s law, the equilibrium relations (Equation 2.12) for components 1 and 2 at the bubble point, and the vapor mole fraction summation equation, are written as... 

For either scheme, the total number of data points digitised, acquisition times and spectral widths are identical, so the resulting spectra are largely equivalent. The most obvious difference is in the appearance of aliased signals, that is, those that violate the Nyquist condition, as described below. Experimentally, flatter baselines are observed for the simultaneous method as a result of the symmetrical sampling of the initial data points in the FID and this is the recommended protocol. 

Phosphorus has been determined40 in phosphosilico organic compounds (POSi bonds) by dissolving the sample in water, leaving for 4-5 h to hydrolyse (or for 30-40 min at 80°) and titrating the alkylphosphinic acid so produced with 0.1 M-alkali to the thymolphthalein end-point. A method has been described41 for the simultaneous determination of free phosphorus, combined phosphorus and silicon in the reaction products of tetraalkoxysilanes with phosphorus halides. 

A comparison of the two-compartment first-order absorption model fit to measured plasma concentration data from a traditional method of residuals analysis and a nonlinear regression analysis is provided in Figure 10.99. This figure illustrates the fact that both methods offer a very reasonable fit to the measured data. It also demonstrates that there is not a large difference between the fit provided by the two different techniques. Close examination does reveal, however, that the nonlinear regression analysis does provide a more universal fit to all the data points. This is likely due to the fact that nonlinear regression fits all the points simultaneously, whereas the method of residuals analysis fits the data in a piecewise manner with different data points used for different regions of the curve. 

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