Point-simultaneous

Corrosion starts rapidly at many points simultaneously. The maximum penetration rate, however, depends on the type of soil and the existence of foreign cathodic structures [16,17]. 

In a point release attack, the agent would be released in a plume from a single point (or perhaps several discrete points simultaneously). Individuals near the point of release would be exposed to agent at high levels, whereas those farther away would be exposed at lower doses. Initially, the spread of agent would be confined to the space in which it was... 

A multipass marching solution is used in COBRA IIIC (Rowe, 1973). The inlet flow division between subchannels is fixed as a boundary condition, and an iterated solution is obtained to satisfy the other boundary solution of zero pressure differential at the channel exit. The procedure is to guess a pattern of subchannel boundary pressure differentials for all mesh points simultaneously, and from this to compute, without further iteration, the corresponding pattern of crossflows using a marching technique up the channel. The pressure differentials are updated during each pass, and the overall channel iteration is completed when the fractional change in subchannel flows is less than a preset amount. 

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... 

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). 

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... 

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) ... 

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. 

There are two different procedures by which the iteration may be carried out. One may substitute each improved coefficient, along with all the others, into the succeeding equation. This is known as the point successive procedure. Alternatively, one may determine all the unknown coefficients in all the equations using only the set of coefficients determined from the previous iteration. This is known as the point simultaneous procedure. Undesirable convergence properties of the point simultaneous procedure motivated its abandonment early in the research, however. The former procedure converges almost always when the tolerance is not chosen exceedingly small. The tolerance determines when the iteration is stopped. The iterations continue until the maximum change in any coefficient over its... 

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. 

The application of the point simultaneous procedure to two-dimensional equations yields adequate restorations for many interesting two-dimensional... 

If the CFM is formed by a complex pair, the unit circle can be crossed at some point (actually at two points simultaneously) off the real axis. Generally this will correspond to the bifurcation from a stable limit cycle to a quasi-periodic motion on a torus. In special cases, however, where the crossing point corresponds to the fcth complex root of - 1, the limit cycle bifurcates to a phase-locked cycle (closed loop on the torus) corresponding to period-fc resonance such as that described ip 13.3. 

Fe,v-Fe2+ system in the presence of a trace of dissolved oxygen. The measured zero-current potential is that value where the rate of 02 reduction at the electrode surface is equal to the rate of Fe2+ oxidation rather than the value of since at the latter point simultaneous 02 reduction produces excess cathodic current. In addition, because the net reaction of Em converts Fe2+ to Fe >>+, the measured potential exhibits a slow drift. Such mixed potentials are of little worth in determining equilibrium Eh values. 

The structure of Aristotelian extension can be captured first by characterizing a richer structure (arguably latent in common sense) and then by abstracting certain features from that structure. Let us begin, therefore, by appealing to a four-dimensional structure M of points.6 The four dimensions are the three spatial dimensions and the temporal dimension. To give the structure a temporal dimension it is natural and in accord with Aristode s own view of time to suppose that for any two points, p and q, within M there is a well-defined interval of time. As a result, for any point p there will be a set of points simultaneous with p. With these stipulations, the structure M can be thought of as stratified into a succession of instantaneous three-dimensional spaces called hyper-planes of simultaneity each of which bears determinate temporal relations to all the other such spaces in the manifold. 

Reactions of third and higher orders are rare, but there are in fact reactions which are definitely of third and sometimes of higher order. This is due to the fact that the probability of three molecules coming to a single point simultaneously, i.e., probability of trimolecular collisions is much less as compared to unimolecular or bimolecular collisions. 

The resulting equations are solved numerically to compute the film thickness distribution or the interface shapes and the radius of the meniscus at the channel s corner. The implicit conservative scheme was used to obtain the solution. We used non-uniform grid pattern near moving film-meniscus contact point. Simultaneously with calculation of the interface shape, the heat transfer problem has been solved in the symmetry element of the channel. For the case of uniform heat production in the wall we have solved conservation energy equations in channel wall area ... 

A position sensitive detector (PSD) employs the principle of a gas proportional counter, with an added capability to detect the location of a photon absorption event. Hence, unlike the conventional gas proportional counter, the PSD is a line detector that can measure the intensity of the diffracted beam in multiple (usually thousands) points simultaneously. As a result, a powder diffraction experiment becomes much faster, while its quality generally remains nearly identical to that obtained using a standard gas proportional counter. ... 

There are in fact two solutions after the singular solution, X = A, Y — BjA bifurcates, the new solution is a stable limit cycle. Lefever and Nicolis (1971) give these solutions for the point with parameter values A = 1,5 = 3, lying in the parameter plane region corresponding to a limit cycle and an unstable singular point, simultaneously. 

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. 

For this second stage of fitting, a large number of (X, r, T c s,) points are chosen, at many temperatures, and broadly distributed over the available rate surface. The complete rate expression, with the Arrhenius parameters of each constant included, is then fitted to all these points simultaneously. Optimization, as described in Chapter 7, yields the best-fit values of the parameters of this expression. Several attempts to reach the same optimum set of parameters are made, with somewhat different starting values, to make sure that a stable and optimal solution has been found. 

In the X-ray structure of 148 EtOH (R = Bu ), all three t-butylamido substituents are in a triaxial arrangement but their bulkiness does not allow all the three amide carbonyls to point simultaneously to the centre of the annulus. Only two COs point to the annulus and are H-bonded with an indole NH. The third CO points away and is not capable of H-bonding. However, all three indole molecules are H-bonded in... 

Detonated at many points simultaneously, the HE would blow inward. The shock wave from that explosion would squeeze the tamper from all sides, which in trirn would squeeze the core. Squeezing the core would change its geometry from hollow shell to solid ball. What had been subcritical because of its geometry would be squeezed critical far faster and more efficiently than any mere gim could fire. The gim will compress in one dimension, Manley remembers Neddermeyer telling them. Two dimensions would be better. Three dimensions worild be better still. ... 

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