Truncating series

Those involving series truncation. The quantity In (1 - X2) can be represented by the infinite series - [x2 + (1/2) x + (1/3) x - - ]. Truncating this series after the first term is a valid approximation for dilute solutions and also simplifies the form of the equation. It is an optional step, however, and can be avoided or mitigated by simply retaining more terms in the series. 

Gram-Charlier Series This is an infinite series whose coefficients involve the Gaussian distribution and its derivatives (Kendall, Advanced Theory of Statistics, vol. 1, Griffin, 1958). The derivatives, in turn, are expressed in terms of the moments. The series truncated at the coefficient involving the fourth moment is... 

If ACp is independent of temperature, the final term in Eq. (6-16) can be neglected. Clarke and Glew expanded AH in a Taylor s series, truncating at the third derivative of ACp. and obtained Eq. (6-17). 

The expression (8) is very useful in the sense one can control the series truncation. This is so because the parameter in gives the order of the derivatives appearing in the expansion. 

Non-uniform prior prejudice. The dynamic range of the exp( ) map is reduced from 966 to a value of 3.3 when a NUP of spherical valence monopoles is used as a consequence, the size of the Lagrange multipliers is reduced by between one and two orders of magnitudes, and the error due to series truncation in the -map is less than 0.213 e A 3 in absolute value everywhere in the cell, the rms deviation from the model being as low as 0.212 e A 3 (Figure 2(b)).3... 

Since y does not depend linearly on k, and because of the series truncation in Eq. (13) the differences of Eq. (16) represent only improved differences, not optimal ones. This is only the case if the process is iterated until the condition... 

VV is the gradient of V(V(V = 3 V/dx/), F is the matrix of second derivatives of V(Fq = 9 2 V/bXf 9x ). Since the first derivatives of V do not depend linearly on the Cartesian coordinates, and because of the series truncation, the Newton-Raphson process (Eq. (30)) has to be iterated until Eq. (27) is satisfied with sufficient precision. For illustration consider the arbitrary onedimensional function u (v) (Fig. 5). The following holds for this simple example ... 

Series truncation effects due to the experimental resolution limit are reduced when the core- or spherical-atom densities are subtracted from the Fourier summation, as in Eq. (6.9). 

In the point matching method (Oguchi, 1973 Bates, 1975) the fields inside and outside a particle are expanded in vector spherical harmonics and the resulting series truncated the tangential field components are required to be continuous at a finite number of points on the particle boundary. Although easy to describe and to understand, the practical usefulness of this method is limited to nearly spherical particles large demands on computer time and uncertain convergence are also drawbacks (Yeh and Mei, 1980). 

It is expected that a suitable choice of g will improve the convergence of the power series as the singularity of tj/ may be cancelled by the zero of g. In most cases, the Pade approximation /jg provides with a better approximation than the corresponding power series truncated at any order q, especially when /c is comparable to (or even greater than) the convergence radius of the power series [Eq. (342)] (Takeshi, 1999). 

Examination of the various contributors to ES reveals that the dipole-dipole R"3 term is particularly sensitive to angular distortions. Whereas the dipole-quadrupole interactions contained in the R 4 term are also sizable, it is important to note that they behave differently depending upon which molecule is rotated. That is, the R 4 term produces a net stabilization if the donor is turned but adds to the destabilization of R-3 if the rotation occurs in the acceptor. Overall, the multipole series, truncated at R 5, provides a reasonable approximation of the full ES distortion energy, particularly at the longer distance. 

Conceptually, the STO basis is straightforward as it mimics the exact solution for the single electron atom. The exact orbitals for carbon, for example, are not hydrogenic orbitals, but are similar to the hydrogenic orbitals. Unfortunately, with STOs, many of the integrals that need to be evaluated to construct the Fock matrix can only be solved using an infinite series. Truncation of this infinite series results in errors, which can be significant. 

Evaluate the average on the left of Eq. (5.18) assuming a gaussian distribution for the random variable x, thereby establishing that the cumulant series truncated at second order is correct for a gaussian distribution. 

Figure i. 4 Distance dependence of the multipole series of the electrostatic interaction energy, truncated at various orders, for the water dimer. Data from. The values for the series truncated at differ only very slightly from the series and so are not shown explicitly. 

In the experimental works discussed in this book, the real values of asu = E /k T have ranged from about 15 to 30. If so, just the first terms of the above power series 2.30 or 2.31 suffice to provide the precision that is required when treating experimental data. It can be verified that the series truncated this way yield a common approximate formula for the six integrals ... 

Analytic linearization relies on the application of a Taylor series, truncated after the first term. Let us consider the general case of a vector function, g, of several variables, x. When x is displaced by a small deviation x from a constant, centrai value x. 

The series truncates (exactly) after four commutators regardless of the level at which (T ) T is truncated (if at all). This exact truncation is a result of the fact that // contains at most two-electron operators, which involve four general (particle or hole) operators i j lk. Therefore [//, T] contains at most three general operators, [[f/, T], T] contains two, and [[[[ J... 

A drawback of this equation is that it cannot be treated in graphical form. However, by expanding the sine function in a power series truncated after the second term, a more manageable form can be obtained as follows ... 

With the aim of determining the value of I (q), Vonk proposed that the function I iq) can be expanded in a power series truncated at the first or second term, according to... 

In any case, a disorder must be chemically reasonable. Not every significant residual electron density peak is caused by disorder. High residual electron density can also be caused by inadequately corrected absorption, Eourier series truncation errors (for example when strong reflections are missing) or radiation damage. Such artefacts often lead to the accumulation of spurious electron density at special positions. 

Division of the expansion coefficients by smaller and smaller values of a must be stopped and the series truncated or tapered off Co not include amplified noise in the reconstruction. In practice, with the order of noise achievable in real experiments, it turns out that only three or four independent components can be reliably... 

The ZOA provides a time-invariant formulation of the dynamic part of the cutting forces, which is time variant due to the rotation of the cutting tool. This is done by approximating the time varying part by means of a Fourier series, truncated at the zeroth-order constant term. Therefore, a single-frequency solution results, which is limited to slot milling processes. To... 

The second term in the RHS of Eq. 6.38 means that the error of the result of the series truncation to N terms has the order of magnitude as the sequence 6jv+i that is. 

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