Quadrature, definition

The analysis becomes easier if we introduce the following quadrature noise operators [44,45] (for further comparisons, we adjust the phase in quadrature definitions for the second harmonic mode in such a way as to take into account that i) = 2— < )fe = n/2)... 

For certain mathematical functions and operations it is necessary for the physicist to know their context, definition and mathematical properties, which we treat in the book. He does not need to know how to calculate them or to control their calculation. Numerical values of functions such as sinx have traditionally been taken from table books or slide rules. Modern computational facilities have enabled us to extend this concept, for example, to Coulomb functions, associated Legendre polynomials, Clebsch—Gordan and related coefficients, matrix inversion and diagonali-sation and Gaussian quadratures. The subroutine library has replaced the table book. We give references to suitable library subroutines. 

Given a function f x) and an error tolerance e, an automatic quadrature algorithm computes an approximation to the definite integral... 

The construction of the A -point quadrature approximation requires the definition of the following abscissa basis vectors ... 

By using the definition reported in Eq. (3.61) and exploiting the univariate quadrature over the first internal coordinate, the generic mixed moment of order k = [ki,lc2,..., Icm-i, can be calculated as... 

When applying this algorithm, the moment set [mo, m, ..., m2N must be known (and realizable). From the definition of the objective function and the properties of Gaussian quadrature,we have /(O) > 0. Thus, as a first step in the bounded-search algorithm, an upper bound cr+ can be determined such that /(cr+) < 0 under the condition that the moment set mj, m, 2iv-i) found from Eq. (3.93) using cr = is realizable. If no such cr+ exists, then can be chosen such that it minimizes /(cr ) and the moment set... 

The other two collision source vectors, and can be evaluated using the definitions in Eqs. (6.104) and (6.106). As mentioned earlier, will be closed in terms of the moments of order two and lower, and their gradients. In contrast, C will not be closed in terms of any finite set of moments. Nevertheless, it can be approximated using quadrature-based moment methods as described in Section 6.5. In the fluid-particle limit d d2), neither CI2 i or C will contribute terms involving spatial gradients of the fluid properties (i.e. buoyancy, lift, etc.) to the fluid-phase momentum equation. As mentioned earlier, such terms result from the model for gapi i-n) and would appear, for example, on using the expression in Eq. (6.81). With the latter, Eq. (6.161) becomes... 

By extending the concepts of the previous section, it is possible to derive an approximation (which is no longer a real Gaussian quadrature) for the bivariate system. The resulting Appoint quadrature approximation transforms the definition of the general bivariate moment into (Wright et at, 2001b) m = Za=i where, as for the univariate case, Wa... 

If the realizability condition in Eq. (8.52) is satisfied, found from Eq. (8.53) is guaranteed to be realizable. Using the method described above for the PBE (Eq. (8.35)), it is straightforward to extend Eq. (8.53) to second-order time-stepping. The extension to multiple velocity components v in multiple spatial dimensions is a bit more complicated. As described in Yuan Fox (2011) and in Section B.3 of Appendix B, when the CQMOM is used to constmct the multivariate quadratures all permutations must be used in a consistent manner in order to get the correct kinetic energy fiuxes. Nevertheless, for each permutation of the CQMOM, the basic time-stepping formula in Eq. (8.53) is used to update the transported moment set by modifying the definition of K , to include the multivariate moments in the optimal-moment set. Readers interested in more details on multidimensional free transport should consult Yuan Eox (2011) and Section B.3 of Appendix B. 

In this definition, n -ifiy, fi) is reconstructed using extended multivariate quadrature from the transported moment set Since nf-ifiw,fi) has a known functional form, the halfmoment sets in Eq. (B.54) can be evaluated analytically. Once these half-moment sets are known, the Af-node multivariate quadrature is applied to find the two set of weights and abscissas. The overall procedure is as follows ... 

Now, assuming that the two modes are not correlated at time x = 0, it is straightforward to calculate the variances of the quadrature field operators and check, according to the definition (12), whether the field is in a squeezed state. If the initial state of the field is a coherent state of the fundamental mode and a vacuum for the second-harmonic mode, /0) = wa(0)) 0), for which we have... 

Squeezing occurs if one of the quadrature variances ((Ap)2), ((Aqj2) is below the coherent state level. A more general definition deals with single and compound mode principal squeeze variances Xj and V [140]... 

There is another widely used way to obtain a numerical approximation to a definite integral, known as Gauss quadrature. In this method, the integrand function must be evaluated at particular unequally spaced points on the interval of integration. We will not discuss this method, but you can read about it in books on numerical analysis. 

As concerns the electromotive force, a definition of its quadrature and inphase components can be done in two ways. In fact, we can compare either a phase shift of the electromotive force with the current in the transmitter or with the primary electromotive force t Q. In the future, the latter approach will be used and correspondingly one can write (Fig. 2.3b) ... 

The definition of the Gaussian quadrature formula in Eq. E.30 implies that the determination of this formula is determined by the selection of N quadrature points and N quadrature weights that is, we have 2N parameters to be found. With these degrees of freedom (2N parameters), it is possible to fit a polynomial of degree 2N - 1. This means that if Xj and are properly chosen, the Gausssian quadrature formula can exactly integrate a polynomial of degree up to 2N — 1. 

The integrals (with respect to ) in the source term definitions (see (9.87)-(9.90)) can be expressed in terms of a quadrature as a sum of integrals over the sub-intervals. Replacing the corresponding integrals in all the source terms with such a quadrature sum, the macroscopic equation can be formulated as ... 

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