Laplacian expansion

In 2.Vni M it was shown that a determinant may be expanded in the form 2 — y- aikAik, where a/ is an element of the ith row and fcth column and Aik is the minor determinant formed by suppressing the ith row and fcth column in the original determinant. It is possible, however, to form minors by suppressing 2, 3,. .. m rows and an equal munber of colmnns, and as this method of expansion was investigated by Laplace in 1772, it is often called the Laplacian expansion. If the order of the original determinant is n, that of each minor will be n—m. 

The determinant of a square matrix A =. ,) of size n n (see footnote c in Section 3.7.5), denoted IL4II, is a number that is calculated from the elements Ay of the matrix by Laplacian expansion by minors (see textbooks of mathematics). The results for 1 x 1, 2 x 2 and 3x3 determinants are shown below. 

Ferraz-Mello, S. (1994), The convergence domain of the Laplacian expansion of the disturbing function. Cel. Mech. Dynam. Astron. 58, 37-52. 

The equations in this section retain some compact notation, including the substantial derivative operator D/Dt, the divergence of the velocity vector V-V, and the Laplacian operator V2. The expansion of these operations into the various coordinate systems may be found in Appendix A. 

A more complicated situation emerges in motion along nonintersecting surfaces with variable curvatures. If the distance between these surfaces remains finite everywhere, then the field lines do not expand infinitely in the directions normal to the surfaces. In the absence of dissipation this means that there is no unbounded growth of the normal field component. However, introduction of the finite conductivity yields an equation for the normal component which is not decoupled it contains the contribution of the Laplacian of the remaining components. At the same time, it is possible for all other components to increase exponentially with an increment which depends on the conductivity and vanishes for infinite conductivity. The authors called this mechanism of field amplification a slow dynamo, in contrast to the fast dynamo feasible in the three-dimensional case, i.e., the mechanism related only to infinite expansion of the field lines as, for example, in motion with magnetic field loop doubling. In a fast dynamo the characteristic time of the field increase must be of the same order as the characteristic period of the motion s fundamental scale. 

The FOCE method uses a first-order Taylor series expansion around the conditional estimates of the t] values. This means that for each iteration step where population estimates are obtained the respective individual parameter estimates are obtained by the FOCE estimation method. Thus, this method involves minimizations within each minimization step. The interaction option available in FOCE considers the dependency of the residual variability on the interindividual variability. The Laplacian estimation method is similar to the FOCE estimation method but uses a second-order Taylor series expansion around the conditional estimates of the 77 values. This method is especially useful when a high degree of nonlinearity occurs in the model [10]. 

The NONMEM program implements two alternative estimation methods, the first-order conditional estimation and the Laplacian methods. The first-order conditional estimation (FOCE) method uses a first-order expansion about conditional estimates (empirical Bayes estimates) of interindividual random effects, rather than about zero. In this respect, it is like the conditional first-order method of Lindstrom and Bates.f Unlike the latter, which is iterative, a single objective function is minimized, achieving a similar effect as with iteration. The Laplacian method uses second-order expansions about the conditional estimates of the random effects. ... 

The terms in the expansion derived from derivatives with respect to X are identical to those obtained by taking the corresponding derivatives of the charge density p(X) itself. The dyadic Wp is the Hessian matrix of p, whose eigenvectors and eigenvalues determine the properties of the critical points in the charge distribution. The trace of this term is the Laplacian of the charge density, V p. 

The cross Laplacian, Aj, is Important for narrow light beams, for systems with self-focusing. It is worthwhile to stress here tliat most of tlie currently cherished approaches applied for solving Eq. (10), are based on tlie power series expansion (2) of the polarization over the laser field amplitude and on an account of only the coherent contributions to the corresponding polarizabilities. However, this approximation breaks down even for fairly short laser pulses, which is the case... 

The solution of these equations by means of standard eigenfunction expansions can be carried out for any curvilinear, orthogonal coordinate system for which the Laplacian operator V2 is separable. Of course, the most appropriate coordinate system for a particular application will depend on the boundary geometry. In this section we briefly consider the most common cases for 2D flows of Cartesian and circular cylindrical coordinates. 

A typical VMC computation to estimate the energy or other expectation values for a given 4/x(R) might involve the calculation of the wavefunction value, gradient, and Laplacian at several millions points distributed in configuration space. Computationally this is the most expensive part. So a desirable feature of TVR), from the point of view of Monte Carlo, is its compactness. It would be highly impractical to use a trial wavefunction represented, for example, as a Cl expansion of thousands (or more) of Slater determinants. 

FORTRAN source code in which the maximum likelihood is evaluated with one of two different first-order expansions (FO or FOCE) and a second-order expansion about the conditional estimates of the random effects (Laplacian) S-PLUS algorithm utilizing a generalized least-squares (GLS) procedure and Taylor series expansion about the conditional estimates of the interindividual random effects... 

For slowly varying densities, the kinetic energy functional can be represented by one of its gradient expansions. The gradient expansion of the kinetic energy density is not unique since it relies upon different derivations techniques [35], which yield or not a contibution of the laplacian of the density in the second order correction. In the following we will consider the expansion expression which does not involve V /i(r) ... 

This is the generalization of the usual equation connecting the multiplication constant and the Laplacian. In the present case, it is useful only if one can neglect the second and higher powers of A in the expansion of the exponential. 

Thermal diffusivity, angle, coefficient Coefficient of thermal expansion, coefficient Kronecker delta Variation, Laplacian, hlter size... 

Most textbook discussions of relaxation methods for solving partial differential equations use the familiar second-order form for the Laplacian. Finite difference Laplacian representations result from Taylor series (i.e., polynomial) expansions of a function centered on the grid point x. We give the prescription for the second-order formula and then it is apparent how to proceed to higher orders. Expand the function in the positive and negative directions to fourth order ... 

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