Gaussian decomposition

Fig. 21.4 The X-ray-excited luminescence spectrum of Ba3BP30i2 host. The dotted lines are the Gaussian decomposition.

Fig. A. Background subtraction and Gaussian decomposition of the NM Raman spectrum of HaO(as) deposited at 110 K...

The Z-scan theory has been described by different authors. In the thin sample limit the Z-scan measurement is described either through Fresnel integration or through a Gaussian decomposition procedure [3,6]. 

Using the Gaussian decomposition (GD) method [16], in which the complex electric field at the exit plan of the sample is decomposed into a summation of Gaussian beams via a Taylor series expansion of the nonlinear phase term. 

The most frequently used modifications of the basic Gaussian elimination method in finite element analysis are the LU decomposition and frontal solution techniques. 

The LU decomposition is essentially a Gaussian elimination, arranged for maximum efficiency (Ref. 112). The chief reason for doing an LU decomposition is that it takes fewer multiplications than would be needed to find an inverse. Also, once the LU decomposition has been found, it is possible to solve for multiple right-hand sides with little increase in work. The multiphcation count for a.n n X n matrix and m right-hand sides is... 

Steady-state solutions are found by iterative solution of the nonlinear residual equations R(a,P) = 0 using Newton s methods, as described elsewhere (28). Contributions to the Jacobian matrix are formed explicitly in terms of the finite element coefficients for the interface shape and the field variables. Special matrix software (31) is used for Gaussian elimination of the linear equation sets which result at each Newton iteration. This software accounts for the special "arrow structure of the Jacobian matrix and computes an LU-decomposition of the matrix so that qu2usi-Newton iteration schemes can be used for additional savings. 

Figure 36. The scaled distributions of mean, P(H/Y ) (a), and Gaussian, P(K/Y]2) (b), curvatures scaled with the inteface area density, computed at several time intervals of the spindal decomposition of a symmetric blend. There is no scaling at the late times because the amplitude of the thermal undulations does not depend on the average growth of the domains, and therefore the scaled curvature distributions functions broaden with rescaled time.

This result is not as satisfactory as it seems. The reader will notice that the behavior of the Gaussian components appears different from that described on p. 130. MN use the data of H. A. Lindner [Ph. D. dissertation, University of Karlsruhe (1970)], and his spectral decomposition must differ from that of Wal-rafen judging from the frequencies (see Table 2) as well as the temperature dependences of the components (see also the comments by Walrafen 30>). Until this difference is resolved, and better spectra are available, the findings of MN must be viewed with reserve. 

Fig. 2.14. Influence of the reverse activation energy on KER and thus, on peak shapes in metastable ion decompositions, suitable experimental setup as prerequisite. From left no or small reverse barrier causes Gaussian peak shape, whereas medium or yields flat-topped peaks and large For causes dish-shaped peaks.

The Kronecker product with the identity ensures rotational invariance (sphericalness) elliptical Gaussians could be obtained by using a full n x n A matrix. In the former formulation of the basis function, it is difficult to ensure the square integrability of the functions, but this becomes easy in the latter formulation. In this format, all that is required is that the matrix, A, be positive definite. This may be achieved by constructing the matrix from a Cholesky decomposition A), = Later in this work we will use the notation... 

Figure 2 reports the results [34, 35] of a space-scale decomposition of the GC content fluctuations of a 10-Mbp-long fragment of human chromosome 22, when using the Gaussian [Eq. (6)] as smoothing filter. This... 

In physical terms the site inhomogeneous component may be conceived of either as slightly different protein binding sites for chromophores or as a distribution of protein conformational substates [127] at any one site. In operational terms the spectral forms are represented by the sub-bands of a gaussian or lorentzian decomposition analysis of absorption or fluorescence spectra. 

A factorization of O of this form can be carried out using standard matrix algorithms such as LU decomposition or Gaussian elimination with back-substitution [2,34,35]. The main advantage is the straightforward evaluation of the stmcture transformation corresponding to Eq. (3), which is just... 

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