Higher order eigenvalues

An alternative setup is the classical one with two detectors and analyzers which have been used by Kask et al. 1987 and Tsai et al. 2008. Relating to the work of Aragon and Pecora (1975 [14]) the analysis with this setup leads to a complex solution with higher order eigenvalues. Our setup can easily be verified in the confocal setup [15], which started the new era of FCS. 

For Fo < Foc, additional terms in the series solutions must be included. It is therefore necessary to use numerical methods to compute the higher-order eigenvalues 8 that lie in the intervals nn < 8 < (n + 1/2)ji for the plate and (n -1 )rt < 8 < nn for the cylinder and the sphere. Computer algebra systems are very effective in computing the eigenvalues. 

The h-2h calculation needed for the Richardson type extrapolation is also the calculation needed to estimate the error in the solution of a boundary value problem as discussed in the previous section. Thus it seem appropriate to combine these into an eigenvalue solver and such a function has been coded as the function odebveveO which is also available with the require obebv statement. An example of using this function for a higher order eigenvalue and function of the same constant potential problem is shown in Listing 11.10. The difference from Listing... 

It has been assumed above that Eq. (18) has been solved. In principle, the resulting eigenvalues and eigenfunctions can then be substituted in Eq. (19) to yield the first-order corrections, and so on, for higher orders of approximation. 

The superscript est denotes an estimate for the condition p of a perturbed LP, the subscript o refers to the reference LP, and To denotes the adjoint flux for the generalized core response in location 1 . For linear operators, equation (3) is accurate to one higher order than the estimate of the flux and eigenvalue used in the functional. In such cases, linear superposi-... 

Higher-order and non-isolated fixed points at least one eigenvalue is zero. 

These ideas also generalize neatly to higher-order systems. A fixed point of an th-order system is hyperbolic if all the eigenvalues of the linearization lie off the imaginary axis, i.e., Re(Aj iO for / = ,. . ., . The important Hartman-Grobman theorem states that the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait of the linearization in particular, the stability type of the fixed point is faithfully captured by the linearization. Here topologically equivalent means that there i s a homeomorphism (a continuous deformation with a continuous inverse) that maps one local phase portrait onto the other, such that trajectories map onto trajectories and the sense of time (the direction of the arrows) is preserved. 

The —> leading eigenvalues obtained from the geometric distance/topological distance quotient matrix and its higher order matrices were proposed to describe the sequence. In any case, the degeneracy of this approach still remains large. 

The Douglas-Kroll transformation can be carried out to higher orders, if desired (Barysz et al. 1997). In this way, arbitrary accuracy with respect to the eigenvalues of D can be achieved. 

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