Eigenvalue after

Fig. 3. A comparison of the eigenvalues of the outermost valence electrons for Pu using relativistic, semi-relativistic and non-relativistic kinematics and the local density approximation (LSD). Dirac-Fock eigenvalues after Desclaux are also shown. The total energies of the atoms (minus sign omitted), calculated with relativistic and non-relativistic kinematics are also shown. HF means Hartree Fock...

Any basis set, consisting of a complete orthonormal set of functions, should produce the correct eigenvalues after variational minimization, e.g. 

The eigenvalues of this matrix represent the variances of the principal components. PCA reduces the dimensionality by eliminating variables that contribute the least to the variance of the data, i.e., those with the smallest eigenvalues. After diagonaliza-tion of the covariance matrix, the original data can be transformed by, in (17). 

It is instructive to compare the eigenvalues of transition matrices obtained for different lag times t. That is, we do not count transitions on a timescale of O.lps which means to observe transitions from one instance of the time series to the next, but count transitions on a timescale of, say, ps which is between every tenth step in the global Viterbi path. For all time lags. Fig. 5 clearly indicates two dominant eigenvalues after which we find a gap, followed by other gaps after 4, 9, or 16 eigenvalues. This yields 2, respectively 4, 9, or 16 metastable sets. To avoid confusion we call these metastable sets (molecular) conformations. 

After one step back down the chain we have an eigenvalue of 1, given by dividing 2 by the arity, and another eigenvalue of 1, added to the list in the usual way. After two steps we have eigenvalues of 1/2,1/2 by dividing these by the arity plus a unit eigenvalue. After three the list is 1/4,1/4,1/2,1, and after four 1/8,1/8,1/4,1/2,1. 

The appropriate SVD-derived spectral and temporal eigenvectors were selected and the temporal vectors were modeled. Ideally, the temporal vectors are the kinetic traces of individual components, each one being associated with a spectrum of a pure component Le., the spectral vector). Once the temporal vectors had been modeled the pure component spectra were reconstructed as a function of the pre-exponential multiplier obtained from the analysis, SVD determined spectral eigenvectors, and the corresponding eigenvalues. After the spectra of the component species were determined, the extinction profile was calculated and used along with the calculated decay times to construct a linear combination of the pure component species contributions to the observed... 

E are the orbital eigenvalues after chemisorption, and E. the orbital eigenvalues before interaction. 

To simplify FECO evaluation, it is conmion practice to experimentally filter out one of the components by the use of a linear polarizer after the interferometer. Mica bireftingence can, however, be useftil to study thin films of birefringent molecules [49] between the surfaces. Rabinowitz [53] has presented an eigenvalue analysis of birefringence in the multiple beam interferometer. 

Figure 3. Floquet band structure for a threefold cyclic barrier (a) in the plane wave case after using Eq. (A.l 1) to fold the band onto the interval —I < and (b) in the presence of a threefold potential barrier. Open circles in case (b) mark the eigenvalues at = 0, 1, consistent with periodic boundary conditions. Closed circles mark those at consistent with sign-changing...

After the assembling of the stochastic matrix Pd we have to solve the associated non-selfadjoint eigenvalue problem. Our present numerical results have been computed using the code speig by Radke AND S0RENSEN in Matlab,... 

It is possible (see, for example, J. Nichols, H. E. Taylor, P. Schmidt, and J. Simons, J. Chem. Phys. 92, 340 (1990) and references therein) to remove from H the zero eigenvalues that correspond to rotation and translation and to thereby produce a Hessian matrix whose eigenvalues correspond only to internal motions of the system. After doing so, the number of negative eigenvalues of H can be used to characterize the nature of the... 

Note that even after requiring normalization, there is still an indeterminaney in the sign of v(3). The eigenvalue equation, as we reeall, only speeifies a direetion in spaee. The sense or sign is not determined. We ean ehoose either sign we prefer. 

After the first measurement is made (say for operator R), the wavefunetion beeomes an eigenfunetion of R with a well defined R-eigenvalue (say X) = /( i). 

We now prove several identities that are needed to diseover the information about the eigenvalues and eigenfunetions of general angular momenta that we are after. Later in this Appendix, the essential results are summarized. 

Figure 3 Flow of a distance geometry calculation. On the left is shown the development of the data on the right, the operations, d , is the distance between atoms / and j Z. , and Ujj are lower and upper bounds on the distance Z. and ZZj, are the smoothed bounds after application of the triangle inequality is the distance between atom / and the geometric center N is the number of atoms (Mj,) is the metric matrix is the positional vector of atom / 2, is the first eigenvector of (M ,) with eigenvalue Xf,. V , r- , and ate the y-, and -coordinates of atom /. (1-5 correspond to the numbered list on pg. 258.)...

In the next step, which is numerically the most demanding, the differential equations (3) are solved. Two possible strategies using a variational expansion of the single particle wave functions, /., are described below. After the eigenvalues and eigenfunctions have been found, a new ("output") charge density can be... 

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