Analysis using eigenvalues

The linear differential system which describes the local behaviour in the vicinity of a point (cj, c, ..Cj), is associated with the system (1)  

Putting into equation (51) and multiplying the left-hand side by V gives  

As the matrix of eigenvalues A is diagonal, the differential system (53) is completely uncoupled, which allows the differential equations associated with the shortest characteristic times to be replaced by algebraic equations, which define an Intrinsic Low-Dimension Manifold ILDM. The dynamics of the system can thus be described by the dynamics on this manifold. 

With respect to the previous methods, it can be remarked that an eigenvalue analysis allows the fast and the slow variables to be defined, without having to identify the quasi-stationary species or the reactions at quasi-equilibrium. 

Carry out the analysis of the following mechanism, using eigenvalues  

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The —> physico-chemical properties used to derive BC(DEF) descriptors are activity coefficient in water, —> octanol-water partition coefficient, hoiling point, —> molarr activity, liquid state molar volume, and heat of vaporization. The eigenvalues and corresponding cumulative explained variances of the five principal properties (denoted by B, C, D, E, and F) are reported in Table B1. It can be noted that the first two principal properties B and C already explain 95.7% of the original variance of the six physico-chemical properties further analysis using different compounds and properties showed B and C to be independent of the data set used in their derivation, identifying them as measures of molecular bulk and cohesiveness, respectively. The other three parameters, D, E, and F, are of minor importance, however they were... 

The existence of multiple solutions does not ensure that these solutions are physically attainable. In order for these solutions to be physically attainable, they must be stable. The linear stability analysis presented here provides the necessary conditions for the stability. The method of Lyapunov s fimction can also be used to assess the stability and the magnitude of the permissible pertiffbations so that the reactor returns to the steady state. In the case of Unear stability analysis, the eigenvalues of a differential operator determine the stability. An excellent account of the stabihty analysis of chemical reactors can be found in Perlmutter (1972). 

Canonical Correlation Analysis. Our method of canonical variate analysis uses factor scores as input. Since using the minimum number of data reduction factors (eigenvalue > 1.0, 1 in the PMR data) constrained the CV space to too low a dimensionality to take full advantage of all the data, the first 6 factors from each data set were used. It can be seen from the last colvimn of Table II that the PMR data input contained more factors than were significant, and that the MS data used somewhat too few factors (6 out of 9) based on the eigenvalue criterion. Table III shows the canonical variate... 

Previous work using eigenvalue tracking (ET) as a method of spectral association has been successfully applied for the purposes of dynamic analysis and model reduction. ET uses homo-topy methods that transform a system with known eigenvalue-to-state association into the final system and track the eigenvalue associations as the system is transformed. 

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. 

The method of vibrational analysis presented here ean work for any polyatomie moleeule. One knows the mass-weighted Hessian and then eomputes the non-zero eigenvalues whieh then provide the squares of the normal mode vibrational frequeneies. Point group symmetry ean be used to bloek diagonalize this Hessian and to label the vibrational modes aeeording to symmetry. 

Let us calculate the relaxation time of particles in this potential (escape time over a barrier) which agrees with inverse of the lowest nonvanishing eigenvalue Yj. Using the method of eigenfunction analysis as presented in detail in Refs. 2, 15, 17, and 18 we search for the solution of the Fokker-Planck equation in the... 

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