Variational matrix

By assuming and maintaining the orbital coefficient variation matrix in block diagonal form. 

Accurate quantitation of analytes requires consistent measurements independent from matrix lot variation. Matrix lot to lot variation becomes critical for successfully implementing DBS. In addition to the analyte distribution across dried blood spots, matrix effect was also evaluated. In addition to the blood pool used for standards and... 

The BSSE is avoided by assuming, and then maintaining, the orbital coefficient variation matrix in a block diagonal form. The stationary condition 8E = 0 is equivalent to K secular problems... 

If all of the eigenvalues of the variational matrix have negative real parts, then y is an asymptotically stable rest point of (3.1). When this happens it is possible to find an arbitrarily small neighborhood around the rest point such that, on the boundary of the neighborhood, all trajectories cross the boundary from outside to inside. 

If an omega limit set contains an asymptotically stable rest point P, then that point is the entire omega limit set. If all of the eigenvalues of the variational matrix have positive real part then the rest point is said to be a repeller such a rest point cannot be in the omega limit set of any trajectory other than itself. If k eigenvalues have positive real part and n-k eigenvalues have negative real part then there exist two sets M P), called the stable manifold and defined by... 

Both eigenvalues are positive and the origin is a repeller. In particular, the origin is not in the omega limit set of any trajectory (other than itself). At (1 - Ai, 0), the variational matrix is of the form... 

Since 0 < Ai < A2 and 2 > 1, both eigenvalues are negative. Thus is (locally) asymptotically stable. At (0,1-A2), the variational matrix takes the form... 

The local stability is determined by the eigenvalues of the variational matrix... 

The next step is to analyze the stability of the interior rest point. To do this one considers the variational matrix at... 

We turn now to computing the local stability of the rest points of the full system. The arguments are based on standard linearization techniques, but the size of the variational matrix makes some of the computations difficult. The variational matrix for (2.4) takes the form... 

Comparing this with (3.4), it is clear that a 4 must satisfy (3.4). Therefore a 4 = fx, since /x is the only real root of (3.4). It follows that there is only one equilibrium of (5.1) and it is given in the first line of the proof. A direct calculation shows that the trace of the variational matrix at this equilibrium is negative and the determinant is positive. Therefore, the equilibrium is locally asymptotically stable. 

Theorem A. 10 allows us to conclude stability if the matrix is the variational matrix evaluated at a rest point. An important result is that the test in Theorem A.10 will work for some matrices that are not symmetric, not in the sense of being negative definite but in the sense of yielding stability based on the sign of the real parts of the eigenvalues. The type of matrix is closely associated with the orderings and monotone flow discussed in Appendices B and C. 

These restrictions obviously permit the complete elimination of BSSE in an a priori fashion, by assuming and maintaining the orbital coefficient variation matrix in block diagonal form ... 

In his 1986 text Aitchison proves (for the mathematically literate reader) that die covariance structure of log-ratios is superior to the covariance structure of a percentage array (the crude covariance structure, as it is termed in his text). The covariance structure of log-ratios is free from the problems of the negative bias and of subcompositions which bedevil percentage data. In detail he shows that there are three ways in which the compositional covariance structure can be specified. Each is illustrated in Table 2.5. Firsdy, it can be presented as a variation matrix in which the log-ratio variances are plotted for every variable ratioed to every other variable. This matrix provides a measure of the relative variation of every pair of variables and can be used in a descriptive sense to identify relationships within the data array and in a comparative mode between data arrays. 

The variation matrix shows that there is the greatest relative variation between MgO (indicative of olivine — the fractionating phase) and the elements excluded from olivine and concentrated in the melt — K, Ti, P, Na, Ca and... 

High values in a variation matrix will identify the element pairs which show the greatest variability. In igneous rocks this may be between a crystallizing mineral and the melt or between two or more crystallizing minerals. 

Large negative values in the two covariance matrices tend to confinn the variability indicated in the variation matrix. 

Dorfan, Komamiya, Snyder Filling in the Detector Variation Matrix May 9.1990... 

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