Applied Multivariate Techniques - download pdf or read online

By Subhash Sharma

ISBN-10: 0471310646

ISBN-13: 9780471310648

This e-book specializes in whilst to take advantage of a number of the analytic recommendations and the way to interpret the ensuing output from the main typical statistical applications (e.g., SAS, SPSS).

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Example text

The points of the phase space from which the target can be reached (in any time) are called the domain of attainability. Fig. 49 47 The boundary of the attainability domain can have singularities even when neither the target nor the field of the indicatrices at the various points of phase space have them. We shall give below a classification of the singularities of the attainability boundary in a generic controlled system on a phase plane for the case where the indicatrices and the target are smooth curves (due to A.

With a decrease in viscosity) is described mathematically by just such complex attractors in the phase space of the fluid. The dimension of this attractor, it appears, is finite for any Reynolds number (Yu. S. Il'yashenko recently obtained an upper bound for this dimension of the order of Re 4 in the case of flows on closed 2-manifolds without boundary; for 2-manifolds with boundary the upper bound exp(Re 4 has been obtained by M. I. Vishik and A. V. Babin for the attractor dimension 2), but tends to infinity as Re tends to infinity.

For a generic system such an event can only occur at isolated points of the boundary K of the domain of complete controllability. In Fig. 51 it happens at point 3, where the tangent to K is horizontal. e. of the target and of the indicatrix field, point 3 will move slightly but will not vanish. Let us now consider the network of limiting curves near the point 3. Both fields of limiting directions are smooth nearby. By a choice of a suitable system of coordinates we may straighten one of them. In Fig.

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Applied Multivariate Techniques by Subhash Sharma


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