By Magri F., Morosi C.
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Additional info for A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds (Quaderno 19-1984, Univ. of Milan)
The range of values that pr assumes as a function of y is identical with the range of values of p in Eq. 10 and Fig. 1. The network classification in the three cases illustrated in Fig. 1 should now be restated in terms of the T-matrix representation. This is easily accomplished by considering Eq. 23. According to it, the matrices -t] [l Qt~^ [l 1 -t]^ and -(Z 4- Z^) are related by a colinear I transformation, A colinear transformation preserves the signature of a matrix. 1. The same conclusion may be reached from the facts that Nr and N^ have the same eigenvalues, and both EE^ and 85^ are positive definite.
It is the single stationary value of Pe,i and also the negative of the sole eigenvalue of N. Another simple but quite different case arises if the original (nonreciprocal) network is a passive one with dissipation, (Z -|- Z^) positive Then the definite, in thermal equilibrium at absolute temperature T. operations defined by Fig. 4 must, on thermodynamic grounds, always lead to Pe,i = kT A/ in a frequency band A/, where k is Boltzmann's constant. No matter what form the variable lossless network may take.
4. This second imbedding network is completely variable. One possible variation removes the first 2w-terminal-pair imbedding. Accordingly, the stationary values of the exchangeable power at the (w -f l)th terminal pair in Fig. 4 do not change when a 2 w- terminal-pair lossless transformation is interposed between the two networks shown. It follows that: The eigenvalues of the characteristic noise matrix N are invariant to a lossless transformation that preserves the number of terminal pairs. At this point appHcation of our results to two familiar examples of networks helps to establish further significance for the characteristic-noise matrix and its eigenvalues.
A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds (Quaderno 19-1984, Univ. of Milan) by Magri F., Morosi C.