Minimal positive realizations of transfer functions with nonnegative multiple poles

This note concerns a particular case of the minimality problem in positive system theory. A standard result in linear system theory states that any nth-order rational transfer function of a discrete time-invariant linear single-input-single-output (SISO) system admits a realization of order n. In some applications, however, one is restricted to realizations with nonnegative entries (i.e., a positive system), and it is known that this restriction may force the order N of realizations to be strictly larger than n. A general solution to the minimality problem (i.e., determining the smallest possible value of N) is not known. In this note, we consider the case of transfer functions with nonnegative multiple poles, and give sufficient conditions for the existence of positive realizations of order N = n. With the help of our results we also give an improvement of an existing result in positive system theory

Frequencies and resonances around L4L_4 in the elliptic restricted three-body problem

The stability of the Lagrangian point $L_4$ is investigated in the elliptic restricted three-body problem by using Floquet's theory. Stable and unstable domains are determined in the parameter plane of the mass parameter and the eccentricity by computing the characteristic exponents. Frequencies of motion around $L_4$ have been determined both in the stable and unstable domains and fitting functions for the frequencies are derived depending on the mass parameter and the eccentricity. Resonances between the frequencies are studied in the whole parameter plane. It is shown that the 1:1 resonances are not restricted only to single curves but extend to the whole unstable domain. In the unstable domains longer escape times of the test particle from the neighbourhood of $L_4$ are related to certain resonances, but changing the parameters the same resonances may lead to faster escape

<saponifiable and nonsaponifiable soxlet and cold solvent extracts of a number of soils, recent sediment cores from the pacific ocean, and the orgueil carbonaceous meteorite< semiannual progress report, nov. 1964 - may 1965

Saponifiable and nonsaponifiable soxlet and cold solvent extracts of soil, carbonaceous meteorite, and sedimentary rocks studied by thin layer chromatography and spectroscop

Positive decomposition of transfer functions with multiple poles

We present new results on decomposing the transfer function t(z) of a linear, asymptotically stable, discrete-time SISO system as a difference t(z) = t(1)(z) - t(2)(z) of two positive linear systems. We extend the results of [4] to a class of transfer functions t(z) with multiple poles. One of the appearing positive systems is always 1-dimensional, while the other has dimension corresponding to the location and order of the poles of t(z). Recently, in [11], a universal approach was found, providing a decomposition for any asymptotically stable t(z). Our approach here gives lower dimensions than [11] in certain cases but, unfortunately, at present it can only be applied to a relatively small class of transfer functions, and it does not yield a general algorithm

Quantum-classical transition in the Caldeira-Leggett model

The quantum-classical transition in the Caldeira-Leggett model is investigated in the framework of the functional renormalization group method. It is shown that a divergent quadratic term arises in the action due to the heat bath in the model. By removing the divergence with a frequency cutoff we considered the critical behavior of the model. The critical exponents belonging to the susceptibility and the correlation length are determined and their independence of the frequency cutoff and the renormalization scheme is shown.Comment: 8 pages, 4 figure