The theorem that was none - I. Early history
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The early history of the Vogt-Russell theorem is retraced following its route starting at the realization of a correlation between mass and luminosity of binary and pulsating stars, through the embossing of this observation into a theorem, and finally to the emerging first signs of its failure to serve as a theorem in the strict mathematical sense of the word.Keywords:
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In the literature, three different types of nonnegative realization problem have been addressed: the nonnegative realization problem for strictly proper rational functions [L. Benvenuti and L. Farina, 2004], the nonnegative realization problem for autonomous behaviors [J.W. Nieuwenhuis, 1998], [M.E. Valcher, 2001] and the K-realization problem [H. Maeda and S. Kodama, 1981]. In this paper these problems are compared and mutually related, and some results obtained within the behavioral setting are applied to the solution of the nonnegative realization problem for rational functions.
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The Dyson realization of the quantum group SU(1,1)q is constructed and the transformation matrix, which turns the Dyson realization into the Holstein–Primakoff realization, is given. The coherent state for the Dyson realization is discussed and is found to be the q-deformed Weyl–Heisenberg coherent state.
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