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∗-Algebras in Several Complex Variables
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
The associated cone Λ=Hr+(K) consists of all positive definite matrices. For K = R, C we have N(x) = det x, whereas for quaternions N(x) is related to the Pfaffian determinant. For r = 3 one can also take the (non-associative) division algebra O of octonions (Cayley numbers), yielding the exceptional Jordan algebraℋ3(O) of dimension 27.
The Basic Formalism of Field Theory
Published in A.N. Vasiliev, Patricia A. Millard, Functional Methods in Quantum Field Theory and Statistical Physics, 2019
A.N. Vasiliev, Patricia A. Millard
Generalizing (1.157), we can express the Pfaffian of a two-by-two matrix in terms of the determinant of K up to a sign and obtain () ∫DψDψ+exp[ψ+Kψ+ψ+A+A+ψ]==εdet(K2π)exp(−A+K−1A).Equations (1.145) and (1.159) can be combined to form the single equation () ∫Dφexp[−12φKφ+φA]=εdet(K2π)−ϰ/2exp[ϰ2AK−1A].
Minimal realisations of odd transfer functions for first-degree nD systems
Published in International Journal of Control, 2020
This paper is organised as follows. In the next section, some necessary preliminaries are presented; the notion of the skew-symmetric matrix and the calculation of the skew-symmetric determinant by the Pfaffian function are recalled. In Section 3, the formulation of the absolutely minimal realisation problem is shown, and the systems of equations satisfied by the absolutely minimal realisation of the transfer function for the first-degree nD SISO discrete systems are obtained. In Section 4, the conditions for existence and construction of the absolutely minimal realisation for the lack of items and not missing two cases are derived. In Section 5, two numerical examples for 3D and 4D systems are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure. Finally, concluding remarks are made in Section 6.