Applied Koopman Operator Theory for Power Systems Technology

Koopman operator is a composition operator defined for a dynamical system described by nonlinear differential or difference equation. Although the original system is nonlinear and evolves on a finite-dimensional state space, the Koopman operator itself is linear but infinite-dimensional (evolves on a function space). This linear operator captures the full information of the dynamics described by the original nonlinear system. In particular, spectral properties of the Koopman operator play a crucial role in analyzing the original system. In the first part of this paper, we review the so-called Koopman operator theory for nonlinear dynamical systems, with emphasis on modal decomposition and computation that are direct to wide applications. Then, in the second part, we present a series of applications of the Koopman operator theory to power systems technology. The applications are established as data-centric methods, namely, how to use massive quantities of data obtained numerically and experimentally, through spectral analysis of the Koopman operator: coherency identification of swings in coupled synchronous generators, precursor diagnostic of instabilities in the coupled swing dynamics, and stability assessment of power systems without any use of mathematical models. Future problems of this research direction are identified in the last concluding part of this paper.Comment: 31 pages, 11 figure

Matrix geometries and fuzzy spaces as finite spectral triples

A class of real spectral triples that are similar in structure to a Riemannian manifold but have a finite-dimensional Hilbert space is defined and investigated, determining a general form for the Dirac operator. Examples include fuzzy spaces defined as real spectral triples. Fuzzy 2-spheres are investigated in detail, and it is shown that the fuzzy analogues correspond to two spinor fields on the commutative sphere. In some cases it is necessary to add a mass mixing matrix to the commutative Dirac operator to get a precise agreement for the eigenvalues.Comment: 39 pages, final versio

Why is Quantum Physics Based on Complex Numbers?

The modern quantum theory is based on the assumption that quantum states are represented by elements of a complex Hilbert space. It is expected that in future quantum theory the number field will be not postulated but derived from more general principles. We consider the choice of the number field in quantum theory based on a Galois field (GFQT) discussed in our previous publications. Since any Galois field is not algebraically closed, in the general case there is no guarantee that even a Hermitian operator necessarily has eigenvalues. We assume that the symmetry algebra is the Galois field analog of the de Sitter algebra so(1,4) and consider spinless irreducible representations of this algebra. It is shown that the Galois field analog of complex numbers is the minimal extension of the residue field modulo $p$ for which the representations are fully decomposable.Comment: Latex, 27 pages, no figures, minor correction