Analytic perturbation of Sylvester and Lyapunov matrix equations
We consider an analytic perturbation of the Sylvester matrix equation. Mainly we are interested in the singular case, that is, when the null space of the unperturbed Sylvester operator is not trivial, but the perturbed equation has a unique solution. In this case, the solution of the perturbed equation can be given in terms of a Laurent series. We provide a necessary and sufficient condition for the existence of a Laurent series with a first order pole. An efficient recursive procedure for the calculation of the Laurent series' coefficients is given. Finally, we show that in the particular, but practically important case of semisimple eigenvalues, the recursive procedure can be written in a compact matrix form.
Mathematical Foundations of the State Lumping of Large Systems
Introduction. 1. Classes of Linear Operators. 2. Semigroups of Operators and Markov Processes. 3. Perturbations of Invertibly Reducible Operators. 4. Singular Perturbations of Holomorphic Semigroups. 5. Asymptotic Expansions and Limit Theorems. 6. Asymptotic Phase Lumping of Markov and Semi-Markov Processes. 7. Applications of the Theory of Singularly Perturbed Semigroups. References. Subject Index.
The Invariance of the Vacuum is the Invariance of the World
The following theorem is demonstrated: If the vacuum is invariant under the group generated by the space integral of the time component of a local vector current, then the Hamiltonian is invariant also. A similar theorem holds for the group generated by the space integral of a space component of a local axial-vector current.