The Resonance Phenomena Associated with the Time Asymmetry in Non-Hermitian Quantum Mechanics

Resonances are defined as the poles of the scattering matrix. The poles are associated with the complex eigenvalues of the Hamiltonian which are embedded in the lower half of the complex plane. The asymptotes of the corresponding eigenfunctions are exponentially diverged. Therefore, the resonance eigenfunctions are not embedded in the Hermitian domain of the Hamiltonian. The time asymmetric problem is discussed for these types of non-Hermitian Hamiltonians and several solutions of this problem are proposed.     

On the stability of some analytic results arising in the characterization of distributions

Zinger characterization describes a class of probability distributions that have finite moments of all orders. The stability problem is considered by the properties of independence of statistics in this characterization. It was proved by method of Azlarov that order statistics satisfying a certain ε-regression condition also have finite moments of all orders. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

The investigation of the photonic waveguide structure made of silicon carbide by means of the method of the singular integral equations

Here we present the analyses of the open (without a metal screen) ridged and photonic waveguide structures by means of the electrodynamical rigorous method of the Singular Integral Equations (SIE). The waveguides are made of a lossy silicon carbide (SiC) material. We have discovered peculiarities of the dispersion characteristics. We have found the numerical solutions to the complex wave equations. The dispersion characteristics of both waveguide structures are numerically analyzed and compared with each other. It was found that the losses of modes propagating in the waveguide structure with air openings (channels) are smaller than the losses in the waveguide structure without air openings. We came to the conclusion that it is possible to optimize the dispersion characteristics by adding openings of different shapes and sizes into the waveguide structures.     

Truss optimization under stiffness, stability constraints and random loading

An actual design of light-weight structures must evaluate strength, stiffness and stability constraints as well as the nature of external loading. A designed structure must satisfy optimality and safety criterions per prescribed maintenance period. One faces the known difficulties when trying to implement several from the above mentioned requirements into optimization problem for further successful numerical realization. A method to formulate the optimization problem, incorporating all above described criterions, the mathematical model and algorithm to solve it numerically, taking into account the stochastic nature of external loading, are presented for elastic–plastic truss-type structure.