Parts Of Eigenvalues Research Articles

On some classes of solutions of two-dimensional multifrequency quasilinear differential systems

where the entries of the matrix P = (pjk(t, e))j,k=1,2 belong to S, the components of the vector f = col (f1, f2) belong to F , and x = col (x1, x2) ∈ D ⊂ C. The eigenvalues of P have the form ±iω(t, e) (ω ∈ R). We analyze the existence of particular solutions of system (1) in the class F . The corresponding result for single-frequency systems was obtained in [3]. A multifrequency system was also considered in [5], but only under the assumption that the real parts of eigenvalues of the matrix P are bounded away from zero. We consider system (1) under the following assumptions. A. The inequality infG |p12| > 0 is valid.

Optimization of Piezoactuator Configuration in Control of Flexible Structures

In order to find the optimum dimensions and locations of piezoactuators in the active control systems, a novel procedure for simultaneous dimension and location optimization of piezoactuators for active control of flexible structures is introduced. Three optimization criteria are investigated and compared which leads to an optimal configuration of the actuators particularly relevant when the weight and geometry limitations are important. In the first criterion we seek a configuration of the piezoactuator which maximizes the overall damping ratio of the closed loop system. Minimization of the overall real part of eigenvalues of the controlled system is the second criterion. For the third criterion, we seek the minimum LQR cost function over possible dimensions and locations. The optimization indicates that the best configuration of piezoactuators in active control systems is the shortest and widest.

Radiative relaxation time of quasinormal optical modes in small dielectric particles

Abstract Radiative relaxation times of optical states in small dielectric particles of arbitrary shape are calculated in the quasistatic limit up to the second order of perturbation theory, assuming that the polarization distribution for the optical states is known. The concept of antisymmetrical optical states, previously developed for a system of discrete dipoles, is generalized for the case of a bulk dielectric particle. We use the integral form of Maxwell's equations to obtain a general expansion of solutions for the polarization function inside a particle in terms of eigenfunctions of the integral interaction operator. Then we calculate imaginary parts of corresponding eigenvalues, which determine the radiative relaxation times of optical excitations, treating the non-Hermitian part of the interaction operator as a perturbation. The imaginary parts of eigenvalues are expanded in terms of total multipole moments of corresponding eigenmodes. Particles with special properties of symmetry can possess pol...

On the complex spectra of second order elliptic systems

A basic problem in the theory of partial differential equations is that of describing the spectra of elliptic operators. If an operator is self-adjoint or symmetric in the appropriate sense, the spectrum is real; but general elliptic operators may have complex spectrum. The object of this article is to study the complex spectra of second order elliptic systems. If the elliptic system arises from linearization about a steady state of a reaction-diffusion system, information about the signs of the real parts of complex eigenvalues is relevant to questions of stability and bifurcation. We give bounds for the real parts of eigenvalues which prevent them from changing sign, and discuss some examples where complex eigenvalues may cross the imaginary axis as a parameter in the system varies. We also discuss briefly the extent to which the theory of positive operators can be used to give bounds for complex eigenvalues. For the case of a single second order elliptic operator of general form on a bounded domain, the fundamental bounds on the spectrum were obtained by Protter and Weinberger in [17]. They considered the problem n n (1.1 ) Cu == L a;j{x)uZiZ; + L b;{x)uz; + c{x)u = Am{x)u. ';,;=1 ;=1 in 0, u = ° on ao, with (a;j) positive definite and symmetric, c ~ 0, m > 0, and ° ~Rn bounded. They showed that for any smooth function w that is positive on n and any eigenvalue A,

On convection in a fluid filling the cavity of a moving solid body

The problem of simultaneous motion about a fixed point O of a solid body and unevenly heated viscous incompressible fluid completely filling a finite cavity of the body is considered in linear formulation. The center of mass of the system body plus fluid in the state of mechanical equilibrium is assumed to coincide with point O. The theorem on solvability of the Cauchy problem for small unsteady equilibrium perturbations is proved, and normal perturbations and the spectrum of the problem arising in the analysis of such perturbations are investigated. It is shown that the whole spectrum consists of normal eigenvalues and lies in some half-band containing the real axis. It is shown that the respective system of root vectors is complete. Properties of the spectrum and the dependence on Rayleigh numbers are investigated. Rayleigh numbers for which the real parts of eigenvalues are positive,i.e. when the generated oscillating normal perturbations are damped in time, are evaluated in the case of the fluid being heated from below and above.

Laser-Induced Rate Processes in Gases: Dynamics, Unimolecular Decay, and Scattered Field

We have recast a linear quantum-mechanical Boltzmann equation, describing the dynamics of a gas of $N$-level atoms absorbing light between an arbitrary pair of levels, into the form of a generalized master equation. The atoms are permitted to undergo both phase-changing and energy-changing collisions with a heat bath of inert molecules. The possibility of unimolecular decay is incorporated into the Boltzmann equation. The solutions of the generalized master equation give an accurate description of the system for short as well as long times and for arbitrary light-field intensity. Eigenvalues obtained from the solution of the $N$-level master equation are shown to have particular significance with regard to the scattered electromagnetic field, with the imaginary parts of the complex eigenvalues providing the location and the real parts of the eigenvalues giving the widths of scattered radiation bands. The dynamics and scattered field for a two-level system is discussed in detail. Two collision parameters giving the frequency of phase-changing and energy-changing collisions are important. For special values of these parameters, our results reduce to scattered field spectra calculated previously. When unimolecular decay is included, one identifies the smallest eigenvalue of the generalized master equation with the macroscopic rate constant. Observation of the scattered field for different incident field intensities would permit separation of radiative and thermal effects in an enhanced reaction rate, without requiring an accurate temperature measurement.