Compact operators without extended eigenvalues

A complex number lambda is called an extended eigenvalue of a bounded linear operator T on a Banach space B if there exists a non-zero bounded linear operator X acting on B such that XT = lambda TX. We show that there are compact quasinilpotent operators on a separable Hilbert space, for which the set of extended eigenvalues is the one-point set {1}.

Eigenvalues of the time-delay matrix in overlapping resonances

We discuss the properties of the lifetime or the time-delay matrix Q(E) for multichannel scattering, which is related to the scattering matrix S(E) by Q = i?S(dS†/dE). For two overlapping resonances occurring at energies E with widths G(? = 1, 2), with an energy-independent background, only two eigenvalues of Q(E) are proved to be different from zero and to show typical avoided-crossing behaviour. These eigenvalues are expressible in terms of the four resonance parameters (E , G) and a parameter representing the strength of the interaction of the resonances. An example of the strong and weak interaction in an overlapping double resonance is presented for the positronium negative ion. When more than two resonances overlap (? = 1, ..., N), no simple representation of each eigenvalue has been found. However, the formula for the trace of the Q-matrix leads to the expression d(E) = -?arctan[(G/2)/(E - E)] + d(E) for the eigenphase sum d(E) and the background eigenphase sum d(E), in agreement with the known form of the state density. The formulae presented in this paper are useful in a parameter fitting of overlapping resonances. © 2006 IOP Publishing Ltd.

A Wavelet Approach to the Selective Computation of Eigenvalues for Small Signal Stability Analysis

This paper proposes a method to assess the small signal stability of a power system network by selective determination of the modal eigenvalues. This uses an accelerating polynomial transform, designed using approximate eigenvalues
obtained from a wavelet approximation. Application to the IEEE 14 bus network model produced computational savings of 20%,over the QR algorithm.

A Wavelet Approach to the Selective Computation of Domninant Eigenvalues

This paper introduces an algorithm that calculates the dominant eigenvalues (in terms of system stability) of a linear model and neglects the exact computation of the non-dominant eigenvalues. The method estimates all of the eigenvalues using wavelet based compression techniques. These estimates are used to find a suitable invariant subspace such that projection by this subspace will provide one containing the eigenvalues of interest. The proposed algorithm is exemplified by application to a power system model.

A variational approach to the relaxation of single domain magnetic particles based on Brown's model

Brown's model for the relaxation of the magnetization of a single domain ferromagnetic particle is considered. This model results in the Fokker-Planck equation of the process. The solution of this equation in the cases of most interest is non- trivial. The probability density of orientations of the magnetization in the Fokker-Planck equation can be expanded in terms of an infinite set of eigenfunctions and their corresponding eigenvalues where these obey a Sturm-Liouville type equation. A variational principle is applied to the solution of this equation in the case of an axially symmetric potential. The first (non-zero) eigenvalue, corresponding to the largest time constant, is considered. From this we obtain two new results. Firstly, an approximate minimising trial function is obtained which allows calculation of a rigorous upper bound. Secondly, a new upper bound formula is derived based on the Euler-Lagrange condition. This leads to very accurate calculation of the eigenvalue but also, interestingly, from this, use of the simplest trial function yields an equivalent result to the correlation time of Coffey et at. and the integral relaxation time of Garanin. (C) 2004 Elsevier B.V. All rights reserved.

The resonance spectrum of the cusp map in the space of analytic functions

We prove that the Frobenius-Perron operator $U$ of the cusp map $F:[-1,1]\to [-1,1]$, $F(x)=1-2 x^{1/2}$ (which is an approximation of the Poincare section of the Lorenz attractor) has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1. We also prove that for any $q\in (0,1)$ the spectrum of $U$ in the Hardy space in the disk $\{z\in C:|z-q|

Resonances of the cusp family

We study a family of chaotic maps with limit cases-the tent map and the cusp map (the cusp family). We discuss the spectral properties of the corresponding Frobenius-Perron operator in different function spaces including spaces of analytical functions and study numerically the eigenvalues and eigenfunctions.

Disturbance attenuation in linear systems via dynamical compensators: A parametric eigenstructure assignment approach

A simple approach is proposed for disturbance attenuation in multivariable linear systems via dynamical output compensators based on complete parametric eigenstructure assignment. The basic idea is to minimise the H-2 norm of the disturbance-output transfer function using the design freedom provided by eigenstructure assignment. For robustness, the closed-loop system is restricted to be nondefective. Besides the design parameters, the closed-loop eigenvalues are also optimised within desired regions on the left-half complex plane to ensure both closed-loop stability and dynamical performance. With the proposed approach, additional closed-loop specifications can be easily achieved. As a demonstration, robust pole assignment, in the sense that the closed-loop eigenvalues are as insensitive as possible to open-loop system parameter perturbations, is treated. Application of the proposed approach to robust control of a magnetic bearing with a pair of opposing electromagnets and a rigid rotor is discussed.

Simulating Coxian Phase-type Distributions for Patient Survival

Coxian phase-type distributions are a special type of Markov model that can be used to represent survival times in terms of phases through which an individual may progress until they eventually leave the system completely. Previous research has considered the Coxian phase-type distribution to be ideal in representing patient survival in hospital. However, problems exist in fitting the distributions. This paper investigates the problems that arise with the fitting process by simulating various Coxian phase-type models for the representation of patient survival and examining the estimated parameter values and eigenvalues obtained. The results indicate that numerical methods previously used for fitting the model parameters do not always converge. An alternative technique is therefore considered. All methods are influenced by the choice of initial parameter values. The investigation uses a data set of 1439 elderly patients and models their survival time, the length of time they spend in a UK hospital.

Moving window kernel PCA for adaptive monitoring of nonlinear processes

This paper discusses the monitoring of complex nonlinear and time-varying processes. Kernel principal component analysis (KPCA) has gained significant attention as a monitoring tool for nonlinear systems in recent years but relies on a fixed model that cannot be employed for time-varying systems. The contribution of this article is the development of a numerically efficient and memory saving moving window KPCA (MWKPCA) monitoring approach. The proposed technique incorporates an up- and downdating procedure to adapt (i) the data mean and covariance matrix in the feature space and (ii) approximates the eigenvalues and eigenvectors of the Gram matrix. The article shows that the proposed MWKPCA algorithm has a computation complexity of O(N2), whilst batch techniques, e.g. the Lanczos method, are of O(N3). Including the adaptation of the number of retained components and an l-step ahead application of the MWKPCA monitoring model, the paper finally demonstrates the utility of the proposed technique using a simulated nonlinear time-varying system and recorded data from an industrial distillation column.

Dynamics of Gompertzian tumour growth under environmental fluctuations

We investigate the effect of correlated additive and multiplicative Gaussian white noise oil the Gompertzian growth of tumours. Our results are obtained by Solving numerically the time-dependent Fokker-Planck equation (FPE) associated with the stochastic dynamics. In Our numerical approach we have adopted B-spline functions as a truncated basis to expand the approximated eigenfunctions. The eigenfunctions and eigenvalues obtained using this method are used to derive approximate solutions of the dynamics under Study. We perform simulations to analyze various aspects, of the probability distribution. of the tumour cell populations in the transient- and steady-state regimes. More precisely, we are concerned mainly with the behaviour of the relaxation time (tau) to the steady-state distribution as a function of (i) of the correlation strength (lambda) between the additive noise and the multiplicative noise and (ii) as a function of the multiplicative noise intensity (D) and additive noise intensity (alpha). It is observed that both the correlation strength and the intensities of additive and multiplicative noise, affect the relaxation time.

On the Performance of Eigenvalue-Based Cooperative Spectrum Sensing for Cognitive Radio

In this paper, the distribution of the ratio of extreme eigenvalues of a complex Wishart matrix is studied in order to calculate the exact decision threshold as a function of the desired probability of false alarm for the maximum-minimum eigenvalue (MME) detector. In contrast to the asymptotic analysis reported in the literature, we consider a finite number of cooperative receivers and a finite number of samples and derive the exact decision threshold for the probability of false alarm. The proposed exact formulation is further reduced to the case of two receiver-based cooperative spectrum sensing. In addition, an approximate closed-form formula of the exact threshold is derived in terms of a desired probability of false alarm for a special case having equal number of receive antennas and signal samples. Finally, the derived analytical exact decision thresholds are verified with Monte-Carlo simulations. We show that the probability of detection performance using the proposed exact decision thresholds achieves significant performance gains compared to the performance of the asymptotic decision threshold.

Entanglement Spectrum, Critical Exponents and Order Parameters in Quantum Spin Chains

We investigate the entanglement spectrum near criticality in finite quantum spin chains. Using finite size scaling we show that when approaching a quantum phase transition, the Schmidt gap, i.e., the difference between the two largest eigenvalues of the reduced density matrix ?1, ?2, signals the critical point and scales with universal critical exponents related to the relevant operators of the corresponding perturbed conformal field theory describing the critical point. Such scaling behavior allows us to identify explicitly the Schmidt gap as a local order parameter.