Complex Eigenvalues Research Articles

What is the resonant state in open quantum systems?

The article reviews the theory of open quantum system from a perspective of the non-Hermiticity that emerges from the environment with an infinite number of degrees of freedom. The non-Hermiticity produces resonant states with complex eigenvalues, resulting in peak structures in scattering amplitudes and transport coefficients. After introducing the definition of resonant states with complex eigenvalues, we answer typical questions regarding the non-Hermiticity of open quantum systems. What is the physical meaning of the complex eigenmomenta and eigenenergies? How and why do the resonant states break the time-reversal symmetry that the system observes? Can we make the probabilistic interpretation of the resonant states with diverging wave functions? What is the physical meaning of the divergence of the wave functions? We also present an alternative way of finding resonant states, namely the Feshbach formalism, in which we eliminate the infinite number of the environmental degrees of freedom. In this formalism, we attribute the non-Hermiticity to the introduction of the retarded and advanced Green’s functions.

Estimation of gassy sediment parameters in shallow water using shipping noise signals in frequency domain near cut-off

Modal structure of shipping noise in a shallow water waveguide is studied experimentally and theoretically. Experiments were carried out in the Sea of Galilee (Lake Kinneret, Israel) having maximal depth of 40.5 m and remarkable concentration of methane bubbles in the upper sedimentary layer. Receiving system consisted of 27-m vertical line arrays (VLAs) with ten hydrophones at each. They spanned a lower part of the waveguide with interval ∼3 m. The VLAs were fixed in the centre of the lake with the distance ∼40 m between them. R/V “Hermona” was moving along a straight line connecting VLAs at the range of up to 1 km from the arrays. Modal filtration technique on two synchronized VLAs with a phase correction was proposed to extract complex eigenvalues of each mode, including modal attenuation, which strongly increase in the proximity of cut-off frequency for all normal modes. Based on frequency dependences of attenuation coefficients of lower modes, sound speed in gas-saturated bottom layer was estimated. For application of this technique, there is no need to know exact coordinate and speed of noise source, which seems to us to be an advantage of the method. [Work was supported by RFBR, Grant No. 20-05-00119.]

Exceptional topological insulators

We introduce the exceptional topological insulator (ETI), a non-Hermitian topological state of matter that features exotic non-Hermitian surface states which can only exist within the three-dimensional topological bulk embedding. We show how this phase can evolve from a Weyl semimetal or Hermitian three-dimensional topological insulator close to criticality when quasiparticles acquire a finite lifetime. The ETI does not require any symmetry to be stabilized. It is characterized by a bulk energy point gap, and exhibits robust surface states that cover the bulk gap as a single sheet of complex eigenvalues or with a single exceptional point. The ETI can be induced universally in gapless solid-state systems, thereby setting a paradigm for non-Hermitian topological matter.

Improvement in Drum-Brake Squeal Noise Prediction by Including the Effect of Brake-Liner Wear

<div class="section abstract"><div class="htmlview paragraph">Drum brake squeal noise has been under investigation by automotive companies for decades due to consistent customer complaints and high warranty costs. Unstable natural frequencies of a brake system are closely linked with brake squeal noise. Complex eigenvalues are widely used to predict unstable frequencies of brake systems. Unstable frequencies are very sensitive to various system parameters such as vehicle speed, coefficient of friction, contact pressure distribution at the liner-drum interface, brake pressure, etc.</div><div class="htmlview paragraph">In this paper, the author has focused on the brake pad wear effects on pressure distribution and eventually on the squeal noise propensity of a commercial vehicle drum-brake system. Wear at brake pads interface is simulated using simplified wear rate formula followed by complex eigenvalue analysis. Predicted unstable frequencies considering the effect of pad wear are found closer to the frequencies of measured noise in physical testing.</div><div class="htmlview paragraph">After successful prediction of unstable frequencies, another challenge in understanding and solving brake squeal issues is to be able to generate insights into this complex phenomenon. Analytical techniques like component contribution factor (CCF) and component mode contribution factor (CMCF) are very useful in this task.</div><div class="htmlview paragraph">In this paper, the effect of brake-liner wear on the prediction accuracy of squeal noise is discussed with the help of a live case. Subsequently how to make effective use of techniques like CCF and CMCF to identify the root cause of the problem is also discussed.</div></div>

Generation of test matrices with specified eigenvalues using floating-point arithmetic

This paper concerns test matrices for numerical linear algebra using an error-free transformation of floating-point arithmetic. For specified eigenvalues given by a user, we propose methods of generating a matrix whose eigenvalues are exactly known based on, for example, Schur or Jordan normal form and a block diagonal form. It is also possible to produce a real matrix with specified complex eigenvalues. Such test matrices with exactly known eigenvalues are useful for numerical algorithms in checking the accuracy of computed results. In particular, exact errors of eigenvalues can be monitored. To generate test matrices, we first propose an error-free transformation for the product of three matrices YSX. We approximate S by {S^{prime }} to compute {YS^{prime }X} without a rounding error. Next, the error-free transformation is applied to the generation of test matrices with exactly known eigenvalues. Note that the exactly known eigenvalues of the constructed matrix may differ from the anticipated given eigenvalues. Finally, numerical examples are introduced in checking the accuracy of numerical computations for symmetric and unsymmetric eigenvalue problems.

Light duty automotive drum brake squeal analysis using the finite-element method

The method presented in this work intends to analyze drum brake design parameters of a light duty automotive drum brake system. The main objective of this work is to correlate brake materials and unstability parameters to identify which condition will effectively reduce squeal propensity. The methodology involves (a) the finite-element method of the brake components, namely, drum, shoes, and frictional linings, (b) static calculations to get a pre-stress state around which (c) is computed the complex eigenvalues of the system. Hence, positive real parts indicate dynamic instabilities which are explored by varying parameters, namely, the modulus of elasticity of the materials and the friction coefficient at the contact of the shoes with the drum. According to calculations, it was observed that there exist a given range of values for Young’s modulus and friction coefficient that are favorable to reduce drum brake squeal occurrence. In addition, the method proposed delivered results that match with brake squeal literature.

A New Method for Landmark-Based Studies of the Dynamic Stability of Growth, with Implications for Evolutionary Analyses

A matrix manipulation new to the quantitative study of develomental stability reveals unexpected morphometric patterns in a classic data set of landmark-based calvarial growth. There are implications for evolutionary studies. Among organismal biology’s fundamental postulates is the assumption that most aspects of any higher animal’s growth trajectories are dynamically stable, resilient against the types of small but functionally pertinent transient perturbations that may have originated in genotype, morphogenesis, or ecophenotypy. We need an operationalization of this axiom for landmark data sets arising from longitudinal data designs. The present paper introduces a multivariate approach toward that goal: a method for identification and interpretation of patterns of dynamical stability in longitudinally collected landmark data. The new method is based in an application of eigenanalysis unfamiliar to most organismal biologists: analysis of a covariance matrix of Boas coordinates (Procrustes coordinates without the size standardization) against their changes over time. These eigenanalyses may yield complex eigenvalues and eigenvectors (terms involving i=sqrt{-1}); the paper carefully explains how these are to be scattered, gridded, and interpreted by their real and imaginary canonical vectors. For the Vilmann neurocranial octagons, the classic morphometric data set used as the running example here, there result new empirical findings that offer a pattern analysis of the ways perturbations of growth are attenuated or otherwise modified over the course of developmental time. The main finding, dominance of a generalized version of dynamical stability (negative autoregressions, as announced by the negative real parts of their eigenvalues, often combined with shearing and rotation in a helpful canonical plane), is surprising in its strength and consistency. A closing discussion explores some implications of this novel pattern analysis of growth regulation. It differs in many respects from the usual way covariance matrices are wielded in geometric morphometrics, differences relevant to a variety of study designs for comparisons of development across species.

Joint image clustering and feature selection with auto-adjoined learning for high-dimensional data

Due to the rapid development of modern multimedia techniques, high-dimensional image data are frequently encountered in many image analysis communities, such as clustering and feature learning. K-means (KM) is one of the widely-used and efficient tools for clustering high-dimensional data. However, as the commonly contained irrelevant features or noise, conventional KMs suffer from degraded performance for high-dimensional data. Recent studies try to overcome this problem by combining KMs with subspace learning. Nevertheless, they usually depend on complex eigenvalue decomposition, which needs expensive computation resources. Besides, their clustering models also ignore the local manifold structure among data, failing to utilize the underlying adjacent information. Two points are critical for clustering high-dimensional image data: efficient feature selecting and clear adjacency exploring. Based on the above consideration, we propose an auto-adjoined subspace clustering. Concretely, to efficiently locate the redundant features, we impose an extremely sparse feature selection matrix into KM, which is easy to be optimized. Besides, to accurately encode the local adjacency among data without the influence of noise, we propose to automatically assign the connectivity of each sample in the low-dimensional feature space. Compared with several state-of-the-art clustering methods, the proposed method constantly improves the clustering performance on six publicly available benchmark image datasets, demonstrating the effectiveness of our method.

Concentrated matrix exponential distributions with real eigenvalues

AbstractConcentrated random variables are frequently used in representing deterministic delays in stochastic models. The squared coefficient of variation ($\mathrm {SCV}$) of the most concentrated phase-type distribution of order $N$ is $1/N$. To further reduce the $\mathrm {SCV}$, concentrated matrix exponential (CME) distributions with complex eigenvalues were investigated recently. It was obtained that the $\mathrm {SCV}$ of an order $N$ CME distribution can be less than $n^{-2.1}$ for odd $N=2n+1$ orders, and the matrix exponential distribution, which exhibits such a low $\mathrm {SCV}$ has complex eigenvalues. In this paper, we consider CME distributions with real eigenvalues (CME-R). We present efficient numerical methods for identifying a CME-R distribution with smallest SCV for a given order $n$. Our investigations show that the $\mathrm {SCV}$ of the most concentrated CME-R of order $N=2n+1$ is less than $n^{-1.85}$. We also discuss how CME-R can be used for numerical inverse Laplace transformation, which is beneficial when the Laplace transform function is impossible to evaluate at complex points.

Dynamic response and stability of a flexible foil with special emphasis on the flutter mechanism via the reduced-order model

The objective of this paper is to study the dynamic response and stability of a flexible foil with emphasis on flutter mechanism. The hybrid coupled fluid-structure algorithm is applied to investigate the effect of solid-to-fluid mass ratio on the dynamic response and stability of a 2-D flexible foil. The results show that, with increase of the solid-to-fluid mass ratio, the failure will be triggered in turn, such as the static divergence, the dynamic divergence, and the flutter divergence. The predicted divergence velocity boundary, increasing with the increase of solid-to-fluid mass ratio, is in accordance with the experimental results. The reduced-order model (ROM) is applied to investigate the flutter mechanism. The flutter boundary estimated by the ROM-based FSI model shows good agreement with the experimental and numerical results. The results obtained from the complex eigenvalue analysis show that the instability depends on the fluid mode and the structural mode, which consist of plunging mode and pitching mode. Flutter is triggered by the unstable pitching mode, meanwhile the fluid mode and plunging mode are stable.

Multi-fold binary Darboux transformation and mixed solitons of a three-component Gross-Pitaevskii system in the spinor Bose-Einstein condensate

The Bose-Einstein-condensation applications give rise to the superfluidity in the liquid helium and superconductivity in the metals. In this paper, we work on a three-component Gross-Pitaevskii system, which describes the matter waves in an spin-1 spinor Bose-Einstein condensate. We construct a multi-fold binary Darboux transformation with the zero seed solutions to describe the three vertical spin projection of the spin-1 spinor BEC, which is different from all the existing Darboux-type ones for the same system, and derive three types of the exponential-and-rational mixed soliton solutions associated with two conjugate complex eigenvalues. For such mixed solitons, we give their asymptotic expressions, indicating that they consist of the Ieda-Miyakawa-Wadati (IMW)-polar-state or IMW-ferromagnetic solitons but possess the time-dependent velocities. Asymptotically and graphically, interaction mechanisms between the mixed and exponential solitons are classified in six cases, and we exhibit the inelastic and elastic interactions through calculating the modifications of the polarization matrices and phase shifts for the two interacting solitons. We find that both the IMW-polar-state solitons, including the mixed and exponential solitons, can not alter the other soliton’s intensity distribution during the interaction, while the mixed or exponential soliton in the IMW-ferromagnetic state does.

On eigenvalues of a matrix arising in energy-preserving/dissipative continuous-stage Runge-Kutta methods

Abstract In this short note, we define an s × s matrix Ks constructed from the Hilbert matrix H s = ( 1 i + j - 1 ) i , j = 1 s {H_s} = \left( {{1 \over {i + j - 1}}} \right)_{i,j = 1}^s and prove that it has at least one pair of complex eigenvalues when s ≥ 2. Ks is a matrix related to the AVF collocation method, which is an energy-preserving/dissipative numerical method for ordinary differential equations, and our result gives a matrix-theoretical proof that the method does not have large-grain parallelism when its order is larger than or equal to 4.

Complex TE-modes in Slotted Waveguides of Arbitrary Cross Section

Implementation of the rigorous Analytical Regularization Method is presented for plane wave scattering by arbitrary two-dimensional cavities with longitudinal slit. The general solution of the Neumann boundary value problem for Helmholtz equation is obtained in a form of two coupled infinite systems of linear algebraic equations of the second kind. Compactness of the matrix operator provides efficient numerical solution of the joint system by truncation method. Fast convergence of the truncated system of linear algebraic equations makes this method an effective instrument for accurate analysis of spectral and diffraction problems. Considering the homogeneous version of equations and finding the complex roots of their determinants, we calculate the complex eigenvalues of complex oscillations. Employing spline interpolation and mollifier function the problem geometry is generalized.

ON FRACTIONAL ORDER MAPS AND THEIR SYNCHRONIZATION

We study the stability of linear fractional order maps. We show that in the stable region, the evolution is described by Mittag-Leffler functions and a well-defined effective Lyapunov exponent can be obtained in these cases. For one-dimensional systems, this exponent can be related to the corresponding fractional differential equation. A fractional equivalent of map [Formula: see text] is stable for [Formula: see text] where [Formula: see text] is a fractional order parameter and [Formula: see text]. For coupled linear fractional maps, we can obtain ‘normal modes’ and reduce the evolution to an effective one-dimensional system. If the coefficient matrix has real eigenvalues, the stability of the coupled system is dictated by the stability of effective one-dimensional normal modes. If the coefficient matrix has complex eigenvalues, we obtain a much richer picture. However, in the stable region, evolution is dictated by a complex effective Lyapunov exponent. For larger [Formula: see text], the effective Lyapunov exponent is determined by modulus of eigenvalues. We extend these studies to fixed points of fractional nonlinear maps.

Basitleştirilmiş Tekerlek/Ray Temas Modelinin ve Kalker Doğrusal Teorisinin Demiryolu Taşıtı Sinüs Hareketi Kararlılığı Açısından Değerlendirilmesi

This paper investigates the performance of a simplified wheel/rail contact modeling approach that can be implemented to dynamics analysis of railway vehicles. The scope is limited to a two-axle bogie system and the problem is investigated from the hunting instability perspective. Accordingly, a mathematical model of a two-axle bogie with elastically connected wheelsets is developed. The governing equations of the proposed four degree of freedom mathematical model are obtained. These equations are then first solved analytically by considering a simplified approach for the interfacial forces at the wheel/rail contact interface. Second, the contact forces are calculated by using Kalker’s linear theory, and the corresponding nonlinear governing equations are solved numerically. Based on the results, it is observed that the analytical and numerical solutions show a good match. Hence, it is concluded that the simplified wheel/rail contact modeling approach can represent the dynamics of the system successfully, especially when the system is in a stable state for hunting behavior. Third, the corresponding eigenvalue problem is formulated from the linear equations, and complex eigenvalues are calculated over the operational speed range of the bogie. It is observed that the natural frequencies of the system are speed-dependent, and either one or both of the wheelsets may become unstable at certain parameter sets. Finally, stability maps are generated for several parameters, and a better understanding of the effects of system parameters on the stability of two-axle bogie is obtained.

The effect of the friction block installation direction on the tribological behavior and vibrational response of the high-speed train brake interface

In this study, to investigate the effect of the hexagon friction block installation direction on the surface tribology and friction-induced vibration and noise (FIVN), tribological tests were carried out on a self-manufactured high-speed train brake dynamometer under different installation directions. Accordingly, a finite element (FE) model with material parameters and boundary conditions coinciding with the test rig was used to conduct complex eigenvalue analysis (CEA), and the Archard wear formula was employed to simultaneously study the wear and vibration. The results demonstrate that the installation direction has little effect on the frequency and mode shape of unstable modes, but the FIVN intensity, the phase-space characteristics of the friction noise, and the damping ratio of the brake system are changed. The contact stress and wear are found to be mainly distributed in the leading edge and the inner area of the friction block, leading to the severe eccentric wear phenomenon, and the friction heat is mainly concentrated in this area and the inner region of the disc friction area. The degree of uneven wear of the block is found to vary with the change of its installation direction, which also causes differences in the amount of wear debris flowing in the brake interface and the damage to the interface. Under the joint action of wear debris, friction heat, and contact stress, the brake interfaces of the brake systems with different block installation directions are found to exhibit different tribological behaviors, which affects the intensity of FIVN and the phase-space characteristics of friction noise. Therefore, it is necessary to consider the influence of the block installation direction on the tribological and dynamic behaviors of the brake interface of high-speed trains when designing the brake pad, which can aid in the development of a brake pad with excellent interface tribological behavior and friendly FIVN characteristics.

New universality classes of the non-Hermitian Dirac operator in QCD-like theories

In non-Hermitian random matrix theory there are three universality classes for local spectral correlations: the Ginibre class and the nonstandard classes ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${\mathrm{AII}}^{\ifmmode\dagger\else\textdagger\fi{}}$. We show that the continuum Dirac operator in two-color QCD coupled to a chiral U(1) gauge field or an imaginary chiral chemical potential falls in class ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ (${\mathrm{AII}}^{\ifmmode\dagger\else\textdagger\fi{}}$) for fermions in pseudoreal (real) representations of SU(2). We introduce the corresponding chiral random matrix theories and verify our predictions in lattice simulations with staggered fermions, for which the correspondence between representation and universality class is reversed. Specifically, we compute the complex eigenvalue spacing ratios introduced recently. We also derive novel spectral sum rules.

Prediction of Squeal Instabilities of a Finite Element Model Automotive Brake With Uncertain Structural and Environmental Parameters With a Hybrid Surrogate Model

Abstract This study focuses on the prediction of the stability behavior of an industrial automotive brake system under structural and environmental uncertainties. Uncertainties are modeled with a random distribution or an interval and are propagated with a hybrid surrogate model associating polynomial chaos and kriging. The objective is to create a surrogate model of each eigenvalue computed with the complex eigenvalue analysis (CEA). As the modes can be tracked only when unstable, the effective size of the training sets can become extremely small. Despite this limitation, it is shown the hybrid meta-model is still able to predict the stability of the brake system. Moreover, the hybrid meta-model gives a direct access to the mean and variance of the eigenvalues with respect to the design parameters without any additional Monte Carlo simulations (MCS). By considering different probability density function for the friction coefficient, it is shown it has a high influence on the stability and the latter should be accurately estimated.

Directional population control beyond the exceptional point in a non-Hermitian system

In the context of dynamical control, non-Hermitian systems offer unique opportunities and challenges. In the simplest case of two interacting levels with gain and/or loss, the existence of an exceptional point (EP) of degeneracy in the chiral complex eigenvalue landscape fundamentally changes how the system responds to adiabatic changes in the coupling $\ensuremath{\gamma}$ and energy separation $\ensuremath{\delta}$ of the bare levels. In particular, previous studies have shown that selective population transfer can be achieved by adiabatically steering the system around closed paths that encircle the EP with the proper helicity in the $\ensuremath{\gamma}$ vs $\ensuremath{\delta}$ parameter space. Here we show that efficient state-selective population transfer can, in some cases, be achieved even with control loops that do not enclose the EP.

Solitons to the derivative nonlinear Schrödinger equation: Double Wronskians and reductions

In this paper, we derive solutions to the derivative nonlinear Schrödinger equation, which are associated to real and complex discrete eigenvalues of the Kaup–Newell spectral problem. These solutions are obtained by investigating double Wronskian solutions of the coupled Kaup–Newell equations and their reductions by means of bilinear method and a reduction technique. The reduced equations include the derivative nonlinear Schrödinger equation and its nonlocal version. Some obtained solutions allow not only periodic behavior, but also solitons on periodic background. Dynamics are illustrated.