Journal articles: 'Time eigenvalue'

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VITANCOL, ROBERTO S., and ERIC A. GALAPON. "APPLICATION OF CLENSHAW–CURTIS METHOD IN CONFINED TIME OF ARRIVAL OPERATOR EIGENVALUE PROBLEM." International Journal of Modern Physics C 19, no. 05 (May 2008): 821–44. http://dx.doi.org/10.1142/s0129183108012534.

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The Clenshaw–Curtis method in discretizing a Fredholm integral operator is applied to solving the confined time of arrival operator eigenvalue problem. The accuracy of the method is measured against the known analytic solutions for the noninteracting case, and its performance compared against the well-known Nystrom method. It is found that Clenshaw–Curtis's is superior to Nystrom's. In particular, Nystrom method yields at most five correct decimal places for the eigenvalues and eigenfunctions, while Clenshaw–Curtis yields eigenvalues correct to 16 decimal places and eigenfunctions up to 15 decimal places for the same number of quadrature points. Moreover, Clenshaw–Curtis's accuracy in the eigenvalues is uniform over a determinable range of the computed eigenvalues for a given number of quadrature abscissas. Clenshaw–Curtis is then applied to the harmonic oscillator confined time of arrival operator eigenvalue problem.

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Petrie, Adam, and Xiaopeng Zhao. "Estimating eigenvalues of dynamical systems from time series with applications to predicting cardiac alternans." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2147 (July 4, 2012): 3649–66. http://dx.doi.org/10.1098/rspa.2012.0098.

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The stability of a dynamical system can be indicated by eigenvalues of its underlying mathematical model. However, eigenvalue analysis of a complicated system (e.g. the heart) may be extremely difficult because full models may be intractable or unavailable. We develop data-driven statistical techniques, which are independent of any underlying dynamical model, that use principal components and maximum-likelihood methods to estimate the dominant eigenvalues and their standard errors from the time series of one or a few measurable quantities, e.g. transmembrane voltages in cardiac experiments. The techniques are applied to predicting cardiac alternans that is characterized by an eigenvalue approaching −1. Cardiac alternans signals a vulnerability to ventricular fibrillation, the leading cause of death in the USA.

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Hsu, Jung Chang, and Shao Shu Chu. "An Innovative Eigenvalue Problem Solver by Using Adomian Decomposition Approach." Advanced Materials Research 433-440 (January 2012): 6742–50. http://dx.doi.org/10.4028/www.scientific.net/amr.433-440.6742.

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The paper deals with eigenvalues and normalized eigenfunctions for a Strum-Liouville eigenvalue problem. The technique we have used is based on applying a Adomian decomposition method (ADM) to our eigenvalue problems. Doing some simple mathematical operations on the method, we can obtain ith eigenvalues and eigenfunctions one at a time. The computed results agree well with those analytical results given in the literature.

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Chen, Shinn-Horng, Jyh-Horng Chou, and Liang-An Zheng. "Robust Regional Eigenvalue-Clustering Analysis for Linear Discrete Singular Time-Delay Systems With Structured Parameter Uncertainties." Journal of Dynamic Systems, Measurement, and Control 129, no. 1 (April 27, 2006): 83–90. http://dx.doi.org/10.1115/1.2397156.

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In this paper, the regional eigenvalue-clustering robustness problem of linear discrete singular time-delay systems with structured (elemental) parameter uncertainties is investigated. Under the assumptions that the linear nominal discrete singular time-delay system is regular and causal, and has all its finite eigenvalues lying inside certain specified regions, two new sufficient conditions are proposed to preserve the assumed properties when the structured parameter uncertainties are added into the linear nominal discrete singular time-delay system. When all the finite eigenvalues are just required to locate inside the unit circle, the proposed criteria will become the stability robustness criteria. For the case of eigenvalue clustering in a specified circular region, one proposed sufficient condition is mathematically proved to be less conservative than those reported very recently in the literature. Another new sufficient condition is also proposed for guaranteeing that the linear discrete singular time-delay system with both structured (elemental) and unstructured (norm-bounded) parameter uncertainties holds the properties of regularity, causality, and eigenvalue clustering in a specified region. An example is given to demonstrate the applicability of the proposed sufficient conditions.

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Zhang, Chao, and Shurong Sun. "Eigenvalue Comparisons for Second-Order Linear Equations with Boundary Value Conditions on Time Scales." Journal of Applied Mathematics 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/486230.

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This paper studies the eigenvalue comparisons for second-order linear equations with boundary conditions on time scales. Using results from matrix algebras, the existence and comparison results concerning eigenvalues are obtained.

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Manolis, George D., and Georgios I. Dadoulis. "On the Numerical Treatment of the Temporal Discontinuity Arising from a Time-Varying Point Mass Attachment on a Waveguide." Algorithms 16, no. 1 (January 3, 2023): 26. http://dx.doi.org/10.3390/a16010026.

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A vibrating pylon, modeled as a waveguide, with an attached point mass that is time-varying poses a numerically challenging problem regarding the most efficient way for eigenvalue extraction. The reason is three-fold, starting with a heavy mass attachment that modifies the original eigenvalue problem for the stand-alone pylon, plus the fact that the point attachment results in a Dirac delta function in the mixed-type boundary conditions, and finally the eigenvalue problem becomes time-dependent and must be solved for a sequence of time steps until the time interval of interests is covered. An additional complication is that the eigenvalues are now complex quantities. Following the formulation of the eigenvalue problem as a system of first-order, time-dependent matrix differential equations, two eigenvalue extraction methods are implemented and critically examined, namely the Laguerre and the QR algorithms. The aim of the analysis is to identify the most efficient technique for interpreting time signals registered at a given pylon as a means for detecting damage, a procedure which finds application in structural health monitoring of civil engineering infrastructure.

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Liu, Hao, Ranran Li, and Yingying Ding. "Partial Eigenvalue Assignment for Gyroscopic Second-Order Systems with Time Delay." Mathematics 8, no. 8 (July 27, 2020): 1235. http://dx.doi.org/10.3390/math8081235.

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In this paper, the partial eigenvalue assignment problem of gyroscopic second-order systems with time delay is considered. We propose a multi-step method for solving this problem in which the undesired eigenvalues are moved to desired values and the remaining eigenvalues are required to remain unchanged. Using the matrix vectorization and Hadamard product, we transform this problem into a linear systems of lower order, and analysis the computational costs of our method. Numerical results exhibit the efficiency of our method.

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Li, Meng-lei, Ji-jun Ao, and Hai-yan Zhang. "Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions." Open Mathematics 20, no. 1 (January 1, 2022): 1215–28. http://dx.doi.org/10.1515/math-2022-0507.

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Abstract In this article, we study the eigenvalue dependence of Sturm-Liouville problems on time scales with spectral parameter in the boundary conditions. We obtain that the eigenvalues not only continuously but also smoothly depend on the parameters of the problem. Moreover, the differential expressions of the eigenvalues with respect to the data are given.

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Ali, Syed Sajjad, Chang Liu, Jialong Liu, Minglu Jin, and Jae Moung Kim. "On the Eigenvalue Based Detection for Multiantenna Cognitive Radio System." Mobile Information Systems 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/3848734.

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Eigenvalue based spectrum sensing can make detection by catching correlation features in space and time domains, which can not only reduce the effect of noise uncertainty, but also achieve high detection probability. Hence, the eigenvalue based detection is always a hot topic in spectrum sensing area. However, most existing algorithms only consider part of eigenvalues rather than all the eigenvalues, which does not make full use of correlation of eigenvalues. Motivated by this, this paper focuses on multiantenna system and makes all the eigenvalues weighted for detection. Through the analysis of system model, we transfer the eigenvalue weighting issue to an optimal problem and derive the theoretical expression of detection threshold and probability of false alarm and obtain the close form expression of optimal solution. Finally, we propose new weighting schemes to give promotions of the detection performance. Simulations verify the efficiency of the proposed algorithms.

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Röbenack, Klaus, and Daniel Gerbet. "Full and Partial Eigenvalue Placement for Minimum Norm Static Output Feedback Control." SYSTEM THEORY, CONTROL AND COMPUTING JOURNAL 2, no. 1 (June 30, 2022): 22–33. http://dx.doi.org/10.52846/stccj.2022.2.1.32.

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The controller design for linear time-invariant state space systems seems to be straightforward and well established. This is not true for static output feedback control, which is still a challenging task. This paper deals with controller design based on eigenvalue assignment. We consider the placement of distinct as well as multiple real eigenvalues or complex conjugate pairs. The desired eigenvalue configurations are characterised in terms of algebraic divisibility of the characteristic polynomial of the closed-loop system. We also consider the problem of partial eigenvalue placement, where not all eigenvalues are fixed by feedback. Degrees of freedom in the controller design are used for the minimization of various matrix norms of the feedback gain matrix.

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Filiciotto, F., A. Jinaphanh, and A. Zoia. "SUPER-HISTORY METHODS FOR ADJOINT-WEIGHTED TALLIES IN MONTE CARLO TIME EIGENVALUE CALCULATIONS." EPJ Web of Conferences 247 (2021): 04008. http://dx.doi.org/10.1051/epjconf/202124704008.

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Time eigenvalues emerge in several key applications related to neutron transport problems, including reactor start-up and reactivity measurements. In this context, experimental validation and uncertainty quantification would demand to assess the variation of the dominant time eigenvalue in response to a variation of nuclear data. Recently, we proposed the use of a Generalized Iterated Fission Probability method (G-IFP) to compute adjoint-weighted tallies, such as kinetic parameters, perturbations and sensitivity coefficients, for Monte Carlo time (or alpha) eigenvalue calculations. With the massive use of parallel Monte Carlo calculations, it would be therefore useful to trade the memory burden of the G-IFP method (which is comparable to that of the standard IFP method for k-eigenvalue problems) for computation time and to rely on history-based schemes for such adjoint-weighted tallies. For this purpose, we investigate the use of the super-history method as applied to estimating adjoint-weighted tallies within the α-k power iteration, based on previous work on k-eigenvalue problems. Verification of the algorithms is performed on some simple preliminary tests where analytic solutions exist. In addition, the performances of the proposed method are assessed by comparing the super-history and the G-IFP methods for the same sets of benchmark problems.

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Araújo, José Mário. "Partial eigenvalue assignment in linear time-invariant systems using state-derivative feedback and a left eigenvectors parametrization." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 233, no. 8 (November 7, 2018): 1085–89. http://dx.doi.org/10.1177/0959651818811010.

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In this brief, a novel parametrized state-derivative feedback is given to achieve the well-known partial eigenvalue assignment in linear time-invariant systems. In particular, the parametrized matrix for linear feedback is shown to depend only on the measured (known) left eigenvectors and its corresponding eigenvalues that must be reassigned. The solution is proven to have no spillover, an appreciable feature for the cases in which most of the eigenstructure is unmeasured (unknown). Two numerical examples are given to see that the obtained formulas are valid for partial eigenvalue assignment using only measured information of the eigenstructure.

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Kaczorek, T. "Stability of fractional positive continuous-time linear systems with state matrices in integer and rational powers." Bulletin of the Polish Academy of Sciences Technical Sciences 65, no. 3 (June 27, 2017): 305–11. http://dx.doi.org/10.1515/bpasts-2017-0034.

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AbstractThe stability of fractional standard and positive continuous-time linear systems with state matrices in integer and rational powers is addressed. It is shown that the fractional systems are asymptotically stable if and only if the eigenvalues of the state matrices satisfy some conditions imposed on the phases of the eigenvalues. The fractional standard systems are unstable if the state matrices have at least one positive eigenvalue.

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Leible, Benedikt, Daniel Plabst, and Norbert Hanik. "Back-to-Back Performance of the Full Spectrum Nonlinear Fourier Transform and Its Inverse." Entropy 22, no. 10 (October 6, 2020): 1131. http://dx.doi.org/10.3390/e22101131.

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In this paper, data-transmission using the nonlinear Fourier transform for jointly modulated discrete and continuous spectra is investigated. A recent method for purely discrete eigenvalue removal at the detector is extended to signals with additional continuous spectral support. At first, the eigenvalues are sequentially detected and removed from the jointly modulated received signal. After each successful removal, the time-support of the resulting signal for the next iteration can be narrowed, until all eigenvalues are removed. The resulting truncated signal, ideally containing only continuous spectral components, is then recovered by a standard NFT algorithm. Numerical simulations without a fiber channel show that, for jointly modulated discrete and continuous spectra, the mean-squared error between transmitted and received eigenvalues can be reduced using the eigenvalue removal approach, when compared to state-of-the-art detection methods. Additionally, the computational complexity for detection of both spectral components can be decreased when, by the choice of the modulated eigenvalues, the time-support after each removal step can be reduced. Numerical simulations are also carried out for transmission over a Raman-amplified, lossy SSMF channel. The mutual information is approximated and the eigenvalue removal method is shown to result in achievable rate improvements.

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Gao, Yingjie, Jinhai Zhang, and Zhenxing Yao. "Removing the stability limit of the explicit finite-difference scheme with eigenvalue perturbation." GEOPHYSICS 83, no. 6 (November 1, 2018): A93—A98. http://dx.doi.org/10.1190/geo2018-0447.1.

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The explicit finite-difference scheme is popular for solving the wave equation in the field of seismic exploration due to its simplicity in numerical implementation. However, its maximum time step is strictly restricted by the Courant-Friedrichs-Lewy (CFL) stability limit, which leads to a heavy computational burden in the presence of small-scale structures and high-velocity targets. We remove the CFL stability limit of the explicit finite-difference scheme using the eigenvalue perturbation, which allows us to use a much larger time step beyond the CFL stability limit. For a given time step that is within the CFL stability limit, the eigenvalues of the update matrix would be distributed along the unit circle; otherwise, some eigenvalues would be distributed outside of the unit circle, which introduces unstable phenomena. The eigenvalue perturbation can normalize the unstable eigenvalues and guarantee the stability of the update matrix by using an arbitrary time step. The update matrix can be preprocessed before the numerical simulation, thus retaining the computational efficiency well. We further incorporate the forward time-dispersion transform (FTDT) and the inverse time-dispersion transform (ITDT) to reduce the time-dispersion error caused by using an unusually large time step. Our numerical experiments indicate that the combination of the eigenvalue perturbation, the FTDT method, and the ITDT method can simulate highly accurate waveforms when applying a time step beyond the CFL stability limit. The time step can be extended even toward the Nyquist limit. This means that we could save many iteration steps without suffering from time-dispersion error and stability problems.

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Saracco, P., N. Abrate, M. Burrone, S. Dulla, and P. Ravetto. "STUDY OF THE EIGENVALUE SPECTRA OF THE NEUTRON TRANSPORT PROBLEM IN PN APPROXIMATION." EPJ Web of Conferences 247 (2021): 03018. http://dx.doi.org/10.1051/epjconf/202124703018.

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The study of the steady-state solutions of neutron transport equation requires the introduction of appropriate eigenvalues: this can be done in various different ways by changing each of the operators in the transport equation; such modifications can be physically viewed as a variation of the corresponding macroscopic cross sections only, so making the different (generalized) eigenvalue problems non-equivalent. In this paper the eigenvalue problem associated to the time-dependent problem (α eigenvalue), also in the presence of delayed emissions is evaluated. The properties of associated spectra can give different insight into the physics of the problem.

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Ashokkumar, Chimpalthradi R., George WP York, and Scott F. Gruber. "Proportional–integral–derivative controller family for pole placement." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231, no. 20 (May 20, 2016): 3791–97. http://dx.doi.org/10.1177/0954406216651893.

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In this paper, linear time-invariant square systems are considered. A procedure to design infinitely many proportional–integral–derivative controllers, all of them assigning closed-loop poles (or closed-loop eigenvalues), at desired locations fixed in the open left half plane of the complex plane is presented. The formulation accommodates partial pole placement features. The state-space realization of the linear system incorporated with a proportional–integral–derivative controller boils down to the generalized eigenvalue problem. The generalized eigenvalue-eigenvector constraint is transformed into a system of underdetermined linear homogenous set of equations whose unknowns include proportional–integral–derivative parameters. Hence, the proportional–integral–derivative solution sets are infinitely many for the chosen closed-loop eigenvalues in the eigenvalue-eigenvector constraint. The solution set is also useful to reduce the tracking errors and improve the performance. Three examples are illustrated.

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Xiaodan, Yu, Jia Hongjie, Wang Chengshan, and Jiang Yilang. "A Method to Determine Oscillation Emergence Bifurcation in Time-Delayed LTI System with Single Lag." Journal of Applied Mathematics 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/823937.

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One type of bifurcation named oscillation emergence bifurcation (OEB) found in time-delayed linear time invariant (abbr. LTI) systems is fully studied. The definition of OEB is initially put forward according to the eigenvalue variation. It is revealed that a real eigenvalue splits into a pair of conjugated complex eigenvalues when an OEB occurs, which means the number of the system eigenvalues will increase by one and a new oscillation mode will emerge. Next, a method to determine OEB bifurcation in the time-delayed LTI system with single lag is developed based on Lambert W function. A one-dimensional (1-dim) time-delayed system is firstly employed to explain the mechanism of OEB bifurcation. Then, methods to determine the OEB bifurcation in 1-dim, 2-dim, and high-dimension time-delayed LTI systems are derived. Finally, simulation results validate the correctness and effectiveness of the presented method. Since OEB bifurcation occurs with a new oscillation mode emerging, work of this paper is useful to explore the complex phenomena and the stability of time-delayed dynamic systems.

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Davis, John M., Johnny Henderson, K. Rajendra Prasad, and William Yin. "Solvability of a nonlinear second order conjugate eigenvalue problem on a time scale." Abstract and Applied Analysis 5, no. 2 (2000): 91–99. http://dx.doi.org/10.1155/s108533750000018x.

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We consider the nonlinear second order conjugate eigenvalue problem on a time scale:y ΔΔ(t)+λa(t)f(y(σ(t)))=0,t∈[0,1],y(0)=0=y(σ(1)). Values of the parameterλ(eigenvalues) are determined for which this problem has a positive solution. The methods used here extend recent results by allowing for a broader class of functions fora(t).

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MICCICHÈ, SALVATORE, FABRIZIO LILLO, and ROSARIO N. MANTEGNA. "THE ROLE OF UNBOUNDED TIME-SCALES IN GENERATING LONG-RANGE MEMORY IN ADDITIVE MARKOVIAN PROCESSES." Fluctuation and Noise Letters 12, no. 02 (June 2013): 1340002. http://dx.doi.org/10.1142/s0219477513400026.

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Any additive stationary and continuous Markovian process described by a Fokker–Planck equation can also be described in terms of a Schrödinger equation with an appropriate quantum potential. By using such analogy, it has been proved that a power-law correlated stationary Markovian process can stem from a quantum potential that (i) shows an x-2 decay for large x values and (ii) whose eigenvalue spectrum admits a null eigenvalue and a continuum part of positive eigenvalues attached to it. In this paper we show that such two features are both necessary. Specifically, we show that a potential with tails decaying like x-μ with μ < 2 gives rise to a stationary Markovian process which is not power-law autocorrelated, despite the fact that the process has an unbounded set of time scales. Moreover, we present an exactly solvable example where the potential decays as x-2 but there is a gap between the continuum spectrum of eigenvalues and the null eigenvalue. We show that the process is not power law autocorrelated, but by decreasing the gap one can arbitrarily well approximate it. A crucial role in obtaining a power-law autocorrelated process is played by the weights [Formula: see text] giving the contribution of each time-scale contribute to the autocorrelation function. In fact, we will see that such weights must behave like a power-law for small energy values λ. This is only possible if the potential VS(x) shows a x-2 decay to zero for large x values.

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Hu, Zhiying. "Estimation and application of matrix eigenvalues based on deep neural network." Journal of Intelligent Systems 31, no. 1 (January 1, 2022): 1246–61. http://dx.doi.org/10.1515/jisys-2022-0126.

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Abstract In today’s era of rapid development in science and technology, the development of digital technology has increasingly higher requirements for data processing functions. The matrix signal commonly used in engineering applications also puts forward higher requirements for processing speed. The eigenvalues of the matrix represent many characteristics of the matrix. Its mathematical meaning represents the expansion of the inherent vector, and its physical meaning represents the spectrum of vibration. The eigenvalue of a matrix is the focus of matrix theory. The problem of matrix eigenvalues is widely used in many research fields such as physics, chemistry, and biology. A neural network is a neuron model constructed by imitating biological neural networks. Since it was proposed, the application research of its typical models, such as recurrent neural networks and cellular neural networks, has become a new hot spot. With the emergence of deep neural network theory, scholars continue to combine deep neural networks to calculate matrix eigenvalues. This article aims to study the estimation and application of matrix eigenvalues based on deep neural networks. This article introduces the related methods of matrix eigenvalue estimation based on deep neural networks, and also designs experiments to compare the time of matrix eigenvalue estimation methods based on deep neural networks and traditional algorithms. It was found that under the serial algorithm, the algorithm based on the deep neural network reduced the calculation time by about 7% compared with the traditional algorithm, and under the parallel algorithm, the calculation time was reduced by about 17%. Experiments are also designed to calculate matrix eigenvalues with Obj and recurrent neural networks (RNNS) models, which proves that the Oja algorithm is only suitable for calculating the maximum eigenvalues of non-negative matrices, while RNNS is commonly used in general models.

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Yi, S., P. W. Nelson, and A. G. Ulsoy. "Eigenvalue Assignment via the Lambert W Function for Control of Time-delay Systems." Journal of Vibration and Control 16, no. 7-8 (June 2010): 961–82. http://dx.doi.org/10.1177/1077546309341102.

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In this paper, we consider the problem of feedback controller design via eigenvalue assignment for linear time-invariant systems of linear delay differential equations (DDEs) with a single delay. Unlike ordinary differential equations (ODEs), DDEs have an infinite eigenspectrum, and it is not feasible to assign all closed-loop eigenvalues. However, we can assign a critical subset of them using a solution to linear systems of DDEs in terms of the matrix Lambert W function. The solution has an analytical form expressed in terms of the parameters of the DDE, and is similar to the state transition matrix in linear ODEs. Hence, one can extend controller design methods developed based upon the solution form of systems of ODEs to systems of DDEs, including the design of feedback controllers via eigenvalue assignment. We present such an approach here, illustrate it using some examples, and compare with other existing methods.

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Nauchi, Yasushi, Alexis Jinaphanh, and Andrea Zoia. "ANALYSIS OF TIME-EIGENVALUE AND EIGENFUNCTIONS IN THE CROCUS BENCHMARK." EPJ Web of Conferences 247 (2021): 04011. http://dx.doi.org/10.1051/epjconf/202124704011.

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Time-dependent neutron transport in non-critical state can be expressed by the natural mode equation. In order to estimate the dominant eigenvalue and eigenfunction of the natural mode, CEA had extended the α-k method and developed the generalized iterated fission probability method (G-IFP) in the TRIPOLI-4® code. CRIEPI has chosen to compute those quantities by a time-dependent neutron transport calculation, and has thus developed a time-dependent neutron transport technique based on k-power iteration (TDPI) in MCNP-5. In this work, we compare the two approaches by computing the dominant eigenvalue and the direct and adjoint eigenfunctions for the CROCUS benchmark. The model has previously been qualified for keffs and kinetic parameters by TRIPOLI-4 and MCNP-5. The eigenvalues of the natural mode equations by α-k and TDPI are in good agreement with each other, and closely follow those predicted by the inhour equation. Neutron spectra and spatial distributions (flux and fission neutron emission) obtained by the two methods are also in good agreement. Similar results are also obtained for the adjoint fundamental eigenfunctions. These findings substantiate the coherence of both calculation strategies for natural mode.

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Gin, Craig, and Prabir Daripa. "Stability results on radial porous media and Hele-Shaw flows with variable viscosity between two moving interfaces." IMA Journal of Applied Mathematics 86, no. 2 (March 18, 2021): 294–319. http://dx.doi.org/10.1093/imamat/hxab001.

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Abstract We perform a linear stability analysis of three-layer radial porous media and Hele-Shaw flows with variable viscosity in the middle layer. A nonlinear change of variables results in an eigenvalue problem that has time-dependent coefficients and eigenvalue-dependent boundary conditions. We study this eigenvalue problem and find upper bounds on the spectrum. We also give a characterization of the eigenvalues and prescribe a measure for which the eigenfunctions form an orthonormal basis of the corresponding $L^2$ space. This allows for any initial perturbation of the interfaces and viscosity profile to be easily expanded in terms of the eigenfunctions by using the inner product of the $L^2$ space, thus providing an efficient method for simulating the growth of the perturbations via the linear theory. The limit as the viscosity gradient goes to zero is compared with previous results on multi-layer radial flows. We then numerically compute the eigenvalues and obtain, among other results, optimal profiles within certain classes of functions.

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Zhang, Yuan, Ying Zhang, and Gang Wang. "Exclusion sets in the S-type eigenvalue localization sets for tensors." Open Mathematics 17, no. 1 (October 13, 2019): 1136–46. http://dx.doi.org/10.1515/math-2019-0090.

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Abstract In this paper, we break the index set N into disjoint subsets S and its complement, and propose two S-type exclusion sets that all the eigenvalues do not belong to them. Furthermore, we establish new S-type eigenvalue inclusion sets, which can reduce computations and obtain more accurate numerical results. At the same time, we give two criteria for identifying nonsingular tensors. Finally, new S-type eigenvalue inclusion sets are shown to be sharper than existing results via two examples.

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CONLON, T., H. J. RUSKIN, and M. CRANE. "MULTISCALED CROSS-CORRELATION DYNAMICS IN FINANCIAL TIME-SERIES." Advances in Complex Systems 12, no. 04n05 (August 2009): 439–54. http://dx.doi.org/10.1142/s0219525909002325.

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The cross-correlation matrix between equities comprises multiple interactions between traders with varying strategies and time horizons. In this paper, we use the Maximum Overlap Discrete Wavelet Transform to calculate correlation matrices over different time–scales and then explore the eigenvalue spectrum over sliding time-windows. The dynamics of the eigenvalue spectrum at different times and scales provides insight into the interactions between the numerous constituents involved. Eigenvalue dynamics are examined for both medium, and high-frequency equity returns, with the associated correlation structure shown to be dependent on both time and scale. Additionally, the Epps effect is established using this multivariate method and analyzed at longer scales than previously studied. A partition of the eigenvalue time-series demonstrates, at very short scales, the emergence of negative returns when the largest eigenvalue is greatest. Finally, a portfolio optimization shows the importance of time–scale information in the context of risk management.

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Warrier, Latha S. "Preparation of Approximate Eigenvector by Unitary Operations on Eigenstate in Abrams-Lloyd Quantum Algorithm." Journal of Computational Methods in Physics 2013 (September 11, 2013): 1–5. http://dx.doi.org/10.1155/2013/235624.

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The Abrams-Lloyd quantum algorithm computes an eigenvalue and the corresponding eigenstate of a unitary matrix from an approximate eigenvector Va. The eigenstate is a basis vector in the orthonormal eigenspace. Finding another eigenvalue, using a random approximate eigenvector, may require many trials as the trial may repeatedly result in the eigenvalue measured earlier. We present a method involving orthogonalization of the eigenstate obtained in a trial. It is used as the Va for the next trial. Because of the orthogonal construction, Abrams-Lloyd algorithm will not repeat the eigenvalue measured earlier. Thus, all the eigenvalues are obtained in sequence without repetitions. An operator that anticommutes with a unitary operator orthogonalizes the eigenvectors of the unitary. We implemented the method on the programming language model of quantum computation and tested it on a unitary matrix representing the time evolution operator of a small spin chain. All the eigenvalues of the operator were obtained sequentially. Another use of the first eigenvector from Abrams-Lloyd algorithm is preparing a state that is the uniform superposition of all the eigenvectors. This is possible by nonorthogonalizing the first eigenvector in all dimensions and then applying the Abrams-Lloyd algorithm steps stopping short of the last measurement.

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Komijani, Javad. "First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior." Journal of Physics A: Mathematical and Theoretical 54, no. 46 (November 3, 2021): 465202. http://dx.doi.org/10.1088/1751-8121/ac2e29.

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Abstract Eigenvalue problems arise in many areas of physics, from solving a classical electromagnetic problem to calculating the quantum bound states of the hydrogen atom. In textbooks, eigenvalue problems are defined for linear problems, particularly linear differential equations such as time-independent Schrödinger equations. Eigenfunctions of such problems exhibit several standard features independent of the form of the underlying equations. As discussed in Bender et al (2014 J. Phys. A: Math. Theor. 47 235204), separatrices of nonlinear differential equations share some of these features. In this sense, they can be considered eigenfunctions of nonlinear differential equations, and the quantized initial conditions that give rise to the separatrices can be interpreted as eigenvalues. We introduce a first-order nonlinear eigenvalue problem involving a general class of functions and obtain the large-eigenvalue limit by reducing it to a random walk problem on a half-line. The introduced general class of functions covers many special functions such as the Bessel and Airy functions, which are themselves solutions of second-order differential equations. For instance, in a special case involving the Bessel functions of the first kind, i.e. for y′(x) = J ν (xy), we show that the eigenvalues asymptotically grow as 241/42 n 1/4. We also introduce and discuss nonlinear eigenvalue problems involving the reciprocal gamma and the Riemann zeta functions, which are not solutions to simple differential equations. With the reciprocal gamma function, i.e. for y′(x) = 1/Γ(−xy), we show that the nth eigenvalue grows factorially fast as ( 1 − 2 n ) / Γ ( r 2 n − 1 ) , where r k is the kth root of the digamma function.

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Banova, T., W. Ackermann, and T. Weiland. "Eigenvalue extraction from time domain computations." Advances in Radio Science 11 (July 4, 2013): 23–29. http://dx.doi.org/10.5194/ars-11-23-2013.

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Abstract. In this paper we address a fast approach for an accurate eigenfrequency extraction, taken into consideration the evaluated electric field computations in time domain of a superconducting resonant structure. Upon excitation of the cavity, the electric field intensity is recorded at different detection probes inside the cavity. Thereafter, we perform Fourier analysis of the recorded signals and by means of fitting techniques with the theoretical cavity response model (in support of the applied excitation) we extract the requested eigenfrequencies by finding the optimal model parameters in least square sense. The major challenges posed by our work are: first, the ability of the approach to tackle the large scale eigenvalue problem and second, the capability to extract many, i.e. order of thousands, eigenfrequencies for the considered cavity. At this point, we demonstrate that the proposed approach is able to extract many eigenfrequencies of a closed resonator in a relatively short time. In addition to the need to ensure a high precision of the calculated eigenfrequencies, we compare them side by side with the reference data available from CEM3D eigenmode solver. Furthermore, the simulations have shown high accuracy of this technique and good agreement with the reference data. Finally, all of the results indicate that the suggested technique can be used for precise extraction of many eigenfrequencies based on time domain field computations.

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Dryden, Emily B., Jeffrey J. Langford, and Patrick McDonald. "Exit time moments and eigenvalue estimates." Bulletin of the London Mathematical Society 49, no. 3 (March 29, 2017): 480–90. http://dx.doi.org/10.1112/blms.12045.

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Bracher, C., and M. Kleber. "Reflection time as an eigenvalue problem." Annalen der Physik 507, no. 7 (1995): 696–717. http://dx.doi.org/10.1002/andp.19955070707.

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Miller, U., S. Bograd, A. Schmidt, and L. Gaul. "Eigenpath Following for Systems with Symmetric Complex-Valued Stiffness Matrices." Shock and Vibration 17, no. 4-5 (2010): 397–405. http://dx.doi.org/10.1155/2010/849840.

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A vibration analysis of a structure with joints is performed. The simulation is conducted with finite element software capable of performing a numeric modal analysis with hysteretic damping assumption. The joints are modeled with thin layer elements, representing dissipation and stiffness of the joints. The matrices describing the system consist of the mass, as well as real and complex-valued stiffness matrices. If the eigenvalues of this system are found in one step, due to the mode crossing occurring for the closely spaced modes, it is difficult and time consuming to assign calculated modal damping factors to the corresponding undamped eigenvalues. In order to avoid this problem, an eigenvalue following method is used. The outcome of the solution is the graphical presentation of continuous eigenvalue paths, showing the change in the eigenvalues from the undamped to the fully damped case. For every undamped eigenvalue exists its equivalent eigenfrequency and damping factor that can be used for further numerical analysis.In scope of this article a Predictor-Corrector and a Rayleigh-Quotient Iteration algorithms are applied to the problem. The algorithms are tested specifically on the type of matrices resulting from the weakly damped hysteretic formulation arising from the simulation of metallic structures with joints.

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He, Shu-An, and I.-Kong Fong. "Stability and Performance of First-Order Linear Time-Delay Feedback Systems: An Eigenvalue Approach." Journal of Control Science and Engineering 2011 (2011): 1–8. http://dx.doi.org/10.1155/2011/719730.

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Linear time-delay systems with transcendental characteristic equations have infinitely many eigenvalues which are generally hard to compute completely. However, the spectrum of first-order linear time-delay systems can be analyzed with the Lambert function. This paper studies the stability and state feedback stabilization of first-order linear time-delay system in detail via the Lambert function. The main issues concerned are the rightmost eigenvalue locations, stability robustness with respect to delay time, and the response performance of the closed-loop system. Examples and simulations are presented to illustrate the analysis results.

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Yang, Ouyang, Zhang, Zhang, Hu, and Liu. "Receptance-Based Dominant Eigenvalues Computation of Controlled Vibrating Systems with Multiple Time-Delays Using a Contour Integral Method." Applied Sciences 9, no. 23 (December 3, 2019): 5263. http://dx.doi.org/10.3390/app9235263.

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The computation of dominant eigenvalues of second-order linear control systems with multiple time-delays is tackled by using a contour integral method. The proposed approach depends on a reduced characteristic function and the associated characteristic matrix comprised of measured open-loop receptances. This reduced characteristic function is derived from the original characteristic function of the second-order time delayed systems based on the reasonable assumption that eigenvalues of the closed-loop system are distinct from those of the open-loop system, and has the same eigenvalues as those of the original. Then, the eigenvalues computation is equivalent to solve a nonlinear eigenvalue problem of the associated characteristic matrix by using a contour integral method. The proposed approach also utilizes the spectrum distribution features of the retarded time-delay systems. Finally, numerical examples are given to illustrate the effectiveness of the proposed approach.

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Ao, Ji-Jun, and Juan Wang. "Finite spectrum of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on time scales." Filomat 33, no. 6 (2019): 1747–57. http://dx.doi.org/10.2298/fil1906747a.

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The spectral analysis of a class of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on bounded time scales is investigated. By partitioning the bounded time scale such that the coefficients of Sturm-Liouville equation satisfy certain conditions on the adjacent subintervals, the finite eigenvalue results are obtained. The results show that the number of eigenvalues not only depend on the partition of the bounded time scale, but also depend on the eigenparameter-dependent boundary conditions. Both of the self-adjoint and non-self-adjoint cases are considered in this paper.

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Li, Chongtao, Gengfeng Li, Cong Wang, and Zhengchun Du. "Eigenvalue Sensitivity and Eigenvalue Tracing of Power Systems With Inclusion of Time Delays." IEEE Transactions on Power Systems 33, no. 4 (July 2018): 3711–19. http://dx.doi.org/10.1109/tpwrs.2017.2787713.

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Li, Xue, Huai Wu, and Yikang Yang. "Consensus of Second-Order Multiagent Systems with Fixed Topology and Time-Delay." Mathematical Problems in Engineering 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/138276.

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We are concerned with the consensus problems for networks of second-order agents, where each agent can only access the relative position information from its neighbours. We aim to find the largest tolerable input delay such that the system consensus can be reached. We introduce a protocol with time-delay and fixed topology. A sufficient and necessary condition is given to guarantee the consensus. By using eigenvector-eigenvalue method and frequency domain method, it is proved that the largest tolerable time-delay is only related to the eigenvalues of the graph Laplacian. And simulation results are also provided to demonstrate the effectiveness of our theoretical results.

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Liu, Chein-Shan, Jiang-Ren Chang, Jian-Hung Shen, and Yung-Wei Chen. "A Boundary Shape Function Method for Computing Eigenvalues and Eigenfunctions of Sturm–Liouville Problems." Mathematics 10, no. 19 (October 9, 2022): 3689. http://dx.doi.org/10.3390/math10193689.

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In the paper, we transform the general Sturm–Liouville problem (SLP) into two canonical forms: one with the homogeneous Dirichlet boundary conditions and another with the homogeneous Neumann boundary conditions. A boundary shape function method (BSFM) was constructed to solve the SLPs of these two canonical forms. Owing to the property of the boundary shape function, we could transform the SLPs into an initial value problem for the new variable with initial values that were given definitely. Meanwhile, the terminal value at the right boundary could be entirely determined by using a given normalization condition for the uniqueness of the eigenfunction. In such a manner, we could directly determine the eigenvalues as the intersection points of an eigenvalue curve to the zero line, which was a horizontal line in the plane consisting of the zero values of the target function with respect to the eigen-parameter. We employed a more delicate tuning technique or the fictitious time integration method to solve an implicit algebraic equation for the eigenvalue curve. We could integrate the Sturm–Liouville equation using the given initial values to obtain the associated eigenfunction when the eigenvalue was obtained. Eight numerical examples revealed a great advantage of the BSFM, which easily obtained eigenvalues and eigenfunctions with the desired accuracy.

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Liu, Chein-Shan, Jiang-Ren Chang, Jian-Hung Shen, and Yung-Wei Chen. "A New Quotient and Iterative Detection Method in an Affine Krylov Subspace for Solving Eigenvalue Problems." Journal of Mathematics 2023 (March 6, 2023): 1–17. http://dx.doi.org/10.1155/2023/9859889.

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The paper solves the eigenvalues of a symmetric matrix by using three novel algorithms developed in the m-dimensional affine Krylov subspace. The n-dimensional eigenvector is superposed by a constant shifting vector and an m-vector. In the first algorithm, the m-vector is derived as a function of eigenvalue by maximizing the Rayleigh quotient to generate the first characteristic equation, which, however, is not easy to determine the eigenvalues since its roots are not of simple ones, exhibiting turning points, spikes, and even no intersecting point to the zero line. To overcome that difficulty by the first algorithm, we propose the second characteristic equation through a new quotient with the inner product of the shifting vector to the eigen-equation. The Newton method and the fictitious time integration method are convergent very fast due to the simple roots of the second characteristic equation. For both symmetric and nonsymmetric eigenvalue problems solved by the third algorithm, we develop a simple iterative detection method to maximize the Euclidean norm of the eigenvector in terms of the eigen-parameter, of which the peaks of the response curve correspond to the eigenvalues. Through a few finer tunings to the smaller intervals sequentially, a very accurate eigenvalue and eigenvector can be obtained. The efficiency and accuracy of the proposed iterative algorithms are verified and compared to the Lanczos algorithm and the Rayleigh quotient iteration method.

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Chen, S.-H. "Robust D-stability analysis for linear discrete-time singular systems with structured parameter uncertainties and delayed perturbations." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 217, no. 1 (February 1, 2003): 1–5. http://dx.doi.org/10.1177/095965180321700102.

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In this paper, the robust D-stability problem (i.e. robust eigenvalue-clustering in a specified circular region problem) of linear discrete-time singular systems with structured (elemental) parameter uncertainties and delayed perturbations is investigated. Under the assumptions that the nominal discrete-time singular system is regular and impulse free and has all its finite eigenvalues lying inside a specified circular region, a new sufficient condition is proposed to preserve the assumed properties when the structured (elemental) parameter uncertainties and delayed perturbations are added into the nominal discrete-time singular system. When all the finite eigenvalues lie inside the unit circle of the z plane, the proposed criterion will become the stability robustness criterion. An example is given to demonstrate the applicability of the proposed sufficient condition.

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41

Mao, Xiaobin, and Hua Dai. "Partial eigenvalue assignment with time delay robustness." Numerical Algebra, Control & Optimization 3, no. 2 (2013): 207–21. http://dx.doi.org/10.3934/naco.2013.3.207.

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42

Somma, Rolando D. "Quantum eigenvalue estimation via time series analysis." New Journal of Physics 21, no. 12 (December 16, 2019): 123025. http://dx.doi.org/10.1088/1367-2630/ab5c60.

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43

HATAKE, Susumu, Takatsugu OCHIAI, and Kyouji TAKAHARA. "Calculation time of large scale eigenvalue analysis." Transactions of the Japan Society of Mechanical Engineers Series C 52, no. 481 (1986): 2309–12. http://dx.doi.org/10.1299/kikaic.52.2309.

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Agarwal, Ravi P., Martin Bohner, and Patricia J. Y. Wong. "Sturm-Liouville eigenvalue problems on time scales." Applied Mathematics and Computation 99, no. 2-3 (March 1999): 153–66. http://dx.doi.org/10.1016/s0096-3003(98)00004-6.

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LI, ZHENYANG, and PAWEŁ GÓRA. "Instability of the isolated spectrum for W-shaped maps." Ergodic Theory and Dynamical Systems 33, no. 4 (May 30, 2012): 1052–59. http://dx.doi.org/10.1017/s0143385712000223.

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AbstractIn this note we consider the W-shaped map $W_0=W_{s_1,s_2}$ with ${1}/{s_1}+{1}/{s_2}=1$ and show that the eigenvalue $1$ is not stable. We do this in a constructive way. For each perturbing map $W_a$ we show the existence of a ‘second’ eigenvalue $\lambda _a$, such that $\lambda _a\to 1$ as $a\to 0$, which proves instability of the isolated spectrum of $W_0$. At the same time, the existence of second eigenvalues close to 1 causes the maps $W_a$to behave in a metastable way. There are two almost-invariant sets, and the system spends long periods of consecutive iterations in each of them, with infrequent jumps from one to the other.

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Korohoda, Przemysław, and Daniel Schneditz. "Analytical Solution of Multicompartment Solute Kinetics for Hemodialysis." Computational and Mathematical Methods in Medicine 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/654726.

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Objective.To provide an exact solution for variable-volume multicompartment kinetic models with linear volume change, and to apply this solution to a 4-compartment diffusion-adjusted regional blood flow model for both urea and creatinine kinetics in hemodialysis.Methods.A matrix-based approach applicable to linear models encompassing any number of compartments is presented. The procedure requires the inversion of a square matrix and the computation of its eigenvaluesλ, assuming they are all distinct. This novel approach bypasses the evaluation of the definite integral to solve the inhomogeneous ordinary differential equation.Results.For urea two out of four eigenvalues describing the changes of concentrations in time are about 105times larger than the other eigenvalues indicating that the 4-compartment model essentially reduces to the 2-compartment regional blood flow model. In case of creatinine, however, the distribution of eigenvalues is more balanced (a factor of 102between the largest and the smallest eigenvalue) indicating that all four compartments contribute to creatinine kinetics in hemodialysis.Interpretation.Apart from providing an exact analytic solution for practical applications such as the identification of relevant model and treatment parameters, the matrix-based approach reveals characteristic details on model symmetry and complexity for different solutes.

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Chesnel, Lucas, Xavier Claeys, and Sergei A. Nazarov. "Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 4 (July 2018): 1285–313. http://dx.doi.org/10.1051/m2an/2016080.

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We investigate the eigenvalue problem −div(σ∇u) = λu (P) in a 2D domain Ω divided into two regions Ω±. We are interested in situations where σ takes positive values on Ω+ and negative ones on Ω−. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [L. Chesnel, X. Claeys and S.A. Nazarov, Asymp. Anal. 88 (2014) 43–74], we highlighted an unusual instability phenomenon for the source term problem associated with (P): for certain configurations, when the interface between the subdomains Ω± presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem (P). We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments.

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TRAN, MINH T., and M. E. SAWAN. "Exact eigenvalue placement for two-time-scale discrete-time systems." International Journal of Systems Science 19, no. 3 (January 1988): 453–58. http://dx.doi.org/10.1080/00207728808967617.

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Song, Di, Qi Feng, Shengyao Chen, Feng Xi, and Zhong Liu. "Random Matrix Theory-Based Reduced-Dimension Space-Time Adaptive Processing under Finite Training Samples." Remote Sensing 14, no. 16 (August 15, 2022): 3959. http://dx.doi.org/10.3390/rs14163959.

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Space-time adaptive processing (STAP) is a fundamental topic in airborne radar applications due to its clutter suppression ability. Reduced-dimension (RD)-STAP can release the requirement of the number of training samples and reduce the computational load from traditional STAP, which attracts much attention. However, under the situation that training samples are severely deficient, RD-STAP will become poor like the traditional STAP. To enhance RD-STAP performance in such cases, this paper develops a novel RD-STAP algorithm using random matrix theory (RMT), RMT-RD-STAP. By minimizing the output clutter-plus-noise power, the estimate of the inversion of clutter plus noise covariance matrix (CNCM) can be obtained through optimally manipulating its eigenvalues, thus producing the optimal STAP weight vector. Specifically, the clutter-related eigenvalues are estimated according to the clutter-related sample eigenvalues via RMT, and the noise-related eigenvalue is optimally selected from the noise-related sample eigenvalues. It is found that RMT-RD-STAP significantly outperforms the RD-STAP algorithm when the RMB rule cannot be satisfied. Theoretical analyses and numerical results demonstrate the effectiveness and the performance advantages of the proposed RMT-RD-STAP algorithm.

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Han, Rui-Qi, Wen-Jie Xie, Xiong Xiong, Wei Zhang, and Wei-Xing Zhou. "Market Correlation Structure Changes Around the Great Crash: A Random Matrix Theory Analysis of the Chinese Stock Market." Fluctuation and Noise Letters 16, no. 02 (May 2017): 1750018. http://dx.doi.org/10.1142/s0219477517500183.

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The correlation structure of a stock market contains important financial contents, which may change remarkably due to the occurrence of financial crisis. We perform a comparative analysis of the Chinese stock market around the occurrence of the 2008 crisis based on the random matrix analysis of high-frequency stock returns of 1228 Chinese stocks. Both raw correlation matrix and partial correlation matrix with respect to the market index in two time periods of one year are investigated. We find that the Chinese stocks have stronger average correlation and partial correlation in 2008 than in 2007 and the average partial correlation is significantly weaker than the average correlation in each period. Accordingly, the largest eigenvalue of the correlation matrix is remarkably greater than that of the partial correlation matrix in each period. Moreover, each largest eigenvalue and its eigenvector reflect an evident market effect, while other deviating eigenvalues do not. We find no evidence that deviating eigenvalues contain industrial sectorial information. Surprisingly, the eigenvectors of the second largest eigenvalues in 2007 and of the third largest eigenvalues in 2008 are able to distinguish the stocks from the two exchanges. We also find that the component magnitudes of the some largest eigenvectors are proportional to the stocks’ capitalizations.

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Journal articles on the topic 'Time eigenvalue'

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