Multiple Eigenvalues Research Articles

Locating eigenvalues of unbalanced unicyclic signed graphs

A signed graph is a pair Γ=(G,σ), where G is a graph, and σ:E(G)⟶{+1,−1} is a signature of the edges of G. A signed graph Γ is said to be unbalanced if there exists a cycle in Γ with an odd number of negatively signed edges. In this paper it is presented a linear time algorithm which computes the inertia indices of an unbalanced unicyclic signed graph. Additionally, the algorithm computes the number of eigenvalues in a given real interval by operating directly on the graph, so that the adjacency matrix is not needed explicitly. As an application, the algorithm is employed to check the integrality of some infinite families of unbalanced unicyclic graphs. In particular, the multiplicities of eigenvalues of signed circular caterpillars are studied, getting a geometric characterization of those which are non-singular and sufficient conditions for them to be non-integral. Finally, the algorithm is also used to retrieve the spectrum of bidegreed signed circular caterpillars, none of which turns out to be integral.

THE DISCRETE SPECTRUM OF TWO PARTICLE HAMILTONIAN ON TWO-DIMENSIONAL LATTICE

We consider a system of two arbitrary quantum particles on a two-dimensional lattice with special dispersion functions (describing site-to-site particle transport), where the particles interact by a chosen attraction potential. We study how the number of eigenvalues of operator family ℎ(?) depends on the particle interaction energy and the total quasi-momentum ? ∈ ? 2 (where ? 2 is a two-dimensional torus). Subject to the particle inter-action energy, we obtain conditions for existence of multiple eigenvalues below the essential spectrum of operator ℎ(?).

Eigenvalue-based cooperative spectrum sensing using kernel fuzzy c-means clustering

In this paper, novel techniques of eigenvalue-based cooperative spectrum sensing (CSS) using Kernel fuzzy c-means (KFCM) clustering are proposed. Test vectors derived from measured eigenvalues are categorized into channel available and unavailable class by performing clustering in two/three dimensional space. This is in contrast to existing eigenvalue-based spectrum sensing techniques, where sensing decision is made on the basis of a single test statistic in one dimension. Though multiple eigenvalues are used in those techniques, finally a single test statistic is computed and used for spectrum sensing. It is shown that proposed CSS using KFCM clustering in multidimensional space provides improvement in detection performance. Three different spectrum sensing techniques, utilizing different combinations of eigenvalues of the signal covariance matrix, are proposed and studied. Proposed algorithm is then compared to some of the recent techniques for spectrum sensing in literature. Extensive simulation results are given which highlight improvement in detection performance offered by the proposed algorithm.

On the stability of equilibrium states of the dynamical systems in critical cases

The stability of equilibrium states described by an autonomous system of linear differential equations with constant real coefficients is studied. The cases when, among the eigenvalues of the matrix of coefficients of the system, there are simple or multiple zero eigenvalues or complex eigenvalues with zero real part (critical cases of the Lyapunov’s general theory of stability) are discussed in detail. Stability criteria for the equilibrium states in critical cases are formulated using only information about the order of the matrix of coefficients, rank of the corresponding λ-matrix, and the multiplicity of the corresponding eigenvalues without reducing to the Jordan canonical form. Particular examples of the use of the proposed stability criteria are given. The dynamics of phase transitions due to thermal fluctuations for systems described by the Landau-type kinetic potential is discussed separately. An identity is given for matrices with a zero determinant and a given rank, which can be used to analyze the stability of solutions of linear dynamical systems.

A hybrid approach based on multiple Eigenvalues selection (MES) for the automated grading of a brain tumor using MRI

Background and objective:The manual segmentation, identification, and classification of brain tumor using magnetic resonance (MR) images are essential for making a correct diagnosis. It is, however, an exhausting and time consuming task performed by clinical experts and the accuracy of the results is subject to their point of view. Computer aided technology has therefore been developed to computerize these procedures.Methods:In order to improve the outcomes and decrease the complications involved in the process of analysing medical images, this study has investigated several methods. These include: a Local Difference in Intensity - Means (LDI-Means) based brain tumor segmentation, Mutual Information (MI) based feature selection, Singular Value Decomposition (SVD) based dimensionality reduction, and both Support Vector Machine (SVM) and Multi-Layer Perceptron (MLP) based brain tumor classification. Also, this study has presented a new method named Multiple Eigenvalues Selection (MES) to choose the most meaningful features as inputs to the classifiers. This combination between unsupervised and supervised techniques formed an effective system for the grading of brain glioma.Results:The experimental results of the proposed method showed an excellent performance in terms of accuracy, recall, specificity, precision, and error rate. They are 91.02%,86.52%, 94.26%, 87.07%, and 0.0897 respectively.Conclusion:The obtained results prove the significance and effectiveness of the proposed method in comparison to other state-of-the-art techniques and it can have in the contribution to an early diagnosis of brain glioma.

Axial Piston Pump Fault Diagnosis Method Based on Symmetrical Polar Coordinate Image and Fuzzy C-Means Clustering Algorithm

In this paper, a fault diagnosis method based on symmetric polar coordinate image and Fuzzy C-Means clustering algorithm is proposed to solve the problem that the fault signal of axial piston pump is not intuitive under the time-domain waveform diagram. In this paper, the sampled vibration signals of axial piston pump were denoised firstly by the combination of ensemble empirical mode decomposition and Pearson correlation coefficient. Secondly, the data, after noise reduction, was converted into images, called snowflake images, according to symmetric polar coordinate method. Different fault types of axial piston pump can be identified by observing the snowflake images. After that, in order to evaluate the research results objectively, the obtained images were converted into Gray-Level Cooccurrence Matrixes. Their multiple eigenvalues were extracted, and the eigenvectors consisting of multiple eigenvalues were classified by Fuzzy C-Means clustering algorithm. Finally, according to the accuracy of classification results, the feasibility of applying the symmetric polar coordinate method to axial piston pump fault diagnosis has been validated.

Asymptotics for Averages over Classical Orthogonal Ensembles

AbstractWe study the averages of multiplicative eigenvalue statistics in ensembles of orthogonal Haar-distributed matrices, which can alternatively be written as Toeplitz+Hankel determinants. We obtain new asymptotics for symbols with Fisher–Hartwig singularities in cases where some of the singularities merge together and for symbols with a gap or an emerging gap. We obtain these asymptotics by relying on known analogous results in the unitary group and on asymptotics for associated orthogonal polynomials on the unit circle. As consequences of our results, we derive asymptotics for gap probabilities in the circular orthogonal and symplectic ensembles and an upper bound for the global eigenvalue rigidity in the orthogonal ensembles.

On the tightness of SDP relaxations of QCQPs

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the projected epigraph of the SDP gives the convex hull of the epigraph in the original QCQP. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.

Path-connectedness in global bifurcation theory

<p style='text-indent:20px;'>A celebrated result in bifurcation theory is that, when the operators involved are compact, global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem. This paper presents a simple example in which the hypotheses of the global bifurcation theorem are satisfied, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differentiable and classical bifurcation occurs locally at a simple eigenvalue, the global continua may not be path-connected away from the bifurcation point. A third example shows that the non-trivial solutions which bifurcate at non-zero eigenvalues, irrespective of multiplicity when the problem has gradient structure, may not be connected and may not contain any paths except singletons.</p>

Stability switching and its directions in cournot duopoly game with three delays

<p style='text-indent:20px;'>A three-delay duopoly is considered where the firms have identical implementation delays with different information delays. The equilibrium is locally asymptotically stable without delays however this stability is lost with increasing values of the delays. The stability properties of the equilibrium depend on the common implementation delay of the firms and on the sum of the two information delays. The stability switching curves are first analytically characterized and illustrated, and then the direction of the stability switching is determined at each point of the curves. The possibility of multiple pure imaginary eigenvalues is also discussed when the directions of the stability switches cannot be determined. Simulation examples illustrate the theoretical results.</p>

Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues

<p style='text-indent:20px;'>In this paper, we study the dynamic phase transition for one dimensional Brusselator model. By the linear stability analysis, we define two critical numbers <inline-formula><tex-math id="M1">\begin{document}$ {\lambda}_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ {\lambda}_1 $\end{document}</tex-math></inline-formula> for the control parameter <inline-formula><tex-math id="M3">\begin{document}$ {\lambda} $\end{document}</tex-math></inline-formula> in the equation. Motivated by [<xref ref-type="bibr" rid="b9">9</xref>], we assume that <inline-formula><tex-math id="M4">\begin{document}$ {\lambda}_0&lt; {\lambda}_1 $\end{document}</tex-math></inline-formula> and the linearized operator at the trivial solution has multiple critical eigenvalues <inline-formula><tex-math id="M5">\begin{document}$ \beta_N^+ $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \beta_{N+1}^+ $\end{document}</tex-math></inline-formula>. Then, we show that as <inline-formula><tex-math id="M7">\begin{document}$ {\lambda} $\end{document}</tex-math></inline-formula> passes through <inline-formula><tex-math id="M8">\begin{document}$ {\lambda}_0 $\end{document}</tex-math></inline-formula>, the trivial solution bifurcates to an <inline-formula><tex-math id="M9">\begin{document}$ S^1 $\end{document}</tex-math></inline-formula>-attractor <inline-formula><tex-math id="M10">\begin{document}$ {\mathcal A}_N $\end{document}</tex-math></inline-formula>. We verify that <inline-formula><tex-math id="M11">\begin{document}$ {\mathcal A}_N $\end{document}</tex-math></inline-formula> consists of eight steady state solutions and orbits connecting them. We compute the leading coefficients of each steady state solution via the center manifold analysis. We also give numerical results to explain the main theorem.

MPACT VERIFICATION WITH MAGNOX REACTOR NEUTRONICS PROGRESSION PROBLEMS

MPACT is a state-of-the-art core simulator designed to perform high-fidelity analysis using whole-core, three-dimensional, pin-resolved neutron transport calculations on modern parallel computing hardware. MPACT was originally developed to model light water reactors, and its capabilities are being extended to simulate gas-cooled, graphite-moderated cores such as Magnox reactors. To verify MPACT’s performance in this new application, the code is being formally benchmarked using representative problems. Progression problems are a series of example models that increase in complexity designed to test a code’s performance. The progression problems include both beginning-of-cycle and depletion calculations. Reference solutions for each progression problem have been generated using Serpent 2, a continuous-energy Monte Carlo reactor physics burnup calculation code. Using the neutron multiplication eigenvalue ke_ as a metric, MPACT’s performance is assessed on each of the progression problems. Initial results showed that MPACT’s multigroup cross section libraries, originally developed for pressurized water reactor problems, were not sufficient to accurately solve Magnox problems. MPACT’s improved performance on the progression problems is demonstrated using this new optimized cross section library.

Spectral problem of fullerene vibrations

Small vibrations of a graph of fullerene (truncated icosahedron) is considered each edge of which is a so-called Stieltjes string (a massless thread bearing finite number of point masses) symmetric with respect to its midpoint. The spectral problem is obtained by imposing the continuity and balance of forces conditions at the vertices. It is shown that when all the edges of the graph are the same then due to the symmetry of the problem there are multiple eigenvalues. The maximal multiplicity of an eigenvalue of such problem is 32, exactly the value which is maximal for cyclically connected graphs, i.e. $\mu+1$ where $\mu$ is the cyclomatic number of the graph.

Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

We are concerned with the study of some classical spectral collocation methods, mainly Chebyshev and sinc as well as with the new software system Chebfun in computing high order eigenpairs of singular and regular Schrödinger eigenproblems. We want to highlight both the qualities as well as the shortcomings of these methods and evaluate them in conjunction with the usual ones. In order to resolve a boundary singularity, we use Chebfun with domain truncation. Although it is applicable with spectral collocation, a special technique to introduce boundary conditions as well as a coordinate transform, which maps an unbounded domain to a finite one, are the special ingredients. A challenging set of “hard”benchmark problems, for which usual numerical methods (f. d., f. e. m., shooting, etc.) fail, were analyzed. In order to separate “good”and “bad”eigenvalues, we have estimated the drift of the set of eigenvalues of interest with respect to the order of approximation and/or scaling of domain parameter. It automatically provides us with a measure of the error within which the eigenvalues are computed and a hint on numerical stability. We pay a particular attention to problems with almost multiple eigenvalues as well as to problems with a mixed spectrum.

Eigenvalue Formulations for the PN Approximation to the Neutron Transport Equation

The study of the eigenvalues of the neutron transport operator yields an important insight into the physical features of the neutronic phenomena taking place in a nuclear reactor. Although the multiplication eigenvalue is the most popular because of its implication in the engineering design of multiplying structures, alternative interesting formulations are possible. In this paper the interest is focused on the multiplication, collision and time eigenvalues. The transport model is considered in the spherical harmonics approximation and the study is restricted to the one-dimensional plane geometry in the monokinetic case. The spectra of the different eigenvalues are investigated using a numerical code, validating its performance against the results available in the literature. The observation of the convergence trends allows to establish the performance of even- and odd-order approximations. It is shown that in general even-order approximations yield slightly less accurate results, nevertheless they appear to converge to the reference values. The effect of the choice of the boundary conditions according to the methodologies proposed by either Mark or Marshak is also investigated. The analysis of all the results presented allows to characterize the convergence properties of the spherical harmonics approach to neutron transport. The spectrum of the time eigenvalues retains a very rich physical meaning, as they are the actual time constants of the time-dependent solution of the transport problem. Therefore, in the last part of the paper the behavior of the pattern of the spectrum of the time eigenvalues when changing the scattering ratio and the order of the approximation is examined.

On extremal eigenvalues of the graph Laplacian * *A preprint appears in the PhD thesis of the author [

Upper and lower estimates of eigenvalues of the Laplacian on a metric graph have been established in 2017 by Berkolaiko, Kennedy, Kurasov, and Mugnolo. Both these estimates can be achieved at the same time only by highly degenerate eigenvalues which we call maximally degenerate. By comparison with the maximal eigenvalue multiplicity proved by Kac and Pivovarchik in 2011, we characterize the graphs exhibiting maximally degenerate eigenvalues which are the figure-of-eight graph, the 3-watermelon graph, and the lasso trees—namely, trees decorated with lasso graphs.

Two-scale and three-scale asymptotic computations of the Neumann-type eigenvalue problems for hierarchically perforated materials

A top-down strategy is proposed for analyzing the elliptic eigenvalue problems of the hierarchically perforated materials with three-scale periodic configurations. The heterogeneous structure considered is composed of perforated cells in the mesoscopic scale and composite cells in the microscopic scale, and Neumann boundary conditions are imposed on the boundaries of the cavities. By using the classical two-scale asymptotic expansion method, the homogenized eigenfunctions and eigenvalues are obtained and the first- and second-order auxiliary cell functions are defined firstly in the mesoscale. Then, the two-scale asymptotic analysis is furtherly applied to the mesoscopic cell problems and by expanding the meso cell functions to the second-order terms, the homogenized cell functions are derived and the relations between the homogenized coefficients and the coefficients of constituent materials in the three scale levels are established. Finally, the second-order three-scale asymptotic approximations of the eigenfunctions are presented and by the idea of “corrector equation”, the three-scale expressions of the eigenvalues are obtained. The corresponding finite element algorithm is established and the successively up-scaling procedures are established. Typical two-dimensional numerical examples are performed, and both the two-scale and three-scale computed approximations of the eigenvalues are compared with the ones obtained in the classical computation. By the least squares technique, it is demonstrated that the three-scale asymptotic solutions of the eigenfunctions are good approximations of the original eigensolutions corresponding to both the simple and multiple eigenvalues. This study offers an alternative approach to describe the physical and mechanical behaviors of the hierarchically structures with more than two scales and it is indicated that the second-order terms plays an important role not only in the derivation of the expansions but also in the practical computations to capture the local oscillations within the cells.

Newton‐like and inexact Newton‐like methods for a parameterized generalized inverse eigenvalue problem

In this paper, we establish the Newton‐like and inexact Newton‐like based methods for solving a type of parameterized generalized inverse eigenvalue problem. This type of parameterized generalized inverse eigenvalue problem, including multiplicative and additive inverse eigenvalue problems, appears in many applications. We show that the direction produced by the Newton‐like method does not depend explicitly on the eigenvalues. Also, the inexact version can minimize the oversolving problem of Newton‐like methods and hence improve efficiency. We discuss the convergence properties of the presented methods. Finally, the performance and effectiveness of the algorithms are tested on three numerical examples and compared to the Newton algorithm.

Analytical parameters for equal mode damping ratio inducing TMDs for seismic response reduction

This paper’s objective is to obtain exact analytical parameters for Tuned Mass Dampers (TMDs) that derive their efficiency for controlling the seismic response by inducing two modes with equal damping ratios, when affixed to single-degree-of-freedom mechanical systems. The parameters that are derived are intrinsic to the systems and are independent of the ground motion. It is seen that for TMDs tuned with this criterion, as damping is increased a critical condition of multiplicity of eigenvalues is reached. The damping provided by the TMDs calibrated in this manner is effective only in the sub-critical range. As the critical and supercritical conditions are reached, the TMDs lose their efficiency, and are not adequate to reduce the response to seismic excitation. Nevertheless, in this paper the exact analytical parameters are provided for all conditions. It is shown that at the critical condition the derivation of parameters results in a commonly referenced work by others, who also proposed using TMDs for the purpose of seismic response abatement. Finally, some calculations in the frequency and time domains are presented to illustrate the utilization of the parameters that are derived, and to illustrate the effect of the TMD mass on the seismic response.

Study properties of Eigenvalues, Eigenvectors and Eigenspace of matrices by using MatLab

The aim of this paper is to study the properties of eigenvalues with types (real, Repeated, complex) of matrices (2x2), (3x3), (4x4), eigenvectors and eigenspace of matrices by using Matlab. It also investigates the left and right eigenvectors, characteristic equation, existence and multiplicity of eigenvalues, gives some examples about the computation of eigenvalues, eigenvectors, and solves the Dynamic Systems of First Order Linear Differential Equations of (2 dim and 3 dim) as well as the Numerical solution to Dynamic systems by using program Matlab.