Multiple Eigenvalues Research Articles

Even order uniform hypergraph via the Einstein product

We propose the algebraic connectivity of an undirected 2m-uniform hypergraph under the Einstein product. We generalize the algebraic connectivity to a directed 2m-uniform hypergraph and reveal the relationship between the vertex connectivity and the algebraic connectivity. We present some results on the eigenvalue multiplicity of the -splitting of tensors by the hypergraph method under the Einstein product.

Small divisors in discrete local holomorphic dynamics

We give an introduction to the study of local dynamics of iterated holomorphic mappings near a fixed point via local conjugations in one and several complex variables. Starting with the systematic construction of formal conjugations to Poincaré–Dulac normal form in general and formal linearisations in particular, we discuss conditions for convergence of the normalising series in terms of the linear part. The convergence is closely related to the size of denominators that show up in the normalising series and depend only on the linear part of the mapping. Hence, we speak of a small-divisor problem. The central result on the linearisation problem is the Brjuno condition, that ensures holomorphic linearisability. The first part of these notes is a survey of the local dynamics in one variable from the viewpoint of local normalisations. In this case, the picture is fairly complete. In the second part we introduce the additional obstacles to both formal and holomorphic normalisations that emerge from interactions of multiple eigenvalues, such as resonances, that preclude the Brjuno condition in particular. For these cases, we proceed with several generalisations of the Brjuno condition, that allow us to find convergent conjugations to at least partial normalisations. The last part reviews a recent application of such a partial normalisation to illuminate the local dynamics near certain one-resonant fixed points.

Recognition of DC01 Mild Steel Laser Welding Penetration Status Based on Photoelectric Signal and Neural Network

Achieving online inspection and recognition of laser welding quality is essential for intelligent industrial manufacturing. The weld penetration status is an important indicator for assessing the welding quality, and the optical signal is the most common changing feature in the laser welding process. This paper proposes a new method based on a photoelectric signal and neural network for laser welding penetration status identification. A laser welding experimental system platform based on a photoelectric sensor is built, the laser welding experimental material is DC01 mild steel, and the photoelectric signal in the laser welding process is collected. The collected signal is then processed, and features are extracted using wavelet packet transform and probability density analyses. The mapping relationship between the signal features and weld penetration status is investigated. A deep learning convolutional neural network (CNN)-based weld penetration status recognition model is constructed, with multiple eigenvalue vectors as input, and the model training and recognition results are analyzed and compared. The experimental results show that the photoelectric signal features are highly correlated with the weld penetration status, and the constructed CNN weld penetration status recognition model has an accuracy of up to 98.5% on the test set, demonstrating excellent performance in identifying the quality of the laser welding. This study provides the basis for the online inspection and intelligent identification of laser welding quality.

Spectral Properties of the Operator in the Problem of Oscillations in a Mixture of Viscous Compressible Fluids

We study the problem of normal oscillations of a homogeneous mixture of several viscous compressible fluids that fills a bounded domain in three-dimensional space with infinitely smooth boundary. We prove that the essential spectrum of the problem is a finite set of segments located on the real axis. The remaining spectrum consists of isolated eigenvalues of finite algebraic multiplicity and is located on the real axis, with the possible exception of finitely many complex conjugate eigenvalues. The spectrum of the problem contains a subsequence of eigenvalues with a limit point at infinity and a power-law asymptotic distribution.

SF-SGL: Solver-Free Spectral Graph Learning From Linear Measurements

This work introduces a highly-scalable spectral graph densification framework (SGL) for learning resistor networks with linear measurements, such as node voltages and currents. We show that the proposed graph learning approach is equivalent to solving the classical graphical Lasso problems with Laplacian-like precision matrices. We prove that given $O(\log N)$ pairs of voltage and current measurements, it is possible to recover sparse $N$-node resistor networks that can well preserve the effective resistance distances on the original graph. In addition, the learned graphs also preserve the structural (spectral) properties of the original graph, which can potentially be leveraged in many circuit design and optimization tasks. To achieve more scalable performance, we also introduce a solver-free method (SF-SGL) that exploits multilevel spectral approximation of the graphs and allows for a scalable and flexible decomposition of the entire graph spectrum (to be learned) into multiple different eigenvalue clusters (frequency bands). Such a solver-free approach allows us to more efficiently identify the most spectrally-critical edges for reducing various ranges of spectral embedding distortions. Through extensive experiments for a variety of real-world test cases, we show that the proposed approach is highly scalable for learning sparse resistor networks without sacrificing solution quality. We also introduce a data-driven EDA algorithm for vectorless power/thermal integrity verifications to allow estimating worst-case voltage/temperature (gradient) distributions across the entire chip by leveraging a few voltage/temperature measurements.

Spherical Two-Distance Sets and Eigenvalues of Signed Graphs

We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let N_{alpha ,beta }(d) denote the maximum number of unit vectors in {mathbb {R}}^d where all pairwise inner products lie in {alpha ,beta }. For fixed -1le beta<0le alpha <1, we propose a conjecture for the limit of N_{alpha ,beta }(d)/d as d rightarrow infty in terms of eigenvalue multiplicities of signed graphs. We determine this limit when alpha +2beta <0 or (1-alpha )/(alpha -beta ) in {1, sqrt{2}, sqrt{3}}.Our work builds on our recent resolution of the problem in the case of alpha = -beta (corresponding to equiangular lines). It is the first determination of lim _{d rightarrow infty } N_{alpha ,beta }(d)/d for any nontrivial fixed values of alpha and beta outside of the equiangular lines setting.

Smoothing inertial method for worst-case robust topology optimization under load uncertainty

We consider a worst-case robust topology optimization problem under load uncertainty, which can be formulated as a minimization problem of the maximum eigenvalue of a symmetric matrix. The objective function is nondifferentiable where the multiplicity of maximum eigenvalues occurs. Nondifferentiability often causes some numerical instabilities in an optimization algorithm such as oscillation of the generated sequence and convergence to a non-optimal point. We use a smoothing method to tackle these issues. The proposed method is guaranteed to converge to a point satisfying the first-order optimality condition. In addition, it is a simple first-order optimization method and thus has low computational cost per iteration even in a large-scale problem. In numerical experiments, we show that the proposed method suppresses oscillation and converges faster than other existing methods.

Local solvability and stability of the generalized inverse Robin–Regge problem with complex coefficients

Abstract We prove local solvability and stability of the inverse Robin–Regge problem in the general case, taking eigenvalue multiplicities into account. We develop the new approach based on the reduction of this inverse problem to the recovery of the Sturm–Liouville potential from the Cauchy data

A Generalized Eigenvalue Formulation for Core-Design Applications

The adoption of multiplication eigenvalue is a well-established approach for the design of nuclear reactors. However, despite its popularity and nice physico-mathematical properties, this eigenvalue formulation is not able to provide quantitative information about what parameters the designer has to modify. In this paper, a novel generalized eigenvalue formulation is introduced to disclose the full potential of the neutron transport equation for core design applications. To illustrate the advantages of this new design-oriented approach with respect to traditional methods, some relevant problems arising in the physics of reactors are solved, such as the determination of the absorber density in the control rods and of the fissile concentration in the molten salt fast reactor.

The Numerical Solution of Large-Scale Generalized Eigenvalue Problems Arising from Finite-Element Modeling of Electroelastic Materials

The generalized eigenvalue problem for a symmetric definite matrix pencil obtained from finite-element modeling of electroelastic materials is solved numerically by the Lanczos algorithm. The mass matrix is singular in the considered problem, and therefore the process proceeds with the semi-inner product defined by this matrix. The shift-and-invert Lanczos algorithm is used to find multiple eigenvalues closest to some shift and the corresponding eigenvectors. The results of the numerical experiments are presented.

On graphs with a few distinct reciprocal distance Laplacian eigenvalues

&lt;abstract&gt;&lt;p&gt;For a $ \nu $-vertex connected graph $ \Gamma $, we consider the reciprocal distance Laplacian matrix defined as $ RD^L(\Gamma) = RT(\Gamma)-RD(\Gamma) $, i.e., $ RD^L(\Gamma) $ is the difference between the diagonal matrix of the reciprocal distance degrees $ RT(\Gamma) $ and the Harary matrix $ RD(\Gamma) $. In this article, we determine the graphs with exactly two distinct reciprocal distance Laplacian eigenvalues.We completely characterize the graph classes with a $ RD^L $ eigenvalue of multiplicity $ \nu-2 $. Moreover, we characterize families of graphs with reciprocal distance Laplacian eigenvalue whose multiplicity is $ \nu-3 $.&lt;/p&gt;&lt;/abstract&gt;

The multiplicity and asymptotic forms of eigenvalues of vectorial diffusion equations with some certain assumptions

The main motivation point of this study is to obtain some novel results on the multiplicity of eigenvalues of diffusion equations. A diffusion equation with some boundary and jump conditions have been analyzed and integral equations have been obtained for the solution under certain initial conditions. Later, integral representations of these solutions have been provided. Finally, asymptotic formulas of eigenvalues with zeros of the characteristic have been considered. A brief conclusion has been given.

Jordan–Schur Algorithms for Computing the Matrix Exponential

In this paper, two new versions of the Schur algorithm for computing the matrix exponential of an n × n complex matrix A are presented. Instead of the Schur form, these algorithms use the Jordan–Schur form of A. The Jordan–Schur form is found by less computation and it is determined more reliable than the reduction to Jordan form since it is obtained using only unitary similarity transformations. In contrast to the known methods, the diagonal blocks of the matrix exponential are obtained by using finite Taylor series. This improves the accuracy and avoids the decisions made about the termination of the series expansion. The off‐diagonal blocks of the exponential are determined by modifications of the Schur–Parlett or Schur–Fréchet method, which takes advantage of the Jordan–Schur form of the matrix. The numerical features of the new algorithms are discussed, revealing their advantages and disadvantages in comparison with the other methods for computing the matrix exponential. Computational experiments show that using the new algorithms, the matrix exponential is determined in certain cases with higher accuracy than some widely used methods, however, at the price of an increase in the computational cost which is of order n4. It is shown that the Jordan–Schur algorithms for computing the matrix exponential are appropriate for matrices with multiple eigenvalues and are especially efficient in cases of large Weyr characteristics.

NUMERICALLY STABLE SOLUTION TO THE 6R PROBLEM OF INVERSE KINEMATICS.

In this paper, a stable and accurate algorithm to compute all solutions of the inverse kinematics problem of a 6 revolute manipulator chain is presented. A system of equations is constructed based on the fundamental closure conditions, leading to a closed algebraic system of 20 equations involving 16 quantities, composed of trigonometric functions of five among the six unknown joint angles. Two among these five are stably eliminated using singular value decomposition (SVD) avoiding the need to consider special cases. The resulting system of equations involving three unknowns is solved by conversion to a generalized eigenvalue problem. The remaining three unknown angles are obtained using the previously computed pseudoinverse. In this formulation we exploit the inherently complex form of the system reducing it to 10 complex equations in 9 quantities, which substantially accelerates the SVD computation. The method's robustness is demonstrated through a comparison to current methods and several examples including known problematic cases where some axis or link lengths vanish, or some joint angles are 180 degrees, as well as cases where multiple eigenvalues arise.

Eigenvalue bounds for double saddle-point systems

Abstract We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. Analysis for approximations of Schur complements is included. Some numerical experiments validate our analytical findings.

A relation between multiplicity of nonzero eigenvalues of trees and their matching numbers

Using star complement technique, Rowlinson [Linear Algebra Appl. 2010] proved that if a tree T, with diameter at least 4, has 1 as an eigenvalue of multiplicity k, then it has k+1 pendant edges that form an induced matching. In this paper, we apply the Parter-Wiener Theorem to generalize Rowlinson's result by replacing eigenvalue 1 with an arbitrary nonzero eigenvalue. Moreover, we describe the trees with a maximum matching or a maximum induced matching consisting of k+1 edges.

Verified eigenvalue and eigenvector computations using complex moments and the Rayleigh–Ritz procedure for generalized Hermitian eigenvalue problems

We propose a verified computation method for eigenvalues in a region and the corresponding eigenvectors of generalized Hermitian eigenvalue problems. The proposed method uses complex moments to extract the eigencomponents of interest from a random matrix and uses the Rayleigh–Ritz procedure to project a given eigenvalue problem into a reduced eigenvalue problem. The complex moment is given by contour integral and approximated using numerical quadrature. We split the error in the complex moment into the truncation error of the quadrature and rounding errors and evaluate each. This idea for error evaluation inherits our previous Hankel matrix approach, whereas the proposed method enables verification of eigenvectors and requires half the number of quadrature points for the previous approach to reduce the truncation error to the same order. Moreover, the Rayleigh–Ritz procedure approach forms a transformation matrix that enables verification of the eigenvectors. Numerical experiments show that the proposed method is faster than previous methods while maintaining verification performance and works even for nearly singular matrix pencils and in the presence of multiple and nearly multiple eigenvalues.

Approximation Technique for Solving High-Order Non-Oscillatory Vibration Systems with Slowly Changing Coefficients Represented by Strong Non-Linearity and Multiple Integrated Roots

A formula for the asymptotic solution of an over-damped strongly nonlinear vibration system combining the extended KBM method and the harmonic balance (HB) method with slowly varying coefficients is proposed. This article aims to establish a slowly time-varying solution of an over-damped nonlinear vibration system where one eigenvalue is an integer multiple (greater than two hundred times) of the other eigenvalues. The integrated multiple eigenvalues can provide a better result than other eigenvalues for strong linearity (even if). We found solutions by considering initial conditions and comparing the percentage error of present solutions with unified solutions by using this procedure in examples. Finally, the findings are addressed, especially to improve the physical prospects and shown graphically by using Excel, Dev C++, MATHEMATICA, and MATLAB software.

A bound for the p-domination number of a graph in terms of its eigenvalue multiplicities

Let G be a connected graph of order n with domination number γ(G). Wang, Yan, Fang, Geng and Tian [Linear Algebra Appl. 607 (2020), 307-318] showed that for any Laplacian eigenvalue λ of G with multiplicity mG(λ), it holds that γ(G)≤n−mG(λ). Using techniques from the theory of star sets, in this work we prove that the same bound holds when λ is an arbitrary adjacency eigenvalue of a non-regular graph, and we characterize the cases of equality. Moreover, we show a result that gives a relationship between start sets and the p-domination number, and we apply it to extend the aforementioned spectral bound to the p-domination number using the adjacency and Laplacian eigenvalue multiplicities.

A Jordan-Schur Algorithm for Solving Sylvester and Lyapunov Matrix Equations

This paper presents a version of the Bartels-Stewart algorithm for solving the Sylvester and Lyapunov equations that utilizes the Jordan-Schur form of the equation matrices. The Jordan-Schur form is a type of Schur form which contains additional information about the Jordan structure of the corresponding matrix. This information can be used to solve more efficiently the Sylvester and Lyapunov equations in some cases. A two-level algorithm is implemented which allows us to find directly non-scalar blocks of the solution matrix. These blocks have sizes that are determined by the Weyr characteristics associated with the eigenvalues of the equation matrices. In the case of large elements of the Weyr characteristics associated with multiple eigenvalues, the determination of the solution blocks can be done more efficiency. Also, the blocks equations can be more appropriate in solving the Sylvester and Lyapunov equations in the case of parallel computations. Results obtained from numerical experiments confirm that the accuracy of the new algorithm is comparable with the accuracy of the Bartels-Stewart algorithm.