Approximate Eigenvalues Research Articles

Study of scalar particles through the Klein–Gordon equation under rainbow gravity effects in Bonnor–Melvin-Lambda space-time

In our investigation, we explore the quantum dynamics of charge-free scalar particles through the Klein–Gordon equation within the framework of rainbow gravity, considering the Bonnor–Melvin-Lambda (BML) space-time background. The BML solution is characterized by the magnetic field strength along the axis of the symmetry direction which is related to the cosmological constant Λ and the topological parameter α of the geometry. The behavior of charge-free scalar particles described by the Klein–Gordon equation is investigated, utilizing two sets of rainbow functions: (i) f(χ)=(eβχ−1)βχ , h(χ) = 1 and (ii) f(χ) = 1, h(χ)=1+βχ2 . Here 0<χ=∣E∣Ep≤1 with E representing the particle’s energy, E p is the Planck’s energy, and β is the rainbow parameter. We obtain the approximate analytical solutions for the scalar particles and conduct a thorough analysis of the obtained results. Afterwards, we study the quantum dynamics of quantum oscillator fields within this BML space-time, employing the Klein–Gordon oscillator. Here also, we choose the same sets of rainbow functions and obtain the approximate eigenvalue solution for the oscillator fields. Notably, we demonstrate that the relativistic approximate energy profiles of charge-free scalar particles and oscillator fields get influenced by the topology of the geometry and the cosmological constant. Furthermore, we show that the energy profiles of scalar particles receive modifications from the rainbow parameter and the quantum oscillator fields by both the rainbow parameter and the frequency of oscillation.

Estimates for low Steklov eigenvalues of surfaces with several boundary components

R\'esum\'eIn this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $$\sigma _1$$ σ 1 of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.

An asymptotically exact estimate of the median noise eigenvalue of sample covariance matrices

The Dominant Mode Rejection beamformer [Abraham &amp; Owsley (1990)] averages the noise subspace eigenvalues of the sample covariance matrix (SCM) to estimate the background noise power. This noise power estimate from snapshot deficient SCMs can be unreliable for large arrays in time-varying environments with limited snapshots. Median filtering offers robustness to outliers and subspace mismatch for these challenging problems. Recent numerical experiments [Campos Anchieta &amp; Buck (2022)] identified a simple regression relating the median sample eigenvalue to the true background power for snapshot deficient SCMs. However, the complicated expression of the Marchenko-Pastur (MP) distribution for the noise eigenvalues of the SCM thwarted prior attempts to find a closed form estimator for the MP distribution median. This talk exploits a coordinate transform to obtain a closed-form median of the MP distribution that is exact for the leading term in the power series expansion of the distribution. The new median estimator is more accurate than the previous numerical regression when the ratio of sensors to snapshots is 2 or more. Given the central role of SCM eigenvalues in principal component analysis and constant false alarm rate detectors, the new median expression should find application in other data science algorithms for underwater acoustics. [Work supported by ONR Code 321US.]

Rotation number and eigenvalues of two-component modified Camassa–Holm equations

In this paper, we study the spectral problem y1y2x=−1212λm−12λn12y1y2for the two-component modified Camassa–Holm equation, where m,n are two potentials. By introducing the rotation number ρ(λ) and studying its properties, we prove that for any integer k, the periodic or anti-periodic eigenvalues are the endpoints of the interval λ∈R:ρ(λ)=−k2. Moreover, we prove that as nonlinear functionals of potentials, such eigenvalues are continuous in potentials with respect to the weak topologies in the Lebesgue space L1[0,T]. Finally, we apply the trace formula to give some estimates of the periodic eigenvalues when the L1 norms of potentials are given.

Eigenvalue estimates for the magnetic Hodge Laplacian on differential forms

In this paper we introduce the magnetic Hodge Laplacian, which is a generalization of the magnetic Laplacian on functions to differential forms. We consider various spectral results, which are known for the magnetic Laplacian on functions or for the Hodge Laplacian on differential forms, and discuss similarities and differences of this new “magnetic-type” operator.

Fast multi-level iteration schemes with compression technique for eigen-problems of compact integral operators

In this paper, we present multi-level iteration schemes to solve the eigen-problems of compact integral operators based on the multiscale Galerkin methods. By using the compression technique, the wavelet coefficient matrix is truncated into sparse. This technique results in a fast algorithm. To further minimize computational complexity, we establish Jacobi, Gauss-Seidel and L-H iteration schemes for solving the final algebraic eigenvalue problems obtained from discretizing the integral equation using fast multiscale Galerkin methods. We show that the proposed schemes require only linear computational complexity and obtain the desired convergence order for the approximate eigenvalues and eigenvectors. Numerical experiments are conducted and analyzed, with comparisons to the standard routine and the power iteration algorithm included. Ultimately, our multi-level iteration schemes yield faster solutions with fewer iterations, which is independent of the dimension of approximate space.

The a posteriori error estimates of the Ciarlet-Raviart mixed finite element method for the biharmonic eigenvalue problem

&lt;abstract&gt;&lt;p&gt;The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics and has wide applications. In this paper, for the biharmonic eigenvalue problem, based on the work of Gudi [&lt;italic&gt;Numer. Methods Partial Differ. Equ.&lt;/italic&gt;, &lt;bold&gt;27&lt;/bold&gt; (2011), 315-328], we study the a posteriori error estimates of the approximate eigenpairs obtained by the Ciarlet-Raviart mixed finite element method. We prove the reliability and efficiency of the error estimator of the approximate eigenfunction and analyze the reliability of the error estimator of the approximate eigenvalues. We also implement the adaptive calculation and exhibit the numerical experiments which show that our method is efficient and can get an approximate solution with high accuracy.&lt;/p&gt;&lt;/abstract&gt;

The eigenvalues of $ \beta $-Laplacian of slant submanifolds in complex space forms

&lt;abstract&gt;&lt;p&gt;In this paper, we provided various estimates of the first nonzero eigenvalue of the $ \beta $-Laplacian operator on closed orientated $ p $-dimensional slant submanifolds of a $ 2m $-dimensional complex space form $ \widetilde{\mathbb{V}}^{2m}(4\epsilon) $ with constant holomorphic sectional curvature $ 4\epsilon $. As applications of our results, we generalized the Reilly-inequality for the Laplacian to the $ \beta $-Laplacian on slant submanifolds of a complex Euclidean space and a complex projective space.&lt;/p&gt;&lt;/abstract&gt;

Smoothing algorithm for the maximal eigenvalue of non-defective positive matrices

&lt;abstract&gt;&lt;p&gt;This paper introduced a smoothing algorithm for calculating the maximal eigenvalue of non-defective positive matrices. Two special matrices were constructed to provide monotonically increasing lower-bound estimates and monotonically decreasing upper-bound estimates of the maximal eigenvalue. The monotonicity and convergence of these estimations was also proven. Finally, the effectiveness of the algorithm was demonstrated with numerical examples.&lt;/p&gt;&lt;/abstract&gt;

Localized Phase for the Erdős–Rényi Graph

We analyse the eigenvectors of the adjacency matrix of the Erdős–Rényi graph G(N,d/N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {G}}(N,d/N)$$\\end{document} for logN≪d≲logN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{\\log N} \\ll d \\lesssim \\log N$$\\end{document}. We show the existence of a localized phase, where each eigenvector is exponentially localized around a single vertex of the graph. This complements the completely delocalized phase previously established in Alt et al. (Commun Math Phys 388(1):507–579, 2021). For large enough d, we establish a mobility edge by showing that the localized phase extends up to the boundary of the delocalized phase. We derive explicit asymptotics for the localization length up to the mobility edge and characterize its divergence near the phase boundary. The proof is based on a rigorous verification of Mott’s criterion for localization, comparing the tunnelling amplitude between localization centres with the eigenvalue spacing. The first main ingredient is a new family of global approximate eigenvectors, for which sharp enough estimates on the tunnelling amplitude can be established. The second main ingredient is a lower bound on the spacing of approximate eigenvalues. It follows from an anticoncentration result for these discrete random variables, obtained by a recursive application of a self-improving anticoncentration estimate due to Kesten.

Finite element modeling of the eigenvibrations of the square plate with attached oscillator

For the problem on eigenvibrations of the plate with an attached oscillator, the new symmetric linear variational statement is proposed. It is established the existence of the sequence of positive eigenvalues of finite multiplicity with limit point at infinity and the corresponding complete orthonormal system of eigenvectors. The new symmetric scheme of the finite element method with Hermite finite elements is formulated. Error estimates consistent with the solution smoothness for the approximate eigenvalues and approximate eigenvectors are proved. The results of numerical experiments illustrating the influence of the solution smoothness on the computation accuracy are presented.

On the distribution of Born transmission eigenvalues in the complex plane

We analyze an approximate interior transmission eigenvalue problem in R d for d = 2 or d = 3, motivated by the transmission problem of a transformation optics-based cloaking scheme and obtained by replacing the refractive index with its first order approximation, which is an unbounded function. Using the radial symmetry we show the existence of (infinitely many) complex transmission eigenvalues and prove their discreteness. Moreover, it is shown that there exists a horizontal strip in the complex plane around the real axis, that does not contain any transmission eigenvalues.

Neural network-aided receivers for soliton communication impaired by solitonic interaction.

In this paper, different neural network-based methods are proposed to improve the achievable information rate in amplitude-modulated soliton communication systems. The proposed methods use simulated data to learn effective soliton detection by suppressing nonlinear impairments beyond amplifier noise, including intrinsic inter-soliton interaction, Gordon-Haus effect-induced timing jitter, and their combined impact. We first present a comprehensive study of these nonlinear impairments based on numerical simulations. Then, two neural network designs are developed based on a regression network and a classifier. We estimate the achievable information rates of the proposed learning-based soliton detection schemes as well as two model-based benchmark schemes, including the nonlinear Fourier transform eigenvalue estimation and continuous spectrum-aided eigenvalue estimation schemes. Our results demonstrate that both learning-based designs lead to substantial performance gains when compared to the benchmark schemes. Importantly, we highlight that exploiting the channel memory, introduced by solitonic interactions, can yield additional gains in the achievable information rate. Through a comparative analysis of the two neural network designs, we establish that the classifier design exhibits superior adaptability to interaction impairment and is more suitable for symbol detection tasks in the context of the investigated scenarios.

Space–Time Adaptive Processing Using Deep Neural Network-Based Shrinkage Algorithm Under Small Training Samples

This correspondence discusses a space-time adaptive processing (STAP) method using deep neural network (DNN) and shrinkage algorithm (SA) under small training samples. A DNN-based SA is developed to estimate the clutter-plus-noise covariance matrix (CNCM) from the sample covariance matrix. This technique establishes a learning-based nonlinear mapping between the sample eigenvalues and the optimal shrunk eigenvalues and thus provides an accurate CNCM estimation. Specifically, a DNN consisting of cascaded convolutional neural network (CNN) and fully connected network (FCN) is designed, where the CNN performs the feature extraction and the FCN derives the estimation of the shrunk eigenvalues. To improve the CNN performance, a deep residual learning strategy is introduced into CNN. Due to the lack of labeled data, an airborne radar system is simulated for the construction of dataset consisting of sample eigenvalues and optimal shrunk eigenvalues. Numerical results demonstrate that the proposed DNN-based SA STAP has superior performance in comparison with the state-of-the-art SA STAP methods.

Hybrid quantum algorithms for flow problems.

For quantum computing (QC) to emerge as a practically indispensable computational tool, there is a need for quantum protocols with end-to-end practical applications-in this instance, fluid dynamics. We debut here a high-performance quantum simulator which we term QFlowS (Quantum Flow Simulator), designed for fluid flow simulations using QC. Solving nonlinear flows by QC generally proceeds by solving an equivalent infinite dimensional linear system as a result of linear embedding. Thus, we first choose to simulate two well-known flows using QFlowS and demonstrate a previously unseen, full gate-level implementation of a hybrid and high precision Quantum Linear Systems Algorithms (QLSA) for simulating such flows at low Reynolds numbers. The utility of this simulator is demonstrated by extracting error estimates and power law scaling that relates [Formula: see text] (a parameter crucial to Hamiltonian simulations) to the condition number [Formula: see text] of the simulation matrix and allows the prediction of an optimal scaling parameter for accurate eigenvalue estimation. Further, we include two speedup preserving algorithms for a) the functional form or sparse quantum state preparation and b) in situ quantum postprocessing tool for computing nonlinear functions of the velocity field. We choose the viscous dissipation rate as an example, for which the end-to-end complexity is shown to be [Formula: see text], where [Formula: see text] is the size of the linear system of equations, [Formula: see text] is the solution error, and [Formula: see text] is the error in postprocessing. This work suggests a path toward quantum simulation of fluid flows and highlights the special considerations needed at the gate-level implementation of QC.

Tiling and PCA Strategy for Clustering-Based High-Dimensional Gaussian Filtering

Edge-preserving filtering is an essential tool for image processing applications and has various types of filtering. High-dimensional Gaussian filtering (HDGF) supports a wide range of edge-preserving filtering. This paper approximates HDGF by clustering with Nyström approximation, tiling, and principal component analysis (PCA) to accelerate HDGF. Also, we compare it with the conventional HDGF approximations and clarify its effective range. We accelerate HDGF by clustering-based constant-time algorithm, which has O(K) order for convolution, where K is the number of clusters. First, we perform PCA for dimensionality reduction and then cluster signals with k-means++. HDGF is decomposed to Gaussian filtering by approximate eigenvalue decomposition of Nyström approximation using the clusters. The Gaussian filtering is performed in a constant-time algorithm. The process is further accelerated by the tiling strategy cooperating with PCA. In our experimental results, we compared three approximated HDGFs: clustering-based HDGF, permutohedral lattice, and Gaussian KD-tree. Also, we evaluated six types of high dimensional signals: RGB, RGB-IR, RGB-D, flash/no-flash, hyperspectral image, and non-local means. The proposed clustering-based HDGF was effective for low/middle-dimensional cases: RGB, RGB-IR, RGB-D, flash/no-flash, and hyperspectral images. Also, tiling with PCA strategy is effective for the conventional permutohedral lattice and Gaussian KD-tree. In the approximation of HDGF, the clustering-based HDGF is the better solution for low/middle-dimensional signals. For the higher-dimensional case of non-local means filtering, the conventional HDGF of the permutohedral lattice with the proposed PCA tiling is effective. Our code and dataset are available at https://norishigefukushima.github.io/TilingPCA4CHDGF/.

Model order estimation based on the correntropy of observation eigenvalues

This paper presents a novel approach for estimating the model order in the presence of observation errors. The proposed method is based on the correntropy estimation of eigenvalues in the observation space. By utilizing the bootstrap method to resample the observations, more precise estimations can be obtained. The next step in the algorithm involves partitioning the observation space, generated by the covariance matrix of mixtures, into two subspaces: the signal subspace and the noise subspace. The order of the model, corresponding to the dimension of the signal subspace, is determined using a correntropy estimator based on kernel functions. Theoretical analysis demonstrates the consistency of the proposed algorithm. Comparative evaluations against existing methods in the literature highlight the superior performance of this information-theoretic approach.

Sequential stabilizing spline algorithm for linear systems: Eigenvalue approximation and polishing

The sequential stabilizing spline (SSS) algorithm is a remarkable algorithm for identifying linear dynamical systems. It can guarantee the stability of an estimated model by polishing the maximum modulus roots of an equation. Therefore, the root structure of an equation plays an important role in the SSS algorithm. Although the traditional power iterative method can be applied to approximate the extremal eigenvalues which have the largest modulus, it has two limitations: (1) the extremal eigenvalues must be real numbers, and (2) the structures of the extremal eigenvalues should be known a priori. In this paper, a novel power iterative method is proposed to approximate the extremal eigenvalues. Compared with the traditional power iterative method, the method in this paper can (1) determine the types of the extremal eigenvalues without prior knowledge of the matrix; (2) approximate the true values of the extremal eigenvalues regardless of their type; (3) become a worthy addition to SSS algorithm and gradient descent algorithm. Simulation examples demonstrate the effectiveness of the proposed method.

Computing Dirichlet eigenvalues of the Schrödinger operator with a PT-symmetric optical potential

We provide estimates for the eigenvalues of non-self-adjoint Sturm–Liouville operators with Dirichlet boundary conditions for a shift of the special potential 4cos ^{2}x+4iVsin 2x that is a PT-symmetric optical potential, especially when |c|=|sqrt{1-4V^{2}}|<2 or correspondingly 0leq V<sqrt {5}/2. We obtain some useful equations for calculating Dirichlet eigenvalues also for |c|geq 2 or equally Vgeq sqrt{5}/2. We discuss our results by comparing them with the periodic and antiperiodic eigenvalues of the Schrödinger operator. We even approximate complex eigenvalues by the roots of some polynomials derived from some iteration formulas. Moreover, we give a numerical example with error analysis.

The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices

The numerical approximation of both eigenvalues and singular values corresponding to a class of totally positive Bernstein–Vandermonde matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices are well studied and investigated in the literature. We aim to present some new results for the numerical approximation of the largest singular values corresponding to Bernstein–Vandermonde, Bernstein–Bezoutian, Cauchy—polynomial-Vandermonde and quasi-rational Bernstein–Vandermonde structured matrices. The numerical approximation for the reciprocal of the largest singular values returns the structured singular values. The new results for the numerical approximation of bounds from below for structured singular values are accomplished by computing the largest singular values of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices. Furthermore, we present the spectral properties of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and structured quasi-rational Bernstein–Vandermonde matrices by computing the eigenvalues, singular values, structured singular values and its lower and upper bounds and condition numbers.