Approximate Eigenvalues Research Articles

Topological defects with generalized Hulthen–Coulomb-inverse quadratic Yukawa potential on eigenvalue solution under Aharonov–Bohm flux field

In this work, we solve the radial Schrödinger wave equation in three dimensions under Aharonov–Bohm (AB)-flux field with potential superposition of generalized q-deformed Hulthen potential, Coulomb potential, and inverse quadratic Yukawa potential in a point-like defect. We determine the approximate eigenvalue solution using the parametric Nikiforov–Uvarov (NU) method and analyze the effects of topological defect and the magnetic flux field with this superposed potential. We show an analogous of the AB effect because the eigenvalue solution depends on the geometric quantum phase and bound state solutions are possible under condition. Finally, we utilize the approximate eigenvalue solution to some molecular potential models, such as Deng–Fan potential and inverse quadratic Hulthen potential and analyze the effects on the energy levels and the radial wave functions.

Fast asymptotic algorithm for real-time causal connectivity analysis of multivariate systems and signals

The purpose of this work is to optimize the current state-of-the-art asymptotic algorithm of connectivity Granger-causality measures in the frequency-domain, such as the Directed Transfer Function (DTF), Partial Directed Coherence (PDC), and their variants. These measures stem from the modeling of multidimensional time series by multivariate autoregressive model. Surrogate and asymptotic analysis are the most frequently used methods to quantify the statistical significance of such derived interactions, a critical step for validation of the results. The current asymptotic algorithms run fairly fast on low-dimensional datasets but become impractical for high-dimensional datasets due to the involved computational time and memory demand. This is a huge limitation in the application of these connectivity measures to the fields dealing with numerous concurrently acquired signals from probing of complex systems such as the human brain. Here, we optimized the current algorithms for the fast asymptotic analysis of these connectivity measures and achieved a reduction of their computation time by at least three orders of magnitude. The optimizations were accomplished by decreasing the dimension of the involved matrices, eliminating the complicated functions (e.g., eigenvalue estimation and Cholesky factorization), and variable separation. The superior performance of the proposed optimized algorithm is shown with simulation examples.

H-type foliations

With a view toward sub-Riemannian geometry, we introduce and study H-type foliations. These structures are natural generalizations of K-contact geometries which encompass as special cases K-contact manifolds, twistor spaces, 3K-contact manifolds and H-type groups. Under an horizontal Ricci curvature lower bound on these structures, we prove a sub-Riemannian diameter upper bounds and first eigenvalue estimates for the sub-Laplacian. Then, using a result by Moroianu-Semmelmann [38], we classify the H-type foliations that carry a parallel horizontal Clifford structure. Finally, we prove an horizontal Einstein property and compute the horizontal Ricci curvature of these spaces in codimension more than 2.

Heisenberg-limited quantum phase estimation of multiple eigenvalues with few control qubits

Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentially-large sparse matrices.The maximum rate at which these eigenvalues may be learned, –known as the Heisenberg limit–, is constrained by bounds on the circuit complexity required to simulate an arbitrary Hamiltonian. Single-control qubit variants of quantum phase estimation that do not require coherence between experiments have garnered interest in recent years due to lower circuit depth and minimal qubit overhead. In this work we show that these methods can achieve the Heisenberg limit, also when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a `phase function' g(k)=∑jAjeiϕjk with unknown eigenphases ϕj and overlaps Aj at quantum cost O(k), we show how to estimate the phases {ϕj} with (root-mean-square) error δ for total quantum cost T=O(δ−1). Our scheme combines the idea of Heisenberg-limited multi-order quantum phase estimation for a single eigenvalue phase [Higgins et al (2009) and Kimmel et al (2015)] with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series analysis for the QEEP problem [Somma (2019)] or the matrix pencil method. For our algorithm which adaptively fixes the choice for k in g(k) we prove Heisenberg-limited scaling when we use the time-series/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well.

Spectral estimation for Hamiltonians: a comparison between classical imaginary-time evolution and quantum real-time evolution

We consider the task of spectral estimation of local quantum Hamiltonians. The spectral estimation is performed by estimating the oscillation frequencies or decay rates of signals representing the time evolution of states. We present a classical Monte Carlo (MC) scheme which efficiently estimates an imaginary-time, decaying signal for stoquastic (i.e. sign-problem-free) local Hamiltonians. The decay rates in this signal correspond to Hamiltonian eigenvalues (with associated eigenstates present in an input state) and can be extracted using a classical signal processing method like ESPRIT. We compare the efficiency of this MC scheme to its quantum counterpart in which one extracts eigenvalues of a general local Hamiltonian from a real-time, oscillatory signal obtained through quantum phase estimation circuits, again using the ESPRIT method. We prove that the ESPRIT method can resolve S = poly(n) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) quantum and classical effort through the quantum phase estimation (QPE) circuits, assuming efficient preparation of the input state. We prove that our MC scheme plus the ESPRIT method can resolve S = O(1) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) purely classical effort for stoquastic Hamiltonians, requiring some access structure to the input state. However, we also show that under these assumptions, i.e. S = O(1) eigenvalues, assuming a 1/poly(n) gap between them and some access structure to the input state, one can achieve this with poly(n) purely classical effort for general local Hamiltonians. These results thus quantify some opportunities and limitations of MC methods for spectral estimation of Hamiltonians. We numerically compare the MC eigenvalue estimation scheme (for stoquastic Hamiltonians) and the quantum-phase-estimation-based eigenvalue estimation scheme by implementing them for an archetypal stoquastic Hamiltonian system: the transverse field Ising chain.

Inverse Nodal Problem for p-Laplacian String Equation with Prüfer Substitution

We consider an inverse nodal problem for p-Laplacian string equation under some boundary conditions. Asymptotic formulas for eigenvalues and nodal parameters are constructed by modified Pr¨ufer substitution. The most important process is to apply modified Pr¨ufer substitution to get an exhaustive asymptotic estimate for eigenvalues. Moreover, a reconstruction formula for density function of p-aplacian string equation is obtained by nodal parameters. Generated outcomes are the generalization of the known string problem.

Existence and multiplicity of solutions to Dirichlet problem for semilinear subelliptic equation with a free perturbation

This paper is concerned with the existence and multiplicity results for the semilinear subelliptic equation with free perturbation term. By using the degenerate Rellich-Kondrachov compact embedding theorem, the precise lower bound estimate of Dirichlet eigenvalues for the finitely degenerate elliptic operator and minimax method, we obtain the existence and multiplicity of weak solutions for the problem.

Approximating Matrix Eigenvalues by Subspace Iteration with Repeated Random Sparsification

Traditional numerical methods for calculating matrix eigenvalues are prohibitively expensive for high-dimensional problems. Iterative random sparsification methods allow for the estimation of a single dominant eigenvalue at reduced cost by leveraging repeated random sampling and averaging. We present a general approach to extending such methods for the estimation of multiple eigenvalues and demonstrate its performance for several benchmark problems in quantum chemistry.

Approximate eigenvalue solutions with diatomic molecular potential under topological defects and Aharonov-Bohm flux field: application for some known potentials

In this paper, we determine the approximate eigenvalue bound state solution of the radial Schrodinger equation in three-dimension in the presence of the Aharonov-Bohm flux field with two-term diatomic molecular potential in point-like global monopole (PGM) defect. We analyse the effects of factors, such as topological defects, background curvature associated with topological defects as well as the potential. We also show that the energy levels shift due to the presence of the magnetic flux which gives us an analogue of the Aharonov-Bohm effect for the bound state. Finally, we apply the quantum system for some known potential models, such as q-deformed Hulthen-type potential and Manning-Rosen potential, and discussed the effects of various factors on the eigenvalue solutions.

Estimates of eigenvalues of an elliptic differential system in divergence form

In this paper, we compute universal estimates of eigenvalues of a coupled system of elliptic differential equations in divergence form on a bounded domain in Euclidean space. As an application, we show an interesting case of rigidity inequalities of the eigenvalues of the Laplacian, more precisely, we consider a countable family of bounded domains in Gaussian shrinking soliton that makes the behavior of known estimates of the eigenvalues of the Laplacian invariant by a first-order perturbation of the Laplacian. We also address the Gaussian expanding soliton case in two different settings. We finish with the special case of divergence-free tensors which is closely related to the Cheng–Yau operator.

Eigenvalues of the Birman-Schwinger operator for singular measures: The noncritical case

For operators of the form T=TA,P=A⁎PA with pseudodifferential operator A of negative order −l in a domain in RN, 2l≠N, and a singular measure P, an estimate of eigenvalues is found with an order depending on the dimensional characteristics of the measure P and the coefficient depending on an integral norm of the density of P with respect to the Hausdorff measure of an appropriate dimension. These estimates are used to establish asymptotic formulas for the eigenvalues of T for the case when P is supported on a Lipschitz surface of some codimension and on certain sets of a more complicated structure. In one of applications, a version of the CLR estimate for singular measures is proved.

Diagnosis of Broken Bars in V/F Control Induction Motor Drive Using Wavelets and EEV Estimation for Electric Vehicle Applications

The induction motor (IM) defect diagnosis has been an important field of research in recent years. The development in control circuits for IM has piqued the interest of industrialists and researchers. This paper presents a method for detecting and quantifying broken rotor bar (BRB) faults via wavelets and energy Eigen value (EEV) estimation in voltage/frequency control-fed IM. The fast Fourier transform (FFT) extracts the signal’s amplitude and frequency components, while the discrete wavelet transform (DWT) decomposes it. In this paper, the energy estimation for each level of breakdown and the method to overcome the diagnose faults are explained. The EEV of the motor current of the signal determines the fault’s severity and provides a better method for identifying the faults. The usage of a single current sensor is a gain of this technology. With a fluctuating load, we can identify the issue and the number of broken bars via online. After processing of DWT, the faulty BRB’s stator current signal is suppressed to 91% in amplitude when compared to existing techniques. Simulation and experimental results have proved that the proposed method’s stability, durability, and resilience.

An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem

AbstractIn this article, an effective finite element method based on dimension reduction scheme is proposed for a fourth-order Steklov eigenvalue problem in a circular domain. By using the Fourier basis function expansion and variable separation technique, the original problem is transformed into a series of radial one-dimensional eigenvalue problems with boundary eigenvalue. Then we introduce essential polar conditions and establish the discrete variational form for each radial one-dimensional eigenvalue problem. Based on the minimax principle and the approximation property of the interpolation operator, we prove the error estimates of approximation eigenvalues. Finally, some numerical experiments are provided, and the numerical results show the efficiency of the proposed algorithm.

Dynamic Investigation and Resonant Frequency Determination of Shaft Bending Fatigue Testing Machine

The testing machine of shaft bending fatigue is widely used to test the fatigue limit and life of various shaft-type parts in mechanical engineering. Machine dynamic characteristics and system bending resonant frequency are very important for the design, testing, and application of the testing machine, but existing complex finite element calculation and aimless large-range frequency scanning in testing are the available methods to solve these two problems helplessly and inefficiently. A novel analytical model of nonlinear coupling dynamic system is built for the fatigue testing machine in this paper. The nonlinear dynamic model is simulated and investigated numerically, then it is reasonably linearized and reduced as a linear one, and the corresponding error analysis and application range are presented. By using eigenvalue approximation, the shaft bending resonant frequency formula is obtained, then the formula is compared and verified with the results from the nonlinear system simulation and finite element software, and it can improve the machine design and testing efficiency obviously. The analytical dynamic modeling and linearizing, and the given mode resonant frequency approximating, also provide a reference for many similar vibration machines.

Global Properties of Eigenvalues of Parametric Rank One Perturbations for Unstructured and Structured Matrices II

We present in this note a correction to Theorem 17 in Ran and Wojtylak (Compl. Anal. Oper. Theory 15:44, 2021) and sharpen the estimates for eigenvalues of parametric rank one perturbations given in that theorem.

Stochastic approximation of eigenvectors and eigenvalues of the Q-symmetric expectation of a random matrix

We establish an almost sure convergence theorem of the stochastic approximation process of Oja for estimating eigenvectors of the Q-symmetric expectation of a random matrix, under a correlation model between the incoming random matrices. This theorem generalizes previous theorems and extends them to the case where the metric Q is unknown and estimated online in parallel. We apply it to streaming principal component analysis of a random vector Z, when a mini-batch of observations of Z is used at each step or all the observations up to the current step. We deal with the case of streaming generalized canonical correlation analysis, with a metric estimated online in parallel.

New Fourth Order Postprocessing Techniques for Plate Bending Eigenvalues by Morley Element

In this paper, we propose and analyze the extrapolation method and asymptotically exact a posterior error estimate for eigenvalues of the Morley element. We establish an asymptotic expansion of eigenvalues and prove an optimal result for this expansion and the corresponding extrapolation method. We also design an asymptotically exact a posterior error estimate and propose new approximate eigenvalues with higher accuracy by utilizing this a posteriori error estimate. Finally, several numerical experiments are considered to confirm the theoretical results and compare the performance of the proposed methods.

Evaluation Algorithm of Volleyball Players’ Competitive Ability Based on the Random Matrix Model

It is the trend of the development of modern competitive sports to put scientific and technological analysis methods and means into the study of volleyball, and it is also one of the powerful guarantee ways to promote the competitive level of all countries. The random matrix model algorithm has unique advantages to construct the team’s collective technical and tactical ability structure model. The quantitative relationship of the model describes the relationship between the technical and tactical ability structure and the result of victory and defeat and makes the advantages and disadvantages of the team clear, which is conducive to the subsequent targeted training and improvement. The technical and tactical abilities of the teams in different seasons were input to verify the prediction accuracy of the model for the teams in different seasons. In the face of the rapidly changing game situation, the coach team timely transmits the adjusted technical and tactical strategies to the players on the field and deals with the changes accurately and effectively. After the game, the opponent’s strengths and weaknesses should be clarified, and the team’s daily training details should be summarized to provide reference for the cultivation of collective technical and tactical consciousness. The random sample covariance matrix of the random monitoring matrix is constructed and the maximum and minimum eigenvalues of the sample covariance matrix are solved. The ratio of characteristic values is used to construct the detection index of characteristic values, and the detection threshold algorithm of characteristic values is determined to judge the competitive ability of volleyball players. In the case of false alarm rate and matrix size, based on Tracy-Widom distribution characteristics, the maximum eigenvalue and minimum eigenvalue approximations of sample covariance matrix are used to improve the eigenvalue index detection threshold algorithm, and the influence of false alarm rate, matrix size, and other parameters on the improved eigenvalue index detection threshold is further studied. Then, Iris data set was used to verify the effectiveness of the algorithm in terms of accuracy, recall rate, and comprehensive effective value, and the validation results proved that the accuracy of the algorithm reached more than 90%.

Density function of numerical solution of splitting AVF scheme for stochastic Langevin equation

In this article, we study the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. We first show the existence of the density function of the numerical solution by proving its exponential integrability property, Malliavin differentiability and the almost surely non-degeneracy of the associated Malliavin covariance matrix. Then the smoothness of the density function is obtained through a lower bound estimate of the smallest eigenvalue of the corresponding Malliavin covariance matrix. Meanwhile, we derive the optimal strong convergence rate in every Malliavin–Sobolev norm of the numerical solution via Malliavin calculus. Combining the strong convergence result and the smoothness of the density functions, we prove that the convergence order of the density function of the numerical scheme coincides with its strong convergence order.

Virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media

In this paper, we propose a conforming virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media. By using [Formula: see text]-coercivity theory, the spectral approximation theory of compact operator and the projection and interpolation error estimates, we prove the spectral convergence of the discrete scheme and the optimal a priori error estimates for the discrete eigenvalues and eigenfunctions. The virtual element method has great flexibility in handling polygonal meshes, which motivates us to construct a fully computable a posteriori error estimator for the virtual element method. Then the upper bound of the approximation error is derived from the residual equation and the inf-sup condition. In turn, the related lower bound is established by using the bubble function strategy. Finally, we provide numerical examples to verify the theoretical results, including the optimal convergence of the virtual element scheme on uniformly refined meshes and the efficiency of the estimator on adaptively refined meshes.