Order Eigenvalue Research Articles

First order structure-preserving perturbation theory for eigenvalues of symplectic matrices: Part II

The first order eigenvalue perturbation theory for structure-preserving perturbations of real or complex symplectic matrices started in [21] is continued and completed. While in [21] all eigenvalues were considered that qualitatively behave similarly under structure-preserving and structure-ignoring perturbations, the focus of this paper is on one particular case when the behavior under structure-preserving perturbations differs significantly from the one under structure-ignoring perturbations. This case occurs for the exceptional eigenvalues ±1 and is caused by restrictions in the possible Jordan structure of symplectic matrices. The main result gives asymptotic expansions of specific perturbed eigenvalues that are not yet covered by the results from [21]. While the main theorem requires that the rank of the perturbation is equal to the rank of its first order term, it is also discussed what can be said if this hypothesis is not satisfied.

KASP-IEva: an intelligent typing evaluation model for KASP primers.

KASP marker technology has been used in molecular marker-assisted breeding because of its high efficiency and flexibility, and an intelligent evaluation model of KASP marker primer typing results is essential to improve the efficiency of marker development on a large scale. To this end, this paper proposes a gene population delineation method based on NTC identification module and data distribution judgment module to improve the accuracy of K-Means clustering, and introduces a decision tree to construct the KASP-IEva primer typing evaluation model. The model firstly designs the NTC identification module and data distribution judgment module to extract four types of data, grouping and categorizing to achieve the improvement of the distinguishability of amplification product signals; secondly, the K-Means algorithm is used to aggregate and classify the data, to visualize the five aggregated clusters and to obtain the morphology location eigenvalues; lastly, the evaluation criteria for the typing effect level are constructed, and the logical decision tree is used to make conditional discrimination on the eigenvalues in order to realize the score prediction. The performance of the model was tested by the KASP marker typing test results of 2519 groups of cotton varieties, and the following conclusions were obtained: the model is able to visualize the aggregation and classification effects of the amplification products of NTC, pure genotypes, heterozygous genotypes, and untyped genotypes, enabling rapid and accurate KASP marker typing evaluation. Comparing and analyzing the model evaluation results with the expert evaluation results, the average accuracy rate of the four grades evaluated by the model was 87%, and the overall evaluation results showed an uneven distribution of the grades with significant differential characteristics. When evaluating 2519 KASP fractal maps, the expert evaluation consumes 15 hours, and the model evaluation only uses 8min27.45s, which makes the model intelligent evaluation significantly better than the expert evaluation from the perspective of time. The establishment of the model will further enhance the application of KASP markers in molecular marker-assisted breeding and provide technical support for the large-scale screening and identification of excellent genotypes.

Exploring universality of the β-Gaussian ensemble in complex networks via intermediate eigenvalue statistics.

The eigenvalue statistics are an important tool to capture localization to delocalization transition in physical systems. Recently, a β-Gaussian ensemble is being proposed as a single parameter to describe the intermediate eigenvalue statistics of many physical systems. It is critical to explore the universality of a β-Gaussian ensemble in complex networks. In this work, we study the eigenvalue statistics of various network models, such as small-world, Erdős-Rényi random, and scale-free networks, as well as in comparing the intermediate level statistics of the model networks with that of a β-Gaussian ensemble. It is found that the nearest-neighbor eigenvalue statistics of all the model networks are in excellent agreement with the β-Gaussian ensemble. However, the β-Gaussian ensemble fails to describe the intermediate level statistics of higher order eigenvalue statistics, though there is qualitative agreement till n<4. Additionally, we show that the nearest-neighbor eigenvalue statistics of the β-Gaussian ensemble is in excellent agreement with the intermediate higher order eigenvalue statistics of model networks.

Linear stability of a compressible flow in a channel

Summary Modal instabilities in a compressible flow through a channel at high Reynolds numbers are studied for three-dimensional (3D) perturbations. In addition to the Tollmien–Schlichting (TS) mode, there exist compressible modes in a channel flow that do not have a counterpart in the incompressible limit. The stability characteristics of these compressible modes, obtained through numerical calculations, are compared with boundary layer and Couette flows that have been previously studied. The dominant compressible instabilities in a channel flow are shown to be viscous in nature, in contrast to compressible boundary layer modes. For general compressible bounded-domain flows, a necessary condition for the existence of neutral modes in the inviscid limit is obtained, and this criterion is used to determine critical Mach numbers below which the compressible modes remain stable. This criterion also delineates a range of wave-angles which could go unstable at a specified Mach number. Asymptotic analysis is carried out for the lower and upper branches of the stability curve in the limit of high Reynolds number for both the T-S and the compressible modes. A common set of relations are identified for the scaling exponents, and the leading order eigenvalues for the unstable modes are obtained through an adjoint-based procedure. The asymptotic analysis shows that the stability boundaries for 3D perturbations at high Reynolds numbers can be calculated from the strain rate and the temperature of the base flow at the wall.

Pole decomposition of BFKL eigenvalue at zero conformal spin and the real part of digamma function

We consider the powers of leading order eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation at zero conformal spin. Using reflection identities of harmonic sums we demonstrate how involved generalized polygamma functions are introduced by pole separation of a rather simple digamma function. This generates higher weight generalized polygamma functions at any given order of perturbative expansion. As a byproduct of our analysis we develop a general technique for calculating powers of the real part of digamma function in a pole separated form.

Behavior of the Poincaré constant along the Polchinski renormalization flow

We control the behavior of the Poincaré constant along the Polchinski renormalization flow using a dynamic version of [Formula: see text]-calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by Klartag and Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general [Formula: see text]-measures and discuss the interpretation in terms of transport maps.

Estimates for eigenvalues of the Neumann and Steklov problems

Abstract We prove Li-Yau-Kröger-type bounds for Neumann-type eigenvalues of the biharmonic operator on bounded domains in a Euclidean space. We also prove sharp estimates for lower order eigenvalues of a biharmonic Steklov problem and of the Laplacian, which directly implies two sharp Reilly-type inequalities for the corresponding first nonzero eigenvalue.

Cheeger estimates of Dirichlet-to-Neumann operators on infinite subgraphs of graphs

In this paper, we study the Dirichlet-to-Neumann operators on infinite subgraphs of graphs. For an infinite subgraph, we prove Cheeger-type estimates for the bottom spectrum of the Dirichlet-to-Neumann operator, and the higher order Cheeger estimates for higher order eigenvalues of the Dirichlet-to-Neumann operator.

Order determination for spiked-type models with a divergent number of spikes

For large dimensional spiked models, the order (number of spikes) determination is an important issue for dimension reduction. The authors propose a generic criterion to estimate the order when the dimension is proportional to the sample size and the order is divergent as the dimension goes to infinity. To handle the divergence of the order, the criterion is defined by location-shift truncated eigenvalues, unlike the existing criteria. They suggest two versions of the criterion: the first defines an objective function that is a sequence of ridge ratios of the defined eigenvalues in order to have a clear separation between the ratio at the true order and other ratios; and the second uses an objective function of double ridge ratios to enhance such a separation. To alleviate the effect of the bias in the scale estimation when the order is large, an iterative procedure is utilized for the estimation. Numerical studies are conducted on spiked population models and spiked Fisher matrices to examine the finite sample performances of the proposed methods.

Orthonormal Bernstein Galerkin technique for computations of higher order eigenvalue problems

The numerical approximation of eigenvalues of higher even order boundary value problems has sparked a lot of interest in recent years. However, it is always difficult to deal with higher-order BVPs because of the presence of boundary conditions. The objective of this work is to investigate a few higher order eigenvalue (Rayleigh numbers) problems utilizing the method of Galerkin weighted residual (MWR) and the effect of solution due to direct implementation of polynomial bases. The proposed method develops a precise matrix formulation for the eighth order eigenvalue and linear electro-hydrodynamic (EHD) stability problems.•The article explores the same for tenth and twelfth order eigenvalue problems.•This method involves computing numerical eigenvalues using Bernstein polynomials as the basis functions.•The novel weighted residual Galerkin technique's performance is numerically validated by comparing it to other numerical/analytical approaches in the literature.

Broadband S-Parameter Simulation Based on the Mixed Finite-Element Method and Symmetric Lanczos Algorithm

In this paper, the symmetric Lanczos algorithm and mixed finite element method are combined to improve the efficiency and accuracy of S-parameter simulation in a broad frequency band. The mixed finite element method based on tree-cotree splitting is adopted to overcome the low-frequency (high-density) breakdown problem, thus ensuring the accuracy of low frequency (high integration density) results. The modified symmetric Lanczos algorithm for second order eigenvalue problem is employed for fast frequency sweep by one matrix inversion, then the field in the entire frequency band is obtained rapidly by solving the reduced order matrix equation. The accuracy of the high frequency results is improved by increasing the number of Lanczos vectors. Numerical simulations of two integrated passive filters (IPD filter) and an interconnect model are shown to verify the hybrid method along with commercial software HFSS.

Spectral stability of multi-solitons for generalized Hamiltonian system I: The Caudrey–Dodd–Gibbon–Sawada–Kotera equation

We consider a generalized Hamiltonian system called the Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation, which is a natural extension of the Korteweg–de Vries (KdV) equation. Our main results are two-fold. Firstly, spectral stability of solitons and multi-solitons are presented by employing some inverse scattering transform techniques. The main difficulty arises from the third order eigenvalue problem of the equation. The second result is a Hamilton–Krein index identity, which verifies a stability criterion established by Maddocks and Sachs (1993) in showing the Lyapunov stability of KdV N-solitons. However, due to generalized Hamiltonian system nature of the CDGSK equation, it does not possess the L2 conservation law. The linearized operators around the multi-solitons have no spectra gap any more. The main ingredient of the proof is new operator identities for higher order linearized Hamiltonians motivated by the Sylvester Law of Inertia. Such operator identities are shown by employing the recursion operators of the CDGSK equation.

Stability of higher order eigenvalues in dimension one

We study stability of the eigenvalues of the generator of a one dimensional reversible diffusion process satisfying some natural conditions. The proof is based on Stein’s method. In particular, these results are applied to the Normal distribution (via the Ornstein–Uhlenbeck process), to Gamma distributions (via the Laguerre process) and to Beta distributions (via the Jacobi process).

When bullwhip increases in the lead time: An eigenvalue analysis of ARMA demand

Problem definition. The impact of lead times on the bullwhip effect produced by the order-up-to (OUT) replenishment policy is studied. Practical relevance. Under general auto-regressive moving average (ARMA) demand, we investigate when the OUT policy possesses an always-increasing-in-the-lead-time bullwhip effect and when it does not. Methodology. A bullwhip measure based on the difference between the demand and order variance is combined with a novel analysis based on the eigenvalues and impulse response of the ARMA demand process. Contribution. We show a positive demand impulse response is a necessary and sufficient condition for an increasing in the lead time bullwhip effect. The ordering of zeros and poles (the eigenvalue ordering) of the z-transform transfer function of the demand process reveals when the demand impulse is positive. To provide further insight, we study ARMA(2,2) demand, which contains six different eigenvalue orderings. Two of these orderings satisfy a sufficient condition (positive demand eigenvalues in a particular order) for a positive impulse response. Two orderings satisfy the inverse of this sufficient condition and do not possess a positive impulse response. The final two orderings do not satisfy the sufficient condition, nor its inverse, but do contain positive impulse responses. Managerial implications. Our findings are important as reducing lead-times is often advocated as an improvement action to reduce the bullwhip effect. By identifying the demand characteristics that lead to a bullwhip effect that increases in the lead time we offer prescriptive advice on when, and when not, to invest in lead time reduction.

The high order spectrum of a graph and its applications in graph colouring and clique counting

D. Cvetković, M. Doob and H. Sachs considered the high order eigenvalue problems of graphs. The high order eigenvalues of a graph G are solutions of the high degree homogeneous polynomial equations derived from G. We propose the adjacency tensor of a graph G and show that the high order eigenvalues of G can be regarded as eigenvalues of . Some results of the spectrum of the adjacency matrix are extended to the spectrum of by using the spectral theory of nonnegative tensors. An upper bound of chromatic number is given via the spectral radius of . Our upper bound is a generalization of Wilf's bound (where is the spectral radius of the adjacency matrix of a graph G) and sharper than the bound of Wilf in some classes of graphs. A formula of the number of cliques of fixed size which involve the spectrum of is obtained.

Isoperimetric bounds for lower-order eigenvalues

New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for further study are also proposed.

Characteristics of Eigenfunctions and Eigenvalues of a Fourth Order Differential Equations with Spectral Parameter in the Boundary Conditions

The characteristics of eigenvalues, and their positions on the real axis are discussed in this article for a fourth order eigenvalue problem with a spectral parameter in the boundary conditions. We have demonstrated that our described problem has a unique solution, as well as, it is shown the characteristics of eigenfunctions and their derivatives in the space

Approximate solutions of the Fourth-Order Eigenvalue Problem

In this paper, the differential transformation (DTM) and the Adomian decomposition (ADM) methods are proposed for solving fourth order eigenvalue problem. This fourth order eigenvalue problem has nonstrongly regular boundary conditions. This the fourth order problem has been examined for p(t) = t, B = 0, a = 0,01 where p(t) ≠ 0 is a complex valued and a ≠ 0 The differential transformation and the Adomian decomposition methods are briefly described. An approximate solution is obtained by performing seven iterations with the Adomian decomposition method. The same number of iterations have been made in the differential transformation method. The approximation results obtained by both methods have been compared with each other. These data have been presented in table. The ADM and the DTM approximation solutions have been shown by plotting in Figure 1. Here, the approaches obtained by using the two methods are found to be in high agreement. Consequently, highly accurate approximate solutions of fourth order eigenvalue problem are obtained. Such good results also revealed that the Adomian decomposition and the differential transformation methods are fast, economical and motivating. The exact solution of the fourth order eigenvalue problem for nonstrongly regular can not be found in the literature. Therefore, this study will give an important idea to determine approximate solution behavior of this fourth order problem.

On a second order eigenvalue problem and its application

A linear eigenvalue problem governed by a second order differential equation with separate and general boundary conditions is considered and a new monotonicity result on the principal eigenvalue with respect to the coefficient of the advection term is established. The main approach is based on the functional proposed by Liu and Lou [16] and a key finding lies in the nice properties of the associated Fréchet operator when confined at suitable points and function spaces. As an application, this monotonicity result is used to study a class of competitive parabolic systems and the so-called “competitive exclusion principle” is observed in a larger parameter region than several existing works, which is a nontrivial improvement.

High Excursions of a Quadratic form for a Gaussian Stationary Vector Process

Exact asymptotic behavior is given for high excursion probabilities of a quadratic form for a zero-mean Gaussian stationary vector process with Pickands’ type covariance matrix in the vicinity of zero. The case of a quadratic form with a positive maximum eigenvalue of order 1 is considered.