scholarly journals An eigenvector-eigenvalue-identity for matrices with a non-semi-simple eigenvalue

2021 ◽  
pp. 1241-1256
Author(s):  
Jia en Ding ◽  
Jiu Ding
Keyword(s):  
Simple Eigenvalue

scholarly journals Improving the condition number of a simple eigenvalue by a rank one matrix

2016 ◽  
Vol 58 ◽  
pp. 7-12
Author(s):  
Rafael Bru ◽  
Rafael Cantó ◽  
Ana M. Urbano
Keyword(s):  
Condition Number ◽  
Simple Eigenvalue ◽  
Rank One

scholarly journals Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

2010 ◽  
Vol 2010 ◽  
pp. 1-23 ◽  
Author(s):  
Josef Diblík ◽  
Denys Ya. Khusainov ◽  
Irina V. Grytsay ◽  
Zdenĕk Šmarda
Keyword(s):  
Lyapunov Functions ◽  
Critical Case ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Simple Eigenvalue ◽  
Discrete Equations ◽  
The Matrix ◽  
The Stability ◽  
Important Characterization

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalueλ=1of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

Simple Eigenvalue Inclusions

2011 ◽  
pp. 129-134
Author(s):  
Krešimir Veselić
Keyword(s):  
Simple Eigenvalue

Perturbation of two-point boundary value problems with eigenvalue parameter in the boundary conditions

1983 ◽  
Vol 94 (3-4) ◽  
pp. 213-219 ◽  
Author(s):  
Lawrence Turyn
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Banach Spaces ◽  
Boundary Value ◽  
Simple Eigenvalue ◽  
Point Boundary ◽  
Eigenvalue Parameter

SynopsisWe discuss smooth changes of eigenvalues under perturbation of the boundary value problems given in the title. The simple eigenvalue criterion is developed in the setting of Banach spaces, so very general perturbations of both the differential equation and the boundary conditions are allowed. Further, we need no assumptions about self-adjointness of the original or perturbed problems. The discussion is concluded with the application of the simple eigenvalue criterion to two examples.

Stability of convection in containers of arbitrary shape

1971 ◽  
Vol 47 (2) ◽  
pp. 257-282 ◽  
Author(s):  
Daniel D. Joseph
Keyword(s):  
Bounded Domain ◽  
Arbitrary Shape ◽  
Side Wall ◽  
Simple Eigenvalue ◽  
Stable Form ◽  
Temperature Contrast ◽  
The Difference ◽  
Infinite Multiplicity ◽  
The Stability ◽  
Linear Stability Problem

When a container of fluid of arbitrary shape is heated from below and the temperature gradient exceeds a critical value ([Rscr ]c2) the conduction solution with no motion becomes unstable and is replaced by convection. The convection may have two forms: one with ‘upflow’ at the centre of the container and one with ‘downflow’ there. Here we study the stability of the two forms of convection. Both forms are here shown to be stable to infinitesimal disturbances. When the viscosity varies with the temperature or the conduction profile is not linear, etc., the steady convection can be driven with finite amplitudes |ε| at subcritical values of the temperature contrast ([Rscr ]2< [Rscr ]c2). This subcritical convection is stable when the convection is strong (|ε| > |ε*| > 0) but is unstable when the convection is feeble (|ε| > |ε*|). Hence, when |ε| > |ε*| and [Rscr ]2< [Rscr ]c2either ‘upflow’ or ‘downflow’, but not both, is stable. When [Rscr ]2> [Rscr ]c2, however, both the ‘upflow’ and the ‘downflow’ can be stable. This contrasts with the corresponding situation which is known to hold when the container is an unbounded layer. In the layer there is only one stable form of convection. The difference between the bounded domain with two forms of convection and the layer with just one stable form is traced to the mathematical property of simplicity of [Rscr ]c2when viewed as an eigenvalue of the linear stability problem for the conduction solution. It is argued that [Rscr ]c2is a simple eigenvalue in most domains, but in the layer [Rscr ]c2can have infinite multiplicity. The explanation of the transition from the bounded domain to the unbounded layer is sought (1) in the chaotic conditions which frequently prevail at the edges of a ‘bounded’ layer and (2) in the fact that in the layer of large horizontal extent, the higher eigenvalues crowd [Rscr ]c2. In the course of the explanation, a new exact solution of the linear Bénard problem in a cylinder with a rigid side wall and a stress-free top and bottom is derived.

scholarly journals Simple eigenvalues and bifurcation for a multiparameter problem

1988 ◽  
Vol 31 (1) ◽  
pp. 77-88 ◽  
Author(s):  
D. F. McGhee ◽  
M. H. Sallam
Keyword(s):  
Banach Spaces ◽  
Simple Eigenvalue ◽  
Linear Mapping ◽  
Spectral Parameters ◽  
Non Linear ◽  
Multiparameter Problem ◽  
Image Position ◽  
Bifurcation Of Solutions

We are concerned with the problem of bifurcation of solutions of a non-linear multiparameter problem at a simple eigenvalue of the linearised problem.Let X and Y be real Banach spaces, and let A, Bi, i = 1, …, n∈B(X, Y). Let : Rn × X → Y be a non-linear mapping. We consider the equationwhereand λ=(λ1, λ2,…,λn) ∈ Rn is an n-tuple of spectral parameters.

A simple eigenvalue formula for the quartic anharmonic oscillator

1985 ◽  
Vol 63 (3) ◽  
pp. 311-313 ◽  
Author(s):  
Richard L. Hall
Keyword(s):  
Lower Bound ◽  
Anharmonic Oscillator ◽  
Simple Eigenvalue ◽  
Approximate Eigenvalue

The eigenvalues Enl(λ) of the Hamiltonian H = −Δ + r2 + λr4 are analysed with the help of "potential envelopes" and "kinetic potentials." The result is the following simple approximate eigenvalue formiula:[Formula: see text]where E ≥ P = (4n + 2l − 1) and Q = 3(An + Bl − C)4/322/3. E is a lower bound to Enl if (A, B, C) = (1, 1/2, 1/4) and a good approximation if (A, B, C) = (1.125, 0.509, 0.218).

Simple Eigenvalue-Equalizing Preprocessor for Broadband Power Inversion Array

1985 ◽  
Vol AES-21 (1) ◽  
pp. 117-127 ◽  
Author(s):  
C.c. Ko
Keyword(s):  
Simple Eigenvalue ◽  
Power Inversion

Analysis of shear lag in cylindrical tubes of arbitrary cross section

1983 ◽  
Vol 389 (1797) ◽  
pp. 259-277
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Bending Moment ◽  
General Purpose ◽  
Simple Eigenvalue ◽  
Arbitrary Cross Section ◽  
Longitudinal Distribution ◽  
Shear Lag ◽  
Cylindrical Tubes ◽  
Polynomial Distribution

A very general analysis is given of the phenomenon of shear lag in thin-walled cylindrical tubes, with single-cell cross sections of arbitrary shape, containing any number of concentrated longitudinal booms that carry direct stress only, and subjected to any longitudinal distribution of bending moment and torque. Two equations relating the distributions of direct and shearing stresses on the cross section are derived for the most general case where the tube is non-uniform because of an arbitrary longitudinal variation of wall thicknesses and boom areas. These equa­tions, which are remarkably simple in view of their generality, incor­porate all the requirements of equilibrium and compatibility and provide corrections to the stresses, curvature and twist calculated from the engineers’ theory of bending and torsion. They also govern the distri­bution of stresses arising from the application of self-equilibrating systems of tractions to the end cross sections. Exact solutions are ob­tained for the case of a uniform, but otherwise arbitrary, cross section under any polynomial distribution of bending moment and torque, and it is shown how conditions at the end cross sections can be satisfied with the aid of solutions of a simple eigenvalue problem. The equations are in a particularly ideal form for incorporating into a general purpose com­puter program for the automatic numerical solution of any problem of this type.

Sign in / Sign up

Export Citation Format

Share Document