Polynomial Characteristic Equation Research Articles

A Theory of High-Order Statistics-Based Virtual Dimensionality for Hyperspectral Imagery

Virtual dimensionality (VD) has received considerable interest in its use of specifying the number of spectrally distinct signatures present in hyperspectral data. Unfortunately, it never defines what such a signature is. For example, various targets of interest, such as anomalies and endmembers, should be considered as different types of spectrally distinct signatures and have their own different values of VD. Specifically, these targets are insignificant in terms of signal energies due to their relatively small populations. Accordingly, their contributions to second-order statistics (2OS) are rather limited. In this case, 2OS-based methods such as eigen-approaches to determine VD may not be effective in determining how many such type of signal sources as spectrally distinct signatures are. This paper develops a new theory that expands 2OS-VD theory to a high-order statistics (HOS)-based VD, called HOS-VD theory. Since there is no counterpart of the characteristic polynomial equation used to find eigenvalues in 2OS available for HOS, a direct extension is inapplicable. This paper re-invents a wheel by finding actual targets directly from the data rather than eigenvectors/singular vectors used in 2OS-VD theory which do not represent any real targets in the data. Consequently, comparing to 2OS-VD theory which can only be used to estimate the value of VD without finding real targets, the developed HOS-VD theory can accomplish both of tasks at the same time, i.e., determining the value of VD as well as finding actual targets directly from the data.

Stability and Bifurcation Analysis in Leukopoiesis Models with Two Delays

We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of Hopf bifurcations which leads to periodic solutions.

A simple approach for determining the eigenvalues of the fourth-order Sturm–Liouville problem with variable coefficients

In this paper, a simple and efficient approach is presented to compute the eigenvalues of the fourth-order Sturm–Liouville equations with variable coefficients. By transforming the governing differential equation to a system of algebraic equation, we can get the corresponding polynomial characteristic equations for kinds of boundary conditions based on the polynomial expansion and integral technique. Moreover, the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples for estimating eigenvalues are given. The convergence and effectiveness of the method are confirmed by comparing numerical results with the exact and other existing numerical results.

New Bifurcation Critical Criterion of Flip‐Neimark‐Sacker Bifurcations for Two‐Parameterized Family of n‐Dimensional Discrete Systems

A new bifurcation critical criterion of flip‐Neimark‐Sacker bifurcation is proposed for detecting or anticontrolling this type of codimension‐two bifurcation of discrete systems in a general sense. The criterion is built on the properties of coefficients of characteristic equations instead of the properties of eigenvalues of Jacobian matrix of nonlinear system, which is formulated using a set of simple equalities and inequalities consisting of the coefficients of characteristic polynomial equation. The inequality conditions enable us to easily pick off the fake parameter domain whereas the equality conditions are used to accurately locate the critical bifurcation point. In particular, after the bifurcation parameter piont is determined, the inequality conditions can be used to figure out the feasible region of other system parameters. Thus, the criterion is suitable for two‐parameterized family of n‐dimensional discrete systems. As compared with the classical critical criterion (or definition) of flip‐Neimark‐Sacker bifurcation stated in terms of the properties of eigenvalues, the proposed criterion is preferable in anticontrolling or detecting the existence of flip‐Neimark‐Sacker bifurcation in high‐dimension nonlinear systems, due to its explicit parameter mechanism of the bifurcation.

Approximating characteristic equations for autonomous systems of differential equations with aftereffect

In this paper we construct approximating polynomial characteristic equations for a linear autonomous system with aftereffect. The procedures for constructing approximating characteristic equations use analytic representations of resolvents of infinitesimal operators and the theory of characteristic determinants and perturbation determinants in a separable Hilbert space.

Transient Waves Due to Thermal Sources in a Generalized Piezothermoelastic Half-Space

This paper is devoted to the study of disturbances due to impact and continuous strip thermal sources, tem- perature or temperature gradient input acting on the rigidly fixed and charge free(open circuit) surface of a homogeneous, transversely isotropic, thermally conducting, generalized piezothermoelastic half-space. The Laplace and Fourier transforms technique have been employed to solve the model consisting of partial dif- ferential equations and boundary conditions in the transformed domain. In order to obtain the results in the physical domain the quadratic complex polynomial characteristic equation corresponding to the associated system of coupled ordinary differential equations has been solved by using DesCartes’ algorithm with the help of irreducible Cardano’s method. The inverse transform integrals are evaluated by using numerical technique consisting of Fourier series approximation and Romberg integration. The temperature change, stresses and electric potential so obtained in the physical domain are computed numerically and presented graphically for cadmium selenide (CdSe) material. The study may find applications in smart structures, pie- zoelectric filters, resonators, transducers, sensing devices and vibration control.

Buckling load of tapered fibers subjected to axially distributed load

This paper studies buckling behavior of non-uniform fiber columns under axially distributed compressive load and presents a new approach for determining buckling load. For free-built-in fibers, a Fredholm equation is derived from solving the governing equation with end constraints. Critical load and critical length can be evaluated by seeking the lowest eigenvalue of the resulting equation. For the cases of the cross-sectional moment of inertia and axially distributed load as power functions with respect to the axial distance from the free end, a necessary condition for buckling is derived, and a polynomial characteristic equation is then obtained and solved. The effect of the weight and axial profile of tapered fibers on the critical length is discussed.

Stability and Hopf Bifurcation for a Cell Population Model with State-Dependent Delay

We propose a mathematical model describing the dynamics of a hematopoietic stem cell population. The method of characteristics reduces the age-structured model to a system of differential equations with a state-dependent delay. A detailed stability analysis is performed. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained using a Lyapunov–Razumikhin function. A unique positive steady state is shown to appear through a transcritical bifurcation of the trivial steady state. The analysis of the positive steady state behavior, through the study of a first order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation and gives criteria for stability switches. A numerical analysis confirms the results and stresses the role of each parameter involved in the system on the stability of the positive steady state.

Elasto-thermodiffusive (ETNP) surface waves in semiconductor materials

The present article is devoted to investigate the propagation of elasto-thermodiffusive (ETNP) surface waves in a homogeneous isotropic, thermally conducting semiconductor material of half-space with relaxation of heat and charge carrier fields. The secular equation, a more general functional relation, that governs the propagation of elasto-thermodiffusive (ETNP) surface waves in homogeneous isotropic, thermoelastic semiconductor material halfspace with relaxation of heat and charge carrier fields has been derived by solving a system of coupled partial differential equations. A hybrid numerical technique consisting of Descartes algorithm for solving complex polynomial characteristic equation along with functional iteration scheme has been successfully used to solve the secular equation in order to obtain dispersion curves, attenuation coefficient and specific loss factor of energy dissipation for p-type germanium (Ge) semiconductor. Some particular forms of the general secular equation governing the propagation of elasto-thermodiffusive (ETN/ETP), thermoelastic (ET), elastodiffusive (EP/EN) and thermodiffusive (TP/TN) surface waves have been also deduced and discussed. In order to illustrate the analytical development, the numerical solution of the secular equation and other relevant relations under different situations is also carried out for Ge semiconductor materials to characterize the elasto-thermodiffusive (ETP) and thermodiffusive (TP) surface waves. The computer simulated results have been presented graphically in respect of the dispersion curves, attenuation coefficient and specific loss factor.

Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch

A nonlinear system of two delay differential equations is proposed to model hematopoietic stem cell dynamics. Each equation describes the evolution of a sub-population, either proliferating or nonproliferating. The nonlinearity accounting for introduction of nonproliferating cells in the proliferating phase is assumed to depend upon the total number of cells. Existence and stability of steady states are investigated. A Lyapunov functional is built to obtain the global asymptotic stability of the trivial steady state. The study of eigenvalues of a second degree exponential polynomial characteristic equation allows to conclude to the existence of stability switches for the unique positive steady state. A numerical analysis of the role of each parameter on the appearance of stability switches completes this analysis.

Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch

A nonlinear system of two delay differential equations is proposed to model hematopoietic stem cell dynamics. Each equation describes the evolution of a sub-population, either proliferating or nonproliferating. The nonlinearity accounting for introduction of nonproliferating cells in the proliferating phase is assumed to depend upon the total number of cells. Existence and stability of steady states are investigated. A Lyapunov functional is built to obtain the global asymptotic stability of the trivial steady state. The study of eigenvalues of a second degree exponential polynomial characteristic equation allows to conclude to the existence of stability switches for the unique positive steady state. A numerical analysis of the role of each parameter on the appearance of stability switches completes this analysis.

Modelling Hematopoiesis Mediated by Growth Factors With Applications to Periodic Hematological Diseases

Hematopoiesis is a complex biological process that leads to the production and regulation of blood cells. It is based upon differentiation of stem cells under the action of growth factors. A mathematical approach of this process is proposed to understand some blood diseases characterized by very long period oscillations in circulating blood cells. A system of three differential equations with delay, corresponding to the cell cycle duration, is proposed and analyzed. The existence of a Hopf bifurcation at a positive steady-state is obtained through the study of an exponential polynomial characteristic equation with delay-dependent coefficients. Numerical simulations show that long-period oscillations can be obtained in this model, corresponding to a destabilization of the feedback regulation between blood cells and growth factors, for reasonable cell cycle durations. These oscillations can be related to observations on some periodic hematological diseases (such as chronic myelogenous leukemia, for example).

Periodic oscillations in leukopoiesis models with two delays

The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analysing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients.

On the Coefficients of the Max-Algebraic Characteristic Polynomial and Equation

No polynomial algorithms are known for finding the coefficients of the characteristic polynomial and characteristic equation of a matrix in max-algebra. The following are proved: (1) The task of finding the max-algebraic characteristic polynomial for permutation matrices is NP-complete. (2) The task of finding the lowest order finite term for a {0, -∞} matrix can be converted to the assignment problem. (3) The task of finding the max-algebraic characteristic equation of a {0, -∞} matrix can be converted to that of finding the conventional characteristic equation for a {0, 1} matrix and thus it is solvable in polynomial time.

Analytical results for crystalline electric field eigenvalues of trivalent rare-earth ions using computer algebra: application to the magnetism of PrX2 ( X = Mg, Al, Ru, Rh, Pt)

The eigenvalues of the Crystalline Electric Field (CEF) Hamiltonian with cubic symmetry are analytically obtained for trivalent rare-earth ions of ground state J= 5 2 , 7 2 , 4, 9 2 , 6, 15 2 and 8, via a Computer Algebra approach. In the presence of both CEF and an effective exchange field, Computer Algebra still allows a partial factorization of the characteristic polynomial equation associated to the total Hamiltonian, a result of interest to the study of the magnetic behavior of rare-earth intermetallics. An application to the PrX 2 intermetallic compounds ( X = Mg, Al, Ru, Rh, Pt) is reported.

Two real critical constraints for real parameter margin computation

A new approach to computing real parameter margins of stabilized dynamical systems with multilinear!} uncertain parameters is presented. A concept of two real critical constraints is introduced to solve the problem of determining the largest stable hypercube in parameter space that touches the stability boundary on one of its corners. The proposed approach is based on sufficient conditions for checking for critical instability only in the corner directions of the parameter space hypercube. Several examples are used to illustrate the proposed concept and approach. HIS paper is concerned with the problem of computing the structured singular values /i for uncertain dynamical systems.1 In particular, the problem of computing oo-norm real parameter margins (or real /*) is investigated, which is of much current research interest.2'14 The real /z problem is essentially the same as the problem of determining the largest stable hypercube in the uncertain parameter space. We exploit the concept of separating the real and imaginary parts of a characteristic polynomial equation. The resulting two equations are referred to as the two real critical constraints, and the vertex (or corner) property of each constraint equation is utilized. The paper is organized as follows: In Sec. II, we introduce the two real critical constraints, the two-constraint real /*, and the single-constraint real /z. In Sec. Ill, we show that the single-constraint real /* always attains its value at a corner of the parameter space hypercube of multilinearly uncertain systems at a given frequency. Section III contains the main result: a sufficient condition for the critical instability to occur at a corner of the parameter space hypercube of multilinearly uncertain systems. In Sec. IV, we apply the results of Sees. II and III to the problem of computing real parameter margins of several different types of characteristic polynomial. In Sec. V, several examples in the literature are used to illustrate the proposed concept and approach.

Defective eigensolutions and stiffness matrices

The equilibrium configuration of a structural member is governed by a set of partial differential equations, which can be reduced to a set of ordinary differential equations depending on one spatial parameter alone by means of the Kantorovich method. The analytical solutions of the resulting boundary value problem are not straight forward when the associated eigenproblem is defective due to the insufficient number of eigenvectors. For the defective case, solution methods for uniform members are suggested. When the analytical solutions are used as shape functions, exact stiffness matrices are obtained. These matrices may be parametric to produce a dynamic stiffness matrix and stability matrix. The whole process from eigensolutions (for shape functions) to element matrix formulation is automated. The characteristic polynomial equation is first obtained by an analytical expansion method. The solution of the polynomial equation for the eigenvalues is standard. The ranks are checked and the generalized vectors are found. Finally, the element matrices are formed. The element matrices are free from all difficulties associated with the assumed shape function approach, e.g., rigid body modes, constant strains, spurious zero-energy modes, slow convergence, etc. A spatial helix is taken as an example.

An Improved Method for Stability and Damped Critical Speeds of Rotor-Bearing Systems

Many computer programs are available for stability and critical speed analysis of rotor-bearing systems. The iterative search-transfer matrix method is widely used for such programs. However, this method sometimes fails to converge or may miss some critical speeds. The polynomial method, which derives the characteristic polynomial and solves for the critical speeds, can partly overcome these shortcomings. However, the advantage of the polynomial method disappears as the number of elements in the system increases. This is because the computational time required to find a characteristic polynomial increases exponentially with the number of elements. This paper describes an improved technique based on the transfer matrix-polynomial method, which reduces the computational time significantly and completely eliminates the possibility of missing some critical speeds. A new technique is developed for the derivation of the characteristic polynomial. The characteristic polynomial equation is then converted into an eigenvalue problem of its companion matrix, whose eigenvalues are identical with the roots of the polynomial equation. The process, which can find only some dominant eigenvalues, eliminates the possibility of missing some eigenvalues without any penalties in the computational time and accuracy. The results from the method presented here are compared with those from some other methods.

On a Specialised Case of the Routh-Hurwitz Stability Criteria

SummaryThe well-known Routh-Hurwitz stability criteria apply to polynomial characteristic equations with real coefficients. Lesser known criteria were devised by Hurwitz for the case of polynomials with complex coefficients. However, when polynomials with real coefficients and even powers only are directly subjected to the first-mentioned criteria, all the test determinants obtained vanish identically and thus fail to indicate the state of stability. In the present paper, the stability test functions for such characteristic polynomials are derived.

The Routh and Routh‐Hurwitz Stability Criteria

THE classic Routh stability criteria1 are applicable only to polynomial characteristic equations with real coefficients. The writer recently came across co‐related criteria in a Russian treatise2 on the stability of rotating shafts which criteria are applicable to polynomial equations with complex coefficients. These criteria were apparently devised by Hurwitz as extensions of the Routh criteria and are thus known as Routh‐Hurwitz stability criteria. They are, however, not well known to British students of stability analysis nor are they readily accessible in British publications.