Characteristic Polynomial Research Articles

Floquet theory and stability analysis for Hamiltonian PDEs

Abstract We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

Exponential stability of positive implicit dynamic equations with constant coefficients

In this paper, the problem of positivity and stability for linear time-invariant implicit dynamic equations is generally studied. We provide necessary and sufficient conditions for positivity of these equations. This characterization can be considered as a unification and generalization for some previous results. On the other hand, we study the exponential stability of positive implicit dynamic equations. Previously, this issue was not completely addressed. By using Krein–Rutman theorem, we show that a positive implicit dynamic equation on a time scale is uniformly exponentially stable if and only if the characteristic polynomial of the matrix pair defining the equation has all its coefficients of the same sign.

Extended Special Linear group ESL2(F) and matrix equations in SL2(F), SL2(Z) and GL2(Fp)

The problem of roots existence for different classes of matrix such as simple and semisimple matrices from SL2(F), SL2(Z) and GL2(F) are solved. For this purpose, we first introduced the concept of an extended special linear group ESL2(F), which is generalisation of the matrix group SL2(F), where F is arbitrary perfect field. The group of unimodular matrices and extended symplectic group ESp2(R) are generalised by us, their structures are found. Our criterion oriented on a general class of matrix depending of the form of minimal and characteristic polynomials, moreover a proposed criterion holds in GL2(F), where F is an arbitrary field. The method of matrix factorisation is outlined. We show that ESL2(F) is a set of all square matrix roots from SL2(F) except of that established in our root existence criterion.

THE GENERAL SYMMETRIC DEGREE AND GENERAL SYMMETRIC DEGREE ADJACENCY POLYNOMIAL OF A GRAPH AND ITS ENERGY

The spectral graph theory is concerned with the relationship between the spectra of specific matrices associated with a graph and the structural properties of that graph. This paper introduces a general symmetric degree adjacency of a graph G, general symmetric degree of a graph G, and complete weighted graph of a graph. Here we explore its characteristic polynomial using Sachs theorem. We also investigate the bounds for index and energy of general symmetric degree adjacency of a graph. We give two algorithms, first one to find general symmetric degree adjacency of a graph G and second one to find general symmetric degree of a graph G.

Polynomials Counting Nowhere-Zero Chains Associated with Homomorphisms

A regular matroid M on a finite set E is represented by a totally unimodular matrix. The set of vectors from ZE orthogonal to rows of the matrix form a regular chain group N. Assume that ψ is a homomorphism from N into a finite additive Abelian group A and let Aψ[N] be the set of vectors g from (A−0)E, such that ∑e∈Eg(e)·f(e)=ψ(f) for each f∈N (where · is a scalar multiplication). We show that |Aψ[N]| can be evaluated by a polynomial function of |A|. In particular, if ψ(f)=0 for each f∈N, then the corresponding assigning polynomial is the classical characteristic polynomial of M.

Attacking trapdoors from matrix products

Recently, Geraud-Stewart and Naccache proposed two trapdoors based on matrix products. In this paper, we answer the call for cryptanalysis. We explore how using the trace and determinant of a matrix can be used to attack their constructions. We fully break their first construction in a polynomial-time attack. We show an information leak in the second construction using characteristic polynomials, and provide two attacks that decrease the bit security by about half.

Robust Controller Design Ensuring the Desired Aperiodic Stability Degree of a Control System with Affine Uncertainty

This paper considers a system whose characteristic polynomial coefficients are linear combinations of the interval parameters of a plant forming a parametric polytope. A linear robust controller is parametrically designed to place a dominant pole of the system within the desired interval of the negative real semi-axis and ensure an aperiodic transient in the system. The parametric design procedure involves a low-order controller with dependent and free parameters: the former serve to place the dominant pole within the desired interval on the complex plane whereas the latter to shift the other poles to some localization regions beyond a given bound (to the left of the dominant pole to satisfy the pole dominance principle). To evaluate the dependent parameters of the controller, the originals of the interval bounds of the dominant pole are determined for the plant’s parametric polytope based on a corresponding theorem (see below). The free parameters of the controller are chosen using the robust vertex or edge D-partition method, depending on the boundary edge branches of the localization regions of the free poles. A numerical example of the parametric design procedure is provided: a PID controller is built to ensure an acceptable aperiodic transient time in a load-lifting mechanism with interval values of cable length and load weight.

Statistics of ranks, determinants and characteristic polynomials of rational matrices

Abstract We consider the set of m × n {m\times n} matrices with rational entries having numerator and denominator of size at most H and obtain various upper bounds on the number of such matrices of a given rank, or with a given determinant, or a given characteristic polynomial. We also consider similar questions for matrices whose entries are Egyptian fractions.

On the matching arrangement of a graph and properties of its characteristic polynomial

On the matching arrangement of a graph and properties of its characteristic polynomial

On the trace-zero doubly stochastic matrices of order 5

We propose a graph theoretic approach to determine the trace of the product of two permutation matrices through a weighted digraph representation for a pair of permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order 5 whose k-th power is also a trace-zero DS matrix for k∈{2,3,4,5}. Then, we determine necessary conditions for the coefficients of a generic polynomial of degree 5 to be realizable as the characteristic polynomial of a trace-zero DS matrix of order 5. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order 5.

Persistent components in Canny's generalized characteristic polynomial

Persistent components in Canny's generalized characteristic polynomial

On arbitrary matrix coefficient assignment for the characteristic matrix polynomial of block matrix linear control systems

For block matrix linear control systems, we study the property of arbitrary matrix coefficient assignability for the characteristic matrix polynomial. This property is a generalization of the property of eigenvalue spectrum assignability or arbitrary coefficient assignability for the characteristic polynomial from system with scalar $(s=1)$ block matrices to systems with block matrices of higher dimensions $(s>1)$. Compared to the scalar case $(s=1)$, new features appear in the block cases of higher dimensions $(s>1)$ that are absent in the scalar case. New properties of arbitrary (upper triangular, lower triangular, diagonal) matrix coefficient assignability for the characteristic matrix polynomial are introduced. In the scalar case, all the described properties are equivalent to each other, but in block matrix cases of higher dimensions this is not the case. Implications between these properties are established.

Common values of linear recurrences related to Shank's simplest cubics

Let A,B,C∈Z not all zeroes and let F(u,n)=F(A,B,C,u,n) be the linear recursive sequence, which is defined by the initial terms F(u,0)=A,F(u,1)=B,F(u,2)=C and whose characteristic polynomial is Daniel Shanks simplest cubic Su(X)=X3−(u−1)X2−(u+2)X−1,u∈Z. We prove that there exists an effectively computable constant c depending only on L=max⁡{|A|,|B|,|C|} such that if |F(A,B,C,u,n)|=|F(A,B,C,u,m)| holds for some integers u,n,m with n≠m then |n|,|m|<c. For the choices (A,B,C)∈{(0,0,1),(1,−1,1)} we solve the above equations completely. At the end we give an outlook to the equation F(0,0,1,u,n)=F(0,0,1,v,m) for some fixed integers n,m.

On a class of degenerate hypoelliptic polynomials

This work is devoted to finding conditions on “lower-order terms”, the addition of which does not violate the hypoellipticity of a given operator (of the characteristic polynomial of this operator). Necessary and sufficient conditions for the hypoellipticity of “two-layered polynomials” are obtained. In terms of comparing strength, power, and the upper bound of the ratio of comparable polynomials, conditions are obtained under which the added polynomials preserve the hypoellipticity of the original polynomial. Examples are given that illustrate the obtained results.

Schur Function Expansion in Non-Hermitian Ensembles and Averages of Characteristic Polynomials

Schur Function Expansion in Non-Hermitian Ensembles and Averages of Characteristic Polynomials

On hybrid dynamical systems of differential–difference equations

In this paper, we define and study a class of linear hybrid dynamical systems characterized by differential–difference equations. We introduce two operators that facilitate the analysis of these systems and derive explicit formulas for their solutions. We examine the transfer function matrix and characteristic polynomial to assess stability. Our theoretical findings are supported by numerical examples, demonstrating their application in power systems stability analysis. Specifically, we substantiate our theory within the context of power systems stability analysis, incorporating elements of discrete behavior.

STABILIZATION OF DISCRETE 2D LINEAR SYSTEMS IN FORNASINI-MARCHESINI MODEL

This paper is concerned with the stabilization problem of discrete 2D linear systems described by the second Fornasini-Marchesini model. A necessary and sufficient condition involving the characteristic polynomial is first quoted by which the unforced system is structurally or exponentially stable. On the basis of the derived stability condition, a tractable condition is formulated in the form of linear matrix inequality for obtaining the controller gain of a desired stabilizing state-feedback controller.

A number‐theoretic classification of toroidal solenoids

AbstractWe classify toroidal solenoids defined by nonsingular ‐matrices with integer coefficients by studying associated first Ĉech cohomology groups. In a previous work, we classified the groups in the case using generalized ideal classes in the splitting field of the characteristic polynomial of . In this paper, we explore the classification problem for an arbitrary .

A Low-Frequency Oscillation Suppression Method for Regional Interconnected Power Systems with High-Permeability Wind Power.

With the integration of large-scale wind power into the power grid, the impact on system stability, especially the issue of low-frequency oscillations caused by small disturbances, is becoming increasingly prominent. Therefore, this paper proposes a damping quantitative analysis method for regional interconnected power systems incorporating large-scale wind power. Using the cross-entropy particle swarm optimization (CE-PSO) algorithm, the control parameters of wind turbines are optimized to suppress low-frequency oscillations in interconnected systems. The method begins with the state equation of the interconnected power system in two regions; it deduces the characteristic polynomial of the interconnected system, including wind farms, and takes into account the influence of wind power integration on the electrical connectivity of the system. Subsequently, the influence of wind turbine control parameters on the system is quantified, and a quantitative analysis model of the impact of wind power integration on system damping characteristics is constructed. Based on this, an optimization model for wind turbine control parameters is established, and the CE-PSO algorithm is utilized to achieve suppression of low-frequency oscillations in interconnected power grids with wind power integration. Finally, the accuracy and effectiveness of the proposed method are verified through a typical electromagnetic transient simulation model of the two-region interconnected power system.

Rational solutions of the matrix equation $p(X) = A$

We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA, 258:187-194, 1997. Let $p(\lambda)$ be any polynomial over $\mathbb{Q}$, and let $A\in M_n(\mathbb{Q})$ have irreducible characteristic polynomial $f(\lambda)$ with degree $n$. We provide necessary and sufficient conditions for the existence of a solution $X\in M_n(\mathbb{Q})$ of the polynomial matrix equation $p(X) = A.$ Specifically, we find necessary and sufficient conditions for $f(p(\lambda))$ to have a factor of degree $n$ over $\mathbb{Q}.$