The Study of Root Subspace Decomposition between Characteristic Polynomials and Minimum Polynomial

Let Abe the linear transformation on the linear space V in the field P, Vλibe the root subspace corresponding to the characteristic polynomial of the eigenvalue λi, and Wλibe the root subspace corresponding to the minimum polynomial of λi. Consider the problem of whether Vλiand Wλiare equal under the condition that the characteristic polynomial of Ahas the same eigenvalue as the minimum polynomial (see Theorem 1, 2). This article uses the method of mutual inclusion to prove that Vλi=Wλi. Compared to previous studies and proofs, the results of this research can be directly cited in related works. For instance, they can be directly cited in Daoji Meng’s book “Introduction to Differential Geometry.”

Trace Formulae of Characteristic Polynomial and Cayley-HamUton's Theorem, and Applications to Chiral Perturbation Theory and General Relativity

By using combinatorics, we give a new proof for the recurrence relations of the characteristic polynomial coefficients, and we further obtain an explicit expression for the generic term of the coefficient sequence, which yields the trace formulae of the Cayley-Hamilton＇s theorem with all coefficients explicitly given. This implies a byproduct, a complete expression for the determinant of any finite-dimensional matrix in terms of the traces of its successive powers. And we discuss some of their applications to ehiral perturbation theory and general relativity.

Circular β ensembles,CMV representation,characteristic polynomials

In this note we first briefly review some recent progress in the study of the circular β ensemble on the unit circle,where β > 0 is a model parameter.In the special cases β = 1,2 and 4,this ensemble describes the joint probability density of eigenvalues of random orthogonal,unitary and sympletic matrices,respectively.For general β,Killip and Nenciu discovered a five-diagonal sparse matrix model,the CMV representation.This representation is new even in the case β = 2;and it has become a powerful tool for studying the circular β ensemble.We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle.

Lattices Generated by Joins of the Flats in Orbits under Finite Affine-singular Symplectic Group and its Characteristic Polynomials

Let ASG（2v ＋ l, v; Fq） be the （2v ＋l）-dimensional affine-singular symplectic space over the finite field Fq and ASp2v＋l,v（Fq） be the affine-singular symplectic group of degree 2v ＋ l over Fq. Let O be any orbit of flats under ASp2v＋l,v（Fq）. Denote by J the set of all flats which are joins of flats in O such that O LJ and assume the join of the empty set of flats in ASG（2v ＋ l, v;Fq） is φ. Ordering LJ by ordinary or reverse inclusion, then two lattices axe obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice LJ, when the lattices form geometric lattice, lastly gives the characteristic polynomial of LJ.

Rank-generating functions and Poincaré polynomials of lattices in finite orthogonal space of even characteristic

The geometry of classical groups over finite fields is widely used in many fields.In this paper,we study the rank-generating function,the characteristic polynomial,and the Poincarépolynomial of lattices generated by the orbits of subspaces under finite orthogonal groups of even characteristic.We also determine their expressions.

EIGENVALUES OF A SPECIAL KIND OF SYMMETRIC BLOCK CIRCULANT MATRICES

In this paper, the spectrum and characteristic polynomial for a special kind of symmetric block circulant matrices are given.

Analysis of the effect on control systems of order variation for fractional-orderPI~λD~μcontrollers

This paper is concerned with fractional-order PI~λD~μcontrollers. The definitions and properties of fractional calculus are introduced. The mathematical descriptions of a fractional-order controller and fractional-order control systems are outlined. The effects on control systems of order variation for fractional-order PI~λD~μ controllers are investigated by qualitative analysis and simulation. The conclusions and simulation examples are given. The results show the fractional-order PI~λD~μ controller is not sensitive to variation of its order.

Robust Stability and Performance Comparison of PID and PPI Control

Predictive PI (PPI) control form, capable of time delay compensation, has been put forward recently. This control algorithm is essentially a PI controller with enhanced derivative action, which is not only suitable for long time delay process, but also of simple structure and excellent robust stability. The performance of PPI controller was demonstrated and compared with that of traditional PID controller by different tuning methods.

A CRITERION FOR ORTHOGONALITY OF REFINABLE FUNCTIONS

In this note, a criterion for orthonormality of refinable functions via characteristic polynomial of a matrix is given.

Deriving Energy-Gap of Some Nonlinear Hamiltonians by Invariant Eigen-operator Method

By virtue of the invariant eigen-operator method we search for the invariant eigen-operators for someHamiltonians describing nonlinear processes in particle physics.In this way the energy-gap of the Hamiltonians can benaturally obtained.The characteristic polynomial theory has been fully employed in our derivation.

Some Properties of Eigenvalues and Eigenvectors of Wilkinson Matrices

Some properties of characteristic polynomials, eigenvalues, and eigenvectors of the Wilkinson matrices W2n＋1 and W2n＋1 are presented. It is proved that the eigenvalues of W2n＋1 just are the eigenvalues of its leading principal submatrix Vn and a bordered matrix of Vn. Recurrence formula are given for the characteristic polynomial of W2＋n＋1 . The eigenvectors of W2＋n＋1 are proved to be symmetric or skew symmetric. For W2n＋1 , it is found that its eigenvalues are zero and the square roots of the eigenvalues of a bordered matrix of Vn2. And the eigenvectors of W2n＋1 , which the corresponding eigenvahies are opposite in pairs, have close relationship.

On the McKay Quiver

Using the connection between McKay quiver and Loewy matrix, and the properties of characteristic polynomial of Loewy matrix, we give a generalized way to determine the McKay quiver for a finite subgroup of a generalized linear group.

Applying Linear Controls to Chaotic Continuous Dynamical Systems

In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-dimensional chaotic dynamical systems is generally a difficult problem, Musielak and Musielak, [1]. We numerically illustrate that sometimes elementary approaches can yield the desired numerical results with two different continuous higher order dynamical systems that exhibit chaotic behavior, the Lorenz equations and the Rössler attractor.

Generalization of Multiplication M-Sequences over Fp and Its Recip­rocal Sequences

M_(p)-Sequences or M-Sequence over Fp not used so much in current time as binary M-Sequences and it is pending with the difficult to construct there coders and decoders of M_(p)-Sequences further these reasons there is expensive values to construct them but the progress in the technical methods will be lead to fast using these sequences in different life’s ways,and these sequences give more collection of information and distribution them on the input and output links of the communication channels,building new systems with more complexity,larger period,and security.In current article we will study the construction of the multiplication M_(p)-Sequence{z_(n)}and its linear equivalent,this sequences are as multiple two sequences,the first sequence{Sn}is an arbitrary M_(p)-Sequence and the second sequence{ζ_(n)}reciprocal sequence of the first sequence{S_(n)},length of the sequence{z_(n)},period,orthogonal and the relations between the coefficients and roots of the characteristic polynomial of f(x)and it’s reciprocal polynomial g(x)and compare these properties with corresponding properties in M-Sequences.

Analysis of Eigenvalues for Molecular Structures

In this article,we study different molecular structures such as Polythiophene network,PLY(n)for n=1,2,and 3,Orthosilicate(Nesosilicate)SiO4,Pyrosilicates(Sorosilicates)Si2O7,Chain silicates(Pyroxenes)(SiO3)n,and Cyclic silicates(Ring Silicates)Si3O9 for their cardinalities,chromatic numbers,graph variations,eigenvalues obtained from the adjacency matrices which are square matrices in order and their corresponding characteristics polynomials.We convert the general structures of these chemical networks in to mathematical graphical structures.We transform the molecular structures of these chemical networks which are mentioned above,into a simple and undirected planar graph and sketch them with various techniques of mathematics.The matrices obtained from these simple undirected graphs are symmetric.We also label the molecular structures by assigning different colors.Their graphs have also been studied for analysis.For a connected graph,the eigenvalue that shows its peak point(largest value)obtained from the adjacency matrix has multiplicity 1.Therefore,the gap between the largest and its smallest eigenvalues is interlinked with some form of“connectivity measurement of the structural graph”.We also note that the chemical structures,Orthosilicate(Nesosilicate)SiO4,Pyrosilicates(Sorosilicates)Si2O7,Chain silicates(Pyroxenes)(SiO3)n,and Cyclic silicates(Ring Silicates)Si3O9 generally have two same eigenvalues.Adjacency matrices have great importance in the field of computer science.

Vibration and buckling analyses of nanobeams embedded in an elastic medium

Boundary characteristic orthogonal polynomials are used as shape functions in the Rayleigh–Ritz method to investigate vibration and buckling of nanobeams embedded in an elastic medium. The present formulation is based on the nonlocal Euler–Bernoulli beam theory. The eigen value equation is developed for the buckling and vibration analyses. The orthogonal property of these polynomials makes the computation easier with less computational effort. It is observed that the frequency and critical buckling load parameters are dependent on the temperature, elastic medium, small scale coefficient,and length-to-diameter ratio. These observations are useful in the mechanical design of devices that use carbon nanotubes.

Stability Analysis of Multi-Dimensional Linear Discrete System and Root Distribution Using Sign Criterion with Real Coefficients

A new idea was proposed to find out the stability and root location of multi-dimensional linear time invariant discrete system (LTIDS) for real coefficient polynomials. For determining stability the sign criterion is synthesized from the Jury’s method for stability which is derived from the characteristic polynomial coefficients of the discrete system. The number of roots lying inside or outside the unit circle and hence on the unit circle is directly determined from the proposed single modified Jury tabulation and the sign criterion. The proposed scheme is simple and the examples are given to bring out the merits of the proposed scheme which is also applicable for the singular and non-singular cases.

The RCH Method for Computing Minimal Polynomials of Polynomial Matrices

In this paper,a randomized Cayley-Hamilton theorem based method(abbreviated by RCH method) for computing the minimal polynomial of a polynomial matrix is presented.It determines the coefficient polynomials term by term from lower to higher degree.By using a random vector and randomly shifting,it requires no condition on the input matrix and works with probability one.In the case that coefficients of entries of the given polynomial matrix are all integers and that the algorithm is performed in exact computation,by using the modular technique,a parallelized version of the RCH method is also given.Comparisons with other algorithms in both theoretical complexity analysis and computational tests are given to show its effectiveness.

Polynomial Time Algorithm for the Two-Side Exponentiation Problem about Ergodic Matrices over Finite Field

By using the minimal polynomial of ergodic matrix and the property of polynomial over finite field,we present a polynomial time algorithm for the two-side exponentiation problem about ergodic matrices over finite field （TSEPEM）,and analyze the time and space complexity of the algorithm.According to this algorithm,the public key scheme based on TSEPEM is not secure.

Quadratic Starlike Trees

We introduce the notion of a quadratic graph,which is a graph whose eigenvalues are integral or quadratic algebraic integral,and we determine nine infinite families of quadratic starlike trees,which are just all the quadratic starlike trees including integral starlike trees.Thus,the quadratic starlike trees are completely characterized.The expressions for characteristic polynomials of quadratic starlike trees are also given.