Roots Of Characteristic Polynomial Research Articles

Kemeny's constant and global mean first passage time of random walks on octagonal cell network

This paper investigates the random walks of octagonal cell network. By using the Laplacian spectrum method, we obtain the mean first passage time and Kemeny's constant between nodes. On one hand, the mean first passage time is explicitly studied in terms of the eigenvalues of a Laplacian matrix. On the other hand, Kemeny's constant is introduced to measure node strength and to determine the scaling of the random walks. We provide an explicit expression of Kemeny's constant and mean first passage time for octagonal cell network, by their Laplacian eigenvalues and the correlation among roots of characteristic polynomial. Based on the achieved results, comparative studies are also performed for and . This work also deliver an inclusive approach for exploring random walks of networks, particularly biased random walks, which likewise support to better understand and tackle some practical problems such as search and routing on networks.

English

We discuss the eigenvalue problem in the max-plus algebra. For a max-plus square matrix, the roots of its characteristic polynomial are not its eigenvalues. In this paper, we give the notion of algebraic eigenvectors associated with the roots of characteristic polynomials. Algebraic eigenvectors are the analogues of the usual eigenvectors in the following three senses: (1) An algebraic eigenvector satisfies an equation similar to the equation A ⊗ x = λ ⊗ x for usual eigenvectors. Under a suitable assumption, the equation has a nontrivial solution if and only if λ is a root of the characteristic polynomial. (2) The set of algebraic eigenvectors forms a max-plus subspace called algebraic eigenspace. (3) The dimension of each algebraic eigenspace is at most the multiplicity of the corresponding root of the characteristic polynomial.

Synthesis of Discrete Modal SISO Controllers by Engineering Performance Indices

The linear discrete minimum-phase plant with one control input, one measured output and with an unmeasured external disturbance is considered. The problem is stated of the design of the discrete output controller that provides the preset engineering performance indices: control error, settling time, stability margins radius. Analytical formulas are obtained that allow to relate the engineering performance indices with the desired closed system characteristic polynomial roots in the modal control problem when choosing real non-negative roots. At the same time, the specified control error and the settling time lead to the constraints on the maximum absolute value of the desired roots, while the values of the desired roots, providing the given stability margins radius, may not satisfy these constraints, so the specified values of the performance indices should be revised. Examples are given showing the effectiveness of the approach proposed.

On the Adjacency-Jacobsthal numbers

In this article, we define the adjacency-Jacobsthal sequence and then we obtain the combinatorial representations and the sums of adjacency-Jacobsthal numbers by the aid of generating function and generating matrix of the adjacency-Jacobsthal sequence. Also, we derive the determinantal and the permanental representations of adjacency-Jacobsthal numbers by using certain matrices which are obtained from generating matrix of adjacency-Jacobsthal numbers. Furthermore, using the roots of characteristic polynomial of the adjacency-Jacobsthal sequence, we produce the Binet formula for adjacency-Jacobsthal numbers. Finally, we give the relationships between adjacency-Jacobsthal numbers and Fibonacci, Pell, and Jacobsthal numbers.

Vector Control for an AC Drive with a Multilevel Frequency Converter Based on an Н-Bridge

Using multilevel topologies for a stand-alone inverter, which is a part of induction or synchronous ac drives, makes it possible to achieve their operation under some fault conditions related to, for example, the failure of any element in a power circuit or short-term blackout in operation of distribution line of centralized power supply system. In the last case, due to switching the speed loop into the mode of voltage control of capacitor filters, it turns out during short-term interval to maintain at the set level the magnetic condition and mechanical coordinates of electrical machine without direct disconnection of converter by corresponding signal feeding from a security unit. The work considers the design formulas for synthesis of special algorithms of vector control by an ac actuating motor under the absence of power supply based on the set distribution of the roots of characteristic polynomial on the complex plane. As well as the reduction of logarithmic amplitude–phase–frequency characteristic of the loop in open condition to the normalized form is presented in the work. Within the framework of these formulas, the regeneration of electric energy in the stand-alone inverter with series-connected H-bridges is carried out to stabilizing the voltage of capacitor bank in each power module owing to when workability recovery of three-phase network, the ac drive returns to normal operation without significant degradation of quality of processes by rotor rotational speed.

КОМПЕНСАЦІЯ ЛІНІЙНОГО ЗБУРЕННЯ СИСТЕМИ СТАБІЛІЗАЦІЇ РАКЕТИ

For disturbed motion in one plane with taking in account the inertia of an executive device the connection between a law of control determined by method of modal control and error matrixes is established. The roots of characteristic polynomial are according to the Batterwort’s scheme on hemicycle of a given radius even distributed. The results can be used in system design as an alternative option of a law of control determinationbased on the requirements for accuracy figures.

Roots of characteristic polynomials and intersection points of line arrangements

We study a relation between roots of characteristic polynomials and intersection points of line arrangements. Using these results, we obtain a lot of applications for line arrangements. Namely, we give (i) a generalized addition theorem for line arrangements, (ii) a generalization of Faenzi-Valles' theorem over a field of arbitrary characteristic, (iii) a partial result on the conjecture of Terao of line arrangements, and (iv) a new sufficient condition for freeness over finite fields. Also, a higher dimensional version of main results is considered.

Method of solving the generalized problem on eigenvalues of multidimensional circuits, I

This work is devoted to the development of a new method for solving the complete generalized problem of large-scale electrical circuit eigenvalues. We prove that the determinant of the conductance node matrix of an arbitrary k-loop LC-circuit is the Weinstein function for the loop impendance matrix of this circuit, and vice versa. A recurrent method of imposing constraints on the basic problem was defined, which permitted us to separate roots of characteristic polynomials in the recurrent process. Also a condition for conservativity of multiple eigenvalues was defined. A solution algorithm for the problem of defining a full range of eigenvalues of an oscillatory system with a finite number of degrees of freedom was suggested.

Time Series Analysis of Monte Carlo Fission Sources—III: Coarse-Mesh Projection

A coarse-mesh projection method has been developed for the Monte Carlo calculation of dominant eigenvalue ratio [dominance ratio (DR)]. The first step of the method consists of the regression analysis of the multivariate time series from the coarse-mesh binning of the Monte Carlo fission source distribution. The second step is computation of the eigenvectors of the adjoint matrix of noise propagation. In general, projections on these eigenvectors can be utilized to compute important characteristics of the eigenmodes of fission source distribution. In this work, it has been proven that if the eigenvector corresponding to the largest eigenvalue of the aforementioned adjoint matrix is taken to be the vector for projection, the projected scalar time series follows the autoregressive process of order one with the root of characteristic polynomial, i.e., the autocorrelation coefficient, being the DR of fission source distribution. Numerical results are presented for four problems including one-energy-group checkerboard-type problems, a one-energy-group cube problem and a continuous-energy pressurized water reactor core problem. The strength of the method is twofold; (a) the elimination of the use of autoregressive moving average fitting, and (b) no need to optimize the order of fitting.

Some classes of Chebyshev systems

Let L be a linear real space of solutions of a linear homogeneous ordinary differential equation with constant coefficients. This space is a Chebyshev space on any interval of length at most π/ω, where ω is maximal among the imaginary parts of the roots of characteristic polynomial. It turns out that this estimate for the length of the interval is exact in a certain sense. Bibliography: 2 titles.

Existence and stability of periodic solutions of semilinear neutral type systems

Forced periodic oscillations of coupled systems of semilinear neutral type functional differential equations are investigated. Explicit conditions for existence, uniqueness and stability of periodic solutions are derived. These conditions are formulated in terms of the roots of characteristic polynomials. In addition, estimates for periodic solutions and their derivatives are established.

Stability-Optimal One-Dimensional Control Systems. I

Sufficient conditions for the optimality of real or nonmultiple complex-conjugate right-most roots of characteristic polynomials in a maximally stable system are formulated. The problem is solved for cases in which the controlled object is described by a transfer function containing a polynomial of degree greater than zero in the numerator.