Characteristic Eigenvalues Research Articles

ON MATRICES ARISING IN FINITE FIELD HYPERGEOMETRIC FUNCTIONS

Abstract Lehmer [‘On certain character matrices’, Pacific J. Math.6 (1956), 491–499, and ‘Power character matrices’, Pacific J. Math.10 (1960), 895–907] defines four classes of matrices constructed from roots of unity for which the characteristic polynomials and the kth powers can be determined explicitly. We study a class of matrices which arise naturally in transformation formulae of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors and kth powers of these matrices. The eigenvalues are natural families of products of Jacobi sums.

Model Predator-Prey Leslie-Gower dengan Fungsi Respon Sokol-Howell dan Perilaku Anti Predator

This study discusses the Leslie-Gower predator-prey model with the Sokol-Howell response function and anti-predator behavior. It is assumed that prey has anti-predator behavior that aims to reduce the risk of predation and not as an attempt by prey to find food. This study aims to formulate a Leslie-Gower predator-prey model with the Sokol-Howell response function and anti-predator behavior, analyze the model's equilibrium point and model interpretation. Stability analysis was carried out using the linearization method. The type of stability is determined based on the characteristic eigenvalues ​​obtained using Routh-Hurwitz criteria. The results of the analysis of the equilibrium point show that prey populations will exist and predators will become extinct if the anti-predator coefficient is greater than the intrinsic growth coefficient of predators, while prey and predator populations will always exist if the intrinsic growth coefficient of predators is greater than the anti-predator coefficient and fulfills other conditions required. Based on the numerical simulations performed, the interpretation is that an enlarged anti-predator coefficient increases the number of prey populations until they approach the carrying capacity, while predator populations decrease significantly and over time experience extinction.

Integral Operator-Based Characteristic Mode Theory for Conducting, Material, and Lossy Structures

Integral operator-based theory of characteristic modes (CMs) for conducting, material, and lossy structures is reviewed. CMs are defined as solutions of a generalized eigenvalue equation (GEE). These GEEs are presented for various surface and volume integral operators and material structures. Interpretation of the characteristic eigenvalues in terms of electromagnetic power is studied based on the Mie expansion and integral operator formalism. Orthogonality and diagonalizing properties of the CMs are summarized. Challenges related to dielectric-magnetic bodies, lossy structures and spurious modes are discussed, as well as differences between the surface and volume operator approaches.

Eigenproblem Basics and Algorithms

Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.

McKay Matrices for Pointed Rank One Hopf Algebras of Nilpotent Type

Let [Formula: see text] be a finite-dimensional pointed rank one Hopf algebra of nilpotent type over a finite group [Formula: see text]. In this paper, we investigate the McKay matrix [Formula: see text] of [Formula: see text] for tensoring with the 2-dimensional indecomposable [Formula: see text]-module [Formula: see text]. It turns out that the characteristic polynomial, eigenvalues and eigenvectors of [Formula: see text] are related to the character table of the finite group [Formula: see text] and a kind of generalized Fibonacci polynomial. Moreover, we construct some eigenvectors of each eigenvalue for [Formula: see text] by using the factorization of the generalized Fibonacci polynomial. As an example, we explicitly compute the characteristic polynomial and eigenvalues of [Formula: see text] and give all eigenvectors of each eigenvalue for [Formula: see text] when [Formula: see text] is a dihedral group of order [Formula: see text].

Механизм диагностики эксплуатации малоинтенсивных железнодорожных линий на основе нечеткого когнитивного моделирования

For to identify functioning problem of railway lines various mechanisms are used, including the mechanism of low-intensity railway line exploitation diagnostics which allows to raise their functioning efficiency. The proposed mechanism allows to evaluate low-intensity railway line current state and to establish cause-and-effect relationships leading to line unprofitability, to give analysis of indicators which to the greatest extent influence negatively the line functioning. Cognitive modeling method is used for this purpose. The article develops enlarged cognitive map of low-intensity railway line functioning that’s used for the diagnostics of line exploitation existing state. Cognitive map consists of concepts, links between concepts and links’ weight values. On cognitive map basis, adjacency matrix is made up which allows to evaluate reliability and sustainability of fuzzy cognitive map according to internal and external disturbances. Characteristic equation and matrix eigenvalues are calculated which allow to give the assessment of cognitive map indicators in the form of the influence of concepts on system and of system on concepts. Fuzzy cognitive map indicators are established, namely, system influence consonance, system influence dissonance, concept influence on system which give the most complete representation of low-intensity line internal structure. “Technical resources”, “stations’ and section’s staff”, “normative base” among concepts touch on the largest positive impact on system. The most significant positive influence on the listed concepts will allow to rise operation efficiency of low-intensity railway lines. “Traffic volume” concept among concepts causes the most negative influence on the system because of the lack of quantity of train pairs, passing along the line, a day. The plan of experiments for to define the changes in the parameters of “Productivity and effectiveness” concepts, when exerting impulse influences on “Operational management”, “Technical resources” and “Stations’ and section’s staff” concepts with the purpose of the evaluation of their influence on “Productivity” and “Efficiency” concepts, is made up. When introducing impulses into “Staff” concept, productivity and efficiency growth is observed that testifies to the possibility of efficiency rise of low-intensity railway line functioning by the way of positive impulse on “Staff” concept.

Analisis Model Mangsa Pemangsa dengan Adanya Penyakit dan Pemanenan pada Pemangsa

The interaction between two populations that are prey and predator can be described in a prey-predator model. In fact, in the interaction of prey and predators it can occur that when the density of prey is low, the effect of predation is also low, but if the size of the prey population increases, predation will be more intense which is stated in the Holling III Type response function model. In addition, it can also be found in an environment where there are populations of sick predator that result in death from the disease. This study aims to analyze the stability of the prey-predator model with the Holling III Type response function which consists of three subpopulations namely prey, healthy predator and sick predator. The analysis was carried out using the linearization method and then the type of stability was determined based on the characteristic eigenvalues obtained using the Routh-Hurwitz criteria. From this research it was found that population extinction is not possible while prey exists, extinction of diseased predator and populations exist is still possible if the required conditions are satisfy. Numerical simulations show that an increased harvesting rate in healthy predator populations results in a decrease in healthy predator populations, an increase in prey populations and a decrease in diseased predator populations. Meanwhile, a reduced harvesting rate for sick predator populations does not have a significant effect on the number of diseased predator populations, but results in an increase in the number of healthy predator populations and a decrease in prey populations

Pulsed-Neutron Die-Away Experiments for Plastics and Neutron Thermal Scattering Laws

Pulsed-neutron die-away (PNDA) experiments can be useful benchmarks to validate neutron thermal scattering laws (TSLs). The experiment uses a neutron generator to impinge a short (∼10−4 s), mono-energetic neutron pulse on a target sample. After the pulse, the neutron population within the sample moderates and reaches thermal equilibrium with a fundamental spatial mode and characteristic decay-time eigenvalue. The eigenvalue can be extracted from the experimental measurements of the neutron flux and used as an integral parameter in validation. For certain materials and geometric configurations, the eigenvalue is heavily influenced by thermal neutron scattering of only the target material. For that reason, a PNDA experiment can have a higher sensitivity to TSLs than is commonly available in critical experiments. Herein, we present results for a series of new PNDA experiments conducted at Lawrence Livermore National Laboratory with plastic materials, e.g. high-density polyethylene and Lucite. We compare the experimental integral parameters to simulated results and report trends in the biases. We evaluate the bias with different Monte Carlo transport codes (MCNP6.2 and Mercury) and show no significant differences between the results of the two codes.

An investigation of even ordered magic squares (4, 6, and 8): characteristic polynomials, eigenvalues, and encryption

In this paper, we discuss and mathematically compute the eigenvalues and the characteristic polynomials of special even square matrices of orders 4x4 and 8x8. Also, we introduce two 8th order compound magic squares. The computed values are verified using Maple software. First, for the 4th order square matrix, the characteristic polynomial was derived to be: λ(λ-2s)( λ²+4Θ) with the eigenvalues: 0,2 s, and two other conjugates. In further analysis, we performed numerical classification of the squares for the matrices of order 4. Second, for the 8th order magic square, the characteristic polynomial was obtained in the form: λ<sup>3</sup>(λ-4s)(λ<sup>4</sup>+Ωλ<sup>2</sup>+θ) where Ω,Θ are constants; the eigenvalues are 0,4 s, ∓√λ<sub>1</sub>, ∓√λ<sub>2</sub>; where λ<sub>1</sub>, λ<sub>2</sub> are the roots of the quadratic equation: λ<sup>2</sup>+Ωλ+Θ=0. Third, for the franklin square, we obtained the eigenvalues 0,4 s, and the roots of the equation: λ<sup>2</sup>+aλ+b. Finally, we suggested a hybrid image encryption technique based on Franklin magic square matrices and improved substitution technique. The proposed a grayscale image encryption/ decryption algorithm uses circular rotation of bits and Franklin magic squares’ properties in conjunction with substitution techniques to obtain a very secure algorithm against attacks.

IDENTIFICATION OF LOGISTIC ELEMENTS OF CUSTOMER SERVICE USING PRINCIPAL COMPONENT ANALYSIS

Logistics activities are considered as a priority by the management in everyday business, while having an adequate customer service as a powerful tool helps achieving the competitiveness of the company to continuously grow and develop in a global scale. The purpose of this paper is to identify the key logistic elements of customer service by aggregating the most important components in the sample of customers for retail stores in Bosnia and Herzegovina. The principal component analysis of the logistic elements of customer service was employed in the research, within the principal components method with the application of orthogonal rotation of VARIMAX with Kaiser normalization. The analysis of the main components extracted six components with characteristic eigenvalues greater than 1, where 24 claims were grouped around them. Orthogonal VARIMAX rotation was performed to facilitate the interpretation of the obtained components. The rotated solution showed a simpler structure of components that explains 64.08% of the variance. The results of the study confirmed a six-dimensional component structure. The extracted components are given the following names: possibility of ordering and delivery of products, convenience of delivery, packaging and assortment of products, location of retail store and payment terms, loyalty card benefits, spaciousness of stores, improvement of sales and product complaints and correction of potential errors.

Antenna Element Design Using Characteristic Mode Analysis: Insights and research directions

This article provides a comprehensive review of recent applications of characteristic mode analysis (CMA) to innovative antenna element designs, including multiport, circularly polarized, wideband, reconfigurable, and dielectric resonator antennas (DRAs). Emphasis is placed on the interpretation of the characteristic modes (CMs) for those unfamiliar with the method and physical insights gained from the characteristic eigenvalues and eigenvectors of an antenna. In addition, we review CMA-based design strategies and specific design examples that highlight the application of CMA to various types of antennas. Ultimately, this article seeks to demonstrate the value of CMA-based design insights for antenna engineering and look toward promising new research directions for CMA and antenna research.

What we can learn from Bohemian Matrices?

This Maple Workbook explores a new topic in linear algebra, which is called "Bohemian Matrices". The topic is accessible to people who have had even just one linear algebra course, or have arrived at the point in their course where they have touched "eigenvalues". We use only the concepts of characteristic polynomial and eigenvalue. Even so, we will see some open questions, things that no-one knows for sure; even better, this is quite an exciting new area and we haven't even finished asking the easy questions yet! So it is possible that the reader will have found something new by the time they have finished going through this workbook. Reading this workbook is not like reading a paper: you will want to execute the code, and change things, and try alternatives. You will want to read the code, as well. I have tried to make it self-explanatory.
 We will begin with some pictures, and then proceed to show how to make such pictures using Maple (or, indeed, many other computational tools). Then we start asking questions about the pictures, and about other things.

Properties of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph

A directed cyclic graph is a directed graph that has at least one directed cycle graph, that is a cycle graph in which all edges are oriented, so that the direction passes through each vertex once, except the end of vertex. The directed unicyclic tadpole graph is the graph created by concatenating a cycle and a path with an edge from any vertex of to a pendant of for integers m ≥ 3 and n ≥ 1. Antiadjacency matrix is a directed graph representation matrix based on whether or not there is a relation between one vertex and the others. This paper gives the general form of coefficients of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph. To find the general form of coefficients of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph we need to do the grouping of induced subgraphs into acyclic and cyclic and verify with related theorems. After that, the characteristic polynomial is factorized and the roots are calculatedto find its eigenvalues. The coefficients of the characteristic polynomial consist of three distinct values and the eigenvalues are divided into odd case and even case.

Characteristic polynomial and eigenvalues of anti-adjacency matrix of directed unicyclic corona graph

A directed graph can be represented by several matrix representations, such as the anti-adjacency matrix. This paper discusses the general form of characteristic polynomial and eigenvaluesof the anti-adjacencymatrix of directed unicyclic corona graph. The characteristic polynomial of the anti-adjacency matrix can be found by counting the sum of the determinant of the anti-adjacency matrix of the directed cyclic inducedsubgraphs and the directed acyclic induced subgraphs from the graph. The eigenvalues of the anti-adjacency matrix can be real or complex numbers. We prove that the coefficient of the characteristic polynomial and the eigenvalues of the anti-adjacency matrix of directed unicyclic corona graph can be expressed in the function form that depends on the number of subgraphs contained in thedirected unicyclic corona graphs.

Support position optimization with minimum stiffness for plate structures including support mass

The optimum position and minimum restraint stiffness of a flexible point support to raise a natural frequency of a thin bending plate is investigated, with the inclusion of the corresponding additional support mass. First the derivatives of the natural frequencies of the plate structure are derived with respect to the support movement using a finite element model. Second, the minimum support stiffness is analyzed to raise a plate's natural frequency to a target value by solving a characteristic eigenvalue problem. Then the optimal support design is studied to find the optimal attachment point and the associated minimum stiffness. Several typical examples of plate systems are analyzed with addition of the point supports with non-negligible mass. It appears that including the support mass in the plate vibration analysis can significantly increase the minimum support stiffness required to raise a given natural frequency to its target, whereas the optimal support position remains consistent with the massless support design case.

Characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph

This research discussed the characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph. The entries of the antiadjacency matrix of a directed graph represent the presence or the absence of a directed arc from one vertex to the others. If A is the adjacency matrix of a graph , then the antiadjacency matrix B of graph is B = J − A, where J is the square matrix with all entries equal to one. The general form of characteristic polynomial coefficients of the antiadjacency matrix of directed unicyclic flower vase graph can be obtained by calculating the sum of determinants of the antiadjacency matrices of all induced cyclic and acyclic subgraphs, while the eigenvalues were obtained by using polynomial factorization and Horner’s method. This paper gives the characteristic polynomial coefficients and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph. The characteristic polynomial can be considered as a function that depends on the number of vertices.

Properties of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic helm graph

A directed unicyclic graph is a directed graph that has only one directed cycle subgraph. A directed unicyclic helm graph is obtained from a directed wheel graph by adjoining a directed pendant edge at each vertex of the cycle. A directed graph can be represented into several matrix representations, one of them is the antiadjacency matrix. The antiadjacency matrix is a matrix in which the entries represent whether there is a directed edge from one vertex to another. This paper discusses the general form of the coefficients of the characteristic polynomial that obtained by adding all of the determinants of antiadjacency matrix from each induced acyclic and cyclic subgraphs. The eigenvalues of the antiadjacency matrix of the directed unicyclic helm graph obtained by polynomial factorization. The result obtained denotes that the coefficients of the characteristic polynomial and eigenvalues of the antiadjacency matrix depend on the number of vertices of the cycle subgraphs of directed unicyclic helm graph.

Comparison Of Tests For Isomorphism In Planetary Gear Trains

There are plenty of ways available for synthesis and analysis of Planetary Gear Trains (PGTs) of one DOF. However, every method has its own shortcomings. In this paper a comparison is made between Characteristic polynomial, Eigenvalues and Eigenvectors, Hamming Number Method and Modified Path Matrix (MPM) method. There are many methods available to test isomorphism in PGTs, out of which these four methods wasanalyzed in this paper. For a given PGT with a number of links and a single Degree of Freedom (DOF), adjacency matrix is enough to find out the Eigenvalues and Eigenvectors. Isomorphism of PGT is determined using the Eigenvalues. If Eigenvalues are similar then the PGTs are isomorphic. Similarly, if the characteristic polynomials of two PGTs are samethen it represents the isomorphic PGTs. Characteristic polynomials are determined from the Adjacency matrix. Hamming method also uses adjacency matrix to generate Hamming matrix. Hamming strings are developed from Hamming Matrix. Uniform hamming strings of two PGTsindicates isomorphism in PGTs. Along with isomorphism, Symmetry also known from the Hamming method. Modified path matrix method uses a connectivity matrix to generate MPM. If train values of two PGTs are same then they will be isomorphic otherwise non isomorphic. As per literature as the number of links increases the results may not be accurate with Characteristic coefficients and Eigenvalues methods, though all these methods are used to detect isomorphism among a group of PGTs. Whereas with the Hamming number approach, one can detect isomorphism, symmetry and number of possible level combinations of an PGT with a single hamming matrix.

Bifurcation and optimal control analysis of a delayed drinking model

Alcoholism is a social phenomenon that affects all social classes and is a chronic disorder that causes the person to drink uncontrollably, which can bring a series of social problems. With this motivation, a delayed drinking model including five subclasses is proposed in this paper. By employing the method of characteristic eigenvalue and taking the temporary immunity delay for alcoholics under treatment as a bifurcation parameter, a threshold value of the time delay for the local stability of drinking-present equilibrium and the existence of Hopf bifurcation are found. Then the length of delay has been estimated to preserve stability using the Nyquist criterion. Moreover, optimal strategies to lower down the number of drinkers are proposed. Numerical simulations are presented to examine the correctness of the obtained results and the effects of some parameters on dynamics of the drinking model.

A Fast Computational Method for Characteristic Modes and Eigenvalues of Array Antennas

In this article, an efficient and systematic method for fast and accurate computation of characteristic modes (CMs) and eigenvalues of the entire array antenna as a whole is proposed. This technique benefits from the symmetry and Toeplitz property of the method of moments (MOM) impedance matrix of the array structure. Using this technique, we are able to directly obtain CMs of a large-scale array structure with minimal computational resources. First, the self-impedance matrix of a radiating element and the mutual-impedance matrices between that element and the remaining ones are calculated numerically using MOM. Then, the theory of characteristic modes (TCM) is applied to the entire array. Using the proposed method, we decompose the coefficient matrix of the CM equation into smaller matrices. In addition, interesting and meaningful relations between currents flowing on elements associated with each CM of an array are derived and used to significantly reduce the computational complexity in the CM extraction. Moreover, numerical results and computation time are reported to validate the proposed method.