Matrix Formula Research Articles

Perrin’s bivariate and complex polynomials

In this article, a study is carried out around the Perrin sequence, these numbers marked by their applicability and similarity with Padovan’s numbers. With that, we will present the recurrence for Perrin’s polynomials and also the definition of Perrin’s complex bivariate polynomials. From this, the recurrence of these numbers, their generating function, generating matrix and Binet formula are defined.

The Matrices of Some of the Plane Graphs and Their Dual

The importance of Graph Theory comes from being Combinatorics. It can study this subject and its relationship to the topological using plane graphs where they are embedded graphs. Also, its relationship to linear algebra which can study the using of matrices. This article describes the adjacency matrices of adjacent vertices of plane graphs and their duals by finding the formula of the adjacency matrix of some types of plane graphs and their duals.

Optimal Strategy Estimation of Random Evolutionary Boolean Games.

The optimal strategy estimation of random evolutionary Boolean games (REBGs) is discussed in this article. First, using the minimum mean square error criterion, the optimal strategy estimator is proposed for REBGs. Then, a matrix approach is developed to calculate the optimal strategy estimator by the aid of a semitensor product of matrices, which includes the prediction matrix, updating distribution, and strategy iterative formula. Finally, an elucidative example is included to show the obtained results are valid.

Laguerre polynomial solutions of linear fractional integro-differential equations

In this paper, the numerical solutions of the linear fractional Fredholm–Volterra integro-differential equations have been investigated. For this purpose, Laguerre polynomials have been used to develop an approximation method. Precisely, using the suitable collocation points, a system of linear algebraic equations arises which is resulted by the transformation of the integro-differential equation. The fractional derivative is considered in the conformable sense, and the conformable fractional derivative of the Laguerre polynomials is obtained in terms of Laguerre polynomials. Additionally, for the first time in the literature, the exact matrix formula of the conformable derivatives of the Laguerre polynomials is established. Furthermore, the results of the proposed method have been given applying the method to some various examples.

Spherical kinematics in 3-dimensional generalized space

In this study, we obtained generalized Cayley formula, Rodrigues equation and Euler parameters of an orthogonal matrix in 3-dimensional generalized space [Formula: see text]. It is shown that unit generalized quaternion, which is defined by the generalized Euler parameters, corresponds to a rotation in [Formula: see text] space.We found that the rotation in matrix equation forms using matrix form of the generalized quaternion product. Besides, in [Formula: see text] space, we obtained the rotations determined by the unit quaternions and unit split quaternions, which are special cases of generalized quaternions for [Formula: see text] in 3-dimensional Eulidean space [Formula: see text] in 3-dimensional Lorentzian space [Formula: see text] respectively.

An efficient algorithm of three-dimensional explicit electromagnetic sensitivity matrix in marine controlled source electromagnetic measurements

In this paper, an efficient algorithm of three-dimensional (3D) explicit sensitivity (or called Fréchet derivatives) matrix for marine controlled source electromagnetic measurements is established by combining an electric coupled potential finite volume method with a direct solver PARDISO direct method. Firstly, on the Yee’s staggered grids, the coupled potential Helmholtz equations are discretized to form a large, sparse and complex linear system which is excited by mobile transmitters. Secondly, through the inversion of the discrete matrix and 3D linear interpolation formula, the interpolation operators and projection operators are established for each receiver at different positions. Because these interpolation operators and projection operators are unrelated to transmitters, they can be computed in advance according to the positions of all receivers. Then the multiple projection operators with discrete vector of each transmitter source can efficiently produce the electromagnetic (EM) responses. On the basis, the goal conductivity of block model and pixel model is expressed as a piece-wise constant function. By perturbing the goal conductivity, the scattered electric current density can be decomposed into a series of electric current elements distributed on Yee’s grids. Each scattered current element is equal to the product of relative perturbation of conductivity and the electric intensity on the grid. The discrete vector on the right-hand side is computed by integrating each scattered current element on the Yee’s grid and then being multiplied with the project operator. Then the linear relationship between the changes in EM field and the relative conductivity perturbation on each block or pixel can be fast produced to obtain the explicit sensitivity matrix about EM responses. Finally, numerical results demonstrate the efficiency and accuracy of this method. The characteristics of 3D sensitivity in three different cases are further investigated.

Intervención educativa de enfermería sobre quemaduras en escolares de una población mexiquense.

Evaluar las intervenciones educativas de enfermería en quemaduras sobre escolares de 8 a 10 años con una población mexiquense. Investigación cuantitativa, no experimental tipo transversal y descriptivo. De 185 alumnos, se determinan 95 alumnos como muestra a partir de una fórmula de matrices con poblaciones finitas. Se utilizó un cuestionario de 11 ítems dividido en dos partes, contiene preguntas con respuestas de opción múltiple (5 respuestas en su mayoría), 64% de la población conoce que es una quemadura, 51% la forma correcta de clasificar su nivel de gravedad, cifra que coincide con quienes identifican las quemaduras del tipo I (53%). Pero solo 35% sabe como tratarlas.

Process convergence of fluctuations of linear eigenvalue statistics of random circulant matrices

We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of random circulant matrices with independent Brownian motion entries, as the dimension of the matrix tends to [Formula: see text]. Our derivation is based on the trace formula of circulant matrix, method of moments and some combinatorial techniques.

A matrix approach to the Beveridge-Nelson decomposition of Markov-switching processes with applications to business cycle

ABSTRACT We provide simple matrix formulas for calculation of the Beveridge–Nelson decomposition in the Markov-switching multivariate case, where series are generated by vector ARIMA models with Markov-switching intercept term. The treatment is immediately extended for regime changes in the mean, distributed lags in the regime and cointegrated models with Markov switching. We apply the method to real data from an example representative of recently industrialized economies and from a classical study on the US business cycle.

Fractional order of Legendre-type matrix polynomials

Recently, special functions of fractional order calculus have had many applications in various areas of mathematical analysis, physics, probability theory, optimization theory, graph theory, control systems, earth sciences, and engineering. Very recently, Zayed et al. (Mathematics 8:136, 2020) introduced the shifted Legendre-type matrix polynomials of arbitrary fractional orders and their various applications utilizing Rodrigues matrix formulas. In this line of research, we use the fractional order of Rodrigues formula to provide further investigation on such Legendre polynomials from a different point of view. Some properties, such as hypergeometric representations, continuation properties, recurrence relations, and differential equations, are derived. Moreover, Laplace’s first integral form and orthogonality are obtained.

Multichannel filter based on tunable fibonacci quasi-periodic structure for one-dimensional magnetized ternary plasma photonic crystal

In this paper, we propose an one-dimensional magnetized plasma photonic crystal based on the ternary Fibonacci quasi-period structure, and use the transmission matrix method (TMM) to study the terahertz band comb-like multichannel filtering properties, which is different from the periodic structure multichannel properties introduced by the plasma defect or dielectric defect. Second, we give the calculation formula of the transmission matrix of the model, third, we find the numerical relationship between the number of Fibonacci sequences and the number of channels. Higher relative refractive index improves the filter performance of comb-like channels. The effect of the length of each layer on the number of channels and the frequency of adjacent comb-like channels is different from that of traditional periodic multichannel filters. The detuning of magnetic field can increase the number of filtering channels, and different incident angles have different phenomena.

Cellular Legendrian contact homology for surfaces, part I

We give a computation of the Legendrian contact homology (LCH) DGA for an arbitrary generic Legendrian surface L in the 1-jet space of a surface. As input we require a suitable cellular decomposition of the base projection of L. A collection of generators is associated to each cell, and the differential is given by explicit matrix formulas. In the present article, we prove that the equivalence class of this cellular DGA does not depend on the choice of decomposition, and in the sequel papers [35,36] we use this result to show that the cellular DGA is equivalent to the usual Legendrian contact homology DGA defined via holomorphic curves. Extensions are made to allow Legendrians with non-generic cone-point singularities. We apply our approach to compute the LCH DGA for several examples including an infinite family, and to give general formulas for DGAs of front spinnings allowing for the axis of symmetry to intersect L.

Quantized feedback particle filter for unmanned aerial vehicles tracking with quantized measurements

Many existing state estimation approaches assume that the measurement noise of sensors is Gaussian. However, in unmanned aerial vehicles tracking applications with distributed passive radar array, the measurements suffer from quantization noise due to limited communication bandwidth. In this paper, a novel state estimation algorithm referred to as the quantized feedback particle filter is proposed to solve unmanned aerial vehicles tracking with quantized measurements, which is an improvement of the feedback particle filter (FPF) for the case of quantization noise. First, a bearing-only quantized measurement model is presented based on the midriser quantizer. The relationship between quantized measurements and original measurements is analyzed. By assuming that the quantization satisfies [Formula: see text], Sheppard’s correction is used for calculating the variances of the measurement noise. Then, a set of controlled particles is used to approximate the posterior distribution. To cope with the quantization noise of passive radars, a new formula of the gain matrix is derived by modifying the measurement noise covariance. Finally, a typical two-passive radar unmanned aerial vehicles tracking scenario is performed by QFPF and compared with the three other algorithms. Simulation results verify the superiority of the proposed algorithm.

Further Expressions on the Drazin Inverse for Block Matrix

This article addresses the problem of developing new expressions for the Drazin inverse of complex block matrix $$M=\left( \begin{array}{cc} A &{} B \\ C &{} D \\ \end{array} \right) \in {\mathbb {C}}^{n\times n}$$ (where A and D are square matrices but not necessarily of the same size) in terms of the Drazin inverse of matrix A and of its generalized Schur complement $$S=D-CA^DB$$ which is not necessarily invertible. This formula is the extension of the well-known Banachiewicz inversion formula of complex block matrix M. In addition, we provide representation for the Drazin inverse of complex block matrix M without any restriction on the generalized Schur complement S and under different conditions than those used in some current literature on this subject. Finally, several illustrative numerical examples are considered to demonstrate our results.

Covariance matrix of maximum likelihood estimators in censored exponential regression models

The censored exponential regression model is commonly used for modeling lifetime data. In this paper, we derive a simple matrix formula for the second-order covariance matrix of the maximum likelihood estimators in this class of regression models. The general matrix formula covers many types of censoring commonly encountered in practice. Also, the formula only involves simple operations on matrices and hence is quite suitable for computer implementation. Monte Carlo simulations are provided to show that the second-order covariances can be quite pronounced in small to moderate sample sizes. Additionally, based on the second-order covariance matrix, we propose an alternative Wald statistic to test hypotheses in this class of regression models. Monte Carlo simulation experiments reveal that the alternative Wald test exhibits type I error probability closer to the chosen nominal level. We also present an empirical application for illustrative purposes.

Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

Matrix Expression of Shapley Value in Graphical Cooperative Games

This paper studies a class of cooperative games, called graphical cooperative games, where the internal topology of the coalition depends on a prescribed communication graph among players. First, using the semitensor product of matrices, the value function of graphical cooperative games can be expressed as a pseudo-Boolean function. Then, a simple matrix formula is provided to calculate the Shapley value of graphical cooperative games. Finally, some practical examples are presented to illustrate the application of graphical cooperative games in communication-based coalitions and establish the significance of the Shapley value in different communication networks.

Second order optimality on orthogonal Stiefel manifolds

The main tool to study a second order optimality problem is the Hessian operator associated to the cost function that defines the optimization problem. By regarding an orthogonal Stiefel manifold as a constraint manifold embedded in an Euclidean space we obtain a concise matrix formula for the Hessian of a cost function defined on such a manifold. We introduce an explicit local frame on an orthogonal Stiefel manifold in order to compute the components of the Hessian matrix of a cost function. We present some important properties of this frame. As applications we rediscover second order conditions of optimality for the Procrustes and the Penrose regression problems (previously found in the literature). For the Brockett problem we find necessary and sufficient conditions for a critical point to be a local minimum. Since many optimization problems are approached using numerical algorithms, we give an explicit description of the Newton algorithm on orthogonal Stiefel manifolds.

Viscoacoustic least-squares migration with a blockwise Hessian matrix: an effective Q approach

Abstract We present an explicit inverse approach using a Hessian matrix for least-squares migration (LSM) with Q compensation. The scheme is developed by incorporating an effective Q-based solution of the viscoacoustic wave equation into a blockwise approximation to the Hessian in LSM, which is implemented after the so-called deabsorption prestack time migration (PSTM). The effective Q model used fully accounts for frequency-dependent traveltime and amplitude at the same imaging location. We can extract the effective Q parameters by scanning during previous deabsorption PSTM. This avoids the challenging task of building the Q model. The blockwise Hessian matrix approach decomposes the full Hessian matrix into a series of computationally tractable small-sized matrices using a localised approach. We derive the explicit formula of the offset-dependent Hessian matrix using an analytical Green's function obtained from deabsorption PSTM. In this way, we can approximate a reflectivity imaging for the targeted zone by a spatial deconvolution of the migrated result with an explicit inverse. The resulting scheme broadens the frequency-band of imaging by deabsorption, and improves the subsurface illumination and spatial resolution through the inverse Hessian. A high-resolution, true-amplitude migrated gather can then be obtained. Synthetic and field data sets demonstrate the proposed blockwise LS-QPSTM.

Propagation characteristics of non-diffracting Lommel beams in a gradient-index medium

We investigate the propagation of the Lommel beams in a gradient-index medium. Based on the ABCD transfer matrix and Collins diffraction integral formula, the analytical expressions of the complex amplitude of the Lommel beams through the gradient-index medium are derived. The intensity distribution and the Poynting vector of the Lommel beams are studied analytically and numerically. The results show that the Lommel beam is focused at the singularities and the corresponding periodic beam profile is induced by the symmetrical refractive index pattern during the beam propagating in the gradient-index medium. The effects of the asymmetry factor and the topological charge on the propagation characteristics of the Lommel beams in the gradient-index medium are discussed in detail. Especially, the propagation trajectories of the Lommel beams become visible as the topological charge increases and the symmetry of the Lommel beam’s transverse intensity pattern can be controlled by varying the asymmetry factor. It is expected that the results obtained in this paper may be useful for potential applications of optical control and trapping.