Jacobi Matrix Research Articles

A family of orthogonal polynomials corresponding to Jacobi matrices with a trace class inverse

Assume that {an;n≥0} is a sequence of positive numbers and ∑an−1<∞. Let αn=kan, βn=an+k2an−1 where k∈(0,1) is a parameter, and let {Pn(x)} be an orthonormal polynomial sequence defined by the three-term recurrence α0P1(x)+(β0−x)P0(x)=0,αnPn+1(x)+(βn−x)Pn(x)+αn−1Pn−1(x)=0for n≥1, with P0(x)=1. Let J be the corresponding Jacobi (tridiagonal) matrix, i.e. Jn,n=βn, Jn,n+1=Jn+1,n=αn for n≥0. Then J−1 exists and belongs to the trace class. We derive an explicit formula for Pn(x) as well as for the characteristic function of J and describe the orthogonality measure for the polynomial sequence. As a particular case, the modified q-Laguerre polynomials are introduced and studied.

About essential spectra of unbounded Jacobi matrices

We study spectral properties of unbounded Jacobi matrices with periodically modulated or blended entries. Our approach is based on uniform asymptotic analysis of generalized eigenvectors. We determine when the studied operators are self-adjoint. We identify regions where the point spectrum has no accumulation points. This allows us to completely describe the essential spectrum of these operators.

System static voltage stability analysis considering load characteristics of electrolytic aluminum

The load characteristics have a significant impact on the static voltage stability analysis of the power grid. Wenshan power grid has a large amount of relatively centralized electrolytic aluminum load, accounting for 65% of the total local load. Due to the different load characteristics of electrolytic aluminum shown in the big closed-loop feedback current stabilizing state, the detailed load characteristics of electrolytic aluminum must be considered when analyzing the static voltage stability of the Wenshan power grid. Firstly, based on the actual production process of electrolytic aluminum, an accurate load characteristic model of electrolytic aluminum under the big closed-loop feedback current stabilizing state is built; secondly, a static voltage stability analysis model on account of the electrolytic aluminum load characteristic is established, and solving this model can obtain the generalized Jacobi matrix J and reduced dimensional Jacobi matrix J1 which considers comprehensively the influence of system active and reactive power on the static voltage stability, based on the singular value decomposition theory, the static voltage stability margins MSV-J and MSV-J1 decomposed from the matrix J and J1 respectively can be obtained to describe the stability degree of system static voltage accurately. And the effect of adjusting the current stabilizing device on the load characteristics of electrolytic aluminum and the static voltage stability of the system is also studied. Finally, the influence of load characteristics of electrolytic aluminum on the system static voltage stability is analyzed by some examples, and the results indicate that the static voltage stability margin indexes MSV-J and MSV-J1 can well describe the static voltage stability of the grid.

A fast matrix generation method for solving entry trajectory optimization problems under the pseudospectral method

A fast matrix generation method based on correlations is proposed for the pseudospectral method (PSM) to solve the entry trajectory optimization problem. First, the optimization problem is analyzed to obtain some assumptions of performance indicators and constraints. Then, the chain rule and correlations among the variables are used to simplify the gradient calculations, resulting in a fast generation method for the gradient vector. Meanwhile, based on the previous assumptions, a fast generation method for the Jacobi matrix is derived semi-analytically using the correlation information among the variables. Finally, the trajectory optimization problem is studied using a typical aircraft. The results are verified via comparisons with the well-known GPOPS II software. Furthermore, the superiority of the proposed method in terms of the computational efficiency is demonstrated by comparing the relevant time indexes. The performance of the proposed method in calculating the Jacobian matrix and gradient matches that of the GPOPS, combined with a relatively short solution time.

Time-Division-Multiplexed Online Gauss-Newton-Based Multi-Echo Decomposition Method for Real-Time In-Situ Laser Ranging

In forestry and agriculture investigations, the laser spot of FW-LiDAR may simultaneously illuminate multiple targets, which reflect multiple echoes. To realize the real-time in-situ laser ranging of multiple targets, a time-division-multiplexed online Gauss-Newton-based multi-echo decomposition method (GMOT) is proposed for decomposing the multi-echo in an online manner. In waveform preprocessing, a local-region linear analysis method was employed in GMOT to recognize the echo components of multi-target. Gauss-Newton method was then used to decompose the multi-echo by fitting the waveform. In the waveform fitting, a sparse Jacobi matrix was employed to improve the target extraction ratio. Due to the ability of the parallel computation, a Kintex-7 FPGA was employed as the hardware platform. GMOT was implemented on the embedded FPGA with the parallel-pipeline architecture and time-division multiplexing method to improve the measurement rate for decomposing the multi-echo in real-time. Simulations proved that GMOT achieves a measurement rate of 238.1 kHz, approximately 3.5 times as fast as online Gauss-Newton-based multi-echo decomposition method. An FPGA-based in-situ ranging system was built to evaluate the ranging performance of GMOT by experiments. The results revealed that GMOT increases the target extraction ratio by 11.4%, compared with the Gauss-Newton-based multi-echo post-processing decomposition method (GMP). The ranging error of GMOT is on the same scale of GMP. The proposed method will be used for the real-time multi-target ranging in the forestry survey and intelligent driving.

On the construction of de Branges spaces for dynamical systems associated with finite Jacobi matrices

On the construction of de Branges spaces for dynamical systems associated with finite Jacobi matrices

Deficiency indices of block Jacobi matrices and Miura transformation

Abstract We study the infinite Jacobi block matrices under the discrete Miura-type transformations which relate matrix Volterra and Toda lattice systems to each other and the situations when the deficiency indices of the corresponding operators are the same. A special attention is paid to the completely indeterminate case (i.e., then the deficiency indices of the corresponding block Jacobi operators are maximal). It is shown that there exists a Miura transformation which retains the complete indeterminacy of Jacobi block matrices appearing in the Lax representation for such systems, namely, if the Lax matrix of Volterra system is completely indeterminate, then so is the Lax matrix of the corresponding Toda system, and vice versa. We consider an implication of the obtained results to the study of matrix orthogonal polynomials as well as to the analysis of self-adjointness of scalar Jacobi operators.

DESIGN AND EXPERIMENT OF A SAFFLOWER PICKING ROBOT BASED ON A PARALLEL MANIPULATOR

To mitigate the large demand for safflower picking labour and the low efficiency of manual picking, a safflower picking robot that is based on a parallel manipulator is designed. The whole robot mainly consists of a walking device, parallel manipulator device, vision device, picking device, filament collection device, control system, and motor drive system. The forward and inverse kinematics analysis of the parallel manipulator is analysed via a geometric method, the kinematics model is established, and the computational analysis of the parallel manipulator working speed is based on the Jacobi matrix. For driving angle errors of 0.001∼0.005°, the moving platform's motion accuracy is 0.2174–0.9387 mm, which meets the position accuracy requirements of the test stage. Based on MATLAB software, the Monte Carlo method is employed to solve the working space of the parallel manipulator, which is approximately an equilateral triangle with a side length of 0.35 m, which satisfies the safflower growth space picking conditions. The prototype is constructed, and safflower picking experiments are carried out under laboratory conditions. The average picking period of each flower bulb was 16 s, and the average net picking rate of the filaments was 87.91%. The experimental results verified the applicability of the safflower picking robot.

The Inverse Eigenvalue Problem for a Class of Sub-periodic Generalized Jacobi Matrices

The inverse eigenvalue problem of matrix has practical significance in engineering and scientific calculation. In this paper, the inverse eigenvalue problem of sub-periodic generalized Jacobi matrices is studied. Two eigenvalues and corresponding eigenvectors are used to construct generalized Jacobi matrices. The existence and uniqueness conditions of the solution are discussed, the solution algorithm for the inverse problem is given, and the effectiveness of the algorithm is verified by a numerical example.

К ОБОСНОВАНИЮ ОДНОЗНАЧНОСТИ ПРОДОЛЖЕНИЯ РЕШЕНИЯ ЗАДАЧ О ДЕФОРМИРОВАНИИ МЯГКИХ ОБОЛОЧЕК МЕТОДОМ ДИФФЕРЕНЦИРОВАНИЯ ПО ПАРАМЕТРУ

Criterion of hyperelastic soft shell deforming nonlinear problem numerical solution continuation uniqueness assessment is suggested in the work. The criterion can be used when carrying out calculations. It is based on investigation of properties of Jacobi matrix of linear algebraic equations system which is formed when using parameter differentiation method. This method allows reducing nonlinear boundary value problem solution to a couple of quasilinear boundary value and nonlinear initial problems and applying initial parameters method of solving linear boundary value problems. For assessment of solution continuation uniqueness in each point of integration interval one needs controlling magnitudes of resolving differential equation system right-hand sides vector components, as well as calculating determinant and rank of Jacobi matrix of algebraic equations system formed as a result of initial parameters method using, and expanded Jacobi matrix rank calculation. For testing suggested criterion the problem of neohookean material hemisphere static inflation by uniformly applied pressure is considered. Solution of this problem as certain values of numerical algorithm parameters leads to various calculation difficulties – calculation stability loss, big errors of calculation results, non-uniqueness of solution which reason requires additional investigations. It is shown that in the points, where mentioned difficulties are met, determinateness conditions for the function of right-hand sides of differential equation system formulated within the framework of the suggested criterion are broken.

Isolated periodic wave trains in a generalized Burgers–Huxley equation

We study the isolated periodic wave trains in a class of modified generalized Burgers–Huxley equation. The planar systems with a degenerate equilibrium arising after the traveling transformation are investigated. By finding certain positive definite Lyapunov functions in the neighborhood of the degenerate singular points and the Hopf bifurcation points, the number of possible limit cycles in the corresponding planar systems is determined. The existence of isolated periodic wave trains in the equation is established, which is universal for any positive integer n in this model. Within the process, one interesting example is obtained, namely a series of limit cycles bifurcating from a semi-hyperbolic singular point with one zero eigenvalue and one non-zero eigenvalue for its Jacobi matrix.

МАЛОЦИКЛОВАЯ УСТАЛОСТЬ ОБРАЗЦОВ С КОЛЬЦЕВОЙ ВЫТОЧКОЙ ПРИ ЖЕСТКОМ НАГРУЖЕНИИ

The results of tests of cylindrical specimens of alloy 01570C for low-cycle fatigue are presented. The tests were carried out on a smooth sample and samples with an annular recess. Specimens with an annular groove were tested under rigid cyclic alternating loading with a constant range of elongation of the working part. To simulate the experiments performed, a variant of the plasticity model based on the theory of flow under combined hardening is used. In the chosen mathematical model, a memory surface is introduced that separates monotonous and cyclic loading. This division allows one to take into account various features of isotropic and anisotropic hardening of the material. Anisotropic hardening is described by the sum of backstresses of three different types, which make it possible to describe the effects of fitting and stepping out of the elastoplastic hysteresis loop. The plasticity model makes it possible to assess the damaged state of a material using an equation that describes the process of damage accumulation based on the energy criterion of durability. According to the results of testing a smooth sample, the parameters of the plasticity model were determined. The finite element method was used to carry out mathematical modeling of experiments on loading samples with an annular groove. The material behavior model was implemented in the SIMULIA Abaqus software package, for which the model equations were linearized and the Jacobi matrix was calculated, which determines the change in each of the stress increment components caused by an infinitesimal change in each component of the strain increment tensor. Based on the results of mathematical modeling, the dependences of the load applied to the sample on the elongation of its working part were obtained. The article compares the dependences of the load applied to the sample, obtained from the results of the calculation, with the experimental ones, as well as the number of cycles until the destruction of the sample.

On the similarity of complex symmetric operators to perturbations of restrictions of normal operators

In this paper we consider a problem of the similarity of complex symmetric operators to perturbations of restrictions of normal operators. For a subclass of cyclic complex symmetric operators in a finite-dimensional Hilbert space we prove the similarity to rank-one perturbations of restrictions of normal operators. The main tools are a truncated moment problem in $\mathbb{C}$, and some objects similar to objects from the theory of spectral problems for Jacobi matrices.

Spectral resolutions for non-self-adjoint block convolution operators

The paper concerns the spectral theory for a class of non-self-adjoint block convolution operators. We mainly discuss the spectral representations of such operators. It is considered the general case of operators defined on Banach spaces. The main results are applied to periodic Jacobi matrices.

Mathematical modeling of malignant ovarian tumors

The article explores modeling the development of ovarian cancer, the treatment of this oncologicaldisease in women, the assessment of the time to achieve remission, and the assessment of the time of the onset of relapse. The relevance of the study is that ovarian cancer is one of the most common cancers in women and has the highest mortality rate among all gynecological diseases. Modeling the process of the development of the disease makes it possible to better understand the mechanism of the development of the disease, as well as the time frame of the onset of each stage, as well as the assessment of the survival time. The aim of the work is to develop a model of an ovarian tumor. It is based on a model of competition between two types of cells: epithelial cells (normal cells) and tumor cells (dividing cells). The mathematical interpretation of the competition model is the Cauchy problem for a system of ordinary differential equations. Treatment is seen as the direct destruction of tumor cells by drugs. The behavior of solutions in the vicinity of stationary points is investigated by the eigenvalues of the Jacobi matrix of the right side of the equations. On the basis of this model, the distribution of conditional patients by four stages of the disease is proposed. Biochemical processes that stimulate the accelerated growth of the tumor cell population are modeled by a factor that allows tumor cells to gain an advantage in a competitive relationship with epithelial cells. The spatio-temporal dynamics of an ovarian tumor leads to a modification of the competition model due to the introduction of additional factors into it, taking into account the presence of increased nutrition of ovarian tumors, the exit of the tumor from the plane of the ovary, as well as the effect of treatment on tumor cells. The new model describes the interaction conditions with a system of second-order partial differential equations. The results of computer modeling demonstrate an assessment of the distribution of conditional patients by stages of the disease, the time of onset of relapse, the duration of remission, the obtained theoretical results of modeling are compared with the real data.

A Study on the Number of Preferable Neighborhood Vectors When Using the Improved Jacobi Matrix Estimation Method for Roll Motion Prediction of Small Boat

A Study on the Number of Preferable Neighborhood Vectors When Using the Improved Jacobi Matrix Estimation Method for Roll Motion Prediction of Small Boat

A reduction algorithm for reconstructing periodic Jacobi matrices in Minkowski spaces

The periodic Jacobi inverse eigenvalue problem concerns the reconstruction of a periodic Jacobi matrix from prescribed spectral data. In Minkowski spaces, with a given signature operator H=diag(1,1,…,1,−1), the corresponding matrix is a periodic pseudo-Jacobi matrix. The inverse eigenvalue problem for such matrices consists in the reconstruction of pseudo-Jacobi matrices, with the same order and signature operator H. In this paper we solve this problem by applying Sylvester’s identity and Householder transformation. The solution number and the corresponding reconstruction algorithm are here exhibited, and illustrative numerical examples are given. Comparing this approach with the known Lanczos algorithm for reconstructing pseudo-Jacobi matrices, our method is shown to be more stable and effective.

КОМПЛЕКСИРОВАНИЕ ДАННЫХ СИСТЕМЫ УПРАВЛЕНИЯ МОБИЛЬНЫМ ГУСЕНИЧНЫМ РОБОТОМ

Integration of information – the process of combining data to determine or predict the state of an object. The integration provides an increase in the robustness of the robot control and the accuracy of machine perception of information. The paper discusses one of the methods for integrating of the robot onboard data using the example of the mobile tracked robot data. One of the aim is showing how to obtain the Jacobi matrix of the process function and the Jacobi matrix of the observation function of the mobile robot system for subsequent data integration by the state observer, built on the basis of the Kalman filter, that can convert the robot onboard data into the elements of the state vector of the mobile robot system. Each nonzero element of the Jacobi matrix of the observation function of the system is a weight coefficient. It determines the contribution of an element of the output vector of the system corresponding to this weight coefficient to the result of information integration calculated by the Kalman filter – the state observer. The Kalman filter is a sequential recurrent algorithm for filtering information from discrete dynamical systems specified in the state space. A feature of the Kalman filter as an observer of the state of the system is the assumption that the observed system has the effect of white Gaussian noise (characterized by zero mathematical expectation) on its state.

Deficiency Indices of Block Jacobi Matrices: Survey

The paper is a survey and concerns with infinite symmetric block Jacobi matrices J with mm-matrix entries. We discuss several results on general block Jacobi matrices to be either self-adjoint or have maximal as well as intermediate deficiency indices. We also discuss several conditions for J to have discrete spectrum.

Associated orthogonal polynomials of the first kind and Darboux transformations

Let u be a quasi-definite linear functional defined on the linear space of polynomials P with complex coefficients. For such a functional we can define a sequence of monic orthogonal polynomials (SMOP in short) (Pn)n∈N, which satisfies a three term recurrence relation. If we increase the indices of the recurrence relation by one unity, then we get the sequence of associated polynomials of the first kind as solution. These polynomials will be orthogonal with respect to a linear functional denoted by u(1). In the literature two special transformations of the functional u are studied, the canonical Christoffel transformation u˜=(x−c)u and the canonical Geronimus transformation uˆ=(x−c)−1u+Mδc, where c is a fixed complex number, M is a free complex parameter and δc is the linear functional defined on P as 〈δc,p(x)〉=p(c). For the Christoffel transformation with SMOP (P˜n)n∈N, we are interested in analyzing the relation between the linear functionals u(1) and u˜(1). The super index denotes the linear functionals associated with the orthogonal polynomial sequences of the first kind (Pn(1))n∈N and (P˜n(1))n∈N, respectively. This problem is also studied for Geronimus transformations. Here we give close relations between their corresponding monic Jacobi matrices by using the LU and UL factorizations. To get this result, we first need to study the relation between u−1 (the inverse functional) and u(1) which can be given from a quadratic Geronimus transformation.