Verified Error Bounds for Real Eigenvalues of Real Symmetric and Persymmetric Matrices

This paper mainly investigates the verification of real eigenvalues of the real symmetric and persymmetric matrices. For a real symmetric or persymmetric matrix, we use eig code in Matlab to obtain its real eigenvalues on the basis of numerical computation and provide an algorithm to compute verified error bound such that there exists a perturbation matrix of the same type within the computed error bound whose exact real eigenvalues are the computed real eigenvalues.

Computing Nearest Correlation Matrix via Low-Rank ODE’s Based Technique

For n-dimensional real-valued matrix A, the computation of nearest correlation matrix; that is, a symmetric, positive semi-definite, unit diagonal and off-diagonal entries between −1 and 1 is a problem that arises in the finance industry where the correlations exist between the stocks. The proposed methodology presented in this article computes the admissible perturbation matrix and a perturbation level to shift the negative spectrum of perturbed matrix to become non-negative or strictly positive. The solution to optimization problems constructs a gradient system of ordinary differential equations that turn over the desired perturbation matrix. Numerical testing provides enough evidence for the shifting of the negative spectrum and the computation of nearest correlation matrix.

Fast Path Planning for On-Water Automatic Rescue Intelligent Robot Based on Constant Thrust Artificial Fluid Method

In this paper, fast path planning of on-water automatic rescue intelligent robot is studied based on the constant thrust artificial fluid method. First, a three-dimensional environment model is established, and then the kinematics equation of the robot is given. The constant thrust artificial fluid method based on the isochronous interpolation method is proposed, and a novel algorithm of constant thrust fitting is researched through the impulse compensation. The effect of obstacles on original fluid field is quantified by the perturbation matrix, and the streamlines can be regarded as the planned path. Simulation results demonstrate the effectiveness of this method by comparing with A-star algorithm and ant colony algorithm. It is proved that the planned path can avoid all obstacles smoothly and swiftly and reach the destination eventually.

Three-Dimensional Path Planning of Constant Thrust Unmanned Aerial Vehicle Based on Artificial Fluid Method

In this paper, a three-dimensional path planning problem of an unmanned aerial vehicle under constant thrust is studied based on the artificial fluid method. The effect of obstacles on the original fluid field is quantified by the perturbation matrix, the streamlines can be regarded as the planned path for the unmanned aerial vehicle, and the tangential vector and the disturbance matrix of the artificial fluid method are improved. In particular, this paper addresses a novel algorithm of constant thrust fitting which is proposed through the impulse compensation, and then the constant thrust switching control scheme based on the isochronous interpolation method is given. It is proved that the planned path can avoid all obstacles smoothly and swiftly and reach the destination eventually. Simulation results demonstrate the effectiveness of this method.

Dynamical system analysis for a scalar–tensor model with Gauss–Bonnet coupling

In this paper, we study the dynamics of a scalar–tensor model of dark energy in which a scalar field that plays the role of dark energy, non-minimally coupled to the Gauss–Bonnet invariant in four dimensions. We utilize the dynamical system method to extract the critical points of the model and to conclude about their stability, we investigate the sign of the corresponding eigenvalues of the perturbation matrix at each point numerically. For exponential form of the scalar field potential and coupling function, we find five stable points among the critical points of the autonomous system. We also find four scaling attractor solutions with the property that the ratio of dark energy to dark matter density parameters are of order one. These solutions give the hope to alleviate the well-known coincidence problem in cosmology.

Minimum property of condition numbers for the Drazin inverse and singular linear equations

For a singular linear equation Ax = b,x ? R(AD), a small perturbation matrix E and a vector ?b are given to A and b, respectively. We then have the perturbed singular linear equation (A+E)~x = b+?b, ~x ? R[(A+E)D]. This note is devoted to show the minimum property of the condition numbers on the Drazin inverse AD and the Drazin-inverse solution ADb.

Study on Truss Structure Damage Identification Base on Residual Force Vector

By using the intrinsic character of the truss structure, a new direct damage identification method of the truss structure based on residual force vector is presented. The stiffness connectivity matrix is got by sensitivity analysis, and it can be used to expand the stiffness perturbation matrix of damage structure, then, a new residual force vector is got, it can be solved directly, so the stiffness perturbation of structure is got, by it, the damaged locations and the extents of the truss structure are identified. At last, a hyper static truss structure example is given, the identification result shows that, only needed the first-order mode of the damaged structure, the damage can be identified accurately.

A Random Matrix Approach to Manipulator Jacobian

Traditional kinematic analysis of manipulators, built upon a deterministic articulated kinematic modeling often proves inadequate to capture uncertainties affecting the performance of the real robotic systems. While a probabilistic framework is necessary to characterize the system response variability, the random variable/vector based approaches are unable to effectively and efficiently characterize the system response uncertainties. Hence in this paper, we propose a random matrix formulation for the Jacobian matrix of a robotic system. It facilitates characterization of the uncertainty model using limited system information in addition to taking into account the structural inter-dependencies and kinematic complexity of the manipulator. The random Jacobian matrix is modeled such that it adopts a symmetric positive definite random perturbation matrix. The maximum entropy principle permits characterization of this perturbation matrix in the form of a Wishart distribution with specific parameters. Comparing to the random variable/vector based schemes, the benefits now include: incorporating the kinematic configuration and complexity in the probabilistic formulation, achieving the uncertainty model using limited system information (mean and dispersion parameter), and realizing a faster simulation process. A case study of a 6R serial manipulator (PUMA 560) is presented to highlight the critical aspects of the process. A Monte Carlo analysis is performed to capture the deviations of distal path from the desired trajectory and the statistical analysis on the realizations of the end effector position and orientation shows how the uncertainty propagates throughout the system.

Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

We establish a new asymptotic theorem for the nth order nonautonomous dynamic equation by its transformation to the almost diagonal system and applying Levinson's asymptotic theorem. Our transformation is given in the terms of unknown phase functions and is chosen in such a way that the entries of the perturbation matrix are the weighted characteristic functions. The characteristic function is defined in the terms of the phase functions and their choice is exible. Further applying this asymptotic theorem we prove the new oscillation and nonoscillation theorems for the solutions of the nth order linear nonautonomous differential equation with complex-valued coefficients. We show that the existence of the oscillatory solutions is connected with the existence of the special pairs of phase functions.

Application of graph theory and topological models for the determination of fundamentals of the aromatic character of pi-conjugated hydrocarbons

Application of topological analysis and graph theory to benzenoid hydrocarbons leads to the determination of fundamentals of aromaticity: the Hückel rule and the Clar rule. The approach, based on a treatment of the adjacency matrix, allows resonance energy (RE)-like characteristics to be estimated with quite good accuracy, and magnetic aromaticity indices to be derived for both the individual rings and the whole molecules. It also allows an effective approach for interpreting ring current formation in molecules when exposed to an external magnetic field. The transformation of the perturbation matrix into a form describing the canonical structures allows their gradation and determination of their stabilizing/destabilizing character.