A nonsmooth Newton’s method for control-state constrained optimal control problems

We investigate optimal control problems subject to mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer–Burmeister function the minimum principle is transformed into an equivalent nonlinear and nonsmooth equation in appropriate Banach spaces. This nonlinear and nonsmooth equation is solved by a nonsmooth Newton’s method. We will show the local quadratic convergence under certain regularity conditions and suggest a globalization strategy based on the minimization of the squared residual norm. A numerical example for the Rayleigh problem concludes the article.     

Information transmittal, Newton’s law of gravitation, and tensor approach to general relativity

The special relativity considered in [A. Einstein, Zur Elektrodynamik der bewegte Körper. Ann. Physik, 17 (1905) 891-921] is based on the concept of finite speed of information transmittal by the available signals (rays of light). It is demonstrated that the same concept applies to Newton’s law of universal gravitation since the magnitude of distances between attracting masses can be physically defined (carried, accounted in acting forces of gravity) only by signals (physical processes) propagating at finite velocities. It follows that the speed of propagation of gravity is finite. The linear transformations of special relativity are applied to Newton’s law of gravitation to take into account the relativistic effects of information transmittal in a field of central forces of attraction. Relativistic representations of Newton’s law are obtained with respect to the center of gravity exposing illusory effects that appear at high velocities. It is verified that in atomic physics the effect of Newtonian gravitation on the motion of elementary particles at high velocities is negligible also in relativistic consideration. Computational methods are developed to measure the intensity of gravitation at a distant space-time location using a body that travels in space, emitting uniform pulses of light that are received by the observer at a different space-time location. It is demonstrated that the tensor approach to the general relativity and the united theory of space, time and gravitation in which the geometrical properties (metric) of the four-dimensional space-time continuum depend on the distribution of gravitating masses in space and their motion represent a transformed Lorentz invariant with a new type of inertia in the field of forces changing in space and time. Real physical processes evolve according to the forces represented in the tensor form by this invariant which is equivalent to the coordinate-free local invariant of relativistic dynamics that defines the field and the motion of a body whose velocities and accelerations can be measured by relativistic identification methods at a point, time and direction of interest. The results open new avenues for research in the general relativity and can be used for software development, field measurements and experimental studies in application to distant or fast moving systems.     

Élie Cartan’s geometrical vision or how to avoid expression swell

The aim of the paper is to demonstrate the superiority of Cartan’s method over direct methods based on differential elimination for handling otherwise intractable equivalence problems. In this sense, using our implementation of Cartan’s method, we establish two new equivalence results. We establish when a system of second order ODEs is equivalent to flat system (second derivations are zero), and when a system of holomorphic PDEs with two independent variables and one dependent variable is flat. We consider the problem of finding transformation that brings a given equation to the target one. We shall see that this problem becomes algebraic when the symmetry pseudogroup of the target equation is zerodimensional. We avoid the swelling of the expressions, by using non-commutative derivations adapted to the problem.     

A semi-local convergence theorem for a robust revised Newton’s method

It is well known that Newton’s iteration will abort due to the overflow if the derivative of the function at an iterate is singular or almost singular. In this paper, we study a robust revised Newton’s method for solving nonlinear equations, which can be carried out with a starting point with a degenerate derivative at an iterative step. It is proved that the method is convergent under the conditions of the Newton–Kantorovich theorem, which implies a larger convergence domain of the method. We also show that our method inherits the fast convergence of Newton’s method. Numerical experiments are performed to show the robustness of the proposed method in comparison with the standard Newton’s method.     

Poisson’s equation for discrete-time quasi-birth-and-death processes

We consider Poisson’s equation for quasi-birth-and-death processes (QBDs) and we exploit the special transition structure of QBDs to obtain its solutions in two different forms. One is based on a decomposition through first passage times to lower levels, and the other is based on a recursive expression for the deviation matrix.     

Approximate analytic solution of the Dirichlet problems for Laplace’s equation in planar domains by a perturbation method

In this paper, we propose a regular perturbation method to obtain approximate analytic solutions of exterior and interior Dirichlet problems for Laplace’s equation in planar domains. This method, starting from a geometrical perturbation of these planar domains, reduces our problems to a family of classical Dirichlet problems for Laplace’s equation in a circle. Numerical examples are given and comparisons are made with the solutions obtained by other approximation methods.     

Semi-analytical solution of Laplace’s equation in non-equilibrating unbounded problems

Some two-dimensional problems of elastostatics are governed by Laplace’s equation. Using the terminology of elastostatics, if the face loads and body loads are not self-equilibrating, even when the displacement at infinity is restricted to zero, displacements in the near field will be infinite. However, the stress field within the domain is well behaved, and is of practical interest. In this paper the semi-analytical scaled boundary finite-element method is extended to permit the analysis of such problems. The solutions in the primary variable so obtained include an infinite component, but the difference in value between any two points in the domain can be computed accurately. The method is also extended to solve the non-homogeneous form of Laplace’s equation.     

Higher-order correction to the FDTD method based on the integral form of Maxwell’s equations

We propose a higher-order correction to the finite-difference time-domain (FDTD) method based on the integral form of Maxwell’s equations. We calculate the errors between the numerical and analytic solutions. Numerical solutions are obtained by the original method and our corrected FDTD method to show that the accuracy and reliability of our corrected FDTD method is superior to that of the original FDTD method.     

一类离散时间代数Riccati矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在线性二次优化问题中遇到的含未知矩阵之逆的离散时间代数Riccati矩阵方程(DTARME)转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求DTARME的对称解的双迭代算法。双迭代算法仅要求DTARME有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定。数值算例表明双迭代算法是有效的。     

The effect of nonhuman’s external regulation on young children’s self-regulation to regulate their own process of learning

The present study sought to examine the effect of nonhuman’s external regulation on children’s self-regulation to regulate their own process of learning and to what extent did children succeed in terms of application when they talk and think while act alone with nonhuman’s external regulator. The Aginian’s methodology (, , ,  and ) that used an isolated, computer-based learning system and acts as a standalone learning environment with special set of tasks was used by hundred healthy preschool children. The results showed that young children were able to regulate their own process of learning and engage with their full free-will without the need of their real teacher’s regulation. The conclusion provided evidence that the verbalization of thinking aloud should occur spontaneously by nature, the nonhuman’s external regulation has a positive effect on young children’s development when they act with their full free-will, and has a positive effect on their behavior either.     

The method of fundamental solutions for Helmholtz-type equations in composite materials

In this paper, we introduce and develop the method of fundamental solutions (MFS) for solving Helmholtz-type elliptic partial differential equations in composite materials. This study builds upon the previous developments and applications of the MFS to linear and nonlinear heat conduction, elasticity, and functionally graded composite layered materials. Numerical results are presented and discussed for four examples involving both the modified Helmholtz and the Helmholtz equations in two-dimensional or three-dimensional, bounded or unbounded, smooth or non-smooth composite domains. It was found that the method produces numerical results which are in good agreement with the analytical solutions, where available.     

The effect of nonhuman’s external regulation on children’s responses to detect children with developmental problems (DP) associated with the natural development of self-regulation during learning tasks

The present study sought to examine the effect of the nonhuman’s external regulation on children’s responses during learning tasks to detect children with developmental problems (DP) associated with the natural development process of self-regulation. The material was an isolated, computer-based learning system that acts as a standalone learning environment and used by 100 preschool children, which were randomly selected from ten preschools without revising their medical files. Participants were classified by the system itself during learning progression in three essential groups based on Aginian’s zone of children regulation (ZCR), which is “the equilibrium point in the self-regulation’s development process that controls the child to be either a self-Vygotskyian’s learner, self-Piagetian’s learner, or self-Aginian’s learner during learning tasks” ( Agina, Kommers, & Steehouder, 2011d). The results showed that the preschool children can spontaneously do diagnostic tests during learning tasks and the nonhuman external regulator was able to analysis children’s responses that, in turn, used for detecting those children with DP. This result was practically confirmed by revising all children’s medical files that matched the final judgment of the nonhuman external regulator. However, the results confirmed that the natural development of self-regulation was fluctuated among three paradoxical views (Vygotskyian vs. Piagetian vs. Aginian).     

Solution of non-linear fuzzy systems by decomposition of incremental fuzzy numbers

This paper presents a proposal for decomposing large uncertainties associated with fuzzy numbers, which can be useful for solving non-linear fuzzy problems with crisp coefficients. The proposed procedure is based on a property of incremental fuzzy numbers and improves the quality of solution of the linearization process. The methodology is illustrated by some numeric studies.     

一种利用FFT自适应阈值处理失真杂散信号的方法

在图像信号传输过程中,由于传输网络以及各种数模转换设备和数字处理设备的影响,可能会造成失真杂散信号的产生,其随机不确定性将导致某些专业图像检测处理设备工作的异常。文中首先分析了图像失真杂散信号产生的原因及其特征,然后针对此特征提出了一种FFT自适应阈值屏蔽算法,最后利用MATLAB仿真验证了此算法且进行了实际的FPGA实验。     

FFT处理机的形式化模型及正确性验证

快速傅立叶变换的应用领域非常广泛,其硬件实现方法多种多样,验证这些电路的正确性具有很强的实用价值。传统的电路正确性验证的方法是模拟,这种方法的主要缺点是随着参与运算的点数的增加,穷尽模拟全部输入情况所耗费的时间越来越长,甚至难以实现。而形式化方法使用纯数学手段证明电路的正确性,克服了传统方法的缺点。首先用重写系统给出了任意N=2M点的基2的流水式快速傅里叶变换处理机的形式化模型,然后给出它的正确性验证,探索了验证处理复数的复杂电路正确性的方法。     

Stability and non-standard finite difference method of the generalized Chua’s circuit

In this paper, we develop a framework to obtain approximate numerical solutions of the fractional-order Chua’s circuit with Memristor using a non-standard finite difference method. Chaotic response is obtained with fractional-order elements as well as integer-order elements. Stability analysis and the condition of oscillation for the integer-order system are discussed. In addition, the stability analyses for different fractional-order cases are investigated showing a great sensitivity to small order changes indicating the poles’ locations inside the physical s-plane. The Grünwald-Letnikov method is used to approximate the fractional derivatives. Numerical results are presented graphically and reveal that the non-standard finite difference scheme is an effective and convenient method to solve fractional-order chaotic systems, and to validate their stability.