Improving Performance of Computer Algebra Systems

Computer Algebra Systems have been extensively used in higher education. The reasons are many e.g., visualize mathematical problems, correlate real-world problems on a conceptual level, are flexible, simple to use, accessible from anywhere, etc. However, there is still room for improvement. Computer algebra system (CAS) optimization is the set of best practices and techniques to keep the CAS running optimally. Best practices are related to how to carry out a mathematical task or configure your system. In this paper, we are going to examine these techniques. The documentation sheets of CASs are the source of data that we used to compare them and examine their characteristics. The research results reveal that there are many tips that we can follow to accelerate performance.

CONVERGENCE ANALYSIS OF RUNGE-KUTTA METHODS FOR A CLASS OF RETARDED DIFFERENTIAL ALGEBRAIC SYSTEMS

This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. A new convergence concept, called DA-convergence, is introduced. The DA-convergence result for the methods is derived. At the end, a numerical example is given to verify the computational effectiveness and the theoretical result.

On the Order of the Solutions of Systems of Complex Algebraic Differential Equations

This paper is concerned with the order of the solutions of systems of high-order complex algebraic differential equations.By means of Zalcman Lemma,the systems of equations of[1]is extended to more general form.

AN ALGEBRAIC METHOD FOR POLE PLACEMENT IN MULTIVARIABLE SYSTEMS

This paper considers the pole placement in multivariable systems involving known delays by using dynamic controllers subject to multirate sampling. The controller parameterizations are calculated from algebraic equations which are solved by using the Kronecker product of matrices. It is pointed out that the sampling periods can be selected in a convenient way for the solvability of such equations under rather weak conditions provided that the continuous plant is spectrally controllable. Some overview about the use of nonuniform sampling is also given in order to improve the system's performance.

A Back Propagation-Type Neural Network Architecture for Solving the Complete n ×n Nonlinear Algebraic System of Equations

The objective of this research is the presentation of a neural network capable of solving complete nonlinear algebraic systems of n equations with n unknowns. The proposed neural solver uses the classical back propagation algorithm with the identity function as the output function, and supports the feature of the adaptive learning rate for the neurons of the second hidden layer. The paper presents the fundamental theory associated with this approach as well as a set of experimental results that evaluate the performance and accuracy of the proposed method against other methods found in the literature.

Advantages & Disadvantages of Online CAS

Online Computer Algebra Systems (CAS) have become increasingly popular among students and teachers. The reasons are many such as being more flexible, simple to use, accessible from anywhere, etc. However, as with any educational tool, they also have some disadvantages that we should know. The purpose of this study is to analyze advantages and disadvantages of online CASs and propose some techniques to optimize CAS performance in order to reduce weaknesses. The research results reveal that online CAS versions are on the rise but lag in some capabilities in comparison with desktop versions.

Feedback control of nonlinear differential algebraic systems using Hamiltonian function method

The stabilization and H∞ control of nonlinear differential algebraic systems （NDAS） are investigated using the Hamiltonian function method. Firstly, we put forward a novel dissipative Hamiltonian realization （DHR） structure and give the condition to complete the Hamiltonian realization. Then, based on the DHR, we present a criterion for the stability analysis of NDAS and construct a stabilization controller for NDAS in absence of disturbances. Finally, for NDAS in presence of disturbances, the L2 gain is analyzed via generalized Hamilton-Jacobi inequality and an H∞ control strategy is constructed. The proposed stabilization and robust controller can effectively take advantage of the structural characteristics of NDAS and is simple in form.

Banach Algebra Dynamical Systems

Abstract This paper generalizes the C＊-dynamical system to the Banach algebra dynam- ical system （A, G, α） and define the crossed product A αG of a given Banach algebra dynamical system （A, G,α）. Then the representation of A α G is described when A ad- mits a bounded left approximate identity. In a natural way, the authors define the reduced crossed product A αG and discuss the question when A α G coincides with A αG.

Hybrid extended Fourier series for optimal control of nonlinear algebraic dynamical systems

The paper introduces a new method for finding optimal control of algebraic dynamic systems. The structure of algebraic dynamical systems is nonlinear with quadratic and bilinear terms. A new hybrid extended Fourier series is introduced, and state and control variables of the system are expanded by this series. Moreover, properties of new series are presented, and integration and product operational matrices are obtained. Using operational matrices, optimal control of the systems is converted to a set of simultaneous nonlinear algebraic relations. An illustrative example is included to compare our results with those in the literature.

A Simple Jerky Dynamics,Genesio System

The third order explicit autonomous differential equations named as jerk equations represent an interesting subclass of dynamical systems that can exhibit many major features of the regular and chaotic motion. In this paper, we show that an algebraically simple system, the Genesio system can be recast into a jerky dynamics and its jerk equation can be derived from one-dimensional Newtonian equation. We also investigate the global dynamical properties of the corresponding jerk system.

Robust excitation control of multi-machine multi-load power systems using Hamiltonian function method

Using an energy-based Hamiltonian function method,this paper investigates the robust excitation control of multi-machine multi-load power systems described by a set of uncertain differential algebraic equations.First,we complete the dissipative Hamiltonian realization of the power system and adjust its operating point by the means of pre-feedback control.Then,based on the obtained Hamiltonian realization,we discuss the robust excitation control of the power system and put forward an H1 excitation control strategy.Simulation results demonstrate the effectiveness of the control scheme.

Reduced Crossed Products of Pro-Banach Algebras

This paper gives the concept of the reduced pro-Banach algebra crossed product associated with inversely pro-Banach algebra dynamical system,and shows that the reduced crossed product is an inverse limit of an inverse system of Banach algebra crossed products.Also,the authors show that if the locally compact group is amenable,then the crossed product and the reduced crossed product are isometrically isomorphic.

A Novel Method to Solve Nonlinear Klein-Gordon Equation Arising in Quantum Field Theory Based on Bessel Functions and Jacobian Free Newton-Krylov Sub-Space Methods

The Klein-Cordon equation arises in many scientific areas of quantum mechanics and quantum field theory. In this paper a novel method based on spectral method and Jacobian free Newton method composed by generalized minimum residual （JFNGMRes） method with adaptive preconditioner will be introduced to solve nonlinear Klein-Gordon equation. In this work the nonlinear Klein-Gordon equation has been converted to a nonlinear system of algebraic equations using collocation method based on Bessel functions without any linearization, discretization and getting help of any other methods. Finally, by using JFNGMRes, solution of the nonlinear algebraic system will be achieved. To illustrate the reliability and efficiency of the proposed method, we solve some examples of the Klein-Gordon equation and compare our results with other methods.

Numerical Integration over Pyramids

Pyramidal elements are often used to connect tetrahedral and hexahedral elements in the finite element method.In this paper we derive three new higher order numerical cubature formulae for pyramidal elements.