Search

We introduce a topic in the intersection of symbolic mathematics and computation, concerning topics in multivariable Optimization and Dynamic Analysis. Our computational approach gives emphasis to mathematical methodology and aims at both symbolic and numerical results as implemented by a powerful digital mathematical tool, CAS software Xcas. This work could be used as guidance to develop course contents in advanced calculus curricula, to conduct individual or collaborative projects for programming related objectives, as Xcas is freely available to users and institutions. Furthermore, it could assist educators to reproduce calculus methodologies by generating automatically, in one entry, abstract calculus formulations.

This paper presents results concerning the nonlinear analysis of the mean annual value temperature time series corresponding to the Earth’s global climate for the time period of 713 – 2004. The nonlinear analysis consists of the application of several filtering methods, the estimation of geometrical and dynamical characteristics in the reconstructed phase space, techniques of discrimination between nonlinear low dimensional and linear high dimensional (stochastic) dynamics and tests for serial dependence and nonlinear structure. All study results converge to the conclusion of nonlinear stochastic and complex nature of the global earth climate.

Modern microeconomics and macroeconomics study dynamic phenomena. Dynamics could predict future states of an economy based on its structural characteristics. Economic dynamics are modeled in discrete and continuous time context, mainly via autonomous difference and differential equations. In this study, we use Xcas and Mathematica as software tools, in order to generate results concerning the dynamic properties of the solutions of the difference and differential equation(s) models and determine whether an economic equilibrium exists. Our computational approach does not require solving the difference or differential equation(s) and makes no assumptions for initial conditions. The results provide quantitative information based on the qualitative properties of the mathematical solutions. The computer codes are fully presented and can be reproduced as they are in computational-based research practice and education. The relevant output of CAS software is created in a way as to be interpreted without the knowledge of advanced mathematics.

Computations with large matrices work out faster with computer software, even faster creating automatically the matrix of the size and pattern needed. In this paper we propose free computer algebra system Xcas resources to display particular matrices that can be called up directly. Our computer codes provide shortcuts for entering random block diagonal matrices, random triangular matrices, random and specialized band matrices, elementary matrices Eij, Fourier matrices. As for matrices needed in the study of mathematical issues concerning the properties of the roots of a polynomial, we create features with polynomial coefficients. We also propose codes for immediate construction of functional matrices such as Jacobian, bordered Hessian and Wronskian. The computer codes proposed provide visual representation of the matrix pattern (which is traditionally explained using indices and numerals), infinite number of examples using random numbers and immediate construction of large matrices of various forms.