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Derive
Published in Paul W. Ross, The Handbook of Software for Engineers and Scientists, 2018
In the last three decades Computer Algebra, the field that deals with the development of techniques, algorithms, and creation of software for symbolic manipulation of mathematical expressions, formulas, and equations, has experienced enormous success. Currently, several softwares are available for computer algebra (see de Souza, Chapter 42, in this volume). With the new generation of processors and the high memory capability of the microcomputers, the use of computer algebra systems is no longer restricted to academia, but rather has spread to all kind of professionals who make use of mathematics as a tool for their job.
Numerical Methods for Elliptic PDEs
Published in Victor G. Ganzha, Evgenii V. Vorozhtsov, Numerical Solutions for Partial Differential Equations, 2017
Victor G. Ganzha, Evgenii V. Vorozhtsov
Since the algebraic computations needed for the approximation study of scheme (4.7.21) are rather laborious (although the underlying idea of these computations is very simple), it is reasonable to use the symbolic manipulations on a computer (the computer algebra) for the implementation of this study. We mention here the computer algebra systems REDUCE, Mathematica and MAPLE among the most popular systems. Now we describe the steps of the algorithm for the local approximation study of difference schemes approximating the Euler equation system (4.7.20).
Introduction and Review
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
There are many programming languages and software packages that can be used to determine numerical solutions to algebraic equations or differential equations. For example, CAS (Computer Algebra Systems) such as Maple and Mathematica are available. Open source packages such as Maxima, which has been around for a while, Mathomatic, and the SAGE Project, do exist as alternatives. One can use built-in routines and do some programming. The main features are that they can produce symbolic solutions. Generally, they are slow in generating numerical solutions.
A classic recursive sequence calculus task at the secondary-tertiary level in France
Published in International Journal of Mathematical Education in Science and Technology, 2022
Macarena Flores González, Fabrice Vandebrouck, Laurent Vivier
In the following, we refer to the notion of paradigms (Houdement & Kuzniak, 2006). This notion allows us to identify different types of work that are internally coherent. In the domain of analysis, Montoya Delgadillo and Vivier (2016, pp. 742–743) distinguish three paradigms: Arithmetic/geometric analysis [A1]. This supports interpretations that draw upon implicit assumptions based on geometry, arithmetic calculations or the real world. Although argumentation plays a role, work is grounded on the visualization of signs, possibly produced by software or a calculator. Visualizing the curve of a function or a table of values is admissible as a proof in [A1].Calculation analysis [A2]. The rules of the calculation are defined more-or-less explicitly and are applied independently of any reflection on the existence and nature of the objects in question. Calculations are often based on an algorithmic approach, along with formal expressions that have a representative role. These routines are executed without being aware of the nature of the mathematical objects. Work is oriented toward the production of proofs and demonstrations, using the properties of the objects and theorems, in the mathematical tradition. Visualizing the curve of a function or a table of values is not admissible in [A2]. However, tools, such as computer algebra system software, may be used (to compute a derivative, for instance).
Joint replacement and production control for a queueing system with different failures
Published in Journal of Industrial and Production Engineering, 2022
We highlight three critical issues in terms of solution method, parameter setting, and contribution. (1). Solution method: When we use PGF, we have to ensure that there are sufficient real roots to form enough equations for deriving boundary probabilities. Further, we have to guarantee that the set of linear equations formed by the distinct roots are independent such that a unique solution for boundary probabilities may be obtained. [18] and [20] conjectured that the set of linear equations formed by the distinct roots are independent and a unique solution for boundary probabilities may be obtained. However, we cannot guarantee the above procedure will always work. Further, PGF relies heavily on the implementation of a computer algebra system. For example, [16] used alternative approach different from the above procedure when the authors know the above method does not produce sufficiently informative equations to calculate the boundary probabilities. Alternatively, since MGM is a method based on numerical computation it can solve problem size larger than PGF within a time frame of few minutes [16].
Integrability and linearizability of a family of three-dimensional quadratic systems
Published in Dynamical Systems, 2021
Waleed Aziz, Azad Amen, Chara Pantazi
Integrability and linearizability conditions of system (3). We will first seek necessary conditions for integrability and linearizability at the origin of system (9). So we consider and in the form (10) and we write them as power series up to degree 15. In order to find the necessary conditions, we compute the obstructions to form first integrals which are known as resonant focus quantity, for more details see [16]. Then, a factorized Gröbner basis was found using the Computer Algebra system Reduce. Finally, the minAssGTZ algorithm of the Computer Algebra system Singular [15,33] was used to check that the conditions found were irreducible. To prove the sufficiency of the conditions, we exhibit first integrals of the form (10).