Computer Algebra Systems and its Computational Application to the Thermo-Elastic Problem

Incorporating visual representations, rather than strictly relying on symbolic representations, is a research-based strategy for supporting fraction learning. However, students must also make transitions between visual and symbolic fraction representations to apply the conceptual understanding they gained from visual representations to symbolic fraction computation. Virtual manipulatives (VMs) offer opportunities for supporting students in making these transitions, as many fraction VMs integrate visual and symbolic representations into one manipulative. Some VMs also dynamically link the representations, so learners can observe how changes to one representation impact the other. For these features to support students in transitioning among representations, teachers must orchestrate opportunities for students to use and reflect on their use of the features. This study examined how six fourth- and fifth-grade teachers orchestrated opportunities in lesson plans for students to make transitions among the visual and symbolic representations within and beyond fraction VMs. Results showed that teachers used two strategies for orchestrating these transitions: VM choice and direct teacher intervention. Implications of teachers’ uses of these strategies are discussed in terms of what kinds of transition opportunities were made available to students and what professional learning experiences could be needed to support teachers in orchestrating transitions.

Maple is a computer algebra system and programming language that can be used for numerical computations, solving equations, manipulating symbolic expressions, plotting, and programming, just to name a few of the basics. APPL is, simply, a set of supplementary Maple commands and procedures that augments the existing computer algebra system. In effect, APPL takes the capabilities of Maple and turns it into a computer algebra system for computational probability. This chapter contains guidelines for using Maple, and discusses the Maple commands that are used in APPL programming. After reading this chapter, an APPL user will have the knowledge necessary to modify the APPL code to meet his or her particular needs. We will start with a discussion of basic numeric computation, then advance to defining variables, symbolic computations, functions, data types, solving equations, calculus and graphing. Then we will discuss the programming features of Maple that facilitate building the APPL language: loops, conditions, and procedures.

This article introduces and explains a computer algebra system (CAS) wxMaxima for Calculus teaching and learning at the tertiary level. The didactic reasoning behind this approach is the need to implement an element of technology into classrooms to enhance students' understanding of Calculus concepts. For many mathematics educators who have been using CAS, this material is of great interest, particularly for secondary teachers and university instructors who plan to introduce an alternative CAS into their classrooms. By highlighting both the strengths and limitations of the software, we hope that it will stimulate further debate not only among mathematics educators and software users but also also among symbolic computation and software developers.

On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity

Exploring Third-Order Chromatic Aberrations of Electron Lenses with Computer Algebra

Mathematical models of biochemical reaction networks involve nonlinear differential equations with a number of parameters whose values are generally not known a priori. Analysis of such networks includes both analytical and numerical work. The analytical part of such analysis often requires manipulation with massive algebraic expressions that may become challenging due to the inherent nonlinearities and unknown parameters in the model. Computer algebra systems (CASs), which are also capable of performing calculations with symbols and to generate ready‐to‐use computer codes for numerical computations, are excellent tools in such analysis. The present work gives a brief description of how we can benefit from computer algebra to overcome such difficulties and combine symbolic and numeric computation techniques in such analysis with several illustrative biochemical reaction networks. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Owing to its superb drawing utilities and symbolic computation capabilities, computer algebra system (CAS) has become one of the most preferred tools for drawing scientific artworks. However, according to the result of our questionnaire survey, many collegiate mathematics teachers in Japan feel difficulty in effectively using the graphics generated by CAS in printed materials. For instance, quite a number of mathematics teachers in universities are using TeX as their daily tool for editing high-quality mathematical documents, but the use of graphics generated by CAS in TeX documents seems not to be so easy task for usual TeX users. This difficulty seems to prevent many teachers from using CAS graphics. To fill this gap, some dynamic geometry softwares (DGS) and others have been equipped with the capability to export TeX-readable code of their graphical output. Also the authors have developed a macro package KETpic designed for CAS to supply high-quality graphics in TeX documents. In this paper, we will propose a scheme for demonstrating and improving the effect of CAS use in mathematics education. Through this scheme, it is shown that we can greatly improve the educational effect of our mathematics class materials by appropriately using the precise 2D-graphics generated by CAS and KETpic.

Mathematics is a ubiquitous foundation of science, technology, and engineering. Specific areas of mathematics, such as numeric and symbolic computation or logics, enjoy considerable software support. Working mathematicians have recently started to adopt Web 2.0 environments, such as blogs and wikis, but these systems lack machine support for knowledge organization and reuse, and they are disconnected from tools such as computer algebra systems or interactive proof assistants. We argue that such scenarios will benefit from Semantic Web technology.Conversely, mathematics is still underrepresented on the Web of [Linked] Data. There are mathematics-related Linked Data, for example statistical government data or scientific publication databases, but their mathematical semantics has not yet been modeled. We argue that the services for the Web of Data will benefit from a deeper representation of mathematical knowledge.Mathematical knowledge comprises structures given in a logical language --formulae, statements e.g. axioms, and theo-ries --, a mixture of rigorous natural language and symbolic notation in documents, application-specific metadata, and discussions about conceptualizations, formalizations, proofs, and counter-examples. Our review of vocabularies for representing these structures covers ontologies for mathematical problems, proofs, interlinked scientific publications, scientific discourse, as well as mathematical metadata vocabularies and domain knowledge from pure and applied mathematics.Many fields of mathematics have not yet been implemented as proper Semantic Web ontologies; however, we show that MathML and OpenMath, the standard XML-based exchange languages for mathematical knowledge, can be fully integrated with RDF representations in order to contribute existing mathematical knowledge to the Web of Data.We conclude with a roadmap for getting the mathematical Web of Data started: what datasets to publish, how to interlink them, and how to take advantage of these new connections.

Describes a prototype agent for automating the performance of symbolic computations in engineering design. The input to the system comprises a mathematical model of a design artifact and the queries or transformations desired on this model by the engineer. The system automatically does the mathematics required to achieve the input goals by intelligently using a computer algebra system. The system performance is demonstrated by an example from sensitivity analysis. >

Computer algebra-perturbation solution to a nonlinear wave equation

Surprisingly, computer algebra systems have not been used for the analysis of discrete-time nonlinear control systems. The only exceptions seem to be papers based on Reduce and on Maple. This paper presents certain modelling problems of discrete-time nonlinear system as an example of what can be done with symbolic computation in nonlinear discrete-time control with Mathematica.

Computer algebra systems are currently successively used in several areas of engineering, such as VLSI, robotics, non-linear dynamics, or control systems. Recently, there has been a growing interest in the application of the computer algebra systems to control design and analysis. Symbolic computation has been used for modeling, simulation, and synthesis of systems. This paper presents some of the key benefits and applications of symbolic computation techniques from the viewpoint of the symbolic system simulation.

Automata networks are a class of fully discrete dynamical systems, which have received considerable interest in various different areas. This brief addresses the observability of automata networks and switched automata networks in a unified framework, and proposes simple necessary and sufficient conditions for observability. The results are achieved by employing methods from symbolic computation, and are suited for implementation using computer algebra systems. Several examples are presented to demonstrate the application of the results.

Using Fourier series to analyse mass imperfections in vibratory gyroscopes

This paper examines the process through which students learn to make functional use of computer algebra systems (CAS), and the interaction between that process and the wider mathematical development of students. The result of ‘instrumentalising‘ a device to become a mathematical tool and correspondingly ‘instrumenting’ mathematical activity through use of that tool is not only to extend students' mathematical technique but to shape their sense of the mathematical entities involved. These ideas have been developed within a French programme of research – as reported by Artigue in this issue of the journal – which has explored the integration of CAS – typically in the form of symbolic calculators – into the everyday practice of mathematics classrooms. The French research –influenced by socio-psychological theorisation of the development of conceptual systems- seeks to take account of the cultural and cognitive facets of these issues, noting how mathematical norms – or their absence – shape the mental schemes which students form as they appropriate CAS as tools. Instrumenting graphic and symbolic reasoning through using CAS influences the range and form of the tasks and techniques experienced by students, and so the resources available for more explicit codification and theorisation of such reasoning. This illuminates an influential North American study– conducted by Heid – which French researchers have seen as taking a contrasting view of the part played by technical activity in developing conceptual understanding. Reconsidered from this perspective, it appears that while teaching approaches which ‘resequence skills and concepts’ indeed defer – and diminish –attention to routinised skills, the tasks introduced in their place depend on another –albeit less strongly codified – system of techniques, supporting more extensive and active theorisation. The French research high lights important challenges which arise in instrumenting classroom mathematical activity and correspondingly instrumentalising CAS. In particular, it reveals fundamental constraints on human-machine interaction which may limit the capacity of the present generation of CAS to scaffold the mathematical thinking and learning of students.