Maple Computer Algebra System

To gain more insight into the process of teaching and learning mathematics, we experimented during the Fall of 2003 and the spring of 2004 with some calculus chat bots. These bots know an encyclope- dic glossary of mathematics definitions as well as general knowledge background and can solve simple mathematical template problems posed in colloquial language. The bot "Sofia" can also interact with computer algebra systems like Pari or Mathematica to solve simple pre-wired calculus problems.

Abstract: A group G is said to be (2, 3, t)-generated if it can be generated by two elements x and y such that o (x)= 2, o (y)= 3 and o (xy)= t. In this paper, we determine (2, 3, t)-generations of the Tits simple group T∼= 2F4 (2)′ where t is divisor of| T|. Most of the ...

We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + αx + βy + γ for q ∈ K * and α, β, γ ∈ K. We prove, that among such algebras there are precisely five isomorphism classes. The representatives of these classes, which are ubiquitous operator algebras, are called model algebras. We derive explicit multiplication formulas for y m · x n in terms of standard monomials x i y j for many algebras of the considered type. Such formulas are used in e. g. establishing formulas of binomial type and in an implementation of non-commutative multiplication in a computer algebra system. By using the formulas we also study centers and ring-theoretic properties of the non-commutative model algebras.

This paper describes the recent development of hand-held algebraic calculators and evaluates their significance for secondary education. Sophisticated computer algebra systems (CAS) have been available to mathematicians for some years now but have been too powerful, too sophisticated and too expensive and have required too much access to powerful computers to have had much impact on teaching and learning elementary

In this paper, the solution for a differential-algebraic equation (DAE) can be expanded up to arbitrary order using MAPLE computer algebra systems (Frank, MAPLE V, CRC Press, Boca Raton, FL, 1996). First we calculate Power series of the given equations system then transform it into Pad e e series form, which gives an arbitrary order for solving DAE numerically. .tr (M. Bayram). 0096-3003/02/$ -see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 -3 0 0 3 ( 0 2 ) 0 0 0 9 3 -0 Applied Mathematics and Computation 137 www.elsevier.com/locate/amc

The use of Computer Algebra Systems (CAS) in years 9 and 10 classrooms as a tool to support learning or in preparation for senior secondary mathematics is controversial. This paper presents an analysis of the positive and negative aspects of using CAS identified in the literature related to these year levels, along with the perceptions of 12 experienced secondary teachers who were working with years 9 and 10 students. The literature review shows that CAS is valued for calculation and manipulation capabilities, the option of alternative representations, the opportunity for systematic exploration and for prompting rich discussion. However, the technical overhead, initial workload for the teacher and unresolved questions about the perceived relative contribution of machine and by-hand work to learning currently pose obstacles to teaching with CAS in the middle secondary years. The teachers who contributed data to this study perceived that using CAS in their teaching is, on balance, worth the effort. However, they believed that CAS is of most benefit to their high ability students and may present an obstacle to their low ability students' learning of mathematics.

Synopsis: Although there are a number of technologies related to school algebra, it is only the personal technology of the graphics calculator that seems likely to be available widely enough to influence curriculum design and implementation on a large scale. The algebra curriculum of the past is overburdened with symbolic manipulation at the expense of understanding for most students. But

In this paper, employing the general Doolittle factorization, an efficient algorithm is developed to find the inverse of a general pentadiagonal matrix which is suitable for implementation using computer algebra systems software such as Matlab and Maple. Examples are given to illustrate the efficiency of the algorithm.

This project is an attempt to understand general relativity and its relation with quantum field theory by assimilating the singularities of Klein-Gordon equation to Heun functions. In this report, derivations of Einstein equations, Klein-Gordon equation and Heun equation are explained as background information. After that, two softwares, COMSOL and Maple is introduced, however any application in COMSOL is abandoned since it was not suitable for this project. Maple and its general relativity package is used for defining the differential equations that we are dealing with, solving them and working on their singularities.

Contemporary computer tools can generate a tremendous amount of numerical data so the user might easily lose insight into the phenomenon being investigated. Those who use powerful computer algebra systems must thoroughly understand the assumptions that underlie the software. In this paper, the role and importance of symbolic computation in control engineering and signal processing is exemplified. Real-life application examples are presented in which systems are symbolically solved and simulated with Mathematica. We introduce an original approach to algorithm development, system design and symbolic processing that successfully overcomes some problems encountered in the traditional approach. Benefits of symbolic methods and the role of computer algebra systems are highlighted from the viewpoint of both academia and industry.

A universal template for a 4-node quadrilateral in plane stress is constructed using a combination of old and new techniques. The use of natural quantities, strongly advocated by J. H. Argyris since his Continua and Discontinua 1965 exposition, is further expanded. The qualifier 'supernatural' means that all governing equations: kinematic, constitutive and equilibrium, are expressed in both Cartesian and natural forms. These two sets are used to build different components of the template. With timely help from a computer algebra system, a template that include all possible quadrilateral elements that pass the Individual Element Test of Bergan and Hanssen emerges. It yields an infinite number of hitherto undiscovered instances that may be customized to fit special needs. A striking example is the construction of a two-trapezoid macroelement that is bending exact about one direction, for any amount of distortion. This concludes a five decade search that begins with the formulation of the wing-cover rectangular panel in Argyris' 1954 seminal serial on Energy Methods and Structural Analysis.

Contemporary computer tools can generate a tremendous amount of numerical data so the user might easily loose the insight into the phenomenon he is looking at. Those who are using powerful computer algebra systems have thoroughly to understand what assumptions underlie the software that they are using. In this paper, we present the high-speed filter design in which we symbolically solve and simulate filters with Mathematica. We expose original approach to filter design and symbolic processing.

I have added support for predicate dispatching, a powerful generalization of other dispatching mechanisms, to the Common Lisp Object System (CLOS). To demonstrate its utility, I used predicate dispatching to enhance Weyl, a computer algebra system which doubles as a CLOS library. My result is Dispatching-Enhanced Weyl (DEW), a computer algebra system that I have demonstrated to be well suited for both users and programmers.

In this paper we demonstrate the use of formal methods tools to provide a semantics for the type hierarchy of the AXIOM computer algebra system, and a methodology for Aldor program analysis and veri cation. We give examples of abstract speci cations of AXIOM primitives, and provide an interface between these abstractions and Aldor code.

In the last years we have witnessed an impressive advance in the efficiency of boolean solving techniques, which has brought large previously intractable problems at the reach of state-of-the-art solvers. Unfortunately, simple boolean expressions are not expressive enough for representing many real-world problems, which require handling also integer or real values and operators. On the other hand, mathematical solvers, like computer-algebra systems or constraint solvers, cannot handle efficiently problems involving heavy boolean search, or do not handle them at all. In this paper we present the foundations and the basic algorithms for a new class of procedures for solving boolean combinations of mathematical propositions, which combine boolean and mathematical solvers, and we highlight the main requirements that boolean and mathematical solvers must fulfill in order to achieve the maximum benefits from their integration. Finally we show how existing systems are captured by our framework.

A fundamental task in geodesy is solving systems 1 of equations. Many geodetic problems are represented as sys-2 tems of multivariate polynomials. A common problem in 3 solving such systems is improper initial starting values for 4 iterative methods, leading to convergence to solutions with 5 no physical meaning, or to convergence that requires global 6 methods. Though symbolic methods such as Groebner bases 7 or resultants have been shown to be very efficient, i.e., provid-8 ing solutions for determined systems such as 3-point prob-9 lem of 3D affine transformation, the symbolic algebra can be 10 very time consuming, even with special Computer Algebra 11 Systems (CAS). This study proposes the Linear Homotopy 12 method that can be implemented easily in high-level com-13 puter languages like C++ and Fortran that are faster than CAS 14 B. Paláncz (B) by at least two orders of magnitude. Using Mathematica, the 15 power of Homotopy is demonstrated in solving three nonlin-16 ear geodetic problems: resection, GPS positioning, and affine 17 transformation. The method enlarging the domain of conver-18 gence is found to be efficient, less sensitive to rounding of 19 numbers, and has lower complexity compared to other local 20 methods like Newton-Raphson. 21 Keywords Homotopy · Nonlinear systems of equations · 22 GPS positioning · Resection · Affine transformation 23 Journal: 190 MS: 0346 TYPESET DISK LE CP Disp.:2009/9/15 Pages: 17 Layout: Large Author Proof u n c o r r e c t e d p r o o f B. Paláncz et al. (Bates et al. 2008), HOM4PS (Lee et al. 2008), HOMPACK 47 (Watson et al. 1997), PHCpack (Verschelde 1999), and PHoM 48 (Gunji et al. 2004). Other codes are written in Maple (see, 49 e.g., Leykin and Verschelde 2004) and Matlab (Decarolis 50 et al. 2002). Recently, codes for fixed point and Newton 51 homotopy were developed in Mathematica, too, see Binous 52 (2007a,b). 53 In geodesy, several algebraic procedures have been put 54 forward for solving nonlinear systems of equations (see, e.g., 55 Awange and Grafarend 2005; Paláncz et al. 2008a). The pro-56 cedures suggested in these studies, such as Groebner bases 57 and resultant approaches, are, however, normally restricted 58 by the size of the nonlinear systems of equations involved. 59 In most cases the symbolic computations are time consum-60 ing. These symbolic computations were necessitated by the 61 failure to obtain suitable starting values for numerical itera-62 tive procedures. In situations where large systems of equa-63 tions are to be solved (e.g., affine transformation), and where 64 symbolic methods are insufficient, there exists the need to 65 investigate the suitability of other alternatives. One such alter-66 native is the linear homotopy. 67 In this study linear homotopy continuation methods are 68 introduced (see Kotsireas 2001). The definitions and basic 69 ideas are considered and illustrated. The general algorithms 70 of linear homotopy method are considered: iterated solution 71 of homotopy equations via Newton-Raphson method; and as 72 an initial value problem of a system of ordinary differential 73 equations. The efficiency of these methods are illustrated here 74 by solving polynomial equation systems in geodesy, namely 75 solution of 3D resection, GPS navigation, and 3D affine trans-76 formation problems. Our computations were carried out with 77 Mathematica and Fermat computer algebra systems on HP 78 workstation xw 4100 with XP operation system, 3 GHz P4 79 Intel processor and 1 GB RAM. The details can be found 80 in MathSource (Wolfram Research Inc.) in Paláncz (2008). 81 Although the mathematical background of the algorithms is 82 discussed in this paper, it can be also found in the literature, 83 e.g., Sommese et al. (2005). The application of the imple-84 mented Mathematica functions is illustrated in the Appendix. 85 The study is organized as follows: In Sect. 2, we pres-86 ent the definition and basic concepts of homotopy required 87 to understand the solution of nonlinear equations presented 88 in Sect. 3. For simplicity purposes, a quadratic and a third-89 degree polynomial equation are used to illustrate the 90 approach. Once the example has been made clear, Sect. 4 91 demonstrates how the technique is applied to solve three non-92 linear geodetic problems. The results are discussed in Sect. 5 93 and concluded in Appendix.

El cálculo de equipos de bombeo es una tarea que generalmente se realiza a través de las curvas características que obtiene cada fabricante para sus equipos; por tanto, dicho cálculo es laborioso y susceptible a errores. En el presente trabajo, se realiza una propuesta del cálculo automatizado de bombas sumergibles, basado en sistemas algebraicos computacionales (SAC), de tal manera que el tiempo de cálculo y especialmente el margen de error, se reducen considerablemente. Adicionalmente, esta forma de realizar el cálculo mencionado, permite que los estudiantes de ingeniería integren los conocimientos que han adquirido en los semestres iniciales de su formación profesional.

We present a verifiable symbolic definite integral table look-up: a system which matches a query, comprising a definite integral with parameters and side conditions, against an entry in a verifiable table and uses a call to a library of facts about the reals in the theorem prover PVS to aid in the transformation of the table entry into an answer. Our system is able to obtain correct answers in cases where standard techniques implemented in computer algebra systems fail. We present the full model of such a system as well as a description of our prototype implementation showing the efficacy of such a system: for example, the prototype is able to obtain correct answers in cases where computer algebra systems [CAS] do not. We extend upon Fateman's web-based table by including parametric limits of integration and queries with side conditions.

En los últimos años los ordenadores han incrementado de forma drástica su capacidad para resolver grandes problemas procedentes de los más diversos campos de la Ciencia debido, de un lado al portentoso avance que ha sufrido el hardware (ordenadores más potentes y rápidos) y de otro al reciente desarrollo de software con un elevado nivel de sofisticación. Como parte de este software están los sistemas de Cálculo Científico que permiten llevar a cabo no sólo cálculos numéricos complicados sino manipulaciones analíticas y tratamientos gráficos de los problemas.

Son múltiples los sistemas de este tipo, mencionaremos algunos como DERIVE, REDUCE, MACSIMA, Mathematica, Maple. MuPAD o AXIOM, que están entre los de propósito general. Citamos también otros, más dirigidos al cálculo numérico, como Mathcad o Matlab que han incorporado el núcleo algebraico de Maple para manipulaciones analíticas.

Integrals and sums involving special functions are in constant demand in applied mathematics. Rather than refer to a handbook of integrals or to a computer algebra system, we present a do-it-yourself systematic approach that shows how the evaluation of such integrals and sums can be made as simple as possible. Illustrating our method, we present several examples of integrals of Poisson type, Fourier transform, as well as integrals involving product of Bessel functions. We also obtain a new identity involving the sums of 2 F 1 .

In signal processing, a pulse means a rapid change in the amplitude of a signal from a baseline value to a higher or lower value, followed by a rapid return to the baseline value. A square wave function may be viewed as a pulse that repeats its occurrence periodically but the return to the baseline value takes some time to happen. When these periodic functions act as inputs in dynamic systems, the standard tool commonly used to solve the associated initial value problem IVP is Laplace transform and its inverse. We show how a computer algebra system may also provide the solution of these IVP straight forwardly by adequately introducing the periodic input.

The main goal of this paper is to give a general method to compute (via computer algebra systems) an explicit set of generators of the ideals of the projective embeddings of some ruled surfaces, namely projective line bundles over curves such that the fibres are embedded as smooth rational curves.

In the present thesis I discuss the hard spectator interaction amplitude in $B\to\pi\pi$ at NLO i.e. at $\mathcal{O}(\alpha_s^2)$. This special part of the amplitude, whose LO starts at $\mathcal{O}(\alpha_s)$, is defined in the framework of QCD factorization. QCD factorization allows to separate the short- and the long-distance physics in leading power in an expansion in $\lqcd/m_b$, where the short-distance physics

In this paper, the solution for a chemical differential algebraic equation (DAE) can be expanded up to arbitrary order using MAPLE computer algebra systems. First we calculate power series of the given equations system then transform it into Pad e e series form, which give an arbitrary order for solving chemical DAE numerically.

Every day more companies and organizations insist not only on the quality of the processes that are followed in software development, but also on the quality of the products purchased and / or developed. The quality of software is perceived mainly by the flaws found in the product and by gravity they represent for their business. These failures directly affect the quality of non-functional requirements, which are mostly overlooked during the life cycle of software development. In this paper presents an investigation to determine the degree of importance of the non-functional requirements, especially for the case of usability. For this was used as a research method literature review in journals and digital libraries coming from Cuban institutions, using the CORE and Journal Citation Report (JCR) methods for evaluating of quality of publications, that from the definitions of chains of searches, a questionnaire of objective and subjective questions, criteria of inclusion and exclusion, obtained excellent publications for the exhaustive study.Finally the results of the study are analyzed and the conclusive criterion is emitted on the found results.

Education (CAME) has been established. Further to this (but not because of this!!) some of us have 'lost our innocence' along the way; youthful dreams of enhanced learning have been replaced by realism that all tools have their constraints as well as their enablements. CAME Symposia have been forums for these discussions. The rationale and goals for CAME Symposia are to serve as a bridge between two communities, the CAS research community and the main mathematics education community; to facilitate the dissemination and exchange of information on research and development in the use of computer algebra in mathematics education; to facilitate access to international expertise in the use of computer algebra in mathematics education; to promote the study of the use of computer algebra in mathematics education. The rationale for this discussion group is to present the products of CAME Symposia and to discuss their potential contribution to the PME community. In this paper we first describe briefly issues that were dealt with in each symposium. We then outline the specific topics for discussion. Regev. We wanted to address two issues in particular:

This article suggests a framework to organise a cluster of variables that are associated with students' effective use of computer algebra systems (CAS) in mathematics learning. Based on a review of the literature and from the authors' own teaching experience, the framework identifies the main characteristics of students' interactions with CAS technology and how these may be used to monitor students' developing use of CAS; from this, the framework may be used to plan teaching in order to gain greater benefit from the availability of CAS. Four case studies describing students' development over a semester are reported. These demonstrate a variety of combinations of technical competencies and personal attributes. They indicate the importance of both the technical and personal aspects but suggest that negative attitudes rather than technical difficulties can limit the effective use of CAS. Finally practical suggestions are given for teaching strategies which may promote effective use of CAS.

Modern computer algebra systems provide means for exact experimental calculations (including manipulations on formulas) that enable one to obtain a solution of a problem under examination for certain not very large initial data. The results can provide a basis for a conjecture concerning the general solution of the problem. A computer algebra system can also be helpful in verifying the conjecture.

Abstract. This paper describes the recent convergence of four topics: polynomial systems, flexibility of three dimensional objects, computa-tional chemistry, and computer algebra. We discuss a way to solve sys-tems of polynomial equations with resultants. Using ideas of Bricard, we find a system of polynomial equations that models a configuration of quadrilaterals that is equivalent to some three dimensional structures. These structures are of interest in computational chemistry, as they rep-resent molecules. We then describe an algorithm that examines the re-sultant and determines ways that the structure can be flexible. 1

In this paper, the method of approximating solutions of partial differential equations with variable coefficients is studied. This is done by considering heat flow through a one-dimensional model with variable cross-sections. Two cases are considered. The first one involves quadratic approximation of the variable coefficient by direct integration. This case is studied using a conic domain. The second case approximates the variable coefficient quadratically and by step functions. The solution of the problem in each case is expressed using Green's function, and the results are compared. By the suitable use of a computer algebra system (CAS) all of these ideas can easily enough be introduced at the advanced undergraduate level.

Unambiguous representation of mathematics is crucial for communications among humans or among computer systems. OpenMath is a standard aimed at supporting a semantically rich interchange of mathematics among varied computational software tools such as computer algebra systems, theorem provers, and tools for visualizing or editing mathematical text. MathML is a W3C Recommendation for the encoding of mathematics `on the web' which also includes mechanisms for encoding mathematical semantics. We introduce each of these two languages and describe their relationships.

Our research Group is developing methods of producing practical reference documentation for software products and components. Our document contents are defined by a relational model in which each document is required to be a representation of a specified relation. In effect, we are using mathematical descriptions of relations to provide specifications and descriptions of programs written in conventional languages.

Ane size-change analysis has been used for termination analysis of eager functional programming languages. The same style of analysis is also capable of compactly recording and calculating other properties of programs, including their runtime, maximum stack depth, and (relative) path time costs. In this paper we show how precise poly- nomial bounds on such costs may be calculated on programs,

Abstract. Polynomial interpretations are a standard technique used in almost all tools for proving termination of term rewrite systems (TRSs) automatically. Traditionally, one applies interpretations with polynomials over the naturals. But recently, it was shown that interpretations with polynomials over the rationals can be significantly more powerful. However, searching for such interpretations is considerably more difficult than for natural polynomials. Moreover, while there exist highly efficient SAT-based techniques for finding natural polynomials, no such techniques had been developed for rational polynomials yet. In this paper, we tackle the two main problems when applying rational polynomial interpretations in practice: (1) We develop new criteria to decide when to use rational instead of natural polynomial interpretations. (2) Afterwards, we present SAT-based methods for finding rational polynomial interpretations and evaluate them empirically. Topics. computer algebra syst...