NEWTON: An Interactive Environment for Exploring Mathematics

We have developed a unique computational environment for use in teaching introductory mathematics. Our system, calledNewton, runs on Macintosh computers and consists of a user-friendly interface to the symbolic mathematics package Maple, supplemented by an extensive library of our own Maple code. Formulas are easily constructed and modified, appearing like those in textbooks; multiple windows allow users to see and work with several formulas at once. Formulas, graphs, and text can be intermixed in collapsible sections on worksheets. Users do not interact with Maple directly and need know nothing of Maple's syntax and command structure. Mathematical operations are selected from menus, with the added bonus that this makes it possible to document solutions. Originally designed for use in conjunction with the two semester introductory calculus sequence, the system has been extended for multivariate and vector calculus, linear algebra, numerical analysis, and differential equations.     

POINCARÉ CODE: A package of open-source implements for normalization and computer algebra reduction near equilibria of coupled ordinary differential equations

The Poincaré code is a Maple project package that aims to gather significant computer algebra normal form (and subsequent reduction) methods for handling nonlinear ordinary differential equations. As a first version, a set of fourteen easy-to-use Maple  commands is introduced for symbolic creation of (improved variants of Poincaré’s) normal forms as well as their associated normalizing transformations. The software is the implementation by the authors of carefully studied and followed up selected normal form procedures from the literature, including some authors’ contributions to the subject. As can be seen, joint-normal-form programs involving Lie-point symmetries are of special interest and are published in CPC Program Library for the first time, Hamiltonian variants being also very useful as they lead to encouraging results when applied, for example, to models from computational physics like Hénon–Heiles.     

The diatomic anharmonic oscillator according to matrix mechanics

We determine the energies of states of the diatomic anharmonic oscillator by matrix mechanics in its original form as developed by Heisenberg, Born and Jordan using perturbation theory in successive orders. We present exact formulae for the second-, fourth-, and sixth-order contributions to the energy that were computed with Maple. The calculations involve matrices of finite rank with symbolic entries. We include the Maple programs.     

The Complete Stoichiometer

The Complete Stoichiometer, a computer program used in solving stoichiometric problems in chemical, biochemical or metabolic systems, is described. The program contains general facilities for handling the stoichiometric and real matrices used in chemical network problems. The program contains a number of utility and application subprograms. The utility programs provide for input of matrices in both standard form and symbolic equation form, editing and combination of matrices; storing and retrieving stoichiometric matrices from the disk; printing of reports; a BlackBoard subprogram which allows stoichiometric matrices to be manipulated on the screen with a command-driven language. The application programs are: standard processing of network stoichiometry to obtain reaction trees and reaction loops, the independent driving potentials, and the conserved moieties; the standard processing of an atomic composition matrix to obtain the set of independent reactions interconnecting the reactants, and a possible moiety structure; factoring of an atomic composition matrix to obtain the structure of reactants in terms of moieties and of moieties in terms of atoms; generation of reduced equivalent networks when some reactions are in rapid equilibrium and some reactants are in steady-state; a robust mass action chemical equilibrium solver.     

SPECTRA – a Maple library for solving linear matrix inequalities in exact arithmetic

This document describes our freely distributed Maple library spectra, for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities with symbolic computation in exact arithmetic and it is targeted to small-size, possibly degenerate problems for which symbolic infeasibility or feasibility certificates are required.     

The CHEASE code for toroidal MHD equilibria

The CHEASE code (Cubic Hermite Element Axisymmetric Static Equilibrium) solves the Grad-Shafranov equation for toroidal MHD equilibria using a Hermite bicubic finite element discretization with pressure, current profiles and plasma boundaries specified by analytical forms or sets of experimental data points. Moreover, CHEASE allows the automatic generation of pressure profiles marginally stable to ballooning modes or with a prescribed fraction of bootstrap current. The code provides equilibrium quantities for several stability and global wave propagation codes.     

MOPS: Multivariate orthogonal polynomials (symbolically)

In this paper we present a Maple library (MOPS) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of random matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities.     

Object-Oriented Symbolic Derivation and Automatic Programming of Finite Elements in Mechanics

Symbolic approaches to assist in the development of finite element formulations have been used since the late 1970s. Today, symbolic mathematical software such as Mathematica, Maple, etc., has proved to be helpful when testing formulations. In earlier work, the authors introduced a new way of integrating naturally symbolic concepts in numerical finite element codes, taking advantage of an objectoriented code organization. In this paper, we wish to prove on practical examples that the proposed approach is very attractive and promising today, leading to an alternative way of conceiving finite element codes. After presenting a state-of- the-art of symbolic approaches for finite element developments, we first give a practical application of symbolic developments (for discontinuous space-time formulations), and then examples of Computer Aided Software Engineering tools that can be introduced into such a finite element environment.     

一种新型计算机代数系统编译器的设计与实现

计算机代数系统（Computer Algebra System,CAS）是集大整数运算和符号运算于一体的计算平台,这方面的代表软件有Maple和Mathematica。然而,在计算机代数系统的设计中,编译器的设计与实现始终是一个关键和难点。GiNaC是Linux平台上开放源码的符号计算包,该文以它为基础提出了一种新型CAS编译器的设计方法,并在Linux平台上实现。新的CAS编译器兼容Maple编程语言, 对比测试结果显示它的效率并不逊色于Maple。     

A reaction compiler for electrochemical kinetics

A computer code serving for an automatic translation of user-written source texts of electrochemical reaction mechanisms into corresponding target texts of mathematical equations that govern the kinetics of electrochemical systems under transient conditions is reported. The rules of the language enabling symbolic specification of the reaction mechanisms, the compiler options, and conventions regarding the target formulae are outlined and illustrated by examples. A considerable diversity of reaction mechanisms involving equilibrium, non-equilibrium reversible or irreversible reactions that can be electrochemical, heterogeneous non-electrochemical or homogeneous, is permitted. The reactions may involve bulk species (distributed in the electrolyte volume) and interfacial species (localized at the electrodes) of variable or constant concentrations, and electrons. The transient conditions may correspond to a number of electrochemical techniques, including potential-step method, linear potential scan voltammetry and chronopotentiometry. For kinetic problems in one-dimensional space geometry the generated governing equations take the general form of the reaction-advection-diffusion partial differential equations for the concentrations of bulk species (with initial and boundary conditions), optionally coupled with algebraic, ordinary differential or differential-algebraic equations for the concentrations of interfacial species. The governing equations can be obtained in the form of ELSIM problem definitions, enabling further solution by means of this simulation program.     

Prediction of multi-phase equilibria in alloy systems by the technique of successive partial equilibration

The technique of successive partial equilibria was recently employed to predict the precipitation of carbides in equilibrium with a solution phase. This involved predicting the equilibrium between a phase (the solution) whose composition could change with the overall composition of the system, and a phase which had a fixed stoichiometry with respect to the ratio of the number of atoms in the cationic sub-lattice to that in the anionic sub-lattice. This is a special case of predicting phase equilibria using this technique. In the present communication, this technique is extended to predict equilibria involving two or more phases whose compositions can vary over a range, such as that between two solutions phases. This technique is illustrated with respect to the Ag-Cu system and some salient features of the algorithm are discussed.     

Evolving Mazes from Images

We propose a novel reaction diffusion (RD) simulator to evolve image-resembling mazes. The evolved mazes faithfully preserve the salient interior structures in the source images. Since it is difficult to control the generation of desired patterns with traditional reaction diffusion, we develop our RD simulator on a different computational platform, cellular neural networks. Based on the proposed simulator, we can generate the mazes that exhibit both regular and organic appearance, with uniform and/or spatially varying passage spacing. Our simulator also provides high controllability of maze appearance. Users can directly and intuitively “paint” to modify the appearance of mazes in a spatially varying manner via a set of brushes. In addition, the evolutionary nature of our method naturally generates maze without any obvious seam even though the input image is a composite of multiple sources. The final maze is obtained by determining a solution path that follows the user-specified guiding curve. We validate our method by evolving several interesting mazes from different source images.     

A reconstruction and extension of Maple’s assume facility via constraint contextual rewriting

Maple’s symbolic evaluator, together with a feature that is usually known as the assume facility, implements a powerful form of conditional rewriting. In a previous paper the authors showed that Maple’s evaluation process can be recast as constraint contextual rewriting (CCR), a form of conditional rewriting that incorporates the services provided by a decision procedure through a well-specified interface. In the present paper, this analysis is extended to a component of the assume facility that deals with problems beyond linear arithmetic and that we call the general solver. This led to the discovery of a fault that causes Maple to return wrong results with some contexts. The reason for this is that the facility wrongly assumes that the general solver is complete in the sense that it uses all the available assumptions in the context. While a simple fix to this problem would reduce the logical strength of the assume facility, we show that a more general approach inspired by techniques available in CCR does not suffer from the problem and naturally leads to stronger forms of simplification.     

Applying static analysis for automated extraction of database interactions in web applications

Database interactions are among the most essential functional features in web applications. Therefore, for the testing and maintenance of a web application, it is important that the web engineer could identify all the database interactions in the web application. However, the highly dynamic nature of web applications makes it challenging to extract all the possible database interactions from source code.In this paper, we propose an automated approach to extract database interactions from source code by using symbolic execution and inference rules. Our approach automatically identifies all the possible database interaction points. After that, all the program paths, which pass through each interaction point, are also computed. Each of these paths is then symbolically executed following our proposed symbolic evaluation rules. We also develop inference rules to deduce the interaction types from the set of symbolic expressions derived during the symbolic execution. Experiments have been conducted to evaluate the performance and usefulness of the proposed approach. The results indicate that even with some limitations in handling function calls, pointers and polymorphism, our approach still gives an average precision of 79.2%, which is 45.4% more than that of the conservative approach.     

Maple在椭圆曲线密码体制中的应用

Maple是功能强大的符号处理和数值分析工具,作为强大的交互式计算软件,Maple提供了强大的编程接口和工具包来帮助完成复杂的编程工作。利用Maple编程求出椭圆曲线上有理点,用Maple实现椭圆曲线上两点的加法、点的数乘运算及求某个基点阶数的算法,利用Maple实现椭圆曲线密码体制的加密及解密。相比C语言,Maple语言更接近于平时说话的语法。同时,Maple语言可以方便地转化成C语言。效率分析表明,对于数学公式比较多的程序,用Maple要比C语言简洁很多,这为编程带来了方便。     

PDEBellII: A Maple package for finding bilinear forms,bilinear Bäcklund transformations,Lax pairs and conservation laws of the KdV-type equations

Based on the Bell polynomials scheme, this paper presents a Maple computer algebra program PDEBellII which can automatically construct the bilinear forms, bilinear Bäcklund transformations, Lax pairs and conservation laws of the KdV-type soliton equations. Some examples are given to verify the validity of our program.     

An integrated performance analysis tool for SPMD data-parallel programs

A new generation of data parallel languages have been proposed whereby a user specifies how data structures are to be distributed amongst the processor nodes of a distributed-memory machine. Based on this information, the compiler then generates code for the parallel application. Although this approach significantly simplifies the development of the initial version of a parallel application, selection of good data distributions leading to efficient computations is often quite difficult. Therefore, performance debuggers are needed to yield insights into the data distribution effects. On the other hand, most of the existing approaches to performance debugging are very general and thus do not provide the user feedback in terms of the high level programming model or the source code of the parallel application. In this paper, we describe a novel approach to performance debugging of data parallel programs. The design and implementation of a visual performance debugger is described that is specifically targeted to meet the performance debugging requirements of a data-parallel programming model based on user-specified data distributions. The performance debugger is part of an integrated programming environment, called EPPP, which also supports a data parallel compiler and a parallel architecture simulator. Thus development and performance debugging of an application may be done either on the real hardware or by using the simulator. The code for EPPP can be obtained free of cost (contact web address, http://www.crim.ca/apar).     

An overview on phase equilibria and thermodynamic modeling in multicomponent Al alloys: Focusing on the Al-Cu-Fe-Mg-Mn-Ni-Si-Zn system

Knowledge of thermodynamics and phase diagram is a prerequisite for understanding many scientific and technological disciplines. To establish a reliable thermodynamic database, an integrated approach of key experiments and thermodynamic modeling, supplemented with first-principles calculations, can be utilized. In this paper, first investigations of phase diagram and thermodynamics of technologically important Al alloys (focusing on the Al-Cu-Fe-Mg-Mn-Ni-Si-Zn system, which covers the major elements in most commercial Al alloys) is reviewed with an emphasis on the need of the integrated approach. Second, the major experimental methods (X-ray diffraction, metallography, electron probe microanalysis, differential thermal analysis, diffusion couple method, and calorimetry), which are widely employed to provide phase diagram and thermodynamic data, are briefly described. Third, the basics of the first-principles calculations and CALPHAD are presented focusing on the integration of these two computational approaches. Case study for the representative Al-Fe-Ni ternary system is then demonstrated, followed by a thermodynamic modeling of the quinary Al-Fe-Mg-Mn-Si system and a brief summary to our recent activities on investigations of phase equilibria in multicomponent Al alloys.     

基于隐实体建模模式的分布式约束数据库系统

在传统对象关系数据库中采用扩充基于隐实体建模模式的空间逻辑运算和空间分析运算功能,建立了用于虚拟现实应用的分布式约束数据库系统。隐实体建模模式采用符号表达式方法表达空间物体,提高了数据的表现力,消除了数据冗余,减少了记录的数量,处理空间数据的实时性显著增强。采用与关系数据一致的空间数据表达形式可将空间数据与关系数据一同存储,保证了数据的完整性。构造树方式管理隐实体数据使数据库中物体间的空间逻辑关系的表达更清晰,空间分析和运算更简洁。