Parametric Computation in Axiom Towards Indefinite Symbolic Computing

Axiom 中的参数计算走向不定符号计算

• 批准号: 0430722
• 负责人: Gilbert Baumslag
• 金额: $ 17万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0430722&HistoricalAwards=false
• 关键词: Parametric; Computation; Axiom; Towards; Indefinite

Current research in computation focuses on how to program a computer to perform algebraic calculations that involve concrete mathematical objects. In a concrete situation, explicit numerical data is available to effect and control the calculations at run time. No such information is available when the objects are unspecified (indefinite), and yet mathematicians routinely carry out manual calculations with them as preliminary steps to obtain insightful theorems. For the most part, computations involving indefinites have not been studied in depth. This research explores both the scope and the methods by which algebraic manipulations of indefinite objects can be automated and lays the foundation for an entirely new level of abstraction in symbolic computation.Computer implementation of indefinite computation considerably expands the scope of computer use in algebra and is an intellectual challenge of the highest order. Every tiny inroad into understanding and developing methods to do such computations will have very wide applications. The planned research consists of several phases: (1) Investigation and analysis of examples. (2) Experimental prototypes in specific cases. (3) Defining a practical and categorical framework for indefinite computation. (4) Eventually full implementation in the open source system Axiom. (5) Applications to open problems. The investigators will apply state of the art algorithms in algebra and in computer science to define the framework. Cutting edge methods for solving parametric equations, symbolic summation, recurrence equations, Grobner basis, cylindrical algebraic decomposition, dynamical evaluation, and lazy evaluation will be integrated as necessary. Specific open problems of great importance in group theory and differential algebra will be studied in order to better understand some of the basic difficulties. Students will participate through courses, seminars, and hands-on implementation.

当前计算领域的研究集中在如何对计算机进行编程以执行涉及具体数学对象的代数计算。在具体情况下，显式数值数据可用于在运行时影响和控制计算。当对象是未指定的（不确定的）时，没有这样的信息可用，然而数学家们通常用它们进行手工计算，作为获得有见地的定理的初步步骤。在大多数情况下，涉及不定式的计算还没有被深入研究。本研究探讨的范围和方法，代数操作的不确定的对象可以自动化，奠定了基础，一个全新的抽象层次的符号computation.Computer实现不确定的计算大大扩展了计算机使用代数的范围，是一个智力挑战的最高秩序。在理解和开发这种计算方法方面的每一个微小进展都将有非常广泛的应用。本研究分为以下几个阶段：（1）实例调查与分析。(2)在特定情况下的实验原型。(3)为不确定计算定义一个实用的分类框架。(4)最终在开源系统Axiom中全面实现。(5)应用于开放的问题。研究人员将应用代数和计算机科学中最先进的算法来定义框架。用于求解参数方程、符号求和、递归方程、Grobner基、圆柱代数分解、动态评估和惰性评估的前沿方法将在必要时集成。具体开放的问题，非常重要的群论和微分代数将研究，以更好地了解一些基本的困难。学生将通过课程，研讨会和动手实施参与。