Computer Algebra: Algorithms and Applications

计算机代数系统是大型软件系统，它允许科学家、工程师、数学家和任何其他使用数学的学科轻松有效地进行数学计算。这些系统能做的远不止数值计算。它们被编入了大量的数学知识，它们可以处理符号量（高中生的x，y,z）和数字。系统的数学知识使他们能够执行高级任务，例如解方程或简化复杂的表达式。它们可以处理冗长的表达式，而人类数学家要花一页多的时间才能手写出来，它们还可以处理如此庞大的表达式，而不会出现人类数学家不可避免会出现的错误。因此，它们使科学家和工程师能够利用原本无法获得的专业数学知识。代数学系统的用户可以将问题引用到系统中，并受益于几代数学家和计算机科学家的专业知识，而不是研究者通过图书馆或互联网来学习如何解决一些困难的方程，然后不得不花时间将发现的知识应用到研究者的特定问题上。科学研究可以大大提高数学水平和生产力。其中一个这样的计算机系统是加拿大的。它被恰当地命名为枫树。本提案支持的研究结果将公开发表，但结果将与改进Maple有关。该项目将通过增加现有计算机代数系统的数学知识库来增强其能力。将研究系统已知的数学函数，并发现有助于系统解决用户问题的新属性。对系统及其用户来说，特别重要的是与复数有关的性质。复数是用- 1的平方根构成的。处理复数函数，并简化由此产生的数学表达式，是一个长期存在的难题，至今仍未得到充分解决。本项目旨在加强系统在这一领域的能力。

Computer algebra systems are large software systems that allow scientists, engineers, mathematicians, and any other discipline that uses mathematics, to conduct mathematical calculations easily and efficiently. These systems can do much more than numerical computations. They have been programmed with large amounts of mathematical knowledge, and they can work with symbolic quantities (the high school student's x,y,z) as well as numbers. The system's mathematical knowledge allows them to perform high-level tasks, such as solving equations or simplifying complicated expressions. They can handle lengthy expressions that would take a human mathematician more than a page to write down by hand, and manipulate such large expressions without the errors that a human mathematician would inevitably introduce. Thus they allow scientists and engineers to take advantage of specialist mathematical knowledge that would otherwise not be available to them. Instead of the researcher hunting through the library or internet to learn how to solve some difficult equation, and then having to spend time applying the discovered knowledge to the researcher's particular problem, the algebra system user can refer the problem to the system and benefit from the expert knowledge of generations of developer mathematicians and computer scientists. Scientific investigations can gain greatly in mathematical level and productivity.One such computer system is Canadian. It is named, appropriately, Maple. The results of research supported in this proposal will be openly published, but the results will relevant to improving Maple. This project will increase the power of existing computer algebra systems by adding to their mathematical knowledge base. The mathematical functions known to the systems will be investigated and new properties discovered that will help systems solve users' problems. Of particular importance to the systems and their users are properties connected with complex numbers. Complex numbers are those built using the square root of negative one. Working with functions of complex numbers, and simplifying the mathematical expressions which arise, is a long standing difficulty, which is still not adequately solved. This project aims to strengthen the abilities of the systems in this area.