Computer Algebra: Algorithms and Applications

计算机代数系统是允许科学家、工程师、数学家和任何其他使用数学的学科轻松高效地进行数学计算的大型软件系统。这些系统可以做的远不止数值计算。他们已经用大量的数学知识进行了编程，他们可以处理符号量(高中生的x，y，z)和数字。该系统的数学知识使他们能够执行高级任务，如求解方程或简化复杂的表达式。它们可以处理人类数学家需要一页多的时间才能手写下来的冗长的表达式，并且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地会引入的错误。因此，它们允许科学家和工程师利用他们原本无法获得的专业数学知识。与其研究人员在图书馆或互联网上寻找如何解决一些困难的方程，然后不得不花费时间将所发现的知识应用于研究人员的特定问题，代数系统用户可以将问题提交给系统，并受益于几代开发数学家和计算机科学家的专业知识。科学研究可以极大地提高数学水平和生产力。 加拿大就是一个这样的计算机系统。它被恰当地命名为Maple。这项提案支持的研究结果将公开发表，但结果将与改良Maple相关。这个项目将通过增加现有计算机代数系统的数学知识库来增加它们的能力。将对系统已知的数学函数进行调查，并发现有助于系统解决用户问题的新特性。对系统及其用户特别重要的是与复数相关的性质。复数是用负一的平方根构成的数。处理复数函数，简化由此产生的数学表达式，是一个长期存在的困难，至今仍未得到充分解决。该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.