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Mathematical Sciences: "Solution of Equations and Paths of Equations with Homotopies and Infinitesmals"

数学科学：“同伦和无穷方程的解和方程的路径”

基本信息

批准号：
9502892
负责人：
B. Curtis Eaves
金额：
$ 9万
依托单位：
Stanford University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
1995
资助国家：
美国
起止时间：
1995-07-01 至 1998-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9502892&HistoricalAwards=false
关键词：
Mathematical Sciences Solution Equations Paths

项目摘要

DMS-9502892 Eaves This research is to be directed along three closely related concepts, namely, the solution of equations and paths of equations with homotopies, the solution of parametric or short paths of equations with infinitesimals and ordered fields, and the formalisms of a linear problem and linear algorithm. The essence of the first concept is following the path of solutions where there is one dimension of freedom, while perhaps, using approximations, subdivisions, and/or Newton's method. The essence of the second concept is doing arithmetic with infinitesimals and, transferring algorithms, problems and solutions from one ordered field to another. For certain problems and algorithms a parametric version of the problem can be solved by lifting it to an ordered field containing an infinitesimal to obtain a single problem, the original algorithm is run in the new ordered field, and the solution is lowered to the original field to obtain a short path of solutions to the parametric problem. The essence of the third concept is the precise identification of what a linear problem is and what type of algorithms solve them. This research is to be directed along the concepts related to the solution of systems of equations; the pertinent systems of equations are widely applicable in engineering, science, and management.

DMS-9502892屋檐这项研究是直接沿着三个密切相关的概念，即，方程和方程的同伦路径的解决方案，参数或短路径的方程与无穷小和有序领域的解决方案，以及线性问题和线性算法的形式。第一个概念的本质是遵循有一维自由的解决方案的路径，同时可能使用近似，细分和/或牛顿方法。第二个概念的本质是用无穷小做算术，将算法、问题和解决方案从一个有序域转移到另一个有序域。对于某些问题和算法可以通过将问题提升到包含无穷小的有序域来求解该问题的参数版本以获得单个问题，在新的有序域中运行原始算法，并且将解降低到原始域以获得参数问题的解的短路径。第三个概念的本质是精确识别什么是线性问题以及什么类型的算法可以解决它们。本研究将沿着与方程组解相关的概念进行，相关的方程组在工程、科学和管理中有着广泛的应用。