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

本研究将围绕三个密切相关的概念展开，即：同伦方程的方程和路径的解，具有无穷小和有序域的方程的参数或短路径的解，线性问题的形式化和线性算法。第一个概念的本质是遵循解的路径，其中有一个维度的自由，而可能，使用近似，细分，和/或牛顿的方法。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，可以通过将问题的参数化版本提升到包含无穷小的有序域以得到单个问题，在新的有序域中运行原算法，并将解降低到原域以获得参数化问题的短路径解。第三个概念的本质是精确地识别什么是线性问题，以及用什么类型的算法来解决它们。这项研究将沿着与方程组解有关的概念进行；相关方程组广泛应用于工程、科学和管理领域。

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.