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

这项研究将沿着三个密切相关的概念进行，即具有同伦的方程和方程的路径的求解，具有无穷小和有序域的方程的参数路径或短路的求解，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个自由维度的解的路径，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进行算术运算，并将算法、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，可以通过将其提升到包含无穷小的有序域来求解问题的参数版本以获得单个问题，原始算法在新的有序域中运行，并且将解降低到原始域以获得参数问题的解的最短路径。第三个概念的本质是准确地识别什么是线性问题以及什么类型的算法可以解决它们。这项研究将沿着与求解方程组有关的概念进行；相关的方程组在工程、科学和管理中都有广泛的应用。

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.