Mathematical Sciences: Computation Using Homotopies and Ordered Fields

本研究探讨了两种密切相关的方法， 解决方程，同伦的使用和使用 无穷小，在有序域和分段 线性 同伦方法涉及使用单个参数来 将方程组变形为辅助方程组， 解的系统，然后将解追溯到 原始系统。 应用包括经济学中的问题 与工程学 具体而言，一个不完全的经济模型 市场将进行分析。 由此产生的方程组具有 一个格拉斯曼流形作为域--一个特征， 新的障碍和机遇交织在一起。 无穷小方法涉及的解决方案， 参数化系统族 通过将参数替换为 一个无穷小的变量和解决系统相对于一个 包含无穷小的更大的域，一族解 以有效的方式获得。 应用包括线性 不等式组，线性规划问题，有界变量，线性规划 整数规划、线性互补问题和二次规划 程序.

This research explores two closely related methods for solving equations, the use of homotopies and the use of infinitesimals, in the context of ordered fields and piecewise linearity. The homotopy approach involves using a single parameter to deform a system of equations into an auxiliary and more easily solved system, then tracing the solution back to the solution of the original system. Applications include problems in economics and engineering. Specifically, an economic model with incomplete markets will be analyzed. The resulting system of equations has a Grassmanian manifold as the domain -- a feature which interjects new obstacles and opportunities. The infinitesimal approach involves solutions of a parametrized family of systems. By replacing the parameter with an infinitesimal variable and solving the system relative to a larger field containing the infinitesimal, a family of solutions are obtained in an efficient way. Applications include linear inequality systems, linear programming problems, bounded variable integer programs, linear complementary problems, and quadratic programs.