Mathematical Sciences: Computation Using Homotopies and Ordered Fields

在有序域和分段线性的背景下，这项研究探索了两种密切相关的求解方程的方法，同伦的使用和无穷小的使用。同伦方法涉及到使用单个参数将方程组变形为更容易求解的辅助系统，然后将解追溯到原始系统的解。应用包括经济学和工程学方面的问题。具体地说，将分析具有不完全市场的经济模型。由此产生的方程系统有一个格拉斯曼流形作为域--这一特征插入了新的障碍和机会。无穷小方法涉及到一类系统的参数化解。通过将参数替换为无穷小变量，并相对于包含无穷小变量的较大域求解系统，以一种有效的方式获得了一族解。应用包括线性不等式系统、线性规划问题、有界变量整数规划、线性互补问题和二次规划。

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.