Mathematical Sciences: Solutions of Equations and Paths of Equations With Homotopies and Infinitesimals

这项研究将沿着两条密切相关的途径进行，即求解方程和同伦方程的路径，以及求解具有无穷小的方程的参数路径或短路。第一条大道的本质是沿着一个点的前像走的路，也许是以细分的方式。第二种方法的本质是对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。将参数问题提升到包含无穷小的有序域以获得单个问题，在新的有序域中运行原始算法，并将解降低到原始域以获得参数问题的解的最短路径；这个序列被称为提升、求解和更低。为了说明这两条途径的相关性，我们注意到分段仿射同伦方法经常使用无穷小，并且可以用来求解参数问题，并且提升、求解和下序列适用于分段仿射同伦方法。在这些标题下要研究的具体任务和项目包括无限维问题的变维算法，参数线性规划，同伦和单纯形法的平均分析，矩阵缩放，设计居中，计算动态曲线，有序域中计算的各个方面和问题表征，有序域中的转移原理，一个用于计算无穷小、经济模型的解和组合分析以及参数生产问题的算法的编译器。

This research is to be directed along two closely related avenues, namely, the solution of equations and paths of equations with homotopies, and the solution of parametric or short paths of equations with infinitesimals. The essence of the first avenue is path following the preimage of a point,perhaps, in subdivisions. The essence of the second is doing arithmetric with infinitesimals and, transferring algorithms, problems and solutions from one ordered field to another. The parametric problem is lifted 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 problems; this sequence is referred to as a lift, solve, and lower. To see that the two avenues are related we note that piecewise affine homotopy methods regularly employ infinitesimals and can be used to solve parametric problems, and the lift, solve, and lower sequence applies to piecewise affine homotopy methods. Specific tasks and projects to be studied under these headings include variable dimension algorithms for infinite dimensional problems, parametric linear programs, average analysis of homotopy and simplex methods, matrix scaling, design centering, computing dynamic curves, various aspects of computing in ordered fields and problem characterization, transfer principles in ordered fields, a compiler for algorithms computing with an infinitesimal, solution and combinatorial analysis of economic models, and parametric production problems.