Computational Refinement and Existence Proofs for Singularities

9701540 R·贝克·科尔福特在过去的二十年里，人们对奇点、分叉和相关现象以及与物理世界的相关性的理解有所增加。同时，边界离散化和舍入误差的工具也得到了改进，允许基于经典不动点理论的自动结果验证。传统上，这种验证是通过线性化，然后用区间算术定界来进行的。然而，在迄今开发的方法中，当系统在解的奇异性时，存在和唯一性验证必然失败。这个项目的目标是(1)对现有的验证技术进行新的、简单的和自然的修改，以证明当非线性系统是奇异的时存在和唯一；(2)调查所产生的工具的力量，首先是有限维问题(代数系统)，然后是无限维问题(微分方程式和积分方程式)；(3)增加对奇点的理解。从承重土木工程结构到生态系统再到人类心脏节律，科学和工程中的许多复杂系统都是由表现出奇异性和分叉的非线性方程组来建模的。这种分叉对应于作为条件(负载、温度等)的行为的变化。变化。近年来，人们对这类现象的了解越来越多，但分析问题依然存在。这项研究将提供工具，以实现高效、自动化、但在数学上无懈可击的分析。这反过来将在物理世界中带来更多的可预测性。

9701540 R. Baker Kearfott During the past two decades, understanding of singularities, bifurcations and related phenomena, and relevance to the physical world have increased. At the same time, tools for bounding discretization and roundoff error have improved, to allow automatic result verification based on classical fixed point theory. Traditionally, such verification proceeds by linearization, then bounding with interval arithmetic. However, in methods developed to date, existence and uniqueness verification necessarily must fail when the system is singular at solutions. Goals of this project are to (1) develop new, simple and natural modifications to existing verification techniques, to prove existence and uniqueness when the nonlinear system is singular; (2) investigate the power of the resulting tools, first for finite-dimensional problems (algebraic systems), then for infinite-dimensional problems (differential and integral equations); (3) increase understanding of singularities. Many complex systems throughout the sciences and engineering, ranging from load-bearing civil engineering structures to ecological systems to human heart rythms, are modeled by sets of nonlinear equations that exhibit singularities and bifurcations. Such bifurcations correspond to changes in behavior as conditions (the load, a temperature, etc.) change. Understanding of such phenomena has blossomed in recent years, but analytical problems remain. This study will provide tools to enable an efficient and automated, yet mathematically airtight analysis. This in turn will lead to more predictability in the physical world.