FRG: Collaborative Research: Model Theory of Differential and Difference Equations with Applications

该项目涉及微分方程，它将一个量与其相对于连续时间的变化率联系起来;差分方程，它将一个量与其相对于离散时间的变化率联系起来;以及这些方程的组合。这些方程的系统描述或模型的行为和现象在整个科学;流行病学，人口动态学，力学等该项目将使用模型理论的方法，数学逻辑的一个分支，以开发或改进程序，以显示一致性（存在的解决方案）的系统，这些方程的系统，确定辅助参数，并消除未知数从这些系统。该项目还将扩展现有的解决方案集的分类，从超定（或有限维）系统的情况下，欠定（或无限维系统）。在这个重点研究小组项目中，研究人员团队从数学逻辑意义上的模型论的角度研究微分，差分和微分差分方程。其目的之一是提供决策程序或改进有关一致性和消除的现有决策程序。预计各种物理系统的建模问题的应用。拟议的研究有三个方面。第一个是开发有效的算法的情况下，现有的方法从理论上的微分或差场提供算法的原则。第二是将现有的有限维系统解空间的模型理论分类扩展到无限维解空间，以及开发将分类应用于实践的方法。第三个奖项是介绍和研究新的一阶环理论与运营商，其决策程序将适用于新的例子中出现的应用。这个奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的智力价值和更广泛的影响审查标准。

This project is concerned with differential equations, which relate a quantity to its rate of change with respect to continuous time; difference equations, which relate a quantity to its rate of change with respect to discrete time; and combinations of these. Systems of such equations describe or model behavior and phenomena throughout the sciences; epidemiology, population dynamics, mechanics etc. The project will use methods from model theory, a branch of mathematical logic, to develop or improve procedures for showing consistency (existence of solutions) of systems of such equations, for identifying auxiliary parameters in systems of such equations, and for eliminating unknowns from such systems. The project will also extend the existing classification of sets of solutions from the case of over-determined (or finite-dimensional) systems to under-determined (or infinite-dimensional systems). In this focused research group project, the team of researchers studies differential, difference, and differential-difference equations, from the point of view of model theory in the sense of mathematical logic. Among the aims is to provide decision procedures or improve existing decision procedures regarding consistency and elimination. Applications to a variety of modelling problems for physical systems are expected. The proposed research is threefold. The first is to develop efficient algorithms in cases where the existing methods from theories of differential or difference fields provide algorithms in principle. The second is to extend the existing model-theoretic classification of finite-dimensional solution spaces of systems to infinite-dimensional solution spaces, as well as developing methods to apply the classification in practice. The third is to introduce and study new first order theories of rings with operators, whose decision procedures will apply to new examples arising in applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.