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

这个项目涉及微分方程，微分方程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，它将一个量与它相对于离散时间的变化率联系起来；以及它们的组合。这些方程的系统描述或模拟了整个科学中的行为和现象；流行病学、人口动力学、力学等。该项目将使用模型理论（数学逻辑的一个分支）的方法来开发或改进程序，用于显示此类方程系统的一致性（解的存在性），用于识别此类方程系统中的辅助参数，以及用于从此类系统中消除未知数。该项目还将扩展现有的解决方案集分类，从超定（或有限维）系统到欠定（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理论的角度研究微分、差分和微分-差分方程。目标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。预计将应用于物理系统的各种建模问题。拟议的研究有三个方面。首先是在现有的微分或差分理论方法原则上提供算法的情况下，开发有效的算法。二是将现有的系统有限维解空间的模型理论分类推广到无限维解空间，并开发将该分类应用于实践的方法。三是引入和研究新的一阶带算子环理论，其决策过程将应用于应用中出现的新实例。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.