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

FRG：协作研究：微分方程和差分方程的模型理论及其应用

• 批准号: 1760413
• 负责人: Thomas Scanlon
• 金额: $ 30万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-15 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1760413&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Model; Theory

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.

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，将一个量与其相对于离散时间的变化率联系起来；以及这些的组合。这些方程组描述或模拟了整个科学领域的行为和现象；流行病学、人口动力学、力学等。该项目将使用数学逻辑的一个分支——模型论的方法来开发或改进程序，以显示此类方程组的一致性（解的存在性）、识别此类方程组中的辅助参数以及消除此类系统中的未知数。该项目还将现有的解集分类从超定（或有限维）系统的情况扩展到欠定（或无限维系统）的情况。在这个重点研究小组项目中，研究小组从数理逻辑意义上的模型论的角度研究微分方程、差分方程和微分-差分方程。其目标之一是提供决策程序或改进现有决策程序的一致性和消除性。预计可应用于物理系统的各种建模问题。拟议的研究有三个方面。第一个是在微分或差分场理论的现有方法原则上提供算法的情况下开发有效的算法。第二个是将系统有限维解空间的现有模型理论分类扩展到无限维解空间，并开发在实践中应用该分类的方法。第三个是与算子一起介绍和研究新的环一阶理论，其决策程序将适用于应用中出现的新例子。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

A variant of the Mordell–Lang conjecture

莫德尔·朗猜想的一种变体

• DOI: 10.4310/mrl.2019.v26.n5.a7
• 发表时间: 2019
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: Ghioca, Dragos;Hu, Fei;Scanlon, Thomas;Zannier, Umberto
• 通讯作者: Zannier, Umberto

Multi-experiment Parameter Identifiability of ODEs and Model Theory

常微分方程的多实验参数可辨识性和模型理论

• DOI: 10.1137/21m1389845
• 发表时间: 2022
• 期刊: SIAM Journal on Applied Algebra and Geometry
• 影响因子: 1.2
• 作者: Ovchinnikov, Alexey;Pillay, Anand;Pogudin, Gleb;Scanlon, Thomas
• 通讯作者: Scanlon, Thomas

Berezin integral as a limit of Riemann sum

Berezin 积分作为黎曼和的极限

• DOI: 10.1063/1.5144877
• 发表时间: 2020
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Scanlon, Thomas;Sverdlov, Roman
• 通讯作者: Sverdlov, Roman

Computing all identifiable functions of parameters for ODE models

计算 ODE 模型参数的所有可识别函数

• DOI: 10.1016/j.sysconle.2021.105030
• 发表时间: 2021
• 期刊: Systems & Control Letters
• 影响因子: 2.6
• 作者: Ovchinnikov, Alexey;Pillay, Anand;Pogudin, Gleb;Scanlon, Thomas
• 通讯作者: Scanlon, Thomas

SOLVING DIFFERENCE EQUATIONS IN SEQUENCES: UNIVERSALITY AND UNDECIDABILITY

• DOI: 10.1017/fms.2020.14
• 发表时间: 2019-09
• 期刊: Forum of Mathematics, Sigma
• 作者: G. Pogudin;T. Scanlon;M. Wibmer