高阶系统的叠加规则及其应用

混合叠加规则与Riccati层次

Mixed superposition rules and the Riccati hierarchy

混合叠加规则，即，研究了用一阶系统的一般特解族和一些常数描述一阶微分方程组通解的函数。其主要成就是著名的李-谢弗定理的推广，刻画了系统的混合叠加规则。这个出乎意料的结果表明，这样的系统确实是Lie系统，即，他们承认一个标准的叠加法则。这为寻找李系统提供了一个新的、强有力的工具，本文将其应用于研究Riccati族，并以一种更有效、更简单的方式检索一些已知的结果。

Milne-Pinney方程解的一个非线性叠加法则

A nonlinear superposition rule for solutions of the Milne--Pinney equation

导出了Milne-Pinney方程两个解的叠加法则.

叠加法则、李定理与偏微分方程

Superposition rules, Lie theorem and partial differential equations

给出了非自治常微分方程解的非线性叠加规则的李定理的严格几何证明，填补了现有文献中的空白。证明是基于一个替代的，但等效的叠加规则的定义：它被认为是一个叶理与一些合适的属性。解决了叠加函数的唯一性问题，其关键是由给定的向量场李代数构造的叶理的余维数。最后，作为一个更有说服力的论据支持使用这种替代定义的叠加规则，它表明，这种定义允许立即推广的李定理的情况下，系统的偏微分方程。

Graph Reconstruction, Functorial Feynman Rules and Superposition Principles

In this article functorial Feynman rules are introduced as large generalizations of physicists Feynman rules, in the sense that they can be applied to arbitrary classes of hypergraphs, possibly endowed with any kind of structure on their vertices and hyperedges. We show that the reconstruction conjecture for classes of (possibly structured) hypergraphs admit a sheaf-theoretic characterization, allowing us to consider analogous conjectures. We propose an axiomatization for the notion of superposition principle and prove that the functorial Feynman rules work as a bridge between reconstruction c…

叠加法则与二阶微分方程

Superposition rules and second-order differential equations

本文的主要目的是介绍和分析一阶微分方程的各种叠加规则在二阶微分方程中的推广。结果，我们找到了一种方法来应用李和拟李系统的理论来分析二阶微分方程。为了说明我们的结果，我们分析了几个出现在物理和数学文献中的二阶微分方程，并利用我们的方法导出了这些方程的一些叠加规则。

Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra

The formalism for Poisson-Hopf (PH) deformations of Lie-Hamilton systems is refined in one of its crucial points concerning applications, namely the obtention of effective and computationally feasible PH deformed superposition rules for prolonged PH deformations of Lie-Hamilton systems. The two new notions here proposed are a generalization of the standard superposition rules and the concept of diagonal prolongations for Lie systems, which are consistently recovered under the non-deformed limit. Using a technique from superintegrability theory, we obtain a maximal number of functionally indepe…