三度树结构与Magnus展开式
The tridendriform structure of a Magnus expansion
AI总结：
本文基于树结构和Magnus展开式，探讨了自由三度树代数中Magnus展开式的闭式公式，并推导了离散的Mielnik-Plebanski-Strichartz公式。
摘要：
树的概念在Butcher的B级数中起着重要作用。最近，通过使用根树，人们对Magnus展开式背后的代数和组合结构有了更深入的理解。我们进一步发展了观察，即线性一阶差分方程解的对数可以表示为在预李代数中进行的Magnus展开式。通过在平面减少树上进行基本组合学，我们推导出自由三度树代数中Magnus展开式的闭式公式。三度树代数结构在词准对称函数上允许我们推导该对数的离散Mielnik-Plebanski-Strichartz公式。
英文摘要：
The notion of trees plays an important role in Butcher's B-series. More recently, a refined understanding of algebraic and combinatorial structures underlying the Magnus expansion has emerged thanks to the use of rooted trees. We follow these ideas by further developing the observation that the logarithm of the solution of a lihear first-order finite-difference equation can be written in terms of the Magnus expansion taking place in a pre-Lie algebra. By using basic combinatorics on planar reduced trees we derive a closed formula for the Magnus expansion in the context of free tridendriform algebra. The tridendriform algebra structure on word quasi-symmetric functions permits us to derive a discrete analogue of the Mielnik-Plebanski-Strichartz formula for this logarithm.
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