论文标题
林前指数，马格努斯的操作员和累积肿瘤关系的森林公式
A forest formula for pre-Lie exponentials, Magnus' operator and cumulant-cumulant relations
论文作者
论文摘要
在量子场理论中概括了Zimmermann的森林公式的森林公式，以计算对抗前代代数代数的双重代数。在这项工作中，在很大程度上是由穆鲁亚（Murua）对贝克 - 贝克（Baker-Campbell-Hausdorff）公式进行分析的动机，我们表明，相同的思想和技术概括并提供有效的工具来处理这些代数中的计算，这在数值分析和相关领域中至关重要。我们通过研究自由前代数中的lie指数和马格努斯操作员的作用来说明我们的结果，并在源自自由概率的词语前代数中的前代代数。后一个示例提供了组合公式，该公式将不同的累积品品牌在非交通概率中。
Forest formulas that generalize Zimmermann's forest formula in quantum field theory have been obtained for the computation of the antipode in the dual of enveloping algebras of pre-Lie algebras. In this work, largely motivated by Murua's analysis of the Baker-Campbell-Hausdorff formula, we show that the same ideas and techniques generalize and provide effective tools to handle computations in these algebras, which are of utmost importance in numerical analysis and related areas. We illustrate our results by studying the action of the pre-Lie exponential and the Magnus operator in the free pre-Lie algebra and in a pre-Lie algebra of words originating in free probability. The latter example provides combinatorial formulas relating the different brands of cumulants in non-commutative probability.
