Elementary differentials from multi-indices to rooted trees
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912671527337984 |
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| author | Bruned, Yvain Laubie, Paul |
| author_facet | Bruned, Yvain Laubie, Paul |
| contents | Rooted trees are essential for describing numerical schemes via the so-called B-series. They have also been used extensively in rough analysis for expanding solutions of singular Stochastic Partial Differential Equations (SPDEs). When one considers scalar-valued equations, the most efficient combinatorial set is multi-indices. In this paper, we investigate the existence of intermediate combinatorial sets that will lie between multi-indices and rooted trees. We provide a negative result stating that there is no combinatorial set encoding elementary differentials in dimension $d\neq 1$, and compatible with the rooted trees and the multi-indices aside from the rooted trees. This does not close the debate of the existence of such combinatorial sets, but it shows that it cannot be obtained via a naive and natural approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elementary differentials from multi-indices to rooted trees Bruned, Yvain Laubie, Paul Numerical Analysis Combinatorics Probability Representation Theory Rooted trees are essential for describing numerical schemes via the so-called B-series. They have also been used extensively in rough analysis for expanding solutions of singular Stochastic Partial Differential Equations (SPDEs). When one considers scalar-valued equations, the most efficient combinatorial set is multi-indices. In this paper, we investigate the existence of intermediate combinatorial sets that will lie between multi-indices and rooted trees. We provide a negative result stating that there is no combinatorial set encoding elementary differentials in dimension $d\neq 1$, and compatible with the rooted trees and the multi-indices aside from the rooted trees. This does not close the debate of the existence of such combinatorial sets, but it shows that it cannot be obtained via a naive and natural approach. |
| title | Elementary differentials from multi-indices to rooted trees |
| topic | Numerical Analysis Combinatorics Probability Representation Theory |
| url | https://arxiv.org/abs/2509.13118 |