arXiv subjects

Pierre Catoire

Publications and source records attributed to Pierre Catoire.

6 recordsLinked to original sources

arXiv2026

Regularised double shuffle relations for planar Arborified Zeta Values

We endow spaces of decorated planar rooted trees with new dendriform and tridendrifrom algebra structures and provide their combinatorial description. We then show that the planar counterparts of Arborified Zeta Values are algebra morphisms for these shuffle and quasi-shuffle products of planar rooted trees. We also prove an arborified version of Hoffman's regularisation relation for Arborified Zeta Values. We conjecture that those give every rational relation between Arborified Zeta Values and show that this conjecture implies the regularised double shuffle conjecture for Multiple Zeta Values.

math.NT↗

arXiv2026

Effective approach of the tridendriform Schroeder tree algebra

We introduce a primitive computation problem in the free tridendriform algebra generated by one element which is a Hopf algebra based on Schroeder trees. We know a complex way to generate all of them. To understand it clearer, we want to implement this method on a computer. However, we need to create some tools to implement Schroeder trees and the multiplications over this algebra to be able to compute the primitive elements. We also checked numerically that they are all primitive elements. In this paper, we detail how we made the problem mathematically understandable for a computer and how we implement it.

cs.DM↗

arXiv2026

Prediction with Missing Data: Target Probabilities and Missingness Mechanisms

Conditions ensuring optimal parameter estimation in the presence of missing data are well established in inference, typically relying on the Missing-at-Random (MAR) assumption. In prediction, similar principles are often assumed to apply. However, methods considered biased in inference, such as pattern sub-modelling or unconditional imputation, have been shown to achieve optimal predictive performance under any missingness mechanism, including non-MAR (MNAR). To explain this apparent contradiction, we introduce a new formal framework for describing missingness in prediction. Central to this framework is a distinction between two prediction targets, defined according to whether or not the indicator of observation of the predictors is exploited to predict the outcome. This distinction leads to a classification of the missingness mechanisms describing the conditions under which these targets are equal, and when consistent prediction of each is achievable. A key result is that both targets may be consistently predicted under conditions weaker than MAR. We discuss the implications of this paradigm for handling missing data in prediction, distinguishing between missingness at development, validation and deployment of a forecaster. The findings are illustrated using simulated data and a real-world application with the prediction of significant injury after trauma upon arrival at the emergency department.

stat.ME↗

arXiv2025

Tridendriform and dendriform Zeta Values from Schroeder trees

To build new generalisations of Multiple Zeta Values, we define new spaces of formal series and formal integrals. We show that they are tridendriform and dendriform algebras. This allows us to reinterpret the fact that Multiple Zeta Values are algebra morphisms for shuffles of words in terms of finer tridendriform and dendriform structures. Applying universal properties of Schroeder trees we obtain generalisations of Multiple Zeta Values that are algebra morphisms for associative products. Hence we find new properties of Arborified Zeta Values and state how this enables the computation of some Shintani Zeta Values.

math.CO↗

arXiv2024

The Cartier-Quillen-Milnor-Moore theorem in the Post-Hopf case

We give the definition of left/right Post-Lie algebras and left/right Post-Hopf algebras and establish a link between those objects. We get a Cartier-Quillen-Milnor-Moore theorem for Post-Hopf algebras. We give another description for free Post-Lie algebras and a description for Post-Lie algebras obtained from an associative product.

math.CO↗

arXiv2023

Tridendriform structures

We first study tensor products of tridendriform algebras in order to introduce the notion of tridendriform bialgebra. We shall need for this a notion of augmented tridendriform algebras. Inspired by the work of J-L. Loday and M. Ronco, we build free tridendriform algebras over reduced trees and show that they have a coproduct satisfying some compatibilities with the tridendriform products. Such an object will be called a (3, 1)--dendriform algebra. Studying the free (3, 1)--dendriform bialgebra over one generator, we describe its products and coproduct in a combinatorial way. The products are described by branches shuffle and the coproduct by admissible cuts. We compare it with quasi-shuffle algebras over words. Its graded dual is the bialgebra TSym introduced by N. Bergeron and al which is described by the lightening splitting of a tree. As a consequence, this shows that TSym has a (1, 3)--dendriform bialgebra structure. This means that its coproduct can be split in three parts with convenient compatibilities. This can be extended to (3, 1)-bialgebras over an arbitrary number of generators. Finally, we introduce the notion of (3, 2)--dendriform bialgebra. This is a Hopf algebra, where we can split the product in three pieces and the coproduct in two with Hopf compatibilities. We give an example of such an algebra built on the free (3, 1)-dendriform bialgebra with one generator. We describe and generate its codendriform primitives and count its coassociative primitives thanks to L. Foissy's work. We end this paper by showing that a quotient of this (3, 2)-dendriform bialgebra is the Loday-Ronco bialgebra.

math.CO↗