Random planar trees and the Jacobian conjecture

We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formulated in terms of a problem involving labellings of rooted trees; we give a new probabilistic derivation of this formulation using multi-type branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a problem about shuffling subtrees of $d$-Catalan trees, i.e. planar $d$-ary trees. We also show that, if one can construct a certain Markov chain on large $d$-Catalan trees which updates its value by randomly shuffling certain nearby subtrees, and in such a way that the stationary distribution of this chain is uniform, then the Jacobian conjecture is true. Finally, we show that the subtree shuffling conjecture is true in a certain asymptotic sense, and thereafter use our machinery to prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high degree terms.Comment: 36 pages, 4 figures. Section 2.5 added, Section 3 expanded, further minor edit

Análise de conglomerados em curvas de aprendizado para formação de agrupamentos homogêneos de trabalhadores Cluster analysis of learning curves for grouping workers with homogeneous learning profiles

Em diversos setores da indústria é desejado que trabalhadores reunidos em uma estação de trabalho apresentem perfil de aprendizado similar. O presente artigo apresenta um método de agrupamento de trabalhadores utilizando modelagem por curva de aprendizado e técnicas de clusterização. O método modela dados de desempenho de trabalhadores por intermédio de diversos modelos de curvas de aprendizado; os parâmetros de aprendizado dos modelos testados permitem predizer o desempenho dos trabalhadores em intervalos de tempo pré-determinados. Os valores preditos são agrupados através de ferramentas de clusterização. O maior índice de ajuste (IA), gerado a partir do Silhouette Index e do coeficiente de determinação, indica o modelo de curva mais consistente em termos de aderência aos dados e qualidade de agrupamento de perfis de aprendizado. Ao ser aplicado em dados de uma indústria de calçados, o método gerou agrupamentos consistentes de trabalhadores com base nos distintos perfis de aprendizado.<br>In many industrial segments, it is desirable to allocate workers with similar learning profiles in the same workstation. This paper presents a method that groups workers based on learning curve modeling and clustering techniques. Workers' performance data are modeled through several learning curve models; learning parameters allow for workers' performance prediction at intervals of predetermined time. The predicted values are then grouped by clustering techniques. The largest Adjustment Index (AI), derived from the Silhouette Index and Coefficient of Determination, indicates the model yielding superior adherence to data and better clustering of learning profiles. When applied to a shoe manufacturing process, the method generated consistent groups of workers based on their learning profiles