Some results on L-dendriform algebras

We introduce a notion of L-dendriform algebra due to several different motivations. L-dendriform algebras are regarded as the underlying algebraic structures of pseudo-Hessian structures on Lie groups and the algebraic structures behind the $\mathcal O$-operators of pre-Lie algebras and the related $S$-equation. As a direct consequence, they provide some explicit solutions of $S$-equation in certain pre-Lie algebras constructed from L-dendriform algebras. They also fit into a bigger framework as Lie algebraic analogues of dendriform algebras. Moreover, we introduce a notion of $\mathcal O$-operator of an L-dendriform algebra which gives an algebraic equation regarded as an analogue of the classical Yang-Baxter equation in a Lie algebra.Comment: 15 page

Infants can infer the presence of hidden objects from referential gaze information

Infants' apparent failure in gaze-following tasks is often interpreted as a sign of lack of understanding the referential nature of looking. In the present study, 8- and 12-month-old infants followed the gaze of a model to one of two locations hidden from their view by occluders. When the occluders were removed, an object was revealed either at the location where the model had looked or at the other side. Infants at both ages looked longer at the empty location when it had been indicated by the model's looking behaviour, and this effect held up even when their first look after gaze following was discounted. This result demonstrates that even young infants hold referential expectations when they follow others' gaze and infer the location of hidden objects accordingly