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A meaningless symbol that repeatedly co-occurs with emotionally salient faces (US) can transform into a valenced symbol (CS). US-to-CS valence transformations have been observed for CS that have been directly (US→CS0) and indirectly (US→CS0→CS1→CS2) linked with face US. The structure of a US→CS0→CS1→CS2 series may be conceptualized in terms of “nodal distance,” where CS0, CS1, and CS2 are 0, 1, and 2 nodes from the US respectively. Increasing nodal distance between an evaluated CS and its linked US can reduce magnitude of observed CS valence transformations. We explored currently whether nodal distance can influence CS valence extinction, which describes reductions in CS valence following repeated exposures to CS without any accompanying US. In our study, faces with happy/neutral/sad expressions (US) were directly linked with nonsense words (US→CS0). The directly linked CS0 was concurrently linked with other words (CS0→CS1, CS1→CS2). Subjects evaluated all stimuli before and after conditioning, then continued to provide CS evaluations twice a week for 6 weeks. Bayesian factors provided credible evidence for the transformation and extinction of CS valences that were 0 and 1 nodes from US (all BF10’s > 100). The variability across post-conditioning CS evaluations provides indirect evidence for context-sensitive/propositional and structural/associative operations during CS evaluations.

Let $r_k(G)$ denote the set of all $k$-roots of the identity in a Lie group $G$. We show that $r_k(G)$ is always an embedded submanifold of $G$, having the conjugacy classes of its elements as open submanifolds. These conjugacy classes are examples of $k$-symmetric spaces and we show, more generally, that every $k$-symmetric space of a Lie group $G$ is a covering manifold of an embedded submanifold $Orb$ of $G$. We compute also the Hessian of the inclusions of $r_k(G)$ and $Orb$ into $G$, relative to the natural connection on the domain and to the symmetric connection on $G$. [ABSTRACT FROM AUTHOR]