A large literature proposes that preferences for exaggerated sex typicality in human faces (masculinity/femininity) reflect a long evolutionary history of sexual and social selection. This proposal implies that dimorphism was important to judgments of attractiveness and personality in ancestral environments. It is difficult to evaluate, however, because most available data come from largescale, industrialized, urban populations. Here, we report the results for 12 populations with very diverse levels of economic development. Surprisingly, preferences for exaggerated sex-specific traits are only found in the novel, highly developed environments. Similarly, perceptions that masculine males look aggressive increase strongly with development and, specifically, urbanization. These data challenge the hypothesis that facial dimorphism was an important ancestral signal of heritable mate value. One possibility is that highly developed environments provide novel opportunities to discern relationships between facial traits and behavior by exposing individuals to large numbers of unfamiliar faces, revealing patterns too subtle to detect with smaller samples.facial attractiveness | evolution | cross-cultural | aggression | stereotyping

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a b s t r a c tLet Θ(n, k) be the set of digraphs of order n that have at most one walk of length k with the same endpoints. Let θ (n, k) be the maximum number of arcs of a digraph in Θ(n, k). We prove that if n ≥ 5 and k ≥ n − 1 then θ (n, k) = n(n − 1)/2 and this maximum number is attained at D if and only if D is a transitive tournament. θ (n, n − 2) and θ (n, n − 3) are also determined.

Abstract. In this paper, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ on mn × mn Hermitian matrices such that φ(A ⊗ B) and A ⊗ B have the same spectrum for any m × m Hermitian A and n × n Hermitian B. Such a map has the form A ⊗ B → U (ϕ1(A) ⊗ ϕ2(B))U * for mn × mn Hermitian matrices in tensor form A ⊗ B, where U is a unitary matrix, and for j ∈ {1, 2}, ϕj is the identity map X → X or the transposition map X → X t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A ⊗ B is also obtained. The results are connected bipartite (quantum) systems and are extended to multipartite systems.2010 Math. Subj. Class.: 15A69, 15A86, 15B57, 15A18.

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