A Central Limit Theorem for intransitive dice

Intransive dice $D^{(1)}, \ldots, D^{(\ell)}$ are dice such that $D^{(1)}$ has advantage with respect to $D^{(2)}$, dice $D^{(2)}$ has advantage with respect to $D^{(3)}$ and so on, up to $D^{(\ell)}$, which has advantage over $D^{(1)}$. In this twofold work, we present: first, (deterministic) results on existence of general intransitive dice. Second and mainly, a central limit theorem for the vector of normalized victories of a die against the next one in the list when the faces of a die are i.i.d.\ random variables and all dice are independent, but different dice may have distinct distributions associated to, as well as they may have distinct number of faces. From this central limit theorem we derive a criteria to assure that the asymptotic probability of observing intransitive dice is null, which applies for many cases, including all continuous distributions and many discrete ones.Comment: 37 pages, 3 figure

Quantifying garnet-melt trace element partitioning using lattice-strain theory: New crystal-chemical and thermodynamic constraints

Many geochemical models of major igneous differentiation events on the Earth, the Moon, and Mars invoke the presence of garnet or its high-pressure majoritic equivalent as a residual phase, based on its ability to fractionate critical trace element pairs (Lu/Hf, U/Th, heavy REE/light REE). As a result, quantitative descriptions of mid-ocean ridge and hot spot magmatism, and lunar, martian, and terrestrial magma oceans require knowledge of garnet-melt partition coefficients over a wide range of conditions. In this contribution, we present new crystal-chemical and thermodynamic constraints on the partitioning of rare earth elements (REE), Y and Sc between garnet and anhydrous silicate melt as a function of pressure (P), temperature (T), and composition (X). Our approach is based on the interpretation of experimentally determined values of partition coefficients D using lattice-strain theory. In this and a companion paper (Draper and van Westrenen this issue) we derive new predictive equations for the ideal ionic radius of the dodecahedral garnet X-site,

Gendered self-views across 62 countries: a test of competing models

Social role theory posits that binary gender gaps in agency and communion should be larger in less egalitarian countries, reflecting these countries’ more pronounced sex-based power divisions. Conversely, evolutionary and self-construal theorists suggest that gender gaps in agency and communion should be larger in more egalitarian countries, reflecting the greater autonomy support and flexible self-construction processes present in these countries. Using data from 62 countries (N = 28,640), we examine binary gender gaps in agentic and communal self-views as a function of country-level objective gender equality (the Global Gender Gap Index) and subjective distributions of social power (the Power Distance Index). Findings show that in more egalitarian countries, gender gaps in agency are smaller and gender gaps in communality are larger. These patterns are driven primarily by cross-country differences in men’s self-views and by the Power Distance Index (PDI) more robustly than the Global Gender Gap Index (GGGI). We consider possible causes and implications of these findings

Psychometric Properties and Correlates of Precarious Manhood Beliefs in 62 Nations

Precarious manhood beliefs portray manhood, relative to womanhood, as a social status that is hard to earn, easy to lose, and proven via public action. Here, we present cross-cultural data on a brief measure of precarious manhood beliefs (the Precarious Manhood Beliefs scale [PMB]) that covaries meaningfully with other cross-culturally validated gender ideologies and with country-level indices of gender equality and human development. Using data from university samples in 62 countries across 13 world regions (N = 33,417), we demonstrate: (1) the psychometric isomorphism of the PMB (i.e., its comparability in meaning and statistical properties across the individual and country levels); (2) the PMB’s distinctness from, and associations with, ambivalent sexism and ambivalence toward men; and (3) associations of the PMB with nation-level gender equality and human development. Findings are discussed in terms of their statistical and theoretical implications for understanding widely-held beliefs about the precariousness of the male gender role