Extremal measures maximizing functionals based on simplicial volumes

We consider functionals measuring the dispersion of a d-dimensional distribution which are based on the volumes of simplices of dimension k ≤ d formed by k + 1 independent copies and raised to some power δ. We study properties of extremal measures that maximize these functionals. In particular, for positive δ we characterize their support and for negative δ we establish connection with potential theory and motivate the application to space-filling design for computer experiments. Several illustrative examples are presented

The Molecular Clockwork of the Fire Ant Solenopsis invicta

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Towards Uniform Online Spherical Tessellations

The problem of uniformly placing N points onto a sphere finds applications in many areas. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The proposed online algorithm of Chen et al. was upper-bounded by 5.99 and then improved to 3.69, which is achieved by considering a circumscribed dodecahedron followed by a recursive decomposition of each face. We analyse a simple tessellation technique based on the regular icosahedron, which decreases the upper-bound for the online version of this problem to around 2.84. Moreover, we show that the lower bound for the gap ratio of placing up to three points is 1+5√2≈1.618 . The uniform distribution of points on a sphere also corresponds to uniform distribution of unit quaternions which represent rotations in 3D space and has numerous applications in many areas

Global urban environmental change drives adaptation in white clover.

Urbanization transforms environments in ways that alter biological evolution. We examined whether urban environmental change drives parallel evolution by sampling 110,019 white clover plants from 6169 populations in 160 cities globally. Plants were assayed for a Mendelian antiherbivore defense that also affects tolerance to abiotic stressors. Urban-rural gradients were associated with the evolution of clines in defense in 47% of cities throughout the world. Variation in the strength of clines was explained by environmental changes in drought stress and vegetation cover that varied among cities. Sequencing 2074 genomes from 26 cities revealed that the evolution of urban-rural clines was best explained by adaptive evolution, but the degree of parallel adaptation varied among cities. Our results demonstrate that urbanization leads to adaptation at a global scale