A panorama of positivity. I: dimension free

This survey contains a selection of topics unified by the concept of positive semidefiniteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical references. Modern applications to high-dimensional covariance estimation and regularization are included

Are solitary and gregarious Mormon crickets (<em>Anabrus simplex</em>, Orthoptera, Tettigoniidae) genetically distinct?

Phase polyphenisms are usually thought to reflect plastic responses of species, independent of genetic differences; however, phase differences could correlate with genetic differentiation for various reasons. Mormon crickets appear to occur in two phases that differ in morphology and behaviour. Solitary individuals are cryptic and sedentary whereas gregarious individuals form bands, migrate, and are aposematically coloured. These traits have been thought to be phenotypically plastic and induced by environmental conditions. However, there has been no previous investigation of the extent of genetic differences between solitary and gregarious populations of this widespread North American species. We sequenced two mitochondrial genes, COII and COIII, in samples of Mormon crickets from gregarious populations west of the continental divide and solitary mountain populations primarily east of the divide. Sequencing revealed two genetically distinct clades that broadly correspond with the solitary eastern populations and the mainly gregarious western populations. We used coalescent modelling to test the hypothesis that the species consists of two deep genetic clades, as opposed to a series of equally distinct populations. Results allowed us to reject the null hypothesis that a radiation independent of phase produced these clades, and molecular clock estimates indicate the time of divergence to be approximately 2 million years ago. This work establishes that the solitary populations found in the mountains on the eastern slope are part of a clade that is genetically distinct from the western populations, which are primarily gregarious, and the implications of this apparent correlation between phase and genetic differentiation are discussed.</p