Geometric Poisson brackets on Grassmannians and conformal spheres

In this paper we relate the geometric Poisson brackets on the Grassmannian of 2-planes in R^4 and on the (2,2) Moebius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the Moebius sphere does not restrict to the space of differential invariants of Schwarzian type. But when the concept of conformal natural frame is transported from the conformal sphere into the Grassmannian, and the Poisson bracket is written in terms of the Grassmannian natural frame, it restricts and results into either a decoupled system or a complexly coupled system of KdV equations, depending on the character of the invariants. We also show that the biHamiltonian Grassmannian geometric brackets are equivalent to the non-commutative KdV biHamiltonian structure. Both integrable systems and Hamiltonian structure can be brought back to the conformal sphere.Comment: 33 page

Constant Curvature Coefficients and Exact Solutions in Fractional Gravity and Geometric Mechanics

We study fractional configurations in gravity theories and Lagrange mechanics. The approach is based on Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists in a proof that for corresponding classes of nonholonomic distributions a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.Comment: latex2e, 11pt, 27 pages, the variant accepted to CEJP; added and up-dated reference

Between but not within species variation in the distribution of fitness effects

New mutations provide the raw material for evolution and adaptation. The distribution of fitness effects (DFE) describes the spectrum of effects of new mutations that can occur along a genome, and is therefore of vital interest in evolutionary biology. Recent work has uncovered striking similarities in the DFE between closely related species, prompting us to ask whether there is variation in the DFE among populations of the same species, or among species with different degrees of divergence, i.e., whether there is variation in the DFE at different levels of evolution. Using exome capture data from six tree species sampled across Europe we characterised the DFE for multiple species, and for each species, multiple populations, and investigated the factors potentially influencing the DFE, such as demography, population divergence and genetic background. We find statistical support for there being variation in the DFE at the species level, even among relatively closely related species. However, we find very little difference at the population level, suggesting that differences in the DFE are primarily driven by deep features of species biology, and that evolutionarily recent events, such as demographic changes and local adaptation, have little impact