20 research outputs found
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