Autonomous three dimensional Newtonian systems which admit Lie and Noether point symmetries

We determine the autonomous three dimensional Newtonian systems which admit Lie point symmetries and the three dimensional autonomous Newtonian Hamiltonian systems, which admit Noether point symmetries. We apply the results in order to determine the two dimensional Hamiltonian dynamical systems which move in a space of constant non-vanishing curvature and are integrable via Noether point symmetries. The derivation of the results is geometric and can be extended naturally to higher dimensions.Comment: Accepted for publication in Journal of Physics A: Math. and Theor.,13 page

Two dimensional dynamical systems which admit Lie and Noether symmetries

We prove two theorems which relate the Lie point symmetries and the Noether symmetries of a dynamical system moving in a Riemannian space with the special projective group and the homothetic group of the space respectively. The theorems are applied to classify the two dimensional Newtonian dynamical systems, which admit a Lie point/Noether symmetry. Two cases are considered, the non-conservative and the conservative forces. The use of the results is demonstrated for the Kepler - Ermakov system, which in general is non-conservative and for potentials similar to the H\`enon Heiles potential. Finally it is shown that in a FRW background with no matter present, the only scalar cosmological model which is integrable is the one for which 3-space is flat and the potential function of the scalar field is exponential. It is important to note that in all applications the generators of the symmetry vectors are found by reading the appropriate entry in the relevant tables.Comment: 25 pages, 17 table

Generalizing the autonomous Kepler Ermakov system in a Riemannian space

We generalize the two dimensional autonomous Hamiltonian Kepler Ermakov dynamical system to three dimensions using the sl(2,R) invariance of Noether symmetries and determine all three dimensional autonomous Hamiltonian Kepler Ermakov dynamical systems which are Liouville integrable via Noether symmetries. Subsequently we generalize the autonomous Kepler Ermakov system in a Riemannian space which admits a gradient homothetic vector by the requirements (a) that it admits a first integral (the Riemannian Ermakov invariant) and (b) it has sl(2,R) invariance. We consider both the non-Hamiltonian and the Hamiltonian systems. In each case we compute the Riemannian Ermakov invariant and the equations defining the dynamical system. We apply the results in General Relativity and determine the autonomous Hamiltonian Riemannian Kepler Ermakov system in the spatially flat Friedman Robertson Walker spacetime. We consider a locally rotational symmetric (LRS) spacetime of class A and discuss two cosmological models. The first cosmological model consists of a scalar field with exponential potential and a perfect fluid with a stiff equation of state. The second cosmological model is the f(R) modified gravity model of {\Lambda}_{bc}CDM. It is shown that in both applications the gravitational field equations reduce to those of the generalized autonomous Riemannian Kepler Ermakov dynamical system which is Liouville integrable via Noether integrals.Comment: Reference [25] update, 21 page

Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.Comment: 19 page

Neonatal DNA methylation profile in human twins is specified by a complex interplay between intrauterine environmental and genetic factors, subject to tissue-specific influence

Comparison between groups of monozygotic (MZ) and dizygotic (DZ) twins enables an estimation of the relative contribution of genetic and shared and nonshared environmental factors to phenotypic variability. Using DNA methylation profiling of ∼20,000 CpG sites as a phenotype, we have examined discordance levels in three neonatal tissues from 22 MZ and 12 DZ twin pairs. MZ twins exhibit a wide range of within-pair differences at birth, but show discordance levels generally lower than DZ pairs. Within-pair methylation discordance was lowest in CpG islands in all twins and increased as a function of distance from islands. Variance component decomposition analysis of DNA methylation in MZ and DZ pairs revealed a low mean heritability across all tissues, although a wide range of heritabilities was detected for specific genomic CpG sites. The largest component of variation was attributed to the combined effects of nonshared intrauterine environment and stochastic factors. Regression analysis of methylation on birth weight revealed a general association between methylation of genes involved in metabolism and biosynthesis, providing further support for epigenetic change in the previously described link between low birth weight and increasing risk for cardiovascular, metabolic, and other complex diseases. Finally, comparison of our data with that of several older twins revealed little evidence for genome-wide epigenetic drift with increasing age. This is the first study to analyze DNA methylation on a genome scale in twins at birth, further highlighting the importance of the intrauterine environment on shaping the neonatal epigenome

Integrating stakeholder knowledge through modular cooperative participatory processes for marine spatial planning outcomes (CORPORATES)

This research was funded by the UK Natural Environment Research Council NERC (Knowledge Exchange Award 2014-2016) RC Grant reference: NE/M000184/1Peer reviewedPostprin