Erratum to: Determination of metal ion content of beverages and estimation of target hazard quotients: a comparative study

This is a correction to the following paper: Hague T, Petroczi A, Andrews PR, Barker J, Naughton DP: Determination of metal ion content of beverages and estimation of target hazard quotients: a comparative study. Chem Central J 2008, 2:13.     

Openness of momentum maps and persistence of extremal relative equilibria

We prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e. images of G-invariant open sets are open). As an application we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries, extremal relative equilibria persist for every perturbation of the value of the momentum map, provided the isotropy subgroup of this value is compact. We also demonstrate how this persistence result applies to an example of ellipsoidal figures of rotating fluid. We also provide an example with plane point vortices which shows how the compactness assumption is related to persistence.     

Superradiant phase transition in a lattice model with minimal coupling

A gauge invariant formulation of a lattice model of the interaction between radiation field and two-level atoms is given in terms of minimal coupling. Through a real space renormalization group procedure the conditions for the occurrence of macroscopic atomic polarization are determined.     

Relative Equilibria of Molecules

Summary. We describe a method for finding the families of relative equilibria of molecules that bifurcate from an equilibrium point as the angular momentum is increased from 0 . Relative equilibria are steady rotations about a stationary axis during which the shape of the molecule remains constant. We show that the bifurcating families correspond bijectively to the critical points of a function h on the two-sphere which is invariant under an action of the symmetry group of the equilibrium point. From this it follows that for each rotation axis of the equilibrium configuration there is a bifurcating family of relative equilibria for which the molecule rotates about that axis. In addition, for each reflection plane there is a family of relative equilibria for which the molecule rotates about an axis perpendicular to the plane. We also show that if the equilibrium is nondegenerate and stable, then the minima, maxima, and saddle points of h correspond respectively to relative equilibria which are (orbitally) Liapounov stable, linearly stable, and linearly unstable. The stabilities of the bifurcating branches of relative equilibria are computed explicitly for XY ₂ , X ₃ , and XY ₄ molecules. These existence and stability results are corollaries of more general theorems on relative equilibria of G -invariant Hamiltonian systems that bifurcate from equilibria with finite isotropy subgroups as the momentum is varied. In the general case, the function h is defined on the Lie algebra dual {\frak g} ^* and the bifurcating relative equilibria correspond to critical points of the restrictions of h to the coadjoint orbits in {\frak g} ^* . Received June 9, 1997; second revision received December 15, 1997; final revision received January 19, 1998     

On Generic Composites of Maps

We show that (in the nice dimensions) if a stable map is restrictedto a generic submanifold of the source manifold, then the resultingmap will also be stable. At the same time, the analogous resultfor versal unfoldings is proved. A few examples of applicationsto the extrinsic geometry of submanifolds of Euclidean spaceare discussed.     

Neutrino wave equation in the Robertson-Walker geometry

The massless Dirac equation is separated in the Robertson-Walker geometry. The Schrödinger-like one-dimensional equation to which the problem is reduced is shown to admit a discrete positive spectrum. The existence or nonexistence of the discrete neutrino energy spectrum is connected, in the case of the standard cosmology, with the assumption that the universe is closed or not.     

Determination of metal ion content of beverages and estimation of target hazard quotients: a comparative study

Background  

Considerable research has been directed towards the roles of metal ions in nutrition with metal ion toxicity attracting particular attention. The aim of this study is to measure the levels of metal ions found in selected beverages (red wine, stout and apple juice) and to determine their potential detrimental effects via calculation of the Target Hazard Quotients (THQ) for 250 mL daily consumption.     

Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology

In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary ∂M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field XM on M, Witten defines an inhomogeneous coboundary operator dXM=d+ιXM on invariant forms on M. The main purpose is to adapt Belishev-Sharafutdinov?s boundary data to invariant forms in terms of the operator dXM in order to investigate to what extent the equivariant topology of a manifold is determined by the corresponding variant of the DN map. We define an operator ΛXM on invariant forms on the boundary which we call the XM-DN map and using this we recover the XM-cohomology groups from the generalized boundary data (∂M,ΛXM). This shows that for a Zariski-open subset of the Lie algebra, ΛXM determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the ring structure of XM-cohomology groups from ΛXM. These results explain to what extent the equivariant topology of the manifold in question is determined by ΛXM.