The Lie-Poisson structure of the reduced n-body problem

The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. As a result we obtain a reduced system with a Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced n-body problem using a splitting method.Comment: 26 pages, 2 figure

Poisson Geometry in Constrained Systems

Constrained Hamiltonian systems fall into the realm of presymplectic geometry. We show, however, that also Poisson geometry is of use in this context. For the case that the constraints form a closed algebra, there are two natural Poisson manifolds associated to the system, forming a symplectic dual pair with respect to the original, unconstrained phase space. We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf in one of those. In the second class case the original constrained system may be reformulated equivalently as an abelian first class system in an extended phase space by these methods. Inspired by the relation of the Dirac bracket of a general second class constrained system to the original unconstrained phase space, we address the question of whether a regular Poisson manifold permits a leafwise symplectic embedding into a symplectic manifold. Necessary and sufficient for this is the vanishing of the characteristic form-class of the Poisson tensor, a certain element of the third relative cohomology.Comment: 41 pages, more detailed abstract in paper; v2: minor corrections and an additional referenc

Symplectic geometry on moduli spaces of J-holomorphic curves

Let (M,\omega) be a symplectic manifold, and Sigma a compact Riemann surface. We define a 2-form on the space of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M,\omega) that ensure that the restriction of the form to the moduli space of simple immersed J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Sigma-curves in a given homology class for a generic J.Comment: 16 page

On the geometric quantization of twisted Poisson manifolds

We study the geometric quantization process for twisted Poisson manifolds. First, we introduce the notion of Lichnerowicz-twisted Poisson cohomology for twisted Poisson manifolds and we use it in order to characterize their prequantization bundles and to establish their prequantization condition. Next, we introduce a polarization and we discuss the quantization problem. In each step, several examples are presented

Multi-mode bosonic Gaussian channels

A complete analysis of multi-mode bosonic Gaussian channels is proposed. We clarify the structure of unitary dilations of general Gaussian channels involving any number of bosonic modes and present a normal form. The maximum number of auxiliary modes that is needed is identified, including all rank deficient cases, and the specific role of additive classical noise is highlighted. By using this analysis, we derive a canonical matrix form of the noisy evolution of n-mode bosonic Gaussian channels and of their weak complementary counterparts, based on a recent generalization of the normal mode decomposition for non-symmetric or locality constrained situations. It allows us to simplify the weak-degradability classification. Moreover, we investigate the structure of some singular multi-mode channels, like the additive classical noise channel that can be used to decompose a noisy channel in terms of a less noisy one in order to find new sets of maps with zero quantum capacity. Finally, the two-mode case is analyzed in detail. By exploiting the composition rules of two-mode maps and the fact that anti-degradable channels cannot be used to transfer quantum information, we identify sets of two-mode bosonic channels with zero capacity.Comment: 37 pages, 3 figures (minor editing), accepted for publication in New Journal of Physic

Classical field theory on Lie algebroids: Variational aspects

The variational formalism for classical field theories is extended to the setting of Lie algebroids. Given a Lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to be morphisms of Lie algebroids. In addition to the standard case, our formalism includes as particular examples the case of systems with symmetry (covariant Euler-Poincare and Lagrange Poincare cases), Sigma models or Chern-Simons theories.Comment: Talk deliverd at the 9th International Conference on Differential Geometry and its Applications, Prague, September 2004. References adde

Groupoids and an index theorem for conical pseudo-manifolds

We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold $M$. A main ingredient is a non-commutative algebra that plays in our setting the role of $C_0(T^*M)$. We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in $K$-theory. We then give a new proof of the Atiyah-Singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds

Imbedding HACCP principles in dairy herd health and production management: case report on calf rearing

Driven by consumer demands, European legislation has suggested the use of HACCP (Hazard Analysis Critical Control Point) as the quality risk management programme for the whole dairy chain. Until now, an exception has been made for primary producers, but as regulations evolve, on-farm HACCP-like programmes should be ready to assure food safety as well as animal health and animal welfare. In our field experiment, the HACCP-concept was used to combine both optimal farm management and formalisation of quality assurance in an on-farm situation in the Netherlands. The process of young stock rearing was chosen, since its importance for the future of the farm is often underestimated. Hazards and their associated risk factors can be controlled within the farm-specific standards and tolerances, as targets can be controlled by corrective measures and by implementation of farm-specific worksheets. The veterinarian is pivotal for the facility-based HACCP team, since he/she has knowledge about on-farm risk assessment and relations between clinical pathology, feed and farm management. The HACCP concept in combination with veterinary herd health and production management programmes offers a promising approach to optimise on-farm production processes (i.e., young stock rearing) in addition to a structural approach for quality risk management on dairy farms

The Weyl bundle as a differentiable manifold

Construction of an infinite dimensional differentiable manifold ${\mathbb R}^{\infty}$ not modelled on any Banach space is proposed. Definition, metric and differential structures of a Weyl algebra and a Weyl algebra bundle are presented. Continuity of the $\circ$-product in the Tichonov topology is proved. Construction of the $*$-product of the Fedosov type in terms of theory of connection in a fibre bundle is explained.Comment: 31 pages; revised version - some typoes have been eliminated, notation has been simplifie

Communication in production animal medicine: modelling a complex interaction with the example of dairy herd health medicine

<p>Abstract</p> <p>Background</p> <p>The importance of communication skills in veterinary medicine is increasingly recognised. Appropriate communication skills towards the client are of utmost importance in both companion animal practice and production animal field and consultancy work. The need for building a relationship with the client, alongside developing a structure for the consultation is widely recognised and applies to both types of veterinary practice.</p> <p>Results</p> <p>Veterinary advisory practice in production animal medicine is, however, characterised by a more complex communication on different levels. While the person-orientated communication is a permanent process between veterinarian and client with a rather personal perspective and defines the roles of interaction, the problem-orientated communication deals with emerging difficulties; the objective is to solve an acute health problem. The solution - orientated communication is a form of communication in which both veterinarian and client address longstanding situations or problems with the objective to improve herd health and subsequently productivity performance. All three forms of communication overlap.</p> <p>Conclusions</p> <p>Based on this model, it appears useful for a veterinary practice to offer both a curative and an advisory service, but to keep these two separated when deemed appropriate. In veterinary education, the strategies and techniques necessary for solution orientated communication should be included in the teaching of communication skills.</p