Modular classes of Poisson-Nijenhuis Lie algebroids

The modular vector field of a Poisson-Nijenhuis Lie algebroid $A$ is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian $A$-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson-Nijenhuis structure.Comment: To appear in Letters in Mathematical Physic

(Un-) Coupling gaze and attention outside central vision

In normal vision, shifts of attention and gaze are tightly coupled. Here we ask if this coupling affects performance also when central vision is not available. To this aim, we trained normal-sighted participants to perform a visual search task while vision was restricted to a gaze-contingent viewing window ("forced field location") either in the left, right, upper, or lower visual field. Gaze direction was manipulated within a continuous visual search task that required leftward, rightward, upward, or downward eye movements. We found no general performance advantage for a particular part of the visual field or for a specific gaze direction. Rather, performance depended on the coordination of visual attention and eye movements, with impaired performance when sustainedattention and gaze have to be moved in oppositedirections. Our results suggest that during early stages of central visual field loss, the optimal location for the substitution of foveal vision does not depend on the particular retinal location alone, as has previously been thought, but also on the gaze direction required by the task the patient wishes to perform

A supergeometric approach to Poisson reduction

This work introduces a unified approach to the reduction of Poisson manifolds using their description by graded symplectic manifolds. This yields a generalization of the classical Poisson reduction by distributions (Marsden-Ratiu reduction). Further it allows one to construct actions of strict Lie 2-groups and to describe the corresponding reductions.Comment: 40 pages. Final version accepted for publicatio

In situ electron microscopy studies of electric field assisted sintering of oxide ceramics

A wide range of studies shows a dramatic effect of applied electric fields or currents on the sintering behavior of oxide ceramic powders. However, the mechanisms accounting for the so-called flash sintering remain elusive despite the wide application potential. Using in-situ scanning and transmission electron microscopy, we aim to gain insight into the atomic origins of sintering behavior, as well as of the high conductivity states that occur in conjunction with flash events during field-assisted sintering. We investigate the sintering dynamics of ZnO green bodies with a density between 50% and 70% and ZnO thin films with and without electric fields and under different oxidizing and reducing gas pressures. Specifically, we use a specially designed SEM heating stage to study the evolution of microstructure and morphology, including grain/void morphology, segregation, and precipitation, both with and without applied fields and with and without gas pressures up to 2 mbar. The in-situ TEM sintering studies, also under controlled electric field and gas pressure, allow us to detect chemical segregation and valence changes (using EDX and EELS) near the sintering boundaries. By gaining access to structural and chemical information down to the atomic scale, we hope to determine how the electric field causes flash sintering. Please click Additional Files below to see the full abstract

Modular classes of skew algebroid relations

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E* which is invariant with respect to all Hamiltonian vector fields if and only if E is modular, i.e. mod(E)=0. Further, relative modular class of a subalgebroid is introduced and studied together with its application to holonomy, as well as modular class of a skew algebroid relation. These notions provide, in particular, a unified approach to the concepts of a modular class of a Lie algebroid morphism and that of a Poisson map.Comment: 20 page

Jacobi structures revisited

Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra associated with a vector bundle which satisfy a property similar to that of the Jacobi brackets, are introduced. They turn out to be equivalent to generalized Lie algebroids in the sense of Iglesias and Marrero and can be viewed also as odd Jacobi brackets on the supermanifolds associated with the vector bundles. Jacobi bialgebroids are defined in the same manner. A lifting procedure of elements of this Grassmann algebra to multivector fields on the total space of the vector bundle which preserves the corresponding brackets is developed. This gives the possibility of associating canonically a Lie algebroid with any local Lie algebra in the sense of Kirillov.Comment: 20 page

Gardner's deformations of the N=2 supersymmetric a=4-KdV equation

We prove that P.Mathieu's Open problem on constructing Gardner's deformation for the N=2 supersymmetric a=4-Korteweg-de Vries equation has no supersymmetry invariant solutions, whenever it is assumed that they retract to Gardner's deformation of the scalar KdV equation under the component reduction. At the same time, we propose a two-step scheme for the recursive production of the integrals of motion for the N=2, a=4-SKdV. First, we find a new Gardner's deformation of the Kaup-Boussinesq equation, which is contained in the bosonic limit of the super-hierarchy. This yields the recurrence relation between the Hamiltonians of the limit, whence we determine the bosonic super-Hamiltonians of the full N=2, a=4-SKdV hierarchy. Our method is applicable towards the solution of Gardner's deformation problems for other supersymmetric KdV-type systems.Comment: Extended version of the talks given by A.V.K. at 8th International conference `Symmetry in Nonlinear Mathematical Physics' (June 20-27, 2009, Kiev, Ukraine) and 9th International workshop `Supersymmetry and Quantum Symmetries' (July 29 - August 3, 2009, JINR, Dubna, Russia); 22 page

Integration of Holomorphic Lie Algebroids

We prove that a holomorphic Lie algebroid is integrable if, and only if, its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic-Fernandes do also apply in the holomorphic context without any modification. As a consequence we give another proof of the following theorem: a holomorphic Poisson manifold is integrable if, and only if, its real (or imaginary) part is integrable as a real Poisson manifold.Comment: 26 pages, second part of arXiv:0707.4253 which was split into two, v2: example 3.19 and section 3.7 adde

Algebraic Bethe Ansatz for deformed Gaudin model

The Gaudin model based on the sl_2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a recurrence relation. The commutation relations between the generating function t(\lambda) of the integrals of motion and the creation operators are calculated and therefore the algebraic Bethe Ansatz is fully implemented. The energy spectrum as well as the corresponding Bethe equations of the system coincide with the ones of the sl_2-invariant Gaudin model. As opposed to the sl_2-invariant case, the operator t(\lambda) and the Gaudin Hamiltonians are not hermitian. Finally, the inner products and norms of the Bethe states are studied.Comment: 23 pages; presentation improve