Jacobi Structures in R3\mathbb{R}^3

The most general Jacobi brackets in $\mathbb{R}^3$ are constructed after solving the equations imposed by the Jacobi identity. Two classes of Jacobi brackets were identified, according to the rank of the Jacobi structures. The associated Hamiltonian vector fields are also constructed

Geometrical aspects of integrable systems

We review some basic theorems on integrability of Hamiltonian systems, namely the Liouville-Arnold theorem on complete integrability, the Nekhoroshev theorem on partial integrability and the Mishchenko-Fomenko theorem on noncommutative integrability, and for each of them we give a version suitable for the noncompact case. We give a possible global version of the previous local results, under certain topological hypotheses on the base space. It turns out that locally affine structures arise naturally in this setting.Comment: It will appear on International Journal of Geometric Methods in Modern Physics vol.5 n.3 (May 2008) issu

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

Integration of Dirac-Jacobi structures

We study precontact groupoids whose infinitesimal counterparts are Dirac-Jacobi structures. These geometric objects generalize contact groupoids. We also explain the relationship between precontact groupoids and homogeneous presymplectic groupoids. Finally, we present some examples of precontact groupoids.Comment: 10 pages. Brief changes in the introduction. References update

Adoptive immunotherapy monitored by micro-MRI in experimental colorectal liver metastasis

In this study we used the colon carcinoma DHDK12 cell line and generated single metastasis after subcapsular injection in BDIX rats as an experimental tumor model. The aim of the work was to set up in vitro experimental conditions to prepare immune effector cells and in vivo conditions for monitoring the effects of such cells injected as adoptive immunotherapy. Dendritic cells can process tumor cell antigens, induce a T-cell response and be used ex vivo to prepare activated lymphocytes. Lymphocytes were harvested from mesenteric lymph nodes and cocultured with bone marrow-derived autologous dendritic cells previously loaded with irradiated tumor cells. In vitro, the coculture: 1) induced the proliferation of lymphocytes, 2) expanded a preferential subpopulation of T CD8 lymphocytes, and 3) was in favor of lymphocyte cytotoxic activity against the DHDK12 tumor cell line. Activated lymphocytes were injected in the tumor-bearing rat portal vein. Parameters could be set to monitor tumor volume by micro MRI. This monitoring before and after treatment and immunohistochemical examinations revealed that: 1) micro MRI is an appropriate tool to survey metastasis growth in rat, 2) injected lymphocytes increase lesional infiltration with T CD8 cells even 15 days after treatment, 3) a dose of 50 millions lymphocytes is not sufficient to act on the course of the tumor

Is magnetic resonance imaging texture analysis a useful tool for cell therapy in vivo monitoring?

Assessment of anti-tumor treatment efficiency is usually done by measuring tumor size. Treatment may however induce changes in the tumor other than tumor size. Magnetic Resonance Imaging Texture Analysis (MRI-TA) is presently used to follow activated lymphocyte cell therapy. We used a 7T microimager to acquire high-resolution MR images of an experimental liver metastasis from colon carcinoma in rats treated (n = 4) or not (n = 3) with a cell therapy product. MRI-TA was then performed with Linear Discriminant Analysis and showed: i) a significant variation of tumor texture with tumor growth and ii) a significant modification in the texture of tumors treated with activated lymphocytes compared with untreated tumors. T2-weighted images or volume calculation did not evidence any difference. MRI-TA appears as a promising method for early detection and follow-up of response to cell therapy

Poisson-Jacobi reduction of homogeneous tensors

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold $M$, homogeneous with respect to a vector field $\Delta$ on $M$, and first-order polydifferential operators on a closed submanifold $N$ of codimension 1 such that $\Delta$ is transversal to $N$. This correspondence relates the Schouten-Nijenhuis bracket of multivector fields on $M$ to the Schouten-Jacobi bracket of first-order polydifferential operators on $N$ and generalizes the Poissonization of Jacobi manifolds. Actually, it can be viewed as a super-Poissonization. This procedure of passing from a homogeneous multivector field to a first-order polydifferential operator can be also understood as a sort of reduction; in the standard case -- a half of a Poisson reduction. A dual version of the above correspondence yields in particular the correspondence between $\Delta$-homogeneous symplectic structures on $M$ and contact structures on $N$.Comment: 19 pages, minor corrections, final version to appear in J. Phys. A: Math. Ge

The graded Jacobi algebras and (co)homology

Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the spirit of E.Witten's gauging of exterior derivative) is developed. One constructs a corresponding Cartan differential calculus (graded commutative one) in a natural manner. This, in turn, gives canonical generating operators for triangular Jacobi algebroids. One gets, in particular, the Lichnerowicz-Jacobi homology operators associated with classical Jacobi structures. Courant-Jacobi brackets are obtained in a similar way and use to define an abstract notion of a Courant-Jacobi algebroid and Dirac-Jacobi structure. All this offers a new flavour in understanding the Batalin-Vilkovisky formalism.Comment: 20 pages, a few typos corrected; final version to be published in J. Phys. A: Math. Ge