Hamiltonian structures on foliations

We discuss hamiltonian structures of the Gelfand-Dorfman complex of projectable vector fields and differential forms on a foliated manifold. Such a structure defines a Poisson structure on the algebra of foliated functions, and embeds the given foliation into a larger, generalized foliation with presymplectic leaves. In a so-called tame case, the structure is induced by a Poisson structure of the manifold. Cohomology spaces and classes relevant to geometric quantization are also considered.Comment: LaTex, 18 page

Lie algebroid structures on a class of affine bundles

We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibred over the real numbers. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is further shown that the affine Lie algebroid structure gives rise to a coboundary operator on such forms. The concept of admissible curves and dynamical systems whose integral curves are admissible, brings an associated affine bundle into the picture, on which one can define in a natural way a prolongation of the original affine Lie algebroid structure.Comment: 28 page

Lie algebroid structures and Lagrangian systems on affine bundles

As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dynamics on an affine Lie algebroid.Comment: 28 pages, Late

Adapted connections on metric contact manifolds

In this paper, we describe the space of adapted connections on a metric contact manifold through the space of their torsion tensors. The torsion tensor is an element of the space of TM-valued two-forms, which splits into various subspaces. We study the parts of the torsion tensor according to this splitting to completely describe the space of adapted connections. We use this description to obtain characterizations of the generalized Tanaka-Webster connection and to describe the Dirac operators of adapted connections.Comment: 25 pages; some remarks added, minor correction

Wave packet evolution in non-Hermitian quantum systems

The quantum evolution of the Wigner function for Gaussian wave packets generated by a non-Hermitian Hamiltonian is investigated. In the semiclassical limit $\hbar\to 0$ this yields the non-Hermitian analog of the Ehrenfest theorem for the dynamics of observable expectation values. The lack of Hermiticity reveals the importance of the complex structure on the classical phase space: The resulting equations of motion are coupled to an equation of motion for the phase space metric---a phenomenon having no analog in Hermitian theories.Comment: Example added, references updated, 4 pages, 2 figure

Generalized Reduction Procedure: Symplectic and Poisson Formalism

We present a generalized reduction procedure which encompasses the one based on the momentum map and the projection method. By using the duality between manifolds and ring of functions defined on them, we have cast our procedure in an algebraic context. In this framework we give a simple example of reduction in the non-commutative setting.Comment: 39 pages, Latex file, Vienna ESI 28 (1993

The Synergistic Regulatory Effect of Runx2 and MEF Transcription Factors on Osteoblast Differentiation Markers

Purpose: Bone tissues for clinical application can be improved by studies on osteoblast differentiation. Runx2 is known to be an important transcription factor for osteoblast differentiation. However, bone morphogenetic protein (BMP)-2 treatment to stimulate Runx2 is not sufficient to acquire enough bone formation in osteoblasts. Therefore, it is necessary to find other regulatory factors which can improve the transcriptional activity of Runx2. The erythroblast transformation-specific (ETS) transcription factor family is reported to be involved in various aspects of cellular proliferation and differentiation. Methods: We have noticed that the promoters of osteoblast differentiation markers such as alkaline phosphatase (Alp), osteopontin (Opn), and osteocalcin (Oc) contain Ets binding sequences which are also close to Runx2 binding elements. Luciferase assays were performed to measure the promoter activities of these osteoblast differentiation markers after the transfection of Runx2, myeloid Elf-1-like factor (MEF), and Runxs+MEF. Reverse-transcription polymerase chain reaction was also done to check the mRNA levels of Opn after Runx2 and MEF transfection into rat osteoblast (ROS) cells. Results: We have found that MEF, an Ets transcription factor, increased the transcriptional activities of Alp, Opn, and Oc. The addition of Runx2 resulted in the 2- to 6-fold increase of the activities. This means that these two transcription factors have a synergistic effect on the osteoblast differentiation markers. Furthermore, early introduction of these two Runx2 and MEF factors significantly elevated the expression of the Opn mRNA levels in ROS cells. We also showed that Runx2 and MEF proteins physically interact with each other. Conclusions: Runx2 interacts with MEF proteins and binds to the promoters of the osteoblast markers such as Opn nearby MEF to increase its transcriptional activity. Our results also imply that osteoblast differentiation and bone formation can be increased by activating MEF to elicit the synergistic effect of Runx2 and MEF

Clifford-Finsler Algebroids and Nonholonomic Einstein-Dirac Structures

We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off-diagonal metrics and linear and nonlinear connections define different types of Finsler, Lagrange and/or Riemann-Cartan spaces. A generalization to spinor fields and Dirac operators on nonholonomic manifolds motivates the theory of Clifford algebroids defined as Clifford bundles, in general, enabled with nonintegrable distributions defining the nonlinear connection. In this work, we elaborate the algebroid spinor differential geometry and formulate the (scalar, Proca, graviton, spinor and gauge) field equations on Lie algebroids. The paper communicates new developments in geometrical formulation of physical theories and this approach is grounded on a number of previous examples when exact solutions with generic off-diagonal metrics and generalized symmetries in modern gravity define nonholonomic spacetime manifolds with uncompactified extra dimensions.Comment: The manuscript was substantially modified following recommendations of JMP referee. The former Chapter 2 and Appendix were elliminated. The Introduction and Conclusion sections were modifie

Contact complete integrability

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical $\R\ltimes \R^{n-1}$ structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of $n$ contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures. We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories

Poisson brackets with prescribed Casimirs

We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$, while, in the case where $M$ is of odd dimension, our objective is achieved by using a convenient almost cosymplectic structure. Several examples and applications are presented.Comment: 24 page