VI 3.7 Metrical N-linear connections 3.8 Gravitational and electromagnetic fields 3.9 The Lagrange space of electrodynamics 3.10 Generalized Lagrange spaces The geometry of cotangent bundle 4.1 The bundle (T*M,ir*,M) 4.2 The Poisson brackets. The Hamiltonian systems 4.3 Homogeneity 4.4 Nonlinear connections 4.5 Distinguished vector and covector fields 4.6 The almost product structure F. The metrical structure G. The almost complex structure IF 4.7 d-tensor algebra. N-linear connections 4.8 Torsion and curvature 4.9 The coefficients of an N-linear connection 4.10 The local expressions of d-tensors of torsion and curvature 4.11 Parallelism. Horizontal and vertical paths 4.12 Structure equations of an N-linear connection. Bianchi identities . . . Hamilton spaces 5.1 The spaces GH n 5.2 N-metrical connections i n 121 5.3 The N-lift of GH n 5.4 Hamilton spaces 5.5 Canonical nonlinear connection of the space H n 5.6 The canonical metrical connection of Hamilton space H n 5.7 Structure equations of CT(JV). Bianchi identities 5.8 Parallelism. Horizontal and vertical paths 5.9 The Hamilton spaces of electrodynamics 5.10 The almost Kählerian model of an Hamilton space Cartan spaces 6.1 The notion of Cartan space 6.2 Properties of the fundamental function K of Cartan space C n 6.3 Canonical nonlinear connection of a Cartan space 6.4 The canonical metrical connection 6.5 Structure equations. Bianchi identities 6.6 Special N-linear connections of Cartan space C n 6.7 Some special Cartan spaces 6.8 Parallelism in Cartan space. Horizontal and vertical paths VII 6.9 The almost Kählerian model of a Cartan space 7 The duality between Lagrange and Hamilton spaces 7.1 The Lagrange-Hamilton duality 7.2 -dual nonlinear connections 7.3 -du ald-connections 7.4 The Finsler-Cartan -duality 7.5 Berwald connection for Cartan spaces. Landsberg and Berwald spaces. Locally Minkowski spaces 7.6 Applications of the -duality 8 Symplectic transformations of the differential geometry of T*M 8.1 Connection-pairs on cotangent bundle 8.2 Special Linear Connections on T*M 8.3 The homogeneous case 8.4 f-related connection-pairs 8.5 f-related connections 8.6 The geometry of a homogeneous contact transformation 8.7 Examples 9 The dual bundle of a k-osculator bundle 9.1 The (T* k M,K* k ,M) bundle 9.2 The dual of the 2-osculator bundle 9.3 Dual semisprays on . 9.4 Homogeneity 9.5 Nonlinear connections 9.6 Distinguished vector and covector fields 9.7 Lie brackets. Exterior differentials 9.8 The almost product structure P. The almost contact structure . . . 9.9 The Riemannian structures on T* 2 M

The Kosambi-Cartan-Chern (KCC) theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach one describes the evolution of a dynamical system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a non-linear connection and a Berwald type connection to the dynamical system, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In the present paper we review the basic mathematical formalism of the KCC theory, and present some specific applications of this method in general relativity, cosmology and astrophysics. In particular we investigate the Jacobi stability of the general relativistic static fluid sphere with a linear barotropic equation of state, of the vacuum in the brane world models, of a dynamical dark energy model, and of the Lane-Emden equation, respectively. It is shown that the Jacobi stability analysis offers a powerful and simple method for constraining the physical properties of different systems, described by second order differential equations.

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The present paper studies globally defined Kropina metrics as solutions of the Zermelo's navigation problem. Moreover, we characterize the Kropina metrics of constant flag curvature showing that up to local isometry, there are only two model spaces of them: the Euclidean space and the odd-dimensional spheres. * Mathematics Subject Classification (2010) : 53C60, 53C22. †

Recent whole-genome studies and in-depth expressed sequence tag (EST) analyses have identified most of the developmentally relevant genes in the urochordate, Ciona intestinalis. In this study, we made use of a large-scale oligo-DNA microarray to further investigate and identify genes with specific or correlated expression profiles, and we report global gene expression profiles for about 66% of all the C. intestinalis genes that are expressed during its life cycle. We succeeded in categorizing the data set into 5 large clusters and 49 sub-clusters based on the expression profile of each gene. This revealed the higher order of gene expression profiles during the developmental and aging stages. Furthermore, a combined analysis of microarray data with the EST database revealed the gene groups that were expressed at a specific stage or in a specific organ of the adult. This study provides insights into the complex structure of ascidian gene expression, identifies co-expressed gene groups and marker genes and makes predictions for the biological roles of many uncharacterized genes. This large-scale oligo-DNA microarray for C. intestinalis should facilitate the understanding of global gene expression and gene networks during the development and aging of a basal chordate.