Singularities of flat extensions from generic surfaces with boundaries

We solve the problem on flat extensions of a generic surface with boundary in Euclidean 3-space, relating it to the singularity theory of the envelope generated by the boundary. We give related results on Legendre surfaces with boundaries via projective duality and observe the duality on boundary singularities. Moreover we give formulae related to remote singularities of the boundary-envelope.Comment: Several typos are corrected (2010. Jan.

Global classification of curves on the symplectic plane

We consider the global symplectic classification problem of plane curves. First we give the exact classification result under symplectomorphisms, for the case of generic plane curves, namely immersions with transverse self-intersections. Then the set of symplectic classes form the symplectic moduli space which we completely describe by its global topological term. For the general plane curves with singularities, the difference between symplectomorphism and diffeomorphism classifications is clearly described by local symplectic moduli spaces of singularities and a global topological term. We introduce the symplectic moduli space of a global plane curve and the local symplectic moduli space of a plane curve singularity as quotients of mapping spaces, and we endow them with differentiable structures in a natural way.Comment: 25 pages 3 figure

THE EFFECTS OF THE TRANSPORTATION COSTS IN R&D TECHNOLOGY SECTOR ON THE ENDOGENOUS GRWOTH

The paper centers on investigating theoretically how the transportation costs of R&D technology, none of the transportation costs in final goods and intermediate inputs, affect the long-run endogenous economic growth. The basic ideas adopted in this paper are different from well-known models in the sense that the prices of R&D technology are influenced by the transportation cost in R&D technology sector and the accumulated profit of the intermediate inputs over time is equal to the price of R&D technology, and thus the transportation costs indirectly influence the endogenous growth. That is, the larger are only the transportation costs of R&D technology, the higher is the price of R&D technology and the slower is endogenous economic growth.Transportation Costs, R&D Technology, Endogenous Growth

Monge-Amp\`ere Systems with Lagrangian Pairs

The classes of Monge-Amp\`ere systems, decomposable and bi-decomposable Monge-Amp\`ere systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables $\geq 3$. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homogeneous Monge-Amp\`ere systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge-Amp\`ere systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3)