Narrow Results

We present a novel $C^0$-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of $C^0$-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from $J$-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a $C^0$-characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of $C^0$-rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape.

In many image analysis applications, such as image retrieval, the shape of an object is of primary importance. In this paper, a new shape descriptor, namely the Normalized Wavelet Descriptor (NWD), which is a generalization and extension of the Wavelet Descriptor (WD), is introduced. The NWD is compared to the Fourier Descriptor (FD), which in image retrieval experiments conducted by Zhang and Lu, outperformed even the Curvature Scale Space Descriptor (CSSD). Image retrieval experiments have been conducted using a dataset containing 2D-contours of 1400 objects extracted from the standard MPEG7 database. For the chosen dataset, our experimental results show that the NWD outperforms the FD.

The paper deals with a wind wave, which is a day-ahead wave healing of a power grid with lots of wind farms and a transmission fault. This wave healing restores synchronization and shape of a power grid. The synchronization restoration is achieved by Casimir force, adjusted by an energy spike. The shape restoration is achieved via an energy spike, adjusted by the change of the free energy of the grid.

This paper contains selected topics from four lectures given at the Abdus Salam International Centre for Theoretical Physics in May 2009. We introduce the study of the influence of knotting and linking on the spatial characteristics of linear and ring polymer chains with examples of scientific interest. We describe a few basic concepts of the geometry and topology of knots and measures of the spatial shape of open and closed polymer chains. We then present some fundamental mathematical results concerning them. Next we discuss random sampling methods of collections of open and closed chains that are employed to provide estimates of the spatial properties of the chains. Finally, we discuss implementations of the sampling algorithms, survey consequences of theoretical and experimental results, and discuss some interesting problems deserving further research.