Narrow Results

Watermarking techniques for vector graphics dislocate vertices in order to embed imperceptible, yet detectable, statistical features into the input data. The embedding process may result in a change of the topology of the input data, e.g., by introducing self-intersections, which is undesirable or even disastrous for many applications. In this paper we present a watermarking framework for two-dimensional vector graphics that employs conventional watermarking techniques but still provides the guarantee that the topology of the input data is preserved. The geometric part of this framework computes so-called maximum perturbation regions (MPR) of vertices. We propose two efficient algorithms to compute MPRs based on Voronoi diagrams and constrained triangulations. Furthermore, we present two algorithms to conditionally correct the watermarked data in order to increase the watermark embedding capacity and still guarantee topological correctness. While we focus on the watermarking of input formed by straight-line segments, one of our approaches can also be extended to circular arcs. We conclude the paper by demonstrating and analyzing the applicability of our framework in conjunction with two well-known watermarking techniques.

We study the transition graph of generic Hamiltonian surface flows, whose vertices are the topological equivalence classes of generic Hamiltonian surface flows and whose edges are the generic transitions. Using the transition graph, we can describe time evaluations of generic Hamiltonian surface flows (e.g., fluid phenomena) as walks on the graph. We propose a method for constructing the complete transition graph of all generic Hamiltonian flows. In fact, we construct two complete transition graphs of Hamiltonian surface flows having three and four genus elements. Moreover, we demonstrate that a lower bound on the transition distance between two Hamiltonian surface flows with any number of genus elements can be calculated by solving an integer programming problem using vector representations of Hamiltonian surface flows.

In this work we propose a new procedure on how to extract global information of a space-time. We consider a space-time immersed in a higher dimensional space and formulate the equations of Einstein through the Frobenius conditions of immersion. Through an algorithm and implementation into algebraic computing system we calculate normal vectors from the immersion to find the second fundamental form. We make an application for a static space-time with spherical symmetry. We solve Einstein's equations in the vacuum and obtain space-times with different topologies.

It is shown that topological changes in space-time are necessary to make General Relativity compatible with the Newtonian limit and to solve the hierarchy of the fundamental interactions. We detail how topology and topological changes appear in General Relativity and how it leaves an observable footprint in space-time. In cosmology we show that such topological observable is the cosmic radiation produced by the acceleration of the universe. The cosmological constant is a very particular case which occurs when the expansion of the universe into the vacuum occurs only in the direction of the cosmic time flow.

We investigate the topology of Schwarzschild's black holes through the immersion of this space-time in space of higher dimension. Through the immersions of Kasner and Fronsdal we calculate the extension of the Schwarzschilds black hole.

We present a laboratory developed in the mathematics activities during the lessons of the research project Mathematical High School at the University of Salerno. We consider a continuous location optimization problem, where an optimal location is found in a continuum on a plane, using a topological approach involving the Voronoi diagram and the Delaunay triangulation to find the equilibrium point.