Globally Linked Pairs of Vertices in Equivalent Realizations of Graphs

A two-dimensional framework (G,p) is a graph G = (V,E) together with a map p: V → ℝ². We view (G,p) as a straight line realization of G in ℝ². Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u,v} is globally linked in G if %and for all equivalent frameworks (G,q), the distance between the points corresponding to u and v is the same in all pairs of equivalent generic realizations of G. The graph G is globally rigid if all of its pairs of vertices are globally linked. We extend the characterization of globally rigid graphs given by the first two authors [13] by characterizing globally linked pairs in M-connected graphs, an important family of rigid graphs. As a byproduct we simplify the proof of a result of Connelly [6] which is a key step in the characterization of globally rigid graphs. We also determine the number of distinct realizations of an M-connected graph, each of which is equivalent to a given generic realization. Bounds on this number for minimally rigid graphs were obtained by Borcea and Streinu in [3].     

Electronic inhomogeneities,electron-lattice and pairing interactions in high-T c superconductors

The presence of electronic inhomogeneities strongly reduces the screening of the electron-ion interaction in high-temperature superconductors. This implies the existence of an non-totally screened long-range contribution to the electron-lattice coupling and opens an additional channel for the formation of copper pairs. We calculate the superconducting order parameter taking into account a) the longrange and the short-range parts of the electron-lattice interaction and b) the Coulomb repulsion between charge-carriers. We show that whereas the long-range electron-lattice coupling determines the anisotropy of the order parameter, the Coulomb repulsion and the short-range interactions determine its symmetry. Thus, different high-T _c superconductors may have s- or d-wave symmetry, depending on the relative strength of the interactions involved in the pairing.     

Fabrication of magnetically separable mesostructured silica with an open pore system

Magnetically separable mesostructured silica with an unobstructed pore system was fabricated through the deposition of cobalt nanoparticles on the outer surface of the submicron-sized silica particles. These cobalt nanoparticles were further protected by a nanometer-thick carbon shell against acid erosion. Due to the fact that the magnetic particles are grafted on the outer surface of the porous silica, the pores are still accessible for further modification, which could widen the application range of porous silica.