Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional II: Smooth case

The paper is centered around a new proof of the infinitesimal rigidity of smooth closed surfaces with everywhere positive Gauss curvature. We use a reformulation that replaces deformation of an embedding by deformation of the metric inside the body bounded by the surface. The proof is obtained by studying derivatives of the Hilbert-Einstein functional with boundary term. This approach is in a sense dual to proving the Gauss infinitesimal rigidity, that is rigidity with respect to the Gauss curvature parametrized by the Gauss map, by studying derivatives of the volume bounded by the surface. We recall that Blaschke's classical proof of the infinitesimal rigidity is also related to the Gauss infinitesimal rigidity, but in a different way: while Blaschke uses Gauss rigidity of the same surface, we use the Gauss rigidity of the polar dual. In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volume, as well as between both kinds of rigidity. We also indicate directions for future research, including the infinitesimal rigidity of convex cores of hyperbolic 3--manifolds.Comment: 60 page

Gamma spectrometric method to control activity and nuclide composition of gaseous radioactive waste formed at operation of nuclear power plants

Gamma spectrometric method was developed to monitor continuously and on line radioactivity and nuclide composition of inert radioactive gases, radioactive aerosols and iodine in gas aerosol emissions from power reactor facilities. This method is based on continuous representative sampling of gas aerosol samples and quasi-continuous automated recording of nuclide composition and radioactive material emission rate. Low detectable level of the method is about 0,1 Bq/m3, highest detectable level for noble gases (Ar_41, isotopes Xe and Kr) is about 105 Bq/m

Semi-empirical control method of solid wastes of medium and high activity

Semi-empirical method has been developed to monitor medium and high level solid radioactive wastes based on direct measurement of wastes radioactivity and nuclide composition in lorry body. The energy range of measurements was from 80 tо 3000 keV. The radioactive waste activity was from 106 to 1012 Bq. The proposed method was certified and measurement of basic errors were determined that not exceeding 60 %

Non-rigidity of spherical inversive distance circle packings

We give a counterexample of Bowers-Stephenson's conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances.Comment: 6 pages, one pictur

Rigid ball-polyhedra in Euclidean 3-space

A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a standard ball-polyhedron if its vertex-edge-face structure is a lattice (with respect to containment). To each edge of a ball-polyhedron one can assign an inner dihedral angle and say that the given ball-polyhedron is locally rigid with respect to its inner dihedral angles if the vertex-edge-face structure of the ball-polyhedron and its inner dihedral angles determine the ball-polyhedron up to congruence locally. The main result of this paper is a Cauchy-type rigidity theorem for ball-polyhedra stating that any simple and standard ball-polyhedron is locally rigid with respect to its inner dihedral angles.Comment: 11 pages, 2 figure

Gamma spectrometric method to control activity and nuclide composition of gaseous radioactive waste formed at operation of nuclear power plants

Gamma spectrometric method was developed to monitor continuously and on line radioactivity and nuclide composition of inert radioactive gases, radioactive aerosols and iodine in gas aerosol emissions from power reactor facilities. This method is based on continuous representative sampling of gas aerosol samples and quasi-continuous automated recording of nuclide composition and radioactive material emission rate. Low detectable level of the method is about 0,1 Bq/m3, highest detectable level for noble gases (Ar_41, isotopes Xe and Kr) is about 105 Bq/m

Gamma-spectrometric control method of activity and nuclide composition of low-active solid radioactive waste

The gamma-spectrometric control method of low-active solid radioactive waste, based on direct measurement of activity and nuclide composition has been developed. The measurements were carried out in the geometry of standard steel container of 200 l. volume, where low-active wastes were placed. To take into account distribution non-homogeneities of solid waste over the geometry measured a special rotating platform was used, the low-active radioactive wastes being placed on it. Metrological certification was performed and the main errors of this method for 95 % of confidence probability were determined

Shapes of polyhedra, mixed volumes and hyperbolic geometry

We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to π/2

Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes

We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of generalized convex polytopes with positive singular curvature. This Hessian is shown to be equal to the Hessian of the volume of the dual generalized polyhedron. We prove the non-degeneracy by applying the technique used in the proof of Alexandrov-Fenchel inequality. Our construction of a convex polytope from a given metric is implemented in a computer program