Andreev's Theorem on hyperbolic polyhedra

In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev's Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing $C$ with the assigned dihedral angles. Andreev's Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Andreev's Theorem is both an interesting statement about the geometry of hyperbolic 3-dimensional space, as well as a fundamental tool used in the proof for Thurston's Hyperbolization Theorem for 3-dimensional Haken manifolds. It is also remarkable to what level the proof of Andreev's Theorem resembles (in a simpler way) the proof of Thurston. We correct a fundamental error in Andreev's proof of existence and also provide a readable new proof of the other parts of the proof of Andreev's Theorem, because Andreev's paper has the reputation of being ``unreadable''.Comment: To appear les Annales de l'Institut Fourier. 47 pages and many figures. Revision includes significant modification to section 4, making it shorter and more rigorous. Many new references include

Future research to underpin successful peste des petits ruminants virus (PPRV) eradication

Peste des petits ruminants virus (PPRV) is a significant pathogen of small ruminants and is prevalent in much of Africa, the Near and Middle East and Asia. Despite the availability of an efficacious and cheap live-attenuated vaccine, the virus has continued to spread, with its range stretching from Morocco in the west to China and Mongolia in the east. Some of the world’s poorest communities rely on small ruminant farming for subsistence and the continued endemicity of PPRV is a constant threat to their livelihoods. Moreover, PPRV’s effects on the world’s population are felt broadly across many economic, agricultural and social situations. This far-reaching impact has prompted the Food and Agriculture Organization of the United Nations (FAO) and the World Organisation for Animal Health (OIE) to develop a global strategy for the eradication of this virus and its disease. PPRV is a morbillivirus and, given the experience of these organizations in eradicating the related rinderpest virus, the eradication of PPRV should be feasible. However, there are many critical areas where basic and applied virological research concerning PPRV is lacking. The purpose of this review is to highlight areas where new research could be performed in order to guide and facilitate the eradication programme. These areas include studies on disease transmission and epidemiology, the existence of wildlife reservoirs and the development of next-generation vaccines and diagnostics. With the support of the international virology community, the successful eradication of PPRV can be achieved

Tessellations and Pattern Formation in Plant Growth and Development

The shoot apical meristem (SAM) is a dome-shaped collection of cells at the apex of growing plants from which all above-ground tissue ultimately derives. In Arabidopsis thaliana (thale cress), a small flowering weed of the Brassicaceae family (related to mustard and cabbage), the SAM typically contains some three to five hundred cells that range from five to ten microns in diameter. These cells are organized into several distinct zones that maintain their topological and functional relationships throughout the life of the plant. As the plant grows, organs (primordia) form on its surface flanks in a phyllotactic pattern that develop into new shoots, leaves, and flowers. Cross-sections through the meristem reveal a pattern of polygonal tessellation that is suggestive of Voronoi diagrams derived from the centroids of cellular nuclei. In this chapter we explore some of the properties of these patterns within the meristem and explore the applicability of simple, standard mathematical models of their geometry.Comment: Originally presented at: "The World is a Jigsaw: Tessellations in the Sciences," Lorentz Center, Leiden, The Netherlands, March 200

Six Peaks Visible in the Redshift Distribution of 46,400 SDSS Quasars Agree with the Preferred Redshifts Predicted by the Decreasing Intrinsic Redshift Model

The redshift distribution of all 46,400 quasars in the Sloan Digital Sky Survey (SDSS) Quasar Catalog III, Third Data Release, is examined. Six Peaks that fall within the redshift window below z = 4, are visible. Their positions agree with the preferred redshift values predicted by the decreasing intrinsic redshift (DIR) model, even though this model was derived using completely independent evidence. A power spectrum analysis of the full dataset confirms the presence of a single, significant power peak at the expected redshift period. Power peaks with the predicted period are also obtained when the upper and lower halves of the redshift distribution are examined separately. The periodicity detected is in linear z, as opposed to log(1+z). Because the peaks in the SDSS quasar redshift distribution agree well with the preferred redshifts predicted by the intrinsic redshift relation, we conclude that this relation, and the peaks in the redshift distribution, likely both have the same origin, and this may be intrinsic redshifts, or a common selection effect. However, because of the way the intrinsic redshift relation was determined it seems unlikely that one selection effect could have been responsible for both.Comment: 12 pages, 12 figure, accepted for publication in the Astrophysical Journa

Development and Testing of Operational Dual-Polarimetric Radar Based Lightning Initiation Forecast Techniques

Lightning is one of Earth s natural dangers, destructive not only to life but also physical property. According to the National Weather Service, there are on average 58 lightning fatalities each year, with over 300 related injuries (NWS 2010). The ability to forecast lightning is critical to a host of activities ranging from space vehicle launch operations to recreational sporting events. For example a single lightning strike to a Space Shuttle could cause billions of dollars of damage and possible loss of life. While forecasting that provides longer lead times could provide sporting officials with more time to respond to possible threatening weather events, thus saving the lives of player and bystanders. Many researchers have developed and tested different methods and tools of first flash forecasting, however few have done so using dual-polarimetric radar variables and products on an operational basis. The purpose of this study is to improve algorithms for the short-term prediction of lightning initiation through development and testing of operational techniques that rely on parameters observed and diagnosed using C-band dual-polarimetric radar