The road to deterministic matrices with the restricted isometry property

The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.Comment: 24 page

The last dinosaurs of Brazil: The Bauru Group and its implications for the end-Cretaceous mass extinction

ABSTRACT The non-avian dinosaurs died out at the end of the Cretaceous, ~66 million years ago, after an asteroid impact. The prevailing hypothesis is that the effects of the impact suddenly killed the dinosaurs, but the poor fossil record of latest Cretaceous (Campanian-Maastrichtian) dinosaurs from outside Laurasia (and even more particularly, North America) makes it difficult to test specific extinction scenarios. Over the past few decades, a wealth of new discoveries from the Bauru Group of Brazil has revealed a unique window into the evolution of terminal Cretaceous dinosaurs from the southern continents. We review this record and demonstrate that there was a diversity of dinosaurs, of varying body sizes, diets, and ecological roles, that survived to the very end of the Cretaceous (Maastrichtian: 72-66 million years ago) in Brazil, including a core fauna of titanosaurian sauropods and abelisaurid and carcharodontosaurid theropods, along with a variety of small-to-mid-sized theropods. We argue that this pattern best fits the hypothesis that southern dinosaurs, like their northern counterparts, were still diversifying and occupying prominent roles in their ecosystems before the asteroid suddenly caused their extinction. However, this hypothesis remains to be tested with more refined paleontological and geochronological data, and we give suggestions for future work

Evolução da vassoura-de-bruxa e avaliação da resistência em progênies de cupuaçuzeiro Evolution of the witch's broom disease and evaluation of resistance in cupuassu progenies

Com o objetivo de conhecer a evolução da doença vassoura-de-bruxa, a taxa de segregação e estimar parâmetros genéticos, foi conduzido um experimento com 21 progênies de irmãos completos de cupuaçuzeiro, instalado em Belém, Pará. As progênies foram obtidas pelo cruzamento controlado de clones resistentes com clones resistentes, clones resistentes com clones suscetíveis e, clones suscetíveis com clones suscetíveis. Foram avaliadas também três progênies de meios-irmãos como testemunhas. Os experimentos foram avaliados ao nível de indivíduos, e as análises foram conduzidas via metodologia de modelos lineares mistos (procedimento REMl/BlUP), como delineamento em blocos incompletos desbalanceados, com tratamentos comuns entre dois experimentos. As variáveis avaliadas foram: percentagem de plantas resistentes ao ataque da doença no ramo, inflorescência, fruto imaturo e maduro, no período de 2002 a 2007. Os resultados mostraram que a emissão de vassoura vegetativa é especialmente importante entre julho e setembro. A evolução da doença evidencia que as podas fitossanitárias devem ser realizadas no final da safra, nos meses de maio/junho, e repassadas em setembro/outubro. A população estudada oferece excelente possibilidade de seleção e ganho genético, respaldada pelos elevados índices de variabilidade genética e herdabilidade dos caracteres de resistência. As progênies segregaram tanto para os sintomas de vassoura nos ramos, nas inflorescências, como nos frutos imaturos e maduros. Para o controle integrado desta doença, os resultados mostram a importância da associação de materiais genéticos resistentes ou moderadamente resistentes à vassoura-de-bruxa com as podas fitossanitárias.<br>This paper aimed to study the evolution and individual segregation of the witch's broom disease and estimate the genetic parameters in trials with 21 full sib families of Cupuaçú established in Belém, Pará. The progenies were obtained through control pollination involving resistant and susceptible parents in all three combinations. Three half-sib families were also evaluated as checks. The experiments were evaluated at individual level and all the analyses were based on the mixed model methodology (REMl/ BlUP procedure) with unbalanced incomplete blocks design and common checks across trials. The evaluated traits comprised the percentage of resistant plants concerning the disease on the branches, on the flowers and on the mature and immature fruits from 2002 to 2007. The results showed that the disease is especially important between July and September. The disease evolution showed that pruning against the disease must be conducted at the end of the productive season in May/June and also in September/October. The breeding population can be successfully improved due to the high levels of genetic variability and heritability associated to the resistance traits. There were genetic segregations for all the traits. The integrated control of the disease, associating resistant genotypes and pruning is recommended