An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

E7(7) formulation of N=2 backgrounds

In this paper we reformulate N=2 supergravity backgrounds arising in type II string theory in terms of quantities transforming under the U-duality group E7(7). In particular we combine the Ramond--Ramond scalar degrees of freedom together with the O(6,6) pure spinors which govern the Neveu-Schwarz sector by considering an extended version of generalised geometry. We give E7(7)-invariant expressions for the Kahler and hyperkahler potentials describing the moduli space of vector and hypermultiplets, demonstrating that both correspond to standard E7(7) coset spaces. We also find E7(7) expressions for the Killing prepotentials defining the scalar potential, and discuss the equations governing N=1 vacua in this formalism.Comment: 40 pages, final version to appear in JHE

Efeito do fósforo, fumigação do substrato e fungo micorrízico arbuscular sobre o crescimento de plantas de mamoeiro Effect of phosphorus, soil fumigation and mycorrhizal colonization on papaya growth

Estudou-se o efeito da inoculação com o fungo micorrízico arbuscular (FMA), Glomus macrocarpum, da fumigação do substrato e da adição de fósforo solúvel (60, 120, 240 e 480 mg kg-1 de P no solo) sobre as variáveis altura, número de folhas e diâmetro do caule de plantas de mamoeiro cv. Sunrise Solo.O FMA edoses crescentes de fósforo, isoladamente, exerceram efeitos significativos sobre essas variáveis. Não houve efeito significativo do fator fumigação do substrato. O efeito da inoculação foi mais acentuado no tratamento com adição de 60 mg kg-1 de P no solo. A inoculação com G. macrocarpum reduziu a necessidade de fósforo para o mamoeiro, tanto que as variáveis estudadas em plantas inoculadas na ausência de adubação fosfática não diferiram de plantas não inoculadas em substrato adicionado de mais de 240 mg kg-1 de P no solo.<br>The effects of inoculation with arbuscular mycorrhizal fungus (AMF), substratum fumigation, and addition of soluble phosphate on papaya plant cultivar "Sunrise Solo" parameters (height, leaf number, and stem diameter) were studied. Separate treatments with AMF inoculation (Glomus macrocarpum) and increasing phosphorus levels (60, 120, 240, and 480 mg kg-1 of P in soil) influenced these parameters significantly, while the factor substratum fumigation had no significant effect. The inoculation effect was most expressive when 60 mg kg-1 of P in soil was added. Such was the reduction in phosphorus requirement of papaya by Glomus macrocarpum inoculation, that the studied parameters between inoculated plants without phosphate addition and not-inoculated ones in substratum enriched with over 240 mg kg-1 of P in soil did not differ

Dual gravity and matter

We consider the problem of finding a dual formulation of gravity in the presence of non-trivial matter couplings. In the absence of matter a dual graviton can be introduced only for linearised gravitational interactions. We show that the coupling of linearised gravity to matter poses obstructions to the usual construction and comment on possible resolutions of this difficulty.SCOPUS: ar.jinfo:eu-repo/semantics/publishe