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Register a new userFiltrations de stratification de quelques varietes de Shimura simples
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Pascal Boyer
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We define and study new filtrations called of stratification of a perverse sheaf on a scheme; beside the cases of the weight or monodromy filtrations, these filtrations are available whatever are the ring of coefficients. We illustrate these constructions in the geometric situation of the simple unitary Shimura varieties of Harris and Taylors book for the perverse sheaves of Harris-Taylor and the complex of vanishing cycles, introduced and studied in my 2009 paper at inventiones. In the situation studied in loc. cit., we show how to use these filtrations to simplify the principal step of this paper; the cases of finite field or ring of integer of a local field will be studied in the next published paper.
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We introduce a notion of positive pair of contact structures on a 3-manifold which generalizes a previous definition of Eliashberg-Thurston and Mitsumatsu. Such a pair gives rise to a locally integrable plane field $lambda$. We prove that if $lambda$ is uniquely integrable and if both structures of the pair are tight, then the integral foliation of $lambda$ doesnt contain any Reeb component whose core curve is homologous to zero. Moreover, the ambient manifold carries a Reebless foliation. We also show a stability theorem `a la Reeb for positive pairs of tight contact structures.
We give some remarks on some manifolds K3 surfaces, Complex projective spaces, real projective space and Torus and the classification of two dimensional Riemannian surfaces, Green functions and the Stokes formula. We also, talk about traces of Sobolev spaces, the distance function, the notion of degree and a duality theorem, the variational formulation and conformal map in dimension 2, the metric on the boundary of a Lipschitz domain and polar geodesic coordinates and the Gauss-Bonnet formula and the positive mass theorem in dimension 3 and in the locally conformally flat case. And the Ricci flow. And fields and their relation to the equations.
The legacy of Jordans canonical form on Poincares algebraic practices. This paper proposes a transversal overview on Henri Poincares early works (1878-1885). Our investigations start with a case study of a short note published by Poincare on 1884 : Sur les nombres complexes. In the perspective of todays mathematical disciplines - especially linear algebra -, this note seems completely isolated in Poincares works. This short paper actually exemplifies that the categories used today for describing some collective organizations of knowledge fail to grasp both the collective dimensions and individual specificity of Poincares work. It also highlights the crucial and transversal role played in Poincares works by a specific algebraic practice of classification of linear groups by reducing the analytical representation of linear substitution to their Jordans canonical forms. We then analyze in detail this algebraic practice as well as the roles it plays in Poincares works. We first provide a micro-historical analysis of Poincares appropriation of Jordans approach to linear groups through the prism of the legacy of Hermites works on algebraic forms between 1879 and 1881. This mixed legacy illuminates the interrelations between all the papers published by Poincare between 1878 and 1885 ; especially between some researches on algebraic forms and the development of the theory of Fuchsian functions. Moreover, our investigation sheds new light on how the notion of group came to play a key role in Poincares approach. The present paper also offers a historical account of the statement by Jordan of his canonical form theorem. Further, we analyze how Poincare transformed this theorem by appealing to Hermites
Let ${rm F}$ be a rank-2 semi-stable sheaf on the projective plane, with Chern classes $c_{1}=0,c_{2}=n$. The curve $beta_{rm F}$ of jumping lines of ${rm F}$, in the dual projective plane, has degree $n$. Let ${rm M}_{n}$ be the moduli space of equivalence classes of semi-stables sheaves of rank 2 and Chern classes $(0,n)$ on the projective plane and ${cal C}_{n}$ be the projective space of curves of degree $n$ in the dual projective plane. The Barth morphism $$beta: {rm M}_{n}longrightarrow{cal C}_{n}$$ associates the point $beta_{rm F}$ to the class of the sheaf ${rm F}$. We prove that this morphism is generically injective for $ngeq 4.$ The image of $beta$ is a closed subvariety of dimension $4n-3$ of ${cal C}_{n}$; as a consequence of our result, the degree of this image is given by the Donaldson number of index $4n-3$ of the projective plane.
The effect of leverage on liquidity is a tool for analysing the level of liquidity for a given production process. It measures the sensitivity of the level of liquidity that results from changes in the volume of production and unit operating margin. A commercial activity is liquid at the moment when all costs are covered by revenues. However, not all of the cash flows from production influence liquidity levels. The estimated costs do not directly influence the level of liquidity. Therefore, two indicators are to be taken into consideration: the elasticity of ongoing liquidity - fixed costs include estimated costs, and, the elasticity of immediate liquidity - fixed costs only include costs that are payable. The coefficients of leverage of ongoing liquidity and of leverage of immediate liquidity in relation to the operating margin have a behaviour that is identical to that calculated in relation to production. If the productive capacity remains unchanged, the regulation of the change in elasticity of the costs and of its influence on the unitary operating margin is the sole parameter available to the entrepreneur to maintain the liquidity of the company at the desired level. But, if the productive capacity is variable, the entrepreneur can use the volume of sales to control liquidity but then the transformation of the production process must be analysed so as to adjust the relevant elements to retain in the operating structure the degree of liquidity wished for.
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