245 research outputs found

    Covering rough sets based on neighborhoods: An approach without using neighborhoods

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    Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighborhoods can be defined equivalently without using neighborhoods. To analyze the coverings themselves, we introduce unary and composition operations on coverings. A notion of homomorphismis provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.Comment: 13 pages; to appear in International Journal of Approximate Reasonin

    Local inequalities for plurisubharmonic functions

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    The main objective of this paper is to prove a new inequality for plurisubharmonic functions estimating their supremum over a ball by their supremum over a measurable subset of the ball. We apply this result to study local properties of polynomial, algebraic and analytic functions. The paper has much in common with an earlier paper of the author.Comment: 23 pages, published versio

    Flows on homogeneous spaces and Diophantine approximation on manifolds

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    We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. Sprindzhuk in 1964. We also prove several related hypotheses of A. Baker and V. Sprindzhuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on non-divergence of unipotent flows on the space of lattices.Comment: 19 pages. To appear in Annals of Mathematic

    Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: Aleksandrov solutions

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    We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge-Ampere equation using finite dimensional spaces of piecewise polynomial C0 or C1 functions. Standard discretizations of the type considered in this paper have been previous analyzed in the case the equation has a smooth solution and numerous numerical evidence of convergence were given in the case of non smooth solutions. Our convergence result is valid for non smooth solutions, is given in the setting of Aleksandrov solutions, and consists in discretizing the equation in a subdomain with the boundary data used as an approximation of the solution in the remaining part of the domain. Our result gives a theoretical validation for the use of a non monotone finite element method for the Monge-Amp\`ere equation
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