245 research outputs found
Covering rough sets based on neighborhoods: An approach without using neighborhoods
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
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
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
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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