38,247 research outputs found
Matroidal approaches to rough sets via closure operators
AbstractThis paper studies rough sets from the operator-oriented view by matroidal approaches. We firstly investigate some kinds of closure operators and conclude that the Pawlak upper approximation operator is just a topological and matroidal closure operator. Then we characterize the Pawlak upper approximation operator in terms of the closure operator in Pawlak matroids, which are first defined in this paper, and are generalized to fundamental matroids when partitions are generalized to coverings. A new covering-based rough set model is then proposed based on fundamental matroids and properties of this model are studied. Lastly, we refer to the abstract approximation space, whose original definition is modified to get a one-to-one correspondence between closure systems (operators) and concrete models of abstract approximation spaces. We finally examine the relations of four kinds of abstract approximation spaces, which correspond exactly to the relations of closure systems
The Bing-Borsuk and the Busemann Conjectures
We present two classical conjectures concerning the characterization of
manifolds: the Bing Borsuk Conjecture asserts that every -dimensional
homogeneous ANR is a topological -manifold, whereas the Busemann Conjecture
asserts that every -dimensional -space is a topological -manifold. The
key object in both cases are so-called {\it generalized manifolds}, i.e. ENR
homology manifolds. We look at the history, from the early beginnings to the
present day. We also list several open problems and related conjectures.Comment: We have corrected three small typos on pages 8 and
Generalized Sums over Histories for Quantum Gravity I. Smooth Conifolds
This paper proposes to generalize the histories included in Euclidean
functional integrals from manifolds to a more general set of compact
topological spaces. This new set of spaces, called conifolds, includes
nonmanifold stationary points that arise naturally in a semiclasssical
evaluation of such integrals; additionally, it can be proven that sequences of
approximately Einstein manifolds and sequences of approximately Einstein
conifolds both converge to Einstein conifolds. Consequently, generalized
Euclidean functional integrals based on these conifold histories yield
semiclassical amplitudes for sequences of both manifold and conifold histories
that approach a stationary point of the Einstein action. Therefore sums over
conifold histories provide a useful and self-consistent starting point for
further study of topological effects in quantum gravity. Postscript figures
available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file
gen1.ps.Comment: 81pp., plain TeX, To appear in Nucl. Phys.
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
Approximation results for a general class of Kantorovich type operators
We introduce and study a family of integral operators in the Kantorovich
sense for functions acting on locally compact topological groups. We obtain
convergence results for the above operators with respect to the pointwise and
uniform convergence and in the setting of Orlicz spaces with respect to the
modular convergence. Moreover, we show how our theory applies to several
classes of integral and discrete operators, as the sampling, convolution and
Mellin type operators in the Kantorovich sense, thus obtaining a simultaneous
approach for discrete and integral operators. Further, we derive our general
convergence results for particular cases of Orlicz spaces, as spaces,
interpolation spaces and exponential spaces. Finally we construct some concrete
example of our operators and we show some graphical representations.Comment: 23 pages, 5 figure
The Generalized Hartle-Hawking Initial State: Quantum Field Theory on Einstein Conifolds
Recent arguments have indicated that the sum over histories formulation of
quantum amplitudes for gravity should include sums over conifolds, a set of
histories with more general topology than that of manifolds. This paper
addresses the consequences of conifold histories in gravitational functional
integrals that also include scalar fields. This study will be carried out
explicitly for the generalized Hartle-Hawking initial state, that is the
Hartle-Hawking initial state generalized to a sum over conifolds. In the
perturbative limit of the semiclassical approximation to the generalized
Hartle-Hawking state, one finds that quantum field theory on Einstein conifolds
is recovered. In particular, the quantum field theory of a scalar field on de
Sitter spacetime with spatial topology is derived from the generalized
Hartle-Hawking initial state in this approximation. This derivation is carried
out for a scalar field of arbitrary mass and scalar curvature coupling.
Additionally, the generalized Hartle-Hawking boundary condition produces a
state that is not identical to but corresponds to the Bunch-Davies vacuum on
de Sitter spacetime. This result cannot be obtained from the original
Hartle-Hawking state formulated as a sum over manifolds as there is no Einstein
manifold with round boundary.Comment: Revtex 3, 31 pages, 4 epsf figure
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