862 research outputs found
Quantale Modules and their Operators, with Applications
The central topic of this work is the categories of modules over unital
quantales. The main categorical properties are established and a special class
of operators, called Q-module transforms, is defined. Such operators - that
turn out to be precisely the homomorphisms between free objects in those
categories - find concrete applications in two different branches of image
processing, namely fuzzy image compression and mathematical morphology
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
Homomorphisms between fuzzy information systems revisited
Recently, Wang et al. discussed the properties of fuzzy information systems
under homomorphisms in the paper [C. Wang, D. Chen, L. Zhu, Homomorphisms
between fuzzy information systems, Applied Mathematics Letters 22 (2009)
1045-1050], where homomorphisms are based upon the concepts of consistent
functions and fuzzy relation mappings. In this paper, we classify consistent
functions as predecessor-consistent and successor-consistent, and then proceed
to present more properties of consistent functions. In addition, we improve
some characterizations of fuzzy relation mappings provided by Wang et al.Comment: 10 page
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
Retraction and Generalized Extension of Computing with Words
Fuzzy automata, whose input alphabet is a set of numbers or symbols, are a
formal model of computing with values. Motivated by Zadeh's paradigm of
computing with words rather than numbers, Ying proposed a kind of fuzzy
automata, whose input alphabet consists of all fuzzy subsets of a set of
symbols, as a formal model of computing with all words. In this paper, we
introduce a somewhat general formal model of computing with (some special)
words. The new features of the model are that the input alphabet only comprises
some (not necessarily all) fuzzy subsets of a set of symbols and the fuzzy
transition function can be specified arbitrarily. By employing the methodology
of fuzzy control, we establish a retraction principle from computing with words
to computing with values for handling crisp inputs and a generalized extension
principle from computing with words to computing with all words for handling
fuzzy inputs. These principles show that computing with values and computing
with all words can be respectively implemented by computing with words. Some
algebraic properties of retractions and generalized extensions are addressed as
well.Comment: 13 double column pages; 3 figures; to be published in the IEEE
Transactions on Fuzzy System
Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series
Topology based analysis of time-series data from dynamical systems is
powerful: it potentially allows for computer-based proofs of the existence of
various classes of regular and chaotic invariant sets for high-dimensional
dynamics. Standard methods are based on a cubical discretization of the
dynamics and use the time series to construct an outer approximation of the
underlying dynamical system. The resulting multivalued map can be used to
compute the Conley index of isolated invariant sets of cubes. In this paper we
introduce a discretization that uses instead a simplicial complex constructed
from a witness-landmark relationship. The goal is to obtain a natural
discretization that is more tightly connected with the invariant density of the
time series itself. The time-ordering of the data also directly leads to a map
on this simplicial complex that we call the witness map. We obtain conditions
under which this witness map gives an outer approximation of the dynamics, and
thus can be used to compute the Conley index of isolated invariant sets. The
method is illustrated by a simple example using data from the classical H\'enon
map.Comment: laTeX, 9 figures, 32 page
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