14,404 research outputs found
Fuzzy Galois connections on fuzzy sets
In fairly elementary terms this paper presents how the theory of preordered
fuzzy sets, more precisely quantale-valued preorders on quantale-valued fuzzy
sets, is established under the guidance of enriched category theory. Motivated
by several key results from the theory of quantaloid-enriched categories, this
paper develops all needed ingredients purely in order-theoretic languages for
the readership of fuzzy set theorists, with particular attention paid to fuzzy
Galois connections between preordered fuzzy sets.Comment: 30 pages, final versio
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
Non-associative gauge theory and higher spin interactions
We give a framework to describe gauge theory on a certain class of
commutative but non-associative fuzzy spaces. Our description is in terms of an
Abelian gauge connection valued in the algebra of functions on the cotangent
bundle of the fuzzy space. The structure of such a gauge theory has many formal
similarities with that of Yang-Mills theory. The components of the gauge
connection are functions on the fuzzy space which transform in higher spin
representations of the Lorentz group. In component form, the gauge theory
describes an interacting theory of higher spin fields, which remains
non-trivial in the limit where the fuzzy space becomes associative. In this
limit, the theory can be viewed as a projection of an ordinary non-commutative
Yang-Mills theory. We describe the embedding of Maxwell theory in this extended
framework which follows the standard unfolding procedure for higher spin gauge
theories.Comment: 1+49 pages, LaTeX; references and clarifying remarks adde
Fuzzy Fluid Mechanics in Three Dimensions
We introduce a rotation invariant short distance cut-off in the theory of an
ideal fluid in three space dimensions, by requiring momenta to take values in a
sphere. This leads to an algebra of functions in position space is
non-commutative. Nevertheless it is possible to find appropriate analogues of
the Euler equations of an ideal fluid. The system still has a hamiltonian
structure. It is hoped that this will be useful in the study of possible
singularities in the evolution of Euler (or Navier-Stokes) equations in three
dimensions.Comment: Additional reference
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