120,490 research outputs found
Vagueness and Roughness
The paper proposes a new formal approach to vagueness and vague sets taking inspirations from Pawlak’s rough set theory. Following a brief introduction to the problem of vagueness, an approach to conceptualization and representation of vague knowledge is presented from a number of different perspectives: those of logic, set theory, algebra, and computer science. The central notion of the vague set, in relation to the rough set, is defined as a family of sets approximated by the so called lower and upper limits. The family is simultaneously considered as a family of all denotations of sharp terms representing a suitable vague term, from the agent’s point of view. Some algebraic operations on vague sets and their properties are defined. Some important conditions concerning the membership relation for vague sets, in connection to Blizard’s multisets and Zadeh’s fuzzy sets, are established as well. A classical outlook on a logic of vague sentences (vague logic) based on vague sets is also discussed
Differential form representation of stochastic electromagnetic fields
In this work, we revisit the theory of stochastic electromagnetic fields
using exterior differential forms. We present a short overview as well as a
brief introduction to the application of differential forms in
electromagnetic theory. Within the framework of exterior calculus we derive
equations for the second order moments, describing stochastic electromagnetic
fields. Since the resulting objects are continuous quantities in space, a
discretization scheme based on the Method of Moments (MoM) is introduced for
numerical treatment. The MoM is applied in such a way, that the notation of
exterior calculus is maintained while we still arrive at the same set of
algebraic equations as obtained for the case of formulating the theory using
the traditional notation of vector calculus. We conclude with an analytic
calculation of the radiated electric field of two Hertzian dipole, excited by uncorrelated random currents
Brief introduction to tropical geometry
The paper consists of lecture notes for a mini-course given by the authors at
the G\"okova Geometry \& Topology conference in May 2014. We start the
exposition with tropical curves in the plane and their applications to problems
in classical enumerative geometry, and continue with a look at more general
tropical varieties and their homology theories.Comment: 75 pages, 37 figures, many examples and exercise
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