18,744 research outputs found
On Fuzzy Concepts
In this paper we try to combine two approaches. One is the theory of knowledge graphs in which concepts are represented by graphs. The other is the axiomatic theory of fuzzy sets (AFS).
The discussion will focus on the idea of fuzzy concept. It will be argued that the fuzziness of a concept in natural language is mainly due to the difference in interpretation that people give to a certain word. As different interpretations lead to different knowledge graphs, the notion of fuzzy concept should be describable in terms of sets of graphs. This leads to a natural introduction of membership values for elements of graphs. Using these membership values we apply AFS theory as well as an alternative approach to calculate fuzzy decision trees, that can be used to determine the most relevant elements of a concept
Micro-Macro Analysis of Complex Networks
Complex systems have attracted considerable interest because of their wide range of applications, and are often studied via a \u201cclassic\u201d approach: study a specific system, find a complex network behind it, and analyze the corresponding properties. This simple methodology has produced a great deal of interesting results, but relies on an often implicit underlying assumption: the level of detail on which the system is observed. However, in many situations, physical or abstract, the level of detail can be one out of many, and might also depend on intrinsic limitations in viewing the data with a different level of abstraction or precision. So, a fundamental question arises: do properties of a network depend on its level of observability, or are they invariant? If there is a dependence, then an apparently correct network modeling could in fact just be a bad approximation of the true behavior of a complex system. In order to answer this question, we propose a novel micro-macro analysis of complex systems that quantitatively describes how the structure of complex networks varies as a function of the detail level. To this extent, we have developed a new telescopic algorithm that abstracts from the local properties of a system and reconstructs the original structure according to a fuzziness level. This way we can study what happens when passing from a fine level of detail (\u201cmicro\u201d) to a different scale level (\u201cmacro\u201d), and analyze the corresponding behavior in this transition, obtaining a deeper spectrum analysis. The obtained results show that many important properties are not universally invariant with respect to the level of detail, but instead strongly depend on the specific level on which a network is observed. Therefore, caution should be taken in every situation where a complex network is considered, if its context allows for different levels of observability
Imprint of quantum gravity in the dimension and fabric of spacetime
We here conjecture that two much-studied aspects of quantum gravity,
dimensional flow and spacetime fuzziness, might be deeply connected. We
illustrate the mechanism, providing first evidence in support of our
conjecture, by working within the framework of multifractional theories, whose
key assumption is an anomalous scaling of the spacetime dimension in the
ultraviolet and a slow change of the dimension in the infrared. This sole
ingredient is enough to produce a scale-dependent deformation of the
integration measure with also a fuzzy spacetime structure. We also compare the
multifractional correction to lengths with the types of Planckian uncertainty
for distance and time measurements that was reported in studies combining
quantum mechanics and general relativity heuristically. This allows us to fix
two free parameters of the theory and leads, in one of the scenarios we
contemplate, to a value of the ultraviolet dimension which had already found
support in other quantum-gravity analyses. We also formalize a picture such
that fuzziness originates from a fundamental discrete scale invariance at short
scales and corresponds to a stochastic spacetime geometry.Comment: 6 pages; v2: phenomenology section adde
Quantum Mechanics as a Framework for Dealing with Uncertainty
Quantum uncertainty is described here in two guises: indeterminacy with its
concomitant indeterminism of measurement outcomes, and fuzziness, or
unsharpness. Both features were long seen as obstructions of experimental
possibilities that were available in the realm of classical physics. The birth
of quantum information science was due to the realization that such
obstructions can be turned into powerful resources. Here we review how the
utilization of quantum fuzziness makes room for a notion of approximate joint
measurement of noncommuting observables. We also show how from a classical
perspective quantum uncertainty is due to a limitation of measurability
reflected in a fuzzy event structure -- all quantum events are fundamentally
unsharp.Comment: Plenary Lecture, Central European Workshop on Quantum Optics, Turku
2009
Continuous Fuzzy Measurement of Energy for a Two-Level System
A continuous measurement of energy which is sharp (perfect) leads to the
quantum Zeno effect (freezing of the state). Only if the quantum measurement is
fuzzy, continuous monitoring gives a readout E(t) from which information about
the dynamical development of the state vector of the system may be obtained in
certain cases. This is studied in detail. Fuzziness is thereby introduced with
the help of restricted path integrals equivalent to non-Hermitian Hamiltonians.
For an otherwise undisturbed multilevel system it is shown that this
measurement represents a model of decoherence. If it lasts long enough, the
measurement readout discriminates between the energy levels and the von Neumann
state reduction is obtained. For a two-level system under resonance influence
(which undergoes in absence of measurement Rabi oscillations between the
levels) different regimes of measurement are specified depending on its
duration and fuzziness: 1) the Zeno regime where the measurement results in a
freezing of the transitions between the levels and 2) the Rabi regime when the
transitions maintain. It is shown that in the Rabi regime at the border to the
Zeno regime a correlation exists between the time dependent measurement readout
and the modified Rabi oscillations of the state of the measured system.
Possible realizations of continuous fuzzy measurements of energy are sketched.Comment: 29 pages in LATEX, 1 figure in EPS, to be published in Physical
Review
Hausdorff dimension of a quantum string
In the path integral formulation of quantum mechanics, Feynman and Hibbs
noted that the trajectory of a particle is continuous but nowhere
differentiable. We extend this result to the quantum mechanical path of a
relativistic string and find that the ``trajectory'', in this case, is a
fractal surface with Hausdorff dimension three. Depending on the resolution of
the detecting apparatus, the extra dimension is perceived as ``fuzziness'' of
the string world-surface. We give an interpretation of this phenomenon in terms
of a new form of the uncertainty principle for strings, and study the
transition from the smooth to the fractal phase.Comment: 18 pages, non figures, ReVTeX 3.0, in print on Phys.Rev.
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