This contribution deals with developments in the history of philosophy, logic, and mathematics during the time before and up to the beginning of fuzzy logic. Even though the term “fuzzy” was introduced by Lotfi A. Zadeh in 1964/1965, it should be noted that older concepts of “vagueness” and “haziness” had previously been discussed in philosophy, logic, mathematics, applied sciences, and medicine. This paper delineates some specific paths through the history of the use of these “loose concepts”. Vagueness was avidly discussed in the fields of logic and philosophy during the first decades of the 20th century—particularly in Vienna, at Cambridge and in Warsaw and Lvov. An interesting sequel to these developments can be seen in the work of the Polish physician and medical philosopher Ludwik Fleck. Haziness and fuzziness were concepts of interest in mathematics and engineering during the second half of the 1900s. The logico-philosophical history presented here covers the work of Bertrand Russell, Max Black, and others. The mathematical–technical history deals with the theories founded by Karl Menger and Lotfi Zadeh. Menger's concepts of probabilistic metrics, hazy sets (ensembles flous) and micro-geometry as well as Zadeh's theory of fuzzy sets paved the way for the establishment of soft computing methods using vague concepts that connote the nonexistence of sharp boundaries.
The structure of loop corrections is examined in a scalar field theory on a three-dimensional space whose spatial coordinates are noncommutative and satisfy SU(2) Lie algebra. In particular, the 2- and 4-point functions in scalar theory are calculated at the 1-loop order. The theory is UV-finite as the momentum space is compact. It is shown that the non-planar corrections are proportional to a one-dimensional -function, rather than a three-dimensional one, so that in transition rates only the planar corrections contribute.
The core of soft computing consists of fuzzy sets and systems (FSS) computing with words (CW) and the computational theory of perceptions (CTP). In the introduction of this paper we give a brief presentation of that subject. The second section of the paper focuses on a general view on fuzziness in evolutionary biology. The view on evolutionary biology is a meta-scientific reflection: The theory of FSS complemented by CW and CTP builds a stack of methodologies to help bridge the gap between systems and phenomena in the real world and scientific theories. Lotfi A. Zadeh established the theory of FSS to bridging this gap in the 1960s when he compared living systems that are very complex and man-made systems e.g. in electrical engineering. We use Zadeh's stack of soft computing methodologies in this tradition to reflect the field of evolutionary biology in the view of the evolutionary biologist, historian and philosopher of biology Ernst Mayr who emphasized that most theories in biology are based not on laws but on concepts.
Uncertainty measures model different types of uncertainty that are inherent in complex information systems. Measures that model either fuzzy or probabilistic uncertainty types have been explored in the literature. This paper shows that a combination of fuzzy and probabilistic uncertainty types, combined with the generalized maximum uncertainty principle, can be applied to time-series sequence classification and analysis. We present a novel algorithm that selects a wavelet from a wavelet library such that it best represents a time-series sequence, in a maximum uncertainty sense. Transformation coefficients are combined together in feature vectors that capture sequence temporal trends. A neural network is trained and tested using extracted gait sequence temporal features. Results have shown that models that combine together fuzzy and probabilistic uncertainty types better classify time-series gait sequences.
Quantum mechanics of models is considered which are constructed in spaces with Lie algebra type commutation relations between spatial coordinates. The case is specialized to that of the group SU(2), for which the formulation of the problem via the Euler parameterization is also presented. SU(2)-invariant systems are discussed, and the corresponding eigenvalue problem for the Hamiltonian is reduced to an ordinary differential equation, as is the case with such models on commutative spaces.
Evaluation of a retinal image is widely employed to help doctors diagnose many diseases, such as diabetes or hypertension. From acquisition process, retinal images often have low grey level contrast and dynamic range. This paper proposes indices of fuzziness for solving the problems of retinal image enhancement. Histogram of the retinal image is stretched by estimating the optimal parameters of the indices of fuzziness in S-function, which is designed for contrasting a field of view in the images. Our algorithm can achieve a number of significant properties for contrast stretching, such as compressed the noise in background and produced a high contrast in a field of view or foreground that provides feasibility in diagnosis from a retinal image.
2008 International Workshop on Education Technology and Training & 2008 International Workshop on Geoscience and Remote Sensing ISSN:9780769535630,0769535631 Publication-Date:2008-12 Volume:1 Page:614-617
In present paper, based on the classical fuzzy entropy,the fuzziness of variable precision rough sets in generalized approximation space are discussed by the axiomatized method. And the same time, a general framework that is studied the fuzziness of the rough sets in generalized approximation space is given. Meanwhile, the probability measure is introduced to the generalized approximation space and the generalized variable precision probability rough sets model is established.Furthermore, several properties of the model are studied.Finally, the fuzziness of the generalized variable precision rough sets is discussed by the method that is proposed in this paper, too.
The paper studies the fuzziness measure in fuzzy rough sets. By making use of the support set of fuzzy sets, a rough membership function for fuzzy sets based on fuzzy relation is introduced. Simultaneously, a fuzziness measure of fuzzy rough sets from total mean fuzzy degree is proposed. And then, it is proved that the fuzziness measure of fuzzy rough sets, denoted by, equals to zero if the set is crisp and definable.
The structure of transition amplitudes in field theory in a three-dimensional space whose spatial coordinates are noncommutative and satisfy the SU(2) Lie algebra commutation relations is examined. In particular, the basic notions for constructing the observables of the theory as well as subtleties related to the proper treatment of δ distributions (corresponding to conservation laws) are introduced. Explicit examples are given for scalar field theory amplitudes in the lowest order of perturbation.
In critical care, patients are surrounded by a multitude of devices for monitoring, diagnostics, and therapy. These devices include patient monitors to measure and monitor human vital parameters, therapeutic devices to support or replace impaired or failing organs and also to administer medications and fluids for the patients. In this special environment the complexity of biological systems makes traditional quantitative approaches of analysis inappropriate. There is an unavoidable substantial degree of fuzziness in the description of the behavior of biological systems as well as their characteristics. Fuzzy logic provides a suitable basis for the ability to summarize and extract from masses of data impinging upon the human brain those facts that are related to the performance of the task at hand. Therefore, this approach may be very suitable for intensive care medicine, where experience and intuition play an important role in decision-making. This paper surveys the utilisation of the fuzzy set theory in medical sciences in general, as well as on the basis of two concrete medical fuzzy applications.