This work aims to contribute to a rethinking of the computer simulation used in the high education in Engineering. In order to design a set of simulation laboratory activities, a pedagogical proposal is presented on basis of Kolb's Experiential Learning Theory and Belhot's Learning Cycle. The chosen content to be taught is the Ant Colony Optimization (ACO) technique that is adapted and implemented in RoboMind software. The pedagogical approach presented in this paper can act as a reference point for debates in Engineering Education area, considering the use of the Kolb's theory as a model for development of teaching-learning process and computer simulations as a didactic tool. Finally, some recommendations are offered in order to help future works, as well as to consolidate the implementation of this pedagogical proposal in real case studies. ß 2015 Wiley Periodicals, Inc. Comput Appl Eng Educ 24:79-88, 2016; View this article online at wileyonlinelibrary.com/journal/cae;
In this paper, we apply the concepts of fuzzy sets to Lie algebras in order to introduce and to study the notions of solvable and nilpotent fuzzy radicals. We present conditions to prove the existence and uniqueness of such radicals.MSC 2010: 17B99, 08A72
IntroductionLie algebras were discovered by Sophus Lie [4]. There are many applications of Lie algebras in several branches of physics. The notion of fuzzy sets was introduced by Zadeh [8]. Fuzzy set theory has been developed in many directions by many scholars and has evoked a great interest among mathematicians working in different fields of mathematics. Many mathematicians have been involved in extending the concepts and results of abstract algebra. The notions of fuzzy ideals and fuzzy subalgebras of Lie algebras over a field were first introduced by Yehia in [7]. In this paper, we introduce the notion of solvable and nilpotent fuzzy radical of a fuzzy algebra of Lie algebras and investigate some of their properties. The results presented in this paper are strongly connected with the results proved in [1,2,3].
Fuzzy setsIn this section, we present the basic concepts on fuzzy sets which will be used throughout this paper. A new notion is introduced and results are proved for guiding the construction of the main theorems of this work. Definition 1. A mapping of a non-empty set X into the closed unit interval [0, 1] is called a fuzzy set of X. Let μ be any fuzzy set of X, then the set {μ(x) | x ∈ X} is called the image of μ and is denoted by μ(X). The set {x | x ∈ X, μ(x) > 0} is called the support of μ and is denoted by μ * . In particular, μ is called a finite fuzzy set if μ * is a finite set, and an infinite fuzzy set otherwise. For all real t ∈ [0, 1] the subsetDefinition 2. Let X be a non-empty set and {ν i } i∈I an arbitrary family of fuzzy sets of X. One defines the fuzzy set of X i∈I ν i , called union, asRemark 3. Let us note that if {ν i } i∈I is a family of fuzzy sets of X, then i∈I [ν i ] t ⊆ i∈I ν i t , for all t ∈]0, 1]. Definition 4. Let X be a non-empty set. One says that a family of fuzzy sets of X {ν i } i∈I satisfies the second sup property if for all x ∈ X there is an index iThus, a family of fuzzy sets of X {ν i } i∈I satisfies the second sup property if, and only if, i∈I ν i (x) ∈ {ν i (x) | i ∈ I}, for all x ∈ X. Proposition 5. Let X be a non-empty set and {ν i } i∈I an arbitrary family of fuzzy sets of X. Then i∈I ν i t = i∈I [ν i ] t for all t ∈]0, 1] if, and only if, the family {ν i } i∈I satisfies the second sup property.
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