Alternative Determines Positivity of Hexagonal Fuzzy Numbers and Their Alternative Arithmetic
Susmitha Harun,
Mashadi Mashadi,
Sri Gemawati
Issue:
Volume 6, Issue 1, March 2020
Pages:
1-7
Received:
1 April 2020
Accepted:
28 April 2020
Published:
9 June 2020
Abstract: There are quite a lot of arithmetic operations for hexagonal fuzzy numbers, most of them only define positive fuzzy numbers and few are discussing negative fuzzy numbers. And rarely found inverse of a fuzzy hexagonal number. So, often the results obtained in a hexagonal fuzzy linear equation system are not compatible. In this paper, we will discuss arithmetic alternatives on fuzzy hexagonal numbers. In this paper will definitions of positive and negative fuzzy numbers based on the concept of wide area covered by hexagonal fuzzy numbers in quadrant I and in quadrant II (right and left segments called r). From the concept of positivity and negativity the hexagonal fuzzy numbers will be constructed arithmetic alternatives for hexagonal fuzzy numbers. At the final part be given an inverse for a hexagonal fuzzy number so that, so for any fuzzy number there is an inverse hexagonal fuzzy number and its multiplication produces an identity.
Abstract: There are quite a lot of arithmetic operations for hexagonal fuzzy numbers, most of them only define positive fuzzy numbers and few are discussing negative fuzzy numbers. And rarely found inverse of a fuzzy hexagonal number. So, often the results obtained in a hexagonal fuzzy linear equation system are not compatible. In this paper, we will discus...
Show More
Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems
Issue:
Volume 6, Issue 1, March 2020
Pages:
8-13
Received:
26 April 2020
Accepted:
1 June 2020
Published:
15 June 2020
Abstract: In this paper, we introduced the concept of pseudo-convex open covering of topological spaces and entropy for topological spaces with such covering. Entropy was previously defined only for open coverings of compact topological spaces. It is shown by examples that the classes of topological spaces for which the concept of entropy is defined is quite wide. A discrete random process describing the evolution of the phase space of closed dynamical systems is built. Two random processes are constructed, one in which the elements of the transition matrices depend on the first indices, and the second Markov`s process, one in which the elements of the transition matrices do not depend from the first indices. The construction of transition matrices is based on the fact that the probability of a change in the phase space of a system to another space is proportional to the entropy of this other space. Based on the concept of entropy of topological spaces and on the well-known construction of an infinite product of probability measures which is also probabilistic introduced the concept entropy of trajectory of evolution of phase space of system. Previously, the entropy of the trajectory was defined only for the motion of a structureless material point, in this article is defined entropy of trajectory of evolution of structured objects represented in the form of topological spaces. On the basis of the concept entropy of trajectory, a method is determined for finding the most probable trajectory of evolution of the phase space of a closed system.
Abstract: In this paper, we introduced the concept of pseudo-convex open covering of topological spaces and entropy for topological spaces with such covering. Entropy was previously defined only for open coverings of compact topological spaces. It is shown by examples that the classes of topological spaces for which the concept of entropy is defined is quite...
Show More
