Short CommunicationGeneralized uniform fuzzy partition: The solution to Holčapek's open problem
Introduction
Let be the set of real numbers. A function is said to be a generating function, if K is an even function that is non-increasing on and iff holds true. A generating function K is said to be normal if .
Triangular and raised cosine functions are typical examples of normal generating functions: and
Let K be a generating function, h and r be positive real numbers and . A system of fuzzy sets on defined by is said to be a generalized uniform fuzzy partition (GUFP) of the real line determined by the quadruplet if the Ruspini condition is satisfied: holds for any . The parameters and are called the bandwidth, shift and the central node, respectively.
In [2], Holčapek et al. have proved a full characterization of generalized uniform fuzzy partitions by using the sum of suitable integrals.
Lemma 1 (See [2].) A quadruplet determines a generalized uniform fuzzy partition iff the quadruplet determines it as well.
Corollary 1 (See [2].) Let be a real number. A triplet determines a generalized uniform fuzzy partition iff determines it as well.
Let K be a normal generating function. Define for , where is the common product of real numbers. The necessary and sufficient condition for GUFPs can be significantly simplified in the cases of triangular and raised cosine generating functions have proved in [2].
Theorem 1 (See [2].) Let . Then, determines a GUFP iff .
Section snippets
Counterexamples
Below we present Problem 7.1 from [3], which was posed by Holčapek et al. during the conference FSTA 2014.
Problem 1 (See [2].) Let K be a normal generating function satisfying the symmetry condition. Then determines a GUFP iff .
The following counterexamples show that the necessary and sufficient condition for GUFP do not hold under the symmetric condition given by (2).
Counterexample 1 The counterexample of necessary condition Let K be a normal generating function defined by
Modified result
In this section, we modify the symmetry condition and subsequently propose a sufficient condition. It is noted that if K is continuous and satisfies the symmetry condition (2), then or equivalently
Triangular and raised cosine functions are good examples of satisfying the condition (3).
Note. If a normal generating function K satisfies the condition (3), then the graph of K on has rotational symmetry with respect to the point
Acknowledgements
We would like to thank the reviewers for taking the time to carefully read the paper and for providing some very valuable feedback and suggestions. This work was supported Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0021089).
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