Symmetrical Piecewise Linear Functions Composed by Absolute Value Function

[1] Didier Dubois and Henri Prade. Operations on fuzzy numbers. International Journal of Systems Science, 9(6):613–626, 1978. doi:10.1080/00207727808941724. Search in Google Scholar

[2] Didier Dubois and Henri Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980. Search in Google Scholar

[3] Ronald E. Giachetti and Robert E. Young. A parametric representation of fuzzy numbers and their arithmetic operators. Fuzzy Sets and Systems, 91(2):185–202, 1997. doi:10.1016/S0165-0114(97)00140-1. Search in Google Scholar

[4] Eikou Gonda, Hitoshi Miyata, and Masaaki Ohkita. Self-turning of fuzzy rules with di erent types of MSFs (in Japanese). Journal of Japan Society for Fuzzy Theory and Intelligent Informatics, 16(6):540–550, 2004. doi:10.3156/jsoft.16.540. Search in Google Scholar

[5] Adam Grabowski. The formal construction of fuzzy numbers. Formalized Mathematics, 22(4):321–327, 2014. doi:10.2478/forma-2014-0032. Search in Google Scholar

[6] Adam Grabowski. Fuzzy implications in the Mizar system. In 30th IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2021, Luxembourg, July 11–14, 2021, pages 1–6. IEEE, 2021. doi:10.1109/FUZZ45933.2021.9494593. Search in Google Scholar

[7] Adam Grabowski. On the computer certification of fuzzy numbers. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, 2013 Federated Conference on Computer Science and Information Systems (FedCSIS), Federated Conference on Computer Science and Information Systems, pages 51–54, 2013. Search in Google Scholar

[8] Adam Grabowski and Takashi Mitsuishi. Initial comparison of formal approaches to fuzzy and rough sets. In Leszek Rutkowski, Marcin Korytkowski, Rafal Scherer, Ryszard Tadeusiewicz, Lotfi A. Zadeh, and Jacek M. Zurada, editors, Artificial Intelligence and Soft Computing – 14th International Conference, ICAISC 2015, Zakopane, Poland, June 14-18, 2015, Proceedings, Part I, volume 9119 of Lecture Notes in Computer Science, pages 160–171. Springer, 2015. doi:10.1007/978-3-319-19324-3 15. Search in Google Scholar

[9] Adam Grabowski and Christoph Schwarzweller. On duplication in mathematical repositories. In Serge Autexier, Jacques Calmet, David Delahaye, Patrick D. F. Ion, Laurence Rideau, Renaud Rioboo, and Alan P. Sexton, editors, Intelligent Computer Mathematics, 10th International Conference, AISC 2010, 17th Symposium, Calculemus 2010, and 9th International Conference, MKM 2010, Paris, France, July 5–10, 2010. Proceedings, volume 6167 of Lecture Notes in Computer Science, pages 300–314. Springer, 2010. doi:10.1007/978-3-642-14128-7 26. Search in Google Scholar

[10] Adam Grabowski and Christoph Schwarzweller. Translating mathematical vernacular into knowledge repositories. In Michael Kohlhase, editor, Mathematical Knowledge Management, volume 3863 of Lecture Notes in Computer Science, pages 49–64. Springer, 2006. doi:10.1007/11618027 4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15–17, 2005, Revised Selected Papers. Search in Google Scholar

[11] Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Mizar in a nutshell. Journal of Formalized Reasoning, 3(2):153–245, 2010. Search in Google Scholar

[12] Tetsuro Katafuchi, Kiyoji Asai, and Hiroshi Fujita. Investigation of de uzification in fuzzy inference: Proposal of a new defuzzification method (in Japanese). Medical Imaging and Information Sciences, 18(1):19–30, 2001. doi:10.11318/mii1984.18.19. Search in Google Scholar

[13] Ebrahim H. Mamdani. Application of fuzzy algorithms for control of simple dynamic plant. IEE Proceedings, 121:1585–1588, 1974. Search in Google Scholar

[14] Takashi Mitsuishi. Some properties of membership functions composed of triangle functions and piecewise linear functions. Formalized Mathematics, 29(2):103–115, 2021. doi:10.2478/forma-2021-0011. Search in Google Scholar

[15] Takashi Mitsuishi. Definition of centroid method as defuzzification. Formalized Mathematics, 30(2):125–134, 2022. doi:10.2478/forma-2022-0010. Search in Google Scholar

[16] Takashi Mitsuishi. Isosceles triangular and isosceles trapezoidal membership functions using centroid method. Formalized Mathematics, 31(1):59–66, 2023. doi:10.2478/forma-2023-0006. Search in Google Scholar

[17] Takashi Mitsuishi, Takanori Terashima, Nami Shimada, Toshimichi Homma, and Yasunari Shidama. Approximate reasoning using LR fuzzy number as input for sensorless fuzzy control. In 2016 IEEE Symposium on Sensorless Control for Electrical Drives (SLED), pages 1–5, 2016. doi:10.1109/SLED.2016.7518804. Search in Google Scholar

[18] Masaharu Mizumoto. Improvement of fuzzy control (IV)-case by product-sum-gravity method. In Proc. 6th Fuzzy System Symposium, 1990, pages 9–13, 1990. Search in Google Scholar

[19] Timothy J. Ross. Fuzzy Logic with Engineering Applications. John Wiley and Sons Ltd, 2010. Search in Google Scholar

[20] Luciano Stefanini and Laerte Sorini. Fuzzy arithmetic with parametric LR fuzzy numbers. In Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, pages 600–605, 2009. Search in Google Scholar

[21] Werner Van Leekwijck and Etienne E. Kerre. Defuzzification: Criteria and classification. Fuzzy Sets and Systems, 108(2):159–178, 1999. Search in Google Scholar

[22] Lotfi Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. doi:10.1016/S0019-9958(65)90241-X. Search in Google Scholar