Applications of Trigonometic Measures of Fuzzy Entropy to Geometry

Authors: Om Parkash, C.P.Gandhi

Abstract:

In the literature of fuzzy measures, there exist many well known parametric and non-parametric measures, each with its own merits and limitations. But our main emphasis is on applications of these measures to a variety of disciplines. To extend the scope of applications of these fuzzy measures to geometry, we need some special fuzzy measures. In this communication, we have introduced two new fuzzy measures involving trigonometric functions and simultaneously provided their applications to obtain the basic results already existing in the literature of geometry.

Keywords: Entropy, Uncertainty, Fuzzy Entropy, Concavity, Symmetry.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1060145

References:

[1] Bhandari, D. and Pal, N.R. (1993): "Some new information measures for fuzzy sets", Inform. Sci., 67, 209-228.
[2] De Luca, A. and Termini, S. (1972): "A definition of non-probabilistic entropy in setting of fuzzy set theory ", Inform. and Control, 20, 301-312.
[3] Ebanks, B.R.(1983): "On measures of fuzziness and thei representations", Jour. of Mathematical Analysis and Applications, 94, 24-37.
[4] Guo, X. Z. and Xin, X. L. (2006): "Some new generalized entropy formulas of fuzzy sets", J. Northwest Univ.,36 (4), 529-532.
[5] Havrada, J.H. and Charvat, F. (1967): "Quantification methods of classification process: Concept of structural ─▒-entropy", Kybernetika, 3, 30-35.
[6] Hu,Q.and Yu, D. (2004): "Entropies of fuzzy indiscernibility relation and its operations", Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 12 (5), 575-589.
[7] Kapur, J.N. (1997): "Measures of Fuzzy Information", Mathematical Sciences Trust Society, New Delhi.
[8] Klir, G.J. and Folger, T.A. (1988): "Fuzzy Sets, Uncertainty and Indetermination", Prentice Hall, New York.
[9] Liu, S.T. and Kao, C. (2002): "Fuzzy measures for correlation coefficient of fuzzy numbers", Fuzzy Sets and Systems, 128, 267-275.
[10] Parkash, O. (1998): "A new parametric measure of fuzzy entropy," Inform. Process. and Management of Uncertainty, 2, pp. 1732-1737.
[11] Parkash, O. and Gandhi, C. P. (2005): "Generating measures of fuzzy entropy through fuzzy directed divergence", Ultra Scientist of Physical Sci., 17(3), 419-428.
[12] Parkash, O. and Sharma, P. K. (2004): "Noiseless coding theorems corresponding to fuzzy entropies", Southeast Asian Bulletin of Maths., 27, 1073-1080.
[13] Parkash, O., Sharma, P. K.and Mahajan, R (2008): "New measures of weighted fuzzy entropy and their applications for the study of maximum weighted fuzzy entropy principle", Information Sciences, 178, 2389-2395.
[14] Parkash, O., Sharma, P. K. and Mahajan, R (2008): "Optimization principle for weighted fuzzy entropy using unequal constraints", Southeast Asian Bulletin Maths.(Accepted).
[15] Renyi, A. (1961): "On measures of entropy and information", Proc. 4th Ber. Symp. Math. Stat. and Prob., 1, 547-561.
[16] Rudas, I.J.(2001):"Measures of fuzziness : theory and applications ," Advance in fuzzy systems and evolutionary computation, 187-192, Artfi. Intell. Series (Athens) World Science Eng. Soc. Press, Athens.
[17] Shannon, C. E. (1948):"A mathematical theory of communication", Bell. Sys. Tech. Jr., 27, 379-423, 623-659.
[18] Zadeh,L.A.(1968): "Probability measures of fuzzy events", Jr. Math. Ann. Appli., 23, 421-427.