Applications of Trigonometic Measures of Fuzzy Entropy to Geometry
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:54050", author = "Om Parkash and C.P.Gandhi", title = "Applications of Trigonometic Measures of Fuzzy Entropy to Geometry", 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.", volume = "4", number = "1", pages = "29-4", }