Trignometric Inequations and Fuzzy Information Theory

Transcription

1 Iteratoal Joural of Scetfc ad Iovatve Mathematcal Reearch (IJSIMR) Volume, Iue, Jauary - 0, PP ISSN 7-07X (Prt) & ISSN 7- (Ole) Trgometrc Iequato ad Fuzzy Iformato Theory P.K. Sharma, ead, Pot Graduate Dept. of Mathematc, du College, mrtar-00, pk_harma7@redffmal.com Ndh Joh, tat Profeor, Dept. of Mathematc, rya College, Ludhaa dh.aryacollege@gmal.com btract: Some ew trgoometrc meaure of fuzzy etropy volvg trgoometrc fucto have bee provded ad ther valdty checked by tudyg ther eetal properte ad certa equalte volvg trgoometrc agle of a covex polygo of de have bee proved by makg ue of ome cocept from fuzzy formato theory. Key word: Trgoometry, fuzzy formato theory, etropy, fuzzy etropy trgoometrc equalte. INTRODUCTION For the probablty dtrbuto P = ( p, p,..., p ), Shao [8] obtaed the meaure of etropy a : S (P) = p log p (.) Whch a cocave fucto ad ha maxmum value whe p p... p Correpodg to (.), Deluca ad Term [] uggeted the meaure of fuzzy etropy a () = ( x )log ( x ) ( ( x )) log ( x ) Whch a cocave fucto ad ha maxmum value whe mot fuzzy et. Rey [7] probabltc meaure of etropy gve by (P) log p (.), > 0, (.) Correpodg to (.), Bhadar ad Pal [] uggeted meaure of fuzzy etropy a () log ( x ) ( x ) Correpodg to avrada ad Charvat [] probabltc meaure of etropy, (P) p Kapur [] uggeted the followg meaure of fuzzy etropy : ; > 0, (.) ; > 0,, (.) RC Page 00

2 P.K. Sharma & Ndh Joh () ( x ) ( ( x )) ; > 0, (.6) pparetly, there eem to be o relato betwee trgoometry ad fuzzy formato theory. Neverthele oe relatohp are a meaure of fuzzy etropy ued fuzzy formato theory are cocave fucto ad ome trgoometrc fucto are alo cocave fucto. Our teret to explot th relatohp ad etablh ome equalte betwee agle of a covex polygo. meaure of etropy havg thee properte ad volvg trgoometrc fucto ha bee gve by Kapur ad Trpath [ ]. Th meaure gve by S ( P) p (.7) Correpodg to (.7), Kapur [ ] gave meaure of fuzzy etropy a ( ) ( x ) ( ( x )) ( x ) (.8) other meaure of etropy havg ame properte ad volvg logarthmc fucto gve by S ( P) log p Correpodg to (.9), Parkah ad Sharma [6] gave meaure of fuzzy etropy a ( ) log ( x ) log ( x ) (.9) (.0). MESURES OF FUZZY ENTROPY ND TEIR VLIDITY We, ow propoe ew meaure of fuzzy etropy a follow: ( ) = d ( ) ( x ) ( ) ( ( x )) { } { } ( ) ( ) ( x ) ( ) ( ( x )) ( ) = ta{ } ta{ } ( ) ta ta{ } (a) Dfferetate (.) w.r.t. ( x ) ( ) ( x ), we get (.) (.) Iteratoal Joural of Scetfc ad Iovatve Mathematcal Reearch (IJSIMR) Page 0

3 Trgometrc Iequato ad Fuzzy Iformato Theory For Thu ( ) ( x ) ( ) ( ( x )) ( ) { } { } ( ) ( x ) 0 ad ce 0 whe 0 ( ) < 0 ad ( ) ( x ) a cocave fucto of ( x ) ad t maxmum are whe ( x ) = Thu we have the followg reult : () ( ) a cocave fucto of (x ). () ( ) doe t chage whe (x ) chaged to (x ). () ( ) a creag fucto of (x ) whe 0 (x ) /. (v) ( ) a decreag fucto of (x ) whe (v) ( ) =0 whe (x ) = 0 or. ece ( ) a vald meaure of fuzzy etropy. (b) Dfferetate (.) w.r.t. ( x ) ( ) = ( x ), we get (x ). ( ) ( x ) ( ) ( x ) ( ) ec { }ta{ } ( ) ( ( x )) ( ) ( ( x )) ( ) ec { }ta{ } For ( x ) <, we have ( ) ( x ) < < f So that ( ) < 0 ad ( ) a cocave fucto of ( x ) ( x ) Thu we have the followg reult : () ( ) a cocave fucto of (x ). () ( ) doe t chage whe (x ) chaged to (x ). Iteratoal Joural of Scetfc ad Iovatve Mathematcal Reearch (IJSIMR) Page 0

4 P.K. Sharma & Ndh Joh () ( ) a creag fucto of (x ) whe 0 (x ) /. (v) ( ) a decreag fucto of (x ) whe (v) ( ) = 0 whe (x ) = 0 or. ece ( ) a vald meaure of fuzzy etropy.. BSIC TRIGNOMETRIC INQULITIES (x ). Let,,,..., be the agle meaured rada of a covex polygo of de, o that... ( ), Let ( x ) = ( ), =,,,.. Now coder the meaure of fuzzy etropy ( ) = ( ) ( ) ( ) ( ( )) { x } { x } ( ) (.) Where ad are two parameter atfyg 0 < < ad 0 < < Now each < ce each a agle of a covex polygo. ( ) ( x ) ( ) ( x ) < or ( x ) < ( ) < = ( ) Sce ( ) a cocave fucto of ( x ), ( x ), ( x ),..., ( x ) ad ha maxmum value for the mot fuzzy et o that ( ) ( ) (.) ( x ), ( x ), ( x ),... ( x ),,,..., ( ) ( x ) ( ) ( ( x )) { } { } { ( ) } { ( ) } ( ) ( ) ( ) (.) Iteratoal Joural of Scetfc ad Iovatve Mathematcal Reearch (IJSIMR) Page 0

5 Trgometrc Iequato ad Fuzzy Iformato Theory Moreover the equalty g (.) hold oly whe ( x ) ( ) g (.) hold whe = =...= = for each o that equalty Iequalty (.) our bac equalty volvg trgoometrc fucto of the agle,,,..., of a covex polygo of de. The equalty g (.) wll hold whe all the agle are equal.. SPECIL CSES The equalty (.) repreet a trple fty of equalte ce t volve three parameter, ad. Frtly, we ca gve ay tegral value to. Secodly, we ca gve ay real value to lyg betwee o ad. Thrdly, correpodg to ay value of, we ca gve ay potve value to k le tha -. We ca get pecal equalte by gvg partcular value to,,. CSE I N = For a tragle wth agle,,, (.) gve ( ) ( ) ( ) ( ) Th gve the equalte ; 0< (.) co co co CSE II N = For a quadrlateral wth agle,,,, we have ( ) ( ) ( ) ( ) ( ) 0< Th gve the equalte, Iteratoal Joural of Scetfc ad Iovatve Mathematcal Reearch (IJSIMR) Page 0

6 P.K. Sharma & Ndh Joh co co co co Sce ( ) a cocave fucto of ( x ), ( x ), ( x ),..., ( x ) maxmum value for the mot fuzzy et o that ( ) ( ) that ad ha ( x ), ( x ), ( x ),... ( x ),,,..., o that Or ( ) ( x ) ta{ } ( ) ta{ } ta( ) ( ) ta{ } Th our bac equalty volvg taget of agle of a covex polygo. ga the equalty (.) gve a trple fty of equalte ce t volve three parameter, ad,. The parameter ca take all tegral value. ca take all real value le (.) tha ad ca take all real value le tha. Partcular cae For =0 ta ta( ) ; < For tragle,, =, = ta ta ta ta 9 o that For quadrlateral, ta ta ta ta ta 8. CONCLUSION Mmum rea of a Tragle wth Gve Permeter lthough there are may proof of the reult that the area of a tragle wth gve permeter maxmum whe the tragle equlateral. lo we have may proof of the reult that the permeter of a tragle wth area mmum whe the tragle equlateral. ere we gve proof of thee reult by makg ue of Fuzzy formato theoretc approach From ero formula Iteratoal Joural of Scetfc ad Iovatve Mathematcal Reearch (IJSIMR) Page 0

7 Trgometrc Iequato ad Fuzzy Iformato Theory ( a)( b)( c ) Where a,b,c are the legth of the de of the tragle, o that log log log( a) log( b) log( c ) a( a) b( b) c( c) log log log log log a log b log c log a b c a b c 7log log abc log log log log log log 7 log log ( x ) log{ ( x )} log ( x ) log{ ( x )} log ( x ) log{ ( x )} abc 7 log log ( x ) log{ ( x )} abc 7 log ( ) abc, where ( ) fuzzy etropy correpodg to Shao [8] probabltc meaure of a etropy. ad ( x ), ( ) b x, ( ) c x, a b c ( x ) Now f S gve log maxmum whe ( ) maxmum. lo ( ) maxmum whe ( x ) ( x ) ( x ) ), that, whe a=b=c ga f gve, log mmum whe ( ) maxmum that whe the tragle equlateral. Thu we have proved both the reult tated above.now for a geeral quadrlateral, a formula lke (.) ot avalable. owever, for cyclc quadrlateral, we have Brahmagupta` formula = ( a)( b)( c)( d ), a b c d log = log( a) log( b) log( c) log( d ) = a ( a ) b ( b log log ) c log ( c ) d log ( d ) log abcd log 8 log ( ) abcd probabltc meaure of fuzzy etropy, where ( ) fuzzy etropy correpodg to Shao [8] That out of all cyclc quadrlateral wth a gve permeter, the quare ha maxmum area ad out of all cyclc quadrlateral wth a gve area, the quare ha a mmum permeter. Iteratoal Joural of Scetfc ad Iovatve Mathematcal Reearch (IJSIMR) Page 06

8 P.K. Sharma & Ndh Joh REFERENCES [] Bhadar, D. ad Pal, N.R. (99), Some ew formato meaure of fuzzy et, Iformato Scece, 67, [] Deluca, ad Term, S. (97), defto of o-probabltc etropy the ettg of fuzzy et theory, Iformato ad Cotrol, 0, 0. [] avrada, J.. ad Charvat, F. (967), Quatfcato method of clafcato procee, cocept of tructural etropy, Kyberetka,, 0. [] Kapur, J.N.ad Trpath,G.P. (990), O Trgometrc Meaure of Iformato, Jour. Math. Phy. Sc.,,-0. [] Kapur, J.N. (997), Meaure of Fuzzy Iformato, Mathematcal Scece Trut Socety, New Delh. [6] Parkah,O. ad Sharma, P.K.(00), Meaure of Fuzzy Etropy ad ther Relato, Iter. Jour. Of Mgt. ad Sy., 0,, 6-7. [7] Rey,. (96), Meaure of etropy ad formato, Proc. th Berkeley Symp. Prob. Stat.,, 7 6. [8] Shao, C.E. (98), The mathematcal theory of commucato, Bell Syt. Tech. Jour., 7, 67. Iteratoal Joural of Scetfc ad Iovatve Mathematcal Reearch (IJSIMR) Page 07