On generalized fuzzy mean code word lengths. Department of Mathematics, Jaypee University of Engineering and Technology, Guna, Madhya Pradesh, India

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1 merca Joural of ppled Mathematcs 04; (4): 7-34 Publshed ole ugust 30, 04 ( do: 0.648/.aam ISSN: (Prt); ISSN: X (Ole) O geeralzed fuzzy mea code word legths hara Sgh Hooda, ruodaya Ra Mshra, vya Ja epartmet of Mathematcs, Jaypee Uversty of Egeerg ad Techology, Gua, Madhya Pradesh, Ida Emal address: ds_hooda@redffmal.com (. S. Hooda), aruodaya87@outlook.com (. R. Mshra), agrawal_dvya00@yahoo.co. (. Ja) To cte ths artcle: hara Sgh Hooda, ruodaya Ra Mshra, vya Ja. O Geeralzed Fuzzy Mea Code Word Legths. merca Joural of ppled Mathematcs. Vol., No. 4, 04, pp do: 0.648/.aam bstract: I preset commucato, a geeralzed fuzzy mea code word legth of degree has bee defed ad ts bouds the term of geeralzed fuzzy formato measure have bee studed. Further we have defed the fuzzy mea code word legth of type, ad ts bouds have also bee studed. Mootoc behavor of these fuzzy mea code word legths have bee llustrated graphcally by takg some emprcal data. Keywords: Etropy, Fuzzy Etropy, Codeword Legth, ecpherable Code, Crsp Set, Hölder s Iequalty. Itroducto Let X be a dscrete radom varable wth probablty a dstrbuto P {( p, p,..., p) : p 0; p } epermet. [5] gave a mathematcal formulato to measure ucertaty of the radomess a probablty dstrbuto ad formato cotaed a epermet as H( P) p log p, () whch s called Shao s etropy. I may applcatos of the ucertaty fucto, the ma problem geerated by researchers s that of effcet codg of message to be set over a oseless chael, ad to mamze the umber of messages that ca be set over the chael a gve tme. Let us cosder that the messages to be trasmtted are geerated by a radom varable X {,,..., } wth the probablty dstrbuto P ( p, p,..., p ): p 0; p. Each s called source symbol or alphabet ad s represeted by a fte sequece of symbols select from the set,,...,. The set s kow as code alphabet { } a a a or set of code characters ad the sequece assged to each ;,,..., s called code word. The umber of code character used for a code word s called code word legth. Let be the code word legth of, the mea code word legth s gve L p, () p s the probablty of occurrece of satsfyg Kraft s equalty, (3) s the sze of code alphabet. I evaluatg log ru effcecy of commucatos, we choose codes to mmze average code word legth (). For uquely decpherable codes, [5] oseless codg theorem states that H ( P) H ( P) L +, (4) log log whch determes the lower ad upper bouds o L terms of [5] etropy. To prove oseless codg theorem, [] equalty plays a mportat role ad s uquely determed by the codto for uquely decpherablty. To tackle such stuatos, stead of takg the probablty, the dea of fuzzess ca be eplored. [0] Itroduced the cocept of fuzzy set whch mprecse kowledge ca be used to defe a evet. ecause of ther capablty to model o-statstcal mprecso fuzzy set plays a mportat role may systems fuzzy set s a subset of X ad s defed as

2 merca Joural of ppled Mathematcs 04; (4): {( ( )) : ( ) [ 0, ]; }, X, represets the degree of membershp ad s defed as 0, f adtheresoambguty ( ), f adtheresoambguty 0.5,theresmamumambguty whether or The dea of measurg fuzzy ucertaty wthout referece to probabltes bega 97 wth the work of [] who defed the etropy of P ( X) usg Shao s etropy as [ ] H ( )log ( ) + ( ( ))log( ( )) (5) Ths s called fuzzy formato measure. The fuzzy formato measure has foud wde applcatos to Egeerg, Fuzzy traffc cotrol, Fuzzy arcraft cotrol, Computer sceces, maagemet ad ecso makg, etc. ad those have already bee studed by varous authors. [6] Itroduced a ew measure of fuzzy dvergece eplag ts applcato to clusterg problems ad to a obect etracto problem. I preset paper, we defe fuzzy mea code word legths secto. I secto 3, we also study the bouds of the geeralzed fuzzy mea code word legth of degree terms of fuzzy formato measure ad we study the bouds of the ew fuzzy mea code word legth of type (, ) terms of fuzzy formato measure secto 4. We dscuss the mootoc behavor of geeralzed fuzzy mea code word legths secto 5.. Fuzzy Mea Code Word Legths [] defed fuzzy mea code word legth as gve below: ( ( )) ( ) L log. They studed the lower ad upper bouds ofl term of (5). Further based o [], they geeralzed (6) as gve below: { ( ) } + L log ; > 0,, whch was called fuzzy mea code word legth of order. Its lower ad upper bouds were obtaed term of the followg fuzzy formato measure characterzed by [6]: (6) (7) H log ( ) ( ( )) ; 0,. + > (8) [] also defed fuzzy mea code word legth of degree as gve below: L { ( ) ( ( )) } ; 0,, + > (9) whch was called fuzzy mea code word legth of degree ad studed ts lower ad upper bouds term of fuzzy formato defed by [9] ad s gve as H ( ) ( ( )) ; 0,. + > (0) Correspodg to [3], [8] proposed ad studed the measure of fuzzy etropy as gve below: ( ) { log ( )log( ) } + H, > 0,. () Correspodg to [4], [7] studed geeralzed sub addtve fuzzy formato of type (, ) gve by H ( ) ( ( )) ( ) ( ( )), + 0 < <, or 0 < <,. () 3. ouds of a Geeralzed Fuzzy Mea Code Word Legth of egree It may be oted that (6) ca be geeralzed varous ways; however, we cosder the followg geeralzato: ( ( )) ( ) log L ; (3) > 0, ad study ts bouds terms of (). Theorem. For all uquely decpherable codes, oseless codg theorem states: H L H + ( ) ( ), (4) wth equalty f ad oly f

3 9 hara Sgh Hooda et al.: O Geeralzed Fuzzy Mea Code Word Legths, H s gve by (). Proof. [7] have gve the followg epresso for drected dvergece: log ( ) ( )log ( ) (, ) 0 I + (5) log log ( ) log( ) ( )log( ) (, ) 0. I + Takg,, log log ( ) log( ) ( )log (, ) 0, I + log log ( )log. H (6) Usg Kraft s equalty, that s, we get ( ) log H. H L For uquely decpherable code, [5] oseless codg theorem for fuzzy formato measure as. H L H + The we have L H +. It mples ( ) log log ( )log( ) + + ( ) log log ( )log( ). + + It mples ( ) log log ( )log( ) + +

4 merca Joural of ppled Mathematcs 04; (4): ( ) log ( ) ( )log ( ) ( ( ))log( ( )) + + ( ) log ( )log( ) + L + { } ( ) ( )log ( )log( ) + L Hece { } + L H + ( ) ( ). H L H + ( ) ( ). Partcular Case Whe, L reduces to (6). ( ( )) ( ) L log. Set Thus (3) ca be called the geeralzed fuzzy mea code word legth of degree. 4. ouds of Fuzzy Mea Code Word Legth of Type (, ) I ths secto, we defed fuzzy mea code word legth of type (, ). L { ( ) ( ( )) } { ( ) ( ( )) } < <, or 0 < <,. (7) Theorem. For all uquely decpherable codes, oseless codg theorem states: H ( ) L H + (8) H s gve by (). Proof. y Hölder s equalty, we have p p q q y y ;0 p, q 0 or 0 q, p 0, < < < < < < ad p p q (9) q y y ;, p, q ad, y 0. p + q > > (0) From (9), we have p p q q y y ;0 p, q 0or 0 q, p 0. < < < < < < t (, ) t, (, ) t f y f ad p t, q. + t The t t t f ( ( ), ( )) f ( ( ), ( )) by Kraft s equalty t t t f ( ( ), ( )) f ( ( ), ( )) or t f ( ( ), ( )) f ( ( ), ( )). t { f(, ) } { (, ) } f. Settg, > 0, t ad + t The f ( ( ), ( )) ( ) + ( ( )). Subtractg from both sdes, we get

5 3 hara Sgh Hooda et al.: O Geeralzed Fuzzy Mea Code Word Legths ( ) + ( ) ( ( )). + Chagg to, we get () Smlarly, we ca prove that L H + It mples. (6) ( ) + () ( ) ( ( )). + ddg () ad (), we get ( ) ( ) + + { ( ) } { ( ) } + + ( ) ( ) + { ( ) } { ( ) }. + + Hece H L. Now, we have to prove L H + (3) L H + (4) ( ) + ( ( )) { } ( ) ( ( )). + + It mples ( ) + ( ( )) { } ( ) ( ( )). + + (5) ad ( ) + ( ( )) { } + ( ) +. From (5) ad (7), we have ( ) + ( ( )) { } ( ) ( ( )). + + { ( ) } + ( ). + + ddg (8) ad (9), we have { ( ) + ( ( )) } { ( ) } + ( ) ( ( )) ( ) ( ( )) Hece H ( ) L H +. Partcular Cases. Whe ad, L reduces to (7) (8) (9) L ( )log ( ) ( ( ))log( ( )) log log +.

6 merca Joural of ppled Mathematcs 04; (4): Whe ad, L reduces to L decreases wth respect to. L ( )log ( ) ( ( ))log( ( )) log log + Thus (7) ca be called fuzzy mea code word legth of type(, ). 5. Mootoc ehavor of Fuzzy Mea Code Word Legths I ths secto, we study aalytcally the mootoc behavor of fuzzy mea code word legth L. From equato (3), we have ( ( )) ( ) log L, (30) Fgure. Relato betwee L ad ( < 0). Case : Whe >, we have dl 0, d < whch shows that L s a mootocally decreasg fucto of. The above result s verfed by plottg the graphs o MTL for dfferet values of ad >. Equato (30) ca be rewrtte as N L,, > 0, (3) ( ( )) ( ) N log 0. fferetatg (3) wth respect to, we have N { } d L N log N { } +. d ( ) ( ) dl N N { + log }. d ( ) Here two cases arse: dl Case : Whe <, we have 0, d < whch shows that L s a mootocally decreasg fucto of ad <. The above result s verfed by plottg the graphs o MTL for dfferet values of ad <. From below fgure we ca geeralze that the value of Fgure. Relato betwee L ad ( > 0). From above fgure we ca geeralze that the value of L mootocally decreases wth respect to ad >. We have also studed the mootoc behavor of fuzzy mea code word legth L. From equato (7), we have { ( ) ( ( )) + } L. { ( ) } + Equato (3) ca be rewrtte as (3) L, L L (33)

7 33 hara Sgh Hooda et al.: O Geeralzed Fuzzy Mea Code Word Legths L { ( ) ( ( )) } + ad L { ( ) ( ( )) }. + fferetatg (33) wth respect to, we have L log ( ) L L ( ) dl + ( ) L. d L L {log } + L ( ) L log ( ) dl +. d Here two cases arse: Case : Whe 0 < <, >, L > L, we get L > 0, whch shows that L s a mootocally creasg fucto of. The above result s verfed by plottg the graphs o MTL for dfferet values of ( < ) ad fed value of. whch shows that L s a mootocally decreasg fucto of. The above result s verfed by plottg the graphs o MTL for dfferet values of ( > ) ad fed values of. Fgure 4. Relato betwee L ad ( > 0). ga dfferetatg (33) wth respect to, we have L log ( ) L L ( ) dl ( ) L. d L L {( )log } + L L log ( ) dl. d Here two cases arse: ( ) Case : Whe 0 < <, >, L > L, we get L > 0, whch shows that L s a mootocally creasg fucto of. Case : Whe >,0 < <, L < L, we have Fgure 3. Relato betwee L ad ( < 0). Case : Whe >,0 < <, L < L, we have L > 0, L > 0, whch shows that L s a mootocally creasg fucto of.

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