Fig. 1 Initial Iteration of RSFC

Transcription

1 Jeyabhaath et al. Iteatoal Joal of vace Egeeg Techology E-ISSN 09-9 Reseach Pae YERS-LM STBILITY OF PRIK VECTORS FOR FINITE WORDS IN RECTNGLR SPCE FILLING CRVE Jeyabhaath S # Thagaaja K # Navaeetha K # ess fo Coesoece # ssocate Pofesso Deatet of Matheatcs Thagaaja College of Egeeg Maa. # ssocate Pofesso Deatet of Matheatcs PSN College of Egeeg & Techology Dgl. # Reseach scholaph.d-cb JL0-09 Reseach a Develoet Cete Bhaatha vesty Cobatoe. BSTRCT Recece elato fo the seqece of Pakh vectos fo fte wos Rectagla Sace Fllg Cves s establshe. yes-la stablty of the fst oe lea o-hoogeeos atx ffeece eqato s tate a scsse wth a exale obtag a vecto that s eaest to the Pakh vecto. Moeove Geealze Pakh vectos fo the fte wos have bee aalyse. KEYWORDS Rectagla Sace Fllg Cves Pakh vectos Posto Vectos Recece elatos Stablty ooohs. I. INTRODCTION Sace Fllg Cves SFC wee a toc of teest fo leag e atheatcas the late 9th cety a the Fst gahcal eesetato of oe was gve by Dav lbet 9. The goal of the eseach esete ths ae s to exte the cocet of Sace Fllg Cves o sqae fae to Sace Fllg Cves o ectagla fae. Jst as SFCs ae covolte les that fll a sqae these SFCs ae caeflly elaboate to fll a ectagle. The atho has toce the oto of lbet wos []. Cotg occeces of soe attes was oe []. I [] the sffcet cotos fo exstece of eoc solto of o-hoogeeos lea ffeece eqatos ae esete. Masoo he a Shah Bokha [] exlae abot Sace Fllg Sfaces. The thos [] vestgate the jectvty of the Pakh atx ag aly o the bay alhabet. lah Sael [] scsse abot soe eslts a oetes of Geealze Pakh Matx fo fte wos. The atho has gve the ecessay a sffcet coto fo - Boe Soltos of No-hoogeeos Lea Dffeece Eqatos []. The athos of [] [9] a [0] ae scsse abot the stablty a ostablty of ffeece eqatos. fte eesetg Rectagla Sace Fllg Cve RSFC fte wos fo ths Rectagla SFC s gve Secto. The Recece elato fo Pakh vectos ae gve a ts stabltes ae scsse secto. Fally the Geealze Pakh vectos fo the fte wos ae calclate. II. GEOMETRIC GENERTION OF RECTNGLR SFC Sace Fllg Cve as a -esoal sace oto a hghe-esoal sace e.g. the t teval oto the t sqae. geoetc geeato cle fo the exteso of lbet cve costcto to fll a ectagle ego s sggeste. The etho s a wok ot ecsve thkg a ca be se a few les. Let s asse that the t teval I ca be ae cotosly oto the ectagle Ω 0 x0.. Ital ag: If I s attoe to sx coget sbtevals the t shol be ossble to atto Ω to sx coget sb sqaes sch that each sbteval wll be ae cotosly oto oe of the sb sqaes. B. Iteato ag: If the sbtevals ae attoe to e coget sbtevals the t shol be ossble to atto the sb sqaes to e coget sb sqaes sch that each sbteval wll be ae cotosly oto oe of the sb sqaes. Ths easog ca be eeate by aga attog each sbteval to e coget sbtevals a og the sae fo the esectve sb sqaes Whe eeatg ths ocee ake se that the sb sqaes ae aage sch a way that ajacet sb sqaes coeso to ajacet sbtevals. Lke ths the oveall cotty of the ag s eseve. If a teval coesos to a sqae the ts sbtevals st coeso to the sb sqaes of that sqae. Ths clso elatosh asses that a ag of the th teato eseves the ag of the - th teato. Fg. Ital Iteato of RSFC Fg. Fst Iteato of RSFC It J v Egg Tech/Vol. VII/Isse II/l-Je0/-9

2 Jeyabhaath et al. Iteatoal Joal of vace Egeeg Techology E-ISSN 09-9 Fg. Seco Iteato of RSFC III. REPRESENTTION OF TE RECTNGLR SFC TROG GRMMR I the costcto of ths lbet Rectagla cve the fo telates B a C llstate the fge ae se evey teatos. These telates ae sbsttte evey teato ste to a fst teato of ths Sace Fllg Cve. These fxe sbsttto ocee ca be escbe by a gaa G VTPS whee V B C a P s efe by T S [C C C ] [B B B ] C [ C C C B B B] B [ B B B C C C ] Fg. Telates B C a sbsttto fo the The Lagage geeate by G calle LG s LG w T S w * / * G LG s the set of stgs ove Tteal sybols that ca be eve fo the tal sybol S zeo o oe stes sg the octos of G sltaeosly to all o-teals that ae esete g evatos. ccogly LG cotas a as fst two wos gahs Fte wos to eesets the fte aoxato of ths Rectagla Sace Fllg Cve s gog to be foe the followg secto. IV. FINITE WORD FOR TE RECTNGLR SFC The costcto of the lbet Rectagla cve s obtae by awg wthot eovg the e fo the sface of the ae to fll the ectagle of sze x a fte sccesso of t les a oble t les left ght o ow. Ths ths sccesso ca be eesete by a wo ove the alhabet l l whee stas fo t le stas fo t le ow stas fo t le ght stas fo t le left stas fo oble ts le stas fo oble ts le ow stas fo oble ts le ght stas fo oble ts le left. Let s exla the algoth of ectagla lbet Cve. The geeal ea s to ve at ste the ectagle to 9 - eqal sb sqaes each of the cotag a eqal legth at of the cve excet the fst a the last oes whch cota a at of legth ½. The cve so obtae s the ecte by a wo of legth [ 9 ] a calle as the th ectagla lbet wo. Theefoe It J v Egg Tech/Vol. VII/Isse II/l-Je0/-9

3 Jeyabhaath et al. Iteatoal Joal of vace Egeeg Techology E-ISSN 09-9 Slaly s scsse the evos secto the lagage geeate by G s LG {.} Now let s efe o thee lteal ohss f t a t by f f f f f f f f t t t t t t t t t t t t t t t t These thee ohss eeset a vetcal fl a qate t left otato a a qate t ght otato of the telate esectvely. sg these ohss the ecece elato fo fte wo s geeate by whee t f a t f Theoe : Fo ay a f f s s eve o f s eve f s o 0 eas that thee s zeo occece of. It J v Egg Tech/Vol. VII/Isse II/l-Je0/-9

4 .. Poof: Jeyabhaath et al. Iteatoal Joal of vace Egeeg Techology E-ISSN 09-9 f s o f s eve f s o f s eve The ecece elato fo s gve by 9 +. Solvg ths eqato wth tal coto the eqalty s obtae. The eqaltes fo to 0 ca be obtae fo the efto of a the foato of. Othe eqaltes ca be ove by cto o. Reak : Fo to of the theoe t s otce that the total execto te to f the occeces of each alhabet s exoetal. Coollay : The ecece elatos fo by.... +*- * f s o f s eve f s o * f s eve - *- 0 0 a ae gve V. PRELIMINRIES Defto : Pakh Vecto Let Σ { a < a <.<a k } be a oee alhabet. The Pakh ag s a k ag : * N efe as w w w... w whee w s the be of a a a k a k occeces of a k w. Pakh vecto of w s w w... w. Defto : Posto Vecto o Geealze Pakh Vecto Fo each the Geealze Pakh vecto eote by s gve by... whee whee a N a occs. k a k a a a k a cotas all the ostos whee a VI. RECRRENCE RELTION FOR PRIK VECTORS OF N Let the alhabet Σ s oee by. The sg theoe Pakh vecto of s gve by ψ Theefoe ψ ψ 0 a so o. Ths thee s a seqece of Pakh vectos. Theoe : The seqece of Pakh vectos of ae foe by a lea o-hoogeeos oatooos ecece elato. Poof: Fo coollay Pakh vecto of ca be ecsvely wtte as P F whee 0 f s o wth tal coto F f s eve The ecece eqato s lea o-hoogeeos o-atooos eqato. ece the theoe.. Stablty of Recece Relato fo Pakh Vectos Theoe : The Recece elato of Pakh vectos fo s stable. Poof: Pakh vectos ae ecsvely efe by the elato F The chaactestc eqato fo the ecece elato s o 9. Ths the ege vale s 9 whch geate tha ty absolte vale. ece the ecece elato fo { } s stable. It J v Egg Tech/Vol. VII/Isse II/l-Je0/-9

5 Jeyabhaath et al. Iteatoal Joal of vace Egeeg Techology E-ISSN 09-9 B. yes-la Stablty of the Fst Oe No-hoogeeos Matx Dffeece Eqatos Geeally a fctoal eqato s sa to be stable yes-la sese f fo evey solto of the eqato thee exsts a solto of the eqato that ffes eglgbly fo the solto of the eqato. Reesetg as atx ffeece eqato f whee T f F T a 9I I s etty x atx the yes la stablty of ths eqato s gve by the theoe below. Theoe : Gve a fxe ostve tege let R. It J v Egg Tech/Vol. VII/Isse II/l-Je0/-9 x a R. be eal oe saces whose eleets ae col vectos x eal atces esectvely wth the oety B B a X X. sse that the tasto atx s o-sgla a N s a seqece of oegatve eal bes. If a seqece q q f fo f R N x q fo R satsfes the eqalty fo R of the thee exsts a solto the fst oe o-hoogeeos atx ffeece eqato f sch that k 0 k q k fo. Poof: Let q q f By cto oe obtas k q q f k k Defe q Theefoe f Now q k k k k k k k f k k k k k ece the theoe. Exale : avg the seqece N of Pakh vectos let s f a ew seqece q N that s aoxate solto of a ffes fo N sgfcatly as. Let s choose c a q q whee k eotes the x col c c vecto whose all etes ae the costat k a 0. Let x axx x... x a 0 ax a j j q q f 9 f c c 0 9 c c c Moeove q c c Coollay : Takg c a q q q q f 9 Moeove q k 9 c c VII. POSITION VECTOR OF Now kowg to cot the be of occeces of alhabets as Pakh vectos let s cocetate o the ostos whee they ae lace. Fo that let s efe Posto set fo a alhabet as follows. Defto : The Posto Set w a of a alhabet a a wo w s efe as the set of all ostos of a wheeve t s aeae w. Exale: If w abaa w a {} w b {} Oeato o w a N

6 Jeyabhaath et al. Iteatoal Joal of vace Egeeg Techology E-ISSN 09-9 It J v Egg Tech/Vol. VII/Isse II/l-Je0/-9 Defto : The set c + w a o w a+c ae efe as the esltg set by ag the costat c to each eleet of w a a t s ety set f w a s ety set. Exale: I the above exale w a + {9} Posto Sets Fo lhabets of Now let s f the Posto set of lettes ecsvely. Postos of a alhabet ee o the fctos ρ λ fo a the lettes. Ths ca be exlae teatvely by exales. {} } { } { {} {} {} {} {}. {} { + x+} { + x+} { + x + { } {} Slaly { 9} a { 9} Now { } { } { } Ths Posto sets fo all the lettes of ca be exesse teatvely as follows. {}. l } { } { {} {}. {} }. { {} 9 It s estaable that } { {} a a N fo all a avg Posto sets of all lettes of the Posto vecto of s gve by whee VIII. FTRE WORK The oto of Gaa geeatg fte wos of Rectagla Sace Fllg Cve has to be extee to Two esoal Gaa. IX. CONCLSION The cocet of lbet fte wo has bee extee to Rectagla SFC. The costcto of Rectagla SFC has bee exlae. s SFC ae foe by teatve etho Pakh vectos ae calclate thogh Recece

7 Jeyabhaath et al. Iteatoal Joal of vace Egeeg Techology E-ISSN 09-9 elato a the stabltes ae aalyse. toally the Posto set of each alhabet has bee exesse ecsvely a Geealze Pakh vectos of the fte wos have bee gve. CKNOWLEDGMENT The athos wol lke to thak D. Poaal Nataaja woke as Decto of Reseach a vesty Chea Ia a D. K. Sakes foe vce chacello sta vesty Chea Ia fo the cogtve eas a yac scssos wth esect to the ae s cotbto. REFERENCES. a taas Calos Mat-Ve lexa Mateesc O the jectvty of the Pakh atx ag Fa. Ifo el Daaesc Exstece of -Boe Soltos of Nohoogeeos Lea Dffeece Eqatos le Matheatcs E-otes lah Sael O Geealze Pakh Matces fo fte a fte wos It. J. Co.l Jasz BzekDoa Poa Bg X O ostablty of the lea ecece of oe oe J.Math.al.l Klaa Jaglajew Ewa Scheel Peocty of soltos of ohoogeeos lea ffeece eqatos v.dffe.eq. 0//9. Masoo he Shah Bokha Mag wth Sace Fllg Sfaces Paallel a Dstbte Systes Patce Seebol Tag systes fo the lbet cve Dscete Maths. & Theo. Co. Sc. 9: Patce Seebol Segey KtaevTofk Maso Geeatg the Peao cve a cotg occeces of soe attes J. of toata Lagages a Cobatocs Pegag Wag MegW 0 -boeess a 0 -stablty of ffeece eqatos Co.a Maths. Wth Soo-Mo Jg yes-la st ablty of the fst-oe atx ffeece eqatos v.dffe.eq. 0:0 It J v Egg Tech/Vol. VII/Isse II/l-Je0/-9