Occurrences of ordered patterns in rectangular space filling curve through homomorphism

Transcription

1 AUSTRALIAN JOURNAL OF BASI AND APPLIED SIENES ISSN:99- EISSN: 9- Jo hoe ge: Occece o odeed tte ectg ce g cve thogh hoooh K.Nveeth K. Thg S. Jeybhth Po Nt Reech cho Ph.D-B JUL-69 Detet o Mthetc Reech d Deveoet ete Bhth Uvety Id. Aocte Poeo Detet o Mthetc PSNA oege o Egeeg & Techoogy Id. Aocte Poeo Detet o Mthetc Thg oege o Egeeg Id. Foe Decto Detet o Mthetc Reech d Deveoet A Uvety he Id. Adde Fo oeodece: K.Nveeth Reech cho Ph.D-B JUL-69 Detet o Mthetc Reech d Deveoet ete Bhth Uvety Id. E-: veedh@g.co A R T I L E I N F O Atce htoy: Receved Feby 6 Acceted Mch 6 Avbe oe Mch 6 Keywod: Fte wod G Odeed Ptte Rectg Sce Fg ve Recece eto. A B S T R A T Fte Wod coeodg to te oxto o Rectg Sce Fg ve e oed. By odeg hbet o the te wod the be o occece o cet odeed tte thee wod e vetgted. Moeove DOL yte geetg thee wod w etbhed d occece o odeed tte wee dced thoght the hoooh o the DOL yte. INTRODUTION The e o Sce Fg ve SF h od y cto ch otg yte e cotg ge oceg d dt be. The go o the eech eeted th e to exted the cocet o Sce Fg ve o qe e to Sce Fg ve o ectg e. Jt SF e covoted e tht qe thee SF e cey eboted to ectge. The tho h dced Ktev S. bot occece o oe tte beqece d b wod geqece. obtoc o wod h bee yzed De Lc A. 999 d Bete J. d D. Pe. Mood Ahed d Shhd Boh Ahed M. S. Boh exed bot Sce Fg Sce. The tho h todced the oto o bet wod Sιιbod P.. I Ktev S. d T. Mo cotg occece o oe tte Peo wod w doe. otg odeed tte wod geeted by oh w doe Ktev S. et.. Fte wod whch coeod to te oxto o the Rectg SF e odced ecto te eetg the geeto o Rectg Sce Fg ve by geoetcy d thogh by g. A odeg gve to the hbet o te wod d dced bot e d decet the ext ecto. Fy the be o occece o odeed tte the te wod h bee exected. II. Geoetc geeto o ectg SF: A Sce Fg ve -deo ce oto hghe-deo ce e.g. the t tev oto the t qe. A geoetc geeto ce o the exteo o bet cve cotcto to Oe Acce Jo Pbhed BY AENSI Pbcto 6 AENSI Pbhe A ght eeved Th wo ceed de the etve oo Attbto Iteto Lcee BY. htt://cetvecoo.og/cee/by/./ To te Th Atce: K.Nveeth K. Thg S. Jeybhth Po Nt. Occece o odeed tte ectg ce g cve thogh hoooh. At. J. Bc & A. Sc. :

2 55 K.Nveeth et. 6 ectge ego ggeted. The ethod wo ot ecve thg d c be ed ew e. Let e tht the t tev I c be ed cotoy oto the ectge Ω x[ ]... It g: I I ttoed to x coget btev the t hod be obe to tto Ω to x coget b qe ch tht ech btev w be ed cotoy oto oe o the b qe... Iteto g: I the btev e ttoed to e coget btev the t hod be obe to tto the b qe to e coget b qe ch tht ech btev w be ed cotoy oto oe o the b qe. Th eog c be eeted by g ttog ech btev to e coget btev d dog the e o the eectve b qe. Whe eetg th ocede e e tht the b qe e ged ch wy tht dcet b qe coeod to dcet btev. Le th the ove cotty o the g eeved. I tev coeod to qe the t btev t coeod to the b qe o tht qe. Th co etoh e tht g o the th teto eeve the g o the - th teto. Fg. : It Iteto Fg. : Ft teto Fg. : Secod teto III. Reeetto o the ectg SF thogh g: I the cotcto o th bet Rectg cve the o tete A B d tted the ge e ed evey teto. Thee tete e bttted evey teto te to t teto o th Sce Fg ve. Thee xed bttto ocede c be decbed by g G V T P S A B d P deed by whee V { } T { } S { } [ A A A] A [B B B A A A ]

3 55 K.Nveeth et. 6 [ B B B] B [A A A B B B ] IV. Fte Wod o the Rectg SF: The cotcto o the bet Rectg cve obted by dwg wthot eovg the e o the ce o the e to the ectge o ze x te cceo o t e d dobe t e et ght o dow. Th th cceo c be eeeted by wod ove the hbet { d d } whee td o t e d td o t e dow td o t e ght td o t e et td o dobe t e d td o dobe t e dow td o dobe t e ght td o dobe t e et. Let ex the goth o ectg bet ve. The gee de to dvde t te the ectge to 69 - eq b qe ech o the cotg eq egth t o the cve excet the t d the t oe whch cot t o egth ½. The cve o obted the dected by wod o egth [ 9 ] whch ced the th ectg bet wod. Theeoe d Sy d d d d d d dd d d d d d d d d d d d d d d d d d d d d d d d d dd dd d d d dd dd dd d d d dd dd dd d d d d d dd dd d d d d d d d d dd dd d d d d d d d d dd dd d d d Now et dee o thee te oh t d t by d d d d t t d t t d t t d t t d t t d t d t t t d t d t

4 55 K.Nveeth et. 6 Thee thee oh eeet vetc qte t et otto d qte t ght otto eectvey. Ug thee oh the ecece eto o te wod geeted by ρ ρ ρ dλ dλ dλ ρ d λ t o whee t o.. Theoe: Fo y d d d d d d d d d d d eve odd. d e tht thee zeo occece o. d. odd eve. odd eve Poo The ecece eto o gve by 9. Sovg th eqto wth t codto the eqty obted. The eqte o to c be obted o the deto o ρ λ d the oto o. Othe eqte c be oved by dcto o. eve odd V. Re d decet : Let ode the hbet o < < < < d < d << 5.. Pooto: Let x.x be o-ety wod ove { } y y.y - o-ety wod ove { d d } wod ove { } d y wod ove { d } x y..x y the x d y be x є. O y є o both. Let w x y

5 55 K.Nveeth et. 6 R ρw D w D w D w x y othewe ε x ε d y ε Dρw R w D w R w x x y ε ε d y othewe ε Rλw Dw d Dλw Rw whee R w d D w deote the be o e d decet eectvey wod w. Poo ey D w [ D x D y ] x y ε d R w [ R x R y ] x ε d y ε othewe Let ρ w x' y' x ' y '... x ' y ' Fo the deto o ρ R x D x D x R x R y D y D y R y x ' y ' ε R ρ w [ R x ' R y '] x' ε d y ' ε othewe x y ε [ D x D y ] x ε d y ε othewe D w x y ε Th e tht R ρ w D w x ε d y ε D w othewe Sce λ the te oh whch vet the ode o the ette we get R λ w D w d D λ w R w 5.. Theoe: Fo y N D D R { } { } { }. R Poo Fo the cotcto o D 6R D 5 D 6R D R R 6D 9 6D R R. Ad g the bove ooto 5. the theoe c be oved.

6 555 K.Nveeth et. 6 VI. Occece o odeed tte wthot te hhe : The te wod c be deed by the oowg dctve chee. d B d D d B B B dd dd dd B B B B D dd dd d B B B D dd dd Th whee D Let X deote the be o occece o the tte X whee X { A B } 6.. Deto: Soe wod W AB whee A d B e oe wod o the e egth d ette. The Kee o ode o the wod W to be the bwod cotg o the - ghtot ette o A the ette d the etot ette o B. It deoted by κ W. Fo exe κ. I A < the κ W ε. Ao W deote the be o occece o the tte. 6.. Theoe: Let... be bty geezed tte wthot te dhe tht cot o be. Soe og B B D dd o { } ch tht M x x x x B B B D x5 x6 x B x x x D dd D d x9 x whee x c be ethe - o deed o whethe Odd o Eve d - The o > M M B M M M D M eve M M M M M M M M B D [ P P P P ] M M M M M M M M B D [ P P P P ] M M M M M M M M D B [ P P P P ] M M M M M M M M M D B [ P P P P ] e M x X X X X X Whe

7 556 K.Nveeth et. 6 x X X X X 6 X x X X X X X x X X X X X X { } Poo B B B dd dd dd κ B B dd κ D dd. So Soe >. I th ce A occece o the tte ethe B - o - o - o κ B o κ o κ o B D x whee X c be o - deed o Odd o Eve. Sy we c hve B A B x D B x D A x D B D The bove yte o eqto c be wtte tx o B D X X X X Th we eed to ove ecece eto o the o Y AY PX. A VDV Th eto c be oved by dgozto o the tx tht by wtg dgo tx cotg o egeve o A d the co o V e ege vecto o A. ece the oo o the theoe oow o the oto o Exe Let. Th B D 5 X X X X { o X }. Th 9. Sy the ce o B d D c be doe Exe B D 9 Let.. Th 5 Th R. { } R The ce o B d D e.. VII. Occece o tte # d #: Let deote d d d 56 d eectvey. Let X deote the be o occece o the ette coeodg to {... } X. Fo to o Theoe c be ed to d It ce tht B B 5 B B B B 5 6 D D o 5 whee D

8 55 K.Nveeth et. 6 D D D D 5 Th o to o Theoe c be ed g to d Let deote o X X Let dc bot the be o occece # We hve Thee e x ce. X whee X { B D} B B B dd dd dd e c e ethe y oe o B - o - o D -. Th B D e The ette coeodg to the c be o o d. So thee e cobto o the occece o the ette coeodg to othe o o d. Moeove the ette coeodg to c e ethe B - o - o D -. I th ce B 9 A 9 D A 6 D D 6 e The ette coeodg to the c e y oe o B -. I th ce B [ B A D ] A get c be ed to othe two ce. obg thee ce we hve the oowg theoe... Theoe: Let #. Sy # o Fo > ecece. B D whee h B D h b c d B D B B d D e gve by the oowg [ A D ] B A [ A D ] [ D ] D B 9 9 D 6 D 6 D b B A B [ ] A B [ B ] 9 B 9 B 6 6 [ ] 5

9 55 K.Nveeth et. 6 c B D [ D B ] D [ B ] [ B ] B D 9 9 B 6 B 6 B 6 d D D A [ ] D [ D A ] 9 D 9 D 6 6 A [ A ] A VIII. otg occece o odeed tte thogh hoooh: The geeto o RSF thogh by g h bee exed ecto. I the teete ABd ow ed the cotcto o RSF e ed eectvey by the ette whee S the the RSF c be geeted by the DOL yte S X whee Iveo d o-veo o Let be o-egtve tege. The vecto o o-veo o : occece o the odeed tte # the wod RG # The vecto o veo o the -vecto x co tx whoe th ety the be o.e. occece o the odeed tte # the wod DG # the -vecto x co tx whoe th ety the be o.e... Reetto o oe ette: Let be o-egtve tege d otve tege. The vecto o -eetto wth g o oe ette o the -vecto whoe -th ety the be o occece o the odeed tte # the wod..e. R G #

10 559 K.Nveeth et. 6.. Odeed tte wth o g d oh:... Re Decet d cbe o : The vecto o e o the wod the -vecto whoe th ety the be o occece o the odeed tte..e. R The vecto o decet o tte the wod The vecto o cbe o tte the wod... Icdece Mtx o the -vecto whoe th ety the be o occece o the odeed..e. D the -vecto whoe th ety the be o occece o the odeed..e. R : Let be o-egtve tege.. The cdece tx o the x tx M whee wod.e.. So we c deote Let deote. ey t detdbe tht the be o occece o the ette the Ug thee otto d geezg the theoe 6 we hve the oowg theoe... Theoe: # # # # # # 5 # # # # # 6 # 5 # #

11 56 K.Nveeth et # # # 6 # 5 # # # # # 6 # # # # # 6 6 6

12 56 K.Nveeth et # # # # # # 5 # The ecve eto o e. # # d # c be exeed.5. Theoe: Let The the t ette o L the Ft ette o F K L L L < F > F F 5 K K K 5 K 5 K K K K

13 56 K.Nveeth et K - K - 5 K - 5 K K K K K.6. Theoe: The cdece tx o whe eve. whe odd. # G R eve

14 56 K.Nveeth et oco: I th e we obeved ecve occece o e decet d odeed tte the te wod whch eeet te oxto o the Rectg Sce Fg ve. REFERENES De Lc A O the cobtoc o te wod. Theoetc ote Scece : -9. Bete J. d D. Pe. The og o cobtoc o wod. Eoe Jo o obtoc : Ahed M. S. Boh. Mg wth ce g ce. Pe d Dtbted Syte IEEE Tcto o 9: Sιιbod P.. Tg-yte o the bet cve. Dcete Mthetc & Theoetc ote Scece 9. Ktev S. d T. Mo. The Peo cve d cotg occece o oe tte. Xv et th/6. Ktev S.. The g-eqece d cotg occece o oe tte beqece d bwod. Xv et th/6. Ktev S. T. Mo d P. Séébod. otg odeed tte wod geeted by oh. I Lgge d Atot Theoy d Acto : -9. Sge Be edebeg.