. The second term denotes the transition [n 1, n 2-1] [n 1, n 2 ] and leads to an increased p ( n1,

Transcription

1 . THE MSTE EQUTION OCH Te mase euao oesos o e saeme a e obab of beg a gve sae ages eeg o e obabes of aso o a fom a oe sae e ssem. I oves e fu obab sbuo we a be e sove. Ufouae s s o ofe e ase so we mus see fo some of e momes of e sbuo. Tese ae eas obae fom e geeag fuo so we w wo w e mase euao a fom w ees o e geeag fuo ae a e sbuo. Te gee ewo s efe b N sae vaabes.. N a M ae osas.. M. Te vaabes eoe e umbe of oes of a ea ema sees su as mns o oes. Befoe ag e mase euao aoa o eeme e ose oees of a gee ewo we w sa b obag e mase euao e geeag fuo fom fo some eemea ema euaos: Sess fom a emae I umeous gee eaos su as aso a asao mns a oes ae sese fom a emae DN a mn eseve. fe sess e umbe of emaes s o age. Te oesog eao s eefoe: B Moeue oues moeue B a a ae us of oeao me. Te mase euao esbes ow e obab o be sae [ ] moeues B moeues a me ages me. o e eao above: Te fs em efes a aso fom sae [ ] o sae [ ] a eefoe eas o a eease. Te seo em eoes e aso [ -] [ ] a eas o a ease. Te mase euao above s ea a a be sove fo e momes b osug e mome geeag fuo. I geea fo N ssem vaabes e mome geeag fuo s gve b:... N N... N... N... N

2 wee e sum us ove a ossbe saes fo ea s ase fom o. Ts fuo as e foowg usefu oees: wee meas a e fuo s evauae a fo a. Tese exessos usf e ame mome geeag : we a oba e momes of e obab sbuo b evauag e aa evaves of e fuo. Mug e mase euao above b o bo ses gves: Ts euao a be smfe sgfa b eag a: ' ' ' wee e age e owe m of e sum fo s aowe beause. Ts eas o: I e sea ase of sess fom a fxe umbe of emaes e euao fo e mome geeag fuo eues o: Ts euao a be ex sove bu sef oes o eese e fu oess. We eefoe w oba e exessos fo e oe ems befoe ombg em o moe a ea suao. Degaao Now ose e egaao eao:

3 B. Ts eao a eese wo ffee oesses: egaao wee moeue B s ovee o a sees w s o a of e subse of ees a uo wee s sa seaae fom e voume of ees. I ae oex s e egaao ae a e af-fe of e moeue. Te mase euao fo s eao s: Usg e same saeg as above e me evouo of e mome geeao fuo es: owa eao osevao of oa umbe of moeues Now ose e eao: B wee o os. Se e oa umbe s oseve e ssem s efe b o oe vaabe. We w use as e sge vaabe o efe s ssem. o e eao above: Ts eas o: Base o ese eemea eaos age ema ewos a be bu u. Te esus above ae summae Tabe. I s ase e sums o go u o sea of. Howeve e exa ems a aea we ag e age of vaabes ae w ea oe. 3

4 I eao Te B II B III B os. ems of.iv B Tabe. Mome geeag fuo euaos fo eemea eaos. Te mase euao fo ea eao e oues ffee ems w a be ombe o moe moe omex oesses. Nose oees of a osuve exesse gee Base o e esus fo ese eemea eaos e euao fo e mome geeag fuos of moe omex ewos a be eas eue. s e us ose a osuve exesse gee a sge o e omosome of a baeum. I s ase e sae of s ssem a a me s efe b e umbe of mn moeues a umbe of oes fo a gee. mn moeues ae sese off e emae DN sa a a ae a ae asae a a ae. Te mn a oe egaao ae esbe b e esuo aes a eseve g.. gue. Bas moe fo osuve exesso of a sge gee. O fou vua eaos ae osee: eao of mn fom a DN emae eao of oes fom vua mn moeues a e egaaouo of bo sees. 4

5 5 Base o e esus Tabe e mome geeag fuo a be eue e: Te fs wo ems ae e aso a asao eaos Tabe e I a e as wo ems moe egaao of mn a oes eseve Tabe e II. Beow e euao w be sove fo e momes e sea sae. I s ase: Te mea mn eve a oe eve ae fou b ag e evave w ese o a eseve: [ ] [ ] evauag bo exessos a gves: Te esus ae osse w e euvae eems ssem: Te fuuaos mn a oes eve ae fou b ffeeag e above euaos aga w ese o a a evauag a :

6 ue momes a be obae seuea s mae. Noe a fo a aom vaabe w ossoa sbuo a momes ae eua so e vaae ove e mea euas oe as s e ase fo e mn s moe. Te oe umbe fuuaes w a ge a ossoa ose e oeo eeme ma b e em e bus se w oesos o e aveage umbe of oes oue e mn [9]. I sme ases e s e momes a aso be obae as a fuo of me. o a sge gee e ose ou of eubum a be 4% age a s sea sae vaue e m of so mn femes [9]. moe eae moeg of s oess ou ue moe emeae eae oesses su as e aom ses a a bosome aes aog a mn bu mos u ou o ave e effe we omae smuaos. Howeve we a eesso o avao s ese s bg a ubg mg ave o be ue e moe fo s a be a mao soue of ose. I s s oex a e ems sow Tabe e III ae eee. uemoe e eesso oeao sef mg be fuuag w ase we ave o ose e ee ssem of gees. Leae max fomuao Te meo above a aso be use fo eag ssems of gees bu sovg s o sagfowa uess e oeos ae ea. eave f e ssem s a a sabe o sea sae e eao a be eae aou e sea sae vaue. aa wa of wg s ou s max fom. Te aso obabes fo sees x ae gve b f x x...x fo eao a fo esuo a a ae e maes efe b f a x x x.... Leg x be e veo of ema. sees e eae maoso euaos ae e gve b x x Noe a se ma ases e maoso euaos ue osa eao ems. If e ssem s ea mg be eessa o ue a aoa vaabe w s o fuuag a aows e uso of e osa ems e oma max fom. s a usao of s e maes fo e sge gee ase ae. wee e sae veo s x T wee s e gee o umbe. Ts osa sae ooae ees o o eese a aua ema; fo a ssem wee ma sees ave a osa eao ae ese aes a a be ae e fs oum of seg a. exame of s s e max fo e ase of wo 6

7 7 eag gees eae aou sea sae w fxe gee o umbes a eseve a wee e fs gee eesses e seo w asfe fuo f : f wee f f. We ems of ese maes e mase euao geeag fuo fom wou be. sea sae a ag e evave w ese o we oba: Seg a we ave fo ea x oesog o e maoso esu. Sma ffeeag aga a evauag a T T T Θ Θ

8 wee Θ. Tese ea euaos a be sove fo e meas vaaes a oeaos. Ts aoa s ve geea a e esug max euaos a be sove e. Howeve eve fo e ase of us wo eag gees s eues a 5x5 max ssem as sow so ges umbesome fo age ssems eve oug mos ees ae eo. Usg smbo max mauao sofwae s sagfowa o oba e ese exessos so fo ow aamees s s a goo meo fo obag vaues wou fue aoxmaos.. THE LNGEVIN OCH aeae aoa a aows fo a moe sagfowa eeao a saes eas o ffee eves of ea s e use of a Lagev euao. Te Lagev aoa osss essea of ag a ose em o e eems euaos. Ts ose em a eese e effe of e s fuuaos [] o e exea us of e ssem []. o x e oeao of some ema sees x f x x f x x wee e aom vaabe s eeme b s sasa oees. oma s a be a aom oess bu ae we assume we-ose sass w w gve aoxmae vaues fo e fs wo momes. Te oos fo we ose ae: τ τ wee eoes a esembe aveage. Se we ae eese e sea sae fuuaos we w assume e oeffe of e ose em o be osa 3.e. evauae a x. ss o e ase of ou bas moe of e sge gee we ave wo maoso euaos eeseg mn a oe eao eseve: Ts a be summae a ve aa wa [3] ems of e ogam gas o oba a euao w efes e esug omoes of e ose. 3 o e ase wee x s o osa e soas ffeea euao w be uesoo o foow e Saoov eeao [9]. Ts aows a geea oe-a euao o be we s fom bu w o be eessa e ases of ees. 8

9 9 wee e oeffes of e ose ems ae o be eeme. Cea a fom e oo of eo mea fo e ose em. Te ffeee w e sea sae foows e euao oue asfomg we oba so afe mug b e omex ougae a ag e aveage Te sea sae fuuaos ae gve b e vese oue asfom w 4 : x x π π. Bu se e ouo of mn s s moe a sge se eee aom oess as a osso sbuo so e vaae euas e mea w mes. o e umbe of oes we ave w bu s ase we aso ee o oe a * * w w se ese ae wo eee aom oesses w eo mea. So s ase. 4 om e Wee-Ke eoem; see [3].

10 Usg a se s eeses e ea ose a fo a fxe umbe of mns e ouo of oes s aso a ossoa oess. efomg e vese asfom 5 π fo omaso w e evous esu oe a so s a be ewe as Ts s ea o e esu obae b e mase euao. Ts meo a be ea geeae fo ma eag gees we e ssem s fuuag aou a sea sae. s a exame we w aae e ase wee oe gee eesses a seo gee. Le be e oe umbes of ea gee a e f be e ae of eao of e seo oe as a fuo of e fs. Ts meas a e euaos esbg s ssem ae f. Noe a e euaos ue e ee oess of oug a oe so mn eves ae o oge ex auae. Iug e Lagev ose em a oog a e fuuaos fom sea sae f f f wee a ea ose em as e same oos as befoe. Ts eaao s va a ea sabe o bu o fo asos bewee ffee sabe os o fo m es. o ve sma umbes of emas s aso beas ow beause se s oesses ae mos ossoa e fuuaos ae of oe so a Tao exaso mg o be va. oue asfomg a ag e suae a e aveage as befoe we ge 5 π π wee a π.

11 Te oeao bewee e gees a aso be auae fom * * w w wee e em * vases beause e fuuaos e fs gee ae eee fom e ea fuuaos e seo gee. I ma ases e ea me w be eeme ma b e uo me so w be e same fo a gees. Ts assumo smfes e exessos a ae obae uo asfomg ba: π π π π wee e aoa a of e ega vases beause of a. om ou evous esus we ow a fo a sge gee b b o o o wee b s e bus se fo gee. o bas H-e eesso Y Y Y Y Y B Y f wee B s e maxmum eao ae Y s e af uo o s e H oeffe a B s e basa aso eve. ssumg a e ea ose fo e seo gee aoe as e same fom

12 b e vaae a oeao a be ex we as b Y b. Y b Noe a we ee e aamees of e maoso euaos us a ea aamee fo ea gee b w ees o e aamees of e maoso euaos fo ea gee.