International Mathematical Forum, Vol. 9, 2014, no. 13, HIKARI Ltd,

Transcription

1 Ieol Mhemcl oum Vol. 9 4 o HIKARI Ld hp//d.do.o/.988/m.4.43 Some Recuece Relo ewee he Sle Doule d Tple Mome o Ode Sc om Iveed mm Duo d hceo S. M. Ame * ollee o Scece d Hume Quwh Shq Uve Sud A Mo. El Weel Hhe Iue o ompue Iomo d Meme Techolo T Ep. oph 4 S. M. Ame d Mo. El Weel. Th ope cce cle dued ude he eve ommo Auo Lcee whch pem ueced ue duo d epoduco medum povded he ol wo popel ced. Ac I h ppe we deve ome ecuece elo ewee le doule d ple mome o ode c om veed mm I duo he complee d doul uced ce. Alo we chcee I duo houh codol mome. Kewod Ode c; Recuece elo; veed mm duo Tuced duo; le doule d ple mome d hceo

2 66 S. M. Ame d Mo. El Weel. Ioduco The ecuece elo d dee o he mome o ode c hve umed codele ee ce he educe he mou o dec compuo. The mome hve ee uled que eevel o evel duo. o eeve uve ee o emple Aold e l 3] d Blh d Sul 4]. M uho hve deved ome ecuece elo o oh le d doule mome ee o emple Al ] d Kh e l. 6] d Mhmoud e l. 89] uded Ode c om vee weull duo o chceo d oced eece popecve le duo m wo h ee doe o o emple L e l. 7] hve dcued he mmum lelhood emo o he pmee o I duo oehe wh he emo o he coepod ell uco Aj e l. ] hve uded he oue o equel e o he me o I duo d o d llo 5] hve dcued he covoluo o I duo wh pplco o Behe-he duo. Le... e dom mple o e om I duo wh p.d. e > d >. Γ whee hpe pmee d Γ. mm uco. The c.d. ve Γ. Γ whee Γ complee mm uco deed Γ e d.3 The -h mome o I duo. oed o e Γ E Γ >..4 whee Γ. mm uco. d vce ve V >..5 The ell d hd uco e ve R Γ Γ > >.6 d

3 Recuece elo 67 e Γ > >..7 I e o how h lm d lm. Th me h he hed e omoooc uco ue d 3 how he pol de ell d hd uco o I duo 3 4 d 5. ue.. Pol de uco o I duo. ue. Rell uco o I duo ue.3 Hd uco o I duo The p.d. o he doul uced I duo c e we om. Γ Whee e < <.8

4 68 S. M. Ame d Mo. El Weel Γ Γ..9 Γ Γ The c.d. o doul uced I duo c e oed om. Γ e d < <. whee d e ve.9. The p.d. o I duo ve. e he ollow elo. whee. he p.d. ve. d he devve o wh epec o. The p.d. o doul uced I duo ve.8 e he ollow elo ]. whee. he p.d. ve.8 d he devve o wh epec o. Le.. e mple o ode c o e om I duo ve..the he p.d. o -h ode c ve Dvd98] ] ] < <.3 whee. d. e ve.d. epecvel d!!!..4 The jo p.d. o he wo ode c d < ve ] ] ] < < <.5! whee!!!.6 Whle he jo p.d. o d o < < ve ] ] ] ] < < < <.7! whee!!!!..8. Recuece Relo ewee Mome o Ode Sc om I Duo I h eco we elhed ome ecuece elo ewee he le doule d ple mome o ode c om I duo ce o complee d doul uced ce. The -h le mome o he -h ode c deoed µ... ve

5 Recuece elo 69 d E ] ] µ. whee ve.4. The doule -h] mome o d deoed µ... < e ve ] ] ] dd E µ. whee ve.6. The ple mome o ode c d e ve ] ] ] ] ddd E µ < < < < -.3 whee ve.8. The doul uced le doule d ple mome c e oed eplc he el lm o e o he om ] ] P Q d.4 whee ve.4 < P Q ] ] ] dd... <.5 whee ve.6 d ] ] ] ] ddd E P Q Q Q µ Q < < P.6 whee ve.8.. Relo ed o complee ce The le doule d ple mome o ode c ve.. d.3 e he ollow ecuece elo

6 6 S. M. Ame d Mo. El Weel Relo. o < d > µ µ µ µ µ ] Poo.ee Apped A Relo. o < > µ ] µ..7 3 µ µ µ ] µ µ Poo.ee Apped B ] µ..8 µ. Relo ed o doul uced ce I h ueco we deve ome ecuece elo ewee he le doule d ple mome o ode c om DTI duo. The le doule d ple mome o ode c ve.4.5 d.6 e he ollow ecuece elo Relo.3 o d < we hve ]..9 Poo.ee Apped Relo.4 o < d > he mome o ode c om DTI duo.8 3 ] ]

7 Recuece elo 6 ].. Poo. See Apped D Rem o d oh o Relo.3 d.4 educe o he coepod Relo ve.7 d.8 epecvel. 3. hceo Bed o Ode Sc I h eco we hcee I duo u he ollow wo heoem Theoem 3. Le... e dom mple om oluel couou populo wh c.d. d p.d. d le... deoe he ode c oed om h mple. The he codol duo o ve h o < he me he duo o he -h ode c oed om mple o e om populo whoe duo mpl uced o he le ee Aold e l. 99]. So om he Theoem 3. we hve! ] ]!! ] d <. Theoem 3. Le... e dom mple om oluel couou populo wh c.d. d p.d. d le... deoe he ode c oed om h mple. The he codol duo o ve h o > he me he duo o he -h ode c mple o e om populo whoe duo mpl uced o he h ; ee Aold e l. 99]. So om he heoem 3. we hve 3.

8 6 S. M. Ame d Mo. El Weel! ] ]!! d <. ] hceo o I duo I h ueco we chcee I duo ed o Theoem 3. d 3. Theoem 3.3 A ece d uce codo o dom vle o hve veed mm duo E I < 3.3 Γ B Γ ] whee I d 3.4 d Γ deed.3 o he le uced Poo o he ece codo u Theoem 3. we e ] E ] d. 3.5 U eo p e o deeo d he e ed o eo we o E ] d ] 3.6 U. he ece codo poved. To pove he uce codo. om Theoem3. d 3.3 we e ] ] d I 3.7 ] whee I ve 3.4. Dee oh de o 3.7 wh epec o d mpl we e Γ. 3.8 Γ whch he c.d. o he veed mm duo ve.. Theoem 3.4 A ece d uce codo o dom vle o hve veed mm duo E I < 3.9 Γ B ]

9 Recuece elo 63 whee Γ B Γ ] I d 3. d Γ deed.3 o he h uced Poo ollow he me w Theoem 3.3 APPENDI A. Poo o Relo. om. d. we hve ] ] µ µ d I A. whee ve.4 d I ] ] d. A. Upo u eo p e o eo d he e eed o deeo we o whee d I ] ] d ] ] d I I A.3 I ] ] d I ] ] d. A.4 om. we hve d d. A.5 Thu vew o A.3 d A.5 we hve

10 64 S. M. Ame d Mo. El Weel ] ] dd I ] ] dd. A.6 Sce ] c e epeed ] ] ]. A.7 Theeoe ] ] I dd ] ] ] dd ] A.8 Smll we hve ] ] I dd ] ] ] dd ]. A.9 B uu om A.8 d A.9 A.3 Relo. c e oed. B. Poo o Relo. om. d. we hve µ µ ] d I B. whee ve.6 d d I ] ]. B. B u eo p e o eo d he e eed o deeo we hve d I ] ]

11 Recuece elo 65 d ] ] d ] ] B.3 om B.3 B. d. we o µ µ ] ddd ] ] ] ] ddd] ] ] d d d ] ] ] ] ddd] ] ] ] d d ]. B.4 Sce ] ] ]. B.5 B mpl B.4 u B.5 Relo. poved. Poo o Relo.3 B u.4 d. we e ] ] d I < <. whee ve.4 d

12 66 S. M. Ame d Mo. El Weel ] ] d I.. M ue o eo p e o eo d he e ed o deeo eld d I ] ] d ] ] I I.3 whee d I ] ] d d I ] ].4 om. we we d d..5 om.4 d.5 we o d I ] ] d d ] ] dd ] ].6 ce ] c e epeed ] ] ].7 The.6 c e we ] ] d I

13 Recuece elo 67 ] ] dd ] ] ] d d ].8 Smll we hve d I ] ] ] ] d d ] ] ] d d ].9 om..3.8 d.9 we e ] ] d d ] ] ] ] dd ] ] ] ] d d ] ] dd ] ] ] ] d d ] ].

14 68 S. M. Ame d Mo. El Weel Relo.3 ollow upo mpl.. D. Poo o Relo.4 B u.5 d. we hve ] d I D. whee ] ] d I. D. Upo u eo p e o deeo d he e ed o deeo we hve d I ] ] d ] ] d ] ]. D.3 om D. D.3 d.5 we e ] d d ] ] ] ] d d d ] ] ] ddd ] ] ] d d ] ] ] ] d d d ]

15 Recuece elo 69 ] ] ddd] ] ] ] ] dd. D.4 B u.7 D.4 d mpl Relo.4 poved. Reeece ] Aj. Aj K. d Pml. Roue o equel e o he me o veed mm duo Joul o he Id Scl Aoco No ] Al A.A. d Kh A.H. Recuece elo o he epeced vlue o he ce uco o wo ode c Meo LVI No ] Aold B.. Blh N. d Nju H.N.. A oue Ode Sc Joh Wle & So New Yo. 99 4] Blh N. d Sul K.S. Recuece elo d dee o mome o ode c Hdoo o Sc Ode Sc-I Theo d Mehod pp Ed. N. Blh d.r. Ro Noh-Hold Amedm. 5] o.j. d llo. A oe o he covoluo o veed mm duo wh pplco o Behe-he duo Acdem-de-ec-Ec-c-- Nule.-See-A.-Memc. 95. No ] Kh A.H. Yqu M. d Pve S. Recuece elo ewee mome o ode c Nvel Re. Lo. Qu ] L.T. Du B. S. Lew T. O. Iveed mm le duo. Mcoeleco. Rel No ] Mhmoud M.A.W. Sul K.S. d Ame S.M. Ode c om vee weull duo d chceo. Meo Ieol Joul o Sc. Vol LI.3 No

16 6 S. M. Ame d Mo. El Weel 9] Mhmoud M. A.W. Sul K.S. d Ame S.M. Ode c om vee Weull duo d oced eece ompuol Sc d D Al Receved eu 5 4