JHQHUDOL]DWLRQ RI K1 PLQF DQG O1 VDWKUH*V LQHTXDOLW\ 6,B "W # "WNWB wN Ot|iBW| W? |i @h|U*ic @? ?i^@*|) Lu O ?U @?_ w 5@|hi EhLU ,_?Mh} @| 5LU e EbSe%SDc eeS t }i?ih@*3i_G wi| ? @?_ 6 Mi ?@|h@* ?4Mihtc & @ ?L??i}@|i ?|i}ihc |i? i @i s q E? n &-*&?n& q.ps ?n6n& E? n 6 n &-*&- 6hL4 |tc tL4i ULhL**@hit @hi _i_Ui_ | *@t|c @? LTi? ThLM*i4 t ThLTLti_ Lw lv nqrzq wkdw/ iru q 5 Q/ wkh iroorzlqj lqhtxdolwlhv zhuh jlyhq lq ^6`= s q q$ q s ? q.4 ? 4= q.4 +q . 4,$ Lq ^4`/ wkh ohiw lqhtxdolw| lq +4, zdv uhqhg e| # q ! s q.4 $4@u q q 4[ u q$ 4 [ u s ? l l ? q.4 q.4 q l@4 q . 4 l@4 +q . 4,$ +4, +5, iru doo srvlwlyh uhdo qxpehuv u1 Erwk erxqgv duh ehvw srvvleoh1 Lq wklv duwlfoh/ xvlqj dqdo|wlf phwkrg/ zh rewdlq Wkhruhp1 Ohw q dqg p eh qdwxudo qxpehuv/ n d qrqqhjdwlyh lqwhjhu1 Wkhq zh kdyh s q +q . n,$@n$ q.n s ? q. p ? 4= q.p.n +q . p . n,$@n$ +6, Surri1 Wkh xsshu erxqg lv rewdlqhg lpphgldwho| iurp %# q.n $p /# q.p.n $q &4@q+q.p, s q +q . n,$@n$ \ \ s l l ? 4= @ q.p +q . p . n,$@n$ [email protected] [email protected] Wkh ohiw lqhtxdolw| lq +6, fdq eh uhduudqjhg dv q.n q.p.n s s ? q. p > q +q . n,$@n$ +q . p . n,$@n$ wklv lv htxlydohqw wr q.n.4 q.n s s ? q.4 = q +q . n,$@n$ +q . n . 4,$@n$ bb @|i4@|Ut 5MiU| *@ttU@|L? h4@h) 2S#D ki) Lh_t @?_ Th@tit Bi?ih@*3@|L?c O ?U @?_ w 5@|hi<t ?i^@*|) Ai ht| @|Lh @t tTTLh|i_ ? T@h| M) 56 Lu Oi?@? hL?Uic Ai iLT*i<t +iTM*U Lu ?@ At T@Tih @t |)Titi| t?} }4wA,j 4 +7, 6,B "W # "WNWB wN 5 Zkhq n @ 3/ lqhtxdolw| +7, iroorzv iurp wkh ohiw lqhtxdolw| lq +4,1 Zkhq n4/ wkh lqhtxdolw| +7, fdq eh uhzulwwhq dv 4@q +q . n,q.4 +q . n,$ A = +8, n$ +q . n . 4,q Lq ^7/ s1 4;7`/ wkh iroorzlqj lqhtxdolwlhv zhuh jlyhq iru q 5 Q q q q q s s 4 = +9, 5q ? q$ ? 5q h{s h h 45q E| vxevwlwxwlqj wkh lqhtxdolwlhv lq +9, lqwr wkh ohiw whup ri lqhtxdolw| +8,/ zh vhh wkdw lw lv vxflhqw wr suryh s n q.n 4@q 4@q q.n +q . n,q.4 s n 4 5+q . n, A 5n h{s = +:, h +q . n . 4,q h 45n Vlpsoli|lqj +:, gluhfwo| dqg vwdqgdug dujxphqwv ohdgv wr q 4 5n . 4 4 oq 4 . 4 A 3= q oq 4 . . q.n 5q n 45nq +;, Lq ^5/ ss1 69:~69;`/ ^7/ ss1 5:6~5:7` dqg ^;`/ zh kdyh iru w A 3 4 5 = oq 4 . A w 5w . 4 Wkxv/ wr jhw lqhtxdolw| +;,/ lw vxfhv wr vkrz 5n . 4 5q 4 5q . 4 A 3= 5+q . n, . 4 5q 5n . q 45nq Exw wklv lv htxlydohqw wr 5+45n5 4,q5 . +45nq 4,q . 7+9q 4,n5 . 5+6q 4,n A 3= Wkh surri lv frpsohwh1 Fruroodu| 41 Iru dq| jlyhq qrqqhjdwlyh lqwhjhu n/ wkh vhtxhqfhv s q.n q +q . n,$@n$> s > q +q . n,$@n$ s s +q . n, q.4 +q . n . 4,$@n$ +q . n . 4, q +q . n,$@n$ s s > q +q . n,$@n$ q.4 +q . n . 4,$@n$ duh vwulfwo| lqfuhdvlqj zlwk uhvshfw wr q 5 Q1 Fruroodu| 51 Iru dq| jlyhq q 5 Q/ wkh vhtxhqfhv s q.4 s q +q . n,$@n$> +q . n,s +q . n . 4,$@n$ > q +q . n,$@n$ s +q . n . 4, q +q . n,$@n$ s q.4 +q . n . 4,$@n$ duh vwulfwo| lqfuhdvlqj zlwk uhvshfw wr wkh qrqqhjdwlyh lqwhjhuv n1 Uhpdun1 Uhfhqwo|/ wkh uvw dxwkru lq ^8` dqg ^:`/ dprqj rwkhu wklqjv/ jhqhudol}hg wkh ohiw lqhtxdolw| lq +5, lq qhz gluhfwlrqv dqg jrw wkdw/ li q dqg p duh qdwxudo qxpehuv/ n lv d qrqqhjdwlyh lqwhjhu/ wkhq # q.n ! $4@u q.[ p.n 4 [ u q.n 4 u ? l l > +<, q.p.n q q.p [email protected] [email protected] B,,+wW~AW 6 O W # w 5AO+,<5 W,"NwWAv 6 6,B "W # "WNWB wN zkhuh u lv dq| jlyhq srvlwlyh uhdo qxpehu1 Wkh orzhu erxqg lv ehvw srvvleoh1 Lq ^9`/ wkh uvw dxwkru ixuwkhu suhvhqwhg wkdw/ ohw q dqg p eh qdwxudo qxpehuv/ vxssrvh d @ +d4 > d5 > = = = , lv d srvlwlyh dqg lqfuhdvlqj vhtxhqfh vdwlvi|lqj d5n.4 dn dn.5 > n.4 n.5 dn.4 dn pd{ > d5n.4 dn dn.5 dn.4 dn.5 iru n 5 Q/ wkhq wkh lqhtxdolw| dq # dq.p q 4[ u d q l@4 l ! q[ .p 4 du q . p l@4 l +43, +44, $4@u = +45, krogv iru dq| jlyhq srvlwlyh uhdo qxpehu u 5 U1 Wkh orzhu erxqg ri +45, lv ehvw srvvleoh1 Xvlqj O*Krvslwdo uxoh |lhogv # q.n ! $4@u s q.[ p.n q +q . n,$@n$ 4 [ u 4 u s olp l l @ q.p > u$3 q q.p +q . p . n,$@n$ [email protected] [email protected] +46, wkxv/ zh sursrvh wkh iroorzlqj Rshq Sureohp1 Ohw q dqg p eh qdwxudo qxpehuv/ n d qrqqhjdwlyh lqwhjhu1 Wkhq/ iru doo uhdo qxpehuv u A 3/ zh kdyh # q.n ! $4@u s q.[ p.n q +q . n,$@n$ 4 [ u 4 u s l l ? q.p = +47, q q.p +q . p . n,$@n$ [email protected] [email protected] Wkh xsshu erxqg lv ehvw srvvleoh1 -jujijAWjt do O *3ihc ? @? ?i^@*|) Lu O ?U @?_ w 5@|hic a @| ?@* TT* .b Ebbc bSef2 d2o a@?} k@?}c TT*i_ W?i^@*|itc 2?_ i_|L?c O?@? ,_U@|L? hittc @?}t@c ?@c bb E?iti do O ?U @?_ w 5@|hic 5L4i ?i^@*|it ?L*?} Eo *o e ?@*)|U W?i^@*|it ? @*}iMh@U ?i^@*|) 6h|ih }i?ih@*3@|L? Lu O *3ih<t ?i^@*|) Bi?ih@*3@|L? Lu O *3ih<t ?i^@*|) L?L|L?U|it Lu |L ti^i?Uit - deo # 5 |h?L Uc dDo 6i?} "c c hLU ,_?Mh} @| 5LU EbSe%SDc eeS c 5Th?}ihVih*@}c i vLh!%Oi_i*Mih}%ih*?c b.f c tM4||i_ uLh TM*U@|L? dSo 6i?} "c c tM4||i_ uLh TM*U@|L? d.o 6i?} "c c tM4||i_ uLh TM*U@|L? dHo 6i?} " @?_ @L?} i?c c @|i4@|Ut @?_ W?uLh4@|Ut "@h|ih*) Ebbbc |L @TTi@h $jVBi|6jA| Nu B|j6B|Wte cBN5N YAt|||j Nu CjWAN,N}+e cBN5N |+e QjABA gFghhhe Cj jNV,j?t -jVO,W Nu AB ,4@* @__hitt Zqf={71zf[Q= $jVBi|6jA| Nu iNBaWBt|Cj,jtNA CjBWA}e cBN5N PAjit|+e cBN5N |+e QjABA gFgFe Cj jNV,j?t -jVO,W Nu AB G B,,+wW~AW 6 O W # w 5AO+,<5 W,"NwWAv
© Copyright 2025 Paperzz