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J@8<5@K 8A?>;L7>20=8O =5G5B:>9 ;>38:8&& V48=328AB8G5A:0O ?5@5<5==0O ?@2 (?>;=>5): 8=328AB8G5A:0O ?5@5<5==0O - =01>@ < b,T,X,G,M >, 345 b  =08<5=>20=85 ;8=328AB8G5A:>9 ?5@5<5==>9; "  <=>65AB2> 55 7=0G5=89 (107>2>5 B5@<-<=>65AB2>), ?@54AB02;ONI8E A>1>9 =08<5=>20=8O =5G5B:8E ?5@5<5==KE =0 >1;0AB8 >?@545;5=8O X; G  A8=B0:A8G5A:0O ?@>F54C@0, ?>72>;ONI0O 35=5@8@>20BL =>2K5 B5@<K. =>65AB2> T G(T), 345 G(T) - <=>65AB2> A35=5@8@>20==KE B5@<>2,  @0AH8@5==>5 B5@<-<=>65AB2>;   A5<0=B8G5A:0O ?@>F54C@0, ?@52@0I0NI0O :064K9 B5@< 87 G(T) 2 =5G5B:CN ?5@5<5==CN.4FP.0P2(0Z2U0P2 -3OO63TH - S     6  !0G !0XY&5GQB:0O ?5@5<5==0O H5G5B:0O ?5@5<5==0O E0@0:B5@87C5BAO B@>9:>9 < a,X,A >, 345 a - =08<5=>20=85 ?5@5<5==>9, X - C=825@A0;L=>5 <=>65AB2> (>1;0ABL >?@545;5=8O), A  DC=:F8O ?@8=04;56=>AB8, >?@545;Q==0O 4;O 2A5E M;5<5=B>2 xX 8 3>2>@OI0O > AB5?5=8 C25@5==>AB8 2 B><, GB> E O2;O5BAO 7=0G5=85< 40==>9 ?5@5<5==>9.=Z$0Z2$Z 32 Y<5GQB:>5 <=>65AB2> (fuzzy set)   : 5G5B:>5 <=>65AB2>  ?>4<=>65AB2> M;5<5=B>2  87 C=825@A0  B0:>5, GB> :064><C M;5<5=BC A>?>AB02;5=0 AB5?5=L ?@8=04;56=>AB8 MB>3> M;5<5=B0 <=>65AB2C A. 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Z 4 &6 3    2   C 2) 5@5A5G5=85 42CE >B=>H5=89 R1 )" R2 @8<5@: xR1y  <>4C;L @07=>AB8 |x-y| 1;87>: : , xR2y  <>4C;L @07=>AB8 |x-y| 1;87>: : , x(R1)"R2) y  <>4C;L @07=>AB8 |x-y| 1;87>: :  8 . P ,, " .P             F D><?>78F8O 42CE =5GQB:8E >B=>H5=89##   ,5G5B:85 2KA:07K20=8O  $0  $&m  .Ip>38:>-;8=328AB8G5A:>5 >?8A0=85 A8AB5<, =5G5B:85 <>45;8 99 Jp>38:>-;8=328AB8G5A:>5 >?8A0=85 A8AB5<, =5G5B:85 <>45;8  97:!>2>:C?=>ABL 8<?;8:0F89 {L1, L2, ..., Lk} >B@0605B DC=:F8>=0;L=CN 2708<>A2O7L 2E>4=KE 8 2KE>4=KE ?5@5<5==KE 8 O2;O5BAO >A=>2>9 ?>AB@>5=8O =5G5B:>3> >B=>H5=8O XRY, 7040==>3> =0 ?@>872545=88 X x Y C=825@A0;L=KE <=>65AB2 2E>4=KE 8 2KE>4=KE ?5@5<5==KE. B=>H5=85 R AB@>8BAO :0: .rZucB& [ Kp>38:>-;8=328AB8G5A:>5 >?8A0=85 A8AB5<, =5G5B:85 <>45;8  97 >A=>25 ?>AB@>5=8O ;>38:>-;8=328AB8G5A:8E A8AB5< ;568B :><?>78F8>==>5 ?@028;> 2K2>40. A;8 =0 <=>65AB25 X 7040=> =5G5B:>5 <=>65AB2> A, B> :><?>78F8>==>5 ?@028;> 2K2>40 B = AR, 345 R  =5:>B>@>5 =5G5B:>5 >B=>H5=85,>?@545;O5B =0 Y =5G5B:>5 <=>65AB2> B A DC=:F859 ?@8=04;56=>AB8: "0:8< >1@07><, :><?>78F8>==>5 ?@028;> 2K2>40 2 MB>< A;CG05 70405B 70:>= DC=:F8>=8@>20=8O =5G5B:>9 <>45;8 A8AB5<K.VZ85HV ` ` rL5GQB:89 2K2>4 ?@:5G5B:8< ;>38G5A:8< 2K2>4>< (fuzzy logic inference) =07K205BAO 0??@>:A8<0F8O 7028A8<>AB8 Y = f(X1,X2& Xn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{KA>:0O, !@54=OO, 87:0O}. $C=:F88 ?@8=04;56=>AB8 B5@<>2 8<5NB A;54CNI89 284: KA>:0O !@54=OO 87:0O 02;5=85. #=82Q@A  >B@57>: [0,100]. 0G0;L=>5 <=>65AB2> B5@<>2 {KA>:>5, !@54=55, 87:>5} $C=:F88 ?@8=04;56=>AB8 B5@<>2 8<5NB A;54CNI89 284: KA>:>5 !@54=55 87:>5 0AE>4. #=82Q@A  >B@57>: [0,8]. 0G0;L=>5 <=>65AB2> B5@<>2 {>;LH>9, !@54=89, 0;K9} $C=:F88 ?@8=04;56=>AB8 B5@<>2 8<5NB A;54CNI89 284 >;LH>9 !@54=89 0;K9 PaPPPGPPP         } K z !P j -B0? D0778D8:0F88 ! ?><>ILN DC=:F89 ?@8=04;56=>AB8 2A5E B5@<>2 2E>4=KE ;8=328AB8G5A:8E ?5@5<5==KE 8 =0 >A=>20=88 7040205<KE G5B:8E 7=0G5=89 87 C=825@A>2 2E>4=KE ;8=328AB8G5A:8E ?5@5<5==KE >?@545;ONBAO AB5?5=8 C25@5==>AB8 2 B><, GB> 2KE>4=0O ;8=328AB8G5A:0O ?5@5<5==0O ?@8=8<05B 7=0G5=85  :>=:@5B=K9 B5@<. -B0 AB5?5=L C25@5==>AB8 5ABL >@48=0B0 B>G:8 ?5@5A5G5=8O 3@0D8:0 DC=:F88 ?@8=04;56=>AB8 B5@<0 8 ?@O<>9 x = G5B:>5 7=0G5=85 .jZ" Z" ZX   } Q 2) -B0? =5?>A@54AB25==>3> =5GQB:>3> 2K2>40 0 >A=>20=88 =01>@0 ?@028;  =5G5B:>9 107K 7=0=89  2KG8A;O5BAO 7=0G5=85 8AB8==>AB8 4;O ?@54?>AK;:8 :064>3> ?@028;0 =0 >A=>20=88 :>=:@5B=KE =5G5B:8E >?5@0F89, A>>B25BAB2CNI8E :>=JN=:F88 8;8 487JN=:F88 B5@<>2 2 ;52>9 G0AB8 ?@028;. A?>;L7CO >48= 87 A?>A>1>2 ?>AB@>5=8O =5GQB:>9 8<?;8:0F88 <K ?>;CG8< =5GQB:CN ?5@5<5==CN, A>>B25BAB2CNICN 2KG8A;5==><C 7=0G5=8N AB5?5=8 C25@5==>AB8 2 ;52>9 G0AB8 ?@028;0 8 =5G5B:><C <=>65AB2C 2 ?@02>9 G0AB8 ?@028;0. 8Z*RS  T  U ^4) -B0? 45D0778D8:0F88 (=5>1O70B5;L=K9) A?>;L7C5BAO :>340 =5>1E>48<> >B ?>;CG5==>3> =5GQB:>3> <=>65AB20 ?5@59B8 : :>=:@5B=><C G8A;>2><C 7=0G5=8N. ;O 40==>3> MB0?0 ACI5AB2C5B 1>;55 30 <5B>4>2. 20 87 >1I8E <5B>4>2 - MB> <5B>4K ?>;=>9 8=B5@?@5B0F88 8 ?> <0:A8<C<C.  <5B>45 ?>;=>9 8=B5@?@5B0F88, B>G=>5 7=0G5=85 2K2>48<>9 ?5@5<5==>9 2KG8A;O5BAO :0: 7=0G5=85 "F5=B@0 BO65AB8" DC=:F88 ?@8=04;56=>AB8 4;O =5G5B:>3> 7=0G5=8O.  <5B>45 <0:A8<C<0 2 :0G5AB25 B>G=>3> 7=0G5=8O 2K2>48<>9 ?5@5<5==>9 ?@8=8<05BAO <0:A8<0;L=>5 7=0G5=85 DC=:F88 ?@8=04;56=>AB8.60P[\  ]  ^  _  `a  b  c  d  e  f  g  h  ij/VWX Y Z PM+  ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>>0 F(    6쪼 " `}  V"1@075F 703>;>2:0 j  0 " `  1@075F B5:AB0 B>@>9 C@>25=L "@5B89 C@>25=L '5B25@BK9 C@>25=L OBK9 C@>25=L M  0 "^ `  X*  0 "^   Z*  0Ȼ "Y  ^*l  FAH޽h ?2"@` 33___PPT10i.>Ї+D=' = @B + .D>@<;5=85 ?> C<>;G0=8N0 zr`{  (  |   0\ P   \ P*    0 \    \ R*  d  c $ ?  \  0h\  0 \ RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  6J\ _P  \ P*    6hL\ _  \ R*  H | 0޽h ? 3380___PPT10.5< pc (  d   0 P    X*   08H     Z*   6G _P   X*   6P _   Z* H d 0޽h ? 3380___PPT10.5     0  (  2 @ N耰a ~8653>@>4A:89 3>AC40@AB25==K9 C=825@A8B5B 8<. .. >10G52A:>3>@@f3 @ N j8=8AB5@AB2> >1@07>20=8O 8 =0C:8 >AA89A:>9 $545@0F8866f3  NX7' @f3   NdD b$0:C;LB5B 2KG8A;8B5;L=>9 <0B5<0B8:8 8 :815@=5B8:822f3:   Tݼ ?"6@ NNN?N   x8038AB@0=BK 1 3>40 >1CG5=8O:$r   TT ?"6@ NNN?N }  6 D8<>2 .!. 8 >@Q=>2 ..<  $  T ?"6@ NNN?N\ b.86=89 >23>@>4, 20043.n 0 T ?"6@ NNN?NPq  hA=>2K =5G5B:>9 ;>38:8 >38:>-;8=328AB8G5A:85 <>45;8(54(f3(  Nl ?"6@ NNN?NM,4 #G51=>-8AA;54>20B5;LA:0O ;01>@0B>@8O  =D>@<0F8>==K5 B5E=>;>388 "A0(Af34%  P  s *޽h ?"@` 33___PPT10i.>+D=' = @B +' @D(  D D Hϰ ?D l:A=>2K B5>@88 =5G5B:>9 ;>38:8*( D TӰ ?"6@ NNN?NZ=  f'0ABL I( : D Tװ ?"6@ NNN?NIi0 x A53> 19 A;094>2 H D 0޽h ? 33y___PPT10Y+D=' = @B +[  ZR@ (    T0/4 ?"6@ NNN?N :  N =>65AB20, 4;O :>B>@KE DC=:F8O ?@8=04;56=>AB8 ?@54AB02;O5B A>1>9 =5 65AB:89 ?>@>3 (?@8=04;568B/=5 ?@8=04;568B), 0 ?;02=CN :@82CN (G0AB> C?@>I05<CN ;><0=>9 ;8=859), ?@>1530NICN 2A5 7=0G5=8O >B =C;O 4> 548=8FK  =5G5B:85 <=>65AB20 (fuzzy sets)     T64 ?"6@ NNN?NR >;LH0O G0ABL 8A?>;L7C5<KE =0<8 ?>=OB89 ?> A2>59 ?@8@>45 =5G5B:8 8 @07<KBK:  ?>AB@>8BL ?>@>3>2CN DC=:F8N ?@8=04;56=>AB8 4;O <=>65AB2 27@>A;K9, E>;>4=K9, :0G5AB25==K9, 1KAB@K9 8 B.4. ???>W~333Z  T4>4 ?"6@ NNN?N9 ]J 0@B >A:>, >48= 87 :;0AA8:>2 =5G5B:>9 ;>38:8:  8=0@=0O ;>38:0 - =5 1>;55 G5< @>:>20O >H81:0 0=B8G=>9 F828;870F88. * 0FffL&  C   B$B4 'g  4 4   HF4 ?"6@ NNN?N<)} ~NN48, 2 A8;C ?@8@>4=>3> M3>87<0, A:;>==K >F5=820BL ;N1K5 40==K5 A B>G:8 7@5=8O ?>;57=>AB8 4;O !/: AB0=>2OBAO 206=K =5 A0<8 :>;8G5AB25==K5 40==K5, 0 8E 8=B5@?@5B0F8OZ   T(M4 ?"6@ NNN?N]7D d ?>=OB8O, :0B53>@88 8A?>;L7>20BL 4;O MB>3>???3332   B?"6@`NNN?N@ "R  C *Aej0149481jB  HD8c?"0@NNN?N 2  B?"6@`NNN?N   N ?"6@ NNN?N9 q<H  0޽h ? 33___PPT10i.+D=' = @B +L ' KC0 (  \   B 4 'g  4 T \ T  ?"6@ NNN?N  X5G5B:0O ;>38:0 - =04<=>65AB2> 1C;52>9 ;>38:8, @0AH8@5==>9 A F5;LN >1@01>B:8 7=0G5=89 8AB8==>AB8 <564C ?>;=>ABLN 8AB8==K< 8 ?>;=>ABLN ;>6=K< =0 >A=>25 =5G5B:8E <=>65AB2 \ T  ?"6@ NNN?NS \A>740=85 0??0@0B0, A?>A>1=>3> <>45;8@>20BL @0AAC645=8O =0 >A=>25 A;>6=KE ?@8G8==>-A;54AB25==KE A2O759  =5G5B:>9 ;>38:8 8 =5G5B:>3> 2K2>40 *-h" \ T4 ?"6@ NNN?N ' ~  .*@07@01>B0=0 ?@>D5AA>@>< 0;8D>@=89A:>3> C=825@A8B5B0 5@:;8 >BD8 . 045 (Lotfi A. Zadeh) 2 @01>B0E: Fuzzy sets (19653) 8 Fuzzy logic (19753).*< M<           B \ HD8c?"0@NNN?N   Tt4 ?"6@ NNN?Njk  =0 >A=>25 B0:8E ?>=OB89 (?@54AB02;5==KE =5G5B:8<8 <=>65AB20<8) A<>45;8@>20BL ?@>F5AA G5;>25G5A:8E @0AAC645=89 ???ww32  B?"6@`NNN?NM ]   N ?"6@ NNN?N ' V  C .AOzadehsmall6 C R  C *ANbuildingc o Y  Hd 4 ?"6@ NNN?N  a Professor Lotfi A. Zadeh H      d  H|%4 ?"6@ NNN?Nu O~ $ University of California, Berkeley%%P  H \ 0޽h ? 33___PPT10i.9+D=' = @B +S  ' R J   (    B  I     H$4 ?"6@ NNN?N7  ,2865=85< ?@83>@>4=KE ?>574>2 4> O?>=A:>3> 3>@>40 !5=409, =0G8=0O A 1987 3>40 C?@02;O5B A8AB5<0, >A=>20==0O =0 =5G5B:>9 ;>38:5 !>740=85 C?@02;ONI53> <8:@>?@>F5AA>@0 =0 >A=>25 =5G5B:>9 ;>38:8, A?>A>1=>3> 02B><0B8G5A:8 @5H0BL 7040GC > A>10:5, 4>3>=ONI59 :>B0 (8=8AB5@AB2> >1>@>=K !() Matsuhita 2 D52@0;5 1991 3>40 0=>=A8@>20;0 ?5@2CN 8=B5;;5:BC0;L=CN AB8@0;L=CN <0H8=C, 2 A8AB5<5 C?@02;5=8O :>B>@>9 ?@8<5=O;0AL =5G5B:0O ;>38:0 5H5=8O A;>6=59H8E 7040G ?@>3=>78@>20=8O @07;8G=KE D8=0=A>2KE 8=48:0B>@>2 (O?>=A:0O :>@?>@0F8O Yamaichi) 48= 87 >B5G5AB25==KE ?@>3@0<<=KE ?@>4C:B>2 - ?0:5B  87=5A-?@>3=>7 4;O >F5=:8 ?@81K;L=>AB8 8=25AB8F8>==KE ?@>5:B>2.Cq3Z>q3ZBq3Z^q3ZGq3ZNq3Z>q3Z,q3Z5q3ZBq3Z@1R2  zkkMM  <X4 ?"6@ NNN?NV -D 0<5G0=85: 5G5B:85 A8AB5<K C?@02;5=8O 8 ?@>3=>78@>20=8O >A=>20=K =0 =5G5B:>9 1075 7=0=89 8 8A?>;L7>20=88 ;8=328AB8G5A:8E ?5@5<5==KE.  >A=>2C DC=:F8>=8@>20=8O ?>;>65= <5E0=87< =5G5B:>3> 2K2>40.V g3133  N ?"6@ NNN?NV ,,H  0޽h ? 33___PPT10i.cP7>+D=' = @B +w  vn` (  P x  c $ I9     s *H }D   ,r?@ABz  T  ?"6@ NNN?N R?@1 (C?@>I5==>5): 8=328AB8G5A:0O ?5@5<5==0O - ?5@5<5==0O, ?@8=8<0NI0O 7=0G5=8O 87 <=>65AB20 A;>2 8;8 A;>2>A>G5B0=89 =5:>B>@>3> 5AB5AB25==>3> 8;8 8A:CAAB25==>3> O7K:0. 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J@8<5@K 8A?>;L7>20=8O =5G5B:>9 ;>38:8&& V48=328AB8G5A:0O ?5@5<5==0O ?@2 (?>;=>5): 8=328AB8G5A:0O ?5@5<5==0O - =01>@ < b,T,X,G,M >, 345 b  =08<5=>20=85 ;8=328AB8G5A:>9 ?5@5<5==>9; "  <=>65AB2> 55 7=0G5=89 (107>2>5 B5@<-<=>65AB2>), ?@54AB02;ONI8E A>1>9 =08<5=>20=8O =5G5B:8E ?5@5<5==KE =0 >1;0AB8 >?@545;5=8O X; G  A8=B0:A8G5A:0O ?@>F54C@0, ?>72>;ONI0O 35=5@8@>20BL =>2K5 B5@<K. =>65AB2> T G(T), 345 G(T) - <=>65AB2> A35=5@8@>20==KE B5@<>2,  @0AH8@5==>5 B5@<-<=>65AB2>;   A5<0=B8G5A:0O ?@>F54C@0, ?@52@0I0NI0O :064K9 B5@< 87 G(T) 2 =5G5B:CN ?5@5<5==CN.4FP.0P2(0Z2U0P2 -3OO63TH - S     6  !0G !0XY&5GQB:0O ?5@5<5==0O H5G5B:0O ?5@5<5==0O E0@0:B5@87C5BAO B@>9:>9 < a,X,A >, 345 a - =08<5=>20=85 ?5@5<5==>9, X - C=825@A0;L=>5 <=>65AB2> (>1;0ABL >?@545;5=8O), A  DC=:F8O ?@8=04;56=>AB8, >?@545;Q==0O 4;O 2A5E M;5<5=B>2 xX 8 3>2>@OI0O > AB5?5=8 C25@5==>AB8 2 B><, GB> E O2;O5BAO 7=0G5=85< 40==>9 ?5@5<5==>9.=Z$0Z2$Z 32 Y<5GQB:>5 <=>65AB2> (fuzzy set)   : 5G5B:>5 <=>65AB2>  ?>4<=>65AB2> M;5<5=B>2  87 C=825@A0  B0:>5, GB> :064><C M;5<5=BC A>?>AB02;5=0 AB5?5=L ?@8=04;56=>AB8 MB>3> M;5<5=B0 <=>65AB2C A. Bq3V 0   Y  D?5@0F88 =04 =5GQB:8<8 <=>65AB20<8## N0=K =5G5B:85 <=>65AB20  8  A DC=:F8O<8 ?@8=04;56=>AB8: m(u) 8 mB(u), B>340 @57C;LB0B >?5@0F89 - =5G5B:>5 <=>65AB2> A DC=:F859 ?@8=04;56=>AB8 mC(u), ?@8G5<: 5A;8 ! =  , B> mC(u) = min(m(u), mB(u)); 5A;8 ! =  , B> mC(u) = max(m(u), mB(u)); 5A;8 ! = , B> mC(u) = 1 - m(u). 2UP:N     )    O !         4V!2>9AB20 >?5@0F89 =04 =5GQB:8<8 <=>65AB20<8,, CABL , , ! - =5G5B:85 <=>65AB20, B>340 2K?>;=ONBAO A;54CNI85 A2>9AB20: 1)><<CB0B82=>ABL: 2)AA>F80B82=>ABL: 3)45<?>B5=B=>ABL: 4)8AB@81CB82=>ABL: 5)0:>=K 45 >@30=0: 6) , 345 - ?CAB>5 <=>65AB2>, B.5 . 7) 8)`JZ=?$5GQB:85 >B=>H5=8O(!B5?5=L ?@8=04;56=>AB8 ?>:07K205B AB5?5=L 2K?>;=5=8O >B=>H5=8O R <564C M;5<5=B0<8  A;CG05 :>=5G=KE 8;8 AG5B=KE C=825@A>2 8=B5@?@5B0F8O 18=0@=>3> =5G5B:>3> >B=>H5=8O 2 2845 2725H5==>3> 3@0D0 A 25A0<8 R(xi, yj) 8;8 <0B@8F59k0q3Z2q3ZX    ( 0 P    BVA=>2=K5 >?5@0F88 =04 =5GQB:8<8 >B=>H5=8O<8,,  1) 1J548=5=85 42CE >B=>H5=89 R1 R2. @8<5@: xR1y - 459AB28B5;L=K5 G8A;0 x 8 y >G5=L 1;87:85, xR2y - G8A;0 x 8 y >G5=L @07;8G=K, x(R1R2)y - "G8A;0 x 8 y >G5=L 1;87:85 8;8 >G5=L @07;8G=K5". 0<5G0=85: $C=:F88 ?@8=04;56=>AB8 >B=>H5=89 7040=K =0 |x-y|.  Z 4 &6 3    2   C 2) 5@5A5G5=85 42CE >B=>H5=89 R1 )" R2 @8<5@: xR1y  <>4C;L @07=>AB8 |x-y| 1;87>: : , xR2y  <>4C;L @07=>AB8 |x-y| 1;87>: : , x(R1)"R2) y  <>4C;L @07=>AB8 |x-y| 1;87>: :  8 . P ,, " .P             F D><?>78F8O 42CE =5GQB:8E >B=>H5=89##   ,5G5B:85 2KA:07K20=8O  $0  $&m  .Ip>38:>-;8=328AB8G5A:>5 >?8A0=85 A8AB5<, =5G5B:85 <>45;8 99 Jp>38:>-;8=328AB8G5A:>5 >?8A0=85 A8AB5<, =5G5B:85 <>45;8  97:!>2>:C?=>ABL 8<?;8:0F89 {L1, L2, ..., Lk} >B@0605B DC=:F8>=0;L=CN 2708<>A2O7L 2E>4=KE 8 2KE>4=KE ?5@5<5==KE 8 O2;O5BAO >A=>2>9 ?>AB@>5=8O =5G5B:>3> >B=>H5=8O XRY, 7040==>3> =0 ?@>872545=88 X x Y C=825@A0;L=KE <=>65AB2 2E>4=KE 8 2KE>4=KE ?5@5<5==KE. B=>H5=85 R AB@>8BAO :0: .rZucB& [ Kp>38:>-;8=328AB8G5A:>5 >?8A0=85 A8AB5<, =5G5B:85 <>45;8  97 >A=>25 ?>AB@>5=8O ;>38:>-;8=328AB8G5A:8E A8AB5< ;568B :><?>78F8>==>5 ?@028;> 2K2>40. 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