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("030=@>3, 2005), <564C=0@>4=>9 =0CG=>9 :>=D5@5=F88 &8D@>2K5 <5B>4K 8 B5E=>;>388 ("030=@>3, 2005), 10-9 564C=0@>4=>9 " ABC45=B>2 8 0A?8@0=B>2 048>M;5:B@>=8:0, M;5:B@>B5E=8:0 8 M=5@35B8:0 C1;8:0F8O. > B5<5 48AA5@B0F88 >?C1;8:>20=0 <>=>3@0D8O 2 A>02B>@AB25, B@8 AB0BL8, B@8=04F0BL B578A>2 4>:;04>2 =0 =0CG=KE :>=D5@5=F8OE @07=>3> C@>2=O. !B@C:BC@0 8 >1J5< @01>BK. 8AA5@B0F8O A>45@68B 177 AB@0=8F <0H8=>?8A=>3> B5:AB0, 2:;NG0O 22545=85, G5BK@5 @0745;0, 70:;NG5=85, ?@8;>65=85 A>45@68B 29 AB@0=8F, A?8A>: 8AB>G=8:>2 87 105 =08<5=>20=89, 93 @8AC=:0, 14 B01;8F.  " !  !! "& > 22545=88 >1>A=>20=0 0:BC0;L=>ABL 8AA;54>20=8O, AD>@<C;8@>20=K F5;L, =0CG=0O =>287=0, ?@0:B8G5A:0O F5==>ABL @01>BK, >A=>2=K5 ?>;>65=8O, 2K=>A8<K5 =0 70I8BC, 4>AB>25@=>ABL 8 >1>A=>20==>ABL =0CG=KE ?>;>65=89 48AA5@B0F88, 0?@>10F8O @01>BK.  ?5@2>< @0745;5 @07@01>B0= <5B>4 D>@<0;870F88 8=D>@<0F8>==KE ?0@0<5B@>2 <548F8=A:>3> CG@5645=8O. A5 <548F8=A:85 CG@5645=8O :;0AA8D8F8@>20=K ?> B8?C <548F8=A:>3> CG@5645=8O 8 ?> ?@8=F8?0< @01>BK. K45;5=K >A=>2=K5 7040G8 <548F8=A:8E CG@5645=89 @07;8G=>3> B8?0. =0;87 H8@>:> 8725AB=KE >?@545;5=89 A8AB5<K, >A>15==>AB59 >@30=870F88 8 ?@8=F8?>2 DC=:F8>=8@>20=8O <548F8=A:8E CG@5645=89 ?>72>;8; D>@<0;L=> >?@545;8BL <548F8=A:>5 CG@5645=85 2 2845 =01>@0: S(<A, QA, R, QR, B, Z, CU, (T, N, LN>. (1) 345:A={a1,a2,& ,an} <=>65AB2> M;5<5=B>2; QA={q(a1),q(a2),& ,q(an)} <=>65AB2> A2>9AB2 M;5<5=B>2, R={r1,r2,& ,rm} <=>65AB2> DC=:F8>=0;L=KE 8 8=D>@<0F8>==KE A2O759 <564C M;5<5=B0<8; QR={q(r1),q(r2),& ,q(rm)} <=>65AB2> A2>9AB2 A2O759 M;5<5=B>2; B={B1,B2,& ,Bn} 25:B>@ :>=AB@C:B82=KE ?0@0<5B@>2 <548F8=A:>3> CG@5645=8O; Z F5;L, =0?@02;5==0O =0 4>AB865=85 M:AB@5<0;L=KE 7=0G5=89 :@8B5@852 MDD5:B82=>AB8 DC=:F8>=8@>20=8O; U CA;>28O F5;5>1@07>20=8O; (T 8=B5@20; 2@5<5=8, 2 B5G5=85 :>B>@>3> 1C45B ACI5AB2>20BL <548F8=A:>5 CG@5645=85; N ;8F0 8;8 A>2>:C?=>ABL ;8F, ?@8=8<0NI8E @5H5=8O, LN O7K: >1I5=8O =01;N40B5;59. 5:B>@ :>=AB@C:B82=KE ?0@0<5B@>2 <548F8=A:>3> CG@5645=8O - B={B1,B2,& ,Bn}, 345 Bi  i-9 M;5<5=B CG@5645=8O, >?8AK205<K9 B0:65 25:B>@><  EMBED Equation.2 . 7<5@8<>5 A>AB>O=85 i-3> M;5<5=B0 <548F8=A:>3> CG@5645=8O >?@545;O5B :><?>=5=B  EMBED Equation.2  25:B>@0 Bi. @54AB02;5=8O > F5;8 8 D>@<C;8@>2:8 F5;8 7028AOB >B 7040G C?@02;5=8O >1J5:B0<8 <548F8=A:>3> CG@5645=8O, 0 ?> <5@5 @5H5=8O 7040G F5;L <>65B 1KBL 87<5=5=0. $>@<C;8@>2:8 F5;8 7028AOB >B 2=5H=8E ?> >B=>H5=8N : A8AB5<5 8 2=CB@5==8E D0:B>@>2. 725AB=K @07;8G=K5 <5B>4K A8AB5<=>3> 0=0;870, >?@545;ONI85 ?>@O4>:, <5B>4K D>@<8@>20=8O 8 >F5=:8 ?@8>@8B5B>2 M;5<5=B>2 AB@C:BC@ F5;59. 4=0:> :064K9 87 <5B>4>2 8<55B A2>8 =54>AB0B:8, 2:;NG0O >@85=B0F8N =0 >?8A0=85 AB0B8G5A:>3> A>AB>O=8O <548F8=A:>3> CG@5645=8O.  <5B>40E =5B A@54AB2, :>B>@K5 ?><>30;8 1K 2KO28BL =>2K5 >1J5:BK, =>2K5 DC=:F88, B0:85, :0: B5E=>;>388 ;5G5=8O ?0F85=B>2, >1A;C6820=8O ?0F85=B>2, =>2>22545=8O 2 C?@02;5=G5A:>9 45OB5;L=>AB8. @54;0305BAO <5B>4 D>@<8@>20=8O 8 0=0;870 AB@C:BC@ F5;59 8 DC=:F89, :>B>@K9 2:;NG05B <5B>4K AB@C:BC@870F88, <5B>4K >F5=:8 AB@C:BC@ 8 ?@54CA<0B@8205B 2>7<>6=>ABL 2K1>@0 ?5@2>=0G0;L=>3> 20@80=B0 AB@C:BC@K 8 <5B>4>2 53> >F5=:8, =081>;55 ?>4E>4OI8E 4;O A>>B25BAB2CNI53> ?5@8>40 @0728B8O <548F8=A:>3> CG@5645=8O. !B@C:BC@0 F5;59 8 DC=:F89 D>@<8@C5BAO 2 2845 <>45;8, 7040==>9 3@0D>< (@8A. 1)  EMBED Word.Picture.6  8A.1 AE>4=K<8 40==K<8 4;O @5H5=8O 7040G8 AB@C:BC@870F88 F5;59 O2;ONBAO: 3;>10;L=0O F5;L Z <548F8=A:>3> CG@5645=8O; <=>65AB2> :@8B5@852 K={k1,k2,& ,km}, >?@545;ONI8E MDD5:B82=>ABL DC=:F8>=8@>20=8O <548F8=A:>3> CG@5645=8O; <=>65AB2> M;5<5=B>2 8 ?>4A8AB5< <548F8=A:>3> CG@5645=8O S={s1,s2,& ,sk}; <=>65AB2> @5H05<KE 7040G F={f1,f2,& ,fr}; <=>65AB2> D={d1,d2,& ,db}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quation.2 =<{(V(U(PP(W(S)((Z1(Z2(& (Zn)}, (K(PW(R(RP),  EMBED Equation.2 >, (3) 345  EMBED Equation.2  - =5G5B:89 3@0D8: =5G5B:>3> A>>B25BAB28O  EMBED Equation.2 . 564C M;5<5=B0<8 <=>65AB20 Z,- M;5<5=B0<8 <=>65AB2 X 8 Z ACI5AB2C5B A>>B25BAB285 (DC=:F8O ?5@5E>4>2), :>B>@>5 2 >1I5< A;CG05 B0:65 A;54C5B @0AA<0B@820BL :0: =5G5B:>5 A>>B25BAB285  EMBED Equation.2 =<{(V(U(PP(W(S)((Z1(Z2(& (Zn)}, (Z1(Z2(& (Zn),  EMBED Equation.2 >, (4) 345  EMBED Equation.2  - =5G5B:89 3@0D8: =5G5B:>3> A>>B25BAB28O  EMBED Equation.2 . 0AA<>B@5=K ?@8=F8?K ?>AB@>5=8O 8=D>@<0F8>==>9 <>45;8, 7040G8 C?@02;5=8O <548F8=A:8< CG@5645=85<, AB@C:BC@=K9 0=0;87 <548F8=A:>3> CG@5645=8O. @54;>65= <5B>4 @07@01>B:8 8=D>@<0F8>==>9 <>45;8, >B;8G0NI89AO A8AB5<0B870F859 @57C;LB0B>2 >?@>A0 M:A?5@B>2 8 0;3>@8B<870F859 8=D>@<0F8>==KE ?>B>:>2 8 >1@01>B:8 8=D>@<0F88. > 2B>@>< @0745;5 @07@01>B0=0 B5>@5B8:>-<=>65AB25==0O <>45;L ?@8=OB8O @5H5=89, :>B>@0O <>65B 1KBL ?@8<5=5=0 4;O >?@545;5=8O A>AB>O=8O ?0F85=B0 8 2K1>@0 4;O =53> B5@0?52B8G5A:>3> :C@A0. A>1K9 8=B5@5A ?@8 ?>AB@>5=88 <>45;8 ?@8=OB8O @5H5=89 8<55B >1>1I5=85 @5H5=89 <=>65AB20 M:A?5@B>2 - 3@C??>2>5 @5H5=85. >4 3@C??>2K< ?@8=OB85< @5H5=89 2 <548F8=A:>9 ?@0:B8:5 1C45< ?>=8<0BL 2K@01>B:C A>3;0A>20==>3> 3@C??>2>3> @5H5=8O > ?>@O4:5 ?@54?>GB8B5;L=>AB8 >1J5:B>2 2K1>@0, A>AB02;ONI8E ?@54<5B=CN >1;0ABL <548F8=A:>9 ?@0:B8:8, =0 >A=>25 8=48284C0;L=KE <=5=89 G;5=>2 3@C??K (2@0G59-M:A?5@B>2). @>1;5<0 3@C??>2>3> ?@8=OB8O @5H5=89 2 <548F8=A:>9 ?@0:B8:5 A2>48BAO : 03@538@>20=8N 8=48284C0;L=KE ?@54?>GB5=89 f1,f2,& ,fn n-G8A;0 2@0G59-M:A?5@B>2 2 548=>5 (3@C??>2>5) ?@54?>GB5=85 f. @8 >?8A0=88 <>45;8 ?@8=OB8O @5H5=8O >B=>A8B5;L=> A>AB>O=8O ?0F85=B0 A CG5B>< 3@C??>2>3> 2K1>@0 8A?>;L7>20=> ?>=OB85 :2078A5@88 8 >B=>H5=8O =5@07;8G8<>AB8. CABL A - <=>65AB2> B5@0?52B8G5A:8E :C@A>2, f(a)->F5=:0 B5@0?52B8G5A:>3> :C@A0, :>B>@K9 <>6=> =07=0G8BL ?0F85=BC ?@8 >?@545;5==>9 =>7>;>388. ! DC=:F859 f A2O70=> >B=>H5=85 Pf, A>AB>OI55 87 2A5E ?0@ (0,b), 4;O :>B>@KE :C@A a ;CGH5 :C@A0 b: (a,b)(Pf(f(a)>f(b). (5) 0AA<>B@8< >B=>H5=85 =5@07;8G8<>AB8  EMBED Equation.3 . "0: :0: (a,b)(If(f(a)=f(b), (6) B> >BAN40 A;54C5B B@0=78B82=>ABL >B=>H5=8O If : [(a,b)(If 8(b,c)(If] ((a,c)(If. (7) B=>H5=85 If @5D;5:A82=> ((a,a)(If) 8 A8<<5B@8G=> ((a,b)(If((b,a)(If).  @50;L=>9 687=8 8A?>;L7CNBAO >B=>H5=8O ?@54?>GB5=8O 8 =5@07;8G8<>AB8 ?> =5A:>;L:8< ?>:070B5;O<. CABL 701>;520=85 ?0F85=B0 a(A E0@0:B5@87C5BAO n ?>:070B5;O<8 f1(a),& ,fn(a). C45< AG8B0BL, GB> aRb!(f1(a),& ,fn(a))e"(f1(b),& ,fn(b))!fi(a)e"fi(b) (i=1,& ,n) 1>7=0G8< G5@57 Ri ;8=59=K9 :2078?>@O4>:, ?>@>6405<K9 ?>:070B5;5< fi: (a,b)(Ri(fi(a)(fi(b). ">340 R=R1)"R2)"& )"Rn. (8) A;8 A>AB>O=8N ?0F85=B0 A>>B25BAB2CNB >B=>H5=8O Rk (k(K) =0 <=>65AB25 :;8=8G5A:8E ?@>O2;5=89 , B> MB8 >B=>H5=8O =5 >1O70B5;L=> 1C4CB 18=0@=K<8. 0?@8<5@, ?@8 87<5@5=88 ?@54?>GB5=8O R <>65B A>AB>OBL 87 B5E 8 B>;L:> B5E C?>@O4>G5==KE G5B25@>: (a,b,A,d), 4;O :>B>@KE 0 1>;55 ?@54?>GB8B5;L=> ?> A@02=5=8N A b, G5< A ?> A@02=5=8N A d.  MB>< A;CG05 R(A(A(A(A=A4. > 2@0G51=>9 ?@0:B8:5 <>3CB 1KBL ?@8<5=5=K @07=K5 <>45;8 =5G5B:>3> ;>38G5A:>3> 2K2>40: <>45;L :;0AA8D8:0F88, <>45;L 2KG8A;5=8O AB5?5=8 8AB8==>AB8 =5G5B:8E ?@028; 2K2>40, A8BC0F8>==0O <>45;L ?@8=OB8O @5H5=89. 725AB5= ?>4E>4 : @07@01>B:5 <>45;59 ?@8=OB8O @5H5=89, 2 :>B>@KE ?@8=8<05BAO @5H5=85 =0 >A=>25 0=0;870 @50;L=KE =5G5B:8E A8BC0F89 8 A>?>AB02;5=8O 8E A MB0;>==K<8 =5G5B:8<8 A8BC0F8O<8. 0==K5 =5G5B:85 A8BC0F88 >?@545;ONB M:A?5@BK 8 8< A>?>AB02;ONB @5H5=8O. @8<5=8B5;L=> : 2@0G51=>9 ?@0:B8:5 7040G0 A2>48BAO : A>?>AB02;5=8N @50;L=>3> A>AB>l < > ʿʴթ՛th_h_TTTIhkhS&CJaJhkh*@CJaJhQU5CJaJhkhu5CJaJhkh#CJaJh#CJaJhuCJaJhQUhu5CJaJhQUh*@5CJ\aJhkhRCJaJhQUhuCJ aJ hQUh6mCJ aJ hQUh*@CJ aJ hkhuCJaJhQUCJaJhQUhQUCJ\hQUhQU5CJ\aJl , + a^+ `agdQU $`a$gd9`gd9 $`a$gd9$d&dPa$gdQU$&dPa$gdQUv4w, 2 h j l n p r t v x 8:<j $`a$gd9 $`a$gd9`gd9$4 7$8$^4 a$gdQU h j l r t  4 6 T V h &(68:<fhjlȽȽȽȶȽȜ{tmf_ h%rh# h%rhH h%rhb h%rhCm h%rhVn h%rhh%rh6h%rhS&mH sH  h%rh6mhAv h%rh!\h%rhu6 h%rh; h%rhh 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