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"01;8F0 2.7 1>7=0G5=8508<5=>20=850:8<8 725=LO<8 >1@07>20=0;0AA%0@0:B5@8AB8:001@0I0B5;L=0O>;5A> 1 - AB>9:0 05;>A:0O, =87H0OC1G0B0O>;5A> 1 - :>;5A> 24;>A:0O, 2KAH0OC;0G:>20OC;0G>: 2 - @>;8: 74;>A:0O, 2KAH0O!@0I0B5;L=0O >;8: 7 - :>@><KA;> 85;>A:0O, =87H0O08@0I0B5;L=0O>@><KA;> 8 - AB>9:0 05;>A:0O, =87H0O02@0I0B5;L=0OC;0G>: 2 - AB>9:0 05;>A:0O, =87H0O@0I0B5;L=0OC;0G>: 2 - :C;8A0 35;>A:0O, =87H0OD>ABC?0B5;L=0OC;8A0 3 - :>@><KA;> 45;>A:0O, =87H0O04@0I0B5;L=0O>@><KA;> 4 - AB>9:0 05;>A:0O, =87H0OF@0I0B5;L=0O>@><KA;> 4 - H0BC= 55;>A:0O, =87H0OJ@0I0B5;L=0O(0BC= 5 - ?>;7C= 65;>A:0O, =87H0O>ABC?0B5;L=0O>;7C= 6 - AB>9:0 05;>A:0O, =87H0O !B5?5=L ?>4286=>AB8 <5E0=87<0: W = 3((9 ( 1) ( 2(10 ( 1(2 = 2, ?>;CG5==K5 425 AB5?5=8 A2>1>4K >?@545;ONB: >A=>2=CN DC=:F8N <5E0=87<0 ?@5>1@07>20=85 2E>4=>3> 42865=8O f1 2 420 DC=:F8>=0;L=> 2708<>A2O70==KE f8 8 S6. 0AA82=0O 2KAH0O :8=5<0B8G5A:0O ?0@0 70<5=O5B 2 ?0@5 :C;0G>: - B>;:0B5;L B@5=85 A:>;L65=8O B@5=85< :0G5=8O 57C;LB0BK AB@C:BC@=>3> 0=0;870 87>1@065=K =0 @8A. 2.17. 25=> 7 8 ?0@0 ! 22545=K 2 AB@C:BC@C <5E0=87<0 A F5;LN 70<5=K B@5=8O A:>;L65=8O B@5=85< :0G5=8O. 5E0=87< 8<55B >4=C >A=>2=CN ?>4286=>ABL 8, A;54>20B5;L=>, >48= =0G0;L=K9 <5E0=87<, A>AB>OI89 87 725=L52 1 8 0.  C;0G:>2K9 <5E0=87< W = 3((4  1) - 2(3 - 1(1 = 2; 157 CG5B0 ?0AA82=>3> 725=0 7 W = 3((3  1) - 2(2 - 1(1 = 1 02     8  ! 08 2 7  @C??0 AAC@0 5-6 @C??0 AAC@0 3-4 6 J W3@ = 3(2 - 2(3 = 0; 3 C 5 F 4 D K W3@ = 3(2 - 2(3 = 0; E 0G0;L=K9 <5E0=87<  1  01  W?< = 3((2-1) - 2(1 = 1; 8A. 2.17 0<5=0 2 ?;>A:8E <5E0=87<0E 2KAH8E ?0@ =87H8<8 ;>A:85 <5E0=87<K <>3CB A>AB>OBL 87 725=L52 2E>4OI8E :0: 2 =87H85, B0: 8 2KAH85 ?0@K. @8 87CG5=88 AB@C:BC@K 8 :8=5<0B8:8 ?;>A:8E <5E0=87<>2 C4>1=> 70<5=OBL 2KAH85 :8=5<0B8G5A:85 ?0@K :8=5<0B8G5A:8<8 F5?O<8, 8;8 725=LO<8 2E>4OI8<8 B>;L:> 2 =87H85 2@0I0B5;L=K5 8 ?>ABC?0B5;L=K5 ?0@K 5-3> :;0AA0. @8 B0:>9 70<5=5 4>;6=K 2K?>;=OBLAO CA;>28O: GB>1K <5E0=87<, ?>;CG5==K9 ?>A;5 B0:>9 70<5=K, >1;040; ?@56=59 AB5?5=LN ?>4286=>AB8 8 GB>1K A>E@0=8;8AL >B=>A8B5;L=K5 2 @0AA<0B@8205<>< ?>;>65=88 42865=8O 2A5E 53> 725=L52. 0AA<>B@8< ?;>A:89 <5E0=87< @8A. 2.18: >= A>AB>8B 87 3-E 725=L52 8 3-E :8=5<0B8G5A:8E ?0@ (0-1, 2-0  5-3> :;0AA0, 2@0I0B5;L=K5; 1-2  4-3> :;0AA0 2KAH0O, A;>6=0O), 0 8 2  M;5<5=BK 725=L52 1 8 2 >:@C6=>AB8 @048CA0<8 01 8 02! A>>B25BAB25==>  8A. 2.18 !B5?5=L ?>4286=>AB8 <5E0=87<0: W = 3((3 ( 1) ( 2(2 ( 1(1 = 1, 0AA<0B@8205<K9 <5E0=87< <>6=> 70<5=8BL M:2820;5=B=K< 5<C H0@=8@=K< G5BK@QE725==K< <5E0=87<>< 01!02. KAH0O :8=5<0B8G5A:0O ?0@0 4-3> :;0AA0 2 B>G:5  70<5=O5BAO 725=>< 3, >1@07CNI8< 2 B>G:0E  8 ! 2@0I0B5;L=K5 ?0@K 5-3> :;0AA0. 5E0=87< 01!02 =07K20NB 70<5I0NI8<, 53> AB5?5=L ?>4286=>AB8: W = 3((4 ( 1) ( 2(4 = 1, B.:. M;5<5=BK 0 8 2 725=L52 1 8 2 O2;ONBAO >:@C6=>ABO<8 A F5=B@0<8 2 B>G:0E  8 !, B> 4;8=0 ! 725=0 3 O2;O5BAO ?>AB>O==>9, 4;8=K 01 8 !02 O2;ONBAO ?>AB>O==K<8, >BAN40 >B=>A8B5;L=K5 42865=8O 725=L52 1 8 2 A>E@0=OBAO. >=B@>;L=K5 2>?@>AK 0?8H8B5 D>@<C;K 4;O >?@545;5=8O AB5?5=8 ?>4286=>AB8 ?@>AB@0=AB25==>3> 8 ?;>A:>3> <5E0=87<>2? >G5<C B@51C5BAO ?@8 0=0;875 2KO2;OBL 2 AB@C:BC@5 <5E0=87<0 ?0AA82=K5 725=LO? 'B> B0:>5 3@C??0 AAC@0? 09B5 >?@545;5=85 :;0AA0, ?>@O4:0 8 2840 3@C??K AAC@0? @82548B5 ?>A;54>20B5;L=>ABL AB@C:BC@=>3> 0=0;870 <5E0=87<>2?  G5< 70:;NG0NBAO CA;>28O 70<5=K 2KAH8E ?0@ =87H8<8 2 ?;>A:8E <5E0=87<0E? 5:F8O ! 3 8=5<0B8G5A:89 0=0;87 <5E0=87<>2. 040G8 :8=5<0B8G5A:>3> 0=0;870. >=OB85 > 35><5B@8G5A:8E 8 :8=5<0B8G5A:8E E0@0:B5@8AB8:0E <5E0=87<>2 (DC=:F8O ?>;>65=8O 8 5Q ?@>872>4=K5 ?> 2@5<5=8 8 ?> >1>1I5==>9 :>>@48=0B5). @0D8G5A:85 <5B>4K :8=5<0B8G5A:>3> 0=0;870: <5B>4 ?;0=>2 8 4803@0<<. &8:; 8 F8:;>2K5 3@0D8:8. !2O7L <564C :8=5<0B8G5A:8<8 8 35><5B@8G5A:8<8 ?0@0<5B@0<8. 8=5<0B8G5A:>5 8AA;54>20=85 B8?>2KE <5E0=87<>2. A=>2=K5 7040G8 :8=5<0B8G5A:>3> 8AA;54>20=8O <5E0=87<>2  @0745;5 87CG05BAO 42865=85 >B45;L=KE 725=L52 <5E0=87<0 157 CG5B0 D0:B>@>2 >1CA;02;820NI8E 8E 42865=85, :0:8<8 O2;ONBAO A8;K, 459AB2CNI85 2 <5E0=87<5. AO:>5 42865=85 B5;0 E0@0:B5@87C5BAO ?5@5<5I5=85< 53> 2 ?@>AB@0=AB25, A:>@>ABLN 8 CA:>@5=85< 42865=8O 53> B>G5:. 8=5<0B8G5A:89 0=0;87 <5E0=87<>2 70:;NG05BAO 2 >?@545;5=88 ?0@0<5B@>2 42865=8O 725=L52 <5E0=87<0 ?> 7040==><C 70:>=C 42865=8O =0G0;L=>3> 725=0 (157 CG5B0 A8;, >1CA;02;820NI8E MB> 42865=85) 8 ?@54CA<0B@8205B @5H5=85 A;54CNI8E >A=>2=KE 7040G: >?@545;5=85 :>>@48=0B 8 @07<5B:0 B@05:B>@88 42865=8O 2A5E E0@0:B5@=KE B>G5: <5E0=87<0, GB> ?>72>;O5B @0F8>=0;L=> A?@>5:B8@>20BL :>@?CA=K5 45B0;8 <5E0=87<0; >?@545;5=85 A:>@>AB59 E0@0:B5@=KE B>G5: <5E0=87<0 2 @07;8G=KE 53> ?>;>65=8OE, GB> ?>72>;O5B >?@545;8BL :8=5B8G5A:CN M=5@38N 2A5E ?>4286=KE 725=L52 <5E0=87<0; >?@545;5=85 CA:>@5=89 E0@0:B5@=KE B>G5: <5E0=87<0, GB> ?>72>;O5B >?@545;8BL A8;K 8=5@F88 4286CI8EAO 725=L52. !CI5AB2C5B =5A:>;L:> <5B>4>2 :8=5<0B8G5A:>3> 0=0;870: -:A?5@8<5=B0;L=K9; @0D8G5A:89 (=5 >1;0405B 1>;LH>9 B>G=>ABLN, => 1KAB@ 2 8A?>;=5=88); @0D>0=0;8B8G5A:89; =0;8B8G5A:89 (B>G=K9, => >G5=L A;>6=K9 4065 4;O ?@>AB59H8E AE5<). >=OB85 > 35><5B@8G5A:8E 8 :8=5<0B8G5A:8E E0@0:B5@8AB8:0E <5E0=87<>2 $C=:F859 ?>;>65=8O <5E0=87<0 =07K205BAO 7028A8<>ABL C3;>2>3> 8;8 ;8=59=>3> ?5@5<5I5=8O B>G:8 8;8 725=0 <5E0=87<0 >B 2@5<5=8 8;8 >1>1I5==>9 :>>@48=0BK (@8A. 3.1).  dP(q)/dq 5@20O dP2(q)/dq2 B>@0O ?5@540B>G=0O ?5@540B>G=0O P(q) DC=:F8O DC=:F8O (q, wq aq, eq $C=:F8O ?>;>65=8O (, w a, e P(t) !:>@>ABL #A:>@5=85 dP(t)/dt dP2(t)/dt2 8A. 3.1 8=5<0B8G5A:8<8 ?5@540B>G=K<8 DC=:F8O<8 <5E0=87<0 =07K20NBAO ?@>872>4=K5 >B DC=:F88 ?>;>65=8O ?> >1>1I5==>9 :>>@48=0B5. 5@20O ?@>872>4=0O =07K205BAO ?5@2>9 ?5@540B>G=>9 DC=:F859 8;8 0=0;>3>< A:>@>AB8 (>1>7=0G05BAO (q , wq), 2B>@0O - 2B>@>9 ?5@540B>G=>9 DC=:F859 8;8 0=0;>3>< CA:>@5=8O (>1>7=0G05BAO aq, eq). 8=5<0B8G5A:8<8 E0@0:B5@8AB8:0<8 <5E0=87<0 =07K20NBAO ?@>872>4=K5 >B DC=:F88 ?>;>65=8O ?> 2@5<5=8. 5@20O ?@>872>4=0O =07K205BAO A:>@>ABLN (>1>7=0G05BAO (, w ), 2B>@0O - CA:>@5=85< (>1>7=0G05BAO a, e ). 5E0=87< A >4=>9 ?>4286=>ABLN 8<55B >4=> 7040==>5 2E>4=>5 42865=85 8 15AG8A;5==>5 <=>65AB2> 2KE>4=KE (42865=85 ;N1>3> 725=0 8;8 B>G:8 <5E0=87<0). 5@540B>G=K5 DC=:F88 B5E 42865=89, :>B>@K5 2 40==>< A;CG05 8A?>;L7CNBAO :0: 2KE>4=K5, =07K20NBAO 3;02=K<8, >AB0;L=K5 - 2A?><>30B5;L=K<8. 0AA<>B@8< AE5<C <5E0=8G5A:>9 A8AB5<K >1@07>20==>9 ?>A;54>20B5;L=K< 8 ?0@0;;5;L=K< A>548=5=85< B8?>2KE <5E0=87<>2. !E5<0 2:;NG05B 2E>4=>5 725=>, 7C1G0BCN ?5@540GC, :C;0G:>2K9 8 @KG06=K9 <5E0=87<K 8 8<55B 420 2KE>4=KE 725=0 (@8A. 3.2). 5  !  2  6 1    D A P O    0 K  E  Q 4 3 8A. 3.2. !E5<0 <5E0=8G5A:>9 A8AB5<K   f2 f3 C;0G:>2K9 <5E0=87< - P3(f2)  f1 C1G0BK9  <5E0=87< P2(f1)  f2 '5BK@5EH0@=8@=K9 f6  <5E0=87< - P6(f2) 8A. 3.3 0 @8A. 3.3. ?@54AB02;5=0 DC=:F8>=0;L=0O AE5<0 <0H8=K. $C=:F88 ?>;>65=89 <5E0=87<>2 ?@82545=K =0 @8A. 3.4. $C=:F88 ?>;>65=8O   P3 (f1)  ;02=K5  E>4=>5 P6 (f1) ?5@5<5I5=85  f1 P2 (f1)  A?><>30B5;L=K5 P3 (f2)  P6 (f2) 8A. 3.4. $C=:F88 ?>;>65=8O 2 <5E0=87<0E !2O7L :8=5<0B8G5A:8E 8 ?5@540B>G=KE DC=:F89 8=59=K5 A:>@>AB8 8 CA:>@5=8O (L = dSL/ dt = (dSL/df1)((df1/dt) = (qL ( w1; aL = d((ql ( w1)/dt = (d(qL/df1)((df1/dt)(w1 + (qL( e1 = aqL( w12 + (qL( e1; #3;>2K5 A:>@>AB8 8 CA:>@5=8O wi = dfi/ dt = (dfi /df1)((df1/dt) = wqi ( w1; ei = d(wqi(w1)/dt = (dwi/df1)((df1/dt)(w1 + wqi ( e1 = eqi( w12 + wqi ( ei . "0: :0: 40==K5 D>@<C;K ?>;CG5=K :0: ?@>872>4=K5 >B A:0;O@=KE 25;8G8=, B> ?@8 >?5@0F8OE A 25:B>@=K<8 25;8G8=0<8 >=8 ?@8<5=8<K B>;L:> 4;O ?@>5:F89 MB8E 25;8G8= =0 >A8 :>>@48=0B. =0;8B8G5A:85 <5B>4K :8=5<0B8G5A:>3> 0=0;870 1.1. 5B>4 ?@>5:F89 25:B>@=>3> :>=BC@0 (@KG06=K5 <5E0=87<K) 0AA<>B@8< ?@>AB59H89 :C;8A=K9 <5E0=87< (@8A. 3.5).   EMBED Word.Picture.8  8A. 3.5 0<5=8< :8=5<0B8G5A:CN AE5<C <5E0=87<0 M:2820;5=B=K< 25:B>@=K< :>=BC@><. ">340 C@02=5=85 70<:=CB>AB8 25:B>@=>3> :>=BC@0 70?8H5BAO  EMBED Equation.3  040G0 > ?>;>65=8OE 725=L52 <5E0=87<0 @>5F8@C5< 25:B>@=K9 :>=BC@ =0 >A8 :>>@48=0B 8 ?>;CG05< :>>@48=0BK B>G:8  <5E0=87<0: xB = lAB ( cos (f1) = lAD( cos (p) + lDB ( cos (f3); yB = lAB ( sin (f1) = lAD( sin (p) + lDB ( sin (f3); 87 @5H5=8O MB>9 A8AB5<K C@02=5=89 >?@545;O5< =58725AB=K5 25;8G8=K f3 8 lDB, :>B>@K5 >?@545;ONB ?>;>65=85 725=L52 8 B>G5: <5E0=87<0 tg (f3) = sin (f3) / cos (f3) = lAB ( sin (f1) / (lAB ( cos (f1) - lAD( cos (p)); lDB = (lAB ( sin (f1)) / sin (f3); 040G0 > ?5@2KE :8=5<0B8G5A:8E ?5@540B>G=KE DC=:F8OE <5E0=87<0 @>48DD5@5=F8@C5< C@02=5=8O ?@>5:F89 25:B>@=>3> :>=BC@0 ?> >1>1I5==>9 :>>@48=0B5 8 ?>;CG8< (qBx = - lAB ( sin (f1) = (qDB ( cos (f3) - lDB ( wq3 ( sin (f3); (qBy = lAB ( cos (f1) = (qDB ( sin (f3) + lDB ( wq3 ( cos (f3). 7 MB>9 A8AB5<K C@02=5=89 >?@545;O5< ?5@2K5 ?5@540B>G=K5 DC=:F88 (qB 8 wq3. 040G0 > 2B>@KE ?5@540B>G=KE DC=:F8OE <5E0=87<0 B>@8G=> ?@>48DD5@5=F8@C5< C@02=5=8O ?@>5:F89 25:B>@=>3> :>=BC@0 ?> >1>1I5==>9 :>>@48=0B5 8 ?>;CG8< aqBx= - lAB(cos (f1) = aqDB(cos (f3) -2((qDB(wq3 ( sin (f3) - lDB(e q3(sin (f3) - - lDB(wq32( cos (f3); aqBy = - lAB ( sin (f1) = aqDB ( sin (f3) + 2 ( (qDB ( wq3 ( cos (f3) + lDB (e q3 ( cos (f3) - lDB ( wq32( sin (f3); 7 MB>9 A8AB5<K C@02=5=89 >?@545;O5< 2B>@K5 ?5@540B>G=K5 DC=:F88 aqB 8 eq3. ;O :8=5<0B8G5A:>3> 0=0;870 @57C;LB0BK F5;5A>>1@07=55 ?@54AB02;OBL 2 2845 :8=5<0B8G5A:8E 4803@0<<. 803@0<<0 DC=:F88 ?>;>65=8O  f3,@04   0 p 2p f1,@04  803@0<<0 ?5@2>9 ?5@540B>G=>9 DC=:F88  wq3, @04/A  0 p 2p f1,@04  803@0<<0 2B>@>9 ?5@540B>G=>9 DC=:F88  eq3, A-2  0 p 2p f1,@04 8A. 3.6 &8:;>2K5 :8=5<0B8G5A:85 (35><5B@8G5A:85) 4803@0<<K 4;O :C;8A=>3> <5E0=87<0 (@8A. 3.6). &8:;>< =07K205BAO ?5@8>4 2@5<5=8 8;8 87<5=5=8O >1>1I5==>9 :>>@48=0BK ?> 8AB5G5=88, :>B>@>3> 2A5 ?0@0<5B@K A8AB5<K ?@8=8<0NB ?5@2>=0G0;L=K5 7=0G5=8O. >MB><C 7=0G5=8O 25;8G8= 2 =0G0;5 8 2 :>=F5 F8:;0 >48=0:>2K. 1.2. 5B>4 F5=B@>84 (7C1G0BK5 ?5@540G8) @8A. 3.7 &5=B@>84>9 (?>;>84>9) =07K205BAO 35><5B@8G5A:>5 <5AB> F5=B@>2 (?>;NA>2) >B=>A8B5;L=>3> 2@0I5=8O 2 A8AB5<0E :>>@48=0B A2O70==KE A> 725=LO<8 <5E0=87<0.  7C1G0B>< <5E0=87<5 ?@8 ?5@540G5 42865=8O F5=B@>84K :>;5A ?5@5:0BK20NBAO 4@C3 ?> 4@C3C 157 A:>;L65=8O. >25@=5< 254CI55 :>;5A> =0 <0;K9 C3>; df1, B>340 254><>5 :>;5A> ?>25@=5BAO =0 C3>; df2. "0: :0: F5=B@>84K 8;8 =0G0;L=K5 >:@C6=>AB8 :>;5A ?5@5:0BK20NBAO 4@C3 ?> 4@C3C 157 A:>;L65=8O, B> 4C30 dSw1 1C45B @02=0 4C35 dSw2. ">340 <>6=> 70?8A0BL A;54CNI55 @025=AB2>  EMBED Equation.3 , 345 dSw1 = rw1 ( df1, dSw2 = rw2 ( df2. B:C40 i 21 = df2/df1 = rw1/rw2 = const. $C=:F8O ?>;>65=8O 4;O 2KE>4=>3> 725=0 7C1G0B>9 ?5@540G8 df2= i 21 ( df1 , >B:C40  EMBED Equation.3 .  w2  w1 02 P o1   df2 df1  rw1  rw2   dSw1 = dSw2 = dSw 8A. 3.7. !E5<0 7C1G0B>3> <5E0=87<0 B>@0O ?5@540B>G=0O DC=:F8O 4;O 2KE>4=>3> 725=0 7C1G0B>9 ?5@540G8 eq2 = d i21/df1 = 0 . 803@0<<0 DC=:F88 ?>;>65=8O  f3, @04  0 p 2p f1,@04  803@0<<0 ?5@2>9 ?5@540B>G=>9 DC=:F88  wq2, @04/A   0 p 2p f1,@04 803@0<<0 2B>@>9 ?5@540B>G=>9 DC=:F88  eq2, A-2  0 p 2p f1,@04 8A. 3.8 5E0=87< 7C1G0B>9 ?5@540G8 =5 O2;O5BAO F8:;>2K< <5E0=87<><, B0: :0: C3;>2>5 ?5@5<5I5=85 2KE>4=>3> 725=0 C25;8G8205BAO ?@8 C25;8G5=88 C3;>2>3> ?5@5<5I5=8O 2E>4=>3>. >MB><C :8=5<0B8G5A:85 4803@0<<K ?@8=OB> AB@>8BL 4;O >4=>3> >1>@>B0 2E>4=>3> 725=0 (@8A 3.8). 803@0<<K DC=:F88 ?>;>65=8O 8 ?5@540B>G=KE DC=:F89 4;O 7C1G0B>9 ?5@540G8. @0D8G5A:85 <5B>4K :8=5<0B8G5A:>3> 0=0;870 5B>4 ?;0=>2 (@0AA<>B@8< =0 ?@8<5@5 :@82>H8?=>-?>;7C==>3> <5E0=87<0): >AB@>5=85 :8=5<0B8G5A:8E AE5< (?;0=>2 ?>;>65=89): A=>2=0O 7040G0 ?>AB@>5=8O :8=5<0B8G5A:8E AE5<, 70:;NG05BAO 2 B><, GB>1K 87>1@078BL =0 1C<035 AE5<C, 40NICN ?@54AB02;5=85 > :8=5<0B8G5A:8E 8 35><5B@8G5A:8E 7028A8<>ABOE >B45;L=KE 725=L52 <5E0=87<0. ;O MB>3> =5B =5>1E>48<>AB8 87>1@060BL <5E0=87< >B>1@060O A;>6=K5 :>=AB@C:B82=K5 D>@<K, 4>AB0B>G=> 87>1@065=85 <5E0=87<0 2 2845 ?@>AB59H8E ;8=89, CG8BK20O, GB> >= A>AB02;5= 87 65AB:8E =587<5=O5<KE 725=L52. 8=5<0B8G5A:85 AE5<K 2K?>;=ONBAO 2 <0AHB015 (<0AHB01=>< :>MDD8F85=B5):  EMBED Equation.3  >AB@>5==K9 @O4 ?>A;54>20B5;L=KE ?;0=>2 ?>;>65=89 <5E0=87<0 ?>72>;O5B ?>;CG8BL B@05:B>@88 42865=8O B>G5: 725=L52 <5E0=87<0, 0 B0:65 8E ?5@5<5I5=8O, @0AA<>B@8< ?>A;54>20B5;L=>ABL ?>AB@>5=89 4;O :@82>H8?=>-?>;7C==>3> <5E0=87<0 (@8A. 3.9, 0). 07<5B:0 B@05:B>@89 42865=8O 2A5E 725=L52 <5E0=87<0 >ACI5AB2;O5BAO <5B>4>< 70A5G5:. ! MB>9 F5;LN C3>; ?>2>@>B0 :@82>H8?0 @0718205BAO 12 @02=KE G0AB59, 8 AB@>OBAO B5:CI85 ?>;>65=8O :@82>H8?0 1i (70 =0G0;> >BAG5B0 C4>1=59 ?@8=OBL 2=5H=55 ?@545;L=>5 ?>;>65=85 :@82>H8?0 8 H0BC=0 A>>B25BAB2CNI55 =86=59 <5@B2>9 B>G:5 ?>;7C=0). 7 ?>;CG5==KE B>G5: i F8@:C;5<, @0AAB>O=85, <564C =>6:0<8 :>B>@>3> @02=> 4;8=5 H0BC=0  2 <0AHB015 ?>AB@>5=8O, 45;0NBAO 70A5G:8 =0 B@05:B>@88 42865=8O ?>;7C=0 (?@O<0O %%), B.5. ?>;CG05< B5:CI85 ?>;>65=8O ?>;7C=0 (B>G:0 i), A>548=82 :>B>@K5 A A>>B25BAB2CNI8<8 B>G:0<8 i, ?>;CG0NB ?@><56CB>G=K5 ?>;>65=8O H0BC=0. 0 ?;0=5 ?>;>65=89 <5E0=87<0 >?@545;O5< B5:CI85 ?>;>65=8O F5=B@>2 BO65AB8 :@82>H8?0 8 H0BC=0 (B>G:8 S1 8 S2). "5:CI85 7=0G5=8O ?5@5<5I5=89 ?>;7C=0 <>6=> >?@545;8BL 87 ?;0=0 ?>;>65=89 <5E0=87<0, :0: @0AAB>O=85 >B :@09=53> =86=53> ?>;>65=8O ?>;7C=0 (B>G:0 0) 4> B5:CI53> ?>;>65=8O (B>G:8 i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quation.3 , 345 (:@ .- C3;>20O A:>@>ABL :@82>H8?0, @04/A; r  @048CA :@82>H8?0, <. 725AB=K9 ?> 25;8G8=5 8 =0?@02;5=8N 25:B>@ A:>@>AB8 (( AB@>OB 2 2845 >B@57:0 ?@>872>;L=>9 4;8=K @(0, 87 2K1@0==>3> ?>;NA0 @( - ?;0=0 A:>@>AB59 (@8A. 3.9, 1).  MB>< A;CG05 <0AHB01 ?;0=0 A:>@>AB59:  EMBED Equation.3 ,  EMBED Equation.3 . @8 >?@545;5=88 A:>@>AB8 B>G:8  A;54C5B >B<5B8BL, GB> ?>;7C= A>25@H05B 2>72@0B=>-?>ABC?0B5;L=>5 42865=85, B.5. B@05:B>@859 53> 42865=8O O2;O5BAO ?@O<0O ;8=8O, 0 25:B>@ 5Q A:>@>AB8 =0?@02;5= ?0@0;;5;L=> ;8=88 ?5@5<5I5=8O. ".:. B>G:0  >4=>2@5<5==> ?@8=04;568B 8 ?>;7C=C, 8 H0BC=C, B> 4;O 40;L=59H53> ?>AB@>5=8O ?;0=0 A:>@>AB59 A;54C5B 2>A?>;L7>20BLAO 25:B>@=K< C@02=5=85<, 2K@060NI8< A2O7L <564C A:>@>ABO<8 B>G5:  8  H0BC=0:  EMBED Equation.3 , 345  EMBED Equation.3   25:B>@ 01A>;NB=>9 A:>@>AB8 B>G:8 ;  EMBED Equation.3   25:B>@ A:>@>AB8 ?5@5=>A=>3> 42865=8O, A:>@>AB8 ?>;NA0 2 :0G5AB25 :>B>@>3> ?@8=OB0 B>G:0 ;  EMBED Equation.3   25:B>@ >B=>A8B5;L=>9 A:>@>AB8 B>G:8  ?> >B=>H5=8N : B>G:5  (25:B>@ 2@0I0B5;L=>9 A:>@>AB8 B>G:8  2>:@C3 ?>;NA0  B>G:8 ). =8<0=85: GB>1K >B;>68BL ;N1>9 25:B>@ =C6=> 7=0BL 53> 25;8G8=C 8 =0?@02;5=85, ?>MB><C, 745AL, 8 40;55 25:B>@, 8725AB=K9 ?> 25;8G8=5 8 =0?@02;5=8N, ?>4G5@:=CB 42C<O ;8=8O<8, 0 25:B>@ 8725AB=K9 B>;L:> ?> =0?@02;5=8N, ?>4G5@:=CB >4=>9 ;8=859.  8A. 3.9  25:B>@=>< @025=AB25 425 =58725AB=K5 25;8G8=K: A:>@>ABL ( 8 >B=>A8B5;L=0O (2@0I0B5;L=0O) A:>@>ABL (. 5:B>@ 01A>;NB=>9 A:>@>AB8  EMBED Equation.3  =0?@02;5= ?0@0;;5;L=> ;8=88 ?5@5<5I5=8O ?>;7C=0 %%, 0 25:B>@ >B=>A8B5;L=>9 A:>@>AB8  EMBED Equation.3   ?5@?5=48:C;O@=> @048CAC 2@0I5=8O, B.5. ?5@?5=48:C;O@=> B5:CI5<C ?>;>65=8N H0BC=0 . 5:B>@ ?5@5=>A=>9 A:>@>AB8  EMBED Equation.3  (A:>@>AB8 ?>;NA0) =0 ?;0=5 A:>@>AB59 ?@54AB02;5= >B@57:>< @(0, ?>MB><C 40==>5 25:B>@=>5 @025=AB2> <>6=> @5H8BL 3@0D8G5A:8< ?CB5<. '5@57 B>G:C 0 25:B>@0 @(0 ?@>2>4OB ;8=8N 459AB28O A:>@>AB8 ( ?5@?5=48:C;O@=> . 0;55, 2 A>>B25BAB288 A 25:B>@=K< C@02=5=85<, G5@57 ?>;NA ?;0=0 A:>@>AB59 @( ?@>2>4OB ;8=8N 459AB28O A:>@>AB8 ( ?0@0;;5;L=> ;8=88 %% ?5@5<5I5=8O ?>;7C=0. 0 ?5@5A5G5=88 ;8=89 459AB28O A:>@>AB59 ( 8 ( =0E>48< B>G:C 2, @0AAB>O=85 >B :>B>@>9 4> ?>;NA0 ?;0=0 2 <0AHB015 8 >?@545;O5B 25;8G8=K A:>@>AB59, </A:  EMBED Equation.3 ;  EMBED Equation.3 . =0O >B=>A8B5;L=CN A:>@>ABL B>G:8  2>:@C3 ?>;NA0 B>G:8 , <>6=> >?@545;8BL C3;>2CN A:>@>ABL H0BC=0, @04/A:  EMBED Equation.3 , 345 ( - </A; l  4;8=0 H0BC=0, <. "5>@5<0 ?>4>18O D83C@ 4;O ?;0=>2 A:>@>AB59: D83C@K =0 ?;0=5 ?>;>65=89 8 =0 ?;0=5 A:>@>AB59 >1@07>20==K5 25:B>@0<8 >B=>A8B5;L=KE A:>@>AB59 ?>4>1=K. 0AA<>B@8< B@5C3>;L=8:8 =0 ?;0=5 ?>;>65=89 (01 8 =0 ?;0=5 A:>@>AB59 (@(02, >=8 O2;ONBAO ?>4>1=K<8 :0: 8<5NI8<8 425 AB>@>=K 2708<=> ?5@?5=48:C;O@=K5 4@C3 4@C3C 8 >4=C ?0@0;;5;L=CN. ;O >?@545;5=8O A:>@>AB59 F5=B@>2 BO65AB8 725=L52 A;54C5B =09B8 ?>;>65=8O B>G5: S1 8 S2 =0 ?;0=5 A:>@>AB59, 2>A?>;L7>202H8AL B5>@5<>9 ?>4>18O A>AB0282 A>>B=>H5=8O:  EMBED Equation.3  8  EMBED Equation.3 , B.5. 01A>;NB=0O A:>@>ABL  EMBED Equation.3 . 'B>1K ?>;CG8BL 01A>;NB=CN A:>@>ABL B>G:8 S2 A;54C5B A>548=8BL B>G:C s2 A ?>;NA>< ?;0=0 A:>@>AB59 @( , 8 B>340 >B@57>: @(s2 >?@545;8B 2 <0AHB015 ?;0=0 (( A:>@>AB59 01A>;NB=CN A:>@>ABL F5=B@0 BO65AB8 H0BC=0. AB8==>5 7=0G5=85 01A>;NB=>9 A:>@>AB8 B>G:8 S2 >?@545;O5<, </A:  EMBED Equation.3 . >AB@>5=85 ?;0=0 CA:>@5=89: >AB@>5=85 ?;0=0 CA:>@5=89 =5>1E>48<> =0G0BL A 2KG8A;5=8O 8 =0=5A5=8O =0 ?;0= CA:>@5=8O B.  :@82>H8?0.  >1I5< A;CG05 ?>;=>5 CA:>@5=85 B.  A:;04K205BAO 87 =>@<0;L=>3> (F5=B@>AB@5<8B5;L=>3>)  EMBED Equation.3  8 :0A0B5;L=>3>  EMBED Equation.3  CA:>@5=89:  EMBED Equation.3 . '8A;5==>5 7=0G5=85 =>@<0;L=>3> CA:>@5=8O >?@545;ONB ?> D>@<C;5, </A2:  EMBED Equation.3 . 0?@02;5=> MB> CA:>@5=85 ?0@0;;5;L=> >B@57:C 1 >B B>G:8  : F5=B@C 2@0I5=8O 1. 0A0B5;L=>5 CA:>@5=85 >?@545;O5BAO ?> D>@<C;5, </A2:  EMBED Equation.3 , 345 (:@  C3;>2>5 CA:>@5=85 :@82>H8?0, A-2; r  4;8=0 :@82>H8?0 <. 0?@02;5=> CA:>@5=85  EMBED Equation.3 ?5@?5=48:C;O@=> >B@57:C 1, ;81> ?> =0?@02;5=8N 25:B>@0 A:>@>AB8(( (CA:>@5==>5 2@0I5=85), ;81> ?@>B82(( (70<54;5==>5 2@0I5=85). !:;04K20O 35><5B@8G5A:8 =>@<0;L=>5 8 :0A0B5;L=>5 CA:>@5=8O, =0945< ?>;=>5 CA:>@5=85 B>G:8 :  EMBED Equation.3  @8 @02=><5@=>< 2@0I5=88 :@82>H8?0 ((:@=const) 53> C3;>2>5 CA:>@5=85  EMBED Equation.3 , A;54>20B5;L=>, ?>;=>5 CA:>@5=85 B>G:8  1C45B >?@545;OBLAO B>;L:> =>@<0;L=>9 A>AB02;ONI59  EMBED Equation.3 , 8<5NI59 G8A;5==>5 7=0G5=85:  EMBED Equation.3 , KG8A;5==>5 =>@<0;L=>5 CA:>@5=85 87>1@0605< =0 ?;0=5 CA:>@5=89 2 2845 >B@57:0 @00 ?@>872>;L=>9 4;8=K, 87 2K1@0==>3> ?>;NA0 @0 ?;0=0 CA:>@5=89 B0:, GB>1K >= 1K; ?0@0;;5;5= B5:CI5<C ?>;>65=8N :@82>H8?0 1 8 =0?@02;5= >B B>G:8  : B>G:5 1 (@8A. 3.9, 2). ">340 <0AHB01 ?;0=0 CA:>@5=89:  EMBED Equation.3 ,  EMBED Equation.3 . 0;55 ?5@5E>4OB : >?@545;5=8N CA:>@5=8O B>G:8 .  25:B>@=>< 2845:  EMBED Equation.3 , 345(0  25:B>@ ?>;=>3> CA:>@5=8O B>G:8  ?>;7C=0 8 H0BC=0;(0 - 25:B>@ ?>;=>3> CA:>@5=8O B>G:8  :@82>H8?0;(0 - 25:B>@ >B=>A8B5;L=>3> CA:>@5=8O 42865=8O B>G:8  H0BC=0 ?> >B=>H5=8N : B>G:5  :@82>H8?0, :>B>@>5 <>6=> @07;>68BL =0 =>@<0;L=CN ( EMBED Equation.3 ) 8 :0A0B5;L=CN ( EMBED Equation.3 ) A>AB02;ONI85, =0?@02;5=8O :>B>@KE 8725AB=K (25:B>@  EMBED Equation.3  =0?@02;5= ?0@0;;5;L=> ?>;>65=8N H0BC=0  >B B>G:8  : B>G:5 , 25:B>@  EMBED Equation.3 =0?@02;5= ?5@?5=48:C;O@=> 25:B>@C =>@<0;L=>3> CA:>@5=8O). @8 8725AB=>9 2@0I0B5;L=>9 A:>@>AB8 B>G:8  2>:@C3 ?>;NA0  ((), G8A;5==>5 7=0G5=85 =>@<0;L=>3> CA:>@5=8O >?@545;ONB ?> D>@<C;5, </A2:  EMBED Equation.3 , 745AL ( - 2 </A; l  4;8=0 H0BC=0 2 <. '5@B56=>5 7=0G5=85 4;8=K 25:B>@0  EMBED Equation.3  @02=>  EMBED Equation.3 , <<. !;54>20B5;L=>, 25:B>@=>5 C@02=5=85 <>65B 1KBL @5H5=> 3@0D8G5A:8< ?CBQ<. '5@57 B>G:C 0 ?@>2>4OB ?@O<CN, ?0@0;;5;L=CN B5:CI5<C ?>;>65=8N H0BC=0 ii , 8 >B:;04K205< =0 =59 25:B>@  EMBED Equation.3 2 =0?@02;5=88 >B B>G:8 i : B>G:5 i. 0B5< G5@57 B>G:C 01 ?@>2>4OB ;8=8N 459AB28O :0A0B5;L=>3> CA:>@5=8O, ?5@?5=48:C;O@=CN 40==><C ?>;>65=8N H0BC=0. 7 ?>;NA0 ?;0=0 CA:>@5=89 @0 ?@>2>4OB ;8=8N 459AB28O ?>;=>3> CA:>@5=8O B>G:8 , ?0@0;;5;L=CN ;8=88 %% ?5@5<5I5=8O ?>;7C=0. 0AAB>O=85 >B B>G:8 2, ?5@5A5G5=8O ;8=89 459AB28O 42CE ?>A;54=8E CA:>@5=89 4> ?>;NA0 8 B>G:8 01 >?@545;O5B 2 <0AHB015 7=0G5=8O CA:>@5=89, </A2:  EMBED Equation.3 ;  EMBED Equation.3 . !>548=82 B>G:8 0 8 2 25:B>@><  EMBED Equation.3 , ?>;CG05< ?>;=>5 CA:>@5=85 B>G:8  2 >B=>A8B5;L=>< 42865=88 ?> >B=>H5=8N : ?>;NAC B>G:5 , B.5.:  EMBED Equation.3 , </A2. ;O >?@545;5=8O CA:>@5=89 F5=B@>2 BO65AB8 725=L52 A;54C5B =09B8 ?>;>65=8O B>G5: S1 8 S2 =0 ?;0=5 CA:>@5=89, 2>A?>;L7>202H8AL A>>B=>H5=8O<8:  EMBED Equation.3  8  EMBED Equation.3 , 1A>;NB=K5 7=0G5=8O CA:>@5=89 F5=B@>2 BO65AB8 725=L52, </A2:  EMBED Equation.3  8  EMBED Equation.3 . =0O 25;8G8=C :0A0B5;L=>3> CA:>@5=8O  EMBED Equation.3 , <>6=> >?@545;8BL C3;>2>5 CA:>@5=85 H0BC=0, A-2:  EMBED Equation.3 . 'B>1K >?@545;8BL, :0:>5 42865=85 A>25@H05B H0BC= (CA:>@5==>5 8;8 70<54;5==>5), =5>1E>48<> 7=0BL =0?@02;5=85 C3;>2>9 A:>@>AB8 (2 8 C3;>2>3> CA:>@5=8O (2 2 40==K9 <><5=B 2@5<5=8. ;O MB>3> 25:B>@K 2@0I0B5;L=>9 A:>@>AB8 (( A ?;0=0 A:>@>AB59 8 :0A0B5;L=>3> CA:>@5=8O  EMBED Equation.3  A ?;0=0 CA:>@5=89 ?5@5=>AOBAO ?0@0;;5;L=> 2 A>>B25BAB2CNICN B>G:C  ?;0=0 ?>;>65=89 <5E0=87<0. E =0?@02;5=85 >B=>A8B5;L=> B>G:8  8 >?@545;8B =0?@02;5=85 C3;>2KE A:>@>AB8 8 CA:>@5=8O. A;8 =0?@02;5=85 C3;>2>9 A:>@>AB8 A>2?0405B A =0?@02;5=85< C3;>2>3> CA:>@5=8O, B> 42865=85 H0BC=0 1C45B CA:>@5==K< 8 =0>1>@>B. ;0=K A:>@>AB59 8 CA:>@5=89 ?>AB@>5==K5 4;O 40==>3> ?>;>65=8O <5E0=87<0, 40NB 2>7<>6=>ABL CAB0=>28BL ;8HL 7=0G5=8O <3=>25==KE A:>@>AB59 8 CA:>@5=89 @07;8G=KE B>G5: 725=L52 <5E0=87<0, 2 A>>B25BAB2CNI55 2@5<O. >AB@>5=85 ?>A;54>20B5;L=KE ?;0=>2 ?>;>65=89, A:>@>AB59 8 CA:>@5=89 <5E0=87<0 ?>72>;O5B ?>;CG8BL :8=5<0B8G5A:85 4803@0<<K S = S(t), ( = ((t), a = a(t) A>>B25BAB25==>. 8=5<0B8G5A:85 4803@0<<K AB@>OBAO 2 ?@O<>C3>;L=>9 A8AB5<5 :>>@48=0B: ?> >A8 01AF8AA >B:;04K205< >B@57>: l  4;8=0 :>B>@>3> 2 <0AHB015 (t, ?@8=8<05BAO @02=K< 2@5<5=8 >4=>3> >1>@>B0 (?5@8>40) =0G0;L=>3> 725=0. >=B@>;L=K5 2>?@>AK 'B> B0:>5 :8=5<0B8G5A:85 E0@0:B5@8AB8:8 <5E0=87<0?  G5< 70:;NG05BAO <5B>4 ?@>5:F89 :8=5<0B8G5A:>3> 0=0;870 @KG06=>3> <5E0=87<0? ;O G53> AB@>8BAO :8=5<0B8G5A:0O AE5<0 <5E0=87<0? 0 G5< >A=>20=> >?@545;5=85 A:>@>AB59 B>G5: 725=L52, A>25@H0NI8E A;>6=>5 (?;>A:>?0@0;;5;L=>5) 42865=85?  G5< 70:;NG05BAO 8A?>;L7C5<K9 ?@8 ?>AB@>5=88 ?;0=>2 A:>@>AB59 8 CA:>@5=89 <5B>4 ?>4>18O? 5:F8O 4 @0D8G5A:>5 48DD5@5=F8@>20=85. 040G8 ?@>5:B8@>20=8O <5E0=87<>2. 5E0=8G5A:85 ?5@540G8: :;0AA8D8:0F8O, ?0@0<5B@K. 5B>4 :8=5<0B8G5A:8E 4803@0<< @0D8G5A:89 A?>A>1 :8=5<0B8G5A:>3> 0=0;870 <5B>4>< ?>AB@>5=8O 4803@0<< >B;8G05BAO ?@>AB>B>9 2K?>;=5=8O 8 =03;O4=>ABLN ?@54AB02;5=8O @57C;LB0B>2. B>3>< <5B>40 O2;ONBAO 3@0D8:8 ?5@5<5I5=89, A:>@>AB59 8 CA:>@5=89 2 7028A8<>AB8 >B 2@5<5=8 8;8 C3;0 ?>2>@>B0 =0G0;L=>3> 725=0, 40==K5 3@0D8:8 ?>;CG8;8 =0720=85 :8=5<0B8G5A:8E 4803@0<<. @8 :8=5<0B8G5A:>< 0=0;875 8A?>;L7CNB A?>A>1 3@0D8G5A:>3> 48DD5@5=F8@>20=8O (ACI5AB2C5B A?>A>1 E>@4 8 A?>A>1 :0A0B5;L=KE, @0AA<>B@8< ?5@2K9: 45;05BAO 4>?CI5=85, GB> E>@40 4C38 >:@C6=>AB8 045:20B=0 :0A0B5;L=>9). CABL 7040= 3@0D8: S = S(t), B@51C5BAO ?>AB@>8BL ( = ((t) 8 a = a(t). ;O B>G:8  :@82>9 (@8A. 4.1) <>6=> 70?8A0BL: S = C ((S , t = x ((t ">340:  EMBED Equation.3 , 345 ( - C3>; =0:;>=0 :0A0B5;L=>9 : @0AA<0B@8205<>9 B>G:5  (?@54?>;0305<, GB> :0A0B5;L=0O ?0@0;;5;L=0 E>@45 4C38 >:@C6=>AB8 >:>;> @0AA<0B@8205<>9 B>G:8).  8A. 4.1 5B>48:0 3@0D8G5A:>3> 48DD5@5=F8@>20=8O (@8A. 4.2): 803@0<<0 A:>@>AB59. Q AB@>OB 3@0D8G5A:8< 48DD5@5=F8@>20=85< 4803@0<<K ?5@5<5I5=89 ?> <5B>4C E>@4: @0745;ONB >AL 01AF8AA 4803@0<<K ?5@5<5I5=89 =0 ?@>872>;L=>5 G8A;> >48=0:>2KE G0AB59; G5@57 B>G:8 45;5=8O 1, 2, 3& ?@>2>4OB :>>@48=0B=CN A5B:C; B>G:8 ?5@5A5G5=8O >@48=0B A 3@0D8:>< ?5@5<5I5=89 A>548=ONB >B@57:0<8 (E>@40<8) 0((1(, 0((2(, 0((3(, ( (G5< 1>;LH5 B>G5: 45;5=8O, B5< E>@4K 1C4CB 1;865 : 8AB8==>9 :@82>9); AB@>OB A8AB5<C :>>@48=0B ( = ((t), A?@020 >B =0G0;0 :>>@48=0B >B:;04K205< >B@57>: 0@(., =07K205<K9 ?>;NA=K< @0AAB>O=85<, 4;8=>9 1 8 >B<5G0NB ?>;NA 4803@0<<K A:>@>AB59 @(; @8<5G0=85: 5;8G8=0 1 2K18@05BAO 2 7028A8<>AB8 >B 65;05<>3> @07<0E0 4803@0<<K ( = ((t). 87 ?>;NA0 @( ?@>2>4OB ;CG8, ?0@0;;5;L=K5 A>>B25BAB2CNI8< E>@40< =0 4803@0<<5 ?5@5<5I5=89 4> ?5@5A5G5=8O A >ALN >@48=0B, ?>;CG0NB B>G:8 1((, 2(, ( 87 ?>;CG5==KE B>G5: ?@>2>4OB 3>@87>=B0;L=K5 ;CG8 4> ?5@5A5G5=8O A 25@B8:0;L=K<8 ?@O<K<8, >?CI5==K<8 87 A5@548= A>>B25BAB2CNI8E >B@57:>2 =0 4803@0<<5 ?5@5<5I5=89. >;CG5==K5 B>G:8 1(, 2(,( A>548=ONB ?;02=>9 :@82>9 8 ?>;CG0NB 4803@0<<C 87<5=5=8O A:>@>AB8 (?5@2>5 ?@81;865=85), 2 <0AHB015  EMBED Equation.3 , 345 (S - <0AHB01 4803@0<<K ?5@5<5I5=89, </<<; (( - <0AHB01 C3;0 ?>2>@>B0 =0G0;L=>3> 725=0, @04/<<; (1- C3;>20O A:>@>ABL =0G0;L=>3> 725=0, @04/A; (t - <0AHB01 2@5<5=8, A/<<; 1- ?>;NA=>5 @0AAB>O=85, 27OB>5 A G5@B560, <<. @>45;02 0=0;>38G=K5 >?5@0F88 A 4803@0<<>9 ( = ((t), ?@5420@8B5;L=> 2>AAB0=>282 B>G:8 1, 2, 3, 8 B.4., ?>;CG05< 7028A8<>ABL CA:>@5=8O 0 = ((t) 2 <0AHB015  EMBED Equation.3 , 345 2 - ?>;NA=>5 @0AAB>O=85 4;O 4803@0<<K 0 = ((t), <<.  8A. 4.2 -:A?5@8<5=B0;L=K9 <5B>4 :8=5<0B8G5A:>3> 8AA;54>20=8O @8 M:A?5@8<5=B0;L=>< 8AA;54>20=88 :8=5<0B8:8 <5E0=87<>2 :8=5<0B8G5A:85 E0@0:B5@8AB8:8 725=L52 8 B>G5: <5E0=87<0 >?@545;ONBAO 8 @538AB@8@CNBAO A ?><>ILN GC2AB28B5;L=KE M;5<5=B>2 - 40BG8:>2, :>B>@K5 8A?>;L7CO @07;8G=K5 D878G5A:85 MDD5:BK ?@5>1@07CNB :8=5<0B8G5A:85 ?0@0<5B@K 2 ?@>?>@F8>=0;L=K5 M;5:B@8G5A:85 A83=0;K. -B8 A83=0;K @538AB@8@CNBAO 87<5@8B5;L=K<8 A0<>?8HCI8<8 ?@81>@0<8 (A0<>?8AF0<8, >AF8;;>3@0D0<8 8 4@.).  ?>A;54=55 2@5<O 4;O @538AB@0F88 8 >1@01>B:8 M:A?5@8<5=B0;L=KE 40==KE 2A5 1>;55 H8@>:> 8A?>;L7CNBAO A?5F80;L=K5 8;8 C=825@A0;L=K5 :><?LNB5@K. ;O ?@8<5@0 @0AA<>B@8< M:A?5@8<5=B0;L=CN CAB0=>2:C 4;O 8AA;54>20=8O :8=5<0B8G5A:8E E0@0:B5@8AB8: A8=CA=>3> <5E0=87<0 (@8A. 4.3):   0BG8: ?5@5<5I5=8O  1 2   w1 B, C R  SD = f(t)  A D    0BG8: 0BG8:  0 A:>@>AB8 N S CA:>@5=8O   3 "5=7><5B@8G5A:89 (D= f(t) CA8;8B5;L  8A. 4.3 aD = f(t)  MB>9 M:A?5@8<5=B0;L=>9 CAB0=>2:5: 4;O 87<5@5=8O ?5@5<5I5=8O 2KE>4=>3> 725=0 8A?>;L7C5BAO ?>B5=F8><5B@8G5A:89 40BG8: ?5@5<5I5=8O, 2 :>B>@>< ?@>?>@F8>=0;L=> ?>;>65=8N 4286:0 ?>B5=F8><5B@0 87<5=O5BAO 53> A>?@>B82;5=85; 4;O 87<5@5=8O A:>@>AB8 2KE>4=>3> 725=0 8A?>;L7C5BAO 8=4C:F8>==K9 40BG8: A:>@>AB8, 2 :>B>@>< =0?@O65=85 =0 :>=F0E :0BCH:8 4286CI59AO 2 ?>;5 ?>AB>O==>3> <03=8B0 ?@>?>@F8>=0;L=> A:>@>AB8 :0BCH:8; 4;O 87<5@5=8O CA:>@5=8O 2KE>4=>3> 725=0 8A?>;L7C5BAO B5=7><5B@8G5A:889 0:A5;5@><5B@. = A>AB>8B 87 ?;0AB8=G0B>9 ?@C68=K, >48= :>=5F :>B>@>9 70:@5?;5= =0 2KE>4=>< 725=5 <5E0=87<0, 0 =0 2B>@>< 70:@5?;5=0 <0AA0. 0 ?;0AB8=C =0:;55=K ?@>2>;>G=K5 B5=7>?@5>1@07>20B5;8. @8 42865=88 2KE>4=>3> 725=0 A CA:>@5=85< 8=5@F8>==>ABL <0AAK 2K7K205B 87381 ?;0AB8=K, 45D>@<0F8N B5=7>?@5>1@07>20B5;59 8 87<5=5=85 8E A>?@>B82;5=8O ?@>?>@F8>=0;L=>5 CA:>@5=8N 2KE>4=>3> 725=0. @>5:B8@>20=85 B8?>2KE ?;>A:8E 8 ?@>AB@0=AB25==KE <5E0=87<>2 040G8 ?@>5:B8@>20=8O @8 ?@>5:B8@>20=88 <5E0=87<>2 @07;8G0NB B@8 MB0?0: ?5@2K< MB0?>< O2;O5BAO CAB0=>2;5=85 :8=5<0B8G5A:>9 AE5<K <5E0=87<0, :>B>@0O >15A?5G8;0 1K B@51C5<K9 284 8 70:>= 42865=8O; 2B>@>9 MB0?  @07@01>B:0 :>=AB@C:B82=KE D>@< <5E0=87<0, >15A?5G820NI8E ?@>G=>ABL, 4>;3>25G=>ABL, 2KA>:89 :.?.4. 8 B.4.; B@5B89 MB0?  @07@01>B:0 B5E=>;>38G5A:8E 8 B5E=8:>-M:>=><8G5A:8E ?>:070B5;59 ?@>5:B8@C5<>3> <5E0=87<0, >?@545;O5<KE M:A?;C0B0F859 2 ?@>872>4AB25, @5<>=B0<8 8 B.4. "5>@8O <5E0=87<>2 8 <0H8= 70=8<05BAO ?5@2K< MB0?><. 0745; ", ?>A2OI5==K9 <5B>40< ?@>5:B8@>20=8O ?> 7040==K< :8=5<0B8G5A:8< CA;>28O< AE5< <5E0=87<>2, ?>;CG8; =0720=85  A8=B570 <5E0=87<>2. A=>2=K5 7040G8 A8=B570 <5E0=87<>2: ?@5>1@07>20=85 2@0I0B5;L=>3> 42865=8O 2>:@C3 >4=>9 >A8 2> 2@0I0B5;L=>5 42865=85 2>:@C3 4@C3>9 >A8; ?@5>1@07>20=85 2@0I0B5;L=>3> 42865=8O 2>:@C3 >4=>9 >A8 2 ?>ABC?0B5;L=>5 42865=85 24>;L =5:>B>@>9 7040==>9 ?@O<>9 8 =0>1>@>B; ?@5>1@07>20=85 ?>ABC?0B5;L=>3> 42865=8O 24>;L >4=>9 7040==>9 ?@O<>9 2 ?>ABC?0B5;L=>5 42865=85 24>;L 4@C3>9 7040==>9 ?@O<>9; 2>A?@>872545=85 >4=>9 87 B>G5: 725=L52 @KG06=>3> <5E0=87<0 B@51C5<>9 B@05:B>@88, 2>A?@>872545=85 7040==KE C3;>2 ?>2>@>B0 2KE>4=>3> 725=0, 42865=85 2KE>4=>3> 725=0 A >AB0=>20<8. 5E0=87<K ?5@540G8 @54=07=0G5=K 4;O ?5@540G8 2@0I0B5;L=>3> 42865=8O <564C 725=LO<8. 5@540G0 2@0I0B5;L=>3> 42865=8O <>65B ?@>872>48BLAO A 87<5=5=85< C3;>2>9 A:>@>AB8 2@0I5=8O, A A>E@0=5=85< 8;8 87<5=5=85< =0?@02;5=8O 2@0I5=8O. 0@0<5B@, E0@0:B5@87CNI89 ?@8 ?5@540G5 42865=8O 87<5=5=85 A:>@>AB8 8 =0?@02;5=8O, =07K20NB ?5@540B>G=K< >B=>H5=85<: EMBED Equation.3, 8;8 EMBED Equation.3, 345 7=0: (+) 5A;8 2@0I5=85 725=L52 2 >4=>< =0?@02;5=88; 7=0: (() 5A;8 2@0I5=85 725=L52 2 ?@>B82>?>;>6=KE =0?@02;5=8OE; n1, (1 ( A>>B25BAB25==> G0AB>B0, >1/<8=. 8 A:>@>ABL 2@0I5=8O, @04/A, 20;0 1; n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i1n, 2:;NG0NI53> =5A:>;L:> ?>A;54>20B5;L=> A>548=5==KE 4@C3 A 4@C3>< ?@>ABKE (>4=>ABC?5=G0BKE) <5E0=87<>2, @02=> ?@>872545=8N ?5@540B>G=KE >B=>H5=89 >B45;L=KE ?@>ABKE <5E0=87<>2 (ABC?5=59), 2E>4OI8E 2 53> A>AB02, B.5.: EMBED Equation.3 "0:, 4;O <5E0=87<0, ?@54AB02;5==>3> =0 @8A. 4.5 >1I55 ?5@540B>G=>5 >B=>H5=85 @02=>: EMBED Equation.3.  8A. 4.5 @><5 >4=>ABC?5=G0BKE 8 <=>3>ABC?5=G0BKE ?@8<5=ONBAO ?;0=5B0@=K5 8 2>;=>2K5 ?5@540G8. >=B@>;L=K5 2>?@>AK 0 G5< >A=>20=> 3@0D8G5A:>5 48DD5@5=F8@>20=85 ?> <5B>4C E>@4? !D>@<C;8@C9B5 >A=>2=K5 7040G8, @5H05<K5 " ?@8 ?@>5:B8@>20=88 <5E0=87<0? 'B> B0:>5 ?5@540B>G=>5 >B=>H5=85? 'B> >7=0G05B 7=0: ?5@540B>G=>3> >B=>H5=8O? 'B> B0:>5 ABC?5=L 2 7C1G0B>< <5E0=87<5? 0: >?@545;O5BAO ?5@540B>G=>5 >B=>H5=85 A;>6=>3> 7C1G0B>3> <5E0=87<0? 5:F8O 5 $@8:F8>==K5 ?5@540G8. 0@80B>@ A:>@>AB8. C1G0BK5 <5E0=87<K. A=>2=0O B5>@5<0 70F5?;5=8O. 5><5B@8G5A:85 ?0@0<5B@K 7C1G0BKE :>;5A. $@8:F8>==K5 ?5@540G8 $@8:F8>==K5 ?5@540G8 ?@54=07=0G5=K 4;O ?5@540G8 2@0I0B5;L=>3> 42865=8O 70 AG5B A8; B@5=8O (@8A. 5.1).  8A. 5.1 #A;>285< ?5@540G8 42865=8O 2> D@8:F8>==>< <5E0=87<5 O2;O5BAO:  EMBED Equation.3 , 345 21, 12  ?5@540205<>5 CA8;85; FB@  A8;0 B@5=8O 2 <5AB5 :>=B0:B0 20;:>2 1 8 2, >?@545;O5<0O ?> D>@<C;5:  EMBED Equation.3 , 345 Q  =>@<0;L=>5 CA8;85 ?@868<0 20;:>2; f  :>MDD8F85=B B@5=8O A:>;L65=8O. "0: :0: G0I5 2A53> f ( 0,2& 0,3, B> 4;O ?5@540G8 42865=8O 2> D@8:F8>==>9 ?5@540G5 CA8;85 ?@868<0 4>;6=> 1KBL ?@81;878B5;L=> 2 5 @07 1>;LH5, G5< ?5@540205<>5 CA8;85. 0AA<>B@8< >1ICN 4;O 20;:>2 B>G:C :>=B0:B0 20;:>2 , 5Q >:@C6=0O A:>@>ABL @02=0: 4;O ?5@2>3> 20;:0 EMBED Equation.3 , 4;O 2B>@>3> 20;:0  EMBED Equation.3 . @8 >BACBAB288 ?@>A:0;L7K20=8O 20;:>2  EMBED Equation.3 , B>340:  EMBED Equation.3  (  EMBED Equation.3 . &8;8=4@K ?5@5:0BK20NI85AO >48= >B=>A8B5;L=> 4@C3>3> 157 ?@>A:0;L7K20=8O =07K20NBAO =0G0;L=K<8. :@C6=>AB8, ?>;CG0NI85AO 2 ?5@5A5G5=88 MB8E F8;8=4@>2 ?;>A:>ABLN ?5@?5=48:C;O@=>9 8E >AO< =07K20NBAO =0G0;L=K<8 (@048CAK r1 8 r2). 5@540B>G=>5 >B=>H5=85 8<55B 7=0:: - :>340 =0?@02;5=8O 42865=8O =5 A>2?040NB (B.5. :>340 :0A0=85 2=5H=55, @8A. 5.2, 0); + :>340 =0?@02;5=8O 42865=8O A>2?040NB (B.5. :>340 :0A0=85 2=CB@5==55, @8A. 5.2, 1).  8A. 5.2 >1>20O D@8:F8>==0O ?5@540G0 (20@80B>@ A:>@>AB8) >1>20O D@8:F8>==0O ?5@540G0 ?>72>;O5B ?;02=>5 87<5=5=85 A:>@>AB8 2KE>4=>3> 725=0 70 AG5B 87<5=5=8O @048CA0 =0G0;L=>9 >:@C6=>AB8 (@8A. 5.3).  8A. 5.3 5@540B>G=>5 >B=>H5=85 53> <>6=> =09B8 87 ?>;CG5==>9 D>@<C;K:  EMBED Equation.3  "0: :0: @0AAB>O=85 E <>6=> 87<5=OBL, B> 8 ?5@540B>G=>5 >B=>H5=85 ?@>?>@F8>=0;L=> 1C45B <5=OBLAO. >AB>8=AB20 D@8:F8>==KE ?5@540G: @>AB>B0 :>=AB@C:F88 8 873>B>2;5=8O; 5AHC<=>ABL @01>BK; 5 ?5@540G0 >B >4=>3> 725=0 4@C3><C A;CG09=KE ?5@53@C7>: 70 AG5B ?@>A:0;L7K20=8O; >7<>6=>ABL 15AABC?5=G0B>3> @53C;8@>20=8O A:>@>AB8 ?@8 =5:>B>@>< :>=AB@C:B82=>< 8A?>;=5=88. 54>AB0B:8 D@8:F8>==KE ?5@540G: >;LH0O =03@C7:0 =0 20;K 8 >?>@K >B CA8;8O ?@860B8O; >7<>6=>ABL =0@CH5=8O :8=5<0B8G5A:>9 A2O78, 2A;54AB285 ?@>A:0;L7K20=8O; 52KA>:0O =03@C7>G=0O A?>A>1=>ABL. C1G0BK5 <5E0=87<K (?5@540G8) 1I85 ?>;>65=8O C1G0BK9 <5E0=87< - <5E0=87<, 2KAH0O ?0@0 :>B>@>3> >1@07>20=0 7C1G0BK<8 725=LO<8 (:>;5A0<8). C1G0BK5 <5E0=87<K A;C60B 3;02=K< >1@07>< 4;O ?5@540G8 2@0I0B5;L=>3> 42865=8O <564C 42C<O :0:8<8-;81> >AO<8 A 87<5=5=85< C3;>2>9 A:>@>AB8 254><>3> 20;0. 0;, >B :>B>@>3> ?5@5405BAO 42865=85, =07K205BAO 254CI8<; 20;, :>B>@><C ?5@5405BAO 42865=85, =07K205BAO 254><K<. @>AB>9 7C1G0BK9 <5E0=87<, 8;8 ?@>AB0O 7C1G0B0O ?5@540G0, ?@54AB02;O5B A>1>9 B@5E725==K9 <5E0=87<, >10 ?>4286=KE 725=0 :>B>@>3> O2;ONBAO 7C1G0BK<8 :>;5A0<8. C1G0BK5 :>;5A0 >1@07CNB A> AB>9:>9 425 =87H85 2@0I0B5;L=K5 ?0@K, 0 <564C A>1>9 2KAHCN (7C1G0B>5 70F5?;5=85).  7028A8<>AB8 >B @0A?>;>65=8O >A59 7C1G0BKE :>;5A @07;8G0NB 7C1G0BK5 ?5@540G8 A ?0@0;;5;L=K<8 >AO<8 8;8 F8;8=4@8G5A:85 (@8A. 5.4), A ?5@5A5:0NI8<8AO >AO<8, 8;8 :>=8G5A:85 (@8A. 5.5), 8 A ?5@5:@5I820NI8<8AO >AO<8, 8;8 38?5@1>;>84=K5, 20@80=B0<8 :>B>@KE O2;ONBAO 28=B>2K5 (@8A. 5.6, 0), G5@2OG=K5 (@8A. 5.6, 1) 8 38?>84=K5 (@8A. 5.6, 2) ?5@540G8. ><8<> MB>3> 2A5 ?5@540G8 45;OBAO =0 ?5@540G8 A 2=5H=8<, 2=CB@5==8< 8 @55G=K< 70F5?;5=8O<8. @87=0:>< ?5@540G8 A 2=5H=8< 70F5?;5=85< (@8A. 5.4, 0) O2;O5BAO 2@0I5=85 5Q 7C1G0BKE :>;5A 2 ?@>B82>?>;>6=K5 AB>@>=K, 0 ?5@540G8 A 2=CB@5==8< 70F5?;5=85< (@8A. 5.4, 6) ( 2 >4=>< =0?@02;5=88. 5@540G0 A @55G=K< 70F5?;5=85< (@8A. 5.4, 2) A>AB>8B 87 :>;5A0 8 @59:8.  8A. 5.4  8A. 5.5  8A. 5.6  7028A8<>AB8 >B B>3>, B@51C5BAO ;8 >15A?5G8BL ?>AB>O==>5 >B=>H5=85 C3;>2KE A:>@>AB59 254CI53> 8 254><>3> 7C1G0BKE :>;5A 8;8, =0>1>@>B, ?5@5<5==>5 ?> 7040==><C 70:>=C, 7C1G0BK5 ?5@540G8 2K?>;=ONB A>>B25BAB25==> 87 :@C3;KE (@8A. 5.4, 0, 1) 8 =5:@C3;KE (@8A. 5.7) :>;5A.  8A. 5.7 !;>6=K5 7C1G0BK5 <5E0=87<K 45;OBAO =0 7C1G0BK5 ?5@540G8 A ?@><56CB>G=K<8 :>;5A0<8 8 20;0<8 (<=>3>:@0B=K5 8;8 <=>3>ABC?5=G0BK5) 8 ?;0=5B0@=K5 <5E0=87<K. =>3>:@0B=>9 7C1G0B>9 ?5@540G59 =07K205BAO <5E0=87<, ?@54AB02;ONI89 A>1>9 A>548=5=85 =5A:>;L:8E ?@>ABKE 7C1G0BKE ?5@540G (ABC?5=59) A =5?>4286=K<8 (?> >B=>H5=8N : AB>9:5) >AO<8 :>;5A (@8A. 5.8, 0). 25=> 2 <5E0=87<0 A>AB>8B 87 42CE 7C1G0BKE :>;5A 2" 8 2', =0A065==KE 65AB:> =0 >48= >1I89 20;. 0 @8A. 5.8 7C1G0BK5 :>;5A0 CA;>2=> ?>:070=K 2 2845 F8;8=4@>2, <5E0=87< >1;0405B >4=>9 AB5?5=LN A2>1>4K 8 =>A8B =0720=85 @54C:B>@0. @><56CB>G=>5 :>;5A> ?>;CG05B 42865=85 >B ?@54K4CI53> :>;5A0 8 ?5@5405B A;54CNI5<C. @><56CB>G=K9 20; ?>;CG05B 42865=85 G5@57 >4=C ?0@C :>;5A 8 ?5@5405B G5@57 4@C3CN (=0 @8A. 5.8, 0 MB> A@54=89 20;).  8A. 5.8 5@540B>G=K< >B=>H5=85< 7C1G0B>3> <5E0=87<0 =07K205BAO >B=>H5=85 G0AB>B 2@0I5=8O 7C1G0BKE :>;5A: 5@540B>G=>5 >B=>H5=85 B@5E725==>3> 7C1G0B>3> <5E0=87<0 <>6=> 2K@078BL =5 B>;L:> G5@57 >B=>H5=85 G0AB>B 2@0I5=8O 7C1G0BKE :>;5A (n1 8 n2), => 8 G5@57 >B=>H5=85: C3;>2KE A:>@>AB59 ((1 8 (2); G8A5; 7C1L52 :>;5A (Z1 8 Z2); >B=>H5=85 480<5B@>2 =0G0;L=KE >:@C6=>AB59 (dw1 8 dw2, :@><5 G5@2OG=KE): EMBED Equation.3 345 7=0: (+) 4;O 2=CB@5==53> 70F5?;5=8O (2@0I5=85 725=L52 2 >4=>< =0?@02;5=88); 7=0: (-) 4;O 2=5H=53> 70F5?;5=8O (2@0I5=85 725=L52 2 ?@>B82>?>;>6=KE =0?@02;5=8OE). 5@540B>G=>5 >B=>H5=85 2 =0?@02;5=88 A8;>2>3> ?>B>:0, B.5. >B=>H5=85 C3;>2>9 A:>@>AB8 254CI53> 20;0 : C3;>2>9 A:>@>AB8 254><>3>, =07K205BAO ?5@540B>G=K< G8A;><. 5@540B>G=>5 G8A;> 2A5340 1>;LH5 8;8 @02=> 548=8F5. A=>2=0O B5>@5<0 70F5?;5=8O 0AA<>B@8< B5;0 1 8 2, A>25@H0NI85 2@0I0B5;L=K5 42865=8O, A>>B25BAB25==> 2>:@C3 F5=B@>2 01 8 02 A C3;>2K<8 A:>@>ABO<8 (1 8 (2 , 8 >1@07CNI8E <564C A>1>9 2KAHCN :8=5<0B8G5A:CN ?0@C  (@8A. 5.9). 1I0O =>@<0;L n-n ?@>D8;59 2 B>G:5 :>=B0:B0  ?5@5A5:05B ;8=8N <56F5=B@>2>3> @0AAB>O=8O 0102 B>G:5 , =07K205<>9 ?>;NA>< 70F5?;5=8O, 8 2 :8=5<0B8G5A:>< >B=>H5=88, O2;ONI59AO F5=B@>< <3=>25==>3> 2@0I5=8O 2 >B=>A8B5;L=>< 42865=88 725=L52 2KAH59 :8=5<0B8G5A:>9 ?0@K.  ?;>A:>< <5E0=87<5 >15A?5G5=88 ?5@540G8 7040==>3> 42865=8O 7028A8B >B 35><5B@88 A>?@O65==KE ?@>D8;59 (I 8 II =0 @8A. 5.9). '0AB> =0 ?@0:B8:5 35><5B@8N A>?@O65==KE ?@>D8;59 ?>418@0NB B0:, GB>1K >=0 >15A?5G820;0 70:>= 42865=8O, E0@0:B5@87C5<K9 ?>AB>O=AB2>< ?5@540B>G=>3> >B=>H5=8O <564C 725=LO<8 1 8 2 2KAH59 ?0@K, B.5. i12 = const.  @0AA<0B@8205<K9 <><5=B 2@5<5=8 A:>@>AB8 B>G:8  @02=K: - 2 A8AB5<5 :>;5A0 1:  EMBED Equation.3 , - 2 A8AB5<5 :>;5A0 2:  EMBED Equation.3 , 345 (1 8 (1  @048CA 25:B>@K (@0AAB>O=8O A>>B25BAB25==> >B F5=B>2 2@0I5=8O 01 8 02 4> B>G:8 ).  8A. 5.9 @>5:F88  EMBED Equation.3 8  EMBED Equation.3  =0 =>@<0;L n-n 4>;6=K 1KBL @02=K:  EMBED Equation.3 ( EMBED Equation.3  (1) =0G5, 8;8 7C1LO 1C4CB 2=54@OBLAO >48= 2 4@C3>9, 8;8 :>;5A0 2K94CB 87 70F5?;5=8O. @>5:F88  EMBED Equation.3 8  EMBED Equation.3  =0 :0A0B5;L=CN t-t =5 @02=K <564C A>1>9, ?>MB><C 2 2KAH59 :8=5<0B8G5A:>9 ?0@5 2>7=8:05B ?@>A:0;L7K20=85 ?@>D8;59. @>5F8@C5< F5=B@K 2@0I5=8O 01 8 02 =0 =>@<0;L n-n ?>;CG05< B>G:8 N1 8 N2. 7 B@5C3>;L=8:0 (01N1:  EMBED Equation.3 . 7 B@5C3>;L=8:0 (02N2:  EMBED Equation.3 . !;54>20B5;L=>, A CG5B>< 2K@065=8O 1:  EMBED Equation.3 , 8;8  EMBED Equation.3 . 0AA<0B@820O B@5C3>;L=8:8 (01 N1 8 (02 N2, CAB0=02;8205<, GB> >=8 ?>4>1=K5 (8<5NB B@8 AB>@>=K 2708<=> ?0@0;;5;L=K5 4@C3 4@C3C), A>AB02;O5< ?@>?>@F8N:  EMBED Equation.3 , 345 r1 8 r2  @048CAK =0G0;L=KE >:@C6=>AB59. A;8 >=8 8A?>;L7CNBAO 2 :0G5AB25 ?@>872>4AB25==KE >:@C6=>AB59 2 ?@>F5AA5 =0@570=8O :>;5A, B> >=8 ?>;CG0NB 5IQ =0720=85 45;8B5;L=KE >:@C6=>AB59. !;54>20B5;L=> ?5@540B>G=>5 >B=>H5=85 2 >1I5< 2845 <>65B 1KBL 70?8A0=>:  EMBED Equation.3 ( >A=>2=0O B5>@5<0 70F5?;5=8O. ".5. >1I0O =>@<0;L n-n : A>?@8:0A0NI8<AO (A>?@O65==K<) ?@>D8;O< 7C1L52 45;8BL <56>A52>5 @0AAB>O=85 0102 =0 G0AB8 >1@0B=> ?@>?>@F8>=0;L=K5 C3;>2K< A:>@>ABO< (?5@540B>G=><C >B=>H5=8N). !;54>20B5;L=>, 4;O >15A?5G5=8O ?>AB>O=AB20 ?5@540B>G=>3> >B=>H5=8O :>=B0:B8@CNI85 CG0AB:8 ?@>D8;59 4>;6=K 1KBL >G5@G5=K ?> B0:8< :@82K<, GB>1K 2 ;N1>9 <><5=B A>?@8:>A=>25=8O 8E >1I0O =>@<0;L 2 B>G:5 :>=B0:B0 ?@>E>48;0 G5@57 >4=C 8 BC 65 B>G:C  (?>;NA 70F5?;5=8O), =0 ;8=88 F5=B@>2, B.5. ?>;NA 70F5?;5=8O 2 ?@>F5AA5 ?5@5:0BK20=8O 7C1L52 =5 <5=O5B A2>53> ?>;>65=8O. 56>A52>5 @0AAB>O=85 <>6=> >?@545;8BL:  EMBED Equation.3 , 8;8  EMBED Equation.3 ;  EMBED Equation.3 . #3>; (W, A>AB02;5==K9 >1I59 =>@<0;LN n-n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r1 8 r2, 2A5340 @02=K, A;54>20B5;L=>, 5A;8 ?5@20O >:@C6=>ABL >AB0=>2;5=0, B> 2B>@0O 1C45B :0B8BLAO ?> ?5@2>9 157 A:>;L65=8O, 0 B>G:0  AB0=5B <3=>25==K< F5=B@>< A:>@>AB59 2B>@>3> B5;0. ;O >?@545;5=8O <3=>25==>9 A:>@>AB8 ( 70?8H5< A:>@>ABL >A8 02 2 2845 @025=AB20:  EMBED Equation.3 . "0: :0:  EMBED Equation.3 , 0  EMBED Equation.3 , B>:  EMBED Equation.3 . B0:, ?>A;5 >AB0=>2:8 ?5@2>3> B5;0 2B>@>5 1C45B 2@0I0BLAO 2>:@C3 ?>;NA0 A <3=>25==>9 C3;>2>9 A:>@>ABLN  EMBED Equation.3 . BAN40 A;54C5B, GB> B>G:0  :>=B0:B0 2B>@>3> B5;0 1C45B A:>;L78BL ?> ?>25@E=>AB8 ?5@2>3> B5;0 A> A:>@>ABLN,  EMBED Equation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quation.3 , 345 s  B>;I8=0 7C10; s2  H8@8=0 2?048=K. 5;8G8=0  EMBED Equation.3 , << =07K205BAO <>4C;5< 70F5?;5=8O. >;CG8< D>@<C;C 4;O >?@545;5=8O @048CA0 45;8B5;L=>9 >:@C6=>AB8 rw. ;8=0 45;8B5;L=>9 >:@C6=>AB8 :>;5A0 @02=0:  EMBED Equation.3 , 8;8 2 H030E  EMBED Equation.3 . BAN40:  EMBED Equation.3 (  EMBED Equation.3 .  n  ra  s   s2  N  p  ry (  (y n r     rf 0 rb 8A. 5.10 5;8B5;L=0O >:@C6=>ABL  MB> >:@C6=>ABL 4;O :>B>@>9 H03 405B 2 ?5@5AG5B5 AB0=40@B=>5 7=0G5=85 <>4C;O. ;O =>@<0;L=KE :>;5A =0E>4OI8EAO 2 70F5?;5=88 45;8B5;L=K5 >:@C6=>AB8 A>2?040NB A =0G0;L=K<8 r = rw. 5;8B5;L=0O >:@C6=>ABL 45;8B 7C1 =0 3>;>2:C 8 =>6:C. KA>B0 7C10 @02=0: h = h( + h(, 345 h(  2KA>B0 3>;>2:8 7C10, h( = m; h( - 2KA>B0 =>6:8 7C10, h( = 1,25( m. >;=0O 2KA>B0 7C10 h =2,25( m. 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">G=55 45;8B5;L=>9 =07K205BAO B0:0O >:@C6=>ABL 7C1G0B>3> :>;5A0, =0 :>B>@>9 <>4C;L 8 H03 ?@8=8<0NB AB0=40@B=>5 7=0G5=85. :@C6=K< H03>< 8;8 H03>< =07K205BAO @0AAB>O=85 ?> 4C35 >:@C6=>AB8 <564C >4=>8<5==K<8 B>G:0<8 ?@>D8;59 A>A54=8E 7C1L52 (?>4 >4=>8<5==K<8 ?>=8<0NBAO ?@02K5 8;8 ;52K5 ?@>D8;8 7C10). #3;>2>9 H03 ( - F5=B@0;L=K9 C3>; A>>B25BAB2CNI89 4C35 p - >:@C6=><C H03C ?> 45;8B5;L=>9 >:@C6=>AB8. @8<5G0=85: !>3;0A=> !" >A=>2=K5 M;5<5=BK 7C1G0B>3> :>;5A0 >1>7=0G0NBAO ?> A;54CNI8< ?@028;0<: ;8=59=K5 25;8G8=K - AB@>G=K<8 1C:20<8 ;0B8=A:>3> 0;D028B0, C3;>2K5 - 3@5G5A:8<8 1C:0<8; CAB0=>2;5=K 8=45:AK 4;O 25;8G8=: ?> >:@C6=>ABO<: 45;8B5;L=>9 - 157 8=45:A0, 25@H8= - a , 2?048= - f , >A=>2=0O - b , =0G0;L=0O - w ; ?> A5G5=8O<: =>@<0;L=>5 A5G5=85 - n , B>@F52>5 A5G5=85 - t , >A52>5 A5G5=85 - x; >B=>AOI8EAO : 7C1>@57=><C 8=AB@C<5=BC - 0 . 048CA >A=>2=>9 >:@C6=>AB8 4;O M2>;L25=BK: rb = r( cos (. 345 ( - C3>; ?@>D8;O =0 45;8B5;L=>9 >:@C6=>AB8, #3;>< ?@>D8;O =07K205BAO >AB@K9 C3>; <564C :0A0B5;L=>9 : ?@>D8;N 2 40==>9 B>G:8 8 @048CA>< - 25:B>@><, ?@>2545==K< 2 40==CN B>G:C 87 F5=B@0 :>;5A0. >=B@>;L=K5 2>?@>AK @>0=0;878@C9B5 4>AB>8=AB20 8 =54>AB0B:8 D@8:F8>==>9 ?5@540G8? 'B> B0:>5 D@8:F8>==K9 20@80B>@ A:>@>AB8? 07>28B5 >A=>2=K5 284K 7C1G0BKE <5E0=87<>2? !D>@<C;8@C9B5 8 4>:068B5 >A=>2=CN B5>@5<C 70F5?;5=8O? 'B> =07K205BAO H03>< 70F5?;5=8O? 'B> =07K205BAO <>4C;5< 70F5?;5=8O, 70G5< 53> 3>AB8@CNB? '5<C @025= 480<5B@ 45;8B5;L=>9 >:@C6=>AB8? '5<C 2 4>;OE <>4C;O @02=0 2KA>B0 =>6:8 8 3>;>2:8 7C10 =>@<0;L=>3> :>;5A0? 5:F8O 6 !;>6=K5 7C1G0BK5 ?5@540G8. 5@540B>G=>5 >B=>H5=85 ?5@540G A ?@><56CB>G=K<8 :>;5A0<8 8 20;0<8. ;0=5B0@=K5 <5E0=87<K. $>@<C;0 8;;8A0. 8=5<0B8G5A:>5 8AA;54>20=85 B8?>2KE ?;0=5B0@=KE <5E0=87<>2 0=0;8B8G5A:8< <5B>40<. !;>6=K5 7C1G0BK5 <5E0=87<K  1>;LH8=AB25 A>2@5<5==KE <0H8= 8 ?@81>@>2 2>7=8:05B =5>1E>48<>ABL >ACI5AB2;OBL ?5@540GC 2@0I5=8O >B 254CI53> 20;0 : 254><><C A 1>;LH8< ?5@540B>G=K< >B=>H5=85< (?@8 7=0G8B5;L=KE <56>A52KE @0AAB>O=8OE) 8 AB@>3>< A>3;0A>20=88 A:>@>AB59 2@0I5=8O >B45;L=KE 20;>2.  MB>< A;CG05 ?@8<5=ONB 7C1G0BK5 <5E0=87<K ;81> 70<54;ONI85 ((254CI53> ( (254><>3>), B0: =07K205<K5 @54C:B>@K, ;81> CA:>@ONI85 ((254CI53> ( (254><>3>) ( <C;LB8?;8:0B>@K. "0:85 <5E0=87<K >1;040NB >4=>9 AB5?5=LN A2>1>4K (;5:F8O 5, @8A 5.4, 0, 1). "0: :0: 2 <0H8=>AB@>5=88 G0I5 2>7=8:05B =5>1E>48<>ABL 2 C<5=LH5=88 A:>@>AB59 2@0I5=8O, B> 70<54;ONI85 <5E0=87<K (8;8 @54C:B>@K) =0H;8 1>;55 H8@>:>5 ?@8<5=5=85 =0 ?@0:B8:5. 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B5E A;CG0OE, :>340 7040==>5 ?5@540B>G=>5 >B=>H5=85 ?@52KH05B F5;5A>>1@07=>5 4;O >4=>9 ?0@K :>;5A 8;8 :>340 B@51C5BAO >15A?5G8BL 1>;LH>5 <56>A52>5 @0AAB>O=85, 8A?>;L7CNB A;>6=K5 7C1G0BK5 <5E0=87<K, A>AB>OI85 87 =5A:>;L:8E ?0@0;;5;L=> 8;8 ?>A;54>20B5;L=> A>548=5==KE 4@C3 A 4@C3>< 7C1G0BKE ?5@540G. 07;8G0NB 420 2840 B0:8E <5E0=87<>2: A;>6=K5 7C1G0BK5 <5E0=87<K A =5?>4286=K<8 >AO<8 (<=>3>:@0B=K5 7C1G0BK5 ?5@540G8 ;5:F8O 5, @8A 5.8, 0) 8 ?;0=5B0@=K5 (M?8F8:;8G5A:85) 7C1G0BK5 <5E0=87<K (>A8 >B45;L=KE :>;5A <>3CB ?5@5<5I0BLAO >B=>A8B5;L=> AB>9:8, :0: =0 @8A 5.8, 1, 2 ;5:F88 5). 064K9 87 MB8E 284>2 A;>6=KE 7C1G0BKE <5E0=87<>2 <>65B 1KBL A>AB02;5= =5 B>;L:> 87 >4=>@>4=KE :>;5A (F8;8=4@8G5A:8E 8;8 :>=8G5A:8E) 8 ?5@540G (A =5?>4286=K<8 >AO<8 8;8 ?;0=5B0@=KE), => 8 87 8E A>G5B0=8O (@8A. 6.1). 081>;LH55 @0A?@>AB@0=5=85 ?>;CG8;8 A;>6=K5 7C1G0BK5 <5E0=87<K, A>AB02;5==K5 87 F8;8=4@8G5A:8E :>;5A A ?@O<K<8 7C1LO<8 (@565 :>AK<8) A @02=>45;5==K< H03><. A;8 C 7C1G0BKE <5E0=87<>2 >A8 254CI53> 8 254><>3> 725=L52 @0A?>;030NBAO ?> >4=>9 ?@O<>9 (@8A 5.8, 1, 2), B> >=8 =07K20NBAO A>>A=K<8.  8A. 6.1 1I55 ?5@540B>G=>5 >B=>H5=85 ;N1>3> B8?0 A;>6=>3> 7C1G0B>3> <5E0=87<0 i1n, 2:;NG0NI53> =5A:>;L:> ?>A;54>20B5;L=> A>548=5==KE 4@C3 A 4@C3>< ?@>ABKE <5E0=87<>2 (=5 ?;0=5B0@=KE ABC?5=59, ?;0=5B0@=KE 8;8 8E :><18=0F89) 87 n :>;5A, @02=> ?@>872545=8N ?5@540B>G=KE >B=>H5=89 >B45;L=KE ?@>ABKE <5E0=87<>2 (ABC?5=59), 2E>4OI8E 2 53> A>AB02, B.5.: EMBED Equation.3, "0:, 4;O <5E0=87<0, ?@54AB02;5==>3> =0 @8A. 6.1, A>AB>OI53> 87 ABC?5=8 F8;8=4@8G5A:>9 ?5@540G8 A :>;5A0<8 1(2 A =5?>4286=K<8 >AO<8, :>=8G5A:>9 ?5@540G8 3(4, ?;0=5B0@=>9 ABC?5=8 5(6(7, :>=8G5A:>9 ?5@540G8 A =5?>4286=K<8 >AO<8 7(8, >1I55 ?5@540B>G=>5 >B=>H5=85 @02=>: EMBED Equation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quation.3 345 i12 ( ?5@540B>G=>5 >B=>H5=85 ?5@2>9 ?0@K AF5?;ONI8EAO 7C1G0BKE :>;5A 2=5H=53> 70F5?;5=8O: EMBED Equation.3  8A. 6.2 =0: <8=CA ?>AB02;5= ?>B><C, GB> :>;5A0 1 8 2 2@0I0NBAO 2 ?@>B82>?>;>6=KE =0?@02;5=8OE. ;O 2B>@>9 ?0@K: EMBED Equation.3 ;O B@5BL59 ?0@K: EMBED Equation.3, ">340 8A:><>5 ?5@540B>G=>5 >B=>H5=85: EMBED Equation.3  >1I5< A;CG05 ?@8 n :>;5A0E 2 <5E0=87<5: EMBED Equation.3 1I55 ?5@540B>G=>5 >B=>H5=85 @O4>2>3> 7C1G0B>3> <5E0=87<0 @02=> >1@0B=><C >B=>H5=8N G8A5; 7C1L52 8;8 @048CA>2 :@09=8E :>;5A. =0: ?5@540B>G=>3> >B=>H5=8O >?@545;O5BAO <=>68B5;5< ((1)n, 345 n ( G8A;> ?5@540G 2=5H=53> 70F5?;5=8O. @8 n G5B=>< i ( 0, B.5. 254><>5 8 254CI55 725=LO @54C:B>@0 8;8 <C;LB8?;8:0B>@0 2@0I0NBAO 2 >4=>< =0?@02;5=88; ?@8 =5G5B=>< n ( 2 @07=KE =0?@02;5=8OE. =0;878@CO ?@82545==K5 ?@8<5@K CAB0=02;8205<, GB> G8A;> 7C1L52 ?@><56CB>G=KE :>;5A 2 8 3, =0E>4OI8EAO >4=>2@5<5==> 2 70F5?;5=88 A 42C<O 4@C38<8 :>;5A0<8, =5 2;8O5B =0 25;8G8=C >1I53> ?5@540B>G=>3> >B=>H5=8O <5E0=87<0. > CAB0=>2:0 B0:8E ?@><56CB>G=KE :>;5A ?>72>;O5B 87<5=OBL =0?@02;5=85 2@0I5=8O 254><>3> 725=0. @8 G5B=>< G8A;5 ?@><56CB>G=KE :>;5A =0?@02;5=85 2@0I5=8O 254CI53> 8 254><>3> 725=L52 ?@>B82>?>;>6=K, ?@8 =5G5B=>< ( >48=0:>2K. @8<5=ONB MB8 :>;5A0 3;02=K< >1@07>< B0<, 345 =5>1E>48<> 87<5=8BL =0?@02;5=85 2@0I5=8O 254><>3> 20;0 ?@8 =587<5==>< =0?@02;5=88 2@0I5=8O 254CI53> (<5E0=87< B@5=75;O B>:0@=>3> AB0=:0, <5E0=87< 704=53> E>40 02B><>18;L=>9 :>@>1:8 ?5@540G 8 4@.), ;81> B0<, 345 =5>1E>48<> >15A?5G8BL ?5@540GC 42865=8O ?@8 1>;LH8E <56>A52KE @0AAB>O=8OE (:>340 =5;L7O C25;8G820BL @07<5@K 254CI8E 8 254><KE :>;5A 87-70 8E 1>;LH8E 3010@8B>2). AA;54>20=85 7C1G0BKE <5E0=87<>2 A ?@><56CB>G=K<8 20;0<8 !;>6=K5 7C1G0BK5 <5E0=87<K A ?@><56CB>G=K<8 20;0<8 ?@54AB02;ONB A>1>9 ?>A;54>20B5;L=>5 A>548=5=85 =5A:>;L:8E ?0@ :>;5A, =0 :064K9 87 20;>2 :>B>@>3> ?><5I5=> 1>;55 >4=>3> :>;5A0 (:@><5 20;>2 254CI53> 8 254><>3> :>;5A). 0 @8A. 6.3 ?@54AB02;5= B0:>9 B@QEABC?5=G0BK9 <5E0=87< 4;O ?@5>1@07>20=8O 42865=8O <564C ?0@0;;5;L=K<8 20;0<8, :>B>@K9 A>AB>8B 87 42CE ABC?5=59 2=5H=53> 70F5?;5=8O A F8;8=4@8G5A:8<8 :>;5A0<8 (1(2 8 3(4) 8 >4=>9 ABC?5=8 2=CB@5==53> > 70F5?;5=8O (:>;5A0 5(6). >;5A0 2(3 8 4(5 A>548=5=K 2<5AB5, >1@07CO 725=LO.  8A. 6.3 5@540B>G=>5 >B=>H5=85 ?5@2>9 ABC?5=8 @02=>: EMBED Equation.3, 2B>@>9: EMBED Equation.3 B@5BL59: EMBED Equation.3 5@5<=>60O MB8 7=0G5=8O ?5@540B>G=KE >B=>H5=89, ?>;CG05< EMBED Equation.3 #G8BK20O, GB> , (2 = (3; (4 = (5, ?>A;5 A>:@0I5=8O ?>;CG05< EMBED Equation.3 1I55 ?5@540B>G=>5 >B=>H5=85 ABC?5=G0B>9 ?5@540G8 @02=> ?@>872545=8N ?5@540B>G=KE >B=>H5=89 ABC?5=59, 2E>4OI8E 2 A>AB02 <5E0=87<0, 8;8 @02=> >B=>H5=8N ?@>872545=8O G8A5; 7C1L52 (?>;>84=KE @048CA>2) 254><KE :>;5A : ?@>872545=8N G8A5; 7C1L52 (@048CA>2) 254CI8E :>;55, 27OBKE A> A2>8<8 7=0:0<8. "0: :0: ?5@540B>G=>5 >B=>H5=85 MB>3> <5E0=87<0 (2 >B;8G85 >B @O4>2>3>) 7028A8B >B G8A;0 7C1L52 2A5E 2E>4OI8E 2 53> A>AB02 :>;5A, B> ?CB5< A>>B25BAB2CNI53> ?>41>@0 G8A5; 7C1L52 :>;5A <>6=> ?>;CG8BL 1>;LH85 ?5@540B>G=K5 >B=>H5=8O.  >1I5< A;CG05 ?@8 n :>;5A0E q 2=5H=8E 70F5?;5=89, >1I55 ?5@540B>G=>5 >B=>H5=85 @02=>: EMBED Equation.3  A;CG05 A>>A=>3> <5E0=87<0, A>AB02;5==>3> 87 =C;52KE :>;5A (@8A. 6.4) 4>;6=> C4>2;5B2>@OBLAO CA;>285 A>>A=>AB8 (@025=AB2> <56>A52KE @0AAB>O=89):  EMBED Equation.3 , 8;8  EMBED Equation.3 , 345 m12 8 m34 ( A>>B25BAB25==> <>4C;8 70F5?;5=89 ?5@2>9 8 2B>@>9 ABC?5=59.  8A. 6.4 ;0=5B0@=K5 <5E0=87<K !;>6=K5 7C1G0BK5 <5E0=87<K, 2 :>B>@KE >AL E>BO 1K >4=>3> :>;5A0 ?>4286=0, =07K20NBAO ?;0=5B0@=K<8 <5E0=87<0<8.  B8?>2K< ?;0=5B0@=K< <5E0=87<0< >B=>AOBAO: >4=>@O4=K9 ?;0=5B0@=K9 <5E0=87<; 42CE@O4=K9 ?;0=5B0@=K9 <5E0=87< A >4=8< 2=5H=8< 8 >4=8< 2=CB@5==8< 70F5?;5=85<; 42CE@O4=K9 ?;0=5B0@=K9 <5E0=87< A 42C<O 2=5H=8<8 70F5?;5=8O<8; 42CE@O4=K9 ?;0=5B0@=K9 <5E0=87< A 42C<O 2=CB@5==8<8 70F5?;5=8O<8. -;5<5=BK ?;0=5B0@=>3> <5E0=87<0 8<5NB A?5F80;L=K5 =0720=8O: 7C1G0BK5 :>;5A0 >A8 :>B>@KE =5?>4286=K =07K20NBAO F5=B@0;L=K<8. >;5A> A 2=5H=8<8 7C1LO<8, @0A?>;>65==>5 2 F5=B@5 <5E0=87<0 =07K205BAO A>;=5G=K<, :>;5A> A 2=CB@5==8<8 7C1LO<8 =07K20NB :>@>=>9 8;8 M?8F8:;><; :>;5A0, >A8 :>B>@KE ?>4286=K, =07K20NB ?;0=5B=K<8 8;8 A0B5;;8B0<8; ?>4286=>5 725=>, A>548=ONI55 >A8 F5=B@0;L=KE :>;5A 8 A0B5;;8B>2 =07K20NB 2>48;><. >48;> ?@8=OB> >1>7=0G0BL =5 F8D@>9, 0 ;0B8=A:>9 1C:2>9 h, 8;8 @CAA:>9 2.  B01;8F5 6.1 ?@82545=K AB@C:BC@=K5 AE5<K B8?>2KE ?;0=5B0@=KE <5E0=87<>2, 0 B0:65 480?07>=K @5:><5=4C5<KE ?5@540B>G=KE >B=>H5=89 8 >@85=B8@>2>G=K5 7=0G5=8O  ?@8 MB8E ?5@540B>G=KE >B=>H5=8OE. "01;8F0 6.1 "8?>2K5 AE5<K ?;0=5B0@=KE <5E0=87<>2 ! !B@C:BC@=0O AE5<0 <5E0=87<0u(1. 2 3 2 (1 (23& 10  EMBED Equation.3 0,97& 0,99 2  2 3 1 2 (1 (27& 16  EMBED Equation.3 0,96& 0,983 2 3 1 2 (1 (225& 300  EMBED Equation.3 0,9& 0,34 2 3 1 2 (1 (230& 300  EMBED Equation.3 0,9& 0,3 A;8 AB5?5=L ?>4286=>AB8 ?;0=5B0@=>3> <5E0=87<0 2 8 1>;55, B> 53> =07K20NB 48DD5@5=F80;L=K<. ;O <5E0=87<0 2K?>;=5==>3> ?> AE5<5 1 B01;. 6.1 AB5?5=L ?>4286=>AB8 ?@8 70:@5?;Q==>< :>;5A5 3 @02=0: W = 3( (4  1) - 2(3  1( 2 = 1. @8 A2>1>4=>< :>;5A5 3: W = 3( (5  1) - 2(4  1( 2 = 2. >AB>8=AB20 ?;0=5B0@=KE <5E0=87<>2: 1>;LH85 ?5@540B>G=K5 >B=>H5=8O ?@8 <0;KE 3010@8B=KE @07<5@0E; <>6=> ?@8<5=OBL 4;O A;>65=8O, 8;8 @0745;5=8O 42865=8O. 5B>4 8;;8A0 (0=0;8B8G5A:89 <5B>4 :8=5<0B8G5A:>3> 0=0;870) 5B>4 8;;8A0 >A=>20= =0 A?>A>15 >1@0I5=8O 42865=8O (A?>A>1 <KA;5==>9 >AB0=>2:8 2>48;0): 2A5< 725=LO< <5E0=87<0 <KA;5==> ?@8405< 2@0I5=85 A C3;>2>9 A:>@>ABLN @02=>9 C3;>2>9 A:>@>AB8 2>48;0, B>;L:> 2 >1@0B=>< =0?@02;5=88. B=>A8B5;L=K5 42865=8O 725=L52 ?@8 MB>< =5 87<5=ONBAO, 01A>;NB=K5 1C4CB A;54CNI8<8 (4;O AE5<K 1, B01;. 6.1):  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 . 0AH8D@C5< ?@8=OBK5 >1>7=0G5=8O: (1, (2, (3, (2  8AB8==K5 7=0G5=8O C3;>2KE A:>@>AB59 725=L52;  EMBED Equation.3 - C3;>20O A:>@>ABL F5=B@0;L=>3> :>;5A0 1 ?@8 >AB0=>2;5==>< 2>48;5;  EMBED Equation.3 - C3;>20O A:>@>ABL A0B5;;8B0 2 ?@8 >AB0=>2;5==>< 2>48;5;  EMBED Equation.3 - C3;>20O A:>@>ABL F5=B@0;L=>3> :>;5A0 3 ?@8 >AB0=>2;5==>< 2>48;5 (@02=0 - (2);  EMBED Equation.3 - C3;>20O A:>@>ABL 2>48;0 ?@8 >AB0=>2;5==>< 2>48;5 (@02=0 =C;N). ">340 ?>;CG05<, GB> 2A5 :>;5A0 A>25@H0NB 2@0I0B5;L=K5 42865=8O 2>:@C3 =5?>4286=KE >A59 8 >1I55 ?5@540B>G=>5 >B=>H5=85 <>6=> =09B8 ?> D>@<C;5:  EMBED Equation.3 ( D>@<C;0 8;;8A0. >A;5 ?@8<5=5=8O <5B>40 >1@0I5=8O 42865=8O, @0AA<0B@8205<K9 <5E0=87< <>6=> @0AA<0B@820BL :0: A;>6=K9 42CEABC?5=G0BK9 A ?@><56CB>G=K< :>;5A><:  EMBED Equation.3 , B>340:  EMBED Equation.3 . >=B@>;L=K5 2>?@>AK >G5<C ?@8 >?@545;5=88 ?5@540B>G=>3> >B=>H5=8O 7C1G0B>3> A;>6=>3> <5E0=87<0 A ?@><56CB>G=K<8 :>;5A0<8 <>6=> =5 CG8BK20BL :>;8G5AB2> 7C1L52 ?@><56CB>G=KE :>;5A? 'B> B0:>5 ?@><56CB>G=K9 20;? 09B5 E0@0:B5@8AB8:C 725=L52 2E>4OI8E 2 ?;0=5B0@=K9 <5E0=87<? K2548B5 D>@<C;C 8;;8A0 4;O 0=0;870 ?;0=5B0@=>3> <5E0=87<0? 5:F8O 7 @>5:B8@>20=85 ?;0=5B0@=KE 7C1G0BKE <5E0=87<>2. >AB0=>2:0 7040G8 A8=B570. #A;>28O ?>41>@0 G8A5; 7C1L52. K2>4 @0AG5B=KE D>@<C; 4;O CA;>289 A>>A=>AB8, A>A54AB20 8 A1>@:8. 040G8 A8=B570 70F5?;5=89. -2>;L25=B0. -2>;L25=B=>5 70F5?;5=85. @>5:B8@>20=85 ?;0=5B0@=KE <5E0=87<>2. >AB0=>2:0 7040G8 A8=B570 @8 ?@>5:B8@>20=88 <=>3>?>B>G=KE ?;0=5B0@=KE <5E0=87<>2 =5>1E>48<>, :@><5 B@51>20=89 B5E=8G5A:>3> 7040=8O, 2K?>;=OBL @O4 CA;>289 A2O70==KE A >A>15==>ABO<8 ?;0=5B0@=KE <5E0=87<>2. 040G59 A8=B570, O2;O5BAO ?>;CG5=85 B@51C5<>3> ?5@540B>G=>3> >B=>H5=8O, 2K1>@ AE5<K >B25G0NI59 B@51>20=8O< =081>;55 2KA>:>3>  8 3010@8B=K<8 @07<5@0<8 8 B.4. >A;5 2K1>@0 AE5<K <5E0=87<0 =5>1E>48<> >?@545;8BL A>G5B0=85 G8A5; 7C1L52 53> :>;5A, :>B>@K5 >15A?5G0B 2K?>;=5=85 CA;>289 B5E=8G5A:>3> 7040=8O  4;O @54C:B>@0 MB> ?5@540B>G=>5 >B=>H5=85 8 25;8G8=0 <><5=B0 A>?@>B82;5=8O =0 2KE>4=>< 20;C. 5@540B>G=>5 >B=>H5=85 70405B CA;>28O 2K1>@0 >B=>A8B5;L=KE @07<5@>2 7C1G0BKE :>;5A  G8A5; 7C1L52 :>;5A, :@CBOI89 <><5=B 70405B CA;>28O 2K1>@0 01A>;NB=KE @07<5@>2  <>4C;59 7C1G0BKE 70F5?;5=89. "0: :0: 4;O >?@545;5=8O <>4C;O =5>1E>48<> 2K1@0BL <0B5@80; 7C1G0B>9 ?0@K 8 284 53> B5@<>>1@01>B:8, B> =0 ?5@2KE MB0?0E ?@>5:B8@>20=8O ?@8=8<0NB <>4C;L 7C1G0BKE :>;5A @02=K< 548=8F5, B> 5ABL @5H0NB 7040GC :8=5<0B8G5A:>3> A8=B570 <5E0=87<0 2 >B=>A8B5;L=KE 25;8G8=0E. 1KG=> ?@8 >?@545;5=88 @07<5@>2 725=L52 <5E0=87<>2, B.5. ?@8 ?>41>@5 G8A5; 7C1L52 :>;5A 2K?>;=ONB CA;>285 3-E A: A>>A=>AB8, A>A54AB20 8 A1>@:8. #A;>28O ?>41>@0 G8A5; 7C1L52. K2>4 @0AG5B=KE D>@<C; 4;O CA;>289 A>>A=>AB8, A>A54AB20 8 A1>@:8. #A;>28O, :>B>@K5 =5>1E>48<> 2K?>;=8BL ?@8 ?>41>@5 G8A5; 7C1L52 :>;5A B8?>2>3> ?;0=5B0@=>3> <5E0=87<0: 7040==>5 ?5@540B>G=>5 >B=>H5=85 A B@51C5<>9 B>G=>ABLN; A>>A=>ABL 2E>4=>3> 8 2KE>4=>3> 20;>2 <5E0=87<0; A2>1>4=>5 @07<5I5=85 (A>A54AB2>) A0B5;;8B>2; A1>@:C <5E0=87<0 ?@8 2K1@0==KE G8A;0E 7C1L52 :>;5A; <8=8<0;L=K5 >B=>A8B5;L=K5 3010@8BK <5E0=87<0. 0AA<>B@8< MB8 CA;>28O ?>4@>1=55 =0 ?@8<5@5 42CE@O4=>3> ?;0=5B0@=>3> <5E0=87<0 A >4=8< 2=5H=8< 8 >4=8< 2=CB@5==8< 70F5?;5=85< (@8A. 7.1). 15A?5G5=85 7040==>3> ?5@540B>G=>3> >B=>H5=8O A B@51C5<>9 B>G=>ABLN. @8=8<05< B@51C5<CN B>G=>ABL ( 5%, B>340 4;O @0AA<0B@8205<>9 AE5<K <5E0=87<0  EMBED Equation.3  15A?5G5=85 CA;>28O A>>A=>AB8 2E>4=>3> 8 2KE>4=>3> 20;>2. ;O MB>3> =5>1E>48<> GB>1K <56>A52>5 @0AAB>O=85 2 ?5@540G5 2=5H=53> 70F5?;5=8O (?5@2K9 @O4) @02=O;>AL <56>A52><C @0AAB>O=8N 2 ?5@540G5 2=CB@5==53> 70F5?;5=8O (2B>@>9 @O4), B> 5ABL:  EMBED Equation.3 ;  EMBED Equation.3 ;  EMBED Equation.3 . 1KG=> 2 ?;0=5B0@=KE <5E0=87<0E ?@8<5=ONBAO 7C1G0BK5 :>;5A0 157 A<5I5=8O, 4;O :>B>@KE xi = 0 8 rwi = ri = Zi ( m / 2. ">340  EMBED Equation.3  @8=8<05<, GB> mI = mII = m, 8 ?>;CG05< CA;>285 A>>A=>AB8 4;O 40==>9 AE5<K <5E0=87<0  EMBED Equation.3 .  3  !1  2 C  B 1   awI A awII  A1  0 (1 (2 B2 2 3 B3  1 C2 C3  2 (2 8A. 7.1 15A?5G5=85 CA;>28O A>A54AB20 A0B5;;8B>2 70:;NG05BAO 2 B><, GB>1K A0B5;;8BK ?>AB02;5==K5 4;O ?>2KH5=8O 65AB:>AB8, ?@>G=>AB8, 0 B0:65 C@02=>25H820=8O <0AA =5 704520;8 4@C3 4@C30 (?@8 G8A;5 A0B5;;8B>2 k > 1). !0B5;;8BK @07<5I0NBAO =0 >:@C6=>AB8 @048CA0 aw. 5@H8=K 7C1L52 A0B5;;8B>2 =5 1C4CB <5H0BL 42865=8N 4@C3 4@C30, 5A;8 2K?>;=O5BAO CA;>285:  EMBED Equation.3  ;O 7C1G0BKE :>;5A 157 A<5I5=8O <0:A8<0;L=K9 87 480<5B@>2 A0B5;;8B>2 @025=:  EMBED Equation.3 . 0AAB>O=85 <564C >AO<8 A0B5;;8B>2  EMBED Equation.3 , 345 (2  C3>; <564C 42C<O A>A54=8<8 A0B5;;8B0<8. >4AB028< ?>;CG5==K5 2K@065=8O 2 =5@025=AB2> 8 ?>;CG8< CA;>285 A>A54AB20:  EMBED Equation.3 ,  EMBED Equation.3 . 4. #A;>285 A1>@:8  CA;>285 @02=KE C3;>2 <564C A0B5;;8B0<8, 70:;NG05BAO 2 B><, GB> ?@8 ?>AB0=>2:5 1-3> A0B5;;8B0 F5=B@0;L=K5 :>;5A0 709<CB 2?>;=5 >?@545;Q==>5 2708<=>5 @0A?>;>65=85 8 >AB0;L=K5 A0B5;;8BK <>3CB 1KBL 22545=K 2 70F5?;5=85 B>;L:> ?@8 >?@545;Q==>< A>>B=>H5=88 <564C G8A;>< 8E 7C1L52. 0945< 4;8=K 4C3 <564C 42C<O A0B5;;8B0<8 :>;5A0 3 8 1: - 2 H030E:  EMBED Equation.3  - G5@57 :>;8G5AB2> 7C1L52 =0 4C35:  EMBED Equation.3 , (F5;>5 G8A;> 7C1L52 ?;NA :CA>G5: 7C10). !;>68< ;52K5 8 ?@02K5 G0AB8 2K@065=89:  EMBED Equation.3 , 3@C??8@C5<:  EMBED Equation.3 , ?>A;5 45;5=8O ?>;CG05<:  EMBED Equation.3 . 848<, GB> 2 ;52>9 ?>;>28=5 F5;>5 G8A;>, A;54>20B5;L=> 8 AC<<0 2 ?@02>9 ?>;>28=5 2K@065=8O 4>;6=0 40BL F5;>5 G8A;>, B.:. ?5@2>5 A;0305<>5 B>65 405B 2A5340 F5;>5 G8A;>, B>:  EMBED Equation.3  -B> 2K?>;=O5BAO :>340:  EMBED Equation.3 . !;54>20B5;L=>, ?>;CG05<:  EMBED Equation.3 , 345  EMBED Equation.3 ,   ;N1>5 F5;>5 G8A;>. BAN40 CA;>285 A1>@:8 ?@8=8<05B 284:  EMBED Equation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quation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rb). 0AA<>B@8< ?>AB@>5=85 M2>;L25=BK  (@8A. 7.3).  ?@>872>;L=>9 B>G:5 M2>;L25=BK  ?@>2545< =>@<0;L, :>B>@0O :0A05BAO >A=>2=>9 >:@C6=>AB8 2 B>G:5 , ?>;CG05< @048CA :@8287=K M2>;L25=BK (.  8A. 7.3 7 ?@O<>C3>;L=>3> B@5C3>;L=8:0 ( =0945< :0B5B :  EMBED Equation.3 . 7 CA;>28O >1@07>20=8O M2>;L25=BK @048CA :@8287=K  4>;65= 1KBL @025= 4;8=5 @0725@BK205<>9 4C38  >A=>2=>9 >:@C6=>AB8: ( = rb(((+(),  EMBED Equation.3 , 345 ( ( ?>;O@=K9 C3>; =0:;>=0 @048CA 25:B>@0; ( C3>; <564C =0?@02;5=85< @048CA 25:B>@0 8 =0?@02;5=85< @048CA0 >A=>2=>9 >:@C6=>AB8 ?@>2545==>3> 2 B>G:5 :0A0=8O =>@<0;8. BAN40:  EMBED Equation.3 . 07=>ABL B0=35=A0 8 C3;0 ?@54AB02;O5B A>1>9 M2>;L25=B=CN DC=:F8N =07K205<CN 8=2>;NB>9. =2>;NB0 O2;O5BAO ?0@0<5B@>< 4;O 35><5B@8G5A:8E @0AG5B>2 7C1G0BKE <5E0=87<>2. !2>9AB20 M2>;L25=BK: M2>;L25=B0 =5 8<55B B>G5: 2=CB@8 >A=>2=>9 >:@C6=>AB8; =>@<0;L : ;N1>9 B>G:5 M2>;L25=BK =0?@02;5=0 ?> :0A0B5;L=>9 : >A=>2=>9 >:@C6=>AB8; F5=B@ :@8287=K M2>;L25=BK ;568B 2 B>G:5 :0A0=8O =>@<0;8 A >A=>2=>9 >:@C6=>ABLN. -2>;L25=B=>5 70F5?;5=85 8 53> A2>9AB20 7 A2>9AB2 M2>;L25=BK 2KB5:0NB A2>9AB20 M2>;L25=B=>3> 70F5?;5=8O. CABL ?@>D8;L 7C10 :>;5A0 1 (@8A. 7.4) >G5@G5= ?> M>;L25=B5 >A=>2=>9 >:@C6=>AB8 A @048CA>< rb1, 0 ?@>D8;L 7C10 :>;5A0 2  ?> M2>;L25=B5 >A=>2=>9 >:@C6=>AB8 @048CA0 rb2. ><5AB8< F5=B@K MB8E >:@C6=>AB59 2 F5=B@K 2@0I5=8O 1 8 2. >@<0;L : M2>;L25=B5 ?5@2>3> :>;5A0 4>;6=0 1KBL :0A0B5;L=>9 : >A=>2=>9 >:@C6=>AB8 ?5@2>3> :>;5A0, 0 =>@<0;L : M2>;L25=B5 2B>@>3> :>;5A0 4>;6=0 1KBL :0A0B5;L=>9 : >A=>2=>9 >:@C6=>AB8 2B>@>3> :>;5A0.  B>G:5 :0A0=8O M2>;L25=B =>@<0;L 4>;6=0 1KBL >1I59 : >1>8< ?@>D8;O<, 8, A;54>20B5;L=>, B>G:0 :>=B0:B0 ;568B =0 >1I59 :0A0B5;L=>9 : >A=>2=K< >:@C6=>ABO<. @8 2@0I5=88 254CI53> :>;5A0 1 ?@>B82 G0A>2>9 AB@5;:8, 0 254><>3> :>;5A0 2  ?> G0A>2>9 (@8A. 7.4, 0) B>G:0 :0A0=8O M2>;L25=B ?5@5<5I05BAO ?> >B@57:C 12 MB>9 :0A0B5;L=>9, B.:. 2=5 >B@57:0 12 M2>;L25=BK =5 <>3CB :0A0BLAO, B.5. 8<5BL >1ICN =>@<0;L; 12 O2;O5BAO ;8=859 70F5?;5=8O. ">G:0 ?5@5A5G5=8O >1I59 =>@<0;8 : M2>;L25=B0< A ;8=859 <56>A52>3> @0AAB>O=8O 12 O2;O5BAO ?>;NA>< 70F5?;5=8O  8 70=8<05B =587<5==>5 ?>;>65=85. A;8 =0?@02;5=85 2@0I5=85 254CI53> :>;5A0 1 8 254><>3> :>;5A0 2 87<5=8BAO, B> ;8=8O 70F5?;5=8O 12, ?> :>B>@>9 ?5@5<5I05BAO B>G:0 :>=B0:B0, 709<5B =>2>9 ?>;>65=85 (@8A. 7.4, 1). #3>; <564C ;8=859 70F5?;5=8O 12 8 ?@O<>9, ?5@?5=48:C;O@=>9 ;8=88 <56>A52>3> @0AAB>O=8O, =07K205BAO C3;>< 70F5?;5=8O 8 >1>7=0G05BAO G5@57 (w. #3;K 11 8 22 @02=K C3;C 70F5?;5=8O (w :0: C3;K A A>>B25BAB25==> ?5@?5=48:C;O@=K<8 AB>@>=0<8. >A:>;L:C 1 = rw1, 0 2 = rw2, B>  EMBED Equation.3 . !;54>20B5;L=>, ?@8 M2>;L25=B=>< 70F5?;5=88 ?5@540B>G=>5 >B=>H5=85 <>65B 1KBL 2K@065=> G5@57 >B=>H5=85 @048CA>2 >A=>2=KE >:@C6=>AB59:  EMBED Equation.3 , ?@8G5< 7=0: ?;NA >B=>A8BAO : 2=CB@5==5<C 70F5?;5=8N, 0 7=0: <8=CA  : 2=5H=5<C. 7 D>@<C;K 284=>, GB> ?@8 M2>;L25=B=>< 70F5?;5=88 87<5=5=85 <56>A52>3> @0AAB>O=8O =5 2;8O5B =0 7=0G5=85 ?5@540B>G=>3> >B=>H5=8O 2A;54AB285 =587<5==>AB8 @048CA>2 >A=>2=KE >:@C6=>AB59. @8 87<5=5=88 <56>A52>3> @0AAB>O=8O 87<5=ONBAO ;8HL @048CAK =0G0;L=KE >:@C6=>AB59 8 C3>; 70F5?;5=8O.  >=B@>;L=K5 2>?@>AK !D>@<C;8@C9B5 >A=>2=K5 7040G8 A8=B570 ?;0=5B0@=KE <5E0=87<>2?  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BAN40 C<5=LH05BAO ?;02=>ABL @01>BK ?5@540G8 8 ?>O2;ONBAO C40@K. 0 @8A 8.6 2, A?;>H=>9 ;8=859 ?>:070= ?@>D8;L ?@8 ( = 20(, 0 ?C=:B8@><  ?@8 1>;LH5< C3;5. > 2B>@>< A;CG05 ?@>D8;L 1>;55 ?>;>389 8 @01>G0O G0ABL ;8=88 70F5?;5=8O (N( <5=LH5 ?> 25;8G8=5.  8A. 8.6 !<5H0==>5 :>@@538@>20=85 (@8A. 8.6 3). -B>B 284 :>@@538@>20=8O ?>;CG8; =081>;LH55 @0A?@>AB@0=5=85. @8 =0@570=88 703>B>2:8 @56CI89 8=AB@C<5=B A<5I05BAO =0 7C1>@57=>< AB0=:5. 0 @8A. 8.6 3, ?C=:B8@>< ?>:070=> AB0=40@B=>5 70F5?;5=85 :>;5A0 A @59:>9, 0 A<5I5==>5 =0 25;8G8=C 2  :>=BC@=K<8 ;8=8O<8. @8 ?>;>68B5;L=>< A<5I5=88 B>;I8=0 7C10 @59:8 ?> ;8=88 AA, ?@>E>4OI59 G5@57 ?>;NA 70F5?;5=8O , 1C45B <5=LH5, G5< =0 ;8=88 A(A(, ?>MB><C H8@8=C 2?048=K H5AB5@=8 2> 871560=85 C40@>2 C<5=LH0NB, B.5. ?@8 :064>< B0:>< H035 B>;I8=C 5Q 7C10 C25;8G820NB. ">;I8=0 =0@5705<>3> 7C10 C >A=>20=8O C25;8G8205BAO, GB> C?@>G=O5B 7C1. >MDD8F85=B B>@F52>3> ?5@5:@KB8O. 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(03>< 70F5?;5=8O @( =07K205BAO @0AAB>O=85 ?> :>=B0:B=>9 =>@<0;8 (=>@<0;L : 3;02=K< ?@>D8;O< 2 B>G:5 8E :0A0=8O) <564C 42C<O :>=B0:B=K<8 B>G:0<8 >4=>8<5==KE 3;02=KE ?@>D8;59 A>A54=8E 7C1L52: @( = m ( cos( . ;8=0 0:B82=>9 ;8=88 70F5?;5=8O g(:  EMBED Equation.3  45AL @048CA >A=>2=>9 >:@C6=>AB8 rb ?>;CG5= 87 ?@O<>C3>;L=>3> B@5C3>;L=8:0 (@8A. 8.8), 345 38?>B5=C70  @048CA 45;8B5;L=>9 >:@C6=>AB8 (r = mZ/2), 0 ?@8;560I89 :0B5B  @048CA >A=>2=>9 >:@C6=>AB8:   EMBED Equation.3 . :>=G0B5;L=>  EMBED Equation.3  . >4AB02;OO D>@<C;K 4;O @( 8 g( 2 ((, ?>;CG05< 2K@065=85 4;O 2KG8A;5=8O :>MDD8F85=B0 B>@F>2>3> ?5@5:@KB8O:  EMBED Equation.3  . ;O ?@O<>7C1KE 7C1G0BKE :>;5A >1KG=> (( < 1,7. ;O C25;8G5=8O :>MDD8F85=B0 ?5@5:@KB8O 8A?>;L7CNB :>A>7C1K5 :>;5A0, B>340 4>102;O5BAO :>MDD8F85=B >A52>3> ?5@5:@KB8O ((, :>B>@K9 <>65B 1KBL 2KG8A;5= :0: >B=>H5=85 @01>G59 H8@8=K 25=F0 ?5@540G8 bw : >A52><C H03C @E (@8A. 8.9):  EMBED Equation.3  , 345 mn  @0AG5B=K9 8;8 =>@<0;L=K9 <>4C;L, B.5. <>4C;L 2 =>@<0;L=>< A5G5=88 nn.  0G5AB25==K5 E0@0:B5@8AB8:8 ?5@540G8 0AA<>B@8< 35><5B@8G5A:85 8 :8=5<0B8G5A:85 E0@0:B5@8AB8:8 7C1G0B>9 ?5@540G8, 7028AOI85 >B 8AE>4=KE ?0@0<5B@>2 ?5@540G8 Z1, Z2, m, x1, x2 8 2;8ONI85 =0 M:A?;C0B0F8>==K5 :0G5AB20 ?5@540G8. @82545==K9 @048CA :@8287=K. #AB0;>AB=>5 2K:@0H820=85 O2;O5BAO >A=>2=K< 284>< @07@CH5=8O 0:B82=>9 ?>25@E=>AB8 7C1L52 70:@KBKE 8 E>@>H> A<070==KE 7C1G0BKE ?5@540G. K:@0H820=85 70:;NG05BAO 2 B><, GB> 2A;54AB285 <=>3>:@0B=>3> 2>7=8:=>25=8O :>=B0:B=KE =0?@O65=89 =0 ?>25@E=>AB8 7C1L52 21;878 ?>;NA0 ?>O2;ONBAO <8:@>A:>?8G5A:85 B@5I8=K, :>B>@K5, @072820OAL 8 >1J548=OOAL, ?@82>4OB : >B45;5=8N <5;:8E G0AB8F <5B0;;0 8 >1@07>20=8N O<>:. ;O ?@54>B2@0I5=8O 2K:@0H820=8O =5>1E>48<>, GB>1K :>=B0:B=K5 =0?@O65=8O =0 0:B82=KE ?>25@E=>ABOE =5 ?@52KH0;8 4>?CAB8<KE. A;8 M2>;L25=BK 2 ?>;NA5 70F5?;5=8O 70<5=8BL 4C30<8 >:@C6=>AB8 A @048CA0<8 (1 8 (2, @02=K< @048CA0< :@8287=K M2>;L25=B 2 ?>;NA5, B> :>=B0:B=K5 =0?@O65=8O <>6=> ?@81;865==> >?@545;8BL ?> D>@<C;5 5@F0-5;O520 4;O 42CE :>=B0:B8@CNI8E F8;8=4@>2 (@8A. 8.10).  EMBED Equation.3 , 345 q  C45;L=0O =03@C7:0; (?@  ?@8254Q==K9 @048CA :@8287=K; E?@  ?@82545==K9 <>4C;L C?@C3>AB8; ( ( :>MDD8F85=B C0AA>=0. 7 MB>9 D>@<C;K, 2 G0AB=>AB8, A;54C5B, GB> :>=B0:B=K5 =0?@O65=8O >1@0B=> ?@>?>@F8>=0;L=K  EMBED Equation.3 . @82545==K9 @048CA :@8287=K (?@ @025=:  EMBED Equation.3 .  !;54>20B5;L=>, :>=B0:B=K5 =0?@O65=8O C<5=LH0NBAO A C25;8G5=85< (?@, :>B>@K9, 2 A2>N >G5@54L, @0AB5B A C25;8G5=85< (1 8 (2 . >6=> ?>:070BL, GB> @048CA :@8287=K @0AB5B A C25;8G5=85< C3;0 70F5?;5=8O. !;54>20B5;L=>, <0:A8<0;L=0O :>=B0:B=0O ?@>G=>ABL 4>AB8305BAO <0:A8<0;L=>9 AC<<>9 :>MDD8F85=B>2 A<5I5=8O E( = E1 + E2. 5>1E>48<> >B<5B8BL, GB> 2 7C1G0B>9 ?5@540G5 2=CB@5==53> 70F5?;5=8O (@8A. 8.11) ?@82545==K9 @048CA :@8287=K 7=0G8B5;L=> 1>;LH5, G5< 2 ?5@540G5 2=5H=53> 70F5?;5=8O:  EMBED Equation.3 .   A>>B25BAB288 A @0=55 A:070==K< :>=B0:B=K5 =0?@O65=8O 2 ?5@540G5 2=CB@5==53> 70F5?;5=8O 7=0G8B5;L=> <5=LH5, G5< 2 ?5@540G5 2=5H=53> 70F5?;5=8O. -2>;L25=B=>5 70F5?;5=85  =5 548=AB25==>5, E>BO 8 A0<>5 @0A?@>AB@0=5==>5. !CI5AB2C5B, =0?@8<5@, 70F5?;5=85 >28:>20 (@8A. 8.12), 2 :>B>@>< ?@82545==K9 @048CA :@8287=K, B0:65 7=0G8B5;L=> 1>;LH5, G5< 2 M2>;L25=B=KE :>;5A0E 2=5H=53> 70F5?;5=8O. A;54AB285 MB>3> 7=0G8B5;L=> <5=LH5 :>=B0:B=K5 =0?@O65=8O 8 2KH5 =03@C7>G=0O A?>A>1=>ABL. A>15==>ABL 70F5?;5=8O >28:>20  B>@F>2K9 :>MDD8F85=B ?5@5:@KB8O (( = 0, ?>MB><C (( = ((, B.5. 70F5?;5=85 @01>B>A?>A>1=> B>;L:> 2 :>A>7C1>< 8A?>;=5=88. 54>AB0B>: 70F5?;5=8O >28:>20  GC2AB28B5;L=>ABL : B>G=>AB8 873>B>2;5=8O, :>B>@>5 O2;O5BAO 4>AB0B>G=> A;>6=K<. 2. >MDD8F85=B, CG8BK20NI89 D>@<C 7C10. >4 459AB285< ?@8;>65==KE =03@C7>: <>65B ?@>87>9B8 ?>;><:0 7C1L52. ;O ?@54>B2@0I5=8O MB>3> 7C1LO 4>;6=K 1KBL @0AAG8B0=K =0 87381=CN ?@>G=>ABL. 0 @8A. 8.13 ?@54AB02;5=> ?>?5@5G=>5 A5G5=85 7C10 ?@O<>7C1>3> :>;5A0, AE5<0 459AB28O A8; 8 M?N@K =0?@O65=89. 45AL A8;0 R  @50:F8O A> AB>@>=K A>?@O65==>3> :>;5A0, @07;>65==0O =0 425 A>AB02;ONI85: >:@C6=CN A8;C , A>740NICN :@CBOI89 <><5=B =0 :>;5A5 8 2K7K20NICN ?>O2;5=85 2 A5G5=88 87381=KE =0?@O65=89, 8 @0480;L=CN A8;C F, A68<0NICN 7C1. 04048<AO 2>?@>A><, 2 :0:>9 >1;0AB8 =081>;55 25@>OB=> @07@CH5=85 7C10?  !@02=820O AC<<0@=K5 =0?@O65=8O (( 7>=K  8 , ?@8E>48< : 2K2>4C, GB> =081>;55 >?0A=>9 O2;O5BAO 7>=0 , B.:. 8<5==> 2 =59 =081>;LH85 ?>;>68B5;L=K5 =0?@O65=8O.  7>=5  AC<<0@=K5 =0?@O65=8O ?> <>4C;N 1>;LH5, => >=8  >B@8F0B5;L=K5, B.5. A>>B25BAB2CNB 45D>@<0F88 A60B8O. 7 A>?@>B82;5=8O <0B5@80;>2 8725AB=>, GB> G@572KG09=> B@C4=> @07@CH8BL >1@075F, ?>425@305<K9 A60B8N. 0:A8<0;L=>5 7=0G5=85 =>@<0;L=KE =0?@O65=89 2 >?0A=>< A5G5=88 7C10 >1@0B=> ?@>?>@F8>=0;L=> :>MDD8F85=BC YF, CG8BK20NI5<C D>@<C 7C10. ! C25;8G5=85< MB>3> :>MDD8F85=B0 =0?@O65=8O >B 873810 C<5=LH0NBAO. >MDD8F85=B YF 7028A8B 2 G8A;5 ?@>G53> >B :>MDD8F85=B0 A<5I5=8O 8 >B B>3>, O2;O5BAO ;8 7C1G0B>5 :>;5A> 2 ?5@540G5 254CI8< 8;8 254><K<. >2KH5=85 87381=>9 ?@>G=>AB8 4>AB8305BAO ?>41>@>< B0:8E :>MDD8F85=B>2 A<5I5=8O, :>B>@>5 >15A?5G8205B ?>;CG5=85 <0:A8<0;L=KE :>MDD8F85=B>2 YF ?@8 CA;>288 @02=>?@>G=>AB8 7C1L52 >1>8E 7C1G0BKE :>;5A ?5@540G8. B<5B8<, GB> MB> ?>2KH5=85 =01;N405BAO ?@8 C25;8G5=88 4> >?@545;5==KE 7=0G5=89 AC<<K :>MDD8F85=B>2 A<5I5=8O. #45;L=>5 A:>;L65=85. 7=>A 7C1L52 ?@>8AE>48B 2A;54AB285 >B=>A8B5;L=>3> A:>;L65=8O 8E 0:B82=KE ?>25@E=>AB59 8 =0;8G8O 01@0782=KE G0AB8F <564C =8<8. = ?@82>48B : 8A:065=8N ?>25@E=>AB59 8, A;54>20B5;L=>, : ?>O2;5=8N 4>?>;=8B5;L=KE 48=0<8G5A:8E =03@C7>: 8 HC<0. 7=>A ?>25@E=>AB59 7C1L52 ?>;CG05BAO, :0: ?>:07K205B ?@0:B8:0, =5>48=0:>2K< ?> 2KA>B5 7C10 8 2 ?5@2>< ?@81;865=88 <>65B E0@0:B5@87>20BLAO C45;L=K< A:>;L65=85<. CABL ?@8 ?>2>@>B5 7C1G0BKE :>;5A =0 C3;K d(i 8 d(j >1I0O B>G:0 :>=B0:B0 ?@>D8;59 ?5@5<5AB8;0AL ?> >4=><C ?@>D8;N =0 4;8=C 4C38 dsi, 0 ?> 4@C3><C  =0 4;8=C 4C38 dsj. 07=>ABL dsi - dsj ?@54AB02;O5B A>1>9 01A>;NB=>5 A:>;L65=85 ?@>D8;59. 5;8G8=C  EMBED Equation.3 , =07K20NB C45;L=K< A:>;L65=85<, >B=5A5==K< : ?@>D8;N 7C10 i-3> 7C1G0B>3> :>;5A0. 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B=>H5=85 d(j / d(i 2K@0605B ?5@540B>G=>5 >B=>H5=85 7C1G0B>9 ?5@540G8, A;54>20B5;L=>,  EMBED Equation.3  . =0;878@CO ?>;CG5==>5 2K@065=85, CAB0=02;8205<, GB> 2 ?>;NA5  C45;L=K5 A:>;L65=8O @02=K 0, B.5. ?@>D8;8 =5 A:>;L7OB, 0 ?5@5:0BK20NBAO 4@C3 ?> 4@C3C; =0 =0G0;L=KE 3>;>2:0E, :>B>@K5 @0A?>;030NBAO <564C >:@C6=>ABLN 25@H8= 8 =0G0;L=>9 >:@C6=>ABLN, C45;L=K5 A:>;L65=8O =525;8:8; =0 =0G0;L=KE =>6:0E (<564C =0G0;L=>9 >:@C6=>ABLN 8 >:@C6=>ABLN 2?048=) C45;L=K5 A:>;L65=8O 70<5B=> 1>;LH5, G5< =0 =0G0;L=KE 3>;>2:0E. !>>B25BAB2CNI8< 2K1>@>< :>MDD8F85=B>2 A<5I5=8O <>6=> C<5=LH8BL 8 A45;0BL >48=0:>2K<8 <0:A8<0;L=K5 C45;L=K5 A:>;L65=8O =0 =0G0;L=KE =>6:0E 7C1L52 ?5@2>3> 8 2B>@>3> 7C1G0BKE :>;5A. >MDD8F85=B ?5@5:@KB8O 7C1G0B>9 ?5@540G8. = E0@0:B5@87C5B A@54=55 G8A;> ?0@ 7C1L52, =0E>4OI8EAO >4=>2@5<5==> 2 70F5?;5=88. ;O 1>;55 ?;02=>9 8 A?>:>9=>9 @01>BK >= 4>;65= 1KBL 2>7<>6=> 1>;LH8<  >1KG=> =5 <5=55 1,2. 3> 2KG8A;ONB ?> D>@<C;0<, @0AA<>B@5==K< 2 ;5:F88 8. =0O :>MDD8F85=B A<5I5=8O, <>6=> ?>;=>ABLN @0AAG8B0BL 35><5B@8G5A:85 ?0@0<5B@K 8 @07<5@K ?5@540G8. 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A=>2=K5 B8?K @57L1: @O<>C3>;L=0O (@8A. 9.7). @>D8;L @57L1K  :204@0B. 7 2A5E @57L1 8<5NB A0<K9 2KA>:89 :.?.4., B0: :0: C3>; ?@>D8;O @57L1K ( = 0(. 1;040NB ?>=865==>9 ?@>G=>ABLN. @8 87=>A5 >1@07CNBAO >A52K5 707>@K, :>B>@K5 B@C4=> CAB@0=8BL.  =0AB>OI55 2@5<O =5 AB0=40@B878@>20=K. <5NB >3@0=8G5==>5 ?@8<5=5=85.  8A. 9.7 A=>2=K5 ?0@0<5B@K @57L1K: d, d1, d2 ( A>>B25BAB25==> =0@C6=K9, A@54=89 8 2=CB@5==89 480<5B@ @57L1K; @  H03 @57L1K, @0AAB>O=85 <564C 42C<O >4=>8<Q==K<8 AB>@>=0<8 42CE A>A54=8E 28B:>2 2 >A52>< =0?@02;5=88; S  E>4 @57L1K, @0AAB>O=85 <564C 42C<O >4=>8<Q==K<8 AB>@>=0<8 >4=>3> 8 B>3> 65 28B:0 2 >A52>< =0?@02;5=88 ( EMBED Equation.3 , 345 Z  G8A;> 70E>4>2 @57L1K); ( ( C3>; ?@>D8;O @57L1K; ( ( C3>; ?>4JQ<0 @57L1K. "@0?5F5840;L=0O A8<<5B@8G=0O (@8A. 9.8). @>D8;L @57L1K  @02=>1>G=0O B@0?5F8O A C3;>< ( = 30(. %0@0:B5@87CNBAO =51>;LH8<8 ?>B5@O<8 =0 B@5=85, B5E=>;>38G=K. @8<5=O5BAO 4;O ?5@540G8 @525@A82=>3> 42865=8O ?>4 =03@C7:>9.  8A. 9.8 "@0?5F5840;L=0O =5A8<<5B@8G=0O, 8;8 C?>@=0O (@8A. 9.9). @>D8;L @57L1K  =5@02=>1>G=0O B@0?5F8O A C3;>< ( = 27(. ;O 2>7<>6=>AB8 873>B>2;5=8O @57L1K D@575@>20=85< @01>G0O AB>@>=0 ?@>D8;O 8<55B C3>; =0:;>=0 3(. .?.4. 2KH5, G5< C B@0?5F5840;L=>9 A8<<5B@8G=>9. 0:@C3;5=85 2?048= ?>2KH05B ?@>G=>ABL. @8<5=ONBAO ?@58<CI5AB25==> ?@8 2KA>:8E >4=>AB>@>==8E =03@C7:0E.  8A. 9.9 >=B@>;L=K5 2>?@>AK 0: D>@<8@C5BAO ?@>D8;L 7C10 F8:;>840;L=>3> 70F5?;5=8O. 09B5 E0@0:B5@8AB8:C G0A>2>3> 8 F52>G=>3> F8:;>840;L=KE 70F5?;5=89. 07>28B5 >A=>2=K5 35><5B@8G5A:85 ?0@0<5B@K G5@2O:0 8 G5@2OG=>3> :>;5A0. @>0=0;878@C9B5 :8=5<0B8:C G5@2OG=KE ?5@540G. @>0=0;878@C9B5 8 A>?>AB02LB5 <564C A>1>9 >A=>2=K5 284K 28=B>2KE ?5@540G. 5:F8O 10 C;0G:>2K5 <5E0=87<K: >1I85 A2545=8O, :;0AA8D8:0F8O, :8=5<0B8G5A:89 0=0;87 8 A8=B57, >?@545;5=85 <8=8<0;L=>-4>?CAB8<KE @07<5@>2 :C;0G:0. 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"0: :0: ?@8 :8=5<0B8G5A:>< 0=0;875 :C;0G:>2K9 <5E0=87< 7040=, B> 8725AB=0 53> :8=5<0B8G5A:0O AE5<0 8 D>@<0 :>=AB@C:B82=>3> ?@>D8;O :C;0G:0. >AB@>5=85 4803@0<<K ?5@5<5I5=89 ?@>2>48BAO 2 A;54CNI59 ?>A;54>20B5;L=>AB8: AB@>8BAO :8=5<0B8G5A:0O AE5<0 :C;0G:>2>3> <5E0=87<0; 2 ?>;CG5==K9 F5=B@>2>9 ?@>D8;L 2?8AK20NBAO >:@C6=>AB8 @048CA>2 r0 8 r0 + Smax , >?@545;O5BAO 25;8G8=0 M:AF5=B@8A8B5B0 5; ?> 25;8G8=5 CG0AB:>2, =5 A>2?040NI8E A 4C30<8 >:@C6=>AB59 @048CA>2 r0 8 r0 + Smax, >?@545;ONBAO D07>2K5 C3;K (@01 , (C , (22 8 (? ; 4C30 >:@C6=>AB8 r, A>>B25BAB2CNI0O @01>G5<C D07>2><C C3;C, @0718205BAO =0 =5A:>;L:> @02=KE CG0AB:>2; G5@57 B>G:8 @07185=8O ?@>2>4OBAO :0A0B5;L=> : >:@C6=>AB8 @048CA0 M:AF5=B@8A8B5B0 ?@O<K5 ;8=88, (MB8 ;8=88 A>>B25BAB2CNB ?>;>65=8O< >A8 B>;:0B5;O 2 53> 42865=88 >B=>A8B5;L=> :C;0G:0); =0 MB8E ?@O<KE 87<5@ONBAO >B@57:8 @0A?>;>65==K5 <564C F5=B@>2K< ?@>D8;5< 8 >:@C6=>ABLN @048CA0 r0 ; MB8 >B@57:8 A>>B25BAB2CNB ?5@5<5I5=8O< F5=B@0 @>;8:0 B>;:0B5;O Si ; ?> ?>;CG5==K< ?5@5<5I5=8O< Si AB@>8BAO 4803@0<<0 DC=:F88 ?>;>65=8O F5=B@0 @>;8:0 B>;:0B5;O Si = f ((1 ); <5B>4>< 3@0D8G5A:>3> 48DD5@5=F8@>20=8O 4803@0<<K ?5@5<5I5=89 ?>;CG0NB 4803@0<<K (i = f((1)8 0i = f((1).  C Si , < ; (S , <</<   Si  yS0i Smax K 0 0 (1 (1i (( , <</@04   (1i 01 (1C (1?  (@01  8A.10.5 0 @8A. 10.5 ?>:070=0 AE5<0 ?>AB@>5=8O DC=:F88 ?>;>65=8O 4;O :C;0G:>2>3> <5E0=87<0 A F5=B@0;L=K< (5 = 0) ?>ABC?0B5;L=> 4286CI8<AO @>;8:>2K< B>;:0B5;5<. !8=B57 :C;0G:>2>3> <5E0=87<0. -B0?K A8=B570 @8 A8=B575 :C;0G:>2>3> <5E0=87<0, :0: 8 ?@8 A8=B575 ;N1>3> <5E0=87<0, @5H05BAO @O4 7040G 87 :>B>@KE 2 :C@A5 " @0AA<0B@820NBAO 425: 2K1>@ AB@C:BC@=>9 AE5<K 8 >?@545;5=85 >A=>2=KE @07<5@>2 725=L52 <5E0=87<0 (2:;NG0O ?@>D8;L :C;0G:0). 5@2K9 MB0? A8=B570  AB@C:BC@=K9. !B@C:BC@=0O AE5<0 >?@545;O5B G8A;> 725=L52 <5E0=87<0; G8A;>, 284 8 ?>4286=>ABL :8=5<0B8G5A:8E ?0@; G8A;> 871KB>G=KE A2O759 8 <5AB=KE ?>4286=>AB59. @8 AB@C:BC@=>< A8=B575 =5>1E>48<> >1>A=>20BL 22545=85 2 AE5<C <5E0=87<0 :064>9 871KB>G=>9 A2O78 8 <5AB=>9 ?>4286=>AB8. ?@545;ONI8<8 CA;>28O<8 ?@8 2K1>@5 AB@C:BC@=>9 AE5<K O2;ONBAO: 7040==K9 284 ?@5>1@07>20=8O 42865=8O, @0A?>;>65=85 >A59 2E>4=>3> 8 2KE>4=>3> 725=L52. E>4=>5 42865=85 2 <5E0=87<5 ?@5>1@07C5BAO 2 2KE>4=>5, =0?@8<5@, 2@0I0B5;L=>5 2> 2@0I0B5;L=>5, 2@0I0B5;L=>5 2 ?>ABC?0B5;L=>5 8 B.?. A;8 >A8 ?0@0;;5;L=K, B> 2K18@05BAO ?;>A:0O AE5<0 <5E0=87<0. @8 ?5@5A5:0NI8EAO 8;8 ?5@5:@5I820NI8EAO >AOE =5>1E>48<> 8A?>;L7>20BL ?@>AB@0=AB25==CN AE5<C.  :8=5<0B8G5A:8E <5E0=87<0E =03@C7:8 <0;K, ?>MB><C <>6=> 8A?>;L7>20BL B>;:0B5;8 A 70>AB@5==K< =0:>=5G=8:><.  A8;>2KE <5E0=87<0E 4;O ?>2KH5=8O 4>;3>25G=>AB8 8 C<5=LH5=8O 87=>A0 2 AE5<C <5E0=87<0 22>4OB @>;8: 8;8 C25;8G820NB ?@82545==K9 @048CA :@8287=K :>=B0:B8@CNI8E ?>25@E=>AB59 2KAH59 ?0@K. B>@>9 MB0? A8=B570  <5B@8G5A:89. 0 MB>< MB0?5 >?@545;ONBAO >A=>2=K5 @07<5@K 725=L52 <5E0=87<0, :>B>@K5 >15A?5G820NB 7040==K9 70:>= ?@5>1@07>20=8O 42865=8O 2 <5E0=87<5 8;8 7040==CN ?5@540B>G=CN DC=:F8N. 0: >B<5G0;>AL 2KH5, ?5@540B>G=0O DC=:F8O O2;O5BAO G8AB> 35><5B@8G5A:>9 E0@0:B5@8AB8:>9 <5E0=87<0, 0, A;54>20B5;L=>, 7040G0 <5B@8G5A:>3> A8=B570 G8AB> 35><5B@8G5A:0O 7040G0, =57028AOI0O >B 2@5<5=8 8;8 A:>@>AB59. A=>2=K5 :@8B5@88, :>B>@K<8 @C:>2>4AB2C5BAO ?@>5:B8@>2I8:, ?@8 @5H5=88 7040G <5B@8G5A:>3> A8=B570: <8=8<870F8O 3010@8B>2, 0, A;54>20B5;L=>, 8 <0AAK; <8=8<870F8O C3;0 402;5=8O 2 2KH59 ?0@5; ?>;CG5=85 B5E=>;>38G=>9 D>@<K ?@>D8;O :C;0G:0. ?@545;5=85 <8=8<0;L=>-4>?CAB8<KE @07<5@>2 :C;0G:>2KE <5E0=87<>2 07<5@K :C;0G:>2>3> <5E0=87<0 >?@545;ONBAO <8=8<0;L=K< @048CA>< :C;0G:0. 48= 8 B>B 65 70:>= 42865=8O B>;:0B5;O <>65B 1KBL 2>A?@>87254Q= :C;0G:>< A @07;8G=K<8 <8=8<0;L=K<8 @048CA0<8. 0: ?@028;>, 65;0B5;L=> ?>;CG8BL <5E0=87< =08<5=LH8E @07<5@>2, => C<5=LH5=85 @07<5@>2 :C;0G:0 ?@82>48B : C25;8G5=8N A>?@>B82;5=8O 2 :C;0G:>2>9 ?0@5 8 2 :@09=5< A;CG05 : 2>7<>6=>AB8 70:;8=820=8O B>;:0B5;O. 8=8<0;L=>-4>?CAB8<K5 @07<5@K :C;0G:0 >?@545;ONBAO 87 CA;>28O >15A?5G5=8O 4>?CA:05<KE C3;>2 402;5=8O.  :0G5AB25 ?@8<5@0 @0AA<>B@8< :C;0G:>2K9 <5E0=87< A> A<5I5=85< B>;:0B5;O (@8A. 10.6, 0).  8A. 10.6  <5AB5 :>=B0:B0 B>;:0B5;O 8 :C;0G:0 B>G:0 , 2>7=8:05B @50:F8O 12 :C;0G:0 =0 B>;:0B5;L, =0?@02;5==0O ?> =>@<0;8 n-n ?@>254Q==>9 : ?@>D8;N :C;0G:0. 07;>68< ?>;=CN @50:F8N =0 ?@>5:F88  EMBED Equation.3  8  EMBED Equation.3 , C3>; <564C 12 8 ;8=859 42865=8O B>;:0B5;O O2;O5BAO C3;>< 402;5=8O (. '5< 1>;LH5 C3>; 402;5=8O, B5< 1>;LH5 A>?@>B82;5=85 42865=8N, B5< <5=LH5 :.?.4. ;O =>@<0;L=>9 @01>BK :C;0G:>2>3> <5E0=87<0 =5>1E>48<>, GB>1K <0:A8<0;L=K9 C3>; 402;5=8O =5 ?@52>AE>48; C3;0 ?5@540G8 42865=8O (. #AB0=>28< 7028A8<>ABL C3;0 402;5=8O >B 35><5B@8G5A:8E 8 :8=5<0B8G5A:8E ?0@0<5B@>2 :C;0G:>2>3> <5E0=87<0, 4;O G53> @0AA<>B@8< B@5C3>;L=8: :  EMBED Equation.3 . ;O >:>=G0B5;L=>3> 2K@065=8O 2K@078< >B@57>: 1, 4;O G53> ?>AB@>8< ?;0= A:>@>AB59 :C;0G:>2>3> <5E0=87<0 (@8A. 10.6, 1).  EMBED Equation.3 , 345  EMBED Equation.3   25:B>@ A:>@>AB8 ?5@5=>A=>3> 42865=8O (>:@C6=0O A:>@>ABL :C;0G:0) =0?@02;5= ?5@?5=48:C;O@=> @048CA 25:B>@C  EMBED Equation.3  ?> =0?@02;5=8N 2@0I5=8O :C;0G:0 (1.  EMBED Equation.3 .  EMBED Equation.3   25:B>@ A:>@>AB8 >B=>A8B5;L=>3> 42865=8O (A:>@>ABL A:>;L65=8O B>;:0B5;O ?> :C;0G:C) =0?@02;5=0 ?> =0?@02;5=8N A:>;L65=8O B.5. ?0@0;;5;L=> :0A0B5;L=>9 t-t ?@>2545==>9 2 B>G:5 :>=B0:B0  : ?@>D8;N :C;0G:0;  EMBED Equation.3   25:B>@ 01A>;NB=>9 A:>@>AB8 B>;:0B5;O =0?@02;5= ?> =0?@02;5=8N 42865=8O B>;:0B5;O. 0AA<>B@8< B@5C3>;L=8:8 (1 8 (@(0102 , CAB0=02;8205<, GB> >=8 ?>4>1=K, B.:. A>>B25BAB25==> 8<5NB 425 AB>@>=K ?5@?5=48:C;O@=K5 4@C3 4@C3C 8 >4=C ?0@0;;5;L=CN. !>AB028< ?@>?>@F8N:  EMBED Equation.3 , >B:C40  EMBED Equation.3 . #AB0=>28;8, GB> >B@57>: 1 O2;O5BAO 0=0;>3>< A:>@>AB8 B>;:0B5;O :C;0G:>2>3> <5E0=87<0, A;54>20B5;L=>:  EMBED Equation.3 . =0;878@CO ?>;CG5==>5 2K@065=85 CAB0=02;8205<, GB> A C<5=LH5=85< <8=8<0;L=>3> @048CA0 :C;0G:0 C3>; 402;5=8O 2>7@0AB05B, 22545=85 A<5I5=8O B>;:0B5;O ?>72>;O5B C<5=LH8BL @07<5@K :C;0G:0 ?@8 >4=>< 8 B>< 65 C3;5 402;5=8O. K1>@ 70:>=0 42865=8O B>;:0B5;O A;8 2 7040=88 =0 ?@>5:B8@>20=85 =5 40= 70:>= 42865=8O, B> :>=AB@C:B>@ 4>;65= 2K1@0BL 53> 87 =01>@0 B8?>2KE 70:>=>2 42865=8O, =5>1E>48<>, GB>1K CA:>@5=8O B>;:0B5;O =5 ?@82>48;8 : 1>;LH8< 8=5@F8>==K< =03@C7:0<, 0 8<5NI0OAO =0 ?@54?@8OB88 B5E=>;>38O ?>72>;8;0 1K 873>B>28BL ?@>D8;L A 4>AB0B>G=>9 B>G=>ABLN. "8?>2K5 70:>=K 42865=8O 45;OBAO =0 70:>=K A 65AB:8<8, <O3:8<8 C40@0<8 8 157C40@=K5. ! B>G:8 7@5=8O 48=0<8G5A:8E =03@C7>:, 65;0B5;L=K 157C40@=K5 70:>=K. 4=0:> :C;0G:8 A B0:8<8 70:>=0<8 42865=8O B5E=>;>38G5A:8 1>;55 A;>6=K, B0: :0: B@51CNB 1>;55 B>G=>3> 8 A;>6=>3> >1>@C4>20=8O, ?>MB><C 8E 873>B>2;5=85 ACI5AB25==> 4>@>65. 0:>=K A 65AB:8<8 C40@0<8 8<5NB 25AL<0 >3@0=8G5==>5 ?@8<5=5=85 8 8A?>;L7CNBAO 2 =5>B25BAB25==KE <5E0=87<0E ?@8 =87:8E A:>@>ABOE 42865=8O 8 =52KA>:>9 4>;3>25G=>AB8. C;0G:8 A 157C40@=K<8 70:>=0<8 F5;5A>>1@07=> ?@8<5=OBL 2 <5E0=87<0E 2KA>:8<8 A:>@>ABO<8 42865=8O ?@8 65AB:8E B@51>20=8OE : B>G=>AB8 8 4>;3>25G=>AB8. 081>;LH55 @0A?@>AB@0=5=85 ?>;CG8;8 70:>=K 42865=8O A <O3:8<8 C40@0<8, A ?><>ILN :>B>@KE <>6=> >15A?5G8BL @0F8>=0;L=>5 A>G5B0=85 AB>8<>AB8 873>B>2;5=8O 8 M:A?;C0B0F8>==KE E0@0:B5@8AB8: <5E0=87<0. 0AA<>B@8< G5BK@5 70:>=0 42865=8O B>;:0B5;O (@8A. 10.8):  8A. 10.8 1. 02=><5@=>5 42865=85 B>;:0B5;O (@8A. 10.8, 0) MB> =081>;55 ?@>AB>9 70:>= 42865=8O. C;0G>: 8<55B =5A;>6=K9 ?@>D8;L. 4=0:> 4;O 1KAB@>E>4=KE :C;0G:>2KE <5E0=87<>2 >= =5 ?@83>45=, B0: :0: >= A2O70= A> A:0G:0<8 A:>@>AB8 2 =0G0;5 8 2 :>=F5 E>40 B>;:0B5;O, :>B>@K5 ?@82>4OB : 2>7=8:=>25=8N CA:>@5=89 =5 >3@0=8G5==KE ?> 25;8G8=5 (+(; - ():  EMBED Equation.3 .  =0G0;5 8 2 :>=F5 E>40 B>;:0B5;O, A;54>20B5;L=>, A8;K 8=5@F88 4>AB83;8 1K 15A:>=5G=> 1>;LH>9 25;8G8=K, 8<5NB <5AB> 65AB:85 C40@K. AE>4O 87 C:070==KE A>>1@065=89, @02=><5@=>5 42865=85 B>;:0B5;O <>6=> ?@8<5=OBL ;8HL 4;O :C;0G:>2KE <5E0=87<>2 ?@8 <0;KE A:>@>ABOE 8 <0;KE <>I=>ABOE. 2. 02=>CA:>@5==>5 42865=85 B>;:0B5;O (@8A. 10.8, 1) A:>@>ABL =0 ?5@2>9 G0AB8 E>40  EMBED Equation.3  @02=><5@=> 2>7@0AB05B, 0 70B5< =0 2B>@>< CG0AB:5 E>40 @02=><5@=> C1K205B 4> =C;O. 0 ?@>BO65=88 CG0AB:>2 E>40 CA:>@5=85 >48=0:>2>5. #G0AB:8 @073>=0 8 70<54;5=8O G0AB> 45;0NB =5>48=0:>2K<8, GB>1K C<5=LH8BL CA:>@5=85 8 A8;K 8=5@F88 =0 >4=>< 87 =8E. 02=>CA:>@5==>5 42865=85, E0@0:B5@87C5<>5 ?@O<>C3>;L=>9 4803@0<<>9 CA:>@5=89, =5 A>?@>2>6405BAO C40@0<8, A:0G:>2 A:>@>AB8 =5B, CA:>@5=8O 8, A;54>20B5;L=>, A8;K 8=5@F88 >AB0NBAO >3@0=8G5==K<8. 4=0:> 2 1KAB@>E>4=KE :C;0G:>2KE <5E0=87<0E MB>B 70:>= 42865=8O 2K7K205B ?>2KH5==CN 281@0F8N 8 87=>A. @8G8=>9 MB>3> O2;O5BAO 87<5=5=85 CA:>@5=8O B>;:0B5;O A:0G:><, 2K7K20NI55 <3=>25==>5 (70 >G5=L :>@>B:89 ?@><56CB>: 2@5<5=8) ?@8;>65=85 : B>;:0B5;N 1>;LH8E A8;. -B> O2;5=85 =07K20NB <O3:8< C40@><. 3. !3;065==>5 @02=>CA:>@5==>5 42865=85 B>;:0B5;O (@8A. 10.8, 2). >AB>8=AB2>  =08<5=LH0O 25;8G8=0 <0:A8<0;L=>3> CA:>@5=8O B>;:0B5;O. 803@0<<0 CA:>@5=89 8<55B D>@<C B@0?5F88, GB> ?>72>;O5B 871560BL A:0G:>2 CA:>@5=8O 8 <O3:8E C40@>2. "0:>9 70:>= 42865=8O <>65B ?@8<5=OBLAO 8 4;O 1KAB@>E>4=KE :C;0G:>2KE <5E0=87<>2. 4. !8=CA>840;L=K9 70:>= 42865=8O B>;:0B5;O (@8A. 10.8, 3) ?>72>;O5B ?>;CG8BL =081>;LHCN ?;02=>ABL 42865=8O, >BACBAB2CNB C40@K. -B>B 70:>= 42865=8O =081>;55 ?@54?>GB8B5;L=> ?@8<5=OBL 2 1KAB@>E>4=KE <5E0=87<0E. ;02=K< =54>AB0B:>< A8=CA>840;L=>3> (8 B@0?5F5840;L=>3>) O2;O5BAO 2KA>:0O B>G=>ABL ?@>D8;O :C;0G:0. >=B@>;L=K5 2>?@>AK ;0AA8D8:0F8O 8 =07=0G5=85 :C;0G:>2KE <5E0=87<>2. A=>2=K5 ?0@0<5B@K :C;0G:>2KE <5E0=87<>2. 0: ?@>872>48BAO :8=5<0B8G5A:89 0=0;87 :C;0G:>2KE <5E0=87<>2. ?@545;5=85 <8=8<0;L=>-4>?CAB8<KE @07<5@>2 :C;0G:>2KE <5E0=87<>2. 0: 2K18@05BAO =0 AB0488 ?@>5:B8@>20=8O 70:>= 42865=8O B>;:0B5;O. @>0=0;878@C9B5 >A=>2=K5 284K. 5:F8O 11 8=0<8:0 <5E0=87<>2 8 <0H8=. 040G8 A8;>2>3> 0=0;870 <5E0=87<>2. !8;K 8 8E :;0AA8D8:0F8O. #A;>285 :8=5B>AB0B8G5A:>9 >?@545;8<>AB8 :8=5<0B8G5A:8E F5?59. @0D>-0=0;8B8G5A:89 A?>A>1 A8;>2>3> 0=0;870 (<5B>4 ?;0=>2). 8=0<8:0 <5E0=87<>2 8 <0H8= 8=0<8:0 87CG05B 70:>=><5@=>AB8 42865=8O 725=L52 <5E0=87<0 ?>4 459AB285< ?@8;>65==KE A8;, ?@8 MB>< @0AA<0B@820NB 425 7040G8: @O<0O 7040G0 48=0<8:8 - >?@545;5=85 70:>=0 42865=8O A8AB5<K ?@8 7040==>< A8;>2>< 2>7459AB288. 1@0B=0O 7040G0 48=0<8:8 - >?@545;5=85 B@51C5<>3> A8;>2>3> 2>7459AB28O, >15A?5G820NI53> 7040==K9 70:>= 42865=8O A8AB5<K.  >1I59 ?>AB0=>2:5 48=0<8:0 - 87CG5=85 :0:8E-;81> ?@>F5AA>2 8;8 O2;5=89 2 DC=:F88 2@5<5=8. 8=0<8G5A:0O <>45;L - <>45;L A8AB5<K, ?@54=07=0G5==0O 4;O 8AA;54>20=8O 55 A2>9AB2 2 DC=:F88 2@5<5=8 (8;8 <>45;L A8AB5<K, ?@54=07=0G5==0O 4;O 8AA;54>20=8O 2 =59 48=0<8G5A:8E O2;5=89). 5B>4K A>AB02;5=8O C@02=5=89 (48=0<8G5A:>9 <>45;8 A8AB5<K): M=5@35B8G5A:89 (C@02=5=8O M=5@35B8G5A:>3> @02=>25A8O - 70:>= A>E@0=5=8O M=5@38O); :8=5B>AB0B8G5A:89 (C@02=5=8O A8;>2>3> @02=>25A8O A CG5B>< A8; 8=5@F88 ?> ?@8=F8?C  ;0<15@0). ;0AA8D8:0F8O A8;, 459AB2CNI8E 2 <5E0=87<0E A5 A8;K, 459AB2CNI85 2 <5E0=87<0E, CA;>2=> 45;OBAO =0: 2=5H=85, 459AB2CNI85 =0 8AA;54C5<CN A8AB5<C A> AB>@>=K 2=5H=8E A8AB5< 8 A>25@H0NI85 @01>BC =04 A8AB5<>9. -B8 A8;K 2 A2>N >G5@54L ?>4@0745;ONBAO =0: 4286CI85  MB> A8;K :>B>@K5 CA:>@ONB 42865=85 725=L52 8 A>25@H0NB ?>;>68B5;L=CN @01>BC (C25;8G8205B M=5@38N A8AB5<K); A>?@>B82;5=8O, @01>B0 :>B>@KE >B@8F0B5;L=0 (C<5=LH05B M=5@38N A8AB5<K). !8;K A>?@>B82;5=8O 45;OBAO =0: A8;K ?>;57=>3> (?@>872>4AB25==>3>) A>?@>B82;5=8O - 2>7=8:0NI85 ?@8 2K?>;=5=88 <5E0=8G5A:>9 A8AB5<>9 55 >A=>2=KE DC=:F89 (2K?>;=5=85 B@51C5<>9 @01>BK ?> 87<5=5=8N :>>@48=0B, D>@<K 8;8 A2>9AB2 8745;8O 8 B.?., A>25@H0NB >B@8F0B5;L=CN @01>BC); A8;K 2@54=>3> A>?@>B82;5=8O  MB> A8;K B@5=8O 2>7=8:0NI85 2 <5AB5 A2O78 2  8 >?@545;O5<K5 CA;>28O<8 D878:>-<5E0=8G5A:>3> 2708<>459AB28O <564C 725=LO<8 8 A8;K A>?@>B82;5=8O A@54K (@01>B0 2A5340 >B@8F0B5;L=0); 2708<>459AB28O A ?>B5=F80;L=K<8 ?>;O<8 (?>78F8>==K5) - 2>7=8:0NB ?@8 @07<5I5=88 >1J5:B0 2 ?>B5=F80;L=>< ?>;5, 25;8G8=0 7028A8B >B ?>B5=F80;0 B>G:8, 2 :>B>@>9 @07<5I05BAO B5;> (@01>B0 ?@8 ?5@5<5I5=88 87 B>G:8 A =87:8< ?>B5=F80;>< 2 B>G:C A 1>;55 2KA>:8< - ?>;>68B5;L=0; 70 F8:;, B.5. ?@8 2>72@0B5 2 8AE>4=>5 ?>;>65=85, @01>B0 @02=0 =C;N). >B5=F80;L=>5 ?>;5 - A8;K BO65AB8 8;8 25A0. !CI5AB2CNB M;5:B@><03=8B=K5, M;5:B@>AB0B8G5A:85 8 4@C385 ?>;O. 2=CB@5==85, 459AB2CNI85 <564C 725=LO<8 <5E0=8G5A:>9 A8AB5<K. 01>B0 MB8E A8; =5 87<5=O5B M=5@388 A8AB5<K.  <5E0=8G5A:8E A8AB5<0E MB8 A8;K =07K20NBAO @50:F8O<8 2 :8=5<0B8G5A:8E ?0@0E. @0AG5B=K5 (B5>@5B8G5A:85) - A8;K, :>B>@K5 =5 ACI5AB2CNB 2 @50;L=>AB8, 0 B>;L:> 8A?>;L7CNBAO 2 @07;8G=KE @0AG5B0E A F5;LN 8E C?@>I5=8O: A8;K 8=5@F88 - ?@54;>65=K  ;0<15@>< 4;O A8;>2>3> @0AG5B0 ?>4286=KE <5E0=8G5A:8E A8AB5<. @8 4>102;5=88 MB8E A8; : 2=5H=8< A8;0<, 459AB2CNI8< =0 A8AB5<C, CAB0=02;8205BAO :2078AB0B8G5A:>5 @02=>25A85 A8AB5<K 8 55 <>6=> @0AAG8BK20BL, 8A?>;L7CO C@02=5=8O AB0B8:8 (<5B>4 :8=5B>AB0B8:8). ?@82545==K5 (>1>1I5==K5) A8;K - A8;K A>25@H0NI85 @01>BC ?> >1>1I5==>9 :>>@48=0B5 @02=CN @01>B5 A>>B25BAB2CNI59 @50;L=>9 A8;K =0 M:2820;5=B=>< ?5@5<5I5=88 B>G:8 55 ?@8;>65=8O. 5>1E>48<> >B<5B8BL, GB> ?>4 A8;0<8 ?>=8<0NBAO @02=>459AB2CNI85 A>>B25BAB2CNI8E @0A?@545;5==KE 2 <5AB5 :>=B0:B0 :8=5<0B8G5A:8E ?0@ =03@C7>:. A5 2KH5A:070==>5 >B=>A8B5;L=> A8; @0A?@>AB@0=O5BAO 8 =0 <><5=BK A8;. 286CI85 8 A8;K ?>;57=KE A>?@>B82;5=89 G0I5 2A53> 7040NB 2 2845 <5E0=8G5A:8E E0@0:B5@8AB8: <0H8=K. 5E0=8G5A:85 E0@0:B5@8AB8:8 <0H8= 5E0=8G5A:>9 E0@0:B5@8AB8:>9 <0H8=K =07K205BAO 7028A8<>ABL A8;K 8;8 <><5=B0 =0 2KE>4=>< 20;C 8;8 @01>G5< >@30=5 <0H8=K >B A:>@>AB8 8;8 ?5@5<5I5=8O B>G:8 8;8 725=0 55 ?@8;>65=8O. 0AA<>B@8< ?@8<5@K <5E0=8G5A:8E E0@0:B5@8AB8: @07;8G=KE <0H8=. '5BK@QEB0:B=K9 42830B5;L 2=CB@5==53> A3>@0=8O (!) 2 :0G5AB25 4286CI59 A8;K 2KABC?05B A8;0 402;5=8O A3>@05<KE 307>2 2 F8;8=4@5. =48:0B>@=0O 4803@0<<0 ( 3@0D8G5A:>5 87>1@065=85 7028A8<>AB8 402;5=8O 2 F8;8=4@5 ?>@H=52>9 <0H8=K >B E>40 ?>@H=O (@8A. 11.1).   @, 0 ab - @0AH8@5=85 (@01>G89 E>4); bc - 2KE;>?; a pmax cd - 2A0AK20=85; b da - A60B85. c 0 d S, < H  8A. 11.1 -;5:B@>42830B5;8: 0A8=E@>==K9 M;5:B@>42830B5;L ?5@5<5==>3> B>:0 (<5E0=8G5A:0O E0@0:B5@8AB8:0 ?@82545=0 =0 @8A. 11.2): 0 4803@0<<5: 4? - ?CA:>2>9 <><5=B; 4= ( =><8=0;L=K9 :@CBOI89 <><5=B; 4: 8;8 4max ( :@8B8G5A:89 8;8 <0:A8<0;L=K9 <><5=B; (4= ( =><8=0;L=0O :@C3>20O G0AB>B0 2@0I5=8O 20;0 42830B5;O; (4EE 8;8 (4A ( G0AB>B0 2@0I5=8O 20;0 42830B5;O E>;>AB>3> E>40 8;8 A8=E@>==0O. #@02=5=85 AB0B8G5A:>9 E0@0:B5@8AB8:8 0A8=E@>==>3> M;5:B@>42830B5;O =0 ;8=50@87>20==>< CG0AB:5 CAB>9G82>9 G0AB8: 4 = b1 + k1((4 , 345 4 ( 4286CI89 <><5=B =0 20;C 42830B5;O, (4 ( :@C3>20O G0AB>B0 20;0 42830B5;O: b1 8 k1  :>MDD8F85=BK >?@545;O5<K5 A>>B25BAB25==> ?> D>@<C;0<: b1 = 4= ( (4 /((4A - (4= ) , k1 = - 4= / ((4A - (4= ).  4 ,( <  b ab - =5CAB>9G82K9  a CG0AB>: E0@0:B5@8AB8:8; A bd -CAB>9G82K9 4= 4max CG0AB>: E0@0:B5@8AB8:8. 4?  d 0 (4: (4= (4A (4 , @04/A 8A. 11.2 !B0B8G5A:0O E0@0:B5@8AB8:0 0A8=E@>==>3> 42830B5;O, 2K@060NI0O 7028A8<>ABL =03@C7:8 >B A:>;L65=8O, >?@545;O5BAO D>@<C;>9 ;>AA0: 4 = 2( 4: ( (S/S: + S:/S ), 345 S = 1 - (4 /(4A; S: = 1 - (4: /(4A , (4 ( (4A . 42830B5;L ?>AB>O==>3> B>:0 A =57028A8<K< 2>71C645=85< (<5E0=8G5A:0O E0@0:B5@8AB8:0 ?@82545=0 =0 @8A. 11.3): 4 ,( <  a ac - 2=5H=OO E0@0:B5@8AB8:0 @53C;8@>2>G=K5 E0@0:B5@8AB8:8  4? 4= b c 0 (4= (4EE (4 , @04/A 8A. 11.3 #@02=5=85 AB0B8G5A:>9 E0@0:B5@8AB8:8 4;O 42830B5;O ?>AB>O==>3> B>:0 A =57028A8<K< 2>71C645=85<: 4 = M4= + k( ((4= - (4 ), 345 k = 4= /((4EE - (4= ).  M;5:B@8G5A:8E ?0@0<5B@0E E0@0:B5@8AB8:0 70?8AK205BAO 2 A;54CNI5< 2845 4 = kM ( (UO - k( ((4)/ RO , 345 kM ( :>MDD8F85=B <><5=B0: kM = M4=/IO= , k( - :>MDD8F85=B ?@>B82>M;5:B@>4286CI59 A8;K: k( = (UO= - RO( IO= ) / (4= UO  =><8=0;L=>5 =0?@O65=85 2 F5?8 O:>@O; RO ( A>?@>B82;5=85 F5?8 O:>@O. JO ( =><8=0;L=0O A8;0 B>:0 2 F5?8 O:>@O. 01>G85 (8A?>;=8B5;L=K5) <0H8=K ?>@H=52>9 =0A>A (8=48:0B>@=0O 4803@0<<0 ?@82545=0 =0 @8A. 11.4):  @, 0 ab - =03=5B0=85; a b cd - 2A0AK20=85. pmax ;8=8O 0B<>AD5@=>3> 402;5=8O pmin 0  d c S, < H  8A. 11.4 ?>@H=52>9 :><?@5AA>@ (8=48:0B>@=0O 4803@0<<0 ?@82545=0 =0 @8A. 11.5): @, 0 a ba  =03=5B0=85; b cb - A60B85; dc - 2A0AK20=85; ad  @0AH8@5=85 >AB0B>G=>3> 2>74CE0. pmax ;8=8O 0B<>AD5@=>3> 402;5=8O  pmin  0 d c S, < H  8A. 11.5 8=88 bc 8 ad - ;8=88 A60B8O 8 @0AH8@5=8O 3070 (2>74CE0) >?@545;ONBAO ?0@0<5B@0<8 3070 (>1J5<><, 402;5=85< 8 B5<?5@0BC@>9) 8 2 >1I5< 2845 >?8AK20NBAO C@02=5=85< ?>;8B@>?K p( Vn = const , 345 n - ?>:070B5;L ?>;8B@>?K (1( n ( 0). 5E0=8G5A:85 E0@0:B5@8AB8:8 >?@545;ONB 2=5H=85 A8;K 8 <><5=BK, 459AB2CNI85 =0 2E>4=K5 8 2KE>4=K5 725=LO, @0AA<0B@8205<>9 <5E0=8G5A:>9 A8AB5<K A> AB>@>=K 2708<>459AB2CNI8E A =59 2=5H=8E A8AB5< 8 >:@C60NI59 A@54K. %0@0:B5@8AB8:8 >?@545;ONBAO M:A?5@8<5=B0;L=>, ?> @57C;LB0B0< M:A?5@8<5=B>2 ?>;CG0NB @53@5AA8>==K5 M<?8@8G5A:85 <>45;8, :>B>@K5 2 40;L=59H5< 8A?>;L7CNBAO ?@8 ?@>2545=88 48=0<8G5A:8E @0AG5B>2 <0H8= 8 <5E0=87<>2. !8;K 8=5@F88 725=L52 !8;K 8=5@F88 725=L52 @0AA<0B@820NBAO :0: @50:F88 725=0 =0 87<5=5=85 53> A:>@>AB8 ?> 25;8G8=5 8 =0?@02;5=8N. !CI5AB2>20=85 A8; 8=5@F88 >1CA;>2;5=> 42C<O >1AB>OB5;LAB20<8: D0:B>< =0;8G8O C 725=L52 <0AAK 8 D0:B>< 42865=8O 725=L52, A>?@>2>640NI53>AO 2 >1I5< A;CG05 CA:>@5=8O<8 >B45;L=KE B>G5: 8 2A53> 725=0 2 F5;><, B0: :0: 8725AB=> 87 B5>@5B8G5A:>9 <5E0=8:8, GB> <5@>9 A8; 8=5@F88 O2;O5BAO ?@>872545=85 <0AAK =0 CA:>@5=85. 7 :C@A0 B5>@5B8G5A:>9 <5E0=8:8 8725AB=>, GB> A8AB5<C A8; 8=5@F88 2 >1I5< A;CG05 <>6=> ?@825AB8 : A8;5  3;02=><C 25:B>@C A8; 8=5@F88  EMBED Equation.3  ?@8;>65==>3> 2 F5=B@5 <0AA s 725=0 (@8A. 11.6) 8 : ?0@5 A8;, <><5=B :>B>@>9 =07K205BAO 3;02=K< <><5=B>< A8; 8=5@F88  EMBED Equation.3 .  8A. 11.6 ;02=K9 25:B>@ A8; 8=5@F88 >?@545;ONB ?> D>@<C;5:  EMBED Equation.3 . ;02=K9 <><5=B A8; 8=5@F88 >?@545;ONB ?> D>@<C;5:  EMBED Equation.3 , 345 m  <0AA0 725=0, :3; 0s  CA:>@5=85 F5=B0 <0AA, </A2; Js  <><5=B 8=5@F88 725=0 >B=>A8B5;L=> >A8 ?@>E>4OI59 G5@57 F5=B@ <0AA ?5@?5=48:C;O@=>9 ?;>A:>AB8 42865=8O, :3/<2; ( - C3;>2>5 CA:>@5=85 725=0, A-2. =0: ( C:07K205B =0 B>, GB> 25:B>@K EMBED Equation.3 8 EMBED Equation.3  A>>B25BAB25==> =0?@02;5=K ?@>B82>?>;>6=> 0s 8 (. !8;K 8=5@F88 725=L52 A>25@H0NI8E 2@0I0B5;L=>5 42865=85 @8 @02=><5@=>< 2@0I0B5;L=>< 42865=88 725=L52 8<5NI8E F8;8=4@8G5A:CN D>@<C (@8A. 11.7, 0) 8<55<:  EMBED Equation.3  8  EMBED Equation.3 , B0: :0: A>>B25BAB25==> 0s = 0 8 ( = 0. @8 =5@02=><5@=>< 2@0I5=88 725=L52 8<5NI8E F8;8=4@8G5A:CN D>@<C 8<55<:  EMBED Equation.3  B0: :0: 0s = 0 8  EMBED Equation.3 , B.:. ( ( 0.  8A. 11.7 @8 @02=><5@=>< 2@0I5=88 :@82>H8?0 (@8A. 11.7, 1) 8<55<:  EMBED Equation.3  B0: :0: 0s ( 0 8  EMBED Equation.3 , B.:. ( = 0. @8 =5@02=><5@=>< 2@0I5=88 :@82>H8?0 (@8A. 11.7, 2) 8<55<:  EMBED Equation.3  B0: :0: 0s ( 0 8  EMBED Equation.3 , B.:. ( ( 0. ;O C4>1AB20 @0AG5B>2 40==CN A8AB5<C ?@8=OB> 70<5=OBL >4=>9 @57C;LB8@CNI59 A8;>9 8=5@F88  EMBED Equation.3  ?@8;>65==>9 2 F5=B@5 :0G0=8O , @0A?>;>65=85 :>B>@>9 >?@545;ONB 87 2K@065=8O:  EMBED Equation.3 . !8;K 8=5@F88 725=L52 A>25@H0NI8E ?>ABC?0B5;L=>5 42865=85 A;8 725=> A>25@H05B B>;L:> ?>ABC?0B5;L=>5 42865=85 (@8A. 11.8) B>:  EMBED Equation.3  8  EMBED Equation.3 , B0: :0: ( = 0.  8A. 11.8 !8;K 8=5@F88 725=L52 A>25@H0NI8E ?;>A:>-?0@0;;5;L=>5 42865=85 @8 A;>6=>< ?;>A:>-?0@0;;5;L=>< 42865=88 725=0, =0?@8<5@ H0BC=0 2 :@82>H8?=>-?>;7C==>< <5E0=87<5 (@8A. 11.9), 2>7=8:0NB 3;02=K9 25:B>@ A8; 8=5@F88  EMBED Equation.3  8 3;02=K9 <><5=B A8; 8=5@F88  EMBED Equation.3 . ;O C4>1AB20 @0AG5B>2 40==CN A8AB5<C ?@8=OB> 70<5=OBL >4=>9 @57C;LB8@CNI59 A8;>9 8=5@F88  EMBED Equation.3  ?@8;>65==>9 2 F5=B@5 :0G0=8O , 8<5NI59 ?;5G> >B=>A8B5;L=> F5=B@0 <0AA @02=>5  EMBED Equation.3  8 A>740NI59 <><5=B 2 =0?@02;5=88 >1@0B=>< C3;>2><C CA:>@5=8N H0BC=0 (2.  8A. 11.9 #A;>285 :8=5B>AB0B8G5A:>9 >?@545;8<>AB8 :8=5<0B8G5A:8E F5?59 !8;0, :0: 25:B>@=0O 25;8G8=0 E0@0:B5@87C5BAO >B=>A8B5;L=> 725=L52 <5E0=87<0 B@5<O ?0@0<5B@0<8: :>>@48=0B0<8 B>G:8 ?@8;>65=8O, 25;8G8=>9 8 =0?@02;5=85<. 0AA<>B@8< A MB8E ?>78F89 @50:F88 2 :8=5<0B8G5A:8E ?0@0E ?;>A:8E <5E0=87<>2. 1. >ABC?0B5;L=0O :8=5<0B8G5A:0O ?0@0.  ?>ABC?0B5;L=>9 :8=5<0B8G5A:>9 ?0@5 A2O78, =0;>65==K5 =0 >B=>A8B5;L=>5 42865=85 725=L52 70?@5I0NB >B=>A8B5;L=>5 ?>ABC?0B5;L=>5 42865=85 ?> >A8 y 8 >B=>A8B5;L=>5 2@0I5=85. 0<5=OO MB8 A2O78 @50:F8O<8, ?>;CG8< @50:F8N R12 (@8A. 11.10).  y (n)  R12 x   Pj Pj A 1 2 R12 n 8A. 11.10 @8 A8;>2>< @0AG5B5 ?>ABC?0B5;L=>9 :8=5<0B8G5A:>9 ?0@K >?@545;ONB 25;8G8=C @50:F88 R12 8 B>G:C 5Q ?@8;>65=8O, ?@8 MB>< 8725AB=> =0?@02;5=85 - =>@<0;L : :>=B0:B8@CNI8< ?>25@E=>ABO< 725=L52. '8A;> A2O759 (>3@0=8G5=89 42865=89) 2 :8=5<0B8G5A:>9 ?0@5 2, G8A;> @07@5H05<KE 42865=89 ( 1, G8A;> =58725AB=KE ?@8 A8;>2>< @0AG5B5 ( 2. 2. @0I0B5;L=0O :8=5<0B8G5A:0O ?0@0. > 2@0I0B5;L=>9 :8=5<0B8G5A:>9 ?0@5 A2O78, =0;>65==K5 =0 >B=>A8B5;L=>5 42865=85 725=L52 70?@5I0NB >B=>A8B5;L=K5 ?>ABC?0B5;L=K5 42865=8O ?> >AO< y 8 x. 0<5=OO MB8 A2O78 @50:F8O<8, ?>;CG8< @50:F8N R12 (@8A. 11.11).  y x  i (  j   1 B 2 R12 8A. 11.11 @8 A8;>2>< @0AG5B5 2@0I0B5;L=>9 :8=5<0B8G5A:>9 ?0@K >?@545;O5BAO =0?@02;5=85 8 25;8G8=0 @50:F88 R12, ?@8 8725AB=>9 B>G:5 ?@8;>65=8O A8;K - 35><5B@8G5A:><C F5=B@C :8=5<0B8G5A:>9 ?0@K B. '8A;> A2O759 (>3@0=8G5=89 42865=89) 2 :8=5<0B8G5A:>9 ?0@5 2, G8A;> @07@5H05<KE 42865=89 ( 1, G8A;> =58725AB=KE ?@8 A8;>2>< @0AG5B5 2. 3. KAH0O :8=5<0B8G5A:0O ?0@0.  2KAH59 ?0@5 A2O78, =0;>65==K5 =0 >B=>A8B5;L=>5 42865=85 725=L52, 70?@5I0NB 42865=85 2 =0?@02;5=88 =>@<0;8 (nn) : :>=B0:B8@CNI8< ?>25@E=>ABO< (>AL y). 0<5=OO MBC A2O7L @50:F859, ?>;CG8< @50:F8N R12 (@8A. 11.12).  y (n)   i ( x t  j  1 ! t R12 2 n 8A. 11.12 @8 A8;>2>< @0AG5B5 2 2KAH59 :8=5<0B8G5A:>9 ?0@5 >?@545;ONB 25;8G8=C @50:F88 R12 ?> 8725AB=K< B>G:5 ?@8;>65=8O A8;K (B>G:0 :>=B0:B0 @01>G8E ?@>D8;59 :8=5<0B8G5A:>9 ?0@K !) 8 =0?@02;5=8N 25:B>@0 A8;K ( =>@<0;L : ?@>D8;O<. '8A;> A2O759 (>3@0=8G5=89 42865=89) 2 :8=5<0B8G5A:>9 ?0@5 1, G8A;> @07@5H05<KE 42865=89 ( 2, G8A;> =58725AB=KE ?@8 A8;>2>< @0AG5B5 1. 0AA<>B@8< ?;>A:89 <5E0=87< A>AB>OI89 87 n 725=L52, A>548=Q==KE 2 :8=5<0B8G5A:85 ?0@K: 5 :;0AA0 2 :>;8G5AB25 @5 8 4 :;0AA0 2 :>;8G5AB25 @4. '8A;> C@02=5=89 AB0B8:8 :>B>@K5 <K <>65< A>AB028BL  3, >1I55 G8A;> C@02=5=89 - 3(n. 0640O :8=5<0B8G5A:0O ?0@0 5 :;0AA0 A>45@68B 2 =58725AB=K5 > @50:F88, 4 :;0AA0 1 =58725AB=>5, B>340 >1I55 G8A;> =58725AB=KE  EMBED Equation.3 . ">340 CA;>285 :8=5B>AB0B8G5A:>9 >?@545;8<>AB8 ?;>A:>3> <5E0=87<0 <>6=> 70?8A0BL :0::  EMBED Equation.3 . ".5. 4;O AB0B8G5A:8 >?@545;8<KE <5E0=87<>2 AB5?5=L ?>4286=>AB8 @02=0 =C;N. ;O @KG06=KE <5E0=87<>2  EMBED Equation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f m ( i = 0; ( Mi = 0; i=1 i=1 345 i ( 2=5H=85 A8;K, ?@8;>65==K5 : <5E0=87<C 8;8 53> 725=LO<; Mi( 2=5H=85 <><5=BK A8;, ?@8;>65==K5 : <5E0=87<C 8;8 53> 725=LO<. :8=5B>AB0B8G5A:89 ( 4;O 4286CI8EAO <5E0=87<>2 ?@8 8725AB=KE <0AA0E 8 <><5=B0E 8=5@F88 725=L52, :>340 ?@5=51@565=85 8=5@F8>==K<8 A8;0<8 ?@82>48B : ACI5AB25==K< ?>3@5H=>ABO<; #@02=5=8O :8=5B>AB0B8G5A:>3> @02=>25A8O: f n m k ( i + ( 8i = 0; ( Mi + ( M8i = 0; i=1 i=1 i=1 i=1 345 8i ( 8=5@F8>==K5 A8;K, ?@8;>65==K5 : 725=LO<; M8i( <><5=BK A8; 8=5@F88, ?@8;>65==K5 : 725=LO<. :8=5B>AB0B8G5A:89 A CG5B>< B@5=8O - <>65B 1KBL ?@>2545= :>340 >?@545;5=K E0@0:B5@8AB8:8 B@5=8O 2 :8=5<0B8G5A:8E ?0@0E 8 @07<5@K M;5<5=B>2 ?0@. ?@545;5=85 G8A;0 =58725AB=KE ?@8 A8;>2>< @0AG5B5. ;O >?@545;5=8O G8A;0 =58725AB=KE, 0, A;54>20B5;L=>, 8 G8A;0 =57028A8<KE C@02=5=89, ?@8 A8;>2KE @0AG5B0E =5>1E>48<> ?@>25AB8 AB@C:BC@=K9 0=0;87 <5E0=87<0 8 >?@545;8BL G8A;> 8 :;0AAK :8=5<0B8G5A:8E ?0@, G8A;> >A=>2=KE ?>4286=>AB59 <5E0=87<0, G8A;> 871KB>G=KE A2O759. 'B>1K A8;>2>9 @0AG5B <>6=> 1K;> ?@>25AB8, 8A?>;L7CO B>;L:> C@02=5=8O :8=5B>AB0B8:8, =5>1E>48<> CAB@0=8BL 2 =5< 871KB>G=K5 A2O78. "0: :0: :0640O A2O7L 2 :8=5<0B8G5A:>9 ?0@5 <5E0=87<0 A>>B25BAB2C5B >4=>9 :><?>=5=B5 @50:F88, B> G8A;> =58725AB=KE :><?>=5=B @50:F89 @02=> AC<<0@=><C G8A;C A2O759 =0:;04K205<KE :8=5<0B8G5A:8<8 ?0@0<8 <5E0=87<0. >=B@>;L=K5 2>?@>AK ;0AA8D8:0F8O A8; 459AB2CNI8E =0 <5E0=87<. @82548B5 ?@8<5@K <5E0=8G5A:8E E0@0:B5@8AB8: <0H8=. !8;K 8=5@F88 725=L52 A>25@H0NI8E 2@0I0B5;L=>5, ?>ABC?0B5;L=>5 8 ?;>A:>-?0@0;;5;L=>5 42865=85. #A;>285 :8=5B>AB0B8G5A:>9 >?@545;8<>AB8 :8=5<0B8G5A:8E F5?59. 5:F8O 12 !8;>2>9 0=0;87 @KG06=KE <5E0=87<>2. 568<K 42865=8O <5E0=87<>2. #@02=5=85 42865=8O <5E0=87<>2. 8=0<8G5A:0O <>45;L <5E0=87<0. @82545=85 A8; 8 <0AA 2 <5E0=87<0E. 8=0<8G5A:0O <>45;L. !8;>2>9 0=0;87 @KG06=>3> <5E0=87<0 <5B>4>< ?;0=>2 A8; (157 CG5B0 B@5=8O 2 :8=5<0B8G5A:8E ?0@0E) 8=5B>AB0B8G5A:89 <5B>4 @0AG5B0 ?>72>;O5B =0E>48BL @50:F88 2 :8=5<0B8G5A:8E ?0@0E, 0 B0:65 >?@545;8BL C@02=>25H820NICN A8;C (8;8 C@02=>25H820NI89 <><5=B ?0@K A8;). >4 C@02=>25H820NI8<8 A8;0<8 ?>=8<0NB A8;K, ?@8;>65==K5 : 254CI8< 725=LO<, :>B>@K5 C@02=>25H820NB A8AB5<C 2A5E 2=5H=8E A8; 8 ?0@ A8; 8 2A5E A8; 8=5@F88 8 ?0@ A8; 8=5@F88. A;8 <5E0=87< 8<55B =5A:>;L:> AB5?5=59 A2>1>4K, B> 4;O 53> @02=>25A8O =5>1E>48<> AB>;L:> C@02=>25H820NI8E A8; 8;8 ?0@ A8;, A:>;L:> 8<55BAO AB5?5=59 A2>1>4K. @0D8G5A:>5 >?@545;5=85 @50:F89 2 :8=5<0B8G5A:8E ?0@0E ?;>A:8E <5E0=87<>2 A ?><>ILN ?;0=>2 A8; ?@8<5=O5BAO =5 B>;L:> 2A;54AB285 =03;O4=>AB8, => 8 ?>B><C, GB> 2=5H=85 A8;K, 459AB2CNI85 =0 725=LO <5E0=87<0, >1KG=> 8725AB=K ;8HL ?@81;86Q==>, 8 B>G=>ABL ?@>AB59H8E 3@0D8G5A:8E ?>AB@>5=89 >:07K205BAO 2?>;=5 4>AB0B>G=>9. !8;>2>9 0=0;87 <5E0=87<>2 <5B>4>< ?>AB@>5=8O ?;0=>2 A8; @0AA<>B@8< =0 ?@8<5@5 H0@=8@=>3> G5BK@QE725==>3> <5E0=87<0 (@8A. 12.1). !G8B05<, GB> ?> 7040==><C 70:>=C 42865=8O =0G0;L=>3> 725=0 1 2K?>;=5= :8=5<0B8G5A:89 0=0;87 8 >?@545;5=K A8;K 8 ?0@K A8; 8=5@F88: :@82>H8?0 1 81; H0BC=0 2 82, 82; :>@><KA;0 3 83, 83. 5H5=85 7040G8 =0G8=0NB A ?>AB@>5=8O :8=5<0B8G5A:>9 AE5<K <5E0=87<0 (@8A. 12.1, 0) A ?@8;>65==K<8 A8;0<8. !8;>2>9 0=0;87 ?@>2>4OB 2 ?>@O4:5 >BA>548=5=8O 3@C?? AC@0. !8;>2>9 0=0;87 3@C??K AAC@0 (@8A. 12.1, 1) =0;87 =0G8=05< A @0AA<>B@5=8O 3@C??K AAC@0 (2:;NG0NI59 H0BC= 2 8 :>@><KA;> 3), =0 :>B>@CN 459AB2CNB A8;K: 25A0 H0BC=0 G2; 25A0 :>@><KA;0 G3; A8;K 8 <><5=BK A8; 8=5@F88 H0BC=0 8 :>@><KA;0, A>>B25BAB25==> 82, 82, 8 83, 83; @50:F88 2 H0@=8@0E (>?>@0E) R03, R12(A>>B25BAB25==>: AB>9:8 0 =0 :>@><KA;> 3; :@82>H8?0 1 =0 H0BC= 2). !B@>8< 2 <0AHB015 (l (</<<) 3@C??C AAC@0.  A>>B25BAB2CNI85 B>G:8 ?@8:;04K205< 2=5H=85 A8;K ?0@0;;5;L=> 8E 459AB28N, ?@8 MB>< AC<<0@=>5 459AB285 =0 725=> A8;K 8 <><5=B0 A8;K 8=5@F88 70<5=O5< >4=>9 @57C;LB8@CNI59 A8;>9 8=5@F88, A>740NI59 <><5=B, 459AB2CNI89 2 >1@0B=>< =0?@02;5=88 C3;>2><C CA:>@5=8N, 8 ?@8;>65==>9 2 F5=B@5 :0G0=8O: B>G:5  4;O :>@><KA;0 3, ;560I59 =0 @0AAB>O=88 l3 >B >A8 2@0I5=8O 3  EMBED Equation.3 , 345 l3S3 ( @0AAB>O=85 >B >A8 2@0I5=8O :>@><KA;0 3 4> 53> F5=B@0 BO65AB8, <. 4;O H0BC=0 2, >BAB>OI59 >B ;8=88 459AB28O A8;K 8=5@F88 82 =0 @0AAB>O=88  EMBED Equation.3 .  H0@=8@0E  8 3 ?@8:;04K205< @50:F88 R12 8 R 03, @0A:;04K20O 8E =0 =>@<0;L=K5 8 :0A0B5;L=K5 A>AB02;ONI85. >@<0;L=K5 A>AB02;ONI85  EMBED Equation.3  8  EMBED Equation.3  =0?@02;O5< ?0@0;;5;L=> A>>B25BAB25==> 725=LO< 3 8 2, :0A0B5;L=K5  EMBED Equation.3  8  EMBED Equation.3  ( ?5@?5=48:C;O@=> 725=LO<.  8A. 12.1 !>AB02;O5< C@02=5=85 <><5=B>2 A8; >B=>A8B5;L=> B>G:8  4;O 2B>@>3> 725=0 (=0 @8A. 12.1, 1 >B<5G05< ?;5G8 A8;): ( 2( i) = 0;  EMBED Equation.3  >;CG5==>5 >B@8F0B5;L=>5 7=0G5=85 A8;K 3>2>@8B > B><, GB> =0?@02;5=85 A8;K A;54C5B 87<5=8BL =0 ?@>B82>?>;>6=>5, ?5@5G5@:=C2 :@5AB>< =0 AE5<5 8AE>4=K9 25:B>@. =0G5=8O ?;5G 27OBKE A G5@B560, 2 C@02=5=85 <><5=B>2, <>6=> ?>4AB02;OBL 2 <8;;8<5B@0E, B.:. C@02=5=85 =5 A>45@68B <><5=B>2 A8; 2 G8AB>< 2845 (i). !>AB02;O5< C@02=5=85 <><5=B>2 A8; >B=>A8B5;L=> B>G:8  4;O B@5BL53> 725=0 ( 3 ( i) = 0;  EMBED Equation.3  !>AB02;O5< 25:B>@=>5 C@02=5=85 A8;, 459AB2CNI8E =0 3@C??C AAC@0, 345 =58725AB=K5 70?8AK205< 2 :>=F5 (=>@<0;L=K5 A>AB02;ONI85 @50:F89  EMBED Equation.3  8  EMBED Equation.3 ): (( i = 0; ( EMBED Equation.3 . @>872>48< 3@0D8G5A:>5 A;>65=85 25:B>@>2 2 <0AHB015 (  (@8A. 12.1, 2). >A;54=89 25:B>@  EMBED Equation.3  >B:;04K205< 87 ?>;NA0 ?;0=0 A8;. 0 ?;0=5 ?>;CG05< =0?@02;5=8O 8 7=0G5=8O A8; 2 <0AHB015  EMBED Equation.3  8  EMBED Equation.3 . 5:B>@=> A:;04K20O :0A0B5;L=K5 8 =>@<0;L=K5 A>AB02;ONI85, ?>;CG05< 01A>;NB=K5 7=0G5=8O @50:F89 (=0 @8A. 12.1, 2 ?@54AB02;5=K ?C=:B8@><): A>548=OO B>G:8 1 8 2 ?>;CG05< ( EMBED Equation.3 ,  EMBED Equation.3 , ; A>548=OO B>G:8 3 8 2 ?>;CG05< ( EMBED Equation.3 ,  EMBED Equation.3 , . ;O >?@545;5=8O @50:F88 2 H0@=8@5  A;54C5B 25:B>@=> A;>68BL 2A5 A8;K, 459AB2CNI85 =0 725=> 2 8;8 3, =0?@8<5@, 4;O 725=0 2  EMBED Equation.3  0 @8A. 12.1, 2 A>548=82 B>G:8 4 8 2, ?>;CG05< =0?@02;5=85 459AB28O @50:F88 R32 :>@><KA;0 3 =0 H0BC= 2. >A;5 @0AA<>B@5=8O CA;>289 @02=>25A8O 3@C??K AC@0 ?5@5E>48< : >?@545;5=8N A8;, 459AB2CNI8E =0 =0G0;L=K9 <5E0=87<. !8;>2>9 0=0;87 =0G0;L=>3> <5E0=87<0 !B@>8< :8=5<0B8G5A:CN AE5<C =0G0;L=>3> <5E0=87<0 2 <0AHB015 (@8A. 12.1, 3), 2 A>>B25BAB2CNI85 B>G:8 ?@8:;04K205< A8;K: 8=5@F88 :@82>H8?0 1 81; 25A0 :@82>H8?0 1 G1; @50:F88 2 H0@=8@0E (>?>@0E) R21 - H0BC=0 2 =0 :@82>H8? 1; R01 - AB>9:8 0 =0 :@82>H8? 1; C@02=>25H820NICN A8;C C. 50:F8O H0BC=0 2 =0 :>@><KA;> 1, R21 >?@545;5=0 ?@8 @0AA<>B@5=88 A8;>2>3> 0=0;870 3@C??K AAC@0 (=> B0< >?@545;5=0 @50:F8O :@82>H8?0 1 =0 H0BC= 2, ?>MB><C ?@8 ?@8;>65=88 5Q =5>1E>48<> 87<5=8BL =0?@02;5=85 =0 ?@>B82>?>;>6=>5); #@02=>25H820NI0O A8;0 C. (@50:F8O 42830B5;O =0 <5E0=87<), =58725AB=0O 25;8G8=0, ?@8:;04K205BAO 2 H0@=8@5  ?5@?5=48:C;O@=> 1. #:07K205< ?;5G8 459AB28O A8; >B=>A8B5;L=> H0@=8@0 1 8 A>AB02;O5< C@02=5=85 <><5=B>2 2A5E A8; >B=>A8B5;L=> 1: ( 1( i)= 0;  EMBED Equation.3 . ><5=B C@02=>25H820NI59 A8;K  EMBED Equation.3 (745AL r:@  @048CA :@82>H8?0, <). 50:F8N 2 H0@=8@5 1, R01, >?@545;O5< 87 25:B>@=>3> C@02=5=8O @02=>25A8O 2A5E A8;, 459AB2CNI8E =0 725=> 1:  EMBED Equation.3 . !B@>8< ?;0= A8; (@8A. 12.1, 4) 2 <0AHB015 A8; (@, /<<, 345 70<K:0NI89 25:B>@ >?@545;O5B =0?@02;5=85 8 25;8G8=C >?>@=>9 @50:F88 R01, 5Q 7=0G5=85  EMBED Equation.3 . ?@545;5=85 C@02=>25H820NI59 A8;K <5B>4>< .. C:>2A:>3> @8 >?@545;5=88 <>I=>AB8 42830B5;O 8 CAB0=>2;5=88 53> B8?0, @0AG5B5 <0E>2>3> :>;5A0, A>AB02;5=88 E0@0:B5@8AB8:8 @53C;OB>@>2 8 2 @O45 4@C38E A;CG052 =5>1E>48<> 7=0BL B>;L:> C@02=>25H820NI89 <><5=B 8;8 C@02=>25H820NICN A8;C, @50:F88 2 :8=5<0B8G5A:8E ?0@0E 8AA;54C5<>3> <5E0=87<0 ?@8 MB>< <>3CB >AB0BLAO =58725AB=K<8.  MB>< A;CG05 C4>1=55 8A?>;L7>20BL B5>@5<C C:>2A:>3>: 5A;8 :0:>9-;81> <5E0=87< ?>4 459AB285< A8AB5<K A8;, =0E>48BAO 2 A>AB>O=88 @02=>25A8O, B> ?>2Q@=CBK9 =0 90( 2 :0:CN-;81> AB>@>=C ?;0= A:>@>AB59, @0AA<0B@8205<K9 :0: B2Q@4>5 B5;>, 2@0I0NI55AO 2>:@C3 ?>;NA0 ?;0=0 8 =03@C65==>5 B5<8 65 A8;0<8, ?@8;>65==K<8 2 A>>B25BAB2CNI85 B>G:8 ?;0=0, B0:65 =0E>48BAO 2 @02=>25A88. "5>@5<C C:>2A:>3> <>6=> ?@8<5=8BL 8 : A8AB5<5, =5 =0E>4OI59AO 2 @02=>25A88. ;O MB>3> 4>AB0B>G=>, :@><5 459AB2CNI8E A8; ?@8;>68BL 8 A8;K 8=5@F88. ;O 4>:070B5;LAB20 B5>@5<K 2>A?>;L7C5<AO ?@8=F8?>< 2>7<>6=KE ?5@5<5I5=89: 5A;8 A8AB5<0 =0E>48BAO 2 @02=>25A88, B> AC<<0 M;5<5=B0@=KE @01>B =0 2>7<>6=KE ?5@5<5I5=8OE @02=0 =C;N (2>7<>6=K5 ?5@5<5I5=8O  MB> ?5@5<5I5=8O 4>?CA:05<K5 A2O7O<8):  EMBED Equation.3 , 8;8 @0745;82 =0 dt,  EMBED Equation.3 , >;CG05<:  EMBED Equation.3 , 345 i  7040205<K5 A8;K; (i  A:>@>AB8 B>G5: ?@8;>65=8O i; (j  A:>@>AB8 2@0I5=8O 725=L52 : :>B>@K< ?@8;>65=K <><5=BK A8; j; Ni, Nj  <>I=>AB8 A>>B25BAB25==> A8; i 8 <><5=B>2 A8; j. @54?>;>68<, GB> 2 :0:>9 B> B>G:5 725=0 ?@8;>65=0 A8;0 i ?5@5=5AQ==0O ?0@0;;5;L=> A0<>9 A515 2 A>>B25BAB2CNICN B>G:C ?>2Q@=CB>3> =0 90( ?;0=0 A:>@>AB59. >I=>ABL MB>9 A8;K <>6=> 2K@078BL A;54CNI8< >1@07><:  EMBED Equation.3 , 345 hi  ?5@?5=48:C;O@, >?CI5==K9 87 ?>;NA0 ?;0=0 A:>@>AB59 =0 ;8=8N 459AB28O A8;K i. "0: :0: ?>;CG5==>5 2KH5 C@02=5=85, >?@545;ONI55 25;8G8=C Ni, 8<55B <5AB> 4;O 2A5E A8; i, 459AB2CNI8E =0 4@C385 725=LO <5E0=87<0, B> ?>;CG05<:  EMBED Equation.3 . >A:>;L:C  EMBED Equation.3 , B>:  EMBED Equation.3 , GB> 8 O2;O5BAO 4>:070B5;LAB2>< B5>@5<K. @8<5=8< <5B>4 C:>2A:>3> : =0E>645=8N ?@82545==>9, 8;8 C@02=>25H820NI59 A8;K C. 0AA<>B@8< H0@=8@=K9 G5BK@QE725==K9 <5E0=87< (@8A. 12.2, 0) =0E>4OI89AO 2 A>AB>O=88 @02=>25A8O ?>4 459AB285< A8;: 25A0 :@82>H8?0 1 G1, H0BC=0 2 G2 8 :>@><KA;0 3 G3; 8=5@F88: :@82>H8?0 1 81; H0BC=0 2 82, 82; :>@><KA;0 3 83, 83. !C<<0@=>5 459AB285 =0 725=> A8;K 8 <><5=B0 A8;K 8=5@F88 70<5=O5< >4=>9 @57C;LB8@CNI59 A8;>9 8=5@F88, A>740NI59 <><5=B, 459AB2CNI89 2 >1@0B=>< =0?@02;5=88 C3;>2><C CA:>@5=8N, 8 ?@8;>65==>9 2 F5=B@5 :0G0=8O (4;O H0BC=0 2  K2, :>@><KA;0 3  K3).  8A. 12.2 ;O ?@82545=8O <5E0=87<0 2 @02=>25A=>5 A>AB>O=85 =5>1E>48<>, 2 :0:>9 ;81> B>G:5 <5E0=87<0 ?@8;>68BL C@02=>25H820NICN A8;C C. 0 B>G:C ?@8;>65=8O C@02=>25H820NI59 A8;K G0I5 2A53> ?@8=8<0NB B>G:C  =0G0;L=>3> 725=0, =0?@02;OO 5Q ?5@?5=48:C;O@=> : 1. !B@>8< 2 ?@>872>;L=>< <0AHB015 ?>25@=CBK9 =0 90( ?;0= A:>@>AB59 <5E0=87<0 (@8A. 12.2, 1) 8 ?5@5=>A8< 2 A>>B25BAB2CNI85 B>G:8 25:B>@0 2=5H=8E A8;, 0 B0:65 C@02=>25H820NICN A8;C ?0@0;;5;L=> 8E 459AB28N. @8=8<0O ?;0= A:>@>AB59 70 @KG03, =03@C65==K9 A8;0<8 G1, G2, G3, 81, 82, 83 8 C, A>AB02;O5< C@02=5=85 <><5=B>2 MB8E A8; >B=>A8B5;L=> ?>;NA0 ?;0=0 A:>@>AB59 @(:  EMBED Equation.3 . 7 MB>3> C@02=5=8O >?@545;ONB 25;8G8=C C@02=>25H820NI59 A8;K, 5A;8 >=0 ?>;CG8;0AL ?>;>68B5;L=>9, B> =0?@02;5=85 5Q 459AB28O 2K1@0=> ?@028;L=>. @8 >B@8F0B5;L=>< 7=0G5=88 C =5>1E>48<> 87<5=8BL 5Q =0?@02;5=85 =0 ?@>B82>?>;>6=>5. #@02=>25H820NI0O A8;0 O2;O5BAO CA;>2=>9, 8 5Q 8A?>;L7CNB ;8HL 4;O 2>?@>A>2, A2O70==KE A >?@545;5=85< <>I=>AB8 8;8 @01>BK <0H8=K. 568<K 42865=8O <5E0=87<>2  7028A8<>AB8 >B B>3> :0:CN @01>BC A>25@H0NB 2=5H=85 A8;K <0H8=K @07;8G0NB B@8 @568<0 42865=8O: @073>= (@07153, ?CA:), B>@<>65=85 (2K153, >AB0=>2) 8 CAB0=>282H55AO 42865=85 (@8A. 12.3).  (1, @04/A TF    (1A@ = const  (10  0 t, c. 073>= #AB0=>282H55AO 42865=85 K153  8A. 12.3 #AB0=>282H8<AO 42865=85< <5E0=87<0 =07K20NB B0:>5 42865=85, ?@8 :>B>@>< 53> >1>1I5==0O A:>@>ABL 8 :8=5B8G5A:0O M=5@38O O2;ONBAO ?5@8>48G5A:8<8 DC=:F8O<8 2@5<5=8. 8=8<0;L=K9 ?@><56CB>: 2 =0G0;5 8 2 :>=F5 :>B>@>3> ?>2B>@ONBAO 7=0G5=8O :8=5B8G5A:>9 M=5@388 8 >1>1I5==>9 A:>@>AB8 <5E0=87<0  =07K20NB 2@5<5=5< F8:;0 CAB0=>282H53>AO 42865=8O. ;O 8450;L=>9 <5E0=8G5A:>9 A8AB5<K, 2 :>B>@>9 =5B ?>B5@L M=5@388 8 725=LO 01A>;NB=> 65AB:85 ?@8 ?>;CG5=88 C@02=5=89 42865=8O <5E0=87<0 <>6=> 2>A?>;L7>20BLAO B5>@5<>9 >1 87<5=5=88 :8=5B8G5A:89 M=5@388: @07=>ABL M=5@388 70 :0:>9 ;81> ?@><56CB>: 2@5<5=8 @02=0 @01>B5 A8; 70 B>B 65 ?@><56CB>: 2@5<5=8.  EMBED Equation.3 , 345 4.A.  @01>B0 4286CI8E A8;; ?.A.  @01>B0 A8; ?@>872>4AB25==KE A>?@>B82;5=89; 2.A.  @01>B0 A8; 2@54=KE A>?@>B82;5=89 (B@5=8O 8 2=5H=59 A@54K); G  @01>B0 A8; 25A0. ;O @568<0 @073>=0: (i0 = 0, ?.A. = 0, B>340:  EMBED Equation.3 . 01>B0 4286CI8E A8; ?@8 @073>=5 @0AE>4C5BAO :8=5B8G5A:CN M=5@38N, @01>BC A8; 2@54=KE A>?@>B82;5=89 8 25A0. @8 CAB0=>282H5<AO 42865=88 70 :064K9 F8:; 42865=8O @01>B0 2A5E 2=5H=8E A8; @02=0 =C;N EMBED Equation.3 . ;O @568<0 2K1530: (i = 0, 4.A. = 0, ?.A. = 0 B>340:  EMBED Equation.3 . 0?0AQ==0O :8=5B8G5A:0O M=5@38O ?@8 2K1535 B@0B8BAO =0 ?@5>4>;5=85 @01>B A8; 2@54=KE A>?@>B82;5=89 8 25A0. 568<K @073>=0 8 2K1530 =07K20NB @568<0<8 =5CAB0=>282H53>AO 42865=8O. A=>2=K5 D>@<K C@02=5=8O 42865=8O <5E0=87<0 (?@O<0O 7040G0 48=0<8:8) @O<0O 7040G0 48=0<8:8 <0H8=K @5H05B 2>?@>AK 0=0;870 - >?@545;5=85 70:>=0 42865=8O <5E0=8G5A:>9 A8AB5<K ?>4 459AB285< 7040==KE 2=5H=8E A8;. @8 @5H5=88 MB>9 7040G8 ?0@0<5B@K <0H8==>3> 03@530B0 8 459AB2CNI85 =0 =53> 2=5H=85 A8;K 8725AB=K, =5>1E>48<> >?@545;8BL 70:>= 42865=8O: A:>@>AB8 8 CA:>@5=8O 2 DC=:F88 2@5<5=8 8;8 >1>1I5==>9 :>>@48=0BK. =0G5 MBC 7040GC <>6=> AD>@<C;8@>20BL B0:: 7040=K C?@02;ONI85 A8;K 8 A8;K 2=5H=53> A>?@>B82;5=8O, >?@545;8BL >15A?5G8205<K9 8<8 70:>= 42865=8O <0H8=K. #@02=5=85 42865=8O <0H8=K, 8;8 <5E0=87<0 40QB 2>7<>6=>ABL >F5=8BL 8E 48=0<8G5A:85 :0G5AB20 2 =5A:>;L:> C?@>I5==>< 2845 8 A25AB8 MB> 8AA;54>20=85 : @0AA<>B@5=8N 42865=8O :0:>3> ;81> >4=>3> 725=0 (2 1>;LH8=AB25 A;CG052 =0G0;L=>3>), B.5. 2>A?@8=8<0NI53> =5?>A@54AB25==> <>I=>ABL 42830B5;O. ;O MB>3> : MB><C 725=C (2 40;L=59H5< 1C45< =07K20BL 53> 725=>< ?@82545=8O), ?@82>4OB 2A5 2=5H=85 A8;K, 459AB2CNI85 =0 <5E0=87< 8 <0AAK 725=L52. #@02=5=85 42865=8O <5E0=87<0 2 48DD5@5=F80;L=>< 2845 !>45@68B 2B>@K5 ?@>872>4=K5 >B :>>@48=0B ?> 2@5<5=8. 7<5=5=85 :8=5B8G5A:>9 M=5@388 <5E0=87<0 @02=> ?@8@0I5=8N @01>B A8; 459AB2CNI8E =0 <5E0=87<:  EMBED Equation.3 .  A;CG05 5A;8 =0G0;L=>5 725=> A>25@H05B 2@0I0B5;L=>5 42865=85:  EMBED Equation.3 . ">340:  EMBED Equation.3 ,  EMBED Equation.3 , @5>1@07C5< 2B>@>5 A;0305<>5 A CG5B><:  EMBED Equation.3 . >4AB02;OO ?>;CG05<:  EMBED Equation.3 .  A;CG05 5A;8 J?@ = const (<0E>2>5 :>;5A>, @>B>@ 42830B5;O 8 B.?.)  EMBED Equation.3  ?>;CG05<  EMBED Equation.3  (2B>@>9 70:>= LNB>=0 4;O 2@0I0B5;L=>3> 42865=8O). A;8 =0G0;L=>5 725=> A>25@H05B ?>ABC?0B5;L=>5 42865=85 ?>;CG05<:  EMBED Equation.3 .  A;CG05 5A;8 m?@ = const ?>;CG05<  EMBED Equation.3 . 8=0<8G5A:0O <>45;L <5E0=87<0 8=0<8G5A:0O <>45;L <5E0=87<0, 8;8 <0H8=K ?@54AB02;O5B A>1>9 C@02=5=85 42865=8O 725=0 ?@82545=8O, : :>B>@><C ?@82545=K 2A5 A8;K 8 <0AAK 725=L52.  A;CG05 5A;8 725=> ?@82545=8O A>25@H05B 2@0I0B5;L=>5 42865=85 (=0?@8<5@ :@82>H8?, @8A. 12. 3, 0) B> C@02=5=85 42865=8O ?@8=8<05B 284:  EMBED Equation.3 , 345 J?@  ?@82545==K9 <><5=B 8=5@F88 725=0 ?@82545=8O; ?@  ?@82545==K9 <><5=B A8; 725=0 ?@82545=8O.  8A. 12.3  A;CG05 5A;8 725=> ?@82545=8O A>25@H05B ?>ABC?0B5;L=>5 42865=85 (?>;7C=, @8A. 12.3, 1) C@02=5=85 42865=8O 8<55B 284:  EMBED Equation.3 . 345 m?@  ?@82545==0O <0AA0 725=0 ?@82545=8O; ?@  ?@82545==0O A8;0 725=0 ?@82545=8O. @82545=85 A8; 8 <><5=B>2 A8; : 725=C ?@82545=8O (>?@545;5=85 ?0@0<5B@>2 48=0<8G5A:>9 <>45;8) 0 725=LO <5E0=87<0 459AB2CNB A8;K 8 <><5=BK A8;, @072820NI85 A>>B25BAB2CNI85 <>I=>AB8. "0:8< >1@07><, <>I=>ABL 2A5E 7040205<KE A8; A>AB>8B 87 42CE G0AB59:  EMBED Equation.3 , 345 N  ( <>I=>ABL, @0728205<0O A8;0<8, ?@8;>65==K<8 2 @07;8G=KE B>G:0E 725=L52, A>25@H0NI8E ?>ABC?0B5;L=>5 8;8 A;>6=>5 ?;>A:>5 42865=85; N ( <>I=>ABL, @0728205<0O <><5=B0<8 A8;, ?@8;>65==K<8 : 2@0I0NI8<AO 725=LO<. >I=>ABL N  <>65B 1KBL 2KG8A;5=0 ?> D>@<C;5:  EMBED Equation.3 , 345 i ( A8;K, ?@8;>65==K5 : i-< 725=LO< <5E0=87<0; (i ( A:>@>AB8 B>G5: ?@8;>65=8O A8;; (i( C3;K, >1@07>20==K5 =0?@02;5=85< A8; 8 A:>@>AB59 8E B>G5: ?@8;>65=8O. >I=>ABL N 2KG8A;O5BAO ?> D>@<C;5:  EMBED Equation.3 , 345 Mk ( <><5=B, 459AB2CNI89 =0 k-e 2@0I0NI85AO 725=LO; (k ( C3;>2K5 A:>@>AB8 MB8E 725=L52. >4AB02;OO 7=0G5=8O N  8 N ?>;CG8<:  EMBED Equation.3 . -BC <>I=>ABL, @0728205<CN A8;0<8 8 <><5=B0<8 A8;, ?@8;>65==K<8 :> 2A5< ?>4286=K< 725=LO< <5E0=87<0, <>6=> ?@8;>68BL : ;N1><C 2K1@0==><C 725=C ?@82545=8O. A;8 725=> ?@82545=8O A>25@H05B 2@0I0B5;L=>5 42865=85, B> 53> <>I=>ABL 1C45B ?@54AB02;5=0 A;54CNI8< 2K@065=85<:  EMBED Equation.3 , 345 (1 ( C3;>20O A:>@>ABL 725=0 ?@82545=8O. "0: :0: ;52K5 G0AB8 C@02=5=89 @02=K, B>:  EMBED Equation.3 . "0:8< >1@07><, ?@82545==K< <><5=B>< A8; =07K205BAO <><5=B (?@), ?@8;>65==K9 : 725=C ?@82545=8O 8 @072820NI89 <>I=>ABL, @02=CN AC<<5 <>I=>AB59 2A5E A8; 8 <><5=B>2 A8;, ?@8;>65==KE : 725=LO< <5E0=87<0. 0 >A=>20=88 C@02=5=8O 8<55<:  EMBED Equation.3 . >;CG5==>5 C@02=5=85 G0I5 ?@8<5=ONB : H0@=8@=K< 8 :C;0G:>2K< <5E0=87<0<, 284=>, GB> ?@ 7028A8B >B >B=>H5=89 A:>@>AB59, G8A;>2K5 7=0G5=8O :>B>@KE <5=ONBAO 2 7028A8<>AB8 >B 25;8G8=K ( C3;0 ?>2>@>B0 725=0 ?@82545=8O (. "0:8< >1@07><, Mnp = f((). ;O >?@545;5=8O >B=>H5=89 A:>@>AB59 =5>1E>48<> ?>AB@>8BL ?;0=K A:>@>AB59 4;O =5A:>;L:8E ?>;>65=89 <5E0=87<0. "0: :0: >B=>H5=85 A:>@>AB59 =5 1C45B 7028A5BL >B <0AHB010, B> ?@8 ?>AB@>5=88 8E <>6=> ?@8=OBL (1 = 1 @04/A5:. ;O <5E0=87<>2, ?@5>1@07CNI8E B>;L:> 2@0I0B5;L=>5 42865=85 A ?>AB>O==K< >B=>H5=85< C3;>2KE A:>@>AB59, ?@82545==K9 <><5=B A8;:  EMBED Equation.3 . B=>H5=8O  EMBED Equation.3 , ?@54AB02;ONB A>1>9 ?5@540B>G=K5 >B=>H5=8O. ">340:  EMBED Equation.3 . A;8 Mk = const, B> ?@82545==K9 <><5=B A8; B0:65 O2;O5BAO ?>AB>O==>9 25;8G8=>9, =5 7028AOI59 >B C3;0 ?>2>@>B0 725=0 ?@82545=8O. @82545==K9 <><5=B 4286CI8E A8; =0?@02;5= 2 AB>@>=C 2@0I5=8O 725=0 ?@82545=8O, ?@82545==K9 <><5=B A8; A>?@>B82;5=8O =0?@02;5= 2 AB>@>=C, ?@>B82>?>;>6=CN =0?@02;5=8N 2@0I5=8O 725=0 ?@82545=8O. A;8 ?@82>48BL : 725=C ?@82545=8O 2A5 7040205<K5 A8;K, B> ?@82545==K9 <><5=B A8; ?@54AB02;O5B A>1>N @07=>ABL <564C ?@82545==K<8 <><5=B0<8 A8; 4286CI8E (4.A.) 8 A8; A>?@>B82;5=8O (A.A.), B. 5.  EMBED Equation.3 . A;8 725=> ?@82545=8O A>25@H05B ?>ABC?0B5;L=>5 42865=85, B> 53> <>I=>ABL 1C45B ?@54AB02;5=0 A;54CNI8< 2K@065=85<:  EMBED Equation.3 , 345 (1 ( A:>@>ABL 725=0 ?@82545=8O. @8254Q==>9 A8;>9 =07K205BAO A8;0 ( ?@), ?@8;>65==0O : 725=C ?@82545=8O 8 A>740NI0O <>I=>ABL, @02=CN AC<<5 <>I=>AB59 2A5E A8; 8 <><5=B>2 A8;, ?@8;>65==KE : 725=LO< <5E0=87<0, B.5.:  EMBED Equation.3 . @82545=85 <0AA 8 <><5=B>2 8=5@F88 725=L52 ;O ?@82545=8O <0AA 8 <><5=B>2 8=5@F88 8A?>;L7C5BAO ?>=OB85 > :8=5B8G5A:>9 M=5@388 725=L52. B<5B8<, :0: 2KG8A;O5BAO :8=5B8G5A:0O M=5@38O 725=L52 ?@8 @07;8G=KE 2840E 8E 42865=8O. ;O 725=0, A>25@H0NI53> ?>ABC?0B5;L=>5 42865=85, :8=5B8G5A:0O M=5@38O >?@545;O5BAO ?> A;54CNI59 D>@<C;5:  EMBED Equation.3 , 345 m (<0AA0 725=0; ( ( A:>@>ABL ;N1>9 B>G:8 725=0, </A5:. A;8 725=> A>25@H05B 2@0I0B5;L=>5 42865=85, B> :8=5B8G5A:0O M=5@38O:  EMBED Equation.3 , 345 J ( <><5=B 8=5@F88 725=0 >B=>A8B5;L=> >A8 53> 2@0I5=8O, :3(<2; ( ( C3;>20O A:>@>ABL 725=0, @04/A5:. ;O 725=0, A>25@H0NI53> A;>6=>5 ?;>A:>5 42865=85, :8=5B8G5A:0O M=5@38O A>AB>8B 87 :8=5B8G5A:>9 M=5@388 2 ?>ABC?0B5;L=>< 42865=88 2<5AB5 A F5=B@>< BO65AB8 8 :8=5B8G5A:>9 M=5@388 2> 2@0I0B5;L=>< 42865=88 2>:@C3 >A8, ?@>E>4OI59 G5@57 F5=B@ BO65AB8:  EMBED Equation.3 , 345 (s ( A:>@>ABL F5=B@0 BO65AB8 725=0; Js ( <><5=B 8=5@F88 725=0 >B=>A8B5;L=> >A8, ?@>E>4OI59 G5@57 53> F5=B@ BO65AB8. 1>7=0G8< G8A;> 725=L52 <5E0=87<0, A>25@H0NI8E ?>ABC?0B5;L=>5, 2@0I0B5;L=>5 8 A;>6=>-?;>A:>5 42865=8O, A>>B25BAB25==> G5@57 @, k 8 q. ">340 C@02=5=85 :8=5B8G5A:>9 M=5@388 ?@8<5B A;54CNI89 284:  EMBED Equation.3 . 8=5B8G5A:CN M=5@38N <5E0=87<0 <>6=> ?@54AB028BL :0: :8=5B8G5A:CN M=5@38N 2@0I0NI53>AO 725=0 1 ?@82545=8O, B. 5.  EMBED Equation.3 . BAN40  EMBED Equation.3  !;54>20B5;L=>:  EMBED Equation.3 . "0:8< >1@07><, ?@82545==K9 <><5=B 8=5@F88 Jnp ?@54AB02;O5B A>1>9 <><5=B 8=5@F88 725=0 ?@82545=8O, >1;040NI89 :8=5B8G5A:>9 M=5@3859, @02=>9 AC<<5 :8=5B8G5A:8E M=5@389 2A5E 4286CI8EAO 725=L52 <5E0=87<0. $>@<C;0 ?@8<5=O5BAO 3;02=K< >1@07>< 4;O ?;>A:8E H0@=8@=KE <5E0=87<>2.  MB>< A;CG05 Jnp 7028A8B >B ?>;>65=8O <5E0=87<0, B0: :0: 4;O :064>3> 53> ?>;>65=8O >B=>H5=8O A:>@>AB59  EMBED Equation.3  1C4CB <5=OBLAO. B=>H5=85 A:>@>AB59 A;54C5B >?@545;OBL 87 ?;0=0 A:>@>AB59. A;8 <5E0=87< A>AB>8B B>;L:> 87 2@0I0NI8EAO 725=L52 (=0?@8<5@, @07;8G=K5 284K ?5@540G), B> C@02=5=85 ?@8=8<05B A;54CNI89 284:  EMBED Equation.3 . 0<5=OO >B=>H5=85 C3;>2KE A:>@>AB59 A>>B25BAB2CNI8< ?5@540B>G=K< >B=>H5=85<, ?>;CG8<:  EMBED Equation.3 . "0: :0: 4;O ?5@540B>G=KE <5E0=87<>2 7=0G5=8O i1k ?>AB>O==K, B> ?@82545==K9 <><5=B 8=5@F88 2 MB>< A;CG05 B0:65 O2;O5BAO ?>AB>O==K<. B<5B8<, GB> 2 @O45 A;CG052, =0?@8<5@ 2 A;54OI8E CAB@>9AB20E, =C6=> 2K1@0BL 42830B5;L, :>B>@K9 >15A?5G8; 1K <5E0=87<C =5>1E>48<>5 ?> CA;>28O< M:A?;C0B0F88 2@5<O A@010BK20=8O. 5>1E>48<0O ?CA:>20O <>I=>ABL <>65B 1KBL >?@545;5=0 ?> ?CA:>2><C <><5=BC, :>B>@K9 @025= ?@>872545=8N ?@82545==>3> <><5=B0 8=5@F88 =0 C3;>2>5 CA:>@5=85. >=B@>;L=K5 2>?@>AK !8;>2>9 0=0;87 @KG06=>3> <5E0=87<0 <5B>4>< ?;0=>2 A8;.  G5< 70:;NG05BAO <5B>4 .. C:>2A:>3> 4;O >?@545;5=8O C@02=>25H820NI59 A8;K. A=>2=K5 @568<K 8 C@02=5=8O 42865=8O <5E0=87<0. #@02=5=85 42865=8O <5E0=87<0 2 48DD5@5=F80;L=>< 2845. 8=0<8G5A:0O <>45;L <0H8==>3> 03@530B0. @82545=85 A8; 2 <5E0=87<0E. @82545=85 <0AA 2 <5E0=87<0E. 5:F8O 13 #AB0=>282H55AO 42865=88 <5E0=87<0. 5@02=><5@=>ABL 42865=8O. 0AG5B <0E>2>3> :>;5A0. #AB0=>282H55AO 42865=85 <0H8==>3> 03@530B0 5@02=><5@=>ABL 42865=8O #AB0=>282H8<AO @568<>< 42865=8O =07K20NB @568<, C :>B>@>3> >1>1I5==0O A:>@>ABL 725=0 ?@82545=8O 5ABL ?5@8>48G5A:0O DC=:F8O 2> 2@5<5=8 (@8A. 13.1). (1, @04/A t?5@8>40  ((1   (1min (1A@ = const (1max  0 t, A5: 8A. 13.1 0 2@5<O >4=>3> ?5@8>40 (i0 = (i, 8 :0: A;54AB285 ( = 0, G = 0. ">340 87 70:>=0 87<5=5=8O :8=5B8G5A:>9 M=5@388 ?>;CG05<:  EMBED Equation.3 . A;8 @0AA<0B@820BL CAB0=>282H55AO 42865=85 2=CB@8 ?5@8>40 A;54C5B 8A?>;L7>20BL C@02=5=85:  EMBED Equation.3 .  ?@545;0E ?5@8>40 B5:CI55 7=0G5=85 AC<<0@=>9 @01>BK =5 @02=> =C;N. 01>B0 <>65B 1KBL B> ?>;>68B5;L=>9, B> >B@8F0B5;L=>9. @8 ?>;>68B5;L=>9 25;8G8=5 @01>BK <0H8=0 C25;8G8205B A2>N :8=5B8G5A:CN M=5@38N 70 AG5B C25;8G5=8O A:>@>AB8, B> 5ABL @073>=O5BAO. 0 CG0AB:0E, 345 AC<<0@=0O @01>B0 >B@8F0B5;L=0, :8=5B8G5A:0O M=5@38O 8 A:>@>ABL <0H8=K C<5=LH05BAO, <0H8=0 ?@8B>@<068205BAO.  CAB0=>282H5<AO @568<5 25;8G8=K C25;8G5=8O A:>@>AB8 =0 CG0AB:0E @073>=0 8 A=865=8O =0 CG0AB:0E B>@<>65=8O 70 F8:; @02=K, ?>MB><C A@54=OO A:>@>ABL 42865=8O (1A@ = const ?>AB>O==0.  <0H8=0E ?@82545==K9 <><5=B 8=5@F88 :>B>@KE 7028A8B >B >1>1I5==>9 :>>@48=0BK, =0 =5@02=><5@=>ABL 42865=8O >:07K205B 2;8O=85 25;8G8=0 87<5=5=8O ?@82545==>3> <><5=B0 8=5@F88. >;510=8O A:>@>AB8 87<5=5=8O >1>1I5==>9 :>>@48=0BK <0H8=K =5 >:07K20NB ?@O<>3> 2;8O=8O =0 DC=40<5=B <0H8=K. >MB><C MB8 :>;510=8O 8 2K7K20NI85 8E ?@8G8=K >?@545;ONB, B0: =07K205<CN, 2=CB@5==NN 281@>0:B82=>ABL <0H8=K. 5;8G8=0 0<?;8BC4K :>;510=89 A:>@>AB8 ((1 >?@545;O5BAO @07=>ABLN <564C <0:A8<0;L=>9 (1max 8 <8=8<0;L=>9 (1min A:>@>ABO<8. 0 <5@C 87<5@5=8O :>;510=89 A:>@>AB8 2 CAB0=>282H5<AO @568<5 ?@8=OB0 >B=>A8B5;L=0O 25;8G8=0, =07K205<0O :>MDD8F85=B>< =5@02=><5@=>AB8 42865=8O (=5@02=><5@=>AB8 E>40):  EMBED Equation.3 , 345  EMBED Equation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quation.3 . CABL 70 2@5<O ?5@8>40 @01>B0 4286CI8E A8; =5 @02=0 @01>B5 A8; A>?@>B82;5=8O B>340 <0:A8<0;L=K5 8 <8=8<0;L=K5 C3;>2K5 A:>@>AB8 1C4CB A>>B25BAB2>20BL <0:A8<0;L=K< 8 <8=8<0;L=K< M=5@38O<:  EMBED Equation.3 , 345 871.<0:A.  <0:A8<0;L=0O 871KB>G=0O @01>B0; 4.A.  @01>B0 4286CI8E A8;; A.A.  @01>B0 A8; A>?@>B82;5=89; Emax  <0:A8<0;L=0O :8=5B8G5A:0O M=5@38O <5E0=87<0; Emin  <8=8<0;L=0O :8=5B8G5A:0O M=5@38O <5E0=87<0; J?@  ?@8254Q==K9 <><5=B 8=5@F88 <0AA; J<0E  <><5=B 8=5@F88 <0E>2>3> :>;5A0. ">340 ?@5>1@07CO:  EMBED Equation.3 . :>=G0B5;L=> ?>;CG05<:  EMBED Equation.3 .  (1, @04/A 157 <0E>28:0     ((1     ((1* (1A@ = const A <0E>28:><   0 t, A5: 8A. 13.2 2. 1I89 A;CG09 (4;O <0H8= A ?5@5<5==K< ?@8254Q==K< <><5=B>< 8=5@F88  EMBED Equation.3 ), ?>;>65=8O A <0:A8<0;L=K<8 8 <8=8<0;L=K<8 7=0G5=8O<8 A:>@>AB8 =5 A>2?040NB A ?>;>65=8O<8 ?@8 :>B>@KE <0:A8<0;L=K5 8 <8=8<0;L=K5 7=0G5=8O M=5@388. 0AA<>B@8< 3@0D8G5A:89 A?>A>1 ?> <5B>4C 8BB5=10CM@0 (<5B>4 ?>AB@>5=8O 4803@0<<K M=5@3>-<0AA). >=B@>;L=K5 2>?@>AK #AB0=>282H89AO @568< 42865=8O <0H8==>3> 03@530B0. @8G8=K 2K7K20NI85 =5@02=><5@=>ABL 42865=8O. 0: >F5=8205BAO 8 @53C;8@C5BAO =5@02=><5@=>ABL 42865=8O. ?@545;5=85 <><5=B0 8=5@F88 <0E>2>3> :>;5A0 4;O <0H8= A ?>AB>O==K< ?@82545==K< <><5=B>< 8=5@F88. 5:F8O 14 81@0F88 8 :>;510=8O 2 <0H8=0E 8 <5E0=87<0E, 281@>0:B82=>ABL 8 281@>70I8B0. >=OB85 > =5C@02=>25H5==>AB8 725=0 8 <5E0=87<0. !B0B8G5A:>5 C@02=>25H820=85 @KG06=KE <5E0=87<>2. 5B>4 70<5I0NI8E <0AA. >;=>5 8 G0AB8G=>5 C@02=>25H820=85 <5E0=87<0. >B>@ 8 284K 53> C@02=>25H820=8O: AB0B8G5A:>5, 48=0<8G5A:>5. 0;0=A8@>2:0. 81@0F88 8 :>;510=8O 2 <0H8=0E 8 <5E0=87<0E @8 42865=88 <5E0=8G5A:>9 A8AB5<K ?>4 459AB285< 2=5H=8E A8; 2 =59 <>3CB 2>7=8:0BL <5E0=8G5A:85 :>;510=8O 8;8 281@0F88. @8G8=0<8 2>7=8:=>25=8O 281@0F89 <>3CB 1KBL ?5@8>48G5A:85 87<5=5=8O A8; (A8;>2>5 2>7<CI5=85), ?5@5<5I5=89 (:8=5<0B8G5A:>5 2>7<CI5=85) 8;8 8=5@F8>==KE E0@0:B5@8AB8: (?0@0<5B@8G5A:>5 2>7<CI5=85). 81@0F859 (>B ;0B. vibratio ( :>;510=85) =07K20NB <5E0=8G5A:85 :>;510=8O 2 <0H8=0E 8;8 <5E0=87<0E. >;510=85 ( 42865=85 8;8 87<5=5=85 A>AB>O=8O, >1;040NI85 B>9 8;8 8=>9 AB5?5=LN ?>2B>@O5<>AB8 8;8 ?5@8>48G=>ABLN. A;8 8AB>G=8: 2>7=8:=>25=8O 281@0F89 >?@545;O5BAO 2=CB@5==8<8 A2>9AB20<8 <0H8=K 8;8 <5E0=87<0, B> 3>2>@OB >1 53> 281@>0:B82=>AB8. 'B>1K 281@0F88 <5E0=87<0 =5 @0A?@>AB@0=O;8AL =0 >:@C60NI85 53> A8AB5<K 8;8 GB>1K 70I8B8BL <5E0=87< >B 281@0F89, 2>7459AB2CNI8E =0 =53> A> AB>@>=K 2=5H=8E A8AB5<, ?@8<5=ONBAO @07;8G=K5 <5B>4K 281@>70I8BK. 07;8G0NB 2=5H=NN 8 2=CB@5==NN 281@>0:B82=>ABL. >4 2=CB@5==59 281@>0:B82=>ABLN ?>=8<0NB :>;510=8O 2>7=8:0NI85 2=CB@8 <5E0=87<0 8;8 <0H8=K, :>B>@K5 ?@>8AE>4OB ?> 53> ?>4286=>ABO< 8;8 >1>1I5==K< :>>@48=0B0<. -B8 :>;510=8O =5 >:07K20NB =5?>A@54AB25==>3> 2;8O=8O =0 >:@C60NICN A@54C. @8 2=5H=59 281@>0:B82=>AB8 87<5=5=85 ?>;>65=8O <5E0=87<0 ?@82>48B : 87<5=5=8N @50:F89 2 >?>@0E (B.5. A2O7OE <5E0=87<0 A >:@C60NI59 A@54>9) 8 =5?>A@54AB25==><C 281@0F8>==><C 2>7459AB28N =0 A2O70==K5 A =8< A8AB5<K. 4=0 8 >A=>2=KE ?@8G8= 2=5H=59 281@>0:B82=>AB8 - =5C@02=>25H5==>ABL 53> 725=L52 8 <5E0=87<0 2 F5;><. >=OB85 > =5C@02=>25H5==>AB8 <5E0=87<0 (725=0) 5C@02=>25H5==K< 1C45< =07K20BL B0:>9 <5E0=87< (8;8 53> 725=>), 2 :>B>@>< ?@8 42865=88 F5=B@ <0AA <5E0=87<0 (8;8 725=0) 42865BAO A CA:>@5=85<. "0: :0: CA:>@5==>5 42865=85 A8AB5<K 2>7=8:05B B>;L:> 2 A;CG05, 5A;8 @02=>459AB2CNI0O 2=5H=8E A8;>2KE 2>7459AB289 =5 @02=0 =C;N. !>3;0A=> ?@8=F8?C  ;0<15@0, 4;O C@02=>25H820=8O 2=5H=8E A8; : A8AB5<5 4>102;ONBAO @0AG5B=K5 A8;K - A8;K 8 <><5=BK A8; 8=5@F88. >MB><C C@02=>25H5==K< 1C45< AG8B0BL <5E0=87<, 2 :>B>@>< 3;02=K5 25:B>@0 8 <><5=BK A8; 8=5@F88 @02=K =C;N, 0 =5C@02=>25H5==K< <5E0=87<, 2 :>B>@>< MB8 A8;K =5@02=K =C;N. ;O ?@8<5@0 @0AA<>B@8< G5BK@5EH0@=8@=K9 <5E0=87< (@8A. 14.1). 5E0=87< 1C45B =0E>48BAO 2 A>AB>O=88 :8=5B>AB0B8G5A:>3> @02=>25A8O, 5A;8 AC<<0 459AB2CNI8E =0 =53> 2=5H=8E A8; 8 <><5=B>2 A8; (2:;NG0O A8;K 8 <><5=BK A8; 8=5@F88) 1C45B @02=0 =C;N.  EMBED Equation.3 ;  EMBED Equation.3 . #@02=>25H5==>ABL O2;O5BAO A2>9AB2>< 8;8 E0@0:B5@8AB8:>9 <5E0=87<0 8 =5 4>;6=0 7028A5BL >B 459AB2CNI8E =0 =53> 2=5H=8E A8;. A;8 8A:;NG8BL 87 @0AA<>B@5=8O 2A5 2=5H=85 A8;K, B> 2 C@02=5=88 @02=>25A8O >AB0=CBAO B>;L:> 8=5@F8>==K5 A>AB02;ONI85, :>B>@K5 >?@545;ONBAO 8=5@F8>==K<8 ?0@0<5B@0<8 <5E0=87<0 ( <0AA0<8 8 <><5=B0<8 8=5@F88 8 70:>=>< 42865=8O (=0?@8<5@, F5=B@0 <0AA A8AB5<K S<), ?>MB><C C@02=>25H5==K< AG8B05BAO <5E0=87< 4;O :>B>@>3> 3;02=K9 25:B>@ 8 3;02=K9 <><5=B A8; 8=5@F88 @02=K =C;N:  EMBED Equation.3 .  y 2 83 82  S2 C 3  1 B  P41 rS2 83  (1 S1 G2 S3 rS1 81 rS3 G3 (1 G1 rS< S< A3  A 83 D x 81  0 8A. 14.1 5C@02=>25H5==>ABL ( B0:>5 A>AB>O=85 <5E0=87<0, ?@8 :>B>@>< 3;02=K9 25:B>@ 8;8 3;02=K9 <><5=B A8; 8=5@F88 =5 @02=K =C;N. 07;8G0NB: AB0B8G5A:CN =5C@02=>25H5==>ABL  EMBED Equation.3 ; <><5=B=CN =5C@02=>25H5==>ABL  EMBED Equation.3 ; 48=0<8G5A:CN =5C@02=>25H5==>ABL  EMBED Equation.3 . @8 AB0B8G5A:>< C@02=>25H820=88 <5E0=87<0 =5>1E>48<> >15A?5G8BL:  EMBED Equation.3  -B> CA;>285 2K?>;=O5BAO 5A;8 A:>@>ABL F5=B@0 <0AA <5E0=87<0 @02=0 =C;N  EMBED Equation.3 , 8;8 >=0 ?>AB>O==0 ?> 25;8G8=5 8 =0?@02;5=8N EMBED Equation.3 . 15A?5G8BL 2K?>;=5=85 CA;>28O  EMBED Equation.3  2 <5E0=87<5 ?@0:B8G5A:8 =52>7<>6=>, ?>MB><C ?@8 AB0B8G5A:>< C@02=>25H820=88 >15A?5G820NB 2K?>;=5=85 CA;>28O  EMBED Equation.3 . -B> 2>7<>6=>, :>340 F5=B@ <0AA <5E0=87<0 ;568B =0 >A8 2@0I5=8O 725=0 1 ( rS<= 0 8;8 :>340 >= =5?>42865= rS<= const, B>340:  EMBED Equation.3 . 0 ?@0:B8:5 =081>;55 G0AB> AB0B8G5A:>5 C@02=>25H820=85 ?@>2>4OB A;54CNI8<8 <5B>40<8: 2K18@0O A8<<5B@8G=K5 AE5<K <5E0=87<0 (@8A. 14.2); CAB0=02;820O =0 725=LOE <5E0=87<0 ?@>B82>25AK (8;8 :>@@5:B8@CNI85 <0AAK); @07<5I0O ?@>B82>25AK =0 4>?>;=8B5;L=KE 725=LOE 8;8 :8=5<0B8G5A:8E F5?OE. 3 2 B 1 D,C A,S1,S< E,Q  0 K 4 5 8A. 14.2 5B>4 70<5I0NI8E <0AA mi, JSi A Si B 25=> A @0A?@545;5==>9 <0AA>9  A miA Si miB  B >45;L A B>G5G=K<8  <0AA0<8 lASi miSi lAB  8A. 14.3 @8 8A?>;L7>20=88 <5B>40 70<5I0NI8E <0AA, 725=> <5E0=87<0 A @0A?@545;5==>9 <0AA>9 70<5=O5BAO @0AG5B=>9 <>45;LN, :>B>@0O A>AB>8B 87 B>G5G=KE <0AA. #A;>28O<8 ?5@5E>40 >B 725=0 A @0A?@545;5==>9 <0AA>9 : <>45;8 A B>G5G=K<8 <0AA0<8 O2;ONBAO: !>E@0=5=85 <0AAK 725=0: miA + miB = mi; !>E@0=5=85 ?>;>65=8O F5=B@0 <0AA: lASi = const;  EMBED Equation.3 . !>E@0=5=85 <><5=B0 8=5@F88:  EMBED Equation.3 . 4=>2@5<Q==>5 2K?>;=5=85 2A5E B@QE CA;>289 A8AB5<>9 A 42C<O <0AA0<8 =52>7<>6=>, ?>MB><C ?@8 AB0B8G5A:>< C@02=>25H820=88 <5E0=87<>2 >3@0=8G820NBAO 2K?>;=5=85< B>;L:> 42CE ?5@2KE CA;>289. 'B>1K >15A?5G8BL 2K?>;=5=85 2A5E B@5E CA;>289 =5>1E>48<> 225AB8 B@5BLN <0AAC miSi. 0AA<>B@8< ?@8<5=5=85 <5B>40 70<5I0NI8E <0AA ?@8 ?>;=>< 8 G0AB8G=>< AB0B8G5A:>< C@02=>25H820=88 :@82>H8?=>-?>;7C==>3> <5E0=87<0. >;=>5 AB0B8G5A:>5 C@02=>25H820=85 :@82>H8?=>- ?>;7C==>3> <5E0=87<0 (@8A. 14.4) 0=>: lAB, lBC, lAS1, lBS2, lCS3=0, m1, m2, m3. ?@545;8BL: mk1, mk2.  mk2 lk2  mB 2, l2 3  1, l1 B  C, S3  S1 S2 A mA mC   mk1 lk1 0 8A. 14.4 ;O >?@545;5=8O 25;8G8=K A>A@54>B>G5==KE <0AA @0A?@545;8< <0AAK 725=L52 ?> <5B>4C 70<5I0NI8E <0AA, A>A@54>B>G82 8E 2 F5=B@0E H0@=8@>2 A,B,C. ">340  EMBED Equation.3  m1 = mA1 + mB1 ( <0AA0 ?5@2>3> 725=0, @0A?@545;5==0O <564C <0AA0<8, A>A@54>B>G5==K<8 2 B>G:0E ; m2 = m2 + m!2 ( <0AA0 2B>@>3> 725=0, @0A?@545;5==0O <564C <0AA0<8, A>A@54>B>G5==K<8 2 B>G:0E  8 ! . =0G0;5 ?@>2545< C@02=>25H820=85 <0AAK mC :>@@5:B8@CNI59 <0AA>9 mk2. !>AB028< C@02=5=85 AB0B8G5A:8E <><5=B>2 >B=>A8B5;L=> B>G:8  4;O 725=L52 2 8 3:  EMBED Equation.3 . 0405<AO 25;8G8=>9 lk2 8 ?>;CG05< :>@@5:B8@CNICN <0AAC:  EMBED Equation.3 . 0B5< C@02=>25H8205< <0AAK F5=B@0, :>B>@K9 ?>A;5 CAB0=>2:8 :>@@5:B8@CNI59 <0AAK @0A?>;>68;AO 2 B>G:5 :  EMBED Equation.3 . !>AB02;O5< C@02=5=85 AB0B8G5A:8E <><5=B>2 >B=>A8B5;L=> B>G:8 :  EMBED Equation.3 . 0405<AO 25;8G8=>9 lk1 8 ?>;CG05< :>@@5:B8@CNICN <0AAC:  EMBED Equation.3 . '0AB8G=>5 AB0B8G5A:>5 C@02=>25H820=85 :@82>H8?=>-?>;7C==>3> <5E0=87<0  mB 2, l2 3  1, l1 B  C,S3   S1 S2  mA mC rS< S<  mk1 lk1 0 8A. 14.5 1. #@02=>25H820=85 25@B8:0;L=>9 A>AB02;ONI59 3;02=>3> 25:B>@0 A8; 8=5@F88 (@8A. 14.5). 0=>: lAB, lBC, lAS1, lBS2, lCS3=0, m1, m2, m3. ?@545;8BL: mk1  MB>< A;CG05 =5>1E>48<> 4>18BLAO, GB>1K F5=B@ <0AA <5E0=87<0 S< ?@8 42865=88 ?5@5<5I0;AO 24>;L =0?@02;ONI59 ?>;7C=0 (4;O AE5<K =0 @8A. 14.5 ?> 3>@87>=B0;8). ;O MB>3> 4>AB0B>G=> C@02=>25A8BL B>;L:> <0AAC mB. !>AB02;O5< C@02=5=85 AB0B8G5A:8E <><5=B>2 >B=>A8B5;L=> B>G:8 :  EMBED Equation.3 . 0405<AO 25;8G8=>9 lk1 8 ?>;CG05< :>@@5:B8@CNICN <0AAC:  EMBED Equation.3 . 2. #@02=>25H820=85 3>@87>=B0;L=>9 A>AB02;ONI59 3;02=>3> 25:B>@0 A8; 8=5@F88.  mB 2, l2 3  1, l1 B  C,S3   S1 S2 mA rS<** mC  mk1* S<**  Sk y x 0 lk1 8A. 14.6 0=>: lAB, lBC, lAS1, lBS2, lCS3=0, m1, m2, m3. ?@545;8BL: mk1  MB>< A;CG05 =5>1E>48<> 4>18BLAO, GB>1K F5=B@ <0AA <5E0=87<0 ?@8 42865=88 ?5@5<5I0;AO ?> 4C35 >:@C6=>AB8 @048CA0  EMBED Equation.3  (@8A. 14.6). 0AG5B :>@@5:B8@CNI59 <0AAK 2545BAO 2 420 MB0?0.  =0G0;5 ?5@2>9 A>AB02;ONI59 :>@@5:B8@CNI59 <0AAK  EMBED Equation.3  C@02=>25H8205BAO <0AA0 mB. !>AB02;O5BAO, :0: 8 2 ?@54K4CI5< ?@8<5@5, C@02=5=85 AB0B8G5A:8E <><5=B>2 >B=>A8B5;L=> B>G:8 :  EMBED Equation.3 . 0405BAO 25;8G8=0 lk1 8 @0AAG8BK205BAO :>@@5:B8@CNI0O <0AA0:  EMBED Equation.3 . 0B5< A ?><>ILN 2B>@>9 A>AB02;ONI59 :>@@5:B8@CNI59 <0AAK  EMBED Equation.3  F5=B@ <0AAK m!. ?5@5<5I05BAO 2 B>G:C  EMBED Equation.3 . 5;8G8=0  EMBED Equation.3  >?@545;O5BAO A;54CNI8< >1@07><: F5=B@ H0@=8@0 ! A>548=O5BAO ?@O<>9 A :>=F>< >B@57:0 lk1 B>G:>9 Sk. 048CA  EMBED Equation.3  ?@>2>48BAO ?0@0;;5;L=> >B@57:C B!. ">340,  EMBED Equation.3  ?>4>15=  EMBED Equation.3  8  EMBED Equation.3 . !B0B8G5A:89 <><5=B >B=>A8B5;L=> B>G:8  EMBED Equation.3 :  EMBED Equation.3 ,  EMBED Equation.3 . 048CA-25:B>@  EMBED Equation.3  >?@545;O5BAO 87 ?@>?>@F89 A>>B25BAB2CNI8E AB>@>= B@5C3>;L=8:>2:  EMBED Equation.3 ,  EMBED Equation.3 , >B:C40  EMBED Equation.3 . >@@5:B8@CNI0O <0AA0, >15A?5G820NI0O C@02=>25H820=85 3>@87>=B0;L=>9 A>AB02;ONI59 3;02=>3> 25:B>@0 A8; 8=5@F88 :@82>H8?=>-?>;7C==>3> <5E0=87<0, @07<5I05BAO =0 ?5@2>< 725=5 <5E0=87<0 8 @02=0 AC<<5 A>AB02;ONI8E:  EMBED Equation.3 . &5=B@ <0AAK <5E0=87<0 ?@8 B0:>< C@02=>25H820=88 @0A?>;>65= 2 B>G:5 S<, :>B>@0O 42865BAO ?> 4C35 @048CA0 rS<:  EMBED Equation.3 . (m1 + mC2 +  EMBED Equation.3 ) A S< (m3 + mC2 + EMBED Equation.3 ) rS<  EMBED Equation.3   EMBED Equation.3  8A. 14.7 !E5<0 @0A?@545;5=8O <0AA 2 <5E0=87<5 ?>A;5 C@02=>25H820=8O 40=0 =0 @8A. 14.7. 0;0=A8@>2:0 @>B>@>2 1I85 A2545=8O > 10;0=A8@>2:5. >B>@, =5C@02=>25H5==>ABL @>B>@0 8 55 284K. 040G8 10;0=A8@>2:8 >B>@>< =07K20NB 725=LO <5E0=87<>2, A>25@H0NI85 2@0I0B5;L=>5 42865=85 8 C45@68205<K5 ?@8 MB>< A2>8<8 =5ACI8<8 ?>25@E=>ABO<8 2 >?>@0E. A;8 <0AA0 @>B>@0 @0A?@545;5=0 >B=>A8B5;L=> >A8 2@0I5=8O @02=><5@=>, B> 3;02=0O F5=B@0;L=0O >AL 8=5@F88 x-x A>2?0405B A >ALN 2@0I5=8O 8 @>B>@ O2;O5BAO C@02=>25H5==K< 8;8 8450;L=K<. @8 =5A>2?045=88 >A8 2@0I5=8O A >ALN x-x, @>B>@ 1C45B =5C@02=>25H5==K< 8 2 53> >?>@0E ?@8 2@0I5=88 2>7=8:=CB ?5@5<5==K5 @50:F88, 2K720==K5 459AB285< 8=5@F8>==KE A8; 8 <><5=B>2 A8; (B>G=55, 42865=85< F5=B@0 <0AA A CA:>@5=85<).  7028A8<>AB8 >B 2708<=>3> @0A?>;>65=8O >A8 2@0I5=8O 8 3;02=>9 F5=B@0;L=>9 >A8 8=5@F88 x-x , @07;8G0NB A;54CNI85 284K =5C@02=>25H5==>AB8 @>B>@>2 (@8A. 14.8): ( AB0B8G5A:CN, :>340 MB8 >A8 ?0@0;;5;L=K (@8A. 14. 8, 0); ( <><5=B=CN, :>340 >A8 ?5@5A5:0NBAO 2 F5=B@5 <0AA @>B>@0 S (@8A. 14. 8, 1); ( 48=0<8G5A:CN, :>340 >A8 ;81> ?5@5A5:0NBAO 2=5 F5=B@0 <0AA, ;81> =5 ?5@5A5:0NBAO, 0 ?5@5:@5I820NBAO 2 ?@>AB@0=AB25 (@8A. 14. 8, 2).  0) e 1) 2) e  x x 8   x S x S S x x 8 8 8A. 14.8 5C@02=>25H5==>ABL >?@545;O5BAO :>=AB@C:B82=K<8 E0@0:B5@8AB8:0<8 @>B>@0 8;8 <5E0=87<0 8 =5 7028A8B >B ?0@0<5B@>2 42865=8O. >MB><C ?@8 10;0=A8@>2:5 >?5@8@CNB =5 8=5@F8>==K<8 A8;0<8, 0 48A10;0=A0<8. 8A10;0=A ( <5@0 AB0B8G5A:>9 =5C@02=>25H5==>AB8 @>B>@0, 25:B>@=0O 25;8G8=0, @02=0O ?@>872545=8N =5C@02=>25H5==>9 <0AAK m =0 55 M:AF5=B@8A8B5B e, 345 M:AF5=B@8A8B5B e ( @048CA-25:B>@ F5=B@0 MB>9 <0AAK >B=>A8B5;L=> >A8 @>B>@0. 0?@02;5=85 3;02=>3> 25:B>@0 48A10;0=A0 D A>2?0405B A =0?@02;5=85< 3;02=>3> 25:B>@0 A8; 8=5@F88 8, 459AB2CNI8E =0 @>B>@ ?@8 2@0I5=88:  EMBED Equation.3 . ><5=B=0O =5C@02=>25H5==>ABL E0@0:B5@87C5BAO 3;02=K< <><5=B>< 48A10;0=A>2 @>B>@0 MD , :>B>@K9 ?@>?>@F8>=0;5= 3;02=><C <><5=BC A8; 8=5@F88 (@8A. 14.9):  EMBED Equation.3 . ;02=K9 <><5=B 48A10;0=A>2 @>B>@0 ?>;=>ABLN >?@545;O5BAO <><5=B>< ?0@K @02=KE ?> 25;8G8=5 8 ?@>B82>?>;>6=KE ?> =0?@02;5=8N 48A10;0=A>2 DM1 + DM2 = DM, @0A?>;>65==KE 2 42CE ?@>872>;L=KE ?;>A:>ABOE (I 8 II), ?5@?5=48:C;O@=KE >A8 2@0I5=8O @>B>@0. 8A10;0=A 8 <><5=B 48A10;0=A>2 =5 7028AOB >B G0AB>BK 2@0I5=8O, >=8 ?>;=>ABLN >?@545;ONBAO :>=AB@C:F859 @>B>@0 8 B>G=>ABLN 53> 873>B>2;5=8O. 0;0=A8@>2:>9 =07K20NB ?@>F5AA >?@545;5=8O 7=0G5=89 8 C3;>2KE :>>@48=0B 48A10;0=A>2 @>B>@0 8 8E C<5=LH5=8O A ?><>ILN :>@@5:B8@>2:8 @07<5I5=8O 53> <0AA. 0;0=A8@>2:0 M:2820;5=B=0 C@02=>25H820=8N A8AB5<K 8=5@F8>==KE A8;, ?@8:;04K205<KE : ?>4286=><C @>B>@C 4;O 53> @02=>25A8O.  D z ( D2 x  MD S Dc1 Dc2 D1 ek1 0 y DM2  DM1 mk1 e II x x Dk1 ek2 l I mk2 Dk2 8A. 14.9 0==CN A8AB5<C, :0: 8 ;N1CN ?@>872>;L=CN A8AB5<C A8;, <>6=> 70<5=8BL @02=>459AB2CNI8<8 ( 3;02=K< 25:B>@>< 8 3;02=K< <><5=B>< 8;8 42C<O 25:B>@0<8, @0A?>;>65==K<8 2 ?@>872>;L=KE ?0@0;;5;L=KE ?;>A:>ABOE. ;O C@02=>25H820=8O A8AB5<K A8; 4>AB0B>G=> C@02=>25A8BL MB8 @02=>459AB2CNI85. @8 10;0=A8@>2:5 >?5@0F88 =04 A8;0<8 70<5=ONB 459AB28O<8 =04 48A10;0=A0<8. >MB><C 4;O 65AB:8E @>B>@>2 2KH5A:070==>5 <>6=> AD>@<C;8@>20BL B0:: 65AB:89 @>B>@ <>6=> C@02=>25A8BL 42C<O :>@@5:B8@CNI8<8 <0AA0<8, @0A?>;>65==K<8 2 42CE ?@>872>;L=> 2K1@0==KE ?;>A:>ABOE, ?5@?5=48:C;O@=KE >A8 53> 2@0I5=8O. -B8 ?;>A:>AB8 =07K20NB ?;>A:>ABO<8 :>@@5:F88. 040G0 10;0=A8@>2:8 @>B>@0 70:;NG05BAO 2 >?@545;5=88, 2 2K1@0==KE ?;>A:>ABOE :>@@5:F88, 7=0G5=89 8 C3;>2 48A10;0=A>2 8 @07<5I5=88 2 MB8E ?;>A:>ABOE :>@@5:B8@CNI8E <0AA, 48A10;0=AK :>B>@KE @02=K ?> 25;8G8=5 8 ?@>B82>?>;>6=K ?> =0?@02;5=8N =0945==K< 48A10;0=A0< @>B>@0. 0 ?@0:B8:5 10;0=A8@>2:C ?@>2>4OB: ?@8 :>=AB@C8@>20=88 ( @0AG5B=K<8 <5B>40<8, 2 ?@>F5AA5 873>B>2;5=8O 45B0;59 8 C7;>2 ( M:A?5@8<5=B0;L=> =0 A?5F80;L=KE 10;0=A8@>2>G=KE AB0=:0E. 0;0=A8@>2:0 =0 AB0=:0E O2;O5BAO 1>;55 B>G=K< 8 =0456=K< <5B>4><, ?> A@02=5=8N A @0AG5B=K<8. >MB><C >=0 ?@8<5=O5BAO 4;O >B25BAB25==KE 45B0;59 A 2KA>:8<8 @01>G8<8 G0AB>B0<8 2@0I5=8O. >@@5:B8@>2:0 <0AA @>B>@0 >ACI5AB2;O5BAO ;81> ?@8A>548=5=85< : =5<C 4>?>;=8B5;L=KE :>@@5:B8@CNI8E <0AA (=0?;02;5=85<, =020@:>9 8;8 ?@828=G820=85< ?@>B82>25A>2), ;81> C40;5=85< G0AB8 <0AAK @>B>@0 A BO65;>9 AB>@>=K (D@575@>20=85< 8;8 2KA25@;820=85<). ">G=>ABL 10;0=A8@>2:8 E0@0:B5@87C5BAO 25;8G8=>9 >AB0B>G=>3> 48A10;0=A0 D0 @>B>@0 2 :064>9 87 ?;>A:>AB59 :>@@5:F88. 5;8G8=0 D0 =5 4>;6=0 ?@52KH0BL 4>?CAB8<KE 4;O 40==>3> :;0AA0 B>G=>AB8 7=0G5=89, @53;0<5=B8@C5<KE !". 0;0=A8@>2:0 @>B>@>2 ?@8 @07;8G=KE 2840E =5C@02=>25H5==>AB8 !B0B8G5A:0O =5C@02=>25H5==>ABL Dk  5 mk  ek  x S x Dc m 8A. 14.10 @8 AB0B8G5A:>9 =5C@02=>25H5==>AB8 (@8A. 14.10) 3;02=0O F5=B@0;L=0O >AL 8=5@F88 ?0@0;;5;L=0 >A8 2@0I5=8O @>B>@0, 3;02=K9 25:B>@ 48A10;0=A>2 1>;LH5 =C;O, 0 3;02=K9 <><5=B 48A10;0=A>2 @025= =C;N ( EMBED Equation.3 ), B.5. =5>1E>48<> C@02=>25A8BL B>;L:> 25:B>@ DA = m ( e. ;O MB>3> 4>AB0B>G=> CAB0=>28BL =0 @>B>@5 B>;L:> >4=C :>@@5:B8@CNICN <0AAC mk, 25;8G8=0 :>B>@>9 >?@545;O5BAO 87 @025=AB20:  EMBED Equation.3 , 345 25;8G8=>9 ek 7040NBAO 87 A>>1@065=89 C4>1AB20 @07<5I5=8O ?@>B82>25A>2. 0?@02;5=85 25:B>@0 Dk ?@>B82>?>;>6=> =0?@02;5=8N Dc. "0:8< >1@07><, CA;>285 AB0B8G5A:>9 C@02=>25H5==>AB8 @>B>@0 70:;NG05BAO 2  EMBED Equation.3 . ><5=B=0O =5C@02=>25H5==>ABL Dk MDk m mk  ek x D  S x ek  lk lk mk   l Dk  8A. 14.11 @8 <><5=B=>9 =5C@02=>25H5==>AB8 (@8A. 14.11) 3;02=0O F5=B@0;L=0O >AL 8=5@F88 ?5@5A5:05B >AL 2@0I5=8O 2 F5=B@5 <0AA @>B>@0 B>G:5 S, 3;02=K9 25:B>@ 48A10;0=A>2 DA @025= =C;N, 3;02=K9 <><5=B 48A10;0=A>2 D =5 @025= =C;N ( EMBED Equation.3 ), B.5. =5>1E>48<> C@02=>25A8BL B>;L:> <><5=B 48A10;0=A>2 D. ;O MB>3> 4>AB0B>G=> @07<5AB8BL =0 @>B>@5 425 >48=0:>2KE :>@@5:B8@CNI8E <0AAK mk =0 @02=KE @0AAB>O=8OE >B >A8 2@0I5=8O ek 8 >B F5=B@0 <0AA S ( lk. 0AAK 2K18@0NBAO 8 @07<5I0NBAO B0:, GB>1K <><5=B 8E 48A10;0=A>2 MDk 1K; ?> 25;8G8=5 @025=, 0 ?> =0?@02;5=8N ?@>B82>?>;>65= <><5=BC 48A10;0=A>2 @>B>@0 D:  EMBED Equation.3 , 345 Dk  48A10;0=A :>@@5:B8@CNI59 <0AAK,  EMBED Equation.3 .  MB8E 7028A8<>ABOE 25;8G8=0<8 lk 8 ek 7040NBAO ?> CA;>28O< C4>1AB20 @07<5I5=8O ?@>B82>25A>2 =0 @>B>@5, 0 25;8G8=C mk @0AAG8BK20NB. 5>1E>48<> >B<5B8BL, GB> 25;8G8=K Dk 2 ?;>A:>ABOE :>@@5:F88 =5>1O70B5;L=> 4>;6=K 1KBL @02=K<8, =5>1E>48<> 2K?>;=OBL B>;L:> =587<5==>ABL ?>;>65=8O F5=B@0 <0AA - >= 4>;65= >AB020BLAO =0 >A8 2@0I5=8O. "0:8< >1@07><, CA;>285 <><5=B=>9 C@02=>25H5==>AB8 @>B>@0 70:;NG05BAO 2  EMBED Equation.3 . 8=0<8G5A:0O =5C@02=>25H5==>ABL Dk1 MDk m  mk1  ek1 D x  e S  ek2  x Dc lk1 lk2 mk2   l Dk2  8A. 14.12 @8 48=0<8G5A:>9 =5C@02=>25H5==>AB8 (@8A. 14.12) 3;02=0O F5=B@0;L=0O >AL 8=5@F88 ?5@5A5:05B >AL 2@0I5=8O =5 2 F5=B@5 <0AA @>B>@0 B>G:5 S, ;81> ?5@5:@5I8205BAO A =59; 8 3;02=K9 25:B>@ 48A10;0=A>2 DA, 8 3;02=K9 <><5=B 48A10;0=A>2 D =5 @02=K =C;N ( EMBED Equation.3 ), B.5. =5>1E>48<> C@02=>25A8BL 25:B>@ DA 8 <><5=B 48A10;0=A>2 D . ;O MB>3> 4>AB0B>G=> @07<5AB8BL =0 @>B>@5 425 :>@@5:B8@CNI8E <0AAK mk1 8 mk2 =0 @0AAB>O=8OE >B >A8 2@0I5=8O ek1 8 ek2, 0 >B F5=@0 <0AA S, A>>B25BAB25==> =0 lk1 8 lk2. 0AAK 2K18@0NBAO 8 @07<5I0NBAO B0:, GB>1K <><5=B 8E 48A10;0=A>2 MDk 1K; ?> 25;8G8=5 @025=, 0 ?> =0?@02;5=8N ?@>B82>?>;>65= <><5=BC 48A10;0=A>2 @>B>@0 D:  EMBED Equation.3 , 345 Dk1 8 Dk2  48A10;0=AK :>@@5:B8@CNI8E <0AA,  EMBED Equation.3  8  EMBED Equation.3 . 5:B>@=0O AC<<0 48A10;0=A>2 ?@8 MB>< 4>;6=0 1KBL @02=0 8 ?@>B82>?>;>6=> =0?@02;5=0 25:B>@C DA:  EMBED Equation.3 .  MB8E 7028A8<>ABOE 25;8G8=0<8 lki 8 eki 7040NBAO ?> CA;>28O< C4>1AB20 @07<5I5=8O ?@>B82>25A>2 =0 @>B>@5, 0 25;8G8=K mki @0AAG8BK20NB. "0:8< >1@07><, CA;>285 48=0<8G5A:>9 C@02=>25H5==>AB8 @>B>@0 70:;NG05BAO 2  EMBED Equation.3 . >=B@>;L=K5 2>?@>AK 81@0F88 8 :>;510=8O 2 <0H8=0E 8 <5E0=87<0E. 'B> B0:>5 =5C@02=>25H5==>ABL <5E0=87<0, 5Q @07=>284=>AB8. 5B>4 70<5I0NI8E <0AA. >;=>5 AB0B8G5A:>5 C@02=>25H820=85 :@82>H8?=>-?>;7C==>3> <5E0=87<0 <5B>4>< 70<5I0NI8E <0AA. '0AB8G=>5 AB0B8G5A:>5 C@02=>25H820=85 :@82>H8?=>-?>;7C==>3> <5E0=87<0 <5B>4>< 70<5I0NI8E <0AA. 0;0=A8@>2:0 @>B>@>2. 0: @5H0NBAO 7040G8 C@02=>25H820=8O ?@8 AB0B8G5A:>9, <><5=B=>9 8 48=0<8G5A:>9 =5C@02=>25H5==>AB8. 5:F8O 15 #@02=>25H820=85 @>B>@>2 ?@8 ?@>5:B8@>20=88. 81@>70I8B0 <0H8= 8 <5E0=87<>2. 5B>4K 281@>70I8BK. >4@5AA>@820=85 8 281@>87>;OF8O. 8=0<8G5A:>5 30H5=85 :>;510=89. #@02=>25H820=85 @>B>@>2 ?@8 ?@>5:B8@>20=88 !B0B8G5A:>5 C@02=>25H820=85 ?@8 ?@>5:B8@>20=88 @8 ?@>5:B8@>20=88 AB0B8G5A:8 C@02=>25H820NB 45B0;8, 8<5NI85 =51>;LH85 >A52K5 @07<5@K 8 :>=AB@C:B82=> =5C@02=>25H5==K5, =0?@8<5@, 48A:>2K5 :C;0G:8 (@8A. 15.1). >340 :C;0G>: =5?>42865= (1 = 0, @50:F8O 2 >?>@5 R10 = ( G. @8 2@0I5=88 :C;0G:0 (1 ( 0, @50:F8O 2 >?>@5 @02=0 25:B>@=>9 AC<<5 A8; BO65AB8 8 F5=B@>156=>9 A8;K 8=5@F88 R10 = ( (G + P8), 345 P8 = ( m1( e1 ((12. @8 ?@>5:B8@>20=88 45B0;8 B8?0 :C;0G:0 C@02=>25H820=85 ?@>872>48BAO B0:: 2 45B0;L A F5=B@>< =0 >A8 2@0I5=8O 2?8AK205BAO >:@C6=>ABL, ?>4AG8BK20NBAO ?;>I048 >3@0=8G5==K5 :>=BC@>< :C;0G:0 8 @0A?>;>65==K5 2=5 8;8 2=CB@8 >:@C6=>AB8, >?@545;O5BAO <0AAK 8 F5=B@K <0AA Sn =5C@02=>25H5==KE G0AB59 :C;0G:0, =0E>48BAO M:AF5=B@8A8B5B e1 F5=B@0 <0AA S1 :C;0G:0 ?> 25;8G8=5 8 =0?@02;5=8N 8 >?@545;O5BAO 53> 48A10;0=A D1 = m1( e1, A ?><>ILN :>@@5:B8@CNI59 <0AAK mk, @07<5I05<>9 =0 M:AF5=B@8A8B5B5 ek, A>7405BAO 48A10;0=A Dk @02=K9 ?> 25;8G8=5 8 ?@>B82>?>;>6=K9 ?> =0?@02;5=8N D1.   1 1 8  (1 Sn R e S A ek A  R10 G 0 0 mk P8k 8A. 15.1 8=0<8G5A:>5 C@02=>25H820=85 ?@8 ?@>5:B8@>20=88 8=0<8G5A:>5 C@02=>25H820=85 ?@8 ?@>5:B8@>20=88 ?@>2>4OB A 45B0;O<8 8 C7;0<8, 2 :>B>@KE <0AAK @0A?@545;5=K >B=>A8B5;L=> >A8 2@0I5=8O =5@02=><5@=>, =0?@8<5@, 45B0;8 B8?0 :>;5=G0B>3> 20;0. -B8 45B0;8 45;OB =0 =5A:>;L:> 48A:>2 8 2 :064>< 48A:5, B0:65 :0: ?@8 AB0B8G5A:>< C@02=>25H820=88, >?@545;ONB 25;8G8=C 8 =0?@02;5=85 48A10;0=A0 Di. 0 45B0;8 2K18@0NB 425 ?;>A:>AB8 :>@@5:F88 8 :064K9 25:B>@ 48A10;0=A0 @0A:;04K20NB =0 425 A>AB02;ONI85, @0A?>;>65==K5 2 ?;>A:>ABOE :>@@5:F88. 0B5< A>AB02;ONI85 25:B>@K 48A10;0=A>2 2 ?;>A:>ABOE :>@@5:F88 AC<<8@CNBAO 8 8E @02=>459AB2CNI89 48A10;0=A, =0?@8<5@, DI, C@02=>25H8205BAO A>>B25BAB2CNI59 :>@@5:B8@CNI59 <0AA>9 mIk. @8<5@ B0:>3> C@02=>25H820=8O 87>1@065= =0 @8A. 15.2.  D1 I II m1 D11 D11 D31 D12 DI e1 DII  pD DI D21 D21 D31 D22 D32  e3 m3  l1 m2  D3 D2 l2 D12 D32 l3 pD  l DII D22  8A. 15.2. !E5<0 >?@545;5=8O 25:B>@>2 @02=>459AB2CNI8E 48A10;0=A>2 !E5<0 @07<5I5=8O :>@@5:B8@CNI8E <0AA 2 ?;>A:>ABOE :>@@5:F88 ?@82545=0 =0 @8A. 15.3.  I II DII DI eII 0I 0II   eI DkI mkII mkI DkII 8A. 15.3 81@>70I8B0 <0H8= 8 <5E0=87<>2 @8 42865=88 <5E0=8G5A:>9 A8AB5<K ?>4 459AB285< 2=5H=8E A8; 2 =59 2>7=8:0NB <5E0=8G5A:85 :>;510=8O 8;8 281@0F88. -B8 281@0F88 >:07K20NB 2;8O=85 =0 DC=:F8>=8@>20=85 <5E0=87<0 8 G0AB> CEC4H0NB 53> M:A?;C0B0F8>==K5 E0@0:B5@8AB8:8: A=860NB B>G=>ABL, C<5=LH0NB  8 4>;3>25G=>ABL <0H8=K, C25;8G820NB =03@52 45B0;59, A=860NB 8E ?@>G=>ABL, >:07K20NB 2@54=>5 2>7459AB285 =0 G5;>25:0->?5@0B>@0. ;O A=865=8O 2;8O=8O 281@0F89 8A?>;L7CNB @07;8G=K5 <5B>4K 1>@L1K A 281@0F859. ! >4=>9 AB>@>=K ?@8 ?@>5:B8@>20=88 <0H8=K ?@8=8<0NB <5@K 4;O A=865=8O 55 281@>0:B82=>AB8 (C@02=>25H820=85 8 10;0=A8@>2:0 <5E0=87<>2), A 4@C3>9 ( ?@54CA<0B@820NBAO A@54AB20 70I8BK :0: <0H8=K >B 281@0F89, 8AE>4OI8E >B 4@C38E <0H8= (4;O @0AA<0B@8205<>9 <0H8=K >B A@54K), B0: A@54K 8 >?5@0B>@>2 >B 281@0F89 40==>9 <0H8=K. !CI5AB2C5B 420 A?>A>10 281@>70I8BK: 281@>30H5=85 8 281@>87>;OF8O. 81@>30H5=85 >A=>20=> =0 ?@8A>548=5=88 : :>;51;NI8<AO 725=LO< <0H8=K 4>?>;=8B5;L=KE :>;510B5;L=KE A8AB5< (48=0<8G5A:8E 281@>30A8B5;59). 81@>87>;OF8O >A=>20=0 =0 @0745;5=88 8AE>4=>9 A8AB5<K =0 425 G0AB8 8 A>548=5=8O MB8E G0AB59 ?>A@54AB2>< 281@>87>;OB>@>2 8;8 0<>@B870B>@>2. 8=0<8G5A:85 30A8B5;8 8;8 0=B8281@0B>@K, >A=>20=K =0 87<5=5=88 A>>B=>H5=8O <564C A>1AB25==K<8 G0AB>B0<8 A8AB5<K 8 G0AB>B0<8 2>7<CI0NI8E A8; 4;O CAB@0=5=8O @57>=0=A=KE :>;510=89. 81@>87>;OB>@K, 2 :>B>@KE 70 AG5B 8E C?@C38E 8 45<?D8@CNI8E A2>9AB2 C<5=LH05BAO 0<?;8BC40 :>;510=89 :0: =0 @57>=0=A=KE 8 =5@57>=0=A=KE @568<0E. 708<>459AB285 42CE ?>4286=KE 725=L52 0AA<>B@8< <5E0=8G5A:CN A8AB5<C (@8A. 15.4), A>AB>OICN 87 42CE ?>4286=KE 725=L52, >1@07CNI8E <564C A>1>9 :8=5<0B8G5A:CN ?0@C. ;O C?@>I5=8O ?@54?>;>68<, GB> 42865=85 725=L52 2>7<>6=> B>;L:> ?> >4=>9 :>>@48=0B5 x. 0AA0 ?5@2>3> 725=0 m1, 2B>@>3> ( m2. 0 725=> 2 459AB2C5B ?5@8>48G5A:0O 2=5H=OO A8;0 2 = R20( sin (t , 459AB285< A8; 25A0 ?@5=51@5305<. #@02=5=8O 42865=8O 725=L52 8<5NB 284:  EMBED Equation.3 . A;8 AG8B0BL, GB> :>=B0:B <564C 725=LO<8 2 ?@>F5AA5 42865=8O =5 =0@CH05BAO 8 B5;0 01A>;NB=> 65AB:85, B>  EMBED Equation.3 . ! CG5B>< R21 = ( R12, >?@545;8< @50:F8N 2 B>G:5 :>=B0:B0 <564C 725=LO<8:  1 m1    K x1 2 m2    x2  2 = R20( sin (t 8A. 15.4  EMBED Equation.3 . B:C40  EMBED Equation.3 , 8 ?>A;5 ?@5>1@07>20=89 ?>;CG05<:  EMBED Equation.3 . @>0=0;878@C5< MBC 7028A8<>ABL: 5A;8 m1 ( 0, B> R21 ( 0; 5A;8 m2 ( 0, B> R21 ( ( P2; 5A;8 m2 = m1 = m, B> R21 ( ( 0,5 ( 2 ; 5A;8 m2 ( (, B> R21 ( 0; 5A;8 m1 ( (, B> R21 ( ( P2. =0;87 ?>:07K205B, GB> @50:F8O 2708<>459AB28O <564C 725=LO<8 7028A8B >B A>>B=>H5=8O 8E <0AA 8 25;8G8=K 2=5H=59 A8;K. @8 MB>< :8=5B8G5A:0O M=5@38O A8AB5<K A>AB02;O5B:  EMBED Equation.3 , 0 ?>B5=F80;L=0O @02=0 =C;N. >4@5AA>@820=85 8;8 281@>87>;OF8O @8 281@>87>;OF88 <564C @0AA<0B@8205<K<8 725=LO<8 CAB0=02;820NB ;8=59=K9 8;8 =5;8=59=K9 281@>87>;OB>@, :>B>@K9 >1KG=> A>AB>8B 87 C?@C3>3> 8 45<?D8@CNI53> M;5<5=B>2 (@8A. 15.5).  1 m1    x1   c k  2 m2     x2 2 = R20( sin (t 8A. 15.5  MB>9 <5E0=8G5A:>9 A8AB5<5 x2 ( x1 (?@54?>;>68<, GB> x2 > x1) 8 (x = x2 - x1, B>340 :8=5B8G5A:0O M=5@38O A8AB5<K  EMBED Equation.3 , 0 ?>B5=F80;L=0O  EMBED Equation.3 . "> 5ABL 2 A8AB5<5 A 281@>87>;OB>@>< B>;L:> G0ABL @01>BK 2=5H=59 A8;K @0AE>4C5BAO =0 87<5=5=85 :8=5B8G5A:>9 M=5@388. '0ABL MB>9 @01>BK ?5@5E>48B 2 ?>B5=F80;L=CN M=5@38N C?@C3>3> M;5<5=B0, 8 G0ABL @0AA58205BAO 45<?D5@>< (?5@5E>48B 2 B5?;> 8 @0AA58205BAO 2 >:@C60NI59 A@545). #@02=5=8O 42865=8O:  EMBED Equation.3 ,  EMBED Equation.3 . 5H5=85 MB>9 A8AB5<K C@02=5=89 ?>4@>1=> @0AA<0B@8205BAO 2 :C@A5 B5>@88 :>;510=89, ?>MB><C >3@0=8G8<AO B>;L:> 0=0;87>< 0<?;8BC4=>-G0AB>B=>9 E0@0:B5@8AB8:8 (@8A. 15.6). %0@0:B5@8AB8:C ?>AB@>8< 2 >B=>A8B5;L=KE :>>@48=0B0E (x>B= = x/xAB , 345 xAB - AB0B8G5A:0O 45D>@<0F8O C?@C3>3> M;5<5=B0. k2 > k1 k1 (x>B= k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quation.3 , 345  EMBED Equation.3  ( A>1AB25==0O G0AB>B0 30A8B5;O, m3 ( <0AA0 30A8B5;O, A3 ( 65AB:>ABL ?@C68=K 30A8B5;O. #@02=5=8O 42865=8O A8AB5<K A 48=0<8G5A:8< 30A8B5;5<, AE5<0 :>B>@>3> 87>1@065=0 =0 @8A. 15.7 8<5NB 284:    0 A 2 = R20( sin (t 1    x1 m1 c3 k3   3 m3    x3 8A. 15. 7  EMBED Equation.3 ,  EMBED Equation.3 , 345 (x = x - x3 - 45D>@<0F8O ?@C68=K 30A8B5;O. 0 @8A. 15.8 ?@82545=K 0<?;8BC4=>-G0AB>B=K5 E0@0:B5@8AB8:8 MB>9 A8AB5<K 157 48=0<8G5A:>3> 30A8B5;O 8 A 48=0<8G5A:8< 30A8B5;5<. 0: 284=> 87 MB8E E0@0:B5@8AB8:, ?@8 CAB0=>2:5 48=0<8G5A:>3> 30A8B5;O 0<?;8BC40 =0 G0AB>B5 =0AB@>9:8 @57:> A=8605BAO, >4=0:> 2 A8AB5<5 2<5AB> >4=>9 A>1AB25==>9 G0AB>BK 2>7=8:05B 425. >MB><C 48=0<8G5A:85 30A8B5;8 MDD5:B82=K B>;L:> 2 C7:>< 480?07>=5 G0AB>B 21;878 G0AB>BK =0AB@>9:8 30A8B5;O. 7>1@065==K5 =0 @8AC=:5 :@82K5 1 8 2 >B=>AOBAO : 48=0<8G5A:><C 30A8B5;N 157 45<?D8@>20=8O. @8 =0;8G88 2 A8AB5<5 45<?D5@>2 D>@<0 :@82>9 87<5=O5BAO (:@820O 3): 0<?;8BC4K 2 7>=0E 30H5=8O C25;8G820NBAO, 0 7>=0E @57>=0=A0 - C<5=LH0NBAO. 2 (x>B= 1   1;0AB8 MDD5:B82=>AB8 48=0<8G5A:>3>  3 30A8B5;O  1   0 (01 (0 (02 (, @04/A 8A. 15.8 >=B@>;L=K5 2>?@>AK !B0B8G5A:>5 8 48=0<8G5A:>5 C@02=>25H820=85 ?@8 ?@>5:B8@>20=88 45B0;59.  G5< @07;8G85 <564C 281@>30H5=85< 8 281@>87>;OF859. !CI=>ABL ?>4@5AA>@820=8O, 8;8 281@>87>;OF88. 8=0<8G5A:>5 30H5=85 :>;510=89. 5:F8O 16 "@5=85 2 :8=5<0B8G5A:8E ?0@0E. "@5=85 A:>;L65=8O: A8;0 B@5=8O, C3>; 8 :>=CA B@5=8O, 42865=85 =0 3>@87>=B0;L=>9 ?;>A:>AB8, B@5=85 =0 =0:;>==>9 ?;>A:>AB8, B@5=85 :;8=G0B>3> ?>;7C=0, B@5=85 2> 2@0I0B5;L=KE ?0@0E, B@5=85 2 ?OB0E. "@5=85 2 :8=5<0B8G5A:8E ?0@0E /2;5=85 B@5=8O 8<55B <5AB> 2A5340, :>340 ;N1K5 B5;0  B2Q@4K5, 684:85 8;8 307>>1@07=K5, =0E>4OI85AO 2 A>?@8:>A=>25=88 >4=> A 4@C38<, 4286CBAO >B=>A8B5;L=> 4@C3 4@C30, 8;8 ?>425@30NBAO 2>7459AB28N A8;, AB@5<OI8EAO 2K720BL 8E >B=>A8B5;L=>5 42865=85. "@5=85 2 <0H8=0E 83@05B 25AL<0 ACI5AB25==CN @>;L.  ?5@540B>G=KE <5E0=87<0E  D@8:F8>==KE, @5<5==KE 8 4@. 70 AG5B A8;K B@5=8O >ACI5AB2;O5BAO ?5@540G0 42865=8O >B 254CI53> 725=0 : 254><><C.  4@C38E A;CG0OE B@5=85 =565;0B5;L=>, B0: :0: >=> ?@5?OBAB2C5B 42865=8N 8 =0 ?@5>4>;5=85 A8; B@5=8O 70B@0G8205BAO 7=0G8B5;L=0O G0ABL @01>BK 4286CI8E A8;  B0: =07K205BAO @01>B0 A8; 2@54=KE A>?@>B82;5=89. "@5=85 2K7K205B 87=>A B@CI8EAO G0AB59 <0H8=K, GB> ACI5AB25==> ?@5?OBAB2C5B ?>2KH5=8N A:>@>AB59 42865=8O, 0 A;54>20B5;L=> 8 ?@>872>48B5;L=>AB8 <0H8=.  40;L=59H5< <K 1C45< 8<5BL 2 284C B@5=85 B>;L:> =0 ?>25@E=>ABOE A>?@8:0A0NI8EAO B2Q@4KE B5;. !?>A>1=>ABL :>=B0:B8@CNI8E ?>25@E=>AB59 725=L52 A>?@>B82;OBLAO 8E >B=>A8B5;L=><C 42865=8N =07K205BAO 2=5H=8< B@5=85<. "@5=85 >1CA;>2;5=> =58450;L=K< A>AB>O=85< :>=B0:B8@CNI8E ?>25@E=>AB59 (<8:@>=5@>2=>AB8, 703@O7=5=8O, >:8A=K5 ?;5=:8 8 B.?.) 8 A8;0<8 <56<>;5:C;O@=>3> AF5?;5=8O. ?KB ?>:07K205B, GB> ?@8 >B=>A8B5;L=>< 42865=88 42CE A>?@8:0A0NI8EAO B2Q@4KE B5;, ?@860BKE >4=> : 4@C3><C =5:>B>@>9 A8;>9, =0 ?>25@E=>AB8 A>?@8:>A=>25=8O 459AB2C5B A8;0, A>?@>B82;ONI0OAO >B=>A8B5;L=><C 42865=8N  A8;0 B@5=8O. 07;8G0NB A;54CNI85 284K B@5=8O: B@5=85 ?>:>O ?@>O2;O5BAO 2 <><5=B, :>340 420 B5;0 =0E>4OI85AO 2 A>AB>O=88 >B=>A8B5;L=>3> ?>:>O =0G8=0NB >B=>A8B5;L=>5 42865=85 (:0A0B5;L=CN A>AB02;ONICN, 2>7=8:0NICN 2 7>=5 :>=B0:B0 4> 2>7=8:=>25=8O >B=>A8B5;L=>3> 42865=8O, 2 CA;>28OE :>340 >=0 <5=LH5 A8;K B@5=8O ?>:>O, 1C45< =07K20BL A8;>9 AF5?;5=8O; <0:A8<0;L=0O 25;8G8=0 A8;K AF5?;5=8O @02=0 A8;5 B@5=8O ?>:>O); B@5=85 A:>;L65=8O ?>O2;O5BAO 2 :8=5<0B8G5A:>9 ?0@5 ?@8 =0;8G88 >B=>A8B5;L=>3> 42865=8O 725=L52; B@5=85 :0G5=8O ?>O2;O5BAO 2 2KAH8E :8=5<0B8G5A:8E ?0@0E ?@8 =0;8G88 >B=>A8B5;L=>3> 2@0I0B5;L=>3> 42865=8O 725=L52 2>:@C3 >A8 8;8 B>G:8 :>=B0:B0; B@5=85 25@G5=8O 2>7=8:05B ?@8 2708<>459AB288 B>@F52KE ?>25@E=>AB59 725=L52 2@0I0B5;L=KE :8=5<0B8G5A:8E ?0@ (?>4?OB=8:8). ?KB ?>:07K205B B0:65, GB> A8;0 B@5=8O C<5=LH05BAO, 5A;8 A>?@8:0A0NI85AO ?>25@E=>AB8 B2Q@4KE B5; A<070=K 684:>ABLN. > =0;8G8N 8 284C ?@8<5=O5<KE A<07>G=KE <0B5@80;>2 @07;8G0NB ACE>5, 3@0=8G=>5, 684:>AB=>5 B@5=85 8 A 307>2>9 A<07:>9. !=865=85 B@5=8O 2 ?>4286=KE A>548=5=8OE <0H8= B@51C5B 4>?>;=8B5;L=KE @5H5=89, MDD5:B82=>ABL :>B>@KE 2 1>;LH>9 AB5?5=8 7028A8B >B @>445@60=8O CA;>289 M:A?;C0B0F88 (@8A. 16.1).  8A. 16.1. CB8 A=865=8O B@5=8O "@5=85 A:>;L65=8O =5A<070==KE B5; !8;0 B@5=8O A:>;L65=8O F 2>7=8:05B 2 ?;>A:>AB8 :0A0=8O (A:>;L65=8O) 42CE A>?@8:0A0NI8EAO ?@860BKE A8;>9 Fn 4@C3 : 4@C3C B5; ?@8 8E >B=>A8B5;L=>< ?5@5<5I5=88. ;O ?@5>4>;5=8O A8;K B@5=8O =5>1E>48<> ?@8;>68BL A42830NICN A8;C F(, @0A?>;>65==CN 2 ?;>A:>AB8 A:>;L65=8O. !8;C B@5=8O A:>;L65=8O <>6=> 2K@078BL A;54CNI59 ?@81;865==>9 D>@<C;>9 (70:>= C;>=0-<>=B>=0):  EMBED Equation.3 , 345 F  A8;0 B@5=8O A:>;L65=8O, B.5. A8;0, ?@5?OBAB2CNI0O >B=>A8B5;L=><C 42865=8N 42CE A>?@8:0A0NI8EAO B5;; f  :>MDD8F85=B ?@>?>@F8>=0;L=>AB8 4;O :>=:@5B=KE CA;>289 B@5=8O, =07K205<K9 :>MDD8F85=B>< B@5=8O A:>;L65=8O (:>MDD8F85=B B@5=8O 1-@>40); Fn  @57C;LB8@CNI0O A8;0 =>@<0;L=KE 402;5=89, =0?@02;5==0O ?> >1I59 =>@<0;8 : ?>25@E=>AB8 A>?@8:0A0NI8EAO B5;, A>>B25BAB25==> =>@<0;L=0O @50:F8O Rn @02=0 Fn. ">340  EMBED Equation.3 . "@5=85 ?>O2;O5BAO =5 B>;L:> 2> 2@5<O 42865=8O B5;, => 8 B>340, :>340 B5;0 =0E>4OBAO 2 ?>:>5. "@CI85AO B5;0 1 8 2 (@8A. 16.2) A 1>;LH59 8;8 <5=LH59 A8;>9 AF5?;5=K >4=> A 4@C38<. CABL : B5;C 1 ?@8;>65=0 A8;0, AB@5<OI0OAO A428=CBL 53> A;520 =0?@02>. @8 C25;8G5=88 A8;K 1C45B 8<5BL <5AB> =52848<K9 3;07><, => >1=0@C68205<K9 B>G=K<8 ?@81>@0<8 C?@C389 A4283 >4=>9 B@CI59AO ?>25@E=>AB8 >B=>A8B5;L=> 4@C3>9. @8 C25;8G5=88 A8;K 45D>@<0F8O A42830 1C45B C25;8G820BLAO 8 ?@8 =5:>B>@>< ?@545;L=>< 7=0G5=88 A42830NI59 A8;K =0G=5BAO 2848<>5 42865=85 B5;0 1 >B=>A8B5;L=> B5;0 2. ?KBK ?>:07K20NB, GB> 4;O ?@82545=8O 2 42865=85 B5;0 1 B@51C5BAO A8;0, =5A:>;L:> 1>;LH0O B>9, :>B>@CN ?@8E>48BAO ?@5>4>;520BL ?@8 ?>A;54CNI5< @02=><5@=>< 42865=88. @545;L=0O A8;0, A>?@>B82;ONI0OAO 2 =0G0;L=K9 <><5=B 42865=8O B@CI53>AO B5;0, =07K205BAO A8;>9 B@5=8O ?>:>O 8;8 A8;>9 AF5?;5=8O, 0 A8;0 A>?@>B82;5=8O, 2>7=8:0NI0O 2> 2@5<O 42865=8O, ?>;CG8;0 =0720=85 A8;K B@5=8O 42865=8O. ;O B>3>, GB>1K B5;> 1 (@8A. 16.2) 2K25AB8 87 A>AB>O=8O ?>:>O, : =5<C =C6=> ?@8;>68BL 4286CICN A8;C  EMBED Equation.3 , @02=CN  EMBED Equation.3 .  8A. 16.2 #AB0=>2;5=>, GB> B@5=85 ?@54AB02;O5B A;>6=K9 :><?;5:A <5E0=8G5A:8E, E8<8G5A:8E 8 4@. O2;5=89. 4=0:> :>;8G5AB25==K9 CG5B MB8E O2;5=89 2 =0AB>OI55 2@5<O ?@>8725AB8 =5 ?@54AB02;O5BAO 2>7<>6=K<. ?KBK ?>:07K20NB, GB> 25;8G8=0 A8;K B@5=8O G@572KG09=> 87<5=G820. ><8<> <0B5@80;0, G8AB>BK >1@01>B:8 ?>25@E=>AB8, A<07:8, C45;L=>3> 402;5=8O 8 >B=>A8B5;L=>9 A:>@>AB8 ?5@542865=8O B@CI8EAO B5; =0 25;8G8=C A8;K B@5=8O >:07K20NB 2;8O=85 B0:85, :070;>AL 1K, <0;>7=0G8<K5 D0:B>@K, :0: =0;8G85 8;8 >BACBAB285 >:A84=>9 ?;Q=:8, 2;06=>ABL, B5<?5@0BC@0 8 B.?. AQ MB> ?@82>48B : B><C, GB> =0C:0 4> A8E ?>@ 5I5 =5 8<55B 4> :>=F0 @07@01>B0==>9 B5>@88 B@5=8O 8 87=>A0. >MB><C 2 8=65=5@=KE @0AG5B0E ?@8E>48BAO ?>;L7>20BLAO =5:>B>@K<8 ?@81;878B5;L=K<8 (CA@54=5==K<8) 25;8G8=0<8 :>MDD8F85=B>2 B@5=8O, 7=0G5=8O :>B>@KE ?>;CG5=K :0: @57C;LB0B >1>1I5=8O 1>;LH>3> M:A?5@8<5=B0;L=>3> <0B5@80;0.  1>;LH8=AB25 A?@02>G=8:>2 7=0G5=8O :>MDD8F85=B>2 B@5=8O A:>;L65=8O ?@82>4OB 2 7028A8<>AB8 >B <0B5@80;0 A>?@8:0A0NI8EAO B5;, G8AB>BK >1@01>B:8 ?>25@E=>AB8 8 4;O A;CG0O ACE>3> B@5=8O. >MDD8F85=B f0 =07K205BAO :>MDD8F85=B>< B@5=8O ?>:>O. #3>; (0, >1@07>20==K9 =0?@02;5=8O<8 ?>;=>9 @50:F88 R0 8 =>@<0;L=>9 @50:F88  EMBED Equation.3 , =07K205BAO C3;>< B@5=8O ?>:>O. ;O B>3>, GB>1K B5;> ?@>4>;60;> 42865=85 A ?>AB>O==>9 A:>@>ABLN, =5>1E>48<> ?@8;>68BL A8;C  EMBED Equation.3 , =5A:>;L:> <5=LHCN A8;K  EMBED Equation.3  8 @02=CN  EMBED Equation.3 . 7 @8A. 16.2 ?>;CG8<  EMBED Equation.3 ;  EMBED Equation.3 ; B0: :0:  EMBED Equation.3 > EMBED Equation.3 , B> F0 > EMBED Equation.3 , f0 ( f, (0 ( (, B.5. :>MDD8F85=B B@5=8O ?>:>O 1>;LH5 :>MDD8F85=B0 B@5=8O 42865=8O. @8 42865=88 B5;0 1 2 @07=KE =0?@02;5=8OE ?> ?;>A:>AB8 2 @02=>459AB2CNI0O R21 1C45B >B:;>=OBLAO >B =>@<0;L=>9 @50:F88 Rn =0 C3>; ( 2 AB>@>=C ?@>B82>?>;>6=CN >B=>A8B5;L=><C 42865=8N, >1@07CO :>=CA A C3;>< 2( ?@8 25@H8=5  =07K205<K9 :>=CA>< B@5=8O (@8A. 16.3). >=CA B@5=8O  ?>25@E=>ABL >?8AK205<0O @02=>459AB2CNI59 (A8;K B@5=8O 8 =>@<0;L=>9 @50:F88) ?@8 2@0I5=88 5Q 2>:@C3 =>@<0;L=>9 @50:F88, 8;8 :>=CA A C3;>< ?@8 25@H8=5 @02=K< 42>9=><C C3;C B@5=8O.  8A. 16.3 "@5=85 =0 3>@87>=B0;L=>9 ?;>A:>AB8 2865=85 B5;0 =0 ?;>A:>AB8 <>6=> @0AA<0B@820BL ?>4 459AB285< B>;L:> >4=>9 A8;K, B.:. =5A:>;L:> 459AB2CNI8E A8; =0 B5;> <>6=> ?@825AB8 : >4=>9 ( @02=>459AB2CNI59. @54?>;>68<, GB> =0 B5;> 459AB2C5B A8;0  (@8A. 16.4) ?>4 C3;>< ( : =>@<0;8. 5@5=5AQ< A8;C  2 B>G:C ?5@5A5G5=8O 5Q 25:B>@0 A ?5@?5=48:C;O@>< : ?;>A:>AB8 A>?@8:>A=>25=8O 8 @07;>68< 5Q ?> 42C< 2708<=> ?5@?5=48:C;O@=K< =0?@02;5=8O<. >@<0;L=0O A>AB02;ONI0O n, ?@868<0O B5;> : ?;>A:>AB8, A>7405B ?@>B82>459AB2CNICN 42865=8N B5;0 A8;C F0 B@5=8O ?>:>O; B0=35=F80;L=0O A>AB02;ONI0O ( AB@5<8BAO A428=CBL B5;>.  8A. 16.4 ;O 2>7<>6=>AB8 42865=8O =5>1E>48<> 2K?>;=5=85 CA;>28O:  EMBED Equation.3 . "0: :0:  EMBED Equation.3  ">  EMBED Equation.3 , >B:C40  EMBED Equation.3 , 345 (0 ( C3>; B@5=8O ?>:>O. 7 MB>3> A;54C5B, GB> A8;0, 459AB2CNI0O 2=CB@8 :>=CA0 B@5=8O, =57028A8<> >B 5Q 25;8G8=K, ?@825AB8 2 42865=85 B5;> =5 <>65B. >4AB0282 2 ?@8254Q==K5 2KH5 =5@025=AB20 F 2<5AB> F0, f 8 ( 2<5AB> f0 8 (0 8 70<5=82 2> 2A5E =5@025=AB20E 7=0: > 7=0:0<8 (, ?@84Q< : 70:;NG5=8N, GB> 42865=85 B5;0 =0 ?;>A:>AB8 2>7<>6=> ;8HL ?@8 A>1;N45=88 CA;>28O  EMBED Equation.3 . @8  EMBED Equation.3  ?>;CG05<  EMBED Equation.3  8 42865=85 A ?>AB>O==>9 A:>@>ABLN, ?@8  EMBED Equation.3  ?>;CG05<  EMBED Equation.3  8 42865=85 A CA:>@5=85<. @8  EMBED Equation.3  2>7=8:=>25=85 42865=8O =52>7<>6=>, ?@8  EMBED Equation.3  2K720==>5 :0:8<8-;81> A8;0<8 8 ?@>4>;60NI55AO ?> 8=5@F88 42865=85 B5;0 ?@8 459AB288 =0 =53> A8;K  1C45B ?@>8AE>48BL A 70<54;5=85<, B0: :0: 2 MB>< A;CG05 A>740205<0O A8;>9 n ?@>B82>459AB2CNI0O 42865=8N A8;0 B@5=8O 1C45B 1>;LH5 A8;K (, ?>445@6820NI59 42865=85. "@5=85 =0 =0:;>==>9 ?;>A:>AB8 0AA<>B@8< A;CG09, :>340 =0 B5;>, =0E>4OI55AO =0 =0:;>==>9 ?;>A:>AB8 A C3;>< ( : 3>@87>=BC, 459AB2C5B >4=0 A8;0, ?5@?5=48:C;O@=0O >A=>20=8N ?;>A:>AB8 (@8A 16.5), =0?@8<5@ A8;0 25A0.  8A. 16.5  MB>< A;CG05 2>7<>6=> 42865=85 B>;L:> 2=87 ?@8 A>1;N45=88 CA;>28O  EMBED Equation.3 . @8  EMBED Equation.3  ?>4 459AB285< A8;K, ?5@?5=48:C;O@=>9 >A=>20=8N ?;>A:>AB8, 42865=85, =57028A8<> >B 25;8G8=K A8;K, =5 <>65B =8 2>7=8:=CBL, =8 ?@>4>;60BLAO 157 70<54;5=8O. ;>A:>ABL A C3;>< =0:;>=0, <5=LH8< C3;0 B@5=8O, =07K205BAO A0<>B>@<>7OI59. 0AA<>B@8< A;CG09, :>340 =0 B5;>, =0E>4OI55AO =0 ?;>A:>AB8, =0:;>=5==>9 ?>4 C3;>< ( : 3>@87>=BC, 459AB2CNB 425 A8;K, >4=0 87 :>B>@KE ?5@?5=48:C;O@=0 >A=>20=8N ?;>A:>AB8, 0 4@C30O ?0@0;;5;L=0.  MB>< A;CG05 2>7<>6=> 42865=85 B5;0 225@E, 8;8 2=87 ?> =0:;>==>9 ?;>A:>AB8. 2865=85 B5;0 225@E ?> =0:;>==>9 ?;>A:>AB8 (@8A. 16.6).  8A. 16.6  MB>< A;CG05 ?0@0;;5;L=0O >A=>20=8N ?;>A:>AB8 A8;0  4>;6=0 1KBL =0?@02;5=0 2?@02>. ;O 2>7<>6=>AB8 42865=8O 225@E ?> ?;>A:>AB8 =5>1E>48<>, GB>1K @02=>459AB2CNI0O R A8; P 8 Q A>AB02;O;0 C3>; ( A ?5@?5=48:C;O@>< : ?;>A:>AB8 =5 <5=LH5 C3;0 B@5=8O (, B.5. =5>1E>48<> 2K?>;=5=85 CA;>28O  EMBED Equation.3 . 7 ?>AB@>5=8O A;54C5B  EMBED Equation.3 . 7 ?@82545==KE 2KH5 =5@025=AB20 8 @025=AB20 2KB5:05B A;54CNI55 CA;>285 4;O 2>7<>6=>AB8 42865=8O B5;0 225@E ?> =0:;>==>9 ?;>A:>AB8:  EMBED Equation.3 . 2865=85 B5;0 2=87 ?> =0:;>==>9 ?;>A:>AB8 (@8A. 16.7).  MB>< A;CG05 87 ?>AB@>5=8O A;54C5B  EMBED Equation.3  8, A;54>20B5;L=>, 42865=8O B5;0 2=87 ?> =0:;>==>9 ?;>A:>AB8 >:07K205BAO ?@8  EMBED Equation.3  2>7<>6=K< ?@8 A>1;N45=88 CA;>28O  EMBED Equation.3 .  8A. 16.7 0 @8A. 16.7 A8;0  =0?@02;5=0 2?@02> 8 O2;O5BAO A8;>9, ?@>B82>459AB2CNI59 42865=8N. 5>1E>48<>ABL 2 A8;5 , =0?@02;5==>9 2;52> (@8A. 16.8), 4;O 42865=8O B5;0 2=87 1C45B B>;L:> 2 B>< A;CG05, 5A;8 ?>4 459AB285< B>;L:> >4=>9 A8;K Q 42865=85 =5 1C45B ?@>8AE>48BL (A0<>B>@<>7OI0O ?;>A:>ABL). 7 ?>AB@>5=8O (@8A. 16.8) ?>;CG05<  EMBED Equation.3  @8  EMBED Equation.3  =5>1E>48<0O 4;O 42865=8O A8;0  4>;6=0 C4>2;5B2>@OBL CA;>28N  EMBED Equation.3 .  8A. 16.8 "@5=85 :;8=G0B>3> ?>;7C=0 0 @8A. 16.9 87>1@06Q= :;8=G0BK9 ?>;7C=, 8<5NI89 2 ?>?5@5G=>< A5G5=88 D>@<C B@0?5F88, :>B>@K9 ?@868<05BAO A8;>9 Q : 42C< ?>25@E=>ABO< B@5=8O, =0:;>=5==K< :0640O : ;8=88 459AB28O A8;K Q ?>4 C3;>< (. @8 42865=88 :;8=G0B>3> ?>;7C=0 2 =0?@02;5=88, ?5@?5=48:C;O@=>< : ?;>A:>AB8 G5@B560, 25;8G8=0 A8;K B@5=8O:  EMBED Equation.3 . @8=O2 2> 2=8<0=85, GB> AC<<0 ?@>5:F89 A8; =0 25@B8:0;L @02=0 =C;N, ?>;CG05<  EMBED Equation.3 . 254O >1>7=0G5=85  EMBED Equation.3 , ?>;CG05<  EMBED Equation.3 , 345 25;8G8=0 f( =07K205BAO D8:B82=K< 8;8 ?@8254Q==K< :>MDD8F85=B>< B@5=8O.  8A. 16.9 "@5=85 2> 2@0I0B5;L=KE ?0@0E @0I0B5;L=0O :8=5<0B8G5A:0O ?0@0 >1@07C5BAO F0?D>9 (>?>@=>9 G0ABLN 20;0) 8 >E20BK20NI8< 5Q ?>4H8?=8:><. ;O B>3> GB>1K F0?D0, =0E>4OI0OAO ?>4 459AB285< =5A:>;L:8E ?@8;>65==KE : =59 A8;, <>3;0 2@0I0BLAO, =5>1E>48<>, GB>1K @02=>459AB2CNI0O  MB8E A8; (@8A. 16.10) A>74020;0 <><5=B =5 <5=LH89 <><5=B0 A8;K B@5=8O.  8A. 16.10 07;>682 A8;C  =0 =>@<0;L=CN n 8 B0=35=F80;L=CN ( A>AB02;ONI85 8 >1>7=0G82 G5@57: r ?;5G> 459AB28O A8;K  >B=>A8B5;L=> >A8 2@0I5=8O F0?DK; R  @048CA F0?DK; ( - C3>; <564C ;8=859 459AB28O A8;K  8 @048CA><, ?@>254Q==K< 2 B>G:C ?@8;>65=8O A8;K P, ?>;CG8<: <><5=B, 2@0I0NI89 F0?DC, @025=  EMBED Equation.3 ; <><5=B A8;K F B@5=8O @025=  EMBED Equation.3 . ;O 2>7<>6=>AB8 42865=8O =5>1E>48<> A>1;N45=85 CA;>28O  EMBED Equation.3 , >B:C40  EMBED Equation.3 , 8 ?>MB><C  EMBED Equation.3 . !;54>20B5;L=>, <><5=B A8;K  =5 <>65B 2@0I0BL F0?DK, 5A;8 ;8=8O 459AB28O A8;K  ?@>E>48B 2=CB@8 :@C30 A @048CA><  EMBED Equation.3 . "0:>9 :@C3 ?>;CG8; =0720=85  :@C30 B@5=8O. "@5=85 2 ?OB0E (B@5=85 25@G5=8O) "@5=85 2 ?OB0E 8<55B <5AB> =0 ?>25@E=>AB8 A>?@8:>A=>25=8O 42CE ?@860BKE >4=> : 4@C3><C B5;, 2@0I0NI8EAO >4=> >B=>A8B5;L=> 4@C3>3> 2>:@C3 =>@<0;8 : ?>25@E=>AB8 A>?@8:>A=>25=8O (@8A. 16.11, 0).  8A. 16.11 @8 CG5B5 A>?@>B82;5=8O, >:07K205<>3> A8;0<8 B@5=8O <><5=BC, 2@0I0NI5<C >4=> B5;> >B=>A8B5;L=> 4@C3>3>, ?@8E>48BAO >?@545;OBL @02=>459AB2CNI89 <><5=B A8; B@5=8O, 70<5=OO M;5<5=B0@=K5 A8;K B@5=8O =0 2A59 ?>25@E=>AB8 B@5=8O @02=>459AB2CNI59 A8;>9, ?@8;>65==>9 =0 >:@C6=>AB8 A @048CA>< R. ?@545;8< 25;8G8=C @048CA0 R, ?@54?>;>682, GB> ?>25@E=>ABL B@5=8O ?@54AB02;O5B A>1>9 :>;LF52CN ?;>I04L A 2=5H=8< @048CA>< r1 8 2=CB@5==8< @048CA>< r2 8 GB> 2@0I5=85 ?@>8AE>48B 2>:@C3 >1I53> F5=B@0 >:@C6=>AB59 (@8A. 16.11, 1). >;=0O A8;0 B@5=8O A:>;L65=8O @02=0:  EMBED Equation.3 , 345 @  402;5=85 =0 ?>25@E=>AB8 B@5=8O, B.5. A8;0, ?@8E>4OI0OAO =0 548=8FC ?>25@E=>AB8 B@5=8O;  EMBED Equation.3  ( ?;>I04L ?>25@E=>AB8 A>?@8:>A=>25=8O. ><5=B ?>;=>9 A8;K B@5=8O  EMBED Equation.3 , 345 R  @048CA >:@C6=>AB8, =0 :>B>@>9 ?@8;>65=0 @02=>459AB2CNI0O M;5<5=B0@=KE A8; B@5=8O. -;5<5=B0@=0O A8;0 B@5=8O, @0A?@545;Q==0O ?> :>;LFC A ?;>I04LN  EMBED Equation.3 , @02=0  EMBED Equation.3 . ><5=B M;5<5=B0@=>9 A8;K B@5=8O @025=  EMBED Equation.3 . !C<<0@=K9 <><5=B  EMBED Equation.3 . ">340 ?@8@02=820O ?@02K5 ?>;>28=K C@02=5=89, ?>;CG05< @025=AB2>  EMBED Equation.3 , >B:C40  EMBED Equation.3 . @8 r2 = 0, ?>;CG05<  EMBED Equation.3 . >=B@>;L=K5 2>?@>AK 'B> B0:>5 B@5=85, 8 >B G53> >=> 7028A8B. "@5=85 A:>;L65=8O: A8;0, C3>; 8 :>=CA B@5=8O. "@5=85 =0 3>@87>=B0;L=>9 ?;>A:>AB8. "@5=85 =0 =0:;>==>9 ?;>A:>AB8. "@5=85 :;8=G0B>3> ?>;7C=0, ?@82545==K9 :>MDD8F85=B B@5=8O. "@5=85 2> 2@0I0B5;L=KE ?0@0E. "@5=85 2 ?OB0E. 5:F8O 17 "@5=85 381:8E B5;. "@5=85 :0G5=8O. >MDD8F85=B ?>;57=>3> 459AB28O <5E0=87<>2: >1I85 A2545=8O,  ?@8 ?>A;54>20B5;L=>< 8 ?0@0;;5;L=>< A>548=5=88 725=L52,  28=B>2>9 ?5@540G8. "@5=85 381:8E B5; 0AA<>B@8< 8450;L=> 381:>5 8 =5@0ABO68<>5 B5;>, B.5. B5;>, A>25@H5==> =5 45D>@<8@CNI55AO ?>4 459AB285< @0ABO3820NI59 A8;K 8 =5 >:07K20NI55 =8:0:>3> A>?@>B82;5=8O ?@8 53> ?5@53815, :>B>@>5 >38105B =5?>4286=K9 F8;8=4@. E20B F8;8=4@0 ?@>8AE>48B ?> 4C35 ab A F5=B@0;L=K< C3;>< ( (@8A. 17.1). 0 B5;> 459AB2CNB A8;K S1 8 S2, 4;O 42865=8O 53> ?> F8;8=4@C A @02=><5@=>9 A:>@>ABLN =5>1E>48<> A>1;N45=85 CA;>28O  EMBED Equation.3 , 345 F  A8;0 B@5=8O <564C 381:8< B5;>< 8 F8;8=4@><.  8A. 17.1 "0: :0: A8;0 B@5=8O @0A?@545;5=0 =0 4C35 ab A>?@8:>A=>25=8O 381:>3> B5;0 A F8;8=4@><, B> =0BO65=85 381:>3> B5;0 >B B>G:8 a =01530=8O 53> =0 F8;8=4@ 4> B>G:8 b A1530=8O 53> A F8;8=4@0 2>7@0AB05B ?> =5:>B>@><C 70:>=C >B S2 4> S1.  B>G:5 A 15A:>=5G=> <0;>9 4C38 cd =0BO65=85 4>AB8305B =5:>B>@>9 25;8G8=K S, 2 B>G:5 d =0BO65=85 C25;8G8205BAO 4> S + dS. 5A:>=5G=> <0;0O A8;0 B@5=8O =0 4C35 cd, >1CA;02;820NI0O C25;8G5=85 =0BO65=8O =0 25;8G8=C dS 8 ?>MB><C @02=0O dS, <>65B 1KBL 2K@065=0 A;54CNI8< >1@07><:  EMBED Equation.3 , 345 f  :>MDD8F85=B B@5=8O; dRn  15A:>=5G=> <0;0O A8;0, =>@<0;L=0O : ?>25@E=>AB8 B@5=8O. 5;8G8=0 A8;K dRn A>7405BAO ?@>5:F8O<8 =0BO65=89 S 8 S + dS =0 @048CA, ?@>2545==K9 2 A5@548=C 4C38 cd:  EMBED Equation.3 . "0: :0:  EMBED Equation.3  8  EMBED Equation.3 , :0: G;5= 2KAH53> ?>@O4:0 <0;>AB8, B>  EMBED Equation.3 ;  EMBED Equation.3 . 0745;82, ?>;CG05<  EMBED Equation.3 . =B53@8@CO, 0 70B5<, 48DD5@5=F8@CO, ?>;CG05<  EMBED Equation.3 ;  EMBED Equation.3 ;  EMBED Equation.3 ;  EMBED Equation.3 . 028A8<>ABL <564C A8;>9 S1, 4286CI59 8450;L=> 381:>5 8 =5@0ABO68<>5 B5;>, >E20BK20NI55 F8;8=4@, 8 A8;>9 S2, A>?@>B82;ONI59AO 42865=8N 1K;0 ?>;CG5=0 . -9;5@><. 0 >A=>20=88 D>@<C;K -9;5@0 A8;0 B@5=8O =0 ?>25@E=>AB8 A>?@8:>A=>25=8O 381:>3> B5;0 8 >E20BK205<>3> 8< F8;8=4@0 @02=0  EMBED Equation.3 . =0;878@CO D>@<C;C, 2848<, GB> 25;8G8=0 A8;K B@5=8O 2 7=0G8B5;L=>9 AB5?5=8 7028A8B 8 >B :>MDD8F85=B0 B@5=8O 8 >B C3;0 >E20B0: ?@8 f = 0,35 >1<>B02 381:>5 B5;> 2>:@C3 F8;8=4@0 =0 420 ?>;=KE >1>@>B0 (( = 4(), A8;>9 10 , <>6=> C@02=>25A8BL A8;C 800 . 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