. . . . . .
| 06.06.2014 | |
| 1,1 M |
. ,
, , , , .
ࡱ > + R ~ ] Z R _ a | ~ [ bjbj ΐ ΐ | 2 ( ( D6 D6 D6 D6 D6 6 6 6 7 < 4= L 6 d B N] N] N] N] A^ W^ c^ $ R D6 k^ =^ A^ k^ k^ D6 D6 N] N] b k^ D6 N] D6 N] k^ K N] @#ϋ 6 Qg . ۤ > x 0 A x ( D6 _ ( k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ k^ ( ( 5 : !"# 8?;><=89 ?@>5:B =0 B5<C: 030B>@VG=0 <V=;82VABL ?>2B>@N20=>ABV B0 :V;L:>ABV >?04V2 2 <VABV 45A0 28:>=0=89 =0 :0D54@V DV78:8 0B<>AD5@8 B0 :;V<0B>;>3VW ?V4 :5@V2=8FB2>< 4>F5=B0 :0D54@8, :. 35>3@. =. 0=>2>W ". . , ?@54AB02;OT 4>A;V465==O ACG0A=8E 7<V= :V;L:>ABV B0 ?>2B>@N20=>ABV >?04V2 2 <VABV 45A0, O:V ?@>2>48;8AO 70 4>?><>3>N AB0B8AB8G=8E <5B>4V2. @8 ?V43>B>2FV 48?;><=>3> ?@>5:BC ?> 4>A;V465==N <V=;82>ABV :V;L:>ABV >?04V2 B0 WE ?>2B>@N20=>ABV, 28:>@8AB>2C20;8AO 40=V A?>AB5@565=L 3V4@><5B5>@>;>3VG=>W AB0=FVW 70 ?5@V>4 7 1 9 7 1 ?> 2 0 1 2 @@. 7 VA?0=AL:>3> :;V<0B8G=>3> A09BC h t t p : / / w w w . t u t i e m p o . n e t / c l i m a . h t m 2 <. 45A0. :BC0;L=VABL B5<8 >1C<>2;5=0 B8<, I> ?@>1;5<0 7<V= :;V<0BC >4=0 7 =090:BC0;L=VH8E ?@>1;5< ACG0A=>ABV. !B>ACTBLAO 2>=0 2AVE =0C:>28E =0?@O<:V2, B><C I> <0T =5 BV;L:8 =0C:>289 V ?@0:B8G=89 V=B5@5A, 0;5 9 A>FV0;L=89: 7<V=8 :;V<0BC ?@87254CBL 4> 7<V=8 728G=8E C<>2 68BBO V 4VO;L=>ABV ;N48=8. I>1 @>7@>18B8 45@602=C ?@>3@0<C 040?B0FVW ACA?V;LAB20 4> >GV:C20=8E 7<V= :;V<0BC B@510 45B0;L=> 282G8B8 ?@8G8=8, E0@0:B5@ V <>6;82V =0A;V4:8 F8E 7<V=. 06;828< 5:>;>3VG=8< =0A;V4:>< 3;>10;L=5 ?>B5?;V==O :;V<0BC T B@0=AD>@<0FVO =5 BV;L:8 ?>;O B5<?5@0BC@8, 0;5 V ?>;O 0B<>AD5@=8E >?04V2. >A;V465==O 45O:8E 02B>@V2 4>2>4OBL ?@> A?5F8DVG=89 5D5:B A57>==> 35>3@0DVG=>3> 28@V2=N20==O 0B<>AD5@=8E >?04V2, O:5 2V41C20;>AO ?@>BO3>< <8=C;>3> AB>;VBBO =0 B5@8B>@VW #:@0W=8 2 C<>20E 3;>10;L=>3> ?>B5?;V==O. &59 5D5:B ?@>O2;O2AO 2 B><C, I> 70 >AB0==V AB> @>:V2 2 @09>=0E 7 =54>AB0B=V< 72>;>65==O< ( ?V245==>- AEV4=V >1;0ABV #:@0W=8) WE @VG=0 AC<0 ACBBT2> 71V;LH8;0AL, 0 2 @09>=0E 7 =04;8H:>28< 72>;>65==O< ( ?V2=VG=>- 70EV4=V >1;0ABV #:@0W=8) 7<5=H8;0AL. > B>3> 6 02B>@8 2:07CNBL =0 7=0G=89 2=5A>: 2 FV 7<V=8 >?04V2 E>;>4=>3> ?V2@VGGO. @>BO3>< %% AB>@VGGO 2V41C20;8AL ?><VB=V 7<V=8 2 @568<V >?04V2 =0 B5@8B>@VW #:@0W=8, 0;5 7 @V7=>N WE V=B5=A82=VABN O: 2 >:@5<8E @53V>=0E, B0: V 2 >:@5<V ?5@V>48. AB0==T ?@872>48BL 4> 7<V= 2 A?V22V4=>H5==V >?04V2 B5?;>3> V E>;>4=>3> ?5@V>4V2, =091V;LH>W B0 =09<5=H>W WE <VAOG=>W :V;L:>ABV 2 @>FV, B>1B> 4> 7<V= WE @VG=>3> E>4C. &5 A2V4G8BL ?@> 7<5=H5==O :>=B8=5=B0;L=>ABV :;V<0BC #:@0W=8 =0?@8:V=FV %% AB>@VG, I> >A>1;82> ?><VB=> =0 ?V24=V #:@0W=8. ;5 B5@8B>@VN #:@0W=8 =5<>6;82> @>73;O40B8 2V4>:@5<;5=> 2V4 >B>GCNG8E WW B5@8B>@V9 V=H8E 45@602, B><C =091V;LH89 V=B5@5A ?@54AB02;OT ?@>1;5<0 282G5==O AB0B8AB8G=8E E0@0:B5@8AB8: >?04V2 2 CAL><C @53V>=V. 5B0 48?;><=>3> ?@>5:BC @>73;O=CB8 AB0B8AB8G=V E0@0:B5@8AB8:8 :V;L:>ABV >?04V2, G8A;0 4=V2 7 4>I5< B0 G8A;0 4=V2 7V A=V3>< 2 <VABV 45A0 70 ?5@V>4 1 9 7 3 - 2 0 1 2 @>:8. 87=0G8B8 48=0<V:C :V;L:>ABV >?04V2 B0 G8A;0 4=V2 7 4>I5< B0 A=V3>< 70 >AB0==V A>@>: @>:V2. 8O28B8 B5=45=FVN ?@8B0<0==C :V;L:>ABV B0 ?>2B>@N20=>ABV >?04V2. 1 , % " !" &! # # / % % " #" / 1 . 1 >03C;OFV9=5 7@>AB0==O :@0?5;L C E<0@V 1 . 1 . 1 #AB0;5=0 H284:VABL ?04V==O :@0?5;L @0?;V 2>48 V :@8AB0;8 ;L>4C, O:V 28?040NBL 7 E<0@ =0 75<=C ?>25@E=N, =07820NBL >?040<8. @>F5A @>728B:C E<0@ B0 CB2>@5==O >?04V2 7 O:VA=>W AB>@>=8 <>6=0 @>74V;8B8 =0 42V AB04VW. ?048, I> 28?040NBL =0 75<=C ?>25@E=N =0A0<?5@54 <>6=0 @>74V;8B8 =0 B25@4V, @V4:V B0 7<VH0=V. # ?>G0B:>2V9 AB04VW @>728B:C E<0@8 >A=>2=C @>;L 2 C:@C?=5==V 70@>4:>28E E<0@=8E 5;5<5=BV2 2V4V3@0T ?@>F5A :>=45=A0FVW 2>4O=>W ?0@8. >=45=A0FVO 2V41C20TBLAO 7024O:8 =525;8:><C ?5@5A8G5==N 2>4O=>W ?0@8 2V4=>A=> ?>25@E=V E<0@=8E :@0?5;L. A>1;82> H284:> ?>G8=0TBLAO 7@>AB0==O G0AB8=>: C E<0@V ?VA;O B>3>, O: ?>@O4 7 ?5@5>E>;>465=8<8 :@0?;O<8 7 O2;ONBLAO 9 :@8AB0;8 ;L>4C. 0 F8E C<>2 ?>G8=0TBLAO ?5@53>=:0 2>4O=>W ?0@8 7 ?5@5>E>;>465=8E :@0?5;L =0 :@8AB0;8 ;L>4C 2=0A;V4>: B>3>, I> B8A: =0A8G5=>W 2>4O=>W ?0@8 =04 2>4>N 1V;LH5, =V6 =04 ;L>4><. &59 ?@>F5A 4>:;04=> @>73;O402AO 28I5. # 4@C3V9 AB04VW, ?VA;O B>3>, O: :@0?;V B0 :@8AB0;8 ;L>4C 7@>AB0NBL 4> E M B E D M a t h T y p e 2 0 & 6 0 <:<, >A=>2=C @>;L ?>G8=0T 2V4V3@020B8 ?@>F5A 7;8BBO ( :>03C;OFVW) E<0@=8E 5;5<5=BV2. >03C;OFVO >1C<>2;5=0 3>;>2=8< G8=>< @V7=>N H284:VABN ?04V==O :@0?5;L V :@8AB0;V2 ( 3@02VB0FV9=0 :>03C;OFVO) . 5O:5 7=0G5==O <0T :>03C;OFVO, I> >1C<>2;NTBLAO BC@1C;5=B=8< V 1@>C=V2AL:8< @CE0<8 ?>2VB@O, 5;5:B@>AB0B8G=8<8 A8;0<8 B>I>. 024O:8 :>03C;OFVW :@0?;V 9 :@8AB0;8 7@>AB0NBL 2V4 45AOB:V2 <V:@><5B@V2 4> 45:V;L:>E <V;V<5B@V2, 0 A=V3>2V ?;0ABV2FV B0 G0AB8=:8 3@04C 4> 45:V;L:>E A0=B8<5B@V2. @0?;V @V7=8E @>7<V@V2 ?040NBL ?V4 4VTN A8;8 2038 7 @V7=>N H284:VABN, 2 @57C;LB0BV G>3> 2>=8 A?V2C40@ONBLAO. ;5 4>:8 :@0?;V 4@V1=V, WE 7VB:=5==O B0 7;8BBO <0;> V<>2V@=5. V 7@>AB0==O< :@0?5;L @V7=8FO H284:>AB59 ?04V==O 71V;LHCTBLAO, I> 70157?5GCT A?@8OB;82V C<>28 4;O WE :>03C;OFVW. [ 1 ] C65 25;8:5 7=0G5==O 4;O 7@>AB0==O E<0@=8E 5;5<5=BV2 9 CB2>@5==O >?04V2 <0NBL 25@B8:0;L=V @CE8 C A5@548=V E<0@8. >- ?5@H5, ?@8 28AEV4=8E @CE0E 2V41C20TBLAO 7<5=H5==O B5<?5@0BC@8 ?>2VB@O, I> A?@8OT CB2>@5==N ?5@5A8G5==O 2>4O=>W ?0@8 B0 :>=45=A0FV9=><C 7@>AB0==N :@0?5;L C E<0@V 4> @>7<V@V2, ?@8 O:8E 0:B82=C @>;L ?>G8=0T 2V4V3@020B8 :>03C;OFVO. >- 4@C35, :@0?;V ?V4 4VTN 25@B8:0;L=8E B5GV9 ?@>E>4OBL 25;8:V B>2IV E<0@8. 024O:8 FL><C 2>=8 7@>AB0NBL 4> 25;8:8E @>7<V@V2 70 @0EC=>: :>03C;OFVW. )5 >48= ?@>F5A <>65 1CB8 5D5:B82=8< C E<0@V, O:0 <0T 25;8:C 25@B8:0;L=C ?@>BO6=VABL. V= ?>;O30T C B><C, I> :@0?;V, O:V ?V4 4VTN 25@B8:0;L=8E @CEV2 ?>?040NBL =0 45O:89 @V25=L 7 @V7=8E 28A>B, <0NBL @V7=V B5<?5@0BC@8. &V @V7=8FV B5<?5@0BC@ =57=0G=V ( 45AOBV G0AB:8 3@04CA0) , 0;5 ?@8 2V4=>A=> 28A>:8E 4>40B=8E B5<?5@0BC@0E, I> A?>AB5@V30NBLAO 2 =87L:8E H8@>B0E, <>6CBL 2V4V3@0B8 ?><VB=C @>;L. G5284=>, @V7=8FV B5<?5@0BC@ ?@82>4OBL 4> 28=8:=5==O @V7=8FV B8A:V2 =0A8G5==O =04 ?>25@E=O<8 F8E :@0?5;L V, O:I>, =0?@8:;04, 1V;LH B5?;0 :@0?;O 7=0E>48BLAO C AB0=V @V2=>2038 7 >B>GCNG8< ?0@>?>2VB@O=8< A5@54>28I5<, =04 ?>25@E=5N 1V;LH E>;>4=>W :@0?;V CB2>@NTBLAO ?5@5A8G5==O. 024O:8 FL><C 2V41C20TBLAO ?5@53>=:0 2>48 7 B5?;>W :@0?;V =0 E>;>4=C. $V78G=89 <5E0=V7< B0:>3> ?@>F5AC T 0=0;>3VG=8< DV78G=><C <5E0=V7<C 270T<>4VW <V6 ?5@5>E>;>465=>N :@0?;5N 9 G0AB:>N ;L>4C. ;O B>3>, I>1 7 O282AO 5D5:B :>=45=A0FV9=>3> 7@>AB0==O 1V;LH E>;>4=>W :@0?;V, ?>@V2=O==89 4> 5D5:BC 270T<>4VW @V4:>W D078 7 G0AB:>N ;L>4C C E<0@V, =5>1EV4=>, I>1 @V7=8FO B5<?5@0BC@, ?@> O:C 945BLAO, 4>@V2=N20;0 45:V;L:>< 3@04CA0< ?@8 =87L:8E B5<?5@0BC@0E V ;8H5 G0AB:0< 3@04CA0 ?@8 28A>:8E 4>40B=8E B5<?5@0BC@0E. A:V;L:8 25;8:V @V7=8FV B5<?5@0BC@ =0 287=0G=8E @V2=OE C E<0@V =5 7CAB@VG0NBLAO, F59 5D5:B =5 <>65 2V4V3@020B8 7=0G=C @>;L C ?><V@=8E V 28A>:8E H8@>B0E, 45 E<0@8 <0NBL, O: ?@028;>, =87L:C B5<?5@0BC@C, 9 T ACBBT28< C =87L:8E H8@>B0E, 45 =86=O G0AB8=0 E<0@8 7=0E>48BLAO 2 >1;0ABV 28A>:8E 4>40B=8E B5<?5@0BC@. 7- 70 @V7=>W @50:FVW :@0?5;L =0 4VN 3@02VB0FV9=8E, 5;5:B@8G=8E 01> 05@>48=0<VG=8E A8; <>6CBL 2V41C20B8AO WE=V 7VB:=5==O. @8 FL><C, :@0?;V <>6CBL 7;820B8AO. ;820==O :@0?5;L ?@8 WE ?04V==V =07820NBL :>03C;OFVTN. 4=8< 7 >A=>2=8E 5D5:BV2 V7 ?5@5;VG5=8E 28I5, T 3@02VB0FV9=89 5D5:B, O:89 ?@82>48BL 4> ?04V==O :@0?5;L @V7=8E @>7<V@V2 7 @V7=>N H284:VABN. "><C 206;828< T ?8B0==O ?@> CAB0;5=C H284:VABL ?04V==O :@0?5;L C E<0@V. 040NG0 ?V4 4VTN A8;8 2038 E M B E D M a t h T y p e , :@0?;O 707=0T 4VN A8;8 05@>48=0<VG=>3> >?>@C E M B E D M a t h T y p e , O:0 4;O :@0?;V, I> <0T AD5@8G=C D>@<C 4>@V2=NT: E M B E D M a t h T y p e , ( 1 . 1 ) 45 E M B E D M a t h T y p e - @04VCA AD5@8, E M B E D M a t h T y p e - 3CAB8=0 ?>2VB@O, E M B E D M a t h T y p e - H284:VABL ?04V==O :@0?;V, E M B E D M a t h T y p e - :>5DVFVT=B 05@>48=0<VG=>3> >?>@C. /:I> 28:>@8AB0B8 G8A;> 59=>;L40 E M B E D M a t h T y p e , ( 1 . 2 ) 45 E M B E D M a t h T y p e - 48=0<VG=0 2 O7:VABL ?>2VB@O, B> D>@<C; <>6=0 70?8A0B8 C D>@<V E M B E D M a t h T y p e . ( 1 . 3 ) !8;0 2038, I> 4VT =0 AD5@C, >G5284=>, 4>@V2=NT E M B E D M a t h T y p e , ( 1 . 4 ) 45 E M B E D M a t h T y p e - 3CAB8=0 :@0?;V. @8 ?04V==V FO A8;0 G0AB:>2> :><?5=ACTBLAO A8;>N @EV<540 E M B E D M a t h T y p e , ( 1 . 5 ) 45 E M B E D M a t h T y p e - 3CAB8=0 ?>2VB@O. V2=>4VNG0 42>E >AB0==VE A8; T E M B E D M a t h T y p e . ( 1 . 6 ) ;O 2>4O=>W :@0?;V E M B E D M a t h T y p e . "><C, 2;0A=5 :06CG8, 4VTN A8;8 @EV<540 <>6=0 7=5EBC20B8 V 22060B8, I> E M B E D M a t h T y p e . ( 1 . 7 ) @0?;O ?040T 2 ?>2VB@V 7 CAB0;5=>N H284:VABN, O:I> E M B E D M a t h T y p e . 0 B0:8E C<>2 E M B E D M a t h T y p e ( 1 . 8 ) 01>, O:I> 28:>@8AB0B8 D>@<C;C ( 1 . 3 ) E M B E D M a t h T y p e . ( 1 . 9 ) !B>:A 287=0G82, I> 4;O ?>;O B5GVW =02:>;> AD5@8 ?@8 4C65 <0;8E G8A;0E 59=>;L40 E M B E D M a t h T y p e . # B0:><C @07V E M B E D M a t h T y p e , ( 1 . 1 0 ) 45 E M B E D M a t h T y p e . &O :204@0B8G=0 70;56=VABL H284:>ABV ?04V==O 2V4 @>7<V@V2 =07820TBLAO 70:>=>< !B>:A0. >=0 A?@0254;820 4;O :@0?5;L, I> <0NBL @04VCA E M B E D M a t h T y p e <:<. :A?5@8<5=B8 7 ?040NG8<8 AD5@0<8 ?>:07CNBL, I> ?@8 25;8:8E G8A;0E 59=>;L40 E M B E D M a t h T y p e =5 70;568BL 2V4 E M B E D M a t h T y p e V ?@81;87=> 4>@V2=NT 0 , 4 5 . V4AB0282H8 F5 9>3> 7=0G5==O 2 D>@<C;C ( 1 . 8 ) , >B@8<0T<>: E M B E D M a t h T y p e , ( 1 . 1 1 ) E M B E D M a t h T y p e . ( 1 . 1 2 ) # D>@<C;V ( 1 . 8 ) E M B E D M a t h T y p e - 3CAB8=0 ?>2VB@O, 2 O:><C ?040T :@0?;O, ( E M B E D M a t h T y p e - 3CAB8=0 ?>2VB@O ?@8 E M B E D M a t h T y p e 3 0 V E M B E D M a t h T y p e ) . 040NG8 C ?>2VB@V, 4>I>2V :@0?;V E0@0:B5@87CNBLAO 25;8:8<8 G8A;0<8 59=>;L40, 0;5 WE=O D>@<0 2V4@V7=OTBLAO 2V4 V450;L=> AD5@8G=>W. B65, D>@<C;0 ( 1 . 1 1 ) ?>@V2=O=> 4>1@5 >?8ACT H284:VABL ?04V==O :@0?5;L 4>IC ;8H5 2 >1<565=><C V=B5@20;V @>7<V@V2. @8 ?04V==V :@0?5;L @04VCA>< 0 , 5 << V 1V;LH5 2V41C20TBLAO WE 45D>@<0FVO. ;O B0:8E :@0?5;L 40=V ?@> H284:VABL ?04V==O <>6=0 >B@8<0B8 ;8H5 5:A?5@8<5=B0;L=8< H;OE><. @8 FL><C 22>48BLAO ?>=OBBO 5:2V20;5=B=>3> @04VCA0 ( 01> 4V0<5B@0) :@0?;V. V4 5:2V20;5=B=8< @04VCA>< @>7C<VTBLAO @04VCA B0:>3> H0@C, >1 T< O:>3> 4>@V2=NT >1 T<C 45D>@<>20=>W :@0?;V. :A?5@8<5=B0;L=V 40=V ?@> H284:>ABV ?04V==O 4>I>28E :@0?5;L >45@60=V 0==>< V V=F5@>< 70 =>@<0;L=8E 0B<>AD5@=8E C<>2, 70 O:V ?@89<0;8AO: E M B E D M a t h T y p e 3 0 V E M B E D M a t h T y p e . >=8 ?@82>4OBLAO C B01;. 1 . 1 . A:V;L:8 3CAB8=0 ?>2VB@O 7<5=HCTBLAO 7 28A>B>N, =0 45O:V9 28A>BV :@0?;O ?040T H284H5, =V6 1V;O 75<=>W ?>25@E=V, ?@81;87=> 2V4?>2V4=> 70:>=C :204@0B=>3> :>@5=N ( 1 . 1 1 ) . V;O 75<=>W ?>25@E=V =091V;LH 25;8:V 4>I>2V :@0?;V ?040NBL 7 H284:VABN, I> =5 ?5@51V;LHCT ?@81;87=> 9 </ A, =0 @V2=V ?>25@E=V 5 0 0 3 0 25@E=O 3@0=8FO H284:>ABV ?04V==O :@0?5;L 4>@V2=NT 1 3 </ A. "01;8FO 1 . 1 0;56=VABL CAB0;5=>W H284:>ABV ?04V==O :@0?;V 2V4 WW @>7<V@V2 ( 4V0<5B@0 5:2V20;5=B=>W AD5@8) 70 0==>< V V=F5@>< V0<5B@, << (284:VABL ?04V==O, </ A V0<5B@, << (284:VABL ?04V==O, </ A V0<5B@, << (284:VABL ?04V==O, </ A 0 , 1 0 , 2 7 1 , 6 5 , 6 5 4 , 0 8 , 8 3 0 , 2 0 , 7 2 1 , 8 6 , 0 9 4 , 2 8 , 9 2 0 , 3 1 , 1 7 2 , 0 6 , 4 9 4 , 4 8 , 9 8 0 , 4 1 , 6 2 2 , 2 6 , 9 0 4 , 6 9 , 0 3 0 , 5 2 , 0 6 2 , 4 7 , 2 7 4 , 8 9 , 0 7 0 , 6 2 , 4 7 2 , 6 7 , 5 7 5 , 0 9 , 0 9 0 , 7 2 , 8 7 2 , 8 7 , 8 2 5 , 2 9 , 1 2 0 , 8 3 , 2 7 3 , 0 8 , 0 6 5 , 4 9 , 1 4 0 , 9 3 , 6 7 3 , 2 8 , 2 6 5 , 6 9 , 1 6 1 , 0 4 , 0 3 3 , 4 8 , 4 4 5 , 8 9 , 1 7 1 , 2 4 , 6 4 3 , 6 8 , 6 0 - - 1 , 4 5 , 1 7 3 , 8 8 , 7 2 - - 0=V, I> ?@82>4OBLAO C B01;. 2 . 1 , ?>:07CNBL, I> D>@<C;0 ( 1 . 1 1 ) 70157?5GCT 4>?CAB8<C 0?@>:A8<0FVN H284:>AB59 ?04V==O 2 V=B5@20;V @04VCAV2 E M B E D M a t h T y p e <<, O:I> E M B E D M a t h T y p e . # ?@><V6=><C V=B5@20;V @>7<V@V2 <V6 >1;0ABO<8, 45 70:>= !B>:A0 V 70:>= :204@0B=>3> :>@5=O T A?@0254;828<, <>65 28:>@8AB>2C20B8AO D>@<C;0 E M B E D M a t h T y p e , ( 1 . 1 3 ) E M B E D M a t h T y p e . # !( 70 40=8<8 A?>AB5@565=L C 3V@AL:V9 >1A5@20B>@VW =0 28A>BV 3 3 5 0 < 28O2;5=0 B0:0 70;56=VABL H284:>ABV ?04V==O G0AB>: B25@4>W D078 ( A=V3>2>W :@C?8) 2V4 WE=L>3> 4V0<5B@0 E M B E D M a t h T y p e : E M B E D M a t h T y p e . ( 1 . 1 4 ) # 28?04:C G0AB>: 35:A03>=0;L=>W D>@<8 H284:VABL ?04V==O 287=0G0TBLAO D>@<C;>N E M B E D M a t h T y p e . ( 1 . 1 5 ) 1 . 1 . 2 >5DVFVT=B 70E20BC ( 70E>?;5==O) # :C1VG=><C A0=B8<5B@V E<0@8 <VABOBLAO A>B=V :@0?5;L @V7=8E @>7<V@V2. 5;8:V :@0?;V ?040NBL H284H5 4@V1=8E, 4>30=ONBL WE V 4>A8BL G0AB> 7HB>2ECNBLAO >4=0 7 >4=>N. VB:=5==O I5 =5 30@0=BCT 7;820==O :@0?5;L. @8 7HB>2EC20==V ?0@8 :@0?5;L <>6;82V 270T<>4VW :V;L:>E B8?V2: 0) >;8 7HB>2ECNBLAO 25;8:V :@0?;V ?@8 25;8:8E :CB0E <V6 WE=V<8 B@0T:B>@VO<8, 2>=8 <>6CBL 2V4A:>G8B8 >4=0 2V4 >4=>W; 1) /:I> 7HB>2ECNBLAO :@0?;V 7 @04VCA><, <5=H8< 70 1 0 0 <:<, B> <>65 2V41CB8AO WE=T 7;8BBO; 2) @0?;V <>6CBL B8<G0A>2> >1 T4=0B8AO, 0 ?>BV< @>7V9B8AO, 715@V30NG8 A2>N V=482V4C0;L=VABL; 3) VA;O B8<G0A>2>3> >1 T4=0==O :@0?;V <>6CBL @>74@>18B8AO =0 @O4 1V;LH 4@V1=8E :@0?5;L. "8?8 270T<>4VW 70;560BL 2V4 @>7<V@V2 :@0?5;L V WE B@0T:B>@V9, 0 B0:>6 2V4 =0O2=>ABV 5;5:B@8G=8E A8; V 2V4 V=H8E D0:B>@V2. ;O B>3>, I>1 ?>7=09><8B8AL 7 >A=>2=8<8 ?>=OBBO<8 B5>@VW :>03C;OFV9=>3> 7@>AB0==O :@0?5;L, @>73;O=5<> V450;V7>20=89 28?04>: ?04V==O :@0?;V @04VCA0 E M B E D M a t h T y p e ( 1C45<> WW =07820B8 :0?;5N- :>;5:B>@><) :@V7L A?>:V9=C ( =5 BC@1C;V7>20=C) <>=>48A?5@A=C E<0@C, I> A:;040TBLAO 7 :@0?5;L @04VCA>< E M B E D M a t h T y p e , :>=F5=B@0FVO O:8E 4>@V2=NT E M B E D M a t h T y p e . >7=0G8<> G5@57 E M B E D M a t h T y p e G8A;> 7VB:=5=L 70 G0A E M B E D M a t h T y p e :@0?;V- :>;5:B>@0, O:0 ?040T G5@57 >4=>@V4=C E<0@C :@0?5;L E M B E D M a t h T y p e . >=> 4>@V2=NT E M B E D M a t h T y p e . ( 1 . 1 6 ) >5DVFVT=B ?@>?>@FV9=>ABV E M B E D M a t h T y p e 70;568BL 2V4 @>7<V@V2 :@0?5;L V =07820TBLAO :>5DVFVT=B>< :>03C;OFVW. V= <0T @>7<V@=VABL E M B E D M a t h T y p e ( :>=F5=B@0FVO :@0?5;L F5 :V;L:VABL WE C E M B E D M a t h T y p e , B>1B> <0T @>7<V@=VABL E M B E D M a t h T y p e ) V G8A5;L=> 4>@V2=NT B><C >1 T<C, 7 O:>3> 2AV :@0?;V @04VCA>< E M B E D M a t h T y p e 70 1 A5:C=4C 7VB:=CBLAO 7 :0?;5N- :>;5:B>@>< E M B E D M a t h T y p e . V4:@5A;8<>, I>, :06CG8 ?@> >1 T< ?@8 287=0G5==V E M B E D M a t h T y p e , <8 @>7C<VT<> =5 ?@>ABV@ O:>W- =51C4L D>@<8, 0 <V@C :V;L:>ABV :@0?5;L, @V2=><V@=> @>7?>4V;5=8E C FL><C >1 T<V. 5;8G8=0 E M B E D M a t h T y p e E0@0:B5@87CT G8A;> :@0?5;L @04VCA>< E M B E D M a t h T y p e , O:V =0 ?@>BO7V 1 A 7VB:=CBLAO 7 :0?;5N- :>;5:B>@><. &N 25;8G8=C =07820NBL ?>B>:>< :@0?5;L @04VCA>< E M B E D M a t h T y p e =0 :@0?;N @04VCA>< E M B E D M a t h T y p e E M B E D M a t h T y p e . ( 1 . 1 7 ) @8?CAB8<>, I> C ?>2VB@V BC@1C;5=B=VABL 2V4ACB=O, =5 4VNBL 5;5:B@8G=V A8;8 9 I> 1@>C=V2AL:8<8 @CE0<8 <>6=0 7=5EBC20B8. ">4V :@0?;V 1C4CBL ?040B8 25@B8:0;L=> 2=87 ?V4 4VTN A8;8 2038. =H8<8 A;>20<8, 1C45<> 22060B8, I> 7VB:=5==O >1C<>2;5=V BV;L:8 @V7=8F5N H284:>AB59 ?04V==O :@0?5;L @V7=8E @>7<V@V2, B>1B> I> 2V41C20TBLAO 3@02VB0FV9=0 :>03C;OFVO. >;8 1 :@0?;V =5 270T<>4VO;8 <V6 A>1>N, B>, >G5284=>, 25;8G8=0 E M B E D M a t h T y p e 287=0G0;0AL 18 ?@>AB8< A?V22V4=>H5==O< E M B E D M a t h T y p e , ( 1 . 1 8 ) 45 E M B E D M a t h T y p e V E M B E D M a t h T y p e - H284:VABL ?04V==O :@0?;V- :>;5:B>@0 V :@0?;V @04VCA0 E M B E D M a t h T y p e 2V4?>2V4=>. @>B5 G8A;> 7VB:=5=L <>65 1CB8 7=0G=> <5=H8<. &5 >1C<>2;NTBLAO B8<, I> @CE8 :@0?5;L ?@8 WE=L><C ?04V==V =5<>6;82> @>73;O40B8 =570;56=8<8. V6 =8<8 VA=CT 3V4@>48=0<VG=0 270T<>4VO. @8 ?04V==V :@0?;V- :>;5:B>@0 ?>2VB@O, I> <0T 2 O7:VABL E M B E D M a t h T y p e , =V18 B> @>7A>2CTBLAO :@0?;5N, V 9>3> G0AB:8 ?@8410NBL ?>?5@5G=C A:;04>2C H284:>ABV 9 >1BV:0NBL :@0?;N- :>;5:B>@. O7:V A8;8 4VNBL =0 :@0?;V @04VCA>< E M B E D M a t h T y p e V ?@8<CHCNBL WE @CE0B8AO ?> B@0T:B>@VOE, 1;87L:8E 4> B@0T:B>@V9 G0AB>: ?>2VB@O. # @57C;LB0BV 7 :0?;5N- :>;5:B>@>< 7VB:=CBLAO :@0?;V E M B E D M a t h T y p e , F5=B@8 O:8E 7=0E>4OBLAO =5 2 F8;V=4@V 7 ?5@5@V7>< E M B E D M a t h T y p e , 0 2 F8;V=4@V <5=H>3> ?5@5@V7C E M B E D M a t h T y p e . V4=>H5==O D0:B8G=>3> G8A;0 7VB:=5=L 4> G8A;0 7VB:=5=L, O:5 1 <0;> <VAF5 ?@8 ?>2=><C 35><5B@8G=><C 70E>?;5==V, =07820TBLAO :>5DVFVT=B>< 7VB:=5==O E M B E D M a t h T y p e . V= 70;568BL 3>;>2=8< G8=>< 2V4 @>7<V@C :@0?;V- :>;5:B>@0 V @>7<V@V2 :@0?5;L, I> 70E>?;NNBLAO E M B E D M a t h T y p e . ( 1 . 1 9 ) /: 1C;> 707=0G5=>, =5 702648 :@0?;V, I> 7VB:=C;8AO, 7;820NBLAO. V4=>H5==O G8A;0 7;820=L 4> G8A;0 7VB:=5=L =07820TBLAO :>5DVFVT=B>< 7;8BBO. >03C;OFV9=5 7@>AB0==O :@0?5;L 287=0G0TBLAO :>5DVFVT=B>< 70E>?;5==O, O:89 4>@V2=NT 4>1CB:C :>5DVFVT=BV2 7VB:=5==O V 7;8BBO. 01>@0B>@=V 4>A;V465==O 7VB:=5==O 4@V1=8E :@0?5;L ?>:070;8, I> :>5DVFVT=B 7;8BBO 1;87L:89 4> >48=8FV, O:I> :@0?;V 70@O465=V 01> O:I> VA=CT 5;5:B@8G=5 ?>;5. A:V;L:8 2 @50;L=8E E<0@0E 702648 VA=CNBL A;01:V ?>;O B0 70@O48, ?@89<0NBL, I> :>5DVFVT=B 70E>?;5==O 4>@V2=NT :>5DVFVT=BC 7VB:=5==O 9 287=0G0TBLAO D>@<C;>N ( 1 . 1 9 ) . B65, 7 C@0EC20==O< 707=0G5=>W 3V4@>48=0<VG=>W 270T<>4VW :>5DVFVT=B :>03C;OFVW 287=0G0TBLAO @V2=O==O< E M B E D M a t h T y p e ( 1 . 2 0 ) ( V=45:A E M B E D M a t h T y p e 707=0G0T, I> 945BLAO ?@> 3@02VB0FV9=C :>03C;OFVN) . 5;8G8=0 E M B E D M a t h T y p e 2V4V3@0T 4C65 206;82C @>;L C B5>@VW CB2>@5==O E<0@ B0 >?04V2. B5>@5B8G=5 9 5:A?5@8<5=B0;L=5 287=0G5==O ?>2 O70=5 7 25;8:8<8 B@C4=>I0<8, O:V 2 >A=>2=><C 1C;8 ?>4>;0=V 2 7 0 - E @>:0E. =0G5==O :>5DVFVT=B0 70E>?;5==O 4;O @V7=8E @>7<V@V2 :@0?5;L- :>;5:B>@V2 V :@0?5;L @04VCA0 E M B E D M a t h T y p e 7>1@06CNBLAO =0 @8A. 1 . 1 . =L>3> <>6=0 7@>18B8 28A=>2>:, I> :>5DVFVT=B 70E>?;5==O T 7@>AB0NG0 DC=:FVO E M B E D M a t h T y p e V E M B E D M a t h T y p e . @8 7=0G5==OE E M B E D M a t h T y p e <:< FO DC=:FVO C 1V;LH><C AB5?5=V 70;568BL 2V4 E M B E D M a t h T y p e . # B01;. 1 . 2 ?@82>4OBLAO 7=0G5==O :>5DVFVT=B0 70E>?;5==O 4;O :@0?5;L @04VCAV2 E M B E D M a t h T y p e V E M B E D M a t h T y p e . ;0A=5 :06CG8, E M B E D M a t h T y p e T DC=:FVTN =5 BV;L:8 E M B E D M a t h T y p e V E M B E D M a t h T y p e , 0;5 V G8A;0 E M B E D M a t h T y p e . '8A;> 59=>;L40, 2 A2>N G5@3C, 4;O :@0?;V E M B E D M a t h T y p e 70;568BL =5 BV;L:8 2V4 WW @04VCA0 0;5 9 2V4 :V=5<0B8G=>W 2 O7:>ABV E M B E D M a t h T y p e , O:0 45I> 7<V=NTBLAO 7V 7<V=N20==O< B5<?5@0BC@8. 4=0:, 4;O =86=L>W B@>?>AD5@8 FO 70;56=VABL ?@>O2;OTBLAO =5 4C65 A8;L=>, B><C @V7=8FO C<>2 C E<0@0E V C<>2 ;01>@0B>@=8E 5:A?5@8<5=BV2, 70 4>?><>3>N O:8E 1C;8 >B@8<0=V 7=0G5==O :>5DVFVT=B0 70E>?;5==O, I> ?@82>4OBLAO 28I5, ?@>O2;OTBLAO =5 4C65 7=0G=> V ?>2 O70=V 7 F8< 7<V=N20==O< E M B E D M a t h T y p e =5 28E>4OBL 70 3@0=8FV ?>E81>: @>7@0EC=:>28E B0 5:A?5@8<5=B0;L=8E <5B>4V2 287=0G5==O. 8A. 1 . 1 0;56=VABL :>5DVFVT=B0 70E>?;5==O E M B E D M a t h T y p e 2V4 @04VCAV2 :@0?5;L, O:V AB8:0NBLAO ( F8D@8 1V;O :@828E ?>7=0G0NBL E M B E D M a t h T y p e , <:<) "01;8FO 1 . 2 =0G5==O :>5DVFVT=B0 70E>?;5==O E M B E D M a t h T y p e 4;O :@0?5;L @04VCA>< E M B E D M a t h T y p e , O:V ?040NBL :@V7L E<0@C :@0?5;L E M B E D M a t h T y p e <:< E M B E D M a t h T y p e <:< 1 0 1 5 2 0 3 0 4 0 5 0 6 0 8 0 1 0 0 3 0 , 1 8 - < 0 , 0 1 < 0 , 0 1 < 0 , 0 1 0 , 0 1 1 0 , 0 2 5 0 , 0 6 0 0 , 0 9 0 5 0 , 0 2 7 0 , 0 2 4 0 , 0 2 2 0 , 0 3 3 0 , 0 5 6 0 , 1 0 0 , 1 5 0 , 2 8 0 , 3 6 1 0 0 , 0 2 6 0 , 0 5 4 0 , 0 7 7 0 , 3 2 0 , 4 9 0 , 6 1 0 , 7 0 0 , 8 0 0 , 8 6 1 5 - 0 , 0 2 8 0 , 0 8 2 0 , 5 0 0 , 7 4 0 , 8 3 0 , 8 7 0 , 9 2 0 , 9 4 2 0 - - 0 , 1 0 0 , 5 0 0 , 7 7 0 , 8 6 0 , 9 0 0 , 9 3 0 , 9 6 1 . 2 @>AB0==O :@0?5;L E<0@ ?V4 2?;82>< 3@02VB0FV9=>W :>03C;OFVW @8?CAB8<>, I> 2 >48=8FN G0AC 2 ?>;V48A?5@A=V9 E<0@V 7 @>7?>4V;>< :@0?5;L 70 @>7<V@0<8 E M B E D M a t h T y p e V 7 CAB0;5=>N H284:VABN E M B E D M a t h T y p e ?040T :@0?;O- :>;5:B>@ E M B E D M a t h T y p e G5@57 A:C?G5==O 1V;LH 4@V1=8E :@0?5;L. 1 T<, C O:><C 70 >48=8FN G0AC <>6;825 7VB:=5==O :@0?;V- :>;5:B>@0, 7 :@0?;O<8 @04VCA0 E M B E D M a t h T y p e , 4>@V2=NT E M B E D M a t h T y p e . ( 1 . 2 1 ) /: 1C;> ?>:070=>, :@0?;O- :>;5:B>@ 7HB>2ECTBLAO =5 7 CAV<0 :@0?;O<8, 0 ;8H5 7 WW G0AB8=>N, O:0 287=0G0TBLAO :>5DVFVT=B>< 70E>?;5==O E M B E D M a t h T y p e . B65, A5@54=T G8A;> :@0?5;L @04VCA>< 2V4 E M B E D M a t h T y p e 4> E M B E D M a t h T y p e , 70E>?;5=8E :@0?;5N- :>;5:B>@>< 70 >48=8FN G0AC 2 >48=8FV >1 T<C, T E M B E D M a t h T y p e . ( 1 . 2 2 ) 030;L=C H284:VABL 7@>AB0==O >1 T<C :@0?;V- :>;5:B>@0 E M B E D M a t h T y p e <>6=0 >B@8<0B8, O:I> ?><=>68<> >AB0==T A?V22V4=>H5==O =0 >1 T< 70E>?;5=>W :@0?;V B0 ?@> V=B53@CT<> 9>3> ?> 2AVE @>7<V@0E 70E>?;5=8E :@0?5;L E M B E D M a t h T y p e . ( 1 . 2 3 ) A:V;L:8 E M B E D M a t h T y p e , B> E M B E D M a t h T y p e . ( 1 . 2 4 ) E M B E D M a t h T y p e . ( 1 . 2 5 ) V2=O==O ( 1 . 2 2 ) 2@0E>2CT @>7<V@8 9 H284:VABL ?04V==O :@0?5;L, I> 70E>?;NNBLAO :@0?;5N- :>;5:B>@><. /:I> @>7<V@8 2AVE F8E :@0?5;L =01030B> <5=H89, =V6 @>7<V@ :@0?;V- :>;5:B>@0, B> <>6=0 22060B8, I> E M B E D M a t h T y p e V E M B E D M a t h T y p e . ">4V 1C45<> <0B8 E M B E D M a t h T y p e . ( 1 . 2 6 ) 0 >A=>2V 2V4><>W B5>@5<8 ?@> A5@54=T <>6=0 7 ?V4 7=0:0 V=B53@0;0 28=5AB8 A5@54=T 7=0G5==O :>5DVFVT=B0 70E>?;5==O E M B E D M a t h T y p e . /:I> 70 B0:>W C<>28 @>74V;8B8 V ?><=>68B8 ?@02C G0AB8=C @V2=O==O =0 E M B E D M a t h T y p e , B> ?@8945<> 4> @V2=O==O E M B E D M a t h T y p e . ( 1 . 2 7 ) /: >G5284=> =B53@0; ?@02>W G0AB8=8 @V2=O==O ( 1 . 2 4 ) <0T A5=A <0A8 2>48, O:0 CB@8<CTBLAO 2 >48=8FV >1 T<C E<0@8. &O 25;8G8=0 =07820TBLAO, O: 2V4><>, 2>4=VABN E<0@8 E M B E D M a t h T y p e . ( 1 . 2 8 ) "0:8< G8=><, 70 AD>@<C;L>20=8E C<>2 E M B E D M a t h T y p e . ( 1 . 2 9 ) 2545<> ?@>ABC >?5@0FVN E M B E D M a t h T y p e . ( 1 . 3 0 ) >EV4=0 E M B E D M a t h T y p e , <0T A5=A 7<V=5==O 70 G0A>< 25@B8:0;L=>W :>>@48=0B8 :@0?;V- :>;5:B>@0, B>1B> 25@B8:0;L=>W H284:>ABV WW ?04V==O. /:I> C E<0@V A?>AB5@V30NBLAO 25@B8:0;L=V B>:8 ?>2VB@O H284:VABN E M B E D M a t h T y p e , B>, >G5284=>, E M B E D M a t h T y p e . ( 1 . 3 1 ) <V=5==O @>7<V@V2 :@0?;V- :>;5:B>@0 7 28A>B>N ?@8 WW ?04V==V :@V7L ?>;V48A?5@A=C E<0@C :@0?5;L 287=0G0TBLAO D>@<C;>N E M B E D M a t h T y p e . ( 1 . 3 2 ) @8 2V4ACB=>ABV 25@B8:0;L=8E @CEV2 E M B E D M a t h T y p e E M B E D M a t h T y p e . ( 1 . 3 3 ) V4=>A=> 7=0:0 <V=CA C >AB0==V9 D>@<C;V B@510 <0B8 =0 C207V, I> C WW ;V2V9 G0AB8=V E M B E D M a t h T y p e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a t h T y p e B0:8E :@0?5;L @04VCA>< E M B E D M a t h T y p e =0 :@0?;N @04VCA>< E M B E D M a t h T y p e 287=0G0TBLAO 70:>=0<8 728G09=>W 48DC7VW. B65, 0=0;>3VG=> D>@<C;V, I> 287=0G0T 48DC7VN <>;5:C; ?0@8 =0 :@0?;N, 4;O ?>B>:C E M B E D M a t h T y p e <0T<> E M B E D M a t h T y p e , ( 1 . 3 4 ) 45 E M B E D M a t h T y p e - 2V4=>A=89 :>5DVFVT=B 48DC7VW, A:;04>2V O:>3> >?8ACNBLAO D>@<C;>N E M B E D M a t h T y p e , ( 1 . 3 5 ) 45 E M B E D M a t h T y p e - AB0;0 >;LF<0=0; E M B E D M a t h T y p e - @CE;82VABL :@0?5;L @04VCA>< E M B E D M a t h T y p e ; E M B E D M a t h T y p e - 48=0<VG=0 2 O7:VABL ?>2VB@O; E M B E D M a t h T y p e - :>=F5=B@0FVO :@0?5;L @04VCA0 E M B E D M a t h T y p e . ">4V 4;O :>5DVFVT=B0 1@>C=V2AL:>W :>03C;OFVW <0T<>: E M B E D M a t h T y p e . ( 1 . 3 6 ) # 28?04:C G0AB>: @V2=8E @>7<V@V2 :>5DVFVT=B :>03C;OFVW =5 70;568BL 2V4 @>7<V@V2 :@0?5;L E M B E D M a t h T y p e . ( 1 . 3 7 ) >BV: <0A8 C FL><C 28?04:C B56 =5 70;568BL 2V4 @04VCA0 :@0?5;L V 287=0G0TBLAO BV;L:8 2>4=VABN E<0@8 E M B E D M a t h T y p e E M B E D M a t h T y p e . ( 1 . 3 8 ) @V< 1@>C=V2AL:8E @CEV2 :@0?5;L =0 V=B5=A82=VABL :>03C;OFVW 2?;820NBL V 5;5:B@8G=V A8;8, I> 4VNBL =0 :@0?;V. "0:0 :>03C;OFVO, O: 707=0G0;>AO 28I5, =>A8BL =072C 5;5:B@>AB0B8G=>W. >=0 >1C<>2;NTBLAO 01> 270T<>4VTN G0AB8=8 :@0?5;L, O:V =5ACBL =0 A>1V 5;5:B@8G=89 70@O4, 01> 270T<>4VTN :@0?5;L 7 5;5:B@8G=8< ?>;5<, O:I> 2>=> 2V4@V7=OTBLAO 2V4 =C;O. # ?5@H><C 28?04:C 945BLAO ?@> 5;5:B@>AB0B8G=C :>03C;OFVN. "CB ACBBT28<8 28O2;ONBLAO BV;L:8 ?0@=V 270T<>4VW :@0?5;L, O:V =5ACBL =0 A>1V 70@O48 @V7=8E 7=0:V2. #@0E>2CNG8 BV;L:8 :C;>=V2AL:V A8;8, <>6=0 22060B8, I> ?>BV: E M B E D M a t h T y p e :@0?5;L @04VCA>< E M B E D M a t h T y p e =0 :@0?;N @04VCA>< E M B E D M a t h T y p e T AB0FV>=0@=8<. ">4V =0 2V4AB0=V E M B E D M a t h T y p e , O:C ?@>E>48BL :@0?;O 70 G0A E M B E D M a t h T y p e , A8;C 270T<>4VW :@0?5;L 7 70@O40<8 E M B E D M a t h T y p e V E M B E D M a t h T y p e E M B E D M a t h T y p e ( 1 . 3 9 ) <>6=0 22060B8 ?>ABV9=>N, >A:V;L:8 E M B E D M a t h T y p e , ( 1 . 4 0 ) E M B E D M a t h T y p e . ( 1 . 4 1 ) 02VBL ?@8 <V=V<0;L=><C E M B E D M a t h T y p e , V E M B E D M a t h T y p e <:< ( E M B E D M a t h T y p e 1 , 2 ) , F5 A?V22V4=>H5==O <0T ?>@O4>: E M B E D M a t h T y p e . 28G09=>, 7VB:=5==O <V6 :@0?;O<8 ?>28==V 1CB8 =0ABV;L:8 @V4:8<8, I>1 ?@><V6>: G0AC <V6 =8<8 1C2 7=0G=> 1V;LH89 70 E M B E D M a t h T y p e . # V=H><C 28?04:C 7040G0 AB0T =5AB0FV>=0@=>N G5@57 7<V=5==O @>7<V@V2 V 70@O4C :@0?;V. !B0FV>=0@=89 ?>BV: E M B E D M a t h T y p e 287=0G0TBLAO D>@<C;>N E M B E D M a t h T y p e , ( 1 . 4 2 ) 45 E M B E D M a t h T y p e - :>=F5=B@0FVO :@0?5;L @04VCA0 E M B E D M a t h T y p e , 0 E M B E D M a t h T y p e , >A:V;L:8 70@O48 E M B E D M a t h T y p e V E M B E D M a t h T y p e <0NBL @V7=V 7=0:8. B65, :>5DVFVT=B :>03C;OFVW, >1C<>2;5=89 270T<>4VTN 42>E 70@O465=8E :@0?5;L, 4>@V2=NT E M B E D M a t h T y p e . ( 1 . 4 3 ) 0O2=VABL 5;5:B@8G=>3> ?>;O ?@82>48BL, ?>- ?5@H5, 4> ?5@5<VI5==O :@0?5;L, ?>- 4@C35 4> V=4C:C20==O 4>40B:>28E 48?>;L=8E <><5=BV2 C :@0?5;L V, ?>- B@5BT, WE 270T<>4VW. >;L 4@C3>3> 5D5:BC C AB8<C;N20==V 7VB:=5=L :@0?5;L, :>;8 <0NBL <VAF5 @50;L=V 5;5:B@8G=V ?>;O =53@>7>28E E<0@, =5EB>2=> <0;0. 5@H89 5D5:B B0:>6 <0T =525;8:5 7=0G5==O. /: ?>:07CNBL >FV=:8, :>;8 =0?@C65=VABL 5;5:B@8G=>3> ?>;O E M B E D M a t h T y p e B/ <, :>5DVFVT=B :>03C;OFVW E M B E D M a t h T y p e <0T 4C65 <0;5 7=0G5==O V =5 ?5@51V;LHCT E M B E D M a t h T y p e . @8?CAB8<> B5?5@, I> ?>BV: ?>2VB@O, 45 7=0E>4OBLAO :@0?;V, T BC@1C;5=B=8<8. ">4V 420 D0:B>@8 A?@8G8=ONBL 7VB:=5==O :@0?5;L. 5@H89 7 =8E ?>2 O70=89 7 V=5@FVTN :@0?5;L B0 WE=L>N @V7=>N <V@>N 70E>?;5==O BC@1C;5=B=8<8 28E>@0<8. V4?>2V4=89 BC@1C;5=B=89 <5E0=V7< 7VB:=5==O :@0?5;L =07820NBL V=5@FV9=8<. >3> 4VO 0=0;>3VG=0 <5E0=V7<C 3@02VB0FV9=>W :>03C;OFVW. @8 FL><C ?@8A:>@5==O 2V;L=>3> ?04V==O 2V4V3@0T BC@1C;5=B=5 ?@8A:>@5==O E M B E D M a t h T y p e , ( 1 . 4 4 ) 45 E M B E D M a t h T y p e - H284:VABL 48A8?0FVW BC@1C;5=B=>W 5=5@3VW =0 >48=8FN <0A8 ?>2VB@O ( E M B E D M a t h T y p e ) , E M B E D M a t h T y p e - :>5DVFVT=B. =H>N ?@8G8=>N 7VB:=5==O :@0?5;L C BC@1C;5=B=><C ?>B>FV T <>6;820 @V7=8FO H284:>AB59 ?>2VB@O=8E G0AB>: C ACAV4=VE B>G:0E ?@>AB>@C, I> ?@82>48BL 4> 28=8:=5==O 2V4=>A=8E @CEV2 :@0?5;L. &59 <5E0=V7< <0T =072C 48DC7V9=>3>. 70T<>4VO V=5@FV9=>3> B0 48DC7V9=>3> <5E0=V7<V2 >1C<>2;NT BC@1C;5=B=89 :>5DVFVT=B :>03C;OFVW C 283;O4V E M B E D M a t h T y p e , ( 1 . 4 5 ) V= =02>48BLAO 7 B>G=VABN 4> G;5=V2 ?>@O4:C E M B E D M a t h T y p e V E M B E D M a t h T y p e . 57@>7<V@=V <=>6=8:8 E M B E D M a t h T y p e , E M B E D M a t h T y p e 287=0G0NBLAO B5>@5B8G=8< H;OE>< V 4>@V2=NNBL E M B E D M a t h T y p e , E M B E D M a t h T y p e . 1 . 4 #B2>@5==O 4>IC 2 :@0?;8==8E E<0@0E !?>AB5@565==O ?>:07CNBL, I> 4>I <>65 @>72820B8AO 2 B5?;8E :C?G0AB8E E<0@0E 70 ?>@V2=O=> :>@>B:89 G0A ?VA;O B>3>, O: 2>=8 ?>G0;8 D>@<C20B8AO ?@81;87=> 70 1 5 E28;8=. 030;L=> ?@89=OB0 4C<:0 ?>;O30T 2 B><C, I> F59 ?@>F5A ?@8?8=OTBLAO 3@02VB0FV9=>N :>03C;OFVTN :@0?5;L. ;5 4;O FL>3> ?>B@V1=>, I>1 A?5:B@ @>7<V@V2 :@0?5;L 1C2 4>A8BL H8@>:8<. &5 70157?5G8BL 28A>:C G0AB>BC 7VB:=5=L. >03C;OFV9=><C 7@>AB0==N :@0?5;L A5@9>7=> ?5@5H:>460T B5, I> :>5DVFVT=B8 7VB:=5==O <V6 4@V1=8<8 :@0?;O<8 4C65 <0;V. V;LH B>3>, :>=45=A0FV9=>- 48DC7V9=89 ?@>F5A, O:89 T ?5@52060NG8< ?@>F5A>< 7@>AB0==O :@0?5;L C ?>G0B:>2V9 AB04VW, ?@82>48BL 4> 72C6C20==O A?5:B@C :@0?5;L, I>, >G5284=>, B56 CB@C4=NT @>728B>: :>03C;OFVW. B65, B5>@VO ?>28==0 ?>OA=N20B8 CB2>@N20==O 4>I>28E :@0?5;L 70 @50;L=89 G0A 2AC?5@5G <0;8< :>5DVFVT=B0< 7VB:=5==O 9 ?@81;87=> ?0@01>;VG=V9 D>@<V 70:>=C 48DC7V9=>3> 7@>AB0==O. A=CT 45:V;L:0 <>45;59, O:V 704>2>;L=ONBL FV9 28<>7V. >73;O=5<> 45O:V 7 =8E [ 5 , 6 ] . >C5=>< ?@8 4>A;V465==V ?@>F5AC @>728B:C 4>IC 2 B5?;8E E<0@0E 1C;> 28:>@8AB0=5 @V2=O==O :>03C;OFV9=>3> 7@>AB0==O @07>< 7 @V2=O==O< 48DC7V9=>3> 7@>AB0==O :@0?5;L. 2060;>AO, I> 2 >4=>@V4=><C 28AEV4=><C ?>B>FV ?>2VB@O ?V4=V<0TBLAO E<0@0 :@0?5;L >4=0:>2>3> @>7<V@C B0 V A5@54 =8E ?@8ACB=O =525;8:0 :V;L:VABL 1V;LH8E 70 @>7<V@>< :@0?5;L, O:V <>3;8 CB2>@8B8AL 7024O:8 28?04:>2><C 7;8BBN 42>E E<0@=8E :@0?5;L. # <>45;V >C5=0 206;828<8 ?0@0<5B@0<8 T H284:VABL 28AEV4=>3> ?>B>:C 9 2>4=VABL E<0@8. @0?;O 1C45 ?V4=V<0B8AO 4> B8< 1V;LH8E @V2=V2, G8< 1V;LH0 H284:VABL 28AEV4=>3> ?>B>:C. # FL><C @07V 2>=0 28?045 :@V7L =86=N 3@0=8FN E<0@8, <0NG8 25;8:89 @>7<V@, >A:V;L:8 7=0G=> 71V;LHCTBLAO V<>2V@=VABL 7VB:=5==O WW 7 4@V1=8<8 :@0?;O<8 ?@8 ?04V==V 2 E<0@V. /:I> 71V;LHCTBLAO 2>4=VABL ?@8 287=0G5=V9 H284:>ABV 28AEV4=>3> ?>B>:C, B> 7@>AB0==O :@0?;V 1C45 1V;LH V=B5=A82=8<, 0;5 2>=0 ?@>E>48B8<5 C E<0@V ?@8 =87L:V9 B@0T:B>@VW. @8 @>7@0EC=:0E 22060;>AL, I> :>5DVFVT=B 70E>?;5==O E M B E D M a t h T y p e . ;5 B@510 <0B8 =0 C207V, I> ?@8 @>7<V@0E 4@V1=8E :@0?5;L E M B E D M a t h T y p e <:< V E M B E D M a t h T y p e <:<, O:V @>73;O40NBLAO C <>45;V C ?>G0B:>289 <><5=B, :>5DVFVT=B 70E>?;5==O <0T <0;5 7=0G5==O, V :@0?;O =5 <>65 7=0G=> 71V;LH8B8AL. 02VBL C 4>1@5 ?5@5<VH0=V9 E<0@V, O:0 <0T >4=C 9 BC 6 A5@54=N :>=F5=B@0FVN :@0?5;L, 1C4CBL VA=C20B8 <VAF52V 2V4E8;5==O :>=F5=B@0FVW 2V4 A5@54=L>W. /:I> ?>7=0G8B8 G5@57 E M B E D M a t h T y p e A5@54=N :>=F5=B@0FVN :@0?5;L C 45O:><C V=B5@20;V @>7<V@V2, B> <>6;825 G8A;> E M B E D M a t h T y p e :@0?5;L C FL><C V=B5@20;V 2 >1 T<V E M B E D M a t h T y p e <0T ?C0A>=V2AL:89 @>7?>4V; 9<>2V@=>AB59 E M B E D M a t h T y p e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a t h T y p e 01> E M B E D M a t h T y p e WE=L>3> G8A;0 =0 A0<><C ?>G0B:C AB0T 1V;LH>N, =V6 ACAV4=V :@0?;V, 9 ?>BV< 71V;LHCTBLAO 4> @>7<V@V2 4>I>2>W :@0?;V H;OE>< 3@02VB0FV9=>W :>03C;OFVW. "@510 ?@825@=CB8 C203C I5 4> >4=>3> 206;82>3> <><5=BC. V2=O==O 7@>AB0==O @>7<V@V2 :@0?5;L >?8ACT ?@>F5A :>03C;OFVW O: 157?5@5@2=V9 B0 ?;02=89. # 4V9A=>ABV :@0?;V 7@>AB0NBL C @57C;LB0BV 48A:@5B=8E 7VB:=5=L V 7;8BBO. "5;D>@4 AD>@<C;N202 <>45;L :>03C;OFVW, 45 2@0E>2CNBLAO 48A:@5B=0 ?@8@>40 ?@>F5AC 7@>AB0==O V AB0B8AB8G=V D;C:BC0FVW :>=F5=B@0FVW :@0?5;L. V= ?@8?CAB82, I> 2AV :@0?;V E<0@8, O:V 70E>?;NNBLAO, <0NBL >4=0:>289 @>7<V@ V I> >1 T<8 :@0?5;L- :>;5:B>@V2 C 2 @078 1V;LHV, =V6 :@0?5;L E<0@8. V= 7@>182 28A=>2>:, I> AB0B8AB8G=89 48A:@5B=89 ?@>F5A 70E>?;5==O T >A>1;82> 206;828< C @0==VE AB04VOE CB2>@5==O 4>IC, 0 ?@024>?>4V1=89 A?5:B@ 4>I>28E :@0?5;L @>72820TBLAO ?@>BO3>< 45:V;L:>E 45AOB:V2 E28;8=. 0 >A=>2V <>45;V "5;D>@40 1C;> ?@>2545=> 45:V;L:0 @>7@0EC=:V2 ?@8 @04VCA0E :@0?5;L E<0@8 8 - 1 4 <:< V ?@8 ?>G0B:>28E @04VCA0E :@0?5;L- :>;5:B>@V2 2V4 2 0 4> 4 0 <:<. C;> 287=0G5=>, I> G8A;> 7VB:=5=L, 4>AB0B=T 4;O 287=0G5==O CAB0=>2;5==O FL>3> 3@0=8G=>3> 7=0G5==O, 4>@V2=NT ?@81;87=> 6 ?@8 E M B E D M a t h T y p e <:<, 4 0 ?@8 E M B E D M a t h T y p e <:< V 1 0 0 ?@8 E M B E D M a t h T y p e <:<. @0=8G=V 7=0G5==O E M B E D M a t h T y p e 7<5=HCNBLAO 7V 71V;LH5==O< E M B E D M a t h T y p e . >=8 =5EB>2=> <0;V 4;O :@0?5;L @04VCA>< 1V;LH 4 0 <:<. &5 >7=0G0T I> ?@8 B0:8E @>7<V@0E :@0?5;L- :>;5:B>@V2 <>6CBL 28:>@8AB>2C20B8AO @V2=O==O 157?5@5@2=>3> 7@>AB0==O. ;O @04VCAV2 E M B E D M a t h T y p e <:< :>5DVFVT=B8 7VB:=5==O 4C65 <0;V 4;O B>3>, I>1 2V41C;>AO 7@>AB0==O :@0?5;L <@OFV <5=H5 =V6 70 45:V;L:0 3>48=, =02VBL O:I> T 7=0G=8<8 AB0B8AB8G=V 2V4E8;8 2V4 ?>254V=:8 A5@54=L>W :@0?;V. >73;O40NG8 7@>AB0==O :@0?;V, >C5= V "5;D>@4 ?>1C4C20;8 <>45;L =0 >A=>2V AC:C?=>ABV 4@V1=8E :@0?5;L, I> 70E>?;NNBLAO, 9 45:V;L:>E :@0?5;L, O:V 2V4@V7=ONBLAO 2V4 =8E A2>W<8 @>7<V@0<8 V 22060NBLAO :>;5:B>@0<8 7 A0<>3> ?>G0B:C. ;5 ?@>F5A C 4V9A=>ABV ?@>E>48BL ?>- V=H><C. 57?5@5@2=89 A?5:B@ @>7<V@V2 :@0?5;L, O:89 D>@<CTBLAO 7024O:8 WE=L><C :>=45=A0FV9=><C 7@>AB0==N, B@0=AD>@<CTBLAO G5@57 28?04:>2V 7VB:=5==O, A?>G0B:C 4C65 @V4:V, ?>H8@NNG8AL C AB>@>=C 1V;LH :@C?=8E :@0?5;L. B65, I>1 7@>7C<VB8 >A>1;82>ABV ?@>F5AC CB2>@N20==O 4>IC, 9>3> B@510 @>73;O40B8 =5 7 B>G:8 7>@C 7@>AB0==O 45O:>W ?V43@C?8 :@0?5;L :>;5:B>@V2, 0 O: 52>;NFVN 2AL>3> A?5:B@C :@0?5;L. 8I5 4;O E0@0:B5@8AB8:8 V=B5=A82=>ABV :>03C;OFV9=>3> 7@>AB0==O :@0?5;L 1C2 70AB>A>20=89 :>5DVFVT=B :>03C;OFVW E M B E D M a t h T y p e , O:89 ?@8 C@0EC20==V BV;L:8 5D5:BV2, >1C<>2;5=8E 3@02VB0FVTN, 287=0G0TBLAO @V2=O==O< E M B E D M a t h T y p e . ( 1 . 4 7 ) @8 AB0B8AB8G=><C ?V4E>4V :>5DVFVT=B :>03C;OFVW <0T A<8A; V<>2V@=>ABV B>3>, I> :@0?;O @04VCA>< E M B E D M a t h T y p e 70E>?8BL :@0?;N @04VCA>< E M B E D M a t h T y p e C >48=8FN G0AC 70 C<>28, I> :@0?;V B>3> 9 V=H>3> A>@BC <0NBL >48=8G=V :>=F5=B@0FVW. >5DVFVT=B :>03C;OFVW <>6=0 70?8A0B8 =5 G5@57 @04VCA8 :@0?5;L, 0 G5@57 WE=V9 >1 T<. >7=0G82H8 G5@57 E M B E D M a t h T y p e V E M B E D M a t h T y p e >1 T<8 :@0?5;L- :>;5:B>@V2 V :@0?5;L, O:V =8<8 70E>?;NNBLAO, 2V4?>2V4=>, 1C45<> <0B8 E M B E D M a t h T y p e , ( 1 . 4 8 ) 45 E M B E D M a t h T y p e G8A;>289 :>5DVFVT=B. >5DVFVT=B :>03C;OFVW B5?5@ <0T A<8A; V<>2V@=>ABV B>3>, I> :@0?;O >1 T<>< E M B E D M a t h T y p e 70E>?;NT :@0?;N >1 T<>< E M B E D M a t h T y p e C >48=8FN G0AC. @8?CAB8<>, I> A?5:B@ :@0?5;L E0@0:B5@87CTBLAO DC=:FVTN @>7?>4V;C :@0?5;L 70 @>7<V@0<8 ( B5?5@ 265 ?> >1 T<0E) E M B E D M a t h T y p e . ">4V A5@54=T G8A;> :@0?5;L E M B E D M a t h T y p e C >48=8G=><C >1 T<V ?@>AB>@C, O:V <0NBL >1 T<8 2V4 E M B E D M a t h T y p e 4> E M B E D M a t h T y p e , 4>@V2=NT E M B E D M a t h T y p e . ( 1 . 4 9 ) %0@0:B5@ :>03C;OFV9=>3> ?@>F5AC C E<0@V, O:89 1C45 ?@82>48B8 4> B@0=AD>@<0FVW A?5:B@C E<0@=8E :@0?5;L, <>6=0 28@078B8 G5@57 H284:VABL 7<V=N20==O E M B E D M a t h T y p e , B>1B> G5@57 E M B E D M a t h T y p e . >73;O=5<> ?V4 2?;82>< O:8E 5D5:BV2 B0:0 B@0=AD>@<0FVO A?5:B@C 1C45 2V41C20B8AL. >- ?5@H5, :@0?;V 7 FL>3> V=B5@20;C 1C4CBL 70E>?;N20B8AL 1V;LH :@C?=8<8 :@0?;O<8, O:V <0NBL >1 T<8 2 V=B5@20;V E M B E D M a t h T y p e . '8A;> B0:8E 70E>?;5=L T 4>1CB>: E M B E D M a t h T y p e =0 V<>2V@=VABL 7VB:=5==O 7 :@0?;O<8 1V;LH8E >1 T<V2 E M B E D M a t h T y p e . ;5, >G5284=>, E M B E D M a t h T y p e . ( 1 . 5 0 ) >- 4@C35, 1C45 2V41C20B8AL 7VB:=5==O 9 7;8BBO F8E :@0?5;L 7 1V;LH 4@V1=8<8 :@0?;O<8, B>1B> 7 :@0?;O<8 V7 V=B5@20;C E M B E D M a t h T y p e . '8A;> WE 1C45 E M B E D M a t h T y p e , 45 E M B E D M a t h T y p e . ( 1 . 5 1 ) B65, 70 @0EC=>: >1>E :>03C;OFV9 =8E 5D5:BV2 G8A;> :@0?5;L 7 V=B5@20;C >1 T<V2 E M B E D M a t h T y p e , E M B E D M a t h T y p e 1C45 7<5=HC20B8AL, V 70 >48=8FN G0AC 7030;L=5 7<5=H5==O 1C45 4>@V2=N20B8 E M B E D M a t h T y p e ( 1 . 5 2 ) ;5, ?>- B@5BT, 2 V=B5@20;V 2V4 E M B E D M a t h T y p e 4> E M B E D M a t h T y p e G8A;> :@0?5;L 1C45 71V;LHC20B8AL 7024O:8 7;8BBN CAVE ?0@ 1V;LH 4@V1=8E :@0?5;L, AC<0 >1 T<V2 O:8E 4>@V2=NT E M B E D M a t h T y p e . '8A;> B0:8E ?0@ 70 >48=8FN G0AC 2 V=B5@20;V E M B E D M a t h T y p e 70;568BL 2V4 V<>2V@=>ABV :@0?5;L V=B5@20;C 2V4 E M B E D M a t h T y p e 4> E M B E D M a t h T y p e 2 V=B5@20;V E M B E D M a t h T y p e , O:0 4>@V2=NT E M B E D M a t h T y p e , B0 V<>2V@=>ABV B>3>, I> :@0?;V @>7<V@V2 E M B E D M a t h T y p e 7=0E>4OBLAO 2 V=B5@20;V E M B E D M a t h T y p e . AB0==O, >G5284=>, 4>@V2=NT E M B E D M a t h T y p e . A:V;L:8 707=0G5=V 28?04:>2V ?@>F5A8 =570;56=V, 7030;L=0 V<>2V@=VABL FVTW A:;04=>W 28?04:>2>W ?>4VW 4>@V2=NT, O: 2V4><>, 4>1CB:C 9<>2V@=>AB59 E M B E D M a t h T y p e , 45 E M B E D M a t h T y p e . ">4V 7 C@0EC20==O< :>5DVFVT=B0 :>03C;OFVW E M B E D M a t h T y p e G8A;> :@0?5;L, I> 71V;LHCTBLAO 70 >48=8FN G0AC 2 V=B5@20;V E M B E D M a t h T y p e 7024O:8 707=0G5=><C ?@>F5AC, 4>@V2=NT E M B E D M a t h T y p e . ( 1 . 5 3 ) >5DVFVT=B E M B E D M a t h T y p e ?>B@V15= 4;O B>3>, I>1 =5 ?5@5@0E>2C20B8 42VGV :>6=C :><1V=0FVN 70E>?;5==O. V4AC<>2CNG8 2AV 5D5:B8, O:V >?8A0=V 28I5, <0T<> E M B E D M a t h T y p e ( 1 . 5 4 ) E M B E D M a t h T y p e . ( 1 . 5 5 ) &V @V2=O==O =07820NBL :V=5B8G=8< @V2=O==O<, @V2=O==O< :>03C;OFVW, @V2=O==O< AB>E0AB8G=>W :>03C;OFVW. >72 O7:>< @V2=O==O< ( 1 . 5 5 ) T A?5:B@ :@0?5;L E M B E D M a t h T y p e C <><5=BV G0AC E M B E D M a t h T y p e , O:89 7024O:8 :>03C;OFVW @>72820TBLAO 7 ?>G0B:>2>3> A?5:B@C E M B E D M a t h T y p e . "@510 <0B8 =0 C207V, I> <8 <0T<> 4V;> 7 AB>E0AB8G=8< ?@>F5A><, V @V2=O==O ( 1 . 5 5 ) T AB>E0AB8G=5 48D5@5=FV0;L=5 @V2=O==O. "><C ?>G0B:>289 @>7?>4V; E M B E D M a t h T y p e ?@82>48BL 4> FV;>W AV< W @>72 O7:V2 E M B E D M a t h T y p e , :>6=89 V7 :>B@8E 2V41820T >4=C @50;V70FVN ?@>F5AC :>03C;OFVW. 5B5@<V=VAB8G=89 @>72 O7>: @V2=O==O( 1 . 5 5 ) ?@82>48BL 4> A5@54=L>W DC=:FVW E M B E D M a t h T y p e @>7?>4V;C :@0?5;L 70 @>7<V@0<8, B>1B> DC=:FVW, >A5@54=5=>W 70 1030BL<0 @50;V70FVO<8. /: V A;V4 1C;> G5:0B8, @>7?>4V; :@0?5;L @>72820TBLAO =09H284H5 ?@8 35><5B@8G=V9 V<>2V@=>ABV 70E>?;5==O ( E M B E D M a t h T y p e ) . @8 V=H8E 7=0G5==OE :>5DVFVT=B0 70E>?;5==O, @>728B>: A?5:B@C T <5=H V=B5=A82=8<, 0;5 4@C30 <>40 =0 :@82V9 @>7?>4V;C 2A5 6 B0:8 =0<VG0TBLAO. 0 70:V=G5==O =03040T<>, I> @V2=O==O ( 1 . 5 5 ) C@0E>2CT BV;L:8 5D5:B 3@02VB0FV9=>W :>03C;OFVW. 0 B5?5@VH=L>3> G0AC 282545=V :V=5B8G=V @V2=O==O :>03C;OFV9=>3> 7@>AB0==O :@0?5;L C E<0@V, O:V 2 BV9 01> V=HV9 ?>2=>BV C@0E>2CNBL 2?;82 =0 B@0=AD>@<0FVN A?5:B@C 3>@87>=B0;L=8E V 25@B8:0;L=8E C?>@O4:>20=8E ?>B>:V2 ?>2VB@O, BC@1C;5=B=8E @CEV2 C E<0@V, 2>4=>ABV E<0@8, 7<V=5==O 7 @>7<V@0<8 :@0?5;L ?@8 40=V9 DC=:FVW @>7?>4V;C, :>5DVFVT=B0 :>03C;OFVW. 1 . 5 #<>28 CB2>@5==O >?04V2 @0?;V 2>48 V :@8AB0;8 ;L>4C, O:V 28?040NBL 7 E<0@ =0 75<=C ?>25@E=N, =07820NBL >?040<8. @>F5A @>728B:C E<0@ B0 CB2>@5==O >?04V2 7 O:VA=>W AB>@>=8 <>6=0 @>74V;8B8 =0 42V AB04VW. ?048, I> 28?040NBL =0 75<=C ?>25@E=N =0A0<?5@54 <>6=0 @>74V;8B8 =0 B25@4V, @V4:V B0 7<VH0=V. > B25@48E >?04V2 2V4=>AOBLAO B0:V >A=>2=V WE=V D>@<8: 1 . !=V3 >?048 C 283;O4V :@8AB0;V2 ( A=V68=>:) @V7=><0=VB=>W D>@<8, =09G0ABVH5 C D>@<V 7V@>G>:, O:V G0AB> >1 T4=CNBLAO 2 ?;0ABV2FV 25;8:>3> @>7<V@C. !?>AB5@V30TBLAO 25;8:0 @V7=><0=VB=VABL D>@< A=V68=>:: 3>;:8, AB>2?G8:8, ?;0AB8=:8; 3>;G0BV 7V@:8; ?;0AB8=G0BV 7V@:8; W60:8, O:V A:;040NBLAO 7 45:V;L:>E AB>2?G8:V2; AB>2?G8:8 7 ?;0AB8=:0<8 01> 7V@:0<8 =0 :V=FOE B>I>. >7<V@8 >:@5<8E A=V68=>: <>6CBL 1CB8 4C65 @V7=8<8. 091V;LHV ;V=V9=V @>7<V@8 <0NBL 3>;G0BV 7V@:8. E=V @04VCA8 <>6CBL 4>AO30B8 4 - 5 <<. # 7<VH0=8E E<0@0E G0AB> A?>AB5@V30TBLAO >175@=5==O A=V68=>:. >=> T @57C;LB0B>< 70<5@70==O ?5@5>E>;>465=8E :@0?5;L ?@8 A?V2C40@0E 7 A=V68=:>N. "0:0 G0AB:0 <0T <0B>289 2V4BV=>:. ;0ABV2FV A=V3C <>6CBL <0B8 @V7=V @>7<V@8: 2V4 0 , 5 << 4> 5 A<. !?>AB5@V30;8AL ?;0ABV2FV 7 @04VCA>< 4> 1 5 - 2 0 A<. ;0ABV2FV A=V3C F5 G0AB5 O28I5. >=8 A?>AB5@V30NBLAO C 1 4 % 28?04:V2 ?@8 A;01:8E V 2 9 2 % 28?04:V2 ?@8 A8;L=8E A=V3>?040E. #B2>@5==N A=V3>28E ?;0ABV2FV2 A?@8ONBL 2V4=>A=> 28A>:0 B5<?5@0BC@0 ?>2VB@O, 25;8:0 3CAB>B0 A=V3>?04C, 25;8:V 2V4AB0=V, I> ?@>E>4OBL A=V68=:8 ?@8 ?04V==V, B0 V=HV D0:B>@8. 2 . >:@89 A=V3 >?048 C 283;O4V A=V68=>: V :@0?5;L 01> A=V68=>:, O:V B0=CBL. !=V3 AB0T <>:@8<, :>;8 ?>1;87C 2V4 75<=>W ?>25@E=V B5<?5@0BC@0 1;87L:0 4> E M B E D M a t h T y p e 01> I5 28I>N. 3 . @C?0 F5 >?048, O:V A:;040NBLAO 7 ;L>4O=8E G0AB>: V 4C65 >175@=5=8E A=V68=>:. '0AB8=:8 :@C?8 <0NBL @04VCA 2V4 G0AB>: <V;V<5B@V2 4> 7 , 5 <<. @C?0 CB2>@NTBLAO H;OE>< 70<5@70==O ?5@5>E>;>465=8E :@0?5;L 2>48 9 >175@=5==O A=V68=>:. # 70;56=>ABV 2V4 A?V22V4=>H5==O <V6 A=V3>2>N B0 ;L>4O=>N D@0:FVO<8 :@C?8, F59 284 >?04V2 ?V4@>74V;ONBL =0 A=V3>2V 75@=0, A=V3>2C 9 ;L>4O=C :@C?C. 4 . @04 G0AB:8 :C;OAB>W D>@<8 7 ;L>4O=8<8 ?@>H0@:0<8 @V7=>W 3CAB8=8. >7<V@8 3@04C <>6CBL 1CB8 4C65 @V7=8<8: 9>3> @04VCA =09G0ABVH5 1C20T 2V4 1 4> 2 5 <<. ;5 V=:>;8 28?040NBL G0AB:8 3@04C 7 @04VCA>< 4> 1 5 A< V =02VBL 1V;LH8<. 5;8:V G0AB:8 3@04C <0NBL H0@C20BC 1C4>2C. # F5=B@V @>7B0H>2CTBLAO <0B>25 1V;5 O4@>, O:5 AE>65 =0 A=V3>2C :@C?C. /4@> >1BO3CTBLAO H0@>< ?@>7>@>3> ;L>4C. 0;V 94CBL ?>?5@5<V==> =5?@>7>@V 9 ?@>7>@V H0@8 ;L>4C. @04 CB2>@NTBLAO 2 :C?G0AB>- 4>I>28E E<0@0E C @57C;LB0BV 7;8BBO ?5@5>E>;>465=8E :@0?5;L 2>48 7 G0AB:0<8 :@C?8 9 WE 70<5@70==O. V4:V >?048 A:;040NBLAO 7 B0:8E D>@<: 1 . >I @V4:V 2>4O=V >?048 C D>@<V :@0?5;L 7 @04VCA>< 1V;LH8< 0 , 2 5 <<. /: ?>:07CNBL A?>AB5@565==O, :@0?5;L 1V;LH8E =V6 2 , 5 - 3 , 2 <<, =5 7CAB@VG0TBLAO, >A:V;L:8 25;8:V :@0?;V A?;NICNBLAO 9 @>71820NBLAO =0 45:V;L:0 4@V1=8E. 5@5206=V @04VCA8 1@87>: 4>@V2=NNBL 0 , 7 5 - 1 <<. (284:VABL ?04V==O :@0?5;L 4>IC 4>AO30T 8 - 1 0 </ A. >I 28?040T 7 H0@C20B>- 4>I>28E ( E M B E D M a t h T y p e ) V :C?G0AB>- 4>I>28E ( E M B E D M a t h T y p e ) , 0 V=>4V 9 7 28A>:>H0@C20B8E ( E M B E D M a t h T y p e ) E<0@. 2 . @O:0 >4=>@V4=V >?048, I> A:;040NBLAO 7 4@V1=8E :@0?5;L, @04VCA><, <5=H8< 70 0 , 2 5 <<. >=8 <0965 =5 <0NBL A?@O<>20=>3> @CEC. !B2>@NTBLAO 2@065==O, I> :@0?;V <@O:8 ?;020NBL C ?>2VB@V. @O:0 28?040T 7 H0@C20B8E ( E M B E D M a t h T y p e ) V H0@C20B>- :C?G0AB8E ( E M B E D M a t h T y p e ) E<0@. @0?;V <@O:8 V=:>;8 CB2>@NNBLAO 2 3CAB><C BC<0=V. =B5=A82=VABL >?04V2 C 28?04:C <@O:8 =5 ?5@528ICT 0 , 2 5 <</ G, H284:VABL ?04V==O :@0?5;L C =5@CE><><C ?>2VB@V <5=H0 70 0 , 3 </ A. ;0A8DV:0FVN 70 D>@<>N =07820NBL <>@D>;>3VG=>N. A=CT I5 35=5B8G=0 :;0A8DV:0FVO >?04V2, B>1B> :;0A8DV:0FVO 70 DV78G=8<8 C<>20<8 CB2>@5==O. V4?>2V4=> 4> FVTW :;0A8DV:0FVW >?048 ?V4@>74V;ONBLAO =0 B0:V 2848: 1 . 1;>6=V >?048 F5 B@820;V 9 @>7?>2AN465==V =0 25;8:V ?;>IV >?048 A5@54=L>W V=B5=A82=>ABV, O:V 28?040NBL V7 E<0@ A8AB5<8 E M B E D M a t h T y p e C 283;O4V 4>IC 01> A=V3C, V=>4V <>:@>3>. 2 . ;82>2V >?048 >?048, O:V 28?040NBL 7 :C?G0AB>- 4>I>28E E<0@ C 283;O4V 4>IC, A=V3C, :@C?8, 3@04C. >=8 @0?B>2> ?>G8=0NBLAO 9 70:V=GCNBLAO. =B5=A82=VABL F8E >?04V2 @V7:> 7<V=NTBLAO 70 G0A><. ;82>2V >?048 G0AB> AC?@>2>46CNBLAO 3@>70<8 B0 H:20;0<8. 3 . @OG=V >?048 F5 >?048, O:V 28?040NBL 7 3CAB8E H0@C20B8E V H0@C20B>- :C?G0AB8E E<0@. >=8 CB2>@NNBLAO C ABV9:> AB@0B8DV:>20=8E ?>2VB@O=8E <0A0E. @8 2V4 T<=8E =87L:8E B5<?5@0BC@0E B0:V E<0@8 40NBL >?048 C D>@<V ;L>4O=8E :@8AB0;V2. ?048 E0@0:B5@87CNBLAO 42><0 25;8G8=0<8: :V;L:VABN 9 V=B5=A82=VABN. V;L:VABL >?04V2 28<V@NNBL B>2I8=>N H0@C 2>48 2 <V;V<5B@0E, O:89 <V3 18 CB2>@N20B8AO ?VA;O 28?04V==O >?04V2 =0 3>@87>=B0;L=C =5?@>=8:;82C ?>25@E=N. 48= <V;V<5B@ H0@C 2>48 =0 ?>25@E=V ?;>I5N E M B E D M a t h T y p e 2V4?>2V40T <0AV >?04V2 1 :3. =B5=A82=VABN >?04V2 =07820NBL :V;L:VABL >?04V2, I> 28?0;8 70 >48=8FN G0AC ( =0?@8:;04, 1 3>48=C) . [ 1 ] # ?>G0B:>2V9 AB04VW @>728B:C E<0@8 >A=>2=C @>;L 2 C:@C?=5==V 70@>4:>28E E<0@=8E 5;5<5=BV2 2V4V3@0T ?@>F5A :>=45=A0FVW 2>4O=>W ?0@8. >=45=A0FVO 2V41C20TBLAO 7024O:8 =525;8:><C ?5@5A8G5==N 2>4O=>W ?0@8 2V4=>A=> ?>25@E=V E<0@=8E :@0?5;L. A>1;82> H284:> ?>G8=0TBLAO 7@>AB0==O G0AB8=>: C E<0@V ?VA;O B>3>, O: ?>@O4 7 ?5@5>E>;>465=8<8 :@0?;O<8 7 O2;ONBLAO 9 :@8AB0;8 ;L>4C. 0 F8E C<>2 ?>G8=0TBLAO ?5@53>=:0 2>4O=>W ?0@8 7 ?5@5>E>;>465=8E :@0?5;L =0 :@8AB0;8 ;L>4C 2=0A;V4>: B>3>, I> B8A: =0A8G5=>W 2>4O=>W ?0@8 =04 2>4>N 1V;LH5, =V6 =04 ;L>4><. &59 ?@>F5A 4>:;04=> @>73;O402AO 28I5. # 4@C3V9 AB04VW, ?VA;O B>3>, O: :@0?;V B0 :@8AB0;8 ;L>4C 7@>AB0NBL 4> E M B E D M a t h T y p e 2 0 & 6 0 <:<, >A=>2=C @>;L ?>G8=0T 2V4V3@020B8 ?@>F5A 7;8BBO ( :>03C;OFVW) E<0@=8E 5;5<5=BV2. >03C;OFVO >1C<>2;5=0 3>;>2=8< G8=>< @V7=>N H284:VABN ?04V==O :@0?5;L V :@8AB0;V2 ( 3@02VB0FV9=0 :>03C;OFVO) . 5O:5 7=0G5==O <0T :>03C;OFVO, I> >1C<>2;NTBLAO BC@1C;5=B=8< V 1@>C=V2AL:8< @CE0<8 ?>2VB@O, 5;5:B@>AB0B8G=8<8 A8;0<8 B>I>. 024O:8 :>03C;OFVW :@0?;V 9 :@8AB0;8 7@>AB0NBL 2V4 45AOB:V2 <V:@><5B@V2 4> 45:V;L:>E <V;V<5B@V2, 0 A=V3>2V ?;0ABV2FV B0 G0AB8=:8 3@04C 4> 45:V;L:>E A0=B8<5B@V2. @0?;V @V7=8E @>7<V@V2 ?040NBL ?V4 4VTN A8;8 2038 7 @V7=>N H284:VABN, 2 @57C;LB0BV G>3> 2>=8 A?V2C40@ONBLAO. ;5 4>:8 :@0?;V 4@V1=V, WE 7VB:=5==O B0 7;8BBO <0;> V<>2V@=5. V 7@>AB0==O< :@0?5;L @V7=8FO H284:>AB59 ?04V==O 71V;LHCTBLAO, I> 70157?5GCT A?@8OB;82V C<>28 4;O WE :>03C;OFVW. C65 25;8:5 7=0G5==O 4;O 7@>AB0==O E<0@=8E 5;5<5=BV2 9 CB2>@5==O >?04V2 <0NBL 25@B8:0;L=V @CE8 C A5@548=V E<0@8. >- ?5@H5, ?@8 28AEV4=8E @CE0E 2V41C20TBLAO 7<5=H5==O B5<?5@0BC@8 ?>2VB@O, I> A?@8OT CB2>@5==N ?5@5A8G5==O 2>4O=>W ?0@8 B0 :>=45=A0FV9=><C 7@>AB0==N :@0?5;L C E<0@V 4> @>7<V@V2, ?@8 O:8E 0:B82=C @>;L ?>G8=0T 2V4V3@020B8 :>03C;OFVO. >- 4@C35, :@0?;V ?V4 4VTN 25@B8:0;L=8E B5GV9 ?@>E>4OBL 25;8:V B>2IV E<0@8. 024O:8 FL><C 2>=8 7@>AB0NBL 4> 25;8:8E @>7<V@V2 70 @0EC=>: :>03C;OFVW. )5 >48= ?@>F5A <>65 1CB8 5D5:B82=8< C E<0@V, O:0 <0T 25;8:C 25@B8:0;L=C ?@>BO6=VABL. V= ?>;O30T C B><C, I> :@0?;V, O:V ?V4 4VTN 25@B8:0;L=8E @CEV2 ?>?040NBL =0 45O:89 @V25=L 7 @V7=8E 28A>B, <0NBL @V7=V B5<?5@0BC@8. &V @V7=8FV B5<?5@0BC@ =57=0G=V ( 45AOBV G0AB:8 3@04CA0) , 0;5 ?@8 2V4=>A=> 28A>:8E 4>40B=8E B5<?5@0BC@0E, I> A?>AB5@V30NBLAO 2 =87L:8E H8@>B0E, <>6CBL 2V4V3@0B8 ?><VB=C @>;L. G5284=>, @V7=8FV B5<?5@0BC@ ?@82>4OBL 4> 28=8:=5==O @V7=8FV B8A:V2 =0A8G5==O =04 ?>25@E=O<8 F8E :@0?5;L V, O:I>, =0?@8:;04, 1V;LH B5?;0 :@0?;O 7=0E>48BLAO C AB0=V @V2=>2038 7 >B>GCNG8< ?0@>?>2VB@O=8< A5@54>28I5<, =04 ?>25@E=5N 1V;LH E>;>4=>W :@0?;V CB2>@NTBLAO ?5@5A8G5==O. 024O:8 FL><C 2V41C20TBLAO ?5@53>=:0 2>48 7 B5?;>W :@0?;V =0 E>;>4=C. $V78G=89 <5E0=V7< B0:>3> ?@>F5AC T 0=0;>3VG=8< DV78G=><C <5E0=V7<C 270T<>4VW <V6 ?5@5>E>;>465=>N :@0?;5N 9 G0AB:>N ;L>4C. V= 4>:;04=> @>73;O402AO 28I5. ;O B>3>, I>1 7 O282AO 5D5:B :>=45=A0FV9=>3> 7@>AB0==O 1V;LH E>;>4=>W :@0?;V, ?>@V2=O==89 4> 5D5:BC 270T<>4VW @V4:>W D078 7 G0AB:>N ;L>4C C E<0@V, =5>1EV4=>, I>1 @V7=8FO B5<?5@0BC@, ?@> O:C 945BLAO, 4>@V2=N20;0 45:V;L:>< 3@04CA0< ?@8 =87L:8E B5<?5@0BC@0E V ;8H5 G0AB:0< 3@04CA0 ?@8 28A>:8E 4>40B=8E B5<?5@0BC@0E. A:V;L:8 25;8:V @V7=8FV B5<?5@0BC@ =0 287=0G=8E @V2=OE C E<0@V =5 7CAB@VG0NBLAO, F59 5D5:B =5 <>65 2V4V3@020B8 7=0G=C @>;L C ?><V@=8E V 28A>:8E H8@>B0E, 45 E<0@8 <0NBL, O: ?@028;>, =87L:C B5<?5@0BC@C, 9 T ACBBT28< C =87L:8E H8@>B0E, 45 =86=O G0AB8=0 E<0@8 7=0E>48BLAO 2 >1;0ABV 28A>:8E 4>40B=8E B5<?5@0BC@. !5@54 E<0@ 25@B8:0;L=>3> @>728B:C 28:;8:0NBL =091V;LH89 V=B5@5A :C?G0AB>- 4>I>2V E<0@8, >A:V;L:8 7 =8<8 ?>2 O70=V 0B<>AD5@=V O28I0, O:V 2 45O:8E 28?04:0E <0NBL =5157?5G=89 E0@0:B5@: 7;828, H:20;8, 3@>78, 3@04. C?G0AB>- 4>I>2V E<0@8 CB2>@NNBLAO 7 <>3CB=L>- :C?G0AB8E, :>;8 WE 25@H8=8 ?@>=8:0NBL C H0@8 0B<>AD5@8, 45 B5<?5@0BC@0 =01030B> =86G0 2V4 0 , 0 !, 9 70;545=V20NBL. 0;045=V2H0 25@E=O G0AB8=0 :C?G0AB>- 4>I>2>W E<0@8 E M B E D M a t h T y p e <>65 70;8H0B8AO :C?>;>?>4V1=>N E M B E D M a t h T y p e ( ;8A5) , 0;5 G0ABVH5 7 =5W 28:840NBLAO ?CG:8 ?5@8AB8E E<0@ C 284V ?0@0A>;V 01> =0 :>20;L=V. "0: 28=8:0NBL E M B E D M a t h T y p e ( 2>;>A0BV) , @V7=>284=VABN O:8E T :C?G0AB>- 4>I>2V 7 =0 :>20;L=5N E M B E D M a t h T y p e . C?G0AB>- 4>I>2V E<0@8 A:;040NBLAO 7 >4=VTW 01> 45:V;L:>E :>=25:B82=8E >A5@54:V2. >=25:B82=V >A5@54:8 A:;040TBLAO 7 >1;0ABV 28AEV4=>3> B0 :><?5=A0FV9=>3> =87EV4=>3> ?>B>:V2. # 1V;LH>ABV :C?G0AB>- 4>I>28E E<0@ :>=25:B82=V >A5@54:8 <0NBL ?>4V1=V AB@C:BC@8 V <>6CBL @>73;O40B8AL O: :;0A :>=25:B82=8E O28I. 8BBT289 F8:; >A5@54:C ?>4V;ONBL =0 B@8 AB04VW 2 70;56=>ABV 2V4 ?5@5206=8E =0?@O<:V2 25@B8:0;L=>3> @CEC ?>2VB@O B0 9>3> V=B5=A82=>ABV. 1 ) AB04VO :C?G0AB>W E<0@8 E0@0:B5@87CTBLAO 28AEV4=8<8 @CE0<8 2 1V;LHV9 G0AB8=V >A5@54:C; 2 ) AB04VO 7@V;>ABV E0@0:B5@87CTBLAO =0O2=VABN O: 28AEV4=8E, B0: V =87EV4=8E @CEV2; 3 ) AB04VO @>7?04C E0@0:B5@87CTBLAO A;01:8<8 =87EV4=8<8 @CE0<8 C 1V;LHV9 G0AB8=V >A5@54:C. &V AB04VW 9 2V4?>2V4=V @V7=>284=>ABV :C?G0AB>- 4>I>28E E<0@ 7>1@065=V =0 @8A. 1 . 2 . # AB04VW :C?G0AB>W E<0@8, :>;8 28AEV4=V @CE8 G8=OBL 7@>AB0==O E<0@8, G5@57 WW 1>:>2V ?>25@E=V 70BO3CTBLAO ?>2VB@O ( 5D5:B 70BO3C20==O, I> @>73;O40TBLAO 28I5) 9 7<VHCTBLAO 7 ?>2VB@O< C 28AEV4=><C ?>B>FV. A:V;L:8 @CE 225@E B@820T, 2V41C20TBLAO :>=F5=B@0FVO 25;8:>W :V;L:>ABV 2>4O=>W ?0@8 9 CB2>@5==O :@0?5;L, 0 28I5 =C;L>2>W V7>B5@<8 9 :@8AB0;V2. @8 2V4?>2V4=8E C<>20E ?>G8=0NBL 28?040B8 >?048, O:V AB8<C;NNBL =87EV4=V @CE8, >1C<>2;5=V 2 O7:8< >?>@>< ?>2VB@O 9 >E>;>465==O< >AB0==L>3> ?@8 28?0@>2C20==V :@0?5;L. &5 9 T ?>G0B:>< AB04VW 7@V;>ABV E<0@8. 87EV4=89 ?>BV: ?>2VB@O 4>AO30T 75<=>W ?>25@E=V C 284V 2V4=>A=> E>;>4=>3> O4@0 7>=8 28?040==O 4>IC 9 @>7BV:0TBLAO =04 =5N, 7<V=NNG8 ?@875<=89 =0?@O<>: 2VB@C. 0 =86=VE @V2=OE E<0@8 =87EV4=89 ?>BV: =0?;820T =0 28AEV4=89 V 2V4B8A:CT >AB0==V9 2V4 >1;0ABV 9>3> 70@>465==O. VA;O FL>3> ?>G8=0TBLAO AB04VO @>7?04C >A5@54:C. 0) AB04VO c u m u l u s ( AB04VO 7@>AB0==O) , 1) AB04VO 7@V;>ABV, 2) AB04VO 48A8?0FVW, 8A. 1 . 2 2>;NFVO C b 70 09T@A>< B0 @590<>< !B04VO :C?G0AB>W E<0@8 B@820T C A5@54=L><C 1 0 - 1 5 E28;8=, AB04VO 7@V;>ABV 1 5 - 3 0 E28;8=, 0 @>7?04 E<0@8 2V41C20TBLAO ?@>BO3>< 3 0 E28;8=. C?G0AB>- 4>I>2C E<0@C, C O:V9 A?>AB5@V30TBLAO >48= :>=25:B82=89 >A5@54>:, =07820NBL >4=>>A5@54:>2>N. ;5 G0ABVH5 E<0@8 :C?G0AB>- 4>I>2V CB@8<CNBL 45:V;L:0 :>=25:B82=8E >A5@54:V2. "0:V E M B E D M a t h T y p e =07820NBLAO 1030B>>A5@54:>28<8. :@5<V >A5@54:8 ?>A;V4>2=> @>72820NBLAO 7 ?@02>W AB>@>=8 25;8G57=>W E<0@=>W A8AB5<8, O:>N O2;ONBL 1030B>>A5@54:>28< E M B E D M a t h T y p e . 572060NG8 =0 B5, I> :>65= >A5@54>: <0T >1<565=89 68BBT289 F8:;, 1030B>@07>289 @>728B>: =>28E ?@82>48BL 4> B@820;>3> VA=C20==O E M B E D M a t h T y p e . !E5<0B8G=5 7>1@065==O 1030B>>A5@54:>2>W :C?G0AB>- 4>I>2>W E<0@8 =02>48BLAO =0 @8A. 1 . 4 . ?>G0B:>289 <><5=B G0AC E<0@0 A:;040TBLAO 7 G>B8@L>E >A5@54:V2, O:V 7=0E>4OBLAO =0 @V7=8E AB04VOE @>728B:C. @V< B>3>, =0 @8A. 1 . 3 ?>:07CTBLAO @>728B>: =09<>;>4H>3> ( ?V245==>3>) >A5@54:C 2 ?>A;V4>2=V <><5=B8 G0AC 9 25@B8:0;L=89 ?@>DV;L 2VB@C. 8A. 1 . 3 !E5<0B8G=5 7>1@065==O 1030B>>A5@54:>2>W :C?G0AB>- 4>I>2>W E<0@8 04728G09=> <>3CB=V<8 T AC?5@>A5@4:>2V :C?G0AB>- 4>I>2V E<0@8. >=8 <0NBL 9 =091V;LHC B@820;VABL VA=C20==O, O:0 4>AO30T 45:V;L:>E 3>48=. &V E<0@8 A:;040NBLAO 7 >4=>3> :207VAB0FV>=0@=>3> :>=25:B82=>3> >A5@54:C. V0<5B@ 9>3> V=:>;8 <>65 4>AO30B8 1 0 - 1 5 :<. A5@54>: >1 T4=CT 28AEV4=89 V =87EV4=89 ?>B>:8, O:V @>7B0H>2CNBLAO >48= 1V;O >4=>3>. 0 2V4<V=C 2V4 1030B>>A5@54:>28E :C?G0AB>- 4>I>28E E<0@, C O:8E >A5@54:8 H284:> 7<V=NNBL A2>W E0@0:B5@8AB8:8, 2 AC?5@>A5@4:>28E E<0@0E 2>=8 70;8H0NBLAO ?@0:B8G=> =57<V==8<8 =0 ?@>BO7V 45AOB:V2 E28;8=. !C?5@>A5@4:>2V E M B E D M a t h T y p e E0@0:B5@87CNBLAO 4C65 25;8:8<8 H284:>ABO<8 28AEV4=8E ?>B>:V2. /:I> C >4=>>A5@54:>28E V 1030B>>A5@54:>28E E<0@0E 2>=8 4>AO30NBL 45:V;L:>E </ A V V=>4V 1 0 - 2 0 </ A, B> C AC?5@>A5@4:>2>W E<0@V 3 0 - 5 0 </ A V =02VBL 1V;LH5. 5@B8:0;L=89 . @ D Z r 4 H H J < $ b ۽۽۽ۮ۟۟۟ېېېہہr h^ hk B* CJ aJ ph h^ h, B* CJ aJ ph h^ h^ B* CJ aJ ph h^ h-5 B* CJ aJ ph h^ hS B* CJ aJ ph h^ h9 B* CJ aJ ph h^ hl B* CJ aJ ph h^ h B* CJ aJ ph h^ h aJ h^ h aJ mH"sH"( t , F H H J J% , / f; > $ dh -D M `a$gd^ $dh -D M `a$gd^ N dh -D M `gd^ 0 2 4 F H J L N T V % % % % % % > > ? ? ᱤ
. . Գ- . .
[7,1 M], 10.09.2015, . . .
[104,7 K], 29.05.2010. . . . .
[803,5 K], 18.07.2014. . . . .
[37,8 K], 25.11.2015, . . . , .
[19,7 K], 28.10.2010, . . .
[19,3 K], 08.11.2010. . . . . . .
[639,1 K], 04.11.2015, . .
[120,6 K], 01.11.2010. , . , , , , , .
[2,1 M], 02.02.2014. . . . . , , .
[55,6 K], 30.03.2011
