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AM /2M w2HH#BH/2`M r2`/2M /;2;2M mMi2`b+?B2/HB+?2 P#D2Fi2 ;2bm+?iX q ?`2M/ BM /2M JQbBFm7M?K2M /B2 .SA@;27 `#i2M w2HHF2`M2 bB+?i#` bBM/- x2B;2M /B2 m7M?K2M /2` GM;x2Bi#2Q#+?imM; /B2 w2HH2M Hb S?b2MFQMi`bi@ Q/2` a+?` ;HB+?i#BH/- rB2 BM

RR8

ETBi2H eX EQK#BMiBQM pQM :2M2HQ;B2M mM/ 6HmQ`2bx2MxMHvb2M

##X eX9 ;2x2B;iX .b #2/2mi2i- /bb /B2 EQMbi2HHiBQM2M BM /2M a+?` ;HB+?i#BH/2`M /m`+?

/B2 a+?r2`TmMFi2 /2` w2HH2M mM/ BM /2M JQbBF#BH/2`M /m`+? /B2 a+?r2`TmMFi2 /2`

w2HHF2`M2 /2}MB2`i r2`/2MX 1b Bbi /?2` /pQM mbxm;2?2M- /bb bB+? /B2 SQbBiBQM2M /2`

w2HH2M mM/ w2HHF2`M2 mMi2`b+?2B/2M- rQ#2B xr2B 6FiQ`2M 2BM2M r2b2MiHB+? 1BM~mbb m7 /B2b2 o2`b+?B2#mM; ?#2M,

Ç .B2 "2r2;mM; /2` w2HH2MX .B2 GM;x2Bim7M?K2M r2`/2M pQM H2#2M/2M w2HH2M m7;2MQKK2M- /B2 BM /2` G;2 bBM/- B?`2 SQbBiBQM xm p2` M/2`MX

Ç o2`b+?B2#mM; /2` w2HHF2`M2X .B2 SQbBiBQM /2` w2HH2M mM/ w2HHF2`M2 rB`/ Ƀ#2` B?`2 JBii2HTmMFi2 #2biBKKiX >B2`#2B Bbi Mm` bB+?2`;2bi2HHi- /bb bB+? /2` w2HHF2`M BM /2` w2HH2 #2}M/2i- MB+?i #2`- /bb 2` BM /2K w2Mi`mK /2` w2HH2 HB2;iX

.2` `2bmHiB2`2M/2 o2`bix /2` w2HH2M mM/ w2HHF2`M2 rB`/ Hb _mb+?2M BMi2`T`2iB2`i- KBi /2K /B2 ;2bm+?i2M Jmbi2` p2`x2``i r2`/2M UbB2?2 ##X eX9VX JBi /2K Jmbi2` r2`/2M m+? /B2 2BM/2miB;2M "2b+?`2B#mM;2M- /B2 ;2QK2i`Bb+?2M >b?r2`i2- p2``mb+?i UqQH7b@

QM mM/ _B;QmibQb(Rky)VX .b ?i xm` 6QH;2- /bb M+? mM ?MHB+?2`2M EQMbi2HHiBQM2M Hb BM bi`QMQKBb+?2M Mb ix2M ;2bm+?i r2`/2M Kmbb mM/ bB+? /2` am+?`mK 2MibT`2@

+?2M/ p2`;`ƺĽ2`iX 1b ;2?2M m+? 2BMB;2 >b?r2`i2 p2`HQ`2M #xrX r2`/2M mM;ɃHiB;X oB2H 7QH;2M`2B+?2` #2` Bbi- /bb KBi /2M >b?r2`i2M m+? /B2 h`Mb7Q`KiBQM2M p2``mb+?i r2`/2M- /B2 Hb S`QD2FiBQM pQM 2BM2` EQMbi2HHiBQM mb /2K h2bi#BH/ m7 2BM2 S`QD2FiBQM BM /b _272`2Mx#BH/ 2`bi2HHi r2`/2MX .b #2/2mi2i- /bb bB+? /B2 o2`b+?B2#mM; /2` w2HHF2`@

M2 BM /2K T`QDBxB2`i2M "BH/ p2`;`ƺĽ2`i- / /B2 xm;`mM/2HB2;2M/2 h`Mb7Q`KiBQM 2BM2M bvbi2KiBb+?2M 62?H2` 2Mi? HiX lK 2BM/2miB; xrBb+?2M /2K _mb+?2M /2` w2HHF2`M2 BM /2M _272`2Mxm7M?K2M mM/ /2K _mb+?2M /2` T`QDBxB2`i2M .i2M xm mMi2`b+?2B/2M-rB`/ /b _272`2Mx`mb+?2M Ƀ#2` /B2 o`BMx σ<sup>2</sup> /2` xm;`mM/2HB2;2M/2M LQ`KHp2`i2B@

HmM; M;2;2#2MX .B2 o`BMx /2` o2`i2BHmM; /2` T`QDBxB2`i2M .i2M rB`/ /;2;2M Hbσ²_F

#2x2B+?M2iX .B2 H2ixi;2MMMi2 o`BMx HB272`i /2M 2Mib+?2B/2M/2M S`K2i2` 7Ƀ` /b bT @ i2`2 oQ`/2`;`mM/KQ/2HHF- KBi /2K /B2 o2`i2BHmM; /2` w2HHF2`M2 KQ/2HHB2`i rB`/X >B2`@

#2B rB`/ /B2 #b+? ixmM;σ<sub>F</sub> ≈2σ ;2MmixiX 1BM2 ;2Mm2`2 1`FH `mM; Bbi BM :HX UeXRjV m7 a2Bi2 RjR ;2;2#2MX wmbKK2M;27bbi p2`b+?H2+?i2`i bB+? /b ;2bi2HHi2 S`Q#H2K BK o2`;H2B+? xm /2M bi`QMQKBb+?2M Mb ix2MX

oQ`i2BH?7i BM /2K pQ`;2bi2HHi2M ax2M`BQ Bbi- /bb /B2 Mx?H /2` w2HH2M BM 2BM2`

S2i`Bb+?H2 BK :2;2Mbix xm /2M ai2`M2M b2?` #2;`2Mxi Bbi, AK bB+?i#`2M lMBp2`bmK

#2}M/2M bB+?≈6×10²²ai2`M2 U:Qii 2i HX (jN)V- r ?`2M/ BM 2BM2` S2i`Bb+?H2 KtBKH 10⁶ w2HH2M #2Q#+?i2i r2`/2MX .b rB2/2`mK 2`Hm#i 2b- /B2 #2MƺiB;i2M "2`2+?MmM;2M m7 2BM2K aiM/`/@S* mbxm7Ƀ?`2M mM/ 2b rB`/ F2BM _2+?2M+Hmbi2` #2MƺiB;i- rB2 2`

BK S`QD2Fibi`QK2i`vXM2i p2`r2M/2i rB`/ UGM; 2i HX (ej)VX eXRXkX >b?7mMFiBQM2M 7Ƀ` .`2B@- oB2`@- mM/ 6ɃM72+F2

wm2`bi r2`/2M /B2 p2`r2M/2i2M >b?7mMFiBQM2M pQ`;2bi2HHi- /B2 7Ƀ` 2BM2 ;2;2#2M2 EQM@

bi2HHiBQM mb /`2B- pB2` Q/2` 7ɃM7 w2HH2M 2BM2M 2BM/2miB;2M >b?r2`i #2`2+?M2MX .B2b2

>b?r2`i2 bBM/ BMp`BMi ;2;2MɃ#2` _QiiBQM- aFHB2`mM; mM/ h`MbHiBQM- /X?X /B2b2H@

#2M /`2B- pB2` Q/2` 7ɃM7 w2HH2M r2`/2M KBi /2Kb2H#2M >b?r2`i #2b+?`B2#2M- mM#? M;B;

pQM /2` p2`r2M/2i2M o2`;`ƺĽ2`mM; /2b JBF`QbFQTb Q/2` /2` SQbBiBQM mM/ mb`B+?imM;

/2` S2i`Bb+?H2X .B2 "2`2+?MmM; /2` >b?r2`i2 Bbi BM ##X eX8 /`;2bi2HHi mM/ rB`/

M+? /2K 7QH;2M/2M a+?2K /m`+?;27Ƀ?`i,

RRe

eXRX :2QK2i`Bb+?2b >b?BM;

##BH/mM; eX8X, .B2 ;2QK2i`Bb+?2M >b?7mMFiBQM2M 7Ƀ` .`2B@- oB2`@- mM/ 6ɃM72+F2 bBM/ /m`+? /b HQFH2 EQQ`/BMi2Mbvbi2K /2}MB2`i- r2H+?2b /B2 w2HH2M A, B KBi /2K

;`ƺĽi2M #biM/ /m`+? A = (0,0) mM/ B = (1,0) m7bTMMiX AM /B2b2K EQQ`@

/BMi2Mbvbi2K FƺMM2M /B2 >b?r2`i2 Hb EQQ`/BMi2M /2` w2HH2M C, D mM/ E Hb h= [cx, cy, dx, dy, ex, ey]#;2H2b2M r2`/2MX 6Ƀ` /`2B- pB2` mM/ 7ɃM7 w2HH2M 2`;2#2M bB+?

2MibT`2+?2M/ xr2B@- pB2`@ Q/2` b2+?b@/BK2MbBQMH2 >b?r2`i2hX

Ç 6ɃM72+F2X wm2`bi rB`/ 7Ƀ` 7ɃM7 ;2;2#2M2 w2HH2M /b S` KBi /2K ;`ƺĽi2M #@

biM/ ;2bm+?iX .B2b2 w2HH2M r2`/2M KBi A mM/ B #2x2B+?M2i mM/ /2}MB2`2M 2BM HQFH2b EQQ`/BMi2Mbvbi2K KBiA= (0,0)mM/B= (1,0)X .B2 Ƀ#`B;2M w2HH2M KɃb@

b2M bB+? BM 2BM2K E`2Bb KBi .m`+?K2bb2` ∥B−A∥2 #2}M/2M- /2` m+? /B2 w2HH2M AmM/B b2H#bi 2Mi? Hi- bB2?2 ##X eX8X .B2b2 w2HH2M r2`/2M Hb C= (cx, cy), D= (dx, dy), E = (ex, ey) KBi cx < ex < dx #2x2B+?M2iX JBi /B2b2` MQ`/MmM; Bbi MQ+? F2BM2 2BM/2miB;2 "2b+?`2B#mM; ;2;2#2M- / /B2 w2HH2M A mM/ B ;2imb+?i r2`/2M FƺMMi2M mM/ bQKBi xr2B ;ɃHiB;2 >b?r2`i2 ;2M2`B2`i r2`/2M rɃ`/2MX lK 2BM2M 2BM/2miB;2M >b?r2`i xm 2`?Hi2M- Kmbb /B2 "2/BM;mM; |cy| > |dy| 2`7ɃHHi b2BM- /X?X /B2 w2HH2 DKmbb bB+? M ?2` M /2` x− +?b2 #2}M/2M Hb w2HH2CX Abi /b MB+?i /2` 6HH- r2`/2M /B2 w2HH2MA mM/ B ;2imb+?i UmM/ KBi B?M2M C mM/

DV- mK 2BM2 ;ɃHiB;2 EQMbi2HHiBQM xm 2`?Hi2MX AM /2K bQ ;2M2`B2`i2M- HQFH2M EQQ`/BMi2Mbvbi2K rB`/ /2` e@/BK2MbBQMH2 >b?r2`i /m`+? /B2 w2HHFQQ`/BMi2M h= [cx, cy, dx, dy, ex, ey]/2}MB2`iX

Ç oB2`2+F2X .B2 "2`2+?MmM; /2b >b?r2`i2b 7Ƀ` pB2` w2HH2M 2`7QH;i MHQ;X 1b rB`/ /B2b2H#2 "2x2B+?MmM; ;2r ?Hi mM/ 2b r2`/2M /B2b2H#2M "2/BM;mM;2M M /B2 w2HH2M A, B, C mM/ D ;2bi2HHi rB2 xmpQ`X . /B2bKH /B2 xr2B EQQ`/BMi2M /2`

7ɃM7i2M w2HH2 E 72?H2M- rB`/ 2BM oB2`2+F KBi 2BM2K pB2`/BK2MbBQMH2M >b?r2`i h= [cx, cy, dx, dy]#2b+?`B2#2MX

Ç .`2B2+F2X6Ƀ` /`2B ;2;2#2M2 w2HH2M rB`/ /B2b2H#2 .2}MBiBQM /2` w2HH2MA, B mM/

CrB2 xmpQ` ;2r ?HiX 1BM2M 2BM/2miB;2M >b?r2`ih= [cx, cy]2`? Hi KM KBi /2`

"2/BM;mM;cx < ¹₂X

.B2b2 >b?7mMFiBQM2M r2`/2M ;2Mmixi- mK pQHHbi M/B;2 >b?i#2HH2M 7Ƀ` /B2 ;2;2#2M2M w2HH#BH/2` xm #2`2+?M2M- BM /2M2M /B2 >b?r2`i2 HH2` EQMbi2HHiBQM2M KBi /`2B- pB2`

Q/2` 7ɃM7 w2HH2M 2Mi?Hi2M bBM/X 1BM2 ;2Mm2 "2b+?`2B#mM; /B2b2` >b?i#2HH2M Bbi BM /2` .2}MBiBQM m7 a2Bi2 RRN xm }M/2MX

RRd

ETBi2H eX EQK#BMiBQM pQM :2M2HQ;B2M mM/ 6HmQ`2bx2MxMHvb2M .B2 >b?7mMFiBQM2M- /B2 BM /2M `#2Bi2M pQM GM; 2i HX (ej) mM/ qQH7bQM M/ _B@

;QmibQb (Rky) pQ`;2bi2HHi rm`/2M- ;`2B72M m7  ?MHB+?2 >b?7mMFiBQM2M xm`Ƀ+F- rB2 /B2

?B2` p2`r2M/2i2 J2i?Q/2X qQH7bQM ;B#i 2BM2 ;2M2`2HH2 "2`2+?MmM; 7Ƀ` N− 1+F2 M-BM /2` 2M-BM HQFH2b EQQ`/M-BMi2Mbvbi2K /m`+? xr2B SmMFi2 M-BM (−¹₂,0) mM/ (¹₂,0) m7;2@

bTMMi rB`/X AM /2` J2i?Q/2 pQM GM; 2i HX (ej) rB`/ 2BM2 2tTHBxBi2 "2`2+?MmM; 7Ƀ`

2BM2 >b?7mMFiBQM M;2;2#2M- BM /2` pB2` ai2`M2 ;2Mmixi r2`/2MX 6Ƀ` /B2 "2`2+?MmM;

/2` 2MibT`2+?2M/2M >b?r2`i2 rB`/ 2BM HQFH2b EQQ`/BMi2Mbvbi2K /m`+? xr2B ai2`M2 BM(0,0)mM/(1,1)#2biBKKiX .B2 ?B2` p2`r2M/2i2 J2i?Q/2 rm`/2 ;2r ?Hi- r2BH bB2 2BM2 b+?M2HH2 mM/ 2BM7+?2 "2`2+?MmM; /2` >b?r2`i2 2`Kƺ;HB+?iX

AMp`BMx /2` >b?r2`i2

1BM2 rB+?iB;2 oQ`mbb2ixmM; /2` p2`r2M/2i2M >b?7mMFiBQM2M Bbi /B2 AMp`BMx ;2;2M@

Ƀ#2` HBM2`2M h`Mb7Q`KiBQM2M- /X?X /bb /B2 >b?r2`i2 mM#? M;B; pQM /2` aFHB2@

`mM;- _QiiBQM mM/ h`MbHiBQM /2` w2HH#BH/2` M;2;2#2M r2`/2MX .B2 ;27Q`/2`i2 AMp`B@

Mx Bbi 7Ƀ` /B2 #2i`+?i2i2M ;2QK2i`Bb+?2M 6B;m`2M /m`+? /b HQFH2 EQQ`/BMi2Mbvbi2K

;2;2#2M- r2H+?2b BM ##X eX8 /`;2bi2HHi BbiX .B2 w2HH2MA, B KBi /2K ;`ƺĽi2M #biM/

/2}MB2`2M KBiA= (0,0), B= (1,0)/b HQFH2 EQQ`/BMi2Mbvbi2K- rQKBi 2BM2 2BM/2m@

iB;2 mb`B+?imM; U_QiiBQMbBMp`BMxV mM/ 2BM?2BiHB+?2 LQ`KB2`mM; UaFHB2`mM;b@ mM/

h`MbHiBQMbBMp`BMxV M;2;2#2M rB`/X lK /B2 AMp`BMx M+?xmr2Bb2M- rB`/ 2BM2 2t@

THBxBi2 _2+?2MpQ`b+?`B7i 7Ƀ` /B2 >b?r2`i2 #2i`+?i2iX >B2`xm r2`/2M /`2B SmMFi2A, B mM/ C KBi /2` Q#B;2M "2x2B+?MmM; ;2r ?HiX JBi /2M o2FiQ`2M AB = B−A

∥B−A∥2 mM/

AC= C−A

∥B−A∥2 FƺMM2M /B2 >b?r2`i2h= [cx, cy]Ƀ#2` /B2 aFH`T`Q/mFi2 cx = < AB, AC > und

cy = < AB, AC^⊥>

#2`2+?M2i r2`/2M- rQ#2B AC^⊥ /2` xm AC Q`i?Q;QMH2 o2FiQ` ;H2B+?2` G M;2 Bbi- /2`

mK 90^◦ BK l?`x2B;2`bBMM ;2/`2?i BbiX .B2 "2`2+?MmM; /2` >b?r2`i2 7Ƀ` oB2`@ #xrX 6ɃM72+F2 2`7QH;i MHQ;X mb /2M 1B;2Mb+?7i2M /2b aFH`T`Q/mFi2b 7QH;i /B`2Fi /B2 AMp`BMx ;2;2MɃ#2` HBM2`2M h`Mb7Q`KiBQM2MX

RR3

eXRX :2QK2i`Bb+?2b >b?BM;

.B2 >b?i#2HH2M

JBi /2M ;2;2#2M2M >b?7mMFiBQM2M FMM 2BM pQHHbi M/B;2` aix pQM >b?r2`i2M 7Ƀ` 2BM ;2;2#2M2b w2HH#BH/ #2`2+?M2i r2`/2M- /2` HH2 EQMbi2HHiBQM2M pQM /`2B-pB2` Q/2` 7ɃM7 w2HH2M 2Mi? HiX .B2 >b?r2`i2 7Ƀ` /b _272`2Mx#BH/ I^R mM/ /b h2bi#BH/I^T r2`/2M Hb>b?i#2HH2M

H^T mit h^T_l ∈H^T, l= 1, . . . , N^T mM/

H^R mit h^R_k ∈H^R, k= 1, . . . , N^R

#2x2B+?M2i- rQ#2B N^R mM/ N^T /B2 :`ƺĽ2 /2` >b?i#2HH2M #xrX /B2 Mx?H /2`

#2`2+?M2i2M >b?r2`i2 M;2#2MX .B2b2 :`ƺĽ2 Bbi 2Mib+?2B/2M/ 7Ƀ` /B2 Gm7x2Bi /2` am+?2 UM+?  ?MHB+?2M EQMbi2HHiBQM2MV mM/ ? M;i /B`2Fi pQM /2` w?H M /2` w2HH2M BK "BH/ #, N^R #xrX N^T r+?b2M KBi _Mk!(M−M^{M! k})! = ^-_M^Mk

.- rQ#2B M^k∈3,4,5/B2 w?H /2` p2`r2M/2i2M w2HH2M #xrX /B2 EQMbi2HHiBQMb;`ƺĽ2 U.`2B@-oB2`@- mM/ 6ɃM72+F2V M;B#iX

AM /2K pQ`;2bi2HHi2M Mbix Bbi 2b MB+?i 2`7Q`/2`HB+?- 2BM2 pQHHbi M/B;2 >b?i@

#2HH2 xm #2`2+?M2M, aQHM;2 bB+?2`;2bi2HHi rB`/- /bb /B2 EQMbi2HHiBQM2M h<sup>T</sup><sub>l</sub> mb /2K h2bi#BH/ m+? BM /2` _272`2Mxi#2HH2H<sup>R</sup>2Mi?Hi2M bBM/- FMM /B2 :`ƺĽ2 /B2@

b2` h#2HH2M 2BM;2b+?` MFi r2`/2MX AM /2K pQ`;2bi2HHi2M Mbix rB`/ /B2 :`ƺĽ2

#2;`2Mxi- BM/2K KBi

dmin < ∥B−A∥2 < dmax UeXRV 2BM2 KBMBKH2 mM/ KtBKH2 G M;2 /2` EQMbi2HHiBQM2M pQ`;2;2#2M rB`/- rQ#2BA mM/B /B2 SQbBiBQM /2` w2HH2M BK "BH/`mK M;2#2MX >B2`#2B Bbi xm #2+?i2M- /bb /B2 q?H pQM d_min, d_max Mm` pQ`;2MQKK2M r2`/2M FMM- r2MM /B2 aFHB2`mM;

/2` "BH//i2M #2FMMi BbiX . /B2b2 "2/BM;mM; 7Ƀ` JBF`QbFQT#BH/2` pQ`mb;2b2ixi r2`/2M FMM- bi2HHi /B2 q?H pQMdminmM/dmaxF2BM2 1BMb+?` MFmM; /`- bQM/2`M

`2/mxB2`i /2M #2MƺiB;i2M _2+?2Mm7rM/ 2MQ`KX

.B2 h`Mb7Q`KiBQM2M

.B2 am+?2 M+?  ?MHB+?2M EQMbi2HHiBQM2M FMM KBi >BH72 /2` ;2;2#2M2M >b?i#2HH2M H<sup>R</sup>, H<sup>T</sup> mb /2` Q#B;2M wmbKK2M7bbmM; Hb 2BM2 am+?2 M+?  ?MHB+?2M >b?r2`i2M mb;27Ƀ?`i r2`/2MX 6Ƀ` 2BM2 ;2;2#2M2 EQMbi2HHiBQM mb /2K h2bi#BH/ I^T KBi /2K

>b?r2`ih<sup>T</sup><sub>l</sub> ∈H<sup>T</sup> bBM/  ?MHB+?2 EQMbi2HHiBQM2M /2b _272`2Mx#BH/2b Ƀ#2` /B2 >b?r2`i2 Hl := ^/h^R_k |∥h^T_l −h^R_k∥2<), h^R_k ∈H^R0, UeXkV

;2;2#2MX >B2`#2B #2b+?`2B#iHl /B2 J2M;2 HH2` >b?r2`i2 /2` _272`2Mxi#2HH2H^R- /B2 2BM2  ?MHB+?2 EQMbi2HHiBQM rB2h^T_l #2b+?`2B#2MX C2/2b hmT2H pQM >b?r2`i2M^&h^R_k, h^T_l ^', h^R_k ∈Hl /2}MB2`i 2BM2 Kƺ;HB+?2 h`Mb7Q`KiBQMTk,l :I^T →I^R pQK h2bi#BH/ I^T BM /b

RRN

ETBi2H eX EQK#BMiBQM pQM :2M2HQ;B2M mM/ 6HmQ`2bx2MxMHvb2M _272`2Mx#BH/I^RX .B2b2 h`Mb7Q`KiBQM Tk,l rB`/ mb /2M EQQ`/BMi2M /2` w2HH2M #xrX w2HHKBii2HTmMFi2 /2` EQMbi2HHiBQM2Mh^R_k, h^T_l /2}MB2`iX JBiTk,lBbi 2b Kƺ;HB+?- HH2 w2HH2M /2b h2bi#BH/2bc^Z_m, m= 1, . . . , M^T KBi

Tk,l(c^Z_m) = c^T_m UeXjV

BM /b EQQ`/BMi2Mbvbi2K /2b _272`2Mx#BH/2bI^Rxm i`Mb72`B2`2MX 6Ƀ` 2BM2 EQMbi2HHiBQM h^T_l ∈ H^T 2tBbiB2`2M /?2` pB2H2 Kƺ;HB+?2 h`Mb7Q`KiBQM2M T_k,l- pQM /2M2M KtBKH 2BM2 /B2 `B+?iB;2 "BH/TQbBiBQM #2b+?`2B#iX lK xm 2Mib+?2B/2M- r2H+?2b k /B2 `B+?iB;2 h`Mb7Q`KiBQM2M M;B#i- rB`/ BK 6QH;2M/2M 2BM "v2b@h2bi ;2Mmixi- mK 2BM2 ;HQ#H2 S`Ƀ7mM;KBi HH2M T`QDBxB2`i2M w2HHF2`M2M c^T_m,1, . . . , M^T /m`+?xm7Ƀ?`2MX

UV _272`2Mx#BH/I<sup>T</sup>X U#V Ÿ?MHB+?2 EQMbi2HHiBQM2MX U+V Jƺ;HB+?2 h`Mb7Q`KiBQ@

M2MX

##BH/mM; eXeX, JBi >BH72 /2` a+?r2`TmMFi2 /2` w2HH2M mM/ w2HHF2`M2 r2`/2M w2HH@

FQMbi2HHiBQM2M KBi 2BM2K 2BM/2miB;2M mM/ BMp`BMi2M >b?r2`i #2b+?`B2#2MX 6Ƀ` /B2 EQMbi2HHiBQM KBi /2K >b?r2`ih^T₁ /2b h2bi#BH/2b I^T BM UV ;B#i 2b xr2B  ?MHB+?2 EQM@

bi2HHiBQM2M H1 = ^/h^R₁, h^R₁⁰ BM /2K _272`2Mx#BH/ I^R U#VX C2/2b hmT2H ^&h^T₁, h^R₁^' mM/

&

h^T₁, h^R₂^' /2}MB2`i 2BM2 h`Mb7Q`KiBQM pQM "BH/ I<sup>T</sup> BM /2M _mK /2b _272`2Mx#BH/2b I^R U+VX

eXRXjX oHB/B2`2M /2` h`Mb7Q`KiBQM2M

6Ƀ` /B2 xr2Bi2 S?b2 rB`/ 2BM2 oHB/B2`mM; BM 6Q`K 2BM2b "v2b@h2bib 2BM;27Ƀ?`i /2`

2Mib+?2B/2i- Q# 2BM2 ;2;2#2M2 h`Mb7Q`KiBQM /B2 `B+?iB;2 "BH/TQbBiBQM #2b+?`2B#iX 1b rB`/ /2K Mbix pQM _B;QmibQb 2i HX (N8- Ne) ;27QH;i- BM /2K 2BM2 "v2bb+?2 6Q`KmHB2@

`mM; 7Ƀ` /b EHbbB}FiBQMbT`Q#H2K pQM ;2QK2i`Bb+?2M >b?p2`7?`2M ;2;2#2M rm`/2X

Rky

eXRX :2QK2i`Bb+?2b >b?BM;

AK 6QH;2M/2M r2`/2M /B2 #2MƺiB;i2M JQ/2HH2 2BM;27Ƀ?`i mM/ m7#m2M/ m7 /2K aix pQM "v2b rB`/ /b 1Mib+?2B/mM;bT`Q#H2K m7 2BM a+?r2HHr2`iT`Q#H2K xm`Ƀ+F;27Ƀ?`i mM/ 2BM2 `Q#mbi2 oHB/B2`mM; /2` h`Mb7Q`KiBQM 2`Hm#iX

"v2bb+?2@JQ/2HHB2`mM;

AM /2` "v2bb+?2M@JQ/2HHB2`mM; rB`/ KBi 2BM2K oQ`/2`;`mM/KQ/2HHF /B2 q?`@

b+?2BMHB+?F2Bi #2`2+?M2i- KBi /2` 2BM2 Kƺ;HB+?2 h`Mb7Q`KiBQM Tk,l /B2 `B+?iB@

;2 "BH/TQbBiBQM M;B#iX .b >BMi2`;`mM/KQ/2HH B #2b+?`2B#i /;2;2M /B2 q?`@

b+?2BMHB+?F2Bi /7Ƀ`- /bb T<sub>k,l</sub> F2BM2 `B+?iB;2 SQbBiBQM x2B;iX .B2 2MibT`2+?2M/2M q?`b+?2BMHB+?F2Bi r2`/2M KBi /2K aix pQM "v2b Hb

p(F|Tk,l) = p(T_k,l|F)P(F)

p(Tk,l) für das Vordergrundmodell und p(B|Tk,l) = p(Tk,l|B)P(B)

p(T_k,l) für das Hintergrundmodell

M;2;2#2MX >B2`#2B #2b+?`2B#2M p(F|Tk,l) mM/ p(B|Tk,l) /B2 #2/BM;i2M Q/2` @ TQbi2`BQ`B q?`b+?2BMHB+?F2Bi2M- r2MM /B2 h2bix2HH2M KBi /2` h`Mb7Q`KiBQMTk,l

T`QDBxB2`i rm`/2MXp(F), p(B)mM/p(Tk,l)#2x2B+?M2M /B2 UmM#? M;B;2MV@T`BQ`B q?`b+?2BMHB+?F2Bi2M U"`#2` (Rk)VX

lK /B2 JQ/2HH2 F mM/ B xm p2`;H2B+?2M mM/ /b r?`b+?2BMHB+?2`2 JQ/2HH mb@

xmr ?H2M- r2`/2M KBi >BH72 /2` @TQbi2`BQ`B q?`b+?2BMHB+?F2Bi2M p(F|Tk,l) mM/

p(B|Tk,l) /B2 /i2M#? M;B;2M- #2/BM;i2M q?`b+?2BMHB+?F2Bi2M p(F|Tk,l)

M;2;2#2MX .B2@T`BQ`Bq?`b+?2BMHB+?F2Bi2Mp(F)mM/p(B);2#2M M- rB2 r?`@

b+?2BMHB+? 2b Bbi- xm7 HHB; 2BM2 ;ɃHiB;2 #xrX mM;ɃHiB;2 h`Mb7Q`KiBQM2M xm }M/2MX . 2b 7Ƀ` D2/2 EQMbi2HHiBQM Mm` 2BM2 `B+?iB;2- #2` b2?` pB2H2 7Hb+?2T<sub>k,l</sub> ;B#i- ;BHi p(F)<< p(B)X .B2 FQMFm``B2`2M/2M JQ/2HH2F mM/BFƺMM2M M?M/ /2b"v2b@

6FiQ`b K = p(T_k,l|F)

p(Tk,l|B) p2`;HB+?2M r2`/2M- /2` pQM C2z`2vb (8R) 2BM;27Ƀ?`i mM/

#2x2B+?M2i rm`/2X aQKBi rB`/ /b 1Mib+?2B/mM;bT`Q#H2K- Q# 2BM2 h`Mb7Q`KiBQM Tk,l `B+?iB; Bbi Q/2` MB+?i- m7 /b a+?r2HHr2`iT`Q#H2K

K >θK UeX9V

xm`Ƀ+F;27Ƀ?`iX q2MM /2` "v2b@6FiQ` K Ƀ#2` 2BM2K pQ`/2}MB2`i2M a+?r2HHr2`i θK HB2;i- rB`/ /B2 /m`+?Tk,l #2b+?`B2#2M2 SQbBiBQM Hb FQ``2Fi Fx2TiB2`iX

RkR

ETBi2H eX EQK#BMiBQM pQM :2M2HQ;B2M mM/ 6HmQ`2bx2MxMHvb2M

JQ/2HHB2`mM;

.B2 p2`r2M/2i2M :`mM/H;2M /2` JQ/2HHB2`mM; bBM/ BM /2` "2b+?`2B#mM; /2` "v2bb+?2M@

6Q`KmHB2`mM; m7 a2Bi2 RkR xmbKK2M;27bbiX >B2` rB`/ /B2 2Mib+?2B/2M/2 6`;2- Q# 2BM2 h`Mb7Q`KiBQMTk,l /B2 FQ``2Fi2 SQbBiBQM x2B;i- KBi >BH72 /2` #2/BM;i2M q?`b+?2BMHB+?@

F2Bi2Mp(F|Tk,l)mM/p(B|Tk,l)KQ/2HHB2`iX >B2`#2B #2x2B+?M2iF /b oQ`/2`;`mM/KQ/2HH U`B+?iB;2 h`Mb7Q`KiBQMV mM/B /b >BMi2`;`mM/KQ/2HH U7Hb+?2 h`Mb7Q`KiBQMVX M@

b+?HB2Ľ2M/ rB`/ /2` aix pQM "v2b ;2Mmixi- mK /B2b2 6`;2 m7 /b a+?r2HHr2`iT`Q#H2K mb :HX UeX9V xm `2/mxB2`2MX AM /2` ;2;2#2M2M 6Q`KmHB2`mM; ? M;i /B2 1Mib+?2B/mM;- Q#

/B2 h`Mb7Q`KiBQMT_k,l `B+?iB; Bbi Q/2` MB+?i- pQM /2K "v2b@6FiQ`K #X .B2b2` rB2@

/2`mK rB`/ pQM /2M JQ/2HH2MF mM/B #2biBKKi- /B2 /B2 q?`b+?2BMHB+?F2Bi2M 7Ƀ` 2BM2 FQ``2Fi2 SQbBiBQM Q/2` 2BM2 mM;ɃHiB;2 h`Mb7Q`KiBQM #2b+?`2B#2MX lK /B2b2 2Mib+?2B@

/2M/2M JQ/2HH2 2tTHBxBi Mxm;2#2M- rB`/ /2` Mbix pQM GM; 2i HX (ej) p2`7QH;iX L2#2M /2K "v2b@6FiQ` K- Bbi /2` a+?r2HHr2`i θ_K /2` xr2Bi2 2Mib+?2B/2M/2 S`@

K2i2`X 1BM2 2`bi2- b2?` ;`Q#2 #b+? ixmM; 7Ƀ` θK HB272`i /b am+?bx2M`BQ b2H#bi, 1b Bbi #2FMMi- /bb 7Ƀ` D2/2 EQMbi2HHiBQMh^T_l b2?` pB2H2  ?MHB+?2 >b?r2`i2h^R_k 2tBbiB2`2M-pQM /2M2M ?ƺ+?bi2Mb 2BM2 `B+?iB; b2BM FMMX .b #2/2mi2i rB2/2`mK- /bb /B2 @T`BQ`B q?`b+?2BMHB+?F2Bi p(F) 7Ƀ` 2BM2 `B+?iB;2 h`Mb7Q`KiBQM b2?` pB2H FH2BM2` Bbi Hb /B2 7Ƀ` 2BM2 7Hb+?2- /X?X 2b ;BHi p(F) ) p(B)X Hb EQMb2[m2Mx bQHHi2 /2` a+?r2HHr2`i θ<sub>K</sub> 2MibT`2+?2M/ ;`QĽ ;2r ?Hi r2`/2MX qB2 2BM2 ;mi2 q?H 7Ƀ`θ<sub>K</sub> ;2i`Qz2M r2`/2M FMM-rB`/ BM /2K bT i2`2M #b+?MBii eXkXj ;2x2B;iX

oQ`/2`;`mM/KQ/2HH F

.b oQ`/2`;`mM/KQ/2HHF ;B#i /B2 q?`b+?2BMHB+?F2Bi M- KBi /2` 2BM2 ;2;2#2M2 h`Mb@

7Q`KiBQM /B2 `B+?iB;2 "BH/TQbBiBQM #2b+?`2B#iX .b JQ/2HH xm F b2ixi bB+? mb xr2B h2`K2M xmbKK2MX AM /2K 2`bi2M h2`K r2`/2M HH2 w2HHF2`M2 xmbKK2M;27bbi- /B2 BM #2B/2M "BH/2`M ;27mM/2M rm`/2MX >B2`xm rB`/ 7Ƀ` D2/2 i`Mb7Q`KB2`i2 w2HH2 c^T_m /B2

7Q`KiBQM /B2 `B+?iB;2 "BH/TQbBiBQM #2b+?`2B#iX .b JQ/2HH xm F b2ixi bB+? mb xr2B h2`K2M xmbKK2MX AM /2K 2`bi2M h2`K r2`/2M HH2 w2HHF2`M2 xmbKK2M;27bbi- /B2 BM #2B/2M "BH/2`M ;27mM/2M rm`/2MX >B2`xm rB`/ 7Ƀ` D2/2 i`Mb7Q`KB2`i2 w2HH2 c^T_m /B2

Im Dokument Algorithmen zur automatisierten Analyse von in vitro-Stammzellpopulationen (Seite 121-148)