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UkV q2 +QmH/ K�F2 i?2 bm#biBimiBQMb 7Q` i?2 HQr2` �M/ mTT2` HBKBib b?QrM BM UkV #mi }M/ Bi 2�bB2` iQ H2i i?2 +QKTmi2` /Q BiX � THQi Q7 - (,) Bb b?QrM BM 6B;m`2 jX h?2 7mM+iBQM Bb bi`B+iHv BM+`2�bBM;X h?Bb Bb �HbQ 2pB/2Mi 7`QK 6B;m`2 k r?2`2 i?2 b?�/2/ �`2� +H2�`Hv BM+`2�b2b �b , BM+`2�b2b `2�+?BM; � K�tBKmK p�Hm2 Q7 k ;BpBM; � ?�H7 ;`�xBM; �`2� Q7 & 2 . h?2`2 �`2 K�Mv i2+?MB[m2b iQ bQHp2 i?2 7Q` , BM UkV r?2M - (,) = & 4 X L2riQMǶb K2i?Q/- i�m;?i BM BMi`Q/m+iQ`v +�H+m@ Hmb- +QK2b iQ KBM/X � +QKTmi�iBQM�HHv bBKTH2` �TT`Q�+?- MQi M22/BM; i?2 /2`Bp�iBp2 `2[mB`2/ #v L2riQMǶb K2i?Q/- Bb BMi2`@ p�H ?�HpBM; �HbQ FMQrM �b #BM�`v b2�`+?X ke aQHmiBQM Q7 i?2 :`�xBM; :Q�i S`Q#H2K, � *QM~B+i #2ir22M "2�miv �M/ S`�;K�iBbK h?2 B/2� Bb bBKTH2 �b Bb b22M BM i?2 bi`�B;?i7Q`r�`/ �H;Q`Bi?K ?2`2, RX a2i ./0 = 0 kX a2i 1231 = 2 jX 1p�Hm�i2 i?2 KB/TQBMi , = ?B;?+HQr 2 9X A7 %,,/, = - (,) # & 4 > 0, b2i ?B;? = , 8X Pi?2`rBb2- b2i HQr = , eX :Q iQ ai2T j h?2 Bi2`�iBQM +QMiBMm2b mMiBH i?2 /2bB`2/ �++m`�+v- Q` r?2M i?2 2``Q`- +QMbi`�BM2/ iQ i?2 +QKTmi�iBQM�H �++m`�+v Q7 i?2 +QKTmi2`- Bb x2`QX GQQFBM; �i 6B;m`2 j- �M 2v2#�HH 2biBK�i2 Q7 - (,) = & 4 $ yXdN b�72Hv TH�+2b , #2ir22M ./0 = 1.00 �M/ 1231 = 1.25 bQ i?2 BMi2`p�H ?�HpBM; �H;Q`Bi?K +QmH/ #2 BMBiB�i2/ i?2`2X "mi i?2 BMi2`p�H ?�HpBM; �H;Q`Bi?K +QMp2`;2b [mB+FHv �M/ #`Q�/2` BMB@ iB�HBx�iBQM QMHv `2[mB`2 � 72r KQ`2 Bi2`�iBQMbX 1p2`v Bi2`�iBQM `2/m+2b i?2 BMi2`p�H QM r?B+? i?2 bQHmiBQM #v ?�H7X AMBiB�HHv- i?2 bQHmiBQM Bb FMQrM iQ #2 QM i?2 BMi2`p�H Uy-kVX h?2 b2+QM/ Bi2`@ �iBQM `2/m+2b i?2 BMi2`p�H iQ UR-kVX h?2M iQ UR-RX8V- UR-RXk8V �M/ URXRk8-RXk8VX h?2 H2M;i? Q7 i?2b2 BMi2`p�Hb `2bT2+iBp2Hv �`2 + 2, 1, 1 2 , 1 4 . 1 8 , . A7 4 = 2 Bb i?2 rB/i? Q7 i?2 BMBiB�H BMi2`p�H- i?2 rB/i? Q7 i?2 BMi2`p�H �7i2` i?2 5i? Bi2`�iBQM Bb �++m`�+v = 4%2#' UjV h?2 �H;Q`Bi?K xQQKb BM QM i?2 +Q``2+i �Mbr2` [mB+FHvX h?2 Bi2`�iBQM 7Q` i?2 ;`�xBM; ;Q�i T`Q#H2K +QMp2`;2b iQ , =RXR83dk39djyR3Rk �7i2` 9N Bi2`�iBQMb iQ x2`Q 2``Q` r?2M i?2 �H;Q`Bi?K Bb T2`7Q`K2/ rBi? i?2 �++m`�+v Q7 � Ja 1t+2H bT`2�/ b?22iX h?2 bQHmiBQM mb2b R9 /B;Bib iQ i?2 `B;?i Q7 i?2 /2+BK�H +QMbBbi2Mi rBi? 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BM � bQHmiBQM K�v MQi #2 #2�miB7mHX *HQb2/ 7Q`K bQHmiBQMb �`2 #2�miB7mH #mi BM i?2 +�b2 Q7 �++m@ `�i2 bQHmiBQMb iQ K�i?2K�iB+�H T`Q#H2Kb- HBF2 i?2 ;`�xBM; ;Q�i T`Q#H2K- i`mi? M22/ MQi #2 #2�miB7mHX oQHmK2 j- Abbm2 k _161_1L*1a kd _272`2M+2b �M/`22b+m- hBim �M/ :�#`B2H .QbTBM2b+m UkyRyVX S`Q#H2Kb 7`QK i?2 "QQFX kM/ 1/BiBQMX suw S`2bbX "Q`r2BM- CQM�i?�M J �M/ _B+?�`/ 1 *`�M/�HH UkyRjVX dz*HQb2/ 6Q`Kb, q?�i h?2v �`2 �M/ q?v q2 *�`2ǴX AM, LQiB+2b Q7 i?2 �Ja eyXR- TTX 8yĜe8X m`H, ?iiTb,ff+Bi2b22`tX BbiXTbmX2/mfpB2r/Q+f/QrMHQ�/\/QB4RyXRXRXRNyX 93RdW8*�`2T4`2TRW8*�ivT24T/7X *?Qr- hBKQi?v u URNNNVX dzq?�i Bb � *HQb2/@6Q`K LmK#2`\Ǵ AM, h?2 �K2`B+�M J�i?2K�iB+�H JQMi?Hv RyeX8- TTX 99yĜ 993X m`H, ?iiTb,ffrrrXi�M/7QMHBM2X+QKf/QBfT/7f RyXRy3yfyyykN3NyXRNNNXRkyy8yee\+�b�niQF2M4 oNEp;#?92tJ����� , +oG.M_lxQl_NBqR9djN`lj#K n EwDdM8SSQL#7>h;Cqa2p[NTQ�>JabdQCBL n hiGCa89htt;BQiJyX >2Hbi`QK- * q URN3NVX dz.2i2+i�#BHBiv Q7 aB;M�Hb BM G�TH�+2 LQBb2ǴX AM, A111 h`�Mb�+iBQMb QM �2`QbT�+2 �M/ 1H2+i`QMB+ avbi2Kb �1a@k8Xk- TTX RNyĜRNeX J�`Fb AA- _ C 2i �HX URNd3VX dz.2i2+iBQM BM G�TH�+2 LQBb2ǴX AM, A111 h`�Mb�+iBQMb QM �2`QbT�+2 �M/ 1H2+i`QMB+ avb@ i2Kb �1a@R9- TTX 3eeĜ3dkX /QB, RyXRRyNfh�1aXRNd3X jy388yX m`H, ?iiTb , f f B222tTHQ`2 X B222 X Q`; f /Q+mK2Mif9Rykyd8X L�/Bb- ai2p2 UkykyVX dz�7i2` *2Mim`B2b- � a22KBM;Hv aBKTH2 J�i? S`Q#H2K :2ib �M 1t�+i aQHmiBQMǴX AM, Zm�Mi� J�;@ �xBM2X m`H, ?iiTb , f f rrr X [m�Mi�K�;�xBM2 X Q`; f K�i?2K�iB+B�M@ bQHp2b@ +2Mim`B2b@ QH/@ ;`�xBM;@ ;Q�i@T`Q#H2K@2t�+iHv@kykyRkyNfX lHHBb+?- AM;Q UkykyVX dz� *HQb2/@6Q`K aQHmiBQM iQ i?2 :2Q@ K2i`B+ :Q�i S`Q#H2KǴX AM, J�i?2K�iB+�H AMi2HHB;2M+2` 9k- TTX RĜ8X /QB, RyXRyydfbyyk3j@ yky@ yNNee@ yX m`H, ?iiTb,ffHBMFXbT`BM;2`X+QKf�`iB+H2fRyXRyydW k6byyk3j@yky@yNNee@yX https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.190.4817%5C&rep=rep1%5C&type=pdf https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.190.4817%5C&rep=rep1%5C&type=pdf https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.190.4817%5C&rep=rep1%5C&type=pdf https://www.tandfonline.com/doi/pdf/10.1080/00029890.1999.12005066?casa_token=V9Kvgbh4exMAAAAA:cVLDnRUzoUR9iW14739rU3bm_KZj7n5PPoNbfHTgJWSevq9poaHMSs7oJiN_TtLJS54TxxgiotM0 https://www.tandfonline.com/doi/pdf/10.1080/00029890.1999.12005066?casa_token=V9Kvgbh4exMAAAAA:cVLDnRUzoUR9iW14739rU3bm_KZj7n5PPoNbfHTgJWSevq9poaHMSs7oJiN_TtLJS54TxxgiotM0 https://www.tandfonline.com/doi/pdf/10.1080/00029890.1999.12005066?casa_token=V9Kvgbh4exMAAAAA:cVLDnRUzoUR9iW14739rU3bm_KZj7n5PPoNbfHTgJWSevq9poaHMSs7oJiN_TtLJS54TxxgiotM0 https://www.tandfonline.com/doi/pdf/10.1080/00029890.1999.12005066?casa_token=V9Kvgbh4exMAAAAA:cVLDnRUzoUR9iW14739rU3bm_KZj7n5PPoNbfHTgJWSevq9poaHMSs7oJiN_TtLJS54TxxgiotM0 https://www.tandfonline.com/doi/pdf/10.1080/00029890.1999.12005066?casa_token=V9Kvgbh4exMAAAAA:cVLDnRUzoUR9iW14739rU3bm_KZj7n5PPoNbfHTgJWSevq9poaHMSs7oJiN_TtLJS54TxxgiotM0 https://doi.org/10.1109/TAES.1978.308550 https://doi.org/10.1109/TAES.1978.308550 https://ieeexplore.ieee.org/document/4102075 https://ieeexplore.ieee.org/document/4102075 https://www.quantamagazine.org/mathematician-solves-centuries-old-grazing-goat-problem-exactly-20201209/ https://www.quantamagazine.org/mathematician-solves-centuries-old-grazing-goat-problem-exactly-20201209/ https://www.quantamagazine.org/mathematician-solves-centuries-old-grazing-goat-problem-exactly-20201209/ https://doi.org/10.1007/s00283-020-09966-0 https://link.springer.com/article/10.1007%2Fs00283-020-09966-0 https://link.springer.com/article/10.1007%2Fs00283-020-09966-0 About This Journal The Purpose of the Journal Paper Submission Policies Other Journal Content From the Editors Eugenio DarbesioEugenio DarbesioA Corollary of the Conant-Ashby Theorem Applied to Abiogenesis Robert SheldonRobert SheldonComets, Water, and Big Bang Nucleosynthesis Introduction Benefits of a Magnetized Big Bang and Dark Matter Comets Consequences of Primordial Comets Analysis Conclusions Acknowledgments Robert J Marks IIRobert J Marks IISolution of the Grazing Goat Problem: A Conflict between Beauty and Pragmatism Background Grazing Goat Problem Solution Grazing Goat Problem Inversion Final Thought Letters and Notes Eric Holloway2D Puzzle Visualizations of Boolean Formulae Jonathan BartlettThe Products of Hyperreal Series and the Limitations of Cauchy Products Tom and Elizabeth SiewertFollowing the Science News