id sid tid token lemma pos american_scientific_journal-1433 1 1 9 9 NUM american_scientific_journal-1433 1 2 american american ADJ american_scientific_journal-1433 1 3 scientific scientific ADJ american_scientific_journal-1433 1 4 research research NOUN american_scientific_journal-1433 1 5 journal journal NOUN american_scientific_journal-1433 1 6 for for ADP american_scientific_journal-1433 1 7 engineering engineering NOUN american_scientific_journal-1433 1 8 , , PUNCT american_scientific_journal-1433 1 9 technology technology NOUN american_scientific_journal-1433 1 10 , , PUNCT american_scientific_journal-1433 1 11 and and CCONJ american_scientific_journal-1433 1 12 sciences science NOUN american_scientific_journal-1433 1 13 ( ( PUNCT american_scientific_journal-1433 1 14 asrjets asrjet NOUN american_scientific_journal-1433 1 15 ) ) PUNCT american_scientific_journal-1433 1 16 issn issn PROPN american_scientific_journal-1433 1 17 ( ( PUNCT american_scientific_journal-1433 1 18 print print NOUN american_scientific_journal-1433 1 19 ) ) PUNCT american_scientific_journal-1433 1 20 2313 2313 NUM american_scientific_journal-1433 1 21 - - SYM american_scientific_journal-1433 1 22 4410 4410 NUM american_scientific_journal-1433 1 23 , , PUNCT american_scientific_journal-1433 1 24 issn issn PROPN american_scientific_journal-1433 1 25 ( ( PUNCT american_scientific_journal-1433 1 26 online online ADJ american_scientific_journal-1433 1 27 ) ) PUNCT american_scientific_journal-1433 1 28 2313 2313 NUM american_scientific_journal-1433 1 29 - - SYM american_scientific_journal-1433 1 30 4402 4402 NUM american_scientific_journal-1433 1 31 Β© Β© PROPN american_scientific_journal-1433 1 32 global global ADJ american_scientific_journal-1433 1 33 society society NOUN american_scientific_journal-1433 1 34 of of ADP american_scientific_journal-1433 1 35 scientific scientific ADJ american_scientific_journal-1433 1 36 research research NOUN american_scientific_journal-1433 1 37 and and CCONJ american_scientific_journal-1433 1 38 researchers researcher NOUN american_scientific_journal-1433 1 39 http://asrjetsjournal.org/ http://asrjetsjournal.org/ VERB american_scientific_journal-1433 1 40 on on ADP american_scientific_journal-1433 1 41 a a DET american_scientific_journal-1433 1 42 generalized generalized ADJ american_scientific_journal-1433 1 43 𝛃𝛃𝐑𝐑 𝛃𝛃𝐑𝐑 NUM american_scientific_journal-1433 1 44 βˆ’ βˆ’ NOUN american_scientific_journal-1433 1 45 birecurrent birecurrent ADJ american_scientific_journal-1433 1 46 finsler finsler NOUN american_scientific_journal-1433 1 47 space space NOUN american_scientific_journal-1433 1 48 fahmi fahmi PROPN american_scientific_journal-1433 1 49 yaseen yaseen PROPN american_scientific_journal-1433 1 50 abdo abdo PROPN american_scientific_journal-1433 1 51 qasema qasema VERB american_scientific_journal-1433 1 52 * * PUNCT american_scientific_journal-1433 1 53 , , PUNCT american_scientific_journal-1433 1 54 wafa'a wafa'a PROPN american_scientific_journal-1433 1 55 hadi hadi PROPN american_scientific_journal-1433 1 56 ali ali PROPN american_scientific_journal-1433 1 57 hadib hadib PROPN american_scientific_journal-1433 1 58 adepartment adepartment NOUN american_scientific_journal-1433 1 59 of of ADP american_scientific_journal-1433 1 60 mathematics mathematic NOUN american_scientific_journal-1433 1 61 , , PUNCT american_scientific_journal-1433 1 62 faculty faculty NOUN american_scientific_journal-1433 1 63 of of ADP american_scientific_journal-1433 1 64 education education NOUN american_scientific_journal-1433 1 65 - - PUNCT american_scientific_journal-1433 1 66 aden aden PROPN american_scientific_journal-1433 1 67 , , PUNCT american_scientific_journal-1433 1 68 university university PROPN american_scientific_journal-1433 1 69 of of ADP american_scientific_journal-1433 1 70 aden aden PROPN american_scientific_journal-1433 1 71 , , PUNCT american_scientific_journal-1433 1 72 khormaksar khormaksar PROPN american_scientific_journal-1433 1 73 , , PUNCT american_scientific_journal-1433 1 74 aden aden PROPN american_scientific_journal-1433 1 75 , , PUNCT american_scientific_journal-1433 1 76 yemen yemen PROPN american_scientific_journal-1433 1 77 bdepartment bdepartment NOUN american_scientific_journal-1433 1 78 of of ADP american_scientific_journal-1433 1 79 mathematics mathematics PROPN american_scientific_journal-1433 1 80 , , PUNCT american_scientific_journal-1433 1 81 community community NOUN american_scientific_journal-1433 1 82 college college PROPN american_scientific_journal-1433 1 83 , , PUNCT american_scientific_journal-1433 1 84 dar dar PROPN american_scientific_journal-1433 1 85 saad saad PROPN american_scientific_journal-1433 1 86 , , PUNCT american_scientific_journal-1433 1 87 aden aden PROPN american_scientific_journal-1433 1 88 , , PUNCT american_scientific_journal-1433 1 89 yemen yemen PROPN american_scientific_journal-1433 1 90 aemail aemail NOUN american_scientific_journal-1433 1 91 : : PUNCT american_scientific_journal-1433 1 92 fahmiyassen1@gmail.com fahmiyassen1@gmail.com X american_scientific_journal-1433 2 1 bemail bemail NOUN american_scientific_journal-1433 2 2 : : PUNCT american_scientific_journal-1433 2 3 wf_hadi@yahoo.com wf_hadi@yahoo.com X american_scientific_journal-1433 2 4 abstract abstract NOUN american_scientific_journal-1433 2 5 in in ADP american_scientific_journal-1433 2 6 the the DET american_scientific_journal-1433 2 7 present present ADJ american_scientific_journal-1433 2 8 paper paper NOUN american_scientific_journal-1433 2 9 , , PUNCT american_scientific_journal-1433 2 10 we we PRON american_scientific_journal-1433 2 11 introduced introduce VERB american_scientific_journal-1433 2 12 a a DET american_scientific_journal-1433 2 13 finsler finsler NOUN american_scientific_journal-1433 2 14 space space NOUN american_scientific_journal-1433 2 15 whose whose DET american_scientific_journal-1433 2 16 cartan cartan PROPN american_scientific_journal-1433 2 17 's 's PART american_scientific_journal-1433 2 18 third third ADJ american_scientific_journal-1433 2 19 curvature curvature NOUN american_scientific_journal-1433 2 20 tensor tensor NOUN american_scientific_journal-1433 2 21 π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 2 22 satisfies satisfy VERB american_scientific_journal-1433 2 23 β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 2 24 = = PUNCT american_scientific_journal-1433 2 25 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– X american_scientific_journal-1433 2 26 + + NUM american_scientific_journal-1433 2 27 π‘π‘π‘šπ‘šπ‘›π‘› π‘π‘π‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 2 28 οΏ½ οΏ½ PROPN american_scientific_journal-1433 2 29 𝛿𝛿𝑗𝑗𝑖𝑖 𝛿𝛿𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 2 30 π‘”π‘”π‘—π‘—β„Ž π‘”π‘”π‘—π‘—β„Ž NOUN american_scientific_journal-1433 2 31 βˆ’ βˆ’ PROPN american_scientific_journal-1433 3 1 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 3 2 𝑔𝑔𝑗𝑗𝑗𝑗 𝑔𝑔𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 3 3 οΏ½ οΏ½ PROPN american_scientific_journal-1433 3 4 βˆ’ βˆ’ NUM american_scientific_journal-1433 3 5 2π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘Ÿ(𝛿𝛿𝑗𝑗𝑖𝑖 2π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘Ÿ(𝛿𝛿𝑗𝑗𝑖𝑖 PROPN american_scientific_journal-1433 3 6 πΆπΆπ‘—π‘—β„Žπ‘šπ‘š πΆπΆπ‘—π‘—β„Žπ‘šπ‘š PROPN american_scientific_journal-1433 3 7 βˆ’ βˆ’ PROPN american_scientific_journal-1433 3 8 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 3 9 πΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š πΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š NOUN american_scientific_journal-1433 3 10 ) ) PUNCT american_scientific_journal-1433 3 11 . . PUNCT american_scientific_journal-1433 4 1 where where SCONJ american_scientific_journal-1433 4 2 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 4 3 and and CCONJ american_scientific_journal-1433 4 4 π‘π‘π‘šπ‘šπ‘›π‘› π‘π‘π‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 4 5 are be AUX american_scientific_journal-1433 4 6 nonzero nonzero PROPN american_scientific_journal-1433 4 7 covariant covariant ADJ american_scientific_journal-1433 4 8 tensor tensor NOUN american_scientific_journal-1433 4 9 fields field NOUN american_scientific_journal-1433 4 10 of of ADP american_scientific_journal-1433 4 11 second second ADJ american_scientific_journal-1433 4 12 order order NOUN american_scientific_journal-1433 4 13 called call VERB american_scientific_journal-1433 4 14 recurrence recurrence NOUN american_scientific_journal-1433 4 15 tensor tensor NOUN american_scientific_journal-1433 4 16 fields field NOUN american_scientific_journal-1433 4 17 , , PUNCT american_scientific_journal-1433 4 18 such such ADJ american_scientific_journal-1433 4 19 space space NOUN american_scientific_journal-1433 4 20 is be AUX american_scientific_journal-1433 4 21 called call VERB american_scientific_journal-1433 4 22 as as ADP american_scientific_journal-1433 4 23 a a DET american_scientific_journal-1433 4 24 generalized generalized ADJ american_scientific_journal-1433 4 25 𝛽𝛽𝑅𝑅 𝛽𝛽𝑅𝑅 PROPN american_scientific_journal-1433 4 26 βˆ’birecurrent βˆ’birecurrent NOUN american_scientific_journal-1433 4 27 finsler finsler NOUN american_scientific_journal-1433 4 28 space space NOUN american_scientific_journal-1433 4 29 . . PUNCT american_scientific_journal-1433 5 1 the the DET american_scientific_journal-1433 5 2 curvature curvature NOUN american_scientific_journal-1433 5 3 tensor tensor NOUN american_scientific_journal-1433 5 4 π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 5 5 , , PUNCT american_scientific_journal-1433 5 6 the the DET american_scientific_journal-1433 5 7 torsion torsion NOUN american_scientific_journal-1433 5 8 tensor tensor NOUN american_scientific_journal-1433 5 9 π»π»π‘—π‘—β„Žπ‘–π‘– π»π»π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 5 10 , , PUNCT american_scientific_journal-1433 5 11 the the DET american_scientific_journal-1433 5 12 deviation deviation NOUN american_scientific_journal-1433 5 13 tensor tensor NOUN american_scientific_journal-1433 5 14 π»π»β„Žπ‘–π‘– π»π»β„Žπ‘–π‘– PROPN american_scientific_journal-1433 5 15 , , PUNCT american_scientific_journal-1433 5 16 the the DET american_scientific_journal-1433 5 17 ricci ricci PROPN american_scientific_journal-1433 5 18 tensors tensor NOUN american_scientific_journal-1433 5 19 ( ( PUNCT american_scientific_journal-1433 5 20 𝐻𝐻𝑗𝑗𝑗𝑗 𝐻𝐻𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 5 21 , , PUNCT american_scientific_journal-1433 5 22 𝑅𝑅𝑗𝑗𝑗𝑗 𝑅𝑅𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 5 23 ) ) PUNCT american_scientific_journal-1433 5 24 , , PUNCT american_scientific_journal-1433 5 25 the the DET american_scientific_journal-1433 5 26 vector vector NOUN american_scientific_journal-1433 5 27 𝐻𝐻𝑗𝑗 𝐻𝐻𝑗𝑗 PROPN american_scientific_journal-1433 5 28 and and CCONJ american_scientific_journal-1433 5 29 the the DET american_scientific_journal-1433 5 30 scalar scalar ADJ american_scientific_journal-1433 5 31 curvature curvature NOUN american_scientific_journal-1433 5 32 tensor tensor NOUN american_scientific_journal-1433 5 33 𝐻𝐻of 𝐻𝐻of PROPN american_scientific_journal-1433 5 34 such such ADJ american_scientific_journal-1433 5 35 space space NOUN american_scientific_journal-1433 5 36 are be AUX american_scientific_journal-1433 5 37 non non ADJ american_scientific_journal-1433 5 38 - - ADJ american_scientific_journal-1433 5 39 vanishing vanishing ADJ american_scientific_journal-1433 5 40 . . PUNCT american_scientific_journal-1433 6 1 under under ADP american_scientific_journal-1433 6 2 certain certain ADJ american_scientific_journal-1433 6 3 conditions condition NOUN american_scientific_journal-1433 6 4 , , PUNCT american_scientific_journal-1433 6 5 a a DET american_scientific_journal-1433 6 6 generalized generalize VERB american_scientific_journal-1433 6 7 𝛽𝛽𝑅𝑅 𝛽𝛽𝑅𝑅 PROPN american_scientific_journal-1433 6 8 βˆ’birecurrent βˆ’birecurrent NOUN american_scientific_journal-1433 6 9 finsler finsler NOUN american_scientific_journal-1433 6 10 space space NOUN american_scientific_journal-1433 6 11 becomes become VERB american_scientific_journal-1433 6 12 landsberg landsberg PROPN american_scientific_journal-1433 6 13 space space NOUN american_scientific_journal-1433 6 14 . . PUNCT american_scientific_journal-1433 7 1 some some DET american_scientific_journal-1433 7 2 conditions condition NOUN american_scientific_journal-1433 7 3 have have AUX american_scientific_journal-1433 7 4 been be AUX american_scientific_journal-1433 7 5 pointed point VERB american_scientific_journal-1433 7 6 out out ADP american_scientific_journal-1433 7 7 which which PRON american_scientific_journal-1433 7 8 reduce reduce VERB american_scientific_journal-1433 7 9 a a DET american_scientific_journal-1433 7 10 generalized generalized ADJ american_scientific_journal-1433 7 11 𝛽𝛽𝑅𝑅 𝛽𝛽𝑅𝑅 PROPN american_scientific_journal-1433 7 12 βˆ’birecurrent βˆ’birecurrent NOUN american_scientific_journal-1433 7 13 finsler finsler NOUN american_scientific_journal-1433 7 14 space space NOUN american_scientific_journal-1433 7 15 𝐹𝐹𝑛𝑛(𝑛𝑛 𝐹𝐹𝑛𝑛(𝑛𝑛 X american_scientific_journal-1433 7 16 > > X american_scientific_journal-1433 7 17 2 2 X american_scientific_journal-1433 7 18 ) ) PUNCT american_scientific_journal-1433 7 19 into into ADP american_scientific_journal-1433 7 20 finsler finsler NOUN american_scientific_journal-1433 7 21 space space NOUN american_scientific_journal-1433 7 22 of of ADP american_scientific_journal-1433 7 23 scalar scalar ADJ american_scientific_journal-1433 7 24 curvature curvature NOUN american_scientific_journal-1433 7 25 . . PUNCT american_scientific_journal-1433 8 1 keywords keyword NOUN american_scientific_journal-1433 8 2 : : PUNCT american_scientific_journal-1433 8 3 finsler finsler NOUN american_scientific_journal-1433 8 4 space space NOUN american_scientific_journal-1433 8 5 ; ; PUNCT american_scientific_journal-1433 8 6 generalized generalize VERB american_scientific_journal-1433 8 7 𝛽𝛽𝑅𝑅 𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 8 8 βˆ’birecurrent βˆ’birecurrent NOUN american_scientific_journal-1433 8 9 finsler finsler NOUN american_scientific_journal-1433 8 10 space space NOUN american_scientific_journal-1433 8 11 ; ; PUNCT american_scientific_journal-1433 8 12 ricci ricci PROPN american_scientific_journal-1433 8 13 tensor tensor NOUN american_scientific_journal-1433 8 14 ; ; PUNCT american_scientific_journal-1433 8 15 landsberg landsberg PROPN american_scientific_journal-1433 8 16 space space NOUN american_scientific_journal-1433 8 17 ; ; PUNCT american_scientific_journal-1433 8 18 finsler finsler NOUN american_scientific_journal-1433 8 19 space space NOUN american_scientific_journal-1433 8 20 of of ADP american_scientific_journal-1433 8 21 scalar scalar ADJ american_scientific_journal-1433 8 22 curvature curvature NOUN american_scientific_journal-1433 8 23 . . PUNCT american_scientific_journal-1433 9 1 1 1 X american_scientific_journal-1433 9 2 . . X american_scientific_journal-1433 9 3 introduction introduction PROPN american_scientific_journal-1433 9 4 h.s h.s PROPN american_scientific_journal-1433 9 5 . . PUNCT american_scientific_journal-1433 9 6 ruse[4 ruse[4 PROPN american_scientific_journal-1433 9 7 ] ] PUNCT american_scientific_journal-1433 9 8 considered consider VERB american_scientific_journal-1433 9 9 a a DET american_scientific_journal-1433 9 10 three three NUM american_scientific_journal-1433 9 11 dimensional dimensional ADJ american_scientific_journal-1433 9 12 riemannian riemannian ADJ american_scientific_journal-1433 9 13 space space NOUN american_scientific_journal-1433 9 14 having have VERB american_scientific_journal-1433 9 15 the the DET american_scientific_journal-1433 9 16 recurrent recurrent NOUN american_scientific_journal-1433 9 17 of of ADP american_scientific_journal-1433 9 18 curvature curvature NOUN american_scientific_journal-1433 9 19 tensor tensor NOUN american_scientific_journal-1433 9 20 and and CCONJ american_scientific_journal-1433 9 21 he he PRON american_scientific_journal-1433 9 22 called call VERB american_scientific_journal-1433 9 23 such such ADJ american_scientific_journal-1433 9 24 space space NOUN american_scientific_journal-1433 9 25 as as ADP american_scientific_journal-1433 9 26 riemannian riemannian ADJ american_scientific_journal-1433 9 27 space space NOUN american_scientific_journal-1433 9 28 of of ADP american_scientific_journal-1433 9 29 recurrent recurrent ADJ american_scientific_journal-1433 9 30 curvature curvature NOUN american_scientific_journal-1433 9 31 . . PUNCT american_scientific_journal-1433 10 1 this this DET american_scientific_journal-1433 10 2 idea idea NOUN american_scientific_journal-1433 10 3 was be AUX american_scientific_journal-1433 10 4 extended extend VERB american_scientific_journal-1433 10 5 to to ADP american_scientific_journal-1433 10 6 n n CCONJ american_scientific_journal-1433 10 7 - - PUNCT american_scientific_journal-1433 10 8 dimensional dimensional ADJ american_scientific_journal-1433 10 9 riemannian riemannian NOUN american_scientific_journal-1433 10 10 and and CCONJ american_scientific_journal-1433 10 11 nonriemannian nonriemannian ADJ american_scientific_journal-1433 10 12 space space NOUN american_scientific_journal-1433 10 13 by by ADP american_scientific_journal-1433 10 14 a.g a.g PROPN american_scientific_journal-1433 10 15 . . PROPN american_scientific_journal-1433 10 16 walker walker PROPN american_scientific_journal-1433 11 1 [ [ X american_scientific_journal-1433 11 2 1],y.c 1],y.c NUM american_scientific_journal-1433 11 3 . . PUNCT american_scientific_journal-1433 11 4 worg worg ADJ american_scientific_journal-1433 11 5 [ [ X american_scientific_journal-1433 11 6 14 14 NUM american_scientific_journal-1433 11 7 ] ] PUNCT american_scientific_journal-1433 11 8 , , PUNCT american_scientific_journal-1433 11 9 y.c y.c PROPN american_scientific_journal-1433 11 10 . . PROPN american_scientific_journal-1433 11 11 worg worg PROPN american_scientific_journal-1433 11 12 and and CCONJ american_scientific_journal-1433 11 13 k. k. PROPN american_scientific_journal-1433 11 14 yano yano PROPN american_scientific_journal-1433 12 1 [ [ X american_scientific_journal-1433 12 2 15 15 NUM american_scientific_journal-1433 12 3 ] ] PUNCT american_scientific_journal-1433 12 4 and and CCONJ american_scientific_journal-1433 12 5 others other NOUN american_scientific_journal-1433 12 6 . . PUNCT american_scientific_journal-1433 13 1 ----------------------------------------------------------------- ----------------------------------------------------------------- PUNCT american_scientific_journal-1433 14 1 * * PUNCT american_scientific_journal-1433 14 2 corresponding correspond VERB american_scientific_journal-1433 14 3 author author NOUN american_scientific_journal-1433 14 4 . . PUNCT american_scientific_journal-1433 15 1 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PROPN american_scientific_journal-1433 15 2 american american PROPN american_scientific_journal-1433 15 3 scientific scientific PROPN american_scientific_journal-1433 15 4 research research NOUN american_scientific_journal-1433 15 5 journal journal NOUN american_scientific_journal-1433 15 6 for for ADP american_scientific_journal-1433 15 7 engineering engineering NOUN american_scientific_journal-1433 15 8 , , PUNCT american_scientific_journal-1433 15 9 technology technology NOUN american_scientific_journal-1433 15 10 , , PUNCT american_scientific_journal-1433 15 11 and and CCONJ american_scientific_journal-1433 15 12 sciences science NOUN american_scientific_journal-1433 15 13 ( ( PUNCT american_scientific_journal-1433 15 14 asrjets asrjet NOUN american_scientific_journal-1433 15 15 ) ) PUNCT american_scientific_journal-1433 15 16 ( ( PUNCT american_scientific_journal-1433 15 17 2016 2016 NUM american_scientific_journal-1433 15 18 ) ) PUNCT american_scientific_journal-1433 15 19 volume volume NOUN american_scientific_journal-1433 15 20 19 19 NUM american_scientific_journal-1433 15 21 , , PUNCT american_scientific_journal-1433 15 22 no no DET american_scientific_journal-1433 15 23 1 1 NUM american_scientific_journal-1433 15 24 , , PUNCT american_scientific_journal-1433 15 25 pp pp ADV american_scientific_journal-1433 15 26 9 9 NUM american_scientific_journal-1433 15 27 - - SYM american_scientific_journal-1433 15 28 18 18 NUM american_scientific_journal-1433 15 29 10 10 NUM american_scientific_journal-1433 15 30 s. s. PROPN american_scientific_journal-1433 15 31 dikshit dikshit PROPN american_scientific_journal-1433 16 1 [ [ X american_scientific_journal-1433 16 2 13 13 NUM american_scientific_journal-1433 16 3 ] ] PUNCT american_scientific_journal-1433 16 4 introduced introduce VERB american_scientific_journal-1433 16 5 a a DET american_scientific_journal-1433 16 6 finsler finsler NOUN american_scientific_journal-1433 16 7 space space NOUN american_scientific_journal-1433 16 8 whose whose DET american_scientific_journal-1433 16 9 berwald berwald NOUN american_scientific_journal-1433 16 10 curvature curvature NOUN american_scientific_journal-1433 16 11 tensor tensor NOUN american_scientific_journal-1433 17 1 π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 17 2 satisfies satisfy VERB american_scientific_journal-1433 17 3 recurrence recurrence VERB american_scientific_journal-1433 17 4 property property NOUN american_scientific_journal-1433 17 5 in in ADP american_scientific_journal-1433 17 6 the the DET american_scientific_journal-1433 17 7 sense sense NOUN american_scientific_journal-1433 17 8 of of ADP american_scientific_journal-1433 17 9 berwald berwald NOUN american_scientific_journal-1433 17 10 , , PUNCT american_scientific_journal-1433 17 11 f.y.a.qasem f.y.a.qasem PROPN american_scientific_journal-1433 17 12 and and CCONJ american_scientific_journal-1433 17 13 a.a.m.saleem a.a.m.saleem PROPN american_scientific_journal-1433 18 1 [ [ X american_scientific_journal-1433 18 2 3 3 X american_scientific_journal-1433 18 3 ] ] PUNCT american_scientific_journal-1433 18 4 discussed discuss VERB american_scientific_journal-1433 18 5 general general ADJ american_scientific_journal-1433 18 6 finsler finsler NOUN american_scientific_journal-1433 18 7 space space NOUN american_scientific_journal-1433 18 8 for for ADP american_scientific_journal-1433 18 9 the the DET american_scientific_journal-1433 18 10 β„Žπ‘£π‘£ β„Žπ‘£π‘£ PROPN american_scientific_journal-1433 18 11 βˆ’curvature βˆ’curvature PROPN american_scientific_journal-1433 18 12 tensor tensor NOUN american_scientific_journal-1433 18 13 π‘ˆπ‘ˆπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘ˆπ‘ˆπ‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 18 14 satisfies satisfy VERB american_scientific_journal-1433 18 15 the the DET american_scientific_journal-1433 18 16 birecurrence birecurrence NOUN american_scientific_journal-1433 18 17 property property NOUN american_scientific_journal-1433 18 18 with with ADP american_scientific_journal-1433 18 19 respect respect NOUN american_scientific_journal-1433 18 20 to to ADP american_scientific_journal-1433 18 21 berwald berwald NOUN american_scientific_journal-1433 18 22 's 's PART american_scientific_journal-1433 18 23 coefficient coefficient NOUN american_scientific_journal-1433 18 24 𝐺𝐺𝑗𝑗𝑗𝑗𝑖𝑖 𝐺𝐺𝑗𝑗𝑗𝑗𝑖𝑖 PROPN american_scientific_journal-1433 18 25 and and CCONJ american_scientific_journal-1433 18 26 they they PRON american_scientific_journal-1433 18 27 called call VERB american_scientific_journal-1433 18 28 it it PRON american_scientific_journal-1433 18 29 ubrfinsler ubrfinsler ADJ american_scientific_journal-1433 18 30 space space NOUN american_scientific_journal-1433 18 31 . . PUNCT american_scientific_journal-1433 19 1 p.n.pandey p.n.pandey PROPN american_scientific_journal-1433 19 2 , , PUNCT american_scientific_journal-1433 19 3 s.saxena s.saxena NOUN american_scientific_journal-1433 19 4 and and CCONJ american_scientific_journal-1433 19 5 a.goswami a.goswami NOUN american_scientific_journal-1433 19 6 [ [ X american_scientific_journal-1433 19 7 8 8 NUM american_scientific_journal-1433 19 8 ] ] PUNCT american_scientific_journal-1433 19 9 introduced introduce VERB american_scientific_journal-1433 19 10 a a DET american_scientific_journal-1433 19 11 finsler finsler NOUN american_scientific_journal-1433 19 12 space space NOUN american_scientific_journal-1433 19 13 whose whose DET american_scientific_journal-1433 19 14 berwald berwald NOUN american_scientific_journal-1433 19 15 curvature curvature NOUN american_scientific_journal-1433 19 16 tensor tensor NOUN american_scientific_journal-1433 20 1 π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 20 2 satisfies satisfy VERB american_scientific_journal-1433 20 3 generalized generalized ADJ american_scientific_journal-1433 20 4 recurrence recurrence NOUN american_scientific_journal-1433 20 5 property property NOUN american_scientific_journal-1433 20 6 in in ADP american_scientific_journal-1433 20 7 the the DET american_scientific_journal-1433 20 8 sense sense NOUN american_scientific_journal-1433 20 9 of of ADP american_scientific_journal-1433 20 10 berwald berwald NOUN american_scientific_journal-1433 20 11 they they PRON american_scientific_journal-1433 20 12 called call VERB american_scientific_journal-1433 20 13 such such ADJ american_scientific_journal-1433 20 14 space space NOUN american_scientific_journal-1433 20 15 generalized generalize VERB american_scientific_journal-1433 20 16 𝐻𝐻-recurrent 𝐻𝐻-recurrent NOUN american_scientific_journal-1433 20 17 finsler finsler NOUN american_scientific_journal-1433 20 18 space space NOUN american_scientific_journal-1433 20 19 . . PUNCT american_scientific_journal-1433 21 1 ( ( PUNCT american_scientific_journal-1433 21 2 1.1 1.1 NUM american_scientific_journal-1433 21 3 ) ) PUNCT american_scientific_journal-1433 21 4 a a DET american_scientific_journal-1433 21 5 ) ) PUNCT american_scientific_journal-1433 21 6 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 21 7 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 21 8 = = PUNCT american_scientific_journal-1433 21 9 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 21 10 b b PROPN american_scientific_journal-1433 21 11 ) ) PUNCT american_scientific_journal-1433 21 12 g𝑖𝑖𝑗𝑗 g𝑖𝑖𝑗𝑗 PROPN american_scientific_journal-1433 21 13 = = SYM american_scientific_journal-1433 21 14 οΏ½ οΏ½ PROPN american_scientific_journal-1433 21 15 Μ‡ Μ‡ PROPN american_scientific_journal-1433 21 16 οΏ½ οΏ½ PROPN american_scientific_journal-1433 21 17 πœ•π‘–π‘–π‘¦π‘¦π‘—π‘— πœ•π‘–π‘–π‘¦π‘¦π‘—π‘— PART american_scientific_journal-1433 21 18 = = SYM american_scientific_journal-1433 21 19 οΏ½ οΏ½ PROPN american_scientific_journal-1433 21 20 Μ‡ Μ‡ PROPN american_scientific_journal-1433 21 21 οΏ½ οΏ½ PROPN american_scientific_journal-1433 21 22 πœ•π‘—π‘—π‘¦π‘¦π‘–π‘– πœ•π‘—π‘—π‘¦π‘¦π‘–π‘– PROPN american_scientific_journal-1433 21 23 c c PROPN american_scientific_journal-1433 21 24 ) ) PUNCT american_scientific_journal-1433 21 25 ℬ𝑗𝑗 ℬ𝑗𝑗 NOUN american_scientific_journal-1433 21 26 𝑦𝑦𝑖𝑖= 𝑦𝑦𝑖𝑖= NUM american_scientific_journal-1433 21 27 0 0 SYM american_scientific_journal-1433 22 1 d d X american_scientific_journal-1433 22 2 ) ) PUNCT american_scientific_journal-1433 22 3 𝐢𝐢 𝐢𝐢 PROPN american_scientific_journal-1433 22 4 𝑖𝑖𝑗𝑗𝑗𝑗 𝑖𝑖𝑗𝑗𝑗𝑗 NOUN american_scientific_journal-1433 22 5 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 22 6 = = SYM american_scientific_journal-1433 22 7 𝐢𝐢 𝐢𝐢 NOUN american_scientific_journal-1433 22 8 𝑗𝑗𝑖𝑖𝑗𝑗 𝑗𝑗𝑖𝑖𝑗𝑗 NOUN american_scientific_journal-1433 22 9 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 22 10 = = SYM american_scientific_journal-1433 22 11 𝐢𝐢 𝐢𝐢 PROPN american_scientific_journal-1433 22 12 𝑗𝑗𝑗𝑗𝑖𝑖 𝑗𝑗𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 22 13 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 22 14 = = SYM american_scientific_journal-1433 22 15 0 0 NUM american_scientific_journal-1433 22 16 e e NOUN american_scientific_journal-1433 22 17 ) ) PUNCT american_scientific_journal-1433 22 18 ℬ𝑗𝑗 ℬ𝑗𝑗 NOUN american_scientific_journal-1433 22 19 𝑔𝑔𝑖𝑖𝑗𝑗 𝑔𝑔𝑖𝑖𝑗𝑗 NOUN american_scientific_journal-1433 22 20 = = SYM american_scientific_journal-1433 22 21 βˆ’2πΆπΆπ‘–π‘–π‘—π‘—π‘—π‘—β„Žπ‘¦π‘¦β„Ž βˆ’2πΆπΆπ‘–π‘–π‘—π‘—π‘—π‘—β„Žπ‘¦π‘¦β„Ž NOUN american_scientific_journal-1433 23 1 = = PUNCT american_scientific_journal-1433 23 2 2π‘¦π‘¦β„Žβ„¬ 2π‘¦π‘¦β„Žβ„¬ X american_scientific_journal-1433 23 3 β„ŽπΆπΆπ‘–π‘–π‘—π‘—π‘—π‘— β„ŽπΆπΆπ‘–π‘–π‘—π‘—π‘—π‘— NOUN american_scientific_journal-1433 23 4 f f X american_scientific_journal-1433 23 5 ) ) PUNCT american_scientific_journal-1433 23 6 𝐺𝐺 𝐺𝐺 NOUN american_scientific_journal-1433 23 7 π‘—π‘—π‘—π‘—β„Ž π‘—π‘—π‘—π‘—β„Ž NOUN american_scientific_journal-1433 23 8 𝑖𝑖 𝑖𝑖 INTJ american_scientific_journal-1433 23 9 𝑦𝑦 𝑦𝑦 INTJ american_scientific_journal-1433 23 10 𝑗𝑗 𝑗𝑗 NOUN american_scientific_journal-1433 23 11 = = PUNCT american_scientific_journal-1433 23 12 𝐺𝐺 𝐺𝐺 PROPN american_scientific_journal-1433 23 13 β„Žπ‘—π‘—π‘—π‘— β„Žπ‘—π‘—π‘—π‘— NOUN american_scientific_journal-1433 23 14 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-1433 23 15 𝑦𝑦 𝑦𝑦 INTJ american_scientific_journal-1433 23 16 𝑗𝑗 𝑗𝑗 NOUN american_scientific_journal-1433 23 17 = = PUNCT american_scientific_journal-1433 23 18 𝐺𝐺 𝐺𝐺 PROPN american_scientific_journal-1433 23 19 π‘—π‘—β„Žπ‘—π‘— π‘—π‘—β„Žπ‘—π‘— NOUN american_scientific_journal-1433 23 20 𝑖𝑖 𝑖𝑖 INTJ american_scientific_journal-1433 23 21 𝑦𝑦 𝑦𝑦 INTJ american_scientific_journal-1433 23 22 𝑗𝑗 𝑗𝑗 VERB american_scientific_journal-1433 23 23 = = NOUN american_scientific_journal-1433 23 24 0 0 PROPN american_scientific_journal-1433 23 25 . . PUNCT american_scientific_journal-1433 24 1 the the DET american_scientific_journal-1433 24 2 unit unit NOUN american_scientific_journal-1433 24 3 vector vector NOUN american_scientific_journal-1433 24 4 Ι­ Ι­ PROPN american_scientific_journal-1433 24 5 𝑖𝑖 𝑖𝑖 X american_scientific_journal-1433 24 6 and and CCONJ american_scientific_journal-1433 24 7 the the DET american_scientific_journal-1433 24 8 associate associate ADJ american_scientific_journal-1433 24 9 vector vector NOUN american_scientific_journal-1433 24 10 Ι­ Ι­ X american_scientific_journal-1433 24 11 i i PRON american_scientific_journal-1433 24 12 is be AUX american_scientific_journal-1433 24 13 defined define VERB american_scientific_journal-1433 24 14 by by ADP american_scientific_journal-1433 24 15 ( ( PUNCT american_scientific_journal-1433 24 16 1.2 1.2 NUM american_scientific_journal-1433 24 17 ) ) PUNCT american_scientific_journal-1433 24 18 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-1433 24 19 ) ) PUNCT american_scientific_journal-1433 24 20 Ι­ Ι­ NOUN american_scientific_journal-1433 24 21 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-1433 24 22 = = PUNCT american_scientific_journal-1433 24 23 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 24 24 𝐹𝐹 𝐹𝐹 PROPN american_scientific_journal-1433 24 25 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-1433 24 26 ) ) PUNCT american_scientific_journal-1433 24 27 Ι­ Ι­ NOUN american_scientific_journal-1433 24 28 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-1433 25 1 = = SYM american_scientific_journal-1433 25 2 g g PROPN american_scientific_journal-1433 25 3 𝑖𝑖𝑗𝑗ɭ𝑗𝑗 𝑖𝑖𝑗𝑗ɭ𝑗𝑗 NOUN american_scientific_journal-1433 25 4 = = SYM american_scientific_journal-1433 25 5 οΏ½ οΏ½ PROPN american_scientific_journal-1433 25 6 Μ‡ Μ‡ PROPN american_scientific_journal-1433 25 7 οΏ½ οΏ½ PROPN american_scientific_journal-1433 25 8 πœ•π‘–π‘– πœ•π‘–π‘– NOUN american_scientific_journal-1433 25 9 𝐹𝐹 𝐹𝐹 NOUN american_scientific_journal-1433 25 10 = = SYM american_scientific_journal-1433 25 11 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 25 12 𝐹𝐹 𝐹𝐹 PROPN american_scientific_journal-1433 25 13 . . PUNCT american_scientific_journal-1433 26 1 the the DET american_scientific_journal-1433 26 2 processes process NOUN american_scientific_journal-1433 26 3 of of ADP american_scientific_journal-1433 26 4 berwald berwald NOUN american_scientific_journal-1433 26 5 's 's PART american_scientific_journal-1433 26 6 covariant covariant ADJ american_scientific_journal-1433 26 7 differentiation differentiation NOUN american_scientific_journal-1433 26 8 and and CCONJ american_scientific_journal-1433 26 9 the the DET american_scientific_journal-1433 26 10 partial partial ADJ american_scientific_journal-1433 26 11 differentiation differentiation NOUN american_scientific_journal-1433 26 12 commute commute NOUN american_scientific_journal-1433 26 13 according accord VERB american_scientific_journal-1433 26 14 to to ADP american_scientific_journal-1433 26 15 ( ( PUNCT american_scientific_journal-1433 26 16 1.3 1.3 NUM american_scientific_journal-1433 26 17 ) ) PUNCT american_scientific_journal-1433 26 18 ( ( PUNCT american_scientific_journal-1433 26 19 οΏ½ οΏ½ PROPN american_scientific_journal-1433 26 20 Μ‡ Μ‡ PROPN american_scientific_journal-1433 26 21 οΏ½ οΏ½ PROPN american_scientific_journal-1433 26 22 πœ• πœ• PROPN american_scientific_journal-1433 26 23 π‘—π‘—β„¬β„Ž π‘—π‘—β„¬β„Ž PROPN american_scientific_journal-1433 26 24 βˆ’ βˆ’ PROPN american_scientific_journal-1433 26 25 ℬ𝑗𝑗 ℬ𝑗𝑗 PROPN american_scientific_journal-1433 26 26 οΏ½ οΏ½ PROPN american_scientific_journal-1433 26 27 Μ‡ Μ‡ NOUN american_scientific_journal-1433 26 28 οΏ½ οΏ½ PROPN american_scientific_journal-1433 26 29 πœ• πœ• PROPN american_scientific_journal-1433 26 30 β„Ž β„Ž PROPN american_scientific_journal-1433 26 31 ) ) PUNCT american_scientific_journal-1433 26 32 𝑇𝑇𝑗𝑗𝑖𝑖 𝑇𝑇𝑗𝑗𝑖𝑖 ADJ american_scientific_journal-1433 26 33 = = PUNCT american_scientific_journal-1433 26 34 π‘‡π‘‡π‘—π‘—π‘Ÿπ‘ŸπΊπΊπ‘—π‘—β„Žπ‘Ÿπ‘Ÿπ‘–π‘– π‘‡π‘‡π‘—π‘—π‘Ÿπ‘ŸπΊπΊπ‘—π‘—β„Žπ‘Ÿπ‘Ÿπ‘–π‘– NOUN american_scientific_journal-1433 26 35 βˆ’ βˆ’ NOUN american_scientific_journal-1433 26 36 π‘‡π‘‡π‘Ÿπ‘Ÿπ‘–π‘–πΊπΊπ‘—π‘—β„Žπ‘—π‘—π‘Ÿπ‘Ÿ π‘‡π‘‡π‘Ÿπ‘Ÿπ‘–π‘–πΊπΊπ‘—π‘—β„Žπ‘—π‘—π‘Ÿπ‘Ÿ ADJ american_scientific_journal-1433 26 37 . . PUNCT american_scientific_journal-1433 27 1 the the DET american_scientific_journal-1433 27 2 tensor tensor NOUN american_scientific_journal-1433 27 3 π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 27 4 satisfies satisfy VERB american_scientific_journal-1433 27 5 the the DET american_scientific_journal-1433 27 6 relation relation NOUN american_scientific_journal-1433 27 7 ( ( PUNCT american_scientific_journal-1433 27 8 1.4 1.4 NUM american_scientific_journal-1433 27 9 ) ) PUNCT american_scientific_journal-1433 27 10 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 27 11 π‘—π‘—π‘—π‘—β„Ž π‘—π‘—π‘—π‘—β„Ž NOUN american_scientific_journal-1433 27 12 𝑖𝑖 𝑖𝑖 INTJ american_scientific_journal-1433 27 13 𝑦𝑦 𝑦𝑦 INTJ american_scientific_journal-1433 27 14 𝑗𝑗 𝑗𝑗 VERB american_scientific_journal-1433 27 15 = = PUNCT american_scientific_journal-1433 27 16 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 28 1 π‘—π‘—β„Ž π‘—π‘—β„Ž INTJ american_scientific_journal-1433 28 2 𝑖𝑖 𝑖𝑖 INTJ american_scientific_journal-1433 28 3 . . PUNCT american_scientific_journal-1433 29 1 ( ( PUNCT american_scientific_journal-1433 29 2 1.5 1.5 NUM american_scientific_journal-1433 29 3 ) ) PUNCT american_scientific_journal-1433 29 4 𝐻𝐻 𝐻𝐻 ADJ american_scientific_journal-1433 29 5 π‘—π‘—π‘—π‘—β„Ž π‘—π‘—π‘—π‘—β„Ž NOUN american_scientific_journal-1433 29 6 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-1433 29 7 = = PUNCT american_scientific_journal-1433 29 8 οΏ½ οΏ½ PROPN american_scientific_journal-1433 29 9 Μ‡ Μ‡ PROPN american_scientific_journal-1433 29 10 οΏ½ οΏ½ PROPN american_scientific_journal-1433 29 11 πœ•π‘—π‘—π»π» πœ•π‘—π‘—π»π» NOUN american_scientific_journal-1433 29 12 π‘—π‘—β„Ž π‘—π‘—β„Ž INTJ american_scientific_journal-1433 29 13 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-1433 29 14 . . PUNCT american_scientific_journal-1433 30 1 the the DET american_scientific_journal-1433 30 2 torsion torsion NOUN american_scientific_journal-1433 30 3 tensor tensor NOUN american_scientific_journal-1433 30 4 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 30 5 π‘—π‘—β„Ž π‘—π‘—β„Ž NOUN american_scientific_journal-1433 30 6 𝑖𝑖 𝑖𝑖 X american_scientific_journal-1433 30 7 satisfies satisfie NOUN american_scientific_journal-1433 30 8 ( ( PUNCT american_scientific_journal-1433 30 9 1.6 1.6 NUM american_scientific_journal-1433 30 10 ) ) PUNCT american_scientific_journal-1433 30 11 𝐻𝐻 𝐻𝐻 NOUN american_scientific_journal-1433 30 12 π‘—π‘—β„Ž π‘—π‘—β„Ž NUM american_scientific_journal-1433 30 13 𝑖𝑖 𝑖𝑖 INTJ american_scientific_journal-1433 30 14 𝑦𝑦 𝑦𝑦 INTJ american_scientific_journal-1433 30 15 𝑗𝑗 𝑗𝑗 VERB american_scientific_journal-1433 30 16 = = PUNCT american_scientific_journal-1433 30 17 𝐻𝐻 𝐻𝐻 ADJ american_scientific_journal-1433 30 18 β„Ž β„Ž NOUN american_scientific_journal-1433 30 19 𝑖𝑖 𝑖𝑖 INTJ american_scientific_journal-1433 30 20 , , PUNCT american_scientific_journal-1433 30 21 ( ( PUNCT american_scientific_journal-1433 30 22 1.7 1.7 NUM american_scientific_journal-1433 30 23 ) ) PUNCT american_scientific_journal-1433 30 24 𝑅𝑅 𝑅𝑅 PROPN american_scientific_journal-1433 30 25 π‘—π‘—π‘—π‘—β„Ž π‘—π‘—π‘—π‘—β„Ž PROPN american_scientific_journal-1433 30 26 𝑖𝑖 𝑖𝑖 INTJ american_scientific_journal-1433 30 27 𝑦𝑦 𝑦𝑦 INTJ american_scientific_journal-1433 30 28 𝑗𝑗 𝑗𝑗 VERB american_scientific_journal-1433 30 29 = = PUNCT american_scientific_journal-1433 30 30 𝐻𝐻 𝐻𝐻 NOUN american_scientific_journal-1433 30 31 π‘—π‘—β„Ž π‘—π‘—β„Ž INTJ american_scientific_journal-1433 30 32 𝑖𝑖 𝑖𝑖 INTJ american_scientific_journal-1433 30 33 , , PUNCT american_scientific_journal-1433 30 34 ( ( PUNCT american_scientific_journal-1433 30 35 1.8 1.8 NUM american_scientific_journal-1433 30 36 ) ) PUNCT american_scientific_journal-1433 30 37 𝐻𝐻 𝐻𝐻 NOUN american_scientific_journal-1433 30 38 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 NOUN american_scientific_journal-1433 30 39 = = SYM american_scientific_journal-1433 30 40 𝐻𝐻 𝐻𝐻 NOUN american_scientific_journal-1433 30 41 𝑗𝑗𝑗𝑗𝑖𝑖 𝑗𝑗𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 30 42 𝑖𝑖 𝑖𝑖 X american_scientific_journal-1433 30 43 , , PUNCT american_scientific_journal-1433 30 44 ( ( PUNCT american_scientific_journal-1433 30 45 1.9 1.9 NUM american_scientific_journal-1433 30 46 ) ) PUNCT american_scientific_journal-1433 30 47 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 30 48 𝑗𝑗 𝑗𝑗 NOUN american_scientific_journal-1433 30 49 = = SYM american_scientific_journal-1433 30 50 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 30 51 𝑗𝑗𝑖𝑖 𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 30 52 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-1433 30 53 , , PUNCT american_scientific_journal-1433 30 54 and and CCONJ american_scientific_journal-1433 30 55 ( ( PUNCT american_scientific_journal-1433 30 56 1.10 1.10 NUM american_scientific_journal-1433 30 57 ) ) PUNCT american_scientific_journal-1433 30 58 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 30 59 = = SYM american_scientific_journal-1433 30 60 1 1 NUM american_scientific_journal-1433 30 61 π‘›π‘›βˆ’1 π‘›π‘›βˆ’1 NOUN american_scientific_journal-1433 30 62 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 30 63 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-1433 30 64 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-1433 30 65 . . PUNCT american_scientific_journal-1433 31 1 where where SCONJ american_scientific_journal-1433 31 2 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 31 3 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 NOUN american_scientific_journal-1433 31 4 and and CCONJ american_scientific_journal-1433 31 5 𝐻𝐻 𝐻𝐻 NOUN american_scientific_journal-1433 31 6 are be AUX american_scientific_journal-1433 31 7 called call VERB american_scientific_journal-1433 31 8 β„Ž β„Ž ADP american_scientific_journal-1433 31 9 -ricci -ricci PROPN american_scientific_journal-1433 31 10 tensor tensor NOUN american_scientific_journal-1433 31 11 [ [ NOUN american_scientific_journal-1433 31 12 7 7 NUM american_scientific_journal-1433 31 13 ] ] PUNCT american_scientific_journal-1433 31 14 and and CCONJ american_scientific_journal-1433 31 15 curvature curvature VERB american_scientific_journal-1433 31 16 scalar scalar ADJ american_scientific_journal-1433 31 17 respectively respectively ADV american_scientific_journal-1433 31 18 . . PUNCT american_scientific_journal-1433 32 1 since since SCONJ american_scientific_journal-1433 32 2 contraction contraction NOUN american_scientific_journal-1433 32 3 of of ADP american_scientific_journal-1433 32 4 the the DET american_scientific_journal-1433 32 5 american american ADJ american_scientific_journal-1433 32 6 scientific scientific PROPN american_scientific_journal-1433 32 7 research research PROPN american_scientific_journal-1433 32 8 journal journal NOUN american_scientific_journal-1433 32 9 for for ADP american_scientific_journal-1433 32 10 engineering engineering NOUN american_scientific_journal-1433 32 11 , , PUNCT american_scientific_journal-1433 32 12 technology technology NOUN american_scientific_journal-1433 32 13 , , PUNCT american_scientific_journal-1433 32 14 and and CCONJ american_scientific_journal-1433 32 15 sciences science NOUN american_scientific_journal-1433 32 16 ( ( PUNCT american_scientific_journal-1433 32 17 asrjets asrjet NOUN american_scientific_journal-1433 32 18 ) ) PUNCT american_scientific_journal-1433 32 19 ( ( PUNCT american_scientific_journal-1433 32 20 2016 2016 NUM american_scientific_journal-1433 32 21 ) ) PUNCT american_scientific_journal-1433 32 22 volume volume NOUN american_scientific_journal-1433 32 23 19 19 NUM american_scientific_journal-1433 32 24 , , PUNCT american_scientific_journal-1433 32 25 no no DET american_scientific_journal-1433 32 26 1 1 NUM american_scientific_journal-1433 32 27 , , PUNCT american_scientific_journal-1433 32 28 pp pp ADV american_scientific_journal-1433 32 29 9 9 NUM american_scientific_journal-1433 32 30 - - SYM american_scientific_journal-1433 32 31 18 18 NUM american_scientific_journal-1433 32 32 11 11 NUM american_scientific_journal-1433 32 33 indices index NOUN american_scientific_journal-1433 32 34 does do AUX american_scientific_journal-1433 32 35 not not PART american_scientific_journal-1433 32 36 affect affect VERB american_scientific_journal-1433 32 37 the the DET american_scientific_journal-1433 32 38 homogeneity homogeneity NOUN american_scientific_journal-1433 32 39 in in ADP american_scientific_journal-1433 32 40 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 32 41 , , PUNCT american_scientific_journal-1433 32 42 hence hence ADV american_scientific_journal-1433 32 43 the the DET american_scientific_journal-1433 32 44 tensors tensor NOUN american_scientific_journal-1433 32 45 𝐻𝐻 𝐻𝐻 NOUN american_scientific_journal-1433 32 46 π‘Ÿπ‘Ÿπ‘—π‘— π‘Ÿπ‘Ÿπ‘—π‘— NOUN american_scientific_journal-1433 32 47 , , PUNCT american_scientific_journal-1433 32 48 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 32 49 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 32 50 and and CCONJ american_scientific_journal-1433 32 51 the the DET american_scientific_journal-1433 32 52 scalar scalar ADJ american_scientific_journal-1433 32 53 𝐻𝐻 𝐻𝐻 NOUN american_scientific_journal-1433 32 54 are be AUX american_scientific_journal-1433 32 55 also also ADV american_scientific_journal-1433 32 56 homogeneous homogeneous ADJ american_scientific_journal-1433 32 57 of of ADP american_scientific_journal-1433 32 58 degree degree NOUN american_scientific_journal-1433 32 59 zero zero NUM american_scientific_journal-1433 32 60 , , PUNCT american_scientific_journal-1433 32 61 one one NUM american_scientific_journal-1433 32 62 and and CCONJ american_scientific_journal-1433 32 63 two two NUM american_scientific_journal-1433 32 64 in in ADP american_scientific_journal-1433 32 65 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 32 66 respectively respectively ADV american_scientific_journal-1433 32 67 . . PUNCT american_scientific_journal-1433 33 1 the the DET american_scientific_journal-1433 33 2 above above ADJ american_scientific_journal-1433 33 3 tensors tensor NOUN american_scientific_journal-1433 33 4 are be AUX american_scientific_journal-1433 33 5 also also ADV american_scientific_journal-1433 33 6 connected connect VERB american_scientific_journal-1433 33 7 by by ADP american_scientific_journal-1433 33 8 ( ( PUNCT american_scientific_journal-1433 33 9 1.11 1.11 NUM american_scientific_journal-1433 33 10 ) ) PUNCT american_scientific_journal-1433 33 11 𝐻𝐻 𝐻𝐻 NOUN american_scientific_journal-1433 33 12 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 NOUN american_scientific_journal-1433 33 13 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 NOUN american_scientific_journal-1433 33 14 = = PUNCT american_scientific_journal-1433 33 15 𝐻𝐻 𝐻𝐻 NOUN american_scientific_journal-1433 33 16 𝑗𝑗 𝑗𝑗 NOUN american_scientific_journal-1433 33 17 , , PUNCT american_scientific_journal-1433 33 18 ( ( PUNCT american_scientific_journal-1433 33 19 1.12 1.12 NUM american_scientific_journal-1433 33 20 ) ) PUNCT american_scientific_journal-1433 33 21 𝐻𝐻 𝐻𝐻 NOUN american_scientific_journal-1433 33 22 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 NOUN american_scientific_journal-1433 33 23 = = SYM american_scientific_journal-1433 33 24 οΏ½ οΏ½ PROPN american_scientific_journal-1433 33 25 Μ‡ Μ‡ NOUN american_scientific_journal-1433 33 26 οΏ½ οΏ½ NOUN american_scientific_journal-1433 33 27 πœ•π‘—π‘— πœ•π‘—π‘— PART american_scientific_journal-1433 33 28 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 33 29 𝑗𝑗 𝑗𝑗 NOUN american_scientific_journal-1433 33 30 , , PUNCT american_scientific_journal-1433 33 31 ( ( PUNCT american_scientific_journal-1433 33 32 1.13 1.13 NUM american_scientific_journal-1433 33 33 ) ) PUNCT american_scientific_journal-1433 33 34 𝐻𝐻 𝐻𝐻 NOUN american_scientific_journal-1433 33 35 𝑗𝑗 𝑗𝑗 NOUN american_scientific_journal-1433 33 36 𝑦𝑦 𝑦𝑦 NOUN american_scientific_journal-1433 33 37 𝑗𝑗 𝑗𝑗 VERB american_scientific_journal-1433 33 38 = = PUNCT american_scientific_journal-1433 33 39 ( ( PUNCT american_scientific_journal-1433 33 40 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-1433 33 41 βˆ’ βˆ’ PROPN american_scientific_journal-1433 33 42 1)𝐻𝐻 1)𝐻𝐻 NUM american_scientific_journal-1433 33 43 . . PUNCT american_scientific_journal-1433 34 1 ( ( PUNCT american_scientific_journal-1433 34 2 1.14 1.14 NUM american_scientific_journal-1433 34 3 ) ) PUNCT american_scientific_journal-1433 34 4 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 34 5 π‘—π‘—β„Ž π‘—π‘—β„Ž NOUN american_scientific_journal-1433 34 6 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-1433 34 7 = = PUNCT american_scientific_journal-1433 34 8 οΏ½ οΏ½ PROPN american_scientific_journal-1433 34 9 Μ‡ Μ‡ PROPN american_scientific_journal-1433 34 10 οΏ½ οΏ½ NOUN american_scientific_journal-1433 34 11 πœ•π‘—π‘— πœ•π‘—π‘— PART american_scientific_journal-1433 34 12 𝐻𝐻 𝐻𝐻 ADJ american_scientific_journal-1433 34 13 β„Ž β„Ž NOUN american_scientific_journal-1433 34 14 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-1433 34 15 . . PUNCT american_scientific_journal-1433 35 1 the the DET american_scientific_journal-1433 35 2 necessary necessary ADJ american_scientific_journal-1433 35 3 and and CCONJ american_scientific_journal-1433 35 4 sufficient sufficient ADJ american_scientific_journal-1433 35 5 condition condition NOUN american_scientific_journal-1433 35 6 for for ADP american_scientific_journal-1433 35 7 a a DET american_scientific_journal-1433 35 8 finsler finsler NOUN american_scientific_journal-1433 35 9 space space NOUN american_scientific_journal-1433 35 10 𝐹𝐹𝑛𝑛(𝑛𝑛 𝐹𝐹𝑛𝑛(𝑛𝑛 X american_scientific_journal-1433 35 11 > > X american_scientific_journal-1433 35 12 2 2 X american_scientific_journal-1433 35 13 ) ) PUNCT american_scientific_journal-1433 35 14 to to PART american_scientific_journal-1433 35 15 be be AUX american_scientific_journal-1433 35 16 a a DET american_scientific_journal-1433 35 17 finsler finsler NOUN american_scientific_journal-1433 35 18 space space NOUN american_scientific_journal-1433 35 19 of of ADP american_scientific_journal-1433 35 20 scalar scalar ADJ american_scientific_journal-1433 35 21 curvature curvature NOUN american_scientific_journal-1433 35 22 is be AUX american_scientific_journal-1433 35 23 given give VERB american_scientific_journal-1433 35 24 by by ADP american_scientific_journal-1433 35 25 ( ( PUNCT american_scientific_journal-1433 35 26 1.15 1.15 NUM american_scientific_journal-1433 35 27 ) ) PUNCT american_scientific_journal-1433 35 28 π»π»β„Žπ‘–π‘– π»π»β„Žπ‘–π‘– PROPN american_scientific_journal-1433 35 29 = = SYM american_scientific_journal-1433 35 30 𝐹𝐹2𝑅𝑅 𝐹𝐹2𝑅𝑅 PROPN american_scientific_journal-1433 35 31 ( ( PUNCT american_scientific_journal-1433 35 32 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 35 33 βˆ’ βˆ’ PROPN american_scientific_journal-1433 35 34 Ι­ Ι­ X american_scientific_journal-1433 35 35 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-1433 35 36 Ι­ Ι­ X american_scientific_journal-1433 35 37 β„Ž β„Ž PROPN american_scientific_journal-1433 35 38 ) ) PUNCT american_scientific_journal-1433 35 39 . . PUNCT american_scientific_journal-1433 36 1 a a DET american_scientific_journal-1433 36 2 finsler finsler NOUN american_scientific_journal-1433 36 3 space space NOUN american_scientific_journal-1433 36 4 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 36 5 is be AUX american_scientific_journal-1433 36 6 said say VERB american_scientific_journal-1433 36 7 to to PART american_scientific_journal-1433 36 8 be be AUX american_scientific_journal-1433 36 9 landsberg landsberg PROPN american_scientific_journal-1433 36 10 space space NOUN american_scientific_journal-1433 36 11 if if SCONJ american_scientific_journal-1433 36 12 satisfies satisfie NOUN american_scientific_journal-1433 36 13 ( ( PUNCT american_scientific_journal-1433 36 14 1.16 1.16 NUM american_scientific_journal-1433 36 15 ) ) PUNCT american_scientific_journal-1433 36 16 π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘–π‘–π‘—π‘—π‘—π‘—π‘Ÿπ‘Ÿ π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘–π‘–π‘—π‘—π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 36 17 = = SYM american_scientific_journal-1433 36 18 0 0 NUM american_scientific_journal-1433 36 19 . . PUNCT american_scientific_journal-1433 37 1 the the DET american_scientific_journal-1433 37 2 ricci ricci PROPN american_scientific_journal-1433 37 3 tensor tensor NOUN american_scientific_journal-1433 37 4 𝑅𝑅𝑗𝑗𝑗𝑗 𝑅𝑅𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 37 5 is be AUX american_scientific_journal-1433 37 6 given give VERB american_scientific_journal-1433 37 7 by by ADP american_scientific_journal-1433 37 8 ( ( PUNCT american_scientific_journal-1433 37 9 1.17 1.17 NUM american_scientific_journal-1433 37 10 ) ) PUNCT american_scientific_journal-1433 37 11 𝑅𝑅 𝑅𝑅 PROPN american_scientific_journal-1433 37 12 𝑗𝑗𝑗𝑗𝑖𝑖 𝑗𝑗𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 37 13 𝑖𝑖 𝑖𝑖 X american_scientific_journal-1433 38 1 = = SYM american_scientific_journal-1433 38 2 𝑅𝑅 𝑅𝑅 PROPN american_scientific_journal-1433 38 3 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 NOUN american_scientific_journal-1433 38 4 . . PUNCT american_scientific_journal-1433 39 1 2 2 X american_scientific_journal-1433 39 2 . . NUM american_scientific_journal-1433 39 3 generalized generalize VERB american_scientific_journal-1433 39 4 πœ·πœ·π‘Ήπ‘Ή πœ·πœ·π‘Ήπ‘Ή PROPN american_scientific_journal-1433 39 5 βˆ’birecurrent βˆ’birecurrent NOUN american_scientific_journal-1433 39 6 finsler finsler NOUN american_scientific_journal-1433 39 7 space space NOUN american_scientific_journal-1433 39 8 a a DET american_scientific_journal-1433 39 9 finsler finsler NOUN american_scientific_journal-1433 39 10 space space NOUN american_scientific_journal-1433 39 11 whose whose DET american_scientific_journal-1433 39 12 cartan cartan PROPN american_scientific_journal-1433 39 13 's 's PART american_scientific_journal-1433 39 14 third third ADJ american_scientific_journal-1433 39 15 curvature curvature NOUN american_scientific_journal-1433 39 16 tensor tensor NOUN american_scientific_journal-1433 39 17 π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 39 18 satisfies satisfie NOUN american_scientific_journal-1433 39 19 ( ( PUNCT american_scientific_journal-1433 39 20 2.1 2.1 NUM american_scientific_journal-1433 39 21 ) ) PUNCT american_scientific_journal-1433 39 22 β„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– β„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 39 23 = = SYM american_scientific_journal-1433 39 24 πœ†πœ†π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– πœ†πœ†π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 39 25 + + CCONJ american_scientific_journal-1433 39 26 πœ‡πœ‡π‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 πœ‡πœ‡π‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 ADJ american_scientific_journal-1433 39 27 π‘”π‘”π‘—π‘—β„Ž π‘”π‘”π‘—π‘—β„Ž NOUN american_scientific_journal-1433 39 28 βˆ’ βˆ’ PROPN american_scientific_journal-1433 39 29 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 39 30 𝑔𝑔𝑗𝑗𝑗𝑗 𝑔𝑔𝑗𝑗𝑗𝑗 NOUN american_scientific_journal-1433 39 31 ) ) PUNCT american_scientific_journal-1433 39 32 , , PUNCT american_scientific_journal-1433 39 33 π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 39 34 β‰  β‰  PROPN american_scientific_journal-1433 39 35 0 0 NUM american_scientific_journal-1433 39 36 , , PUNCT american_scientific_journal-1433 39 37 where where SCONJ american_scientific_journal-1433 39 38 πœ†πœ†π‘›π‘› πœ†πœ†π‘›π‘› NOUN american_scientific_journal-1433 39 39 and and CCONJ american_scientific_journal-1433 39 40 πœ‡πœ‡π‘›π‘› πœ‡πœ‡π‘›π‘› NOUN american_scientific_journal-1433 39 41 are be AUX american_scientific_journal-1433 39 42 non non ADJ american_scientific_journal-1433 39 43 - - ADJ american_scientific_journal-1433 39 44 zero zero NUM american_scientific_journal-1433 39 45 covariant covariant ADJ american_scientific_journal-1433 39 46 vector vector NOUN american_scientific_journal-1433 39 47 fields field NOUN american_scientific_journal-1433 39 48 and and CCONJ american_scientific_journal-1433 39 49 called call VERB american_scientific_journal-1433 39 50 the the DET american_scientific_journal-1433 39 51 recurrence recurrence NOUN american_scientific_journal-1433 39 52 vector vector NOUN american_scientific_journal-1433 39 53 fields field NOUN american_scientific_journal-1433 39 54 , , PUNCT american_scientific_journal-1433 39 55 we we PRON american_scientific_journal-1433 39 56 shall shall AUX american_scientific_journal-1433 39 57 call call VERB american_scientific_journal-1433 39 58 such such ADJ american_scientific_journal-1433 39 59 finsler finsler NOUN american_scientific_journal-1433 39 60 space space NOUN american_scientific_journal-1433 39 61 as as ADP american_scientific_journal-1433 39 62 a a DET american_scientific_journal-1433 39 63 generalized generalized ADJ american_scientific_journal-1433 39 64 𝑅𝑅recurrent 𝑅𝑅recurrent PROPN american_scientific_journal-1433 39 65 finsler finsler NOUN american_scientific_journal-1433 39 66 space space NOUN american_scientific_journal-1433 39 67 . . PUNCT american_scientific_journal-1433 40 1 differentiating differentiate VERB american_scientific_journal-1433 40 2 ( ( PUNCT american_scientific_journal-1433 40 3 2.1 2.1 NUM american_scientific_journal-1433 40 4 ) ) PUNCT american_scientific_journal-1433 40 5 covariantly covariantly ADV american_scientific_journal-1433 40 6 with with ADP american_scientific_journal-1433 40 7 respect respect NOUN american_scientific_journal-1433 40 8 to to ADP american_scientific_journal-1433 40 9 π‘₯π‘₯π‘šπ‘š π‘₯π‘₯π‘šπ‘š NOUN american_scientific_journal-1433 40 10 in in ADP american_scientific_journal-1433 40 11 the the DET american_scientific_journal-1433 40 12 sense sense NOUN american_scientific_journal-1433 40 13 of of ADP american_scientific_journal-1433 40 14 berwald berwald NOUN american_scientific_journal-1433 40 15 and and CCONJ american_scientific_journal-1433 40 16 using use VERB american_scientific_journal-1433 40 17 ( ( PUNCT american_scientific_journal-1433 40 18 1.1e 1.1e NUM american_scientific_journal-1433 40 19 ) ) PUNCT american_scientific_journal-1433 40 20 , , PUNCT american_scientific_journal-1433 40 21 we we PRON american_scientific_journal-1433 40 22 get get VERB american_scientific_journal-1433 40 23 ( ( PUNCT american_scientific_journal-1433 40 24 2.2 2.2 NUM american_scientific_journal-1433 40 25 ) ) PUNCT american_scientific_journal-1433 40 26 β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 40 27 = = PUNCT american_scientific_journal-1433 40 28 ( ( PUNCT american_scientific_journal-1433 40 29 β„¬π‘šπ‘šπœ†πœ†π‘›π‘› β„¬π‘šπ‘šπœ†πœ†π‘›π‘› PROPN american_scientific_journal-1433 40 30 + + CCONJ american_scientific_journal-1433 40 31 πœ†πœ†π‘›π‘› πœ†πœ†π‘›π‘› NOUN american_scientific_journal-1433 40 32 πœ†πœ†π‘šπ‘š)π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– πœ†πœ†π‘šπ‘š)π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PRON american_scientific_journal-1433 41 1 + + CCONJ american_scientific_journal-1433 41 2 ( ( PUNCT american_scientific_journal-1433 41 3 πœ†πœ†π‘›π‘›πœ‡πœ‡π‘šπ‘š πœ†πœ†π‘›π‘›πœ‡πœ‡π‘šπ‘š NOUN american_scientific_journal-1433 41 4 + + CCONJ american_scientific_journal-1433 41 5 β„¬π‘šπ‘šπœ‡πœ‡π‘›π‘›) β„¬π‘šπ‘šπœ‡πœ‡π‘›π‘›) VERB american_scientific_journal-1433 41 6 οΏ½ οΏ½ NOUN american_scientific_journal-1433 41 7 𝛿𝛿𝑗𝑗𝑖𝑖 𝛿𝛿𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 41 8 π‘”π‘”π‘—π‘—β„Ž π‘”π‘”π‘—π‘—β„Ž NOUN american_scientific_journal-1433 41 9 βˆ’ βˆ’ PROPN american_scientific_journal-1433 41 10 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 41 11 𝑔𝑔𝑗𝑗𝑗𝑗 𝑔𝑔𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 41 12 οΏ½ οΏ½ PROPN american_scientific_journal-1433 41 13 βˆ’2π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘Ÿ(𝛿𝛿𝑗𝑗𝑖𝑖 βˆ’2π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘Ÿ(𝛿𝛿𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 41 14 πΆπΆπ‘—π‘—β„Žπ‘šπ‘š πΆπΆπ‘—π‘—β„Žπ‘šπ‘š PROPN american_scientific_journal-1433 41 15 βˆ’ βˆ’ PROPN american_scientific_journal-1433 41 16 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 41 17 πΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š πΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š NOUN american_scientific_journal-1433 41 18 ) ) PUNCT american_scientific_journal-1433 41 19 . . PUNCT american_scientific_journal-1433 42 1 which which PRON american_scientific_journal-1433 42 2 can can AUX american_scientific_journal-1433 42 3 be be AUX american_scientific_journal-1433 42 4 written write VERB american_scientific_journal-1433 42 5 as as ADP american_scientific_journal-1433 42 6 ( ( PUNCT american_scientific_journal-1433 42 7 2.3 2.3 NUM american_scientific_journal-1433 42 8 ) ) PUNCT american_scientific_journal-1433 42 9 β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 42 10 = = PUNCT american_scientific_journal-1433 42 11 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– X american_scientific_journal-1433 42 12 + + NUM american_scientific_journal-1433 42 13 π‘π‘π‘šπ‘šπ‘›π‘› π‘π‘π‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 42 14 οΏ½ οΏ½ PROPN american_scientific_journal-1433 42 15 𝛿𝛿𝑗𝑗𝑖𝑖 𝛿𝛿𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 42 16 π‘”π‘”π‘—π‘—β„Ž π‘”π‘”π‘—π‘—β„Ž NOUN american_scientific_journal-1433 42 17 βˆ’ βˆ’ PROPN american_scientific_journal-1433 43 1 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 43 2 𝑔𝑔𝑗𝑗𝑗𝑗 𝑔𝑔𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 43 3 οΏ½ οΏ½ PROPN american_scientific_journal-1433 43 4 βˆ’ βˆ’ NUM american_scientific_journal-1433 43 5 2π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘Ÿ(𝛿𝛿𝑗𝑗𝑖𝑖 2π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘Ÿ(𝛿𝛿𝑗𝑗𝑖𝑖 PROPN american_scientific_journal-1433 43 6 πΆπΆπ‘—π‘—β„Žπ‘šπ‘š πΆπΆπ‘—π‘—β„Žπ‘šπ‘š PROPN american_scientific_journal-1433 43 7 βˆ’ βˆ’ PROPN american_scientific_journal-1433 43 8 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 43 9 πΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š πΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š NOUN american_scientific_journal-1433 43 10 ) ) PUNCT american_scientific_journal-1433 43 11 . . PUNCT american_scientific_journal-1433 44 1 american american PROPN american_scientific_journal-1433 44 2 scientific scientific ADJ american_scientific_journal-1433 44 3 research research PROPN american_scientific_journal-1433 44 4 journal journal NOUN american_scientific_journal-1433 44 5 for for ADP american_scientific_journal-1433 44 6 engineering engineering NOUN american_scientific_journal-1433 44 7 , , PUNCT american_scientific_journal-1433 44 8 technology technology NOUN american_scientific_journal-1433 44 9 , , PUNCT american_scientific_journal-1433 44 10 and and CCONJ american_scientific_journal-1433 44 11 sciences science NOUN american_scientific_journal-1433 44 12 ( ( PUNCT american_scientific_journal-1433 44 13 asrjets asrjet NOUN american_scientific_journal-1433 44 14 ) ) PUNCT american_scientific_journal-1433 44 15 ( ( PUNCT american_scientific_journal-1433 44 16 2016 2016 NUM american_scientific_journal-1433 44 17 ) ) PUNCT american_scientific_journal-1433 44 18 volume volume NOUN american_scientific_journal-1433 44 19 19 19 NUM american_scientific_journal-1433 44 20 , , PUNCT american_scientific_journal-1433 44 21 no no DET american_scientific_journal-1433 44 22 1 1 NUM american_scientific_journal-1433 44 23 , , PUNCT american_scientific_journal-1433 44 24 pp pp ADV american_scientific_journal-1433 44 25 9 9 NUM american_scientific_journal-1433 44 26 - - SYM american_scientific_journal-1433 44 27 18 18 NUM american_scientific_journal-1433 44 28 12 12 NUM american_scientific_journal-1433 45 1 where where SCONJ american_scientific_journal-1433 45 2 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 45 3 = = SYM american_scientific_journal-1433 45 4 β„¬π‘šπ‘šπœ†πœ†π‘›π‘› β„¬π‘šπ‘šπœ†πœ†π‘›π‘› PROPN american_scientific_journal-1433 45 5 + + CCONJ american_scientific_journal-1433 45 6 πœ†πœ†π‘›π‘› πœ†πœ†π‘›π‘› NOUN american_scientific_journal-1433 45 7 πœ†πœ†π‘šπ‘š πœ†πœ†π‘šπ‘š PROPN american_scientific_journal-1433 45 8 and and CCONJ american_scientific_journal-1433 45 9 π‘π‘π‘šπ‘šπ‘›π‘› π‘π‘π‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 45 10 = = SYM american_scientific_journal-1433 45 11 πœ†πœ†π‘›π‘›πœ‡πœ‡π‘šπ‘š πœ†πœ†π‘›π‘›πœ‡πœ‡π‘šπ‘š NOUN american_scientific_journal-1433 45 12 + + CCONJ american_scientific_journal-1433 45 13 β„¬π‘šπ‘šπœ‡πœ‡π‘›π‘› β„¬π‘šπ‘šπœ‡πœ‡π‘›π‘› PROPN american_scientific_journal-1433 45 14 are be AUX american_scientific_journal-1433 45 15 non non ADJ american_scientific_journal-1433 45 16 - - ADJ american_scientific_journal-1433 45 17 zero zero NUM american_scientific_journal-1433 45 18 covariant covariant ADJ american_scientific_journal-1433 45 19 tensor tensor NOUN american_scientific_journal-1433 45 20 fields field NOUN american_scientific_journal-1433 45 21 of of ADP american_scientific_journal-1433 45 22 second second ADJ american_scientific_journal-1433 45 23 order order NOUN american_scientific_journal-1433 45 24 . . PUNCT american_scientific_journal-1433 46 1 definition definition NOUN american_scientific_journal-1433 46 2 2.1 2.1 NUM american_scientific_journal-1433 46 3 . . PUNCT american_scientific_journal-1433 47 1 a a DET american_scientific_journal-1433 47 2 finsler finsler NOUN american_scientific_journal-1433 47 3 space space NOUN american_scientific_journal-1433 47 4 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PRON american_scientific_journal-1433 47 5 whose whose DET american_scientific_journal-1433 47 6 cartan cartan PROPN american_scientific_journal-1433 47 7 's 's PART american_scientific_journal-1433 47 8 third third ADJ american_scientific_journal-1433 47 9 curvature curvature NOUN american_scientific_journal-1433 47 10 tensor tensor NOUN american_scientific_journal-1433 47 11 π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘…π‘…π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 47 12 satisfies satisfy VERB american_scientific_journal-1433 47 13 the the DET american_scientific_journal-1433 47 14 condition condition NOUN american_scientific_journal-1433 47 15 ( ( PUNCT american_scientific_journal-1433 47 16 2.3 2.3 NUM american_scientific_journal-1433 47 17 ) ) PUNCT american_scientific_journal-1433 47 18 will will AUX american_scientific_journal-1433 47 19 be be AUX american_scientific_journal-1433 47 20 called call VERB american_scientific_journal-1433 47 21 generalized generalized ADJ american_scientific_journal-1433 47 22 𝛽𝛽𝑅𝑅-birecurrent 𝛽𝛽𝑅𝑅-birecurrent DET american_scientific_journal-1433 47 23 finsler finsler NOUN american_scientific_journal-1433 47 24 space space NOUN american_scientific_journal-1433 47 25 , , PUNCT american_scientific_journal-1433 47 26 we we PRON american_scientific_journal-1433 47 27 shall shall AUX american_scientific_journal-1433 47 28 denote denote VERB american_scientific_journal-1433 47 29 it it PRON american_scientific_journal-1433 47 30 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 PRON american_scientific_journal-1433 48 1 βˆ’ βˆ’ NOUN american_scientific_journal-1433 48 2 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 49 1 βˆ’ βˆ’ PROPN american_scientific_journal-1433 49 2 𝐹𝐹𝑛𝑛. 𝐹𝐹𝑛𝑛. NOUN american_scientific_journal-1433 49 3 transvecting transvecte VERB american_scientific_journal-1433 49 4 ( ( PUNCT american_scientific_journal-1433 49 5 2.3 2.3 NUM american_scientific_journal-1433 49 6 ) ) PUNCT american_scientific_journal-1433 49 7 by by ADP american_scientific_journal-1433 49 8 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 PRON american_scientific_journal-1433 49 9 , , PUNCT american_scientific_journal-1433 49 10 using use VERB american_scientific_journal-1433 49 11 ( ( PUNCT american_scientific_journal-1433 49 12 1.1c 1.1c NUM american_scientific_journal-1433 49 13 ) ) PUNCT american_scientific_journal-1433 49 14 , , PUNCT american_scientific_journal-1433 49 15 ( ( PUNCT american_scientific_journal-1433 49 16 1.7 1.7 NUM american_scientific_journal-1433 49 17 ) ) PUNCT american_scientific_journal-1433 49 18 and and CCONJ american_scientific_journal-1433 49 19 ( ( PUNCT american_scientific_journal-1433 49 20 1.1d 1.1d NUM american_scientific_journal-1433 49 21 ) ) PUNCT american_scientific_journal-1433 49 22 , , PUNCT american_scientific_journal-1433 49 23 we we PRON american_scientific_journal-1433 49 24 get get VERB american_scientific_journal-1433 49 25 ( ( PUNCT american_scientific_journal-1433 49 26 2.4 2.4 NUM american_scientific_journal-1433 49 27 ) ) PUNCT american_scientific_journal-1433 49 28 β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 49 29 = = SYM american_scientific_journal-1433 49 30 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 49 31 + + CCONJ american_scientific_journal-1433 49 32 π‘π‘π‘šπ‘šπ‘›π‘› π‘π‘π‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 49 33 οΏ½ οΏ½ NOUN american_scientific_journal-1433 49 34 𝛿𝛿𝑗𝑗𝑖𝑖 𝛿𝛿𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 49 35 π‘¦π‘¦β„Ž π‘¦π‘¦β„Ž NOUN american_scientific_journal-1433 49 36 βˆ’ βˆ’ PROPN american_scientific_journal-1433 49 37 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 49 38 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 PROPN american_scientific_journal-1433 49 39 οΏ½ οΏ½ PROPN american_scientific_journal-1433 49 40 . . PUNCT american_scientific_journal-1433 50 1 further further PROPN american_scientific_journal-1433 50 2 transvecting transvecte VERB american_scientific_journal-1433 50 3 ( ( PUNCT american_scientific_journal-1433 50 4 2.4 2.4 NUM american_scientific_journal-1433 50 5 ) ) PUNCT american_scientific_journal-1433 50 6 by by ADP american_scientific_journal-1433 50 7 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 NOUN american_scientific_journal-1433 50 8 , , PUNCT american_scientific_journal-1433 50 9 using use VERB american_scientific_journal-1433 50 10 ( ( PUNCT american_scientific_journal-1433 50 11 1.1c 1.1c NUM american_scientific_journal-1433 50 12 ) ) PUNCT american_scientific_journal-1433 50 13 , , PUNCT american_scientific_journal-1433 50 14 ( ( PUNCT american_scientific_journal-1433 50 15 1.6 1.6 NUM american_scientific_journal-1433 50 16 ) ) PUNCT american_scientific_journal-1433 50 17 and and CCONJ american_scientific_journal-1433 50 18 ( ( PUNCT american_scientific_journal-1433 50 19 1.1a 1.1a NOUN american_scientific_journal-1433 50 20 ) ) PUNCT american_scientific_journal-1433 50 21 , , PUNCT american_scientific_journal-1433 50 22 we we PRON american_scientific_journal-1433 50 23 get get VERB american_scientific_journal-1433 50 24 ( ( PUNCT american_scientific_journal-1433 50 25 2.5 2.5 NUM american_scientific_journal-1433 50 26 ) ) PUNCT american_scientific_journal-1433 50 27 β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»β„Žπ‘–π‘– β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»β„Žπ‘–π‘– NOUN american_scientific_journal-1433 50 28 = = PUNCT american_scientific_journal-1433 50 29 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»β„Žπ‘–π‘– π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»β„Žπ‘–π‘– X american_scientific_journal-1433 51 1 + + SYM american_scientific_journal-1433 51 2 π‘π‘π‘šπ‘šπ‘›π‘› π‘π‘π‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 51 3 οΏ½ οΏ½ NOUN american_scientific_journal-1433 51 4 π‘¦π‘¦π‘–π‘–π‘¦π‘¦β„Ž π‘¦π‘¦π‘–π‘–π‘¦π‘¦β„Ž NOUN american_scientific_journal-1433 51 5 βˆ’ βˆ’ PROPN american_scientific_journal-1433 51 6 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– PROPN american_scientific_journal-1433 51 7 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 51 8 οΏ½ οΏ½ PROPN american_scientific_journal-1433 51 9 . . PUNCT american_scientific_journal-1433 52 1 thus thus ADV american_scientific_journal-1433 52 2 , , PUNCT american_scientific_journal-1433 52 3 we we PRON american_scientific_journal-1433 52 4 have have AUX american_scientific_journal-1433 52 5 theorem theorem VERB american_scientific_journal-1433 52 6 2.1 2.1 NUM american_scientific_journal-1433 52 7 . . PUNCT american_scientific_journal-1433 53 1 in in ADP american_scientific_journal-1433 53 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 53 3 βˆ’ βˆ’ PROPN american_scientific_journal-1433 53 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 53 5 βˆ’ βˆ’ NUM american_scientific_journal-1433 53 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 53 7 , , PUNCT american_scientific_journal-1433 53 8 berwald berwald ADJ american_scientific_journal-1433 53 9 second second ADJ american_scientific_journal-1433 53 10 covariant covariant ADJ american_scientific_journal-1433 53 11 derivative derivative NOUN american_scientific_journal-1433 53 12 of of ADP american_scientific_journal-1433 53 13 the the DET american_scientific_journal-1433 53 14 β„Ž(𝑣𝑣 β„Ž(𝑣𝑣 NUM american_scientific_journal-1433 53 15 ) ) PUNCT american_scientific_journal-1433 53 16 βˆ’torsion βˆ’torsion NOUN american_scientific_journal-1433 53 17 tensor tensor NOUN american_scientific_journal-1433 53 18 π»π»π‘—π‘—β„Žπ‘–π‘– π»π»π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 53 19 and and CCONJ american_scientific_journal-1433 53 20 the the DET american_scientific_journal-1433 53 21 deviation deviation NOUN american_scientific_journal-1433 53 22 tensor tensor NOUN american_scientific_journal-1433 53 23 π»π»β„Žπ‘–π‘– π»π»β„Žπ‘–π‘– PROPN american_scientific_journal-1433 53 24 is be AUX american_scientific_journal-1433 53 25 given give VERB american_scientific_journal-1433 53 26 by by ADP american_scientific_journal-1433 53 27 the the DET american_scientific_journal-1433 53 28 equations equation NOUN american_scientific_journal-1433 53 29 ( ( PUNCT american_scientific_journal-1433 53 30 2.4 2.4 NUM american_scientific_journal-1433 53 31 ) ) PUNCT american_scientific_journal-1433 53 32 and and CCONJ american_scientific_journal-1433 53 33 ( ( PUNCT american_scientific_journal-1433 53 34 2.5 2.5 NUM american_scientific_journal-1433 53 35 ) ) PUNCT american_scientific_journal-1433 53 36 , , PUNCT american_scientific_journal-1433 53 37 respectively respectively ADV american_scientific_journal-1433 53 38 . . PUNCT american_scientific_journal-1433 54 1 contracting contract VERB american_scientific_journal-1433 54 2 the the DET american_scientific_journal-1433 54 3 indices index NOUN american_scientific_journal-1433 54 4 𝑖𝑖 𝑖𝑖 X american_scientific_journal-1433 54 5 and and CCONJ american_scientific_journal-1433 54 6 β„Ž β„Ž PROPN american_scientific_journal-1433 54 7 in in ADP american_scientific_journal-1433 54 8 ( ( PUNCT american_scientific_journal-1433 54 9 2.3 2.3 NUM american_scientific_journal-1433 54 10 ) ) PUNCT american_scientific_journal-1433 54 11 , , PUNCT american_scientific_journal-1433 54 12 ( ( PUNCT american_scientific_journal-1433 54 13 2.4 2.4 NUM american_scientific_journal-1433 54 14 ) ) PUNCT american_scientific_journal-1433 54 15 and and CCONJ american_scientific_journal-1433 54 16 ( ( PUNCT american_scientific_journal-1433 54 17 2.5 2.5 NUM american_scientific_journal-1433 54 18 ) ) PUNCT american_scientific_journal-1433 54 19 , , PUNCT american_scientific_journal-1433 54 20 respectively respectively ADV american_scientific_journal-1433 54 21 and and CCONJ american_scientific_journal-1433 54 22 using use VERB american_scientific_journal-1433 54 23 ( ( PUNCT american_scientific_journal-1433 54 24 1.17 1.17 NUM american_scientific_journal-1433 54 25 ) ) PUNCT american_scientific_journal-1433 54 26 , , PUNCT american_scientific_journal-1433 54 27 ( ( PUNCT american_scientific_journal-1433 54 28 1.9 1.9 NUM american_scientific_journal-1433 54 29 ) ) PUNCT american_scientific_journal-1433 54 30 and and CCONJ american_scientific_journal-1433 54 31 ( ( PUNCT american_scientific_journal-1433 54 32 1.10 1.10 NUM american_scientific_journal-1433 54 33 ) ) PUNCT american_scientific_journal-1433 54 34 , , PUNCT american_scientific_journal-1433 54 35 we we PRON american_scientific_journal-1433 54 36 get get VERB american_scientific_journal-1433 54 37 ( ( PUNCT american_scientific_journal-1433 54 38 2.6 2.6 NUM american_scientific_journal-1433 54 39 ) ) PUNCT american_scientific_journal-1433 54 40 β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘— β„¬π‘šπ‘šβ„¬π‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘— NOUN american_scientific_journal-1433 54 41 = = SYM american_scientific_journal-1433 54 42 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘— π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘…π‘…π‘—π‘—π‘—π‘— PROPN american_scientific_journal-1433 55 1 + + CCONJ american_scientific_journal-1433 55 2 ( ( PUNCT american_scientific_journal-1433 55 3 1 1 NUM american_scientific_journal-1433 55 4 βˆ’ βˆ’ NOUN american_scientific_journal-1433 55 5 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘— 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘— NOUN american_scientific_journal-1433 56 1 βˆ’ βˆ’ NUM american_scientific_journal-1433 56 2 2(1 2(1 NUM american_scientific_journal-1433 56 3 βˆ’ βˆ’ NOUN american_scientific_journal-1433 56 4 𝑛𝑛)π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘ŸπΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š. 𝑛𝑛)π‘¦π‘¦π‘Ÿπ‘Ÿπœ‡πœ‡π‘›π‘›β„¬π‘Ÿπ‘ŸπΆπΆπ‘—π‘—π‘—π‘—π‘šπ‘š. NOUN american_scientific_journal-1433 56 5 ( ( PUNCT american_scientific_journal-1433 56 6 2.7 2.7 NUM american_scientific_journal-1433 56 7 ) ) PUNCT american_scientific_journal-1433 56 8 β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘— β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘— NOUN american_scientific_journal-1433 56 9 = = SYM american_scientific_journal-1433 56 10 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘— π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘— NOUN american_scientific_journal-1433 56 11 + + CCONJ american_scientific_journal-1433 56 12 ( ( PUNCT american_scientific_journal-1433 56 13 1 1 NUM american_scientific_journal-1433 56 14 βˆ’ βˆ’ NOUN american_scientific_journal-1433 56 15 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘¦π‘¦π‘—π‘— 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘¦π‘¦π‘—π‘— NOUN american_scientific_journal-1433 56 16 . . PUNCT american_scientific_journal-1433 57 1 and and CCONJ american_scientific_journal-1433 57 2 ( ( PUNCT american_scientific_journal-1433 57 3 2.8 2.8 NUM american_scientific_journal-1433 57 4 ) ) PUNCT american_scientific_journal-1433 57 5 β„¬π‘šπ‘šβ„¬π‘›π‘›π»π» β„¬π‘šπ‘šβ„¬π‘›π‘›π»π» X american_scientific_journal-1433 57 6 = = SYM american_scientific_journal-1433 57 7 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π» π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π» PROPN american_scientific_journal-1433 57 8 βˆ’ βˆ’ PROPN american_scientific_journal-1433 57 9 π‘π‘π‘šπ‘šπ‘›π‘›πΉπΉ2 π‘π‘π‘šπ‘šπ‘›π‘›πΉπΉ2 PROPN american_scientific_journal-1433 57 10 . . PUNCT american_scientific_journal-1433 58 1 thus thus ADV american_scientific_journal-1433 58 2 , , PUNCT american_scientific_journal-1433 58 3 we we PRON american_scientific_journal-1433 58 4 have have AUX american_scientific_journal-1433 58 5 theorem theorem VERB american_scientific_journal-1433 58 6 2.2 2.2 NUM american_scientific_journal-1433 58 7 . . PUNCT american_scientific_journal-1433 59 1 in in ADP american_scientific_journal-1433 59 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 59 3 βˆ’ βˆ’ PROPN american_scientific_journal-1433 59 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 59 5 βˆ’ βˆ’ NUM american_scientific_journal-1433 59 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 59 7 , , PUNCT american_scientific_journal-1433 59 8 the the DET american_scientific_journal-1433 59 9 𝑅𝑅 𝑅𝑅 PROPN american_scientific_journal-1433 59 10 βˆ’ βˆ’ PROPN american_scientific_journal-1433 59 11 ricci ricci PROPN american_scientific_journal-1433 59 12 tensor tensor NOUN american_scientific_journal-1433 59 13 𝑅𝑅𝑗𝑗𝑗𝑗 𝑅𝑅𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 59 14 , , PUNCT american_scientific_journal-1433 59 15 the the DET american_scientific_journal-1433 59 16 curvature curvature NOUN american_scientific_journal-1433 59 17 vector vector NOUN american_scientific_journal-1433 59 18 𝐻𝐻𝑗𝑗 𝐻𝐻𝑗𝑗 PROPN american_scientific_journal-1433 59 19 and and CCONJ american_scientific_journal-1433 59 20 the the DET american_scientific_journal-1433 59 21 scalar scalar ADJ american_scientific_journal-1433 59 22 curvature curvature NOUN american_scientific_journal-1433 59 23 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 59 24 are be AUX american_scientific_journal-1433 59 25 non non ADJ american_scientific_journal-1433 59 26 - - ADJ american_scientific_journal-1433 59 27 vanishing vanishing ADJ american_scientific_journal-1433 59 28 . . PUNCT american_scientific_journal-1433 60 1 differentiating differentiate VERB american_scientific_journal-1433 60 2 ( ( PUNCT american_scientific_journal-1433 60 3 2.7 2.7 NUM american_scientific_journal-1433 60 4 ) ) PUNCT american_scientific_journal-1433 60 5 partially partially ADV american_scientific_journal-1433 60 6 with with ADP american_scientific_journal-1433 60 7 respect respect NOUN american_scientific_journal-1433 60 8 to to ADP american_scientific_journal-1433 60 9 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 PRON american_scientific_journal-1433 60 10 and and CCONJ american_scientific_journal-1433 60 11 using use VERB american_scientific_journal-1433 60 12 ( ( PUNCT american_scientific_journal-1433 60 13 1.1b 1.1b NUM american_scientific_journal-1433 60 14 ) ) PUNCT american_scientific_journal-1433 60 15 , , PUNCT american_scientific_journal-1433 60 16 we we PRON american_scientific_journal-1433 60 17 get get VERB american_scientific_journal-1433 60 18 ( ( PUNCT american_scientific_journal-1433 60 19 2.9 2.9 NUM american_scientific_journal-1433 60 20 ) ) PUNCT american_scientific_journal-1433 60 21 οΏ½ οΏ½ PROPN american_scientific_journal-1433 60 22 Μ‡ Μ‡ PROPN american_scientific_journal-1433 60 23 οΏ½ οΏ½ NOUN american_scientific_journal-1433 60 24 πœ•π‘—π‘—(β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘— πœ•π‘—π‘—(β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘— NUM american_scientific_journal-1433 60 25 ) ) PUNCT american_scientific_journal-1433 60 26 = = SYM american_scientific_journal-1433 60 27 ( ( PUNCT american_scientific_journal-1433 60 28 οΏ½ οΏ½ PROPN american_scientific_journal-1433 60 29 Μ‡ Μ‡ NOUN american_scientific_journal-1433 60 30 οΏ½ οΏ½ NOUN american_scientific_journal-1433 60 31 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)𝐻𝐻𝑗𝑗 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)𝐻𝐻𝑗𝑗 NOUN american_scientific_journal-1433 60 32 + + CCONJ american_scientific_journal-1433 60 33 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 60 34 οΏ½ οΏ½ PROPN american_scientific_journal-1433 60 35 οΏ½ οΏ½ PROPN american_scientific_journal-1433 60 36 Μ‡ Μ‡ PROPN american_scientific_journal-1433 60 37 οΏ½ οΏ½ PROPN american_scientific_journal-1433 60 38 πœ•π‘—π‘—π»π»π‘—π‘— πœ•π‘—π‘—π»π»π‘—π‘— PROPN american_scientific_journal-1433 60 39 οΏ½ οΏ½ PROPN american_scientific_journal-1433 60 40 + + CCONJ american_scientific_journal-1433 60 41 ( ( PUNCT american_scientific_journal-1433 60 42 1 1 NUM american_scientific_journal-1433 60 43 βˆ’ βˆ’ NUM american_scientific_journal-1433 60 44 𝑛𝑛) 𝑛𝑛) NUM american_scientific_journal-1433 60 45 οΏ½ οΏ½ PROPN american_scientific_journal-1433 60 46 οΏ½ οΏ½ PROPN american_scientific_journal-1433 60 47 Μ‡ Μ‡ PROPN american_scientific_journal-1433 60 48 οΏ½ οΏ½ NOUN american_scientific_journal-1433 60 49 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 60 50 οΏ½ οΏ½ NOUN american_scientific_journal-1433 60 51 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 NOUN american_scientific_journal-1433 60 52 + + CCONJ american_scientific_journal-1433 60 53 ( ( PUNCT american_scientific_journal-1433 60 54 1 1 NUM american_scientific_journal-1433 60 55 βˆ’ βˆ’ NOUN american_scientific_journal-1433 60 56 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘—. 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘—. NOUN american_scientific_journal-1433 60 57 using use VERB american_scientific_journal-1433 60 58 commutation commutation NOUN american_scientific_journal-1433 60 59 formula formula NOUN american_scientific_journal-1433 60 60 exhibited exhibit VERB american_scientific_journal-1433 60 61 by by ADP american_scientific_journal-1433 60 62 ( ( PUNCT american_scientific_journal-1433 60 63 1.3 1.3 NUM american_scientific_journal-1433 60 64 ) ) PUNCT american_scientific_journal-1433 60 65 for for ADP american_scientific_journal-1433 60 66 ( ( PUNCT american_scientific_journal-1433 60 67 ℬ𝑛𝑛𝐻𝐻𝑗𝑗 ℬ𝑛𝑛𝐻𝐻𝑗𝑗 NOUN american_scientific_journal-1433 60 68 ) ) PUNCT american_scientific_journal-1433 60 69 in in ADP american_scientific_journal-1433 60 70 ( ( PUNCT american_scientific_journal-1433 60 71 2.9 2.9 NUM american_scientific_journal-1433 60 72 ) ) PUNCT american_scientific_journal-1433 60 73 and and CCONJ american_scientific_journal-1433 60 74 using use VERB american_scientific_journal-1433 60 75 ( ( PUNCT american_scientific_journal-1433 60 76 1.12 1.12 NUM american_scientific_journal-1433 60 77 ) ) PUNCT american_scientific_journal-1433 60 78 , , PUNCT american_scientific_journal-1433 60 79 we we PRON american_scientific_journal-1433 60 80 get get VERB american_scientific_journal-1433 60 81 ( ( PUNCT american_scientific_journal-1433 60 82 2.10 2.10 NUM american_scientific_journal-1433 60 83 ) ) PUNCT american_scientific_journal-1433 60 84 β„¬π‘šπ‘š β„¬π‘šπ‘š PROPN american_scientific_journal-1433 60 85 οΏ½ οΏ½ PROPN american_scientific_journal-1433 60 86 Μ‡ Μ‡ PROPN american_scientific_journal-1433 60 87 οΏ½ οΏ½ PROPN american_scientific_journal-1433 60 88 πœ•π‘—π‘—(ℬ𝑛𝑛𝐻𝐻𝑗𝑗 πœ•π‘—π‘—(ℬ𝑛𝑛𝐻𝐻𝑗𝑗 PROPN american_scientific_journal-1433 60 89 ) ) PUNCT american_scientific_journal-1433 61 1 βˆ’ βˆ’ PROPN american_scientific_journal-1433 62 1 ( ( PUNCT american_scientific_journal-1433 62 2 β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 62 3 βˆ’ βˆ’ PROPN american_scientific_journal-1433 62 4 ( ( PUNCT american_scientific_journal-1433 62 5 β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 62 6 = = SYM american_scientific_journal-1433 62 7 οΏ½ οΏ½ PROPN american_scientific_journal-1433 62 8 οΏ½ οΏ½ PROPN american_scientific_journal-1433 62 9 Μ‡ Μ‡ PROPN american_scientific_journal-1433 62 10 οΏ½ οΏ½ PROPN american_scientific_journal-1433 62 11 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 62 12 οΏ½ οΏ½ PROPN american_scientific_journal-1433 62 13 𝐻𝐻𝑗𝑗 𝐻𝐻𝑗𝑗 PROPN american_scientific_journal-1433 62 14 + + PROPN american_scientific_journal-1433 62 15 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘— π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘— PROPN american_scientific_journal-1433 62 16 + + X american_scientific_journal-1433 62 17 ( ( PUNCT american_scientific_journal-1433 62 18 1 1 NUM american_scientific_journal-1433 62 19 βˆ’ βˆ’ NUM american_scientific_journal-1433 62 20 𝑛𝑛) 𝑛𝑛) NUM american_scientific_journal-1433 62 21 οΏ½ οΏ½ PROPN american_scientific_journal-1433 62 22 οΏ½ οΏ½ PROPN american_scientific_journal-1433 62 23 Μ‡ Μ‡ PROPN american_scientific_journal-1433 62 24 οΏ½ οΏ½ NOUN american_scientific_journal-1433 62 25 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 62 26 οΏ½ οΏ½ NOUN american_scientific_journal-1433 62 27 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 NOUN american_scientific_journal-1433 62 28 + + CCONJ american_scientific_journal-1433 62 29 ( ( PUNCT american_scientific_journal-1433 62 30 1 1 NUM american_scientific_journal-1433 62 31 βˆ’ βˆ’ NOUN american_scientific_journal-1433 62 32 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘— 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘— NOUN american_scientific_journal-1433 62 33 . . PUNCT american_scientific_journal-1433 63 1 american american PROPN american_scientific_journal-1433 63 2 scientific scientific ADJ american_scientific_journal-1433 63 3 research research PROPN american_scientific_journal-1433 63 4 journal journal NOUN american_scientific_journal-1433 63 5 for for ADP american_scientific_journal-1433 63 6 engineering engineering NOUN american_scientific_journal-1433 63 7 , , PUNCT american_scientific_journal-1433 63 8 technology technology NOUN american_scientific_journal-1433 63 9 , , PUNCT american_scientific_journal-1433 63 10 and and CCONJ american_scientific_journal-1433 63 11 sciences science NOUN american_scientific_journal-1433 63 12 ( ( PUNCT american_scientific_journal-1433 63 13 asrjets asrjet NOUN american_scientific_journal-1433 63 14 ) ) PUNCT american_scientific_journal-1433 63 15 ( ( PUNCT american_scientific_journal-1433 63 16 2016 2016 NUM american_scientific_journal-1433 63 17 ) ) PUNCT american_scientific_journal-1433 63 18 volume volume NOUN american_scientific_journal-1433 63 19 19 19 NUM american_scientific_journal-1433 63 20 , , PUNCT american_scientific_journal-1433 63 21 no no DET american_scientific_journal-1433 63 22 1 1 NUM american_scientific_journal-1433 63 23 , , PUNCT american_scientific_journal-1433 63 24 pp pp ADV american_scientific_journal-1433 63 25 9 9 NUM american_scientific_journal-1433 63 26 - - SYM american_scientific_journal-1433 63 27 18 18 NUM american_scientific_journal-1433 63 28 13 13 NUM american_scientific_journal-1433 63 29 again again ADV american_scientific_journal-1433 63 30 applying apply VERB american_scientific_journal-1433 63 31 the the DET american_scientific_journal-1433 63 32 commutation commutation NOUN american_scientific_journal-1433 63 33 formula formula NOUN american_scientific_journal-1433 63 34 exhibited exhibit VERB american_scientific_journal-1433 63 35 by by ADP american_scientific_journal-1433 63 36 ( ( PUNCT american_scientific_journal-1433 63 37 1.3 1.3 NUM american_scientific_journal-1433 63 38 ) ) PUNCT american_scientific_journal-1433 63 39 for for ADP american_scientific_journal-1433 63 40 ( ( PUNCT american_scientific_journal-1433 63 41 𝐻𝐻𝑗𝑗 𝐻𝐻𝑗𝑗 PROPN american_scientific_journal-1433 63 42 ) ) PUNCT american_scientific_journal-1433 63 43 , , PUNCT american_scientific_journal-1433 63 44 we we PRON american_scientific_journal-1433 63 45 get get VERB american_scientific_journal-1433 63 46 ( ( PUNCT american_scientific_journal-1433 63 47 2.11 2.11 NUM american_scientific_journal-1433 63 48 ) ) PUNCT american_scientific_journal-1433 63 49 β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘— β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘— PROPN american_scientific_journal-1433 63 50 βˆ’ βˆ’ PROPN american_scientific_journal-1433 63 51 β„¬π‘šπ‘š β„¬π‘šπ‘š PROPN american_scientific_journal-1433 63 52 οΏ½ οΏ½ PROPN american_scientific_journal-1433 63 53 π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 63 54 οΏ½ οΏ½ PROPN american_scientific_journal-1433 63 55 βˆ’ βˆ’ PROPN american_scientific_journal-1433 63 56 ( ( PUNCT american_scientific_journal-1433 63 57 β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 63 58 βˆ’ βˆ’ PROPN american_scientific_journal-1433 63 59 ( ( PUNCT american_scientific_journal-1433 63 60 β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 63 61 = = SYM american_scientific_journal-1433 63 62 οΏ½ οΏ½ PROPN american_scientific_journal-1433 63 63 οΏ½ οΏ½ PROPN american_scientific_journal-1433 63 64 Μ‡ Μ‡ PROPN american_scientific_journal-1433 63 65 οΏ½ οΏ½ PROPN american_scientific_journal-1433 63 66 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 63 67 οΏ½ οΏ½ PROPN american_scientific_journal-1433 63 68 𝐻𝐻𝑗𝑗 𝐻𝐻𝑗𝑗 PROPN american_scientific_journal-1433 63 69 + + PROPN american_scientific_journal-1433 64 1 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘— π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘— PROPN american_scientific_journal-1433 64 2 + + CCONJ american_scientific_journal-1433 64 3 ( ( PUNCT american_scientific_journal-1433 64 4 1 1 NUM american_scientific_journal-1433 64 5 βˆ’ βˆ’ NUM american_scientific_journal-1433 64 6 𝑛𝑛) 𝑛𝑛) NUM american_scientific_journal-1433 64 7 οΏ½ οΏ½ PROPN american_scientific_journal-1433 64 8 οΏ½ οΏ½ PROPN american_scientific_journal-1433 64 9 Μ‡ Μ‡ PROPN american_scientific_journal-1433 64 10 οΏ½ οΏ½ NOUN american_scientific_journal-1433 64 11 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 64 12 οΏ½ οΏ½ NOUN american_scientific_journal-1433 64 13 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 NOUN american_scientific_journal-1433 64 14 + + CCONJ american_scientific_journal-1433 64 15 ( ( PUNCT american_scientific_journal-1433 64 16 1 1 NUM american_scientific_journal-1433 64 17 βˆ’ βˆ’ NOUN american_scientific_journal-1433 64 18 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘— 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘— NOUN american_scientific_journal-1433 64 19 . . PUNCT american_scientific_journal-1433 65 1 this this PRON american_scientific_journal-1433 65 2 shows show VERB american_scientific_journal-1433 65 3 that that SCONJ american_scientific_journal-1433 65 4 ( ( PUNCT american_scientific_journal-1433 65 5 2.12 2.12 NUM american_scientific_journal-1433 65 6 ) ) PUNCT american_scientific_journal-1433 65 7 β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘— β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘— PROPN american_scientific_journal-1433 65 8 = = SYM american_scientific_journal-1433 65 9 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘— π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘— PROPN american_scientific_journal-1433 65 10 + + CCONJ american_scientific_journal-1433 65 11 ( ( PUNCT american_scientific_journal-1433 65 12 1 1 NUM american_scientific_journal-1433 65 13 βˆ’ βˆ’ PROPN american_scientific_journal-1433 65 14 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘—. 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘”π‘”π‘—π‘—π‘—π‘—. NUM american_scientific_journal-1433 65 15 if if SCONJ american_scientific_journal-1433 65 16 and and CCONJ american_scientific_journal-1433 65 17 only only ADV american_scientific_journal-1433 65 18 if if SCONJ american_scientific_journal-1433 65 19 ( ( PUNCT american_scientific_journal-1433 65 20 2.13 2.13 NUM american_scientific_journal-1433 65 21 ) ) PUNCT american_scientific_journal-1433 65 22 βˆ’β„¬π‘šπ‘š βˆ’β„¬π‘šπ‘š PROPN american_scientific_journal-1433 65 23 οΏ½ οΏ½ PROPN american_scientific_journal-1433 65 24 π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 65 25 οΏ½ οΏ½ PROPN american_scientific_journal-1433 65 26 βˆ’ βˆ’ PROPN american_scientific_journal-1433 65 27 ( ( PUNCT american_scientific_journal-1433 65 28 β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 65 29 βˆ’ βˆ’ PROPN american_scientific_journal-1433 65 30 ( ( PUNCT american_scientific_journal-1433 65 31 β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 65 32 = = SYM american_scientific_journal-1433 65 33 οΏ½ οΏ½ PROPN american_scientific_journal-1433 65 34 οΏ½ οΏ½ PROPN american_scientific_journal-1433 65 35 Μ‡ Μ‡ PROPN american_scientific_journal-1433 65 36 οΏ½ οΏ½ PROPN american_scientific_journal-1433 65 37 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 65 38 οΏ½ οΏ½ PROPN american_scientific_journal-1433 65 39 𝐻𝐻𝑗𝑗 𝐻𝐻𝑗𝑗 PROPN american_scientific_journal-1433 65 40 + + CCONJ american_scientific_journal-1433 65 41 ( ( PUNCT american_scientific_journal-1433 65 42 1 1 NUM american_scientific_journal-1433 65 43 βˆ’ βˆ’ NUM american_scientific_journal-1433 65 44 𝑛𝑛) 𝑛𝑛) NUM american_scientific_journal-1433 65 45 οΏ½ οΏ½ PROPN american_scientific_journal-1433 65 46 οΏ½ οΏ½ PROPN american_scientific_journal-1433 65 47 Μ‡ Μ‡ PROPN american_scientific_journal-1433 65 48 οΏ½ οΏ½ PROPN american_scientific_journal-1433 65 49 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 65 50 οΏ½ οΏ½ PROPN american_scientific_journal-1433 65 51 𝑦𝑦𝑗𝑗. 𝑦𝑦𝑗𝑗. NOUN american_scientific_journal-1433 65 52 thus thus ADV american_scientific_journal-1433 65 53 , , PUNCT american_scientific_journal-1433 65 54 we we PRON american_scientific_journal-1433 65 55 have have AUX american_scientific_journal-1433 65 56 theorem theorem VERB american_scientific_journal-1433 65 57 2.3 2.3 NUM american_scientific_journal-1433 65 58 . . PUNCT american_scientific_journal-1433 66 1 in in ADP american_scientific_journal-1433 66 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 66 3 βˆ’ βˆ’ PROPN american_scientific_journal-1433 66 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 66 5 βˆ’ βˆ’ NUM american_scientific_journal-1433 66 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 NOUN american_scientific_journal-1433 66 7 , , PUNCT american_scientific_journal-1433 66 8 the the DET american_scientific_journal-1433 66 9 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 66 10 βˆ’ricci βˆ’ricci NOUN american_scientific_journal-1433 66 11 tensor tensor NOUN american_scientific_journal-1433 66 12 𝐻𝐻𝑗𝑗𝑗𝑗 𝐻𝐻𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 66 13 is be AUX american_scientific_journal-1433 66 14 given give VERB american_scientific_journal-1433 66 15 by by ADP american_scientific_journal-1433 66 16 equation equation NOUN american_scientific_journal-1433 66 17 ( ( PUNCT american_scientific_journal-1433 66 18 2.12 2.12 NUM american_scientific_journal-1433 66 19 ) ) PUNCT american_scientific_journal-1433 66 20 if if SCONJ american_scientific_journal-1433 67 1 and and CCONJ american_scientific_journal-1433 67 2 only only ADV american_scientific_journal-1433 67 3 if if SCONJ american_scientific_journal-1433 67 4 ( ( PUNCT american_scientific_journal-1433 67 5 2.13 2.13 NUM american_scientific_journal-1433 67 6 ) ) PUNCT american_scientific_journal-1433 67 7 holds hold VERB american_scientific_journal-1433 67 8 good good ADJ american_scientific_journal-1433 67 9 . . PUNCT american_scientific_journal-1433 68 1 transvecting transvecte VERB american_scientific_journal-1433 68 2 ( ( PUNCT american_scientific_journal-1433 68 3 2.11 2.11 NUM american_scientific_journal-1433 68 4 ) ) PUNCT american_scientific_journal-1433 68 5 by by ADP american_scientific_journal-1433 68 6 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 PRON american_scientific_journal-1433 68 7 , , PUNCT american_scientific_journal-1433 68 8 using use VERB american_scientific_journal-1433 68 9 ( ( PUNCT american_scientific_journal-1433 68 10 1.1c 1.1c NUM american_scientific_journal-1433 68 11 ) ) PUNCT american_scientific_journal-1433 68 12 , , PUNCT american_scientific_journal-1433 68 13 ( ( PUNCT american_scientific_journal-1433 68 14 1.1f 1.1f NUM american_scientific_journal-1433 68 15 ) ) PUNCT american_scientific_journal-1433 68 16 , , PUNCT american_scientific_journal-1433 68 17 ( ( PUNCT american_scientific_journal-1433 68 18 1.11 1.11 NUM american_scientific_journal-1433 68 19 ) ) PUNCT american_scientific_journal-1433 68 20 ( ( PUNCT american_scientific_journal-1433 68 21 1.13 1.13 NUM american_scientific_journal-1433 68 22 ) ) PUNCT american_scientific_journal-1433 68 23 and and CCONJ american_scientific_journal-1433 68 24 ( ( PUNCT american_scientific_journal-1433 68 25 1.1a 1.1a NOUN american_scientific_journal-1433 68 26 ) ) PUNCT american_scientific_journal-1433 68 27 , , PUNCT american_scientific_journal-1433 68 28 we we PRON american_scientific_journal-1433 68 29 get get VERB american_scientific_journal-1433 68 30 ( ( PUNCT american_scientific_journal-1433 68 31 2.14 2.14 NUM american_scientific_journal-1433 68 32 ) ) PUNCT american_scientific_journal-1433 68 33 β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘— β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘— NOUN american_scientific_journal-1433 68 34 βˆ’ βˆ’ PROPN american_scientific_journal-1433 69 1 ( ( PUNCT american_scientific_journal-1433 69 2 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-1433 69 3 βˆ’ βˆ’ PROPN american_scientific_journal-1433 69 4 1)(β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ 1)(β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 70 1 = = SYM american_scientific_journal-1433 70 2 ( ( PUNCT american_scientific_journal-1433 70 3 𝑛𝑛 𝑛𝑛 AUX american_scientific_journal-1433 70 4 βˆ’ βˆ’ ADP american_scientific_journal-1433 70 5 1) 1) NUM american_scientific_journal-1433 70 6 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 7 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 8 Μ‡ Μ‡ PROPN american_scientific_journal-1433 70 9 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 10 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 70 11 οΏ½ οΏ½ NOUN american_scientific_journal-1433 70 12 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 70 13 + + CCONJ american_scientific_journal-1433 70 14 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘— π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘— NOUN american_scientific_journal-1433 70 15 + + CCONJ american_scientific_journal-1433 70 16 ( ( PUNCT american_scientific_journal-1433 70 17 1 1 NUM american_scientific_journal-1433 70 18 βˆ’ βˆ’ NUM american_scientific_journal-1433 70 19 𝑛𝑛) 𝑛𝑛) NUM american_scientific_journal-1433 70 20 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 21 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 22 Μ‡ Μ‡ PROPN american_scientific_journal-1433 70 23 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 24 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 70 25 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 26 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 70 27 + + PROPN american_scientific_journal-1433 70 28 ( ( PUNCT american_scientific_journal-1433 70 29 1 1 NUM american_scientific_journal-1433 70 30 βˆ’ βˆ’ NOUN american_scientific_journal-1433 70 31 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘¦π‘¦π‘—π‘—. 𝑛𝑛)π‘π‘π‘šπ‘šπ‘›π‘›π‘¦π‘¦π‘—π‘—. PROPN american_scientific_journal-1433 70 32 using use VERB american_scientific_journal-1433 70 33 ( ( PUNCT american_scientific_journal-1433 70 34 2.7 2.7 NUM american_scientific_journal-1433 70 35 ) ) PUNCT american_scientific_journal-1433 70 36 in in ADP american_scientific_journal-1433 70 37 ( ( PUNCT american_scientific_journal-1433 70 38 2.14 2.14 NUM american_scientific_journal-1433 70 39 ) ) PUNCT american_scientific_journal-1433 70 40 , , PUNCT american_scientific_journal-1433 70 41 we we PRON american_scientific_journal-1433 70 42 get get VERB american_scientific_journal-1433 70 43 ( ( PUNCT american_scientific_journal-1433 70 44 2.15 2.15 NUM american_scientific_journal-1433 70 45 ) ) PUNCT american_scientific_journal-1433 70 46 ( ( PUNCT american_scientific_journal-1433 70 47 β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 70 48 = = PUNCT american_scientific_journal-1433 70 49 βˆ’ βˆ’ PROPN american_scientific_journal-1433 70 50 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 51 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 52 Μ‡ Μ‡ PROPN american_scientific_journal-1433 70 53 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 54 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 70 55 οΏ½ οΏ½ SYM american_scientific_journal-1433 70 56 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 70 57 + + SYM american_scientific_journal-1433 70 58 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 59 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 60 Μ‡ Μ‡ PROPN american_scientific_journal-1433 70 61 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 62 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 70 63 οΏ½ οΏ½ PROPN american_scientific_journal-1433 70 64 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 70 65 . . PUNCT american_scientific_journal-1433 71 1 suppose suppose VERB american_scientific_journal-1433 71 2 ( ( PUNCT american_scientific_journal-1433 71 3 β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 71 4 = = SYM american_scientific_journal-1433 71 5 0 0 NUM american_scientific_journal-1433 71 6 , , PUNCT american_scientific_journal-1433 71 7 in in ADP american_scientific_journal-1433 71 8 view view NOUN american_scientific_journal-1433 71 9 of of ADP american_scientific_journal-1433 71 10 ( ( PUNCT american_scientific_journal-1433 71 11 2.15 2.15 NUM american_scientific_journal-1433 71 12 ) ) PUNCT american_scientific_journal-1433 71 13 , , PUNCT american_scientific_journal-1433 71 14 we we PRON american_scientific_journal-1433 71 15 get get VERB american_scientific_journal-1433 71 16 ( ( PUNCT american_scientific_journal-1433 71 17 2.16 2.16 NUM american_scientific_journal-1433 71 18 ) ) PUNCT american_scientific_journal-1433 72 1 βˆ’ βˆ’ ADP american_scientific_journal-1433 72 2 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 3 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 4 Μ‡ Μ‡ PROPN american_scientific_journal-1433 72 5 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 6 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 72 7 οΏ½ οΏ½ SYM american_scientific_journal-1433 72 8 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 72 9 + + SYM american_scientific_journal-1433 72 10 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 11 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 12 Μ‡ Μ‡ PROPN american_scientific_journal-1433 72 13 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 14 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 72 15 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 16 𝐹𝐹2 𝐹𝐹2 NOUN american_scientific_journal-1433 72 17 = = SYM american_scientific_journal-1433 72 18 0 0 PROPN american_scientific_journal-1433 72 19 . . NOUN american_scientific_journal-1433 72 20 which which PRON american_scientific_journal-1433 72 21 can can AUX american_scientific_journal-1433 72 22 be be AUX american_scientific_journal-1433 72 23 written write VERB american_scientific_journal-1433 72 24 as as ADP american_scientific_journal-1433 72 25 ( ( PUNCT american_scientific_journal-1433 72 26 2.17 2.17 NUM american_scientific_journal-1433 72 27 ) ) PUNCT american_scientific_journal-1433 72 28 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 29 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 30 Μ‡ Μ‡ PROPN american_scientific_journal-1433 72 31 οΏ½ οΏ½ NOUN american_scientific_journal-1433 72 32 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 72 33 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 34 = = SYM american_scientific_journal-1433 72 35 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 36 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 37 Μ‡ Μ‡ PROPN american_scientific_journal-1433 72 38 οΏ½ οΏ½ PROPN american_scientific_journal-1433 72 39 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘šπ‘š πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘šπ‘š NOUN american_scientific_journal-1433 72 40 οΏ½ οΏ½ NOUN american_scientific_journal-1433 72 41 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 72 42 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 72 43 . . PUNCT american_scientific_journal-1433 73 1 if if SCONJ american_scientific_journal-1433 73 2 the the DET american_scientific_journal-1433 73 3 covariant covariant PROPN american_scientific_journal-1433 73 4 tensor tensor NOUN american_scientific_journal-1433 73 5 field field NOUN american_scientific_journal-1433 73 6 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 73 7 is be AUX american_scientific_journal-1433 73 8 independent independent ADJ american_scientific_journal-1433 73 9 of of ADP american_scientific_journal-1433 73 10 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 73 11 , , PUNCT american_scientific_journal-1433 73 12 equation equation NOUN american_scientific_journal-1433 73 13 ( ( PUNCT american_scientific_journal-1433 73 14 2.17 2.17 NUM american_scientific_journal-1433 73 15 ) ) PUNCT american_scientific_journal-1433 73 16 shows show VERB american_scientific_journal-1433 73 17 that that SCONJ american_scientific_journal-1433 73 18 the the DET american_scientific_journal-1433 73 19 covariant covariant PROPN american_scientific_journal-1433 73 20 tensor tensor NOUN american_scientific_journal-1433 73 21 field field NOUN american_scientific_journal-1433 73 22 π‘π‘π‘šπ‘šπ‘›π‘› π‘π‘π‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 73 23 is be AUX american_scientific_journal-1433 73 24 independent independent ADJ american_scientific_journal-1433 73 25 of of ADP american_scientific_journal-1433 73 26 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 73 27 . . PUNCT american_scientific_journal-1433 74 1 conversely conversely ADV american_scientific_journal-1433 74 2 , , PUNCT american_scientific_journal-1433 74 3 if if SCONJ american_scientific_journal-1433 74 4 the the DET american_scientific_journal-1433 74 5 covariant covariant ADJ american_scientific_journal-1433 74 6 tensor tensor NOUN american_scientific_journal-1433 74 7 π‘π‘π‘šπ‘šπ‘›π‘› π‘π‘π‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 74 8 is be AUX american_scientific_journal-1433 74 9 independent independent ADJ american_scientific_journal-1433 74 10 of of ADP american_scientific_journal-1433 74 11 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 74 12 , , PUNCT american_scientific_journal-1433 74 13 we we PRON american_scientific_journal-1433 74 14 get get VERB american_scientific_journal-1433 74 15 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 74 16 οΏ½ οΏ½ PROPN american_scientific_journal-1433 74 17 οΏ½ οΏ½ PROPN american_scientific_journal-1433 74 18 Μ‡ Μ‡ PROPN american_scientific_journal-1433 74 19 οΏ½ οΏ½ PROPN american_scientific_journal-1433 74 20 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 74 21 οΏ½ οΏ½ PROPN american_scientific_journal-1433 74 22 = = SYM american_scientific_journal-1433 74 23 0 0 PROPN american_scientific_journal-1433 74 24 . . PUNCT american_scientific_journal-1433 75 1 in in ADP american_scientific_journal-1433 75 2 view view NOUN american_scientific_journal-1433 75 3 theorem2.2 theorem2.2 NOUN american_scientific_journal-1433 75 4 , , PUNCT american_scientific_journal-1433 75 5 the the DET american_scientific_journal-1433 75 6 condition condition NOUN american_scientific_journal-1433 75 7 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 75 8 οΏ½ οΏ½ PROPN american_scientific_journal-1433 75 9 οΏ½ οΏ½ PROPN american_scientific_journal-1433 75 10 Μ‡ Μ‡ PROPN american_scientific_journal-1433 75 11 οΏ½ οΏ½ PROPN american_scientific_journal-1433 75 12 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 75 13 οΏ½ οΏ½ X american_scientific_journal-1433 75 14 = = SYM american_scientific_journal-1433 75 15 0 0 NUM american_scientific_journal-1433 75 16 implies imply VERB american_scientific_journal-1433 75 17 οΏ½ οΏ½ PROPN american_scientific_journal-1433 75 18 Μ‡ Μ‡ VERB american_scientific_journal-1433 75 19 οΏ½ οΏ½ NOUN american_scientific_journal-1433 75 20 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› X american_scientific_journal-1433 75 21 = = SYM american_scientific_journal-1433 75 22 0 0 NUM american_scientific_journal-1433 75 23 , , PUNCT american_scientific_journal-1433 75 24 i.e. i.e. X american_scientific_journal-1433 75 25 the the DET american_scientific_journal-1433 75 26 covariant covariant ADJ american_scientific_journal-1433 75 27 tensor tensor NOUN american_scientific_journal-1433 75 28 field field NOUN american_scientific_journal-1433 75 29 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 75 30 is be AUX american_scientific_journal-1433 75 31 also also ADV american_scientific_journal-1433 75 32 independent independent ADJ american_scientific_journal-1433 75 33 of of ADP american_scientific_journal-1433 75 34 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 75 35 . . PUNCT american_scientific_journal-1433 76 1 this this PRON american_scientific_journal-1433 76 2 leads lead VERB american_scientific_journal-1433 76 3 to to ADP american_scientific_journal-1433 76 4 american american ADJ american_scientific_journal-1433 76 5 scientific scientific ADJ american_scientific_journal-1433 76 6 research research NOUN american_scientific_journal-1433 76 7 journal journal NOUN american_scientific_journal-1433 76 8 for for ADP american_scientific_journal-1433 76 9 engineering engineering NOUN american_scientific_journal-1433 76 10 , , PUNCT american_scientific_journal-1433 76 11 technology technology NOUN american_scientific_journal-1433 76 12 , , PUNCT american_scientific_journal-1433 76 13 and and CCONJ american_scientific_journal-1433 76 14 sciences science NOUN american_scientific_journal-1433 76 15 ( ( PUNCT american_scientific_journal-1433 76 16 asrjets asrjet NOUN american_scientific_journal-1433 76 17 ) ) PUNCT american_scientific_journal-1433 76 18 ( ( PUNCT american_scientific_journal-1433 76 19 2016 2016 NUM american_scientific_journal-1433 76 20 ) ) PUNCT american_scientific_journal-1433 76 21 volume volume NOUN american_scientific_journal-1433 76 22 19 19 NUM american_scientific_journal-1433 76 23 , , PUNCT american_scientific_journal-1433 76 24 no no DET american_scientific_journal-1433 76 25 1 1 NUM american_scientific_journal-1433 76 26 , , PUNCT american_scientific_journal-1433 76 27 pp pp ADV american_scientific_journal-1433 76 28 9 9 NUM american_scientific_journal-1433 76 29 - - SYM american_scientific_journal-1433 76 30 18 18 NUM american_scientific_journal-1433 76 31 14 14 NUM american_scientific_journal-1433 76 32 theorem theorem VERB american_scientific_journal-1433 76 33 2.4 2.4 NUM american_scientific_journal-1433 76 34 . . PUNCT american_scientific_journal-1433 77 1 in in ADP american_scientific_journal-1433 77 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 77 3 βˆ’ βˆ’ PROPN american_scientific_journal-1433 77 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 77 5 βˆ’ βˆ’ NUM american_scientific_journal-1433 77 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 NOUN american_scientific_journal-1433 77 7 , , PUNCT american_scientific_journal-1433 77 8 the the DET american_scientific_journal-1433 77 9 covariant covariant ADJ american_scientific_journal-1433 77 10 tensor tensor NOUN american_scientific_journal-1433 77 11 field field NOUN american_scientific_journal-1433 77 12 π‘π‘π‘šπ‘šπ‘›π‘› π‘π‘π‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 77 13 is be AUX american_scientific_journal-1433 77 14 independent independent ADJ american_scientific_journal-1433 77 15 of of ADP american_scientific_journal-1433 77 16 the the DET american_scientific_journal-1433 77 17 directional directional ADJ american_scientific_journal-1433 77 18 arguments argument NOUN american_scientific_journal-1433 77 19 if if SCONJ american_scientific_journal-1433 78 1 and and CCONJ american_scientific_journal-1433 78 2 only only ADV american_scientific_journal-1433 78 3 if if SCONJ american_scientific_journal-1433 78 4 the the DET american_scientific_journal-1433 78 5 covariant covariant PROPN american_scientific_journal-1433 78 6 tensor tensor NOUN american_scientific_journal-1433 78 7 field field NOUN american_scientific_journal-1433 78 8 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 78 9 is be AUX american_scientific_journal-1433 78 10 independent independent ADJ american_scientific_journal-1433 78 11 of of ADP american_scientific_journal-1433 78 12 the the DET american_scientific_journal-1433 78 13 directional directional ADJ american_scientific_journal-1433 78 14 arguments argument NOUN american_scientific_journal-1433 78 15 provided provide VERB american_scientific_journal-1433 78 16 ( ( PUNCT american_scientific_journal-1433 78 17 β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 78 18 = = SYM american_scientific_journal-1433 78 19 0 0 PROPN american_scientific_journal-1433 78 20 . . PUNCT american_scientific_journal-1433 79 1 suppose suppose VERB american_scientific_journal-1433 79 2 the the DET american_scientific_journal-1433 79 3 tensor tensor NOUN american_scientific_journal-1433 79 4 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 79 5 is be AUX american_scientific_journal-1433 79 6 not not PART american_scientific_journal-1433 79 7 independent independent ADJ american_scientific_journal-1433 79 8 of of ADP american_scientific_journal-1433 79 9 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖 NOUN american_scientific_journal-1433 79 10 and and CCONJ american_scientific_journal-1433 79 11 in in ADP american_scientific_journal-1433 79 12 view view NOUN american_scientific_journal-1433 79 13 of of ADP american_scientific_journal-1433 79 14 ( ( PUNCT american_scientific_journal-1433 79 15 2.11 2.11 NUM american_scientific_journal-1433 79 16 ) ) PUNCT american_scientific_journal-1433 79 17 , , PUNCT american_scientific_journal-1433 79 18 ( ( PUNCT american_scientific_journal-1433 79 19 2.12 2.12 NUM american_scientific_journal-1433 79 20 ) ) PUNCT american_scientific_journal-1433 79 21 and and CCONJ american_scientific_journal-1433 79 22 ( ( PUNCT american_scientific_journal-1433 79 23 2.17 2.17 NUM american_scientific_journal-1433 79 24 ) ) PUNCT american_scientific_journal-1433 79 25 , , PUNCT american_scientific_journal-1433 79 26 we we PRON american_scientific_journal-1433 79 27 get get VERB american_scientific_journal-1433 79 28 ( ( PUNCT american_scientific_journal-1433 79 29 2.18 2.18 NUM american_scientific_journal-1433 79 30 ) ) PUNCT american_scientific_journal-1433 79 31 βˆ’β„¬π‘šπ‘š βˆ’β„¬π‘šπ‘š PROPN american_scientific_journal-1433 79 32 οΏ½ οΏ½ PROPN american_scientific_journal-1433 79 33 π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 79 34 οΏ½ οΏ½ PROPN american_scientific_journal-1433 79 35 βˆ’ βˆ’ PROPN american_scientific_journal-1433 80 1 ( ( PUNCT american_scientific_journal-1433 80 2 β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 80 3 βˆ’ βˆ’ PROPN american_scientific_journal-1433 80 4 ( ( PUNCT american_scientific_journal-1433 80 5 β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 80 6 = = PROPN american_scientific_journal-1433 80 7 οΏ½ οΏ½ PROPN american_scientific_journal-1433 80 8 Μ‡ Μ‡ PROPN american_scientific_journal-1433 80 9 οΏ½ οΏ½ PROPN american_scientific_journal-1433 80 10 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›(𝐻𝐻𝑗𝑗 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›(𝐻𝐻𝑗𝑗 PROPN american_scientific_journal-1433 80 11 βˆ’ βˆ’ PROPN american_scientific_journal-1433 80 12 ( ( PUNCT american_scientific_journal-1433 80 13 π‘›π‘›βˆ’1 π‘›π‘›βˆ’1 NOUN american_scientific_journal-1433 80 14 ) ) PUNCT american_scientific_journal-1433 81 1 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 81 2 𝐻𝐻𝑦𝑦𝑗𝑗 𝐻𝐻𝑦𝑦𝑗𝑗 PROPN american_scientific_journal-1433 81 3 ) ) PUNCT american_scientific_journal-1433 81 4 . . PUNCT american_scientific_journal-1433 82 1 transvecting transvecte VERB american_scientific_journal-1433 82 2 ( ( PUNCT american_scientific_journal-1433 82 3 2.18 2.18 NUM american_scientific_journal-1433 82 4 ) ) PUNCT american_scientific_journal-1433 82 5 by by ADP american_scientific_journal-1433 82 6 π‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 82 7 , , PUNCT american_scientific_journal-1433 82 8 using use VERB american_scientific_journal-1433 82 9 ( ( PUNCT american_scientific_journal-1433 82 10 1.1c 1.1c NUM american_scientific_journal-1433 82 11 ) ) PUNCT american_scientific_journal-1433 82 12 and and CCONJ american_scientific_journal-1433 82 13 ( ( PUNCT american_scientific_journal-1433 82 14 1.1f 1.1f NUM american_scientific_journal-1433 82 15 ) ) PUNCT american_scientific_journal-1433 82 16 , , PUNCT american_scientific_journal-1433 82 17 we we PRON american_scientific_journal-1433 82 18 get get VERB american_scientific_journal-1433 82 19 ( ( PUNCT american_scientific_journal-1433 82 20 2.19 2.19 NUM american_scientific_journal-1433 82 21 ) ) PUNCT american_scientific_journal-1433 82 22 βˆ’β„¬π‘šπ‘š βˆ’β„¬π‘šπ‘š ADP american_scientific_journal-1433 82 23 οΏ½ οΏ½ PROPN american_scientific_journal-1433 82 24 π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 82 25 οΏ½ οΏ½ PROPN american_scientific_journal-1433 82 26 π‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 82 27 = = SYM american_scientific_journal-1433 82 28 ( ( PUNCT american_scientific_journal-1433 82 29 οΏ½ οΏ½ PROPN american_scientific_journal-1433 82 30 Μ‡ Μ‡ PROPN american_scientific_journal-1433 82 31 οΏ½ οΏ½ PROPN american_scientific_journal-1433 82 32 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π‘¦π‘¦π‘šπ‘š(𝐻𝐻𝑗𝑗 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π‘¦π‘¦π‘šπ‘š(𝐻𝐻𝑗𝑗 PROPN american_scientific_journal-1433 82 33 βˆ’ βˆ’ PROPN american_scientific_journal-1433 82 34 ( ( PUNCT american_scientific_journal-1433 82 35 π‘›π‘›βˆ’1 π‘›π‘›βˆ’1 NOUN american_scientific_journal-1433 82 36 ) ) PUNCT american_scientific_journal-1433 83 1 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 84 1 𝐻𝐻𝑦𝑦𝑗𝑗 𝐻𝐻𝑦𝑦𝑗𝑗 PROPN american_scientific_journal-1433 84 2 ) ) PUNCT american_scientific_journal-1433 84 3 . . PUNCT american_scientific_journal-1433 85 1 which which PRON american_scientific_journal-1433 85 2 implies imply VERB american_scientific_journal-1433 85 3 ( ( PUNCT american_scientific_journal-1433 85 4 2.20 2.20 NUM american_scientific_journal-1433 85 5 ) ) PUNCT american_scientific_journal-1433 85 6 βˆ’β„¬π‘šπ‘š βˆ’β„¬π‘šπ‘š PROPN american_scientific_journal-1433 85 7 οΏ½ οΏ½ PROPN american_scientific_journal-1433 85 8 π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 85 9 οΏ½ οΏ½ PROPN american_scientific_journal-1433 85 10 π‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 85 11 = = SYM american_scientific_journal-1433 85 12 ( ( PUNCT american_scientific_journal-1433 85 13 οΏ½ οΏ½ PROPN american_scientific_journal-1433 85 14 Μ‡ Μ‡ PROPN american_scientific_journal-1433 85 15 οΏ½ οΏ½ PROPN american_scientific_journal-1433 85 16 πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› NOUN american_scientific_journal-1433 85 17 βˆ’ βˆ’ PROPN american_scientific_journal-1433 85 18 π‘Žπ‘Žπ‘—π‘—π‘›π‘›)(𝐻𝐻𝑗𝑗 π‘Žπ‘Žπ‘—π‘—π‘›π‘›)(𝐻𝐻𝑗𝑗 PUNCT american_scientific_journal-1433 85 19 βˆ’ βˆ’ PROPN american_scientific_journal-1433 85 20 ( ( PUNCT american_scientific_journal-1433 85 21 π‘›π‘›βˆ’1 π‘›π‘›βˆ’1 NOUN american_scientific_journal-1433 85 22 ) ) PUNCT american_scientific_journal-1433 85 23 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 85 24 𝐻𝐻𝑦𝑦𝑗𝑗 𝐻𝐻𝑦𝑦𝑗𝑗 PROPN american_scientific_journal-1433 85 25 ) ) PUNCT american_scientific_journal-1433 85 26 . . PUNCT american_scientific_journal-1433 86 1 where where SCONJ american_scientific_journal-1433 86 2 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘¦π‘¦π‘šπ‘š π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π‘¦π‘¦π‘šπ‘š PROPN american_scientific_journal-1433 86 3 = = PROPN american_scientific_journal-1433 86 4 π‘Žπ‘Žπ‘›π‘› π‘Žπ‘Žπ‘›π‘› PROPN american_scientific_journal-1433 86 5 suppose suppose VERB american_scientific_journal-1433 86 6 β„¬π‘šπ‘š β„¬π‘šπ‘š PROPN american_scientific_journal-1433 86 7 οΏ½ οΏ½ PROPN american_scientific_journal-1433 86 8 π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ π»π»π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 86 9 οΏ½ οΏ½ PROPN american_scientific_journal-1433 86 10 π‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 86 11 = = SYM american_scientific_journal-1433 86 12 0 0 NUM american_scientific_journal-1433 86 13 , , PUNCT american_scientific_journal-1433 86 14 equation equation NOUN american_scientific_journal-1433 86 15 ( ( PUNCT american_scientific_journal-1433 86 16 2.20 2.20 NUM american_scientific_journal-1433 86 17 ) ) PUNCT american_scientific_journal-1433 86 18 has have VERB american_scientific_journal-1433 86 19 at at ADV american_scientific_journal-1433 86 20 least least ADJ american_scientific_journal-1433 86 21 one one NUM american_scientific_journal-1433 86 22 of of ADP american_scientific_journal-1433 86 23 the the DET american_scientific_journal-1433 86 24 following follow VERB american_scientific_journal-1433 86 25 conditions condition NOUN american_scientific_journal-1433 86 26 ( ( PUNCT american_scientific_journal-1433 86 27 2.21 2.21 NUM american_scientific_journal-1433 86 28 ) ) PUNCT american_scientific_journal-1433 86 29 a a PRON american_scientific_journal-1433 86 30 ) ) PUNCT american_scientific_journal-1433 86 31 π‘Žπ‘Žπ‘—π‘—π‘›π‘› π‘Žπ‘Žπ‘—π‘—π‘›π‘› NOUN american_scientific_journal-1433 86 32 = = SYM american_scientific_journal-1433 86 33 οΏ½ οΏ½ PROPN american_scientific_journal-1433 86 34 Μ‡ Μ‡ PROPN american_scientific_journal-1433 86 35 οΏ½ οΏ½ PROPN american_scientific_journal-1433 86 36 πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› PROPN american_scientific_journal-1433 86 37 , , PUNCT american_scientific_journal-1433 87 1 b b X american_scientific_journal-1433 87 2 ) ) PUNCT american_scientific_journal-1433 87 3 𝐻𝐻𝑗𝑗 𝐻𝐻𝑗𝑗 PROPN american_scientific_journal-1433 87 4 = = PRON american_scientific_journal-1433 87 5 ( ( PUNCT american_scientific_journal-1433 87 6 π‘›π‘›βˆ’1 π‘›π‘›βˆ’1 NOUN american_scientific_journal-1433 87 7 ) ) PUNCT american_scientific_journal-1433 88 1 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 88 2 𝐻𝐻𝑦𝑦𝑗𝑗 𝐻𝐻𝑦𝑦𝑗𝑗 PROPN american_scientific_journal-1433 88 3 . . PUNCT american_scientific_journal-1433 89 1 thus thus ADV american_scientific_journal-1433 89 2 , , PUNCT american_scientific_journal-1433 89 3 we we PRON american_scientific_journal-1433 89 4 have have AUX american_scientific_journal-1433 89 5 theorem theorem VERB american_scientific_journal-1433 89 6 2.5 2.5 NUM american_scientific_journal-1433 89 7 . . PUNCT american_scientific_journal-1433 90 1 in in ADP american_scientific_journal-1433 90 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 90 3 βˆ’ βˆ’ PROPN american_scientific_journal-1433 90 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 90 5 βˆ’ βˆ’ NUM american_scientific_journal-1433 90 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 90 7 , , PUNCT american_scientific_journal-1433 90 8 which which PRON american_scientific_journal-1433 90 9 the the DET american_scientific_journal-1433 90 10 covariant covariant ADJ american_scientific_journal-1433 90 11 tensor tensor NOUN american_scientific_journal-1433 90 12 field field NOUN american_scientific_journal-1433 90 13 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 90 14 is be AUX american_scientific_journal-1433 90 15 not not PART american_scientific_journal-1433 90 16 independent independent ADJ american_scientific_journal-1433 90 17 of of ADP american_scientific_journal-1433 90 18 the the DET american_scientific_journal-1433 90 19 directional directional ADJ american_scientific_journal-1433 90 20 argument argument NOUN american_scientific_journal-1433 90 21 at at ADP american_scientific_journal-1433 90 22 least least ADJ american_scientific_journal-1433 90 23 one one NUM american_scientific_journal-1433 90 24 of of ADP american_scientific_journal-1433 90 25 the the DET american_scientific_journal-1433 90 26 conditions(2.21a conditions(2.21a NOUN american_scientific_journal-1433 90 27 ) ) PUNCT american_scientific_journal-1433 90 28 and and CCONJ american_scientific_journal-1433 90 29 ( ( PUNCT american_scientific_journal-1433 90 30 2.21b 2.21b NUM american_scientific_journal-1433 90 31 ) ) PUNCT american_scientific_journal-1433 90 32 hold hold NOUN american_scientific_journal-1433 90 33 . . PUNCT american_scientific_journal-1433 91 1 suppose suppose VERB american_scientific_journal-1433 91 2 ( ( PUNCT american_scientific_journal-1433 91 3 2.21b 2.21b NUM american_scientific_journal-1433 91 4 ) ) PUNCT american_scientific_journal-1433 91 5 holds hold VERB american_scientific_journal-1433 91 6 , , PUNCT american_scientific_journal-1433 91 7 then then ADV american_scientific_journal-1433 91 8 ( ( PUNCT american_scientific_journal-1433 91 9 2.18 2.18 NUM american_scientific_journal-1433 91 10 ) ) PUNCT american_scientific_journal-1433 91 11 implies imply VERB american_scientific_journal-1433 91 12 ( ( PUNCT american_scientific_journal-1433 91 13 2.22 2.22 NUM american_scientific_journal-1433 91 14 ) ) PUNCT american_scientific_journal-1433 91 15 βˆ’β„¬π‘šπ‘š βˆ’β„¬π‘šπ‘š X american_scientific_journal-1433 91 16 οΏ½ οΏ½ PROPN american_scientific_journal-1433 91 17 ( ( PUNCT american_scientific_journal-1433 91 18 π‘›π‘›βˆ’1)𝐻𝐻 π‘›π‘›βˆ’1)𝐻𝐻 NOUN american_scientific_journal-1433 92 1 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 92 2 π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 92 3 οΏ½ οΏ½ PROPN american_scientific_journal-1433 92 4 βˆ’ βˆ’ PROPN american_scientific_journal-1433 92 5 οΏ½ οΏ½ PROPN american_scientific_journal-1433 92 6 β„¬π‘Ÿπ‘Ÿ β„¬π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 92 7 ( ( PUNCT american_scientific_journal-1433 92 8 π‘›π‘›βˆ’1)𝐻𝐻 π‘›π‘›βˆ’1)𝐻𝐻 NOUN american_scientific_journal-1433 92 9 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 92 10 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 NOUN american_scientific_journal-1433 92 11 οΏ½ οΏ½ PROPN american_scientific_journal-1433 93 1 πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 93 2 βˆ’ βˆ’ PROPN american_scientific_journal-1433 93 3 οΏ½ οΏ½ PROPN american_scientific_journal-1433 93 4 ℬ𝑛𝑛 ℬ𝑛𝑛 PROPN american_scientific_journal-1433 93 5 ( ( PUNCT american_scientific_journal-1433 93 6 π‘›π‘›βˆ’1)𝐻𝐻 π‘›π‘›βˆ’1)𝐻𝐻 NOUN american_scientific_journal-1433 93 7 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 93 8 π‘¦π‘¦π‘Ÿπ‘Ÿ π‘¦π‘¦π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 93 9 οΏ½ οΏ½ PROPN american_scientific_journal-1433 93 10 πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 93 11 = = SYM american_scientific_journal-1433 93 12 0 0 X american_scientific_journal-1433 93 13 . . PUNCT american_scientific_journal-1433 94 1 transvecting transvecte VERB american_scientific_journal-1433 94 2 ( ( PUNCT american_scientific_journal-1433 94 3 2.22 2.22 NUM american_scientific_journal-1433 94 4 ) ) PUNCT american_scientific_journal-1433 94 5 by by ADP american_scientific_journal-1433 94 6 π‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 94 7 and and CCONJ american_scientific_journal-1433 94 8 using use VERB american_scientific_journal-1433 94 9 ( ( PUNCT american_scientific_journal-1433 94 10 1.1c 1.1c NUM american_scientific_journal-1433 94 11 ) ) PUNCT american_scientific_journal-1433 94 12 and and CCONJ american_scientific_journal-1433 94 13 ( ( PUNCT american_scientific_journal-1433 94 14 1.1f 1.1f NUM american_scientific_journal-1433 94 15 ) ) PUNCT american_scientific_journal-1433 94 16 , , PUNCT american_scientific_journal-1433 94 17 we we PRON american_scientific_journal-1433 94 18 get get VERB american_scientific_journal-1433 94 19 ( ( PUNCT american_scientific_journal-1433 94 20 2.23 2.23 NUM american_scientific_journal-1433 94 21 ) ) PUNCT american_scientific_journal-1433 94 22 β„¬π‘šπ‘š(𝐻𝐻)π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘šπ‘š(𝐻𝐻)π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 95 1 π‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 95 2 + + CCONJ american_scientific_journal-1433 95 3 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 95 4 οΏ½ οΏ½ PROPN american_scientific_journal-1433 95 5 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 95 6 οΏ½ οΏ½ NOUN american_scientific_journal-1433 95 7 π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 95 8 = = NOUN american_scientific_journal-1433 95 9 0 0 X american_scientific_journal-1433 95 10 . . PUNCT american_scientific_journal-1433 96 1 if if SCONJ american_scientific_journal-1433 96 2 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 96 3 οΏ½ οΏ½ PROPN american_scientific_journal-1433 96 4 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 96 5 οΏ½ οΏ½ NOUN american_scientific_journal-1433 96 6 π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 96 7 = = SYM american_scientific_journal-1433 96 8 0 0 NUM american_scientific_journal-1433 96 9 , , PUNCT american_scientific_journal-1433 96 10 the the DET american_scientific_journal-1433 96 11 equation equation NOUN american_scientific_journal-1433 96 12 ( ( PUNCT american_scientific_journal-1433 96 13 2.23 2.23 NUM american_scientific_journal-1433 96 14 ) ) PUNCT american_scientific_journal-1433 96 15 implies imply VERB american_scientific_journal-1433 96 16 ( ( PUNCT american_scientific_journal-1433 96 17 2.24 2.24 NUM american_scientific_journal-1433 96 18 ) ) PUNCT american_scientific_journal-1433 96 19 π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ π‘¦π‘¦π‘Ÿπ‘ŸπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 97 1 = = NOUN american_scientific_journal-1433 97 2 0 0 NUM american_scientific_journal-1433 97 3 , , PUNCT american_scientific_journal-1433 97 4 since since SCONJ american_scientific_journal-1433 97 5 β„¬π‘šπ‘š(𝐻𝐻)π‘¦π‘¦π‘šπ‘š β„¬π‘šπ‘š(𝐻𝐻)π‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 97 6 β‰  β‰  PROPN american_scientific_journal-1433 97 7 0 0 NUM american_scientific_journal-1433 97 8 therefore therefore ADV american_scientific_journal-1433 97 9 the the DET american_scientific_journal-1433 97 10 space space NOUN american_scientific_journal-1433 97 11 is be AUX american_scientific_journal-1433 97 12 landsberg landsberg PROPN american_scientific_journal-1433 97 13 space space NOUN american_scientific_journal-1433 97 14 thus thus ADV american_scientific_journal-1433 97 15 , , PUNCT american_scientific_journal-1433 97 16 we we PRON american_scientific_journal-1433 97 17 have have VERB american_scientific_journal-1433 97 18 american american ADJ american_scientific_journal-1433 97 19 scientific scientific ADJ american_scientific_journal-1433 97 20 research research NOUN american_scientific_journal-1433 97 21 journal journal NOUN american_scientific_journal-1433 97 22 for for ADP american_scientific_journal-1433 97 23 engineering engineering NOUN american_scientific_journal-1433 97 24 , , PUNCT american_scientific_journal-1433 97 25 technology technology NOUN american_scientific_journal-1433 97 26 , , PUNCT american_scientific_journal-1433 97 27 and and CCONJ american_scientific_journal-1433 97 28 sciences science NOUN american_scientific_journal-1433 97 29 ( ( PUNCT american_scientific_journal-1433 97 30 asrjets asrjet NOUN american_scientific_journal-1433 97 31 ) ) PUNCT american_scientific_journal-1433 97 32 ( ( PUNCT american_scientific_journal-1433 97 33 2016 2016 NUM american_scientific_journal-1433 97 34 ) ) PUNCT american_scientific_journal-1433 97 35 volume volume NOUN american_scientific_journal-1433 97 36 19 19 NUM american_scientific_journal-1433 97 37 , , PUNCT american_scientific_journal-1433 97 38 no no DET american_scientific_journal-1433 97 39 1 1 NUM american_scientific_journal-1433 97 40 , , PUNCT american_scientific_journal-1433 97 41 pp pp ADV american_scientific_journal-1433 97 42 9 9 NUM american_scientific_journal-1433 97 43 - - SYM american_scientific_journal-1433 97 44 18 18 NUM american_scientific_journal-1433 97 45 15 15 NUM american_scientific_journal-1433 97 46 theorem theorem VERB american_scientific_journal-1433 97 47 2.6 2.6 NUM american_scientific_journal-1433 97 48 . . PUNCT american_scientific_journal-1433 98 1 an an DET american_scientific_journal-1433 98 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 98 3 βˆ’ βˆ’ NOUN american_scientific_journal-1433 98 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 NOUN american_scientific_journal-1433 98 5 βˆ’ βˆ’ PROPN american_scientific_journal-1433 98 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 98 7 is be AUX american_scientific_journal-1433 98 8 landsberg landsberg PROPN american_scientific_journal-1433 98 9 space space NOUN american_scientific_journal-1433 98 10 if if SCONJ american_scientific_journal-1433 98 11 condition condition NOUN american_scientific_journal-1433 98 12 ( ( PUNCT american_scientific_journal-1433 98 13 2.21b 2.21b NUM american_scientific_journal-1433 98 14 ) ) PUNCT american_scientific_journal-1433 98 15 holds hold VERB american_scientific_journal-1433 98 16 and and CCONJ american_scientific_journal-1433 98 17 provided provide VERB american_scientific_journal-1433 98 18 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 98 19 οΏ½ οΏ½ PROPN american_scientific_journal-1433 98 20 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 98 21 οΏ½ οΏ½ NOUN american_scientific_journal-1433 98 22 π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 98 23 = = NOUN american_scientific_journal-1433 98 24 0 0 PROPN american_scientific_journal-1433 98 25 . . PUNCT american_scientific_journal-1433 99 1 if if SCONJ american_scientific_journal-1433 99 2 the the DET american_scientific_journal-1433 99 3 covariant covariant PROPN american_scientific_journal-1433 99 4 tensor tensor NOUN american_scientific_journal-1433 99 5 field field NOUN american_scientific_journal-1433 99 6 π‘Žπ‘Žπ‘—π‘—π‘›π‘› π‘Žπ‘Žπ‘—π‘—π‘›π‘› PROPN american_scientific_journal-1433 99 7 β‰  β‰  PROPN american_scientific_journal-1433 99 8 οΏ½ οΏ½ PROPN american_scientific_journal-1433 99 9 Μ‡ Μ‡ PROPN american_scientific_journal-1433 99 10 οΏ½ οΏ½ NOUN american_scientific_journal-1433 99 11 πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› NOUN american_scientific_journal-1433 99 12 , , PUNCT american_scientific_journal-1433 99 13 in in ADP american_scientific_journal-1433 99 14 view view NOUN american_scientific_journal-1433 99 15 of of ADP american_scientific_journal-1433 99 16 theorem2.5 theorem2.5 PROPN american_scientific_journal-1433 99 17 , , PUNCT american_scientific_journal-1433 99 18 ( ( PUNCT american_scientific_journal-1433 99 19 2.21b 2.21b NUM american_scientific_journal-1433 99 20 ) ) PUNCT american_scientific_journal-1433 99 21 holds hold VERB american_scientific_journal-1433 99 22 good good ADJ american_scientific_journal-1433 99 23 . . PUNCT american_scientific_journal-1433 100 1 in in ADP american_scientific_journal-1433 100 2 view view NOUN american_scientific_journal-1433 100 3 of of ADP american_scientific_journal-1433 100 4 this this DET american_scientific_journal-1433 100 5 fact fact NOUN american_scientific_journal-1433 100 6 , , PUNCT american_scientific_journal-1433 100 7 we we PRON american_scientific_journal-1433 100 8 may may AUX american_scientific_journal-1433 100 9 rewrite rewrite VERB american_scientific_journal-1433 100 10 theorem2.6 theorem2.6 NOUN american_scientific_journal-1433 100 11 in in ADP american_scientific_journal-1433 100 12 the the DET american_scientific_journal-1433 100 13 following following NOUN american_scientific_journal-1433 100 14 theorem theorem ADJ american_scientific_journal-1433 100 15 2.7 2.7 NUM american_scientific_journal-1433 100 16 . . PUNCT american_scientific_journal-1433 101 1 an an DET american_scientific_journal-1433 101 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 101 3 βˆ’ βˆ’ NOUN american_scientific_journal-1433 101 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 NOUN american_scientific_journal-1433 101 5 βˆ’ βˆ’ PROPN american_scientific_journal-1433 101 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 101 7 is be AUX american_scientific_journal-1433 101 8 necessarily necessarily ADV american_scientific_journal-1433 101 9 landsberg landsberg ADJ american_scientific_journal-1433 101 10 space space NOUN american_scientific_journal-1433 101 11 provided provide VERB american_scientific_journal-1433 101 12 π‘Žπ‘Žπ‘—π‘—π‘›π‘› π‘Žπ‘Žπ‘—π‘—π‘›π‘› NOUN american_scientific_journal-1433 101 13 β‰  β‰  PROPN american_scientific_journal-1433 101 14 οΏ½ οΏ½ PROPN american_scientific_journal-1433 101 15 Μ‡ Μ‡ PROPN american_scientific_journal-1433 101 16 οΏ½ οΏ½ NOUN american_scientific_journal-1433 101 17 πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› NOUN american_scientific_journal-1433 101 18 and and CCONJ american_scientific_journal-1433 101 19 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 101 20 οΏ½ οΏ½ PROPN american_scientific_journal-1433 101 21 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 101 22 οΏ½ οΏ½ NOUN american_scientific_journal-1433 101 23 π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 101 24 = = SYM american_scientific_journal-1433 101 25 0 0 NUM american_scientific_journal-1433 101 26 . . PUNCT american_scientific_journal-1433 102 1 differentiating differentiate VERB american_scientific_journal-1433 102 2 ( ( PUNCT american_scientific_journal-1433 102 3 2.4 2.4 NUM american_scientific_journal-1433 102 4 ) ) PUNCT american_scientific_journal-1433 102 5 partially partially ADV american_scientific_journal-1433 102 6 with with ADP american_scientific_journal-1433 102 7 respect respect NOUN american_scientific_journal-1433 102 8 to to ADP american_scientific_journal-1433 102 9 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 PRON american_scientific_journal-1433 102 10 , , PUNCT american_scientific_journal-1433 102 11 using use VERB american_scientific_journal-1433 102 12 ( ( PUNCT american_scientific_journal-1433 102 13 1.5 1.5 NUM american_scientific_journal-1433 102 14 ) ) PUNCT american_scientific_journal-1433 102 15 and and CCONJ american_scientific_journal-1433 102 16 ( ( PUNCT american_scientific_journal-1433 102 17 1.1b 1.1b NUM american_scientific_journal-1433 102 18 ) ) PUNCT american_scientific_journal-1433 102 19 , , PUNCT american_scientific_journal-1433 102 20 we we PRON american_scientific_journal-1433 102 21 get get VERB american_scientific_journal-1433 102 22 ( ( PUNCT american_scientific_journal-1433 102 23 2.25 2.25 NUM american_scientific_journal-1433 102 24 ) ) PUNCT american_scientific_journal-1433 102 25 οΏ½ οΏ½ PROPN american_scientific_journal-1433 102 26 Μ‡ Μ‡ PROPN american_scientific_journal-1433 102 27 οΏ½ οΏ½ PROPN american_scientific_journal-1433 102 28 πœ•π‘—π‘— πœ•π‘—π‘— X american_scientific_journal-1433 102 29 οΏ½ οΏ½ PROPN american_scientific_journal-1433 102 30 β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 102 31 οΏ½ οΏ½ PROPN american_scientific_journal-1433 102 32 = = SYM american_scientific_journal-1433 102 33 οΏ½ οΏ½ PROPN american_scientific_journal-1433 102 34 οΏ½ οΏ½ PROPN american_scientific_journal-1433 102 35 Μ‡ Μ‡ PROPN american_scientific_journal-1433 102 36 οΏ½ οΏ½ PROPN american_scientific_journal-1433 102 37 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 102 38 οΏ½ οΏ½ PROPN american_scientific_journal-1433 103 1 π»π»π‘—π‘—β„Žπ‘–π‘– π»π»π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 103 2 + + CCONJ american_scientific_journal-1433 103 3 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 103 4 + + CCONJ american_scientific_journal-1433 103 5 ( ( PUNCT american_scientific_journal-1433 103 6 οΏ½ οΏ½ PROPN american_scientific_journal-1433 103 7 Μ‡ Μ‡ NOUN american_scientific_journal-1433 103 8 οΏ½ οΏ½ PROPN american_scientific_journal-1433 103 9 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) NOUN american_scientific_journal-1433 103 10 οΏ½ οΏ½ NOUN american_scientific_journal-1433 103 11 𝛿𝛿𝑗𝑗𝑖𝑖 𝛿𝛿𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 103 12 π‘¦π‘¦β„Ž π‘¦π‘¦β„Ž NOUN american_scientific_journal-1433 103 13 βˆ’ βˆ’ PROPN american_scientific_journal-1433 103 14 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 103 15 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 NOUN american_scientific_journal-1433 103 16 οΏ½ οΏ½ PROPN american_scientific_journal-1433 104 1 + + SYM american_scientific_journal-1433 104 2 π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 PROPN american_scientific_journal-1433 104 3 π‘”π‘”π‘—π‘—β„Ž π‘”π‘”π‘—π‘—β„Ž NOUN american_scientific_journal-1433 104 4 βˆ’ βˆ’ PROPN american_scientific_journal-1433 104 5 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 104 6 𝑔𝑔𝑗𝑗𝑗𝑗 𝑔𝑔𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 104 7 ) ) PUNCT american_scientific_journal-1433 104 8 . . PUNCT american_scientific_journal-1433 105 1 using use VERB american_scientific_journal-1433 105 2 commutation commutation NOUN american_scientific_journal-1433 105 3 formula formula NOUN american_scientific_journal-1433 105 4 exhibited exhibit VERB american_scientific_journal-1433 105 5 by by ADP american_scientific_journal-1433 105 6 ( ( PUNCT american_scientific_journal-1433 105 7 1.3 1.3 NUM american_scientific_journal-1433 105 8 ) ) PUNCT american_scientific_journal-1433 105 9 for for ADP american_scientific_journal-1433 105 10 ( ( PUNCT american_scientific_journal-1433 105 11 β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 105 12 ) ) PUNCT american_scientific_journal-1433 105 13 in in ADP american_scientific_journal-1433 105 14 ( ( PUNCT american_scientific_journal-1433 105 15 2.25 2.25 NUM american_scientific_journal-1433 105 16 ) ) PUNCT american_scientific_journal-1433 105 17 , , PUNCT american_scientific_journal-1433 105 18 we we PRON american_scientific_journal-1433 105 19 get get VERB american_scientific_journal-1433 105 20 ( ( PUNCT american_scientific_journal-1433 105 21 2.26 2.26 NUM american_scientific_journal-1433 105 22 ) ) PUNCT american_scientific_journal-1433 105 23 β„¬π‘šπ‘š β„¬π‘šπ‘š PROPN american_scientific_journal-1433 105 24 οΏ½ οΏ½ PROPN american_scientific_journal-1433 105 25 οΏ½ οΏ½ PROPN american_scientific_journal-1433 105 26 Μ‡ Μ‡ VERB american_scientific_journal-1433 105 27 οΏ½ οΏ½ NOUN american_scientific_journal-1433 105 28 πœ•π‘—π‘—β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– πœ•π‘—π‘—β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 105 29 οΏ½ οΏ½ NOUN american_scientific_journal-1433 105 30 βˆ’ βˆ’ PROPN american_scientific_journal-1433 106 1 ( ( PUNCT american_scientific_journal-1433 106 2 β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 106 3 ) ) PUNCT american_scientific_journal-1433 106 4 πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 106 5 + + CCONJ american_scientific_journal-1433 106 6 ( ( PUNCT american_scientific_journal-1433 106 7 β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 106 8 ) ) PUNCT american_scientific_journal-1433 106 9 πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– NOUN american_scientific_journal-1433 106 10 βˆ’ βˆ’ NOUN american_scientific_journal-1433 106 11 ( ( PUNCT american_scientific_journal-1433 106 12 β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– PROPN american_scientific_journal-1433 106 13 ) ) PUNCT american_scientific_journal-1433 106 14 πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 106 15 βˆ’(β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– βˆ’(β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– ADJ american_scientific_journal-1433 106 16 ) ) PUNCT american_scientific_journal-1433 106 17 πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 106 18 = = SYM american_scientific_journal-1433 106 19 ( ( PUNCT american_scientific_journal-1433 106 20 οΏ½ οΏ½ PROPN american_scientific_journal-1433 106 21 Μ‡ Μ‡ NOUN american_scientific_journal-1433 106 22 οΏ½ οΏ½ NOUN american_scientific_journal-1433 106 23 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 106 24 + + NOUN american_scientific_journal-1433 106 25 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 106 26 + + CCONJ american_scientific_journal-1433 106 27 ( ( PUNCT american_scientific_journal-1433 106 28 οΏ½ οΏ½ PROPN american_scientific_journal-1433 106 29 Μ‡ Μ‡ NOUN american_scientific_journal-1433 106 30 οΏ½ οΏ½ PROPN american_scientific_journal-1433 106 31 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) NOUN american_scientific_journal-1433 106 32 οΏ½ οΏ½ NOUN american_scientific_journal-1433 106 33 𝛿𝛿𝑗𝑗𝑖𝑖 𝛿𝛿𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 106 34 π‘¦π‘¦β„Ž π‘¦π‘¦β„Ž NOUN american_scientific_journal-1433 106 35 βˆ’ βˆ’ PROPN american_scientific_journal-1433 106 36 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 106 37 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 NOUN american_scientific_journal-1433 106 38 οΏ½ οΏ½ PROPN american_scientific_journal-1433 107 1 + + SYM american_scientific_journal-1433 107 2 π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 PROPN american_scientific_journal-1433 107 3 π‘”π‘”π‘—π‘—β„Ž π‘”π‘”π‘—π‘—β„Ž NOUN american_scientific_journal-1433 107 4 βˆ’ βˆ’ PROPN american_scientific_journal-1433 107 5 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 107 6 𝑔𝑔𝑗𝑗𝑗𝑗 𝑔𝑔𝑗𝑗𝑗𝑗 NOUN american_scientific_journal-1433 107 7 ) ) PUNCT american_scientific_journal-1433 107 8 . . PUNCT american_scientific_journal-1433 108 1 again again ADV american_scientific_journal-1433 108 2 applying apply VERB american_scientific_journal-1433 108 3 the the DET american_scientific_journal-1433 108 4 commutation commutation NOUN american_scientific_journal-1433 108 5 formula formula NOUN american_scientific_journal-1433 108 6 exhibited exhibit VERB american_scientific_journal-1433 108 7 by by ADP american_scientific_journal-1433 108 8 ( ( PUNCT american_scientific_journal-1433 108 9 1.3 1.3 NUM american_scientific_journal-1433 108 10 ) ) PUNCT american_scientific_journal-1433 108 11 for for ADP american_scientific_journal-1433 108 12 ( ( PUNCT american_scientific_journal-1433 108 13 π»π»π‘—π‘—β„Žπ‘–π‘– π»π»π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 108 14 ) ) PUNCT american_scientific_journal-1433 108 15 in in ADP american_scientific_journal-1433 108 16 ( ( PUNCT american_scientific_journal-1433 108 17 2.26 2.26 NUM american_scientific_journal-1433 108 18 ) ) PUNCT american_scientific_journal-1433 108 19 and and CCONJ american_scientific_journal-1433 108 20 using use VERB american_scientific_journal-1433 108 21 ( ( PUNCT american_scientific_journal-1433 108 22 1.5 1.5 NUM american_scientific_journal-1433 108 23 ) ) PUNCT american_scientific_journal-1433 108 24 , , PUNCT american_scientific_journal-1433 108 25 we we PRON american_scientific_journal-1433 108 26 get get VERB american_scientific_journal-1433 108 27 β„¬π‘šπ‘š β„¬π‘šπ‘š NOUN american_scientific_journal-1433 108 28 οΏ½ οΏ½ NOUN american_scientific_journal-1433 108 29 β„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– β„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 108 30 + + CCONJ american_scientific_journal-1433 108 31 π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 108 32 πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 108 33 βˆ’ βˆ’ PROPN american_scientific_journal-1433 108 34 π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– PROPN american_scientific_journal-1433 108 35 πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 108 36 βˆ’ βˆ’ ADP american_scientific_journal-1433 108 37 π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 108 38 πΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 108 39 οΏ½ οΏ½ PROPN american_scientific_journal-1433 108 40 βˆ’ βˆ’ PROPN american_scientific_journal-1433 108 41 ( ( PUNCT american_scientific_journal-1433 108 42 β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 108 43 ) ) PUNCT american_scientific_journal-1433 109 1 πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 110 1 + + PROPN american_scientific_journal-1433 110 2 ( ( PUNCT american_scientific_journal-1433 110 3 β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 110 4 ) ) PUNCT american_scientific_journal-1433 110 5 πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– NOUN american_scientific_journal-1433 110 6 βˆ’ βˆ’ NOUN american_scientific_journal-1433 110 7 ( ( PUNCT american_scientific_journal-1433 110 8 β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– PROPN american_scientific_journal-1433 110 9 ) ) PUNCT american_scientific_journal-1433 110 10 πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 110 11 βˆ’ βˆ’ PROPN american_scientific_journal-1433 110 12 ( ( PUNCT american_scientific_journal-1433 110 13 β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– NOUN american_scientific_journal-1433 110 14 ) ) PUNCT american_scientific_journal-1433 111 1 πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 111 2 = = SYM american_scientific_journal-1433 111 3 ( ( PUNCT american_scientific_journal-1433 111 4 οΏ½ οΏ½ PROPN american_scientific_journal-1433 111 5 Μ‡ Μ‡ NOUN american_scientific_journal-1433 111 6 οΏ½ οΏ½ NOUN american_scientific_journal-1433 111 7 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 111 8 + + NOUN american_scientific_journal-1433 111 9 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 111 10 + + CCONJ american_scientific_journal-1433 111 11 ( ( PUNCT american_scientific_journal-1433 111 12 οΏ½ οΏ½ PROPN american_scientific_journal-1433 111 13 Μ‡ Μ‡ NOUN american_scientific_journal-1433 111 14 οΏ½ οΏ½ PROPN american_scientific_journal-1433 111 15 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) NOUN american_scientific_journal-1433 111 16 οΏ½ οΏ½ NOUN american_scientific_journal-1433 111 17 𝛿𝛿𝑗𝑗𝑖𝑖 𝛿𝛿𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 111 18 π‘¦π‘¦β„Ž π‘¦π‘¦β„Ž NOUN american_scientific_journal-1433 111 19 βˆ’ βˆ’ PROPN american_scientific_journal-1433 111 20 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 111 21 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 NOUN american_scientific_journal-1433 111 22 οΏ½ οΏ½ INTJ american_scientific_journal-1433 111 23 + + CCONJ american_scientific_journal-1433 111 24 π‘π‘π‘šπ‘šπ‘›π‘› π‘π‘π‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 111 25 οΏ½ οΏ½ PROPN american_scientific_journal-1433 111 26 𝛿𝛿𝑗𝑗𝑖𝑖 𝛿𝛿𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 111 27 π‘”π‘”π‘—π‘—β„Ž π‘”π‘”π‘—π‘—β„Ž NOUN american_scientific_journal-1433 111 28 βˆ’ βˆ’ PROPN american_scientific_journal-1433 111 29 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 111 30 𝑔𝑔𝑗𝑗𝑗𝑗 𝑔𝑔𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 111 31 οΏ½ οΏ½ PROPN american_scientific_journal-1433 111 32 . . PUNCT american_scientific_journal-1433 111 33 above above ADP american_scientific_journal-1433 111 34 equation equation NOUN american_scientific_journal-1433 111 35 can can AUX american_scientific_journal-1433 111 36 be be AUX american_scientific_journal-1433 111 37 written write VERB american_scientific_journal-1433 111 38 as as ADP american_scientific_journal-1433 111 39 ( ( PUNCT american_scientific_journal-1433 111 40 2.27 2.27 NUM american_scientific_journal-1433 111 41 ) ) PUNCT american_scientific_journal-1433 111 42 β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 111 43 + + CCONJ american_scientific_journal-1433 111 44 ( ( PUNCT american_scientific_journal-1433 111 45 β„¬π‘šπ‘šπ»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ β„¬π‘šπ‘šπ»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 111 46 ) ) PUNCT american_scientific_journal-1433 111 47 πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 111 48 + + CCONJ american_scientific_journal-1433 111 49 π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 111 50 ( ( PUNCT american_scientific_journal-1433 111 51 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– ADJ american_scientific_journal-1433 111 52 ) ) PUNCT american_scientific_journal-1433 111 53 βˆ’ βˆ’ PROPN american_scientific_journal-1433 112 1 ( ( PUNCT american_scientific_journal-1433 112 2 β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– ADJ american_scientific_journal-1433 112 3 ) ) PUNCT american_scientific_journal-1433 112 4 πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 112 5 βˆ’π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– βˆ’π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– PROPN american_scientific_journal-1433 112 6 ( ( PUNCT american_scientific_journal-1433 112 7 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 112 8 ) ) PUNCT american_scientific_journal-1433 112 9 βˆ’ βˆ’ PROPN american_scientific_journal-1433 112 10 οΏ½ οΏ½ PROPN american_scientific_journal-1433 112 11 β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 112 12 οΏ½ οΏ½ NOUN american_scientific_journal-1433 112 13 πΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 112 14 βˆ’ βˆ’ PROPN american_scientific_journal-1433 112 15 π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 112 16 ( ( PUNCT american_scientific_journal-1433 112 17 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 112 18 ) ) PUNCT american_scientific_journal-1433 112 19 βˆ’ βˆ’ PROPN american_scientific_journal-1433 112 20 ( ( PUNCT american_scientific_journal-1433 112 21 β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 112 22 ) ) PUNCT american_scientific_journal-1433 113 1 πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 113 2 + + PROPN american_scientific_journal-1433 113 3 ( ( PUNCT american_scientific_journal-1433 113 4 β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 113 5 ) ) PUNCT american_scientific_journal-1433 113 6 πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– NOUN american_scientific_journal-1433 113 7 βˆ’ βˆ’ NOUN american_scientific_journal-1433 113 8 ( ( PUNCT american_scientific_journal-1433 113 9 β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– PROPN american_scientific_journal-1433 113 10 ) ) PUNCT american_scientific_journal-1433 113 11 πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 113 12 βˆ’ βˆ’ PROPN american_scientific_journal-1433 113 13 ( ( PUNCT american_scientific_journal-1433 113 14 β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– NOUN american_scientific_journal-1433 113 15 ) ) PUNCT american_scientific_journal-1433 114 1 πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 114 2 = = SYM american_scientific_journal-1433 114 3 ( ( PUNCT american_scientific_journal-1433 114 4 οΏ½ οΏ½ PROPN american_scientific_journal-1433 114 5 Μ‡ Μ‡ NOUN american_scientific_journal-1433 114 6 οΏ½ οΏ½ NOUN american_scientific_journal-1433 114 7 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 114 8 + + NOUN american_scientific_journal-1433 114 9 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 114 10 + + CCONJ american_scientific_journal-1433 114 11 ( ( PUNCT american_scientific_journal-1433 114 12 οΏ½ οΏ½ PROPN american_scientific_journal-1433 114 13 Μ‡ Μ‡ NOUN american_scientific_journal-1433 114 14 οΏ½ οΏ½ PROPN american_scientific_journal-1433 114 15 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) NOUN american_scientific_journal-1433 114 16 οΏ½ οΏ½ NOUN american_scientific_journal-1433 114 17 𝛿𝛿𝑗𝑗𝑖𝑖 𝛿𝛿𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 114 18 π‘¦π‘¦β„Ž π‘¦π‘¦β„Ž NOUN american_scientific_journal-1433 114 19 βˆ’ βˆ’ PROPN american_scientific_journal-1433 114 20 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 114 21 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 NOUN american_scientific_journal-1433 114 22 οΏ½ οΏ½ PROPN american_scientific_journal-1433 114 23 + + CCONJ american_scientific_journal-1433 114 24 π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 PROPN american_scientific_journal-1433 114 25 π‘”π‘”π‘—π‘—β„Ž π‘”π‘”π‘—π‘—β„Ž NOUN american_scientific_journal-1433 114 26 βˆ’ βˆ’ PROPN american_scientific_journal-1433 114 27 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 114 28 𝑔𝑔𝑗𝑗𝑗𝑗 𝑔𝑔𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 114 29 ) ) PUNCT american_scientific_journal-1433 114 30 . . PUNCT american_scientific_journal-1433 115 1 american american PROPN american_scientific_journal-1433 115 2 scientific scientific ADJ american_scientific_journal-1433 115 3 research research PROPN american_scientific_journal-1433 115 4 journal journal NOUN american_scientific_journal-1433 115 5 for for ADP american_scientific_journal-1433 115 6 engineering engineering NOUN american_scientific_journal-1433 115 7 , , PUNCT american_scientific_journal-1433 115 8 technology technology NOUN american_scientific_journal-1433 115 9 , , PUNCT american_scientific_journal-1433 115 10 and and CCONJ american_scientific_journal-1433 115 11 sciences science NOUN american_scientific_journal-1433 115 12 ( ( PUNCT american_scientific_journal-1433 115 13 asrjets asrjet NOUN american_scientific_journal-1433 115 14 ) ) PUNCT american_scientific_journal-1433 115 15 ( ( PUNCT american_scientific_journal-1433 115 16 2016 2016 NUM american_scientific_journal-1433 115 17 ) ) PUNCT american_scientific_journal-1433 115 18 volume volume NOUN american_scientific_journal-1433 115 19 19 19 NUM american_scientific_journal-1433 115 20 , , PUNCT american_scientific_journal-1433 115 21 no no DET american_scientific_journal-1433 115 22 1 1 NUM american_scientific_journal-1433 115 23 , , PUNCT american_scientific_journal-1433 115 24 pp pp ADV american_scientific_journal-1433 115 25 9 9 NUM american_scientific_journal-1433 115 26 - - SYM american_scientific_journal-1433 115 27 18 18 NUM american_scientific_journal-1433 115 28 16 16 NUM american_scientific_journal-1433 115 29 this this PRON american_scientific_journal-1433 115 30 shows show VERB american_scientific_journal-1433 115 31 that that SCONJ american_scientific_journal-1433 115 32 ( ( PUNCT american_scientific_journal-1433 115 33 2.28 2.28 NUM american_scientific_journal-1433 115 34 ) ) PUNCT american_scientific_journal-1433 115 35 β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– β„¬π‘šπ‘šβ„¬π‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 115 36 = = SYM american_scientific_journal-1433 115 37 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 115 38 + + CCONJ american_scientific_journal-1433 115 39 π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 π‘π‘π‘šπ‘šπ‘›π‘›(𝛿𝛿𝑗𝑗𝑖𝑖 PROPN american_scientific_journal-1433 115 40 π‘”π‘”π‘—π‘—β„Ž π‘”π‘”π‘—π‘—β„Ž NOUN american_scientific_journal-1433 115 41 βˆ’ βˆ’ PROPN american_scientific_journal-1433 115 42 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 115 43 𝑔𝑔𝑗𝑗𝑗𝑗 𝑔𝑔𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 115 44 ) ) PUNCT american_scientific_journal-1433 115 45 . . PUNCT american_scientific_journal-1433 116 1 if if SCONJ american_scientific_journal-1433 116 2 and and CCONJ american_scientific_journal-1433 116 3 only only ADV american_scientific_journal-1433 116 4 if if SCONJ american_scientific_journal-1433 116 5 ( ( PUNCT american_scientific_journal-1433 116 6 2.29 2.29 NUM american_scientific_journal-1433 116 7 ) ) PUNCT american_scientific_journal-1433 116 8 ( ( PUNCT american_scientific_journal-1433 116 9 β„¬π‘šπ‘šπ»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ β„¬π‘šπ‘šπ»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 116 10 ) ) PUNCT american_scientific_journal-1433 116 11 πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 116 12 + + CCONJ american_scientific_journal-1433 116 13 π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 116 14 ( ( PUNCT american_scientific_journal-1433 116 15 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– ADJ american_scientific_journal-1433 116 16 ) ) PUNCT american_scientific_journal-1433 116 17 βˆ’ βˆ’ PROPN american_scientific_journal-1433 116 18 ( ( PUNCT american_scientific_journal-1433 116 19 β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– ADJ american_scientific_journal-1433 116 20 ) ) PUNCT american_scientific_journal-1433 116 21 πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ ADP american_scientific_journal-1433 116 22 βˆ’ βˆ’ PROPN american_scientific_journal-1433 117 1 π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– PROPN american_scientific_journal-1433 117 2 ( ( PUNCT american_scientific_journal-1433 117 3 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 117 4 ) ) PUNCT american_scientific_journal-1433 117 5 βˆ’ βˆ’ NOUN american_scientific_journal-1433 117 6 οΏ½ οΏ½ PROPN american_scientific_journal-1433 117 7 β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– NOUN american_scientific_journal-1433 117 8 οΏ½ οΏ½ NOUN american_scientific_journal-1433 117 9 πΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 117 10 βˆ’ βˆ’ PROPN american_scientific_journal-1433 117 11 π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 117 12 ( ( PUNCT american_scientific_journal-1433 117 13 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 117 14 ) ) PUNCT american_scientific_journal-1433 117 15 βˆ’ βˆ’ PROPN american_scientific_journal-1433 117 16 ( ( PUNCT american_scientific_journal-1433 117 17 β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– β„¬π‘Ÿπ‘Ÿπ»π»π‘—π‘—β„Žπ‘–π‘– NOUN american_scientific_journal-1433 117 18 ) ) PUNCT american_scientific_journal-1433 117 19 πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 117 20 + + CCONJ american_scientific_journal-1433 117 21 ( ( PUNCT american_scientific_journal-1433 117 22 β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ β„¬π‘›π‘›π»π»π‘—π‘—β„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 117 23 ) ) PUNCT american_scientific_journal-1433 117 24 πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 117 25 βˆ’(β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– βˆ’(β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘—π‘—π‘–π‘– VERB american_scientific_journal-1433 117 26 ) ) PUNCT american_scientific_journal-1433 117 27 πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 117 28 βˆ’ βˆ’ PROPN american_scientific_journal-1433 117 29 ( ( PUNCT american_scientific_journal-1433 117 30 β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿπ‘–π‘– NOUN american_scientific_journal-1433 117 31 ) ) PUNCT american_scientific_journal-1433 117 32 πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘—π‘—π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 117 33 = = SYM american_scientific_journal-1433 117 34 ( ( PUNCT american_scientific_journal-1433 117 35 οΏ½ οΏ½ PROPN american_scientific_journal-1433 117 36 Μ‡ Μ‡ NOUN american_scientific_journal-1433 117 37 οΏ½ οΏ½ NOUN american_scientific_journal-1433 117 38 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘›)π»π»π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 117 39 + + X american_scientific_journal-1433 117 40 ( ( PUNCT american_scientific_journal-1433 117 41 οΏ½ οΏ½ PROPN american_scientific_journal-1433 117 42 Μ‡ Μ‡ PROPN american_scientific_journal-1433 117 43 οΏ½ οΏ½ PROPN american_scientific_journal-1433 117 44 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) NOUN american_scientific_journal-1433 117 45 οΏ½ οΏ½ NOUN american_scientific_journal-1433 117 46 𝛿𝛿𝑗𝑗𝑖𝑖 𝛿𝛿𝑗𝑗𝑖𝑖 VERB american_scientific_journal-1433 117 47 π‘¦π‘¦β„Ž π‘¦π‘¦β„Ž NOUN american_scientific_journal-1433 117 48 βˆ’ βˆ’ PROPN american_scientific_journal-1433 117 49 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 117 50 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 PROPN american_scientific_journal-1433 117 51 οΏ½ οΏ½ PROPN american_scientific_journal-1433 117 52 . . PUNCT american_scientific_journal-1433 118 1 thus thus ADV american_scientific_journal-1433 118 2 , , PUNCT american_scientific_journal-1433 118 3 we we PRON american_scientific_journal-1433 118 4 have have AUX american_scientific_journal-1433 118 5 theorem theorem VERB american_scientific_journal-1433 118 6 2.8 2.8 NUM american_scientific_journal-1433 118 7 . . PUNCT american_scientific_journal-1433 119 1 in in ADP american_scientific_journal-1433 119 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 119 3 βˆ’ βˆ’ PROPN american_scientific_journal-1433 119 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 119 5 βˆ’ βˆ’ NUM american_scientific_journal-1433 119 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 NOUN american_scientific_journal-1433 119 7 , , PUNCT american_scientific_journal-1433 119 8 the the DET american_scientific_journal-1433 119 9 berwald berwald NOUN american_scientific_journal-1433 119 10 curvature curvature NOUN american_scientific_journal-1433 119 11 tensor tensor NOUN american_scientific_journal-1433 119 12 π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 119 13 is be AUX american_scientific_journal-1433 119 14 non non ADJ american_scientific_journal-1433 119 15 - - ADJ american_scientific_journal-1433 119 16 vanishing vanishing ADJ american_scientific_journal-1433 119 17 if if SCONJ american_scientific_journal-1433 119 18 and and CCONJ american_scientific_journal-1433 119 19 only only ADV american_scientific_journal-1433 119 20 if if SCONJ american_scientific_journal-1433 119 21 ( ( PUNCT american_scientific_journal-1433 119 22 2 2 NUM american_scientific_journal-1433 119 23 . . NOUN american_scientific_journal-1433 119 24 29 29 NUM american_scientific_journal-1433 119 25 ) ) PUNCT american_scientific_journal-1433 119 26 holds hold VERB american_scientific_journal-1433 119 27 good good ADJ american_scientific_journal-1433 119 28 . . PUNCT american_scientific_journal-1433 120 1 transvecting transvecte VERB american_scientific_journal-1433 120 2 ( ( PUNCT american_scientific_journal-1433 120 3 2.29 2.29 NUM american_scientific_journal-1433 120 4 ) ) PUNCT american_scientific_journal-1433 120 5 by by ADP american_scientific_journal-1433 120 6 𝑦𝑦𝑗𝑗 𝑦𝑦𝑗𝑗 PRON american_scientific_journal-1433 120 7 , , PUNCT american_scientific_journal-1433 120 8 using use VERB american_scientific_journal-1433 120 9 ( ( PUNCT american_scientific_journal-1433 120 10 1.1c 1.1c NUM american_scientific_journal-1433 120 11 ) ) PUNCT american_scientific_journal-1433 120 12 , , PUNCT american_scientific_journal-1433 120 13 ( ( PUNCT american_scientific_journal-1433 120 14 1.1f 1.1f NUM american_scientific_journal-1433 120 15 ) ) PUNCT american_scientific_journal-1433 120 16 , , PUNCT american_scientific_journal-1433 120 17 ( ( PUNCT american_scientific_journal-1433 120 18 1.1a 1.1a NOUN american_scientific_journal-1433 120 19 ) ) PUNCT american_scientific_journal-1433 120 20 and and CCONJ american_scientific_journal-1433 120 21 ( ( PUNCT american_scientific_journal-1433 120 22 1.6 1.6 NUM american_scientific_journal-1433 120 23 ) ) PUNCT american_scientific_journal-1433 120 24 , , PUNCT american_scientific_journal-1433 120 25 we we PRON american_scientific_journal-1433 120 26 get get VERB american_scientific_journal-1433 120 27 ( ( PUNCT american_scientific_journal-1433 120 28 2.30 2.30 NUM american_scientific_journal-1433 120 29 ) ) PUNCT american_scientific_journal-1433 120 30 ( ( PUNCT american_scientific_journal-1433 120 31 β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– X american_scientific_journal-1433 120 32 + + CCONJ american_scientific_journal-1433 120 33 π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– X american_scientific_journal-1433 120 34 ) ) PUNCT american_scientific_journal-1433 121 1 βˆ’ βˆ’ PROPN american_scientific_journal-1433 122 1 ( ( PUNCT american_scientific_journal-1433 122 2 β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 122 3 βˆ’π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ βˆ’π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 122 4 ) ) PUNCT american_scientific_journal-1433 123 1 βˆ’ βˆ’ PROPN american_scientific_journal-1433 123 2 ( ( PUNCT american_scientific_journal-1433 123 3 β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– NOUN american_scientific_journal-1433 123 4 ) ) PUNCT american_scientific_journal-1433 124 1 πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 124 2 + + CCONJ american_scientific_journal-1433 124 3 ( ( PUNCT american_scientific_journal-1433 124 4 β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 124 5 βˆ’(β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ βˆ’(β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 124 6 = = PUNCT american_scientific_journal-1433 124 7 οΏ½ οΏ½ PROPN american_scientific_journal-1433 124 8 οΏ½ οΏ½ PROPN american_scientific_journal-1433 124 9 Μ‡ Μ‡ PROPN american_scientific_journal-1433 124 10 οΏ½ οΏ½ PROPN american_scientific_journal-1433 124 11 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 124 12 οΏ½ οΏ½ NOUN american_scientific_journal-1433 124 13 π»π»β„Žπ‘–π‘– π»π»β„Žπ‘–π‘– PROPN american_scientific_journal-1433 124 14 βˆ’ βˆ’ PROPN american_scientific_journal-1433 124 15 ( ( PUNCT american_scientific_journal-1433 124 16 οΏ½ οΏ½ PROPN american_scientific_journal-1433 124 17 Μ‡ Μ‡ PROPN american_scientific_journal-1433 124 18 οΏ½ οΏ½ PROPN american_scientific_journal-1433 124 19 πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) πœ•π‘—π‘—π‘π‘π‘šπ‘šπ‘›π‘›) NOUN american_scientific_journal-1433 124 20 οΏ½ οΏ½ NOUN american_scientific_journal-1433 124 21 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 124 22 𝐹𝐹2 𝐹𝐹2 PROPN american_scientific_journal-1433 124 23 βˆ’ βˆ’ PROPN american_scientific_journal-1433 124 24 π‘¦π‘¦π‘–π‘–π‘¦π‘¦β„Ž π‘¦π‘¦π‘–π‘–π‘¦π‘¦β„Ž PROPN american_scientific_journal-1433 124 25 οΏ½ οΏ½ PROPN american_scientific_journal-1433 124 26 . . PUNCT american_scientific_journal-1433 125 1 in in ADP american_scientific_journal-1433 125 2 view view NOUN american_scientific_journal-1433 125 3 of of ADP american_scientific_journal-1433 125 4 ( ( PUNCT american_scientific_journal-1433 125 5 2.17 2.17 NUM american_scientific_journal-1433 125 6 ) ) PUNCT american_scientific_journal-1433 125 7 and and CCONJ american_scientific_journal-1433 125 8 ( ( PUNCT american_scientific_journal-1433 125 9 2.30 2.30 NUM american_scientific_journal-1433 125 10 ) ) PUNCT american_scientific_journal-1433 125 11 , , PUNCT american_scientific_journal-1433 125 12 we we PRON american_scientific_journal-1433 125 13 get get VERB american_scientific_journal-1433 125 14 ( ( PUNCT american_scientific_journal-1433 125 15 2.31 2.31 NUM american_scientific_journal-1433 125 16 ) ) PUNCT american_scientific_journal-1433 125 17 ( ( PUNCT american_scientific_journal-1433 125 18 β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– X american_scientific_journal-1433 125 19 + + CCONJ american_scientific_journal-1433 125 20 π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– X american_scientific_journal-1433 125 21 ) ) PUNCT american_scientific_journal-1433 125 22 βˆ’ βˆ’ PROPN american_scientific_journal-1433 126 1 ( ( PUNCT american_scientific_journal-1433 126 2 β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 126 3 βˆ’π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ βˆ’π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 126 4 ) ) PUNCT american_scientific_journal-1433 127 1 βˆ’ βˆ’ PROPN american_scientific_journal-1433 127 2 ( ( PUNCT american_scientific_journal-1433 127 3 β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– NOUN american_scientific_journal-1433 127 4 ) ) PUNCT american_scientific_journal-1433 128 1 πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 128 2 + + CCONJ american_scientific_journal-1433 128 3 ( ( PUNCT american_scientific_journal-1433 128 4 β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 128 5 βˆ’(β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ βˆ’(β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 128 6 = = PUNCT american_scientific_journal-1433 128 7 οΏ½ οΏ½ PROPN american_scientific_journal-1433 128 8 οΏ½ οΏ½ PROPN american_scientific_journal-1433 128 9 Μ‡ Μ‡ PROPN american_scientific_journal-1433 128 10 οΏ½ οΏ½ PROPN american_scientific_journal-1433 128 11 πœ•π‘—π‘—π‘Žπ‘Žβ„“π‘šπ‘š πœ•π‘—π‘—π‘Žπ‘Žβ„“π‘šπ‘š PROPN american_scientific_journal-1433 128 12 οΏ½ οΏ½ PROPN american_scientific_journal-1433 128 13 [π»π»β„Žπ‘–π‘– [π»π»β„Žπ‘–π‘– PUNCT american_scientific_journal-1433 128 14 βˆ’ βˆ’ PROPN american_scientific_journal-1433 128 15 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 128 16 οΏ½ οΏ½ PROPN american_scientific_journal-1433 128 17 π›Ώπ›Ώβ„Žπ‘–π‘– π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 128 18 βˆ’ βˆ’ PROPN american_scientific_journal-1433 128 19 Ι­ Ι­ PRON american_scientific_journal-1433 128 20 𝑖𝑖ɭ 𝑖𝑖ɭ NOUN american_scientific_journal-1433 128 21 β„Ž β„Ž VERB american_scientific_journal-1433 128 22 οΏ½ οΏ½ X american_scientific_journal-1433 128 23 ] ] PUNCT american_scientific_journal-1433 128 24 . . PUNCT american_scientific_journal-1433 129 1 if if SCONJ american_scientific_journal-1433 129 2 ( ( PUNCT american_scientific_journal-1433 129 3 2.32 2.32 NUM american_scientific_journal-1433 129 4 ) ) PUNCT american_scientific_journal-1433 129 5 ( ( PUNCT american_scientific_journal-1433 129 6 β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– X american_scientific_journal-1433 129 7 + + CCONJ american_scientific_journal-1433 129 8 π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– X american_scientific_journal-1433 129 9 ) ) PUNCT american_scientific_journal-1433 129 10 βˆ’ βˆ’ PROPN american_scientific_journal-1433 129 11 ( ( PUNCT american_scientific_journal-1433 129 12 β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ ADJ american_scientific_journal-1433 129 13 βˆ’ βˆ’ PROPN american_scientific_journal-1433 129 14 π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 129 15 ) ) PUNCT american_scientific_journal-1433 129 16 βˆ’(β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– βˆ’(β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– NOUN american_scientific_journal-1433 129 17 ) ) PUNCT american_scientific_journal-1433 129 18 πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 129 19 + + CCONJ american_scientific_journal-1433 129 20 ( ( PUNCT american_scientific_journal-1433 129 21 β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– NOUN american_scientific_journal-1433 129 22 βˆ’ βˆ’ NOUN american_scientific_journal-1433 129 23 ( ( PUNCT american_scientific_journal-1433 129 24 β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 129 25 = = SYM american_scientific_journal-1433 129 26 0 0 NUM american_scientific_journal-1433 129 27 . . PUNCT american_scientific_journal-1433 130 1 we we PRON american_scientific_journal-1433 130 2 have have VERB american_scientific_journal-1433 130 3 at at ADV american_scientific_journal-1433 130 4 least least ADJ american_scientific_journal-1433 130 5 one one NUM american_scientific_journal-1433 130 6 of of ADP american_scientific_journal-1433 130 7 the the DET american_scientific_journal-1433 130 8 following follow VERB american_scientific_journal-1433 130 9 conditions condition NOUN american_scientific_journal-1433 130 10 ( ( PUNCT american_scientific_journal-1433 130 11 2.33 2.33 NUM american_scientific_journal-1433 130 12 ) ) PUNCT american_scientific_journal-1433 130 13 a a NOUN american_scientific_journal-1433 130 14 ) ) PUNCT american_scientific_journal-1433 130 15 ( ( PUNCT american_scientific_journal-1433 130 16 οΏ½ οΏ½ PROPN american_scientific_journal-1433 130 17 Μ‡ Μ‡ PROPN american_scientific_journal-1433 130 18 οΏ½ οΏ½ NOUN american_scientific_journal-1433 130 19 πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› X american_scientific_journal-1433 130 20 ) ) PUNCT american_scientific_journal-1433 131 1 = = SYM american_scientific_journal-1433 131 2 0 0 NUM american_scientific_journal-1433 131 3 , , PUNCT american_scientific_journal-1433 131 4 b b NOUN american_scientific_journal-1433 131 5 ) ) PUNCT american_scientific_journal-1433 131 6 π»π»β„Žπ‘–π‘– π»π»β„Žπ‘–π‘– PROPN american_scientific_journal-1433 131 7 = = PUNCT american_scientific_journal-1433 132 1 𝐻𝐻(π›Ώπ›Ώβ„Žπ‘–π‘– 𝐻𝐻(π›Ώπ›Ώβ„Žπ‘–π‘– ADJ american_scientific_journal-1433 132 2 βˆ’ βˆ’ ADP american_scientific_journal-1433 132 3 Ι­ Ι­ SYM american_scientific_journal-1433 132 4 𝑖𝑖ɭ 𝑖𝑖ɭ NOUN american_scientific_journal-1433 132 5 β„Ž β„Ž PROPN american_scientific_journal-1433 132 6 ) ) PUNCT american_scientific_journal-1433 132 7 . . PUNCT american_scientific_journal-1433 133 1 american american PROPN american_scientific_journal-1433 133 2 scientific scientific ADJ american_scientific_journal-1433 133 3 research research PROPN american_scientific_journal-1433 133 4 journal journal NOUN american_scientific_journal-1433 133 5 for for ADP american_scientific_journal-1433 133 6 engineering engineering NOUN american_scientific_journal-1433 133 7 , , PUNCT american_scientific_journal-1433 133 8 technology technology NOUN american_scientific_journal-1433 133 9 , , PUNCT american_scientific_journal-1433 133 10 and and CCONJ american_scientific_journal-1433 133 11 sciences science NOUN american_scientific_journal-1433 133 12 ( ( PUNCT american_scientific_journal-1433 133 13 asrjets asrjet NOUN american_scientific_journal-1433 133 14 ) ) PUNCT american_scientific_journal-1433 133 15 ( ( PUNCT american_scientific_journal-1433 133 16 2016 2016 NUM american_scientific_journal-1433 133 17 ) ) PUNCT american_scientific_journal-1433 133 18 volume volume NOUN american_scientific_journal-1433 133 19 19 19 NUM american_scientific_journal-1433 133 20 , , PUNCT american_scientific_journal-1433 133 21 no no DET american_scientific_journal-1433 133 22 1 1 NUM american_scientific_journal-1433 133 23 , , PUNCT american_scientific_journal-1433 133 24 pp pp ADV american_scientific_journal-1433 133 25 9 9 NUM american_scientific_journal-1433 133 26 - - SYM american_scientific_journal-1433 133 27 18 18 NUM american_scientific_journal-1433 133 28 17 17 NUM american_scientific_journal-1433 133 29 putting put VERB american_scientific_journal-1433 133 30 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 133 31 = = SYM american_scientific_journal-1433 133 32 𝐹𝐹2𝑅𝑅 𝐹𝐹2𝑅𝑅 PROPN american_scientific_journal-1433 133 33 , , PUNCT american_scientific_journal-1433 133 34 𝑅𝑅 𝑅𝑅 PROPN american_scientific_journal-1433 133 35 β‰  β‰  PROPN american_scientific_journal-1433 133 36 0 0 NUM american_scientific_journal-1433 133 37 , , PUNCT american_scientific_journal-1433 133 38 ( ( PUNCT american_scientific_journal-1433 133 39 2.33 2.33 NUM american_scientific_journal-1433 133 40 ) ) PUNCT american_scientific_journal-1433 133 41 may may AUX american_scientific_journal-1433 133 42 be be AUX american_scientific_journal-1433 133 43 written write VERB american_scientific_journal-1433 133 44 as as ADP american_scientific_journal-1433 133 45 ( ( PUNCT american_scientific_journal-1433 133 46 2.34 2.34 NUM american_scientific_journal-1433 133 47 ) ) PUNCT american_scientific_journal-1433 133 48 π»π»β„Žπ‘–π‘– π»π»β„Žπ‘–π‘– PROPN american_scientific_journal-1433 134 1 = = PUNCT american_scientific_journal-1433 135 1 𝐹𝐹2𝑅𝑅(π›Ώπ›Ώβ„Žπ‘–π‘– 𝐹𝐹2𝑅𝑅(π›Ώπ›Ώβ„Žπ‘–π‘– NOUN american_scientific_journal-1433 135 2 βˆ’ βˆ’ NOUN american_scientific_journal-1433 135 3 Ι­ Ι­ PRON american_scientific_journal-1433 135 4 𝑖𝑖ɭ 𝑖𝑖ɭ NOUN american_scientific_journal-1433 135 5 β„Ž β„Ž PROPN american_scientific_journal-1433 135 6 ) ) PUNCT american_scientific_journal-1433 135 7 . . PUNCT american_scientific_journal-1433 136 1 therefore therefore ADV american_scientific_journal-1433 136 2 the the DET american_scientific_journal-1433 136 3 space space NOUN american_scientific_journal-1433 136 4 is be AUX american_scientific_journal-1433 136 5 a a DET american_scientific_journal-1433 136 6 finsler finsler NOUN american_scientific_journal-1433 136 7 space space NOUN american_scientific_journal-1433 136 8 of of ADP american_scientific_journal-1433 136 9 scalar scalar ADJ american_scientific_journal-1433 136 10 curvature curvature NOUN american_scientific_journal-1433 136 11 thus thus ADV american_scientific_journal-1433 136 12 , , PUNCT american_scientific_journal-1433 136 13 we we PRON american_scientific_journal-1433 136 14 have have AUX american_scientific_journal-1433 136 15 theorem theorem VERB american_scientific_journal-1433 136 16 2.9 2.9 NUM american_scientific_journal-1433 136 17 . . PUNCT american_scientific_journal-1433 137 1 an an DET american_scientific_journal-1433 137 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 137 3 βˆ’ βˆ’ NOUN american_scientific_journal-1433 137 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 137 5 βˆ’ βˆ’ NUM american_scientific_journal-1433 137 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 137 7 , , PUNCT american_scientific_journal-1433 137 8 for for ADP american_scientific_journal-1433 137 9 ( ( PUNCT american_scientific_journal-1433 137 10 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-1433 137 11 > > X american_scientific_journal-1433 137 12 2 2 X american_scientific_journal-1433 137 13 ) ) PUNCT american_scientific_journal-1433 137 14 admitting admit VERB american_scientific_journal-1433 137 15 ( ( PUNCT american_scientific_journal-1433 137 16 β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 137 17 + + CCONJ american_scientific_journal-1433 137 18 π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– X american_scientific_journal-1433 137 19 ) ) PUNCT american_scientific_journal-1433 138 1 βˆ’ βˆ’ PROPN american_scientific_journal-1433 138 2 ( ( PUNCT american_scientific_journal-1433 138 3 β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ ADJ american_scientific_journal-1433 138 4 βˆ’ βˆ’ PROPN american_scientific_journal-1433 138 5 π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 138 6 ) ) PUNCT american_scientific_journal-1433 138 7 βˆ’ βˆ’ PROPN american_scientific_journal-1433 138 8 ( ( PUNCT american_scientific_journal-1433 138 9 β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– NOUN american_scientific_journal-1433 138 10 ) ) PUNCT american_scientific_journal-1433 139 1 πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 139 2 + + CCONJ american_scientific_journal-1433 139 3 ( ( PUNCT american_scientific_journal-1433 139 4 β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– NOUN american_scientific_journal-1433 139 5 βˆ’ βˆ’ NOUN american_scientific_journal-1433 139 6 ( ( PUNCT american_scientific_journal-1433 139 7 β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 139 8 = = SYM american_scientific_journal-1433 139 9 0 0 NUM american_scientific_journal-1433 139 10 is be AUX american_scientific_journal-1433 139 11 a a DET american_scientific_journal-1433 139 12 finsler finsler NOUN american_scientific_journal-1433 139 13 space space NOUN american_scientific_journal-1433 139 14 of of ADP american_scientific_journal-1433 139 15 scalar scalar ADJ american_scientific_journal-1433 139 16 curvature curvature NOUN american_scientific_journal-1433 139 17 provided provide VERB american_scientific_journal-1433 139 18 𝑅𝑅 𝑅𝑅 PROPN american_scientific_journal-1433 139 19 β‰ 0 β‰ 0 NOUN american_scientific_journal-1433 139 20 and and CCONJ american_scientific_journal-1433 139 21 the the DET american_scientific_journal-1433 139 22 covariant covariant PROPN american_scientific_journal-1433 139 23 tensor tensor NOUN american_scientific_journal-1433 139 24 filed file VERB american_scientific_journal-1433 139 25 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 139 26 is be AUX american_scientific_journal-1433 139 27 not not PART american_scientific_journal-1433 139 28 independent independent ADJ american_scientific_journal-1433 139 29 of of ADP american_scientific_journal-1433 139 30 the the DET american_scientific_journal-1433 139 31 directional directional ADJ american_scientific_journal-1433 139 32 arguments argument NOUN american_scientific_journal-1433 139 33 . . PUNCT american_scientific_journal-1433 140 1 3 3 X american_scientific_journal-1433 140 2 . . X american_scientific_journal-1433 140 3 conclusion conclusion NOUN american_scientific_journal-1433 140 4 1 1 NUM american_scientific_journal-1433 140 5 . . PUNCT american_scientific_journal-1433 141 1 the the DET american_scientific_journal-1433 141 2 space space NOUN american_scientific_journal-1433 141 3 whose whose DET american_scientific_journal-1433 141 4 defined define VERB american_scientific_journal-1433 141 5 by by ADP american_scientific_journal-1433 141 6 condition condition NOUN american_scientific_journal-1433 141 7 ( ( PUNCT american_scientific_journal-1433 141 8 2.3 2.3 NUM american_scientific_journal-1433 141 9 ) ) PUNCT american_scientific_journal-1433 141 10 is be AUX american_scientific_journal-1433 141 11 called call VERB american_scientific_journal-1433 141 12 generalized generalized ADJ american_scientific_journal-1433 141 13 𝛽𝛽𝑅𝑅 𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 141 14 βˆ’ βˆ’ NOUN american_scientific_journal-1433 141 15 birecurrent birecurrent ADJ american_scientific_journal-1433 141 16 finsler finsler NOUN american_scientific_journal-1433 141 17 space space NOUN american_scientific_journal-1433 141 18 . . PUNCT american_scientific_journal-1433 142 1 2 2 X american_scientific_journal-1433 142 2 . . X american_scientific_journal-1433 142 3 in in ADP american_scientific_journal-1433 142 4 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 142 5 βˆ’ βˆ’ PROPN american_scientific_journal-1433 142 6 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 142 7 βˆ’ βˆ’ NUM american_scientific_journal-1433 142 8 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 142 9 , , PUNCT american_scientific_journal-1433 142 10 berwald berwald ADJ american_scientific_journal-1433 142 11 second second ADJ american_scientific_journal-1433 142 12 covariant covariant ADJ american_scientific_journal-1433 142 13 derivative derivative NOUN american_scientific_journal-1433 142 14 of of ADP american_scientific_journal-1433 142 15 the the DET american_scientific_journal-1433 142 16 β„Ž(𝑣𝑣 β„Ž(𝑣𝑣 NUM american_scientific_journal-1433 142 17 ) ) PUNCT american_scientific_journal-1433 142 18 βˆ’torsion βˆ’torsion NOUN american_scientific_journal-1433 142 19 tensor tensor NOUN american_scientific_journal-1433 142 20 π»π»π‘—π‘—β„Žπ‘–π‘– π»π»π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 142 21 and and CCONJ american_scientific_journal-1433 142 22 the the DET american_scientific_journal-1433 142 23 deviation deviation NOUN american_scientific_journal-1433 142 24 tensor tensor NOUN american_scientific_journal-1433 142 25 π»π»β„Žπ‘–π‘– π»π»β„Žπ‘–π‘– PROPN american_scientific_journal-1433 142 26 is be AUX american_scientific_journal-1433 142 27 given give VERB american_scientific_journal-1433 142 28 by by ADP american_scientific_journal-1433 142 29 the the DET american_scientific_journal-1433 142 30 equations equation NOUN american_scientific_journal-1433 142 31 ( ( PUNCT american_scientific_journal-1433 142 32 2.4 2.4 NUM american_scientific_journal-1433 142 33 ) ) PUNCT american_scientific_journal-1433 142 34 and and CCONJ american_scientific_journal-1433 142 35 ( ( PUNCT american_scientific_journal-1433 142 36 2.5 2.5 NUM american_scientific_journal-1433 142 37 ) ) PUNCT american_scientific_journal-1433 142 38 , , PUNCT american_scientific_journal-1433 142 39 respectively respectively ADV american_scientific_journal-1433 142 40 . . PUNCT american_scientific_journal-1433 143 1 3 3 X american_scientific_journal-1433 143 2 . . X american_scientific_journal-1433 143 3 in in ADP american_scientific_journal-1433 143 4 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 143 5 βˆ’ βˆ’ PROPN american_scientific_journal-1433 143 6 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 143 7 βˆ’ βˆ’ NUM american_scientific_journal-1433 143 8 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 143 9 , , PUNCT american_scientific_journal-1433 143 10 the the DET american_scientific_journal-1433 143 11 𝑅𝑅 𝑅𝑅 PROPN american_scientific_journal-1433 143 12 βˆ’ βˆ’ PROPN american_scientific_journal-1433 143 13 ricci ricci PROPN american_scientific_journal-1433 143 14 tensor tensor NOUN american_scientific_journal-1433 143 15 𝑅𝑅𝑗𝑗𝑗𝑗 𝑅𝑅𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 143 16 , , PUNCT american_scientific_journal-1433 143 17 the the DET american_scientific_journal-1433 143 18 curvature curvature NOUN american_scientific_journal-1433 143 19 vector vector NOUN american_scientific_journal-1433 143 20 𝐻𝐻𝑗𝑗 𝐻𝐻𝑗𝑗 PROPN american_scientific_journal-1433 143 21 and and CCONJ american_scientific_journal-1433 143 22 the the DET american_scientific_journal-1433 143 23 scalar scalar ADJ american_scientific_journal-1433 143 24 curvature curvature NOUN american_scientific_journal-1433 143 25 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 143 26 are be AUX american_scientific_journal-1433 143 27 non non ADJ american_scientific_journal-1433 143 28 - - ADJ american_scientific_journal-1433 143 29 vanishing vanishing ADJ american_scientific_journal-1433 143 30 . . PUNCT american_scientific_journal-1433 144 1 4 4 X american_scientific_journal-1433 144 2 . . X american_scientific_journal-1433 144 3 in in ADP american_scientific_journal-1433 144 4 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 144 5 βˆ’ βˆ’ PROPN american_scientific_journal-1433 144 6 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 144 7 βˆ’ βˆ’ NUM american_scientific_journal-1433 144 8 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 NOUN american_scientific_journal-1433 144 9 , , PUNCT american_scientific_journal-1433 144 10 the the DET american_scientific_journal-1433 144 11 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 144 12 βˆ’ricci βˆ’ricci NOUN american_scientific_journal-1433 144 13 tensor tensor NOUN american_scientific_journal-1433 144 14 𝐻𝐻𝑗𝑗𝑗𝑗 𝐻𝐻𝑗𝑗𝑗𝑗 PROPN american_scientific_journal-1433 144 15 is be AUX american_scientific_journal-1433 144 16 given give VERB american_scientific_journal-1433 144 17 by by ADP american_scientific_journal-1433 144 18 equation equation NOUN american_scientific_journal-1433 144 19 ( ( PUNCT american_scientific_journal-1433 144 20 2.12 2.12 NUM american_scientific_journal-1433 144 21 ) ) PUNCT american_scientific_journal-1433 144 22 if if SCONJ american_scientific_journal-1433 145 1 and and CCONJ american_scientific_journal-1433 145 2 only only ADV american_scientific_journal-1433 145 3 if if SCONJ american_scientific_journal-1433 145 4 ( ( PUNCT american_scientific_journal-1433 145 5 2.13 2.13 NUM american_scientific_journal-1433 145 6 ) ) PUNCT american_scientific_journal-1433 145 7 holds hold VERB american_scientific_journal-1433 145 8 good good ADJ american_scientific_journal-1433 145 9 . . PUNCT american_scientific_journal-1433 146 1 5 5 X american_scientific_journal-1433 146 2 . . X american_scientific_journal-1433 146 3 in in ADP american_scientific_journal-1433 146 4 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 146 5 βˆ’ βˆ’ PROPN american_scientific_journal-1433 146 6 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 146 7 βˆ’ βˆ’ NUM american_scientific_journal-1433 146 8 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 NOUN american_scientific_journal-1433 146 9 , , PUNCT american_scientific_journal-1433 146 10 the the DET american_scientific_journal-1433 146 11 covariant covariant ADJ american_scientific_journal-1433 146 12 tensor tensor NOUN american_scientific_journal-1433 146 13 field field NOUN american_scientific_journal-1433 146 14 π‘π‘π‘šπ‘šπ‘›π‘› π‘π‘π‘šπ‘šπ‘›π‘› NOUN american_scientific_journal-1433 146 15 is be AUX american_scientific_journal-1433 146 16 independent independent ADJ american_scientific_journal-1433 146 17 of of ADP american_scientific_journal-1433 146 18 the the DET american_scientific_journal-1433 146 19 directional directional ADJ american_scientific_journal-1433 146 20 arguments argument NOUN american_scientific_journal-1433 146 21 if if SCONJ american_scientific_journal-1433 147 1 and and CCONJ american_scientific_journal-1433 147 2 only only ADV american_scientific_journal-1433 147 3 if if SCONJ american_scientific_journal-1433 147 4 the the DET american_scientific_journal-1433 147 5 covariant covariant PROPN american_scientific_journal-1433 147 6 tensor tensor NOUN american_scientific_journal-1433 147 7 field field NOUN american_scientific_journal-1433 147 8 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 147 9 is be AUX american_scientific_journal-1433 147 10 independent independent ADJ american_scientific_journal-1433 147 11 of of ADP american_scientific_journal-1433 147 12 the the DET american_scientific_journal-1433 147 13 directional directional ADJ american_scientific_journal-1433 147 14 arguments argument NOUN american_scientific_journal-1433 147 15 provided provide VERB american_scientific_journal-1433 147 16 ( ( PUNCT american_scientific_journal-1433 147 17 β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ β„¬π‘Ÿπ‘Ÿπ»π»)πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 147 18 = = SYM american_scientific_journal-1433 147 19 0 0 NUM american_scientific_journal-1433 147 20 . . NOUN american_scientific_journal-1433 148 1 6 6 NUM american_scientific_journal-1433 148 2 . . X american_scientific_journal-1433 149 1 in in ADP american_scientific_journal-1433 149 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 149 3 βˆ’ βˆ’ PROPN american_scientific_journal-1433 149 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 149 5 βˆ’ βˆ’ NUM american_scientific_journal-1433 149 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 149 7 , , PUNCT american_scientific_journal-1433 149 8 which which PRON american_scientific_journal-1433 149 9 the the DET american_scientific_journal-1433 149 10 covariant covariant ADJ american_scientific_journal-1433 149 11 tensor tensor NOUN american_scientific_journal-1433 149 12 field field NOUN american_scientific_journal-1433 149 13 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 149 14 is be AUX american_scientific_journal-1433 149 15 not not PART american_scientific_journal-1433 149 16 independent independent ADJ american_scientific_journal-1433 149 17 of of ADP american_scientific_journal-1433 149 18 the the DET american_scientific_journal-1433 149 19 directional directional ADJ american_scientific_journal-1433 149 20 argument argument NOUN american_scientific_journal-1433 149 21 at at ADP american_scientific_journal-1433 149 22 least least ADJ american_scientific_journal-1433 149 23 one one NUM american_scientific_journal-1433 149 24 of of ADP american_scientific_journal-1433 149 25 the the DET american_scientific_journal-1433 149 26 conditions condition NOUN american_scientific_journal-1433 149 27 ( ( PUNCT american_scientific_journal-1433 149 28 2.21a 2.21a NUM american_scientific_journal-1433 149 29 ) ) PUNCT american_scientific_journal-1433 149 30 and and CCONJ american_scientific_journal-1433 149 31 ( ( PUNCT american_scientific_journal-1433 149 32 2.21b 2.21b NUM american_scientific_journal-1433 149 33 ) ) PUNCT american_scientific_journal-1433 149 34 hold hold NOUN american_scientific_journal-1433 149 35 . . PUNCT american_scientific_journal-1433 150 1 7 7 X american_scientific_journal-1433 150 2 . . X american_scientific_journal-1433 150 3 an an DET american_scientific_journal-1433 150 4 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NOUN american_scientific_journal-1433 150 5 βˆ’ βˆ’ NOUN american_scientific_journal-1433 150 6 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 NOUN american_scientific_journal-1433 150 7 βˆ’ βˆ’ PROPN american_scientific_journal-1433 150 8 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 150 9 is be AUX american_scientific_journal-1433 150 10 landsberg landsberg PROPN american_scientific_journal-1433 150 11 space space NOUN american_scientific_journal-1433 150 12 if if SCONJ american_scientific_journal-1433 150 13 condition condition NOUN american_scientific_journal-1433 150 14 ( ( PUNCT american_scientific_journal-1433 150 15 2.21b 2.21b NUM american_scientific_journal-1433 150 16 ) ) PUNCT american_scientific_journal-1433 150 17 holds hold VERB american_scientific_journal-1433 150 18 and and CCONJ american_scientific_journal-1433 150 19 provided provide VERB american_scientific_journal-1433 150 20 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 150 21 οΏ½ οΏ½ PROPN american_scientific_journal-1433 150 22 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 150 23 οΏ½ οΏ½ NOUN american_scientific_journal-1433 150 24 π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 150 25 = = SYM american_scientific_journal-1433 150 26 0 0 NUM american_scientific_journal-1433 150 27 8 8 NUM american_scientific_journal-1433 150 28 . . PUNCT american_scientific_journal-1433 151 1 an an DET american_scientific_journal-1433 151 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 151 3 βˆ’ βˆ’ NOUN american_scientific_journal-1433 151 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 NOUN american_scientific_journal-1433 151 5 βˆ’ βˆ’ PROPN american_scientific_journal-1433 151 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 151 7 is be AUX american_scientific_journal-1433 151 8 necessarily necessarily ADV american_scientific_journal-1433 151 9 landsberg landsberg ADJ american_scientific_journal-1433 151 10 space space NOUN american_scientific_journal-1433 151 11 provided provide VERB american_scientific_journal-1433 151 12 π‘Žπ‘Žπ‘—π‘—π‘›π‘› π‘Žπ‘Žπ‘—π‘—π‘›π‘› NOUN american_scientific_journal-1433 151 13 β‰  β‰  PROPN american_scientific_journal-1433 151 14 οΏ½ οΏ½ PROPN american_scientific_journal-1433 151 15 Μ‡ Μ‡ PROPN american_scientific_journal-1433 151 16 οΏ½ οΏ½ NOUN american_scientific_journal-1433 151 17 πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› πœ•π‘—π‘—π‘Žπ‘Žπ‘›π‘› NOUN american_scientific_journal-1433 151 18 and and CCONJ american_scientific_journal-1433 151 19 𝐻𝐻 𝐻𝐻 PROPN american_scientific_journal-1433 151 20 οΏ½ οΏ½ PROPN american_scientific_journal-1433 151 21 β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘—π‘—π‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 151 22 οΏ½ οΏ½ NOUN american_scientific_journal-1433 151 23 π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š π‘¦π‘¦π‘Ÿπ‘Ÿπ‘¦π‘¦π‘šπ‘š NOUN american_scientific_journal-1433 151 24 = = SYM american_scientific_journal-1433 151 25 0 0 NUM american_scientific_journal-1433 151 26 9 9 NUM american_scientific_journal-1433 151 27 . . PUNCT american_scientific_journal-1433 152 1 in in ADP american_scientific_journal-1433 152 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 152 3 βˆ’ βˆ’ PROPN american_scientific_journal-1433 152 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 152 5 βˆ’ βˆ’ NUM american_scientific_journal-1433 152 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 NOUN american_scientific_journal-1433 152 7 , , PUNCT american_scientific_journal-1433 152 8 the the DET american_scientific_journal-1433 152 9 berwald berwald NOUN american_scientific_journal-1433 152 10 curvature curvature NOUN american_scientific_journal-1433 152 11 tensor tensor NOUN american_scientific_journal-1433 152 12 π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– π»π»π‘—π‘—π‘—π‘—β„Žπ‘–π‘– PROPN american_scientific_journal-1433 152 13 is be AUX american_scientific_journal-1433 152 14 non non ADJ american_scientific_journal-1433 152 15 - - ADJ american_scientific_journal-1433 152 16 vanishing vanishing ADJ american_scientific_journal-1433 152 17 if if SCONJ american_scientific_journal-1433 152 18 and and CCONJ american_scientific_journal-1433 152 19 only only ADV american_scientific_journal-1433 152 20 if if SCONJ american_scientific_journal-1433 152 21 ( ( PUNCT american_scientific_journal-1433 152 22 2 2 NUM american_scientific_journal-1433 152 23 . . NOUN american_scientific_journal-1433 152 24 29 29 NUM american_scientific_journal-1433 152 25 ) ) PUNCT american_scientific_journal-1433 152 26 holds hold VERB american_scientific_journal-1433 152 27 good good ADJ american_scientific_journal-1433 152 28 . . PUNCT american_scientific_journal-1433 153 1 10 10 NUM american_scientific_journal-1433 153 2 . . PUNCT american_scientific_journal-1433 154 1 an an DET american_scientific_journal-1433 154 2 𝐺𝐺𝛽𝛽𝑅𝑅 𝐺𝐺𝛽𝛽𝑅𝑅 NUM american_scientific_journal-1433 154 3 βˆ’ βˆ’ NOUN american_scientific_journal-1433 154 4 𝐡𝐡𝑅𝑅 𝐡𝐡𝑅𝑅 PROPN american_scientific_journal-1433 154 5 βˆ’ βˆ’ NUM american_scientific_journal-1433 154 6 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 PROPN american_scientific_journal-1433 154 7 , , PUNCT american_scientific_journal-1433 154 8 for for ADP american_scientific_journal-1433 154 9 ( ( PUNCT american_scientific_journal-1433 154 10 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-1433 154 11 > > X american_scientific_journal-1433 154 12 2 2 X american_scientific_journal-1433 154 13 ) ) PUNCT american_scientific_journal-1433 154 14 admitting admit VERB american_scientific_journal-1433 154 15 ( ( PUNCT american_scientific_journal-1433 154 16 β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘šπ‘šπ»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– PROPN american_scientific_journal-1433 154 17 + + CCONJ american_scientific_journal-1433 154 18 π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– π»π»β„Žπ‘Ÿπ‘Ÿ(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›π‘Ÿπ‘Ÿπ‘–π‘– X american_scientific_journal-1433 154 19 ) ) PUNCT american_scientific_journal-1433 155 1 βˆ’ βˆ’ PROPN american_scientific_journal-1433 155 2 ( ( PUNCT american_scientific_journal-1433 155 3 β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ β„¬π‘šπ‘šπ»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ ADJ american_scientific_journal-1433 155 4 βˆ’ βˆ’ PROPN american_scientific_journal-1433 155 5 π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ π»π»π‘Ÿπ‘Ÿπ‘–π‘–(β„¬π‘šπ‘šπΊπΊπ‘—π‘—π‘›π‘›β„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 155 6 ) ) PUNCT american_scientific_journal-1433 155 7 βˆ’ βˆ’ PROPN american_scientific_journal-1433 155 8 ( ( PUNCT american_scientific_journal-1433 155 9 β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– β„¬π‘Ÿπ‘Ÿπ»π»β„Žπ‘–π‘– NOUN american_scientific_journal-1433 155 10 ) ) PUNCT american_scientific_journal-1433 156 1 πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ πΊπΊπ‘—π‘—π‘šπ‘šπ‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-1433 156 2 + + CCONJ american_scientific_journal-1433 156 3 ( ( PUNCT american_scientific_journal-1433 156 4 β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– β„¬π‘›π‘›π»π»β„Žπ‘Ÿπ‘Ÿ)πΊπΊπ‘—π‘—π‘šπ‘šπ‘Ÿπ‘Ÿπ‘–π‘– NOUN american_scientific_journal-1433 156 5 βˆ’ βˆ’ NOUN american_scientific_journal-1433 156 6 ( ( PUNCT american_scientific_journal-1433 156 7 β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ β„¬π‘›π‘›π»π»π‘Ÿπ‘Ÿπ‘–π‘–)πΊπΊπ‘—π‘—π‘šπ‘šβ„Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-1433 156 8 = = SYM american_scientific_journal-1433 156 9 0 0 NUM american_scientific_journal-1433 156 10 is be AUX american_scientific_journal-1433 156 11 a a DET american_scientific_journal-1433 156 12 finsler finsler NOUN american_scientific_journal-1433 156 13 space space NOUN american_scientific_journal-1433 156 14 of of ADP american_scientific_journal-1433 156 15 scalar scalar ADJ american_scientific_journal-1433 156 16 curvature curvature NOUN american_scientific_journal-1433 156 17 provided provide VERB american_scientific_journal-1433 156 18 𝑅𝑅 𝑅𝑅 PROPN american_scientific_journal-1433 156 19 β‰ 0 β‰ 0 NOUN american_scientific_journal-1433 156 20 and and CCONJ american_scientific_journal-1433 156 21 the the DET american_scientific_journal-1433 156 22 covariant covariant PROPN american_scientific_journal-1433 156 23 tensor tensor NOUN american_scientific_journal-1433 156 24 filed file VERB american_scientific_journal-1433 156 25 π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› π‘Žπ‘Žπ‘šπ‘šπ‘›π‘› PROPN american_scientific_journal-1433 156 26 is be AUX american_scientific_journal-1433 156 27 not not PART american_scientific_journal-1433 156 28 independent independent ADJ american_scientific_journal-1433 156 29 of of ADP american_scientific_journal-1433 156 30 the the DET american_scientific_journal-1433 156 31 directional directional ADJ american_scientific_journal-1433 156 32 arguments argument NOUN american_scientific_journal-1433 156 33 . . PUNCT american_scientific_journal-1433 157 1 4 4 X american_scientific_journal-1433 157 2 . . X american_scientific_journal-1433 157 3 recommendations recommendation NOUN american_scientific_journal-1433 157 4 the the DET american_scientific_journal-1433 157 5 authors author NOUN american_scientific_journal-1433 157 6 recommend recommend VERB american_scientific_journal-1433 157 7 the the DET american_scientific_journal-1433 157 8 research research NOUN american_scientific_journal-1433 157 9 should should AUX american_scientific_journal-1433 157 10 be be AUX american_scientific_journal-1433 157 11 continued continue VERB american_scientific_journal-1433 157 12 in in ADP american_scientific_journal-1433 157 13 the the DET american_scientific_journal-1433 157 14 finsler finsler NOUN american_scientific_journal-1433 157 15 spaces space NOUN american_scientific_journal-1433 157 16 because because SCONJ american_scientific_journal-1433 157 17 it it PRON american_scientific_journal-1433 157 18 has have VERB american_scientific_journal-1433 157 19 many many ADJ american_scientific_journal-1433 157 20 applications application NOUN american_scientific_journal-1433 157 21 in in ADP american_scientific_journal-1433 157 22 biology biology NOUN american_scientific_journal-1433 157 23 , , PUNCT american_scientific_journal-1433 157 24 relativity relativity NOUN american_scientific_journal-1433 157 25 physics physics NOUN american_scientific_journal-1433 157 26 and and CCONJ american_scientific_journal-1433 157 27 other other ADJ american_scientific_journal-1433 157 28 fields field NOUN american_scientific_journal-1433 157 29 . . PUNCT american_scientific_journal-1433 158 1 american american ADJ american_scientific_journal-1433 158 2 scientific scientific ADJ american_scientific_journal-1433 158 3 research research PROPN american_scientific_journal-1433 158 4 journal journal NOUN american_scientific_journal-1433 158 5 for for ADP american_scientific_journal-1433 158 6 engineering engineering NOUN american_scientific_journal-1433 158 7 , , PUNCT american_scientific_journal-1433 158 8 technology technology NOUN american_scientific_journal-1433 158 9 , , PUNCT american_scientific_journal-1433 158 10 and and CCONJ american_scientific_journal-1433 158 11 sciences science NOUN american_scientific_journal-1433 158 12 ( ( PUNCT american_scientific_journal-1433 158 13 asrjets asrjet NOUN american_scientific_journal-1433 158 14 ) ) PUNCT american_scientific_journal-1433 158 15 ( ( PUNCT american_scientific_journal-1433 158 16 2016 2016 NUM american_scientific_journal-1433 158 17 ) ) PUNCT american_scientific_journal-1433 158 18 volume volume NOUN american_scientific_journal-1433 158 19 19 19 NUM american_scientific_journal-1433 158 20 , , PUNCT american_scientific_journal-1433 158 21 no no DET american_scientific_journal-1433 158 22 1 1 NUM american_scientific_journal-1433 158 23 , , PUNCT american_scientific_journal-1433 158 24 pp pp ADV american_scientific_journal-1433 158 25 9 9 NUM american_scientific_journal-1433 158 26 - - SYM american_scientific_journal-1433 158 27 18 18 NUM american_scientific_journal-1433 158 28 18 18 NUM american_scientific_journal-1433 158 29 references reference NOUN american_scientific_journal-1433 158 30 [ [ X american_scientific_journal-1433 158 31 1 1 NUM american_scientific_journal-1433 158 32 ] ] PUNCT american_scientific_journal-1433 158 33 . . PUNCT american_scientific_journal-1433 159 1 a.g a.g PROPN american_scientific_journal-1433 159 2 . . PROPN american_scientific_journal-1433 159 3 walker walker PROPN american_scientific_journal-1433 159 4 . . PUNCT american_scientific_journal-1433 160 1 on on ADP american_scientific_journal-1433 160 2 ruse ruse NOUN american_scientific_journal-1433 160 3 's 's PART american_scientific_journal-1433 160 4 space space NOUN american_scientific_journal-1433 160 5 of of ADP american_scientific_journal-1433 160 6 recurrent recurrent ADJ american_scientific_journal-1433 160 7 curvature curvature NOUN american_scientific_journal-1433 160 8 , , PUNCT american_scientific_journal-1433 160 9 proc proc NOUN american_scientific_journal-1433 160 10 . . PUNCT american_scientific_journal-1433 161 1 land land NOUN american_scientific_journal-1433 161 2 math math PROPN american_scientific_journal-1433 161 3 . . PUNCT american_scientific_journal-1433 162 1 soc soc PROPN american_scientific_journal-1433 162 2 . . PUNCT american_scientific_journal-1433 162 3 , , PUNCT american_scientific_journal-1433 162 4 52 52 NUM american_scientific_journal-1433 162 5 , , PUNCT american_scientific_journal-1433 162 6 1950,pp 1950,pp NUM american_scientific_journal-1433 162 7 . . PUNCT american_scientific_journal-1433 163 1 36 36 NUM american_scientific_journal-1433 163 2 - - SYM american_scientific_journal-1433 163 3 64 64 NUM american_scientific_journal-1433 163 4 . . PUNCT american_scientific_journal-1433 164 1 [ [ X american_scientific_journal-1433 164 2 2 2 NUM american_scientific_journal-1433 164 3 ] ] PUNCT american_scientific_journal-1433 164 4 . . PUNCT american_scientific_journal-1433 164 5 b.b.sinha b.b.sinha NOUN american_scientific_journal-1433 164 6 and and CCONJ american_scientific_journal-1433 164 7 s.p.singh s.p.singh NOUN american_scientific_journal-1433 164 8 '' '' PUNCT american_scientific_journal-1433 164 9 on on ADP american_scientific_journal-1433 164 10 recurrent recurrent ADJ american_scientific_journal-1433 164 11 spaces space NOUN american_scientific_journal-1433 164 12 of of ADP american_scientific_journal-1433 164 13 second second ADJ american_scientific_journal-1433 164 14 order order NOUN american_scientific_journal-1433 164 15 in in ADP american_scientific_journal-1433 164 16 finsler finsler NOUN american_scientific_journal-1433 164 17 space space NOUN american_scientific_journal-1433 164 18 . . PUNCT american_scientific_journal-1433 164 19 '' '' PUNCT american_scientific_journal-1433 165 1 yakohama yakohama PROPN american_scientific_journal-1433 165 2 math math NOUN american_scientific_journal-1433 165 3 . . PUNCT american_scientific_journal-1433 166 1 j. j. PROPN american_scientific_journal-1433 166 2 ,vol ,vol PROPN american_scientific_journal-1433 166 3 . . PUNCT american_scientific_journal-1433 167 1 18,pp 18,pp NUM american_scientific_journal-1433 167 2 . . PUNCT american_scientific_journal-1433 168 1 27 27 NUM american_scientific_journal-1433 168 2 - - SYM american_scientific_journal-1433 168 3 32 32 NUM american_scientific_journal-1433 168 4 , , PUNCT american_scientific_journal-1433 168 5 1970 1970 NUM american_scientific_journal-1433 168 6 . . PUNCT american_scientific_journal-1433 169 1 [ [ X american_scientific_journal-1433 169 2 3 3 NUM american_scientific_journal-1433 169 3 ] ] PUNCT american_scientific_journal-1433 169 4 . . PUNCT american_scientific_journal-1433 170 1 f.y.a f.y.a NOUN american_scientific_journal-1433 170 2 .qasem .qasem NOUN american_scientific_journal-1433 170 3 and and CCONJ american_scientific_journal-1433 170 4 a.a.m.saleem a.a.m.saleem PROPN american_scientific_journal-1433 170 5 '' '' PUNCT american_scientific_journal-1433 170 6 on on ADP american_scientific_journal-1433 170 7 ubirecurrent ubirecurrent ADJ american_scientific_journal-1433 170 8 finsler finsler NOUN american_scientific_journal-1433 170 9 space space NOUN american_scientific_journal-1433 170 10 . . PUNCT american_scientific_journal-1433 170 11 '' '' PUNCT american_scientific_journal-1433 171 1 univ univ PROPN american_scientific_journal-1433 171 2 . . PUNCT american_scientific_journal-1433 172 1 aden aden PROPN american_scientific_journal-1433 172 2 j. j. PROPN american_scientific_journal-1433 172 3 nat nat PROPN american_scientific_journal-1433 172 4 . . PUNCT american_scientific_journal-1433 173 1 and and CCONJ american_scientific_journal-1433 173 2 appl appl NOUN american_scientific_journal-1433 173 3 . . PUNCT american_scientific_journal-1433 174 1 sc.vol.14 sc.vol.14 PROPN american_scientific_journal-1433 174 2 no.3 no.3 PROPN american_scientific_journal-1433 174 3 , , PUNCT american_scientific_journal-1433 174 4 december december PROPN american_scientific_journal-1433 174 5 . . PUNCT american_scientific_journal-1433 175 1 2010 2010 NUM american_scientific_journal-1433 175 2 . . PUNCT american_scientific_journal-1433 176 1 [ [ X american_scientific_journal-1433 176 2 4 4 NUM american_scientific_journal-1433 176 3 ] ] PUNCT american_scientific_journal-1433 176 4 . . PUNCT american_scientific_journal-1433 177 1 h.s h.s PROPN american_scientific_journal-1433 177 2 .ruse .ruse NOUN american_scientific_journal-1433 177 3 . . PUNCT american_scientific_journal-1433 178 1 three three NUM american_scientific_journal-1433 178 2 dimensional dimensional ADJ american_scientific_journal-1433 178 3 spaces space NOUN american_scientific_journal-1433 178 4 of of ADP american_scientific_journal-1433 178 5 recurrent recurrent ADJ american_scientific_journal-1433 178 6 curvature curvature NOUN american_scientific_journal-1433 178 7 : : PUNCT american_scientific_journal-1433 178 8 proc proc NOUN american_scientific_journal-1433 178 9 . . PUNCT american_scientific_journal-1433 179 1 lond lond PROPN american_scientific_journal-1433 179 2 . . PUNCT american_scientific_journal-1433 180 1 math math NOUN american_scientific_journal-1433 180 2 . . PUNCT american_scientific_journal-1433 181 1 soc soc PROPN american_scientific_journal-1433 181 2 . . PUNCT american_scientific_journal-1433 181 3 , , PUNCT american_scientific_journal-1433 181 4 50 50 NUM american_scientific_journal-1433 181 5 , , PUNCT american_scientific_journal-1433 181 6 1949,pp 1949,pp NUM american_scientific_journal-1433 181 7 . . PUNCT american_scientific_journal-1433 182 1 438 438 NUM american_scientific_journal-1433 182 2 - - SYM american_scientific_journal-1433 182 3 446 446 NUM american_scientific_journal-1433 182 4 . . PUNCT american_scientific_journal-1433 183 1 [ [ X american_scientific_journal-1433 183 2 5 5 NUM american_scientific_journal-1433 183 3 ] ] PUNCT american_scientific_journal-1433 183 4 . . PUNCT american_scientific_journal-1433 184 1 p. p. NOUN american_scientific_journal-1433 184 2 n. n. PROPN american_scientific_journal-1433 184 3 pandey pandey PROPN american_scientific_journal-1433 184 4 . . PUNCT american_scientific_journal-1433 185 1 a a DET american_scientific_journal-1433 185 2 recurrent recurrent ADJ american_scientific_journal-1433 185 3 finsler finsler NOUN american_scientific_journal-1433 185 4 manifold manifold NOUN american_scientific_journal-1433 185 5 admitting admit VERB american_scientific_journal-1433 185 6 special special ADJ american_scientific_journal-1433 185 7 recurrent recurrent ADJ american_scientific_journal-1433 185 8 transformation transformation NOUN american_scientific_journal-1433 185 9 : : PUNCT american_scientific_journal-1433 185 10 prog prog PROPN american_scientific_journal-1433 185 11 . . PUNCT american_scientific_journal-1433 186 1 math math NOUN american_scientific_journal-1433 186 2 . . PUNCT american_scientific_journal-1433 187 1 ,13 ,13 PUNCT american_scientific_journal-1433 187 2 ( ( PUNCT american_scientific_journal-1433 187 3 1 1 NUM american_scientific_journal-1433 187 4 - - SYM american_scientific_journal-1433 187 5 2 2 NUM american_scientific_journal-1433 187 6 ) ) PUNCT american_scientific_journal-1433 187 7 , , PUNCT american_scientific_journal-1433 187 8 1979 1979 NUM american_scientific_journal-1433 187 9 , , PUNCT american_scientific_journal-1433 187 10 pp pp ADJ american_scientific_journal-1433 187 11 . . PUNCT american_scientific_journal-1433 188 1 85 85 NUM american_scientific_journal-1433 188 2 - - SYM american_scientific_journal-1433 188 3 98 98 NUM american_scientific_journal-1433 188 4 . . PUNCT american_scientific_journal-1433 189 1 [ [ X american_scientific_journal-1433 189 2 6 6 NUM american_scientific_journal-1433 189 3 ] ] PUNCT american_scientific_journal-1433 189 4 . . PUNCT american_scientific_journal-1433 190 1 p. p. NOUN american_scientific_journal-1433 190 2 n. n. PROPN american_scientific_journal-1433 190 3 pandey pandey PROPN american_scientific_journal-1433 190 4 . . PUNCT american_scientific_journal-1433 191 1 a a DET american_scientific_journal-1433 191 2 note note NOUN american_scientific_journal-1433 191 3 on on ADP american_scientific_journal-1433 191 4 recurrence recurrence NOUN american_scientific_journal-1433 191 5 vector vector NOUN american_scientific_journal-1433 191 6 : : PUNCT american_scientific_journal-1433 191 7 proc proc NOUN american_scientific_journal-1433 191 8 . . PUNCT american_scientific_journal-1433 192 1 nat nat PROPN american_scientific_journal-1433 192 2 . . PUNCT american_scientific_journal-1433 193 1 acad acad PROPN american_scientific_journal-1433 193 2 . . PUNCT american_scientific_journal-1433 194 1 sci sci PROPN american_scientific_journal-1433 194 2 . . PROPN american_scientific_journal-1433 194 3 51a 51a PROPN american_scientific_journal-1433 194 4 , , PUNCT american_scientific_journal-1433 194 5 i i PRON american_scientific_journal-1433 194 6 , , PUNCT american_scientific_journal-1433 194 7 1981,pp 1981,pp NUM american_scientific_journal-1433 194 8 . . PUNCT american_scientific_journal-1433 195 1 6 6 NUM american_scientific_journal-1433 195 2 - - SYM american_scientific_journal-1433 195 3 8 8 NUM american_scientific_journal-1433 195 4 . . PUNCT american_scientific_journal-1433 196 1 [ [ X american_scientific_journal-1433 196 2 7 7 NUM american_scientific_journal-1433 196 3 ] ] PUNCT american_scientific_journal-1433 196 4 . . PUNCT american_scientific_journal-1433 197 1 p. p. NOUN american_scientific_journal-1433 197 2 n. n. PROPN american_scientific_journal-1433 197 3 pandey pandey PROPN american_scientific_journal-1433 197 4 . . PUNCT american_scientific_journal-1433 197 5 '' '' PUNCT american_scientific_journal-1433 198 1 some some DET american_scientific_journal-1433 198 2 problems problem NOUN american_scientific_journal-1433 198 3 in in ADP american_scientific_journal-1433 198 4 finsler finsler NOUN american_scientific_journal-1433 198 5 spaces space NOUN american_scientific_journal-1433 198 6 . . PUNCT american_scientific_journal-1433 198 7 '' '' PUNCT american_scientific_journal-1433 199 1 d.sc d.sc NOUN american_scientific_journal-1433 199 2 . . PUNCT american_scientific_journal-1433 199 3 thesis thesis NOUN american_scientific_journal-1433 199 4 , , PUNCT american_scientific_journal-1433 199 5 university university NOUN american_scientific_journal-1433 199 6 of of ADP american_scientific_journal-1433 199 7 allahabad allahabad PROPN american_scientific_journal-1433 199 8 , , PUNCT american_scientific_journal-1433 199 9 india india PROPN american_scientific_journal-1433 199 10 , , PUNCT american_scientific_journal-1433 199 11 1993 1993 NUM american_scientific_journal-1433 199 12 . . PUNCT american_scientific_journal-1433 200 1 [ [ X american_scientific_journal-1433 200 2 8 8 NUM american_scientific_journal-1433 200 3 ] ] PUNCT american_scientific_journal-1433 200 4 . . PUNCT american_scientific_journal-1433 201 1 p.n p.n PROPN american_scientific_journal-1433 201 2 . . PROPN american_scientific_journal-1433 201 3 pandey pandey PROPN american_scientific_journal-1433 201 4 , , PUNCT american_scientific_journal-1433 201 5 s. s. PROPN american_scientific_journal-1433 201 6 saxena saxena PROPN american_scientific_journal-1433 201 7 , , PUNCT american_scientific_journal-1433 201 8 and and CCONJ american_scientific_journal-1433 201 9 a. a. NOUN american_scientific_journal-1433 201 10 goswani goswani NOUN american_scientific_journal-1433 201 11 '' '' PUNCT american_scientific_journal-1433 201 12 on on ADP american_scientific_journal-1433 201 13 a a DET american_scientific_journal-1433 201 14 generalized generalized ADJ american_scientific_journal-1433 201 15 h h ADJ american_scientific_journal-1433 201 16 - - PUNCT american_scientific_journal-1433 201 17 recurrent recurrent ADJ american_scientific_journal-1433 201 18 space space NOUN american_scientific_journal-1433 201 19 . . PUNCT american_scientific_journal-1433 201 20 '' '' PUNCT american_scientific_journal-1433 202 1 journal journal PROPN american_scientific_journal-1433 202 2 of of ADP american_scientific_journal-1433 202 3 international international PROPN american_scientific_journal-1433 202 4 academy academy PROPN american_scientific_journal-1433 202 5 of of ADP american_scientific_journal-1433 202 6 physical physical ADJ american_scientific_journal-1433 202 7 science science NOUN american_scientific_journal-1433 202 8 , , PUNCT american_scientific_journal-1433 202 9 vol vol NOUN american_scientific_journal-1433 202 10 . . PROPN american_scientific_journal-1433 202 11 15 15 NUM american_scientific_journal-1433 202 12 , , PUNCT american_scientific_journal-1433 202 13 pp pp ADJ american_scientific_journal-1433 202 14 . . PUNCT american_scientific_journal-1433 203 1 201 201 NUM american_scientific_journal-1433 203 2 - - SYM american_scientific_journal-1433 203 3 211 211 NUM american_scientific_journal-1433 203 4 , , PUNCT american_scientific_journal-1433 203 5 2011 2011 NUM american_scientific_journal-1433 203 6 . . PUNCT american_scientific_journal-1433 204 1 [ [ X american_scientific_journal-1433 204 2 9 9 NUM american_scientific_journal-1433 204 3 ] ] PUNCT american_scientific_journal-1433 204 4 . . PUNCT american_scientific_journal-1433 205 1 r r NOUN american_scientific_journal-1433 205 2 .b .b PROPN american_scientific_journal-1433 205 3 . . PUNCT american_scientific_journal-1433 206 1 misra misra ADJ american_scientific_journal-1433 206 2 . . PUNCT american_scientific_journal-1433 207 1 on on ADP american_scientific_journal-1433 207 2 a a DET american_scientific_journal-1433 207 3 recurrent recurrent ADJ american_scientific_journal-1433 207 4 finsler finsler NOUN american_scientific_journal-1433 207 5 spaces space NOUN american_scientific_journal-1433 207 6 : : PUNCT american_scientific_journal-1433 207 7 rev rev VERB american_scientific_journal-1433 207 8 .roumhaine .roumhaine ADP american_scientific_journal-1433 207 9 math math NOUN american_scientific_journal-1433 207 10 . . PUNCT american_scientific_journal-1433 208 1 pures pure NOUN american_scientific_journal-1433 208 2 appl.18 appl.18 PROPN american_scientific_journal-1433 208 3 , , PUNCT american_scientific_journal-1433 208 4 1973,pp.701 1973,pp.701 PROPN american_scientific_journal-1433 208 5 712 712 NUM american_scientific_journal-1433 208 6 . . PUNCT american_scientific_journal-1433 209 1 [ [ X american_scientific_journal-1433 209 2 10 10 NUM american_scientific_journal-1433 209 3 ] ] PUNCT american_scientific_journal-1433 209 4 . . PUNCT american_scientific_journal-1433 210 1 r.s.d r.s.d NOUN american_scientific_journal-1433 210 2 . . PUNCT american_scientific_journal-1433 211 1 dubey dubey PROPN american_scientific_journal-1433 211 2 and and CCONJ american_scientific_journal-1433 211 3 a.k a.k PROPN american_scientific_journal-1433 211 4 . . PROPN american_scientific_journal-1433 211 5 srivastava srivastava PROPN american_scientific_journal-1433 211 6 . . PUNCT american_scientific_journal-1433 212 1 on on ADP american_scientific_journal-1433 212 2 recurrent recurrent ADJ american_scientific_journal-1433 212 3 finsler finsler NOUN american_scientific_journal-1433 212 4 spaces space NOUN american_scientific_journal-1433 212 5 : : PUNCT american_scientific_journal-1433 212 6 bull bull NOUN american_scientific_journal-1433 212 7 . . PUNCT american_scientific_journal-1433 213 1 soc.math soc.math NOUN american_scientific_journal-1433 213 2 . . PUNCT american_scientific_journal-1433 214 1 belgique belgique PROPN american_scientific_journal-1433 214 2 , , PUNCT american_scientific_journal-1433 214 3 1981,pp 1981,pp NUM american_scientific_journal-1433 214 4 . . PUNCT american_scientific_journal-1433 215 1 283 283 NUM american_scientific_journal-1433 215 2 - - SYM american_scientific_journal-1433 215 3 288 288 NUM american_scientific_journal-1433 215 4 . . PUNCT american_scientific_journal-1433 216 1 [ [ X american_scientific_journal-1433 216 2 11 11 NUM american_scientific_journal-1433 216 3 ] ] PUNCT american_scientific_journal-1433 216 4 . . PUNCT american_scientific_journal-1433 217 1 r.s r.s PROPN american_scientific_journal-1433 217 2 . . PROPN american_scientific_journal-1433 217 3 mishra mishra PROPN american_scientific_journal-1433 217 4 and and CCONJ american_scientific_journal-1433 217 5 h.d h.d PROPN american_scientific_journal-1433 217 6 . . PROPN american_scientific_journal-1433 217 7 pande pande PROPN american_scientific_journal-1433 217 8 '' '' PUNCT american_scientific_journal-1433 217 9 recurrent recurrent ADJ american_scientific_journal-1433 217 10 finsler finsler NOUN american_scientific_journal-1433 217 11 space space NOUN american_scientific_journal-1433 217 12 . . PUNCT american_scientific_journal-1433 217 13 '' '' PUNCT american_scientific_journal-1433 218 1 j j PROPN american_scientific_journal-1433 218 2 , , PUNCT american_scientific_journal-1433 218 3 india india PROPN american_scientific_journal-1433 218 4 math math PROPN american_scientific_journal-1433 218 5 . . PUNCT american_scientific_journal-1433 219 1 soc soc PROPN american_scientific_journal-1433 219 2 . . PUNCT american_scientific_journal-1433 220 1 n.s n.s PROPN american_scientific_journal-1433 220 2 . . PROPN american_scientific_journal-1433 220 3 vol.32 vol.32 PROPN american_scientific_journal-1433 220 4 , , PUNCT american_scientific_journal-1433 220 5 pp.17 pp.17 ADJ american_scientific_journal-1433 220 6 - - SYM american_scientific_journal-1433 220 7 22 22 NUM american_scientific_journal-1433 220 8 , , PUNCT american_scientific_journal-1433 220 9 1968 1968 NUM american_scientific_journal-1433 220 10 . . PUNCT american_scientific_journal-1433 221 1 [ [ X american_scientific_journal-1433 221 2 12 12 NUM american_scientific_journal-1433 221 3 ] ] PUNCT american_scientific_journal-1433 221 4 . . PUNCT american_scientific_journal-1433 222 1 r. r. PROPN american_scientific_journal-1433 222 2 verma verma PROPN american_scientific_journal-1433 222 3 '' '' PUNCT american_scientific_journal-1433 222 4 some some DET american_scientific_journal-1433 222 5 transformations transformation NOUN american_scientific_journal-1433 222 6 in in ADP american_scientific_journal-1433 222 7 finsler finsler NOUN american_scientific_journal-1433 222 8 spaces space NOUN american_scientific_journal-1433 222 9 . . PUNCT american_scientific_journal-1433 222 10 '' '' PUNCT american_scientific_journal-1433 223 1 ph.d ph.d PROPN american_scientific_journal-1433 223 2 . . PUNCT american_scientific_journal-1433 224 1 thesis thesis NOUN american_scientific_journal-1433 224 2 , , PUNCT american_scientific_journal-1433 224 3 university university NOUN american_scientific_journal-1433 224 4 of of ADP american_scientific_journal-1433 224 5 allahabad allahabad PROPN american_scientific_journal-1433 224 6 , , PUNCT american_scientific_journal-1433 224 7 , , PUNCT american_scientific_journal-1433 224 8 india india PROPN american_scientific_journal-1433 224 9 , , PUNCT american_scientific_journal-1433 224 10 1991 1991 NUM american_scientific_journal-1433 224 11 . . PUNCT american_scientific_journal-1433 225 1 [ [ X american_scientific_journal-1433 225 2 13 13 NUM american_scientific_journal-1433 225 3 ] ] PUNCT american_scientific_journal-1433 225 4 . . PUNCT american_scientific_journal-1433 226 1 s. s. PROPN american_scientific_journal-1433 226 2 dikshit dikshit PROPN american_scientific_journal-1433 226 3 '' '' PUNCT american_scientific_journal-1433 226 4 certain certain ADJ american_scientific_journal-1433 226 5 types type NOUN american_scientific_journal-1433 226 6 of of ADP american_scientific_journal-1433 226 7 recurrences recurrence NOUN american_scientific_journal-1433 226 8 in in ADP american_scientific_journal-1433 226 9 finsler finsler NOUN american_scientific_journal-1433 226 10 spaces space NOUN american_scientific_journal-1433 226 11 . . PUNCT american_scientific_journal-1433 226 12 '' '' PUNCT american_scientific_journal-1433 227 1 ph.d ph.d PROPN american_scientific_journal-1433 227 2 . . PUNCT american_scientific_journal-1433 228 1 thesis thesis NOUN american_scientific_journal-1433 228 2 , , PUNCT american_scientific_journal-1433 228 3 university university NOUN american_scientific_journal-1433 228 4 of of ADP american_scientific_journal-1433 228 5 allahabad allahabad PROPN american_scientific_journal-1433 228 6 , , PUNCT american_scientific_journal-1433 228 7 1992 1992 NUM american_scientific_journal-1433 228 8 . . PUNCT american_scientific_journal-1433 229 1 [ [ X american_scientific_journal-1433 229 2 14 14 NUM american_scientific_journal-1433 229 3 ] ] PUNCT american_scientific_journal-1433 229 4 . . PUNCT american_scientific_journal-1433 230 1 y y PROPN american_scientific_journal-1433 230 2 .c .c PROPN american_scientific_journal-1433 230 3 . . PUNCT american_scientific_journal-1433 231 1 worg worg ADJ american_scientific_journal-1433 231 2 . . PUNCT american_scientific_journal-1433 232 1 linear linear ADJ american_scientific_journal-1433 232 2 connections connection NOUN american_scientific_journal-1433 232 3 with with ADP american_scientific_journal-1433 232 4 zero zero NUM american_scientific_journal-1433 232 5 torsion torsion NOUN american_scientific_journal-1433 232 6 and and CCONJ american_scientific_journal-1433 232 7 recurrent recurrent ADJ american_scientific_journal-1433 232 8 curvature curvature NOUN american_scientific_journal-1433 232 9 : : PUNCT american_scientific_journal-1433 232 10 trans trans PROPN american_scientific_journal-1433 232 11 . . PUNCT american_scientific_journal-1433 233 1 amer amer PROPN american_scientific_journal-1433 233 2 . . PUNCT american_scientific_journal-1433 234 1 math math NOUN american_scientific_journal-1433 234 2 . . PUNCT american_scientific_journal-1433 235 1 soc soc NOUN american_scientific_journal-1433 235 2 . . PUNCT american_scientific_journal-1433 236 1 , , PUNCT american_scientific_journal-1433 236 2 102 102 NUM american_scientific_journal-1433 236 3 , , PUNCT american_scientific_journal-1433 236 4 1962 1962 NUM american_scientific_journal-1433 236 5 , , PUNCT american_scientific_journal-1433 236 6 pp.471 pp.471 NOUN american_scientific_journal-1433 236 7 – – PUNCT american_scientific_journal-1433 236 8 506 506 NUM american_scientific_journal-1433 236 9 . . PUNCT american_scientific_journal-1433 237 1 [ [ X american_scientific_journal-1433 237 2 15 15 NUM american_scientific_journal-1433 237 3 ] ] PUNCT american_scientific_journal-1433 237 4 . . PUNCT american_scientific_journal-1433 238 1 y y PROPN american_scientific_journal-1433 238 2 .c .c PROPN american_scientific_journal-1433 239 1 .worg .worg PROPN american_scientific_journal-1433 239 2 and and CCONJ american_scientific_journal-1433 239 3 k. k. PROPN american_scientific_journal-1433 239 4 yano yano PROPN american_scientific_journal-1433 239 5 . . PUNCT american_scientific_journal-1433 240 1 projectively projectively ADV american_scientific_journal-1433 240 2 flat flat ADJ american_scientific_journal-1433 240 3 spaces space NOUN american_scientific_journal-1433 240 4 with with ADP american_scientific_journal-1433 240 5 recurrent recurrent ADJ american_scientific_journal-1433 240 6 curvature curvature NOUN american_scientific_journal-1433 240 7 : : PUNCT american_scientific_journal-1433 240 8 comment comment NOUN american_scientific_journal-1433 240 9 math math NOUN american_scientific_journal-1433 240 10 . . PUNCT american_scientific_journal-1433 241 1 helv helv PROPN american_scientific_journal-1433 241 2 .,35 .,35 PROPN american_scientific_journal-1433 241 3 , , PUNCT american_scientific_journal-1433 241 4 1961 1961 NUM american_scientific_journal-1433 241 5 , , PUNCT american_scientific_journal-1433 241 6 pp pp ADJ american_scientific_journal-1433 241 7 . . PUNCT american_scientific_journal-1433 241 8 223 223 NUM american_scientific_journal-1433 241 9 – – PUNCT american_scientific_journal-1433 241 10 232 232 NUM american_scientific_journal-1433 241 11 . . PUNCT