id sid tid token lemma pos cana-590 1 1 communications communication NOUN cana-590 1 2 on on ADP cana-590 1 3 applied apply VERB cana-590 1 4 nonlinear nonlinear ADJ cana-590 1 5 analysis analysis NOUN cana-590 1 6 issn issn NOUN cana-590 1 7 : : PUNCT cana-590 1 8 1074 1074 NUM cana-590 1 9 - - PUNCT cana-590 1 10 133x 133x NUM cana-590 1 11 vol vol NOUN cana-590 1 12 31 31 NUM cana-590 1 13 no no NOUN cana-590 1 14 . . PUNCT cana-590 2 1 2s 2s NUM cana-590 2 2 ( ( PUNCT cana-590 2 3 2024 2024 NUM cana-590 2 4 ) ) PUNCT cana-590 2 5 26 26 NUM cana-590 2 6 https://internationalpubls.com https://internationalpubls.com X cana-590 2 7 on on ADP cana-590 2 8 the the DET cana-590 2 9 geometry geometry NOUN cana-590 2 10 of of ADP cana-590 2 11 grassmannian grassmannian ADJ cana-590 2 12 manifold manifold NOUN cana-590 2 13 [ [ PUNCT cana-590 2 14 t. t. NOUN cana-590 2 15 venkatesh venkatesh PROPN cana-590 2 16 1 1 NUM cana-590 2 17 , , PUNCT cana-590 2 18 shruti shruti PROPN cana-590 2 19 kamalakar kamalakar PROPN cana-590 2 20 govekar govekar PROPN cana-590 2 21 2 2 NUM cana-590 2 22 1department 1department NUM cana-590 2 23 of of ADP cana-590 2 24 mathematics mathematic NOUN cana-590 2 25 , , PUNCT cana-590 2 26 and and CCONJ cana-590 2 27 director director NOUN cana-590 2 28 , , PUNCT cana-590 2 29 mathematical mathematical ADJ cana-590 2 30 sciences sciences PROPN cana-590 2 31 institute institute PROPN cana-590 2 32 belagavi belagavi PROPN cana-590 2 33 , , PUNCT cana-590 2 34 karnataka karnataka PROPN cana-590 2 35 , , PUNCT cana-590 2 36 india india PROPN cana-590 2 37 . . PUNCT cana-590 3 1 e e X cana-590 3 2 - - NOUN cana-590 3 3 mail mail NOUN cana-590 3 4 : : PUNCT cana-590 3 5 tmathvenky@yahoo.co.in tmathvenky@yahoo.co.in PROPN cana-590 3 6 2department 2department NUM cana-590 3 7 of of ADP cana-590 3 8 mathematics mathematic NOUN cana-590 3 9 , , PUNCT cana-590 3 10 rani rani PROPN cana-590 3 11 channamma channamma PROPN cana-590 3 12 university university PROPN cana-590 3 13 , , PUNCT cana-590 3 14 belagavi belagavi VERB cana-590 3 15 , , PUNCT cana-590 3 16 karnataka karnataka PROPN cana-590 3 17 , , PUNCT cana-590 3 18 india india PROPN cana-590 3 19 e e PROPN cana-590 3 20 - - NOUN cana-590 3 21 mail mail NOUN cana-590 3 22 : : PUNCT cana-590 3 23 shru16_shruti@rediffmail.com shru16_shruti@rediffmail.com PROPN cana-590 3 24 article article PROPN cana-590 3 25 history history NOUN cana-590 3 26 : : PUNCT cana-590 3 27 received receive VERB cana-590 3 28 : : PUNCT cana-590 3 29 26 26 NUM cana-590 3 30 - - PUNCT cana-590 3 31 02 02 NUM cana-590 3 32 - - PUNCT cana-590 3 33 2024 2024 NUM cana-590 3 34 revised revise VERB cana-590 3 35 : : PUNCT cana-590 3 36 28 28 NUM cana-590 3 37 - - PUNCT cana-590 3 38 04 04 NUM cana-590 3 39 - - PUNCT cana-590 3 40 2024 2024 NUM cana-590 3 41 accepted accept VERB cana-590 3 42 : : PUNCT cana-590 3 43 12 12 NUM cana-590 3 44 - - PUNCT cana-590 3 45 05 05 NUM cana-590 3 46 - - PUNCT cana-590 3 47 2024 2024 NUM cana-590 3 48 abstract abstract NOUN cana-590 3 49 : : PUNCT cana-590 3 50 we we PRON cana-590 3 51 start start VERB cana-590 3 52 with with ADP cana-590 3 53 basics basic NOUN cana-590 3 54 in in ADP cana-590 3 55 differential differential NOUN cana-590 3 56 manifold manifold NOUN cana-590 3 57 such such ADJ cana-590 3 58 as as ADP cana-590 3 59 complex complex ADJ cana-590 3 60 manifolds manifold NOUN cana-590 3 61 , , PUNCT cana-590 3 62 tangent tangent ADJ cana-590 3 63 space space NOUN cana-590 3 64 to to ADP cana-590 3 65 a a DET cana-590 3 66 manifold manifold ADJ cana-590 3 67 , , PUNCT cana-590 3 68 complex complex ADJ cana-590 3 69 sub sub NOUN cana-590 3 70 manifolds manifold NOUN cana-590 3 71 and and CCONJ cana-590 3 72 sub sub NOUN cana-590 3 73 varieties variety NOUN cana-590 3 74 and and CCONJ cana-590 3 75 specified specify VERB cana-590 3 76 their their PRON cana-590 3 77 generalizations generalization NOUN cana-590 3 78 to to PART cana-590 3 79 projective projective VERB cana-590 3 80 spaces space NOUN cana-590 3 81 with with ADP cana-590 3 82 its its PRON cana-590 3 83 rich rich ADJ cana-590 3 84 topological topological ADJ cana-590 3 85 and and CCONJ cana-590 3 86 smooth smooth ADJ cana-590 3 87 manifold manifold ADJ cana-590 3 88 structure structure NOUN cana-590 3 89 . . PUNCT cana-590 4 1 keywords keyword NOUN cana-590 4 2 : : PUNCT cana-590 4 3 complex complex ADJ cana-590 4 4 manifolds manifold NOUN cana-590 4 5 , , PUNCT cana-590 4 6 tangent tangent ADJ cana-590 4 7 space space NOUN cana-590 4 8 , , PUNCT cana-590 4 9 sub sub NOUN cana-590 4 10 manifold manifold ADJ cana-590 4 11 , , PUNCT cana-590 4 12 sub sub NOUN cana-590 4 13 varieties variety NOUN cana-590 4 14 . . PUNCT cana-590 5 1 1 1 X cana-590 5 2 . . X cana-590 5 3 introduction introduction NOUN cana-590 5 4 the the DET cana-590 5 5 core core NOUN cana-590 5 6 group group NOUN cana-590 5 7 of of ADP cana-590 5 8 compact compact ADJ cana-590 5 9 complex complex ADJ cana-590 5 10 manifolds manifold NOUN cana-590 5 11 is be AUX cana-590 5 12 known know VERB cana-590 5 13 as as ADP cana-590 5 14 the the DET cana-590 5 15 grassmannians grassmannian NOUN cana-590 5 16 . . PUNCT cana-590 6 1 " " PUNCT cana-590 6 2 they they PRON cana-590 6 3 could could AUX cana-590 6 4 be be AUX cana-590 6 5 viewed view VERB cana-590 6 6 as as ADP cana-590 6 7 an an DET cana-590 6 8 extension extension NOUN cana-590 6 9 of of ADP cana-590 6 10 projective projective ADJ cana-590 6 11 space space NOUN cana-590 6 12 as as ADV cana-590 6 13 well well ADV cana-590 6 14 . . PUNCT cana-590 7 1 we we PRON cana-590 7 2 'll will AUX cana-590 7 3 define define VERB cana-590 7 4 grassmannians grassmannian NOUN cana-590 7 5 formally formally ADV cana-590 7 6 here here ADV cana-590 7 7 . . PUNCT cana-590 8 1 understanding understand VERB cana-590 8 2 their their PRON cana-590 8 3 individual individual ADJ cana-590 8 4 structures structure NOUN cana-590 8 5 is be AUX cana-590 8 6 crucial crucial ADJ cana-590 8 7 for for ADP cana-590 8 8 both both CCONJ cana-590 8 9 geometric geometric ADJ cana-590 8 10 and and CCONJ cana-590 8 11 topological topological ADJ cana-590 8 12 analyses analysis NOUN cana-590 8 13 of of ADP cana-590 8 14 grassmannians grassmannian NOUN cana-590 8 15 . . PUNCT cana-590 9 1 1.1 1.1 NUM cana-590 9 2 definition definition NOUN cana-590 9 3 : : PUNCT cana-590 9 4 for for ADP cana-590 9 5 this this DET cana-590 9 6 purpose purpose NOUN cana-590 9 7 , , PUNCT cana-590 9 8 we we PRON cana-590 9 9 may may AUX cana-590 9 10 write write VERB cana-590 9 11 𝐺(π‘˜ 𝐺(π‘˜ PROPN cana-590 9 12 , , PUNCT cana-590 9 13 𝑛 𝑛 NOUN cana-590 9 14 ) ) PUNCT cana-590 9 15 for for ADP cana-590 9 16 𝐺(π‘˜ 𝐺(π‘˜ PROPN cana-590 9 17 , , PUNCT cana-590 9 18 𝐢𝑛 𝐢𝑛 PROPN cana-590 9 19 ) ) PUNCT cana-590 9 20 and and CCONJ cana-590 9 21 define define VERB cana-590 9 22 the the DET cana-590 9 23 grassmannians grassmannian NOUN cana-590 9 24 𝐺(π‘˜ 𝐺(π‘˜ ADP cana-590 9 25 , , PUNCT cana-590 9 26 𝑉 𝑉 PROPN cana-590 9 27 ) ) PUNCT cana-590 9 28 as as ADP cana-590 9 29 the the DET cana-590 9 30 set set NOUN cana-590 9 31 of of ADP cana-590 9 32 π‘˜-dimensional π‘˜-dimensional ADJ cana-590 9 33 linear linear ADJ cana-590 9 34 subspaces subspace NOUN cana-590 9 35 of of ADP cana-590 9 36 𝑉. 𝑉. NOUN cana-590 9 37 let let VERB cana-590 9 38 𝑉 𝑉 PROPN cana-590 9 39 be be AUX cana-590 9 40 a a DET cana-590 9 41 complex complex ADJ cana-590 9 42 vector vector NOUN cana-590 9 43 space space NOUN cana-590 9 44 of of ADP cana-590 9 45 size size NOUN cana-590 9 46 n. n. NOUN cana-590 9 47 to to PART cana-590 9 48 represent represent VERB cana-590 9 49 Ξ» Ξ» PROPN cana-590 9 50 in in ADP cana-590 9 51 𝐢𝑛 𝐢𝑛 PROPN cana-590 9 52 , , PUNCT cana-590 9 53 one one PRON cana-590 9 54 may may AUX cana-590 9 55 use use VERB cana-590 9 56 a a DET cana-590 9 57 collection collection NOUN cana-590 9 58 of of ADP cana-590 9 59 π‘˜-row π‘˜-row PROPN cana-590 9 60 vectors vector NOUN cana-590 9 61 in in ADP cana-590 9 62 𝐢𝑛that 𝐢𝑛that PROPN cana-590 9 63 span span VERB cana-590 9 64 a a DET cana-590 9 65 particular particular ADJ cana-590 9 66 k k ADJ cana-590 9 67 - - NOUN cana-590 9 68 plane plane NOUN cana-590 9 69 𝐴. 𝐴. PROPN cana-590 9 70 ie ie X cana-590 9 71 by by ADP cana-590 9 72 a a DET cana-590 9 73 π‘˜π‘₯𝑛 π‘˜π‘₯𝑛 NOUN cana-590 9 74 matrix matrix NOUN cana-590 9 75 ( ( PUNCT cana-590 9 76 𝑣11 𝑣11 NUM cana-590 9 77 … … SYM cana-590 9 78 … … PUNCT cana-590 9 79 … … PUNCT cana-590 9 80 … … PUNCT cana-590 9 81 … … PUNCT cana-590 9 82 … … PUNCT cana-590 9 83 … … PUNCT cana-590 9 84 … … PUNCT cana-590 9 85 … … PUNCT cana-590 9 86 𝑣1𝑛 𝑣1𝑛 NOUN cana-590 9 87 π‘£π‘˜π‘› π‘£π‘˜π‘› NOUN cana-590 9 88 … … SYM cana-590 9 89 … … PUNCT cana-590 9 90 … … PUNCT cana-590 9 91 … … PUNCT cana-590 9 92 … … PUNCT cana-590 9 93 … … PUNCT cana-590 9 94 … … PUNCT cana-590 9 95 … … PUNCT cana-590 9 96 … … PUNCT cana-590 9 97 π‘£π‘˜π‘› π‘£π‘˜π‘› NOUN cana-590 9 98 ) ) PUNCT cana-590 9 99 of of ADP cana-590 9 100 rank rank PROPN cana-590 9 101 k. k. PROPN cana-590 10 1 if if SCONJ cana-590 10 2 ∧ ∧ PROPN cana-590 10 3 = = NOUN cana-590 10 4 gβˆ§β€² gβˆ§β€² PROPN cana-590 10 5 for for ADP cana-590 10 6 some some DET cana-590 10 7 g g PROPN cana-590 10 8 ∈ ∈ PROPN cana-590 10 9 πΊπΏπ‘˜ πΊπΏπ‘˜ NOUN cana-590 10 10 then then ADV cana-590 10 11 𝐴 𝐴 PROPN cana-590 10 12 and and CCONJ cana-590 10 13 𝐴 𝐴 PROPN cana-590 10 14 ' ' PART cana-590 10 15 , , PUNCT cana-590 10 16 two two NUM cana-590 10 17 such such ADJ cana-590 10 18 matrices matrix NOUN cana-590 10 19 , , PUNCT cana-590 10 20 may may AUX cana-590 10 21 both both PRON cana-590 10 22 represent represent VERB cana-590 10 23 for for ADP cana-590 10 24 the the DET cana-590 10 25 same same ADJ cana-590 10 26 π‘˜ π‘˜ PRON cana-590 10 27 βˆ’ βˆ’ NOUN cana-590 10 28 𝑛 𝑛 DET cana-590 10 29 element element NOUN cana-590 10 30 in in ADP cana-590 10 31 𝐺. 𝐺. NOUN cana-590 10 32 any any DET cana-590 10 33 matrix matrix NOUN cana-590 10 34 of of ADP cana-590 10 35 this this DET cana-590 10 36 kind kind NOUN cana-590 10 37 clearly clearly ADV cana-590 10 38 represents represent VERB cana-590 10 39 a a DET cana-590 10 40 point point NOUN cana-590 10 41 or or CCONJ cana-590 10 42 an an DET cana-590 10 43 element element NOUN cana-590 10 44 of of ADP cana-590 10 45 𝐺(π‘˜ 𝐺(π‘˜ NOUN cana-590 10 46 , , PUNCT cana-590 10 47 𝑛 𝑛 NOUN cana-590 10 48 ) ) PUNCT cana-590 10 49 . . PUNCT cana-590 11 1 let let VERB cana-590 11 2 𝐼 𝐼 PROPN cana-590 11 3 = = SYM cana-590 11 4 { { PUNCT cana-590 11 5 𝑖1 𝑖1 PROPN cana-590 11 6 , , PUNCT cana-590 11 7 … … PUNCT cana-590 11 8 …… …… X cana-590 11 9 π‘–π‘˜ π‘–π‘˜ ADP cana-590 11 10 } } PUNCT cana-590 11 11 βŠ‚ βŠ‚ X cana-590 11 12 { { PUNCT cana-590 11 13 1 1 NUM cana-590 11 14 , , PUNCT cana-590 11 15 … … PUNCT cana-590 11 16 … … PUNCT cana-590 11 17 . . PUNCT cana-590 11 18 . . PUNCT cana-590 12 1 , , PUNCT cana-590 12 2 𝑛 𝑛 X cana-590 12 3 } } PUNCT cana-590 12 4 of of ADP cana-590 12 5 cardinality cardinality PROPN cana-590 12 6 π‘˜ π‘˜ ADP cana-590 12 7 , , PUNCT cana-590 12 8 then then ADV cana-590 12 9 for for ADP cana-590 12 10 𝑉iβŠ‚ 𝑉iβŠ‚ ADJ cana-590 12 11 𝐢n 𝐢n PROPN cana-590 12 12 , , PUNCT cana-590 12 13 a a DET cana-590 12 14 ( ( PUNCT cana-590 12 15 𝑛 𝑛 PROPN cana-590 12 16 βˆ’ βˆ’ PROPN cana-590 12 17 π‘˜ π‘˜ NOUN cana-590 12 18 ) ) PUNCT cana-590 12 19 – – PUNCT cana-590 12 20 plane plane NOUN cana-590 12 21 in in ADP cana-590 12 22 𝐢n 𝐢n PROPN cana-590 12 23 be be AUX cana-590 12 24 spanned span VERB cana-590 12 25 by by ADP cana-590 12 26 the the DET cana-590 12 27 vectors vector NOUN cana-590 12 28 { { PUNCT cana-590 12 29 𝑒𝑗 𝑒𝑗 NOUN cana-590 12 30 : : PUNCT cana-590 12 31 𝑗 𝑗 PROPN cana-590 12 32 βˆ‰ βˆ‰ PROPN cana-590 12 33 𝐼 𝐼 PROPN cana-590 12 34 } } PUNCT cana-590 12 35 . . PUNCT cana-590 13 1 2 2 X cana-590 13 2 . . PUNCT cana-590 13 3 complex complex ADJ cana-590 13 4 manifolds manifold NOUN cana-590 13 5 and and CCONJ cana-590 13 6 examples example NOUN cana-590 13 7 : : PUNCT cana-590 13 8 we we PRON cana-590 13 9 define define VERB cana-590 13 10 a a DET cana-590 13 11 complex complex ADJ cana-590 13 12 manifold manifold NOUN cana-590 13 13 and and CCONJ cana-590 13 14 provide provide VERB cana-590 13 15 some some DET cana-590 13 16 important important ADJ cana-590 13 17 examples example NOUN cana-590 13 18 of of ADP cana-590 13 19 complex complex ADJ cana-590 13 20 manifolds manifold NOUN cana-590 13 21 . . PUNCT cana-590 14 1 2.1 2.1 NUM cana-590 14 2 definition definition NOUN cana-590 14 3 : : PUNCT cana-590 14 4 a a DET cana-590 14 5 differentiable differentiable ADJ cana-590 14 6 manifold manifold NOUN cana-590 14 7 is be AUX cana-590 14 8 a a DET cana-590 14 9 multilayered multilayered ADJ cana-590 14 10 manifold manifold NOUN cana-590 14 11 . . PUNCT cana-590 15 1 𝑀 𝑀 PROPN cana-590 15 2 allows allow VERB cana-590 15 3 coordinate coordinate NOUN cana-590 15 4 mappings mapping NOUN cana-590 15 5 πœ‘π›Ό:π‘ˆπ›ΌβŸΆπΆ πœ‘π›Ό:π‘ˆπ›ΌβŸΆπΆ NUM cana-590 15 6 𝑛 𝑛 PRON cana-590 15 7 and and CCONJ cana-590 15 8 an an DET cana-590 15 9 open open ADJ cana-590 15 10 cover cover NOUN cana-590 15 11 { { PUNCT cana-590 15 12 π‘ˆπ›Ό π‘ˆπ›Ό NOUN cana-590 15 13 : : PUNCT cana-590 15 14 π›Όπœ–π›¬ π›Όπœ–π›¬ PROPN cana-590 15 15 } } PUNCT cana-590 15 16 , , PUNCT cana-590 15 17 𝐢 𝐢 PROPN cana-590 15 18 n n PRON cana-590 15 19 such such ADJ cana-590 15 20 that that SCONJ cana-590 15 21 , , PUNCT cana-590 15 22 for for ADP cana-590 15 23 all all DET cana-590 15 24 𝛼 𝛼 PROPN cana-590 15 25 , , PUNCT cana-590 15 26 𝛽 𝛽 NOUN cana-590 15 27 such such ADJ cana-590 15 28 that that DET cana-590 15 29 πœ‘π›Όπ‘œπœ‘π›½ πœ‘π›Όπ‘œπœ‘π›½ NOUN cana-590 15 30 -1 -1 INTJ cana-590 15 31 is be AUX cana-590 15 32 holomorphic holomorphic ADJ cana-590 15 33 on on ADP cana-590 15 34 πœ‘π›½(π‘ˆπ›Ό πœ‘π›½(π‘ˆπ›Ό PROPN cana-590 15 35 ∩ ∩ ADJ cana-590 15 36 π‘ˆπ›½ π‘ˆπ›½ PROPN cana-590 15 37 ) ) PUNCT cana-590 15 38 βŠ‚ βŠ‚ PROPN cana-590 15 39 𝐢 𝐢 PROPN cana-590 15 40 n n PRON cana-590 15 41 communications communication NOUN cana-590 15 42 on on ADP cana-590 15 43 applied apply VERB cana-590 15 44 nonlinear nonlinear ADJ cana-590 15 45 analysis analysis NOUN cana-590 15 46 issn issn NOUN cana-590 15 47 : : PUNCT cana-590 15 48 1074 1074 NUM cana-590 15 49 - - PUNCT cana-590 15 50 133x 133x NUM cana-590 15 51 vol vol NOUN cana-590 15 52 31 31 NUM cana-590 15 53 no no NOUN cana-590 15 54 . . PUNCT cana-590 16 1 2s 2s NUM cana-590 16 2 ( ( PUNCT cana-590 16 3 2024 2024 NUM cana-590 16 4 ) ) PUNCT cana-590 16 5 27 27 NUM cana-590 16 6 https://internationalpubls.com https://internationalpubls.com X cana-590 16 7 βˆͺ βˆͺ ADP cana-590 16 8 π‘ˆπ›Ό π‘ˆπ›Ό PROPN cana-590 16 9 ∩ ∩ ADJ cana-590 17 1 π‘ˆπ›½ π‘ˆπ›½ PROPN cana-590 17 2 βŠ‚ βŠ‚ PROPN cana-590 17 3 π‘ˆπ›Ό π‘ˆπ›Ό VERB cana-590 17 4 πœ‘π›Ό πœ‘π›Ό ADP cana-590 17 5 β†’ β†’ SYM cana-590 17 6 𝐢nβŠƒ 𝐢nβŠƒ VERB cana-590 17 7 πœ‘π›Ό(π‘ˆπ›Ό πœ‘π›Ό(π‘ˆπ›Ό PROPN cana-590 17 8 βˆ©π‘ˆπ›½ βˆ©π‘ˆπ›½ NOUN cana-590 17 9 ) ) PUNCT cana-590 17 10 πœ‘π›½ πœ‘π›½ PROPN cana-590 17 11 ↓ ↓ PROPN cana-590 17 12 ↓ ↓ PROPN cana-590 17 13 πœ‘π›Όπ‘œπœ‘π›½ πœ‘π›Όπ‘œπœ‘π›½ PROPN cana-590 17 14 -1 -1 PUNCT cana-590 17 15 𝐢nβŠƒ 𝐢nβŠƒ PROPN cana-590 17 16 πœ‘π›½(π‘ˆπ›Ό πœ‘π›½(π‘ˆπ›Ό PROPN cana-590 17 17 ∩ ∩ ADJ cana-590 17 18 π‘ˆπ›½ π‘ˆπ›½ PROPN cana-590 17 19 ) ) PUNCT cana-590 17 20 our our PRON cana-590 17 21 intension intension NOUN cana-590 17 22 , , PUNCT cana-590 17 23 to to PART cana-590 17 24 define define VERB cana-590 17 25 holomorphic holomorphic ADJ cana-590 17 26 maps map NOUN cana-590 17 27 on on ADP cana-590 17 28 𝑀 𝑀 PROPN cana-590 17 29 into into ADP cana-590 17 30 𝑁 𝑁 PROPN cana-590 17 31 , , PUNCT cana-590 17 32 where where SCONJ cana-590 17 33 𝑀 𝑀 PROPN cana-590 17 34 and and CCONJ cana-590 17 35 𝑁 𝑁 PROPN cana-590 17 36 smooth smooth ADJ cana-590 17 37 complex complex ADJ cana-590 17 38 manifolds manifold NOUN cana-590 17 39 . . PUNCT cana-590 17 40 2.2 2.2 NUM cana-590 17 41 definition definition NOUN cana-590 17 42 : : PUNCT cana-590 17 43 on on ADP cana-590 17 44 an an DET cana-590 17 45 open open ADJ cana-590 17 46 set set NOUN cana-590 17 47 π‘ˆ π‘ˆ PROPN cana-590 17 48 βŠ‚ βŠ‚ PROPN cana-590 17 49 𝑀 𝑀 PROPN cana-590 17 50 , , PUNCT cana-590 17 51 a a DET cana-590 17 52 function function NOUN cana-590 17 53 𝑓 𝑓 NOUN cana-590 17 54 is be AUX cana-590 17 55 holomorphic holomorphic ADJ cana-590 17 56 if if SCONJ cana-590 17 57 and and CCONJ cana-590 17 58 only only ADV cana-590 17 59 if if SCONJ cana-590 17 60 π‘“π‘œπœ‘π›Ό π‘“π‘œπœ‘π›Ό VERB cana-590 17 61 -1 -1 PUNCT cana-590 17 62 is be AUX cana-590 17 63 holomorphic holomorphic ADJ cana-590 17 64 on on ADP cana-590 17 65 πœ‘π›Ό πœ‘π›Ό NOUN cana-590 17 66 ( ( PUNCT cana-590 17 67 π‘ˆβ‹‚π‘ˆπ›Ό)βŠ‚ π‘ˆβ‹‚π‘ˆπ›Ό)βŠ‚ NUM cana-590 17 68 𝐢n 𝐢n VERB cana-590 17 69 for for ADP cana-590 17 70 all all DET cana-590 17 71 𝛼. 𝛼. NOUN cana-590 17 72 similarly similarly ADV cana-590 17 73 , , PUNCT cana-590 17 74 a a DET cana-590 17 75 collection collection NOUN cana-590 17 76 𝑍 𝑍 NOUN cana-590 17 77 = = SYM cana-590 17 78 ( ( PUNCT cana-590 17 79 𝑧1 𝑧1 NOUN cana-590 17 80 , , PUNCT cana-590 17 81 … … PUNCT cana-590 17 82 … … PUNCT cana-590 17 83 . . PUNCT cana-590 17 84 . . PUNCT cana-590 18 1 , , PUNCT cana-590 18 2 𝑧𝑛 𝑧𝑛 PROPN cana-590 18 3 ) ) PUNCT cana-590 18 4 of of ADP cana-590 18 5 functions function NOUN cana-590 18 6 on on ADP cana-590 18 7 π‘ˆ π‘ˆ PROPN cana-590 18 8 βŠ‚ βŠ‚ PROPN cana-590 18 9 𝑀 𝑀 PROPN cana-590 18 10 is be AUX cana-590 18 11 called call VERB cana-590 18 12 a a DET cana-590 18 13 holomorphic holomorphic ADJ cana-590 18 14 coordinate coordinate NOUN cana-590 18 15 system system NOUN cana-590 18 16 if if SCONJ cana-590 18 17 and and CCONJ cana-590 18 18 only only ADV cana-590 18 19 if if SCONJ cana-590 18 20 πœ‘π›Όπ‘œπ‘§ πœ‘π›Όπ‘œπ‘§ NOUN cana-590 18 21 βˆ’1 βˆ’1 NOUN cana-590 18 22 and and CCONJ cana-590 18 23 are be AUX cana-590 18 24 π‘§π‘œπœ‘π›Ό π‘§π‘œπœ‘π›Ό ADJ cana-590 18 25 -1 -1 PUNCT cana-590 18 26 holomorphic holomorphic ADJ cana-590 18 27 on on ADP cana-590 18 28 𝑧(π‘ˆβ‹‚π‘ˆπ›Ό 𝑧(π‘ˆβ‹‚π‘ˆπ›Ό PROPN cana-590 18 29 ) ) PUNCT cana-590 18 30 and and CCONJ cana-590 18 31 πœ‘π›Ό πœ‘π›Ό NOUN cana-590 18 32 ( ( PUNCT cana-590 18 33 π‘ˆβ‹‚π‘ˆπ›Ό π‘ˆβ‹‚π‘ˆπ›Ό NOUN cana-590 18 34 ) ) PUNCT cana-590 18 35 respectively respectively ADV cana-590 18 36 , , PUNCT cana-590 18 37 for for ADP cana-590 18 38 𝛼. 𝛼. NOUN cana-590 18 39 2.3 2.3 NUM cana-590 18 40 definition definition NOUN cana-590 18 41 : : PUNCT cana-590 18 42 a a DET cana-590 18 43 complex complex ADJ cana-590 18 44 manifold manifold ADJ cana-590 18 45 map map NOUN cana-590 18 46 𝑓:𝑀 𝑓:𝑀 PUNCT cana-590 18 47 ⟢ ⟢ NOUN cana-590 18 48 𝑁 𝑁 PROPN cana-590 18 49 is be AUX cana-590 18 50 holomorphic holomorphic ADJ cana-590 18 51 if if SCONJ cana-590 18 52 and and CCONJ cana-590 18 53 only only ADV cana-590 18 54 if if SCONJ cana-590 18 55 holomorphic holomorphic ADJ cana-590 18 56 functions function NOUN cana-590 18 57 provide provide VERB cana-590 18 58 local local ADJ cana-590 18 59 holomorphic holomorphic ADJ cana-590 18 60 coordinates coordinate NOUN cana-590 18 61 on on ADP cana-590 18 62 𝑁. 𝑁. PROPN cana-590 18 63 ( ( PUNCT cana-590 18 64 i i NOUN cana-590 18 65 ) ) PUNCT cana-590 18 66 a a DET cana-590 18 67 complex complex ADJ cana-590 18 68 manifold manifold NOUN cana-590 18 69 that that PRON cana-590 18 70 is be AUX cana-590 18 71 one one NUM cana-590 18 72 - - PUNCT cana-590 18 73 dimensional dimensional ADJ cana-590 18 74 is be AUX cana-590 18 75 referred refer VERB cana-590 18 76 to to ADP cana-590 18 77 as as ADP cana-590 18 78 a a DET cana-590 18 79 riemann riemann PROPN cana-590 18 80 surface surface NOUN cana-590 18 81 . . PUNCT cana-590 19 1 ( ( PUNCT cana-590 19 2 ii ii X cana-590 19 3 ) ) PUNCT cana-590 19 4 the the DET cana-590 19 5 set set NOUN cana-590 19 6 of of ADP cana-590 19 7 all all DET cana-590 19 8 lines line NOUN cana-590 19 9 in in ADP cana-590 19 10 𝐢𝑛+1 𝐢𝑛+1 NOUN cana-590 19 11 that that PRON cana-590 19 12 intersect intersect NOUN cana-590 19 13 at at ADP cana-590 19 14 the the DET cana-590 19 15 origin origin NOUN cana-590 19 16 is be AUX cana-590 19 17 denoted denote VERB cana-590 19 18 as as ADP cana-590 19 19 𝑃𝑛. 𝑃𝑛. NOUN cana-590 19 20 for for ADP cana-590 19 21 any any DET cana-590 19 22 𝑧 𝑧 PRON cana-590 19 23 β‰  β‰  PROPN cana-590 19 24 0 0 NUM cana-590 19 25 ∈ ∈ NOUN cana-590 19 26 𝑙 𝑙 NOUN cana-590 19 27 we we PRON cana-590 19 28 can can AUX cana-590 19 29 write write VERB cana-590 19 30 , , PUNCT cana-590 19 31 𝑝n 𝑝n PROPN cana-590 19 32 = = PRON cana-590 19 33 { { PUNCT cana-590 19 34 [ [ X cana-590 19 35 𝑧 𝑧 X cana-590 19 36 ] ] X cana-590 19 37 β‰  β‰  PROPN cana-590 19 38 0 0 NUM cana-590 19 39 ∈ ∈ PROPN cana-590 19 40 𝐢𝑛+1 𝐢𝑛+1 NOUN cana-590 19 41 } } PUNCT cana-590 19 42 / / PUNCT cana-590 20 1 [ [ X cana-590 20 2 𝑧]~[πœ†π‘§ 𝑧]~[πœ†π‘§ X cana-590 20 3 ] ] X cana-590 20 4 determines determine VERB cana-590 20 5 a a DET cana-590 20 6 line line NOUN cana-590 20 7 𝑙 𝑙 X cana-590 20 8 in in ADP cana-590 20 9 𝐢𝑛+1 𝐢𝑛+1 NOUN cana-590 20 10 . . PUNCT cana-590 21 1 a a DET cana-590 21 2 bijective bijective ADJ cana-590 21 3 map map NOUN cana-590 21 4 πœ‘π‘– πœ‘π‘– NOUN cana-590 21 5 to to ADP cana-590 21 6 𝐢n 𝐢n PROPN cana-590 21 7 is be AUX cana-590 21 8 given give VERB cana-590 21 9 by by ADP cana-590 21 10 πœ‘π‘–([𝑧0 πœ‘π‘–([𝑧0 NOUN cana-590 21 11 , , PUNCT cana-590 21 12 … … PUNCT cana-590 21 13 . . PUNCT cana-590 22 1 , , PUNCT cana-590 22 2 𝑧𝑛 𝑧𝑛 PROPN cana-590 22 3 ] ] X cana-590 22 4 ) ) PUNCT cana-590 22 5 = = SYM cana-590 23 1 ( ( PUNCT cana-590 23 2 𝑧0 𝑧0 PROPN cana-590 23 3 𝑧1 𝑧1 PROPN cana-590 23 4 , , PUNCT cana-590 23 5 … … PUNCT cana-590 23 6 … … PUNCT cana-590 23 7 , , PUNCT cana-590 23 8 𝑧 𝑧 X cana-590 23 9 οΏ½ οΏ½ PROPN cana-590 23 10 Μ‚ Μ‚ SYM cana-590 23 11 οΏ½ οΏ½ PROPN cana-590 23 12 𝑧𝑖 𝑧𝑖 NUM cana-590 23 13 , , PUNCT cana-590 23 14 … … PUNCT cana-590 23 15 . . PUNCT cana-590 24 1 𝑧𝑛 𝑧𝑛 INTJ cana-590 24 2 𝑧𝑖 𝑧𝑖 INTJ cana-590 24 3 ) ) PUNCT cana-590 24 4 and and CCONJ cana-590 24 5 a a DET cana-590 24 6 subset subset NOUN cana-590 24 7 π‘ˆπ‘– π‘ˆπ‘– PROPN cana-590 24 8 of of ADP cana-590 24 9 𝑝n 𝑝n PROPN cana-590 24 10 looks look VERB cana-590 24 11 like like ADP cana-590 24 12 , , PUNCT cana-590 25 1 π‘ˆπ‘– π‘ˆπ‘– PROPN cana-590 25 2 = = PUNCT cana-590 25 3 { { PUNCT cana-590 25 4 [ [ X cana-590 25 5 𝑧 𝑧 X cana-590 25 6 ] ] X cana-590 25 7 : : PUNCT cana-590 25 8 𝑧𝑖 𝑧𝑖 INTJ cana-590 25 9 β‰  β‰  PROPN cana-590 25 10 0 0 NUM cana-590 25 11 } } PUNCT cana-590 25 12 βŠ‚ βŠ‚ PRON cana-590 25 13 𝑝n 𝑝n X cana-590 25 14 of of ADP cana-590 25 15 lines line NOUN cana-590 25 16 not not PART cana-590 25 17 contained contain VERB cana-590 25 18 in in ADP cana-590 25 19 the the DET cana-590 25 20 hyper hyper ADJ cana-590 25 21 plane plane NOUN cana-590 25 22 ( ( PUNCT cana-590 25 23 𝑧𝑖 𝑧𝑖 NOUN cana-590 25 24 = = NOUN cana-590 25 25 0 0 NUM cana-590 25 26 ) ) PUNCT cana-590 25 27 . . PUNCT cana-590 26 1 on on ADP cana-590 26 2 ( ( PUNCT cana-590 26 3 𝑧𝑗 𝑧𝑗 INTJ cana-590 26 4 β‰  β‰  PROPN cana-590 26 5 0 0 NUM cana-590 26 6 ) ) PUNCT cana-590 26 7 = = SYM cana-590 26 8 πœ‘π‘–(π‘ˆπ‘–β‹‚π‘ˆπ‘—)βŠ‚ πœ‘π‘–(π‘ˆπ‘–β‹‚π‘ˆπ‘—)βŠ‚ PROPN cana-590 26 9 𝐢 𝐢 PROPN cana-590 26 10 n n CCONJ cana-590 26 11 , , PUNCT cana-590 26 12 πœ‘π‘—π‘œπœ‘π‘– πœ‘π‘—π‘œπœ‘π‘– PROPN cana-590 26 13 -1(𝑧1 -1(𝑧1 NOUN cana-590 26 14 , , PUNCT cana-590 26 15 … … PUNCT cana-590 26 16 . . PUNCT cana-590 26 17 , , PUNCT cana-590 26 18 𝑧𝑛 𝑧𝑛 PROPN cana-590 26 19 ) ) PUNCT cana-590 26 20 = = SYM cana-590 27 1 ( ( PUNCT cana-590 27 2 𝑧1 𝑧1 NOUN cana-590 27 3 𝑧𝑗 𝑧𝑗 VERB cana-590 27 4 , , PUNCT cana-590 27 5 … … PUNCT cana-590 27 6 … … PUNCT cana-590 27 7 , , PUNCT cana-590 27 8 𝑧 𝑧 X cana-590 27 9 οΏ½ οΏ½ PROPN cana-590 27 10 Μ‚ Μ‚ SYM cana-590 27 11 οΏ½ οΏ½ NOUN cana-590 27 12 𝑧𝑗 𝑧𝑗 PROPN cana-590 27 13 , , PUNCT cana-590 27 14 … … PUNCT cana-590 27 15 1 1 NUM cana-590 27 16 𝑧𝑗 𝑧𝑗 VERB cana-590 27 17 , , PUNCT cana-590 27 18 … … PUNCT cana-590 27 19 , , PUNCT cana-590 27 20 𝑧𝑛 𝑧𝑛 INTJ cana-590 27 21 𝑧𝑗 𝑧𝑗 VERB cana-590 27 22 ) ) PUNCT cana-590 27 23 is be AUX cana-590 27 24 clearly clearly ADV cana-590 27 25 holomorphic holomorphic ADJ cana-590 27 26 . . PUNCT cana-590 28 1 as as ADP cana-590 28 2 a a DET cana-590 28 3 consequence consequence NOUN cana-590 28 4 of of ADP cana-590 28 5 this this PRON cana-590 28 6 , , PUNCT cana-590 28 7 n n PRON cana-590 28 8 is be AUX cana-590 28 9 a a DET cana-590 28 10 complicated complicated ADJ cana-590 28 11 manifold manifold NOUN cana-590 28 12 that that PRON cana-590 28 13 has have VERB cana-590 28 14 the the DET cana-590 28 15 structure structure NOUN cana-590 28 16 of of ADP cana-590 28 17 a a DET cana-590 28 18 complex complex ADJ cana-590 28 19 projective projective ADJ cana-590 28 20 space space NOUN cana-590 28 21 . . PUNCT cana-590 29 1 the the DET cana-590 29 2 coordinates coordinate NOUN cana-590 29 3 that that PRON cana-590 29 4 are be AUX cana-590 29 5 supplied supply VERB cana-590 29 6 by by ADP cana-590 29 7 the the DET cana-590 29 8 mappings mapping NOUN cana-590 29 9 πœ‘π‘– πœ‘π‘– ADV cana-590 29 10 are be AUX cana-590 29 11 referred refer VERB cana-590 29 12 to to ADP cana-590 29 13 as as ADP cana-590 29 14 euclidean euclidean ADJ cana-590 29 15 coordinates coordinate NOUN cana-590 29 16 , , PUNCT cana-590 29 17 whereas whereas SCONJ cana-590 29 18 the the DET cana-590 29 19 '' '' PUNCT cana-590 29 20 coordinates coordinate NOUN cana-590 29 21 '' '' PUNCT cana-590 29 22 𝑧 𝑧 NOUN cana-590 29 23 = = NOUN cana-590 30 1 [ [ X cana-590 30 2 𝑧1 𝑧1 NOUN cana-590 30 3 , , PUNCT cana-590 30 4 … … PUNCT cana-590 30 5 . . PUNCT cana-590 30 6 , , PUNCT cana-590 30 7 𝑧𝑛 𝑧𝑛 PROPN cana-590 30 8 ] ] X cana-590 30 9 are be AUX cana-590 30 10 known know VERB cana-590 30 11 as as ADP cana-590 30 12 homogeneous homogeneous ADJ cana-590 30 13 coordinates coordinate NOUN cana-590 30 14 on on ADP cana-590 30 15 𝑝n 𝑝n PROPN cana-590 30 16 . . PUNCT cana-590 31 1 given give VERB cana-590 31 2 that that SCONJ cana-590 31 3 there there PRON cana-590 31 4 is be VERB cana-590 31 5 a a DET cana-590 31 6 continuous continuous ADJ cana-590 31 7 surjective surjective ADJ cana-590 31 8 map map NOUN cana-590 31 9 from from ADP cana-590 31 10 the the DET cana-590 31 11 unit unit NOUN cana-590 31 12 sphere sphere ADV cana-590 31 13 in in ADP cana-590 31 14 𝐢𝑛+1 𝐢𝑛+1 NOUN cana-590 31 15 to to ADP cana-590 31 16 𝑝n 𝑝n PROPN cana-590 31 17 , , PUNCT cana-590 31 18 it it PRON cana-590 31 19 is be AUX cana-590 31 20 necessary necessary ADJ cana-590 31 21 to to PART cana-590 31 22 conclude conclude VERB cana-590 31 23 that that PRON cana-590 31 24 pn pn PROPN cana-590 31 25 is be AUX cana-590 31 26 compact compact ADJ cana-590 31 27 . . PUNCT cana-590 32 1 it it PRON cana-590 32 2 should should AUX cana-590 32 3 be be AUX cana-590 32 4 noted note VERB cana-590 32 5 that that SCONJ cana-590 32 6 β„™1 β„™1 PROPN cana-590 32 7 is be AUX cana-590 32 8 simply simply ADV cana-590 32 9 the the DET cana-590 32 10 riemann riemann PROPN cana-590 32 11 surface surface PROPN cana-590 32 12 𝐢⋃{𝛼 𝐢⋃{𝛼 PROPN cana-590 32 13 } } PUNCT cana-590 32 14 . . PUNCT cana-590 33 1 any any DET cana-590 33 2 reference reference NOUN cana-590 33 3 to to ADP cana-590 33 4 an an DET cana-590 33 5 inclusion inclusion NOUN cana-590 33 6 is be AUX cana-590 33 7 caused cause VERB cana-590 33 8 by by ADP cana-590 33 9 πΆπ‘˜+1⟢ πΆπ‘˜+1⟢ PROPN cana-590 33 10 𝐢𝑛+1 𝐢𝑛+1 NOUN cana-590 33 11 . . PUNCT cana-590 34 1 the the DET cana-590 34 2 picture picture NOUN cana-590 34 3 of of ADP cana-590 34 4 a a DET cana-590 34 5 map map NOUN cana-590 34 6 such such ADJ cana-590 34 7 as as ADP cana-590 34 8 β„™π‘˜ β„™π‘˜ PROPN cana-590 34 9 ⟢ ⟢ NOUN cana-590 34 10 ℙ𝑛 ℙ𝑛 PROPN cana-590 34 11 is be AUX cana-590 34 12 a a DET cana-590 34 13 linear linear ADJ cana-590 34 14 subspace subspace NOUN cana-590 34 15 of of ADP cana-590 34 16 𝑝n 𝑝n PROPN cana-590 34 17 . . PUNCT cana-590 35 1 in in ADP cana-590 35 2 general general ADJ cana-590 35 3 , , PUNCT cana-590 35 4 a a DET cana-590 35 5 π‘˜-plane π‘˜-plane NOUN cana-590 35 6 is be AUX cana-590 35 7 the the DET cana-590 35 8 image image NOUN cana-590 35 9 of of ADP cana-590 35 10 πΆπ‘˜+1οƒŒ πΆπ‘˜+1οƒŒ NOUN cana-590 35 11 𝐢𝑛+1 𝐢𝑛+1 NOUN cana-590 35 12 , , PUNCT cana-590 35 13 a a DET cana-590 35 14 line line NOUN cana-590 35 15 is be AUX cana-590 35 16 the the DET cana-590 35 17 image image NOUN cana-590 35 18 of of ADP cana-590 35 19 a a DET cana-590 35 20 2plane 2plane NUM cana-590 35 21 𝐢2οƒŒ 𝐢2οƒŒ NOUN cana-590 35 22 cn+1 cn+1 NOUN cana-590 35 23 , , PUNCT cana-590 35 24 and and CCONJ cana-590 35 25 a a DET cana-590 35 26 hyperplane hyperplane NOUN cana-590 35 27 is be AUX cana-590 35 28 the the DET cana-590 35 29 image image NOUN cana-590 35 30 of of ADP cana-590 35 31 a a DET cana-590 35 32 hyperplane hyperplane NOUN cana-590 35 33 in in ADP cana-590 35 34 𝐢𝑛+1again 𝐢𝑛+1again PROPN cana-590 35 35 . . PUNCT cana-590 36 1 now now ADV cana-590 36 2 , , PUNCT cana-590 36 3 we we PRON cana-590 36 4 can can AUX cana-590 36 5 discuss discuss VERB cana-590 36 6 linear linear NOUN cana-590 36 7 relations relation NOUN cana-590 36 8 between between ADP cana-590 36 9 points point NOUN cana-590 36 10 in in ADP cana-590 36 11 𝑝n 𝑝n PROPN cana-590 36 12 in in ADP cana-590 36 13 this this DET cana-590 36 14 particular particular ADJ cana-590 36 15 situation situation NOUN cana-590 36 16 . . PUNCT cana-590 37 1 2.4 2.4 NUM cana-590 37 2 example example NOUN cana-590 37 3 : : PUNCT cana-590 37 4 if if SCONJ cana-590 37 5 the the DET cana-590 37 6 span span NOUN cana-590 37 7 of of ADP cana-590 37 8 a a DET cana-590 37 9 line line NOUN cana-590 37 10 in in ADP cana-590 37 11 𝑝n 𝑝n PROPN cana-590 37 12 is be AUX cana-590 37 13 a a DET cana-590 37 14 ( ( PUNCT cana-590 37 15 π‘˜ π‘˜ PRON cana-590 37 16 βˆ’ βˆ’ PROPN cana-590 37 17 1 1 NUM cana-590 37 18 ) ) PUNCT cana-590 37 19 -plane -plane NOUN cana-590 37 20 , , PUNCT cana-590 37 21 then then ADV cana-590 37 22 the the DET cana-590 37 23 line line NOUN cana-590 37 24 in in ADP cana-590 37 25 𝐢𝑛+1 𝐢𝑛+1 NOUN cana-590 37 26 is be AUX cana-590 37 27 said say VERB cana-590 37 28 to to PART cana-590 37 29 be be AUX cana-590 37 30 linearly linearly ADV cana-590 37 31 independent independent ADJ cana-590 37 32 of of ADP cana-590 37 33 k k PROPN cana-590 37 34 points point NOUN cana-590 37 35 . . PUNCT cana-590 38 1 assuming assume VERB cana-590 38 2 that that SCONJ cana-590 38 3 the the DET cana-590 38 4 image image NOUN cana-590 38 5 of of ADP cana-590 38 6 the the DET cana-590 38 7 subspace subspace NOUN cana-590 38 8 in in ADP cana-590 38 9 𝐢𝑛+1 𝐢𝑛+1 NOUN cana-590 38 10 that that SCONJ cana-590 38 11 the the DET cana-590 38 12 lines line NOUN cana-590 38 13 πœ‹βˆ’1(𝑝𝑖 πœ‹βˆ’1(𝑝𝑖 NOUN cana-590 38 14 ) ) PUNCT cana-590 38 15 cover cover NOUN cana-590 38 16 in in ADP cana-590 38 17 ℙ𝑛 ℙ𝑛 PROPN cana-590 38 18 is be AUX cana-590 38 19 the the DET cana-590 38 20 span span NOUN cana-590 38 21 of of ADP cana-590 38 22 a a DET cana-590 38 23 collection collection NOUN cana-590 38 24 of of ADP cana-590 38 25 points point NOUN cana-590 38 26 𝑝𝑖 𝑝𝑖 NOUN cana-590 38 27 in in ADP cana-590 38 28 ℙ𝑛 ℙ𝑛 PROPN cana-590 38 29 , , PUNCT cana-590 38 30 it it PRON cana-590 38 31 is be AUX cana-590 38 32 assumed assume VERB cana-590 38 33 that that SCONJ cana-590 38 34 the the DET cana-590 38 35 span span NOUN cana-590 38 36 of of ADP cana-590 38 37 ℙ𝑛 ℙ𝑛 PROPN cana-590 38 38 is be AUX cana-590 38 39 the the DET cana-590 38 40 image image NOUN cana-590 38 41 of of ADP cana-590 38 42 the the DET cana-590 38 43 subspace subspace NOUN cana-590 38 44 . . PUNCT cana-590 39 1 2.5 2.5 NUM cana-590 39 2 remark remark NOUN cana-590 39 3 : : PUNCT cana-590 39 4 as as ADP cana-590 39 5 a a DET cana-590 39 6 result result NOUN cana-590 39 7 , , PUNCT cana-590 39 8 the the DET cana-590 39 9 set set NOUN cana-590 39 10 of of ADP cana-590 39 11 hyper hyper ADJ cana-590 39 12 planes plane NOUN cana-590 39 13 in in ADP cana-590 39 14 ℙ𝑛 ℙ𝑛 PROPN cana-590 39 15 is be AUX cana-590 39 16 a a DET cana-590 39 17 projective projective ADJ cana-590 39 18 space space NOUN cana-590 39 19 in in ADP cana-590 39 20 and and CCONJ cana-590 39 21 of of ADP cana-590 39 22 itself itself PRON cana-590 39 23 ; ; PUNCT cana-590 39 24 it it PRON cana-590 39 25 is be AUX cana-590 39 26 known know VERB cana-590 39 27 as as ADP cana-590 39 28 the the DET cana-590 39 29 dual dual ADJ cana-590 39 30 projective projective ADJ cana-590 39 31 space space NOUN cana-590 39 32 and and CCONJ cana-590 39 33 is be AUX cana-590 39 34 represented represent VERB cana-590 39 35 by by ADP cana-590 39 36 β„™π‘›βˆ—. β„™π‘›βˆ—. VERB cana-590 39 37 it it PRON cana-590 39 38 corresponds correspond VERB cana-590 39 39 to to ADP cana-590 39 40 the the DET cana-590 39 41 set set NOUN cana-590 39 42 𝐢𝑛+1 𝐢𝑛+1 NOUN cana-590 39 43 βˆ’ βˆ’ NOUN cana-590 39 44 { { PUNCT cana-590 39 45 0 0 NUM cana-590 39 46 } } PUNCT cana-590 39 47 , , PUNCT cana-590 39 48 of of ADP cana-590 39 49 non non ADJ cana-590 39 50 zero zero NUM cana-590 39 51 linear linear ADJ cana-590 39 52 functionals functional NOUN cana-590 39 53 on on ADP cana-590 39 54 𝐢𝑛+1 𝐢𝑛+1 NOUN cana-590 39 55 modulo modulo NOUN cana-590 39 56 scalar scalar ADJ cana-590 39 57 multiplication multiplication NOUN cana-590 39 58 . . PUNCT cana-590 40 1 communications communication NOUN cana-590 40 2 on on ADP cana-590 40 3 applied apply VERB cana-590 40 4 nonlinear nonlinear ADJ cana-590 40 5 analysis analysis NOUN cana-590 40 6 issn issn NOUN cana-590 40 7 : : PUNCT cana-590 40 8 1074 1074 NUM cana-590 40 9 - - PUNCT cana-590 40 10 133x 133x NUM cana-590 40 11 vol vol NOUN cana-590 40 12 31 31 NUM cana-590 40 13 no no NOUN cana-590 40 14 . . PUNCT cana-590 41 1 2s 2s NUM cana-590 41 2 ( ( PUNCT cana-590 41 3 2024 2024 NUM cana-590 41 4 ) ) PUNCT cana-590 41 5 28 28 NUM cana-590 41 6 https://internationalpubls.com https://internationalpubls.com X cana-590 41 7 sometimes sometimes ADV cana-590 41 8 it it PRON cana-590 41 9 is be AUX cana-590 41 10 advantageous advantageous ADJ cana-590 41 11 to to PART cana-590 41 12 visualize visualize VERB cana-590 41 13 ℙ𝑛 ℙ𝑛 PROPN cana-590 41 14 on on ADP cana-590 41 15 the the DET cana-590 41 16 compactifization compactifization NOUN cana-590 41 17 of of ADP cana-590 41 18 𝐢𝑛 𝐢𝑛 PROPN cana-590 41 19 that that PRON cana-590 41 20 results result VERB cana-590 41 21 from from ADP cana-590 41 22 appending append VERB cana-590 41 23 the the DET cana-590 41 24 hyper hyper ADJ cana-590 41 25 plane plane NOUN cana-590 41 26 h h NOUN cana-590 41 27 at at ADP cana-590 41 28 infinity infinity NOUN cana-590 41 29 . . PUNCT cana-590 42 1 coordinates coordinate NOUN cana-590 42 2 - - PUNCT cana-590 42 3 wise wise ADJ cana-590 42 4 , , PUNCT cana-590 42 5 the the DET cana-590 42 6 inclusion inclusion NOUN cana-590 42 7 𝐢n 𝐢n PROPN cana-590 42 8 ⟢ ⟢ NOUN cana-590 42 9 ℙ𝑛 ℙ𝑛 PROPN cana-590 42 10 is be AUX cana-590 42 11 [ [ X cana-590 42 12 𝑧1 𝑧1 NOUN cana-590 42 13 , , PUNCT cana-590 42 14 … … PUNCT cana-590 42 15 . . PUNCT cana-590 43 1 , , PUNCT cana-590 43 2 𝑧𝑛 𝑧𝑛 PROPN cana-590 43 3 ] ] PUNCT cana-590 43 4 ⟢ ⟢ NOUN cana-590 43 5 [ [ X cana-590 43 6 1 1 NUM cana-590 43 7 , , PUNCT cana-590 43 8 𝑧1 𝑧1 NOUN cana-590 43 9 , , PUNCT cana-590 43 10 … … PUNCT cana-590 43 11 . . PUNCT cana-590 44 1 , , PUNCT cana-590 44 2 𝑧𝑛 𝑧𝑛 ADP cana-590 44 3 ] ] X cana-590 44 4 ; ; PUNCT cana-590 44 5 here here ADV cana-590 44 6 h h NOUN cana-590 44 7 has have VERB cana-590 44 8 identification identification NOUN cana-590 44 9 ( ( PUNCT cana-590 44 10 z0=0 z0=0 NUM cana-590 44 11 ) ) PUNCT cana-590 44 12 and and CCONJ cana-590 44 13 the the DET cana-590 44 14 identification identification NOUN cana-590 44 15 𝐻 𝐻 PROPN cana-590 44 16 ⋍ ⋍ VERB cana-590 44 17 ℙ𝑛-1 ℙ𝑛-1 VERB cana-590 44 18 is be AUX cana-590 44 19 obtained obtain VERB cana-590 44 20 by by ADP cana-590 44 21 taking take VERB cana-590 44 22 the the DET cana-590 44 23 hyper hyper ADJ cana-590 44 24 plane plane NOUN cana-590 44 25 at at ADP cana-590 44 26 infinity infinity NOUN cana-590 44 27 as as SCONJ cana-590 44 28 the the DET cana-590 44 29 directions direction NOUN cana-590 44 30 that that PRON cana-590 44 31 go go VERB cana-590 44 32 to to ADP cana-590 44 33 infinity infinity NOUN cana-590 44 34 in in ADP cana-590 44 35 𝐢n 𝐢n PROPN cana-590 44 36 . . PROPN cana-590 44 37 let let AUX cana-590 44 38 be be AUX cana-590 44 39 a a DET cana-590 44 40 lattice lattice NOUN cana-590 44 41 𝛬 𝛬 NOUN cana-590 44 42 = = SYM cana-590 44 43 𝑍kβŠ‚ 𝑍kβŠ‚ PROPN cana-590 44 44 𝐢n 𝐢n PROPN cana-590 44 45 after after ADP cana-590 44 46 that that PRON cana-590 44 47 , , PUNCT cana-590 44 48 the the DET cana-590 44 49 projective projective ADJ cana-590 44 50 map map NOUN cana-590 44 51 πœ‹ πœ‹ NOUN cana-590 44 52 : : PUNCT cana-590 44 53 𝐢n⟢ 𝐢n⟢ NOUN cana-590 44 54 𝐢n 𝐢n PROPN cana-590 44 55 /𝛬 /𝛬 PUNCT cana-590 44 56 induces induce VERB cana-590 44 57 a a DET cana-590 44 58 complex complex ADJ cana-590 44 59 manifold manifold ADJ cana-590 44 60 structure structure NOUN cana-590 44 61 in in ADP cana-590 44 62 the the DET cana-590 44 63 quotient quotient NOUN cana-590 44 64 group group NOUN cana-590 44 65 𝐢𝑛/𝛬. 𝐢𝑛/𝛬. VERB cana-590 44 66 only only ADV cana-590 44 67 if if SCONJ cana-590 44 68 π‘˜ π‘˜ PROPN cana-590 44 69 = = SYM cana-590 44 70 2𝑛 2𝑛 PROPN cana-590 44 71 does do AUX cana-590 44 72 it it PRON cana-590 44 73 qualify qualify VERB cana-590 44 74 as as ADP cana-590 44 75 compact compact ADJ cana-590 44 76 ; ; PUNCT cana-590 44 77 in in ADP cana-590 44 78 this this DET cana-590 44 79 instance instance NOUN cana-590 44 80 , , PUNCT cana-590 44 81 πΆβˆ—/𝛬 πΆβˆ—/𝛬 ADJ cana-590 44 82 is be AUX cana-590 44 83 referred refer VERB cana-590 44 84 to to ADP cana-590 44 85 as as ADP cana-590 44 86 a a DET cana-590 44 87 complex complex ADJ cana-590 44 88 torus torus NOUN cana-590 44 89 . . PUNCT cana-590 45 1 when when SCONJ cana-590 45 2 πœ‹ πœ‹ X cana-590 45 3 : : PUNCT cana-590 45 4 𝑀 𝑀 PROPN cana-590 45 5 ⟢ ⟢ AUX cana-590 45 6 𝑁 𝑁 PROPN cana-590 45 7 is be AUX cana-590 45 8 a a DET cana-590 45 9 complex complex ADJ cana-590 45 10 manifold manifold NOUN cana-590 45 11 , , PUNCT cana-590 45 12 it it PRON cana-590 45 13 is be AUX cana-590 45 14 common common ADJ cana-590 45 15 for for SCONJ cana-590 45 16 𝑁 𝑁 PROPN cana-590 45 17 to to PART cana-590 45 18 inherit inherit VERB cana-590 45 19 the the DET cana-590 45 20 structure structure NOUN cana-590 45 21 of of ADP cana-590 45 22 𝑀. 𝑀. PROPN cana-590 45 23 it it PRON cana-590 45 24 is be AUX cana-590 45 25 possible possible ADJ cana-590 45 26 for for SCONJ cana-590 45 27 𝑁 𝑁 PROPN cana-590 45 28 to to PART cana-590 45 29 acquire acquire VERB cana-590 45 30 the the DET cana-590 45 31 structure structure NOUN cana-590 45 32 of of ADP cana-590 45 33 a a DET cana-590 45 34 complex complex ADJ cana-590 45 35 manifold manifold NOUN cana-590 45 36 due due ADP cana-590 45 37 to to ADP cana-590 45 38 the the DET cana-590 45 39 fact fact NOUN cana-590 45 40 that that SCONJ cana-590 45 41 it it PRON cana-590 45 42 is be AUX cana-590 45 43 both both CCONJ cana-590 45 44 a a DET cana-590 45 45 complex complex ADJ cana-590 45 46 manifold manifold NOUN cana-590 45 47 and and CCONJ cana-590 45 48 a a DET cana-590 45 49 topological topological ADJ cana-590 45 50 covering covering NOUN cana-590 45 51 space space NOUN cana-590 45 52 . . PUNCT cana-590 46 1 however however ADV cana-590 46 2 , , PUNCT cana-590 46 3 this this PRON cana-590 46 4 is be AUX cana-590 46 5 only only ADV cana-590 46 6 the the DET cana-590 46 7 case case NOUN cana-590 46 8 if if SCONJ cana-590 46 9 𝑀 𝑀 PROPN cana-590 46 10 is be AUX cana-590 46 11 also also ADV cana-590 46 12 a a DET cana-590 46 13 complex complex ADJ cana-590 46 14 manifold manifold NOUN cana-590 46 15 and and CCONJ cana-590 46 16 its its PRON cana-590 46 17 deck deck NOUN cana-590 46 18 transformations transformation NOUN cana-590 46 19 are be AUX cana-590 46 20 holomorphic holomorphic ADJ cana-590 46 21 . . PUNCT cana-590 47 1 3 3 X cana-590 47 2 . . X cana-590 47 3 tangent tangent ADJ cana-590 47 4 space space NOUN cana-590 47 5 to to ADP cana-590 47 6 a a DET cana-590 47 7 manifold manifold ADJ cana-590 47 8 m m VERB cana-590 47 9 we we PRON cana-590 47 10 offer offer VERB cana-590 47 11 a a DET cana-590 47 12 complex complex ADJ cana-590 47 13 manifold manifold ADJ cana-590 47 14 𝑀 𝑀 PROPN cana-590 47 15 as as ADV cana-590 47 16 well well ADV cana-590 47 17 as as ADP cana-590 47 18 a a DET cana-590 47 19 holomorphic holomorphic ADJ cana-590 47 20 coordinate coordinate NOUN cana-590 47 21 system system NOUN cana-590 47 22 𝑧 𝑧 NOUN cana-590 47 23 = = PUNCT cana-590 47 24 ( ( PUNCT cana-590 47 25 𝑧1 𝑧1 NOUN cana-590 47 26 , , PUNCT cana-590 47 27 … … PUNCT cana-590 47 28 . . PUNCT cana-590 48 1 , , PUNCT cana-590 48 2 𝑧𝑛 𝑧𝑛 PROPN cana-590 48 3 ) ) PUNCT cana-590 48 4 that that PRON cana-590 48 5 revolves revolve VERB cana-590 48 6 around around ADP cana-590 48 7 𝑝. 𝑝. VERB cana-590 48 8 a a DET cana-590 48 9 tangent tangent ADJ cana-590 48 10 space space NOUN cana-590 48 11 to to ADP cana-590 48 12 𝑀 𝑀 PROPN cana-590 48 13 at at ADP cana-590 48 14 𝑝 𝑝 PROPN cana-590 48 15 is be AUX cana-590 48 16 defined define VERB cana-590 48 17 in in ADP cana-590 48 18 this this DET cana-590 48 19 article article NOUN cana-590 48 20 in in ADP cana-590 48 21 three three NUM cana-590 48 22 distinct distinct ADJ cana-590 48 23 ways way NOUN cana-590 48 24 . . PUNCT cana-590 49 1 a a DET cana-590 49 2 ) ) PUNCT cana-590 49 3 𝑇𝑅,𝑝(𝑀 𝑇𝑅,𝑝(𝑀 NOUN cana-590 49 4 ) ) PUNCT cana-590 49 5 we we PRON cana-590 49 6 define define VERB cana-590 49 7 𝑀 𝑀 PROPN cana-590 49 8 to to PART cana-590 49 9 be be AUX cana-590 49 10 a a DET cana-590 49 11 real real ADJ cana-590 49 12 manifold manifold NOUN cana-590 49 13 of of ADP cana-590 49 14 size size NOUN cana-590 49 15 2𝑛. 2𝑛. NOUN cana-590 49 16 ( ( PUNCT cana-590 49 17 𝑀 𝑀 PROPN cana-590 49 18 ) ) PUNCT cana-590 49 19 is be AUX cana-590 49 20 the the DET cana-590 49 21 normal normal ADJ cana-590 49 22 real real ADJ cana-590 49 23 tangent tangent NOUN cana-590 49 24 space space NOUN cana-590 49 25 at at ADP cana-590 49 26 𝑀 𝑀 PROPN cana-590 49 27 at at ADP cana-590 49 28 𝑝. 𝑝. PROPN cana-590 49 29 if if SCONJ cana-590 49 30 we we PRON cana-590 49 31 write write VERB cana-590 49 32 𝑧𝑖 𝑧𝑖 NOUN cana-590 50 1 = = SYM cana-590 50 2 π‘₯𝑖 π‘₯𝑖 PROPN cana-590 51 1 + + CCONJ cana-590 51 2 𝑖𝑦𝑖 𝑖𝑦𝑖 INTJ cana-590 51 3 , , PUNCT cana-590 51 4 𝑇𝑅,𝑝 𝑇𝑅,𝑝 PROPN cana-590 51 5 ( ( PUNCT cana-590 51 6 𝑀 𝑀 PROPN cana-590 51 7 ) ) PUNCT cana-590 51 8 = = PROPN cana-590 51 9 𝑅 𝑅 PROPN cana-590 51 10 { { PUNCT cana-590 51 11 πœ• πœ• PROPN cana-590 51 12 πœ•π‘₯𝑖 πœ•π‘₯𝑖 PUNCT cana-590 51 13 , , PUNCT cana-590 51 14 πœ• πœ• PROPN cana-590 51 15 πœ•π‘¦π‘– πœ•π‘¦π‘– PUNCT cana-590 51 16 } } PUNCT cana-590 51 17 then then ADV cana-590 51 18 𝑇𝑅,𝑝 𝑇𝑅,𝑝 PROPN cana-590 51 19 ( ( PUNCT cana-590 51 20 𝑀 𝑀 PROPN cana-590 51 21 ) ) PUNCT cana-590 51 22 may may AUX cana-590 51 23 be be AUX cana-590 51 24 expressed express VERB cana-590 51 25 as as ADP cana-590 51 26 the the DET cana-590 51 27 set set NOUN cana-590 51 28 of of ADP cana-590 51 29 all all DET cana-590 51 30 𝑅-linear 𝑅-linear PROPN cana-590 51 31 derivations derivation NOUN cana-590 51 32 on on ADP cana-590 51 33 the the DET cana-590 51 34 set set NOUN cana-590 51 35 of of ADP cana-590 51 36 all all DET cana-590 51 37 c^∞ c^∞ ADJ cana-590 51 38 functions function NOUN cana-590 51 39 with with ADP cana-590 51 40 real real ADJ cana-590 51 41 values value NOUN cana-590 51 42 that that PRON cana-590 51 43 are be AUX cana-590 51 44 close close ADJ cana-590 51 45 to to ADP cana-590 51 46 p. p. NOUN cana-590 51 47 b b NOUN cana-590 51 48 ) ) PUNCT cana-590 51 49 𝑇𝑐,𝑝(𝑀 𝑇𝑐,𝑝(𝑀 NUM cana-590 51 50 ) ) PUNCT cana-590 52 1 = = PRON cana-590 52 2 𝑇𝑅,𝑝(𝑀)⨂𝑅𝐢 𝑇𝑅,𝑝(𝑀)⨂𝑅𝐢 NOUN cana-590 52 3 we we PRON cana-590 52 4 refer refer VERB cana-590 52 5 to to ADP cana-590 52 6 the the DET cana-590 52 7 complexified complexifie VERB cana-590 52 8 tangent tangent NOUN cana-590 52 9 space space NOUN cana-590 52 10 to to ADP cana-590 52 11 𝑀 𝑀 PROPN cana-590 52 12 as as ADP cana-590 52 13 at at ADP cana-590 52 14 𝑝. 𝑝. NOUN cana-590 52 15 the the DET cana-590 52 16 space space NOUN cana-590 52 17 of of ADP cana-590 52 18 𝐢 𝐢 PROPN cana-590 52 19 βˆ’linear βˆ’linear NOUN cana-590 52 20 derivations derivation NOUN cana-590 52 21 in in ADP cana-590 52 22 the the DET cana-590 52 23 ring ring NOUN cana-590 52 24 of of ADP cana-590 52 25 complex complex NOUN cana-590 52 26 - - PUNCT cana-590 52 27 valued value VERB cana-590 52 28 𝐢∞ 𝐢∞ PROPN cana-590 52 29 functions function NOUN cana-590 52 30 on on ADP cana-590 52 31 𝑀 𝑀 PROPN cana-590 52 32 around around ADP cana-590 52 33 𝑝 𝑝 PROPN cana-590 52 34 is be AUX cana-590 52 35 one one NUM cana-590 52 36 way way NOUN cana-590 52 37 to to PART cana-590 52 38 realise realise VERB cana-590 52 39 it it PRON cana-590 52 40 . . PUNCT cana-590 53 1 we we PRON cana-590 53 2 may may AUX cana-590 53 3 write write VERB cana-590 53 4 𝑇𝑐,𝑝(𝑀)=𝐢 𝑇𝑐,𝑝(𝑀)=𝐢 PROPN cana-590 53 5 { { PUNCT cana-590 53 6 πœ• πœ• PROPN cana-590 53 7 πœ•π‘₯𝑖 πœ•π‘₯𝑖 PUNCT cana-590 53 8 , , PUNCT cana-590 53 9 πœ• πœ• PROPN cana-590 53 10 πœ•π‘¦π‘– πœ•π‘¦π‘– PUNCT cana-590 53 11 } } PUNCT cana-590 53 12 = = SYM cana-590 53 13 𝐢 𝐢 PROPN cana-590 53 14 { { PUNCT cana-590 53 15 πœ• πœ• PROPN cana-590 53 16 πœ•π‘§π‘– πœ•π‘§π‘– NOUN cana-590 53 17 , , PUNCT cana-590 53 18 πœ• πœ• PROPN cana-590 53 19 πœ• πœ• PROPN cana-590 53 20 οΏ½ οΏ½ PROPN cana-590 53 21 Μ… Μ… NOUN cana-590 53 22 οΏ½ οΏ½ NOUN cana-590 53 23 𝑖 𝑖 SYM cana-590 53 24 } } PUNCT cana-590 53 25 c c X cana-590 53 26 ) ) PUNCT cana-590 53 27 the the DET cana-590 53 28 space space NOUN cana-590 53 29 of of ADP cana-590 53 30 holomorphic holomorphic ADJ cana-590 53 31 tangents tangent NOUN cana-590 53 32 to to ADP cana-590 53 33 𝑀 𝑀 PROPN cana-590 53 34 at at ADP cana-590 53 35 point point NOUN cana-590 53 36 𝑝 𝑝 PROPN cana-590 53 37 is be AUX cana-590 53 38 represented represent VERB cana-590 53 39 by by ADP cana-590 53 40 an an DET cana-590 53 41 𝑇𝑝(m 𝑇𝑝(m NOUN cana-590 53 42 ) ) PUNCT cana-590 53 43 = = SYM cana-590 53 44 𝐢 𝐢 PROPN cana-590 53 45 { { PUNCT cana-590 53 46 πœ• πœ• PROPN cana-590 53 47 πœ•π‘§π‘– πœ•π‘§π‘– ADV cana-590 53 48 } } PUNCT cana-590 53 49 βŠ‚ βŠ‚ PROPN cana-590 53 50 𝑇𝑐,𝑝(m 𝑇𝑐,𝑝(m NOUN cana-590 53 51 ) ) PUNCT cana-590 53 52 . . PUNCT cana-590 54 1 due due ADP cana-590 54 2 to to ADP cana-590 54 3 the the DET cana-590 54 4 fact fact NOUN cana-590 54 5 that that SCONJ cana-590 54 6 the the DET cana-590 54 7 subspace subspace NOUN cana-590 54 8 of of ADP cana-590 54 9 𝑇𝐢,𝑝(𝑀 𝑇𝐢,𝑝(𝑀 PROPN cana-590 54 10 ) ) PUNCT cana-590 54 11 is be AUX cana-590 54 12 composed compose VERB cana-590 54 13 of of ADP cana-590 54 14 derivations derivation NOUN cana-590 54 15 that that PRON cana-590 54 16 vanish vanish VERB cana-590 54 17 on on ADP cana-590 54 18 antiholomorphic antiholomorphic ADJ cana-590 54 19 functions function NOUN cana-590 54 20 or or CCONJ cana-590 54 21 functions function NOUN cana-590 54 22 such such ADJ cana-590 54 23 that that SCONJ cana-590 54 24 f f PROPN cana-590 54 25 is be AUX cana-590 54 26 holomorphic holomorphic ADJ cana-590 54 27 , , PUNCT cana-590 54 28 the the DET cana-590 54 29 coordinate coordinate NOUN cana-590 54 30 system system NOUN cana-590 54 31 that that PRON cana-590 54 32 was be AUX cana-590 54 33 communications communication NOUN cana-590 54 34 on on ADP cana-590 54 35 applied apply VERB cana-590 54 36 nonlinear nonlinear ADJ cana-590 54 37 analysis analysis NOUN cana-590 54 38 issn issn NOUN cana-590 54 39 : : PUNCT cana-590 54 40 1074 1074 NUM cana-590 54 41 - - PUNCT cana-590 54 42 133x 133x NUM cana-590 54 43 vol vol NOUN cana-590 54 44 31 31 NUM cana-590 54 45 no no NOUN cana-590 54 46 . . PUNCT cana-590 55 1 2s 2s NUM cana-590 55 2 ( ( PUNCT cana-590 55 3 2024 2024 NUM cana-590 55 4 ) ) PUNCT cana-590 55 5 29 29 NUM cana-590 55 6 https://internationalpubls.com https://internationalpubls.com X cana-590 55 7 selected select VERB cana-590 55 8 ( ( PUNCT cana-590 55 9 𝑧1 𝑧1 NOUN cana-590 55 10 , , PUNCT cana-590 55 11 … … PUNCT cana-590 55 12 . . PUNCT cana-590 55 13 , , PUNCT cana-590 55 14 𝑧𝑛 𝑧𝑛 PROPN cana-590 55 15 ) ) PUNCT cana-590 55 16 is be AUX cana-590 55 17 not not PART cana-590 55 18 valid valid ADJ cana-590 55 19 . . PUNCT cana-590 56 1 the the DET cana-590 56 2 subspace subspace NOUN cana-590 56 3 is be AUX cana-590 56 4 shown show VERB cana-590 56 5 by by ADP cana-590 56 6 the the DET cana-590 56 7 anti anti PROPN cana-590 56 8 holomorphic holomorphic PROPN cana-590 56 9 tangent tangent ADJ cana-590 56 10 space space NOUN cana-590 56 11 to to ADP cana-590 56 12 𝑀 𝑀 PROPN cana-590 56 13 at at ADP cana-590 56 14 p p PROPN cana-590 56 15 is be AUX cana-590 56 16 𝑇′′𝑝(m 𝑇′′𝑝(m NOUN cana-590 56 17 ) ) PUNCT cana-590 56 18 = = NOUN cana-590 56 19 𝐢 𝐢 PROPN cana-590 56 20 { { PUNCT cana-590 56 21 πœ• πœ• PROPN cana-590 56 22 πœ• πœ• PROPN cana-590 56 23 οΏ½ οΏ½ PROPN cana-590 56 24 Μ… Μ… NOUN cana-590 56 25 οΏ½ οΏ½ NOUN cana-590 56 26 𝑖 𝑖 X cana-590 56 27 } } PUNCT cana-590 56 28 . . PUNCT cana-590 57 1 clearly clearly ADV cana-590 57 2 , , PUNCT cana-590 57 3 actually actually ADV cana-590 57 4 , , PUNCT cana-590 57 5 a a DET cana-590 57 6 holomorphic holomorphic ADJ cana-590 57 7 map map NOUN cana-590 57 8 𝑓:𝑀 𝑓:𝑀 PUNCT cana-590 57 9 ⟢ ⟢ NOUN cana-590 57 10 𝑁 𝑁 PROPN cana-590 57 11 exists exist VERB cana-590 57 12 for for ADP cana-590 57 13 each each DET cana-590 57 14 p p NOUN cana-590 57 15 in in ADP cana-590 57 16 m m ADJ cana-590 57 17 only only ADV cana-590 57 18 if if SCONJ cana-590 57 19 πΉβˆ—(𝑇 πΉβˆ—(𝑇 NUM cana-590 57 20 β€² β€² NUM cana-590 57 21 𝑝(m 𝑝(m NOUN cana-590 57 22 ) ) PUNCT cana-590 57 23 ) ) PUNCT cana-590 58 1 βŠ‚ βŠ‚ PROPN cana-590 58 2 𝑇′𝑓(𝑝)(n 𝑇′𝑓(𝑝)(n PROPN cana-590 58 3 ) ) PUNCT cana-590 58 4 exists exist VERB cana-590 58 5 . . PUNCT cana-590 59 1 observe observe VERB cana-590 59 2 that that SCONJ cana-590 59 3 , , PUNCT cana-590 59 4 because because SCONJ cana-590 59 5 , , PUNCT cana-590 59 6 𝑇𝐢,𝑝(𝑀 𝑇𝐢,𝑝(𝑀 NOUN cana-590 59 7 ) ) PUNCT cana-590 59 8 is be AUX cana-590 59 9 provided provide VERB cana-590 59 10 naturally naturally ADV cana-590 59 11 as as SCONJ cana-590 59 12 the the DET cana-590 59 13 real real ADJ cana-590 59 14 vector vector NOUN cana-590 59 15 space space NOUN cana-590 59 16 , , PUNCT cana-590 59 17 𝑇𝑅 𝑇𝑅 PROPN cana-590 59 18 , , PUNCT cana-590 59 19 𝑝(𝑀 𝑝(𝑀 NOUN cana-590 59 20 ) ) PUNCT cana-590 59 21 tensored tensore VERB cana-590 59 22 with with ADP cana-590 59 23 , , PUNCT cana-590 59 24 the the DET cana-590 59 25 statement statement NOUN cana-590 59 26 conjugation conjugation NOUN cana-590 59 27 sending send VERB cana-590 59 28 πœ• πœ• NOUN cana-590 59 29 πœ•π‘§π‘– πœ•π‘§π‘– PRON cana-590 59 30 π‘‘π‘œ π‘‘π‘œ PROPN cana-590 59 31 πœ• πœ• PROPN cana-590 59 32 πœ• πœ• PROPN cana-590 59 33 οΏ½ οΏ½ PROPN cana-590 59 34 Μ… Μ… NOUN cana-590 59 35 οΏ½ οΏ½ NOUN cana-590 59 36 𝑖 𝑖 NOUN cana-590 59 37 is be AUX cana-590 59 38 well well ADV cana-590 59 39 defined define VERB cana-590 59 40 and and CCONJ cana-590 59 41 𝑇′′𝑝(𝑀 𝑇′′𝑝(𝑀 NOUN cana-590 59 42 ) ) PUNCT cana-590 59 43 = = NOUN cana-590 59 44 𝑇′𝑝(𝑀 𝑇′𝑝(𝑀 NOUN cana-590 59 45 ) ) PUNCT cana-590 59 46 . . PUNCT cana-590 60 1 the the DET cana-590 60 2 projection projection NOUN cana-590 60 3 𝑇𝑅,𝑝(m 𝑇𝑅,𝑝(m NOUN cana-590 60 4 ) ) PUNCT cana-590 60 5 ⟢ ⟢ NOUN cana-590 60 6 𝑇𝐢,𝑝(m)⟢ 𝑇𝐢,𝑝(m)⟢ NOUN cana-590 60 7 𝑇′𝑝(m 𝑇′𝑝(m NOUN cana-590 60 8 ) ) PUNCT cana-590 60 9 is be AUX cana-590 60 10 therefore therefore ADV cana-590 60 11 an an DET cana-590 60 12 r r NOUN cana-590 60 13 - - PUNCT cana-590 60 14 linear linear NOUN cana-590 60 15 isomorphism isomorphism NOUN cana-590 60 16 of of ADP cana-590 60 17 this this DET cana-590 60 18 type type NOUN cana-590 60 19 . . PUNCT cana-590 61 1 therefore therefore ADV cana-590 61 2 , , PUNCT cana-590 61 3 this this PRON cana-590 61 4 meant mean VERB cana-590 61 5 that that SCONJ cana-590 61 6 the the DET cana-590 61 7 only only ADJ cana-590 61 8 place place NOUN cana-590 61 9 where where SCONJ cana-590 61 10 we we PRON cana-590 61 11 could could AUX cana-590 61 12 '' '' PUNCT cana-590 61 13 do do VERB cana-590 61 14 geometry geometry NOUN cana-590 61 15 " " PUNCT cana-590 61 16 was be AUX cana-590 61 17 in in ADP cana-590 61 18 the the DET cana-590 61 19 holomorphic holomorphic ADJ cana-590 61 20 tangent tangent NOUN cana-590 61 21 space space NOUN cana-590 61 22 . . PUNCT cana-590 62 1 3.1 3.1 NUM cana-590 62 2 example example NOUN cana-590 62 3 : : PUNCT cana-590 62 4 assume assume VERB cana-590 62 5 that that SCONJ cana-590 62 6 𝑧(𝑑 𝑧(𝑑 PROPN cana-590 62 7 ) ) PUNCT cana-590 62 8 , , PUNCT cana-590 62 9 where where SCONJ cana-590 62 10 0 0 NUM cana-590 62 11 ≀ ≀ NUM cana-590 62 12 𝑑 𝑑 VERB cana-590 62 13 ≀ ≀ NOUN cana-590 62 14 1 1 NUM cana-590 62 15 , , PUNCT cana-590 62 16 is be AUX cana-590 62 17 a a DET cana-590 62 18 smooth smooth ADJ cana-590 62 19 arc arc NOUN cana-590 62 20 in in ADP cana-590 62 21 the the DET cana-590 62 22 complex complex ADJ cana-590 62 23 z z NOUN cana-590 62 24 - - PUNCT cana-590 62 25 plane plane NOUN cana-590 62 26 . . PUNCT cana-590 63 1 then then ADV cana-590 63 2 𝑧(𝑑 𝑧(𝑑 PROPN cana-590 63 3 ) ) PUNCT cana-590 63 4 = = SYM cana-590 63 5 π‘₯(𝑑 π‘₯(𝑑 NOUN cana-590 63 6 ) ) PUNCT cana-590 63 7 + + NUM cana-590 63 8 βˆšβˆ’1𝑦(𝑑 βˆšβˆ’1𝑦(𝑑 NUM cana-590 63 9 ) ) PUNCT cana-590 63 10 and and CCONJ cana-590 63 11 the the DET cana-590 63 12 tangent tangent NOUN cana-590 63 13 to to ADP cana-590 63 14 the the DET cana-590 63 15 arc arc NOUN cana-590 63 16 may may AUX cana-590 63 17 be be AUX cana-590 63 18 calculated calculate VERB cana-590 63 19 as as SCONJ cana-590 63 20 follows follow VERB cana-590 63 21 : : PUNCT cana-590 63 22 π‘₯β€²(t π‘₯β€²(t X cana-590 63 23 ) ) PUNCT cana-590 63 24 πœ• πœ• NOUN cana-590 63 25 πœ•π‘₯ πœ•π‘₯ NOUN cana-590 63 26 + + CCONJ cana-590 63 27 𝑦′(t 𝑦′(t NOUN cana-590 63 28 ) ) PUNCT cana-590 63 29 πœ• πœ• NOUN cana-590 63 30 πœ•π‘¦ πœ•π‘¦ NOUN cana-590 63 31 in in ADP cana-590 63 32 𝑇𝑅(𝐢 𝑇𝑅(𝐢 NOUN cana-590 63 33 ) ) PUNCT cana-590 63 34 or or CCONJ cana-590 63 35 𝑧′(t 𝑧′(t PROPN cana-590 63 36 ) ) PUNCT cana-590 63 37 πœ• πœ• PROPN cana-590 63 38 πœ•π‘§ πœ•π‘§ PROPN cana-590 63 39 in in ADP cana-590 63 40 𝑇′(𝐢 𝑇′(𝐢 NOUN cana-590 63 41 ) ) PUNCT cana-590 63 42 and and CCONJ cana-590 63 43 these these DET cana-590 63 44 two two NUM cana-590 63 45 coincide coincide NOUN cana-590 63 46 under under ADP cana-590 63 47 the the DET cana-590 63 48 projection projection NOUN cana-590 63 49 𝑇𝑅(𝐢)⟢ 𝑇𝑅(𝐢)⟢ NOUN cana-590 63 50 𝑇′(𝐢 𝑇′(𝐢 NOUN cana-590 63 51 ) ) PUNCT cana-590 63 52 . . PUNCT cana-590 64 1 suppose suppose VERB cana-590 64 2 that that SCONJ cana-590 64 3 𝑀 𝑀 PROPN cana-590 64 4 and and CCONJ cana-590 64 5 𝑁 𝑁 PROPN cana-590 64 6 are be AUX cana-590 64 7 complex complex ADJ cana-590 64 8 manifolds manifold NOUN cana-590 64 9 . . PUNCT cana-590 65 1 the the DET cana-590 65 2 set set NOUN cana-590 65 3 of of ADP cana-590 65 4 holomorphic holomorphic ADJ cana-590 65 5 coordinates coordinate NOUN cana-590 65 6 centred centre VERB cana-590 65 7 at at ADP cana-590 65 8 p∈m p∈m NOUN cana-590 65 9 is be AUX cana-590 65 10 denoted denote VERB cana-590 65 11 as as ADP cana-590 65 12 𝑧 𝑧 PROPN cana-590 65 13 = = PUNCT cana-590 65 14 ( ( PUNCT cana-590 65 15 𝑧1 𝑧1 NOUN cana-590 65 16 , , PUNCT cana-590 65 17 … … PUNCT cana-590 65 18 . . PUNCT cana-590 66 1 , , PUNCT cana-590 66 2 𝑧𝑛 𝑧𝑛 PROPN cana-590 66 3 ) ) PUNCT cana-590 66 4 . . PUNCT cana-590 67 1 a a DET cana-590 67 2 set set NOUN cana-590 67 3 of of ADP cana-590 67 4 holomorphic holomorphic ADJ cana-590 67 5 coordinates coordinate NOUN cana-590 67 6 centred centre VERB cana-590 67 7 at at ADP cana-590 67 8 π‘ž π‘ž PROPN cana-590 67 9 ∈ ∈ PROPN cana-590 67 10 𝑁 𝑁 PROPN cana-590 67 11 is be AUX cana-590 67 12 denoted denote VERB cana-590 67 13 as as ADP cana-590 67 14 𝑀 𝑀 ADP cana-590 67 15 =( =( PROPN cana-590 67 16 𝑀1 𝑀1 PROPN cana-590 67 17 , , PUNCT cana-590 67 18 … … PUNCT cana-590 67 19 . . PUNCT cana-590 67 20 , , PUNCT cana-590 67 21 𝑀𝑛 𝑀𝑛 PROPN cana-590 67 22 ) ) PUNCT cana-590 67 23 . . PUNCT cana-590 68 1 we we PRON cana-590 68 2 have have VERB cana-590 68 3 different different ADJ cana-590 68 4 thoughts thought NOUN cana-590 68 5 on on ADP cana-590 68 6 the the DET cana-590 68 7 jacobian jacobian NOUN cana-590 68 8 of of ADP cana-590 68 9 𝑓 𝑓 PRON cana-590 68 10 , , PUNCT cana-590 68 11 we we PRON cana-590 68 12 refer refer VERB cana-590 68 13 to to ADP cana-590 68 14 [ [ X cana-590 68 15 𝐺 𝐺 PROPN cana-590 68 16 , , PUNCT cana-590 68 17 𝐻 𝐻 PROPN cana-590 68 18 ] ] PUNCT cana-590 68 19 . . PUNCT cana-590 69 1 with with ADP cana-590 69 2 𝑓(𝑝 𝑓(𝑝 PROPN cana-590 69 3 ) ) PUNCT cana-590 69 4 = = SYM cana-590 69 5 π‘ž π‘ž PROPN cana-590 69 6 , , PUNCT cana-590 69 7 the the DET cana-590 69 8 holomorphic holomorphic ADJ cana-590 69 9 map map NOUN cana-590 69 10 𝑓:𝑀 𝑓:𝑀 NUM cana-590 69 11 ⟢ ⟢ NOUN cana-590 69 12 𝑁 𝑁 PROPN cana-590 69 13 corresponds correspond NOUN cana-590 69 14 to to ADP cana-590 69 15 the the DET cana-590 69 16 different different ADJ cana-590 69 17 tangent tangent NOUN cana-590 69 18 spaces space VERB cana-590 69 19 to to ADP cana-590 69 20 𝑀 𝑀 PROPN cana-590 69 21 and and CCONJ cana-590 69 22 𝑁 𝑁 PROPN cana-590 69 23 at at ADP cana-590 69 24 𝑝 𝑝 PROPN cana-590 69 25 and and CCONJ cana-590 69 26 π‘ž π‘ž PROPN cana-590 69 27 , , PUNCT cana-590 69 28 respectively respectively ADV cana-590 69 29 . . PUNCT cana-590 70 1 4 4 X cana-590 70 2 . . X cana-590 70 3 sub sub NOUN cana-590 70 4 manifolds manifold NOUN cana-590 70 5 and and CCONJ cana-590 70 6 sub sub NOUN cana-590 70 7 varieties variety NOUN cana-590 70 8 4.1 4.1 NUM cana-590 70 9 definition definition NOUN cana-590 70 10 : : PUNCT cana-590 70 11 specified specify VERB cana-590 70 12 sub sub NOUN cana-590 70 13 manifold manifold PROPN cana-590 70 14 s s PROPN cana-590 70 15 of of ADP cana-590 70 16 a a DET cana-590 70 17 complicated complicated ADJ cana-590 70 18 manifold manifold ADJ cana-590 70 19 m m NOUN cana-590 70 20 with with ADP cana-590 70 21 complex complex ADJ cana-590 70 22 characteristics characteristic NOUN cana-590 70 23 a a DET cana-590 70 24 finite finite ADJ cana-590 70 25 collection collection NOUN cana-590 70 26 of of ADP cana-590 70 27 holomorphic holomorphic ADJ cana-590 70 28 functions function NOUN cana-590 70 29 𝑓1 𝑓1 PROPN cana-590 70 30 , , PUNCT cana-590 70 31 𝑓2 𝑓2 NOUN cana-590 70 32 , , PUNCT cana-590 70 33 . . PUNCT cana-590 70 34 . . PUNCT cana-590 71 1 . . PUNCT cana-590 72 1 , , PUNCT cana-590 72 2 π‘“π‘˜ π‘“π‘˜ VERB cana-590 72 3 with with ADP cana-590 72 4 rank rank NOUN cana-590 72 5 𝑔(𝑓 𝑔(𝑓 PROPN cana-590 72 6 ) ) PUNCT cana-590 73 1 = = PUNCT cana-590 73 2 π‘˜ π‘˜ NOUN cana-590 73 3 is be AUX cana-590 73 4 represented represent VERB cana-590 73 5 locally locally ADV cana-590 73 6 by by ADP cana-590 73 7 the the DET cana-590 73 8 subset subset PROPN cana-590 73 9 𝑀 𝑀 PROPN cana-590 73 10 , , PUNCT cana-590 73 11 which which PRON cana-590 73 12 is be AUX cana-590 73 13 known know VERB cana-590 73 14 as as ADP cana-590 73 15 the the DET cana-590 73 16 zeros zero NOUN cana-590 73 17 . . PUNCT cana-590 74 1 the the DET cana-590 74 2 sign sign PROPN cana-590 74 3 π‘‰βˆ— π‘‰βˆ— NOUN cana-590 74 4 is be AUX cana-590 74 5 used use VERB cana-590 74 6 to to PART cana-590 74 7 denote denote VERB cana-590 74 8 the the DET cana-590 74 9 location location NOUN cana-590 74 10 of of ADP cana-590 74 11 smooth smooth ADJ cana-590 74 12 points point NOUN cana-590 74 13 in in ADP cana-590 74 14 the the DET cana-590 74 15 curve curve NOUN cana-590 74 16 𝑉. 𝑉. PROPN cana-590 74 17 on on ADP cana-590 74 18 the the DET cana-590 74 19 other other ADJ cana-590 74 20 hand hand NOUN cana-590 74 21 , , PUNCT cana-590 74 22 a a DET cana-590 74 23 singular singular ADJ cana-590 74 24 point point NOUN cana-590 74 25 of of ADP cana-590 74 26 𝑉 𝑉 PROPN cana-590 74 27 is be AUX cana-590 74 28 defined define VERB cana-590 74 29 as as ADP cana-590 74 30 𝑝 𝑝 PROPN cana-590 74 31 ∈ ∈ PROPN cana-590 74 32 𝑉 𝑉 PROPN cana-590 74 33 π‘‰βˆ— π‘‰βˆ— NOUN cana-590 74 34 , , PUNCT cana-590 74 35 and and CCONJ cana-590 74 36 the the DET cana-590 74 37 singular singular ADJ cana-590 74 38 locus locus NOUN cana-590 74 39 of of ADP cana-590 74 40 𝑉 𝑉 PROPN cana-590 74 41 is be AUX cana-590 74 42 denoted denote VERB cana-590 74 43 by by ADP cana-590 74 44 𝑉𝑠. 𝑉𝑠. NOUN cana-590 74 45 the the DET cana-590 74 46 only only ADJ cana-590 74 47 conditions condition NOUN cana-590 74 48 under under ADP cana-590 74 49 which which PRON cana-590 74 50 𝑉 𝑉 PROPN cana-590 74 51 is be AUX cana-590 74 52 deemed deem VERB cana-590 74 53 smooth smooth ADJ cana-590 74 54 or or CCONJ cana-590 74 55 nonsingular nonsingular ADJ cana-590 74 56 are be AUX cana-590 74 57 those those PRON cana-590 74 58 in in ADP cana-590 74 59 which which PRON cana-590 74 60 it it PRON cana-590 74 61 is be AUX cana-590 74 62 a a DET cana-590 74 63 sub sub NOUN cana-590 74 64 - - ADJ cana-590 74 65 manifold manifold ADJ cana-590 74 66 of of ADP cana-590 74 67 𝑀 𝑀 PROPN cana-590 74 68 or or CCONJ cana-590 74 69 if if SCONJ cana-590 74 70 𝑉 𝑉 PROPN cana-590 74 71 is be AUX cana-590 74 72 equal equal ADJ cana-590 74 73 to to PART cana-590 74 74 π‘‰βˆ—. π‘‰βˆ—. VERB cana-590 74 75 specifically specifically ADV cana-590 74 76 , , PUNCT cana-590 74 77 for for ADP cana-590 74 78 every every DET cana-590 74 79 point point NOUN cana-590 74 80 𝑝 𝑝 NOUN cana-590 74 81 on on ADP cana-590 74 82 an an DET cana-590 74 83 analytic analytic ADJ cana-590 74 84 hyper hyper ADJ cana-590 74 85 surface surface NOUN cana-590 74 86 𝑉 𝑉 PROPN cana-590 74 87 βŠ‚ βŠ‚ PROPN cana-590 74 88 𝑀 𝑀 PROPN cana-590 74 89 defined define VERB cana-590 74 90 in in ADP cana-590 74 91 terms term NOUN cana-590 74 92 of of ADP cana-590 74 93 local local ADJ cana-590 74 94 coordinates coordinate NOUN cana-590 74 95 𝑍 𝑍 VERB cana-590 74 96 by by ADP cana-590 74 97 the the DET cana-590 74 98 function function NOUN cana-590 74 99 𝑓 𝑓 X cana-590 74 100 , , PUNCT cana-590 74 101 we we PRON cana-590 74 102 say say VERB cana-590 74 103 that that SCONJ cana-590 74 104 the the DET cana-590 74 105 multiplicity multiplicity NOUN cana-590 74 106 π‘šπ‘’π‘™π‘(𝑉 π‘šπ‘’π‘™π‘(𝑉 NOUN cana-590 74 107 ) ) PUNCT cana-590 74 108 is be AUX cana-590 74 109 the the DET cana-590 74 110 degree degree NOUN cana-590 74 111 to to PART cana-590 74 112 which which PRON cana-590 74 113 𝑓 𝑓 DET cana-590 74 114 eliminates eliminate NOUN cana-590 74 115 at at ADP cana-590 74 116 𝑝 𝑝 NOUN cana-590 74 117 , , PUNCT cana-590 74 118 or or CCONJ cana-590 74 119 the the DET cana-590 74 120 greatest great ADJ cana-590 74 121 number number NOUN cana-590 74 122 m m VERB cana-590 74 123 for for ADP cana-590 74 124 which which PRON cana-590 74 125 all all DET cana-590 74 126 partial partial ADJ cana-590 74 127 derivatives derivative NOUN cana-590 74 128 are be AUX cana-590 74 129 identical identical ADJ cana-590 74 130 πœ•π‘˜π‘“ πœ•π‘˜π‘“ ADV cana-590 74 131 πœ•π‘§π‘–1, πœ•π‘§π‘–1, X cana-590 74 132 … … PUNCT cana-590 74 133 … … PUNCT cana-590 74 134 … … PUNCT cana-590 74 135 πœ•π‘§π‘–π‘˜ πœ•π‘§π‘–π‘˜ NOUN cana-590 74 136 ( ( PUNCT cana-590 74 137 𝑃 𝑃 NOUN cana-590 74 138 ) ) PUNCT cana-590 74 139 = = SYM cana-590 74 140 0 0 NUM cana-590 74 141 , , PUNCT cana-590 74 142 π‘˜ π‘˜ DET cana-590 74 143 ≀ ≀ NOUN cana-590 74 144 π‘š π‘š ADP cana-590 74 145 βˆ’ βˆ’ PROPN cana-590 74 146 1 1 NUM cana-590 74 147 . . PUNCT cana-590 75 1 communications communication NOUN cana-590 75 2 on on ADP cana-590 75 3 applied apply VERB cana-590 75 4 nonlinear nonlinear ADJ cana-590 75 5 analysis analysis NOUN cana-590 75 6 issn issn NOUN cana-590 75 7 : : PUNCT cana-590 75 8 1074 1074 NUM cana-590 75 9 - - PUNCT cana-590 75 10 133x 133x NUM cana-590 75 11 vol vol NOUN cana-590 75 12 31 31 NUM cana-590 75 13 no no NOUN cana-590 75 14 . . PUNCT cana-590 76 1 2s 2s NUM cana-590 76 2 ( ( PUNCT cana-590 76 3 2024 2024 NUM cana-590 76 4 ) ) PUNCT cana-590 76 5 30 30 NUM cana-590 76 6 https://internationalpubls.com https://internationalpubls.com X cana-590 76 7 when when SCONJ cana-590 76 8 dealing deal VERB cana-590 76 9 with with ADP cana-590 76 10 a a DET cana-590 76 11 family family NOUN cana-590 76 12 of of ADP cana-590 76 13 objects object NOUN cana-590 76 14 that that PRON cana-590 76 15 are be AUX cana-590 76 16 parametrized parametrize VERB cana-590 76 17 locally locally ADV cana-590 76 18 by by ADP cana-590 76 19 a a DET cana-590 76 20 complex complex ADJ cana-590 76 21 manifold manifold NOUN cana-590 76 22 or or CCONJ cana-590 76 23 an an DET cana-590 76 24 analytic analytic ADJ cana-590 76 25 subvariety subvariety NOUN cana-590 76 26 of of ADP cana-590 76 27 a a DET cana-590 76 28 complex complex ADJ cana-590 76 29 manifold manifold NOUN cana-590 76 30 , , PUNCT cana-590 76 31 the the DET cana-590 76 32 statement statement NOUN cana-590 76 33 '' '' PUNCT cana-590 76 34 a a DET cana-590 76 35 generic generic ADJ cana-590 76 36 relative relative NOUN cana-590 76 37 has have VERB cana-590 76 38 a a DET cana-590 76 39 certain certain ADJ cana-590 76 40 property property NOUN cana-590 76 41 '' '' PUNCT cana-590 76 42 indicates indicate VERB cana-590 76 43 that that SCONJ cana-590 76 44 the the DET cana-590 76 45 set set NOUN cana-590 76 46 of of ADP cana-590 76 47 objects object NOUN cana-590 76 48 in in ADP cana-590 76 49 the the DET cana-590 76 50 family family NOUN cana-590 76 51 that that PRON cana-590 76 52 do do AUX cana-590 76 53 not not PART cana-590 76 54 possess possess VERB cana-590 76 55 the the DET cana-590 76 56 property property NOUN cana-590 76 57 is be AUX cana-590 76 58 contained contain VERB cana-590 76 59 in in ADP cana-590 76 60 a a DET cana-590 76 61 sub sub NOUN cana-590 76 62 variety variety NOUN cana-590 76 63 of of ADP cana-590 76 64 strictly strictly ADV cana-590 76 65 smaller small ADJ cana-590 76 66 dimension dimension NOUN cana-590 76 67 . . PUNCT cana-590 77 1 this this PRON cana-590 77 2 is be AUX cana-590 77 3 the the DET cana-590 77 4 case case NOUN cana-590 77 5 when when SCONJ cana-590 77 6 dealing deal VERB cana-590 77 7 with with ADP cana-590 77 8 a a DET cana-590 77 9 complex complex ADJ cana-590 77 10 manifold manifold NOUN cana-590 77 11 . . PUNCT cana-590 78 1 in in ADP cana-590 78 2 most most ADJ cana-590 78 3 cases case NOUN cana-590 78 4 , , PUNCT cana-590 78 5 the the DET cana-590 78 6 proper proper ADJ cana-590 78 7 way way NOUN cana-590 78 8 to to PART cana-590 78 9 parametrize parametrize VERB cana-590 78 10 items item NOUN cana-590 78 11 in in ADP cana-590 78 12 our our PRON cana-590 78 13 family family NOUN cana-590 78 14 will will AUX cana-590 78 15 be be AUX cana-590 78 16 obvious obvious ADJ cana-590 78 17 . . PUNCT cana-590 79 1 in in ADP cana-590 79 2 ℙ𝑛 ℙ𝑛 PROPN cana-590 79 3 , , PUNCT cana-590 79 4 the the DET cana-590 79 5 generic generic ADJ cana-590 79 6 π‘˜-plane π‘˜-plane PROPN cana-590 79 7 is be AUX cana-590 79 8 an an DET cana-590 79 9 exception exception NOUN cana-590 79 10 ( ( PUNCT cana-590 79 11 this this DET cana-590 79 12 one one NOUN cana-590 79 13 will will AUX cana-590 79 14 be be AUX cana-590 79 15 again again ADV cana-590 79 16 revisited revisit VERB cana-590 79 17 in in ADP cana-590 79 18 the the DET cana-590 79 19 sections section NOUN cana-590 79 20 on on ADP cana-590 79 21 grassmannians grassmannian NOUN cana-590 79 22 in in ADP cana-590 79 23 the the DET cana-590 79 24 work work NOUN cana-590 79 25 that that PRON cana-590 79 26 follows follow VERB cana-590 79 27 later later ADV cana-590 79 28 ) ) PUNCT cana-590 79 29 . . PUNCT cana-590 80 1 4.2 4.2 NUM cana-590 80 2 remark remark NOUN cana-590 80 3 : : PUNCT cana-590 80 4 grassmannians grassmannian NOUN cana-590 80 5 are be AUX cana-590 80 6 a a DET cana-590 80 7 family family NOUN cana-590 80 8 of of ADP cana-590 80 9 compact compact ADJ cana-590 80 10 complex complex ADJ cana-590 80 11 manifolds manifold NOUN cana-590 80 12 . . PUNCT cana-590 81 1 to to PART cana-590 81 2 be be AUX cana-590 81 3 specific specific ADJ cana-590 81 4 are be AUX cana-590 81 5 generalizations generalization NOUN cana-590 81 6 to to PART cana-590 81 7 projective projective VERB cana-590 81 8 spaces space NOUN cana-590 81 9 . . PUNCT cana-590 82 1 they they PRON cana-590 82 2 have have VERB cana-590 82 3 rich rich ADJ cana-590 82 4 topological topological ADJ cana-590 82 5 and and CCONJ cana-590 82 6 smooth smooth ADJ cana-590 82 7 manifold manifold ADJ cana-590 82 8 structure structure NOUN cana-590 82 9 . . PUNCT cana-590 83 1 before before SCONJ cana-590 83 2 we we PRON cana-590 83 3 define define VERB cana-590 83 4 them they PRON cana-590 83 5 formally formally ADV cana-590 83 6 some some DET cana-590 83 7 background background NOUN cana-590 83 8 of of ADP cana-590 83 9 affine affine NOUN cana-590 83 10 and and CCONJ cana-590 83 11 projective projective ADJ cana-590 83 12 vector vector NOUN cana-590 83 13 spaces space NOUN cana-590 83 14 are be AUX cana-590 83 15 defined define VERB cana-590 83 16 . . PUNCT cana-590 84 1 the the DET cana-590 84 2 setting setting NOUN cana-590 84 3 is be AUX cana-590 84 4 complex complex ADJ cana-590 84 5 . . PUNCT cana-590 85 1 5 5 X cana-590 85 2 . . X cana-590 85 3 vector vector NOUN cana-590 85 4 spaces space NOUN cana-590 85 5 : : PUNCT cana-590 85 6 these these PRON cana-590 85 7 are be AUX cana-590 85 8 rich rich ADJ cana-590 85 9 algebraic algebraic ADJ cana-590 85 10 structures structure NOUN cana-590 85 11 under under ADP cana-590 85 12 additive additive ADJ cana-590 85 13 binary binary ADJ cana-590 85 14 operation operation NOUN cana-590 85 15 they they PRON cana-590 85 16 form form VERB cana-590 85 17 abelian abelian ADJ cana-590 85 18 group group NOUN cana-590 85 19 and and CCONJ cana-590 85 20 multiplication/ multiplication/ NUM cana-590 85 21 scalar scalar ADJ cana-590 85 22 distribution distribution NOUN cana-590 85 23 operation operation NOUN cana-590 85 24 ensures ensure VERB cana-590 85 25 distribution distribution NOUN cana-590 85 26 ensures ensure VERB cana-590 85 27 distribution distribution NOUN cana-590 85 28 properties property NOUN cana-590 85 29 . . PUNCT cana-590 86 1 one one NUM cana-590 86 2 main main ADJ cana-590 86 3 thing thing NOUN cana-590 86 4 about about ADP cana-590 86 5 them they PRON cana-590 86 6 is be AUX cana-590 86 7 that that SCONJ cana-590 86 8 they they PRON cana-590 86 9 appear appear VERB cana-590 86 10 as as ADP cana-590 86 11 pairs pair NOUN cana-590 86 12 of of ADP cana-590 86 13 fields field NOUN cana-590 86 14 . . PUNCT cana-590 87 1 thus thus ADV cana-590 87 2 every every DET cana-590 87 3 field field NOUN cana-590 87 4 is be AUX cana-590 87 5 a a DET cana-590 87 6 vector vector NOUN cana-590 87 7 space space NOUN cana-590 87 8 over over ADP cana-590 87 9 itself itself PRON cana-590 87 10 . . PUNCT cana-590 88 1 the the DET cana-590 88 2 best good ADJ cana-590 88 3 examples example NOUN cana-590 88 4 are be AUX cana-590 88 5 the the DET cana-590 88 6 real real ADJ cana-590 88 7 field field NOUN cana-590 88 8 𝑅 𝑅 PROPN cana-590 88 9 is be AUX cana-590 88 10 a a DET cana-590 88 11 vector vector NOUN cana-590 88 12 space space NOUN cana-590 88 13 projected project VERB cana-590 88 14 onto onto ADP cana-590 88 15 itself itself PRON cana-590 88 16 . . PUNCT cana-590 89 1 𝐢 𝐢 PROPN cana-590 89 2 , , PUNCT cana-590 89 3 the the DET cana-590 89 4 selfdescribing selfdescribing NOUN cana-590 89 5 complex complex ADJ cana-590 89 6 field field NOUN cana-590 89 7 vector vector NOUN cana-590 89 8 space space NOUN cana-590 89 9 . . PUNCT cana-590 90 1 next next ADJ cana-590 90 2 , , PUNCT cana-590 90 3 is be AUX cana-590 90 4 their their PRON cana-590 90 5 higher high ADJ cana-590 90 6 analogues analogue NOUN cana-590 90 7 𝑅𝑛 𝑅𝑛 PROPN cana-590 90 8 over over ADP cana-590 90 9 𝑅 𝑅 PROPN cana-590 90 10 , , PUNCT cana-590 90 11 𝐢𝑛 𝐢𝑛 NOUN cana-590 90 12 over over ADP cana-590 90 13 𝐢. 𝐢. PROPN cana-590 90 14 if if SCONJ cana-590 90 15 we we PRON cana-590 90 16 choose choose VERB cana-590 90 17 a a DET cana-590 90 18 vector vector NOUN cana-590 90 19 space space NOUN cana-590 90 20 with with ADP cana-590 90 21 n n ADP cana-590 90 22 dimensions dimension NOUN cana-590 90 23 that that PRON cana-590 90 24 is be AUX cana-590 90 25 isomorphic isomorphic ADJ cana-590 90 26 to to ADP cana-590 90 27 𝑅2𝑛 𝑅2𝑛 PROPN cana-590 90 28 , , PUNCT cana-590 90 29 then then ADV cana-590 90 30 n n PRON cana-590 90 31 is be AUX cana-590 90 32 its its PRON cana-590 90 33 dimension dimension NOUN cana-590 90 34 . . PUNCT cana-590 91 1 5.1 5.1 NUM cana-590 91 2 proposition proposition NOUN cana-590 91 3 : : PUNCT cana-590 91 4 𝐹𝑛 𝐹𝑛 PROPN cana-590 91 5 is be AUX cana-590 91 6 isomorphic isomorphic ADJ cana-590 91 7 to to ADP cana-590 91 8 any any DET cana-590 91 9 finite finite ADJ cana-590 91 10 dimensional dimensional ADJ cana-590 91 11 vector vector NOUN cana-590 91 12 space space NOUN cana-590 91 13 of of ADP cana-590 91 14 size size NOUN cana-590 91 15 𝑛 𝑛 PROPN cana-590 91 16 , , PUNCT cana-590 91 17 where where SCONJ cana-590 91 18 𝐹 𝐹 PROPN cana-590 91 19 is be AUX cana-590 91 20 a a DET cana-590 91 21 scalar scalar ADJ cana-590 91 22 field field NOUN cana-590 91 23 . . PUNCT cana-590 92 1 the the DET cana-590 92 2 real real ADJ cana-590 92 3 field field NOUN cana-590 92 4 𝐹 𝐹 PROPN cana-590 92 5 = = SYM cana-590 92 6 𝑅 𝑅 PROPN cana-590 92 7 is be AUX cana-590 92 8 defined define VERB cana-590 92 9 as as ADP cana-590 92 10 a a DET cana-590 92 11 real real ADJ cana-590 92 12 vector vector NOUN cana-590 92 13 space space NOUN cana-590 92 14 𝑉 𝑉 PROPN cana-590 92 15 ′𝑠of ′𝑠of PROPN cana-590 92 16 size size NOUN cana-590 92 17 n n CCONJ cana-590 92 18 over over ADV cana-590 92 19 𝑅. 𝑅. ADJ cana-590 92 20 proof proof NOUN cana-590 92 21 : : PUNCT cana-590 92 22 each each DET cana-590 92 23 vector vector NOUN cana-590 92 24 space space NOUN cana-590 92 25 has have VERB cana-590 92 26 a a DET cana-590 92 27 basis basis NOUN cana-590 92 28 , , PUNCT cana-590 92 29 which which PRON cana-590 92 30 is be AUX cana-590 92 31 not not PART cana-590 92 32 unique unique ADJ cana-590 92 33 ; ; PUNCT cana-590 92 34 two two NUM cana-590 92 35 bases basis NOUN cana-590 92 36 of of ADP cana-590 92 37 v v NOUN cana-590 92 38 can can AUX cana-590 92 39 be be AUX cana-590 92 40 equivalent equivalent ADJ cana-590 92 41 , , PUNCT cana-590 92 42 but but CCONJ cana-590 92 43 the the DET cana-590 92 44 dimension dimension NOUN cana-590 92 45 of of ADP cana-590 92 46 n n PROPN cana-590 92 47 is be AUX cana-590 92 48 unique unique ADJ cana-590 92 49 . . PUNCT cana-590 93 1 we we PRON cana-590 93 2 take take VERB cana-590 93 3 these these DET cana-590 93 4 vital vital ADJ cana-590 93 5 facts fact NOUN cana-590 93 6 for for SCONJ cana-590 93 7 granted grant VERB cana-590 93 8 , , PUNCT cana-590 93 9 despite despite SCONJ cana-590 93 10 their their PRON cana-590 93 11 importance importance NOUN cana-590 93 12 . . PUNCT cana-590 94 1 an an DET cana-590 94 2 instance instance NOUN cana-590 94 3 of of ADP cana-590 94 4 an an DET cana-590 94 5 inner inner ADJ cana-590 94 6 product product NOUN cana-590 94 7 on on ADP cana-590 94 8 𝑅𝑛 𝑅𝑛 PROPN cana-590 94 9 is be AUX cana-590 94 10 the the DET cana-590 94 11 bilinear bilinear NOUN cana-590 94 12 map map NOUN cana-590 95 1 𝑓 𝑓 PRON cana-590 95 2 : : PUNCT cana-590 95 3 𝑅𝑋𝑅 𝑅𝑋𝑅 PROPN cana-590 95 4 β†’ β†’ SYM cana-590 95 5 𝑅 𝑅 PROPN cana-590 95 6 , , PUNCT cana-590 95 7 which which PRON cana-590 95 8 is be AUX cana-590 95 9 positive positive ADJ cana-590 95 10 definite definite ADJ cana-590 95 11 , , PUNCT cana-590 95 12 linear linear ADJ cana-590 95 13 in in ADP cana-590 95 14 the the DET cana-590 95 15 slot slot NOUN cana-590 95 16 , , PUNCT cana-590 95 17 and and CCONJ cana-590 95 18 compatible compatible ADJ cana-590 95 19 with with ADP cana-590 95 20 scalar scalar ADJ cana-590 95 21 multiplication multiplication NOUN cana-590 95 22 . . PUNCT cana-590 96 1 in in ADP cana-590 96 2 reality reality NOUN cana-590 96 3 , , PUNCT cana-590 96 4 the the DET cana-590 96 5 only only ADJ cana-590 96 6 differences difference NOUN cana-590 96 7 are be AUX cana-590 96 8 that that SCONJ cana-590 96 9 the the DET cana-590 96 10 inner inner ADJ cana-590 96 11 product product NOUN cana-590 96 12 is be AUX cana-590 96 13 hermitian hermitian ADJ cana-590 96 14 and and CCONJ cana-590 96 15 complex complex ADJ cana-590 96 16 numbers number NOUN cana-590 96 17 have have VERB cana-590 96 18 conjugates conjugate NOUN cana-590 96 19 . . PUNCT cana-590 97 1 a a DET cana-590 97 2 metric metric ADJ cana-590 97 3 induces induce VERB cana-590 97 4 an an DET cana-590 97 5 inner inner ADJ cana-590 97 6 product product NOUN cana-590 97 7 space space NOUN cana-590 97 8 , , PUNCT cana-590 97 9 which which PRON cana-590 97 10 in in ADP cana-590 97 11 turn turn NOUN cana-590 97 12 is be AUX cana-590 97 13 a a DET cana-590 97 14 metric metric ADJ cana-590 97 15 space space NOUN cana-590 97 16 . . PUNCT cana-590 98 1 grassmannians grassmannian NOUN cana-590 98 2 are be AUX cana-590 98 3 obviously obviously ADV cana-590 98 4 of of ADP cana-590 98 5 importance importance NOUN cana-590 98 6 to to ADP cana-590 98 7 us we PRON cana-590 98 8 . . PUNCT cana-590 99 1 we we PRON cana-590 99 2 briefly briefly ADV cana-590 99 3 touch touch VERB cana-590 99 4 on on ADP cana-590 99 5 affine affine NOUN cana-590 99 6 projective projective PROPN cana-590 99 7 geometry geometry NOUN cana-590 99 8 as as ADP cana-590 99 9 a a DET cana-590 99 10 digression digression NOUN cana-590 99 11 before before ADP cana-590 99 12 moving move VERB cana-590 99 13 on on ADP cana-590 99 14 to to ADP cana-590 99 15 the the DET cana-590 99 16 primary primary ADJ cana-590 99 17 topic topic NOUN cana-590 99 18 . . PUNCT cana-590 100 1 observe observe VERB cana-590 100 2 that that SCONJ cana-590 100 3 𝑅𝑛 𝑅𝑛 NOUN cana-590 100 4 can can AUX cana-590 100 5 be be AUX cana-590 100 6 decomposed decompose VERB cana-590 100 7 by by ADP cana-590 100 8 writing write VERB cana-590 100 9 it it PRON cana-590 100 10 as as ADP cana-590 100 11 π‘…π‘˜(π‘›βˆ’π‘˜ π‘…π‘˜(π‘›βˆ’π‘˜ ADJ cana-590 100 12 ) ) PUNCT cana-590 100 13 . . PUNCT cana-590 101 1 if if SCONJ cana-590 101 2 it it PRON cana-590 101 3 were be AUX cana-590 101 4 𝐢 𝐢 PROPN cana-590 101 5 then then ADV cana-590 101 6 πΆπ‘˜(π‘›βˆ’π‘˜ πΆπ‘˜(π‘›βˆ’π‘˜ NOUN cana-590 101 7 ) ) PUNCT cana-590 101 8 . . PUNCT cana-590 102 1 π‘˜ π‘˜ DET cana-590 102 2 rows row NOUN cana-590 102 3 and and CCONJ cana-590 102 4 ( ( PUNCT cana-590 102 5 𝑛 𝑛 PROPN cana-590 102 6 βˆ’ βˆ’ PROPN cana-590 102 7 π‘˜ π‘˜ NOUN cana-590 102 8 ) ) PUNCT cana-590 102 9 column column NOUN cana-590 102 10 matrices matrix NOUN cana-590 102 11 are be AUX cana-590 102 12 imaginable imaginable ADJ cana-590 102 13 in in ADP cana-590 102 14 the the DET cana-590 102 15 language language NOUN cana-590 102 16 of of ADP cana-590 102 17 matrices matrix NOUN cana-590 102 18 π‘˜. π‘˜. NOUN cana-590 103 1 ( ( PUNCT cana-590 103 2 𝑛 𝑛 PROPN cana-590 103 3 βˆ’ βˆ’ PROPN cana-590 103 4 π‘˜ π‘˜ NOUN cana-590 103 5 ) ) PUNCT cana-590 103 6 . . PUNCT cana-590 104 1 when when SCONJ cana-590 104 2 a a DET cana-590 104 3 metric metric NOUN cana-590 104 4 is be AUX cana-590 104 5 represented represent VERB cana-590 104 6 by by ADP cana-590 104 7 a a DET cana-590 104 8 π‘˜-row π‘˜-row NOUN cana-590 104 9 vector vector NOUN cana-590 104 10 and and CCONJ cana-590 104 11 vice vice NOUN cana-590 104 12 versa versa ADV cana-590 104 13 , , PUNCT cana-590 104 14 two two NUM cana-590 104 15 matrices matrix NOUN cana-590 104 16 a a PRON cana-590 104 17 and and CCONJ cana-590 104 18 b b NOUN cana-590 104 19 are be AUX cana-590 104 20 declared declare VERB cana-590 104 21 to to PART cana-590 104 22 have have VERB cana-590 104 23 the the DET cana-590 104 24 same same ADJ cana-590 104 25 mean mean NOUN cana-590 104 26 if if SCONJ cana-590 104 27 there there PRON cana-590 104 28 is be VERB cana-590 104 29 a a DET cana-590 104 30 scalar scalar ADJ cana-590 104 31 𝑔 𝑔 NOUN cana-590 104 32 such such ADJ cana-590 104 33 that that DET cana-590 104 34 𝐴 𝐴 PROPN cana-590 104 35 = = PUNCT cana-590 105 1 𝑔. 𝑔. PROPN cana-590 105 2 𝐡. 𝐡. PROPN cana-590 105 3 this this DET cana-590 105 4 connection connection NOUN cana-590 105 5 splits split VERB cana-590 105 6 the the DET cana-590 105 7 set set NOUN cana-590 105 8 into into ADP cana-590 105 9 equivalent equivalent ADJ cana-590 105 10 classes class NOUN cana-590 105 11 and and CCONJ cana-590 105 12 is be AUX cana-590 105 13 an an DET cana-590 105 14 equivalent equivalent ADJ cana-590 105 15 relation relation NOUN cana-590 105 16 . . PUNCT cana-590 106 1 communications communication NOUN cana-590 106 2 on on ADP cana-590 106 3 applied apply VERB cana-590 106 4 nonlinear nonlinear ADJ cana-590 106 5 analysis analysis NOUN cana-590 106 6 issn issn NOUN cana-590 106 7 : : PUNCT cana-590 106 8 1074 1074 NUM cana-590 106 9 - - PUNCT cana-590 106 10 133x 133x NUM cana-590 106 11 vol vol NOUN cana-590 106 12 31 31 NUM cana-590 106 13 no no NOUN cana-590 106 14 . . PUNCT cana-590 107 1 2s 2s NUM cana-590 107 2 ( ( PUNCT cana-590 107 3 2024 2024 NUM cana-590 107 4 ) ) PUNCT cana-590 107 5 31 31 NUM cana-590 107 6 https://internationalpubls.com https://internationalpubls.com X cana-590 107 7 since since SCONJ cana-590 107 8 π‘₯ π‘₯ PROPN cana-590 107 9 and and CCONJ cana-590 107 10 𝑦 𝑦 NOUN cana-590 107 11 in in ADP cana-590 107 12 the the DET cana-590 107 13 real real ADJ cana-590 107 14 plane plane NOUN cana-590 107 15 𝑅2 𝑅2 NOUN cana-590 107 16 are be AUX cana-590 107 17 equivalent equivalent ADJ cana-590 107 18 if if SCONJ cana-590 108 1 and and CCONJ cana-590 108 2 only only ADV cana-590 108 3 if if SCONJ cana-590 108 4 𝑦 𝑦 NOUN cana-590 108 5 = = X cana-590 108 6 π‘Žπ‘₯ π‘Žπ‘₯ NOUN cana-590 108 7 for for ADP cana-590 108 8 some some DET cana-590 108 9 constant constant ADJ cana-590 108 10 a a PRON cana-590 108 11 , , PUNCT cana-590 108 12 we we PRON cana-590 108 13 can can AUX cana-590 108 14 assert assert VERB cana-590 108 15 that that SCONJ cana-590 108 16 π‘₯ π‘₯ PROPN cana-590 108 17 and and CCONJ cana-590 108 18 𝑦 𝑦 NOUN cana-590 108 19 are be AUX cana-590 108 20 equivalent equivalent ADJ cana-590 108 21 . . PUNCT cana-590 109 1 the the DET cana-590 109 2 equivalence equivalence NOUN cana-590 109 3 class class NOUN cana-590 109 4 determined determine VERB cana-590 109 5 by by ADP cana-590 109 6 𝑋 𝑋 PROPN cana-590 109 7 is be AUX cana-590 109 8 indicated indicate VERB cana-590 109 9 by by ADP cana-590 109 10 [ [ X cana-590 109 11 𝑋 𝑋 PROPN cana-590 109 12 ] ] PUNCT cana-590 109 13 , , PUNCT cana-590 109 14 allowing allow VERB cana-590 109 15 𝑅2 𝑅2 NOUN cana-590 109 16 to to PART cana-590 109 17 be be AUX cana-590 109 18 divided divide VERB cana-590 109 19 . . PUNCT cana-590 110 1 in in ADP cana-590 110 2 other other ADJ cana-590 110 3 words word NOUN cana-590 110 4 , , PUNCT cana-590 110 5 the the DET cana-590 110 6 line line NOUN cana-590 110 7 passing pass VERB cana-590 110 8 through through ADP cana-590 110 9 the the DET cana-590 110 10 origin origin NOUN cana-590 110 11 sets set VERB cana-590 110 12 a a DET cana-590 110 13 constant constant ADJ cana-590 110 14 slope slope NOUN cana-590 110 15 . . PUNCT cana-590 111 1 thus thus ADV cana-590 111 2 , , PUNCT cana-590 111 3 one one PRON cana-590 111 4 can can AUX cana-590 111 5 set set VERB cana-590 111 6 , , PUNCT cana-590 111 7 𝑦 𝑦 NOUN cana-590 111 8 = = X cana-590 111 9 π‘Žπ‘₯. π‘Žπ‘₯. NOUN cana-590 111 10 how how SCONJ cana-590 111 11 to to PART cana-590 111 12 realise realise VERB cana-590 111 13 non non ADJ cana-590 111 14 trivial trivial ADJ cana-590 111 15 examples example NOUN cana-590 111 16 where where SCONJ cana-590 111 17 projectivization projectivization NOUN cana-590 111 18 crop crop VERB cana-590 111 19 up up ADP cana-590 111 20 . . PUNCT cana-590 112 1 the the DET cana-590 112 2 stereographic stereographic ADJ cana-590 112 3 projections projection NOUN cana-590 112 4 of of ADP cana-590 112 5 the the DET cana-590 112 6 sphere sphere NOUN cana-590 112 7 and and CCONJ cana-590 112 8 circle circle NOUN cana-590 112 9 are be AUX cana-590 112 10 the the DET cana-590 112 11 best good ADJ cana-590 112 12 examples example NOUN cana-590 112 13 . . PUNCT cana-590 113 1 6 6 X cana-590 113 2 . . X cana-590 113 3 conclusion conclusion NOUN cana-590 113 4 imagine imagine VERB cana-590 113 5 a a DET cana-590 113 6 line line NOUN cana-590 113 7 passing pass VERB cana-590 113 8 through through ADP cana-590 113 9 origin origin NOUN cana-590 113 10 in in ADP cana-590 113 11 𝑅2 𝑅2 NOUN cana-590 113 12 and and CCONJ cana-590 113 13 𝑅3 𝑅3 PROPN cana-590 113 14 with with ADP cana-590 113 15 vectors vector NOUN cana-590 113 16 originating originate VERB cana-590 113 17 from from ADP cana-590 113 18 the the DET cana-590 113 19 origin origin NOUN cana-590 113 20 of of ADP cana-590 113 21 unit unit NOUN cana-590 113 22 length length NOUN cana-590 113 23 then then ADV cana-590 113 24 we we PRON cana-590 113 25 are be AUX cana-590 113 26 done do VERB cana-590 113 27 . . PUNCT cana-590 114 1 differential differential ADJ cana-590 114 2 geometry geometry NOUN cana-590 114 3 methods method NOUN cana-590 114 4 vividly vividly ADV cana-590 114 5 capture capture VERB cana-590 114 6 them they PRON cana-590 114 7 as as ADP cana-590 114 8 one one NUM cana-590 114 9 and and CCONJ cana-590 114 10 two two NUM cana-590 114 11 dimensional dimensional ADJ cana-590 114 12 smooth smooth ADJ cana-590 114 13 compact compact ADJ cana-590 114 14 manifolds manifold NOUN cana-590 114 15 . . PUNCT cana-590 115 1 a a DET cana-590 115 2 natural natural ADJ cana-590 115 3 inclusive inclusive ADJ cana-590 115 4 tower tower NOUN cana-590 115 5 of of ADP cana-590 115 6 subspaces subspace NOUN cana-590 115 7 of of ADP cana-590 115 8 𝑅𝑛 𝑅𝑛 PROPN cana-590 115 9 is be AUX cana-590 115 10 𝑅1contained 𝑅1containe VERB cana-590 115 11 in in ADP cana-590 115 12 𝑅2 𝑅2 ADP cana-590 115 13 … … PUNCT cana-590 115 14 … … PUNCT cana-590 115 15 … … PUNCT cana-590 115 16 𝑅(π‘›βˆ’1 𝑅(π‘›βˆ’1 NOUN cana-590 115 17 ) ) PUNCT cana-590 115 18 and and CCONJ cana-590 115 19 contained contain VERB cana-590 115 20 in in ADP cana-590 115 21 𝑅𝑛. 𝑅𝑛. PROPN cana-590 115 22 therefore therefore ADV cana-590 115 23 π‘…π‘˜ π‘…π‘˜ VERB cana-590 115 24 with with ADP cana-590 115 25 π‘˜ π‘˜ DET cana-590 115 26 row row NOUN cana-590 115 27 vectors vector NOUN cana-590 115 28 and and CCONJ cana-590 115 29 π‘˜ π‘˜ DET cana-590 115 30 Γ— Γ— NOUN cana-590 115 31 𝑛 𝑛 DET cana-590 115 32 matrix matrix NOUN cana-590 115 33 and and CCONJ cana-590 115 34 totality totality NOUN cana-590 115 35 of of ADP cana-590 115 36 them they PRON cana-590 115 37 would would AUX cana-590 115 38 be be AUX cana-590 115 39 aright aright ADJ cana-590 115 40 frame frame NOUN cana-590 115 41 work work NOUN cana-590 115 42 for for ADP cana-590 115 43 imagining imagine VERB cana-590 115 44 grassmannians grassmannian NOUN cana-590 115 45 . . PUNCT cana-590 116 1 references reference NOUN cana-590 116 2 [ [ X cana-590 116 3 1 1 NUM cana-590 116 4 ] ] X cana-590 116 5 harris harris PROPN cana-590 116 6 , , PUNCT cana-590 116 7 algebraic algebraic ADJ cana-590 116 8 geometry geometry NOUN cana-590 116 9 , , PUNCT cana-590 116 10 a a DET cana-590 116 11 first first ADJ cana-590 116 12 course course NOUN cana-590 116 13 , , PUNCT cana-590 116 14 springer springer NOUN cana-590 116 15 - - PUNCT cana-590 116 16 verlag,1995 verlag,1995 PROPN cana-590 116 17 . . PUNCT cana-590 117 1 lakshmibai lakshmibai PROPN cana-590 117 2 and and CCONJ cana-590 117 3 gonciulea gonciulea ADJ cana-590 117 4 flag flag NOUN cana-590 117 5 varieties variety NOUN cana-590 117 6 , , PUNCT cana-590 117 7 hermann,2001 hermann,2001 NOUN cana-590 117 8 . . PUNCT cana-590 118 1 [ [ X cana-590 118 2 2 2 NUM cana-590 118 3 ] ] PUNCT cana-590 118 4 hatcher hatcher NOUN cana-590 118 5 , , PUNCT cana-590 118 6 allen allen PROPN cana-590 118 7 ( ( PUNCT cana-590 118 8 2003).vector 2003).vector NUM cana-590 118 9 bundles bundle NOUN cana-590 118 10 & & CCONJ cana-590 118 11 k k NOUN cana-590 118 12 - - NOUN cana-590 118 13 theory theory NOUN cana-590 118 14 . . PUNCT cana-590 119 1 [ [ X cana-590 119 2 3 3 X cana-590 119 3 ] ] X cana-590 119 4 jeanpierre jeanpierre PROPN cana-590 119 5 serre serre PROPN cana-590 119 6 , , PUNCT cana-590 119 7 trees tree NOUN cana-590 119 8 , , PUNCT cana-590 119 9 springer springer NOUN cana-590 119 10 mongr.math mongr.math PROPN cana-590 119 11 . . PROPN cana-590 119 12 , , PUNCT cana-590 119 13 springer springer NOUN cana-590 119 14 - - PUNCT cana-590 119 15 verlag verlag PROPN cana-590 119 16 , , PUNCT cana-590 119 17 berlin berlin PROPN cana-590 119 18 , , PUNCT cana-590 119 19 2003 2003 NUM cana-590 119 20 . . PUNCT cana-590 120 1 [ [ X cana-590 120 2 4 4 NUM cana-590 120 3 ] ] X cana-590 120 4 peter peter PROPN cana-590 120 5 buser buser PROPN cana-590 120 6 , , PUNCT cana-590 120 7 geometry geometry NOUN cana-590 120 8 and and CCONJ cana-590 120 9 spectra spectra NOUN cana-590 120 10 of of ADP cana-590 120 11 compact compact ADJ cana-590 120 12 rieman rieman NOUN cana-590 120 13 surfaces surface NOUN cana-590 120 14 , , PUNCT cana-590 120 15 progress progress NOUN cana-590 120 16 in in ADP cana-590 120 17 mathematics mathematic NOUN cana-590 120 18 , , PUNCT cana-590 120 19 vol.106,birkhouser vol.106,birkhous ADJ cana-590 120 20 boston boston PROPN cana-590 120 21 , , PUNCT cana-590 120 22 inc inc PROPN cana-590 120 23 . . PROPN cana-590 120 24 ,boston ,boston PROPN cana-590 120 25 , , PUNCT cana-590 120 26 ma,1992.mr ma,1992.mr PROPN cana-590 120 27 11833224 11833224 NUM cana-590 120 28 . . PUNCT cana-590 121 1 [ [ X cana-590 121 2 5 5 NUM cana-590 121 3 ] ] PUNCT cana-590 121 4 m m VERB cana-590 121 5 arym arym VERB cana-590 121 6 mirza mirza PROPN cana-590 121 7 khani khani PROPN cana-590 121 8 and and CCONJ cana-590 121 9 bram bram PROPN cana-590 121 10 petri petri PROPN cana-590 121 11 , , PUNCT cana-590 121 12 lengths length NOUN cana-590 121 13 of of ADP cana-590 121 14 closed closed ADJ cana-590 121 15 jodesics jodesic NOUN cana-590 121 16 on on ADP cana-590 121 17 random random ADJ cana-590 121 18 surfaces surface NOUN cana-590 121 19 of of ADP cana-590 121 20 large large ADJ cana-590 121 21 genus genus NOUN cana-590 121 22 , , PUNCT cana-590 121 23 comment.math.helv comment.math.helv PROPN cana-590 121 24 . . PUNCT cana-590 122 1 94(2019),no.4,869 94(2019),no.4,869 NUM cana-590 122 2 - - SYM cana-590 122 3 889,doi 889,doi NUM cana-590 122 4 10.4171 10.4171 NUM cana-590 122 5 / / SYM cana-590 122 6 cmh/477.mr4046008 cmh/477.mr4046008 NOUN cana-590 122 7 . . PUNCT cana-590 123 1 [ [ X cana-590 123 2 6 6 NUM cana-590 123 3 ] ] PUNCT cana-590 123 4 thomas thomas PROPN cana-590 123 5 bendokat bendokat PROPN cana-590 123 6 , , PUNCT cana-590 123 7 ralf ralf PROPN cana-590 123 8 zimmermann zimmermann PROPN cana-590 123 9 and and CCONJ cana-590 123 10 p.-a p.-a PROPN cana-590 123 11 . . PUNCT cana-590 124 1 absil absil NOUN cana-590 124 2 -basic -basic PROPN cana-590 124 3 geometry geometry NOUN cana-590 124 4 and and CCONJ cana-590 124 5 computational computational ADJ cana-590 124 6 aspects aspect NOUN cana-590 124 7 . . PUNCT cana-590 125 1 [ [ X cana-590 125 2 7 7 NUM cana-590 125 3 ] ] PUNCT cana-590 125 4 p.-a p.-a NOUN cana-590 125 5 . . PUNCT cana-590 125 6 absil absil PROPN cana-590 125 7 , , PUNCT cana-590 125 8 r. r. PROPN cana-590 125 9 mahony mahony PROPN cana-590 125 10 , , PUNCT cana-590 125 11 and and CCONJ cana-590 125 12 r. r. PROPN cana-590 125 13 sepulchre sepulchre PROPN cana-590 125 14 . . PUNCT cana-590 126 1 riemannian riemannian ADJ cana-590 126 2 geometry geometry NOUN cana-590 126 3 of of ADP cana-590 126 4 grassmann grassmann PROPN cana-590 126 5 manifolds manifold NOUN cana-590 126 6 with with ADP cana-590 126 7 a a DET cana-590 126 8 view view NOUN cana-590 126 9 on on ADP cana-590 126 10 algorithmic algorithmic ADJ cana-590 126 11 computation computation NOUN cana-590 126 12 . . PUNCT cana-590 127 1 acta acta PROPN cana-590 127 2 applicandae applicandae PROPN cana-590 127 3 mathematica mathematica PROPN cana-590 127 4 , , PUNCT cana-590 127 5 80(2):199–220 80(2):199–220 PROPN cana-590 127 6 , , PUNCT cana-590 127 7 2004 2004 NUM cana-590 127 8 . . PUNCT cana-590 128 1 [ [ X cana-590 128 2 8 8 NUM cana-590 128 3 ] ] PUNCT cana-590 128 4 a. a. NOUN cana-590 128 5 a. a. PROPN cana-590 128 6 borisenko borisenko PROPN cana-590 128 7 and and CCONJ cana-590 128 8 yu yu PROPN cana-590 128 9 . . PUNCT cana-590 128 10 a. a. PROPN cana-590 128 11 nikolaevski˘ı nikolaevski˘ı PROPN cana-590 128 12 . . PUNCT cana-590 128 13 grassmann grassmann PROPN cana-590 128 14 manifolds manifold NOUN cana-590 128 15 and and CCONJ cana-590 128 16 grassmann grassmann NOUN cana-590 128 17 image image NOUN cana-590 128 18 of of ADP cana-590 128 19 submanifolds submanifold NOUN cana-590 128 20 . . PUNCT cana-590 129 1 uspekhi uspekhi PROPN cana-590 129 2 mat mat PROPN cana-590 129 3 . . PUNCT cana-590 129 4 nauk nauk PROPN cana-590 129 5 , , PUNCT cana-590 129 6 46(2(278)):41–83 46(2(278)):41–83 NUM cana-590 129 7 , , PUNCT cana-590 129 8 240 240 NUM cana-590 129 9 , , PUNCT cana-590 129 10 1991 1991 NUM cana-590 129 11 . . PUNCT cana-590 130 1 [ [ X cana-590 130 2 9 9 NUM cana-590 130 3 ] ] X cana-590 130 4 j. j. PROPN cana-590 130 5 m. m. PROPN cana-590 130 6 lee lee PROPN cana-590 130 7 . . PROPN cana-590 130 8 introduction introduction NOUN cana-590 130 9 to to ADP cana-590 130 10 smooth smooth ADJ cana-590 130 11 manifolds manifold NOUN cana-590 130 12 . . PUNCT cana-590 131 1 graduate graduate NOUN cana-590 131 2 texts text NOUN cana-590 131 3 in in ADP cana-590 131 4 mathematics mathematic NOUN cana-590 131 5 . . PUNCT cana-590 132 1 springer springer PROPN cana-590 132 2 new new PROPN cana-590 132 3 york york PROPN cana-590 132 4 , , PUNCT cana-590 132 5 2012 2012 NUM cana-590 132 6 . . PUNCT cana-590 133 1 [ [ X cana-590 133 2 10 10 NUM cana-590 133 3 ] ] X cana-590 133 4 a. a. NOUN cana-590 133 5 machado machado PROPN cana-590 133 6 and and CCONJ cana-590 133 7 i. i. PROPN cana-590 133 8 salavessa salavessa PROPN cana-590 133 9 . . PUNCT cana-590 134 1 grassmannian grassmannian PROPN cana-590 134 2 manifolds manifold NOUN cana-590 134 3 as as ADP cana-590 134 4 subsets subset NOUN cana-590 134 5 of of ADP cana-590 134 6 euclidean euclidean ADJ cana-590 134 7 spaces space NOUN cana-590 134 8 . . PUNCT cana-590 135 1 in in ADP cana-590 135 2 differential differential ADJ cana-590 135 3 geometry geometry NOUN cana-590 135 4 ( ( PUNCT cana-590 135 5 santiago santiago PROPN cana-590 135 6 de de PROPN cana-590 135 7 compostela compostela PROPN cana-590 135 8 , , PUNCT cana-590 135 9 1984 1984 NUM cana-590 135 10 ) ) PUNCT cana-590 135 11 , , PUNCT cana-590 135 12 volume volume NOUN cana-590 135 13 131 131 NUM cana-590 135 14 of of ADP cana-590 135 15 res re NOUN cana-590 135 16 . . PUNCT cana-590 136 1 notes note NOUN cana-590 136 2 in in ADP cana-590 136 3 math math NOUN cana-590 136 4 . . PUNCT cana-590 137 1 , , PUNCT cana-590 137 2 pages page VERB cana-590 137 3 85–102 85–102 PROPN cana-590 137 4 . . PUNCT cana-590 137 5 pitman pitman PROPN cana-590 137 6 , , PUNCT cana-590 137 7 boston boston PROPN cana-590 137 8 , , PUNCT cana-590 137 9 ma ma PROPN cana-590 137 10 , , PUNCT cana-590 137 11 1985 1985 NUM cana-590 137 12 . . PUNCT cana-590 138 1 [ [ X cana-590 138 2 11 11 NUM cana-590 138 3 ] ] X cana-590 138 4 l. l. PROPN cana-590 138 5 qiu qiu PROPN cana-590 138 6 , , PUNCT cana-590 138 7 y. y. PROPN cana-590 138 8 zhang zhang PROPN cana-590 138 9 , , PUNCT cana-590 138 10 and and CCONJ cana-590 138 11 c.-k c.-k PROPN cana-590 138 12 . . PUNCT cana-590 139 1 li li PROPN cana-590 139 2 . . PROPN cana-590 139 3 unitarily unitarily ADV cana-590 139 4 invariant invariant ADJ cana-590 139 5 metrics metric NOUN cana-590 139 6 on on ADP cana-590 139 7 the the DET cana-590 139 8 grassmann grassmann PROPN cana-590 139 9 space space NOUN cana-590 139 10 . . PUNCT cana-590 140 1 siam siam PROPN cana-590 140 2 journal journal PROPN cana-590 140 3 on on ADP cana-590 140 4 matrix matrix NOUN cana-590 140 5 analysis analysis NOUN cana-590 140 6 and and CCONJ cana-590 140 7 applications application NOUN cana-590 140 8 , , PUNCT cana-590 140 9 27(2):507–25 27(2):507–25 NUM cana-590 140 10 , , PUNCT cana-590 140 11 2005 2005 NUM cana-590 140 12 . . PUNCT