id sid tid token lemma pos cana-1707 1 1 communications communication NOUN cana-1707 1 2 on on ADP cana-1707 1 3 applied apply VERB cana-1707 1 4 nonlinear nonlinear ADJ cana-1707 1 5 analysis analysis NOUN cana-1707 1 6 issn issn NOUN cana-1707 1 7 : : PUNCT cana-1707 1 8 1074 1074 NUM cana-1707 1 9 - - PUNCT cana-1707 1 10 133x 133x NUM cana-1707 1 11 vol vol NOUN cana-1707 1 12 32 32 NUM cana-1707 1 13 no no NOUN cana-1707 1 14 . . NOUN cana-1707 1 15 2 2 NUM cana-1707 1 16 ( ( PUNCT cana-1707 1 17 2025 2025 NUM cana-1707 1 18 ) ) PUNCT cana-1707 1 19 53 53 NUM cana-1707 1 20 https://internationalpubls.com https://internationalpubls.com X cana-1707 1 21 extension extension NOUN cana-1707 1 22 and and CCONJ cana-1707 1 23 generalization generalization NOUN cana-1707 1 24 of of ADP cana-1707 1 25 banach banach ADJ cana-1707 1 26 contraction contraction NOUN cana-1707 1 27 in in ADP cana-1707 1 28 metric metric ADJ cana-1707 1 29 and and CCONJ cana-1707 1 30 in in ADP cana-1707 1 31 menger menger PROPN cana-1707 1 32 space space NOUN cana-1707 1 33 ajay ajay PROPN cana-1707 1 34 kumar kumar PROPN cana-1707 1 35 chaudhary1 chaudhary1 PROPN cana-1707 1 36 , , PUNCT cana-1707 1 37 chet chet PROPN cana-1707 1 38 raj raj PROPN cana-1707 1 39 bhatta2 bhatta2 PROPN cana-1707 1 40 , , PUNCT cana-1707 1 41 and and CCONJ cana-1707 1 42 uday uday PROPN cana-1707 1 43 kumar kumar PROPN cana-1707 1 44 karn3 karn3 PROPN cana-1707 1 45 * * PROPN cana-1707 1 46 1department 1department NUM cana-1707 1 47 of of ADP cana-1707 1 48 mathematics mathematic NOUN cana-1707 1 49 , , PUNCT cana-1707 1 50 tri tri PROPN cana-1707 1 51 - - ADJ cana-1707 1 52 chandra chandra PROPN cana-1707 1 53 multiple multiple ADJ cana-1707 1 54 campus campus PROPN cana-1707 1 55 , , PUNCT cana-1707 1 56 tribhuvan tribhuvan PROPN cana-1707 1 57 university university PROPN cana-1707 1 58 , , PUNCT cana-1707 1 59 kathmandu kathmandu NOUN cana-1707 1 60 , , PUNCT cana-1707 1 61 nepal nepal ADJ cana-1707 1 62 2central 2central NUM cana-1707 1 63 department department NOUN cana-1707 1 64 of of ADP cana-1707 1 65 mathematics mathematics PROPN cana-1707 1 66 , , PUNCT cana-1707 1 67 kirtipur kirtipur NOUN cana-1707 1 68 , , PUNCT cana-1707 1 69 tribhuvan tribhuvan PROPN cana-1707 1 70 university university PROPN cana-1707 1 71 , , PUNCT cana-1707 1 72 kathmandu kathmandu NOUN cana-1707 1 73 , , PUNCT cana-1707 1 74 nepal nepal NOUN cana-1707 1 75 3department 3department NUM cana-1707 1 76 of of ADP cana-1707 1 77 mathematics mathematic NOUN cana-1707 1 78 , , PUNCT cana-1707 1 79 patan patan ADJ cana-1707 1 80 multiple multiple ADJ cana-1707 1 81 campus campus NOUN cana-1707 1 82 , , PUNCT cana-1707 1 83 and and CCONJ cana-1707 1 84 phd phd NOUN cana-1707 1 85 scholar scholar NOUN cana-1707 1 86 of of ADP cana-1707 1 87 tribhuvan tribhuvan PROPN cana-1707 1 88 university university PROPN cana-1707 1 89 , , PUNCT cana-1707 1 90 kathmandu kathmandu NOUN cana-1707 1 91 , , PUNCT cana-1707 1 92 nepal nepal NOUN cana-1707 1 93 * * PUNCT cana-1707 1 94 corresponding correspond VERB cana-1707 1 95 e e NOUN cana-1707 1 96 - - NOUN cana-1707 1 97 mail mail NOUN cana-1707 1 98 : : PUNCT cana-1707 1 99 udaykarn984@gmail.com udaykarn984@gmail.com X cana-1707 1 100 article article NOUN cana-1707 1 101 history history NOUN cana-1707 1 102 : : PUNCT cana-1707 1 103 received receive VERB cana-1707 1 104 : : PUNCT cana-1707 1 105 21 21 NUM cana-1707 1 106 - - SYM cana-1707 1 107 07 07 NUM cana-1707 1 108 - - PUNCT cana-1707 1 109 2024 2024 NUM cana-1707 1 110 revised revise VERB cana-1707 1 111 : : PUNCT cana-1707 1 112 01 01 NUM cana-1707 1 113 - - SYM cana-1707 1 114 09 09 NUM cana-1707 1 115 - - PUNCT cana-1707 1 116 2024 2024 NUM cana-1707 1 117 accepted accept VERB cana-1707 1 118 : : PUNCT cana-1707 1 119 14 14 NUM cana-1707 1 120 - - SYM cana-1707 1 121 09 09 NUM cana-1707 1 122 - - PUNCT cana-1707 1 123 2024 2024 NUM cana-1707 1 124 abstract abstract NOUN cana-1707 1 125 : : PUNCT cana-1707 1 126 the the DET cana-1707 1 127 root root NOUN cana-1707 1 128 of of ADP cana-1707 1 129 metric metric ADJ cana-1707 1 130 fixed fix VERB cana-1707 1 131 point point NOUN cana-1707 1 132 theory theory NOUN cana-1707 1 133 is be AUX cana-1707 1 134 stefen stefen PROPN cana-1707 1 135 banach banach NOUN cana-1707 1 136 's 's PART cana-1707 1 137 contraction contraction NOUN cana-1707 1 138 mapping mapping NOUN cana-1707 1 139 , , PUNCT cana-1707 1 140 a a DET cana-1707 1 141 research research NOUN cana-1707 1 142 source source NOUN cana-1707 1 143 for for ADP cana-1707 1 144 shrinking shrink VERB cana-1707 1 145 the the DET cana-1707 1 146 distance distance NOUN cana-1707 1 147 between between ADP cana-1707 1 148 two two NUM cana-1707 1 149 points point NOUN cana-1707 1 150 in in ADP cana-1707 1 151 space space NOUN cana-1707 1 152 . . PUNCT cana-1707 2 1 as as ADP cana-1707 2 2 a a DET cana-1707 2 3 source source NOUN cana-1707 2 4 , , PUNCT cana-1707 2 5 many many ADJ cana-1707 2 6 authors author NOUN cana-1707 2 7 have have AUX cana-1707 2 8 introduced introduce VERB cana-1707 2 9 many many ADJ cana-1707 2 10 contraction contraction NOUN cana-1707 2 11 mappings mapping NOUN cana-1707 2 12 as as ADP cana-1707 2 13 extensions extension NOUN cana-1707 2 14 and and CCONJ cana-1707 2 15 generalizations generalization NOUN cana-1707 2 16 of of ADP cana-1707 2 17 banach banach NOUN cana-1707 2 18 contraction contraction NOUN cana-1707 2 19 and and CCONJ cana-1707 2 20 established establish VERB cana-1707 2 21 fixed fix VERB cana-1707 2 22 point point NOUN cana-1707 2 23 theorems theorem NOUN cana-1707 2 24 under under ADP cana-1707 2 25 the the DET cana-1707 2 26 property property NOUN cana-1707 2 27 that that PRON cana-1707 2 28 each each DET cana-1707 2 29 such such ADJ cana-1707 2 30 mapping mapping NOUN cana-1707 2 31 in in ADP cana-1707 2 32 complete complete ADJ cana-1707 2 33 metric metric ADJ cana-1707 2 34 and and CCONJ cana-1707 2 35 menger menger PROPN cana-1707 2 36 space space NOUN cana-1707 2 37 has have VERB cana-1707 2 38 a a DET cana-1707 2 39 unique unique ADJ cana-1707 2 40 fixed fix VERB cana-1707 2 41 point point NOUN cana-1707 2 42 . . PUNCT cana-1707 3 1 this this DET cana-1707 3 2 article article NOUN cana-1707 3 3 presents present VERB cana-1707 3 4 updated update VERB cana-1707 3 5 results result NOUN cana-1707 3 6 of of ADP cana-1707 3 7 banach banach NOUN cana-1707 3 8 contraction contraction NOUN cana-1707 3 9 generalization generalization NOUN cana-1707 3 10 and and CCONJ cana-1707 3 11 extension extension NOUN cana-1707 3 12 forms form NOUN cana-1707 3 13 in in ADP cana-1707 3 14 metric metric ADJ cana-1707 3 15 and and CCONJ cana-1707 3 16 menger menger PROPN cana-1707 3 17 space space NOUN cana-1707 3 18 which which PRON cana-1707 3 19 helps help VERB cana-1707 3 20 the the DET cana-1707 3 21 comparative comparative ADJ cana-1707 3 22 and and CCONJ cana-1707 3 23 interrelationship interrelationship NOUN cana-1707 3 24 study study NOUN cana-1707 3 25 in in ADP cana-1707 3 26 these these DET cana-1707 3 27 spaces space NOUN cana-1707 3 28 . . PUNCT cana-1707 4 1 keywords keyword NOUN cana-1707 4 2 : : PUNCT cana-1707 4 3 banach banach NOUN cana-1707 4 4 contraction contraction NOUN cana-1707 4 5 , , PUNCT cana-1707 4 6 distribution distribution NOUN cana-1707 4 7 function function NOUN cana-1707 4 8 , , PUNCT cana-1707 4 9 triangular triangular NOUN cana-1707 4 10 norm norm NOUN cana-1707 4 11 and and CCONJ cana-1707 4 12 menger menger PROPN cana-1707 4 13 space space NOUN cana-1707 4 14 . . PUNCT cana-1707 5 1 1 1 X cana-1707 5 2 . . X cana-1707 5 3 introduction introduction NOUN cana-1707 5 4 in in ADP cana-1707 5 5 the the DET cana-1707 5 6 nineteenth nineteenth ADJ cana-1707 5 7 century century NOUN cana-1707 5 8 , , PUNCT cana-1707 5 9 existence existence NOUN cana-1707 5 10 theorems theorem NOUN cana-1707 5 11 emerged emerge VERB cana-1707 5 12 in in ADP cana-1707 5 13 analysis analysis NOUN cana-1707 5 14 when when SCONJ cana-1707 5 15 basic basic ADJ cana-1707 5 16 mathematical mathematical ADJ cana-1707 5 17 facts fact NOUN cana-1707 5 18 were be AUX cana-1707 5 19 considered consider VERB cana-1707 5 20 critically critically ADV cana-1707 5 21 . . PUNCT cana-1707 6 1 the the DET cana-1707 6 2 first first ADJ cana-1707 6 3 mathematician mathematician NOUN cana-1707 6 4 to to PART cana-1707 6 5 demonstrate demonstrate VERB cana-1707 6 6 the the DET cana-1707 6 7 existence existence NOUN cana-1707 6 8 of of ADP cana-1707 6 9 a a DET cana-1707 6 10 theorem theorem NOUN cana-1707 6 11 for for ADP cana-1707 6 12 differential differential ADJ cana-1707 6 13 equation equation NOUN cana-1707 6 14 systems system NOUN cana-1707 6 15 with with ADP cana-1707 6 16 analytic analytic ADJ cana-1707 6 17 right right ADJ cana-1707 6 18 - - PUNCT cana-1707 6 19 hand hand NOUN cana-1707 6 20 sides side NOUN cana-1707 6 21 was be AUX cana-1707 6 22 a. a. NOUN cana-1707 6 23 l. l. PROPN cana-1707 6 24 cauchy cauchy PROPN cana-1707 7 1 [ [ X cana-1707 7 2 9 9 NUM cana-1707 7 3 ] ] PUNCT cana-1707 7 4 . . PUNCT cana-1707 8 1 e. e. PROPN cana-1707 8 2 picard picard PROPN cana-1707 9 1 [ [ X cana-1707 9 2 43 43 NUM cana-1707 9 3 ] ] PUNCT cana-1707 9 4 suggested suggest VERB cana-1707 9 5 the the DET cana-1707 9 6 method method NOUN cana-1707 9 7 of of ADP cana-1707 9 8 successive successive ADJ cana-1707 9 9 approximations approximation NOUN cana-1707 9 10 to to PART cana-1707 9 11 prove prove VERB cana-1707 9 12 the the DET cana-1707 9 13 existence existence NOUN cana-1707 9 14 of of ADP cana-1707 9 15 the the DET cana-1707 9 16 theorems theorem NOUN cana-1707 9 17 . . PUNCT cana-1707 10 1 in in ADP cana-1707 10 2 1922 1922 NUM cana-1707 10 3 , , PUNCT cana-1707 10 4 birkhoff birkhoff NOUN cana-1707 10 5 and and CCONJ cana-1707 10 6 kellogg[4 kellogg[4 PROPN cana-1707 10 7 ] ] X cana-1707 10 8 gave give VERB cana-1707 10 9 proof proof NOUN cana-1707 10 10 of of ADP cana-1707 10 11 the the DET cana-1707 10 12 classical classical ADJ cana-1707 10 13 existence existence NOUN cana-1707 10 14 theorem theorem VERB cana-1707 10 15 for for ADP cana-1707 10 16 the the DET cana-1707 10 17 equation equation NOUN cana-1707 10 18 𝑑𝑦 𝑑𝑦 ADP cana-1707 10 19 𝑑π‘₯ 𝑑π‘₯ NOUN cana-1707 10 20 = = PUNCT cana-1707 10 21 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-1707 10 22 , , PUNCT cana-1707 10 23 𝑦 𝑦 NOUN cana-1707 10 24 ) ) PUNCT cana-1707 10 25 in in ADP cana-1707 10 26 function function NOUN cana-1707 10 27 spaces space NOUN cana-1707 10 28 . . PUNCT cana-1707 11 1 however however ADV cana-1707 11 2 , , PUNCT cana-1707 11 3 the the DET cana-1707 11 4 most most ADV cana-1707 11 5 elementary elementary NOUN cana-1707 11 6 and and CCONJ cana-1707 11 7 by by ADP cana-1707 11 8 far far ADV cana-1707 11 9 the the DET cana-1707 11 10 most most ADV cana-1707 11 11 fruitful fruitful ADJ cana-1707 11 12 method method NOUN cana-1707 11 13 for for ADP cana-1707 11 14 proving prove VERB cana-1707 11 15 theorems theorem NOUN cana-1707 11 16 on on ADP cana-1707 11 17 the the DET cana-1707 11 18 existence existence NOUN cana-1707 11 19 and and CCONJ cana-1707 11 20 uniqueness uniqueness NOUN cana-1707 11 21 of of ADP cana-1707 11 22 solutions solution NOUN cana-1707 11 23 is be AUX cana-1707 11 24 the the DET cana-1707 11 25 principle principle NOUN cana-1707 11 26 formulated formulate VERB cana-1707 11 27 and and CCONJ cana-1707 11 28 proved prove VERB cana-1707 11 29 by by ADP cana-1707 11 30 s. s. PROPN cana-1707 11 31 banach banach PROPN cana-1707 12 1 [ [ X cana-1707 12 2 1]1920 1]1920 NUM cana-1707 12 3 in in ADP cana-1707 12 4 his his PRON cana-1707 12 5 phd phd NOUN cana-1707 12 6 thesis thesis NOUN cana-1707 12 7 published publish VERB cana-1707 12 8 in in ADP cana-1707 12 9 1922 1922 NUM cana-1707 12 10 . . PUNCT cana-1707 13 1 although although SCONJ cana-1707 13 2 the the DET cana-1707 13 3 idea idea NOUN cana-1707 13 4 of of ADP cana-1707 13 5 successive successive ADJ cana-1707 13 6 approximations approximation NOUN cana-1707 13 7 in in ADP cana-1707 13 8 some some DET cana-1707 13 9 concrete concrete ADJ cana-1707 13 10 situations situation NOUN cana-1707 13 11 ( ( PUNCT cana-1707 13 12 solving solve VERB cana-1707 13 13 differential differential NOUN cana-1707 13 14 and and CCONJ cana-1707 13 15 integral integral ADJ cana-1707 13 16 equations equation NOUN cana-1707 13 17 ) ) PUNCT cana-1707 13 18 appears appear VERB cana-1707 13 19 in in ADP cana-1707 13 20 some some DET cana-1707 13 21 works work NOUN cana-1707 13 22 of of ADP cana-1707 13 23 e. e. PROPN cana-1707 13 24 picard picard PROPN cana-1707 13 25 , , PUNCT cana-1707 13 26 r. r. PROPN cana-1707 13 27 caccioppoli caccioppoli PROPN cana-1707 13 28 , , PUNCT cana-1707 13 29 et et PROPN cana-1707 13 30 al al PROPN cana-1707 13 31 . . PUNCT cana-1707 14 1 [ [ X cana-1707 14 2 10 10 NUM cana-1707 14 3 ] ] PUNCT cana-1707 14 4 , , PUNCT cana-1707 14 5 it it PRON cana-1707 14 6 was be AUX cana-1707 14 7 banach banach ADV cana-1707 14 8 who who PRON cana-1707 14 9 placed place VERB cana-1707 14 10 it it PRON cana-1707 14 11 in in ADP cana-1707 14 12 the the DET cana-1707 14 13 right right ADJ cana-1707 14 14 abstract abstract ADJ cana-1707 14 15 setting setting NOUN cana-1707 14 16 , , PUNCT cana-1707 14 17 making make VERB cana-1707 14 18 it it PRON cana-1707 14 19 suitable suitable ADJ cana-1707 14 20 for for ADP cana-1707 14 21 a a DET cana-1707 14 22 wide wide ADJ cana-1707 14 23 range range NOUN cana-1707 14 24 of of ADP cana-1707 14 25 applications application NOUN cana-1707 14 26 . . PUNCT cana-1707 15 1 this this DET cana-1707 15 2 principle principle ADJ cana-1707 15 3 results result VERB cana-1707 15 4 from from ADP cana-1707 15 5 the the DET cana-1707 15 6 geometric geometric ADJ cana-1707 15 7 interpretation interpretation NOUN cana-1707 15 8 of of ADP cana-1707 15 9 picard picard PROPN cana-1707 15 10 's 's PART cana-1707 15 11 method method NOUN cana-1707 15 12 of of ADP cana-1707 15 13 successive successive ADJ cana-1707 15 14 approximations approximation NOUN cana-1707 15 15 . . PUNCT cana-1707 16 1 several several ADJ cana-1707 16 2 generalizations generalization NOUN cana-1707 16 3 of of ADP cana-1707 16 4 this this DET cana-1707 16 5 principle principle NOUN cana-1707 16 6 have have AUX cana-1707 16 7 appeared appear VERB cana-1707 16 8 and and CCONJ cana-1707 16 9 many many ADJ cana-1707 16 10 authors author NOUN cana-1707 16 11 have have AUX cana-1707 16 12 done do VERB cana-1707 16 13 their their PRON cana-1707 16 14 comparative comparative ADJ cana-1707 16 15 study study NOUN cana-1707 16 16 see see VERB cana-1707 16 17 references reference NOUN cana-1707 16 18 [ [ X cana-1707 16 19 [ [ X cana-1707 16 20 35 35 NUM cana-1707 16 21 ] ] PUNCT cana-1707 16 22 , , PUNCT cana-1707 16 23 [ [ X cana-1707 16 24 43 43 NUM cana-1707 16 25 ] ] PUNCT cana-1707 16 26 , , PUNCT cana-1707 16 27 [ [ X cana-1707 16 28 44 44 NUM cana-1707 16 29 ] ] PUNCT cana-1707 16 30 , , PUNCT cana-1707 16 31 [ [ X cana-1707 16 32 48 48 NUM cana-1707 16 33 ] ] X cana-1707 16 34 ] ] PUNCT cana-1707 16 35 in in ADP cana-1707 16 36 metric metric ADJ cana-1707 16 37 space space NOUN cana-1707 16 38 . . PUNCT cana-1707 17 1 menger menger PROPN cana-1707 18 1 [ [ X cana-1707 18 2 38 38 NUM cana-1707 18 3 ] ] PUNCT cana-1707 18 4 introduced introduce VERB cana-1707 18 5 the the DET cana-1707 18 6 probabilistic probabilistic ADJ cana-1707 18 7 metric metric ADJ cana-1707 18 8 space space NOUN cana-1707 18 9 in in ADP cana-1707 18 10 1942 1942 NUM cana-1707 18 11 to to PART cana-1707 18 12 generalize generalize VERB cana-1707 18 13 frechet frechet PROPN cana-1707 18 14 's 's PART cana-1707 18 15 [ [ X cana-1707 18 16 25 25 NUM cana-1707 18 17 ] ] X cana-1707 18 18 metric metric ADJ cana-1707 18 19 space space NOUN cana-1707 18 20 by by ADP cana-1707 18 21 replacing replace VERB cana-1707 18 22 the the DET cana-1707 18 23 distance distance NOUN cana-1707 18 24 function function NOUN cana-1707 18 25 with with ADP cana-1707 18 26 the the DET cana-1707 18 27 distribution distribution NOUN cana-1707 18 28 function function NOUN cana-1707 18 29 . . PUNCT cana-1707 19 1 it it PRON cana-1707 19 2 helps help VERB cana-1707 19 3 to to PART cana-1707 19 4 solve solve VERB cana-1707 19 5 uncertainty uncertainty NOUN cana-1707 19 6 cases case NOUN cana-1707 19 7 regarding regard VERB cana-1707 19 8 the the DET cana-1707 19 9 distance distance NOUN cana-1707 19 10 between between ADP cana-1707 19 11 two two NUM cana-1707 19 12 points point NOUN cana-1707 19 13 in in ADP cana-1707 19 14 space space NOUN cana-1707 19 15 . . PUNCT cana-1707 20 1 this this DET cana-1707 20 2 probabilistic probabilistic ADJ cana-1707 20 3 metric metric ADJ cana-1707 20 4 space space NOUN cana-1707 20 5 became become VERB cana-1707 20 6 active active ADJ cana-1707 20 7 for for ADP cana-1707 20 8 mathematicians mathematician NOUN cana-1707 20 9 after after ADP cana-1707 20 10 the the DET cana-1707 20 11 great great ADJ cana-1707 20 12 contribution contribution NOUN cana-1707 20 13 in in ADP cana-1707 20 14 this this DET cana-1707 20 15 space space NOUN cana-1707 20 16 from from ADP cana-1707 20 17 schweizer schweizer PROPN cana-1707 20 18 and and CCONJ cana-1707 20 19 a. a. NOUN cana-1707 20 20 sklar sklar PROPN cana-1707 20 21 . . PUNCT cana-1707 21 1 [ [ X cana-1707 21 2 49 49 NUM cana-1707 21 3 ] ] PUNCT cana-1707 21 4 communications communication NOUN cana-1707 21 5 on on ADP cana-1707 21 6 applied apply VERB cana-1707 21 7 nonlinear nonlinear ADJ cana-1707 21 8 analysis analysis NOUN cana-1707 21 9 issn issn NOUN cana-1707 21 10 : : PUNCT cana-1707 21 11 1074 1074 NUM cana-1707 21 12 - - PUNCT cana-1707 21 13 133x 133x NUM cana-1707 21 14 vol vol NOUN cana-1707 21 15 32 32 NUM cana-1707 21 16 no no NOUN cana-1707 21 17 . . NOUN cana-1707 21 18 2 2 NUM cana-1707 21 19 ( ( PUNCT cana-1707 21 20 2025 2025 NUM cana-1707 21 21 ) ) PUNCT cana-1707 21 22 54 54 NUM cana-1707 21 23 https://internationalpubls.com https://internationalpubls.com X cana-1707 21 24 in in ADP cana-1707 21 25 1966 1966 NUM cana-1707 21 26 , , PUNCT cana-1707 21 27 sehgal sehgal PROPN cana-1707 22 1 [ [ X cana-1707 22 2 50 50 NUM cana-1707 22 3 ] ] PUNCT cana-1707 22 4 introduced introduce VERB cana-1707 22 5 the the DET cana-1707 22 6 first first ADJ cana-1707 22 7 generalized generalized ADJ cana-1707 22 8 form form NOUN cana-1707 22 9 of of ADP cana-1707 22 10 banach banach NOUN cana-1707 22 11 contraction contraction NOUN cana-1707 22 12 in in ADP cana-1707 22 13 probabilistic probabilistic ADJ cana-1707 22 14 metric metric ADJ cana-1707 22 15 space space NOUN cana-1707 22 16 and and CCONJ cana-1707 22 17 established establish VERB cana-1707 22 18 the the DET cana-1707 22 19 fixed fix VERB cana-1707 22 20 point point NOUN cana-1707 22 21 theorem theorem VERB cana-1707 22 22 in in ADP cana-1707 22 23 complete complete ADJ cana-1707 22 24 menger menger PROPN cana-1707 22 25 probabilistic probabilistic VERB cana-1707 22 26 metric metric ADJ cana-1707 22 27 space space NOUN cana-1707 22 28 [ [ X cana-1707 22 29 51 51 NUM cana-1707 22 30 ] ] PUNCT cana-1707 22 31 , , PUNCT cana-1707 22 32 and and CCONJ cana-1707 22 33 hicks hick NOUN cana-1707 23 1 [ [ X cana-1707 23 2 32 32 NUM cana-1707 23 3 ] ] PUNCT cana-1707 23 4 defined define VERB cana-1707 23 5 another another DET cana-1707 23 6 contraction contraction NOUN cana-1707 23 7 in in ADP cana-1707 23 8 1983 1983 NUM cana-1707 23 9 . . PUNCT cana-1707 24 1 after after SCONJ cana-1707 24 2 that that DET cana-1707 24 3 many many ADJ cana-1707 24 4 authors author NOUN cana-1707 24 5 worked work VERB cana-1707 24 6 in in ADP cana-1707 24 7 this this DET cana-1707 24 8 space space NOUN cana-1707 24 9 and and CCONJ cana-1707 24 10 introduced introduce VERB cana-1707 24 11 different different ADJ cana-1707 24 12 variants variant NOUN cana-1707 24 13 of of ADP cana-1707 24 14 contraction contraction NOUN cana-1707 24 15 in in ADP cana-1707 24 16 this this DET cana-1707 24 17 space space NOUN cana-1707 24 18 , , PUNCT cana-1707 24 19 see see VERB cana-1707 24 20 references reference NOUN cana-1707 24 21 [ [ X cana-1707 24 22 [ [ X cana-1707 24 23 13 13 NUM cana-1707 24 24 – – PUNCT cana-1707 24 25 17 17 NUM cana-1707 24 26 ] ] PUNCT cana-1707 24 27 , , PUNCT cana-1707 24 28 [ [ X cana-1707 24 29 20 20 NUM cana-1707 24 30 ] ] PUNCT cana-1707 24 31 , , PUNCT cana-1707 24 32 [ [ X cana-1707 24 33 26 26 NUM cana-1707 24 34 ] ] PUNCT cana-1707 24 35 , , PUNCT cana-1707 24 36 [ [ X cana-1707 24 37 28 28 NUM cana-1707 24 38 ] ] PUNCT cana-1707 24 39 , , PUNCT cana-1707 24 40 [ [ X cana-1707 24 41 39 39 NUM cana-1707 24 42 ] ] PUNCT cana-1707 24 43 , , PUNCT cana-1707 24 44 [ [ X cana-1707 24 45 41 41 NUM cana-1707 24 46 ] ] X cana-1707 24 47 ] ] PUNCT cana-1707 24 48 . . PUNCT cana-1707 25 1 in in ADP cana-1707 25 2 this this DET cana-1707 25 3 paper paper NOUN cana-1707 25 4 , , PUNCT cana-1707 25 5 we we PRON cana-1707 25 6 shall shall AUX cana-1707 25 7 update update VERB cana-1707 25 8 the the DET cana-1707 25 9 study study NOUN cana-1707 25 10 of of ADP cana-1707 25 11 various various ADJ cana-1707 25 12 general general ADJ cana-1707 25 13 contractive contractive ADJ cana-1707 25 14 results result NOUN cana-1707 25 15 to to ADP cana-1707 25 16 the the DET cana-1707 25 17 banach banach NOUN cana-1707 25 18 contraction contraction NOUN cana-1707 25 19 principle principle NOUN cana-1707 25 20 . . PUNCT cana-1707 26 1 each each DET cana-1707 26 2 mapping mapping NOUN cana-1707 26 3 has have VERB cana-1707 26 4 a a DET cana-1707 26 5 property property NOUN cana-1707 26 6 to to PART cana-1707 26 7 establish establish VERB cana-1707 26 8 unique unique ADJ cana-1707 26 9 fixed fix VERB cana-1707 26 10 points point NOUN cana-1707 26 11 in in ADP cana-1707 26 12 complete complete ADJ cana-1707 26 13 metric metric ADJ cana-1707 26 14 and and CCONJ cana-1707 26 15 menger menger PROPN cana-1707 26 16 space space NOUN cana-1707 26 17 . . PUNCT cana-1707 27 1 2 2 X cana-1707 27 2 . . X cana-1707 27 3 preliminaries preliminary NOUN cana-1707 27 4 : : PUNCT cana-1707 27 5 definition definition NOUN cana-1707 27 6 2.1 2.1 NUM cana-1707 28 1 [ [ X cana-1707 28 2 25 25 NUM cana-1707 28 3 ] ] PUNCT cana-1707 28 4 : : PUNCT cana-1707 28 5 let let VERB cana-1707 28 6 𝑋 𝑋 NOUN cana-1707 28 7 be be AUX cana-1707 28 8 an an DET cana-1707 28 9 abstract abstract ADJ cana-1707 28 10 set set NOUN cana-1707 28 11 and and CCONJ cana-1707 28 12 𝑑 𝑑 AUX cana-1707 28 13 be be VERB cana-1707 28 14 a a DET cana-1707 28 15 distance distance NOUN cana-1707 28 16 function function NOUN cana-1707 28 17 from from ADP cana-1707 28 18 𝑋 𝑋 PROPN cana-1707 28 19 Γ— Γ— NOUN cana-1707 28 20 𝑋 𝑋 NOUN cana-1707 28 21 β†’ β†’ SYM cana-1707 28 22 ℝ+ ℝ+ PROPN cana-1707 28 23 . . PUNCT cana-1707 29 1 then then ADV cana-1707 29 2 , , PUNCT cana-1707 29 3 an an DET cana-1707 29 4 ordered order VERB cana-1707 29 5 pair pair NOUN cana-1707 29 6 ( ( PUNCT cana-1707 29 7 𝑋 𝑋 PROPN cana-1707 29 8 , , PUNCT cana-1707 29 9 𝑑 𝑑 NOUN cana-1707 29 10 ) ) PUNCT cana-1707 29 11 is be AUX cana-1707 29 12 said say VERB cana-1707 29 13 to to PART cana-1707 29 14 be be AUX cana-1707 29 15 metric metric ADJ cana-1707 29 16 space space NOUN cana-1707 29 17 if if SCONJ cana-1707 29 18 𝑑 𝑑 PRON cana-1707 29 19 satisfies satisfy VERB cana-1707 29 20 the the DET cana-1707 29 21 following follow VERB cana-1707 29 22 conditions condition NOUN cana-1707 29 23 for for ADP cana-1707 29 24 all all PRON cana-1707 29 25 π‘₯ π‘₯ PROPN cana-1707 29 26 , , PUNCT cana-1707 29 27 𝑦 𝑦 NOUN cana-1707 29 28 , , PUNCT cana-1707 29 29 𝑧 𝑧 DET cana-1707 29 30 ∈ ∈ PROPN cana-1707 29 31 𝑋 𝑋 NOUN cana-1707 29 32 : : PUNCT cana-1707 29 33 ( ( PUNCT cana-1707 29 34 i i NOUN cana-1707 29 35 ) ) PUNCT cana-1707 29 36 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 29 37 , , PUNCT cana-1707 29 38 𝑦 𝑦 NOUN cana-1707 29 39 ) ) PUNCT cana-1707 29 40 β‰₯ β‰₯ NOUN cana-1707 29 41 0 0 NUM cana-1707 29 42 ; ; PUNCT cana-1707 29 43 ( ( PUNCT cana-1707 29 44 ii ii NOUN cana-1707 29 45 ) ) PUNCT cana-1707 29 46 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 29 47 , , PUNCT cana-1707 29 48 𝑦 𝑦 X cana-1707 29 49 ) ) PUNCT cana-1707 29 50 = = SYM cana-1707 29 51 0 0 PUNCT cana-1707 30 1 if if SCONJ cana-1707 30 2 and and CCONJ cana-1707 30 3 only only ADV cana-1707 30 4 if if SCONJ cana-1707 30 5 π‘₯ π‘₯ PROPN cana-1707 30 6 = = SYM cana-1707 30 7 𝑦 𝑦 NUM cana-1707 30 8 ; ; PUNCT cana-1707 30 9 ( ( PUNCT cana-1707 30 10 iii iii X cana-1707 30 11 ) ) PUNCT cana-1707 30 12 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 30 13 , , PUNCT cana-1707 30 14 𝑦 𝑦 X cana-1707 30 15 ) ) PUNCT cana-1707 30 16 = = SYM cana-1707 30 17 𝑑(𝑦 𝑑(𝑦 NOUN cana-1707 30 18 , , PUNCT cana-1707 30 19 π‘₯ π‘₯ NOUN cana-1707 30 20 ) ) PUNCT cana-1707 30 21 ; ; PUNCT cana-1707 30 22 and and CCONJ cana-1707 30 23 ( ( PUNCT cana-1707 30 24 iv iv X cana-1707 30 25 ) ) PUNCT cana-1707 30 26 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 30 27 , , PUNCT cana-1707 30 28 𝑦 𝑦 NOUN cana-1707 30 29 ) ) PUNCT cana-1707 30 30 ≀ ≀ NOUN cana-1707 30 31 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 30 32 , , PUNCT cana-1707 30 33 𝑧 𝑧 NOUN cana-1707 30 34 ) ) PUNCT cana-1707 30 35 + + X cana-1707 30 36 𝑑(𝑧 𝑑(𝑧 ADJ cana-1707 30 37 , , PUNCT cana-1707 30 38 𝑦 𝑦 NOUN cana-1707 30 39 ) ) PUNCT cana-1707 30 40 here here ADV cana-1707 30 41 , , PUNCT cana-1707 30 42 ( ( PUNCT cana-1707 30 43 i i NOUN cana-1707 30 44 ) ) PUNCT cana-1707 30 45 and and CCONJ cana-1707 30 46 ( ( PUNCT cana-1707 30 47 ii ii NOUN cana-1707 30 48 ) ) PUNCT cana-1707 30 49 guarantee guarantee VERB cana-1707 30 50 that that SCONJ cana-1707 30 51 the the DET cana-1707 30 52 distance distance NOUN cana-1707 30 53 between between ADP cana-1707 30 54 any any DET cana-1707 30 55 two two NUM cana-1707 30 56 points point NOUN cana-1707 30 57 of of ADP cana-1707 30 58 𝑋 𝑋 NOUN cana-1707 30 59 is be AUX cana-1707 30 60 always always ADV cana-1707 30 61 positive positive ADJ cana-1707 30 62 and and CCONJ cana-1707 30 63 only only ADV cana-1707 30 64 zero zero NUM cana-1707 30 65 when when SCONJ cana-1707 30 66 the the DET cana-1707 30 67 points point NOUN cana-1707 30 68 coincide coincide NOUN cana-1707 30 69 . . PUNCT cana-1707 31 1 ( ( PUNCT cana-1707 31 2 iii iii X cana-1707 31 3 ) ) PUNCT cana-1707 31 4 assures assure VERB cana-1707 31 5 that that SCONJ cana-1707 31 6 the the DET cana-1707 31 7 order order NOUN cana-1707 31 8 of of ADP cana-1707 31 9 measurement measurement NOUN cana-1707 31 10 of of ADP cana-1707 31 11 the the DET cana-1707 31 12 distance distance NOUN cana-1707 31 13 between between ADP cana-1707 31 14 two two NUM cana-1707 31 15 points point NOUN cana-1707 31 16 is be AUX cana-1707 31 17 insignificant insignificant ADJ cana-1707 31 18 . . PUNCT cana-1707 32 1 ( ( PUNCT cana-1707 32 2 iv iv X cana-1707 32 3 ) ) PUNCT cana-1707 32 4 is be AUX cana-1707 32 5 a a DET cana-1707 32 6 statement statement NOUN cana-1707 32 7 of of ADP cana-1707 32 8 the the DET cana-1707 32 9 familiar familiar ADJ cana-1707 32 10 triangular triangular NOUN cana-1707 32 11 inequality inequality NOUN cana-1707 32 12 . . PUNCT cana-1707 33 1 definition definition NOUN cana-1707 33 2 2.2 2.2 NUM cana-1707 34 1 [ [ X cana-1707 34 2 15 15 NUM cana-1707 34 3 ] ] X cana-1707 34 4 : : PUNCT cana-1707 34 5 a a DET cana-1707 34 6 mapping mapping NOUN cana-1707 34 7 f f X cana-1707 34 8 : : PUNCT cana-1707 34 9 ℝ ℝ PROPN cana-1707 34 10 β†’ β†’ X cana-1707 34 11 ℝ+ ℝ+ PUNCT cana-1707 34 12 is be AUX cana-1707 34 13 said say VERB cana-1707 34 14 to to PART cana-1707 34 15 be be AUX cana-1707 34 16 a a DET cana-1707 34 17 distribution distribution NOUN cana-1707 34 18 function function NOUN cana-1707 34 19 if if SCONJ cana-1707 34 20 it it PRON cana-1707 34 21 is be AUX cana-1707 34 22 non non ADJ cana-1707 34 23 - - ADJ cana-1707 34 24 decreasing decrease VERB cana-1707 34 25 and and CCONJ cana-1707 34 26 left leave VERB cana-1707 34 27 continuous continuous ADJ cana-1707 34 28 with with ADP cana-1707 34 29 𝑖𝑛𝑓π‘₯βˆˆβ„ 𝑖𝑛𝑓π‘₯βˆˆβ„ INTJ cana-1707 34 30 𝐹 𝐹 PROPN cana-1707 34 31 ( ( PUNCT cana-1707 34 32 π‘₯ π‘₯ NOUN cana-1707 34 33 ) ) PUNCT cana-1707 34 34 = = SYM cana-1707 34 35 0 0 NUM cana-1707 34 36 , , PUNCT cana-1707 34 37 and and CCONJ cana-1707 34 38 𝑠𝑒𝑝π‘₯βˆˆβ„ 𝑠𝑒𝑝π‘₯βˆˆβ„ INTJ cana-1707 34 39 𝐹 𝐹 PROPN cana-1707 34 40 ( ( PUNCT cana-1707 34 41 π‘₯ π‘₯ NOUN cana-1707 34 42 ) ) PUNCT cana-1707 34 43 = = SYM cana-1707 34 44 1 1 NUM cana-1707 34 45 , , PUNCT cana-1707 34 46 where where SCONJ cana-1707 34 47 ℝ+ ℝ+ PUNCT cana-1707 34 48 denotes denote VERB cana-1707 34 49 the the DET cana-1707 34 50 set set NOUN cana-1707 34 51 of of ADP cana-1707 34 52 non non ADJ cana-1707 34 53 - - ADJ cana-1707 34 54 negative negative ADJ cana-1707 34 55 real real ADJ cana-1707 34 56 numbers number NOUN cana-1707 34 57 . . PUNCT cana-1707 35 1 definition definition NOUN cana-1707 35 2 2.3 2.3 NUM cana-1707 36 1 [ [ X cana-1707 36 2 15 15 NUM cana-1707 36 3 ] ] PUNCT cana-1707 36 4 : : PUNCT cana-1707 36 5 let let VERB cana-1707 36 6 𝑋 𝑋 NOUN cana-1707 36 7 be be AUX cana-1707 36 8 a a DET cana-1707 36 9 non non ADJ cana-1707 36 10 - - ADJ cana-1707 36 11 empty empty ADJ cana-1707 36 12 set set NOUN cana-1707 36 13 and and CCONJ cana-1707 36 14 𝐹 𝐹 PROPN cana-1707 36 15 : : PUNCT cana-1707 36 16 𝑋 𝑋 NOUN cana-1707 36 17 Γ— Γ— NOUN cana-1707 36 18 𝑋 𝑋 PROPN cana-1707 36 19 β†’ β†’ SYM cana-1707 36 20 𝐿 𝐿 PROPN cana-1707 36 21 ( ( PUNCT cana-1707 36 22 set set NOUN cana-1707 36 23 of of ADP cana-1707 36 24 all all DET cana-1707 36 25 distribution distribution NOUN cana-1707 36 26 functions function NOUN cana-1707 36 27 ) ) PUNCT cana-1707 36 28 be be AUX cana-1707 36 29 a a DET cana-1707 36 30 distribution distribution NOUN cana-1707 36 31 function function NOUN cana-1707 36 32 . . PUNCT cana-1707 37 1 then then ADV cana-1707 37 2 , , PUNCT cana-1707 37 3 a a DET cana-1707 37 4 pair pair NOUN cana-1707 37 5 ( ( PUNCT cana-1707 37 6 𝑋 𝑋 PROPN cana-1707 37 7 , , PUNCT cana-1707 37 8 𝐹 𝐹 PROPN cana-1707 37 9 ) ) PUNCT cana-1707 37 10 is be AUX cana-1707 37 11 said say VERB cana-1707 37 12 to to PART cana-1707 37 13 be be AUX cana-1707 37 14 a a DET cana-1707 37 15 probabilistic probabilistic ADJ cana-1707 37 16 metric metric ADJ cana-1707 37 17 space space NOUN cana-1707 37 18 ( ( PUNCT cana-1707 37 19 abbreviated abbreviate VERB cana-1707 37 20 as as ADP cana-1707 37 21 pm pm NOUN cana-1707 37 22 - - PUNCT cana-1707 37 23 space space NOUN cana-1707 37 24 ) ) PUNCT cana-1707 37 25 if if SCONJ cana-1707 37 26 the the DET cana-1707 37 27 distribution distribution NOUN cana-1707 37 28 function function VERB cana-1707 37 29 𝐹 𝐹 PROPN cana-1707 37 30 ( ( PUNCT cana-1707 37 31 π‘₯ π‘₯ PROPN cana-1707 37 32 , , PUNCT cana-1707 37 33 𝑦 𝑦 NOUN cana-1707 37 34 ) ) PUNCT cana-1707 37 35 , , PUNCT cana-1707 37 36 also also ADV cana-1707 37 37 denoted denote VERB cana-1707 37 38 by by ADP cana-1707 37 39 𝐹π‘₯,𝑦 𝐹π‘₯,𝑦 PROPN cana-1707 37 40 satisfies satisfy VERB cana-1707 37 41 the the DET cana-1707 37 42 following follow VERB cana-1707 37 43 conditions condition NOUN cana-1707 37 44 : : PUNCT cana-1707 37 45 ( ( PUNCT cana-1707 37 46 i i NOUN cana-1707 37 47 ) ) PUNCT cana-1707 37 48 𝐹π‘₯,𝑦(𝑑 𝐹π‘₯,𝑦(𝑑 X cana-1707 37 49 ) ) PUNCT cana-1707 37 50 = = SYM cana-1707 37 51 1 1 NUM cana-1707 37 52 for for ADP cana-1707 37 53 every every DET cana-1707 37 54 𝑑 𝑑 X cana-1707 37 55 > > X cana-1707 37 56 0 0 PUNCT cana-1707 38 1 if if SCONJ cana-1707 38 2 and and CCONJ cana-1707 38 3 only only ADV cana-1707 38 4 if if SCONJ cana-1707 38 5 π‘₯ π‘₯ PROPN cana-1707 38 6 = = SYM cana-1707 38 7 𝑦 𝑦 NUM cana-1707 38 8 ; ; PUNCT cana-1707 38 9 ( ( PUNCT cana-1707 38 10 ii ii NOUN cana-1707 38 11 ) ) PUNCT cana-1707 38 12 𝐹π‘₯,𝑦(0 𝐹π‘₯,𝑦(0 X cana-1707 38 13 ) ) PUNCT cana-1707 39 1 = = SYM cana-1707 39 2 0 0 NUM cana-1707 39 3 for for ADP cana-1707 39 4 every every DET cana-1707 39 5 π‘₯ π‘₯ PROPN cana-1707 39 6 , , PUNCT cana-1707 39 7 𝑦 𝑦 NOUN cana-1707 39 8 ∈ ∈ PROPN cana-1707 39 9 𝑋 𝑋 NOUN cana-1707 39 10 ; ; PUNCT cana-1707 39 11 ( ( PUNCT cana-1707 39 12 iii iii NOUN cana-1707 39 13 ) ) PUNCT cana-1707 39 14 𝐹π‘₯,𝑦(𝑑 𝐹π‘₯,𝑦(𝑑 NUM cana-1707 39 15 ) ) PUNCT cana-1707 39 16 = = SYM cana-1707 40 1 𝐹𝑦,π‘₯(𝑑 𝐹𝑦,π‘₯(𝑑 VERB cana-1707 40 2 ) ) PUNCT cana-1707 40 3 for for ADP cana-1707 40 4 every every DET cana-1707 40 5 π‘₯ π‘₯ PROPN cana-1707 40 6 , , PUNCT cana-1707 40 7 𝑦 𝑦 NOUN cana-1707 40 8 ∈ ∈ PROPN cana-1707 40 9 𝑋 𝑋 NOUN cana-1707 40 10 ; ; PUNCT cana-1707 40 11 and and CCONJ cana-1707 40 12 ( ( PUNCT cana-1707 40 13 iv iv X cana-1707 40 14 ) ) PUNCT cana-1707 40 15 𝐹π‘₯,𝑧(𝑝 𝐹π‘₯,𝑧(𝑝 NOUN cana-1707 40 16 + + CCONJ cana-1707 40 17 π‘ž π‘ž X cana-1707 40 18 ) ) PUNCT cana-1707 40 19 = = SYM cana-1707 40 20 1 1 NUM cana-1707 40 21 if if SCONJ cana-1707 40 22 and and CCONJ cana-1707 40 23 only only ADV cana-1707 40 24 if if SCONJ cana-1707 40 25 𝐹π‘₯,𝑦(𝑝 𝐹π‘₯,𝑦(𝑝 NOUN cana-1707 40 26 ) ) PUNCT cana-1707 40 27 = = SYM cana-1707 40 28 1 1 NUM cana-1707 40 29 and and CCONJ cana-1707 40 30 𝐹𝑦,𝑧(π‘ž 𝐹𝑦,𝑧(π‘ž NUM cana-1707 40 31 ) ) PUNCT cana-1707 40 32 = = SYM cana-1707 40 33 1 1 X cana-1707 40 34 . . PUNCT cana-1707 40 35 example example NOUN cana-1707 40 36 2.1 2.1 NUM cana-1707 40 37 : : PUNCT cana-1707 40 38 let let VERB cana-1707 40 39 ( ( PUNCT cana-1707 40 40 𝑋 𝑋 NOUN cana-1707 40 41 , , PUNCT cana-1707 40 42 𝑑 𝑑 NOUN cana-1707 40 43 ) ) PUNCT cana-1707 40 44 be be AUX cana-1707 40 45 metric metric ADJ cana-1707 40 46 space space NOUN cana-1707 40 47 where where SCONJ cana-1707 40 48 𝑋 𝑋 NOUN cana-1707 40 49 = = PUNCT cana-1707 41 1 [ [ X cana-1707 41 2 0 0 NUM cana-1707 41 3 , , PUNCT cana-1707 41 4 5]with 5]with ADJ cana-1707 41 5 usual usual ADJ cana-1707 41 6 metric metric ADJ cana-1707 41 7 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 41 8 , , PUNCT cana-1707 41 9 𝑦 𝑦 X cana-1707 41 10 ) ) PUNCT cana-1707 41 11 = = SYM cana-1707 41 12 |π‘₯ |π‘₯ ADP cana-1707 41 13 βˆ’ βˆ’ PROPN cana-1707 41 14 𝑦| 𝑦| NOUN cana-1707 41 15 and and CCONJ cana-1707 41 16 distribution distribution NOUN cana-1707 41 17 function function NOUN cana-1707 41 18 𝐹 𝐹 PRON cana-1707 41 19 defined define VERB cana-1707 41 20 as as ADP cana-1707 41 21 : : PUNCT cana-1707 41 22 𝐹π‘₯,𝑦(𝑑 𝐹π‘₯,𝑦(𝑑 X cana-1707 41 23 ) ) PUNCT cana-1707 41 24 = = PRON cana-1707 41 25 { { PUNCT cana-1707 41 26 𝑒 𝑒 PROPN cana-1707 41 27 |π‘₯βˆ’π‘¦| |π‘₯βˆ’π‘¦| NUM cana-1707 41 28 𝑑 𝑑 PROPN cana-1707 41 29 , , PUNCT cana-1707 41 30 𝑖𝑓 𝑖𝑓 NUM cana-1707 41 31 𝑑 𝑑 X cana-1707 41 32 > > X cana-1707 41 33 0 0 NUM cana-1707 41 34 , , PUNCT cana-1707 41 35 0 0 NUM cana-1707 41 36 , , PUNCT cana-1707 41 37 𝑖𝑓 𝑖𝑓 NUM cana-1707 41 38 𝑑 𝑑 X cana-1707 41 39 = = NOUN cana-1707 41 40 0 0 X cana-1707 41 41 . . PUNCT cana-1707 42 1 for for ADP cana-1707 42 2 all all DET cana-1707 42 3 π‘₯ π‘₯ PROPN cana-1707 42 4 , , PUNCT cana-1707 42 5 𝑦 𝑦 PRON cana-1707 42 6 ∈ ∈ PROPN cana-1707 42 7 𝑋. 𝑋. PROPN cana-1707 42 8 then then ADV cana-1707 42 9 , , PUNCT cana-1707 42 10 ( ( PUNCT cana-1707 42 11 𝑋 𝑋 PROPN cana-1707 42 12 , , PUNCT cana-1707 42 13 𝐹 𝐹 PROPN cana-1707 42 14 ) ) PUNCT cana-1707 42 15 be be VERB cana-1707 42 16 pm pm NOUN cana-1707 42 17 space space NOUN cana-1707 42 18 . . PUNCT cana-1707 43 1 definition definition NOUN cana-1707 43 2 2.4 2.4 NUM cana-1707 44 1 [ [ X cana-1707 44 2 28 28 NUM cana-1707 44 3 ] ] X cana-1707 44 4 : : PUNCT cana-1707 44 5 a a DET cana-1707 44 6 function function NOUN cana-1707 44 7 𝑇 𝑇 PROPN cana-1707 44 8 ∢ ∢ NOUN cana-1707 44 9 [ [ X cana-1707 44 10 0 0 NUM cana-1707 44 11 , , PUNCT cana-1707 44 12 1 1 NUM cana-1707 44 13 ] ] SYM cana-1707 44 14 Γ— Γ— NOUN cana-1707 45 1 [ [ X cana-1707 45 2 0 0 NUM cana-1707 45 3 , , PUNCT cana-1707 45 4 1 1 NUM cana-1707 45 5 ] ] PUNCT cana-1707 45 6 β†’ β†’ PUNCT cana-1707 45 7 [ [ X cana-1707 45 8 0 0 NUM cana-1707 45 9 , , PUNCT cana-1707 45 10 1 1 NUM cana-1707 45 11 ] ] PUNCT cana-1707 45 12 is be AUX cana-1707 45 13 referred refer VERB cana-1707 45 14 to to ADP cana-1707 45 15 as as ADP cana-1707 45 16 a a DET cana-1707 45 17 triangular triangular NOUN cana-1707 45 18 norm norm NOUN cana-1707 45 19 communications communication NOUN cana-1707 45 20 on on ADP cana-1707 45 21 applied apply VERB cana-1707 45 22 nonlinear nonlinear ADJ cana-1707 45 23 analysis analysis NOUN cana-1707 45 24 issn issn NOUN cana-1707 45 25 : : PUNCT cana-1707 45 26 1074 1074 NUM cana-1707 45 27 - - PUNCT cana-1707 45 28 133x 133x NUM cana-1707 45 29 vol vol NOUN cana-1707 45 30 32 32 NUM cana-1707 45 31 no no NOUN cana-1707 45 32 . . NOUN cana-1707 45 33 2 2 NUM cana-1707 45 34 ( ( PUNCT cana-1707 45 35 2025 2025 NUM cana-1707 45 36 ) ) PUNCT cana-1707 45 37 55 55 NUM cana-1707 45 38 https://internationalpubls.com https://internationalpubls.com X cana-1707 45 39 ( ( PUNCT cana-1707 45 40 shortly shortly ADV cana-1707 45 41 t t NOUN cana-1707 45 42 - - PUNCT cana-1707 45 43 norm norm NOUN cana-1707 45 44 ) ) PUNCT cana-1707 45 45 if if SCONJ cana-1707 45 46 it it PRON cana-1707 45 47 satisfies satisfy VERB cana-1707 45 48 the the DET cana-1707 45 49 following follow VERB cana-1707 45 50 conditions condition NOUN cana-1707 45 51 : : PUNCT cana-1707 45 52 t1 t1 NOUN cana-1707 45 53 : : PUNCT cana-1707 45 54 𝑇 𝑇 PROPN cana-1707 45 55 ( ( PUNCT cana-1707 45 56 0 0 NUM cana-1707 45 57 , , PUNCT cana-1707 45 58 0 0 NUM cana-1707 45 59 ) ) PUNCT cana-1707 45 60 = = SYM cana-1707 45 61 0 0 NUM cana-1707 45 62 ; ; PUNCT cana-1707 45 63 t2 t2 NOUN cana-1707 45 64 : : PUNCT cana-1707 45 65 𝑇 𝑇 PROPN cana-1707 45 66 ( ( PUNCT cana-1707 45 67 π‘Ž π‘Ž NOUN cana-1707 45 68 , , PUNCT cana-1707 45 69 1 1 NUM cana-1707 45 70 ) ) PUNCT cana-1707 45 71 = = NOUN cana-1707 46 1 π‘Ž π‘Ž NOUN cana-1707 46 2 for for ADP cana-1707 46 3 all all DET cana-1707 46 4 π‘Ž π‘Ž PRON cana-1707 46 5 ∈ ∈ NOUN cana-1707 47 1 [ [ X cana-1707 47 2 0 0 NUM cana-1707 47 3 , , PUNCT cana-1707 47 4 1 1 NUM cana-1707 47 5 ] ] PUNCT cana-1707 47 6 ; ; PUNCT cana-1707 47 7 t3 t3 NOUN cana-1707 47 8 : : PUNCT cana-1707 47 9 𝑇 𝑇 PROPN cana-1707 47 10 ( ( PUNCT cana-1707 47 11 π‘Ž π‘Ž NOUN cana-1707 47 12 , , PUNCT cana-1707 47 13 𝑏 𝑏 NOUN cana-1707 47 14 ) ) PUNCT cana-1707 47 15 = = SYM cana-1707 48 1 𝑑 𝑑 PROPN cana-1707 48 2 ( ( PUNCT cana-1707 48 3 𝑏 𝑏 NOUN cana-1707 48 4 , , PUNCT cana-1707 48 5 π‘Ž π‘Ž NOUN cana-1707 48 6 ) ) PUNCT cana-1707 48 7 for for ADP cana-1707 48 8 all all DET cana-1707 48 9 π‘Ž π‘Ž NOUN cana-1707 48 10 , , PUNCT cana-1707 48 11 𝑏 𝑏 PROPN cana-1707 48 12 ∈ ∈ PROPN cana-1707 49 1 [ [ X cana-1707 49 2 0 0 NUM cana-1707 49 3 , , PUNCT cana-1707 49 4 1 1 NUM cana-1707 49 5 ] ] PUNCT cana-1707 49 6 ; ; PUNCT cana-1707 49 7 t4 t4 PROPN cana-1707 49 8 : : PUNCT cana-1707 49 9 𝑖𝑓 𝑖𝑓 NUM cana-1707 49 10 π‘Ž π‘Ž X cana-1707 49 11 ≀ ≀ NUM cana-1707 49 12 𝑐 𝑐 NOUN cana-1707 49 13 , , PUNCT cana-1707 49 14 𝑏 𝑏 PROPN cana-1707 49 15 ≀ ≀ NOUN cana-1707 49 16 𝑑 𝑑 PROPN cana-1707 49 17 then then ADV cana-1707 49 18 𝑇 𝑇 PROPN cana-1707 49 19 ( ( PUNCT cana-1707 49 20 π‘Ž π‘Ž PROPN cana-1707 49 21 , , PUNCT cana-1707 49 22 𝑏 𝑏 NOUN cana-1707 49 23 ) ) PUNCT cana-1707 49 24 ≀ ≀ NOUN cana-1707 49 25 𝑇(𝑐 𝑇(𝑐 NUM cana-1707 49 26 , , PUNCT cana-1707 49 27 𝑑 𝑑 NOUN cana-1707 49 28 ) ) PUNCT cana-1707 49 29 and and CCONJ cana-1707 49 30 t5 t5 PROPN cana-1707 49 31 : : PUNCT cana-1707 49 32 𝑇 𝑇 PROPN cana-1707 49 33 ( ( PUNCT cana-1707 49 34 𝑑 𝑑 PROPN cana-1707 49 35 ( ( PUNCT cana-1707 49 36 π‘Ž π‘Ž X cana-1707 49 37 , , PUNCT cana-1707 49 38 𝑏 𝑏 NOUN cana-1707 49 39 ) ) PUNCT cana-1707 49 40 , , PUNCT cana-1707 49 41 𝑐 𝑐 NOUN cana-1707 49 42 ) ) PUNCT cana-1707 49 43 = = SYM cana-1707 49 44 𝑇 𝑇 PROPN cana-1707 49 45 ( ( PUNCT cana-1707 49 46 π‘Ž π‘Ž X cana-1707 49 47 , , PUNCT cana-1707 49 48 𝑑 𝑑 X cana-1707 49 49 ( ( PUNCT cana-1707 49 50 𝑏 𝑏 NOUN cana-1707 49 51 , , PUNCT cana-1707 49 52 𝑐 𝑐 NOUN cana-1707 49 53 ) ) PUNCT cana-1707 49 54 ) ) PUNCT cana-1707 49 55 , , PUNCT cana-1707 49 56 where where SCONJ cana-1707 49 57 π‘Ž π‘Ž X cana-1707 49 58 , , PUNCT cana-1707 49 59 𝑏 𝑏 NOUN cana-1707 49 60 , , PUNCT cana-1707 49 61 𝑐 𝑐 PROPN cana-1707 49 62 , , PUNCT cana-1707 49 63 𝑑 𝑑 PROPN cana-1707 49 64 ∈ ∈ PROPN cana-1707 50 1 [ [ X cana-1707 50 2 0 0 NUM cana-1707 50 3 , , PUNCT cana-1707 50 4 1 1 NUM cana-1707 50 5 ] ] PUNCT cana-1707 50 6 . . PUNCT cana-1707 51 1 definition definition NOUN cana-1707 51 2 2.5 2.5 NUM cana-1707 51 3 . . PUNCT cana-1707 52 1 [ [ X cana-1707 52 2 17 17 NUM cana-1707 52 3 ] ] PUNCT cana-1707 52 4 a a DET cana-1707 52 5 triplet triplet NOUN cana-1707 52 6 ( ( PUNCT cana-1707 52 7 𝑋 𝑋 PROPN cana-1707 52 8 , , PUNCT cana-1707 52 9 𝐹 𝐹 PROPN cana-1707 52 10 , , PUNCT cana-1707 52 11 𝑇 𝑇 PROPN cana-1707 52 12 ) ) PUNCT cana-1707 52 13 is be AUX cana-1707 52 14 said say VERB cana-1707 52 15 to to PART cana-1707 52 16 be be AUX cana-1707 52 17 menger menger NOUN cana-1707 52 18 space space NOUN cana-1707 52 19 , , PUNCT cana-1707 52 20 where where SCONJ cana-1707 52 21 𝑋 𝑋 PROPN cana-1707 52 22 is be AUX cana-1707 52 23 a a DET cana-1707 52 24 non non ADJ cana-1707 52 25 - - ADJ cana-1707 52 26 empty empty ADJ cana-1707 52 27 set set NOUN cana-1707 52 28 , , PUNCT cana-1707 52 29 𝐹 𝐹 PRON cana-1707 52 30 be be VERB cana-1707 52 31 a a DET cana-1707 52 32 distribution distribution NOUN cana-1707 52 33 function function NOUN cana-1707 52 34 , , PUNCT cana-1707 52 35 and and CCONJ cana-1707 52 36 𝑇 𝑇 PROPN cana-1707 52 37 is be AUX cana-1707 52 38 a a DET cana-1707 52 39 𝑑-norm 𝑑-norm NOUN cana-1707 52 40 such such ADJ cana-1707 52 41 that that SCONJ cana-1707 52 42 the the DET cana-1707 52 43 following following NOUN cana-1707 52 44 are be AUX cana-1707 52 45 satisfied satisfied ADJ cana-1707 52 46 for for ADP cana-1707 52 47 every every DET cana-1707 52 48 𝑑 𝑑 NOUN cana-1707 52 49 , , PUNCT cana-1707 52 50 𝑠 𝑠 PROPN cana-1707 52 51 > > X cana-1707 52 52 0 0 PROPN cana-1707 52 53 & & CCONJ cana-1707 52 54 π‘₯ π‘₯ PROPN cana-1707 52 55 , , PUNCT cana-1707 52 56 𝑦 𝑦 NOUN cana-1707 52 57 , , PUNCT cana-1707 52 58 𝑧 𝑧 DET cana-1707 52 59 ∈ ∈ PROPN cana-1707 52 60 𝑋 𝑋 NOUN cana-1707 52 61 : : PUNCT cana-1707 52 62 ( ( PUNCT cana-1707 52 63 i i NOUN cana-1707 52 64 ) ) PUNCT cana-1707 52 65 𝐹π‘₯,𝑦(𝑑 𝐹π‘₯,𝑦(𝑑 X cana-1707 52 66 ) ) PUNCT cana-1707 52 67 = = SYM cana-1707 52 68 1 1 NUM cana-1707 52 69 for for ADP cana-1707 52 70 every every DET cana-1707 52 71 π‘₯ π‘₯ PROPN cana-1707 52 72 > > X cana-1707 52 73 0 0 PUNCT cana-1707 53 1 if if SCONJ cana-1707 53 2 and and CCONJ cana-1707 53 3 only only ADV cana-1707 53 4 if if SCONJ cana-1707 53 5 π‘₯ π‘₯ PRON cana-1707 53 6 = = SYM cana-1707 53 7 𝑦 𝑦 NOUN cana-1707 53 8 , , PUNCT cana-1707 53 9 ( ( PUNCT cana-1707 53 10 ii ii NOUN cana-1707 53 11 ) ) PUNCT cana-1707 53 12 𝐹π‘₯,𝑦(0 𝐹π‘₯,𝑦(0 X cana-1707 53 13 ) ) PUNCT cana-1707 54 1 = = SYM cana-1707 54 2 0 0 NUM cana-1707 54 3 ; ; PUNCT cana-1707 54 4 ( ( PUNCT cana-1707 54 5 iii iii NOUN cana-1707 54 6 ) ) PUNCT cana-1707 54 7 𝐹π‘₯,𝑦(𝑑 𝐹π‘₯,𝑦(𝑑 NUM cana-1707 54 8 ) ) PUNCT cana-1707 54 9 = = PUNCT cana-1707 54 10 𝐹𝑦,π‘₯(𝑑 𝐹𝑦,π‘₯(𝑑 NOUN cana-1707 54 11 ) ) PUNCT cana-1707 54 12 , , PUNCT cana-1707 54 13 and and CCONJ cana-1707 54 14 ( ( PUNCT cana-1707 54 15 iv iv X cana-1707 54 16 ) ) PUNCT cana-1707 54 17 𝐹π‘₯,𝑧(𝑑 𝐹π‘₯,𝑧(𝑑 PUNCT cana-1707 55 1 + + NUM cana-1707 55 2 𝑠 𝑠 X cana-1707 55 3 ) ) PUNCT cana-1707 55 4 β‰₯ β‰₯ NOUN cana-1707 55 5 𝑇(𝐹π‘₯,𝑦(𝑑 𝑇(𝐹π‘₯,𝑦(𝑑 VERB cana-1707 55 6 ) ) PUNCT cana-1707 55 7 , , PUNCT cana-1707 55 8 𝐹𝑦,𝑧(𝑠 𝐹𝑦,𝑧(𝑠 X cana-1707 55 9 ) ) PUNCT cana-1707 55 10 ) ) PUNCT cana-1707 55 11 . . PUNCT cana-1707 56 1 definition definition NOUN cana-1707 56 2 2.6.[36 2.6.[36 NUM cana-1707 56 3 ] ] PUNCT cana-1707 56 4 let let VERB cana-1707 56 5 𝑇 𝑇 PROPN cana-1707 56 6 : : PUNCT cana-1707 56 7 𝑋 𝑋 PROPN cana-1707 56 8 β†’ β†’ SYM cana-1707 56 9 𝑋 𝑋 PROPN cana-1707 56 10 be be VERB cana-1707 56 11 a a DET cana-1707 56 12 mapping mapping NOUN cana-1707 56 13 in in ADP cana-1707 56 14 metric metric ADJ cana-1707 56 15 space space NOUN cana-1707 56 16 ( ( PUNCT cana-1707 56 17 𝑋 𝑋 PROPN cana-1707 56 18 , , PUNCT cana-1707 56 19 𝑑 𝑑 NOUN cana-1707 56 20 ) ) PUNCT cana-1707 56 21 . . PUNCT cana-1707 57 1 given give VERB cana-1707 57 2 π‘₯ π‘₯ DET cana-1707 57 3 ∈ ∈ PROPN cana-1707 57 4 𝑋 𝑋 PROPN cana-1707 57 5 , , PUNCT cana-1707 57 6 𝑂(π‘₯ 𝑂(π‘₯ NUM cana-1707 57 7 ) ) PUNCT cana-1707 57 8 = = SYM cana-1707 57 9 { { PUNCT cana-1707 57 10 𝑓𝑛 𝑓𝑛 NOUN cana-1707 57 11 : : PUNCT cana-1707 57 12 𝑛 𝑛 PRON cana-1707 57 13 ∈ ∈ PROPN cana-1707 57 14 𝑁 𝑁 PROPN cana-1707 57 15 } } PUNCT cana-1707 57 16 , , PUNCT cana-1707 57 17 οΏ½ οΏ½ PROPN cana-1707 57 18 Μ… Μ… NOUN cana-1707 57 19 οΏ½ οΏ½ NOUN cana-1707 57 20 (π‘₯ (π‘₯ VERB cana-1707 57 21 ) ) PUNCT cana-1707 57 22 be be AUX cana-1707 57 23 its its PRON cana-1707 57 24 closure closure NOUN cana-1707 57 25 . . PUNCT cana-1707 58 1 a a DET cana-1707 58 2 point point NOUN cana-1707 58 3 π‘₯ π‘₯ PRON cana-1707 58 4 ∈ ∈ NOUN cana-1707 58 5 𝑋 𝑋 NOUN cana-1707 58 6 is be AUX cana-1707 58 7 said say VERB cana-1707 58 8 to to PART cana-1707 58 9 be be AUX cana-1707 58 10 regular regular ADJ cana-1707 58 11 for for ADP cana-1707 58 12 𝑇 𝑇 PROPN cana-1707 58 13 if if SCONJ cana-1707 58 14 π‘‘π‘–π‘Žπ‘š π‘‘π‘–π‘Žπ‘š NOUN cana-1707 58 15 𝑂(π‘₯ 𝑂(π‘₯ VERB cana-1707 58 16 ) ) PUNCT cana-1707 58 17 < < X cana-1707 58 18 ∞. ∞. PROPN cana-1707 58 19 given give VERB cana-1707 58 20 , , PUNCT cana-1707 58 21 π‘₯ π‘₯ PROPN cana-1707 58 22 , , PUNCT cana-1707 58 23 𝑦 𝑦 NOUN cana-1707 58 24 ∈ ∈ PROPN cana-1707 58 25 𝑋 𝑋 NOUN cana-1707 58 26 , , PUNCT cana-1707 58 27 let let VERB cana-1707 58 28 π‘š(π‘₯ π‘š(π‘₯ NOUN cana-1707 58 29 , , PUNCT cana-1707 58 30 𝑦 𝑦 NOUN cana-1707 58 31 ) ) PUNCT cana-1707 58 32 = = SYM cana-1707 58 33 π‘šπ‘Žπ‘₯{𝑑(π‘₯ π‘šπ‘Žπ‘₯{𝑑(π‘₯ PROPN cana-1707 58 34 , , PUNCT cana-1707 58 35 𝑦 𝑦 NOUN cana-1707 58 36 ) ) PUNCT cana-1707 58 37 , , PUNCT cana-1707 58 38 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 58 39 , , PUNCT cana-1707 58 40 𝑓π‘₯ 𝑓π‘₯ NOUN cana-1707 58 41 ) ) PUNCT cana-1707 58 42 , , PUNCT cana-1707 58 43 𝑑(𝑦 𝑑(𝑦 PROPN cana-1707 58 44 , , PUNCT cana-1707 58 45 𝑓𝑦 𝑓𝑦 NOUN cana-1707 58 46 ) ) PUNCT cana-1707 58 47 , , PUNCT cana-1707 58 48 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 58 49 , , PUNCT cana-1707 58 50 𝑓𝑦 𝑓𝑦 NOUN cana-1707 58 51 ) ) PUNCT cana-1707 58 52 , , PUNCT cana-1707 58 53 𝑑(𝑦 𝑑(𝑦 PROPN cana-1707 58 54 , , PUNCT cana-1707 58 55 𝑓π‘₯ 𝑓π‘₯ NOUN cana-1707 58 56 ) ) PUNCT cana-1707 58 57 } } PUNCT cana-1707 58 58 , , PUNCT cana-1707 58 59 and and CCONJ cana-1707 58 60 𝛿(π‘₯ 𝛿(π‘₯ PROPN cana-1707 58 61 , , PUNCT cana-1707 58 62 𝑦 𝑦 NOUN cana-1707 58 63 ) ) PUNCT cana-1707 58 64 = = SYM cana-1707 58 65 π‘‘π‘–π‘Žπ‘š{𝑂(π‘₯ π‘‘π‘–π‘Žπ‘š{𝑂(π‘₯ NOUN cana-1707 58 66 ) ) PUNCT cana-1707 58 67 βˆͺ βˆͺ ADP cana-1707 58 68 𝑂(𝑦 𝑂(𝑦 NUM cana-1707 58 69 ) ) PUNCT cana-1707 58 70 } } PUNCT cana-1707 58 71 . . PUNCT cana-1707 59 1 definition definition NOUN cana-1707 59 2 2.7.[1 2.7.[1 NUM cana-1707 59 3 ] ] PUNCT cana-1707 59 4 let let VERB cana-1707 59 5 ( ( PUNCT cana-1707 59 6 𝑋 𝑋 PROPN cana-1707 59 7 , , PUNCT cana-1707 59 8 𝑑 𝑑 NOUN cana-1707 59 9 ) ) PUNCT cana-1707 59 10 be be VERB cana-1707 59 11 a a DET cana-1707 59 12 metric metric ADJ cana-1707 59 13 space space NOUN cana-1707 59 14 . . PUNCT cana-1707 60 1 then then ADV cana-1707 60 2 , , PUNCT cana-1707 60 3 a a DET cana-1707 60 4 mapping mapping NOUN cana-1707 60 5 𝑇 𝑇 NOUN cana-1707 60 6 : : PUNCT cana-1707 60 7 𝑋 𝑋 PROPN cana-1707 60 8 β†’ β†’ SYM cana-1707 60 9 𝑋 𝑋 PROPN cana-1707 60 10 is be AUX cana-1707 60 11 said say VERB cana-1707 60 12 to to PART cana-1707 60 13 be be AUX cana-1707 60 14 contraction contraction NOUN cana-1707 60 15 mapping mapping NOUN cana-1707 60 16 if if SCONJ cana-1707 60 17 there there PRON cana-1707 60 18 exists exist VERB cana-1707 60 19 a a DET cana-1707 60 20 number number NOUN cana-1707 60 21 Ξ» Ξ» X cana-1707 60 22 ∈ ∈ NOUN cana-1707 61 1 [ [ X cana-1707 61 2 0,1 0,1 NUM cana-1707 61 3 ) ) PUNCT cana-1707 61 4 such such ADJ cana-1707 61 5 that that PRON cana-1707 61 6 for for ADP cana-1707 61 7 every every DET cana-1707 61 8 π‘₯ π‘₯ PROPN cana-1707 61 9 , , PUNCT cana-1707 61 10 𝑦 𝑦 NOUN cana-1707 61 11 ∈ ∈ PROPN cana-1707 61 12 𝑋 𝑋 PROPN cana-1707 61 13 , , PUNCT cana-1707 61 14 we we PRON cana-1707 61 15 have have VERB cana-1707 61 16 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 61 17 , , PUNCT cana-1707 61 18 𝑇𝑦 𝑇𝑦 PROPN cana-1707 61 19 ) ) PUNCT cana-1707 61 20 ≀ ≀ NUM cana-1707 62 1 Ξ» Ξ» PROPN cana-1707 62 2 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 62 3 , , PUNCT cana-1707 62 4 𝑦 𝑦 NOUN cana-1707 62 5 ) ) PUNCT cana-1707 62 6 . . PUNCT cana-1707 63 1 ( ( PUNCT cana-1707 63 2 contraction contraction NOUN cana-1707 63 3 , , PUNCT cana-1707 63 4 in in ADP cana-1707 63 5 1922 1922 NUM cana-1707 63 6 ) ) PUNCT cana-1707 63 7 example example NOUN cana-1707 64 1 2.2 2.2 NUM cana-1707 64 2 [ [ X cana-1707 64 3 16 16 NUM cana-1707 64 4 ] ] X cana-1707 64 5 : : PUNCT cana-1707 64 6 let let VERB cana-1707 64 7 function function VERB cana-1707 64 8 𝑓 𝑓 NOUN cana-1707 64 9 : : PUNCT cana-1707 64 10 [ [ X cana-1707 64 11 0,2 0,2 NUM cana-1707 64 12 ] ] X cana-1707 64 13 β†’ β†’ SYM cana-1707 64 14 [ [ X cana-1707 64 15 0,2 0,2 NUM cana-1707 64 16 ] ] PUNCT cana-1707 64 17 be be AUX cana-1707 64 18 defined define VERB cana-1707 64 19 by by ADP cana-1707 64 20 f(x f(x PROPN cana-1707 64 21 ) ) PUNCT cana-1707 64 22 = = SYM cana-1707 64 23 0 0 NUM cana-1707 65 1 for for ADP cana-1707 65 2 x x PROPN cana-1707 65 3 ∈ ∈ PROPN cana-1707 65 4 [ [ X cana-1707 65 5 0,1 0,1 NUM cana-1707 65 6 ] ] SYM cana-1707 65 7 1 1 NUM cana-1707 65 8 for for ADP cana-1707 65 9 x x PROPN cana-1707 65 10 ∈ ∈ PROPN cana-1707 65 11 ( ( PUNCT cana-1707 65 12 1 1 NUM cana-1707 65 13 , , PUNCT cana-1707 65 14 2 2 NUM cana-1707 65 15 ] ] PUNCT cana-1707 65 16 then then ADV cana-1707 65 17 , , PUNCT cana-1707 65 18 𝑓2 𝑓2 PROPN cana-1707 65 19 is be AUX cana-1707 65 20 a a DET cana-1707 65 21 contraction contraction NOUN cana-1707 65 22 but but CCONJ cana-1707 65 23 𝑓 𝑓 PRON cana-1707 65 24 is be AUX cana-1707 65 25 not not PART cana-1707 65 26 a a DET cana-1707 65 27 contraction contraction NOUN cana-1707 65 28 . . PUNCT cana-1707 66 1 3 3 X cana-1707 66 2 . . X cana-1707 66 3 various various ADJ cana-1707 66 4 generalizations generalization NOUN cana-1707 66 5 and and CCONJ cana-1707 66 6 extensions extension NOUN cana-1707 66 7 of of ADP cana-1707 66 8 banach banach NOUN cana-1707 66 9 contraction contraction NOUN cana-1707 66 10 are be AUX cana-1707 66 11 : : PUNCT cana-1707 66 12 definition definition NOUN cana-1707 66 13 3.1 3.1 NUM cana-1707 66 14 . . PUNCT cana-1707 67 1 a a DET cana-1707 67 2 mapping mapping NOUN cana-1707 67 3 𝑇 𝑇 NOUN cana-1707 67 4 : : PUNCT cana-1707 67 5 𝑋 𝑋 PROPN cana-1707 67 6 β†’ β†’ SYM cana-1707 67 7 𝑋 𝑋 PROPN cana-1707 67 8 of of ADP cana-1707 67 9 a a DET cana-1707 67 10 metric metric ADJ cana-1707 67 11 space space NOUN cana-1707 67 12 ( ( PUNCT cana-1707 67 13 𝑋 𝑋 PROPN cana-1707 67 14 , , PUNCT cana-1707 67 15 𝑑 𝑑 NOUN cana-1707 67 16 ) ) PUNCT cana-1707 67 17 is be AUX cana-1707 67 18 said say VERB cana-1707 67 19 to to PART cana-1707 67 20 be be AUX cana-1707 67 21 contractive contractive ADJ cana-1707 67 22 if if SCONJ cana-1707 67 23 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 67 24 , , PUNCT cana-1707 67 25 𝑇𝑦 𝑇𝑦 NOUN cana-1707 67 26 ) ) PUNCT cana-1707 67 27 < < X cana-1707 67 28 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 67 29 , , PUNCT cana-1707 67 30 𝑦 𝑦 NOUN cana-1707 67 31 ) ) PUNCT cana-1707 67 32 , , PUNCT cana-1707 67 33 for for ADP cana-1707 67 34 every every DET cana-1707 67 35 π‘₯ π‘₯ PROPN cana-1707 67 36 β‰  β‰  PROPN cana-1707 67 37 𝑦 𝑦 PRON cana-1707 67 38 ∈ ∈ PROPN cana-1707 67 39 𝑋. 𝑋. PROPN cana-1707 67 40 ( ( PUNCT cana-1707 67 41 edelstein edelstein PROPN cana-1707 67 42 in in ADP cana-1707 67 43 1962 1962 NUM cana-1707 67 44 , , PUNCT cana-1707 67 45 [ [ X cana-1707 67 46 24 24 NUM cana-1707 67 47 ] ] PUNCT cana-1707 67 48 ) ) PUNCT cana-1707 67 49 it it PRON cana-1707 67 50 is be AUX cana-1707 67 51 extended extend VERB cana-1707 67 52 forms form NOUN cana-1707 67 53 as as ADP cana-1707 67 54 : : PUNCT cana-1707 67 55 ( ( PUNCT cana-1707 67 56 i i NOUN cana-1707 67 57 ) ) PUNCT cana-1707 67 58 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 67 59 , , PUNCT cana-1707 67 60 𝑇𝑦 𝑇𝑦 PROPN cana-1707 67 61 ) ) PUNCT cana-1707 67 62 < < X cana-1707 67 63 π‘š(π‘₯ π‘š(π‘₯ PROPN cana-1707 67 64 , , PUNCT cana-1707 67 65 𝑦 𝑦 NOUN cana-1707 67 66 ) ) PUNCT cana-1707 67 67 ( ( PUNCT cana-1707 67 68 rhodes rhode NOUN cana-1707 67 69 in in ADP cana-1707 67 70 1977 1977 NUM cana-1707 67 71 , , PUNCT cana-1707 67 72 [ [ X cana-1707 67 73 48 48 NUM cana-1707 67 74 ] ] SYM cana-1707 67 75 ) ) PUNCT cana-1707 67 76 ( ( PUNCT cana-1707 67 77 ii ii NOUN cana-1707 67 78 ) ) PUNCT cana-1707 67 79 if if SCONJ cana-1707 67 80 π‘₯ π‘₯ PROPN cana-1707 67 81 and and CCONJ cana-1707 67 82 𝑦 𝑦 NOUN cana-1707 67 83 are be AUX cana-1707 67 84 regular regular ADJ cana-1707 67 85 , , PUNCT cana-1707 67 86 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADV cana-1707 67 87 , , PUNCT cana-1707 67 88 𝑇𝑦 𝑇𝑦 PROPN cana-1707 67 89 ) ) PUNCT cana-1707 67 90 < < X cana-1707 67 91 𝛿(π‘₯ 𝛿(π‘₯ PROPN cana-1707 67 92 , , PUNCT cana-1707 67 93 𝑦 𝑦 NOUN cana-1707 67 94 ) ) PUNCT cana-1707 67 95 . . PUNCT cana-1707 68 1 ( ( PUNCT cana-1707 68 2 park park NOUN cana-1707 68 3 in in ADP cana-1707 68 4 1980 1980 NUM cana-1707 68 5 , , PUNCT cana-1707 68 6 [ [ X cana-1707 68 7 43 43 NUM cana-1707 68 8 ] ] SYM cana-1707 68 9 ) ) PUNCT cana-1707 68 10 communications communication NOUN cana-1707 68 11 on on ADP cana-1707 68 12 applied apply VERB cana-1707 68 13 nonlinear nonlinear ADJ cana-1707 68 14 analysis analysis NOUN cana-1707 68 15 issn issn NOUN cana-1707 68 16 : : PUNCT cana-1707 68 17 1074 1074 NUM cana-1707 68 18 - - PUNCT cana-1707 68 19 133x 133x NUM cana-1707 68 20 vol vol NOUN cana-1707 68 21 32 32 NUM cana-1707 68 22 no no NOUN cana-1707 68 23 . . NOUN cana-1707 68 24 2 2 NUM cana-1707 68 25 ( ( PUNCT cana-1707 68 26 2025 2025 NUM cana-1707 68 27 ) ) PUNCT cana-1707 68 28 56 56 NUM cana-1707 68 29 https://internationalpubls.com https://internationalpubls.com NUM cana-1707 68 30 remark remark NOUN cana-1707 68 31 1 1 NUM cana-1707 68 32 . . PUNCT cana-1707 69 1 a a DET cana-1707 69 2 contractive contractive ADJ cana-1707 69 3 map map NOUN cana-1707 69 4 is be AUX cana-1707 69 5 continuous continuous ADJ cana-1707 69 6 ; ; PUNCT cana-1707 69 7 if if SCONJ cana-1707 69 8 such such DET cana-1707 69 9 a a DET cana-1707 69 10 mapping mapping NOUN cana-1707 69 11 has have VERB cana-1707 69 12 a a DET cana-1707 69 13 fixed fix VERB cana-1707 69 14 point point NOUN cana-1707 69 15 , , PUNCT cana-1707 69 16 then then ADV cana-1707 69 17 this this DET cana-1707 69 18 fixed fix VERB cana-1707 69 19 point point NOUN cana-1707 69 20 is be AUX cana-1707 69 21 unique unique ADJ cana-1707 69 22 . . PUNCT cana-1707 70 1 while while SCONJ cana-1707 70 2 the the DET cana-1707 70 3 condition condition NOUN cana-1707 70 4 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 70 5 , , PUNCT cana-1707 70 6 𝑇𝑦 𝑇𝑦 PROPN cana-1707 70 7 ) ) PUNCT cana-1707 70 8 < < X cana-1707 70 9 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 70 10 , , PUNCT cana-1707 70 11 𝑦 𝑦 X cana-1707 70 12 ) ) PUNCT cana-1707 70 13 is be AUX cana-1707 70 14 sufficient sufficient ADJ cana-1707 70 15 to to PART cana-1707 70 16 assure assure VERB cana-1707 70 17 that that SCONJ cana-1707 70 18 𝑇 𝑇 PROPN cana-1707 70 19 has have VERB cana-1707 70 20 a a DET cana-1707 70 21 fixed fix VERB cana-1707 70 22 point point NOUN cana-1707 70 23 but but CCONJ cana-1707 70 24 it it PRON cana-1707 70 25 is be AUX cana-1707 70 26 too too ADV cana-1707 70 27 weak weak ADJ cana-1707 70 28 to to PART cana-1707 70 29 guarantee guarantee VERB cana-1707 70 30 the the DET cana-1707 70 31 existence existence NOUN cana-1707 70 32 of of ADP cana-1707 70 33 one one NUM cana-1707 70 34 as as SCONJ cana-1707 70 35 will will AUX cana-1707 70 36 be be AUX cana-1707 70 37 seen see VERB cana-1707 70 38 from from ADP cana-1707 70 39 the the DET cana-1707 70 40 following following ADJ cana-1707 70 41 examples example NOUN cana-1707 70 42 : : PUNCT cana-1707 70 43 example example NOUN cana-1707 71 1 3.1 3.1 NUM cana-1707 71 2 . . PUNCT cana-1707 72 1 let let VERB cana-1707 72 2 𝑇 𝑇 PROPN cana-1707 72 3 : : PUNCT cana-1707 72 4 ℝ ℝ PROPN cana-1707 72 5 β†’ β†’ SYM cana-1707 72 6 ℝ ℝ PROPN cana-1707 72 7 be be AUX cana-1707 72 8 defined define VERB cana-1707 72 9 by by ADP cana-1707 72 10 𝑇(π‘₯ 𝑇(π‘₯ NOUN cana-1707 72 11 ) ) PUNCT cana-1707 72 12 = = SYM cana-1707 73 1 π‘₯ π‘₯ PROPN cana-1707 74 1 + + CCONJ cana-1707 74 2 πœ‹ πœ‹ NOUN cana-1707 74 3 2 2 NUM cana-1707 74 4 βˆ’ βˆ’ NOUN cana-1707 74 5 π‘‡π‘Žπ‘›βˆ’1π‘₯ π‘‡π‘Žπ‘›βˆ’1π‘₯ PROPN cana-1707 74 6 since since SCONJ cana-1707 74 7 π‘‡π‘Žπ‘›βˆ’1π‘₯ π‘‡π‘Žπ‘›βˆ’1π‘₯ PROPN cana-1707 74 8 < < X cana-1707 74 9 πœ‹ πœ‹ X cana-1707 74 10 2 2 NUM cana-1707 74 11 for for ADP cana-1707 74 12 every every DET cana-1707 74 13 π‘₯ π‘₯ PROPN cana-1707 74 14 , , PUNCT cana-1707 74 15 the the DET cana-1707 74 16 operator operator NOUN cana-1707 74 17 𝑇 𝑇 PROPN cana-1707 74 18 has have AUX cana-1707 74 19 no no DET cana-1707 74 20 fixed fix VERB cana-1707 74 21 point point NOUN cana-1707 74 22 although although SCONJ cana-1707 74 23 𝑇 𝑇 PROPN cana-1707 74 24 is be AUX cana-1707 74 25 a a DET cana-1707 74 26 contractive contractive ADJ cana-1707 74 27 map map NOUN cana-1707 74 28 for for ADP cana-1707 74 29 𝑇′(π‘₯ 𝑇′(π‘₯ PROPN cana-1707 74 30 ) ) PUNCT cana-1707 74 31 = = SYM cana-1707 75 1 1 1 NUM cana-1707 75 2 βˆ’ βˆ’ NUM cana-1707 75 3 1 1 NUM cana-1707 75 4 1+π‘₯2 1+π‘₯2 NOUN cana-1707 75 5 < < X cana-1707 75 6 1 1 NUM cana-1707 75 7 . . PUNCT cana-1707 75 8 example example NOUN cana-1707 75 9 3.2 3.2 NUM cana-1707 75 10 . . PUNCT cana-1707 76 1 let let VERB cana-1707 76 2 𝑇 𝑇 PROPN cana-1707 76 3 : : PUNCT cana-1707 76 4 ℝ ℝ PROPN cana-1707 76 5 β†’ β†’ SYM cana-1707 76 6 ℝ ℝ PROPN cana-1707 76 7 be be AUX cana-1707 76 8 defined define VERB cana-1707 76 9 by by ADP cana-1707 76 10 𝑇(π‘₯ 𝑇(π‘₯ NOUN cana-1707 76 11 ) ) PUNCT cana-1707 76 12 = = PUNCT cana-1707 77 1 ln(1 ln(1 NOUN cana-1707 77 2 + + NUM cana-1707 77 3 𝑒π‘₯ 𝑒π‘₯ NOUN cana-1707 77 4 ) ) PUNCT cana-1707 77 5 , , PUNCT cana-1707 77 6 differentiating differentiating NOUN cana-1707 77 7 , , PUNCT cana-1707 77 8 we we PRON cana-1707 77 9 obtain obtain VERB cana-1707 77 10 𝑇′(π‘₯ 𝑇′(π‘₯ NOUN cana-1707 77 11 ) ) PUNCT cana-1707 78 1 = = SYM cana-1707 78 2 𝑒π‘₯ 𝑒π‘₯ NOUN cana-1707 79 1 1+𝑒π‘₯ 1+𝑒π‘₯ NUM cana-1707 79 2 < < X cana-1707 79 3 1 1 NUM cana-1707 79 4 . . PUNCT cana-1707 79 5 i.e. i.e. X cana-1707 79 6 𝑇 𝑇 PROPN cana-1707 79 7 is be AUX cana-1707 79 8 a a DET cana-1707 79 9 contractive contractive ADJ cana-1707 79 10 mapping mapping NOUN cana-1707 79 11 , , PUNCT cana-1707 79 12 and and CCONJ cana-1707 79 13 it it PRON cana-1707 79 14 is be AUX cana-1707 79 15 easy easy ADJ cana-1707 79 16 to to PART cana-1707 79 17 see see VERB cana-1707 79 18 that that SCONJ cana-1707 79 19 𝑇 𝑇 PROPN cana-1707 79 20 has have VERB cana-1707 79 21 no no DET cana-1707 79 22 fixed fix VERB cana-1707 79 23 point point NOUN cana-1707 79 24 . . PUNCT cana-1707 80 1 definition definition NOUN cana-1707 80 2 3.2 3.2 NUM cana-1707 80 3 . . PUNCT cana-1707 81 1 let let VERB cana-1707 81 2 𝑇 𝑇 PROPN cana-1707 81 3 : : PUNCT cana-1707 81 4 𝑋 𝑋 PROPN cana-1707 81 5 β†’ β†’ SYM cana-1707 81 6 𝑋 𝑋 PROPN cana-1707 81 7 be be VERB cana-1707 81 8 a a DET cana-1707 81 9 mapping mapping NOUN cana-1707 81 10 of of ADP cana-1707 81 11 a a DET cana-1707 81 12 metric metric ADJ cana-1707 81 13 space space NOUN cana-1707 81 14 ( ( PUNCT cana-1707 81 15 𝑋 𝑋 PROPN cana-1707 81 16 , , PUNCT cana-1707 81 17 𝑑 𝑑 NOUN cana-1707 81 18 ) ) PUNCT cana-1707 81 19 . . PUNCT cana-1707 82 1 and and CCONJ cana-1707 82 2 𝛼(= 𝛼(= PROPN cana-1707 82 3 𝛼 𝛼 PROPN cana-1707 82 4 [ [ X cana-1707 82 5 π‘œπ‘Ÿ π‘œπ‘Ÿ X cana-1707 82 6 ( ( PUNCT cana-1707 82 7 π‘₯ π‘₯ PROPN cana-1707 82 8 , , PUNCT cana-1707 82 9 𝑦 𝑦 NOUN cana-1707 82 10 ) ) PUNCT cana-1707 82 11 = = SYM cana-1707 82 12 𝛼𝑑(π‘₯ 𝛼𝑑(π‘₯ NUM cana-1707 82 13 , , PUNCT cana-1707 82 14 𝑦 𝑦 NOUN cana-1707 82 15 ) ) PUNCT cana-1707 82 16 ] ] PUNCT cana-1707 82 17 : : PUNCT cana-1707 83 1 [ [ X cana-1707 83 2 0 0 NUM cana-1707 83 3 , , PUNCT cana-1707 83 4 ∞ ∞ PROPN cana-1707 83 5 ) ) PUNCT cana-1707 83 6 β†’ β†’ PUNCT cana-1707 83 7 [ [ X cana-1707 83 8 0,1 0,1 NUM cana-1707 83 9 ] ] PUNCT cana-1707 83 10 be be VERB cana-1707 83 11 a a DET cana-1707 83 12 monotone monotone NOUN cana-1707 83 13 decreasing decrease VERB cana-1707 83 14 function function NOUN cana-1707 83 15 . . PUNCT cana-1707 84 1 then then ADV cana-1707 84 2 , , PUNCT cana-1707 84 3 𝑇 𝑇 PROPN cana-1707 84 4 is be AUX cana-1707 84 5 a a DET cana-1707 84 6 rakotch rakotch NOUN cana-1707 84 7 contractive contractive ADJ cana-1707 84 8 mapping mapping NOUN cana-1707 84 9 if if SCONJ cana-1707 84 10 , , PUNCT cana-1707 84 11 for for ADP cana-1707 84 12 every every DET cana-1707 84 13 π‘₯ π‘₯ PROPN cana-1707 84 14 β‰  β‰  PROPN cana-1707 84 15 𝑦 𝑦 PRON cana-1707 84 16 ∈ ∈ PROPN cana-1707 84 17 𝑋 𝑋 PROPN cana-1707 84 18 , , PUNCT cana-1707 84 19 we we PRON cana-1707 84 20 have have VERB cana-1707 84 21 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 84 22 , , PUNCT cana-1707 84 23 𝑇𝑦 𝑇𝑦 NOUN cana-1707 84 24 ) ) PUNCT cana-1707 84 25 ≀ ≀ NOUN cana-1707 84 26 𝛼𝑑(π‘₯ 𝛼𝑑(π‘₯ NUM cana-1707 84 27 , , PUNCT cana-1707 84 28 𝑦 𝑦 NOUN cana-1707 84 29 ) ) PUNCT cana-1707 84 30 . . PUNCT cana-1707 85 1 ( ( PUNCT cana-1707 85 2 rakotch rakotch VERB cana-1707 85 3 in in ADP cana-1707 85 4 1962 1962 NUM cana-1707 85 5 , , PUNCT cana-1707 85 6 [ [ X cana-1707 85 7 45 45 NUM cana-1707 85 8 ] ] SYM cana-1707 85 9 ) ) PUNCT cana-1707 85 10 definition definition NOUN cana-1707 85 11 3.3 3.3 NUM cana-1707 85 12 . . PUNCT cana-1707 86 1 a a DET cana-1707 86 2 mapping mapping NOUN cana-1707 86 3 𝑇 𝑇 NOUN cana-1707 86 4 : : PUNCT cana-1707 86 5 𝑋 𝑋 PROPN cana-1707 86 6 β†’ β†’ SYM cana-1707 86 7 𝑋 𝑋 PROPN cana-1707 86 8 in in ADP cana-1707 86 9 metric metric ADJ cana-1707 86 10 space space NOUN cana-1707 86 11 ( ( PUNCT cana-1707 86 12 𝑋 𝑋 PROPN cana-1707 86 13 , , PUNCT cana-1707 86 14 𝑑 𝑑 NOUN cana-1707 86 15 ) ) PUNCT cana-1707 86 16 is be AUX cana-1707 86 17 said say VERB cana-1707 86 18 to to PART cana-1707 86 19 be be AUX cana-1707 86 20 caristi caristi ADJ cana-1707 86 21 contractions contraction NOUN cana-1707 86 22 if if SCONJ cana-1707 86 23 there there PRON cana-1707 86 24 is be VERB cana-1707 86 25 βˆ… βˆ… NOUN cana-1707 86 26 : : PUNCT cana-1707 86 27 𝑋 𝑋 NOUN cana-1707 86 28 β†’ β†’ SYM cana-1707 86 29 [ [ X cana-1707 86 30 0 0 NUM cana-1707 86 31 , , PUNCT cana-1707 86 32 ∞ ∞ NOUN cana-1707 86 33 ) ) PUNCT cana-1707 86 34 lower low ADJ cana-1707 86 35 semi semi ADJ cana-1707 86 36 - - ADJ cana-1707 86 37 continuous continuous ADJ cana-1707 86 38 function function NOUN cana-1707 86 39 such such ADJ cana-1707 86 40 that that SCONJ cana-1707 86 41 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 86 42 , , PUNCT cana-1707 86 43 𝑇π‘₯ 𝑇π‘₯ PROPN cana-1707 86 44 ) ) PUNCT cana-1707 86 45 ≀ ≀ NOUN cana-1707 86 46 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-1707 86 47 ) ) PUNCT cana-1707 86 48 βˆ’ βˆ’ PROPN cana-1707 86 49 𝑇(π‘₯ 𝑇(π‘₯ NOUN cana-1707 86 50 ) ) PUNCT cana-1707 86 51 . . PUNCT cana-1707 87 1 ( ( PUNCT cana-1707 87 2 caristi caristi VERB cana-1707 87 3 in in ADP cana-1707 87 4 1966 1966 NUM cana-1707 87 5 , , PUNCT cana-1707 87 6 [ [ X cana-1707 87 7 8 8 NUM cana-1707 87 8 ] ] SYM cana-1707 87 9 ) ) PUNCT cana-1707 87 10 note note VERB cana-1707 87 11 that that SCONJ cana-1707 87 12 every every DET cana-1707 87 13 banach banach NOUN cana-1707 87 14 contraction contraction NOUN cana-1707 87 15 𝑇 𝑇 PROPN cana-1707 87 16 satisfies satisfy VERB cana-1707 87 17 caristi caristi VERB cana-1707 87 18 contractions contraction NOUN cana-1707 87 19 if if SCONJ cana-1707 87 20 for for ADP cana-1707 87 21 some some DET cana-1707 87 22 Ξ» Ξ» NOUN cana-1707 87 23 ∈ ∈ PROPN cana-1707 88 1 [ [ X cana-1707 88 2 0,1 0,1 NUM cana-1707 88 3 ) ) PUNCT cana-1707 88 4 , , PUNCT cana-1707 88 5 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-1707 88 6 ) ) PUNCT cana-1707 88 7 = = PUNCT cana-1707 88 8 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 88 9 , , PUNCT cana-1707 88 10 𝑇π‘₯ 𝑇π‘₯ NOUN cana-1707 88 11 ) ) PUNCT cana-1707 88 12 1 1 NUM cana-1707 88 13 βˆ’ βˆ’ PROPN cana-1707 88 14 Ξ» Ξ» PROPN cana-1707 88 15 . . PUNCT cana-1707 89 1 definition definition NOUN cana-1707 89 2 3.4 3.4 NUM cana-1707 89 3 . . PUNCT cana-1707 90 1 let let VERB cana-1707 90 2 𝑇 𝑇 PROPN cana-1707 90 3 : : PUNCT cana-1707 90 4 𝑋 𝑋 PROPN cana-1707 90 5 β†’ β†’ SYM cana-1707 90 6 𝑋 𝑋 PROPN cana-1707 90 7 be be VERB cana-1707 90 8 a a DET cana-1707 90 9 contractive contractive ADJ cana-1707 90 10 mapping mapping NOUN cana-1707 90 11 of of ADP cana-1707 90 12 a a DET cana-1707 90 13 metric metric ADJ cana-1707 90 14 space space NOUN cana-1707 90 15 ( ( PUNCT cana-1707 90 16 𝑋 𝑋 PROPN cana-1707 90 17 , , PUNCT cana-1707 90 18 𝑑 𝑑 NOUN cana-1707 90 19 ) ) PUNCT cana-1707 90 20 . . PUNCT cana-1707 91 1 then then ADV cana-1707 91 2 , , PUNCT cana-1707 91 3 𝑇 𝑇 PROPN cana-1707 91 4 is be AUX cana-1707 91 5 said say VERB cana-1707 91 6 to to PART cana-1707 91 7 be be AUX cana-1707 91 8 kanan kanan PROPN cana-1707 91 9 contraction contraction NOUN cana-1707 91 10 if if SCONJ cana-1707 91 11 there there PRON cana-1707 91 12 exists exist VERB cana-1707 91 13 𝛼 𝛼 PRON cana-1707 91 14 ∈ ∈ PROPN cana-1707 91 15 ( ( PUNCT cana-1707 91 16 0 0 NUM cana-1707 91 17 , , PUNCT cana-1707 91 18 1 1 NUM cana-1707 91 19 2 2 NUM cana-1707 91 20 ) ) PUNCT cana-1707 91 21 such such ADJ cana-1707 91 22 that that PRON cana-1707 91 23 , , PUNCT cana-1707 91 24 we we PRON cana-1707 91 25 have have VERB cana-1707 91 26 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 91 27 , , PUNCT cana-1707 91 28 𝑇𝑦 𝑇𝑦 PROPN cana-1707 91 29 ) ) PUNCT cana-1707 91 30 ≀ ≀ NOUN cana-1707 91 31 𝛼[𝑑(π‘₯ 𝛼[𝑑(π‘₯ PROPN cana-1707 91 32 , , PUNCT cana-1707 91 33 𝑇π‘₯ 𝑇π‘₯ PROPN cana-1707 91 34 ) ) PUNCT cana-1707 92 1 + + CCONJ cana-1707 92 2 ( ( PUNCT cana-1707 92 3 𝑦 𝑦 NOUN cana-1707 92 4 , , PUNCT cana-1707 92 5 𝑇𝑦 𝑇𝑦 PROPN cana-1707 92 6 ) ) PUNCT cana-1707 92 7 , , PUNCT cana-1707 92 8 for for ADP cana-1707 92 9 every every DET cana-1707 92 10 π‘₯ π‘₯ PROPN cana-1707 92 11 , , PUNCT cana-1707 92 12 𝑦 𝑦 NOUN cana-1707 92 13 ∈ ∈ NOUN cana-1707 92 14 𝑋 𝑋 PROPN cana-1707 92 15 ( ( PUNCT cana-1707 92 16 kanan kanan PROPN cana-1707 92 17 in in ADP cana-1707 92 18 1968 1968 NUM cana-1707 92 19 , , PUNCT cana-1707 92 20 [ [ X cana-1707 92 21 37 37 NUM cana-1707 92 22 ] ] PUNCT cana-1707 92 23 ) ) PUNCT cana-1707 92 24 the the DET cana-1707 92 25 following follow VERB cana-1707 92 26 example example NOUN cana-1707 92 27 shows show VERB cana-1707 92 28 that that SCONJ cana-1707 92 29 𝑇 𝑇 PROPN cana-1707 92 30 is be AUX cana-1707 92 31 not not PART cana-1707 92 32 continuous continuous ADJ cana-1707 92 33 but but CCONJ cana-1707 92 34 𝑇 𝑇 PROPN cana-1707 92 35 is be AUX cana-1707 92 36 a a DET cana-1707 92 37 kanan kanan PROPN cana-1707 92 38 contraction contraction NOUN cana-1707 92 39 at at ADP cana-1707 92 40 𝛼 𝛼 NOUN cana-1707 92 41 = = SYM cana-1707 92 42 1 1 NUM cana-1707 92 43 5 5 NUM cana-1707 92 44 : : PUNCT cana-1707 92 45 example example NOUN cana-1707 92 46 3.3 3.3 NUM cana-1707 92 47 : : PUNCT cana-1707 92 48 let let VERB cana-1707 92 49 𝑋 𝑋 NOUN cana-1707 92 50 = = SYM cana-1707 92 51 ℝ ℝ PROPN cana-1707 92 52 be be AUX cana-1707 92 53 a a DET cana-1707 92 54 usual usual ADJ cana-1707 92 55 metric metric NOUN cana-1707 92 56 and and CCONJ cana-1707 92 57 mapping mapping NOUN cana-1707 92 58 𝑇 𝑇 PROPN cana-1707 92 59 : : PUNCT cana-1707 92 60 𝑋 𝑋 PROPN cana-1707 92 61 β†’ β†’ SYM cana-1707 92 62 𝑋 𝑋 PROPN cana-1707 92 63 be be AUX cana-1707 92 64 defined define VERB cana-1707 92 65 by by ADP cana-1707 92 66 𝑇(π‘₯ 𝑇(π‘₯ NOUN cana-1707 92 67 ) ) PUNCT cana-1707 92 68 = = NOUN cana-1707 92 69 { { PUNCT cana-1707 92 70 0 0 NUM cana-1707 92 71 , , PUNCT cana-1707 92 72 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-1707 92 73 π‘₯ π‘₯ PRON cana-1707 92 74 ∈ ∈ PROPN cana-1707 92 75 ( ( PUNCT cana-1707 92 76 βˆ’βˆž βˆ’βˆž NOUN cana-1707 92 77 , , PUNCT cana-1707 92 78 2 2 NUM cana-1707 92 79 ] ] SYM cana-1707 92 80 1 1 NUM cana-1707 92 81 2 2 NUM cana-1707 92 82 , , PUNCT cana-1707 92 83 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-1707 92 84 π‘₯ π‘₯ DET cana-1707 92 85 ∈ ∈ PROPN cana-1707 92 86 ( ( PUNCT cana-1707 92 87 2 2 NUM cana-1707 92 88 , , PUNCT cana-1707 92 89 + + NOUN cana-1707 92 90 ∞ ∞ NOUN cana-1707 92 91 ) ) PUNCT cana-1707 92 92 . . PUNCT cana-1707 93 1 bianchini bianchini PROPN cana-1707 93 2 extends extend VERB cana-1707 93 3 the the DET cana-1707 93 4 kanan kanan PROPN cana-1707 93 5 contractions contraction NOUN cana-1707 93 6 as as ADP cana-1707 93 7 : : PUNCT cana-1707 93 8 definition definition NOUN cana-1707 93 9 3.5 3.5 NUM cana-1707 93 10 . . PUNCT cana-1707 94 1 a a DET cana-1707 94 2 mapping mapping NOUN cana-1707 94 3 𝑇 𝑇 NOUN cana-1707 94 4 : : PUNCT cana-1707 94 5 𝑋 𝑋 PROPN cana-1707 94 6 β†’ β†’ SYM cana-1707 94 7 𝑋 𝑋 PROPN cana-1707 94 8 in in ADP cana-1707 94 9 metric metric ADJ cana-1707 94 10 space space NOUN cana-1707 94 11 ( ( PUNCT cana-1707 94 12 𝑋 𝑋 PROPN cana-1707 94 13 , , PUNCT cana-1707 94 14 𝑑 𝑑 NOUN cana-1707 94 15 ) ) PUNCT cana-1707 94 16 is be AUX cana-1707 94 17 said say VERB cana-1707 94 18 to to PART cana-1707 94 19 be be AUX cana-1707 94 20 bianchini bianchini NOUN cana-1707 94 21 contractions contraction NOUN cana-1707 94 22 if if SCONJ cana-1707 94 23 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 94 24 , , PUNCT cana-1707 94 25 𝑇𝑦 𝑇𝑦 PROPN cana-1707 94 26 ) ) PUNCT cana-1707 94 27 ≀ ≀ NOUN cana-1707 94 28 π‘Ÿ π‘Ÿ X cana-1707 94 29 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ X cana-1707 94 30 { { PUNCT cana-1707 94 31 ( ( PUNCT cana-1707 94 32 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 94 33 , , PUNCT cana-1707 94 34 𝑇π‘₯ 𝑇π‘₯ PROPN cana-1707 94 35 ) ) PUNCT cana-1707 94 36 , , PUNCT cana-1707 94 37 𝑑(𝑦 𝑑(𝑦 PROPN cana-1707 94 38 , , PUNCT cana-1707 94 39 𝑇𝑦 𝑇𝑦 PROPN cana-1707 94 40 ) ) PUNCT cana-1707 94 41 } } PUNCT cana-1707 94 42 for for ADP cana-1707 94 43 all all PRON cana-1707 94 44 π‘₯ π‘₯ PROPN cana-1707 94 45 , , PUNCT cana-1707 94 46 𝑦 𝑦 NOUN cana-1707 94 47 ∈ ∈ PROPN cana-1707 94 48 𝑋 𝑋 PROPN cana-1707 94 49 , , PUNCT cana-1707 94 50 and and CCONJ cana-1707 94 51 π‘Ÿ π‘Ÿ PRON cana-1707 94 52 ∈ ∈ NOUN cana-1707 94 53 [ [ X cana-1707 94 54 0,1 0,1 NUM cana-1707 94 55 ) ) PUNCT cana-1707 94 56 . . PUNCT cana-1707 95 1 ( ( PUNCT cana-1707 95 2 bianchini bianchini NOUN cana-1707 95 3 in in ADP cana-1707 95 4 1972 1972 NUM cana-1707 95 5 , , PUNCT cana-1707 95 6 [ [ X cana-1707 95 7 6 6 NUM cana-1707 95 8 ] ] SYM cana-1707 95 9 ) ) PUNCT cana-1707 95 10 communications communication NOUN cana-1707 95 11 on on ADP cana-1707 95 12 applied apply VERB cana-1707 95 13 nonlinear nonlinear ADJ cana-1707 95 14 analysis analysis NOUN cana-1707 95 15 issn issn NOUN cana-1707 95 16 : : PUNCT cana-1707 95 17 1074 1074 NUM cana-1707 95 18 - - PUNCT cana-1707 95 19 133x 133x NUM cana-1707 95 20 vol vol NOUN cana-1707 95 21 32 32 NUM cana-1707 95 22 no no NOUN cana-1707 95 23 . . NOUN cana-1707 95 24 2 2 NUM cana-1707 95 25 ( ( PUNCT cana-1707 95 26 2025 2025 NUM cana-1707 95 27 ) ) PUNCT cana-1707 95 28 57 57 NUM cana-1707 95 29 https://internationalpubls.com https://internationalpubls.com X cana-1707 95 30 definition definition NOUN cana-1707 95 31 3.6 3.6 NUM cana-1707 95 32 . . PUNCT cana-1707 96 1 a a DET cana-1707 96 2 mapping mapping NOUN cana-1707 96 3 𝑇 𝑇 NOUN cana-1707 96 4 : : PUNCT cana-1707 96 5 𝑋 𝑋 PROPN cana-1707 96 6 β†’ β†’ SYM cana-1707 96 7 𝑋 𝑋 PROPN cana-1707 96 8 in in ADP cana-1707 96 9 metric metric ADJ cana-1707 96 10 space space NOUN cana-1707 96 11 ( ( PUNCT cana-1707 96 12 𝑋 𝑋 PROPN cana-1707 96 13 , , PUNCT cana-1707 96 14 𝑑 𝑑 NOUN cana-1707 96 15 ) ) PUNCT cana-1707 96 16 is be AUX cana-1707 96 17 said say VERB cana-1707 96 18 to to PART cana-1707 96 19 be be AUX cana-1707 96 20 reich reich PROPN cana-1707 96 21 contractions contraction NOUN cana-1707 96 22 if if SCONJ cana-1707 96 23 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 96 24 , , PUNCT cana-1707 96 25 𝑇𝑦 𝑇𝑦 NOUN cana-1707 96 26 ) ) PUNCT cana-1707 96 27 ≀ ≀ NOUN cana-1707 96 28 π‘Ž π‘Ž PRON cana-1707 96 29 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 96 30 , , PUNCT cana-1707 96 31 𝑇π‘₯ 𝑇π‘₯ NOUN cana-1707 96 32 ) ) PUNCT cana-1707 96 33 + + NUM cana-1707 96 34 𝑏 𝑏 PROPN cana-1707 96 35 𝑑(𝑦 𝑑(𝑦 NOUN cana-1707 96 36 , , PUNCT cana-1707 96 37 𝑇𝑦 𝑇𝑦 PROPN cana-1707 96 38 ) ) PUNCT cana-1707 96 39 + + CCONJ cana-1707 96 40 𝑐 𝑐 PROPN cana-1707 96 41 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 96 42 , , PUNCT cana-1707 96 43 𝑦 𝑦 NOUN cana-1707 96 44 ) ) PUNCT cana-1707 96 45 , , PUNCT cana-1707 96 46 where where SCONJ cana-1707 96 47 π‘Ž π‘Ž X cana-1707 96 48 , , PUNCT cana-1707 96 49 𝑏 𝑏 NOUN cana-1707 96 50 , , PUNCT cana-1707 96 51 𝑐 𝑐 PROPN cana-1707 96 52 are be AUX cana-1707 96 53 nonnegative nonnegative ADJ cana-1707 96 54 and and CCONJ cana-1707 96 55 satisfy satisfy VERB cana-1707 96 56 a a DET cana-1707 96 57 + + NOUN cana-1707 96 58 b b NOUN cana-1707 96 59 + + CCONJ cana-1707 96 60 c c NOUN cana-1707 96 61 < < X cana-1707 96 62 1 1 NUM cana-1707 96 63 and and CCONJ cana-1707 96 64 for for ADP cana-1707 96 65 all all DET cana-1707 96 66 π‘₯ π‘₯ PROPN cana-1707 96 67 , , PUNCT cana-1707 96 68 𝑦 𝑦 PROPN cana-1707 96 69 ∈ ∈ PROPN cana-1707 96 70 𝑋. 𝑋. PROPN cana-1707 96 71 ( ( PUNCT cana-1707 96 72 reich reich PROPN cana-1707 96 73 in in ADP cana-1707 96 74 1971 1971 NUM cana-1707 96 75 , , PUNCT cana-1707 96 76 [ [ X cana-1707 96 77 46 46 NUM cana-1707 96 78 ] ] PUNCT cana-1707 96 79 ) ) PUNCT cana-1707 96 80 note note VERB cana-1707 96 81 that that SCONJ cana-1707 96 82 π‘Ž π‘Ž X cana-1707 96 83 = = SYM cana-1707 96 84 𝑏 𝑏 NOUN cana-1707 96 85 = = SYM cana-1707 96 86 0 0 NUM cana-1707 96 87 yields yield NOUN cana-1707 96 88 banach banach NOUN cana-1707 96 89 's 's PART cana-1707 96 90 fixed fix VERB cana-1707 96 91 point point NOUN cana-1707 96 92 theorem theorem ADJ cana-1707 96 93 , , PUNCT cana-1707 96 94 while while SCONJ cana-1707 96 95 π‘Ž π‘Ž PROPN cana-1707 96 96 = = SYM cana-1707 96 97 𝑏 𝑏 NOUN cana-1707 96 98 , , PUNCT cana-1707 96 99 𝑐 𝑐 NOUN cana-1707 96 100 = = SYM cana-1707 96 101 0 0 NUM cana-1707 96 102 yields yield VERB cana-1707 96 103 kannan kannan PROPN cana-1707 96 104 's 's PART cana-1707 96 105 theorem theorem PROPN cana-1707 97 1 . . PROPN cana-1707 97 2 definition definition NOUN cana-1707 97 3 3.7 3.7 NUM cana-1707 97 4 . . PUNCT cana-1707 98 1 let let VERB cana-1707 98 2 𝑇 𝑇 PROPN cana-1707 98 3 : : PUNCT cana-1707 98 4 𝑋 𝑋 PROPN cana-1707 98 5 β†’ β†’ SYM cana-1707 98 6 𝑋 𝑋 PROPN cana-1707 98 7 be be VERB cana-1707 98 8 a a DET cana-1707 98 9 contractive contractive ADJ cana-1707 98 10 mapping mapping NOUN cana-1707 98 11 of of ADP cana-1707 98 12 a a DET cana-1707 98 13 metric metric ADJ cana-1707 98 14 space space NOUN cana-1707 98 15 ( ( PUNCT cana-1707 98 16 𝑋 𝑋 PROPN cana-1707 98 17 , , PUNCT cana-1707 98 18 𝑑 𝑑 NOUN cana-1707 98 19 ) ) PUNCT cana-1707 98 20 . . PUNCT cana-1707 99 1 then then ADV cana-1707 99 2 , , PUNCT cana-1707 99 3 𝑇 𝑇 PROPN cana-1707 99 4 is be AUX cana-1707 99 5 said say VERB cana-1707 99 6 to to PART cana-1707 99 7 be be AUX cana-1707 99 8 chatterjee chatterjee NOUN cana-1707 99 9 contraction contraction NOUN cana-1707 99 10 if if SCONJ cana-1707 99 11 there there PRON cana-1707 99 12 exists exist VERB cana-1707 99 13 𝛼 𝛼 PRON cana-1707 99 14 ∈ ∈ PROPN cana-1707 99 15 ( ( PUNCT cana-1707 99 16 0 0 NUM cana-1707 99 17 , , PUNCT cana-1707 99 18 1 1 NUM cana-1707 99 19 2 2 NUM cana-1707 99 20 ) ) PUNCT cana-1707 99 21 such such ADJ cana-1707 99 22 that that PRON cana-1707 99 23 , , PUNCT cana-1707 99 24 we we PRON cana-1707 99 25 have have VERB cana-1707 99 26 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 99 27 , , PUNCT cana-1707 99 28 𝑇𝑦 𝑇𝑦 PROPN cana-1707 99 29 ) ) PUNCT cana-1707 99 30 ≀ ≀ NUM cana-1707 99 31 𝛼[𝑑(π‘₯ 𝛼[𝑑(π‘₯ PROPN cana-1707 99 32 , , PUNCT cana-1707 99 33 𝑇𝑦 𝑇𝑦 PROPN cana-1707 99 34 ) ) PUNCT cana-1707 100 1 + + CCONJ cana-1707 100 2 ( ( PUNCT cana-1707 100 3 𝑦 𝑦 NOUN cana-1707 100 4 , , PUNCT cana-1707 100 5 𝑇π‘₯ 𝑇π‘₯ NOUN cana-1707 100 6 ) ) PUNCT cana-1707 100 7 , , PUNCT cana-1707 100 8 for for ADP cana-1707 100 9 every every DET cana-1707 100 10 π‘₯ π‘₯ PROPN cana-1707 100 11 , , PUNCT cana-1707 100 12 𝑦 𝑦 NOUN cana-1707 100 13 ∈ ∈ PROPN cana-1707 100 14 𝑋 𝑋 PROPN cana-1707 100 15 ( ( PUNCT cana-1707 100 16 chatterjee chatterjee PROPN cana-1707 100 17 in in ADP cana-1707 100 18 1972 1972 NUM cana-1707 100 19 , , PUNCT cana-1707 100 20 [ [ X cana-1707 100 21 12 12 NUM cana-1707 100 22 ] ] PUNCT cana-1707 100 23 ) ) PUNCT cana-1707 100 24 definition definition NOUN cana-1707 100 25 3.8 3.8 NUM cana-1707 100 26 . . PUNCT cana-1707 101 1 let let VERB cana-1707 101 2 𝑇 𝑇 PROPN cana-1707 101 3 : : PUNCT cana-1707 101 4 𝑋 𝑋 PROPN cana-1707 101 5 β†’ β†’ SYM cana-1707 101 6 𝑋 𝑋 PROPN cana-1707 101 7 be be VERB cana-1707 101 8 a a DET cana-1707 101 9 contractive contractive ADJ cana-1707 101 10 mapping mapping NOUN cana-1707 101 11 of of ADP cana-1707 101 12 a a DET cana-1707 101 13 metric metric ADJ cana-1707 101 14 space space NOUN cana-1707 101 15 ( ( PUNCT cana-1707 101 16 𝑋 𝑋 PROPN cana-1707 101 17 , , PUNCT cana-1707 101 18 𝑑 𝑑 NOUN cana-1707 101 19 ) ) PUNCT cana-1707 101 20 . . PUNCT cana-1707 102 1 then then ADV cana-1707 102 2 , , PUNCT cana-1707 102 3 𝑇 𝑇 PROPN cana-1707 102 4 is be AUX cana-1707 102 5 said say VERB cana-1707 102 6 to to PART cana-1707 102 7 be be AUX cana-1707 102 8 t. t. PROPN cana-1707 102 9 zamfirescu zamfirescu PROPN cana-1707 102 10 contraction contraction PROPN cana-1707 102 11 if if SCONJ cana-1707 102 12 there there PRON cana-1707 102 13 exist exist VERB cana-1707 102 14 real real ADJ cana-1707 102 15 numbers number NOUN cana-1707 102 16 π‘Ž π‘Ž ADP cana-1707 102 17 , , PUNCT cana-1707 102 18 𝑏 𝑏 NOUN cana-1707 102 19 , , PUNCT cana-1707 102 20 𝑐 𝑐 DET cana-1707 102 21 satisfying satisfy VERB cana-1707 102 22 0 0 NUM cana-1707 102 23 ≀ ≀ NOUN cana-1707 103 1 π‘Ž π‘Ž PRON cana-1707 103 2 < < X cana-1707 103 3 1 1 NUM cana-1707 103 4 , , PUNCT cana-1707 103 5 0 0 NUM cana-1707 103 6 ≀ ≀ NUM cana-1707 103 7 𝑏 𝑏 NOUN cana-1707 103 8 , , PUNCT cana-1707 103 9 𝑐 𝑐 X cana-1707 103 10 < < X cana-1707 103 11 1 1 NUM cana-1707 103 12 2 2 NUM cana-1707 103 13 , , PUNCT cana-1707 104 1 such such ADJ cana-1707 104 2 that that PRON cana-1707 104 3 for for ADP cana-1707 104 4 every every DET cana-1707 104 5 π‘₯ π‘₯ PROPN cana-1707 104 6 , , PUNCT cana-1707 104 7 𝑦 𝑦 NOUN cana-1707 104 8 ∈ ∈ PROPN cana-1707 104 9 𝑋 𝑋 PROPN cana-1707 104 10 , , PUNCT cana-1707 104 11 at at ADP cana-1707 104 12 least least ADJ cana-1707 104 13 one one NUM cana-1707 104 14 of of ADP cana-1707 104 15 the the DET cana-1707 104 16 following follow VERB cana-1707 104 17 is be AUX cana-1707 104 18 true true ADJ cana-1707 104 19 : : PUNCT cana-1707 104 20 ( ( PUNCT cana-1707 104 21 zamfirescu zamfirescu X cana-1707 104 22 in in ADP cana-1707 104 23 1972 1972 NUM cana-1707 104 24 , , PUNCT cana-1707 104 25 [ [ X cana-1707 104 26 57 57 NUM cana-1707 104 27 ] ] PUNCT cana-1707 104 28 ) ) PUNCT cana-1707 104 29 ( ( PUNCT cana-1707 104 30 i i NOUN cana-1707 104 31 ) ) PUNCT cana-1707 104 32 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 104 33 , , PUNCT cana-1707 104 34 𝑇𝑦 𝑇𝑦 PROPN cana-1707 104 35 ) ) PUNCT cana-1707 104 36 ≀ ≀ NOUN cana-1707 104 37 π‘Ž[𝑑(π‘₯ π‘Ž[𝑑(π‘₯ NOUN cana-1707 104 38 , , PUNCT cana-1707 104 39 𝑦 𝑦 NOUN cana-1707 104 40 ) ) PUNCT cana-1707 104 41 ; ; PUNCT cana-1707 104 42 ( ( PUNCT cana-1707 104 43 ii ii NOUN cana-1707 104 44 ) ) PUNCT cana-1707 104 45 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 104 46 , , PUNCT cana-1707 104 47 𝑇𝑦 𝑇𝑦 PROPN cana-1707 104 48 ) ) PUNCT cana-1707 104 49 ≀ ≀ NUM cana-1707 104 50 𝑏[𝑑(π‘₯ 𝑏[𝑑(π‘₯ PROPN cana-1707 104 51 , , PUNCT cana-1707 104 52 𝑇π‘₯ 𝑇π‘₯ PROPN cana-1707 104 53 ) ) PUNCT cana-1707 104 54 + + CCONJ cana-1707 104 55 ( ( PUNCT cana-1707 104 56 𝑦 𝑦 NOUN cana-1707 104 57 , , PUNCT cana-1707 104 58 𝑇𝑦 𝑇𝑦 PROPN cana-1707 104 59 ) ) PUNCT cana-1707 104 60 ] ] PUNCT cana-1707 104 61 ; ; PUNCT cana-1707 104 62 ( ( PUNCT cana-1707 104 63 iii iii X cana-1707 104 64 ) ) PUNCT cana-1707 104 65 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 104 66 , , PUNCT cana-1707 104 67 𝑇𝑦 𝑇𝑦 PROPN cana-1707 104 68 ) ) PUNCT cana-1707 104 69 ≀ ≀ NOUN cana-1707 104 70 𝑐[𝑑(π‘₯ 𝑐[𝑑(π‘₯ NUM cana-1707 104 71 , , PUNCT cana-1707 104 72 𝑇𝑦 𝑇𝑦 PROPN cana-1707 104 73 ) ) PUNCT cana-1707 105 1 + + CCONJ cana-1707 105 2 ( ( PUNCT cana-1707 105 3 𝑦 𝑦 NOUN cana-1707 105 4 , , PUNCT cana-1707 105 5 𝑇π‘₯ 𝑇π‘₯ NOUN cana-1707 105 6 ) ) PUNCT cana-1707 105 7 . . PUNCT cana-1707 106 1 definition definition NOUN cana-1707 106 2 3.9 3.9 NUM cana-1707 106 3 . . PUNCT cana-1707 107 1 a a DET cana-1707 107 2 mapping mapping NOUN cana-1707 107 3 𝑇 𝑇 NOUN cana-1707 107 4 : : PUNCT cana-1707 107 5 𝑋 𝑋 PROPN cana-1707 107 6 β†’ β†’ SYM cana-1707 107 7 𝑋 𝑋 PROPN cana-1707 107 8 in in ADP cana-1707 107 9 metric metric ADJ cana-1707 107 10 space space NOUN cana-1707 107 11 ( ( PUNCT cana-1707 107 12 𝑋 𝑋 PROPN cana-1707 107 13 , , PUNCT cana-1707 107 14 𝑑 𝑑 NOUN cana-1707 107 15 ) ) PUNCT cana-1707 107 16 is be AUX cana-1707 107 17 said say VERB cana-1707 107 18 to to PART cana-1707 107 19 be be AUX cana-1707 107 20 sehgal sehgal ADJ cana-1707 107 21 contractions contraction NOUN cana-1707 107 22 if if SCONJ cana-1707 107 23 , , PUNCT cana-1707 107 24 for for ADP cana-1707 107 25 every every DET cana-1707 107 26 π‘₯ π‘₯ PROPN cana-1707 107 27 , , PUNCT cana-1707 107 28 𝑦 𝑦 NOUN cana-1707 107 29 ∈ ∈ PROPN cana-1707 107 30 𝑋 𝑋 PROPN cana-1707 107 31 , , PUNCT cana-1707 107 32 π‘₯ π‘₯ PROPN cana-1707 107 33 β‰  β‰  PROPN cana-1707 107 34 𝑦 𝑦 NOUN cana-1707 107 35 , , PUNCT cana-1707 107 36 we we PRON cana-1707 107 37 have have VERB cana-1707 107 38 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 107 39 , , PUNCT cana-1707 107 40 𝑇𝑦 𝑇𝑦 PROPN cana-1707 107 41 ) ) PUNCT cana-1707 107 42 < < X cana-1707 107 43 max max PROPN cana-1707 107 44 { { PUNCT cana-1707 107 45 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 107 46 , , PUNCT cana-1707 107 47 𝑇π‘₯ 𝑇π‘₯ PROPN cana-1707 107 48 ) ) PUNCT cana-1707 107 49 , , PUNCT cana-1707 107 50 𝑑(𝑦 𝑑(𝑦 PROPN cana-1707 107 51 , , PUNCT cana-1707 107 52 𝑇𝑦 𝑇𝑦 PROPN cana-1707 107 53 ) ) PUNCT cana-1707 107 54 , , PUNCT cana-1707 107 55 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 107 56 , , PUNCT cana-1707 107 57 𝑦 𝑦 NOUN cana-1707 107 58 ) ) PUNCT cana-1707 107 59 } } PUNCT cana-1707 107 60 ( ( PUNCT cana-1707 107 61 sehgal sehgal PROPN cana-1707 107 62 in in ADP cana-1707 107 63 1972 1972 NUM cana-1707 107 64 , , PUNCT cana-1707 107 65 [ [ X cana-1707 107 66 52 52 NUM cana-1707 107 67 ] ] SYM cana-1707 107 68 ) ) PUNCT cana-1707 107 69 definition definition NOUN cana-1707 107 70 3.10 3.10 NUM cana-1707 107 71 . . PUNCT cana-1707 108 1 a a DET cana-1707 108 2 mapping mapping NOUN cana-1707 108 3 𝑇 𝑇 NOUN cana-1707 108 4 : : PUNCT cana-1707 108 5 𝑋 𝑋 PROPN cana-1707 108 6 β†’ β†’ SYM cana-1707 108 7 𝑋 𝑋 PROPN cana-1707 108 8 in in ADP cana-1707 108 9 metric metric ADJ cana-1707 108 10 space space NOUN cana-1707 108 11 ( ( PUNCT cana-1707 108 12 𝑋 𝑋 PROPN cana-1707 108 13 , , PUNCT cana-1707 108 14 𝑑 𝑑 NOUN cana-1707 108 15 ) ) PUNCT cana-1707 108 16 is be AUX cana-1707 108 17 said say VERB cana-1707 108 18 to to PART cana-1707 108 19 be be AUX cana-1707 108 20 rhodes rhodes PROPN cana-1707 108 21 contractions contraction NOUN cana-1707 108 22 if if SCONJ cana-1707 108 23 , , PUNCT cana-1707 108 24 for for ADP cana-1707 108 25 each each DET cana-1707 108 26 π‘₯ π‘₯ PROPN cana-1707 108 27 , , PUNCT cana-1707 108 28 𝑦 𝑦 NOUN cana-1707 108 29 ∈ ∈ PROPN cana-1707 108 30 𝑋 𝑋 PROPN cana-1707 108 31 , , PUNCT cana-1707 108 32 π‘₯ π‘₯ PROPN cana-1707 108 33 β‰  β‰  PROPN cana-1707 108 34 𝑦 𝑦 PROPN cana-1707 108 35 , , PUNCT cana-1707 108 36 ( ( PUNCT cana-1707 108 37 i i NOUN cana-1707 108 38 ) ) PUNCT cana-1707 108 39 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 108 40 , , PUNCT cana-1707 108 41 𝑇𝑦 𝑇𝑦 PROPN cana-1707 108 42 ) ) PUNCT cana-1707 108 43 < < X cana-1707 108 44 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ X cana-1707 108 45 { { PUNCT cana-1707 108 46 ( ( PUNCT cana-1707 108 47 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 108 48 , , PUNCT cana-1707 108 49 𝑇𝑦 𝑇𝑦 PROPN cana-1707 108 50 ) ) PUNCT cana-1707 108 51 , , PUNCT cana-1707 108 52 𝑑(𝑦 𝑑(𝑦 PROPN cana-1707 108 53 , , PUNCT cana-1707 108 54 𝑇π‘₯ 𝑇π‘₯ NOUN cana-1707 108 55 ) ) PUNCT cana-1707 108 56 } } PUNCT cana-1707 108 57 . . PUNCT cana-1707 109 1 ( ( PUNCT cana-1707 109 2 rhodes rhode NOUN cana-1707 109 3 in in ADP cana-1707 109 4 1977 1977 NUM cana-1707 109 5 , , PUNCT cana-1707 109 6 [ [ X cana-1707 109 7 48 48 NUM cana-1707 109 8 ] ] SYM cana-1707 109 9 ) ) PUNCT cana-1707 109 10 ( ( PUNCT cana-1707 109 11 ii ii NOUN cana-1707 109 12 ) ) PUNCT cana-1707 109 13 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 109 14 , , PUNCT cana-1707 109 15 𝑇𝑦 𝑇𝑦 PROPN cana-1707 109 16 ) ) PUNCT cana-1707 109 17 ≀ ≀ NUM cana-1707 109 18 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ ADJ cana-1707 109 19 { { PUNCT cana-1707 109 20 ( ( PUNCT cana-1707 109 21 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 109 22 , , PUNCT cana-1707 109 23 𝑇𝑦 𝑇𝑦 PROPN cana-1707 109 24 ) ) PUNCT cana-1707 109 25 , , PUNCT cana-1707 109 26 𝑑(𝑦 𝑑(𝑦 PROPN cana-1707 109 27 , , PUNCT cana-1707 109 28 𝑇π‘₯ 𝑇π‘₯ PROPN cana-1707 109 29 ) ) PUNCT cana-1707 109 30 , , PUNCT cana-1707 109 31 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 109 32 , , PUNCT cana-1707 109 33 𝑦 𝑦 NOUN cana-1707 109 34 ) ) PUNCT cana-1707 109 35 } } PUNCT cana-1707 109 36 . . PUNCT cana-1707 110 1 ( ( PUNCT cana-1707 110 2 rhodes rhode NOUN cana-1707 110 3 in in ADP cana-1707 110 4 1977 1977 NUM cana-1707 110 5 , , PUNCT cana-1707 110 6 [ [ X cana-1707 110 7 48 48 NUM cana-1707 110 8 ] ] SYM cana-1707 110 9 ) ) PUNCT cana-1707 110 10 ( ( PUNCT cana-1707 110 11 iii iii NOUN cana-1707 110 12 ) ) PUNCT cana-1707 110 13 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 110 14 , , PUNCT cana-1707 110 15 𝑇𝑦 𝑇𝑦 PROPN cana-1707 110 16 ) ) PUNCT cana-1707 110 17 ≀ ≀ NUM cana-1707 110 18 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ ADJ cana-1707 110 19 { { PUNCT cana-1707 110 20 ( ( PUNCT cana-1707 110 21 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 110 22 , , PUNCT cana-1707 110 23 𝑦 𝑦 NOUN cana-1707 110 24 ) ) PUNCT cana-1707 110 25 , , PUNCT cana-1707 110 26 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 110 27 , , PUNCT cana-1707 110 28 𝑇π‘₯ 𝑇π‘₯ PROPN cana-1707 110 29 ) ) PUNCT cana-1707 110 30 , , PUNCT cana-1707 110 31 𝑑(𝑦 𝑑(𝑦 PROPN cana-1707 110 32 , , PUNCT cana-1707 110 33 𝑇𝑦 𝑇𝑦 PROPN cana-1707 110 34 ) ) PUNCT cana-1707 110 35 , , PUNCT cana-1707 110 36 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 110 37 , , PUNCT cana-1707 110 38 𝑇𝑦 𝑇𝑦 PROPN cana-1707 110 39 ) ) PUNCT cana-1707 110 40 , , PUNCT cana-1707 110 41 𝑑(𝑦 𝑑(𝑦 PROPN cana-1707 110 42 , , PUNCT cana-1707 110 43 𝑇π‘₯ 𝑇π‘₯ NOUN cana-1707 110 44 ) ) PUNCT cana-1707 110 45 } } PUNCT cana-1707 110 46 ( ( PUNCT cana-1707 110 47 rhodes rhode NOUN cana-1707 110 48 in in ADP cana-1707 110 49 1977 1977 NUM cana-1707 110 50 , , PUNCT cana-1707 110 51 [ [ X cana-1707 110 52 48 48 NUM cana-1707 110 53 ] ] SYM cana-1707 110 54 ) ) PUNCT cana-1707 110 55 definition definition NOUN cana-1707 110 56 3.11 3.11 NUM cana-1707 110 57 . . PUNCT cana-1707 111 1 a a DET cana-1707 111 2 mapping mapping NOUN cana-1707 111 3 𝑇 𝑇 NOUN cana-1707 111 4 : : PUNCT cana-1707 111 5 𝑋 𝑋 PROPN cana-1707 111 6 β†’ β†’ SYM cana-1707 111 7 𝑋 𝑋 PROPN cana-1707 111 8 in in ADP cana-1707 111 9 metric metric ADJ cana-1707 111 10 space space NOUN cana-1707 111 11 ( ( PUNCT cana-1707 111 12 𝑋 𝑋 PROPN cana-1707 111 13 , , PUNCT cana-1707 111 14 𝑑 𝑑 NOUN cana-1707 111 15 ) ) PUNCT cana-1707 111 16 is be AUX cana-1707 111 17 said say VERB cana-1707 111 18 to to PART cana-1707 111 19 be be AUX cana-1707 111 20 ciric ciric ADJ cana-1707 111 21 contractions contraction NOUN cana-1707 111 22 if if SCONJ cana-1707 111 23 there there PRON cana-1707 111 24 exists exist VERB cana-1707 111 25 a a DET cana-1707 111 26 non non ADJ cana-1707 111 27 - - ADJ cana-1707 111 28 negative negative ADJ cana-1707 111 29 number number NOUN cana-1707 111 30 π‘ž(π‘₯ π‘ž(π‘₯ PROPN cana-1707 111 31 , , PUNCT cana-1707 111 32 𝑦 𝑦 NOUN cana-1707 111 33 ) ) PUNCT cana-1707 111 34 , , PUNCT cana-1707 111 35 π‘Ÿ(π‘₯ π‘Ÿ(π‘₯ PROPN cana-1707 111 36 , , PUNCT cana-1707 111 37 𝑦 𝑦 NOUN cana-1707 111 38 ) ) PUNCT cana-1707 111 39 , , PUNCT cana-1707 111 40 𝑠(π‘₯ 𝑠(π‘₯ PROPN cana-1707 111 41 , , PUNCT cana-1707 111 42 𝑦 𝑦 NOUN cana-1707 111 43 ) ) PUNCT cana-1707 111 44 , , PUNCT cana-1707 111 45 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-1707 111 46 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 111 47 , , PUNCT cana-1707 111 48 𝑦 𝑦 NOUN cana-1707 111 49 ) ) PUNCT cana-1707 111 50 such such ADJ cana-1707 111 51 that that DET cana-1707 111 52 sup sup NOUN cana-1707 111 53 π‘₯,π‘¦βˆˆπ‘‹ π‘₯,π‘¦βˆˆπ‘‹ ADV cana-1707 111 54 { { PUNCT cana-1707 111 55 π‘ž(π‘₯ π‘ž(π‘₯ PROPN cana-1707 111 56 , , PUNCT cana-1707 111 57 𝑦 𝑦 NOUN cana-1707 111 58 ) ) PUNCT cana-1707 112 1 + + CCONJ cana-1707 112 2 π‘Ÿ(π‘₯ π‘Ÿ(π‘₯ PROPN cana-1707 112 3 , , PUNCT cana-1707 112 4 𝑦 𝑦 NOUN cana-1707 112 5 ) ) PUNCT cana-1707 112 6 , , PUNCT cana-1707 113 1 + + ADJ cana-1707 113 2 𝑠(π‘₯ 𝑠(π‘₯ PROPN cana-1707 113 3 , , PUNCT cana-1707 113 4 𝑦 𝑦 NOUN cana-1707 113 5 ) ) PUNCT cana-1707 113 6 + + CCONJ cana-1707 113 7 2 2 NUM cana-1707 113 8 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 113 9 , , PUNCT cana-1707 113 10 𝑦 𝑦 NOUN cana-1707 113 11 ) ) PUNCT cana-1707 113 12 } } PUNCT cana-1707 113 13 < < X cana-1707 113 14 1 1 NUM cana-1707 113 15 and and CCONJ cana-1707 113 16 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADV cana-1707 113 17 , , PUNCT cana-1707 113 18 𝑇𝑦 𝑇𝑦 PROPN cana-1707 113 19 ) ) PUNCT cana-1707 113 20 ≀ ≀ NUM cana-1707 113 21 q(x q(x PROPN cana-1707 113 22 , , PUNCT cana-1707 113 23 y y NOUN cana-1707 113 24 ) ) PUNCT cana-1707 113 25 d(x d(x PROPN cana-1707 113 26 , , PUNCT cana-1707 113 27 y y NOUN cana-1707 113 28 ) ) PUNCT cana-1707 113 29 + + CCONJ cana-1707 113 30 r(x r(x PROPN cana-1707 113 31 , , PUNCT cana-1707 113 32 y y NOUN cana-1707 113 33 ) ) PUNCT cana-1707 113 34 d(x d(x PROPN cana-1707 113 35 , , PUNCT cana-1707 113 36 tx tx PROPN cana-1707 113 37 ) ) PUNCT cana-1707 113 38 + + CCONJ cana-1707 113 39 s(x s(x PROPN cana-1707 113 40 , , PUNCT cana-1707 113 41 y y NOUN cana-1707 113 42 ) ) PUNCT cana-1707 113 43 d(y d(y PROPN cana-1707 113 44 , , PUNCT cana-1707 113 45 ty ty INTJ cana-1707 113 46 ) ) PUNCT cana-1707 113 47 + + CCONJ cana-1707 113 48 t(x t(x PROPN cana-1707 113 49 , , PUNCT cana-1707 113 50 y y NOUN cana-1707 113 51 ) ) PUNCT cana-1707 113 52 d(x d(x PROPN cana-1707 113 53 , , PUNCT cana-1707 113 54 ty ty INTJ cana-1707 113 55 ) ) PUNCT cana-1707 113 56 + + CCONJ cana-1707 113 57 𝑑(𝑦 𝑑(𝑦 NOUN cana-1707 113 58 , , PUNCT cana-1707 113 59 𝑇π‘₯ 𝑇π‘₯ NOUN cana-1707 113 60 ) ) PUNCT cana-1707 113 61 for for ADP cana-1707 113 62 every every DET cana-1707 113 63 π‘₯ π‘₯ PROPN cana-1707 113 64 , , PUNCT cana-1707 113 65 𝑦 𝑦 PROPN cana-1707 113 66 ∈ ∈ PROPN cana-1707 113 67 𝑋. 𝑋. PROPN cana-1707 113 68 ( ( PUNCT cana-1707 113 69 ciric ciric ADJ cana-1707 113 70 in in ADP cana-1707 113 71 1971 1971 NUM cana-1707 113 72 , , PUNCT cana-1707 113 73 [ [ X cana-1707 113 74 19 19 NUM cana-1707 113 75 ] ] SYM cana-1707 113 76 ) ) PUNCT cana-1707 113 77 definition definition NOUN cana-1707 113 78 3.12 3.12 NUM cana-1707 113 79 . . PUNCT cana-1707 114 1 a a DET cana-1707 114 2 mapping mapping NOUN cana-1707 114 3 𝑇 𝑇 NOUN cana-1707 114 4 : : PUNCT cana-1707 114 5 𝑋 𝑋 PROPN cana-1707 114 6 β†’ β†’ SYM cana-1707 114 7 𝑋 𝑋 PROPN cana-1707 114 8 in in ADP cana-1707 114 9 metric metric ADJ cana-1707 114 10 space space NOUN cana-1707 114 11 ( ( PUNCT cana-1707 114 12 𝑋 𝑋 PROPN cana-1707 114 13 , , PUNCT cana-1707 114 14 𝑑 𝑑 NOUN cana-1707 114 15 ) ) PUNCT cana-1707 114 16 is be AUX cana-1707 114 17 said say VERB cana-1707 114 18 to to PART cana-1707 114 19 be be AUX cana-1707 114 20 hardy hardy ADJ cana-1707 114 21 and and CCONJ cana-1707 114 22 roger roger PROPN cana-1707 114 23 contractions contraction NOUN cana-1707 114 24 if if SCONJ cana-1707 114 25 there there PRON cana-1707 114 26 exists exist VERB cana-1707 114 27 π‘Ž π‘Ž PROPN cana-1707 114 28 , , PUNCT cana-1707 114 29 𝑏 𝑏 NOUN cana-1707 114 30 , , PUNCT cana-1707 114 31 𝑐 𝑐 PROPN cana-1707 114 32 , , PUNCT cana-1707 114 33 𝑒 𝑒 PROPN cana-1707 114 34 , , PUNCT cana-1707 114 35 𝑓 𝑓 DET cana-1707 114 36 monotonically monotonically ADV cana-1707 114 37 decreasing decrease VERB cana-1707 114 38 function function NOUN cana-1707 114 39 from from ADP cana-1707 114 40 [ [ X cana-1707 114 41 0 0 NUM cana-1707 114 42 , , PUNCT cana-1707 114 43 ∞ ∞ PROPN cana-1707 114 44 ) ) PUNCT cana-1707 114 45 π‘‘π‘œ π‘‘π‘œ PROPN cana-1707 114 46 ( ( PUNCT cana-1707 114 47 0,1 0,1 NUM cana-1707 114 48 ) ) PUNCT cana-1707 114 49 , , PUNCT cana-1707 114 50 and and CCONJ cana-1707 114 51 π‘Ž π‘Ž X cana-1707 114 52 + + NOUN cana-1707 114 53 𝑏 𝑏 NOUN cana-1707 114 54 + + NOUN cana-1707 114 55 𝑐 𝑐 PROPN cana-1707 114 56 + + PUNCT cana-1707 114 57 𝑒 𝑒 PUNCT cana-1707 114 58 + + ADP cana-1707 114 59 𝑓 𝑓 PRON cana-1707 114 60 < < X cana-1707 114 61 1 1 NUM cana-1707 114 62 such such ADJ cana-1707 114 63 that that SCONJ cana-1707 114 64 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 114 65 , , PUNCT cana-1707 114 66 𝑇𝑦 𝑇𝑦 PROPN cana-1707 114 67 ) ) PUNCT cana-1707 114 68 ≀a ≀a NOUN cana-1707 114 69 d(x d(x PROPN cana-1707 114 70 , , PUNCT cana-1707 114 71 tx)+b tx)+b PROPN cana-1707 114 72 d(y d(y PROPN cana-1707 114 73 , , PUNCT cana-1707 114 74 ty)+c ty)+c NOUN cana-1707 114 75 d(x d(x PROPN cana-1707 114 76 , , PUNCT cana-1707 114 77 ty)+𝑒𝑑(𝑦 ty)+𝑒𝑑(𝑦 PROPN cana-1707 114 78 , , PUNCT cana-1707 114 79 𝑇π‘₯ 𝑇π‘₯ PROPN cana-1707 114 80 ) ) PUNCT cana-1707 114 81 + + NUM cana-1707 114 82 𝑓𝑑(π‘₯ 𝑓𝑑(π‘₯ NOUN cana-1707 114 83 , , PUNCT cana-1707 114 84 𝑦 𝑦 NOUN cana-1707 114 85 ) ) PUNCT cana-1707 114 86 , , PUNCT cana-1707 114 87 for for ADP cana-1707 114 88 every every DET cana-1707 114 89 π‘₯ π‘₯ PROPN cana-1707 114 90 , , PUNCT cana-1707 114 91 𝑦 𝑦 NOUN cana-1707 114 92 ∈ ∈ NOUN cana-1707 114 93 𝑋,with 𝑋,with NOUN cana-1707 114 94 π‘Ž π‘Ž NOUN cana-1707 114 95 = = SYM cana-1707 114 96 π‘Ž(𝑑(π‘₯ π‘Ž(𝑑(π‘₯ PROPN cana-1707 114 97 , , PUNCT cana-1707 114 98 𝑦 𝑦 NOUN cana-1707 114 99 ) ) PUNCT cana-1707 114 100 ) ) PUNCT cana-1707 114 101 , , PUNCT cana-1707 115 1 𝑏 𝑏 PROPN cana-1707 115 2 = = SYM cana-1707 115 3 𝑏(𝑑(π‘₯ 𝑏(𝑑(π‘₯ PROPN cana-1707 115 4 , , PUNCT cana-1707 115 5 𝑦 𝑦 NOUN cana-1707 115 6 ) ) PUNCT cana-1707 115 7 ) ) PUNCT cana-1707 115 8 , , PUNCT cana-1707 115 9 𝑐 𝑐 PROPN cana-1707 115 10 = = SYM cana-1707 115 11 𝑐(𝑑(π‘₯ 𝑐(𝑑(π‘₯ PROPN cana-1707 115 12 , , PUNCT cana-1707 115 13 𝑦 𝑦 NOUN cana-1707 115 14 ) ) PUNCT cana-1707 115 15 ) ) PUNCT cana-1707 115 16 , , PUNCT cana-1707 115 17 𝑒 𝑒 PROPN cana-1707 115 18 = = SYM cana-1707 115 19 𝑒(𝑑(π‘₯ 𝑒(𝑑(π‘₯ PROPN cana-1707 115 20 , , PUNCT cana-1707 115 21 𝑦 𝑦 NOUN cana-1707 115 22 ) ) PUNCT cana-1707 115 23 ) ) PUNCT cana-1707 115 24 , , PUNCT cana-1707 115 25 𝑓 𝑓 PROPN cana-1707 115 26 = = SYM cana-1707 115 27 𝑓(𝑑(π‘₯ 𝑓(𝑑(π‘₯ PROPN cana-1707 115 28 , , PUNCT cana-1707 115 29 𝑦 𝑦 NOUN cana-1707 115 30 ) ) PUNCT cana-1707 115 31 ) ) PUNCT cana-1707 115 32 ( ( PUNCT cana-1707 115 33 hardy hardy ADJ cana-1707 115 34 and and CCONJ cana-1707 115 35 roger roger NOUN cana-1707 115 36 in in ADP cana-1707 115 37 1973 1973 NUM cana-1707 115 38 , , PUNCT cana-1707 115 39 [ [ X cana-1707 115 40 31 31 NUM cana-1707 115 41 ] ] SYM cana-1707 115 42 ) ) PUNCT cana-1707 115 43 communications communication NOUN cana-1707 115 44 on on ADP cana-1707 115 45 applied apply VERB cana-1707 115 46 nonlinear nonlinear ADJ cana-1707 115 47 analysis analysis NOUN cana-1707 115 48 issn issn NOUN cana-1707 115 49 : : PUNCT cana-1707 115 50 1074 1074 NUM cana-1707 115 51 - - PUNCT cana-1707 115 52 133x 133x NUM cana-1707 115 53 vol vol NOUN cana-1707 115 54 32 32 NUM cana-1707 115 55 no no NOUN cana-1707 115 56 . . NOUN cana-1707 115 57 2 2 NUM cana-1707 115 58 ( ( PUNCT cana-1707 115 59 2025 2025 NUM cana-1707 115 60 ) ) PUNCT cana-1707 115 61 58 58 NUM cana-1707 115 62 https://internationalpubls.com https://internationalpubls.com X cana-1707 116 1 the the DET cana-1707 116 2 browder browder NOUN cana-1707 116 3 contraction contraction NOUN cana-1707 116 4 [ [ X cana-1707 116 5 7 7 NUM cana-1707 116 6 ] ] PUNCT cana-1707 116 7 was be AUX cana-1707 116 8 extended extend VERB cana-1707 116 9 as as SCONJ cana-1707 116 10 follows follow VERB cana-1707 116 11 : : PUNCT cana-1707 116 12 definition definition NOUN cana-1707 116 13 3.13 3.13 NUM cana-1707 116 14 . . PUNCT cana-1707 117 1 let let VERB cana-1707 117 2 ( ( PUNCT cana-1707 117 3 𝑋 𝑋 NOUN cana-1707 117 4 , , PUNCT cana-1707 117 5 𝑑 𝑑 NOUN cana-1707 117 6 ) ) PUNCT cana-1707 117 7 be be VERB cana-1707 117 8 a a DET cana-1707 117 9 metric metric ADJ cana-1707 117 10 space space NOUN cana-1707 117 11 and and CCONJ cana-1707 117 12 𝑇 𝑇 PROPN cana-1707 117 13 : : PUNCT cana-1707 117 14 𝑋 𝑋 PROPN cana-1707 117 15 β†’ β†’ SYM cana-1707 117 16 𝑋 𝑋 PROPN cana-1707 117 17 be be VERB cana-1707 117 18 a a DET cana-1707 117 19 mapping mapping NOUN cana-1707 117 20 . . PUNCT cana-1707 118 1 there there PRON cana-1707 118 2 exists exist VERB cana-1707 118 3 a a DET cana-1707 118 4 non non ADJ cana-1707 118 5 - - ADJ cana-1707 118 6 decreasing decrease VERB cana-1707 118 7 right right ADJ cana-1707 118 8 continuous continuous ADJ cana-1707 118 9 function function NOUN cana-1707 118 10 πœ™ πœ™ PROPN cana-1707 118 11 ∢ ∢ NOUN cana-1707 118 12 [ [ X cana-1707 118 13 0 0 NUM cana-1707 118 14 , , PUNCT cana-1707 118 15 ∞ ∞ PROPN cana-1707 118 16 ) ) PUNCT cana-1707 118 17 β†’ β†’ PUNCT cana-1707 119 1 [ [ X cana-1707 119 2 0 0 NUM cana-1707 119 3 , , PUNCT cana-1707 119 4 ∞ ∞ NOUN cana-1707 119 5 ) ) PUNCT cana-1707 119 6 such such ADJ cana-1707 119 7 that that SCONJ cana-1707 119 8 πœ™(𝑑 πœ™(𝑑 PROPN cana-1707 119 9 ) ) PUNCT cana-1707 119 10 < < X cana-1707 120 1 𝑑 𝑑 X cana-1707 120 2 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-1707 120 3 𝑑 𝑑 X cana-1707 120 4 > > X cana-1707 120 5 0 0 PUNCT cana-1707 121 1 and and CCONJ cana-1707 121 2 , , PUNCT cana-1707 121 3 for for ADP cana-1707 121 4 any any DET cana-1707 121 5 π‘₯ π‘₯ PROPN cana-1707 121 6 , , PUNCT cana-1707 121 7 𝑦 𝑦 NOUN cana-1707 121 8 ∈ ∈ PROPN cana-1707 121 9 𝑋 𝑋 PROPN cana-1707 121 10 , , PUNCT cana-1707 121 11 ( ( PUNCT cana-1707 121 12 i i NOUN cana-1707 121 13 ) ) PUNCT cana-1707 121 14 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 121 15 , , PUNCT cana-1707 121 16 𝑇𝑦 𝑇𝑦 PROPN cana-1707 121 17 ) ) PUNCT cana-1707 121 18 ≀ ≀ NUM cana-1707 121 19 πœ™(𝑑(π‘₯ πœ™(𝑑(π‘₯ PROPN cana-1707 121 20 , , PUNCT cana-1707 121 21 𝑦 𝑦 NOUN cana-1707 121 22 ) ) PUNCT cana-1707 121 23 ) ) PUNCT cana-1707 121 24 ( ( PUNCT cana-1707 121 25 browder browder X cana-1707 121 26 [ [ X cana-1707 121 27 7 7 NUM cana-1707 121 28 ] ] NUM cana-1707 121 29 ) ) PUNCT cana-1707 121 30 ( ( PUNCT cana-1707 121 31 ii ii NOUN cana-1707 121 32 ) ) PUNCT cana-1707 121 33 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 121 34 , , PUNCT cana-1707 121 35 𝑇𝑦 𝑇𝑦 PROPN cana-1707 121 36 ) ) PUNCT cana-1707 121 37 ≀ ≀ NOUN cana-1707 121 38 πœ™(π‘š(π‘₯ πœ™(π‘š(π‘₯ PROPN cana-1707 121 39 , , PUNCT cana-1707 121 40 𝑦 𝑦 NOUN cana-1707 121 41 ) ) PUNCT cana-1707 121 42 ) ) PUNCT cana-1707 122 1 ( ( PUNCT cana-1707 122 2 danes dane NOUN cana-1707 122 3 [ [ X cana-1707 122 4 21 21 NUM cana-1707 122 5 ] ] SYM cana-1707 122 6 ) ) PUNCT cana-1707 122 7 ( ( PUNCT cana-1707 122 8 iii iii NOUN cana-1707 122 9 ) ) PUNCT cana-1707 122 10 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 122 11 , , PUNCT cana-1707 122 12 𝑇𝑦 𝑇𝑦 PROPN cana-1707 122 13 ) ) PUNCT cana-1707 122 14 ≀ ≀ NOUN cana-1707 123 1 πœ™(𝛿(π‘₯ πœ™(𝛿(π‘₯ NUM cana-1707 123 2 , , PUNCT cana-1707 123 3 𝑦 𝑦 NOUN cana-1707 123 4 ) ) PUNCT cana-1707 123 5 ) ) PUNCT cana-1707 124 1 if if SCONJ cana-1707 124 2 π‘₯ π‘₯ PROPN cana-1707 124 3 , , PUNCT cana-1707 124 4 𝑦 𝑦 PRON cana-1707 124 5 are be AUX cana-1707 124 6 regular regular ADJ cana-1707 124 7 ( ( PUNCT cana-1707 124 8 kasahara kasahara X cana-1707 124 9 [ [ X cana-1707 124 10 36 36 NUM cana-1707 124 11 ] ] SYM cana-1707 124 12 ) ) PUNCT cana-1707 124 13 definition definition NOUN cana-1707 124 14 3.14 3.14 NUM cana-1707 124 15 . . PUNCT cana-1707 125 1 let let VERB cana-1707 125 2 ( ( PUNCT cana-1707 125 3 𝑋 𝑋 NOUN cana-1707 125 4 , , PUNCT cana-1707 125 5 𝑑 𝑑 NOUN cana-1707 125 6 ) ) PUNCT cana-1707 125 7 be be VERB cana-1707 125 8 a a DET cana-1707 125 9 metric metric ADJ cana-1707 125 10 space space NOUN cana-1707 125 11 , , PUNCT cana-1707 125 12 and and CCONJ cana-1707 125 13 let let VERB cana-1707 125 14 𝑇 𝑇 PROPN cana-1707 125 15 : : PUNCT cana-1707 125 16 𝑋 𝑋 PROPN cana-1707 125 17 β†’ β†’ SYM cana-1707 125 18 𝑋 𝑋 PROPN cana-1707 125 19 be be VERB cana-1707 125 20 a a DET cana-1707 125 21 mapping mapping NOUN cana-1707 125 22 . . PUNCT cana-1707 126 1 then then ADV cana-1707 126 2 , , PUNCT cana-1707 126 3 t t PROPN cana-1707 126 4 is be AUX cana-1707 126 5 said say VERB cana-1707 126 6 to to PART cana-1707 126 7 be be AUX cana-1707 126 8 meir meir PROPN cana-1707 126 9 keeler keeler PROPN cana-1707 126 10 's 's PART cana-1707 126 11 contraction contraction NOUN cana-1707 126 12 if if SCONJ cana-1707 126 13 for for ADP cana-1707 126 14 any any DET cana-1707 126 15 νœ€ νœ€ NOUN cana-1707 126 16 > > X cana-1707 126 17 0 0 NUM cana-1707 126 18 , , PUNCT cana-1707 126 19 βˆƒπ›Ώ βˆƒπ›Ώ PROPN cana-1707 126 20 > > X cana-1707 126 21 0 0 PROPN cana-1707 126 22 , , PUNCT cana-1707 126 23 such such ADJ cana-1707 126 24 that that SCONJ cana-1707 126 25 νœ€ νœ€ DET cana-1707 126 26 ≀ ≀ X cana-1707 126 27 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 126 28 , , PUNCT cana-1707 126 29 𝑦 𝑦 NOUN cana-1707 126 30 ) ) PUNCT cana-1707 126 31 < < X cana-1707 127 1 νœ€ νœ€ X cana-1707 127 2 + + X cana-1707 127 3 𝛿 𝛿 ADJ cana-1707 127 4 implies implie NOUN cana-1707 127 5 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 127 6 , , PUNCT cana-1707 127 7 𝑇𝑦 𝑇𝑦 PROPN cana-1707 127 8 ) ) PUNCT cana-1707 127 9 ≀ ≀ NUM cana-1707 127 10 νœ€ νœ€ NOUN cana-1707 127 11 for for ADP cana-1707 127 12 all all PRON cana-1707 127 13 π‘₯ π‘₯ PROPN cana-1707 127 14 , , PUNCT cana-1707 127 15 𝑦 𝑦 NOUN cana-1707 127 16 ∈ ∈ NOUN cana-1707 127 17 𝑋 𝑋 NOUN cana-1707 127 18 π‘€π‘–π‘‘β„Ž π‘€π‘–π‘‘β„Ž NOUN cana-1707 127 19 π‘₯ π‘₯ PROPN cana-1707 127 20 β‰  β‰  PROPN cana-1707 127 21 𝑦. 𝑦. PROPN cana-1707 127 22 ( ( PUNCT cana-1707 127 23 meir meir PROPN cana-1707 127 24 keeler keeler PROPN cana-1707 127 25 in in ADP cana-1707 127 26 1969 1969 NUM cana-1707 127 27 , , PUNCT cana-1707 127 28 [ [ X cana-1707 127 29 40 40 NUM cana-1707 127 30 ] ] PUNCT cana-1707 127 31 ) ) PUNCT cana-1707 127 32 its its PRON cana-1707 127 33 extended extended ADJ cana-1707 127 34 forms form NOUN cana-1707 127 35 are be AUX cana-1707 127 36 as as ADP cana-1707 127 37 : : PUNCT cana-1707 127 38 ( ( PUNCT cana-1707 127 39 i i NOUN cana-1707 127 40 ) ) PUNCT cana-1707 127 41 νœ€ νœ€ PRON cana-1707 127 42 ≀ ≀ NUM cana-1707 127 43 π‘š(π‘₯ π‘š(π‘₯ NOUN cana-1707 127 44 , , PUNCT cana-1707 127 45 𝑦 𝑦 NOUN cana-1707 127 46 ) ) PUNCT cana-1707 127 47 < < X cana-1707 128 1 νœ€ νœ€ X cana-1707 128 2 + + PUNCT cana-1707 128 3 𝛿 𝛿 ADJ cana-1707 128 4 π‘–π‘šπ‘π‘™π‘–π‘’π‘  π‘–π‘šπ‘π‘™π‘–π‘’π‘  ADJ cana-1707 128 5 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ X cana-1707 128 6 , , PUNCT cana-1707 128 7 𝑇𝑦 𝑇𝑦 PROPN cana-1707 128 8 ) ) PUNCT cana-1707 128 9 < < X cana-1707 128 10 νœ€ νœ€ X cana-1707 128 11 ; ; PUNCT cana-1707 128 12 ( ( PUNCT cana-1707 128 13 ii ii NOUN cana-1707 128 14 ) ) PUNCT cana-1707 128 15 νœ€ νœ€ PROPN cana-1707 128 16 ≀ ≀ NOUN cana-1707 128 17 𝛿(π‘₯ 𝛿(π‘₯ PROPN cana-1707 128 18 , , PUNCT cana-1707 128 19 𝑦 𝑦 NOUN cana-1707 128 20 ) ) PUNCT cana-1707 128 21 < < X cana-1707 129 1 νœ€ νœ€ X cana-1707 129 2 + + PUNCT cana-1707 129 3 𝛿 𝛿 ADJ cana-1707 129 4 π‘–π‘šπ‘π‘™π‘–π‘’π‘  π‘–π‘šπ‘π‘™π‘–π‘’π‘  ADJ cana-1707 129 5 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ X cana-1707 129 6 , , PUNCT cana-1707 129 7 𝑇𝑦 𝑇𝑦 PROPN cana-1707 129 8 ) ) PUNCT cana-1707 129 9 < < X cana-1707 129 10 νœ€ νœ€ X cana-1707 129 11 . . PUNCT cana-1707 129 12 ( ( PUNCT cana-1707 129 13 park park NOUN cana-1707 129 14 in in ADP cana-1707 129 15 1980 1980 NUM cana-1707 129 16 , , PUNCT cana-1707 129 17 [ [ X cana-1707 129 18 43 43 NUM cana-1707 129 19 ] ] SYM cana-1707 129 20 ) ) PUNCT cana-1707 129 21 definition definition NOUN cana-1707 129 22 3.15 3.15 NUM cana-1707 129 23 . . PUNCT cana-1707 130 1 let let VERB cana-1707 130 2 ( ( PUNCT cana-1707 130 3 𝑋 𝑋 NOUN cana-1707 130 4 , , PUNCT cana-1707 130 5 𝑑 𝑑 NOUN cana-1707 130 6 ) ) PUNCT cana-1707 130 7 be be VERB cana-1707 130 8 a a DET cana-1707 130 9 metric metric ADJ cana-1707 130 10 space space NOUN cana-1707 130 11 , , PUNCT cana-1707 130 12 and and CCONJ cana-1707 130 13 let let VERB cana-1707 130 14 𝑇 𝑇 PROPN cana-1707 130 15 : : PUNCT cana-1707 130 16 𝑋 𝑋 PROPN cana-1707 130 17 β†’ β†’ SYM cana-1707 130 18 𝑋 𝑋 PROPN cana-1707 130 19 be be VERB cana-1707 130 20 a a DET cana-1707 130 21 mapping mapping NOUN cana-1707 130 22 . . PUNCT cana-1707 131 1 then then ADV cana-1707 131 2 , , PUNCT cana-1707 131 3 t t PROPN cana-1707 131 4 is be AUX cana-1707 131 5 said say VERB cana-1707 131 6 to to PART cana-1707 131 7 be be AUX cana-1707 131 8 park park NOUN cana-1707 131 9 contraction contraction NOUN cana-1707 131 10 if if SCONJ cana-1707 131 11 for for ADP cana-1707 131 12 any any DET cana-1707 131 13 πœ– πœ– X cana-1707 131 14 > > X cana-1707 131 15 0 0 PROPN cana-1707 131 16 , , PUNCT cana-1707 131 17 βˆƒπœ–0 βˆƒπœ–0 NOUN cana-1707 131 18 < < X cana-1707 131 19 πœ– πœ– PROPN cana-1707 131 20 , , PUNCT cana-1707 131 21 and and CCONJ cana-1707 131 22 𝛿0 𝛿0 NOUN cana-1707 131 23 > > X cana-1707 131 24 0 0 NUM cana-1707 131 25 such such ADJ cana-1707 131 26 that that PRON cana-1707 131 27 for for ADP cana-1707 131 28 any any DET cana-1707 131 29 π‘₯ π‘₯ PROPN cana-1707 131 30 , , PUNCT cana-1707 131 31 𝑦 𝑦 NOUN cana-1707 131 32 ∈ ∈ PROPN cana-1707 131 33 𝑋 𝑋 PROPN cana-1707 131 34 , , PUNCT cana-1707 131 35 ( ( PUNCT cana-1707 131 36 i i NOUN cana-1707 131 37 ) ) PUNCT cana-1707 131 38 πœ– πœ– PROPN cana-1707 131 39 ≀ ≀ PROPN cana-1707 131 40 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 131 41 , , PUNCT cana-1707 131 42 𝑦 𝑦 NOUN cana-1707 131 43 ) ) PUNCT cana-1707 131 44 < < X cana-1707 131 45 πœ– πœ– PROPN cana-1707 132 1 + + NUM cana-1707 132 2 𝛿0 𝛿0 NOUN cana-1707 132 3 implies imply VERB cana-1707 132 4 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 132 5 , , PUNCT cana-1707 132 6 𝑇𝑦 𝑇𝑦 PROPN cana-1707 132 7 ) ) PUNCT cana-1707 132 8 ≀ ≀ NOUN cana-1707 132 9 πœ–0 πœ–0 PROPN cana-1707 132 10 . . PUNCT cana-1707 133 1 ( ( PUNCT cana-1707 133 2 park park NOUN cana-1707 133 3 in in ADP cana-1707 133 4 1980 1980 NUM cana-1707 133 5 , , PUNCT cana-1707 133 6 [ [ X cana-1707 133 7 43 43 NUM cana-1707 133 8 ] ] SYM cana-1707 133 9 ) ) PUNCT cana-1707 133 10 ( ( PUNCT cana-1707 133 11 ii ii NOUN cana-1707 133 12 ) ) PUNCT cana-1707 133 13 πœ– πœ– PROPN cana-1707 133 14 ≀ ≀ PROPN cana-1707 133 15 π‘š(π‘₯ π‘š(π‘₯ NOUN cana-1707 133 16 , , PUNCT cana-1707 133 17 𝑦 𝑦 NOUN cana-1707 133 18 ) ) PUNCT cana-1707 133 19 < < X cana-1707 133 20 πœ– πœ– PROPN cana-1707 134 1 + + NUM cana-1707 134 2 𝛿0 𝛿0 NOUN cana-1707 134 3 implies imply VERB cana-1707 134 4 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 134 5 , , PUNCT cana-1707 134 6 𝑇𝑦 𝑇𝑦 PROPN cana-1707 134 7 ) ) PUNCT cana-1707 134 8 ≀ ≀ NOUN cana-1707 134 9 πœ–0 πœ–0 PROPN cana-1707 134 10 . . PUNCT cana-1707 135 1 ( ( PUNCT cana-1707 135 2 park park NOUN cana-1707 135 3 in in ADP cana-1707 135 4 1980 1980 NUM cana-1707 135 5 , , PUNCT cana-1707 135 6 [ [ X cana-1707 135 7 43 43 NUM cana-1707 135 8 ] ] SYM cana-1707 135 9 ) ) PUNCT cana-1707 135 10 ( ( PUNCT cana-1707 135 11 iii iii X cana-1707 135 12 ) ) PUNCT cana-1707 135 13 πœ– πœ– PROPN cana-1707 135 14 ≀ ≀ NOUN cana-1707 135 15 𝛿(π‘₯ 𝛿(π‘₯ PROPN cana-1707 135 16 , , PUNCT cana-1707 135 17 𝑦 𝑦 NOUN cana-1707 135 18 ) ) PUNCT cana-1707 135 19 < < X cana-1707 135 20 πœ– πœ– PROPN cana-1707 136 1 + + NUM cana-1707 136 2 𝛿0 𝛿0 NOUN cana-1707 136 3 implies imply VERB cana-1707 136 4 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADP cana-1707 136 5 , , PUNCT cana-1707 136 6 𝑇𝑦 𝑇𝑦 PROPN cana-1707 136 7 ) ) PUNCT cana-1707 136 8 ≀ ≀ NOUN cana-1707 136 9 πœ–0 πœ–0 PROPN cana-1707 136 10 . . PUNCT cana-1707 137 1 ( ( PUNCT cana-1707 137 2 hegedlis hegedlis PROPN cana-1707 137 3 - - PUNCT cana-1707 137 4 szilagyi szilagyi PROPN cana-1707 138 1 [ [ X cana-1707 138 2 29 29 NUM cana-1707 138 3 ] ] SYM cana-1707 138 4 ) ) PUNCT cana-1707 138 5 . . PUNCT cana-1707 139 1 definition definition NOUN cana-1707 139 2 3.16 3.16 NUM cana-1707 139 3 . . PUNCT cana-1707 140 1 let let VERB cana-1707 140 2 ( ( PUNCT cana-1707 140 3 𝑋 𝑋 NOUN cana-1707 140 4 , , PUNCT cana-1707 140 5 𝑑 𝑑 NOUN cana-1707 140 6 ) ) PUNCT cana-1707 140 7 be be VERB cana-1707 140 8 a a DET cana-1707 140 9 metric metric ADJ cana-1707 140 10 space space NOUN cana-1707 140 11 . . PUNCT cana-1707 141 1 then then ADV cana-1707 141 2 a a DET cana-1707 141 3 mapping mapping NOUN cana-1707 141 4 𝑇 𝑇 NOUN cana-1707 141 5 : : PUNCT cana-1707 141 6 𝑋 𝑋 PROPN cana-1707 141 7 β†’ β†’ SYM cana-1707 141 8 𝑋 𝑋 PROPN cana-1707 141 9 is be AUX cana-1707 141 10 said say VERB cana-1707 141 11 to to PART cana-1707 141 12 be be AUX cana-1707 141 13 contractive contractive ADJ cana-1707 141 14 if if SCONJ cana-1707 141 15 there there PRON cana-1707 141 16 exists exist VERB cana-1707 141 17 𝛼 𝛼 PROPN cana-1707 141 18 , , PUNCT cana-1707 141 19 𝛽 𝛽 NOUN cana-1707 141 20 > > X cana-1707 141 21 0 0 PUNCT cana-1707 141 22 with with ADP cana-1707 141 23 𝛼 𝛼 PROPN cana-1707 141 24 + + NOUN cana-1707 141 25 𝛽 𝛽 NOUN cana-1707 141 26 < < X cana-1707 141 27 1 1 NUM cana-1707 141 28 satisfying satisfying NOUN cana-1707 141 29 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 141 30 , , PUNCT cana-1707 141 31 𝑇𝑦 𝑇𝑦 PROPN cana-1707 141 32 ) ) PUNCT cana-1707 141 33 ≀ ≀ NUM cana-1707 142 1 𝛼 𝛼 X cana-1707 142 2 𝑑(𝑦,𝑇𝑦)(1 𝑑(𝑦,𝑇𝑦)(1 NOUN cana-1707 142 3 + + CCONJ cana-1707 142 4 𝑑(π‘₯,𝑇π‘₯ 𝑑(π‘₯,𝑇π‘₯ NOUN cana-1707 142 5 ) ) PUNCT cana-1707 142 6 ) ) PUNCT cana-1707 142 7 1 1 NUM cana-1707 143 1 + + NUM cana-1707 143 2 𝑑(π‘₯,𝑦 𝑑(π‘₯,𝑦 NUM cana-1707 143 3 ) ) PUNCT cana-1707 144 1 + + CCONJ cana-1707 144 2 𝛽𝑑(π‘₯ 𝛽𝑑(π‘₯ NUM cana-1707 144 3 , , PUNCT cana-1707 144 4 𝑦 𝑦 NOUN cana-1707 144 5 ) ) PUNCT cana-1707 144 6 , , PUNCT cana-1707 144 7 βˆ€π‘₯ βˆ€π‘₯ PROPN cana-1707 144 8 , , PUNCT cana-1707 144 9 𝑦 𝑦 NOUN cana-1707 144 10 ∈ ∈ PROPN cana-1707 144 11 𝑋. 𝑋. PROPN cana-1707 144 12 ( ( PUNCT cana-1707 144 13 das das PROPN cana-1707 144 14 and and CCONJ cana-1707 144 15 gupta gupta PROPN cana-1707 144 16 in1975 in1975 PROPN cana-1707 144 17 , , PUNCT cana-1707 144 18 [ [ X cana-1707 144 19 22 22 NUM cana-1707 144 20 ] ] SYM cana-1707 144 21 ) ) PUNCT cana-1707 144 22 definition definition NOUN cana-1707 144 23 3.17 3.17 NUM cana-1707 144 24 . . PUNCT cana-1707 145 1 a a DET cana-1707 145 2 mapping mapping NOUN cana-1707 145 3 𝑇 𝑇 NOUN cana-1707 145 4 : : PUNCT cana-1707 145 5 𝑋 𝑋 PROPN cana-1707 145 6 β†’ β†’ SYM cana-1707 145 7 𝑋 𝑋 PROPN cana-1707 145 8 is be AUX cana-1707 145 9 said say VERB cana-1707 145 10 to to PART cana-1707 145 11 be be AUX cana-1707 145 12 contractive contractive ADJ cana-1707 145 13 in in ADP cana-1707 145 14 metric metric ADJ cana-1707 145 15 space space NOUN cana-1707 145 16 ( ( PUNCT cana-1707 145 17 𝑋 𝑋 PROPN cana-1707 145 18 , , PUNCT cana-1707 145 19 𝑑 𝑑 NOUN cana-1707 145 20 ) ) PUNCT cana-1707 145 21 if if SCONJ cana-1707 145 22 for for ADP cana-1707 145 23 all all PRON cana-1707 145 24 π‘₯ π‘₯ PROPN cana-1707 145 25 , , PUNCT cana-1707 145 26 𝑦 𝑦 NOUN cana-1707 145 27 ∈ ∈ PROPN cana-1707 145 28 𝑋 𝑋 PROPN cana-1707 145 29 , , PUNCT cana-1707 145 30 π‘₯ π‘₯ PROPN cana-1707 145 31 β‰  β‰  PROPN cana-1707 145 32 𝑦 𝑦 NOUN cana-1707 145 33 and and CCONJ cana-1707 145 34 some some DET cana-1707 145 35 𝛼 𝛼 NOUN cana-1707 145 36 , , PUNCT cana-1707 145 37 𝛽 𝛽 PROPN cana-1707 145 38 ∈ ∈ PROPN cana-1707 146 1 [ [ X cana-1707 146 2 0 0 NUM cana-1707 146 3 , , PUNCT cana-1707 146 4 1 1 NUM cana-1707 146 5 ) ) PUNCT cana-1707 146 6 with with ADP cana-1707 146 7 𝛼 𝛼 PROPN cana-1707 146 8 + + NOUN cana-1707 146 9 𝛽 𝛽 NOUN cana-1707 146 10 < < X cana-1707 146 11 1 1 NUM cana-1707 146 12 satisfying satisfying NOUN cana-1707 146 13 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 146 14 , , PUNCT cana-1707 146 15 𝑇𝑦 𝑇𝑦 PROPN cana-1707 146 16 ) ) PUNCT cana-1707 146 17 ≀ ≀ NUM cana-1707 147 1 𝛼 𝛼 NUM cana-1707 147 2 𝑑(π‘₯,𝑇π‘₯)𝑑(𝑦,𝑇𝑦 𝑑(π‘₯,𝑇π‘₯)𝑑(𝑦,𝑇𝑦 NOUN cana-1707 147 3 ) ) PUNCT cana-1707 147 4 𝑑(π‘₯,𝑦 𝑑(π‘₯,𝑦 NUM cana-1707 147 5 ) ) PUNCT cana-1707 148 1 + + CCONJ cana-1707 148 2 𝛽𝑑(π‘₯ 𝛽𝑑(π‘₯ NUM cana-1707 148 3 , , PUNCT cana-1707 148 4 𝑦 𝑦 NOUN cana-1707 148 5 ) ) PUNCT cana-1707 148 6 ( ( PUNCT cana-1707 148 7 jaggi jaggi NOUN cana-1707 148 8 in in ADP cana-1707 148 9 1977 1977 NUM cana-1707 148 10 , , PUNCT cana-1707 148 11 [ [ X cana-1707 148 12 33 33 NUM cana-1707 148 13 ] ] SYM cana-1707 148 14 ) ) PUNCT cana-1707 148 15 definition definition NOUN cana-1707 148 16 3.18 3.18 NUM cana-1707 148 17 . . PUNCT cana-1707 149 1 consider consider VERB cana-1707 149 2 a a DET cana-1707 149 3 right right ADJ cana-1707 149 4 upper upper ADJ cana-1707 149 5 semi semi ADJ cana-1707 149 6 - - ADJ cana-1707 149 7 continuous continuous ADJ cana-1707 149 8 function function NOUN cana-1707 149 9 πœ“ πœ“ NOUN cana-1707 149 10 : : PUNCT cana-1707 149 11 οΏ½ οΏ½ NOUN cana-1707 149 12 Μ… Μ… NOUN cana-1707 149 13 οΏ½ οΏ½ NOUN cana-1707 149 14 β†’ β†’ SYM cana-1707 149 15 [ [ AUX cana-1707 149 16 0 0 NUM cana-1707 149 17 , , PUNCT cana-1707 149 18 ∞ ∞ NUM cana-1707 149 19 ) ) PUNCT cana-1707 149 20 satisfying satisfy VERB cana-1707 149 21 πœ“(𝑑 πœ“(𝑑 PROPN cana-1707 149 22 ) ) PUNCT cana-1707 149 23 < < X cana-1707 150 1 𝑑 𝑑 X cana-1707 150 2 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-1707 150 3 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ X cana-1707 150 4 𝑑 𝑑 PROPN cana-1707 150 5 ∈ ∈ PROPN cana-1707 150 6 οΏ½ οΏ½ PROPN cana-1707 150 7 Μ… Μ… NOUN cana-1707 150 8 οΏ½ οΏ½ PROPN cana-1707 150 9 \ \ PUNCT cana-1707 150 10 { { PUNCT cana-1707 150 11 0 0 NUM cana-1707 150 12 } } PUNCT cana-1707 150 13 , , PUNCT cana-1707 150 14 where where SCONJ cana-1707 150 15 p p NOUN cana-1707 150 16 is be AUX cana-1707 150 17 the the DET cana-1707 150 18 range range NOUN cana-1707 150 19 of of ADP cana-1707 150 20 d. d. PROPN cana-1707 150 21 a a DET cana-1707 150 22 mapping mapping NOUN cana-1707 150 23 𝑇 𝑇 PROPN cana-1707 150 24 : : PUNCT cana-1707 150 25 𝑋 𝑋 PROPN cana-1707 150 26 β†’ β†’ SYM cana-1707 150 27 𝑋 𝑋 PROPN cana-1707 150 28 in in ADP cana-1707 150 29 metric metric ADJ cana-1707 150 30 space space NOUN cana-1707 150 31 ( ( PUNCT cana-1707 150 32 𝑋 𝑋 PROPN cana-1707 150 33 , , PUNCT cana-1707 150 34 𝑑 𝑑 NOUN cana-1707 150 35 ) ) PUNCT cana-1707 150 36 is be AUX cana-1707 150 37 said say VERB cana-1707 150 38 to to PART cana-1707 150 39 be be AUX cana-1707 150 40 boyd boyd PROPN cana-1707 150 41 - - PUNCT cana-1707 150 42 wong wong PROPN cana-1707 150 43 contractions contraction NOUN cana-1707 150 44 if if SCONJ cana-1707 150 45 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 150 46 , , PUNCT cana-1707 150 47 𝑇𝑦 𝑇𝑦 PROPN cana-1707 150 48 ) ) PUNCT cana-1707 150 49 ≀ ≀ NOUN cana-1707 150 50 πœ“ πœ“ X cana-1707 150 51 ( ( PUNCT cana-1707 150 52 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 150 53 , , PUNCT cana-1707 150 54 𝑦 𝑦 NOUN cana-1707 150 55 ) ) PUNCT cana-1707 150 56 ) ) PUNCT cana-1707 150 57 for for ADP cana-1707 150 58 every every DET cana-1707 150 59 π‘₯ π‘₯ PROPN cana-1707 150 60 , , PUNCT cana-1707 150 61 𝑦 𝑦 PROPN cana-1707 150 62 ∈ ∈ PROPN cana-1707 150 63 𝑋. 𝑋. PROPN cana-1707 150 64 ( ( PUNCT cana-1707 150 65 boydwong boydwong NOUN cana-1707 150 66 in in ADP cana-1707 150 67 1969 1969 NUM cana-1707 150 68 , , PUNCT cana-1707 150 69 [ [ X cana-1707 150 70 5 5 NUM cana-1707 150 71 ] ] SYM cana-1707 150 72 ) ) PUNCT cana-1707 150 73 definition definition NOUN cana-1707 150 74 3.19 3.19 NUM cana-1707 150 75 . . PUNCT cana-1707 151 1 a a DET cana-1707 151 2 mapping mapping NOUN cana-1707 151 3 𝑇 𝑇 NOUN cana-1707 151 4 : : PUNCT cana-1707 151 5 𝑋 𝑋 PROPN cana-1707 151 6 β†’ β†’ SYM cana-1707 151 7 𝑋 𝑋 PROPN cana-1707 151 8 in in ADP cana-1707 151 9 metric metric ADJ cana-1707 151 10 space space NOUN cana-1707 151 11 ( ( PUNCT cana-1707 151 12 𝑋 𝑋 PROPN cana-1707 151 13 , , PUNCT cana-1707 151 14 𝑑 𝑑 NOUN cana-1707 151 15 ) ) PUNCT cana-1707 151 16 is be AUX cana-1707 151 17 said say VERB cana-1707 151 18 to to PART cana-1707 151 19 be be AUX cana-1707 151 20 singh singh PROPN cana-1707 151 21 contractions contraction NOUN cana-1707 151 22 if if SCONJ cana-1707 151 23 there there PRON cana-1707 151 24 exists exist VERB cana-1707 151 25 a a DET cana-1707 151 26 positive positive ADJ cana-1707 151 27 number number NOUN cana-1707 151 28 π‘š π‘š NOUN cana-1707 151 29 and and CCONJ cana-1707 151 30 a a DET cana-1707 151 31 number number NOUN cana-1707 151 32 𝛼 𝛼 NOUN cana-1707 151 33 ∈ ∈ PROPN cana-1707 151 34 ( ( PUNCT cana-1707 151 35 0 0 NUM cana-1707 151 36 , , PUNCT cana-1707 151 37 1 1 NUM cana-1707 151 38 2 2 NUM cana-1707 151 39 ) ) PUNCT cana-1707 151 40 , , PUNCT cana-1707 151 41 for for ADP cana-1707 151 42 each each DET cana-1707 151 43 π‘₯ π‘₯ PROPN cana-1707 151 44 , , PUNCT cana-1707 151 45 𝑦 𝑦 NOUN cana-1707 151 46 ∈ ∈ PROPN cana-1707 151 47 𝑋 𝑋 PROPN cana-1707 151 48 , , PUNCT cana-1707 151 49 we we PRON cana-1707 151 50 have have VERB cana-1707 151 51 𝑑(π‘“π‘š(π‘₯ 𝑑(π‘“π‘š(π‘₯ NOUN cana-1707 151 52 ) ) PUNCT cana-1707 151 53 , , PUNCT cana-1707 151 54 π‘“π‘š(𝑦 π‘“π‘š(𝑦 NUM cana-1707 151 55 ) ) PUNCT cana-1707 151 56 ) ) PUNCT cana-1707 152 1 ≀ ≀ NUM cana-1707 152 2 𝛼[𝑑(π‘₯ 𝛼[𝑑(π‘₯ PROPN cana-1707 152 3 , , PUNCT cana-1707 152 4 π‘“π‘š(π‘₯ π‘“π‘š(π‘₯ NUM cana-1707 152 5 ) ) PUNCT cana-1707 152 6 ) ) PUNCT cana-1707 153 1 + + CCONJ cana-1707 153 2 𝑑(𝑦 𝑑(𝑦 NOUN cana-1707 153 3 , , PUNCT cana-1707 153 4 π‘“π‘š(𝑦 π‘“π‘š(𝑦 NUM cana-1707 153 5 ) ) PUNCT cana-1707 153 6 ) ) PUNCT cana-1707 153 7 . . PUNCT cana-1707 154 1 ( ( PUNCT cana-1707 154 2 singh singh PROPN cana-1707 154 3 in in ADP cana-1707 154 4 1969 1969 NUM cana-1707 154 5 , , PUNCT cana-1707 154 6 [ [ X cana-1707 154 7 53 53 NUM cana-1707 154 8 ] ] PUNCT cana-1707 154 9 ) ) PUNCT cana-1707 154 10 communications communication NOUN cana-1707 154 11 on on ADP cana-1707 154 12 applied apply VERB cana-1707 154 13 nonlinear nonlinear ADJ cana-1707 154 14 analysis analysis NOUN cana-1707 154 15 issn issn NOUN cana-1707 154 16 : : PUNCT cana-1707 154 17 1074 1074 NUM cana-1707 154 18 - - PUNCT cana-1707 154 19 133x 133x NUM cana-1707 154 20 vol vol NOUN cana-1707 154 21 32 32 NUM cana-1707 154 22 no no NOUN cana-1707 154 23 . . NOUN cana-1707 154 24 2 2 NUM cana-1707 154 25 ( ( PUNCT cana-1707 154 26 2025 2025 NUM cana-1707 154 27 ) ) PUNCT cana-1707 154 28 59 59 NUM cana-1707 154 29 https://internationalpubls.com https://internationalpubls.com X cana-1707 154 30 definition definition NOUN cana-1707 154 31 3.20 3.20 NUM cana-1707 154 32 . . PUNCT cana-1707 155 1 a a DET cana-1707 155 2 mapping mapping NOUN cana-1707 155 3 𝑇 𝑇 NOUN cana-1707 155 4 : : PUNCT cana-1707 155 5 𝑋 𝑋 PROPN cana-1707 155 6 β†’ β†’ SYM cana-1707 155 7 𝑋 𝑋 PROPN cana-1707 155 8 in in ADP cana-1707 155 9 metric metric ADJ cana-1707 155 10 space space NOUN cana-1707 155 11 ( ( PUNCT cana-1707 155 12 𝑋 𝑋 PROPN cana-1707 155 13 , , PUNCT cana-1707 155 14 𝑑 𝑑 NOUN cana-1707 155 15 ) ) PUNCT cana-1707 155 16 is be AUX cana-1707 155 17 said say VERB cana-1707 155 18 to to PART cana-1707 155 19 be be AUX cana-1707 155 20 yen yen NOUN cana-1707 155 21 contractions contraction NOUN cana-1707 155 22 if if SCONJ cana-1707 155 23 there there PRON cana-1707 155 24 exists exist VERB cana-1707 155 25 a a DET cana-1707 155 26 positive positive ADJ cana-1707 155 27 integer integer NOUN cana-1707 155 28 π‘š π‘š PROPN cana-1707 155 29 and and CCONJ cana-1707 155 30 𝑛 𝑛 PROPN cana-1707 155 31 and and CCONJ cana-1707 155 32 a a DET cana-1707 155 33 number number NOUN cana-1707 155 34 𝛼 𝛼 NOUN cana-1707 155 35 ∈ ∈ PROPN cana-1707 155 36 ( ( PUNCT cana-1707 155 37 0,1 0,1 NOUN cana-1707 155 38 ) ) PUNCT cana-1707 155 39 , , PUNCT cana-1707 155 40 such such ADJ cana-1707 155 41 that that PRON cana-1707 155 42 for for ADP cana-1707 155 43 each each DET cana-1707 155 44 π‘₯ π‘₯ PROPN cana-1707 155 45 , , PUNCT cana-1707 155 46 𝑦 𝑦 NOUN cana-1707 155 47 ∈ ∈ PROPN cana-1707 155 48 𝑋 𝑋 PROPN cana-1707 155 49 , , PUNCT cana-1707 155 50 we we PRON cana-1707 155 51 have have VERB cana-1707 155 52 𝑑(π‘“π‘š(π‘₯ 𝑑(π‘“π‘š(π‘₯ NOUN cana-1707 155 53 ) ) PUNCT cana-1707 155 54 , , PUNCT cana-1707 155 55 𝑓𝑛(𝑦 𝑓𝑛(𝑦 NOUN cana-1707 155 56 ) ) PUNCT cana-1707 155 57 ) ) PUNCT cana-1707 155 58 ≀ ≀ NOUN cana-1707 155 59 𝛼𝑑(π‘₯ 𝛼𝑑(π‘₯ NUM cana-1707 155 60 , , PUNCT cana-1707 155 61 𝑦 𝑦 NOUN cana-1707 155 62 ) ) PUNCT cana-1707 155 63 . . PUNCT cana-1707 156 1 ( ( PUNCT cana-1707 156 2 yen yen NOUN cana-1707 156 3 in in ADP cana-1707 156 4 1972 1972 NUM cana-1707 156 5 , , PUNCT cana-1707 156 6 [ [ X cana-1707 156 7 56 56 NUM cana-1707 156 8 ] ] SYM cana-1707 156 9 ) ) PUNCT cana-1707 156 10 definition definition NOUN cana-1707 156 11 3.21 3.21 NUM cana-1707 156 12 . . PUNCT cana-1707 157 1 a a DET cana-1707 157 2 mapping mapping NOUN cana-1707 157 3 𝑇 𝑇 NOUN cana-1707 157 4 : : PUNCT cana-1707 157 5 𝑋 𝑋 PROPN cana-1707 157 6 β†’ β†’ SYM cana-1707 157 7 𝑋 𝑋 PROPN cana-1707 157 8 in in ADP cana-1707 157 9 metric metric ADJ cana-1707 157 10 space space NOUN cana-1707 157 11 ( ( PUNCT cana-1707 157 12 𝑋 𝑋 PROPN cana-1707 157 13 , , PUNCT cana-1707 157 14 𝑑 𝑑 NOUN cana-1707 157 15 ) ) PUNCT cana-1707 157 16 is be AUX cana-1707 157 17 said say VERB cana-1707 157 18 to to PART cana-1707 157 19 be be AUX cana-1707 157 20 guseman guseman ADJ cana-1707 157 21 contractions contraction NOUN cana-1707 157 22 if if SCONJ cana-1707 157 23 there there PRON cana-1707 157 24 exists exist VERB cana-1707 157 25 a a DET cana-1707 157 26 positive positive ADJ cana-1707 157 27 integer integer NOUN cana-1707 157 28 π‘š π‘š X cana-1707 157 29 and and CCONJ cana-1707 157 30 a a DET cana-1707 157 31 number number NOUN cana-1707 157 32 𝛼 𝛼 NOUN cana-1707 157 33 ∈ ∈ PROPN cana-1707 157 34 ( ( PUNCT cana-1707 157 35 0,1 0,1 NOUN cana-1707 157 36 ) ) PUNCT cana-1707 157 37 , , PUNCT cana-1707 157 38 such such ADJ cana-1707 157 39 that that PRON cana-1707 157 40 for for ADP cana-1707 157 41 each each DET cana-1707 157 42 π‘₯ π‘₯ PROPN cana-1707 157 43 , , PUNCT cana-1707 157 44 𝑦 𝑦 NOUN cana-1707 157 45 ∈ ∈ PROPN cana-1707 157 46 𝑋 𝑋 PROPN cana-1707 157 47 , , PUNCT cana-1707 157 48 we we PRON cana-1707 157 49 have have VERB cana-1707 157 50 𝑑(π‘“π‘š(π‘₯ 𝑑(π‘“π‘š(π‘₯ NOUN cana-1707 157 51 ) ) PUNCT cana-1707 157 52 , , PUNCT cana-1707 157 53 π‘“π‘š(𝑦 π‘“π‘š(𝑦 NUM cana-1707 157 54 ) ) PUNCT cana-1707 157 55 ) ) PUNCT cana-1707 157 56 ≀ ≀ NOUN cana-1707 157 57 𝛼𝑑(π‘₯ 𝛼𝑑(π‘₯ NUM cana-1707 157 58 , , PUNCT cana-1707 157 59 𝑦 𝑦 NOUN cana-1707 157 60 ) ) PUNCT cana-1707 157 61 . . PUNCT cana-1707 158 1 ( ( PUNCT cana-1707 158 2 guseman guseman NOUN cana-1707 158 3 in in ADP cana-1707 158 4 1970 1970 NUM cana-1707 158 5 , , PUNCT cana-1707 158 6 [ [ X cana-1707 158 7 27 27 NUM cana-1707 158 8 ] ] SYM cana-1707 158 9 ) ) PUNCT cana-1707 158 10 definition definition NOUN cana-1707 158 11 3.22 3.22 NUM cana-1707 158 12 . . PUNCT cana-1707 159 1 a a DET cana-1707 159 2 mapping mapping NOUN cana-1707 159 3 𝑇 𝑇 NOUN cana-1707 159 4 : : PUNCT cana-1707 159 5 𝑋 𝑋 PROPN cana-1707 159 6 β†’ β†’ SYM cana-1707 159 7 𝑋 𝑋 PROPN cana-1707 159 8 in in ADP cana-1707 159 9 metric metric ADJ cana-1707 159 10 space space NOUN cana-1707 159 11 ( ( PUNCT cana-1707 159 12 𝑋 𝑋 PROPN cana-1707 159 13 , , PUNCT cana-1707 159 14 𝑑 𝑑 NOUN cana-1707 159 15 ) ) PUNCT cana-1707 159 16 is be AUX cana-1707 159 17 said say VERB cana-1707 159 18 to to PART cana-1707 159 19 be be AUX cana-1707 159 20 bailey bailey NOUN cana-1707 159 21 contractions contraction NOUN cana-1707 159 22 if if SCONJ cana-1707 159 23 there there PRON cana-1707 159 24 exists exist VERB cana-1707 159 25 a a DET cana-1707 159 26 positive positive ADJ cana-1707 159 27 integer integer NOUN cana-1707 159 28 π‘š π‘š NOUN cana-1707 159 29 and and CCONJ cana-1707 159 30 0 0 NUM cana-1707 159 31 < < X cana-1707 159 32 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 159 33 , , PUNCT cana-1707 159 34 𝑦 𝑦 NOUN cana-1707 159 35 ) ) PUNCT cana-1707 159 36 , , PUNCT cana-1707 159 37 such such ADJ cana-1707 159 38 that that PRON cana-1707 159 39 for for ADP cana-1707 159 40 each each DET cana-1707 159 41 π‘₯ π‘₯ PROPN cana-1707 159 42 , , PUNCT cana-1707 159 43 𝑦 𝑦 NOUN cana-1707 159 44 ∈ ∈ PROPN cana-1707 159 45 𝑋 𝑋 PROPN cana-1707 159 46 , , PUNCT cana-1707 159 47 we we PRON cana-1707 159 48 have have VERB cana-1707 159 49 𝑑(π‘“π‘š(π‘₯ 𝑑(π‘“π‘š(π‘₯ NOUN cana-1707 159 50 ) ) PUNCT cana-1707 159 51 , , PUNCT cana-1707 159 52 π‘“π‘š(𝑦 π‘“π‘š(𝑦 NUM cana-1707 159 53 ) ) PUNCT cana-1707 159 54 ) ) PUNCT cana-1707 160 1 ≀ ≀ NUM cana-1707 160 2 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 160 3 , , PUNCT cana-1707 160 4 𝑦 𝑦 NOUN cana-1707 160 5 ) ) PUNCT cana-1707 160 6 . . PUNCT cana-1707 161 1 ( ( PUNCT cana-1707 161 2 bailey bailey NOUN cana-1707 161 3 in in ADP cana-1707 161 4 1966 1966 NUM cana-1707 161 5 , , PUNCT cana-1707 161 6 [ [ X cana-1707 161 7 2 2 NUM cana-1707 161 8 ] ] PUNCT cana-1707 161 9 ) ) PUNCT cana-1707 161 10 definition definition NOUN cana-1707 161 11 3.23 3.23 NUM cana-1707 161 12 . . PUNCT cana-1707 162 1 a a DET cana-1707 162 2 mapping mapping NOUN cana-1707 162 3 𝑇 𝑇 NOUN cana-1707 162 4 : : PUNCT cana-1707 162 5 𝑋 𝑋 PROPN cana-1707 162 6 β†’ β†’ SYM cana-1707 162 7 𝑋 𝑋 PROPN cana-1707 162 8 in in ADP cana-1707 162 9 metric metric ADJ cana-1707 162 10 space space NOUN cana-1707 162 11 ( ( PUNCT cana-1707 162 12 𝑋 𝑋 PROPN cana-1707 162 13 , , PUNCT cana-1707 162 14 𝑑 𝑑 NOUN cana-1707 162 15 ) ) PUNCT cana-1707 162 16 is be AUX cana-1707 162 17 said say VERB cana-1707 162 18 to to PART cana-1707 162 19 be be AUX cana-1707 162 20 berinde berinde VERB cana-1707 162 21 contractions contraction NOUN cana-1707 162 22 if if SCONJ cana-1707 162 23 there there PRON cana-1707 162 24 exists exist VERB cana-1707 162 25 𝛼 𝛼 PRON cana-1707 162 26 ∈ ∈ NOUN cana-1707 162 27 [ [ X cana-1707 162 28 0,1 0,1 NUM cana-1707 162 29 ) ) PUNCT cana-1707 162 30 , , PUNCT cana-1707 162 31 and and CCONJ cana-1707 162 32 Ξ» Ξ» X cana-1707 162 33 β‰₯ β‰₯ NOUN cana-1707 162 34 0 0 NUM cana-1707 162 35 such such ADJ cana-1707 162 36 that that PRON cana-1707 162 37 for for ADP cana-1707 162 38 each each DET cana-1707 162 39 π‘₯ π‘₯ PROPN cana-1707 162 40 , , PUNCT cana-1707 162 41 𝑦 𝑦 NOUN cana-1707 162 42 ∈ ∈ PROPN cana-1707 162 43 𝑋 𝑋 PROPN cana-1707 162 44 , , PUNCT cana-1707 162 45 we we PRON cana-1707 162 46 have have AUX cana-1707 162 47 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ NOUN cana-1707 162 48 , , PUNCT cana-1707 162 49 𝑇𝑦 𝑇𝑦 NOUN cana-1707 162 50 ) ) PUNCT cana-1707 162 51 ≀ ≀ NOUN cana-1707 162 52 𝛼𝑑(π‘₯ 𝛼𝑑(π‘₯ NUM cana-1707 162 53 , , PUNCT cana-1707 162 54 𝑦 𝑦 NOUN cana-1707 162 55 ) ) PUNCT cana-1707 162 56 + + NUM cana-1707 162 57 Ξ»d(y Ξ»d(y NOUN cana-1707 162 58 , , PUNCT cana-1707 162 59 tx tx PROPN cana-1707 162 60 ) ) PUNCT cana-1707 162 61 ( ( PUNCT cana-1707 162 62 berinde berinde VERB cana-1707 162 63 weak weak ADJ cana-1707 162 64 contractions contraction NOUN cana-1707 162 65 in in ADP cana-1707 162 66 2004 2004 NUM cana-1707 162 67 , , PUNCT cana-1707 162 68 [ [ X cana-1707 162 69 3 3 NUM cana-1707 162 70 ] ] SYM cana-1707 162 71 ) ) PUNCT cana-1707 162 72 definition definition NOUN cana-1707 162 73 3.24 3.24 NUM cana-1707 162 74 . . PUNCT cana-1707 163 1 a a DET cana-1707 163 2 mapping mapping NOUN cana-1707 163 3 𝑇 𝑇 NOUN cana-1707 163 4 : : PUNCT cana-1707 163 5 𝑋 𝑋 PROPN cana-1707 163 6 β†’ β†’ SYM cana-1707 163 7 𝑋 𝑋 PROPN cana-1707 163 8 in in ADP cana-1707 163 9 metric metric ADJ cana-1707 163 10 space space NOUN cana-1707 163 11 ( ( PUNCT cana-1707 163 12 𝑋 𝑋 PROPN cana-1707 163 13 , , PUNCT cana-1707 163 14 𝑑 𝑑 NOUN cana-1707 163 15 ) ) PUNCT cana-1707 163 16 is be AUX cana-1707 163 17 said say VERB cana-1707 163 18 to to PART cana-1707 163 19 be be AUX cana-1707 163 20 a a DET cana-1707 163 21 contractive contractive ADJ cana-1707 163 22 mapping mapping NOUN cana-1707 163 23 for for ADP cana-1707 163 24 every every DET cana-1707 163 25 π‘₯ π‘₯ PROPN cana-1707 163 26 , , PUNCT cana-1707 163 27 𝑦 𝑦 NOUN cana-1707 163 28 ∈ ∈ PROPN cana-1707 163 29 𝑋 𝑋 PROPN cana-1707 163 30 , , PUNCT cana-1707 163 31 πœ‘(𝑑(𝑇π‘₯ πœ‘(𝑑(𝑇π‘₯ PROPN cana-1707 163 32 , , PUNCT cana-1707 163 33 𝑇𝑦 𝑇𝑦 PROPN cana-1707 163 34 ) ) PUNCT cana-1707 163 35 ) ) PUNCT cana-1707 163 36 ≀ ≀ NUM cana-1707 163 37 πœ‘(𝑑(π‘₯ πœ‘(𝑑(π‘₯ PROPN cana-1707 163 38 , , PUNCT cana-1707 163 39 𝑦 𝑦 NOUN cana-1707 163 40 ) ) PUNCT cana-1707 163 41 ) ) PUNCT cana-1707 164 1 βˆ’ βˆ’ PROPN cana-1707 164 2 βˆ…(𝑑(π‘₯ βˆ…(𝑑(π‘₯ INTJ cana-1707 164 3 , , PUNCT cana-1707 164 4 𝑦 𝑦 NOUN cana-1707 164 5 ) ) PUNCT cana-1707 164 6 ) ) PUNCT cana-1707 164 7 , , PUNCT cana-1707 164 8 where where SCONJ cana-1707 164 9 πœ‘ πœ‘ NOUN cana-1707 164 10 , , PUNCT cana-1707 164 11 βˆ… βˆ… NOUN cana-1707 164 12 : : PUNCT cana-1707 164 13 [ [ X cana-1707 164 14 0 0 NUM cana-1707 164 15 , , PUNCT cana-1707 164 16 ∞ ∞ PROPN cana-1707 164 17 ) ) PUNCT cana-1707 164 18 β†’ β†’ PUNCT cana-1707 164 19 [ [ X cana-1707 164 20 0 0 NUM cana-1707 164 21 , , PUNCT cana-1707 164 22 ∞ ∞ NUM cana-1707 164 23 ) ) PUNCT cana-1707 164 24 are be AUX cana-1707 164 25 continuous continuous ADJ cana-1707 164 26 , , PUNCT cana-1707 164 27 non non ADJ cana-1707 164 28 - - ADJ cana-1707 164 29 decreasing decrease VERB cana-1707 164 30 , , PUNCT cana-1707 164 31 and and CCONJ cana-1707 164 32 πœ‘βˆ’1({0 πœ‘βˆ’1({0 NOUN cana-1707 164 33 } } PUNCT cana-1707 164 34 ) ) PUNCT cana-1707 165 1 = = SYM cana-1707 165 2 βˆ…βˆ’1({0 βˆ…βˆ’1({0 NOUN cana-1707 165 3 } } PUNCT cana-1707 165 4 ) ) PUNCT cana-1707 166 1 = = PUNCT cana-1707 166 2 { { PUNCT cana-1707 166 3 0 0 NUM cana-1707 166 4 } } PUNCT cana-1707 166 5 . . PUNCT cana-1707 167 1 ( ( PUNCT cana-1707 167 2 dutta dutta PROPN cana-1707 167 3 and and CCONJ cana-1707 167 4 choudhury choudhury PROPN cana-1707 167 5 in in ADP cana-1707 167 6 2008 2008 NUM cana-1707 167 7 , , PUNCT cana-1707 167 8 [ [ X cana-1707 167 9 23 23 NUM cana-1707 167 10 ] ] SYM cana-1707 167 11 ) ) PUNCT cana-1707 167 12 definition definition NOUN cana-1707 167 13 3.35 3.35 NUM cana-1707 167 14 . . PUNCT cana-1707 168 1 let let VERB cana-1707 168 2 ( ( PUNCT cana-1707 168 3 𝑋 𝑋 NOUN cana-1707 168 4 , , PUNCT cana-1707 168 5 𝑑 𝑑 NOUN cana-1707 168 6 ) ) PUNCT cana-1707 168 7 be be VERB cana-1707 168 8 a a DET cana-1707 168 9 metric metric ADJ cana-1707 168 10 space space NOUN cana-1707 168 11 and and CCONJ cana-1707 168 12 𝑇 𝑇 PROPN cana-1707 168 13 : : PUNCT cana-1707 168 14 𝑋 𝑋 PROPN cana-1707 168 15 β†’ β†’ SYM cana-1707 168 16 𝑋 𝑋 PROPN cana-1707 168 17 be be VERB cana-1707 168 18 a a DET cana-1707 168 19 mapping mapping NOUN cana-1707 168 20 . . PUNCT cana-1707 169 1 define define VERB cana-1707 169 2 a a DET cana-1707 169 3 non non ADJ cana-1707 169 4 - - ADJ cana-1707 169 5 increasing increasing ADJ cana-1707 169 6 function function NOUN cana-1707 169 7 πœƒ πœƒ NOUN cana-1707 169 8 : : PUNCT cana-1707 169 9 [ [ X cana-1707 169 10 0 0 NUM cana-1707 169 11 , , PUNCT cana-1707 169 12 1 1 NUM cana-1707 169 13 ) ) PUNCT cana-1707 169 14 β†’ β†’ SYM cana-1707 169 15 ( ( PUNCT cana-1707 169 16 1 1 NUM cana-1707 169 17 2 2 NUM cana-1707 169 18 , , PUNCT cana-1707 169 19 1 1 NUM cana-1707 169 20 ] ] PUNCT cana-1707 169 21 by by ADP cana-1707 169 22 πœƒ(π‘Ÿ πœƒ(π‘Ÿ NOUN cana-1707 169 23 ) ) PUNCT cana-1707 170 1 = = PRON cana-1707 170 2 { { PUNCT cana-1707 170 3 1 1 NUM cana-1707 170 4 , , PUNCT cana-1707 170 5 if if SCONJ cana-1707 170 6 0 0 NUM cana-1707 170 7 ≀ ≀ NUM cana-1707 170 8 r r NOUN cana-1707 170 9 ≀ ≀ NOUN cana-1707 170 10 √5βˆ’ √5βˆ’ NOUN cana-1707 170 11 1 1 NUM cana-1707 170 12 2 2 NUM cana-1707 170 13 , , PUNCT cana-1707 170 14 ( ( PUNCT cana-1707 170 15 1 1 NUM cana-1707 170 16 βˆ’ βˆ’ NOUN cana-1707 170 17 r)rβˆ’2 r)rβˆ’2 NOUN cana-1707 170 18 if if SCONJ cana-1707 170 19 √5βˆ’ √5βˆ’ NOUN cana-1707 170 20 1 1 NUM cana-1707 170 21 2 2 NUM cana-1707 170 22 ≀ ≀ NOUN cana-1707 170 23 r r NOUN cana-1707 170 24 ≀ ≀ NUM cana-1707 170 25 2βˆ’ 2βˆ’ NUM cana-1707 170 26 1 1 NUM cana-1707 170 27 2 2 NUM cana-1707 170 28 ( ( PUNCT cana-1707 170 29 1 1 NUM cana-1707 170 30 + + NOUN cana-1707 170 31 r)βˆ’1 r)βˆ’1 NOUN cana-1707 170 32 if if SCONJ cana-1707 170 33 2βˆ’ 2βˆ’ NUM cana-1707 170 34 1 1 NUM cana-1707 170 35 2 2 NUM cana-1707 170 36 ≀ ≀ NOUN cana-1707 170 37 r r NOUN cana-1707 170 38 < < X cana-1707 170 39 1 1 NUM cana-1707 170 40 . . PUNCT cana-1707 170 41 assume assume VERB cana-1707 170 42 that that SCONJ cana-1707 170 43 there there PRON cana-1707 170 44 exists exist VERB cana-1707 170 45 π‘Ÿ π‘Ÿ X cana-1707 170 46 ∈ ∈ PROPN cana-1707 171 1 [ [ X cana-1707 171 2 0 0 NUM cana-1707 171 3 , , PUNCT cana-1707 171 4 1 1 NUM cana-1707 171 5 ) ) PUNCT cana-1707 171 6 such such ADJ cana-1707 171 7 that that SCONJ cana-1707 171 8 πœƒ(π‘Ÿ πœƒ(π‘Ÿ NOUN cana-1707 171 9 ) ) PUNCT cana-1707 171 10 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 171 11 , , PUNCT cana-1707 171 12 𝑇 𝑇 PROPN cana-1707 171 13 π‘₯ π‘₯ NOUN cana-1707 171 14 ) ) PUNCT cana-1707 171 15 ≀ ≀ NOUN cana-1707 171 16 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 171 17 , , PUNCT cana-1707 171 18 𝑦 𝑦 NOUN cana-1707 171 19 ) ) PUNCT cana-1707 171 20 implies imply VERB cana-1707 171 21 𝑑(𝑇 𝑑(𝑇 PROPN cana-1707 171 22 π‘₯ π‘₯ PROPN cana-1707 171 23 , , PUNCT cana-1707 171 24 𝑇 𝑇 PROPN cana-1707 171 25 𝑦 𝑦 NOUN cana-1707 171 26 ) ) PUNCT cana-1707 171 27 ≀ ≀ NOUN cana-1707 171 28 π‘Ÿ π‘Ÿ NOUN cana-1707 171 29 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 171 30 , , PUNCT cana-1707 171 31 𝑦 𝑦 NOUN cana-1707 171 32 ) ) PUNCT cana-1707 171 33 for for ADP cana-1707 171 34 all all PRON cana-1707 171 35 π‘₯ π‘₯ PROPN cana-1707 171 36 , , PUNCT cana-1707 171 37 𝑦 𝑦 PROPN cana-1707 171 38 ∈ ∈ PROPN cana-1707 171 39 𝑋. 𝑋. PROPN cana-1707 171 40 ( ( PUNCT cana-1707 171 41 suzuki suzuki NOUN cana-1707 171 42 contraction contraction NOUN cana-1707 171 43 in in ADP cana-1707 171 44 2008 2008 NUM cana-1707 171 45 , , PUNCT cana-1707 171 46 [ [ X cana-1707 171 47 54 54 NUM cana-1707 171 48 ] ] SYM cana-1707 171 49 ) ) PUNCT cana-1707 171 50 definition definition NOUN cana-1707 171 51 3.36 3.36 NUM cana-1707 171 52 . . PUNCT cana-1707 172 1 a a DET cana-1707 172 2 mapping mapping NOUN cana-1707 172 3 𝑇 𝑇 NOUN cana-1707 172 4 : : PUNCT cana-1707 172 5 𝑋 𝑋 PROPN cana-1707 172 6 β†’ β†’ SYM cana-1707 172 7 𝑋 𝑋 PROPN cana-1707 172 8 in in ADP cana-1707 172 9 metric metric ADJ cana-1707 172 10 space space NOUN cana-1707 172 11 ( ( PUNCT cana-1707 172 12 𝑋 𝑋 PROPN cana-1707 172 13 , , PUNCT cana-1707 172 14 𝑑 𝑑 NOUN cana-1707 172 15 ) ) PUNCT cana-1707 172 16 is be AUX cana-1707 172 17 said say VERB cana-1707 172 18 to to PART cana-1707 172 19 be be AUX cana-1707 172 20 𝐹 𝐹 PROPN cana-1707 172 21 contractions contraction NOUN cana-1707 172 22 if if SCONJ cana-1707 172 23 there there PRON cana-1707 172 24 exist exist VERB cana-1707 172 25 𝜏 𝜏 DET cana-1707 172 26 > > X cana-1707 172 27 0 0 NUM cana-1707 172 28 such such ADJ cana-1707 172 29 that that DET cana-1707 172 30 for for ADP cana-1707 172 31 every every DET cana-1707 172 32 π‘₯ π‘₯ PROPN cana-1707 172 33 , , PUNCT cana-1707 172 34 𝑦 𝑦 NOUN cana-1707 172 35 ∈ ∈ PROPN cana-1707 172 36 𝑋 𝑋 PROPN cana-1707 172 37 , , PUNCT cana-1707 172 38 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADV cana-1707 172 39 , , PUNCT cana-1707 172 40 𝑇𝑦 𝑇𝑦 PROPN cana-1707 172 41 ) ) PUNCT cana-1707 172 42 > > X cana-1707 172 43 0 0 NUM cana-1707 173 1 π‘–π‘šπ‘π‘™π‘–π‘’π‘  π‘–π‘šπ‘π‘™π‘–π‘’π‘  NOUN cana-1707 173 2 𝜏 𝜏 X cana-1707 174 1 + + NUM cana-1707 174 2 f f X cana-1707 174 3 ( ( PUNCT cana-1707 174 4 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ X cana-1707 174 5 , , PUNCT cana-1707 174 6 𝑇𝑦 𝑇𝑦 PROPN cana-1707 174 7 ) ) PUNCT cana-1707 174 8 ) ) PUNCT cana-1707 174 9 ≀ ≀ NUM cana-1707 174 10 𝐹𝑑(π‘₯ 𝐹𝑑(π‘₯ NOUN cana-1707 174 11 , , PUNCT cana-1707 174 12 𝑦 𝑦 NOUN cana-1707 174 13 ) ) PUNCT cana-1707 174 14 , , PUNCT cana-1707 174 15 where where SCONJ cana-1707 174 16 𝐹 𝐹 PROPN cana-1707 174 17 : : PUNCT cana-1707 174 18 ℝ+ ℝ+ ADP cana-1707 174 19 β†’ β†’ PUNCT cana-1707 174 20 ℝ ℝ PROPN cana-1707 174 21 be be AUX cana-1707 174 22 a a DET cana-1707 174 23 mapping mapping NOUN cana-1707 174 24 satisfying satisfying ADJ cana-1707 174 25 : : PUNCT cana-1707 174 26 ( ( PUNCT cana-1707 174 27 i i NOUN cana-1707 174 28 ) ) PUNCT cana-1707 174 29 𝐹 𝐹 PROPN cana-1707 174 30 is be AUX cana-1707 174 31 strictly strictly ADV cana-1707 174 32 increasing increase VERB cana-1707 174 33 i.e. i.e. ADV cana-1707 174 34 for for ADP cana-1707 174 35 all all DET cana-1707 174 36 𝛼 𝛼 PROPN cana-1707 174 37 , , PUNCT cana-1707 174 38 𝛽 𝛽 NOUN cana-1707 174 39 ∈ ∈ PROPN cana-1707 174 40 ℝ+ ℝ+ PUNCT cana-1707 174 41 such such ADJ cana-1707 174 42 that that SCONJ cana-1707 174 43 𝛼 𝛼 PROPN cana-1707 174 44 < < X cana-1707 174 45 𝛽 𝛽 PROPN cana-1707 174 46 , , PUNCT cana-1707 174 47 𝐹(𝛼 𝐹(𝛼 NUM cana-1707 174 48 ) ) PUNCT cana-1707 174 49 < < X cana-1707 174 50 𝐹(𝛽 𝐹(𝛽 X cana-1707 174 51 ) ) PUNCT cana-1707 174 52 ; ; PUNCT cana-1707 174 53 ( ( PUNCT cana-1707 174 54 ii ii NOUN cana-1707 174 55 ) ) PUNCT cana-1707 174 56 for for ADP cana-1707 174 57 each each DET cana-1707 174 58 sequence sequence NOUN cana-1707 174 59 { { PUNCT cana-1707 174 60 π‘₯𝑛 π‘₯𝑛 NOUN cana-1707 174 61 } } PUNCT cana-1707 174 62 , , PUNCT cana-1707 174 63 𝑛 𝑛 PRON cana-1707 174 64 ∈ ∈ NOUN cana-1707 174 65 𝑁 𝑁 NOUN cana-1707 174 66 of of ADP cana-1707 174 67 positive positive ADJ cana-1707 174 68 numbers number NOUN cana-1707 174 69 π‘™π‘–π‘š π‘™π‘–π‘š VERB cana-1707 174 70 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1707 174 71 π‘₯𝑛 π‘₯𝑛 VERB cana-1707 174 72 = = SYM cana-1707 174 73 0 0 PUNCT cana-1707 175 1 if if SCONJ cana-1707 175 2 and and CCONJ cana-1707 175 3 only only ADV cana-1707 175 4 if if SCONJ cana-1707 175 5 π‘™π‘–π‘š π‘™π‘–π‘š VERB cana-1707 175 6 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1707 175 7 𝐹(π‘₯𝑛 𝐹(π‘₯𝑛 NUM cana-1707 175 8 ) ) PUNCT cana-1707 175 9 = = PUNCT cana-1707 176 1 βˆ’βˆž βˆ’βˆž PROPN cana-1707 176 2 ; ; PUNCT cana-1707 176 3 ( ( PUNCT cana-1707 176 4 iii iii X cana-1707 176 5 ) ) PUNCT cana-1707 176 6 there there PRON cana-1707 176 7 exists exist VERB cana-1707 176 8 π‘˜ π‘˜ PRON cana-1707 176 9 ∈ ∈ PROPN cana-1707 176 10 ( ( PUNCT cana-1707 176 11 0,1 0,1 NOUN cana-1707 176 12 ) ) PUNCT cana-1707 176 13 such such DET cana-1707 177 1 that that SCONJ cana-1707 177 2 π‘™π‘–π‘š π‘™π‘–π‘š VERB cana-1707 177 3 𝛼→0 𝛼→0 ADJ cana-1707 177 4 + + CCONJ cana-1707 177 5 π›Όπ‘˜πΉ(𝛼 π›Όπ‘˜πΉ(𝛼 NOUN cana-1707 177 6 ) ) PUNCT cana-1707 177 7 = = SYM cana-1707 178 1 0 0 X cana-1707 178 2 . . PUNCT cana-1707 179 1 ( ( PUNCT cana-1707 179 2 f f PROPN cana-1707 179 3 contractions contraction NOUN cana-1707 179 4 wardowaski wardowaski NOUN cana-1707 179 5 in in ADP cana-1707 179 6 2012 2012 NUM cana-1707 179 7 , , PUNCT cana-1707 179 8 [ [ X cana-1707 179 9 55 55 NUM cana-1707 179 10 ] ] SYM cana-1707 179 11 ) ) PUNCT cana-1707 179 12 communications communication NOUN cana-1707 179 13 on on ADP cana-1707 179 14 applied apply VERB cana-1707 179 15 nonlinear nonlinear ADJ cana-1707 179 16 analysis analysis NOUN cana-1707 179 17 issn issn NOUN cana-1707 179 18 : : PUNCT cana-1707 179 19 1074 1074 NUM cana-1707 179 20 - - PUNCT cana-1707 179 21 133x 133x NUM cana-1707 179 22 vol vol NOUN cana-1707 179 23 32 32 NUM cana-1707 179 24 no no NOUN cana-1707 179 25 . . NOUN cana-1707 179 26 2 2 NUM cana-1707 179 27 ( ( PUNCT cana-1707 179 28 2025 2025 NUM cana-1707 179 29 ) ) PUNCT cana-1707 179 30 60 60 NUM cana-1707 179 31 https://internationalpubls.com https://internationalpubls.com X cana-1707 179 32 definition definition NOUN cana-1707 179 33 3.37 3.37 NUM cana-1707 179 34 . . PUNCT cana-1707 180 1 a a DET cana-1707 180 2 mapping mapping NOUN cana-1707 180 3 𝑇 𝑇 NOUN cana-1707 180 4 : : PUNCT cana-1707 180 5 𝑋 𝑋 PROPN cana-1707 180 6 β†’ β†’ SYM cana-1707 180 7 𝑋 𝑋 PROPN cana-1707 180 8 in in ADP cana-1707 180 9 metric metric ADJ cana-1707 180 10 space space NOUN cana-1707 180 11 ( ( PUNCT cana-1707 180 12 𝑋 𝑋 PROPN cana-1707 180 13 , , PUNCT cana-1707 180 14 𝑑 𝑑 NOUN cana-1707 180 15 ) ) PUNCT cana-1707 180 16 is be AUX cana-1707 180 17 said say VERB cana-1707 180 18 to to PART cana-1707 180 19 be be AUX cana-1707 180 20 ΞΈ ΞΈ NOUN cana-1707 180 21 - - NOUN cana-1707 180 22 contraction contraction NOUN cana-1707 180 23 if if SCONJ cana-1707 180 24 there there PRON cana-1707 180 25 exist exist VERB cana-1707 180 26 π‘˜ π‘˜ DET cana-1707 180 27 ∈ ∈ PROPN cana-1707 180 28 ( ( PUNCT cana-1707 180 29 0,1 0,1 NOUN cana-1707 180 30 ) ) PUNCT cana-1707 180 31 such such ADJ cana-1707 180 32 that that PRON cana-1707 180 33 for for SCONJ cana-1707 180 34 every every DET cana-1707 180 35 π‘₯ π‘₯ PROPN cana-1707 180 36 , , PUNCT cana-1707 180 37 𝑦 𝑦 NOUN cana-1707 180 38 ∈ ∈ PROPN cana-1707 180 39 𝑋 𝑋 PROPN cana-1707 180 40 , , PUNCT cana-1707 180 41 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ ADV cana-1707 180 42 , , PUNCT cana-1707 180 43 𝑇𝑦 𝑇𝑦 PROPN cana-1707 180 44 ) ) PUNCT cana-1707 180 45 β‰  β‰  PROPN cana-1707 180 46 0 0 NUM cana-1707 180 47 β‡’ β‡’ X cana-1707 180 48 ΞΈ(𝑑(𝑇π‘₯ ΞΈ(𝑑(𝑇π‘₯ PROPN cana-1707 180 49 , , PUNCT cana-1707 180 50 𝑇𝑦 𝑇𝑦 PROPN cana-1707 180 51 ) ) PUNCT cana-1707 180 52 ) ) PUNCT cana-1707 180 53 ≀ ≀ NOUN cana-1707 181 1 [ [ X cana-1707 181 2 ΞΈ(𝑑(π‘₯ ΞΈ(𝑑(π‘₯ PROPN cana-1707 181 3 , , PUNCT cana-1707 181 4 𝑦))]π‘˜ 𝑦))]π‘˜ PROPN cana-1707 181 5 , , PUNCT cana-1707 181 6 where where SCONJ cana-1707 181 7 πœƒ πœƒ X cana-1707 181 8 : : PUNCT cana-1707 181 9 ( ( PUNCT cana-1707 181 10 0 0 NUM cana-1707 181 11 , , PUNCT cana-1707 181 12 ∞ ∞ PROPN cana-1707 181 13 ) ) PUNCT cana-1707 181 14 β†’ β†’ SYM cana-1707 181 15 ( ( PUNCT cana-1707 181 16 1 1 NUM cana-1707 181 17 , , PUNCT cana-1707 181 18 ∞ ∞ PROPN cana-1707 181 19 ) ) PUNCT cana-1707 181 20 is be AUX cana-1707 181 21 a a DET cana-1707 181 22 function function NOUN cana-1707 181 23 satisfying satisfy VERB cana-1707 181 24 the the DET cana-1707 181 25 following follow VERB cana-1707 181 26 conditions condition NOUN cana-1707 181 27 : : PUNCT cana-1707 181 28 ( ( PUNCT cana-1707 181 29 i i NOUN cana-1707 181 30 ) ) PUNCT cana-1707 181 31 ΞΈ ΞΈ PROPN cana-1707 181 32 is be AUX cana-1707 181 33 non non ADJ cana-1707 181 34 - - ADJ cana-1707 181 35 decreasing decrease VERB cana-1707 181 36 ; ; PUNCT cana-1707 181 37 ( ( PUNCT cana-1707 181 38 ii ii NOUN cana-1707 181 39 ) ) PUNCT cana-1707 181 40 for for ADP cana-1707 181 41 each each DET cana-1707 181 42 sequence,{𝑑𝑛 sequence,{𝑑𝑛 NOUN cana-1707 181 43 } } PUNCT cana-1707 181 44 βŠ‚ βŠ‚ PROPN cana-1707 181 45 ( ( PUNCT cana-1707 181 46 0 0 NUM cana-1707 181 47 , , PUNCT cana-1707 181 48 ∞ ∞ PROPN cana-1707 181 49 ) ) PUNCT cana-1707 181 50 , , PUNCT cana-1707 181 51 π‘™π‘–π‘š π‘™π‘–π‘š VERB cana-1707 181 52 π‘›β†’βˆž π‘›β†’βˆž NOUN cana-1707 181 53 ΞΈ(𝑑𝑛 ΞΈ(𝑑𝑛 ADJ cana-1707 181 54 ) ) PUNCT cana-1707 182 1 = = SYM cana-1707 182 2 1if 1if NOUN cana-1707 182 3 and and CCONJ cana-1707 182 4 only only ADV cana-1707 182 5 if if SCONJ cana-1707 182 6 π‘™π‘–π‘š π‘™π‘–π‘š VERB cana-1707 182 7 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1707 182 8 𝑑𝑛 𝑑𝑛 VERB cana-1707 182 9 = = SYM cana-1707 182 10 0 0 NUM cana-1707 182 11 + + NOUN cana-1707 182 12 ; ; PUNCT cana-1707 182 13 ( ( PUNCT cana-1707 182 14 iii iii X cana-1707 182 15 ) ) PUNCT cana-1707 182 16 ΞΈ ΞΈ PROPN cana-1707 182 17 is be AUX cana-1707 182 18 continuous continuous ADJ cana-1707 182 19 on on ADP cana-1707 182 20 ( ( PUNCT cana-1707 182 21 0 0 NUM cana-1707 182 22 , , PUNCT cana-1707 182 23 ∞ ∞ PROPN cana-1707 182 24 ) ) PUNCT cana-1707 182 25 ; ; PUNCT cana-1707 182 26 ( ( PUNCT cana-1707 182 27 iv iv X cana-1707 182 28 ) ) PUNCT cana-1707 182 29 there there PRON cana-1707 182 30 exist exist VERB cana-1707 182 31 π‘Ÿ π‘Ÿ PRON cana-1707 182 32 ∈ ∈ PROPN cana-1707 182 33 ( ( PUNCT cana-1707 182 34 0,1 0,1 NOUN cana-1707 182 35 ) ) PUNCT cana-1707 182 36 , , PUNCT cana-1707 182 37 and and CCONJ cana-1707 182 38 𝑙 𝑙 PRON cana-1707 182 39 ∈ ∈ PROPN cana-1707 182 40 ( ( PUNCT cana-1707 182 41 0 0 NUM cana-1707 182 42 , , PUNCT cana-1707 182 43 ∞]such ∞]such NOUN cana-1707 182 44 that that PRON cana-1707 182 45 π‘™π‘–π‘š π‘™π‘–π‘š VERB cana-1707 182 46 𝑑→0 𝑑→0 ADJ cana-1707 182 47 + + CCONJ cana-1707 182 48 πœƒ(𝑑)βˆ’1 πœƒ(𝑑)βˆ’1 NOUN cana-1707 182 49 π‘‘π‘Ÿ π‘‘π‘Ÿ NOUN cana-1707 182 50 = = PUNCT cana-1707 182 51 𝑙. 𝑙. NOUN cana-1707 182 52 ( ( PUNCT cana-1707 182 53 jleli jleli ADJ cana-1707 182 54 and and CCONJ cana-1707 182 55 samet samet VERB cana-1707 182 56 in in ADP cana-1707 182 57 2014 2014 NUM cana-1707 182 58 , , PUNCT cana-1707 182 59 [ [ X cana-1707 182 60 34 34 NUM cana-1707 182 61 ] ] SYM cana-1707 182 62 ) ) PUNCT cana-1707 182 63 definition definition NOUN cana-1707 182 64 3.38 3.38 NUM cana-1707 182 65 . . PUNCT cana-1707 183 1 let let VERB cana-1707 183 2 ( ( PUNCT cana-1707 183 3 𝑋 𝑋 NOUN cana-1707 183 4 , , PUNCT cana-1707 183 5 𝑑 𝑑 NOUN cana-1707 183 6 ) ) PUNCT cana-1707 183 7 be be VERB cana-1707 183 8 a a DET cana-1707 183 9 metric metric ADJ cana-1707 183 10 space space NOUN cana-1707 183 11 , , PUNCT cana-1707 183 12 and and CCONJ cana-1707 183 13 𝑇 𝑇 PROPN cana-1707 183 14 : : PUNCT cana-1707 183 15 𝑋 𝑋 PROPN cana-1707 183 16 β†’ β†’ SYM cana-1707 183 17 𝑋 𝑋 PROPN cana-1707 183 18 be be VERB cana-1707 183 19 a a DET cana-1707 183 20 mapping mapping NOUN cana-1707 183 21 . . PUNCT cana-1707 184 1 then then ADV cana-1707 184 2 for for ADP cana-1707 184 3 all all DET cana-1707 184 4 x x NOUN cana-1707 184 5 , , PUNCT cana-1707 184 6 y y PROPN cana-1707 184 7 ∈ ∈ PROPN cana-1707 184 8 x x X cana-1707 184 9 , , PUNCT cana-1707 184 10 we we PRON cana-1707 184 11 denote denote VERB cana-1707 184 12 π‘š(𝑇π‘₯ π‘š(𝑇π‘₯ ADP cana-1707 184 13 , , PUNCT cana-1707 184 14 𝑇𝑦 𝑇𝑦 NOUN cana-1707 184 15 ) ) PUNCT cana-1707 184 16 = = PUNCT cana-1707 184 17 π‘Žπ‘‘(π‘₯ π‘Žπ‘‘(π‘₯ X cana-1707 184 18 , , PUNCT cana-1707 184 19 𝑦 𝑦 NOUN cana-1707 184 20 ) ) PUNCT cana-1707 184 21 + + CCONJ cana-1707 184 22 𝑏 𝑏 PROPN cana-1707 184 23 π‘šπ‘Žπ‘₯{𝑑(π‘₯ π‘šπ‘Žπ‘₯{𝑑(π‘₯ PROPN cana-1707 184 24 , , PUNCT cana-1707 184 25 𝑇π‘₯ 𝑇π‘₯ PROPN cana-1707 184 26 ) ) PUNCT cana-1707 184 27 , , PUNCT cana-1707 184 28 𝑑(𝑦 𝑑(𝑦 PROPN cana-1707 184 29 , , PUNCT cana-1707 184 30 𝑇𝑦 𝑇𝑦 PROPN cana-1707 184 31 ) ) PUNCT cana-1707 184 32 } } PUNCT cana-1707 184 33 + + CCONJ cana-1707 184 34 𝑐[𝑑(π‘₯ 𝑐[𝑑(π‘₯ NUM cana-1707 184 35 , , PUNCT cana-1707 184 36 𝑇𝑦 𝑇𝑦 PROPN cana-1707 184 37 ) ) PUNCT cana-1707 184 38 + + NUM cana-1707 184 39 𝑑(𝑦 𝑑(𝑦 NOUN cana-1707 184 40 , , PUNCT cana-1707 184 41 𝑇π‘₯ 𝑇π‘₯ PROPN cana-1707 184 42 ) ) PUNCT cana-1707 184 43 ] ] PUNCT cana-1707 184 44 , , PUNCT cana-1707 184 45 where where SCONJ cana-1707 184 46 π‘Ž π‘Ž X cana-1707 184 47 , , PUNCT cana-1707 184 48 𝑏 𝑏 NOUN cana-1707 184 49 , , PUNCT cana-1707 184 50 and and CCONJ cana-1707 184 51 𝑐 𝑐 PROPN cana-1707 184 52 are be AUX cana-1707 184 53 non non ADJ cana-1707 184 54 - - ADJ cana-1707 184 55 negative negative ADJ cana-1707 184 56 reals real NOUN cana-1707 184 57 such such ADJ cana-1707 184 58 that that SCONJ cana-1707 184 59 π‘Ž π‘Ž PROPN cana-1707 184 60 + + NOUN cana-1707 184 61 𝑏 𝑏 NOUN cana-1707 184 62 + + NUM cana-1707 184 63 2𝑐 2𝑐 NUM cana-1707 185 1 = = PUNCT cana-1707 185 2 π‘Ÿ π‘Ÿ NOUN cana-1707 185 3 π‘€π‘–π‘‘β„Ž π‘€π‘–π‘‘β„Ž VERB cana-1707 185 4 π‘Ÿ π‘Ÿ X cana-1707 185 5 ∈ ∈ PROPN cana-1707 186 1 [ [ X cana-1707 186 2 0 0 NUM cana-1707 186 3 , , PUNCT cana-1707 186 4 1 1 NUM cana-1707 186 5 ) ) PUNCT cana-1707 186 6 . . PUNCT cana-1707 187 1 now now ADV cana-1707 187 2 , , PUNCT cana-1707 187 3 we we PRON cana-1707 187 4 consider consider VERB cana-1707 187 5 the the DET cana-1707 187 6 following follow VERB cana-1707 187 7 generalized generalize VERB cana-1707 187 8 contractive contractive ADJ cana-1707 187 9 condition condition NOUN cana-1707 187 10 : : PUNCT cana-1707 187 11 πœƒ(π‘Ÿ πœƒ(π‘Ÿ NOUN cana-1707 187 12 ) ) PUNCT cana-1707 187 13 π‘šπ‘–π‘›{𝑑(π‘₯ π‘šπ‘–π‘›{𝑑(π‘₯ PROPN cana-1707 187 14 , , PUNCT cana-1707 187 15 𝑇π‘₯ 𝑇π‘₯ PROPN cana-1707 187 16 ) ) PUNCT cana-1707 187 17 , , PUNCT cana-1707 187 18 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 187 19 , , PUNCT cana-1707 187 20 𝑇𝑦 𝑇𝑦 PROPN cana-1707 187 21 ) ) PUNCT cana-1707 187 22 } } PUNCT cana-1707 187 23 ≀ ≀ NUM cana-1707 187 24 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1707 187 25 , , PUNCT cana-1707 187 26 𝑦 𝑦 NOUN cana-1707 187 27 ) ) PUNCT cana-1707 187 28 π‘–π‘šπ‘π‘™π‘–π‘’π‘  π‘–π‘šπ‘π‘™π‘–π‘’π‘  NOUN cana-1707 187 29 𝑑(𝑇π‘₯ 𝑑(𝑇π‘₯ X cana-1707 187 30 , , PUNCT cana-1707 187 31 𝑇𝑦 𝑇𝑦 PROPN cana-1707 187 32 ) ) PUNCT cana-1707 187 33 ≀ ≀ NOUN cana-1707 187 34 π‘š(𝑇π‘₯ π‘š(𝑇π‘₯ ADP cana-1707 187 35 , , PUNCT cana-1707 187 36 𝑇𝑦 𝑇𝑦 PROPN cana-1707 187 37 ) ) PUNCT cana-1707 187 38 . . PUNCT cana-1707 188 1 ( ( PUNCT cana-1707 188 2 chandra chandra PROPN cana-1707 188 3 , , PUNCT cana-1707 188 4 joshi joshi PROPN cana-1707 188 5 , , PUNCT cana-1707 188 6 and and CCONJ cana-1707 188 7 joshi joshi PROPN cana-1707 188 8 in in ADP cana-1707 188 9 2022 2022 NUM cana-1707 188 10 , , PUNCT cana-1707 188 11 ] ] PUNCT cana-1707 188 12 11 11 NUM cana-1707 188 13 ] ] SYM cana-1707 188 14 ) ) PUNCT cana-1707 188 15 4 4 X cana-1707 188 16 . . X cana-1707 188 17 generalization generalization NOUN cana-1707 188 18 of of ADP cana-1707 188 19 contraction contraction NOUN cana-1707 188 20 mapping mapping NOUN cana-1707 188 21 in in ADP cana-1707 188 22 menger menger PROPN cana-1707 188 23 space space NOUN cana-1707 188 24 : : PUNCT cana-1707 188 25 sehgal sehgal PROPN cana-1707 189 1 [ [ X cana-1707 189 2 50 50 NUM cana-1707 189 3 ] ] PUNCT cana-1707 189 4 first first ADV cana-1707 189 5 defined define VERB cana-1707 189 6 probabilistic probabilistic ADJ cana-1707 189 7 contraction contraction NOUN cana-1707 189 8 in in ADP cana-1707 189 9 his his PRON cana-1707 189 10 phd phd NOUN cana-1707 189 11 dissertation dissertation NOUN cana-1707 189 12 in in ADP cana-1707 189 13 1966 1966 NUM cana-1707 189 14 as as ADP cana-1707 189 15 : : PUNCT cana-1707 189 16 definition definition NOUN cana-1707 189 17 4.1 4.1 NUM cana-1707 189 18 : : PUNCT cana-1707 189 19 let let VERB cana-1707 189 20 ( ( PUNCT cana-1707 189 21 𝑋 𝑋 PROPN cana-1707 189 22 , , PUNCT cana-1707 189 23 𝐹 𝐹 PROPN cana-1707 189 24 ) ) PUNCT cana-1707 189 25 be be VERB cana-1707 189 26 a a DET cana-1707 189 27 probabilistic probabilistic ADJ cana-1707 189 28 metric metric ADJ cana-1707 189 29 space space NOUN cana-1707 189 30 . . PUNCT cana-1707 190 1 a a DET cana-1707 190 2 mapping mapping NOUN cana-1707 190 3 𝑇 𝑇 NOUN cana-1707 190 4 : : PUNCT cana-1707 190 5 𝑋 𝑋 PROPN cana-1707 190 6 β†’ β†’ SYM cana-1707 190 7 𝑋 𝑋 PROPN cana-1707 190 8 is be AUX cana-1707 190 9 a a DET cana-1707 190 10 probabilistic probabilistic ADJ cana-1707 190 11 contraction contraction NOUN cana-1707 190 12 or or CCONJ cana-1707 190 13 sehgal sehgal ADJ cana-1707 190 14 contraction contraction NOUN cana-1707 190 15 if if SCONJ cana-1707 190 16 there there PRON cana-1707 190 17 exists exist VERB cana-1707 190 18 π‘˜ π‘˜ PRON cana-1707 190 19 ∈ ∈ PROPN cana-1707 190 20 ( ( PUNCT cana-1707 190 21 0,1 0,1 NOUN cana-1707 190 22 ) ) PUNCT cana-1707 190 23 such such ADJ cana-1707 190 24 that that SCONJ cana-1707 190 25 𝐹𝑇𝑝,π‘‡π‘ž(π‘˜π‘‘ 𝐹𝑇𝑝,π‘‡π‘ž(π‘˜π‘‘ NOUN cana-1707 190 26 ) ) PUNCT cana-1707 190 27 β‰₯ β‰₯ NOUN cana-1707 190 28 𝐹𝑝,π‘ž(𝑑 𝐹𝑝,π‘ž(𝑑 VERB cana-1707 190 29 ) ) PUNCT cana-1707 190 30 for for ADP cana-1707 190 31 all all DET cana-1707 190 32 𝑝 𝑝 NOUN cana-1707 190 33 , , PUNCT cana-1707 190 34 π‘ž π‘ž PROPN cana-1707 190 35 ∈ ∈ PROPN cana-1707 190 36 𝑋 𝑋 PROPN cana-1707 190 37 , , PUNCT cana-1707 190 38 and and CCONJ cana-1707 190 39 𝑑 𝑑 X cana-1707 190 40 > > X cana-1707 190 41 0 0 X cana-1707 190 42 . . PUNCT cana-1707 190 43 hicks hicks PROPN cana-1707 191 1 [ [ X cana-1707 191 2 32 32 NUM cana-1707 191 3 ] ] PUNCT cana-1707 191 4 defined define VERB cana-1707 191 5 another another DET cana-1707 191 6 contraction contraction NOUN cana-1707 191 7 mapping mapping NOUN cana-1707 191 8 in in ADP cana-1707 191 9 probabilistic probabilistic ADJ cana-1707 191 10 metric metric ADJ cana-1707 191 11 space space NOUN cana-1707 191 12 in in ADP cana-1707 191 13 1983 1983 NUM cana-1707 191 14 : : PUNCT cana-1707 191 15 definition definition NOUN cana-1707 191 16 4.2 4.2 NUM cana-1707 191 17 : : PUNCT cana-1707 191 18 a a DET cana-1707 191 19 mapping mapping NOUN cana-1707 191 20 𝑇 𝑇 NOUN cana-1707 191 21 : : PUNCT cana-1707 191 22 𝑋 𝑋 PROPN cana-1707 191 23 β†’ β†’ SYM cana-1707 191 24 𝑋 𝑋 PROPN cana-1707 191 25 in in ADP cana-1707 191 26 probabilistic probabilistic ADJ cana-1707 191 27 metric metric ADJ cana-1707 191 28 space space NOUN cana-1707 191 29 ( ( PUNCT cana-1707 191 30 𝑋 𝑋 PROPN cana-1707 191 31 , , PUNCT cana-1707 191 32 𝐹 𝐹 PROPN cana-1707 191 33 ) ) PUNCT cana-1707 191 34 is be AUX cana-1707 191 35 said say VERB cana-1707 191 36 to to PART cana-1707 191 37 be be AUX cana-1707 191 38 hicks hicks PROPN cana-1707 191 39 contraction contraction NOUN cana-1707 191 40 or or CCONJ cana-1707 191 41 c c NOUN cana-1707 191 42 - - PUNCT cana-1707 191 43 contraction contraction NOUN cana-1707 191 44 if if SCONJ cana-1707 191 45 there there PRON cana-1707 191 46 exists exist VERB cana-1707 191 47 π‘˜ π‘˜ PRON cana-1707 191 48 ∈ ∈ PROPN cana-1707 191 49 ( ( PUNCT cana-1707 191 50 0,1 0,1 NOUN cana-1707 191 51 ) ) PUNCT cana-1707 191 52 such such ADJ cana-1707 191 53 that that PRON cana-1707 191 54 for for ADP cana-1707 191 55 every every DET cana-1707 191 56 𝑝 𝑝 NOUN cana-1707 191 57 , , PUNCT cana-1707 191 58 π‘ž π‘ž PROPN cana-1707 191 59 ∈ ∈ PROPN cana-1707 191 60 𝑋 𝑋 PROPN cana-1707 191 61 , , PUNCT cana-1707 191 62 and and CCONJ cana-1707 191 63 every every DET cana-1707 191 64 𝑑 𝑑 X cana-1707 191 65 > > X cana-1707 191 66 0 0 NUM cana-1707 191 67 : : PUNCT cana-1707 191 68 𝐹𝑝,π‘ž(𝑑 𝐹𝑝,π‘ž(𝑑 ADJ cana-1707 191 69 ) ) PUNCT cana-1707 191 70 > > SYM cana-1707 191 71 1 1 NUM cana-1707 191 72 βˆ’ βˆ’ PROPN cana-1707 191 73 𝑑 𝑑 PROPN cana-1707 191 74 β‡’ β‡’ NOUN cana-1707 191 75 𝐹𝑇𝑝 𝐹𝑇𝑝 NOUN cana-1707 191 76 , , PUNCT cana-1707 191 77 π‘‡π‘ž(π‘˜π‘‘ π‘‡π‘ž(π‘˜π‘‘ PROPN cana-1707 191 78 ) ) PUNCT cana-1707 191 79 > > X cana-1707 191 80 1 1 NUM cana-1707 191 81 βˆ’ βˆ’ PROPN cana-1707 191 82 π‘˜π‘‘. π‘˜π‘‘. PROPN cana-1707 191 83 a a DET cana-1707 191 84 weaker weak ADJ cana-1707 191 85 form form NOUN cana-1707 191 86 of of ADP cana-1707 191 87 hicks hicks PROPN cana-1707 191 88 contraction contraction PROPN cana-1707 191 89 was be AUX cana-1707 191 90 introduced introduce VERB cana-1707 191 91 by by ADP cana-1707 191 92 d. d. PROPN cana-1707 191 93 mihet mihet PROPN cana-1707 192 1 [ [ X cana-1707 192 2 39 39 NUM cana-1707 192 3 ] ] PUNCT cana-1707 192 4 in in ADP cana-1707 192 5 2005 2005 NUM cana-1707 192 6 as as ADP cana-1707 192 7 : : PUNCT cana-1707 192 8 definition definition NOUN cana-1707 192 9 4.3 4.3 NUM cana-1707 192 10 : : PUNCT cana-1707 192 11 a a DET cana-1707 192 12 mapping mapping NOUN cana-1707 192 13 𝑇 𝑇 NOUN cana-1707 192 14 : : PUNCT cana-1707 192 15 𝑋 𝑋 PROPN cana-1707 192 16 β†’ β†’ SYM cana-1707 192 17 𝑋 𝑋 PROPN cana-1707 192 18 is be AUX cana-1707 192 19 said say VERB cana-1707 192 20 to to PART cana-1707 192 21 be be AUX cana-1707 192 22 weak weak ADJ cana-1707 192 23 hicks hick NOUN cana-1707 192 24 contraction contraction NOUN cana-1707 192 25 ( ( PUNCT cana-1707 192 26 w w NOUN cana-1707 192 27 - - PUNCT cana-1707 192 28 h h NOUN cana-1707 192 29 contraction contraction NOUN cana-1707 192 30 ) ) PUNCT cana-1707 192 31 if if SCONJ cana-1707 192 32 there there PRON cana-1707 192 33 exists exist VERB cana-1707 192 34 π‘˜ π‘˜ PRON cana-1707 192 35 ∈ ∈ PROPN cana-1707 192 36 ( ( PUNCT cana-1707 192 37 0,1 0,1 NOUN cana-1707 192 38 ) ) PUNCT cana-1707 192 39 such such ADJ cana-1707 192 40 that that SCONJ cana-1707 192 41 , , PUNCT cana-1707 192 42 for for ADP cana-1707 192 43 all all DET cana-1707 192 44 𝑝 𝑝 NOUN cana-1707 192 45 , , PUNCT cana-1707 192 46 π‘ž π‘ž X cana-1707 192 47 ∈ ∈ PROPN cana-1707 192 48 𝑆. 𝑆. PROPN cana-1707 192 49 ( ( PUNCT cana-1707 192 50 𝑀 𝑀 ADP cana-1707 192 51 βˆ’ βˆ’ PROPN cana-1707 192 52 𝐻 𝐻 PROPN cana-1707 192 53 ): ): PUNCT cana-1707 192 54 𝑑 𝑑 PROPN cana-1707 192 55 ∈ ∈ PROPN cana-1707 192 56 ( ( PUNCT cana-1707 192 57 0,1 0,1 NOUN cana-1707 192 58 ) ) PUNCT cana-1707 192 59 , , PUNCT cana-1707 192 60 𝐹𝑝,π‘ž(𝑑 𝐹𝑝,π‘ž(𝑑 NOUN cana-1707 192 61 ) ) PUNCT cana-1707 192 62 > > SYM cana-1707 192 63 1 1 NUM cana-1707 192 64 βˆ’ βˆ’ PROPN cana-1707 192 65 𝑑 𝑑 PROPN cana-1707 192 66 β‡’ β‡’ NOUN cana-1707 192 67 𝐹𝑇𝑝 𝐹𝑇𝑝 NOUN cana-1707 192 68 , , PUNCT cana-1707 192 69 π‘‡π‘ž(π‘˜π‘‘ π‘‡π‘ž(π‘˜π‘‘ PROPN cana-1707 192 70 ) ) PUNCT cana-1707 192 71 > > X cana-1707 192 72 1 1 NUM cana-1707 192 73 βˆ’ βˆ’ PROPN cana-1707 192 74 π‘˜π‘‘. π‘˜π‘‘. PROPN cana-1707 192 75 example example NOUN cana-1707 192 76 4.1 4.1 NUM cana-1707 192 77 : : PUNCT cana-1707 192 78 let let VERB cana-1707 192 79 𝑋 𝑋 PROPN cana-1707 192 80 = = PUNCT cana-1707 193 1 [ [ X cana-1707 193 2 0 0 NUM cana-1707 193 3 , , PUNCT cana-1707 193 4 ∞ ∞ PROPN cana-1707 193 5 ) ) PUNCT cana-1707 193 6 , , PUNCT cana-1707 193 7 and and CCONJ cana-1707 193 8 𝐹𝑝,π‘ž(𝑑 𝐹𝑝,π‘ž(𝑑 ADJ cana-1707 193 9 ) ) PUNCT cana-1707 193 10 = = SYM cana-1707 193 11 min min PROPN cana-1707 193 12 ( ( PUNCT cana-1707 193 13 𝑝,π‘ž 𝑝,π‘ž NOUN cana-1707 193 14 ) ) PUNCT cana-1707 193 15 max max PROPN cana-1707 193 16 ( ( PUNCT cana-1707 193 17 𝑝,π‘ž 𝑝,π‘ž NOUN cana-1707 193 18 ) ) PUNCT cana-1707 193 19 , , PUNCT cana-1707 193 20 βˆ€ βˆ€ X cana-1707 193 21 𝑝 𝑝 NOUN cana-1707 193 22 , , PUNCT cana-1707 193 23 π‘ž π‘ž PROPN cana-1707 193 24 ∈ ∈ PROPN cana-1707 193 25 𝑋 𝑋 PROPN cana-1707 193 26 , , PUNCT cana-1707 193 27 𝑝 𝑝 PROPN cana-1707 193 28 β‰  β‰  PROPN cana-1707 193 29 π‘ž. π‘ž. NOUN cana-1707 193 30 then then ADV cana-1707 193 31 , , PUNCT cana-1707 193 32 ( ( PUNCT cana-1707 193 33 𝑋 𝑋 PROPN cana-1707 193 34 , , PUNCT cana-1707 193 35 𝐹 𝐹 PROPN cana-1707 193 36 , , PUNCT cana-1707 193 37 𝑇 𝑇 PROPN cana-1707 193 38 ) ) PUNCT cana-1707 193 39 be be VERB cana-1707 193 40 a a DET cana-1707 193 41 complete complete ADJ cana-1707 193 42 menger menger NOUN cana-1707 193 43 space space NOUN cana-1707 193 44 under under ADP cana-1707 193 45 triangular triangular NOUN cana-1707 193 46 norm norm NOUN cana-1707 193 47 𝑇 𝑇 PROPN cana-1707 193 48 = = SYM cana-1707 193 49 𝑇𝑃 𝑇𝑃 PROPN cana-1707 193 50 > > X cana-1707 193 51 𝑇𝐿. 𝑇𝐿. ADP cana-1707 193 52 it it PRON cana-1707 193 53 can can AUX cana-1707 193 54 be be AUX cana-1707 193 55 seen see VERB cana-1707 193 56 that that SCONJ cana-1707 193 57 the the DET cana-1707 193 58 mapping mapping NOUN cana-1707 193 59 𝑔 𝑔 NOUN cana-1707 193 60 : : PUNCT cana-1707 193 61 𝑋 𝑋 PROPN cana-1707 193 62 β†’ β†’ SYM cana-1707 193 63 𝑋 𝑋 PROPN cana-1707 193 64 , , PUNCT cana-1707 193 65 𝑔(π‘₯ 𝑔(π‘₯ PROPN cana-1707 193 66 ) ) PUNCT cana-1707 193 67 = = PRON cana-1707 193 68 { { PUNCT cana-1707 194 1 1,𝑖𝑓 1,𝑖𝑓 INTJ cana-1707 194 2 π‘₯>0 π‘₯>0 PROPN cana-1707 194 3 0,𝑖𝑓 0,𝑖𝑓 PROPN cana-1707 195 1 π‘₯=0 π‘₯=0 X cana-1707 195 2 is be AUX cana-1707 195 3 a a DET cana-1707 195 4 𝑀 𝑀 NOUN cana-1707 195 5 βˆ’ βˆ’ NOUN cana-1707 196 1 𝐻 𝐻 PROPN cana-1707 196 2 contraction contraction NOUN cana-1707 196 3 for for ADP cana-1707 196 4 π‘˜ π‘˜ PROPN cana-1707 196 5 ∈ ∈ PROPN cana-1707 196 6 ( ( PUNCT cana-1707 196 7 0,1 0,1 NUM cana-1707 196 8 ) ) PUNCT cana-1707 196 9 . . PUNCT cana-1707 197 1 communications communication NOUN cana-1707 197 2 on on ADP cana-1707 197 3 applied apply VERB cana-1707 197 4 nonlinear nonlinear ADJ cana-1707 197 5 analysis analysis NOUN cana-1707 197 6 issn issn NOUN cana-1707 197 7 : : PUNCT cana-1707 197 8 1074 1074 NUM cana-1707 197 9 - - PUNCT cana-1707 197 10 133x 133x NUM cana-1707 197 11 vol vol NOUN cana-1707 197 12 32 32 NUM cana-1707 197 13 no no NOUN cana-1707 197 14 . . NOUN cana-1707 197 15 2 2 NUM cana-1707 197 16 ( ( PUNCT cana-1707 197 17 2025 2025 NUM cana-1707 197 18 ) ) PUNCT cana-1707 197 19 61 61 NUM cana-1707 197 20 https://internationalpubls.com https://internationalpubls.com SYM cana-1707 197 21 4.1 4.1 NUM cana-1707 197 22 generalized generalized ADJ cana-1707 197 23 form form NOUN cana-1707 197 24 of of ADP cana-1707 197 25 probabilistic probabilistic ADJ cana-1707 197 26 contraction contraction NOUN cana-1707 197 27 : : PUNCT cana-1707 197 28 a a DET cana-1707 197 29 probabilistic probabilistic ADJ cana-1707 197 30 ( ( PUNCT cana-1707 197 31 m m PROPN cana-1707 197 32 , , PUNCT cana-1707 197 33 k k NOUN cana-1707 197 34 ) ) PUNCT cana-1707 197 35 contraction contraction NOUN cana-1707 197 36 is be AUX cana-1707 197 37 a a DET cana-1707 197 38 generalization generalization NOUN cana-1707 197 39 of of ADP cana-1707 197 40 sehgal sehgal ADJ cana-1707 197 41 contraction contraction NOUN cana-1707 197 42 , , PUNCT cana-1707 197 43 where where SCONJ cana-1707 197 44 π‘š π‘š PROPN cana-1707 197 45 β‰₯ β‰₯ NUM cana-1707 197 46 1 1 NUM cana-1707 197 47 and and CCONJ cana-1707 197 48 π‘˜ π‘˜ PRON cana-1707 197 49 ∈ ∈ PROPN cana-1707 197 50 ( ( PUNCT cana-1707 197 51 0,1 0,1 NUM cana-1707 197 52 ) ) PUNCT cana-1707 197 53 and and CCONJ cana-1707 197 54 is be AUX cana-1707 197 55 defined define VERB cana-1707 197 56 as as ADP cana-1707 197 57 : : PUNCT cana-1707 197 58 definition definition NOUN cana-1707 197 59 4.1.1 4.1.1 NUM cana-1707 197 60 : : PUNCT cana-1707 198 1 [ [ X cana-1707 198 2 16 16 NUM cana-1707 198 3 ] ] X cana-1707 198 4 if if SCONJ cana-1707 198 5 ( ( PUNCT cana-1707 198 6 𝑋 𝑋 PROPN cana-1707 198 7 , , PUNCT cana-1707 198 8 𝐹 𝐹 PROPN cana-1707 198 9 ) ) PUNCT cana-1707 198 10 is be AUX cana-1707 198 11 a a DET cana-1707 198 12 pm pm NOUN cana-1707 198 13 space space NOUN cana-1707 198 14 , , PUNCT cana-1707 198 15 π‘š π‘š X cana-1707 198 16 β‰₯ β‰₯ NUM cana-1707 198 17 1 1 NUM cana-1707 198 18 and and CCONJ cana-1707 198 19 π‘˜ π‘˜ DET cana-1707 198 20 ∈ ∈ PROPN cana-1707 198 21 ( ( PUNCT cana-1707 198 22 0,1 0,1 NOUN cana-1707 198 23 ) ) PUNCT cana-1707 198 24 , , PUNCT cana-1707 198 25 a a DET cana-1707 198 26 function function NOUN cana-1707 198 27 𝑓 𝑓 NOUN cana-1707 198 28 : : PUNCT cana-1707 198 29 𝑋 𝑋 PROPN cana-1707 198 30 β†’ β†’ SYM cana-1707 198 31 𝑋 𝑋 PROPN cana-1707 198 32 is be AUX cana-1707 198 33 called call VERB cana-1707 198 34 probabilistic probabilistic ADJ cana-1707 198 35 ( ( PUNCT cana-1707 198 36 m m PROPN cana-1707 198 37 , , PUNCT cana-1707 198 38 k)-contraction k)-contraction NOUN cana-1707 198 39 if if SCONJ cana-1707 198 40 for for ADP cana-1707 198 41 any any DET cana-1707 198 42 𝑝 𝑝 NOUN cana-1707 198 43 , , PUNCT cana-1707 198 44 π‘ž π‘ž PROPN cana-1707 198 45 ∈ ∈ PROPN cana-1707 198 46 𝑋 𝑋 PROPN cana-1707 198 47 there there PRON cana-1707 198 48 is be VERB cana-1707 198 49 an an DET cana-1707 198 50 𝑖 𝑖 NOUN cana-1707 198 51 with with ADP cana-1707 198 52 1 1 NUM cana-1707 198 53 < < X cana-1707 198 54 𝑖 𝑖 X cana-1707 198 55 < < X cana-1707 198 56 π‘š π‘š ADP cana-1707 198 57 such such ADJ cana-1707 198 58 that that PRON cana-1707 198 59 for for SCONJ cana-1707 198 60 every every DET cana-1707 198 61 𝑑 𝑑 X cana-1707 198 62 > > X cana-1707 198 63 0 0 NUM cana-1707 198 64 , , PUNCT cana-1707 198 65 𝐹𝑓𝑖 𝐹𝑓𝑖 NOUN cana-1707 198 66 𝑝,π‘“π‘–π‘ž(π‘˜π‘–π‘‘ 𝑝,π‘“π‘–π‘ž(π‘˜π‘–π‘‘ NOUN cana-1707 198 67 ) ) PUNCT cana-1707 198 68 β‰₯ β‰₯ NOUN cana-1707 198 69 𝐹𝑝,π‘ž(𝑑 𝐹𝑝,π‘ž(𝑑 NOUN cana-1707 198 70 ) ) PUNCT cana-1707 198 71 . . PUNCT cana-1707 199 1 if if SCONJ cana-1707 199 2 π‘š π‘š PROPN cana-1707 199 3 = = SYM cana-1707 199 4 1 1 NUM cana-1707 199 5 and and CCONJ cana-1707 199 6 π‘˜ π‘˜ PRON cana-1707 199 7 ∈ ∈ PROPN cana-1707 199 8 ( ( PUNCT cana-1707 199 9 0,1 0,1 NUM cana-1707 199 10 ) ) PUNCT cana-1707 199 11 then then ADV cana-1707 199 12 a a DET cana-1707 199 13 probabilistic probabilistic ADJ cana-1707 199 14 ( ( PUNCT cana-1707 199 15 1 1 NUM cana-1707 199 16 , , PUNCT cana-1707 199 17 π‘˜)-sehgal π‘˜)-sehgal ADJ cana-1707 199 18 contraction contraction NOUN cana-1707 199 19 , , PUNCT cana-1707 199 20 𝑓 𝑓 PRON cana-1707 199 21 is be AUX cana-1707 199 22 a a DET cana-1707 199 23 probabilistic probabilistic ADJ cana-1707 199 24 sehgal sehgal ADJ cana-1707 199 25 contraction contraction NOUN cana-1707 199 26 . . PUNCT cana-1707 200 1 following follow VERB cana-1707 200 2 is be AUX cana-1707 200 3 a a DET cana-1707 200 4 generalization generalization NOUN cana-1707 200 5 of of ADP cana-1707 200 6 hicks hicks PROPN cana-1707 200 7 c c NOUN cana-1707 200 8 - - PUNCT cana-1707 200 9 contraction contraction NOUN cana-1707 200 10 : : PUNCT cana-1707 200 11 definition definition NOUN cana-1707 200 12 4.1.2 4.1.2 NUM cana-1707 200 13 : : PUNCT cana-1707 201 1 [ [ X cana-1707 201 2 16 16 NUM cana-1707 201 3 ] ] X cana-1707 201 4 if if SCONJ cana-1707 201 5 ( ( PUNCT cana-1707 201 6 𝑋 𝑋 PROPN cana-1707 201 7 , , PUNCT cana-1707 201 8 πœ‘ πœ‘ PROPN cana-1707 201 9 ) ) PUNCT cana-1707 201 10 is be AUX cana-1707 201 11 a a DET cana-1707 201 12 pm pm NOUN cana-1707 201 13 space space NOUN cana-1707 201 14 , , PUNCT cana-1707 201 15 π‘š π‘š X cana-1707 201 16 β‰₯ β‰₯ NUM cana-1707 201 17 1 1 NUM cana-1707 201 18 and and CCONJ cana-1707 201 19 π‘˜ π‘˜ DET cana-1707 201 20 ∈ ∈ PROPN cana-1707 201 21 ( ( PUNCT cana-1707 201 22 0,1 0,1 NOUN cana-1707 201 23 ) ) PUNCT cana-1707 201 24 , , PUNCT cana-1707 201 25 a a DET cana-1707 201 26 function function NOUN cana-1707 201 27 𝑓 𝑓 NOUN cana-1707 201 28 : : PUNCT cana-1707 201 29 𝑋 𝑋 PROPN cana-1707 201 30 β†’ β†’ SYM cana-1707 201 31 𝑋 𝑋 PROPN cana-1707 201 32 is be AUX cana-1707 201 33 called call VERB cana-1707 201 34 a a DET cana-1707 201 35 ( ( PUNCT cana-1707 201 36 m m NOUN cana-1707 201 37 , , PUNCT cana-1707 201 38 k)-c k)-c ADJ cana-1707 201 39 - - PUNCT cana-1707 201 40 contraction contraction NOUN cana-1707 201 41 if if SCONJ cana-1707 201 42 for for ADP cana-1707 201 43 any any DET cana-1707 201 44 𝑝 𝑝 NOUN cana-1707 201 45 , , PUNCT cana-1707 201 46 π‘ž π‘ž PROPN cana-1707 201 47 ∈ ∈ PROPN cana-1707 201 48 𝑋 𝑋 PROPN cana-1707 201 49 there there PRON cana-1707 201 50 is be VERB cana-1707 201 51 an an DET cana-1707 201 52 𝑖 𝑖 NOUN cana-1707 201 53 with with ADP cana-1707 201 54 𝑙 𝑙 X cana-1707 201 55 < < X cana-1707 201 56 𝑖 𝑖 X cana-1707 201 57 < < X cana-1707 201 58 π‘š π‘š ADP cana-1707 201 59 such such ADJ cana-1707 201 60 that that PRON cana-1707 201 61 for for ADP cana-1707 201 62 every every DET cana-1707 201 63 𝑑 𝑑 X cana-1707 201 64 > > X cana-1707 201 65 0 0 NUM cana-1707 201 66 . . PUNCT cana-1707 201 67 𝐹𝑝,π‘ž(𝑑 𝐹𝑝,π‘ž(𝑑 NOUN cana-1707 201 68 ) ) PUNCT cana-1707 201 69 > > SYM cana-1707 201 70 1 1 NUM cana-1707 201 71 βˆ’ βˆ’ PROPN cana-1707 201 72 𝑑 𝑑 PROPN cana-1707 201 73 β‡’ β‡’ NOUN cana-1707 201 74 𝐹𝑓𝑖 𝐹𝑓𝑖 PROPN cana-1707 201 75 𝑝,π‘“π‘–π‘ž(π‘˜π‘–π‘‘ 𝑝,π‘“π‘–π‘ž(π‘˜π‘–π‘‘ NUM cana-1707 201 76 ) ) PUNCT cana-1707 201 77 > > X cana-1707 201 78 1 1 NUM cana-1707 201 79 βˆ’ βˆ’ NOUN cana-1707 201 80 π‘˜π‘–π‘‘. π‘˜π‘–π‘‘. NOUN cana-1707 201 81 if if SCONJ cana-1707 201 82 π‘š π‘š PROPN cana-1707 201 83 = = SYM cana-1707 201 84 1 1 NUM cana-1707 201 85 and and CCONJ cana-1707 201 86 π‘˜ π‘˜ PRON cana-1707 201 87 ∈ ∈ PROPN cana-1707 201 88 ( ( PUNCT cana-1707 201 89 0,1 0,1 NUM cana-1707 201 90 ) ) PUNCT cana-1707 201 91 then then ADV cana-1707 201 92 a a DET cana-1707 201 93 probabilistic probabilistic ADJ cana-1707 201 94 ( ( PUNCT cana-1707 201 95 1 1 NUM cana-1707 201 96 , , PUNCT cana-1707 201 97 π‘˜)-c π‘˜)-c NOUN cana-1707 201 98 - - PUNCT cana-1707 201 99 contraction contraction NOUN cana-1707 201 100 𝑓 𝑓 NOUN cana-1707 201 101 is be AUX cana-1707 201 102 a a DET cana-1707 201 103 probabilistic probabilistic ADJ cana-1707 201 104 c c NOUN cana-1707 201 105 - - PUNCT cana-1707 201 106 contraction contraction NOUN cana-1707 201 107 . . PUNCT cana-1707 202 1 definition definition NOUN cana-1707 202 2 4.1.3 4.1.3 NUM cana-1707 202 3 : : PUNCT cana-1707 203 1 [ [ X cana-1707 203 2 26 26 NUM cana-1707 203 3 ] ] PUNCT cana-1707 203 4 let let VERB cana-1707 203 5 𝑓 𝑓 PRON cana-1707 203 6 , , PUNCT cana-1707 203 7 𝑔 𝑔 PROPN cana-1707 203 8 be be AUX cana-1707 203 9 two two NUM cana-1707 203 10 mappings mapping NOUN cana-1707 203 11 defined define VERB cana-1707 203 12 on on ADP cana-1707 203 13 a a DET cana-1707 203 14 menger menger NOUN cana-1707 203 15 space space NOUN cana-1707 203 16 ( ( PUNCT cana-1707 203 17 𝑋 𝑋 PROPN cana-1707 203 18 , , PUNCT cana-1707 203 19 𝐹 𝐹 PROPN cana-1707 203 20 , , PUNCT cana-1707 203 21 𝑇 𝑇 PROPN cana-1707 203 22 ) ) PUNCT cana-1707 203 23 with with ADP cana-1707 203 24 values value NOUN cana-1707 203 25 into into ADP cana-1707 203 26 itself itself PRON cana-1707 203 27 , , PUNCT cana-1707 203 28 and and CCONJ cana-1707 203 29 let let VERB cana-1707 203 30 us we PRON cana-1707 203 31 suppose suppose VERB cana-1707 203 32 that that SCONJ cana-1707 203 33 𝑔 𝑔 PROPN cana-1707 203 34 is be AUX cana-1707 203 35 bijective bijective ADJ cana-1707 203 36 . . PUNCT cana-1707 204 1 the the DET cana-1707 204 2 mapping mapping NOUN cana-1707 204 3 𝑓 𝑓 NOUN cana-1707 204 4 is be AUX cana-1707 204 5 called call VERB cana-1707 204 6 a a DET cana-1707 204 7 probabilistic probabilistic ADJ cana-1707 204 8 g g NOUN cana-1707 204 9 - - PUNCT cana-1707 204 10 contraction contraction NOUN cana-1707 204 11 with with ADP cana-1707 204 12 a a DET cana-1707 204 13 constant constant ADJ cana-1707 204 14 π‘˜ π‘˜ PRON cana-1707 204 15 ∈ ∈ PROPN cana-1707 204 16 ( ( PUNCT cana-1707 204 17 0,1 0,1 NUM cana-1707 204 18 ) ) PUNCT cana-1707 204 19 if if SCONJ cana-1707 204 20 𝑑 𝑑 X cana-1707 204 21 > > X cana-1707 204 22 0 0 NUM cana-1707 204 23 and and CCONJ cana-1707 204 24 𝐹𝑔(π‘₯),𝑔(𝑦)(𝑑 𝐹𝑔(π‘₯),𝑔(𝑦)(𝑑 NOUN cana-1707 204 25 ) ) PUNCT cana-1707 204 26 > > X cana-1707 204 27 1 1 NUM cana-1707 204 28 βˆ’ βˆ’ PROPN cana-1707 204 29 𝑑 𝑑 PROPN cana-1707 204 30 impies impi NOUN cana-1707 204 31 𝐹𝑓(π‘₯),𝑓(𝑦)(π‘˜π‘‘ 𝐹𝑓(π‘₯),𝑓(𝑦)(π‘˜π‘‘ NOUN cana-1707 204 32 ) ) PUNCT cana-1707 204 33 > > X cana-1707 205 1 1 1 NUM cana-1707 205 2 βˆ’ βˆ’ PROPN cana-1707 205 3 π‘˜π‘‘. π‘˜π‘‘. PROPN cana-1707 205 4 conclusion conclusion NOUN cana-1707 205 5 : : PUNCT cana-1707 205 6 this this DET cana-1707 205 7 study study NOUN cana-1707 205 8 discusses discuss VERB cana-1707 205 9 only only ADV cana-1707 205 10 banach banach NOUN cana-1707 205 11 contraction contraction NOUN cana-1707 205 12 's 's PART cana-1707 205 13 extended extended ADJ cana-1707 205 14 and and CCONJ cana-1707 205 15 generalized generalized ADJ cana-1707 205 16 form form NOUN cana-1707 205 17 in in ADP cana-1707 205 18 complete complete ADJ cana-1707 205 19 metric metric ADJ cana-1707 205 20 space space NOUN cana-1707 205 21 and and CCONJ cana-1707 205 22 menger menger NOUN cana-1707 205 23 space space NOUN cana-1707 205 24 so so SCONJ cana-1707 205 25 that that SCONJ cana-1707 205 26 each each DET cana-1707 205 27 mapping mapping NOUN cana-1707 205 28 establishes establish VERB cana-1707 205 29 unique unique ADJ cana-1707 205 30 fixed fix VERB cana-1707 205 31 point point NOUN cana-1707 205 32 theorems theorem NOUN cana-1707 205 33 . . PUNCT cana-1707 206 1 it it PRON cana-1707 206 2 helps help VERB cana-1707 206 3 in in ADP cana-1707 206 4 comparative comparative ADJ cana-1707 206 5 studies study NOUN cana-1707 206 6 and and CCONJ cana-1707 206 7 may may AUX cana-1707 206 8 solve solve VERB cana-1707 206 9 many many ADJ cana-1707 206 10 related relate VERB cana-1707 206 11 open open ADJ cana-1707 206 12 problems problem NOUN cana-1707 206 13 . . PUNCT cana-1707 207 1 acknowledgment acknowledgment NOUN cana-1707 207 2 : : PUNCT cana-1707 208 1 authors author NOUN cana-1707 208 2 are be AUX cana-1707 208 3 grateful grateful ADJ cana-1707 208 4 to to ADP cana-1707 208 5 the the DET cana-1707 208 6 anonymous anonymous ADJ cana-1707 208 7 referees referee NOUN cana-1707 208 8 for for ADP cana-1707 208 9 their their PRON cana-1707 208 10 precise precise ADJ cana-1707 208 11 remarks remark NOUN cana-1707 208 12 , , PUNCT cana-1707 208 13 and and CCONJ cana-1707 208 14 suggestions suggestion NOUN cana-1707 208 15 which which PRON cana-1707 208 16 led lead VERB cana-1707 208 17 to to ADP cana-1707 208 18 the the DET cana-1707 208 19 improvement improvement NOUN cana-1707 208 20 of of ADP cana-1707 208 21 the the DET cana-1707 208 22 paper paper NOUN cana-1707 208 23 . . PUNCT cana-1707 209 1 and and CCONJ cana-1707 209 2 the the DET cana-1707 209 3 corresponding corresponding ADJ cana-1707 209 4 author author NOUN cana-1707 209 5 is be AUX cana-1707 209 6 highly highly ADV cana-1707 209 7 grateful grateful ADJ cana-1707 209 8 for for ADP cana-1707 209 9 their their PRON cana-1707 209 10 financial financial ADJ cana-1707 209 11 support support NOUN cana-1707 209 12 for for ADP cana-1707 209 13 researchers researcher NOUN cana-1707 209 14 . . PUNCT cana-1707 210 1 references reference NOUN cana-1707 210 2 : : PUNCT cana-1707 211 1 [ [ X cana-1707 211 2 1 1 X cana-1707 211 3 ] ] X cana-1707 211 4 banach banach NOUN cana-1707 211 5 s. s. PROPN cana-1707 211 6 ( ( PUNCT cana-1707 211 7 1922 1922 NUM cana-1707 211 8 ) ) PUNCT cana-1707 211 9 , , PUNCT cana-1707 211 10 sur sur PROPN cana-1707 211 11 les les PROPN cana-1707 211 12 operations operation NOUN cana-1707 211 13 dans dan NOUN cana-1707 211 14 les le NOUN cana-1707 211 15 ensembles ensemble NOUN cana-1707 211 16 abstraits abstrait NOUN cana-1707 211 17 et et PROPN cana-1707 211 18 leur leur PROPN cana-1707 211 19 applications applications PROPN cana-1707 211 20 aux aux PROPN cana-1707 211 21 equations equations PROPN cana-1707 211 22 integral integral ADJ cana-1707 211 23 . . PUNCT cana-1707 212 1 fund fund PROPN cana-1707 212 2 . . PUNCT cana-1707 213 1 math math NOUN cana-1707 213 2 . . PUNCT cana-1707 214 1 3 3 NUM cana-1707 214 2 , , PUNCT cana-1707 214 3 133 133 NUM cana-1707 214 4 - - SYM cana-1707 214 5 181 181 NUM cana-1707 214 6 . . PUNCT cana-1707 215 1 [ [ X cana-1707 215 2 2 2 NUM cana-1707 215 3 ] ] SYM cana-1707 215 4 bailey bailey PROPN cana-1707 215 5 d. d. PROPN cana-1707 215 6 f. f. PROPN cana-1707 215 7 ( ( PUNCT cana-1707 215 8 1966 1966 NUM cana-1707 215 9 ) ) PUNCT cana-1707 215 10 , , PUNCT cana-1707 215 11 some some DET cana-1707 215 12 theorems theorem NOUN cana-1707 215 13 on on ADP cana-1707 215 14 contractive contractive ADJ cana-1707 215 15 mappings mapping NOUN cana-1707 215 16 , , PUNCT cana-1707 215 17 j. j. PROPN cana-1707 215 18 london london PROPN cana-1707 215 19 math math PROPN cana-1707 215 20 . . PUNCT cana-1707 216 1 soc soc PROPN cana-1707 216 2 . . PUNCT cana-1707 217 1 41 41 NUM cana-1707 217 2 , , PUNCT cana-1707 217 3 101 101 NUM cana-1707 217 4 - - SYM cana-1707 217 5 106 106 NUM cana-1707 217 6 . . PUNCT cana-1707 218 1 [ [ X cana-1707 218 2 3 3 X cana-1707 218 3 ] ] X cana-1707 218 4 berinde berinde NOUN cana-1707 218 5 v. v. CCONJ cana-1707 218 6 ( ( PUNCT cana-1707 218 7 2004 2004 NUM cana-1707 218 8 ) ) PUNCT cana-1707 218 9 , , PUNCT cana-1707 218 10 approximating approximate VERB cana-1707 218 11 fixed fix VERB cana-1707 218 12 points point NOUN cana-1707 218 13 of of ADP cana-1707 218 14 weak weak ADJ cana-1707 218 15 contractions contraction NOUN cana-1707 218 16 using use VERB cana-1707 218 17 the the DET cana-1707 218 18 picard picard NOUN cana-1707 218 19 iteration iteration NOUN cana-1707 218 20 , , PUNCT cana-1707 218 21 nonlinear nonlinear ADJ cana-1707 218 22 anal anal NOUN cana-1707 218 23 . . PUNCT cana-1707 219 1 forum forum NOUN cana-1707 219 2 9 9 NUM cana-1707 219 3 , , PUNCT cana-1707 219 4 45–53 45–53 NUM cana-1707 219 5 . . PUNCT cana-1707 220 1 [ [ X cana-1707 220 2 4 4 NUM cana-1707 220 3 ] ] SYM cana-1707 220 4 birkhoff birkhoff NOUN cana-1707 220 5 , , PUNCT cana-1707 220 6 g.d g.d PROPN cana-1707 220 7 . . PROPN cana-1707 220 8 , , PUNCT cana-1707 220 9 kellogg kellogg PROPN cana-1707 220 10 , , PUNCT cana-1707 220 11 o.d o.d PROPN cana-1707 220 12 . . PROPN cana-1707 220 13 ( ( PUNCT cana-1707 220 14 1922 1922 NUM cana-1707 220 15 ) ) PUNCT cana-1707 220 16 , , PUNCT cana-1707 220 17 invariant invariant ADJ cana-1707 220 18 points point NOUN cana-1707 220 19 in in ADP cana-1707 220 20 function function NOUN cana-1707 220 21 space space NOUN cana-1707 220 22 . . PUNCT cana-1707 221 1 trans trans AUX cana-1707 221 2 . . PROPN cana-1707 221 3 am be AUX cana-1707 221 4 . . PUNCT cana-1707 222 1 math math NOUN cana-1707 222 2 . . PUNCT cana-1707 223 1 soc soc PROPN cana-1707 223 2 . . PUNCT cana-1707 224 1 23 23 NUM cana-1707 224 2 , , PUNCT cana-1707 224 3 96–115 96–115 NUM cana-1707 224 4 . . PUNCT cana-1707 225 1 [ [ X cana-1707 225 2 5 5 X cana-1707 225 3 ] ] PUNCT cana-1707 225 4 boyd boyd PROPN cana-1707 225 5 d. d. PROPN cana-1707 225 6 , , PUNCT cana-1707 225 7 and and CCONJ cana-1707 225 8 wong wong PROPN cana-1707 225 9 j. j. PROPN cana-1707 225 10 ( ( PUNCT cana-1707 225 11 1969 1969 NUM cana-1707 225 12 ) ) PUNCT cana-1707 225 13 , , PUNCT cana-1707 225 14 on on ADP cana-1707 225 15 nonlinear nonlinear ADJ cana-1707 225 16 contractions contraction NOUN cana-1707 225 17 , , PUNCT cana-1707 225 18 proc proc NOUN cana-1707 225 19 . . PUNCT cana-1707 226 1 amer amer PROPN cana-1707 226 2 . . PUNCT cana-1707 226 3 math math PROPN cana-1707 226 4 . . PUNCT cana-1707 227 1 soc soc PROPN cana-1707 227 2 . . PUNCT cana-1707 228 1 20 20 NUM cana-1707 228 2 , , PUNCT cana-1707 228 3 458–464 458–464 NUM cana-1707 228 4 . . PUNCT cana-1707 229 1 [ [ X cana-1707 229 2 6 6 NUM cana-1707 229 3 ] ] SYM cana-1707 229 4 bianchini bianchini PROPN cana-1707 229 5 , , PUNCT cana-1707 229 6 r. r. PROPN cana-1707 229 7 m. m. PROPN cana-1707 229 8 t. t. PROPN cana-1707 229 9 ( ( PUNCT cana-1707 229 10 1972 1972 NUM cana-1707 229 11 ) ) PUNCT cana-1707 229 12 . . PUNCT cana-1707 230 1 , , PUNCT cana-1707 230 2 su su PROPN cana-1707 230 3 un un PROPN cana-1707 230 4 problema problema PROPN cana-1707 230 5 di di PROPN cana-1707 230 6 s. s. PROPN cana-1707 230 7 reich reich PROPN cana-1707 230 8 riguardante riguardante PROPN cana-1707 230 9 la la PROPN cana-1707 230 10 teoria teoria PROPN cana-1707 230 11 dei dei PROPN cana-1707 230 12 punti punti PROPN cana-1707 230 13 fissi fissi NOUN cana-1707 230 14 , , PUNCT cana-1707 230 15 bolletrino bolletrino NOUN cana-1707 230 16 u.m.i u.m.i PROPN cana-1707 230 17 . . PUNCT cana-1707 231 1 ( ( PUNCT cana-1707 231 2 4 4 NUM cana-1707 231 3 ) ) PUNCT cana-1707 231 4 5 5 NUM cana-1707 231 5 , , PUNCT cana-1707 231 6 103 103 NUM cana-1707 231 7 - - SYM cana-1707 231 8 108 108 NUM cana-1707 231 9 . . PUNCT cana-1707 232 1 [ [ X cana-1707 232 2 7 7 X cana-1707 232 3 ] ] PUNCT cana-1707 232 4 browder browder PROPN cana-1707 232 5 e. e. PROPN cana-1707 232 6 ( ( PUNCT cana-1707 232 7 1968 1968 NUM cana-1707 232 8 ) ) PUNCT cana-1707 232 9 , , PUNCT cana-1707 232 10 on on ADP cana-1707 232 11 the the DET cana-1707 232 12 convergence convergence NOUN cana-1707 232 13 of of ADP cana-1707 232 14 successive successive ADJ cana-1707 232 15 approximations approximation NOUN cana-1707 232 16 for for ADP cana-1707 232 17 nonlinear nonlinear ADJ cana-1707 232 18 functional functional ADJ cana-1707 232 19 equations equation NOUN cana-1707 232 20 , , PUNCT cana-1707 232 21 indag indag PROPN cana-1707 232 22 . . PUNCT cana-1707 233 1 math math NOUN cana-1707 233 2 . . PUNCT cana-1707 234 1 30 30 NUM cana-1707 234 2 , , PUNCT cana-1707 234 3 27 27 NUM cana-1707 234 4 - - SYM cana-1707 234 5 35 35 NUM cana-1707 234 6 . . PUNCT cana-1707 235 1 [ [ X cana-1707 235 2 8 8 NUM cana-1707 235 3 ] ] X cana-1707 235 4 caristi caristi PROPN cana-1707 235 5 j. j. PROPN cana-1707 235 6 ( ( PUNCT cana-1707 235 7 1976 1976 NUM cana-1707 235 8 ) ) PUNCT cana-1707 235 9 , , PUNCT cana-1707 235 10 fixed fix VERB cana-1707 235 11 point point NOUN cana-1707 235 12 theorems theorem NOUN cana-1707 235 13 for for ADP cana-1707 235 14 mappings mapping NOUN cana-1707 235 15 satisfying satisfy VERB cana-1707 235 16 inwardness inwardness NOUN cana-1707 235 17 conditions condition NOUN cana-1707 235 18 . . PUNCT cana-1707 236 1 trans trans PROPN cana-1707 236 2 . . PUNCT cana-1707 237 1 amer amer PROPN cana-1707 237 2 . . PUNCT cana-1707 237 3 math math PROPN cana-1707 237 4 . . PUNCT cana-1707 238 1 soc soc PROPN cana-1707 238 2 . . PUNCT cana-1707 239 1 215 215 NUM cana-1707 239 2 , , PUNCT cana-1707 239 3 241–251 241–251 NUM cana-1707 239 4 . . PUNCT cana-1707 240 1 communications communication NOUN cana-1707 240 2 on on ADP cana-1707 240 3 applied apply VERB cana-1707 240 4 nonlinear nonlinear ADJ cana-1707 240 5 analysis analysis NOUN cana-1707 240 6 issn issn NOUN cana-1707 240 7 : : PUNCT cana-1707 240 8 1074 1074 NUM cana-1707 240 9 - - PUNCT cana-1707 240 10 133x 133x NUM cana-1707 240 11 vol vol NOUN cana-1707 240 12 32 32 NUM cana-1707 240 13 no no NOUN cana-1707 240 14 . . NOUN cana-1707 240 15 2 2 NUM cana-1707 240 16 ( ( PUNCT cana-1707 240 17 2025 2025 NUM cana-1707 240 18 ) ) PUNCT cana-1707 240 19 62 62 NUM cana-1707 240 20 https://internationalpubls.com https://internationalpubls.com X cana-1707 241 1 [ [ X cana-1707 241 2 9 9 NUM cana-1707 241 3 ] ] X cana-1707 241 4 cauchy cauchy PROPN cana-1707 241 5 a. a. PROPN cana-1707 241 6 l. l. PROPN cana-1707 241 7 ( ( PUNCT cana-1707 241 8 1884 1884 NUM cana-1707 241 9 ) ) PUNCT cana-1707 241 10 , , PUNCT cana-1707 241 11 a. a. PROPN cana-1707 241 12 l. l. PROPN cana-1707 241 13 , , PUNCT cana-1707 241 14 lecons lecon NOUN cana-1707 241 15 de de PROPN cana-1707 241 16 calcul calcul PROPN cana-1707 241 17 di_erentiel di_erentiel PROPN cana-1707 241 18 et et PROPN cana-1707 241 19 de de PROPN cana-1707 241 20 calcul calcul PROPN cana-1707 241 21 intgral intgral PROPN cana-1707 241 22 , , PUNCT cana-1707 241 23 vol vol NOUN cana-1707 241 24 . . PROPN cana-1707 241 25 2 2 NUM cana-1707 241 26 , , PUNCT cana-1707 241 27 mallet mallet NOUN cana-1707 241 28 - - PUNCT cana-1707 241 29 bachelier bachelier NOUN cana-1707 241 30 , , PUNCT cana-1707 241 31 paris paris PROPN cana-1707 241 32 . . PUNCT cana-1707 242 1 [ [ X cana-1707 242 2 10 10 NUM cana-1707 242 3 ] ] X cana-1707 242 4 caccippoli caccippoli PROPN cana-1707 242 5 r. r. PROPN cana-1707 242 6 ( ( PUNCT cana-1707 242 7 1930 1930 NUM cana-1707 242 8 ) ) PUNCT cana-1707 242 9 , , PUNCT cana-1707 242 10 un un PROPN cana-1707 242 11 teorema teorema PROPN cana-1707 242 12 generale generale PROPN cana-1707 242 13 sull’eslstenza sull’eslstenza PROPN cana-1707 242 14 di di X cana-1707 242 15 elemcnti elemcnti PROPN cana-1707 242 16 uniti uniti PROPN cana-1707 242 17 in in ADP cana-1707 242 18 um um INTJ cana-1707 242 19 trasformazione trasformazione NOUN cana-1707 242 20 funzionale funzionale NOUN cana-1707 242 21 , , PUNCT cana-1707 242 22 atti atti PROPN cana-1707 242 23 accad accad PROPN cana-1707 242 24 . . PUNCT cana-1707 243 1 naz naz PROPN cana-1707 243 2 . . PUNCT cana-1707 244 1 lincei lincei NOUN cana-1707 244 2 rend rend VERB cana-1707 244 3 . . PUNCT cana-1707 245 1 cl cl NOUN cana-1707 245 2 . . PUNCT cana-1707 246 1 sci sci PROPN cana-1707 246 2 . . PROPN cana-1707 246 3 fis fis PROPN cana-1707 246 4 . . PUNCT cana-1707 246 5 mat mat PROPN cana-1707 246 6 . . PUNCT cana-1707 246 7 natur natur PROPN cana-1707 246 8 . . PUNCT cana-1707 247 1 11 11 NUM cana-1707 247 2 , , PUNCT cana-1707 247 3 794 794 NUM cana-1707 247 4 - - SYM cana-1707 247 5 799 799 NUM cana-1707 247 6 . . PUNCT cana-1707 248 1 [ [ X cana-1707 248 2 11 11 NUM cana-1707 248 3 ] ] X cana-1707 248 4 chandra chandra PROPN cana-1707 248 5 n. n. PROPN cana-1707 248 6 , , PUNCT cana-1707 248 7 joshi joshi PROPN cana-1707 248 8 b. b. PROPN cana-1707 248 9 , , PUNCT cana-1707 248 10 and and CCONJ cana-1707 248 11 joshi joshi PROPN cana-1707 248 12 m. m. PROPN cana-1707 248 13 c. c. PROPN cana-1707 248 14 , , PUNCT cana-1707 248 15 generalized generalize VERB cana-1707 248 16 fixed fix VERB cana-1707 248 17 point point NOUN cana-1707 248 18 theorems theorem NOUN cana-1707 248 19 on on ADP cana-1707 248 20 metric metric ADJ cana-1707 248 21 spaces space NOUN cana-1707 248 22 , , PUNCT cana-1707 248 23 mathematica mathematica PROPN cana-1707 248 24 moravica moravica PROPN cana-1707 248 25 26(2 26(2 NUM cana-1707 248 26 ) ) PUNCT cana-1707 248 27 ( ( PUNCT cana-1707 248 28 2022)85–101 2022)85–101 NUM cana-1707 249 1 [ [ X cana-1707 249 2 12 12 NUM cana-1707 249 3 ] ] X cana-1707 249 4 chhaterjea chhaterjea PROPN cana-1707 249 5 s. s. PROPN cana-1707 249 6 k. k. PROPN cana-1707 249 7 ( ( PUNCT cana-1707 249 8 1972 1972 NUM cana-1707 249 9 ) ) PUNCT cana-1707 249 10 , , PUNCT cana-1707 249 11 fixed fix VERB cana-1707 249 12 point point NOUN cana-1707 249 13 theorems theorem NOUN cana-1707 249 14 , , PUNCT cana-1707 249 15 comptes compte VERB cana-1707 249 16 rendus rendus PROPN cana-1707 249 17 del del PROPN cana-1707 249 18 ’ ' PUNCT cana-1707 249 19 academie academie PROPN cana-1707 249 20 bulgare bulgare PROPN cana-1707 249 21 des des PROPN cana-1707 249 22 sciences sciences PROPN cana-1707 249 23 , , PUNCT cana-1707 249 24 tome tome NOUN cana-1707 249 25 25 25 NUM cana-1707 249 26 ( ( PUNCT cana-1707 249 27 6 6 NUM cana-1707 249 28 ) ) PUNCT cana-1707 249 29 , , PUNCT cana-1707 249 30 727 727 NUM cana-1707 249 31 - - SYM cana-1707 249 32 730 730 NUM cana-1707 249 33 . . PUNCT cana-1707 250 1 [ [ X cana-1707 250 2 13 13 NUM cana-1707 250 3 ] ] X cana-1707 250 4 chaudhary chaudhary PROPN cana-1707 250 5 a.k a.k PROPN cana-1707 250 6 . . PROPN cana-1707 250 7 , , PUNCT cana-1707 250 8 manandhar manandhar VERB cana-1707 250 9 k.b k.b PROPN cana-1707 250 10 . . PROPN cana-1707 250 11 , , PUNCT cana-1707 250 12 jha jha PROPN cana-1707 250 13 k. k. PROPN cana-1707 250 14 , , PUNCT cana-1707 250 15 and and CCONJ cana-1707 250 16 murthy murthy ADJ cana-1707 250 17 , , PUNCT cana-1707 250 18 p. p. NOUN cana-1707 250 19 p. p. NOUN cana-1707 250 20 ( ( PUNCT cana-1707 250 21 2021 2021 NUM cana-1707 250 22 ) ) PUNCT cana-1707 250 23 , , PUNCT cana-1707 250 24 a a DET cana-1707 250 25 common common ADJ cana-1707 250 26 fixed fix VERB cana-1707 250 27 point point NOUN cana-1707 250 28 theorem theorem VERB cana-1707 250 29 in in ADP cana-1707 250 30 menger menger PROPN cana-1707 250 31 space space NOUN cana-1707 250 32 with with ADP cana-1707 250 33 compatible compatible ADJ cana-1707 250 34 mapping mapping NOUN cana-1707 250 35 of of ADP cana-1707 250 36 type type NOUN cana-1707 250 37 ( ( PUNCT cana-1707 250 38 p p NOUN cana-1707 250 39 ) ) PUNCT cana-1707 250 40 , , PUNCT cana-1707 250 41 international international ADJ cana-1707 250 42 journal journal NOUN cana-1707 250 43 of of ADP cana-1707 250 44 math math NOUN cana-1707 250 45 . . PUNCT cana-1707 251 1 sci sci PROPN cana-1707 251 2 . . PROPN cana-1707 251 3 & & CCONJ cana-1707 251 4 engg engg PROPN cana-1707 251 5 . . PUNCT cana-1707 252 1 appls appls PROPN cana-1707 252 2 . . PUNCT cana-1707 252 3 , , PUNCT cana-1707 252 4 15(2 15(2 NUM cana-1707 252 5 ) ) PUNCT cana-1707 252 6 , , PUNCT cana-1707 252 7 59 59 NUM cana-1707 252 8 - - SYM cana-1707 252 9 70 70 NUM cana-1707 252 10 . . PUNCT cana-1707 253 1 [ [ X cana-1707 253 2 14 14 NUM cana-1707 253 3 ] ] X cana-1707 253 4 chaudhary chaudhary PROPN cana-1707 253 5 a.k a.k PROPN cana-1707 253 6 . . PROPN cana-1707 253 7 , , PUNCT cana-1707 253 8 manandhar manandhar VERB cana-1707 253 9 k.b k.b PROPN cana-1707 253 10 . . PROPN cana-1707 253 11 , , PUNCT cana-1707 253 12 and and CCONJ cana-1707 253 13 jha jha PROPN cana-1707 253 14 k. k. PROPN cana-1707 253 15 ( ( PUNCT cana-1707 253 16 2022 2022 NUM cana-1707 253 17 ) ) PUNCT cana-1707 253 18 , , PUNCT cana-1707 253 19 a a DET cana-1707 253 20 common common ADJ cana-1707 253 21 fixed fix VERB cana-1707 253 22 point point NOUN cana-1707 253 23 theorem theorem VERB cana-1707 253 24 in in ADP cana-1707 253 25 menger menger PROPN cana-1707 253 26 space space NOUN cana-1707 253 27 with with ADP cana-1707 253 28 compatible compatible ADJ cana-1707 253 29 mapping mapping NOUN cana-1707 253 30 of of ADP cana-1707 253 31 type type NOUN cana-1707 253 32 ( ( PUNCT cana-1707 253 33 k k NOUN cana-1707 253 34 ) ) PUNCT cana-1707 253 35 , , PUNCT cana-1707 253 36 advances advance NOUN cana-1707 253 37 in in ADP cana-1707 253 38 mathematics mathematic NOUN cana-1707 253 39 : : PUNCT cana-1707 253 40 scientific scientific ADJ cana-1707 253 41 journal journal NOUN cana-1707 253 42 , , PUNCT cana-1707 253 43 11(10 11(10 NUM cana-1707 253 44 ) ) PUNCT cana-1707 253 45 , , PUNCT cana-1707 253 46 883 883 NUM cana-1707 253 47 - - SYM cana-1707 253 48 892 892 NUM cana-1707 253 49 . . PUNCT cana-1707 254 1 [ [ X cana-1707 254 2 15 15 NUM cana-1707 254 3 ] ] X cana-1707 254 4 chaudhary chaudhary PROPN cana-1707 254 5 a.k a.k PROPN cana-1707 254 6 . . PROPN cana-1707 254 7 , , PUNCT cana-1707 254 8 occasionally occasionally ADV cana-1707 254 9 weakly weakly ADJ cana-1707 254 10 compatible compatible ADJ cana-1707 254 11 mappings mapping NOUN cana-1707 254 12 and and CCONJ cana-1707 254 13 common common ADJ cana-1707 254 14 fixed fix VERB cana-1707 254 15 points point NOUN cana-1707 254 16 in in ADP cana-1707 254 17 menger menger PROPN cana-1707 254 18 space space NOUN cana-1707 254 19 , , PUNCT cana-1707 254 20 results result VERB cana-1707 254 21 in in ADP cana-1707 254 22 nonlinear nonlinear ADJ cana-1707 254 23 analysis analysis NOUN cana-1707 254 24 6 6 NUM cana-1707 254 25 ( ( PUNCT cana-1707 254 26 2023 2023 NUM cana-1707 254 27 ) ) PUNCT cana-1707 255 1 no no NOUN cana-1707 255 2 . . NOUN cana-1707 255 3 4 4 NUM cana-1707 255 4 , , PUNCT cana-1707 255 5 47–54 47–54 NUM cana-1707 255 6 . . PUNCT cana-1707 256 1 [ [ X cana-1707 256 2 16 16 NUM cana-1707 256 3 ] ] X cana-1707 256 4 chaudhary chaudhary PROPN cana-1707 256 5 a. a. PROPN cana-1707 256 6 k. k. PROPN cana-1707 256 7 , , PUNCT cana-1707 256 8 and and CCONJ cana-1707 256 9 jha jha PROPN cana-1707 256 10 k. k. PROPN cana-1707 256 11 , , PUNCT cana-1707 256 12 contraction contraction NOUN cana-1707 256 13 conditions condition NOUN cana-1707 256 14 in in ADP cana-1707 256 15 probabilistic probabilistic ADJ cana-1707 256 16 metric metric ADJ cana-1707 256 17 space space NOUN cana-1707 256 18 , , PUNCT cana-1707 256 19 american american ADJ cana-1707 256 20 journal journal PROPN cana-1707 256 21 of of ADP cana-1707 256 22 mathematics mathematics PROPN cana-1707 256 23 and and CCONJ cana-1707 256 24 statistics statistic NOUN cana-1707 256 25 ( ( PUNCT cana-1707 256 26 2019 2019 NUM cana-1707 256 27 ) ) PUNCT cana-1707 256 28 , , PUNCT cana-1707 256 29 9(5 9(5 NUM cana-1707 256 30 ) ) PUNCT cana-1707 256 31 , , PUNCT cana-1707 256 32 199 199 NUM cana-1707 256 33 - - SYM cana-1707 256 34 202 202 NUM cana-1707 256 35 . . PUNCT cana-1707 257 1 [ [ X cana-1707 257 2 17 17 NUM cana-1707 257 3 ] ] X cana-1707 257 4 chaudhary chaudhary PROPN cana-1707 257 5 a. a. PROPN cana-1707 257 6 k. k. PROPN cana-1707 257 7 , , PUNCT cana-1707 257 8 a a DET cana-1707 257 9 common common ADJ cana-1707 257 10 fixed fix VERB cana-1707 257 11 point point NOUN cana-1707 257 12 result result NOUN cana-1707 257 13 in in ADP cana-1707 257 14 menger menger PROPN cana-1707 257 15 space space NOUN cana-1707 257 16 , , PUNCT cana-1707 257 17 communications communication NOUN cana-1707 257 18 on on ADP cana-1707 257 19 applied apply VERB cana-1707 257 20 nonlinear nonlinear ADJ cana-1707 257 21 analysis analysis NOUN cana-1707 257 22 , , PUNCT cana-1707 257 23 vol vol NOUN cana-1707 257 24 31 31 NUM cana-1707 257 25 , , PUNCT cana-1707 257 26 5s 5s NUM cana-1707 257 27 , , PUNCT cana-1707 257 28 458 458 NUM cana-1707 257 29 - - SYM cana-1707 257 30 465 465 NUM cana-1707 257 31 . . PUNCT cana-1707 258 1 [ [ X cana-1707 258 2 18 18 NUM cana-1707 258 3 ] ] X cana-1707 258 4 ciric ciric PROPN cana-1707 258 5 lj lj PROPN cana-1707 258 6 . . PUNCT cana-1707 259 1 ( ( PUNCT cana-1707 259 2 1974 1974 NUM cana-1707 259 3 ) ) PUNCT cana-1707 259 4 , , PUNCT cana-1707 259 5 a a DET cana-1707 259 6 generalization generalization NOUN cana-1707 259 7 of of ADP cana-1707 259 8 banach banach NOUN cana-1707 259 9 's 's PART cana-1707 259 10 contraction contraction NOUN cana-1707 259 11 principle principle NOUN cana-1707 259 12 , , PUNCT cana-1707 259 13 proc proc PROPN cana-1707 259 14 . . PUNCT cana-1707 260 1 amer amer PROPN cana-1707 260 2 . . PUNCT cana-1707 260 3 math math PROPN cana-1707 260 4 . . PUNCT cana-1707 261 1 soc soc PROPN cana-1707 261 2 . . PUNCT cana-1707 262 1 45 45 NUM cana-1707 262 2 , , PUNCT cana-1707 262 3 267 267 NUM cana-1707 262 4 - - SYM cana-1707 262 5 273 273 NUM cana-1707 262 6 . . PUNCT cana-1707 263 1 [ [ X cana-1707 263 2 19 19 NUM cana-1707 263 3 ] ] X cana-1707 263 4 ciric ciric ADJ cana-1707 263 5 lj lj PROPN cana-1707 263 6 ( ( PUNCT cana-1707 263 7 1971 1971 NUM cana-1707 263 8 ) ) PUNCT cana-1707 263 9 . . PUNCT cana-1707 264 1 , , PUNCT cana-1707 264 2 on on ADP cana-1707 264 3 contraction contraction NOUN cana-1707 264 4 type type NOUN cana-1707 264 5 mappings mapping NOUN cana-1707 264 6 , , PUNCT cana-1707 264 7 math math NOUN cana-1707 264 8 . . PUNCT cana-1707 264 9 balkanika balkanika PROPN cana-1707 264 10 , , PUNCT cana-1707 264 11 1 1 NUM cana-1707 264 12 , , PUNCT cana-1707 264 13 52 52 NUM cana-1707 264 14 - - SYM cana-1707 264 15 57 57 NUM cana-1707 264 16 . . PUNCT cana-1707 265 1 [ [ X cana-1707 265 2 20 20 NUM cana-1707 265 3 ] ] SYM cana-1707 265 4 ciric ciric ADJ cana-1707 265 5 , , PUNCT cana-1707 265 6 lj lj PROPN cana-1707 265 7 . . PROPN cana-1707 265 8 b. b. PROPN cana-1707 265 9 ( ( PUNCT cana-1707 265 10 1975 1975 NUM cana-1707 265 11 ) ) PUNCT cana-1707 265 12 , , PUNCT cana-1707 265 13 on on ADP cana-1707 265 14 fixed fix VERB cana-1707 265 15 points point NOUN cana-1707 265 16 of of ADP cana-1707 265 17 generalized generalized ADJ cana-1707 265 18 contractions contraction NOUN cana-1707 265 19 on on ADP cana-1707 265 20 probabilistic probabilistic ADJ cana-1707 265 21 metric metric ADJ cana-1707 265 22 spaces space NOUN cana-1707 265 23 , , PUNCT cana-1707 266 1 pub pub NOUN cana-1707 266 2 . . PUNCT cana-1707 266 3 inst inst PROPN cana-1707 266 4 . . PUNCT cana-1707 267 1 math math NOUN cana-1707 267 2 . . PUNCT cana-1707 268 1 beograd beograd PROPN cana-1707 268 2 , , PUNCT cana-1707 268 3 18(32 18(32 NUM cana-1707 268 4 ) ) PUNCT cana-1707 268 5 , , PUNCT cana-1707 268 6 71 71 NUM cana-1707 268 7 - - SYM cana-1707 268 8 78 78 NUM cana-1707 268 9 . . PUNCT cana-1707 269 1 [ [ X cana-1707 269 2 21 21 NUM cana-1707 269 3 ] ] PUNCT cana-1707 269 4 danes danes PROPN cana-1707 269 5 j. j. PROPN cana-1707 269 6 ( ( PUNCT cana-1707 269 7 1976 1976 NUM cana-1707 269 8 ) ) PUNCT cana-1707 269 9 , , PUNCT cana-1707 269 10 two two NUM cana-1707 269 11 fixed fix VERB cana-1707 269 12 point point NOUN cana-1707 269 13 theorems theorem NOUN cana-1707 269 14 in in ADP cana-1707 269 15 topological topological ADJ cana-1707 269 16 and and CCONJ cana-1707 269 17 metric metric ADJ cana-1707 269 18 spaces space NOUN cana-1707 269 19 , , PUNCT cana-1707 269 20 bull bull NOUN cana-1707 269 21 . . PUNCT cana-1707 270 1 austral austral PROPN cana-1707 270 2 . . PUNCT cana-1707 271 1 math math NOUN cana-1707 271 2 . . PUNCT cana-1707 272 1 soc soc PROPN cana-1707 272 2 . . PUNCT cana-1707 273 1 14 14 NUM cana-1707 273 2 , , PUNCT cana-1707 273 3 259 259 NUM cana-1707 273 4 - - SYM cana-1707 273 5 265 265 NUM cana-1707 273 6 . . PUNCT cana-1707 274 1 [ [ X cana-1707 274 2 22 22 NUM cana-1707 274 3 ] ] X cana-1707 274 4 dass dass PROPN cana-1707 274 5 b.k b.k PROPN cana-1707 274 6 . . PROPN cana-1707 274 7 , , PUNCT cana-1707 274 8 and and CCONJ cana-1707 274 9 gupta gupta PROPN cana-1707 274 10 s. s. PROPN cana-1707 274 11 ( ( PUNCT cana-1707 274 12 1975 1975 NUM cana-1707 274 13 ) ) PUNCT cana-1707 274 14 , , PUNCT cana-1707 274 15 an an DET cana-1707 274 16 extension extension NOUN cana-1707 274 17 of of ADP cana-1707 274 18 banach banach NOUN cana-1707 274 19 contraction contraction NOUN cana-1707 274 20 principle principle NOUN cana-1707 274 21 through through ADP cana-1707 274 22 rational rational ADJ cana-1707 274 23 expressions expression NOUN cana-1707 274 24 , , PUNCT cana-1707 274 25 indian indian ADJ cana-1707 274 26 j. j. PROPN cana-1707 274 27 pure pure PROPN cana-1707 274 28 appl appl PROPN cana-1707 274 29 . . PUNCT cana-1707 274 30 math math PROPN cana-1707 274 31 . . PUNCT cana-1707 275 1 , , PUNCT cana-1707 275 2 6 6 NUM cana-1707 275 3 ( ( PUNCT cana-1707 275 4 1975 1975 NUM cana-1707 275 5 ) ) PUNCT cana-1707 275 6 1455–1458 1455–1458 NUM cana-1707 275 7 . . PUNCT cana-1707 276 1 [ [ X cana-1707 276 2 23 23 NUM cana-1707 276 3 ] ] X cana-1707 276 4 dutta dutta PROPN cana-1707 276 5 , , PUNCT cana-1707 276 6 p. p. PROPN cana-1707 276 7 n. n. NOUN cana-1707 276 8 , , PUNCT cana-1707 276 9 and and CCONJ cana-1707 276 10 choudhury choudhury PROPN cana-1707 276 11 , , PUNCT cana-1707 276 12 b.s b.s PROPN cana-1707 276 13 . . PROPN cana-1707 276 14 ( ( PUNCT cana-1707 276 15 2008 2008 NUM cana-1707 276 16 ) ) PUNCT cana-1707 276 17 , , PUNCT cana-1707 276 18 a a DET cana-1707 276 19 generalization generalization NOUN cana-1707 276 20 of of ADP cana-1707 276 21 contraction contraction NOUN cana-1707 276 22 principle principle NOUN cana-1707 276 23 in in ADP cana-1707 276 24 metric metric ADJ cana-1707 276 25 spaces space NOUN cana-1707 276 26 . . PUNCT cana-1707 277 1 fixed fix VERB cana-1707 277 2 point point NOUN cana-1707 277 3 theory theory NOUN cana-1707 277 4 appl appl PROPN cana-1707 277 5 2008 2008 NUM cana-1707 277 6 , , PUNCT cana-1707 277 7 8 8 NUM cana-1707 277 8 . . PUNCT cana-1707 278 1 [ [ X cana-1707 278 2 24 24 NUM cana-1707 278 3 ] ] X cana-1707 278 4 edelstein edelstein PROPN cana-1707 278 5 , , PUNCT cana-1707 278 6 m. m. NOUN cana-1707 278 7 ( ( PUNCT cana-1707 278 8 1962 1962 NUM cana-1707 278 9 ) ) PUNCT cana-1707 278 10 . . PUNCT cana-1707 278 11 , , PUNCT cana-1707 278 12 on on ADP cana-1707 278 13 fixed fixed ADJ cana-1707 278 14 and and CCONJ cana-1707 278 15 periodic periodic ADJ cana-1707 278 16 points point NOUN cana-1707 278 17 under under ADP cana-1707 278 18 contractive contractive ADJ cana-1707 278 19 mappings mapping NOUN cana-1707 278 20 , , PUNCT cana-1707 278 21 journal journal NOUN cana-1707 278 22 london london PROPN cana-1707 278 23 math math PROPN cana-1707 278 24 . . PUNCT cana-1707 279 1 sot sot PROPN cana-1707 279 2 . . PROPN cana-1707 279 3 , , PUNCT cana-1707 279 4 37 37 NUM cana-1707 279 5 , , PUNCT cana-1707 279 6 7479 7479 NUM cana-1707 279 7 . . PUNCT cana-1707 280 1 [ [ X cana-1707 280 2 25 25 NUM cana-1707 280 3 ] ] X cana-1707 280 4 frechet frechet NOUN cana-1707 280 5 , , PUNCT cana-1707 280 6 m. m. NOUN cana-1707 280 7 , , PUNCT cana-1707 280 8 sur sur PROPN cana-1707 280 9 quelques quelques PROPN cana-1707 280 10 points point NOUN cana-1707 280 11 du du PROPN cana-1707 280 12 calcul calcul PROPN cana-1707 280 13 fonctionnel fonctionnel PROPN cana-1707 280 14 , , PUNCT cana-1707 280 15 rendic rendic ADJ cana-1707 280 16 . . PUNCT cana-1707 281 1 circ circ PROPN cana-1707 281 2 . . PUNCT cana-1707 282 1 mat mat PROPN cana-1707 282 2 . . PUNCT cana-1707 282 3 palermo palermo PROPN cana-1707 282 4 ( ( PUNCT cana-1707 282 5 1906),1 1906),1 NUM cana-1707 282 6 - - SYM cana-1707 282 7 74 74 NUM cana-1707 282 8 . . PUNCT cana-1707 283 1 [ [ X cana-1707 283 2 26 26 NUM cana-1707 283 3 ] ] PUNCT cana-1707 283 4 golet golet VERB cana-1707 283 5 i. i. PROPN cana-1707 283 6 ( ( PUNCT cana-1707 283 7 2004 2004 NUM cana-1707 283 8 ) ) PUNCT cana-1707 283 9 on on ADP cana-1707 283 10 contractions contraction NOUN cana-1707 283 11 in in ADP cana-1707 283 12 probabilistic probabilistic ADJ cana-1707 283 13 metric metric ADJ cana-1707 283 14 space space NOUN cana-1707 283 15 , , PUNCT cana-1707 283 16 radovi radovi PROPN cana-1707 283 17 mathematic mathematic PROPN cana-1707 283 18 ki ki PROPN cana-1707 283 19 . . PROPN cana-1707 283 20 , , PUNCT cana-1707 283 21 13 13 NUM cana-1707 283 22 , , PUNCT cana-1707 283 23 87 87 NUM cana-1707 283 24 - - SYM cana-1707 283 25 92 92 NUM cana-1707 283 26 . . PUNCT cana-1707 284 1 [ [ X cana-1707 284 2 27 27 NUM cana-1707 284 3 ] ] PUNCT cana-1707 284 4 guseman guseman PROPN cana-1707 284 5 l. l. PROPN cana-1707 284 6 f. f. PROPN cana-1707 284 7 , , PUNCT cana-1707 284 8 jr jr PROPN cana-1707 284 9 ( ( PUNCT cana-1707 284 10 1970 1970 NUM cana-1707 284 11 ) ) PUNCT cana-1707 284 12 , , PUNCT cana-1707 284 13 fixed fix VERB cana-1707 284 14 point point NOUN cana-1707 284 15 theorems theorem NOUN cana-1707 284 16 for for ADP cana-1707 284 17 mappings mapping NOUN cana-1707 284 18 with with ADP cana-1707 284 19 a a DET cana-1707 284 20 contractive contractive ADJ cana-1707 284 21 iterate iterate NOUN cana-1707 284 22 at at ADP cana-1707 284 23 a a DET cana-1707 284 24 point point NOUN cana-1707 284 25 , , PUNCT cana-1707 284 26 [ [ X cana-1707 284 27 28 28 NUM cana-1707 284 28 ] ] PUNCT cana-1707 284 29 proc.a proc.a ADP cana-1707 284 30 mer.m mer.m PROPN cana-1707 284 31 ath ath NOUN cana-1707 284 32 . . PUNCT cana-1707 285 1 soc.2 soc.2 ADJ cana-1707 285 2 6 6 NUM cana-1707 285 3 ( ( PUNCT cana-1707 285 4 1970)6 1970)6 NUM cana-1707 285 5 , , PUNCT cana-1707 285 6 15 15 NUM cana-1707 285 7 - - SYM cana-1707 285 8 618 618 NUM cana-1707 285 9 , , PUNCT cana-1707 285 10 m m VERB cana-1707 285 11 r r NOUN cana-1707 285 12 42 42 NUM cana-1707 285 13 , , PUNCT cana-1707 285 14 919 919 NUM cana-1707 285 15 . . PUNCT cana-1707 286 1 [ [ X cana-1707 286 2 29 29 NUM cana-1707 286 3 ] ] X cana-1707 286 4 hadzic hadzic PROPN cana-1707 286 5 o o X cana-1707 286 6 and and CCONJ cana-1707 286 7 pap pap NOUN cana-1707 286 8 e e NOUN cana-1707 286 9 ( ( PUNCT cana-1707 286 10 2010 2010 NUM cana-1707 286 11 ) ) PUNCT cana-1707 286 12 probabilistic probabilistic ADJ cana-1707 286 13 fixed fix VERB cana-1707 286 14 - - PUNCT cana-1707 286 15 point point NOUN cana-1707 286 16 theory theory NOUN cana-1707 286 17 in in ADP cana-1707 286 18 probabilistic probabilistic ADJ cana-1707 286 19 metric metric ADJ cana-1707 286 20 space space NOUN cana-1707 286 21 . . PUNCT cana-1707 287 1 kluwer kluwer NOUN cana-1707 287 2 academic academic PROPN cana-1707 287 3 publisher publisher NOUN cana-1707 287 4 , , PUNCT cana-1707 287 5 london london PROPN cana-1707 287 6 . . PUNCT cana-1707 288 1 536 536 NUM cana-1707 289 1 [ [ SYM cana-1707 289 2 30 30 NUM cana-1707 289 3 ] ] X cana-1707 289 4 hegediis hegediis PROPN cana-1707 289 5 m m PROPN cana-1707 289 6 , , PUNCT cana-1707 289 7 a a DET cana-1707 289 8 new new ADJ cana-1707 289 9 generalization generalization NOUN cana-1707 289 10 of of ADP cana-1707 289 11 banach banach NOUN cana-1707 289 12 's 's PART cana-1707 289 13 contraction contraction NOUN cana-1707 289 14 principle principle NOUN cana-1707 289 15 , , PUNCT cana-1707 289 16 acta acta PROPN cana-1707 289 17 . . PUNCT cana-1707 290 1 sci sci PROPN cana-1707 290 2 . . PROPN cana-1707 290 3 math math PROPN cana-1707 290 4 . . PUNCT cana-1707 291 1 ( ( PUNCT cana-1707 291 2 szeged szeged PROPN cana-1707 291 3 ) ) PUNCT cana-1707 291 4 , , PUNCT cana-1707 291 5 ( ( PUNCT cana-1707 291 6 to to PART cana-1707 291 7 appear appear VERB cana-1707 291 8 ) ) PUNCT cana-1707 291 9 . . PUNCT cana-1707 292 1 [ [ X cana-1707 292 2 31 31 NUM cana-1707 292 3 ] ] PUNCT cana-1707 292 4 hegediis hegediis ADJ cana-1707 292 5 m. m. NOUN cana-1707 292 6 and and CCONJ cana-1707 292 7 szilagyi szilagyi PROPN cana-1707 292 8 t. t. PROPN cana-1707 292 9 , , PUNCT cana-1707 292 10 equivalent equivalent ADJ cana-1707 292 11 conditions condition NOUN cana-1707 292 12 and and CCONJ cana-1707 292 13 a a DET cana-1707 292 14 new new ADJ cana-1707 292 15 fixed fix VERB cana-1707 292 16 point point NOUN cana-1707 292 17 theorem theorem VERB cana-1707 292 18 in in ADP cana-1707 292 19 the the DET cana-1707 292 20 theory theory NOUN cana-1707 292 21 of of ADP cana-1707 292 22 contractive contractive ADJ cana-1707 292 23 type type NOUN cana-1707 292 24 mappings mapping NOUN cana-1707 292 25 , , PUNCT cana-1707 292 26 math math NOUN cana-1707 292 27 . . PUNCT cana-1707 293 1 sem sem PROPN cana-1707 293 2 . . PUNCT cana-1707 294 1 notes notes PROPN cana-1707 294 2 , , PUNCT cana-1707 294 3 ( ( PUNCT cana-1707 294 4 to to PART cana-1707 294 5 appear appear VERB cana-1707 294 6 ) ) PUNCT cana-1707 294 7 . . PUNCT cana-1707 295 1 [ [ X cana-1707 295 2 32 32 NUM cana-1707 295 3 ] ] SYM cana-1707 295 4 hardy hardy ADJ cana-1707 295 5 , , PUNCT cana-1707 295 6 g. g. PROPN cana-1707 295 7 e. e. PROPN cana-1707 295 8 & & CCONJ cana-1707 295 9 rogers rogers PROPN cana-1707 295 10 , , PUNCT cana-1707 295 11 t. t. PROPN cana-1707 295 12 d. d. PROPN cana-1707 295 13 ( ( PUNCT cana-1707 295 14 1973 1973 NUM cana-1707 295 15 ) ) PUNCT cana-1707 295 16 . . PUNCT cana-1707 296 1 , , PUNCT cana-1707 296 2 a a DET cana-1707 296 3 generalisation generalisation NOUN cana-1707 296 4 of of ADP cana-1707 296 5 a a DET cana-1707 296 6 fixed fix VERB cana-1707 296 7 point point NOUN cana-1707 296 8 theorem theorem NOUN cana-1707 296 9 of of ADP cana-1707 296 10 reich reich PROPN cana-1707 296 11 , , PUNCT cana-1707 296 12 canad canad PROPN cana-1707 296 13 . . PUNCT cana-1707 297 1 math math NOUN cana-1707 297 2 . . PUNCT cana-1707 298 1 bull bull NOUN cana-1707 298 2 . . PUNCT cana-1707 299 1 vol vol NOUN cana-1707 299 2 . . PUNCT cana-1707 300 1 16 16 NUM cana-1707 300 2 ( ( PUNCT cana-1707 300 3 2 2 NUM cana-1707 300 4 ) ) PUNCT cana-1707 300 5 , , PUNCT cana-1707 300 6 201 201 NUM cana-1707 300 7 - - SYM cana-1707 300 8 206 206 NUM cana-1707 300 9 . . PUNCT cana-1707 301 1 [ [ X cana-1707 301 2 33 33 NUM cana-1707 301 3 ] ] PUNCT cana-1707 301 4 hicks hicks PROPN cana-1707 301 5 t.l t.l PROPN cana-1707 301 6 . . PUNCT cana-1707 301 7 ( ( PUNCT cana-1707 301 8 1983 1983 NUM cana-1707 301 9 ) ) PUNCT cana-1707 301 10 , , PUNCT cana-1707 301 11 fixed fix VERB cana-1707 301 12 point point NOUN cana-1707 301 13 theory theory NOUN cana-1707 301 14 in in ADP cana-1707 301 15 probabilistic probabilistic ADJ cana-1707 301 16 metric metric ADJ cana-1707 301 17 spaces space NOUN cana-1707 301 18 , , PUNCT cana-1707 301 19 review review NOUN cana-1707 301 20 of of ADP cana-1707 301 21 research research NOUN cana-1707 301 22 faculty faculty NOUN cana-1707 301 23 of of ADP cana-1707 301 24 novi novi PROPN cana-1707 301 25 sad sad ADJ cana-1707 301 26 13 13 NUM cana-1707 301 27 , , PUNCT cana-1707 301 28 63 63 NUM cana-1707 301 29 - - SYM cana-1707 301 30 72 72 NUM cana-1707 301 31 . . PUNCT cana-1707 302 1 [ [ X cana-1707 302 2 34 34 NUM cana-1707 302 3 ] ] PUNCT cana-1707 302 4 jaggi jaggi PROPN cana-1707 302 5 d. d. PROPN cana-1707 302 6 s. s. PROPN cana-1707 302 7 ( ( PUNCT cana-1707 302 8 1977 1977 NUM cana-1707 302 9 ) ) PUNCT cana-1707 302 10 , , PUNCT cana-1707 302 11 some some DET cana-1707 302 12 unique unique ADJ cana-1707 302 13 fixed fix VERB cana-1707 302 14 point point NOUN cana-1707 302 15 theorems theorem NOUN cana-1707 302 16 , , PUNCT cana-1707 302 17 indian indian ADJ cana-1707 302 18 j. j. PROPN cana-1707 302 19 pure pure PROPN cana-1707 302 20 appl appl PROPN cana-1707 302 21 . . PUNCT cana-1707 302 22 math math NOUN cana-1707 302 23 . . PUNCT cana-1707 303 1 8 8 NUM cana-1707 303 2 , , PUNCT cana-1707 303 3 223–230 223–230 NUM cana-1707 303 4 . . PUNCT cana-1707 304 1 [ [ X cana-1707 304 2 35 35 NUM cana-1707 304 3 ] ] X cana-1707 304 4 jleli jleli ADJ cana-1707 304 5 m. m. NOUN cana-1707 304 6 and and CCONJ cana-1707 304 7 samet samet PROPN cana-1707 304 8 b. b. PROPN cana-1707 304 9 ( ( PUNCT cana-1707 304 10 2014 2014 NUM cana-1707 304 11 ) ) PUNCT cana-1707 304 12 , , PUNCT cana-1707 304 13 a a DET cana-1707 304 14 new new ADJ cana-1707 304 15 generalization generalization NOUN cana-1707 304 16 of of ADP cana-1707 304 17 the the DET cana-1707 304 18 banach banach NOUN cana-1707 304 19 contraction contraction NOUN cana-1707 304 20 principle principle NOUN cana-1707 304 21 , , PUNCT cana-1707 304 22 journal journal NOUN cana-1707 304 23 of of ADP cana-1707 304 24 inequalities inequality NOUN cana-1707 304 25 and and CCONJ cana-1707 304 26 applications application NOUN cana-1707 304 27 38 38 NUM cana-1707 304 28 ( ( PUNCT cana-1707 304 29 2014 2014 NUM cana-1707 304 30 ) ) PUNCT cana-1707 304 31 , , PUNCT cana-1707 304 32 8 8 NUM cana-1707 304 33 pages page NOUN cana-1707 304 34 . . PUNCT cana-1707 305 1 [ [ X cana-1707 305 2 36 36 NUM cana-1707 305 3 ] ] X cana-1707 305 4 kincses kincses PROPN cana-1707 305 5 j j PROPN cana-1707 305 6 , , PUNCT cana-1707 305 7 and and CCONJ cana-1707 305 8 totik totik PROPN cana-1707 305 9 v. v. ADP cana-1707 305 10 ( ( PUNCT cana-1707 305 11 1990 1990 NUM cana-1707 305 12 ) ) PUNCT cana-1707 305 13 , , PUNCT cana-1707 305 14 theorems theorem NOUN cana-1707 305 15 and and CCONJ cana-1707 305 16 counter counter ADJ cana-1707 305 17 examples example NOUN cana-1707 305 18 on on ADP cana-1707 305 19 contraction contraction NOUN cana-1707 305 20 mapping mapping NOUN cana-1707 305 21 , , PUNCT cana-1707 305 22 mathematic mathematic PROPN cana-1707 305 23 balkanica balkanica PROPN cana-1707 305 24 , , PUNCT cana-1707 305 25 new new ADJ cana-1707 305 26 series series NOUN cana-1707 305 27 vol vol NOUN cana-1707 305 28 4 4 NUM cana-1707 305 29 , , PUNCT cana-1707 305 30 fasc fasc PROPN cana-1707 305 31 1 1 NUM cana-1707 305 32 . . PUNCT cana-1707 306 1 [ [ X cana-1707 306 2 37 37 NUM cana-1707 306 3 ] ] PUNCT cana-1707 306 4 kasahara kasahara PROPN cana-1707 306 5 s. s. PROPN cana-1707 306 6 , , PUNCT cana-1707 306 7 generalizations generalization NOUN cana-1707 306 8 of of ADP cana-1707 306 9 hegedus hegedus NOUN cana-1707 306 10 ' ' PUNCT cana-1707 306 11 fixed fixed ADJ cana-1707 306 12 point point NOUN cana-1707 306 13 theorem theorem VERB cana-1707 306 14 , , PUNCT cana-1707 306 15 math math NOUN cana-1707 306 16 . . PUNCT cana-1707 307 1 sem sem PROPN cana-1707 307 2 . . PUNCT cana-1707 308 1 notes note NOUN cana-1707 308 2 , , PUNCT cana-1707 308 3 7(1979 7(1979 NUM cana-1707 308 4 ) ) PUNCT cana-1707 308 5 , , PUNCT cana-1707 308 6 107 107 NUM cana-1707 308 7 - - SYM cana-1707 308 8 111 111 NUM cana-1707 308 9 . . PUNCT cana-1707 309 1 [ [ X cana-1707 309 2 38 38 NUM cana-1707 309 3 ] ] X cana-1707 309 4 kanan kanan PROPN cana-1707 309 5 , , PUNCT cana-1707 309 6 r. r. PROPN cana-1707 309 7 ( ( PUNCT cana-1707 309 8 1969 1969 NUM cana-1707 309 9 ) ) PUNCT cana-1707 309 10 , , PUNCT cana-1707 309 11 some some PRON cana-1707 309 12 results result NOUN cana-1707 309 13 on on ADP cana-1707 309 14 fixed fix VERB cana-1707 309 15 points point NOUN cana-1707 309 16 - - PUNCT cana-1707 309 17 ii ii NOUN cana-1707 309 18 , , PUNCT cana-1707 309 19 amer amer PROPN cana-1707 309 20 . . PROPN cana-1707 309 21 math math PROPN cana-1707 309 22 . . PUNCT cana-1707 310 1 monthly monthly ADJ cana-1707 310 2 76,405 76,405 NUM cana-1707 310 3 - - SYM cana-1707 310 4 408 408 NUM cana-1707 310 5 . . PUNCT cana-1707 311 1 [ [ X cana-1707 311 2 39 39 NUM cana-1707 311 3 ] ] PUNCT cana-1707 311 4 menger menger PROPN cana-1707 311 5 k. k. PROPN cana-1707 311 6 ( ( PUNCT cana-1707 311 7 1942 1942 NUM cana-1707 311 8 ) ) PUNCT cana-1707 311 9 statistical statistical ADJ cana-1707 311 10 matrices matrix NOUN cana-1707 311 11 , , PUNCT cana-1707 311 12 proceedings proceeding NOUN cana-1707 311 13 of of ADP cana-1707 311 14 national national PROPN cana-1707 311 15 academy academy PROPN cana-1707 311 16 of of ADP cana-1707 311 17 sciences sciences PROPN cana-1707 311 18 of of ADP cana-1707 311 19 usa usa PROPN cana-1707 311 20 , , PUNCT cana-1707 311 21 28 28 NUM cana-1707 311 22 , , PUNCT cana-1707 311 23 535 535 NUM cana-1707 311 24 - - SYM cana-1707 311 25 537 537 NUM cana-1707 311 26 . . PUNCT cana-1707 312 1 [ [ X cana-1707 312 2 40 40 NUM cana-1707 312 3 ] ] X cana-1707 312 4 mihet mihet PROPN cana-1707 312 5 d d NOUN cana-1707 312 6 ( ( PUNCT cana-1707 312 7 2005 2005 NUM cana-1707 312 8 ) ) PUNCT cana-1707 312 9 , , PUNCT cana-1707 312 10 weak weak ADJ cana-1707 312 11 hicks hick NOUN cana-1707 312 12 contractions contraction NOUN cana-1707 312 13 , , PUNCT cana-1707 312 14 fixed fix VERB cana-1707 312 15 point point NOUN cana-1707 312 16 theory theory NOUN cana-1707 312 17 6(1)71 6(1)71 PROPN cana-1707 312 18 - - SYM cana-1707 312 19 78 78 NUM cana-1707 312 20 . . PUNCT cana-1707 313 1 [ [ X cana-1707 313 2 41 41 NUM cana-1707 313 3 ] ] X cana-1707 313 4 meir meir PROPN cana-1707 313 5 a a PROPN cana-1707 313 6 and and CCONJ cana-1707 313 7 keeler keeler PROPN cana-1707 313 8 e. e. PROPN cana-1707 313 9 ( ( PUNCT cana-1707 313 10 1969 1969 NUM cana-1707 313 11 ) ) PUNCT cana-1707 313 12 , , PUNCT cana-1707 313 13 a a DET cana-1707 313 14 theorem theorem NOUN cana-1707 313 15 on on ADP cana-1707 313 16 contraction contraction NOUN cana-1707 313 17 mappings mapping NOUN cana-1707 313 18 , , PUNCT cana-1707 313 19 math math NOUN cana-1707 313 20 . . PUNCT cana-1707 314 1 anal anal PROPN cana-1707 314 2 . . PUNCT cana-1707 315 1 appl.28 appl.28 PROPN cana-1707 315 2 , , PUNCT cana-1707 315 3 326 326 NUM cana-1707 315 4 - - SYM cana-1707 315 5 329 329 NUM cana-1707 315 6 . . PUNCT cana-1707 316 1 communications communication NOUN cana-1707 316 2 on on ADP cana-1707 316 3 applied apply VERB cana-1707 316 4 nonlinear nonlinear ADJ cana-1707 316 5 analysis analysis NOUN cana-1707 316 6 issn issn NOUN cana-1707 316 7 : : PUNCT cana-1707 316 8 1074 1074 NUM cana-1707 316 9 - - PUNCT cana-1707 316 10 133x 133x NUM cana-1707 316 11 vol vol NOUN cana-1707 316 12 32 32 NUM cana-1707 316 13 no no NOUN cana-1707 316 14 . . NOUN cana-1707 316 15 2 2 NUM cana-1707 316 16 ( ( PUNCT cana-1707 316 17 2025 2025 NUM cana-1707 316 18 ) ) PUNCT cana-1707 316 19 63 63 NUM cana-1707 316 20 https://internationalpubls.com https://internationalpubls.com X cana-1707 317 1 [ [ X cana-1707 317 2 42 42 NUM cana-1707 317 3 ] ] X cana-1707 317 4 mishra mishra PROPN cana-1707 317 5 s.n s.n PROPN cana-1707 317 6 . . PROPN cana-1707 317 7 ( ( PUNCT cana-1707 317 8 1991 1991 NUM cana-1707 317 9 ) ) PUNCT cana-1707 317 10 , , PUNCT cana-1707 317 11 common common ADJ cana-1707 317 12 fixed fix VERB cana-1707 317 13 points point NOUN cana-1707 317 14 of of ADP cana-1707 317 15 compatible compatible ADJ cana-1707 317 16 mappings mapping NOUN cana-1707 317 17 in in ADP cana-1707 317 18 probabilistic probabilistic ADJ cana-1707 317 19 metric metric ADJ cana-1707 317 20 space space NOUN cana-1707 317 21 , , PUNCT cana-1707 317 22 math.japon.36 math.japon.36 PROPN cana-1707 317 23 , , PUNCT cana-1707 317 24 283289 283289 NUM cana-1707 317 25 . . PUNCT cana-1707 318 1 [ [ X cana-1707 318 2 43 43 NUM cana-1707 318 3 ] ] X cana-1707 318 4 picard picard PROPN cana-1707 318 5 e. e. PROPN cana-1707 318 6 ( ( PUNCT cana-1707 318 7 1890 1890 NUM cana-1707 318 8 ) ) PUNCT cana-1707 318 9 , , PUNCT cana-1707 318 10 memoire memoire PROPN cana-1707 318 11 sur sur PROPN cana-1707 318 12 la la PROPN cana-1707 318 13 theorie theorie PROPN cana-1707 318 14 des des PROPN cana-1707 318 15 equations equations PROPN cana-1707 318 16 aux aux PROPN cana-1707 318 17 derivees derivee NOUN cana-1707 318 18 partielleset partielleset VERB cana-1707 318 19 la la PRON cana-1707 318 20 methode methode PROPN cana-1707 318 21 des des PROPN cana-1707 318 22 approximations approximations PROPN cana-1707 318 23 successives successive NOUN cana-1707 318 24 , , PUNCT cana-1707 318 25 journal journal PROPN cana-1707 318 26 de de PROPN cana-1707 318 27 math_ematiques math_ematiques PROPN cana-1707 318 28 pures pure NOUN cana-1707 318 29 et et PROPN cana-1707 318 30 appliquees applique VERB cana-1707 318 31 6 6 NUM cana-1707 318 32 , , PUNCT cana-1707 318 33 145 145 NUM cana-1707 318 34 - - SYM cana-1707 318 35 210 210 NUM cana-1707 318 36 . . PUNCT cana-1707 319 1 [ [ X cana-1707 319 2 44 44 NUM cana-1707 319 3 ] ] PUNCT cana-1707 319 4 park park NOUN cana-1707 319 5 s. s. PROPN cana-1707 319 6 ( ( PUNCT cana-1707 319 7 1980 1980 NUM cana-1707 319 8 ) ) PUNCT cana-1707 319 9 , , PUNCT cana-1707 319 10 on on ADP cana-1707 319 11 general general ADJ cana-1707 319 12 contractive contractive ADJ cana-1707 319 13 type type NOUN cana-1707 319 14 conditions condition NOUN cana-1707 319 15 , , PUNCT cana-1707 319 16 journal journal NOUN cana-1707 319 17 korean korean PROPN cana-1707 319 18 mathematical mathematical PROPN cana-1707 319 19 society society NOUN cana-1707 319 20 , , PUNCT cana-1707 319 21 vol vol NOUN cana-1707 319 22 . . PROPN cana-1707 319 23 17 17 NUM cana-1707 319 24 , , PUNCT cana-1707 319 25 1 1 NUM cana-1707 319 26 , , PUNCT cana-1707 319 27 131 131 NUM cana-1707 319 28 - - SYM cana-1707 319 29 140 140 NUM cana-1707 319 30 . . PUNCT cana-1707 320 1 [ [ X cana-1707 320 2 45 45 NUM cana-1707 320 3 ] ] PUNCT cana-1707 320 4 park park NOUN cana-1707 320 5 s. s. PROPN cana-1707 320 6 ( ( PUNCT cana-1707 320 7 2024 2024 NUM cana-1707 320 8 ) ) PUNCT cana-1707 320 9 , , PUNCT cana-1707 320 10 improving improve VERB cana-1707 320 11 many many ADJ cana-1707 320 12 metrics metric NOUN cana-1707 320 13 fixed fix VERB cana-1707 320 14 point point NOUN cana-1707 320 15 theorems theorem NOUN cana-1707 320 16 , , PUNCT cana-1707 320 17 letters letter NOUN cana-1707 320 18 in in ADP cana-1707 320 19 nonlinear nonlinear ADJ cana-1707 320 20 analysis analysis NOUN cana-1707 320 21 and and CCONJ cana-1707 320 22 its its PRON cana-1707 320 23 applications application NOUN cana-1707 320 24 2 2 NUM cana-1707 320 25 , , PUNCT cana-1707 320 26 no no INTJ cana-1707 320 27 . . NOUN cana-1707 320 28 2 2 NUM cana-1707 320 29 , , PUNCT cana-1707 320 30 35–61 35–61 NUM cana-1707 320 31 . . PUNCT cana-1707 321 1 [ [ X cana-1707 321 2 46 46 NUM cana-1707 321 3 ] ] X cana-1707 321 4 rakotech rakotech PROPN cana-1707 321 5 e. e. PROPN cana-1707 321 6 ( ( PUNCT cana-1707 321 7 1962 1962 NUM cana-1707 321 8 ) ) PUNCT cana-1707 321 9 , , PUNCT cana-1707 321 10 a a DET cana-1707 321 11 note note NOUN cana-1707 321 12 on on ADP cana-1707 321 13 contractive contractive ADJ cana-1707 321 14 mappings mapping NOUN cana-1707 321 15 , , PUNCT cana-1707 321 16 proc proc NOUN cana-1707 321 17 . . PUNCT cana-1707 322 1 amer amer PROPN cana-1707 322 2 . . PUNCT cana-1707 322 3 marh marh PROPN cana-1707 322 4 . . PUNCT cana-1707 323 1 sot sot PROPN cana-1707 323 2 . . PROPN cana-1707 323 3 13 13 NUM cana-1707 323 4 , , PUNCT cana-1707 323 5 459 459 NUM cana-1707 323 6 - - SYM cana-1707 323 7 465 465 NUM cana-1707 323 8 . . PUNCT cana-1707 324 1 [ [ X cana-1707 324 2 47 47 NUM cana-1707 324 3 ] ] X cana-1707 324 4 reich reich PROPN cana-1707 324 5 , , PUNCT cana-1707 324 6 s. s. PROPN cana-1707 324 7 ( ( PUNCT cana-1707 324 8 1971 1971 NUM cana-1707 324 9 ) ) PUNCT cana-1707 324 10 . . PUNCT cana-1707 325 1 , , PUNCT cana-1707 325 2 kannan kannan PROPN cana-1707 325 3 ’s ’s PART cana-1707 325 4 fixed fix VERB cana-1707 325 5 point point NOUN cana-1707 325 6 theorem theorem VERB cana-1707 325 7 , , PUNCT cana-1707 325 8 bollettino bollettino PROPN cana-1707 325 9 u.m.i u.m.i PROPN cana-1707 325 10 . . PUNCT cana-1707 326 1 ( ( PUNCT cana-1707 326 2 4 4 NUM cana-1707 326 3 ) ) PUNCT cana-1707 326 4 4 4 NUM cana-1707 326 5 , , PUNCT cana-1707 326 6 l-11 l-11 ADJ cana-1707 326 7 . . PUNCT cana-1707 327 1 [ [ X cana-1707 327 2 48 48 NUM cana-1707 327 3 ] ] SYM cana-1707 327 4 reich reich PROPN cana-1707 327 5 , , PUNCT cana-1707 327 6 s. s. PROPN cana-1707 327 7 ( ( PUNCT cana-1707 327 8 1971 1971 NUM cana-1707 327 9 ) ) PUNCT cana-1707 327 10 . . PUNCT cana-1707 328 1 , , PUNCT cana-1707 328 2 some some DET cana-1707 328 3 remarks remark NOUN cana-1707 328 4 concerning concern VERB cana-1707 328 5 contraction contraction NOUN cana-1707 328 6 mappings mapping NOUN cana-1707 328 7 , , PUNCT cana-1707 328 8 canad canad PROPN cana-1707 328 9 . . PUNCT cana-1707 329 1 math math NOUN cana-1707 329 2 . . PUNCT cana-1707 330 1 bull bull PROPN cana-1707 330 2 . . PUNCT cana-1707 330 3 , , PUNCT cana-1707 330 4 vol vol NOUN cana-1707 330 5 . . PROPN cana-1707 331 1 14 14 NUM cana-1707 331 2 ( ( PUNCT cana-1707 331 3 1 1 NUM cana-1707 331 4 ) ) PUNCT cana-1707 331 5 . . PUNCT cana-1707 332 1 121 121 NUM cana-1707 332 2 - - SYM cana-1707 332 3 124 124 NUM cana-1707 332 4 . . PUNCT cana-1707 333 1 [ [ X cana-1707 333 2 49 49 NUM cana-1707 333 3 ] ] X cana-1707 333 4 rhoades rhoades PROPN cana-1707 333 5 b. b. PROPN cana-1707 333 6 e. e. PROPN cana-1707 333 7 ( ( PUNCT cana-1707 333 8 1977 1977 NUM cana-1707 333 9 ) ) PUNCT cana-1707 333 10 , , PUNCT cana-1707 333 11 a a DET cana-1707 333 12 comparison comparison NOUN cana-1707 333 13 of of ADP cana-1707 333 14 various various ADJ cana-1707 333 15 definitions definition NOUN cana-1707 333 16 of of ADP cana-1707 333 17 contractive contractive ADJ cana-1707 333 18 mappings mapping NOUN cana-1707 333 19 . . PUNCT cana-1707 334 1 trans trans PROPN cana-1707 334 2 . . PUNCT cana-1707 335 1 [ [ X cana-1707 335 2 50 50 NUM cana-1707 335 3 ] ] SYM cana-1707 335 4 amer amer PROPN cana-1707 335 5 . . PROPN cana-1707 335 6 math math PROPN cana-1707 335 7 . . PUNCT cana-1707 336 1 soc soc PROPN cana-1707 336 2 . . PUNCT cana-1707 337 1 226 226 NUM cana-1707 337 2 , , PUNCT cana-1707 337 3 257 257 NUM cana-1707 337 4 - - SYM cana-1707 337 5 290 290 NUM cana-1707 337 6 . . PUNCT cana-1707 338 1 [ [ X cana-1707 338 2 51 51 NUM cana-1707 338 3 ] ] X cana-1707 338 4 sklar sklar NOUN cana-1707 338 5 a a PROPN cana-1707 338 6 and and CCONJ cana-1707 338 7 schweizer schweizer PROPN cana-1707 338 8 b b PROPN cana-1707 338 9 ( ( PUNCT cana-1707 338 10 2005 2005 NUM cana-1707 338 11 ) ) PUNCT cana-1707 338 12 probabilistic probabilistic ADJ cana-1707 338 13 metric metric ADJ cana-1707 338 14 space space NOUN cana-1707 338 15 . . PUNCT cana-1707 339 1 dover dover PROPN cana-1707 339 2 publications publications PROPN cana-1707 339 3 , , PUNCT cana-1707 339 4 inc inc PROPN cana-1707 339 5 , , PUNCT cana-1707 339 6 mineola mineola PROPN cana-1707 339 7 , , PUNCT cana-1707 339 8 new new PROPN cana-1707 339 9 york york PROPN cana-1707 339 10 . . PUNCT cana-1707 340 1 [ [ X cana-1707 340 2 52 52 NUM cana-1707 340 3 ] ] SYM cana-1707 340 4 sehgal sehgal PROPN cana-1707 340 5 v. v. ADP cana-1707 340 6 m. m. NOUN cana-1707 340 7 ( ( PUNCT cana-1707 340 8 1966 1966 NUM cana-1707 340 9 ) ) PUNCT cana-1707 340 10 , , PUNCT cana-1707 340 11 some some DET cana-1707 340 12 common common ADJ cana-1707 340 13 fixed fix VERB cana-1707 340 14 point point NOUN cana-1707 340 15 theorem theorem VERB cana-1707 340 16 in in ADP cana-1707 340 17 functional functional ADJ cana-1707 340 18 analysis analysis NOUN cana-1707 340 19 and and CCONJ cana-1707 340 20 probability probability NOUN cana-1707 340 21 , , PUNCT cana-1707 340 22 phd phd NOUN cana-1707 340 23 thesis thesis NOUN cana-1707 340 24 , , PUNCT cana-1707 340 25 wayne wayne PROPN cana-1707 340 26 state state PROPN cana-1707 340 27 university university PROPN cana-1707 340 28 , , PUNCT cana-1707 340 29 usa usa PROPN cana-1707 340 30 . . PUNCT cana-1707 341 1 [ [ X cana-1707 341 2 53 53 NUM cana-1707 341 3 ] ] PUNCT cana-1707 341 4 sehgal sehgal PROPN cana-1707 341 5 v.m v.m PROPN cana-1707 341 6 . . PROPN cana-1707 342 1 and and CCONJ cana-1707 342 2 bharucha bharucha ADV cana-1707 342 3 - - PUNCT cana-1707 342 4 reid reid PROPN cana-1707 342 5 a.t a.t PROPN cana-1707 342 6 . . PUNCT cana-1707 342 7 ( ( PUNCT cana-1707 342 8 1972 1972 NUM cana-1707 342 9 ) ) PUNCT cana-1707 342 10 , , PUNCT cana-1707 342 11 fixed fix VERB cana-1707 342 12 point point NOUN cana-1707 342 13 contraction contraction NOUN cana-1707 342 14 mapping mapping NOUN cana-1707 342 15 in in ADP cana-1707 342 16 probabilistic probabilistic ADJ cana-1707 342 17 metric metric ADJ cana-1707 342 18 space space NOUN cana-1707 342 19 . . PUNCT cana-1707 343 1 math math NOUN cana-1707 343 2 system system NOUN cana-1707 343 3 theory theory NOUN cana-1707 343 4 , , PUNCT cana-1707 343 5 6 6 NUM cana-1707 343 6 , , PUNCT cana-1707 343 7 97 97 NUM cana-1707 343 8 - - SYM cana-1707 343 9 102 102 NUM cana-1707 343 10 . . PUNCT cana-1707 344 1 [ [ X cana-1707 344 2 54 54 NUM cana-1707 344 3 ] ] SYM cana-1707 344 4 sehgal sehgal PROPN cana-1707 344 5 , , PUNCT cana-1707 344 6 v.m(1972 v.m(1972 PROPN cana-1707 344 7 ) ) PUNCT cana-1707 344 8 , , PUNCT cana-1707 344 9 on on ADP cana-1707 344 10 fixed fixed ADJ cana-1707 344 11 and and CCONJ cana-1707 344 12 periodic periodic ADJ cana-1707 344 13 points point NOUN cana-1707 344 14 for for ADP cana-1707 344 15 a a DET cana-1707 344 16 class class NOUN cana-1707 344 17 of of ADP cana-1707 344 18 mappings mapping NOUN cana-1707 344 19 , , PUNCT cana-1707 344 20 j. j. PROPN cana-1707 344 21 london london PROPN cana-1707 344 22 math math PROPN cana-1707 344 23 . . PUNCT cana-1707 345 1 sot sot PROPN cana-1707 345 2 . . PUNCT cana-1707 346 1 ( ( PUNCT cana-1707 346 2 2 2 NUM cana-1707 346 3 ) ) PUNCT cana-1707 346 4 5 5 NUM cana-1707 346 5 , , PUNCT cana-1707 346 6 571 571 NUM cana-1707 346 7 - - SYM cana-1707 346 8 576 576 NUM cana-1707 346 9 . . PUNCT cana-1707 347 1 [ [ X cana-1707 347 2 55 55 NUM cana-1707 347 3 ] ] X cana-1707 347 4 singh singh PROPN cana-1707 347 5 s. s. PROPN cana-1707 347 6 p. p. PROPN cana-1707 347 7 ( ( PUNCT cana-1707 347 8 1969 1969 NUM cana-1707 347 9 ) ) PUNCT cana-1707 347 10 , , PUNCT cana-1707 347 11 some some DET cana-1707 347 12 results result NOUN cana-1707 347 13 of of ADP cana-1707 347 14 fixed fix VERB cana-1707 347 15 point point NOUN cana-1707 347 16 theorems theorem NOUN cana-1707 347 17 , , PUNCT cana-1707 347 18 yokohama yokohama PROPN cana-1707 347 19 math math PROPN cana-1707 347 20 . . PUNCT cana-1707 348 1 j. j. PROPN cana-1707 348 2 17 17 PROPN cana-1707 348 3 , , PUNCT cana-1707 348 4 6 6 NUM cana-1707 348 5 1 1 NUM cana-1707 348 6 - - SYM cana-1707 348 7 64 64 NUM cana-1707 348 8 . . PUNCT cana-1707 349 1 mr mr PROPN cana-1707 350 1 [ [ X cana-1707 350 2 56 56 NUM cana-1707 350 3 ] ] SYM cana-1707 350 4 41 41 NUM cana-1707 350 5 , , PUNCT cana-1707 350 6 2488 2488 NUM cana-1707 351 1 [ [ X cana-1707 351 2 57 57 NUM cana-1707 351 3 ] ] PUNCT cana-1707 351 4 suzuki suzuki PROPN cana-1707 351 5 t. t. PROPN cana-1707 351 6 ( ( PUNCT cana-1707 351 7 2008 2008 NUM cana-1707 351 8 ) ) PUNCT cana-1707 351 9 , , PUNCT cana-1707 351 10 a a DET cana-1707 351 11 generalized generalized ADJ cana-1707 351 12 banach banach NOUN cana-1707 351 13 contraction contraction NOUN cana-1707 351 14 principle principle NOUN cana-1707 351 15 that that PRON cana-1707 351 16 characterizes characterize VERB cana-1707 351 17 metric metric ADJ cana-1707 351 18 completeness completeness NOUN cana-1707 351 19 , , PUNCT cana-1707 351 20 proc proc NOUN cana-1707 351 21 . . PUNCT cana-1707 352 1 amer amer PROPN cana-1707 352 2 . . PUNCT cana-1707 352 3 math math PROPN cana-1707 352 4 . . PUNCT cana-1707 353 1 soc soc PROPN cana-1707 353 2 . . PUNCT cana-1707 354 1 136 136 NUM cana-1707 354 2 1861–1869 1861–1869 NUM cana-1707 354 3 . . PUNCT cana-1707 355 1 [ [ X cana-1707 355 2 58 58 NUM cana-1707 355 3 ] ] PUNCT cana-1707 355 4 wardowski wardowski PROPN cana-1707 355 5 d. d. PROPN cana-1707 355 6 ( ( PUNCT cana-1707 355 7 2012 2012 NUM cana-1707 355 8 ) ) PUNCT cana-1707 355 9 , , PUNCT cana-1707 355 10 fixed fix VERB cana-1707 355 11 points point NOUN cana-1707 355 12 of of ADP cana-1707 355 13 a a DET cana-1707 355 14 new new ADJ cana-1707 355 15 type type NOUN cana-1707 355 16 of of ADP cana-1707 355 17 contractive contractive ADJ cana-1707 355 18 mappings mapping NOUN cana-1707 355 19 in in ADP cana-1707 355 20 complete complete ADJ cana-1707 355 21 metric metric ADJ cana-1707 355 22 spaces space NOUN cana-1707 355 23 , , PUNCT cana-1707 355 24 fixed fix VERB cana-1707 355 25 point point NOUN cana-1707 355 26 theory theory NOUN cana-1707 355 27 and and CCONJ cana-1707 355 28 applications application NOUN cana-1707 355 29 2012(94 2012(94 NUM cana-1707 355 30 ) ) PUNCT cana-1707 355 31 . . PUNCT cana-1707 356 1 [ [ X cana-1707 356 2 59 59 NUM cana-1707 356 3 ] ] SYM cana-1707 356 4 yen yen PROPN cana-1707 356 5 c. c. PROPN cana-1707 356 6 l. l. PROPN cana-1707 356 7 ( ( PUNCT cana-1707 356 8 1972 1972 NUM cana-1707 356 9 ) ) PUNCT cana-1707 356 10 , , PUNCT cana-1707 356 11 remark remark NOUN cana-1707 356 12 on on ADP cana-1707 356 13 common common ADJ cana-1707 356 14 fixed fix VERB cana-1707 356 15 points point NOUN cana-1707 356 16 , , PUNCT cana-1707 356 17 tamkang tamkang NOUN cana-1707 356 18 j. j. PROPN cana-1707 356 19 math.3,9 math.3,9 PROPN cana-1707 356 20 5 5 NUM cana-1707 356 21 - - SYM cana-1707 356 22 96 96 NUM cana-1707 356 23 . . PUNCT cana-1707 357 1 [ [ X cana-1707 357 2 60 60 NUM cana-1707 357 3 ] ] PUNCT cana-1707 357 4 zamfirescu zamfirescu NOUN cana-1707 357 5 , , PUNCT cana-1707 357 6 t. t. PROPN cana-1707 357 7 ( ( PUNCT cana-1707 357 8 1972 1972 NUM cana-1707 357 9 ) ) PUNCT cana-1707 357 10 , , PUNCT cana-1707 357 11 fix fix NOUN cana-1707 357 12 point point NOUN cana-1707 357 13 theorems theorem NOUN cana-1707 357 14 in in ADP cana-1707 357 15 metric metric ADJ cana-1707 357 16 spaces space NOUN cana-1707 357 17 , , PUNCT cana-1707 357 18 arch arch NOUN cana-1707 357 19 . . PUNCT cana-1707 358 1 math math NOUN cana-1707 358 2 . . PUNCT cana-1707 359 1 ( ( PUNCT cana-1707 359 2 basel basel PROPN cana-1707 359 3 ) ) PUNCT cana-1707 359 4 23 23 NUM cana-1707 359 5 , , PUNCT cana-1707 359 6 292 292 NUM cana-1707 359 7 - - SYM cana-1707 359 8 298 298 NUM cana-1707 359 9 . . PUNCT