id sid tid token lemma pos iajs-2679 1 1 67 67 NUM iajs-2679 1 2 common common ADJ iajs-2679 1 3 diskcyclic diskcyclic ADJ iajs-2679 1 4 vectors vector NOUN iajs-2679 1 5 nareen nareen VERB iajs-2679 1 6 bamerni bamerni PROPN iajs-2679 1 7 department department PROPN iajs-2679 1 8 of of ADP iajs-2679 1 9 mathematics mathematics PROPN iajs-2679 1 10 , , PUNCT iajs-2679 1 11 university university NOUN iajs-2679 1 12 of of ADP iajs-2679 1 13 duhok duhok PROPN iajs-2679 1 14 , , PUNCT iajs-2679 1 15 kurdistan kurdistan PROPN iajs-2679 1 16 region region NOUN iajs-2679 1 17 , , PUNCT iajs-2679 1 18 iraq iraq PROPN iajs-2679 1 19 nareen.sabih@uod.ac nareen.sabih@uod.ac NOUN iajs-2679 1 20 abstract abstract NOUN iajs-2679 1 21 in in ADP iajs-2679 1 22 this this DET iajs-2679 1 23 paper paper NOUN iajs-2679 1 24 , , PUNCT iajs-2679 1 25 we we PRON iajs-2679 1 26 study study VERB iajs-2679 1 27 the the DET iajs-2679 1 28 common common ADJ iajs-2679 1 29 diskcyclic diskcyclic ADJ iajs-2679 1 30 vectors vector NOUN iajs-2679 1 31 for for ADP iajs-2679 1 32 a a DET iajs-2679 1 33 path path NOUN iajs-2679 1 34 of of ADP iajs-2679 1 35 diskcyclic diskcyclic ADJ iajs-2679 1 36 operators operator NOUN iajs-2679 1 37 . . PUNCT iajs-2679 2 1 in in ADP iajs-2679 2 2 particular particular ADJ iajs-2679 2 3 , , PUNCT iajs-2679 2 4 if if SCONJ iajs-2679 2 5 { { PUNCT iajs-2679 2 6 𝑇𝑑 𝑇𝑑 ADJ iajs-2679 2 7 : : PUNCT iajs-2679 2 8 𝑑 𝑑 PROPN iajs-2679 2 9 ∈ ∈ PROPN iajs-2679 2 10 [ [ X iajs-2679 2 11 π‘Ž π‘Ž X iajs-2679 2 12 , , PUNCT iajs-2679 2 13 𝑏 𝑏 NOUN iajs-2679 2 14 ] ] PUNCT iajs-2679 2 15 } } PUNCT iajs-2679 2 16 is be AUX iajs-2679 2 17 a a DET iajs-2679 2 18 path path NOUN iajs-2679 2 19 of of ADP iajs-2679 2 20 diskcyclic diskcyclic PROPN iajs-2679 2 21 operators operator NOUN iajs-2679 2 22 , , PUNCT iajs-2679 2 23 we we PRON iajs-2679 2 24 show show VERB iajs-2679 2 25 that that SCONJ iajs-2679 2 26 under under ADP iajs-2679 2 27 certain certain ADJ iajs-2679 2 28 conditions condition NOUN iajs-2679 2 29 the the DET iajs-2679 2 30 intersection intersection NOUN iajs-2679 2 31 of of ADP iajs-2679 2 32 diskcyclic diskcyclic ADJ iajs-2679 2 33 vectors vector NOUN iajs-2679 2 34 for for ADP iajs-2679 2 35 theses thesis NOUN iajs-2679 2 36 operators operator NOUN iajs-2679 2 37 is be AUX iajs-2679 2 38 a a DET iajs-2679 2 39 dense dense ADJ iajs-2679 2 40 gΞ΄ gΞ΄ NOUN iajs-2679 2 41 set set NOUN iajs-2679 2 42 . . PUNCT iajs-2679 3 1 keywords keyword NOUN iajs-2679 3 2 : : PUNCT iajs-2679 3 3 diskcyclic diskcyclic PROPN iajs-2679 3 4 operators operator NOUN iajs-2679 3 5 , , PUNCT iajs-2679 3 6 common common ADJ iajs-2679 3 7 diskcyclic diskcyclic ADJ iajs-2679 3 8 vectors vector NOUN iajs-2679 3 9 , , PUNCT iajs-2679 3 10 weighted weight VERB iajs-2679 3 11 shift shift NOUN iajs-2679 3 12 operators operator NOUN iajs-2679 3 13 . . PUNCT iajs-2679 4 1 1 1 X iajs-2679 4 2 . . X iajs-2679 4 3 introduction introduction NOUN iajs-2679 4 4 π‘‚π‘Ÿπ‘(𝑇 π‘‚π‘Ÿπ‘(𝑇 NOUN iajs-2679 4 5 , , PUNCT iajs-2679 4 6 π‘₯ π‘₯ NOUN iajs-2679 4 7 ) ) PUNCT iajs-2679 4 8 = = PRON iajs-2679 4 9 { { PUNCT iajs-2679 4 10 𝑇𝑛π‘₯ 𝑇𝑛π‘₯ NOUN iajs-2679 4 11 : : PUNCT iajs-2679 4 12 𝑛 𝑛 PROPN iajs-2679 4 13 ∈ ∈ PROPN iajs-2679 4 14 β„• β„• PROPN iajs-2679 4 15 } } PUNCT iajs-2679 4 16 which which PRON iajs-2679 4 17 is be AUX iajs-2679 4 18 dense dense ADJ iajs-2679 4 19 in in ADP iajs-2679 4 20 𝑋. 𝑋. PROPN iajs-2679 4 21 the the DET iajs-2679 4 22 study study NOUN iajs-2679 4 23 of of ADP iajs-2679 4 24 hypercyclic hypercyclic ADJ iajs-2679 4 25 operators operator NOUN iajs-2679 4 26 on on ADP iajs-2679 4 27 a a DET iajs-2679 4 28 banach banach NOUN iajs-2679 4 29 space space NOUN iajs-2679 4 30 goes go VERB iajs-2679 4 31 back back ADV iajs-2679 4 32 to to ADP iajs-2679 4 33 a a DET iajs-2679 4 34 1969 1969 NUM iajs-2679 4 35 paper paper NOUN iajs-2679 4 36 of of ADP iajs-2679 4 37 rolewi rolewi NOUN iajs-2679 4 38 let let VERB iajs-2679 4 39 𝑋 𝑋 NOUN iajs-2679 4 40 be be AUX iajs-2679 4 41 a a DET iajs-2679 4 42 banach banach NOUN iajs-2679 4 43 space space NOUN iajs-2679 4 44 and and CCONJ iajs-2679 4 45 𝐡(𝑋 𝐡(𝑋 PROPN iajs-2679 4 46 ) ) PUNCT iajs-2679 4 47 be be VERB iajs-2679 4 48 the the DET iajs-2679 4 49 space space NOUN iajs-2679 4 50 of of ADP iajs-2679 4 51 all all DET iajs-2679 4 52 bounded bound VERB iajs-2679 4 53 linear linear PROPN iajs-2679 4 54 operators operator NOUN iajs-2679 4 55 on𝑋. on𝑋. VERB iajs-2679 4 56 an an DET iajs-2679 4 57 operator operator NOUN iajs-2679 4 58 𝑇 𝑇 PROPN iajs-2679 4 59 ∈ ∈ PROPN iajs-2679 4 60 𝐡(𝑋 𝐡(𝑋 PROPN iajs-2679 4 61 ) ) PUNCT iajs-2679 4 62 is be AUX iajs-2679 4 63 called call VERB iajs-2679 4 64 hypercyclic hypercyclic ADJ iajs-2679 4 65 if if SCONJ iajs-2679 4 66 there there PRON iajs-2679 4 67 is be VERB iajs-2679 4 68 a a DET iajs-2679 4 69 vector vector NOUN iajs-2679 4 70 π‘₯ π‘₯ PRON iajs-2679 4 71 ∈ ∈ NOUN iajs-2679 4 72 𝑋 𝑋 NOUN iajs-2679 4 73 called call VERB iajs-2679 4 74 hypercyclic hypercyclic ADJ iajs-2679 4 75 vector vector NOUN iajs-2679 4 76 for for ADP iajs-2679 4 77 𝑇 𝑇 PROPN iajs-2679 4 78 such such DET iajs-2679 4 79 that that DET iajs-2679 4 80 cz cz NOUN iajs-2679 4 81 [ [ X iajs-2679 4 82 1 1 NUM iajs-2679 4 83 ] ] PUNCT iajs-2679 4 84 that that PRON iajs-2679 4 85 proves prove VERB iajs-2679 4 86 if if SCONJ iajs-2679 4 87 𝐡 𝐡 NOUN iajs-2679 4 88 is be AUX iajs-2679 4 89 the the DET iajs-2679 4 90 backward backward ADJ iajs-2679 4 91 shift shift NOUN iajs-2679 4 92 on on ADP iajs-2679 4 93 the the DET iajs-2679 4 94 sequence sequence NOUN iajs-2679 4 95 space space NOUN iajs-2679 4 96 𝑙𝑝(β„• 𝑙𝑝(β„• PROPN iajs-2679 4 97 ) ) PUNCT iajs-2679 4 98 of of ADP iajs-2679 4 99 then then ADV iajs-2679 4 100 πœ†π΅ πœ†π΅ PUNCT iajs-2679 4 101 is be AUX iajs-2679 4 102 hypercyclic hypercyclic ADJ iajs-2679 4 103 whenever whenever SCONJ iajs-2679 4 104 πœ† πœ† NOUN iajs-2679 4 105 is be AUX iajs-2679 4 106 a a DET iajs-2679 4 107 scalar scalar NOUN iajs-2679 4 108 of of ADP iajs-2679 4 109 modulus modulus NOUN iajs-2679 4 110 > > X iajs-2679 4 111 1 1 NUM iajs-2679 4 112 . . PUNCT iajs-2679 5 1 perhaps perhaps ADV iajs-2679 5 2 , , PUNCT iajs-2679 5 3 inspired inspire VERB iajs-2679 5 4 by by ADP iajs-2679 5 5 rolewicz rolewicz ADJ iajs-2679 5 6 example example NOUN iajs-2679 5 7 , , PUNCT iajs-2679 5 8 hilden hilden PROPN iajs-2679 5 9 and and CCONJ iajs-2679 5 10 wallen wallen PROPN iajs-2679 6 1 [ [ X iajs-2679 6 2 2 2 NUM iajs-2679 6 3 ] ] PUNCT iajs-2679 6 4 considered consider VERB iajs-2679 6 5 the the DET iajs-2679 6 6 scaled scale VERB iajs-2679 6 7 orbit orbit NOUN iajs-2679 6 8 of of ADP iajs-2679 6 9 an an DET iajs-2679 6 10 operator operator NOUN iajs-2679 6 11 . . PUNCT iajs-2679 7 1 an an DET iajs-2679 7 2 operator operator NOUN iajs-2679 7 3 𝑇 𝑇 PROPN iajs-2679 7 4 is be AUX iajs-2679 7 5 supercyclic supercyclic ADJ iajs-2679 7 6 if if SCONJ iajs-2679 7 7 there there PRON iajs-2679 7 8 is be VERB iajs-2679 7 9 a a DET iajs-2679 7 10 vector vector NOUN iajs-2679 7 11 π‘₯ π‘₯ PRON iajs-2679 7 12 ∈ ∈ NOUN iajs-2679 7 13 𝑋 𝑋 NOUN iajs-2679 7 14 called call VERB iajs-2679 7 15 supercyclic supercyclic ADJ iajs-2679 7 16 vector vector NOUN iajs-2679 7 17 for for ADP iajs-2679 7 18 𝑇 𝑇 PROPN iajs-2679 7 19 such such ADJ iajs-2679 7 20 that that SCONJ iajs-2679 7 21 β„‚π‘‚π‘Ÿπ‘(𝑇 β„‚π‘‚π‘Ÿπ‘(𝑇 ADV iajs-2679 7 22 , , PUNCT iajs-2679 7 23 π‘₯ π‘₯ NOUN iajs-2679 7 24 ) ) PUNCT iajs-2679 7 25 = = SYM iajs-2679 7 26 { { PUNCT iajs-2679 7 27 πœ†π‘‡π‘›π‘₯ πœ†π‘‡π‘›π‘₯ PROPN iajs-2679 7 28 : : PUNCT iajs-2679 7 29 πœ† πœ† PROPN iajs-2679 7 30 ∈ ∈ PROPN iajs-2679 7 31 β„‚ β„‚ PROPN iajs-2679 7 32 , , PUNCT iajs-2679 7 33 𝑛 𝑛 PRON iajs-2679 7 34 ∈ ∈ PROPN iajs-2679 7 35 β„• β„• PROPN iajs-2679 7 36 } } PUNCT iajs-2679 7 37 is be AUX iajs-2679 7 38 dense dense ADJ iajs-2679 7 39 in in ADP iajs-2679 7 40 𝑋. 𝑋. PROPN iajs-2679 7 41 also also ADV iajs-2679 7 42 , , PUNCT iajs-2679 7 43 an an DET iajs-2679 7 44 operator operator NOUN iajs-2679 7 45 𝑇 𝑇 PROPN iajs-2679 7 46 is be AUX iajs-2679 7 47 called call VERB iajs-2679 7 48 diskcyclic diskcyclic ADJ iajs-2679 7 49 if if SCONJ iajs-2679 7 50 there there PRON iajs-2679 7 51 is be VERB iajs-2679 7 52 a a DET iajs-2679 7 53 vector vector NOUN iajs-2679 7 54 π‘₯ π‘₯ PRON iajs-2679 7 55 ∈ ∈ NOUN iajs-2679 7 56 𝑋 𝑋 NOUN iajs-2679 7 57 called call VERB iajs-2679 7 58 diskcyclic diskcyclic ADJ iajs-2679 7 59 vector vector NOUN iajs-2679 7 60 for for ADP iajs-2679 7 61 𝑇 𝑇 PROPN iajs-2679 7 62 such such ADJ iajs-2679 7 63 that that SCONJ iajs-2679 7 64 the the DET iajs-2679 7 65 disk disk NOUN iajs-2679 7 66 orbit orbit NOUN iajs-2679 7 67 π”»π‘‚π‘Ÿπ‘(𝑇 π”»π‘‚π‘Ÿπ‘(𝑇 NOUN iajs-2679 7 68 , , PUNCT iajs-2679 7 69 π‘₯ π‘₯ NOUN iajs-2679 7 70 ) ) PUNCT iajs-2679 7 71 = = SYM iajs-2679 7 72 { { PUNCT iajs-2679 7 73 πœ†π‘‡π‘›π‘₯ πœ†π‘‡π‘›π‘₯ PROPN iajs-2679 7 74 : : PUNCT iajs-2679 7 75 πœ† πœ† PROPN iajs-2679 7 76 ∈ ∈ PROPN iajs-2679 7 77 β„‚ β„‚ PROPN iajs-2679 7 78 , , PUNCT iajs-2679 7 79 |πœ†| |πœ†| ADP iajs-2679 7 80 ≀ ≀ NOUN iajs-2679 7 81 1 1 NUM iajs-2679 7 82 , , PUNCT iajs-2679 7 83 𝑛 𝑛 DET iajs-2679 7 84 ∈ ∈ PROPN iajs-2679 7 85 β„• β„• PROPN iajs-2679 7 86 } } PUNCT iajs-2679 7 87 is be AUX iajs-2679 7 88 dense dense ADJ iajs-2679 7 89 in in ADP iajs-2679 7 90 𝑋 𝑋 PROPN iajs-2679 7 91 [ [ X iajs-2679 7 92 3 3 NUM iajs-2679 7 93 ] ] PUNCT iajs-2679 7 94 . . PUNCT iajs-2679 8 1 for for ADP iajs-2679 8 2 more more ADJ iajs-2679 8 3 information information NOUN iajs-2679 8 4 on on ADP iajs-2679 8 5 these these DET iajs-2679 8 6 operators operator NOUN iajs-2679 8 7 , , PUNCT iajs-2679 8 8 the the DET iajs-2679 8 9 reader reader NOUN iajs-2679 8 10 may may AUX iajs-2679 8 11 refer refer VERB iajs-2679 8 12 to to ADP iajs-2679 8 13 [ [ X iajs-2679 8 14 46 46 NUM iajs-2679 8 15 ] ] PUNCT iajs-2679 8 16 . . PUNCT iajs-2679 9 1 recently recently ADV iajs-2679 9 2 , , PUNCT iajs-2679 9 3 the the DET iajs-2679 9 4 orbit orbit NOUN iajs-2679 9 5 of of ADP iajs-2679 9 6 an an DET iajs-2679 9 7 operator operator NOUN iajs-2679 9 8 in in ADP iajs-2679 9 9 subspaces subspace NOUN iajs-2679 9 10 was be AUX iajs-2679 9 11 studied study VERB iajs-2679 9 12 . . PUNCT iajs-2679 10 1 more more ADV iajs-2679 10 2 precisely precisely ADV iajs-2679 10 3 , , PUNCT iajs-2679 10 4 if if SCONJ iajs-2679 10 5 the the DET iajs-2679 10 6 orbit orbit NOUN iajs-2679 10 7 of of ADP iajs-2679 10 8 an an DET iajs-2679 10 9 operator operator NOUN iajs-2679 10 10 is be AUX iajs-2679 10 11 dense dense ADJ iajs-2679 10 12 in in ADP iajs-2679 10 13 a a DET iajs-2679 10 14 subspace subspace NOUN iajs-2679 10 15 , , PUNCT iajs-2679 10 16 then then ADV iajs-2679 10 17 such such DET iajs-2679 10 18 an an DET iajs-2679 10 19 operator operator NOUN iajs-2679 10 20 is be AUX iajs-2679 10 21 called call VERB iajs-2679 10 22 subspace subspace NOUN iajs-2679 10 23 - - PUNCT iajs-2679 10 24 hypercyclic hypercyclic NOUN iajs-2679 10 25 . . PUNCT iajs-2679 11 1 by by ADP iajs-2679 11 2 , , PUNCT iajs-2679 11 3 the the DET iajs-2679 11 4 same same ADJ iajs-2679 11 5 manner manner NOUN iajs-2679 11 6 if if SCONJ iajs-2679 11 7 the the DET iajs-2679 11 8 scaled scale VERB iajs-2679 11 9 orbit orbit NOUN iajs-2679 11 10 ( ( PUNCT iajs-2679 11 11 disk disk NOUN iajs-2679 11 12 orbit orbit NOUN iajs-2679 11 13 ) ) PUNCT iajs-2679 11 14 of of ADP iajs-2679 11 15 an an DET iajs-2679 11 16 operator operator NOUN iajs-2679 11 17 is be AUX iajs-2679 11 18 dense dense ADJ iajs-2679 11 19 in in ADP iajs-2679 11 20 a a DET iajs-2679 11 21 subspace subspace NOUN iajs-2679 11 22 , , PUNCT iajs-2679 11 23 then then ADV iajs-2679 11 24 such such DET iajs-2679 11 25 an an DET iajs-2679 11 26 operator operator NOUN iajs-2679 11 27 is be AUX iajs-2679 11 28 called call VERB iajs-2679 11 29 subspace subspace NOUN iajs-2679 11 30 - - PUNCT iajs-2679 11 31 supercyclic supercyclic NOUN iajs-2679 11 32 ( ( PUNCT iajs-2679 11 33 subspace subspace NOUN iajs-2679 11 34 - - PUNCT iajs-2679 11 35 diskcyclic diskcyclic PROPN iajs-2679 11 36 ) ) PUNCT iajs-2679 11 37 respectively respectively ADV iajs-2679 11 38 . . PUNCT iajs-2679 12 1 for for ADP iajs-2679 12 2 more more ADJ iajs-2679 12 3 information information NOUN iajs-2679 12 4 on on ADP iajs-2679 12 5 these these DET iajs-2679 12 6 operators operator NOUN iajs-2679 12 7 , , PUNCT iajs-2679 12 8 the the DET iajs-2679 12 9 reader reader NOUN iajs-2679 12 10 may may AUX iajs-2679 12 11 refer refer VERB iajs-2679 12 12 to to ADP iajs-2679 12 13 [ [ X iajs-2679 12 14 710 710 NUM iajs-2679 12 15 ] ] PUNCT iajs-2679 12 16 . . PUNCT iajs-2679 13 1 ibn ibn PROPN iajs-2679 13 2 al al PROPN iajs-2679 13 3 haitham haitham PROPN iajs-2679 13 4 journal journal PROPN iajs-2679 13 5 for for ADP iajs-2679 13 6 pure pure ADJ iajs-2679 13 7 and and CCONJ iajs-2679 13 8 applied apply VERB iajs-2679 13 9 science science NOUN iajs-2679 13 10 journal journal PROPN iajs-2679 13 11 homepage homepage NOUN iajs-2679 13 12 : : PUNCT iajs-2679 13 13 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2679 13 14 doi doi NOUN iajs-2679 13 15 : : PUNCT iajs-2679 13 16 10.30526/34.3.2679 10.30526/34.3.2679 PROPN iajs-2679 13 17 article article NOUN iajs-2679 13 18 history history NOUN iajs-2679 13 19 : : PUNCT iajs-2679 13 20 received receive VERB iajs-2679 13 21 5 5 NUM iajs-2679 13 22 november november PROPN iajs-2679 13 23 2020 2020 NUM iajs-2679 13 24 , , PUNCT iajs-2679 13 25 accepted accept VERB iajs-2679 13 26 11 11 NUM iajs-2679 13 27 april april PROPN iajs-2679 13 28 2021 2021 NUM iajs-2679 13 29 , , PUNCT iajs-2679 13 30 published publish VERB iajs-2679 13 31 in in ADP iajs-2679 13 32 2021 2021 NUM iajs-2679 13 33 . . PUNCT iajs-2679 14 1 mailto:nareen.sabih@uod.ac mailto:nareen.sabih@uod.ac VERB iajs-2679 14 2 ibn ibn PROPN iajs-2679 14 3 al al PROPN iajs-2679 14 4 - - PUNCT iajs-2679 14 5 haitham haitham PROPN iajs-2679 14 6 jour jour X iajs-2679 14 7 . . PROPN iajs-2679 15 1 for for ADP iajs-2679 15 2 pure pure ADJ iajs-2679 15 3 & & CCONJ iajs-2679 15 4 appl appl PROPN iajs-2679 15 5 . . PUNCT iajs-2679 16 1 sci sci PROPN iajs-2679 16 2 . . PROPN iajs-2679 17 1 34(3)2021 34(3)2021 NUM iajs-2679 17 2 68 68 NUM iajs-2679 17 3 of of ADP iajs-2679 17 4 real real ADJ iajs-2679 17 5 numbers number NOUN iajs-2679 17 6 , , PUNCT iajs-2679 17 7 then then ADV iajs-2679 17 8 , , PUNCT iajs-2679 17 9 the the DET iajs-2679 17 10 set set NOUN iajs-2679 17 11 { { PUNCT iajs-2679 17 12 𝑇𝑑 𝑇𝑑 ADJ iajs-2679 17 13 : : PUNCT iajs-2679 17 14 𝑑 𝑑 PROPN iajs-2679 17 15 ∈ ∈ PROPN iajs-2679 17 16 [ [ X iajs-2679 17 17 π‘Ž π‘Ž X iajs-2679 17 18 , , PUNCT iajs-2679 17 19 𝑏 𝑏 NOUN iajs-2679 17 20 ] ] PUNCT iajs-2679 17 21 } } PUNCT iajs-2679 17 22 is be AUX iajs-2679 17 23 called call VERB iajs-2679 17 24 a a DET iajs-2679 17 25 path path NOUN iajs-2679 17 26 of of ADP iajs-2679 17 27 operators operator NOUN iajs-2679 17 28 if if SCONJ iajs-2679 17 29 𝑇𝑑 𝑇𝑑 PROPN iajs-2679 17 30 ∈ ∈ PROPN iajs-2679 17 31 𝐡(𝑋 𝐡(𝑋 PROPN iajs-2679 17 32 ) ) PUNCT iajs-2679 17 33 for for SCONJ iajs-2679 17 34 it it PRON iajs-2679 17 35 has have AUX iajs-2679 17 36 been be AUX iajs-2679 17 37 studied study VERB iajs-2679 17 38 that that SCONJ iajs-2679 17 39 under under ADP iajs-2679 17 40 certain certain ADJ iajs-2679 17 41 conditions condition NOUN iajs-2679 17 42 , , PUNCT iajs-2679 17 43 an an DET iajs-2679 17 44 uncountable uncountable ADJ iajs-2679 17 45 family family NOUN iajs-2679 17 46 of of ADP iajs-2679 17 47 hypercyclic hypercyclic ADJ iajs-2679 17 48 operators operator NOUN iajs-2679 17 49 ( ( PUNCT iajs-2679 17 50 or or CCONJ iajs-2679 17 51 supercyclic supercyclic ADJ iajs-2679 17 52 operators operator NOUN iajs-2679 17 53 ) ) PUNCT iajs-2679 17 54 has have VERB iajs-2679 17 55 a a DET iajs-2679 17 56 dense dense ADJ iajs-2679 17 57 gΞ΄ gΞ΄ NOUN iajs-2679 17 58 set set NOUN iajs-2679 17 59 of of ADP iajs-2679 17 60 common common ADJ iajs-2679 17 61 hypercyclic hypercyclic ADJ iajs-2679 17 62 vectors vector NOUN iajs-2679 17 63 ( ( PUNCT iajs-2679 17 64 or or CCONJ iajs-2679 17 65 supercyclic supercyclic ADJ iajs-2679 17 66 vectors vector NOUN iajs-2679 17 67 , , PUNCT iajs-2679 17 68 respectively respectively ADV iajs-2679 17 69 ) ) PUNCT iajs-2679 17 70 . . PUNCT iajs-2679 18 1 for for ADP iajs-2679 18 2 example example NOUN iajs-2679 18 3 , , PUNCT iajs-2679 18 4 [ [ X iajs-2679 18 5 11 11 NUM iajs-2679 18 6 ] ] X iajs-2679 18 7 ( ( PUNCT iajs-2679 18 8 [ [ X iajs-2679 18 9 12 12 NUM iajs-2679 18 10 ] ] PUNCT iajs-2679 18 11 ) ) PUNCT iajs-2679 18 12 gave give VERB iajs-2679 18 13 some some DET iajs-2679 18 14 conditions condition NOUN iajs-2679 18 15 on on ADP iajs-2679 18 16 a a DET iajs-2679 18 17 path path NOUN iajs-2679 18 18 of of ADP iajs-2679 18 19 supercyclic supercyclic NOUN iajs-2679 18 20 ( ( PUNCT iajs-2679 18 21 hypercyclic hypercyclic ADJ iajs-2679 18 22 ) ) PUNCT iajs-2679 18 23 operators operator NOUN iajs-2679 18 24 to to PART iajs-2679 18 25 have have AUX iajs-2679 18 26 a a DET iajs-2679 18 27 common common ADJ iajs-2679 18 28 supercyclic supercyclic NOUN iajs-2679 18 29 ( ( PUNCT iajs-2679 18 30 or or CCONJ iajs-2679 18 31 hypercyclic hypercyclic ADJ iajs-2679 18 32 , , PUNCT iajs-2679 18 33 respectively respectively ADV iajs-2679 18 34 ) ) PUNCT iajs-2679 18 35 vectors vector NOUN iajs-2679 18 36 . . PUNCT iajs-2679 19 1 for for ADP iajs-2679 19 2 more more ADJ iajs-2679 19 3 information information NOUN iajs-2679 19 4 on on ADP iajs-2679 19 5 common common ADJ iajs-2679 19 6 hypercyclic hypercyclic ADJ iajs-2679 19 7 and and CCONJ iajs-2679 19 8 supercyclic supercyclic ADJ iajs-2679 19 9 vectors vector NOUN iajs-2679 19 10 , , PUNCT iajs-2679 19 11 the the DET iajs-2679 19 12 reader reader NOUN iajs-2679 19 13 may may AUX iajs-2679 19 14 refer refer VERB iajs-2679 19 15 to to ADP iajs-2679 19 16 [ [ X iajs-2679 19 17 1319 1319 NUM iajs-2679 19 18 ] ] PUNCT iajs-2679 19 19 . . PUNCT iajs-2679 20 1 now now ADV iajs-2679 20 2 , , PUNCT iajs-2679 20 3 since since SCONJ iajs-2679 20 4 the the DET iajs-2679 20 5 set set NOUN iajs-2679 20 6 of of ADP iajs-2679 20 7 all all DET iajs-2679 20 8 diskcyclic diskcyclic ADJ iajs-2679 20 9 vectors vector NOUN iajs-2679 20 10 is be AUX iajs-2679 20 11 dense dense ADJ iajs-2679 20 12 gΞ΄ gΞ΄ NOUN iajs-2679 20 13 , , PUNCT iajs-2679 20 14 then then ADV iajs-2679 20 15 a a DET iajs-2679 20 16 countable countable ADJ iajs-2679 20 17 collection collection NOUN iajs-2679 20 18 of of ADP iajs-2679 20 19 diskcyclic diskcyclic PROPN iajs-2679 20 20 operators operator NOUN iajs-2679 20 21 , , PUNCT iajs-2679 20 22 by by ADP iajs-2679 20 23 applying apply VERB iajs-2679 20 24 the the DET iajs-2679 20 25 baire baire NOUN iajs-2679 20 26 category category NOUN iajs-2679 20 27 theorem theorem VERB iajs-2679 20 28 , , PUNCT iajs-2679 20 29 has have VERB iajs-2679 20 30 a a DET iajs-2679 20 31 dense dense ADJ iajs-2679 20 32 gΞ΄ gΞ΄ NOUN iajs-2679 20 33 set set NOUN iajs-2679 20 34 of of ADP iajs-2679 20 35 common common ADJ iajs-2679 20 36 diskcyclic diskcyclic ADJ iajs-2679 20 37 vectors vector NOUN iajs-2679 20 38 . . PUNCT iajs-2679 21 1 however however ADV iajs-2679 21 2 , , PUNCT iajs-2679 21 3 it it PRON iajs-2679 21 4 is be AUX iajs-2679 21 5 unknown unknown ADJ iajs-2679 21 6 in in ADP iajs-2679 21 7 which which PRON iajs-2679 21 8 cases case VERB iajs-2679 21 9 an an DET iajs-2679 21 10 uncountable uncountable ADJ iajs-2679 21 11 family family NOUN iajs-2679 21 12 of of ADP iajs-2679 21 13 diskcyclic diskcyclic PROPN iajs-2679 21 14 operators operators PROPN iajs-2679 21 15 has have VERB iajs-2679 21 16 a a DET iajs-2679 21 17 common common ADJ iajs-2679 21 18 diskcyclic diskcyclic ADJ iajs-2679 21 19 vectors vector NOUN iajs-2679 21 20 . . PUNCT iajs-2679 22 1 therefore therefore ADV iajs-2679 22 2 , , PUNCT iajs-2679 22 3 in in ADP iajs-2679 22 4 this this DET iajs-2679 22 5 paper paper NOUN iajs-2679 22 6 , , PUNCT iajs-2679 22 7 we we PRON iajs-2679 22 8 study study VERB iajs-2679 22 9 the the DET iajs-2679 22 10 common common ADJ iajs-2679 22 11 diskcyclic diskcyclic ADJ iajs-2679 22 12 vectors vector NOUN iajs-2679 22 13 for for ADP iajs-2679 22 14 some some DET iajs-2679 22 15 uncountable uncountable ADJ iajs-2679 22 16 families family NOUN iajs-2679 22 17 of of ADP iajs-2679 22 18 diskcyclic diskcyclic PROPN iajs-2679 22 19 operators operator NOUN iajs-2679 22 20 . . PUNCT iajs-2679 23 1 in in ADP iajs-2679 23 2 particular particular ADJ iajs-2679 23 3 , , PUNCT iajs-2679 23 4 we we PRON iajs-2679 23 5 give give VERB iajs-2679 23 6 a a DET iajs-2679 23 7 sufficient sufficient ADJ iajs-2679 23 8 condition condition NOUN iajs-2679 23 9 for for ADP iajs-2679 23 10 a a DET iajs-2679 23 11 path path NOUN iajs-2679 23 12 of of ADP iajs-2679 23 13 operators operator NOUN iajs-2679 23 14 to to PART iajs-2679 23 15 have have VERB iajs-2679 23 16 a a DET iajs-2679 23 17 dense dense ADJ iajs-2679 23 18 gΞ΄ gΞ΄ NOUN iajs-2679 23 19 set set NOUN iajs-2679 23 20 of of ADP iajs-2679 23 21 common common ADJ iajs-2679 23 22 diskcyclic diskcyclic ADJ iajs-2679 23 23 vectors vector NOUN iajs-2679 23 24 , , PUNCT iajs-2679 23 25 every every DET iajs-2679 23 26 operator operator NOUN iajs-2679 23 27 in in ADP iajs-2679 23 28 this this DET iajs-2679 23 29 path path NOUN iajs-2679 23 30 satisfies satisfy VERB iajs-2679 23 31 diskcyclic diskcyclic ADJ iajs-2679 23 32 criterion criterion NOUN iajs-2679 23 33 . . PUNCT iajs-2679 24 1 then then ADV iajs-2679 24 2 , , PUNCT iajs-2679 24 3 we we PRON iajs-2679 24 4 study study VERB iajs-2679 24 5 a a DET iajs-2679 24 6 path path NOUN iajs-2679 24 7 of of ADP iajs-2679 24 8 unilateral unilateral ADJ iajs-2679 24 9 weighted weight VERB iajs-2679 24 10 backward backward ADJ iajs-2679 24 11 shifts shift NOUN iajs-2679 24 12 with with ADP iajs-2679 24 13 common common ADJ iajs-2679 24 14 diskcyclic diskcyclic ADJ iajs-2679 24 15 vectors vector NOUN iajs-2679 24 16 . . PUNCT iajs-2679 25 1 first first ADV iajs-2679 25 2 , , PUNCT iajs-2679 25 3 we we PRON iajs-2679 25 4 recall recall VERB iajs-2679 25 5 the the DET iajs-2679 25 6 following follow VERB iajs-2679 25 7 definition definition NOUN iajs-2679 25 8 from from ADP iajs-2679 25 9 [ [ X iajs-2679 25 10 12 12 NUM iajs-2679 25 11 ] ] PUNCT iajs-2679 25 12 . . PUNCT iajs-2679 26 1 definition definition NOUN iajs-2679 26 2 1.1 1.1 NUM iajs-2679 26 3 . . PUNCT iajs-2679 27 1 let let VERB iajs-2679 27 2 𝑋 𝑋 NOUN iajs-2679 27 3 be be AUX iajs-2679 27 4 a a DET iajs-2679 27 5 banach banach NOUN iajs-2679 27 6 space space NOUN iajs-2679 27 7 and and CCONJ iajs-2679 27 8 [ [ X iajs-2679 27 9 π‘Ž π‘Ž X iajs-2679 27 10 , , PUNCT iajs-2679 27 11 𝑏 𝑏 NOUN iajs-2679 27 12 ] ] PUNCT iajs-2679 27 13 be be AUX iajs-2679 27 14 an an DET iajs-2679 27 15 interval interval NOUN iajs-2679 27 16 all all DET iajs-2679 27 17 𝑑 𝑑 ADP iajs-2679 27 18 ∈ ∈ PROPN iajs-2679 28 1 [ [ X iajs-2679 28 2 π‘Ž π‘Ž X iajs-2679 28 3 , , PUNCT iajs-2679 28 4 𝑏 𝑏 NOUN iajs-2679 28 5 ] ] PUNCT iajs-2679 28 6 and and CCONJ iajs-2679 28 7 if if SCONJ iajs-2679 28 8 the the DET iajs-2679 28 9 map map NOUN iajs-2679 28 10 𝑇 𝑇 PROPN iajs-2679 28 11 : : PUNCT iajs-2679 28 12 [ [ X iajs-2679 28 13 π‘Ž π‘Ž X iajs-2679 28 14 , , PUNCT iajs-2679 28 15 𝑏 𝑏 NOUN iajs-2679 28 16 ] ] X iajs-2679 28 17 β†’ β†’ PUNCT iajs-2679 28 18 ( ( PUNCT iajs-2679 28 19 𝐡(𝑋 𝐡(𝑋 PROPN iajs-2679 28 20 ) ) PUNCT iajs-2679 28 21 , , PUNCT iajs-2679 28 22 β€–. β€–. X iajs-2679 28 23 β€– β€– PROPN iajs-2679 28 24 ) ) PUNCT iajs-2679 28 25 defined define VERB iajs-2679 28 26 by by ADP iajs-2679 28 27 𝑇(𝑑 𝑇(𝑑 NOUN iajs-2679 28 28 ) ) PUNCT iajs-2679 28 29 = = PUNCT iajs-2679 29 1 𝑇𝑑 𝑇𝑑 NOUN iajs-2679 29 2 is be AUX iajs-2679 29 3 continuous continuous ADJ iajs-2679 29 4 with with ADP iajs-2679 29 5 respect respect NOUN iajs-2679 29 6 to to ADP iajs-2679 29 7 both both CCONJ iajs-2679 29 8 the the DET iajs-2679 29 9 operator operator NOUN iajs-2679 29 10 norm norm NOUN iajs-2679 29 11 topology topology NOUN iajs-2679 29 12 on on ADP iajs-2679 29 13 𝐡(𝑋 𝐡(𝑋 PROPN iajs-2679 29 14 ) ) PUNCT iajs-2679 29 15 and and CCONJ iajs-2679 29 16 the the DET iajs-2679 29 17 usual usual ADJ iajs-2679 29 18 topology topology NOUN iajs-2679 29 19 on on ADP iajs-2679 29 20 ℝ. ℝ. PROPN iajs-2679 29 21 2 2 NUM iajs-2679 29 22 . . PUNCT iajs-2679 29 23 main main ADJ iajs-2679 29 24 results result NOUN iajs-2679 29 25 definition definition NOUN iajs-2679 29 26 2.1 2.1 NUM iajs-2679 29 27 . . PUNCT iajs-2679 30 1 let let VERB iajs-2679 30 2 { { PUNCT iajs-2679 30 3 𝛼𝐹𝑑 𝛼𝐹𝑑 PROPN iajs-2679 30 4 𝑛 𝑛 NOUN iajs-2679 30 5 : : PUNCT iajs-2679 30 6 𝛼 𝛼 PRON iajs-2679 30 7 ∈ ∈ PROPN iajs-2679 30 8 𝔻 𝔻 PROPN iajs-2679 30 9 , , PUNCT iajs-2679 30 10 𝑛 𝑛 DET iajs-2679 30 11 β‰₯ β‰₯ NOUN iajs-2679 30 12 1 1 NUM iajs-2679 30 13 } } PUNCT iajs-2679 30 14 βŠ‚ βŠ‚ PROPN iajs-2679 30 15 𝐡(𝑋 𝐡(𝑋 PROPN iajs-2679 30 16 ) ) PUNCT iajs-2679 30 17 for for ADP iajs-2679 30 18 each each DET iajs-2679 30 19 𝑑 𝑑 PROPN iajs-2679 30 20 ∈ ∈ PROPN iajs-2679 31 1 [ [ X iajs-2679 31 2 π‘Ž π‘Ž X iajs-2679 31 3 , , PUNCT iajs-2679 31 4 𝑏 𝑏 NOUN iajs-2679 31 5 ] ] PUNCT iajs-2679 31 6 . . PUNCT iajs-2679 32 1 let let VERB iajs-2679 32 2 𝑑 𝑑 PRON iajs-2679 32 3 β†’ β†’ SYM iajs-2679 32 4 𝛼𝐹𝑑 𝛼𝐹𝑑 PROPN iajs-2679 32 5 𝑛 𝑛 PART iajs-2679 32 6 be be AUX iajs-2679 32 7 a a DET iajs-2679 32 8 path path NOUN iajs-2679 32 9 of of ADP iajs-2679 32 10 operators operator NOUN iajs-2679 32 11 on on ADP iajs-2679 32 12 [ [ X iajs-2679 32 13 π‘Ž π‘Ž X iajs-2679 32 14 , , PUNCT iajs-2679 32 15 𝑏 𝑏 NOUN iajs-2679 32 16 ] ] X iajs-2679 32 17 , , PUNCT iajs-2679 32 18 then then ADV iajs-2679 32 19 the the DET iajs-2679 32 20 set set NOUN iajs-2679 32 21 of of ADP iajs-2679 32 22 common common ADJ iajs-2679 32 23 diskcyclic diskcyclic ADJ iajs-2679 32 24 vectors vector NOUN iajs-2679 32 25 for for ADP iajs-2679 32 26 the the DET iajs-2679 32 27 path path NOUN iajs-2679 32 28 of of ADP iajs-2679 32 29 operators operator NOUN iajs-2679 32 30 is be AUX iajs-2679 32 31 defined define VERB iajs-2679 32 32 as as SCONJ iajs-2679 32 33 follows follow VERB iajs-2679 32 34 : : PUNCT iajs-2679 32 35 β‹‚ β‹‚ PROPN iajs-2679 32 36 𝐷𝐢(𝐹𝑑 𝐷𝐢(𝐹𝑑 X iajs-2679 32 37 ) ) PUNCT iajs-2679 33 1 = = PRON iajs-2679 33 2 { { PUNCT iajs-2679 33 3 π‘₯ π‘₯ PUNCT iajs-2679 33 4 ∈ ∈ PROPN iajs-2679 33 5 𝑋 𝑋 NOUN iajs-2679 33 6 : : PUNCT iajs-2679 33 7 π‘‚π‘Ÿπ‘(𝐹𝑑 π‘‚π‘Ÿπ‘(𝐹𝑑 PROPN iajs-2679 33 8 , , PUNCT iajs-2679 33 9 π‘₯ π‘₯ NOUN iajs-2679 33 10 ) ) PUNCT iajs-2679 33 11 , , PUNCT iajs-2679 33 12 𝑑 𝑑 PROPN iajs-2679 33 13 ∈ ∈ PROPN iajs-2679 33 14 [ [ X iajs-2679 33 15 π‘Ž π‘Ž X iajs-2679 33 16 , , PUNCT iajs-2679 33 17 𝑏 𝑏 NOUN iajs-2679 33 18 ] ] PUNCT iajs-2679 33 19 } } PUNCT iajs-2679 33 20 π‘‘βˆˆ[π‘Ž,𝑏 π‘‘βˆˆ[π‘Ž,𝑏 NOUN iajs-2679 33 21 ] ] PUNCT iajs-2679 33 22 is be AUX iajs-2679 33 23 dense dense ADJ iajs-2679 33 24 in in ADP iajs-2679 33 25 𝑋. 𝑋. PROPN iajs-2679 33 26 theorem theorem VERB iajs-2679 33 27 2.2 2.2 NUM iajs-2679 33 28 . . PUNCT iajs-2679 34 1 the the DET iajs-2679 34 2 set set NOUN iajs-2679 34 3 β‹‚ β‹‚ PROPN iajs-2679 34 4 𝐷𝐢(𝐹𝑑)π‘‘βˆˆ[π‘Ž,𝑏 𝐷𝐢(𝐹𝑑)π‘‘βˆˆ[π‘Ž,𝑏 NOUN iajs-2679 34 5 ] ] PUNCT iajs-2679 34 6 of of ADP iajs-2679 34 7 common common ADJ iajs-2679 34 8 diskcyclic diskcyclic ADJ iajs-2679 34 9 vectors vector NOUN iajs-2679 34 10 is be AUX iajs-2679 34 11 dense dense ADJ iajs-2679 34 12 gΞ΄ gΞ΄ NOUN iajs-2679 34 13 set set VERB iajs-2679 34 14 in in ADP iajs-2679 34 15 𝑋 𝑋 PROPN iajs-2679 34 16 if if SCONJ iajs-2679 34 17 and and CCONJ iajs-2679 34 18 only only ADV iajs-2679 34 19 if if SCONJ iajs-2679 34 20 for for ADP iajs-2679 34 21 each each DET iajs-2679 34 22 nonempty nonempty ADV iajs-2679 34 23 open open ADJ iajs-2679 34 24 sets set NOUN iajs-2679 34 25 π‘ˆ π‘ˆ PROPN iajs-2679 34 26 and and CCONJ iajs-2679 34 27 𝑉 𝑉 PROPN iajs-2679 34 28 , , PUNCT iajs-2679 34 29 there there PRON iajs-2679 34 30 exists exist VERB iajs-2679 34 31 a a DET iajs-2679 34 32 partition partition NOUN iajs-2679 34 33 𝑃 𝑃 NOUN iajs-2679 34 34 = = SYM iajs-2679 34 35 { { PUNCT iajs-2679 34 36 π‘Ž π‘Ž NOUN iajs-2679 34 37 = = SYM iajs-2679 34 38 𝑑0 𝑑0 NOUN iajs-2679 34 39 < < X iajs-2679 34 40 𝑑1 𝑑1 X iajs-2679 34 41 < < X iajs-2679 34 42 β‹― β‹― X iajs-2679 34 43 < < X iajs-2679 34 44 π‘‘π‘˜ π‘‘π‘˜ PROPN iajs-2679 34 45 = = SYM iajs-2679 34 46 𝑏 𝑏 NOUN iajs-2679 34 47 } } PUNCT iajs-2679 34 48 of of ADP iajs-2679 34 49 [ [ X iajs-2679 34 50 π‘Ž π‘Ž X iajs-2679 34 51 , , PUNCT iajs-2679 34 52 𝑏 𝑏 NOUN iajs-2679 34 53 ] ] X iajs-2679 34 54 , , PUNCT iajs-2679 34 55 𝛼1 𝛼1 NOUN iajs-2679 34 56 , , PUNCT iajs-2679 34 57 𝛼2 𝛼2 ADJ iajs-2679 34 58 , , PUNCT iajs-2679 34 59 … … PUNCT iajs-2679 34 60 , , PUNCT iajs-2679 34 61 π›Όπ‘˜ π›Όπ‘˜ NOUN iajs-2679 34 62 ∈ ∈ NOUN iajs-2679 34 63 𝔻 𝔻 PROPN iajs-2679 34 64 , , PUNCT iajs-2679 34 65 𝑛1 𝑛1 NOUN iajs-2679 34 66 , , PUNCT iajs-2679 34 67 𝑛2 𝑛2 NOUN iajs-2679 34 68 , , PUNCT iajs-2679 34 69 … … PUNCT iajs-2679 34 70 , , PUNCT iajs-2679 34 71 π‘›π‘˜ π‘›π‘˜ ADP iajs-2679 34 72 ∈ ∈ PROPN iajs-2679 34 73 β„• β„• PROPN iajs-2679 34 74 , , PUNCT iajs-2679 34 75 and and CCONJ iajs-2679 34 76 an an DET iajs-2679 34 77 open open ADJ iajs-2679 34 78 set set NOUN iajs-2679 34 79 𝐺 𝐺 PROPN iajs-2679 35 1 such such ADJ iajs-2679 35 2 that that SCONJ iajs-2679 35 3 if if SCONJ iajs-2679 35 4 1 1 NUM iajs-2679 35 5 ≀ ≀ NOUN iajs-2679 35 6 𝑖 𝑖 ADP iajs-2679 35 7 ≀ ≀ NOUN iajs-2679 35 8 π‘˜ π‘˜ PRON iajs-2679 35 9 and and CCONJ iajs-2679 35 10 𝑑 𝑑 PROPN iajs-2679 35 11 ∈ ∈ PROPN iajs-2679 36 1 [ [ X iajs-2679 36 2 π‘‘π‘–βˆ’1 π‘‘π‘–βˆ’1 NOUN iajs-2679 36 3 , , PUNCT iajs-2679 36 4 𝑑𝑖 𝑑𝑖 VERB iajs-2679 36 5 ] ] X iajs-2679 36 6 then then ADV iajs-2679 36 7 𝐺 𝐺 PROPN iajs-2679 36 8 βŠ† βŠ† PROPN iajs-2679 36 9 π‘ˆ π‘ˆ PROPN iajs-2679 36 10 and and CCONJ iajs-2679 36 11 𝛼𝑖𝐹𝑑 𝛼𝑖𝐹𝑑 PROPN iajs-2679 36 12 ( ( PUNCT iajs-2679 36 13 𝑛𝑖 𝑛𝑖 NOUN iajs-2679 36 14 ) ) PUNCT iajs-2679 36 15 𝐺 𝐺 PROPN iajs-2679 36 16 βŠ† βŠ† NUM iajs-2679 36 17 𝑉. 𝑉. NOUN iajs-2679 36 18 proof proof NOUN iajs-2679 36 19 . . PUNCT iajs-2679 37 1 the the DET iajs-2679 37 2 proof proof NOUN iajs-2679 37 3 follows follow VERB iajs-2679 37 4 the the DET iajs-2679 37 5 same same ADJ iajs-2679 37 6 idea idea NOUN iajs-2679 37 7 of of ADP iajs-2679 37 8 [ [ X iajs-2679 37 9 12 12 NUM iajs-2679 37 10 , , PUNCT iajs-2679 37 11 theorem theorem VERB iajs-2679 37 12 2.1 2.1 NUM iajs-2679 37 13 ] ] PUNCT iajs-2679 37 14 , , PUNCT iajs-2679 37 15 therefore therefore ADV iajs-2679 37 16 we we PRON iajs-2679 37 17 omit omit VERB iajs-2679 37 18 the the DET iajs-2679 37 19 details detail NOUN iajs-2679 37 20 . . PUNCT iajs-2679 38 1 the the DET iajs-2679 38 2 following follow VERB iajs-2679 38 3 theorem theorem NOUN iajs-2679 38 4 shows show VERB iajs-2679 38 5 that that SCONJ iajs-2679 38 6 in in ADP iajs-2679 38 7 some some DET iajs-2679 38 8 cases case NOUN iajs-2679 38 9 , , PUNCT iajs-2679 38 10 an an DET iajs-2679 38 11 uncountable uncountable ADJ iajs-2679 38 12 family family NOUN iajs-2679 38 13 of of ADP iajs-2679 38 14 diskcyclic diskcyclic PROPN iajs-2679 38 15 operators operators PROPN iajs-2679 38 16 has have VERB iajs-2679 38 17 a a DET iajs-2679 38 18 common common ADJ iajs-2679 38 19 diskcyclic diskcyclic ADJ iajs-2679 38 20 vectors vector NOUN iajs-2679 38 21 which which PRON iajs-2679 38 22 is be AUX iajs-2679 38 23 a a DET iajs-2679 38 24 dense dense ADJ iajs-2679 38 25 gΞ΄ gΞ΄ NOUN iajs-2679 38 26 set set NOUN iajs-2679 38 27 . . PUNCT iajs-2679 39 1 theorem theorem VERB iajs-2679 39 2 2.3 2.3 NUM iajs-2679 39 3 . . PUNCT iajs-2679 40 1 let let VERB iajs-2679 40 2 𝑋 𝑋 NOUN iajs-2679 40 3 be be AUX iajs-2679 40 4 a a DET iajs-2679 40 5 separable separable ADJ iajs-2679 40 6 , , PUNCT iajs-2679 40 7 infinite infinite ADJ iajs-2679 40 8 dimensional dimensional ADJ iajs-2679 40 9 banach banach NOUN iajs-2679 40 10 space space NOUN iajs-2679 40 11 , , PUNCT iajs-2679 40 12 and and CCONJ iajs-2679 40 13 let let VERB iajs-2679 40 14 ibn ibn PROPN iajs-2679 40 15 al al PROPN iajs-2679 40 16 - - PUNCT iajs-2679 40 17 haitham haitham PROPN iajs-2679 40 18 jour jour X iajs-2679 40 19 . . PROPN iajs-2679 41 1 for for ADP iajs-2679 41 2 pure pure ADJ iajs-2679 41 3 & & CCONJ iajs-2679 41 4 appl appl PROPN iajs-2679 41 5 . . PUNCT iajs-2679 42 1 sci sci PROPN iajs-2679 42 2 . . PROPN iajs-2679 43 1 34(3)2021 34(3)2021 NUM iajs-2679 43 2 69 69 NUM iajs-2679 43 3 { { PUNCT iajs-2679 43 4 𝐹𝑙 𝐹𝑙 PROPN iajs-2679 43 5 : : PUNCT iajs-2679 43 6 𝑙 𝑙 X iajs-2679 43 7 ∈ ∈ PROPN iajs-2679 44 1 [ [ X iajs-2679 44 2 π‘Ž π‘Ž X iajs-2679 44 3 , , PUNCT iajs-2679 44 4 𝑏 𝑏 NOUN iajs-2679 44 5 ] ] PUNCT iajs-2679 44 6 } } PUNCT iajs-2679 44 7 be be AUX iajs-2679 44 8 a a DET iajs-2679 44 9 path path NOUN iajs-2679 44 10 of of ADP iajs-2679 44 11 non non ADJ iajs-2679 44 12 - - ADJ iajs-2679 44 13 trivial trivial ADJ iajs-2679 44 14 bounded bound VERB iajs-2679 44 15 linear linear PROPN iajs-2679 44 16 operators operator NOUN iajs-2679 44 17 on on ADP iajs-2679 44 18 𝑋. 𝑋. PROPN iajs-2679 44 19 if if SCONJ iajs-2679 44 20 there there PRON iajs-2679 44 21 exists exist VERB iajs-2679 44 22 a a DET iajs-2679 44 23 dense dense ADJ iajs-2679 44 24 set set VERB iajs-2679 44 25 𝐷1 𝐷1 NOUN iajs-2679 44 26 such such ADJ iajs-2679 44 27 that that PRON iajs-2679 44 28 for for ADP iajs-2679 44 29 every every DET iajs-2679 44 30 𝑦 𝑦 PROPN iajs-2679 44 31 ∈ ∈ NOUN iajs-2679 44 32 𝐷1 𝐷1 NOUN iajs-2679 44 33 and and CCONJ iajs-2679 44 34 νœ€ νœ€ NOUN iajs-2679 44 35 > > X iajs-2679 44 36 0 0 NUM iajs-2679 44 37 , , PUNCT iajs-2679 44 38 there there PRON iajs-2679 44 39 exists exist VERB iajs-2679 44 40 𝛿 𝛿 PROPN iajs-2679 44 41 > > X iajs-2679 44 42 0 0 NUM iajs-2679 44 43 , , PUNCT iajs-2679 44 44 a a DET iajs-2679 44 45 dense dense ADJ iajs-2679 44 46 set set NOUN iajs-2679 44 47 𝐷2 𝐷2 NOUN iajs-2679 44 48 , , PUNCT iajs-2679 44 49 an an DET iajs-2679 44 50 increasing increase VERB iajs-2679 44 51 sequence sequence NOUN iajs-2679 44 52 of of ADP iajs-2679 44 53 positive positive ADJ iajs-2679 44 54 integers integer NOUN iajs-2679 44 55 { { PUNCT iajs-2679 44 56 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 44 57 } } PUNCT iajs-2679 44 58 𝑗=1 𝑗=1 PROPN iajs-2679 44 59 ∞ ∞ NUM iajs-2679 44 60 , , PUNCT iajs-2679 44 61 and and CCONJ iajs-2679 44 62 a a DET iajs-2679 44 63 set set NOUN iajs-2679 44 64 of of ADP iajs-2679 44 65 maps map NOUN iajs-2679 44 66 { { PUNCT iajs-2679 44 67 𝑆𝑙,𝑗 𝑆𝑙,𝑗 PROPN iajs-2679 44 68 : : PUNCT iajs-2679 44 69 𝐷1 𝐷1 PROPN iajs-2679 44 70 β†’ β†’ SYM iajs-2679 44 71 𝑋 𝑋 PROPN iajs-2679 44 72 : : PUNCT iajs-2679 44 73 𝑙 𝑙 X iajs-2679 44 74 ∈ ∈ PROPN iajs-2679 45 1 [ [ X iajs-2679 45 2 π‘Ž π‘Ž X iajs-2679 45 3 , , PUNCT iajs-2679 45 4 𝑏 𝑏 NOUN iajs-2679 45 5 ] ] PUNCT iajs-2679 45 6 , , PUNCT iajs-2679 45 7 𝑗 𝑗 X iajs-2679 45 8 β‰₯ β‰₯ NOUN iajs-2679 45 9 1}such 1}such NUM iajs-2679 45 10 that that SCONJ iajs-2679 45 11 1 1 X iajs-2679 45 12 . . X iajs-2679 45 13 for for ADP iajs-2679 45 14 each each DET iajs-2679 45 15 𝑝 𝑝 PROPN iajs-2679 45 16 ∈ ∈ PROPN iajs-2679 46 1 [ [ X iajs-2679 46 2 π‘Ž π‘Ž X iajs-2679 46 3 , , PUNCT iajs-2679 46 4 𝑏 𝑏 NOUN iajs-2679 46 5 ] ] PUNCT iajs-2679 46 6 and and CCONJ iajs-2679 46 7 π‘₯ π‘₯ DET iajs-2679 46 8 ∈ ∈ PROPN iajs-2679 46 9 𝐷2 𝐷2 NOUN iajs-2679 46 10 , , PUNCT iajs-2679 46 11 the the DET iajs-2679 46 12 sequence sequence NOUN iajs-2679 46 13 ‖𝐹𝑙 ‖𝐹𝑙 NOUN iajs-2679 46 14 ( ( PUNCT iajs-2679 46 15 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 46 16 ) ) PUNCT iajs-2679 46 17 π‘₯β€– π‘₯β€– VERB iajs-2679 46 18 ‖𝑆𝑝,𝑗𝑦‖ ‖𝑆𝑝,𝑗𝑦‖ NOUN iajs-2679 46 19 β†’ β†’ SYM iajs-2679 46 20 0 0 NUM iajs-2679 46 21 for for ADP iajs-2679 46 22 all all PRON iajs-2679 46 23 𝑙 𝑙 PRON iajs-2679 46 24 ∈ ∈ NOUN iajs-2679 47 1 [ [ X iajs-2679 47 2 π‘Ž π‘Ž X iajs-2679 47 3 , , PUNCT iajs-2679 47 4 𝑏 𝑏 NOUN iajs-2679 47 5 ] ] PUNCT iajs-2679 47 6 and and CCONJ iajs-2679 47 7 𝑗 𝑗 INTJ iajs-2679 47 8 β†’ β†’ SYM iajs-2679 47 9 ∞ ∞ PROPN iajs-2679 47 10 , , PUNCT iajs-2679 47 11 2 2 NUM iajs-2679 47 12 . . X iajs-2679 48 1 for for ADP iajs-2679 48 2 each each DET iajs-2679 48 3 𝑝 𝑝 PROPN iajs-2679 48 4 ∈ ∈ PROPN iajs-2679 49 1 [ [ X iajs-2679 49 2 π‘Ž π‘Ž X iajs-2679 49 3 , , PUNCT iajs-2679 49 4 𝑏 𝑏 NOUN iajs-2679 49 5 ] ] PUNCT iajs-2679 49 6 , , PUNCT iajs-2679 49 7 ‖𝑆𝑝,𝑗𝑦‖ ‖𝑆𝑝,𝑗𝑦‖ NOUN iajs-2679 49 8 β†’ β†’ SYM iajs-2679 49 9 0 0 PUNCT iajs-2679 49 10 as as ADP iajs-2679 49 11 𝑗 𝑗 PROPN iajs-2679 49 12 β†’ β†’ SYM iajs-2679 49 13 ∞ ∞ PROPN iajs-2679 49 14 , , PUNCT iajs-2679 49 15 3 3 NUM iajs-2679 49 16 . . X iajs-2679 50 1 for for ADP iajs-2679 50 2 each each DET iajs-2679 50 3 𝑝 𝑝 PROPN iajs-2679 50 4 ∈ ∈ PROPN iajs-2679 50 5 [ [ X iajs-2679 50 6 π‘Ž π‘Ž X iajs-2679 50 7 , , PUNCT iajs-2679 50 8 𝑏 𝑏 NOUN iajs-2679 50 9 ] ] PUNCT iajs-2679 50 10 and and CCONJ iajs-2679 50 11 integer integer PROPN iajs-2679 50 12 𝑐 𝑐 PROPN iajs-2679 50 13 β‰₯ β‰₯ NUM iajs-2679 50 14 1 1 NUM iajs-2679 50 15 , , PUNCT iajs-2679 50 16 there there PRON iajs-2679 50 17 exists exist VERB iajs-2679 50 18 𝑗 𝑗 PRON iajs-2679 50 19 β‰₯ β‰₯ NUM iajs-2679 50 20 𝑐 𝑐 NOUN iajs-2679 50 21 such such ADJ iajs-2679 50 22 that that SCONJ iajs-2679 50 23 if if SCONJ iajs-2679 50 24 |𝑙 |𝑙 VERB iajs-2679 50 25 βˆ’ βˆ’ PROPN iajs-2679 50 26 𝑝| 𝑝| PROPN iajs-2679 50 27 < < X iajs-2679 50 28 𝛿 𝛿 X iajs-2679 50 29 then then ADV iajs-2679 50 30 ‖𝐹𝑙 ‖𝐹𝑙 ADP iajs-2679 50 31 ( ( PUNCT iajs-2679 50 32 π‘šπ‘— π‘šπ‘— ADJ iajs-2679 50 33 ) ) PUNCT iajs-2679 50 34 𝑆𝑝,𝑗𝑦 𝑆𝑝,𝑗𝑦 NOUN iajs-2679 50 35 βˆ’ βˆ’ PROPN iajs-2679 50 36 𝑦‖ 𝑦‖ PROPN iajs-2679 50 37 < < X iajs-2679 50 38 νœ€ νœ€ X iajs-2679 50 39 . . PUNCT iajs-2679 50 40 then then ADV iajs-2679 50 41 β‹‚ β‹‚ PROPN iajs-2679 50 42 dc(ft)t∈[a dc(ft)t∈[a PROPN iajs-2679 50 43 , , PUNCT iajs-2679 50 44 b b NOUN iajs-2679 50 45 ] ] PUNCT iajs-2679 50 46 of of ADP iajs-2679 50 47 common common ADJ iajs-2679 50 48 diskcyclic diskcyclic ADJ iajs-2679 50 49 vectors vector NOUN iajs-2679 50 50 is be AUX iajs-2679 50 51 dense dense ADJ iajs-2679 50 52 gΞ΄ gΞ΄ NOUN iajs-2679 50 53 . . PUNCT iajs-2679 51 1 proof proof NOUN iajs-2679 51 2 . . PUNCT iajs-2679 52 1 let let VERB iajs-2679 52 2 u1and u1and PROPN iajs-2679 52 3 u2 u2 PROPN iajs-2679 52 4 be be AUX iajs-2679 52 5 two two NUM iajs-2679 52 6 non non ADJ iajs-2679 52 7 - - ADJ iajs-2679 52 8 trivial trivial ADJ iajs-2679 52 9 open open ADJ iajs-2679 52 10 sets set NOUN iajs-2679 52 11 in in ADP iajs-2679 52 12 x. x. PROPN iajs-2679 52 13 pick pick VERB iajs-2679 52 14 y y PROPN iajs-2679 52 15 ∈ ∈ PROPN iajs-2679 52 16 d1{0 d1{0 PROPN iajs-2679 52 17 } } PUNCT iajs-2679 52 18 and and CCONJ iajs-2679 52 19 Οƒ Οƒ NOUN iajs-2679 52 20 > > X iajs-2679 52 21 0 0 NUM iajs-2679 52 22 such such ADJ iajs-2679 52 23 that that SCONJ iajs-2679 52 24 b(y b(y PROPN iajs-2679 52 25 , , PUNCT iajs-2679 52 26 Οƒ Οƒ PROPN iajs-2679 52 27 ) ) PUNCT iajs-2679 52 28 βŠ† βŠ† NUM iajs-2679 52 29 u2 u2 NOUN iajs-2679 52 30 . . PUNCT iajs-2679 53 1 then then ADV iajs-2679 53 2 there there PRON iajs-2679 53 3 is be VERB iajs-2679 53 4 an an DET iajs-2679 53 5 increasing increase VERB iajs-2679 53 6 sequence sequence NOUN iajs-2679 53 7 { { PUNCT iajs-2679 53 8 mj}j=1 mj}j=1 NOUN iajs-2679 53 9 ∞ ∞ PROPN iajs-2679 53 10 of of ADP iajs-2679 53 11 positive positive ADJ iajs-2679 53 12 integers integer NOUN iajs-2679 53 13 , , PUNCT iajs-2679 53 14 a a DET iajs-2679 53 15 dense dense ADJ iajs-2679 53 16 set set NOUN iajs-2679 53 17 d2 d2 PROPN iajs-2679 53 18 , , PUNCT iajs-2679 53 19 Ξ΄ Ξ΄ PROPN iajs-2679 53 20 > > X iajs-2679 53 21 0 0 PUNCT iajs-2679 53 22 and and CCONJ iajs-2679 53 23 a a DET iajs-2679 53 24 set set NOUN iajs-2679 53 25 of of ADP iajs-2679 53 26 maps map NOUN iajs-2679 53 27 sl sl INTJ iajs-2679 53 28 , , PUNCT iajs-2679 53 29 j j NOUN iajs-2679 53 30 : : PUNCT iajs-2679 53 31 d1 d1 PROPN iajs-2679 53 32 β†’ β†’ SYM iajs-2679 53 33 x x X iajs-2679 53 34 which which PRON iajs-2679 53 35 satisfy satisfy VERB iajs-2679 53 36 the the DET iajs-2679 53 37 conditions condition NOUN iajs-2679 53 38 ( ( PUNCT iajs-2679 53 39 1 1 NUM iajs-2679 53 40 ) ) PUNCT iajs-2679 53 41 , , PUNCT iajs-2679 53 42 ( ( PUNCT iajs-2679 53 43 2 2 X iajs-2679 53 44 ) ) PUNCT iajs-2679 53 45 and and CCONJ iajs-2679 53 46 ( ( PUNCT iajs-2679 53 47 3 3 X iajs-2679 53 48 ) ) PUNCT iajs-2679 53 49 with with ADP iajs-2679 53 50 respect respect NOUN iajs-2679 53 51 to to ADP iajs-2679 53 52 the the DET iajs-2679 53 53 vector vector NOUN iajs-2679 53 54 y y PROPN iajs-2679 53 55 and and CCONJ iajs-2679 53 56 Ξ΅ Ξ΅ PROPN iajs-2679 53 57 = = SYM iajs-2679 53 58 min min PROPN iajs-2679 53 59 { { PUNCT iajs-2679 53 60 Οƒ Οƒ PROPN iajs-2679 53 61 3 3 NUM iajs-2679 53 62 , , PUNCT iajs-2679 53 63 β€–yβ€– β€–yβ€– PROPN iajs-2679 53 64 2 2 NUM iajs-2679 53 65 } } PUNCT iajs-2679 53 66 . . PUNCT iajs-2679 54 1 let let VERB iajs-2679 54 2 p p NOUN iajs-2679 54 3 = = X iajs-2679 54 4 { { PUNCT iajs-2679 54 5 a a NOUN iajs-2679 54 6 = = X iajs-2679 54 7 l0 l0 PROPN iajs-2679 54 8 < < X iajs-2679 54 9 l1 l1 PROPN iajs-2679 54 10 < < X iajs-2679 54 11 β‹― β‹― X iajs-2679 54 12 < < X iajs-2679 54 13 lk lk PROPN iajs-2679 54 14 = = PUNCT iajs-2679 54 15 b b AUX iajs-2679 54 16 } } PUNCT iajs-2679 54 17 be be AUX iajs-2679 54 18 a a DET iajs-2679 54 19 partition partition NOUN iajs-2679 54 20 of of ADP iajs-2679 54 21 [ [ X iajs-2679 54 22 a a X iajs-2679 54 23 , , PUNCT iajs-2679 54 24 b b NOUN iajs-2679 54 25 ] ] X iajs-2679 55 1 where where SCONJ iajs-2679 55 2 max{|li max{|li NOUN iajs-2679 55 3 βˆ’ βˆ’ PROPN iajs-2679 55 4 liβˆ’1| liβˆ’1| NOUN iajs-2679 55 5 : : PUNCT iajs-2679 55 6 1 1 NUM iajs-2679 55 7 ≀ ≀ NUM iajs-2679 55 8 i i PRON iajs-2679 55 9 ≀ ≀ NOUN iajs-2679 55 10 n n CCONJ iajs-2679 55 11 } } PUNCT iajs-2679 55 12 < < X iajs-2679 55 13 Ξ΄ Ξ΄ PROPN iajs-2679 55 14 . . PUNCT iajs-2679 55 15 claim claim NOUN iajs-2679 55 16 . . PUNCT iajs-2679 56 1 let let VERB iajs-2679 56 2 𝑖 𝑖 PRON iajs-2679 56 3 be be AUX iajs-2679 56 4 an an DET iajs-2679 56 5 integer integer NOUN iajs-2679 56 6 such such ADJ iajs-2679 56 7 that that SCONJ iajs-2679 56 8 1 1 NUM iajs-2679 56 9 ≀ ≀ NUM iajs-2679 56 10 𝑖 𝑖 SYM iajs-2679 56 11 ≀ ≀ PROPN iajs-2679 56 12 𝑛 𝑛 PROPN iajs-2679 56 13 , , PUNCT iajs-2679 56 14 𝐺 𝐺 PROPN iajs-2679 56 15 be be VERB iajs-2679 56 16 a a DET iajs-2679 56 17 nontrivial nontrivial ADJ iajs-2679 56 18 open open ADJ iajs-2679 56 19 set set NOUN iajs-2679 56 20 , , PUNCT iajs-2679 56 21 and and CCONJ iajs-2679 56 22 𝑐 𝑐 PROPN iajs-2679 56 23 β‰₯ β‰₯ NUM iajs-2679 56 24 1 1 NUM iajs-2679 56 25 be be AUX iajs-2679 56 26 an an DET iajs-2679 56 27 integer integer NOUN iajs-2679 56 28 . . PUNCT iajs-2679 57 1 then then ADV iajs-2679 57 2 there there PRON iajs-2679 57 3 exists exist VERB iajs-2679 57 4 a a DET iajs-2679 57 5 nonempty nonempty ADJ iajs-2679 57 6 open open ADJ iajs-2679 57 7 set set VERB iajs-2679 57 8 𝐺′ 𝐺′ PROPN iajs-2679 57 9 βŠ† βŠ† NUM iajs-2679 57 10 𝐺 𝐺 PROPN iajs-2679 57 11 , , PUNCT iajs-2679 57 12 a a DET iajs-2679 57 13 number number NOUN iajs-2679 57 14 0 0 NUM iajs-2679 57 15 < < X iajs-2679 57 16 πœ† πœ† X iajs-2679 57 17 ≀ ≀ NUM iajs-2679 57 18 1 1 NUM iajs-2679 57 19 , , PUNCT iajs-2679 57 20 and and CCONJ iajs-2679 57 21 an an DET iajs-2679 57 22 integer integer NOUN iajs-2679 57 23 𝑗 𝑗 INTJ iajs-2679 57 24 β‰₯ β‰₯ NOUN iajs-2679 57 25 𝑐 𝑐 NOUN iajs-2679 57 26 such such ADJ iajs-2679 57 27 that that SCONJ iajs-2679 57 28 πœ†πΉπ‘™ πœ†πΉπ‘™ NOUN iajs-2679 57 29 ( ( PUNCT iajs-2679 57 30 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 57 31 ) ) PUNCT iajs-2679 57 32 𝐺′ 𝐺′ PROPN iajs-2679 57 33 βŠ† βŠ† NUM iajs-2679 57 34 π‘ˆ2 π‘ˆ2 NOUN iajs-2679 57 35 , , PUNCT iajs-2679 57 36 whenever whenever SCONJ iajs-2679 57 37 𝑙 𝑙 X iajs-2679 57 38 ∈ ∈ PROPN iajs-2679 57 39 [ [ X iajs-2679 57 40 liβˆ’1 liβˆ’1 PROPN iajs-2679 57 41 , , PUNCT iajs-2679 57 42 li li PROPN iajs-2679 57 43 ] ] X iajs-2679 57 44 . . PUNCT iajs-2679 58 1 proof proof NOUN iajs-2679 58 2 of of ADP iajs-2679 58 3 claim claim NOUN iajs-2679 58 4 . . PUNCT iajs-2679 59 1 let let VERB iajs-2679 59 2 𝑀 𝑀 PART iajs-2679 59 3 ∈ ∈ VERB iajs-2679 59 4 𝐷2 𝐷2 NOUN iajs-2679 59 5 and and CCONJ iajs-2679 59 6 π‘˜ π‘˜ PRON iajs-2679 59 7 be be AUX iajs-2679 59 8 a a DET iajs-2679 59 9 small small ADJ iajs-2679 59 10 enough enough ADJ iajs-2679 59 11 positive positive ADJ iajs-2679 59 12 integer integer NOUN iajs-2679 59 13 such such DET iajs-2679 59 14 that that DET iajs-2679 59 15 𝐡(𝑀 𝐡(𝑀 PROPN iajs-2679 59 16 , , PUNCT iajs-2679 59 17 π‘˜ π‘˜ NOUN iajs-2679 59 18 ) ) PUNCT iajs-2679 59 19 βŠ† βŠ† NUM iajs-2679 59 20 𝐺.by 𝐺.by NOUN iajs-2679 59 21 putting put VERB iajs-2679 59 22 𝑝 𝑝 NOUN iajs-2679 59 23 = = PRON iajs-2679 59 24 𝑙𝑖 𝑙𝑖 NOUN iajs-2679 59 25 in in ADP iajs-2679 59 26 conditions condition NOUN iajs-2679 59 27 ( ( PUNCT iajs-2679 59 28 1 1 NUM iajs-2679 59 29 ) ) PUNCT iajs-2679 59 30 , , PUNCT iajs-2679 59 31 ( ( PUNCT iajs-2679 59 32 2 2 X iajs-2679 59 33 ) ) PUNCT iajs-2679 59 34 and and CCONJ iajs-2679 59 35 ( ( PUNCT iajs-2679 59 36 3 3 NUM iajs-2679 59 37 ) ) PUNCT iajs-2679 59 38 , , PUNCT iajs-2679 59 39 one one PRON iajs-2679 59 40 can can AUX iajs-2679 59 41 see see VERB iajs-2679 59 42 that that SCONJ iajs-2679 59 43 there there PRON iajs-2679 59 44 exists exist VERB iajs-2679 59 45 𝑗 𝑗 PRON iajs-2679 59 46 β‰₯ β‰₯ NUM iajs-2679 59 47 𝑐 𝑐 NOUN iajs-2679 59 48 such such ADJ iajs-2679 59 49 that that SCONJ iajs-2679 59 50 ‖𝐹𝑙 ‖𝐹𝑙 ADP iajs-2679 59 51 ( ( PUNCT iajs-2679 59 52 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 59 53 ) ) PUNCT iajs-2679 59 54 𝑀‖ 𝑀‖ ADJ iajs-2679 59 55 ‖𝑆𝑙𝑖,𝑗𝑦‖ ‖𝑆𝑙𝑖,𝑗𝑦‖ VERB iajs-2679 59 56 < < X iajs-2679 59 57 πœ€π‘˜ πœ€π‘˜ PROPN iajs-2679 59 58 2 2 NUM iajs-2679 59 59 for for ADP iajs-2679 59 60 all all DET iajs-2679 59 61 𝑙 𝑙 PRON iajs-2679 59 62 ∈ ∈ PROPN iajs-2679 60 1 [ [ X iajs-2679 60 2 liβˆ’1 liβˆ’1 PROPN iajs-2679 60 3 , , PUNCT iajs-2679 60 4 li li PROPN iajs-2679 60 5 ] ] X iajs-2679 60 6 , , PUNCT iajs-2679 60 7 ( ( PUNCT iajs-2679 60 8 1 1 X iajs-2679 60 9 ) ) PUNCT iajs-2679 60 10 ‖𝑆𝑙𝑖,𝑗𝑦‖ ‖𝑆𝑙𝑖,𝑗𝑦‖ VERB iajs-2679 60 11 < < X iajs-2679 60 12 π‘˜ π‘˜ PRON iajs-2679 60 13 2 2 NUM iajs-2679 60 14 , , PUNCT iajs-2679 60 15 ( ( PUNCT iajs-2679 60 16 2 2 NUM iajs-2679 60 17 ) ) PUNCT iajs-2679 60 18 and and CCONJ iajs-2679 60 19 ‖𝐹𝑙 ‖𝐹𝑙 NOUN iajs-2679 60 20 ( ( PUNCT iajs-2679 60 21 π‘šπ‘— π‘šπ‘— NUM iajs-2679 60 22 ) ) PUNCT iajs-2679 60 23 𝑆𝑙𝑖,𝑗𝑦 𝑆𝑙𝑖,𝑗𝑦 PRON iajs-2679 60 24 βˆ’ βˆ’ PROPN iajs-2679 60 25 𝑦‖ 𝑦‖ PROPN iajs-2679 60 26 < < X iajs-2679 60 27 νœ€ νœ€ X iajs-2679 60 28 for for ADP iajs-2679 60 29 all all DET iajs-2679 60 30 𝑙[liβˆ’1 𝑙[liβˆ’1 PROPN iajs-2679 60 31 , , PUNCT iajs-2679 60 32 li li PROPN iajs-2679 60 33 ] ] X iajs-2679 60 34 . . PUNCT iajs-2679 61 1 ( ( PUNCT iajs-2679 61 2 3 3 X iajs-2679 61 3 ) ) PUNCT iajs-2679 61 4 since since SCONJ iajs-2679 61 5 Ξ΅ Ξ΅ PROPN iajs-2679 61 6 = = SYM iajs-2679 61 7 min min PROPN iajs-2679 61 8 { { PUNCT iajs-2679 61 9 Οƒ Οƒ PROPN iajs-2679 61 10 3 3 NUM iajs-2679 61 11 , , PUNCT iajs-2679 61 12 β€–yβ€– β€–yβ€– PROPN iajs-2679 61 13 2 2 NUM iajs-2679 61 14 } } PUNCT iajs-2679 61 15 and and CCONJ iajs-2679 61 16 𝑦 𝑦 NOUN iajs-2679 61 17 β‰  β‰  PROPN iajs-2679 61 18 0 0 NUM iajs-2679 61 19 , , PUNCT iajs-2679 61 20 then then ADV iajs-2679 61 21 𝐹𝑙 𝐹𝑙 PROPN iajs-2679 61 22 ( ( PUNCT iajs-2679 61 23 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 61 24 ) ) PUNCT iajs-2679 61 25 𝑆𝑙𝑖,𝑗𝑦 𝑆𝑙𝑖,𝑗𝑦 NOUN iajs-2679 61 26 β‰  β‰  PROPN iajs-2679 61 27 0 0 NUM iajs-2679 61 28 by by ADP iajs-2679 61 29 ( ( PUNCT iajs-2679 61 30 3 3 NUM iajs-2679 61 31 ) ) PUNCT iajs-2679 61 32 . . PUNCT iajs-2679 62 1 now now ADV iajs-2679 62 2 , , PUNCT iajs-2679 62 3 let let VERB iajs-2679 62 4 πœ† πœ† NOUN iajs-2679 62 5 = = SYM iajs-2679 62 6 2 2 NUM iajs-2679 62 7 π‘˜ π‘˜ PRON iajs-2679 62 8 ‖𝑆𝑙𝑖,𝑗𝑦‖ ‖𝑆𝑙𝑖,𝑗𝑦‖ NOUN iajs-2679 62 9 , , PUNCT iajs-2679 62 10 then then ADV iajs-2679 62 11 it it PRON iajs-2679 62 12 is be AUX iajs-2679 62 13 clear clear ADJ iajs-2679 62 14 that that SCONJ iajs-2679 62 15 0 0 PUNCT iajs-2679 62 16 < < X iajs-2679 62 17 πœ† πœ† X iajs-2679 62 18 ≀ ≀ ADV iajs-2679 62 19 1 1 NUM iajs-2679 62 20 by by ADP iajs-2679 62 21 ( ( PUNCT iajs-2679 62 22 2 2 NUM iajs-2679 62 23 ) ) PUNCT iajs-2679 62 24 . . PUNCT iajs-2679 63 1 let let VERB iajs-2679 63 2 = = PRON iajs-2679 63 3 𝑀 𝑀 ADP iajs-2679 63 4 + + NUM iajs-2679 63 5 1 1 NUM iajs-2679 63 6 πœ† πœ† NOUN iajs-2679 63 7 𝑆𝑙𝑖,𝑗𝑦 𝑆𝑙𝑖,𝑗𝑦 PRON iajs-2679 63 8 , , PUNCT iajs-2679 63 9 𝛼 𝛼 X iajs-2679 63 10 = = PUNCT iajs-2679 63 11 𝑠𝑒𝑝 𝑠𝑒𝑝 NOUN iajs-2679 63 12 { { PUNCT iajs-2679 63 13 πœ†πΉπ‘™ πœ†πΉπ‘™ NOUN iajs-2679 63 14 ( ( PUNCT iajs-2679 63 15 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 63 16 ) ) PUNCT iajs-2679 63 17 : : PUNCT iajs-2679 64 1 𝑙 𝑙 X iajs-2679 64 2 ∈ ∈ PROPN iajs-2679 65 1 [ [ X iajs-2679 65 2 liβˆ’1 liβˆ’1 PROPN iajs-2679 65 3 , , PUNCT iajs-2679 65 4 li li PROPN iajs-2679 65 5 ] ] X iajs-2679 65 6 } } PUNCT iajs-2679 65 7 > > X iajs-2679 65 8 0 0 PUNCT iajs-2679 65 9 and and CCONJ iajs-2679 65 10 𝐺′ 𝐺′ NUM iajs-2679 65 11 = = SYM iajs-2679 65 12 𝐡(𝑧 𝐡(𝑧 NUM iajs-2679 65 13 , , PUNCT iajs-2679 65 14 πœ€ πœ€ X iajs-2679 65 15 𝛼 𝛼 X iajs-2679 65 16 ) ) PUNCT iajs-2679 65 17 ∩ ∩ PROPN iajs-2679 65 18 𝐺 𝐺 PROPN iajs-2679 65 19 βŠ† βŠ† NUM iajs-2679 65 20 𝐺. 𝐺. NOUN iajs-2679 65 21 to to PART iajs-2679 65 22 prove prove VERB iajs-2679 65 23 that that SCONJ iajs-2679 65 24 the the DET iajs-2679 65 25 open open ADJ iajs-2679 65 26 set set NOUN iajs-2679 65 27 𝐺′ 𝐺′ NUM iajs-2679 65 28 is be AUX iajs-2679 65 29 nonempty nonempty ADJ iajs-2679 65 30 , , PUNCT iajs-2679 65 31 it it PRON iajs-2679 65 32 is be AUX iajs-2679 65 33 clear clear ADJ iajs-2679 65 34 that that SCONJ iajs-2679 65 35 ‖𝑧 ‖𝑧 PROPN iajs-2679 65 36 βˆ’ βˆ’ PROPN iajs-2679 66 1 𝑀‖ 𝑀‖ X iajs-2679 66 2 = = PUNCT iajs-2679 66 3 β€– β€– PROPN iajs-2679 66 4 1 1 NUM iajs-2679 66 5 πœ† πœ† X iajs-2679 66 6 𝑆𝑙𝑖,𝑗𝑦 𝑆𝑙𝑖,𝑗𝑦 PRON iajs-2679 66 7 β€– β€– PROPN iajs-2679 66 8 = = NOUN iajs-2679 66 9 π‘˜ π‘˜ ADP iajs-2679 66 10 2 2 NUM iajs-2679 66 11 < < X iajs-2679 66 12 π‘˜ π‘˜ NOUN iajs-2679 66 13 , , PUNCT iajs-2679 66 14 therefore therefore ADV iajs-2679 66 15 𝑧 𝑧 PRON iajs-2679 66 16 ∈ ∈ PROPN iajs-2679 66 17 𝐡 𝐡 NOUN iajs-2679 66 18 ( ( PUNCT iajs-2679 66 19 𝑧 𝑧 PROPN iajs-2679 66 20 , , PUNCT iajs-2679 66 21 πœ€ πœ€ X iajs-2679 66 22 𝛼 𝛼 SYM iajs-2679 66 23 ) ) PUNCT iajs-2679 66 24 ∩ ∩ PROPN iajs-2679 66 25 𝐡(𝑀 𝐡(𝑀 NUM iajs-2679 66 26 , , PUNCT iajs-2679 66 27 π‘˜ π‘˜ NOUN iajs-2679 66 28 ) ) PUNCT iajs-2679 66 29 βŠ† βŠ† NUM iajs-2679 66 30 𝐡 𝐡 NOUN iajs-2679 66 31 ( ( PUNCT iajs-2679 66 32 𝑧 𝑧 PROPN iajs-2679 66 33 , , PUNCT iajs-2679 66 34 πœ€ πœ€ X iajs-2679 66 35 𝛼 𝛼 SYM iajs-2679 66 36 ) ) PUNCT iajs-2679 66 37 ∩ ∩ ADJ iajs-2679 66 38 𝐺 𝐺 NOUN iajs-2679 66 39 = = PUNCT iajs-2679 66 40 𝐺′. 𝐺′. NOUN iajs-2679 66 41 now now ADV iajs-2679 66 42 , , PUNCT iajs-2679 66 43 if if SCONJ iajs-2679 66 44 𝑙 𝑙 PROPN iajs-2679 66 45 ∈ ∈ PROPN iajs-2679 67 1 [ [ X iajs-2679 67 2 liβˆ’1 liβˆ’1 PROPN iajs-2679 67 3 , , PUNCT iajs-2679 67 4 li li PROPN iajs-2679 67 5 ] ] X iajs-2679 67 6 , , PUNCT iajs-2679 67 7 then then ADV iajs-2679 67 8 β€–πœ†πΉπ‘™ β€–πœ†πΉπ‘™ PROPN iajs-2679 67 9 ( ( PUNCT iajs-2679 67 10 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 67 11 ) ) PUNCT iajs-2679 67 12 𝑧 𝑧 PRON iajs-2679 67 13 βˆ’ βˆ’ NOUN iajs-2679 67 14 𝑦‖ 𝑦‖ PROPN iajs-2679 67 15 = = SYM iajs-2679 67 16 β€–πœ†πΉπ‘™ β€–πœ†πΉπ‘™ PROPN iajs-2679 67 17 ( ( PUNCT iajs-2679 67 18 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 67 19 ) ) PUNCT iajs-2679 67 20 𝑀 𝑀 ADP iajs-2679 68 1 + + PROPN iajs-2679 68 2 𝐹𝑙 𝐹𝑙 PROPN iajs-2679 68 3 ( ( PUNCT iajs-2679 68 4 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 68 5 ) ) PUNCT iajs-2679 68 6 𝑆𝑙𝑖,𝑗𝑦 𝑆𝑙𝑖,𝑗𝑦 PRON iajs-2679 68 7 βˆ’ βˆ’ PROPN iajs-2679 68 8 𝑦‖ 𝑦‖ PROPN iajs-2679 69 1 ≀ ≀ ADV iajs-2679 69 2 πœ† πœ† PRON iajs-2679 69 3 ‖𝐹𝑙 ‖𝐹𝑙 ADP iajs-2679 69 4 ( ( PUNCT iajs-2679 69 5 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 69 6 ) ) PUNCT iajs-2679 69 7 𝑀‖ 𝑀‖ NOUN iajs-2679 70 1 + + CCONJ iajs-2679 70 2 ‖𝐹𝑙 ‖𝐹𝑙 NOUN iajs-2679 70 3 ( ( PUNCT iajs-2679 70 4 π‘šπ‘— π‘šπ‘— NUM iajs-2679 70 5 ) ) PUNCT iajs-2679 70 6 𝑆𝑙𝑖,𝑗𝑦 𝑆𝑙𝑖,𝑗𝑦 PRON iajs-2679 70 7 βˆ’ βˆ’ PROPN iajs-2679 70 8 𝑦‖ 𝑦‖ PROPN iajs-2679 70 9 < < X iajs-2679 70 10 πœ† πœ† DET iajs-2679 70 11 ‖𝐹𝑙 ‖𝐹𝑙 ADP iajs-2679 70 12 ( ( PUNCT iajs-2679 70 13 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 70 14 ) ) PUNCT iajs-2679 70 15 𝑀‖ 𝑀‖ NOUN iajs-2679 71 1 + + CCONJ iajs-2679 71 2 νœ€ νœ€ X iajs-2679 71 3 by by ADP iajs-2679 71 4 ( ( PUNCT iajs-2679 71 5 3 3 NUM iajs-2679 71 6 ) ) PUNCT iajs-2679 71 7 = = SYM iajs-2679 71 8 2 2 NUM iajs-2679 71 9 π‘˜ π‘˜ PRON iajs-2679 71 10 ‖𝐹𝑙 ‖𝐹𝑙 ADP iajs-2679 71 11 ( ( PUNCT iajs-2679 71 12 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 71 13 ) ) PUNCT iajs-2679 71 14 𝑀‖ 𝑀‖ ADJ iajs-2679 71 15 ‖𝑆𝑙𝑖,𝑗𝑦‖ ‖𝑆𝑙𝑖,𝑗𝑦‖ VERB iajs-2679 72 1 + + CCONJ iajs-2679 72 2 νœ€ νœ€ X iajs-2679 72 3 < < X iajs-2679 72 4 2 2 NUM iajs-2679 72 5 π‘˜ π‘˜ PRON iajs-2679 72 6 πœ€π‘˜ πœ€π‘˜ PROPN iajs-2679 72 7 2 2 NUM iajs-2679 72 8 + + CCONJ iajs-2679 72 9 νœ€ νœ€ X iajs-2679 72 10 by by ADP iajs-2679 72 11 ( ( PUNCT iajs-2679 72 12 1 1 NUM iajs-2679 72 13 ) ) PUNCT iajs-2679 72 14 = = NOUN iajs-2679 72 15 2νœ€ 2νœ€ NOUN iajs-2679 72 16 . . PUNCT iajs-2679 73 1 and and CCONJ iajs-2679 73 2 so so ADV iajs-2679 73 3 if if SCONJ iajs-2679 73 4 𝑔 𝑔 PROPN iajs-2679 73 5 ∈ ∈ PROPN iajs-2679 73 6 𝐺′ 𝐺′ NOUN iajs-2679 73 7 and and CCONJ iajs-2679 73 8 𝑙 𝑙 PRON iajs-2679 73 9 ∈ ∈ PROPN iajs-2679 73 10 [ [ X iajs-2679 73 11 liβˆ’1 liβˆ’1 PROPN iajs-2679 73 12 , , PUNCT iajs-2679 73 13 li li PROPN iajs-2679 73 14 ] ] X iajs-2679 73 15 , , PUNCT iajs-2679 73 16 then then ADV iajs-2679 73 17 ibn ibn PROPN iajs-2679 73 18 al al PROPN iajs-2679 73 19 - - PUNCT iajs-2679 73 20 haitham haitham PROPN iajs-2679 73 21 jour jour X iajs-2679 73 22 . . PROPN iajs-2679 74 1 for for ADP iajs-2679 74 2 pure pure ADJ iajs-2679 74 3 & & CCONJ iajs-2679 74 4 appl appl PROPN iajs-2679 74 5 . . PUNCT iajs-2679 75 1 sci sci PROPN iajs-2679 75 2 . . PROPN iajs-2679 76 1 34(3)2021 34(3)2021 NUM iajs-2679 76 2 70 70 NUM iajs-2679 76 3 β€–πœ†πΉπ‘™ β€–πœ†πΉπ‘™ PROPN iajs-2679 76 4 ( ( PUNCT iajs-2679 76 5 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 76 6 ) ) PUNCT iajs-2679 76 7 𝑔 𝑔 PROPN iajs-2679 76 8 βˆ’ βˆ’ PROPN iajs-2679 76 9 𝑦‖ 𝑦‖ PROPN iajs-2679 76 10 = = SYM iajs-2679 76 11 β€–πœ†πΉπ‘™ β€–πœ†πΉπ‘™ PROPN iajs-2679 76 12 ( ( PUNCT iajs-2679 76 13 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 76 14 ) ) PUNCT iajs-2679 76 15 𝑔 𝑔 ADP iajs-2679 76 16 βˆ’ βˆ’ NOUN iajs-2679 76 17 πœ†πΉπ‘™ πœ†πΉπ‘™ NOUN iajs-2679 76 18 ( ( PUNCT iajs-2679 76 19 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 76 20 ) ) PUNCT iajs-2679 76 21 𝑧 𝑧 NOUN iajs-2679 77 1 + + NUM iajs-2679 77 2 πœ†πΉπ‘™ πœ†πΉπ‘™ PROPN iajs-2679 77 3 ( ( PUNCT iajs-2679 77 4 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 77 5 ) ) PUNCT iajs-2679 77 6 𝑧 𝑧 PRON iajs-2679 77 7 βˆ’ βˆ’ NOUN iajs-2679 77 8 𝑦‖ 𝑦‖ PROPN iajs-2679 77 9 ≀ ≀ PROPN iajs-2679 77 10 πœ† πœ† DET iajs-2679 77 11 β€– β€– PROPN iajs-2679 77 12 𝐹𝑙 𝐹𝑙 PROPN iajs-2679 77 13 ( ( PUNCT iajs-2679 77 14 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 77 15 ) ) PUNCT iajs-2679 77 16 β€– β€– ADJ iajs-2679 77 17 ‖𝑔 ‖𝑔 ADJ iajs-2679 77 18 βˆ’ βˆ’ NOUN iajs-2679 77 19 𝑧‖ 𝑧‖ NOUN iajs-2679 77 20 + + X iajs-2679 77 21 β€–πœ†πΉπ‘™ β€–πœ†πΉπ‘™ PROPN iajs-2679 77 22 ( ( PUNCT iajs-2679 77 23 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 77 24 ) ) PUNCT iajs-2679 77 25 𝑧 𝑧 PRON iajs-2679 77 26 βˆ’ βˆ’ NOUN iajs-2679 77 27 𝑦‖ 𝑦‖ PROPN iajs-2679 77 28 < < X iajs-2679 77 29 𝑠𝑒𝑝 𝑠𝑒𝑝 NOUN iajs-2679 77 30 { { PUNCT iajs-2679 77 31 πœ† πœ† PROPN iajs-2679 77 32 β€– β€– PROPN iajs-2679 77 33 𝐹𝑙 𝐹𝑙 PROPN iajs-2679 77 34 ( ( PUNCT iajs-2679 77 35 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 77 36 ) ) PUNCT iajs-2679 77 37 β€– β€– PROPN iajs-2679 77 38 : : PUNCT iajs-2679 77 39 𝑙 𝑙 X iajs-2679 77 40 ∈ ∈ PROPN iajs-2679 78 1 [ [ X iajs-2679 78 2 liβˆ’1 liβˆ’1 PROPN iajs-2679 78 3 , , PUNCT iajs-2679 78 4 li li PROPN iajs-2679 78 5 ] ] X iajs-2679 78 6 } } PUNCT iajs-2679 78 7 νœ€ νœ€ DET iajs-2679 78 8 𝛼 𝛼 NOUN iajs-2679 78 9 + + NOUN iajs-2679 78 10 2νœ€ 2νœ€ NUM iajs-2679 78 11 < < X iajs-2679 78 12 3νœ€ 3νœ€ X iajs-2679 78 13 < < X iajs-2679 78 14 𝜎. 𝜎. PROPN iajs-2679 78 15 therefore therefore ADV iajs-2679 78 16 , , PUNCT iajs-2679 78 17 πœ†πΉπ‘™ πœ†πΉπ‘™ PROPN iajs-2679 78 18 ( ( PUNCT iajs-2679 78 19 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 78 20 ) ) PUNCT iajs-2679 78 21 ( ( PUNCT iajs-2679 78 22 𝐺′ 𝐺′ NOUN iajs-2679 78 23 ) ) PUNCT iajs-2679 78 24 βŠ† βŠ† NUM iajs-2679 78 25 𝐡(𝑦 𝐡(𝑦 ADJ iajs-2679 78 26 , , PUNCT iajs-2679 78 27 𝜎 𝜎 NOUN iajs-2679 78 28 ) ) PUNCT iajs-2679 78 29 βŠ† βŠ† NUM iajs-2679 78 30 π‘ˆ2 π‘ˆ2 NOUN iajs-2679 78 31 , , PUNCT iajs-2679 78 32 whenever whenever SCONJ iajs-2679 78 33 𝑙 𝑙 X iajs-2679 78 34 ∈ ∈ PROPN iajs-2679 79 1 [ [ X iajs-2679 79 2 liβˆ’1 liβˆ’1 PROPN iajs-2679 79 3 , , PUNCT iajs-2679 79 4 li li PROPN iajs-2679 79 5 ] ] X iajs-2679 79 6 . . PUNCT iajs-2679 80 1 the the DET iajs-2679 80 2 claim claim NOUN iajs-2679 80 3 is be AUX iajs-2679 80 4 proved prove VERB iajs-2679 80 5 . . PUNCT iajs-2679 81 1 returning return VERB iajs-2679 81 2 to to ADP iajs-2679 81 3 the the DET iajs-2679 81 4 proof proof NOUN iajs-2679 81 5 of of ADP iajs-2679 81 6 the the DET iajs-2679 81 7 theorem theorem NOUN iajs-2679 81 8 , , PUNCT iajs-2679 81 9 let let VERB iajs-2679 81 10 𝐺0 𝐺0 ADV iajs-2679 81 11 be be AUX iajs-2679 81 12 an an DET iajs-2679 81 13 open open ADJ iajs-2679 81 14 ball ball NOUN iajs-2679 81 15 with with ADP iajs-2679 81 16 center center PROPN iajs-2679 81 17 𝑏 𝑏 PROPN iajs-2679 81 18 , , PUNCT iajs-2679 81 19 𝑏 𝑏 PROPN iajs-2679 81 20 ∈ ∈ PROPN iajs-2679 81 21 𝐷2 𝐷2 NOUN iajs-2679 81 22 such such ADJ iajs-2679 81 23 that that PRON iajs-2679 81 24 𝐺0 𝐺0 ADJ iajs-2679 81 25 βŠ† βŠ† NUM iajs-2679 81 26 π‘ˆ1 π‘ˆ1 NOUN iajs-2679 81 27 . . PUNCT iajs-2679 82 1 then then ADV iajs-2679 82 2 by by ADP iajs-2679 82 3 claim claim NOUN iajs-2679 82 4 , , PUNCT iajs-2679 82 5 there there PRON iajs-2679 82 6 are be VERB iajs-2679 82 7 a a DET iajs-2679 82 8 scalar scalar ADJ iajs-2679 82 9 πœ†1such πœ†1such NOUN iajs-2679 82 10 that that SCONJ iajs-2679 82 11 0 0 PUNCT iajs-2679 82 12 < < X iajs-2679 82 13 πœ†1 πœ†1 X iajs-2679 82 14 ≀ ≀ ADV iajs-2679 82 15 1 1 NUM iajs-2679 82 16 , , PUNCT iajs-2679 82 17 a a DET iajs-2679 82 18 nonempty nonempty ADV iajs-2679 82 19 open open NOUN iajs-2679 82 20 set set VERB iajs-2679 82 21 𝐺1 𝐺1 NOUN iajs-2679 82 22 βŠ† βŠ† NUM iajs-2679 82 23 𝐺0 𝐺0 NUM iajs-2679 82 24 , , PUNCT iajs-2679 82 25 and and CCONJ iajs-2679 82 26 a a DET iajs-2679 82 27 positive positive ADJ iajs-2679 82 28 integer integer NOUN iajs-2679 82 29 𝑗1 𝑗1 PROPN iajs-2679 82 30 β‰₯ β‰₯ NOUN iajs-2679 82 31 1 1 NUM iajs-2679 82 32 such such ADJ iajs-2679 82 33 that that DET iajs-2679 82 34 πœ†1𝐹𝑙 πœ†1𝐹𝑙 NOUN iajs-2679 82 35 ( ( PUNCT iajs-2679 82 36 π‘šπ‘—1 π‘šπ‘—1 NOUN iajs-2679 82 37 ) ) PUNCT iajs-2679 82 38 ( ( PUNCT iajs-2679 82 39 𝐺1 𝐺1 NOUN iajs-2679 82 40 ) ) PUNCT iajs-2679 82 41 βŠ† βŠ† NUM iajs-2679 82 42 π‘ˆ2 π‘ˆ2 NOUN iajs-2679 82 43 whenever whenever SCONJ iajs-2679 82 44 𝑙 𝑙 X iajs-2679 82 45 ∈ ∈ PROPN iajs-2679 82 46 [ [ X iajs-2679 82 47 l0 l0 PROPN iajs-2679 82 48 , , PUNCT iajs-2679 82 49 l1 l1 PROPN iajs-2679 82 50 ] ] PUNCT iajs-2679 82 51 . . PUNCT iajs-2679 83 1 again again ADV iajs-2679 83 2 , , PUNCT iajs-2679 83 3 by by ADP iajs-2679 83 4 the the DET iajs-2679 83 5 claim claim NOUN iajs-2679 83 6 , , PUNCT iajs-2679 83 7 there there PRON iajs-2679 83 8 is be VERB iajs-2679 83 9 a a DET iajs-2679 83 10 scalar scalar ADJ iajs-2679 83 11 πœ†2such πœ†2such NOUN iajs-2679 83 12 that that SCONJ iajs-2679 83 13 0 0 NUM iajs-2679 83 14 < < X iajs-2679 83 15 πœ†2 πœ†2 NOUN iajs-2679 83 16 ≀ ≀ NUM iajs-2679 83 17 1 1 NUM iajs-2679 83 18 , , PUNCT iajs-2679 83 19 a a DET iajs-2679 83 20 nonempty nonempty ADJ iajs-2679 83 21 open open NOUN iajs-2679 83 22 set set VERB iajs-2679 83 23 𝐺2 𝐺2 ADJ iajs-2679 83 24 βŠ† βŠ† NUM iajs-2679 83 25 𝐺1 𝐺1 NOUN iajs-2679 83 26 , , PUNCT iajs-2679 83 27 and and CCONJ iajs-2679 83 28 a a DET iajs-2679 83 29 positive positive ADJ iajs-2679 83 30 integer integer NOUN iajs-2679 83 31 𝑗2 𝑗2 PROPN iajs-2679 83 32 β‰₯ β‰₯ NUM iajs-2679 83 33 𝑗1 𝑗1 VERB iajs-2679 83 34 such such ADJ iajs-2679 83 35 that that SCONJ iajs-2679 83 36 πœ†2𝐹𝑙 πœ†2𝐹𝑙 NOUN iajs-2679 83 37 ( ( PUNCT iajs-2679 83 38 π‘šπ‘—2 π‘šπ‘—2 NOUN iajs-2679 83 39 ) ) PUNCT iajs-2679 83 40 ( ( PUNCT iajs-2679 83 41 𝐺2 𝐺2 ADJ iajs-2679 83 42 ) ) PUNCT iajs-2679 83 43 βŠ† βŠ† NUM iajs-2679 83 44 π‘ˆ2 π‘ˆ2 NOUN iajs-2679 83 45 whenever whenever SCONJ iajs-2679 83 46 𝑙 𝑙 X iajs-2679 84 1 ∈ ∈ PROPN iajs-2679 85 1 [ [ X iajs-2679 85 2 l1 l1 PROPN iajs-2679 85 3 , , PUNCT iajs-2679 85 4 l2 l2 NOUN iajs-2679 85 5 ] ] PUNCT iajs-2679 85 6 . . PUNCT iajs-2679 86 1 by by ADP iajs-2679 86 2 applying apply VERB iajs-2679 86 3 this this DET iajs-2679 86 4 process process NOUN iajs-2679 86 5 𝑛 𝑛 DET iajs-2679 86 6 times time NOUN iajs-2679 86 7 , , PUNCT iajs-2679 86 8 we we PRON iajs-2679 86 9 get get VERB iajs-2679 86 10 open open ADJ iajs-2679 86 11 sets set NOUN iajs-2679 87 1 𝐺𝑛 𝐺𝑛 ADP iajs-2679 87 2 βŠ† βŠ† NUM iajs-2679 87 3 πΊπ‘›βˆ’1 πΊπ‘›βˆ’1 NUM iajs-2679 87 4 βŠ† βŠ† NUM iajs-2679 87 5 β‹― β‹― ADP iajs-2679 87 6 βŠ† βŠ† NUM iajs-2679 87 7 𝐺1 𝐺1 NOUN iajs-2679 87 8 βŠ† βŠ† NUM iajs-2679 87 9 𝐺0 𝐺0 ADJ iajs-2679 87 10 , , PUNCT iajs-2679 87 11 scalars scalar VERB iajs-2679 87 12 0 0 PUNCT iajs-2679 87 13 < < X iajs-2679 87 14 πœ†1 πœ†1 PROPN iajs-2679 87 15 , , PUNCT iajs-2679 87 16 πœ†2 πœ†2 PROPN iajs-2679 87 17 , , PUNCT iajs-2679 87 18 … … PUNCT iajs-2679 87 19 , , PUNCT iajs-2679 87 20 πœ†π‘› πœ†π‘› ADP iajs-2679 87 21 ≀ ≀ NUM iajs-2679 87 22 1 1 NUM iajs-2679 87 23 , , PUNCT iajs-2679 87 24 and and CCONJ iajs-2679 87 25 integers integer NOUN iajs-2679 87 26 1 1 NUM iajs-2679 87 27 ≀ ≀ NUM iajs-2679 87 28 𝑗1 𝑗1 X iajs-2679 87 29 < < X iajs-2679 87 30 𝑗2 𝑗2 PROPN iajs-2679 87 31 < < X iajs-2679 87 32 β‹― β‹― X iajs-2679 87 33 < < X iajs-2679 87 34 𝑗𝑛 𝑗𝑛 ADP iajs-2679 87 35 such such ADJ iajs-2679 87 36 that that SCONJ iajs-2679 87 37 πœ†π‘–πΉπ‘™ πœ†π‘–πΉπ‘™ NOUN iajs-2679 87 38 ( ( PUNCT iajs-2679 87 39 π‘šπ‘—π‘– π‘šπ‘—π‘– ADV iajs-2679 87 40 ) ) PUNCT iajs-2679 87 41 ( ( PUNCT iajs-2679 87 42 𝐺𝑖 𝐺𝑖 PROPN iajs-2679 87 43 ) ) PUNCT iajs-2679 87 44 βŠ† βŠ† NUM iajs-2679 87 45 π‘ˆ2 π‘ˆ2 NOUN iajs-2679 87 46 whenever whenever SCONJ iajs-2679 87 47 1 1 NUM iajs-2679 87 48 ≀ ≀ NUM iajs-2679 87 49 𝑖 𝑖 SYM iajs-2679 87 50 ≀ ≀ NUM iajs-2679 87 51 𝑛 𝑛 PRON iajs-2679 87 52 and and CCONJ iajs-2679 87 53 𝑙 𝑙 PRON iajs-2679 87 54 ∈ ∈ PROPN iajs-2679 88 1 [ [ X iajs-2679 88 2 liβˆ’1 liβˆ’1 PROPN iajs-2679 88 3 , , PUNCT iajs-2679 88 4 li li PROPN iajs-2679 88 5 ] ] X iajs-2679 88 6 . . PUNCT iajs-2679 89 1 so so ADV iajs-2679 89 2 , , PUNCT iajs-2679 89 3 𝐺𝑛 𝐺𝑛 PROPN iajs-2679 89 4 βŠ† βŠ† NUM iajs-2679 89 5 π‘ˆ1 π‘ˆ1 NOUN iajs-2679 89 6 and and CCONJ iajs-2679 89 7 πœ†π‘–πΉπ‘™ πœ†π‘–πΉπ‘™ NOUN iajs-2679 89 8 ( ( PUNCT iajs-2679 89 9 π‘šπ‘—π‘– π‘šπ‘—π‘– ADV iajs-2679 89 10 ) ) PUNCT iajs-2679 89 11 ( ( PUNCT iajs-2679 89 12 𝐺𝑛 𝐺𝑛 PROPN iajs-2679 89 13 ) ) PUNCT iajs-2679 89 14 βŠ† βŠ† NUM iajs-2679 89 15 πœ†π‘–πΉπ‘™ πœ†π‘–πΉπ‘™ NOUN iajs-2679 89 16 ( ( PUNCT iajs-2679 89 17 π‘šπ‘—π‘– π‘šπ‘—π‘– ADV iajs-2679 89 18 ) ) PUNCT iajs-2679 89 19 ( ( PUNCT iajs-2679 89 20 𝐺𝑖 𝐺𝑖 PROPN iajs-2679 89 21 ) ) PUNCT iajs-2679 89 22 βŠ† βŠ† NUM iajs-2679 89 23 π‘ˆ2 π‘ˆ2 NOUN iajs-2679 90 1 whenever whenever SCONJ iajs-2679 90 2 𝑙 𝑙 X iajs-2679 90 3 ∈ ∈ PROPN iajs-2679 91 1 [ [ X iajs-2679 91 2 liβˆ’1 liβˆ’1 PROPN iajs-2679 91 3 , , PUNCT iajs-2679 91 4 li li PROPN iajs-2679 91 5 ] ] X iajs-2679 91 6 . . PUNCT iajs-2679 92 1 the the DET iajs-2679 92 2 proof proof NOUN iajs-2679 92 3 follows follow VERB iajs-2679 92 4 from from ADP iajs-2679 92 5 theorem theorem ADJ iajs-2679 92 6 2.2 2.2 NUM iajs-2679 92 7 . . PUNCT iajs-2679 93 1 even even ADV iajs-2679 93 2 though though SCONJ iajs-2679 93 3 not not PART iajs-2679 93 4 every every DET iajs-2679 93 5 diskcyclic diskcyclic ADJ iajs-2679 93 6 operator operator NOUN iajs-2679 93 7 satisfies satisfy VERB iajs-2679 93 8 diskcyclic diskcyclic ADJ iajs-2679 93 9 criterion criterion NOUN iajs-2679 93 10 , , PUNCT iajs-2679 93 11 the the DET iajs-2679 93 12 following follow VERB iajs-2679 93 13 proposition proposition NOUN iajs-2679 93 14 shows show VERB iajs-2679 93 15 that that SCONJ iajs-2679 93 16 in in ADP iajs-2679 93 17 some some DET iajs-2679 93 18 cases case NOUN iajs-2679 93 19 every every DET iajs-2679 93 20 diskcyclic diskcyclic ADJ iajs-2679 93 21 operator operator NOUN iajs-2679 93 22 satisfies satisfy VERB iajs-2679 93 23 diskcyclic diskcyclic ADJ iajs-2679 93 24 criterion criterion NOUN iajs-2679 93 25 . . PUNCT iajs-2679 94 1 proposition proposition NOUN iajs-2679 94 2 2.4 2.4 NUM iajs-2679 94 3 . . PUNCT iajs-2679 95 1 if if SCONJ iajs-2679 95 2 a a DET iajs-2679 95 3 path path NOUN iajs-2679 95 4 { { PUNCT iajs-2679 95 5 𝐹𝑑 𝐹𝑑 PROPN iajs-2679 95 6 : : PUNCT iajs-2679 95 7 𝑙 𝑙 X iajs-2679 95 8 ∈ ∈ PROPN iajs-2679 96 1 [ [ X iajs-2679 96 2 π‘Ž π‘Ž X iajs-2679 96 3 , , PUNCT iajs-2679 96 4 𝑏 𝑏 NOUN iajs-2679 96 5 ] ] PUNCT iajs-2679 96 6 } } PUNCT iajs-2679 96 7 satisfies satisfy VERB iajs-2679 96 8 the the DET iajs-2679 96 9 conditions condition NOUN iajs-2679 96 10 of of ADP iajs-2679 96 11 theorem theorem ADJ iajs-2679 96 12 2.3 2.3 NUM iajs-2679 96 13 . . PUNCT iajs-2679 97 1 , , PUNCT iajs-2679 97 2 then then ADV iajs-2679 97 3 every every DET iajs-2679 97 4 that that SCONJ iajs-2679 97 5 { { PUNCT iajs-2679 97 6 πœ†π‘– πœ†π‘– PROPN iajs-2679 97 7 ∈ ∈ PROPN iajs-2679 97 8 𝔻\{0 𝔻\{0 PROPN iajs-2679 97 9 } } PUNCT iajs-2679 97 10 : : PUNCT iajs-2679 97 11 𝑖 𝑖 PRON iajs-2679 97 12 β‰₯ β‰₯ NOUN iajs-2679 97 13 1 1 NUM iajs-2679 97 14 } } PUNCT iajs-2679 97 15 be be AUX iajs-2679 97 16 a a DET iajs-2679 97 17 countable countable ADJ iajs-2679 97 18 set set NOUN iajs-2679 97 19 . . PUNCT iajs-2679 98 1 suppose suppose VERB iajs-2679 98 2 that that SCONJ iajs-2679 98 3 the the DET iajs-2679 98 4 sequence sequence NOUN iajs-2679 98 5 { { PUNCT iajs-2679 98 6 𝑇𝑛 𝑇𝑛 PROPN iajs-2679 98 7 } } PUNCT iajs-2679 98 8 is be AUX iajs-2679 98 9 an an DET iajs-2679 98 10 enumeration enumeration NOUN iajs-2679 98 11 of of ADP iajs-2679 98 12 the the DET iajs-2679 98 13 set set NOUN iajs-2679 98 14 { { PUNCT iajs-2679 98 15 πœ†π‘–πΉπ‘™ πœ†π‘–πΉπ‘™ NOUN iajs-2679 98 16 𝑗 𝑗 X iajs-2679 98 17 : : PUNCT iajs-2679 98 18 𝑖 𝑖 X iajs-2679 98 19 , , PUNCT iajs-2679 98 20 𝑗 𝑗 X iajs-2679 98 21 β‰₯ β‰₯ NOUN iajs-2679 98 22 1 1 NUM iajs-2679 98 23 } } PUNCT iajs-2679 98 24 for for ADP iajs-2679 98 25 each each DET iajs-2679 98 26 𝑙 𝑙 X iajs-2679 98 27 ∈ ∈ PROPN iajs-2679 98 28 [ [ X iajs-2679 98 29 π‘Ž π‘Ž NOUN iajs-2679 98 30 , , PUNCT iajs-2679 98 31 𝑏].now 𝑏].now ADV iajs-2679 98 32 , , PUNCT iajs-2679 98 33 suppose suppose VERB iajs-2679 98 34 that that SCONJ iajs-2679 98 35 { { PUNCT iajs-2679 98 36 𝑇𝑛 𝑇𝑛 NOUN iajs-2679 98 37 } } PUNCT iajs-2679 98 38 satisfies satisfy VERB iajs-2679 98 39 the the DET iajs-2679 98 40 operator operator NOUN iajs-2679 98 41 𝐹𝑙 𝐹𝑙 PROPN iajs-2679 98 42 satisfies satisfy VERB iajs-2679 98 43 the the DET iajs-2679 98 44 diskcyclicity diskcyclicity NOUN iajs-2679 98 45 criterion criterion NOUN iajs-2679 98 46 . . PUNCT iajs-2679 99 1 proof proof NOUN iajs-2679 99 2 . . PUNCT iajs-2679 100 1 suppose suppose VERB iajs-2679 100 2 universality universality NOUN iajs-2679 100 3 criterion criterion NOUN iajs-2679 100 4 , , PUNCT iajs-2679 100 5 and and CCONJ iajs-2679 100 6 then then ADV iajs-2679 100 7 each each DET iajs-2679 100 8 𝐹𝑙 𝐹𝑙 PROPN iajs-2679 100 9 satisfies satisfy VERB iajs-2679 100 10 the the DET iajs-2679 100 11 diskcyclic diskcyclic ADJ iajs-2679 100 12 criterion criterion NOUN iajs-2679 100 13 ; ; PUNCT iajs-2679 100 14 see see VERB iajs-2679 100 15 [ [ X iajs-2679 100 16 20 20 NUM iajs-2679 100 17 , , PUNCT iajs-2679 100 18 definition definition NOUN iajs-2679 100 19 1.1 1.1 NUM iajs-2679 100 20 ] ] PUNCT iajs-2679 100 21 , , PUNCT iajs-2679 101 1 [ [ X iajs-2679 101 2 5 5 NUM iajs-2679 101 3 , , PUNCT iajs-2679 101 4 theorem theorem VERB iajs-2679 101 5 1.2 1.2 NUM iajs-2679 101 6 , , PUNCT iajs-2679 101 7 theorem theorem VERB iajs-2679 101 8 2.6 2.6 NUM iajs-2679 101 9 , , PUNCT iajs-2679 101 10 proposition proposition NOUN iajs-2679 101 11 2.8 2.8 NUM iajs-2679 101 12 ] ] PUNCT iajs-2679 101 13 . . PUNCT iajs-2679 102 1 thus thus ADV iajs-2679 102 2 , , PUNCT iajs-2679 102 3 it it PRON iajs-2679 102 4 is be AUX iajs-2679 102 5 enough enough ADJ iajs-2679 102 6 to to PART iajs-2679 102 7 prove prove VERB iajs-2679 102 8 that that SCONJ iajs-2679 102 9 { { PUNCT iajs-2679 102 10 𝑇𝑛 𝑇𝑛 NOUN iajs-2679 102 11 } } PUNCT iajs-2679 102 12 satisfies satisfy VERB iajs-2679 102 13 the the DET iajs-2679 102 14 universality universality NOUN iajs-2679 102 15 criterion criterion NOUN iajs-2679 102 16 which which PRON iajs-2679 102 17 is be AUX iajs-2679 102 18 equivalent equivalent ADJ iajs-2679 102 19 to to ADP iajs-2679 102 20 showing show VERB iajs-2679 102 21 that that PRON iajs-2679 102 22 for for ADP iajs-2679 102 23 every every DET iajs-2679 102 24 nonempty nonempty ADJ iajs-2679 102 25 open open ADJ iajs-2679 102 26 sets set NOUN iajs-2679 102 27 π‘ˆ π‘ˆ PROPN iajs-2679 102 28 , , PUNCT iajs-2679 102 29 𝐺 𝐺 PROPN iajs-2679 102 30 , , PUNCT iajs-2679 102 31 π‘Š π‘Š NOUN iajs-2679 102 32 with with ADP iajs-2679 102 33 0 0 NUM iajs-2679 102 34 ∈ ∈ PROPN iajs-2679 102 35 π‘Š π‘Š PROPN iajs-2679 102 36 , , PUNCT iajs-2679 102 37 we we PRON iajs-2679 102 38 have have VERB iajs-2679 102 39 𝑇𝑛(π‘ˆ 𝑇𝑛(π‘ˆ VERB iajs-2679 102 40 ) ) PUNCT iajs-2679 102 41 ∩ ∩ NOUN iajs-2679 102 42 π‘Š π‘Š PROPN iajs-2679 102 43 β‰  β‰  PROPN iajs-2679 102 44 βˆ… βˆ… NOUN iajs-2679 102 45 and and CCONJ iajs-2679 102 46 𝑇𝑛(π‘Š 𝑇𝑛(π‘Š NOUN iajs-2679 102 47 ) ) PUNCT iajs-2679 102 48 ∩ ∩ PROPN iajs-2679 102 49 𝐺 𝐺 PROPN iajs-2679 102 50 β‰  β‰  PROPN iajs-2679 102 51 βˆ… βˆ… NOUN iajs-2679 102 52 ; ; PUNCT iajs-2679 102 53 see see VERB iajs-2679 102 54 [ [ X iajs-2679 102 55 20 20 NUM iajs-2679 102 56 , , PUNCT iajs-2679 102 57 theorem theorem VERB iajs-2679 102 58 3.4 3.4 NUM iajs-2679 102 59 ] ] PUNCT iajs-2679 102 60 . . PUNCT iajs-2679 103 1 now now ADV iajs-2679 103 2 , , PUNCT iajs-2679 103 3 by by ADP iajs-2679 103 4 theorem theorem NOUN iajs-2679 103 5 2.3 2.3 NUM iajs-2679 103 6 , , PUNCT iajs-2679 103 7 for for ADP iajs-2679 103 8 any any DET iajs-2679 103 9 dense dense ADJ iajs-2679 103 10 set set NOUN iajs-2679 103 11 𝐷1 𝐷1 NOUN iajs-2679 103 12 , , PUNCT iajs-2679 103 13 we we PRON iajs-2679 103 14 can can AUX iajs-2679 103 15 choose choose VERB iajs-2679 103 16 𝑦 𝑦 NUM iajs-2679 103 17 ∈ ∈ NOUN iajs-2679 103 18 𝐷1{0 𝐷1{0 PROPN iajs-2679 103 19 } } PUNCT iajs-2679 103 20 and and CCONJ iajs-2679 103 21 0 0 NUM iajs-2679 103 22 < < X iajs-2679 103 23 νœ€ νœ€ X iajs-2679 103 24 < < X iajs-2679 103 25 ‖𝑦‖ ‖𝑦‖ PROPN iajs-2679 103 26 such such ADJ iajs-2679 104 1 that that SCONJ iajs-2679 104 2 𝐡(𝑦 𝐡(𝑦 NOUN iajs-2679 104 3 , , PUNCT iajs-2679 104 4 νœ€ νœ€ NOUN iajs-2679 104 5 ) ) PUNCT iajs-2679 104 6 βŠ† βŠ† NUM iajs-2679 104 7 𝐺 𝐺 NOUN iajs-2679 104 8 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2679 104 9 𝐡(0 𝐡(0 PROPN iajs-2679 104 10 , , PUNCT iajs-2679 104 11 νœ€ νœ€ NOUN iajs-2679 104 12 ) ) PUNCT iajs-2679 104 13 βŠ† βŠ† NUM iajs-2679 104 14 π‘Š. π‘Š. PROPN iajs-2679 104 15 then then ADV iajs-2679 104 16 there there PRON iajs-2679 104 17 exists exist VERB iajs-2679 104 18 an an DET iajs-2679 104 19 increasing increase VERB iajs-2679 104 20 sequence sequence NOUN iajs-2679 104 21 { { PUNCT iajs-2679 104 22 mj}j=1 mj}j=1 NOUN iajs-2679 104 23 ∞ ∞ PROPN iajs-2679 104 24 of of ADP iajs-2679 104 25 positive positive ADJ iajs-2679 104 26 integers integer NOUN iajs-2679 104 27 , , PUNCT iajs-2679 104 28 a a DET iajs-2679 104 29 dense dense ADJ iajs-2679 104 30 set set NOUN iajs-2679 104 31 𝐷2 𝐷2 NOUN iajs-2679 104 32 , , PUNCT iajs-2679 104 33 a a DET iajs-2679 104 34 small small ADJ iajs-2679 104 35 positive positive ADJ iajs-2679 104 36 number number NOUN iajs-2679 104 37 𝛿 𝛿 NOUN iajs-2679 104 38 > > X iajs-2679 104 39 0 0 NUM iajs-2679 104 40 , , PUNCT iajs-2679 104 41 and and CCONJ iajs-2679 104 42 maps map NOUN iajs-2679 104 43 { { PUNCT iajs-2679 104 44 𝑆𝑐,𝑗 𝑆𝑐,𝑗 ADJ iajs-2679 104 45 : : PUNCT iajs-2679 104 46 𝐷1 𝐷1 PROPN iajs-2679 104 47 β†’ β†’ SYM iajs-2679 104 48 𝑋 𝑋 PROPN iajs-2679 104 49 : : PUNCT iajs-2679 104 50 𝑗 𝑗 PROPN iajs-2679 104 51 β‰₯ β‰₯ NOUN iajs-2679 104 52 1 1 NUM iajs-2679 104 53 , , PUNCT iajs-2679 104 54 𝑐 𝑐 PROPN iajs-2679 104 55 ∈ ∈ PROPN iajs-2679 105 1 [ [ X iajs-2679 105 2 π‘Ž π‘Ž X iajs-2679 105 3 , , PUNCT iajs-2679 105 4 𝑏 𝑏 NOUN iajs-2679 105 5 ] ] X iajs-2679 105 6 } } PUNCT iajs-2679 105 7 satisfying satisfy VERB iajs-2679 105 8 all all DET iajs-2679 105 9 conditions condition NOUN iajs-2679 105 10 of of ADP iajs-2679 105 11 theorem theorem ADJ iajs-2679 105 12 2.3 2.3 NUM iajs-2679 105 13 . . PUNCT iajs-2679 106 1 thus thus ADV iajs-2679 106 2 , , PUNCT iajs-2679 106 3 we we PRON iajs-2679 106 4 can can AUX iajs-2679 106 5 choose choose VERB iajs-2679 106 6 π‘₯ π‘₯ PRON iajs-2679 106 7 ∈ ∈ PROPN iajs-2679 106 8 𝐷2 𝐷2 NOUN iajs-2679 106 9 ∩ ∩ PROPN iajs-2679 106 10 π‘ˆ π‘ˆ PROPN iajs-2679 106 11 , , PUNCT iajs-2679 106 12 an an DET iajs-2679 106 13 integer integer NOUN iajs-2679 106 14 𝑗 𝑗 INTJ iajs-2679 106 15 β‰₯ β‰₯ NUM iajs-2679 106 16 1 1 NUM iajs-2679 106 17 such such ADJ iajs-2679 106 18 that that SCONJ iajs-2679 106 19 ‖𝐹𝑙 ‖𝐹𝑙 NOUN iajs-2679 106 20 ( ( PUNCT iajs-2679 106 21 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 106 22 ) ) PUNCT iajs-2679 106 23 π‘₯β€– π‘₯β€– VERB iajs-2679 106 24 ‖𝑆𝑙,𝑗𝑦‖ ‖𝑆𝑙,𝑗𝑦‖ NOUN iajs-2679 106 25 < < X iajs-2679 106 26 πœ€2 πœ€2 NOUN iajs-2679 106 27 2 2 NUM iajs-2679 106 28 , , PUNCT iajs-2679 106 29 ‖𝐹𝑙 ‖𝐹𝑙 ADP iajs-2679 106 30 ( ( PUNCT iajs-2679 106 31 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 106 32 ) ) PUNCT iajs-2679 106 33 𝑆𝑙,𝑗𝑦 𝑆𝑙,𝑗𝑦 NOUN iajs-2679 106 34 βˆ’ βˆ’ PROPN iajs-2679 107 1 𝑦‖ 𝑦‖ PROPN iajs-2679 107 2 < < X iajs-2679 107 3 νœ€ νœ€ NOUN iajs-2679 107 4 and and CCONJ iajs-2679 107 5 ‖𝑆𝑙,𝑗𝑦‖ ‖𝑆𝑙,𝑗𝑦‖ NOUN iajs-2679 107 6 β†’ β†’ SYM iajs-2679 107 7 0 0 NUM iajs-2679 107 8 let let VERB iajs-2679 107 9 πœ† πœ† DET iajs-2679 107 10 ∈ ∈ PROPN iajs-2679 107 11 { { PUNCT iajs-2679 107 12 πœ†π‘– πœ†π‘– NOUN iajs-2679 107 13 : : PUNCT iajs-2679 107 14 𝑖 𝑖 PRON iajs-2679 107 15 β‰₯ β‰₯ NOUN iajs-2679 107 16 1 1 NUM iajs-2679 107 17 } } PUNCT iajs-2679 107 18 , , PUNCT iajs-2679 107 19 since since SCONJ iajs-2679 107 20 πœ† πœ† DET iajs-2679 107 21 < < X iajs-2679 107 22 1 1 NUM iajs-2679 107 23 we we PRON iajs-2679 107 24 can can AUX iajs-2679 107 25 assume assume VERB iajs-2679 107 26 that that SCONJ iajs-2679 107 27 νœ€πœ† νœ€πœ† PROPN iajs-2679 107 28 < < X iajs-2679 107 29 2‖𝑆𝑙,𝑗𝑦‖ 2‖𝑆𝑙,𝑗𝑦‖ NUM iajs-2679 107 30 < < X iajs-2679 107 31 2νœ€πœ† 2νœ€πœ† ADJ iajs-2679 107 32 ( ( PUNCT iajs-2679 107 33 4 4 NUM iajs-2679 107 34 ) ) PUNCT iajs-2679 107 35 then then ADV iajs-2679 107 36 πœ†πΉπ‘™ πœ†πΉπ‘™ NOUN iajs-2679 107 37 ( ( PUNCT iajs-2679 107 38 π‘šπ‘— π‘šπ‘— NUM iajs-2679 107 39 ) ) PUNCT iajs-2679 107 40 = = PUNCT iajs-2679 108 1 𝑇𝑛 𝑇𝑛 ADV iajs-2679 108 2 for for ADP iajs-2679 108 3 some some DET iajs-2679 108 4 𝑛 𝑛 DET iajs-2679 108 5 β‰₯ β‰₯ NOUN iajs-2679 108 6 1 1 NUM iajs-2679 108 7 . . PUNCT iajs-2679 109 1 since since SCONJ iajs-2679 109 2 π‘₯ π‘₯ PROPN iajs-2679 109 3 ∈ ∈ PROPN iajs-2679 109 4 π‘ˆ π‘ˆ PROPN iajs-2679 109 5 , , PUNCT iajs-2679 109 6 then then ADV iajs-2679 109 7 by by ADP iajs-2679 109 8 equation equation NOUN iajs-2679 109 9 ( ( PUNCT iajs-2679 109 10 4 4 NUM iajs-2679 109 11 ) ) PUNCT iajs-2679 109 12 , , PUNCT iajs-2679 109 13 we we PRON iajs-2679 109 14 get get VERB iajs-2679 109 15 ibn ibn PROPN iajs-2679 109 16 al al PROPN iajs-2679 109 17 - - PUNCT iajs-2679 109 18 haitham haitham PROPN iajs-2679 109 19 jour jour X iajs-2679 109 20 . . PROPN iajs-2679 110 1 for for ADP iajs-2679 110 2 pure pure ADJ iajs-2679 110 3 & & CCONJ iajs-2679 110 4 appl appl PROPN iajs-2679 110 5 . . PUNCT iajs-2679 111 1 sci sci PROPN iajs-2679 111 2 . . PROPN iajs-2679 112 1 34(3)2021 34(3)2021 NUM iajs-2679 112 2 71 71 NUM iajs-2679 112 3 ‖𝑇𝑛π‘₯β€– ‖𝑇𝑛π‘₯β€– NOUN iajs-2679 112 4 = = SYM iajs-2679 112 5 β€–πœ†πΉπ‘™ β€–πœ†πΉπ‘™ PROPN iajs-2679 112 6 ( ( PUNCT iajs-2679 112 7 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 112 8 ) ) PUNCT iajs-2679 112 9 π‘₯β€– π‘₯β€– VERB iajs-2679 112 10 < < X iajs-2679 112 11 2 2 NUM iajs-2679 112 12 πœ€ πœ€ NOUN iajs-2679 112 13 ‖𝐹𝑙 ‖𝐹𝑙 ADP iajs-2679 112 14 ( ( PUNCT iajs-2679 112 15 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 112 16 ) ) PUNCT iajs-2679 112 17 π‘₯β€– π‘₯β€– VERB iajs-2679 112 18 ‖𝑆𝑙,𝑗𝑦‖ ‖𝑆𝑙,𝑗𝑦‖ NOUN iajs-2679 112 19 < < X iajs-2679 112 20 2 2 NUM iajs-2679 112 21 πœ€ πœ€ NOUN iajs-2679 112 22 πœ€2 πœ€2 NOUN iajs-2679 112 23 2 2 NUM iajs-2679 113 1 = = SYM iajs-2679 113 2 νœ€ νœ€ NOUN iajs-2679 113 3 then then ADV iajs-2679 113 4 , , PUNCT iajs-2679 113 5 𝑇𝑛(π‘ˆ 𝑇𝑛(π‘ˆ X iajs-2679 113 6 ) ) PUNCT iajs-2679 113 7 ∩ ∩ NOUN iajs-2679 113 8 π‘Š π‘Š PROPN iajs-2679 113 9 β‰  β‰  NOUN iajs-2679 113 10 βˆ…. βˆ…. VERB iajs-2679 113 11 again again ADV iajs-2679 113 12 from from ADP iajs-2679 113 13 equation equation NOUN iajs-2679 113 14 ( ( PUNCT iajs-2679 113 15 4 4 NUM iajs-2679 113 16 ) ) PUNCT iajs-2679 113 17 , , PUNCT iajs-2679 113 18 we we PRON iajs-2679 113 19 have have VERB iajs-2679 113 20 1 1 NUM iajs-2679 113 21 πœ† πœ† NOUN iajs-2679 113 22 ‖𝑆𝑙,𝑗𝑦‖ ‖𝑆𝑙,𝑗𝑦‖ NOUN iajs-2679 113 23 < < X iajs-2679 113 24 νœ€ νœ€ X iajs-2679 113 25 which which PRON iajs-2679 113 26 means mean VERB iajs-2679 113 27 that that SCONJ iajs-2679 113 28 𝑆𝑙,𝑗𝑦 𝑆𝑙,𝑗𝑦 PROPN iajs-2679 113 29 ∈ ∈ PROPN iajs-2679 113 30 π‘Š. π‘Š. PROPN iajs-2679 113 31 also also ADV iajs-2679 113 32 , , PUNCT iajs-2679 113 33 we we PRON iajs-2679 113 34 have have VERB iajs-2679 113 35 ‖𝑇𝑛(𝑆𝑙,𝑗𝑦 ‖𝑇𝑛(𝑆𝑙,𝑗𝑦 PUNCT iajs-2679 113 36 ) ) PUNCT iajs-2679 113 37 βˆ’ βˆ’ NOUN iajs-2679 113 38 𝑦‖ 𝑦‖ PROPN iajs-2679 114 1 = = SYM iajs-2679 114 2 ‖𝐹𝑙 ‖𝐹𝑙 ADP iajs-2679 114 3 ( ( PUNCT iajs-2679 114 4 π‘šπ‘— π‘šπ‘— NOUN iajs-2679 114 5 ) ) PUNCT iajs-2679 114 6 𝑆𝑙,𝑗𝑦 𝑆𝑙,𝑗𝑦 NOUN iajs-2679 114 7 βˆ’ βˆ’ PROPN iajs-2679 114 8 𝑦‖ 𝑦‖ PROPN iajs-2679 114 9 < < X iajs-2679 114 10 νœ€ νœ€ X iajs-2679 114 11 so so ADV iajs-2679 114 12 , , PUNCT iajs-2679 114 13 𝑇𝑛(π‘Š 𝑇𝑛(π‘Š PUNCT iajs-2679 114 14 ) ) PUNCT iajs-2679 114 15 ∩ ∩ PROPN iajs-2679 114 16 𝐺 𝐺 PROPN iajs-2679 114 17 β‰  β‰  PROPN iajs-2679 114 18 βˆ… βˆ… NOUN iajs-2679 114 19 , , PUNCT iajs-2679 114 20 which which PRON iajs-2679 114 21 gives give VERB iajs-2679 114 22 the the DET iajs-2679 114 23 proof proof NOUN iajs-2679 114 24 . . PUNCT iajs-2679 115 1 corollary corollary ADJ iajs-2679 115 2 2.5 2.5 NUM iajs-2679 115 3 . . PUNCT iajs-2679 116 1 an an DET iajs-2679 116 2 operator operator NOUN iajs-2679 116 3 𝑇 𝑇 PROPN iajs-2679 116 4 ∈ ∈ PROPN iajs-2679 116 5 𝐡(𝑋 𝐡(𝑋 PROPN iajs-2679 116 6 ) ) PUNCT iajs-2679 116 7 satisfies satisfy VERB iajs-2679 116 8 diskcyclic diskcyclic ADJ iajs-2679 116 9 criterion criterion NOUN iajs-2679 116 10 if if SCONJ iajs-2679 116 11 and and CCONJ iajs-2679 116 12 only only ADV iajs-2679 116 13 if if SCONJ iajs-2679 116 14 there there PRON iajs-2679 116 15 exists exist VERB iajs-2679 116 16 a a DET iajs-2679 116 17 dense dense ADJ iajs-2679 116 18 set set VERB iajs-2679 116 19 𝐷1 𝐷1 NOUN iajs-2679 116 20 such such ADJ iajs-2679 116 21 that that PRON iajs-2679 116 22 for for ADP iajs-2679 116 23 each each DET iajs-2679 116 24 𝑦 𝑦 PROPN iajs-2679 116 25 ∈ ∈ PROPN iajs-2679 116 26 𝐷1and 𝐷1and PROPN iajs-2679 116 27 a a DET iajs-2679 116 28 small small ADJ iajs-2679 116 29 positive positive ADJ iajs-2679 116 30 number number NOUN iajs-2679 116 31 νœ€ νœ€ NOUN iajs-2679 116 32 > > X iajs-2679 116 33 0 0 NUM iajs-2679 116 34 , , PUNCT iajs-2679 116 35 there there PRON iajs-2679 116 36 is be VERB iajs-2679 116 37 an an DET iajs-2679 116 38 increasing increase VERB iajs-2679 116 39 sequence sequence NOUN iajs-2679 116 40 { { PUNCT iajs-2679 116 41 mj}j=1 mj}j=1 NOUN iajs-2679 116 42 ∞ ∞ PROPN iajs-2679 116 43 of of ADP iajs-2679 116 44 positive positive ADJ iajs-2679 116 45 integers integer NOUN iajs-2679 116 46 , , PUNCT iajs-2679 116 47 a a DET iajs-2679 116 48 dense dense ADJ iajs-2679 116 49 set set NOUN iajs-2679 116 50 𝐷2 𝐷2 NOUN iajs-2679 116 51 and and CCONJ iajs-2679 116 52 maps maps PROPN iajs-2679 116 53 𝑆𝑗 𝑆𝑗 PROPN iajs-2679 116 54 : : PUNCT iajs-2679 116 55 𝐷1 𝐷1 NOUN iajs-2679 116 56 β†’ β†’ PUNCT iajs-2679 116 57 𝑋 𝑋 NOUN iajs-2679 116 58 satisfying satisfy VERB iajs-2679 116 59 1 1 NUM iajs-2679 116 60 . . PUNCT iajs-2679 117 1 for for ADP iajs-2679 117 2 each each DET iajs-2679 117 3 π‘₯ π‘₯ PROPN iajs-2679 117 4 ∈ ∈ PROPN iajs-2679 117 5 𝐷2 𝐷2 NOUN iajs-2679 117 6 , , PUNCT iajs-2679 117 7 the the DET iajs-2679 117 8 sequence sequence NOUN iajs-2679 117 9 β€– β€– ADJ iajs-2679 117 10 π‘‡π‘šπ‘—π‘₯‖‖𝑆𝑗𝑦‖ π‘‡π‘šπ‘—π‘₯‖‖𝑆𝑗𝑦‖ NOUN iajs-2679 117 11 β†’ β†’ SYM iajs-2679 117 12 0 0 PUNCT iajs-2679 117 13 as as ADP iajs-2679 117 14 𝑗 𝑗 PROPN iajs-2679 117 15 β†’ β†’ SYM iajs-2679 117 16 ∞ ∞ PROPN iajs-2679 117 17 , , PUNCT iajs-2679 117 18 2 2 NUM iajs-2679 117 19 . . PUNCT iajs-2679 118 1 ‖𝑆𝑗𝑦‖ ‖𝑆𝑗𝑦‖ PUNCT iajs-2679 118 2 β†’ β†’ SYM iajs-2679 118 3 0 0 PUNCT iajs-2679 118 4 as as ADP iajs-2679 118 5 𝑗 𝑗 PROPN iajs-2679 118 6 β†’ β†’ SYM iajs-2679 118 7 ∞ ∞ PROPN iajs-2679 118 8 , , PUNCT iajs-2679 118 9 3 3 NUM iajs-2679 118 10 . . X iajs-2679 118 11 for for ADP iajs-2679 118 12 each each DET iajs-2679 118 13 integer integer PROPN iajs-2679 118 14 𝑐 𝑐 PROPN iajs-2679 118 15 β‰₯ β‰₯ NUM iajs-2679 118 16 1 1 NUM iajs-2679 118 17 , , PUNCT iajs-2679 118 18 there there PRON iajs-2679 118 19 exists exist VERB iajs-2679 118 20 𝑗 𝑗 PRON iajs-2679 118 21 β‰₯ β‰₯ NUM iajs-2679 118 22 𝑐 𝑐 NOUN iajs-2679 119 1 such such ADJ iajs-2679 119 2 that that DET iajs-2679 119 3 β€–π‘‡π‘šπ‘—π‘†π‘—π‘¦ β€–π‘‡π‘šπ‘—π‘†π‘—π‘¦ PROPN iajs-2679 119 4 βˆ’ βˆ’ PROPN iajs-2679 119 5 𝑦‖ 𝑦‖ PROPN iajs-2679 119 6 < < X iajs-2679 119 7 νœ€ νœ€ X iajs-2679 119 8 . . NOUN iajs-2679 119 9 since since SCONJ iajs-2679 119 10 a a DET iajs-2679 119 11 unilateral unilateral ADJ iajs-2679 119 12 shift shift NOUN iajs-2679 119 13 𝑇 𝑇 PROPN iajs-2679 119 14 is be AUX iajs-2679 119 15 diskcyclic diskcyclic ADJ iajs-2679 119 16 if if SCONJ iajs-2679 120 1 and and CCONJ iajs-2679 120 2 only only ADV iajs-2679 120 3 if if SCONJ iajs-2679 120 4 it it PRON iajs-2679 120 5 is be AUX iajs-2679 120 6 hypercyclic hypercyclic ADJ iajs-2679 120 7 with with ADP iajs-2679 120 8 𝐻𝐢(𝑇 𝐻𝐢(𝑇 NOUN iajs-2679 120 9 ) ) PUNCT iajs-2679 120 10 = = SYM iajs-2679 120 11 𝐷𝐢(𝑇 𝐷𝐢(𝑇 NOUN iajs-2679 120 12 ) ) PUNCT iajs-2679 121 1 [ [ X iajs-2679 121 2 5 5 NUM iajs-2679 121 3 , , PUNCT iajs-2679 121 4 corollary corollary ADJ iajs-2679 121 5 3.6 3.6 NUM iajs-2679 121 6 ] ] PUNCT iajs-2679 121 7 , , PUNCT iajs-2679 121 8 then then ADV iajs-2679 121 9 the the DET iajs-2679 121 10 following follow VERB iajs-2679 121 11 propositions proposition NOUN iajs-2679 121 12 follow follow VERB iajs-2679 121 13 immediately immediately ADV iajs-2679 121 14 by by ADP iajs-2679 121 15 [ [ X iajs-2679 121 16 12 12 NUM iajs-2679 121 17 , , PUNCT iajs-2679 121 18 theorem theorem VERB iajs-2679 121 19 3.5 3.5 NUM iajs-2679 121 20 ] ] PUNCT iajs-2679 121 21 and and CCONJ iajs-2679 121 22 [ [ X iajs-2679 121 23 12 12 NUM iajs-2679 121 24 , , PUNCT iajs-2679 121 25 theorem theorem VERB iajs-2679 121 26 4.1 4.1 NUM iajs-2679 121 27 ] ] PUNCT iajs-2679 121 28 respectively respectively ADV iajs-2679 121 29 . . PUNCT iajs-2679 122 1 proposition proposition NOUN iajs-2679 122 2 2.6 2.6 NUM iajs-2679 122 3 . . PUNCT iajs-2679 122 4 suppose suppose VERB iajs-2679 122 5 that that SCONJ iajs-2679 122 6 𝑇 𝑇 PROPN iajs-2679 122 7 and and CCONJ iajs-2679 122 8 𝑆 𝑆 PROPN iajs-2679 122 9 are be AUX iajs-2679 122 10 two two NUM iajs-2679 122 11 diskcyclic diskcyclic PROPN iajs-2679 122 12 weighted weight VERB iajs-2679 122 13 backward backward ADJ iajs-2679 122 14 shifts shift NOUN iajs-2679 122 15 . . PUNCT iajs-2679 123 1 then then ADV iajs-2679 123 2 , , PUNCT iajs-2679 123 3 there there PRON iajs-2679 123 4 exists exist VERB iajs-2679 123 5 a a DET iajs-2679 123 6 path path NOUN iajs-2679 123 7 of of ADP iajs-2679 123 8 unilateral unilateral ADJ iajs-2679 123 9 weighted weight VERB iajs-2679 123 10 backward backward ADJ iajs-2679 123 11 shifts shift NOUN iajs-2679 123 12 between between ADP iajs-2679 123 13 𝑇 𝑇 PROPN iajs-2679 123 14 and and CCONJ iajs-2679 123 15 𝑆 𝑆 PROPN iajs-2679 123 16 , , PUNCT iajs-2679 123 17 such such DET iajs-2679 123 18 a a DET iajs-2679 123 19 path path NOUN iajs-2679 123 20 has have VERB iajs-2679 123 21 a a DET iajs-2679 123 22 dense dense ADJ iajs-2679 123 23 gΞ΄ gΞ΄ NOUN iajs-2679 123 24 set set NOUN iajs-2679 123 25 of of ADP iajs-2679 123 26 common common ADJ iajs-2679 123 27 diskcyclic diskcyclic ADJ iajs-2679 123 28 vectors vector NOUN iajs-2679 123 29 . . PUNCT iajs-2679 124 1 proposition proposition NOUN iajs-2679 124 2 2.7 2.7 NUM iajs-2679 124 3 . . PUNCT iajs-2679 124 4 suppose suppose VERB iajs-2679 124 5 that that SCONJ iajs-2679 124 6 𝑇 𝑇 PROPN iajs-2679 124 7 and and CCONJ iajs-2679 124 8 𝑆 𝑆 PROPN iajs-2679 124 9 are be AUX iajs-2679 124 10 two two NUM iajs-2679 124 11 diskcyclic diskcyclic PROPN iajs-2679 124 12 weighted weight VERB iajs-2679 124 13 backward backward ADJ iajs-2679 124 14 shifts shift NOUN iajs-2679 124 15 . . PUNCT iajs-2679 125 1 then then ADV iajs-2679 125 2 , , PUNCT iajs-2679 125 3 there there PRON iajs-2679 125 4 exists exist VERB iajs-2679 125 5 a a DET iajs-2679 125 6 path path NOUN iajs-2679 125 7 of of ADP iajs-2679 125 8 unilateral unilateral ADJ iajs-2679 125 9 weighted weight VERB iajs-2679 125 10 backward backward ADJ iajs-2679 125 11 shifts shift NOUN iajs-2679 125 12 between between ADP iajs-2679 125 13 𝑇 𝑇 PROPN iajs-2679 125 14 and and CCONJ iajs-2679 125 15 𝑆 𝑆 PROPN iajs-2679 125 16 , , PUNCT iajs-2679 125 17 such such DET iajs-2679 125 18 a a DET iajs-2679 125 19 path path NOUN iajs-2679 125 20 has have VERB iajs-2679 125 21 no no DET iajs-2679 125 22 any any DET iajs-2679 125 23 common common ADJ iajs-2679 125 24 diskcyclic diskcyclic ADJ iajs-2679 125 25 vector vector NOUN iajs-2679 125 26 . . PUNCT iajs-2679 126 1 conclusion conclusion NOUN iajs-2679 126 2 we we PRON iajs-2679 126 3 have have AUX iajs-2679 126 4 given give VERB iajs-2679 126 5 a a DET iajs-2679 126 6 sufficient sufficient ADJ iajs-2679 126 7 condition condition NOUN iajs-2679 126 8 for for ADP iajs-2679 126 9 a a DET iajs-2679 126 10 path path NOUN iajs-2679 126 11 of of ADP iajs-2679 126 12 operators operator NOUN iajs-2679 126 13 to to PART iajs-2679 126 14 have have AUX iajs-2679 126 15 a a DET iajs-2679 126 16 dense dense ADJ iajs-2679 126 17 gΞ΄ gΞ΄ NOUN iajs-2679 126 18 set set NOUN iajs-2679 126 19 of of ADP iajs-2679 126 20 common common ADJ iajs-2679 126 21 diskcyclic diskcyclic ADJ iajs-2679 126 22 vectors vector NOUN iajs-2679 126 23 such such ADJ iajs-2679 126 24 that that SCONJ iajs-2679 126 25 every every DET iajs-2679 126 26 operator operator NOUN iajs-2679 126 27 in in ADP iajs-2679 126 28 that that DET iajs-2679 126 29 path path NOUN iajs-2679 126 30 satisfies satisfy VERB iajs-2679 126 31 diskcyclic diskcyclic ADJ iajs-2679 126 32 criterion criterion NOUN iajs-2679 126 33 . . PUNCT iajs-2679 127 1 references reference NOUN iajs-2679 127 2 1 1 NUM iajs-2679 127 3 . . PUNCT iajs-2679 127 4 rolewicz rolewicz PROPN iajs-2679 127 5 , , PUNCT iajs-2679 127 6 s. s. PROPN iajs-2679 127 7 on on ADP iajs-2679 127 8 orbits orbit NOUN iajs-2679 127 9 of of ADP iajs-2679 127 10 elements element NOUN iajs-2679 127 11 . . PUNCT iajs-2679 128 1 studia studia PROPN iajs-2679 128 2 mathematica mathematica PROPN iajs-2679 128 3 , , PUNCT iajs-2679 128 4 1969 1969 NUM iajs-2679 128 5 , , PUNCT iajs-2679 128 6 1(32):17 1(32):17 NOUN iajs-2679 128 7 - - SYM iajs-2679 128 8 22 22 NUM iajs-2679 128 9 , , PUNCT iajs-2679 128 10 . . PUNCT iajs-2679 129 1 2 2 X iajs-2679 129 2 . . X iajs-2679 129 3 hilden hilden PROPN iajs-2679 129 4 , , PUNCT iajs-2679 129 5 h. h. PROPN iajs-2679 129 6 ; ; PUNCT iajs-2679 129 7 wallen wallen PROPN iajs-2679 129 8 , , PUNCT iajs-2679 129 9 l. l. NOUN iajs-2679 129 10 some some DET iajs-2679 129 11 cyclic cyclic ADJ iajs-2679 129 12 and and CCONJ iajs-2679 129 13 non non ADJ iajs-2679 129 14 - - ADJ iajs-2679 129 15 cyclic cyclic ADJ iajs-2679 129 16 vectors vector NOUN iajs-2679 129 17 of of ADP iajs-2679 129 18 certain certain ADJ iajs-2679 129 19 operators operator NOUN iajs-2679 129 20 . . PUNCT iajs-2679 130 1 indiana indiana PROPN iajs-2679 130 2 university university PROPN iajs-2679 130 3 mathematics mathematics PROPN iajs-2679 130 4 journal journal NOUN iajs-2679 130 5 , , PUNCT iajs-2679 130 6 1974 1974 NUM iajs-2679 130 7 , , PUNCT iajs-2679 130 8 23(7):557 23(7):557 NUM iajs-2679 130 9 - - SYM iajs-2679 130 10 565 565 NUM iajs-2679 130 11 . . PUNCT iajs-2679 131 1 3 3 NUM iajs-2679 131 2 . . X iajs-2679 131 3 zeana zeana PROPN iajs-2679 131 4 , , PUNCT iajs-2679 131 5 j. j. PROPN iajs-2679 131 6 cyclic cyclic PROPN iajs-2679 131 7 phenomena phenomenon NOUN iajs-2679 131 8 of of ADP iajs-2679 131 9 operators operator NOUN iajs-2679 131 10 on on ADP iajs-2679 131 11 hilbert hilbert NOUN iajs-2679 131 12 space space NOUN iajs-2679 131 13 . . PUNCT iajs-2679 132 1 phd phd NOUN iajs-2679 132 2 thesis thesis NOUN iajs-2679 132 3 , , PUNCT iajs-2679 132 4 thesis thesis NOUN iajs-2679 132 5 , , PUNCT iajs-2679 132 6 university university NOUN iajs-2679 132 7 of of ADP iajs-2679 132 8 baghdad baghdad PROPN iajs-2679 132 9 , , PUNCT iajs-2679 132 10 2002 2002 NUM iajs-2679 132 11 . . PUNCT iajs-2679 133 1 4 4 X iajs-2679 133 2 . . X iajs-2679 133 3 bamerni bamerni PROPN iajs-2679 133 4 , , PUNCT iajs-2679 133 5 n. n. NOUN iajs-2679 133 6 ; ; PUNCT iajs-2679 133 7 kilicman kilicman NOUN iajs-2679 133 8 , , PUNCT iajs-2679 133 9 a. a. NOUN iajs-2679 133 10 operators operator NOUN iajs-2679 133 11 with with ADP iajs-2679 133 12 diskcyclic diskcyclic ADJ iajs-2679 133 13 vectors vector NOUN iajs-2679 133 14 subspaces subspace NOUN iajs-2679 133 15 . . PUNCT iajs-2679 134 1 journal journal PROPN iajs-2679 134 2 of of ADP iajs-2679 134 3 taibah taibah PROPN iajs-2679 134 4 university university PROPN iajs-2679 134 5 for for ADP iajs-2679 134 6 science science NOUN iajs-2679 134 7 , , PUNCT iajs-2679 134 8 2015 2015 NUM iajs-2679 134 9 , , PUNCT iajs-2679 134 10 9(3):414 9(3):414 NUM iajs-2679 134 11 - - SYM iajs-2679 134 12 419 419 NUM iajs-2679 134 13 , , PUNCT iajs-2679 134 14 5 5 NUM iajs-2679 134 15 . . X iajs-2679 135 1 bamerni bamerni PROPN iajs-2679 135 2 , , PUNCT iajs-2679 135 3 n. n. NOUN iajs-2679 135 4 ; ; PUNCT iajs-2679 135 5 kilicman kilicman NOUN iajs-2679 135 6 , , PUNCT iajs-2679 135 7 a. a. NOUN iajs-2679 135 8 ; ; PUNCT iajs-2679 135 9 noorani noorani ADJ iajs-2679 135 10 , , PUNCT iajs-2679 135 11 m. m. NOUN iajs-2679 135 12 s. s. PROPN iajs-2679 135 13 m. m. VERB iajs-2679 135 14 a a DET iajs-2679 135 15 review review NOUN iajs-2679 135 16 of of ADP iajs-2679 135 17 some some DET iajs-2679 135 18 works work NOUN iajs-2679 135 19 in in ADP iajs-2679 135 20 the the DET iajs-2679 135 21 theory theory NOUN iajs-2679 135 22 of of ADP iajs-2679 135 23 diskcyclic diskcyclic PROPN iajs-2679 135 24 operators operator NOUN iajs-2679 135 25 . . PUNCT iajs-2679 136 1 bulletin bulletin NOUN iajs-2679 136 2 of of ADP iajs-2679 136 3 the the DET iajs-2679 136 4 malaysian malaysian PROPN iajs-2679 136 5 mathematical mathematical PROPN iajs-2679 136 6 sciences sciences PROPN iajs-2679 136 7 society society NOUN iajs-2679 136 8 , , PUNCT iajs-2679 136 9 2015 2015 NUM iajs-2679 136 10 , , PUNCT iajs-2679 136 11 1 1 NUM iajs-2679 136 12 - - SYM iajs-2679 136 13 17 17 NUM iajs-2679 136 14 . . PUNCT iajs-2679 137 1 6 6 NUM iajs-2679 137 2 . . NUM iajs-2679 137 3 , , PUNCT iajs-2679 137 4 bayart bayart PROPN iajs-2679 137 5 f. f. PROPN iajs-2679 137 6 ; ; PUNCT iajs-2679 137 7 matheron matheron PROPN iajs-2679 137 8 , , PUNCT iajs-2679 137 9 e. e. PROPN iajs-2679 137 10 dynamics dynamic NOUN iajs-2679 137 11 of of ADP iajs-2679 137 12 linear linear PROPN iajs-2679 137 13 operators operator NOUN iajs-2679 137 14 , , PUNCT iajs-2679 137 15 volume volume NOUN iajs-2679 137 16 . . PUNCT iajs-2679 138 1 cambridge cambridge PROPN iajs-2679 138 2 university university PROPN iajs-2679 138 3 press press NOUN iajs-2679 138 4 , , PUNCT iajs-2679 138 5 2009 2009 NUM iajs-2679 138 6 , , PUNCT iajs-2679 138 7 179 179 NUM iajs-2679 138 8 . . NOUN iajs-2679 139 1 7 7 X iajs-2679 139 2 . . X iajs-2679 139 3 bamerni bamerni PROPN iajs-2679 139 4 , , PUNCT iajs-2679 139 5 n. n. NOUN iajs-2679 139 6 ; ; PUNCT iajs-2679 139 7 kadets kadet NOUN iajs-2679 139 8 , , PUNCT iajs-2679 139 9 v. v. PROPN iajs-2679 139 10 ; ; PUNCT iajs-2679 139 11 kilicman kilicman NOUN iajs-2679 139 12 , , PUNCT iajs-2679 139 13 a. a. PROPN iajs-2679 139 14 hypercyclic hypercyclic PROPN iajs-2679 139 15 operators operator NOUN iajs-2679 139 16 are be AUX iajs-2679 139 17 subspace subspace NOUN iajs-2679 139 18 hypercyclic hypercyclic NOUN iajs-2679 139 19 . . PUNCT iajs-2679 140 1 journal journal PROPN iajs-2679 140 2 of of ADP iajs-2679 140 3 mathematical mathematical ADJ iajs-2679 140 4 analysis analysis NOUN iajs-2679 140 5 and and CCONJ iajs-2679 140 6 applications application NOUN iajs-2679 140 7 , , PUNCT iajs-2679 140 8 2016 2016 NUM iajs-2679 140 9 , , PUNCT iajs-2679 140 10 435,2,1812 435,2,1812 NUM iajs-2679 140 11 - - SYM iajs-2679 140 12 1815 1815 NUM iajs-2679 140 13 . . PUNCT iajs-2679 141 1 ibn ibn PROPN iajs-2679 141 2 al al PROPN iajs-2679 141 3 - - PUNCT iajs-2679 141 4 haitham haitham PROPN iajs-2679 141 5 jour jour X iajs-2679 141 6 . . PROPN iajs-2679 141 7 for for ADP iajs-2679 141 8 pure pure ADJ iajs-2679 141 9 & & CCONJ iajs-2679 141 10 appl appl PROPN iajs-2679 141 11 . . PUNCT iajs-2679 142 1 sci sci PROPN iajs-2679 142 2 . . PROPN iajs-2679 143 1 34(3)2021 34(3)2021 NUM iajs-2679 143 2 72 72 NUM iajs-2679 143 3 8 8 NUM iajs-2679 143 4 . . PUNCT iajs-2679 144 1 bamerni bamerni PROPN iajs-2679 144 2 , , PUNCT iajs-2679 144 3 n. n. NOUN iajs-2679 144 4 ; ; PUNCT iajs-2679 144 5 kilicman kilicman NOUN iajs-2679 144 6 , , PUNCT iajs-2679 144 7 a. a. NOUN iajs-2679 144 8 on on ADP iajs-2679 144 9 subspaces subspace NOUN iajs-2679 144 10 diskcyclicity diskcyclicity NOUN iajs-2679 144 11 , , PUNCT iajs-2679 144 12 arab arab ADJ iajs-2679 144 13 journal journal PROPN iajs-2679 144 14 of of ADP iajs-2679 144 15 mathematical mathematical ADJ iajs-2679 144 16 sciences science NOUN iajs-2679 144 17 , , PUNCT iajs-2679 144 18 2015 2015 NUM iajs-2679 144 19 , , PUNCT iajs-2679 144 20 23(2),133 23(2),133 NUM iajs-2679 144 21 - - SYM iajs-2679 144 22 140 140 NUM iajs-2679 144 23 , , PUNCT iajs-2679 144 24 . . PUNCT iajs-2679 145 1 9 9 X iajs-2679 145 2 . . X iajs-2679 145 3 madore madore NOUN iajs-2679 145 4 , , PUNCT iajs-2679 145 5 b. b. PROPN iajs-2679 145 6 f. f. PROPN iajs-2679 145 7 ; ; PUNCT iajs-2679 146 1 martinez martinez PROPN iajs-2679 146 2 - - PUNCT iajs-2679 146 3 avendano avendano PROPN iajs-2679 146 4 , , PUNCT iajs-2679 146 5 r. r. PROPN iajs-2679 146 6 a. a. PROPN iajs-2679 146 7 subspace subspace PROPN iajs-2679 146 8 hypercyclicity hypercyclicity NOUN iajs-2679 146 9 . . PUNCT iajs-2679 147 1 journal journal PROPN iajs-2679 147 2 of of ADP iajs-2679 147 3 mathematical mathematical ADJ iajs-2679 147 4 analysis analysis NOUN iajs-2679 147 5 and and CCONJ iajs-2679 147 6 applications application NOUN iajs-2679 147 7 , , PUNCT iajs-2679 147 8 2011 2011 NUM iajs-2679 147 9 , , PUNCT iajs-2679 147 10 373(2):502 373(2):502 NUM iajs-2679 147 11 - - PUNCT iajs-2679 147 12 511 511 NUM iajs-2679 147 13 , , PUNCT iajs-2679 147 14 . . PUNCT iajs-2679 148 1 10 10 NUM iajs-2679 148 2 . . PUNCT iajs-2679 149 1 xian xian NOUN iajs-2679 149 2 - - PUNCT iajs-2679 149 3 feng feng PROPN iajs-2679 149 4 , , PUNCT iajs-2679 149 5 z. z. PROPN iajs-2679 149 6 ; ; PUNCT iajs-2679 149 7 yong yong PROPN iajs-2679 149 8 - - PUNCT iajs-2679 149 9 lu lu PROPN iajs-2679 149 10 , , PUNCT iajs-2679 149 11 s. s. PROPN iajs-2679 149 12 ; ; PUNCT iajs-2679 149 13 yun yun PROPN iajs-2679 149 14 - - PUNCT iajs-2679 149 15 hua hua PROPN iajs-2679 149 16 . . PUNCT iajs-2679 150 1 z. z. PROPN iajs-2679 150 2 subspace subspace PROPN iajs-2679 150 3 - - PUNCT iajs-2679 150 4 supercyclicity supercyclicity NOUN iajs-2679 150 5 and and CCONJ iajs-2679 150 6 common common ADJ iajs-2679 150 7 subspace subspace NOUN iajs-2679 150 8 supercyclic supercyclic PROPN iajs-2679 150 9 vectors vector NOUN iajs-2679 150 10 . . PUNCT iajs-2679 150 11 , , PUNCT iajs-2679 150 12 journal journal PROPN iajs-2679 150 13 of of ADP iajs-2679 150 14 east east PROPN iajs-2679 150 15 china china PROPN iajs-2679 150 16 normal normal ADJ iajs-2679 150 17 university university NOUN iajs-2679 150 18 , , PUNCT iajs-2679 150 19 2012 2012 NUM iajs-2679 150 20 , , PUNCT iajs-2679 150 21 1(1):106112 1(1):106112 NUM iajs-2679 150 22 . . PROPN iajs-2679 151 1 11 11 NUM iajs-2679 151 2 . . X iajs-2679 152 1 chan chan PROPN iajs-2679 152 2 , , PUNCT iajs-2679 152 3 k. k. PROPN iajs-2679 152 4 c. c. PROPN iajs-2679 152 5 ; ; PUNCT iajs-2679 152 6 sanders sanders PROPN iajs-2679 152 7 , , PUNCT iajs-2679 152 8 r. r. PROPN iajs-2679 152 9 common common PROPN iajs-2679 152 10 supercyclic supercyclic PROPN iajs-2679 152 11 vectors vector NOUN iajs-2679 152 12 for for ADP iajs-2679 152 13 a a DET iajs-2679 152 14 path path NOUN iajs-2679 152 15 of of ADP iajs-2679 152 16 operators operator NOUN iajs-2679 152 17 . . 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