id sid tid token lemma pos ejpam-2159 1 1 compile compile NOUN ejpam-2159 1 2 / / SYM ejpam-2159 1 3 output.dvi output.dvi NOUN ejpam-2159 1 4 european european ADJ ejpam-2159 1 5 journal journal NOUN ejpam-2159 1 6 of of ADP ejpam-2159 1 7 pure pure ADJ ejpam-2159 1 8 and and CCONJ ejpam-2159 1 9 applied apply VERB ejpam-2159 1 10 mathematics mathematic NOUN ejpam-2159 1 11 vol vol NOUN ejpam-2159 1 12 . . PROPN ejpam-2159 1 13 8 8 NUM ejpam-2159 1 14 , , PUNCT ejpam-2159 1 15 no no INTJ ejpam-2159 1 16 . . NOUN ejpam-2159 1 17 1 1 NUM ejpam-2159 1 18 , , PUNCT ejpam-2159 1 19 2015 2015 NUM ejpam-2159 1 20 , , PUNCT ejpam-2159 1 21 135 135 NUM ejpam-2159 1 22 - - SYM ejpam-2159 1 23 151 151 NUM ejpam-2159 1 24 issn issn PROPN ejpam-2159 1 25 1307 1307 NUM ejpam-2159 1 26 - - SYM ejpam-2159 1 27 5543 5543 NUM ejpam-2159 1 28 – – PUNCT ejpam-2159 1 29 www.ejpam.com www.ejpam.com X ejpam-2159 1 30 observations observation NOUN ejpam-2159 1 31 on on ADP ejpam-2159 1 32 some some DET ejpam-2159 1 33 special special ADJ ejpam-2159 1 34 matrices matrix NOUN ejpam-2159 1 35 and and CCONJ ejpam-2159 1 36 polynomials polynomial NOUN ejpam-2159 1 37 moawwad moawwad PROPN ejpam-2159 1 38 el el PROPN ejpam-2159 1 39 - - PUNCT ejpam-2159 1 40 mikkawy mikkawy NOUN ejpam-2159 1 41 ! ! PUNCT ejpam-2159 1 42 , , PUNCT ejpam-2159 1 43 faiz faiz PROPN ejpam-2159 1 44 atlan atlan PROPN ejpam-2159 1 45 mathematics mathematics PROPN ejpam-2159 1 46 department department PROPN ejpam-2159 1 47 , , PUNCT ejpam-2159 1 48 faculty faculty NOUN ejpam-2159 1 49 of of ADP ejpam-2159 1 50 science science NOUN ejpam-2159 1 51 , , PUNCT ejpam-2159 1 52 mansoura mansoura PROPN ejpam-2159 1 53 university university NOUN ejpam-2159 1 54 , , PUNCT ejpam-2159 1 55 mansoura mansoura PROPN ejpam-2159 1 56 35516 35516 NUM ejpam-2159 1 57 , , PUNCT ejpam-2159 1 58 egypt egypt PROPN ejpam-2159 1 59 abstract abstract PROPN ejpam-2159 1 60 . . PUNCT ejpam-2159 2 1 in in ADP ejpam-2159 2 2 the the DET ejpam-2159 2 3 current current ADJ ejpam-2159 2 4 paper paper NOUN ejpam-2159 2 5 we we PRON ejpam-2159 2 6 focus focus VERB ejpam-2159 2 7 on on ADP ejpam-2159 2 8 the the DET ejpam-2159 2 9 study study NOUN ejpam-2159 2 10 of of ADP ejpam-2159 2 11 three three NUM ejpam-2159 2 12 special special ADJ ejpam-2159 2 13 matrices matrix NOUN ejpam-2159 2 14 and and CCONJ ejpam-2159 2 15 two two NUM ejpam-2159 2 16 symmetric symmetric ADJ ejpam-2159 2 17 polynomials polynomial NOUN ejpam-2159 2 18 . . PUNCT ejpam-2159 3 1 as as ADP ejpam-2159 3 2 a a DET ejpam-2159 3 3 consequence consequence NOUN ejpam-2159 3 4 , , PUNCT ejpam-2159 3 5 a a DET ejpam-2159 3 6 recurrence recurrence NOUN ejpam-2159 3 7 relation relation NOUN ejpam-2159 3 8 satisfied satisfy VERB ejpam-2159 3 9 by by ADP ejpam-2159 3 10 the the DET ejpam-2159 3 11 entries entry NOUN ejpam-2159 3 12 of of ADP ejpam-2159 3 13 the the DET ejpam-2159 3 14 n n NOUN ejpam-2159 3 15 " " PUNCT ejpam-2159 3 16 n n CCONJ ejpam-2159 3 17 inverse inverse NOUN ejpam-2159 3 18 matrix matrix NOUN ejpam-2159 3 19 , , PUNCT ejpam-2159 3 20 qn qn NOUN ejpam-2159 3 21 of of ADP ejpam-2159 3 22 the the DET ejpam-2159 3 23 n n CCONJ ejpam-2159 3 24 " " PUNCT ejpam-2159 3 25 n n CCONJ ejpam-2159 3 26 symmetric symmetric ADJ ejpam-2159 3 27 pascal pascal ADJ ejpam-2159 3 28 matrix matrix NOUN ejpam-2159 3 29 , , PUNCT ejpam-2159 3 30 pn pn PROPN ejpam-2159 3 31 is be AUX ejpam-2159 3 32 obtained obtain VERB ejpam-2159 3 33 . . PUNCT ejpam-2159 4 1 moreover moreover ADV ejpam-2159 4 2 , , PUNCT ejpam-2159 4 3 a a DET ejpam-2159 4 4 new new ADJ ejpam-2159 4 5 proof proof NOUN ejpam-2159 4 6 for for ADP ejpam-2159 4 7 el el PROPN ejpam-2159 4 8 - - PUNCT ejpam-2159 4 9 mikkawy mikkawy NOUN ejpam-2159 4 10 conjecture conjecture NOUN ejpam-2159 4 11 [ [ X ejpam-2159 4 12 14 14 NUM ejpam-2159 4 13 ] ] PUNCT ejpam-2159 4 14 is be AUX ejpam-2159 4 15 investigated investigate VERB ejpam-2159 4 16 . . PUNCT ejpam-2159 5 1 finally finally ADV ejpam-2159 5 2 , , PUNCT ejpam-2159 5 3 some some DET ejpam-2159 5 4 identities identity NOUN ejpam-2159 5 5 are be AUX ejpam-2159 5 6 discovered discover VERB ejpam-2159 5 7 . . PUNCT ejpam-2159 6 1 2010 2010 NUM ejpam-2159 6 2 mathematics mathematic NOUN ejpam-2159 6 3 subject subject NOUN ejpam-2159 6 4 classifications classification NOUN ejpam-2159 6 5 : : PUNCT ejpam-2159 6 6 15a23 15a23 NUM ejpam-2159 6 7 , , PUNCT ejpam-2159 6 8 15bxx 15bxx NOUN ejpam-2159 6 9 , , PUNCT ejpam-2159 6 10 11b73 11b73 NUM ejpam-2159 6 11 , , PUNCT ejpam-2159 6 12 05e05 05e05 NUM ejpam-2159 6 13 , , PUNCT ejpam-2159 6 14 68w30 68w30 NUM ejpam-2159 6 15 , , PUNCT ejpam-2159 6 16 65f05 65f05 NUM ejpam-2159 6 17 key key ADJ ejpam-2159 6 18 words word NOUN ejpam-2159 6 19 and and CCONJ ejpam-2159 6 20 phrases phrase NOUN ejpam-2159 6 21 : : PUNCT ejpam-2159 6 22 pascal pascal ADJ ejpam-2159 6 23 matrix matrix NOUN ejpam-2159 6 24 , , PUNCT ejpam-2159 6 25 vandermonde vandermonde ADJ ejpam-2159 6 26 matrix matrix NOUN ejpam-2159 6 27 , , PUNCT ejpam-2159 6 28 stirling stirling NOUN ejpam-2159 6 29 matrix matrix NOUN ejpam-2159 6 30 , , PUNCT ejpam-2159 6 31 permutation permutation NOUN ejpam-2159 6 32 matrix matrix NOUN ejpam-2159 6 33 , , PUNCT ejpam-2159 6 34 divided divided ADJ ejpam-2159 6 35 difference difference NOUN ejpam-2159 6 36 , , PUNCT ejpam-2159 6 37 symmetric symmetric ADJ ejpam-2159 6 38 polynomials polynomial NOUN ejpam-2159 6 39 , , PUNCT ejpam-2159 6 40 maple maple NOUN ejpam-2159 6 41 1 1 NUM ejpam-2159 6 42 . . PUNCT ejpam-2159 7 1 introduction introduction NOUN ejpam-2159 7 2 and and CCONJ ejpam-2159 7 3 basic basic ADJ ejpam-2159 7 4 definitions definition NOUN ejpam-2159 7 5 matrices matrix NOUN ejpam-2159 7 6 play play VERB ejpam-2159 7 7 an an DET ejpam-2159 7 8 important important ADJ ejpam-2159 7 9 role role NOUN ejpam-2159 7 10 in in ADP ejpam-2159 7 11 all all DET ejpam-2159 7 12 branches branch NOUN ejpam-2159 7 13 of of ADP ejpam-2159 7 14 science science NOUN ejpam-2159 7 15 , , PUNCT ejpam-2159 7 16 engineering engineering NOUN ejpam-2159 7 17 , , PUNCT ejpam-2159 7 18 social social ADJ ejpam-2159 7 19 science science NOUN ejpam-2159 7 20 and and CCONJ ejpam-2159 7 21 management management NOUN ejpam-2159 7 22 . . PUNCT ejpam-2159 8 1 matrices matrix NOUN ejpam-2159 8 2 are be AUX ejpam-2159 8 3 used use VERB ejpam-2159 8 4 for for ADP ejpam-2159 8 5 data datum NOUN ejpam-2159 8 6 classification classification NOUN ejpam-2159 8 7 by by ADP ejpam-2159 8 8 which which PRON ejpam-2159 8 9 many many ADJ ejpam-2159 8 10 problems problem NOUN ejpam-2159 8 11 are be AUX ejpam-2159 8 12 solved solve VERB ejpam-2159 8 13 using use VERB ejpam-2159 8 14 computers computer NOUN ejpam-2159 8 15 . . PUNCT ejpam-2159 9 1 there there PRON ejpam-2159 9 2 are be VERB ejpam-2159 9 3 many many ADJ ejpam-2159 9 4 special special ADJ ejpam-2159 9 5 types type NOUN ejpam-2159 9 6 of of ADP ejpam-2159 9 7 matrices matrix NOUN ejpam-2159 9 8 such such ADJ ejpam-2159 9 9 as as ADP ejpam-2159 9 10 the the DET ejpam-2159 9 11 pascal pascal NOUN ejpam-2159 10 1 [ [ X ejpam-2159 10 2 1 1 NUM ejpam-2159 10 3 ] ] PUNCT ejpam-2159 10 4 , , PUNCT ejpam-2159 10 5 vandermonde vandermonde VERB ejpam-2159 10 6 [ [ X ejpam-2159 10 7 10 10 NUM ejpam-2159 10 8 ] ] PUNCT ejpam-2159 10 9 , , PUNCT ejpam-2159 10 10 stirling stirling NOUN ejpam-2159 11 1 [ [ X ejpam-2159 11 2 12 12 NUM ejpam-2159 11 3 ] ] PUNCT ejpam-2159 11 4 , , PUNCT ejpam-2159 11 5 and and CCONJ ejpam-2159 11 6 others other NOUN ejpam-2159 11 7 . . PUNCT ejpam-2159 12 1 these these DET ejpam-2159 12 2 matrices matrix NOUN ejpam-2159 12 3 are be AUX ejpam-2159 12 4 of of ADP ejpam-2159 12 5 specific specific ADJ ejpam-2159 12 6 importance importance NOUN ejpam-2159 12 7 in in ADP ejpam-2159 12 8 many many ADJ ejpam-2159 12 9 scientific scientific ADJ ejpam-2159 12 10 and and CCONJ ejpam-2159 12 11 engineering engineering NOUN ejpam-2159 12 12 applications application NOUN ejpam-2159 12 13 . . PUNCT ejpam-2159 13 1 for for ADP ejpam-2159 13 2 example example NOUN ejpam-2159 13 3 the the DET ejpam-2159 13 4 pascal pascal ADJ ejpam-2159 13 5 matrix matrix NOUN ejpam-2159 13 6 , , PUNCT ejpam-2159 13 7 which which PRON ejpam-2159 13 8 has have AUX ejpam-2159 13 9 been be AUX ejpam-2159 13 10 known know VERB ejpam-2159 13 11 since since SCONJ ejpam-2159 13 12 1303 1303 NUM ejpam-2159 13 13 , , PUNCT ejpam-2159 13 14 but but CCONJ ejpam-2159 13 15 has have AUX ejpam-2159 13 16 been be AUX ejpam-2159 13 17 studied study VERB ejpam-2159 13 18 carefully carefully ADV ejpam-2159 13 19 only only ADV ejpam-2159 13 20 recently recently ADV ejpam-2159 13 21 [ [ X ejpam-2159 13 22 1 1 NUM ejpam-2159 13 23 ] ] PUNCT ejpam-2159 13 24 , , PUNCT ejpam-2159 13 25 appears appear VERB ejpam-2159 13 26 in in ADP ejpam-2159 13 27 combinatorics combinatoric NOUN ejpam-2159 13 28 , , PUNCT ejpam-2159 13 29 image image NOUN ejpam-2159 13 30 processing processing NOUN ejpam-2159 13 31 , , PUNCT ejpam-2159 13 32 signal signal NOUN ejpam-2159 13 33 processing processing NOUN ejpam-2159 13 34 , , PUNCT ejpam-2159 13 35 numerical numerical ADJ ejpam-2159 13 36 analysis analysis NOUN ejpam-2159 13 37 , , PUNCT ejpam-2159 13 38 probability probability NOUN ejpam-2159 13 39 and and CCONJ ejpam-2159 13 40 surface surface NOUN ejpam-2159 13 41 reconstruction reconstruction NOUN ejpam-2159 13 42 . . PUNCT ejpam-2159 14 1 much much ADJ ejpam-2159 14 2 researches research NOUN ejpam-2159 14 3 has have AUX ejpam-2159 14 4 been be AUX ejpam-2159 14 5 devoted devote VERB ejpam-2159 14 6 to to PART ejpam-2159 14 7 deal deal VERB ejpam-2159 14 8 with with ADP ejpam-2159 14 9 such such ADJ ejpam-2159 14 10 matrices matrix NOUN ejpam-2159 14 11 and and CCONJ ejpam-2159 14 12 relations relation NOUN ejpam-2159 14 13 between between ADP ejpam-2159 14 14 them they PRON ejpam-2159 14 15 ( ( PUNCT ejpam-2159 14 16 see see VERB ejpam-2159 14 17 for for ADP ejpam-2159 14 18 instance instance NOUN ejpam-2159 14 19 , , PUNCT ejpam-2159 14 20 [ [ X ejpam-2159 14 21 3–5 3–5 NUM ejpam-2159 14 22 , , PUNCT ejpam-2159 14 23 7 7 NUM ejpam-2159 14 24 , , PUNCT ejpam-2159 14 25 9 9 NUM ejpam-2159 14 26 , , PUNCT ejpam-2159 14 27 19 19 NUM ejpam-2159 14 28 , , PUNCT ejpam-2159 14 29 21–25 21–25 NUM ejpam-2159 14 30 , , PUNCT ejpam-2159 14 31 29–37 29–37 NOUN ejpam-2159 14 32 ] ] PUNCT ejpam-2159 14 33 ) ) PUNCT ejpam-2159 14 34 . . PUNCT ejpam-2159 15 1 the the DET ejpam-2159 15 2 main main ADJ ejpam-2159 15 3 objectives objective NOUN ejpam-2159 15 4 of of ADP ejpam-2159 15 5 the the DET ejpam-2159 15 6 current current ADJ ejpam-2159 15 7 paper paper NOUN ejpam-2159 15 8 is be AUX ejpam-2159 15 9 to to PART ejpam-2159 15 10 introduce introduce VERB ejpam-2159 15 11 a a DET ejpam-2159 15 12 recurrence recurrence NOUN ejpam-2159 15 13 relation relation NOUN ejpam-2159 15 14 for for ADP ejpam-2159 15 15 the the DET ejpam-2159 15 16 inverse inverse NOUN ejpam-2159 15 17 of of ADP ejpam-2159 15 18 the the DET ejpam-2159 15 19 symmetric symmetric ADJ ejpam-2159 15 20 pascal pascal ADJ ejpam-2159 15 21 matrix matrix NOUN ejpam-2159 15 22 , , PUNCT ejpam-2159 15 23 pn pn NOUN ejpam-2159 15 24 of of ADP ejpam-2159 15 25 order order NOUN ejpam-2159 15 26 n n CCONJ ejpam-2159 15 27 , , PUNCT ejpam-2159 15 28 some some DET ejpam-2159 15 29 new new ADJ ejpam-2159 15 30 identities identity NOUN ejpam-2159 15 31 and and CCONJ ejpam-2159 15 32 to to PART ejpam-2159 15 33 give give VERB ejpam-2159 15 34 another another DET ejpam-2159 15 35 proof proof NOUN ejpam-2159 15 36 for for ADP ejpam-2159 15 37 el el PROPN ejpam-2159 15 38 - - PUNCT ejpam-2159 15 39 mikkawy mikkawy NOUN ejpam-2159 15 40 conjecture conjecture NOUN ejpam-2159 15 41 [ [ X ejpam-2159 15 42 14 14 NUM ejpam-2159 15 43 ] ] PUNCT ejpam-2159 15 44 . . PUNCT ejpam-2159 16 1 the the DET ejpam-2159 16 2 paper paper NOUN ejpam-2159 16 3 is be AUX ejpam-2159 16 4 organized organize VERB ejpam-2159 16 5 as as SCONJ ejpam-2159 16 6 follows follow VERB ejpam-2159 16 7 : : PUNCT ejpam-2159 16 8 the the DET ejpam-2159 16 9 main main ADJ ejpam-2159 16 10 results result NOUN ejpam-2159 16 11 are be AUX ejpam-2159 16 12 given give VERB ejpam-2159 16 13 in in ADP ejpam-2159 16 14 the the DET ejpam-2159 16 15 next next ADJ ejpam-2159 16 16 section section NOUN ejpam-2159 16 17 . . PUNCT ejpam-2159 17 1 in in ADP ejpam-2159 17 2 section section NOUN ejpam-2159 17 3 3 3 NUM ejpam-2159 17 4 , , PUNCT ejpam-2159 17 5 we we PRON ejpam-2159 17 6 present present VERB ejpam-2159 17 7 a a DET ejpam-2159 17 8 new new ADJ ejpam-2159 17 9 proof proof NOUN ejpam-2159 17 10 for for ADP ejpam-2159 17 11 el el NOUN ejpam-2159 17 12 - - PUNCT ejpam-2159 17 13 mikkawwy mikkawwy ADJ ejpam-2159 17 14 conjecture conjecture NOUN ejpam-2159 17 15 [ [ X ejpam-2159 17 16 14 14 NUM ejpam-2159 17 17 ] ] PUNCT ejpam-2159 17 18 . . PUNCT ejpam-2159 18 1 moreover moreover ADV ejpam-2159 18 2 , , PUNCT ejpam-2159 18 3 new new ADJ ejpam-2159 18 4 identities identity NOUN ejpam-2159 18 5 are be AUX ejpam-2159 18 6 obtained obtain VERB ejpam-2159 18 7 . . PUNCT ejpam-2159 19 1 throughout throughout ADP ejpam-2159 19 2 this this DET ejpam-2159 19 3 paper paper NOUN ejpam-2159 19 4 , , PUNCT ejpam-2159 19 5 ! ! PUNCT ejpam-2159 20 1 i i PRON ejpam-2159 20 2 j j PROPN ejpam-2159 20 3 is be AUX ejpam-2159 20 4 the the DET ejpam-2159 20 5 kronecker kronecker NOUN ejpam-2159 20 6 delta delta NOUN ejpam-2159 20 7 which which PRON ejpam-2159 20 8 is be AUX ejpam-2159 20 9 equal equal ADJ ejpam-2159 20 10 to to ADP ejpam-2159 20 11 1 1 NUM ejpam-2159 20 12 or or CCONJ ejpam-2159 20 13 0 0 NUM ejpam-2159 20 14 according accord VERB ejpam-2159 20 15 as as ADP ejpam-2159 20 16 i i PROPN ejpam-2159 20 17 = = SYM ejpam-2159 20 18 j j PROPN ejpam-2159 20 19 or or CCONJ ejpam-2159 20 20 not not PART ejpam-2159 20 21 . . PUNCT ejpam-2159 21 1 also also ADV ejpam-2159 21 2 ! ! PUNCT ejpam-2159 22 1 n n CCONJ ejpam-2159 22 2 r r NOUN ejpam-2159 22 3 " " PUNCT ejpam-2159 22 4 denotes denote VERB ejpam-2159 22 5 the the DET ejpam-2159 22 6 binomial binomial ADJ ejpam-2159 22 7 coefficient coefficient NOUN ejpam-2159 22 8 and and CCONJ ejpam-2159 22 9 x x ADJ ejpam-2159 22 10 denotes denote VERB ejpam-2159 22 11 the the DET ejpam-2159 22 12 set set NOUN ejpam-2159 22 13 { { PUNCT ejpam-2159 22 14 x1 x1 PROPN ejpam-2159 22 15 , , PUNCT ejpam-2159 22 16 x2 x2 PROPN ejpam-2159 22 17 , , PUNCT ejpam-2159 22 18 . . PUNCT ejpam-2159 22 19 . . PUNCT ejpam-2159 23 1 . . PUNCT ejpam-2159 24 1 , , PUNCT ejpam-2159 24 2 xn xn PROPN ejpam-2159 24 3 } } PUNCT ejpam-2159 24 4 . . PUNCT ejpam-2159 24 5 ! ! PUNCT ejpam-2159 25 1 corresponding correspond VERB ejpam-2159 25 2 author author NOUN ejpam-2159 25 3 . . PUNCT ejpam-2159 26 1 email email NOUN ejpam-2159 26 2 addresses address NOUN ejpam-2159 26 3 : : PUNCT ejpam-2159 26 4 m#elmikkawy@yahoo.com m#elmikkawy@yahoo.com ADP ejpam-2159 26 5 , , PUNCT ejpam-2159 26 6 ( ( PUNCT ejpam-2159 26 7 m. m. NOUN ejpam-2159 26 8 el el PROPN ejpam-2159 26 9 - - PUNCT ejpam-2159 26 10 mikkawy mikkawy PROPN ejpam-2159 26 11 ) ) PUNCT ejpam-2159 26 12 , , PUNCT ejpam-2159 26 13 faizatlan11@yahoo.com faizatlan11@yahoo.com PROPN ejpam-2159 26 14 ( ( PUNCT ejpam-2159 26 15 f. f. PROPN ejpam-2159 26 16 atlan atlan PROPN ejpam-2159 26 17 ) ) PUNCT ejpam-2159 26 18 http://www.ejpam.com http://www.ejpam.com X ejpam-2159 27 1 135 135 NUM ejpam-2159 27 2 c$ c$ NOUN ejpam-2159 27 3 2015 2015 NUM ejpam-2159 27 4 ejpam ejpam VERB ejpam-2159 27 5 all all DET ejpam-2159 27 6 rights right NOUN ejpam-2159 27 7 reserved reserve VERB ejpam-2159 27 8 . . PUNCT ejpam-2159 28 1 m. m. PROPN ejpam-2159 28 2 el el PROPN ejpam-2159 28 3 - - PUNCT ejpam-2159 28 4 mikkawy mikkawy PROPN ejpam-2159 28 5 , , PUNCT ejpam-2159 28 6 f. f. PROPN ejpam-2159 28 7 atlan atlan PROPN ejpam-2159 28 8 / / SYM ejpam-2159 28 9 eur eur PROPN ejpam-2159 28 10 . . PUNCT ejpam-2159 29 1 j. j. PROPN ejpam-2159 29 2 pure pure PROPN ejpam-2159 29 3 appl appl PROPN ejpam-2159 29 4 . . PROPN ejpam-2159 29 5 math math PROPN ejpam-2159 29 6 , , PUNCT ejpam-2159 29 7 8 8 NUM ejpam-2159 29 8 ( ( PUNCT ejpam-2159 29 9 2015 2015 NUM ejpam-2159 29 10 ) ) PUNCT ejpam-2159 29 11 , , PUNCT ejpam-2159 29 12 135 135 NUM ejpam-2159 29 13 - - SYM ejpam-2159 29 14 151 151 NUM ejpam-2159 29 15 136 136 NUM ejpam-2159 29 16 definition definition NOUN ejpam-2159 29 17 1 1 NUM ejpam-2159 29 18 . . PUNCT ejpam-2159 30 1 [ [ X ejpam-2159 30 2 6 6 NUM ejpam-2159 30 3 ] ] PUNCT ejpam-2159 30 4 . . PUNCT ejpam-2159 31 1 for for ADP ejpam-2159 31 2 integer integer NOUN ejpam-2159 31 3 numbers number NOUN ejpam-2159 31 4 n n PRON ejpam-2159 31 5 and and CCONJ ejpam-2159 31 6 k k X ejpam-2159 31 7 with with ADP ejpam-2159 31 8 n% n% PROPN ejpam-2159 32 1 k k NOUN ejpam-2159 32 2 % % NOUN ejpam-2159 32 3 0 0 NUM ejpam-2159 32 4 , , PUNCT ejpam-2159 32 5 the the DET ejpam-2159 32 6 stirling stirling NOUN ejpam-2159 32 7 numbers number NOUN ejpam-2159 32 8 of of ADP ejpam-2159 32 9 the the DET ejpam-2159 32 10 first first ADJ ejpam-2159 32 11 kind kind NOUN ejpam-2159 32 12 , , PUNCT ejpam-2159 32 13 s(n s(n PROPN ejpam-2159 32 14 , , PUNCT ejpam-2159 32 15 k k NOUN ejpam-2159 32 16 ) ) PUNCT ejpam-2159 32 17 and and CCONJ ejpam-2159 32 18 of of ADP ejpam-2159 32 19 the the DET ejpam-2159 32 20 second second ADJ ejpam-2159 32 21 kind kind NOUN ejpam-2159 32 22 , , PUNCT ejpam-2159 32 23 s(n s(n PROPN ejpam-2159 32 24 , , PUNCT ejpam-2159 32 25 k k NOUN ejpam-2159 32 26 ) ) PUNCT ejpam-2159 32 27 are be AUX ejpam-2159 32 28 defined define VERB ejpam-2159 32 29 respectively respectively ADV ejpam-2159 32 30 by by ADP ejpam-2159 32 31 : : PUNCT ejpam-2159 32 32 ( ( PUNCT ejpam-2159 32 33 x)n x)n PUNCT ejpam-2159 32 34 = = PUNCT ejpam-2159 33 1 [ [ X ejpam-2159 33 2 x x X ejpam-2159 33 3 # # NOUN ejpam-2159 33 4 n+ n+ NOUN ejpam-2159 33 5 1]n 1]n NUM ejpam-2159 33 6 = = SYM ejpam-2159 33 7 n n PRON ejpam-2159 33 8 # # NOUN ejpam-2159 33 9 k=0 k=0 PROPN ejpam-2159 33 10 s(n s(n PROPN ejpam-2159 33 11 , , PUNCT ejpam-2159 33 12 k)xk k)xk PROPN ejpam-2159 33 13 , , PUNCT ejpam-2159 33 14 ( ( PUNCT ejpam-2159 33 15 1 1 NUM ejpam-2159 33 16 ) ) PUNCT ejpam-2159 33 17 and and CCONJ ejpam-2159 33 18 xn xn X ejpam-2159 33 19 = = SYM ejpam-2159 33 20 n n PRON ejpam-2159 33 21 # # NOUN ejpam-2159 33 22 k=0 k=0 PROPN ejpam-2159 33 23 s(n s(n PROPN ejpam-2159 33 24 , , PUNCT ejpam-2159 33 25 k)(x)k k)(x)k PRON ejpam-2159 33 26 , , PUNCT ejpam-2159 33 27 ( ( PUNCT ejpam-2159 33 28 2 2 X ejpam-2159 33 29 ) ) PUNCT ejpam-2159 33 30 where where SCONJ ejpam-2159 33 31 the the DET ejpam-2159 33 32 falling fall VERB ejpam-2159 33 33 factorial factorial NOUN ejpam-2159 33 34 of of ADP ejpam-2159 33 35 x x PRON ejpam-2159 33 36 , , PUNCT ejpam-2159 33 37 ( ( PUNCT ejpam-2159 33 38 x)n x)n PUNCT ejpam-2159 33 39 and and CCONJ ejpam-2159 33 40 the the DET ejpam-2159 33 41 rising rise VERB ejpam-2159 33 42 factorial factorial NOUN ejpam-2159 33 43 of of ADP ejpam-2159 33 44 x x PRON ejpam-2159 33 45 , , PUNCT ejpam-2159 33 46 [ [ X ejpam-2159 33 47 x]n x]n PRON ejpam-2159 33 48 are be AUX ejpam-2159 33 49 given give VERB ejpam-2159 33 50 respectively respectively ADV ejpam-2159 33 51 by by ADP ejpam-2159 33 52 : : PUNCT ejpam-2159 33 53 ( ( PUNCT ejpam-2159 33 54 x)n x)n PUNCT ejpam-2159 33 55 = = SYM ejpam-2159 33 56 $ $ SYM ejpam-2159 33 57 1 1 NUM ejpam-2159 33 58 if if SCONJ ejpam-2159 33 59 n= n= ADJ ejpam-2159 33 60 0 0 NUM ejpam-2159 33 61 , , PUNCT ejpam-2159 33 62 x(x x(x PROPN ejpam-2159 33 63 # # NOUN ejpam-2159 33 64 1)(x 1)(x NUM ejpam-2159 33 65 # # NOUN ejpam-2159 33 66 2 2 NUM ejpam-2159 33 67 ) ) PUNCT ejpam-2159 33 68 . . PUNCT ejpam-2159 33 69 . . PUNCT ejpam-2159 33 70 . . PUNCT ejpam-2159 34 1 ( ( PUNCT ejpam-2159 34 2 x x X ejpam-2159 34 3 # # NOUN ejpam-2159 34 4 n+ n+ ADP ejpam-2159 34 5 1 1 NUM ejpam-2159 34 6 ) ) PUNCT ejpam-2159 34 7 if if SCONJ ejpam-2159 34 8 n% n% ADJ ejpam-2159 34 9 1 1 NUM ejpam-2159 34 10 , , PUNCT ejpam-2159 34 11 ( ( PUNCT ejpam-2159 34 12 3 3 NUM ejpam-2159 34 13 ) ) PUNCT ejpam-2159 34 14 and and CCONJ ejpam-2159 34 15 [ [ X ejpam-2159 34 16 x]n x]n NOUN ejpam-2159 34 17 = = SYM ejpam-2159 34 18 $ $ SYM ejpam-2159 34 19 1 1 NUM ejpam-2159 34 20 if if SCONJ ejpam-2159 34 21 n= n= ADJ ejpam-2159 34 22 0 0 NUM ejpam-2159 34 23 , , PUNCT ejpam-2159 34 24 x(x x(x PROPN ejpam-2159 35 1 + + CCONJ ejpam-2159 35 2 1)(x 1)(x NUM ejpam-2159 35 3 + + CCONJ ejpam-2159 35 4 2 2 NUM ejpam-2159 35 5 ) ) PUNCT ejpam-2159 35 6 . . PUNCT ejpam-2159 36 1 . . PUNCT ejpam-2159 36 2 . . PUNCT ejpam-2159 37 1 ( ( PUNCT ejpam-2159 37 2 x x X ejpam-2159 37 3 + + NOUN ejpam-2159 37 4 n n CCONJ ejpam-2159 37 5 # # SYM ejpam-2159 37 6 1 1 NUM ejpam-2159 37 7 ) ) PUNCT ejpam-2159 37 8 if if SCONJ ejpam-2159 37 9 n% n% NOUN ejpam-2159 37 10 1 1 NUM ejpam-2159 37 11 . . PUNCT ejpam-2159 38 1 ( ( PUNCT ejpam-2159 38 2 4 4 X ejpam-2159 38 3 ) ) PUNCT ejpam-2159 38 4 it it PRON ejpam-2159 38 5 is be AUX ejpam-2159 38 6 well well ADV ejpam-2159 38 7 known know VERB ejpam-2159 38 8 that that SCONJ ejpam-2159 38 9 for for ADP ejpam-2159 38 10 integers integer NOUN ejpam-2159 38 11 n n CCONJ ejpam-2159 38 12 , , PUNCT ejpam-2159 38 13 k k PROPN ejpam-2159 38 14 % % NOUN ejpam-2159 38 15 0 0 NUM ejpam-2159 38 16 , , PUNCT ejpam-2159 38 17 the the DET ejpam-2159 38 18 s(n s(n PROPN ejpam-2159 38 19 , , PUNCT ejpam-2159 38 20 k k NOUN ejpam-2159 38 21 ) ) PUNCT ejpam-2159 38 22 and and CCONJ ejpam-2159 38 23 s(n s(n PROPN ejpam-2159 38 24 , , PUNCT ejpam-2159 38 25 k k NOUN ejpam-2159 38 26 ) ) PUNCT ejpam-2159 38 27 satisfy satisfy VERB ejpam-2159 38 28 the the DET ejpam-2159 38 29 following follow VERB ejpam-2159 38 30 pascal pascal ADJ ejpam-2159 38 31 - - PUNCT ejpam-2159 38 32 type type NOUN ejpam-2159 38 33 recurrence recurrence NOUN ejpam-2159 38 34 relations relation NOUN ejpam-2159 38 35 : : PUNCT ejpam-2159 38 36 s(n s(n PROPN ejpam-2159 38 37 , , PUNCT ejpam-2159 38 38 k k NOUN ejpam-2159 38 39 ) ) PUNCT ejpam-2159 38 40 = = PUNCT ejpam-2159 39 1 s(n s(n NOUN ejpam-2159 39 2 # # NOUN ejpam-2159 39 3 1 1 NUM ejpam-2159 39 4 , , PUNCT ejpam-2159 39 5 k k NOUN ejpam-2159 39 6 # # NOUN ejpam-2159 39 7 1 1 NUM ejpam-2159 39 8 ) ) PUNCT ejpam-2159 39 9 # # NOUN ejpam-2159 39 10 ( ( PUNCT ejpam-2159 39 11 n n CCONJ ejpam-2159 39 12 # # NOUN ejpam-2159 39 13 1)s(n 1)s(n NUM ejpam-2159 39 14 # # NOUN ejpam-2159 39 15 1 1 NUM ejpam-2159 39 16 , , PUNCT ejpam-2159 39 17 k k NOUN ejpam-2159 39 18 ) ) PUNCT ejpam-2159 39 19 , , PUNCT ejpam-2159 39 20 ( ( PUNCT ejpam-2159 39 21 5 5 NUM ejpam-2159 39 22 ) ) PUNCT ejpam-2159 39 23 and and CCONJ ejpam-2159 39 24 s(n s(n PROPN ejpam-2159 39 25 , , PUNCT ejpam-2159 39 26 k k NOUN ejpam-2159 39 27 ) ) PUNCT ejpam-2159 39 28 = = PUNCT ejpam-2159 40 1 s(n s(n NOUN ejpam-2159 40 2 # # NOUN ejpam-2159 40 3 1 1 NUM ejpam-2159 40 4 , , PUNCT ejpam-2159 40 5 k k NOUN ejpam-2159 40 6 # # NOUN ejpam-2159 40 7 1 1 NUM ejpam-2159 40 8 ) ) PUNCT ejpam-2159 40 9 + + NOUN ejpam-2159 40 10 ks(n ks(n NOUN ejpam-2159 40 11 # # NOUN ejpam-2159 40 12 1 1 NUM ejpam-2159 40 13 , , PUNCT ejpam-2159 40 14 k k NOUN ejpam-2159 40 15 ) ) PUNCT ejpam-2159 40 16 , , PUNCT ejpam-2159 40 17 ( ( PUNCT ejpam-2159 40 18 6 6 X ejpam-2159 40 19 ) ) PUNCT ejpam-2159 40 20 subject subject NOUN ejpam-2159 40 21 to to ADP ejpam-2159 40 22 : : PUNCT ejpam-2159 40 23 s(k s(k ADV ejpam-2159 40 24 , , PUNCT ejpam-2159 40 25 k k NOUN ejpam-2159 40 26 ) ) PUNCT ejpam-2159 40 27 = = SYM ejpam-2159 40 28 s(k s(k PROPN ejpam-2159 40 29 , , PUNCT ejpam-2159 40 30 k k NOUN ejpam-2159 40 31 ) ) PUNCT ejpam-2159 40 32 = = SYM ejpam-2159 40 33 1,1 1,1 NUM ejpam-2159 40 34 & & CCONJ ejpam-2159 40 35 k k PROPN ejpam-2159 40 36 & & CCONJ ejpam-2159 40 37 n n PROPN ejpam-2159 40 38 ( ( PUNCT ejpam-2159 40 39 7 7 NUM ejpam-2159 40 40 ) ) PUNCT ejpam-2159 40 41 and and CCONJ ejpam-2159 40 42 s(n s(n PROPN ejpam-2159 40 43 , , PUNCT ejpam-2159 40 44 k k NOUN ejpam-2159 40 45 ) ) PUNCT ejpam-2159 41 1 = = PUNCT ejpam-2159 41 2 s(n s(n PROPN ejpam-2159 41 3 , , PUNCT ejpam-2159 41 4 k k NOUN ejpam-2159 41 5 ) ) PUNCT ejpam-2159 41 6 = = PUNCT ejpam-2159 41 7 ! ! PUNCT ejpam-2159 42 1 nk nk PROPN ejpam-2159 42 2 , , PUNCT ejpam-2159 42 3 if if SCONJ ejpam-2159 42 4 k k PROPN ejpam-2159 42 5 = = SYM ejpam-2159 42 6 0 0 NUM ejpam-2159 42 7 or or CCONJ ejpam-2159 42 8 n= n= ADJ ejpam-2159 42 9 0 0 NUM ejpam-2159 42 10 . . PUNCT ejpam-2159 43 1 ( ( PUNCT ejpam-2159 43 2 8) 8) NUM ejpam-2159 43 3 replace replace NOUN ejpam-2159 43 4 x x PUNCT ejpam-2159 43 5 by by ADP ejpam-2159 43 6 # # SYM ejpam-2159 43 7 x x PRON ejpam-2159 43 8 in in ADP ejpam-2159 43 9 ( ( PUNCT ejpam-2159 43 10 1 1 X ejpam-2159 43 11 ) ) PUNCT ejpam-2159 43 12 gives give VERB ejpam-2159 43 13 [ [ PUNCT ejpam-2159 43 14 x]n x]n NOUN ejpam-2159 43 15 = = SYM ejpam-2159 43 16 n n NUM ejpam-2159 43 17 # # NOUN ejpam-2159 43 18 k=0 k=0 PROPN ejpam-2159 43 19 c(n c(n PROPN ejpam-2159 43 20 , , PUNCT ejpam-2159 43 21 k)xk k)xk PROPN ejpam-2159 43 22 , , PUNCT ejpam-2159 43 23 ( ( PUNCT ejpam-2159 43 24 9 9 NUM ejpam-2159 43 25 ) ) PUNCT ejpam-2159 43 26 where where SCONJ ejpam-2159 43 27 c(n c(n PROPN ejpam-2159 43 28 , , PUNCT ejpam-2159 43 29 k k NOUN ejpam-2159 43 30 ) ) PUNCT ejpam-2159 43 31 = = SYM ejpam-2159 44 1 ( ( PUNCT ejpam-2159 44 2 # # SYM ejpam-2159 44 3 1)n#ks(n 1)n#ks(n NUM ejpam-2159 44 4 , , PUNCT ejpam-2159 44 5 k k NOUN ejpam-2159 44 6 ) ) PUNCT ejpam-2159 44 7 is be AUX ejpam-2159 44 8 called call VERB ejpam-2159 44 9 the the DET ejpam-2159 44 10 unsigned unsigned ADJ ejpam-2159 44 11 or or CCONJ ejpam-2159 44 12 signless signless ADJ ejpam-2159 44 13 stirling stirling NOUN ejpam-2159 44 14 number number NOUN ejpam-2159 44 15 of of ADP ejpam-2159 44 16 the the DET ejpam-2159 44 17 first first ADJ ejpam-2159 44 18 kind kind NOUN ejpam-2159 44 19 . . PUNCT ejpam-2159 45 1 it it PRON ejpam-2159 45 2 can can AUX ejpam-2159 45 3 be be AUX ejpam-2159 45 4 shown show VERB ejpam-2159 45 5 that that SCONJ ejpam-2159 45 6 the the DET ejpam-2159 45 7 rising rise VERB ejpam-2159 45 8 factorial factorial NOUN ejpam-2159 45 9 , , PUNCT ejpam-2159 45 10 [ [ X ejpam-2159 45 11 x]n x]n PROPN ejpam-2159 45 12 and and CCONJ ejpam-2159 45 13 the the DET ejpam-2159 45 14 falling fall VERB ejpam-2159 45 15 factorial factorial NOUN ejpam-2159 45 16 , , PUNCT ejpam-2159 45 17 ( ( PUNCT ejpam-2159 45 18 x)n x)n X ejpam-2159 45 19 are be AUX ejpam-2159 45 20 also also ADV ejpam-2159 45 21 related relate VERB ejpam-2159 45 22 by by ADP ejpam-2159 45 23 : : PUNCT ejpam-2159 46 1 [ [ X ejpam-2159 46 2 x]n x]n NOUN ejpam-2159 46 3 = = SYM ejpam-2159 46 4 n n PRON ejpam-2159 46 5 # # NOUN ejpam-2159 46 6 k=0 k=0 PROPN ejpam-2159 46 7 h(n h(n PROPN ejpam-2159 46 8 , , PUNCT ejpam-2159 46 9 k)(x)k k)(x)k PRON ejpam-2159 46 10 , , PUNCT ejpam-2159 46 11 ( ( PUNCT ejpam-2159 46 12 10 10 NUM ejpam-2159 46 13 ) ) PUNCT ejpam-2159 47 1 where where SCONJ ejpam-2159 47 2 h(n h(n PROPN ejpam-2159 47 3 , , PUNCT ejpam-2159 47 4 k k NOUN ejpam-2159 47 5 ) ) PUNCT ejpam-2159 47 6 = = SYM ejpam-2159 47 7 n n X ejpam-2159 47 8 ! ! PUNCT ejpam-2159 47 9 k k X ejpam-2159 47 10 ! ! PUNCT ejpam-2159 47 11 ! ! PUNCT ejpam-2159 48 1 n n CCONJ ejpam-2159 48 2 # # SYM ejpam-2159 48 3 1 1 NUM ejpam-2159 48 4 k k NOUN ejpam-2159 48 5 # # NOUN ejpam-2159 48 6 1 1 NUM ejpam-2159 48 7 " " PUNCT ejpam-2159 48 8 . . PUNCT ejpam-2159 49 1 ( ( PUNCT ejpam-2159 49 2 11 11 NUM ejpam-2159 49 3 ) ) PUNCT ejpam-2159 49 4 m. m. NOUN ejpam-2159 49 5 el el PROPN ejpam-2159 49 6 - - PUNCT ejpam-2159 49 7 mikkawy mikkawy PROPN ejpam-2159 49 8 , , PUNCT ejpam-2159 49 9 f. f. PROPN ejpam-2159 49 10 atlan atlan PROPN ejpam-2159 49 11 / / SYM ejpam-2159 49 12 eur eur PROPN ejpam-2159 49 13 . . PUNCT ejpam-2159 50 1 j. j. PROPN ejpam-2159 50 2 pure pure PROPN ejpam-2159 50 3 appl appl PROPN ejpam-2159 50 4 . . PROPN ejpam-2159 50 5 math math PROPN ejpam-2159 50 6 , , PUNCT ejpam-2159 50 7 8 8 NUM ejpam-2159 50 8 ( ( PUNCT ejpam-2159 50 9 2015 2015 NUM ejpam-2159 50 10 ) ) PUNCT ejpam-2159 50 11 , , PUNCT ejpam-2159 50 12 135 135 NUM ejpam-2159 50 13 - - SYM ejpam-2159 50 14 151 151 NUM ejpam-2159 50 15 137 137 NUM ejpam-2159 50 16 the the DET ejpam-2159 50 17 coefficients coefficient NOUN ejpam-2159 50 18 , , PUNCT ejpam-2159 50 19 h(n h(n PROPN ejpam-2159 50 20 , , PUNCT ejpam-2159 50 21 k k NOUN ejpam-2159 50 22 ) ) PUNCT ejpam-2159 50 23 in in ADP ejpam-2159 50 24 ( ( PUNCT ejpam-2159 50 25 10 10 NUM ejpam-2159 50 26 ) ) PUNCT ejpam-2159 50 27 are be AUX ejpam-2159 50 28 called call VERB ejpam-2159 50 29 the the DET ejpam-2159 50 30 lah lah PROPN ejpam-2159 50 31 numbers number NOUN ejpam-2159 50 32 . . PUNCT ejpam-2159 51 1 these these DET ejpam-2159 51 2 numbers number NOUN ejpam-2159 51 3 satisfy satisfy VERB ejpam-2159 51 4 the the DET ejpam-2159 51 5 following follow VERB ejpam-2159 51 6 pascal pascal ADJ ejpam-2159 51 7 - - PUNCT ejpam-2159 51 8 type type NOUN ejpam-2159 51 9 recurrence recurrence NOUN ejpam-2159 51 10 relation relation NOUN ejpam-2159 51 11 : : PUNCT ejpam-2159 52 1 h(n h(n PROPN ejpam-2159 52 2 , , PUNCT ejpam-2159 52 3 k k NOUN ejpam-2159 52 4 ) ) PUNCT ejpam-2159 52 5 = = PUNCT ejpam-2159 53 1 h(n h(n NOUN ejpam-2159 53 2 # # NOUN ejpam-2159 53 3 1 1 NUM ejpam-2159 53 4 , , PUNCT ejpam-2159 53 5 k k NOUN ejpam-2159 53 6 # # NOUN ejpam-2159 53 7 1 1 NUM ejpam-2159 53 8 ) ) PUNCT ejpam-2159 53 9 + + CCONJ ejpam-2159 53 10 ( ( PUNCT ejpam-2159 53 11 n+ n+ X ejpam-2159 54 1 k k X ejpam-2159 54 2 # # NOUN ejpam-2159 54 3 1)h(n 1)h(n NUM ejpam-2159 54 4 # # NOUN ejpam-2159 54 5 1 1 NUM ejpam-2159 54 6 , , PUNCT ejpam-2159 54 7 k k NOUN ejpam-2159 54 8 ) ) PUNCT ejpam-2159 54 9 . . PUNCT ejpam-2159 55 1 ( ( PUNCT ejpam-2159 55 2 12 12 NUM ejpam-2159 55 3 ) ) PUNCT ejpam-2159 55 4 the the DET ejpam-2159 55 5 lah lah PROPN ejpam-2159 55 6 numbers number NOUN ejpam-2159 55 7 also also ADV ejpam-2159 55 8 satisfy satisfy VERB ejpam-2159 55 9 : : PUNCT ejpam-2159 56 1 h(i h(i PROPN ejpam-2159 56 2 , , PUNCT ejpam-2159 56 3 j j NOUN ejpam-2159 56 4 ) ) PUNCT ejpam-2159 56 5 = = SYM ejpam-2159 57 1 n n PRON ejpam-2159 57 2 # # NOUN ejpam-2159 57 3 k=1 k=1 NOUN ejpam-2159 57 4 ( ( PUNCT ejpam-2159 57 5 # # SYM ejpam-2159 57 6 1)i+ks(i 1)i+ks(i NUM ejpam-2159 57 7 , , PUNCT ejpam-2159 57 8 k)s(k k)s(k PROPN ejpam-2159 57 9 , , PUNCT ejpam-2159 57 10 j j PROPN ejpam-2159 57 11 ) ) PUNCT ejpam-2159 57 12 , , PUNCT ejpam-2159 57 13 1 1 NUM ejpam-2159 57 14 & & CCONJ ejpam-2159 57 15 i i PRON ejpam-2159 57 16 , , PUNCT ejpam-2159 57 17 j j PROPN ejpam-2159 57 18 & & CCONJ ejpam-2159 57 19 n. n. PROPN ejpam-2159 57 20 ( ( PUNCT ejpam-2159 57 21 13 13 NUM ejpam-2159 57 22 ) ) PUNCT ejpam-2159 57 23 definition definition NOUN ejpam-2159 57 24 2 2 NUM ejpam-2159 57 25 ( ( PUNCT ejpam-2159 57 26 [ [ X ejpam-2159 57 27 6 6 NUM ejpam-2159 57 28 ] ] NUM ejpam-2159 57 29 ) ) PUNCT ejpam-2159 57 30 . . PUNCT ejpam-2159 58 1 the the DET ejpam-2159 58 2 stirling stirling NOUN ejpam-2159 58 3 matrix matrix NOUN ejpam-2159 58 4 of of ADP ejpam-2159 58 5 the the DET ejpam-2159 58 6 first first ADJ ejpam-2159 58 7 kind kind NOUN ejpam-2159 58 8 , , PUNCT ejpam-2159 58 9 sn sn PROPN ejpam-2159 58 10 and and CCONJ ejpam-2159 58 11 of of ADP ejpam-2159 58 12 the the DET ejpam-2159 58 13 second second ADJ ejpam-2159 58 14 kind kind NOUN ejpam-2159 58 15 , , PUNCT ejpam-2159 58 16 sn sn PROPN ejpam-2159 58 17 are be AUX ejpam-2159 58 18 defined define VERB ejpam-2159 58 19 respectively respectively ADV ejpam-2159 58 20 by by ADP ejpam-2159 58 21 : : PUNCT ejpam-2159 58 22 sn sn PROPN ejpam-2159 58 23 = = SYM ejpam-2159 58 24 $ $ SYM ejpam-2159 58 25 s(i s(i PROPN ejpam-2159 58 26 , , PUNCT ejpam-2159 58 27 j j PROPN ejpam-2159 58 28 ) ) PUNCT ejpam-2159 58 29 for for ADP ejpam-2159 58 30 i i PRON ejpam-2159 58 31 % % PROPN ejpam-2159 58 32 j j PROPN ejpam-2159 58 33 , , PUNCT ejpam-2159 58 34 0 0 NUM ejpam-2159 59 1 otherwise otherwise ADV ejpam-2159 59 2 , , PUNCT ejpam-2159 59 3 ( ( PUNCT ejpam-2159 59 4 14 14 NUM ejpam-2159 59 5 ) ) PUNCT ejpam-2159 59 6 and and CCONJ ejpam-2159 59 7 sn sn NOUN ejpam-2159 59 8 = = SYM ejpam-2159 59 9 $ $ SYM ejpam-2159 59 10 s(i s(i PROPN ejpam-2159 59 11 , , PUNCT ejpam-2159 59 12 j j PROPN ejpam-2159 59 13 ) ) PUNCT ejpam-2159 59 14 for for ADP ejpam-2159 59 15 i i PRON ejpam-2159 59 16 % % PROPN ejpam-2159 59 17 j j PROPN ejpam-2159 59 18 , , PUNCT ejpam-2159 59 19 0 0 PUNCT ejpam-2159 60 1 otherwise otherwise ADV ejpam-2159 60 2 . . PUNCT ejpam-2159 61 1 ( ( PUNCT ejpam-2159 61 2 15 15 NUM ejpam-2159 61 3 ) ) PUNCT ejpam-2159 61 4 for for ADP ejpam-2159 61 5 example example NOUN ejpam-2159 61 6 s5 s5 PROPN ejpam-2159 61 7 = = SYM ejpam-2159 61 8 % % PROPN ejpam-2159 61 9 & & CCONJ ejpam-2159 61 10 & & CCONJ ejpam-2159 61 11 & & CCONJ ejpam-2159 61 12 & & CCONJ ejpam-2159 61 13 ' ' PUNCT ejpam-2159 61 14 1 1 NUM ejpam-2159 61 15 0 0 NUM ejpam-2159 61 16 0 0 NUM ejpam-2159 61 17 0 0 NUM ejpam-2159 61 18 0 0 NUM ejpam-2159 62 1 # # SYM ejpam-2159 62 2 1 1 NUM ejpam-2159 62 3 1 1 NUM ejpam-2159 62 4 0 0 NUM ejpam-2159 62 5 0 0 NUM ejpam-2159 62 6 0 0 NUM ejpam-2159 62 7 2 2 NUM ejpam-2159 62 8 # # SYM ejpam-2159 62 9 3 3 NUM ejpam-2159 62 10 1 1 NUM ejpam-2159 62 11 0 0 NUM ejpam-2159 62 12 0 0 NUM ejpam-2159 62 13 # # SYM ejpam-2159 62 14 6 6 NUM ejpam-2159 62 15 11 11 NUM ejpam-2159 63 1 # # SYM ejpam-2159 63 2 6 6 NUM ejpam-2159 63 3 1 1 NUM ejpam-2159 63 4 0 0 NUM ejpam-2159 63 5 24 24 NUM ejpam-2159 63 6 # # SYM ejpam-2159 63 7 50 50 NUM ejpam-2159 63 8 35 35 NUM ejpam-2159 63 9 # # SYM ejpam-2159 63 10 10 10 NUM ejpam-2159 63 11 1 1 NUM ejpam-2159 63 12 ( ( PUNCT ejpam-2159 63 13 ) ) PUNCT ejpam-2159 63 14 ) ) PUNCT ejpam-2159 63 15 ) ) PUNCT ejpam-2159 63 16 ) ) PUNCT ejpam-2159 64 1 * * PUNCT ejpam-2159 64 2 and and CCONJ ejpam-2159 64 3 s5 s5 PROPN ejpam-2159 64 4 = = PUNCT ejpam-2159 64 5 % % PROPN ejpam-2159 64 6 & & CCONJ ejpam-2159 64 7 & & CCONJ ejpam-2159 64 8 & & CCONJ ejpam-2159 64 9 & & CCONJ ejpam-2159 64 10 ' ' PUNCT ejpam-2159 64 11 1 1 NUM ejpam-2159 64 12 0 0 NUM ejpam-2159 64 13 0 0 NUM ejpam-2159 64 14 0 0 NUM ejpam-2159 64 15 0 0 NUM ejpam-2159 64 16 1 1 NUM ejpam-2159 64 17 1 1 NUM ejpam-2159 64 18 0 0 NUM ejpam-2159 64 19 0 0 NUM ejpam-2159 64 20 0 0 NUM ejpam-2159 64 21 1 1 NUM ejpam-2159 64 22 3 3 NUM ejpam-2159 64 23 1 1 NUM ejpam-2159 64 24 0 0 NUM ejpam-2159 64 25 0 0 NUM ejpam-2159 64 26 1 1 NUM ejpam-2159 64 27 7 7 NUM ejpam-2159 64 28 6 6 NUM ejpam-2159 64 29 1 1 NUM ejpam-2159 64 30 0 0 NUM ejpam-2159 64 31 1 1 NUM ejpam-2159 64 32 15 15 NUM ejpam-2159 64 33 25 25 NUM ejpam-2159 64 34 10 10 NUM ejpam-2159 64 35 1 1 NUM ejpam-2159 64 36 ( ( PUNCT ejpam-2159 64 37 ) ) PUNCT ejpam-2159 64 38 ) ) PUNCT ejpam-2159 64 39 ) ) PUNCT ejpam-2159 64 40 ) ) PUNCT ejpam-2159 65 1 * * PUNCT ejpam-2159 65 2 . . PUNCT ejpam-2159 66 1 it it PRON ejpam-2159 66 2 is be AUX ejpam-2159 66 3 known know VERB ejpam-2159 66 4 that that SCONJ ejpam-2159 66 5 snsn snsn NOUN ejpam-2159 66 6 = = SYM ejpam-2159 66 7 snsn snsn NOUN ejpam-2159 66 8 = = PUNCT ejpam-2159 66 9 in in ADP ejpam-2159 66 10 , , PUNCT ejpam-2159 66 11 where where SCONJ ejpam-2159 66 12 in in ADP ejpam-2159 66 13 is be AUX ejpam-2159 66 14 the the DET ejpam-2159 66 15 n n CCONJ ejpam-2159 66 16 " " PUNCT ejpam-2159 66 17 n n PRON ejpam-2159 66 18 identity identity NOUN ejpam-2159 66 19 matrix matrix NOUN ejpam-2159 66 20 . . PUNCT ejpam-2159 67 1 in in ADP ejpam-2159 67 2 other other ADJ ejpam-2159 67 3 words word NOUN ejpam-2159 67 4 , , PUNCT ejpam-2159 67 5 the the DET ejpam-2159 67 6 matrices matrix NOUN ejpam-2159 67 7 , , PUNCT ejpam-2159 67 8 sn sn PROPN ejpam-2159 67 9 and and CCONJ ejpam-2159 67 10 sn sn PROPN ejpam-2159 67 11 are be AUX ejpam-2159 67 12 inverse inverse ADJ ejpam-2159 67 13 to to ADP ejpam-2159 67 14 each each DET ejpam-2159 67 15 other other ADJ ejpam-2159 67 16 . . PUNCT ejpam-2159 68 1 if if SCONJ ejpam-2159 68 2 fi(x fi(x NUM ejpam-2159 68 3 ) ) PUNCT ejpam-2159 69 1 = = PUNCT ejpam-2159 70 1 x x PUNCT ejpam-2159 70 2 i i PRON ejpam-2159 70 3 , , PUNCT ejpam-2159 70 4 1 1 NUM ejpam-2159 70 5 & & CCONJ ejpam-2159 70 6 i i PROPN ejpam-2159 70 7 & & CCONJ ejpam-2159 70 8 n n CCONJ ejpam-2159 70 9 , , PUNCT ejpam-2159 70 10 then then ADV ejpam-2159 70 11 s(i s(i PROPN ejpam-2159 70 12 , , PUNCT ejpam-2159 70 13 k k NOUN ejpam-2159 70 14 ) ) PUNCT ejpam-2159 70 15 = = SYM ejpam-2159 70 16 fi[0,1,2 fi[0,1,2 ADJ ejpam-2159 70 17 , , PUNCT ejpam-2159 70 18 . . PUNCT ejpam-2159 70 19 . . PUNCT ejpam-2159 70 20 . . PUNCT ejpam-2159 71 1 , , PUNCT ejpam-2159 71 2 k k X ejpam-2159 71 3 ] ] X ejpam-2159 71 4 = = PUNCT ejpam-2159 71 5 ! ! PUNCT ejpam-2159 72 1 k k PROPN ejpam-2159 72 2 fi(0 fi(0 PROPN ejpam-2159 72 3 ) ) PUNCT ejpam-2159 73 1 k k NOUN ejpam-2159 73 2 ! ! PROPN ejpam-2159 73 3 , , PUNCT ejpam-2159 73 4 where where SCONJ ejpam-2159 73 5 fi[0,1,2 fi[0,1,2 ADJ ejpam-2159 73 6 , , PUNCT ejpam-2159 73 7 . . PUNCT ejpam-2159 73 8 . . PUNCT ejpam-2159 74 1 . . PUNCT ejpam-2159 75 1 , , PUNCT ejpam-2159 75 2 k k X ejpam-2159 75 3 ] ] X ejpam-2159 75 4 is be AUX ejpam-2159 75 5 the the DET ejpam-2159 75 6 k k NOUN ejpam-2159 75 7 - - PUNCT ejpam-2159 75 8 th th X ejpam-2159 75 9 divided divide VERB ejpam-2159 75 10 difference difference NOUN ejpam-2159 75 11 of of ADP ejpam-2159 75 12 the the DET ejpam-2159 75 13 function function NOUN ejpam-2159 75 14 fi(x fi(x NUM ejpam-2159 75 15 ) ) PUNCT ejpam-2159 76 1 = = PUNCT ejpam-2159 77 1 x x PUNCT ejpam-2159 77 2 i i PRON ejpam-2159 77 3 which which PRON ejpam-2159 77 4 lies lie VERB ejpam-2159 77 5 on on ADP ejpam-2159 77 6 the the DET ejpam-2159 77 7 top top NOUN ejpam-2159 77 8 of of ADP ejpam-2159 77 9 each each DET ejpam-2159 77 10 column column NOUN ejpam-2159 77 11 and and CCONJ ejpam-2159 77 12 ! ! PUNCT ejpam-2159 77 13 is be AUX ejpam-2159 77 14 the the DET ejpam-2159 77 15 forward forward ADJ ejpam-2159 77 16 difference difference NOUN ejpam-2159 77 17 operator operator NOUN ejpam-2159 77 18 . . PUNCT ejpam-2159 78 1 for for ADP ejpam-2159 78 2 example example NOUN ejpam-2159 78 3 , , PUNCT ejpam-2159 78 4 for for ADP ejpam-2159 78 5 f5(x f5(x NOUN ejpam-2159 78 6 ) ) PUNCT ejpam-2159 78 7 = = SYM ejpam-2159 78 8 x5 x5 PROPN ejpam-2159 78 9 , , PUNCT ejpam-2159 78 10 we we PRON ejpam-2159 78 11 have have VERB ejpam-2159 78 12 : : PUNCT ejpam-2159 78 13 table table VERB ejpam-2159 78 14 1 1 NUM ejpam-2159 78 15 : : PUNCT ejpam-2159 78 16 divided divide VERB ejpam-2159 78 17 di!erences di!erences PROPN ejpam-2159 78 18 and and CCONJ ejpam-2159 78 19 stirling stirling NOUN ejpam-2159 78 20 numbers number NOUN ejpam-2159 78 21 of of ADP ejpam-2159 78 22 the the DET ejpam-2159 78 23 second second ADJ ejpam-2159 78 24 kind kind NOUN ejpam-2159 78 25 x x NOUN ejpam-2159 78 26 f5(x f5(x NOUN ejpam-2159 78 27 ) ) PUNCT ejpam-2159 78 28 first first ADV ejpam-2159 78 29 dd dd VERB ejpam-2159 78 30 second second ADJ ejpam-2159 78 31 dd dd NOUN ejpam-2159 78 32 third third ADJ ejpam-2159 78 33 dd dd NOUN ejpam-2159 78 34 fourth fourth ADJ ejpam-2159 78 35 dd dd INTJ ejpam-2159 78 36 fifth fifth ADJ ejpam-2159 78 37 dd dd NOUN ejpam-2159 78 38 0 0 NUM ejpam-2159 78 39 0 0 NUM ejpam-2159 78 40 1 1 NUM ejpam-2159 78 41 = = SYM ejpam-2159 78 42 s(5,1 s(5,1 NOUN ejpam-2159 78 43 ) ) PUNCT ejpam-2159 78 44 15 15 NUM ejpam-2159 78 45 = = SYM ejpam-2159 78 46 s(5,2 s(5,2 NOUN ejpam-2159 78 47 ) ) PUNCT ejpam-2159 78 48 25 25 NUM ejpam-2159 78 49 = = SYM ejpam-2159 78 50 s(5,3 s(5,3 PROPN ejpam-2159 78 51 ) ) PUNCT ejpam-2159 78 52 10 10 NUM ejpam-2159 78 53 = = SYM ejpam-2159 78 54 s(5,4 s(5,4 NOUN ejpam-2159 78 55 ) ) PUNCT ejpam-2159 78 56 1 1 NUM ejpam-2159 78 57 = = SYM ejpam-2159 78 58 s(5,5 s(5,5 NOUN ejpam-2159 78 59 ) ) PUNCT ejpam-2159 78 60 1 1 NUM ejpam-2159 78 61 1 1 NUM ejpam-2159 78 62 31 31 NUM ejpam-2159 78 63 90 90 NUM ejpam-2159 78 64 65 65 NUM ejpam-2159 78 65 15 15 NUM ejpam-2159 78 66 2 2 NUM ejpam-2159 78 67 32 32 NUM ejpam-2159 78 68 211 211 NUM ejpam-2159 78 69 285 285 NUM ejpam-2159 78 70 125 125 NUM ejpam-2159 78 71 3 3 NUM ejpam-2159 78 72 243 243 NUM ejpam-2159 78 73 781 781 NUM ejpam-2159 78 74 660 660 NUM ejpam-2159 78 75 4 4 NUM ejpam-2159 78 76 1024 1024 NUM ejpam-2159 78 77 2101 2101 NUM ejpam-2159 78 78 5 5 NUM ejpam-2159 78 79 3125 3125 NUM ejpam-2159 78 80 thus thus ADV ejpam-2159 78 81 , , PUNCT ejpam-2159 78 82 f5(x f5(x NOUN ejpam-2159 78 83 ) ) PUNCT ejpam-2159 78 84 = = SYM ejpam-2159 78 85 x5 x5 NOUN ejpam-2159 78 86 = = SYM ejpam-2159 78 87 ( ( PUNCT ejpam-2159 78 88 x)1 x)1 PROPN ejpam-2159 78 89 + + CCONJ ejpam-2159 78 90 15(x)2 15(x)2 VERB ejpam-2159 78 91 + + CCONJ ejpam-2159 78 92 25(x)3 25(x)3 NOUN ejpam-2159 78 93 + + CCONJ ejpam-2159 78 94 10(x)4 10(x)4 NOUN ejpam-2159 78 95 + + CCONJ ejpam-2159 78 96 ( ( PUNCT ejpam-2159 78 97 x)5 x)5 PROPN ejpam-2159 78 98 m. m. PROPN ejpam-2159 78 99 el el PROPN ejpam-2159 78 100 - - PUNCT ejpam-2159 78 101 mikkawy mikkawy PROPN ejpam-2159 78 102 , , PUNCT ejpam-2159 78 103 f. f. PROPN ejpam-2159 78 104 atlan atlan PROPN ejpam-2159 78 105 / / SYM ejpam-2159 78 106 eur eur PROPN ejpam-2159 78 107 . . PUNCT ejpam-2159 79 1 j. j. PROPN ejpam-2159 79 2 pure pure PROPN ejpam-2159 79 3 appl appl PROPN ejpam-2159 79 4 . . PROPN ejpam-2159 79 5 math math PROPN ejpam-2159 79 6 , , PUNCT ejpam-2159 79 7 8 8 NUM ejpam-2159 79 8 ( ( PUNCT ejpam-2159 79 9 2015 2015 NUM ejpam-2159 79 10 ) ) PUNCT ejpam-2159 79 11 , , PUNCT ejpam-2159 79 12 135 135 NUM ejpam-2159 79 13 - - SYM ejpam-2159 79 14 151 151 NUM ejpam-2159 79 15 138 138 NUM ejpam-2159 79 16 and and CCONJ ejpam-2159 79 17 ! ! PUNCT ejpam-2159 80 1 f5(0 f5(0 PROPN ejpam-2159 80 2 ) ) PUNCT ejpam-2159 81 1 = = SYM ejpam-2159 81 2 1,!2 1,!2 NUM ejpam-2159 81 3 f5(0 f5(0 NOUN ejpam-2159 81 4 ) ) PUNCT ejpam-2159 81 5 = = SYM ejpam-2159 82 1 30,!3 30,!3 NUM ejpam-2159 82 2 f5(0 f5(0 NOUN ejpam-2159 82 3 ) ) PUNCT ejpam-2159 83 1 = = SYM ejpam-2159 84 1 150,!4 150,!4 NUM ejpam-2159 84 2 f5(0 f5(0 PROPN ejpam-2159 84 3 ) ) PUNCT ejpam-2159 84 4 = = SYM ejpam-2159 84 5 240 240 NUM ejpam-2159 84 6 , , PUNCT ejpam-2159 84 7 and and CCONJ ejpam-2159 84 8 ! ! PUNCT ejpam-2159 84 9 5 5 NUM ejpam-2159 84 10 f5(0 f5(0 NOUN ejpam-2159 84 11 ) ) PUNCT ejpam-2159 84 12 = = SYM ejpam-2159 85 1 120 120 NUM ejpam-2159 85 2 . . PUNCT ejpam-2159 85 3 definition definition NOUN ejpam-2159 85 4 3 3 NUM ejpam-2159 85 5 ( ( PUNCT ejpam-2159 85 6 [ [ X ejpam-2159 85 7 16 16 NUM ejpam-2159 85 8 ] ] PUNCT ejpam-2159 85 9 ) ) PUNCT ejpam-2159 85 10 . . PUNCT ejpam-2159 86 1 the the DET ejpam-2159 86 2 elementary elementary ADJ ejpam-2159 86 3 symmetric symmetric ADJ ejpam-2159 86 4 polynomial polynomial NOUN ejpam-2159 86 5 " " PUNCT ejpam-2159 86 6 ( ( PUNCT ejpam-2159 86 7 n)r n)r X ejpam-2159 86 8 and and CCONJ ejpam-2159 86 9 the the DET ejpam-2159 86 10 complete complete ADJ ejpam-2159 86 11 symmetric symmetric ADJ ejpam-2159 86 12 polynomial polynomial ADJ ejpam-2159 86 13 # # SYM ejpam-2159 86 14 ( ( PUNCT ejpam-2159 86 15 n)r n)r X ejpam-2159 86 16 in in ADP ejpam-2159 86 17 x1 x1 PROPN ejpam-2159 86 18 , , PUNCT ejpam-2159 86 19 x2 x2 PROPN ejpam-2159 86 20 , , PUNCT ejpam-2159 86 21 . . PUNCT ejpam-2159 86 22 . . PUNCT ejpam-2159 86 23 . . PUNCT ejpam-2159 87 1 , , PUNCT ejpam-2159 87 2 xn xn PROPN ejpam-2159 87 3 are be AUX ejpam-2159 87 4 defined define VERB ejpam-2159 87 5 respectively respectively ADV ejpam-2159 87 6 by by ADP ejpam-2159 87 7 : : PUNCT ejpam-2159 87 8 " " PUNCT ejpam-2159 87 9 ( ( PUNCT ejpam-2159 87 10 n)r n)r X ejpam-2159 87 11 ( ( PUNCT ejpam-2159 87 12 x x X ejpam-2159 87 13 ) ) PUNCT ejpam-2159 87 14 : : PUNCT ejpam-2159 88 1 = = PUNCT ejpam-2159 88 2 + + CCONJ ejpam-2159 88 3 , , PUNCT ejpam-2159 88 4 , , PUNCT ejpam-2159 88 5 , , PUNCT ejpam-2159 88 6 , , PUNCT ejpam-2159 88 7 . . PUNCT ejpam-2159 88 8 0 0 PUNCT ejpam-2159 89 1 if if SCONJ ejpam-2159 89 2 r r NOUN ejpam-2159 89 3 > > X ejpam-2159 89 4 n n NOUN ejpam-2159 89 5 or or CCONJ ejpam-2159 89 6 n n CCONJ ejpam-2159 89 7 < < X ejpam-2159 89 8 0 0 NUM ejpam-2159 89 9 or or CCONJ ejpam-2159 89 10 r r NOUN ejpam-2159 89 11 < < X ejpam-2159 89 12 0 0 NUM ejpam-2159 89 13 , , PUNCT ejpam-2159 89 14 1 1 NUM ejpam-2159 89 15 if if SCONJ ejpam-2159 89 16 r r NOUN ejpam-2159 89 17 = = SYM ejpam-2159 89 18 0 0 NUM ejpam-2159 89 19 , , PUNCT ejpam-2159 89 20 / / SYM ejpam-2159 90 1 1&i1 1&i1 NUM ejpam-2159 90 2 < < X ejpam-2159 90 3 i2< i2< NOUN ejpam-2159 90 4 ... ... PUNCT ejpam-2159 90 5 > X ejpam-2159 94 16 0 0 NUM ejpam-2159 94 17 ) ) PUNCT ejpam-2159 94 18 , , PUNCT ejpam-2159 94 19 1 1 NUM ejpam-2159 94 20 if if SCONJ ejpam-2159 94 21 r r NOUN ejpam-2159 94 22 = = SYM ejpam-2159 94 23 0 0 NUM ejpam-2159 94 24 , , PUNCT ejpam-2159 94 25 / / SYM ejpam-2159 94 26 1&i1&i2& 1&i1&i2& NUM ejpam-2159 94 27 ... ... PUNCT ejpam-2159 94 28 &ir&n &ir&n NOUN ejpam-2159 94 29 xi1 xi1 PROPN ejpam-2159 94 30 xi2 xi2 PROPN ejpam-2159 94 31 . . PUNCT ejpam-2159 94 32 . . PUNCT ejpam-2159 94 33 . . PUNCT ejpam-2159 95 1 xir xir NOUN ejpam-2159 96 1 if if SCONJ ejpam-2159 96 2 r r NOUN ejpam-2159 96 3 % % NOUN ejpam-2159 96 4 1 1 NUM ejpam-2159 96 5 . . PUNCT ejpam-2159 97 1 ( ( PUNCT ejpam-2159 97 2 17 17 NUM ejpam-2159 97 3 ) ) PUNCT ejpam-2159 97 4 it it PRON ejpam-2159 97 5 should should AUX ejpam-2159 97 6 be be AUX ejpam-2159 97 7 noticed notice VERB ejpam-2159 97 8 that that SCONJ ejpam-2159 97 9 each"(n)r each"(n)r PROPN ejpam-2159 97 10 has have VERB ejpam-2159 97 11 ! ! PUNCT ejpam-2159 98 1 n n CCONJ ejpam-2159 98 2 r r NOUN ejpam-2159 98 3 " " PUNCT ejpam-2159 98 4 terms term NOUN ejpam-2159 98 5 and and CCONJ ejpam-2159 98 6 each each DET ejpam-2159 98 7 # # NOUN ejpam-2159 98 8 ( ( PUNCT ejpam-2159 98 9 n)r n)r X ejpam-2159 98 10 has have VERB ejpam-2159 98 11 ! ! PUNCT ejpam-2159 99 1 n+ n+ X ejpam-2159 99 2 r r NOUN ejpam-2159 99 3 # # NOUN ejpam-2159 99 4 1 1 NUM ejpam-2159 99 5 r r NOUN ejpam-2159 99 6 " " PUNCT ejpam-2159 99 7 terms term NOUN ejpam-2159 99 8 . . PUNCT ejpam-2159 100 1 moreover moreover ADV ejpam-2159 100 2 these these DET ejpam-2159 100 3 polynomials polynomial NOUN ejpam-2159 100 4 can can AUX ejpam-2159 100 5 be be AUX ejpam-2159 100 6 expressed express VERB ejpam-2159 100 7 as as ADP ejpam-2159 100 8 : : PUNCT ejpam-2159 100 9 " " PUNCT ejpam-2159 100 10 ( ( PUNCT ejpam-2159 100 11 n)r n)r X ejpam-2159 100 12 ( ( PUNCT ejpam-2159 100 13 x x X ejpam-2159 100 14 ) ) PUNCT ejpam-2159 100 15 : : PUNCT ejpam-2159 100 16 = = PUNCT ejpam-2159 100 17 # # NOUN ejpam-2159 100 18 k1+k2+ k1+k2+ NOUN ejpam-2159 100 19 ... ... PUNCT ejpam-2159 100 20 +kn +kn NOUN ejpam-2159 100 21 = = SYM ejpam-2159 100 22 r r NOUN ejpam-2159 100 23 k1,k2, k1,k2, NOUN ejpam-2159 100 24 ... ... PUNCT ejpam-2159 100 25 ,kn'{0,1 ,kn'{0,1 SYM ejpam-2159 100 26 } } PUNCT ejpam-2159 100 27 x x SYM ejpam-2159 100 28 k1 k1 NOUN ejpam-2159 100 29 1 1 NUM ejpam-2159 100 30 x x SYM ejpam-2159 100 31 k2 k2 PROPN ejpam-2159 100 32 2 2 NUM ejpam-2159 100 33 . . PUNCT ejpam-2159 100 34 . . PUNCT ejpam-2159 100 35 . . PUNCT ejpam-2159 101 1 xkn xkn X ejpam-2159 101 2 n n CCONJ ejpam-2159 101 3 , , PUNCT ejpam-2159 101 4 0 0 NUM ejpam-2159 101 5 & & CCONJ ejpam-2159 101 6 r r PROPN ejpam-2159 101 7 & & CCONJ ejpam-2159 101 8 n. n. PROPN ejpam-2159 101 9 ( ( PUNCT ejpam-2159 101 10 18 18 NUM ejpam-2159 101 11 ) ) PUNCT ejpam-2159 101 12 and and CCONJ ejpam-2159 101 13 # # SYM ejpam-2159 101 14 ( ( PUNCT ejpam-2159 101 15 n)r n)r X ejpam-2159 101 16 ( ( PUNCT ejpam-2159 101 17 x x X ejpam-2159 101 18 ) ) PUNCT ejpam-2159 101 19 : : PUNCT ejpam-2159 101 20 = = PUNCT ejpam-2159 101 21 # # NOUN ejpam-2159 101 22 d1+d2+ d1+d2+ NOUN ejpam-2159 101 23 ... ... PUNCT ejpam-2159 101 24 +dn +dn X ejpam-2159 101 25 = = NOUN ejpam-2159 101 26 r r NOUN ejpam-2159 101 27 d1,d2, d1,d2, NOUN ejpam-2159 101 28 ... ... PUNCT ejpam-2159 101 29 ,dn'{0,1, ,dn'{0,1, PUNCT ejpam-2159 101 30 ... ... PUNCT ejpam-2159 101 31 ,r ,r SYM ejpam-2159 101 32 } } PUNCT ejpam-2159 101 33 x x SYM ejpam-2159 101 34 d1 d1 NOUN ejpam-2159 101 35 1 1 NUM ejpam-2159 101 36 x x SYM ejpam-2159 101 37 d2 d2 PROPN ejpam-2159 101 38 2 2 NUM ejpam-2159 101 39 . . PUNCT ejpam-2159 101 40 . . PUNCT ejpam-2159 101 41 . . PUNCT ejpam-2159 102 1 xdn xdn PROPN ejpam-2159 102 2 n n PROPN ejpam-2159 102 3 , , PUNCT ejpam-2159 102 4 r r NOUN ejpam-2159 102 5 % % NOUN ejpam-2159 102 6 0 0 NUM ejpam-2159 102 7 . . PUNCT ejpam-2159 103 1 ( ( PUNCT ejpam-2159 103 2 19 19 NUM ejpam-2159 103 3 ) ) PUNCT ejpam-2159 103 4 the the DET ejpam-2159 103 5 falling fall VERB ejpam-2159 103 6 factorial factorial NOUN ejpam-2159 103 7 ( ( PUNCT ejpam-2159 103 8 x)n x)n PUNCT ejpam-2159 103 9 in in ADP ejpam-2159 103 10 ( ( PUNCT ejpam-2159 103 11 3 3 X ejpam-2159 103 12 ) ) PUNCT ejpam-2159 103 13 can can AUX ejpam-2159 103 14 be be AUX ejpam-2159 103 15 written write VERB ejpam-2159 103 16 in in ADP ejpam-2159 103 17 the the DET ejpam-2159 103 18 form form NOUN ejpam-2159 103 19 : : PUNCT ejpam-2159 103 20 ( ( PUNCT ejpam-2159 103 21 x)n x)n PUNCT ejpam-2159 103 22 = = SYM ejpam-2159 104 1 n n NUM ejpam-2159 104 2 # # NOUN ejpam-2159 104 3 k=0 k=0 PROPN ejpam-2159 104 4 ( ( PUNCT ejpam-2159 104 5 # # SYM ejpam-2159 104 6 1)n#k 1)n#k NUM ejpam-2159 104 7 " " PUNCT ejpam-2159 104 8 ( ( PUNCT ejpam-2159 104 9 n n CCONJ ejpam-2159 104 10 ) ) PUNCT ejpam-2159 105 1 n#k n#k NOUN ejpam-2159 105 2 ( ( PUNCT ejpam-2159 105 3 0,1 0,1 NUM ejpam-2159 105 4 , , PUNCT ejpam-2159 105 5 . . PUNCT ejpam-2159 105 6 . . PUNCT ejpam-2159 106 1 . . PUNCT ejpam-2159 107 1 , , PUNCT ejpam-2159 108 1 n n CCONJ ejpam-2159 108 2 # # NOUN ejpam-2159 108 3 1)xk 1)xk PROPN ejpam-2159 108 4 . . PUNCT ejpam-2159 109 1 ( ( PUNCT ejpam-2159 109 2 20 20 NUM ejpam-2159 109 3 ) ) PUNCT ejpam-2159 109 4 comparing compare VERB ejpam-2159 109 5 the the DET ejpam-2159 109 6 coefficients coefficient NOUN ejpam-2159 109 7 of of ADP ejpam-2159 109 8 xk xk PROPN ejpam-2159 109 9 in in ADP ejpam-2159 109 10 ( ( PUNCT ejpam-2159 109 11 1 1 NUM ejpam-2159 109 12 ) ) PUNCT ejpam-2159 109 13 and and CCONJ ejpam-2159 109 14 ( ( PUNCT ejpam-2159 109 15 20 20 NUM ejpam-2159 109 16 ) ) PUNCT ejpam-2159 109 17 , , PUNCT ejpam-2159 109 18 yields yield NOUN ejpam-2159 109 19 c(n c(n PROPN ejpam-2159 109 20 , , PUNCT ejpam-2159 109 21 k k NOUN ejpam-2159 109 22 ) ) PUNCT ejpam-2159 109 23 = = PRON ejpam-2159 109 24 " " PUNCT ejpam-2159 109 25 ( ( PUNCT ejpam-2159 109 26 n n CCONJ ejpam-2159 109 27 ) ) PUNCT ejpam-2159 109 28 n#k n#k NOUN ejpam-2159 109 29 ( ( PUNCT ejpam-2159 109 30 0,1 0,1 NUM ejpam-2159 109 31 , , PUNCT ejpam-2159 109 32 . . PUNCT ejpam-2159 109 33 . . PUNCT ejpam-2159 110 1 . . PUNCT ejpam-2159 111 1 , , PUNCT ejpam-2159 111 2 n n CCONJ ejpam-2159 111 3 # # NOUN ejpam-2159 111 4 1 1 NUM ejpam-2159 111 5 ) ) PUNCT ejpam-2159 111 6 = = PRON ejpam-2159 111 7 " " PUNCT ejpam-2159 111 8 ( ( PUNCT ejpam-2159 111 9 n#1 n#1 NOUN ejpam-2159 111 10 ) ) PUNCT ejpam-2159 111 11 n#k n#k NOUN ejpam-2159 111 12 ( ( PUNCT ejpam-2159 111 13 1,2 1,2 NUM ejpam-2159 111 14 , , PUNCT ejpam-2159 111 15 . . PUNCT ejpam-2159 111 16 . . PUNCT ejpam-2159 112 1 . . PUNCT ejpam-2159 113 1 , , PUNCT ejpam-2159 113 2 n n CCONJ ejpam-2159 113 3 # # NOUN ejpam-2159 113 4 1 1 NUM ejpam-2159 113 5 ) ) PUNCT ejpam-2159 113 6 . . PUNCT ejpam-2159 114 1 ( ( PUNCT ejpam-2159 114 2 21 21 NUM ejpam-2159 114 3 ) ) PUNCT ejpam-2159 114 4 it it PRON ejpam-2159 114 5 can can AUX ejpam-2159 114 6 also also ADV ejpam-2159 114 7 be be AUX ejpam-2159 114 8 shown show VERB ejpam-2159 114 9 that that SCONJ ejpam-2159 114 10 s(n s(n PROPN ejpam-2159 114 11 , , PUNCT ejpam-2159 114 12 k k NOUN ejpam-2159 114 13 ) ) PUNCT ejpam-2159 115 1 = = SYM ejpam-2159 115 2 # # NOUN ejpam-2159 115 3 ( ( PUNCT ejpam-2159 115 4 k k NOUN ejpam-2159 115 5 ) ) PUNCT ejpam-2159 115 6 n#k n#k NOUN ejpam-2159 115 7 ( ( PUNCT ejpam-2159 115 8 1,2 1,2 NUM ejpam-2159 115 9 , , PUNCT ejpam-2159 115 10 . . PUNCT ejpam-2159 115 11 . . PUNCT ejpam-2159 115 12 . . PUNCT ejpam-2159 116 1 , , PUNCT ejpam-2159 116 2 k k X ejpam-2159 116 3 ) ) PUNCT ejpam-2159 116 4 . . PUNCT ejpam-2159 117 1 ( ( PUNCT ejpam-2159 117 2 22 22 NUM ejpam-2159 117 3 ) ) PUNCT ejpam-2159 117 4 also also ADV ejpam-2159 117 5 , , PUNCT ejpam-2159 117 6 s(n s(n PROPN ejpam-2159 117 7 , , PUNCT ejpam-2159 117 8 k k NOUN ejpam-2159 117 9 ) ) PUNCT ejpam-2159 117 10 = = SYM ejpam-2159 117 11 1 1 NUM ejpam-2159 117 12 k k NOUN ejpam-2159 117 13 ! ! PUNCT ejpam-2159 118 1 k k PROPN ejpam-2159 118 2 # # PROPN ejpam-2159 118 3 j=0 j=0 PROPN ejpam-2159 118 4 ( ( PUNCT ejpam-2159 118 5 # # SYM ejpam-2159 118 6 1)k 1)k NUM ejpam-2159 118 7 # # NOUN ejpam-2159 118 8 j j NOUN ejpam-2159 118 9 ! ! PUNCT ejpam-2159 119 1 k k PROPN ejpam-2159 119 2 j j PROPN ejpam-2159 119 3 " " PUNCT ejpam-2159 119 4 jn jn PROPN ejpam-2159 119 5 . . PROPN ejpam-2159 120 1 ( ( PUNCT ejpam-2159 120 2 23 23 NUM ejpam-2159 120 3 ) ) PUNCT ejpam-2159 120 4 m. m. NOUN ejpam-2159 120 5 el el PROPN ejpam-2159 120 6 - - PUNCT ejpam-2159 120 7 mikkawy mikkawy PROPN ejpam-2159 120 8 , , PUNCT ejpam-2159 120 9 f. f. PROPN ejpam-2159 120 10 atlan atlan PROPN ejpam-2159 120 11 / / SYM ejpam-2159 120 12 eur eur PROPN ejpam-2159 120 13 . . PUNCT ejpam-2159 121 1 j. j. PROPN ejpam-2159 121 2 pure pure PROPN ejpam-2159 121 3 appl appl PROPN ejpam-2159 121 4 . . PROPN ejpam-2159 121 5 math math PROPN ejpam-2159 121 6 , , PUNCT ejpam-2159 121 7 8 8 NUM ejpam-2159 121 8 ( ( PUNCT ejpam-2159 121 9 2015 2015 NUM ejpam-2159 121 10 ) ) PUNCT ejpam-2159 121 11 , , PUNCT ejpam-2159 121 12 135 135 NUM ejpam-2159 121 13 - - SYM ejpam-2159 121 14 151 151 NUM ejpam-2159 121 15 139 139 NUM ejpam-2159 121 16 for for ADP ejpam-2159 121 17 any any DET ejpam-2159 121 18 x x SYM ejpam-2159 121 19 j j PROPN ejpam-2159 121 20 ' ' PUNCT ejpam-2159 121 21 x x NOUN ejpam-2159 121 22 , , PUNCT ejpam-2159 121 23 we we PRON ejpam-2159 121 24 have have VERB ejpam-2159 121 25 " " PUNCT ejpam-2159 121 26 ( ( PUNCT ejpam-2159 121 27 n n CCONJ ejpam-2159 121 28 ) ) PUNCT ejpam-2159 121 29 i i PRON ejpam-2159 121 30 ( ( PUNCT ejpam-2159 121 31 x1 x1 PROPN ejpam-2159 121 32 , , PUNCT ejpam-2159 121 33 x2 x2 PROPN ejpam-2159 121 34 , , PUNCT ejpam-2159 121 35 . . PUNCT ejpam-2159 121 36 . . PUNCT ejpam-2159 121 37 . . PUNCT ejpam-2159 122 1 , , PUNCT ejpam-2159 122 2 xn xn X ejpam-2159 122 3 ) ) PUNCT ejpam-2159 122 4 = = PRON ejpam-2159 122 5 " " PUNCT ejpam-2159 122 6 ( ( PUNCT ejpam-2159 122 7 n#1 n#1 NOUN ejpam-2159 122 8 ) ) PUNCT ejpam-2159 122 9 i i PRON ejpam-2159 122 10 ( ( PUNCT ejpam-2159 122 11 x1 x1 PROPN ejpam-2159 122 12 , , PUNCT ejpam-2159 122 13 x2 x2 PROPN ejpam-2159 122 14 , , PUNCT ejpam-2159 122 15 . . PUNCT ejpam-2159 122 16 . . PUNCT ejpam-2159 122 17 . . PUNCT ejpam-2159 123 1 , , PUNCT ejpam-2159 123 2 x x PUNCT ejpam-2159 123 3 j#1 j#1 NOUN ejpam-2159 123 4 , , PUNCT ejpam-2159 123 5 x x PUNCT ejpam-2159 123 6 j+1 j+1 NOUN ejpam-2159 123 7 , , PUNCT ejpam-2159 123 8 . . PUNCT ejpam-2159 123 9 . . PUNCT ejpam-2159 123 10 . . PUNCT ejpam-2159 124 1 , , PUNCT ejpam-2159 124 2 xn xn X ejpam-2159 124 3 ) ) PUNCT ejpam-2159 125 1 + + CCONJ ejpam-2159 125 2 x x PUNCT ejpam-2159 125 3 j j NOUN ejpam-2159 125 4 " " PUNCT ejpam-2159 125 5 ( ( PUNCT ejpam-2159 125 6 n#1 n#1 NOUN ejpam-2159 125 7 ) ) PUNCT ejpam-2159 125 8 i#1 i#1 PROPN ejpam-2159 125 9 ( ( PUNCT ejpam-2159 125 10 x1 x1 PROPN ejpam-2159 125 11 , , PUNCT ejpam-2159 125 12 x2 x2 PROPN ejpam-2159 125 13 , , PUNCT ejpam-2159 125 14 . . PUNCT ejpam-2159 125 15 . . PUNCT ejpam-2159 125 16 . . PUNCT ejpam-2159 126 1 , , PUNCT ejpam-2159 126 2 x x PUNCT ejpam-2159 126 3 j#1 j#1 NOUN ejpam-2159 126 4 , , PUNCT ejpam-2159 126 5 x x PUNCT ejpam-2159 126 6 j+1 j+1 NOUN ejpam-2159 126 7 , , PUNCT ejpam-2159 126 8 . . PUNCT ejpam-2159 126 9 . . PUNCT ejpam-2159 126 10 . . PUNCT ejpam-2159 127 1 , , PUNCT ejpam-2159 127 2 xn xn PROPN ejpam-2159 127 3 ) ) PUNCT ejpam-2159 127 4 . . PUNCT ejpam-2159 128 1 ( ( PUNCT ejpam-2159 128 2 24 24 NUM ejpam-2159 128 3 ) ) PUNCT ejpam-2159 128 4 partial partial ADJ ejpam-2159 128 5 differentiation differentiation NOUN ejpam-2159 128 6 for for ADP ejpam-2159 128 7 both both DET ejpam-2159 128 8 sides side NOUN ejpam-2159 128 9 of of ADP ejpam-2159 128 10 ( ( PUNCT ejpam-2159 128 11 24 24 NUM ejpam-2159 128 12 ) ) PUNCT ejpam-2159 128 13 with with ADP ejpam-2159 128 14 respect respect NOUN ejpam-2159 128 15 to to ADP ejpam-2159 128 16 x x PROPN ejpam-2159 128 17 j j PROPN ejpam-2159 128 18 , , PUNCT ejpam-2159 128 19 j j PROPN ejpam-2159 128 20 ' ' PUNCT ejpam-2159 128 21 { { PUNCT ejpam-2159 128 22 1,2 1,2 NUM ejpam-2159 128 23 , , PUNCT ejpam-2159 128 24 . . PUNCT ejpam-2159 128 25 . . PUNCT ejpam-2159 129 1 . . PUNCT ejpam-2159 130 1 , , PUNCT ejpam-2159 130 2 n n CCONJ ejpam-2159 130 3 } } PUNCT ejpam-2159 130 4 gives give VERB ejpam-2159 130 5 $ $ SYM ejpam-2159 130 6 $ $ SYM ejpam-2159 130 7 x x SYM ejpam-2159 130 8 j j NOUN ejpam-2159 130 9 " " PUNCT ejpam-2159 130 10 ( ( PUNCT ejpam-2159 130 11 n n CCONJ ejpam-2159 130 12 ) ) PUNCT ejpam-2159 130 13 i i PRON ejpam-2159 130 14 = = PUNCT ejpam-2159 130 15 " " PUNCT ejpam-2159 130 16 ( ( PUNCT ejpam-2159 130 17 n n CCONJ ejpam-2159 130 18 ) ) PUNCT ejpam-2159 131 1 i i PROPN ejpam-2159 131 2 , , PUNCT ejpam-2159 131 3 j j PROPN ejpam-2159 131 4 = = PUNCT ejpam-2159 131 5 " " PUNCT ejpam-2159 131 6 ( ( PUNCT ejpam-2159 131 7 n#1 n#1 NOUN ejpam-2159 131 8 ) ) PUNCT ejpam-2159 131 9 i#1 i#1 PROPN ejpam-2159 131 10 ( ( PUNCT ejpam-2159 131 11 x1 x1 PROPN ejpam-2159 131 12 , , PUNCT ejpam-2159 131 13 x2 x2 PROPN ejpam-2159 131 14 , , PUNCT ejpam-2159 131 15 . . PUNCT ejpam-2159 131 16 . . PUNCT ejpam-2159 131 17 . . PUNCT ejpam-2159 132 1 , , PUNCT ejpam-2159 132 2 x x PUNCT ejpam-2159 132 3 j#1 j#1 NOUN ejpam-2159 132 4 , , PUNCT ejpam-2159 132 5 x x PUNCT ejpam-2159 132 6 j+1 j+1 NOUN ejpam-2159 132 7 , , PUNCT ejpam-2159 132 8 . . PUNCT ejpam-2159 132 9 . . PUNCT ejpam-2159 132 10 . . PUNCT ejpam-2159 133 1 , , PUNCT ejpam-2159 133 2 xn xn PROPN ejpam-2159 133 3 ) ) PUNCT ejpam-2159 133 4 . . PUNCT ejpam-2159 134 1 ( ( PUNCT ejpam-2159 134 2 25 25 NUM ejpam-2159 134 3 ) ) PUNCT ejpam-2159 134 4 therefore therefore ADV ejpam-2159 134 5 by by ADP ejpam-2159 134 6 using use VERB ejpam-2159 134 7 ( ( PUNCT ejpam-2159 134 8 16 16 NUM ejpam-2159 134 9 ) ) PUNCT ejpam-2159 134 10 , , PUNCT ejpam-2159 134 11 we we PRON ejpam-2159 134 12 obtain obtain VERB ejpam-2159 134 13 " " PUNCT ejpam-2159 134 14 ( ( PUNCT ejpam-2159 134 15 n n CCONJ ejpam-2159 134 16 ) ) PUNCT ejpam-2159 135 1 i i PRON ejpam-2159 135 2 j j X ejpam-2159 135 3 ( ( PUNCT ejpam-2159 135 4 x x X ejpam-2159 135 5 ) ) PUNCT ejpam-2159 135 6 = = SYM ejpam-2159 136 1 + + CCONJ ejpam-2159 136 2 , , PUNCT ejpam-2159 136 3 , , PUNCT ejpam-2159 136 4 , , PUNCT ejpam-2159 136 5 , , PUNCT ejpam-2159 136 6 , , PUNCT ejpam-2159 136 7 , , PUNCT ejpam-2159 136 8 . . PUNCT ejpam-2159 136 9 0 0 PUNCT ejpam-2159 137 1 if if SCONJ ejpam-2159 137 2 i i PRON ejpam-2159 137 3 > > X ejpam-2159 137 4 n n CCONJ ejpam-2159 137 5 or or CCONJ ejpam-2159 137 6 i i PRON ejpam-2159 137 7 < < X ejpam-2159 137 8 0 0 NUM ejpam-2159 137 9 or or CCONJ ejpam-2159 137 10 n n CCONJ ejpam-2159 137 11 < < X ejpam-2159 137 12 0 0 NUM ejpam-2159 137 13 , , PUNCT ejpam-2159 137 14 1 1 NUM ejpam-2159 137 15 if if SCONJ ejpam-2159 137 16 i i PRON ejpam-2159 137 17 = = NOUN ejpam-2159 137 18 1 1 NUM ejpam-2159 137 19 , , PUNCT ejpam-2159 137 20 / / SYM ejpam-2159 137 21 1&r1 1&r1 NUM ejpam-2159 137 22 < < X ejpam-2159 137 23 r2< r2< NOUN ejpam-2159 137 24 ... ... PUNCT ejpam-2159 137 25 > X ejpam-2159 171 4 j j PROPN ejpam-2159 171 5 , , PUNCT ejpam-2159 171 6 i#1 i#1 PROPN ejpam-2159 171 7 5 5 NUM ejpam-2159 171 8 r=1 r=1 NOUN ejpam-2159 171 9 ( ( PUNCT ejpam-2159 171 10 x x X ejpam-2159 171 11 j j PROPN ejpam-2159 171 12 # # NOUN ejpam-2159 171 13 xr xr PROPN ejpam-2159 171 14 ) ) PUNCT ejpam-2159 171 15 for for ADP ejpam-2159 171 16 i i PROPN ejpam-2159 171 17 & & CCONJ ejpam-2159 171 18 j j PROPN ejpam-2159 171 19 , , PUNCT ejpam-2159 171 20 1 1 NUM ejpam-2159 171 21 & & CCONJ ejpam-2159 171 22 i i PRON ejpam-2159 171 23 , , PUNCT ejpam-2159 171 24 j j PROPN ejpam-2159 171 25 & & CCONJ ejpam-2159 171 26 n n PROPN ejpam-2159 171 27 ( ( PUNCT ejpam-2159 171 28 36 36 NUM ejpam-2159 171 29 ) ) PUNCT ejpam-2159 171 30 the the DET ejpam-2159 171 31 inverse inverse NOUN ejpam-2159 171 32 matrices matrix NOUN ejpam-2159 171 33 l#1 l#1 VERB ejpam-2159 171 34 and and CCONJ ejpam-2159 171 35 u#1 u#1 ADP ejpam-2159 171 36 of of ADP ejpam-2159 171 37 the the DET ejpam-2159 171 38 matrices matrix NOUN ejpam-2159 171 39 l l NOUN ejpam-2159 171 40 and and CCONJ ejpam-2159 171 41 u u NOUN ejpam-2159 171 42 in in ADP ejpam-2159 171 43 ( ( PUNCT ejpam-2159 171 44 35 35 NUM ejpam-2159 171 45 ) ) PUNCT ejpam-2159 171 46 and and CCONJ ejpam-2159 171 47 ( ( PUNCT ejpam-2159 171 48 36 36 NUM ejpam-2159 171 49 ) ) PUNCT ejpam-2159 171 50 are be AUX ejpam-2159 171 51 given give VERB ejpam-2159 171 52 respectively respectively ADV ejpam-2159 171 53 by by ADP ejpam-2159 171 54 : : PUNCT ejpam-2159 171 55 l#1 l#1 PROPN ejpam-2159 171 56 = = SYM ejpam-2159 171 57 6 6 NUM ejpam-2159 171 58 ( ( PUNCT ejpam-2159 171 59 # # SYM ejpam-2159 171 60 1)i 1)i NOUN ejpam-2159 171 61 # # NOUN ejpam-2159 171 62 j j NOUN ejpam-2159 171 63 " " PUNCT ejpam-2159 171 64 ( ( PUNCT ejpam-2159 171 65 i#1 i#1 X ejpam-2159 171 66 ) ) PUNCT ejpam-2159 172 1 i i PROPN ejpam-2159 172 2 # # NOUN ejpam-2159 172 3 j j PROPN ejpam-2159 172 4 ( ( PUNCT ejpam-2159 172 5 x1 x1 PROPN ejpam-2159 172 6 , , PUNCT ejpam-2159 172 7 x2 x2 PROPN ejpam-2159 172 8 , , PUNCT ejpam-2159 172 9 . . PUNCT ejpam-2159 172 10 . . PUNCT ejpam-2159 172 11 . . PUNCT ejpam-2159 173 1 , , PUNCT ejpam-2159 173 2 xi#1 xi#1 X ejpam-2159 173 3 ) ) PUNCT ejpam-2159 173 4 7n 7n NOUN ejpam-2159 174 1 i i PRON ejpam-2159 174 2 , , PUNCT ejpam-2159 174 3 j=1 j=1 PROPN ejpam-2159 174 4 ( ( PUNCT ejpam-2159 174 5 37 37 NUM ejpam-2159 174 6 ) ) PUNCT ejpam-2159 174 7 and and CCONJ ejpam-2159 174 8 u#1 u#1 ADJ ejpam-2159 174 9 = = NUM ejpam-2159 174 10 6 6 NUM ejpam-2159 174 11 1 1 NUM ejpam-2159 174 12 j j PROPN ejpam-2159 174 13 5 5 NUM ejpam-2159 174 14 r=1 r=1 NOUN ejpam-2159 174 15 r r NOUN ejpam-2159 174 16 (= (= NOUN ejpam-2159 174 17 i i PRON ejpam-2159 174 18 ( ( PUNCT ejpam-2159 174 19 xi xi PROPN ejpam-2159 174 20 # # PROPN ejpam-2159 174 21 xr xr PROPN ejpam-2159 174 22 ) ) PUNCT ejpam-2159 174 23 7n 7n NOUN ejpam-2159 175 1 i i PRON ejpam-2159 175 2 , , PUNCT ejpam-2159 175 3 j=1 j=1 PROPN ejpam-2159 175 4 . . PUNCT ejpam-2159 176 1 ( ( PUNCT ejpam-2159 176 2 38 38 NUM ejpam-2159 176 3 ) ) PUNCT ejpam-2159 176 4 in in ADP ejpam-2159 176 5 particular particular ADJ ejpam-2159 176 6 , , PUNCT ejpam-2159 176 7 if if SCONJ ejpam-2159 176 8 xk xk PROPN ejpam-2159 176 9 = = SYM ejpam-2159 176 10 k k PROPN ejpam-2159 176 11 , , PUNCT ejpam-2159 176 12 1 1 NUM ejpam-2159 176 13 & & CCONJ ejpam-2159 176 14 k k PROPN ejpam-2159 176 15 & & CCONJ ejpam-2159 176 16 n n CCONJ ejpam-2159 176 17 , , PUNCT ejpam-2159 176 18 then then ADV ejpam-2159 176 19 we we PRON ejpam-2159 176 20 have have VERB ejpam-2159 176 21 [ [ X ejpam-2159 176 22 6 6 NUM ejpam-2159 176 23 ] ] SYM ejpam-2159 176 24 : : PUNCT ejpam-2159 176 25 v v X ejpam-2159 176 26 = = SYM ejpam-2159 176 27 sn sn PROPN ejpam-2159 176 28 l̃t l̃t PROPN ejpam-2159 176 29 , , PUNCT ejpam-2159 176 30 where where SCONJ ejpam-2159 176 31 sn sn PROPN ejpam-2159 176 32 is be AUX ejpam-2159 176 33 the the DET ejpam-2159 176 34 stirling stirling NOUN ejpam-2159 176 35 matrix matrix NOUN ejpam-2159 176 36 of of ADP ejpam-2159 176 37 the the DET ejpam-2159 176 38 second second ADJ ejpam-2159 176 39 kind kind NOUN ejpam-2159 176 40 and and CCONJ ejpam-2159 176 41 l̃ l̃ PROPN ejpam-2159 176 42 = = PUNCT ejpam-2159 176 43 ( ( PUNCT ejpam-2159 176 44 l̃i l̃i PROPN ejpam-2159 176 45 j j PROPN ejpam-2159 176 46 ) ) PUNCT ejpam-2159 177 1 n n PROPN ejpam-2159 177 2 i i PRON ejpam-2159 177 3 , , PUNCT ejpam-2159 177 4 j=1 j=1 PROPN ejpam-2159 177 5 is be AUX ejpam-2159 177 6 an an DET ejpam-2159 177 7 n"n n"n ADV ejpam-2159 177 8 lower low ADJ ejpam-2159 177 9 triangular triangular NOUN ejpam-2159 177 10 matrix matrix NOUN ejpam-2159 177 11 given give VERB ejpam-2159 177 12 by by ADP ejpam-2159 177 13 : : PUNCT ejpam-2159 177 14 l̃i l̃i PROPN ejpam-2159 177 15 j j X ejpam-2159 177 16 = = PUNCT ejpam-2159 177 17 $ $ SYM ejpam-2159 178 1 [ [ X ejpam-2159 178 2 i i PRON ejpam-2159 178 3 # # NOUN ejpam-2159 178 4 1 1 NUM ejpam-2159 178 5 ] ] X ejpam-2159 178 6 j#1 j#1 NOUN ejpam-2159 178 7 for for ADP ejpam-2159 178 8 i i PRON ejpam-2159 178 9 % % VERB ejpam-2159 178 10 j j PROPN ejpam-2159 178 11 0 0 PROPN ejpam-2159 179 1 for for ADP ejpam-2159 179 2 i i PRON ejpam-2159 179 3 < < X ejpam-2159 179 4 j j PROPN ejpam-2159 179 5 , , PUNCT ejpam-2159 179 6 1 1 NUM ejpam-2159 179 7 & & CCONJ ejpam-2159 179 8 i i PRON ejpam-2159 179 9 , , PUNCT ejpam-2159 179 10 j j PROPN ejpam-2159 179 11 & & CCONJ ejpam-2159 179 12 n n PROPN ejpam-2159 179 13 ( ( PUNCT ejpam-2159 179 14 39 39 NUM ejpam-2159 179 15 ) ) PUNCT ejpam-2159 179 16 the the DET ejpam-2159 179 17 inverse inverse NOUN ejpam-2159 179 18 of of ADP ejpam-2159 179 19 the the DET ejpam-2159 179 20 matrix matrix NOUN ejpam-2159 179 21 l̃ l̃ PROPN ejpam-2159 179 22 is be AUX ejpam-2159 179 23 given give VERB ejpam-2159 179 24 by by ADP ejpam-2159 179 25 : : PUNCT ejpam-2159 179 26 l̃#1 l̃#1 PROPN ejpam-2159 179 27 = = SYM ejpam-2159 179 28 6 6 NUM ejpam-2159 179 29 ( ( PUNCT ejpam-2159 179 30 # # SYM ejpam-2159 179 31 1)i 1)i NOUN ejpam-2159 179 32 # # NOUN ejpam-2159 179 33 j j NOUN ejpam-2159 179 34 1 1 NUM ejpam-2159 179 35 ( ( PUNCT ejpam-2159 179 36 i i NOUN ejpam-2159 179 37 # # NOUN ejpam-2159 179 38 1 1 NUM ejpam-2159 179 39 ) ) PUNCT ejpam-2159 179 40 ! ! PUNCT ejpam-2159 179 41 ! ! PUNCT ejpam-2159 180 1 i i PRON ejpam-2159 180 2 # # NOUN ejpam-2159 180 3 1 1 NUM ejpam-2159 180 4 j j NOUN ejpam-2159 180 5 # # NOUN ejpam-2159 180 6 1 1 NUM ejpam-2159 180 7 " " PUNCT ejpam-2159 180 8 7n 7n NOUN ejpam-2159 181 1 i i PRON ejpam-2159 181 2 , , PUNCT ejpam-2159 181 3 j=1 j=1 PROPN ejpam-2159 181 4 . . PUNCT ejpam-2159 182 1 definition definition NOUN ejpam-2159 182 2 5 5 NUM ejpam-2159 182 3 ( ( PUNCT ejpam-2159 182 4 [ [ X ejpam-2159 182 5 18 18 NUM ejpam-2159 182 6 ] ] NUM ejpam-2159 182 7 ) ) PUNCT ejpam-2159 182 8 . . PUNCT ejpam-2159 183 1 an an DET ejpam-2159 183 2 n n CCONJ ejpam-2159 183 3 " " PUNCT ejpam-2159 183 4 n n PRON ejpam-2159 183 5 matrix matrix NOUN ejpam-2159 183 6 a a PRON ejpam-2159 183 7 is be AUX ejpam-2159 183 8 called call VERB ejpam-2159 183 9 totally totally ADV ejpam-2159 183 10 positive positive ADJ ejpam-2159 183 11 if if SCONJ ejpam-2159 183 12 all all DET ejpam-2159 183 13 its its PRON ejpam-2159 183 14 minors minor NOUN ejpam-2159 183 15 of of ADP ejpam-2159 183 16 all all DET ejpam-2159 183 17 sizes size NOUN ejpam-2159 183 18 are be AUX ejpam-2159 183 19 positive positive ADJ ejpam-2159 183 20 . . PUNCT ejpam-2159 184 1 m. m. PROPN ejpam-2159 184 2 el el PROPN ejpam-2159 184 3 - - PUNCT ejpam-2159 184 4 mikkawy mikkawy PROPN ejpam-2159 184 5 , , PUNCT ejpam-2159 184 6 f. f. PROPN ejpam-2159 184 7 atlan atlan PROPN ejpam-2159 184 8 / / SYM ejpam-2159 184 9 eur eur PROPN ejpam-2159 184 10 . . PUNCT ejpam-2159 185 1 j. j. PROPN ejpam-2159 185 2 pure pure PROPN ejpam-2159 185 3 appl appl PROPN ejpam-2159 185 4 . . PROPN ejpam-2159 185 5 math math PROPN ejpam-2159 185 6 , , PUNCT ejpam-2159 185 7 8 8 NUM ejpam-2159 185 8 ( ( PUNCT ejpam-2159 185 9 2015 2015 NUM ejpam-2159 185 10 ) ) PUNCT ejpam-2159 185 11 , , PUNCT ejpam-2159 185 12 135 135 NUM ejpam-2159 185 13 - - SYM ejpam-2159 185 14 151 151 NUM ejpam-2159 185 15 141 141 NUM ejpam-2159 185 16 definition definition NOUN ejpam-2159 185 17 6 6 NUM ejpam-2159 185 18 ( ( PUNCT ejpam-2159 185 19 [ [ X ejpam-2159 185 20 8 8 NUM ejpam-2159 185 21 ] ] NUM ejpam-2159 185 22 ) ) PUNCT ejpam-2159 185 23 . . PUNCT ejpam-2159 186 1 the the DET ejpam-2159 186 2 symmetric symmetric ADJ ejpam-2159 186 3 matrix matrix NOUN ejpam-2159 186 4 a= a= NOUN ejpam-2159 186 5 ( ( PUNCT ejpam-2159 186 6 ai ai VERB ejpam-2159 186 7 j j PROPN ejpam-2159 186 8 ) ) PUNCT ejpam-2159 186 9 n n PROPN ejpam-2159 186 10 i i PRON ejpam-2159 186 11 , , PUNCT ejpam-2159 186 12 j=1 j=1 PROPN ejpam-2159 186 13 is be AUX ejpam-2159 186 14 called call VERB ejpam-2159 186 15 positive positive ADJ ejpam-2159 186 16 definite definite ADJ ejpam-2159 186 17 if if SCONJ ejpam-2159 186 18 and and CCONJ ejpam-2159 186 19 only only ADV ejpam-2159 186 20 if if SCONJ ejpam-2159 186 21 xt xt PROPN ejpam-2159 186 22 ax ax VERB ejpam-2159 186 23 > > X ejpam-2159 186 24 0 0 NUM ejpam-2159 186 25 , , PUNCT ejpam-2159 186 26 for for ADP ejpam-2159 186 27 all all DET ejpam-2159 186 28 x x NOUN ejpam-2159 186 29 ' ' PUNCT ejpam-2159 186 30 ! ! PUNCT ejpam-2159 187 1 n n CCONJ ejpam-2159 187 2 , , PUNCT ejpam-2159 187 3 x x X ejpam-2159 187 4 (= (= ADP ejpam-2159 187 5 0 0 NUM ejpam-2159 187 6 . . PUNCT ejpam-2159 188 1 definition definition NOUN ejpam-2159 188 2 7 7 NUM ejpam-2159 188 3 . . PUNCT ejpam-2159 189 1 the the DET ejpam-2159 189 2 n n CCONJ ejpam-2159 189 3 " " PUNCT ejpam-2159 189 4 n n CCONJ ejpam-2159 189 5 permutation permutation NOUN ejpam-2159 189 6 matrix matrix NOUN ejpam-2159 189 7 , , PUNCT ejpam-2159 189 8 j j PROPN ejpam-2159 189 9 is be AUX ejpam-2159 189 10 defined define VERB ejpam-2159 189 11 by by ADP ejpam-2159 189 12 : : PUNCT ejpam-2159 189 13 j j PROPN ejpam-2159 189 14 = = SYM ejpam-2159 189 15 8 8 NUM ejpam-2159 189 16 en en X ejpam-2159 189 17 , , PUNCT ejpam-2159 189 18 en#1 en#1 PROPN ejpam-2159 189 19 , , PUNCT ejpam-2159 189 20 . . PUNCT ejpam-2159 189 21 . . PUNCT ejpam-2159 189 22 . . PUNCT ejpam-2159 190 1 , , PUNCT ejpam-2159 190 2 e1 e1 NOUN ejpam-2159 190 3 9 9 NUM ejpam-2159 190 4 , , PUNCT ejpam-2159 190 5 ( ( PUNCT ejpam-2159 190 6 40 40 NUM ejpam-2159 190 7 ) ) PUNCT ejpam-2159 190 8 where where SCONJ ejpam-2159 190 9 ei ei NOUN ejpam-2159 190 10 = = SYM ejpam-2159 190 11 8 8 NUM ejpam-2159 190 12 ! ! PUNCT ejpam-2159 190 13 i1,!i2 i1,!i2 PROPN ejpam-2159 190 14 , , PUNCT ejpam-2159 190 15 . . PUNCT ejpam-2159 190 16 . . PUNCT ejpam-2159 191 1 . . PUNCT ejpam-2159 192 1 , , PUNCT ejpam-2159 192 2 ! ! PUNCT ejpam-2159 193 1 in in ADP ejpam-2159 193 2 9 9 NUM ejpam-2159 193 3 t t NOUN ejpam-2159 193 4 , , PUNCT ejpam-2159 193 5 and and CCONJ ejpam-2159 193 6 ! ! PUNCT ejpam-2159 194 1 i i PRON ejpam-2159 194 2 j j PROPN ejpam-2159 194 3 is be AUX ejpam-2159 194 4 the the DET ejpam-2159 194 5 kronecker kronecker NOUN ejpam-2159 194 6 delta delta NOUN ejpam-2159 194 7 . . PUNCT ejpam-2159 195 1 the the DET ejpam-2159 195 2 permutation permutation NOUN ejpam-2159 195 3 matrix matrix NOUN ejpam-2159 195 4 j j NOUN ejpam-2159 195 5 of of ADP ejpam-2159 195 6 order order NOUN ejpam-2159 195 7 n n PRON ejpam-2159 195 8 enjoys enjoy VERB ejpam-2159 195 9 the the DET ejpam-2159 195 10 following follow VERB ejpam-2159 195 11 properties property NOUN ejpam-2159 195 12 : : PUNCT ejpam-2159 195 13 • • NUM ejpam-2159 195 14 j j X ejpam-2159 195 15 = = SYM ejpam-2159 195 16 j j PROPN ejpam-2159 195 17 t t PROPN ejpam-2159 195 18 = = SYM ejpam-2159 195 19 j#1 j#1 PROPN ejpam-2159 195 20 . . NOUN ejpam-2159 196 1 • • NUM ejpam-2159 196 2 jk jk NOUN ejpam-2159 196 3 = = PUNCT ejpam-2159 196 4 $ $ PROPN ejpam-2159 196 5 in in ADP ejpam-2159 196 6 k k PROPN ejpam-2159 196 7 even even ADV ejpam-2159 196 8 j j PROPN ejpam-2159 196 9 k k PROPN ejpam-2159 196 10 odd odd ADJ ejpam-2159 196 11 , , PUNCT ejpam-2159 196 12 where where SCONJ ejpam-2159 196 13 in in ADP ejpam-2159 196 14 is be AUX ejpam-2159 196 15 the the DET ejpam-2159 196 16 identity identity NOUN ejpam-2159 196 17 matrix matrix NOUN ejpam-2159 196 18 of of ADP ejpam-2159 196 19 order order NOUN ejpam-2159 196 20 n. n. NOUN ejpam-2159 196 21 • • ADP ejpam-2159 196 22 det(j det(j PROPN ejpam-2159 196 23 ) ) PUNCT ejpam-2159 196 24 = = PUNCT ejpam-2159 196 25 ( ( PUNCT ejpam-2159 196 26 # # SYM ejpam-2159 196 27 1 1 NUM ejpam-2159 196 28 ) ) PUNCT ejpam-2159 196 29 n(n#1 n(n#1 NOUN ejpam-2159 196 30 ) ) PUNCT ejpam-2159 196 31 2 2 NUM ejpam-2159 196 32 = = SYM ejpam-2159 196 33 $ $ SYM ejpam-2159 196 34 1 1 NUM ejpam-2159 196 35 if if SCONJ ejpam-2159 196 36 n n CCONJ ejpam-2159 196 37 ) ) PUNCT ejpam-2159 196 38 0 0 NUM ejpam-2159 196 39 or or CCONJ ejpam-2159 196 40 1 1 NUM ejpam-2159 196 41 mod(4 mod(4 NOUN ejpam-2159 196 42 ) ) PUNCT ejpam-2159 196 43 # # SYM ejpam-2159 196 44 1 1 NUM ejpam-2159 196 45 if if SCONJ ejpam-2159 196 46 n n CCONJ ejpam-2159 196 47 ) ) PUNCT ejpam-2159 196 48 2 2 NUM ejpam-2159 196 49 or or CCONJ ejpam-2159 196 50 3 3 NUM ejpam-2159 196 51 mod(4 mod(4 NOUN ejpam-2159 196 52 ) ) PUNCT ejpam-2159 196 53 . . PUNCT ejpam-2159 196 54 . . PUNCT ejpam-2159 197 1 lemma lemma PROPN ejpam-2159 197 2 1 1 NUM ejpam-2159 197 3 . . PUNCT ejpam-2159 198 1 for for ADP ejpam-2159 198 2 nonnegative nonnegative ADJ ejpam-2159 198 3 integer integer NOUN ejpam-2159 198 4 numbers number NOUN ejpam-2159 198 5 n n PRON ejpam-2159 198 6 and and CCONJ ejpam-2159 198 7 k k NOUN ejpam-2159 198 8 , , PUNCT ejpam-2159 198 9 we we PRON ejpam-2159 198 10 have have VERB ejpam-2159 198 11 n n NUM ejpam-2159 198 12 # # NOUN ejpam-2159 198 13 r=1 r=1 NOUN ejpam-2159 198 14 ! ! PUNCT ejpam-2159 199 1 r r NOUN ejpam-2159 199 2 k k NOUN ejpam-2159 199 3 " " PUNCT ejpam-2159 199 4 = = PUNCT ejpam-2159 199 5 ! ! PUNCT ejpam-2159 200 1 n+ n+ NUM ejpam-2159 200 2 1 1 NUM ejpam-2159 200 3 k+ k+ NOUN ejpam-2159 200 4 1 1 NUM ejpam-2159 200 5 " " PUNCT ejpam-2159 200 6 # # NOUN ejpam-2159 200 7 ! ! PUNCT ejpam-2159 200 8 k0 k0 PROPN ejpam-2159 200 9 . . PUNCT ejpam-2159 201 1 ( ( PUNCT ejpam-2159 201 2 41 41 NUM ejpam-2159 201 3 ) ) PUNCT ejpam-2159 201 4 proof proof NOUN ejpam-2159 201 5 . . PUNCT ejpam-2159 202 1 by by ADP ejpam-2159 202 2 using use VERB ejpam-2159 202 3 the the DET ejpam-2159 202 4 following follow VERB ejpam-2159 202 5 pascal pascal PROPN ejpam-2159 202 6 ’s ’s PART ejpam-2159 202 7 rule rule NOUN ejpam-2159 202 8 : : PUNCT ejpam-2159 202 9 ! ! PUNCT ejpam-2159 203 1 r r X ejpam-2159 203 2 k k NOUN ejpam-2159 203 3 " " PUNCT ejpam-2159 203 4 = = PUNCT ejpam-2159 203 5 ! ! PUNCT ejpam-2159 204 1 r r NOUN ejpam-2159 204 2 + + NOUN ejpam-2159 204 3 1 1 NUM ejpam-2159 204 4 k+ k+ NOUN ejpam-2159 204 5 1 1 NUM ejpam-2159 204 6 " " PUNCT ejpam-2159 204 7 # # NOUN ejpam-2159 204 8 ! ! PUNCT ejpam-2159 205 1 r r NOUN ejpam-2159 205 2 k+ k+ NOUN ejpam-2159 205 3 1 1 NUM ejpam-2159 205 4 " " PUNCT ejpam-2159 205 5 , , PUNCT ejpam-2159 205 6 we we PRON ejpam-2159 205 7 get get VERB ejpam-2159 205 8 : : PUNCT ejpam-2159 205 9 n n DET ejpam-2159 205 10 # # NOUN ejpam-2159 205 11 r=1 r=1 NOUN ejpam-2159 205 12 ! ! PUNCT ejpam-2159 206 1 r r NOUN ejpam-2159 206 2 k k NOUN ejpam-2159 206 3 " " PUNCT ejpam-2159 206 4 = = NOUN ejpam-2159 206 5 n n CCONJ ejpam-2159 206 6 # # NOUN ejpam-2159 206 7 r=1 r=1 NOUN ejpam-2159 206 8 6 6 NUM ejpam-2159 206 9 ! ! PUNCT ejpam-2159 207 1 r r NOUN ejpam-2159 207 2 + + NOUN ejpam-2159 207 3 1 1 NUM ejpam-2159 207 4 k+ k+ NOUN ejpam-2159 207 5 1 1 NUM ejpam-2159 207 6 " " PUNCT ejpam-2159 207 7 # # NOUN ejpam-2159 207 8 ! ! PUNCT ejpam-2159 208 1 r r NOUN ejpam-2159 208 2 k+ k+ NOUN ejpam-2159 208 3 1 1 NUM ejpam-2159 208 4 " " PUNCT ejpam-2159 208 5 7 7 NUM ejpam-2159 208 6 = = PUNCT ejpam-2159 208 7 ! ! PUNCT ejpam-2159 209 1 n+ n+ INTJ ejpam-2159 209 2 1 1 NUM ejpam-2159 209 3 k+ k+ NOUN ejpam-2159 209 4 1 1 NUM ejpam-2159 209 5 " " PUNCT ejpam-2159 209 6 # # NOUN ejpam-2159 209 7 ! ! PUNCT ejpam-2159 210 1 1 1 NUM ejpam-2159 210 2 k+ k+ NOUN ejpam-2159 210 3 1 1 NUM ejpam-2159 210 4 " " PUNCT ejpam-2159 210 5 = = PUNCT ejpam-2159 210 6 ! ! PUNCT ejpam-2159 211 1 n+ n+ NUM ejpam-2159 211 2 1 1 NUM ejpam-2159 211 3 k+ k+ NOUN ejpam-2159 211 4 1 1 NUM ejpam-2159 211 5 " " PUNCT ejpam-2159 211 6 # # NOUN ejpam-2159 211 7 ! ! PUNCT ejpam-2159 211 8 k0 k0 PROPN ejpam-2159 211 9 , , PUNCT ejpam-2159 211 10 having having AUX ejpam-2159 211 11 used use VERB ejpam-2159 211 12 the the DET ejpam-2159 211 13 telescoping telescope VERB ejpam-2159 211 14 sum sum NOUN ejpam-2159 211 15 . . PUNCT ejpam-2159 212 1 definition definition NOUN ejpam-2159 212 2 8 8 NUM ejpam-2159 212 3 ( ( PUNCT ejpam-2159 212 4 [ [ X ejpam-2159 212 5 20 20 NUM ejpam-2159 212 6 ] ] NUM ejpam-2159 212 7 ) ) PUNCT ejpam-2159 212 8 . . PUNCT ejpam-2159 213 1 a a DET ejpam-2159 213 2 real real ADJ ejpam-2159 213 3 n n CCONJ ejpam-2159 213 4 " " PUNCT ejpam-2159 213 5 n n PRON ejpam-2159 213 6 matrix matrix NOUN ejpam-2159 213 7 a= a= NOUN ejpam-2159 213 8 ( ( PUNCT ejpam-2159 213 9 ai ai VERB ejpam-2159 213 10 j j PROPN ejpam-2159 213 11 ) ) PUNCT ejpam-2159 213 12 n n PROPN ejpam-2159 214 1 i i PRON ejpam-2159 214 2 , , PUNCT ejpam-2159 214 3 j=1 j=1 PROPN ejpam-2159 214 4 is be AUX ejpam-2159 214 5 called call VERB ejpam-2159 214 6 row row NOUN ejpam-2159 214 7 stochastic stochastic ADJ ejpam-2159 214 8 matrix matrix NOUN ejpam-2159 214 9 if if SCONJ ejpam-2159 214 10 ( ( PUNCT ejpam-2159 214 11 i i NOUN ejpam-2159 214 12 ) ) PUNCT ejpam-2159 214 13 ai ai VERB ejpam-2159 214 14 j j PROPN ejpam-2159 214 15 % % NOUN ejpam-2159 214 16 0 0 NUM ejpam-2159 214 17 , , PUNCT ejpam-2159 214 18 for for ADP ejpam-2159 214 19 1 1 NUM ejpam-2159 214 20 & & CCONJ ejpam-2159 214 21 i i PROPN ejpam-2159 214 22 , , PUNCT ejpam-2159 214 23 j j PROPN ejpam-2159 214 24 & & CCONJ ejpam-2159 214 25 n n PROPN ejpam-2159 214 26 ( ( PUNCT ejpam-2159 214 27 ii ii PROPN ejpam-2159 214 28 ) ) PUNCT ejpam-2159 214 29 /n /n PUNCT ejpam-2159 215 1 j=1 j=1 PROPN ejpam-2159 215 2 ai ai VERB ejpam-2159 215 3 j j PROPN ejpam-2159 215 4 = = SYM ejpam-2159 215 5 1 1 NUM ejpam-2159 215 6 , , PUNCT ejpam-2159 215 7 for for ADP ejpam-2159 215 8 1 1 NUM ejpam-2159 215 9 & & CCONJ ejpam-2159 215 10 i i PROPN ejpam-2159 215 11 & & CCONJ ejpam-2159 215 12 n n PROPN ejpam-2159 215 13 m. m. PROPN ejpam-2159 215 14 el el PROPN ejpam-2159 215 15 - - PUNCT ejpam-2159 215 16 mikkawy mikkawy PROPN ejpam-2159 215 17 , , PUNCT ejpam-2159 215 18 f. f. PROPN ejpam-2159 215 19 atlan atlan PROPN ejpam-2159 215 20 / / SYM ejpam-2159 215 21 eur eur PROPN ejpam-2159 215 22 . . PUNCT ejpam-2159 216 1 j. j. PROPN ejpam-2159 216 2 pure pure PROPN ejpam-2159 216 3 appl appl PROPN ejpam-2159 216 4 . . PROPN ejpam-2159 216 5 math math PROPN ejpam-2159 216 6 , , PUNCT ejpam-2159 216 7 8 8 NUM ejpam-2159 216 8 ( ( PUNCT ejpam-2159 216 9 2015 2015 NUM ejpam-2159 216 10 ) ) PUNCT ejpam-2159 216 11 , , PUNCT ejpam-2159 216 12 135 135 NUM ejpam-2159 216 13 - - SYM ejpam-2159 216 14 151 151 NUM ejpam-2159 216 15 142 142 NUM ejpam-2159 216 16 note note NOUN ejpam-2159 216 17 that that SCONJ ejpam-2159 216 18 ( ( PUNCT ejpam-2159 216 19 ii ii NOUN ejpam-2159 216 20 ) ) PUNCT ejpam-2159 216 21 is be AUX ejpam-2159 216 22 equivalent equivalent ADJ ejpam-2159 216 23 to to ADP ejpam-2159 216 24 ae ae PROPN ejpam-2159 216 25 = = SYM ejpam-2159 217 1 e e PROPN ejpam-2159 217 2 , , PUNCT ejpam-2159 217 3 where where SCONJ ejpam-2159 217 4 e e NOUN ejpam-2159 217 5 = = PUNCT ejpam-2159 218 1 [ [ X ejpam-2159 218 2 1,1 1,1 NUM ejpam-2159 218 3 , , PUNCT ejpam-2159 218 4 . . PUNCT ejpam-2159 218 5 . . PUNCT ejpam-2159 219 1 . . PUNCT ejpam-2159 220 1 , , PUNCT ejpam-2159 220 2 1]t 1]t NUM ejpam-2159 220 3 . . PUNCT ejpam-2159 221 1 definition definition NOUN ejpam-2159 221 2 9 9 NUM ejpam-2159 221 3 ( ( PUNCT ejpam-2159 221 4 [ [ X ejpam-2159 221 5 15 15 NUM ejpam-2159 221 6 ] ] NUM ejpam-2159 221 7 ) ) PUNCT ejpam-2159 221 8 . . PUNCT ejpam-2159 222 1 the the DET ejpam-2159 222 2 symmetric symmetric ADJ ejpam-2159 222 3 pascal pascal ADJ ejpam-2159 222 4 matrix matrix NOUN ejpam-2159 222 5 pn pn NOUN ejpam-2159 222 6 = = SYM ejpam-2159 222 7 ( ( PUNCT ejpam-2159 222 8 pi pi PROPN ejpam-2159 222 9 j j PROPN ejpam-2159 222 10 ) ) PUNCT ejpam-2159 222 11 n n PROPN ejpam-2159 222 12 i i PRON ejpam-2159 222 13 , , PUNCT ejpam-2159 222 14 j=1 j=1 NOUN ejpam-2159 222 15 of of ADP ejpam-2159 222 16 order order NOUN ejpam-2159 222 17 n n X ejpam-2159 222 18 is be AUX ejpam-2159 222 19 a a DET ejpam-2159 222 20 matrix matrix NOUN ejpam-2159 222 21 of of ADP ejpam-2159 222 22 integers integer NOUN ejpam-2159 222 23 defined define VERB ejpam-2159 222 24 by by ADP ejpam-2159 222 25 : : PUNCT ejpam-2159 222 26 pi pi PROPN ejpam-2159 222 27 j j PROPN ejpam-2159 223 1 = = PUNCT ejpam-2159 223 2 ! ! PUNCT ejpam-2159 224 1 i i PRON ejpam-2159 225 1 + + NUM ejpam-2159 226 1 j j PROPN ejpam-2159 226 2 # # NOUN ejpam-2159 226 3 2 2 NUM ejpam-2159 226 4 i i NOUN ejpam-2159 226 5 # # NOUN ejpam-2159 226 6 1 1 NUM ejpam-2159 226 7 " " PUNCT ejpam-2159 226 8 , , PUNCT ejpam-2159 226 9 1 1 NUM ejpam-2159 226 10 & & CCONJ ejpam-2159 226 11 i i PRON ejpam-2159 226 12 , , PUNCT ejpam-2159 226 13 j j PROPN ejpam-2159 226 14 & & CCONJ ejpam-2159 226 15 n. n. PROPN ejpam-2159 226 16 ( ( PUNCT ejpam-2159 226 17 42 42 NUM ejpam-2159 226 18 ) ) PUNCT ejpam-2159 226 19 the the DET ejpam-2159 226 20 pascal pascal ADJ ejpam-2159 226 21 matrix matrix NOUN ejpam-2159 226 22 pn pn PROPN ejpam-2159 226 23 enjoys enjoy VERB ejpam-2159 226 24 the the DET ejpam-2159 226 25 following follow VERB ejpam-2159 226 26 properties property NOUN ejpam-2159 226 27 : : PUNCT ejpam-2159 226 28 ( ( PUNCT ejpam-2159 226 29 i i NOUN ejpam-2159 226 30 ) ) PUNCT ejpam-2159 226 31 pn pn PROPN ejpam-2159 226 32 is be AUX ejpam-2159 226 33 totally totally ADV ejpam-2159 226 34 positive positive ADJ ejpam-2159 226 35 ; ; PUNCT ejpam-2159 226 36 ( ( PUNCT ejpam-2159 226 37 ii ii NOUN ejpam-2159 226 38 ) ) PUNCT ejpam-2159 226 39 pn pn PROPN ejpam-2159 226 40 is be AUX ejpam-2159 226 41 positive positive ADJ ejpam-2159 226 42 definite definite ADJ ejpam-2159 226 43 ; ; PUNCT ejpam-2159 226 44 ( ( PUNCT ejpam-2159 226 45 iii iii X ejpam-2159 226 46 ) ) PUNCT ejpam-2159 226 47 the the DET ejpam-2159 226 48 eigenvalues eigenvalue NOUN ejpam-2159 226 49 of of ADP ejpam-2159 226 50 the the DET ejpam-2159 226 51 matrix matrix NOUN ejpam-2159 226 52 pn pn NOUN ejpam-2159 226 53 are be AUX ejpam-2159 226 54 real real ADJ ejpam-2159 226 55 and and CCONJ ejpam-2159 226 56 positive positive ADJ ejpam-2159 226 57 ; ; PUNCT ejpam-2159 226 58 ( ( PUNCT ejpam-2159 226 59 iv iv X ejpam-2159 226 60 ) ) PUNCT ejpam-2159 227 1 if if SCONJ ejpam-2159 227 2 & & CCONJ ejpam-2159 227 3 (= (= PROPN ejpam-2159 227 4 0 0 NUM ejpam-2159 227 5 is be AUX ejpam-2159 227 6 an an DET ejpam-2159 227 7 eigenvalue eigenvalue NOUN ejpam-2159 227 8 of of ADP ejpam-2159 227 9 pn pn PROPN ejpam-2159 227 10 , , PUNCT ejpam-2159 227 11 then then ADV ejpam-2159 227 12 1 1 NUM ejpam-2159 227 13 & & CCONJ ejpam-2159 227 14 is be AUX ejpam-2159 227 15 also also ADV ejpam-2159 227 16 an an DET ejpam-2159 227 17 eigenvalue eigenvalue NOUN ejpam-2159 227 18 of of ADP ejpam-2159 227 19 pn pn PROPN ejpam-2159 227 20 ; ; PUNCT ejpam-2159 227 21 ( ( PUNCT ejpam-2159 227 22 v v NOUN ejpam-2159 227 23 ) ) PUNCT ejpam-2159 227 24 det(pn det(pn NOUN ejpam-2159 227 25 ) ) PUNCT ejpam-2159 227 26 = = SYM ejpam-2159 227 27 1 1 NUM ejpam-2159 227 28 ; ; PUNCT ejpam-2159 227 29 ( ( PUNCT ejpam-2159 227 30 vi vi X ejpam-2159 227 31 ) ) PUNCT ejpam-2159 227 32 the the DET ejpam-2159 227 33 cholesky cholesky NOUN ejpam-2159 227 34 ’s ’s PART ejpam-2159 227 35 factorization factorization NOUN ejpam-2159 228 1 [ [ X ejpam-2159 228 2 2 2 X ejpam-2159 228 3 ] ] PUNCT ejpam-2159 228 4 of of ADP ejpam-2159 228 5 the the DET ejpam-2159 228 6 matrix matrix NOUN ejpam-2159 228 7 , , PUNCT ejpam-2159 228 8 pn pn PROPN ejpam-2159 228 9 is be AUX ejpam-2159 228 10 given give VERB ejpam-2159 228 11 by by ADP ejpam-2159 228 12 : : PUNCT ejpam-2159 228 13 pn pn NOUN ejpam-2159 228 14 = = PUNCT ejpam-2159 228 15 aat aat X ejpam-2159 228 16 , , PUNCT ejpam-2159 228 17 where where SCONJ ejpam-2159 228 18 a a PRON ejpam-2159 228 19 is be AUX ejpam-2159 228 20 the the DET ejpam-2159 228 21 pascal pascal ADJ ejpam-2159 228 22 matrix matrix NOUN ejpam-2159 228 23 defined define VERB ejpam-2159 228 24 by by ADP ejpam-2159 228 25 : : PUNCT ejpam-2159 228 26 a= a= PROPN ejpam-2159 228 27 6 6 NUM ejpam-2159 228 28 ! ! PUNCT ejpam-2159 229 1 i i PRON ejpam-2159 229 2 # # NOUN ejpam-2159 229 3 1 1 NUM ejpam-2159 229 4 j j NOUN ejpam-2159 229 5 # # NOUN ejpam-2159 229 6 1 1 NUM ejpam-2159 229 7 " " PUNCT ejpam-2159 229 8 7n 7n NOUN ejpam-2159 230 1 i i PRON ejpam-2159 230 2 , , PUNCT ejpam-2159 230 3 j=1 j=1 PROPN ejpam-2159 230 4 ; ; PUNCT ejpam-2159 230 5 ( ( PUNCT ejpam-2159 230 6 43 43 NUM ejpam-2159 230 7 ) ) PUNCT ejpam-2159 230 8 ( ( PUNCT ejpam-2159 230 9 vii vii PROPN ejpam-2159 230 10 ) ) PUNCT ejpam-2159 230 11 the the DET ejpam-2159 230 12 explicit explicit ADJ ejpam-2159 230 13 form form NOUN ejpam-2159 230 14 of of ADP ejpam-2159 230 15 the the DET ejpam-2159 230 16 inverse inverse NOUN ejpam-2159 230 17 matrix matrix NOUN ejpam-2159 230 18 , , PUNCT ejpam-2159 230 19 p#1 p#1 CCONJ ejpam-2159 230 20 n n NOUN ejpam-2159 230 21 = = NOUN ejpam-2159 230 22 qn qn NOUN ejpam-2159 230 23 = = X ejpam-2159 230 24 ( ( PUNCT ejpam-2159 230 25 ' ' PUNCT ejpam-2159 230 26 i i PRON ejpam-2159 230 27 j j NOUN ejpam-2159 230 28 ) ) PUNCT ejpam-2159 231 1 n n PROPN ejpam-2159 231 2 i i PRON ejpam-2159 231 3 , , PUNCT ejpam-2159 231 4 j=1 j=1 PROPN ejpam-2159 231 5 is be AUX ejpam-2159 231 6 given give VERB ejpam-2159 231 7 by by ADP ejpam-2159 231 8 [ [ X ejpam-2159 231 9 15 15 NUM ejpam-2159 231 10 ] ] X ejpam-2159 231 11 : : PUNCT ejpam-2159 231 12 ' ' PUNCT ejpam-2159 231 13 i i PRON ejpam-2159 231 14 j j NOUN ejpam-2159 232 1 = = PUNCT ejpam-2159 232 2 ( ( PUNCT ejpam-2159 232 3 # # SYM ejpam-2159 232 4 1)i+ 1)i+ NUM ejpam-2159 232 5 j j NOUN ejpam-2159 232 6 n n PRON ejpam-2159 232 7 # # NOUN ejpam-2159 232 8 k k PROPN ejpam-2159 232 9 = = PROPN ejpam-2159 232 10 max(i max(i PROPN ejpam-2159 232 11 , , PUNCT ejpam-2159 232 12 j j NOUN ejpam-2159 232 13 ) ) PUNCT ejpam-2159 232 14 ! ! PUNCT ejpam-2159 233 1 k k X ejpam-2159 233 2 # # NOUN ejpam-2159 233 3 1 1 NUM ejpam-2159 233 4 i i NOUN ejpam-2159 233 5 # # NOUN ejpam-2159 233 6 1 1 NUM ejpam-2159 233 7 " " PUNCT ejpam-2159 233 8 ! ! PUNCT ejpam-2159 234 1 k k X ejpam-2159 234 2 # # NOUN ejpam-2159 234 3 1 1 NUM ejpam-2159 234 4 j j NOUN ejpam-2159 234 5 # # NOUN ejpam-2159 234 6 1 1 NUM ejpam-2159 234 7 " " PUNCT ejpam-2159 234 8 ; ; PUNCT ejpam-2159 234 9 ( ( PUNCT ejpam-2159 234 10 44 44 NUM ejpam-2159 234 11 ) ) PUNCT ejpam-2159 234 12 ( ( PUNCT ejpam-2159 234 13 viii viii NOUN ejpam-2159 234 14 ) ) PUNCT ejpam-2159 234 15 the the DET ejpam-2159 234 16 matrix matrix NOUN ejpam-2159 234 17 , , PUNCT ejpam-2159 234 18 pn pn PROPN ejpam-2159 234 19 satisfies satisfie NOUN ejpam-2159 234 20 : : PUNCT ejpam-2159 234 21 pn pn PROPN ejpam-2159 234 22 = = PUNCT ejpam-2159 234 23 ab#1snv ab#1snv PROPN ejpam-2159 234 24 = = SYM ejpam-2159 234 25 t t PROPN ejpam-2159 234 26 v v NOUN ejpam-2159 234 27 , , PUNCT ejpam-2159 234 28 ( ( PUNCT ejpam-2159 234 29 45 45 NUM ejpam-2159 234 30 ) ) PUNCT ejpam-2159 235 1 where where SCONJ ejpam-2159 235 2 t t NOUN ejpam-2159 235 3 = = SYM ejpam-2159 235 4 ( ( PUNCT ejpam-2159 235 5 ti ti X ejpam-2159 235 6 j j PROPN ejpam-2159 235 7 ) ) PUNCT ejpam-2159 235 8 n n PROPN ejpam-2159 236 1 i i PRON ejpam-2159 236 2 , , PUNCT ejpam-2159 236 3 j=1 j=1 PROPN ejpam-2159 236 4 is be AUX ejpam-2159 236 5 an an DET ejpam-2159 236 6 n n CCONJ ejpam-2159 236 7 " " PUNCT ejpam-2159 236 8 n n CCONJ ejpam-2159 236 9 lower low ADJ ejpam-2159 236 10 triangular triangular NOUN ejpam-2159 236 11 stochastic stochastic ADJ ejpam-2159 236 12 matrix matrix NOUN ejpam-2159 236 13 given give VERB ejpam-2159 236 14 by by ADP ejpam-2159 236 15 : : PUNCT ejpam-2159 236 16 ti ti X ejpam-2159 236 17 j j PROPN ejpam-2159 236 18 = = SYM ejpam-2159 236 19 1 1 NUM ejpam-2159 236 20 ( ( PUNCT ejpam-2159 236 21 i i NOUN ejpam-2159 236 22 # # NOUN ejpam-2159 236 23 1 1 NUM ejpam-2159 236 24 ) ) PUNCT ejpam-2159 236 25 ! ! PUNCT ejpam-2159 237 1 c(i c(i NOUN ejpam-2159 237 2 # # NOUN ejpam-2159 237 3 1 1 NUM ejpam-2159 237 4 , , PUNCT ejpam-2159 237 5 j j PROPN ejpam-2159 237 6 # # NOUN ejpam-2159 237 7 1 1 NUM ejpam-2159 237 8 ) ) PUNCT ejpam-2159 237 9 , , PUNCT ejpam-2159 237 10 1 1 NUM ejpam-2159 237 11 & & CCONJ ejpam-2159 237 12 i i PRON ejpam-2159 237 13 , , PUNCT ejpam-2159 237 14 j j PROPN ejpam-2159 237 15 & & CCONJ ejpam-2159 237 16 n n CCONJ ejpam-2159 237 17 , , PUNCT ejpam-2159 237 18 i i PRON ejpam-2159 237 19 % % VERB ejpam-2159 237 20 j j PROPN ejpam-2159 237 21 , , PUNCT ejpam-2159 237 22 ( ( PUNCT ejpam-2159 237 23 46 46 NUM ejpam-2159 237 24 ) ) PUNCT ejpam-2159 237 25 v v NOUN ejpam-2159 237 26 = = PUNCT ejpam-2159 237 27 : : PUNCT ejpam-2159 238 1 ji#1 ji#1 PROPN ejpam-2159 238 2 ; ; PUNCT ejpam-2159 238 3 n n CCONJ ejpam-2159 238 4 i i PRON ejpam-2159 238 5 , , PUNCT ejpam-2159 238 6 j=1 j=1 PROPN ejpam-2159 238 7 , , PUNCT ejpam-2159 238 8 a a PRON ejpam-2159 238 9 is be AUX ejpam-2159 238 10 given give VERB ejpam-2159 238 11 in in ADP ejpam-2159 238 12 ( ( PUNCT ejpam-2159 238 13 43 43 NUM ejpam-2159 238 14 ) ) PUNCT ejpam-2159 238 15 and and CCONJ ejpam-2159 238 16 b b X ejpam-2159 238 17 = = SYM ejpam-2159 238 18 diag(0 diag(0 PROPN ejpam-2159 238 19 ! ! PUNCT ejpam-2159 238 20 , , PUNCT ejpam-2159 238 21 1 1 X ejpam-2159 238 22 ! ! NUM ejpam-2159 238 23 , , PUNCT ejpam-2159 239 1 2 2 NUM ejpam-2159 239 2 ! ! NUM ejpam-2159 239 3 , , PUNCT ejpam-2159 239 4 . . PUNCT ejpam-2159 239 5 . . PUNCT ejpam-2159 239 6 . . PUNCT ejpam-2159 240 1 , , PUNCT ejpam-2159 240 2 ( ( PUNCT ejpam-2159 240 3 n n CCONJ ejpam-2159 240 4 # # NOUN ejpam-2159 240 5 1 1 NUM ejpam-2159 240 6 ) ) PUNCT ejpam-2159 240 7 ! ! PUNCT ejpam-2159 240 8 ) ) PUNCT ejpam-2159 241 1 ( ( PUNCT ejpam-2159 241 2 for for ADP ejpam-2159 241 3 more more ADJ ejpam-2159 241 4 details detail NOUN ejpam-2159 241 5 , , PUNCT ejpam-2159 241 6 see see VERB ejpam-2159 241 7 [ [ X ejpam-2159 241 8 13 13 NUM ejpam-2159 241 9 , , PUNCT ejpam-2159 241 10 14 14 NUM ejpam-2159 241 11 , , PUNCT ejpam-2159 241 12 27 27 NUM ejpam-2159 241 13 , , PUNCT ejpam-2159 241 14 28 28 NUM ejpam-2159 241 15 ] ] PUNCT ejpam-2159 241 16 ) ) PUNCT ejpam-2159 241 17 ; ; PUNCT ejpam-2159 241 18 ( ( PUNCT ejpam-2159 241 19 ix ix X ejpam-2159 241 20 ) ) PUNCT ejpam-2159 241 21 the the DET ejpam-2159 241 22 entries entry NOUN ejpam-2159 241 23 of of ADP ejpam-2159 241 24 the the DET ejpam-2159 241 25 matrix matrix NOUN ejpam-2159 241 26 , , PUNCT ejpam-2159 241 27 pn pn PROPN ejpam-2159 241 28 satisfy satisfy VERB ejpam-2159 241 29 the the DET ejpam-2159 241 30 recurrence recurrence NOUN ejpam-2159 241 31 relation relation NOUN ejpam-2159 241 32 : : PUNCT ejpam-2159 242 1 pi1 pi1 PROPN ejpam-2159 242 2 = = PROPN ejpam-2159 242 3 p1 p1 PROPN ejpam-2159 242 4 j j PROPN ejpam-2159 243 1 = = SYM ejpam-2159 244 1 1 1 PROPN ejpam-2159 244 2 , , PUNCT ejpam-2159 244 3 ( ( PUNCT ejpam-2159 244 4 47 47 NUM ejpam-2159 244 5 ) ) PUNCT ejpam-2159 244 6 and and CCONJ ejpam-2159 244 7 pi pi NOUN ejpam-2159 244 8 j j PROPN ejpam-2159 245 1 = = SYM ejpam-2159 245 2 pi pi PROPN ejpam-2159 245 3 , , PUNCT ejpam-2159 245 4 j#1 j#1 X ejpam-2159 245 5 + + CCONJ ejpam-2159 245 6 pi#1 pi#1 NOUN ejpam-2159 245 7 , , PUNCT ejpam-2159 245 8 j j PROPN ejpam-2159 245 9 . . PUNCT ejpam-2159 246 1 ( ( PUNCT ejpam-2159 246 2 48 48 NUM ejpam-2159 246 3 ) ) PUNCT ejpam-2159 246 4 ( ( PUNCT ejpam-2159 246 5 x x X ejpam-2159 246 6 ) ) PUNCT ejpam-2159 246 7 let let VERB ejpam-2159 246 8 rn rn PART ejpam-2159 246 9 be be AUX ejpam-2159 246 10 the the DET ejpam-2159 246 11 matrix matrix NOUN ejpam-2159 246 12 obtained obtain VERB ejpam-2159 246 13 from from ADP ejpam-2159 246 14 the the DET ejpam-2159 246 15 pascal pascal ADJ ejpam-2159 246 16 matrix matrix NOUN ejpam-2159 246 17 pn pn NOUN ejpam-2159 246 18 by by ADP ejpam-2159 246 19 subtracting subtract VERB ejpam-2159 246 20 one one NUM ejpam-2159 246 21 from from ADP ejpam-2159 246 22 the the DET ejpam-2159 246 23 element element NOUN ejpam-2159 246 24 in in ADP ejpam-2159 246 25 position position NOUN ejpam-2159 246 26 ( ( PUNCT ejpam-2159 246 27 n n CCONJ ejpam-2159 246 28 , , PUNCT ejpam-2159 246 29 n n CCONJ ejpam-2159 246 30 ) ) PUNCT ejpam-2159 246 31 of of ADP ejpam-2159 246 32 pn pn PROPN ejpam-2159 246 33 , , PUNCT ejpam-2159 246 34 then then ADV ejpam-2159 246 35 det(rn det(rn X ejpam-2159 246 36 ) ) PUNCT ejpam-2159 247 1 = = SYM ejpam-2159 247 2 0 0 X ejpam-2159 247 3 . . PUNCT ejpam-2159 247 4 m. m. PROPN ejpam-2159 248 1 el el PROPN ejpam-2159 248 2 - - PUNCT ejpam-2159 248 3 mikkawy mikkawy PROPN ejpam-2159 248 4 , , PUNCT ejpam-2159 248 5 f. f. PROPN ejpam-2159 248 6 atlan atlan PROPN ejpam-2159 248 7 / / SYM ejpam-2159 248 8 eur eur PROPN ejpam-2159 248 9 . . PUNCT ejpam-2159 249 1 j. j. PROPN ejpam-2159 249 2 pure pure PROPN ejpam-2159 249 3 appl appl PROPN ejpam-2159 249 4 . . PROPN ejpam-2159 249 5 math math PROPN ejpam-2159 249 6 , , PUNCT ejpam-2159 249 7 8 8 NUM ejpam-2159 249 8 ( ( PUNCT ejpam-2159 249 9 2015 2015 NUM ejpam-2159 249 10 ) ) PUNCT ejpam-2159 249 11 , , PUNCT ejpam-2159 249 12 135 135 NUM ejpam-2159 249 13 - - SYM ejpam-2159 249 14 151 151 NUM ejpam-2159 249 15 143 143 NUM ejpam-2159 249 16 2 2 NUM ejpam-2159 249 17 . . PUNCT ejpam-2159 249 18 main main ADJ ejpam-2159 249 19 results result NOUN ejpam-2159 249 20 this this DET ejpam-2159 249 21 section section NOUN ejpam-2159 249 22 is be AUX ejpam-2159 249 23 mainly mainly ADV ejpam-2159 249 24 devoted devoted ADJ ejpam-2159 249 25 to to PART ejpam-2159 249 26 study study VERB ejpam-2159 249 27 the the DET ejpam-2159 249 28 n n CCONJ ejpam-2159 249 29 " " PUNCT ejpam-2159 249 30 n n CCONJ ejpam-2159 249 31 inverse inverse NOUN ejpam-2159 249 32 matrix matrix NOUN ejpam-2159 249 33 qn qn NOUN ejpam-2159 249 34 , , PUNCT ejpam-2159 249 35 in in ADP ejpam-2159 249 36 ( ( PUNCT ejpam-2159 249 37 44 44 NUM ejpam-2159 249 38 ) ) PUNCT ejpam-2159 249 39 , , PUNCT ejpam-2159 249 40 of of ADP ejpam-2159 249 41 the the DET ejpam-2159 249 42 n n CCONJ ejpam-2159 249 43 " " PUNCT ejpam-2159 249 44 n n CCONJ ejpam-2159 249 45 symmetric symmetric ADJ ejpam-2159 249 46 pascal pascal ADJ ejpam-2159 249 47 matrix matrix NOUN ejpam-2159 249 48 pn pn PROPN ejpam-2159 249 49 in in ADP ejpam-2159 249 50 ( ( PUNCT ejpam-2159 249 51 42 42 NUM ejpam-2159 249 52 ) ) PUNCT ejpam-2159 249 53 . . PUNCT ejpam-2159 250 1 the the DET ejpam-2159 250 2 main main ADJ ejpam-2159 250 3 object object NOUN ejpam-2159 250 4 is be AUX ejpam-2159 250 5 to to PART ejpam-2159 250 6 find find VERB ejpam-2159 250 7 a a DET ejpam-2159 250 8 recurrence recurrence NOUN ejpam-2159 250 9 relation relation NOUN ejpam-2159 250 10 satisfied satisfy VERB ejpam-2159 250 11 by by ADP ejpam-2159 250 12 the the DET ejpam-2159 250 13 entries entry NOUN ejpam-2159 250 14 of of ADP ejpam-2159 250 15 qn qn NOUN ejpam-2159 250 16 . . PUNCT ejpam-2159 250 17 setting set VERB ejpam-2159 250 18 j j PROPN ejpam-2159 250 19 = = SYM ejpam-2159 250 20 1 1 NUM ejpam-2159 250 21 in in ADP ejpam-2159 250 22 ( ( PUNCT ejpam-2159 250 23 44 44 NUM ejpam-2159 250 24 ) ) PUNCT ejpam-2159 250 25 , , PUNCT ejpam-2159 250 26 we we PRON ejpam-2159 250 27 obtain obtain VERB ejpam-2159 250 28 : : PUNCT ejpam-2159 250 29 ' ' PUNCT ejpam-2159 250 30 i1 i1 PROPN ejpam-2159 250 31 = = PUNCT ejpam-2159 250 32 ( ( PUNCT ejpam-2159 250 33 # # SYM ejpam-2159 250 34 1)i+1 1)i+1 NUM ejpam-2159 250 35 ! ! PUNCT ejpam-2159 251 1 n n CCONJ ejpam-2159 251 2 i i PRON ejpam-2159 251 3 " " PUNCT ejpam-2159 251 4 = = PUNCT ejpam-2159 251 5 ' ' VERB ejpam-2159 251 6 1i 1i NOUN ejpam-2159 251 7 , , PUNCT ejpam-2159 251 8 1 1 NUM ejpam-2159 251 9 & & CCONJ ejpam-2159 251 10 i i PROPN ejpam-2159 251 11 & & CCONJ ejpam-2159 251 12 n n CCONJ ejpam-2159 251 13 , , PUNCT ejpam-2159 251 14 ( ( PUNCT ejpam-2159 251 15 49 49 NUM ejpam-2159 251 16 ) ) PUNCT ejpam-2159 251 17 having having AUX ejpam-2159 251 18 used use VERB ejpam-2159 251 19 lemma lemma PROPN ejpam-2159 251 20 1 1 NUM ejpam-2159 251 21 . . PUNCT ejpam-2159 251 22 putting put VERB ejpam-2159 251 23 j j PROPN ejpam-2159 251 24 = = PUNCT ejpam-2159 252 1 n n PROPN ejpam-2159 252 2 in in ADP ejpam-2159 252 3 ( ( PUNCT ejpam-2159 252 4 44 44 NUM ejpam-2159 252 5 ) ) PUNCT ejpam-2159 252 6 , , PUNCT ejpam-2159 252 7 yields yield VERB ejpam-2159 252 8 : : PUNCT ejpam-2159 252 9 ' ' PUNCT ejpam-2159 252 10 in in ADP ejpam-2159 252 11 = = PUNCT ejpam-2159 252 12 ( ( PUNCT ejpam-2159 252 13 # # SYM ejpam-2159 252 14 1)i+n 1)i+n NUM ejpam-2159 252 15 ! ! PUNCT ejpam-2159 253 1 n n CCONJ ejpam-2159 253 2 # # SYM ejpam-2159 253 3 1 1 NUM ejpam-2159 253 4 i i NOUN ejpam-2159 253 5 # # NOUN ejpam-2159 253 6 1 1 NUM ejpam-2159 253 7 " " PUNCT ejpam-2159 253 8 = = NOUN ejpam-2159 253 9 ' ' NOUN ejpam-2159 253 10 ni ni PROPN ejpam-2159 253 11 , , PUNCT ejpam-2159 253 12 1 1 NUM ejpam-2159 253 13 & & CCONJ ejpam-2159 253 14 i i PROPN ejpam-2159 253 15 & & CCONJ ejpam-2159 253 16 n. n. PROPN ejpam-2159 253 17 ( ( PUNCT ejpam-2159 253 18 50 50 NUM ejpam-2159 253 19 ) ) PUNCT ejpam-2159 253 20 at at ADP ejpam-2159 253 21 this this DET ejpam-2159 253 22 point point NOUN ejpam-2159 253 23 we we PRON ejpam-2159 253 24 may may AUX ejpam-2159 253 25 formulate formulate VERB ejpam-2159 253 26 the the DET ejpam-2159 253 27 following follow VERB ejpam-2159 253 28 result result NOUN ejpam-2159 253 29 . . PUNCT ejpam-2159 254 1 theorem theorem NOUN ejpam-2159 254 2 1 1 NUM ejpam-2159 254 3 . . PUNCT ejpam-2159 255 1 the the DET ejpam-2159 255 2 entries entry NOUN ejpam-2159 255 3 of of ADP ejpam-2159 255 4 the the DET ejpam-2159 255 5 inverse inverse NOUN ejpam-2159 255 6 matrix matrix NOUN ejpam-2159 255 7 qn qn NOUN ejpam-2159 255 8 = = NOUN ejpam-2159 255 9 p#1 p#1 CCONJ ejpam-2159 255 10 n n NOUN ejpam-2159 255 11 = = PUNCT ejpam-2159 255 12 ( ( PUNCT ejpam-2159 255 13 ' ' PUNCT ejpam-2159 255 14 i i PRON ejpam-2159 255 15 j j NOUN ejpam-2159 255 16 ) ) PUNCT ejpam-2159 255 17 n n PROPN ejpam-2159 255 18 i i PRON ejpam-2159 255 19 , , PUNCT ejpam-2159 255 20 j=1 j=1 PROPN ejpam-2159 255 21 , , PUNCT ejpam-2159 255 22 in in ADP ejpam-2159 255 23 ( ( PUNCT ejpam-2159 255 24 44 44 NUM ejpam-2159 255 25 ) ) PUNCT ejpam-2159 255 26 , , PUNCT ejpam-2159 255 27 satisfy satisfy VERB ejpam-2159 255 28 : : PUNCT ejpam-2159 255 29 ' ' PUNCT ejpam-2159 255 30 i i PRON ejpam-2159 255 31 , , PUNCT ejpam-2159 255 32 j j PROPN ejpam-2159 256 1 = = PUNCT ejpam-2159 257 1 ' ' VERB ejpam-2159 257 2 i i NOUN ejpam-2159 257 3 , , PUNCT ejpam-2159 257 4 j+1 j+1 PROPN ejpam-2159 258 1 + + CCONJ ejpam-2159 258 2 ' ' PUNCT ejpam-2159 258 3 i+1 i+1 ADP ejpam-2159 258 4 , , PUNCT ejpam-2159 258 5 j j PROPN ejpam-2159 258 6 + + CCONJ ejpam-2159 258 7 ( ( PUNCT ejpam-2159 258 8 # # SYM ejpam-2159 258 9 1)i+ 1)i+ NUM ejpam-2159 258 10 j j NOUN ejpam-2159 258 11 ! ! PUNCT ejpam-2159 259 1 n n CCONJ ejpam-2159 259 2 i i PRON ejpam-2159 259 3 " " PUNCT ejpam-2159 259 4 ! ! PUNCT ejpam-2159 260 1 n n CCONJ ejpam-2159 260 2 j j PROPN ejpam-2159 260 3 " " PUNCT ejpam-2159 260 4 , , PUNCT ejpam-2159 260 5 1 1 NUM ejpam-2159 260 6 & & CCONJ ejpam-2159 260 7 i i PRON ejpam-2159 260 8 , , PUNCT ejpam-2159 260 9 j j PROPN ejpam-2159 260 10 & & CCONJ ejpam-2159 260 11 n. n. PROPN ejpam-2159 260 12 ( ( PUNCT ejpam-2159 260 13 51 51 NUM ejpam-2159 260 14 ) ) PUNCT ejpam-2159 260 15 proof proof NOUN ejpam-2159 260 16 . . PUNCT ejpam-2159 261 1 it it PRON ejpam-2159 261 2 is be AUX ejpam-2159 261 3 well well ADV ejpam-2159 261 4 - - PUNCT ejpam-2159 261 5 known know VERB ejpam-2159 261 6 that that PRON ejpam-2159 261 7 ! ! PUNCT ejpam-2159 262 1 m m VERB ejpam-2159 262 2 n n PRON ejpam-2159 262 3 " " PUNCT ejpam-2159 262 4 = = PUNCT ejpam-2159 262 5 ! ! PUNCT ejpam-2159 263 1 m m VERB ejpam-2159 263 2 # # NOUN ejpam-2159 263 3 1 1 NUM ejpam-2159 263 4 n n CCONJ ejpam-2159 263 5 # # SYM ejpam-2159 263 6 1 1 NUM ejpam-2159 263 7 " " PUNCT ejpam-2159 263 8 + + CCONJ ejpam-2159 263 9 ! ! PUNCT ejpam-2159 264 1 m m VERB ejpam-2159 264 2 # # NOUN ejpam-2159 264 3 1 1 NUM ejpam-2159 264 4 n n NOUN ejpam-2159 264 5 " " PUNCT ejpam-2159 264 6 . . PUNCT ejpam-2159 265 1 ( ( PUNCT ejpam-2159 265 2 52 52 NUM ejpam-2159 265 3 ) ) PUNCT ejpam-2159 265 4 from from ADP ejpam-2159 265 5 ( ( PUNCT ejpam-2159 265 6 44 44 NUM ejpam-2159 265 7 ) ) PUNCT ejpam-2159 265 8 , , PUNCT ejpam-2159 265 9 we we PRON ejpam-2159 265 10 have have VERB ejpam-2159 265 11 ' ' PUNCT ejpam-2159 265 12 i i PRON ejpam-2159 265 13 , , PUNCT ejpam-2159 265 14 j j PROPN ejpam-2159 266 1 = = PRON ejpam-2159 266 2 ( ( PUNCT ejpam-2159 266 3 # # SYM ejpam-2159 266 4 1)i+ 1)i+ NUM ejpam-2159 266 5 j j NOUN ejpam-2159 266 6 n n PRON ejpam-2159 266 7 # # NOUN ejpam-2159 266 8 k=1 k=1 X ejpam-2159 266 9 ! ! PUNCT ejpam-2159 267 1 k k X ejpam-2159 267 2 # # NOUN ejpam-2159 267 3 1 1 NUM ejpam-2159 267 4 i i NOUN ejpam-2159 267 5 # # NOUN ejpam-2159 267 6 1 1 NUM ejpam-2159 267 7 " " PUNCT ejpam-2159 267 8 ! ! PUNCT ejpam-2159 268 1 k k X ejpam-2159 268 2 # # NOUN ejpam-2159 268 3 1 1 NUM ejpam-2159 268 4 j j NOUN ejpam-2159 268 5 # # NOUN ejpam-2159 268 6 1 1 NUM ejpam-2159 268 7 " " PUNCT ejpam-2159 268 8 . . PUNCT ejpam-2159 269 1 ( ( PUNCT ejpam-2159 269 2 53 53 NUM ejpam-2159 269 3 ) ) PUNCT ejpam-2159 269 4 by by ADP ejpam-2159 269 5 using use VERB ejpam-2159 269 6 the the DET ejpam-2159 269 7 identity identity NOUN ejpam-2159 269 8 ( ( PUNCT ejpam-2159 269 9 52 52 NUM ejpam-2159 269 10 ) ) PUNCT ejpam-2159 269 11 , , PUNCT ejpam-2159 269 12 we we PRON ejpam-2159 269 13 get get VERB ejpam-2159 269 14 ' ' PUNCT ejpam-2159 269 15 i i PRON ejpam-2159 269 16 , , PUNCT ejpam-2159 270 1 j j PROPN ejpam-2159 270 2 =( =( PROPN ejpam-2159 270 3 # # SYM ejpam-2159 270 4 1)i+ 1)i+ NUM ejpam-2159 270 5 j j PROPN ejpam-2159 270 6 6 6 NUM ejpam-2159 270 7 n n NOUN ejpam-2159 270 8 # # NOUN ejpam-2159 270 9 k=1 k=1 NOUN ejpam-2159 271 1 < < X ejpam-2159 271 2 ! ! PUNCT ejpam-2159 272 1 k k PROPN ejpam-2159 272 2 i i PRON ejpam-2159 272 3 " " PUNCT ejpam-2159 272 4 # # NOUN ejpam-2159 272 5 ! ! PUNCT ejpam-2159 273 1 k k X ejpam-2159 273 2 # # NOUN ejpam-2159 273 3 1 1 NUM ejpam-2159 273 4 i i PRON ejpam-2159 273 5 " " PUNCT ejpam-2159 273 6 = = NOUN ejpam-2159 273 7 < < X ejpam-2159 273 8 ! ! PUNCT ejpam-2159 274 1 k k PROPN ejpam-2159 274 2 j j PROPN ejpam-2159 274 3 " " PUNCT ejpam-2159 274 4 # # NOUN ejpam-2159 274 5 ! ! PUNCT ejpam-2159 275 1 k k X ejpam-2159 275 2 # # NOUN ejpam-2159 275 3 1 1 NUM ejpam-2159 275 4 j j NOUN ejpam-2159 275 5 " " PUNCT ejpam-2159 275 6 = = SYM ejpam-2159 275 7 7 7 NUM ejpam-2159 275 8 =( =( NOUN ejpam-2159 275 9 # # SYM ejpam-2159 275 10 1)i+ 1)i+ NUM ejpam-2159 275 11 j j PROPN ejpam-2159 275 12 6 6 NUM ejpam-2159 275 13 n n NOUN ejpam-2159 275 14 # # NOUN ejpam-2159 275 15 k=1 k=1 NOUN ejpam-2159 275 16 ! ! PUNCT ejpam-2159 276 1 k k INTJ ejpam-2159 277 1 i i PRON ejpam-2159 277 2 " " PUNCT ejpam-2159 277 3 ! ! PUNCT ejpam-2159 278 1 k k PROPN ejpam-2159 278 2 j j PROPN ejpam-2159 278 3 " " PUNCT ejpam-2159 278 4 # # NOUN ejpam-2159 278 5 n n PRON ejpam-2159 278 6 # # NOUN ejpam-2159 278 7 k=1 k=1 NOUN ejpam-2159 278 8 ! ! PUNCT ejpam-2159 279 1 k k INTJ ejpam-2159 280 1 i i PRON ejpam-2159 280 2 " " PUNCT ejpam-2159 280 3 ! ! PUNCT ejpam-2159 281 1 k k X ejpam-2159 281 2 # # NOUN ejpam-2159 281 3 1 1 NUM ejpam-2159 281 4 j j NOUN ejpam-2159 281 5 " " PUNCT ejpam-2159 281 6 # # NOUN ejpam-2159 281 7 n n PRON ejpam-2159 281 8 # # NOUN ejpam-2159 281 9 k=1 k=1 NOUN ejpam-2159 281 10 ! ! PUNCT ejpam-2159 282 1 k k X ejpam-2159 282 2 # # NOUN ejpam-2159 282 3 1 1 NUM ejpam-2159 282 4 i i PRON ejpam-2159 282 5 " " PUNCT ejpam-2159 282 6 ! ! PUNCT ejpam-2159 283 1 k k PROPN ejpam-2159 283 2 j j PROPN ejpam-2159 283 3 " " PUNCT ejpam-2159 284 1 + + CCONJ ejpam-2159 284 2 n n PRON ejpam-2159 284 3 # # NOUN ejpam-2159 284 4 k=1 k=1 NOUN ejpam-2159 284 5 ! ! PUNCT ejpam-2159 285 1 k k X ejpam-2159 285 2 # # NOUN ejpam-2159 285 3 1 1 NUM ejpam-2159 285 4 i i PRON ejpam-2159 285 5 " " PUNCT ejpam-2159 285 6 ! ! PUNCT ejpam-2159 286 1 k k X ejpam-2159 286 2 # # NOUN ejpam-2159 286 3 1 1 NUM ejpam-2159 286 4 j j NOUN ejpam-2159 286 5 " " PUNCT ejpam-2159 286 6 7 7 NUM ejpam-2159 286 7 =( =( NOUN ejpam-2159 286 8 # # SYM ejpam-2159 286 9 1)i+ 1)i+ NUM ejpam-2159 286 10 j j PROPN ejpam-2159 286 11 6 6 NUM ejpam-2159 286 12 ! ! PUNCT ejpam-2159 287 1 n n PRON ejpam-2159 287 2 i i PRON ejpam-2159 287 3 " " PUNCT ejpam-2159 287 4 ! ! PUNCT ejpam-2159 288 1 n n CCONJ ejpam-2159 288 2 j j PROPN ejpam-2159 288 3 " " PUNCT ejpam-2159 289 1 + + CCONJ ejpam-2159 289 2 n#1 n#1 ADV ejpam-2159 289 3 # # NOUN ejpam-2159 289 4 k=1 k=1 X ejpam-2159 289 5 ! ! PUNCT ejpam-2159 290 1 k k INTJ ejpam-2159 291 1 i i PRON ejpam-2159 291 2 " " PUNCT ejpam-2159 291 3 ! ! PUNCT ejpam-2159 292 1 k k PROPN ejpam-2159 292 2 j j PROPN ejpam-2159 292 3 " " PUNCT ejpam-2159 292 4 # # NOUN ejpam-2159 292 5 n n PRON ejpam-2159 292 6 # # NOUN ejpam-2159 292 7 k=1 k=1 NOUN ejpam-2159 292 8 ! ! PUNCT ejpam-2159 293 1 k k INTJ ejpam-2159 294 1 i i PRON ejpam-2159 294 2 " " PUNCT ejpam-2159 294 3 ! ! PUNCT ejpam-2159 295 1 k k X ejpam-2159 295 2 # # NOUN ejpam-2159 295 3 1 1 NUM ejpam-2159 295 4 j j NOUN ejpam-2159 295 5 " " PUNCT ejpam-2159 295 6 # # NOUN ejpam-2159 295 7 n n PRON ejpam-2159 295 8 # # NOUN ejpam-2159 295 9 k=1 k=1 NOUN ejpam-2159 295 10 ! ! PUNCT ejpam-2159 296 1 k k X ejpam-2159 296 2 # # NOUN ejpam-2159 296 3 1 1 NUM ejpam-2159 296 4 i i PRON ejpam-2159 296 5 " " PUNCT ejpam-2159 296 6 ! ! PUNCT ejpam-2159 297 1 k k PROPN ejpam-2159 297 2 j j PROPN ejpam-2159 297 3 " " PUNCT ejpam-2159 298 1 + + CCONJ ejpam-2159 298 2 n n PRON ejpam-2159 298 3 # # NOUN ejpam-2159 298 4 k=1 k=1 NOUN ejpam-2159 298 5 ! ! PUNCT ejpam-2159 299 1 k k X ejpam-2159 299 2 # # NOUN ejpam-2159 299 3 1 1 NUM ejpam-2159 299 4 i i PRON ejpam-2159 299 5 " " PUNCT ejpam-2159 299 6 ! ! PUNCT ejpam-2159 300 1 k k X ejpam-2159 300 2 # # NOUN ejpam-2159 300 3 1 1 NUM ejpam-2159 300 4 j j NOUN ejpam-2159 300 5 " " PUNCT ejpam-2159 300 6 7 7 NUM ejpam-2159 300 7 m. m. NOUN ejpam-2159 300 8 el el PROPN ejpam-2159 300 9 - - PUNCT ejpam-2159 300 10 mikkawy mikkawy PROPN ejpam-2159 300 11 , , PUNCT ejpam-2159 300 12 f. f. PROPN ejpam-2159 300 13 atlan atlan PROPN ejpam-2159 300 14 / / SYM ejpam-2159 300 15 eur eur PROPN ejpam-2159 300 16 . . PUNCT ejpam-2159 301 1 j. j. PROPN ejpam-2159 301 2 pure pure PROPN ejpam-2159 301 3 appl appl PROPN ejpam-2159 301 4 . . PROPN ejpam-2159 301 5 math math PROPN ejpam-2159 301 6 , , PUNCT ejpam-2159 301 7 8 8 NUM ejpam-2159 301 8 ( ( PUNCT ejpam-2159 301 9 2015 2015 NUM ejpam-2159 301 10 ) ) PUNCT ejpam-2159 301 11 , , PUNCT ejpam-2159 301 12 135 135 NUM ejpam-2159 301 13 - - SYM ejpam-2159 301 14 151 151 NUM ejpam-2159 301 15 144 144 NUM ejpam-2159 301 16 =( =( NOUN ejpam-2159 301 17 # # SYM ejpam-2159 301 18 1)i+ 1)i+ NUM ejpam-2159 301 19 j j PROPN ejpam-2159 301 20 6 6 NUM ejpam-2159 301 21 ! ! PUNCT ejpam-2159 302 1 n n PRON ejpam-2159 302 2 i i PRON ejpam-2159 302 3 " " PUNCT ejpam-2159 302 4 ! ! PUNCT ejpam-2159 303 1 n n CCONJ ejpam-2159 303 2 j j PROPN ejpam-2159 303 3 " " PUNCT ejpam-2159 304 1 + + CCONJ ejpam-2159 304 2 n n PRON ejpam-2159 304 3 # # NOUN ejpam-2159 304 4 k=1 k=1 NOUN ejpam-2159 304 5 ! ! PUNCT ejpam-2159 305 1 k k X ejpam-2159 305 2 # # NOUN ejpam-2159 305 3 1 1 NUM ejpam-2159 305 4 i i PRON ejpam-2159 305 5 " " PUNCT ejpam-2159 305 6 ! ! PUNCT ejpam-2159 306 1 k k X ejpam-2159 306 2 # # NOUN ejpam-2159 306 3 1 1 NUM ejpam-2159 306 4 j j NOUN ejpam-2159 306 5 " " PUNCT ejpam-2159 306 6 # # NOUN ejpam-2159 306 7 n n PRON ejpam-2159 306 8 # # NOUN ejpam-2159 306 9 k=1 k=1 NOUN ejpam-2159 306 10 ! ! PUNCT ejpam-2159 307 1 k k INTJ ejpam-2159 308 1 i i PRON ejpam-2159 308 2 " " PUNCT ejpam-2159 308 3 ! ! PUNCT ejpam-2159 309 1 k k X ejpam-2159 309 2 # # NOUN ejpam-2159 309 3 1 1 NUM ejpam-2159 309 4 j j NOUN ejpam-2159 309 5 " " PUNCT ejpam-2159 309 6 # # NOUN ejpam-2159 309 7 n n PRON ejpam-2159 309 8 # # NOUN ejpam-2159 309 9 k=1 k=1 NOUN ejpam-2159 309 10 ! ! PUNCT ejpam-2159 310 1 k k PROPN ejpam-2159 310 2 j j PROPN ejpam-2159 311 1 " " PUNCT ejpam-2159 311 2 ! ! PUNCT ejpam-2159 312 1 k k X ejpam-2159 312 2 # # NOUN ejpam-2159 312 3 1 1 NUM ejpam-2159 312 4 i i PRON ejpam-2159 312 5 " " PUNCT ejpam-2159 312 6 + + CCONJ ejpam-2159 312 7 n n PRON ejpam-2159 312 8 # # NOUN ejpam-2159 312 9 k=1 k=1 NOUN ejpam-2159 312 10 ! ! PUNCT ejpam-2159 313 1 k k X ejpam-2159 313 2 # # NOUN ejpam-2159 313 3 1 1 NUM ejpam-2159 313 4 i i PRON ejpam-2159 313 5 " " PUNCT ejpam-2159 313 6 ! ! PUNCT ejpam-2159 314 1 k k X ejpam-2159 314 2 # # NOUN ejpam-2159 314 3 1 1 NUM ejpam-2159 314 4 j j NOUN ejpam-2159 314 5 " " PUNCT ejpam-2159 314 6 7 7 NUM ejpam-2159 314 7 =( =( NOUN ejpam-2159 314 8 # # SYM ejpam-2159 314 9 1)i+ 1)i+ NUM ejpam-2159 314 10 j j PROPN ejpam-2159 314 11 6 6 NUM ejpam-2159 314 12 ! ! PUNCT ejpam-2159 315 1 n n PRON ejpam-2159 315 2 i i PRON ejpam-2159 315 3 " " PUNCT ejpam-2159 315 4 ! ! PUNCT ejpam-2159 316 1 n n CCONJ ejpam-2159 316 2 j j NOUN ejpam-2159 316 3 " " PUNCT ejpam-2159 316 4 # # NOUN ejpam-2159 316 5 n n PRON ejpam-2159 316 6 # # NOUN ejpam-2159 316 7 k=1 k=1 NOUN ejpam-2159 316 8 ! ! PUNCT ejpam-2159 317 1 k k PUNCT ejpam-2159 317 2 # # NOUN ejpam-2159 317 3 1 1 NUM ejpam-2159 317 4 j j NOUN ejpam-2159 317 5 " " PUNCT ejpam-2159 317 6 < < X ejpam-2159 317 7 ! ! PUNCT ejpam-2159 318 1 k k PROPN ejpam-2159 318 2 i i PRON ejpam-2159 318 3 " " PUNCT ejpam-2159 318 4 # # NOUN ejpam-2159 318 5 ! ! PUNCT ejpam-2159 319 1 k k X ejpam-2159 319 2 # # NOUN ejpam-2159 319 3 1 1 NUM ejpam-2159 319 4 i i PRON ejpam-2159 319 5 " " PUNCT ejpam-2159 319 6 = = NOUN ejpam-2159 319 7 # # NOUN ejpam-2159 319 8 n n NOUN ejpam-2159 319 9 # # NOUN ejpam-2159 319 10 k=1 k=1 NOUN ejpam-2159 319 11 ! ! PUNCT ejpam-2159 320 1 k k X ejpam-2159 320 2 # # NOUN ejpam-2159 320 3 1 1 NUM ejpam-2159 320 4 i i PRON ejpam-2159 320 5 " " PUNCT ejpam-2159 320 6 < < X ejpam-2159 320 7 ! ! PUNCT ejpam-2159 321 1 k k PROPN ejpam-2159 321 2 j j PROPN ejpam-2159 321 3 " " PUNCT ejpam-2159 321 4 # # NOUN ejpam-2159 321 5 ! ! PUNCT ejpam-2159 322 1 k k X ejpam-2159 322 2 # # NOUN ejpam-2159 322 3 1 1 NUM ejpam-2159 322 4 j j NOUN ejpam-2159 322 5 " " PUNCT ejpam-2159 322 6 = = SYM ejpam-2159 322 7 7 7 NUM ejpam-2159 322 8 =( =( NOUN ejpam-2159 322 9 # # SYM ejpam-2159 322 10 1)i+ 1)i+ NUM ejpam-2159 322 11 j j PROPN ejpam-2159 322 12 6 6 NUM ejpam-2159 322 13 ! ! PUNCT ejpam-2159 323 1 n n PRON ejpam-2159 323 2 i i PRON ejpam-2159 323 3 " " PUNCT ejpam-2159 323 4 ! ! PUNCT ejpam-2159 324 1 n n CCONJ ejpam-2159 324 2 j j NOUN ejpam-2159 324 3 " " PUNCT ejpam-2159 324 4 # # NOUN ejpam-2159 324 5 n n PRON ejpam-2159 324 6 # # NOUN ejpam-2159 324 7 k=1 k=1 NOUN ejpam-2159 324 8 ! ! PUNCT ejpam-2159 325 1 k k PUNCT ejpam-2159 325 2 # # NOUN ejpam-2159 325 3 1 1 NUM ejpam-2159 325 4 j j NOUN ejpam-2159 325 5 " " PUNCT ejpam-2159 325 6 ! ! PUNCT ejpam-2159 326 1 k k X ejpam-2159 327 1 # # NOUN ejpam-2159 327 2 1 1 NUM ejpam-2159 327 3 i i NOUN ejpam-2159 327 4 # # NOUN ejpam-2159 327 5 1 1 NUM ejpam-2159 327 6 " " PUNCT ejpam-2159 327 7 # # NOUN ejpam-2159 327 8 n n NOUN ejpam-2159 327 9 # # NOUN ejpam-2159 327 10 k=1 k=1 NOUN ejpam-2159 327 11 ! ! PUNCT ejpam-2159 328 1 k k X ejpam-2159 328 2 # # NOUN ejpam-2159 328 3 1 1 NUM ejpam-2159 328 4 i i PRON ejpam-2159 328 5 " " PUNCT ejpam-2159 328 6 ! ! PUNCT ejpam-2159 329 1 k k X ejpam-2159 329 2 # # NOUN ejpam-2159 329 3 1 1 NUM ejpam-2159 329 4 j j NOUN ejpam-2159 329 5 # # NOUN ejpam-2159 329 6 1 1 NUM ejpam-2159 329 7 " " PUNCT ejpam-2159 329 8 7 7 NUM ejpam-2159 329 9 =( =( NOUN ejpam-2159 329 10 # # SYM ejpam-2159 329 11 1)i+ 1)i+ NUM ejpam-2159 329 12 j j PROPN ejpam-2159 329 13 6 6 NUM ejpam-2159 329 14 ! ! PUNCT ejpam-2159 330 1 n n PRON ejpam-2159 330 2 i i PRON ejpam-2159 330 3 " " PUNCT ejpam-2159 330 4 ! ! PUNCT ejpam-2159 331 1 n n PRON ejpam-2159 331 2 j j PROPN ejpam-2159 331 3 " " PUNCT ejpam-2159 332 1 + + CCONJ ejpam-2159 332 2 ( ( PUNCT ejpam-2159 332 3 # # SYM ejpam-2159 332 4 1)i+ 1)i+ NUM ejpam-2159 332 5 j'i j'i NOUN ejpam-2159 332 6 , , PUNCT ejpam-2159 332 7 j+1 j+1 PROPN ejpam-2159 332 8 + + CCONJ ejpam-2159 332 9 ( ( PUNCT ejpam-2159 332 10 # # SYM ejpam-2159 332 11 1)i+ 1)i+ NUM ejpam-2159 332 12 j'i+1 j'i+1 PROPN ejpam-2159 332 13 , , PUNCT ejpam-2159 332 14 j j PROPN ejpam-2159 332 15 7 7 NUM ejpam-2159 332 16 . . PUNCT ejpam-2159 333 1 ( ( PUNCT ejpam-2159 333 2 54 54 NUM ejpam-2159 333 3 ) ) PUNCT ejpam-2159 333 4 therefore therefore ADV ejpam-2159 333 5 ' ' PUNCT ejpam-2159 333 6 i i PRON ejpam-2159 333 7 , , PUNCT ejpam-2159 333 8 j j PROPN ejpam-2159 333 9 = = PUNCT ejpam-2159 333 10 ' ' VERB ejpam-2159 333 11 i i NOUN ejpam-2159 333 12 , , PUNCT ejpam-2159 333 13 j+1 j+1 PROPN ejpam-2159 333 14 + + CCONJ ejpam-2159 333 15 ' ' PUNCT ejpam-2159 333 16 i+1 i+1 ADP ejpam-2159 333 17 , , PUNCT ejpam-2159 333 18 j j PROPN ejpam-2159 333 19 + + CCONJ ejpam-2159 333 20 ( ( PUNCT ejpam-2159 333 21 # # SYM ejpam-2159 333 22 1)i+ 1)i+ NUM ejpam-2159 333 23 j j NOUN ejpam-2159 333 24 ! ! PUNCT ejpam-2159 334 1 n n CCONJ ejpam-2159 334 2 i i PRON ejpam-2159 334 3 " " PUNCT ejpam-2159 334 4 ! ! PUNCT ejpam-2159 335 1 n n PRON ejpam-2159 335 2 j j PROPN ejpam-2159 335 3 " " PUNCT ejpam-2159 335 4 . . PUNCT ejpam-2159 336 1 ( ( PUNCT ejpam-2159 336 2 55 55 NUM ejpam-2159 336 3 ) ) PUNCT ejpam-2159 336 4 using use VERB ejpam-2159 336 5 ( ( PUNCT ejpam-2159 336 6 49 49 NUM ejpam-2159 336 7 ) ) PUNCT ejpam-2159 336 8 , , PUNCT ejpam-2159 336 9 ( ( PUNCT ejpam-2159 336 10 50 50 NUM ejpam-2159 336 11 ) ) PUNCT ejpam-2159 336 12 and and CCONJ ejpam-2159 336 13 ( ( PUNCT ejpam-2159 336 14 55 55 NUM ejpam-2159 336 15 ) ) PUNCT ejpam-2159 336 16 , , PUNCT ejpam-2159 336 17 we we PRON ejpam-2159 336 18 see see VERB ejpam-2159 336 19 that that SCONJ ejpam-2159 336 20 in in ADP ejpam-2159 336 21 order order NOUN ejpam-2159 336 22 to to PART ejpam-2159 336 23 obtain obtain VERB ejpam-2159 336 24 qn qn NOUN ejpam-2159 336 25 , , PUNCT ejpam-2159 336 26 we we PRON ejpam-2159 336 27 only only ADV ejpam-2159 336 28 need need VERB ejpam-2159 336 29 to to PART ejpam-2159 336 30 compute compute VERB ejpam-2159 336 31 the the DET ejpam-2159 336 32 1 1 NUM ejpam-2159 336 33 2(n 2(n NUM ejpam-2159 336 34 # # NOUN ejpam-2159 336 35 1)(n 1)(n NUM ejpam-2159 336 36 # # NOUN ejpam-2159 336 37 2 2 NUM ejpam-2159 336 38 ) ) PUNCT ejpam-2159 336 39 elements element NOUN ejpam-2159 336 40 ' ' NUM ejpam-2159 336 41 22,'32 22,'32 NUM ejpam-2159 336 42 , , PUNCT ejpam-2159 336 43 . . PUNCT ejpam-2159 336 44 . . PUNCT ejpam-2159 336 45 . . PUNCT ejpam-2159 337 1 , , PUNCT ejpam-2159 337 2 ' ' PUNCT ejpam-2159 337 3 n#1,n#1 n#1,n#1 VERB ejpam-2159 337 4 and and CCONJ ejpam-2159 337 5 taking take VERB ejpam-2159 337 6 into into ADP ejpam-2159 337 7 account account NOUN ejpam-2159 337 8 the the DET ejpam-2159 337 9 fact fact NOUN ejpam-2159 337 10 that that SCONJ ejpam-2159 337 11 qn qn PROPN ejpam-2159 337 12 is be AUX ejpam-2159 337 13 symmetric symmetric ADJ ejpam-2159 337 14 . . PUNCT ejpam-2159 338 1 also also ADV ejpam-2159 338 2 , , PUNCT ejpam-2159 338 3 the the DET ejpam-2159 338 4 solution solution NOUN ejpam-2159 338 5 of of ADP ejpam-2159 338 6 the the DET ejpam-2159 338 7 linear linear ADJ ejpam-2159 338 8 system system NOUN ejpam-2159 338 9 of of ADP ejpam-2159 338 10 the the DET ejpam-2159 338 11 pascal pascal ADJ ejpam-2159 338 12 type type NOUN ejpam-2159 338 13 pn[u1u2 pn[u1u2 NOUN ejpam-2159 338 14 . . PUNCT ejpam-2159 338 15 . . PUNCT ejpam-2159 338 16 . . PUNCT ejpam-2159 339 1 un un PROPN ejpam-2159 339 2 ] ] X ejpam-2159 339 3 t t NOUN ejpam-2159 339 4 = = PUNCT ejpam-2159 339 5 [ [ PUNCT ejpam-2159 339 6 f1 f1 NOUN ejpam-2159 339 7 f2 f2 PROPN ejpam-2159 339 8 . . PUNCT ejpam-2159 339 9 . . PUNCT ejpam-2159 339 10 . . PUNCT ejpam-2159 340 1 fn fn CCONJ ejpam-2159 340 2 ] ] X ejpam-2159 340 3 t t PROPN ejpam-2159 340 4 ( ( PUNCT ejpam-2159 340 5 56 56 NUM ejpam-2159 340 6 ) ) PUNCT ejpam-2159 340 7 may may AUX ejpam-2159 340 8 be be AUX ejpam-2159 340 9 obtained obtain VERB ejpam-2159 340 10 in in ADP ejpam-2159 340 11 o(n2 o(n2 ADJ ejpam-2159 340 12 ) ) PUNCT ejpam-2159 340 13 operations operation NOUN ejpam-2159 340 14 by by ADP ejpam-2159 340 15 using use VERB ejpam-2159 340 16 ( ( PUNCT ejpam-2159 340 17 49 49 NUM ejpam-2159 340 18 ) ) PUNCT ejpam-2159 340 19 , , PUNCT ejpam-2159 340 20 ( ( PUNCT ejpam-2159 340 21 50 50 NUM ejpam-2159 340 22 ) ) PUNCT ejpam-2159 340 23 and and CCONJ ejpam-2159 340 24 ( ( PUNCT ejpam-2159 340 25 55 55 NUM ejpam-2159 340 26 ) ) PUNCT ejpam-2159 340 27 . . PUNCT ejpam-2159 341 1 the the DET ejpam-2159 341 2 following follow VERB ejpam-2159 341 3 is be AUX ejpam-2159 341 4 a a DET ejpam-2159 341 5 maple maple NOUN ejpam-2159 341 6 code code NOUN ejpam-2159 341 7 to to PART ejpam-2159 341 8 compute compute VERB ejpam-2159 341 9 the the DET ejpam-2159 341 10 inverse inverse NOUN ejpam-2159 341 11 matrix matrix NOUN ejpam-2159 341 12 of of ADP ejpam-2159 341 13 the the DET ejpam-2159 341 14 pascal pascal ADJ ejpam-2159 341 15 matrix matrix NOUN ejpam-2159 341 16 , , PUNCT ejpam-2159 341 17 pn pn NOUN ejpam-2159 341 18 for for ADP ejpam-2159 341 19 n= n= ADJ ejpam-2159 341 20 6 6 NUM ejpam-2159 341 21 , , PUNCT ejpam-2159 341 22 as as ADP ejpam-2159 341 23 an an DET ejpam-2159 341 24 example example NOUN ejpam-2159 341 25 . . PUNCT ejpam-2159 342 1 maple maple PROPN ejpam-2159 342 2 code code PROPN ejpam-2159 342 3 for for ADP ejpam-2159 342 4 inverting invert VERB ejpam-2159 342 5 pascal pascal ADJ ejpam-2159 342 6 matrix matrix NOUN ejpam-2159 342 7 . . PUNCT ejpam-2159 343 1 > > X ejpam-2159 344 1 r r NOUN ejpam-2159 344 2 e e NOUN ejpam-2159 344 3 s s PROPN ejpam-2159 344 4 t t NOUN ejpam-2159 344 5 a a DET ejpam-2159 344 6 r r NOUN ejpam-2159 344 7 t t NOUN ejpam-2159 344 8 : : PUNCT ejpam-2159 344 9 > > PUNCT ejpam-2159 345 1 n:=6 n:=6 NOUN ejpam-2159 345 2 : : PUNCT ejpam-2159 345 3 q:= q:= VERB ejpam-2159 345 4 array array NOUN ejpam-2159 345 5 ( ( PUNCT ejpam-2159 345 6 1 1 NUM ejpam-2159 345 7 . . PUNCT ejpam-2159 345 8 . . PUNCT ejpam-2159 346 1 n n X ejpam-2159 346 2 , , PUNCT ejpam-2159 346 3 1 1 NUM ejpam-2159 346 4 . . PUNCT ejpam-2159 346 5 . . PUNCT ejpam-2159 347 1 n n X ejpam-2159 347 2 , , PUNCT ejpam-2159 347 3 symmetric symmetric ADJ ejpam-2159 347 4 ) ) PUNCT ejpam-2159 347 5 : : PUNCT ejpam-2159 348 1 > > X ejpam-2159 348 2 f f X ejpam-2159 349 1 o o NOUN ejpam-2159 349 2 r r NOUN ejpam-2159 349 3 i i PRON ejpam-2159 349 4 to to PART ejpam-2159 349 5 n n PRON ejpam-2159 349 6 do do VERB ejpam-2159 349 7 q q PROPN ejpam-2159 350 1 [ [ PUNCT ejpam-2159 350 2 i i PRON ejpam-2159 350 3 , , PUNCT ejpam-2159 350 4 n]:=(#1)^(n+ n]:=(#1)^(n+ ADV ejpam-2159 350 5 i i PRON ejpam-2159 350 6 ) ) PUNCT ejpam-2159 350 7 ! ! PUNCT ejpam-2159 351 1 binomial binomial ADJ ejpam-2159 351 2 ( ( PUNCT ejpam-2159 351 3 n#1 n#1 NOUN ejpam-2159 351 4 , , PUNCT ejpam-2159 351 5 i i PRON ejpam-2159 351 6 # # SYM ejpam-2159 351 7 1 1 NUM ejpam-2159 351 8 ): ): PUNCT ejpam-2159 351 9 q q X ejpam-2159 351 10 [ [ PUNCT ejpam-2159 351 11 i i PRON ejpam-2159 351 12 , , PUNCT ejpam-2159 351 13 1]:=(#1)^ 1]:=(#1)^ PROPN ejpam-2159 351 14 ( ( PUNCT ejpam-2159 351 15 i+1)!binomial i+1)!binomial PROPN ejpam-2159 351 16 ( ( PUNCT ejpam-2159 351 17 n n NOUN ejpam-2159 351 18 , , PUNCT ejpam-2159 351 19 i i PRON ejpam-2159 351 20 ) ) PUNCT ejpam-2159 351 21 : : PUNCT ejpam-2159 352 1 od od X ejpam-2159 352 2 : : PUNCT ejpam-2159 352 3 > > PUNCT ejpam-2159 352 4 f f X ejpam-2159 353 1 o o NOUN ejpam-2159 353 2 r r NOUN ejpam-2159 353 3 i i NOUN ejpam-2159 353 4 from from ADP ejpam-2159 353 5 n#1 n#1 ADV ejpam-2159 353 6 by by ADP ejpam-2159 353 7 # # SYM ejpam-2159 353 8 1 1 NUM ejpam-2159 353 9 to to PART ejpam-2159 353 10 2 2 NUM ejpam-2159 353 11 do do VERB ejpam-2159 353 12 f f NOUN ejpam-2159 354 1 o o NOUN ejpam-2159 354 2 r r NOUN ejpam-2159 354 3 j j PROPN ejpam-2159 354 4 from from ADP ejpam-2159 354 5 i i PRON ejpam-2159 354 6 by by ADP ejpam-2159 354 7 # # SYM ejpam-2159 354 8 1 1 NUM ejpam-2159 354 9 to to PART ejpam-2159 354 10 2 2 NUM ejpam-2159 354 11 do do AUX ejpam-2159 354 12 q q PROPN ejpam-2159 355 1 [ [ PUNCT ejpam-2159 355 2 i i PRON ejpam-2159 355 3 , , PUNCT ejpam-2159 355 4 j j PROPN ejpam-2159 355 5 ] ] X ejpam-2159 355 6 : : PUNCT ejpam-2159 356 1 = = SYM ejpam-2159 356 2 q q X ejpam-2159 356 3 [ [ PUNCT ejpam-2159 356 4 i i PRON ejpam-2159 356 5 , , PUNCT ejpam-2159 356 6 j+1]+q j+1]+q PROPN ejpam-2159 356 7 [ [ PUNCT ejpam-2159 356 8 i+1 i+1 NUM ejpam-2159 356 9 , , PUNCT ejpam-2159 356 10 j j PROPN ejpam-2159 356 11 ] ] X ejpam-2159 356 12 + + PROPN ejpam-2159 356 13 ( ( PUNCT ejpam-2159 356 14 # # SYM ejpam-2159 356 15 1)^ 1)^ NUM ejpam-2159 356 16 ( ( PUNCT ejpam-2159 356 17 i+ i+ NUM ejpam-2159 356 18 j j PROPN ejpam-2159 356 19 ) ) PUNCT ejpam-2159 356 20 ! ! PUNCT ejpam-2159 357 1 binomial binomial ADJ ejpam-2159 357 2 ( ( PUNCT ejpam-2159 357 3 n n NOUN ejpam-2159 357 4 , , PUNCT ejpam-2159 357 5 i i PRON ejpam-2159 357 6 ) ) PUNCT ejpam-2159 357 7 ! ! PUNCT ejpam-2159 358 1 binomial binomial ADJ ejpam-2159 358 2 ( ( PUNCT ejpam-2159 358 3 n n NOUN ejpam-2159 358 4 , , PUNCT ejpam-2159 358 5 j j PROPN ejpam-2159 358 6 ) ) PUNCT ejpam-2159 358 7 : : PUNCT ejpam-2159 359 1 od od X ejpam-2159 359 2 : : PUNCT ejpam-2159 359 3 od od X ejpam-2159 359 4 : : PUNCT ejpam-2159 359 5 > > PUNCT ejpam-2159 359 6 q:=evalm q:=evalm NOUN ejpam-2159 359 7 ( ( PUNCT ejpam-2159 359 8 q q NOUN ejpam-2159 359 9 ) ) PUNCT ejpam-2159 359 10 : : PUNCT ejpam-2159 359 11 > > PUNCT ejpam-2159 359 12 # # NOUN ejpam-2159 359 13 check check NOUN ejpam-2159 359 14 using use VERB ejpam-2159 359 15 the the DET ejpam-2159 359 16 l l NOUN ejpam-2159 360 1 i i PRON ejpam-2159 360 2 n n CCONJ ejpam-2159 360 3 a a DET ejpam-2159 360 4 l l NOUN ejpam-2159 360 5 g g NOUN ejpam-2159 360 6 package package NOUN ejpam-2159 360 7 . . PUNCT ejpam-2159 361 1 > > X ejpam-2159 361 2 p:= p:= PROPN ejpam-2159 362 1 l l PROPN ejpam-2159 363 1 i i PRON ejpam-2159 363 2 n n CCONJ ejpam-2159 363 3 a a DET ejpam-2159 363 4 l l NOUN ejpam-2159 363 5 g g NOUN ejpam-2159 363 6 [ [ PUNCT ejpam-2159 363 7 i i PRON ejpam-2159 363 8 nve nve VERB ejpam-2159 363 9 r r NOUN ejpam-2159 363 10 s s PROPN ejpam-2159 363 11 e e X ejpam-2159 363 12 ] ] PUNCT ejpam-2159 363 13 ( ( PUNCT ejpam-2159 363 14 q q NOUN ejpam-2159 363 15 ) ) PUNCT ejpam-2159 363 16 : : PUNCT ejpam-2159 363 17 m. m. PROPN ejpam-2159 363 18 el el PROPN ejpam-2159 363 19 - - PUNCT ejpam-2159 363 20 mikkawy mikkawy PROPN ejpam-2159 363 21 , , PUNCT ejpam-2159 363 22 f. f. PROPN ejpam-2159 363 23 atlan atlan PROPN ejpam-2159 363 24 / / SYM ejpam-2159 363 25 eur eur PROPN ejpam-2159 363 26 . . PUNCT ejpam-2159 364 1 j. j. PROPN ejpam-2159 364 2 pure pure PROPN ejpam-2159 364 3 appl appl PROPN ejpam-2159 364 4 . . PROPN ejpam-2159 364 5 math math PROPN ejpam-2159 364 6 , , PUNCT ejpam-2159 364 7 8 8 NUM ejpam-2159 364 8 ( ( PUNCT ejpam-2159 364 9 2015 2015 NUM ejpam-2159 364 10 ) ) PUNCT ejpam-2159 364 11 , , PUNCT ejpam-2159 364 12 135 135 NUM ejpam-2159 364 13 - - SYM ejpam-2159 364 14 151 151 NUM ejpam-2159 364 15 145 145 NUM ejpam-2159 364 16 the the DET ejpam-2159 364 17 results result NOUN ejpam-2159 364 18 are be AUX ejpam-2159 364 19 printed print VERB ejpam-2159 364 20 as as ADP ejpam-2159 364 21 shwon shwon VERB ejpam-2159 364 22 . . PUNCT ejpam-2159 365 1 q q PUNCT ejpam-2159 366 1 = = PUNCT ejpam-2159 366 2 % % NOUN ejpam-2159 366 3 & & CCONJ ejpam-2159 366 4 & & CCONJ ejpam-2159 366 5 & & CCONJ ejpam-2159 366 6 & & CCONJ ejpam-2159 366 7 & & CCONJ ejpam-2159 366 8 & & CCONJ ejpam-2159 366 9 ' ' PART ejpam-2159 366 10 6 6 NUM ejpam-2159 366 11 # # SYM ejpam-2159 366 12 15 15 NUM ejpam-2159 366 13 20 20 NUM ejpam-2159 366 14 # # SYM ejpam-2159 366 15 15 15 NUM ejpam-2159 366 16 6 6 NUM ejpam-2159 366 17 # # SYM ejpam-2159 366 18 1 1 NUM ejpam-2159 366 19 # # SYM ejpam-2159 366 20 15 15 NUM ejpam-2159 366 21 55 55 NUM ejpam-2159 366 22 # # SYM ejpam-2159 366 23 85 85 NUM ejpam-2159 366 24 69 69 NUM ejpam-2159 366 25 # # SYM ejpam-2159 366 26 29 29 NUM ejpam-2159 366 27 5 5 NUM ejpam-2159 366 28 20 20 NUM ejpam-2159 366 29 # # SYM ejpam-2159 366 30 85 85 NUM ejpam-2159 366 31 146 146 NUM ejpam-2159 366 32 # # SYM ejpam-2159 366 33 127 127 NUM ejpam-2159 366 34 56 56 NUM ejpam-2159 366 35 # # SYM ejpam-2159 366 36 10 10 NUM ejpam-2159 366 37 # # SYM ejpam-2159 366 38 15 15 NUM ejpam-2159 366 39 69 69 NUM ejpam-2159 366 40 # # SYM ejpam-2159 366 41 127 127 NUM ejpam-2159 366 42 117 117 NUM ejpam-2159 366 43 # # SYM ejpam-2159 366 44 54 54 NUM ejpam-2159 366 45 10 10 NUM ejpam-2159 366 46 6 6 NUM ejpam-2159 366 47 # # SYM ejpam-2159 366 48 29 29 NUM ejpam-2159 366 49 56 56 NUM ejpam-2159 366 50 # # SYM ejpam-2159 366 51 54 54 NUM ejpam-2159 366 52 26 26 NUM ejpam-2159 366 53 # # SYM ejpam-2159 366 54 5 5 NUM ejpam-2159 366 55 # # SYM ejpam-2159 366 56 1 1 NUM ejpam-2159 366 57 5 5 NUM ejpam-2159 366 58 # # SYM ejpam-2159 366 59 10 10 NUM ejpam-2159 366 60 10 10 NUM ejpam-2159 366 61 # # SYM ejpam-2159 366 62 5 5 NUM ejpam-2159 366 63 1 1 NUM ejpam-2159 366 64 ( ( PUNCT ejpam-2159 366 65 ) ) PUNCT ejpam-2159 366 66 ) ) PUNCT ejpam-2159 366 67 ) ) PUNCT ejpam-2159 366 68 ) ) PUNCT ejpam-2159 366 69 ) ) PUNCT ejpam-2159 366 70 ) ) PUNCT ejpam-2159 367 1 * * PUNCT ejpam-2159 368 1 p p X ejpam-2159 368 2 = = X ejpam-2159 368 3 % % NOUN ejpam-2159 368 4 & & CCONJ ejpam-2159 368 5 & & CCONJ ejpam-2159 368 6 & & CCONJ ejpam-2159 368 7 & & CCONJ ejpam-2159 368 8 & & CCONJ ejpam-2159 368 9 & & CCONJ ejpam-2159 368 10 ' ' PART ejpam-2159 368 11 1 1 NUM ejpam-2159 368 12 1 1 NUM ejpam-2159 368 13 1 1 NUM ejpam-2159 368 14 1 1 NUM ejpam-2159 368 15 1 1 NUM ejpam-2159 368 16 1 1 NUM ejpam-2159 368 17 1 1 NUM ejpam-2159 368 18 2 2 NUM ejpam-2159 368 19 3 3 NUM ejpam-2159 368 20 4 4 NUM ejpam-2159 368 21 5 5 NUM ejpam-2159 368 22 6 6 NUM ejpam-2159 368 23 1 1 NUM ejpam-2159 368 24 3 3 NUM ejpam-2159 368 25 6 6 NUM ejpam-2159 368 26 10 10 NUM ejpam-2159 368 27 15 15 NUM ejpam-2159 368 28 21 21 NUM ejpam-2159 368 29 1 1 NUM ejpam-2159 368 30 4 4 NUM ejpam-2159 368 31 10 10 NUM ejpam-2159 368 32 20 20 NUM ejpam-2159 368 33 35 35 NUM ejpam-2159 368 34 56 56 NUM ejpam-2159 368 35 1 1 NUM ejpam-2159 368 36 5 5 NUM ejpam-2159 368 37 15 15 NUM ejpam-2159 368 38 35 35 NUM ejpam-2159 368 39 70 70 NUM ejpam-2159 368 40 126 126 NUM ejpam-2159 368 41 1 1 NUM ejpam-2159 368 42 6 6 NUM ejpam-2159 368 43 21 21 NUM ejpam-2159 368 44 56 56 NUM ejpam-2159 368 45 126 126 NUM ejpam-2159 368 46 252 252 NUM ejpam-2159 368 47 ( ( PUNCT ejpam-2159 368 48 ) ) PUNCT ejpam-2159 368 49 ) ) PUNCT ejpam-2159 368 50 ) ) PUNCT ejpam-2159 368 51 ) ) PUNCT ejpam-2159 368 52 ) ) PUNCT ejpam-2159 368 53 ) ) PUNCT ejpam-2159 369 1 * * PUNCT ejpam-2159 369 2 3 3 X ejpam-2159 369 3 . . PUNCT ejpam-2159 369 4 applications application NOUN ejpam-2159 369 5 in in ADP ejpam-2159 369 6 this this DET ejpam-2159 369 7 section section NOUN ejpam-2159 369 8 we we PRON ejpam-2159 369 9 are be AUX ejpam-2159 369 10 going go VERB ejpam-2159 369 11 to to PART ejpam-2159 369 12 give give VERB ejpam-2159 369 13 a a DET ejpam-2159 369 14 new new ADJ ejpam-2159 369 15 proof proof NOUN ejpam-2159 369 16 for for ADP ejpam-2159 369 17 el el PROPN ejpam-2159 369 18 - - PUNCT ejpam-2159 369 19 mikkawy mikkawy NOUN ejpam-2159 369 20 conjecture conjecture NOUN ejpam-2159 369 21 [ [ X ejpam-2159 369 22 14 14 NUM ejpam-2159 369 23 ] ] PUNCT ejpam-2159 369 24 . . PUNCT ejpam-2159 370 1 moreover moreover ADV ejpam-2159 370 2 , , PUNCT ejpam-2159 370 3 some some DET ejpam-2159 370 4 new new ADJ ejpam-2159 370 5 identities identity NOUN ejpam-2159 370 6 will will AUX ejpam-2159 370 7 be be AUX ejpam-2159 370 8 obtained obtain VERB ejpam-2159 370 9 . . PUNCT ejpam-2159 371 1 3.1 3.1 NUM ejpam-2159 371 2 . . PUNCT ejpam-2159 372 1 a a DET ejpam-2159 372 2 new new ADJ ejpam-2159 372 3 proof proof NOUN ejpam-2159 372 4 for for ADP ejpam-2159 372 5 el el PROPN ejpam-2159 372 6 - - PUNCT ejpam-2159 372 7 mikkawy mikkawy NOUN ejpam-2159 372 8 conjecture conjecture NOUN ejpam-2159 372 9 [ [ X ejpam-2159 372 10 14 14 NUM ejpam-2159 372 11 ] ] PUNCT ejpam-2159 372 12 theorem theorem NOUN ejpam-2159 372 13 2 2 NUM ejpam-2159 372 14 . . PUNCT ejpam-2159 372 15 let let VERB ejpam-2159 372 16 g g NOUN ejpam-2159 372 17 be be AUX ejpam-2159 372 18 the the DET ejpam-2159 372 19 n n CCONJ ejpam-2159 372 20 " " PUNCT ejpam-2159 372 21 n n NOUN ejpam-2159 372 22 matrix matrix NOUN ejpam-2159 372 23 whose whose DET ejpam-2159 372 24 ( ( PUNCT ejpam-2159 372 25 i i PROPN ejpam-2159 372 26 , , PUNCT ejpam-2159 372 27 j j NOUN ejpam-2159 372 28 ) ) PUNCT ejpam-2159 372 29 entry entry NOUN ejpam-2159 372 30 is be AUX ejpam-2159 372 31 given give VERB ejpam-2159 372 32 by by ADP ejpam-2159 372 33 " " PUNCT ejpam-2159 372 34 ( ( PUNCT ejpam-2159 372 35 n n CCONJ ejpam-2159 372 36 ) ) PUNCT ejpam-2159 372 37 i i PROPN ejpam-2159 372 38 , , PUNCT ejpam-2159 372 39 j j PROPN ejpam-2159 372 40 ( ( PUNCT ejpam-2159 372 41 x1 x1 PROPN ejpam-2159 372 42 , , PUNCT ejpam-2159 372 43 x2 x2 PROPN ejpam-2159 372 44 , , PUNCT ejpam-2159 372 45 . . PUNCT ejpam-2159 372 46 . . PUNCT ejpam-2159 373 1 . . PUNCT ejpam-2159 374 1 , , PUNCT ejpam-2159 374 2 xn xn X ejpam-2159 374 3 ) ) PUNCT ejpam-2159 374 4 and and CCONJ ejpam-2159 374 5 let let VERB ejpam-2159 374 6 v v PART ejpam-2159 374 7 be be AUX ejpam-2159 374 8 the the DET ejpam-2159 374 9 n n CCONJ ejpam-2159 374 10 " " PUNCT ejpam-2159 374 11 n n PRON ejpam-2159 374 12 vandermonde vandermonde NOUN ejpam-2159 374 13 matrix matrix NOUN ejpam-2159 374 14 defined define VERB ejpam-2159 374 15 by by ADP ejpam-2159 374 16 ( ( PUNCT ejpam-2159 374 17 27 27 NUM ejpam-2159 374 18 ) ) PUNCT ejpam-2159 374 19 . . PUNCT ejpam-2159 375 1 then then ADV ejpam-2159 375 2 we we PRON ejpam-2159 375 3 have have VERB ejpam-2159 375 4 det(g det(g NOUN ejpam-2159 375 5 ) ) PUNCT ejpam-2159 375 6 = = PUNCT ejpam-2159 375 7 ( ( PUNCT ejpam-2159 375 8 # # SYM ejpam-2159 375 9 1 1 NUM ejpam-2159 375 10 ) ) PUNCT ejpam-2159 375 11 n(n#1 n(n#1 NOUN ejpam-2159 375 12 ) ) PUNCT ejpam-2159 375 13 2 2 NUM ejpam-2159 375 14 det(v det(v NOUN ejpam-2159 375 15 ) ) PUNCT ejpam-2159 375 16 = = SYM ejpam-2159 375 17 $ $ SYM ejpam-2159 375 18 det(v det(v NOUN ejpam-2159 375 19 ) ) PUNCT ejpam-2159 375 20 if if SCONJ ejpam-2159 375 21 n n CCONJ ejpam-2159 375 22 ) ) PUNCT ejpam-2159 375 23 0 0 NUM ejpam-2159 375 24 or or CCONJ ejpam-2159 375 25 1 1 NUM ejpam-2159 375 26 mod(4 mod(4 NOUN ejpam-2159 375 27 ) ) PUNCT ejpam-2159 375 28 , , PUNCT ejpam-2159 375 29 # # SYM ejpam-2159 375 30 det(v det(v NOUN ejpam-2159 375 31 ) ) PUNCT ejpam-2159 375 32 if if SCONJ ejpam-2159 375 33 n n CCONJ ejpam-2159 375 34 ) ) PUNCT ejpam-2159 375 35 2 2 NUM ejpam-2159 375 36 or or CCONJ ejpam-2159 375 37 3 3 NUM ejpam-2159 375 38 mod(4 mod(4 NOUN ejpam-2159 375 39 ) ) PUNCT ejpam-2159 375 40 . . PUNCT ejpam-2159 376 1 ( ( PUNCT ejpam-2159 376 2 57 57 NUM ejpam-2159 376 3 ) ) PUNCT ejpam-2159 376 4 proof proof NOUN ejpam-2159 376 5 . . PUNCT ejpam-2159 377 1 for for ADP ejpam-2159 377 2 i i PRON ejpam-2159 377 3 = = SYM ejpam-2159 377 4 1,2 1,2 NUM ejpam-2159 377 5 , , PUNCT ejpam-2159 377 6 . . PUNCT ejpam-2159 377 7 . . PUNCT ejpam-2159 377 8 . . PUNCT ejpam-2159 378 1 , , PUNCT ejpam-2159 378 2 n n CCONJ ejpam-2159 378 3 , , PUNCT ejpam-2159 378 4 let let VERB ejpam-2159 378 5 fi fi NOUN ejpam-2159 378 6 as as SCONJ ejpam-2159 378 7 given give VERB ejpam-2159 378 8 in in ADP ejpam-2159 378 9 ( ( PUNCT ejpam-2159 378 10 31 31 NUM ejpam-2159 378 11 ) ) PUNCT ejpam-2159 378 12 , , PUNCT ejpam-2159 379 1 d d NOUN ejpam-2159 379 2 and and CCONJ ejpam-2159 379 3 d̃ d̃ PROPN ejpam-2159 379 4 are be AUX ejpam-2159 379 5 n n CCONJ ejpam-2159 379 6 " " PUNCT ejpam-2159 379 7 n n CCONJ ejpam-2159 379 8 diagonal diagonal ADJ ejpam-2159 379 9 matrices matrix NOUN ejpam-2159 379 10 given give VERB ejpam-2159 379 11 by by ADP ejpam-2159 379 12 : : PUNCT ejpam-2159 379 13 d d X ejpam-2159 379 14 = = SYM ejpam-2159 379 15 diag diag NOUN ejpam-2159 379 16 ( ( PUNCT ejpam-2159 379 17 1 1 NUM ejpam-2159 379 18 f1 f1 NOUN ejpam-2159 379 19 , , PUNCT ejpam-2159 379 20 1 1 NUM ejpam-2159 379 21 f2 f2 ADJ ejpam-2159 379 22 , , PUNCT ejpam-2159 379 23 . . PUNCT ejpam-2159 379 24 . . PUNCT ejpam-2159 380 1 . . PUNCT ejpam-2159 381 1 , , PUNCT ejpam-2159 381 2 1 1 NUM ejpam-2159 381 3 fn fn NOUN ejpam-2159 381 4 ) ) PUNCT ejpam-2159 381 5 and and CCONJ ejpam-2159 381 6 d̃ d̃ PROPN ejpam-2159 381 7 = = SYM ejpam-2159 382 1 diag((#1)n#1 diag((#1)n#1 PROPN ejpam-2159 382 2 , , PUNCT ejpam-2159 382 3 ( ( PUNCT ejpam-2159 382 4 # # SYM ejpam-2159 382 5 1)n#2 1)n#2 NUM ejpam-2159 382 6 , , PUNCT ejpam-2159 382 7 . . PUNCT ejpam-2159 382 8 . . PUNCT ejpam-2159 382 9 . . PUNCT ejpam-2159 383 1 , , PUNCT ejpam-2159 383 2 ( ( PUNCT ejpam-2159 383 3 # # NOUN ejpam-2159 383 4 1)0 1)0 NUM ejpam-2159 383 5 ) ) PUNCT ejpam-2159 383 6 . . PUNCT ejpam-2159 384 1 it it PRON ejpam-2159 384 2 can can AUX ejpam-2159 384 3 be be AUX ejpam-2159 384 4 shown show VERB ejpam-2159 384 5 that that SCONJ ejpam-2159 384 6 for for ADP ejpam-2159 384 7 the the DET ejpam-2159 384 8 matrices matrix NOUN ejpam-2159 384 9 d d NOUN ejpam-2159 384 10 , , PUNCT ejpam-2159 384 11 j j PROPN ejpam-2159 384 12 and and CCONJ ejpam-2159 384 13 d̃ d̃ PROPN ejpam-2159 384 14 , , PUNCT ejpam-2159 384 15 we we PRON ejpam-2159 384 16 have have VERB ejpam-2159 384 17 det(d det(d NOUN ejpam-2159 384 18 ) ) PUNCT ejpam-2159 384 19 = = SYM ejpam-2159 385 1 1 1 NUM ejpam-2159 385 2 n n NUM ejpam-2159 385 3 5 5 NUM ejpam-2159 385 4 i=1 i=1 PROPN ejpam-2159 385 5 fi fi NOUN ejpam-2159 385 6 = = NOUN ejpam-2159 385 7 1 1 NUM ejpam-2159 385 8 : : PUNCT ejpam-2159 385 9 ( ( PUNCT ejpam-2159 385 10 # # SYM ejpam-2159 385 11 1 1 NUM ejpam-2159 385 12 ) ) PUNCT ejpam-2159 385 13 n(n#1 n(n#1 NOUN ejpam-2159 385 14 ) ) PUNCT ejpam-2159 385 15 2 2 NUM ejpam-2159 385 16 6 6 NUM ejpam-2159 385 17 det(v det(v NOUN ejpam-2159 385 18 ) ) PUNCT ejpam-2159 385 19 72 72 NUM ejpam-2159 385 20 ) ) PUNCT ejpam-2159 385 21 , , PUNCT ejpam-2159 385 22 ( ( PUNCT ejpam-2159 385 23 58 58 X ejpam-2159 385 24 ) ) PUNCT ejpam-2159 385 25 having having AUX ejpam-2159 385 26 used use VERB ejpam-2159 385 27 ( ( PUNCT ejpam-2159 385 28 28 28 NUM ejpam-2159 385 29 ) ) PUNCT ejpam-2159 385 30 and and CCONJ ejpam-2159 385 31 ( ( PUNCT ejpam-2159 385 32 31 31 NUM ejpam-2159 385 33 ) ) PUNCT ejpam-2159 385 34 , , PUNCT ejpam-2159 385 35 det(j det(j PROPN ejpam-2159 385 36 ) ) PUNCT ejpam-2159 385 37 = = PUNCT ejpam-2159 385 38 ( ( PUNCT ejpam-2159 385 39 # # SYM ejpam-2159 385 40 1 1 NUM ejpam-2159 385 41 ) ) PUNCT ejpam-2159 385 42 n(n#1 n(n#1 NOUN ejpam-2159 385 43 ) ) PUNCT ejpam-2159 385 44 2 2 NUM ejpam-2159 385 45 and and CCONJ ejpam-2159 385 46 det(d̃ det(d̃ NOUN ejpam-2159 385 47 ) ) PUNCT ejpam-2159 386 1 = = PUNCT ejpam-2159 386 2 ( ( PUNCT ejpam-2159 386 3 # # SYM ejpam-2159 386 4 1 1 NUM ejpam-2159 386 5 ) ) PUNCT ejpam-2159 386 6 n(n#1 n(n#1 NOUN ejpam-2159 386 7 ) ) PUNCT ejpam-2159 386 8 2 2 NUM ejpam-2159 386 9 . . PUNCT ejpam-2159 387 1 ( ( PUNCT ejpam-2159 387 2 59 59 NUM ejpam-2159 387 3 ) ) PUNCT ejpam-2159 387 4 by by ADP ejpam-2159 387 5 noticing notice VERB ejpam-2159 387 6 that that SCONJ ejpam-2159 387 7 the the DET ejpam-2159 387 8 matrix matrix NOUN ejpam-2159 387 9 v#1 v#1 NOUN ejpam-2159 387 10 in in ADP ejpam-2159 387 11 ( ( PUNCT ejpam-2159 387 12 30 30 NUM ejpam-2159 387 13 ) ) PUNCT ejpam-2159 387 14 can can AUX ejpam-2159 387 15 be be AUX ejpam-2159 387 16 written write VERB ejpam-2159 387 17 in in ADP ejpam-2159 387 18 the the DET ejpam-2159 387 19 form form NOUN ejpam-2159 387 20 : : PUNCT ejpam-2159 387 21 v#1 v#1 NOUN ejpam-2159 387 22 = = PUNCT ejpam-2159 387 23 dgj dgj PROPN ejpam-2159 387 24 d̃ d̃ PROPN ejpam-2159 387 25 , , PUNCT ejpam-2159 387 26 ( ( PUNCT ejpam-2159 387 27 60 60 NUM ejpam-2159 387 28 ) ) PUNCT ejpam-2159 387 29 the the DET ejpam-2159 387 30 result result NOUN ejpam-2159 387 31 follows follow VERB ejpam-2159 387 32 . . PUNCT ejpam-2159 388 1 m. m. PROPN ejpam-2159 388 2 el el PROPN ejpam-2159 388 3 - - PUNCT ejpam-2159 388 4 mikkawy mikkawy PROPN ejpam-2159 388 5 , , PUNCT ejpam-2159 388 6 f. f. PROPN ejpam-2159 388 7 atlan atlan PROPN ejpam-2159 388 8 / / SYM ejpam-2159 388 9 eur eur PROPN ejpam-2159 388 10 . . PUNCT ejpam-2159 389 1 j. j. PROPN ejpam-2159 389 2 pure pure PROPN ejpam-2159 389 3 appl appl PROPN ejpam-2159 389 4 . . PROPN ejpam-2159 389 5 math math PROPN ejpam-2159 389 6 , , PUNCT ejpam-2159 389 7 8 8 NUM ejpam-2159 389 8 ( ( PUNCT ejpam-2159 389 9 2015 2015 NUM ejpam-2159 389 10 ) ) PUNCT ejpam-2159 389 11 , , PUNCT ejpam-2159 389 12 135 135 NUM ejpam-2159 389 13 - - SYM ejpam-2159 389 14 151 151 NUM ejpam-2159 389 15 146 146 NUM ejpam-2159 389 16 3.2 3.2 NUM ejpam-2159 389 17 . . PUNCT ejpam-2159 390 1 new new ADJ ejpam-2159 390 2 identities identity NOUN ejpam-2159 390 3 in in ADP ejpam-2159 390 4 [ [ X ejpam-2159 390 5 13 13 NUM ejpam-2159 390 6 ] ] X ejpam-2159 390 7 it it PRON ejpam-2159 390 8 is be AUX ejpam-2159 390 9 shown show VERB ejpam-2159 390 10 that that SCONJ ejpam-2159 390 11 the the DET ejpam-2159 390 12 pascal pascal ADJ ejpam-2159 390 13 matrix matrix NOUN ejpam-2159 390 14 , , PUNCT ejpam-2159 390 15 pn pn PROPN ejpam-2159 390 16 and and CCONJ ejpam-2159 390 17 the the DET ejpam-2159 390 18 vandermonde vandermonde ADJ ejpam-2159 390 19 matrix matrix NOUN ejpam-2159 390 20 , , PUNCT ejpam-2159 390 21 v v NOUN ejpam-2159 390 22 = = SYM ejpam-2159 390 23 ( ( PUNCT ejpam-2159 390 24 ji#1)ni ji#1)ni PROPN ejpam-2159 390 25 , , PUNCT ejpam-2159 390 26 j=1 j=1 PROPN ejpam-2159 390 27 are be AUX ejpam-2159 390 28 related relate VERB ejpam-2159 390 29 by by ADP ejpam-2159 390 30 : : PUNCT ejpam-2159 390 31 pn pn PROPN ejpam-2159 390 32 = = PROPN ejpam-2159 390 33 t t PROPN ejpam-2159 390 34 v. v. CCONJ ejpam-2159 390 35 ( ( PUNCT ejpam-2159 390 36 61 61 NUM ejpam-2159 390 37 ) ) PUNCT ejpam-2159 390 38 from from ADP ejpam-2159 390 39 ( ( PUNCT ejpam-2159 390 40 61 61 NUM ejpam-2159 390 41 ) ) PUNCT ejpam-2159 390 42 , , PUNCT ejpam-2159 390 43 we we PRON ejpam-2159 390 44 get get VERB ejpam-2159 390 45 : : PUNCT ejpam-2159 390 46 qn qn NOUN ejpam-2159 390 47 = = PUNCT ejpam-2159 390 48 ( ( PUNCT ejpam-2159 390 49 ' ' PUNCT ejpam-2159 390 50 i i PRON ejpam-2159 390 51 j j NOUN ejpam-2159 390 52 ) ) PUNCT ejpam-2159 391 1 n n PROPN ejpam-2159 391 2 i i PRON ejpam-2159 391 3 , , PUNCT ejpam-2159 391 4 j=1 j=1 NOUN ejpam-2159 391 5 = = PRON ejpam-2159 391 6 p#1 p#1 CCONJ ejpam-2159 391 7 n n NOUN ejpam-2159 391 8 = = PUNCT ejpam-2159 391 9 v#1t#1 v#1t#1 PROPN ejpam-2159 391 10 , , PUNCT ejpam-2159 391 11 ( ( PUNCT ejpam-2159 391 12 62 62 NUM ejpam-2159 391 13 ) ) PUNCT ejpam-2159 391 14 where where SCONJ ejpam-2159 391 15 t#1 t#1 VERB ejpam-2159 391 16 = = SYM ejpam-2159 391 17 ( ( PUNCT ejpam-2159 391 18 ( ( PUNCT ejpam-2159 391 19 i i PROPN ejpam-2159 391 20 j j PROPN ejpam-2159 391 21 ) ) PUNCT ejpam-2159 391 22 n n PROPN ejpam-2159 391 23 i i PRON ejpam-2159 391 24 , , PUNCT ejpam-2159 391 25 j=1 j=1 PROPN ejpam-2159 391 26 , , PUNCT ejpam-2159 391 27 ( ( PUNCT ejpam-2159 391 28 63 63 NUM ejpam-2159 391 29 ) ) PUNCT ejpam-2159 391 30 with with ADP ejpam-2159 391 31 ( ( PUNCT ejpam-2159 391 32 i i PRON ejpam-2159 391 33 j j NOUN ejpam-2159 391 34 = = PUNCT ejpam-2159 391 35 ( ( PUNCT ejpam-2159 391 36 # # SYM ejpam-2159 391 37 1)i+ 1)i+ NUM ejpam-2159 391 38 j j NOUN ejpam-2159 391 39 ( ( PUNCT ejpam-2159 391 40 j j PROPN ejpam-2159 391 41 # # NOUN ejpam-2159 391 42 1)!s(i 1)!s(i NUM ejpam-2159 391 43 # # NOUN ejpam-2159 391 44 1 1 NUM ejpam-2159 391 45 , , PUNCT ejpam-2159 391 46 j j PROPN ejpam-2159 391 47 # # NOUN ejpam-2159 391 48 1 1 NUM ejpam-2159 391 49 ) ) PUNCT ejpam-2159 391 50 , , PUNCT ejpam-2159 391 51 1 1 NUM ejpam-2159 391 52 & & CCONJ ejpam-2159 391 53 i i PRON ejpam-2159 391 54 , , PUNCT ejpam-2159 391 55 j j PROPN ejpam-2159 391 56 & & CCONJ ejpam-2159 391 57 n n CCONJ ejpam-2159 391 58 , , PUNCT ejpam-2159 391 59 i i PRON ejpam-2159 391 60 % % VERB ejpam-2159 391 61 j. j. PROPN ejpam-2159 391 62 ( ( PUNCT ejpam-2159 391 63 64 64 NUM ejpam-2159 391 64 ) ) PUNCT ejpam-2159 391 65 thus thus ADV ejpam-2159 391 66 ' ' PUNCT ejpam-2159 391 67 i i PRON ejpam-2159 391 68 j j NOUN ejpam-2159 391 69 = = PUNCT ejpam-2159 391 70 n n PRON ejpam-2159 391 71 # # NOUN ejpam-2159 392 1 k=1 k=1 NOUN ejpam-2159 393 1 % % INTJ ejpam-2159 393 2 ik(k ik(k PROPN ejpam-2159 394 1 j j PROPN ejpam-2159 394 2 = = PUNCT ejpam-2159 395 1 n n PRON ejpam-2159 395 2 # # NOUN ejpam-2159 395 3 k=1 k=1 NOUN ejpam-2159 395 4 6 6 NUM ejpam-2159 395 5 ( ( PUNCT ejpam-2159 395 6 # # SYM ejpam-2159 395 7 1)i+k 1)i+k NUM ejpam-2159 395 8 ( ( PUNCT ejpam-2159 395 9 n n CCONJ ejpam-2159 395 10 # # NOUN ejpam-2159 395 11 1 1 NUM ejpam-2159 395 12 ) ) PUNCT ejpam-2159 395 13 ! ! PUNCT ejpam-2159 395 14 ! ! PUNCT ejpam-2159 396 1 n n CCONJ ejpam-2159 396 2 # # SYM ejpam-2159 396 3 1 1 NUM ejpam-2159 396 4 i i NOUN ejpam-2159 396 5 # # NOUN ejpam-2159 396 6 1 1 NUM ejpam-2159 396 7 " " PUNCT ejpam-2159 396 8 " " PUNCT ejpam-2159 396 9 ( ( PUNCT ejpam-2159 396 10 n n CCONJ ejpam-2159 396 11 ) ) PUNCT ejpam-2159 396 12 n#k+1,i n#k+1,i NOUN ejpam-2159 396 13 76 76 NUM ejpam-2159 396 14 ( ( PUNCT ejpam-2159 396 15 # # SYM ejpam-2159 396 16 1 1 NUM ejpam-2159 396 17 ) ) PUNCT ejpam-2159 396 18 j+k j+k NUM ejpam-2159 396 19 ( ( PUNCT ejpam-2159 396 20 j j PROPN ejpam-2159 396 21 # # NOUN ejpam-2159 396 22 1)!s(k 1)!s(k NUM ejpam-2159 396 23 # # NOUN ejpam-2159 396 24 1 1 NUM ejpam-2159 396 25 , , PUNCT ejpam-2159 396 26 j j PROPN ejpam-2159 396 27 # # NOUN ejpam-2159 396 28 1 1 NUM ejpam-2159 396 29 ) ) PUNCT ejpam-2159 396 30 7 7 NUM ejpam-2159 396 31 =( =( NOUN ejpam-2159 396 32 # # SYM ejpam-2159 396 33 1)i+ 1)i+ NUM ejpam-2159 396 34 j j NOUN ejpam-2159 396 35 ( ( PUNCT ejpam-2159 396 36 j j PROPN ejpam-2159 396 37 # # NOUN ejpam-2159 396 38 1 1 NUM ejpam-2159 396 39 ) ) PUNCT ejpam-2159 396 40 ! ! PUNCT ejpam-2159 397 1 ( ( PUNCT ejpam-2159 397 2 i i NOUN ejpam-2159 397 3 # # NOUN ejpam-2159 397 4 1)!(n 1)!(n NUM ejpam-2159 397 5 # # NOUN ejpam-2159 397 6 i i NOUN ejpam-2159 397 7 ) ) PUNCT ejpam-2159 397 8 ! ! PUNCT ejpam-2159 398 1 n n PRON ejpam-2159 398 2 # # NOUN ejpam-2159 398 3 k=1 k=1 NOUN ejpam-2159 398 4 " " PUNCT ejpam-2159 398 5 ( ( PUNCT ejpam-2159 398 6 n n CCONJ ejpam-2159 398 7 ) ) PUNCT ejpam-2159 398 8 n#k+1,i n#k+1,i NOUN ejpam-2159 398 9 s(k s(k ADP ejpam-2159 398 10 # # NOUN ejpam-2159 398 11 1 1 NUM ejpam-2159 398 12 , , PUNCT ejpam-2159 398 13 j j PROPN ejpam-2159 398 14 # # NOUN ejpam-2159 398 15 1 1 NUM ejpam-2159 398 16 ) ) PUNCT ejpam-2159 398 17 . . PUNCT ejpam-2159 399 1 ( ( PUNCT ejpam-2159 399 2 65 65 X ejpam-2159 399 3 ) ) PUNCT ejpam-2159 399 4 rewriting rewrite VERB ejpam-2159 399 5 ( ( PUNCT ejpam-2159 399 6 44 44 NUM ejpam-2159 399 7 ) ) PUNCT ejpam-2159 399 8 in in ADP ejpam-2159 399 9 the the DET ejpam-2159 399 10 form form NOUN ejpam-2159 399 11 : : PUNCT ejpam-2159 399 12 ' ' PUNCT ejpam-2159 399 13 i i PRON ejpam-2159 399 14 j j NOUN ejpam-2159 400 1 = = PUNCT ejpam-2159 400 2 ( ( PUNCT ejpam-2159 400 3 # # SYM ejpam-2159 400 4 1)i+ 1)i+ NUM ejpam-2159 400 5 j j NOUN ejpam-2159 400 6 n n PRON ejpam-2159 400 7 # # NOUN ejpam-2159 400 8 k=1 k=1 X ejpam-2159 400 9 ! ! PUNCT ejpam-2159 401 1 k k X ejpam-2159 401 2 # # NOUN ejpam-2159 401 3 1 1 NUM ejpam-2159 401 4 i i NOUN ejpam-2159 401 5 # # NOUN ejpam-2159 401 6 1 1 NUM ejpam-2159 401 7 " " PUNCT ejpam-2159 401 8 ! ! PUNCT ejpam-2159 402 1 k k X ejpam-2159 402 2 # # NOUN ejpam-2159 402 3 1 1 NUM ejpam-2159 402 4 j j NOUN ejpam-2159 402 5 # # NOUN ejpam-2159 402 6 1 1 NUM ejpam-2159 402 7 " " PUNCT ejpam-2159 402 8 , , PUNCT ejpam-2159 402 9 ( ( PUNCT ejpam-2159 402 10 66 66 NUM ejpam-2159 402 11 ) ) PUNCT ejpam-2159 402 12 then then ADV ejpam-2159 402 13 using use VERB ejpam-2159 402 14 ( ( PUNCT ejpam-2159 402 15 65 65 NUM ejpam-2159 402 16 ) ) PUNCT ejpam-2159 402 17 and and CCONJ ejpam-2159 402 18 ( ( PUNCT ejpam-2159 402 19 66 66 NUM ejpam-2159 402 20 ) ) PUNCT ejpam-2159 402 21 , , PUNCT ejpam-2159 402 22 yields yield VERB ejpam-2159 402 23 : : PUNCT ejpam-2159 402 24 n n DET ejpam-2159 402 25 # # NOUN ejpam-2159 402 26 k k PROPN ejpam-2159 402 27 = = X ejpam-2159 402 28 max max X ejpam-2159 402 29 ( ( PUNCT ejpam-2159 402 30 i i PROPN ejpam-2159 402 31 , , PUNCT ejpam-2159 402 32 j j PROPN ejpam-2159 402 33 ) ) PUNCT ejpam-2159 402 34 ! ! PUNCT ejpam-2159 403 1 k k X ejpam-2159 403 2 # # NOUN ejpam-2159 403 3 1 1 NUM ejpam-2159 403 4 i i NOUN ejpam-2159 403 5 # # NOUN ejpam-2159 403 6 1 1 NUM ejpam-2159 403 7 " " PUNCT ejpam-2159 403 8 ! ! PUNCT ejpam-2159 404 1 k k X ejpam-2159 404 2 # # NOUN ejpam-2159 404 3 1 1 NUM ejpam-2159 404 4 j j NOUN ejpam-2159 404 5 # # NOUN ejpam-2159 404 6 1 1 NUM ejpam-2159 404 7 " " PUNCT ejpam-2159 404 8 = = SYM ejpam-2159 404 9 ( ( PUNCT ejpam-2159 404 10 j j PROPN ejpam-2159 404 11 # # NOUN ejpam-2159 404 12 1 1 NUM ejpam-2159 404 13 ) ) PUNCT ejpam-2159 404 14 ! ! PUNCT ejpam-2159 405 1 ( ( PUNCT ejpam-2159 405 2 i i NOUN ejpam-2159 405 3 # # NOUN ejpam-2159 405 4 1)!(n 1)!(n NUM ejpam-2159 405 5 # # NOUN ejpam-2159 405 6 i i NOUN ejpam-2159 405 7 ) ) PUNCT ejpam-2159 405 8 ! ! PUNCT ejpam-2159 406 1 n n PRON ejpam-2159 407 1 # # NOUN ejpam-2159 407 2 k= k= PRON ejpam-2159 407 3 j j PROPN ejpam-2159 407 4 " " PUNCT ejpam-2159 407 5 ( ( PUNCT ejpam-2159 407 6 n n CCONJ ejpam-2159 407 7 ) ) PUNCT ejpam-2159 407 8 n#k+1,i n#k+1,i NOUN ejpam-2159 407 9 s(k s(k ADP ejpam-2159 407 10 # # NOUN ejpam-2159 407 11 1 1 NUM ejpam-2159 407 12 , , PUNCT ejpam-2159 407 13 j j PROPN ejpam-2159 407 14 # # NOUN ejpam-2159 407 15 1 1 NUM ejpam-2159 407 16 ) ) PUNCT ejpam-2159 407 17 . . PUNCT ejpam-2159 408 1 ( ( PUNCT ejpam-2159 408 2 67 67 X ejpam-2159 408 3 ) ) PUNCT ejpam-2159 408 4 setting set VERB ejpam-2159 408 5 i i PRON ejpam-2159 408 6 = = SYM ejpam-2159 408 7 n n CCONJ ejpam-2159 408 8 on on ADP ejpam-2159 408 9 both both DET ejpam-2159 408 10 sides side NOUN ejpam-2159 408 11 of of ADP ejpam-2159 408 12 ( ( PUNCT ejpam-2159 408 13 67 67 NUM ejpam-2159 408 14 ) ) PUNCT ejpam-2159 408 15 gives give VERB ejpam-2159 408 16 ! ! PUNCT ejpam-2159 409 1 n n CCONJ ejpam-2159 409 2 # # SYM ejpam-2159 409 3 1 1 NUM ejpam-2159 409 4 j j NOUN ejpam-2159 409 5 # # NOUN ejpam-2159 409 6 1 1 NUM ejpam-2159 409 7 " " PUNCT ejpam-2159 409 8 = = SYM ejpam-2159 409 9 ( ( PUNCT ejpam-2159 409 10 j j PROPN ejpam-2159 409 11 # # NOUN ejpam-2159 409 12 1 1 NUM ejpam-2159 409 13 ) ) PUNCT ejpam-2159 409 14 ! ! PUNCT ejpam-2159 410 1 ( ( PUNCT ejpam-2159 410 2 n n CCONJ ejpam-2159 410 3 # # NOUN ejpam-2159 410 4 1)!0 1)!0 NUM ejpam-2159 410 5 ! ! PUNCT ejpam-2159 411 1 n n PRON ejpam-2159 412 1 # # NOUN ejpam-2159 412 2 k= k= PRON ejpam-2159 412 3 j j PROPN ejpam-2159 412 4 " " PUNCT ejpam-2159 412 5 ( ( PUNCT ejpam-2159 412 6 n n CCONJ ejpam-2159 412 7 ) ) PUNCT ejpam-2159 412 8 n#k+1,n n#k+1,n NOUN ejpam-2159 412 9 s(k s(k ADV ejpam-2159 412 10 # # NOUN ejpam-2159 412 11 1 1 NUM ejpam-2159 412 12 , , PUNCT ejpam-2159 412 13 j j PROPN ejpam-2159 412 14 # # NOUN ejpam-2159 412 15 1 1 NUM ejpam-2159 412 16 ) ) PUNCT ejpam-2159 412 17 = = PUNCT ejpam-2159 413 1 ( ( PUNCT ejpam-2159 413 2 j j PROPN ejpam-2159 413 3 # # NOUN ejpam-2159 413 4 1 1 NUM ejpam-2159 413 5 ) ) PUNCT ejpam-2159 413 6 ! ! PUNCT ejpam-2159 414 1 ( ( PUNCT ejpam-2159 414 2 n n CCONJ ejpam-2159 414 3 # # NOUN ejpam-2159 414 4 1 1 NUM ejpam-2159 414 5 ) ) PUNCT ejpam-2159 414 6 ! ! PUNCT ejpam-2159 415 1 n n PRON ejpam-2159 416 1 # # NOUN ejpam-2159 416 2 k= k= PRON ejpam-2159 416 3 j j PROPN ejpam-2159 416 4 " " PUNCT ejpam-2159 416 5 ( ( PUNCT ejpam-2159 416 6 n#1 n#1 NOUN ejpam-2159 416 7 ) ) PUNCT ejpam-2159 416 8 n#k n#k NOUN ejpam-2159 416 9 ( ( PUNCT ejpam-2159 416 10 1,2 1,2 NUM ejpam-2159 416 11 , , PUNCT ejpam-2159 416 12 . . PUNCT ejpam-2159 416 13 . . PUNCT ejpam-2159 417 1 . . PUNCT ejpam-2159 418 1 , , PUNCT ejpam-2159 419 1 n n CCONJ ejpam-2159 419 2 # # NOUN ejpam-2159 419 3 1)s(k 1)s(k NUM ejpam-2159 419 4 # # NOUN ejpam-2159 419 5 1 1 NUM ejpam-2159 419 6 , , PUNCT ejpam-2159 419 7 j j PROPN ejpam-2159 419 8 # # NOUN ejpam-2159 419 9 1 1 NUM ejpam-2159 419 10 ) ) PUNCT ejpam-2159 419 11 = = PUNCT ejpam-2159 420 1 ( ( PUNCT ejpam-2159 420 2 j j PROPN ejpam-2159 420 3 # # NOUN ejpam-2159 420 4 1 1 NUM ejpam-2159 420 5 ) ) PUNCT ejpam-2159 420 6 ! ! PUNCT ejpam-2159 421 1 ( ( PUNCT ejpam-2159 421 2 n n CCONJ ejpam-2159 421 3 # # NOUN ejpam-2159 421 4 1 1 NUM ejpam-2159 421 5 ) ) PUNCT ejpam-2159 421 6 ! ! PUNCT ejpam-2159 422 1 n n PRON ejpam-2159 422 2 # # NOUN ejpam-2159 422 3 k= k= NOUN ejpam-2159 422 4 j j PROPN ejpam-2159 423 1 c(n c(n PROPN ejpam-2159 423 2 , , PUNCT ejpam-2159 423 3 k)s(k k)s(k NOUN ejpam-2159 423 4 # # NOUN ejpam-2159 423 5 1 1 NUM ejpam-2159 423 6 , , PUNCT ejpam-2159 423 7 j j PROPN ejpam-2159 423 8 # # NOUN ejpam-2159 423 9 1 1 NUM ejpam-2159 423 10 ) ) PUNCT ejpam-2159 423 11 = = SYM ejpam-2159 423 12 1 1 NUM ejpam-2159 423 13 ( ( PUNCT ejpam-2159 423 14 n n CCONJ ejpam-2159 423 15 # # NOUN ejpam-2159 423 16 j j PROPN ejpam-2159 423 17 ) ) PUNCT ejpam-2159 423 18 ! ! PUNCT ejpam-2159 423 19 ! ! PUNCT ejpam-2159 424 1 n n CCONJ ejpam-2159 424 2 # # SYM ejpam-2159 424 3 1 1 NUM ejpam-2159 424 4 j j NOUN ejpam-2159 424 5 # # NOUN ejpam-2159 424 6 1 1 NUM ejpam-2159 424 7 " " PUNCT ejpam-2159 424 8 n n CCONJ ejpam-2159 424 9 # # NOUN ejpam-2159 424 10 k= k= NOUN ejpam-2159 424 11 j j PROPN ejpam-2159 425 1 c(n c(n PROPN ejpam-2159 425 2 , , PUNCT ejpam-2159 425 3 k)s(k k)s(k NOUN ejpam-2159 425 4 # # NOUN ejpam-2159 425 5 1 1 NUM ejpam-2159 425 6 , , PUNCT ejpam-2159 425 7 j j PROPN ejpam-2159 425 8 # # NOUN ejpam-2159 425 9 1 1 NUM ejpam-2159 425 10 ) ) PUNCT ejpam-2159 425 11 . . PUNCT ejpam-2159 426 1 ( ( PUNCT ejpam-2159 426 2 68 68 NUM ejpam-2159 426 3 ) ) PUNCT ejpam-2159 426 4 m. m. NOUN ejpam-2159 426 5 el el PROPN ejpam-2159 426 6 - - PUNCT ejpam-2159 426 7 mikkawy mikkawy PROPN ejpam-2159 426 8 , , PUNCT ejpam-2159 426 9 f. f. PROPN ejpam-2159 426 10 atlan atlan PROPN ejpam-2159 426 11 / / SYM ejpam-2159 426 12 eur eur PROPN ejpam-2159 426 13 . . PUNCT ejpam-2159 427 1 j. j. PROPN ejpam-2159 427 2 pure pure PROPN ejpam-2159 427 3 appl appl PROPN ejpam-2159 427 4 . . PROPN ejpam-2159 427 5 math math PROPN ejpam-2159 427 6 , , PUNCT ejpam-2159 427 7 8 8 NUM ejpam-2159 427 8 ( ( PUNCT ejpam-2159 427 9 2015 2015 NUM ejpam-2159 427 10 ) ) PUNCT ejpam-2159 427 11 , , PUNCT ejpam-2159 427 12 135 135 NUM ejpam-2159 427 13 - - SYM ejpam-2159 427 14 151 151 NUM ejpam-2159 427 15 147 147 NUM ejpam-2159 427 16 hence hence ADV ejpam-2159 427 17 n n ADP ejpam-2159 427 18 # # NOUN ejpam-2159 427 19 k= k= NOUN ejpam-2159 427 20 j j PROPN ejpam-2159 428 1 c(n c(n PROPN ejpam-2159 428 2 , , PUNCT ejpam-2159 428 3 k)s(k k)s(k NOUN ejpam-2159 428 4 # # NOUN ejpam-2159 428 5 1 1 NUM ejpam-2159 428 6 , , PUNCT ejpam-2159 428 7 j j PROPN ejpam-2159 428 8 # # NOUN ejpam-2159 428 9 1 1 NUM ejpam-2159 428 10 ) ) PUNCT ejpam-2159 428 11 =( =( NOUN ejpam-2159 428 12 n n CCONJ ejpam-2159 428 13 # # NOUN ejpam-2159 428 14 j j PROPN ejpam-2159 428 15 ) ) PUNCT ejpam-2159 428 16 ! ! PUNCT ejpam-2159 428 17 ! ! PUNCT ejpam-2159 429 1 n n CCONJ ejpam-2159 429 2 # # SYM ejpam-2159 429 3 1 1 NUM ejpam-2159 429 4 j j NOUN ejpam-2159 429 5 # # NOUN ejpam-2159 429 6 1 1 NUM ejpam-2159 429 7 " " PUNCT ejpam-2159 429 8 2 2 NUM ejpam-2159 429 9 =( =( NOUN ejpam-2159 429 10 n n CCONJ ejpam-2159 429 11 # # NOUN ejpam-2159 429 12 1)n 1)n NUM ejpam-2159 429 13 # # NOUN ejpam-2159 429 14 j j PROPN ejpam-2159 429 15 ! ! PUNCT ejpam-2159 430 1 n n CCONJ ejpam-2159 430 2 # # SYM ejpam-2159 430 3 1 1 NUM ejpam-2159 430 4 n n NUM ejpam-2159 430 5 # # NOUN ejpam-2159 430 6 j j NOUN ejpam-2159 430 7 " " PUNCT ejpam-2159 430 8 . . PUNCT ejpam-2159 430 9 " " PUNCT ejpam-2159 431 1 ( ( PUNCT ejpam-2159 431 2 69 69 NUM ejpam-2159 431 3 ) ) PUNCT ejpam-2159 431 4 from from ADP ejpam-2159 431 5 ( ( PUNCT ejpam-2159 431 6 61 61 NUM ejpam-2159 431 7 ) ) PUNCT ejpam-2159 431 8 , , PUNCT ejpam-2159 431 9 we we PRON ejpam-2159 431 10 see see VERB ejpam-2159 431 11 that that PRON ejpam-2159 431 12 ! ! PUNCT ejpam-2159 432 1 i i PRON ejpam-2159 432 2 + + NUM ejpam-2159 433 1 j j PROPN ejpam-2159 433 2 # # NOUN ejpam-2159 433 3 2 2 NUM ejpam-2159 433 4 i i NOUN ejpam-2159 433 5 # # NOUN ejpam-2159 433 6 1 1 NUM ejpam-2159 433 7 " " PUNCT ejpam-2159 433 8 = = NOUN ejpam-2159 433 9 n n DET ejpam-2159 433 10 # # NOUN ejpam-2159 433 11 k=1 k=1 PUNCT ejpam-2159 433 12 > > X ejpam-2159 433 13 1 1 NUM ejpam-2159 433 14 ( ( PUNCT ejpam-2159 433 15 i i NOUN ejpam-2159 433 16 # # NOUN ejpam-2159 433 17 1 1 NUM ejpam-2159 433 18 ) ) PUNCT ejpam-2159 433 19 ! ! PUNCT ejpam-2159 434 1 c(i c(i NOUN ejpam-2159 434 2 # # NOUN ejpam-2159 434 3 1 1 NUM ejpam-2159 434 4 , , PUNCT ejpam-2159 434 5 k k NOUN ejpam-2159 434 6 # # NOUN ejpam-2159 434 7 1 1 NUM ejpam-2159 434 8 ) ) PUNCT ejpam-2159 434 9 ? ? PUNCT ejpam-2159 435 1 > > X ejpam-2159 435 2 jk#1 jk#1 PROPN ejpam-2159 435 3 ? ? PUNCT ejpam-2159 435 4 . . PUNCT ejpam-2159 436 1 then then ADV ejpam-2159 436 2 ! ! PUNCT ejpam-2159 437 1 i i PRON ejpam-2159 438 1 + + NUM ejpam-2159 438 2 j j PROPN ejpam-2159 438 3 # # NOUN ejpam-2159 438 4 2 2 NUM ejpam-2159 438 5 i i NOUN ejpam-2159 438 6 # # NOUN ejpam-2159 438 7 1 1 NUM ejpam-2159 438 8 " " PUNCT ejpam-2159 438 9 = = NOUN ejpam-2159 438 10 n n PRON ejpam-2159 438 11 # # NOUN ejpam-2159 438 12 k=1 k=1 PROPN ejpam-2159 439 1 jk#1 jk#1 PROPN ejpam-2159 440 1 ( ( PUNCT ejpam-2159 440 2 i i NOUN ejpam-2159 440 3 # # NOUN ejpam-2159 440 4 1 1 NUM ejpam-2159 440 5 ) ) PUNCT ejpam-2159 440 6 ! ! PUNCT ejpam-2159 441 1 c(i c(i NOUN ejpam-2159 441 2 # # NOUN ejpam-2159 441 3 1 1 NUM ejpam-2159 441 4 , , PUNCT ejpam-2159 441 5 k k NOUN ejpam-2159 441 6 # # NOUN ejpam-2159 441 7 1 1 NUM ejpam-2159 441 8 ) ) PUNCT ejpam-2159 441 9 . . PUNCT ejpam-2159 442 1 ( ( PUNCT ejpam-2159 442 2 70 70 X ejpam-2159 442 3 ) ) PUNCT ejpam-2159 442 4 setting set VERB ejpam-2159 442 5 j j PROPN ejpam-2159 442 6 = = SYM ejpam-2159 442 7 2 2 NUM ejpam-2159 442 8 in in ADP ejpam-2159 442 9 ( ( PUNCT ejpam-2159 442 10 70 70 NUM ejpam-2159 442 11 ) ) PUNCT ejpam-2159 442 12 , , PUNCT ejpam-2159 442 13 we we PRON ejpam-2159 442 14 obtain obtain VERB ejpam-2159 442 15 n n ADV ejpam-2159 442 16 # # NOUN ejpam-2159 442 17 k=1 k=1 NOUN ejpam-2159 443 1 2k#1c(i 2k#1c(i NUM ejpam-2159 443 2 # # NOUN ejpam-2159 443 3 1 1 NUM ejpam-2159 443 4 , , PUNCT ejpam-2159 443 5 k k NOUN ejpam-2159 443 6 # # NOUN ejpam-2159 443 7 1 1 NUM ejpam-2159 443 8 ) ) PUNCT ejpam-2159 443 9 = = PUNCT ejpam-2159 443 10 ( ( PUNCT ejpam-2159 443 11 i i NOUN ejpam-2159 443 12 # # NOUN ejpam-2159 443 13 1 1 NUM ejpam-2159 443 14 ) ) PUNCT ejpam-2159 443 15 ! ! PUNCT ejpam-2159 443 16 ! ! PUNCT ejpam-2159 444 1 i i PRON ejpam-2159 445 1 i i PRON ejpam-2159 445 2 # # NOUN ejpam-2159 445 3 1 1 NUM ejpam-2159 445 4 " " PUNCT ejpam-2159 445 5 = = PROPN ejpam-2159 445 6 i i PROPN ejpam-2159 445 7 ! ! PUNCT ejpam-2159 445 8 . . PUNCT ejpam-2159 446 1 ( ( PUNCT ejpam-2159 446 2 71 71 NUM ejpam-2159 446 3 ) ) PUNCT ejpam-2159 446 4 therefore therefore ADV ejpam-2159 446 5 , , PUNCT ejpam-2159 446 6 for for ADP ejpam-2159 446 7 any any DET ejpam-2159 446 8 positive positive ADJ ejpam-2159 446 9 integers integer NOUN ejpam-2159 446 10 n n PRON ejpam-2159 446 11 and and CCONJ ejpam-2159 446 12 1 1 NUM ejpam-2159 446 13 & & CCONJ ejpam-2159 446 14 i i PROPN ejpam-2159 446 15 & & CCONJ ejpam-2159 446 16 n n CCONJ ejpam-2159 446 17 , , PUNCT ejpam-2159 446 18 we we PRON ejpam-2159 446 19 have have VERB ejpam-2159 446 20 n n ADV ejpam-2159 446 21 # # NOUN ejpam-2159 446 22 k=1 k=1 NOUN ejpam-2159 447 1 2k#1c(i 2k#1c(i NUM ejpam-2159 447 2 # # NOUN ejpam-2159 447 3 1 1 NUM ejpam-2159 447 4 , , PUNCT ejpam-2159 447 5 k k NOUN ejpam-2159 447 6 # # NOUN ejpam-2159 447 7 1 1 NUM ejpam-2159 447 8 ) ) PUNCT ejpam-2159 447 9 = = NOUN ejpam-2159 447 10 i i PROPN ejpam-2159 447 11 ! ! PUNCT ejpam-2159 447 12 . . PUNCT ejpam-2159 447 13 " " PUNCT ejpam-2159 448 1 ( ( PUNCT ejpam-2159 448 2 72 72 X ejpam-2159 448 3 ) ) PUNCT ejpam-2159 448 4 we we PRON ejpam-2159 448 5 list list VERB ejpam-2159 448 6 the the DET ejpam-2159 448 7 following follow VERB ejpam-2159 448 8 additional additional ADJ ejpam-2159 448 9 identities identity NOUN ejpam-2159 448 10 , , PUNCT ejpam-2159 448 11 without without ADP ejpam-2159 448 12 proof proof NOUN ejpam-2159 448 13 , , PUNCT ejpam-2159 448 14 for for ADP ejpam-2159 448 15 the the DET ejpam-2159 448 16 sake sake NOUN ejpam-2159 448 17 of of ADP ejpam-2159 448 18 space space NOUN ejpam-2159 448 19 requirements requirement NOUN ejpam-2159 448 20 . . PUNCT ejpam-2159 449 1 in in ADP ejpam-2159 449 2 all all DET ejpam-2159 449 3 cases case NOUN ejpam-2159 449 4 n n PRON ejpam-2159 449 5 is be AUX ejpam-2159 449 6 a a DET ejpam-2159 449 7 positive positive ADJ ejpam-2159 449 8 integer integer NOUN ejpam-2159 449 9 and and CCONJ ejpam-2159 449 10 i i PRON ejpam-2159 449 11 , , PUNCT ejpam-2159 449 12 j j PROPN ejpam-2159 449 13 are be AUX ejpam-2159 449 14 integers integer NOUN ejpam-2159 449 15 such such ADJ ejpam-2159 449 16 that that SCONJ ejpam-2159 449 17 1 1 NUM ejpam-2159 449 18 & & CCONJ ejpam-2159 449 19 i i PRON ejpam-2159 449 20 , , PUNCT ejpam-2159 449 21 j j PROPN ejpam-2159 449 22 & & CCONJ ejpam-2159 449 23 n. n. PROPN ejpam-2159 449 24 • • PROPN ejpam-2159 449 25 n n PROPN ejpam-2159 449 26 / / SYM ejpam-2159 449 27 r=1 r=1 NOUN ejpam-2159 449 28 ( ( PUNCT ejpam-2159 449 29 # # SYM ejpam-2159 449 30 1)r+1 1)r+1 NUM ejpam-2159 449 31 ! ! PUNCT ejpam-2159 450 1 i i PRON ejpam-2159 450 2 + + NOUN ejpam-2159 451 1 r r NOUN ejpam-2159 451 2 # # NOUN ejpam-2159 451 3 2 2 NUM ejpam-2159 451 4 i i NOUN ejpam-2159 451 5 # # NOUN ejpam-2159 451 6 1 1 NUM ejpam-2159 451 7 " " PUNCT ejpam-2159 451 8 ! ! PUNCT ejpam-2159 452 1 n n CCONJ ejpam-2159 452 2 r r NOUN ejpam-2159 452 3 " " PUNCT ejpam-2159 452 4 = = SYM ejpam-2159 452 5 ! ! PUNCT ejpam-2159 452 6 i1 i1 PROPN ejpam-2159 452 7 . . PUNCT ejpam-2159 453 1 • • NUM ejpam-2159 453 2 n n PROPN ejpam-2159 453 3 / / SYM ejpam-2159 453 4 k=1 k=1 X ejpam-2159 453 5 ( ( PUNCT ejpam-2159 453 6 # # SYM ejpam-2159 453 7 1)k+1ki#1 1)k+1ki#1 NUM ejpam-2159 453 8 ! ! PUNCT ejpam-2159 454 1 n n CCONJ ejpam-2159 454 2 k k NOUN ejpam-2159 454 3 " " PUNCT ejpam-2159 454 4 = = SYM ejpam-2159 454 5 ! ! PUNCT ejpam-2159 454 6 i1 i1 PROPN ejpam-2159 454 7 . . PUNCT ejpam-2159 455 1 • • NUM ejpam-2159 455 2 n n PROPN ejpam-2159 455 3 / / SYM ejpam-2159 456 1 k=1 k=1 X ejpam-2159 456 2 ( ( PUNCT ejpam-2159 456 3 # # SYM ejpam-2159 456 4 1)k+1 1)k+1 NUM ejpam-2159 456 5 " " PUNCT ejpam-2159 456 6 ( ( PUNCT ejpam-2159 456 7 n#1 n#1 NOUN ejpam-2159 456 8 ) ) PUNCT ejpam-2159 456 9 n#k n#k NOUN ejpam-2159 456 10 ( ( PUNCT ejpam-2159 456 11 2,3 2,3 NUM ejpam-2159 456 12 , , PUNCT ejpam-2159 456 13 . . PUNCT ejpam-2159 456 14 . . PUNCT ejpam-2159 457 1 . . PUNCT ejpam-2159 457 2 , , PUNCT ejpam-2159 458 1 n n CCONJ ejpam-2159 458 2 ) ) PUNCT ejpam-2159 459 1 = = SYM ejpam-2159 459 2 ( ( PUNCT ejpam-2159 459 3 n n CCONJ ejpam-2159 459 4 # # NOUN ejpam-2159 459 5 1 1 NUM ejpam-2159 459 6 ) ) PUNCT ejpam-2159 459 7 ! ! PUNCT ejpam-2159 459 8 . . PUNCT ejpam-2159 460 1 • • NUM ejpam-2159 460 2 n n PROPN ejpam-2159 460 3 / / SYM ejpam-2159 460 4 k=1 k=1 PROPN ejpam-2159 460 5 s(n s(n PROPN ejpam-2159 460 6 , , PUNCT ejpam-2159 460 7 k k NOUN ejpam-2159 460 8 ) ) PUNCT ejpam-2159 460 9 jk#1 jk#1 NOUN ejpam-2159 460 10 = = SYM ejpam-2159 460 11 ( ( PUNCT ejpam-2159 460 12 n n CCONJ ejpam-2159 460 13 # # NOUN ejpam-2159 460 14 1)!!n 1)!!n NUM ejpam-2159 460 15 j j PROPN ejpam-2159 460 16 . . PUNCT ejpam-2159 461 1 • • NUM ejpam-2159 461 2 n n CCONJ ejpam-2159 461 3 / / SYM ejpam-2159 461 4 r= r= PROPN ejpam-2159 461 5 j j PROPN ejpam-2159 461 6 ! ! PUNCT ejpam-2159 462 1 n n CCONJ ejpam-2159 462 2 # # SYM ejpam-2159 462 3 1 1 NUM ejpam-2159 462 4 n n CCONJ ejpam-2159 462 5 # # NOUN ejpam-2159 462 6 r r NOUN ejpam-2159 462 7 " " PUNCT ejpam-2159 462 8 ( ( PUNCT ejpam-2159 462 9 n n CCONJ ejpam-2159 462 10 # # NOUN ejpam-2159 462 11 1)n#r 1)n#r NUM ejpam-2159 462 12 s(r s(r NOUN ejpam-2159 462 13 # # NOUN ejpam-2159 462 14 1 1 NUM ejpam-2159 462 15 , , PUNCT ejpam-2159 462 16 j j PROPN ejpam-2159 462 17 # # NOUN ejpam-2159 462 18 1 1 NUM ejpam-2159 462 19 ) ) PUNCT ejpam-2159 462 20 = = SYM ejpam-2159 463 1 c(n c(n PROPN ejpam-2159 463 2 , , PUNCT ejpam-2159 463 3 j j PROPN ejpam-2159 463 4 ) ) PUNCT ejpam-2159 463 5 . . PUNCT ejpam-2159 464 1 • • NUM ejpam-2159 464 2 n n CCONJ ejpam-2159 464 3 / / SYM ejpam-2159 464 4 r= r= ADJ ejpam-2159 464 5 j j PROPN ejpam-2159 464 6 n n PROPN ejpam-2159 464 7 / / SYM ejpam-2159 464 8 k=1 k=1 X ejpam-2159 464 9 ( ( PUNCT ejpam-2159 464 10 # # SYM ejpam-2159 464 11 1 1 NUM ejpam-2159 464 12 ) ) PUNCT ejpam-2159 464 13 j+k j+k NUM ejpam-2159 464 14 ! ! PUNCT ejpam-2159 465 1 n n CCONJ ejpam-2159 465 2 # # SYM ejpam-2159 465 3 1 1 NUM ejpam-2159 465 4 k k NOUN ejpam-2159 465 5 # # NOUN ejpam-2159 465 6 1 1 NUM ejpam-2159 465 7 " " PUNCT ejpam-2159 465 8 " " PUNCT ejpam-2159 465 9 ( ( PUNCT ejpam-2159 465 10 n n CCONJ ejpam-2159 465 11 ) ) PUNCT ejpam-2159 465 12 n#r+1,k n#r+1,k PROPN ejpam-2159 465 13 s(r s(r PROPN ejpam-2159 465 14 # # NOUN ejpam-2159 465 15 1 1 NUM ejpam-2159 465 16 , , PUNCT ejpam-2159 465 17 j j PROPN ejpam-2159 465 18 # # NOUN ejpam-2159 465 19 1 1 NUM ejpam-2159 465 20 ) ) PUNCT ejpam-2159 465 21 = = SYM ejpam-2159 465 22 0 0 X ejpam-2159 465 23 . . NOUN ejpam-2159 465 24 references reference NOUN ejpam-2159 465 25 148 148 NUM ejpam-2159 465 26 • • NUM ejpam-2159 465 27 n n CCONJ ejpam-2159 465 28 / / SYM ejpam-2159 465 29 r=1 r=1 NOUN ejpam-2159 465 30 n n NOUN ejpam-2159 465 31 / / SYM ejpam-2159 465 32 k=1 k=1 X ejpam-2159 465 33 ( ( PUNCT ejpam-2159 465 34 # # SYM ejpam-2159 465 35 1)k 1)k NUM ejpam-2159 465 36 ! ! PUNCT ejpam-2159 466 1 n n CCONJ ejpam-2159 466 2 # # SYM ejpam-2159 466 3 1 1 NUM ejpam-2159 466 4 k k NOUN ejpam-2159 466 5 # # NOUN ejpam-2159 466 6 1 1 NUM ejpam-2159 466 7 " " PUNCT ejpam-2159 466 8 ! ! PUNCT ejpam-2159 467 1 i i PRON ejpam-2159 467 2 + + CCONJ ejpam-2159 468 1 k k ADJ ejpam-2159 468 2 # # NOUN ejpam-2159 468 3 2 2 NUM ejpam-2159 468 4 k k NOUN ejpam-2159 468 5 # # NOUN ejpam-2159 468 6 1 1 NUM ejpam-2159 468 7 " " PUNCT ejpam-2159 468 8 " " PUNCT ejpam-2159 468 9 ( ( PUNCT ejpam-2159 468 10 n n CCONJ ejpam-2159 468 11 ) ) PUNCT ejpam-2159 468 12 n#r+1,k n#r+1,k PROPN ejpam-2159 468 13 = = PUNCT ejpam-2159 469 1 + + CCONJ ejpam-2159 469 2 , , PUNCT ejpam-2159 469 3 , , PUNCT ejpam-2159 469 4 . . PUNCT ejpam-2159 470 1 # # X ejpam-2159 470 2 ( ( PUNCT ejpam-2159 470 3 n n CCONJ ejpam-2159 470 4 # # NOUN ejpam-2159 470 5 1 1 NUM ejpam-2159 470 6 ) ) PUNCT ejpam-2159 470 7 ! ! PUNCT ejpam-2159 471 1 if if SCONJ ejpam-2159 471 2 i i PRON ejpam-2159 471 3 = = SYM ejpam-2159 471 4 1 1 NUM ejpam-2159 471 5 ( ( PUNCT ejpam-2159 471 6 n n CCONJ ejpam-2159 471 7 # # NOUN ejpam-2159 471 8 1 1 NUM ejpam-2159 471 9 ) ) PUNCT ejpam-2159 471 10 ! ! PUNCT ejpam-2159 472 1 if if SCONJ ejpam-2159 472 2 i i PRON ejpam-2159 472 3 = = NOUN ejpam-2159 472 4 2 2 NUM ejpam-2159 472 5 0 0 NUM ejpam-2159 472 6 otherwise otherwise ADV ejpam-2159 472 7 . . PUNCT ejpam-2159 473 1 • • NUM ejpam-2159 473 2 n n CCONJ ejpam-2159 473 3 / / SYM ejpam-2159 473 4 r=1 r=1 NOUN ejpam-2159 473 5 n n NOUN ejpam-2159 473 6 / / SYM ejpam-2159 473 7 k=1 k=1 X ejpam-2159 473 8 ( ( PUNCT ejpam-2159 473 9 # # SYM ejpam-2159 473 10 1)kk2 1)kk2 NUM ejpam-2159 473 11 ! ! PUNCT ejpam-2159 474 1 n n CCONJ ejpam-2159 474 2 k k X ejpam-2159 474 3 " " PUNCT ejpam-2159 474 4 " " PUNCT ejpam-2159 474 5 ( ( PUNCT ejpam-2159 474 6 n n CCONJ ejpam-2159 474 7 ) ) PUNCT ejpam-2159 474 8 n#r+1,k n#r+1,k PROPN ejpam-2159 474 9 = = PROPN ejpam-2159 474 10 n n CCONJ ejpam-2159 474 11 ! ! PUNCT ejpam-2159 474 12 . . PUNCT ejpam-2159 475 1 • • NUM ejpam-2159 475 2 n n CCONJ ejpam-2159 475 3 / / SYM ejpam-2159 475 4 k k X ejpam-2159 475 5 = = SYM ejpam-2159 475 6 max(i max(i PROPN ejpam-2159 475 7 , , PUNCT ejpam-2159 475 8 j j NOUN ejpam-2159 475 9 ) ) PUNCT ejpam-2159 475 10 @ @ ADP ejpam-2159 476 1 a a PRON ejpam-2159 476 2 k k X ejpam-2159 477 1 i i PRON ejpam-2159 477 2 b b PROPN ejpam-2159 477 3 c c X ejpam-2159 477 4 k k X ejpam-2159 477 5 ! ! PUNCT ejpam-2159 478 1 c(k c(k PROPN ejpam-2159 478 2 , , PUNCT ejpam-2159 478 3 j j NOUN ejpam-2159 478 4 ) ) PUNCT ejpam-2159 478 5 = = PUNCT ejpam-2159 478 6 @ @ ADP ejpam-2159 478 7 a a DET ejpam-2159 478 8 n n NOUN ejpam-2159 479 1 i i PRON ejpam-2159 479 2 b b PROPN ejpam-2159 479 3 c c NOUN ejpam-2159 479 4 n n X ejpam-2159 479 5 ! ! PUNCT ejpam-2159 479 6 " " PUNCT ejpam-2159 480 1 ( ( PUNCT ejpam-2159 480 2 n#1 n#1 NOUN ejpam-2159 480 3 ) ) PUNCT ejpam-2159 480 4 n n CCONJ ejpam-2159 480 5 # # NOUN ejpam-2159 480 6 j j PROPN ejpam-2159 480 7 ( ( PUNCT ejpam-2159 480 8 1,2 1,2 NUM ejpam-2159 480 9 , , PUNCT ejpam-2159 480 10 . . PUNCT ejpam-2159 480 11 . . PUNCT ejpam-2159 481 1 . . PUNCT ejpam-2159 482 1 , , PUNCT ejpam-2159 482 2 i i PRON ejpam-2159 482 3 # # NOUN ejpam-2159 482 4 1 1 NUM ejpam-2159 482 5 , , PUNCT ejpam-2159 482 6 i i PRON ejpam-2159 482 7 + + NOUN ejpam-2159 482 8 1 1 NUM ejpam-2159 482 9 , , PUNCT ejpam-2159 482 10 . . PUNCT ejpam-2159 482 11 . . PUNCT ejpam-2159 482 12 . . PUNCT ejpam-2159 482 13 , , PUNCT ejpam-2159 482 14 n n CCONJ ejpam-2159 482 15 ) ) PUNCT ejpam-2159 482 16 ; ; PUNCT ejpam-2159 482 17 • • NUM ejpam-2159 482 18 n n NOUN ejpam-2159 482 19 / / SYM ejpam-2159 482 20 k k X ejpam-2159 482 21 = = SYM ejpam-2159 482 22 max(i max(i PROPN ejpam-2159 482 23 , , PUNCT ejpam-2159 482 24 j j NOUN ejpam-2159 482 25 ) ) PUNCT ejpam-2159 482 26 ( ( PUNCT ejpam-2159 482 27 n n CCONJ ejpam-2159 482 28 # # NOUN ejpam-2159 482 29 i)n#kc(k i)n#kc(k PROPN ejpam-2159 482 30 , , PUNCT ejpam-2159 482 31 j j NOUN ejpam-2159 482 32 ) ) PUNCT ejpam-2159 482 33 = = PRON ejpam-2159 482 34 " " PUNCT ejpam-2159 482 35 ( ( PUNCT ejpam-2159 482 36 n#1 n#1 NOUN ejpam-2159 482 37 ) ) PUNCT ejpam-2159 482 38 n n CCONJ ejpam-2159 482 39 # # NOUN ejpam-2159 482 40 j j PROPN ejpam-2159 482 41 ( ( PUNCT ejpam-2159 482 42 1,2 1,2 NUM ejpam-2159 482 43 , , PUNCT ejpam-2159 482 44 . . PUNCT ejpam-2159 482 45 . . PUNCT ejpam-2159 482 46 . . PUNCT ejpam-2159 483 1 , , PUNCT ejpam-2159 483 2 i i PRON ejpam-2159 483 3 # # NOUN ejpam-2159 483 4 1 1 NUM ejpam-2159 483 5 , , PUNCT ejpam-2159 483 6 i i PRON ejpam-2159 483 7 + + NOUN ejpam-2159 483 8 1 1 NUM ejpam-2159 483 9 , , PUNCT ejpam-2159 483 10 . . PUNCT ejpam-2159 483 11 . . PUNCT ejpam-2159 483 12 . . PUNCT ejpam-2159 483 13 , , PUNCT ejpam-2159 483 14 n n CCONJ ejpam-2159 483 15 ) ) PUNCT ejpam-2159 483 16 ; ; PUNCT ejpam-2159 483 17 • • NUM ejpam-2159 483 18 n n NOUN ejpam-2159 483 19 / / SYM ejpam-2159 483 20 k k NOUN ejpam-2159 483 21 = = NOUN ejpam-2159 483 22 i i X ejpam-2159 483 23 @ @ ADP ejpam-2159 483 24 a a DET ejpam-2159 483 25 n n NOUN ejpam-2159 483 26 # # NOUN ejpam-2159 483 27 i i NOUN ejpam-2159 483 28 n n CCONJ ejpam-2159 483 29 # # NOUN ejpam-2159 483 30 k k PROPN ejpam-2159 483 31 b b PROPN ejpam-2159 484 1 c c NOUN ejpam-2159 484 2 @ @ ADP ejpam-2159 484 3 a a DET ejpam-2159 484 4 n n CCONJ ejpam-2159 484 5 # # SYM ejpam-2159 484 6 1 1 NUM ejpam-2159 484 7 n n CCONJ ejpam-2159 485 1 # # NOUN ejpam-2159 485 2 k k PROPN ejpam-2159 485 3 b b PROPN ejpam-2159 485 4 c c NOUN ejpam-2159 485 5 = = SYM ejpam-2159 485 6 n n PROPN ejpam-2159 485 7 / / SYM ejpam-2159 485 8 k k NOUN ejpam-2159 486 1 = = NOUN ejpam-2159 486 2 i i PROPN ejpam-2159 486 3 ( ( PUNCT ejpam-2159 486 4 n#i)n#k n#i)n#k ADV ejpam-2159 486 5 ( ( PUNCT ejpam-2159 486 6 n#1)n#k n#1)n#k NOUN ejpam-2159 486 7 = = SYM ejpam-2159 486 8 n n PROPN ejpam-2159 486 9 i i PRON ejpam-2159 486 10 ; ; PUNCT ejpam-2159 486 11 • • PRON ejpam-2159 486 12 n n NOUN ejpam-2159 486 13 / / SYM ejpam-2159 486 14 k k NOUN ejpam-2159 486 15 = = NOUN ejpam-2159 486 16 i i X ejpam-2159 486 17 @ @ ADP ejpam-2159 486 18 a a PRON ejpam-2159 486 19 k k X ejpam-2159 487 1 i i PRON ejpam-2159 487 2 b b PROPN ejpam-2159 487 3 c c NOUN ejpam-2159 487 4 k k NOUN ejpam-2159 487 5 = = PUNCT ejpam-2159 487 6 n n PROPN ejpam-2159 487 7 / / SYM ejpam-2159 487 8 k k NOUN ejpam-2159 487 9 = = NOUN ejpam-2159 487 10 i i PROPN ejpam-2159 487 11 ( ( PUNCT ejpam-2159 487 12 k#1)i#1 k#1)i#1 PROPN ejpam-2159 487 13 i i PRON ejpam-2159 487 14 ! ! PUNCT ejpam-2159 488 1 = = PRON ejpam-2159 489 1 @ @ ADP ejpam-2159 489 2 a a DET ejpam-2159 489 3 n n NOUN ejpam-2159 490 1 i i PRON ejpam-2159 490 2 b b X ejpam-2159 490 3 c c NOUN ejpam-2159 491 1 i i PRON ejpam-2159 491 2 ; ; PUNCT ejpam-2159 491 3 • • PRON ejpam-2159 491 4 n n NOUN ejpam-2159 491 5 / / SYM ejpam-2159 491 6 k k NOUN ejpam-2159 491 7 = = NOUN ejpam-2159 491 8 i i PROPN ejpam-2159 491 9 ( ( PUNCT ejpam-2159 491 10 k)i k)i PROPN ejpam-2159 491 11 k k X ejpam-2159 491 12 = = PUNCT ejpam-2159 491 13 ( ( PUNCT ejpam-2159 491 14 n)i n)i PROPN ejpam-2159 491 15 i i INTJ ejpam-2159 491 16 . . PUNCT ejpam-2159 492 1 • • NUM ejpam-2159 492 2 n n INTJ ejpam-2159 492 3 / / PUNCT ejpam-2159 493 1 k=1 k=1 PROPN ejpam-2159 493 2 ks(n+ ks(n+ PROPN ejpam-2159 493 3 1 1 NUM ejpam-2159 493 4 , , PUNCT ejpam-2159 493 5 k+ k+ NOUN ejpam-2159 493 6 1 1 NUM ejpam-2159 493 7 ) ) PUNCT ejpam-2159 493 8 = = SYM ejpam-2159 494 1 ( ( PUNCT ejpam-2159 494 2 # # SYM ejpam-2159 494 3 1)n#1(n 1)n#1(n NUM ejpam-2159 494 4 # # NOUN ejpam-2159 494 5 1)!= 1)!= NUM ejpam-2159 494 6 s(n s(n NOUN ejpam-2159 494 7 , , PUNCT ejpam-2159 494 8 1 1 NUM ejpam-2159 494 9 ) ) PUNCT ejpam-2159 494 10 . . PUNCT ejpam-2159 495 1 acknowledgements acknowledgement NOUN ejpam-2159 495 2 the the DET ejpam-2159 495 3 authors author NOUN ejpam-2159 495 4 are be AUX ejpam-2159 495 5 extremely extremely ADV ejpam-2159 495 6 grateful grateful ADJ ejpam-2159 495 7 to to ADP ejpam-2159 495 8 the the DET ejpam-2159 495 9 editorial editorial ADJ ejpam-2159 495 10 team team NOUN ejpam-2159 495 11 of of ADP ejpam-2159 495 12 ejpam ejpam NOUN ejpam-2159 495 13 and and CCONJ ejpam-2159 495 14 anonymous anonymous ADJ ejpam-2159 495 15 referees referee NOUN ejpam-2159 495 16 for for ADP ejpam-2159 495 17 their their PRON ejpam-2159 495 18 useful useful ADJ ejpam-2159 495 19 comments comment NOUN ejpam-2159 495 20 and and CCONJ ejpam-2159 495 21 suggestions suggestion NOUN ejpam-2159 495 22 that that PRON ejpam-2159 495 23 have have AUX ejpam-2159 495 24 helped help VERB ejpam-2159 495 25 in in ADP ejpam-2159 495 26 improving improve VERB ejpam-2159 495 27 the the DET ejpam-2159 495 28 readability readability NOUN ejpam-2159 495 29 of of ADP ejpam-2159 495 30 this this DET ejpam-2159 495 31 paper paper NOUN ejpam-2159 495 32 . . PUNCT ejpam-2159 496 1 references reference NOUN ejpam-2159 496 2 [ [ X ejpam-2159 496 3 1 1 NUM ejpam-2159 496 4 ] ] PUNCT ejpam-2159 496 5 l. l. PROPN ejpam-2159 496 6 aceto aceto PROPN ejpam-2159 496 7 and and CCONJ ejpam-2159 496 8 d. d. PROPN ejpam-2159 496 9 trigiante trigiante PROPN ejpam-2159 496 10 . . PUNCT ejpam-2159 497 1 the the DET ejpam-2159 497 2 matrices matrix NOUN ejpam-2159 497 3 of of ADP ejpam-2159 497 4 pascal pascal ADJ ejpam-2159 497 5 and and CCONJ ejpam-2159 497 6 other other ADJ ejpam-2159 497 7 greats great NOUN ejpam-2159 497 8 . . PUNCT ejpam-2159 498 1 american american PROPN ejpam-2159 498 2 mathematical mathematical PROPN ejpam-2159 498 3 monthly monthly ADV ejpam-2159 498 4 , , PUNCT ejpam-2159 498 5 108:232–245 108:232–245 NUM ejpam-2159 498 6 , , PUNCT ejpam-2159 498 7 2001 2001 NUM ejpam-2159 498 8 . . PUNCT ejpam-2159 499 1 [ [ X ejpam-2159 499 2 2 2 NUM ejpam-2159 499 3 ] ] PUNCT ejpam-2159 499 4 m. m. PROPN ejpam-2159 499 5 b. b. PROPN ejpam-2159 499 6 allen allen PROPN ejpam-2159 499 7 and and CCONJ ejpam-2159 499 8 e. e. PROPN ejpam-2159 499 9 l. l. PROPN ejpam-2159 499 10 isaacson isaacson PROPN ejpam-2159 499 11 . . PROPN ejpam-2159 500 1 numerical numerical PROPN ejpam-2159 500 2 analysis analysis NOUN ejpam-2159 500 3 for for ADP ejpam-2159 500 4 applied applied ADJ ejpam-2159 500 5 science science NOUN ejpam-2159 500 6 . . PUNCT ejpam-2159 501 1 john john PROPN ejpam-2159 501 2 wiley wiley PROPN ejpam-2159 501 3 & & CCONJ ejpam-2159 501 4 sons sons PROPN ejpam-2159 501 5 , , PUNCT ejpam-2159 501 6 hoboken hoboken PROPN ejpam-2159 501 7 , , PUNCT ejpam-2159 501 8 usa usa PROPN ejpam-2159 501 9 , , PUNCT ejpam-2159 501 10 1997 1997 NUM ejpam-2159 501 11 . . PUNCT ejpam-2159 502 1 [ [ X ejpam-2159 502 2 3 3 X ejpam-2159 502 3 ] ] X ejpam-2159 502 4 r. r. PROPN ejpam-2159 502 5 brawer brawer PROPN ejpam-2159 502 6 and and CCONJ ejpam-2159 502 7 m. m. NOUN ejpam-2159 502 8 pirovino pirovino PROPN ejpam-2159 502 9 . . PUNCT ejpam-2159 503 1 the the DET ejpam-2159 503 2 linear linear PROPN ejpam-2159 503 3 algebra algebra NOUN ejpam-2159 503 4 of of ADP ejpam-2159 503 5 the the DET ejpam-2159 503 6 pascal pascal ADJ ejpam-2159 503 7 matrix matrix NOUN ejpam-2159 503 8 . . PUNCT ejpam-2159 504 1 linear linear ADJ ejpam-2159 504 2 algebra algebra NOUN ejpam-2159 504 3 and and CCONJ ejpam-2159 504 4 its its PRON ejpam-2159 504 5 applications application NOUN ejpam-2159 504 6 , , PUNCT ejpam-2159 504 7 174:13–23 174:13–23 NUM ejpam-2159 504 8 , , PUNCT ejpam-2159 504 9 1992 1992 NUM ejpam-2159 504 10 . . PUNCT ejpam-2159 505 1 references reference NOUN ejpam-2159 505 2 149 149 NUM ejpam-2159 505 3 [ [ X ejpam-2159 505 4 4 4 NUM ejpam-2159 505 5 ] ] X ejpam-2159 505 6 g. g. PROPN ejpam-2159 505 7 s. s. PROPN ejpam-2159 505 8 call call PROPN ejpam-2159 505 9 and and CCONJ ejpam-2159 505 10 d. d. PROPN ejpam-2159 505 11 j. j. PROPN ejpam-2159 505 12 velleman velleman PROPN ejpam-2159 505 13 . . PUNCT ejpam-2159 506 1 pascal pascal PROPN ejpam-2159 506 2 ’s ’s PART ejpam-2159 506 3 matrices matrix NOUN ejpam-2159 506 4 . . PUNCT ejpam-2159 507 1 american american PROPN ejpam-2159 507 2 mathematical mathematical PROPN ejpam-2159 507 3 monthly monthly ADV ejpam-2159 507 4 , , PUNCT ejpam-2159 507 5 100(4):372–376 100(4):372–376 NUM ejpam-2159 507 6 , , PUNCT ejpam-2159 507 7 1993 1993 NUM ejpam-2159 507 8 . . PUNCT ejpam-2159 508 1 [ [ X ejpam-2159 508 2 5 5 X ejpam-2159 508 3 ] ] PUNCT ejpam-2159 508 4 g. g. PROPN ejpam-2159 508 5 s. s. PROPN ejpam-2159 508 6 cheon cheon PROPN ejpam-2159 508 7 and and CCONJ ejpam-2159 508 8 m. m. PROPN ejpam-2159 508 9 el el PROPN ejpam-2159 508 10 - - PUNCT ejpam-2159 508 11 mikkawy mikkawy PROPN ejpam-2159 508 12 . . PUNCT ejpam-2159 509 1 extended extend VERB ejpam-2159 509 2 symmetric symmetric ADJ ejpam-2159 509 3 pascal pascal ADJ ejpam-2159 509 4 matrices matrix NOUN ejpam-2159 509 5 via via ADP ejpam-2159 509 6 hypergeometric hypergeometric ADJ ejpam-2159 509 7 functions function NOUN ejpam-2159 509 8 . . PUNCT ejpam-2159 510 1 applied apply VERB ejpam-2159 510 2 mathematics mathematic NOUN ejpam-2159 510 3 and and CCONJ ejpam-2159 510 4 computation computation NOUN ejpam-2159 510 5 , , PUNCT ejpam-2159 510 6 158:159–168 158:159–168 NUM ejpam-2159 510 7 , , PUNCT ejpam-2159 510 8 2004 2004 NUM ejpam-2159 510 9 . . PUNCT ejpam-2159 511 1 [ [ X ejpam-2159 511 2 6 6 NUM ejpam-2159 511 3 ] ] PUNCT ejpam-2159 511 4 g. g. PROPN ejpam-2159 511 5 s. s. PROPN ejpam-2159 511 6 cheon cheon PROPN ejpam-2159 511 7 and and CCONJ ejpam-2159 511 8 j. j. PROPN ejpam-2159 511 9 s. s. PROPN ejpam-2159 511 10 kim kim PROPN ejpam-2159 511 11 . . PUNCT ejpam-2159 512 1 stirling stirling NOUN ejpam-2159 512 2 matrix matrix NOUN ejpam-2159 512 3 via via ADP ejpam-2159 512 4 pascal pascal ADJ ejpam-2159 512 5 matrix matrix NOUN ejpam-2159 512 6 . . PUNCT ejpam-2159 513 1 linear linear ADJ ejpam-2159 513 2 algebra algebra NOUN ejpam-2159 513 3 and and CCONJ ejpam-2159 513 4 its its PRON ejpam-2159 513 5 applications application NOUN ejpam-2159 513 6 , , PUNCT ejpam-2159 513 7 329:49–59 329:49–59 PROPN ejpam-2159 513 8 , , PUNCT ejpam-2159 513 9 2001 2001 NUM ejpam-2159 513 10 . . PUNCT ejpam-2159 514 1 [ [ X ejpam-2159 514 2 7 7 X ejpam-2159 514 3 ] ] X ejpam-2159 514 4 g. g. PROPN ejpam-2159 514 5 s. s. PROPN ejpam-2159 514 6 cheon cheon PROPN ejpam-2159 514 7 and and CCONJ ejpam-2159 514 8 j. j. PROPN ejpam-2159 514 9 s. s. PROPN ejpam-2159 514 10 kim kim PROPN ejpam-2159 514 11 . . PUNCT ejpam-2159 514 12 factorial factorial PROPN ejpam-2159 514 13 stirling stirling NOUN ejpam-2159 514 14 matrix matrix NOUN ejpam-2159 514 15 and and CCONJ ejpam-2159 514 16 related relate VERB ejpam-2159 514 17 combinatorial combinatorial ADJ ejpam-2159 514 18 sequences sequence NOUN ejpam-2159 514 19 . . PUNCT ejpam-2159 515 1 linear linear PROPN ejpam-2159 515 2 algebra algebra NOUN ejpam-2159 515 3 and and CCONJ ejpam-2159 515 4 its its PRON ejpam-2159 515 5 applications application NOUN ejpam-2159 515 6 , , PUNCT ejpam-2159 515 7 357:247–258 357:247–258 NUM ejpam-2159 515 8 , , PUNCT ejpam-2159 515 9 2002 2002 NUM ejpam-2159 515 10 . . PUNCT ejpam-2159 516 1 [ [ X ejpam-2159 516 2 8 8 X ejpam-2159 516 3 ] ] PUNCT ejpam-2159 516 4 j. j. PROPN ejpam-2159 516 5 w. w. PROPN ejpam-2159 516 6 demmel demmel PROPN ejpam-2159 516 7 . . PUNCT ejpam-2159 517 1 applied apply VERB ejpam-2159 517 2 numerical numerical PROPN ejpam-2159 517 3 linear linear PROPN ejpam-2159 517 4 algebra algebra PROPN ejpam-2159 517 5 . . PUNCT ejpam-2159 518 1 siam siam PROPN ejpam-2159 518 2 , , PUNCT ejpam-2159 518 3 philadelphia philadelphia PROPN ejpam-2159 518 4 , , PUNCT ejpam-2159 518 5 1997 1997 NUM ejpam-2159 518 6 . . PUNCT ejpam-2159 519 1 [ [ X ejpam-2159 519 2 9 9 NUM ejpam-2159 519 3 ] ] PUNCT ejpam-2159 519 4 a. a. NOUN ejpam-2159 519 5 edelman edelman NOUN ejpam-2159 519 6 and and CCONJ ejpam-2159 519 7 g. g. PROPN ejpam-2159 519 8 strangl strangl PROPN ejpam-2159 519 9 . . PUNCT ejpam-2159 520 1 pascal pascal ADJ ejpam-2159 520 2 matrices matrix NOUN ejpam-2159 520 3 . . PUNCT ejpam-2159 521 1 american american PROPN ejpam-2159 521 2 mathematical mathematical PROPN ejpam-2159 521 3 monthly monthly PROPN ejpam-2159 521 4 , , PUNCT ejpam-2159 521 5 cambridge cambridge PROPN ejpam-2159 521 6 , , PUNCT ejpam-2159 521 7 2003 2003 NUM ejpam-2159 521 8 . . PUNCT ejpam-2159 522 1 [ [ X ejpam-2159 522 2 10 10 NUM ejpam-2159 522 3 ] ] PUNCT ejpam-2159 522 4 m. m. NOUN ejpam-2159 522 5 e. e. PROPN ejpam-2159 522 6 a. a. PROPN ejpam-2159 522 7 el el PROPN ejpam-2159 522 8 - - PUNCT ejpam-2159 522 9 mikkawy mikkawy PROPN ejpam-2159 522 10 . . PUNCT ejpam-2159 523 1 an an DET ejpam-2159 523 2 algorithm algorithm NOUN ejpam-2159 523 3 for for ADP ejpam-2159 523 4 solving solve VERB ejpam-2159 523 5 vandermonde vandermonde NOUN ejpam-2159 523 6 systems system NOUN ejpam-2159 523 7 . . PUNCT ejpam-2159 524 1 journal journal PROPN ejpam-2159 524 2 of of ADP ejpam-2159 524 3 institute institute PROPN ejpam-2159 524 4 of of ADP ejpam-2159 524 5 mathematics mathematics PROPN ejpam-2159 524 6 and and CCONJ ejpam-2159 524 7 computer computer NOUN ejpam-2159 524 8 science science NOUN ejpam-2159 524 9 , , PUNCT ejpam-2159 524 10 3(3):293–297 3(3):293–297 NUM ejpam-2159 524 11 , , PUNCT ejpam-2159 524 12 1990 1990 NUM ejpam-2159 524 13 . . PUNCT ejpam-2159 525 1 [ [ X ejpam-2159 525 2 11 11 NUM ejpam-2159 525 3 ] ] PUNCT ejpam-2159 525 4 m. m. PROPN ejpam-2159 525 5 e. e. PROPN ejpam-2159 525 6 a. a. PROPN ejpam-2159 525 7 el el PROPN ejpam-2159 525 8 - - PUNCT ejpam-2159 525 9 mikkawy mikkawy NOUN ejpam-2159 525 10 . . PUNCT ejpam-2159 526 1 explicit explicit ADJ ejpam-2159 526 2 inverse inverse NOUN ejpam-2159 526 3 of of ADP ejpam-2159 526 4 a a DET ejpam-2159 526 5 generalized generalized ADJ ejpam-2159 526 6 vandermonde vandermonde NOUN ejpam-2159 526 7 matrix matrix NOUN ejpam-2159 526 8 . . PUNCT ejpam-2159 527 1 applied apply VERB ejpam-2159 527 2 mathematics mathematic NOUN ejpam-2159 527 3 and and CCONJ ejpam-2159 527 4 computation computation NOUN ejpam-2159 527 5 , , PUNCT ejpam-2159 527 6 146:643–651 146:643–651 NUM ejpam-2159 527 7 , , PUNCT ejpam-2159 527 8 2003 2003 NUM ejpam-2159 527 9 . . PUNCT ejpam-2159 528 1 [ [ X ejpam-2159 528 2 12 12 NUM ejpam-2159 528 3 ] ] PUNCT ejpam-2159 528 4 m. m. NOUN ejpam-2159 528 5 e. e. PROPN ejpam-2159 528 6 a. a. PROPN ejpam-2159 528 7 el el PROPN ejpam-2159 528 8 - - PUNCT ejpam-2159 528 9 mikkawy mikkawy NOUN ejpam-2159 528 10 . . PUNCT ejpam-2159 529 1 on on ADP ejpam-2159 529 2 a a DET ejpam-2159 529 3 connection connection NOUN ejpam-2159 529 4 between between ADP ejpam-2159 529 5 symmetric symmetric ADJ ejpam-2159 529 6 polynomials polynomial NOUN ejpam-2159 529 7 , , PUNCT ejpam-2159 529 8 generalized generalize VERB ejpam-2159 529 9 stirling stirling NOUN ejpam-2159 529 10 numbers number NOUN ejpam-2159 529 11 and and CCONJ ejpam-2159 529 12 the the DET ejpam-2159 529 13 newton newton PROPN ejpam-2159 529 14 general general ADJ ejpam-2159 529 15 divided divide VERB ejpam-2159 529 16 difference difference NOUN ejpam-2159 529 17 interpolation interpolation NOUN ejpam-2159 529 18 polynomial polynomial NOUN ejpam-2159 529 19 . . PUNCT ejpam-2159 530 1 applied apply VERB ejpam-2159 530 2 mathematics mathematic NOUN ejpam-2159 530 3 and and CCONJ ejpam-2159 530 4 computation computation NOUN ejpam-2159 530 5 , , PUNCT ejpam-2159 530 6 138(2):375–385 138(2):375–385 NUM ejpam-2159 530 7 , , PUNCT ejpam-2159 530 8 2003 2003 NUM ejpam-2159 530 9 . . PUNCT ejpam-2159 531 1 [ [ X ejpam-2159 531 2 13 13 NUM ejpam-2159 531 3 ] ] PUNCT ejpam-2159 531 4 m. m. NOUN ejpam-2159 531 5 e. e. PROPN ejpam-2159 531 6 a. a. PROPN ejpam-2159 531 7 el el PROPN ejpam-2159 531 8 - - PUNCT ejpam-2159 531 9 mikkawy mikkawy NOUN ejpam-2159 531 10 . . PUNCT ejpam-2159 532 1 on on ADP ejpam-2159 532 2 a a DET ejpam-2159 532 3 connection connection NOUN ejpam-2159 532 4 between between ADP ejpam-2159 532 5 the the DET ejpam-2159 532 6 pascal pascal NOUN ejpam-2159 532 7 , , PUNCT ejpam-2159 532 8 vandermonde vandermonde NOUN ejpam-2159 532 9 and and CCONJ ejpam-2159 532 10 stirling stirling NOUN ejpam-2159 532 11 matrices matrix NOUN ejpam-2159 532 12 - - PUNCT ejpam-2159 532 13 i. i. NOUN ejpam-2159 532 14 applied apply VERB ejpam-2159 532 15 mathematics mathematic NOUN ejpam-2159 532 16 and and CCONJ ejpam-2159 532 17 computation computation NOUN ejpam-2159 532 18 , , PUNCT ejpam-2159 532 19 145(1):23–32 145(1):23–32 NUM ejpam-2159 532 20 , , PUNCT ejpam-2159 532 21 2003 2003 NUM ejpam-2159 532 22 . . PUNCT ejpam-2159 533 1 [ [ X ejpam-2159 533 2 14 14 NUM ejpam-2159 533 3 ] ] PUNCT ejpam-2159 533 4 m. m. PROPN ejpam-2159 533 5 e. e. PROPN ejpam-2159 533 6 a. a. PROPN ejpam-2159 533 7 el el PROPN ejpam-2159 533 8 - - PUNCT ejpam-2159 533 9 mikkawy mikkawy NOUN ejpam-2159 533 10 . . PUNCT ejpam-2159 534 1 on on ADP ejpam-2159 534 2 a a DET ejpam-2159 534 3 connection connection NOUN ejpam-2159 534 4 between between ADP ejpam-2159 534 5 the the DET ejpam-2159 534 6 pascal pascal NOUN ejpam-2159 534 7 , , PUNCT ejpam-2159 534 8 vandermonde vandermonde NOUN ejpam-2159 534 9 and and CCONJ ejpam-2159 534 10 stirling stirling NOUN ejpam-2159 534 11 matrices matrix NOUN ejpam-2159 534 12 - - PUNCT ejpam-2159 534 13 ii ii NOUN ejpam-2159 534 14 . . PUNCT ejpam-2159 534 15 applied apply VERB ejpam-2159 534 16 mathematics mathematic NOUN ejpam-2159 534 17 and and CCONJ ejpam-2159 534 18 computation computation NOUN ejpam-2159 534 19 , , PUNCT ejpam-2159 534 20 146(2):759–769 146(2):759–769 NUM ejpam-2159 534 21 , , PUNCT ejpam-2159 534 22 2003 2003 NUM ejpam-2159 534 23 . . PUNCT ejpam-2159 535 1 [ [ X ejpam-2159 535 2 15 15 NUM ejpam-2159 535 3 ] ] X ejpam-2159 535 4 m. m. NOUN ejpam-2159 535 5 e. e. PROPN ejpam-2159 535 6 a. a. PROPN ejpam-2159 535 7 el el PROPN ejpam-2159 535 8 - - PUNCT ejpam-2159 535 9 mikkawy mikkawy NOUN ejpam-2159 535 10 . . PUNCT ejpam-2159 536 1 on on ADP ejpam-2159 536 2 solving solve VERB ejpam-2159 536 3 linear linear NOUN ejpam-2159 536 4 systems system NOUN ejpam-2159 536 5 of of ADP ejpam-2159 536 6 the the DET ejpam-2159 536 7 pascal pascal ADJ ejpam-2159 536 8 type type NOUN ejpam-2159 536 9 . . PUNCT ejpam-2159 537 1 applied apply VERB ejpam-2159 537 2 mathematics mathematic NOUN ejpam-2159 537 3 and and CCONJ ejpam-2159 537 4 computation computation NOUN ejpam-2159 537 5 , , PUNCT ejpam-2159 537 6 136:195–202 136:195–202 NUM ejpam-2159 537 7 , , PUNCT ejpam-2159 537 8 2003 2003 NUM ejpam-2159 537 9 . . PUNCT ejpam-2159 538 1 [ [ X ejpam-2159 538 2 16 16 NUM ejpam-2159 538 3 ] ] PUNCT ejpam-2159 538 4 m. m. PROPN ejpam-2159 538 5 e. e. PROPN ejpam-2159 538 6 a. a. PROPN ejpam-2159 538 7 el el PROPN ejpam-2159 538 8 - - PUNCT ejpam-2159 538 9 mikkawy mikkawy PROPN ejpam-2159 538 10 and and CCONJ ejpam-2159 538 11 f. f. PROPN ejpam-2159 538 12 atlan atlan PROPN ejpam-2159 538 13 . . PUNCT ejpam-2159 539 1 remarks remark NOUN ejpam-2159 539 2 on on ADP ejpam-2159 539 3 two two NUM ejpam-2159 539 4 symmetric symmetric ADJ ejpam-2159 539 5 polynomials polynomial NOUN ejpam-2159 539 6 and and CCONJ ejpam-2159 539 7 some some DET ejpam-2159 539 8 matrices matrix NOUN ejpam-2159 539 9 . . PUNCT ejpam-2159 540 1 applied apply VERB ejpam-2159 540 2 mathematics mathematic NOUN ejpam-2159 540 3 and and CCONJ ejpam-2159 540 4 computation computation NOUN ejpam-2159 540 5 , , PUNCT ejpam-2159 540 6 219:8770–8778 219:8770–8778 NUM ejpam-2159 540 7 , , PUNCT ejpam-2159 540 8 2013 2013 NUM ejpam-2159 540 9 . . PUNCT ejpam-2159 541 1 [ [ X ejpam-2159 541 2 17 17 NUM ejpam-2159 541 3 ] ] PUNCT ejpam-2159 541 4 m. m. PROPN ejpam-2159 541 5 e. e. PROPN ejpam-2159 541 6 a. a. PROPN ejpam-2159 541 7 el el PROPN ejpam-2159 541 8 - - PUNCT ejpam-2159 541 9 mikkawy mikkawy PROPN ejpam-2159 541 10 and and CCONJ ejpam-2159 541 11 t. t. NOUN ejpam-2159 541 12 sogabe sogabe NOUN ejpam-2159 541 13 . . PUNCT ejpam-2159 542 1 notes note NOUN ejpam-2159 542 2 on on ADP ejpam-2159 542 3 particular particular ADJ ejpam-2159 542 4 symmetric symmetric ADJ ejpam-2159 542 5 polynomials polynomial NOUN ejpam-2159 542 6 with with ADP ejpam-2159 542 7 applications application NOUN ejpam-2159 542 8 . . PUNCT ejpam-2159 543 1 applied apply VERB ejpam-2159 543 2 mathematics mathematic NOUN ejpam-2159 543 3 and and CCONJ ejpam-2159 543 4 computation computation NOUN ejpam-2159 543 5 , , PUNCT ejpam-2159 543 6 215:3311–3317 215:3311–3317 NUM ejpam-2159 543 7 , , PUNCT ejpam-2159 543 8 2010 2010 NUM ejpam-2159 543 9 . . PUNCT ejpam-2159 544 1 [ [ X ejpam-2159 544 2 18 18 NUM ejpam-2159 544 3 ] ] X ejpam-2159 544 4 s. s. PROPN ejpam-2159 544 5 m. m. PROPN ejpam-2159 544 6 fallat fallat PROPN ejpam-2159 544 7 . . PUNCT ejpam-2159 545 1 bidiagonal bidiagonal ADJ ejpam-2159 545 2 factorizations factorization NOUN ejpam-2159 545 3 of of ADP ejpam-2159 545 4 totally totally ADV ejpam-2159 545 5 nonnegative nonnegative ADJ ejpam-2159 545 6 matrices matrix NOUN ejpam-2159 545 7 . . PUNCT ejpam-2159 546 1 american american PROPN ejpam-2159 546 2 mathematical mathematical PROPN ejpam-2159 546 3 monthly monthly ADJ ejpam-2159 546 4 , , PUNCT ejpam-2159 546 5 108(8):697–712 108(8):697–712 NUM ejpam-2159 546 6 , , PUNCT ejpam-2159 546 7 2001 2001 NUM ejpam-2159 546 8 . . PUNCT ejpam-2159 547 1 [ [ X ejpam-2159 547 2 19 19 NUM ejpam-2159 547 3 ] ] PUNCT ejpam-2159 547 4 t. t. NOUN ejpam-2159 547 5 x. x. NOUN ejpam-2159 547 6 he he PRON ejpam-2159 547 7 and and CCONJ ejpam-2159 547 8 j. j. PROPN ejpam-2159 547 9 s. s. PROPN ejpam-2159 547 10 shiue shiue VERB ejpam-2159 547 11 . . PUNCT ejpam-2159 548 1 a a DET ejpam-2159 548 2 note note NOUN ejpam-2159 548 3 on on ADP ejpam-2159 548 4 horner horner PROPN ejpam-2159 548 5 ’s ’s PART ejpam-2159 548 6 method method PROPN ejpam-2159 548 7 . . PUNCT ejpam-2159 549 1 journal journal NOUN ejpam-2159 549 2 of of ADP ejpam-2159 549 3 concrete concrete ADJ ejpam-2159 549 4 and and CCONJ ejpam-2159 549 5 applicable applicable ADJ ejpam-2159 549 6 mathematics mathematic NOUN ejpam-2159 549 7 , , PUNCT ejpam-2159 549 8 10(1):53–64 10(1):53–64 NUM ejpam-2159 549 9 , , PUNCT ejpam-2159 549 10 2012 2012 NUM ejpam-2159 549 11 . . PUNCT ejpam-2159 550 1 [ [ X ejpam-2159 550 2 20 20 NUM ejpam-2159 550 3 ] ] PUNCT ejpam-2159 550 4 j. j. PROPN ejpam-2159 550 5 g. g. PROPN ejpam-2159 550 6 kemeny kemeny PROPN ejpam-2159 550 7 and and CCONJ ejpam-2159 550 8 j. j. PROPN ejpam-2159 550 9 l. l. PROPN ejpam-2159 550 10 snell snell PROPN ejpam-2159 550 11 . . PUNCT ejpam-2159 551 1 finite finite PROPN ejpam-2159 551 2 markov markov NOUN ejpam-2159 551 3 chains chain NOUN ejpam-2159 551 4 . . PUNCT ejpam-2159 552 1 springer springer NOUN ejpam-2159 552 2 - - PUNCT ejpam-2159 552 3 verlag verlag PROPN ejpam-2159 552 4 , , PUNCT ejpam-2159 552 5 new new PROPN ejpam-2159 552 6 york york PROPN ejpam-2159 552 7 , , PUNCT ejpam-2159 552 8 1976 1976 NUM ejpam-2159 552 9 . . PUNCT ejpam-2159 553 1 references reference NOUN ejpam-2159 553 2 150 150 NUM ejpam-2159 553 3 [ [ X ejpam-2159 553 4 21 21 NUM ejpam-2159 553 5 ] ] PUNCT ejpam-2159 553 6 x. x. NOUN ejpam-2159 553 7 g. g. PROPN ejpam-2159 553 8 lv lv PROPN ejpam-2159 553 9 , , PUNCT ejpam-2159 553 10 t. t. PROPN ejpam-2159 553 11 z. z. PROPN ejpam-2159 553 12 huang huang PROPN ejpam-2159 553 13 , , PUNCT ejpam-2159 553 14 and and CCONJ ejpam-2159 553 15 z. z. PROPN ejpam-2159 553 16 g. g. PROPN ejpam-2159 553 17 ren ren PROPN ejpam-2159 553 18 . . PUNCT ejpam-2159 554 1 a a DET ejpam-2159 554 2 new new ADJ ejpam-2159 554 3 algorithm algorithm NOUN ejpam-2159 554 4 for for ADP ejpam-2159 554 5 linear linear PROPN ejpam-2159 554 6 systems system NOUN ejpam-2159 554 7 of of ADP ejpam-2159 554 8 the the DET ejpam-2159 554 9 pascal pascal ADJ ejpam-2159 554 10 type type NOUN ejpam-2159 554 11 . . PUNCT ejpam-2159 555 1 journal journal PROPN ejpam-2159 555 2 of of ADP ejpam-2159 555 3 computational computational ADJ ejpam-2159 555 4 and and CCONJ ejpam-2159 555 5 applied applied ADJ ejpam-2159 555 6 mathematics mathematic NOUN ejpam-2159 555 7 , , PUNCT ejpam-2159 555 8 225(1):309–315 225(1):309–315 PROPN ejpam-2159 555 9 , , PUNCT ejpam-2159 555 10 2009 2009 NUM ejpam-2159 555 11 . . PUNCT ejpam-2159 556 1 [ [ X ejpam-2159 556 2 22 22 NUM ejpam-2159 556 3 ] ] X ejpam-2159 556 4 c. c. PROPN ejpam-2159 556 5 y. y. PROPN ejpam-2159 556 6 ma ma PROPN ejpam-2159 556 7 and and CCONJ ejpam-2159 556 8 s. s. PROPN ejpam-2159 556 9 l. l. PROPN ejpam-2159 556 10 yang yang PROPN ejpam-2159 556 11 . . PUNCT ejpam-2159 557 1 pascal pascal ADJ ejpam-2159 557 2 type type NOUN ejpam-2159 557 3 matrices matrix NOUN ejpam-2159 557 4 and and CCONJ ejpam-2159 557 5 bernoulli bernoulli NOUN ejpam-2159 557 6 numbers number NOUN ejpam-2159 557 7 . . PUNCT ejpam-2159 558 1 international international ADJ ejpam-2159 558 2 journal journal NOUN ejpam-2159 558 3 of of ADP ejpam-2159 558 4 pure pure ADJ ejpam-2159 558 5 and and CCONJ ejpam-2159 558 6 applied applied ADJ ejpam-2159 558 7 mathematics mathematic NOUN ejpam-2159 558 8 , , PUNCT ejpam-2159 558 9 58(3):249–254 58(3):249–254 NUM ejpam-2159 558 10 , , PUNCT ejpam-2159 558 11 2010 2010 NUM ejpam-2159 558 12 . . PUNCT ejpam-2159 559 1 [ [ X ejpam-2159 559 2 23 23 NUM ejpam-2159 559 3 ] ] PUNCT ejpam-2159 559 4 e. e. PROPN ejpam-2159 559 5 n. n. PROPN ejpam-2159 559 6 onwuchekwa onwuchekwa PROPN ejpam-2159 559 7 . . PUNCT ejpam-2159 560 1 some some DET ejpam-2159 560 2 classes class NOUN ejpam-2159 560 3 of of ADP ejpam-2159 560 4 lower low ADJ ejpam-2159 560 5 triangular triangular NOUN ejpam-2159 560 6 matrices matrix NOUN ejpam-2159 560 7 and and CCONJ ejpam-2159 560 8 their their PRON ejpam-2159 560 9 inverses inverse NOUN ejpam-2159 560 10 . . PUNCT ejpam-2159 561 1 international international ADJ ejpam-2159 561 2 journal journal PROPN ejpam-2159 561 3 of of ADP ejpam-2159 561 4 mathematical mathematical ADJ ejpam-2159 561 5 sciences science NOUN ejpam-2159 561 6 and and CCONJ ejpam-2159 561 7 applications application NOUN ejpam-2159 561 8 , , PUNCT ejpam-2159 561 9 1(3):1169–1180 1(3):1169–1180 NUM ejpam-2159 561 10 , , PUNCT ejpam-2159 561 11 2011 2011 NUM ejpam-2159 561 12 . . PUNCT ejpam-2159 562 1 [ [ X ejpam-2159 562 2 24 24 NUM ejpam-2159 562 3 ] ] X ejpam-2159 562 4 h. h. PROPN ejpam-2159 562 5 oruc oruc PROPN ejpam-2159 562 6 and and CCONJ ejpam-2159 562 7 h. h. PROPN ejpam-2159 562 8 k. k. PROPN ejpam-2159 563 1 akmaz akmaz PROPN ejpam-2159 563 2 . . PUNCT ejpam-2159 564 1 symmetric symmetric ADJ ejpam-2159 564 2 functions function NOUN ejpam-2159 564 3 and and CCONJ ejpam-2159 564 4 the the DET ejpam-2159 564 5 vandermonde vandermonde ADJ ejpam-2159 564 6 matrix matrix NOUN ejpam-2159 564 7 . . PUNCT ejpam-2159 565 1 journal journal NOUN ejpam-2159 565 2 of of ADP ejpam-2159 565 3 computational computational ADJ ejpam-2159 565 4 and and CCONJ ejpam-2159 565 5 applied applied ADJ ejpam-2159 565 6 mathematics mathematic NOUN ejpam-2159 565 7 , , PUNCT ejpam-2159 565 8 172:49–64 172:49–64 NUM ejpam-2159 565 9 , , PUNCT ejpam-2159 565 10 2004 2004 NUM ejpam-2159 565 11 . . PUNCT ejpam-2159 566 1 [ [ X ejpam-2159 566 2 25 25 NUM ejpam-2159 566 3 ] ] PUNCT ejpam-2159 566 4 t. t. NOUN ejpam-2159 566 5 sogabe sogabe NOUN ejpam-2159 566 6 and and CCONJ ejpam-2159 566 7 m. m. PROPN ejpam-2159 566 8 e. e. PROPN ejpam-2159 566 9 a. a. PROPN ejpam-2159 566 10 el el PROPN ejpam-2159 566 11 - - PUNCT ejpam-2159 566 12 mikkawy mikkawy NOUN ejpam-2159 566 13 . . PUNCT ejpam-2159 567 1 on on ADP ejpam-2159 567 2 a a DET ejpam-2159 567 3 problem problem NOUN ejpam-2159 567 4 related relate VERB ejpam-2159 567 5 to to ADP ejpam-2159 567 6 the the DET ejpam-2159 567 7 vandermonde vandermonde NOUN ejpam-2159 567 8 determinant determinant ADJ ejpam-2159 567 9 . . PUNCT ejpam-2159 568 1 discrete discrete ADJ ejpam-2159 568 2 applied apply VERB ejpam-2159 568 3 mathematics mathematic NOUN ejpam-2159 568 4 , , PUNCT ejpam-2159 568 5 157:2997–2999 157:2997–2999 NUM ejpam-2159 568 6 , , PUNCT ejpam-2159 568 7 2009 2009 NUM ejpam-2159 568 8 . . PUNCT ejpam-2159 569 1 [ [ X ejpam-2159 569 2 26 26 NUM ejpam-2159 569 3 ] ] X ejpam-2159 569 4 r. r. PROPN ejpam-2159 569 5 vein vein PROPN ejpam-2159 569 6 and and CCONJ ejpam-2159 569 7 p. p. NOUN ejpam-2159 569 8 dale dale PROPN ejpam-2159 569 9 . . PUNCT ejpam-2159 570 1 determinants determinant NOUN ejpam-2159 570 2 and and CCONJ ejpam-2159 570 3 their their PRON ejpam-2159 570 4 applications application NOUN ejpam-2159 570 5 in in ADP ejpam-2159 570 6 mathematical mathematical ADJ ejpam-2159 570 7 physics physics NOUN ejpam-2159 570 8 . . PUNCT ejpam-2159 571 1 springer springer PROPN ejpam-2159 571 2 , , PUNCT ejpam-2159 571 3 new new PROPN ejpam-2159 571 4 york york PROPN ejpam-2159 571 5 , , PUNCT ejpam-2159 571 6 1999 1999 NUM ejpam-2159 571 7 . . PUNCT ejpam-2159 572 1 [ [ X ejpam-2159 572 2 27 27 NUM ejpam-2159 572 3 ] ] X ejpam-2159 572 4 w. w. PROPN ejpam-2159 572 5 wang wang PROPN ejpam-2159 572 6 and and CCONJ ejpam-2159 572 7 t. t. PROPN ejpam-2159 572 8 wang wang PROPN ejpam-2159 572 9 . . PUNCT ejpam-2159 573 1 commentary commentary NOUN ejpam-2159 573 2 on on ADP ejpam-2159 573 3 an an DET ejpam-2159 573 4 open open ADJ ejpam-2159 573 5 question question NOUN ejpam-2159 573 6 . . PUNCT ejpam-2159 574 1 applied apply VERB ejpam-2159 574 2 mathematics mathematic NOUN ejpam-2159 574 3 and and CCONJ ejpam-2159 574 4 computation computation NOUN ejpam-2159 574 5 , , PUNCT ejpam-2159 574 6 196(1):353–355 196(1):353–355 NUM ejpam-2159 574 7 , , PUNCT ejpam-2159 574 8 2008 2008 NUM ejpam-2159 574 9 . . PUNCT ejpam-2159 575 1 [ [ X ejpam-2159 575 2 28 28 NUM ejpam-2159 575 3 ] ] X ejpam-2159 575 4 w. w. PROPN ejpam-2159 575 5 wang wang PROPN ejpam-2159 575 6 and and CCONJ ejpam-2159 575 7 t. t. PROPN ejpam-2159 575 8 wang wang PROPN ejpam-2159 575 9 . . PUNCT ejpam-2159 576 1 remarks remark NOUN ejpam-2159 576 2 on on ADP ejpam-2159 576 3 two two NUM ejpam-2159 576 4 special special ADJ ejpam-2159 576 5 matrices matrix NOUN ejpam-2159 576 6 . . PUNCT ejpam-2159 577 1 ars ars PROPN ejpam-2159 577 2 combinatoria combinatoria PROPN ejpam-2159 577 3 , , PUNCT ejpam-2159 577 4 94:521–535 94:521–535 PROPN ejpam-2159 577 5 , , PUNCT ejpam-2159 577 6 2010 2010 NUM ejpam-2159 577 7 . . PUNCT ejpam-2159 578 1 [ [ X ejpam-2159 578 2 29 29 NUM ejpam-2159 578 3 ] ] PUNCT ejpam-2159 578 4 x. x. NOUN ejpam-2159 578 5 wang wang PROPN ejpam-2159 578 6 . . PUNCT ejpam-2159 579 1 a a DET ejpam-2159 579 2 stable stable ADJ ejpam-2159 579 3 fast fast ADJ ejpam-2159 579 4 algorithm algorithm NOUN ejpam-2159 579 5 for for ADP ejpam-2159 579 6 solving solve VERB ejpam-2159 579 7 linear linear NOUN ejpam-2159 579 8 systems system NOUN ejpam-2159 579 9 of of ADP ejpam-2159 579 10 the the DET ejpam-2159 579 11 pascal pascal ADJ ejpam-2159 579 12 type type NOUN ejpam-2159 579 13 . . PUNCT ejpam-2159 580 1 journal journal NOUN ejpam-2159 580 2 of of ADP ejpam-2159 580 3 computational computational ADJ ejpam-2159 580 4 analysis analysis NOUN ejpam-2159 580 5 and and CCONJ ejpam-2159 580 6 applications application NOUN ejpam-2159 580 7 , , PUNCT ejpam-2159 580 8 9(4):411–419 9(4):411–419 NUM ejpam-2159 580 9 , , PUNCT ejpam-2159 580 10 2007 2007 NUM ejpam-2159 580 11 . . PUNCT ejpam-2159 581 1 [ [ X ejpam-2159 581 2 30 30 NUM ejpam-2159 581 3 ] ] PUNCT ejpam-2159 581 4 x. x. NOUN ejpam-2159 581 5 wang wang PROPN ejpam-2159 581 6 and and CCONJ ejpam-2159 581 7 l. l. PROPN ejpam-2159 581 8 lu lu PROPN ejpam-2159 581 9 . . PUNCT ejpam-2159 582 1 a a DET ejpam-2159 582 2 fast fast ADJ ejpam-2159 582 3 algorithm algorithm NOUN ejpam-2159 582 4 for for ADP ejpam-2159 582 5 solving solve VERB ejpam-2159 582 6 linear linear NOUN ejpam-2159 582 7 systems system NOUN ejpam-2159 582 8 of of ADP ejpam-2159 582 9 the the DET ejpam-2159 582 10 pascal pascal ADJ ejpam-2159 582 11 type type NOUN ejpam-2159 582 12 . . PUNCT ejpam-2159 583 1 applied apply VERB ejpam-2159 583 2 mathematics mathematic NOUN ejpam-2159 583 3 and and CCONJ ejpam-2159 583 4 computation computation NOUN ejpam-2159 583 5 , , PUNCT ejpam-2159 583 6 175(1):441–451 175(1):441–451 NUM ejpam-2159 583 7 , , PUNCT ejpam-2159 583 8 2006 2006 NUM ejpam-2159 583 9 . . PUNCT ejpam-2159 584 1 [ [ X ejpam-2159 584 2 31 31 NUM ejpam-2159 584 3 ] ] PUNCT ejpam-2159 584 4 s. s. PROPN ejpam-2159 584 5 l. l. PROPN ejpam-2159 584 6 yang yang PROPN ejpam-2159 584 7 . . PUNCT ejpam-2159 585 1 on on ADP ejpam-2159 585 2 the the DET ejpam-2159 585 3 lu lu NOUN ejpam-2159 585 4 factorization factorization NOUN ejpam-2159 585 5 of of ADP ejpam-2159 585 6 the the DET ejpam-2159 585 7 vandermonde vandermonde ADJ ejpam-2159 585 8 matrix matrix NOUN ejpam-2159 585 9 . . PUNCT ejpam-2159 586 1 discrete discrete ADJ ejpam-2159 586 2 applied apply VERB ejpam-2159 586 3 mathematics mathematic NOUN ejpam-2159 586 4 , , PUNCT ejpam-2159 586 5 146:102–105 146:102–105 NUM ejpam-2159 586 6 , , PUNCT ejpam-2159 586 7 2005 2005 NUM ejpam-2159 586 8 . . PUNCT ejpam-2159 587 1 [ [ X ejpam-2159 587 2 32 32 NUM ejpam-2159 587 3 ] ] PUNCT ejpam-2159 587 4 s. s. PROPN ejpam-2159 587 5 l. l. PROPN ejpam-2159 587 6 yang yang PROPN ejpam-2159 587 7 . . PUNCT ejpam-2159 588 1 on on ADP ejpam-2159 588 2 a a DET ejpam-2159 588 3 connection connection NOUN ejpam-2159 588 4 between between ADP ejpam-2159 588 5 the the DET ejpam-2159 588 6 pascal pascal NOUN ejpam-2159 588 7 , , PUNCT ejpam-2159 588 8 stirling stirling NOUN ejpam-2159 588 9 and and CCONJ ejpam-2159 588 10 vandermonde vandermonde ADJ ejpam-2159 588 11 matrices matrix NOUN ejpam-2159 588 12 . . PUNCT ejpam-2159 589 1 discrete discrete ADJ ejpam-2159 589 2 applied apply VERB ejpam-2159 589 3 mathematics mathematic NOUN ejpam-2159 589 4 , , PUNCT ejpam-2159 589 5 155(15):2025–2030 155(15):2025–2030 NUM ejpam-2159 589 6 , , PUNCT ejpam-2159 589 7 2007 2007 NUM ejpam-2159 589 8 . . PUNCT ejpam-2159 590 1 [ [ X ejpam-2159 590 2 33 33 NUM ejpam-2159 590 3 ] ] PUNCT ejpam-2159 590 4 s. s. PROPN ejpam-2159 590 5 l. l. PROPN ejpam-2159 590 6 yang yang PROPN ejpam-2159 590 7 and and CCONJ ejpam-2159 590 8 z. z. PROPN ejpam-2159 590 9 k. k. PROPN ejpam-2159 590 10 qiao qiao PROPN ejpam-2159 590 11 . . PUNCT ejpam-2159 591 1 the the DET ejpam-2159 591 2 bessel bessel ADJ ejpam-2159 591 3 numbers number NOUN ejpam-2159 591 4 and and CCONJ ejpam-2159 591 5 bessel bessel ADJ ejpam-2159 591 6 matrices matrix NOUN ejpam-2159 591 7 . . PUNCT ejpam-2159 592 1 journal journal PROPN ejpam-2159 592 2 of of ADP ejpam-2159 592 3 mathematical mathematical ADJ ejpam-2159 592 4 research research NOUN ejpam-2159 592 5 and and CCONJ ejpam-2159 592 6 exposition exposition NOUN ejpam-2159 592 7 , , PUNCT ejpam-2159 592 8 31(4):627–636 31(4):627–636 NUM ejpam-2159 592 9 , , PUNCT ejpam-2159 592 10 2011 2011 NUM ejpam-2159 592 11 . . PUNCT ejpam-2159 593 1 [ [ X ejpam-2159 593 2 34 34 NUM ejpam-2159 593 3 ] ] X ejpam-2159 593 4 s. s. PROPN ejpam-2159 593 5 l. l. PROPN ejpam-2159 593 6 yang yang PROPN ejpam-2159 593 7 and and CCONJ ejpam-2159 593 8 h. h. PROPN ejpam-2159 593 9 you you PRON ejpam-2159 593 10 . . PUNCT ejpam-2159 594 1 on on ADP ejpam-2159 594 2 a a DET ejpam-2159 594 3 relationship relationship NOUN ejpam-2159 594 4 between between ADP ejpam-2159 594 5 pascal pascal ADJ ejpam-2159 594 6 matrix matrix NOUN ejpam-2159 594 7 and and CCONJ ejpam-2159 594 8 vandermonde vandermonde ADJ ejpam-2159 594 9 matrix matrix NOUN ejpam-2159 594 10 . . PUNCT ejpam-2159 595 1 journal journal PROPN ejpam-2159 595 2 of of ADP ejpam-2159 595 3 mathematical mathematical ADJ ejpam-2159 595 4 research research NOUN ejpam-2159 595 5 and and CCONJ ejpam-2159 595 6 exposition exposition NOUN ejpam-2159 595 7 , , PUNCT ejpam-2159 595 8 26(1):33–39 26(1):33–39 NUM ejpam-2159 595 9 , , PUNCT ejpam-2159 595 10 2006 2006 NUM ejpam-2159 595 11 . . PUNCT ejpam-2159 596 1 [ [ X ejpam-2159 596 2 35 35 NUM ejpam-2159 596 3 ] ] PUNCT ejpam-2159 596 4 z. z. PROPN ejpam-2159 596 5 zeng zeng PROPN ejpam-2159 596 6 , , PUNCT ejpam-2159 596 7 y. y. PROPN ejpam-2159 596 8 tu tu PROPN ejpam-2159 596 9 , , PUNCT ejpam-2159 596 10 and and CCONJ ejpam-2159 596 11 j. j. PROPN ejpam-2159 596 12 xiao xiao PROPN ejpam-2159 596 13 . . PUNCT ejpam-2159 597 1 a a DET ejpam-2159 597 2 neural neural ADJ ejpam-2159 597 3 - - PUNCT ejpam-2159 597 4 network network NOUN ejpam-2159 597 5 method method NOUN ejpam-2159 597 6 based base VERB ejpam-2159 597 7 on on ADP ejpam-2159 597 8 rls rls PROPN ejpam-2159 597 9 algorithm algorithm NOUN ejpam-2159 597 10 for for ADP ejpam-2159 597 11 solving solve VERB ejpam-2159 597 12 special special ADJ ejpam-2159 597 13 linear linear NOUN ejpam-2159 597 14 systems system NOUN ejpam-2159 597 15 of of ADP ejpam-2159 597 16 equations equation NOUN ejpam-2159 597 17 . . PUNCT ejpam-2159 598 1 journal journal NOUN ejpam-2159 598 2 of of ADP ejpam-2159 598 3 computational computational ADJ ejpam-2159 598 4 information information NOUN ejpam-2159 598 5 systems system NOUN ejpam-2159 598 6 , , PUNCT ejpam-2159 598 7 8(7):2915–2920 8(7):2915–2920 NUM ejpam-2159 598 8 , , PUNCT ejpam-2159 598 9 2012 2012 NUM ejpam-2159 598 10 . . PUNCT ejpam-2159 599 1 [ [ X ejpam-2159 599 2 36 36 NUM ejpam-2159 599 3 ] ] PUNCT ejpam-2159 599 4 z. z. PROPN ejpam-2159 599 5 z. z. PROPN ejpam-2159 599 6 zhang zhang PROPN ejpam-2159 599 7 . . PUNCT ejpam-2159 600 1 the the DET ejpam-2159 600 2 linear linear PROPN ejpam-2159 600 3 algebra algebra NOUN ejpam-2159 600 4 of of ADP ejpam-2159 600 5 the the DET ejpam-2159 600 6 generalized generalized ADJ ejpam-2159 600 7 pascal pascal ADJ ejpam-2159 600 8 matrix matrix NOUN ejpam-2159 600 9 . . PUNCT ejpam-2159 601 1 linear linear ADJ ejpam-2159 601 2 algebra algebra NOUN ejpam-2159 601 3 and and CCONJ ejpam-2159 601 4 its its PRON ejpam-2159 601 5 applications application NOUN ejpam-2159 601 6 , , PUNCT ejpam-2159 601 7 250:51–60 250:51–60 NUM ejpam-2159 601 8 , , PUNCT ejpam-2159 601 9 1997 1997 NUM ejpam-2159 601 10 . . PUNCT ejpam-2159 602 1 references reference NOUN ejpam-2159 602 2 151 151 NUM ejpam-2159 603 1 [ [ X ejpam-2159 603 2 37 37 NUM ejpam-2159 603 3 ] ] PUNCT ejpam-2159 603 4 z. z. PROPN ejpam-2159 603 5 z. z. PROPN ejpam-2159 603 6 zhang zhang PROPN ejpam-2159 603 7 and and CCONJ ejpam-2159 603 8 m. m. PROPN ejpam-2159 603 9 liu liu PROPN ejpam-2159 603 10 . . PUNCT ejpam-2159 604 1 an an DET ejpam-2159 604 2 extension extension NOUN ejpam-2159 604 3 of of ADP ejpam-2159 604 4 the the DET ejpam-2159 604 5 generalized generalized ADJ ejpam-2159 604 6 pascal pascal ADJ ejpam-2159 604 7 matrix matrix NOUN ejpam-2159 604 8 and and CCONJ ejpam-2159 604 9 its its PRON ejpam-2159 604 10 algebraic algebraic ADJ ejpam-2159 604 11 properties property NOUN ejpam-2159 604 12 . . PUNCT ejpam-2159 605 1 linear linear ADJ ejpam-2159 605 2 algebra algebra NOUN ejpam-2159 605 3 and and CCONJ ejpam-2159 605 4 its its PRON ejpam-2159 605 5 applications application NOUN ejpam-2159 605 6 , , PUNCT ejpam-2159 605 7 271:169–177 271:169–177 NUM ejpam-2159 605 8 , , PUNCT ejpam-2159 605 9 1998 1998 NUM ejpam-2159 605 10 . . PUNCT