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# Organizations and Social Institutions SOCIOL 110

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This 22 page Class Notes was uploaded by Lionel Will on Thursday October 22, 2015. The Class Notes belongs to SOCIOL 110 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/226564/sociol-110-university-of-california-berkeley in Sociology at University of California - Berkeley.

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Date Created: 10/22/15

Paci c Journal of Mathematics A NOTE ON GENERALIZED BERNOULLI NUMBERS KWANG WU CHEN AND MINKING EIE Volume 199 N0 1 May 2001 PACIFIC JOURNAL OF MATHEMATICS Vol 199 No 1 2001 A NOTE ON GENERALIZED BERNOULLI NUMBERS KWANG WU CHEN AND MINKING EIE In this paper7 we consider the zeta function ZPX s as sociated with a polynomial PX e RX1 Xr and X X1 90 with Xj nontrivial Dirichlet characters7 de ned by 00 cc ZP7X75 Z Z X1n1quot39XrnrPn17 77170757 1111 n1 which is absolutely convergent for su iciently large Re 5 under some conditions on PX We shall prove that the special value ZP x m is completely determined by PmX in a simple way As an immediate application7 we give a closed expression for sums of products of any number of generalized Bernoulli numbers 1 Introduction and Notation As usual7 N denotes the set of positive numbers7 N0 N and R denotes the eld of real numbers Let X be a non trivial Dirichlet character with conductor N The L series attached to X is de ned by 0 LSX Zxnnis Res gt1 n1 It is well known 14 that Ls7 x may be continued analytically to the whole complex s plane Furthermore7 the special values at non positive integers s 7m m O7 17 27 can be expressed by the generalized Bernoulli numbers B n O7 17 27 de ned by N xateat 7 00 ngt t 271 eNtiliz nl7HltN39 11 n0 m1 Indeed7 L7mX 7 as given on Page 30 of 14 The generalized Bernoulli numbers can be expressed in terms of Bernoulli polynomials as N a 13 N7 1 ZxaBn N 11 41 42 KWANG7WU CHEN AND MINKING EIE where the Bernoulli polynomials BAX are de ned by teXt 0 tn t 1 237420 w lt 27139 n0 39 Also n 71 BAX Z BHX k0 where the Bernoulli numbers B n 0 1 2 are de ned by 777 t t at 71 EB ltl lt 27139 n0 39 Consequently7 we can express the generalized Bernoulli numbers in terms of Bernoulli numbers as follows N B Zxa Z Bkanikail a1 n k0 Let PX PX17 7X7 be a polynomial ofr variables with non negative real coefficients such that Pn gt 0 for all n 6 NT and the series 2 Pn 9 Z Z Pn1 n s neNT n11 m1 is absolutely convergent for Res gt a gt 0 X1 7X are non trivial Dirich let characters with conductors N17 NT7 respectively Consider the zeta function associated with P and X X17 7X7 de ned by max Z Z X1ltnlgtx7ltn7gtPltn1 Res gt a n11 nT1 It is the main purpose of this paper to prove the following result Theorem ZPX7 3 de ned above has a merommjnhz39c analytic continua tion to the whole complex s plane For any integer m 2 0 if m1 PmX Z 0amp5 sz pdegP lal0 then mp ZP7er Z CaLea17gtlt1gtLltiamx7gt lal0 mp 39r Bg l 71 C 7 Z aHaj1 lal0 11 A NOTE ON GENERALIZED BERNOULLI NUMBERS 43 Another zeta function ZP7 E 5 de ned by 0 0 2037573 2 Z 1 Lrpn17n39 7717 n11 nT1 was considered by P Cassou Nougues in Her result for the special values of ZP7 E s can be restated as CO CO ZP7 57 7m 121ng 21 215111 gripmneimmmtt n1 nT Here we also have the same formula for the special values of ZP7 x s7 ie7 00 00 7 7 m mtrmtrm ZP7 X7 m 7 151 12 21 X1n1 X7 nTP ne However xn 1121 Xjnj is a multiplicative character while 5 1121 5 is an additive character Hence the treatments are different in some respect As shown in Section 4 P Cassou Nougu es7 formula for the special values of ZP7 E 5 follows from our formula for the special values of ZP7 x s In additon we have another explicit expression for the special values of ZP7 E s A well known relation among the Bernoulli numbers is n71 Z 32kman 7271 1B2m for n 2 2 k1 This was found by many authors7 including Euler ref 57 Dilcher 5 generalized the formula for sums of products of any number of both Bernoulli and Euler numbers Bernoulli and Euler numbers are special cases of the generalized Bernoulli numbers B belonging to a residue class character x However it is not easy to get the generalized formula for generalized Bernoulli numbers At the end of this paper7 we give a closed expression for the case as an immediate application of our main theorem 2 Some Basic Results We need some classical results reproduced in 15 Proposition 1 Suppose that Lplt8gt ZAgtO aAA S ranges over a se quence ofpositiue real numbers tending 00 is a Dirichlet series conuerging for su icientlg large Re 3 ZAgtO aAe At is the corresponding expo nential series Suppose that at t 0 has the asymptotic expansion 2 n20 where p is a xed positiue number Then 44 KWANG7WU CHEN AND MINKING EIE 1 Lplt8gt has a meromorphic continuation to the whole complex plane 2 Lplt8gt has possible simple poles at s inp where n is not a multiple ofp with residue CnP7np and has no other poles lt3 wen Await Note that the above proposition is different from Proposition 2 of 15 However7 it follows from Apsl s Ooot9 1ftdt Res gt o 6 1 i 0 1 259 C t Pdt 259 ftdt n 5 0 00 6s co 2 CW7 t9 1ftdt n0 S E 6 where 6 is a small positive number so that ft 220 Cntnp From the above7 we get our assertions A function fx is called a rapidly decreasing function if it belongs to C X Rn and satis es hm mmw 0 Mace for any 04 and any integer k gt 0 ref 107 or page 245 in 11 The following is a consequence of the Euler Maclaurin summation formula which is also reproduced in 15 Proposition 2 Suppose that f is a rapidly decreasing function on 07 00 and at t 0 f has the power series expansion 00 WHO T t it f M Suppose that gt 2201 Then att 0 gt has the asymptotic expansion 71T B7 WHO with COooftdt r 1l To nd the special value at s 7m of the zeta function 00 00 ZltP7xisgt Z Z X1n1xrmPn i n11 m by Proposition 17 it is equivalent to nd the coef cient of t in the asymp totic expansion at t 0 of the function Z Z X10711 Xrnr exp Pnt39 n11 nT1 A NOTE ON GENERALIZED BERNOULLI NUMBERS 45 It is also equivalent to nd the constant term in the asymptotic expansion at t 0 of the function W Z Z X1011 quot XrerPleeXPi PWltl n11 nT1 For the given polynomial Z7 PX Z AaXD p degP lozl0 we let p QX7 Y Z AaXu YP h lozl0 be the corresponding homogeneous polynomial in r1 variables Oloviously7 c2nt7 t Pntp and so W Z X1011XrmPmneXpPntp nENT Z X1011 gtltrmPmn expQnt7t WENT mp Z 0a Z mm xrltmgtnaexpec2ltntitn lozl0 nENT where mp PmX Z CaX and n 71 lozl0 In the next section7 we shall compute the asymptotic expansion at t 0 of the function Mt Z X1011 Xrmn exp 2nt7t WENT 3 The Proof of the Theorem First we shall prove the case r 1 Indeed this special case plays an important role in our proof of the theorem Lemma 1 Let P be a polynomial with real coe icients such that Pn gt 0 for all n E N and Q be the corresponding polynomial de ned above Let 46 KWANG7WU CHEN AND MINKING EIE hx7 t m exp7Qxtt N a positive integer and 1 g j g N Further more denote 00 mt 2 MN g t k kN 3 quot 8XPQ1 N m 75gt k0 Th en 1 00 71 T 4 W N hm tgtdm Z UL 1wth 7 r0 39 Mt where h7xt is the r th partial derivative with respect to m Proof It follows from the Euler Maclaurin summation formula that Z hkN j t 000Nm 3 quot exp7QNm jt tdm k1 00 DTBHI r 4 r T1 h J7tN 0 00 DTBrJrl r 4 r h td h tN j mm H1 lt37 2 D Proposition 3 Let X be a non trivial character with conductor N Let B Z 0 be an integer and P Q polynomials as given in the previous lemma Suppose that N ZXWW 8XpQnt7t n1 Then N 1 N 739 at 2 who 2 7 N Z w hm was j1 j1 0 N 00 4 ilyBT r 4 r 2X0 full JHWV In particular at t 0 has an asymptotic expansion of the form 00 Z dnt n0 A NOTE ON GENERALIZED BERNOULLI NUMBERS 47 with the constant term do given by B l l d 7X7 L 7 7 0 61 5 X Proof Note that Nk 3 quot expQNk m 25 Ms W X0 MZ x H w H 0 Xjfjt H M2 x H So the rst assertion follows from Lemma 1 by noting that 0 hmtdm 0 0 hmtdm 2 0j Hum 00 N j xj hmtdm7ZXjO hmtdm 21 M2 H u since Zilx j O Also7 from this expression of ft we have a power series expansion of the form 00 2cm n0 A with do M2 mm o 7 iixm jhltz was 739 1 7 N721 0 7 4 00 71HTBT 39r 4 39r 1X0 h070N H MZ N 1 N 00 7 T mm 7 Dani 1 2w 2 71 hltTgtltypogtN j1 j1 70 MZ H 7 Now it remains to compute 71Tj7 0 The Leibniz rule for differentiation yields that 13W t BM exp7Qt7 tm WWW expmm w 48 KWANG7WU CHEN AND MINKING EIE From the above7 we see that l 7 i I 0 mi Ty 1fT B mini 07 otherwise ppm t It follows that N 4 71 47 xiz DIEBWMBMN J 7 70 Note that 31 7 and 4713 73m in 2 1 So 1 4 is N 4 B T a d0 NZXB1717 XltJgtZltT1 lt iT BT1NJ 771 771 770 B 3 139 Our theorem is a direct consequence of the following proposition Proposition 4 Let X X1 7X7 3 31 73 P and Q as given in Section 2 Suppose that Mt Z Z X1011Xrmnquotexp 2nt7t n11 nT1 Then f t has an asymptotic expansion of the form 00 Z n0 with the constant term 10 given by d0 47317 X1 17377 X7 7quot j i l BX HVHaH39 7391 Proof We prove the assertion by induction on r The case r 1 was already proved in the previous proposition Suppose that r 2 2 and the assertion is true for the case of Tel variables Consider the case of r variables Applying A NOTE ON GENERALIZED BERNOULLI NUMBERS 49 the previous proposition to the rst summation of ght7 where m ranges over all positive integers7 we obtain N1 1 N1 739 Mt Mum 7 F1 gxlm z 1hltz7tgtdz N1 00 T 2 my 2 jjf h lt07 2 j1 70 where 00 00 Mm Z Z XZltn2gtxrltmgtn 2nET n21 m1 expec2ltzt7 n22 m at and 00 00 tht N1zjquot1 Z Z gtlt2712 Xrnr 2 n21 7LT1 exp7QN1z jtngt7 771725723 N195 j 1hN1z N Note that 6204t7 ngt 7nrtnf Poz7 n2 n tpl p degPozX27 7X7 for any xed number 04 gt 0 Applying our induction hypothesis to hj7 t7 Mm t7 and hjz t7 we get the asymptotic expansion of ght and the con stant term do is do 1790 N1 1 N1 7 Z X1jj 1hj70 7 E 2 X1 339 z 1hltz7ogtdz j1 j1 N1 00 1TBT1T ZXlU 2 W11 0 0 71 70 To compute I1 Tgt007 we use a trick similar to the one in Proposition 3 for computing 71Tj7 0 The Leibniz rule for differentiation yields that 7 7 DQMW 2 Z Dwle jquot1lD lhNw g tl u0 50 KWANG7WU CHEN AND MINKING EIE From the aloove7 we see that Nr i HM LW l WW If S 31 O7 otherwise Dihjmt It follows that N1 1 N1 7 do Emmi1w 7 F1 2mm z 1hltz7ogtdz j1 j1 r 1m 7 r Since the constant term in the asymptotic expansion of hj7 t or Mm t is 7 N1 i 4 7 71713 EN7 l T E X1JE Hl l 1 W70 j1 0 T Bf 71771 7 7 lt gt PM we have T B zfrl N1 4 j i1 1 N1 7 7 Til 4 419177 104 1 g l gm N1X1J l1 91 lgtlt1j2 71TmX BT1NTj 177l F1 70 71l lirl 39r 391 awn BX B i 3739 1 51 1 72 n1 71 1quot BX 39 3739 1 11 This proves our assertions D Corollary Suppose that Ft Z X1011 X7nTPmne mquot39mt 25gt 0 EN then ZP7 X7 7m 3351 Proof From the notation in our main theorem it follows that Ft Z C F t l l0 A NOTE ON GENERALIZED BERNOULLI NUMBERS 51 with Flea 2 ml x7ltnrgtn altmmmgtt nENT 7 00 H ZXJnm je n j1 n1 Horn 00 co Nj Z X 77157nt Z Z XjaeiakNjt n1 k0 11 1v39 1 7 X161 7N 11 17 e 7 co 7Bj7tn71 nl n1 and differentiating term by term j times with respect to 257 we get 00 7BW 7tni jil 939 int 7 X X 75 7 2 7 mlni jil Consequently we have T Bf l F t 7 7 151 A j1gt 11 n j1 and hence our assertion follows E 4 A Consequence Let PX E RX17 7X7 be a polynomial as given before and E 517 7 ET 6 C7 such that lEJl 1 and 57 7 1 for all j ln 19827 P Cassou Nogues considered the zeta function ZltP757sgt Z 5 Pltngt 9 nENT co co Z Z 51 5 Pltngti Rem n11 m1 and she proved that ZP7E7 7m RPm where RPmT Z Pmmw nENT 52 KWANG7WU CHEN AND MINKING EIE which is a power series and can be realized as a rational function in T Here we change the dummy variable n and reformulate the above result so that we can use our theorem to give a new proof Theorem P Cassou Nogu es Suppose that 00 00 2035 E Z 515LTPltn1 mri Resgta n11 nT1 Then ZPEs has a meromomhic continuation to the whole complex 3 plane and for any integer in Z 0 ZltP7 7 7m tag 2 Emelt7lgteiltnlnnTgttt nENT Proof Recall that in the proof of our result7 we use only the following two properties of the Dirichlet characters X1 7X7 1 Xi is a periodic function7 Xjn Nj Xjn for all n E N 2 X7 is non trivial and Xja 0 Thus7 in particular7 it works for the case Xjn ezmnNi or in general Xjn e27 m 77397 777 is a positive rational number such that 0 lt 777 lt 1 Now we suppose that E 517 75 e27 iq17 7ezmqt with 0 lt qj lt 1 Let nk 777 7779 be a sequence of r tuples of rational numbers such that 1 Oltn klt1foralllgj rk21 2 limknoo nk Consider the sequence of zeta functions Zk de ned by new 2 WWW Res gt a nENT On the half plane Re 3 gt a we have lim ZkP77k75 Z0768 kaoo Also all the zeta function Z P7 mm s and ZP7 E s have analytic continu ation to the whole complex s plane So that lim ZkP77lk7 m ZP7Erm haoo By our result Z P 7 n m 7n1nTt klt 7mm m 151 2 We 7 nENT it follows that ZP 7 1 P Wt39tm 5 m tirgmgf We A NOTE ON GENERALIZED BERNOULLI NUMBERS 53 The special values ZP7 E 7m can be expressed in terms of special values of the L series 0 Lqs Zezwmqnis Res gt17 0 lt q lt1 n1 00 I 00 LqsPs ZezmnqO tsile mdt n1 co eZ 39l thil W dt 7 Res gt 17 0 e 7 e 1 we conclude that Lq m 7 71mml x the coefficient of t in the power series eZM q expansion at t 0 of 5t 7 52mg In other words7 00 I 27riq 0 71 mL 7 tm ZEZaneint t e 2 I q 777 7 t lt 27rq39 e 7 e 7 ml 1 m0 Differentiating the above equality 3 times with respect to 257 we obtain 0 00 7 Zezwmqn fm Z Lq m tm 7 7 V n1 m m m39 Proposition 5 Suppose that PmX i1 CaXD lal0 Then mp Z0376 m Z Cal117O 1 LM CW lal0 Proof Note that Z E Pmneltmmmgtt ca 2 snnaeiltmwmgtt neNT lal0 neNT mp 7 co 2 Ca H 2 eZanjnaj eint 39 lal0 j1 n1 54 KWANG7WU CHEN AND MINKING EIE From 00 27rinq39 or int 7 1 lim E e 7n 7e 7 qu7047 ta0 n1 we get our assertion by the previous theorem D Now we give expressions for Lq7m From the power series expansion eZWiq 7 i itn n527riq t 7 eZWiq W 7 M lt 271 n0 where np 221 Amkpk is the Eulerian polynomials the coefficients Amk are the Eulerian numbers which are the numbers of permutations of the chain 1 lt 2 lt lt n with precisely k 7 1 descents see eg 4 we have 8m627riq Lq m 1 527riqm139 Meanwhile we have the following Proposition 6 Suppose that W PmX Z CaX lal0 then 7 mp T ailt jgt ZP757 m EOCaFl W 5 Sums of Products of Generalized Bernoulli Numbers A well known relation among the Bernoulli numbers is n71 Z ngBgn2k 7271 1BZ for n 2 2 k1 This was found by many authors including Euler ref 5 Dilcher remarked in 5 that it may be of interest to nd formulas of the above type for sums of products of generalized Bernoulli numbers In the following Proposition 7 we give a closed expression for sums of products of generalized Bernoulli numbers Proposition 7 Letr be a positive integer and Xi be a non trivial Dirichlet character with conductor Ni for i 1 2 7 Then for any positive A NOTE ON GENERALIZED BERNOULLI NUMBERS 55 tntegerm i lt m B521 352 mnme plyWm Nf1p11 Nrprwl my pTZO N1 NT 71 r71 Z Z X1lta1quot39Xrar 7 111 W1 r 71l 771 739 4 7 j reliyk Til k EMT M where 6 1 1 and 371 k is the Stirling number of the rst kind Proof Consider the zeta function Z45ZZXlmXm Z T H Ni 71739 m1 rLT1 j1 Substitute m ai Nimi where ai 17 Ni and mi 2 0 for t 17 r Thus Z43 becomes N1 NT 00 00 r r a is 2 zZlt Xim miNnNis Zltmjigt 111 aT1m10 mT0 i1 j1 N Now we let 0 00 r 9 r a is 7 77 zBltsgtizzltl N1 I WM m10 mTO 11 71 Then we can represent the zeta function Zs as N1 NT r ms Z Z HMan 233 111 aT1 i1 From 8 we know that this zeta function Z3 3 has an analytic continuation to the whole complex plane7 and the special values at non positive integers s 7m are given by 7 7 ml a zBem lt11 Np Z 7 H Bp i1 F1quot39FrmT p1 39 39 39pTI j1 7 N7 my ymgt0 56 KWANGrWU CHEN AND MINKING EIE Using the following identity 5 Theorem 3 774 Z Bj1ltz1gtBJltz gt 1Tn 317 39 quot 7 7391 szo 71rilrltgt ikly r 7j71 k B gt57 7Tj MW in M k 71 70 k0 where y 1 z and 3n k are Stirling numbers of the rst kind and we can rewrite ZB7m as 71 H111 NZquot1T 1 T 39 7 1 7 r 7 1 7390 j rilijk erijw ta m 7 ij 7 where 6 1 Now applying our theorem the special values at non positive integers s 7m of the zeta function Zs are T mim Pi1 m N B 29771 2 lt gt HA WHWFW p17 7197 H pi 1 my MZO On the other hand using the equality N1 NT 7 Zr m Z Z HX ltaiZBm 211 m1i1 and the above values of ZT7m and ZB7m we get our assertion D Remark As special cases we state formulas for sums of products of two respectively three generalized Bernoulli numbers 1 Let X1 X2 be non trivial Dirichlet characters with conductors N1 N2 respectively Then for any positive integer m m Br H k NM 1 Ngn m 7 k 1 N1 N2 Z Z X1a1X2a2 111 121 L Q 7 N1 N2 1 B E iBm2lt gt m1 M1 N1 N2 m2 A NOTE ON GENERALIZED BERNOULLI NUMBERS 57 2 Let X1 X2 X3 be non trivial Dirichlet characters with conductors N17 N27 N37 respectively Then for any positive integer m7 we have 1 1 1 lt m gt 3 133 3 W160 297617 N509 1 N q 1 New 1 qu N1 N2 N3 g 2 Z Z X1a1gtlt2a2x3a3 111 121 131 Bm Bm Bm36 2 7 71 7 72 7 7L L2 L3 where6iN1N2N3 As a nal example we consider the Euler numbers En7 0 g n lt 00 We have E2n1 07 n 2 07 while Em7 n 2 O7 is de ned by 00 71 E2n 2n 7r seczizwz n0 The Euler numbers are special cases of the generalized Bernoulli numbers B belonging to a residue class character x In fact we have 2Bn1 wily 3 07 where 7 is the primitive character with conductor 4 If we let T 2 and the characters X1 and X2 in Proposition 7 be the same character 7 the primitive character with conductor 47 then we get an identity which is a special case of Eq 49 in En Proposition 8 For a non negative integer n we have the following iden tity V L 2n 2W2 E E 2zntzi1 g3lt2kgt 2k 2n 2k 2n2 Proof Let r 2 and X1 X2 as indicated above7 ie7 N1 N2 4 Then k1 ik1 Bx B 4 quot i 7 m k0 4 4 m1 11 B 2 ail Z 2 Mb 4 BM1 M m 4 771 2 111 171 iBm1 7 Bm2 23m21 Bm1 7 Bm2 2m1 m2 m2 2m1 m239 58 KWANG7WU CHEN AND MINKING EIE The left hand side of the above identity is exactly 4 1 22 Using some basic properties of the Bernoulli polynomials 3 1 315 dwmtml l 7 1 lin Bn 5 2 7 1B Bn1 90 71 Bn7 the right hand side of the above identity becomes B 2 21 71m272 m 17quot l lm2 The result follows by setting m 2n D References 1 P Cassou Nougues Valeurs aux entiers negatifs des fonctions zeta et fonctions zeta pradiques Invent Math 51 1979 2559 2 7 Valeurs aux entiers negatifs des series de Dirichlet associees a un polynome l J Number Theory 14 1982 3264 3 7 series de Dirichlet et integrales associees a un polynome a deuz indetere minees J Number Theory 23 1986 154 L Comtet Advanced Combinatorics Dordrecht and Boston Reidel 1974 K Dilcher Sums of products of Bernoulli numbers J Number Theory 60 1996 2341 6 Minking Eie On a Dirichlet series associated with a polynomial Proc Amer Math Soc 110 1990 583590 EE 7 7 The special values at negative integers of Dirichlet series associated with polynomials of several variables Proc Amer Math Soc 119 1993 5161 8 7 A note on Bernoulli numbers and Shintani generalized Bernoulli polynomials Trans Amer Math Soc 348 1996 11171136 9 LM Gelfand and GE ShiloV Generalized Functions Vol 1 Academic Press 1964 0 Mathematical Society of Japan Encyclopedic Dictionary of Mathematics Cambridge Mass MIT Press 1993 11 Serge Lang Algebraic Number Theory SpringerVerlag 1994 12 H Rademacher Topics in Analytic Number Theory in Die Grundlehren der Math ematischen Wissenschaften Band 169 SpringerVerlag 1973 13 T Shintani On evaluation of zeta functions of totally real algebraic number elds at nonepositive integers J Fac Sci Univ Tokyo Sect lA Math 23 1976 393417 14 LC Washington Introduction to Cyclotomic Fields SpringerVerlag 1982 A NOTE ON GENERALIZED BERNOULLI NUMBERS 59 15 D Zagier Valem s des fouctz39ons z ta des corps quadratiques Te els aux entiers ne gatz39fs7 Aste risque7 4142 1977 135151 Received July 97 1999 and revised December 307 1999 This Work Was supported by the National Science Foundation of Taiwan Republic of China DEPARTMENT OF ACCOUNTING AND STATISTICS DAHAN INSTITUTE OF TECHNOLOGY SHINeCHENG HUAeLIAN 971 TAIWAN REPUBLIC OF CHINA Eimaz39l address chhen InSO1dahanedutW INSTITUTE OF APPLIED MATHEMATICS NATIONAL CHUNG CHENG UNIVERSITY MINGeIISIUNG CHIAeYI 621 TAIWAN REPUBLIC OF CHINA Eimaz39l address InkeieInathccuedutW

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