行列式17世纪日本与18_19世纪欧洲的行列式_消元式与判别式_英文_

2003年6月 西北大学学报(自然科学版) Jun 2003第33卷第3期 Journal of Northwest U niversity(Natural Science Edit ion) Vo l 33 N o 3

D eterm inan ts,resultan ts and d iscr iminan ts in Japanin the seven teen th cen tury and in Europe in thee igh teen th and n ineteen th cen tur ies

T akefum i GO TO,H iko sabu ro KOM A T SU

(Departm en t of M athem atics, Science U n iversity of Tokyo,W akam iya2cho,Sh injuku2ku,Tokyo, 16220827 Japan)

Abstract: It is by now w ell know n that Japanese mathem atician s in t roduced dete rminan ts in theseventeenth centu ry bu t it is no t necessarily understood well why and how they made u se of de2term inan t s.T heir calcula t ion s are fo llow ed and what they did is show ed,wh ich is essent ia lly thesame as the eliminat ion method of aux ilia ry variab les from systems of algeb raic equat ion s withm o re than one unknow n s as deve lop ed in Eu rope in the eigh teen th and n ineteen th cen tu rie s.

Key words:dete rminan t; resu ltan t; discrim inan t; Japanese mathem atics in the seventeenth centu2ry; Eu ropean m athem atics in the eigh teen th and n ineteen th cen turies

CLC number:NO 9 Documentcode:A Article ID: 10002274X(2003)0320363205ficientsΑi. Since this arrangement comes from the1 Chinese mathematics as the back- Chinese way of writ ing,we will here denote theground same

[Α0 Α1 Αn] (3)

Japan impo rted mathematics fromCh inatw ice of coefficien t s.The f irstp lace fo r the con stan t isin her h isto ry. In the ancien t t ime it was thepracti2 called sh i实, the second fo r the linear te rm fangcal m athem atics rep resen ted by th, ,

The second impo rtat iontookp lace around the Inth is te rmino logy a po lynomia lf(x) cou ldyear 1600 and Japanese learned algeb raic equat ion s no t be distingu ished from the equat ionf(x)=0. Itfrom books Yáng HuīSuàn2Fǎ[杨3 ]辉算法[2] and seems that th is cau sed somet imes carelessness ofSuán2X uãqǐ2M ãng 算学启蒙 . In these books an Japanese mathematician s in the signs of po lynom i2algebraic equation als.f(x)=Α0+Α1x+ +Αnx n=0 (1) Given a′numberΦ,they cou ld calcu la te the co2isdenotedbya column efficientsΑi of the sh′ifted′equation

of calculating rods rep resenting the numerical coef2 smaller and smalle r.W e no te that they knew

Receiveddate:2002- 11-01

Author:後藤武史(1975-) ,男,日本东京人,日本东京理科大学理学院研究生,从事数学史研究。

· 364 · 西北大学学报(自然科学版) 第33卷

Α1′=f′ (Φ)=Α1+ 2Α2Φ+ + nΑnΦn- 1. (5) were inherited in Seki′ sschoolof mathematicians.W

The seventeenth century was thet ime when ithout giving a precise terminology Seki[4]the m ass educat io n started in Japan.T here w as, defines the determinantof n equations of order n-therefo re,a la rge demand of textbook s of mathe2 1

lem s became very comp lica ted.Some needed an e2 that is,quat ion of deg

Thebookst ion s of one unknown w ithnumerical coefficien ts. %

unknowns.He invented exp ressions of po lynomials quat ion by the cofacto r of c20 , , the last equat ion

田中由真(1651—1719) and other Japanese mathe2 T he con stan t te rm of the equat ion is equal to thematician s deve lop ed a systematic theo ry of elimina2 dete rminan t and the o ther terms van ish.There2tion w ith use of determinants. IsekiTomotoki井关 fo re,h is dete rminan t is the same as in Eu rop e.It is知辰pub lished the f irst book[ 6 ]on elim inat io n in clear fromth is defin it ionthat the dete rminant van2

1690. ishes whe′never sy4stem (6) has acommon roo t.

Seki sbook[ ]makes anerro rin the expan sion2 Determinants and resultants of dete rminants of order≥5 as pointed ou tby

M ikam i[ 19 ] ,Ho riuch i[21 ]and m any o thers since the

Sek iclassified mathematical p rob lems in to the year 1715.W e remark,however, th[a6t] the erro r hadfo llow ing th ree: the exp licit p rob lemscan be so lved w ith arithm etic, the imp licitp rob lem s

needs algeb raic equat ion s with mo re than one un2 To eliminate a variab le x,we u sually startknowns,and wrote for each class a bookof solu2 withtwoequat ions:t io)f irst t ime in M)

B ooks ofMa them a tics大成算经[ 7 ]w rit ten in 1683 quat ion s (6)of o rder less than n and then app liesth rough 1710 by Sek i and h is pup ils T akebe the dete rminant.

Kataakira建部贤明( 1661—1716) and Takebe Asexh ibited inMikami[ 19 ]andHo riuch i[21 ] theKatah iro建部贤弘(1664—1739).Unfortunately first transformed equation

)

第3期 · 365 ·

is ob nt-amined by elim inating the top te rms off(x ) as show n by Cauchy[ 14]and m any o thers. T here2andx g(x) , that is, n-m fore, the eliminatedequat ionh1 (x)= bmf(x) - anx g(x). (101) R(f,g)=0 (15)Then, thei2thtransformedequat ion n- 1 gives a necessary and sufficien t condit ion in orderh i(x)=di,0+di, 1x+ +di,n- 1x =0 (9i) that the systemof two equat ion s (7) and (8)havefor 1< i≤m isdefinedby n-m a solutioninanalgebraicallyclosedfieldcontainingh i(x)=xh i- 1 (x)+bm- i+ 1f(x) - an- i+ 1x g(x ) all coefficients of the equat ion s in thesystem.

· 366 · 西北大学学报(自然科学版) 第33卷

R(f,f

This is divisib lebyΑn as seen from( 13). In SEKI′ s CollectedWorks edited with Explanations

(18) ka: Ikedaya and N aganoya, 1690bu t Sek i and h is pup ils paid lit t le atten t io n to the [7] SEK I T akakazu关孝和,TA KEBE Kataak ira建部贤

of[7 ] there is ana lmo stco rrect list of 59te rms of deux lignes des o rdres quelquonques peuven t sethe discrim inantD(f) of a quintic equation. co

T he sub jects of M ethods of E quation Mod if i2 [C] L au sanne:O rell FüssliT urici, 1953 46259cations[5] are first to count the number of po sitive [ 9 ] GABRIEL Cramer Appendices I, II ′ de l′o r rea

cien t of an equat ion w ith real roo t s.Yet, Sek idoes Opera Omnia,Ser IVol6[C]Berlin:T eubner, 1921no t seemto have no t iced the fact that the sign of 1972211the discrim inan t decides the number of real roo ts of [ 1 1 ] BEZOU T E R′ echerches sur le degrãde s ãquat ionsa real quadratic equat io n,w h ich everybody know s rãsultan tes de ′l ãvanou issem′en t des inconnues, et sur

References: 2882338

第3期 · 367 ·

Ph il T ran sRoyal Soc L ondon, 1853, 143:4072548 家并びに支那の算法と の关系及び′ 比较(Relations

[ 17] CA YL EY A M emo ir on the resu ltan t of atwo equat ion s [J ]Ph il T ran s Royal Soc L ondon, 1857, M ethods of M athem atics by M athem2atician s in Kyo to

147: 7032715 andOsaka and thoseinChina)Part [J ]TōyōGakuhō

tono Kankei oyob i H ikaku关孝和の业绩と京阪の算

後藤武史,小松彦三郎

(东京理科大学理学部,东京 16220827,日本)

摘要:认为日本数学家在17世纪引入了行列式的概念,为什么和如何使用行列式却并未被人们所深刻地理解,故通过对和算家当时的具体计算的分析,表明他们所使用的方法正是18—19世纪欧洲的数学家创立的对于超过一个未知量的方程组的辅助变量的消元法。

关 键 词:行列式;消元式;判别式; 17世纪的日本数学; 18—19世纪的欧洲数学

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Compar isonof prokaryot icexpress ionresults of d ifferen t connect ionmanners of Han taan v irus M and S gene segments

LU O W en,XU Zhi2kai,ZHAN G Fang2lin,YAN Yan,

W U X ing2an,L IU Yong,BA IW en2tao′,W AN G H ai2tao

(Departm ent ofM icrobiology,Fourth M ilitary M edical U n iversity,X i an 710032,Shaanx i P rovince,Ch ina)Abstract:To compare the p rokaryotic exp ression results of different connection manners of Hantaan virusglycop ro tein G1 and nucleop r′o tein fragm en t including major an t igen sites,G1 gene segmen t encoded byMgene and 0.7 kb segmentof 5 terminal in Sgene reg ion from H antann viru s 762118 strain were connectedand cloned into pGEX24T2 to construct the prokaryotic fusion exp ression vectors pGEX24T22G1S0. 7 andp GEX 24T22S0.7G1. T he exp ressio n of the fu sion p ro tein GST 2G1S0. 7 o r GST- S0. 7G1 was induced in

E.coli XL 12B lue.A fter IPT G induct ion,EL ISA resu lts showed that bo th fusion p ro teins could b ind specifi2cally to the han taviru s nucleop ro tein specific mA b, and the fu sio n p ro tein GST 2G1S0.7 cou ld b ind H an2taan viru s glycop ro tein specific mA b.W estern blot resu lts showed that the fusion p ro tein GST 2G1S0.7orGST 2S0.7G1 w as exp ressedafter induct ion,and the exp ressionp roduct of chimeric gene G1S0.7degradat2ed relatively less than the one of S0. 7G1.The result s suggest that two differen tchimeric genes both canexp ress b iologically active fu sio np rotein s inE.coli,bu tw ith differen t exp ressio n efficiency. Ithas laid thefoundation for develop ing Hantaan virus genetic engineering vaccine.

Key words:Han taan virus; p rokaryo t ic exp ression; fu sion p ro tein