ALGEBRAIC ANALYSIS AND RELATED TOPICS \noindentBANACHALGEBRAIC CENTER PUBLICATIONS ANALYSIS AND comma RELATED VOLUME TOPICS 5 3 INSTITUTE OF MATHEMATICS \noindentPOLISH ACADEMYBANACH CENTER OF SCIENCES PUBLICATIONS , VOLUME 5 3 WARSZAWA 2 0 0 \ centerlineTWO .. CENTURIES{ALGEBRAICINSTITUTE .. ANALYSIS OF OF THE MATHEMATICS AND TERM RELATED TOPICS} quotedblleftBANACH ALGEBRAIC CENTER ANALYSIS PUBLICATIONS quotedblright , VOLUME 5 3 \ centerlineDANUTA PRZ{POLISH EWORSKA ACADEMY hyphen OF ROLEWICZ SCIENCESINSTITUTE} OF MATHEMATICS Institute of Mathematics comma Polish AcademyPOLISH ACADEMYof Sciences OF SCIENCES \ centerlineS-acute niadeckich{WARSZAWA 8 comma 2 0 0 hyphen0 } 950 WarszawaWARSZAWA comma 2 Poland 0 0 e hyphen mail : rolewicz atTWO impan period CENTURIES gov period pl OF THE TERM \ centerlineAbstract period{TWO The\quad termCENTURIES quotedblleft“ ALGEBRAIC Algebraic\quad OF Analysis THE TERM ANALYSISquotedblright} in ” the last two decades is used in two com- pletely DANUTA PRZ EWORSKA - ROLEWICZ \ centerlinedifferent senses{ ‘ ‘ period ALGEBRAIC It seems ANALYSIS thatInstitute at least of Mathematics ’ one’ } is far away , Polish from Academy its historical of Sciences roots period Thus comma in order to ´ explain this misunderstanding commaS theniadeckich history 8 of , 0this - 950 term Warszawa from its , originsPoland is recalled period \ centerlineThe term quotedblleft{DANUTA PRZ Analyse EWORSKA Alg acute-ee− -ROLEWICZ mail brique : rolewicz quotedblright} @ impan . open gov . parenthesis pl quotedblleft Algebraic Analysis quotedblright closingAbstract parenthesis . 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Let inD ∈ theR(X 1). 9 thLet and 20 th X closing braceRD period⊂ L0(X ....) be Let the R set open of allparenthesis right inverses X closing for D, parenthesisi . e .DR be= theI( set identity of all operator right invertible ) if operators in \noindent century . d R t L open parenthesisR ∈ X RD closing( i . e parenthesis . the periodLeibniz .... - Let Newton D in formulaR open parenthesisholds : d Xt closinga f(s parenthesis)ds = f(t) for period all .... Let R sub D subsetfunctions L subf 0 open parenthesis X closing parenthesis be the set of all right inverses for D comma i period e period\ hspace DR∗{\ = Iffrom openi l l }To theparenthesis space begin under identity with consideration , operator we should closing ) . explainMoreover parenthesis , what dom if D is= RX meant⊕ ker byD. AlgebraicFor all R,R Analysis0 ∈ at present ( cf . R in R sub D open parenthesis i period e period .... the Leibniz hyphen Newton formula holds : d divided by d t integral\noindent sub aEncyclopaedia to the power of t f of open Mathematics parenthesis s closing , [ 1 parenthesis 997 ] ) ds . = f open parenthesis t closing parenthesis for all functions f \ hspacefrom the∗{\ spacef i l l under}The consideration main idea closing of Algebraic parenthesis Analysis period .... Moreover in its commapresent dom , D more = RX strict oplus ker , D sense period derives from the .... For all R comma R to2000 the powerMathematics of prime Subject in Classification : 1 - 0 , 1 A 55 , 1 A 60 , 0 A 20 . \noindenthline factKey words that and the phrases differential: algebraic analysis operator , right invertible $ D = operator\ f r a , c operational{ d }{ d calculus t } .$ The is right invertible in several function spaces . 2000 Mathematicspaper is Subject in final formClassification and no version : 1 hyphen of it will 0 commabe published 1 A 55 elsewhere comma . 1 A 60 comma 0 A 20 period \ hspaceKey words∗{\ f and i l l phrases} Foundations : algebraic of analysis Algebraic comma rightAnalysis invertible are operator the following comma operational : Let calculus $ L period ( X ) $ [47] beThe the paper set is of in finalall form linear and no version of it will be published elsewhere period open square bracket 47 closing square bracket \noindent operators with domains and ranges in a linear space $ X ( $ in general , without any topology )

\noindent over a field $ F $ of scalars with characteristic zero and let $ L { 0 } ( X ) = \{ A \ in L ( X ) :$dom$A =$

\noindent $ X \} . $ \ h f i l l Let $R ( X ) $ be the set of all right invertible operators in $ L ( X ) . $ \ h f i l l Let $ D \ in R ( X ) . $ \ h f i l l Let

\noindent $ R { D }\subset L { 0 } ( X ) $ be the set of all right inverses for $D ,$ i .e $. DR = I ($ identityoperator)if

\noindent $ R \ in R { D } ( $ i . e . \ h f i l l the Leibniz − Newton formula holds $ : \ f r a c { d }{ d t }\ int ˆ{ t } { a } f ( s ) ds = f ( t )$ forallfunctions $ f $

\noindent from the space under consideration ) . \ h f i l l Moreover , dom $D = RX \oplus $ ker $ D . $ \ h f i l l Forall $R , Rˆ{\prime }\ in $

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\ centerline {2000 Mathematics Subject Classification : 1 − 0 ,1A55 ,1A60 ,0A20 . }

\noindent Key words and phrases : algebraic analysis , right invertible operator , operational calculus . The paper is in final form and no version of it will be published elsewhere .

\ [ [ 47 ] \ ] 48 .. D period PRZEWORSKA hyphen ROLEWICZ \noindentR sub D comma48 \quad x in XD comma . PRZEWORSKA Rx minus R− toROLEWICZ the power of prime x in ker D comma i period e period the difference of two primitives of x is a constant period Let \noindentF sub D = open$ R brace{ D F} in L, sub x 0 open\ in parenthesisX , X closing Rx parenthesis− R ˆ{\ : Fprime to the} powerx of 2\ in = F$ semicolon ker $ FX D =, kernel $ i D . and e . exists the sub difference R in R subD of FR two = 0 primitives closing brace period of $ x $ is a constant . Let Any F in F48 sub DD is . PRZEWORSKA said to be an - initial ROLEWICZ operator for D open parenthesis corresponding to an R closing parenthesis \ [F { D } = \{ F 0 \ in L { 0 } ( X ) : F ˆ{ 2 } = F ; FX = period .. One canRD, x ∈ X, Rx − R x ∈ ker D, i . e . the difference of two primitives of x is a constant . Let \kerproveD that any and projection\ exists F to the{ R power\ ofin primeR onto{ D ker}} D isFR an initial = operator 0 \} for D. corresponding\ ] to a 2 right inverse R to the powerFD of= prime{F ∈ =L R0(X minus): F F= toF the; FX power= ker ofD primeand∃R∈R R independentD FR = 0}. of the choice of an R in R sub D period If two right inverses \noindent AnyAny F $∈ F FD is\ in said toF be an{ Dinitial}$operator is said for toD(be corresponding an initial to an operatorR). One can for prove $D ( $ corresponding to an open parenthesis resp period initial0 operators closing parenthesis commute with each other comma then they are equal $ R ) .that $ any\quad projectionOne canF onto ker D is an initial operator for D corresponding to a right inverse period Thus this0 theory 0 prove thatR any= R projection− F R independent $ F of ˆ{\ theprime choice of}$ an R onto∈ RD ker. If two $D$ right inverses is an ( initial resp . initial operator for is essentiallyoperators noncommutative ) commute period with .. each An operator other , then F is they initial are for equal D if and . Thus only this if there theory is is essentially $ Dan $ R in corresponding R sub D such that to F a = I minus RD on dom D period .. The last formula yields open parenthesis by a two noncommutative . An operator F is initial for D if and only if there is an R ∈ RD such hyphenright lines inverse $ R ˆ{\prime } = R − F ˆ{\prime } R $ independent of the choice of an $ R \ in thatR F{ D= I} − RD. $on If dom twoD. rightThe last inverses formula yields ( by a two - lines induction ) the induction closingTaylor parenthesis Formula : the Taylor Formula : (Line resp 1 n . Line initial 2 I = sum operators R to the power ) commute of n FD with to the each power other of n plus , R then to the they power are of n equal D to the .power Thus of this n on theory is essentially noncommutative . \quad An operator $ F $ is initial for $D$ if and only if there is dom D to the power of n open parenthesis n in N closing parenthesis period Line 3n k = 0 an $ R \ in R { D }$ suchthat $F = I − RD$ ondom $D .$ \quad The last formula yields ( by a two − l i n e s With these facts one can obtain CalculusX and solutions to linear equations open parenthesis under ap hyphen I = RnFDn + RnDnondomDn(n ∈ ). inductionpropriate assumptions ) the Taylor on resolving Formula equations : closing parenthesis period If theN field F is algebraically closed then solutions of linear equations with scalar coefficients can be calculated by a decomposik = 0 hyphen \ [ \tionbegin of a{ rationala l i g n e d function} n \\ into vulgar fractions open parenthesis as in Operational Calculus closing parenthesis period With these facts one can obtain Calculus and solutions to linear equations ( under ap - .. IfI X is = a \sum R ˆ{ n } FD ˆ{ n } + R ˆ{ n } D ˆ{ n } on dom D ˆ{ n } ( n propriate assumptions on resolving equations ) . If the field is algebraically closed then \ incommutativeN). algebra\\ with unit e comma F = C and D satisfies the LeibnizF Condition : solutions of linear equations with scalar coefficients can be calculated by a decomposi - tion kD open = parenthesis 0 \end{ xya l i closing g n e d }\ parenthesis] = xDy plus yDx for x comma y in dom D comma of a rational function into vulgar fractions ( as in Operational Calculus ) . If X is a then the Trigonometric Identity holds period Some results can be proved also for left invertible commutative algebra with unit e, = and D satisfies the Leibniz Condition : operators comma even for operators having eitherF C finite nullity or finite deficiency period .. There is a \noindentrich theoryWith of shifts these and periodic facts problemsone can period obtain Recently Calculus comma and logarithms solutions and antilogarithms to linear have equations ( under ap − D(xy) = xDy + yDx forx, y ∈ domD, been introduced and studied open parenthesis even in noncommutative algebras semicolon cf period Przeworska hyphen \noindent propriate assumptions on resolving equations ) . If the field $ F $ is algebraically closed then Rolewicz then the Trigonometric Identity holds . Some results can be proved also for left invertible solutions of linear equations with scalar coefficients can be calculated by a decomposi − open squareoperators bracket 1 , 998 even closing for operators square bracket having either closing finite parenthesis nullity orperiod finite It deficiency means that . Algebraic There is Analysis a is no tion of a rational function into vulgar fractions ( as in Operational Calculus ) . \quad I f more purely linearrich period theory of shifts and periodic problems . Recently , logarithms and antilogarithms have $ X $ i s a Main advantagesbeen introduced of Algebraic and Analysis studied are ( even : in noncommutative algebras ; cf . Przeworska - Rolewicz commutative algebra with unit $ e , F = C$ and $D$ satisfies the Leibniz Condition : bullet s implifications[ 1 998 ] )of . It proofs means due that to Algebraic an algebraic Analysis description is no ofmore problems purely under linear con . hyphen sideration semicolon Main advantages of Algebraic Analysis are : \ [D ( xy ) = xDy + yDx for x , y \ in dom D , \ ] bullet algorithms• fors implifications solving .. quotedblleft of proofs similardue quotedblright to an algebraic .. description problems comma of problems .. although under con these - similarities could be rather far from each other and very formal semicolon \noindentbullet severalthen new the results Trigonometric even in the classical Identity case of thesideration; holds operator . Some d divided results by d t open can parenthesis be proved which also was for comma left invertible indeedoperators comma , even for operators having either finite nullity or finite deficiency . \quad There i s a richunexpected theory closing of• algorithms shifts parenthesis andfor period periodic solving “ problems similar ” .problems Recently , although , logarithms these similarities and antilogarithms could have There are severalbe rather applications far from toeach ordinary other and and very partial formal differential ; equations with scalar \noindent been introduced and studied ( even in noncommutatived algebras ; cf . Przeworska − Rolewicz and variable coefficients• several commanew results functionaleven hyphen in the differential classical case equations of the operator and for discretedt ( which analogues was , indeed of , these equationsunexpected comma for) . instance comma for difference equations period .. There are also some results for \noindentnonlinear equations[There 1 998 are period ] several ) . It applications means that to ordinary Algebraic and partial Analysis differential isequations no more with purely scalar linear and . It should bevariable pointed coefficients out that in, Algebraic functional Analysis - differential a notion equations of convolution and for is discrete not analogues of these \ centerlinenecessary period{equationsMain .. Also advantages , for there instance is no of , need for Algebraic difference to have a equations structure Analysis . of a There arefield comma :are} also like some the results Mikusi for n-acute nonlinear ski field period This commaequations together with. the noncommutativity of right inverses and initial operators comma shows \ hspacethe essential∗{\ f i distinction l lIt} should$ \ bullet of be Algebraic pointed$ out s Analysis implificationsthat in from Algebraic Operational Analysis of proofs Calculus a notion dueperiod of convolution to an algebraicis not necessary description of problems under con − As we have. mentioned Also there at the is no beginning need to have comma a structure .. the term of a ..field quotedblleft, like the AlgebraicMikusi n´ Analysisski field quotedblright. This , .. was first\ begin { a l i g ntogether∗} with the noncommutativity of right inverses and initial operators , shows the essential siderationused in the titdistinction le ; of a book of Algebraic by Joseph Analysis Louis de from Lagrange Operational in 1 797 Calculus : .. Th . acute-e orie des Fonctions \endAnalytiques{ a l i g n ∗} contenantAs we have Les Principesmentioned du at Calcul the beginning Diff acute-e , the rentiel term comma “ Algebraic d e-acute Analysis gag to the” power was first of acute-e s de toute consid e-acuteused in ration the tit le of a book by Joseph Louis de Lagrange in 1 797 : Th e´ orie des Fonctions $ \ bullet $Analytiques algorithms contenant for Les solving Principes\quad du Calcul‘‘ Diff similare´ rentiel ’’ ,\ dquadegag´ problemse´ s de toute consid, \quade´ although these similarities could be rather farration from each other and very formal ;

\ hspace ∗{\ f i l l } $ \ bullet $ several new results even in the classical case of the operator $\ f r a c { d }{ d t } ( $ which was , indeed ,

\noindent unexpected ) .

\noindent There are several applications to ordinary and partial differential equations with scalar and variable coefficients , functional − differential equations and for discrete analogues of these equations , for instance , for difference equations . \quad There are also some results for nonlinear equations .

It should be pointed out that in Algebraic Analysis a notion of convolution is not n e c e s s a r y . \quad Also there is no need to have a structure of a field , like the Mikusi $ \acute{n} $ s k i f i e l d . This , together with the noncommutativity of right inverses and initial operators , shows the essential distinction of Algebraic Analysis from Operational Calculus .

As we have mentioned at the beginning , \quad the term \quad ‘‘ Algebraic Analysis ’’ \quad was f i r s t used in the tit le of a book by Joseph Louis de Lagrange in 1 797 : \quad Th $ \acute{e} $ orie des Fonctions Analytiques contenant Les Principes du Calcul Diff $ \acute{e} $ rentiel ,d $ \acute{e} gag ˆ{\acute{e}}$ s de toute consid $ \acute{e} $ r a t i o n TWO CENTURIES OF quotedblleft ALGEBRAIC ANALYSIS quotedblright .. 49 \ hspaced quoteright∗{\ f i linfiniment l }TWO CENTURIES pe tits comma OF .. ‘ ‘ d ALGEBRAIC quoteright sub ANALYSIS acute-e vanouissans ’ ’ \quad comma49 .. de limites e t de fluxions comma .. e t r e-acute duit a-grave l quoteright analyse alg e-acute hyphen \noindentbrique de quantitd ’ infiniment acute-e s finies pe comma tits 2 , nd\quad revisedd and $ enlarged ’ {\acute ed period{e}} comma$ vanouissansM to the power of, \ mequad V tode the limites e t de fluxions , \quad e t r power$ \acute of e Courcier{e} $ comma duit Imprimeur $ \grave hyphen{a} $ l ’ analyse alg $ \acute{e} − $ briqueLibraire de pour quantit les Math acute-e $ \acute matiques{e} $ comma s finies ParisTWO comma , 2ndCENTURIES 1 8 revised 1 3 openOF “ ALGEBRAIC parenthesis and enlarged ANALYSIS 1 st ed ed period ” $49 hyphen. , 1 797 Mˆ{ me } V ˆ{ e }$ Courcier , Imprimeur −0 closing parenthesisd ’ infinimentopen parenthesis pe tits cf , period d e´ Referencesvanouissans comma ,de open limites square e tbracket de fluxions 1 797 , hyphen e t r1 8e´ 1duit 3 closing square me e bracketLibraire semicolon poura` l ’ analyse les Math alg e´ $− \briqueacute de{e quantit} $ matiquese´ s finies , 2 Paris nd revised , 1 and 8 1 enlarged 3 ( 1 ed st., M edV . − 1 797 ) ( cf . References , [ 1 797 − 1 8 1 3 ] ; seesee also the theCourcier paper paper by , J Imprimeur period by J Synowiec . Synowiec- Libraire in this pour in volume les this Math comma volumee ´ matiques in particular , in , Paris particular comma , 1 8 1 concerning 3 ( 1 , st concerning ed contributions. - 1 797 contributions ofof the the Polish Polish) mathematicians ( cf mathematicians . References J , period [ 1 797 J S-acute - 1 $ 8 1. 3niadecki ] ;\ seeacute also and{S the J} period paper$ niadecki M by period J . Synowiec Hoene and Jhyphen in . this M volume Wro . Hoene acute-n , in− skiWro closing $ \acute{n} $ ´ parenthesiss k i ) . periodparticular , concerning contributions of the Polish mathematicians J .S niadecki and J . M . NeverthelessHoene comma - Wro it seemsn ´ ski that ) . this term had been used much earlier comma for instance comma by NeverthelessEuler and d quoteright ,Nevertheless it seems Alembert , thatit seems period this that .. Interm this the term timehad had of been beenLagrange used used this much much term earlier earlier was , in for use instance , also for by , instance byLacroix Euler comma , by EulerPfaff and and others dand ’ d open Alembert ’ Alembert parenthesis .\quad In cf periodthe timeIn Jahnke the of Lagrange time open square of this Lagrange term bracket was 1 in993 this use closing also term by square Lacroix was bracket in , Pfaff use comma and also Dhombre by Lacroix , openPfaff square and bracketothers 1 ( 992 cf ( . closing cf Jahnke . squareJahnke [ 1 993 bracket ] [ , 1Dhombre 993closing ] [ parenthesis , 1 992Dhombre ] ) . period A [ source 1 A 992 source for ]an ) for algebraic . an A algebraic source treatment treatment for of an algebraic treatment ofof analytic quantitiesanalytic quantities quantities was comma was was no , , doubt no no doubt doubt comma , the , the theLeibniz Leibniz Leibniz symbolic symbolic symbolic calculus calculus periodcalculus. Some.. 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Fenaroli and comma some toPenco Pascal [ 1comma 979 ] ) Fermat . and Huygens open parenthesis cf period Fenaroli and \noindentPenco openPenco squareThe [ bracket title 1 979 could 1 979 ] be ) closing explained . square by bracket the fact closing that at parenthesis that time period the notions of limit , con - The title couldvergence be explained , and so by on the , were fact notthat made at that precise time .the However notions , of the limit main comma reason con was hyphen to point out Thevergence title comma couldthat mostand be so ofexplained on the comma results were by were thenot obtained made fact precise by thatalgebraic period at that Howeveroperations time comma the on analytic the notions main reasonquantities of limit was to . point, con out− vergencethat most of,The andthe next results so book on were ,with obtainedwere the notterm by made“ algebraic algebraic precise operations analysis . ” However on in analytic its title , wasquantities the written main period in reason German was by F to . point out thatThe next most bookB of . withA the . Lembert the results term [ 1quotedblleft 8 were 1 5 ] , obtained according algebraic to by analysis the algebraic library quotedblright catalogue operations of in the its titleformer on was Jacobsonanalytic written inSchule Germanquantities by . TheF period next B bookin period Seesen with A ( period Harz the ) termLembert ( private ‘‘ open communication algebraic square bracket analysis of Professor 1 8 1 5’’ Hans closing in Lausch its square title , Monash bracket was University comma written according , in German to the by libraryF . B catalogue . AClayton . of Lembert the( former Melbourne [ Jacobson1 8 )1 , 5 December Schule ] , according 1 992 ) . to the library catalogue of the former Jacobson Schule in Seesen open parenthesisThe same term Harz as closing a subtitle parenthesis was used open byAugustin parenthesis Louis private Cauchy communication [ 1 82 1 ] .In of Professorhis intro - Hans Lausch comma\noindent Monashinduction University Seesen he wrote comma ( Harz that ) , as ( to private methods communication , he had sought “ of tomake Professor them as Hans rigorous Lausch as those , Monash University , Clayton openof parenthesis geometry , soMelbourne as never closingto have parenthesis recourse to comma j ustifications December drawn 1 992 from closing the generality parenthesis of period \noindentThe sameClayton term as a subtitle ( Melbourne was used ) by , Augustin December Louis 1 992Cauchy ) .open square bracket 1 82 1 closing square bracket period In his intro hyphen 00 Theduction same he term wrote as that a comma subtitle as to was methods used comma by Augustinalgebra he had sought Louis .. quotedblleft Cauchy [ to 1 make 82 1 them ] . as In rigorous his asintro those− ductionof geometry he comma wrote so that as never , as to have to methods recourse to , j he ustifications had sought. drawn\ fromquad the‘‘ generality to make of them as rigorous as those ofCase geometry 1 quotedblright , so asCase never 2 period to have recourse to j ustifications drawn from the generality of This may provoke the idea that the name “ algebraic analysis ” emphasized that the This may provoke the idea that the name quotedblleft algebraic analysis quotedblright emphasized that the analyses under consideration were more or less “ different ” from other concepts of analysis \ beginanalyses{ a l iunder g n ∗} consideration were more or less quotedblleft different quotedblright from other concepts of analysis at that time . And , indeed , it was so . \ l eat f t that. algebra time period\ begin And{ a comma l i g n e d indeed} &’’ comma\\ it was so period Unfortunately , Cauchy was forced by the authorities of L ’ E´ cole Polytechnique to &.Unfortunately\end{ a comma l i g n e d ..}\ Cauchyright was. forced by the authorities of L quoteright E-acute cole Polytechnique to change his way of teaching mathematical analysis so that , finally , Analyse alg e´ brique \endchange{ a l i g his n ∗} way of teaching mathematical analysis so that comma .. finally comma .. Analyse alg acute-e brique completely disappeared after the academic year 1 924 / 25 as an autonomous part of the completely disappeared after the academic year 1 924 slash 25 as an autonomous part of the course ( cf . Gilain [ 1 989 ] ) . Thiscourse may open provoke parenthesis the cf idea period that Gilain the open name square ‘‘ bracket algebraic 1 989 closing analysis square ’’ bracket emphasized closing parenthesis that the period However , outside L ’ E´ cole Polytechnique , Lagrange ’ s book was used as a handbook analysesHowever comma under outside consideration L quoteright were E-acute more cole orPolytechnique less ‘‘ commadifferent Lagrange ’’ from quoteright other s book concepts was used of as aanalysis for several years without regard to these dramatic changes . handbookat that time . And , indeed , it was so . Hans Lausch wrote ( again a private communication ; April , 1 989 ) : for several years without regard to these dramatic changes period ... On the theme “ The occurrence of the t erm ‘ alge b raic analysis ’ in history UnfortunatelyHans Lausch wrote , \ openquad parenthesisCauchy was again forced a private by communication the authorities semicolon of April L comma’ $ \ 1acute 989 closing{E} $ parenthesis cole Polytechnique to ” : I ran across a b i ographical account by the histo rian Alfred Stern ( 1 84 6 - 1 936 ) . : change his way of teaching mathematical analysis so that , \quad f i n a l l y , \quad Analyse a l g Stern te l ls of his $ \periodacute period{e} $ period brique On the theme .. quotedblleft The occurrence of the t erm quoteleft alge b raic analysis quoteright father Moritz Abraham Stern ( 1 807 - 1 894 ) , who toge th er with Riemann succeeded ..completely in history quotedblright disappeared : I ran after the academic year 1 924 / 25 as an autonomous part of the Dirichlet 1 859 in G o¨ ttingen and was the first German Jew to hold a chair . His across a b i ographical account by the histo rian Alfred Stern open parenthesis 1 84 6 hyphen 1 936 closing parenthesis lectures covered a wide area , as his s on reports : “ ... popular astronomy , alge braic period\noindent .. Sterncourse te l ls of ( his cf . Gilain [ 1 989 ] ) . analysis and e lements of analytic geometry , ... ” father Moritz Abraham Stern open parenthesis 1 807 hyphen 1 894 closing parenthesis comma who toge th er with Note that M . A . Stern was obliged to deliver lectures in algebraic analysis , since this was RiemannHowever succeeded , outside Dirichlet L ’ $ \acute{E} $ cole Polytechnique , Lagrange ’ s book was used as a handbook an essential part of the mathematical syllabus of the Prussian educational system according for1 859 several in G dieresis-o years ttingen without and was regard the first to German these Jew dramatic to hold a changes chair period . .. His lectures covered a to the reforms of Wilhelm von Humboldt in 1 809 – 1 0 . This system was obliga - tory until wide area comma as his s on reports : .. quotedblleft period period period popular astronomy comma alge braic the end of the 1 9 th century ( cf . Jahnke [ 1 992 ] , [ 1 993 ] ) . Probably , the textbook analysis\ centerline and e lements{Hans of Lausch wrote ( again a private communication ; April , 1 989 ) : } analytic geometry comma period period period .. quotedblright . .Note . On that the M period theme A\ periodquad Stern‘‘ The wasobliged occurrence to deliver of lectures the t in erm algebraic ‘ alge analysis b raic comma analysis since this ’ \quad in history ’’ : I ran acrosswas an essentiala b i ographical part of the mathematical account by syllabus the of histo the Prussian rian Alfred educational Stern system ( 1 84 6 − 1 936 ) . \quad Stern te l ls of his according to the reforms of Wilhelm von Humboldt in 1 809 endash 1 0 period This system was obliga hyphen \noindenttory until thefather end of Moritz the 1 9 thAbraham century Sternopen parenthesis ( 1 807 cf− period1 894 Jahnke ) , openwho square togeth bracket er with1 992 closing Riemann square succeeded Dirichlet bracket1 859 comma in G open $ \ squareddot{ bracketo} $ 1 ttingen 993 closing and square was bracket the closing first parenthesis German Jew period to Probably hold a comma chair the . textbook\quad His lectures covered a wide area , as his s on reports : \quad ‘‘ . . . popular astronomy , alge braic analysis and e lements of analytic geometry , . . . \quad ’’

Note that M . A . Stern was obliged to deliver lectures in algebraic analysis , since this was an essential part of the mathematical syllabus of the Prussian educational system according to the reforms of Wilhelm von Humboldt in 1 809 −− 1 0 . This system was obliga − tory until the end of the 1 9 th century ( cf . Jahnke [ 1 992 ] , [ 1 993 ] ) . Probably , the textbook 50 .. D period PRZEWORSKA hyphen ROLEWICZ \noindentof Lembert50 was\quad also preparedD . PRZEWORSKA for that reason− ROLEWICZ period C period G period Jacobi during his studies was under strong influence of that trend open parenthesis cf period Knobloch comma Pieper and Pulte open square bracket 1 995\noindent closing squareof Lembert bracket closing was parenthesisalso prepared period for that reason . C . G . Jacobi during his studies was under strongNext the influence title quotedblleft of that algebraic trend analysis ( cf quotedblright . Knobloch was , used Pieper in the and following Pulte books [ 1 : 995 ] ) . bullet Oskar50 SchlD o-dieresis . PRZEWORSKA milch open - ROLEWICZ square bracket 1 845 closing square bracket semicolon 5 th ed period .. 1 873 semicolon\ centerline {ofNext Lembert the was title also prepared ‘‘ algebraic for that reason analysis . C . ’’G . was Jacobi used during in his the studies following was under books : } bullet J periodstrong Dienger influence open of square that trend bracket ( cf 1 . 851 Knobloch closing ,square Pieper bracket and Pulte semicolon [ 1 995 ] ) . \ centerlinebullet W period{ $ Gallenkamp\ bulletNext$ the open Oskar title square “ algebraic Schl bracket $analysis 1\ 860ddot closing ”{ waso} square$used milch in bracket the following [ 1845semicolon books ] ; : 5 th ed . \quad 1 873 ; } bullet M period A period Stern open• Oskar square Schl bracketo ¨ milch 1 [860 1 845 closing ] ; 5 square th ed . bracket 1 873 semicolon ; \ centerlinebullet G period{ $ Novi\ bullet open square$ J bracket . Dienger 1 863• closingJ [ . 1 Dienger 851 square ] [ 1 ; bracket 851} ] ; semicolon bullet Johann Lieblein open parenthesis Professor• W . Gallenkamp of Technical [ 1 University 860 ] ; in Prag closing parenthesis comma open square\ centerline bracket{ 1 867$ \ closingbullet square$ W. bracket Gallenkamp comma• M a. collection A [ . Stern1 860 of [ 1 ] 860 ; ]} ; exercises for the book of O period Schl o-dieresis• milchG . Novi semicolon [ 1 863 ] ; \ centerlinebullet Karl Hattendorf{ $• \Johannbullet open Lieblein$ square M.A. ( Professor bracket Stern 1 of 877 Technical closing [ 1860 square University ] bracket ; } in Prag semicolon ) , [ 1 867 ] , a collection of bullet A periodexercises Capelli for and the G book period of O Garbieri . Schlo ¨ openmilch square ; bracket 1 886 closing square bracket semicolon \ centerlinebullet Salvatore{ $ Pincherle\ bullet open$ squareG. Novi bracket [ 11863 893 closing ] ; square} bracket semicolon bullet A period Capelli open square bracket 1•KarlHattendorf[1877]; 894 closing square bracket comma whose book concerned algebraic $ \ bullet $ Johann Lieblein ( Professor of Technical University in Prag ) , [ 1 867 ] , a collection of curves comma • A . Capelli and G . Garbieri [ 1 886 ] ; exercisesbullet Ernesto for Ces the grave-a book ro ofopen O square . Schl bracket $ \ 1ddot 894 closing{o} $ square milch bracket ; comma who wrote in his Prefazione : period period period Forse un giorno mi decider•SalvatorePincherle[1893]; grave-o a publicare un li bro di .. quotedblleft is tituzioni analitiche quotedblright\ [ \ bullet fondaKarl Hattendorf [ 1 877 ] ; \ ] tre cattedre divers e s otto• iA nomi . Capelli di Algebra [ 1 894 comma ] , whose Geometria book concerned analitica algebraic e Calcolo curves period , .. Per ora comma pure s tando a disagio in un programma• Ernesto necessariamente Cesa ` ro [ 1 894 e t ] erogeneo , who wrote e pieno in his diPrefazione addentellamenti: \ centerlinefittizii con altre{ $... materie\ bulletForse comma un$ giorno A i o . mi Capelli mipropongo decider andG dio` guidarea publicare . ilGarbieri le ttore un li comma bro [ 1di 886con “ mosse ]is tituzioni; } rap ide analitiche a s icura comma aa far larga messe” fonda di fatti tre analitici cattedre comma divers ponendoe s otto a i bas nomi e open di Algebra parenthesis , Geometria non a fine analitica closing parenthesis e Calcolo un. quoteright esposizione\ [ \ bullet rigorosPerSalvatore ora , pure s tando Pincherle a disagio in [ un programma1 893 necessariamente ] ; \ ] e t erogeneo e pieno di dei principiaddentellamenti de l l quoteright Analisi fittizii con alge altre brica materie period , i o mi propongo di guidare il le ttore , con mosse In his book secondrap ide book a s icura open , parenthesis in German comma open square bracket 1 904 closing square bracket closing parenthesis\ centerline Ces{aa a-grave$ far\ largabullet ro wrote messe$ open di A fatti .parenthesis Capelli analitici p , [ponendo period 1 894 683 a ]bas closing , e whose ( non parenthesis a book fine ) :un concerned ’ esposizione algebraic rigoros curves , } period perioddei period principi s o de kann l l ’ man Analisi sagen alge comma brica dass. die Integration die inverse Operation der Differentiation \ centerlineis t period { $ \ bulletIn his book$ second Ernesto book Ces ( in German $ \grave , [ 1{ 904a} ]$ ) Ces roa ` [ro 1 wrote 894 ( ] p . , 683 who ) : wrote in his Prefazione : } This means that... Cess grave-a o kann ro man not sagen only made , dass an die attempt Integration at a die common inverse treatment Operation of der Algebra Differentiation comma . .Linear . Forse Algebrais un t comma . giorno Calculus mi decider and Differential $ \grave Equations{o} $ comma a publicare but also followed un li the bro ideas di of Leibniz\quad ‘‘ is tituzioni analitiche ’’ fonda treand cattedre Lagrange periodThis divers means e that s otto Cesa ` iro nomi not only di made Algebra an attempt , Geometria at a common analitica treatment e of AlgebraCalcolo , . \quad Per ora , pure sbullet tando Heinrich aLinear disagio Burkhardt Algebra in unopen , Calculus programma square and bracket Differential necessariamente 1 903 closing Equations square , e but bracket t also erogeneo followed semicolon e the pieno ideas of di Leibniz addentellamenti fittiziibullet E period conand Ces altre Lagrange grave-a materie . ro opensquare , i o bracket mi propongo 1 904 closing di square guidare bracket il comma le ttore a German , con translation mosse semicolon rap ide a s icura , bullet Salvatore Pincherle open square bracket 1 906 closing square bracket semicolon \noindentbullet D periodaa far O period larga Grave messe open di parenthesis fatti•HeinrichBurkhardt[1903]; analitici Dimitr Aleksandroviq , ponendo Grave a bas comma e (1 863 non endash a fine 1 939 ) comma un ’ esposizione in rigoros Russian 1 9 1 1 and in • E . Cesa ` ro [ 1 904 ] , a German translation ; \noindentRussian anddei Ukrainian principi 1 938 de endash l l 1’ 939 Analisi closing alge parenthesis brica semicolon . Grave devoted his books to an analysis of algebraic •SalvatorePincherle[1906]; \ centerlineproblems which{ In appeared his book in connection second book with systems ( in German of differential , [ 1equations 904 ] describing ) Ces $ \grave{a} $ ro wrote ( p . 683 ) : } movements of three• D bodies. O . Grave period ( Dimitr Aleksandroviq Grave , 1 863 – 1 939 , in Russian 1 9 1 1 and in . .We . should s o kann mentionedRussian man and here sagen Ukrainian a large , dass survey 1 938 die of – Algebraic 1 Integration 939 ) ; Analysis Grave devoted die itself inverse by his Alfred books Operation Pring to an hyphen analysis der of Differentiation alge- i ssheim t . and Georgbraic Faberproblems given which in Encyclop appeared dieresis-a in connection die der with Mathematischen systems of differential Wissenchaften equations open describing square bracket 1 909 hyphen movements of three bodies . This1 921 means closing that squareWe Ces should bracket $ mentioned\ periodgrave{a here} $ a large ro not survey only of Algebraic made an Analysis attempt itself at by a Alfred common Pring treatment - of Algebra , LinearD period Algebra Laugwitzsheim and , in Calculus his Georg book Faber open and given square Differential in bracketEncyclop 1 996a¨ Equationsdie closing der Mathematischen square , bracket but also comma Wissenchaften followed in Section[ the 1 0 909 period ideas 4 period of Leibniz 2and entitled Lagrange Algebraische- . Analysis comma gives an overview1 of 921 Riemann ] . quoteright s contributions in this direction period \ [ In\ bullet the academicHeinrichD years . Laugwitz 1 973 slash in Burkhardt his 74 book and [ 1 1 974 996 [ slash ] , 1 in 75 Section I 903 was delivering 0 . ] 4 . 2 ; entitled lectures\ ] Algebraische for the first and Analysis , second yeargives students an overview at the Cybernetics of Riemann Faculty ’ s contributions of the Military in this Engineering direction . Academy In the academic years 1 973 / 74 and 1 974 / 75 I was delivering lectures for the first and \ centerline {second$ \ bullet year students$ E at . the Ces Cybernetics $ \grave Faculty{a} of$ the ro Military [ 1 904 Engineering ] , a German Academy translation ; }

\ [ \ bullet Salvatore Pincherle [ 1 906 ] ; \ ]

\ hspace ∗{\ f i l l } $ \ bullet $ D . O . Grave ( Dimitr Aleksandroviq Grave , 1 863 −− 1 939 , in Russian 1 9 1 1 and in

\noindent Russian and Ukrainian 1 938 −− 1 939 ) ; Grave devoted his books to an analysis of algebraic problems which appeared in connection with systems of differential equations describing movements of three bodies .

We should mentioned here a large survey of Algebraic Analysis itself by Alfred Pring − sheim and Georg Faber given in Encyclop $ \ddot{a} $ die der Mathematischen Wissenchaften [ 1 909 −

\noindent 1 921 ] .

D . Laugwitz in his book [ 1 996 ] , in Section 0 . 4 . 2 entitled Algebraische Analysis , gives an overview of Riemann ’ s contributions in this direction .

In the academic years 1 973 / 74 and 1 974 / 75 I was delivering lectures for the first and second year students at the Cybernetics Faculty of the Military Engineering Academy TWO CENTURIES OF quotedblleft ALGEBRAIC ANALYSIS quotedblright .. 5 1 \ hspacein Warsaw∗{\ f based i l l }TWO on the CENTURIES idea of Algebraic OF ‘ ‘Analysis ALGEBRAIC period ANALYSIS For that new ’ programme’ \quad 5 comma 1 prepared by mathematicians and engineers from this school working in Operations Research De hyphen \noindentpartment commain Warsaw .. I wrote based some on textbooks the idea period of .. Algebraic One of them Analysishad a t itle : . .. For Algebraic that Analysis new programme and , prepared byDifferential mathematicians Equations openand parenthesis engineers in from Polish this semicolon school Warszawa working comma in 1 st Operations ed period .... Research 1 973 comma De 2− nd ed periodpartment .... 1 974 , \ closingquad parenthesisI wrote some period textbooks Then comma . afterTWO\quad anCENTURIESOne of OF them “ ALGEBRAIC had a ANALYSIS t itle ” : \5quad 1 Algebraic Analysis and essential elaborationin Warsaw comma based my on Polishthe idea book of Algebraic reappeared Analysis in 1 979 . For period that new programme , prepared by \noindentAt the InternationalDifferentialmathematicians Conference andEquations engineerson Generalized ( from in this Functions Polish school ;workingand Warszawa Operational in Operations , Calculi 1 st Research ed . \ Deh f - i partment l l 1 973 , 2nd ed . \ h f i l l 1 974 ) . Then , after an held in Varna, comma I wrote .. some September textbooks 29 . endash One Octoberof them had 6 comma a t itle .. : 1 975Algebraic comma Analysis I had a and talk comma in which I described\noindent the essentialDifferential Equations elaboration( in Polish , my ; Polish Warszawa book , 1 st reappeared ed . 1 973 , 2 in nd 1ed 979 . 1 974 . ) . Then , differences betweenafter an Operational Calculi and the newly born quotedblleft modern quotedblright Algebraic Analysis Atin the the Internationalfollowingessential way :elaboration Conference , my Polish on Generalized book reappeared Functions in 1 979 . and Operational Calculi heldBy Operational in VarnaAt Calculus , the\quad International in aSeptember common Conference s ense 29 −− is meant onOctober Generalized : .. 1 period6 , Functions\quad .. a method1 and 975 Operational of integration , I had Calculi a talk held , in which I described the differenceswhich us esin algebetween Varna braic , properties SeptemberOperational of the 29 derivation –Calculi October operator 6 and , 1 the 975 semicolon , newly I had 2 a periodborn talk , ..‘‘ in applications whichmodern I described ’’ of thisAlgebraic method the Analysis infor the s o lving followingdifferences differential way between equations : Operational comma mainly Calculi o rdinary and the differential newly born equations “ modern with ” Algebraic s calar coef Analysis hyphen ficients periodin the following way : ByThe Operational first alge braicBy Calculus Operational connection in Calculusbetween a common thein a derivation common s ense s and ense is the meant is meantintegration : :\quad is1 . as1 o lda. method as\quad ofa integration method of integration whichthe Calculus us eswhich its alge e lf us period es braic alge .. braic Namely properties properties comma of of .. the G the periodderivation derivation .. W operator period operator .. ; 2 Leibniz . applications o bs; 2 erved . \ ofinquad thisa non methodapplications hyphen published of this method paperfor in s 1 o 675 lvingfor s o differential lving differential equations equations , mainly , mainly o rdinary o rdinary differential differential equations with s equations calar coef - with s calar coef − f ithat c i e n the t s symbol . ficients us ed. by him as a symbol of derivation can be treated as an .. quotedblleft inverse quotedblright of the symbol of integrationThe first alge period braic .. connection He applied between many times the derivation this fact and and h the e wrote integration about is it as in o his ld as the Thequotedblleft first algeCalculus Historia braic its e t e orig connection lf . o calculi Namely Case between , 1 quotedblrightG . the W . derivation Case Leibniz 2 period o bs and erved the in a integration non - published paper is as o ld as theThe Calculus furtherin history 1 its 675 of e Operationalthat lf the . \ symbolquad Calculus usNamely ed comma by him , \ .. asquad in a particular symbolG. \ ofquad comma derivationW. .. in\ canquad the be last treatedLeibniz fi f-tsub as an y o years bs “ erved comma in.. is a non − published paper in 1 675 thatwell known the symbol periodinverse ” us of ed the by symbol him of as integration a symbol . of He derivation applied many can times be this treated fact and as h e an wrote\quad ‘‘ inverse ’’ of the symbol of integration . \quad He applied many times00 this fact and h e wrote about it in his Algebraic Analysisabout it appearsin his “ when Historia open e t parenthesis orig o calculi perioddifferentialis period period closing parenthesis for a right invertible operator‘‘ Historia acting in e alinear t orig o calculi $\ l e f t .di ff erentialis \.begin { a l i g n e d } &’’ \\ &. \end{ a l i g n e d }\ right . $ space one is interestedThe further not only history in one of Operational right inverse Calculus comma but , s in imultaneously particular , comma in the in last the familyfif − ofty all right inversesyears and , the family is well of known initial . operators comma which are comma in general comma non hyphen commutative periodThe further historyAlgebraic of Analysis Operational appears when Calculus ( ... ) , for\quad a rightin invertible particular operator ,acting\quad in ain linear the last $ fi f−tWe{ pointy } out$space thatyears one nothing is , interested\quad like ai notconvolution s only in one is us right ed inverse in Algebraic , but s Analysis imultaneously period , ....in the open family parenthesis of all cf period Przeworskaw e l l known hyphenright . inverses and the family of initial operators , which are , in general , non - commutative Rolewicz open. square bracket 1 979 closing square bracket closing parenthesis AlgebraicThis distinction AnalysisWe point between out appears Operational that nothing when Calculi like ( a . convolution and . . Algebraic ) for is usa Analysis edright in Algebraic was invertible immediately Analysis operator . acting( cf . in a linear spaceadopted one byPrzeworska theis mathematical interested - community not only working in one in right this field inverse and related , topicsbut s period imultaneously , in the family of all rightNote that inverses my first and my papers the family concerning of the initial theory of operators right invertible , whichoperators are and , in general , non − commutative . induced families of initial operators and right inverses appeared in 1 972 open parenthesis cf period Przeworska hyphen \noindentRolewicz commaWe point Studia out Math that period nothing 48 open like parenthesisRolewicz[1979]) a convolution 1 973 closing is parenthesis us ed in comma Algebraic 1 29 endash Analysis 144 closing . \ h f i l l ( cf . Przeworska − parenthesis period \ beginThe{ nexta l i g use n ∗} of theThis term distinction .. quotedblleft between Algebraic Operational Analysis Calculi quotedblright and Algebraic .. Analysis in a book was was immediately open parenthesis not countingRolewicz severaladopted [col hyphen 1 by the 979 mathematical ] ) community working in this field and related topics . \endlections{ a l i g concerningn ∗} Note microlocal that my first analysis my papers semicolon concerning cf period the References theory of closing right invertible parenthesis operators in the andbook of Masaki Kashiwara commainduced families of initial operators and right inverses appeared in 1 972 ( cf . Przeworska - \ hspaceTakahiro∗{\ KawaifRolewicz i l l } commaThis , Studia Tatsuodistinction Math Kimura . 48 open between ( 1 973 square ) , Operational1 bracket 29 – 144 1 )986 . closing Calculi square and bracket Algebraic comma also Analysis concerned was with immediately microlocal analysisThe period next use of the term “ Algebraic Analysis ” in a book was ( not counting several \noindentThe revieweradoptedcol of - thislections last by concerning bookthe mathematicalcomma microlocal J period analysis L community period ; cf Brylinsky . References working open ) in inparenthesis the this book of field Bulletin Masaki and Kashiwara AMS related comma 1 topics 8 open . parenthesis 1 988, Takahiro closing parenthesis Kawai , Tatsuo comma Kimura 1 4 hyphen [ 1 986 1 ] 5 , alsoclosing concerned parenthesis with microlocal analysis . Notebegan that his myreview firstThe with reviewer mythe following papers of this laststatement concerning book , : J . L the . Brylinsky theory ( Bulletin of right AMS invertible , 1 8 ( 1 988 ) operators , 1 4 - 1 5 and inducedquotedblleft families) Algebraic began his of analysis review initial with quotedblright the operators following is a statementterm and coined right : by inverses Mikio Sato appeared period period in period 1 972 ( cf . Przeworska − RolewiczA few months , Studia before comma Math“ on . Algebraic 48the turn( 1 analysis of 973 1 987 ) ” and , is 1 a 1 term 29 988−− therecoined144 was by ) Mikio published . Sato a monograph... of the present authorA few openmonths square before bracket , on the 1 988 turn closing of 1 987 square and bracket 1 988 there period was In published its review a open monograph parenthesis Zbl 696 periodThe next 47002 use closingof theof parenthesis present the term author M\quad period [ 1 988 Z‘‘ ] period . Algebraic In its Nashed review writes Analysis ( Zbl 696 : . 47002 ’’ \quad ) M . Zin . Nashed a book writes was : ( not counting several col − lectionsWhat is .. concerning quotedblleftWhat is Algebraic microlocal “ Algebraic Analysis analysisAnalysis quotedblright ” ? ; cfThe ? . .. name References The name “ Alge .. quotedblleft ) braic in Analysis the Alge book ” braic was of us Analysis Masaki ed quoted- Kashiwara , blrightTakahiro was us Kawai edby by Lagrange Lagrange , Tatsuo Kimura [ 1 986 ] , also concerned with microlocal analysis . in a subtitlein to a th subtitle e s econd to threvis e s ed econd and enlarged revis ed and editio enlarged n of his editio quotedblleft n of his Th “ acute-eTh e´ orie orie des des fonctions fonctions ana hyphen The reviewerana of - lytiquthis es last ” ( 1 book 81 3 ) , . J . The L same. Brylinsky subtitle was ( used Bulletin by Cauchy AMS in ,1 821 1 8 in ( his 1 988 “ ) , 1 4 − 1 5 ) lytiqu es quotedblright open parenthesis´ 1 81 3 closing parenthesis period .. The same subtitle was used by Cauchy in 1began 821 in his his .. quotedblleft reviewCours d ’ with analyse Cours the d de quoteright followingl ’ E cole analyse Royale statement Polytechnique : , 1 re partie , Analyse alg e´ brique de l quoteright” . E-acute In his cole introduction Royale Polytechnique he comma .. 1 re partie comma Analyse alg e-acute brique quotedblright period\ centerline .. In his{ introduction‘‘ Algebraic he analysis ’’ is a term coined by Mikio Sato . . . } A few months before , on the turn of 1 987 and 1 988 there was published a monograph of the present author [ 1 988 ] . In its review ( Zbl 696 . 47002 ) M . Z . Nashed writes :

\ hspace ∗{\ f i l l }What i s \quad ‘‘ Algebraic Analysis ’’ $ ? $ \quad The name \quad ‘‘ Alge braic Analysis ’’ was us ed by Lagrange

\noindent in a subtitle to th e s econd revis ed and enlarged editio n of his ‘‘ Th $ \acute{e} $ orie des fonctions ana − lytiqu es ’’ ( 1 81 3 ) . \quad The same subtitle was used by Cauchy in 1 821 in his \quad ‘‘ Cours d ’ analyse de l ’ $ \acute{E} $ cole Royale Polytechnique , \quad 1 re partie , Analyse alg $ \acute{e} $ brique ’ ’ . \quad In his introduction he 52 D . PRZEWORSKA - ROLEWICZ wrote “ As to methods , I have s ought to make th em as rigo rous as thos e of g eometry 00 , s o as never to have recourse to justifications drawn from the generality of alge bra . The t erm “ alge braic analysis ” appears in th e title of over a dozen books without a clear delineation of what it describes ; o f − t en it is us ed in contexts where the common thread is t enuous o r doesn ’ t exist . Of the o lder books we mention “ Istituzioni di Analysi Algebrica ” by A . Capelli ( Napoli , 1 894 ) ; “ Corso di Analysi Algebrica con Introduzione al Calcolo Infinitesimale ” by E . Ces a` ro ( Torino , 1 894 ) ; “ Elementares Lehrbuch der Algebraischen Analysis und der Infinitesimal Rechnung ” , als o by E . Ces a` ro ( Leipzig , 1 904 ) , “ Course of Algebraic Analysis ” ( in Russian , Kiev , 1 91 1 ) and “ Treatis e on Algebraic Analysis ” ( in Russian and Ukrainian , Kiev , 1 938 - 1 939 ; Zbl . 20 , 1 97 ) by D . O . Grave . Capelli ’ s book concerns alge braic curves , the two books by Grave are devoted to alge braic problems , while Ces a` ro ’ s book is an attempt at a common treatment of Algebra , Linear Algebra , Calculus and Differential Equations , close to what is o f − t en 00 called nowadays “ linear analysis . In 1 988 two volumes entitled “ Algebraic Analysis ” ( Vo l I : Zbl . 665 . 8 ) were pub - lish ed . Edited by M . Kashiwara and T . Kawai , th e two volumes consist of papers dedicated to Professor Mikio Sato , “ the initiator of alge braic analysis in the twentieth century ” , whose res earch s eems to aim at the renaissance of “ Alge braic Analysis ” of Euler , and deals with th e theory of hyperfunctions ( which Sato invented in 1 957 ) and with other top - i cs not re lated to the classical books mentioned earlier . Finally , we mention “ Foundations of Algebraic Analysis ” , Princeton ( 1 986 ; Zbl . 605 . 35001 ) b y M . Kashiwara , T . Kawai and T . Kimura which is concerned with microlocal analysis . The author of the book under review has her own very interesting exp lanation of what led to the type of “ Algebraic Analysis ” considered in h er book . But it is c lear from a bove that “ Algebraic Analysis ” means markedly different things to different authors ; one has to examine th e meaning from the context in which it is us ed . For the present book , this is best highlighted by quoting titles of the main chapters and key phrases : Calculus of right invertible operators , general s o lution of equations with right invertible operators , initial and boundary value problems , well - posed and i l l - posed boundary value problems , periodic operators and e lements , shi f − t operators and shift invariant subspaces ,D− alg e bras , perturbations and nonlinear problems , metric properties in alge braic analysis . The common thread and concepts throughout the book ( 9 chapters ) are the proper def - initio n of initial operators for right invertible operators and their fundamental properties , and “ Calculus in Algebraic Analysis ” b y which th e author means th e theory of right in - vertible operators in linear spaces ( without any topology , in general ) - think of indefinite

integrals!

The first edition of the preprint Short s to ry of t erm “ Algebraic Analysis ” was prepared by the present author ( cf . Przeworska - Rolewicz [ 1 996 ] ) in the following way . Items until 1 940 were found more or less at random . Items from 1 940 on were found in 52 .. D period PRZEWORSKA hyphen ROLEWICZ \noindentwrote .. quotedblleft52 \quad AsD to . methods PRZEWORSKA comma− IROLEWICZ have s ought to make th em as rigo rous as thos e of g eometry comma s o \noindentas never towrote have recourse\quad to‘‘ justifications As to methods drawn from , I the have generality s ought of alge to Case make 1 quotedblright th em as rigo Case 2 rous period as thos e of g eometry , s o asThe never t erm to quotedblleft have recourse alge braic toanalysis justifications quotedblright appears drawn in from th e title the of generality over a dozen books of alge without $\ al cleare f t . bra \ begin { a l i g n e d } & ’’ \\ 00 delineation ofMathematical what it describes Reviews semicolonby means o f-t of MathSciNet en it is us ed asking in contexts for the where term the “ algebraic commonanalysis thread is &.t enuous\end o r{ doesna l i g n quoteright e d }\ right t exist. $ period .. Of the o lder books we mention quotedblleft Istituzioni. di Analysi Algebrica quotedblright 00 Until 1 994 there were 3 1 8 items . Not all of them are of the same “ kind I made a Theby t A erm period ‘‘ .. alge Capelli braic .. open analysis parenthesis ’’ Napoli appears comma in .. 1 th 894 e closing title parenthesis of over semicolon a dozen. .. books quotedblleft without Corso a clear didelineation Analysi Algebricaselection of con what Introduzione it describes al Calcolo ; o $ f−t $ en it is us ed in contexts where the common thread is tInfinitesimale enuous oin r quotedblright the doesn following ’ t way.... exist by . First E period . \ ,quad I cancelled .... CesOf thegrave-a a few o which rolder .... did open books not parenthesis contain we mention this Torino term ‘‘ comma in anyIstituzioni form .... 1 894 closing di Analysi Algebrica ’’ parenthesisby A . \ semicolonquad. NextC a .... , p there e quotedblleftl l i are\quad chosen Elementares( collections Napoli , Lehrbuch which\quad contain der1 894 Algebraischen this ) term ; \ inquad their‘‘ titles Corso . All di individual Analysi Algebrica con Introduzione al Calcolo Analysis undpapers der Infinitesimal in these collections Rechnung are quotedblrightalso cancelled comma. The als remaining o by .... papers E period and books .... Ces contain grave-a ro .... open parenthesis\noindent LeipzigInfinitesimale comma .... 1 904 ’’ closing\ h f i l parenthesis l by E . comma\ h f i l ....l Ces quotedblleft $ \grave Course{a} $ ro \ h f i l l ( Torino , \ h f i l l 1 894 ) ; \ h f i l l ‘‘ Elementares Lehrbuch der Algebraischen of Algebraic Analysis quotedblright open parenthesis in Russian comma Kiev comma 1 91 1 closing parenthesis and quotedblleft\noindent TreatisAnalysis e on Algebraic und der Analysis Infinitesimal quotedblright Rechnung open parenthesis ’’ , als in o by \ h f i l l E. \ h f i l l Ces $ \grave{a} $ ro Russian\ h f i l l and( Ukrainian L e i p z i g comma , \ h f iKiev l l 1 comma 904 ) .. 1 , 938\ h hyphenf i l l ‘ ‘ 1 939 Course semicolon Zbl period .. 20 comma .. 1 97 closing parenthesis by D period .. O period .. Grave period .. 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Kashiwara $ and T period Kawai comma th e two volumes consist of papers dedicated to Professor Mikio Sato comma .. quotedblleft the initiator of alge braic analysis in the twentieth century quotedblright commaIn 1 988 two volumes entitled \quad ‘‘ Algebraic Analysis ’’ \quad ( Vo l I : Zbl . \quad 665 . 8 ) were pub − lishwhose ed res . earch Edited s eems by to M aim .at Kashiwara the renaissance and of T quotedblleft . Kawai , Alge th braic e two Analysis volumes quotedblright consist of of Euler papers comma dedicated .. andto Professor Mikio Sato , \quad ‘‘ the initiator of alge braic analysis in the twentieth century ’’ , whosedeals with res th earch e theory s eems of hyperfunctions to aim at open the parenthesis renaissance which Satoof ‘‘ invented Alge in braic 1 957 Analysisclosing parenthesis ’’ of and Euler with , \quad and otherdeals top with hyphen th e theory of hyperfunctions ( which Sato invented in 1 957 ) and with other top − ii cs cs not not re lated re lated to the classical to the books classical mentioned books earlier periodmentioned .. Finally earlier comma we . \ mentionquad Finally .. quotedblleft , we Foundations mention \quad ‘‘ Foundations ofof Algebraic Analysis Analysis quotedblright ’’ , Princeton comma Princeton ( 1 open 986 parenthesis ; Zbl . \ 1quad 986 semicolon605 . Zbl35001 period ) b.. 605 yM period . Kashiwara 35001 , \quad T . Kawai closingand T parenthesis . 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T microlocal period Kawai analysis . and T period Kimura which is concerned with microlocal analysis period TheThe author author of the the book book under under review review has her ownhas very her interesting own very exp interesting lanation of what exp lanation of what ledled to the the type type of quotedblleft of ‘‘ Algebraic Algebraic Analysis Analysis quotedblright ’’ considered considered in in h h er er book book period . \ ..quad But itBut is cit lear is from c lear from a bove athat bove \quad ‘‘ Algebraic Analysis ’’ means markedly different things to different authors ; one has tothat examine .. quotedblleft th e meaning Algebraic Analysis from the quotedblright context means in which markedly it is different us ed things . \quad to differentFor authors the present semicolon book , this oneis has best highlighted by quoting titles of the main chapters and key phrases : \quad Calculus o f rightto examine invertible th e meaning\quad fromo the p e r context a t o r s in , which\quad itgeneral is us ed period s o .. lution For the present of equations book comma with this right invertible \quad o p e r a t o r s , initialis best highlighted and boundary by quoting value titles problems of the main , chapters\quad andw el key l − phrasesposed : ..and Calculus i ll of− posed boundary value problems , periodicright invertible operators .. operators and comma e lements .. general , sshi o lution $ f of−t equations $ operators with right and invertible shift .. operators invariant comma subspaces $ ,initial D and− boundary$ a l g value e bras problems , comma .. well hyphen posed and i l l hyphen posed boundary value problems commaperturbations and nonlinear problems , metric properties in alge braic analysis . periodic operators and e lements comma shi f-t operators and shift invariant subspaces comma D hyphen alg e bras commaThe common thread and concepts throughout the book ( 9 chapters ) are the proper def − initioperturbations n of and initial nonlinear operators problems comma for right metric invertible properties in alge operators braic analysis and period their fundamental properties , andThe\ commonquad ‘‘ thread Calculus and concepts in Algebraic throughout Analysisthe book open ’’ parenthesis\quad b 9 y chapters which closing th e parenthesisauthor means are the th proper e theory of right in − defvertible hyphen operators in linear spaces ( without any topology , in general ) − think of indefinite initio n of initial operators for right invertible operators and their fundamental properties comma \ beginand ..{ a quotedblleft l i g n ∗} Calculus in Algebraic Analysis quotedblright .. b y which th e author means th e theory of right ini hyphenn t e g r a l s ! \endvertible{ a l i g operators n ∗} in linear spaces open parenthesis without any topology comma in general closing parenthesis hyphen think of indefinite \ hspaceintegrals∗{\ ! f i l l }The first edition of the preprint Short s to ry of t erm ‘‘ Algebraic Analysis ’’ was prepared The first edition of the preprint Short s to ry of t erm quotedblleft Algebraic Analysis quotedblright was prepared \noindentby the presentby authorthe present open parenthesis author cf ( period cf . Przeworska Przeworska hyphen− Rolewicz open [ square 1 996 bracket ] ) in 1 996 the closing following square way . bracket closing parenthesis in the following way period \ hspaceItems until∗{\ f 1 i l 940 l } Items were found until more 1 or 940 less were at random found period more Items or from less 1 940 at on random were found . Items in from 1 940 on were found in Mathematical Reviews by means of MathSciNet asking for the term .. quotedblleft algebraic Case 1 quotedblright Case\noindent 2 periodMathematical Reviews by means of MathSciNet asking for the term \quad ‘‘ algebraic $\ lUntil e f t . 1 a 994n a l ythere s i s \ werebegin 3{ 1a 8 l i items g n e d period} &’’ Not all\\ of them are of the same .. quotedblleft kindRow 1 quotedblright Row&. 2 period\end kind{ a made l i g n e a d selection}\ right . $ Untilin the 1 following 994 there way period were First 3 1 comma 8 items I cancelled . 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Next , there are chosen collections which contain this term in their titles . \quad All individual papers in these collections are also cancelled . \quad The remaining papers and books contain TWO CENTURIES OF quotedblleft ALGEBRAIC ANALYSIS quotedblright .. 53 \ hspacethe term∗{\ quotedblleftf i l l }TWO algebraic CENTURIES analysis OF quotedblright ‘ ‘ ALGEBRAIC either ANALYSIS in an explicit ’ ’ \ formquad in their53 t itles or in their review open parenthesis which is denoted by quotedblleft aa in review quotedblright closing parenthesis or in the author quoteright\noindent s summariesthe term open ‘‘ parenthesis algebraic denoted analysis by .. quotedblleft ’’ either aa in in an explicit form in their t itles or in their review (summary which is quotedblright denoted closingby ‘‘ parenthesis aa in review or comma ’’ in ) a or few in cases the comma author in the ’ cited s summaries introductions ( open denoted parenthesis by \quad ‘ ‘ aa in again denoted similarly closing parenthesis period .. 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However comma , these Thelike present foundationsexcerpts paper of Algebraic ( is since a 1 revised Analysis 940 ) can and and be itemseasily extended from completed Zentralblatt version by means f of u-dieresis of Short MathSciNet r StoryMathematik . , since und it contains , likeihre .. foundations GrenzgebieteThe presentsince of Algebraic .. 1 paper 943 open is a Analysisparenthesis revised and collected and extended items in version the from same of Zentralblatt mannerShort Story comma, since .. f by CompactMATH $it contains\ddot{u} $ closing r Mathematik und parenthesisi h r e \quad .. due,Grenzgebiete like foundations of since Algebraic\quad Analysis1 943 and ( items collected from Zentralblatt in the f sameu¨ r Mathematikmanner , \ undquad by CompactMATH ) \quad due toto the the kind kind helpihre help of Professor Grenzgebiete of Professor Berndsince Wegner Bernd 1 from 943 Wegner ( Technische collected from in Universit the Technische same dieresis-a manner Universit t , Berlin by CompactMATH period $ ..\ddot This{a} $ t B e r l i n . \quad This paperpaper is also also) essentially due essentially to the enriched kind help thanks enriched of Professorto Professor thanks Bernd Ernst to Wegner AlbrechtProfessor from from Technische Ernst Universit Albrecht Universit a-dieresisa ¨ fromtt Berlin Universit . $ \ddot{a} $ t des SaarlandesThis in paper Saarbr is u-dieresis also essentially cken comma enriched who thanks kindly to sent Professor me a xerox Ernst copy Albrecht of the from survey Universit article a ¨ desAlge Saarlandes brais chet des Analysis Saarlandes in Saarbr by Alfred in Saarbr $ Pringsheim\ddotu ¨ cken{u} and ,$ who Georg cken kindly Faber , sent who from me kindly Encyclop a xerox sent copy dieresis-a ofme the a die survey xerox der article copy of the survey article AlgeMathematischen braisAlge che Wissenchaftenbrais Analysis che Analysis by open Alfred squareby Alfred Pringsheimbracket Pringsheim 1 909 endash and and Georg 1 921 Faber closing Faber from square fromEncyclop bracket Encyclopa¨ perioddie der $ \ddot{a} $ d i eThere der are alsoMathematischen added items from Wissenchaften MathSciNet[ up 1 909 to date – 1 921 and ] a . few others found again Mathematischenat random periodThere Wissenchaften are also added items [ 1 from909 −− MathSciNet1 921 ]up . to date and a few others found again at Note that onerandom book . open parenthesis in Spanish closing parenthesis had in 1 960 the fifth edition open parenthesis ! closingThere parenthesis are alsoNote period added that .. Another itemsone book from one ( in open Spanish MathSciNet parenthesis ) had in up in 1 960 to the date fifth and edition a few( ! ) . others Another found one ( again in atSerbo random hyphenSerbo . Kroatian - Kroatian closing ) had parenthesis in 1 970 had the third in 1 970 edition the ( third cf . editionReferences open parenthesis) . I am notcf period able to .. References closing parenthesisfind period earlier .. references I am not , able because to find these books have not been reviewed in Zentralblatt . \ hspaceearlier∗{\ referencesf i l lA}Note comma conclusion that because follows one these bookif you books look ( inhave through Spanish not beenReferences reviewed ) had. in in ZentralblattThe 1 960 term the “ algebraic period fifth analysis edition ” ( ! ) . \quad Another one ( in A conclusionwas follows used ifthrough you look centuries through and References is still used period whenever .. The authors term quotedblleft wish to point algebraic out their analysis algebraic quotedblright \noindentwas used throughSerboapproach− centuriesKroatian to analytic and is problems ) still had used in( or whenever 1, possibly 970 authorsthe , to their third wish far to generalizationsedition point out ( their cf ) . . \ Forquad thatReferences reason , ) . \quad I am not able to find earlieralgebraic referencesapproachone can to find analytic ,in becauseReferences problems thesepapers open parenthesis books in Theoretical have or comma notPhysics been possibly , Logics reviewed comma , Graph to in their Theory Zentralblatt far generalizations , System . closing parenthesis periodTheory .. For , and so on . A conclusionthat reason comma followsAcknowledgements .. one if can you find inlook References . throughThe authorpapers References inwould Theoretical like to . express\ Physicsquad herThe comma appreciation term Logics ‘‘ comma algebraic and grat .. Graph analysis ’’ wasTheory used comma through- itude System to centuries Professors Theory comma Hans and Lausch and is so still on, period Monash used University whenever , authors Clayton ( wish Melbourne to point ) , Ernst out their algebraicAcknowledgements approachAl - brecht period , to Universit .. analytic The authora ¨ t des problems would Saarlandes like to ( ,express or Saarbr , herpossiblyu ¨ cken appreciation , and , Bernd to and their Wegnergrat hyphen far , Technische generalizations ) . \quad For thatitude reason to ProfessorsUniver , \quad -Hans sita ¨ Lauschonet Berlin can comma , for find their .. Monash in support References University in collectingpapers comma material .. in Clayton used Theoretical in open this parenthesis paper Physics . Melbourne , Logics closing , \quad Graph parenthesisTheory , comma System Ernst Theory Al hyphen , and so on . References brecht comma Universit a-dieresis t des Saarlandes comma Saarbr u-dieresis cken comma and Bernd Wegner comma TechnischeAcknowledgements Univer hyphen . \quad The author would like to express her appreciation and grat − itudesit dieresis-a to Professors t Berlin comma Hans for Lausch their support , \quad inCOLLECT collectingMonash material IONS University: used in this , \ paperquad periodClayton ( Melbourne ) , Ernst Al − brechtReferences , Universit $ \ddot{a} $ t des Saarlandes , Saarbr $ \ddot{u} $ cken , and Bernd Wegner , Technische Univer − 1975 s iCOLLECTIONS t $ \ddot{a} : $ t Berlin , for their support in collecting material used in this paper . Daisu kaisekigaku to sono oyo ( in Japanese ) [ Algebraic analysis and its applications ] . Proc 1975 . Conf . , Res . Inst . Math . Sci . , Kyoto Univ . , Kyoto , July 1 - 4 , 1 9 74 . \ centerlineDaisu kaisekigaku{ References to sono oyo} open parenthesis in Japanese closing parenthesis .. open square bracket Algebraic Surikaisekikenkyusho Kokyuroku , 2 26 ( 1 975 ) . analysis and its applications closing square bracket period .. Proc period 1976 \ beginConf{ perioda l i g n comma∗} Res period .. Inst period .. Math period .. Sci period comma Kyoto Univ period comma Kyoto Various problems in algebraic analysis . Proc . Sympos . , Res . Inst . Math . Sci commaCOLLECTIONS July 1 hyphen : 4 comma 1 9 74 period .. Surikaisekikenkyusho . , Kyoto Univ . , Kyoto , 1 9 75 . Surikaisekikenkyusho - Kokyuroku , 266 ( 1 976 ) . Proceedings \endKokyuroku{ a l i g n ∗} comma 2 26 open parenthesis 1 975 closing parenthesis period of the Oji Seminar on Algebraic Analysis and the RIMS Symposium on A lgebraic Analysis . Kyoto 1976 Univ . , Kyoto , Res . Inst . Math . Sci . 1 2 ( 1 976 / 77 ) ; supplement . \noindentVarious problems1975 in algebraic analysis period .. Proc period .. Sympos period comma Res period .. Inst period .. 1978 Math period .. Sci period comma Kyoto Univ period comma A lgebraic Analysis of Quantum Field Theory . Proc . Sympos . , Res . Inst . Math . \noindentKyoto commaDaisu 1 9 kaisekigaku 75 period Surikaisekikenkyusho to sono oyo ( hyphen in Japanese Kokyuroku ) comma\quad 266[ Algebraic open parenthesis analysis 1 976 andclosing its applications ] . \quad Proc . Sci . , Kyoto Univ . , Kyoto , January 30 – February 1 , 1 9 78 ; Surikaisekikenkyusho Kokyuroku , parenthesisConf . , period Res . \quad I n s t . \quad Math . \quad Sci . , Kyoto Univ . , Kyoto , July 1 − 4 , 1 9 74 . \quad Surikaisekikenkyusho 324 ( 1 978 ) . KokyurokuProceedings , of 2 the 26 Oji ( Seminar 1 975 on ) Algebraic. Analysis and the RIMS Symposium on A lgebraic Analysis period Kyoto Univ period comma Kyoto comma Res period Inst period Math period Sci period 1 2 open parenthesis\noindent 1 9761976 slash 77 closing parenthesis semicolon supplement period 1978 \noindentA lgebraicVarious Analysis of problems Quantum Field in algebraicTheory period analysis .. Proc period . \ ..quad SymposProc period . \ commaquad Sympos Res period . .. , Inst Res period . \quad I n s t . \quad Math . \quad Sci . , Kyoto Univ . , ..Kyoto Math period , 1 9 .. Sci75 period . Surikaisekikenkyusho comma Kyoto − Kokyuroku , 266 ( 1 976 ) . ProceedingsUniv period comma of the Kyoto Oji comma Seminar January on Algebraic 30 endash February Analysis 1 comma and the 1 9 RIMS 78 semicolon Symposium Surikaisekikenkyusho on A lgebraic KokyurokuAnalysis comma . Kyoto 324 open Univ parenthesis . , Kyoto 1 978 , closing Res . parenthesis Inst . Math period . Sci . 1 2 ( 1 976 / 77 ) ; supplement . \noindent 1978

\noindent A lgebraic Analysis of Quantum Field Theory . \quad Proc . \quad Sympos . , Res . \quad I n s t . \quad Math . \quad S c i . , Kyoto Univ . , Kyoto , January 30 −− February 1 , 1 9 78 ; Surikaisekikenkyusho Kokyuroku , 324 ( 1 978 ) . 54 .. D period PRZEWORSKA hyphen ROLEWICZ \noindent1979 54 \quad D . PRZEWORSKA − ROLEWICZ Recent Development in A lgebraic Analysis period Proc period Sympos period comma Res period Inst period Math period\noindent Sci period1979 comma Kyoto Univ period comma Kyoto comma July 2 endash 5 comma 1 979 semicolon Surikaisekikenkyusho Kokyuroku comma 361 open parenthesis 1\noindent 9 79 closingRecent parenthesis54 D . Development PRZEWORSKA period - ROLEWICZ in A lgebraic Analysis . Proc . Sympos . , Res . Inst . Math . Sci . , Kyoto Univ . , Kyoto1981 , July1979 2 −− 5 , 1 979 ; Surikaisekikenkyusho Kokyuroku , 361 ( 1 9 79 ) . Microlocal analysisRecent Development for differential in A equations lgebraic Analysis period ... Proc . period Sympos .. . Sympos , Res . Inst period . Math .. Res . Sci period . , Kyoto .. Inst period .. Math\noindent period ..1981Univ Sci period . , Kyoto comma , July Ky 2 hyphen– 5 , 1 979 ; Surikaisekikenkyusho Kokyuroku , 361 ( 1 9 79 ) . oto University1981 comma Kyoto comma January 1 9 endash 2 2 comma 1 98 1 open parenthesis in Japanese closing parenthesis\noindent periodMicrolocalMicrolocal .. RIMS analysis Kokyuroku analysis for differential comma for differential equations431 period. .. Kyoto Proc equations . Sympos . .\quad Res .Proc Inst . .\quad MathSympos . \quad Res . \quad I n s t . \quad Math . \quad S c i . , Ky − otoUniversity University comma. Sci , Res . Kyoto , Kyperiod - ,oto Inst January University period 1 ,Math Kyoto 9 −− period ,2 January 2 Sci , 1 1 period 9 98 – 2 1 2 III (, 1 commain 98 1 Japanese ( in Kyoto Japanese comma ) ) . .\quad 1 RIMS98 1RIMS period Kokyuroku open , 431 . \quad Kyoto parenthesis aa inKokyuroku summary , closing431 . parenthesis Kyoto \ centerline1984 { UniversityUniversity , Res , Res . . Inst Inst . Math . Math . Sci . . III Sci , Kyoto . III , 1 98 , 1 Kyoto . ( aa in summary, 1 98 1) . ( aa in summary ) } A lgebraic analysis1984 period Proc period Sympos period comma Res period Inst period .. Math period Sci period comma Kyoto\noindent Univ period1984A lgebraic comma analysis Kyoto comma. Proc . October Sympos 1 . 7 , hyphen Res . Inst 20 .comma Math . Sci . , Kyoto Univ . , Kyoto , 1 9 83 semicolonOctober Surikaisekikenkyusho 1 7 - 20 , 1 9 83 ; Surikaisekikenkyusho Kokyuroku comma Kokyuroku 533 open parenthesis, 533 ( 1 984 1 ) 984. closing parenthesis period \noindent1986 A1986 lgebraic analysis . Proc . Sympos . , Res . Inst . \quad Math . Sci . , Kyoto Univ . , Kyoto , October 1 7 − 20 , 1Recent 9 83 developments; SurikaisekikenkyushoRecent developments in algebraic in algebraic analysis Kokyuroku analysis period Proc. , Proc 533 period . Sympos( Sympos1 984 . , ) Resperiod . . Inst comma . Math Res . Sciperiod . , Kyoto Inst period Math period Sci periodUniv comma . , Kyoto Kyoto , July Univ 1 0period – 1 3 ,comma 1 985 ; Surikaisekikenkyusho Kokyuroku , 594 ( 1 986 ) . \noindentKyoto comma19861988 July 1 0 endash 1 3 comma 1 985 semicolon Surikaisekikenkyusho Kokyuroku comma 594 open paren- thesis 1 986 closingDevelopments parenthesis of Algebraic period Analysis . Proc . Sympos . , Res . Inst . Math . Sci . , Kyoto Univ . , \noindent1988 RecentKyoto , developments in algebraic analysis . Proc . Sympos . , Res . Inst . Math . Sci . , Kyoto Univ . , KyotoDevelopments , JulyOctober of 1 Algebraic 0 6−− – 9 ,1 1 Analysis 3 986 , ; Surikaisekikenkyusho1 985 period ; Proc Surikaisekikenkyusho period Kokyuroku Sympos , period 638 ( 1 comma 988Kokyuroku ) . ResDaisui period , kaisekigaku 594 Inst ( period 1... 986 Math ) . period Sci period comma( in Kyoto Japanese Univ ) period [ Several comma aspects Kyoto of algebraic comma analysis ] . Proc . Sympos . , Res . Inst \noindentOctober 61988 endash. Math 9 comma . Sci 1 .986 , Kyoto semicolon Univ . Surikaisekikenkyusho , Kyoto , October 28 – Kokyuroku 3 1 , 1 987 ; comma Surikaisekikenkyusho 638 open parenthesis 1 988 closing parenthesisKokyuroku period , 660 ( 1 988 ) . Daisu kaisekigaku no tenbo ( in Japanese ) [ A view of algebraic analysis \noindentDaisui kaisekigakuDevelopments] . Proc period . period Symposof Algebraicperiod . open Res parenthesis . Analysis Inst . Math in . Japanese . ProcSci . , Kyoto . closing Sympos Univ parenthesis .. , Kyoto , Res .. , Aprilopen . Inst 1square 8 – 2 .bracket 1 Math , Several . Sci . , Kyoto Univ . , Kyoto , aspects of algebraic1 988 analysis ; Surikaisekikenkyusho closing square Kokyuroku bracket period , 675 .. ( Proc1 988 period ) . A lgebraic.. Sympos analysis period. comma Vol . I , II \noindentRes periodOctober ... Papers Inst period dedicated 6 −− ..9 Math to , Professor 1 period 986 Mikio ..; Sci Surikaisekikenkyusho Sato period oncomma the occasion Kyoto of hisUniv Kokyuroku sixtieth period birthday comma , 638 ; Kyoto Eds ( . comma1 988 M . October ) . 28 endashDaisui 3 1 kaisekigakucommaKashiwara 1 987 semicolon and . T . . .Kawai ..( Surikaisekikenkyusho in . Japanese Academic Press ) \quad , Boston[ , Several MA , 1 988 aspects , 1 9 89 . of algebraic analysis ] . \quad Proc . \quad Sympos . , ResKokyuroku . \quad comma1989I n s t 660 . open\quad parenthesisMath . 1\quad 988 closingSci parenthesis . , Kyoto period Univ . , Kyoto , October 28 −− 3 1 , 1 987 ; \quad Surikaisekikenkyusho KokyurokuDaisu kaisekigakuA , lgebraic 660 no ( tenboanalysis 1 988 open , ) geometry parenthesis . , and in number Japanese theory closing. Proc parenthesis . JAMI Inaugural open square Conf . bracket Johns Hopkins A view of algebraic analysisDaisu closing kaisekigakuUniv square . Press bracket no , Baltimore tenbo period ..( , Proc MD in ,Japanese period 1 989 . .. SymposA ) lgebraic [ A period viewanalysis .. of Res of nonlinear algebraic period integrable analysis systems ]( . in\quad Proc . \quad Sympos . \quad Res . InstInst period . Math MathJapanese . Sci period ) Proc . Sci , . Kyoto period Symp . comma Univ Res . Inst . Kyoto , . KyotoMath Univ . period ,Sci April . , commaKyoto 1 Univ 8 Kyoto−− . Kyoto2 comma 1 , , January 1 April 988 1 1 ; 8 7 endash –Surikaisekikenkyusho 2 0 , 1 2 1 comma 1 Kokyuroku , 988675 semicolon ( 1 988 Surikaisekikenkyusho989 ) . SRIMS. Kokyuroku Kokyuroku , 694 . Kyoto comma Univ . , Res . Inst . Math . Sci . Kyoto , 1 9 89 . A675 lgebraic open parenthesis1992 analysis 1 988 . closing\quad parenthesisVol . \quad periodI , II . Papers dedicated to Professor Mikio Sato on the occasion of his sixtiethA lgebraic birthday analysisA lgebraic period analysis ; Eds .. Vol and . period\ numberquad ..M I theory comma . Kashiwara. II period Proc . Papers and Sympos T dedicated . Kawai . to Res .Professor\ .quad Inst MikioAcademic . Sato Math on Press . the occasion , Boston , MA , 1 988 , of1 his 9 89 . Sci . , Kyoto Univ . , Kyoto , March 23 – 28 , 1 99 2 . Surikaisekikenkyusho - Kokyuroku 81 0 ( 1 sixtieth birthday992 ) .semicolon Eds period .. M period Kashiwara and T period Kawai period .. Academic Press comma Boston\noindent comma19891994 MA comma 1 988 comma 1 9 89 periodA lgebraic Analysis Meeting . Li e` ge , 1 993 . Bull . Soc . Roy . Sci . Li e` ge , 3 – 4 , 63 ( 1 9 94 t−r \noindent1989 A). lgebraicAnalyse alg analysise´ brique des , geometrype urbations , and singuli numbere` res theory. I . M . Proce´ thodes . resurgentesJAMI Inaugural. Conf . Johns Hopkins UnivA lgebraic . Press analysisEd . , Baltimore Lcomma . Boutet geometry de , Monvel MD comma , . 1 989 Confand . numbere´ rences theory du symposium period Procfranco period - j aponais JAMI sur Inaugural l ’ analyse Conf period JohnsA lgebraic Hopkins alg analysise´ brique des of perturba nonlinear - tions integrable singuli e` res , systems CIRM , Marseille ( in Japanese- Luminy , France ) Proc , October . Symp 2 . Res . Inst . Math . SciUniv . period , Kyoto0 Press – 26 Univ ,comma 1 99 . 1 Baltimore .Kyoto Travaux , encomma January Cours MD 47 1 comma . 7 Hermann−− 12 989 0 , Paris period , 1 , 989 1 9 94 . . SRIMSAnalyse Kokyuroku alg e´ brique , des 694 . Kyoto Univ . , Res . t−r InstA lgebraic . Math analysispe . Sciurbations of nonlinear . Kyoto singuli integrable ,e` 1res 9 . 89 systems II . M. opene´ thodes parenthesis diff e´ rentielles in Japanese. Ed closing . L . Boutet parenthesis de Monvel Proc . period Symp period Res periodConf Inste´ periodrences du Math symposium period franco - j aponais sur l ’ analyse alg e´ brique des perturba - tions \noindentSci period1992 commasinguli Kyotoe` res , Univ CIRM period , Marseille Kyoto - Luminy comma , January France ,1 October 7 endash 2 0 2 – 0 26 comma , 1 99 1 1 . 989 Travaux period en SRIMS Cours Kokyuroku comma 694 period48 . Kyoto Hermann Univ , Paris period , 1 9comma 94 . Res period \noindentInst periodA Math lgebraic period Sci analysis period Kyoto and comma number 1 9 theory 89 period . \quad Proc . \quad Sympos . \quad Res . \quad I n s t . \quad Math . \quad Sci . , Kyoto Univ . , Kyoto1992 , March 23 −− 28 , 1 99 2 . Surikaisekikenkyusho − Kokyuroku 81 0 ( 1 992 ) . A lgebraic analysis and number theory period .. Proc period .. Sympos period .. Res period .. Inst period .. Math period\noindent .. Sci period1994 comma Kyoto Univ period comma Kyoto comma March 23 endash 28 comma 1 99 2 period Surikaisekikenkyusho hyphen Kokyuroku 81 0 open parenthesis 1\noindent 992 closingA parenthesis lgebraic period Analysis Meeting . Li $ \grave{e} $ ge , 1 993 . Bull . Soc . Roy . Sci . Li $ \1994grave{e} $ ge , 3 −− 4 ,63(1994) . 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Leipzig brique comma . 1 860 period 1863 \ centerlineNovi G period{ Paris comma , Analisi 1 82 A 1 lgebrica . } period Firenze comma 1 863 period 1867 \noindentLieblein J1845 period comma Sammlung von Aufgaben aus der Algebraischen Analysis period .. Verlag von H period Carl JSchl period $ \ddot{o} $ milch O . , Handbuch der Algebraischen Analysis . Jena , 1 845 ; 5 . Aufl . 1 873 . Satov comma Prag comma 1 867 period \noindent1877 1851 DiengerHattendorf J K . period , Grundz comma $ A\ lgebraischeddot{u} $ Analysis ge der period Algebraischen Hannover comma Analysis 1 877 period . Karlsruhe , 1 85 1 . \noindent 1860 Gallenkamp W . , Die Elemente der Mathematik . \quad III. \quad T e i l . \quad Die algebraische Analysis und die

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\noindent 1863 Novi G . , Analisi A lgebrica . Firenze , 1 863 .

\noindent 1867 Lieblein J . , Sammlung von Aufgaben aus der Algebraischen Analysis . \quad Verlag von H . Carl J .

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\noindent 1877 Hattendorf K . , A lgebraische Analysis . Hannover , 1 877 . 56 .. D period PRZEWORSKA hyphen ROLEWICZ \noindent1886 56 \quad D . PRZEWORSKA − ROLEWICZ Capelli A period comma Garbieri G period comma Analisi A lgebrica period Padova comma 1 886 period \noindent1893 1886 CapelliPincherle A S .period , Garbieri comma Analisi G . Algebrica , Analisi period A lgebrica Milano comma . Padova 1 893 , 1 886 . 1894 56 D . PRZEWORSKA - ROLEWICZ \noindentCapelli A period18931886 commaCapelli Istituzioni A . , Garbieri di Analysi G ., AlgebricaAnalisi A comma lgebrica Libreria. Padova scientifica , 1 886 . ed industriale D period B period PelleranoPincherle period S1893 . ,Pincherle AnalisiS., AlgebricaAnalisi Algebrica . Milano. Milano , 1 ,893 1 893 Napoli comma1894 1 894Capelli semicolonA., 3 periodIstituzioni ed period di Analysi 1 902 Algebrica period , Libreria scientifica ed industriale D . 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Fratelli Bocca Ces a-grave1904 sub ro ECes perioda` commaro E., ElementaresElementares Lehrbuch Lehrbuch der Algebraischen Analysis Analysis und und der der Infinitesimal Infinitesimal Rechnung \ centerlinemit zahlreichen{RechnungEditori U-dieresis . bungsbeispielen Torino , 1 894 period . B} period G period Teubner comma Leipzig comma 1 904 semicolon ¨ translated from the Italianmit zahlreichen U bungsbeispielen . B . G . Teubner , Leipzig , 1 904 ; translated from the \noindentmanuscript1903 byItalian Doctor Gerhard Kowalewski comma Universit a-dieresis t Greisswald period Burkhardt1906 H . , A lgebraischemanuscript by Doctor Analysis Gerhard . Kowalewski Leipzig , , Universit 1 903a¨ . t Greisswald . Pincherle S1906 periodPincherle comma LezioniS., Lezioni di A lgebra di A lgebra Complementare Complementare period . I . AnalisiI period A Analisilgebrica A. Bologna lgebrica ,period 1 9 Bologna comma\noindent 1 9 6 period19046 . 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Verlag von B period G period\noindent Teubner1906 comma Leipzig comma 1 909 endash 1 9 2 1 . Pincherle1 9 2 1 period S191 . 1 ,Grave LezioniD.O., di ACourse lgebra of A Complementare lgebraic Analysis ( .in RussianI . Analisi ) . Kiev ,A 1 9 lgebrica 1 1 . . Bologna , 1 9 6 . 191 1 1938 – 39 Grave D.O., Treatise on Algebraic Analysis ( in Russian and Ukrainian ) . 2 \noindentGrave D period1909volumes O period . Izd comma . Course of A lgebraic Analysis open parenthesis in Russian closing parenthesis period KievPringsheim comma 1 9 A 1 1 . period , Faber G . ,Ukrain IIC .1 Akad . A . Nauklgebraische , Kiev , 1 938 Analysis – 1 939 . . In : Encyklop $ \ddot{a} $ die1938 der endash Mathematische1943 39 Vajda S., The algebraic analysis of contingency tables . J . Roy . Statist . Soc . ( N . Grave D periodS . ) O 1 6 period ( 1 943 comma ) , .. Treatise on Algebraic Analysis open parenthesis in Russian and Ukrainian closing parenthesis\ hspace ∗{\ periodf i l l ..} Wissenchaften 2 volumes period .. mit Izd period Einschluss333 – ihrer 342 . Anwendungen . \quad Zweiter Band in drei Teilen . \quad Eds . \quad H. Ukrain period1955 AkadElston period F Nauk . G comma., The Kievlast theorem comma of 1 Fermat 938 endash not only 1 939 a problem period of algebraic analysis but \ hspace1943 ∗{\ falso i l l } aBurhardt , M . Wirtinger , R . Fricke und E . Hilb . \quad Verlag von B . G . Teubner , Leipzig , 1 909 −− Vajda S period comma Theprobability algebraic problem analysis? ofMath contingency . Mag . tables 28 ( 1 95 period 5 ) , 1 J 50 period – 1 52 Roy . period Statist period Soc period\ centerline .. open parenthesis{Garcia1 9 2G., 1 N .Rosenblatt period} S periodA., closingAnalisis parenthesis algebraico ( 1 in 6 Spanish open parenthesis ) [ Algebraic 1 943analysis closing ] . Sanmartiparenthesis comma 333 endash 342y period \noindent1955 191 1 Compania , Lima , 1 955 . GraveElston D F .period O1958 . G ,Chang period Course commaC.C., of A TheAlgebraic lgebraic last theorem analysis Analysis of of Fermat many valued ( not in only logics Russian a problem. Trans ) . of .Kiev algebraic Amer , 1 analysis . 9 1 Math 1 but . also a probability problem. Soc ?. , .. 88 Math ( 1 958 period ) , Mag period 28 open parenthesis 1 95 5 closing parenthesis comma 1 50 endash 1\noindent 52 period 1938467 –−− 49039 . GraveGarcia D G .period O1960 . comma , Rey\quad P Rosenblatt . JTreatise . , Pi Calleja A period on P comma Algebraic . , Trejo Analisis C . Analysis A algebraico., Analisis ( open matematico in parenthesis Russian . in and Vol Spanish . Ukrainian I : Analisis closing parenthesis ) . \quad 2 volumes . \quad Izd . open square bracketalgebraico Algebraic . Teoria analysis closing square bracket period .... 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\noindent 467 −− 490 .

\noindent 1960 Rey P . J . , Pi Calleja P . , Trejo C . A . , Analisis matematico . \quad Vol . I : Analisis algebraico . Teoria

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\ hspace ∗{\ f i l l } I : Algebraic analysis . Theory of equations . Infinitesimal calculus of one variable ] . Fifth ed .

\ centerline { Editorial Kapelusz , Buenos Aires , 1 960 . } TWO CENTURIES OF quotedblleft ALGEBRAIC ANALYSIS quotedblright .. 57 \ hspace1969 ∗{\ f i l l }TWO CENTURIES OF ‘ ‘ ALGEBRAIC ANALYSIS ’ ’ \quad 57 Marcu E period T period comma On an operator that is related to Sestier quoteright s operator in contextual algebraic analysis\noindent period1969 MarcuStud period E . T Cerc . , period On an Mat operator period comma that 2 is1 open related parenthesis to Sestier 1 969 closing ’ s parenthesis operator comma in contextual 499 endash 508 algebraic analysis . period TWO CENTURIES OF “ ALGEBRAIC ANALYSIS ” 57 \ centerline1970 {1969StudMarcu . CercE.T., . MatOn . an , operator 2 1 ( that 1 969is related ) , to 499 Sestier−− ’ s508 operator . } in contextual algebraic Blanusa D periodanalysis comma. .. Visa matematika period .. I dio period .. Prvi svezak open parenthesis in Serbo hyphen Croatian\noindent closing1970 parenthesis open squareStud bracket . Cerc Higher . Mat . mathematics , 2 1 ( 1 969 ) period , 499 – 508 . BlanusaVol period D I1970 . period , \quadBlanusa Part OneVisaD., closing matematika squareVisa matematika bracket . \quad period .I AlgebraI dio dio . . i algebarska\ Prviquad svezakPrvi analiza( in svezak Serbo period - Croatian ( .. in open Serbo )square [ − bracketCroatian ) [ Higher mathematics . Algebra and algebraicHigher mathematics analysis closing . square bracket period Third ed period semicolon \ hspaceManualia∗{\ Universitatisf i l l }VolVol . Studiorum . I . .Part Part One Zagrebiensis One] . Algebra ] . period i Algebra algebarska Tehnicka analiza i algebarska Knjiga. comma [ Algebra analiza Zagreb and algebraic comma . \quad analysis 1 970[ period Algebra ] . and algebraic analysis ] . Third ed . ; 1972 Third ed . ; \ centerlineMeyer R period{Manualia KManualia period Universitatis comma Universitatis Routley Studiorum R Studiorum period Zagrebiensis comma Zagrebiensis Algebraic . Tehnicka analysis Knjiga . 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