1. Mathematics

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TRANSLATION OF SPECIALIST TEXTS

SCIENTIFIC DISCOURSE EXCERPTS.PART 1. Mathematics

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Tet 1.A historical look at the evolution of the genres ofmathematical texts suggests that the lexicogrammar ofmathematical symbolism may have evolved from naturallanguage because mathematical texts were initially writtenin the prose form of verbal ``rhetorical algebra.'' Thesetexts contained detailed verbal instructions about what wasto be done for the solution of a problem. In later texts, thereappeared abbreviations for recurring participants andoperations in what is known ``syncopated algebra.'' The use

of variables and signs for participants and mathematicaloperations in the last !! years resulted in ``symbolicalgebra'' and the contemporary lexicogrammar ofmathematics. Thus, we may con"ecture that the grammar ofmodern mathematical symbolism grew directly out of thelexicogrammar of natural language and this may explain thehigh level of integration of symbolic and linguistic forms inmathematical texts.

#$xtracted from %&alloran,Classroom Discourse in

Mathematics, in (inguistics and education, )!!!, *+ *-*,pdf/

Tet !.%ur sources of information on the history of 0reek geometrybefore $uclid consist merely of scattered notices in ancientwriters. The early mathematicians, Thales and 1ythagoras,left behind no written records of their discoveries. A fullhistory of 0reek geometry and astronomy during this

period, written by $udemus, a pupil of Aristotle, has beenlost. It was well known to 1roclus, who, in his commentarieson $uclid, gives a brief account of it. This abstractconstitutes our most reliable information. 2e shall 3uote itfre3uently under the name of $udemian 4ummary.

Tet ".To Thales of 5iletus #67!-76 b.c/,one of the 8 seven wisemen,8 and the founder of the Ionic school, falls the honour

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of having introduced the study of geometry into 0reece.9uring middle life he engaged in commercial pursuits,which took him to $gypt. e is said to have resided there,

and to have studied the physical sciences and mathematicswith the $gyptian priests.1lutarch declares that Thales soon excelled his masters, andama:ed ;ing Amasis by measuring the heights of thepyramids from their shadows. According to 1lutarch, thiswas done by considering that the shadow cast by a verticalsta< of known length bears the same ratio to the shadow ofthe pyramid as the height of the sta< bears to the height ofthe pyramid. This solution presupposes a knowledge of

proportion, and the Ahmes papyrus actually shows that therudiments of proportion were known to the $gyptians.According to 9iogenes (aertius, the pyramids weremeasured by Thales in a dinding thelength of the shadow of the pyramid at the moment whenthe shadow of a sta< was e3ual to its own length.

Tet #.Thales may be said to have created the geometry of lines,essentially abstract in its character, while the $gyptiansstudied only the geometry of surfaces and the rudiments ofsolid geometry, empirical in their character.'2ith Thales begins also the study of scienti>c astronomy. eac3uired great celebrity by the prediction of a solar eclipsein ?.@. 2hether he predicted the day of the occurrence,or simply the year, is not known. It is told of him that whilecontemplating the stars during an evening walk, he fell intoa ditch. The good old woman attending him exclaimed,Bow canst thou know what is doing in the heavens, when

thou seest not what is at thy feetC8

#$xcerpt from @a"ori, Dlorian, E!,A History of Mathematics,downloadable from www.Dorgotten?ooks.org/

Tet $.

2e will model derivations algebraically by using so-called%re polynomials. These have >rst been considered byFystein %re in G%re**H. They are a generalisation of

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ordinary polynomials which oed way ofdescribing linear dirst introduced and proved toexist by @harles ermite in GerEH for non-singular s3uarematrices over the integers. It was later extended to moregeneral matrices. Its main application is solving of9iophantine e3uations. 2e will introduce the ermite formin 9e>nition E) in 4ection ).*.

Tet &.

The 1opov normal form was >rst described by asile 5ihai1opov in G1opJ!, 1opJ)H. Together with similar concepts likerow-reduction that is sometimes also called row-propernessit is widely used in control theoryKsee for example GLer!JH.2e will treat the 1opov normal form in 9e>nition E*.

0rMbner bases have been >rst mentioned in G?uc6H.9evised originally for ideals of #commutative/ multivariatepolynomials they have since then been extended to non-

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commutative domains and also to #free/ modules overthem. 4ee for example G@4H for a treatise and anapplication of 0rMbner bases for %re polynomials. An

introduction to extensions of 0rMbner bases to modules maybe found in the textbooks GA(7H for the commutative caseand similarly in G?0T!*H for non-commutative domains. 2ewill use the latter book extensively in this report.

Tet '.

0rMbner bases are important in modern day computationalalgebra because they provide two useful features+ Dirst,

they solve the ideal membership problem. That is, theyprovide an algorithmic way to check whether a givenpolynomial is contained in a given ideal. 4econd, 0rMbnerbases have an elimination property which makes themuseful for solving systems of polynomial e3uations. Thesame properties carry over to the non-commutative and tothe module case.

In G;NT!JH it was shown that 1opov and ermite normalforms of matrices with #univariate/ polynomial entries are

actually 0rMbner bases for their row span. This result caneasily be generali:ed to %re polynomial rings. 2e do this inTheorems ) and )J.

Tet (.

The FGLM algorithm is an eOcient method to convert0rMbner bases of :ero-dimensional ideals from oneadmissible ordering to another. It was presented in

GD0(5*H for ideals of commutative polynomials. The mainreason for its eOciency is that it manages to translate theproblem from polynomial to linear algebra. The D0(5algorithm may lead to a speed-up of 0rMbner basiscomputations for slow orderings like the lexicographicordering by >rst computing a 0rMbner basis with respect toa faster ordering like a degree ordering and then convertingit to the desired ordering. A short introduction to the D0(5algorithm is given in 4ection *.*.E

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2e will translate the original algorithm from the case ofcommutative ideals to modules over non-commutativedomains in 4ection *.*.7. There, we will in particular deal

with the problem that the modules we consider are notB:ero-dimensionalP, i. e. that their 3uotient spaces can be ofin>nite dimension. 2e will be able to solve this using adegree bound on ermite and 1opov normal forms that issimilar to the results in G0;!H.

#$xtracted from Popov to Hermite via FLGMQ. 5idekke )!E!.pdf/

Tet 1)

Artless innocents an* i+or,-toer so/histicates0Some /ersonalities on the In*ian mathematical

scene1

At the beginning of the )!thcentury, science was still anesoteric pursuit of reclusive intellectuals. The 3uietrevolution in the academic worlds of 0Mttingen,@openhagen, @ambridge and 1aris of the early decadesexploded into global awareness of science with iroshima.

GRRRRRR.H1hysicists dominated this celebrity parade, but there

were chemists and biologists in fair number. In this contextof hype about science and adulation for the scientist, I thinkthe mathematician is described best by a Tamil proverb+ heis the hapless fellow who brought home a copper vesselafter taking part in a raid on ;ubera&s AlakapuriS 5any greatnames in mathematics are entirely unfamiliar to peopleoutside the scienti>c community. This article is about some

mathematicians who have contributed signi>cantly tomathematics in the )!th century. GRRRR.His work on what is now called the Namanu"an tau

function, which evoked only a moderate response at thattime, later proved to be profound and central to what iscalled the Theory of 5odular Dorms. ecke, a great 0erman

1?ased on a public lecture delivered at the Annual 5eeting of the Indian

Academy of 4ciences, held at @handigarh in )!!).

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mathematician who was one of the architects of the theorywas born in the same year EJ as Namanu"an= it doesseem a great pity that the two never met. A con"ecture of

Namanu"an on the tau function was settled in EJ7 by 1ierre9eligne, a leading mathematician of our era. %ne of themost fruitful techni3ues applied successfully to diverseproblems in umber Theory is known as the @ircle 5ethod.Namanu"an&s notebooks are a treasure house of beautifulformulae and identities, set down without details. 1rovidingproper proofs to these has been a challenge and at thesame time, a public service to the mathematical community.$xperience indicates that there is likely to be much more to

many of these results than the formal beauty which by itselfmakes them attractive. It has been said that in the matterof formal manipulations Namanu"an has no e3uals in thehistory of mathematics, other than $uler and Qacobi.

A partition of a positive integer n is an expression of nas a sum n U nE V n) V R V nr, each ni a positive integerwith ! W nE X n) X Y X nr. The number of digures of the twentieth century and, asthe Namanu"an story shows, he was a wonderful humanbeing. e was also exceptionally articulate and a giftedwriter of $nglish prose. is little book, A Mathematicians

Apology, giving his view of his profession, makes delightfulreading. ardy was very much the ivory-tower intellectual,and the book is not so much an Zapology& as an emphaticaOrmation of his belief in the irrelevance of social relevancein the pursuit of pure science. e has also written anaccount of the ?ertrand Nussell a


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ardy was an ardent cricket fan= so ardent indeed thathe calibrated excellence in any >eld by cricketing greats+the highest accolade was to be in the Z?radman class&S

Interesting people were people who had Zspin& in them.ardy was an outstanding analyst as well as a numbertheorist= his work in analysis has in some ways been morein[uential than that in number theory.

It is in the thirties at Annamalai \niversity that 1illai&stalents were in full bloom and he cracked a problem thatwas engaging some of the >nest minds. 9avid ilbert hadshown that for every integer k ] !, there is a smallestinteger g#k/ ] !, such that every positive integer can be

expressed as a sum of g#k/ kth powers. 1illai&s work centredon the exact determination of g#k/. e achieved thecomplete determination for k ^ J, a superb achievement byany reckoning. e went on a little later to tackle the evenmore diOcult case k U 6. owever, a controversy overpriorities involving the American mathematician (. $.9ickson was a cause for some distress to 1illai and hisIndian colleagues. 1illai published his great papers in Indian"ournals which did not have a wide circulation= nevertheless,recognition did come eventually for these outstandingcontributions, but tragedy struck once more before he couldsavour his success. %n *E August E!, 1illai died in an air-crash over $gypt= he was on his way to the \4 his >rst tripabroad to spend a year at the Institute for Advanced 4tudyin 1rinceton, where he had been invited. The news wasreceived with great shock by the many mathematiciansassembled at arvard, where 1illai was to participate in theInternational @ongress of 5athematicians before going on to1rinceton. 1illai&s work on the Z2aring& problem the

determination of the g#k/ is a piece that has given him apermanent place in the history of mathematics. Thedetermination of g#k/ for all k has now been completed, thecase k U 7 was the one that de>ed mathematicians thelongest, till about E! years ago when another Indian, N.?alasubramanian in collaboration with two Drenchmen,9eshouillers and 9ress settled the matter. 1illai hadnumerous other important contributions as well.

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Tet 1!.There is little doubt that 1illai would have achieved a

great deal more if his life had not been cut short so abruptly.

4il5ert6s Theorem 71()(80 Given a positive integerk, there is a positive integer r such that every positiveinteger n is the sum of r kth poers of integers. Inother words, every positive integer n is e3ual to nEk

Vn)kVRVnrk for suitable non-negative integers nE, n),R nr. The same r works for every n. There is evidently aminimal such r for a given k= this minimal r is denotedby g#k/.9arin: Pro5lem0 9etermine g#k/.

g#E/ U E #%bvious/

g#)/ U 7 #(agrange/g#*/ U #2ieferich and ;empner, EE)/4olved by 1illai for k ^ 6 #E*6/

g#/ U *J #@hen, E*6/g#7/ U E #?alasubramanian, 9eshouillers and

9ress, E6/The 9arin: /ro5lem

Z1illai&, in the words of 1rof. ;. @handrasekharan Zwas aperson of genuine modesty and remarkable simplicity. e

possessed that rare 3uality among intellectuals intellectual honesty which endeared him to his friends, butlost him many material advantages&. Ze was&, says@handrasekharan, Zunsophisticated in a peculiar sense&. Inthe early part of the )!thcentury, ?ritish mathematics heldsway over us. It did produce some bene>cial results. ?ut inthe twenties and thirties, the most exciting developments inmathematics were taking place in 1aris and 0Mttingen, not@ambridge. These developments seem to have had noserious immediate impact on the Indian scene.

$xcerpt from @\NN$T 4@I$@$, %(. , %. 7, pp. )6-*6, )A\0\4T )!!*

Tet 1"ever more :ealously and successfully has mathematicsbeen cultivated than in this century. or has progress, as inprevious periods, been con>ned to one or two countries.2hile the Drench and 4wiss, who alone during the preceding

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epoch carried the torch of progress, have continued todevelop mathematics with great success, from othercountries whole armies of enthusiastic workers have

wheeled into the front rank. 0ermany awoke from herlethargy by bringing forward 0auss, Qacobi, and hosts ofmore recent men= 0reat ?ritain produced her 9e 5organ,?oole, amilton, besides champions who are still living=Nussia entered the arena with her (obatchewsky = orwaywith Abel= Italy with @remona = ungary with her two?olyais= the \nited 4tates with ?en"amin 1eirce. Theproductiveness of modern writers has been enormous.BIt is diOcult,8 says 1rofessor @ayley, to give an idea of the

vast extent of modern mathematics. This word 'extent' isnot the right one+ I mean extent crowded with beautifuldetail,8 not an extent of mere uniformity such as anob"ectless plain, but of a tract of beautiful country seen at>rst in the distance, but which will bear to be rambledthrough and studied in every detail of hillside and valley,stream, rock, wood, and [ower.8 It is pleasant to themathematician to think that in his, as in no other science,the achievements of every age remain possessions forever=new discoveries seldom disprove older tenets= seldom isanything lost or wasted.

Tet 1#If it be asked wherein the utility of some modern extensionsof mathematics lies, it must be acknowledged that it is atpresent diOcult to see how they are ever to becomeapplicable to 3uestions of common life or physical science.?ut our inability to do this should not be urged as anargument against the pursuit of such studies. In the >rst

place, we know neither the day nor the hour when theseabstract developments will >nd application in the mechanicarts, in physical science, or in other branches ofmathematics. Dor example, the whole sub"ect of graphicalstatics, so useful to the practical engineer, was made to restupon von 4taudt's 0eometrie der (age= amilton's8principle of varying action8 has its use in astronomy=complex 3uantities, general integrals, and general theoremsin integration o


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and magnetism. 2ho, for instance, would have supposedthat the calculus of forms or the theory of substitutionswould have thrown much light upon ordinary e3uations= or

that Abelian functions and hyperelliptic transcendentswould have told us anything about the properties of curves=or that the calculus of operations would have helped us inany way towards the >gure of the earthC8A second reason in favour of the pursuit of advancedmathematics, even when there is no promise of practicalapplication is that mathematics, like poetry and music,deserves cultivation for its own sake.Translate into En:lish

Tet 1$

;ri:ore ;he5a9l 0heba a fost profesor de matematic_ la 4c nr. !,

pe 1anduri, n timp ce eram n clasele -III= eu am avut altprofesor, tot aa de bun. @red c_ ne-a inut ) ore nlocuindu-l pe profesorul nostru cand a fost bolnav. Nemarcabil om.

1rofesorul 0rigore 0heba scoate o culegere dematematica la E de ani .

@el mai cunoscut autor de culegeri de matematic_ din

Nomnia, profesorul 0rigore 0heba, ateapta n aceste :iles_-i apar_ o nou_ lucrare, BTeme fundamentale n studiulmatematicii - @lasele I-P. @ulegerea, prima pe carevenerabilul profesor nu o testea:_ mai nti pe elevii s_i, arepe coperta B>gura 0hebaP, model geometric ce a limpe:itani de-a rndul minile elevilor.

Intr-o diminea_ nehot_rt_ de prim_var_ din anul alE-lea al existenei sale, 0rigore 0heba a f_cut roat_ cugndul peste ntmpl_rile ce-i umplu >ina+ anii de coal_,

anii de r_:boi i de lag_r, dou_ neveste, copiii, o iubire carel-a readus din mori, altele mai mici ce doar i-au reamintitc_ e n via_. i culegerile de matematica, *7 la numar,tiparite in peste sase milioane de exemplare. Nndurile demai "os ncearc_ s_ recompun_ povestea profesorului0heba. %mul care a scos la tabl_ ara intreag_.

Tet 1%.Anii *e


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5-am ntlnit cu profesorul - i generalul in retragere -0rigore 0heba n sufrageria casei sale i am mp_rit pentructeva ore aceeai mas_ acoperit_ cu sticl_ spart_ la unul

dintre coluri. e desp_rea un m_nunchi de po:e. N_:boiul,matematica, femeile iubite. % via_.

B5-am n_scut ntr-un sat de munte din "udeulrancea, se cheama 1oienia - 9umitresi, ca unul dintre ceiapte copii ani unei familii s_race. 5ama mea, 5aria, puneaga: n lamp_ doar smb_ta i duminica, n restul s_pt_mniiculegeam surcele ca s_ in focul trea: n sob_ i s_ pot facesocoteli cu condeiul pe t_blia din piatr_ moale. (ucramdintr-o carte, BE!!E de probleme de matematic_P se

numea, primit_ de la un unchi, inspector de matematic_P,povesteste profesorul 0heba. ?_iatul nu mplinise nc_ E)ani cnd tia pe dinafara cartea de probleme, strninduimirea unchiului. (a imboldul acestuia, intr-o dimineata,5aria 0heba arunc_ ntr-o traist_ patru oua >erte lng_ unpumn de m_m_lig_ i, cu 0rigore de mn_, o lua pestedealuri, cale de *! de kilometri pe "os, s_-i dea feciorul laliceul din Nmnicu-4arat.

A"unser_ inainte de c_derea soarelui, plini de praf i[_mn:i.

Demeia ncepu s_ plng_ f_r_ s_ poat_ spune de ce. Iiag__ copilului traista pe um_r, f_cu o cruce mare cudreapta i-l mpinse pe poarta colii. 1uiul de _ran intr_ intr-un hohot de rs. \niformele apretate din b_nci se hli:eau deopincile lui. 4e oprir_ doar cnd directorul 0herda i slobo:iglasul c_tre nou-venit+ B@e tii tu din matematici, b_iete SCP.Btiu tot din cartea astaSP, ndra:ni 0rigore, scond dintraist_ c_rulia de la unche. 1este clas_ se l_s_ o t_cerecurioas_. BIa scrie-mi tu pe tabl_ trei milionimiSP. ou

venitul apuc_ creta i scri"eli cu :gomot ascutit... !,!!!.!!*.0herda :mbi, ap_s_ un buton i-n cteva clipe pe u_ se iicapul contabilului colii. BIl iei pe _r_noiul asta, l imbraci lamaga:inul B5ireasaP i-l pui la internat, o s_ >e elevul coliimeleP.

(ui 0rigore i-au trebuit trei s_pt_mni s_ nvee s_mearg_ n ghetele galbene cu toc de patru centimetri pecare le primise de la coal_.

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Tet 1&.Pri>onier la cot3l Don3l3i

Tn_rul 0heba urm_ apoi coala de o>eri n re:erv_

de la ?ac_u i obinu gradul de sublocotenent. Ii amintetec_ a r_mas concentrat permanent la Nmnicu-4arat Bpentruc_ nevestei colonelului Anastasescu, comandantulregimentului, i pl_cea cum dansamP. tia ea, coloneasa,ceva, de vreme ce, tocmai n timpul unui bal,sublocotenentul a cucerit inima unei frumoase nv__toare,(ilica 1opescu, care, n E*6, avea s_-i devin_ soie.N_:boiul l purt_ pe sublocotenentul de artilerie 0rigore0heba in Transnistria, s_ lupte mpotriva ruilor. 1e *

noiembrie E7) c_:u pri:onier la @otul 9onului. B9up_ ceam trecut prin trei lag_re, dintr-un b_rbat voinic, la )6 deani, a"unsesem s_ cntaresc * de kilograme. Atunci aaparut ngerul vieii mele.P

Tet 1'.Mar3sia? in:er3l c3 ochi al5astri$ra medicul lag_rului, avea gradul de c_pitan, era blond_ cuochi albatri. % chema 5arusia Anka. B4-a ae:at lng_patul meu, plngeam, convins c_ mai am cteva ore detr_it. 5-a ntrebat de ce plng, i-am r_spuns c_ nu voi maivedea niciodat_ ara. 5-a mutat ntr-o alt_ camer_ i s-angro:it de oasele mele cnd m-a de:bracat s_ m_ consulte.A revenit cu nite lapte i ou_. u mai v_:usem de doi aniaa ceva. 5ai tr:iu am a[at c_ erau din raia ei...P

Aa ncepu ntre cei doi o poveste de dragoste carecontinu_ tot r_:boiul. (a ndemnul 5arusiei, 0heba s-anscris n 9ivi:ia Tudor ladimirescu i a luptat mpotrivanemilor. A fost de dou_ ori r_nit i decorat de opt ori. In

ciuda acestei evoluii, a avut relaii tensionate cu ctivaconduc_tori sovietici, accentuate imediat dup_ r_:boi, cind,a fost acu:at de antisovietism. BAm p_r_sit toate structurilei m-am nscris la Dacultatea de 5atematic_ din ?ucureti.u mi s-a permis s_ acced la catedre nalte pentru ca amrefu:at s_ m_ nscriu n partid.P In E prima lui soie amurit, iar 0heba s-a rec_s_torit doi ani mai tr:iu cu(ucreia, care-i st_ i ast_:i al_turi.

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la preg_tire i nu a probat pe ei, ca n atitea alte rnduri,metodele de re:olvare. B9ar n-am cum s_ dau gre, eexperiena mea de-o via_ acoloSP

(a plecare l intreb dac_ mai tie ceva de c_pitanul cu:ulu> i ochi albastri. Imi spune, uor soptit, c_ n E, cuacordul soiei i cu *7! de ruble n bu:unar, a :burat la5oscova, s_ dea de urma ngerului. @nd a a"uns n faablocului de pe ?ulevardul 5axim 0orki, unde s-a petrecut onoapte din dragostea lor, i s-a oprit inima n loc. -a maig_sit-o pe 5arusia. A r_mas la locul ei doar p_durea demoli:i de la marginea 5oscovei.

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1. Mathematics