Nonstandard Analysis in Practice PDF
Preview Nonstandard Analysis in Practice
Universitext SSpprriinnggeerr--VVeerrllaagg BBeerrlliinn HHeeiiddeellbbeerrgg GGmmbbHH Francine Diener Marc Diener Editors Nonstandard Analysis in Practice With 34 Figures Springer FFrraanncciinnee DDiieenneerr MMaarrccOOiieenneerr UUnniivveerrssiittee ddee NNiiccee LLaabboorraattooiirree CCNNRRSS ddee MMaatthheemmaattiiqquueess PPaarrcc VVaallrroossee FF--0066110088 NNiiccee CCeeddeexx 22,, FFrraannccee CCIIPP--ddaattaa aapppplliieedd ffoorr DDiiee DDeeuuttsscchhee BBiibblliiootthheekk -- CCIIPP--EEiinnhheeiittssaauuffnnaahhmmee NNoonnssttaannddaarrdd aannaallyyssiiss iinn pprraaccttiiccee // FFrraanncciinnee DDiieenneerr ;; MMaarree DDiieenneerr eedd.. -- BBeerrlliinn;; HHeeiiddeellbbeerrgg :: NNeeww YYoorrkk:: BBaarrcceelloonnaa :: BBuuddaappeesstt :: HHoonngg KKoonngg :: LLoonnddoonn :: MMiillaann :: PPaarriiss :: SSaannttaa CCllaarraa :: SSiinnggaappoorree;; TTookkyyoo:: SSpprriinnggeerr,, 11999955 ((UUnniivveerrssiitteexxtt)) IISSBBNN 997788--33--554400--6600229977--22 IISSBBNN 997788--33--664422--5577775588--11 ((eeBBooookk)) DDOOII 1100..11000077//997788--33--664422--5577775588--11 NNEE:: DDiieenneerr,, FFrraanncciinnee [[HHrrssgg..)) 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AAllII rriigghhttss aarree rreesseerrvveedd,, wwhheetthheerr tthhee wwhhoollee oorr ppaarrtt ooff tthhee mmaatteerriiaall iiss ccoonncceerrnneedd,, ssppeecciiffiiccaaIIllyy tthhee rriigghhttss ooff ttrraannssllaattiioonn,, rreepprriinnttiinngg,, rreeuussee ooff iilllluussttrraattiioonnss,, rreecciittaattiioonn,, bbrrooaadd--ccaassttiinngg,, rreepprroodduuccttiioonn oonn mmiiccrrooffiillmmss oorr iinn aannyy ootthheerr wwaayy,, aanndd ssttoorraaggee iinn ddaattaa bbaannkkss.. DDuupplliiccaattiioonn ooff tthhiiss ppuubblliiccaattiioonn oorr ppaarrttss tthheerreeooff iiss ppeerrmmiitttteedd oooollyy uunnddeerr tthhee pprroovviissiioonnss ooff tthhee GGeerrmmaann CCooppyyrriigghhtt LLaaww ooff SSeepptteemmbbeerr 99,, 11996655,, iinn iittss ccuurrrreenntt vveerrssiioonn,, aanndd ppeerrmmiissssiioonn ffoorr uussee mmuusstt aallwwaayyss bbee oobbttaaiinneedd ffrroomm SSpprriinnggeerr--VVeerrllaagg.. VViiooIIaattiioonnss aarree IIiiaabbllee ffoorr pprroosseeccuuttiioonn uunnddeerr tthhee GGeerrmmaann CCooppyyrriigghhtt LLaaww.. OO SSpprriinnggeerr--VVeerrllaagg BBeerrlliinn HHeeiiddeellbbeerrgg 11999955 SSPPIINN:: 1100550000225566 4411//33114433--554433221100 --PPrriinntteedd oonn aacciidd--ffrreeee ppaappeerr Foreword In the early seventies, Georges Reeb learnt about Robinson's Nonstandard Analysis (NSA). It became quickly obvious to him that a kind ofrevolution had happened in mathematics: the old dream of actual infinitesimals had been realized. This seemed to him a great event. He was a mechanics-minded topologist, in the dynamical tradition of Painleve, Poincare, Cartan, and thetopologicaltraditionofhis master Ehres mann. He got convinced that NSA is exactly the right framework within which to study dynamics with small parameters, including problems of asymptotics and bifurcations. This was thestartingpointofthesecond school hecreated in Strasbourg, the topological school (which was mainly oriented towards foliation theory and dynamical systems) being the first. This new school grew up with the philosophy of using NSA in everyday mathematics, as a tool to get simple and natural proofs and to detect new mathematical phenomena. The first works were focused on differential equations, but other topics, including perturbation problems in algebra, quickly became of interest. The axiomatic presentation of NSA by E. Nelson within Internal Set Theory in 1977 gave a second impulse to the Alsatian school; this was partly because thisformalsettingwasinagreementwithReeb'sphilosophicalconvictionthat infinitesimals were an unexpected benefit ofthe impossibmty offormalising the intuitive feeling that all natural numbers are of the same kind. Calling "naive" the natural numbers obtainable from zero bythe successive addition ofone, he asked everywhere he went in Europe the disturbing question "Les entiers naifs remplissent-ils N ?" One of the most famous achievements of Reeb's school was the discov ery in 1977 ofthe "canard solutions" that appear furtively when some one parameter families of dynamical systems perform a Hopf bifurcation. The phenomenon was brought to light, first theoretically, then by computer, by a group of young mathematicians working at Oran, Algeria. One of them was Jean- Louis Callot, a very bright and clever scientist who unfortunately passed away in August 1993, just three months before Reeb himself. His work in his last years mainlyconcerned differentialequations inone complex variable, a subject that is present in this book because of his inspiration. VI Foreword In 1981 volume 881 of the Springer Lecture Notes series appeared, the first book to give an account of the state of the art in Reeb's school. Its title was "NSA, a practical guide with applications", and I had the pleasure of writing it in cooperation with my friend, Michel Goze, and with strong encouragementfrom Reeb. Thiswasa (pleasant) intellectualadventurewhich highly stimulated our own work on new applications of NSA. In 1992, the community which developed around Reeb's students orga nized a meeting at Luminy. It there became clear that the time was ripe to present to mathematicians our collective knowledge which had greatly increased since the early eighties. In the work that follows, the readers will first find in the tutorial all that they need to become themselves actors. Then nine typical subjects where NSA is deeply helpful are treated with considerable detail. Theycover some, butnot all mattersinwhichthe "reebiannetwork" isinterested. Forinstance, thereisalmostnothingaboutinfinitesimalstroboscopyandaveraging, bound ary value problems in singular perturbations ofODE's, perturbation theory ofalgebraic structures etc... The aim ofthe book is not to be a catalogue of results, but rather, as was already true ofthe Springer Lecture Notes publi cation above, a practical guide that illustrates various possibilities of using NSA. For those who want to introduce NSA in their teaching practice, the last chaptergives a humorous indication, nurtured byeffectiveexperience, of what is typical in this respect. Clearlytheauthorshavewrittentheircontributionstothiscollectivework with the aim of motivating the readers and of helping them share the very particular turn of mind which is typical of nonstandardists among mathe maticians: to seek simple and natural proofs avoiding artificial subterfuges, to be interested in applying techniques to problems rather than adapting problems to techniques, to jump over the boundaries within mathematics. Reeb and CaBot strongly shared this philosophy, which agrees with Abra ham Robinson's original purposes. I am convinced that the present book will incite the readers to use in their everyday mathematical work some pleasant additional tools which, as Leibniz said about his infinitesimals, "facilitent l'art d'inventer". Robert Lutz Universite de Haute Alsace Departement de MatMmatiques 4, rue des Freres Lumiere F- 68093 MULHOUSE lutz~univ-mulhouse.fr Editors' Foreword The projectofwritingthis bookincollaboration arosetwoyearsagofrom the considerationswhich follow. Firstly, we areoftenasked bycolleagueswho are interested in some result ofnonstandard analysis, or simply curious to know moreofthis newapproachto infinitesimals, wheretheymight find apractical introduction to the subject. Someten years ago, wewould have suggested to them to look at volume 881 in the Springer Lecture Notes series, by R. Lutz and M. Goze, did they wish to see a development after the manner in which wework. Thefact that this bookis no longerin printwasonegood reason for writing an updated introduction to NSA in the spirit ofthe French school. The second consideration was that it should comply with the philosophy that we were taught by our mentor, Georges Reeb : NSA must be easy to learn and its useshould beintuitiveand natural. Butwhatisconsideredeasy, intuitive, or natural depends on the person. One might prefer a picturewhile another might feel that such a picture gives too fuzzy an idea and so would prefer a formula. Thus, to fulfil the program, we decided that it would be best to have several points ofview. In the book which emerged from these reflections the reader will find, after a brief "tutorial", examples ofapplications in various domains, written in somewhat different styles. Despite these differences of content and style thereisan underlyingunityofapproachderivedfromthefactthatthevarious authorshavebeenworkingtogetherinthesamespiritfor morethantenyears. Only the first chapter is needed as background for theones that follow which are, by and large, independent ofone another. As promoters of this program, we would like to thank all our friends who agreed to collaborate to bring it to success. We know that they had to be patient to work through the various versions of their contributions, and especially the last one : the translation into English. Special thanks go to Noel Murphy who read all the papers and did his best to improve our poor use ofthis language. We hopethat thisventurewill helpmore peopletoenjoy the use of infinitesimals, now rediscovered. Nice, April 27th, 1995 Francine and Marc Diener Table of Contents 1. Tutorial F. Diener and M. Diener. ............................ .. .. .. 1 1.1 A new view ofold sets. ................................. 1 1.1.1 Standardand infinitesimalrealnumbers,and theLeib- niz rules. ........................................ 2 1.1.2 To be or not to be standard 4 1.1.3 Internal statements (standard or not) and external statements 5 1.1.4 External sets 5 1.2 Using the extended language. ............................ 7 1.2.1 The axioms. ..................................... 8 1.2.2 Application to standard objects .................... 11 1.3 Shadows and S-properties 14 1.3.1 Shadow ofa set. ................................. 14 1.3.2 S-continuity at a point. ........................... 15 1.3.3 Shadow ofa function 15 1.3.4 S-differentiability................................. 17 1.3.5 Notion ofS-theorem .............................. 18 1.4 Permanence principles 18 1.4.1 The Cauchy principle ............................. 18 1.4.2 Fehrele principle 19 2. Complex analysis A. Jilruchard 23 2.1 Introduction........................................... 23 2.2 Tutorial............................................... 28 2.2.1 Proofofthe Robinson-Callot theorem. .............. 28 2.2.2 Applications..................................... 30 2.2.3 Exercises with answers. ........................... 31 2.2.4 Periodic functions. ............................... 31 2.3 Complex iteration. ..................................... 34 2.4 Airy's equation. ........................................ 42 2.4.1 The distinguished solutions 46 2.5 Answers to exercises .................................... 50 X Table ofContents 3. The Vibrating String Pierre Delfini and Claude Lobry ............................. 51 3.1 Introduction........................................... 51 3.2 Fourier analysis of (DEN) 53 3.2.1 Diagonalisation ofA " 53 3.2.2 Interpretation ofN i-large. ........................ 54 3.2.3 Resolution of (DEN) .............................. 57 3.3 An interesting example 58 3.4 Solutions oflimited energy. .............................. 63 3.4.1 A preliminary theorem ............................ 63 3.4.2 Limited energy: S-continuity ofsolution. ............ 65 3.4.3 Limited energy: propagation and reflexion ........... 67 3.4.4 A particular case: comparison with classical model. ... 69 3.5 Conclusion............................................ 70 4. Random walks and stochastic differential equations Eric Benoit. ............................................... 71 4.1 Introduction........................................... 71 4.2 The Wiener walk with infinitesimal steps .................. 71 4.2.1 The law ofWt for a fixed t ......................... 72 4.2.2 Law ofW . .. .. . . .. . . . . . . .. . . . .. . . . . . . . . . . . . . . . . .. 74 4.3 Equivalent processes .................................... 77 4.3.1 The notion ...................................... 77 4.3.2 Macroscopic properties. ........................... 79 4.3.3 The brownian process. ............................ 79 4.4 Diffusions. Stochastic differential equations. ................ 81 4.4.1 Definitions...................................... 81 4.4.2 Theorems....................................... 82 4.4.3 Change ofvariable. ............................... 83 4.5 Probability law ofa diffusion. ............................ 83 4.6 Ito's calculus - Girsanov's theorem ...... 85 4.7 The "density" ofa diffusion. ............................. 87 4.8 Conclusion............................................ 89 5. Infinitesimal algebra and geometry Michel Goze ............................................... 91 5.1 A natural algebraic calculus 91 5.1.1 The Leibniz rules. ................................ 91 5.1.2 Thealgebraic-geometriccalculusunderlyingapointof the plane. ....................................... 91 5.2 A decomposition theorem for a limited point. .............. 92 5.2.1 The decomposition theorem 93 5.2.2 Geometrical approach. ............................ 95 5.2.3 Algebraic approach. .............................. 96 5.3 Infinitesimal riemannian geometry. ....................... 97 TableofContents XI 5.3.1 Orthonormal decomposition ofa point. ............. 97 5.3.2 The Serret-Frenet frame ofa differentiable curve in]R3 97 5.3.3 The curvature and the torsion 98 5.4 The theory ofmoving frames 100 5.4.1 The theory ofmoving frames 100 5.4.2 The moving frame: an infinitesimal approach 102 5.4.3 The Serret-Frenet fibre bundle 104 5.5 Infinitesimal linear algebra 104 5.5.1 Nonstandard vector spaces 104 5.5.2 Perturbation oflinear operators 105 5.5.3 The Jordan reduction ofa complex linear operator 107 6. General topology Tewfik Sari 109 6.1 Halos in topological spaces '" 109 6.1.1 Topological proximity 109 6.1.2 The halo of a point 110 6.1.3 The shadow ofa subset 111 6.1.4 The halo of a subset 112 6.2 What purpose do halos serve ? 113 6.2.1 Comparison oftopologies 114 6.2.2 Continuity 114 6.2.3 Neighbourhoods, open sets and closed sets 114 6.2.4 Separation and compactness 115 6.3 The external definition ofa topology 116 6.3.1 Halic preorders and P-halos 117 6.3.2 The ball ofcentre x and radius 0: 119 6.3.3 Product spaces and function spaces 121 6.4 The power set ofa topological space 123 6.4.1 The Vietoris topology 123 6.4.2 The Choquet topology 124 6.5 Set-valued mappings and limits ofsets 125 6.5.1 Semicontinuous set-valued mappings 125 6.5.2 The topologization ofsemi-continuities 126 6.5.3 Limits ofSets 127 6.5.4 The topologization ofthe notion ofthe limit ofsets 129 6.6 Uniform spaces 130 6.6.1 Uniform proximity 131 6.6.2 Limited, accessible and nearstandard points 132 6.6.3 The external definition ofa uniformity 133 6.7 Answers to the exercises 134
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