41. Cass 1 even more obvious. However, it took about 300 years before it was proven,by the Norwegian mathematician axel thue. It is arguable http://www.ams.org/new-in-math/cover/cass1.html

Packing Pennies in the Plane An illustrated proof of Kepler's conjecture in 2D by Bill Casselman NOTE: This month's contribution contains several Java applets. They may not work on your particular computer, for any of various reasons. If you do not have Java enabled in your browser, for example, you will see only static images representing the animated applets. If you have trouble with viewing the applets even though Java is enabled, or if you want to print out this note, you should disable Java. If Java is enabled and you still have trouble viewing the applets, please let Bill Casselman know about it. 1. "Kepler's Conjecture" This and the other image nearby are from Kepler's pamphlet on snowflakes. Contrary to what one might think at first. they are not of two dimensional objects, but rather an attempt to render on the page three dimensional packings of spheres. In his book De nive sexangula (`On the six-sided snowflake') of 1611, Kepler asserted that the packing in three dimensions made familiar to us by fruit stands (called the face-centred cubic packing by crystallographers) was the tightest possible: Coaptatio fiet arctissima: ut nullo praetera ordine plures globuli in idem vas compingi queant.

42. Beezer's Academic Genealogy Albert Thoralf Skolem TCSGMHMBDM; axel thue TCSGMHM BDM;Marius Sophus Lie MHM; Peter Ludwig Mejdell Sylow MHM. The http://buzzard.ups.edu/genealogy.html

Beezer's Academic Genealogy Here it is the succession of PhD advisers and students that goes backwards in time from my own degree. For the later entries it is not clear that there was a formal advisor/student/degree relationship, but there is evidence that one person was influenced in their education by the other. It seems odd that [TCSG] lists Ore as a student of Skolem, with Ore's degree awarded in 1924 while [BDM] lists Skolem's degree as being given in 1926. Tree Paul Morris Weichsel (Cal Tech 1960) [ MGP Richard Albert Dean (Ohio State 1953) [ MGP Marshall Hall, Jr. (Yale University 1936) [ MGP TCSG MGP TCSG Albert Thoralf Skolem [ TCSG MHM ][BDM] Axel Thue [ TCSG MHM ] [BDM] Marius Sophus Lie [ MHM Peter Ludwig Mejdell Sylow [ MHM The following quotes are from articles in the Biographical Dictionary of Mathematicians [BDM]: Skolem: "In the latter year [1916] he returned to Oslo, where he was made Dozent in 1918. He received his doctorate in 1926." (H. Oettel, p. 2296) Thue: "Thue enrolled at Oslo University in 1883 and became a candidate for the doctorate in 1889." (Viggo Brun, p. 2460)

43. Ma Thèse : Introduction Translate this page La combinatoire des mots a déjà une longue histoire au début de ce siècle (entre1906 et 1914), le mathématicien norvégien axel thue publia une série d http://iml.univ-mrs.fr/~cassaign/these/these04.html

Introduction La combinatoire des mots a déjà une longue histoire : au début de ce siècle (entre 1906 et 1914), le mathématicien norvégien Axel Thue publia une série d'articles sur les problèmes combinatoires soulevés par l'étude des suites de symboles. Deux d'entre eux [61,62] en particulier traitaient des répétitions de facteurs dans les suites, et des moyens d'éviter de telles répétitions. Ainsi, il a montré qu'il est possible de construire une suite infinie sur un alphabet à trois lettres qui ne contienne pas de facteur répété deux fois consécutivement : on dit que c'est une suite sans carré. Par la suite, les résultats de Thue furent plusieurs fois redécouverts, dans des buts divers. Alors que Thue avait fait cette étude pour le développement des sciences logiques et sans application en vue, Arson [3] construisit une suite sans carré en 1937 pour résoudre un problème d'algèbre et Marston Morse et Gustav Hedlund [41] s'intéressèrent à ce problème dans les années 1940 en vue de l'étude des propriétés de surfaces de courbure négative. Citons également des applications aux algèbres universelles, à la théorie des groupes, à la théorie ergodique, mais aussi en physique et bien sûr en informatique où le problème de la recherche de motifs dans un texte est un sujet d'étude important. De nombreux auteurs ont étudié les mots et les morphismes sans carré ou sans cube [1,7,8,20,22,27,29,32,33,37,49,60,64]. En 1979, Bean, Ehrenfeucht et McNulty [6] et indépendamment Zimin [65] proposèrent de généraliser l'étude de répétitions à celle de motifs arbitraires, formés de blocs de différents types disposés dans un ordre précis. Ils ont montré qu'on peut caractériser les motifs qui sont évités par un mot infini sur un alphabet fini mais non précisé; dans le cas où l'alphabet est fixé, aucune caractérisation n'est connue à ce jour. En 1989, Baker, McNulty et Taylor [5] ont poursuivi l'étude de l'évitabilité des motifs sur un alphabet fixé, en donnant entre autres un exemple de motif évitable sur un alphabet à quatre lettres mais pas sur un alphabet plus petit.

44. American Scientist - Computing Science thue, axel. 1912. Über die gegenseitige lage gleicher teile gewisser Zeichenreihen.In Selected Mathematical Papers of axel thue, pp. 413–477. http://www.americanscientist.org/Issues/Comsci01/Compsci2001-11.html

Computing Science November-December, 2001 Third Base Brian Hayes Note: This document is available in other formats People count by tens and machines count by twos—that pretty much sums up the way we do arithmetic on this planet. But there are countless other ways to count. Here I want to offer three cheers for base 3, the ternary system. The numerals in this sequence—beginning 0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101—are not as widely known or widely used as their decimal and binary cousins, but they have charms all their own. They are the Goldilocks choice among numbering systems: When base 2 is too small and base 10 is too big, base 3 is just right. Cheaper by the Threesome Under the skin, numbering systems are all alike. Numerals in various bases may well look different, but the numbers they represent are the same. In decimal notation, the numeral 19 is shorthand for this expression: 1 x 10 + 9 x 10 Likewise the binary numeral 10011 is understood to mean: 1 x 2 + x 2 + x 2 + 1 x 2 + 1 x 2 which adds up to the same value. So does the ternary version, 201:

45. T 0 =0 T 2n =t N E T (2n+1) =t N ' Per Ogni N =0. Translate this page Il matematico axel thue (1863-1922) si chiese se esista una sequenza infinita binaria(fatta di 0 e 1) nella quale non appaiano mai 2 blocchi consecutivi di 3 http://alpha01.dm.unito.it/personalpages/cerruti/Az1/thue.html

Se non le conoscete leggete le istruzioni Thue Azionando l'applet (che conviene rimpicciolire) partono 4 trenini nelle 4 direzioni, producendo degli alianti che collidono. Quello che ci interessa sono le 4 sequenze diagonali di semafori che si leggono partendo dal centro. Il centro è vuoto e la successione comincia con 0; seguono 1 1, cioè vi sono 2 semafori uno di seguito all'altro, poi vi è uno spazio vuoto che leggiamo come 0, poi 1 1 e così via. L'inizio della successione è il seguente: 0110110111110110111110110110... Questa successione è definita in modo ricorsivo, in maniera simile alla famosa successione di Prouhet-Thue-Morse. Il matematico Axel Thue (1863-1922) si chiese se esista una sequenza infinita binaria (fatta di e 1) nella quale non appaiano mai 2 blocchi consecutivi di 3 simboli uguali nè blocchi della forma awawa dove a è od 1 e w un arbitrario blocco binario. In effetti esiste, e comincia così: 0110100110010110100101100110... Essa può essere definita formalmente in molti modi; ne vediamo due. Nel seguito dato un bit t denotiamo con t' il suo complemento : t'=0 se t=1, t'=1 se t=0.

46. Formal Numbers This site is dedicated to the memory of the norwegian mathematician axel thue(18661922), who proved one of the most remarkable result of the twentieth http://www.math.u-bordeaux.fr/~lasjauni/

FORMAL NUMBERS by Alain Lasjaunias Alain.Lasjaunias@math.u-bordeaux.fr (Publications) Click here to enter... This site is dedicated to the memory of the norwegian mathematician Axel Thue (1866-1922), who proved one of the most remarkable result of the twentieth century in number theory Our goal is to present in an elementary way a class of abstract numbers . These numbers have been progressively introduced and studied in the last fifty years. Unlike real numbers, they are of no use to measure physical quantities but will certainly have applications still unknown. The further removed from usefulness or practical application, the more important. Axel Thue

47. Formal Numbers Translate this page Ce site est dédié à la mémoire du mathématicien norvégien axel thue (1866-1922),qui a démontré un des résultats les plus remarquables du vingtième http://www.math.u-bordeaux.fr/~lasjauni/page_fr_0.htm

NOMBRES FORMELS par Alain Lasjaunias Alain.Lasjaunias@libertysurf.fr (Publications) Cliquer ici pour entrer... Axel Thue

48. String Rewriting And The Fibonacci Word String rewriting has been studied for about a century, since the Norwegian logicianand mathematician axel thue devised and perused The Word Problem as an http://www.washingtonart.net/whealton/fibword.html

Steve Whealton String Rewriting and the Fibonacci Word Something that my musical and my visual work have in common is maintaining a proper balance between sameness and randomness. I am forever looking for new, interesting, and different ways to create patterns, to alter patterns, to merge patterns, and to render and manifest patterns in ways audible and visible. Greep Theory . The Thue-Morse Word and the Fibonacci Word seemed to have been invented just for me. Unlike with greeps, however, the strings dealt with in rewriting are of varying, indefinite, or even of a theoretically infinite length. A given set of rules are applied over and over so as to produce, in theory at least, a string that can go on forever! This "infinite" string goes by the provocative name, the "Omega Word." String rewriting has been studied for about a century, since the Norwegian logician and mathematician Axel Thue devised and perused "The Word Problem" as an exercise in logic. Today, a field of study, called "Combinatorics on Words," has grown up from this beginning. It flourishes in France and elsewhere. But my earliest work with strings was visual. Here is how it fell out.

49. Biografisk Register Translate this page 335-395) Thom, A. Thom, AS Thompson, John Griggs (1932-) thue, axel (1863-1922) Torricelli,Evengelista (1608-47) Tsjebysjev, Pafnutij Lvovitsj (1821-94) Turing http://www.geocities.com/CapeCanaveral/Hangar/3736/biografi.htm

Biografisk register Matematikerne er ordnet alfabetisk på bakgrunn av etternavn. Linker angir at personen har en egen artikkel her. Fødsels- og dødsår oppgis der dette har vært tilgjengelig. Abel, Niels Henrik Abu Kamil (ca. 850-930) Ackermann, Wilhelm (1896-1962) Adelard fra Bath (1075-1160) Agnesi, Maria G. (1718-99) al-Karaji (rundt 1000) al-Khwarizmi, Abu Abd-Allah Ibn Musa (ca. 790-850) Anaximander (610-547 f.Kr.) Apollonis fra Perga (ca. 262-190 f.Kr.) Appel, Kenneth Archytas fra Taras (ca. 428-350 f.Kr.) Argand, Jean Robert (1768-1822) Aristoteles (384-322 f.Kr.) Arkimedes (287-212 f.Kr.) Arnauld, Antoine (1612-94) Aryabhata (476-550) Aschbacher, Michael Babbage, Charles (1792-1871) Bachmann, Paul Gustav (1837-1920) Bacon, Francis (1561-1626) Baker, Alan (1939-) Ball, Walter W. R. (1892-1945) Banach, Stéfan (1892-1945) Banneker, Benjamin Berkeley, George (1658-1753) Bernoulli, Jacques (1654-1705) Bernoulli, Jean (1667-1748) Bernstein, Felix (1878-1956) Bertrand, Joseph Louis Francois (1822-1900) Bharati Krsna Tirthaji, Sri (1884-1960)

50. I955: Malene Larsdatter BAGSTAD (1769 - 1844) BAPTISM (dy). Father Iver Olsen HELT Mother Sophie Lauritzdatter HOLGERSEN Family1 Johan Robertsen thue MARRIAGE +Elisabeth Anna thue. axel MOTZFELDT. http://home.online.no/~nermo/slekt/d0004/g0000000.html

Malene Larsdatter BAGSTAD BIRTH : 1769, Bagstad Søre DEATH Family 1 Peder Johannessen ARONSVEEN MARRIAGE Lars Pedersen NERMO Aase PEDERSDATTER Peder Pedersen NERMO Kari Pedersdatter NERMO ... INDEX HTML created by GED2HTML v3.5e-WIN95 (Sep 26 1998) on 09/10/2002 10:30:40 Harald Gudrødsson GRENSKE OCCUPATION : Underkonge i Viken 976-87 BIRTH : 0952, Kvam, Nord-Fron DEATH : 0995, (brent inne) Father: Gudrød BJØRNSSON Family 1 Åsta Gudbrandsdotter KULA MARRIAGE Olav II Haraldsson den HELLIGE _Harald (Luva) HÅRFAGRE Harald Gudrødsson GRENSKE INDEX HTML created by GED2HTML v3.5e-WIN95 (Sep 26 1998) on 09/10/2002 10:30:40 Margrethe Iversdatter HELT BIRTH : CIR1665, (usikre foreldre) Bud, Aukra, MR ? BAPTISM : (d.y.) Father: Iver Olsen HELT Mother: Sophie Lauritzdatter HOLGERSEN Family 1 Johan Robertsen THUE MARRIAGE Elisabeth Anna THUE _Ole HELT _Iver Olsen HELT Margrethe Iversdatter HELT ... INDEX HTML created by GED2HTML v3.5e-WIN95 (Sep 26 1998) on 09/10/2002 10:30:40 Axel MOTZFELDT BIRTH : 1731, Verdal, NT DEATH : 1810, Moss Father: Peter Jacob MOTZFELDT Mother: Thale Marie ARENFELDT Family 1 Bolette BRIX MARRIAGE Axel MOTZFELDT INDEX HTML created by GED2HTML v3.5e-WIN95 (Sep 26 1998)

51. Www.math.uwaterloo.ca/~ljcummin/info/cv.txt 1981, June 11 On Construction of thue Strings , Mathematical Institute, Oxford,UK . 1981, March 3 The work of axel thue , University of Oslo. http://www.math.uwaterloo.ca/~ljcummin/info/cv.txt

52. Www.liacs.nl/~beatcs/toc/beatcs72.txt 252 18. Agenda of Events . . . 253 19. Historical Comments Trees and Term Rewritingin 1910 On a Paper by axel thue (by M. Steinby, W. Thomas) . . . 256 20. http://www.liacs.nl/~beatcs/toc/beatcs72.txt

53. [Thomesd. - Tolnes] Thott, 36 147. Thott, axel Laurenss. 7 276. Thu, Chatrine Helene, 9 359. thue,Andreas, apoteker, 3 99, 21 260. thue, Anna, gm Kaasbøll, 20 370. thue, Anne,29 259. http://www.genealogi.no/nst_reg/T/t-2.htm

T [Thomesd. - Tolnes] oversikt forrige neste Thomesd. Anne Cathrine Thomesd. Giertrud Thomesd. Inger, g.m. Winter Thomesd. Malene, g.m. Frey Thomesd. Maren Thomesd. Tabitta, g.m. Christenss. Thomesen. Thomess. Alf Andreas Thomess. Alf, handelsmann Thomess. Anders Thomess. Anders, fogd Thomess. Anders, ved Halden Thomess. Anna Caroline Thomess. Thomess. Benjamin Olai Angell Thomess. Bernt Thomess. Birgitte Thomess. Birgitte Johanne, Larss. Thomess. Thomess. Christiane Christine Mentzoni Thomess. David Thomess. Elise Johanne (Lisa) Angell, g.m. 1) Jenss., 2) Jacobss. Thomess. Frans, prest til Sem Thomess. Gunnar Thomess. Hans Thomess. Hans (ca. 1627), skibsr. m. m. 10: 9 f Thomess. Hans (ca. 1638-1717) Thomess. Hans, krovert Thomess. Hartvig Bugge, prest Thomess. Helga Thomess. Inge Thomess. Ingeborg Eliasd. Thomess. Ingeborg Jonette Thomess. Ingeborg Sophie, g.m. Ole Thomess. Thomess. Thomess. Jacob Thomess. Thomess. Jan 15: 119 fg Thomess. Jens Thomess. Jens Lind Thomess. Thomess. Johan Thomess. Johanne Andrea, g.m. Jenss. Thomess. Jon Thomess. Jon, borgerm. i Bergen Thomess. Thomess.

54. Literatur Thu06 axel thue. Über unendliche zeichenreihen. Kra. Vidensk. Thu12 axelthue. Über die gegenseitige lage gleicher teile gewisser zeichenreihen. Kra. http://www.informatik.uni-leipzig.de/~joe/edu/ss01/l/l-bib.html

Literatur Julien Cassaigne. Tree Automata Techniques and Applications. http://www.grappa.univ-lille3.fr/tata/ John~H. Conway and Richard~K. Guy. The Book of Numbers. Copernicus/Springer, 1995. Karel Culik~II and Tero Harju. Journal ACM, 31(2):282298, April 1984. Combinatorics of Words, pages 329428. Volker Diekert. Makanin's Algorithm, pages 344391. Vesa Havala and Tero Harju. Some new results on post correspondence problem and its modifications. TUCS Technical Report 338, Turku Centre for Computer Science, January 2001. http://www.tucs.fi/publications/techreports/TR338.html Morphisms, pages 439510. Lila Kari, Grzegorz Rozenberg, and Arto Salomaa. L Systems, pages 253328. Winfriend Kurth. Die Simulation der Baumarchitektur mit Wachstumsgrammatiken. Wissenschaftlicher Verlag Berlin, 1999. http://www.uni-forst.gwdg.de/~wkurth/public.html Aristid Lindenmayer. Mathematical models for cellular interaction in development. J. Theoret. Biology, 18:280315, 1968. Aristid Lindenmayer. Developmental systems without cellular interactions, their languages and grammars. J. Theoret. Biology, 30:455484, 1971.

55. Juristforeningen 1928, Knut Glad, Øivind Rye Florentz, Hans Kristian Skou, Franz Beyer Jersen,axel thue, Arvid Frithjof Rasmussen, 1927, Knut Tvedt, 1926, Birger Motzfeld, JohnLyng, http://www.juristforeningen.no/dekorandi.shtml

Siste nytt Lover Kart Kontakt ... Organisasjonskart Navn Devise Julianne Meling Jon Ole Whist Christian F. Platou En offiser og en genitalmann Thomas D. A. Howard Preben Willoch Are Gauslaa Fra (H)Are til pus Nina Harboe Jensen Henrik Kolderup Radioaktivt Tangohue, nese for fag ble pengetue Anne Hesjedal Saken er Biff! Hege Farnes Hansen Aadel H. T. Heilemann Liten pike, store tanker Thomas Fjeld Heltne Stine N. Johannessen Jeg bare jobber her, jeg... Bendik Christoffersen Henrik Hagberg Mammadalt i full galopp Sjiraff i regnskapsskogen Lars Berge Andersen Thor Martin Dalhaug Walter Martin Tveter Liten skrue kan trekke stort lass Odd-Kaare Oftedal Merete Astrup Svartveit Monica Svendsen Christopher Borch Margrethe Buskerud Carsten Gunnarstorp Henning Harborg Kanarifull Christopher J. Helgeby Frokostkjellerns drillsjersjant, nesten blid en gang i blant Thomas Lia Jan Arild Pedersen Eystein Eriksrud Christen Horn Johannessen Olav Hasaas Festkamerat? Ikke akkurat, demokratisk kamp for Arafat Kjetil Edvardsen Tine Blom Hartvigsen Fest, sier du? Dessverre jeg har time hos tannlegen

56. Thue - 1897 Translate this page Zurück axel thue - 1897 thue benutzt in seinem Beweis den Hauptsatzder Zahlentheorie, also die Eindeutigkeit der Zerlegung der http://www.didaktik.mathematik.uni-wuerzburg.de/veranstaltungen/zahlsys_ws01_02/

Axel Thue - 1897 Thue benutzt in seinem Beweis den Hauptsatz der Zahlentheorie, also die Eindeutigkeit der Zerlegung der natürlichen Zahlen in ihre Primfaktoren. Dieser Beweis gibt sogar als quantitatives Resultat eine untere Schranke für die Anzahl der Primzahlen an, wenn es hier auch mittlerweile bessere Ergebnisse gibt. Seien n und k natürliche Zahlen (ohne die Null !), so daß folgendes gilt: Weiterhin seien nun alle Primzahlen, die erfüllen. Weiterhin sei Nach dem Hauptsatz der Zahlentheorie läßt sich jede natürliche Zahl m mit folgendermaßen eindeutig als Produkt darstellen: Dabei ist Werden in dieser Darstellung alle Möglichkeiten gezählt, dann folgt: und dies ist unmöglich, deshalb folgt: Wird nun gewählt, dann folgt aus daß ist. Es gibt also mindestens k+1 Primzahlen p mit Wird k unendlich groß gewählt, dann folgt sofort, daß es unendlich viele Primzahlen gibt. Paulo Ribenboim, The Book of Prime Number Records, Springer-Verlag 1989, 2. Auflage

57. Read This: How The Other Half Thinks Finally, Chapter 8 solves a problem posed by axel thue in 1912 can we constructarbitrarily long strings in a's, b's and c's which contain no pairs of http://www.maa.org/reviews/otherhalf.html

Read This! The MAA Online book review column How the Other Half Thinks by Sherman Stein Reviewed by Stacy Langton Sherman Stein, author of a calculus textbook, a monograph on the theory of tiling, a study of Archimedes , and Strength in Numbers (the latter two previously reviewed on MAA Online ), here presents another installment of mathematics for the general public. How the Other Half Thinks: Adventures in Mathematical Reasoning consists of eight short chapters, each of which sets up and then solves a nontrivial mathematical problem. Proofs from THE BOOK Chapters 2 and 4 deal with random strings of a's and b's. In Chapter 2, Stein asks how long such a string must be before the number of occurrences of one of the letters exceeds the number of occurrences of the other by 2. The expected value of this length is given by an infinite series. Stein evaluates the series by a clever rearrangement which goes back to the 14th century scholastic Nicole Oresme. The same series occurs in Chapter 4, where Stein computes the expected length of a run of a's or b's. Another problem about probability is treated in Chapter 6: in an election involving two candidates, what is the probability that one candidate will lead during the entire count? The solution here is based on a geometric reflection argument.

58. Elementary Number Theory - Kenneth H. Rosen Page 504 Biographical information about axel thue can be found at the MacTutor Historyof Mathematics Archive at http//wwwgroups.dcs.st-andrews.ac.uk/~history http://www.aw.com/rosen/resourcesc_13.html

Annotated Web Links CHAPTER 13 Some Nonlinear Diophantine Equations Return to Annotated Web Links Home 13.1 Pythagorean Triples Page 482 Biographical information about Pythagoras can be found at the MacTutor History of Mathematics Archive at http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Pythagoras.html (Pythagoras) 13.2 Fermat's Last Theorem Page 488 An excellent survey of the history of Fermat's last theorem can be found at http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html (Fermat's last theorem) Information about Fermat's last theorem can be found at NOVA Web site on the pages accompanying their episode devoted to Wiles's proof : http://www.pbs.org/wgbh/nova/proof (NOVA Online - The Proof) To begin exploring the mathematics behind Wiles's proof of Fermat's last theorem, you should look at a page developed by Charles Daney at http://www.best.com/~cgd/home/flt/flt01.htm

59. NewAbelianSquare-FreeTD0L-LanguagesOver4Letters.nb Abstract In 1906 axel thue 34 started the systematic study of structuresin words. Consequently, he studied basic objects of theoretical http://south.rotol.ramk.fi/keranen/ias2002/NewAbelianSquare-FreeTD0L-LanguagesOv

New Abelian Square-Free TD0L-Languages over 4 Letters Rovaniemi Polytechnic, School of Technology veikko.keranen@ramk.fi http://south.rotol.ramk.fi Abstract on the four letter alphabet a b c d abcd have been based on the structure of this ; see Arturo Carpi [46]. In this paper, we report of a completely new endomorphism of , the iteration of which produces an infinite abelian square-free word. The size of , for which they were directly obtained by permutating letters cyclically. The endomorphism is not an a-2-free endomorphism itself, since it does not preserve the a-2-freeness of all words of length 7. However, can be used together with to produce a-2-free TD0L-languages of unlimited size. Here TD0L-languages mean deterministic context-independent Lindenmayer languages produced by using compositions of endomorphisms so called tables; see [32, p.188]. Indeed, by using Carpi's algorithm [4] for prefixes of and , where does not contain a certain subword pattern, ) and ) are always a-2-free and avoid all undesirable patterns that would, in the case of , lead to an abelian-square in the next iteration step.

60. Repetition Free Words And Computer Algebra The systematic study of word structures (combinatorics on words) was started bya Norwegian mathematician axel thue 7 (18631922) at the beginning of this http://south.rotol.ramk.fi/keranen/research/RepetitionFreeStrings.html

Repetition Free Words Repetition Free Words and Computer Algebra Abstract Words or strings belong to the very basic objects in theoretical computer science. Thus, the investigation of structures in words constitutes a central research topic in this branch of science. The systematic study of word structures (combinatorics on words) was started by a Norwegian mathematician Axel Thue [7] (1863-1922) at the beginning of this century. One of the remarkable discoveries made by Thue is that the consecutive repetitions of non-empty subwords (squares) can be avoided in infinite words over a three letter alphabet. After Thue's time, repetition-free words have been used in various fields of mathematics. For example, in group theory, in formal languages, in connection with unending games, and in symbolic dynamics (which constitutes a tool for studying chaos). Very recently repetition-free words have also aroused interest in the field of music, see eg. Laakso [5]. Let X a b c d g X X * which we found by the aid of computers. This endomorphism g X X g abcd a is g a abcacdcbcdcadcdbdabacabadbabcbdbcbacbcdcacbabdabacadcbcdcacdbcbacbcdcacdcbdcdadbdcbca and the image words of letters b c d , i.e., the words

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