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Thue Axel:     more detail

Selected Mathematical Papers (German Edition) by Axel Thue, 1977-04 Mathématicien Norvégien: Niels Henrik Abel, Sophus Lie, Atle Selberg, Thoralf Skolem, Ludwig Sylow, Kristen Nygaard, Axel Thue, Viggo Brun (French Edition) Spectrum and extended states in a harmonic Chain with controlled disorder: Effects of the thue-Morse symmetry by Françoise Axel, 1989

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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
    
    Extractions: 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. 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
    
    Extractions: 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. 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
    
    Extractions: 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
    
    Extractions: 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
    
    Extractions: 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...

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/
    
    Extractions: 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.

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

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
    
    Extractions: 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

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

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
    
    Extractions: 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

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
    
    Extractions: 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/
    
    Extractions: 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.

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
    
    Extractions: by Sherman Stein 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

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
    
    Extractions: 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
    
    Extractions: Repetition Free Words 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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