1. Axel Thue Axel Thue. Axel Thue was born in Tönsberg, Norway, on February 19, 1863. 20, 1995.G. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk Mat. http://www.student.math.uwaterloo.ca/~cs462/Hall/thue.html

Axel Thue Acta Mathematica from 1916-1922. Thue is sometimes known as the "father of formal languages". He was the first to investigate the problem of finding an infinite sequence over a finite alphabet that did not contain any occurrences of adjacent identical blocks, a property today known as squarefreeness . Unfortunately, because his papers on repetitions in words were published in a relatively obscure Scandinavian journal, his work was forgotten and later rediscovered by many writers, such as Max Euwe and Marston Morse. However, Gustav Hedlund later rediscovered Thue's contribution and popularized it. Thue also made fundamental contributions to number theory (particularly transcendence theory). Thue died in Oslo, Norway, on March 7, 1922. Sources G. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk Mat. Tidskrift T. Nagell, A. Selberg, S. Selberg, and K. Thalberg, eds., Selected Mathematical Papers of Axel Thue , Universitetsforlaget, Oslo, 1977. Back to Theory of Computing Hall of Fame Main Page Back to CS 462 home page cs462@descartes September 10 1997

2. Thue Axel Thue. Born 19 Feb 1863 in Tönsberg, Norway Died 7 March 1922 in Oslo,Norway. Axel Thue was the son of Niels Thue and Nicoline Cathinka Eger. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Thue.html

Axel Thue Born: Died: 7 March 1922 in Oslo, Norway Click the picture above to see two larger pictures Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Axel Thue was the son of Niels Thue and Nicoline Cathinka Eger. He studied at Voss's school in Oslo, where he showed a great interest in physics, completing his studies there in 1883. He then enrolled at the University of Oslo graduating in 1889. He went to Leipzig in 1890 and spent a year studying under Lie . However, [1]:- ... his works do not reveal Lie 's influence, probably because of Thue's inability to follow anyone else's line of thought. He also spent a while in Berlin where he attended lectures by Helmholtz Fuchs and Kronecker . Back in Olso, Thue held a scholarship in mathematics from 1891 to 1894. On 6 July 1894 he married Lucie Collett Lund who was ten years younger than Thue. Then Thue was appointed to Trondheim Technical College where he worked from 1894 until 1903. He was appointed as professor of applied mathematics at Oslo University in 1903. He held this post until his death in 1922. In 1909 he produced an important paper, published in

3. Poster Of Thue Axel Thue. lived from 1863 to 1922. Thue studied Diophantine equations,showing that, for example, y 3 2x 2 = 1 cannot be satisfied http://www-gap.dcs.st-and.ac.uk/~history/Posters2/Thue.html

Axel Thue lived from 1863 to 1922 Thue studied Diophantine equations, showing that, for example, y = 1 cannot be satisfied by infinitely many pairs of integers. Find out more at http://www-history.mcs.st-andrews.ac.uk/history/ Mathematicians/Thue.html

4. Thue Translate this page thue axel norvégien, 1863-1922 Après des études à Oslo (alorsKristiana) et à Leipzig, Thue enseigne à l'université d'Oslo http://www.sciences-en-ligne.com/momo/chronomath/chrono2/Thue.html

THUE Axel (théorie des nombres). Crelle entier non nul, admet au plus un nombre fini de solutions. + y + y Cette Mordell Siegel Un autre théorème, sur l'approximation rationnelle des nombres algébriques, prolongeant ceux de Liouville Roth Fields Young

5. Poster Of Thue Axel Thue. lived from 1863 to 1922. Find out more at http//wwwhistory.mcs.st-andrews.ac.uk/history/Mathematicians/Thue.html. http://www.cmi.univ-mrs.fr/~liardet/Recherche/Pierre/conferences/TransIUFM/Doc/P

Axel Thue lived from 1863 to 1922 Find out more at http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thue.html

6. Axel Thue's Work On Repetitions In Words - Berstel (ResearchIndex) The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of axel thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is http://citeseer.nj.nec.com/berstel92axel.html

Axel Thue's work on repetitions in words (1992) (Make Corrections) (16 citations) J. Berstel Home/Search Context Related View or download: igm.univmlv.fr/~berst montreal.ps.gz Cached: PS.gz PS PDF DjVu ... Help From: igm.univmlv.fr/~bers Recherche (more) Homepages: J.Berstel HPSearch (Update Links) Rate this article: (best) Comment on this article (Enter summary) Abstract: The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched. 1 Introduction In a series of four papers which appeared during the period 19061914, Axel Thue considered several combinatorial problems which arise in the study of sequences of symbols. Two of these papers [44, 46] deal with... (Update) Context of citations to this paper: More ...n 2 P i the binary expansion of n contains an even number of 1. The xed point (1) is the well known Thue Morse word (see e.g. ) Recall that a B uchi automaton is an automaton A = Q; E; I ; F ) where Q is a nite set of states, E Q A Q is the set of transitions

7. Publications Du LaCIM semithue grammar, which is named for (and possibly created by) the Norwegian mathematician axel thue (pronounced "TOO-ay"). http://www.lacim.uqam.ca/Publications/20.html

Publications du Laboratoire de Combinatoire et d' Informatique Volume 20 Jean Berstel Axel Thue's papers on repetitions in words: a translation 1. Preliminaries 2. Thue's First Paper: About infinite sequences of symbols 3. Thue's Second Paper: On the relative position of equal parts in certain sequences of symbols 4. Notes Axel Thue's papers on repetitions in words: a translation The other two papers deal with repetitions in finite and infinite words. Perhaps because these papers were published in a journal with restricted availability, this work of Thue was widely ignored during a long time, and consequently some of his results have been rediscovered again and again. Axel Thue's papers on sequences are now more easily accessible since they are included in the "Selected Papers" which were edited in 1977. It is the purpose of the present text to give a translation into English of Axel Thue's papers on repetitions in sequences, both in more recent terminology and in relation with new results and directions of research. PAFACC, UQAM

8. Thue Biography of axel thue (18631922) axel thue. Born 19 Feb 1863 in Tönsberg, Norway http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thue.html

Axel Thue Born: Died: 7 March 1922 in Oslo, Norway Click the picture above to see two larger pictures Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Axel Thue was the son of Niels Thue and Nicoline Cathinka Eger. He studied at Voss's school in Oslo, where he showed a great interest in physics, completing his studies there in 1883. He then enrolled at the University of Oslo graduating in 1889. He went to Leipzig in 1890 and spent a year studying under Lie . However, [1]:- ... his works do not reveal Lie 's influence, probably because of Thue's inability to follow anyone else's line of thought. He also spent a while in Berlin where he attended lectures by Helmholtz Fuchs and Kronecker . Back in Olso, Thue held a scholarship in mathematics from 1891 to 1894. On 6 July 1894 he married Lucie Collett Lund who was ten years younger than Thue. Then Thue was appointed to Trondheim Technical College where he worked from 1894 until 1903. He was appointed as professor of applied mathematics at Oslo University in 1903. He held this post until his death in 1922. In 1909 he produced an important paper, published in

9. Mordell Translate this page rationnelles. Des avancées importantes sur ce sujet avaient été obtenuspar le mathématicien norvégien axel thue au début du siècle. http://www.sciences-en-ligne.com/momo/chronomath/chrono2/Mordell.html

anglais, 1892-1972 univers Gerd Faltings une , de genre Axel Thue Weil et de Siegel Andrew Wiles conjecture de Fermat dernier (ou grand Fermat Pour en savoir plus Encyclopaedia Universalis Ed. Albin Michel, Paris, 1997 Banach Rademacher

10. Thue Portraits Portraits of axel thue axel thue. JOC/EFR November 2002 http://www-history.mcs.st-and.ac.uk/history/PictDisplay/Thue.html

Axel Thue JOC/EFR November 2002 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Thue.html

11. Citations: Axel Thue's Work On Repetitions In Words - Berstel (ResearchIndex) J. Berstel 1992, axel thue's work on repetitions in words, in S'eries Formelleset Combinatoire Alg'ebrique (P. Leroux and C. Reutenauer, eds.), Publ. http://citeseer.nj.nec.com/context/279024/244961

16 citations found. Retrieving documents... Jean Berstel. Axel Thue's work on repetitions in words . In Pierre Leroux and Christophe Reutenauer, editors, Series formelles et combinatoire algebrique, volume 11 of Publications du LaCIM, pages 65-80. Universite du Quebec a Montreal, June 1992. Home/Search Document Details and Download Summary Related Articles ... Check This paper is cited in the following contexts: On the Density of Critical Factorizations - Harju, Nowotka (2001) (Correct) ....word is not ultimately periodic, both proven di erently in the literature. Section 4 contains the constructions of in nite sequences of words in four letters with density one for every word, in nite sequences of ternary words which has a limit of their densities at one, using square free words , and in nite sequences of binary words which has a limit of their densities at two, using Thue Morse words [11, 10, 1, 2] We also show that these limits are optimal. 2 Preliminaries In this section we x the notations for this paper. We refer to [8, 4] for more basic and general de nitions. ....

12. The Theory Of Computing Hall Of Fame the major figures in the field. axel thue (18631922); David Hilbert(1862-1943); Alan Turing (1913-1954); Stephen Kleene (1909-1994); http://www.student.math.uwaterloo.ca/~cs360/hall.html

Theory of Computation Fall 2002 The Theory of Computing Hall of Fame The purpose of the Theory of Computing Hall of Fame is to give pictures and brief biographies of some of the major figures in the field. Axel Thue David Hilbert Alan Turing Stephen Kleene ... Leonid Levin

13. Stephen Wolfram: A New Kind Of Science -- Index T-z 18601948) and forms in nature, 967 and growth processes, 1010 and leaf shapes,1006 and theoretical biology, 1004 thue, axel (Norway, 1863-1922) and circle http://www.wolframscience.com/nks/index/names/t-z.html?SearchIndex=t-z

14. Stephen Wolfram: A New Kind Of Science -- Index T-ti free parsing, 1103 and pointerbased encoding, 1071 and testing invariances, 1022and testing reversibility, 1017 thue, axel (Norway, 1863-1922) and circle http://www.wolframscience.com/nks/index/t-ti.html?SearchIndex=Theology

15. Biography-center - Letter T Thost, Hermann Arthur www.whonamedit.com/doctor.cfm/1699.html; thue,axel wwwhistory.mcs.st-and.ac.uk/~history/Mathematicians/thue.html; http://www.biography-center.com/t.html

Visit a random biography ! Any language Arabic Bulgarian Catalan Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Norwegian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Turkish T 340 biographies Tabern, Donalee L. www.invent.org/hall_of_fame/142.html Tacca, Ferdinand www.getty.edu/art/collections/bio/a451-1.html Tacquet, Andrea www-history.mcs.st-and.ac.uk/~history/Mathematicians/Tacquet.html Taft, Helen Herron www.whitehouse.gov/WH/glimpse/firstladies/html/ht27.html Taft, William Howard odur.let.rug.nl/~usa/P/wt27/about/taftbio.htm Tagore, Rabindranath www.nobel.se/literature/laureates/1913/tagore-bio.html Tahir, ibn www-history.mcs.st-and.ac.uk/~history/Mathematicians/Al-Baghdadi.html Taiga, Marton www.kirjasto.sci.fi/taiga.htm Taillandier, Vincent www.getty.edu/art/collections/bio/a1075-1.html Tait, Peter Guthrie www-history.mcs.st-and.ac.uk/~history/Mathematicians/Tait.html Takacs, Karoly

16. Thue - 1897 Translate this page axel thue - 1897 thue benutzt in seinem Beweis den Hauptsatz der Zahlentheorie,also die Eindeutigkeit der Zerlegung der natürlichen http://www.turing-maschine.de/daten/mathematic/beweise/primenumbers-thue.html

Axel Thue - 1897 Seien n und k Weiterhin seien nun alle Primzahlen, die m mit Dabei ist Wird nun ist. Es gibt also mindestens k+1 Primzahlen p mit Wird k Paulo Ribenboim, The Book of Prime Number Records, Springer-Verlag 1989, 2. Auflage www.mathematic.de

17. Index For The Letter T Thomborson, Clark; Thompson, Ben; Thompson, Clark; Thompson, Ken; Thomson,JJ; thue, axel; Tienari, Martti; Tiuryn, Jerzy; Tiwari, Prasoon; Toda http://sigact.acm.org/genealogy/index-T.html

This is the index into entries in the TCS Genealogy for names beginning with the letter T. Tagamlitzki, Yaroslav Tai, Kuo-chung Taitslin, Mikhail Takagi, Teiji ... Tzeng, Wen-Guey

18. Www.catseye.mb.ca /projects/thue-2001.0127/ based on the concept of the semithue grammar/process, which is named for (and possiblycreated by) the Norwegian mathematician axel thue (pronounced TOO-ay http://www.catseye.mb.ca/esoteric/thue/thue.txt

www.catseye.mb.ca The Thue Programming Language Conceived and implemented by John Colagioia For want of a nail, the shoe was lost. For want of a shoe, the horse was lost. For want of a horse, the knight was lost. For want of a knight, the battle was lost. So it was a kingdom was lost, all for the want of a nail. George Herbert, Jacula Prudentum (Colloqual Adaptation) The Thue programming language was invented by John Colagioia in early 2000. It is named after the computational regimen it is based on - a pure string-rewriting process called a semi-Thue grammar , which is named for (and possibly created by) the Norwegian mathematician Axel Thue (pronounced "TOO-ay"). Thue represents one of the simplest possible ways to convey constraint-based programming. It is to the constraint-based paradigm what languages like OISC are to the imperative paradigm; in other words, it's a tar pit. projects/ thue-2001.0127/ (thue-2001.0127.tgz) subdirectory archive/ doc/ eg/ src/ Powered by openflax v2003.0327

19. Www.catseye.mb.ca /projects/thue-2001.0127/ a pure stringrewriting process called a semi-thue grammar, which is named for (andpossibly created by) the Norwegian mathematician axel thue (pronounced TOO http://www.catseye.mb.ca/esoteric/thue/

www.catseye.mb.ca The Thue Programming Language Conceived and implemented by John Colagioia For want of a nail, the shoe was lost. For want of a shoe, the horse was lost. For want of a horse, the knight was lost. For want of a knight, the battle was lost. So it was a kingdom was lost, all for the want of a nail. George Herbert, Jacula Prudentum (Colloqual Adaptation) The Thue programming language was invented by John Colagioia in early 2000. It is named after the computational regimen it is based on - a pure string-rewriting process called a semi-Thue grammar , which is named for (and possibly created by) the Norwegian mathematician Axel Thue (pronounced "TOO-ay"). Thue represents one of the simplest possible ways to convey constraint-based programming. It is to the constraint-based paradigm what languages like OISC are to the imperative paradigm; in other words, it's a tar pit. projects/ thue-2001.0127/ (thue-2001.0127.tgz) subdirectory archive/ doc/ eg/ src/ Powered by openflax v2003.0327

20. Jean Berstel axel thue's Work on Repetitions in Words Séries formelles et combinatoire algébrique(P. Leroux, C. Reutenauer eds.) Publications du LaCIM, Université du http://www-igm.univ-mlv.fr/~berstel/Recherche.html

J. Berstel, ps.gz (pdf ) Ecoles des jeunes chercheurs en informatique J. Berstel, Repetitions in words : recent results and open problems (ps.gz (pdf) Ottrott, 23-27 mars 2003 Combinatorics on words - a tutorial, (ps Bull. EATCS (february 2003), pp 178-228. J. Berstel, D. Perrin, Theory of Codes... . Academic Press, 1985. J.-M. Autebert, J. Berstel et L. Boasson Context-free languages and PushDown Automata (ps.gz ps) in: Handbook of formal languages , Vol. 1, p 111174, Springer, Berlin, 1997. J. Berstel, Recent results on Sturmian Words Developments in language theory, II (Magdeburg, 1995), p 1324, World Sci. Publishing, River Edge, NJ, 1996. J. Berstel, Axel Thue's Work on Repetitions in Words (P. Leroux, C. Reutenauer eds.) Publications du LaCIM, J. Berstel, Axel Thue's papers on repetitions in words: a translation Publications du LaCIM, This is a translation, into English and in contemporary terminology, of the two fundamental papers of Axel Thue about repetitions in words. Articles An Exercice on Fibonacci Representations Balanced Grammars and Their Languages (avec Luc Boasson) Coding rotations on intervals (avec Laurent Vuillon) A Scalable Formal Method for Design and Automatic Checking of User Interfaces International Conference on Software Engineering (ICSE 2001) Formal Properties of XML Grammars and Languages XML grammars (avec Luc Boasson), M.Nielsen, B. Rovan (Eds.) Mathematical Foundations of Computer Science 2000, Bratislava

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