21. Doug, Basho, And Axel Thu Douglas J. Shaw, 2002. Historical Note His name was actually axel thue,he was Norwegian, and he died in 1922. But I didn't know that at the time. http://www.dougshaw.com/essays/thu.html

In times of severe stress, we often find out who our real friends are. The Winter of 1991 was the most stressful time in my life so far. I would not have gotten through it without my friends Basho and Axel Thu. Axel Thu was a Chinese mathematician who died in the year 400 AD . Basho was my cat. January 14 was my preliminary examination in Mathematics. The preliminary examination is the biggest hurdle that lies between the hopeful graduate student, and the Ph. D. that is his or her heart's desire. Picture a two hour oral exam consisting of two professors hurling questions at you, where you only have a piece of chalk for self defense. At the University of Michigan, the rule was that if you failed, you got one more chance. If you failed a second time, that's it. You're out. Thanks for playing. You leave college with nothing but debt, misery, and a Master's degree. I started studying six weeks before the date, telling my family that Christmas was canceled for me that year. This is the story of a friendship. Basho the Cat, Axel Thu and I became very close during those six weeks. Unfortunately, this is also the story of a betrayal, and the end of a friendship.

22. Auteur Translate this page AD Thomas, Joy A. Thomasian, AJ Thompson, AC Thompson, D'Arcy Wentworth Thompson,Anthony C. Thorpe, John A. Thorup, Anders thue, A. thue, axel Thuizat, A. et http://bibmath.u-strasbg.fr/SearchAut.htm&Value=A&Range=0210

Accueil Index Auteur Recherche Simple Expert Revues Ouvrages IRMA Index : Auteur - Chercher ? Thalmaier, A. Thalmaier, Anton Thas, J. A. Thayse, A. et al. Thayse, A. Thayse, Andre the Euler International Mathematical Institute of Russian Academy of Sciences Thedy, Armin Thiebault, A. Thimm, Anton Thirioux, A. et al. Thisted, Ronald A. Thoma, A. Thomas, A. D. Thomas, Joy A. Thomasian, A.J. Thompson, A.C. Horaires Mon espace Recherche rapide Identification Identifiant : Mot de passe : A propos de ... Votre identifiant L'interrogation Ce site Alexandrie

23. ResAnet Browse Results doc); Thucydides (17 docs); Thuderoz, Christian (1 doc); thue, axel,18631922 (1 doc); thue, Laura A., 1970- (1 doc); thuen, Trond, 1943 http://www.amicus.nlc-bnc.ca/wbin/resanet/resultsm/s=b/n=NA/l=0/d=1/r=1/e=0/h=10

Thucydide (2 docs) Thucydides (1 doc) Thucydides (17 docs) Thuderoz, Christian (1 doc) Thue, Axel, 1863-1922 (1 doc) Thue, Laura A., 1970- (1 doc) Thuen, Trond, 1943- (1 doc) Thuente, Mary Helen (1 doc) Thuesen, G. J., 1938- (2 docs) Thuesen, G.J., 1938- (1 doc)

24. The Mathematics Genealogy Project - Index Of THU Thu, PhamGia, University of Toronto, 1972. thue, axel, University of Christiania,1889. thuener, Eleanor, The Catholic University of America, 1932. http://genealogy.math.ndsu.nodak.edu/html/letter.phtml?letter=THU

25. Felix.unife.it/Root/d-Mathematics/d-Dynamical-systems/d-Topological-dynamics/d-S (Russian) 7872 Jean Berstel axel thue's work on repetitions in words. Internet1995, 16p. axel thue Ueber unendliche Zeichenreihen. Norske Vid. Selsk. Skr. http://felix.unife.it/Root/d-Mathematics/d-Dynamical-systems/d-Topological-dynam

S. Arshon: Demonstration de l'existence des suites asymetriques infinies. Mat. Sb. 44 (1937), 769-777. (Russian) 7872 Jean Berstel: Axel Thue's work on repetitions in words. Internet 1995, 16p. Ethan Coven: Endomorphisms of substitution minimal sets. Zeitschr. Wtheorie 20 (1971), 129-133. 10478 Ethan Coven/Michael Keane: The structure of substition minimal sets. Trans. AMS 162 (1971), 89-102. 10518 F. Dekking: The spectrum of dynamical systems arising from substitutions of constant length. Z. Wtheorie 41 (1978), 221-239. 10573 F. Dekking/Michael Keane: Mixing properties of substitutions. Z. Wtheorie 42 (1978), 23-33. Aldo De Luca/Stefano Varricchio: Some combinatorial properties of the Thue-Morse sequence and a problem in semigroups. Theor. Comp. Sci. 63 (1989), 333-348. 15419 Andrew Gleason: Semigroups of shift register matrices. Math. Systems Theory 25 (1992), 253-267. From a course given at Princeton in 1960. 7902 Walter Helbig Gottschalk: Substitution minimal sets. Trans. AMS 109 (1963), 467-491. 1031 Walter Helbig Gottschalk/Gustav Arnold Hedlund: Topological dynamics. AMS 1955. 2554 Walter Helbig Gottschalk/Gustav Arnold Hedlund: A characterization of the Morse minimal set. Proc. AMS 15 (1964), 70-74. 10516 Gustav Arnold Hedlund: Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3 (1969), 320-375. 14037 Charles Holton/Luca Zamboni: Geometric realizations of substitutions. Bull. Soc. Math. France 126 (1998), 149-179. 12126 Teturo Kamae: Spectrum of a substitution minimal set. J. Math. Soc. Japan 22 (1970), 567-578. 2555 Michael Keane: Generalized Morse sequences. Zeit. Wtheorie 10 (1968), 335-353. 5716 Harvey Keynes: The proximal relation in a class of substitution minimal sets. Math. Systems Theory 1 (1967), 165-174. 10479 Nelson Markley: Substition-like minimal sets. Israel J. Math. 22 (1975), 332-353. 830 John Martin: Substitution minimal flows. Am. J. Math. 93 (1971), 503-526. 10512 John Martin: Minimal flows arising from substitutions on non-constant length. Math. Systems Th. 7 (1973), 73-82. 10517 John Martin: Generalized Morse sequences in n symbols. Proc. AMS 54 (1976), 379-383. 10519 John Martin: The structure of generalized Morse minimal sets on n symbols. Trans. AMS 232 (1977), 343-356. P. Michel: Stricte ergodicite' d'ensembles minimaux de substitutions. C.R. Ac. Sci. Paris 278 (1974), 811-813. Marston Morse: Recurrent geodesics on a surface of negative curvature. Trans. AMS 22 (1921), 84-100. Marston Morse: A solution of the problem of infinite play in chess. Abstract 360, Bull. AMS 44 (1938), 632. Marston Morse/Gustav Arnold Hedlund: Unending chess, symbolic dynamics and a problem in semigroups. Duke Math. J. 11 (1944), 1-7. G. Rauzy: Nombres algebriques et substitutions. Bull. Soc. Math. 110 (1982), 147-178. Axel Thue: Ueber unendliche Zeichenreihen. Norske Vid. Selsk. Skr. (?) 7 (1906), 1-22. Axel Thue: Ueber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Videnskabsselskabets Skrifter (?) (1906), ... Axel Thue: Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. (?) 1 (1912), 1-67.

26. Felix.unife.it/Root/d-Mathematics/d-Dynamical-systems/d-Topological-dynamics/d-S 7872 Jean Berstel axel thue's work on repetitions in words. Internet 1995, 16p. GustavArnold Hedlund Remarks on the work of axel thue. Nordisk Mat. Tidskr. http://felix.unife.it/Root/d-Mathematics/d-Dynamical-systems/d-Topological-dynam

7871 Jean Berstel: Recent results in Sturmian words. 7872 Jean Berstel: Axel Thue's work on repetitions in words. Internet 1995, 16p. Gustav Arnold Hedlund: Sturmian minimal sets. Am. J. Math. 66 (1944), 605-620. Gustav Arnold Hedlund: Remarks on the work of Axel Thue. Nordisk Mat. Tidskr. 15 (1967), 148-150. Marston Morse/Gustav Arnold Hedlund: Symbolic dynamics II. Sturmian sequences. Am. J. Math. 62 (1940), 1-42.

27. Thue-Morse L-system Morse sequence. axel thue came up with it in 1912 Thu12 in hisstudy of formal languages (not the term he used then). Marston http://www.math.okstate.edu/mathdept/dynamics/lecnotes/node16.html

Next: Paperfolding and the Dragon Up: L-systems Previous: Types of L-systems Thue-Morse L -system As our next example of an L -system, we introduce the Thue-Morse system: The sequence of generations starts as follows: a ab abba abbabaab abbabaabbaababba Table 4: Thue-Morse Generations The number of organisms clearly doubles each generation, so that . The first question we might ask is what is the global process that emerges, if any. To answer this question, we first observe that generation does appear at the beginning of . If we look at what follows, we eventually conclude that it resembles , except that each a has been replaced by a b and each b has been replaced by an a . For any word w in , we define R w ) to be the mirror image word where we replace a by b and b by a . Then the global process is As a challenge, we ask the reader to supply a proof. (It's very similar to what one writes down for the Fibonacci process; there is one extra key point.) Since is the beginning of , this L -system produces in the limit an infinite sequence of a 's and b 's that begins: This sequence has many interesting properties and is known as the Thue-Morse sequence . Axel Thue came up with it in 1912 [ ] in his study of formal languages (not the term he used then). Marston Morse rediscovered the same sequence in 1917 [

28. People At Leogriff Haakon thue Lie. axel Moulin. Tom Ekeberg. Our dedicated IP managerseach have more than 15 years of experience from technologybased http://www.leogriff.no/people.htm

norsk français Haakon Thue Lie Axel Moulin Tom Ekeberg Campus Kjeller, the Science Park and Innovation Centre outside Oslo, is one of the owners of Leogriff. Their competence and network adds to Leogriff’s expertise. Leogriff has as board members and associates leading professionals with background from the Norwegian Patent Office, patent bureaus and business development. Haakon Thue Lie - partner and CEO 38 years, M. Sc. (NTNU - Telematics - Trondheim, Norway), Master of Management (N.S.M.-BI, Oslo, Norway) Haakon is married and the father of three. Axel Moulin partner 39 years, M. Sc. (E.N.S.G.-Geophysics, Nancy, France), M.B.A. (N.S.M.-BI, Oslo, Norway). Axel has started his career in the Oil Industry, and has later worked for the world leader for water treatment engineering as an Export Commissioning Supervisor, a Process Engineer and a Bid Manager for major export projects. Axel has also been the Project Manager for the selection and implementation of a Customer Management and Billing system for a European actor in the Internet Unified Messaging business, and was involved in a strategy consulting project for an e-learning actor.

29. Seminars axel Moulin. Haakon thue Lie. IP management, a business opportunity today, arequirement tomorrow Haakon thue Lie and axel Moulin Partners in Leogriff. http://www.leogriff.no/seminar/

norsk français Past seminars Read about past seminars here Next open seminar will be in March/April 2003. We also arrange internal seminars - please contact us at seminar@leogriff.no Two days course in valuation of IPR in Oslo with Patentstyret. (Course in Norwegian only.) In this two days course (24-25 March and 13-14 November 2003) arranged by Patentstyret, there will be lectures and workshops on valuation with focus on both patents and trademarks. Leogriff and MarkUp will focus on the new model from Norske Finansanalytikeres Forbund and the use of IPscore The first morning includes an optional introduction to IPR. Read more at Patentstyrets website here (opens in a separate window) You may also call +47 22 38 73 33 or e-mail kurs@patentstyret.no Contact : seminar@leogriff.no Questions Haakon Thue Lie, +47 913 500 43, Axel Moulin, +47 41 47 41 10 In-house seminars Contact us to have an in-house seminar or workshop on Intellectual Property!

30. Books Written In Norwegian QA3.T47 1977 Selected mathematical papers of axel thue / with an introduction byCarl Ludwig Siegel ; ed. by Trygve Nagell et al.. thue, axel, 18631922. http://www.waikato.ac.nz/library/resources/lang/lang_nor.html

Books written in Norwegian F PRO Mrs Pepperpot's outing and other stories. Translated by Marianne Helweg. Illus. by Björn Berg. Prøysen, Alf, 1914-1970. [London] Hutchinson Junior Books [1971] 123 p. illus. 23 cm. F PRO Mrs Pepperpot's year; stories by Alf Prøysen. Translated by Marianne Helweg. Illustrations by Björn Berg. Prøysen, Alf, 1914-1970. London, Hutchinson Junior Books [1973] 92 p. illus. 22 cm. F VES Aurora and the little blue car / by Anne-Cath. Vestly ; translated from the Norwegian by Eileen Amos ; illustrated by Gunvor Edwards. Vestly, Anne Catharina. Harmondsworth : Longman Young Books, 1974. 190p : ill ; 21cm. PB PRO Mrs Pepperpot's Christmas / story by Alf Prøysen ; illustrated by Björn Berg ; translated by Marianne Helweg. Prøysen, Alf. London : Hutchinson Junior, 1972. [32] p. : col. ill. ; 22 x 27 cm. 362.3 TAN Michael / Ragnhild Tangen ; photographs by Odd Furenes. Tangen, Ragnhild. Tring, Hertforshire : Lion Publishing ; Sutherland, N.S.W. : Albatross Books, c1983. [30] p. : col. ill. ; 25 cm.

31. Thue Sequences - Lambda Calculus Circular Programs thue sequences, after axel thue, are sequences over an alphabet ofthree {1,2,3} such that no subsequence is immediately repeated. http://www.csse.monash.edu.au/~lloyd/tildeFP/Lambda/Examples/Thue.shtml

[Lambda] Thue Sequences, Lambda Calculus Circular Program. immediately repeated. e.g. 1213121 is a Thue sequence that cannot be extended 121312 any length. (There are no solutions longer than 3 for binary sequences, and finding solutions is easier for alphabets larger than three.) The ``circular program'' below builds a tree of Thue sequences. Note that within function `build', the tree data-structure `T' and functions `toplevel' and `f' are mutually recursive. The program relies on the fact that a partial soln `abcde' can be extended with `f' iff its shadow, `bcdef', is already in tree at the previous level; this avoids repeating many tests on constraints that have already been treated at a higher level. The subtree for `abcde' is the subtree of `bcde' less any `a' nodes. The tree of all solutions is notionally infinite, but the program prints one branch to a finite depth so only a finite part of the tree is evaluated thanks to lazy evaluation. See L. Allison. Applications of Recursively Defined Data Structures . Australian Computer Journal (1) pp14-20, Feb 1993.

32. Thue-Morse L-system Translate this page seqüência de thue-Morse. axel thue apareceu com ela em 1912 no seuestudo de linguagens formais. Marston Morse redescubriu a mesma http://www.di.ufpe.br/~if114/Monografias/L-systems/Gramaticas/node16.html

L-system de Thue-Morse L Thue-Morse a ab abba abbabaab abbabaabbaababba Table 4: , exceto que cada a b e cada b a . Para cada palavra w em R w a por b e b por a , este L de a 's e b Thue-Morse a por 1 e b R onde cada n n +1 consiste deste string seguido por para todo . Parar isto funcionar para qualquer m m com . Por exemplo, . Em geral, onde t m . Um outro exemplo: www em nenhum string finito w de a 's e b 's. aaa ou ababab ou babbabbab EEe onde E e E Seja n um inteiro positivo. Existe um outro inteiro positivo N (maior que n normalmente) tal que, dado algum substring w de comprimento n W de comprimento N w N pode der provado: de k w de comprimento n

33. Squarefree Word -- From MathWorld Selected Mathematical Papers of axel thue. Oslo, Norway Universitetsforlaget,pp. 139158, 1977. Selected Mathematical Papers of axel thue. http://mathworld.wolfram.com/SquarefreeWord.html

Discrete Mathematics Combinatorics Enumeration Math Contributors ... Finch Squarefree Word N.B. A detailed online essay by S. Finch was the starting point for this entry. A "square" word consists of two identical adjacent subwords (for example, acbacb ). A squarefree word contains no square words as subwords (for example, abcacbabcb ). The only squarefree binary words are a b ab ba aba , and bab (since aa bb aaa aab abb baa bba , and bbb contain square identical adjacent subwords a b a a b a b , and b , respectively). However, there are arbitrarily long ternary squarefree words. The number s n ) of ternary squarefree words of length n = 1, 2, ... are 1, 3, 6, 12, 18, 30, 42, 60, ... (Sloane's ), and s n ) is bounded by (Brandenburg 1983). In addition, (Brinkhuis 1983, Noonan and Zeilberger 1997). The number of squarefree quaternary words of length n = 1, 2, ... are 4, 12, 36, 96, 264, 696, ... (Sloane's Alphabet Cubefree Word Overlapfree Word Word References Baake, M.; Elser, V.; and Grimm, U. "The Entropy of Square-Free Words." 8 Sep 1998. http://xxx.lanl.gov/abs/math-ph/9809010/

34. Robert McNaughton For example, they have looked at ways to improve the efficiency of thue systems,a linguistic method developed by Norwegian logician axel thue in 1914. http://www.cs.rpi.edu/people/mcnaughton.html

Robert McNaughton mcnaught@cs.rpi.edu Emeritus Professor Ph.D., Harvard University Automata theory, formal languages, combinatorics of words McNaughton entered computer science in the 1950s after teaching philosophy for six years. His career switch was due to the lean job market more than anything else. Today, however, his training in philosophy holds him in good stead. McNaughton, who is author of the textbook Elementary Computability, Formal Languages and Automata published by Prentice-Hall, is now looking at problems in the combinatorics of words, a branch of formal languages. Formal languages deal with symbolic logic and computer languages as opposed to the natural languages used in human speech and general-purpose writing. His research is being coordinated with computer scientists formerly at the GE Research and Development Center in nearby Niskayuna, New York. This group at GE was called the Theorem Proving Group. Members of this group are now in the Computer Science Department at the University at Albany and in the Computer Science Department at RPI. Their research was concerned with looking at formal linguistic systems for the sake of carrying through proofs on the machine. For example, they have looked at ways to improve the efficiency of Thue systems, a linguistic method developed by Norwegian logician Axel Thue in 1914. Thue systems are useful for computation because they replace strings (connected characters) with other strings, carrying through a rather basic kind of computer operation.

35. Fun With Base 3 The Norwegian mathematican axel thue proved nearly 100 years ago that there doexist arbitrarily long ternary squarefree sequences, and gave a method for http://www.cs.caltech.edu/cgi-bin/cs1/HyperNews/get/labs/17.html?nogifs

36. Standardseite Translate this page Hilbert, David. Kleene, Stephen C. Leibniz, Gottfried W. thue, axel. Turing,Alan. Info3. URL http//10www.ira.uka.de/info3/unterhaltsames.html. 12.01.2001. http://i10www.ira.uka.de/info3/unterhaltsames.html

Lehrstuhl Prof. Dr. P. Deussen ILKD (Lehrstuhl Deussen) Informatik III Dieses und jenes Alan Turing und seine erste richtige Rechenmaschine Theseus im Alter Interaktives (Java, Javascript, CGIs, ...) Collatz-Funktion (Das 3n+1 Problem) Simulation von Turing-Maschinen Personen (aus " The MacTutor History of Mathematics archive ") Church, Alonzo Herbrand, Jacques Hilbert, David Kleene, Stephen C. ... Turing, Alan URL: http:// 10www.ira.uka.de/info3/unterhaltsames.html

37. Citation 2 J. Berstel, Mots sans carte et morphismes itrs, Discrete Math. 29 (1980) 235244.3 J. Berstel, axel thue's Work on repetitions in words (Fourth Conf. http://portal.acm.org/citation.cfm?id=566545&coll=Portal&dl=GUIDE&CFID=11111111&

38. Citation 3 J. Berstel, axel thue's Papers on Repetitions in Words a Translation, Publicationsdu Laboratoire de Combinatoire et d'lnformatique Matbmatique, Number 20 http://portal.acm.org/citation.cfm?id=566251&coll=Portal&dl=GUIDE&CFID=11111111&

39. Norske Matematikere axel thue (1863 1922). Selected Mathematical Papers of axel thue. Universitetsforlaget,Oslo 1977. Sigmund Selberg axel thue. Normat 28 (4) (1980) pp. http://home.hia.no/~aasvaldl/norskem.html

Norske matematikere Navigasjonslinker: Min hjemmeside. Norske matematikere Adresser til norske matematikere Bent Birkeland har samlet en del materiale i "Norske matematikere". Gyldendal Norsk Forlag Professorer i matematikk ved Universitetet i Oslo. Tapir Avdeling for realfag, hjemmeside. Niels Henrik Abel (1802 - 1829) ABELSENTERET i Gjerstad "Niels Henrik Abel - et geni og hans samtid" . Gyldendal Norsk Forlag, Oslo (1954). "Niels Henrik Abel: mathematician extraordinary." University of Minnisota Press (1957). de Niels Henrik Abel. - Nouv. ed. Johnson Reprint Corporation (1965). Bilder og kart (kommer ...). Fr. Lange-Nielsen: "Niels Henrik Abel." Nordisk Mat. Tidsskrift. Bd. 1 (1953) pp. 65 - 90. Karl Egil Aubert: "Niels Henrik Abel." Normat nr. 27 (4) (1979) pp. 129 - 140. Aschehoug forlag Et foranskutt lyn: Niels Henrik Abel og hans samtid Oversettelse til engelsk Springer-Verlag , se anmeldelse av Jeremy J. Gray i MAA Online Arild Stubhaug: " Abelhaugen i Slottsparken." Kronikk i Aftenposten, 25.09.96. Niels Henrik Abel side ved MacTutor History of Mathematicians archive.

40. FILE NOT FOUND! Peter Hjorth _Marie Pedersdatter _ axel Hjorth _Lars sg., Vestermarie. Father thue Larsen Lyster Mother Kirsten Jensdatter http://kofoed.owns1.com/stamtavle/file/jochum/D0012/G0000002.html

LOGIN CONTACT US TECH SUPPORT UPGRADE YOUR SITE File Not Found! The file you are searching for is not located on our servers. Please check your spelling. File names may be case sensitive. Please verify the URL as this is the most common reason for this error. If you would like your own 100 MB FREE WEB SITE with a long list of features simply sign up at owns1.com. LOGIN CONTACT US TECH SUPPORT ... UPGRADE YOUR SITE

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