21. The Mathematics Genealogy Project - Axel Thue According to our current online database, axel thue has 1 students and 229descendants. We welcome any additional information. http://www.genealogy.math.ndsu.nodak.edu/html/id.phtml?id=18236

22. The Mathematics Genealogy Project - Update Data For Axel Thue If you have Mathematics Subject Classifications to submit for an entire group ofindividuals (for instance all those that worked under a particular advisor) http://www.genealogy.math.ndsu.nodak.edu/html/php/submit-update.php?id=18236

23. Publications Du LaCIM axel thue s papers on repetitions in words a translation axel thue s paperson sequences are now more easily accessible since they are included in the 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

24. Transcendental.html axel thue (18631922). thue s Theorem (1909) . Let alpha be any real algebraicnumber of degree n ( at least 3), and let epsilon be any positive constant http://www.spd.dcu.ie/johnbcos/download/Public and other lectures/transcendental

Axel Thue (1863-1922) Thue's Theorem (1909) . Let be any real algebraic number of degree n at least 3), and let be any positive constant ( however small ), then there is a positive constant (i.e. the value of c depends only on and ) such that ... (i') One can only write of the fundamental importance of that result by resorting to hyperbole. I highly recommend Wolfgang Schmidt's AMS Bulletin (1978) review of Selected mathematical papers of Axel Thue , from which I quote: " His greatest work, on approximation to algebraic numbers, appeared in 1908/1909, when he was well in his forties, and when he had been away from the centres of mathematics for over a decade. ... Landau called it [the above theorem, and its application to the related Thue equation the most important discovery in elementary number theory which he had witnessed in his lifetime. He also said ten years after its publication that already ten competent mathematicians had read Thue's paper. [JC comment. Far be it from me to correct Schmidt's English (his first language is German) but I think any reader will believe that what Landau said, ten years after the publication of Thue's paper, was that only ten competent... ]

25. Fermat-Thue 2-squares.html Title. Fermat s 2squares theorem (proved here using the Dirichlet box principle ,together with an idea of axel thue ) http://www.spd.dcu.ie/johnbcos/download/3rd_year/Challenging/Fermat-Thue 2 squar

# Fermat's 2-squares.mws Note to anyone reading this on the web in html format. Not having Maple you may not be familiar with its use of Sections (where you see those [+]s). To read the contents of a section, simply click on the [+] and it will expand into a [-], and you will see text, plus calculations (of course you cannot change the calculations, as you could with the active Maple worksheet). Later, by scrolling back up, you may close the box by clicking on the [-]. Title Fermat's 2-squares theorem (proved here using the Dirichlet box principle together with an idea of Axel Thue Remark (aimed at number theorists who might read this) on the approach I've adopted here . The proof that I give here of Fermat's 2-squares theorem (as part of my 3rd year (BA only) course Challenging Mathematical Puzzles and Problems ) is most certainly not the best, nor the most insightful proof there is of Fermat's theorem; and the proof given here serves mainly to demonstrate a particular use (the Thue result) of the entirely elementary, but quite profound, Dirichlet box principle (where would the work of - say - Roth or Schmidt be without it?!) which my students will encounter is less demanding areas (e.g. Choose any ( ) integers from [1, 2, 3, 4, ... , (

26. Thue-Morse Sequence: Information From Answers.com thueMorse sequence In mathematics and its applications, the thue-Morse did not mention the sequence explicitly; this was left to axel thue in 1906, http://www.answers.com/topic/thue-morse-sequence

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping Thue-Morse sequence Wikipedia Thue-Morse sequence In mathematics and its applications, the Thue-Morse sequence , or Prouhet-Thue-Morse sequence , is a certain binary sequence whose initial segments alternate (in a certain sense). The Thue-Morse sequence begins: on a given ordered pair Definition There are several equivalent ways of defining the Thue-Morse sequence. Recurrence relation The Thue-Morse sequence is the sequence t n satisfying t and t n t n t n t n for all positive integers n Characterization using bitwise negation The Thue-Morse sequence in the form given above, as a sequence of bits , can be defined recursively using the operation of bitwise negation . So, the first element is 0. Then once the first 2 n elements have been specified, forming a string s , then the next 2 n elements must form the bitwise negation of s . Now we have defined the first 2 n elements, and we recurse.

27. Axel Thue S Work On Repetitions In Words - Berstel (ResearchIndex) The purpose of this survey is to present, in contemporary terminology, thefundamental contributions of axel thue to the study of combinatorial properties http://citeseer.ist.psu.edu/berstel92axel.html

28. Citations Probleme Uber Veranderungen Von Zeichenreihen Nach axel thue, Probleme uber Veranderungen von Zeichenreihen nach gegeben Regeln, Skr.Vid. Kritianaia, I. Mat. Naturv. Klasse, No. 10(1914). http://citeseer.ist.psu.edu/context/349552/0

29. Introduction To Thue Apologies to axel thue for mangling the pronunciation of his name for a cheap joke.Apologies to whatever is left of Infocom for (unknowingly) supplying the http://www.safalra.com/programming/thue/introduction/

Safalra's Website Programming Thue Introduction To Thue Introduction To Thue The following is the introduction to Thue from the original distribution, used under the Thue Licence Thue Licence Note that the e-mail address in the licence now longer works. Safalra (Stephen Morley) last changed this page ( http://www.safalra.com/programming/thue/introduction/ ) on 3rd August 2005.

30. Math Lessons - Axel Thue Math Lessons axel thue. He stated in 1914 the so-called Word problem orthue problem, closely related to the halting problem. See also http://www.mathdaily.com/lessons/Axel_Thue

Search Mathematics Encyclopedia and Lessons Lessons Popular Subjects algebra arithmetic calculus equations ... more References applied mathematics mathematical games mathematicians more ... Number theorists Axel Thue Axel Thue 19 February 7 March ) was a Norwegian mathematician , known for highly original work in diophantine approximation , and combinatorics He stated in 1914 the so-called Word problem or Thue problem , closely related to the halting problem See also: Thue-Morse sequence Thue-Siegel-Roth theorem Thue equation External link Categories 1863 births 1922 deaths Mathematicians ... Number theorists Last updated: 09-02-2005 16:50:46 algebra arithmetic calculus equations ... mathematicians

31. Thue-Morse L-system axel thue came up with it in 1912 Thu12 in his study of formal languages (notthe term he used then). Marston Morse rediscovered the same sequence in 1917 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 [

32. Squarefree Word -- From MathWorld Selected Mathematical Papers of axel thue. Oslo, Norway Universitetsforlaget, pp.139158, 1977. thue, A. Über die gegenseitige Lage gleicher Teile http://mathworld.wolfram.com/SquarefreeWord.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Discrete Mathematics Combinatorics Enumeration ... Finch Squarefree Word 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 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 of ternary squarefree words of length , 2, ... are 1, 3, 6, 12, 18, 30, 42, 60, ... (Sloane's ), and is bounded by (Brandenburg 1983). In addition, (Brinkhuis 1983, Noonan and Zeilberger 1997). The number of squarefree quaternary words of length , 2, ... are 4, 12, 36, 96, 264, 696, ... (Sloane's SEE ALSO: Alphabet Cubefree Word Overlapfree Word Word ... [Pages Linking Here] REFERENCES: Baake, M.; Elser, V.; and Grimm, U. "The Entropy of Square-Free Words." 8 Sep 1998.

33. American Scientist Online - Third Base Nevertheless, the Norwegian mathematician axel thue proved almost a thue, axel.1912. ber die gegenseitige lage gleicher teile gewisser Zeichenreihen. http://www.americanscientist.org/template/AssetDetail/assetid/14405/page/6

Home Current Issue Archives Bookshelf ... Subscribe In This Section Search Book Reviews by Issue Issue Index Topical Index ... Classics Site Search Advanced Search Visitor Login Username Password Help with login Forgot your password? Change your username see full issue: November-December 2001 Volume: Number: Page: DOI: COMPUTING SCIENCE Other Formats: Postscript (gzipped) Postscript PDF Third Base Martha Stewart's File Cabinet Some weeks ago, rooting around in files of old clippings and correspondence, I made a discovery of astonishing obviousness and triviality. What I found had nothing to do with the content of the files; it was about their arrangement in the drawer. click for full image and caption A drawer filled with "half-cut" folders, which have just two tab positions, might initially alternate left-right-left-right . The pattern is spoiled, however, as soon as you insert a folder in the middle of the drawer. No matter which type of folder you choose and no matter where you put it (except at the very ends of the sequence), every such insertion generates a conflict. Removing a folder has the same effect. Translated into a binary numeral with

34. Nuwen.net - Random Work The Norwegian mathematican axel thue proved nearly 100 years ago that there doexist arbitrarily long ternary squarefree sequences, and gave a method for http://nuwen.net/work.html

nuwen.net - Random Work You load sixteen tons, what do you get? New And Good: Good: Old But Okay: Old Fun: Old Info: Old Tech: Old Other: Crummy: Mlar: libnuwen Index bwtzip Coloring ... Origami Polyhedra Random Work Space C Intro Science Fiction News: 2003 ... GeForce FX Jump To: Contents Ternary Sequences Hamiltonian Cycles ElGamel ... Lambda Functions Contents All code appearing on this page is mine, and distributed under the GNU GPL. Ternary sequences (This was initially written directed at CS1 students, unrelated to any course work.) I recently read an issue of American Scientist with a cool article about ternary numbers. Here's a description of something in it that interested me, along with some Scheme code for playing around with sequences of ternary numbers. Let's look at sequences of digits in certain bases. For example, in base 10 one could be 1 2 3 4 5 6 7 8 9. (Which I will write as 0123456789 to save space - we won't work with base 11 or higher. :->) We'll call a "square" sequence one that has at least one pair of adjacent identical subsequences. Meaning that it looks like: [beginning of sequence].....[some subsequence of any length][same subsequence]....[end of sequence].

35. Doug, Basho, And Axel Thu Historical Note His name was actually axel thue, he was Norwegian, and he died in The real axel thue said some wonderful things about mathematics The 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.

36. Biografia De Thue, Axel Translate this page thue, axel. (Tonsberg, 1863-Oslo, 1922) Matemático noruego. Se dedicó preferentementea la investigación abstracta y especulativa y, en particular, http://www.biografiasyvidas.com/biografia/t/thue.htm

Inicio Buscador Las figuras clave de la historia Reportajes Los protagonistas de la actualidad Thue, Axel (Tonsberg, 1863-Oslo, 1922) Matemático noruego. Se dedicó preferentemente a la investigación abstracta y especulativa y, en particular, a la teoría de números. Demostró que la ecuación P( x y m , donde P es un polinomio homogéneo de dos variables y de grado mayor que 2, únicamente tiene un número finito de soluciones enteras. Inicio Buscador Recomendar sitio

37. Thue - Esolang The language is presumably named after either Norwegian mathematician axel thue,or the grammar which bears his name (which the language employs.) http://esoteric.voxelperfect.net/wiki/Thue

Thue From Esolang Thue is an esoteric programming language invented by John Colagioia in 2000. It is a matrioshka language based on nondeterministic string rewriting which the author describes as a constraint-programming version of a Turing tarpit Table of contents showTocToggle("show","hide") 1 Computational model 2 Etymology 3 Language description 4 Input and output ... edit Computational model Thue is based on a semi-Thue grammar , which is a restricted form of Thue system in which each association between a pair of strings is only one-way. edit Etymology The language is presumably named after either Norwegian mathematician Axel Thue, or the grammar which bears his name (which the language employs.) It is worth noting that while the mathematician's name is likely pronounced "TOO-eh" or "TOO", the associated documentation explicitly gives the pronounciation of the name of the language as "TOO-ay". edit Language description Each Thue program consists of two parts: a list of substitution rules of the form which is terminated with a line having both and empty: followed by a string representing the initial program state, which may extend over multiple lines.

38. A Symmetry Group Of A Thue - Morse Quasicrystal in Selected Mathematical Papers of axel thue (Oslo, Universitetsforlaget) pp139, 413) 14 axel F and Terauchi H 1991 Phys. Rev. Lett. 66 2223 http://www.iop.org/EJ/article/0305-4470/31/23/001/ja31023l1.html

@import url(http://ej.iop.org/style/nu/EJ.css); Quick guide Site map Athens login IOP login: Password: Create account Alerts Contact us Journals Home ... Content finder J. Phys. A: Math. Gen No 23 (12 June 1998) L435-L440 PII: S0305-4470(98)91800-5 LETTER TO THE EDITOR A symmetry group of a Thue - Morse quasicrystal Jean-Pierre Gazeau and Jacek Miekisz Institute of Applied Mathematics and Mechanics, University of Warsaw, ul Banacha 2, 02-097 Warsaw, Poland Received 17 February 1998, in final form 17 April 1998 Abstract: We present a method of coding general self-similar structures. In particular, we construct a symmetry group of a one-dimensional Thue - Morse quasicrystal, i.e. of a nonperiodic ground state of a certain translation-invariant, exponentially decaying interaction. A symmetry group of a three-dimensional crystal consists of lattice translations, rotations, and reflections. Starting from any point of a crystal, we can reach any other point, successively applying different elements of the symmetry group of the crystal. It was recently shown [ ] that certain one-dimensional quasicrystals can be built by successive applications, on one of its points, of elements of certain discrete affine semigroups. Here we describe a general method, based on ideas contained in [

39. String Rewriting And The Fibonacci Word since the Norwegian logician and mathematician axel thue devised and of this kind provide the reason why people like axel thue invent systems like 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.

40. Thue Sequences, Lambda Calculus Circular Program thue sequences, after axel thue, are sequences over an alphabet of three {1,2,3}such that no subsequence is immediately repeated. eg 1213121 is a thue http://www.csse.monash.edu.au/~lloyd/tildeFP/Lambda/Examples/Thue/

Thue Sequences, Lambda Calculus Circular Program LA home FP Lambda Introduction ... Examples also see: n-queens 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 Feb 1993.

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