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| {{See also|Prouhet–Thue–Morse constant}}
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| [[Image:Morse-Thue sequence.gif|frame|right|This graphic demonstrates the repeating and complementary makeup of the Thue–Morse sequence.]]
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| [[File:Parity relation 0011 1100 0011 1100.svg|thumb|400px|5 [[Logical matrix|logical matrices]] that give the beginning of the T.-M. sequence, when read line by line<br><br>[[Exclusive or|Either]] in set A (vertical index) [[Exclusive or|or]] in set B (horizontal index)<br> is an odd number of elements.]]
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| In [[mathematics]], the '''Thue–Morse sequence''', or '''Prouhet–Thue–Morse sequence''', is the [[binary sequence]] (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the [[Boolean algebra|Boolean complement]] of the sequence obtained thus far. This procedure yields 0 then 01, 0110, 01101001, 0110100110010110, and so on. The infinite sequence begins:
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| :01101001100101101001011001101001.... {{OEIS|id=A010060}}
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| Any other [[ordered pair]] of symbols may be used instead of 0 and 1; the logical structure of the Thue–Morse sequence does not depend on the symbols that are used to represent it.
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| == Definition ==
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| There are several equivalent ways of defining the Thue–Morse sequence.
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| === Direct definition ===
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| To compute the ''n''<sup>th</sup> element ''t<sub>n</sub>'', write the number ''n'' in binary. If the number of ones in this binary expansion is odd then ''t<sub>n</sub>'' = 1, if even then ''t<sub>n</sub>'' = 0.<ref name=AS15>Allouche & Shallit (2003) p.15</ref> For this reason [[John H. Conway]] ''et al''. call numbers ''n'' satisfying ''t<sub>n</sub>'' = 1 ''odious'' (for ''odd'') numbers and numbers for which ''t<sub>n</sub>'' = 0 ''evil'' (for ''even'') numbers.
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| === Recurrence relation ===
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| The Thue–Morse sequence is the sequence ''t<sub>n</sub>'' satisfying
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| :{|
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| | ''t''<sub>0</sub> || = 0,
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| |-
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| | ''t''<sub>2''n''</sub> || = ''t<sub>n</sub>'', and
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| |-
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| | ''t''<sub>2''n''+1</sub>|| = 1 − ''t<sub>n</sub>''.
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| |}
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| for all positive integers ''n''.<ref name=AS15/>
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| === L-system ===
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| The Thue–Morse sequence is a [[morphic word]]:<ref>Lothaire (2011) p.11</ref> it is the output of the following [[Lindenmayer system]]:
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| '''variables''' 0 1
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| '''constants''' none
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| '''start''' 0
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| '''rules''' (0 → 01), (1 → 10)
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| === Characterization using bitwise negation ===
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| The Thue–Morse sequence in the form given above, as a sequence of [[bit]]s, can be defined [[recursion|recursively]] using the operation of [[bitwise negation]].
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| So, the first element is 0.
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| Then once the first 2<sup>''n''</sup> elements have been specified, forming a string ''s'', then the next 2<sup>''n''</sup> elements must form the bitwise negation of ''s''.
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| Now we have defined the first 2<sup>''n''+1</sup> elements, and we recurse.
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| Spelling out the first few steps in detail:
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| * We start with 0.
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| * The bitwise negation of 0 is 1.
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| * Combining these, the first 2 elements are 01.
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| * The bitwise negation of 01 is 10.
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| * Combining these, the first 4 elements are 0110.
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| * The bitwise negation of 0110 is 1001.
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| * Combining these, the first 8 elements are 01101001.
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| * And so on.
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| So
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| * ''T''<sub>0</sub> = 0.
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| * ''T''<sub>1</sub> = 01.
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| * ''T''<sub>2</sub> = 0110.
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| * ''T''<sub>3</sub> = 01101001.
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| * ''T''<sub>4</sub> = 0110100110010110.
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| * ''T''<sub>5</sub> = 01101001100101101001011001101001.
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| * ''T''<sub>6</sub> = 0110100110010110100101100110100110010110011010010110100110010110.
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| * And so on.
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| === Infinite product ===
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| The sequence can also be defined by:
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| : <math> \prod_{i=0}^{\infty} (1 - x^{2^{i}}) = \sum_{j=0}^{\infty} (-1)^{t_j} x^{j} \mbox{,} \! </math>
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| where ''t<sub>j</sub>'' is the ''j''<sup>th</sup> element if we start at ''j'' = 0.
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| == Some properties ==
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| [[Image:Thue-Morse Squares.PNG|thumb|130px|right|Nested squares generated by successive iterations of Thue–Morse]]
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| Because each new block in the Thue–Morse sequence is defined by forming the bitwise negation of the beginning, and this is repeated at the beginning of the next block, the Thue–Morse sequence is filled with ''squares'': consecutive strings that are repeated.
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| That is, there are many instances of ''XX'', where ''X'' is some string.
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| However, there are no ''cubes'': instances of ''XXX''.
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| There are also no ''overlapping squares'': instances of 0''X''0''X''0 or 1''X''1''X''1.<ref name=ACOW113>Lothaire (2011) p.113</ref><ref name=PF103>Pytheas Fogg (2002) p.103</ref> The [[Critical exponent of a word|critical exponent]] is 2.<ref>{{cite book | title=Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, CA, USA, June 26-29, 2006 | volume=4036 | series=Lecture Notes in Computer Science | editor1-first=Oscar H. | editor1-last=Ibarra | editor2-first=Zhe | editor2-last=Dang | publisher=[[Springer-Verlag]] | year=2006 | isbn=3-540-35428-X | first=Dalia | last=Krieger | chapter=On critical exponents in fixed points of non-erasing morphisms | pages=280–291 | zbl=1227.68074 }}</ref>
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| Notice that ''T''<sub>2''n''</sub> is [[Palindromic number|palindrome]] for any ''n'' > 1. Further, let q<sub>''n''</sub> be a word obtain from ''T''<sub>2''n''</sub> by counting ones between consecutive zeros. For instance, ''q''<sub>1</sub> = 2 and ''q''<sub>2</sub> = 2102012 and so on. The words ''T<sub>n</sub>'' do not contain ''overlapping squares'' in consequence, the words ''q<sub>n</sub>'' are palindrome [[squarefree word]]s.
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| The statement above that the Thue–Morse sequence is "filled with squares" can be made precise:
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| It is a ''[[uniformly recurrent word]]'', meaning that given any finite string ''X'' in the sequence, there is some length ''n<sub>X</sub>'' (often much longer than the length of ''X'') such that ''X'' appears in ''every'' block of length ''n''.<ref name=ACOW30>Lothaire (2011) p.30</ref><ref>{{cite book | editor1-last=Berthé | editor1-first=Valérie | editor2-last=Rigo | editor2-first=Michel | title=Combinatorics, automata, and number theory | series=Encyclopedia of Mathematics and its Applications | volume=135 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2010 | isbn=978-0-521-51597-9 | zbl=1197.68006 | page=7 }}</ref>
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| The easiest way to make a recurrent sequence is to form a [[periodic sequence]], one where the sequence repeats entirely after a given number ''m'' of steps.
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| Then ''n<sub>X</sub>'' can be set to any multiple of ''m'' that is larger than twice the length of ''X''.
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| But the Morse sequence is uniformly recurrent ''without'' being periodic, not even eventually periodic (meaning periodic after some nonperiodic initial segment).<ref name=ACOW31>Lothaire (2011) p.31</ref>
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| We define the '''Thue–Morse morphism''' to be the [[function (mathematics)|function]] ''f'' from the [[Set (mathematics)|set]] of binary sequences to itself by replacing every 0 in a sequence with 01 and every 1 with 10.<ref name=BLRS70>Berstel (2009) p.70</ref> Then if ''T'' is the Thue–Morse sequence, then ''f''(''T'') is ''T'' again; that is, ''T'' is a [[fixed point (mathematics)|fixed point]] of ''f''. The function ''f'' is a [[prolongable morphism]] on the [[free monoid]] {0,1}<sup>∗</sup> with ''T'' as fixed point: indeed, ''T'' is essentially the ''only'' fixed point of ''f''; the only other fixed point is the bitwise negation of ''T'', which is simply the Thue–Morse sequence on (1,0) instead of on (0,1). This property may be generalized to the concept of an [[automatic sequence]].
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| The ''generating series'' of ''T'' over the [[binary field]] is the [[formal power series]]
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| :<math>t(Z) = \sum_{n=0}^\infty T(n) Z^n \ . </math>
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| This power series is algebraic over the field of formal power series, satisfying the equation<ref name=BLRS80>Berstel (2009) p.80</ref>
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| :<math>Z + (1+Z)^2 t + (1+Z)^3 t^2 \ . </math>
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| === In combinatorial game theory ===
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| The set of ''evil numbers'' (numbers <math>n</math> with <math>t_n=0</math>) forms a subspace of the nonnegative integers under [[nim-addition]] ([[bitwise operation|bitwise]] [[exclusive or]]). For the game of [[Kayles]], the evil numbers form the [[sparse space]]—the subspace of [[nim-value]]s which occur for few (finitely many) positions in the game—and the odious numbers are the [[common coset]].
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| === The Prouhet–Tarry–Escott problem === <!-- This section is linked to from [[Prouhet–Tarry–Escott problem]]. -->
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| The [[Prouhet–Tarry–Escott problem]] can be defined as: given a positive integer ''N'' and a non-negative integer ''k'', [[Partition of a set|partition]] the set ''S'' = { 0, 1, ..., ''N''-1 } into two [[Disjoint sets|disjoint]] subsets ''S''<sub>0</sub> and ''S''<sub>1</sub> that have equal sums of powers up to k, that is:
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| :<math> \sum_{x \in S_0} x^i = \sum_{x \in S_1} x^i</math> for all integers ''i'' from 1 to ''k''.
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| This has a solution if ''N'' is a multiple of 2<sup>''k''+1</sup>, given by:
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| * ''S''<sub>0</sub> consists of the integers ''n'' in ''S'' for which ''t<sub>n</sub>'' = 0,
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| * ''S''<sub>1</sub> consists of the integers ''n'' in ''S'' for which ''t<sub>n</sub>'' = 1.
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| For example, for ''N'' = 8 and ''k'' = 2,
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| :{{nowrap|1= 0 + 3 + 5 + 6 = 1 + 2 + 4 + 7,}}
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| :{{nowrap|1= 0<sup>2</sup> + 3<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup> = 1<sup>2</sup> + 2<sup>2</sup> + 4<sup>2</sup> + 7<sup>2</sup>.}}
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| The condition requiring that ''N'' be a multiple of 2<sup>''k''+1</sup> is not strictly necessary: there are some further cases for which a solution exists. However, it guarantees a stronger property: if the condition is satisfied, then the set of ''k''<sup>th</sup> powers of any set of ''N'' numbers in [[arithmetic progression]] can be partitioned into two sets with equal sums. This follows directly from the expansion given by the [[binomial theorem]] applied to the binomial representing the ''n''<sup>th</sup> element of an arithmetic progression.
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| === Fractals and Turtle graphics ===
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| A [[Turtle Graphics]] is the curve that is generated if an automaton is programmed with a sequence. | |
| If the Thue–Morse Sequence members are used in order to select program states: | |
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| * If ''t''(''n'') = 0, move ahead by one unit,
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| * If ''t''(''n'') = 1, rotate counterclockwise by an angle of π/3,
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| the resulting curve converges to the [[Koch snowflake]], a fractal curve of
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| infinite length containing a finite area. This illustrates the fractal nature of the Thue–Morse Sequence.
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| ===Equitable sequencing===
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| In their book<ref>{{cite book| title=The Win-Win Solution: Guaranteeing Fair Shares to Everybody |last1 = Brams | first1 = Steven J. | last2 = Taylor | first2 = Alan D. |pages=36–44 |isbn=0-393-04729-6 |publisher=W. W. Norton & Co., Inc. |year=1999}}</ref> on the problem of [[fair division]], [[Steven Brams]] and [[Alan D. Taylor|Alan Taylor]] invoked the Thue–Morse sequence but did not identify it as such. When allocating a contested pile of items between two parties who agree on the items' relative values, Brams and Taylor suggested a method they called ''balanced alternation'', or ''taking turns taking turns taking turns . . . '', as a way to circumvent the favoritism inherent when one party chooses before the other. An example showed how a divorcing couple might reach a fair settlement in the distribution of jointly-owned items. The parties would take turns to be the first chooser at different points in the selection process: Ann chooses one item, then Ben does, then Ben chooses one item, then Ann does.
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| [[Lionel Levine]] and [[Katherine Stange]], in their discussion of how to fairly apportion a shared meal such as an [[Culture of Ethiopia#Cuisine|Ethiopian dinner]], proposed the Thue–Morse sequence as a way to reduce the advantage of moving first.<ref>{{cite journal|last1=Levine|first1=Lionel|last2=Stange| first2=Katherine E.|title=How to Make the Most of a Shared Meal: Plan the Last Bite First|journal=The American Mathematical Monthly|year=2012| volume=119|issue=7|pages=550–565|url=http://math.colorado.edu/~kstange/papers/EthiopianDinner.pdf|accessdate=11 February 2013}}</ref> They suggested that “it would be interesting to quantify the intuition that the Thue-Morse order tends to produce a fair outcome.”
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| [[Robert Richman]] addressed this problem, but he too did not identify the Thue–Morse sequence as such at the time of publication.<ref name="richman">{{cite journal|last=Richman|first=Robert|title=Recursive Binary Sequences of Differences|journal=Complex Systems|year=2001|volume=13|issue=4|pages=381–392|url=http://www.complex-systems.com/pdf/13-4-3.pdf|accessdate=16 January 2012}}</ref> He presented the sequences [[#Characterization using bitwise negation|''T<sub>n</sub>'']] as [[step function]]s on the interval [0,1] and described their relationship to the [[Walsh function|Walsh]] and [[Hans Rademacher|Rademacher]] functions. He showed that the ''n''<sup>th</sup> [[derivative]] can be expressed in terms of ''T<sub>n</sub>''. As a consequence, the step function arising from ''T<sub>n</sub>'' is [[Orthogonality|orthogonal]] to [[polynomial]]s of [[Degree of a polynomial|order]] ''n'' − 1. A consequence of this result is that a resource whose value is expressed as a [[Monotonic function|monotonically]] decreasing [[continuous function]] is most fairly allocated using a sequence that converges to Thue-Morse as the function becomes [[Flat function|flatter]]. An example showed how to pour cups of [[Drip brew|coffee]] of equal strength from a carafe with a [[Nonlinear system|nonlinear]] [[concentration]] [[gradient]], prompting a whimsical article in the popular press.<ref>{{cite news|last=Abrahams|first=Marc|title=How to pour the perfect cup of coffee|url=http://www.guardian.co.uk/education/2010/jul/13/perfect-coffee-improbable-research|accessdate=16 January 2012|newspaper=The Guardian|date=12 July 2010}}</ref>
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| [[Joshua Cooper (mathematician)|Joshua Cooper]] and [[Aaron Dutle]] showed why the Thue-Morse order provides a fair outcome for discrete events.<ref>{{cite journal|last1=Cooper|first1=Joshua|last2=Dutle| first2=Aaron|title=Greedy Galois Games|journal=The American Mathematical Monthly|year=2013| volume=120|issue=5|pages=441–451|url=http://www.math.sc.edu/~cooper/ThueMorseDueling.pdf|accessdate=19 June 2013}}</ref> They considered the fairest way to stage a [[Évariste Galois|Galois]] duel, in which each of the shooters has equally poor shooting skills. Cooper and Dutle postulated that each dueler would demand a chance to fire as soon as the other’s [[a priori probability|''a priori'' probability]] of winning exceeded their own. They proved that, as the duelers’ hitting probability approaches zero, the firing sequence converges to the Thue–Morse sequence. In so doing, they demonstrated that the Thue-Morse order produces a fair outcome not only for sequences [[#Characterization using bitwise negation|''T<sub>n</sub>'']] of length ''2<sup>n</sup>'', but for sequences of any length.
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| Sports competitions form an important class of equitable sequencing problems, because strict alternation often gives an unfair advantage to one team. Richman suggested that the fairest way for “captain A” and “captain B” to choose sides for a [[Streetball|pick-up game of basketball]] mirrors ''T''<sub>3</sub>: captain A has the first, fourth, sixth, and seventh choices, while captain B has the second, third, fifth, and eighth choices.<ref name="richman" /> [[Ignacio Palacios-Huerta]] proposed changing the sequential order to Thue-Morse to improve the ''[[ex post]]'' fairness of various tournament competitions, such as the kicking sequence of a [[Penalty shoot-out (association football)#Procedure|penalty shoot-out]] in soccer, the rotation of color of pieces in a [[Chess tournament#Rules|chess match]], and the serving order in a [[Tennis score#Scoring a tiebreak game|tennis tie-break]].<ref>{{cite journal|last=Palacios-Huerta|first=Ignacio|title=Tournaments, fairness and the prouhet-thue-morse sequence|journal=Economic inquiry|year=2012| volume=50|issue=3|pages=848–849|url=http://www.palacios-huerta.com/docs/EI-Tournaments_and_PTM_sequence.pdf|accessdate=18 February 2013}}</ref>
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| == History ==
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| The Thue–Morse sequence was first studied by [[Eugène Prouhet]] in 1851, who applied it to [[number theory]].
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| However, Prouhet did not mention the sequence explicitly; this was left to [[Axel Thue]] in 1906, who used it to found the study of [[combinatorics on words]].
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| The sequence was only brought to worldwide attention with the work of [[Marston Morse]] in 1921, when he applied it to [[differential geometry]].
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| The sequence has been discovered independently many times, not always by professional research mathematicians; for example, [[Max Euwe]], a [[Grandmaster (chess)|chess grandmaster]], who held the world championship title from 1935 to 1937, and mathematics [[teacher]], discovered it in 1929 in an application to [[chess]]: by using its cube-free property (see above), he showed how to circumvent a rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw.
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| ==References==
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| {{reflist}}
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| *{{cite book | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}
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| * {{cite book | last1=Berstel | first1=Jean | last2=Lauve | first2=Aaron | last3=Reutenauer | first3=Christophe | last4=Saliola | first4=Franco V. | title=Combinatorics on words. Christoffel words and repetitions in words | series=CRM Monograph Series | volume=27 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2009 | isbn=978-0-8218-4480-9 | url=http://www.ams.org/bookpages/crmm-27 | zbl=1161.68043 }}
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| *{{cite book | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-0-521-11169-0 | zbl=pre06066616 }}
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| *{{cite book | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Algebraic combinatorics on words | others=With preface by Jean Berstel and Dominique Perrin | edition=Reprint of the 2002 hardback | series=Encyclopedia of Mathematics and Its Applications | volume=90| publisher=Cambridge University Press | year=2011 | isbn=978-0-521-18071-9 | zbl=1221.68183 }}
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| *{{cite book | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Applied combinatorics on words | others=A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé| series=Encyclopedia of Mathematics and Its Applications | volume=105 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2005 | isbn=0-521-84802-4 | zbl=1133.68067 }}
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| * {{cite book | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}
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| == External links ==
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| *{{MathWorld|urlname=Thue-MorseSequence|title=Thue-Morse Sequence}}
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| *Allouche, J.-P.; Shallit, J. O. [http://www.cs.uwaterloo.ca/~shallit/Papers/ubiq.ps The Ubiquitous Prouhet-Thue-Morse Sequence]. (contains many applications and some history)
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| *Thue–Morse Sequence over (1,2) {{OEIS|id=A001285}}
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| *Odious numbers {{OEIS|id=A000069}}
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| *Evil numbers {{OEIS|id=A001969}}
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| *[http://www2.kenyon.edu/people/holdenerj/StudentResearch/WhenThueMorsemeetsKochJan222005.pdf When Thue-Morse meets Koch]. A paper showing an astonishing similarity between the Thue–Morse Sequence and the [[Koch snowflake]]
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| *[http://www.webcitation.org/query?url=http://www.geocities.com/jan.schat/ThueMorse.PDF&date=2009-10-26+00:40:54 Reducing the influence of DC offset drift in analog IPs using the Thue-Morse Sequence]. A technical application of the Thue–Morse Sequence
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| *[http://reglos.de/musinum MusiNum - The Music in the Numbers]. Freeware to generate self-similar music based on the Thue–Morse Sequence and related number sequences.
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| {{DEFAULTSORT:Thue-Morse Sequence}}
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| [[Category:Binary sequences]]
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| [[Category:Fixed points (mathematics)]]
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| [[Category:Parity]]
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There are several options (check with your dentist): either porcelain veneers or dental bonding. The people who want this treatment, they will require 3 visits to their dentist. For example, Ageless Derma is one company that offers an affordable skin lightening cream. Clean your skin regularly using warm salt water and a clean piece of cloth. certainly lies in the health of the human body as the core of all activities , and the human body will not always be the same, his time will come sooner or later the body experiences the aging process that can reduce the anti-bodies and metabolism of all diseases and the aging process , but there is a solution that has been chosen , and the more has been chosen is " better to prevent than cure ".
Simply lather the soap, apply it to the face and body, and allow it to work on the skin for up to five minutes before rinsing the lather away. You can also spray on tile, in garbage cans and in empty aquariums. It is somewhat ironic that in the United States and Europe, many people light skinned people seek to darken their skin color by tanning, either in the sun, using a tanning bed or employing spray-on tanning products. Poor hygiene will cause dirt and oil to accumulate in your armpits that makes underarms dark. Most of these products are made out of ingredients that can be harsh for the skin which can be very bad particularly for the underarms.
It's a fact that the movie-going, television-watching public would rather look at beautiful people than unattractive people. Even if you're a veteran photo editor, but you're used to working in a different application, it can be a little frustrating when you're trying to locate a specific tool that isn't where you thought it would be. The fastest way to whiten your teeth and In the age of wanting everything in mega fast time the teeth whitening industry can offer a one hour teeth whitening solution that is fast, effective and safe and can remove the staining on the teeth by up to 10 shades. A good number of people would like to have healthy, young looking skin. If you would like a cheap natural whitening agent, lemons (or calamansi for us Filipinos) are your best bet.
Mix together equal parts of lavender and rose flowers, leaves of rosemary, mint and thyme and comfrey root. Avoid smoking or if you can’t seem to kick the habit – brush your teeth after each cigarette or cigar. Even those with fair skin want to keep their skin tone as white as possible, and Asians are fond of whiter skin so they use all kinds of products to skin lighten and whiten much lighter than their natural skin tone. He does all of his operations in a very precise and careful manner. The downside to abrasion is that too much can remove enamel from your teeth, damaging them and making them sensitive and even more susceptible to staining.
The obvious disadvantage of these chemical bleaching creams is definitely the unwanted side effects. But don't hold your breadth; even if it's diagnosed as strongyloides stercoralis, doctors are very unfamiliar with its treatment. Always rinse the mouth thoroughly to remove any lemon juice residue. Be sure to match your lip shade with the lip liner. The epicuren after bath is one of the most popular products which are beneficial for most of the people including men and women.
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