rich sturmian trapezoidal words
play

Rich, Sturmian & trapezoidal words Amy Glen School of Chemical - PowerPoint PPT Presentation

Oct 05, 2023 •236 likes •1.52k views

Rich, Sturmian & trapezoidal words Amy Glen School of Chemical & Mathematical Sciences Murdoch University, Perth, Australia amy.glen@gmail.com http://wwwstaff.murdoch.edu.au/~aglen 54th AustMS Annual Meeting @ The University of

  1. Finite and Infinite Words Words . . . Depending on the problem to be solved, it may be fruitful to study combinatorial and structural properties of the words representing the elements of a particular set or to impose certain combinatorial conditions on such words. Most commonly studied words are those which satisfy one or more strong regularity properties; for instance, words containing many repetitions or palindromes. The extent to which a word exhibits strong regularity properties is generally inversely proportional to its “complexity”. Basic measure: number of distinct blocks (factors) of each length occurring in the word. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 4 / 40

  2. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  3. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  4. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Example √ x = ( 2 ) 2 = 1 . 0110101000001001111 . . . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  5. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Example √ x = ( 2 ) 2 = 1 . 0110101000001001111 . . . F 1 ( x ) = { 0 , 1 } , C x ( 1 ) = 2 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  6. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Example √ x = ( 2 ) 2 = 1 . 0110101000001001111 . . . F 1 ( x ) = { 0 , 1 } , C x ( 1 ) = 2 F 2 ( x ) = { 00 , 01 , 10 , 11 } , C x ( 2 ) = 4 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  7. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Example √ x = ( 2 ) 2 = 1 . 0110101000001001111 . . . F 1 ( x ) = { 0 , 1 } , C x ( 1 ) = 2 F 2 ( x ) = { 00 , 01 , 10 , 11 } , C x ( 2 ) = 4 F 3 ( x ) = { 000 , 001 , 010 , 100 , 101 , 110 , 111 } , C x ( 3 ) = 8 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  8. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Example √ x = ( 2 ) 2 = 1 . 0110101000001001111 . . . F 1 ( x ) = { 0 , 1 } , C x ( 1 ) = 2 F 2 ( x ) = { 00 , 01 , 10 , 11 } , C x ( 2 ) = 4 F 3 ( x ) = { 000 , 001 , 010 , 100 , 101 , 110 , 111 } , C x ( 3 ) = 8 √ Conjecture: C x ( n ) = 2 n for all n as it is believed 2 is normal in base 2. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  9. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  10. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . That is: w is aperiodic ⇔ C w ( n ) ≥ n + 1 for all n ∈ N . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  11. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . That is: w is aperiodic ⇔ C w ( n ) ≥ n + 1 for all n ∈ N . Sturmian words are the aperiodic infinite words of minimal complexity, i.e., an infinite word w is Sturmian if and only if C w ( n ) = n + 1 for each n . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  12. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . That is: w is aperiodic ⇔ C w ( n ) ≥ n + 1 for all n ∈ N . Sturmian words are the aperiodic infinite words of minimal complexity, i.e., an infinite word w is Sturmian if and only if C w ( n ) = n + 1 for each n . Low complexity accounts for many interesting features, as it induces certain regularities in such words without, however, making them periodic. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  13. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . That is: w is aperiodic ⇔ C w ( n ) ≥ n + 1 for all n ∈ N . Sturmian words are the aperiodic infinite words of minimal complexity, i.e., an infinite word w is Sturmian if and only if C w ( n ) = n + 1 for each n . Low complexity accounts for many interesting features, as it induces certain regularities in such words without, however, making them periodic. References in: Combinatorics, Symbolic Dynamics, Number Theory, Discrete Geometry, Theoretical Physics, Theoretical Computer Science. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  14. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . That is: w is aperiodic ⇔ C w ( n ) ≥ n + 1 for all n ∈ N . Sturmian words are the aperiodic infinite words of minimal complexity, i.e., an infinite word w is Sturmian if and only if C w ( n ) = n + 1 for each n . Low complexity accounts for many interesting features, as it induces certain regularities in such words without, however, making them periodic. References in: Combinatorics, Symbolic Dynamics, Number Theory, Discrete Geometry, Theoretical Physics, Theoretical Computer Science. Numerous equivalent definitions & characterisations . . . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  15. Complexity & Sturmian words Palindrome Complexity Given a finite or infinite word w , let P w ( n ) denote the palindromic complexity function of w , which counts the number of palindromic factors of w of each length n ≥ 0. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 7 / 40

  16. Complexity & Sturmian words Palindrome Complexity Given a finite or infinite word w , let P w ( n ) denote the palindromic complexity function of w , which counts the number of palindromic factors of w of each length n ≥ 0. Theorem (Droubay-Pirillo 1999) An infinite word w is Sturmian if and only if � 1 if n is even P w ( n ) = 2 if n is odd Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 7 / 40

  17. Complexity & Sturmian words Palindrome Complexity Given a finite or infinite word w , let P w ( n ) denote the palindromic complexity function of w , which counts the number of palindromic factors of w of each length n ≥ 0. Theorem (Droubay-Pirillo 1999) An infinite word w is Sturmian if and only if � 1 if n is even P w ( n ) = 2 if n is odd Note: Any Sturmian word is over a 2-letter alphabet since it has two distinct factors of length 1. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 7 / 40

  18. Complexity & Sturmian words Palindrome Complexity Given a finite or infinite word w , let P w ( n ) denote the palindromic complexity function of w , which counts the number of palindromic factors of w of each length n ≥ 0. Theorem (Droubay-Pirillo 1999) An infinite word w is Sturmian if and only if � 1 if n is even P w ( n ) = 2 if n is odd Note: Any Sturmian word is over a 2-letter alphabet since it has two distinct factors of length 1. A Sturmian word over the alphabet { a , b } contains either aa or bb , but not both. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 7 / 40

  19. Complexity & Sturmian words Constructing Sturmian words Let’s consider a nice geometric realisation, starting with a special class of finite words . . . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 8 / 40

  20. Complexity & Sturmian words Constructing Sturmian words Let’s consider a nice geometric realisation, starting with a special class of finite words . . . Words over a 2-letter alphabet { a , b } that are factors of (infinite) Sturmian words are called finite Sturmian words – they are the cyclic shifts of Christoffel words obtained via the following construction. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 8 / 40

  21. Complexity & Sturmian words Constructing Sturmian words Let’s consider a nice geometric realisation, starting with a special class of finite words . . . Words over a 2-letter alphabet { a , b } that are factors of (infinite) Sturmian words are called finite Sturmian words – they are the cyclic shifts of Christoffel words obtained via the following construction. Using a similar construction we obtain infinite Sturmian words. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 8 / 40

  22. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 9 / 40

  23. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 10 / 40

  24. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 11 / 40

  25. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 12 / 40

  26. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 13 / 40

  27. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 14 / 40

  28. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 15 / 40

  29. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 16 / 40

  30. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 17 / 40

  31. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 a L(3,5) = a Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 18 / 40

  32. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 a a L(3,5) = aa Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 19 / 40

  33. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 b a a L(3,5) = aab Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 20 / 40

  34. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 a b a a L(3,5) = aaba Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 21 / 40

  35. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 a a b a a L(3,5) = aabaa Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 22 / 40

  36. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 b a a b a a L(3,5) = aabaab Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 23 / 40

  37. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 a b a a b a a L(3,5) = aabaaba Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 24 / 40

  38. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 b a b a a b a a L(3,5) = aabaabab Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 25 / 40

  39. Complexity & Sturmian words Christoffel words: Construction by example Lower & Upper Christoffel words of slope 3 5 a a b b a a a b b a a a b b a a L(3,5) = aabaabab U(3,5) = babaabaa Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 26 / 40

  40. Complexity & Sturmian words From Christoffel words to Sturmian words Sturmian words: Obtained *similarly* by replacing the line segment by a half-line: y = α x + ρ with irrational α ∈ ( 0 , 1 ) , ρ ∈ R . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 27 / 40

  41. Complexity & Sturmian words From Christoffel words to Sturmian words Sturmian words: Obtained *similarly* by replacing the line segment by a half-line: y = α x + ρ with irrational α ∈ ( 0 , 1 ) , ρ ∈ R . √ 5 − 1 Example: y = x − → Fibonacci word 2 b a a b a b a a b a a Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 27 / 40

  42. Complexity & Sturmian words From Christoffel words to Sturmian words Sturmian words: Obtained *similarly* by replacing the line segment by a half-line: y = α x + ρ with irrational α ∈ ( 0 , 1 ) , ρ ∈ R . √ 5 − 1 Example: y = x − → Fibonacci word 2 b a a b a b a a b a a f = abaababaabaababaaba · · · (note: disregard 1st a in construction) Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 27 / 40

  43. Complexity & Sturmian words From Christoffel words to Sturmian words Sturmian words: Obtained *similarly* by replacing the line segment by a half-line: y = α x + ρ with irrational α ∈ ( 0 , 1 ) , ρ ∈ R . √ 5 − 1 Example: y = x − → Fibonacci word 2 b a a b a b a a b a a f = abaababaabaababaaba · · · (note: disregard 1st a in construction) √ 5 − 1 Standard Sturmian word of slope , golden ratio conjugate 2 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 27 / 40

  44. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  45. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. He showed: Theorem (de Luca 1999) If w is a finite Sturmian word of length | w | (i.e., a cyclic shift of a Christoffel word), then the graph of C w ( n ) as a function of n (for 0 ≤ n ≤ | w | ) is that of a regular trapezoid (possibly degenerated to a triangle). Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  46. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. He showed: Theorem (de Luca 1999) If w is a finite Sturmian word of length | w | (i.e., a cyclic shift of a Christoffel word), then the graph of C w ( n ) as a function of n (for 0 ≤ n ≤ | w | ) is that of a regular trapezoid (possibly degenerated to a triangle). That is: C w ( n ) increases by 1 with each n on some interval of length r . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  47. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. He showed: Theorem (de Luca 1999) If w is a finite Sturmian word of length | w | (i.e., a cyclic shift of a Christoffel word), then the graph of C w ( n ) as a function of n (for 0 ≤ n ≤ | w | ) is that of a regular trapezoid (possibly degenerated to a triangle). That is: C w ( n ) increases by 1 with each n on some interval of length r . Then C w ( n ) is constant on some interval of length s . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  48. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. He showed: Theorem (de Luca 1999) If w is a finite Sturmian word of length | w | (i.e., a cyclic shift of a Christoffel word), then the graph of C w ( n ) as a function of n (for 0 ≤ n ≤ | w | ) is that of a regular trapezoid (possibly degenerated to a triangle). That is: C w ( n ) increases by 1 with each n on some interval of length r . Then C w ( n ) is constant on some interval of length s . Finally C w ( n ) decreases by 1 with each n on an interval of length r . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  49. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. He showed: Theorem (de Luca 1999) If w is a finite Sturmian word of length | w | (i.e., a cyclic shift of a Christoffel word), then the graph of C w ( n ) as a function of n (for 0 ≤ n ≤ | w | ) is that of a regular trapezoid (possibly degenerated to a triangle). That is: C w ( n ) increases by 1 with each n on some interval of length r . Then C w ( n ) is constant on some interval of length s . Finally C w ( n ) decreases by 1 with each n on an interval of length r . So if we set D w ( n ) = C w ( n + 1 ) − C w ( n ) for each n with 0 ≤ n ≤ | w | − 1, then the word D w ( 0 ) D w ( 1 ) · · · D w ( | w | − 1 ) takes the form 1 r 0 s ( − 1 ) r . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  50. Trapezoidal words Example Graph of the factor complexity of the Christoffel word L ( 3 , 5 ) = aabaabab Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 29 / 40

  51. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  52. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. For example, aabb is trapezoidal, but not Sturmian. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  53. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. For example, aabb is trapezoidal, but not Sturmian. Note: If w is a trapezoidal word (i.e., its ‘complexity’ graph has the same behaviour as that of Sturmian words), then necessarily C w ( 1 ) = 2. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  54. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. For example, aabb is trapezoidal, but not Sturmian. Note: If w is a trapezoidal word (i.e., its ‘complexity’ graph has the same behaviour as that of Sturmian words), then necessarily C w ( 1 ) = 2. This is because there is 1 factor of length 0, namely the empty word ε . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  55. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. For example, aabb is trapezoidal, but not Sturmian. Note: If w is a trapezoidal word (i.e., its ‘complexity’ graph has the same behaviour as that of Sturmian words), then necessarily C w ( 1 ) = 2. This is because there is 1 factor of length 0, namely the empty word ε . So any trapezoidal word is on a binary alphabet and the family of trapezoidal words properly contains all finite Sturmian words. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  56. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. For example, aabb is trapezoidal, but not Sturmian. Note: If w is a trapezoidal word (i.e., its ‘complexity’ graph has the same behaviour as that of Sturmian words), then necessarily C w ( 1 ) = 2. This is because there is 1 factor of length 0, namely the empty word ε . So any trapezoidal word is on a binary alphabet and the family of trapezoidal words properly contains all finite Sturmian words. F. D’Alessandro (2002): classified all non-Sturmian trapezoidal words. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  57. Trapezoidal words Characterisation of Sturmian palindromes We have shown: Theorem (de Luca-G.-Zamboni) Let w be a palindrome. Then w is Sturmian if and only if w is trapezoidal. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 31 / 40

  58. Trapezoidal words Characterisation of Sturmian palindromes We have shown: Theorem (de Luca-G.-Zamboni) Let w be a palindrome. Then w is Sturmian if and only if w is trapezoidal. Theorem (de Luca-G.-Zamboni) Let w be a trapezoidal word. Then w contains | w | + 1 distinct palindromes (including ε ). Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 31 / 40

  59. Trapezoidal words Characterisation of Sturmian palindromes We have shown: Theorem (de Luca-G.-Zamboni) Let w be a palindrome. Then w is Sturmian if and only if w is trapezoidal. Theorem (de Luca-G.-Zamboni) Let w be a trapezoidal word. Then w contains | w | + 1 distinct palindromes (including ε ). That is, trapezoidal words (and hence finite Sturmian words) are “rich” in palindromes in the sense that they contain the maximum number of distinct palindromic factors since: Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 31 / 40

  60. Trapezoidal words Characterisation of Sturmian palindromes We have shown: Theorem (de Luca-G.-Zamboni) Let w be a palindrome. Then w is Sturmian if and only if w is trapezoidal. Theorem (de Luca-G.-Zamboni) Let w be a trapezoidal word. Then w contains | w | + 1 distinct palindromes (including ε ). That is, trapezoidal words (and hence finite Sturmian words) are “rich” in palindromes in the sense that they contain the maximum number of distinct palindromic factors since: Theorem (Droubay-Justin-Pirillo 2001) A finite word w contains at most | w | + 1 distinct palindromes (including ε ). Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 31 / 40

  61. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  62. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  63. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  64. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  65. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  66. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal ( C ( 1 ) = 2, C ( 2 ) = 4) Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  67. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal ( C ( 1 ) = 2, C ( 2 ) = 4) Definition (G.-Justin 2007) An infinite word is rich iff all of its factors are rich. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  68. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal ( C ( 1 ) = 2, C ( 2 ) = 4) Definition (G.-Justin 2007) An infinite word is rich iff all of its factors are rich. Examples a ω = aaaaaa · · · and ab ω = abbb · · · are rich. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  69. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal ( C ( 1 ) = 2, C ( 2 ) = 4) Definition (G.-Justin 2007) An infinite word is rich iff all of its factors are rich. Examples a ω = aaaaaa · · · and ab ω = abbb · · · are rich. ( ab ) ω = abababab · · · and ( aba ) ω = abaabaaba · · · are rich. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  70. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal ( C ( 1 ) = 2, C ( 2 ) = 4) Definition (G.-Justin 2007) An infinite word is rich iff all of its factors are rich. Examples a ω = aaaaaa · · · and ab ω = abbb · · · are rich. ( ab ) ω = abababab · · · and ( aba ) ω = abaabaaba · · · are rich. abc is rich, but ( abc ) ω = abcabcabc · · · is not rich. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  71. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  72. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  73. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  74. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  75. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  76. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  77. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  78. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  79. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  80. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

Recommend

Abelian returns in Sturmian words S. Puzynina jointly with L. Q. Zamboni S. Puzynina jointly

Abelian returns in Sturmian words S. Puzynina jointly with L. Q. Zamboni S. Puzynina jointly

Abelian returns in Sturmian words S. Puzynina jointly with L. Q. Zamboni S. Puzynina jointly with L. Q. Zamboni Abelian returns in Sturmian words Periodicity alphabet finite words over (right) infinite words over

548 views • 43 slides
Sturmian words, Lecture 3 Standard words Dominique Perrin 1 er d ecembre 2011 Dominique

Sturmian words, Lecture 3 Standard words Dominique Perrin 1 er d ecembre 2011 Dominique

Sturmian words, Lecture 3 Standard words Dominique Perrin 1 er d ecembre 2011 Dominique Perrin Sturmian words, Lecture 3 Standard words Consider two functions and from { 0 , 1 } { 0 , 1 } into itself defined by ( u , v )

306 views • 29 slides
The repetition threshold for binary rich words Lucas Mol Joint work with James D. Currie and

The repetition threshold for binary rich words Lucas Mol Joint work with James D. Currie and

The repetition threshold for binary rich words Lucas Mol Joint work with James D. Currie and Narad Rampersad AMS Special Session on Sequences, Words, and Automata Joint Mathematics Meetings, Denver, CO January 15, 2020 P LAN C RITICAL E

1.65k views • 137 slides
The repetition threshold for binary rich words Lucas Mol Joint work with James D. Currie and

The repetition threshold for binary rich words Lucas Mol Joint work with James D. Currie and

The repetition threshold for binary rich words Lucas Mol Joint work with James D. Currie and Narad Rampersad University of Winnipeg Mathematics and Statistics Seminar September 20, 2019 P LAN W ORDS AND R EPETITIONS R ICH WORDS W ORDS A word

1.66k views • 153 slides
Lesson 3 Approximating Fourier series 1 Last lecture, we saw that the trapezoidal rule was

Lesson 3 Approximating Fourier series 1 Last lecture, we saw that the trapezoidal rule was

Lesson 3 Approximating Fourier series 1 Last lecture, we saw that the trapezoidal rule was an effective method for calculating integrals of periodic functions We used the EulerMcLaurin formula to prove that the error decayed faster

603 views • 35 slides
On the number of palindromically rich words Amy Glen School of Engineering & IT Murdoch

On the number of palindromically rich words Amy Glen School of Engineering & IT Murdoch

On the number of palindromically rich words Amy Glen School of Engineering & IT Murdoch University, Perth, Australia amy.glen@gmail.com http://amyglen.wordpress.com 59th AustMS Annual Meeting @ Flinders University Special Session:

406 views • 21 slides
Recurrence function of Sturmian sequences. A probabilistic study Pablo Rotondo Universidad de la

Recurrence function of Sturmian sequences. A probabilistic study Pablo Rotondo Universidad de la

Recurrence function of Sturmian sequences. A probabilistic study Pablo Rotondo Universidad de la Rep ublica, Uruguay Ongoing work with Val erie Berth e, Eda Cesaratto, Brigitte Vall ee, and Alfredo Viola AofA15 , 812 June,

999 views • 65 slides
Notes Trapezoidal Rule Again Most of assignment 1 hasn t been covered The method: in

Notes Trapezoidal Rule Again Most of assignment 1 hasn t been covered The method: in

Notes Trapezoidal Rule Again Most of assignment 1 hasn t been covered The method: in class yet, but after today you should be ( ) x n + 1 = x n + t 2 v ( x n , t n ) + 1 1 2 v ( x n + 1 , t n + 1 ) able to do a lot of it

406 views • 7 slides
Point Location Geometry Motivation Point Location and Trapezoidal Maps A Randomized

Point Location Geometry Motivation Point Location and Trapezoidal Maps A Randomized

Yazd Univ. Computational Point Location Geometry Motivation Point Location and Trapezoidal Maps A Randomized Incremental 1397-2 Algorithm Search Structure 1 / 38 Motivation: Point location in a map Point location query: Given a map and

1.18k views • 88 slides
A First Investigation of Sturmian Trees Jean Berstel 2 , Luc Boasson 1 Olivier Carton 1 , Isabelle

A First Investigation of Sturmian Trees Jean Berstel 2 , Luc Boasson 1 Olivier Carton 1 , Isabelle

A First Investigation of Sturmian Trees Jean Berstel 2 , Luc Boasson 1 Olivier Carton 1 , Isabelle Fagnot 2 1 LIAFA, CNRS Universit e Paris 7 2 IGM, CNRS Universit e de Marne-la-Vall ee STACS2007, Aachen Olivier Carton (LIAFA)

1.17k views • 40 slides
Towards Improved Usual Solution to the . . . How to Improve the . . . Trapezoidal Approximation

Towards Improved Usual Solution to the . . . How to Improve the . . . Trapezoidal Approximation

Data Fusion: Main . . . Data Fusion for Fuzzy . . . Problem: Intersection . . . Towards Improved Usual Solution to the . . . How to Improve the . . . Trapezoidal Approximation Least Squares Method to Intersection (Fusion) of Problem with

199 views • 17 slides
Counting Words: Type probabilities Population models Type-rich populations, samples, ZM &

Counting Words: Type probabilities Population models Type-rich populations, samples, ZM &

Populations & samples Baroni & Evert The population Counting Words: Type probabilities Population models Type-rich populations, samples, ZM & fZM Sampling from and statistical models the population Random samples Expectation

1.42k views • 110 slides
Synonyms Antonyms Are words Are words that mean the that mean the same opposite

Synonyms Antonyms Are words Are words that mean the that mean the same opposite

Synonyms Antonyms Are words Are words that mean the that mean the same opposite Okay.Practice Time! On your computer open word, make a Table, (3 columns, 20 rows) label them (left) synonym and (right) antonym, and decide which words

572 views • 13 slides
THE GOOD Nutritional value of seafood: Rich source of vitamins Rich source of minerals Rich

THE GOOD Nutritional value of seafood: Rich source of vitamins Rich source of minerals Rich

Mr. Vernell Kea Seafood safety/inspection presentation outline THE GOOD Nutritional value of seafood: Rich source of vitamins Rich source of minerals Rich source of unsaturated fat Benefits of seafood: Reduce risk of heart disease omega 3

301 views • 4 slides
THE MILL ACADEMY READING WORKSHOP ..the more words children know and understand before they

THE MILL ACADEMY READING WORKSHOP ..the more words children know and understand before they

THE MILL ACADEMY READING WORKSHOP ..the more words children know and understand before they start on a systematic programme of phonic work the better they are to succeed......a broad and rich language experience for children ..is the

875 views • 17 slides
MORPHOLOGY A Study of the internal structure of words and the relationships among words

MORPHOLOGY A Study of the internal structure of words and the relationships among words

MORPHOLOGY A Study of the internal structure of words and the relationships among words Definition of Words: Word is an arbitrary pairing of sound and meaning No way to distinguish between words and phrases or sentences It in It is raining has

179 views • 13 slides
2017 EARLY LITERACY SUMMER INSTITUTE Creating a Literacy-Rich

2017 EARLY LITERACY SUMMER INSTITUTE Creating a Literacy-Rich

2017 EARLY LITERACY SUMMER INSTITUTE Creating a Literacy-Rich Environment in the Classroom Name Tents On the Back Write as many words as you can to describe

636 views • 41 slides
The Persecuted Church 8 To the angel of the church in Smyrna write: These are the words of him

The Persecuted Church 8 To the angel of the church in Smyrna write: These are the words of him

The Persecuted Church 8 To the angel of the church in Smyrna write: These are the words of him who is the First and the Last, who died and came to life again. 9 I know your afflictions and your poverty yet you are rich! I know the slander

676 views • 19 slides
THE POOR RICH [ P A R T T W O ] 1 Come now, you rich, weep and howl for your miseries which are

THE POOR RICH [ P A R T T W O ] 1 Come now, you rich, weep and howl for your miseries which are

1 THE POOR RICH [ P A R T T W O ] 1 Come now, you rich, weep and howl for your miseries which are coming upon you. 2 Your riches have rotted and your garments have become moth-eaten. 3 Your gold and your JAMES 5:1-6 silver have rusted; and

524 views • 25 slides
JUST THE MATHS SLIDES NUMBER 17.2 NUMERICAL MATHEMATICS 2 (Approximate integration (A))

JUST THE MATHS SLIDES NUMBER 17.2 NUMERICAL MATHEMATICS 2 (Approximate integration (A))

JUST THE MATHS SLIDES NUMBER 17.2 NUMERICAL MATHEMATICS 2 (Approximate integration (A)) by A.J.Hobson 17.2.1 The trapezoidal rule UNIT 17.2 - NUMERICAL MATHEMATICS 2 APPROXIMATE INTEGRATION (A) 17.2.1 THE TRAPEZOIDAL RULE The

58 views • 5 slides
Simplicity in Practice https://xkcd.com/1349/ Words, words, words. Hamlet, Act 2 Scene

Simplicity in Practice https://xkcd.com/1349/ Words, words, words. Hamlet, Act 2 Scene

Simplicity in Practice https://xkcd.com/1349/ Words, words, words. Hamlet, Act 2 Scene 2 simple, adj. one fold/braid one role, task, concept, dimension antonym: complex, interleaved, of many parts objective measure

481 views • 44 slides
Words, Words, Words AND WHY THEY MATTER IN ADVERTISING AND MARKETING Steve Kaplan Becky

Words, Words, Words AND WHY THEY MATTER IN ADVERTISING AND MARKETING Steve Kaplan Becky

Words, Words, Words AND WHY THEY MATTER IN ADVERTISING AND MARKETING Steve Kaplan Becky Lobenherz June 28, 2016 Partner Associate +1 202 263 1043 +1 202 263 3436 skaplan@mayerbrown.com rlobenherz@mayerbrown.com Mayer Brown is a global

455 views • 31 slides
Words & Pictures Clustering and Bag of Words Many

Words & Pictures Clustering and Bag of Words Many

Words & Pictures Clustering and Bag of Words Many slides adapted from Svetlana Lazebnik, Fei-Fei Li, Rob Fergus, and Antonio Torralba Document

1.03k views • 55 slides
Proverbs Words: The Power of Life and Death Words: The Power of 3. Words: They Can Be

Proverbs Words: The Power of Life and Death Words: The Power of 3. Words: They Can Be

Proverbs Words: The Power of Life and Death Words: The Power of 3. Words: They Can Be Destructive- Life and Death THEY CAN RUIN RELATIONSHIPS Prov. 16:28 A perverse person stirs up conflict, and a gossip separates Part 13 in Proverbs:

355 views • 17 slides

More recommend