(3.238.99.243) 您好!臺灣時間:2021/05/15 18:29
字體大小: 字級放大   字級縮小   預設字形  
回查詢結果

詳目顯示:::

我願授權國圖
: 
twitterline
研究生:廖芳儀
研究生(外文):Fang-Yi Liao
論文名稱:雙邊無窮特徵字字尾之分解
論文名稱(外文):Factorizations of Suffixes of Two-Way Infinite Characteristic Words
指導教授:郭蕙芳
指導教授(外文):Wai-Fong Chuan
學位類別:博士
校院名稱:中原大學
系所名稱:應用數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2014
畢業學年度:102
語文別:英文
論文頁數:91
中文關鍵詞:分解Markov字型雙邊無窮特徵字種子字鄰邊α-字
外文關鍵詞:adjacent α-wordseed wordMarkov word patternfactorizationtwo-way infinite characteristic word
相關次數:
  • 被引用被引用:0
  • 點閱點閱:125
  • 評分評分:
  • 下載下載:0
  • 收藏至我的研究室書目清單書目收藏:0
令α是界在0和1之間的無理數,且其連分數表示法為[0;a1+1,a2,a3,…],其中an ≧1 (n≧1)。定義q-1=1, q0=1, qn=anqn-1+qn-2 (n≧1)。對於每一個大於等於-1的整數k,我們考慮雙邊無窮特徵字字尾H的k階分解,形式如下:H=ukuk+1uk+2…,其中因子ui的長度為qi (i≧k)。我們證明在這樣的分解中,不是所有ui皆為奇異字,就是存在非負整數q使得H的前q個因子為奇異字,其餘的皆為α-字。更進一步地,這些α-字的標籤可被一非負整數的F-表現法所唯一決定,此非負整數與H在雙邊無窮特徵字上的開始位置有關。

接著,我們定義由一對種子字(u,v)所生成的k階Markov字型(k≧1),形式如下:Mk(u,v)=z1z2z3…,其中z1=u, z2=v, zi=(zi-1)ai+k-1-1 zi-2 zi-1 (i≧3)。我們證明每一個雙邊無窮特徵字字尾H皆為Markov字型,且其所包含的種子字皆為鄰邊α-字;我們也找到H的所有身為鄰邊α-字的種子字。另一方面,我們證明每一個鄰邊α-字所生成的Markov字型皆為雙邊無窮特徵字字尾。

最後,我們研究鄰邊α-字所形成的集合V。我們分別以循環移位、字典順序、標籤描述V中的元素。對於每一個α-字w,我們找到所有α-字v使得(w,v)為一對鄰邊α-字,也找到所有α-字u使得(u,w)為一對鄰邊α-字。另外,我們建造有向圖Gα,此圖的點集為所有α-字所形成的集合,邊集為所有鄰邊α-字所形成的集合;我們以此圖作為例證,說明鄰邊α-字的相關結果,以及它們所生成的雙邊無窮特徵字字尾。




Let α be an irrational number between 0 and 1 with continued fraction expansion [0;a1+1,a2,a3,…], where an ≧1 (n≧1). Define a sequence of numbers q-1=1, q0=1, qn=anqn-1+qn-2 (n≧1). For each integer k≧-1, we consider the k-order factorization of each suffix H of a two-way infinite characteristic word ofαof the form: H=ukuk+1uk+2…, where the length of the factor ui is qi (i≧k). We show that in such a factorization, either all ui are singular words, or there exists a nonnegative integer q such that H begins with q singular words, and uk+quk+q+1uk+q+2… areα-words. Moreover, the labels of theseα-words are uniquely determined by the F-representation of a nonnegative integer obtained from the position of H in the two-way infinite characteristic word.

Next, define Markov word pattern of order k (k≧-1) generated by a pair of seed words (u,v) as follows: Mk(u,v)=z1z2z3…, where z1=u, z2=v, and zi=(zi-1)ai+k-1-1 zi-2 zi-1 (i≧3). We show that each suffix H of each two-way infinite characteristic word is a Markov word pattern, and the pairs of its seed words obtained are adjacentα-words; we also find all possible pairs of seed words of H which are pairs of adjacent α-words. On the other hand, we show that each Markov word pattern generated by any pair of adjacentα-words is a suffix of a two-way infinite characteristic word.

Finally, we study the set V of all pairs of adjacentα-words. We describe the elements of V in terms of cyclic shifts, lexicographic order, and labels. For eachα-word w, we find all possible words u such that (u,w) (resp., (w,u)) are pairs of adjacentα-words. Also, we construct a directed graph Gα consisting of the vertex set of allα-words and the edge set V, and use such a graph to illustrate the results about adjacentα-words, and the suffixes of the two-way infinite characteristic words that they generate.




Contents
摘要 I
Abstract II
誌謝辭 III
Contents IV
List of Figures VI
List of Tables VII
List of Symbols VIII
Chapter 1 Introduction 1
Chapter 2 Preliminaries 4
2.1 Words 4
2.2 α-words and singular words 5
2.3 Two-way infinite characteristic words 9
Chapter 3 Factorizations of Suffixes of Two-Way Infinite Characteristic Words 12
3.1 The F-representation of nonnegative integers 13
3.2 Factorizations of suffixes ofα-words 16
3.3 (-1)st-order factorization of suffixes of f and f' 24
3.4 kth-order factorizations (k≧-1) of suffixes of f and f' 33
3.5 The case α=[0;1+1,1,1,…] 37
Chapter 4 Markov Word Patterns in Two-Way Infinite Characteristic Words 40
4.1 Markov word patterns in f and f' 40
4.2 More seed words of suffixes of f and f' 45
Chapter 5 Adjacentα-Words 50
5.1 The V+-set 51
5.1.1 Cyclic shifts 51
5.1.2 Lexicographic order 53
5.1.3 Labels 59
5.2 The V--set 61
5.3 Suffixes of f and f' determined by a singleα-words 66
5.4 The set V 67
5.5 The directed graph Gα 69
References 74
Appendices 78
A.1 Using kth-order factorizations to prove Theorem 4.1.2 78
A.2 Using labels in Zn+1 (resp., Zn-1) to rewrite Theorem 5.1.10 (resp., 5.2.5) 79
A.3 Using known V+-sets to prove Theorem 5.2.1 81

List of Figures
Figure 2.1 The defining tree Tαof α-words with α=[0; a1+1, a2, a3,…]. 7
Figure 5.1 The subgraph of Gαwith the elements of W in cyclic shifts andα=[0;3+1,3,1,2,…]. 70
Figure 5.2 The subgraph of Gα with the elements of W in lexicographic order andα=[0;3+1,3,1,2,…]. 71
Figure 5.3 The directed graph Gα with the elements of W in cyclic shifts andα=[0;3+1,3,1,2,…]. 72
Figure 5.4 The directed graph Gα with the elements of W in lexicographic order andα=[0;3+1,3,1,2,…]. 73

List of Tables
Table 3.1 I(r) for each r in S1∪S2∪S3∪S4 with α=[0;3+1,2,1,2,4,…]. 14
Table 3.2 σ( r) for each r in S1∪S2∪S3∪S4 withα=[0;3+1,2,1,2,4,…]. 21
Table 3.3 (-1)st-order factorizations of fm and f’m for some m in [-122,28] withα=[0;2+1,2,3,1,1,1,…], where ai=1 for all i≧4. 31
Table 3.4 1st-order factorizations of fm and f’m for some m in [-122,28] withα=[0;2+1,2,3,1,1,1,…], where ai=1 for all i≧4. 36
Table 4.1 Seed words (in cyclic shifts) of nth-order MWPs in fm, which are pairs of adjacent α-words, where -1≦n≦3 and -9≦m≦100 with α=[0;2+1,2,4,3,2,…]. 47
Table 4.2 Seed words (in cyclic shifts) of nth-order MWPs in f’m, which are pairs of adjacent α-words, where -1≦n≦3 and -9≦m≦100 with α=[0;2+1,2,4,3,2,…]. 48
Table 5.1 The elements of V+( Ti(xn)) for n≧1 and 0≦i≦q_n-1. 52
Table 5.2 The integers i3,k, j3,k, and d3,k for each k in [0, q3-1] withα=[0;3+1,3,1,2,…]. 69
[1] J. Berstel, P. Seebold, Morphismes de Sturm, B. Belg. Math. Soc. 1 (1994) 175-189.
[2] T.C. Brown, A characterization of the quadratic irrationals, Can. Math. Bull. 34 (1991) 36-41.
[3] T.C. Brown, Descriptions of the characteristic sequence of an irrational, Can. Math. Bull. 36 (1993) 15-21.
[4] W.-T. Cao, Z.-Y. Wen, Some properties of the factors of Sturmian sequences, Theor. Comput. Sci. 304 (2003) 365-385.
[5] W.-F. Chuan, Fibonacci words, Fibonacci Quart. 30.1 (1992) 68-76.
[6] W.-F. Chuan, Embedding Fibonacci words into Fibonacci word patterns, in: G.E. Bergum, A.N. Philippou, A.F. Horadam (Eds.), Applications of Fibonacci Numbers Vol. 5 (Kluwer, Dordrecht, 1993) 113-122.
[7] W.-F. Chuan, Subwords of the golden sequence and the Fibonacci words, in: G.E. Bergum, A.N. Philippou, A.F. Horadam (Eds.), Applications of Fibonacci Numbers, Vol. 6 (Kluwer, Dordrecht, 1996) 73-84.
[8] W.-F. Chuan, α-words and factors of characteristic sequences, Discrete Math. 177 (1997) 33-50.
[9] W.-F. Chuan, A representation theorem of the suffixes of characteristic sequences, Discrete Appl. Math. 85 (1998) 47-57.
[10] W.-F. Chuan, Unbordered factors of the characteristic sequences of irrational numbers, Theor. Comput. Sci. 205 (1998) 337-344.
[11] W.-F. Chuan, Sturmian morphisms and α-words, Theor. Comput. Sci. 225 (1999) 129-148.
[12] W.-F. Chuan, Characterizations of α-words, moments, and determinants, Fibonacci Quart. 41.3 (2003) 194-208.
[13] W.-F. Chuan, C.-H. Chang, Y.-L. Chang, Suffixes of Fibonacci word patterns, Fibonacci Quart. 38.5 (2000) 432-438.
[14] W.-F. Chuan, Factors of characteristic words of irrational numbers, Theor. Comput. Sci. 337 (2005) 169-182.
[15] W.-F. Chuan, H.-L. Ho, Factors of characteristic words: location and decompositions, Theor. Comput. Sci. 411 (2010) 2827-2846.
[16] W.-F. Chuan, H.-L. Ho, Locating factors of a characteristic word via the Zeckendorf representation of numbers, Theor. Comput. Sci. 440-441 (2012) 39-51.
[17] W.-F. Chuan, F.-Y. Liao, The D-representation of nonnegative integers and the Fibonacci factorization of suffixes of infinite Fibonacci words, Discrete Appl. Math., to appear.
[18] W.-F. Chuan, F.-Y. Liao, H.-L. Ho, F. Yu, Fibonacci word patterns in two-way infinite Fibonacci words, Theor. Comput. Sci. 437 (2012) 69-81.
[19] W.-F. Chuan, F. Yu, Three new extraction formulae, Fibonacci Quart. 45.1 (2007) 76-84.
[20] A. de Luca, A division property of the Fibonacci word, Inform. Process. Lett. 54 (1995) 307-312.
[21] N.P. Fogg, V. Berthe, S. Ferenczi, C. Mauduit, A. Siegel, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 1794 (2002), Chapter 6.
[22] A.S. Fraenkel, Systems of numeration, Am. Math. Mon. 92 (1985) 105-114.
[23] A.S. Fraenkel, I. Borosh, A generalization of Wythoff's game, J. Comb. Theory A 15 (1973) 175-191.
[24] A.S. Fraenkel, J. Levitt, M. Shimshoni, Characterization of the set of values f=[nα], n=1, 2,…, Discrete Math. 2 (1972) 335-345.
[25] Amy Glen, Occurrences of palindromes in characteristic Sturmian words, Theor. Comput. Sci. 352 (2006) 31-46.
[26] V.E. Hoggatt, Jr., Fibonacci and Lucas Numbers, New York, Houghton Mifflin Company, 1969.
[27] S. Ito and S. Yasutomi, On continued fractions, substitutions and characteristic sequences [nx+y]-[(n-1)x+y], Jpn. J. Math 16 (1990) 287-306.
[28] M. Lothaire, Combinatorics on Words, Cambridge Univ. Press, 2002, Chapter 2.
[29] G. Melancon, Viennot factorization of infinite words, Inform. Process. Lett. 60 (1996) 53-57.
[30] G. Melancon, Lyndon words and singular factors of Sturmian words, Theor. Comput. Sci. 218 (1999) 41-59.
[31] G. Melancon, Lyndon factorization of Sturmian words, Discrete Math. 210 (2000) 137-149.
[32] H.-J. Shyr, Free Monoids and Languages, Lecture Notes, Institute of Applied Mathematics, National Chung Hsing University, Taichung, Taiwan, ROC (1991).
[33] K. Tognetti, G. Winley, T. van Ravenstein, The Fibonacci Tree, Hofstader and the Golden String, in: G.E. Bergum, A.N. Philippou, A.F. Horadam (Eds.), Applications of Fibonacci numbers, Vol.3 (Kluwer, Dordrecht, 1990) 325-334.
[34] J.C. Turner, Fibonacci word patterns and binary sequences, Fibonacci Quart. 26.3 (1988) 233-246.
[35] J.C. Turner, The alpha and the omega of the Wythoff pairs, Fibonacci Quart. 27.1 (1989) 76-86.
[36] B.A. Venkov, Elementary Number Theory, Wolters-Noorhoff, Groningen, 1970.
[37] Z.-X. Wen, Z.-Y. Wen, Some properties of the singular words of the Fibonacci word, Eur. J. Combin. 15 (1994) 587-598.
QRCODE
 
 
 
 
 
                                                                                                                                                                                                                                                                                                                                                                                                               
第一頁 上一頁 下一頁 最後一頁 top