English  |  正體中文  |  简体中文  |  Post-Print筆數 : 27 |  Items with full text/Total items : 109952/140891 (78%)
Visitors : 46240540      Online Users : 848
RC Version 6.0 © Powered By DSPACE, MIT. Enhanced by NTU Library IR team.
Scope Tips:
  • please add "double quotation mark" for query phrases to get precise results
  • please goto advance search for comprehansive author search
  • Adv. Search
    HomeLoginUploadHelpAboutAdminister Goto mobile version
    政大機構典藏 > 理學院 > 應用數學系 > 學位論文 >  Item 140.119/32561
    Please use this identifier to cite or link to this item: https://nccur.lib.nccu.edu.tw/handle/140.119/32561


    Title: 兩個組合數學的主題: Hadamard 矩陣的建構及有關森林的研究
    Two Combinatorial Topics: Constructions of Hadamard Matrices and Studies of Forests
    Authors: 施耀振
    Shih,Yaio-Zhern
    Contributors: 陳永秋
    李陽明

    施耀振
    Shih,Yaio-Zhern
    Keywords: Kronecker乘積
    Sylvester-Hadamard矩陣
    J_m-Hadamard矩陣
    正交對
    Weighing矩陣
    最小指數
    平面森林
    Catalan數
    Motzkin數
    Riordan數
    Narayana數
    Dyck路徑
    Motzkin路徑
    Chung-Feller定理
    優美標法
    拉丁方陣
    J_m-classes
    n-Caterpillars
    Date: 2006
    Issue Date: 2009-09-17 13:45:12 (UTC+8)
    Abstract: 在這篇論文,我們主要探討兩個獨立的組合數學主題:一個是Hadamard矩陣的建構,一個是有關森林的研究。在第一個主題,所得者又分為二,其一,我們從一個已知的Hadamard矩陣,利用Sylvester的方法去建構名為Jm-Hadamard矩陣。從這個矩陣裡,藉由在Sm上適當的排列,可以獲致其他2mm!-1個Hadamard矩陣。另外,我們引進Jm-class的概念, 將之寫成CJm,並探討當n整除n`時,CJn`是否包含於CJn。關於這個問題,我們得到最初的結論是CJ8 CJ4 CJ2。其二,在已知的t個階數分別是4m1,4m2,…,4mt的Hadamard矩陣,希望獲得一個階數是2km1m2… mt的Hadamard矩陣,使得k值愈小愈好。我們可以找到最小指數的上界,這個數稍好於Craigen及de Launey所得到的值。在第二個主題裡,我們致力於三個目標,首先,我們將平面樹上的一些結果,推廣到平面森林上,諸如Shapiro的結果,葉子的偶數、奇數問題,Catalan數與類似數之間的恒等式。其二,我們用了一個很簡潔的方法去證明Chung-Feller定理,也獲致相關的結果及應用。最後,我們以研究數種n-caterpillars的優美標法,作為本文的結束,最特別的是我們可藉用拉丁方陣去建構2n-caterpillars的優美標法。
    Reference: [1] S.S. Agayan, Hadamard matrices and their applications, Lecture notes in mathematics, Vol. 1168, Springer-Verlag, Berlin, 1985.
    [2] R.E.L. Aldred and B.D. McKay, Graceful and harmonious labellings of trees, Bull. Inst. Combin. Appl. 23 (1998), 69-72.
    [3] R.E.L. Aldred, J. Siran and M. Siran, A note on the number of graceful labellings of paths, Discrete Math. 261 (2003), 27-30.
    [4] M.D. Atkinson and J.R. Sack, Generating binary trees at random, Inform. Process. Lett. 41 (1) (1992), 21-23.
    [5] J.C. Bermond and D. Sotteau, Graph decompositions and G-design, Proc. 5th British Combinatorics Conference, 1975, Congressus Numerantium XV (1976),53-72.
    [6] J.C. Bermond, Graceful graphs, radio antennae and
    French windmills, Graph Theory and Combinatorics, Pitman, London (1979), 18-37.
    [7] F.R. Bernhart, Catalan, Motzkin, Riordan numbers, Discrete Math. 204 (1999), 73-112.
    [8] V. Bhat-Nayak and U. Deshmukh, New families of graceful banana trees,
    Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 201-216.
    [9] C.P. Bonnington and J. Siran, Bipartite labelling of trees with maximum degree three, J. Graph Theory 31 (1999), 7-15.
    [10] L. Brankovic, A. Rosa, and J. Siran, Labelling of trees with maximum degree three and improved bound, preprint.
    [11] H.J. Broersma and C. Hoede, Another equivalent of the graceful tree
    conjecture, Ars Combinatoria 51 (1999), 183-192.
    [12] M. Burzio and G. Ferrarese, The subdivision graph of a graceful tree is a graceful tree, Discrete Math. 181 (1998), 275-281.
    [13] A. Caley, A theorem on trees, Quart. J. Pure Appl. Math. 23 (1889), 376-378.
    [The Collected Mathematical Papers of Arthur Caley, Vol. \\textrm{XIII}
    (Cambrige University Press, 1897), 26-28.]
    [14] D. Callan, A combinatorial derivation of the number of labelled forests, J. Integer
    Sequences Vol. 6 (2003), Article 03.4.7
    [15] W.-C. Chen, H.-I. Lu, and Y.-N. Yeh, Operations of interlaced trees and graceful
    trees, Southeast Asian Bulletin of Math. 21 (1997), 337-348.
    [16] Y.-M. Chen and Y.-Z. Shih, On enumeration of plane forests, preprint.
    [17] Y.-M. Chen, The Chung-Feller Theorem revisited, submitted.
    [18] Y.-M. Chen and Y.-Z. Shih, 2-Caterpillars are graceful, submitted.
    [19] K. L. Chung and W. Feller, On fluctuations in coin-tossing, Proc. Nat. Acad. Sci.
    USA 35 (1949), 605-608.
    [20] R. Craigen, Constructing Hadamard matrices with orthogonal pairs, Ars Combinatoria 33 (1992), 57-64.
    [21] R. Craigen, J. Seberry and Xian-Mo Zhang, Product of four Hadamard matrices, J. Combin. Theory Series A 59 (1992), 318-320.
    [22] W. de Launey, A product for twelve Hadamard matrices, Australasian J. of Combin. 7 (1993), 123-127.
    [23] N. Dershowitz and S. Zaks, Enumerations of ordered trees, Discrete Math. 31 (1980), 9-28.
    [24] N. Dershowitz and S. Zaks, The cycle lemma and some applications, European J. Combinatorics 11 (1990), 35-40.
    [25] E. Deutsch, Dyck path enumeration, Discrete Math. 204 (1999), 167-202.
    [26] J.H. Dinitz and D.R. Stinson, Contemporary Design Theory: A Collection of Surveys, John Wiley and Sons, Inc., 1992.
    [27] R. Donaghey and L.W. Shapiro, Motzkin numbers, J. Combin. Theory Series A 23 (1977), 291-301.
    [28] T. Doslic, Morgan trees and Dyck paths, Croatica Chemica Acta CCACAA 75
    (4) (2002), 881-889.
    [29] S.-P. Eu, On the Quadratic Algebric Generating Functions
    and Combinatorial Structrues , Ph. D. thesis, Department of Mathematics, National Taiwan Normal University, 2003.
    [30] S.-P. Eu, T.-S. Fu, and Y.-N. Yeh, Refined Chung-Feller Theorems
    for lattice paths, J. Combin. Theory Series A 112 (2005), 143-162.
    [31] S.-P. Eu, S.-C. Liu and Y.-N. Yeh, Taylor expansions for Catalan and Motzkin
    numbers, Advances in Applied Math. 29 (2002), 345-357.
    [32] S.-P. Eu, S.-C. Liu, and Y.-N. Yeh, Dyck paths with
    peaks avoiding or restricted to a given set, Studies in Applied Math. 111 (2003), 453-465.
    [33] S.-P. Eu, S.-C. Liu, and Y.-N. Yeh, Odd or even on plane trees,
    Discrete Math. 281 (2004), 189-196.
    [34] J.A. Gallian, A dynamic survey of graph labelling, Electronic J. Combinatorics 5 (2005), #DS6.
    [35] I. Gessel, Counting forests by descents and leaves, Electronic J. Combinatorics 3 (2) (1996), #R8.
    [36] S.W. Golomb, How to number a graph, in Graph Theory and Computing, R. C. Read,
    ed., Academic Press, New York (1972), 23-37.
    [37] J. Hadamard, Resolution d`une question relative aux determinants,
    Bull. des Sci. Math. 17 (1893), 240-246.
    [38] S.M. Hegde and S. Shetty, On graceful trees, Applied Mathematics
    E-Notes 2 (2002), 192-197.
    [39] P. Hrnciar and A. Haviar, All trees of diameter five are graceful, Discrete Math. 233 (2001), 133-150.
    [40] C. Huang, A. Kotzig and A. Rosa, Further results on tree labellings, {\\it Utilitas
    Math. 21c (1982), 31-48.
    [41] H. Izbicki, Uber Unterbaume eines Baumes, Monatshefte f. Math. 74
    (1970), 56-62.
    [42] K.M. Koh, D.G. Rogers, and T. Tan, On graceful
    trees, Nanta Math. 10 (1977), 27-31.
    [43] K.M. Koh, D.G. Rogers, and T. Tan, A graceful
    arboretum: A survey of graceful trees, in Proceedings of Franco-Southeast
    Asian Conference, Singapore, May 1979, 2 278-287.
    [44] J. Labelle and Y.-N. Yeh, Dyck paths of knight moves,
    Discrete Appl. Math. 24 (1989), 213-221.
    [45] J. Labelle and Y.-N. Yeh, Generalized Dyck paths, Discrete Math.
    82 (1990), 1-6.
    [46] O. Marrero, Une caracterisation des matrices de Hadamard,
    Expositiones Mathematicae 17 (1999), 283-288.
    [47] D. Mishra and P. Panigrahi, Graceful lobsters obtained by partitioning and component moving of branches
    of diameter four trees, Computers and Mathematics with Applications 50
    (2005), 367-380.
    [48] D. Morgan, Gracefully labelled trees from Skolem sequences, Congressus
    Numerantium 142 (2000), 41-48.
    [49] D. Morgan, All lobsters with perfect matchings are graceful, Electronic Notes in Discrete Math. 11 (2002), 503-508.
    [50] D. Morgan and R. Rees, Using Skolem and Hooked-Skolem sequences to generate graceful trees, J. Combin. Math. and Combin. Computing 44 (2003), 47-63.
    [51] T.V. Narayana, Cyclic permutation of lattice paths and the Chung-Feller Theorem, Skandinavisk Aktuarietidskrift (1967), 23-30.
    [52] T.V. Narayana, Lattice path combinatorics with statistical applications, Mathematical Expositions No. 23, University of Toronto Press, Toronto, 1979.
    [53] H.K. Ng, Gracefulness of a class of lobsters, Notices AMS 7 (1986), 825-05-294.
    [54] A.M. Pastel and H. Raynaud, Numerotation gracieuse des oliviers, Colloq.
    Grenoble, Publications Universite de Grenoble (1978), 218-223.
    [55] G. Ringel, Problem 25, in Theory of Graphs and its Applications, Proceedings of Symposium in Smolenice 1963}, Prague (1964), 162.
    [56] J. Riordan, Forests of labelled trees, J. Graph Theory 5 (1968), 90-103.
    [57] J. Riordan, A note on Catalan parentheses, Amer. Math. Monthly 80
    (1973), 904-906.
    [58] J. Riordan, Forests of label-increasing trees, J. Graph Theory 3
    (1979), 127-133.
    [59] F.S. Roberts, Applied Combinatorics, Prentice-Hall, Inc., Englewood Cliffs, New
    Jersey 07632, 1984.
    [60] A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (International Symposium, Rome, July 1966), Gordon and Breach, N.Y. and Dunod Paris (1967),
    349-355.
    [61] A. Rosa, Labelling snakes, Ars Combinatoria 3 (1977), 67-74.
    [62] A. Rosa and J. Siran, Bipartite labellings of trees and the gracesize,
    J. Graph Theory 19 (1995), 201-215.
    [63] S. Seo, A pairing of the vertices of ordered trees, Discrete Math., 241 (2001), 471-477.
    [64] L.W. Shapiro, A short proof of an identity of Touchard`s concerning Catalan numbers, J. Combin. Theory Series A 20 (1976), 375-376.
    [65] L.W. Shapiro, Problem 10753, Amer. Math. Monthly 106 (1999), 777.
    [66] L.W. Shapiro, The higher you go, the older it gets, Congressus Numerantium
    138 (1999), 93-96.
    [67] L.W. Shapiro, The higher you go, the older it gets, Congressus Numerantium
    138 (1999), 93-96.
    [68] L.W. Shapiro, Some open questions about random walks
    , involutions, limiting distributions, and generating functions, Advances
    in Applied Math. 27 (2001), 585-596.
    [69] Y.-Z. Shih and E.-T. Tan, On Jm-Hadamard matrices, Expositiones Mathematicae
    23 (2005), 81-88.
    [70] Y.-Z. Shih and E.-T. Tan, On Marrero`s Jm-Hadamard matrices, to appear in Taiwanese J. Math..
    [71] Y.-Z. Shih and E.-T. Tan, On Craigen-de Launey`s constructions of Hadamard matrices, preprint.
    [72] N.J. Sloane, A Library of Hadamard Matrices, \\\\http://www.research.att.com/~njas/hadamard/.
    [73] R.P. Stanley, Enumerative Combinatorics Volume 1, Cambridge University Press,
    1986.
    [74] R.P. Stanley, Enumerative Combinatorics Volume 2, Cambridge University Press,
    1999.
    [75] D. Stanton and D. White, Constructive
    Combinatorics, Springer-Veerlag New York Inc. 1986.
    [76] R. Stanton and C. Zarnke, Labelling of balanced trees, Proc. 4th Southeast Conference of Comb., Graph Theory, Computing (1973), 479-495.
    [77] J.J. Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colors, with applications to Newton`s Rule, ornamental tile-work, and the theory of numbers, Phil. Mag. 34
    (1867), 461-475.
    [78] . Takacs, Counting forests, Discrete Math. 84 (1990), 323-326.
    [79] J. Touchard, Sur certaines equations fonctionnelles, Proc. Int. Math.
    Congress, Toronto (1924), Vol. 1, p.465, (1928).
    [80] F. van Bussel, Relaxed graceful labellings of trees, Electronic J.
    Combinatorics 9 (2002), #R4.
    [81] J.H. van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, 1992.
    [82] J.-G. Wang, D.J. Jin, X.-G. Lu and D. Zhang, The gracefulness of a class
    of lobster trees, Mathematical Computer Modelling 20 (1994), 105-110.
    [83] W. Woan, Uniform partitions of lattice paths and Chung-Feller
    generalizations, Amer. Math. Monthly 108 (2001), 556-559.
    [84] D.B. West, Introduction to Graph Theory, Prentice-Hall, Inc. 1996.
    [85] W.-C. Wu, Graceful labellings of 4-caterpillars, Master Thesis,
    Department of Mathematical Sciences, National Chengchi University, 2006.
    [86] S.L. Zhao, All trees of diameter four are graceful, Annals New York Academy of Sciences (1986), 700-706.
    Description: 博士
    國立政治大學
    應用數學研究所
    89751501
    95
    Source URI: http://thesis.lib.nccu.edu.tw/record/#G0089751501
    Data Type: thesis
    Appears in Collections:[應用數學系] 學位論文

    Files in This Item:

    File Description SizeFormat
    75150101.pdf190KbAdobe PDF2711View/Open
    75150102.pdf338KbAdobe PDF2813View/Open
    75150103.pdf184KbAdobe PDF2808View/Open
    75150104.pdf84KbAdobe PDF21100View/Open
    75150105.pdf148KbAdobe PDF2792View/Open
    75150106.pdf252KbAdobe PDF2938View/Open
    75150107.pdf430KbAdobe PDF21456View/Open
    75150108.pdf438KbAdobe PDF2932View/Open
    75150109.pdf534KbAdobe PDF2930View/Open
    75150110.pdf672KbAdobe PDF2892View/Open
    75150111.pdf693KbAdobe PDF2925View/Open
    75150112.pdf1442KbAdobe PDF2991View/Open
    75150113.pdf80KbAdobe PDF2803View/Open
    75150114.pdf303KbAdobe PDF2975View/Open


    All items in 政大典藏 are protected by copyright, with all rights reserved.


    社群 sharing

    著作權政策宣告 Copyright Announcement
    1.本網站之數位內容為國立政治大學所收錄之機構典藏,無償提供學術研究與公眾教育等公益性使用,惟仍請適度,合理使用本網站之內容,以尊重著作權人之權益。商業上之利用,則請先取得著作權人之授權。
    The digital content of this website is part of National Chengchi University Institutional Repository. It provides free access to academic research and public education for non-commercial use. Please utilize it in a proper and reasonable manner and respect the rights of copyright owners. For commercial use, please obtain authorization from the copyright owner in advance.

    2.本網站之製作,已盡力防止侵害著作權人之權益,如仍發現本網站之數位內容有侵害著作權人權益情事者,請權利人通知本網站維護人員(nccur@nccu.edu.tw),維護人員將立即採取移除該數位著作等補救措施。
    NCCU Institutional Repository is made to protect the interests of copyright owners. If you believe that any material on the website infringes copyright, please contact our staff(nccur@nccu.edu.tw). We will remove the work from the repository and investigate your claim.
    DSpace Software Copyright © 2002-2004  MIT &  Hewlett-Packard  /   Enhanced by   NTU Library IR team Copyright ©   - Feedback