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題名 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
作者 施耀振
Shih,Yaio-Zhern
貢獻者 陳永秋<br>李陽明
施耀振
Shih,Yaio-Zhern
關鍵詞 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
日期 2006
上傳時間 17-Sep-2009 13:45:12 (UTC+8)
摘要 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。
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[6] J.C. Bermond, Graceful graphs, radio antennae and
French windmills, Graph Theory and Combinatorics, Pitman, London (1979), 18-37.
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Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 201-216.
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conjecture, Ars Combinatoria 51 (1999), 183-192.
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(Cambrige University Press, 1897), 26-28.]
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[16] Y.-M. Chen and Y.-Z. Shih, On enumeration of plane forests, preprint.
[17] Y.-M. Chen, The Chung-Feller Theorem revisited, submitted.
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of diameter four trees, Computers and Mathematics with Applications 50
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[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.
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描述 博士
國立政治大學
應用數學研究所
89751501
95
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0089751501
資料類型 thesis
dc.contributor.advisor 陳永秋<br>李陽明zh_TW
dc.contributor.author (Authors) 施耀振zh_TW
dc.contributor.author (Authors) Shih,Yaio-Zhernen_US
dc.creator (作者) 施耀振zh_TW
dc.creator (作者) Shih,Yaio-Zhernen_US
dc.date (日期) 2006en_US
dc.date.accessioned 17-Sep-2009 13:45:12 (UTC+8)-
dc.date.available 17-Sep-2009 13:45:12 (UTC+8)-
dc.date.issued (上傳時間) 17-Sep-2009 13:45:12 (UTC+8)-
dc.identifier (Other Identifiers) G0089751501en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/32561-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學研究所zh_TW
dc.description (描述) 89751501zh_TW
dc.description (描述) 95zh_TW
dc.description.abstract (摘要) 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。zh_TW
dc.description.tableofcontents Abstract
中文摘要
Introduction
I Constructions of Hadamard Matrices
1 On Jm-Hadamard Matrices
1.1 Jm-Hadamard Matrices
1.2 Counterexamples
2 Further Results on Jm-Hadamard Matrices
2.1 Some Properties of Jm-Hadamard Matrices
2.2 Hadamard Matrices in Jm-classes
3 On Craigen-de Launey’s Constructions of Hadamard Matrices
3.1 Generalizations of Craigen’s Theorem and Craigen-Seberry-Zhang’s Theorem
3.2 Minimum Exponent of Hadamard Matrices Resulting from t Hadamard Matrices
II Studies of Forests
4 On Enumeration of Plane Forests
4.1 A Catalan Identity
4.2 Some Results of Leaves
4.3 Generalizations of Motzkin-Catalan Identity
4.4 Some Riordan Families
5 The Chung-Feller Theorem Revisited
5.1 A Simple Proof of Chumg-Feller Theorem
5.2 Bi-color Plane Forests
5.3 Semiatandard Tableaux and Noncrossing Semiordered Pairs
5.4 Motzkin Paths with Flaws and a Labelled Minimum
6. Graceful Labellings of Some n-Caterpillars
6.1 Graceful Labellings of 2-Caterpillars
6.2 Graceful Labellings of (r,m,n)-Caterpillars
6.3 n-Partition with Parameter k
6.4 Latin Squares and Graceful Labellings of 2n-Caterpillars
Concluding Remarks
Bibliograghy		zh_TW
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dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0089751501en_US
dc.subject (關鍵詞) Kronecker乘積zh_TW
dc.subject (關鍵詞) Sylvester-Hadamard矩陣zh_TW
dc.subject (關鍵詞) J_m-Hadamard矩陣zh_TW
dc.subject (關鍵詞) 正交對zh_TW
dc.subject (關鍵詞) Weighing矩陣zh_TW
dc.subject (關鍵詞) 最小指數zh_TW
dc.subject (關鍵詞) 平面森林zh_TW
dc.subject (關鍵詞) Catalan數zh_TW
dc.subject (關鍵詞) Motzkin數zh_TW
dc.subject (關鍵詞) Riordan數zh_TW
dc.subject (關鍵詞) Narayana數zh_TW
dc.subject (關鍵詞) Dyck路徑zh_TW
dc.subject (關鍵詞) Motzkin路徑zh_TW
dc.subject (關鍵詞) Chung-Feller定理zh_TW
dc.subject (關鍵詞) 優美標法zh_TW
dc.subject (關鍵詞) 拉丁方陣zh_TW
dc.subject (關鍵詞) J_m-classesen_US
dc.subject (關鍵詞) n-Caterpillarsen_US
dc.title (題名) 兩個組合數學的主題： Hadamard 矩陣的建構及有關森林的研究zh_TW
dc.title (題名) Two Combinatorial Topics: Constructions of Hadamard Matrices and Studies of Forestsen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] S.S. Agayan, Hadamard matrices and their applications, Lecture notes in mathematics, Vol. 1168, Springer-Verlag, Berlin, 1985.zh_TW
dc.relation.reference (參考文獻) [2] R.E.L. Aldred and B.D. McKay, Graceful and harmonious labellings of trees, Bull. Inst. Combin. Appl. 23 (1998), 69-72.zh_TW
dc.relation.reference (參考文獻) [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.zh_TW
dc.relation.reference (參考文獻) [4] M.D. Atkinson and J.R. Sack, Generating binary trees at random, Inform. Process. Lett. 41 (1) (1992), 21-23.zh_TW
dc.relation.reference (參考文獻) [5] J.C. Bermond and D. Sotteau, Graph decompositions and G-design, Proc. 5th British Combinatorics Conference, 1975, Congressus Numerantium XV (1976),53-72.zh_TW
dc.relation.reference (參考文獻) [6] J.C. Bermond, Graceful graphs, radio antennae andzh_TW
dc.relation.reference (參考文獻) French windmills, Graph Theory and Combinatorics, Pitman, London (1979), 18-37.zh_TW
dc.relation.reference (參考文獻) [7] F.R. Bernhart, Catalan, Motzkin, Riordan numbers, Discrete Math. 204 (1999), 73-112.zh_TW
dc.relation.reference (參考文獻) [8] V. Bhat-Nayak and U. Deshmukh, New families of graceful banana trees,zh_TW
dc.relation.reference (參考文獻) Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 201-216.zh_TW
dc.relation.reference (參考文獻) [9] C.P. Bonnington and J. Siran, Bipartite labelling of trees with maximum degree three, J. Graph Theory 31 (1999), 7-15.zh_TW
dc.relation.reference (參考文獻) [10] L. Brankovic, A. Rosa, and J. Siran, Labelling of trees with maximum degree three and improved bound, preprint.zh_TW
dc.relation.reference (參考文獻) [11] H.J. Broersma and C. Hoede, Another equivalent of the graceful treezh_TW
dc.relation.reference (參考文獻) conjecture, Ars Combinatoria 51 (1999), 183-192.zh_TW
dc.relation.reference (參考文獻) [12] M. Burzio and G. Ferrarese, The subdivision graph of a graceful tree is a graceful tree, Discrete Math. 181 (1998), 275-281.zh_TW
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dc.relation.reference (參考文獻) [The Collected Mathematical Papers of Arthur Caley, Vol. \\textrm{XIII}zh_TW
dc.relation.reference (參考文獻) (Cambrige University Press, 1897), 26-28.]zh_TW
dc.relation.reference (參考文獻) [14] D. Callan, A combinatorial derivation of the number of labelled forests, J. Integerzh_TW
dc.relation.reference (參考文獻) Sequences Vol. 6 (2003), Article 03.4.7zh_TW
dc.relation.reference (參考文獻) [15] W.-C. Chen, H.-I. Lu, and Y.-N. Yeh, Operations of interlaced trees and gracefulzh_TW
dc.relation.reference (參考文獻) trees, Southeast Asian Bulletin of Math. 21 (1997), 337-348.zh_TW
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