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題名 迪菲七邊形
DIFFY HEPTAGON
作者 林亨峰
貢獻者 李陽明
林亨峰
關鍵詞 迪菲七邊形
強數學歸納法
Diffy Heptagon
Strong Induction
日期 2013
上傳時間 2-Dec-2013 17:47:25 (UTC+8)
摘要 在迪菲方塊(Diffy Box)中，所輸入的數字經有限次運算後，皆會收斂到0，而本論文主要是將迪菲方塊的正方形推廣到正七邊形，我們將其定義為迪菲七邊形(Diffy Heptagon)，討論其經運算後是否亦會收斂到0或存在其他的收斂類型？
本論文主要是利用強數學歸納法(Strong Induction)證明，得到迪菲七邊形經過有限次運算後，會收斂到三種收斂類型。接下來再將非負整數型的迪菲七邊形，推廣到整數型及有理數型，而得到無論是非負整數型、整數型或有理數型的迪菲七邊形，經運算後皆會得到同構(isomorphic)或相似的(similar)收斂類型。
在論文最後，提出一些尚待解決的問題及建議，例如：輸入何種型態的數字，會分別產生第Ⅰ或Ⅱ或Ⅲ型的收斂類型？又收斂類型中除了0以外的數字n，與原輸入數字間存在何種關聯？這都值得後續再做更深入的研究。
In a Diffy Box, after limited steps of calculations, the result converges to all zeros. This essay is commissioned to expand Diffy Box’s square to heptagon, which we call “Diffy Heptagon”. We will discuss if Diffy Heptagon shows the same convergence, or the existence of other result?
This article is proved with Strong Induction. The result shows that a Diffy Heptagon will present three types of convergence after limited operations. Moreover, we extend the nonnegative integers to integers and rational numbers. The conclusion is, regardless of the numbers, what we obtain is isomorphic or similar convergence.
Finally, we propose some issues and suggestions. For examples, what type of numbers we label individually will cause class I, class II or class III convergence? In addition to the zero convergence, what is the connection between the labeled numbers and other convergence types? All those questions are worth studying Diffy Heptagon even further.
參考文獻 [1] A. Behn, C. Kribs-Zaleta, and V. Ponomarenko, The Convergence of Difference Boxes. American Math. Monthly, volume 112, 426-438, (1995)
[2] J. Creely, The Length of a Three-Number Game. Fibonacci Quarterly, volume 26, no2, 141-143, (1988)
[3] A. Ehrlich, Periods in Ducci`s n-Number Game of Differences. Fibonacci Quarterly, volume 28, 302-305, (1990)
[4] A. Ludington-Young, Ducci-Processes of 5-Tuples. Fibonacci Quarterly, volume 36, no5, 419-434, (1998)
[5] A. Ludington-Young, Length of the 7-Number Game. Fibonacci Quarterly, volume 26, no3, 195-204, (1988)
[6] A. Ludington-Young, Length of the n-Number Game. Fibonacci Quarterly, volume 28, no3, 259-265, (1990)
[7] G. Schöffl, Ducci-Processes of 4-Tuples. Fibonacci Quarterly, volume 35, no3, 269-276, (1997)
[8] F.B. Wong, Ducci Processes. Fibonacci Quarterly, volume 20, no2, 97-105, (1982)
[9] 黃信弼, Diffy Pentagon. National Chengchi University, (2012)
[10]蔡秀芬, Diffy Box(迪菲方塊). National Chengchi University, (2008)
描述 碩士
國立政治大學
應用數學系數學教學碩士在職專班
100972011
102
資料來源 http://thesis.lib.nccu.edu.tw/record/#G1009720111
資料類型 thesis
dc.contributor.advisor 李陽明zh_TW
dc.contributor.author (Authors) 林亨峰zh_TW
dc.creator (作者) 林亨峰zh_TW
dc.date (日期) 2013en_US
dc.date.accessioned 2-Dec-2013 17:47:25 (UTC+8)-
dc.date.available 2-Dec-2013 17:47:25 (UTC+8)-
dc.date.issued (上傳時間) 2-Dec-2013 17:47:25 (UTC+8)-
dc.identifier (Other Identifiers) G1009720111en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/61995-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學系數學教學碩士在職專班zh_TW
dc.description (描述) 100972011zh_TW
dc.description (描述) 102zh_TW
dc.description.abstract (摘要) 在迪菲方塊(Diffy Box)中，所輸入的數字經有限次運算後，皆會收斂到0，而本論文主要是將迪菲方塊的正方形推廣到正七邊形，我們將其定義為迪菲七邊形(Diffy Heptagon)，討論其經運算後是否亦會收斂到0或存在其他的收斂類型？
本論文主要是利用強數學歸納法(Strong Induction)證明，得到迪菲七邊形經過有限次運算後，會收斂到三種收斂類型。接下來再將非負整數型的迪菲七邊形，推廣到整數型及有理數型，而得到無論是非負整數型、整數型或有理數型的迪菲七邊形，經運算後皆會得到同構(isomorphic)或相似的(similar)收斂類型。
在論文最後，提出一些尚待解決的問題及建議，例如：輸入何種型態的數字，會分別產生第Ⅰ或Ⅱ或Ⅲ型的收斂類型？又收斂類型中除了0以外的數字n，與原輸入數字間存在何種關聯？這都值得後續再做更深入的研究。		zh_TW
dc.description.abstract (摘要) In a Diffy Box, after limited steps of calculations, the result converges to all zeros. This essay is commissioned to expand Diffy Box’s square to heptagon, which we call “Diffy Heptagon”. We will discuss if Diffy Heptagon shows the same convergence, or the existence of other result?
This article is proved with Strong Induction. The result shows that a Diffy Heptagon will present three types of convergence after limited operations. Moreover, we extend the nonnegative integers to integers and rational numbers. The conclusion is, regardless of the numbers, what we obtain is isomorphic or similar convergence.
Finally, we propose some issues and suggestions. For examples, what type of numbers we label individually will cause class I, class II or class III convergence? In addition to the zero convergence, what is the connection between the labeled numbers and other convergence types? All those questions are worth studying Diffy Heptagon even further.		en_US
dc.description.tableofcontents 論文摘要 i
Abstract ii
第一章 前言 1
第一節 研究動機與目的 1
第二節 研究主題-迪菲七邊形(Diffy Heptagon) 1
第二章 理論基礎與基本性質 3
第一節 迪菲七邊形(Diffy Heptagon)的同構(isomorphism)與相似(similarity) 3
第二節 迪菲七邊形(Diffy Heptagon)的循環收斂(cyclic convergence) 6
第三章 研究方法 9
第一節 利用強數學歸納法(Strong Induction)證明 9
第二節 整數型與有理數型的迪菲七邊形(Diffy Heptagon) 37
第四章 結論與未來展望 39
第一節 結論 39
第二節 未來展望 39
參考文獻 42		zh_TW
dc.format.extent 1218337 bytes-
dc.format.mimetype application/pdf-
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G1009720111en_US
dc.subject (關鍵詞) 迪菲七邊形zh_TW
dc.subject (關鍵詞) 強數學歸納法zh_TW
dc.subject (關鍵詞) Diffy Heptagonen_US
dc.subject (關鍵詞) Strong Inductionen_US
dc.title (題名) 迪菲七邊形zh_TW
dc.title (題名) DIFFY HEPTAGONen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] A. Behn, C. Kribs-Zaleta, and V. Ponomarenko, The Convergence of Difference Boxes. American Math. Monthly, volume 112, 426-438, (1995)
[2] J. Creely, The Length of a Three-Number Game. Fibonacci Quarterly, volume 26, no2, 141-143, (1988)
[3] A. Ehrlich, Periods in Ducci`s n-Number Game of Differences. Fibonacci Quarterly, volume 28, 302-305, (1990)
[4] A. Ludington-Young, Ducci-Processes of 5-Tuples. Fibonacci Quarterly, volume 36, no5, 419-434, (1998)
[5] A. Ludington-Young, Length of the 7-Number Game. Fibonacci Quarterly, volume 26, no3, 195-204, (1988)
[6] A. Ludington-Young, Length of the n-Number Game. Fibonacci Quarterly, volume 28, no3, 259-265, (1990)
[7] G. Schöffl, Ducci-Processes of 4-Tuples. Fibonacci Quarterly, volume 35, no3, 269-276, (1997)
[8] F.B. Wong, Ducci Processes. Fibonacci Quarterly, volume 20, no2, 97-105, (1982)
[9] 黃信弼, Diffy Pentagon. National Chengchi University, (2012)
[10]蔡秀芬, Diffy Box(迪菲方塊). National Chengchi University, (2008)		zh_TW