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| Title: | 迪菲七邊形
DIFFY HEPTAGON |
| Authors: | 林亨峰 |
| Contributors: | 李陽明
林亨峰 |
| Keywords: | 迪菲七邊形
強數學歸納法
Diffy Heptagon
Strong Induction |
| Date: | 2013 |
| Issue Date: | 2013-12-02 17:47:25 (UTC+8) |
| Abstract: | 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. |
| Reference: | [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) |
| Description: | 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 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G1009720111 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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Size | Format | |
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| 011101.pdf | 1189Kb | Adobe PDF2 | 503 | View/Open |
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