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| 研究生： |
王佑欣 Yuhsin Wang |
|---|---|
| 論文名稱： |
Combinatorial Argument of Partition with Point, Line, and Space 點線面與空間分割的組合論證法 |
| 指導教授： | 張宜武 |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2006 |
| 畢業學年度： | 91 |
| 語文別： | 英文 |
| 論文頁數： | 34 |
| 外文關鍵詞： | Difference Equation, Euler's Formula, Standard Partition System of n-Dimensional, Partitioner, n-dimensional space, Combinatorial Argument, Bounded Region, Unbounded Region |
| 相關次數： | 點閱：150 下載：70 |
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在這篇論文裡，我們將要討論一類古典的問題，這類問題已經經由許多方法解決，例如：遞迴關係式、差分方程式、尤拉公式等等。接著我們歸納低維度的特性，並藉由定義出一組方程式-標準n維空間分割系統-來推廣這些特性到一般的$n$維度空間中。然後我們利用演算法來提供一個更直接的組合論證法。最後，我們會把問題再細分成有界區域與無界區域的個數。
In this article, we will discuss a class of classical questions had been solved by Recurrence Relation, Difference Equation, and Euler's Formula, etc.. And then, we construct a system of equations -Standard Partition System of n-Dimensional Space- to generalize the properties of maximizing the number of regions made up by k partitioner in an n-dimensional space and look into the construction of each dimension. Also, we provide a more directly Combinatorial Argument by Algorithm for this kind of question. At last, we focus on the number of bounded regions and unbounded regions in sense of maximizing the number of regions.
Abstract i
中文摘要 ii
1 Introduction 1
1.1 Introduction............................................1
1.2 Description of Three Original Questions.................2
2 Solved By Recurrence Relation 3
2.1 Solution by Recurrence Relation for Question 1..........3
2.2 Solution by Recurrence Relation for Question 2..........4
2.3 Solution by Recurrence Relation for Question 3..........6
3 General Question of Higher Dimensional Spaces 8
3.1 Generalizing These Three Classical Questions............8
3.2 The Properties of Point, Line, and 3-D Space............9
3.3 The Properties of General Question and Standard Partition
System of n-Dimensional Space..........................10
3.4 Proof of the Properties................................11
3.5 Solution by Recurrence Relation for General Question...15
4 Solved By Combinatorial Argument 17
4.1 Non-isomorphic of k-Max-Line-Drawing and k-Max-Plane-
Drawing................................................17
4.2 Combinatorial Argument with Algorithm..................19
4.3 Combinatorial Argument for Higher Dimensional Space with
Algorithm..............................................21
4.4 Presentation of Partitions in the Lower Dimensional
Space..................................................22
4.5 A List of All Numbers..................................25
5 Number of Bounded Regions 26
5.1 Number of Bounded Regions in Sense of k-max-point-
drawing................................................26
5.2 Number of Bounded Regions in Sense of k-max-line-
drawing................................................27
5.3 Number of Bounded Regions in Sense of k-max-plane-
drawing................................................28
5.4 Number of Bounded Regions of Higher Dimensional Space..28
6 Number of Unbounded Regions 30
6.1 Number of Unbounded Regions in Sense of k-max-point-
drawing................................................30
6.2 Number of Unbounded Regions in Sense of k-max-line-
drawing................................................31
6.3 Number of Unbounded Regions in Sense of k-max-plane-
drawing................................................31
6.4 Number of Unbounded Regions of Higher Dimensional
Space..................................................32
References 34
[1] Alan Tucker, Applied Combinatorics, 3rd ed., John Wiley &
Sons, New York, 1995, 281-282, 305.
[2] 簡蒼調, 續談觀察歸納法價值, 數學傳播, 第2卷第1期, 頁33-37.
[3] 何景國, 差分法及其在組合學上的應用, 數學傳播, 第10卷第1期,
頁49-51.
[4] 宋秉信, 從尤拉公式到空間的平面分割, 數學傳播, 第22卷第3期,
頁54-60.
[5] Alan Tucker, Applied Combinatorics, 3rd ed., John Wiley &
Sons, New York, 1995, 216-218.
[6] Stephen H. Friedberg, Arnold J. Insel, and Lawrence E.
Spence, Linear Algebra, 3rd ed., Prentice-Hall, 1997, 47-48.