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| 研究生： |
詹佑民 |
|---|---|
| 論文名稱： |
熱帶幾何之圓錐與凸集的生成元素探討 The Generators Of Cone And Convex Set In Tropical Geometry |
| 指導教授： | 蔡炎龍 |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系數學教學碩士在職專班 |
| 論文出版年： | 2010 |
| 畢業學年度： | 98 |
| 語文別： | 中文 |
| 論文頁數： | 32 |
| 中文關鍵詞： | 熱帶幾何 、圓錐 、生成元素 |
| 相關次數： | 點閱：103 下載：26 |
| 分享至: |
| 查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報 |
此篇論文我們主要是探討熱帶幾何下，凸集(convex set)以及圓錐(cone)的生成元素(generator)個數。在第二章中我們對一些基本環境及運算工具做介紹，例如：熱帶半環(tropical semiring)為度量空間(metric space)、極限值的運算性質等等，在第三章中我們探討回收錐(recession cone)、凸集及圓錐的性質，其中包含三者之間的關係，而在第四章中我們探討閉圓錐(closed cone)、緊緻凸集(compact convex set)、閉凸集(closed convex set)三者的生成元素個數，並以實例說明此性質，最後我們將推論出一個方法來找出在二維的熱帶空間底下的有限生成圓錐之生成元素。
In this thesis, I will discuss the generators of cone and convex set in tropical geometry. In Chapter 2, basic environment in tropical geometry and arithmetic tools are introduced here, such as how to find the limit value in tropical geometry or deciding if tropical semiring is a metric space, etc. In Chapter 3, I explore the properties of cone, convex and recession cone, inclusive of the relations of one another. In Chapter 4, the generators of a closed cone, a compact convex set, a closed convex set are provided with illustrations to present the properties. It will finally lead to a method to find the generators of the finitely generated cone in two dimesion space.
1 緒論 1
2 基本介紹 3
2.1 基本運算...............................3
2.2 度量空間...............................5
3 凸集與圓錐 10
3.1 凸集、圓錐與回收錐....................10
3.2 極點與極值產生元......................14
4 生成元素 22
5 結論 26
[1]Andree Gathmann. Tropical algebraic geometry. Jahresber. Deutsch. Math.-verein.,108(1):3–32,2006.
[2]G. L. Litvinov.The maslov dequantization,idempotent and tropical mathematics, Journal of mathematical sciences,140:426–444,2007.
[3]Shuhei Yoshitomi. Generators of modules in tropical geometry. Preprint at arXiv:math.AG/1001.0448v2,2010.
[4]Marianne Akian, Stephane Gaubert, and Alexander Guterman. Linear independence over tropical semiring and beyond. Preprint at arXiv: math.AC/0812.3496v1,2008.
[5]Jerrold E. Marsden, Michael J. Hoffman. 1993. Elementary Classical Analysis(second). New York. W. H. Freeman and Commpany.
[6] S. Gaubert and R. D. Katz. The Minkowski theorem for max-plus convex sets. Linear Algebra and Appl.,421(2-3):356–369, 2007.
[7]Christian Haase, Gregg Musiker,and Josephine Yu. Linear systems on tropical curves. Preprint at arXiv:math.AG/0909.3685v1,2009.
[8]Xavier Allamigeon, Stephane Gaubert, and Ricardo D. Katz. The number of extreme points of tropical polyhedra. Journal of Combinatorial Theory, series A. To appear.
[9] Xavier Allamigeon, Stephane Gaubert, and Ricardo D. Katz. Tropical polar cones, hypergraph transversals, and mean payoff games. CoRR. abs/1004.2778. 2010
[10] Xavier Allamigeon, Stephane Gaubert, and Eric Goubault. The tropical double description method. CoRR. abs/1001.4119. 2010
