姓名 |
黎右強(You-Chiang Li)
查詢紙本館藏 |
畢業系所 |
數學系 |
論文名稱 |
(Diophantine approximation and the Markoff chain)
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相關論文 | |
檔案 |
[Endnote RIS 格式]
[Bibtex 格式]
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摘要(中) |
對於實數$xi$我們定義$||xi||$為最接近$xi$整數。我的論文主要是探討$V={liminf_{qin mathbb{N}}q|q xi |:xi in mathbb{R} setminusmathbb{Q}}.$ 這個集合。此篇論文裡面有三個重要定理,分別是Dirichlet、Hurwitz和Markoff的定理。由Dirichlet的定理我們可證得 $Vsubset[0,1]$。而由Hurwitz的定理,我們更進一步推得 $Vsubset[0,1/sqrt{5}]$,並且$1/sqrt{5}$ 將不能再更小。Markoff的定理則是一個重要的結果,他清楚的說明了集合$V$在 $(1/3, 1/sqrt{5}]$ 這個區間上分布的情形。 |
摘要(英) |
For raal $xi$, we define $||xi||$ be the nearest integer. We are interested in the set $V={liminf_{qin mathbb{N}}q|q xi |:xi in mathbb{R} setminusmathbb{Q}}.$ . Our main theorems are the Dirichlet’’s theorem, the Hurwitz’’s theorem and the Markoff’’s theorem. From Dirichlet’s theorem, we may prove that $Vsubset[0,1]$. And from Hurwitz’s theorem, we may obtain that $Vsubset[0,1/sqrt{5}]$ and $1/sqrt{5}$ cannot be improved. Markoff’’s theorem is an important result. He explained how $V$ distributes over the interval $(1/3, 1/sqrt{5}]$ |
關鍵字(中) |
★ 馬可夫鏈 |
關鍵字(英) |
★ Markoff chain ★ Diophantine approximation |
論文目次 |
0.Introduction 1
1.The Theorem of Dirichlet 2
2.Continued Fractions 2
3.Pell’s Equation 16
4.Limit Inferior of q||qξ|| 20
5.The Markoff chain 25
6.The Markoff equation 40
Reference 49 |
參考文獻 |
[1] Cassels, J. W. S. An Introduction to Diophantine Approximation, London
University Press, 1957.
[2] Davenport, Harold. The Higher Arithmetic-an introduction to the theory of
numbers, Cambridge University Press, 1982.
[3] Ireland, Kenneth F. and Rosen, Michael. extit{A Classical Introduction
to Modern Number Theory, 2nd edition}, New York Springer-Verlag, 1982.
[4] Niven, Ivan and Zuckerman, Herbert S. An Introduction to the Theory of
Numbers, New York Wiley, 1980.
[5] Nathanson, Melvyn B. Approximation by contunued fractions, American
Mathematical Society, Vol.45, No.3, 1974, pp.323-324.
[6] Silverman, Joseph H. The Markoff equation X^ 2+Y^ 2+Z^ 2=aXYZ over
quadratic imaginary fields, J. Number Theory 35 (1990), no. 1,pp.72-104. |
指導教授 |
夏良忠(Liang-Chung Hsia)
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審核日期 |
2006-7-2 |
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