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 Title: Diophantine approximation and the Markoff chain Authors: 黎右強;You-Chiang Li Contributors: 數學研究所 Keywords: 馬可夫鏈;Markoff chain;Diophantine approximation Date: 2006-06-28 Issue Date: 2009-09-22 11:07:55 (UTC+8) Publisher: 國立中央大學圖書館 Abstract: 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。 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}]\$ Appears in Collections: [數學研究所] 博碩士論文

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