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標題: | 隨機多項式的一個普遍性 A Universality of Polynomials with Complex Random Roots |
作者: | I-Shing Hu 胡亦行 |
指導教授: | 張志中 |
關鍵字: | 隨機多項式,隨機矩陣,普遍性, random polynomials,random matrices,universality, |
出版年 : | 2017 |
學位: | 碩士 |
摘要: | Let $ p_n(x) $ be a random polynomial of degree $n$ and
${Z^{(n)}_j}_{j=1}^n$ and ${X^{n, k}_j}_{j=1}^{n-k}, k<n$, be the zeros of $p_n$ and $p_n^{(k)}$, the $k$th derivative of $p_n$, respectively. We show that if the linear statistics $displaystyle{ %L_n (f) &=& frac {1}{a_n} left[ fleft( frac {Z^{(n)}_1}{b_n} ight) + cdots + f left(frac {Z^{(n)}_n}{b_n} ight) ight]}$ associated with ${Z^{(n)}_j}$ has a limit as $n oinfty$ at some mode of convergence, the linear statistics associated with ${X^{n, k}_j}$ converges to the same limit at the same mode. Similar statement also holds for the centered linear statistics associated with the zeros of $p_n$ and $p_n^{(k)}$, provided the zeros ${Z^{(n)}_j}$ and the sequences ${a_n}$ and ${b_n}$ of positive numbers satisfy some mild conditions. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/2584 |
DOI: | 10.6342/NTU201700594 |
全文授權: | 同意授權(全球公開) |
顯示於系所單位: | 數學系 |
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