Research on the spectrum of Perron-Frobenius operator and pseudo random number associated with higher dimensional dynamical system

高维动力系统Perron-Frobenius算子谱及伪随机数研究

• 批准号: 20540139
• 负责人: MORI Makoto
• 金额: $ 2.58万
• 依托单位: Nihon University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2008
• 资助国家: 日本
• 起止时间: 2008 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-20540139/
• 关键词: 確率論; エルゴード理論; 疑似乱数; エルゴート; 議事乱数; スペクトル; 力学系; ペロンフロベニウス作用素; エルゴード; スペクトル半径; 混合生; 力学的ζ関数; pseudo tandom number; ergodic theory; Perron-Frobenius operator; low discrepancy sequence

In 1-dimensional cases, the essential spectrum radius of thePerron-Frobenius operator takes its minimum value, and it is easy to calculate it.However, in higher dimensional cases, it is important to consider the cases when its unessential spectrum radius takes its minimum value, and it is not easy to construct dynamical systems for which the essential spectrum radius attains its minimum value. We construct such a dynamical system which has no eigenvalue except 1 by using algebraic method. From this dynamical system, we construct low discrepancy sequences in any dimension. This makes a new progression to computer science such as simulation and numerical integral in stochastic integrals. On the other hand, this research suggests the difficulty of constructing dynamical system which has minimal essential spectrum radius, and the difficulty of constructing “good” random numbers in higher dimensional cases.

Perron-Frobenius算子的本质谱半径在1维情况下取最小值，计算比较容易，但在高维情况下，考虑非本质谱半径取最小值的情况是很重要的，要构造本质谱半径取最小值的动力系统是不容易的.利用代数方法构造了这样一个除1外没有特征值的动力系统。从这个动力系统，我们构造低差异序列在任何维度。这为计算机科学如随机积分中的模拟和数值积分的研究带来了新的进展。另一方面，本研究也揭示了构造具有最小本质谱半径的动力系统的困难，以及在高维情形下构造“好”随机数的困难。

项目成果

Low‐Discrepancy Sequences

• DOI: 10.1002/9781118593264.ch9
• 发表时间: 2014-06
• 作者: P. Brandimarte

A strategy of constructing d-dimensional low discrepancy sequences

d维低差异序列的构造策略

• 发表时间: 2009
• 作者: Toshimichi Usuba;森真;林晋;Makoto Mori
• 通讯作者: Makoto Mori

Dynamical system generated by algebraic method and low discrepancy sequences

由代数方法和低差异序列生成的动力系统

• 发表时间: 2012
• 期刊: Monte Carlo Methods and Applications
• 影响因子: 0.9
• 作者: T.Nishida;Y.Teramoto;Masaki Mori and Makoto Mori
• 通讯作者: Masaki Mori and Makoto Mori

高次元の疑似乱数の新しい構成(with森真樹)

高维伪随机数的新配置（与 Maki Mori）

• 发表时间: 2011
• 作者: Yasushi Hirata;Nobuyuki Kemoto;森真
• 通讯作者: 森真

高次元の疑似乱数の新しい構成

高维伪随机数的新构造

• 发表时间: 2011
• 作者: 森真;森真樹
• 通讯作者: 森真樹