拉伸-褶皱-剪切图中发电机生长的计算机辅助证明

Pub Date : 2022-12-17 DOI:10.1080/14689367.2022.2139224
Farhana Akond Pramy, Ben Mestel, Robert Hasson, Katrine Rogers
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引用次数: 0

摘要

拉伸-折叠-剪切(SFS)算子是一种函数线性算子,作用于包含在中的某个域上的实变量x的复值函数。它源于运动学发电机理论中的一个程式化模型,其中磁场增长对应于大于1的模量本征值。当剪切参数α为零时,可以精确地确定的谱,并且与非零本征值相对应的本征函数与伯努利多项式有关。的频谱尚未严格确定,尽管该频谱已在数值上近似。在本文中,提出了一个计算机辅助证明,为的前导特征值提供了严格的边界,特别表明对于所有α满足,其特征值的模大于1,从而部分证实了关于SFS算子的一个突出猜想。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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A computer-assisted proof of dynamo growth in the stretch-fold-shear map
The Stretch-Fold-Shear (SFS) operator is a functional linear operator acting on complex-valued functions of a real variable x on some domain containing in It arises from a stylized model in kinematic dynamo theory where magnetic field growth corresponds to an eigenvalue of modulus greater than 1. When the shear parameter α is zero, the spectrum of can be determined exactly, and the eigenfunctions corresponding to non-zero eigenvalues are related to the Bernoulli polynomials. The spectrum for has not been rigorously determined although the spectrum has been approximated numerically. In this paper, a computer-assisted proof is presented to provide rigorous bounds on the leading eigenvalue for , showing inter alia that has an eigenvalue of modulus greater than 1 for all α satisfying , thereby partially confirming an outstanding conjecture on the SFS operator.
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