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The Many Faces of Random Characteristic Polynomials

随机特征多项式的多个面

基本信息

批准号：
EP/N009436/1
负责人：
Yan Fyodorov
金额：
$ 60.89万
依托单位：
King's College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN009436%2F1
关键词：
Many Faces Random Characteristic Polynomials

项目摘要

When a person hits, strikes, plucks or otherwise disturbs a musical instrument it is set into vibrational motion and produces a sound. The motion can occur only in one of the patterns called "natural modes" by which that particular instrument could vibrate. How fast the instrument vibrates in each of the natural modes is characterized by the "frequency" of that mode,so that the higher is the frequency the higher is pitch of the sound we hear. Apart from the frequency each mode is characterized by its "quality factor" which is a measure of duration of sound until it dies out beyond the audibility level. Musical instruments is one of the basic examples of devices called resonators, but resonance phenomena are wide-spread in Nature and in fact underpin a lot of modern Science and Technology. For example, in much the same way as musical instruments, complex atomic nuclei (as for example uranium, plutonium or thorium) "vibrate" in one of their natural modes when hit in experiments by a fast-moving particle used to probe inner structure of the nuclei. The set of all natural frequencies of any physical object is called the "spectrum". Spectra are important characteristics of anyobject and can reveal a lot about its structure and properties. Spectra of musical instruments are very ordered (or "harmonic") to producepleasing sounds, whereas experiments show that spectra of complex atomic nuclei look disordered, or random, which reflects a very complicated,chaotic and unpredictable character of motion of nuclei constituents. But behind all that "spectral disorder" lurks a certain pattern which representsa new "harmony" in the Nature. For example, the same pattern reveals itself in positions of birds perching on a roof or in distances between closelyparked cars. Or indeed in the positions of zeroes of a function introduced by German mathematician Bernhard Riemann to study properties of oneof the main building blocks of Mathematics - the prime numbers. To be able to characterize such ubiquitouspatterns quantitatively mathematicians use as a model idealized objects called random matrices. Spectral properties of such matrices are encapsulated in a function called characteristic polynomial of the matrix. When plotted graphs of such functions show huge random variations by the orders of magnitude reflecting random nature of the associated spectra. To characterize such variations statistically in a rigorous mathematical way is a challenging mathematical problem. The present project aims to address various facets of statistical behaviour of random characteristic polynomials. As almost invariably happens in Mathematics, tools introduced to understand a certain phenomenon help to reveal many other hidden structures in problems not obviously related to the original one. The theory of random matrices is rich with such unexpected connections. In particular, the present research aims to employ random characteristic polynomials for analysing behaviour of systems of differential equations used describe, among other things, ecological communities of many interacting species and possibly dynamics of networks of interacting neurons. Related characteristic polynomialsmay be helpful for understanding properties of spectra of complex atomic nuclei and similar systems, in particular the associated "quality factors" which are known in that context as "widths of resonances". All these questions are to be addressed in the course of the proposed research.

当一个人敲击、敲击、弹拨或以其他方式干扰乐器时，乐器就会产生振动并发出声音。这种运动只能发生在一种称为“自然模式”的模式中，通过这种模式，特定的乐器可以振动。乐器在每一种自然模式下振动的速度由该模式的“频率”来表征，因此频率越高，我们听到的声音的音高就越高。除了频率之外，每种模式的特征在于其“品质因数”，品质因数是声音持续时间的量度，直到声音消失在可听度水平之外。乐器是被称为谐振器的设备的基本示例之一，但谐振现象在自然界中广泛传播，实际上支撑了许多现代科学和技术。例如，复杂的原子核（如铀、钚或钍）在实验中被用来探测原子核内部结构的快速移动粒子击中时，会以其自然模式之一“振动”，这与乐器的方式大致相同。任何物理物体的所有自然频率的集合称为“频谱”。光谱是物体的重要特征，它可以揭示物体的结构和性质.乐器的光谱是非常有序的（或“和谐”），以产生悦耳的声音，而实验表明，复杂的原子核的光谱看起来是无序的，或随机的，这反映了原子核成分运动的非常复杂，混乱和不可预测的特性。但在所有这些“光谱紊乱”的背后，隐藏着一种特定的模式，它代表着自然界中新的“和谐”。例如，在屋顶上栖息的鸟的位置或紧密停放的汽车之间的距离上，也显示出同样的模式。或者实际上是在德国数学家伯恩哈德·黎曼（Bernhard Riemann）介绍的一个函数的零的位置上，以研究数学的主要组成部分之一-素数的性质。为了能够定量地描述这种普遍存在的现象，数学家们使用一种称为随机矩阵的理想化对象作为模型。此类矩阵的谱特性被封装在称为矩阵特征多项式的函数中。当绘制这些函数的图形时，显示出数量级的巨大随机变化，反映了相关光谱的随机性质。用严格的数学方法统计地描述这种变化是一个具有挑战性的数学问题。本项目旨在解决随机特征多项式统计行为的各个方面。正如数学中几乎总是发生的那样，为了理解某种现象而引入的工具有助于揭示与原始问题没有明显关系的问题中的许多其他隐藏结构。随机矩阵理论中充满了这种意想不到的联系。特别是，本研究的目的是采用随机特征多项式分析行为的微分方程系统描述，除其他事项外，生态社区的许多相互作用的物种和可能的动态网络的相互作用的神经元。相关的特征多项式可能有助于理解复杂原子核和类似系统的光谱特性，特别是相关的“质量因子”，在该上下文中称为“共振宽度”。所有这些问题都将在拟议的研究过程中得到解决。