权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Excellence in Research: Numerical Analysis of Quasiperiodic Topology

卓越研究：准周期拓扑的数值分析

基本信息

批准号：
1832126
负责人：
Roberto De Leo
金额：
$ 24.99万
依托单位：
Howard University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1832126&HistoricalAwards=false
关键词：
Excellence Research Numerical Analysis Quasiperiodic

项目摘要

Hamiltonian dynamical systems are of paramount importance in both Mathematics and Physics. These systems are defined by an energy function, called Hamiltonian, and often admit first integrals, namely other functions, independent on the Hamiltonian but that also, just like the Hamiltonian, do not change their value over the solutions of the equations of motion. Since this formalism was introduced by Sir W.R. Hamilton almost two centuries ago, it grew to embrace the physics of most non-dissipative phenomena, as well as several important branches of geometry that arose naturally out of it, and it became one of the main building block of quantum mechanics. Nevertheless, until relatively recently, an important class of Hamiltonian systems was overlooked, namely the case when some first integral is multivalued (an example of multivalued function is the angle coordinate on a circle). Such systems arise naturally from quantum mechanics (in the so-called semiclassical approximation), in particular in the theory of conductivity of metals at low temperature under a strong magnetic field. Experimental data on this phenomenon were collected for many metals starting from seventy years ago but they could not be checked against the theory exactly for the lack of a theory of Hamiltonian systems with multivalued Hamiltonians. Since the Eighties, Fields medalist S.P. Novikov and his school started filling this gap and about ten years ago the experimental data were successfully verified in the simplest cases, namely for Au and Ag. At the same time, a new field of topology, called Quasiperiodic topology, arose as the generalization and formalization of this Hamiltonian system and important theoretical results have been found after the numerical study of the simplest cases. This project aims, on the one hand, at completing the verification of these fundamental experimental data for the several metals still unchecked and, on the other hand, at extending and deepening the numerical study of quasiperiodic topology. The main goal of this project is to continue and deepen the numerical study of Quasiperiodic Topology focusing, in particular, on the topology of level sets of multivalued maps on the n-dimensional tori. Its main subgoal is the contextual analytical study of the properties suggested by the numerical experiments. This will be achieved through the following specific objectives and methods: (1) development and implementation in Perl/Python/C++ of old and new algorithms for the numerical and analytical study of multivalued maps on n-tori; (2) numerical exploration and classification of these maps; (3) analytical study of minor and major properties of such maps. As an application of these results, a further important subgoal of the project is the application of the code generated in (1) to verify from first principles for the first time some important experimental data about conductivity in metals measured in Fifties and Sixties and whose mathematical description involves quasiperiodic functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

哈密顿动力系统在数学和物理中都具有极其重要的意义。这些系统由称为汉密尔顿函数的能量函数定义，并且通常允许第一积分，即独立于汉密尔顿函数的其他函数，但也像汉密尔顿函数一样，不会改变它们在运动方程解上的值。由于这种形式主义是由W. R.汉密尔顿在大约两个世纪前，它逐渐发展到包含大多数非耗散现象的物理学，以及自然产生的几个重要的几何分支，它成为量子力学的主要基石之一。然而，直到最近，一类重要的哈密顿系统被忽略了，即某些第一积分是多值的情况（多值函数的一个例子是圆上的角坐标）。这样的系统自然产生于量子力学（在所谓的半经典近似中），特别是在强磁场下低温下金属导电性的理论中。从70年前开始收集了许多金属的实验数据，但由于缺乏具有多值哈密顿量的哈密顿系统理论，因此无法根据理论对这些数据进行检查。自80年代以来，菲尔兹奖获得者S. P.诺维科夫和他的学校开始填补这一空白，大约十年前，实验数据在最简单的情况下得到了成功验证，即Au和Ag。与此同时，一个新的拓扑学领域，称为准周期拓扑，出现作为推广和形式化的这个哈密顿系统和重要的理论结果已经发现后，最简单的情况下的数值研究。该项目的目的是，一方面，完成这些基本的实验数据的验证，为几种金属仍然未经检查，另一方面，在扩展和深化的准周期拓扑的数值研究。该项目的主要目标是继续和深化准周期拓扑的数值研究，特别是在n维环面上的多值映射的水平集的拓扑。它的主要子目标是上下文分析研究的数值实验所建议的属性。这将通过以下具体目标和方法来实现：（1）在Perl/Python/C++中开发和实现用于n-tori上多值映射的数值和分析研究的新旧算法;（2）这些映射的数值探索和分类;（3）此类映射的次要和主要属性的分析研究。作为这些结果的应用，该项目的另一个重要子目标是应用（1）中生成的代码首次从第一性原理验证了在50年代和60年代测量的关于金属导电性的一些重要实验数据，这些数据的数学描述涉及准周期函数。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的学术价值和更广泛的影响审查标准。

项目成果

期刊论文数量（7）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Infinite towers in the graphs of many dynamical systems

许多动力系统图中的无限塔

DOI：
10.1007/s11071-021-06561-6
发表时间：
2021
期刊：
Nonlinear Dynamics
影响因子：
5.6
作者：
De Leo, Roberto;Yorke, James A.
通讯作者：
Yorke, James A.

Theory of Dynamical Systems and Transport Phenomena in Normal Metals

普通金属的动力系统理论和输运现象

DOI：
10.1134/s106377611910008x
发表时间：
2019
期刊：
Journal of Experimental and Theoretical Physics
影响因子：
1.1
作者：
Novikov, S. P.;De Leo, R.;Dynnikov, I. A.;Maltsev, A. Ya.
通讯作者：
Maltsev, A. Ya.

Backward Asymptotics in S-Unimodal Maps

S-单峰映射中的后向渐近

DOI：
10.1142/s0218127422300130
发表时间：
2022
期刊：
International journal of bifurcation and chaos in applied sciences and engineering
影响因子：
0
作者：
Roberto De Leo
通讯作者：
Roberto De Leo

Dynamics of Newton Maps of Quadratic Polynomial Maps of ℝ2 into Itself

∄2 的二次多项式映射的牛顿映射动力学