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Imperfect Patterns-Existence, Stability, and Essential Spectrum

不完美模式——存在性、稳定性和本质谱

基本信息

批准号：
1815079
负责人：
Qiliang Wu
金额：
$ 12.5万
依托单位：
Ohio University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1815079&HistoricalAwards=false
关键词：
Imperfect Patterns Existence Stability Essential

项目摘要

How do zebras get their stripes? How do cell membranes get their shapes and keep from bursting into pieces? These two seemingly disconnected scientific topics can actually be investigated via a general mathematical framework outlined within this project. Such a study not only applies to the specific topics above, but also to any physical systems arising from similar mathematical models, such as sand patterns in deserts, cloud patterns in the sky, cell fission/fusion processes, cast ionomers in solar cells, and network morphology in soapy water. The aim of this project is to systematically understand imperfections of patterns, which are ubiquitous in nature and pivotal in various scientific settings and technical applications. The investigator studies defects in periodic patterns, with an application to understanding the development of grain boundaries in materials; he also studies defects in bilayers such as arise in cell membranes. Part of the project includes research experience opportunities for undergraduate students.The investigator applies dynamical systems techniques, combined with functional analysis, differential geometry, asymptotic analysis, and large deviation theory, to study pattern formation, with an emphasis on the role of the essential spectrum in the formation and stability of defects of periodic patterns and bilayer interfaces. The most interesting and challenging case occurs when the essential spectrum touches the origin; here he seeks explicit illustrations of formation mechanisms of stripe patterns, amphiphilic structures, and their imperfections. The classical Swift-Hohenberg equation provides a prototype for rigorous studies of periodic patterns. Firstly, the imperfections of periodic patterns to instantaneous, constant, and random perturbations are investigated. Secondly, in the case of instability, the local biases and the rotational symmetry of the system together give rise to various line and point defects, such as grain boundaries, dislocations, and disclinations. The investigator studies grain boundaries, providing a novel functional analytical machinery to construct deformed patterns. He also investigates defects of amphiphilic morphology in the functionalized Cahn-Hilliard FCH) setting, suggesting mechanisms of the formation of lipid rafts, end caps and Y-junctions. Here the essential spectrum of the linearized operator at bilayer interfaces is continuous up to the origin but is not "simple." It serves as a benchmark for understanding the role of essential spectra. Part of the project includes research experience opportunities for undergraduate students.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.

斑马身上的条纹是怎么来的？细胞膜是如何形成它们的形状并防止破裂成碎片的？这两个看似无关的科学主题实际上可以通过该项目中概述的一般数学框架进行研究。这样的研究不仅适用于上述特定主题，而且适用于任何由类似数学模型产生的物理系统，例如沙漠中的沙模式，天空中的云模式，电池裂变/聚变过程，太阳能电池中的离聚物，以及肥皂水中的网络形态。该项目的目的是系统地了解模式的不完美性，这些不完美性在自然界中普遍存在，在各种科学环境和技术应用中至关重要。研究人员研究周期性图案中的缺陷，并应用于理解材料中晶界的发展;他还研究了双层中的缺陷，如细胞膜中出现的缺陷。该项目的一部分包括为本科生提供研究经验的机会。研究者应用动力系统技术，结合泛函分析，微分几何，渐近分析和大偏差理论，研究图案的形成，重点是基本光谱在周期图案和双层界面缺陷的形成和稳定性中的作用。最有趣和最具挑战性的情况下发生的基本光谱触及的起源;在这里，他寻求明确的说明条纹图案的形成机制，两亲性结构，以及它们的缺陷。经典的Swift-Hohenberg方程为周期性模式的严格研究提供了一个原型。首先，研究了周期方向图对瞬时、常值和随机扰动的不完美性。第二，在不稳定的情况下，系统的局部偏置和旋转对称性一起引起各种线和点缺陷，例如晶界、位错和向错。研究员研究晶界，提供了一种新的功能分析机器来构建变形模式。他还研究了功能化Cahn-Hilliard FCH）设置中两亲形态的缺陷，提出了形成脂筏、端帽和Y-连接的机制。在这里，双层界面上线性化算子的本质谱是连续的，直到原点，但不是“简单的”。“它是理解基本光谱作用的基准。该项目的一部分包括为本科生提供研究经验的机会。该奖项反映了NSF的法定使命，并且通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。