Multi-constituent inhibitory systems with self-organizing properties

本项目研究了几种重要的具有自组织性质的二元和三元体系，重点研究了三元体系和通过非局部相互作用或抑制剂变量的更远距离约束机制。将三元系统与二元系统区分开来的一个有趣的特征是三重结现象。在二维情况下，系统的三个组成部分可以在一点上相遇，在三维情况下，可以在曲线上相遇。PI提出了双气泡组装模式的存在，其中每个双气泡中都出现了三重结现象。两个新的技术，限制微扰类和内变量，将在证明中使用。三元系统的第二个特征是长距离相互作用的复杂性，表现为2 × 2的参数矩阵。例如，只有当矩阵的2-2项大于1-2项时，才会出现核壳模式。当多组分抑制系统出现在弯曲空间时，如脂质膜囊泡，黎曼曲率所起的作用也将被研究。图案及其可能的缺陷将显示为生长、抑制和曲率的平衡和妥协。精巧的结构模式出现在许多多成分的物理和生物系统中，作为自组织原则的有序结果。例子包括嵌段共聚物中的形态相、动物皮毛和皮肤色素沉着。在这些模式形成系统中常见的是，对同质性的偏离对其进一步增加具有强烈的正反馈。就其本身而言，它将导致无限的增长和蔓延。此外，模式的形成还需要对局部自增强过程进行较长范围的限制。该项目从各种二进制和三元物理和生物系统的自组织原理中导出几何图案。对这些几何结构的分析是理解这些系统的机械、光学、电学、离子、势垒和其他性质的基本步骤。该项目支持一个文化多元化的环境，在种族多元化的华盛顿特区的课堂和研究环境中培养批判性和独立思考。它产生了深刻而美丽的数学，并提供了对自然世界的深刻见解。

This project investigates several important binary and ternary systems with self-organizing properties, with an emphasis on ternary systems and the longer ranging confinement mechanism through nonlocal interaction or inhibitor variables. One intriguing feature that sets ternary systems apart from binary systems is the triple junction phenomenon. The three constituents of the system may meet at a point in the two dimensional case or at a curve in the three dimensional case. The PI proposes to show the existence of a double bubble assembly pattern, where the triple junction phenomenon occurs in each double bubble. Two novel techniques, restricted perturbation classes and internal variables, will be used in the proof. The second feature in ternary systems is the complexity of the long range interaction manifested in a two by two matrix of parameters. For instance it will be shown that the core-shell pattern appears only if the 2-2 entry is greater than the 1-2 entry of the matrix. When multi-constituent inhibitory systems appear in a curved space, such as a vesicle of lipid membranes, the role played by the Riemann curvature will also be investigated. Patterns and their possible defects will be shown as a balance and compromise of growth, inhibition, and curvature. Exquisitely structured patterns arise in many multi-constituent physical and biological systems as orderly outcomes of self-organization principles. Examples include morphological phases in block copolymers, animal coats, and skin pigmentation. Common in these pattern-forming systems is that a deviation from homogeneity has a strong positive feedback on its further increase. On its own, it would lead to an unlimited increase and spreading. Pattern formation requires in addition a longer ranging confinement of the locally self-enhancing process. This project derives geometrical patterns from self-organization principles in various binary and ternary physical and biological systems. Analysis of these geometric structures is a fundamental step in understanding the mechanical, optical, electrical, ionic, barrier and other properties of these systems. This project supports a culturally diverse environment that fosters critical and independent thinking both in classroom and research settings in the ethnically diverse metropolitan area of Washington, DC. It produces mathematics of depth and beauty, and offers profound insights into the natural world.