Analysis and Synthesis of Systems Sustainable Over Sets

可持续性的复杂性表明，将人的投入纳入评估过程可使其评估受益。“可持续性大于集合”（SOS）的新概念应对了这一挑战。这个概念是灵活的、全面的，并且可以很容易地纳入人工输入。SOS的概念开启了环境和生态可持续性的新篇章。此外，K-12，本科生和研究生的参与通过创建新项目，夏季研究项目中使用的基础设施，环境可持续性课程以及高级数学和优化课程来实现。该项目还为相关人员提供培训，向他们展示高级数学和计算机编程概念、方法和软件。项目研究人员正在促进参与者在研究生院和大学教师职位的安置。加州大学洛杉矶分校工程与多样性卓越中心（CEED）和加州大学洛杉矶分校高中暑期研究计划（HSSRP）提供这些方面的援助。本研究项目着重于可持续系统的分析和综合，通过可持续发展的新概念（SOS）。首先对SOS概念进行正式定义，然后对无内存、空间均匀的系统进行数学量化，并在系统可持续性评估中进行演示。研究重点是SOS概念的持续发展和分析；将其适用性扩展到空间分布和/或具有内存的系统；生态、生物和社会系统的应用和案例研究；以及可持续系统综合方法的发展。SOS很容易量化，因此对“系统是否可持续”这个问题给出了明确的（是或否）答案。不变集的概念在这个量化过程中起着关键作用，导致了可持续性评估的简单数学标准，特别是对于矩形集，对于“是否存在一个系统可持续的集合”这个问题，可以给出明确的（是或否）答案？该项目还通过创建新的项目和基础设施来帮助K-12、本科生和研究生，这些项目和基础设施将用于夏季研究项目、环境可持续性课程、高级数学和优化课程。

The complex nature of sustainability suggests that its assessment can benefit from the incorporation of human input into the assessment process. The novel concept of "sustainability over sets" (SOS) meets this challenge. This concept is flexible, comprehensive, and can readily incorporate human input. The SOS concept opens a new chapter in environmental and ecological sustainability. In addition, the involvement of K-12, undergraduate, and graduate students occurs through the creation of new projects, infrastructures utilized in summer research programs, environmental sustainability courses, and upper level mathematics and optimization classes. The project also provides training for involved individuals, by exposing them to high level mathematics and computer programming concepts, methods, and software. The project researchers are facilitating the placement of participants in graduate schools, and in university faculty positions. Assistance on these fronts is provided by the UCLA Center for Excellence in Engineering and Diversity (CEED) and the UCLA High School Summer Research Program (HSSRP).  This research project focuses on the analysis and synthesis of sustainable systems through the novel concept of sustainability over sets (SOS). The SOS concept is first formally defined, and then mathematically quantified for memory-less, spatially uniform systems, and demonstrated in system sustainability assessment. The research focuses on the continued conceptual development and analysis of the SOS concept; extension of its applicability to systems that are spatially distributed and/or have memory; applications and case studies on ecological, biological, and societal systems; and development of sustainable system synthesis methodologies. SOS is readily quantifiable, thus enabling definitive (yes or no) answers to the question "is a system sustainable?" The concept of invariant sets plays a key role in this quantification process, leading to simple mathematical criteria for sustainability assessment, in particular for rectangular sets for which definitive (yes or no) answers can be given to the question "does there exist a set over which the system is sustainable?"  The project also helps K-12, undergraduate, and graduate students, by creating new projects, and infrastructure to be utilized in summer research programs, environmental sustainability courses, and upper level mathematics and optimization classes.