Analysis and Synthesis of Systems Sustainable Over Sets

可持续性的复杂性质表明，其评估可以受益于将人的投入纳入评估过程。“可持续发展”(SOS)这一新颖的概念迎接了这一挑战。这个概念是灵活的、全面的，并且可以很容易地纳入人的意见。SOS概念开启了环境和生态可持续性的新篇章。此外，K-12、本科生和研究生通过创建新的项目、暑期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参与。该项目还为相关人员提供培训，让他们接触高级数学和计算机编程的概念、方法和软件。项目研究人员正在为参与者在研究生院和大学教员职位的安置提供便利。这些方面的援助由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。该研究项目专注于通过集合可持续发展(SOS)的新概念来分析和综合可持续系统。首先正式定义了SOS概念，然后对无记忆、空间统一的系统进行了数学量化，并在系统可持续性评估中进行了演示。研究的重点是继续发展和分析SOS概念；将其适用范围扩展到空间分布和/或有记忆的系统；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是容易量化的，因此能够明确(是或否)回答“一个系统是可持续的吗？”不变集的概念在这一量化过程中起着关键作用，为可持续发展评估提供了简单的数学标准，特别是对于矩形集，它可以给出明确(是或否)答案的问题“是否存在一个系统是可持续的？”该项目还帮助K-12，本科生和研究生，通过创建新的项目和基础设施，用于暑期研究计划，环境可持续发展课程，以及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.