Level sets of nonsmooth functions, Part 2: Lipschitz and piecewise-differentiable manifolds

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-10-02 DOI:10.1016/j.jmaa.2024.128920

Suzane M. Cavalcanti, Paul I. Barton

引用次数: 0

Abstract

This paper introduces piecewise-differentiable (

P C^{r}

) manifolds according to a unified general framework that also applies to nonsmooth Lipschitz manifolds and smooth manifolds. We present definitions of nonsmooth manifolds and embedded submanifolds for abstract sets as well as for subsets of

R^{n}

, and explore the relationships between them. The

P C^{r}

and Lipschitz Rank Theorems from Part 1 of this series, in terms of the Clarke Jacobian and B-subdifferential generalized derivative sets, are used to characterize level sets of functions between nonsmooth manifolds as embedded submanifolds. We illustrate how the Level Set Theorems developed in this paper can be applied to functions on Euclidean space, including a piecewise-differentiable process model for distillation columns.

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非光滑函数的水平集，第 2 部分： Lipschitz 和片差流形

本文根据一个统一的总体框架介绍了片差流形（PCr），该框架也适用于非光滑的利普齐兹流形和光滑流形。我们提出了抽象集和 Rn 子集的非光滑流形和嵌入子流形的定义，并探讨了它们之间的关系。本系列第一部分中的 PCr 和 Lipschitz 等级定理，以克拉克雅各布和 B 次微分广义导数集为基础，用于描述非光滑流形之间函数的等级集，即嵌入子流形。我们将说明本文中提出的水平集定理如何应用于欧几里得空间上的函数，包括蒸馏塔的片差过程模型。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.