This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old. We relate the cohomology of the output complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincar\'e type inequalities, well-posed Hodge-Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domains.

翻译：本文涉及不同方程式中的各种问题产生的不同复杂体的衍生和特性,这些问题包括连续力学、相对论和其他领域的应用。我们提出了一个系统性程序,从德拉姆综合体等广为人知的不同复杂体开始,产生新的复杂体,并从旧体中推断出新复杂体的特性。我们把产出复杂体与输入综合体的混合体联系起来,并表明新的复杂体具有封闭范围,因此满足了Hodge的分解、Poincar\'e类型不平等、大量存在的Hodge-Laplaceian边界价值问题、常规分解和Lipschitz通用域的紧凑性特性。