Local Homology and Local Cohomology
Let $(R, {\frak m})$ be a local ring, $I$ a proper ideal of $R$ and $M$ a finitely generated $R$-module of dimension $d$. We discuss the local homology modules of $H^d_I(M)$. When $M$ is Cohen-Macaulay, it is proved that $H^d_{{\frak m}}(M)$ is co-Cohen-Macaulay of N.dimension $d$ and $H^{\underline{x}}_d(H^d_{{\frak m}}(M))\cong\hat{M}$ where $\underline{x}=(x_1,...,x_d)$ is a system of parameters for $M$.
Local Cohomology and Sequentially Cohen-Macaulay Modules ⁄
Let (R, 𝔪) be a commutative Noetherian local ring and N a finitely generated R-module with dimM =d. It is shown that M is a sequentially Cohen-Macaulay module if and only if the modules are either 0 or co-Cohen-Macaulay of Noetherian dimension i for all 0 ≤ i ≤ d.
The analysis of nonlinear systems in the frequency domain using Nonlinear Output Frequency Response Functions
Abstract The Nonlinear Output Frequency Response Functions (NOFRFs) are a concept which provides a new extension of the well-known concept of the Frequency Response Function (FRF) of linear systems to the nonlinear case. The present study introduces a NOFRFs based approach for the analysis of nonlinear systems in the frequency domain. It is well known that a nonlinear system can, under rather general conditions, be represented by a polynomial type Nonlinear Auto Regressive with eXogenous input (NARX) model. From the NARX model of a nonlinear system under study, the NOFRFs based approach for the frequency analysis of nonlinear systems involves solving a set of linear difference equations known as the Associated Linear Equations (ALEs) to determine the system nonlinear output responses and then the NOFRFs of the system up to an arbitrary order of nonlinearity of interests. The results enable a representation of the frequency domain characteristics of nonlinear systems by means of a series of Bode diagram like plots that can be used for nonlinear system frequency analyses for various purposes including, for example, condition monitoring, fault diagnosis, and nonlinear modal analysis.