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 $M$.
The Existence of Gorenstein Injective Envelopes
Let R be a commutative Noetherian ring. We prove that every R-module of finite Gorenstein dimension has a Gorenstein injective envelope g:M->G where injective dimension of Ker g is finite and g is injective.
Mathematical Modeling of Scour and Deposition
Electronic computers are opening up the field of sedimentation engineering by permitting the analysis of problems which heretofore have required too much time and effort. Movable stream bed profiles are being calculated in addition to water surface profiles. The initial gradation of sediment material in the stream bed may be prescribed to facilitate the calculation of an armor layer. This gradation is continually changed by the computer program in response to sediment particles being deposited on the stream bed and material on the stream bed being removed to satisfy transport capacity. Hydraulic sorting of sediment by grain size is accomodated. The stream bed does not have to be represented by a single grain size. Retardance of motion because of armor layer is calculated.