It is well-known that, using principal weak flatness property, some important monoids are characterized, such as regular monoids, left almost regular monoids, and so on. In this article, we recall a generalization of principal weak flatness called GP-flatness, and characterize monoids by this property of their right (Rees factor) acts. Also we investigate GP coherent monoids.

An optimal explicit guidance law that maximizes terminal velocity is developed for the reentry of a vehicle to a fixed target. The equations of motion are reduced with differential flatness approach and acceleration commands are related to the parameters of trajectory. An optimal trajectory is determined by solving a real-coded genetic algorithm. For online trajectory generation, optimal trajectory is approximated. The approximated trajectory is compared with the pure proportional navigation and genetic algorithm solutions. The near optimal terminal velocity solution compares very well with these solutions. The approach robustness is examined by Monte Carlo simulation. Other advantages such as trajectory representation with minimum parameters, applicability to any reentry vehicle configuration and any control scheme, and Time-to-Go independency make this guidance approach more favorable.

The concept of free modules, Projective modules, injective modules and the like form an important area in module theory. The notion of free fuzzy modules was introduced by Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameri introduced the concept of Projective and injective L-modules. In this paper we give an alternate definition for Projective L-modules. We prove that every free L-module is a Projective L-module. Also we prove that if $\mu\in L(P)$ is a Projective L-module, and if $0\rightarrow\eta\rightarrow\vartheta\rightarrow\mu\rightarrow 0$ is a short exact sequence of L-modules then $\nu\otimes\mu>\vartheta$. Further it is proved that if $\mu\in L(P)$ is a Projective L-module then $\mu$ is a fuzzy direct summand of a free L-module.

flatness of surface is of great importance in industrial products such as steel sheets. In many cases, the quality of the final product is directly linked to the flatness of its components. One possible defect in the production of hot strip mill steel sheets is the straightness defect which results in flatness asymmetries in the steel sheets, causing a certain curvature on its edges. This defect is more likely to occur at the final stages of the hot strip mill. The deformation may cause stoppages in the rolling process and damage to the machinery. Detection and measurement of these asymmetries can be fed back to the previous stages of the rolling process for proper adjustment of the roller's speed to avoid further defects. Nowadays, a family of methods for surface detection, called topometry, is available for measurement of surface flatness asymmetries. In this paper, a method is proposed based on Interferometric Fringe Projection (IFP) which performs the flatness measurement. The proposed method has the advantage of simplicity and can, hence, be used in real-time applications.

In this paper we study flatness properties (pullback flatness, limit flatness, finite limit flatness) of acts over semigroups. These are defined by requiring preservation of certain limits from the functor of tensor multiplication by a given act. We give a description of firm pullback flat acts using Conditions (P) and (E). We also study pure epimorphisms and their connections to finitely presented acts and pullback flat acts. We study these flatness properties in the category of all acts, as well as in the category of unitary acts and in the category of firm acts, which arise naturally in the Morita theory of semigroups.

Inspired by a recent work of Buchweitz and Flenner, we show that, for a semidualizing bimodule C, C-perfect complexes have the ability to detect when a ring is strongly regular. It is shown that there exists a class of modules which admit minimal resolutions of C-Projective modules.