let $s$ be a monoid. in this paper, we prove every class of $s$-acts having a flatness property is closed underdirected colimits, it extends some known results. furthermore thisresult implies that every $s$-act has a flatness cover if and only if it has a flatness precover.

The constant density, spherically symmetric, perfect fluid solution to Einstein’s equations was given in 1916 by Schwarzschild[1]. Despite being a stalwart of introductory General Relativity, the source of much of the knowledge we have about the Schwarzschild star interior solution is shrouded in obscurity. When he was writing about it in 1971, Buchdahl[2] decried the fact that no standard text...

This paper presents a formulation of differential flatness—a concept originally introduced by Fliess, Levine, Martin, and Rouchon—in terms of absolute equivalence between exterior differential systems. Systems that are differentially flat have several useful properties that can be exploited to generate effective control strategies for nonlinear systems. The original definition of flatness was g...

The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry (RAG). In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this r...

Let $S$ be a monoid. In this paper, we prove every class of $S$-acts having a flatness property is closed underdirected colimits, it extends some known results. Furthermore thisresult implies that every $S$-act has a flatness cover if and only if it has a flatness precover.

In this work we consider the concept of differential flatness defined by Fliess, Levine, Martin and Rouchon. The structural property of k-flatness (the case in which flat outputs depends up to the kth derivative of the inputs) is studied based on the properties of the dynamic extension algorithm and the Cartan-Kähler theorems on the existence of integral manifolds of exterior differential syste...