The main goal of this paper is to introduce a new class in the category modules. It called quasi-invertibility monoform (briefly QI-monoform) This modules generalization Various properties and another characterization QI-monoform are investigated. So, we prove that an R-module M if only for each non-zero homomorphism f:M E(M), kernel not quasi-invertible submodule M. Moreover, cases under which...

The famous shortest path tree lemma states that, for any node s in a graph G = (V,E), there is a subgraph on O(n) edges that preserves all distances between node pairs in the set {s}×V . A very basic question in distance sketching research, with applications to other problems in the field, is to categorize when else graphs admit sparse subgraphs that preserve distances between a set P of p node...

The paper describes various approaches to the invertibility of Toeplitz plus Hankel operators in Hardy and lp-spaces, integral difference Wiener-Hopf generalized operators. Special attention is paid a newly developed method, which allows establish necessary, sufficient also necessary conditions invertibility, one-sided for wide classes derive efficient formulas corresponding inverses. work cont...