The relation between faithful, finitely generated, separated acts and the one-to-one operators was investigated, and the associated S-act of coshT and its attributes have been examined. In this paper, we proved for any bounded Linear operators T, VcoshT is faithful and separated S-act, and if a Banach space V is finite-dimensional, VcoshT is infinitely generated.

In cyber security, the most crucial subject in information security is user authentication. Robust text-based password methods may offer a certain level of protection. Strong passwords are hard to remember, though, so people who use them frequently write them on paper or store them in file for computer .Numerous of computer systems, networks, and Internet-based environments have experimented with using graphical authentication techniques for user authentication in recent years. The two main characteristics of all graphical passwords are their security and usability. Regretfully, none of these methods could adequately address both of these factors concurrently. The ISO usability standards and associated characteristics for graphical

Abstract:
the system of Administrative Control in organizations meets the need to check on the optimal use and proper resources and conservation to achieve the objectives sought by the organization, hence the system of Administrative Control  is part of the overall system in any organization that has undergone evolution always to be able to keep up with progress in the development of other sciences, and that the growth of coherence between subordinates in the organization means the ability to influence the opinions, ideas and attitudes to change it for directions the organization and its values ​​and this is reflected positively on the coherence of the organization, the researcher interest of the imp

Relation on a set is a simple mathematical model to which many real-life data can be connected. A binary relation  on a set  can always be represented by a digraph. Topology on a set  can be generated by binary relations on the set . In this direction, the study will consider different classical categories of topological spaces whose topology is defined by the binary relations adjacency and reachability on the vertex set of a directed graph. This paper analyses some properties of these topologies and studies the properties of closure and interior of the vertex set of subgraphs of a digraph. Further, some applications of topology generated by digraphs in the study of biological systems are cited.

            In this paper, the Normality set  will be investigated. Then, the study highlights some concepts properties and important results. In addition, it will prove that every operator with normality set has non trivial invariant subspace of  .

Vol. 6, Issue 1 (2025)