In this paper, we define some generalizations of topological group namely -topological group, -topological group and -topological group with illustrative examples. Also, we define grill topological group with respect to a grill. Later, we deliberate the quotient on generalizations of topological group in particular -topological group. Moreover, we model a robotic system which relays on the quotient of -topological group.

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.

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

The main purpose of this work is to introduce the concept of higher N-derivation and study this concept into 2-torsion free prime ring we proved that:Let R be a prime ring of char. 2, U be a Jordan ideal of R and be a higher N-derivation of R, then , for all u U , r R , n N .

As regional development, as a matter of course, poses a number of systemic, scientific and political problems. While the issue of development is primarily at the national level to the limits of World War II in the industrialized world and to the 1960s borders in most Third World countries, the increasing awareness of regional disparities has led to the regional issue Were taken into consideration in the early 1960s and 1970s in most industrialized and developing countries alike. The local issue was only introduced in the early 1980s. The awareness of regional disparities and the fact that the regions do not have the same potential and that some regions have the resources to enable them to develop, grow and develop, unlike other r

The importance of government Expenditure policy in economy come from its role leading to the mitigation and adjustment of fluctuations in macroeconomic variables caused by imbalance between aggregate demand and aggregate supply,  It is associated with the efficient management of government Expenditure to reinforcement the relationship between government Expenditure and the overall economic system .

Regarding the Iraqi economy,the increasing in financial rentier after the political change in 2003 has led to finance the budgets Characterized by consumption,The government Expenditure employed to encourage government employment in services jobs, and find different channels for the distribution of