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.

         The aim of this paper is to present a method for solving of system of first order initial value problems of ordinary differential equation by a semi-analytic technique with constructing polynomial solutions for decreasing dangers of lead. The original problem is concerned using two-point osculatory interpolation with the fit equals  numbers of derivatives at the end points of an interval [0 , 1].

The spectroscopic properties, potential energy curve, dipole moments, total charge density, Electrostatic potential as well as the thermodynamic properties of selenium diatomic halides have been studied using code Mopac.7.21 and hyperchem, semi-empirical molecular orbital of MNDO-method (modified neglected of differential overlap) of parameterization PM3 involving quantum mechanical semi-empirical Hamiltonian. The relevant molecular parameters like interatomic distance, bond angle, dihedral angle and net charge were also calculated.

        The set of all (n×n) non-singular matrices over the field F. And this set forms a group under the operation of matrix multiplication. This group is called the general linear group of dimension  over the field F, denoted by . The determinant of these matrices is a homomorphism from  into F* and the kernel of this homomorphism was the special linear group and denoted by  Thus  is the subgroup of  which contains all matrices of determinant one.

The rationally valued characters of the rational representations are written as a linear combination of the induced characters for the groups discussed in this paper. We find the Artin indicator for this group after studying the rationally valued characters of the rational

The set of all (n×n) non-singular matrices over the field F this set forms a group under the operation of matrix multiplication. This group is called the general linear group of dimension  over the field F, denoted by . The determinant of these matrices is a homomorphism from  into F* and the kernel of this homomorphism was the special linear group and denoted by  Thus  is the subgroup of  which contains all matrices of determinant one.

The rational valued characters of the rational representations written as a linear combination of the induced characters for the groups  discuss in this paper and find the Artin indicator for this group after study the rational valued characters of the rational representations and the induce

The technology and modern social communication medias have contributed to changing many of our concepts about education to a large extent; these technological advancements have created new challenges for the Instructor of course, but on the other hand it did provided him with expressive media, models and strategies that the old methods cannot fulfill, including the flipped classroom strategy, which is one of the modern technical solutions and in it's optimal use it can correct the weakness of the traditional education and can develop the communication skills as a whole among the students, in order to create a positive, active and interactive student.
The current research contained - four chapters: the first one included the methodolog

The research demonstrates new species of the games by applying separation axioms via  sets, where the relationships between the various species that were specified and the strategy of winning and losing to any one of the players, and their relationship with the concepts of separation axioms via  sets have been studied.

   The aim of this paper is to introduce and study new class of fuzzy function called fuzzy semi pre homeomorphism in a fuzzy topological space by utilizing fuzzy semi pre-open sets. Therefore, some of their characterization has been proved; In addition to that we define, study and develop corresponding to new class of fuzzy semi pre homeomorphism in fuzzy topological spaces using this new class of functions.