The idea of fuzzy sets has opened new door of research in the world of contemporary Mathematics. The concept of fuzzy sets provided a new approach to model imprecision and uncertainty present in phenomena without sharp boundaries. The fuzzi cation of algebraic structures, play a dynamic role in Mathematics with diverse applications in many other branches such as computer arithmetic''s, control engineering, error correcting codes and formal languages and many more. Moreover, during the course of the last decade, non-associative algebraic structures have gained popularity among the researchers. In this background, many researchers initiated the notion of AG-groupoids, its newly introduced subclasses and its fuzzi cation. The present research is among the very few where non-associative algebraic structures are investigated and fuzzi ed. In this thesis various constructions of AG-groups over theeld Zn are introduced, some related results and example of AG-groups are provided. Further, the structural properties of fuzzy AG-subgroup are introduced and various notions of fuzzy AG-subgroups are investigated, e.g. conjugate of a fuzzy AG-subgroup, fuzzy normal AG-subgroups, relations between fuzzy normal AG-subgroup and commutators in AG-groups and equal-height elements in fuzzy AG-subgroups. Moreover, the notion of fuzzy AG-subgroups is further extended and a fuzzy coset in AG-subgroups is introduced. It is worth mentioning that if A is any fuzzy AGsubgroup of G, thenA(xy) =A(yx) for all x; y 2 G, i.e. each fuzzy left coset is fuzzy right coset and vice versa. Also, fuzzy coset in AG-subgroup could be empty contrary to coset in group theory. However, order of the nonempty fuzzy coset is the same as the index number [G : A] where H is an AG-subgroup of an AG-group G. The notion of fuzzy quotient AG-subgroup, fuzzy AG-subgroup of the quotient (factor) AG-subgroup, fuzzy homomorphism of AG-group and fuzzy Lagrange''s Theorem ofnite AG-group is introduced. Finally, cubic AG-subgroups and its properties are explored.
Withdrawal of US-NATO troops from Afghanistan, remains an issue for military operations of new-old participants in the process of further destabilization of the political situation in Central Asia. The process of political destabilization of the region raises many a questions about new relationship between interests of USA, China, Russia and India in this region. Pronounced influence of radical Islamic movements in the broader area of Asia makes this region tremendously important for the further development of global political relations in this part of the world. The crisis in relations between Russia and the West makes reconfiguration of global strategic interests in the region more complex.
In this thesis Noether symmetries are used for the classi cation of plane symmetric, cylindrically symmetric and spherically symmetric static spacetimes. We consider general metrics for these spacetimes and use their general arc length minimizing Lagrangian densities for the classi cation purpose. The coe cients of the metric in case of plane symmetric static spacetime are general functions of x while the coe cients of cylindrically symmetric and spherically symmetric static spacetimes are general functions of the radial coordinate r. The famous Noether symmetry equation is used for the arc length minimizing Lagrangian densities of these spacetimes. Noether symmetries and particular arc length minimizing Lagrangian densities of plane symmetric, cylindrically symmetric and spherically symmetric static spacetimes are obtained. Once we get the particular Lagrangian densities, we can obtain the corresponding particular spacetimes easily. This thesis not only provides classi cation of the spacetimes but we can also obtainrst integrals corresponding to each Noether symmetry. Theserst integrals can be used to de ne conservation laws in each spacetime. By using general arc length minimizing Lagrangian for plane symmetric, cylindrically symmetric and spherically symmetric static spacetimes in the Noether symmetry equation a system of 19 partial di erential equations is obtained in each case. The solution of the system in each case provides us three important things; the classi cation of the spacetimes, the Noether symmetries and the correspondingrst integrals which can be used for the conservation laws relative to each spacetime. Energy and momentum, the de nitions of which are the focus of many investigations in general relativity, are important quantities in physics. Since there is no invariant de - nitions of energy and momentum in general relativity to de ne these quantities we use the ii iii approximate Noether symmetries of the general geodesic Lagrangian density of the general time conformal plane symmetric spacetime. We use approximate Noether symmetry condition for this purpose to calculate the approximate Noether symmetries of the action of the Lagrangian density of time conformal plane symmetric spacetime. From this approach, those spacetimes are obtained the actions of which admit therst order approximation. The corresponding spacetimes are the approximate gravitational wave spacetimes which give us information and insights for the exact gravitational wave spacetimes. Some of the Noether symmetries obtained here carry approximate parts. These approximate Noether symmetries can further be used tond the correspondingrst integrals which describe the conservation laws in the respective spacetimes. Some of the vacuum solutions of Einsteineld equations for plane symmetric, cylindrically symmetric and spherically symmetric static spacetimes have also been explored.