In this paper, the Seifert -Van Kampen Theorem deals with the situation where a path-connected space may be written as the union of two open path-connected intersection. The fundamental group of the space is isomorphic to the co-limit of the diagram of the fundamental groups of the three subspaces. We also considered the fact that the fundamental groups are all injective, the fundamental group of the space is a classical free product with amalgamation and also a free product with the amalgamation in the category of rings.
Inequalities are essential in the study of Mathematics and are useful tools in the theory of analysis. They have been playing a critical role in the study of the existence and uniqueness properties of solutions of initial and boundary value problems for differential equations as well as difference equations with their bounds. In this paper, we obtain new integral inequalities mainly by using some known inequalities. Various generalizations of Hardy's inequality are special cases of the results therein.
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