Free Quasi-idempotent Differential Rota-baxter Algebras And Gr(?)bner-shirshov Bases

Differential calculus and integral calculus are two very vital topics in mathematics.The algebraic abstraction of differential operator leads to the development of differential algebra.Rota-Baxter operator also can be viewed as an abstraction from integral operator satisfying the integration by part formula.As is well known,differential and integral calculus are linked together by the fundamental theorem of calculus.In this thesis,we introduce a notion of a differential algebra and a notion of a Rota-Baxter algebra in the quasi-idempotent operator contexts.In parallel to the fundamental theorem of calculus,we link them together,leading to a quasi-idempotent differential Rota-Baxter algebra.We mainly develop the construction of free objects in the category of quasi-idempotent differential algebra,quasi-idempotent Rota-Baxter algebra and quasi-idempotent differential Rota-Baxter algebra by the method of Gr(?)bner-Shirshov bases.