1, We introduce the notion of topological direct sum, and get a representative theorem of inductive limit by topological direct sum and quotient space.

2, When a direct sum of lifting modules is lifting is studied here.

3, Competition between supply chains possess the foundation of direct sum and game,(http://sentencedict.com/direct sum.html) and create condition for the Pareto improve of supply chain.

4, According to algebraic property that a direct sum of Galois fields is isomorphic to a finite ring, the linear block code for multiplexing systems is proposed.

5, The direct sum decomposition of the addition of a linear transformation under the coprime polynomial was given, and it was used in the proof of some equality about the rank of idempotent matrix.

6, Besides, direct sum operation of matrices is adopted and expanded to vectors to elegantly represent the functions of quantum circuits on input vectors.

7, In this paper we investigate the characterization of self-adjoint domains of symmetric differential operators with interior singular points in the direct sum spaces.

8, In this paper, we have discussed root subspace decomposition of linear space, and for the linear space have decomposed direct sum of root subspace , a primary proof is given.

9, Introduces the concept of completion functor and discusses its relations with finite direct sum and matrix rings and its simplicity.

10, First, the method for constructing linear stego-codes is proposed by using the direct sum of vector subspaces .