Differential Test for Series of Positive Terms, New Tree-Field Representations in Graph Theory and New Number Field, Extension of Dirac Extraction, and Their Applications

First, a new differential test for series of positive terms is proved. Let f(x) be a positive continuous function corresponded to a series of positive terms , and g(x) is a derivative of reciprocal of f(x), i.e., . Then, if for enough large x, the series converges; if , the series diverges. The rest may make the limit form, and is universal and complete. Next, the tree in the graph theory is extended to a new tree-field representation. It includes two parts: tree and field. A field is a set of legion small trees. They can transform each other between tree and field. This is a unification of simplicity (tree) and complexity (field), and may be applied to various complex systems on science, politics, economy, philosophy and so on. Further, it may be extended to the whole graph theory G=(V,E,F), here F is a set of small graphs. Third, based on a brief review on developments of number system, a new developed pattern is proposed. The quaternion is extended to a matrix form aI+bC+cB+dA, in which the unit matrix I and three special matrices C,B,A correspond to number 1 and three units of imaginary number i,j,k, respectively. They form usually a ring. But some fields may be composed of some special 2-rank, even n-rank matrices, for example, three matrices aI+bC, aI+cB, aI+dA and so on. It is a new type of hypercomplex number fields. The physical applications and possible meaning of the new number system is researched. Finally, the Dirac extraction is extended to any terms whose extraction should be and , etc. Moreover, the general complexity is also discussed.