No abstract uploaded!

A circle family is a group of separate or tangent circles in the plane. In this paper, we study how many parts at most a plane can be divided by several circle families if the circles in a same family must be separate (resp. if the circles can be tangent). We also study the necessary conditions for the intersection of two circle families. Then we primarily discuss the similar problems in higher dimensional space and in the end,raise some conjectures.

The Durfee conjecture, proposed in 1978, relates two important invariants of isolated hypersurface singularities by a famous inequality; however, the inequality in this conjecture is not sharp. In 1995, Yau announced his conjecture which proposed a sharp inequality. The Yau conjecture characterizes the conditions under which an affine hypersurface with an isoalted singularity at the origin is a cone over a nonsingular projective hypersurface; in other words, the conjecture gives a coordinate-free characterization of when a convergent power series is a homogeneous polynomial after a biholomorphic change of variables. In this project, we prove that the Yau conjecture holds for n = 5. As a consequence, we have proved that &5!pg≤μ-p(v)&, where &p(v) = (v-1)^5-v(v-1) ……(v-4)& and pg,μ,and v are, respectively, the geometric genus, the Milnor number, and the multiplicity of the isolated singularity at the origin of a weighted homogeneous polynomial. In the process, we have also defined yet another sharp upper bound for the number of positive integral points within a 5-dimensional simplex.

It is well known that group is a set with an algebra operation, which is often denoted by G . A simple group is the group which has only the 2 trivial normal subgroups. The importance of the simple groups may analogue to the prime numbers in the number theory. The classification of the finite simple groups was the central problem of 20th century’s’ algebra. In this research report, we use Sylow theorem, Burnside theorem, group action and some other elementally group theory methods to obtain some criteria of simple group . By using these criteria we determine the possible simple groups whose order less than 700. Generalizes the results in the reference [1]，[2]， [3]，[4].We conclude that Theorem A group G of order less or equal than 700 could not be simple except for |G|∈{60,168,360,504,660 and all primes}.

No abstract uploaded!