In this paper, the differential equations for copolymerization of multiple monomers are derived based on the steady-status assumption of Alfrey-Goldfinger and Valvassori-Sartori respectively, and a numerical method is implemented in C++ language. The differential equations of copolymer composition with wide application presented in this paper provide a theoretical guidance for industrial production and academic research of multiple free radical copolymerization.

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The subject of counting positive lattice points in n-dimensional simplexes has interested mathematicians for decades due to its applications in singularity theory and number the- ory. Enumerating the lattice points in a right-angled simplex is equivalent to determining the geometric genus of a singularity of a weighted homogeneous complex polynomial. It is also a method to shed insight into large gaps in the sequence of prime numbers. Seeking to contribute to these applications, this research project proves the Yau Geometric Con- jecture in six dimensions, a sharp upper bound for the number of positive lattice points in a six-dimensional tetrahedron. The main method of proof is summing existing sharp upper bounds for the number of points in 5-dimensional simplexes over the cross sections of the six- dimensional simplex. This research project paves the way for the proof of a fully general sharp upper bound for the number of lattice points in a simplex. It also moves the mathematical community one step closer towards proving the Yau Geometric and Yau Number-Theoretic Conjectures in full generality.

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I heard the concept of the ”equal‐sum point” in a math summer camps and found it very interesting. After searching on the Internet,I found that the ”equal‐sum point” was associated with Soddy point. Soddy point is a magic point in plane geometry, which was found by British physicist, chemist Frederick.Soddy. Some research about the Soddy point has been done in foreign countries, but the properties are not comprehensive. In China, Teacher HuasongHuang raised the concept of the “equal‐sum point” and the “equal‐difference point” and drew some properties, I found that these two points were very similar to the Soddy point, but they did not link the two points with Soddy point. In this article ,I will connect the Soddy point with the “equal‐sum point” and the “equal‐difference point” and study their properties more deeply and find some new discoveries.