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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.

This work focuses on the problem of high quality style transfer for Chinese characters, converting Chinese printed with standard typograph into those in calligraphic style. Traditional approaches include employing derivative network structures of CNN, RNN, and GAN, yet generation results are not satisfactory enough. In this paper, we proposed an integrated system of pix2pix structure, auxiliary classifier, and WGAN. Trials were run to evaluate the performance of traditional CNN and GAN. Cross-comparative sets of experiments among GAN, AC-GAN, and our proposed AC-WGAN have been conducted to test the capabilities of our proposed model. Inference and interpolation tests are conducted as well. Our proposed model outperformed existing style transfer system in generation’s delicacy, similarity, and efficiency. Incorporated with WGAN, the model also demonstrated a strong ability in providing a truthful training indicator with its loss. Study also suggested that AC and pix2pix adaption hold huge significance in accelerating the training process. Research validates the viability of fulfilling Chinese character transformation with style transfer. With handwriting recognition network, fast, high-quality Chinese character style transfer forms the basis for instantaneous conversion between printed and written work. Optimization on algorithm may be another direction for further exploration.

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.