In this paper, we established a big data model based on the data analysis method and the endogenous structure mutation theory, and judged from the dimensions of time and space. Additionally, we gave a detailed inspection and analysis among enterprises producing cement of a certain province. The results indicated that this model can identify monopolistic behaviors from multiple dimensions and, thus, improve regulatory efficiency.

Let $p_1,p_2,\dots ,p_7$ be primes. In this paper, we first show that when $k_1=23$, every sufficiently large odd integer can be represented as the sum of one prime square, five prime cubes, one prime biquadrate and at most $k_1$ powers of $2$. We further prove that for $k_2=45$, every pair of sufficiently large odd integers satisfying certain necessary conditions can be represented as a pair of equations involving one prime square, five prime cubes, one prime biquadrate and at most $k_2$ powers of $2$.