Random Tur\'an and counting results for general position sets over finite fields

Let $\alpha(\mathbb{F}_q^d,p)$ denote the maximum size of a general position set in a $p$-random subset of $\mathbb{F}_q^d$. We determine the order of magnitude of $\alpha(\mathbb{F}_q^2,p)$ up to polylogarithmic factors for all possible values of $p$, improving the previous best upper bounds obtained by Roche-Newton--Warren and Bhowmick--Roche-Newton. For $d \ge 3$ we prove upper bounds for $\alpha(\mathbb{F}_q^d,p)$ that are essentially tight within certain intervals of $p$. We establish the upper bound $2^{(1+o(1))q}$ for the number of general position sets in $\mathbb{F}_q^d$, which matches the trivial lower bound $2^{q}$ asymptotically in exponent. We also refine this counting result by proving an asymptotically tight (in exponent) upper bound for the number of general position sets with fixed size. The latter result for $d=2$ improves a result of Roche-Newton--Warren. Our proofs are grounded in the hypergraph container method, and additionally, for $d=2$ we also leverage the pseudorandomness of the point-line incidence bipartite graph of $\mathbb{F}_{q}^2$.Comment: 24 pages(+2 pages for Appendix), 2 figure

Algal proliferation risk assessment using Vine Copula-based coupling methods in the South-to-North Water Diversion Project of China

The Middle Route of the South-to-North Water Diversion Project of China (MRSNWDPC), i.e., the longest inter-basin water diversion project (1,432 km) in the world, has delivered more than 60 billion m3 of water resources to North China and benefiting more than 100 million people since December 2014. However, the abnormal algal proliferation in the main canal under low nutrient background has seriously threatened the water quality safety of this mega project. In this research, 3 years of monitoring data matrix, including water temperature (WT), flow discharge (Q), flow velocity (V), dissolved oxygen (DO), and the algal cell density (ACD), from the main canal of the MRSNWDPC were analyzed. The nonlinear relationships were determined based on multiple regression models, and a composite risk analysis model was constructed by Latin hypercube sampling (LHS) method coupled with Vine Copula function. The impacts of different hydrological and environmental factors on algal proliferation were comprehensively analyzed by Bayesian theory. The results showed that the WT gradually decreased from upstream to downstream, with a narrow range of 16.6–17.4°C, and the annual average concentrations of DO showed a gradual increase from upstream to downstream. The flow velocity of MRSNWDPC had a tendency to increase year by year, and the maximum flow velocity exceeds 0.8 m/s upstream, midstream and downstream by 2018. The ACD accumulated along the main canal, and the annual average ACDs of downstream were the highest, ranging from 366.17 to 462.95 × 104 cells/L. The joint early-warning method considering both water temperature and flow velocity conditions is an effective way for algal proliferation risk warning management. When water temperatures of the upstream, midstream, and downstream were below 26, 26, and 23°C, respectively, the algal proliferation risk can be controlled under 50% by the flow velocity at 0.3 m/s; otherwise, the flow velocity needs to be regulated higher than 0.8 m/s. In order to keep the midstream and downstream avoid abnormal algal proliferation events (ACD ≥ 500 × 104 cells/L), the corresponding ACDs of the upstream and midstream need to be controlled lower than 319 × 104 cells/L and 470 × 104 cells/L, respectively. This study provides a scientific reference for the long-distance water diversion project’s algal control and environmental protection. The proposed coupling Vine Copula models can also be widely applied to multivariate risk analysis fields