On stated SL(n)SL(n)-skein modules

We mainly focus on Classical limit, Splitting map, and Frobenius homomorphism for stated $SL(n)$-skein modules, and Unicity Theorem for stated $SL(n)$-skein algebras. Let $(M,N)$ be a marked three manifold. We use $S_n(M,N,v)$ to denote the stated $SL(n)$-skein module of $(M,N)$ where $v$ is a nonzero complex number. We build a surjective algebra homomorphism from $S_n(M,N,1)$ to the coordinate ring of some algebraic set, and prove it's Kernal consists of all nilpotents. We prove the universal representation algebra of $\pi_1(M)$ is isomorphic to $S_n(M,N,1)$ when $N$ has only one component and $M$ is connected. Furthermore we show $S_n(M,N^{'},1)$ is isomorphic to $S_n(M,N,1)\otimes O(SLn)$, where $N\neq \emptyset$, $M$ is connected, and $N^{'}$ is obtained from $N$ by adding one extra marking. We also prove the splitting map is injective for any marked three manifold when $v=1$, and show that the splitting map is injective (for general $v$) if there exists at least one component of $N$ such that this component and the boundary of the splitting disk belong to the same component of $\partial M$. We also establish the Frobenius homomorphism for $SL(n)$, which is map from $S_n(M,N,1)$ to $S_n(M,N,v)$ when $v$ is a primitive $m$-th root of unity with $m$ being coprime with $2n$ and every component of $M$ contains at least one marking. We also show the commutativity between Frobenius homomorphism and splitting map. When $(M,N)$ is the thickening of an essentially bordered pb surface, we prove the Frobenius homomorphism is injective and it's image lives in the center. We prove the stated $SL(n)$-skein algebra $S_n(\Sigma,v)$ is affine almost Azumaya when $\Sigma$ is an essentially bordered pb surface and $v$ is a primitive $m$-th root of unity with $m$ being coprime with $2n$, which implies the Unicity Theorem for $S_n(\Sigma,v)$.Comment: 66 page

Determinants of the Withdrawal of Foreign-invested Enterprises: Evidence from China

Based on the industry and commerce annual report data, the present study uses the continuous-time nonparametric, parametric, and semi-parametric estimation to investigate the determinants influencing the withdrawal behaviour of foreign-invested enterprises. Through the survival analysis of 3,858 foreign-invested enterprises located in China from 2013 to 2020, the study found that operation profit, enterprise size and enterprise age have significantly negative impacts on the probability of enterprise withdrawal. At the industry-level and region-level, the improvement of industry entry rate and regional business environment ranking can significantly increase the probability of enterprise survival. The rise of the regional GDP growth rate and wage rate can significantly increase the probability of enterprise withdrawal. The study also found that the influence of some variables on enterprise withdrawal varies with different withdrawal patterns. After applying multiple models for estimation, similar results were replicated, which reinforced the validity of the conclusions offered in the present study

Finiteness and dimension of stated skein modules over Frobenius

When the quantum parameter $q^{1/2}$ is a root of unity of odd order. The stated skein module $S_{q^{1/2}}(M,\mathcal{N})$ has an $S_{1}(M,\mathcal{N})$-module structure, where $(M,\mathcal{N})$ is a marked three manifold. We prove $S_{q^{1/2}}(M,\mathcal{N})$ is a finitely generated $S_{1}(M,\mathcal{N})$-module when $M$ is compact, which furthermore indicates the reduced stated skein module for the compact marked three manifold is finite dimensional. We also give an upper bound for the dimension of $S_{q^{1/2}}(M,\mathcal{N})$ over $S_{1}(M,\mathcal{N})$ when $M$ is compact. For a pb surface $\Sigma$, we use $S_{q^{1/2}}(\Sigma)^{(N)}$ to denote the image of the Frobenius map when $q^{1/2}$ is a root of unity of odd order $N$. Then $S_{q^{1/2}}(\Sigma)^{(N)}$ lives in the center of the stated skein algebra $S_{q^{1/2}}(\Sigma)$. Let $\widetilde{S_{q^{1/2}}(\Sigma)^{(N)}}$ be the field of fractions of $S_{q^{1/2}}(\Sigma)^{(N)}$, and $\widetilde{S_{q^{1/2}}(\Sigma)}$ be $S_{q^{1/2}}(\Sigma)\otimes_{S_{q^{1/2}}(\Sigma)^{(N)}} \widetilde{S_{q^{1/2}}(\Sigma)^{(N)}}$. Then we show the dimension of $\widetilde{S_{q^{1/2}}(\Sigma)}$ over $\widetilde{S_{q^{1/2}}(\Sigma)^{(N)}}$ is $N^{3r(\Sigma)}$ where $r(\Sigma)$ equals to the number of boundary components of $\Sigma$ minus the Euler characteristic of $\Sigma$.Comment: 31 page