Lower bound for the Poisson bracket invariant on surfaces

Recently, L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket conjecture by Leonid Polterovich in dimension 2. In this note, instead of open cover constituted of displaceable sets in their work, considering open cover constituted of topological discs we give a sufficient and necessary condition that Poisson bracket invariants of these covers are positive.Comment: 10 page

Understanding the Zhejiang industrial clusters:Questions and re-evaluations

In this study, we have chosen to focus on the cluster of business conglomerates, mainly comprising SMEs engaging in the same type of activity - similar to those observed in Europe (and Italy in particular) some thirty years ago. There are numerous zones of highly specialised development in China. However, among these, Zhejiang province is undoubtedly considered the most remarkable, and is seen as a role model in modern evolution towards a market economy. How have the Zhejiang clusters developed? What are the characteristics of this development? Is it possible to talk about a "third China" along the same lines as the former "3rd Italy" with regard to industrial districts? Do Chinese clusters in fact have any specific characteristics? These are some of the questions that we will look to address today.cluster, Asia

A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs

In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for the kinetic chemotaxis system with random inputs, which will converge to the modified Keller-Segel model with random inputs in the diffusive regime. Based on the generalized Polynomial Chaos (gPC) approach, we design a high order stochastic Galerkin method using implicit-explicit (IMEX) Runge-Kutta (RK) time discretization with a macroscopic penalty term. The new schemes improve the parabolic CFL condition to a hyperbolic type when the mean free path is small, which shows significant efficiency especially in uncertainty quantification (UQ) with multi-scale problems. The stochastic Asymptotic-Preserving property will be shown asymptotically and verified numerically in several tests. Many other numerical tests are conducted to explore the effect of the randomness in the kinetic system, in the aim of providing more intuitions for the theoretic study of the chemotaxis models

Graphs with Diameter n−en-e Minimizing the Spectral Radius

The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix $A(G)$. For a fixed integer $e\ge 1$, let $G^{min}_{n,n-e}$ be a graph with minimal spectral radius among all connected graphs on $n$ vertices with diameter $n-e$. Let $P_{n_1,n_2,...,n_t,p}^{m_1,m_2,...,m_t}$ be a tree obtained from a path of $p$ vertices ($0 \sim 1 \sim 2 \sim ... \sim (p-1)$) by linking one pendant path $P_{n_i}$ at $m_i$ for each $i\in\{1,2,...,t\}$. For $e=1,2,3,4,5$, $G^{min}_{n,n-e}$ were determined in the literature. Cioab\v{a}-van Dam-Koolen-Lee \cite{CDK} conjectured for fixed $e\geq 6$, $G^{min}_{n,n-e}$ is in the family ${\cal P}_{n,e}=\{P_{2,1,...1,2,n-e+1}^{2,m_2,...,m_{e-4},n-e-2}\mid 2<m_2<...<m_{e-4}<n-e-2\}$. For $e=6,7$, they conjectured $G^{min}_{n,n-6}=P^{2,\lceil\frac{D-1}{2}\rceil,D-2}_{2,1,2,n-5}$ and $G^{min}_{n,n-7}=P^{2,\lfloor\frac{D+2}{3}\rfloor,D- \lfloor\frac{D+2}{3}\rfloor, D-2}_{2,1,1,2,n-6}$. In this paper, we settle their three conjectures positively. We also determine $G^{min}_{n,n-8}$ in this paper