Effectiveness of clauses restricting equity transfer in articles of association of limited liability companies using fuzzy AHP: A data analysis of related cases in China from 2010 to 20191

Based on the company cases published in China over the past ten years, both theoretical methods and Artificial intelligence technologies were applied to analysis cases data on the effectiveness of clauses restricting equity transfer in articles of association of limited liability companies (LLCs). With its unique characters based on shareholders and strong vitality, limited liability company (LLC), as the “evergreen tree” among the market players, is a company form adopted by many investors. Nevertheless, due to its prominent closed characteristics, equity transfer has become a bottleneck for the development of LLCs. According to this paper, it is necessary to distinguish between the effectiveness of clauses restricting internal and external equity transfer in articles of association of LLCs. Fuzzy Analytic Hierarchical Process (AHP) is utilized for which involves process of analytic hierarchy modelled with utilizing theory of fuzzy logic. Moreover, instead of being confined to the existing legal norms, the judgment standard of clauses restricting equity transfer in articles of association of LLCs should be comprehensively measured by the golden rules, i.e. “fairness”, “autonomy” and “operability”.

Multiplicity of closed Reeb orbits on dynamically convex $ \mathbb{R}P^{2n-1} $ for $ n\geq2 $

<p style='text-indent:20px;'>In this paper, we prove that there exist at least two non-contractible closed Reeb orbits on every dynamically convex <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}P^{2n-1} $\end{document}</tex-math></inline-formula>, and if all the closed Reeb orbits are non-degenerate, then there are at least <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> closed Reeb orbits, where <inline-formula><tex-math id="M5">\begin{document}$ n\geq2 $\end{document}</tex-math></inline-formula>, the main ingredient is that we generalize some theories developed by I. Ekeland and H. Hofer for closed characteristics on compact convex hypersurfaces in <inline-formula><tex-math id="M6">\begin{document}$ {{\bf R}}^{2n} $\end{document}</tex-math></inline-formula> to symmetric compact star-shaped hypersurfaces. In addition, we use Ekeland-Hofer theory to give a new proof of a theorem recently by M. Abreu and L. Macarini that every dynamically convex symmetric compact star-shaped hypersurface carries an elliptic symmetric closed characteristic.</p>

Rabinowitz Floer Homology for Tentacular Hamiltonians

This paper extends the definition of Rabinowitz Floer homology to non-compact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions on the periodic orbits and the moduli spaces of Floer trajectories. We introduce a class of hypersurfaces arising as the level sets of specific Hamiltonians: strongly tentacular Hamiltonians for which the compactness conditions are satisfied, cf. [ 21], thus enabling us to define the Rabinowitz Floer homology for this class. Rabinowitz Floer homology in turn serves as a tool to address the Weinstein conjecture and establish existence of closed characteristics for non-compact contact manifolds.

Non-hyperbolic P-Invariant Closed Characteristics on Partially Symmetric Compact Convex Hypersurfaces

Let {n\geq 2} be an integer, {P=\mathrm{diag}(-I_{n-\kappa},I_{\kappa},-I_{n-\kappa},I_{\kappa})} for some integer {\kappa\in[0,n]}, and let {\Sigma\subset{\mathbb{R}}^{2n}} be a partially symmetric compact convex hypersurface, i.e., {x\in\Sigma} implies {Px\in\Sigma}, and {(r,R)}-pinched. In this paper, we prove that when {{R/r}<\sqrt{5/3}} and {0\leq\kappa\leq[\frac{n-1}{2}]}, there exist at least {E(\frac{n-2\kappa-1}{2})+E(\frac{n-2\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ. In addition, when {{R/r}<\sqrt{3/2}}, {[\frac{n+1}{2}]\leq\kappa\leq n} and Σ carries exactly nP-invariant closed characteristics, then there exist at least {2E(\frac{2\kappa-n-1}{4})+E(\frac{n-\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ, where the function {E(a)} is defined as {E(a)=\min{\{k\in{\mathbb{Z}}\mid k\geq a\}}} for any {a\in\mathbb{R}}.