论文标题
围绕Efimov的同态差异测试
Around Efimov's differential test for homeomorphism
论文作者
论文摘要
在1968年，n。 $ a（x）> 0 $ and常数$ C_1 \ geqslant 0 $，$ c_2 \ geqslant 0 $ 0 $，使得不等式$ | 1/a（x）-1/a（y）| \ leqslant c_1 | x-y | x-y | x-y |+c_2 $和$ | a（x）| \ operatoRatorname {curl} f（x）|+a^2（x）$保持所有$ x，y \ in \ mathbb {r}^2 $。然后$ f（\ mathbb {r}^2）$是一个凸域，$ f $ maps $ \ mathbb {r}^2 $ to $ f（\ mathbb {r}^2）$ homeomorphally.someomorphally。 $ x \ in \ mathbb {r}^2 $。本文概述了该定理的类似物，其在表面理论中的概括和应用，全球逆函数理论以及雅各布猜想的研究以及动力学系统的全球渐近稳定性的研究。
In 1968, N.\,V.~Efimov proved the following remarkable theorem: \textit{Let $f:\mathbb{R}^2\to\mathbb{R}^2\in C^1$ be such that $\det f'(x)<0$ for all $x\in\mathbb{R}^2$ and let there exist a function $a(x)>0$ and constants $C_1\geqslant 0$, $C_2\geqslant 0$ such that the inequalities $|1/a(x)-1/a(y)|\leqslant C_1 |x-y|+C_2$ and $|\det f'(x)|\geqslant a(x)|\operatorname{curl}f(x)|+a^2(x)$ hold true for all $x, y\in\mathbb{R}^2$. Then $f(\mathbb{R}^2)$ is a convex domain and $f$ maps $\mathbb{R}^2$ onto $f(\mathbb{R}^2)$ homeomorphically.} Here $\operatorname{curl}f(x)$ stands for the curl of $f$ at $x\in\mathbb{R}^2$. This article is an overview of analogues of this theorem, its generalizations and applications in the theory of surfaces, theory of global inverse functions, as well as in the study of the Jacobian Conjecture and the global asymptotic stability of dynamical systems.
