论文标题
有界$ {\ rm hess}^+$补充的功能产品
Products of functions with bounded ${\rm Hess}^+$ complement
论文作者
论文摘要
我们用$ {\ rm hess}^+$表示所有点的集合$ p \ in \ mathbb {r}^n $，以至于$ c^2 $ -SMOTH函数$ c^2 $ -smooth函数$ f：\ mathbb {r}^n \ longrightArrow puttion的hessian matrix $ h_p（f）$ ration在本文中，我们提供一类规范的多项式函数，其中包括大$ {\ rm hess}^+$区域，因为它们的$ {\ rm hess}^+$补充恰好是有限的。还提供了有关特定多项式函数的$ {\ rm hess}^+$区域以及其级别曲线的某些基本属性的详细分析，例如规律性，连接性和凸度。对于此类功能，我们还证明了其水平集的几种属性，例如连接性和凸度，以实现足够大的水平。除了上述示例的来源外，我们还提供了两个功能$ f，g：\ mathbb {r}^2 \ longrightArrow \ Mathbb {r} $的两个功能，并提供有限的$ {\ rm hess}^+$补充其产品$ fg $保持有限$ {\ rm Hess}^+$ refformant的产品$ fg $。
We denote by ${\rm Hess}^+$ the set of all points $p\in\mathbb{R}^n$ such that the Hessian matrix $H_p(f)$ of the $C^2$-smooth function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ is positive definite. In this paper we provide a class of norm-coercive polynomial functions with large ${\rm Hess}^+$ regions, as their ${\rm Hess}^+$ complements happen to be bounded. A detailed analysis concerning the ${\rm Hess}^+$ region of a particular polynomial function along with some basic properties of its level curves, such as regularity, connectedness and convexity, is also provided. For such functions we also prove several properties, such as connectedness and convexity, of their level sets for sufficiently large levels. Apart from the mentioned source of such examples we provide some sufficient conditions on two functions $f,g:\mathbb{R}^2\longrightarrow\mathbb{R}$ with bounded ${\rm Hess}^+$ complements whose product $fg$ keeps having bounded ${\rm Hess}^+$ complement as well.
