We prove that the function Pα(x) =exp(1 − x−α) with α > 0 is 1/α-mild. We apply this result to obtain a uniform 1/α-mild parametrization of the family of curves {xy = 𝜖2 | (x,y) ∈ (0, 1)2} for 𝜖 ∈ (0, 1), which does not have a uniform 0-mild parametrization by work of Yomdin. More generally, we can parametrize families of power-subanalytic curves. This improves a result of Binyamini and Novikov that gives a 2-mild parametrization.

中文翻译：

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关于温和函数族

我们证明函数磷α(X) =经验(1 - X-α)和α > 0是1/α-温和的。我们应用这个结果来获得一个统一的1/α- 曲线族的温和参数化{X是的 = 𝜖2 | (X,是的) ∈ (0, 1)2}为了𝜖 ∈ (0, 1), 没有统一的0- 通过 Yomdin 的工作进行的轻度参数化。更一般地，我们可以参数化幂次解析曲线族。这改进了 Binyamini 和 Novikov 的结果，给出了2- 温和的参数化。