变形对数和指数函数的完全与不完全定义
Complete versus incomplete definitions of the deformed logarithmic and exponential functions
摘要：
玻尔兹曼-吉布斯统计的最近推广在数学上依赖于通过一些变形参数定义的变形对数和指数函数。在本工作中，我们研究变形的对数/指数映射是否是从$\mathbb{R}^+/\mathbb{R}$（正实数/全实数集）到$\mathbb{R}/\mathbb{R}^+$（作为其未变形的对应物）的双射。我们证明了它们的逆映射只存在于上述（共）域的某些子集中。此外，我们提出的条件，广义变形函数必须满足，使最重要的性质的普通功能被保存。这些条件的满足使我们能够确定变形参数的有效区间。最后，我们将我们的分析，Tsallis，Kaniadakis，安倍晋三和Borges-Roditi变形功能。
英文摘要：
The recent generalizations of Boltzmann-Gibbs statistics mathematically relies on the deformed logarithmic and exponential functions defined through some deformation parameters. In the present work, we investigate whether a deformed logarithmic/exponential map is a bijection from $\mathbb{R}^+/\mathbb{R}$ (set of positive real numbers/all real numbers) to $\mathbb{R}/\mathbb{R}^+$, as their undeformed counterparts. We show that their inverse map exists only in some subsets of the aforementioned (co)domains. Furthermore, we present conditions which a generalized deformed function has to satisfy, so that the most important properties of the ordinary functions are preserved. The fulfillment of these conditions permits us to determine the validity interval of the deformation parameters. We finally apply our analysis to Tsallis, Kaniadakis, Abe and Borges-Roditi deformed functions.
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