测量的微分不变量及其与中心矩的关系
Differential invariants of measurements, and their connection to central moments
摘要：
由于最小信息增益的原理，仿射空间$V$中的点的测量确定$V \times V^* \times \mathbb R$的勒让德子流形。这样的勒让德子流形配备了额外的几何结构，这些几何结构来自底层概率分布的中心矩，并且在$V$上的仿射变换群的作用下是不变的。我们调查的行动，这组仿射变换上的勒让德子流形的$V \times V^* \times \mathbb R$给出了详细的概述结构的代数的标量微分不变量。我们展示了如何中心矩可以用来构建标量微分不变量。最后，我们在气体平衡热力学的背景下查看结果，我们注意到热容是微分不变量之一。
英文摘要：
Due to the principle of minimal information gain, the measurement of points in an affine space $V$ determines a Legendrian submanifold of $V \times V^* \times \mathbb R$. Such Legendrian submanifolds are equipped with additional geometric structures that come from the central moments of the underlying probability distributions and are invariant under the action of the group of affine transformations on $V$. We investigate the action of this group of affine transformations on Legendrian submanifolds of $V \times V^* \times \mathbb R$ by giving a detailed overview of the structure of the algebra of scalar differential invariants. We show how the central moments can be used to construct the scalar differential invariants. In the end, we view the results in the context of equilibrium thermodynamics of gases, where we notice that the heat capacity is one of the differential invariants.
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