论文标题
测量的差异不变及其与中心时刻的联系
Differential invariants of measurements, and their connection to central moments
论文作者
论文摘要
由于信息增益最少的原则，仿射空间中的点的测量$ v $确定了$ v \ times v^* \ times \ times \ mathbb r $的legendrian submanifold。这种Legendrian Submanifolds配备了来自基本概率分布的中心时刻的其他几何结构，并且在$ V $上的仿射转换组的行动下是不变的。我们通过详细概述了标量差异不变的代数结构的详细概述，研究了$ v \ times v^* \ times \ mathbb r $的Legendrian submanifolds的动作。我们展示了如何使用中心时刻来构建标量差异不变。最后，我们在气体平衡热力学的背景下查看结果，在那里我们注意到热容量是差异不变的之一。
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.
