标题： 离散动力学与超几何学 显示英文标题

标题： Discrete Dynamics and Supergeometry

摘要： 我们提出了一个关于具有连续和离散自由度的动力学经典系统的几何测量理论。 该方法相对于时钟的选择是协变的，并且自然地包含了实验室。 后者是奇数维辛流形中的嵌入辛子流形。 当适当地定义时，奇数维的辛几何正是协变所需的结构。 一种根本上概率性的观点允许经典的超几何描述离散动力学。 我们解决了在给定（可能是奇数维）超辛结构的情况下如何在超流形上构建概率测度的问题。 这依赖于微分形式的超类比Hodge星以及由凸锥描述的概率。 我们还展示了如何用超几何描述马尔可夫链之类的随机过程。 显示英文摘要

摘要： We formulate a geometric measurement theory of dynamical classical systems possessing both continuous and discrete degrees of freedom. The approach is covariant with respect to choices of clocks and canonically incorporates laboratories. The latter are embedded symplectic submanifolds of an odd-dimensional symplectic structure. When suitably defined, symplectic geometry in odd dimensions is exactly the structure needed for covariance. A fundamentally probabilistic viewpoint allows classical supergeometries to describe discrete dynamics. We solve the problem of how to construct probabilistic measures on supermanifolds given a (possibly odd dimensional) supersymplectic structure. This relies on a superanalog of the Hodge star for differential forms and a description of probabilities by convex cones. We also show how stochastic processes such as Markov chains can be described by supergeometry.