论文标题
Yang-Baxter地图和独立保存财产
Yang-Baxter maps and independence preserving property
论文作者
论文摘要
我们研究了生物功能的两个属性之间的令人惊讶的关系$ f：\ Mathcal {x} \ times \ Mathcal {x} \ to \ Mathcal {x} \ times \ times \ times \ nathcal {x} $，用于set $ \ mathcal {x} $，这些$ \ mathcal {x} $来自不同背景。 One of the property is that $F$ is a Yang-Baxter map, namely it satisfies the "set-theoretical" Yang-Baxter equation, and the other property is the independence preserving property (IP property for short), which means that there exist independent (non-constant) $\mathcal{X}$-valued random variables $X,Y$ such that $U,V$ are also independent with $（u，v）= f（x，y）$。最近，在研究离散集成系统的不变度量的研究中，发现了具有这两种属性的一类功能。在此激励的基础上，我们分析了Yang-baxter地图与IP属性之间的关系，而Yang-Baxter Maps和IP属性从未对此进行研究，重点是$ \ Mathcal {x} = \ Mathbb {r} _+$。我们的第一个主要结果是，所有二次杨 - 巴克斯特映射$ f：\ mathbb {r} _+ \ times \ times \ mathbb {r} _+ \ to \ mathbb {r} _+ \ times \ times \ times \ mathbb {r} {r} _+ $在最有趣的子集中属于独立的属性。特别是，我们发现具有IP属性的新类型的新类别。我们的第二个主要结果是，这些新介绍的双物种在具有IP属性的（已知的）两种属性的类别中至关重要，从某种意义上说，大多数具有IP属性的已知属性通过使用特殊参数或执行某些有限的过程来从这些地图中得出。这表明可以单独研究针对特定功能的IP属性可以以统一的方式理解。
We study a surprising relationship between two properties for bijective functions $F : \mathcal{X} \times \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ for a set $\mathcal{X}$ which are introduced from very different backgrounds. One of the property is that $F$ is a Yang-Baxter map, namely it satisfies the "set-theoretical" Yang-Baxter equation, and the other property is the independence preserving property (IP property for short), which means that there exist independent (non-constant) $\mathcal{X}$-valued random variables $X,Y$ such that $U,V$ are also independent with $(U,V)=F(X,Y)$. Recently in the study of invariant measures for a discrete integrable system, a class of functions having these two properties were found. Motivated by this, we analyze a relationship between the Yang-Baxter maps and the IP property, which has never been studied as far as we are aware, focusing on the case $\mathcal{X}=\mathbb{R}_+$. Our first main result is that all quadrirational Yang-Baxter maps $F : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+ \times \mathbb{R}_+$ in the most interesting subclass have the independence preserving property. In particular, we find new classes of bijections having the IP property. Our second main result is that these newly introduce bijections are fundamental in the class of (known) bijections with the IP property, in the sense that most of known bijections having the IP property are derived from these maps by taking special parameters or performing some limiting procedure. This reveals that the IP property, which has been investigated for specific functions individually, can be understood in a unified manner.
