论文标题
机动空间及其大地路径
Mobility spaces and their geodesic paths
论文作者
论文摘要
我们引入了一个代数系统，该系统可以用作其在任何两个点之间具有大地路径的空间的模型。这种新的代数结构基于移动性代数的概念，该概念最近引入了实数单位间隔的模型。移动性代数与三个$ a $一起，以及三个常数和一个三元操作。对于封闭的单位间隔$ a = [0,1] $，三个常数为0、1和1/2，而三元操作为$ p（x，y，z）= x-yx+yz $。移动性空间与地图$ q \ colon {x \ times a \ times a \ times x \ to x} $在一起，含义$ q（x，x，t，y）$表示沿$ t in a $ in a $ the geodesic path in the geodesiC $ x $ x $ x $的$ q（x，t，t，y）$表示粒子的位置。因此，与移动性代数相对于模块在环上定义的方式相同。我们介绍了移动空间的公理，研究主要特性并给出了例子。我们还建立了代数上下文与具有大地路径的空间之间的联系。简要提到了与仿射空间的连接。
We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for the unit interval of real numbers. Mobility algebras consist on a set $A$ together with three constants and a ternary operation. In the case of the closed unit interval $A=[0,1]$, the three constants are 0, 1 and 1/2 while the ternary operation is $p(x,y,z)=x-yx+yz$. A mobility space is a set $X$ together with a map $q\colon{X\times A\times X\to X}$ with the meaning that $q(x,t,y)$ indicates the position of a particle moving from point $x$ to point $y$ at the instant $t\in A$, along a geodesic path within the space $X$. A mobility space is thus defined with respect to a mobility algebra, in the same way as a module is defined over a ring. We introduce the axioms for mobility spaces, investigate the main properties and give examples. We also establish the connection between the algebraic context and the one of spaces with geodesic paths. The connection with affine spaces is briefly mentioned.
