关于X与$Σ$的对偶--邻近性
On a Duality between Metrics and $Σ$-Proximities
摘要：
在离散结构的研究中，函数经常被用来表示接近性，但不是度量。我们考虑一类这样的功能，其特征在于由一个正规化条件和不等式，起着同样的作用的三角不等式度量。我们证明了所引入的函数$Σ$ -邻近度在一定意义上是度量的对偶：度量与定义在同一有限集上的$Σ$ -邻近度之间存在自然的一一对应关系;与度量相反，$Σ$ -邻近度度量{ \it比较}邻近度;对象越近，$Σ$ -邻近度越大; $Σ$ -邻近矩阵的对角项表征元素的“中心性”。结果被推广到任意的无穷集。
英文摘要：
: In studies of discrete structures, functions are frequently used that express proximity, but are not metrics. We consider a class of such functions that is characterized by a normalization condition and an inequality that plays the same role as the triangle inequality does for metrics. We show that the introduced functions, named $Σ$-proximities, are in a definite sense dual to metrics: there exists a natural one-to-one correspondence between metrics and $Σ$-proximities defined on the same finite set; in contrast to metrics, $Σ$-proximities measure {\it comparative} proximity; the closer the objects, the greater the $Σ$-proximity; diagonal entries of the $Σ$-proximity matrix characterize the ``centrality'' of elements. The results are extended to arbitrary infinite sets.
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