Английская Википедия:Diversity (mathematics)

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In mathematics, a diversity is a generalization of the concept of metric space. The concept was introduced in 2012 by Bryant and Tupper, who call diversities "a form of multi-way metric". The concept finds application in nonlinear analysis.

Given a set <math>X</math>, let <math> \wp_\mbox{fin}(X)</math> be the set of finite subsets of <math>X</math>. A diversity is a pair <math>(X,\delta)</math> consisting of a set <math>X</math> and a function <math>\delta \colon \wp_\mbox{fin}(X) \to \mathbb{R}</math> satisfying

(D1) <math>\delta(A)\geq 0</math>, with <math>\delta(A)=0</math> if and only if <math>\left|A\right|\leq 1</math>

and

(D2) if <math> B\neq\emptyset</math> then <math>\delta(A\cup C)\leq\delta(A\cup B) + \delta(B \cup C)</math>.

Bryant and Tupper observe that these axioms imply monotonicity; that is, if <math>A\subseteq B</math>, then <math>\delta(A)\leq\delta(B)</math>. They state that the term "diversity" comes from the appearance of a special case of their definition in work on phylogenetic and ecological diversities. They give the following examples:

Diameter diversity

Let <math>(X,d)</math> be a metric space. Setting <math>\delta(A)=\max_{a,b\in A} d(a,b)=\operatorname{diam}(A)</math> for all <math>A\in\wp_\mbox{fin}(X)</math> defines a diversity.

<math>L_1</math> diversity

For all finite <math>A\subseteq\mathbb{R}^n</math> if we define <math>\delta(A)=\sum_i\max_{a,b}\left\{\left| a_i-b_i\right|\colon a,b\in A\right\}</math> then <math>(\mathbb{R}^n,\delta)</math> is a diversity.

Phylogenetic diversity

If T is a phylogenetic tree with taxon set X. For each finite <math>A\subseteq X</math>, define <math>\delta(A)</math> as the length of the smallest subtree of T connecting taxa in A. Then <math>(X, \delta)</math> is a (phylogenetic) diversity.

Steiner diversity

Let <math>(X, d)</math> be a metric space. For each finite <math>A\subseteq X</math>, let <math>\delta(A)</math> denote the minimum length of a Steiner tree within X connecting elements in A. Then <math>(X,\delta)</math> is a diversity.

Truncated diversity

Let <math>(X,\delta)</math> be a diversity. For all <math>A\in\wp_\mbox{fin}(X)</math> define <math>\delta^{(k)}(A) = \max\left\{\delta(B)\colon |B|\leq k, B\subseteq A\right\}</math>. Then if <math>k\geq 2</math>, <math>(X,\delta^{(k)})</math> is a diversity.

Clique diversity

If <math>(X,E)</math> is a graph, and <math>\delta(A)</math> is defined for any finite A as the largest clique of A, then <math>(X,\delta)</math> is a diversity.

References

Шаблон:Metric-geometry-stub