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August 5, 2025

(BT) Diversity from (LC) Diversity

Posted by Tom Leinster

Guest post by Mark Meckes

Around 2010, in papers that both appeared in print in 2012, two different mathematical notions were introduced and given the name “diversity”.

One, introduced by Tom Leinster and Christina Cobbold, is already familiar to regular readers of this blog. Say XX is a finite set, and for each x,y∈Xx,y \in X we have a number Z(x,y)=Z(y,x)∈[0,1]Z(x,y) = Z(y,x) \in [0,1] that specifies how “similar” xx and yy are. (Typically we also assume Z(x,x)=1Z(x,x) = 1.) Fix a parameter q∈[0,∞]q \in [0,\infty]. If pp is a probability distribution on XX, then the quantity D q Z(p)=(∑ x∈supp(p)(∑ y∈supp(p)Z(x,y)p(y)) q−1p(x)) 1/(1−q) D_q^Z(p) = \left(\sum_{x\in supp(p)} \left( \sum_{y\in supp(p)} Z(x,y) p(y)\right)^{q-1} p(x)\right)^{1/(1-q)} (with the cases q=1,∞q=1,\infty defined by taking limits) can be interpreted as the “effective number of points” in XX, taking into account both the similarities between points as quantified by ZZ and the weights specified by pp. Its logarithm logD q Z(p)\log D_q^Z(p) is a refinement of the qq-Rényi entropy of pp. The main motivating example is when XX is a set of species of organisms present in an ecosystem, and D q Z(p)D_q^Z(p) quantifies the “effective number of species” in XX, accounting for both similarities between species and their relative abundances. This family of quantities turns out to subsume many of the diversity measures previously introduced in the theoretical ecology literature, and they are now often referred to as Leinster–Cobbold diversities.

Continue reading “(BT) Diversity from (LC) Diversity” …
Posted at 4:34 PM UTC | Permalink | Followups (3)

August 2, 2025

Jack Morava

Posted by John Baez

Today I heard from David Benson that Jack Morava died yesterday. This comes as such a huge shock that I can’t help but hope Benson was somehow misinformed. Morava has been posting comments to the n-Café and sending emails to me even very recently.

This is all I know, now.

Posted at 12:22 PM UTC | Permalink | Followups (1)