Tom Leinster

Measuring diversity: the importance of species similarity

Realistic measures of biodiversity should reflect not only the relative abundances of species, but also the differences between them. We present a natural family of diversity measures taking both factors into account. This is not just another addition to the already long list of diversity indices. Instead, a single formula subsumes many of the most popular indices, including Shannons, Simpsons, species richness, and Raos quadratic entropy. These popular indices can then be used and understood in a unified way, and the relationships between them are made plain. The new measures are, moreover, effective numbers, so that percentage changes and ratio comparisons of diversity value are meaningful. We advocate the use of diversity profiles, which provide a faithful graphical representation of the shape of a community; they show how the perceived diversity changes as the emphasis shifts from rare to common species. Communities can usefully be compared by comparing their diversity profiles. We show by example that this is a far more subtle method than any relying on a single statistic. Some ecologists view diversity indices with suspicion, questioning whether they are biologically meaningful. By dropping the naive assumption that distinct species have nothing in common, working with effective numbers, and using diversity profiles, we arrive at a system of diversity measurement that should lay much of this suspicion to rest.

On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.

Generalized enrichment of categories

Abstract We define the phrase ‘category enriched in an fc -multicategory’ and explore some examples. An fc -multicategory is a very general kind of two-dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fc -multicategory extends the (more or less well-known) theories of enrichment in a monoidal category, in a bicategory, and in a multicategory. Moreover, fc -multicategories provide a natural setting for the bimodules construction, traditionally performed on suitably cocomplete bicategories. Although this paper is elementary and self-contained, we also explain why, from one point of view, fc -multicategories are the natural structures in which to enrich categories.

An objective representation of the Gaussian integers

Abstract A rig is a ring without negatives. We analyse the free rig on a generator x subject to the equivalence x∼1+x+x2, showing that in it the non-constant polynomials form a ring. This ring can be identified with the Gaussian integers, which thus acquire objective meaning.

THE MAGNITUDE OF A METRIC SPACE: FROM CATEGORY THEORY TO GEOMETRIC MEASURE THEORY

Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out to encode many invariants from integral geometry and geometric measure theory, including volume, capacity, dimension, and intrinsic volumes. This paper gives an overview of the theory of magnitude, from its category-theoretic genesis to its connections with these geometric quantities. Some new results are proved, including a geometric formula for the magnitude of a convex body in l n.

Maximizing Diversity in Biology and Beyond

Entropy, under a variety of names, has long been used as a measure of diversity in ecology, as well as in genetics, economics and other fields. There is a spectrum of viewpoints on diversity, indexed by a real parameter q giving greater or lesser importance to rare species. Leinster and Cobbold proposed a one-parameter family of diversity measures taking into account both this variation and the varying similarities between species. Because of this latter feature, diversity is not maximized by the uniform distribution on species. So it is natural to ask: which distributions maximize diversity, and what is its maximum value? In principle, both answers depend on q, but our main theorem is that neither does. Thus, there is a single distribution that maximizes diversity from all viewpoints simultaneously, and any list of species has an unambiguous maximum diversity value. Furthermore, the maximizing distribution(s) can be computed in finite time, and any distribution maximizing diversity from some particular viewpoint q > 0 actually maximizes diversity for all q. Although we phrase our results in ecological terms, they apply very widely, with applications in graph theory and metric geometry.

General self-similarity: An overview

Informal seminar notes explaining the ideas in math.DS/0411344 and math.DS/0411345.

Maths spying: the quandary of working for the spooks

Intelligence agencies hire lots of mathematicians, but would-be employees must realise that their work is misused to snoop on everyone, says Tom Leinster

A multiplicative characterization of the power means

A startlingly simple characterization of the p-norms has recently been found by Aubrun and Nechita (arXiv:1102.2618) and by Fernandez-Gonzalez, Palazuelos and Perez-Garcia. We deduce a simple characterization of the power means of order greater than or equal to 1.