16 research outputs found
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.Comment: 29 page
Spaces of extremal magnitude
Magnitude is a numerical invariant of compact metric spaces. Its theory is
most mature for spaces satisfying the classical condition of being of negative
type, and the magnitude of such a space lies in the interval .
Until now, no example with magnitude was known. We construct some,
thus answering a question open since 2010. We also give a sufficient condition
for the magnitude of a space to converge to 1 as it is scaled down to a point,
unifying and generalizing previously known conditions.Comment: 7 page
Decomposition of the Leinster-Cobbold Diversity Index
The Leinster and Cobbold diversity index possesses a number of merits; in
particular, it generalises many existing indices and defines an effective
number. We present a scheme to quantify the contribution of richness, evenness,
and taxonomic similarity to this index. Compared to the work of van Dam (2019),
our approach gives unbiased estimates of both evenness and similarity in a
non-homogeneous community. We also introduce a notion of taxonomic tree
equilibration which should be of use in the description of community structure.Comment: 10 pages, 1 figur
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Chris Cannings: A Life in Games
Chris Cannings was one of the pioneers of evolutionary game theory. His early work was inspired by the formulations of John Maynard Smith, Geoff Parker and Geoff Price; Chris recognized the need for a strong mathematical foundation both to validate stated results and to give a basis for extensions of the models. He was responsible for fundamental results on matrix games, as well as much of the theory of the important war of attrition game, patterns of evolutionarily stable strategies, multiplayer games and games on networks. In this paper we describe his work, key insights and their influence on research by others in this increasingly important field. Chris made substantial contributions to other areas such as population genetics and segregation analysis, but it was to games that he always returned. This review is written by three of his students from different stages of his career
GAIT: A Geometric Approach to Information Theory
We advocate the use of a notion of entropy that reflects the relative
abundances of the symbols in an alphabet, as well as the similarities between
them. This concept was originally introduced in theoretical ecology to study
the diversity of ecosystems. Based on this notion of entropy, we introduce
geometry-aware counterparts for several concepts and theorems in information
theory. Notably, our proposed divergence exhibits performance on par with
state-of-the-art methods based on the Wasserstein distance, but enjoys a
closed-form expression that can be computed efficiently. We demonstrate the
versatility of our method via experiments on a broad range of domains: training
generative models, computing image barycenters, approximating empirical
measures and counting modes.Comment: Replaces the previous version named "GEAR: Geometry-Aware R\'enyi
Information
Entropic exercises around the Kneser-Poulsen conjecture
We develop an information-theoretic approach to study the Kneser--Poulsen
conjecture in discrete geometry. This leads us to a broad question regarding
whether R\'enyi entropies of independent sums decrease when one of the summands
is contracted by a -Lipschitz map. We answer this question affirmatively in
various cases.Comment: 23 pages, comments welcome! Final version with minor changes, added
Corollary 2.8 (linear contractions decrease intrinsic volumes of convex
bodies
Effective Number Theory: Counting the Identities of a Quantum State
Quantum physics frequently involves a need to count the states, subspaces, measurement outcomes, and other elements of quantum dynamics. However, with quantum mechanics assigning probabilities to such objects, it is often desirable to work with the notion of a “total” that takes into account their varied relevance. For example, such an effective count of position states available to a lattice electron could characterize its localization properties. Similarly, the effective total of outcomes in the measurement step of a quantum computation relates to the efficiency of the quantum algorithm. Despite a broad need for effective counting, a well-founded prescription has not been formulated. Instead, the assignments that do not respect the measure-like nature of the concept, such as versions of the participation number or exponentiated entropies, are used in some areas. Here, we develop the additive theory of effective number functions (ENFs), namely functions assigning consistent totals to collections of objects endowed with probability weights. Our analysis reveals the existence of a minimal total, realized by the unique ENF, which leads to effective counting with absolute meaning. Touching upon the nature of the measure, our results may find applications not only in quantum physics, but also in other quantitative sciences
Magnitude, homology, and the Whitney twist
Magnitude is a numerical invariant of metric spaces and graphs, analogous, in
a precise sense, to Euler characteristic. Magnitude homology is an algebraic
invariant constructed to categorify magnitude. Among the important features of
the magnitude of graphs is its behaviour with respect to an operation known as
the Whitney twist. We give a homological account of magnitude's invariance
under Whitney twists, extending the previously known result to encompass a
substantially wider class of gluings. As well as providing a new tool for the
computation of magnitudes, this is the first new theorem about magnitude to be
proved using magnitude homology.Comment: 24 page