Librería Portfolio

The global biodiversity crisis is one of humanity´s most urgent problems, but even quantifying biological diversity is a difficult mathematical and conceptual challenge. This book brings new mathematical rigour to the ongoing debate. It was born of research in category theory, is given strength by information theory, and is fed by the ancient field of functional equations. It applies the power of the axiomatic method to a biological problem of pressing concern, but it also presents new theorems that stand up as mathematics in their own right, independently of any application. The question ´what is diversity?´ has surprising mathematical depth, and this book covers a wide breadth of mathematics, from functional equations to geometric measure theory, from probability theory to number theory. Despite this range, the mathematical prerequisites are few: the main narrative thread of this book requires no more than an undergraduate course in analysis.
Brings new mathematical rigour to the quest for good ways to measure biological diversity
Assumes only a modest mathematical background
Showcases the connections between many apparently disparate branches of mathematics
Introduces the reader to a new set of mathematical ideas and a new way of understanding the established concept of entropy



Table of Contents

Introduction
1. Fundamental functional equations
2. Shannon entropy
3. Relative entropy
4. Deformations of Shannon entropy
5. Means
6. Species similarity and magnitude
7. Value
8. Mutual information and metacommunities
9. Probabilistic methods
10. Information loss
11. Entropy modulo a prime
12. The categorical origins of entropy
Appendix A. Proofs of background facts
Appendix B. Summary of conditions
References
Index of notation
Subject index.