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E-grāmata: Entropy and Diversity: The Axiomatic Approach

(University of Edinburgh)
  • Formāts: PDF+DRM
  • Izdošanas datums: 22-Apr-2021
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781108962179
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Entropy and Diversity: The Axiomatic ApproachPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 22-Apr-2021
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781108962179
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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.

The biodiversity crisis is one of our most urgent problems. This book develops the mathematics behind the measurement of diversity, spanning a great breadth of subjects: from geometry to logic, from algebra to probability theory. The mathematical prerequisites are few, accessible to mathematicians and biologists with a mathematical background.

Recenzijas

'The book shows that the theory of diversity measurement is fertile soil for new mathematics, just as much as the neighboring but far more thoroughly worked field of information theory.' Hirokazu Nishimura, ZB Math Reviews 'Each new mathematical concept is introduced from fundamental principles, in a lucid, engaging style, keeping the core biological application in the foreground Leinster provides us with the tools to clearly assess what we are losing, so that we may focus better on what we can save.' Benjamin Allen, The Quarterly Review of Biology 'The book is a welcome addition to the theory of biodiversity in that it covers the topics of entropy and diversity in a mathematically rigorous yet fairly accessible way Recommended.' M. Bona, Choice Magazine

Papildus informācija

Discover the mathematical riches of 'what is diversity?' in a book that adds mathematical rigour to a vital ecological debate.
Acknowledgements xi
Note to the Reader xiii
Interdependence of
Chapters		 xiv
Introduction 1(14)
1 Fundamental Functional Equations
15(17)
1.1 Cauchy's Equation
16(7)
1.2 Logarithmic Sequences
23(5)
1.3 The q-Logarithm
28(4)
2 Shannon Entropy
32(30)
2.1 Probability Distributions on Finite Sets
33(6)
2.2 Definition and Properties of Shannon Entropy
39(5)
2.3 Entropy in Terms of Coding
44(8)
2.4 Entropy in Terms of Diversity
52(6)
2.5 The Chain Rule Characterizes Entropy
58(4)
3 Relative Entropy
62(29)
3.1 Definition and Properties of Relative Entropy
63(3)
3.2 Relative Entropy in Terms of Coding
66(4)
3.3 Relative Entropy in Terms of Diversity
70(4)
3.4 Relative Entropy in Measure Theory, Geometry and Statistics
74(11)
3.5 Characterization of Relative Entropy
85(6)
4 Deformations of Shannon Entropy
91(42)
4.1 q-Logarithmic Entropies
92(8)
4.2 Power Means
100(11)
4.3 Renyi Entropies and Hill Numbers
111(8)
4.4 Properties of the Hill Numbers
119(8)
4.5 Characterization of the Hill Number of a Given Order
127(6)
5 Means
133(36)
5.1 Quasiarithmetic Means
135(7)
5.2 Unweighted Means
142(7)
5.3 Strictly Increasing Homogeneous Means
149(6)
5.4 Increasing Homogeneous Means
155(7)
5.5 Weighted Means
162(7)
6 Species Similarity and Magnitude
169(55)
6.1 The Importance of Species Similarity
171(11)
6.2 Properties of the Similarity-Sensitive Diversity Measures
182(10)
6.3 Maximizing Diversity
192(14)
6.4 Introduction to Magnitude
206(11)
6.5 Magnitude in Geometry and Analysis
217(7)
7 Value
224(33)
7.1 Introduction to Value
226(10)
7.2 Value and Relative Entropy
236(4)
7.3 Characterization of Value
240(5)
7.4 Total Characterization of the Hill Numbers
245(12)
8 Mutual Information and Metacommunities
257(46)
8.1 Joint Entropy, Conditional Entropy and Mutual Information
258(11)
8.2 Diversity Measures for Subcommunities
269(4)
8.3 Diversity Measures for Metacommunities
273(10)
8.4 Properties of the Metacommunity Measures
283(11)
8.5 All Entropy Is Relative
294(5)
8.6 Beyond
299(4)
9 Probabilistic Methods
303(26)
9.1 Moment Generating Functions
304(3)
9.2 Large Deviations and Convex Duality
307(8)
9.3 Multiplicative Characterization of the p-Norms
315(7)
9.4 Multiplicative Characterization of the Power Means
322(7)
10 Information Loss
329(14)
10.1 Measure-Preserving Maps
330(6)
10.2 Characterization of Information Loss
336(7)
11 Entropy Modulo a Prime
343(25)
11.1 Fermat Quotients and the Definition of Entropy
344(8)
11.2 Characterizations of Entropy and Information Loss
352(3)
11.3 The Residues of Real Entropy
355(4)
11.4 Polynomial Approach
359(9)
12 The Categorical Origins of Entropy
368(27)
12.1 Operads and Their Algebras
369(8)
12.2 Categorical Algebras and Internal Algebras
377(7)
12.3 Entropy as an Internal Algebra
384(1)
12.4 The Universal Internal Algebra
385(10)
Appendix A Proofs of Background Facts
395(14)
A.1 Forms of the Chain Rule for Entropy
395(3)
A.2 The Expected Number of Species in a Random Sample
398(1)
A.3 The Diversity Profile Determines the Distribution
399(2)
A.4 Affine Functions
401(1)
A.5 Diversity of Integer Orders
402(1)
A.6 The Maximum Entropy of a Coupling
403(3)
A.7 Convex Duality
406(1)
A.8 Cumulant Generating Functions Are Convex
407(1)
A.9 Functions on a Finite Field
408(1)
Appendix B Summary of Conditions
409(3)
References 412(19)
Index of Notation 431(2)
Index 433
Tom Leinster is Professor of Category Theory at the University of Edinburgh, a member of the University of Glasgow's Boyd Orr Centre for Population and Ecosystem Health, and co-author of a highly-cited Ecology article on measuring biodiversity. He was awarded the 2019 Chauvenet Prize for mathematical writing.
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