Discover the mathematical riches of 'what is diversity?' in a book that adds mathematical rigour to a vital ecological debate.

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.

This monograph offers a new foundation for information theory that is based on the notion of information-as-distinctions, being directly measured by logical entropy, and on the re-quantification as Shannon entropy, which is the fundamental concept for the theory of coding and communications. Information is based on distinctions, differences, distinguishability, and diversity. Information sets are defined that express the distinctions made by a partition, e.g., the inverse-image of a random variable so they represent the pre-probability notion of information. Then logical entropy is a probability measure on the information sets, the probability that on two independent trials, a distinction or “dit” of the partition will be obtained. The formula for logical entropy is a new derivation of an old formula that goes back to the early twentieth century and has been re-derived many times in different contexts. As a probability measure, all the compound notions of joint, conditional, and mutual logical entropy are immediate. The Shannon entropy (which is not defined as a measure in the sense of measure theory) and its compound notions are then derived from a non-linear dit-to-bit transform that re-quantifies the distinctions of a random variable in terms of bits—so the Shannon entropy is the average number of binary distinctions or bits necessary to make all the distinctions of the random variable. And, using a linearization method, all the set concepts in this logical information theory naturally extend to vector spaces in general—and to Hilbert spaces in particular—for quantum logical information theory which provides the natural measure of the distinctions made in quantum measurement. Relatively short but dense in content, this work can be a reference to researchers and graduate students doing investigations in information theory, maximum entropy methods in physics, engineering, and statistics, and to all those with a special interest in a new approach to quantum information theory.

Early modern Central Europe was the continent’s most decentralized region politically and its most diverse ethnically and culturally. With the onset of the Reformation, it also became Europe’s most religiously divided territory and potentially its most explosive in terms of confessional conflict and war. Focusing on the Holy Roman Empire and the Polish-Lithuanian Commonwealth, this volume examines the tremendous challenge of managing confessional diversity in Central Europe between 1500 and 1800. Addressing issues of tolerance, intolerance, and ecumenism, each chapter explores a facet of the complex dynamic between the state and the region’s Catholic, Protestant, Orthodox, Utraquist, and Jewish communities. The development of religious toleration—one of the most debated questions of the early modern period—is examined here afresh, with careful consideration of the factors and conditions that led to both confessional concord and religious violence.

This book provides an introduction to the role of diversity in complex adaptive systems. A complex system--such as an economy or a tropical ecosystem--consists of interacting adaptive entities that produce dynamic patterns and structures. Diversity plays a different role in a complex system than it does in an equilibrium system, where it often merely produces variation around the mean for performance measures. In complex adaptive systems, diversity makes fundamental contributions to system performance. Scott Page gives a concise primer on how diversity happens, how it is maintained, and how it affects complex systems. He explains how diversity underpins system level robustness, allowing for multiple responses to external shocks and internal adaptations; how it provides the seeds for large events by creating outliers that fuel tipping points; and how it drives novelty and innovation. Page looks at the different kinds of diversity--variations within and across types, and distinct community compositions and interaction structures--and covers the evolution of diversity within complex systems and the factors that determine the amount of maintained diversity within a system. Provides a concise and accessible introduction Shows how diversity underpins robustness and fuels tipping points Covers all types of diversity The essential primer on diversity in complex adaptive systems

In this text, Sanford, a retired Cornell professor, shows that the "Primary Axiom"--the foundational evolutionary premise that life is merely the result of mutations and natural selection--is false. He strongly refutes the Darwinian concept that man is just the result of a random and pointless natural process.

Life on earth is characterized by three striking phenomena that demand explanation: adaptation—the marvelous fit between organism and environment; diversity—the great variety of organisms; and complexity—the enormous intricacy of their internal structure. Natural selection explains adaptation. But what explains diversity and complexity? Daniel W. McShea and Robert N. Brandon argue that there exists in evolution a spontaneous tendency toward increased diversity and complexity, one that acts whether natural selection is present or not. They call this tendency a biological law—the Zero-Force Evolutionary Law, or ZFEL. This law unifies the principles and data of biology under a single framework and invites a reconceptualization of the field of the same sort that Newton’s First Law brought to physics. Biology’s First Law shows how the ZFEL can be applied to the study of diversity and complexity and examines its wider implications for biology. Intended for evolutionary biologists, paleontologists, and other scientists studying complex systems, and written in a concise and engaging format that speaks to students and interdisciplinary practitioners alike, this book will also find an appreciative audience in the philosophy of science.

This book is about phylogenetic diversity as an approach to reduce biodiversity losses in this period of mass extinction. Chapters in the first section deal with questions such as the way we value phylogenetic diversity among other criteria for biodiversity conservation; the choice of measures; the loss of phylogenetic diversity with extinction; the importance of organisms that are deeply branched in the tree of life, and the role of relict species. The second section is composed by contributions exploring methodological aspects, such as how to deal with abundance, sampling effort, or conflicting trees in analysis of phylogenetic diversity. The last section is devoted to applications, showing how phylogenetic diversity can be integrated in systematic conservation planning, in EDGE and HEDGE evaluations. This wide coverage makes the book a reference for academics, policy makers and stakeholders dealing with biodiversity conservation.

This accessible and timely book provides a comprehensive overview of how to measure biodiversity. The book highlights new developments, including innovative approaches to measuring taxonomic distinctness and estimating species richness, and evaluates these alongside traditional methods such as species abundance distributions, and diversity and evenness statistics. Helps the reader quantify and interpret patterns of ecological diversity, focusing on the measurement and estimation of species richness and abundance. Explores the concept of ecological diversity, bringing new perspectives to a field beset by contradictory views and advice. Discussion spans issues such as the meaning of community in the context of ecological diversity, scales of diversity and distribution of diversity among taxa Highlights advances in measurement paying particular attention to new techniques such as species richness estimation, application of measures of diversity to conservation and environmental management and addressing sampling issues Includes worked examples of key methods in helping people to understand the techniques and use available computer packages more effectively

This Is The First Comprehensive Book About Maximum Entropy Principle And Its Applications To A Diversity Of Fields Like Statistical Mechanics, Thermo-Dynamics, Business, Economics, Insurance, Finance, Contingency Tables, Characterisation Of Probability Distributions (Univariate As Well As Multivariate, Discrete As Well As Continuous), Statistical Inference, Non-Linear Spectral Analysis Of Time Series, Pattern Recognition, Marketing And Elections, Operations Research And Reliability Theory, Image Processing, Computerised Tomography, Biology And Medicine. There Are Over 600 Specially Constructed Exercises And Extensive Historical And Bibliographical Notes At The End Of Each Chapter.The Book Should Be Of Interest To All Applied Mathematicians, Physicists, Statisticians, Economists, Engineers Of All Types, Business Scientists, Life Scientists, Medical Scientists, Radiologists And Operations Researchers Who Are Interested In Applying The Powerful Methodology Based On Maximum Entropy Principle In Their Respective Fields.