Preface to the Princeton Landmarks in Biology Edition vii Preface xi Symbols Used xiii 1. The Importance of Islands 3 2. Area and Number of Speicies 8 3. Further Explanations of the Area-Diversity Pattern 19 4. The Strategy of Colonization 68 5. Invasibility and the Variable Niche 94 6. Stepping Stones and Biotic Exchange 123 7. Evolutionary Changes Following Colonization 145 8. Prospect 181 Glossary 185 References 193 Index 201

The Theory of Island Biogeography

Measuring the Strangeness of Strange Attractors

Complex networks arise in a wide range of biological and sociotechnical systems. Epidemic spreading is central to our understanding of dynamical processes in complex networks, and is of interest to physicists, mathematicians, epidemiologists, and computer and social scientists. This review presents the main results and paradigmatic models in infectious disease modeling and generalized social contagion processes.

Epidemic processes in complex networks

It is rather paradoxical that, although entropy is one of the most important quantities in physics, its main properties are rarely listed in the usual textbooks on statistical mechanics. In this paper we try to fill this gap by discussing these properties, as, for instance, invariance, additivity, concavity, subadditivity, strong subadditivity, continuity, etc., in detail, with reference to their implications in statistical mechanics. In addition, we consider related concepts such as relative entropy, skew entropy, dynamical entropy, etc. Taking into account that statistical mechanics deals with large, essentially infinite systems, we finally will get a glimpse of systems with infinitely many degrees of freedom.

General properties of entropy

A new approach to correlated Fermi systems such as the Hubbard model, the periodic Anderson model etc. is discussed, which makes use of the limit of high spatial dimensions. This limit — which is wellknown in the case of classical as well as localized quantum spin models — is found to be very helpful also in the case of quantum mechanical models with itinerant degrees of freedom. Many investigations, which are prohibitively difficult in lower dimensions, become tractable in this limit. In particular, essential features of systems in d = 3, and even lower dimensions, are very well described by the results in d = ∞ or expansions around this limit. A brief review of the state-of-the-art is presented.

Correlated lattice fermions in d=∞ dimensions

Integrable mappings and soliton equations II

This article traces the development and study of phase transitions from late last century to the present day. We begin with a brief historical sketch and a description of the statistical mechanics of phase transitions. Particular attention is given to the modern era which began in 1944 with Onsager's celebrated solution of the two-dimensional Ising model. Points of development since Onsager, which are highlighted in this article, include the study of critical exponents, the scaling hypothesis, realisation of the universality of critical exponents and the recent renormalization group approach to critical phenomena. The basic idea of the renormalization group method, rather than the detailed application of the recipe to particular cases, is stressed and is discussed critically in some detail.

The Statistical Mechanics of Phase Transitions

Integrable mappings and soliton equations

We prove for Hermitian matrices (or more generally for completely continuous self‐adjoint linear operators in Hilbert space)A and B that Tr (eA+B ) ≤ Tr (eAeB ). The inequality is shown to be sharper than the convexity property (0 ≤ α ≤ 1) Tr (e αA+(1−α)B ) ≤ [Tr (eA )]α[Tr (eB )]1−α, and its possible use for obtaining upper bounds for the partition function is discussed briefly.

Inequality with Applications in Statistical Mechanics

For a one-dimensional system of classical spins with nearest neighbour Heisenberg interaction we derive the equation of motion for each three-dimensional spin vector. In the continuum limit where the spins lie dense on a line this set of equations reduces to a nonlinear partial differential equation. In addition to spin-wave solutions we obtain some other special solutions of this equation. In particular we find solitary waves having total energy localised in a finite region, with velocity of propagation inversely proportional to the width of this region. Solutions of still another type are shown to have a diffusive character. The stability of such solutions and the possibility of interaction of two or more solitary waves have not yet been studied.

Colin J. Thompson

Papers

Integrable mappings and soliton equations II

The Statistical Mechanics of Phase Transitions

Integrable mappings and soliton equations

Inequality with Applications in Statistical Mechanics

On the dynamics of a continuum spin system