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Giving a detailed overview of the subject, this book takes in the results and methods that have arisen since the term 'self-organised criticality' was coined twenty years ago. Providing an overview of numerical and analytical methods, from their theoretical foundation to the actual application and implementation, the book is an easy access point to important results and sophisticated methods. Starting with the famous Bak-Tang-Wiesenfeld sandpile, ten key models are carefully defined, together with their results and applications. Comprehensive tables of numerical results are collected in one volume for the first time, making the information readily accessible to readers. Written for graduate students and practising researchers in a range of disciplines, from physics and mathematics to biology, sociology, finance, medicine and engineering, the book gives a practical, hands-on approach throughout. Methods and results are applied in ways that will relate to the reader's own research.

Scaling laws (Statistical physics) --- System Analysis --- Computer simulation --- System analysis. --- Network analysis --- Network science --- Network theory --- Systems analysis --- System theory --- Mathematical optimization --- Ratio and proportion (Statistical physics) --- Scale invariance (Statistical physics) --- Scaling hypothesis (Statistical physics) --- Scaling phenomena (Statistical physics) --- Physical laws --- Ranking and selection (Statistics) --- Statistical physics --- Computer simulation. --- Scaling laws (Statistical physics) - Computer simulation

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Investigation of the fractal and scaling properties of disordered systems has recently become a focus of great interest in research. Disordered or amorphous materials, like glasses, polymers, gels, colloids, ceramic superconductors and random alloys or magnets, do not have a homogeneous microscopic structure. The microscopic environment varies randomly from site to site in the system and this randomness adds to the complexity and the richness of the properties of these materials. A particularly challenging aspect of random systems is their dynamical behavior. Relaxation in disordered systems

Order-disorder models --- Scaling laws (Statistical physics) --- Statistical physics --- Ratio and proportion (Statistical physics) --- Scale invariance (Statistical physics) --- Scaling hypothesis (Statistical physics) --- Scaling phenomena (Statistical physics) --- Physical laws --- Ranking and selection (Statistics) --- Coniglio, Antonio, --- Coniglio, A.

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Advances in nonlinear dynamics, especially modern multifractal cascade models, allow us to investigate the weather and climate at unprecedented levels of accuracy. Using new stochastic modelling and data analysis techniques, this book provides an overview of the nonclassical, multifractal statistics. By generalizing the classical turbulence laws, emergent higher-level laws of atmospheric dynamics are obtained and are empirically validated over time-scales of seconds to decades and length-scales of millimetres to the size of the planet. In generalizing the notion of scale, atmospheric complexity is reduced to a manageable scale-invariant hierarchy of processes, thus providing a new perspective for modelling and understanding the atmosphere. This synthesis of state-of-the-art data and nonlinear dynamics is systematically compared with other analyses and global circulation model outputs. This is an important resource for atmospheric science researchers new to multifractal theory and is also valuable for graduate students in atmospheric dynamics and physics, meteorology, oceanography and climatology.

Meteorology --- Atmospheric physics --- Fractals --- Scaling laws (Statistical physics) --- Meteorology. --- Atmospheric physics. --- Fractals. --- Ratio and proportion (Statistical physics) --- Scale invariance (Statistical physics) --- Scaling hypothesis (Statistical physics) --- Scaling phenomena (Statistical physics) --- Physical laws --- Ranking and selection (Statistics) --- Statistical physics --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology) --- Aerophysics --- Meteorology, Physical --- Physical meteorology --- Atmospheric science --- Aerology

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This volume constitutes the Proceedings of the IUTAM Symposium on 'Scaling in Solid Mechanics', held in Cardiff from 25th to 29th June 2007, to address topical issues in theoretical, experimental and computational aspects of scaling approaches to solid mechanics and related fields. Scaling is a rapidly expanding area of research which has multidisciplinary applications. The expertise represented at the Symposium was accordingly very wide, and many of the world's greatest authorities in their respective fields participated. Scaling methods apply wherever there is similarity across many scales or a need to bridge different scales, e.g. the nanoscale and macroscale. The emphasis at the Symposium was on fundamental issues such as mathematical foundations of scaling methods based on transformations and connections between multi-scale approaches and transformations. The Symposium remained focussed on fundamental research issues of practical significance. The topics considered included damage accumulation, growth of fatigue cracks, development of patterns of flaws in the earth's core and in ice, abrasiveness of rough surfaces, and so on. The Symposium showed that scaling methods cannot be reduced solely to dimensional analysis and fractal approaches. Modern scaling approaches consist of a great diversity of techniques. These proceedings contain lectures on state-of-the-art developments in self-similar solutions, fractal models, models involving interplay between different scales, size effects in fracture of solids and bundles of fibres, scaling in problems of fracture mechanics, nanomechanics, contact mechanics and testing of materials by indentation, scaling issues in mechanics of agglomeration of adhesive particles, and in biomimetic of adhesive contact.

Mechanical engineering -- Congresses. --- Scaling laws (Statistical physics) -- Congresses. --- Solids -- Mechanical properties -- Congresses. --- Solids --- Scaling laws (Statistical physics) --- Mechanical engineering --- Atomic Physics --- Applied Mathematics --- Materials Science --- Chemical & Materials Engineering --- Engineering & Applied Sciences --- Physics --- Physical Sciences & Mathematics --- Mechanical properties --- Solids. --- Mechanical engineering. --- Engineering, Mechanical --- Engineering. --- Earth sciences. --- Mathematical models. --- Mechanics. --- Applied mathematics. --- Engineering mathematics. --- Continuum mechanics. --- Continuum Mechanics and Mechanics of Materials. --- Earth Sciences, general. --- Mathematical Modeling and Industrial Mathematics. --- Appl.Mathematics/Computational Methods of Engineering. --- Engineering --- Machinery --- Steam engineering --- Solid state physics --- Transparent solids --- Mechanics, Applied. --- Geography. --- Solid Mechanics. --- Classical Mechanics. --- Mathematical and Computational Engineering. --- Engineering analysis --- Mathematical analysis --- Cosmography --- Earth sciences --- World history --- Applied mechanics --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Mathematics --- Models, Mathematical --- Simulation methods --- Geosciences --- Environmental sciences --- Physical sciences

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This book examines in detail the correlations for the two-dimensional Ising model in the infinite volume or thermodynamic limit and the sub- and super-critical continuum scaling limits. Steady progress in recent years has been made in understanding the special mathematical features of certain exactly solvable models in statistical mechanics and quantum field theory, including the scaling limits of the 2-D Ising (lattice) model, and more generally, a class of 2-D quantum fields known as holonomic fields. New results have made it possible to obtain a detailed nonperturbative analysis of the multi-spin correlations. In particular, the book focuses on deformation analysis of the scaling functions of the Ising model. This self-contained work also includes discussions on Pfaffians, elliptic uniformization, the Grassmann calculus for spin representations, Weiner--Hopf factorization, determinant bundles, and monodromy preserving deformations. This work explores the Ising model as a microcosm of the confluence of interesting ideas in mathematics and physics, and will appeal to graduate students, mathematicians, and physicists interested in the mathematics of statistical mechanics and quantum field theory.

Ising model. --- Scaling laws (Statistical physics) --- Ratio and proportion (Statistical physics) --- Scale invariance (Statistical physics) --- Scaling hypothesis (Statistical physics) --- Scaling phenomena (Statistical physics) --- Physical laws --- Ranking and selection (Statistics) --- Statistical physics --- Lenz-Ising model --- Ferromagnetism --- Phase transformations (Statistical physics) --- Mathematics. --- Mathematical physics. --- Statistical physics. --- Applications of Mathematics. --- Complex Systems. --- Mathematical Methods in Physics. --- Statistical Physics and Dynamical Systems. --- Physics --- Mathematical statistics --- Physical mathematics --- Math --- Science --- Statistical methods --- Mathematics --- Applied mathematics. --- Engineering mathematics. --- Dynamical systems. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Engineering --- Engineering analysis --- Mathematical analysis