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Disordered magnetic systems enjoy non-trivial properties which are different and richer than those observed in their pure, non-disordered counterparts. These properties dramatically affect the thermodynamic behaviour and require specific theoretical treatment. This book deals with the theory of magnetic systems in the presence of frozen disorder, in particular paradigmatic and well-known spin models such as the Random Field Ising Model and the Ising Spin Glass. This is a unified presentation using a field theory language which covers mean field theory, dynamics and perturbation expansion within the same theoretical framework. Particular emphasis is given to the connections between different approaches such as statics vs. dynamics, microscopic vs. phenomenological models. The book introduces some useful and little-known techniques in statistical mechanics and field theory. This book will be of great interest to graduate students and researchers in statistical physics and basic field theory.

Random fields. --- Spin glasses. --- Glasses, Magnetic --- Glasses, Spin --- Magnetic glasses --- Magnetic alloys --- Nuclear spin --- Solid state physics --- Fields, Random --- Stochastic processes

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This volume contains twenty-eight refereed research or review papers presented at the 5th Seminar on Stochastic Processes, Random Fields and Applications, which took place at the Centro Stefano Franscini (Monte Verità) in Ascona, Switzerland, from May 30 to June 3, 2005. The seminar focused mainly on stochastic partial differential equations, random dynamical systems, infinite-dimensional analysis, approximation problems, and financial engineering. The book will be a valuable resource for researchers in stochastic analysis and professionals interested in stochastic methods in finance. Contributors: Y. Asai, J.-P. Aubin, C. Becker, M. Benaïm, H. Bessaih, S. Biagini, S. Bonaccorsi, N. Bouleau, N. Champagnat, G. Da Prato, R. Ferrière, F. Flandoli, P. Guasoni, V.B. Hallulli, D. Khoshnevisan, T. Komorowski, R. Léandre, P. Lescot, H. Lisei, J.A. López-Mimbela, V. Mandrekar, S. Méléard, A. Millet, H. Nagai, A.D. Neate, V. Orlovius, M. Pratelli, N. Privault, O. Raimond, M. Röckner, B. Rüdiger, W.J. Runggaldier, P. Saint-Pierre, M. Sanz-Solé, M. Scheutzow, A. Soós, W. Stannat, A. Truman, T. Vargiolu, A.E.P. Villa, A.B. Vizcarra, F.G. Viens, J.-C. Zambrini, B. Zegarlinski.

Stochastic analysis --- Random fields --- Business mathematics --- Fields, Random --- Stochastic processes --- Arithmetic, Commercial --- Business --- Business arithmetic --- Business math --- Commercial arithmetic --- Finance --- Mathematics --- Distribution (Probability theory. --- Probability Theory and Stochastic Processes. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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The author describes the current state of the art in the theory of invariant random fields. This theory is based on several different areas of mathematics, including probability theory, differential geometry, harmonic analysis, and special functions. The present volume unifies many results scattered throughout the mathematical, physical, and engineering literature, as well as it introduces new results from this area first proved by the author. The book also presents many practical applications, in particular in such highly interesting areas as approximation theory, cosmology and earthquake engineering. It is intended for researchers and specialists working in the fields of stochastic processes, statistics, functional analysis, astronomy, and engineering.           .

Mathematics. --- Random fields. --- Spectral theory (Mathematics). --- Stochastic control theory. --- Random fields --- Probabilities --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Probabilities. --- Mathematical physics. --- Cosmology. --- Probability Theory and Stochastic Processes. --- Mathematical Applications in the Physical Sciences. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Fields, Random --- Stochastic processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Astronomy --- Deism --- Metaphysics --- Physical mathematics --- Physics

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This book gives a user friendly tutorial to Fronts in Random Media, an interdisciplinary research topic, to senior undergraduates and graduate students in the mathematical sciences, physical sciences and engineering. Fronts or interface motion occur in a wide range of scientific areas where the physical and chemical laws are expressed in terms of differential equations. Heterogeneities are always present in natural environments: fluid convection in combustion, porous structures, noise effects in material manufacturing to name a few. Stochastic models hence become natural due to the often lack of complete data in applications. The transition from seeking deterministic solutions to stochastic solutions is both a conceptual change of thinking and a technical change of tools. The book explains ideas and results systematically in a motivating manner. It covers multi-scale and random fronts in three fundamental equations (Burgers, Hamilton-Jacobi, and reaction-diffusion-advection equations) and explores their connections and mechanical analogies. It discusses representation formulas, Laplace methods, homogenization, ergodic theory, central limit theorems, large-deviation principles, variational and maximum principles. It shows how to combine these tools to solve concrete problems. Students and researchers will find the step by step approach and the open problems in the book particularly useful. .

Electronic books. -- local. --- Random fields. --- Stochastic processes. --- Stochastic processes --- Wave-motion, Theory of --- Stochastic analysis --- Fluid mechanics --- Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Random processes --- Fields, Random --- Mathematics. --- Partial differential equations. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Partial Differential Equations. --- Probabilities --- Distribution (Probability theory. --- Differential equations, partial. --- Partial differential equations --- Distribution functions --- Frequency distribution --- Characteristic functions --- Wave-motion, Theory of. --- Stochastic analysis. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined. The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities. "Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics.

Mathematics. --- Probability Theory and Stochastic Processes. --- Statistics, general. --- Geometry. --- Mathematical Methods in Physics. --- Distribution (Probability theory). --- Mathematical physics. --- Statistics. --- Mathématiques --- Géométrie --- Distribution (Théorie des probabilités) --- Physique mathématique --- Statistique --- Global differential geometry. --- Random fields. --- Stochastic geometry. --- Random fields --- Global differential geometry --- Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Fields, Random --- Probabilities. --- Physics. --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Euclid's Elements --- Math --- Science --- Geometry, Differential --- Stochastic processes --- Distribution (Probability theory. --- Physical mathematics --- Physics --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Statistics .