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Approximation of Stochastic Invariant Manifolds
This first volume is concerned with the analytic derivation of explicit formulas for the leading-order Taylor approximations of (local) stochastic invariant manifolds associated with a broad class of nonlinear stochastic partial differential equations. These approximations take the form of Lyapunov-Perron integrals, which are further characterized in Volume II as pullback limits associated with some partially coupled backward-forward systems. This pullback characterization provides a useful interpretation of the corresponding approximating manifolds and leads to a simple framework that unifies some other approximation approaches in the literature. A self-contained survey is also included on the existence and attraction of one-parameter families of stochastic invariant manifolds, from the point of view of the theory of random dynamical systems.
Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations
In this second volume, a general approach is developed to provide approximate parameterizations of the "small" scales by the "large" ones for a broad class of stochastic partial differential equations (SPDEs). This is accomplished via the concept of parameterizing manifolds (PMs), which are stochastic manifolds that improve, for a given realization of the noise, in mean square error the partial knowledge of the full SPDE solution when compared to its projection onto some resolved modes. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers as a pullback limit depending on the time-history of the modes with low wave numbers. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. The reduced systems take the form of stochastic differential equations involving random coefficients that convey memory effects. The theory is illustrated on a stochastic Burgers-type equation.
Advances in Nonlinear Geosciences
Advances in Nonlinear Geosciences is a set of contributions from the participants of “30 Years of Nonlinear Dynamics” held July 3-8, 2016 in Rhodes, Greece as part of the Aegean Conferences, as well as from several other experts in the field who could not attend the meeting. The volume brings together up-to-date research from the atmospheric sciences, hydrology, geology, and other areas of geosciences and presents the new advances made in the last 10 years. Topics include chaos synchronization, topological data analysis, new insights on fractals, multifractals and stochasticity, climate dynamics, extreme events, complexity, and causality, among other topics.
Hamilton-Jacobi-Bellman Equations
Optimal feedback control arises in different areas such as aerospace engineering, chemical processing, resource economics, etc. In this context, the application of dynamic programming techniques leads to the solution of fully nonlinear Hamilton-Jacobi-Bellman equations. This book presents the state of the art in the numerical approximation of Hamilton-Jacobi-Bellman equations, including post-processing of Galerkin methods, high-order methods, boundary treatment in semi-Lagrangian schemes, reduced basis methods, comparison principles for viscosity solutions, max-plus methods, and the numerical approximation of Monge-Ampère equations. This book also features applications in the simulation of ...
Probability and Statistical Physics in St. Petersburg
This book brings a reader to the cutting edge of several important directions of the contemporary probability theory, which in many cases are strongly motivated by problems in statistical physics. The authors of these articles are leading experts in the field and the reader will get an exceptional panorama of the field from the point of view of scientists who played, and continue to play, a pivotal role in the development of the new methods and ideas, interlinking it with geometry, complex analysis, conformal field theory, etc., making modern probability one of the most vibrant areas in mathematics.
Theorizing Outdoor Recreation and Ecology
Deciding what user impacts are natural or unnatural has inspired much debate. Biophysically, moose cause similar kinds of soil and vegetation impacts as hikers. Yet moose are the sign of nature while hikers are the sign of damage. The field of outdoor recreation is beset with paradoxes, and this book presents a unique, alternative framework to address these dilemmas. Examining outdoor recreation through the lens of ecological theory, Ryan draws from theorists such as Foucault, Derrida and Latour. The book explores minimum impact strategies designed to protect and enhance ecological integrity, but that also require a disturbing amount of policing of users, which runs counter to the freedom users seek. Recent ecological theory suggests that outdoor recreation's view of nature as balanced when impacts are removed is outdated and incorrect. What is needed, and indeed Ryan presents, is a paradoxical and ecological view of humans as neither natural nor unnatural, a view that embraces some traces in nature.
Climate Change: Multidecadal And Beyond
This book focuses on two major challenges in the climate sciences: 1) to describe the decadal-to-centennial variations in instrumental and proxy records; and 2) to distinguish between anthropogenic variations and natural variability. The National Taiwan University invited some of the world's leading experts across the areas of observational analysis, mathematical theory, and modeling to discuss these two issues. The outcome of the meeting is the 23 chapters in this book that review the state of the art in theoretical, observational and modeling research on internal, unforced and externally forced climate variability. The main conclusion of this research is that internal climate variability on decadal and longer time scales is so large that sidestepping it may lead to false estimates of the climate's sensitivity to anthropogenic forcing.World Scientific Series on Asia-Pacific Weather and Climate is indexed in SCOPUS.
Nonlinear and Stochastic Climate Dynamics
It is now widely recognized that the climate system is governed by nonlinear, multi-scale processes, whereby memory effects and stochastic forcing by fast processes, such as weather and convective systems, can induce regime behavior. Motivated by present difficulties in understanding the climate system and to aid the improvement of numerical weather and climate models, this book gathers contributions from mathematics, physics and climate science to highlight the latest developments and current research questions in nonlinear and stochastic climate dynamics. Leading researchers discuss some of the most challenging and exciting areas of research in the mathematical geosciences, such as the theory of tipping points and of extreme events including spatial extremes, climate networks, data assimilation and dynamical systems. This book provides graduate students and researchers with a broad overview of the physical climate system and introduces powerful data analysis and modeling methods for climate scientists and applied mathematicians.
Mathematical Reviews
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Phase Transition Dynamics
This book is an introduction to a comprehensive and unified dynamic transition theory for dissipative systems and to applications of the theory to a range of problems in the nonlinear sciences. The main objectives of this book are to introduce a general principle of dynamic transitions for dissipative systems, to establish a systematic dynamic transition theory, and to explore the physical implications of applications of the theory to a range of problems in the nonlinear sciences. The basic philosophy of the theory is to search for a complete set of transition states, and the general principle states that dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic and random. The audience for this book includes advanced graduate students and researchers in mathematics and physics as well as in other related fields.
