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MEAN FIELD THEORIES AND DUAL VARIATION
A mathematical theory is introduced in this book to unify a large class of nonlinear partial differential equation (PDE) models for better understanding and analysis of the physical and biological phenomena they represent. The so-called mean field approximation approach is adopted to describe the macroscopic phenomena from certain microscopic principles for this unified mathematical formulation. Two key ingredients for this approach are the notions of “duality” according to the PDE weak solutions and “hierarchy” for revealing the details of the otherwise hidden secrets, such as physical mystery hidden between particle density and field concentration, quantized blow up biological mechanism sealed in chemotaxis systems, as well as multi-scale mathematical explanations of the Smoluchowski–Poisson model in non-equilibrium thermodynamics, two-dimensional turbulence theory, self-dual gauge theory, and so forth. This book shows how and why many different nonlinear problems are inter-connected in terms of the properties of duality and scaling, and the way to analyze them mathematically.
Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model
Mean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews. It is still in progress; fluid mechanics, gauge theory, plasma physics, quantum chemistry, mathematical oncology, non-equilibirum thermodynamics. spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature. The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics.
The Dynamic Magnetosphere
Despite the plethora of monographs published in recent years, few cover recent progress in magnetospheric physics in broad areas of research. While a topical focus is important to in-depth views at a problem, a broad overview of our field is also needed. The volume answers to the latter need. With the collection of articles written by leading scientists, the contributions contained in the book describe latest research results in solar wind-magnetosphere interaction, magnetospheric substorms, magnetosphere-ionosphere coupling, transport phenomena in the plasma sheet, wave and particle dynamics in the ring current and radiation belts, and extra-terrestrial magnetospheric systems. In addition to its breadth and timeliness, the book highlights innovative methods and techniques to study the geospace.
Geometric Properties for Parabolic and Elliptic PDE's
This book collects recent research papers by respected specialists in the field. It presents advances in the field of geometric properties for parabolic and elliptic partial differential equations, an area that has always attracted great attention. It settles the basic issues (existence, uniqueness, stability and regularity of solutions of initial/boundary value problems) before focusing on the topological and/or geometric aspects. These topics interact with many other areas of research and rely on a wide range of mathematical tools and techniques, both analytic and geometric. The Italian and Japanese mathematical schools have a long history of research on PDEs and have numerous active groups collaborating in the study of the geometric properties of their solutions.
Handbook of Differential Equations: Stationary Partial Differential Equations
A collection of self contained state-of-the art surveys. The authors have made an effort to achieve readability for mathematicians and scientists from other fields, for this series of handbooks to be a new reference for research, learning and teaching. - Written by well-known experts in the field - Self contained volume in series covering one of the most rapid developing topics in mathematics - Informed and thoroughly updated for students, academics and researchers
Recent Advances In Elliptic And Parabolic Problems, Proceedings Of The International Conference
The book is an account on recent advances in elliptic and parabolic problems and related equations, including general quasi-linear equations, variational structures, Bose-Einstein condensate, Chern-Simons model, geometric shell theory and stability in fluids. It presents very up-to-date research on central issues of these problems such as maximal regularity, bubbling, blowing-up, bifurcation of solutions and wave interaction. The contributors are well known leading mathematicians and prominent young researchers.The proceedings have been selected for coverage in:• Index to Scientific & Technical Proceedings® (ISTP® / ISI Proceedings)• Index to Scientific & Technical Proceedings (ISTP CDROM version / ISI Proceedings)• CC Proceedings — Engineering & Physical Sciences
Geometric Properties for Parabolic and Elliptic PDE's
The study of qualitative aspects of PDE's has always attracted much attention from the early beginnings. More recently, once basic issues about PDE's, such as existence, uniqueness and stability of solutions, have been understood quite well, research on topological and/or geometric properties of their solutions has become more intense. The study of these issues is attracting the interest of an increasing number of researchers and is now a broad and well-established research area, with contributions that often come from experts from disparate areas of mathematics, such as differential and convex geometry, functional analysis, calculus of variations, mathematical physics, to name a few. This volume collects a selection of original results and informative surveys by a group of international specialists in the field, analyzes new trends and techniques and aims at promoting scientific collaboration and stimulating future developments and perspectives in this very active area of research.
Earth Sciences, Revised Edition
Earth Sciences, Revised Edition describes the evolution of major topics in Earth sciences through the lens of key scientists and researchers in the field. From earthquakes to volcanoes to conserving water, this newly revised edition covers a wide spectrum of all that earth science has to offer, making it an essential read for the earth scientist of today and tomorrow. This resource provides an examination the problems researchers are currently investigating, as well as the methods they have developed to solve them in an effort to protect and better understand our planet. Chapters include: Exploring Earth's Depths Origin and Variability of Earth's Magnetic Field Volcanoes and Hotspots Geothermal Energy—A Furnace Beneath the Soil Water Management—Conserving an Essential Resource Predicting Earthquakes.
