This two-volume monograph is a comprehensive and up-to-date presentation of the theory and applications of kinetic equations. The first volume covers many-particle dynamics, Maxwell models of the Boltzmann equation (including their exact and self-similar solutions), and hydrodynamic limits beyond the Navier-Stokes level.

This open access book presents a comprehensive survey of modern operator techniques for boundary value problems and spectral theory, employing abstract boundary mappings and Weyl functions. It includes self-contained treatments of the extension theory of symmetric operators and relations, spectral characterizations of selfadjoint operators in terms of the analytic properties of Weyl functions, form methods for semibounded operators, and functional analytic models for reproducing kernel Hilbert spaces. Further, it illustrates these abstract methods for various applications, including Sturm-Liouville operators, canonical systems of differential equations, and multidimensional Schroedinger operators, where the abstract Weyl function appears as either the classical Titchmarsh-Weyl coefficient or the Dirichlet-to-Neumann map. The book is a valuable reference text for researchers in the areas of differential equations, functional analysis, mathematical physics, and system theory. Moreover, thanks to its detailed exposition of the theory, it is also accessible and useful for advanced students and researchers in other branches of natural sciences and engineering.

The aim of this third edition is to give an accessible and essentially self-contained account of pseudo-differential operators based on the previous edition. New chapters notwithstanding, the elementary and detailed style of earlier editions is maintained in order to appeal to the largest possible group of readers. The focus of this book is on the global theory of elliptic pseudo-differential operators on Lp(Rn).The main prerequisite for a complete understanding of the book is a basic course in functional analysis up to the level of compact operators. It is an ideal introduction for graduate students in mathematics and mathematicians who aspire to do research in pseudo-differential operators and related topics.

This monograph presents a synopsis of fluid dynamics based on the personal scientific experience of the author who has contributed immensely to the field. The interested reader will also benefit from the general historical context in which the material is presented in the book. The book covers a wide range of relevant topics of the field, and the main tool being rational asymptotic modelling (RAM) approach. The target audience primarily comprises experts in the field of fluid dynamics, but the book may also be beneficial for graduate students.

The present volume develops the theory of integration in Banach spaces, martingales and UMD spaces, and culminates in a treatment of the Hilbert transform, Littlewood-Paley theory and the vector-valued Mihlin multiplier theorem. Over the past fifteen years, motivated by regularity problems in evolution equations, there has been tremendous progress in the analysis of Banach space-valued functions and processes. The contents of this extensive and powerful toolbox have been mostly scattered around in research papers and lecture notes. Collecting this diverse body of material into a unified and accessible presentation fills a gap in the existing literature. The principal audience that we have in mind consists of researchers who need and use Analysis in Banach Spaces as a tool for studying problems in partial differential equations, harmonic analysis, and stochastic analysis. Self-contained and offering complete proofs, this work is accessible to graduate students and researchers with a background in functional analysis or related areas.

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

The primary objective of this monograph is to develop an elementary and se- containedapproachtothemathematicaltheoryofaviscousincompressible?uid n in a domain ? of the Euclidean spaceR , described by the equations of Navier- Stokes. The book is mainly directed to students familiar with basic functional analytic tools in Hilbert and Banach spaces. However, for readers' convenience, in the ?rst two chapters we collect, without proof some fundamental properties of Sobolev spaces, distributions, operators, etc. Another important objective is to formulate the theory for a completely general domain ?. In particular, the theory applies to arbitrary unbounded, non-smooth domains. For this reason, in the nonlinear case, we have to restrict ourselves to space dimensions n=2,3 that are also most signi?cant from the physical point of view. For mathematical generality, we will develop the l- earized theory for all n? 2. Although the functional-analytic approach developed here is, in principle, known to specialists, its systematic treatment is not available, and even the diverseaspectsavailablearespreadoutintheliterature.However,theliterature is very wide, and I did not even try to include a full list of related papers, also because this could be confusing for the student. In this regard, I would like to apologize for not quoting all the works that, directly or indirectly, have inspired this monograph.

Fractional equations and models play an essential part in the description of anomalous dynamics in complex systems. Recent developments in the modeling of various physical, chemical and biological systems have clearly shown that fractional calculus is not just an exotic mathematical theory, as it might have once seemed. The present book seeks to demonstrate this using various examples of equations and models with fractional and generalized operators. Intended for students and researchers in mathematics, physics, chemistry, biology and engineering, it systematically offers a wealth of useful tools for fractional calculus.

Ever since China and Vietnam resumed diplomatic contacts and reopened the border in 1991, the borderland region has become part of the vibrant growing economies of both countries and drawn many from the interior provinces to the borderland for new economic adventures. This book examines Chinese-Vietnamese relationships at the borderland through every day cross-border interaction in trade and tourism activities. It looks into the historical underlining of bilateral relations of the two countries which often shape people's perceptions of the 'other' and interpretation of intentions of acts in their daily interaction. Albeit Chinese and Vietnamese have lived side by side for centuries, their interaction in the space of trade and modern tourism in post-war and post-reform China and Vietnam is something novel to both people. The book provides a 'bottom-up' approach to examine the localized experiences of inter-state relations. It illustrates the changes the vibrant economic process has brought to the borderland communities, and how the revived contacts and interaction have generated a contested space for examining Vietnamese-Chinese relationships and demonstrating trans-border cultural politics. A novel study of the strategic development of the borderland within the new political economy at China-Southeast Asia border region, this book is of interest to academics in the field of Anthropology, Border Studies, Social and Cultural Studies and Asian Studies.

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

Periodically Correlated Solutions to a Class of Stochastic Difference Equations.- On Nonlinear SDE'S whose Densities Evolve in a Finite-Dimensional Family.- Composition of Skeletons and Support Theorems.- Invariant Measure for a Wave Equation on a Riemannian Manifold.- Ergodic Distributed Control for Parameter Dependent Stochastic Semilinear Systems.- Dirichlet Forms, Caccioppoli Sets and the Skorohod Equation Masatoshi Fukushima.- Rate of Convergence of Moments of Spall's SPSA Method.- General Setting for Stochastic Processes Associated with Quantum Fields.- On a Class of Semilinear Stochastic Partial Differential Equations.- Parallel Numerical Solution of a Class of Volterra Integro-Differential Equations.- On the Laws of the Oseledets Spaces of Linear Stochastic Differential Equations.- On Stationarity of Additive Bilinear State-space Representation of Time Series.- On Convergence of Approximations of Ito-Volterra Equations.- Non-isotropic Ornstein-Uhlenbeck Process and White Noise Analysis.- Stochastic Processes with Independent Increments on a Lie Group and their Selfsimilar Properties.- Optimal Damping of Forced Oscillations Discrete-time Systems by Output Feedback.- Forecast of Levy's Brownian Motion as the Observation Domain Undergoes Deformation.- A Maximal Inequality for the Skorohod Integral.- On the Kinematics of Stochastic Mechanics.- Stochastic Equations in Formal Mappings.- On Fisher's Information Matrix of an ARMA Process.- Statistical Analysis of Nonlinear and NonGaussian Time Series.- Bilinear Stochastic Systems with Long Range Dependence in Continuous Time.- On Support Theorems for Stochastic Nonlinear Partial Differential Equations.- Excitation and Performance in Continuous-time Stochastic Adaptive LQ-control.- Invariant Measures for Diffusion Processes in Conuclear Spaces.- Degree Theory on Wiener Space and an Application to a Class of SPDEs.- On the Interacting Measure-Valued Branching Processes.

The study of measure-valued processes in random environments has seen some intensive research activities in recent years whereby interesting nonlinear stochastic partial differential equations (SPDEs) were derived. Due to the nonlinearity and the non-Lipschitz continuity of their coefficients, new techniques and concepts have recently been developed for the study of such SPDEs. These include the conditional Laplace transform technique, the conditional mild solution, and the bridge between SPDEs and some kind of backward stochastic differential equations. This volume provides an introduction to these topics with the aim of attracting more researchers into this exciting and young area of research. It can be considered as the first book of its kind. The tools introduced and developed for the study of measure-valued processes in random environments can be used in a much broader area of nonlinear SPDEs.

This book gives an up-to-date exposition on the theory of oblique derivative problems for elliptic equations. The modern analysis of shock reflection was made possible by the theory of oblique derivative problems developed by the author. Such problems also arise in many other physical situations such as the shape of a capillary surface and problems of optimal transportation. The author begins the book with basic results for linear oblique derivative problems and work through the theory for quasilinear and nonlinear problems. The final chapter discusses some of the applications. In addition, notes to each chapter give a history of the topics in that chapter and suggestions for further reading.

This book is a comprehensive collection of known results about the Lozi map, a piecewise-affine version of the Henon map. Henon map is one of the most studied examples in dynamical systems and it attracts a lot of attention from researchers, however it is difficult to analyze analytically. Simpler structure of the Lozi map makes it more suitable for such analysis. The book is not only a good introduction to the Lozi map and its generalizations, it also summarizes of important concepts in dynamical systems theory such as hyperbolicity, SRB measures, attractor types, and more.

Until now, no book addressed convexity, monotonicity, and variational inequalities together. Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization covers all three topics, including new variational inequality problems defined by a bifunction.

The first part of the book focuses on generalized convexity and generalized monotonicity. The authors investigate convexity and generalized convexity for both the differentiable and nondifferentiable case. For the nondifferentiable case, they introduce the concepts in terms of a bifunction and the Clarke subdifferential.

The second part offers insight into variational inequalities and optimization problems in smooth as well as nonsmooth settings. The book discusses existence and uniqueness criteria for a variational inequality, the gap function associated with it, and numerical methods to solve it. It also examines characterizations of a solution set of an optimization problem and explores variational inequalities defined by a bifunction and set-valued version given in terms of the Clarke subdifferential.

Integrating results on convexity, monotonicity, and variational inequalities into one unified source, this book deepens your understanding of various classes of problems, such as systems of nonlinear equations, optimization problems, complementarity problems, and fixed-point problems. The book shows how variational inequality theory not only serves as a tool for formulating a variety of equilibrium problems, but also provides algorithms for computational purposes.

This monograph is concerned with the mathematical analysis of patterns which are encountered in biological systems. It summarises, expands and relates results obtained in the field during the last fifteen years. It also links the results to biological applications and highlights their relevance to phenomena in nature. Of particular concern are large-amplitude patterns far from equilibrium in biologically relevant models. The approach adopted in the monograph is based on the following paradigms: * Examine the existence of spiky steady states in reaction-diffusion systems and select as observable patterns only the stable ones * Begin by exploring spatially homogeneous two-component activator-inhibitor systems in one or two space dimensions * Extend the studies by considering extra effects or related systems, each motivated by their specific roles in developmental biology, such as spatial inhomogeneities, large reaction rates, altered boundary conditions, saturation terms, convection, many-component systems. Mathematical Aspects of Pattern Formation in Biological Systems will be of interest to graduate students and researchers who are active in reaction-diffusion systems, pattern formation and mathematical biology.

For computer scientists, especially those in the security field, the use of chaos has been limited to the computation of a small collection of famous but unsuitable maps that offer no explanation of why chaos is relevant in the considered contexts. Discrete Dynamical Systems and Chaotic Machines: Theory and Applications shows how to make finite machines, such as computers, neural networks, and wireless sensor networks, work chaotically as defined in a rigorous mathematical framework. Taking into account that these machines must interact in the real world, the authors share their research results on the behaviors of discrete dynamical systems and their use in computer science. Covering both theoretical and practical aspects, the book presents: Key mathematical and physical ideas in chaos theory Computer science fundamentals, clearly establishing that chaos properties can be satisfied by finite state machines Concrete applications of chaotic machines in computer security, including pseudorandom number generators, hash functions, digital watermarking, and steganography Concrete applications of chaotic machines in wireless sensor networks, including secure data aggregation and video surveillance Until the authors' recent research, the practical implementation of the mathematical theory of chaos on finite machines raised several issues. This self-contained book illustrates how chaos theory enables the study of computer security problems, such as steganalysis, that otherwise could not be tackled. It also explains how the theory reinforces existing cryptographically secure tools and schemes.

For more than half a century, stochastic calculus and stochastic differential equations have played a major role in analyzing the dynamic phenomena in the biological and physical sciences, as well as engineering. The advancement of knowledge in stochastic differential equations is spreading rapidly across the graduate and postgraduate programs in universities around the globe. This will be the first available book for use by any undergraduate/graduate stochastic modeling/applied mathematics courses and for use by an interdisciplinary researcher with minimal academic background. "An Introduction to Differential Equations: Volume 2" is a stochastic version of Volume 1 ("An Introduction to Differential Equations: Deterministic Modeling, Methods and Analysis"). Both books have a similar design, but naturally, differ by calculi. Again, both volumes use an innovative style in the presentation of the topics, methods and concepts with adequate preparation in deterministic Calculus.

For more than half a century, stochastic calculus and stochastic differential equations have played a major role in analyzing the dynamic phenomena in the biological and physical sciences, as well as engineering. The advancement of knowledge in stochastic differential equations is spreading rapidly across the graduate and postgraduate programs in universities around the globe. This will be the first available book for use by any undergraduate/graduate stochastic modeling/applied mathematics courses and for use by an interdisciplinary researcher with minimal academic background. "An Introduction to Differential Equations: Volume 2" is a stochastic version of Volume 1 ("An Introduction to Differential Equations: Deterministic Modeling, Methods and Analysis"). Both books have a similar design, but naturally, differ by calculi. Again, both volumes use an innovative style in the presentation of the topics, methods and concepts with adequate preparation in deterministic Calculus.

This monograph is intended to present the fundamentals of the theory of abstract parabolic evolution equations and to show how to apply to various nonlinear dif- sion equations and systems arising in science. The theory gives us a uni?ed and s- tematic treatment for concrete nonlinear diffusion models. Three main approaches are known to the abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. In order to keep the volume of the monograph in reasonable length, we will focus on the semigroup methods. For other two approaches, see the related references in Bibliography. The semigroup methods, which go back to the invention of the analytic se- groups in the middle of the last century, are characterized by precise formulas representing the solutions of the Cauchy problem for evolution equations. The ?tA analytic semigroup e generated by a linear operator ?A provides directly a fundamental solution to the Cauchy problem for an autonomous linear e- dU lution equation, +AU =F(t), 0

To help solve physical and engineering problems, mimetic or compatible algebraic discretization methods employ discrete constructs to mimic the continuous identities and theorems found in vector calculus. Mimetic Discretization Methods focuses on the recent mimetic discretization method co-developed by the first author. Based on the Castillo-Grone operators, this simple mimetic discretization method is invariably valid for spatial dimensions no greater than three. The book also presents a numerical method for obtaining corresponding discrete operators that mimic the continuum differential and flux-integral operators, enabling the same order of accuracy in the interior as well as the domain boundary. After an overview of various mimetic approaches and applications, the text discusses the use of continuum mathematical models as a way to motivate the natural use of mimetic methods. The authors also offer basic numerical analysis material, making the book suitable for a course on numerical methods for solving PDEs. The authors cover mimetic differential operators in one, two, and three dimensions and provide a thorough introduction to object-oriented programming and C++. In addition, they describe how their mimetic methods toolkit (MTK)-available online-can be used for the computational implementation of mimetic discretization methods. The text concludes with the application of mimetic methods to structured nonuniform meshes as well as several case studies. Compiling the authors' many concepts and results developed over the years, this book shows how to obtain a robust numerical solution of PDEs using the mimetic discretization approach. It also helps readers compare alternative methods in the literature.

Weakly Connected Nonlinear Systems: Boundedness and Stability of Motion provides a systematic study on the boundedness and stability of weakly connected nonlinear systems, covering theory and applications previously unavailable in book form. It contains many essential results needed for carrying out research on nonlinear systems of weakly connected equations. After supplying the necessary mathematical foundation, the book illustrates recent approaches to studying the boundedness of motion of weakly connected nonlinear systems. The authors consider conditions for asymptotic and uniform stability using the auxiliary vector Lyapunov functions and explore the polystability of the motion of a nonlinear system with a small parameter. Using the generalization of the direct Lyapunov method with the asymptotic method of nonlinear mechanics, they then study the stability of solutions for nonlinear systems with small perturbing forces. They also present fundamental results on the boundedness and stability of systems in Banach spaces with weakly connected subsystems through the generalization of the direct Lyapunov method, using both vector and matrix-valued auxiliary functions. Designed for researchers and graduate students working on systems with a small parameter, this book will help readers get up to date on the knowledge required to start research in this area.

The monograph is devoted to the study of initial-boundary-value problems for multi-dimensional Sobolev-type equations over bounded domains. The authors consider both specific initial-boundary-value problems and abstract Cauchy problems for first-order (in the time variable) differential equations with nonlinear operator coefficients with respect to spatial variables. The main aim of the monograph is to obtain sufficient conditions for global (in time) solvability, to obtain sufficient conditions for blow-up of solutions at finite time, and to derive upper and lower estimates for the blow-up time. The abstract results apply to a large variety of problems. Thus, the well-known Benjamin-Bona-Mahony-Burgers equation and Rosenau-Burgers equations with sources and many other physical problems are considered as examples. Moreover, the method proposed for studying blow-up phenomena for nonlinear Sobolev-type equations is applied to equations which play an important role in physics. For instance, several examples describe different electrical breakdown mechanisms in crystal semiconductors, as well as the breakdown in the presence of sources of free charges in a self-consistent electric field. The monograph contains a vast list of references (440 items) and gives an overall view of the contemporary state-of-the-art of the mathematical modeling of various important problems arising in physics. Since the list of references contains many papers which have been published previously only in Russian research journals, it may also serve as a guide to the Russian literature.

This book combines the efforts of a distinguished team of authors, who are all renowned mathematicians and expositors, and provides a modern introduction to the calculus of variations. By focusing on the one-dimensional case it remains relatively free of technicalities, and therefore provides a useful overview of the theory at a level suitable for graduate students.