The subject of nonlinear partial differential equations is experiencing a period of intense activity in the study of systems underlying basic theories in geometry, topology and physics. These mathematical models share the property of being derived from variational principles. Understanding the structure of critical configurations and the dynamics of the corresponding evolution problems is of fundamental importance for the development of the physical theories and their applications. This volume contains survey lectures in four different areas, delivered by leading resarchers at the 1995 Barrett Lectures held at The University of Tennessee: nonlinear hyperbolic systems arising in field theory and relativity (S. Klainerman); harmonic maps from Minkowski spacetime (M. Struwe); dynamics of vortices in the Ginzburg-Landau model of superconductivity (F.-H. Lin); the Seiberg-Witten equations and their application to problems in four-dimensional topology (R. Fintushel). Most of this material has not previously been available in survey form. These lectures provide an up-to-date overview and an introduction to the research literature in each of these areas, which should prove useful to researchers and graduate students in mathematical physics, partial differential equations, differential geometry and topology.

The concept of "chaos", and chaos theory, though it is a field of study specifically in the field of mathematics with applications in physics, engineering, economics, management, and education, has also recently taken root in the social sciences. As a method of analyzing the way in which the digital age has connected society more than ever, chaos and complexity theory serves as a tactic to tie world events and cope with the information overload that is associated with heightened social connectivity. The Handbook of Research on Chaos and Complexity Theory in the Social Sciences explores the theories of chaos and complexity as applied to a variety of disciplines including political science, organizational and management science, economics, and education. Presenting diverse research-based perspectives on mathematical patterns in the world system, this publication is an essential reference source for scholars, researchers, mathematicians, social theorists, and graduate-level students in a variety of disciplines.

Multidimensional continued fractions form an area of research within number theory. Recently the topic has been linked to research in dynamical systems, and mathematical physics, which means that some of the results discovered in this area have applications in describing physical systems. This book gives a comprehensive and up to date overview of recent research in the area.

Heun's equation is a second-order differential equation which crops up in a variety of forms in a wide range of problems in applied mathematics. These include integral equations of potential theory, wave propogation, electrostatic oscillation, and Schrodinger's equation. This volume brings together important research work for the first time, providing an important resource for all those interested in this mathematical topic. Both the current theory and the main areas of application are surveyed, and includes contributions from authoritative researchers such as Felix Arscott (Canada), P. Maroni (France), and Gerhard Wolf (Germany).

This book explains the essentials of fractional calculus and demonstrates its application in control system modeling, analysis and design. It presents original research to find high-precision solutions to fractional-order differentiations and differential equations. Numerical algorithms and their implementations are proposed to analyze multivariable fractional-order control systems. Through high-quality MATLAB programs, it provides engineers and applied mathematicians with theoretical and numerical tools to design control systems. Contents Introduction to fractional calculus and fractional-order control Mathematical prerequisites Definitions and computation algorithms of fractional-order derivatives and Integrals Solutions of linear fractional-order differential equations Approximation of fractional-order operators Modelling and analysis of multivariable fractional-order transfer function Matrices State space modelling and analysis of linear fractional-order Systems Numerical solutions of nonlinear fractional-order differential Equations Design of fractional-order PID controllers Frequency domain controller design for multivariable fractional-order Systems Inverse Laplace transforms involving fractional and irrational Operations FOTF Toolbox functions and models Benchmark problems for the assessment of fractional-order differential equation algorithms

In this book the authors show that it is possible to construct efficient computationally oriented models of multi-parameter complex systems by using asymptotic methods, which can, owing to their simplicity, be directly used for controlling processes arising in connection with composite material systems. The book focuses on this asymptotic-modeling-based approach because it allows us to define the most important out of numerous parameters describing the system, or, in other words, the asymptotic methods allow us to estimate the sensitivity of the system parameters. Further, the book addresses the construction of nonlocal and higher-order homogenized models. Local fields on the micro-level and the influence of so-called non-ideal contact between the matrix and inclusions are modeled and investigated. The book then studies composites with non-regular structure and cluster type composite conductivity, and analyzes edge effects in fiber composite materials. Transition of load from a fiber to a matrix for elastic and viscoelastic composites, various types of fiber composite fractures, and buckling of fibers in fiber-reinforced composites is also investigated. Last but not least, the book includes studies on perforated membranes, plates, and shells, as well as the asymptotic modeling of imperfect nonlinear interfaces.

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This second volume collects authoritative chapters covering the mathematical theory of fractional calculus, including ordinary and partial differential equations of fractional order, inverse problems, and evolution equations.

This volume presents the Proceedings of the Joint U.S. / Israel Workshop on Operator Theory and Its Applications, held February 24-28, 1992, at the Ben Gurion University of the Negev, Beersheva. This event was sponsored by the United States / Israel Binational Science Foundation and the Ben Gurion University of the Negev, and many outstanding experts in operator theory took part. The workshop honored Professor Emeritus Moshe Livsic on the occasion of his retirement. The volume contains a selection of papers covering a wide range of topics in modern operator theory and its applications, from abstract operator theory to system theory and computers in operator models. The papers treat linear and nonlinear problems, and study operators from different abstract and concrete classes. Many of the topics concern the area in which contributions of Moshe Livsic were extremely important. This book will appeal to a wide audience of pure and applied mathematicians and engineers.

The present book is the first of the two volume proceedings of the Mark Krein International Conference on Operator Theory and Applications. This conference, which was dedicated to the 90th anniversary of the prominent mathematician Mark Krein, was held in Odessa, Ukraine, from August 18-22, 1997. The conference focused on the main ideas, methods, results, and achievements of M. G. Krein. This first volume is devoted to the theory of differential operators and related topics. It opens with a description of the conference, biographical material and a number of survey papers about the work of M. G. Krein. The main part of the book consists of original research papers presenting the state of the art in the area of differential operators. The second volume of these proceedings, entitled Operator Theory and Related Topics, concerns the other aspects of the conference. The two volumes will be of interest to a wide range of readership in pure and applied mathematics, physics and engineering sciences.

This volume gathers contributions from participants of the Introductory School and the IHP thematic quarter on Numerical Methods for PDE, held in 2016 in Cargese (Corsica) and Paris, providing an opportunity to disseminate the latest results and envisage fresh challenges in traditional and new application fields. Numerical analysis applied to the approximate solution of PDEs is a key discipline in applied mathematics, and over the last few years, several new paradigms have appeared, leading to entire new families of discretization methods and solution algorithms. This book is intended for researchers in the field.

The book is devoted to dynamic inequalities of Hardy type and extensions and generalizations via convexity on a time scale T. In particular, the book contains the time scale versions of classical Hardy type inequalities, Hardy and Littlewood type inequalities, Hardy-Knopp type inequalities via convexity, Copson type inequalities, Copson-Beesack type inequalities, Liendeler type inequalities, Levinson type inequalities and Pachpatte type inequalities, Bennett type inequalities, Chan type inequalities, and Hardy type inequalities with two different weight functions. These dynamic inequalities contain the classical continuous and discrete inequalities as special cases when T = R and T = N and can be extended to different types of inequalities on different time scales such as T = hN, h > 0, T = qN for q > 1, etc.In this book the authors followed the history and development of these inequalities. Each section in self-contained and one can see the relationship between the time scale versions of the inequalities and the classical ones. To the best of the authors' knowledge this is the first book devoted to Hardy-typeinequalities and their extensions on time scales.

This contributed volume contains a collection of articles on the most recent advances in integral methods. The second of two volumes, this work focuses on the applications of integral methods to specific problems in science and engineering. Written by internationally recognized researchers, the chapters in this book are based on talks given at the Fourteenth International Conference on Integral Methods in Science and Engineering, held July 25-29, 2016, in Padova, Italy. A broad range of topics is addressed, such as:* Boundary elements* Transport problems* Option pricing* Gas reservoirs* Electromagnetic scattering This collection will be of interest to researchers in applied mathematics, physics, and mechanical and petroleum engineering, as well as graduate students in these disciplines, and to other professionals who use integration as an essential tool in their work.

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This first volume collects authoritative chapters covering the mathematical theory of fractional calculus, including fractional-order operators, integral transforms and equations, special functions, calculus of variations, and probabilistic and other aspects.

This book treats dynamic stability of structures under nonconservative forces. it is not a mathematics-based, but rather a dynamics-phenomena-oriented monograph, written with a full experimental background. Starting with fundamentals on stability of columns under nonconservative forces, it then deals with the divergence of Euler's column under a dead (conservative) loading from a view point of dynamic stability. Three experiments with cantilevered columns under a rocket-based follower force are described to present the verifiability of nonconservative problems of structural stability. Dynamic stability of columns under pulsating forces is discussed through analog experiments, and by analytical and experimental procedures together with related theories. Throughout the volume the authors retain a good balance between theory and experiments on dynamic stability of columns under nonconservative loading, offering a new window to dynamic stability of structures, promoting student- and scientist-friendly experiments.

The relaxation method has enjoyed an intensive development during many decades and this new edition of this comprehensive text reflects in particular the main achievements in the past 20 years. Moreover, many further improvements and extensions are included, both in the direction of optimal control and optimal design as well as in numerics and applications in materials science, along with an updated treatment of the abstract parts of the theory.

This book presents the applications of fractional calculus, fractional operators of non-integer orders and fractional differential equations in describing economic dynamics with long memory. Generalizations of basic economic concepts, notions and methods for the economic processes with memory are suggested. New micro and macroeconomic models with continuous time are proposed to describe the fractional economic dynamics with long memory as well.

Interest in the area of control of systems defined by partial differential Equations has increased strongly in recent years. A major reason has been the requirement of these systems for sensible continuum mechanical modelling and optimization or control techniques which account for typical physical phenomena. Particular examples of problems on which substantial progress has been made are the control and stabilization of mechatronic structures, the control of growth of thin films and crystals, the control of Laser and semi-conductor devices, and shape optimization problems for turbomachine blades, shells, smart materials and microdiffractive optics. This volume contains original articles by world reknowned experts in the fields of optimal control of partial differential equations, shape optimization, numerical methods for partial differential equations and fluid dynamics, all of whom have contributed to the analysis and solution of many of the problems discussed. The collection provides a state-of-the-art overview of the most challenging and exciting recent developments in the field. It is geared towards postgraduate students and researchers dealing with the theoretical and practical aspects of a wide variety of high technology problems in applied mathematics, fluid control, optimal design, and computer modelling.

This is the second part of a two volume anthology comprising a selection of 49 articles that illustrate the depth, breadth and scope of Nigel Kalton's research. Each article is accompanied by comments from an expert on the respective topic, which serves to situate the article in its proper context, to successfully link past, present and hopefully future developments of the theory and to help readers grasp the extent of Kalton's accomplishments. Kalton's work represents a bridge to the mathematics of tomorrow, and this book will help readers to cross it. Nigel Kalton (1946-2010) was an extraordinary mathematician who made major contributions to an amazingly diverse range of fields over the course of his career.

Numerical partial differential equations (PDEs) are an important part of numerical simulation, the third component of the modern methodology for science and engineering, besides the traditional theory and experiment. This volume contains papers that originated with the collaborative research of the teams that participated in the IMA Workshop for Women in Applied Mathematics: Numerical Partial Differential Equations and Scientific Computing in August 2014.

This book provides the latest competing research results on non-commutative harmonic analysis on homogeneous spaces with many applications. It also includes the most recent developments on other areas of mathematics including algebra and geometry. Lie group representation theory and harmonic analysis on Lie groups and on their homogeneous spaces form a significant and important area of mathematical research. These areas are interrelated with various other mathematical fields such as number theory, algebraic geometry, differential geometry, operator algebra, partial differential equations and mathematical physics. Keeping up with the fast development of this exciting area of research, Ali Baklouti (University of Sfax) and Takaaki Nomura (Kyushu University) launched a series of seminars on the topic, the first of which took place on November 2009 in Kerkennah Islands, the second in Sousse on December 2011, and the third in Hammamet on December 2013. The last seminar, which took place December 18th to 23rd 2015 in Monastir, Tunisia, has promoted further research in all the fields where the main focus was in the area of Analysis, algebra and geometry and on topics of joint collaboration of many teams in several corners. Many experts from both countries have been involved.

Theory of differentiation includes all aspects of various kinds of derivates and derivatives, and the theory of various Perron and Denjoy-Perron integrals. Derivative theorems covered are theorems on unilateral (or Dini) derivates. Through a cohesive format, outstanding problems are resolved, new ones are presented, and developments in this field, both past and present, are covered.

This volume gathers contributions in the field of partial differential equations, with a focus on mathematical models in phase transitions, complex fluids and thermomechanics. These contributions are dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. It particularly develops the following thematic areas: nonlinear dynamic and stationary equations; well-posedness of initial and boundary value problems for systems of PDEs; regularity properties for the solutions; optimal control problems and optimality conditions; feedback stabilization and stability results. Most of the articles are presented in a self-contained manner, and describe new achievements and/or the state of the art in their line of research, providing interested readers with an overview of recent advances and future research directions in PDEs.

This book is devoted to recent developments concerning linear operators, covering topics such as the Cauchy problem, Riesz basis, frames, spectral theory and applications to the Gribov operator in Bargmann space. Also, integral and integro-differential equations as well as applications to problems in mathematical physics and mechanics are discussed. Contents Introduction Linear operators Basic notations and results Bases Semi-groups Discrete operator and denseness of the generalized eigenvectors Frames in Hilbert spaces Summability of series -convergence operators -hypercyclic set of linear operators Analytic operators in Bela Szoekefalvi-Nagy's sense Bases of the perturbed operator T( ) Frame of the perturbed operator T( ) Perturbation method for sound radiation by a vibrating plate in a light fluid Applications to mathematical models Reggeon field theory