Methods of Nonlinear Analysis

The introduction of femtosecond pulse lasers has provided numerous new methods for non-destructive diagnostic analysis of biological samples. This book is the first to provide a focused and systematic treatment of femtosecond biophotonic methods. Each chapter combines theory, practice and applications, walking the reader through imaging, manipulation and fabrication techniques. Beginning with an explanation of nonlinear and multiphoton microscopy, subsequent chapters address the techniques for optical trapping and the development of laser tweezers. In a conclusion that brings together the various topics of the book, the authors discuss the growing field of femtosecond micro-engineering. The wide range of applications for femtosecond biophotonics means this book will appeal to researchers and practitioners in the fields of biomedical engineering, biophysics, life sciences and medicine.

The rapid development of the theories of Volterra integral and functional equations has been strongly promoted by their applications in physics, engineering and biology. This text shows that the theory of Volterra equations exhibits a rich variety of features not present in the theory of ordinary differential equations. The book is divided into three parts. The first considers linear theory and the second deals with quasilinear equations and existence problems for nonlinear equations, giving some general asymptotic results. Part III is devoted to frequency domain methods in the study of nonlinear equations. The entire text analyses n-dimensional rather than scalar equations, giving greater generality and wider applicability and facilitating generalizations to infinite-dimensional spaces. The book is generally self-contained and assumes only a basic knowledge of analysis. The many exercises illustrate the development of the theory and its applications, making this book accessible to res

The rapid development of the theories of Volterra integral and functional equations has been strongly promoted by their applications in physics, engineering and biology. This text shows that the theory of Volterra equations exhibits a rich variety of features not present in the theory of ordinary differential equations. The book is divided into three parts. The first considers linear theory and the second deals with quasilinear equations and existence problems for nonlinear equations, giving some general asymptotic results. Part III is devoted to frequency domain methods in the study of nonlinear equations. The entire text analyses n-dimensional rather than scalar equations, giving greater generality and wider applicability and facilitating generalizations to infinite-dimensional spaces. The book is generally self-contained and assumes only a basic knowledge of analysis. The many exercises illustrate the development of the theory and its applications, making this book accessible to res

Table of contents: Stochastic methods in nonlinear structural dynamics.- Stochastic models of uncertainties in computational structural dynamics and structural acoustics.- The tale of stochastic linea

The area of nonlinear dispersive partial differential equations (PDEs) is a fast developing field which has become exceedingly technical in recent years. With this book, the authors provide a self-contained and accessible introduction for graduate or advanced undergraduate students in mathematics, engineering, and the physical sciences. Both classical and modern methods used in the field are described in detail, concentrating on the model cases that simplify the presentation without compromising the deep technical aspects of the theory, thus allowing students to learn the material in a short period of time. This book is appropriate both for self-study by students with a background in analysis, and for teaching a semester-long introductory graduate course in nonlinear dispersive PDEs. Copious exercises are included, and applications of the theory are also presented to connect dispersive PDEs with the more general areas of dynamical systems and mathematical physics.

The area of nonlinear dispersive partial differential equations (PDEs) is a fast developing field which has become exceedingly technical in recent years. With this book, the authors provide a self-contained and accessible introduction for graduate or advanced undergraduate students in mathematics, engineering, and the physical sciences. Both classical and modern methods used in the field are described in detail, concentrating on the model cases that simplify the presentation without compromising the deep technical aspects of the theory, thus allowing students to learn the material in a short period of time. This book is appropriate both for self-study by students with a background in analysis, and for teaching a semester-long introductory graduate course in nonlinear dispersive PDEs. Copious exercises are included, and applications of the theory are also presented to connect dispersive PDEs with the more general areas of dynamical systems and mathematical physics.

Using real-life applications, this graduate-level textbook introduces different mathematical methods of scientific computation to solve minimization problems using examples ranging from locating an aircraft, finding the best time to replace a computer, analyzing developments on the stock market, and constructing phylogenetic trees. The textbook focuses on several methods, including nonlinear least squares with confidence analysis, singular value decomposition, best basis, dynamic programming, linear programming, and various optimization procedures. Each chapter solves several realistic problems, introducing the modelling optimization techniques and simulation as required. This allows readers to see how the methods are put to use, making it easier to grasp the basic ideas. There are also worked examples, practical notes, and background materials to help the reader understand the topics covered. Interactive exercises are available at www.cambridge.org/9780521849890.

An overview of biomechanical modeling of human soft tissue using nonlinear theoretical mechanics and incremental finite element methods, useful for computer simulation of the human musculoskeletal sys

This book presents a tutorial review of van der Pol model, a universal oscillator model for the analysis of modern RC-oscillators in weak and strong nonlinear regimes. A detailed analysis of the injec

Frequency Domain Criteria for Absolute Stability focuses on recently-developed methods of delay-integral-quadratic constraints to provide criteria for absolute stability of nonlinear control systems.

This book provides a thorough understanding of the concepts of optimization methods from a modern perspective at the conceptual stage of complex technical systems. It focuses on nonlinear optimization

The book deals with modern methods of nonlinear stability theory applied to problems of continuous media mechanics in the presence of interfaces, with applications to materials science, chemical engin

This book presents general methods for the design of economic model predictive control (EMPC) systems for broad classes of nonlinear systems that address key theoretical and practical considerations i

Studying the complex physical systems of stellar jets necessitates the incorporation of nonlinear effects which occur on a wide variety of length and timescales. One of the primary methods used to stu

This book introduces the use of Lie algebra and differential geometry methods to study nonlinear integrable systems of Toda type. Many challenging problems in theoretical physics are related to the solution of nonlinear systems of partial differential equations. One of the most fruitful approaches in recent years has resulted from a merging of group algebraic and geometric techniques. The book gives a comprehensive introduction to this exciting branch of science. Chapters 1 and 2 review basic notions of Lie algebras and differential geometry with an emphasis on further applications to integrable nonlinear systems. Chapter 3 contains a derivation of Toda type systems and their general solutions based on Lie algebra and differential geometry methods. The last chapter examines explicit solutions of the corresponding equations. The book is written in an accessible 'lecture note' style with many examples and exercises to illustrate key points and to reinforce understanding.

This book is devoted to some mathematical methods that arise in two domains of artificial intelligence: neural networks and qualitative physics. Professor Aubin makes use of control and viability theory in neural networks and cognitive systems, regarded as dynamical systems controlled by synaptic matrices, and set-valued analysis that plays a natural and crucial role in qualitative analysis and simulation. This allows many examples of neural networks to be presented in a unified way. In addition, several results on the control of linear and nonlinear systems are used to obtain a 'learning algorithm' of pattern classification problems, such as the back-propagation formula, as well as learning algorithms of feedback regulation laws of solutions to control systems subject to state constraints. This book will be of value to anyone with an interest in neural networks and cognitive systems.

This book is devoted to some mathematical methods that arise in two domains of artificial intelligence: neural networks and qualitative physics. Professor Aubin makes use of control and viability theory in neural networks and cognitive systems, regarded as dynamical systems controlled by synaptic matrices, and set-valued analysis that plays a natural and crucial role in qualitative analysis and simulation. This allows many examples of neural networks to be presented in a unified way. In addition, several results on the control of linear and nonlinear systems are used to obtain a 'learning algorithm' of pattern classification problems, such as the back-propagation formula, as well as learning algorithms of feedback regulation laws of solutions to control systems subject to state constraints. This book will be of value to anyone with an interest in neural networks and cognitive systems.

Hybrid Predictive Control for Dynamic Transport Problems develops methods for the design of predictive control strategies for nonlinear-dynamic hybrid discrete-/continuous-variable systems. The method

Fourier Series in Several Variables with Applications to Partial Differential Equations illustrates the value of Fourier series methods in solving difficult nonlinear partial differential equations (P

There are many methods of stable controller design for nonlinear systems. In seeking to go beyond the minimum requirement of stability, Adaptive Dynamic Programming in Discrete Time approaches the cha

A defining feature of nonlinear hyperbolic equations is the occurrence of shock waves. While the popular shock-capturing methods are easy to implement, shock-fitting techniques provide the most accura

Maximum Principles for the Hill's Equation focuses on the application of these methods to nonlinear equations with singularities (e.g. Brillouin-bem focusing equation, Ermakov-Pinney,…) and for proble

Lattice-gas cellular automata (LGCA) and lattice Boltzmann models (LBM) are relatively new and promising methods for the numerical solution of nonlinear partial differential equations. The book provid

This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to

In this monograph, the authors present some powerful methods for dealing with singularities in elliptic and parabolic partial differential inequalities. Here, the authors take the unique approach of investigating differential inequalities rather than equations, the reason being that the simplest way to study an equation is often to study a corresponding inequality; for example, using sub and superharmonic functions to study harmonic functions. Another unusual feature of the present book is that it is based on integral representation formulae and nonlinear potentials, which have not been widely investigated so far. This approach can also be used to tackle higher order differential equations. The book will appeal to graduate students interested in analysis, researchers in pure and applied mathematics, and engineers who work with partial differential equations. Readers will require only a basic knowledge of functional analysis, measure theory and Sobolev spaces.

Integrates traditional statistical data analysis with new computational experimentation capabilities and concepts of algorithmic complexity and chaotic behavior in nonlinear dynamic systems, offering

This book presents the most recent progress of fundamental nature made in the new developed field of micromechanics: transformation field analysis, variational bounds for nonlinear composites, higher-

Statistical analysis of stock markets and foreign exchange markets has demonstrated the intermittent nature of economic time series. A nonlinear model of business cycles is able to simulate intermitte

This book focuses on the computational analysis of nonlinear vibrations of structural members (beams, plates, panels, shells), where the studied dynamical problems can be reduced to the consideration

This book presents the nonlinear theory of continuum mechanics and demonstrates its use in developing nonlinear computer formulations for large displacement dynamic analysis. Basic concepts used in continuum mechanics are presented and used to develop nonlinear general finite element formulations that can be effectively used in large displacement analysis. The book considers two nonlinear finite element dynamic formulations: a general large deformation finite element formulation and a formulation that can efficiently solve small deformation problems that characterize very stiff structures. The book presents material clearly and systematically, assuming the reader has only basic knowledge in matrix and vector algebra and dynamics. The book is designed for use by advanced undergraduates and first-year graduate students. It is also a reference for researchers, practising engineers, and scientists working in computational mechanics, bio-mechanics, computational biology, multibody system dy

This book presents the nonlinear theory of continuum mechanics and demonstrates its use in developing nonlinear computer formulations for large displacement dynamic analysis. Basic concepts used in continuum mechanics are presented and used to develop nonlinear general finite element formulations that can be effectively used in large displacement analysis. The book considers two nonlinear finite element dynamic formulations: a general large deformation finite element formulation and a formulation that can efficiently solve small deformation problems that characterize very stiff structures. The book presents material clearly and systematically, assuming the reader has only basic knowledge in matrix and vector algebra and dynamics. The book is designed for use by advanced undergraduates and first-year graduate students. It is also a reference for researchers, practising engineers, and scientists working in computational mechanics, bio-mechanics, computational biology, multibody system dy

This book addresses the linear and nonlinear two-phase stability of the one-dimensional Two-Fluid Model (TFM) material waves and the numerical methods used to solve it. The TFM fluid dynamic stability

Written for mathematicians working in nonlinear analysis, calculus of variations, Hamiltonian systems, and symplectic geometry, this study explores existence and multiplicity questions for periodic so

This book provides a comprehensive treatment of the optimal control design and analysis of both linear and nonlinear networked systems without needing the system dynamics and network imperfections. Bo

This book presents the performances of the computational method for system failure prediction by numerical analysis and nonlinear dynamic models. It includes the research of authors on the development

This connects basic earthquake engineering principles with practical structural design. It includes recent developments of RC structures and nonlinear analysis, structural intervention prior or post-e

This research monograph represents an outcome of the cross-fertilization between nonlinear functional analysis and mathematical modelling, and demonstrates its application to solid and contact mechani

An overview of the theory and application of linear and nonlinear mixed-effects models in the analysis of grouped data, such as longitudinal data, repeated measures, and multilevel data. The authors p

Quantile regression is gradually emerging as a unified statistical methodology for estimating models of conditional quantile functions. By complementing the exclusive focus of classical least squares regression on the conditional mean, quantile regression offers a systematic strategy for examining how covariates influence the location, scale and shape of the entire response distribution. This monograph is the first comprehensive treatment of the subject, encompassing models that are linear and nonlinear, parametric and nonparametric. The author has devoted more than 25 years of research to this topic. The methods in the analysis are illustrated with a variety of applications from economics, biology, ecology and finance. The treatment will find its core audiences in econometrics, statistics, and applied mathematics in addition to the disciplines cited above.

Quantile regression is gradually emerging as a unified statistical methodology for estimating models of conditional quantile functions. By complementing the exclusive focus of classical least squares regression on the conditional mean, quantile regression offers a systematic strategy for examining how covariates influence the location, scale and shape of the entire response distribution. This monograph is the first comprehensive treatment of the subject, encompassing models that are linear and nonlinear, parametric and nonparametric. The author has devoted more than 25 years of research to this topic. The methods in the analysis are illustrated with a variety of applications from economics, biology, ecology and finance. The treatment will find its core audiences in econometrics, statistics, and applied mathematics in addition to the disciplines cited above.