In this book, fundamental methods of nonlinear analysis are introduced, discussed and illustrated in straightforward examples. Each method considered is motivated and explained in its general form, but presented in an abstract framework as comprehensively as possible. A large number of methods are applied to boundary value problems for both ordinary and partial differential equations. In this edition we have made minor revisions, added new material and organized the content slightly differently.

In particular, we included evolutionary equations and differential equations on manifolds. The applications to partial differential equations follow every abstract framework of the method in question.

The text is structured in two levels: a self-contained basic level and an advanced level - organized in appendices - for the more experienced reader. The last chapter contains more involved material and can be skipped by those new to the field. This book serves as both a textbook for graduate-level courses and a reference book for mathematicians, engineers and applied scientists

The application of methodological approaches and mathematical formalisms proper to Physics and Engineering to investigate and describe biological processes and design biological structures has led to the development of many disciplines in the context of computational biology and biotechnology. The best known applicative domain is tissue engineering and its branches. Recent domains of interest are in the field of biophysics, e.g.: multiscale mechanics of biological membranes and films and filaments; multiscale mechanics of adhesion; biomolecular motors and force generation.

Modern hypotheses, models, and tools are currently emerging and resulting from the convergence of the methods and phylosophycal apporaches of the different research areas and disciplines. All these emerging approaches share the purpose of disentangling the complexity of organisms, tissues, and cells and mimiking the function of living systems.

The contributions presented in this book are current research highlights of six challenging and representative applicative domains of phyisical, engineering, and computational approaches in medicine and biology, i.e tissue engineering, modelling of molecular structures, cell mechanics and cell adhesion processes, cancer physics, and physico-chemical processes of metabolic interactions. Each chapter presents a compendium or a review of the original results achieved by authors in the last years. Furthermore, the book also wants to pinpoint the questions that are still open and that could propel the future research.

This volume presents selected papers by the brilliant Uruguayan mathematician Ricardo Mane, known for his outstanding contributions to dynamical systems and ergodic theory. He was a student of Jacob Palis at IMPA and pursued his whole scientific career at IMPA. Mane was invited to speak twice in the section of Ordinary Differential Equations and Dynamical Systems, at the International Congress of Mathematics held in Warsaw in 1983 and in Zurich in 1994. He was also a speaker at the Colloquium organized by the Societe Mathematique de France, celebrating R. Thom's 65 anniversary. In 1994, he became a member of the Brazilian Academy of Sciences and was awarded the Third World Academy of Sciences Prize for Mathematics.

Rapid developments in multivariable spectral theory have led to important and fascinating results which also have applications in other mathematical disciplines. In this book, classical results from the cohomology theory of Banach algebras, multidimensional spectral theory, and complex analytic geometry have been freshly interpreted using the language of homological algebra. It has also been used to give in sights into new developments in the spectral theory of linear operators. Various concepts from function theory and complex analytic geometry are drawn together and used to give a new approach to concrete spectral computations. The advantages of this approach are illustrated by a variety of examples, unexpected applications, and conceptually new ideas which should stimulate further research.

Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateaus problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateaus problem have no interior branch points.

The fundamental contributions made by the late Victor Lomonosov in several areas of analysis are revisited in this book, in particular, by presenting new results and future directions from world-recognized specialists in the field. The invariant subspace problem, Burnside's theorem, and the Bishop-Phelps theorem are discussed in detail. This volume is an essential reference to both researchers and graduate students in mathematical analysis.

For one- or two-semester junior orsenior level courses in Advanced Calculus, Analysis I, or Real Analysis. This title is part of the Pearson Modern Classicsseries. This text prepares students for future coursesthat use analytic ideas, such as real and complex analysis, partial andordinary differential equations, numerical analysis, fluid mechanics, anddifferential geometry. This book is designed to challenge advanced studentswhile encouraging and helping weaker students. Offering readability,practicality and flexibility, Wade presents fundamental theorems and ideas froma practical viewpoint, showing students the motivation behind the mathematicsand enabling them to construct their own proofs.

The behavior of materials at the nanoscale is a key aspect of modern nanoscience and nanotechnology. This book presents rigorous mathematical techniques showing that some very useful phenomenological properties which can be observed at the nanoscale in many nonlinear reaction-diffusion processes can be simulated and justified mathematically by means of homogenization processes when a certain critical scale is used in the corresponding framework.

Network Science is the emerging field concerned with the study of large, realistic networks. This interdisciplinary endeavor, focusing on the patterns of interactions that arise between individual components of natural and engineered systems, has been applied to data sets from activities as diverse as high-throughput biological experiments, online trading information, smart-meter utility supplies, and pervasive telecommunications and surveillance technologies. This unique text/reference provides a fascinating insight into the state of the art in network science, highlighting the commonality across very different areas of application and the ways in which each area can be advanced by injecting ideas and techniques from another. The book includes contributions from an international selection of experts, providing viewpoints from a broad range of disciplines. It emphasizes networks that arise in nature-such as food webs, protein interactions, gene expression, and neural connections-and in technology-such as finance, airline transport, urban development and global trade. Topics and Features: begins with a clear overview chapter to introduce this interdisciplinary field; discusses the classic network science of fixed connectivity structures, including empirical studies, mathematical models and computational algorithms; examines time-dependent processes that take place over networks, covering topics such as synchronisation, and message passing algorithms; investigates time-evolving networks, such as the World Wide Web and shifts in topological properties (connectivity, spectrum, percolation); explores applications of complex networks in the physical and engineering sciences, looking ahead to new developments in the field. Researchers and professionals from disciplines as varied as computer science, mathematics, engineering, physics, chemistry, biology, ecology, neuroscience, epidemiology, and the social sciences will all benefit from this topical and broad overview of current activities and grand challenges in the unfolding field of network science.

All the existing books in Infinite Dimensional Complex Analysis focus on the problems of locally convex spaces. However, the theory without convexity condition is covered for the first time in this book. This shows that we are really working with a new, important and interesting field.

Theory of functions and nonlinear analysis problems are widespread in the mathematical modeling of real world systems in a very broad range of applications. During the past three decades many new results from the author have helped to solve multiextreme problems arising from important situations, non-convex and non linear cases, in function theory.

Foundations of Complex Analysis in Non Locally Convex Spaces is a comprehensive book that covers the fundamental theorems in Complex and Functional Analysis and presents much new material.

The book includes generalized new forms of: Hahn-Banach Theorem, Multilinear maps, theory of polynomials, Fixed Point Theorems, p-extreme points and applications in Operations Research, Krein-Milman Theorem, Quasi-differential Calculus, Lagrange Mean-Value Theorems, Taylor series, Quasi-holomorphic and Quasi-analytic maps, Quasi-Analytic continuations, Fundamental Theorem of Calculus, Bolzano's Theorem, Mean-Value Theorem for Definite Integral, Bounding and weakly-bounding (limited) sets, Holomorphic Completions, and Levi problem.

Each chapter contains illustrative examples to help the student and researcher to enhance his knowledge of theory of functions.

The new concept of Quasi-differentiability introduced by the author represents the backbone of the theory of Holomorphy for non-locally convex spaces. In fact it is different but much stronger than the Frechet one.

The book is intended not only for Post-Graduate (M.Sc.& Ph.D.) students and researchers in Complex and Functional Analysis, but for all Scientists in various disciplines whom need nonlinear or non-convex analysis and holomorphy methods without convexity conditions to model and solve problems.

bull; The book contains new generalized versions of:
i) Fundamental Theorem of Calculus, Lagrange Mean-Value Theorem in real and complex cases, Hahn-Banach Theorems, Bolzano Theorem, Krein-Milman Theorem, Mean value Theorem for Definite Integral, and many others.
ii) Fixed Point Theorems of Bruower, Schauder and Kakutani's.

bull; The book contains some applications in Operations research and non convex analysis as a consequence of the new concept p-Extreme points given by the author.

bull; The book contains a complete theory for Taylor Series representations of the different types of holomorphic maps in F-spaces without convexity conditions.

bull; The book contains a general new concept of differentiability stronger than the Frechet one. This implies a new Differentiable Calculus called Quasi-differential (or Bayoumi differential) Calculus. It is due to the author's discovery in 1995.

bull; The book contains the theory of polynomials and Banach Stienhaus theorem in non convex spaces.

New Edition: Fractional Calculus: An Introduction for Physicists (3rd Edition)Fractional calculus is undergoing rapid and ongoing development. We can already recognize, that within its framework new concepts and strategies emerge, which lead to new challenging insights and surprising correlations between different branches of physics.This book is an invitation both to the interested student and the professional researcher. It presents a thorough introduction to the basics of fractional calculus and guides the reader directly to the current state-of-the-art physical interpretation. It is also devoted to the application of fractional calculus on physical problems, in the subjects of classical mechanics, friction, damping, oscillations, group theory, quantum mechanics, nuclear physics, and hadron spectroscopy up to quantum field theory.

Georg Heing, a charming, erudite man, and a ?rst rate mathematician died un- pectedly of a heart attack on May 10, 2005. Georg is survived by his wife Gerti, his daughter Susanne, and his son Peter. We have lost one the leading experts in the ?eld of structured matrices, a wonderful colleague, and a terri?c friend. GeorgHeinig'sresults,approaches,andhisscienti?ctastein?uencedourc- munity of researchers working on structured matrices. In fact, the community's focus grew to re?ect his interdisciplinary vision ranging from applications (e.g., in systems and control theory and signal processing) through fundamental mat- matics(structuredmatrices,periodicJacobi,Toeplitz,andWiener-Hopfoperators, classesofsingularintegraloperators,resultantsandBezoutiansforoperator-valued polynomials and continual analogs thereof) to numerical analysis and fast al- rithms. The broad spectrum of Georg Heinig's interests are represented in this collection. Georg served as an Associate Editor of two top journals: Integral Equations andOperatorTheoryand LinearAlgebraand Its Applications. This volumestarts with two eulogies published earlier by IEOT and LAA. The ?rst one, published in IEOT is by Albrecht Bot .. tcher, Israel Gohberg (who was Georg's advisor during his Ph. D. studies), and Bernd Silbermann. The second one, published in LAA is by Karla Rost who collaborated with Georg during last three decades until day of his death. They have produced together more than 30 papers and a monograph.

Non-Newtonian flows and their numerical simulations have generated an abundant literature, as well as many publications and references to which can be found in this volume s articles. This abundance of publications can be explained by the fact that non-Newtonian fluids occur in many real life situations: the food industry, oil & gas industry, chemical, civil and mechanical engineering, the bio-Sciences, to name just a few. Mathematical and numerical analysis of non-Newtonian fluid flow models provide challenging problems to partial differential equations specialists and applied computational mathematicians alike.

This volume offers investigations. Results and conclusions that will no doubt be useful to engineers and computational and applied mathematicians who are focused on various aspects of non-Newtonian Fluid Mechanics.

New review of well-known computational methods for the simulation viscoelastic and viscoplastic types.; Discusses new numerical methods that have proven to be more efficient and more accurate than traditional methods.; Articles that discuss the numerical simulation of particulate flow for viscoelastic fluids.; "

This book provides a comprehensive guide to analyzing and solving optimal design problems in continuous media by means of the so-called sub-relaxation method. Though the underlying ideas are borrowed from other, more classical approaches, here they are used and organized in a novel way, yielding a distinct perspective on how to approach this kind of optimization problems. Starting with a discussion of the background motivation, the book broadly explains the sub-relaxation method in general terms, helping readers to grasp, from the very beginning, the driving idea and where the text is heading. In addition to the analytical content of the method, it examines practical issues like optimality and numerical approximation. Though the primary focus is on the development of the method for the conductivity context, the book's final two chapters explore several extensions of the method to other problems, as well as formal proofs. The text can be used for a graduate course in optimal design, even if the method would require some familiarity with the main analytical issues associated with this type of problems. This can be addressed with the help of the provided bibliography.

Degenerate and singular parabolic equations have been the subject of extensive research for the last 25 years. Despite important achievements, the issue of the Harnack inequality for non-negative solutions to these equations, both of p-Laplacian and porous medium type, while raised by several authors, has remained basically open. Recently considerable progress has been made on this issue, to the point that, except for the singular sub-critical range, both for the p-laplacian and the porous medium equations, the theory is reasonably complete. It seemed therefore timely to trace a comprehensive overview, that would highlight the main issues and also the problems that still remain open. The authors give a comprehensive treatment of the Harnack inequality for non-negative solutions to p-laplace and porous medium type equations, both in the degenerate (p2 or m1) and in the singular range (1p<2 or 0m<1), starting from the notion of solution and building all the necessary technical tools. The book is self-contained. Building on a similar monograph by the first author, the authors of the present book focus entirely on the Harnack estimates and on their applications: indeed a number of known regularity results are given a new proof, based on the Harnack inequality. It is addressed to all professionals active in the field, and also to advanced graduate students, interested in understanding the main issues of this fascinating research field.

In 1917, Johann Radon published his fundamental work, where he introduced what is now called the Radon transform. Including important contributions by several experts, this book reports on ground-breaking developments related to the Radon transform throughout these years, and also discusses novel mathematical research topics and applications for the next century.

Chapter 1 introduces elementary classical special functions. Gamma, beta, psi, zeta functions, hypergeometric functions and the associated special functions, generalizations to Meijer's G and Fox's H-functions are examined here. Discussion is confined to basic properties and selected applications. Introduction to statistical distribution theory is provided. Some recent extensions of Dirichlet integrals and Dirichlet densities are discussed. A glimpse into multivariable special functions such as Appell's functions and Lauricella functions is part of Chapter 1. Special functions as solutions of differential equations are examined. Chapter 2 is devoted to fractional calculus. Fractional integrals and fractional derivatives are discussed. Their applications to reaction-diffusion problems in physics, input-output analysis, and Mittag-Leffler stochastic processes are developed. Chapter 3 deals with q-hyper-geometric or basic hypergeometric functions. Chapter 4 covers basic hypergeometric functions and Ramanujan's work on elliptic and theta functions. Chapter 5 examines the topic of special functions and Lie groups. Chapters 6 to 9 are devoted to applications of special functions. Applications to stochastic processes, geometric infinite divisibility of random variables, Mittag-Leffler processes, alpha-Laplace processes, density estimation, order statistics and astrophysics problems, are dealt with in Chapters 6 to 9. Chapter 10 is devoted to wavelet analysis. An introduction to wavelet analysis is given. Chapter 11 deals with the Jacobians of matrix transformations. Various types of matrix transformations and the associated Jacobians are provided. Chapter 12 is devoted to the discussion offunctions of matrix argument in the real case. Functions of matrix argument and the pathway models along with their applications are discussed.

Formed of presented papers this volume contains research from the 40th International Conference on Boundary Elements and other Mesh Reduction Methods, recognised as THE international forum for the latest advances in these techniques and their applications in science and engineering. The ongoing success of this series is a result of the strength of research being carried out all over the world and the coverage has continually evolved in line with the latest developments in the field. The books originating from this conference series constitute a record of the development of BEM/MRM, running from the initial successful development of boundary integral techniques into the boundary element method, a technique that eliminates the need for an internal mesh, to the recent and most sophisticated Mesh Reduction and even Meshless Methods. Since these methods are used in many engineering and scientific fields the 2017 book, Boundary Elements and other Mesh Reduction Methods, like the series before, will be of great interest to those working within the areas of numerical analysis, boundary elements and meshless methods. The research papers included in this volume cover: Advanced formulations; Advanced meshless and mesh reduction methods; Structural mechanics applications; Solid mechanics; Heat and mass transfer; Electrical engineering and electromagnetics; Computational methods; Fluid flow modelling; Damage mechanics and fracture; Dynamics and vibrations; Engineering applications; Interfacing with other methods; Coupling with design and manufacturing; Solution of large systems of equations.

This monograph presents a novel method of sliding mode control for switch-regulated nonlinear systems. The Delta Sigma modulation approach allows one to implement a continuous control scheme using one or multiple, independent switches, thus effectively merging the available linear and nonlinear controller design techniques with sliding mode control. Sliding Mode Control: The Delta-Sigma Modulation Approach, combines rigorous mathematical derivation of the unique features of Sliding Mode Control and Delta-Sigma modulation with numerous illustrative examples from diverse areas of engineering. In addition, engineering case studies demonstrate the applicability of the technique and the ease with which one can implement the exposed results. This book will appeal to researchers in control engineering and can be used as graduate-level textbook for a first course on sliding mode control.

Complementarity theory, a relatively new domain in applied mathematics, has deep connections with several aspects of fundamental mathematics and also has many applications in optimization, economics and engineering. The study of variational inequalities is another domain of applied mathematics with many applications to the study of certain problems with unilateral conditions. This book is the first to discuss complementarity theory and variational inequalities using Leray-Schauder type alternatives. The ideas and method presented in this book may be considered as a starting point for new developments.

This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. It is organized in a distinctive, flexible way that would make it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus.

The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;
methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and
methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.
As a result, the book represents a blend of new methods in general computational analysis,
and specific, but also generic, techniques for study of systems theory ant its particular
branches, such as optimal filtering and information compression.
- Best operator approximation,
- Non-Lagrange interpolation,
- Generic Karhunen-Loeve transform
- Generalised low-rank matrix approximation
- Optimal data compression
- Optimal nonlinear filtering

Thepresentbookisamemorialvolumedevotedtoourfriend,colleagueandteacher Peter Jonas who passed away on July 18, 2007. It displays recent advances in modern operator theory in Hilbert and Krein spaces and contains a collection of original research papers written by participants of the 7th Workshop on Operator Theory in Krein Spaces and Spectral Analysis, which was held at the Technische Universit. at Berlin, Germany, December 13 to 16, 2007. The articles in this v- ume contain new results for problems close to the area of research of Peter Jonas: Spectralandperturbationproblemsfor operatorsininner productspaces,gener- ized Nevanlinna functions and de?nitizable functions, scattering theory, extension theory for symmetric operators, ?xed points, hyperbolic matrix polynomials, - ment problems, inde?nite spectral and Sturm-Liouville problems, and invariant subspace problems. It is a pleasure to acknowledge the substantial ?nancial support for the 7th Workshop on Operator Theory in Krein Spaces and Spectral Analysis received from the - Berlin Mathematical School (BMS) - Gesellschaft fur .. Angewandte Mathematik und Mechanik (GAMM) - International Mathematical Union, Commission on Development and Exchanges - Institute of Mathematics of the Technische Universit. at Berlin The Editors Peter Jonas (1941-2007) In Memoriam Peter Jonas (1941-2007) Jussi Behrndt, Karl-Heinz F.. orster and Carsten Trunk Peter Jonas was born on July 18, 1941, in Memel, now Klaipeda, which was at thattime the mosteasterntownofEastPrussia.After the war,PeterJonasmoved with his mother and grandmother to Blankenfelde - a small village near Berlin, where he lived until the end of his school education.

In the field of Dynamical Systems, nonlinear iterative processes play an important role. Nonlinear mappings can be found as immediate models for many systems from different scientific areas, such as engineering, economics, biology, or can also be obtained via numerical methods permitting to solve non-linear differential equations. In both cases, the understanding of specific dynamical behaviors and phenomena is of the greatest interest for scientists. This volume contains papers that were presented at the International Workshop on Nonlinear Maps and their Applications (NOMA 2011) held in Evora, Portugal, on September 15-16, 2011. This kind of collaborative effort is of paramount importance in promoting communication among the various groups that work in dynamical systems and networks in their research theoretical studies as well as for applications. This volume is suitable for graduate students as well as researchers in the field.

This book presents a unified treatise of the theory of measure and integration. In the setting of a general measure space, every concept is defined precisely and every theorem is presented with a clear and complete proof with all the relevant details. Counter-examples are provided to show that certain conditions in the hypothesis of a theorem cannot be simply dropped. The dependence of a theorem on earlier theorems is explicitly indicated in the proof, not only to facilitate reading but also to delineate the structure of the theory. The precision and clarity of presentation make the book an ideal textbook for a graduate course in real analysis while the wealth of topics treated also make the book a valuable reference work for mathematicians.