This volume is based on PDE courses given by the authors at the Courant Institute and at the University of Notre Dame, Indiana. Presented are basic methods for obtaining various a priori estimates for second-order equations of elliptic type with particular emphasis on maximal principles, Harnack inequalities, and their applications. The equations considered in the book are linear; however, the presented methods also apply to nonlinear problems. This second edition has been thoroughly revised and in a new chapter the authors discuss several methods for proving the existence of solutions of primarily the Dirichlet problem for various types of elliptic equations.

Structured population models are transport-type equations often applied to describe evolution of heterogeneous populations of biological cells, animals or humans, including phenomena such as crowd dynamics or pedestrian flows. This book introduces the mathematical underpinnings of these applications, providing a comprehensive analytical framework for structured population models in spaces of Radon measures. The unified approach allows for the study of transport processes on structures that are not vector spaces (such as traffic flow on graphs) and enables the analysis of the numerical algorithms used in applications. Presenting a coherent account of over a decade of research in the area, the text includes appendices outlining the necessary background material and discusses current trends in the theory, enabling graduate students to jump quickly into research.

This book provides an introduction to functional analysis for non-experts in mathematics. As such, it is distinct from most other books on the subject that are intended for mathematicians. Concepts are explained concisely with visual materials, making it accessible for those unfamiliar with graduate-level mathematics. Topics include topology, vector spaces, tensor spaces, Lebesgue integrals, and operators, to name a few. Each chapter explains, concisely, the purpose of the specific topic and the benefit of understanding it. Researchers and graduate students in physics, mechanical engineering, and information science will benefit from this view of functional analysis.

Although the analysis of scattering for closed bodies of simple geometric shape is well developed, structures with edges, cavities, or inclusions have seemed, until now, intractable to analytical methods. This two-volume set describes a breakthrough in analytical techniques for accurately determining diffraction from classes of canonical scatterers with comprising edges and other complex cavity features. It is an authoritative account of mathematical developments over the last two decades that provides benchmarks against which solutions obtained by numerical methods can be verified. The first volume, Canonical Structures in Potential Theory, develops the mathematics, solving mixed boundary potential problems for structures with cavities and edges. The second volume, Acoustic and Electromagnetic Diffraction by Canonical Structures, examines the diffraction of acoustic and electromagnetic waves from several classes of open structures with edges or cavities. Together these volumes present an authoritative and unified treatment of potential theory and diffraction-the first complete description quantifying the scattering mechanisms in complex structures.

Although the analysis of scattering for closed bodies of simple geometric shape is well developed, structures with edges, cavities, or inclusions have seemed, until now, intractable to analytical methods. This two-volume set describes a breakthrough in analytical techniques for accurately determining diffraction from classes of canonical scatterers with comprising edges and other complex cavity features. It is an authoritative account of mathematical developments over the last two decades that provides benchmarks against which solutions obtained by numerical methods can be verified. The first volume, Canonical Structures in Potential Theory, develops the mathematics, solving mixed boundary potential problems for structures with cavities and edges. The second volume, Acoustic and Electromagnetic Diffraction by Canonical Structures, examines the diffraction of acoustic and electromagnetic waves from several classes of open structures with edges or cavities. Together these volumes present an authoritative and unified treatment of potential theory and diffraction-the first complete description quantifying the scattering mechanisms in complex structures.

Basic Analysis: Volumes I-V is written with the aim of balancing theory and abstraction with clear explanations and arguments, so that students and researchers alike who are from a variety of different areas can follow this text and use it profitably for self-study. The first volume is designed for students who have completed the usual calculus and ordinary differential equation sequence and a basic course in linear algebra. This is a critical course in the use of abstraction, but is just first volume in a sequence of courses which prepare students to become practicing scientists. The second volume focuses on differentiation in n-dimensions and important concepts about mappings between finite dimensional Euclidean spaces, such as the inverse and implicit function theorem and change of variable formulae for multidimensional integration. These important topics provide background in important applied and theoretical areas which are no longer covered in mathematical science curricula. Although it follows on from the preceding volume, this is a self-contained book, accessible to undergraduates with a standard course in undergraduate analysis. The third volume is intended as a first course in abstract linear analysis. This textbook covers metric spaces, normed linear spaces and inner product spaces, along with many other deeper abstract ideas such a completeness, operators and dual spaces. These topics act as an important tool in the development of a mathematically trained scientist. The fourth volume introduces students to concepts from measure theory and continues their training in the abstract way of looking at the world. This is a most important skill to have when your life's work will involve quantitative modeling to gain insight into the real world. This text generalizes the notion of integration to a very abstract setting in a variety of ways. We generalize the notion of the length of an interval to the measure of a set and learn how to construct the usual ideas from integration using measures. We discuss carefully the many notions of convergence that measure theory provides. The final volume introduces graduate students in science with concepts from topology and functional analysis, both linear and nonlinear. It is the fifth book in a series designed to train interested readers how to think properly using mathematical abstractions, and how to use the tools of mathematical analysis in applications. It is important to realize that the most difficult part of applying mathematical reasoning to a new problem domain is choosing the underlying mathematical framework to use on the problem. Once that choice is made, we have many tools we can use to solve the problem. However, a different choice would open up avenues of analysis from a different, perhaps more productive perspective. In this volume, the nature of these critical choices is discussed using applications involving the immune system and cognition. Features: Can be used as a supplementary text for anyone whose work requires that they begin to assimilate more abstract mathematical concepts as part of their professional growth Function as a traditional textbook as well as a resource for self-study Suitable for mathematics students and for those in other disciplines such as biology, physics, and economics and others requiring a careful and solid grounding in the use of abstraction in problem solving Emphasizes learning how to understand the consequences of the underlying assumptions used in building a model Regularly uses computation tools to help understand abstract concepts.

This book is not a text devoted to a pedagogical presentation of a specialized topic nor is it a monograph focused on the author's area of research. It accomplishes both these things while providing a rationale for why the reader ought to be interested in learning about fractional calculus. This book is for researchers who has heard about many of these scientifically exotic activities, but could not see how they fit into their own scientific interests, or how they could be made compatible with the way they understand science. It is also for beginners who have not yet decided where their scientific talents could be most productively applied. The book provides insight into the long-term direction of science and show how to develop the skills necessary to successfully do research in the twenty-first century.

This book presents the theory and applications of Fourier series and integrals, eigenfunction expansions, and related topics, on a level suitable for advanced undergraduates. It includes material on Bessel functions, orthogonal polynomials, and Laplace transforms, and it concludes with chapters on generalized functions and Green's functions for ordinary and partial differential equations. The book deals almost exclusively with aspects of these subjects that are useful in physics and engineering, and includes a wide variety of applications. On the theoretical side, it uses ideas from modern analysis to develop the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs.

In Mathematical Analysis and Optimization for Economists, the author aims to introduce students of economics to the power and versatility of traditional as well as contemporary methodologies in mathematics and optimization theory; and, illustrates how these techniques can be applied in solving microeconomic problems. This book combines the areas of intermediate to advanced mathematics, optimization, and microeconomic decision making, and is suitable for advanced undergraduates and first-year graduate students. This text is highly readable, with all concepts fully defined, and contains numerous detailed example problems in both mathematics and microeconomic applications. Each section contains some standard, as well as more thoughtful and challenging, exercises. Solutions can be downloaded from the CRC Press website. All solutions are detailed and complete. Features Contains a whole spectrum of modern applicable mathematical techniques, many of which are not found in other books of this type. Comprehensive and contains numerous and detailed example problems in both mathematics and economic analysis. Suitable for economists and economics students with only a minimal mathematical background. Classroom-tested over the years when the author was actively teaching at the University of Hartford. Serves as a beginner text in optimization for applied mathematics students. Accompanied by several electronic chapters on linear algebra and matrix theory, nonsmooth optimization, economic efficiency, and distance functions available for free on www.routledge.com/9780367759018.

Fourier analysis is a subject that was born in physics but grew up in mathematics. Now it is part of the standard repertoire for mathematicians, physicists and engineers. This diversity of interest is often overlooked, but in this much-loved book, Tom Koerner provides a shop window for some of the ideas, techniques and elegant results of Fourier analysis, and for their applications. These range from number theory, numerical analysis, control theory and statistics, to earth science, astronomy and electrical engineering. The prerequisites are few (a reader with knowledge of second- or third-year undergraduate mathematics should have no difficulty following the text), and the style is lively and entertaining. This edition of Koerner's 1989 text includes a foreword written by Professor Terence Tao introducing it to a new generation of fans.

Although the theory behind solitary waves of strain shows that they hold significant promise in nondestructive testing and a variety of other applications, an enigma has long persisted-the absence of observable elastic solitary waves in practice. Inspired by this apparent contradiction, Strain Solitons in Solids and How to Construct Them refines the existing theory, explores how to construct a powerful deformation pulse in a waveguide without plastic flow or fracture, and proposes a direct method of strain soliton generation, detection, and observation. The author focuses on the theory, simulation, generation, and propagation of strain solitary waves in a nonlinearly elastic, straight cylindrical rod under finite deformations. He introduces the general theory of wave propagation in nonlinearly elastic solids and shows, from first principles, how its main ideas can lead to successful experiments. In doing so, he develops a new approach to solving the corresponding doubly dispersive equation (DDE) with dissipative terms, leading to new explicit and exact solutions. He also shows that the method is applicable to a variety of nonlinear problems. First discovered in virtual reality, nonlinear waves and solitons in solids are finally moving into the genuine reality of physics, mechanics, and engineering. Strain Solitons in Solids and How to Construct Them shows how to balance the mathematics of the problem with the application of the results to experiments and ultimately to generating and observing solitons in solids.

Based on the proceedings of the International Conference on Stochastic Partial Differential Equations and Applications-V held in Trento, Italy, this illuminating reference presents applications in filtering theory, stochastic quantization, quantum probability, and mathematical finance and identifies paths for future research in the field. Stochastic Partial Differential Equations and Applications analyzes recent developments in the study of quantum random fields, control theory, white noise, and fluid dynamics. It presents precise conditions for nontrivial and well-defined scattering, new Gaussian noise terms, models depicting the asymptotic behavior of evolution equations, and solutions to filtering dilemmas in signal processing. With contributions from more than 40 leading experts in the field, Stochastic Partial Differential Equations and Applications is an excellent resource for pure and applied mathematicians; numerical analysts; mathematical physicists; geometers; economists; probabilists; computer scientists; control, electrical, and electronics engineers; and upper-level undergraduate and graduate students in these disciplines.

This book contains papers presented at the Chicago Conference on Harmonic Analysis in 1981. The papers are compiled under topics, namely trigonometric series, singular integrals and pseudodifferential operators, hardy spaces, differentiation theory, and partial differential equations.

Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry.

Far from being separate entities, many social and engineering systems can be considered as complex network systems (CNSs) associated with closely linked interactions with neighbouring entities such as the Internet and power grids. Roughly speaking, a CNS refers to a networking system consisting of lots of interactional individuals, exhibiting fascinating collective behaviour that cannot always be anticipated from the inherent properties of the individuals themselves. As one of the most fundamental examples of cooperative behaviour, consensus within CNSs (or the synchronization of complex networks) has gained considerable attention from various fields of research, including systems science, control theory and electrical engineering. This book mainly studies consensus of CNSs with dynamics topologies - unlike most existing books that have focused on consensus control and analysis for CNSs under a fixed topology. As most practical networks have limited communication ability, switching graphs can be used to characterize real-world communication topologies, leading to a wider range of practical applications. This book provides some novel multiple Lyapunov functions (MLFs), good candidates for analysing the consensus of CNSs with directed switching topologies, while each chapter provides detailed theoretical analyses according to the stability theory of switched systems. Moreover, numerical simulations are provided to validate the theoretical results. Both professional researchers and laypeople will benefit from this book.

An unusual supplement to every calculus textbook, Misteaks and How to Find Them before the Teacher Does is popular with students and teachers alike. Teachers love the way it encourages students to truly think about mathematics rather than simply plugging numbers into equations to crank out answers, and students love the author's straightforward, tongue-in-cheek style. The title of this light-hearted and amusing book might well have been "Going Gray in Elementary Calculus and How to Avoid it." Changing the metaphor, Barry has hit the nail on the finger in hundreds of fine examples. --Philip J. Davis, coauthor of The Mathematical Experience. "How I wish that something like this had been available when I was a student!" --Ralph P. Boas, former editor of The American Mathematical Monthly. Bonus: Solution to LeWitt Puzzle

Aimed primarily at undergraduate level university students, An Illustrative Introduction to Modern Analysis provides an accessible and lucid contemporary account of the fundamental principles of Mathematical Analysis. The themes treated include Metric Spaces, General Topology, Continuity, Completeness, Compactness, Measure Theory, Integration, Lebesgue Spaces, Hilbert Spaces, Banach Spaces, Linear Operators, Weak and Weak* Topologies. Suitable both for classroom use and independent reading, this book is ideal preparation for further study in research areas where a broad mathematical toolbox is required.

This book maps Christopher Isherwood's intellectual and aesthetic reflections from the late 1930s through the late 1970s. Drawing on the queer theory of Eve Sedgwick and the ethical theory of Michel Foucault, Carr illuminates Isherwood's post-war development of a queer ethos through his focus on the aesthetic, social, and historical politics of the 1930s in his novels Prater Violet (1945), The World in the Evening (1954), and Down There on a Visit (1962), and in his memoir, Christopher and His Kind: 1929 1939 (1976)."

Wavelet Analysis: Basic Concepts and Applications provides a basic and self-contained introduction to the ideas underpinning wavelet theory and its diverse applications. This book is suitable for master's or PhD students, senior researchers, or scientists working in industrial settings, where wavelets are used to model real-world phenomena and data needs (such as finance, medicine, engineering, transport, images, signals, etc.). Features: Offers a self-contained discussion of wavelet theory Suitable for a wide audience of post-graduate students, researchers, practitioners, and theorists Provides researchers with detailed proofs Provides guides for readers to help them understand and practice wavelet analysis in different areas

The first book to examine weakly stationary random fields and their connections with invariant subspaces (an area associated with functional analysis). It reviews current literature, presents central issues and most important results within the area. For advanced Ph.D. students, researchers, especially those conducting research on Gaussian theory.

This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines like applied mathematics, physics, engineering. It has evolved while teaching courses on partial differential equations during the last decade at the Politecnico of Milan. The main purpose of these courses was twofold: on the one hand, to train the students to appreciate the interplay between theory and modelling in problems arising in the applied sciences and on the other hand to give them a solid background for numerical methods, such as finite differences and finite elements.

Classification of Lipschitz Mappings presents a systematic, self-contained treatment of a new classification of Lipschitz mappings and its application in many topics of metric fixed point theory. Suitable for readers interested in metric fixed point theory, differential equations, and dynamical systems, the book only requires a basic background in functional analysis and topology. The author focuses on a more precise classification of Lipschitzian mappings. The mean Lipschitz condition introduced by Goebel, Japon Pineda, and Sims is relatively easy to check and turns out to satisfy several principles: Regulating the possible growth of the sequence of Lipschitz constants k(Tn) Ensuring good estimates for k0(T) and k (T) Providing some new results in metric fixed point theory

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.

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.