This book provides a comprehensive study of convex integration theory in immersion-theoretic topology. Convex integration theory, developed originally by M. Gromov, provides general topological methods for solving the h-principle for a wide variety of problems in differential geometry and topology, with applications also to PDE theory and to optimal control theory. Though topological in nature, the theory is based on a precise analytical approximation result for higher order derivatives of functions, proved by M. Gromov. This book is the first to present an exacting record and exposition of all of the basic concepts and technical results of convex integration theory in higher order jet spaces, including the theory of iterated convex hull extensions and the theory of relative h-principles. A second feature of the book is its detailed presentation of applications of the general theory to topics in symplectic topology, divergence free vector fields on 3-manifolds, isometric immersions, totally real embeddings, underdetermined non-linear systems of PDEs, the relaxation theorem in optimal control theory, as well as applications to the traditional immersion-theoretical topics such as immersions, submersions, k-mersions and free maps. The book should prove useful to graduate students and to researchers in topology, PDE theory and optimal control theory who wish to understand the h-principle and how it can be applied to solve problems in their respective disciplines.

The two volumes contain 65 chapters, which are based on talks presented by reputable researchers in the field at the Tenth International Conference on Integral Methods in Science and Engineering. The chapters address a wide variety of methodologies, from the construction of boundary integral methods to the application of integration-based analytic and computational techniques in almost all aspects of today's technological world.

Both volumes are useful references for a broad audience of professionals, including pure and applied mathematicians, physicists, biologists, and mechanical, civil, and electrical engineers, as well as graduate students, who use integration as a fundamental technique in their research.

This book focuses on developments in complex dynamical systems and geometric function theory over the past decade, showing strong links with other areas of mathematics and the natural sciences. Traditional methods and approaches surface in physics and in the life and engineering sciences with increasing frequency - the Schramm-Loewner evolution, Laplacian growth, and quadratic differentials are just a few typical examples. This book provides a representative overview of these processes and collects open problems in the various areas, while at the same time showing where and how each particular topic evolves. This volume is dedicated to the memory of Alexander Vasiliev.

This monograph is part of a larger program, materializing in five volumes, whose principal aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. Volume II is concerned with function spaces measuring size and/or smoothness, such as Hardy spaces, Besov spaces, Triebel-Lizorkin spaces, Sobolev spaces, Morrey spaces, Morrey-Campanato spaces, spaces of functions of Bounded Mean Oscillations, etc., in general geometric settings. Work here also highlights the close interplay between differentiability properties of functions and singular integral operators. The text is intended for researchers, graduate students, and industry professionals interested in harmonic analysis, functional analysis, geometric measure theory, and function space theory.

This volume is part of the collaboration agreement between Springer and the ISAAC society. This is the second in the two-volume series originating from the 2020 activities within the international scientific conference "Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis" (OTHA), Southern Federal University, Rostov-on-Don, Russia. This volume focuses on mathematical methods and applications of probability and statistics in the context of general harmonic analysis and its numerous applications. The two volumes cover new trends and advances in several very important fields of mathematics, developed intensively over the last decade. The relevance of this topic is related to the study of complex multi-parameter objects required when considering operators and objects with variable parameters.

This collection of Heinz Koenig's publications connects to his book of 1997 "Measure and Integration" and presents significant developments in the subject from then up to the present day. The result is a consistent new version of measure theory, including selected applications. The basic step is the introduction of the inner * (bullet) and outer * (bullet) premeasures and their extension to unique maximal measures. New "envelopes" for the initial set function (to replace the traditional Caratheodory outer measures) have been created, which lead to much simpler and more explicit treatment. In view of these new concepts, the main results are unmatched in scope and plainness, as well as in explicitness. Important examples are the formation of products, a unified Daniell-Stone-Riesz representation theorem, and projective limits. Further to the contributions in this volume, after 2011 Heinz Koenig published two more articles that round up his work: On the marginals of probability contents on lattices (Mathematika 58, No. 2, 319-323, 2012), and Measure and integration: the basic extension and representation theorems in terms of new inner and outer envelopes (Indag. Math., New Ser. 25, No. 2, 305-314, 2014).

This book provides a comprehensive analysis of time domain boundary integral equations and their discretisation by convolution quadrature and the boundary element method. Properties of convolution quadrature, based on both linear multistep and Runge-Kutta methods, are explained in detail, always with wave propagation problems in mind. Main algorithms for implementing the discrete schemes are described and illustrated by short Matlab codes; translation to other languages can be found on the accompanying GitHub page. The codes are used to present numerous numerical examples to give the reader a feeling for the qualitative behaviour of the discrete schemes in practice. Applications to acoustic and electromagnetic scattering are described with an emphasis on the acoustic case where the fully discrete schemes for sound-soft and sound-hard scattering are developed and analysed in detail. A strength of the book is that more advanced applications such as linear and non-linear impedance boundary conditions and FEM/BEM coupling are also covered. While the focus is on wave scattering, a chapter on parabolic problems is included which also covers the relevant fast and oblivious algorithms. Finally, a brief description of data sparse techniques and modified convolution quadrature methods completes the book. Suitable for graduate students and above, this book is essentially self-contained, with background in mathematical analysis listed in the appendix along with other useful facts. Although not strictly necessary, some familiarity with boundary integral equations for steady state problems is desirable.

This monograph is devoted to developing a theory of combined measure and shift invariance of time scales with the related applications to shift functions and dynamic equations. The study of shift closeness of time scales is significant to investigate the shift functions such as the periodic functions, the almost periodic functions, the almost automorphic functions, and their generalizations with many relevant applications in dynamic equations on arbitrary time scales. First proposed by S. Hilger, the time scale theory-a unified view of continuous and discrete analysis-has been widely used to study various classes of dynamic equations and models in real-world applications. Measure theory based on time scales, in its turn, is of great power in analyzing functions on time scales or hybrid domains. As a new and exciting type of mathematics-and more comprehensive and versatile than the traditional theories of differential and difference equations-, the time scale theory can precisely depict the continuous-discrete hybrid processes and is an optimal way forward for accurate mathematical modeling in applied sciences such as physics, chemical technology, population dynamics, biotechnology, and economics and social sciences. Graduate students and researchers specializing in general dynamic equations on time scales can benefit from this work, fostering interest and further research in the field. It can also serve as reference material for undergraduates interested in dynamic equations on time scales. Prerequisites include familiarity with functional analysis, measure theory, and ordinary differential equations.

This book provides a modern and up-to-date treatment of the Hilbert transform of distributions and the space of periodic distributions. Taking a simple and effective approach to a complex subject, this volume is a first-rate textbook at the graduate level as well as an extremely useful reference for mathematicians, applied scientists, and engineers.
The author, a leading authority in the field, shares with the reader many new results from his exhaustive research on the Hilbert transform of Schwartz distributions. He describes in detail how to use the Hilbert transform to solve theoretical and physical problems in a wide range of disciplines; these include aerofoil problems, dispersion relations, high-energy physics, potential theory problems, and others.
Innovative at every step, J. N. Pandey provides a new definition for the Hilbert transform of periodic functions, which is especially useful for those working in the area of signal processing for computational purposes. This definition could also form the basis for a unified theory of the Hilbert transform of periodic, as well as nonperiodic, functions.
The Hilbert transform and the approximate Hilbert transform of periodic functions are worked out in detail for the first time in book form and can be used to solve Laplace's equation with periodic boundary conditions. Among the many theoretical results proved in this book is a Paley-Wiener type theorem giving the characterization of functions and generalized functions whose Fourier transforms are supported in certain orthants of Rn.
Placing a strong emphasis on easy application of theory and techniques, the book generalizes the Hilbert problem in higher dimensions and solves it in function spaces as well as in generalized function spaces. It simplifies the one-dimensional transform of distributions; provides solutions to the distributional Hilbert problems and singular integral equations; and covers the intrinsic definition of the testing function spaces and its topology.
The book includes exercises and review material for all major topics, and incorporates classical and distributional problems into the main text. Thorough and accessible, it explores new ways to use this important integral transform, and reinforces its value in both mathematical research and applied science.
The Hilbert transform made accessible with many new formulas and definitions
Written by today's foremost expert on the Hilbert transform of generalized functions, this combined text and reference covers the Hilbert transform of distributions and the space of periodic distributions. The author provides a consistently accessible treatment of this advanced-level subject and teaches techniques that can be easily applied to theoretical and physical problems encountered by mathematicians, applied scientists, and graduate students in mathematics and engineering.
Introducing many new inversion formulas that have been developed and applied by the author and his research associates, the book:
* Provides solutions to the distributional Hilbert problem and singular integral equations
* Focuses on the Hilbert transform of Schwartz distributions, giving intrinsic definitions of the space H(D) and its topology
* Covers the Paley-Wiener theorem and provides many important theoretical results of importance to research mathematicians
* Provides the characterization of functions and generalized functions whose Fourier transforms are supported in certain orthants of Rn
* Offers a new definition of the Hilbert transform of the periodic function that can be used for computational purposes in signal processing
* Develops the theory of the Hilbert transform of periodic distributions and the approximate Hilbert transform of periodic distributions
* Provides exercises at the end of each chapter--useful to professors in planning assignments, tests, and problems

This is the second edition of the book which has two additional new chapters on Maxwell's equations as well as a section on properties of solution spaces of Maxwell's equations and their trace spaces. These two new chapters, which summarize the most up-to-date results in the literature for the Maxwell's equations, are sufficient enough to serve as a self-contained introductory book on the modern mathematical theory of boundary integral equations in electromagnetics. The book now contains 12 chapters and is divided into two parts. The first six chapters present modern mathematical theory of boundary integral equations that arise in fundamental problems in continuum mechanics and electromagnetics based on the approach of variational formulations of the equations. The second six chapters present an introduction to basic classical theory of the pseudo-differential operators. The aforementioned corresponding boundary integral operators can now be recast as pseudo-differential operators. These serve as concrete examples that illustrate the basic ideas of how one may apply the theory of pseudo-differential operators and their calculus to obtain additional properties for the corresponding boundary integral operators. These two different approaches are complementary to each other. Both serve as the mathematical foundation of the boundary element methods, which have become extremely popular and efficient computational tools for boundary problems in applications. This book contains a wide spectrum of boundary integral equations arising in fundamental problems in continuum mechanics and electromagnetics. The book is a major scholarly contribution to the modern approaches of boundary integral equations, and should be accessible and useful to a large community of advanced graduate students and researchers in mathematics, physics, and engineering.

This volume presents the proceedings of the meeting New Trends in One-Dimensional Dynamics, which celebrated the 70th birthday of Welington de Melo and was held at the IMPA, Rio de Janeiro, in November 2016. Highlighting the latest results in one-dimensional dynamics and its applications, the contributions gathered here also celebrate the highly successful meeting, which brought together experts in the field, including many of Welington de Melo's co-authors and former doctoral students. Sadly, Welington de Melo passed away shortly after the conference, so that the present volume became more a tribute to him. His role in the development of mathematics was undoubtedly an important one, especially in the area of low-level dynamics, and his legacy includes, in addition to many articles with fundamental contributions, books that are required reading for all newcomers to the field.

A comprehensive, up-to-date, and accessible introduction to the numerical solution of a large class of integral equations, this book builds an important foundation for the numerical analysis of these equations. It provides a general framework for the degenerate kernel, projection, and Nyström methods and includes an introduction to the numerical solution of boundary integral equations (also known as boundary element methods). It is an excellent resource for graduate students and researchers trying to solve integral equation problems and for engineers using boundary element methods.

This book is among the first concise presentations of the set-valued stochastic integration theory as well as its natural applications, as well as the first to contain complex approach theory of set-valued stochastic integrals. Taking particular consideration of set-valued Ito , set-valued stochastic Lebesgue, and stochastic Aumann integrals, the volume is divided into nine parts. It begins with preliminaries of mathematical methods that are then applied in later chapters containing the main results and some of their applications, and contains many new problems. Methods applied in the book are mainly based on functional analysis, theory of probability processes, and theory of set-valued mappings. The volume will appeal to students of mathematics, economics, and engineering, as well as to mathematics professionals interested in applications of the theory of set-valued stochastic integrals.

This book describes mathematical techniques for integral transforms in a detailed but concise manner. The techniques are subsequently applied to the standard partial differential equations, such as the Laplace equation, the wave equation and elasticity equations. Green's functions for beams, plates and acoustic media are also shown, along with their mathematical derivations. The Cagniard-de Hoop method for double inversion is described in detail and 2D and 3D elastodynamic problems are treated in full. This new edition explains in detail how to introduce the branch cut for the multi-valued square root function. Further, an exact closed form Green's function for torsional waves is presented, as well as an application technique of the complex integral, which includes the square root function and an application technique of the complex integral.

This book offers a thorough and self-contained exposition of the mathematics of time-domain boundary integral equations associated to the wave equation, including applications to scattering of acoustic and elastic waves. The book offers two different approaches for the analysis of these integral equations, including a systematic treatment of their numerical discretization using Galerkin (Boundary Element) methods in the space variables and Convolution Quadrature in the time variable. The first approach follows classical work started in the late eighties, based on Laplace transforms estimates. This approach has been refined and made more accessible by tailoring the necessary mathematical tools, avoiding an excess of generality. A second approach contains a novel point of view that the author and some of his collaborators have been developing in recent years, using the semigroup theory of evolution equations to obtain improved results. The extension to electromagnetic waves is explained in one of the appendices.

Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions. Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces.

This book gives a rigorous and practical treatment of integral equations and aims to tackle the solution of integral equations using a blend of abstract structural results and more direct, down-to-earth mathematics. The interplay between these two approaches is a central feature of the text, and it allows a thorough account to be given of many of the types of integral equation that arise, particularly in numerical analysis and fluid mechanics. Because it is not always possible to find explicit solutions to the problems posed, much attention is devoted to obtaining qualitative information and approximations and the associated error estimates.

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

This monograph provides the theoretical foundations needed for the construction of fundamental solutions and fundamental matrices of (systems of) linear partial differential equations. Many illustrative examples also show techniques for finding such solutions in terms of integrals. Particular attention is given to developing the fundamentals of distribution theory, accompanied by calculations of fundamental solutions. The main part of the book deals with existence theorems and uniqueness criteria, the method of parameter integration, the investigation of quasihyperbolic systems by means of Fourier and Laplace transforms, and the representation of fundamental solutions of homogeneous elliptic operators with the help of Abelian integrals. In addition to rigorous distributional derivations and verifications of fundamental solutions, the book also shows how to construct fundamental solutions (matrices) of many physically relevant operators (systems), in elasticity, thermoelasticity, hexagonal/cubic elastodynamics, for Maxwell's system and others. The book mainly addresses researchers and lecturers who work with partial differential equations. However, it also offers a valuable resource for students with a solid background in vector calculus, complex analysis and functional analysis.

Equations of Mathematical Diffraction Theory focuses on the comparative analysis and development of efficient analytical methods for solving equations of mathematical diffraction theory. Following an overview of some general properties of integral and differential operators in the context of the linear theory of diffraction processes, the authors provide estimates of the operator norms for various ranges of the wave number variation, and then examine the spectral properties of these operators. They also present a new analytical method for constructing asymptotic solutions of boundary integral equations in mathematical diffraction theory for the high-frequency case. Clearly demonstrating the close connection between heuristic and rigorous methods in mathematical diffraction theory, this valuable book provides you with the differential and integral equations that can easily be used in practical applications.

For many years, the subject of functional equations has held a prominent place in the attention of mathematicians. In more recent years this attention has been directed to a particular kind of functional equation, an integral equation, wherein the unknown function occurs under the integral sign. The study of this kind of equation is sometimes referred to as the inversion of a definite integral. While scientists and engineers can already choose from a number of books on integral equations, this new book encompasses recent developments including some preliminary backgrounds of formulations of integral equations governing the physical situation of the problems. It also contains elegant analytical and numerical methods, and an important topic of the variational principles.Primarily intended for senior undergraduate students and first year postgraduate students of engineering and science courses, students of mathematical and physical sciences will also find many sections of direct relevance. The book contains eight chapters, pedagogically organized. This book is specially designed for those who wish to understand integral equations without having extensive mathematical background.Some knowledge of integral calculus, ordinary differential equations, partial differential equations, Laplace transforms, Fourier transforms, Hilbert transforms, analytic functions of complex variables and contour integrations are expected on the part of the reader.

The book is primarily devoted to the Kurzweil-Stieltjes integral and its applications in functional analysis, theory of distributions, generalized elementary functions, as well as various kinds of generalized differential equations, including dynamic equations on time scales. It continues the research that was paved out by some of the previous volumes in the Series in Real Analysis. Moreover, it presents results in a thoroughly updated form and, simultaneously, it is written in a widely understandable way, so that it can be used as a textbook for advanced university or PhD courses covering the theory of integration or differential equations.

This book deals with the numerical solution of integral equations based on approximation of functions and the authors apply wavelet approximation to the unknown function of integral equations. The book's goal is to categorize the selected methods and assess their accuracy and efficiency.

The book assists Calculus students to gain a better understanding and command of integration and its applications. It reaches to students in more advanced courses such as Multivariable Calculus, Differential Equations, and Analysis, where the ability to effectively integrate is essential for their success.Keeping the reader constantly focused on the three principal epistemological questions: 'What for?', 'Why?', and 'How?', the book is designated as a supplementary instructional tool and consists ofThe Answers to all the 192 Problems are provided in the Answer Key. The book will benefit undergraduates, advanced undergraduates, and members of the public with an interest in science and technology, helping them to master techniques of integration at the level expected in a calculus course.

The book assists Calculus students to gain a better understanding and command of integration and its applications. It reaches to students in more advanced courses such as Multivariable Calculus, Differential Equations, and Analysis, where the ability to effectively integrate is essential for their success.Keeping the reader constantly focused on the three principal epistemological questions: 'What for?', 'Why?', and 'How?', the book is designated as a supplementary instructional tool and consists ofThe Answers to all the 192 Problems are provided in the Answer Key. The book will benefit undergraduates, advanced undergraduates, and members of the public with an interest in science and technology, helping them to master techniques of integration at the level expected in a calculus course.